diff --git a/lean/OAI/Geometry/NodalSets/Waves/GaussianFactorBound.lean b/lean/OAI/Geometry/NodalSets/Waves/GaussianFactorBound.lean
new file mode 100644
index 0000000000000000000000000000000000000000..e42f6ce52ba7159ab63737e90e26dadf2f1e213d
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/GaussianFactorBound.lean
@@ -0,0 +1,64 @@
+import Mathlib.Analysis.Normed.Module.FiniteDimension
+import OAI.Geometry.NodalSets.Waves.GaussianFactor
+
+namespace OAI
+
+namespace Yau.Probability
+open MeasureTheory ProbabilityTheory
+open scoped RealInnerProductSpace
+noncomputable section
+variable {ι : Type*} [Fintype ι] [DecidableEq ι]
+variable {F : Type*} [NormedAddCommGroup F] [InnerProductSpace ℝ F]
+  [FiniteDimensional ℝ F]
+local notation "E" => EuclideanSpace ℝ ι
+
+lemma covarianceFactor_norm_sq (T : F →L[ℝ] E) (v : E) :
+    ‖covarianceFactor T v‖^2 = ‖T.adjoint v‖^2 := by
+  simpa only [real_inner_self_eq_norm_sq] using covarianceFactor_inner T v v
+
+lemma covarianceFactor_lower (T : F →L[ℝ] E) (a : ℝ) (ha : 0 < a)
+    (h : ∀ v : E, a*‖v‖^2 ≤ ‖T.adjoint v‖^2) (v : E) :
+    Real.sqrt a * ‖v‖ ≤ ‖covarianceFactor T v‖ := by
+  have hs := Real.sq_sqrt ha.le
+  have hb := h v
+  rw [← covarianceFactor_norm_sq T v] at hb
+  nlinarith [sq_nonneg (Real.sqrt a*‖v‖-‖covarianceFactor T v‖),
+    norm_nonneg (covarianceFactor T v), norm_nonneg v, Real.sqrt_nonneg a]
+
+lemma covarianceFactor_bijective (T : F →L[ℝ] E) (a : ℝ) (ha : 0 < a)
+    (h : ∀ v : E, a*‖v‖^2 ≤ ‖T.adjoint v‖^2) :
+    Function.Bijective (covarianceFactor T) := by
+  have hi : Function.Injective (covarianceFactor T) := by
+    apply (covarianceFactor T).ker_eq_bot.mp
+    rw [LinearMap.ker_eq_bot']
+    intro v hv
+    change covarianceFactor T v = 0 at hv
+    have hb := covarianceFactor_lower T a ha h v
+    rw [hv,norm_zero] at hb
+    have hz : ‖v‖ = 0 := by nlinarith [Real.sqrt_pos.mpr ha, norm_nonneg v]
+    exact norm_eq_zero.mp hz
+  exact ⟨hi, LinearMap.injective_iff_surjective.mp hi⟩
+
+variable [MeasurableSpace F] [BorelSpace F]
+
+lemma exists_gaussian_factor_equiv (T : F →L[ℝ] E) (a : ℝ) (ha : 0 < a)
+    (h : ∀ v : E, a*‖v‖^2 ≤ ‖T.adjoint v‖^2) :
+    ∃ L : E ≃L[ℝ] E,
+      (stdGaussian F).map T = (stdGaussian E).map L ∧
+      ‖L.symm.toContinuousLinearMap‖ ≤ (Real.sqrt a)⁻¹ := by
+  let L := (LinearEquiv.ofBijective (covarianceFactor T).toLinearMap
+    (covarianceFactor_bijective T a ha h)).toContinuousLinearEquiv
+  refine ⟨L, gaussian_covariance_factorization T, ?_⟩
+  apply ContinuousLinearMap.opNorm_le_bound _ (by positivity)
+  intro v
+  have hb := covarianceFactor_lower T a ha h (L.symm v)
+  change Real.sqrt a*‖L.symm v‖ ≤ ‖L (L.symm v)‖ at hb
+  rw [L.apply_symm_apply] at hb
+  change ‖L.symm v‖ ≤ _
+  rw [inv_mul_eq_div]
+  exact (le_div_iff₀ (Real.sqrt_pos.mpr ha)).mpr (by simpa [mul_comm] using hb)
+
+end
+end Yau.Probability
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/GaussianFactorTwoSided.lean b/lean/OAI/Geometry/NodalSets/Waves/GaussianFactorTwoSided.lean
new file mode 100644
index 0000000000000000000000000000000000000000..3f1098e3517edb4d97b3008c6e3f428f1210e718
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/GaussianFactorTwoSided.lean
@@ -0,0 +1,40 @@
+import OAI.Geometry.NodalSets.Waves.GaussianFactorBound
+
+namespace OAI
+
+namespace Yau.Probability
+open MeasureTheory ProbabilityTheory
+noncomputable section
+variable {ι : Type*} [Fintype ι] [DecidableEq ι]
+variable {F : Type*} [NormedAddCommGroup F] [InnerProductSpace ℝ F]
+  [FiniteDimensional ℝ F] [MeasurableSpace F] [BorelSpace F]
+local notation "E" => EuclideanSpace ℝ ι
+
+lemma exists_gaussian_factor_two_sided (T : F →L[ℝ] E) (a B : ℝ)
+    (ha : 0 < a) (hB : 0 ≤ B)
+    (hlo : ∀ v : E, a*‖v‖^2 ≤ ‖T.adjoint v‖^2)
+    (hhi : ∀ v : E, ‖T.adjoint v‖^2 ≤ B^2*‖v‖^2) :
+    ∃ L : E ≃L[ℝ] E,
+      (stdGaussian F).map T = (stdGaussian E).map L ∧
+      ‖L.symm.toContinuousLinearMap‖ ≤ (Real.sqrt a)⁻¹ ∧ ‖L.toContinuousLinearMap‖ ≤ B := by
+  let L := (LinearEquiv.ofBijective (covarianceFactor T).toLinearMap
+    (covarianceFactor_bijective T a ha hlo)).toContinuousLinearEquiv
+  refine ⟨L,gaussian_covariance_factorization T,?_,?_⟩
+  · apply ContinuousLinearMap.opNorm_le_bound _ (by positivity)
+    intro v
+    have hb := covarianceFactor_lower T a ha hlo (L.symm v)
+    change Real.sqrt a*‖L.symm v‖ ≤ ‖L (L.symm v)‖ at hb
+    rw [L.apply_symm_apply] at hb
+    rw [inv_mul_eq_div]
+    exact (le_div_iff₀ (Real.sqrt_pos.mpr ha)).mpr (by simpa [mul_comm] using hb)
+  · apply ContinuousLinearMap.opNorm_le_bound _ hB
+    intro v
+    change ‖covarianceFactor T v‖ ≤ B*‖v‖
+    apply (sq_le_sq₀ (norm_nonneg _) (mul_nonneg hB (norm_nonneg _))).mp
+    rw [covarianceFactor_norm_sq,mul_pow]
+    exact hhi v
+
+end
+end Yau.Probability
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/GaussianGradientSupremum.lean b/lean/OAI/Geometry/NodalSets/Waves/GaussianGradientSupremum.lean
new file mode 100644
index 0000000000000000000000000000000000000000..1182c0ca12e5d0aa40a1ad6bd1a16d7ce015deeb
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/GaussianGradientSupremum.lean
@@ -0,0 +1,99 @@
+import OAI.Geometry.NodalSets.Charts.MetricFrequencyBounds
+import OAI.Geometry.NodalSets.Waves.GaussianCompactSections
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Yau.Probability MeasureTheory ProbabilityTheory
+open scoped ContDiff
+noncomputable section
+variable {ι X : Type*} [Fintype ι] [TopologicalSpace X]
+
+def vectorMeanSection (M : Fin 4 → C(X,ℝ)) : C(X,EuclideanSpace ℝ (Fin 4)) :=
+  ⟨fun x ↦ WithLp.toLp 2 (fun j ↦ M j x),
+    (PiLp.continuous_toLp 2 (fun _ : Fin 4 ↦ ℝ)).comp (continuous_pi (fun j ↦ (M j).continuous))⟩
+
+def vectorCoeffSection (Z : Fin 4 → ι → C(X,ℂ)) (p : ι × Fin 2) :
+    C(X,EuclideanSpace ℝ (Fin 4)) :=
+  vectorMeanSection (fun j ↦ if p.2 = 0 then
+    ⟨fun x ↦ (Z j p.1 x).re, Complex.continuous_re.comp (Z j p.1).continuous⟩ else
+    ⟨fun x ↦ -(Z j p.1 x).im, (Complex.continuous_im.comp (Z j p.1).continuous).neg⟩)
+
+def pairVectorSection (M : Fin 4 → C(X,ℝ)) (Z : Fin 4 → ι → C(X,ℂ))
+    (a : ι × Fin 2 → ℝ) : C(X,EuclideanSpace ℝ (Fin 4)) :=
+  vectorMeanSection M + ∑ p : ι × Fin 2, a p • vectorCoeffSection Z p
+
+lemma pairVectorSection_apply (M : Fin 4 → C(X,ℝ)) (Z : Fin 4 → ι → C(X,ℂ))
+    (a : ι × Fin 2 → ℝ) (x : X) (j : Fin 4) :
+    pairVectorSection M Z a x j = M j x+pairLinearSum (fun i ↦ Z j i x) a := by
+  classical
+  simp only [pairVectorSection,ContinuousMap.add_apply,ContinuousMap.sum_apply,
+    ContinuousMap.smul_apply,PiLp.add_apply,
+    vectorMeanSection,ContinuousMap.coe_mk,pairLinearSum,vectorCoeffSection]
+  congr 1
+  simp only [WithLp.ofLp_sum,WithLp.ofLp_smul,Finset.sum_apply,Pi.smul_apply]
+  apply Finset.sum_congr rfl
+  intro p _
+  split_ifs <;> simp [smul_eq_mul,mul_comm]
+
+lemma pairVectorSection_continuous (M : Fin 4 → C(X,ℝ)) (Z : Fin 4 → ι → C(X,ℂ)) :
+    Continuous (pairVectorSection M Z) := by
+  unfold pairVectorSection
+  exact continuous_const.add (continuous_finsetSum _ (fun p _ ↦
+    (continuous_apply p).smul continuous_const))
+
+variable [CompactSpace X]
+
+lemma pairVectorSection_integrable (M : Fin 4 → C(X,ℝ)) (Z : Fin 4 → ι → C(X,ℂ)) :
+    Integrable (pairVectorSection M Z) gaussianPairs := by
+  let : IsProbabilityMeasure (gaussianPairs (ι := ι)) := by unfold gaussianPairs; infer_instance
+  unfold pairVectorSection
+  apply (integrable_const (μ := gaussianPairs (ι := ι)) (vectorMeanSection M)).add
+  apply integrable_finsetSum
+  intro p _
+  exact (integrable_eval (μ := fun _ : ι × Fin 2 ↦ gaussianReal 0 1)
+    (i := p) IsGaussian.integrable_id).smul_const _
+
+variable (V : ι → Coord → ℂ) (seed : Coord → ℝ)
+    (hV : ∀ i, ContDiff ℝ ∞ (V i)) (hs : ContDiff ℝ ∞ seed)
+    (K : Set Coord) [CompactSpace K]
+
+def gaussianGradientSection (a : ι × Fin 2 → ℝ) : C(K,EuclideanSpace ℝ (Fin 4)) :=
+  pairVectorSection (fun j ↦ derivativeSection seed hs K 1 (fun _ ↦ Pi.single j 1))
+    (fun j i ↦ derivativeSection (V i) (hV i) K 1 (fun _ ↦ Pi.single j 1)) a
+
+omit [CompactSpace K] in
+lemma gaussianGradientSection_apply (a : ι × Fin 2 → ℝ) (x : K) (j : Fin 4) :
+    gaussianGradientSection V seed hV hs K a x j =
+      fderiv ℝ (fun z ↦ seed z+gaussianWaveField V a z) x (Pi.single j 1) := by
+  rw [gaussianGradientSection,pairVectorSection_apply]
+  have hh := seeded_gaussian_iterated_apply V seed hV hs a 1 x (fun _ ↦ Pi.single j 1)
+  simpa only [derivativeSection,ContinuousMap.coe_mk,iteratedFDeriv_one_apply] using hh.symm
+
+def gaussianGradientSup (a : ι × Fin 2 → ℝ) : ℝ :=
+  ‖gaussianGradientSection V seed hV hs K a‖
+
+lemma gaussianGradientSup_continuous : Continuous (gaussianGradientSup V seed hV hs K) :=
+  (pairVectorSection_continuous _ _).norm
+
+lemma gaussianGradientSup_integrable : Integrable (gaussianGradientSup V seed hV hs K) gaussianPairs :=
+  (pairVectorSection_integrable _ _).norm
+
+lemma gaussianGradientSup_le_iff (a : ι × Fin 2 → ℝ) {C : ℝ} (hC : 0 ≤ C) :
+    gaussianGradientSup V seed hV hs K a ≤ C ↔
+      ∀ x ∈ K, sourceEuclideanNorm (fun j ↦
+        fderiv ℝ (fun z ↦ seed z+gaussianWaveField V a z) x (Pi.single j 1)) ≤ C := by
+  unfold gaussianGradientSup
+  rw [ContinuousMap.norm_le _ hC]
+  have he (x : K) : ‖gaussianGradientSection V seed hV hs K a x‖ =
+      sourceEuclideanNorm (fun j ↦
+        fderiv ℝ (fun z ↦ seed z+gaussianWaveField V a z) x (Pi.single j 1)) := by
+    rw [EuclideanSpace.norm_eq]
+    simp only [gaussianGradientSection_apply,Real.norm_eq_abs,sq_abs,sourceEuclideanNorm]
+  simp_rw [he]
+  exact Subtype.forall
+
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/GaussianIntervalBound.lean b/lean/OAI/Geometry/NodalSets/Waves/GaussianIntervalBound.lean
new file mode 100644
index 0000000000000000000000000000000000000000..f679ee78d7650af742d849a7416a372b500a691c
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/GaussianIntervalBound.lean
@@ -0,0 +1,38 @@
+import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
+import Mathlib.Probability.Distributions.Gaussian.Real
+
+namespace OAI
+
+namespace Yau.Probability
+open MeasureTheory ProbabilityTheory Set
+open scoped ENNReal NNReal
+noncomputable section
+
+lemma gaussian_pdf_upper_bound (mean x : ℝ) (v : ℝ≥0) :
+    gaussianPDFReal mean v x ≤ (Real.sqrt (2*Real.pi*v))⁻¹ := by
+  unfold gaussianPDFReal
+  apply mul_le_of_le_one_right (by positivity)
+  apply Real.exp_le_one_iff.mpr
+  exact div_nonpos_of_nonpos_of_nonneg (neg_nonpos.mpr (sq_nonneg _)) (by positivity)
+
+lemma gaussian_interval_bound (mean center radius : ℝ) (hr : 0 ≤ radius)
+    (v : ℝ≥0) (hv : v ≠ 0) :
+    gaussianReal mean v (Icc (center-radius) (center+radius)) ≤
+      ENNReal.ofReal (2*radius/(Real.sqrt (2*Real.pi*v))) := by
+  rw [gaussianReal_apply mean hv]
+  calc
+    _ ≤ ∫⁻ _ in Icc (center-radius) (center+radius),
+        ENNReal.ofReal ((Real.sqrt (2*Real.pi*v))⁻¹) := by
+      apply lintegral_mono
+      intro x
+      exact ENNReal.ofReal_le_ofReal (gaussian_pdf_upper_bound mean x v)
+    _ = _ := by
+      rw [lintegral_const,Measure.restrict_apply_univ,Real.volume_Icc]
+      rw [← ENNReal.ofReal_mul' (by linarith only [hr])]
+      congr 1
+      ring
+
+end
+end Yau.Probability
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/GaussianJet.lean b/lean/OAI/Geometry/NodalSets/Waves/GaussianJet.lean
new file mode 100644
index 0000000000000000000000000000000000000000..9acce191173759ce68e669ab919d18eaab7e3d52
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/GaussianJet.lean
@@ -0,0 +1,91 @@
+import OAI.Geometry.NodalSets.Waves.FiniteGaussianMap
+import OAI.Geometry.NodalSets.Waves.SeededGaussianJet
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Yau.Probability MeasureTheory ProbabilityTheory
+open scoped RealInnerProductSpace
+noncomputable section
+variable {ι : Type*} [Fintype ι]
+
+abbrev JetSpace := EuclideanSpace ℝ (Fin 5)
+
+def normalizedRealJet (f : Coord → ℝ) (N S : ℝ) (x : Coord) : JetSpace :=
+  WithLp.toLp 2 (Fin.cons (Real.exp (-N*S)*f x)
+    (fun i : Fin 4 ↦ Real.exp (-N*S)*N⁻¹*fderiv ℝ f x (Pi.single i 1)))
+
+lemma normalizedRealJet_inner (f : Coord → ℝ) (N S : ℝ) (x : Coord) (v : JetSpace) :
+    ⟪v,normalizedRealJet f N S x⟫ =
+      Real.exp (-N*S)*(v 0*f x+N⁻¹*fderiv ℝ f x (Fin.tail (WithLp.ofLp v))) := by
+  have he : Fin.tail (WithLp.ofLp v) = ∑ i : Fin 4, (v i.succ) • Pi.single i (1:ℝ) := by
+    ext j
+    simp [Pi.single_apply,Fin.tail]
+  rw [he,map_sum]
+  simp only [map_smul,smul_eq_mul]
+  simp only [normalizedRealJet,PiLp.inner_apply,RCLike.inner_apply,conj_trivial,
+    Fin.sum_univ_succ,Fin.cons_zero,Fin.cons_succ]
+  ring
+
+def jetWaveCoefficients (V : ι → Coord → ℂ) (N S : ℝ) (x : Coord) : Fin 5 → ι → ℂ :=
+  Fin.cons (fun i ↦ dualWaveCoefficient N (V i) S x 1 0)
+    (fun j i ↦ dualWaveCoefficient N (V i) S x 0 (Pi.single j 1))
+
+lemma gaussianWaveField_normalizedJet (V : ι → Coord → ℂ) (N S : ℝ) (x : Coord)
+    (hV : ∀ i, DifferentiableAt ℝ (V i) x) (a : ι × Fin 2 → ℝ) :
+    normalizedRealJet (gaussianWaveField V a) N S x =
+      pairJetMap (jetWaveCoefficients V N S x) (WithLp.toLp 2 a) := by
+  ext k
+  rw [pairJetMap_apply]
+  refine Fin.cases ?_ (fun j ↦ ?_) k
+  · simpa [normalizedRealJet,jetWaveCoefficients] using
+      gaussianWaveField_dual V a x 0 hV N S 1
+  · simpa [normalizedRealJet,jetWaveCoefficients,mul_assoc] using
+      gaussianWaveField_dual V a x (Pi.single j 1) hV N S 0
+
+lemma normalizedRealJet_add (f g : Coord → ℝ) (N S : ℝ) (x : Coord)
+    (hf : DifferentiableAt ℝ f x) (hg : DifferentiableAt ℝ g x) :
+    normalizedRealJet (fun y ↦ f y+g y) N S x =
+      normalizedRealJet f N S x+normalizedRealJet g N S x := by
+  ext k
+  refine Fin.cases ?_ (fun j ↦ ?_) k
+  · simp [normalizedRealJet,mul_add]
+  · simp [normalizedRealJet,fderiv_fun_add hf hg,mul_add]
+
+lemma seeded_gaussian_jet_eq (V : ι → Coord → ℂ) (seed : Coord → ℝ)
+    (N S : ℝ) (x : Coord) (hV : ∀ i, DifferentiableAt ℝ (V i) x)
+    (hs : DifferentiableAt ℝ seed x) (a : ι × Fin 2 → ℝ) :
+    normalizedRealJet (fun y ↦ seed y+gaussianWaveField V a y) N S x =
+      normalizedRealJet seed N S x+
+        pairJetMap (jetWaveCoefficients V N S x) (WithLp.toLp 2 a) := by
+  rw [normalizedRealJet_add _ _ _ _ _ hs (gaussianWaveField_differentiable V a x hV),
+    gaussianWaveField_normalizedJet V N S x hV]
+
+lemma seeded_gaussian_jet_law (V : ι → Coord → ℂ) (seed : Coord → ℝ)
+    (N S : ℝ) (x : Coord) (hV : ∀ i, DifferentiableAt ℝ (V i) x)
+    (hs : DifferentiableAt ℝ seed x) :
+    (gaussianPairs (ι := ι)).map (fun a ↦
+      normalizedRealJet (fun y ↦ seed y+gaussianWaveField V a y) N S x) =
+      (stdGaussian (EuclideanSpace ℝ (ι × Fin 2))).map
+        (fun z ↦ normalizedRealJet seed N S x+pairJetMap (jetWaveCoefficients V N S x) z) := by
+  simp_rw [seeded_gaussian_jet_eq V seed N S x hV hs]
+  rw [← map_pi_eq_stdGaussian, Measure.map_map (by fun_prop) (by fun_prop)]
+  rfl
+
+lemma seeded_gaussian_jet_variance (V : ι → Coord → ℂ) (seed : Coord → ℝ)
+    (N S : ℝ) (x : Coord) (hV : ∀ i, DifferentiableAt ℝ (V i) x)
+    (hs : DifferentiableAt ℝ seed x) (v : JetSpace) :
+    Var[fun a : ι × Fin 2 → ℝ ↦
+      ⟪v,normalizedRealJet (fun y ↦ seed y+gaussianWaveField V a y) N S x⟫; gaussianPairs] =
+      ‖(pairJetMap (jetWaveCoefficients V N S x)).adjoint v‖^2 := by
+  let : IsProbabilityMeasure (gaussianPairs (ι := ι)) := by
+    unfold gaussianPairs
+    infer_instance
+  simp_rw [seeded_gaussian_jet_eq V seed N S x hV hs,inner_add_right]
+  rw [variance_const_add (by fun_prop)]
+  exact variance_gaussianMap_inner _ v
+
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/GaussianPairCovariance.lean b/lean/OAI/Geometry/NodalSets/Waves/GaussianPairCovariance.lean
new file mode 100644
index 0000000000000000000000000000000000000000..043f9c104f7002f52d1b5ae11d96f6aacf2284ec
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/GaussianPairCovariance.lean
@@ -0,0 +1,51 @@
+import Mathlib.Probability.Moments.Covariance
+import OAI.Geometry.NodalSets.Waves.GaussianPairMoments
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Probability MeasureTheory ProbabilityTheory
+noncomputable section
+variable {ι : Type*} [Fintype ι]
+
+lemma pairLinearSum_add (z w : ι → ℂ) :
+    pairLinearSum (fun i ↦ z i+w i) = fun a ↦ pairLinearSum z a+pairLinearSum w a := by
+  classical
+  funext a
+  simp only [pairLinearSum,Complex.add_re,Complex.add_im]
+  rw [← Finset.sum_add_distrib]
+  apply Finset.sum_congr rfl
+  intro p _
+  split_ifs <;> ring
+
+lemma covariance_pairLinearSum (z w : ι → ℂ) :
+    cov[pairLinearSum z,pairLinearSum w;gaussianPairs] =
+      ∑ i, ((z i).re*(w i).re+(z i).im*(w i).im) := by
+  let : IsProbabilityMeasure (gaussianPairs (ι := ι)) := by unfold gaussianPairs; infer_instance
+  have hz := pairLinearSum_memLp_two z
+  have hw := pairLinearSum_memLp_two w
+  have hh := variance_add hz hw
+  change Var[fun a ↦ pairLinearSum z a+pairLinearSum w a;gaussianPairs] = _ at hh
+  rw [← pairLinearSum_add] at hh
+  simp only [variance_pairLinearSum] at hh
+  have hn : ∑ i, ‖z i+w i‖^2 = (∑ i, ‖z i‖^2) + (∑ i, ‖w i‖^2) +
+      2 * ∑ i, ((z i).re*(w i).re+(z i).im*(w i).im) := by
+    simp only [← Finset.sum_add_distrib,Finset.mul_sum,Complex.sq_norm,
+      Complex.normSq_apply,Complex.add_re,Complex.add_im]
+    apply Finset.sum_congr rfl
+    intro i _
+    ring
+  linarith
+
+lemma pairLinearSum_covariance_integral (z w : ι → ℂ) :
+    ∫ a, pairLinearSum z a*pairLinearSum w a ∂gaussianPairs =
+      ∑ i, ((z i).re*(w i).re+(z i).im*(w i).im) := by
+  let : IsProbabilityMeasure (gaussianPairs (ι := ι)) := by unfold gaussianPairs; infer_instance
+  have hh := covariance_eq_sub (pairLinearSum_memLp_two z) (pairLinearSum_memLp_two w)
+  rw [pairLinearSum_mean,pairLinearSum_mean,mul_zero,sub_zero,covariance_pairLinearSum] at hh
+  exact hh.symm
+
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/GaussianPairMoments.lean b/lean/OAI/Geometry/NodalSets/Waves/GaussianPairMoments.lean
new file mode 100644
index 0000000000000000000000000000000000000000..0965c355e3d482a220dc93839d9538fc20ed89a5
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/GaussianPairMoments.lean
@@ -0,0 +1,50 @@
+import OAI.Geometry.NodalSets.Coefficients.GaussianCoefficientMSE
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Yau.Probability MeasureTheory ProbabilityTheory
+noncomputable section
+variable {ι : Type*} [Fintype ι]
+
+lemma pairLinearSum_memLp_two (z : ι → ℂ) : MemLp (pairLinearSum z) 2 gaussianPairs := by
+  classical
+  unfold pairLinearSum gaussianPairs
+  apply memLp_finsetSum
+  intro p _
+  have hh : MemLp (id : ℝ → ℝ) 2 (gaussianReal 0 1) := IsGaussian.memLp_two_id
+  exact (hh.comp_measurePreserving (measurePreserving_eval (μ := fun _ : ι × Fin 2 ↦ gaussianReal 0 1) p)).const_mul _
+
+lemma pairLinearSum_mean_square (z : ι → ℂ) (m : ℝ) :
+    ∫ coeff, (m+pairLinearSum z coeff)^2 ∂gaussianPairs = m^2+∑ i, ‖z i‖^2 := by
+  let : IsProbabilityMeasure (gaussianPairs (ι := ι)) := by unfold gaussianPairs; infer_instance
+  have hp := pairLinearSum_memLp_two z
+  have hh := variance_eq_sub ((memLp_const m).add hp)
+  have hm : ∫ coeff, m+pairLinearSum z coeff ∂gaussianPairs = m := by
+    rw [integral_add (integrable_const _) (pairLinearSum_integrable z),pairLinearSum_mean,integral_const]
+    simp
+  change Var[fun coeff ↦ m+pairLinearSum z coeff;gaussianPairs] =
+    (∫ coeff, (m+pairLinearSum z coeff)^2 ∂gaussianPairs) -
+      (∫ coeff, m+pairLinearSum z coeff ∂gaussianPairs)^2 at hh
+  rw [hm,variance_const_add hp.aestronglyMeasurable,variance_pairLinearSum] at hh
+  linarith
+
+lemma rescaledGaussianField_mse (V W : ι → Coord → ℂ) (S0 T0 S : Coord → ℝ)
+    (N s sigma : ℝ) (x v : Coord) :
+    ∫ coeff, (rescaledGaussianField V S0 T0 S N s sigma x coeff v -
+      gaussianWaveField W coeff v)^2 ∂gaussianPairs =
+      (rescaledSeed S0 T0 S N s sigma x v)^2 +
+        ∑ i, ‖normalizedRescaling (V i) S N s sigma x v-W i v‖^2 := by
+  have heq (coeff : ι × Fin 2 → ℝ) :
+      rescaledGaussianField V S0 T0 S N s sigma x coeff v - gaussianWaveField W coeff v =
+        rescaledSeed S0 T0 S N s sigma x v +
+          pairLinearSum (fun i ↦ normalizedRescaling (V i) S N s sigma x v-W i v) coeff := by
+    rw [rescaledGaussianField_decomposition,← gaussianWaveField_sub_pair]
+    ring
+  simp_rw [heq]
+  exact pairLinearSum_mean_square _ _
+
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/GaussianPartialMeasurability.lean b/lean/OAI/Geometry/NodalSets/Waves/GaussianPartialMeasurability.lean
new file mode 100644
index 0000000000000000000000000000000000000000..a88d8becdd95254fc1e0bf14a571d8eb4a83d68c
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/GaussianPartialMeasurability.lean
@@ -0,0 +1,39 @@
+import OAI.Geometry.NodalSets.Elliptic.CoordinatePartialJets
+import OAI.Geometry.NodalSets.Waves.GaussianDerivativeMoments
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Yau.Analysis Yau.Probability MeasureTheory
+open scoped ContDiff
+noncomputable section
+variable {ι : Type*} [Fintype ι]
+
+lemma seeded_partial_joint_continuous (V : ι → Coord → ℂ) (seed : Coord → ℝ)
+    (hV : ∀ i, ContDiff ℝ ∞ (V i)) (hs : ContDiff ℝ ∞ seed)
+    (ds : List (Fin 4)) :
+    Continuous (fun z : (ι × Fin 2 → ℝ) × Coord ↦
+      partialJet (fun x ↦ seed x+gaussianWaveField V z.1 x) ds z.2) := by
+  have hF (a : ι × Fin 2 → ℝ) := hs.add (gaussianWaveField_contDiff V a hV)
+  simp_rw [partialJet_eq_iterated _ (hF _),seeded_gaussian_iterated_apply V seed hV hs]
+  have horder : (ds.length : WithTop ℕ∞) ≤ ∞ := by
+    exact_mod_cast (show (ds.length:ℕ∞) ≤ ⊤ from le_top)
+  have hc {F : Type} [NormedAddCommGroup F] [NormedSpace ℝ F]
+      (f : Coord → F) (hf : ContDiff ℝ ∞ f) :
+      Continuous (fun z : (ι × Fin 2 → ℝ) × Coord ↦
+        iteratedFDeriv ℝ ds.length f z.2 (partialDirs ds)) :=
+    ((hf.continuous_iteratedFDeriv horder).comp continuous_snd).eval continuous_const
+  apply (hc seed hs).add
+  unfold pairLinearSum
+  apply continuous_finsetSum
+  intro p _
+  apply Continuous.mul
+  · split_ifs
+    · exact Complex.continuous_re.comp (hc (V p.1) (hV p.1))
+    · exact (Complex.continuous_im.comp (hc (V p.1) (hV p.1))).neg
+  · exact (continuous_apply p).comp continuous_fst
+
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/GaussianTailBound.lean b/lean/OAI/Geometry/NodalSets/Waves/GaussianTailBound.lean
new file mode 100644
index 0000000000000000000000000000000000000000..16cf2f7156c1c891d6c27399a200c24cb2c7ee59
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/GaussianTailBound.lean
@@ -0,0 +1,56 @@
+import Mathlib.Analysis.SpecialFunctions.Exp
+import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
+import Mathlib.Tactic.GCongr
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.Positivity
+import Mathlib.Tactic.Ring
+
+namespace OAI
+
+namespace Yau.Probability
+open Filter
+open scoped BigOperators Topology
+noncomputable section
+
+lemma finite_gaussian_tail {ι : Type*} [Fintype ι] (d f : ι → ℝ)
+    {a R B r : ℝ} (ha : 0 ≤ a) (hR : 0 ≤ R) (hr : 0 ≤ r)
+    (hd : ∀ i, 0 ≤ d i) (hf : ∀ i, f i ≤ R*Real.exp (-2*a*(d i)^2))
+    (hs : ∑ i, Real.exp (-a*(d i)^2) ≤ B) :
+    ∑ i ∈ Finset.univ.filter (fun i ↦ r < d i), f i ≤
+      R*B*Real.exp (-a*r^2) := by
+  classical
+  calc
+    _ ≤ ∑ i ∈ Finset.univ.filter (fun i ↦ r < d i),
+        (R*Real.exp (-a*r^2))*Real.exp (-a*(d i)^2) := by
+      apply Finset.sum_le_sum
+      intro i hi
+      have hri := (Finset.mem_filter.mp hi).2.le
+      apply (hf i).trans
+      simp only [mul_assoc,← Real.exp_add]
+      gcongr
+      have hh := mul_le_mul_of_nonneg_left ((sq_le_sq₀ hr (hd i)).2 hri) ha
+      nlinarith
+    _ ≤ ∑ i, (R*Real.exp (-a*r^2))*Real.exp (-a*(d i)^2) :=
+      Finset.sum_le_sum_of_subset_of_nonneg (Finset.filter_subset _ _) (fun _ _ _ ↦ by positivity)
+    _ = (R*Real.exp (-a*r^2))*∑ i, Real.exp (-a*(d i)^2) := by rw [Finset.mul_sum]
+    _ ≤ (R*Real.exp (-a*r^2))*B := mul_le_mul_of_nonneg_left hs (by positivity)
+    _ = _ := by ring
+
+lemma main_center_scale (N : ℝ) (hN : 0 < N) :
+    N*(N^(-5/12:ℝ))^2 = N^(1/6:ℝ) := by
+  calc
+    _ = N^(1:ℝ)*N^(-5/6:ℝ) := by
+      rw [← Real.rpow_mul_natCast hN.le]
+      norm_num
+    _ = _ := by rw [← Real.rpow_add hN]; norm_num
+
+lemma tendsto_main_center_tail (a : ℝ) (ha : 0 < a) :
+    Tendsto (fun n : ℕ ↦ Real.exp (-a*(n:ℝ)^(1/6:ℝ))) atTop (𝓝 0) := by
+  have hpow : Tendsto (fun n : ℕ ↦ (n:ℝ)^(1/6:ℝ)) atTop atTop :=
+    (tendsto_rpow_atTop (by norm_num : (0:ℝ) < 1/6)).comp tendsto_natCast_atTop_atTop
+  exact Real.tendsto_exp_atBot.comp ((tendsto_const_mul_atBot_of_neg (neg_neg_of_pos ha)).mpr hpow)
+
+end
+end Yau.Probability
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/GaussianUniformBall.lean b/lean/OAI/Geometry/NodalSets/Waves/GaussianUniformBall.lean
new file mode 100644
index 0000000000000000000000000000000000000000..e2b8516fcc4f18fcb0a70acc01005b217711b379
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/GaussianUniformBall.lean
@@ -0,0 +1,84 @@
+import Mathlib.MeasureTheory.Measure.OpenPos
+import OAI.Geometry.NodalSets.Waves.EuclideanGaussianSmallBall
+
+namespace OAI
+
+namespace Yau.Probability
+open MeasureTheory ProbabilityTheory Set Metric
+open scoped ENNReal
+noncomputable section
+variable {ι : Type*} [Fintype ι]
+local notation "E" => EuclideanSpace ℝ ι
+
+lemma stdGaussian_open_positive : (stdGaussian E).IsOpenPosMeasure := by
+  let : (gaussianReal 0 1).IsOpenPosMeasure :=
+    (gaussianReal_absolutelyContinuous' 0 (by norm_num : (1:NNReal) ≠ 0)).isOpenPosMeasure
+  rw [← map_pi_eq_stdGaussian]
+  apply Continuous.isOpenPosMeasure_map (by fun_prop)
+  exact (WithLp.equiv 2 (ι → ℝ)).symm.surjective
+
+lemma uniform_gaussian_ball (R r : ℝ) (hr : 0 < r) :
+    ∃ p : ℝ≥0∞, 0 < p ∧ ∀ z : E, ‖z‖ ≤ R → p ≤ stdGaussian E (ball z r) := by
+  classical
+  let : (stdGaussian E).IsOpenPosMeasure := stdGaussian_open_positive
+  obtain ⟨s,hs⟩ := (isCompact_closedBall (0:E) R).elim_finite_subcover
+    (fun z : E ↦ ball z (r/2)) (fun _ ↦ isOpen_ball) (by
+      intro z hz
+      exact mem_iUnion.mpr ⟨z,mem_ball_self (by positivity)⟩)
+  let p : ℝ≥0∞ := s.inf (fun z ↦ stdGaussian E (ball z (r/2)))
+  have hp : 0 < p := by
+    dsimp only [p]
+    clear hs p
+    induction s using Finset.induction_on with
+    | empty => simp
+    | @insert z s hz ih =>
+      rw [Finset.inf_insert]
+      exact lt_min (measure_ball_pos _ _ (by positivity)) ih
+  refine ⟨p,hp,?_⟩
+  intro z hz
+  have hz' : z ∈ closedBall (0:E) R := by simpa only [mem_closedBall,dist_zero_right] using hz
+  obtain ⟨y,hy,hyz⟩ := mem_iUnion₂.mp (hs hz')
+  apply (Finset.inf_le hy).trans
+  apply measure_mono
+  intro x hx
+  have hx' : dist x y < r/2 := hx
+  have hz'' : dist z y < r/2 := hyz
+  have hh := dist_triangle x y z
+  rw [dist_comm y z] at hh
+  change dist x z < r
+  linarith
+
+lemma uniform_affine_gaussian_ball (A B M r : ℝ) (hA : 0 ≤ A)
+    (hB : 0 < B) (hM : 0 ≤ M) (hr : 0 < r) (target : E) :
+    ∃ p : ℝ≥0∞, 0 < p ∧ ∀ (L : E ≃L[ℝ] E) (mean : E),
+      ‖L.symm.toContinuousLinearMap‖ ≤ A → ‖L.toContinuousLinearMap‖ ≤ B → ‖mean‖ ≤ M →
+      p ≤ ((stdGaussian E).map (fun z ↦ mean+L z)) (ball target r) := by
+  let radius : NNReal := ⟨A*(‖target‖+M), mul_nonneg hA (add_nonneg (norm_nonneg _) hM)⟩
+  obtain ⟨p,hp,huniform⟩ := uniform_gaussian_ball (ι := ι) radius (r/B) (div_pos hr hB)
+  refine ⟨p,hp,?_⟩
+  intro L mean hLi hL hm
+  let center := L.symm (target-mean)
+  have hc : ‖center‖ ≤ A*(‖target‖+M) := by
+    exact (L.symm.toContinuousLinearMap.le_opNorm _).trans
+      (mul_le_mul hLi ((norm_sub_le _ _).trans (add_le_add le_rfl hm)) (norm_nonneg _) hA)
+  apply (huniform center hc).trans
+  rw [Measure.map_apply (by fun_prop) isOpen_ball.measurableSet]
+  apply measure_mono
+  intro z hz
+  change dist (mean+L z) target < r
+  rw [dist_eq_norm]
+  have he : mean+L z-target = L (z-center) := by
+    simp only [map_sub,center,L.apply_symm_apply]
+    abel
+  rw [he]
+  have hb := L.toContinuousLinearMap.le_opNorm (z-center)
+  have hh : ‖z-center‖ < r/B := by simpa only [mem_ball,dist_eq_norm] using hz
+  have hmul := (lt_div_iff₀ hB).mp hh
+  calc
+    _ ≤ B*‖z-center‖ := hb.trans (mul_le_mul_of_nonneg_right hL (norm_nonneg _))
+    _ < r := by nlinarith
+
+end
+end Yau.Probability
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/GaussianWaveJet.lean b/lean/OAI/Geometry/NodalSets/Waves/GaussianWaveJet.lean
new file mode 100644
index 0000000000000000000000000000000000000000..e2f4b1d4ba125a2f21a24376682aea27c2146db2
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/GaussianWaveJet.lean
@@ -0,0 +1,58 @@
+import OAI.Geometry.NodalSets.Coefficients.DualWaveCoefficient
+import OAI.Geometry.NodalSets.Coefficients.GaussianCoefficientVariance
+import OAI.Geometry.NodalSets.Elliptic.RealPhaseJet
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Yau.Probability Set Filter MeasureTheory ProbabilityTheory
+open scoped ContDiff
+noncomputable section
+variable {ι : Type*} [Fintype ι]
+
+def gaussianCoefficient (a : ι × Fin 2 → ℝ) (i : ι) : ℂ :=
+  (a (i,0):ℂ)+Complex.I*(a (i,1):ℂ)
+
+lemma pairLinearSum_eq_complex (z : ι → ℂ) (a : ι × Fin 2 → ℝ) :
+    pairLinearSum z a = ∑ i, (gaussianCoefficient a i*z i).re := by
+  classical
+  simp only [pairLinearSum,Fintype.sum_prod_type,Fin.sum_univ_succ,Fin.succ_ne_zero,Fin.succ_zero_eq_one,ite_true,ite_false,Fin.sum_univ_zero,add_zero,
+    gaussianCoefficient,Complex.mul_re,Complex.mul_im,Complex.add_re,Complex.add_im,Complex.ofReal_re,
+    Complex.ofReal_im,Complex.I_re,Complex.I_im,zero_mul,one_mul,mul_zero,zero_add]
+  apply Finset.sum_congr rfl
+  intro i _
+  ring
+
+def gaussianWaveField (V : ι → Coord → ℂ) (a : ι × Fin 2 → ℝ) (x : Coord) : ℝ :=
+  (∑ i, gaussianCoefficient a i*V i x).re
+
+lemma gaussianWaveField_dual (V : ι → Coord → ℂ) (a : ι × Fin 2 → ℝ)
+    (x v : Coord) (hV : ∀ i, DifferentiableAt ℝ (V i) x) (N S b : ℝ) :
+    Real.exp (-N*S)*(b*gaussianWaveField V a x+N⁻¹*fderiv ℝ (gaussianWaveField V a) x v) =
+      pairLinearSum (fun i ↦ dualWaveCoefficient N (V i) S x b v) a := by
+  classical
+  let W : Coord → ℂ := fun z ↦ ∑ i, gaussianCoefficient a i*V i z
+  have hW : DifferentiableAt ℝ W x := DifferentiableAt.fun_sum (fun i _ ↦ (hV i).const_mul _)
+  have hd : fderiv ℝ W x v = ∑ i, gaussianCoefficient a i*fderiv ℝ (V i) x v := by
+    dsimp only [W]
+    rw [fderiv_fun_sum (fun i _ ↦ (hV i).const_mul _)]
+    simp only [sum_apply]
+    apply Finset.sum_congr rfl
+    intro i _
+    rw [((hV i).hasFDerivAt.const_mul (gaussianCoefficient a i)).fderiv]
+    rfl
+  have hr := realPart_fderiv W x hW v
+  change fderiv ℝ (gaussianWaveField V a) x v = _ at hr
+  rw [hr,hd,pairLinearSum_eq_complex]
+  simp only [gaussianWaveField,Complex.re_sum,Finset.mul_sum,← Finset.sum_add_distrib]
+  apply Finset.sum_congr rfl
+  intro i _
+  simp only [dualWaveCoefficient,Complex.mul_re,Complex.mul_im,Complex.add_re,Complex.add_im,
+    Complex.mul_im,Complex.ofReal_re,Complex.ofReal_im,← Complex.ofReal_inv,
+    zero_mul,sub_zero,add_zero]
+  ring
+
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/LatticeActualDerivativeEnergy.lean b/lean/OAI/Geometry/NodalSets/Waves/LatticeActualDerivativeEnergy.lean
new file mode 100644
index 0000000000000000000000000000000000000000..a02f9762bb2798a1cc418d6eacc5c1fa073f3f8c
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/LatticeActualDerivativeEnergy.lean
@@ -0,0 +1,74 @@
+import OAI.Geometry.NodalSets.Elliptic.RescaledDerivativeMoments
+import OAI.Geometry.NodalSets.Waves.LatticePlaneWaveEnergy
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Set Filter
+open scoped ContDiff Topology
+noncomputable section
+
+lemma norm_square_le_two_sub_add {E : Type*} [SeminormedAddCommGroup E] (a b : E) :
+    ‖a‖^2 ≤ 2*‖a-b‖^2+2*‖b‖^2 := by
+  have h := norm_add_le (a-b) b
+  rw [sub_add_cancel] at h
+  have hsq := pow_le_pow_left₀ (norm_nonneg a) h 2
+  nlinarith [sq_nonneg (‖a-b‖-‖b‖)]
+
+theorem lattice_actual_derivative_energy
+    (g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ) {H : Set Coord}
+    (hH : IsCompact H) (hg : ContinuousOn g H)
+    (hp : ∀ y ∈ H, ∀ v, v ≠ 0 → 0 < g y v v) (d : ℕ) :
+    ∃ C > 0, ∀ (w S : Coord → ℝ) (D U Q : Set Coord) (m J K k0 : ℕ)
+      (a : LocalCompactWaveData g w S D m J K k0) (_ : H ⊆ D) (hUD : U ⊆ D),
+      U ⊆ H → IsOpen U → Bornology.IsBounded U → IsCompact Q → Q ⊆ U →
+      ContDiff ℝ ∞ S → d ≤ k0 →
+      ∀ᶠ n : ℕ in atTop, ∃ hfin : Fintype (SourceGrid U n), letI := hfin
+        ∀ x ∈ Q, ∀ v : Coord, ‖v‖ ≤ 2 → ∀ k : ℕ, k ≤ d →
+        ∑ i : SourceGrid U n × Fin 3,
+          ‖iteratedFDeriv ℝ k (normalizedRescaling
+            (latticeWave a.cover a.beams hUD n i.1 i.2) S n (sourceSignScale g S x)
+            (a.latticeSigma hUD n x) x) v‖^2 ≤ C := by
+  obtain ⟨C,hC,hplane⟩ := lattice_planeWave_derivative_energy g hH hg hp d
+  refine ⟨2+2*C,by positivity,?_⟩
+  intro w S D U Q m J K k0 a hHD hUD hUH hU hUb hQ hQU hS hd
+  filter_upwards [hplane w S D U Q m J K k0 a hHD hUD hUH hU hUb hQ hQU,
+    a.lattice_full_jet_approximation hUD hU hUb hQ hQU hS 2 (by norm_num) 1 (by norm_num),
+    a.estimates] with n hW he hest
+  obtain ⟨hfin,hW⟩ := hW
+  obtain ⟨hfin',he⟩ := he
+  have heq : hfin' = hfin := Subsingleton.elim _ _
+  subst hfin'
+  let := hfin
+  refine ⟨hfin,?_⟩
+  intro x hx v hv k hk
+  let W := a.latticePlaneWaveCoefficient hUD n x (sourceSignScale g S x)
+  let A := fun i : SourceGrid U n × Fin 3 ↦ normalizedRescaling
+    (latticeWave a.cover a.beams hUD n i.1 i.2) S n (sourceSignScale g S x)
+    (a.latticeSigma hUD n x) x
+  have hAs (i : SourceGrid U n × Fin 3) : ContDiff ℝ ∞ (A i) :=
+    normalizedRescaling_smooth _ (hest (latticeFrame a.cover hUD n i.1,i.2)).1 _ _ _ _ _
+  have hWs (i : SourceGrid U n × Fin 3) : ContDiff ℝ ∞ (W i) :=
+    a.latticePlaneWaveCoefficient_smooth _ _ _ _ _
+  have hk' : (k : WithTop ℕ∞) ≤ ∞ := by exact_mod_cast (show (k:ℕ∞) ≤ ⊤ from le_top)
+  have hid (i : SourceGrid U n × Fin 3) :
+      iteratedFDeriv ℝ k (fun z ↦ A i z-W i z) v =
+        iteratedFDeriv ℝ k (A i) v-iteratedFDeriv ℝ k (W i) v := by
+    change iteratedFDeriv ℝ k (A i-W i) v = _
+    exact iteratedFDeriv_sub_apply ((hAs i).of_le hk').contDiffAt ((hWs i).of_le hk').contDiffAt
+  have herr := he x hx _ (show 1 ≤ sourceSignScale g S x by
+    linarith [a.source_sign_scale_lower x (hUD (hQU hx))]) v hv (⟨k,by omega⟩ : Fin (k0+1))
+  change (∑ i, ‖iteratedFDeriv ℝ k (fun z ↦ A i z-W i z) v‖^2) ≤ 1 at herr
+  simp_rw [hid] at herr
+  have hw := hW x hx v k hk
+  have hb := Finset.sum_le_sum (fun i (_ : i ∈ (Finset.univ : Finset (SourceGrid U n × Fin 3))) ↦
+    norm_square_le_two_sub_add (iteratedFDeriv ℝ k (A i) v) (iteratedFDeriv ℝ k (W i) v))
+  simp only [Finset.sum_add_distrib,← Finset.mul_sum] at hb
+  change (∑ i, ‖iteratedFDeriv ℝ k (A i) v‖^2) ≤ _
+  change (∑ i, ‖iteratedFDeriv ℝ k (W i) v‖^2) ≤ C at hw
+  linarith
+
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/LatticeActualTwoPoint.lean b/lean/OAI/Geometry/NodalSets/Waves/LatticeActualTwoPoint.lean
new file mode 100644
index 0000000000000000000000000000000000000000..2fd10cc55eb5a767870101a901d05f34741b24b4
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/LatticeActualTwoPoint.lean
@@ -0,0 +1,73 @@
+import OAI.Geometry.NodalSets.Coefficients.TwoPointCoefficientEnergy
+import OAI.Geometry.NodalSets.Waves.LatticeTwoPointGap
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Yau.Probability MeasureTheory ProbabilityTheory Set Filter
+open scoped ContDiff Topology
+noncomputable section
+
+theorem lattice_actual_two_point_energy
+    (g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ) {H : Set Coord}
+    (hH : IsCompact H) (hg : ContinuousOn g H)
+    (hp : ∀ y ∈ H, ∀ v, v ≠ 0 → 0 < g y v v) :
+    ∃ tau : ℝ, 0 < tau ∧ tau < 1/4 ∧ ∃ delta > 0,
+      ∀ (w S : Coord → ℝ) (D U Q : Set Coord) (m J K k0 : ℕ)
+        (a : LocalCompactWaveData g w S D m J K k0) (_ : H ⊆ D) (hUD : U ⊆ D),
+        U ⊆ H → IsOpen U → Bornology.IsBounded U → IsCompact Q → Q ⊆ U → ContDiff ℝ ∞ S →
+        ∀ᶠ n : ℕ in atTop, ∃ hfin : Fintype (SourceGrid U n), letI := hfin
+          ∀ x ∈ Q, ∃ j : Fin 4, ∀ u v : ℝ,
+            let coeff := fun i : SourceGrid U n × Fin 3 ↦ normalizedRescaling
+              (latticeWave a.cover a.beams hUD n i.1 i.2) S n
+              (sourceSignScale g S x) (a.latticeSigma hUD n x) x
+            delta*(u^2+v^2) ≤ ∑ i,
+              ‖(u:ℂ)*coeff i 0+(v:ℂ)*coeff i (tau • Pi.single j 1)‖^2 := by
+  obtain ⟨tau,ht,ht4,d,hd,hgaps⟩ := lattice_plane_wave_two_point_gap g hH hg hp
+  refine ⟨tau,ht,ht4,d/4,by positivity,?_⟩
+  intro w S D U Q m J K k0 a hHD hUD hUH hU hUb hQ hQU hS
+  filter_upwards [hgaps w S D U Q m J K k0 a hHD hUD hUH hU hUb hQ hQU,
+    a.lattice_full_jet_approximation hUD hU hUb hQ hQU hS 2 (by norm_num) (d/8) (by positivity)]
+    with n hgap herr
+  obtain ⟨hfin,hgap⟩ := hgap
+  obtain ⟨hfin',herr⟩ := herr
+  have heq : hfin' = hfin := Subsingleton.elim _ _
+  subst hfin'
+  let := hfin
+  refine ⟨hfin,?_⟩
+  intro x hx
+  obtain ⟨j,hM,hM1,hW,hC0,hC⟩ := hgap x hx
+  dsimp only at hM hM1 hW hC0 hC
+  let W := a.latticePlaneWaveCoefficient hUD n x (sourceSignScale g S x)
+  let A := fun i : SourceGrid U n × Fin 3 ↦ normalizedRescaling
+    (latticeWave a.cover a.beams hUD n i.1 i.2) S n
+    (sourceSignScale g S x) (a.latticeSigma hUD n x) x
+  have hgap' : ∀ u v : ℝ, d*(u^2+v^2) ≤
+      ∑ i, ‖(u:ℂ)*W i 0+(v:ℂ)*W i (tau • Pi.single j 1)‖^2 := by
+    intro u v
+    apply two_point_energy_of_gap _ _ _ _ d rfl _ rfl _ _ u v
+    · simpa only [gaussianWaveField_eq_pair] using hW
+    · have hm0 := variance_nonneg (fun coeff ↦ gaussianWaveField W coeff 0) gaussianPairs
+      rw [gaussianWaveField_eq_pair] at hm0
+      simp only [gaussianWaveField_eq_pair] at hC0
+      linarith
+    · simpa only [gaussianWaveField_eq_pair] using hC
+  have hs : 1 ≤ sourceSignScale g S x := by
+    linarith [a.source_sign_scale_lower x (hUD (hQU hx))]
+  have hnorm : ‖(tau • Pi.single j (1:ℝ) : Coord)‖ ≤ 2 := by
+    rw [norm_smul,Real.norm_eq_abs,abs_of_pos ht,Pi.norm_single,norm_one,mul_one]
+    linarith
+  have he0 := herr x hx _ hs 0 (by simp) (0 : Fin (k0+1))
+  have he1 := herr x hx _ hs _ hnorm (0 : Fin (k0+1))
+  simp +instances only [Fin.val_zero,norm_iteratedFDeriv_zero] at he0 he1
+  refine ⟨j,?_⟩
+  intro u v
+  have hh := two_point_energy_perturbation (fun i ↦ W i 0) (fun i ↦ W i (tau • Pi.single j 1))
+    (fun i ↦ A i 0) (fun i ↦ A i (tau • Pi.single j 1)) d (d/8) he0 he1 hgap' u v
+  convert hh using 1
+  ring
+
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/LatticeCount.lean b/lean/OAI/Geometry/NodalSets/Waves/LatticeCount.lean
new file mode 100644
index 0000000000000000000000000000000000000000..de04820db1bbe10e0cd6d6483196c9643a642921
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/LatticeCount.lean
@@ -0,0 +1,75 @@
+import Mathlib.SetTheory.Cardinal.Finite
+import OAI.Geometry.NodalSets.Waves.LatticeGeometry
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Set
+noncomputable section
+
+lemma lattice_scale_eq_inv_sqrt (n : ℕ) : (n:ℝ)^(-1/2:ℝ) = (Real.sqrt n)⁻¹ := by
+  rw [show (-1/2:ℝ) = -(1/2) by ring,Real.rpow_neg,Real.sqrt_eq_rpow]
+  positivity
+
+theorem source_grid_card_bound {U : Set Coord} (hU : Bornology.IsBounded U) :
+    ∃ C > 0, ∀ n : ℕ, 0 < n → (Nat.card (SourceGrid U n) : ℝ) ≤ C*(n:ℝ)^2 := by
+  obtain ⟨B,hB,hbound⟩ := hU.exists_pos_norm_le
+  refine ⟨(2*B+3)^4,by positivity,?_⟩
+  intro n hn
+  have hnR : (1:ℝ) ≤ n := by exact_mod_cast hn
+  have hs : 0 < Real.sqrt n := Real.sqrt_pos.mpr (by exact_mod_cast hn)
+  have hs1 : 1 ≤ Real.sqrt n := Real.one_le_sqrt.mpr hnR
+  have hs2 : (Real.sqrt n)^2 = (n:ℝ) := Real.sq_sqrt (by positivity)
+  let M : ℤ := ⌈B*Real.sqrt n⌉
+  have hM0 : 0 ≤ M := Int.ceil_nonneg (by positivity)
+  have hMc : (M:ℝ) < B*Real.sqrt n+1 := Int.ceil_lt_add_one _
+  have hcoords (z : SourceGrid U n) (i : Fin 4) : z.val i ∈ Icc (-M) M := by
+    have hi := (norm_le_pi_norm (scaledLatticePoint n z) i).trans (hbound _ z.property)
+    change |(n:ℝ)^(-1/2:ℝ)*(z.val i:ℝ)| ≤ B at hi
+    rw [lattice_scale_eq_inv_sqrt,abs_mul,abs_of_pos (inv_pos.mpr hs)] at hi
+    have hmul := mul_le_mul_of_nonneg_left hi hs.le
+    have hcancel : Real.sqrt n * ((Real.sqrt n)⁻¹ * |(z.val i:ℝ)|) = |(z.val i:ℝ)| := by
+      field_simp
+    rw [hcancel] at hmul
+    have ha : |(z.val i:ℝ)| ≤ (M:ℝ) := hmul.trans (by simpa [mul_comm] using Int.le_ceil (B*Real.sqrt n))
+    have hp := abs_le.mp ha
+    exact ⟨by exact_mod_cast hp.1,by exact_mod_cast hp.2⟩
+  let f : SourceGrid U n → (Fin 4 → Icc (-M) M) := fun z i ↦ ⟨z.val i,hcoords z i⟩
+  have hfi : Function.Injective f := by
+    intro z w h
+    apply Subtype.ext
+    funext i
+    exact congrArg Subtype.val (congrFun h i)
+  have hc := Nat.card_le_card_of_injective f hfi
+  have hbox : Nat.card (Fin 4 → Icc (-M) M) = ((2*M+1).toNat)^4 := by
+    rw [Nat.card_fun]
+    simp [Nat.card_eq_fintype_card,Int.card_Icc,show M+1- -M = 2*M+1 by ring]
+  rw [hbox] at hc
+  have hcR : (Nat.card (SourceGrid U n):ℝ) ≤ ((2*M+1).toNat:ℝ)^4 := by exact_mod_cast hc
+  have hcast : ((2*M+1).toNat:ℝ) = 2*(M:ℝ)+1 := by
+    have hi : ((2*M+1).toNat:ℤ) = 2*M+1 := Int.toNat_of_nonneg (by omega)
+    exact_mod_cast hi
+  rw [hcast] at hcR
+  have hmnon : (0:ℝ) ≤ 2*(M:ℝ)+1 := by exact_mod_cast (show (0:ℤ) ≤ 2*M+1 by omega)
+  have hmle : 2*(M:ℝ)+1 ≤ (2*B+3)*Real.sqrt n := by nlinarith
+  have hpow := pow_le_pow_left₀ hmnon hmle 4
+  calc
+    _ ≤ (2*(M:ℝ)+1)^4 := hcR
+    _ ≤ ((2*B+3)*Real.sqrt n)^4 := hpow
+    _ = (2*B+3)^4*(n:ℝ)^2 := by
+      rw [mul_pow,show (Real.sqrt n)^4 = ((Real.sqrt n)^2)^2 by ring,hs2]
+
+theorem source_coefficient_card_bound {U : Set Coord} (hU : Bornology.IsBounded U) :
+    ∃ C > 0, ∀ n : ℕ, 0 < n → (Nat.card (SourceGrid U n × Fin 3):ℝ) ≤ C*(n:ℝ)^2 := by
+  obtain ⟨C,hC,hb⟩ := source_grid_card_bound hU
+  refine ⟨3*C,by positivity,?_⟩
+  intro n hn
+  have h := hb n hn
+  rw [Nat.card_prod]
+  norm_num
+  nlinarith
+
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/LatticeCovariance.lean b/lean/OAI/Geometry/NodalSets/Waves/LatticeCovariance.lean
new file mode 100644
index 0000000000000000000000000000000000000000..559d451c1952648e30f2d0f05c597395bb7e5343
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/LatticeCovariance.lean
@@ -0,0 +1,60 @@
+import OAI.Geometry.NodalSets.Waves.LatticeCovariance2
+import OAI.Geometry.NodalSets.Waves.SeededGaussianJet
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Yau.Probability Set Filter MeasureTheory ProbabilityTheory
+open scoped ContDiff Topology
+noncomputable section
+variable {g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ} {w S : Coord → ℝ}
+variable {D U : Set Coord} {m J K k0 : ℕ}
+namespace LocalCompactWaveData
+variable (a : LocalCompactWaveData g w S D m J K k0)
+
+def latticeSeededRandomDual (hUD : U ⊆ D) (n : ℕ) (hfin : Fintype (SourceGrid U n))
+    (seed : Coord → ℝ) (x : Coord) (b : ℝ) (v : Coord)
+    (coeff : ((SourceGrid U n × Fin 3) × Fin 2) → ℝ) : ℝ :=
+  letI := hfin
+  let V := fun i : SourceGrid U n × Fin 3 ↦ latticeWave a.cover a.beams hUD n i.1 i.2
+  Real.exp (-(n:ℝ)*S x)*(b*(seed x+gaussianWaveField V coeff x)+
+    (n:ℝ)⁻¹*fderiv ℝ (fun z ↦ seed z+gaussianWaveField V coeff z) x v)
+
+lemma variance_latticeSeededRandomDual (hUD : U ⊆ D) (n : ℕ) (hfin : Fintype (SourceGrid U n))
+    (seed : Coord → ℝ) (x : Coord) (hseed : DifferentiableAt ℝ seed x)
+    (hV : ∀ z j, DifferentiableAt ℝ (latticeWave a.cover a.beams hUD n z j) x)
+    (b : ℝ) (v : Coord) :
+    letI := hfin
+    Var[a.latticeSeededRandomDual hUD n hfin seed x b v; gaussianPairs] =
+      a.latticeEnergy hUD n hfin x b v := by
+  let := hfin
+  exact seeded_gaussian_dual_variance
+    (fun i : SourceGrid U n × Fin 3 ↦ latticeWave a.cover a.beams hUD n i.1 i.2)
+    seed x v (fun i ↦ hV i.1 i.2) hseed (n:ℝ) (S x) b
+
+theorem actual_lattice_covariance_lower_bound (hUD : U ⊆ D) (hU : IsOpen U)
+    (hUb : Bornology.IsBounded U) {Q : Set Coord} (hQ : IsCompact Q) (hQU : Q ⊆ U) :
+    ∃ c > 0, ∀ᶠ n : ℕ in atTop, ∃ hfin : Fintype (SourceGrid U n),
+      letI := hfin
+      ∀ (seed : Coord → ℝ) (x : Coord), x ∈ Q → DifferentiableAt ℝ seed x → ∀ (b : ℝ) (v : Coord),
+      c/(n:ℝ)*(b^2+∑ i : Fin 4, (v i)^2) ≤
+        Var[a.latticeSeededRandomDual hUD n hfin seed x b v; gaussianPairs] := by
+  obtain ⟨c,hc,hb⟩ := a.euclidean_lattice_covariance_lower_bound hUD hU hUb hQ hQU
+  refine ⟨c,hc,?_⟩
+  filter_upwards [hb,lattice_wave_estimates a.cover a.beams hUD] with n hn hs
+  obtain ⟨hfin,hbound⟩ := hn
+  let := hfin
+  refine ⟨hfin,?_⟩
+  intro seed x hx hseed b v
+  have hV : ∀ z j, DifferentiableAt ℝ (latticeWave a.cover a.beams hUD n z j) x :=
+    fun z j ↦ ((hs z j).1.differentiable (by simp)).differentiableAt
+  rw [a.variance_latticeSeededRandomDual hUD n hfin seed x hseed hV]
+  have hb := hbound x hx b v
+  rw [a.variance_latticeRandomDual] at hb
+  exact hb
+
+end LocalCompactWaveData
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/LatticeCovariance2.lean b/lean/OAI/Geometry/NodalSets/Waves/LatticeCovariance2.lean
new file mode 100644
index 0000000000000000000000000000000000000000..ec59619c75bb6af24c9c66163356d8b93e7fe11a
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/LatticeCovariance2.lean
@@ -0,0 +1,70 @@
+import OAI.Geometry.NodalSets.Coefficients.GaussianCoefficientVariance
+import OAI.Geometry.NodalSets.Waves.LatticeEnergyLowerBound
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Yau.Probability Set Filter MeasureTheory ProbabilityTheory
+open scoped ContDiff Topology
+noncomputable section
+variable {g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ} {w S : Coord → ℝ}
+variable {D U : Set Coord} {m J K k0 : ℕ}
+namespace LocalCompactWaveData
+variable (a : LocalCompactWaveData g w S D m J K k0)
+
+def latticeRandomDual (hUD : U ⊆ D) (n : ℕ) (hfin : Fintype (SourceGrid U n))
+    (x : Coord) (b : ℝ) (v : Coord) : (((SourceGrid U n × Fin 3) × Fin 2) → ℝ) → ℝ :=
+  letI := hfin
+  pairLinearSum (fun i : SourceGrid U n × Fin 3 ↦
+    dualWaveCoefficient n (latticeWave a.cover a.beams hUD n i.1 i.2) (S x) x b v)
+
+lemma variance_latticeRandomDual (hUD : U ⊆ D) (n : ℕ) (hfin : Fintype (SourceGrid U n))
+    (x : Coord) (b : ℝ) (v : Coord) :
+    letI := hfin
+    Var[a.latticeRandomDual hUD n hfin x b v; gaussianPairs] = a.latticeEnergy hUD n hfin x b v := by
+  let := hfin
+  exact variance_pairLinearSum _
+
+theorem lattice_covariance_lower_bound (hUD : U ⊆ D) (hU : IsOpen U) (hUb : Bornology.IsBounded U)
+    {Q : Set Coord} (hQ : IsCompact Q) (hQU : Q ⊆ U) :
+    ∃ c > 0, ∀ᶠ n : ℕ in atTop, ∃ hfin : Fintype (SourceGrid U n), ∀ x ∈ Q, ∀ b v,
+      letI := hfin
+      c/(n:ℝ)*(b^2+‖v‖^2) ≤ Var[a.latticeRandomDual hUD n hfin x b v; gaussianPairs] := by
+  obtain ⟨c,hc,hb⟩ := a.lattice_energy_lower_bound hUD hU hUb hQ hQU
+  refine ⟨c,hc,?_⟩
+  filter_upwards [hb] with n hn
+  obtain ⟨hfin,hbound⟩ := hn
+  let := hfin
+  refine ⟨hfin,?_⟩
+  intro x hx b v
+  rw [a.variance_latticeRandomDual]
+  exact hbound x hx b v
+
+theorem euclidean_lattice_covariance_lower_bound (hUD : U ⊆ D) (hU : IsOpen U) (hUb : Bornology.IsBounded U)
+    {Q : Set Coord} (hQ : IsCompact Q) (hQU : Q ⊆ U) :
+    ∃ c > 0, ∀ᶠ n : ℕ in atTop, ∃ hfin : Fintype (SourceGrid U n), ∀ x ∈ Q, ∀ b v,
+      letI := hfin
+      c/(n:ℝ)*(b^2+∑ i : Fin 4, (v i)^2) ≤ Var[a.latticeRandomDual hUD n hfin x b v; gaussianPairs] := by
+  obtain ⟨c,hc,hb⟩ := a.lattice_covariance_lower_bound hUD hU hUb hQ hQU
+  refine ⟨c/4,by positivity,?_⟩
+  filter_upwards [hb,eventually_ge_atTop (1:ℕ)] with n hn hn1
+  obtain ⟨hfin,hbound⟩ := hn
+  let := hfin
+  refine ⟨hfin,?_⟩
+  intro x hx b v
+  apply le_trans _ (hbound x hx b v)
+  have hcoord (i : Fin 4) : (v i)^2 ≤ ‖v‖^2 := by
+    have hh := norm_le_pi_norm v i
+    change |v i| ≤ ‖v‖ at hh
+    nlinarith [sq_abs (v i),norm_nonneg v,abs_nonneg (v i)]
+  have hsum := Finset.sum_le_sum (fun i (_ : i ∈ (Finset.univ : Finset (Fin 4))) ↦ hcoord i)
+  norm_num at hsum
+  have hh : b^2+∑ i : Fin 4, (v i)^2 ≤ 4*(b^2+‖v‖^2) := by nlinarith [sq_nonneg b]
+  have hh' := mul_le_mul_of_nonneg_left hh (by positivity : 0 ≤ c/(4*(n:ℝ)))
+  convert hh' using 1 <;> ring
+
+end LocalCompactWaveData
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/LatticeDerivativeMoments.lean b/lean/OAI/Geometry/NodalSets/Waves/LatticeDerivativeMoments.lean
new file mode 100644
index 0000000000000000000000000000000000000000..238c3b621e7afa72648ef4ba570cbcc8863fe472
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/LatticeDerivativeMoments.lean
@@ -0,0 +1,63 @@
+import OAI.Geometry.NodalSets.Elliptic.SignScaleSeed
+import OAI.Geometry.NodalSets.Waves.LatticeActualDerivativeEnergy
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Yau.Probability Set Filter MeasureTheory ProbabilityTheory
+open scoped ContDiff Topology
+noncomputable section
+
+theorem lattice_derivative_second_moments
+    (g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ) {H : Set Coord}
+    (hH : IsCompact H) (hg : ContinuousOn g H)
+    (hp : ∀ y ∈ H, ∀ v, v ≠ 0 → 0 < g y v v) (d : ℕ) :
+    ∃ C > 0, ∀ (w S S0 T0 : Coord → ℝ) (D U Q : Set Coord) (m J K k0 : ℕ)
+      (a : LocalCompactWaveData g w S D m J K k0) (_ : H ⊆ D) (hUD : U ⊆ D),
+      U ⊆ H → IsOpen U → Bornology.IsBounded U → IsCompact Q → Q ⊆ U →
+      ContDiff ℝ ∞ S → ContDiff ℝ ∞ S0 → ContDiff ℝ ∞ T0 → d ≤ k0 →
+      ∀ gamma > 0, (∀ x ∈ Q, S0 x+gamma ≤ S x) →
+      ∀ᶠ n : ℕ in atTop, ∃ hfin : Fintype (SourceGrid U n), letI := hfin
+        ∀ x ∈ Q, ∀ v : Coord, sourceEuclideanNorm v ≤ 2 → ∀ k : ℕ, k ≤ d →
+        ∀ dirs : Fin k → Coord, ‖dirs‖ ≤ 1 →
+        let F := rescaledGaussianField
+          (fun i : SourceGrid U n × Fin 3 ↦ latticeWave a.cover a.beams hUD n i.1 i.2)
+          S0 T0 S n (sourceSignScale g S x) (a.latticeSigma hUD n x) x
+        MemLp (fun coeff ↦ iteratedFDeriv ℝ k (F coeff) v dirs) 2 gaussianPairs ∧
+        (∫ coeff, (iteratedFDeriv ℝ k (F coeff) v dirs)^2 ∂gaussianPairs) ≤ C := by
+  obtain ⟨C,hC,henergy⟩ := lattice_actual_derivative_energy g hH hg hp d
+  refine ⟨1+C,by positivity,?_⟩
+  intro w S S0 T0 D U Q m J K k0 a hHD hUD hUH hU hUb hQ hQU hS hS0 hT0 hd gamma hgamma hgap
+  filter_upwards [henergy w S D U Q m J K k0 a hHD hUD hUH hU hUb hQ hQU hS hd,
+    a.sign_scale_seed_decay hUD hU hUb hQ hQU hS S0 T0 hS0 hT0 gamma hgamma hgap d 1 (by norm_num),
+    a.estimates] with n he hs hest
+  obtain ⟨hfin,he⟩ := he
+  obtain ⟨hfin',hs⟩ := hs
+  have heq : hfin' = hfin := Subsingleton.elim _ _
+  subst hfin'
+  let := hfin
+  refine ⟨hfin,?_⟩
+  intro x hx v hv k hk dirs hdirs
+  let V := fun i : SourceGrid U n × Fin 3 ↦ latticeWave a.cover a.beams hUD n i.1 i.2
+  have hV (i : SourceGrid U n × Fin 3) : ContDiff ℝ ∞ (V i) :=
+    (hest (latticeFrame a.cover hUD n i.1,i.2)).1
+  refine ⟨rescaledGaussianField_derivative_memLp V hV _ _ _ hS0 hT0 _ _ _ _ _ _ _,?_⟩
+  rw [rescaledGaussianField_derivative_moment V hV _ _ _ hS0 hT0]
+  have hseed := hs x hx v hv (⟨k,by omega⟩ : Fin (d+1))
+  have hseed' := (iteratedFDeriv ℝ k
+    (rescaledSeed S0 T0 S n (sourceSignScale g S x) (a.latticeSigma hUD n x) x) v).unit_le_opNorm hdirs
+  have hmean := pow_le_pow_left₀ (norm_nonneg _) (hseed'.trans hseed) 2
+  simp only [Real.norm_eq_abs,sq_abs,one_pow] at hmean
+  have hsum := Finset.sum_le_sum (fun i (_ : i ∈ (Finset.univ : Finset (SourceGrid U n × Fin 3))) ↦
+    pow_le_pow_left₀ (norm_nonneg _)
+      ((iteratedFDeriv ℝ k (normalizedRescaling (V i) S n (sourceSignScale g S x)
+        (a.latticeSigma hUD n x) x) v).unit_le_opNorm hdirs) 2)
+  have hbound := he x hx v ((norm_le_sourceEuclideanNorm v).trans hv) k hk
+  change (∑ i, ‖iteratedFDeriv ℝ k (normalizedRescaling (V i) S n (sourceSignScale g S x)
+    (a.latticeSigma hUD n x) x) v‖^2) ≤ C at hbound
+  linarith
+
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/LatticeDerivativeTail.lean b/lean/OAI/Geometry/NodalSets/Waves/LatticeDerivativeTail.lean
new file mode 100644
index 0000000000000000000000000000000000000000..e915f89b95002c1f7e39c1bb58dfdf301f1dd4f8
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/LatticeDerivativeTail.lean
@@ -0,0 +1,75 @@
+import OAI.Geometry.NodalSets.Waves.GaussianTailBound
+import OAI.Geometry.NodalSets.Waves.LatticeVarianceUpper
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Yau.Probability Set Filter
+open scoped ContDiff Topology
+noncomputable section
+variable {g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ} {w S : Coord → ℝ}
+variable {D U : Set Coord} {m J K k0 : ℕ}
+namespace LocalCompactWaveData
+variable (a : LocalCompactWaveData g w S D m J K k0)
+
+theorem lattice_derivative_tail_bound (hUD : U ⊆ D) :
+    ∃ C > 0, ∀ᶠ n : ℕ in atTop, ∀ [Fintype (SourceGrid U n)], ∀ x : Coord,
+      ∀ k : Fin (k0+1), ∀ r : ℝ, 0 ≤ r →
+      ∑ i ∈ Finset.univ.filter (fun i : SourceGrid U n × Fin 3 ↦
+          r < sourceEuclideanNorm (x-scaledLatticePoint n i.1)),
+        ‖iteratedFDeriv ℝ k.val (latticeWave a.cover a.beams hUD n i.1 i.2) x‖^2 ≤
+        C*(n:ℝ)^(2*k.val)*Real.exp (2*(n:ℝ)*S x)*Real.exp (-a.beams.c*(n:ℝ)*r^2) := by
+  classical
+  obtain ⟨B,hB,hgauss⟩ := source_lattice_gaussian_sum a.beams.c_pos
+  refine ⟨3*a.beams.Cw^2*B,by have := a.beams.Cw_pos; positivity,?_⟩
+  filter_upwards [a.estimates,eventually_gt_atTop (0:ℕ)] with n hn hnpos
+  intro hfin x k r hr
+  let d : SourceGrid U n × Fin 3 → ℝ := fun i ↦ sourceEuclideanNorm (x-scaledLatticePoint n i.1)
+  let R := a.beams.Cw^2*(n:ℝ)^(2*k.val)*Real.exp (2*(n:ℝ)*S x)
+  have hb (i : SourceGrid U n × Fin 3) :
+      ‖iteratedFDeriv ℝ k.val (latticeWave a.cover a.beams hUD n i.1 i.2) x‖^2 ≤
+        R*Real.exp (-2*(a.beams.c*(n:ℝ))*(d i)^2) := by
+    have h := ((hn (latticeFrame a.cover hUD n i.1,i.2)).2.2.2.2 x).1 k
+    simp only [latticeFrame_center] at h
+    change ‖iteratedFDeriv ℝ k.val (latticeWave a.cover a.beams hUD n i.1 i.2) x‖ ≤ _ at h
+    refine (pow_le_pow_left₀ (norm_nonneg _) h 2).trans_eq ?_
+    dsimp [R,d]
+    rw [mul_pow,mul_pow,← Real.exp_nat_mul,← pow_mul]
+    simp only [mul_assoc,← Real.exp_add]
+    congr 2 <;> ring_nf
+  have hg : ∑ i : SourceGrid U n × Fin 3, Real.exp (-(a.beams.c*(n:ℝ))*(d i)^2) ≤ 3*B := by
+    dsimp [d]
+    rw [Fintype.sum_prod_type]
+    simp only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul]
+    rw [← Finset.mul_sum]
+    norm_num only [Nat.cast_ofNat]
+    apply mul_le_mul_of_nonneg_left _ (by norm_num)
+    simpa only [neg_mul] using hgauss U n hnpos x
+  have hh := finite_gaussian_tail d
+    (fun i ↦ ‖iteratedFDeriv ℝ k.val (latticeWave a.cover a.beams hUD n i.1 i.2) x‖^2)
+    (by have := a.beams.c_pos; positivity : 0 ≤ a.beams.c*(n:ℝ))
+    (by dsimp [R]; positivity : 0 ≤ R) hr (fun i ↦ sourceEuclideanNorm_nonneg _) hb hg
+  convert hh using 1
+  dsimp [R]
+  simp only [neg_mul]
+  ring
+
+theorem lattice_nonmain_derivative_bound (hUD : U ⊆ D) :
+    ∃ C > 0, ∀ᶠ n : ℕ in atTop, ∀ [Fintype (SourceGrid U n)], ∀ x : Coord,
+      ∀ k : Fin (k0+1),
+      ∑ i ∈ Finset.univ.filter (fun i : SourceGrid U n × Fin 3 ↦
+          (n:ℝ)^(-5/12:ℝ) < sourceEuclideanNorm (x-scaledLatticePoint n i.1)),
+        ‖iteratedFDeriv ℝ k.val (latticeWave a.cover a.beams hUD n i.1 i.2) x‖^2 ≤
+        C*(n:ℝ)^(2*k.val)*Real.exp (2*(n:ℝ)*S x)*Real.exp (-a.beams.c*(n:ℝ)^(1/6:ℝ)) := by
+  obtain ⟨C,hC,hb⟩ := a.lattice_derivative_tail_bound hUD
+  refine ⟨C,hC,?_⟩
+  filter_upwards [hb,eventually_gt_atTop (0:ℕ)] with n hn hnpos
+  intro hfin x k
+  have h := hn x k ((n:ℝ)^(-5/12:ℝ)) (by positivity)
+  rwa [mul_assoc (-a.beams.c),main_center_scale _ (by exact_mod_cast hnpos)] at h
+
+end LocalCompactWaveData
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/LatticeDirectionalCenters.lean b/lean/OAI/Geometry/NodalSets/Waves/LatticeDirectionalCenters.lean
new file mode 100644
index 0000000000000000000000000000000000000000..fbe451589ca752930a67066d721152c1929c52a5
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/LatticeDirectionalCenters.lean
@@ -0,0 +1,49 @@
+import OAI.Geometry.NodalSets.Elliptic.HalfScaleLemmas
+import OAI.Geometry.NodalSets.Waves.LatticeGeometry
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Set Metric Filter
+noncomputable section
+
+def unitDirection (v : Coord) : Coord := (‖v‖⁻¹:ℝ) • v
+
+lemma norm_unitDirection_le (v : Coord) : ‖unitDirection v‖ ≤ 1 := by
+  by_cases hv : v = 0
+  · simp [unitDirection,hv]
+  have hn : 0 < ‖v‖ := norm_pos_iff.mpr hv
+  simp only [unitDirection,norm_smul,Real.norm_eq_abs,abs_of_pos (inv_pos.mpr hn),inv_mul_cancel₀ hn.ne']
+  exact le_rfl
+
+def directionalLatticeIndex (n : ℕ) (x : Coord) (K : ℝ) (v : Coord) : Fin 4 → ℤ :=
+  roundedLatticeIndex n (x+(K*(n:ℝ)^(-1/2:ℝ)) • unitDirection v)
+
+lemma directional_rounding_error {n : ℕ} (hn : 0 < n) (x : Coord) (K : ℝ) (v : Coord) :
+    ‖scaledLatticePoint n (directionalLatticeIndex n x K v)-
+      (x+(K*(n:ℝ)^(-1/2:ℝ)) • unitDirection v)‖ ≤ 2*(n:ℝ)^(-1/2:ℝ) :=
+  (norm_le_sourceEuclideanNorm _).trans (rounded_lattice_distance hn _)
+
+lemma directional_center_distance {n : ℕ} (hn : 0 < n) (x : Coord) (K : ℝ) (hK : 0 ≤ K) (v : Coord) :
+    ‖scaledLatticePoint n (directionalLatticeIndex n x K v)-x‖ ≤ (K+2)*(n:ℝ)^(-1/2:ℝ) := by
+  let z := x+(K*(n:ℝ)^(-1/2:ℝ)) • unitDirection v
+  have h := norm_sub_le_norm_sub_add_norm_sub (scaledLatticePoint n (directionalLatticeIndex n x K v)) z x
+  have hz : ‖z-x‖ ≤ K*(n:ℝ)^(-1/2:ℝ) := by
+    simp only [z,add_sub_cancel_left,norm_smul,Real.norm_eq_abs,
+      abs_of_nonneg (mul_nonneg hK (lattice_scale_pos hn).le)]
+    exact (mul_le_mul_of_nonneg_left (norm_unitDirection_le v)
+      (mul_nonneg hK (lattice_scale_pos hn).le)).trans_eq (mul_one _)
+  have hr := directional_rounding_error hn x K v
+  dsimp [z] at h hz
+  nlinarith
+
+lemma directional_center_mem {U : Set Coord} {n : ℕ} (hn : 0 < n)
+    (x : Coord) (K : ℝ) (hK : 0 ≤ K) (v : Coord)
+    (hball : closedBall x ((K+2)*(n:ℝ)^(-1/2:ℝ)) ⊆ U) :
+    scaledLatticePoint n (directionalLatticeIndex n x K v) ∈ U :=
+  hball (by simpa only [mem_closedBall,dist_eq_norm] using directional_center_distance hn x K hK v)
+
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/LatticeEnergyLowerBound.lean b/lean/OAI/Geometry/NodalSets/Waves/LatticeEnergyLowerBound.lean
new file mode 100644
index 0000000000000000000000000000000000000000..3eafedeef13ced785dac8882106eeeccff05a5d4
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/LatticeEnergyLowerBound.lean
@@ -0,0 +1,152 @@
+import OAI.Geometry.NodalSets.Elliptic.FiniteJetEnergy
+import OAI.Geometry.NodalSets.Waves.LatticeDirectionalCenters
+import OAI.Geometry.NodalSets.Waves.LocalCovarianceGeometry
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Set Metric Filter
+open scoped ContDiff Topology
+noncomputable section
+
+lemma compact_inner_ball {Q U : Set Coord} (hQ : IsCompact Q) (hU : IsOpen U) (hQU : Q ⊆ U) :
+    ∃ r > 0, ∀ x ∈ Q, closedBall x r ⊆ U := by
+  let : CompactSpace Q := isCompact_iff_compactSpace.mp hQ
+  obtain ⟨r,hr,hri⟩ := compact_family_common_radius isCompact_univ
+    (fun z : Q × Coord ↦ (z.1:Coord)+z.2)
+    ((continuous_subtype_val.comp continuous_fst).add continuous_snd)
+    hU (fun t _ ↦ by simpa only [add_zero] using hQU t.property)
+  refine ⟨r/2,by positivity,?_⟩
+  intro x hx z hz
+  have hn : ‖z-x‖ ≤ r/2 := by simpa only [mem_closedBall,dist_eq_norm] using hz
+  have h := hri ⟨x,hx⟩ (mem_univ _) (z-x) (by linarith)
+  simpa using h
+
+variable {g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ} {w S : Coord → ℝ}
+variable {D U : Set Coord} {m J K k0 : ℕ}
+namespace LocalCompactWaveData
+variable (a : LocalCompactWaveData g w S D m J K k0)
+
+def latticeEnergy (hUD : U ⊆ D) (n : ℕ) (hfin : Fintype (SourceGrid U n))
+    (x : Coord) (b : ℝ) (v : Coord) : ℝ :=
+  letI := hfin
+  ∑ i : SourceGrid U n × Fin 3,
+    ‖dualWaveCoefficient n (latticeWave a.cover a.beams hUD n i.1 i.2) (S x) x b v‖^2
+
+theorem lattice_energy_lower_bound (hUD : U ⊆ D) (hU : IsOpen U) (hUb : Bornology.IsBounded U)
+    {Q : Set Coord} (hQ : IsCompact Q) (hQU : Q ⊆ U) :
+    ∃ c > 0, ∀ᶠ n : ℕ in atTop, ∃ hfin : Fintype (SourceGrid U n), ∀ x ∈ Q, ∀ b v,
+      c/(n:ℝ)*(b^2+‖v‖^2) ≤ a.latticeEnergy hUD n hfin x b v := by
+  obtain ⟨mu,hmu,B,hB,hgeom⟩ := a.original_covariance_geometry
+  obtain ⟨e,he,E,hE,heta⟩ := a.lattice_eta_bounds hUD
+  let R : ℝ := (2*B+2)/mu
+  have hR : 0 < R := by dsimp [R]; positivity
+  let L := R+2
+  have hL : 0 < L := by dsimp [L]; linarith
+  have hlarge : 2*B+1 ≤ R*mu := by dsimp [R]; rw [div_mul_cancel₀ _ hmu.ne']; linarith
+  obtain ⟨Ca,hCa,Cr,hCr,Ci,hCi,s,hs,C,hC,hjets⟩ := a.lattice_logarithmic_jet_estimates hUD L hL
+  let F := B+E*B*L+Cr
+  have hF : 0 ≤ F := by dsimp [F]; positivity
+  let kappa := e^2/(2*(e+F+1)^2)
+  have hkappa : 0 < kappa := by dsimp [kappa]; positivity
+  obtain ⟨r,hr,hinner⟩ := compact_inner_ball hQ hU hQU
+  refine ⟨s^2*kappa/2,by positivity,?_⟩
+  filter_upwards [hjets,eventually_nat_frequency (half_scale_eventually L 1 r hr),
+    eventually_nat_frequency (half_scale_eventually (2*Cr) 1 e he)] with n hj hn hsmall
+  have hn1 : 1 ≤ n := by exact_mod_cast hn.1
+  have hnpos : 0 < n := by omega
+  have hN : 0 < (n:ℝ) := by exact_mod_cast hnpos
+  let h := (n:ℝ)^(-1/2:ℝ)
+  have hh : 0 < h := lattice_scale_pos hnpos
+  have hh1 : h ≤ 1 := Real.rpow_le_one_of_one_le_of_nonpos hn.1 (by norm_num)
+  let : Finite (SourceGrid U n) := finite_source_grid hUb hnpos
+  let hfin : Fintype (SourceGrid U n) := Fintype.ofFinite _
+  refine ⟨hfin,?_⟩
+  intro x hx b v
+  have hxD := hUD (hQU hx)
+  have hball : closedBall x (L*h) ⊆ U := (closedBall_subset_closedBall hn.2.1).trans (hinner x hx)
+  let z (v : Coord) : SourceGrid U n := ⟨directionalLatticeIndex n x R v,
+    directional_center_mem hnpos x R hR.le v hball⟩
+  let V (v : Coord) := latticeWave a.cover a.beams hUD n (z v) 0
+  let yy (v : Coord) := scaledLatticePoint n (z v)
+  have hdist (v : Coord) : ‖x-yy v‖ ≤ L*h := by
+    rw [norm_sub_rev]
+    exact directional_center_distance hnpos x R hR.le v
+  have hjet (v : Coord) := hj (z v) 0 x (hdist v)
+  let Rp := b+(normalizedLogJet n (V v) x v).re
+  let Rm := b+(normalizedLogJet n (V (-v)) x v).re
+  let pp := g (yy v) (x-yy v) v
+  let pm := g (yy (-v)) (x-yy (-v)) v
+  let ep := a.latticeEta hUD n (z v) 0
+  let em := a.latticeEta hUD n (z (-v)) 0
+  have hgp := hgeom (yy v) (hUD (z v).property)
+  have hgm := hgeom (yy (-v)) (hUD (z (-v)).property)
+  have hpp : pp ≤ -h*‖v‖ := rounded_positive_pairing (g (yy v)) mu B R h hmu hB.le hR.le hh.le
+    hgp.1 hgp.2.2 hlarge x (yy v) v (directional_rounding_error hnpos x R v)
+  have hpm : h*‖v‖ ≤ pm := by
+    have hh' := rounded_positive_pairing (g (yy (-v))) mu B R h hmu hB.le hR.le hh.le
+      hgm.1 hgm.2.2 hlarge x (yy (-v)) (-v) (directional_rounding_error hnpos x R (-v))
+    simp only [map_neg,norm_neg] at hh'
+    dsimp [pm]
+    linarith
+  have herr (u : Coord) : |(b+(normalizedLogJet n (V u) x v).re)-
+      ((b+fderiv ℝ S x v)-a.latticeEta hUD n (z u) 0*g (yy u) (x-yy u) v)| ≤ Cr*h^2*‖v‖ := by
+    have hh' := ((hjet u).2.2.2.2.2 v).1
+    have heq : Cr/(n:ℝ)*‖v‖ = Cr*h^2*‖v‖ := by dsimp [h]; rw [half_scale_square hN]; ring
+    rw [heq] at hh'
+    convert hh' using 1
+    congr 1
+    ring
+  have hdiff : e*h*‖v‖ ≤ Rp-Rm := opposite_real_products (b+fderiv ℝ S x v)
+    Rp Rm ep em pp pm h ‖v‖ e Cr hh.le (norm_nonneg _) he.le
+    (heta n (z v) 0).1 (heta n (z (-v)) 0).1 hpp hpm (herr v) (herr (-v)) hsmall.2.1
+  have hprod : |ep*pp| ≤ E*B*L*‖v‖ := by
+    have hpabs := (bilinear_pairing_bound (g (yy v)) (x-yy v) v).trans
+      (show ‖g (yy v)‖*‖x-yy v‖*‖v‖ ≤ B*(L*h)*‖v‖ by
+        gcongr
+        · exact hgp.1
+        · exact hdist v)
+    have hepos : 0 ≤ ep := he.le.trans (heta n (z v) 0).1
+    rw [abs_mul,abs_of_nonneg hepos]
+    calc
+      ep*|pp| ≤ E*(B*(L*h)*‖v‖) := mul_le_mul (heta n (z v) 0).2 hpabs (abs_nonneg _) hE.le
+      _ ≤ E*(B*L*‖v‖) := by
+        gcongr
+        simpa using mul_le_mul_of_nonneg_left hh1 hL.le
+      _ = _ := by ring
+  have hSv : |fderiv ℝ S x v| ≤ B*‖v‖ :=
+    ((fderiv ℝ S x).le_opNorm v).trans (mul_le_mul_of_nonneg_right (hgeom x hxD).2.1 (norm_nonneg v))
+  have hErr : Cr*h^2*‖v‖ ≤ Cr*‖v‖ := by
+    have hh2 := pow_le_pow_left₀ hh.le hh1 2
+    nlinarith [mul_nonneg hCr.le (norm_nonneg v)]
+  have hlog : |(normalizedLogJet n (V v) x v).re| ≤ F*‖v‖ := by
+    have hpR := abs_le.mp ((herr v).trans hErr)
+    have hpP := abs_le.mp hprod
+    have hpS := abs_le.mp hSv
+    dsimp [F,Rp,ep,pp] at *
+    apply abs_le.mpr
+    constructor <;> nlinarith
+  have hb : |b| ≤ |Rp|+F*‖v‖ := by
+    have ht := norm_sub_le (b+(normalizedLogJet n (V v) x v).re) (normalizedLogJet n (V v) x v).re
+    have ht' : |b| ≤ |Rp|+|(normalizedLogJet n (V v) x v).re| := by simpa only [add_sub_cancel_right,Real.norm_eq_abs] using ht
+    exact ht'.trans (add_le_add le_rfl hlog)
+  have hpair := paired_products_control b ‖v‖ Rp Rm h e F (norm_nonneg _) hh hh1 he hF hdiff hb
+  let f (i : SourceGrid U n × Fin 3) := dualWaveCoefficient n
+    (latticeWave a.cover a.beams hUD n i.1 i.2) (S x) x b v
+  have hcoef (u : Coord) : s*|b+(normalizedLogJet n (V u) x v).re| ≤ ‖f (z u,0)‖ :=
+    (mul_le_mul_of_nonneg_right (hjet u).2.2.2.2.1.1 (abs_nonneg _)).trans
+      (real_log_product_le_coefficient n (V u) (S x) x b v (hjet u).2.2.2.1)
+  have henergy := two_coefficient_energy f (z v,0) (z (-v),0) s Rp Rm hs.le (hcoef v) (hcoef (-v))
+  have hpair' := mul_le_mul_of_nonneg_left hpair (sq_nonneg s)
+  have hhbound := hpair'.trans henergy
+  change s^2*(kappa*h^2*(b^2+‖v‖^2)) ≤ 2*a.latticeEnergy hUD n hfin x b v at hhbound
+  calc
+    _ = (s^2*(kappa*h^2*(b^2+‖v‖^2)))/2 := by dsimp [h]; rw [half_scale_square hN]; ring
+    _ ≤ _ := (div_le_iff₀ (by norm_num : (0:ℝ)<2)).mpr (by linarith)
+
+end LocalCompactWaveData
+
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/LatticeFieldBounds.lean b/lean/OAI/Geometry/NodalSets/Waves/LatticeFieldBounds.lean
new file mode 100644
index 0000000000000000000000000000000000000000..4090b4d8159a54bb64c294f45868b339f36a477c
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/LatticeFieldBounds.lean
@@ -0,0 +1,64 @@
+import OAI.Geometry.NodalSets.Waves.FiniteWaveDerivativeBounds
+import OAI.Geometry.NodalSets.Waves.LatticeCount
+import OAI.Geometry.NodalSets.Waves.LocalCompactWaves
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Yau.Probability Set Filter
+open scoped ContDiff Topology
+noncomputable section
+variable {g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ} {w S : Coord → ℝ}
+variable {D U : Set Coord} {m J K k0 : ℕ}
+namespace LocalCompactWaveData
+variable (a : LocalCompactWaveData g w S D m J K k0)
+
+theorem lattice_field_derivative_bound (hUD : U ⊆ D) (hUb : Bornology.IsBounded U) :
+    ∃ C > 0, ∀ᶠ n : ℕ in atTop, ∃ hfin : Fintype (SourceGrid U n),
+      letI := hfin
+      ∀ coeff : ((SourceGrid U n × Fin 3) × Fin 2) → ℝ,
+        coeff ∈ coefficientEvent (n:ℝ) →
+        ContDiff ℝ ∞ (gaussianWaveField
+          (fun i : SourceGrid U n × Fin 3 ↦ latticeWave a.cover a.beams hUD n i.1 i.2) coeff) ∧
+        ∀ x : Coord, ∀ k : Fin (k0+1),
+          ‖iteratedFDeriv ℝ k.val (gaussianWaveField
+            (fun i : SourceGrid U n × Fin 3 ↦ latticeWave a.cover a.beams hUD n i.1 i.2) coeff) x‖ ≤
+          C*(n:ℝ)^(k.val+3)*Real.exp ((n:ℝ)*S x) := by
+  obtain ⟨C,hC,hcard⟩ := source_coefficient_card_bound hUb
+  have hcw := a.beams.Cw_pos
+  have hbc := a.beams.c_pos
+  refine ⟨C*a.beams.Cw,mul_pos hC a.beams.Cw_pos,?_⟩
+  filter_upwards [a.estimates,eventually_gt_atTop (0:ℕ)] with n hn hnpos
+  have := finite_source_grid hUb hnpos
+  let hfin := Fintype.ofFinite (SourceGrid U n)
+  let := hfin
+  refine ⟨hfin,?_⟩
+  let V := fun i : SourceGrid U n × Fin 3 ↦ latticeWave a.cover a.beams hUD n i.1 i.2
+  have hV (i : SourceGrid U n × Fin 3) : ContDiff ℝ ∞ (V i) :=
+    (hn (latticeFrame a.cover hUD n i.1,i.2)).1
+  intro coeff hc
+  refine ⟨gaussianWaveField_contDiff V coeff hV,?_⟩
+  intro x k
+  have hb (i : SourceGrid U n × Fin 3) : ‖iteratedFDeriv ℝ k.val (V i) x‖ ≤
+      a.beams.Cw*(n:ℝ)^k.val*Real.exp ((n:ℝ)*S x) := by
+    have h := ((hn (latticeFrame a.cover hUD n i.1,i.2)).2.2.2.2 x).1 k
+    refine h.trans ?_
+    gcongr
+    exact sub_le_self _ (by positivity : 0 ≤ a.beams.c*(n:ℝ)*
+      (sourceEuclideanNorm (x-coverSourceCenter a.cover (latticeFrame a.cover hUD n i.1)))^2)
+  have h := gaussianWaveField_iterated_bound V coeff hV k.val x (by positivity) hc hb
+  have hcardR : (Fintype.card (SourceGrid U n × Fin 3):ℝ) ≤ C*(n:ℝ)^2 := by
+    simpa only [Nat.card_eq_fintype_card] using hcard n hnpos
+  refine h.trans ?_
+  calc
+    _ ≤ (C*(n:ℝ)^2)*(n:ℝ)*(a.beams.Cw*(n:ℝ)^k.val*Real.exp ((n:ℝ)*S x)) := by
+      gcongr
+    _ = (C*a.beams.Cw)*(n:ℝ)^(k.val+3)*Real.exp ((n:ℝ)*S x) := by
+      rw [pow_add]
+      ring
+
+end LocalCompactWaveData
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/LatticeFrequencyAnnulus.lean b/lean/OAI/Geometry/NodalSets/Waves/LatticeFrequencyAnnulus.lean
new file mode 100644
index 0000000000000000000000000000000000000000..2ff2badd87521c487ea626c37c1c45e0d401ad6e
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/LatticeFrequencyAnnulus.lean
@@ -0,0 +1,39 @@
+import OAI.Geometry.NodalSets.Charts.MetricFrequencyBounds
+import OAI.Geometry.NodalSets.Elliptic.SignScaleContinuity
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Set Filter
+open scoped ContDiff Topology
+noncomputable section
+
+theorem lattice_main_frequency_annulus
+    (g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ) {H : Set Coord}
+    (hH : IsCompact H) (hg : ContinuousOn g H)
+    (hp : ∀ y ∈ H, ∀ v, v ≠ 0 → 0 < g y v v) :
+    ∃ κ > 0, ∃ B > 0, ∀ (w S : Coord → ℝ) (D U : Set Coord) (m J K k0 : ℕ)
+      (a : LocalCompactWaveData g w S D m J K k0) (_ : H ⊆ D) (hUD : U ⊆ D),
+      U ⊆ H → ∀ᶠ n : ℕ in atTop, ∀ x ∈ H, ∀ i : SourceGrid U n × Fin 3,
+      sourceEuclideanNorm (x-scaledLatticePoint n i.1) ≤ (n:ℝ)^(-5/12:ℝ) →
+      κ ≤ sourceEuclideanNorm (sourceFrequency (g (scaledLatticePoint n i.1))
+        (a.cover.triple.q (latticeFrame a.cover hUD n i.1) i.2) (sourceSignScale g S x)) ∧
+      sourceEuclideanNorm (sourceFrequency (g (scaledLatticePoint n i.1))
+        (a.cover.triple.q (latticeFrame a.cover hUD n i.1) i.2) (sourceSignScale g S x)) ≤ B := by
+  obtain ⟨κ,hκ,B,hB,hmetric⟩ := compact_metric_frequency_bounds g hH hg hp
+  refine ⟨κ,hκ,B,hB,?_⟩
+  intro w S D U m J K k0 a hHD hUD hUH
+  filter_upwards [a.source_main_scale_comparison hH hHD] with n hn
+  intro x hx i hi
+  have hy : scaledLatticePoint n i.1 ∈ H := hUH i.1.property
+  have hr := hn x hx _ hy hi
+  have hx0 : 0 < sourceSignScale g S x := by linarith [a.source_sign_scale_lower x (hHD hx)]
+  have hy0 : 0 < sourceSignScale g S (scaledLatticePoint n i.1) := by
+    linarith [a.source_sign_scale_lower _ (hUD i.1.property)]
+  apply hmetric _ hy _ _ _ hx0 hy0 _ hr.1 hr.2
+  simpa only [latticeFrame_center] using a.original_direction_length (latticeFrame a.cover hUD n i.1) i.2
+
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/LatticeFullPlaneWaveApprox.lean b/lean/OAI/Geometry/NodalSets/Waves/LatticeFullPlaneWaveApprox.lean
new file mode 100644
index 0000000000000000000000000000000000000000..0214645698e2bf3de1ba88a6b18fb4645205db9c
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/LatticeFullPlaneWaveApprox.lean
@@ -0,0 +1,72 @@
+import OAI.Geometry.NodalSets.Waves.LatticePlaneWaveJets
+import OAI.Geometry.NodalSets.Waves.RescaledLatticeTail
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Set Filter
+open scoped ContDiff Topology
+noncomputable section
+variable {g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ} {w S : Coord → ℝ}
+variable {D U : Set Coord} {m J K k0 : ℕ}
+namespace LocalCompactWaveData
+variable (a : LocalCompactWaveData g w S D m J K k0)
+
+def latticePlaneWaveCoefficient (hUD : U ⊆ D) (n : ℕ) [Fintype (SourceGrid U n)]
+    (x : Coord) (s : ℝ) (i : SourceGrid U n × Fin 3) (v : Coord) : ℂ :=
+  if sourceEuclideanNorm (x-scaledLatticePoint n i.1) ≤ (n:ℝ)^(-5/12:ℝ) then
+    a.latticeAlpha hUD n x i *
+      planeWave (g (scaledLatticePoint n i.1) (a.cover.triple.q (latticeFrame a.cover hUD n i.1) i.2)) s v
+  else 0
+
+theorem lattice_full_jet_approximation (hUD : U ⊆ D) (hU : IsOpen U)
+    (hUb : Bornology.IsBounded U) {Q : Set Coord} (hQ : IsCompact Q) (hQU : Q ⊆ U)
+    (hS : ContDiff ℝ ∞ S) (R : ℝ) (hR : 0 ≤ R) (e : ℝ) (he : 0 < e) :
+    ∀ᶠ n : ℕ in atTop, ∃ hfin : Fintype (SourceGrid U n), letI := hfin
+      ∀ x ∈ Q, ∀ s : ℝ, 1 ≤ s → ∀ v : Coord, ‖v‖ ≤ R → ∀ k : Fin (k0+1),
+      ∑ i : SourceGrid U n × Fin 3,
+        ‖iteratedFDeriv ℝ k.val (fun z ↦
+          normalizedRescaling (latticeWave a.cover a.beams hUD n i.1 i.2) S n s
+            (a.latticeSigma hUD n x) x z - a.latticePlaneWaveCoefficient hUD n x s i z) v‖^2 ≤ e := by
+  classical
+  filter_upwards [a.lattice_main_jet_approximation hUD hU hUb hQ hQU R hR k0 (e/2) (by positivity),
+    a.rescaled_lattice_nonmain_negligible hUD hU hUb hQ hQU hS R hR (e/2) (by positivity)]
+    with n hmain htail
+  obtain ⟨hfin,hm⟩ := hmain
+  obtain ⟨hfin',ht⟩ := htail
+  have heq : hfin' = hfin := Subsingleton.elim _ _
+  subst hfin'
+  let := hfin
+  refine ⟨hfin,?_⟩
+  intro x hx s hs v hv k
+  let P := fun i : SourceGrid U n × Fin 3 ↦
+    sourceEuclideanNorm (x-scaledLatticePoint n i.1) ≤ (n:ℝ)^(-5/12:ℝ)
+  let err := fun i : SourceGrid U n × Fin 3 ↦
+    ‖iteratedFDeriv ℝ k.val (fun z ↦
+      normalizedRescaling (latticeWave a.cover a.beams hUD n i.1 i.2) S n s
+        (a.latticeSigma hUD n x) x z-a.latticePlaneWaveCoefficient hUD n x s i z) v‖^2
+  have hm' : ∑ i ∈ Finset.univ.filter P, err i ≤ e/2 := by
+    have hh := hm x hx s hs v hv k.val (by omega)
+    convert hh using 1
+    apply Finset.sum_congr rfl
+    intro i hi
+    have hp := (Finset.mem_filter.mp hi).2
+    dsimp only [P] at hp
+    simp only [err,latticePlaneWaveCoefficient,ite_eq_left hp]
+  have ht' : ∑ i ∈ Finset.univ.filter (fun i ↦ ¬P i), err i ≤ e/2 := by
+    have hh := ht x hx s hs v hv k
+    convert hh using 1
+    · apply Finset.sum_congr
+      · ext i; simp only [P,Finset.mem_filter,not_le]
+      · intro i hi
+        have hp := (Finset.mem_filter.mp hi).2
+        simp only [err,latticePlaneWaveCoefficient,ite_eq_right (not_le.mpr hp),sub_zero]
+  have hsplit := Finset.sum_filter_add_sum_filter_not (s := Finset.univ) P err
+  change ∑ i, err i ≤ e
+  linarith
+
+end LocalCompactWaveData
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/LatticeGaussianSmallBall.lean b/lean/OAI/Geometry/NodalSets/Waves/LatticeGaussianSmallBall.lean
new file mode 100644
index 0000000000000000000000000000000000000000..040036f57ec36511519729b4c0b99a54bf741893
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/LatticeGaussianSmallBall.lean
@@ -0,0 +1,81 @@
+import OAI.Geometry.NodalSets.Waves.EuclideanGaussianSmallBall
+import OAI.Geometry.NodalSets.Waves.GaussianJet
+import OAI.Geometry.NodalSets.Waves.LatticeCovariance
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Yau.Probability Set Filter MeasureTheory ProbabilityTheory
+open scoped ContDiff Topology ENNReal RealInnerProductSpace
+noncomputable section
+variable {g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ} {w S : Coord → ℝ}
+variable {D U : Set Coord} {m J K k0 : ℕ}
+namespace LocalCompactWaveData
+variable (a : LocalCompactWaveData g w S D m J K k0)
+
+def latticeSeededJet (hUD : U ⊆ D) (n : ℕ) (hfin : Fintype (SourceGrid U n))
+    (seed : Coord → ℝ) (x : Coord)
+    (coeff : ((SourceGrid U n × Fin 3) × Fin 2) → ℝ) : JetSpace :=
+  letI := hfin
+  let V := fun i : SourceGrid U n × Fin 3 ↦ latticeWave a.cover a.beams hUD n i.1 i.2
+  normalizedRealJet (fun z ↦ seed z+gaussianWaveField V coeff z) (n:ℝ) (S x) x
+
+lemma latticeSeededJet_inner (hUD : U ⊆ D) (n : ℕ) (hfin : Fintype (SourceGrid U n))
+    (seed : Coord → ℝ) (x : Coord) (v : JetSpace)
+    (coeff : ((SourceGrid U n × Fin 3) × Fin 2) → ℝ) :
+    ⟪v,a.latticeSeededJet hUD n hfin seed x coeff⟫ =
+      a.latticeSeededRandomDual hUD n hfin seed x (v 0) (Fin.tail (WithLp.ofLp v)) coeff := by
+  let := hfin
+  unfold latticeSeededJet latticeSeededRandomDual
+  exact normalizedRealJet_inner
+    (fun z ↦ seed z + gaussianWaveField
+      (fun i : SourceGrid U n × Fin 3 ↦ latticeWave a.cover a.beams hUD n i.1 i.2)
+      coeff z) (n : ℝ) (S x) x v
+
+theorem actual_lattice_gaussian_small_ball (hUD : U ⊆ D) (hU : IsOpen U)
+    (hUb : Bornology.IsBounded U) {Q : Set Coord} (hQ : IsCompact Q) (hQU : Q ⊆ U) :
+    ∃ c > 0, ∀ᶠ n : ℕ in atTop, ∃ hfin : Fintype (SourceGrid U n),
+      letI := hfin
+      ∀ (seed : Coord → ℝ) (x : Coord), x ∈ Q → DifferentiableAt ℝ seed x →
+      ∀ r : ℝ, 0 ≤ r →
+        (gaussianPairs.map (a.latticeSeededJet hUD n hfin seed x)) {y | ‖y‖ ≤ r} ≤
+          ENNReal.ofReal ((12*(Real.sqrt (c/(n:ℝ)))⁻¹*r/Real.sqrt (2*Real.pi))^5) := by
+  obtain ⟨c,hc,hcov⟩ := a.actual_lattice_covariance_lower_bound hUD hU hUb hQ hQU
+  refine ⟨c,hc,?_⟩
+  filter_upwards [hcov,lattice_wave_estimates a.cover a.beams hUD,
+    eventually_gt_atTop (0:ℕ)] with n hn hw hnpos
+  obtain ⟨hfin,hbound⟩ := hn
+  let := hfin
+  refine ⟨hfin,?_⟩
+  intro seed x hx hs r hr
+  let V := fun i : SourceGrid U n × Fin 3 ↦ latticeWave a.cover a.beams hUD n i.1 i.2
+  have hV : ∀ i, DifferentiableAt ℝ (V i) x := fun i ↦
+    ((hw i.1 i.2).1.differentiable (by simp)).differentiableAt
+  have hb (v : JetSpace) : c/(n:ℝ)*‖v‖^2 ≤
+      ‖(pairJetMap (jetWaveCoefficients V n (S x) x)).adjoint v‖^2 := by
+    have h := hbound seed x hx hs (v 0) (Fin.tail (WithLp.ofLp v))
+    have he : (fun coeff ↦ ⟪v,a.latticeSeededJet hUD n hfin seed x coeff⟫) =
+        a.latticeSeededRandomDual hUD n hfin seed x (v 0) (Fin.tail (WithLp.ofLp v)) := by
+      funext coeff
+      exact a.latticeSeededJet_inner hUD n hfin seed x v coeff
+    rw [← he] at h
+    change c/(n:ℝ)*((v 0)^2+∑ i : Fin 4, (v i.succ)^2) ≤
+      Var[fun coeff ↦ ⟪v,normalizedRealJet (fun y ↦ seed y+gaussianWaveField V coeff y)
+        (n:ℝ) (S x) x⟫; gaussianPairs] at h
+    rw [seeded_gaussian_jet_variance V seed n (S x) x hV hs v] at h
+    simpa only [EuclideanSpace.real_norm_sq_eq,Fin.sum_univ_succ] using h
+  have hl := seeded_gaussian_jet_law V seed n (S x) x hV hs
+  change gaussianPairs.map (a.latticeSeededJet hUD n hfin seed x) = _ at hl
+  rw [hl]
+  have h := gaussian_map_small_ball (pairJetMap (jetWaveCoefficients V n (S x) x))
+    (normalizedRealJet seed n (S x) x) (c/(n:ℝ)) r (by positivity) hr hb
+  convert h using 1
+  norm_num
+  congr 2
+  ring
+
+end LocalCompactWaveData
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/LatticeGeometry.lean b/lean/OAI/Geometry/NodalSets/Waves/LatticeGeometry.lean
new file mode 100644
index 0000000000000000000000000000000000000000..cbdc75d515cbf2a6f16694ae98706cfaa9c38f46
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/LatticeGeometry.lean
@@ -0,0 +1,74 @@
+import Mathlib.Algebra.Order.Floor.Ring
+import Mathlib.Data.Int.Interval
+import OAI.Geometry.NodalSets.Waves.LatticeSourceWaves
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Set
+noncomputable section
+
+lemma lattice_scale_pos {n : ℕ} (hn : 0 < n) : 0 < (n:ℝ)^(-1/2:ℝ) :=
+  Real.rpow_pos_of_pos (by exact_mod_cast hn) _
+
+lemma scaledLatticePoint_injective {n : ℕ} (hn : 0 < n) : Function.Injective (scaledLatticePoint n) := by
+  intro z w h
+  funext i
+  have hh := congrFun h i
+  change (n:ℝ)^(-1/2:ℝ)*(z i:ℝ) = (n:ℝ)^(-1/2:ℝ)*(w i:ℝ) at hh
+  exact_mod_cast mul_left_cancel₀ (ne_of_gt (lattice_scale_pos hn)) hh
+
+theorem finite_source_grid {U : Set Coord} (hU : Bornology.IsBounded U)
+    {n : ℕ} (hn : 0 < n) : Finite (SourceGrid U n) := by
+  obtain ⟨B,hB,hbound⟩ := hU.exists_pos_norm_le
+  let r : ℝ := (n:ℝ)^(-1/2:ℝ)
+  have hr : 0 < r := lattice_scale_pos hn
+  let M : ℤ := ⌈B/r⌉
+  have hcoords (z : SourceGrid U n) (i : Fin 4) : z.val i ∈ Icc (-M) M := by
+    have hi := (norm_le_pi_norm (scaledLatticePoint n z) i).trans (hbound _ z.property)
+    change |r*(z.val i:ℝ)| ≤ B at hi
+    rw [abs_mul,abs_of_pos hr] at hi
+    have ha : |(z.val i:ℝ)| ≤ B/r := (le_div_iff₀ hr).mpr (by nlinarith)
+    have hM : B/r ≤ (M:ℝ) := Int.le_ceil _
+    have hpair := abs_le.mp (ha.trans hM)
+    constructor
+    · exact_mod_cast hpair.1
+    · exact_mod_cast hpair.2
+  let f : SourceGrid U n → (Fin 4 → Icc (-M) M) := fun z i ↦ ⟨z.val i,hcoords z i⟩
+  apply Finite.of_injective f
+  intro z w h
+  apply Subtype.ext
+  funext i
+  exact congrArg Subtype.val (congrFun h i)
+
+def roundedLatticeIndex (n : ℕ) (x : Coord) : Fin 4 → ℤ :=
+  fun i ↦ ⌊x i / (n:ℝ)^(-1/2:ℝ)⌋
+
+theorem rounded_lattice_distance {n : ℕ} (hn : 0 < n) (x : Coord) :
+    sourceEuclideanNorm (scaledLatticePoint n (roundedLatticeIndex n x)-x) ≤
+      2*(n:ℝ)^(-1/2:ℝ) := by
+  let r : ℝ := (n:ℝ)^(-1/2:ℝ)
+  have hr : 0 < r := lattice_scale_pos hn
+  have hb : ‖scaledLatticePoint n (roundedLatticeIndex n x)-x‖ ≤ r := by
+    apply (pi_norm_le_iff_of_nonneg hr.le).mpr
+    intro i
+    have h1 := Int.floor_le (x i/r)
+    have h2 := Int.lt_floor_add_one (x i/r)
+    have h1' := mul_le_mul_of_nonneg_left h1 hr.le
+    have h2' := mul_lt_mul_of_pos_left h2 hr
+    have he : r*(x i/r) = x i := by field_simp
+    rw [he] at h1' h2'
+    change |r*(⌊x i/r⌋:ℝ)-x i| ≤ r
+    apply abs_le.mpr
+    constructor <;> nlinarith
+  exact (sourceEuclideanNorm_le _).trans (by dsimp [r] at hb ⊢; linarith)
+
+theorem rounded_lattice_mem {U : Set Coord} {n : ℕ} (hn : 0 < n) (x : Coord)
+    (hball : {y | sourceEuclideanNorm (y-x) ≤ 2*(n:ℝ)^(-1/2:ℝ)} ⊆ U) :
+    scaledLatticePoint n (roundedLatticeIndex n x) ∈ U :=
+  hball (rounded_lattice_distance hn x)
+
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/LatticeGoodEvent.lean b/lean/OAI/Geometry/NodalSets/Waves/LatticeGoodEvent.lean
new file mode 100644
index 0000000000000000000000000000000000000000..634609b56c31985c2b664447cf1a3422f392bcc9
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/LatticeGoodEvent.lean
@@ -0,0 +1,58 @@
+import OAI.Geometry.NodalSets.Elliptic.FirstJetGoodEvent
+import OAI.Geometry.NodalSets.Elliptic.FirstJetNonvanishing
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Yau.Probability Set Filter MeasureTheory
+open scoped ContDiff Topology
+noncomputable section
+variable {g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ} {w S : Coord → ℝ}
+    {D U : Set Coord} {m J K k0 : ℕ}
+namespace LocalCompactWaveData
+variable (a : LocalCompactWaveData g w S D m J K k0)
+
+def latticeGoodEvent (hUD : U ⊆ D) (n : ℕ) (hfin : Fintype (SourceGrid U n))
+    (S0 T0 : Coord → ℝ) (Ω : Set Coord) : Set (((SourceGrid U n × Fin 3) × Fin 2) → ℝ) :=
+  letI := hfin
+  coefficientJetGoodEvent
+    (fun i : SourceGrid U n × Fin 3 ↦ latticeWave a.cover a.beams hUD n i.1 i.2)
+    (oscillatorySeed S0 T0 n) n Ω
+    (fun x ↦ ((n:ℝ)^65)⁻¹*max (Real.exp ((n:ℝ)*S x)) (Real.exp ((n:ℝ)*S0 x)))
+
+theorem lattice_good_event_probability (hUD : U ⊆ D) (hU : IsOpen U)
+    (hUb : Bornology.IsBounded U) (hS : ContDiff ℝ ∞ S)
+    (S0 T0 : Coord → ℝ) (hS0 : ContDiff ℝ ∞ S0) (hT0 : ContDiff ℝ ∞ T0)
+    {Ω : Set Coord} (hΩ : IsCompact Ω) (hk0 : 2 ≤ k0)
+    (hq : ∀ x ∈ Ω, fderiv ℝ T0 x ≠ 0)
+    {d : ℝ} (hd : 0 < d) (hgap : ∀ x ∈ Ω, x ∉ U → S x-S0 x ≤ -d) :
+    ∀ ε : ℝ, 0 < ε → ∀ᶠ n : ℕ in atTop, ∃ hfin : Fintype (SourceGrid U n),
+      letI := hfin
+      MeasurableSet (a.latticeGoodEvent hUD n hfin S0 T0 Ω) ∧
+      gaussianPairs (a.latticeGoodEvent hUD n hfin S0 T0 Ω)ᶜ ≤ ENNReal.ofReal ε := by
+  intro ε hε
+  obtain ⟨C,hC,hcoeff⟩ := lattice_coefficient_event_failure hUb
+  have htail := (tendsto_coefficient_tail_rate C).eventually (gt_mem_nhds (by linarith : 0 < ε/2))
+  filter_upwards [a.source_first_jet_failure_tends_zero hUD hU hUb hS S0 T0 hS0 hT0
+    hΩ hk0 hq hd hgap (ε/2) (by linarith),htail,a.estimates,eventually_gt_atTop (0:ℕ)] with n hn ht he hn0
+  obtain ⟨hfin,hjet⟩ := hn
+  let := hfin
+  let V := fun i : SourceGrid U n × Fin 3 ↦ latticeWave a.cover a.beams hUD n i.1 i.2
+  have hV (i : SourceGrid U n × Fin 3) : ContDiff ℝ ∞ (V i) :=
+    (he (latticeFrame a.cover hUD n i.1,i.2)).1
+  refine ⟨hfin,coefficientJetGoodEvent_measurable V (oscillatorySeed S0 T0 n)
+    hV (oscillatorySeed_contDiff S0 T0 hS0 hT0 n) n Ω _,?_⟩
+  apply (coefficientJetGoodEvent_failure_le V (oscillatorySeed S0 T0 n) n Ω _).trans
+  calc
+    _ ≤ ENNReal.ofReal (ε/2)+ENNReal.ofReal (ε/2) :=
+      add_le_add ((hcoeff n hn0 hfin).trans (ENNReal.ofReal_le_ofReal ht.le)) hjet
+    _ = ENNReal.ofReal ε := by
+      rw [← ENNReal.ofReal_add (by linarith : 0 ≤ ε/2) (by linarith : 0 ≤ ε/2)]
+      congr 1
+      ring
+
+end LocalCompactWaveData
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/LatticeGradientExpectation.lean b/lean/OAI/Geometry/NodalSets/Waves/LatticeGradientExpectation.lean
new file mode 100644
index 0000000000000000000000000000000000000000..302edea2d194a8ba28fda65997e1ad0cd8f8fb39
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/LatticeGradientExpectation.lean
@@ -0,0 +1,48 @@
+import OAI.Geometry.NodalSets.Elliptic.RescaledGradientExpectation
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Yau.Probability Set Filter MeasureTheory ProbabilityTheory
+open scoped ContDiff Topology
+noncomputable section
+
+theorem lattice_gradient_expectation
+    (g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ) {H : Set Coord}
+    (hH : IsCompact H) (hg : ContinuousOn g H)
+    (hp : ∀ y ∈ H, ∀ v, v ≠ 0 → 0 < g y v v) :
+    ∃ C > 0, ∀ (w S S0 T0 : Coord → ℝ) (D U Q : Set Coord) (m J K k0 : ℕ)
+      (a : LocalCompactWaveData g w S D m J K k0) (_ : H ⊆ D) (hUD : U ⊆ D),
+      U ⊆ H → IsOpen U → Bornology.IsBounded U → IsCompact Q → Q ⊆ U →
+      ContDiff ℝ ∞ S → ContDiff ℝ ∞ S0 → ContDiff ℝ ∞ T0 → 5 ≤ k0 →
+      ∀ gamma > 0, (∀ x ∈ Q, S0 x+gamma ≤ S x) →
+      ∀ᶠ n : ℕ in atTop, ∃ hfin : Fintype (SourceGrid U n), letI := hfin
+        ∀ x ∈ Q, ∃ G : ((SourceGrid U n × Fin 3) × Fin 2 → ℝ) → ℝ,
+          Continuous G ∧ Integrable G gaussianPairs ∧ (∀ coeff, 0 ≤ G coeff) ∧
+          (∫ coeff, G coeff ∂gaussianPairs) ≤ C ∧
+          ∀ coeff L, 0 ≤ L →
+            (G coeff ≤ L ↔ ∀ v : Coord, sourceEuclideanNorm v ≤ 1 →
+              sourceEuclideanNorm (fun j ↦ fderiv ℝ (rescaledGaussianField
+                (fun i : SourceGrid U n × Fin 3 ↦ latticeWave a.cover a.beams hUD n i.1 i.2)
+                S0 T0 S n (sourceSignScale g S x) (a.latticeSigma hUD n x) x coeff)
+                  v (Pi.single j 1)) ≤ L) := by
+  obtain ⟨B,hB,hmom⟩ := lattice_derivative_second_moments g hH hg hp 5
+  refine ⟨648*(B+1),by positivity,?_⟩
+  intro w S S0 T0 D U Q m J K k0 a hHD hUD hUH hU hUb hQ hQU hS hS0 hT0 hd gamma hgamma hgap
+  filter_upwards [hmom w S S0 T0 D U Q m J K k0 a hHD hUD hUH hU hUb hQ hQU
+    hS hS0 hT0 hd gamma hgamma hgap,a.estimates] with n hn hest
+  obtain ⟨hfin,hn⟩ := hn
+  let := hfin
+  refine ⟨hfin,?_⟩
+  intro x hx
+  let V := fun i : SourceGrid U n × Fin 3 ↦ latticeWave a.cover a.beams hUD n i.1 i.2
+  have hV (i : SourceGrid U n × Fin 3) : ContDiff ℝ ∞ (V i) :=
+    (hest (latticeFrame a.cover hUD n i.1,i.2)).1
+  exact rescaled_gradient_supremum_expectation V hV S0 T0 S hS0 hT0
+    n (sourceSignScale g S x) (a.latticeSigma hUD n x) x hB.le
+    (fun v hv k hk dirs hd ↦ (hn x hx v hv k hk dirs hd).2)
+
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/LatticeLogJets.lean b/lean/OAI/Geometry/NodalSets/Waves/LatticeLogJets.lean
new file mode 100644
index 0000000000000000000000000000000000000000..0ab4f8f4777657116a0e36e0953433bccfc435b8
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/LatticeLogJets.lean
@@ -0,0 +1,44 @@
+import OAI.Geometry.NodalSets.Elliptic.LocalLogJetBounds
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Set Filter
+open scoped ContDiff Topology
+noncomputable section
+variable {g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ} {w S : Coord → ℝ}
+variable {D U : Set Coord} {m J K k0 : ℕ}
+namespace LocalCompactWaveData
+variable (a : LocalCompactWaveData g w S D m J K k0)
+
+def latticeEta (hUD : U ⊆ D) (n : ℕ) (z : SourceGrid U n) (j : Fin 3) : ℝ :=
+  a.originalEta (latticeFrame a.cover hUD n z,j)
+
+theorem lattice_logarithmic_jet_estimates (hUD : U ⊆ D) (L : ℝ) (hL : 0 < L) :
+    ∃ Ca > 0, ∃ Cr > 0, ∃ Ci > 0, ∃ c > 0, ∃ C > 0,
+      ∀ᶠ n : ℕ in atTop, ∀ (z : SourceGrid U n) (j : Fin 3) (x : Coord),
+      ‖x-scaledLatticePoint n z‖ ≤ L*(n:ℝ)^(-1/2:ℝ) →
+      let t := (latticeFrame a.cover hUD n z,j)
+      let V := latticeWave a.cover a.beams hUD n z j
+      ‖a.beams.amplitude n t x-1‖ ≤ Ca*(n:ℝ)^(-1/2:ℝ) ∧
+      (1/2:ℝ) ≤ ‖a.beams.amplitude n t x‖ ∧ ‖a.beams.amplitude n t x‖ ≤ 3/2 ∧ V x ≠ 0 ∧
+      (c ≤ ‖V x‖*Real.exp (-(n:ℝ)*S x) ∧ ‖V x‖*Real.exp (-(n:ℝ)*S x) ≤ C) ∧
+      ∀ v,
+      |(normalizedLogJet n V x v).re -
+        (fderiv ℝ S x v-a.latticeEta hUD n z j*g (scaledLatticePoint n z) (x-scaledLatticePoint n z) v)| ≤ Cr/n*‖v‖ ∧
+      |(normalizedLogJet n V x v).im - g (scaledLatticePoint n z) (a.cover.triple.q t.1 t.2) v| ≤
+        Ci*(n:ℝ)^(-1/2:ℝ)*‖v‖ := by
+  obtain ⟨Ca,hCa,Cr,hCr,Ci,hCi,c,hc,C,hC,hb⟩ := a.original_logarithmic_jet_estimates L hL
+  refine ⟨Ca,hCa,Cr,hCr,Ci,hCi,c,hc,C,hC,?_⟩
+  filter_upwards [hb] with n hn
+  intro z j x hx
+  have hx' : ‖x-coverSourceCenter a.cover (latticeFrame a.cover hUD n z)‖ ≤ L*(n:ℝ)^(-1/2:ℝ) := by
+    simpa only [latticeFrame_center] using hx
+  simpa only [latticeWave,TripleSourceWaveData.wave,latticeEta,latticeFrame_center] using
+    hn (latticeFrame a.cover hUD n z,j) x hx'
+
+end LocalCompactWaveData
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/LatticeMainLogJets.lean b/lean/OAI/Geometry/NodalSets/Waves/LatticeMainLogJets.lean
new file mode 100644
index 0000000000000000000000000000000000000000..5005ae5b14a4220e3949422903440a5c667bd739
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/LatticeMainLogJets.lean
@@ -0,0 +1,66 @@
+import OAI.Geometry.NodalSets.Elliptic.LocalMainLogJets
+import OAI.Geometry.NodalSets.Elliptic.ShiftedCenterDistance
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Set Filter
+open scoped ContDiff Topology
+noncomputable section
+
+lemma eventually_shifted_main_distance (R : ℝ) :
+    ∀ᶠ n : ℕ in atTop, ∀ (s : ℝ), 1 ≤ s → ∀ (x y v : Coord), ‖v‖ ≤ R →
+      sourceEuclideanNorm (x-y) ≤ (n:ℝ)^(-5/12:ℝ) →
+      ‖x+((n:ℝ)*s)⁻¹ • v-y‖ ≤ 2*(n:ℝ)^(-5/12:ℝ) := by
+  have ht : Tendsto (fun n : ℕ ↦ (n:ℝ)^(7/12:ℝ)) atTop atTop :=
+    (tendsto_rpow_atTop (by norm_num : (0:ℝ) < 7/12)).comp tendsto_natCast_atTop_atTop
+  filter_upwards [ht.eventually (eventually_ge_atTop R),eventually_gt_atTop (0:ℕ)]
+    with n hn hnpos
+  intro s hs x y v hv hnear
+  have hn0 : (0:ℝ) < n := by exact_mod_cast hnpos
+  have hn1 : (1:ℝ) ≤ n := by exact_mod_cast hnpos
+  have hd := rescaling_displacement_bound (x := x) hn1 hs hv
+  rw [← main_radius_times_frequency hn0] at hn
+  have hshift : ‖(x+((n:ℝ)*s)⁻¹ • v)-x‖ ≤ (n:ℝ)^(-5/12:ℝ) := by nlinarith
+  have hxy := (norm_le_sourceEuclideanNorm _).trans hnear
+  have he : x+((n:ℝ)*s)⁻¹ • v-y =
+      (x+((n:ℝ)*s)⁻¹ • v-x)+(x-y) := by abel
+  rw [he]
+  exact (norm_add_le _ _).trans (by linarith)
+
+variable {g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ} {w S : Coord → ℝ}
+variable {D U : Set Coord} {m J K k0 : ℕ}
+namespace LocalCompactWaveData
+variable (a : LocalCompactWaveData g w S D m J K k0)
+
+theorem lattice_shifted_main_log_jets (hUD : U ⊆ D) (R : ℝ) :
+    ∃ Ca > 0, ∃ Cr > 0, ∃ Ci > 0,
+      ∀ᶠ n : ℕ in atTop, ∀ (x : Coord) (s : ℝ), 1 ≤ s →
+      ∀ (v : Coord), ‖v‖ ≤ R → ∀ (i : SourceGrid U n × Fin 3),
+      sourceEuclideanNorm (x-scaledLatticePoint n i.1) ≤ (n:ℝ)^(-5/12:ℝ) →
+      let t := (latticeFrame a.cover hUD n i.1,i.2)
+      let z := x+((n:ℝ)*s)⁻¹ • v
+      ‖(a.beams.F t).symm z‖ < (n:ℝ)^(-1/3:ℝ) ∧
+      ‖a.beams.amplitude n t z-1‖ ≤ Ca*(n:ℝ)^(-5/12:ℝ) ∧
+      (1/2:ℝ) ≤ ‖a.beams.amplitude n t z‖ ∧ ‖a.beams.amplitude n t z‖ ≤ 3/2 ∧
+      latticeWave a.cover a.beams hUD n i.1 i.2 z ≠ 0 ∧ ∀ u,
+      |(normalizedLogJet n (latticeWave a.cover a.beams hUD n i.1 i.2) z u).re -
+        (fderiv ℝ S z u-a.originalEta t*g (scaledLatticePoint n i.1)
+          (z-scaledLatticePoint n i.1) u)| ≤ Cr*(n:ℝ)^(-5/6:ℝ)*‖u‖ ∧
+      |(normalizedLogJet n (latticeWave a.cover a.beams hUD n i.1 i.2) z u).im -
+        g (scaledLatticePoint n i.1) (a.cover.triple.q t.1 t.2) u| ≤
+          Ci*(n:ℝ)^(-5/12:ℝ)*‖u‖ := by
+  obtain ⟨Ca,hCa,Cr,hCr,Ci,hCi,hjets⟩ := a.original_main_logarithmic_jet_estimates 2 (by norm_num)
+  refine ⟨Ca,hCa,Cr,hCr,Ci,hCi,?_⟩
+  filter_upwards [hjets,eventually_shifted_main_distance R] with n hj hd
+  intro x s hs v hv i hi
+  have hz := hd s hs x (scaledLatticePoint n i.1) v hv hi
+  have hh := hj (latticeFrame a.cover hUD n i.1,i.2) (x+((n:ℝ)*s)⁻¹ • v)
+    (by simpa only [latticeFrame_center] using hz)
+  simpa only [latticeFrame_center,latticeWave,TripleSourceWaveData.wave] using hh
+
+end LocalCompactWaveData
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/LatticeNeighborhoodJets.lean b/lean/OAI/Geometry/NodalSets/Waves/LatticeNeighborhoodJets.lean
new file mode 100644
index 0000000000000000000000000000000000000000..2eb051633c0e5f272e7240d0736472569c39a476
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/LatticeNeighborhoodJets.lean
@@ -0,0 +1,76 @@
+import Mathlib.Analysis.Calculus.MeanValue
+import OAI.Geometry.NodalSets.Elliptic.NeighborhoodJetBound
+import OAI.Geometry.NodalSets.Waves.LatticeGaussianSmallBall
+import OAI.Geometry.NodalSets.Waves.SeededLatticeFieldBounds
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Yau.Probability Set Filter
+open scoped ContDiff Topology
+noncomputable section
+variable {g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ} {w S : Coord → ℝ}
+variable {D U : Set Coord} {m J K k0 : ℕ}
+namespace LocalCompactWaveData
+variable (a : LocalCompactWaveData g w S D m J K k0)
+
+theorem lattice_neighborhood_jet_bound (hUD : U ⊆ D) (hUb : Bornology.IsBounded U)
+    (hS : ContDiff ℝ ∞ S) (S0 T0 : Coord → ℝ)
+    (hS0 : ContDiff ℝ ∞ S0) (hT0 : ContDiff ℝ ∞ T0)
+    {Ω : Set Coord} (hΩ : IsCompact Ω) (hk0 : 2 ≤ k0) :
+    ∀ᶠ n : ℕ in atTop, ∃ hfin : Fintype (SourceGrid U n),
+      letI := hfin
+      ∀ coeff : ((SourceGrid U n × Fin 3) × Fin 2) → ℝ,
+        coeff ∈ coefficientEvent (n:ℝ) →
+        ∀ x ∈ highEnvelopeRegion Ω S S0 n,
+          ∀ z ∈ Metric.closedBall x ((n:ℝ)^2)⁻¹,
+            ‖fderiv ℝ (fun y ↦ a.latticeSeededJet hUD n hfin
+              (oscillatorySeed S0 T0 n) y coeff) z‖ ≤ (n:ℝ)^13 := by
+  obtain ⟨R,hR,hΩR⟩ := hΩ.isBounded.exists_pos_norm_le
+  let Q := Metric.closedBall (0:Coord) (R+1)
+  have hQ : IsCompact Q := isCompact_closedBall _ _
+  obtain ⟨A,hA,hAb⟩ := (hQ.image (hS.continuous_fderiv (by simp))).isBounded.exists_pos_norm_le
+  have hdS : ContDiff ℝ ∞ (fun z ↦ S0 z-S z) := hS0.sub hS
+  obtain ⟨L,hL,hLb⟩ := (hQ.image (hdS.continuous_fderiv (by simp))).isBounded.exists_pos_norm_le
+  obtain ⟨B,hB,hfield⟩ := a.seeded_lattice_field_derivative_bound hUD hUb S0 T0 hS0 hT0 hQ
+  have hfreq : ∀ᶠ n : ℕ in atTop, max L (6*(A+1)*B*Real.exp 1) ≤ (n:ℝ) :=
+    tendsto_natCast_atTop_atTop.eventually (eventually_ge_atTop _)
+  filter_upwards [hfield,hfreq,eventually_gt_atTop (0:ℕ)] with n hn hsize hnpos
+  obtain ⟨hfin,hb⟩ := hn
+  let := hfin
+  refine ⟨hfin,?_⟩
+  intro coeff hc x hx z hz
+  obtain ⟨hu,hbnd⟩ := hb coeff hc
+  have hn1 : (1:ℝ) ≤ n := by exact_mod_cast hnpos
+  have hnR : (0:ℝ) < n := by exact_mod_cast hnpos
+  have hdist : ‖z-x‖ ≤ ((n:ℝ)^2)⁻¹ := by simpa only [Metric.mem_closedBall,dist_eq_norm] using hz
+  have hrad : ((n:ℝ)^2)⁻¹ ≤ 1 :=
+    (inv_le_one₀ (by positivity)).mpr (one_le_pow₀ hn1)
+  have hxQ : x ∈ Q := by
+    change dist x 0 ≤ R+1
+    rw [dist_zero_right]
+    linarith [hΩR x hx.1]
+  have hzQ : z ∈ Q := by
+    change dist z 0 ≤ R+1
+    rw [dist_zero_right]
+    have ht := norm_le_norm_sub_add z x
+    linarith [hΩR x hx.1,hdist.trans hrad]
+  have hgap : (S0 z-S z)-(S0 x-S x) ≤ L*‖z-x‖ := by
+    have h := (convex_closedBall (0:Coord) (R+1)).norm_image_sub_le_of_norm_fderiv_le
+      (fun _ _ ↦ (hdS.differentiable (by simp)).differentiableAt)
+      (fun y hy ↦ hLb _ ⟨y,hy,rfl⟩) hxQ hzQ
+    exact (le_abs_self _).trans h
+  have h := normalized_jet_neighborhood_derivative_bound
+    (fun y ↦ oscillatorySeed S0 T0 n y+gaussianWaveField
+      (fun i : SourceGrid U n × Fin 3 ↦ latticeWave a.cover a.beams hUD n i.1 i.2) coeff y)
+    S S0 hu hS hnpos hx hA.le hB.le hL.le
+    ((le_max_left _ _).trans hsize) ((le_max_right _ _).trans hsize)
+    hdist hgap (hAb _ ⟨z,hzQ,rfl⟩)
+    (fun k ↦ hbnd z hzQ ⟨k.val,by omega⟩)
+  exact h
+
+end LocalCompactWaveData
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/LatticeNetSmallBall.lean b/lean/OAI/Geometry/NodalSets/Waves/LatticeNetSmallBall.lean
new file mode 100644
index 0000000000000000000000000000000000000000..441ed7246dc4e5be0f01c8301caa20f0921f3413
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/LatticeNetSmallBall.lean
@@ -0,0 +1,79 @@
+import OAI.Geometry.NodalSets.Elliptic.SpatialNet
+import OAI.Geometry.NodalSets.Waves.LatticeGaussianSmallBall
+import OAI.Geometry.NodalSets.Waves.NetProbability
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Yau.Probability Set Filter MeasureTheory ProbabilityTheory
+open scoped ContDiff Topology ENNReal
+noncomputable section
+variable {g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ} {w S : Coord → ℝ}
+variable {D U : Set Coord} {m J K k0 : ℕ}
+namespace LocalCompactWaveData
+variable (a : LocalCompactWaveData g w S D m J K k0)
+
+lemma measurable_latticeSeededJet (hUD : U ⊆ D) (n : ℕ)
+    (hfin : Fintype (SourceGrid U n)) (seed : Coord → ℝ) (x : Coord)
+    (hs : DifferentiableAt ℝ seed x)
+    (hV : ∀ z j, DifferentiableAt ℝ (latticeWave a.cover a.beams hUD n z j) x) :
+    Measurable (a.latticeSeededJet hUD n hfin seed x) := by
+  let := hfin
+  let V := fun i : SourceGrid U n × Fin 3 ↦ latticeWave a.cover a.beams hUD n i.1 i.2
+  have he : a.latticeSeededJet hUD n hfin seed x = fun coeff ↦
+      normalizedRealJet seed n (S x) x +
+        pairJetMap (jetWaveCoefficients V n (S x) x) (WithLp.toLp 2 coeff) := by
+    funext coeff
+    exact seeded_gaussian_jet_eq V seed n (S x) x (fun i ↦ hV i.1 i.2) hs coeff
+  rw [he]
+  fun_prop
+
+theorem actual_lattice_net_small_ball (hUD : U ⊆ D) (hU : IsOpen U)
+    (hUb : Bornology.IsBounded U) {Q : Set Coord} (hQ : IsCompact Q) (hQU : Q ⊆ U) :
+    ∃ c > 0, ∃ C > 0, ∀ᶠ n : ℕ in atTop, ∃ hfin : Fintype (SourceGrid U n),
+      letI := hfin
+      ∀ E : Set Coord, E ⊆ Q → ∃ t : Finset Coord,
+        (↑t : Set Coord) ⊆ E ∧ (t.card : ℝ) ≤ C*(n:ℝ)^276 ∧
+        (∀ x ∈ E, ∃ y ∈ t, ‖x-y‖ ≤ (4*(n:ℝ)^69)⁻¹ ∧
+          sourceEuclideanNorm (x-y) ≤ (2*(n:ℝ)^69)⁻¹) ∧
+        ∀ seed : Coord → ℝ, (∀ x ∈ Q, DifferentiableAt ℝ seed x) →
+          gaussianPairs {coeff | ∃ x ∈ t,
+            ‖a.latticeSeededJet hUD n hfin seed x coeff‖ ≤ 2/(n:ℝ)^56} ≤
+          ENNReal.ofReal (C*(n:ℝ)^276*
+            (12*(Real.sqrt (c/(n:ℝ)))⁻¹*(2/(n:ℝ)^56)/Real.sqrt (2*Real.pi))^5) := by
+  obtain ⟨c,hc,hsmall⟩ := a.actual_lattice_gaussian_small_ball hUD hU hUb hQ hQU
+  obtain ⟨C,hC,hnet⟩ := jet_scale_internal_net hQ.isBounded
+  refine ⟨c,hc,C,hC,?_⟩
+  filter_upwards [hsmall,lattice_wave_estimates a.cover a.beams hUD,
+    eventually_gt_atTop (0:ℕ)] with n hn hw hnpos
+  obtain ⟨hfin,hp⟩ := hn
+  let := hfin
+  refine ⟨hfin,?_⟩
+  intro E hEQ
+  obtain ⟨t,ht,hcard,hcover⟩ := hnet n hnpos E hEQ
+  refine ⟨t,ht,hcard,hcover,?_⟩
+  intro seed hs
+  let b := (12*(Real.sqrt (c/(n:ℝ)))⁻¹*(2/(n:ℝ)^56)/Real.sqrt (2*Real.pi))^5
+  have hb : 0 ≤ b := by dsimp [b]; positivity
+  have hp' (x : Coord) (hx : x ∈ t) :
+      gaussianPairs {coeff | ‖a.latticeSeededJet hUD n hfin seed x coeff‖ ≤ 2/(n:ℝ)^56} ≤
+        ENNReal.ofReal b := by
+    have hxQ := hEQ (ht hx)
+    have h := hp seed x hxQ (hs x hxQ) (2/(n:ℝ)^56) (by positivity)
+    rw [Measure.map_apply (a.measurable_latticeSeededJet hUD n hfin seed x (hs x hxQ)
+      (fun z j ↦ ((hw z j).1.differentiable (by simp)).differentiableAt))
+      (measurableSet_le (by fun_prop) measurable_const)] at h
+    exact h
+  have hu := finite_small_ball_union gaussianPairs t
+    (fun coeff x ↦ a.latticeSeededJet hUD n hfin seed x coeff)
+    (2/(n:ℝ)^56) (ENNReal.ofReal b) hp'
+  refine hu.trans ?_
+  rw [ENNReal.ofReal_mul (by positivity)]
+  gcongr
+  exact_mod_cast (ENNReal.ofReal_le_ofReal hcard)
+
+end LocalCompactWaveData
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/LatticeNonmainDecay.lean b/lean/OAI/Geometry/NodalSets/Waves/LatticeNonmainDecay.lean
new file mode 100644
index 0000000000000000000000000000000000000000..37d3dde6dc533bbb69b0569a0fedd831bae42bff
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/LatticeNonmainDecay.lean
@@ -0,0 +1,72 @@
+import OAI.Geometry.NodalSets.Waves.LatticeNormalizedWeights
+
+namespace OAI
+
+namespace Yau.Probability
+open Filter
+open scoped Topology
+
+lemma tendsto_polynomial_main_center_tail (k : ℕ) (a : ℝ) (ha : 0 < a) :
+    Tendsto (fun n : ℕ ↦ (n:ℝ)^k*Real.exp (-a*(n:ℝ)^(1/6:ℝ))) atTop (𝓝 0) := by
+  have hpow : Tendsto (fun n : ℕ ↦ (n:ℝ)^(1/6:ℝ)) atTop atTop :=
+    (tendsto_rpow_atTop (by norm_num : (0:ℝ) < 1/6)).comp tendsto_natCast_atTop_atTop
+  have ht := (tendsto_rpow_mul_exp_neg_mul_atTop_nhds_zero (6*(k:ℝ)) a ha).comp hpow
+  convert ht using 1
+  funext n
+  simp only [Function.comp_def]
+  rw [← Real.rpow_mul (by positivity : (0:ℝ) ≤ n)]
+  rw [show (1/6:ℝ)*(6*(k:ℝ)) = (k:ℝ) by ring,Real.rpow_natCast]
+
+end Yau.Probability
+
+namespace Yau.Geometry
+open Yau.Jets Yau.Probability Set Filter
+open scoped ContDiff Topology
+noncomputable section
+variable {g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ} {w S : Coord → ℝ}
+variable {D U : Set Coord} {m J K k0 : ℕ}
+namespace LocalCompactWaveData
+variable (a : LocalCompactWaveData g w S D m J K k0)
+
+theorem lattice_nonmain_derivatives_negligible (hUD : U ⊆ D) (hU : IsOpen U)
+    (hUb : Bornology.IsBounded U) {Q : Set Coord} (hQ : IsCompact Q) (hQU : Q ⊆ U)
+    (ε : ℝ) (hε : 0 < ε) :
+    ∀ᶠ n : ℕ in atTop, ∃ hfin : Fintype (SourceGrid U n), letI := hfin
+      ∀ x ∈ Q, ∀ k : Fin (k0+1),
+        (∑ i ∈ Finset.univ.filter (fun i : SourceGrid U n × Fin 3 ↦
+          (n:ℝ)^(-5/12:ℝ) < sourceEuclideanNorm (x-scaledLatticePoint n i.1)),
+          ‖iteratedFDeriv ℝ k.val (latticeWave a.cover a.beams hUD n i.1 i.2) x‖^2) /
+          (a.latticeSigma hUD n x)^2 ≤ ε := by
+  classical
+  obtain ⟨c,hc,hlo⟩ := a.lattice_variance_lower_bound hUD hU hQ hQU
+  obtain ⟨C,hC,htail⟩ := a.lattice_nonmain_derivative_bound hUD
+  have hlim := (tendsto_polynomial_main_center_tail (2*k0) a.beams.c a.beams.c_pos).const_mul (C/c)
+  simp only [mul_zero] at hlim
+  filter_upwards [hlo,htail,hlim.eventually (gt_mem_nhds hε),eventually_gt_atTop (0:ℕ)]
+    with n hl ht hsmall hnpos
+  have := finite_source_grid hUb hnpos
+  let hfin := Fintype.ofFinite (SourceGrid U n)
+  refine ⟨hfin,?_⟩
+  intro x hx k
+  have hv : 0 < a.latticeVariance hUD n x :=
+    lt_of_lt_of_le (mul_pos hc (Real.exp_pos _)) (hl x hx)
+  have hn1 : (1:ℝ) ≤ n := by exact_mod_cast hnpos
+  have hp : (n:ℝ)^(2*k.val) ≤ (n:ℝ)^(2*k0) :=
+    pow_le_pow_right₀ hn1 (by have := k.isLt; omega)
+  rw [a.latticeSigma_sq]
+  apply le_trans _ hsmall.le
+  apply (div_le_iff₀ hv).mpr
+  have hh := mul_le_mul_of_nonneg_left (hl x hx)
+    (by positivity : 0 ≤ (C/c)*((n:ℝ)^(2*k0)*Real.exp (-a.beams.c*(n:ℝ)^(1/6:ℝ))))
+  calc
+    _ ≤ C*(n:ℝ)^(2*k.val)*Real.exp (2*(n:ℝ)*S x)*Real.exp (-a.beams.c*(n:ℝ)^(1/6:ℝ)) := ht x k
+    _ ≤ C*(n:ℝ)^(2*k0)*Real.exp (2*(n:ℝ)*S x)*Real.exp (-a.beams.c*(n:ℝ)^(1/6:ℝ)) := by gcongr
+    _ = ((C/c)*((n:ℝ)^(2*k0)*Real.exp (-a.beams.c*(n:ℝ)^(1/6:ℝ))))*
+        (c*Real.exp (2*(n:ℝ)*S x)) := by field_simp
+    _ ≤ _ := hh
+
+end LocalCompactWaveData
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/LatticeNormalizedWeights.lean b/lean/OAI/Geometry/NodalSets/Waves/LatticeNormalizedWeights.lean
new file mode 100644
index 0000000000000000000000000000000000000000..db919b3fc9a050ab40cfca7403d3bc959500e9b6
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/LatticeNormalizedWeights.lean
@@ -0,0 +1,94 @@
+import OAI.Geometry.NodalSets.Waves.LatticeDerivativeTail
+import OAI.Geometry.NodalSets.Waves.LatticeVarianceComparison
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Yau.Probability Set Filter
+open scoped ContDiff Topology
+noncomputable section
+variable {g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ} {w S : Coord → ℝ}
+variable {D U : Set Coord} {m J K k0 : ℕ}
+namespace LocalCompactWaveData
+variable (a : LocalCompactWaveData g w S D m J K k0)
+
+def latticeAlpha (hUD : U ⊆ D) (n : ℕ) [Fintype (SourceGrid U n)]
+    (x : Coord) (i : SourceGrid U n × Fin 3) : ℂ :=
+  latticeWave a.cover a.beams hUD n i.1 i.2 x / (a.latticeSigma hUD n x:ℂ)
+
+lemma latticeAlpha_sum_sq (hUD : U ⊆ D) (n : ℕ) [Fintype (SourceGrid U n)]
+    (x : Coord) (s : Finset (SourceGrid U n × Fin 3)) :
+    ∑ i ∈ s, ‖a.latticeAlpha hUD n x i‖^2 =
+      (∑ i ∈ s, ‖latticeWave a.cover a.beams hUD n i.1 i.2 x‖^2) / a.latticeVariance hUD n x := by
+  simp only [latticeAlpha,norm_div,Complex.norm_real,Real.norm_eq_abs,
+    abs_of_nonneg (show 0 ≤ a.latticeSigma hUD n x from Real.sqrt_nonneg _),
+    div_pow,a.latticeSigma_sq,Finset.sum_div]
+
+lemma latticeAlpha_total (hUD : U ⊆ D) (n : ℕ) [Fintype (SourceGrid U n)]
+    (x : Coord) (hv : 0 < a.latticeVariance hUD n x) :
+    ∑ i, ‖a.latticeAlpha hUD n x i‖^2 = 1 := by
+  rw [a.latticeAlpha_sum_sq]
+  exact div_self hv.ne'
+
+theorem latticeAlpha_nonmain_bound (hUD : U ⊆ D) (hU : IsOpen U)
+    {Q : Set Coord} (hQ : IsCompact Q) (hQU : Q ⊆ U) :
+    ∃ C > 0, ∀ᶠ n : ℕ in atTop, ∀ [Fintype (SourceGrid U n)], ∀ x ∈ Q,
+      0 < a.latticeVariance hUD n x ∧
+      ∑ i ∈ Finset.univ.filter (fun i : SourceGrid U n × Fin 3 ↦
+          (n:ℝ)^(-5/12:ℝ) < sourceEuclideanNorm (x-scaledLatticePoint n i.1)),
+        ‖a.latticeAlpha hUD n x i‖^2 ≤ C*Real.exp (-a.beams.c*(n:ℝ)^(1/6:ℝ)) := by
+  classical
+  obtain ⟨c,hc,hlo⟩ := a.lattice_variance_lower_bound hUD hU hQ hQU
+  obtain ⟨C,hC,htail⟩ := a.lattice_nonmain_derivative_bound hUD
+  refine ⟨C/c,div_pos hC hc,?_⟩
+  filter_upwards [hlo,htail] with n hl ht
+  intro hfin x hx
+  have hv : 0 < a.latticeVariance hUD n x :=
+    lt_of_lt_of_le (mul_pos hc (Real.exp_pos _)) (hl x hx)
+  refine ⟨hv,?_⟩
+  rw [a.latticeAlpha_sum_sq]
+  apply (div_le_iff₀ hv).mpr
+  have h := ht x (0 : Fin (k0+1))
+  simp only [Fin.val_zero,mul_zero,pow_zero,mul_one] at h
+  change (∑ i ∈ Finset.univ.filter (fun i : SourceGrid U n × Fin 3 ↦
+      (n:ℝ)^(-5/12:ℝ) < sourceEuclideanNorm (x-scaledLatticePoint n i.1)),
+    ‖iteratedFDeriv ℝ 0 (latticeWave a.cover a.beams hUD n i.1 i.2) x‖^2) ≤ _ at h
+  simp only [norm_iteratedFDeriv_zero] at h
+  refine h.trans ?_
+  have hh := mul_le_mul_of_nonneg_left (hl x hx)
+    (by positivity : 0 ≤ (C/c)*Real.exp (-a.beams.c*(n:ℝ)^(1/6:ℝ)))
+  calc
+    _ = ((C/c)*Real.exp (-a.beams.c*(n:ℝ)^(1/6:ℝ)))*(c*Real.exp (2*(n:ℝ)*S x)) := by field_simp
+    _ ≤ _ := hh
+
+theorem latticeAlpha_main_mass (hUD : U ⊆ D) (hU : IsOpen U)
+    (hUb : Bornology.IsBounded U) {Q : Set Coord} (hQ : IsCompact Q) (hQU : Q ⊆ U)
+    (ε : ℝ) (hε : 0 < ε) :
+    ∀ᶠ n : ℕ in atTop, ∃ hfin : Fintype (SourceGrid U n), letI := hfin
+      ∀ x ∈ Q, (∑ i, ‖a.latticeAlpha hUD n x i‖^2 = 1) ∧
+        1-ε ≤ ∑ i ∈ Finset.univ.filter (fun i : SourceGrid U n × Fin 3 ↦
+          sourceEuclideanNorm (x-scaledLatticePoint n i.1) ≤ (n:ℝ)^(-5/12:ℝ)),
+          ‖a.latticeAlpha hUD n x i‖^2 := by
+  classical
+  obtain ⟨C,hC,hb⟩ := a.latticeAlpha_nonmain_bound hUD hU hQ hQU
+  have hlim := (tendsto_main_center_tail a.beams.c a.beams.c_pos).const_mul C
+  simp only [mul_zero] at hlim
+  filter_upwards [hb,hlim.eventually (gt_mem_nhds hε),eventually_gt_atTop (0:ℕ)] with n hn hsmall hnpos
+  have := finite_source_grid hUb hnpos
+  let hfin := Fintype.ofFinite (SourceGrid U n)
+  refine ⟨hfin,?_⟩
+  intro x hx
+  obtain ⟨hv,ht⟩ := hn x hx
+  have hall := a.latticeAlpha_total hUD n x hv
+  refine ⟨hall,?_⟩
+  have he := Finset.sum_filter_add_sum_filter_not (s := (Finset.univ : Finset (SourceGrid U n × Fin 3)))
+    (fun i ↦ sourceEuclideanNorm (x-scaledLatticePoint n i.1) ≤ (n:ℝ)^(-5/12:ℝ))
+    (fun i ↦ ‖a.latticeAlpha hUD n x i‖^2)
+  simp only [not_le,hall] at he
+  linarith
+
+end LocalCompactWaveData
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/LatticePhysicalSignSubballs.lean b/lean/OAI/Geometry/NodalSets/Waves/LatticePhysicalSignSubballs.lean
new file mode 100644
index 0000000000000000000000000000000000000000..060ea3fad5e8405463d592ab54250f766ddbfc84
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/LatticePhysicalSignSubballs.lean
@@ -0,0 +1,71 @@
+import OAI.Geometry.NodalSets.Elliptic.CompactSignEvents
+import OAI.Geometry.NodalSets.Elliptic.RescaledSignTransfer
+import OAI.Geometry.NodalSets.Waves.LatticeSignSubballs
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Yau.Probability Set Filter MeasureTheory ProbabilityTheory
+open scoped ContDiff Topology
+noncomputable section
+
+theorem lattice_physical_sign_subballs
+    (g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ) {H : Set Coord}
+    (hH : IsCompact H) (hg : ContinuousOn g H)
+    (hp : ∀ y ∈ H, ∀ v, v ≠ 0 → 0 < g y v v) :
+    ∃ tau : ℝ, 0 < tau ∧ tau < 1/4 ∧ ∃ r : ℝ, 0 < r ∧
+      r < tau/4 ∧ r < (1-tau)/2 ∧ ∃ p : ℝ, 0 < p ∧
+      ∀ (w S S0 T0 : Coord → ℝ) (D U Q : Set Coord) (m J K k0 : ℕ)
+        (a : LocalCompactWaveData g w S D m J K k0) (_ : H ⊆ D) (hUD : U ⊆ D),
+        U ⊆ H → IsOpen U → Bornology.IsBounded U → IsCompact Q → Q ⊆ U →
+        ContDiff ℝ ∞ S → ContDiff ℝ ∞ S0 → ContDiff ℝ ∞ T0 → 5 ≤ k0 →
+        ∀ gamma > 0, (∀ x ∈ Q, S0 x+gamma ≤ S x) →
+        ∀ᶠ n : ℕ in atTop, ∃ hfin : Fintype (SourceGrid U n), letI := hfin
+          ∀ x ∈ Q, ∃ j : Fin 4,
+            let scale := (n:ℝ)*sourceSignScale g S x
+            let f := fun coeff y ↦ oscillatorySeed S0 T0 n y + gaussianWaveField
+              (fun i : SourceGrid U n × Fin 3 ↦ latticeWave a.cover a.beams hUD n i.1 i.2) coeff y
+            let A : Set (((SourceGrid U n × Fin 3) × Fin 2) → ℝ) := {coeff |
+              (∀ y : Coord, sourceEuclideanNorm (y-x) ≤ r/scale → 0 < f coeff y) ∧
+              (∀ y : Coord, sourceEuclideanNorm (y-(x+scale⁻¹ • (tau • Pi.single j 1))) ≤ r/scale →
+                f coeff y < 0)}
+            0 < scale ∧ MeasurableSet A ∧ p ≤ gaussianPairs.real A := by
+  obtain ⟨tau,ht,ht4,r,hr,hrt,hr1,p,hp0,hprob⟩ := lattice_sign_subballs_probability g hH hg hp
+  refine ⟨tau,ht,ht4,r,hr,hrt,hr1,p,hp0,?_⟩
+  intro w S S0 T0 D U Q m J K k0 a hHD hUD hUH hU hUb hQ hQU hS hS0 hT0 hd gamma hgamma hgap
+  obtain ⟨c,hc,C,hC,hsigma⟩ := a.lattice_sigma_comparison hUD hU hUb hQ hQU
+  filter_upwards [hprob w S S0 T0 D U Q m J K k0 a hHD hUD hUH hU hUb hQ hQU
+    hS hS0 hT0 hd gamma hgamma hgap,hsigma,eventually_gt_atTop (0:ℕ),a.estimates] with n hn hs hn0 hest
+  obtain ⟨hfin,hn⟩ := hn
+  obtain ⟨hfin',hs⟩ := hs
+  have heq : hfin' = hfin := Subsingleton.elim _ _
+  subst hfin'
+  let := hfin
+  let : IsProbabilityMeasure (gaussianPairs (ι := SourceGrid U n × Fin 3)) := by
+    unfold gaussianPairs; infer_instance
+  refine ⟨hfin,?_⟩
+  intro x hx
+  obtain ⟨j,hj⟩ := hn x hx
+  have hsc : 0 < (n:ℝ)*sourceSignScale g S x := mul_pos (by exact_mod_cast hn0)
+    (lt_of_lt_of_le (by norm_num : (0:ℝ) < 2) (a.source_sign_scale_lower x (hUD (hQU hx))))
+  have hsig : 0 < a.latticeSigma hUD n x :=
+    (mul_pos hc (Real.exp_pos _)).trans_le (hs x hx).1
+  let V := fun i : SourceGrid U n × Fin 3 ↦ latticeWave a.cover a.beams hUD n i.1 i.2
+  have hV (i : SourceGrid U n × Fin 3) : ContDiff ℝ ∞ (V i) :=
+    (hest (latticeFrame a.cover hUD n i.1,i.2)).1
+  have hmeas := seeded_gaussian_sign_balls_measurable V (oscillatorySeed S0 T0 n)
+    hV (oscillatorySeed_contDiff S0 T0 hS0 hT0 n)
+    x (x+((n:ℝ)*sourceSignScale g S x)⁻¹ • (tau • Pi.single j 1))
+    (r/((n:ℝ)*sourceSignScale g S x)) (r/((n:ℝ)*sourceSignScale g S x))
+  refine ⟨j,hsc,hmeas,hj.trans ?_⟩
+  apply measureReal_mono
+  · intro coeff hcoeff
+    exact rescaled_sign_subballs_transfer _ S0 T0 S n (sourceSignScale g S x)
+        (a.latticeSigma hUD n x) x coeff hsc hsig tau r j
+        (fun v hv ↦ (hcoeff.1 v hv).2) (fun v hv ↦ (hcoeff.2 v hv).2)
+  · exact measure_ne_top _ _
+
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/LatticePlaneWaveEnergy.lean b/lean/OAI/Geometry/NodalSets/Waves/LatticePlaneWaveEnergy.lean
new file mode 100644
index 0000000000000000000000000000000000000000..39dd3f0683d4d79beb9415ae84ab935a7b28b186
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/LatticePlaneWaveEnergy.lean
@@ -0,0 +1,90 @@
+import OAI.Geometry.NodalSets.Elliptic.FrequencyDerivativeEnergy
+import OAI.Geometry.NodalSets.Waves.LatticeFrequencyAnnulus
+import OAI.Geometry.NodalSets.Waves.LatticeFullPlaneWaveApprox
+import OAI.Geometry.NodalSets.Waves.LatticeNormalizedWeights
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Set Filter
+open scoped ContDiff Topology
+noncomputable section
+namespace LocalCompactWaveData
+variable {g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ} {w S : Coord → ℝ}
+variable {D U : Set Coord} {m J K k0 : ℕ}
+variable (a : LocalCompactWaveData g w S D m J K k0)
+
+lemma latticePlaneWaveCoefficient_smooth (hUD : U ⊆ D) (n : ℕ)
+    [Fintype (SourceGrid U n)] (x : Coord) (s : ℝ) (i : SourceGrid U n × Fin 3) :
+    ContDiff ℝ ∞ (a.latticePlaneWaveCoefficient hUD n x s i) := by
+  classical
+  unfold latticePlaneWaveCoefficient
+  split_ifs
+  · exact contDiff_const.mul (planeWave_smooth _ _)
+  · exact contDiff_const
+
+lemma latticePlaneWaveCoefficient_energy (hUD : U ⊆ D) (n : ℕ)
+    [Fintype (SourceGrid U n)] (x : Coord) (s : ℝ) {B : ℝ} (hB : 1 ≤ B)
+    (hm : ∑ i : SourceGrid U n × Fin 3, ‖a.latticeAlpha hUD n x i‖^2 ≤ 1)
+    (hb : ∀ i : SourceGrid U n × Fin 3,
+      sourceEuclideanNorm (x-scaledLatticePoint n i.1) ≤ (n:ℝ)^(-5/12:ℝ) →
+      sourceEuclideanNorm (sourceFrequency (g (scaledLatticePoint n i.1))
+        (a.cover.triple.q (latticeFrame a.cover hUD n i.1) i.2) s) ≤ B)
+    (k : ℕ) (v : Coord) :
+    ∑ i : SourceGrid U n × Fin 3,
+      ‖iteratedFDeriv ℝ k (a.latticePlaneWaveCoefficient hUD n x s i) v‖^2 ≤
+        (k.factorial*(4*B)^k)^2 := by
+  classical
+  have hbound (i : SourceGrid U n × Fin 3) :
+      ‖iteratedFDeriv ℝ k (a.latticePlaneWaveCoefficient hUD n x s i) v‖ ≤
+        ‖a.latticeAlpha hUD n x i‖*(k.factorial*(4*B)^k) := by
+    unfold latticePlaneWaveCoefficient
+    split_ifs with hi
+    · exact weighted_planeWave_derivative_bound _ _ s hB (hb i hi) k v
+    · simp only [iteratedFDeriv_fun_zero, Pi.zero_apply, norm_zero]
+      positivity
+  calc
+    _ ≤ ∑ i : SourceGrid U n × Fin 3,
+        (‖a.latticeAlpha hUD n x i‖*(k.factorial*(4*B)^k))^2 :=
+      Finset.sum_le_sum (fun i _ ↦ pow_le_pow_left₀ (norm_nonneg _) (hbound i) 2)
+    _ = (∑ i : SourceGrid U n × Fin 3, ‖a.latticeAlpha hUD n x i‖^2)*
+        (k.factorial*(4*B)^k)^2 := by simp only [mul_pow,Finset.sum_mul]
+    _ ≤ _ := by nlinarith [sq_nonneg ((k.factorial:ℝ)*(4*B)^k)]
+end LocalCompactWaveData
+
+theorem lattice_planeWave_derivative_energy
+    (g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ) {H : Set Coord}
+    (hH : IsCompact H) (hg : ContinuousOn g H)
+    (hp : ∀ y ∈ H, ∀ v, v ≠ 0 → 0 < g y v v) (d : ℕ) :
+    ∃ C > 0, ∀ (w S : Coord → ℝ) (D U Q : Set Coord) (m J K k0 : ℕ)
+      (a : LocalCompactWaveData g w S D m J K k0) (_ : H ⊆ D) (hUD : U ⊆ D),
+      U ⊆ H → IsOpen U → Bornology.IsBounded U → IsCompact Q → Q ⊆ U →
+      ∀ᶠ n : ℕ in atTop, ∃ hfin : Fintype (SourceGrid U n), letI := hfin
+        ∀ x ∈ Q, ∀ v : Coord, ∀ k : ℕ, k ≤ d →
+        ∑ i : SourceGrid U n × Fin 3,
+          ‖iteratedFDeriv ℝ k
+            (a.latticePlaneWaveCoefficient hUD n x (sourceSignScale g S x) i) v‖^2 ≤ C := by
+  obtain ⟨κ,hκ,B,hB,hann⟩ := lattice_main_frequency_annulus g hH hg hp
+  let B' := max 1 B
+  let C := ((d.factorial:ℝ)*(4*B')^d)^2
+  have hB' : 1 ≤ B' := le_max_left _ _
+  have h4 : 1 ≤ 4*B' := by linarith
+  refine ⟨C,by dsimp [C]; positivity,?_⟩
+  intro w S D U Q m J K k0 a hHD hUD hUH hU hUb hQ hQU
+  filter_upwards [hann w S D U m J K k0 a hHD hUD hUH,
+    a.latticeAlpha_main_mass hUD hU hUb hQ hQU (1/2) (by norm_num)] with n hn hm
+  obtain ⟨hfin,hm⟩ := hm
+  let := hfin
+  refine ⟨hfin,?_⟩
+  intro x hx v k hk
+  apply (a.latticePlaneWaveCoefficient_energy hUD n x _ hB' (by rw [(hm x hx).1])
+    (fun i hi ↦ ((hn x (hUH (hQU hx)) i hi).2).trans (le_max_right _ _)) k v).trans
+  have hf : (k.factorial:ℝ) ≤ d.factorial := by exact_mod_cast Nat.factorial_le hk
+  have hpow := pow_le_pow_right₀ h4 hk
+  dsimp [C]
+  gcongr
+
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/LatticePlaneWaveJets.lean b/lean/OAI/Geometry/NodalSets/Waves/LatticePlaneWaveJets.lean
new file mode 100644
index 0000000000000000000000000000000000000000..b1496c6aa4a9aa383b96ca2d0bb9e852edeeab1b
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/LatticePlaneWaveJets.lean
@@ -0,0 +1,73 @@
+import OAI.Geometry.NodalSets.Waves.LatticeNormalizedWeights
+import OAI.Geometry.NodalSets.Waves.PlaneWaveJets2
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Set Filter
+open scoped ContDiff Topology
+noncomputable section
+variable {g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ} {w S : Coord → ℝ}
+variable {D U : Set Coord} {m J K k0 : ℕ}
+namespace LocalCompactWaveData
+variable (a : LocalCompactWaveData g w S D m J K k0)
+
+theorem lattice_main_coefficient_jets (hUD : U ⊆ D) (R : ℝ) (hR : 0 ≤ R)
+    (d0 : ℕ) (e : ℝ) (he : 0 < e) :
+    ∀ᶠ n : ℕ in atTop, ∀ x ∈ D, ∀ s : ℝ, 1 ≤ s →
+      ∀ (sigma : ℝ) (v : Coord), ‖v‖ ≤ R → ∀ i : SourceGrid U n × Fin 3,
+      sourceEuclideanNorm (x-scaledLatticePoint n i.1) ≤ (n:ℝ)^(-5/12:ℝ) → ∀ k, k ≤ d0 →
+      ‖iteratedFDeriv ℝ k (fun z ↦ normalizedRescaling (latticeWave a.cover a.beams hUD n i.1 i.2) S n s sigma x z -
+        (latticeWave a.cover a.beams hUD n i.1 i.2 x/(sigma:ℂ))*
+          planeWave (g (scaledLatticePoint n i.1) (a.cover.triple.q (latticeFrame a.cover hUD n i.1) i.2)) s z) v‖ ≤
+        e*‖latticeWave a.cover a.beams hUD n i.1 i.2 x/(sigma:ℂ)‖ := by
+  have hy := continuous_coverSourceCenter a.cover
+  have hp0 t := a.nonzero_gradient _ (a.cover.center t).property
+  filter_upwards [a.beams.uniform_main_coefficient_jets a.smooth_G a.positive_G
+    a.smooth_A hy hp0 R hR d0 e he] with n hn
+  intro x hx s hs sigma v hv i hi k hk
+  have hh := hn (latticeFrame a.cover hUD n i.1,i.2) x s hs
+    (by simpa only [latticeFrame_center] using hi) sigma v hv k hk
+  have hxA := (a.germ x (a.centers_E hx)).2.2
+  have hyG := (a.germ (scaledLatticePoint n i.1) (a.centers_E (hUD i.1.property))).1
+  simpa only [latticeFrame_center,hyG.self_of_nhds,latticeWave,TripleSourceWaveData.wave,
+    normalizedRescaling,rescaledSeedNormalizer,hxA.fderiv_eq] using hh
+
+theorem lattice_main_jet_approximation (hUD : U ⊆ D) (hU : IsOpen U)
+    (hUb : Bornology.IsBounded U) {Q : Set Coord} (hQ : IsCompact Q) (hQU : Q ⊆ U)
+    (R : ℝ) (hR : 0 ≤ R) (d0 : ℕ) (e : ℝ) (he : 0 < e) :
+    ∀ᶠ n : ℕ in atTop, ∃ hfin : Fintype (SourceGrid U n), letI := hfin
+      ∀ x ∈ Q, ∀ s : ℝ, 1 ≤ s → ∀ v : Coord, ‖v‖ ≤ R → ∀ k, k ≤ d0 →
+      ∑ i ∈ Finset.univ.filter (fun i : SourceGrid U n × Fin 3 ↦
+        sourceEuclideanNorm (x-scaledLatticePoint n i.1) ≤ (n:ℝ)^(-5/12:ℝ)),
+        ‖iteratedFDeriv ℝ k (fun z ↦ normalizedRescaling (latticeWave a.cover a.beams hUD n i.1 i.2) S n s
+          (a.latticeSigma hUD n x) x z - a.latticeAlpha hUD n x i *
+          planeWave (g (scaledLatticePoint n i.1) (a.cover.triple.q (latticeFrame a.cover hUD n i.1) i.2)) s z) v‖^2 ≤ e := by
+  classical
+  filter_upwards [a.lattice_main_coefficient_jets hUD R hR d0 (Real.sqrt e) (Real.sqrt_pos.mpr he),
+    a.latticeAlpha_main_mass hUD hU hUb hQ hQU 1 zero_lt_one] with n hn hmass
+  obtain ⟨hfin,hmass⟩ := hmass
+  let := hfin
+  refine ⟨hfin,?_⟩
+  intro x hx s hs v hv k hk
+  let I := Finset.univ.filter (fun i : SourceGrid U n × Fin 3 ↦
+    sourceEuclideanNorm (x-scaledLatticePoint n i.1) ≤ (n:ℝ)^(-5/12:ℝ))
+  have hsum : ∑ i ∈ I, ‖a.latticeAlpha hUD n x i‖^2 ≤ 1 := by
+    rw [← (hmass x hx).1]
+    exact Finset.sum_le_sum_of_subset_of_nonneg (Finset.filter_subset _ _) (fun _ _ _ ↦ sq_nonneg _)
+  calc
+    _ ≤ ∑ i ∈ I, e*‖a.latticeAlpha hUD n x i‖^2 := by
+      apply Finset.sum_le_sum
+      intro i hi
+      have hb := hn x (hUD (hQU hx)) s hs (a.latticeSigma hUD n x) v hv i (Finset.mem_filter.mp hi).2 k hk
+      have hh := pow_le_pow_left₀ (norm_nonneg _) hb 2
+      rw [mul_pow,Real.sq_sqrt he.le] at hh
+      exact hh
+    _ = e*(∑ i ∈ I, ‖a.latticeAlpha hUD n x i‖^2) := by rw [Finset.mul_sum]
+    _ ≤ e := by nlinarith
+
+end LocalCompactWaveData
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/LatticeProbabilityRates.lean b/lean/OAI/Geometry/NodalSets/Waves/LatticeProbabilityRates.lean
new file mode 100644
index 0000000000000000000000000000000000000000..162186ee4deda30b8a5e8c241d7a193496409af6
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/LatticeProbabilityRates.lean
@@ -0,0 +1,65 @@
+import OAI.Geometry.NodalSets.Coefficients.GaussianCoefficientTail
+import OAI.Geometry.NodalSets.Elliptic.NetDecay
+import OAI.Geometry.NodalSets.Waves.LatticeCount
+import OAI.Geometry.NodalSets.Waves.LatticeNetSmallBall
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Yau.Probability Set Filter MeasureTheory ProbabilityTheory
+open scoped Topology ENNReal
+noncomputable section
+
+lemma lattice_coefficient_event_failure {U : Set Coord} (hU : Bornology.IsBounded U) :
+    ∃ C > 0, ∀ n : ℕ, 0 < n → ∀ hfin : Fintype (SourceGrid U n),
+      letI := hfin
+      (gaussianPairs (ι := SourceGrid U n × Fin 3)) (coefficientEvent (n:ℝ))ᶜ ≤
+        ENNReal.ofReal (C*(n:ℝ)^2*Real.exp (-(n:ℝ)^2/8)) := by
+  obtain ⟨C,hC,hcard⟩ := source_coefficient_card_bound hU
+  refine ⟨4*C,by positivity,?_⟩
+  intro n hn hfin
+  let := hfin
+  have h := coefficient_event_failure (ι := SourceGrid U n × Fin 3)
+    (r := (n:ℝ)) (by positivity)
+  refine h.trans (ENNReal.ofReal_le_ofReal ?_)
+  have hc : (Fintype.card (SourceGrid U n × Fin 3):ℝ) ≤ C*(n:ℝ)^2 := by
+    simpa only [Nat.card_eq_fintype_card] using hcard n hn
+  nlinarith [mul_le_mul_of_nonneg_right hc (Real.exp_nonneg (-(n:ℝ)^2/8))]
+
+variable {g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ} {w S : Coord → ℝ}
+variable {D U : Set Coord} {m J K k0 : ℕ}
+namespace LocalCompactWaveData
+variable (a : LocalCompactWaveData g w S D m J K k0)
+
+theorem actual_lattice_net_failure_rate (hUD : U ⊆ D) (hU : IsOpen U)
+    (hUb : Bornology.IsBounded U) {Q : Set Coord} (hQ : IsCompact Q) (hQU : Q ⊆ U) :
+    ∃ B > 0, ∀ᶠ n : ℕ in atTop, ∃ hfin : Fintype (SourceGrid U n),
+      letI := hfin
+      ∀ E : Set Coord, E ⊆ Q → ∃ t : Finset Coord,
+        (↑t : Set Coord) ⊆ E ∧
+        (∀ x ∈ E, ∃ y ∈ t, ‖x-y‖ ≤ (4*(n:ℝ)^69)⁻¹ ∧
+          sourceEuclideanNorm (x-y) ≤ (2*(n:ℝ)^69)⁻¹) ∧
+        ∀ seed : Coord → ℝ, (∀ x ∈ Q, DifferentiableAt ℝ seed x) →
+          gaussianPairs {coeff | ∃ x ∈ t,
+            ‖a.latticeSeededJet hUD n hfin seed x coeff‖ ≤ 2/(n:ℝ)^56} ≤
+          ENNReal.ofReal (B/((n:ℝ)*Real.sqrt n)) := by
+  obtain ⟨c,hc,C,hC,hnet⟩ := a.actual_lattice_net_small_ball hUD hU hUb hQ hQU
+  refine ⟨C*(24/(Real.sqrt c*Real.sqrt (2*Real.pi)))^5,by positivity,?_⟩
+  filter_upwards [hnet,eventually_gt_atTop (0:ℕ)] with n hn hnpos
+  obtain ⟨hfin,hb⟩ := hn
+  let := hfin
+  refine ⟨hfin,?_⟩
+  intro E hEQ
+  obtain ⟨t,ht,_,hcover,hprob⟩ := hb E hEQ
+  refine ⟨t,ht,hcover,?_⟩
+  intro seed hs
+  have h := hprob seed hs
+  rw [net_small_ball_rate C c n hc (by positivity)] at h
+  exact h
+
+end LocalCompactWaveData
+
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/LatticeSignMargin.lean b/lean/OAI/Geometry/NodalSets/Waves/LatticeSignMargin.lean
new file mode 100644
index 0000000000000000000000000000000000000000..ead70b40728139118c0be6267e4e642c8dbbe7b9
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/LatticeSignMargin.lean
@@ -0,0 +1,82 @@
+import OAI.Geometry.NodalSets.Elliptic.SignScaleSeed
+import OAI.Geometry.NodalSets.Waves.LatticeActualTwoPoint
+import OAI.Geometry.NodalSets.Waves.LatticeSignValueBounds
+import OAI.Geometry.NodalSets.Waves.TwoValueSignProbability
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Yau.Probability MeasureTheory ProbabilityTheory Set Filter
+open scoped ContDiff Topology ENNReal
+noncomputable section
+
+theorem lattice_sign_margin_probability
+    (g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ) {H : Set Coord}
+    (hH : IsCompact H) (hg : ContinuousOn g H)
+    (hp : ∀ y ∈ H, ∀ v, v ≠ 0 → 0 < g y v v) :
+    ∃ tau : ℝ, 0 < tau ∧ tau < 1/4 ∧ ∃ p : ℝ≥0∞, 0 < p ∧
+      ∀ (w S S0 T0 : Coord → ℝ) (D U Q : Set Coord) (m J K k0 : ℕ)
+        (a : LocalCompactWaveData g w S D m J K k0) (_ : H ⊆ D) (hUD : U ⊆ D),
+        U ⊆ H → IsOpen U → Bornology.IsBounded U → IsCompact Q → Q ⊆ U →
+        ContDiff ℝ ∞ S → ContDiff ℝ ∞ S0 → ContDiff ℝ ∞ T0 →
+        ∀ gamma > 0, (∀ x ∈ Q, S0 x+gamma ≤ S x) →
+        ∀ᶠ n : ℕ in atTop, ∃ hfin : Fintype (SourceGrid U n), letI := hfin
+          ∀ x ∈ Q, ∃ j : Fin 4,
+            let F := rescaledGaussianField
+              (fun i : SourceGrid U n × Fin 3 ↦ latticeWave a.cover a.beams hUD n i.1 i.2)
+              S0 T0 S n (sourceSignScale g S x) (a.latticeSigma hUD n x) x
+            p ≤ gaussianPairs {coeff | 1 ≤ F coeff 0 ∧ F coeff (tau • Pi.single j 1) ≤ -1} := by
+  obtain ⟨tau,ht,ht4,d,hd,henergy⟩ := lattice_actual_two_point_energy g hH hg hp
+  obtain ⟨p,hp,hprob⟩ := two_value_sign_probability d 2 hd (by norm_num)
+  refine ⟨tau,ht,ht4,p,hp,?_⟩
+  intro w S S0 T0 D U Q m J K k0 a hHD hUD hUH hU hUb hQ hQU hS hS0 hT0 gamma hgamma hgap
+  filter_upwards [henergy w S D U Q m J K k0 a hHD hUD hUH hU hUb hQ hQU hS,
+    a.lattice_sign_coefficient_upper hUD hU hUb hQ hQU hS,
+    a.sign_scale_seed_decay hUD hU hUb hQ hQU hS S0 T0 hS0 hT0 gamma hgamma hgap 0 1 (by norm_num)]
+    with n he hu hs
+  obtain ⟨hfin,he⟩ := he
+  obtain ⟨hfin',hu⟩ := hu
+  obtain ⟨hfin'',hs⟩ := hs
+  have heq : hfin' = hfin := Subsingleton.elim _ _
+  have heq' : hfin'' = hfin := Subsingleton.elim _ _
+  subst hfin'
+  subst hfin''
+  let := hfin
+  refine ⟨hfin,?_⟩
+  intro x hx
+  obtain ⟨j,hj⟩ := he x hx
+  let v : Coord := tau • Pi.single j 1
+  let seed := rescaledSeed S0 T0 S n (sourceSignScale g S x) (a.latticeSigma hUD n x) x
+  let A := fun i : SourceGrid U n × Fin 3 ↦ normalizedRescaling
+    (latticeWave a.cover a.beams hUD n i.1 i.2) S n (sourceSignScale g S x)
+    (a.latticeSigma hUD n x) x
+  let mean : EuclideanSpace ℝ (Fin 2) := WithLp.toLp 2 ![seed 0,seed v]
+  have hvnorm : ‖v‖ = tau := by
+    simp [v,norm_smul,Real.norm_eq_abs,abs_of_pos ht,Pi.norm_single]
+  have hv : sourceEuclideanNorm v ≤ 2 := (sourceEuclideanNorm_le v).trans (by rw [hvnorm]; linarith)
+  have hv0 : sourceEuclideanNorm (0:Coord) ≤ 2 := by simp [sourceEuclideanNorm]
+  have hs0 := hs x hx 0 hv0 (0 : Fin 1)
+  have hs1 := hs x hx v hv (0 : Fin 1)
+  simp only [Fin.val_zero,norm_iteratedFDeriv_zero] at hs0 hs1
+  change ‖seed 0‖ ≤ 1 at hs0
+  change ‖seed v‖ ≤ 1 at hs1
+  have hmean : ‖mean‖ ≤ 2 := by
+    have hsq : ‖mean‖^2 = (seed 0)^2+(seed v)^2 := by
+      simp [mean,EuclideanSpace.real_norm_sq_eq,Fin.sum_univ_two]
+    have h0 := pow_le_pow_left₀ (norm_nonneg _) hs0 2
+    have h1 := pow_le_pow_left₀ (norm_nonneg _) hs1 2
+    simp only [Real.norm_eq_abs,sq_abs,one_pow] at h0 h1
+    nlinarith [norm_nonneg mean]
+  have hb := hprob (fun i ↦ A i 0) (fun i ↦ A i v) mean hmean
+    (hu x hx 0 (by simp)) (hu x hx v (by rw [hvnorm]; linarith)) hj
+  refine ⟨j,?_⟩
+  convert hb using 1
+  congr 1
+  ext coeff
+  simp [rescaledGaussianField_decomposition,mean,seed,A,v,
+    pairLinearSum_eq_complex,gaussianWaveField,Complex.re_sum]
+
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/LatticeSignSubballs.lean b/lean/OAI/Geometry/NodalSets/Waves/LatticeSignSubballs.lean
new file mode 100644
index 0000000000000000000000000000000000000000..3174b7413ef178765f867f86e7a9aaa5a17a6ba2
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/LatticeSignSubballs.lean
@@ -0,0 +1,78 @@
+import OAI.Geometry.NodalSets.Waves.LatticeGradientExpectation
+import OAI.Geometry.NodalSets.Waves.LatticeSignMargin
+import OAI.Geometry.NodalSets.Waves.SignSubballProbability
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Yau.Probability Set Filter MeasureTheory ProbabilityTheory
+open scoped ContDiff Topology
+noncomputable section
+
+theorem lattice_sign_subballs_probability
+    (g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ) {H : Set Coord}
+    (hH : IsCompact H) (hg : ContinuousOn g H)
+    (hp : ∀ y ∈ H, ∀ v, v ≠ 0 → 0 < g y v v) :
+    ∃ tau : ℝ, 0 < tau ∧ tau < 1/4 ∧ ∃ r : ℝ, 0 < r ∧
+      r < tau/4 ∧ r < (1-tau)/2 ∧ ∃ p : ℝ, 0 < p ∧
+      ∀ (w S S0 T0 : Coord → ℝ) (D U Q : Set Coord) (m J K k0 : ℕ)
+        (a : LocalCompactWaveData g w S D m J K k0) (_ : H ⊆ D) (hUD : U ⊆ D),
+        U ⊆ H → IsOpen U → Bornology.IsBounded U → IsCompact Q → Q ⊆ U →
+        ContDiff ℝ ∞ S → ContDiff ℝ ∞ S0 → ContDiff ℝ ∞ T0 → 5 ≤ k0 →
+        ∀ gamma > 0, (∀ x ∈ Q, S0 x+gamma ≤ S x) →
+        ∀ᶠ n : ℕ in atTop, ∃ hfin : Fintype (SourceGrid U n), letI := hfin
+          ∀ x ∈ Q, ∃ j : Fin 4,
+            let F := rescaledGaussianField
+              (fun i : SourceGrid U n × Fin 3 ↦ latticeWave a.cover a.beams hUD n i.1 i.2)
+              S0 T0 S n (sourceSignScale g S x) (a.latticeSigma hUD n x) x
+            p ≤ gaussianPairs.real {coeff |
+              (∀ v : Coord, sourceEuclideanNorm v ≤ r →
+                sourceEuclideanNorm v ≤ 1 ∧ 0 < F coeff v) ∧
+              (∀ v : Coord, sourceEuclideanNorm (v-tau • Pi.single j 1) ≤ r →
+                sourceEuclideanNorm v ≤ 1 ∧ F coeff v < 0)} := by
+  obtain ⟨tau,ht,ht4,p,hp0,hmargin⟩ := lattice_sign_margin_probability g hH hg hp
+  obtain ⟨q,hq,hq1,hqp⟩ := positive_real_probability_floor hp0
+  obtain ⟨C,hC,hmean⟩ := lattice_gradient_expectation g hH hg hp
+  obtain ⟨L,hL,r,hr,hrt,hr1,hrL,hCL⟩ := sign_subball_constants ht (by linarith) hq hC.le
+  refine ⟨tau,ht,ht4,r,hr,hrt,hr1,q/2,by positivity,?_⟩
+  intro w S S0 T0 D U Q m J K k0 a hHD hUD hUH hU hUb hQ hQU hS hS0 hT0 hd gamma hgamma hgap
+  filter_upwards [hmargin w S S0 T0 D U Q m J K k0 a hHD hUD hUH hU hUb hQ hQU
+    hS hS0 hT0 gamma hgamma hgap,
+    hmean w S S0 T0 D U Q m J K k0 a hHD hUD hUH hU hUb hQ hQU
+    hS hS0 hT0 hd gamma hgamma hgap,a.estimates] with n hn hm hest
+  obtain ⟨hfin,hn⟩ := hn
+  obtain ⟨hfin',hm⟩ := hm
+  have heq : hfin' = hfin := Subsingleton.elim _ _
+  subst hfin'
+  let := hfin
+  let : IsProbabilityMeasure (gaussianPairs (ι := SourceGrid U n × Fin 3)) := by
+    unfold gaussianPairs; infer_instance
+  refine ⟨hfin,?_⟩
+  intro x hx
+  obtain ⟨j,hj⟩ := hn x hx
+  obtain ⟨G,hGc,hGi,hG0,hGC,hGe⟩ := hm x hx
+  let V := fun i : SourceGrid U n × Fin 3 ↦ latticeWave a.cover a.beams hUD n i.1 i.2
+  let F := rescaledGaussianField V S0 T0 S n (sourceSignScale g S x) (a.latticeSigma hUD n x) x
+  have hV (i : SourceGrid U n × Fin 3) : ContDiff ℝ ∞ (V i) :=
+    (hest (latticeFrame a.cover hUD n i.1,i.2)).1
+  have hF (coeff) : ContDiff ℝ ∞ (F coeff) := by
+    have he : F coeff = fun z ↦ rescaledSeed S0 T0 S n (sourceSignScale g S x)
+        (a.latticeSigma hUD n x) x z + gaussianWaveField
+          (fun i ↦ normalizedRescaling (V i) S n (sourceSignScale g S x)
+            (a.latticeSigma hUD n x) x) coeff z := by
+      ext z
+      exact rescaledGaussianField_decomposition _ _ _ _ _ _ _ _ _ _
+    rw [he]
+    exact (rescaledSeed_smooth _ _ _ hS0 hT0 _ _ _ _).add
+      (gaussianWaveField_contDiff _ _ (fun i ↦ normalizedRescaling_smooth _ (hV i) _ _ _ _ _))
+  have hqA : q ≤ gaussianPairs.real {coeff | 1 ≤ F coeff 0 ∧ F coeff (tau • Pi.single j 1) ≤ -1} := by
+    have hh := ENNReal.toReal_mono (measure_ne_top _ _) (hqp.trans hj)
+    simpa only [ENNReal.toReal_ofReal hq.le,Measure.real] using hh
+  refine ⟨j,?_⟩
+  exact sign_subballs_probability gaussianPairs F hF G hGi hG0 ht.le hr.le hL
+    (by linarith) hrL hGC hCL (fun coeff hcoeff ↦ (hGe coeff L hL.le).mp hcoeff) j hqA
+
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/LatticeSignValueBounds.lean b/lean/OAI/Geometry/NodalSets/Waves/LatticeSignValueBounds.lean
new file mode 100644
index 0000000000000000000000000000000000000000..f9942ef0839c78674404ac22a1cdd5ab5329b562
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/LatticeSignValueBounds.lean
@@ -0,0 +1,57 @@
+import OAI.Geometry.NodalSets.Coefficients.TwoPointCoefficientEnergy
+import OAI.Geometry.NodalSets.Elliptic.SignScaleSeed
+import OAI.Geometry.NodalSets.Waves.LatticeTwoPointCovariance
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Yau.Probability MeasureTheory ProbabilityTheory Set Filter
+open scoped ContDiff Topology
+noncomputable section
+variable {g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ} {w S : Coord → ℝ}
+variable {D U : Set Coord} {m J K k0 : ℕ}
+namespace LocalCompactWaveData
+variable (a : LocalCompactWaveData g w S D m J K k0)
+
+theorem lattice_sign_coefficient_upper (hUD : U ⊆ D) (hU : IsOpen U)
+    (hUb : Bornology.IsBounded U) {Q : Set Coord} (hQ : IsCompact Q) (hQU : Q ⊆ U)
+    (hS : ContDiff ℝ ∞ S) :
+    ∀ᶠ n : ℕ in atTop, ∃ hfin : Fintype (SourceGrid U n), letI := hfin
+      ∀ x ∈ Q, ∀ v : Coord, ‖v‖ ≤ 2 →
+        ∑ i : SourceGrid U n × Fin 3,
+          ‖normalizedRescaling (latticeWave a.cover a.beams hUD n i.1 i.2)
+            S n (sourceSignScale g S x) (a.latticeSigma hUD n x) x v‖^2 ≤ 4 := by
+  classical
+  filter_upwards [a.lattice_full_jet_approximation hUD hU hUb hQ hQU hS 2 (by norm_num) 1 (by norm_num),
+    a.latticeAlpha_main_mass hUD hU hUb hQ hQU (1/2) (by norm_num)] with n herr hm
+  obtain ⟨hfin,herr⟩ := herr
+  obtain ⟨hfin',hm⟩ := hm
+  have heq : hfin' = hfin := Subsingleton.elim _ _
+  subst hfin'
+  let := hfin
+  refine ⟨hfin,?_⟩
+  intro x hx v hv
+  let W := a.latticePlaneWaveCoefficient hUD n x (sourceSignScale g S x)
+  let A := fun i : SourceGrid U n × Fin 3 ↦ normalizedRescaling
+    (latticeWave a.cover a.beams hUD n i.1 i.2) S n (sourceSignScale g S x)
+    (a.latticeSigma hUD n x) x
+  have hs : 1 ≤ sourceSignScale g S x := by
+    linarith [a.source_sign_scale_lower x (hUD (hQU hx))]
+  have herr' := herr x hx _ hs v hv (0 : Fin (k0+1))
+  simp +instances only [Fin.val_zero,norm_iteratedFDeriv_zero] at herr'
+  have hW : ∑ i, ‖W i v‖^2 ≤ 1 := by
+    rw [← variance_pairLinearSum,← gaussianWaveField_eq_pair,a.lattice_plane_wave_variance,← (hm x hx).1]
+    exact Finset.sum_le_sum_of_subset_of_nonneg (Finset.filter_subset _ _) (fun i _ _ ↦ sq_nonneg _)
+  have hh (i : SourceGrid U n × Fin 3) : ‖A i v‖^2 ≤ 2*‖A i v-W i v‖^2+2*‖W i v‖^2 := by
+    simpa only [sub_add_cancel] using complex_norm_add_sq_bound (A i v-W i v) (W i v)
+  have hsum := Finset.sum_le_sum (fun i (_ : i ∈ (Finset.univ : Finset (SourceGrid U n × Fin 3))) ↦ hh i)
+  simp only [Finset.sum_add_distrib,← Finset.mul_sum] at hsum
+  change (∑ i, ‖A i v‖^2) ≤ 4
+  change (∑ i, ‖A i v-W i v‖^2) ≤ 1 at herr'
+  linarith
+
+end LocalCompactWaveData
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/LatticeSourceWaves.lean b/lean/OAI/Geometry/NodalSets/Waves/LatticeSourceWaves.lean
new file mode 100644
index 0000000000000000000000000000000000000000..0bca450103c1284cf064b7c3e4a58a27b79997d9
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/LatticeSourceWaves.lean
@@ -0,0 +1,82 @@
+import OAI.Geometry.NodalSets.Waves.CompactSourceWaves
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Set Filter
+open scoped ContDiff Topology
+noncomputable section
+
+def scaledLatticePoint (n : ℕ) (z : Fin 4 → ℤ) : Coord :=
+  (n:ℝ)^(-1/2:ℝ) • (fun i ↦ (z i:ℝ))
+
+abbrev SourceGrid (U : Set Coord) (n : ℕ) := {z : Fin 4 → ℤ // scaledLatticePoint n z ∈ U}
+
+variable {g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ} {w S : Coord → ℝ}
+variable {D U : Set Coord} {m J K k0 : ℕ}
+
+def selectFrame (d : SourceCompactFrameCover g S D) (x : D) : d.Parameter :=
+  Classical.choose (d.surjective_center x)
+
+lemma selectFrame_center (d : SourceCompactFrameCover g S D) (x : D) :
+    coverSourceCenter d (selectFrame d x) = x :=
+  congrArg Subtype.val (Classical.choose_spec (d.surjective_center x))
+
+def latticeFrame (d : SourceCompactFrameCover g S D) (hUD : U ⊆ D)
+    (n : ℕ) (z : SourceGrid U n) : d.Parameter :=
+  selectFrame d ⟨scaledLatticePoint n z,hUD z.property⟩
+
+lemma latticeFrame_center (d : SourceCompactFrameCover g S D) (hUD : U ⊆ D)
+    (n : ℕ) (z : SourceGrid U n) :
+    coverSourceCenter d (latticeFrame d hUD n z) = scaledLatticePoint n z :=
+  selectFrame_center d _
+
+def latticeWave (d : SourceCompactFrameCover g S D)
+    (b : TripleSourceWaveData g w S (coverSourceCenter d) d.triple m J K k0)
+    (hUD : U ⊆ D) (n : ℕ) (z : SourceGrid U n) (j : Fin 3) : Coord → ℂ :=
+  chartPushforward (b.F (latticeFrame d hUD n z,j))
+    (coordinateWave b.phi b.A J (n:ℝ) (latticeFrame d hUD n z,j))
+
+theorem lattice_wave_estimates (d : SourceCompactFrameCover g S D)
+    (b : TripleSourceWaveData g w S (coverSourceCenter d) d.triple m J K k0)
+    (hUD : U ⊆ D) :
+    ∀ᶠ n : ℕ in atTop, ∀ (z : SourceGrid U n) (j : Fin 3),
+      ContDiff ℝ ∞ (latticeWave d b hUD n z j) ∧ HasCompactSupport (latticeWave d b hUD n z j) ∧
+      tsupport (latticeWave d b hUD n z j) ⊆
+        {x | sourceEuclideanNorm (x-scaledLatticePoint n z) ≤ b.R*(n:ℝ)^(-1/3:ℝ)} ∧
+      ∀ x : Coord,
+        (∀ k : Fin (k0+1), ‖iteratedFDeriv ℝ k.val (latticeWave d b hUD n z j) x‖ ≤
+          b.Cw*(n:ℝ)^k.val*Real.exp ((n:ℝ)*S x-b.c*(n:ℝ)*(sourceEuclideanNorm (x-scaledLatticePoint n z))^2)) ∧
+        DerivativeBound k0 (fun x ↦ sourceWeightedOperator g w (latticeWave d b hUD n z j) x +
+          ((4:ℂ)*(n:ℂ)^2+6*(n:ℂ))*latticeWave d b hUD n z j x) x
+            (b.Cr*(n:ℝ)^(-(K:ℝ))*Real.exp ((n:ℝ)*S x)) := by
+  filter_upwards [b.estimates] with n hn
+  intro z j
+  simpa only [latticeWave,latticeFrame_center] using hn (latticeFrame d hUD n z,j)
+
+theorem lattice_direction_properties (d : SourceCompactFrameCover g S D)
+    (hUD : U ⊆ D) (n : ℕ) (z : SourceGrid U n) :
+    let x := scaledLatticePoint n z
+    let q := d.triple.q (latticeFrame d hUD n z)
+    LinearIndependent ℝ q ∧ ∀ j,
+      g x (metricGradient g S x) (q j) = 0 ∧
+      g x (q j) (q j) = g x (metricGradient g S x) (metricGradient g S x)+4 ∧
+      0 < sourceHessian g S x (metricGradient g S x) (metricGradient g S x)+sourceHessian g S x (q j) (q j) := by
+  have he := latticeFrame_center d hUD n z
+  refine ⟨d.triple.independent _,?_⟩
+  intro j
+  have hp := d.triple.perpendicular (latticeFrame d hUD n z) j
+  have hl := d.triple.length (latticeFrame d hUD n z) j
+  have hs := d.triple.strict (latticeFrame d hUD n z) j
+  change g (coverSourceCenter d _) (metricGradient g S (coverSourceCenter d _)) _ = 0 at hp
+  change g (coverSourceCenter d _) _ _ = g (coverSourceCenter d _) (metricGradient g S (coverSourceCenter d _))
+    (metricGradient g S (coverSourceCenter d _))+4 at hl
+  change 0 < sourceHessian g S (coverSourceCenter d _) (metricGradient g S (coverSourceCenter d _))
+    (metricGradient g S (coverSourceCenter d _))+sourceHessian g S (coverSourceCenter d _) _ _ at hs
+  rw [he] at hp hl hs
+  exact ⟨hp,hl,hs⟩
+
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/LatticeSphereExtension.lean b/lean/OAI/Geometry/NodalSets/Waves/LatticeSphereExtension.lean
new file mode 100644
index 0000000000000000000000000000000000000000..08c79cf94493964bd6302ae4f9cf704b27870461
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/LatticeSphereExtension.lean
@@ -0,0 +1,46 @@
+import OAI.Geometry.NodalSets.Charts.SeedChartScalarExtension
+import OAI.Geometry.NodalSets.Coefficients.LatticeResidualSupport
+
+namespace OAI
+
+namespace Yau.Target
+open Manifold Yau.Geometry Yau.Jets Yau.Probability Set Filter
+open scoped ContDiff Topology
+noncomputable section
+variable {g : Yau.Jets.Coord → Yau.Jets.Coord →L[ℝ] Yau.Jets.Coord →L[ℝ] ℝ}
+variable {w S : Yau.Jets.Coord → ℝ} {D U : Set Yau.Jets.Coord} {m J K k0 : ℕ}
+variable (b : LocalCompactWaveData g w S D m J K k0)
+
+theorem literal_lattice_sphere_extension (hD : IsCompact D) (hUD : U ⊆ D)
+    (hUb : Bornology.IsBounded U) {Ω : Set Yau.Jets.Coord} (hΩ : IsOpen Ω) (hDΩ : D ⊆ Ω) :
+    ∃ Q : Set Yau.Jets.Coord, IsCompact Q ∧ Q ⊆ Ω ∧
+      ∀ᶠ n : ℕ in atTop, ∃ hfin : Fintype (SourceGrid U n),
+        letI := hfin
+        ∀ coeff : ((SourceGrid U n × Fin 3) × Fin 2) → ℝ,
+          let v := gaussianWaveField
+            (fun i : SourceGrid U n × Fin 3 ↦ latticeWave b.cover b.beams hUD n i.1 i.2) coeff
+          ContDiff ℝ ∞ (fun x ↦ seedCoordinateField n x+v x) →
+          ∃ u : Base → ℝ, ContMDiff (𝓡 4) 𝓘(ℝ,ℝ) ∞ u ∧
+            tsupport (u-sphericalSeed n) ⊆ seedSphereFromCoord '' Q ∧
+            ∀ x, u (seedSphereFromCoord x) = seedCoordinateField n x+v x := by
+  obtain ⟨Q,hQ,hQΩ,hDQ,hbound⟩ := b.lattice_residual_support hD hUD hUb hΩ hDΩ
+  refine ⟨Q,hQ,hQΩ,?_⟩
+  filter_upwards [hbound] with n hn
+  obtain ⟨hfin,hb⟩ := hn
+  let := hfin
+  refine ⟨hfin,?_⟩
+  intro coeff v hu
+  have hv : ContDiff ℝ ∞ v := by
+    have he : v = fun x ↦ (seedCoordinateField n x+v x)-seedCoordinateField n x := by
+      funext x; ring
+    rw [he]
+    exact hu.sub (seedCoordinateField_smooth n)
+  have hs := (hb coeff).1
+  obtain ⟨u,hus,hup,hval⟩ := seed_plus_compact_sphere_field v hv
+    (hQ.of_isClosed_subset isClosed_closure hs) n
+  exact ⟨u,hus,hup.trans (image_mono hs),hval⟩
+
+end
+end Yau.Target
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/LatticeSuccessfulPacking.lean b/lean/OAI/Geometry/NodalSets/Waves/LatticeSuccessfulPacking.lean
new file mode 100644
index 0000000000000000000000000000000000000000..984dd0059b47ebea0fad59666aabbd00b2180fee
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/LatticeSuccessfulPacking.lean
@@ -0,0 +1,95 @@
+import OAI.Geometry.NodalSets.Elliptic.SignScalePacking
+import OAI.Geometry.NodalSets.Elliptic.SuccessfulTestSelection
+import OAI.Geometry.NodalSets.Waves.LatticeGoodEvent
+import OAI.Geometry.NodalSets.Waves.LatticePhysicalSignSubballs
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Yau.Probability Set Filter MeasureTheory
+open scoped ContDiff Topology
+noncomputable section
+
+theorem lattice_successful_packing
+    (g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ) {H : Set Coord}
+    (hH : IsCompact H) (hg : ContinuousOn g H)
+    (hp : ∀ y ∈ H, ∀ v, v ≠ 0 → 0 < g y v v) :
+    ∃ tau : ℝ, 0 < tau ∧ tau < 1/4 ∧ ∃ r : ℝ, 0 < r ∧
+      r < tau/4 ∧ r < (1-tau)/2 ∧ ∃ c : ℝ, 0 < c ∧
+      ∀ (w S S0 T0 : Coord → ℝ) (D U Q Ω : Set Coord) (m J K k0 : ℕ)
+        (a : LocalCompactWaveData g w S D m J K k0) (_ : H ⊆ D) (hUD : U ⊆ D),
+        U ⊆ H → IsOpen U → Bornology.IsBounded U → IsCompact Q → Q ⊆ U →
+        ContDiff ℝ ∞ S → ContDiff ℝ ∞ S0 → ContDiff ℝ ∞ T0 → 5 ≤ k0 →
+        ∀ gamma > 0, (∀ x ∈ Q, S0 x+gamma ≤ S x) →
+        IsCompact Ω → (∀ x ∈ Ω, fderiv ℝ T0 x ≠ 0) →
+        ∀ d > 0, (∀ x ∈ Ω, x ∉ U → S x-S0 x ≤ -d) →
+        0 < (∫ x in Q, sourceSignScale g S x) →
+        ∀ᶠ n : ℕ in atTop, ∃ hfin : Fintype (SourceGrid U n), letI := hfin
+          ∃ t : Finset Coord, ∃ j : t → Fin 4,
+          ∃ coeff ∈ a.latticeGoodEvent hUD n hfin S0 T0 Ω, ∃ u : Finset t,
+            let R := fun x ↦ ((n:ℝ)*sourceSignScale g S x)⁻¹
+            let f := fun y ↦ oscillatorySeed S0 T0 n y + gaussianWaveField
+              (fun i : SourceGrid U n × Fin 3 ↦ latticeWave a.cover a.beams hUD n i.1 i.2) coeff y
+            (↑t : Set Coord) ⊆ Q ∧
+            (↑t : Set Coord).PairwiseDisjoint (fun x ↦ sourceClosedBall x (R x)) ∧
+            (∀ x ∈ t, 0 < R x ∧ sourceClosedBall x (5*R x) ⊆ U) ∧
+            (∀ x ∈ u,
+              (∀ y ∈ sourceClosedBall x (r*R x), 0 < f y) ∧
+              (∀ y ∈ sourceClosedBall ((x : Coord)+R x • (tau • Pi.single (j x) 1)) (r*R x), f y < 0)) ∧
+            c*((n:ℝ)*(∫ x in Q, sourceSignScale g S x)) ≤ ∑ x ∈ u, (R x)^3 := by
+  classical
+  obtain ⟨tau,ht,ht4,r,hr,hrt,hr1,p,hp0,hprob⟩ := lattice_physical_sign_subballs g hH hg hp
+  let C : ℝ := 2*(Real.pi^2/2)*5^4
+  have hC : 0 < C := by dsimp [C]; positivity
+  refine ⟨tau,ht,ht4,r,hr,hrt,hr1,p/(2*C),by positivity,?_⟩
+  intro w S S0 T0 D U Q Ω m J K k0 a hHD hUD hUH hU hUb hQ hQU hS hS0 hT0 hk0
+    gamma hgamma hgap hΩ hq d hd hout hmass
+  filter_upwards [hprob w S S0 T0 D U Q m J K k0 a hHD hUD hUH hU hUb hQ hQU
+    hS hS0 hT0 hk0 gamma hgamma hgap,
+    a.source_sign_scale_packing hQ hU hQU hUD,
+    a.lattice_good_event_probability hUD hU hUb hS S0 T0 hS0 hT0 hΩ (by omega) hq hd hout
+      (p/4) (by positivity)] with n hsign hpack hgood
+  obtain ⟨hfin,hsign⟩ := hsign
+  obtain ⟨hfin',hgood⟩ := hgood
+  have heq : hfin' = hfin := Subsingleton.elim _ _
+  subst hfin'
+  let := hfin
+  let : IsProbabilityMeasure (gaussianPairs (ι := SourceGrid U n × Fin 3)) := by
+    unfold gaussianPairs; infer_instance
+  obtain ⟨hn,t,htQ,htdis,htcov,htU,hvol⟩ := hpack
+  let R := fun x ↦ ((n:ℝ)*sourceSignScale g S x)⁻¹
+  let f := fun coeff y ↦ oscillatorySeed S0 T0 n y + gaussianWaveField
+    (fun i : SourceGrid U n × Fin 3 ↦ latticeWave a.cover a.beams hUD n i.1 i.2) coeff y
+  choose j hj using (fun x : t ↦ hsign x (htQ x.property))
+  let A := fun x : t ↦ {coeff |
+    (∀ y ∈ sourceClosedBall x (r*R x), 0 < f coeff y) ∧
+    (∀ y ∈ sourceClosedBall ((x : Coord)+R x • (tau • Pi.single (j x) 1)) (r*R x), f coeff y < 0)}
+  have hA (x : t) : MeasurableSet (A x) ∧ p ≤ gaussianPairs.real (A x) := by
+    simpa only [A,f,R,sourceClosedBall,mem_ofPred_eq,div_eq_mul_inv] using (hj x).2
+  have hsum : (∑ x : t, (R x)^3) = ∑ x ∈ t, (R x)^3 := by
+    exact Finset.sum_coe_sort t (fun x ↦ (R x)^3)
+  have hvol' : (n:ℝ)*(∫ x in Q, sourceSignScale g S x) ≤ C * ∑ x : t, (R x)^3 := by
+    rw [hsum]
+    exact hvol
+  have hT : 0 < ∑ x : t, (R x)^3 := by
+    have hh : 0 < (n:ℝ)*(∫ x in Q, sourceSignScale g S x) := mul_pos (by exact_mod_cast hn) hmass
+    nlinarith
+  have hfail : (gaussianPairs (a.latticeGoodEvent hUD n hfin S0 T0 Ω)ᶜ).toReal < p/2 := by
+    have hh := ENNReal.toReal_mono ENNReal.ofReal_ne_top hgood.2
+    rw [ENNReal.toReal_ofReal (by positivity : 0 ≤ p/4)] at hh
+    linarith
+  obtain ⟨coeff,hcoeff,u,hu,hw⟩ := exists_good_successful_tests
+    (fun x : t ↦ (R x)^3) (fun x ↦ (pow_pos (htU x x.property).1 3).le) hT
+    A (fun x ↦ (hA x).1) hp0 (fun x ↦ (hA x).2)
+    (a.latticeGoodEvent hUD n hfin S0 T0 Ω) hgood.1 hfail
+  refine ⟨hfin,t,j,coeff,hcoeff,u,htQ,htdis,htU,hu,?_⟩
+  calc
+    p/(2*C)*((n:ℝ)*(∫ x in Q, sourceSignScale g S x)) ≤
+        p/(2*C)*(C*∑ x : t, (R x)^3) := mul_le_mul_of_nonneg_left hvol' (by positivity)
+    _ = p*(∑ x : t, (R x)^3)/2 := by field_simp
+    _ ≤ _ := hw
+
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/LatticeTwoPointCovariance.lean b/lean/OAI/Geometry/NodalSets/Waves/LatticeTwoPointCovariance.lean
new file mode 100644
index 0000000000000000000000000000000000000000..95bf6faf995d77af3bfb7e8e4f22c3b474e7c88b
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/LatticeTwoPointCovariance.lean
@@ -0,0 +1,57 @@
+import OAI.Geometry.NodalSets.Charts.MetricFrequencyBounds
+import OAI.Geometry.NodalSets.Waves.LatticeFullPlaneWaveApprox
+import OAI.Geometry.NodalSets.Waves.PlaneWaveCovariance
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Yau.Probability MeasureTheory ProbabilityTheory Set Filter
+noncomputable section
+variable {g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ} {w S : Coord → ℝ}
+variable {D U : Set Coord} {m J K k0 : ℕ}
+namespace LocalCompactWaveData
+variable (a : LocalCompactWaveData g w S D m J K k0)
+
+lemma lattice_plane_wave_variance (hUD : U ⊆ D) (n : ℕ) [Fintype (SourceGrid U n)]
+    (x v : Coord) (s : ℝ) :
+    Var[fun coeff ↦ gaussianWaveField (a.latticePlaneWaveCoefficient hUD n x s) coeff v;gaussianPairs] =
+      ∑ i ∈ Finset.univ.filter (fun i : SourceGrid U n × Fin 3 ↦
+        sourceEuclideanNorm (x-scaledLatticePoint n i.1) ≤ (n:ℝ)^(-5/12:ℝ)),
+        ‖a.latticeAlpha hUD n x i‖^2 := by
+  classical
+  rw [gaussianWaveField_eq_pair,variance_pairLinearSum,Finset.sum_filter]
+  apply Finset.sum_congr rfl
+  intro i _
+  dsimp only [latticePlaneWaveCoefficient]
+  split_ifs <;> simp [planeWave_norm]
+
+lemma lattice_plane_wave_covariance (hUD : U ⊆ D) (n : ℕ) [Fintype (SourceGrid U n)]
+    (x : Coord) (s tau : ℝ) (j : Fin 4) :
+    cov[(fun coeff ↦ gaussianWaveField (a.latticePlaneWaveCoefficient hUD n x s) coeff 0),
+      (fun coeff ↦ gaussianWaveField (a.latticePlaneWaveCoefficient hUD n x s)
+        coeff (tau • Pi.single j 1));gaussianPairs] =
+      ∑ i ∈ Finset.univ.filter (fun i : SourceGrid U n × Fin 3 ↦
+        sourceEuclideanNorm (x-scaledLatticePoint n i.1) ≤ (n:ℝ)^(-5/12:ℝ)),
+        ‖a.latticeAlpha hUD n x i‖^2 * Real.cos (tau *
+          sourceFrequency (g (scaledLatticePoint n i.1))
+            (a.cover.triple.q (latticeFrame a.cover hUD n i.1) i.2) s j) := by
+  classical
+  rw [gaussianWaveField_eq_pair,gaussianWaveField_eq_pair,covariance_pairLinearSum,Finset.sum_filter]
+  apply Finset.sum_congr rfl
+  intro i _
+  dsimp only [latticePlaneWaveCoefficient]
+  split_ifs
+  · simp only [planeWave_zero,mul_one]
+    rw [show planeWave (g (scaledLatticePoint n i.1)
+        (a.cover.triple.q (latticeFrame a.cover hUD n i.1) i.2)) s (tau • Pi.single j 1) =
+        Complex.exp (Complex.I * (tau * sourceFrequency (g (scaledLatticePoint n i.1))
+          (a.cover.triple.q (latticeFrame a.cover hUD n i.1) i.2) s j : ℝ)) by
+        simp [planeWave,sourceFrequency,map_smul,smul_eq_mul,mul_div_assoc]]
+    exact complex_rotated_pairing _ _
+  · simp
+
+end LocalCompactWaveData
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/LatticeTwoPointGap.lean b/lean/OAI/Geometry/NodalSets/Waves/LatticeTwoPointGap.lean
new file mode 100644
index 0000000000000000000000000000000000000000..89a9398643a7d98d82288607ab658d832c6ec37f
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/LatticeTwoPointGap.lean
@@ -0,0 +1,80 @@
+import OAI.Geometry.NodalSets.Waves.CosineCovarianceBounds
+import OAI.Geometry.NodalSets.Waves.LatticeTwoPointCovariance
+import OAI.Geometry.NodalSets.Waves.PlaneWaveSecondMoment
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Yau.Probability MeasureTheory ProbabilityTheory Set Filter
+open scoped Topology
+noncomputable section
+
+theorem lattice_plane_wave_two_point_gap
+    (g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ) {H : Set Coord}
+    (hH : IsCompact H) (hg : ContinuousOn g H)
+    (hp : ∀ y ∈ H, ∀ v, v ≠ 0 → 0 < g y v v) :
+    ∃ tau : ℝ, 0 < tau ∧ tau < 1/4 ∧ ∃ delta > 0,
+      ∀ (w S : Coord → ℝ) (D U Q : Set Coord) (m J K k0 : ℕ)
+        (a : LocalCompactWaveData g w S D m J K k0) (_ : H ⊆ D) (hUD : U ⊆ D),
+        U ⊆ H → IsOpen U → Bornology.IsBounded U → IsCompact Q → Q ⊆ U →
+        ∀ᶠ n : ℕ in atTop, ∃ hfin : Fintype (SourceGrid U n), letI := hfin
+          ∀ x ∈ Q, ∃ j : Fin 4,
+            let G := fun v coeff ↦ gaussianWaveField
+              (a.latticePlaneWaveCoefficient hUD n x (sourceSignScale g S x)) coeff v
+            let M := Var[G 0;gaussianPairs]
+            let C := cov[G 0,G (tau • Pi.single j 1);gaussianPairs]
+            (1/2:ℝ) ≤ M ∧ M ≤ 1 ∧
+              Var[G (tau • Pi.single j 1);gaussianPairs] = M ∧
+              M/2 ≤ C ∧ C ≤ M-delta := by
+  classical
+  obtain ⟨κ,hκ,B,hB,hann⟩ := lattice_main_frequency_annulus g hH hg hp
+  let tau : ℝ := 1/(8*(B+1))
+  have ht : 0 < tau := by dsimp [tau]; positivity
+  have ht4 : tau < 1/4 := by
+    dsimp [tau]
+    apply (div_lt_iff₀ (by positivity : 0 < 8*(B+1))).mpr
+    linarith
+  have hsmall : tau*B ≤ 1 := by
+    dsimp [tau]
+    rw [one_div,mul_comm,← div_eq_mul_inv]
+    apply (div_le_iff₀ (by positivity : 0 < 8*(B+1))).mpr
+    linarith
+  let delta : ℝ := (2/Real.pi^2)*tau^2*(κ^2/8)
+  have hd : 0 < delta := by dsimp [delta]; positivity
+  refine ⟨tau,ht,ht4,delta,hd,?_⟩
+  intro w S D U Q m J K k0 a hHD hUD hUH hU hUb hQ hQU
+  filter_upwards [hann w S D U m J K k0 a hHD hUD hUH,
+    a.latticeAlpha_main_mass hUD hU hUb hQ hQU (1/2) (by norm_num)] with n hn hm
+  obtain ⟨hfin,hm⟩ := hm
+  let := hfin
+  refine ⟨hfin,?_⟩
+  intro x hx
+  let F := Finset.univ.filter (fun i : SourceGrid U n × Fin 3 ↦
+    sourceEuclideanNorm (x-scaledLatticePoint n i.1) ≤ (n:ℝ)^(-5/12:ℝ))
+  let alpha := a.latticeAlpha hUD n x
+  let k := fun i : SourceGrid U n × Fin 3 ↦ sourceFrequency (g (scaledLatticePoint n i.1))
+    (a.cover.triple.q (latticeFrame a.cover hUD n i.1) i.2) (sourceSignScale g S x)
+  have hmass : (1/2:ℝ) ≤ ∑ i ∈ F, ‖alpha i‖^2 := by
+    have := (hm x hx).2
+    dsimp only [F,alpha]
+    linarith
+  have hmass1 : ∑ i ∈ F, ‖alpha i‖^2 ≤ 1 := by
+    rw [← (hm x hx).1]
+    exact Finset.sum_le_sum_of_subset_of_nonneg (Finset.filter_subset _ _) (fun i _ _ ↦ sq_nonneg _)
+  have hb (i : SourceGrid U n × Fin 3) (hi : i ∈ F) :
+      κ ≤ sourceEuclideanNorm (k i) ∧ sourceEuclideanNorm (k i) ≤ B :=
+    hn x (hUH (hQU hx)) i (Finset.mem_filter.mp hi).2
+  obtain ⟨j,hj⟩ := coordinate_second_moment F alpha k κ hκ.le hmass (fun i hi ↦ (hb i hi).1)
+  have hcoord (i : SourceGrid U n × Fin 3) (hi : i ∈ F) : |k i j| ≤ B :=
+    (norm_le_pi_norm (k i) j).trans ((norm_le_sourceEuclideanNorm _).trans (hb i hi).2)
+  have hbounds := cosine_covariance_bounds F (fun i ↦ ‖alpha i‖^2) (fun i ↦ k i j)
+    tau B (κ^2/8) (fun i hi ↦ sq_nonneg _) ht.le hsmall hcoord hj
+  refine ⟨j,?_⟩
+  dsimp only
+  rw [a.lattice_plane_wave_variance,a.lattice_plane_wave_variance,a.lattice_plane_wave_covariance]
+  exact ⟨hmass,hmass1,rfl,hbounds.1,hbounds.2⟩
+
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/LatticeValueVariance.lean b/lean/OAI/Geometry/NodalSets/Waves/LatticeValueVariance.lean
new file mode 100644
index 0000000000000000000000000000000000000000..ce3ad6a6e0d495b69e1eab99d3242ce7bde04722
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/LatticeValueVariance.lean
@@ -0,0 +1,45 @@
+import OAI.Geometry.NodalSets.Waves.GaussianWaveJet
+import OAI.Geometry.NodalSets.Waves.LatticeNormalizedWeights
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Yau.Probability Set Filter MeasureTheory ProbabilityTheory
+noncomputable section
+
+lemma gaussianWaveField_value_variance {ι : Type*} [Fintype ι]
+    (V : ι → Coord → ℂ) (seed : Coord → ℝ) (x : Coord) :
+    Var[fun coeff ↦ seed x+gaussianWaveField V coeff x; gaussianPairs] =
+      ∑ i, ‖V i x‖^2 := by
+  have he : (fun coeff ↦ gaussianWaveField V coeff x) = pairLinearSum (fun i ↦ V i x) := by
+    funext coeff
+    rw [pairLinearSum_eq_complex]
+    simp only [gaussianWaveField,Complex.re_sum]
+  let : IsProbabilityMeasure (gaussianPairs (ι := ι)) := by
+    change IsProbabilityMeasure (Measure.pi (fun _ : ι × Fin 2 ↦ gaussianReal 0 1))
+    infer_instance
+  have hc : Continuous (pairLinearSum (fun i ↦ V i x)) := by
+    unfold pairLinearSum
+    fun_prop
+  change Var[(fun coeff ↦ seed x+(fun coeff ↦ gaussianWaveField V coeff x) coeff); gaussianPairs] = _
+  rw [he,variance_const_add hc.aestronglyMeasurable,variance_pairLinearSum]
+
+variable {g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ} {w S : Coord → ℝ}
+variable {D U : Set Coord} {m J K k0 : ℕ}
+namespace LocalCompactWaveData
+variable (a : LocalCompactWaveData g w S D m J K k0)
+
+lemma lattice_value_variance (hUD : U ⊆ D) (n : ℕ) [Fintype (SourceGrid U n)]
+    (seed : Coord → ℝ) (x : Coord) :
+    Var[fun coeff ↦ seed x+gaussianWaveField
+      (fun i : SourceGrid U n × Fin 3 ↦ latticeWave a.cover a.beams hUD n i.1 i.2) coeff x;
+      gaussianPairs] = (a.latticeSigma hUD n x)^2 := by
+  rw [a.latticeSigma_sq]
+  exact gaussianWaveField_value_variance _ seed x
+
+end LocalCompactWaveData
+
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/LatticeVarianceComparison.lean b/lean/OAI/Geometry/NodalSets/Waves/LatticeVarianceComparison.lean
new file mode 100644
index 0000000000000000000000000000000000000000..6e38329a5d2c048f4f73c2bc8f9c2e7befc9f472
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/LatticeVarianceComparison.lean
@@ -0,0 +1,84 @@
+import OAI.Geometry.NodalSets.Waves.LatticeEnergyLowerBound
+import OAI.Geometry.NodalSets.Waves.LatticeLogJets
+import OAI.Geometry.NodalSets.Waves.LatticeVarianceUpper
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Set Metric Filter
+open scoped ContDiff Topology
+noncomputable section
+variable {g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ} {w S : Coord → ℝ}
+variable {D U : Set Coord} {m J K k0 : ℕ}
+namespace LocalCompactWaveData
+variable (a : LocalCompactWaveData g w S D m J K k0)
+
+theorem lattice_variance_lower_bound (hUD : U ⊆ D) (hU : IsOpen U)
+    {Q : Set Coord} (hQ : IsCompact Q) (hQU : Q ⊆ U) :
+    ∃ c > 0, ∀ᶠ n : ℕ in atTop, ∀ [Fintype (SourceGrid U n)], ∀ x ∈ Q,
+      c*Real.exp (2*(n:ℝ)*S x) ≤ a.latticeVariance hUD n x := by
+  obtain ⟨Ca,hCa,Cr,hCr,Ci,hCi,c,hc,C,hC,hb⟩ :=
+    a.lattice_logarithmic_jet_estimates hUD 2 (by norm_num)
+  obtain ⟨r,hr,hinner⟩ := compact_inner_ball hQ hU hQU
+  refine ⟨c^2,pow_pos hc 2,?_⟩
+  filter_upwards [hb,eventually_gt_atTop (0:ℕ),
+    eventually_nat_frequency (half_scale_eventually 2 1 r hr)] with n hn hnpos hsmall
+  intro hfin x hx
+  have hd := (norm_le_sourceEuclideanNorm _).trans (rounded_lattice_distance hnpos x)
+  have hmem : scaledLatticePoint n (roundedLatticeIndex n x) ∈ U :=
+    hinner x hx (by simpa only [mem_closedBall,dist_eq_norm] using hd.trans hsmall.2.1)
+  let z : SourceGrid U n := ⟨roundedLatticeIndex n x,hmem⟩
+  have hd' : ‖x-scaledLatticePoint n z‖ ≤ 2*(n:ℝ)^(-1/2:ℝ) := by
+    rw [norm_sub_rev]; exact hd
+  have hl := (hn z 0 x hd').2.2.2.2.1
+  have hle : c*Real.exp ((n:ℝ)*S x) ≤ ‖latticeWave a.cover a.beams hUD n z 0 x‖ := by
+    have hh := mul_le_mul_of_nonneg_right hl.1 (Real.exp_nonneg ((n:ℝ)*S x))
+    simpa only [mul_assoc,← Real.exp_add,neg_mul,neg_add_cancel,Real.exp_zero,mul_one] using hh
+  have hs := pow_le_pow_left₀ (by positivity : 0 ≤ c*Real.exp ((n:ℝ)*S x)) hle 2
+  have he : (c*Real.exp ((n:ℝ)*S x))^2 = c^2*Real.exp (2*(n:ℝ)*S x) := by
+    rw [mul_pow,← Real.exp_nat_mul]
+    congr 2
+    ring
+  rw [he] at hs
+  apply hs.trans
+  unfold latticeVariance
+  exact Finset.single_le_sum (fun i _ ↦ sq_nonneg ‖latticeWave a.cover a.beams hUD n i.1 i.2 x‖)
+    (Finset.mem_univ (z,(0:Fin 3)))
+
+theorem lattice_sigma_comparison (hUD : U ⊆ D) (hU : IsOpen U)
+    (hUb : Bornology.IsBounded U) {Q : Set Coord} (hQ : IsCompact Q) (hQU : Q ⊆ U) :
+    ∃ c > 0, ∃ C > 0, ∀ᶠ n : ℕ in atTop, ∃ hfin : Fintype (SourceGrid U n),
+      letI := hfin
+      ∀ x ∈ Q, c*Real.exp ((n:ℝ)*S x) ≤ a.latticeSigma hUD n x ∧
+        a.latticeSigma hUD n x ≤ C*Real.exp ((n:ℝ)*S x) := by
+  obtain ⟨c,hc,hlo⟩ := a.lattice_variance_lower_bound hUD hU hQ hQU
+  obtain ⟨C,hC,hup⟩ := a.lattice_variance_upper_bound hUD
+  refine ⟨Real.sqrt c,Real.sqrt_pos.mpr hc,Real.sqrt C,Real.sqrt_pos.mpr hC,?_⟩
+  filter_upwards [hlo,hup,eventually_gt_atTop (0:ℕ)] with n hl hu hn
+  have := finite_source_grid hUb hn
+  let hfin := Fintype.ofFinite (SourceGrid U n)
+  refine ⟨hfin,?_⟩
+  intro x hx
+  have hl' := hl x hx
+  have hu' := hu x
+  have hs := a.latticeSigma_sq hUD n x
+  have hnon : 0 ≤ a.latticeSigma hUD n x := Real.sqrt_nonneg _
+  have he : (Real.exp ((n:ℝ)*S x))^2 = Real.exp (2*(n:ℝ)*S x) := by
+    rw [← Real.exp_nat_mul]
+    congr 1
+    ring
+  have hc2 := Real.sq_sqrt hc.le
+  have hC2 := Real.sq_sqrt hC.le
+  constructor
+  · apply (sq_le_sq₀ (by positivity) hnon).mp
+    rw [mul_pow,hc2,he,hs]
+    exact hl'
+  · apply (sq_le_sq₀ hnon (by positivity)).mp
+    rw [mul_pow,hC2,he,hs]
+    exact hu'
+
+end LocalCompactWaveData
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/LatticeVarianceUpper.lean b/lean/OAI/Geometry/NodalSets/Waves/LatticeVarianceUpper.lean
new file mode 100644
index 0000000000000000000000000000000000000000..e3bba1812883416ee6fa601b20dee5f832d96a74
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/LatticeVarianceUpper.lean
@@ -0,0 +1,87 @@
+import OAI.Geometry.NodalSets.Waves.LocalCompactWaves
+import OAI.Geometry.NodalSets.Waves.ScaledLatticeGaussianSum
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Set Filter
+open scoped ContDiff Topology
+noncomputable section
+variable {g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ} {w S : Coord → ℝ}
+variable {D U : Set Coord} {m J K k0 : ℕ}
+namespace LocalCompactWaveData
+variable (a : LocalCompactWaveData g w S D m J K k0)
+
+def latticeVariance (hUD : U ⊆ D) (n : ℕ) [Fintype (SourceGrid U n)] (x : Coord) : ℝ :=
+  ∑ i : SourceGrid U n × Fin 3, ‖latticeWave a.cover a.beams hUD n i.1 i.2 x‖^2
+
+def latticeSigma (hUD : U ⊆ D) (n : ℕ) [Fintype (SourceGrid U n)] (x : Coord) : ℝ :=
+  Real.sqrt (a.latticeVariance hUD n x)
+
+lemma latticeVariance_nonneg (hUD : U ⊆ D) (n : ℕ) [Fintype (SourceGrid U n)] (x : Coord) :
+    0 ≤ a.latticeVariance hUD n x := Finset.sum_nonneg (fun _ _ ↦ sq_nonneg _)
+
+lemma latticeSigma_sq (hUD : U ⊆ D) (n : ℕ) [Fintype (SourceGrid U n)] (x : Coord) :
+    (a.latticeSigma hUD n x)^2 = a.latticeVariance hUD n x :=
+  Real.sq_sqrt (a.latticeVariance_nonneg hUD n x)
+
+theorem lattice_derivative_square_sum_bound (hUD : U ⊆ D) :
+    ∃ C > 0, ∀ᶠ n : ℕ in atTop, ∀ [Fintype (SourceGrid U n)], ∀ x : Coord,
+      ∀ k : Fin (k0+1),
+      ∑ i : SourceGrid U n × Fin 3,
+        ‖iteratedFDeriv ℝ k.val (latticeWave a.cover a.beams hUD n i.1 i.2) x‖^2 ≤
+        C*(n:ℝ)^(2*k.val)*Real.exp (2*(n:ℝ)*S x) := by
+  obtain ⟨B,hB,hgauss⟩ := source_lattice_gaussian_sum (show 0 < 2*a.beams.c by have := a.beams.c_pos; positivity)
+  refine ⟨3*a.beams.Cw^2*B,by have := a.beams.Cw_pos; positivity,?_⟩
+  filter_upwards [a.estimates,eventually_gt_atTop (0:ℕ)] with n hn hnpos
+  intro hfin x k
+  let R := a.beams.Cw^2*(n:ℝ)^(2*k.val)*Real.exp (2*(n:ℝ)*S x)
+  have hR : 0 ≤ R := by dsimp [R]; positivity
+  have hb (i : SourceGrid U n × Fin 3) :
+      ‖iteratedFDeriv ℝ k.val (latticeWave a.cover a.beams hUD n i.1 i.2) x‖^2 ≤
+        R*Real.exp (-(2*a.beams.c)*(n:ℝ)*(sourceEuclideanNorm (x-scaledLatticePoint n i.1))^2) := by
+    have h := ((hn (latticeFrame a.cover hUD n i.1,i.2)).2.2.2.2 x).1 k
+    simp only [latticeFrame_center] at h
+    change ‖iteratedFDeriv ℝ k.val (latticeWave a.cover a.beams hUD n i.1 i.2) x‖ ≤ _ at h
+    have hh := pow_le_pow_left₀ (norm_nonneg _) h 2
+    refine hh.trans_eq ?_
+    dsimp [R]
+    rw [mul_pow,mul_pow,← Real.exp_nat_mul,← pow_mul]
+    simp only [mul_assoc,← Real.exp_add]
+    congr 2 <;> ring_nf
+  calc
+    _ ≤ ∑ i : SourceGrid U n × Fin 3, R*Real.exp (-(2*a.beams.c)*(n:ℝ)*
+        (sourceEuclideanNorm (x-scaledLatticePoint n i.1))^2) := Finset.sum_le_sum (fun i _ ↦ hb i)
+    _ = 3*R*∑ z : SourceGrid U n, Real.exp (-(2*a.beams.c)*(n:ℝ)*
+        (sourceEuclideanNorm (x-scaledLatticePoint n z))^2) := by
+      rw [Fintype.sum_prod_type]
+      simp only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul]
+      rw [← Finset.mul_sum,← Finset.mul_sum,mul_assoc]
+      dsimp [R]
+      ring
+    _ ≤ 3*R*B := mul_le_mul_of_nonneg_left (hgauss U n hnpos x) (by positivity)
+    _ = _ := by dsimp [R]; ring
+
+theorem lattice_variance_upper_bound (hUD : U ⊆ D) :
+    ∃ C > 0, ∀ᶠ n : ℕ in atTop, ∀ [Fintype (SourceGrid U n)], ∀ x : Coord,
+      a.latticeVariance hUD n x ≤ C*Real.exp (2*(n:ℝ)*S x) := by
+  obtain ⟨C,hC,hb⟩ := a.lattice_derivative_square_sum_bound hUD
+  refine ⟨C,hC,?_⟩
+  filter_upwards [hb] with n hn
+  intro hfin x
+  calc
+    a.latticeVariance hUD n x =
+        ∑ i : SourceGrid U n × Fin 3,
+          ‖iteratedFDeriv ℝ 0 (latticeWave a.cover a.beams hUD n i.1 i.2) x‖^2 := by
+      apply Finset.sum_congr rfl
+      intro i hi
+      exact congrArg (fun z : ℝ ↦ z^2)
+        (norm_iteratedFDeriv_zero (𝕜 := ℝ)
+          (f := latticeWave a.cover a.beams hUD n i.1 i.2) (x := x)).symm
+    _ ≤ _ := by simpa only [Fin.val_zero,mul_zero,pow_zero,mul_one] using! hn x 0
+
+end LocalCompactWaveData
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/LocalCompactWaves.lean b/lean/OAI/Geometry/NodalSets/Waves/LocalCompactWaves.lean
new file mode 100644
index 0000000000000000000000000000000000000000..ef4f23fee32192943907f8843b73531c961f6726
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/LocalCompactWaves.lean
@@ -0,0 +1,82 @@
+import OAI.Geometry.NodalSets.Elliptic.CompactSourceRepresentatives
+import OAI.Geometry.NodalSets.Waves.CompactSourceWaves
+import OAI.Geometry.NodalSets.Waves.LocalSourceWaveTransfer
+import OAI.Geometry.NodalSets.Waves.WaveSecondJet
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Set Filter
+open scoped ContDiff Topology
+noncomputable section
+attribute [local instance] clmTopology clmAdd clmModule
+
+structure LocalCompactWaveData
+    (g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ) (w S : Coord → ℝ)
+    (D : Set Coord) (m J K k0 : ℕ) where
+  G : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ
+  W : Coord → ℝ
+  A : Coord → ℝ
+  E : Set Coord
+  smooth_G : ContDiff ℝ ∞ G
+  smooth_W : ContDiff ℝ ∞ W
+  smooth_A : ContDiff ℝ ∞ A
+  symmetric_G : ∀ x u v, G x u v = G x v u
+  positive_G : ∀ x v, v ≠ 0 → 0 < G x v v
+  open_E : IsOpen E
+  centers_E : D ⊆ E
+  germ : ∀ x ∈ E, (G =ᶠ[𝓝 x] g) ∧ (W =ᶠ[𝓝 x] w) ∧ (A =ᶠ[𝓝 x] S)
+  nonzero_gradient : ∀ x ∈ D, metricGradient G A x ≠ 0
+  cover : SourceCompactFrameCover G A D
+  beams : TripleSourceWaveData G W A (coverSourceCenter cover) cover.triple m J K k0
+  estimates : ∀ᶠ n : ℕ in atTop, ∀ t : cover.Parameter × Fin 3,
+    ContDiff ℝ ∞ (beams.wave (n:ℝ) t) ∧ HasCompactSupport (beams.wave (n:ℝ) t) ∧
+    tsupport (beams.wave (n:ℝ) t) ⊆ E ∧
+    tsupport (beams.wave (n:ℝ) t) ⊆
+      {z | sourceEuclideanNorm (z-coverSourceCenter cover t.1) ≤ beams.R*(n:ℝ)^(-1/3:ℝ)} ∧
+    ∀ z : Coord,
+      (∀ k : Fin (k0+1), ‖iteratedFDeriv ℝ k.val (beams.wave (n:ℝ) t) z‖ ≤
+        beams.Cw*(n:ℝ)^k.val*Real.exp ((n:ℝ)*S z-beams.c*(n:ℝ)*(sourceEuclideanNorm (z-coverSourceCenter cover t.1))^2)) ∧
+      DerivativeBound k0 (fun z ↦ sourceWeightedOperator g w (beams.wave (n:ℝ) t) z +
+        ((4:ℂ)*(n:ℂ)^2+6*(n:ℂ))*beams.wave (n:ℝ) t z) z
+          (beams.Cr*(n:ℝ)^(-(K:ℝ))*Real.exp ((n:ℝ)*S z))
+
+theorem construct_local_compact_waves {D U : Set Coord} (hD : IsCompact D)
+    (hU : IsOpen U) (hDU : D ⊆ U) (hUn : U.Nonempty)
+    (g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ) (hg : ContDiffOn ℝ ∞ g U)
+    (hs : ∀ x ∈ U, ∀ u v, g x u v = g x v u)
+    (hp : ∀ x ∈ U, ∀ v, v ≠ 0 → 0 < g x v v)
+    (w S : Coord → ℝ) (hw : ContDiffOn ℝ ∞ w U) (hS : ContDiffOn ℝ ∞ S U)
+    (hwp : ∀ x ∈ U, 0 < w x) (hp0 : ∀ x ∈ D, metricGradient g S x ≠ 0)
+    (hdir : ∀ x ∈ D, ∃ v : Coord, g x (metricGradient g S x) v = 0 ∧ g x v v = 1 ∧
+      0 < sourceHessian g S x (metricGradient g S x) (metricGradient g S x)+
+        (g x (metricGradient g S x) (metricGradient g S x)+4)*sourceHessian g S x v v)
+    (m J K k0 : ℕ) (hm : 3*K+4*k0+6 < m+1) (hJ : K+k0+1 ≤ J) :
+    Nonempty (LocalCompactWaveData g w S D m J K k0) := by
+  obtain ⟨G,W,A,E,hG,hW,hA,hGs,hGp,hE,hDE,hEU,hWp,he⟩ :=
+    compact_source_representatives hD hU hDU hUn g hg hs hp w S hw hS hwp
+  have hn (x : Coord) (hx : x ∈ D) : metricGradient G A x ≠ 0 := by
+    rw [(he x (hDE hx)).2.2.2.1]
+    exact hp0 x hx
+  have hd (x : Coord) (hx : x ∈ D) : ∃ v : Coord,
+      G x (metricGradient G A x) v = 0 ∧ G x v v = 1 ∧
+      0 < sourceHessian G A x (metricGradient G A x) (metricGradient G A x)+
+        (G x (metricGradient G A x) (metricGradient G A x)+4)*sourceHessian G A x v v := by
+    obtain ⟨hge,_,_,hpEq,hHEq⟩ := he x (hDE hx)
+    rw [hge.self_of_nhds,hpEq,hHEq]
+    exact hdir x hx
+  obtain ⟨d,⟨b⟩,_⟩ := construct_compact_source_waves G hG hGs hGp W hW A hA
+    hD hE hDE hWp hn hd m J K k0 hm hJ
+  have hest := local_source_wave_transfer g G w W S A
+    (fun t : d.Parameter × Fin 3 ↦ coverSourceCenter d t.1)
+    ((continuous_coverSourceCenter d).comp continuous_fst) hE
+    (fun t ↦ hDE (d.center t.1).property)
+    (fun x hx ↦ ⟨(he x hx).1,(he x hx).2.1,(he x hx).2.2.1.self_of_nhds⟩)
+    (fun n t ↦ b.wave (n:ℝ) t) k0 K b.c b.Cw b.Cr b.R b.Cw_pos b.Cr_pos b.estimates
+  exact ⟨⟨G,W,A,E,hG,hW,hA,hGs,hGp,hE,hDE,
+    (fun x hx ↦ ⟨(he x hx).1,(he x hx).2.1,(he x hx).2.2.1⟩),hn,d,b,hest⟩⟩
+
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/LocalCovarianceGeometry.lean b/lean/OAI/Geometry/NodalSets/Waves/LocalCovarianceGeometry.lean
new file mode 100644
index 0000000000000000000000000000000000000000..682432405d65d8171b248a5ba6f7cebe5be31e4f
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/LocalCovarianceGeometry.lean
@@ -0,0 +1,53 @@
+import OAI.Geometry.NodalSets.Charts.MetricCoercivity
+import OAI.Geometry.NodalSets.Elliptic.LocalDirectionBounds
+import OAI.Geometry.NodalSets.Waves.LatticeLogJets
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Set Filter
+open scoped ContDiff Topology
+noncomputable section
+variable {g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ} {w S : Coord → ℝ}
+variable {D : Set Coord} {m J K k0 : ℕ}
+namespace LocalCompactWaveData
+variable (a : LocalCompactWaveData g w S D m J K k0)
+
+include a in
+theorem original_covariance_geometry : ∃ mu > 0, ∃ B > 0, ∀ x ∈ D,
+    ‖g x‖ ≤ B ∧ ‖fderiv ℝ S x‖ ≤ B ∧ ∀ v, mu*‖v‖^2 ≤ g x v v := by
+  let y := coverSourceCenter a.cover
+  have hy := continuous_coverSourceCenter a.cover
+  obtain ⟨mu,hmu,B,hB,hg⟩ := uniform_metric_coercivity (a.G ∘ y)
+    (a.smooth_G.continuous.comp hy) (fun t ↦ a.positive_G (y t))
+  have hc : Continuous (fun t ↦ fderiv ℝ a.A (y t)) :=
+    (a.smooth_A.continuous_fderiv (by simp)).comp hy
+  obtain ⟨M,hM,hm⟩ := (isCompact_univ.image hc).isBounded.exists_pos_norm_le
+  refine ⟨mu,hmu,B+M,by positivity,?_⟩
+  intro x hx
+  let t := selectFrame a.cover ⟨x,hx⟩
+  have he : y t = x := selectFrame_center a.cover ⟨x,hx⟩
+  have hG := hg t
+  change ‖a.G (y t)‖ ≤ B ∧ ∀ v, mu*‖v‖^2 ≤ a.G (y t) v v at hG
+  rw [he] at hG
+  have hA := hm _ ⟨t,mem_univ _,rfl⟩
+  change ‖fderiv ℝ a.A (y t)‖ ≤ M at hA
+  rw [he] at hA
+  obtain ⟨hge,_,hae⟩ := a.germ x (a.centers_E hx)
+  rw [hge.self_of_nhds] at hG
+  rw [hae.fderiv_eq] at hA
+  exact ⟨hG.1.trans (by linarith),hA.trans (by linarith),hG.2⟩
+
+theorem lattice_eta_bounds {U : Set Coord} (hUD : U ⊆ D) :
+    ∃ e > 0, ∃ E > 0, ∀ n (z : SourceGrid U n) j,
+      e ≤ a.latticeEta hUD n z j ∧ a.latticeEta hUD n z j ≤ E := by
+  obtain ⟨_,_,e,he,E,hE,hb⟩ := a.original_direction_bounds
+  refine ⟨e,he,E,hE,?_⟩
+  intro n z j
+  exact (hb (latticeFrame a.cover hUD n z)).2 j
+
+end LocalCompactWaveData
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/LocalSourceWaveTransfer.lean b/lean/OAI/Geometry/NodalSets/Waves/LocalSourceWaveTransfer.lean
new file mode 100644
index 0000000000000000000000000000000000000000..aafcaa3f8dbb9de8792963d0872db1dd44ba4a94
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/LocalSourceWaveTransfer.lean
@@ -0,0 +1,76 @@
+import OAI.Geometry.NodalSets.Coefficients.CoefficientLocality
+import OAI.Geometry.NodalSets.Waves.EuclideanSourceWaves
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Set Filter
+open scoped ContDiff Topology
+noncomputable section
+variable {T : Type*} [TopologicalSpace T] [CompactSpace T]
+
+theorem local_source_wave_transfer
+    (g G : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ) (w W S A : Coord → ℝ)
+    (y : T → Coord) (hy : Continuous y) {E : Set Coord} (hE : IsOpen E) (hyE : ∀ t, y t ∈ E)
+    (he : ∀ x ∈ E, (G =ᶠ[𝓝 x] g) ∧ (W =ᶠ[𝓝 x] w) ∧ A x = S x)
+    (u : ℕ → T → Coord → ℂ) (k0 K : ℕ) (c Cw Cr R : ℝ)
+    (hCw : 0 < Cw) (hCr : 0 < Cr)
+    (hest : ∀ᶠ n : ℕ in atTop, ∀ t,
+      ContDiff ℝ ∞ (u n t) ∧ HasCompactSupport (u n t) ∧
+      tsupport (u n t) ⊆ {z | sourceEuclideanNorm (z-y t) ≤ R*(n:ℝ)^(-1/3:ℝ)} ∧
+      ∀ z : Coord,
+        (∀ k : Fin (k0+1), ‖iteratedFDeriv ℝ k.val (u n t) z‖ ≤
+          Cw*(n:ℝ)^k.val*Real.exp ((n:ℝ)*A z-c*(n:ℝ)*(sourceEuclideanNorm (z-y t))^2)) ∧
+        DerivativeBound k0 (fun z ↦ sourceWeightedOperator G W (u n t) z +
+          ((4:ℂ)*(n:ℂ)^2+6*(n:ℂ))*u n t z) z
+            (Cr*(n:ℝ)^(-(K:ℝ))*Real.exp ((n:ℝ)*A z))) :
+    ∀ᶠ n : ℕ in atTop, ∀ t,
+      ContDiff ℝ ∞ (u n t) ∧ HasCompactSupport (u n t) ∧ tsupport (u n t) ⊆ E ∧
+      tsupport (u n t) ⊆ {z | sourceEuclideanNorm (z-y t) ≤ R*(n:ℝ)^(-1/3:ℝ)} ∧
+      ∀ z : Coord,
+        (∀ k : Fin (k0+1), ‖iteratedFDeriv ℝ k.val (u n t) z‖ ≤
+          Cw*(n:ℝ)^k.val*Real.exp ((n:ℝ)*S z-c*(n:ℝ)*(sourceEuclideanNorm (z-y t))^2)) ∧
+        DerivativeBound k0 (fun z ↦ sourceWeightedOperator g w (u n t) z +
+          ((4:ℂ)*(n:ℂ)^2+6*(n:ℂ))*u n t z) z
+            (Cr*(n:ℝ)^(-(K:ℝ))*Real.exp ((n:ℝ)*S z)) := by
+  obtain ⟨r,hr,hball⟩ := compact_family_common_radius isCompact_univ
+    (fun z : T × Coord ↦ y z.1+z.2) ((hy.comp continuous_fst).add continuous_snd)
+    hE (fun t _ ↦ by simpa using hyE t)
+  have hsmall : ∀ᶠ n : ℕ in atTop, R*(n:ℝ)^(-1/3:ℝ) < r := by
+    apply eventually_nat_frequency (P := fun N : ℝ ↦ R*N^(-1/3:ℝ) < r)
+    have ht : Tendsto (fun N : ℝ ↦ R*N^(-1/3:ℝ)) atTop (𝓝 0) := by
+      simpa only [neg_div,mul_zero] using
+        (tendsto_rpow_neg_atTop (by norm_num : (0:ℝ) < 1/3)).const_mul R
+    exact ht.eventually (gt_mem_nhds hr)
+  filter_upwards [hest,hsmall] with n hn hnr
+  intro t
+  obtain ⟨hsm,hcomp,hsup,hb⟩ := hn t
+  have hsE : tsupport (u n t) ⊆ E := by
+    intro z hz
+    have hd := (norm_le_sourceEuclideanNorm (z-y t)).trans (hsup hz)
+    have hh := hball t (mem_univ t) (z-y t) (hd.trans_lt hnr)
+    simpa only [add_sub_cancel] using hh
+  refine ⟨hsm,hcomp,hsE,hsup,?_⟩
+  intro z
+  by_cases hz : z ∈ E
+  · have hsame := he z hz
+    refine ⟨?_,?_⟩
+    · simpa only [hsame.2.2] using (hb z).1
+    · intro k hk
+      rw [← sourceResidual_coefficients_derivative_eq hsame.1 hsame.2.1]
+      simpa only [hsame.2.2] using (hb z).2 k hk
+  · have hnot : z ∉ tsupport (u n t) := fun hh ↦ hz (hsE hh)
+    have hzero := notMem_tsupport_iff_eventuallyEq.mp hnot
+    constructor
+    · intro k
+      rw [(hzero.iteratedFDeriv ℝ k.val).self_of_nhds]
+      simp only [iteratedFDeriv_zero,Pi.zero_apply,norm_zero]
+      positivity
+    · intro k hk
+      rw [source_residual_derivative_zero g w (u n t) _ z hnot k,norm_zero]
+      positivity
+
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/LocalWaveEnvelopeTransfer.lean b/lean/OAI/Geometry/NodalSets/Waves/LocalWaveEnvelopeTransfer.lean
new file mode 100644
index 0000000000000000000000000000000000000000..494bad308cb55ea885f41266e6978e88906cd68e
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/LocalWaveEnvelopeTransfer.lean
@@ -0,0 +1,59 @@
+import OAI.Geometry.NodalSets.Waves.LatticeSourceWaves
+import OAI.Geometry.NodalSets.Waves.LocalCompactWaves
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Set Filter
+open scoped ContDiff Topology
+noncomputable section
+variable {g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ} {w S : Coord → ℝ}
+variable {D : Set Coord} {m J K k0 : ℕ}
+
+def LocalCompactWaveData.withEnvelope (b : LocalCompactWaveData g w S D m J K k0)
+    (S' : Coord → ℝ) {O : Set Coord} (hO : IsOpen O) (hDO : D ⊆ O)
+    (he : ∀ x ∈ O, S' =ᶠ[𝓝 x] S) : LocalCompactWaveData g w S' D m J K k0 where
+  G := b.G
+  W := b.W
+  A := b.A
+  E := b.E ∩ O
+  smooth_G := b.smooth_G
+  smooth_W := b.smooth_W
+  smooth_A := b.smooth_A
+  symmetric_G := b.symmetric_G
+  positive_G := b.positive_G
+  open_E := b.open_E.inter hO
+  centers_E := fun _ hx ↦ ⟨b.centers_E hx,hDO hx⟩
+  germ := fun x hx ↦ ⟨(b.germ x hx.1).1,(b.germ x hx.1).2.1,
+    (b.germ x hx.1).2.2.trans (he x hx.2).symm⟩
+  nonzero_gradient := b.nonzero_gradient
+  cover := b.cover
+  beams := b.beams
+  estimates := local_source_wave_transfer g b.G w b.W S' b.A
+    (fun t : b.cover.Parameter × Fin 3 ↦ coverSourceCenter b.cover t.1)
+    ((continuous_coverSourceCenter b.cover).comp continuous_fst)
+    (b.open_E.inter hO)
+    (fun t ↦ ⟨b.centers_E (b.cover.center t.1).property,hDO (b.cover.center t.1).property⟩)
+    (fun x hx ↦ ⟨(b.germ x hx.1).1,(b.germ x hx.1).2.1,
+      (b.germ x hx.1).2.2.eq_of_nhds.trans (he x hx.2).eq_of_nhds.symm⟩)
+    (fun n t ↦ b.beams.wave (n:ℝ) t) k0 K b.beams.c b.beams.Cw b.beams.Cr b.beams.R
+    b.beams.Cw_pos b.beams.Cr_pos b.beams.estimates
+
+lemma LocalCompactWaveData.withEnvelope_wave (b : LocalCompactWaveData g w S D m J K k0)
+    (S' : Coord → ℝ) {O : Set Coord} (hO : IsOpen O) (hDO : D ⊆ O)
+    (he : ∀ x ∈ O, S' =ᶠ[𝓝 x] S) {U : Set Coord} (hUD : U ⊆ D)
+    (n : ℕ) (z : SourceGrid U n) (j : Fin 3) :
+    latticeWave (b.withEnvelope S' hO hDO he).cover
+      (b.withEnvelope S' hO hDO he).beams hUD n z j =
+      latticeWave b.cover b.beams hUD n z j := rfl
+
+lemma envelope_germ_open (S S' : Coord → ℝ) :
+    IsOpen {x | S' =ᶠ[𝓝 x] S} := by
+  change IsOpen {x | {z | S' z = S z} ∈ 𝓝 x}
+  simpa only [← mem_interior_iff_mem_nhds, ofPred_mem_eq] using
+    (isOpen_interior : IsOpen (interior {z | S' z = S z}))
+
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/LocalWaveHessian.lean b/lean/OAI/Geometry/NodalSets/Waves/LocalWaveHessian.lean
new file mode 100644
index 0000000000000000000000000000000000000000..18203156b4c7a5553ab75cae77b703b952af99da
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/LocalWaveHessian.lean
@@ -0,0 +1,36 @@
+import OAI.Geometry.NodalSets.Waves.LocalWaveJets
+import OAI.Geometry.NodalSets.Waves.WaveHessian
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Filter
+open scoped Topology
+noncomputable section
+variable {g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ} {w S : Coord → ℝ}
+variable {D : Set Coord} {m J K k0 : ℕ}
+namespace LocalCompactWaveData
+variable (a : LocalCompactWaveData g w S D m J K k0)
+
+theorem source_phase_hessian (t : a.cover.Parameter × Fin 3) (u v : Coord) :
+    sourceHessian g (fun z ↦ (a.beams.phase t z).re) (coverSourceCenter a.cover t.1) u v =
+      sourceHessian g S (coverSourceCenter a.cover t.1) u v -
+        sourceDirectionEta (g (coverSourceCenter a.cover t.1)) (sourceHessian g S (coverSourceCenter a.cover t.1))
+          (metricGradient g S (coverSourceCenter a.cover t.1)) (a.cover.triple.q t.1 t.2) *
+          g (coverSourceCenter a.cover t.1) u v := by
+  have h := a.beams.phase_hessian a.smooth_G a.symmetric_G a.positive_G a.smooth_A
+    (fun t ↦ a.nonzero_gradient _ (a.cover.center t).property) t u v
+  obtain ⟨hG,_,_⟩ := a.germ _ (a.centers_E (a.cover.center t.1).property)
+  have he := (sourceHessian_eventuallyEq hG
+    (Filter.EventuallyEq.refl (𝓝 (coverSourceCenter a.cover t.1)) (fun z ↦ (a.beams.phase t z).re))).self_of_nhds
+  change sourceHessian a.G (fun z ↦ (a.beams.phase t z).re) (coverSourceCenter a.cover t.1) =
+    sourceHessian g (fun z ↦ (a.beams.phase t z).re) (coverSourceCenter a.cover t.1) at he
+  rw [he,(a.center_identifications t.1).1,(a.center_identifications t.1).2.2.1,
+    (a.center_identifications t.1).2.2.2] at h
+  exact h
+
+end LocalCompactWaveData
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/LocalWaveJets.lean b/lean/OAI/Geometry/NodalSets/Waves/LocalWaveJets.lean
new file mode 100644
index 0000000000000000000000000000000000000000..2a66da82c35916bb998daed3de8c507e97b243db
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/LocalWaveJets.lean
@@ -0,0 +1,82 @@
+import OAI.Geometry.NodalSets.Waves.LatticeSourceWaves
+import OAI.Geometry.NodalSets.Waves.LocalCompactWaves
+import OAI.Geometry.NodalSets.Waves.WaveGerm
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Set Filter
+open scoped ContDiff Topology
+noncomputable section
+variable {g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ} {w S : Coord → ℝ}
+variable {D : Set Coord} {m J K k0 : ℕ}
+namespace LocalCompactWaveData
+variable (a : LocalCompactWaveData g w S D m J K k0)
+
+lemma center_identifications (t : a.cover.Parameter) :
+    a.G (coverSourceCenter a.cover t) = g (coverSourceCenter a.cover t) ∧
+    a.A (coverSourceCenter a.cover t) = S (coverSourceCenter a.cover t) ∧
+    metricGradient a.G a.A (coverSourceCenter a.cover t) = metricGradient g S (coverSourceCenter a.cover t) ∧
+    sourceHessian a.G a.A (coverSourceCenter a.cover t) = sourceHessian g S (coverSourceCenter a.cover t) := by
+  obtain ⟨hG,hW,hA⟩ := a.germ _ (a.centers_E (a.cover.center t).property)
+  exact ⟨hG.self_of_nhds,hA.self_of_nhds,(metricGradient_eventuallyEq hG hA).self_of_nhds,
+    (sourceHessian_eventuallyEq hG hA).self_of_nhds⟩
+
+lemma source_phase_center (t : a.cover.Parameter × Fin 3) :
+    a.beams.phase t (coverSourceCenter a.cover t.1) = (S (coverSourceCenter a.cover t.1):ℂ) := by
+  rw [a.beams.phase_center,(a.center_identifications t.1).2.1]
+
+lemma source_wave_center (N : ℝ) (hN : 0 < N) (t : a.cover.Parameter × Fin 3) :
+    a.beams.wave N t (coverSourceCenter a.cover t.1) =
+      Complex.exp ((N:ℂ)*(S (coverSourceCenter a.cover t.1):ℂ)) := by
+  rw [a.beams.wave_center N hN,(a.center_identifications t.1).2.1]
+
+lemma source_phase_first (t : a.cover.Parameter × Fin 3) (v : Coord) :
+    fderiv ℝ (a.beams.phase t) (coverSourceCenter a.cover t.1) v =
+      (g (coverSourceCenter a.cover t.1) (metricGradient g S (coverSourceCenter a.cover t.1)) v:ℂ)+
+        Complex.I*(g (coverSourceCenter a.cover t.1) (a.cover.triple.q t.1 t.2) v:ℂ) := by
+  have h := a.beams.phase_first a.positive_G
+    (fun t ↦ a.nonzero_gradient _ (a.cover.center t).property) t v
+  rw [(a.center_identifications t.1).1,(a.center_identifications t.1).2.2.1] at h
+  exact h
+
+lemma source_phase_second (t : a.cover.Parameter × Fin 3) (v : Coord) :
+    sourceHessian g (fun z ↦ (a.beams.phase t z).re) (coverSourceCenter a.cover t.1) v v =
+      sourceHessian g S (coverSourceCenter a.cover t.1) v v -
+        sourceDirectionEta (g (coverSourceCenter a.cover t.1)) (sourceHessian g S (coverSourceCenter a.cover t.1))
+          (metricGradient g S (coverSourceCenter a.cover t.1)) (a.cover.triple.q t.1 t.2) *
+          g (coverSourceCenter a.cover t.1) v v := by
+  have h := a.beams.phase_second a.smooth_G a.symmetric_G a.positive_G a.smooth_A
+    (fun t ↦ a.nonzero_gradient _ (a.cover.center t).property) t v
+  obtain ⟨hG,_,_⟩ := a.germ _ (a.centers_E (a.cover.center t.1).property)
+  have he := (sourceHessian_eventuallyEq hG
+    (Filter.EventuallyEq.refl (𝓝 (coverSourceCenter a.cover t.1)) (fun z ↦ (a.beams.phase t z).re))).self_of_nhds
+  change sourceHessian a.G (fun z ↦ (a.beams.phase t z).re) (coverSourceCenter a.cover t.1) =
+    sourceHessian g (fun z ↦ (a.beams.phase t z).re) (coverSourceCenter a.cover t.1) at he
+  rw [he,(a.center_identifications t.1).1,(a.center_identifications t.1).2.2.1,
+    (a.center_identifications t.1).2.2.2] at h
+  exact h
+
+theorem local_lattice_estimates {U : Set Coord} (hUD : U ⊆ D) :
+    ∀ᶠ n : ℕ in atTop, ∀ (z : SourceGrid U n) (j : Fin 3),
+      let V := latticeWave a.cover a.beams hUD n z j
+      ContDiff ℝ ∞ V ∧ HasCompactSupport V ∧ tsupport V ⊆ a.E ∧
+      tsupport V ⊆ {x | sourceEuclideanNorm (x-scaledLatticePoint n z) ≤ a.beams.R*(n:ℝ)^(-1/3:ℝ)} ∧
+      ∀ x : Coord,
+        (∀ k : Fin (k0+1), ‖iteratedFDeriv ℝ k.val V x‖ ≤
+          a.beams.Cw*(n:ℝ)^k.val*Real.exp ((n:ℝ)*S x-a.beams.c*(n:ℝ)*(sourceEuclideanNorm (x-scaledLatticePoint n z))^2)) ∧
+        DerivativeBound k0 (fun x ↦ sourceWeightedOperator g w V x+((4:ℂ)*(n:ℂ)^2+6*(n:ℂ))*V x) x
+          (a.beams.Cr*(n:ℝ)^(-(K:ℝ))*Real.exp ((n:ℝ)*S x)) := by
+  filter_upwards [a.estimates] with n hn
+  intro z j
+  simpa only [latticeWave,TripleSourceWaveData.wave,latticeFrame_center] using hn (latticeFrame a.cover hUD n z,j)
+
+lemma local_lattice_nonzero {U : Set Coord} (hUD : U ⊆ D) (n : ℕ) (hn : 0 < n)
+    (z : SourceGrid U n) (j : Fin 3) : latticeWave a.cover a.beams hUD n z j ≠ 0 :=
+  a.beams.wave_ne_zero n (by exact_mod_cast hn) (latticeFrame a.cover hUD n z,j)
+
+end LocalCompactWaveData
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/LocalWaveSupportCore.lean b/lean/OAI/Geometry/NodalSets/Waves/LocalWaveSupportCore.lean
new file mode 100644
index 0000000000000000000000000000000000000000..af7becec9075caef92cef0630c3c2a64c8e69e76
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/LocalWaveSupportCore.lean
@@ -0,0 +1,48 @@
+import Mathlib.Topology.MetricSpace.Thickening
+import OAI.Geometry.NodalSets.Waves.LocalCompactWaves
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Set Filter Metric
+open scoped Topology
+noncomputable section
+
+variable {g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ} {w S : Coord → ℝ}
+  {D : Set Coord} {m J K k0 : ℕ}
+
+theorem LocalCompactWaveData.uniform_support_core (a : LocalCompactWaveData g w S D m J K k0)
+    (hD : IsCompact D) {Ω : Set Coord} (hΩ : IsOpen Ω) (hDΩ : D ⊆ Ω) :
+    ∃ C : Set Coord, IsCompact C ∧ C ⊆ Ω ∧ D ⊆ interior C ∧
+      ∀ᶠ n : ℕ in atTop, ∀ t : a.cover.Parameter × Fin 3,
+        tsupport (a.beams.wave (n:ℝ) t) ⊆ C ∧
+        ∀ x ∉ C, ∀ k : ℕ,
+          iteratedFDeriv ℝ k (fun z ↦ sourceWeightedOperator g w (a.beams.wave (n:ℝ) t) z +
+            ((4:ℂ)*(n:ℂ)^2+6*(n:ℂ))*a.beams.wave (n:ℝ) t z) x = 0 := by
+  obtain ⟨C,hC,hDC,hCΩ⟩ := exists_compact_between hD hΩ hDΩ
+  obtain ⟨ε,hε,hth⟩ := hD.exists_cthickening_subset_open isOpen_interior hDC
+  have hsmall : ∀ᶠ n : ℕ in atTop, a.beams.R*(n:ℝ)^(-1/3:ℝ) < ε := by
+    apply eventually_nat_frequency (P := fun N : ℝ ↦ a.beams.R*N^(-1/3:ℝ) < ε)
+    have ht : Tendsto (fun N : ℝ ↦ a.beams.R*N^(-1/3:ℝ)) atTop (𝓝 0) := by
+      simpa only [neg_div,mul_zero] using
+        (tendsto_rpow_neg_atTop (by norm_num : (0:ℝ) < 1/3)).const_mul a.beams.R
+    exact ht.eventually (gt_mem_nhds hε)
+  refine ⟨C,hC,hCΩ,hDC,?_⟩
+  filter_upwards [a.estimates,hsmall] with n hn hnε
+  intro t
+  have hsup : tsupport (a.beams.wave (n:ℝ) t) ⊆ C := by
+    intro x hx
+    have hb := (norm_le_sourceEuclideanNorm (x-coverSourceCenter a.cover t.1)).trans
+      ((hn t).2.2.2.1 hx)
+    have hc : coverSourceCenter a.cover t.1 ∈ D := (a.cover.center t.1).property
+    apply interior_subset (hth (mem_cthickening_of_dist_le x (coverSourceCenter a.cover t.1) ε D hc ?_))
+    rw [dist_eq_norm]
+    exact (hb.trans_lt hnε).le
+  refine ⟨hsup,?_⟩
+  intro x hx k
+  exact source_residual_derivative_zero g w _ _ x (fun hz ↦ hx (hsup hz)) k
+
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/LocalWavesInDomain.lean b/lean/OAI/Geometry/NodalSets/Waves/LocalWavesInDomain.lean
new file mode 100644
index 0000000000000000000000000000000000000000..445a86f93a8dd50338fe7c2d2b75c7c381bb1728
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/LocalWavesInDomain.lean
@@ -0,0 +1,49 @@
+import OAI.Geometry.NodalSets.Waves.LocalCompactWaves
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Set Filter
+open scoped ContDiff Topology
+noncomputable section
+attribute [local instance] clmTopology clmAdd clmModule
+
+theorem construct_local_compact_waves_in_domain {D U : Set Coord} (hD : IsCompact D)
+    (hU : IsOpen U) (hDU : D ⊆ U) (hUn : U.Nonempty)
+    (g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ) (hg : ContDiffOn ℝ ∞ g U)
+    (hs : ∀ x ∈ U, ∀ u v, g x u v = g x v u)
+    (hp : ∀ x ∈ U, ∀ v, v ≠ 0 → 0 < g x v v)
+    (w S : Coord → ℝ) (hw : ContDiffOn ℝ ∞ w U) (hS : ContDiffOn ℝ ∞ S U)
+    (hwp : ∀ x ∈ U, 0 < w x) (hp0 : ∀ x ∈ D, metricGradient g S x ≠ 0)
+    (hdir : ∀ x ∈ D, ∃ v : Coord, g x (metricGradient g S x) v = 0 ∧ g x v v = 1 ∧
+      0 < sourceHessian g S x (metricGradient g S x) (metricGradient g S x)+
+        (g x (metricGradient g S x) (metricGradient g S x)+4)*sourceHessian g S x v v)
+    (m J K k0 : ℕ) (hm : 3*K+4*k0+6 < m+1) (hJ : K+k0+1 ≤ J) :
+    ∃ a : LocalCompactWaveData g w S D m J K k0, a.E ⊆ U := by
+  obtain ⟨G,W,A,E,hG,hW,hA,hGs,hGp,hE,hDE,hEU,hWp,he⟩ :=
+    compact_source_representatives hD hU hDU hUn g hg hs hp w S hw hS hwp
+  have hn (x : Coord) (hx : x ∈ D) : metricGradient G A x ≠ 0 := by
+    rw [(he x (hDE hx)).2.2.2.1]
+    exact hp0 x hx
+  have hd (x : Coord) (hx : x ∈ D) : ∃ v : Coord,
+      G x (metricGradient G A x) v = 0 ∧ G x v v = 1 ∧
+      0 < sourceHessian G A x (metricGradient G A x) (metricGradient G A x)+
+        (G x (metricGradient G A x) (metricGradient G A x)+4)*sourceHessian G A x v v := by
+    obtain ⟨hge,_,_,hpEq,hHEq⟩ := he x (hDE hx)
+    rw [hge.self_of_nhds,hpEq,hHEq]
+    exact hdir x hx
+  obtain ⟨d,⟨b⟩,_⟩ := construct_compact_source_waves G hG hGs hGp W hW A hA
+    hD hE hDE hWp hn hd m J K k0 hm hJ
+  have hest := local_source_wave_transfer g G w W S A
+    (fun t : d.Parameter × Fin 3 ↦ coverSourceCenter d t.1)
+    ((continuous_coverSourceCenter d).comp continuous_fst) hE
+    (fun t ↦ hDE (d.center t.1).property)
+    (fun x hx ↦ ⟨(he x hx).1,(he x hx).2.1,(he x hx).2.2.1.self_of_nhds⟩)
+    (fun n t ↦ b.wave (n:ℝ) t) k0 K b.c b.Cw b.Cr b.R b.Cw_pos b.Cr_pos b.estimates
+  refine ⟨⟨G,W,A,E,hG,hW,hA,hGs,hGp,hE,hDE,
+    (fun x hx ↦ ⟨(he x hx).1,(he x hx).2.1,(he x hx).2.2.1⟩),hn,d,b,hest⟩,hEU⟩
+
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/NetProbability.lean b/lean/OAI/Geometry/NodalSets/Waves/NetProbability.lean
new file mode 100644
index 0000000000000000000000000000000000000000..ec8a9dd415253b91f792164810edc3611b6cead3
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/NetProbability.lean
@@ -0,0 +1,63 @@
+import Mathlib.Analysis.Calculus.MeanValue
+import Mathlib.MeasureTheory.Measure.Basic
+
+namespace OAI
+
+namespace Yau.Probability
+open Set MeasureTheory
+open scoped ENNReal
+noncomputable section
+variable {Ω X F : Type*} [MeasurableSpace Ω] [NormedAddCommGroup F]
+
+lemma finite_small_ball_union (μ : Measure Ω) (t : Finset X) (Z : Ω → X → F)
+    (r : ℝ) (p : ℝ≥0∞) (hp : ∀ x ∈ t, μ {ω | ‖Z ω x‖ ≤ r} ≤ p) :
+    μ {ω | ∃ x ∈ t, ‖Z ω x‖ ≤ r} ≤ t.card*p := by
+  have he : {ω | ∃ x ∈ t, ‖Z ω x‖ ≤ r} = ⋃ x ∈ t, {ω | ‖Z ω x‖ ≤ r} := by
+    ext ω
+    simp
+  rw [he]
+  exact (measure_biUnion_finset_le t _).trans (by
+    simpa using Finset.sum_le_sum hp)
+
+lemma net_failure_le (μ : Measure Ω) (t : Finset X) (E : Set X)
+    (Z : Ω → X → F) (good : Set Ω) (r : ℝ) (p : ℝ≥0∞)
+    (hp : ∀ x ∈ t, μ {ω | ‖Z ω x‖ ≤ 2*r} ≤ p)
+    (happrox : ∀ ω ∈ good, ∀ x ∈ E, ∃ y ∈ t, ‖Z ω y-Z ω x‖ ≤ r) :
+    μ {ω | ∃ x ∈ E, ‖Z ω x‖ < r} ≤ μ goodᶜ+t.card*p := by
+  have hsub : {ω | ∃ x ∈ E, ‖Z ω x‖ < r} ⊆
+      goodᶜ ∪ {ω | ∃ y ∈ t, ‖Z ω y‖ ≤ 2*r} := by
+    intro ω hω
+    by_cases hg : ω ∈ good
+    · obtain ⟨x,hx,hsmall⟩ := hω
+      obtain ⟨y,hy,hd⟩ := happrox ω hg x hx
+      right
+      refine ⟨y,hy,?_⟩
+      have htri : ‖Z ω y‖ ≤ ‖Z ω y-Z ω x‖+‖Z ω x‖ := norm_le_norm_sub_add _ _
+      linarith
+    · exact Or.inl hg
+  exact (measure_mono hsub).trans ((measure_union_le _ _).trans
+    (add_le_add le_rfl (finite_small_ball_union μ t Z (2*r) p hp)))
+
+variable [NormedAddCommGroup X] [NormedSpace ℝ X] [NormedSpace ℝ F]
+
+lemma neighborhood_net_approximation (t : Finset X) (E : Set X) (Z : X → F)
+    {δ R L r : ℝ} (hδ : 0 ≤ δ) (hδR : δ ≤ R) (hL : 0 ≤ L) (hLδ : L*δ ≤ r)
+    (hnet : ∀ x ∈ E, ∃ y ∈ t, ‖x-y‖ ≤ δ)
+    (hdiff : ∀ x ∈ E, ∀ z ∈ Metric.closedBall x R, DifferentiableAt ℝ Z z)
+    (hderiv : ∀ x ∈ E, ∀ z ∈ Metric.closedBall x R, ‖fderiv ℝ Z z‖ ≤ L) :
+    ∀ x ∈ E, ∃ y ∈ t, ‖Z y-Z x‖ ≤ r := by
+  intro x hx
+  obtain ⟨y,hy,hd⟩ := hnet x hx
+  have hxball : x ∈ Metric.closedBall x R := Metric.mem_closedBall_self (hδ.trans hδR)
+  have hyball : y ∈ Metric.closedBall x R := by
+    rw [Metric.mem_closedBall,dist_eq_norm,norm_sub_rev]
+    exact hd.trans hδR
+  have h := (convex_closedBall x R).norm_image_sub_le_of_norm_fderiv_le
+    (hdiff x hx) (hderiv x hx) hxball hyball
+  rw [norm_sub_rev y x] at h
+  exact ⟨y,hy,h.trans ((mul_le_mul_of_nonneg_left hd hL).trans hLδ)⟩
+
+end
+end Yau.Probability
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/NormalFrameWaveJet.lean b/lean/OAI/Geometry/NodalSets/Waves/NormalFrameWaveJet.lean
new file mode 100644
index 0000000000000000000000000000000000000000..6af03fdc5399eed6cf6e6e84a02c1e4a60f85ac2
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/NormalFrameWaveJet.lean
@@ -0,0 +1,60 @@
+import OAI.Geometry.NodalSets.Charts.NormalFrameContinuous
+import OAI.Geometry.NodalSets.Waves.UniformWaveJet
+
+namespace OAI
+
+namespace Yau.Jets
+open MvPolynomial
+open scoped ContDiff
+noncomputable section
+variable {T : Type*} [TopologicalSpace T]
+
+theorem normal_frame_smooth_wave_jets_continuous
+    (a bb S : T → ℝ) (H : T → Fin 4 → Fin 4 → ℝ)
+    (ha : Continuous a) (hbb : Continuous bb) (hS : Continuous S)
+    (hHc : ∀ i j, Continuous (fun t ↦ H t i j))
+    (ha0 : ∀ t, a t ≠ 0) (hbb0 : ∀ t, bb t ≠ 0)
+    (hlen : ∀ t, bb t ^ 2 = a t ^ 2 + 4) (hH : ∀ t i j, H t i j = H t j i)
+    (g : T → Fin 4 → Fin 4 → Coord → ℂ) (b : T → Fin 4 → Coord → ℂ)
+    (G : T → Fin 4 → Fin 4 → CPoly) (B : T → Fin 4 → CPoly)
+    (hG : ∀ i j, ContinuousPolyFamily (fun t ↦ G t i j))
+    (hB : ∀ i, ContinuousPolyFamily (fun t ↦ B t i))
+    (hg : ∀ t i j, ContDiff ℝ ∞ (g t i j)) (hb : ∀ t i, ContDiff ℝ ∞ (b t i))
+    (hmetric : ∀ t i j, homogeneousComponent 0 (G t i j) = if i = j then 1 else 0)
+    (hmetricFirst : ∀ t i j, homogeneousComponent 1 (G t i j) = 0) (m J : ℕ)
+    (hgj : ∀ t i j, FlatAt m (fun x ↦ g t i j x - reval (G t i j) x) 0)
+    (hbj : ∀ t i, FlatAt m (fun x ↦ b t i x - reval (B t i) x) 0) :
+    ∃ (phi : T → CPoly) (A : ℕ → T → CPoly),
+      ContinuousPolyFamily phi ∧ (∀ j, ContinuousPolyFamily (A j)) ∧
+      (∀ t k, k ≤ 2 → homogeneousComponent k (phi t) = initialPhaseJet (S t) (normalFrameVector (a t) (bb t))
+        (normalComplexHessian (a t) (bb t) (H t)) k) ∧
+      (∀ t d, m + 2 * J + 3 < d → homogeneousComponent d (phi t) = 0) ∧
+      (∀ j t, homogeneousComponent 0 (A j t) = if j = 0 then 1 else 0) ∧
+      (∀ j t k, m + 1 + 2 * (J - j) < k → homogeneousComponent k (A j t) = 0) ∧
+      (∀ t, FlatAt m (smoothEikonal (g t) (reval (phi t))) 0) ∧
+      (∀ t, FlatAt m (smoothTransport (smoothBeamVector (g t) (reval (phi t)))
+        (smoothBeamScalar (g t) (b t) (reval (phi t))) (fun _ ↦ 0) (reval (A 0 t))) 0) ∧
+      ∀ j, j < J → ∀ t, FlatAt m
+        (smoothTransport (smoothBeamVector (g t) (reval (phi t)))
+          (smoothBeamScalar (g t) (b t) (reval (phi t)))
+          (fun x ↦ -smoothSecondOrder (g t) (b t) (reval (A j t)) x)
+          (reval (A (j + 1) t))) 0 := by
+  apply smooth_wave_jets_continuous (fun t ↦ normalFrameVector (a t) (bb t))
+    (normalFrameVector_continuous a bb ha hbb) 0
+    (fun t ↦ by simpa [normalFrameVector] using (Complex.ofReal_ne_zero.mpr (ha0 t)))
+    g b G B hG hB hg hb hmetric
+    (fun t ↦ initialPhaseJet (S t) (normalFrameVector (a t) (bb t))
+      (normalComplexHessian (a t) (bb t) (H t)))
+    (fun t k ↦ initialPhaseJet_homogeneous _ _ _ k)
+    (normal_frame_initial_jets_continuous a bb S H ha hbb hS hHc ha0 hbb0)
+    (fun t i ↦ initialPhaseJet_first _ _ _ i)
+    (fun t ↦ normalFrameVector_null (hlen t))
+    (fun t ↦ initialPhaseJet_second _ _ _
+      (normalComplexHessian_symmetric (ha0 t) (hbb0 t) (H t) (hH t))
+      (normalComplexHessian_contract (ha0 t) (hbb0 t) (H t))
+      (fun r i j ↦ homogeneousComponent r (G t i j)) (hmetricFirst t)) m J hgj hbj
+
+end
+end Yau.Jets
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/NormalizedWaveRatio.lean b/lean/OAI/Geometry/NodalSets/Waves/NormalizedWaveRatio.lean
new file mode 100644
index 0000000000000000000000000000000000000000..d1a296aa1304117247847c422200f4ccd598b281
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/NormalizedWaveRatio.lean
@@ -0,0 +1,63 @@
+import OAI.Geometry.NodalSets.Elliptic.NormalizedRescaling
+import OAI.Geometry.NodalSets.Elliptic.RescaledPhaseDefect
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Set
+open scoped ContDiff
+noncomputable section
+
+def planeWave (k : Coord →L[ℝ] ℝ) (s : ℝ) (v : Coord) : ℂ :=
+  Complex.exp (Complex.I*(k v/s:ℝ))
+
+lemma planeWave_norm (k : Coord →L[ℝ] ℝ) (s : ℝ) (v : Coord) :
+    ‖planeWave k s v‖ = 1 := by
+  simp [planeWave,Complex.norm_exp]
+
+lemma normalizedWave_factorization (V amp phi : Coord → ℂ) (S : Coord → ℝ)
+    (k : Coord →L[ℝ] ℝ) (N s sigma : ℝ) (x v : Coord)
+    (hax : amp x ≠ 0)
+    (hx : V x = amp x*Complex.exp ((N:ℂ)*phi x))
+    (hz : V (x+(N*s)⁻¹ • v) = amp (x+(N*s)⁻¹ • v)*Complex.exp ((N:ℂ)*phi (x+(N*s)⁻¹ • v))) :
+    normalizedRescaling V S N s sigma x v =
+      (V x/(sigma:ℂ))*planeWave k s v *
+        ((amp (x+(N*s)⁻¹ • v)/amp x)*
+          Complex.exp (rescaledPhaseDefect phi (frozenPhaseCovector (fderiv ℝ S x) k) N s x v)) := by
+  have hexp : Complex.exp ((N:ℂ)*phi (x+(N*s)⁻¹ • v)) =
+      Complex.exp ((N:ℂ)*phi x) * Complex.exp (fderiv ℝ S x v/s:ℝ) * planeWave k s v *
+        Complex.exp (rescaledPhaseDefect phi (frozenPhaseCovector (fderiv ℝ S x) k) N s x v) := by
+    simp only [planeWave,← Complex.exp_add,rescaledPhaseDefect,frozenPhaseCovector_apply,
+      Complex.real_smul,Complex.ofReal_div,Complex.ofReal_inv]
+    congr 1
+    ring
+  rw [normalizedRescaling_eq_div,hz,hx,hexp]
+  rw [Complex.ofReal_mul,Complex.ofReal_exp]
+  field_simp
+
+lemma ratio_exp_error {a z : ℂ} {e : ℝ} (he : 0 ≤ e) (he1 : e ≤ 1)
+    (ha : ‖a-1‖ ≤ e) (hz : ‖z‖ ≤ e) : ‖a*Complex.exp z-1‖ ≤ 5*e := by
+  have hzexp := Complex.norm_exp_sub_one_le (hz.trans he1)
+  have hnexp : ‖Complex.exp z‖ ≤ 3 := by
+    have h := norm_add_le (Complex.exp z-1) (1:ℂ)
+    simp only [sub_add_cancel,norm_one] at h
+    linarith
+  have heq : a*Complex.exp z-1 = (a-1)*Complex.exp z+(Complex.exp z-1) := by ring
+  rw [heq]
+  apply (norm_add_le _ _).trans
+  rw [norm_mul]
+  have hprod := mul_le_mul ha hnexp (norm_nonneg _) (by linarith : 0 ≤ e)
+  linarith
+
+lemma amplitude_ratio_near_one {a b : ℂ} {e : ℝ} (ha : (1/2:ℝ) ≤ ‖a‖)
+    (hb : ‖b-a‖ ≤ e) : ‖b/a-1‖ ≤ 2*e := by
+  have hane : a ≠ 0 := by intro h; simp [h] at ha; norm_num at ha
+  rw [show b/a-1 = (b-a)/a by field_simp,norm_div]
+  apply (div_le_iff₀ (by linarith : 0 < ‖a‖)).mpr
+  have he : 0 ≤ e := (norm_nonneg _).trans hb
+  nlinarith
+
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/PlacedEnvelopeWaves.lean b/lean/OAI/Geometry/NodalSets/Waves/PlacedEnvelopeWaves.lean
new file mode 100644
index 0000000000000000000000000000000000000000..3db5885af235f6598ede5a92e8382ab9cf94d094
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/PlacedEnvelopeWaves.lean
@@ -0,0 +1,50 @@
+import OAI.Geometry.NodalSets.Elliptic.PlacedEnvelopeDomains
+import OAI.Geometry.NodalSets.Waves.LocalCompactWaves
+
+namespace OAI
+
+namespace Yau.Target
+open Yau.Geometry Yau.Jets Set Metric
+open scoped ContDiff
+noncomputable section
+
+lemma PlacedEnvelopeData.closure_U_branch {g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ}
+    {r a : ℝ} {K : Set Coord} {T : ℝ} (d : PlacedEnvelopeData g r a K T) :
+    closure d.U ⊆ seedCoordBranch :=
+  d.closure_U_Ω.trans (subset_closure.trans (d.closure_Ω_patch.trans
+    (subset_closure.trans (d.patch_V.trans d.V_branch))))
+
+lemma PlacedEnvelopeData.closure_U_V {g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ}
+    {r a : ℝ} {K : Set Coord} {T : ℝ} (d : PlacedEnvelopeData g r a K T) :
+    closure d.U ⊆ d.V :=
+  d.closure_U_Ω.trans (subset_closure.trans (d.closure_Ω_patch.trans
+    (subset_closure.trans d.patch_V)))
+
+theorem PlacedEnvelopeData.waves {g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ}
+    {r a : ℝ} {K : Set Coord} {T : ℝ} (d : PlacedEnvelopeData g r a K T) (ha : 0 ≤ a)
+    (hg : ContDiffOn ℝ ∞ g seedCoordBranch)
+    (hs : ∀ x ∈ seedCoordBranch, ∀ u v, g x u v = g x v u)
+    (hp : ∀ x ∈ seedCoordBranch, ∀ v, v ≠ 0 → 0 < g x v v)
+    (w : Coord → ℝ) (hw : ContDiffOn ℝ ∞ w seedCoordBranch)
+    (hwp : ∀ x ∈ seedCoordBranch, 0 < w x) (k0 K' : ℕ) :
+    Nonempty (LocalCompactWaveData g w d.S (closure d.U)
+      (3*K'+4*k0+6) (K'+k0+1) K' k0) := by
+  have h0 : (0 : Coord) ∈ seedCoordCube a := by
+    constructor <;> intro i <;> simp only [Pi.zero_apply] <;> linarith
+  have h0U : (0 : Coord) ∈ d.U := d.C_U (d.inner_C (Or.inl h0))
+  have h0V : (0 : Coord) ∈ d.V := d.closure_U_V (subset_closure h0U)
+  apply construct_local_compact_waves d.compact_closure_U d.open_V d.closure_U_V ⟨0,h0V⟩
+    g (hg.mono d.V_branch) (fun x hx ↦ hs x (d.V_branch hx))
+    (fun x hx ↦ hp x (d.V_branch hx)) w d.S (hw.mono d.V_branch)
+    (d.smooth.mono d.V_branch) (fun x hx ↦ hwp x (d.V_branch hx))
+    (fun x hx ↦ (d.admissible x (d.closure_U_V hx)).1)
+  · intro x hx
+    obtain ⟨hn,v,hv,hpv,hstr⟩ := d.admissible x (d.closure_U_V hx)
+    exact ⟨v,hpv,hv,hstr⟩
+  · omega
+  · omega
+
+end
+end Yau.Target
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/PlacedFirstJetProbability.lean b/lean/OAI/Geometry/NodalSets/Waves/PlacedFirstJetProbability.lean
new file mode 100644
index 0000000000000000000000000000000000000000..26d64852b09d6598e99fb7e64371c09ee64c4b10
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/PlacedFirstJetProbability.lean
@@ -0,0 +1,86 @@
+import OAI.Geometry.NodalSets.Elliptic.PlacedSmoothRepresentatives
+import OAI.Geometry.NodalSets.Waves.LatticeGoodEvent
+
+namespace OAI
+
+namespace Yau.Target
+open Yau.Geometry Yau.Jets Yau.Probability Set Filter MeasureTheory
+open scoped ContDiff Topology
+noncomputable section
+variable {g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ} {w : Coord → ℝ}
+variable {r a : ℝ} {Kset : Set Coord} {T : ℝ} {m J K k0 : ℕ}
+
+def PlacedEnvelopeData.representativeWaves (d : PlacedEnvelopeData g r a Kset T)
+    (b : LocalCompactWaveData g w d.S (closure d.U) m J K k0)
+    (S : Coord → ℝ) (he : ∀ x ∈ closure d.Ω, S =ᶠ[𝓝 x] d.S) :
+    LocalCompactWaveData g w S (closure d.U) m J K k0 :=
+  b.withEnvelope S (envelope_germ_open d.S S)
+    (fun x hx ↦ he x (subset_closure (d.closure_U_Ω hx))) (fun _ hx ↦ hx)
+
+lemma PlacedEnvelopeData.literal_good_event_eq (d : PlacedEnvelopeData g r a Kset T)
+    (b : LocalCompactWaveData g w d.S (closure d.U) m J K k0)
+    (S S0 T0 : Coord → ℝ)
+    (he : ∀ x ∈ closure d.Ω, S =ᶠ[𝓝 x] d.S ∧ S0 =ᶠ[𝓝 x] seedCoordReal ∧
+      T0 =ᶠ[𝓝 x] seedCoordImag)
+    (n : ℕ) (hfin : Fintype (SourceGrid d.U n)) :
+    (d.representativeWaves b S (fun x hx ↦ (he x hx).1)).latticeGoodEvent
+      subset_closure n hfin S0 T0 (closure d.Ω) =
+    letI := hfin
+    coefficientJetGoodEvent
+      (fun i : SourceGrid d.U n × Fin 3 ↦ latticeWave b.cover b.beams subset_closure n i.1 i.2)
+      (seedCoordinateField n) n (closure d.Ω)
+      (fun x ↦ ((n:ℝ)^65)⁻¹*max (Real.exp ((n:ℝ)*d.S x))
+        (Real.exp ((n:ℝ)*seedCoordReal x))) := by
+  let := hfin
+  ext coeff
+  simp only [LocalCompactWaveData.latticeGoodEvent,coefficientJetGoodEvent,
+    mem_inter_iff,firstJetGoodEvent,mem_ofPred_eq]
+  apply and_congr_right
+  intro _
+  apply forall_congr'
+  intro x
+  apply forall_congr'
+  intro hx
+  have hg := seedCoordinateField_log_germ n (d.closure_Ω_branch hx) S0 T0
+    (he x hx).2.1 (he x hx).2.2
+  have hj := seeded_sourceFirstJetSize_germ (v := gaussianWaveField
+    (fun i : SourceGrid d.U n × Fin 3 ↦ latticeWave b.cover b.beams subset_closure n i.1 i.2) coeff) hg (n:ℝ)
+  change (_ ≤ sourceFirstJetSize _ _ _) ↔ (_ ≤ sourceFirstJetSize _ _ _)
+  rw [hj,(he x hx).1.eq_of_nhds,(he x hx).2.1.eq_of_nhds]
+  rfl
+
+theorem PlacedEnvelopeData.literal_good_event_probability
+    (d : PlacedEnvelopeData g r a Kset T)
+    (b : LocalCompactWaveData g w d.S (closure d.U) m J K k0) (hk0 : 2 ≤ k0)
+    (hq : ∀ x ∈ closure d.Ω, fderiv ℝ seedCoordImag x ≠ 0) :
+    ∀ ε : ℝ, 0 < ε → ∀ᶠ n : ℕ in atTop, ∃ hfin : Fintype (SourceGrid d.U n),
+      letI := hfin
+      let event := coefficientJetGoodEvent
+        (fun i : SourceGrid d.U n × Fin 3 ↦ latticeWave b.cover b.beams subset_closure n i.1 i.2)
+        (seedCoordinateField n) n (closure d.Ω)
+        (fun x ↦ ((n:ℝ)^65)⁻¹*max (Real.exp ((n:ℝ)*d.S x))
+          (Real.exp ((n:ℝ)*seedCoordReal x)))
+      MeasurableSet event ∧ gaussianPairs eventᶜ ≤ ENNReal.ofReal ε := by
+  obtain ⟨S,S0,T0,hS,hS0,hT0,he⟩ := d.smooth_representatives
+  let b' := d.representativeWaves b S (fun x hx ↦ (he x hx).1)
+  have hq' : ∀ x ∈ closure d.Ω, fderiv ℝ T0 x ≠ 0 := by
+    intro x hx
+    rw [(he x hx).2.2.fderiv_eq]
+    exact hq x hx
+  have hgap : ∀ x ∈ closure d.Ω, x ∉ d.U → S x-S0 x ≤ -d.gap := by
+    intro x hx hxu
+    rw [(he x hx).1.eq_of_nhds,(he x hx).2.1.eq_of_nhds]
+    exact d.outside_gap x hxu
+  intro ε hε
+  filter_upwards [b'.lattice_good_event_probability subset_closure d.open_U
+    (d.compact_closure_U.isBounded.subset subset_closure) hS S0 T0 hS0 hT0
+    d.compact_closure_Ω hk0 hq' d.gap_pos hgap ε hε] with n hn
+  obtain ⟨hfin,hn⟩ := hn
+  refine ⟨hfin,?_⟩
+  rw [d.literal_good_event_eq b S S0 T0 he n hfin] at hn
+  exact hn
+
+end
+end Yau.Target
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/PlaneWaveCovariance.lean b/lean/OAI/Geometry/NodalSets/Waves/PlaneWaveCovariance.lean
new file mode 100644
index 0000000000000000000000000000000000000000..c2425599b9d40e1fd27e05812c33f2a0103aae78
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/PlaneWaveCovariance.lean
@@ -0,0 +1,47 @@
+import OAI.Geometry.NodalSets.Waves.GaussianPairCovariance
+import OAI.Geometry.NodalSets.Waves.NormalizedWaveRatio
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Yau.Probability MeasureTheory ProbabilityTheory
+noncomputable section
+variable {ι : Type*} [Fintype ι]
+
+lemma complex_rotated_pairing (z : ℂ) (t : ℝ) :
+    z.re*(z*Complex.exp (Complex.I*(t:ℂ))).re +
+      z.im*(z*Complex.exp (Complex.I*(t:ℂ))).im = ‖z‖^2*Real.cos t := by
+  simp [Complex.exp_re,Complex.exp_im,Complex.mul_re,Complex.mul_im,
+    Complex.sq_norm,Complex.normSq_apply]
+  ring
+
+lemma gaussianWaveField_eq_pair (V : ι → Coord → ℂ) (v : Coord) :
+    (fun a ↦ gaussianWaveField V a v) = pairLinearSum (fun i ↦ V i v) := by
+  funext a
+  rw [pairLinearSum_eq_complex]
+  simp only [gaussianWaveField,Complex.re_sum]
+
+lemma planeWave_zero (k : Coord →L[ℝ] ℝ) (s : ℝ) : planeWave k s 0 = 1 := by
+  simp [planeWave]
+
+lemma planeWave_two_point_covariance (alpha : ι → ℂ) (k : ι → Coord →L[ℝ] ℝ)
+    (s : ℝ) (v : Coord) :
+    cov[(fun a ↦ gaussianWaveField (fun i z ↦ alpha i*planeWave (k i) s z) a 0),
+      (fun a ↦ gaussianWaveField (fun i z ↦ alpha i*planeWave (k i) s z) a v);gaussianPairs] =
+      ∑ i, ‖alpha i‖^2*Real.cos (k i v/s) := by
+  rw [gaussianWaveField_eq_pair,gaussianWaveField_eq_pair,covariance_pairLinearSum]
+  apply Finset.sum_congr rfl
+  intro i _
+  simpa [planeWave] using complex_rotated_pairing (alpha i) (k i v/s)
+
+lemma planeWave_value_variance (alpha : ι → ℂ) (k : ι → Coord →L[ℝ] ℝ)
+    (s : ℝ) (v : Coord) :
+    Var[fun a ↦ gaussianWaveField (fun i z ↦ alpha i*planeWave (k i) s z) a v;gaussianPairs] =
+      ∑ i, ‖alpha i‖^2 := by
+  rw [gaussianWaveField_eq_pair,variance_pairLinearSum]
+  simp only [norm_mul,planeWave_norm,mul_one]
+
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/PlaneWaveJets.lean b/lean/OAI/Geometry/NodalSets/Waves/PlaneWaveJets.lean
new file mode 100644
index 0000000000000000000000000000000000000000..61edc796db50b184e6700285a2eb69bb6555e3e5
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/PlaneWaveJets.lean
@@ -0,0 +1,65 @@
+import OAI.Geometry.NodalSets.Elliptic.RescaledHigherJets
+import OAI.Geometry.NodalSets.Waves.ExponentialWaveBounds
+import OAI.Geometry.NodalSets.Waves.NormalizedWaveRatio
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Set
+open scoped ContDiff
+noncomputable section
+
+lemma planeWave_smooth (k : Coord →L[ℝ] ℝ) (s : ℝ) : ContDiff ℝ ∞ (planeWave k s) := by
+  unfold planeWave
+  have hh : ContDiff ℝ ∞ (fun v ↦ ((k v/s:ℝ):ℂ)) :=
+    Complex.ofRealCLM.contDiff.comp (k.contDiff.div_const s)
+  exact Complex.contDiff_exp.comp (contDiff_const.mul hh)
+
+lemma planeWave_derivative_bound (k : Coord →L[ℝ] ℝ) {s B : ℝ}
+    (hs : 1 ≤ s) (hB : 1 ≤ B) (hk : ‖k‖ ≤ B) (d : ℕ) (v : Coord) :
+    ‖iteratedFDeriv ℝ d (planeWave k s) v‖ ≤ d.factorial*B^d := by
+  let L : Coord →L[ℝ] ℂ := s⁻¹ • (Complex.I • Complex.ofRealCLM.comp k)
+  have he : planeWave k s = Complex.exp ∘ L := by
+    funext z
+    simp [planeWave,L,Complex.real_smul,div_eq_mul_inv,mul_comm,mul_left_comm]
+  have hL : ‖L‖ ≤ B := by
+    apply ContinuousLinearMap.opNorm_le_bound _ (by linarith)
+    intro z
+    have hs0 : 0 < s := lt_of_lt_of_le zero_lt_one hs
+    have hi : s⁻¹ ≤ 1 := (inv_le_one₀ hs0).mpr hs
+    calc
+      ‖L z‖ = s⁻¹*‖k z‖ := by simp [L,abs_of_pos hs0]
+      _ ≤ 1*(B*‖z‖) := mul_le_mul hi ((k.le_opNorm z).trans (by gcongr)) (norm_nonneg _) zero_le_one
+      _ = _ := by ring
+  rw [he]
+  have h := norm_iteratedFDeriv_comp_le (Complex.contDiff_exp (𝕜 := ℝ) (n := ∞))
+    (L.contDiff (n := ∞)) (n := d) (by exact_mod_cast (show (d:ℕ∞) ≤ ⊤ from le_top)) v
+    (C := 1) (D := B)
+    (fun j hj ↦ by
+      rw [norm_iteratedFDeriv_complex_exp]
+      simp [L,Complex.real_smul])
+    (fun j hj hjd ↦ by
+      by_cases hj1 : j = 1
+      · subst j; simpa only [norm_iteratedFDeriv_one,L.fderiv,pow_one] using hL
+      · rw [iteratedFDeriv_linear_high L v (by omega),norm_zero]
+        positivity)
+  simpa only [mul_one] using h
+
+lemma uniform_planeWave_jets {T : Type*} [TopologicalSpace T] [CompactSpace T]
+    (K : T → Coord →L[ℝ] ℝ) (hK : Continuous K) (d : ℕ) :
+    ∃ B > 0, ∀ t s, 1 ≤ s → ∀ v, ∀ j, j ≤ d →
+      ‖iteratedFDeriv ℝ j (planeWave (K t) s) v‖ ≤ B := by
+  obtain ⟨C,hC,hb⟩ := (isCompact_univ.image hK).isBounded.exists_pos_norm_le
+  let D := max 1 C
+  refine ⟨(d.factorial:ℝ)*D^d,by dsimp [D]; positivity,?_⟩
+  intro t s hs v j hj
+  have hnorm : ‖K t‖ ≤ D := (hb _ ⟨t,mem_univ _,rfl⟩).trans (le_max_right _ _)
+  apply (planeWave_derivative_bound (K t) hs (le_max_left _ _) hnorm j v).trans
+  have hfac : (j.factorial:ℝ) ≤ d.factorial := by exact_mod_cast Nat.factorial_le hj
+  have hpow := pow_le_pow_right₀ (show 1 ≤ D from le_max_left _ _) hj
+  gcongr
+
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/PlaneWaveJets2.lean b/lean/OAI/Geometry/NodalSets/Waves/PlaneWaveJets2.lean
new file mode 100644
index 0000000000000000000000000000000000000000..4b81d8d4514f2e8ca96446ee4da36ee9e082bcd3
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/PlaneWaveJets2.lean
@@ -0,0 +1,95 @@
+import OAI.Geometry.NodalSets.Elliptic.ProductErrorJets
+import OAI.Geometry.NodalSets.Elliptic.RescaledJetDecay
+import OAI.Geometry.NodalSets.Waves.PlaneWaveJets
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Set Filter Metric
+open scoped ContDiff Topology
+noncomputable section
+variable {T : Type*} [TopologicalSpace T] [CompactSpace T]
+variable {g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ} {w S : Coord → ℝ}
+variable {y : T → Coord} {d : SourceFrameTriple g S y} {m J K k0 : ℕ}
+namespace TripleSourceWaveData
+variable (b : TripleSourceWaveData g w S y d m J K k0)
+
+theorem uniform_main_coefficient_jets (hg : ContDiff ℝ ∞ g)
+    (hp : ∀ x v, v ≠ 0 → 0 < g x v v) (hS : ContDiff ℝ ∞ S)
+    (hy : Continuous y) (hp0 : ∀ t, metricGradient g S (y t) ≠ 0)
+    (R : ℝ) (hR : 0 ≤ R) (d0 : ℕ) (e : ℝ) (he : 0 < e) :
+    ∀ᶠ n : ℕ in atTop, ∀ t x s, 1 ≤ s →
+      sourceEuclideanNorm (x-y t.1) ≤ (n:ℝ)^(-5/12:ℝ) →
+      ∀ (sigma : ℝ) (v : Coord), ‖v‖ ≤ R → ∀ k, k ≤ d0 →
+      ‖iteratedFDeriv ℝ k (fun z ↦ normalizedRescaling (b.wave n t) S n s sigma x z -
+        (b.wave n t x/(sigma:ℂ))*planeWave (g (y t.1) (d.q t.1 t.2)) s z) v‖ ≤
+        e*‖b.wave n t x/(sigma:ℂ)‖ := by
+  have hq : Continuous (fun t : T × Fin 3 ↦ d.q t.1 t.2) :=
+    continuous_prod_of_discrete_right.mpr (fun j ↦ (continuous_apply j).comp d.continuous_q)
+  have hK : Continuous (fun t : T × Fin 3 ↦ g (y t.1) (d.q t.1 t.2)) :=
+    (hg.continuous.comp (hy.comp continuous_fst)).clm_apply hq
+  obtain ⟨B,hB,hplane⟩ := uniform_planeWave_jets _ hK d0
+  let cost : ℝ := 2^d0*B*(2^d0+2)
+  have hcost : 0 < cost := by dsimp [cost]; positivity
+  let q := min 1 (e/cost)
+  have hqpos : 0 < q := lt_min zero_lt_one (div_pos he hcost)
+  have hq1 : q ≤ 1 := min_le_left _ _
+  have hqcost : cost*q ≤ e := by
+    have := (le_div_iff₀ hcost).mp (min_le_right 1 (e/cost))
+    nlinarith
+  obtain ⟨delta,hdelta,hexp⟩ := small_exp_jet_threshold d0 q hqpos
+  have hmin : 0 < min delta q := lt_min hdelta hqpos
+  filter_upwards [b.uniform_rescaled_factor_jets hg hp hS hy hp0 (R+1) (by linarith) d0
+    (min delta q) hmin,b.uniform_main_amplitude_ratio hg hp hy (R+1) 1 zero_lt_one]
+    with n hfactors hgerms
+  intro t x s hs hnear sigma v hv k hk
+  let U := ball (0:Coord) (R+1)
+  have hU : IsOpen U := isOpen_ball
+  have hvU : v ∈ U := by simpa only [U,mem_ball,dist_zero_right] using (show ‖v‖ < R+1 by linarith)
+  have hnorm (z : Coord) (hz : z ∈ U) : ‖z‖ ≤ R+1 := (by simpa only [U,mem_ball,dist_zero_right] using hz : ‖z‖ < R+1).le
+  let L := frozenPhaseCovector (fderiv ℝ S x) (g (y t.1) (d.q t.1 t.2))
+  let E := rescaledPhaseDefect (b.phase t) L n s x
+  let A := rescaledAmplitudeRatio (b.amplitude n t) n s x
+  let P := planeWave (g (y t.1) (d.q t.1 t.2)) s
+  let alpha := b.wave n t x/(sigma:ℂ)
+  have hEs : ContDiffOn ℝ ∞ E U := fun z hz ↦ (hfactors t x s hs hnear z (hnorm z hz)).1.contDiffWithinAt
+  have hAs : ContDiffOn ℝ ∞ A U := fun z hz ↦ (hfactors t x s hs hnear z (hnorm z hz)).2.1.contDiffWithinAt
+  have hPs : ContDiff ℝ ∞ P := planeWave_smooth _ _
+  have hBs : ContDiffOn ℝ ∞ (fun z ↦ Complex.exp (E z)) U := Complex.contDiff_exp.comp_contDiffOn hEs
+  have hEa (j : ℕ) (hj : j ≤ d0) : ‖iteratedFDeriv ℝ j E v‖ ≤ delta :=
+    ((hfactors t x s hs hnear v (hv.trans (by linarith))).2.2 j hj).1.trans (min_le_left _ _)
+  have hAa (j : ℕ) (hj : j ≤ d0) : ‖iteratedFDeriv ℝ j (fun z ↦ A z-1) v‖ ≤ q :=
+    ((hfactors t x s hs hnear v (hv.trans (by linarith))).2.2 j hj).2.trans (min_le_right _ _)
+  have hExp (j : ℕ) (hj : j ≤ d0) : ‖iteratedFDeriv ℝ j (fun z ↦ Complex.exp (E z)-1) v‖ ≤ q :=
+    hexp U hU E hEs v hvU hEa j hj
+  have hprod (j : ℕ) (hj : j ≤ d0) :
+      ‖iteratedFDeriv ℝ j (fun z ↦ A z*Complex.exp (E z)-1) v‖ ≤ (2^d0+2)*q :=
+    near_one_product_jet_bound hU A (fun z ↦ Complex.exp (E z)) hAs hBs hvU d0 hqpos.le hq1 hAa hExp j hj
+  have hrel := local_mul_jet_bound hU P (fun z ↦ A z*Complex.exp (E z)-1)
+    hPs.contDiffOn ((hAs.mul hBs).sub contDiffOn_const) hvU k hB.le
+    (fun j hj ↦ hplane t s hs v j (hj.trans hk)) (fun j hj ↦ hprod j (hj.trans hk))
+  have hsmall : ‖iteratedFDeriv ℝ k (fun z ↦ P z*(A z*Complex.exp (E z)-1)) v‖ ≤ e := by
+    apply hrel.trans
+    have hpow : (2:ℝ)^k ≤ 2^d0 := pow_le_pow_right₀ (by norm_num) hk
+    calc
+      _ ≤ 2^d0*B*((2^d0+2)*q) := by gcongr
+      _ = cost*q := by dsimp [cost]; ring
+      _ ≤ e := hqcost
+  have heq : (fun z ↦ normalizedRescaling (b.wave n t) S n s sigma x z-alpha*P z) =ᶠ[𝓝 v]
+      (fun z ↦ alpha • (P z*(A z*Complex.exp (E z)-1))) := by
+    filter_upwards [hU.mem_nhds hvU] with z hz
+    obtain ⟨hax,_,hbase,hshift⟩ := hgerms t x s hs hnear z (hnorm z hz)
+    rw [normalizedWave_factorization _ _ _ S (g (y t.1) (d.q t.1 t.2)) n s sigma x z
+      hax hbase.self_of_nhds hshift.self_of_nhds]
+    dsimp [alpha,P,A,E,L,rescaledAmplitudeRatio]
+    ring
+  rw [(heq.iteratedFDeriv (𝕜 := ℝ) k).self_of_nhds,
+    iteratedFDeriv_const_smul_apply' (((hPs.contDiffOn.mul ((hAs.mul hBs).sub contDiffOn_const)).contDiffAt
+      (hU.mem_nhds hvU)).of_le (by exact_mod_cast (show (k:ℕ∞) ≤ ⊤ from le_top))),norm_smul]
+  exact (mul_le_mul_of_nonneg_left hsmall (norm_nonneg alpha)).trans_eq (mul_comm _ _)
+
+end TripleSourceWaveData
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/PlaneWaveSecondMoment.lean b/lean/OAI/Geometry/NodalSets/Waves/PlaneWaveSecondMoment.lean
new file mode 100644
index 0000000000000000000000000000000000000000..76bc277b3140ebf93ca689e35199c9fa91577e65
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/PlaneWaveSecondMoment.lean
@@ -0,0 +1,70 @@
+import OAI.Geometry.NodalSets.Waves.LatticeFrequencyAnnulus
+import OAI.Geometry.NodalSets.Waves.LatticeNormalizedWeights
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Set Filter
+open scoped Topology
+noncomputable section
+
+lemma coordinate_second_moment {ι : Type*} (F : Finset ι) (α : ι → ℂ)
+    (k : ι → Coord) (κ : ℝ) (hκ : 0 ≤ κ)
+    (hmass : (1/2:ℝ) ≤ ∑ i ∈ F, ‖α i‖^2)
+    (hk : ∀ i ∈ F, κ ≤ sourceEuclideanNorm (k i)) :
+    ∃ j : Fin 4, κ^2/8 ≤ ∑ i ∈ F, ‖α i‖^2*(k i j)^2 := by
+  classical
+  have hsq (i : ι) (hi : i ∈ F) : κ^2 ≤ ∑ j : Fin 4, (k i j)^2 := by
+    have hh := pow_le_pow_left₀ hκ (hk i hi) 2
+    have he : (sourceEuclideanNorm (k i))^2 = ∑ j : Fin 4, (k i j)^2 :=
+      Real.sq_sqrt (Finset.sum_nonneg (fun _ _ ↦ sq_nonneg _))
+    rwa [he] at hh
+  have htotal : κ^2/2 ≤ ∑ j : Fin 4, ∑ i ∈ F, ‖α i‖^2*(k i j)^2 := by
+    calc
+      _ ≤ κ^2*(∑ i ∈ F, ‖α i‖^2) := by nlinarith [sq_nonneg κ]
+      _ = ∑ i ∈ F, ‖α i‖^2*κ^2 := by rw [Finset.mul_sum]; congr 1; funext i; ring
+      _ ≤ ∑ i ∈ F, ‖α i‖^2*(∑ j : Fin 4, (k i j)^2) :=
+        Finset.sum_le_sum (fun i hi ↦ mul_le_mul_of_nonneg_left (hsq i hi) (sq_nonneg _))
+      _ = _ := by simp only [Finset.mul_sum]; rw [Finset.sum_comm]
+  by_contra h
+  push Not at h
+  have hh := Finset.sum_lt_sum
+    (fun j (_ : j ∈ (Finset.univ : Finset (Fin 4))) ↦ (h j).le)
+    ⟨0,Finset.mem_univ _,h 0⟩
+  norm_num only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul] at hh
+  linarith
+
+theorem lattice_main_coordinate_moment
+    (g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ) {H : Set Coord}
+    (hH : IsCompact H) (hg : ContinuousOn g H)
+    (hp : ∀ y ∈ H, ∀ v, v ≠ 0 → 0 < g y v v) :
+    ∃ c > 0, ∀ (w S : Coord → ℝ) (D U Q : Set Coord) (m J K k0 : ℕ)
+      (a : LocalCompactWaveData g w S D m J K k0), H ⊆ D → ∀ (hUD : U ⊆ D),
+      U ⊆ H → IsOpen U → Bornology.IsBounded U → IsCompact Q → Q ⊆ U →
+      ∀ᶠ n : ℕ in atTop, ∃ hfin : Fintype (SourceGrid U n), letI := hfin
+        ∀ x ∈ Q, ∃ j : Fin 4,
+        c ≤ ∑ i ∈ Finset.univ.filter (fun i : SourceGrid U n × Fin 3 ↦
+          sourceEuclideanNorm (x-scaledLatticePoint n i.1) ≤ (n:ℝ)^(-5/12:ℝ)),
+          ‖a.latticeAlpha hUD n x i‖^2 *
+            (sourceFrequency (g (scaledLatticePoint n i.1))
+              (a.cover.triple.q (latticeFrame a.cover hUD n i.1) i.2)
+              (sourceSignScale g S x) j)^2 := by
+  classical
+  obtain ⟨κ,hκ,B,hB,hann⟩ := lattice_main_frequency_annulus g hH hg hp
+  refine ⟨κ^2/8,by positivity,?_⟩
+  intro w S D U Q m J K k0 a hHD hUD hUH hU hUb hQ hQU
+  have hm := a.latticeAlpha_main_mass hUD hU hUb hQ hQU (1/2) (by norm_num)
+  filter_upwards [hann w S D U m J K k0 a hHD hUD hUH,hm] with n hn hmass
+  obtain ⟨hfin,hmass⟩ := hmass
+  let := hfin
+  refine ⟨hfin,?_⟩
+  intro x hx
+  apply coordinate_second_moment _ _ _ κ hκ.le
+  · linarith [(hmass x hx).2]
+  · intro i hi
+    exact (hn x (hUH (hQU hx)) i (Finset.mem_filter.mp hi).2).1
+
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/RescaledGaussianField.lean b/lean/OAI/Geometry/NodalSets/Waves/RescaledGaussianField.lean
new file mode 100644
index 0000000000000000000000000000000000000000..f94ee5f8e9e0b3f35e137f9c0c122ca0a71fc981
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/RescaledGaussianField.lean
@@ -0,0 +1,71 @@
+import OAI.Geometry.NodalSets.Elliptic.NormalizedRescaling
+import OAI.Geometry.NodalSets.Waves.LatticeValueVariance
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Yau.Probability MeasureTheory ProbabilityTheory
+noncomputable section
+variable {ι : Type*} [Fintype ι]
+
+def rescaledGaussianField (V : ι → Coord → ℂ) (S0 T0 S : Coord → ℝ)
+    (N s σ : ℝ) (x : Coord) (coeff : ι × Fin 2 → ℝ) (v : Coord) : ℝ :=
+  (oscillatorySeed S0 T0 N (x+(N*s)⁻¹ • v)+
+    gaussianWaveField V coeff (x+(N*s)⁻¹ • v)) /
+      (σ*Real.exp (fderiv ℝ S x v/s))
+
+lemma rescaledGaussianField_decomposition (V : ι → Coord → ℂ) (S0 T0 S : Coord → ℝ)
+    (N s σ : ℝ) (x : Coord) (coeff : ι × Fin 2 → ℝ) (v : Coord) :
+    rescaledGaussianField V S0 T0 S N s σ x coeff v = rescaledSeed S0 T0 S N s σ x v +
+      gaussianWaveField (fun i ↦ normalizedRescaling (V i) S N s σ x) coeff v := by
+  classical
+  simp only [rescaledGaussianField,rescaledSeed,add_div,gaussianWaveField,
+    normalizedRescaling_eq_div,← mul_div_assoc,Complex.re_sum,Complex.div_ofReal_re,
+    Finset.sum_div]
+
+lemma pairLinearSum_integrable (z : ι → ℂ) : Integrable (pairLinearSum z) gaussianPairs := by
+  classical
+  unfold pairLinearSum gaussianPairs
+  apply integrable_finsetSum
+  intro p _
+  exact (integrable_eval (μ := fun _ : ι × Fin 2 ↦ gaussianReal 0 1)
+    (i := p) IsGaussian.integrable_id).const_mul _
+
+lemma pairLinearSum_mean (z : ι → ℂ) : ∫ coeff, pairLinearSum z coeff ∂gaussianPairs = 0 := by
+  classical
+  unfold pairLinearSum gaussianPairs
+  rw [integral_finsetSum]
+  · apply Finset.sum_eq_zero
+    intro p _
+    rw [integral_const_mul,integral_eval]
+    simp
+  · intro p _
+    exact (integrable_eval (μ := fun _ : ι × Fin 2 ↦ gaussianReal 0 1)
+      (i := p) IsGaussian.integrable_id).const_mul _
+
+lemma rescaledGaussianField_mean (V : ι → Coord → ℂ) (S0 T0 S : Coord → ℝ)
+    (N s σ : ℝ) (x v : Coord) :
+    ∫ coeff, rescaledGaussianField V S0 T0 S N s σ x coeff v ∂gaussianPairs =
+      rescaledSeed S0 T0 S N s σ x v := by
+  classical
+  let Z := fun i ↦ normalizedRescaling (V i) S N s σ x v
+  have he : (fun coeff ↦ gaussianWaveField
+      (fun i ↦ normalizedRescaling (V i) S N s σ x) coeff v) = pairLinearSum Z := by
+    funext coeff
+    rw [pairLinearSum_eq_complex]
+    simp only [gaussianWaveField,Complex.re_sum,Z]
+  let : IsProbabilityMeasure (gaussianPairs (ι := ι)) := by
+    unfold gaussianPairs
+    infer_instance
+  simp_rw [rescaledGaussianField_decomposition]
+  change ∫ coeff, rescaledSeed S0 T0 S N s σ x v+
+    (fun coeff ↦ gaussianWaveField (fun i ↦ normalizedRescaling (V i) S N s σ x) coeff v) coeff
+    ∂gaussianPairs = _
+  rw [he,integral_add (integrable_const _) (pairLinearSum_integrable Z),
+    pairLinearSum_mean,add_zero,integral_const]
+  simp
+
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/RescaledLatticeTail.lean b/lean/OAI/Geometry/NodalSets/Waves/RescaledLatticeTail.lean
new file mode 100644
index 0000000000000000000000000000000000000000..753dc526b63ad0a2821f8821d4f33d744b62c228
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/RescaledLatticeTail.lean
@@ -0,0 +1,73 @@
+import OAI.Geometry.NodalSets.Elliptic.NormalizedRescaling
+import OAI.Geometry.NodalSets.Waves.ShiftedLatticeTail
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Set Filter
+open scoped ContDiff Topology
+noncomputable section
+variable {g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ} {w S : Coord → ℝ}
+variable {D U : Set Coord} {m J K k0 : ℕ}
+namespace LocalCompactWaveData
+variable (a : LocalCompactWaveData g w S D m J K k0)
+
+theorem rescaled_lattice_nonmain_negligible (hUD : U ⊆ D) (hU : IsOpen U)
+    (hUb : Bornology.IsBounded U) {Q : Set Coord} (hQ : IsCompact Q) (hQU : Q ⊆ U)
+    (hS : ContDiff ℝ ∞ S) (R : ℝ) (hR : 0 ≤ R) (ε : ℝ) (hε : 0 < ε) :
+    ∀ᶠ n : ℕ in atTop, ∃ hfin : Fintype (SourceGrid U n), letI := hfin
+      ∀ x ∈ Q, ∀ s : ℝ, 1 ≤ s → ∀ v : Coord, ‖v‖ ≤ R → ∀ k : Fin (k0+1),
+      ∑ i ∈ Finset.univ.filter (fun i : SourceGrid U n × Fin 3 ↦
+          (n:ℝ)^(-5/12:ℝ) < sourceEuclideanNorm (x-scaledLatticePoint n i.1)),
+        ‖iteratedFDeriv ℝ k.val
+          (normalizedRescaling (latticeWave a.cover a.beams hUD n i.1 i.2)
+            S n s (a.latticeSigma hUD n x) x) v‖^2 ≤ ε := by
+  classical
+  obtain ⟨A,hA,hamp⟩ := compact_smooth_derivative_bound (rescaledSeedNormalizer S) k0
+    (fun j _ ↦ (smooth_spatial_iteratedFDeriv _ (rescaledSeedNormalizer_smooth S hS) j).continuous)
+    (Q ×ˢ Icc (0:ℝ) 1) (hQ.prod isCompact_Icc) R
+  let C : ℝ := (2^k0*A)^2
+  have hC : 0 < C := by dsimp [C]; positivity
+  have hkpos : (0:ℝ) < k0+1 := by positivity
+  obtain ⟨c,hc,B,hB,hsigma⟩ := a.lattice_sigma_comparison hUD hU hUb hQ hQU
+  have htail := a.lattice_shifted_nonmain_negligible hUD hU hUb hQ hQU hS R hR
+    (ε/(C*((k0:ℝ)+1))) (by positivity)
+  filter_upwards [htail,hsigma,a.estimates,eventually_gt_atTop (0:ℕ)]
+    with n hn hσ hest hnpos
+  obtain ⟨hfin,ht⟩ := hn
+  obtain ⟨hfin',hσ⟩ := hσ
+  have he : hfin' = hfin := Subsingleton.elim _ _
+  subst hfin'
+  let := hfin
+  refine ⟨hfin,?_⟩
+  intro x hx s hs v hv k
+  have hn1 : (1:ℝ) ≤ n := by exact_mod_cast hnpos
+  have hs0 : 0 < s := lt_of_lt_of_le zero_lt_one hs
+  have hσ0 : 0 < a.latticeSigma hUD n x :=
+    lt_of_lt_of_le (mul_pos hc (Real.exp_pos _)) (hσ x hx).1
+  have ha := hamp (x,s⁻¹) ⟨hx,by positivity,(inv_le_one₀ hs0).mpr hs⟩ v hv
+  let F := Finset.univ.filter (fun i : SourceGrid U n × Fin 3 ↦
+    (n:ℝ)^(-5/12:ℝ) < sourceEuclideanNorm (x-scaledLatticePoint n i.1))
+  have hb (i : SourceGrid U n × Fin 3) := normalizedRescaling_square_bound
+    (latticeWave a.cover a.beams hUD n i.1 i.2)
+    (hest (latticeFrame a.cover hUD n i.1,i.2)).1
+    S hS n s _ hn1 hs hσ0 x v k0 A hA.le ha k.val (by omega)
+  apply (Finset.sum_le_sum (fun i (_ : i ∈ F) ↦ hb i)).trans
+  calc
+    _ = C*∑ j : Fin (k0+1), ((∑ i ∈ F,
+        ‖iteratedFDeriv ℝ j.val (latticeWave a.cover a.beams hUD n i.1 i.2)
+          (x+((n:ℝ)*s)⁻¹ • v)‖^2)/(a.latticeSigma hUD n x)^2) := by
+      dsimp only [C]
+      simp only [Finset.mul_sum,Finset.sum_div,mul_div_assoc]
+      rw [Finset.sum_comm]
+    _ ≤ C*∑ _j : Fin (k0+1), ε/(C*((k0:ℝ)+1)) :=
+      mul_le_mul_of_nonneg_left (Finset.sum_le_sum (fun j _ ↦ ht x hx s hs v hv j)) hC.le
+    _ = ε := by
+      simp only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul,Nat.cast_add,Nat.cast_one]
+      field_simp
+
+end LocalCompactWaveData
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/ScaledLatticeGaussianSum.lean b/lean/OAI/Geometry/NodalSets/Waves/ScaledLatticeGaussianSum.lean
new file mode 100644
index 0000000000000000000000000000000000000000..935a8fa4579cd39aaca85794a8842f33870383cf
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/ScaledLatticeGaussianSum.lean
@@ -0,0 +1,45 @@
+import OAI.Geometry.NodalSets.Elliptic.HalfScaleLemmas
+import OAI.Geometry.NodalSets.Waves.LatticeCount
+import OAI.Geometry.NodalSets.Waves.ShiftedGaussianSum
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Yau.Probability
+noncomputable section
+
+lemma scaled_lattice_squared_distance {n : ℕ} (hn : 0 < n) (x : Coord)
+    (z : Fin 4 → ℤ) :
+    (n:ℝ)*(sourceEuclideanNorm (x-scaledLatticePoint n z))^2 =
+      ∑ i, (Real.sqrt n*x i-(z i:ℝ))^2 := by
+  have hs : 0 < Real.sqrt n := Real.sqrt_pos.mpr (by exact_mod_cast hn)
+  have hs2 := Real.sq_sqrt (show (0:ℝ) ≤ n by positivity)
+  rw [sourceEuclideanNorm,Real.sq_sqrt (by positivity),Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro i _
+  change (n:ℝ)*(x i-(n:ℝ)^(-1/2:ℝ)*(z i:ℝ))^2 = _
+  rw [lattice_scale_eq_inv_sqrt]
+  field_simp
+  nlinarith [hs2]
+
+theorem source_lattice_gaussian_sum {a : ℝ} (ha : 0 < a) :
+    ∃ C > 0, ∀ (U : Set Coord) (n : ℕ), 0 < n → ∀ [Fintype (SourceGrid U n)],
+      ∀ x : Coord, ∑ z : SourceGrid U n,
+        Real.exp (-a*(n:ℝ)*(sourceEuclideanNorm (x-scaledLatticePoint n z))^2) ≤ C := by
+  classical
+  obtain ⟨C,hC,hb⟩ := finite_shifted_gaussian_sum_pi (d := 4) ha
+  refine ⟨C,hC,?_⟩
+  intro U n hn hfin x
+  let s : Finset (Fin 4 → ℤ) := Finset.univ.image (fun z : SourceGrid U n ↦ z.val)
+  have h := hb (fun i ↦ Real.sqrt n*x i) s
+  rw [Finset.sum_image (by intro u _ v _ huv; exact Subtype.ext huv)] at h
+  convert h using 1
+  apply Finset.sum_congr rfl
+  intro z _
+  congr 1
+  rw [mul_assoc,scaled_lattice_squared_distance hn]
+
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/SeededGaussianJet.lean b/lean/OAI/Geometry/NodalSets/Waves/SeededGaussianJet.lean
new file mode 100644
index 0000000000000000000000000000000000000000..41ceb142727f7370fffa0a6f2381267a1b01e6e4
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/SeededGaussianJet.lean
@@ -0,0 +1,48 @@
+import OAI.Geometry.NodalSets.Waves.GaussianWaveJet
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Yau.Probability MeasureTheory ProbabilityTheory
+noncomputable section
+variable {ι : Type*} [Fintype ι]
+
+lemma gaussianWaveField_differentiable (V : ι → Coord → ℂ) (a : ι × Fin 2 → ℝ)
+    (x : Coord) (hV : ∀ i, DifferentiableAt ℝ (V i) x) :
+    DifferentiableAt ℝ (gaussianWaveField V a) x := by
+  classical
+  change DifferentiableAt ℝ (Complex.reCLM ∘ (fun z ↦ ∑ i, gaussianCoefficient a i*V i z)) x
+  exact Complex.reCLM.differentiableAt.comp x (DifferentiableAt.fun_sum (fun i _ ↦ (hV i).const_mul _))
+
+lemma seeded_gaussian_dual_variance (V : ι → Coord → ℂ) (seed : Coord → ℝ)
+    (x v : Coord) (hV : ∀ i, DifferentiableAt ℝ (V i) x) (hseed : DifferentiableAt ℝ seed x)
+    (N S b : ℝ) :
+    Var[(fun a : ι × Fin 2 → ℝ ↦ Real.exp (-N*S)*(b*(seed x+gaussianWaveField V a x)+
+      N⁻¹*fderiv ℝ (fun z ↦ seed z+gaussianWaveField V a z) x v)); gaussianPairs] =
+      ∑ i, ‖dualWaveCoefficient N (V i) S x b v‖^2 := by
+  let z := fun i ↦ dualWaveCoefficient N (V i) S x b v
+  let mean := Real.exp (-N*S)*(b*seed x+N⁻¹*fderiv ℝ seed x v)
+  have he : (fun a : ι × Fin 2 → ℝ ↦ Real.exp (-N*S)*(b*(seed x+gaussianWaveField V a x)+
+      N⁻¹*fderiv ℝ (fun z ↦ seed z+gaussianWaveField V a z) x v)) =
+      fun a ↦ mean+pairLinearSum z a := by
+    funext a
+    rw [fderiv_fun_add hseed (gaussianWaveField_differentiable V a x hV),add_apply]
+    have h := gaussianWaveField_dual V a x v hV N S b
+    change _ = mean+pairLinearSum z a
+    rw [← h]
+    dsimp [mean]
+    ring
+  rw [he]
+  let : IsProbabilityMeasure (gaussianPairs (ι := ι)) := by
+    change IsProbabilityMeasure (Measure.pi (fun _ : ι × Fin 2 ↦ gaussianReal 0 1))
+    infer_instance
+  have hc : Continuous (pairLinearSum z) := by
+    unfold pairLinearSum
+    fun_prop
+  rw [variance_const_add hc.aestronglyMeasurable]
+  exact variance_pairLinearSum _
+
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/SeededLatticeFieldBounds.lean b/lean/OAI/Geometry/NodalSets/Waves/SeededLatticeFieldBounds.lean
new file mode 100644
index 0000000000000000000000000000000000000000..7ed7bc385da034c48483fc8e7dcf22e13d4a9f64
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/SeededLatticeFieldBounds.lean
@@ -0,0 +1,71 @@
+import OAI.Geometry.NodalSets.Elliptic.SeedDerivativeBounds
+import OAI.Geometry.NodalSets.Waves.LatticeFieldBounds
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Yau.Probability Set Filter
+open scoped ContDiff Topology
+noncomputable section
+variable {g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ} {w S : Coord → ℝ}
+variable {D U : Set Coord} {m J K k0 : ℕ}
+namespace LocalCompactWaveData
+variable (a : LocalCompactWaveData g w S D m J K k0)
+
+theorem seeded_lattice_field_derivative_bound (hUD : U ⊆ D) (hUb : Bornology.IsBounded U)
+    (S0 T0 : Coord → ℝ) (hS0 : ContDiff ℝ ∞ S0) (hT0 : ContDiff ℝ ∞ T0)
+    {Q : Set Coord} (hQ : IsCompact Q) :
+    ∃ C > 0, ∀ᶠ n : ℕ in atTop, ∃ hfin : Fintype (SourceGrid U n),
+      letI := hfin
+      ∀ coeff : ((SourceGrid U n × Fin 3) × Fin 2) → ℝ,
+        coeff ∈ coefficientEvent (n:ℝ) →
+        let u := fun z ↦ oscillatorySeed S0 T0 n z + gaussianWaveField
+          (fun i : SourceGrid U n × Fin 3 ↦ latticeWave a.cover a.beams hUD n i.1 i.2) coeff z
+        ContDiff ℝ ∞ u ∧ ∀ x ∈ Q, ∀ k : Fin (k0+1),
+          ‖iteratedFDeriv ℝ k.val u x‖ ≤ C*(n:ℝ)^(k.val+3)*
+            max (Real.exp ((n:ℝ)*S x)) (Real.exp ((n:ℝ)*S0 x)) := by
+  have hs (j : Fin (k0+1)) := oscillatorySeed_derivative_bound S0 T0 hS0 hT0 hQ j.val
+  choose b hb hseed using hs
+  let B : ℝ := ∑ j, b j
+  have hB : 0 < B := (hb ⟨0,by omega⟩).trans_le
+    (Finset.single_le_sum (fun j _ ↦ (hb j).le) (Finset.mem_univ _))
+  obtain ⟨C,hC,hfield⟩ := a.lattice_field_derivative_bound hUD hUb
+  refine ⟨B+C,by positivity,?_⟩
+  filter_upwards [hfield,eventually_gt_atTop (0:ℕ)] with n hn hnpos
+  obtain ⟨hfin,hbnd⟩ := hn
+  let := hfin
+  refine ⟨hfin,?_⟩
+  intro coeff hc
+  obtain ⟨hV,hVb⟩ := hbnd coeff hc
+  have hseedSmooth := oscillatorySeed_contDiff S0 T0 hS0 hT0 (n:ℝ)
+  refine ⟨hseedSmooth.add hV,?_⟩
+  intro x hx k
+  have hn1 : (1:ℝ) ≤ n := by exact_mod_cast hnpos
+  have hseed' : ‖iteratedFDeriv ℝ k.val (oscillatorySeed S0 T0 n) x‖ ≤
+      B*(n:ℝ)^(k.val+3)*max (Real.exp ((n:ℝ)*S x)) (Real.exp ((n:ℝ)*S0 x)) := by
+    have h := hseed k n hn1 x hx
+    refine h.trans ?_
+    have hbB : b k ≤ B := Finset.single_le_sum (fun j _ ↦ (hb j).le) (Finset.mem_univ _)
+    gcongr
+    · omega
+    · exact le_max_right _ _
+  have hrand : ‖iteratedFDeriv ℝ k.val (gaussianWaveField
+      (fun i : SourceGrid U n × Fin 3 ↦ latticeWave a.cover a.beams hUD n i.1 i.2) coeff) x‖ ≤
+      C*(n:ℝ)^(k.val+3)*max (Real.exp ((n:ℝ)*S x)) (Real.exp ((n:ℝ)*S0 x)) := by
+    refine (hVb x k).trans ?_
+    gcongr
+    exact le_max_left _ _
+  change ‖iteratedFDeriv ℝ k.val (oscillatorySeed S0 T0 n + gaussianWaveField
+    (fun i : SourceGrid U n × Fin 3 ↦ latticeWave a.cover a.beams hUD n i.1 i.2) coeff) x‖ ≤ _
+  rw [iteratedFDeriv_add_apply (hseedSmooth.of_le
+    (by exact_mod_cast (show (k.val:ℕ∞) ≤ ⊤ from le_top))).contDiffAt
+    (hV.of_le (by exact_mod_cast (show (k.val:ℕ∞) ≤ ⊤ from le_top))).contDiffAt]
+  have h := (norm_add_le _ _).trans (add_le_add hseed' hrand)
+  convert h using 1
+  first | rfl | ring
+
+end LocalCompactWaveData
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/ShiftedGaussianSum.lean b/lean/OAI/Geometry/NodalSets/Waves/ShiftedGaussianSum.lean
new file mode 100644
index 0000000000000000000000000000000000000000..7bbcf94a5ed62e80677d54a8fbae40abe95a0482
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/ShiftedGaussianSum.lean
@@ -0,0 +1,84 @@
+import Mathlib.Algebra.BigOperators.Ring.Finset
+import Mathlib.Algebra.Order.Floor.Ring
+import Mathlib.Analysis.SpecialFunctions.Exp
+import Mathlib.Tactic.GCongr
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.Positivity
+import Mathlib.Topology.Algebra.InfiniteSum.NatInt
+
+namespace OAI
+
+namespace Yau.Probability
+open scoped BigOperators
+noncomputable section
+
+lemma summable_integer_exponential {a : ℝ} (ha : 0 < a) :
+    Summable (fun z : ℤ ↦ Real.exp (-a * |(z:ℝ)|)) := by
+  apply summable_int_iff_summable_nat_and_neg.mpr
+  constructor <;> simpa [mul_comm] using
+    (Real.summable_exp_nat_mul_iff.mpr (neg_neg_of_pos ha))
+
+lemma shifted_square_lower (x : ℝ) (z : ℤ) :
+    |((z-⌊x⌋ : ℤ):ℝ)| ≤ (x-(z:ℝ))^2+2 := by
+  have h0 := Int.floor_le x
+  have h1 := Int.lt_floor_add_one x
+  have ht : |((z-⌊x⌋ : ℤ):ℝ)| ≤ |(z:ℝ)-x|+|x-(⌊x⌋:ℝ)| := by
+    simpa only [Int.cast_sub,sub_add_sub_cancel] using abs_add_le ((z:ℝ)-x) (x-(⌊x⌋:ℝ))
+  rw [abs_of_nonneg (by linarith : 0 ≤ x-(⌊x⌋:ℝ))] at ht
+  nlinarith [sq_nonneg (|(z:ℝ)-x|-1),sq_abs ((z:ℝ)-x)]
+
+theorem finite_shifted_gaussian_sum {a : ℝ} (ha : 0 < a) :
+    ∃ C > 0, ∀ (x : ℝ) (s : Finset ℤ),
+      ∑ z ∈ s, Real.exp (-a*(x-(z:ℝ))^2) ≤ C := by
+  let f : ℤ → ℝ := fun z ↦ Real.exp (-a*|(z:ℝ)|)
+  have hf : Summable f := summable_integer_exponential ha
+  have hsum : 0 < ∑' z, f z := (lt_of_lt_of_le (Real.exp_pos _) (hf.le_tsum 0 (fun _ _ ↦ Real.exp_nonneg _)))
+  refine ⟨Real.exp (2*a)*(∑' z, f z),mul_pos (Real.exp_pos _) hsum,?_⟩
+  intro x s
+  calc
+    _ ≤ ∑ z ∈ s, Real.exp (2*a)*f (z-⌊x⌋) := by
+      apply Finset.sum_le_sum
+      intro z _
+      dsimp [f]
+      rw [← Real.exp_add]
+      apply Real.exp_le_exp.mpr
+      have h := mul_le_mul_of_nonneg_left (shifted_square_lower x z) ha.le
+      nlinarith
+    _ = Real.exp (2*a)*∑ z ∈ s.image (fun z ↦ z-⌊x⌋), f z := by
+      rw [Finset.mul_sum,Finset.sum_image]
+      intro u _ v _ huv
+      exact sub_left_injective huv
+    _ ≤ Real.exp (2*a)*(∑' z, f z) := by
+      gcongr
+      exact hf.sum_le_tsum _ (fun _ _ ↦ Real.exp_nonneg _)
+
+theorem finite_shifted_gaussian_sum_pi {d : ℕ} {a : ℝ} (ha : 0 < a) :
+    ∃ C > 0, ∀ (x : Fin d → ℝ) (s : Finset (Fin d → ℤ)),
+      ∑ z ∈ s, Real.exp (-a*∑ i, (x i-(z i:ℝ))^2) ≤ C := by
+  classical
+  obtain ⟨C,hC,hb⟩ := finite_shifted_gaussian_sum ha
+  refine ⟨C^d,pow_pos hC d,?_⟩
+  intro x s
+  let t : Fin d → Finset ℤ := fun i ↦ s.image (fun z ↦ z i)
+  have hsub : s ⊆ Fintype.piFinset t := by
+    intro z hz
+    exact Fintype.mem_piFinset.mpr (fun i ↦ Finset.mem_image.mpr ⟨z,hz,rfl⟩)
+  calc
+    _ = ∑ z ∈ s, ∏ i, Real.exp (-a*(x i-(z i:ℝ))^2) := by
+      apply Finset.sum_congr rfl
+      intro z _
+      rw [Finset.mul_sum,Real.exp_sum]
+    _ ≤ ∑ z ∈ Fintype.piFinset t, ∏ i, Real.exp (-a*(x i-(z i:ℝ))^2) :=
+      Finset.sum_le_sum_of_subset_of_nonneg hsub (fun _ _ _ ↦ by positivity)
+    _ = ∏ i, ∑ z ∈ t i, Real.exp (-a*(x i-(z:ℝ))^2) :=
+      (Finset.prod_univ_sum t (fun i z ↦ Real.exp (-a*(x i-(z:ℝ))^2))).symm
+    _ ≤ ∏ _i : Fin d, C := by
+      apply Finset.prod_le_prod₀ (fun _ _ ↦ by positivity)
+      intro i _
+      exact hb (x i) (t i)
+    _ = C^d := by simp
+
+end
+end Yau.Probability
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/ShiftedLatticeTail.lean b/lean/OAI/Geometry/NodalSets/Waves/ShiftedLatticeTail.lean
new file mode 100644
index 0000000000000000000000000000000000000000..d24f57603a2c4ab877f930132ed3db663e622669
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/ShiftedLatticeTail.lean
@@ -0,0 +1,109 @@
+import OAI.Geometry.NodalSets.Elliptic.ShiftedCenterDistance
+import OAI.Geometry.NodalSets.Waves.LatticeDerivativeTail
+import OAI.Geometry.NodalSets.Waves.LatticeNonmainDecay
+import OAI.Geometry.NodalSets.Waves.LatticeVarianceComparison
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Yau.Probability Set Filter
+open scoped ContDiff Topology
+noncomputable section
+variable {g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ} {w S : Coord → ℝ}
+variable {D U : Set Coord} {m J K k0 : ℕ}
+namespace LocalCompactWaveData
+variable (a : LocalCompactWaveData g w S D m J K k0)
+
+theorem lattice_shifted_nonmain_bound (hUD : U ⊆ D) (hS : ContDiff ℝ ∞ S)
+    {Q : Set Coord} (hQ : IsCompact Q) (R : ℝ) (hR : 0 ≤ R) :
+    ∃ C > 0, ∀ᶠ n : ℕ in atTop, ∀ [Fintype (SourceGrid U n)],
+      ∀ x ∈ Q, ∀ s : ℝ, 1 ≤ s → ∀ v : Coord, ‖v‖ ≤ R → ∀ k : Fin (k0+1),
+      ∑ i ∈ Finset.univ.filter (fun i : SourceGrid U n × Fin 3 ↦
+          (n:ℝ)^(-5/12:ℝ) < sourceEuclideanNorm (x-scaledLatticePoint n i.1)),
+        ‖iteratedFDeriv ℝ k.val (latticeWave a.cover a.beams hUD n i.1 i.2)
+          (x+((n:ℝ)*s)⁻¹ • v)‖^2 ≤
+        C*(n:ℝ)^(2*k.val)*Real.exp (2*(n:ℝ)*S x)*
+          Real.exp (-(a.beams.c/16)*(n:ℝ)^(1/6:ℝ)) := by
+  classical
+  obtain ⟨B,hB,htail⟩ := a.lattice_derivative_tail_bound hUD
+  obtain ⟨L,hL,hshift⟩ := compact_rescaled_envelope_bound S hS hQ R hR
+  refine ⟨B*Real.exp (2*L),by positivity,?_⟩
+  filter_upwards [htail,eventually_shifted_center_distance R,eventually_gt_atTop (0:ℕ)]
+    with n hn hdist hnpos
+  intro hfin x hx s hs v hv k
+  have hn0 : (0:ℝ) < n := by exact_mod_cast hnpos
+  have hn1 : (1:ℝ) ≤ n := by exact_mod_cast hnpos
+  let z := x+((n:ℝ)*s)⁻¹ • v
+  let r : ℝ := (n:ℝ)^(-5/12:ℝ)
+  have hsub : Finset.univ.filter (fun i : SourceGrid U n × Fin 3 ↦
+      r < sourceEuclideanNorm (x-scaledLatticePoint n i.1)) ⊆
+      Finset.univ.filter (fun i : SourceGrid U n × Fin 3 ↦
+      r/4 < sourceEuclideanNorm (z-scaledLatticePoint n i.1)) := by
+    intro i hi
+    exact Finset.mem_filter.mpr ⟨Finset.mem_univ _,hdist s hs x _ v hv (Finset.mem_filter.mp hi).2⟩
+  have hsum := Finset.sum_le_sum_of_subset_of_nonneg hsub (fun i _ _ ↦
+    sq_nonneg ‖iteratedFDeriv ℝ k.val (latticeWave a.cover a.beams hUD n i.1 i.2) z‖)
+  have he : -a.beams.c*(n:ℝ)*(r/4)^2 = -(a.beams.c/16)*(n:ℝ)^(1/6:ℝ) := by
+    calc
+      _ = -(a.beams.c/16)*((n:ℝ)*r^2) := by ring
+      _ = _ := by rw [main_center_scale _ hn0]
+  have hb := hn z k (r/4) (by dsimp [r]; positivity)
+  rw [he] at hb
+  have hexp : Real.exp (2*(n:ℝ)*S z) ≤ Real.exp (2*(n:ℝ)*S x)*Real.exp (2*L) := by
+    rw [← Real.exp_add]
+    apply Real.exp_le_exp.mpr
+    have hh := (le_abs_self _).trans (hshift n s hn1 hs x hx v hv)
+    dsimp [z]
+    nlinarith
+  apply (hsum.trans hb).trans
+  calc
+    _ ≤ B*(n:ℝ)^(2*k.val)*(Real.exp (2*(n:ℝ)*S x)*Real.exp (2*L))*
+        Real.exp (-(a.beams.c/16)*(n:ℝ)^(1/6:ℝ)) := by gcongr
+    _ = _ := by ring
+
+theorem lattice_shifted_nonmain_negligible (hUD : U ⊆ D) (hU : IsOpen U)
+    (hUb : Bornology.IsBounded U) {Q : Set Coord} (hQ : IsCompact Q) (hQU : Q ⊆ U)
+    (hS : ContDiff ℝ ∞ S) (R : ℝ) (hR : 0 ≤ R) (ε : ℝ) (hε : 0 < ε) :
+    ∀ᶠ n : ℕ in atTop, ∃ hfin : Fintype (SourceGrid U n), letI := hfin
+      ∀ x ∈ Q, ∀ s : ℝ, 1 ≤ s → ∀ v : Coord, ‖v‖ ≤ R → ∀ k : Fin (k0+1),
+      (∑ i ∈ Finset.univ.filter (fun i : SourceGrid U n × Fin 3 ↦
+          (n:ℝ)^(-5/12:ℝ) < sourceEuclideanNorm (x-scaledLatticePoint n i.1)),
+        ‖iteratedFDeriv ℝ k.val (latticeWave a.cover a.beams hUD n i.1 i.2)
+          (x+((n:ℝ)*s)⁻¹ • v)‖^2) / (a.latticeSigma hUD n x)^2 ≤ ε := by
+  classical
+  obtain ⟨c,hc,hlo⟩ := a.lattice_variance_lower_bound hUD hU hQ hQU
+  obtain ⟨C,hC,htail⟩ := a.lattice_shifted_nonmain_bound hUD hS hQ R hR
+  have hlim := (tendsto_polynomial_main_center_tail (2*k0) (a.beams.c/16)
+    (by have := a.beams.c_pos; positivity)).const_mul (C/c)
+  simp only [mul_zero] at hlim
+  filter_upwards [hlo,htail,hlim.eventually (gt_mem_nhds hε),eventually_gt_atTop (0:ℕ)]
+    with n hl ht hsmall hnpos
+  have := finite_source_grid hUb hnpos
+  let hfin := Fintype.ofFinite (SourceGrid U n)
+  let := hfin
+  refine ⟨hfin,?_⟩
+  intro x hx s hs v hv k
+  have hv0 : 0 < a.latticeVariance hUD n x :=
+    lt_of_lt_of_le (mul_pos hc (Real.exp_pos _)) (hl x hx)
+  have hn1 : (1:ℝ) ≤ n := by exact_mod_cast hnpos
+  have hp : (n:ℝ)^(2*k.val) ≤ (n:ℝ)^(2*k0) :=
+    pow_le_pow_right₀ hn1 (by have := k.isLt; omega)
+  rw [a.latticeSigma_sq]
+  apply le_trans _ hsmall.le
+  apply (div_le_iff₀ hv0).mpr
+  have hh := mul_le_mul_of_nonneg_left (hl x hx)
+    (by positivity : 0 ≤ (C/c)*((n:ℝ)^(2*k0)*Real.exp (-(a.beams.c/16)*(n:ℝ)^(1/6:ℝ))))
+  calc
+    _ ≤ C*(n:ℝ)^(2*k.val)*Real.exp (2*(n:ℝ)*S x)*
+        Real.exp (-(a.beams.c/16)*(n:ℝ)^(1/6:ℝ)) := ht x hx s hs v hv k
+    _ ≤ C*(n:ℝ)^(2*k0)*Real.exp (2*(n:ℝ)*S x)*
+        Real.exp (-(a.beams.c/16)*(n:ℝ)^(1/6:ℝ)) := by gcongr
+    _ = ((C/c)*((n:ℝ)^(2*k0)*Real.exp (-(a.beams.c/16)*(n:ℝ)^(1/6:ℝ))))*
+        (c*Real.exp (2*(n:ℝ)*S x)) := by field_simp
+    _ ≤ _ := hh
+
+end LocalCompactWaveData
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/SignSubballProbability.lean b/lean/OAI/Geometry/NodalSets/Waves/SignSubballProbability.lean
new file mode 100644
index 0000000000000000000000000000000000000000..dcccd8fb5f5e5fb978d8e5dc4b5fdb91de8c3164
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/SignSubballProbability.lean
@@ -0,0 +1,59 @@
+import OAI.Geometry.NodalSets.Elliptic.SignEventMarkov
+import OAI.Geometry.NodalSets.Elliptic.StrictSignSubballs
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets MeasureTheory Set
+open scoped ContDiff ENNReal
+noncomputable section
+
+lemma positive_real_probability_floor {p : ℝ≥0∞} (hp : 0 < p) :
+    ∃ q : ℝ, 0 < q ∧ q ≤ 1 ∧ ENNReal.ofReal q ≤ p := by
+  let b := min p 1
+  have hb : 0 < b := lt_min hp (by norm_num)
+  have ht : b ≠ ⊤ := ne_of_lt ((min_le_right p 1).trans_lt (by norm_num))
+  refine ⟨b.toReal,ENNReal.toReal_pos hb.ne' ht,?_,?_⟩
+  · have hh := ENNReal.toReal_mono (by norm_num : (1:ℝ≥0∞) ≠ ⊤) (min_le_right p 1)
+    simpa using hh
+  · rw [ENNReal.ofReal_toReal ht]
+    exact min_le_left p 1
+
+lemma sign_subball_constants {tau p C : ℝ} (ht : 0 < tau) (ht1 : tau < 1)
+    (hp : 0 < p) (hC : 0 ≤ C) :
+    ∃ L > 0, ∃ r > 0, r < tau/4 ∧ r < (1-tau)/2 ∧ 4*L*r < 1 ∧ 2*C ≤ p*L := by
+  let L := 2*C/p+1
+  have hL : 0 < L := by dsimp [L]; positivity
+  obtain ⟨r,hr,hrt,hr1,hrL⟩ := choose_sign_subball_radius ht ht1 hL.le
+  refine ⟨L,hL,r,hr,hrt,hr1,hrL,?_⟩
+  dsimp [L]
+  have he : p*(2*C/p+1) = 2*C+p := by field_simp
+  rw [he]
+  linarith
+
+lemma sign_subballs_probability {Ω : Type*} [MeasurableSpace Ω]
+    (μ : Measure Ω) [IsFiniteMeasure μ] (F : Ω → Coord → ℝ)
+    (hF : ∀ a, ContDiff ℝ ∞ (F a)) (G : Ω → ℝ)
+    (hG : Integrable G μ) (hG0 : ∀ a, 0 ≤ G a)
+    {tau r p C L : ℝ} (ht : 0 ≤ tau) (hr : 0 ≤ r) (hL : 0 < L)
+    (htr : tau+r ≤ 1) (hsmall : 4*L*r < 1) (hmean : (∫ a, G a ∂μ) ≤ C)
+    (hCL : 2*C ≤ p*L)
+    (hgrad : ∀ a, G a ≤ L → ∀ x : Coord, sourceEuclideanNorm x ≤ 1 →
+      sourceEuclideanNorm (fun i ↦ fderiv ℝ (F a) x (Pi.single i 1)) ≤ L)
+    (j : Fin 4) (hsign : p ≤ μ.real {a | 1 ≤ F a 0 ∧ F a (tau • Pi.single j 1) ≤ -1}) :
+    p/2 ≤ μ.real {a |
+      (∀ v : Coord, sourceEuclideanNorm v ≤ r → sourceEuclideanNorm v ≤ 1 ∧ 0 < F a v) ∧
+      (∀ v : Coord, sourceEuclideanNorm (v-tau • Pi.single j 1) ≤ r →
+        sourceEuclideanNorm v ≤ 1 ∧ F a v < 0)} := by
+  have hh := event_inter_gradient_half hG hG0 hL hsign (by linarith : 2*(∫ a, G a ∂μ) ≤ p*L)
+  apply hh.trans
+  apply measureReal_mono
+  · intro a ha
+    exact strict_sign_subballs (F a) (hF a) ht hr hL.le htr hsmall j ha.1.1 ha.1.2
+      (hgrad a ha.2)
+  · exact measure_ne_top _ _
+
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/SmoothWaveOperator.lean b/lean/OAI/Geometry/NodalSets/Waves/SmoothWaveOperator.lean
new file mode 100644
index 0000000000000000000000000000000000000000..b41da71e2b4c096b6f76cee17d96ea544d93c78b
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/SmoothWaveOperator.lean
@@ -0,0 +1,155 @@
+import Mathlib.Analysis.SpecialFunctions.ExpDeriv
+import OAI.Geometry.NodalSets.Elliptic.SmoothBeamJet
+
+namespace OAI
+
+namespace Yau.Jets
+open scoped ContDiff
+noncomputable section
+
+lemma coordPartial_add {f u : Coord → ℂ} (hf : ContDiff ℝ ∞ f) (hu : ContDiff ℝ ∞ u)
+    (i : Fin 4) : coordPartial i (fun x ↦ f x + u x) =
+      fun x ↦ coordPartial i f x + coordPartial i u x := by
+  funext x
+  unfold coordPartial
+  rw [fderiv_fun_add (hf.differentiable (by simp) x) (hu.differentiable (by simp) x)]
+  rfl
+
+lemma coordPartial_mul {f u : Coord → ℂ} (hf : ContDiff ℝ ∞ f) (hu : ContDiff ℝ ∞ u)
+    (i : Fin 4) : coordPartial i (fun x ↦ f x * u x) =
+      fun x ↦ coordPartial i f x * u x + f x * coordPartial i u x := by
+  funext x
+  unfold coordPartial
+  rw [fderiv_fun_mul (hf.differentiable (by simp) x) (hu.differentiable (by simp) x)]
+  simp [smul_eq_mul]
+  ring
+
+lemma coordPartial_const_mul {f : Coord → ℂ} (hf : ContDiff ℝ ∞ f) (a : ℂ) (i : Fin 4) :
+    coordPartial i (fun x ↦ a * f x) = fun x ↦ a * coordPartial i f x := by
+  rw [coordPartial_mul contDiff_const hf]
+  ext x
+  simp [coordPartial]
+
+lemma coordPartial_exp {f : Coord → ℂ} (hf : ContDiff ℝ ∞ f) (i : Fin 4) :
+    coordPartial i (fun x ↦ Complex.exp (f x)) =
+      fun x ↦ coordPartial i f x * Complex.exp (f x) := by
+  funext x
+  unfold coordPartial
+  rw [((hf.differentiable (by simp) x).hasFDerivAt.cexp).fderiv]
+  simp [smul_eq_mul, mul_comm]
+
+lemma coordPartial_twice_mul {f u : Coord → ℂ} (hf : ContDiff ℝ ∞ f) (hu : ContDiff ℝ ∞ u)
+    (i j : Fin 4) : coordPartial i (coordPartial j (fun x ↦ f x * u x)) = fun x ↦
+      coordPartial i (coordPartial j f) x * u x +
+      coordPartial j f x * coordPartial i u x +
+      coordPartial i f x * coordPartial j u x +
+      f x * coordPartial i (coordPartial j u) x := by
+  rw [coordPartial_mul hf hu j,
+    coordPartial_add ((coordPartial_contDiff hf j).mul hu) (hf.mul (coordPartial_contDiff hu j)),
+    coordPartial_mul (coordPartial_contDiff hf j) hu i,
+    coordPartial_mul hf (coordPartial_contDiff hu j) i]
+  ext x
+  ring
+
+theorem smoothSecondOrder_mul (g : Fin 4 → Fin 4 → Coord → ℂ) (b : Fin 4 → Coord → ℂ)
+    {f u : Coord → ℂ} (hf : ContDiff ℝ ∞ f) (hu : ContDiff ℝ ∞ u) (x : Coord) :
+    smoothSecondOrder g b (fun z ↦ f z * u z) x =
+      f x * smoothSecondOrder g b u x + u x * smoothSecondOrder g b f x +
+      ∑ i, ∑ j, g i j x *
+        (coordPartial i f x * coordPartial j u x + coordPartial j f x * coordPartial i u x) := by
+  simp only [smoothSecondOrder]
+  simp_rw [coordPartial_twice_mul hf hu]
+  simp only [coordPartial_mul hf hu,
+    mul_add, Finset.sum_add_distrib, Finset.mul_sum]
+  simp only [mul_left_comm, mul_comm]
+  ring
+
+def waveExp (phi : Coord → ℂ) (N : ℝ) : Coord → ℂ :=
+  fun x ↦ Complex.exp ((N : ℂ) * phi x)
+
+lemma waveExp_contDiff {phi : Coord → ℂ} (hp : ContDiff ℝ ∞ phi) (N : ℝ) :
+    ContDiff ℝ ∞ (waveExp phi N) := (contDiff_const.mul hp).cexp
+
+lemma coordPartial_waveExp {phi : Coord → ℂ} (hp : ContDiff ℝ ∞ phi) (N : ℝ) (i : Fin 4) :
+    coordPartial i (waveExp phi N) =
+      fun x ↦ (N : ℂ) * coordPartial i phi x * waveExp phi N x := by
+  unfold waveExp
+  rw [coordPartial_exp (contDiff_const.mul hp), coordPartial_const_mul hp]
+
+lemma coordPartial_twice_waveExp {phi : Coord → ℂ} (hp : ContDiff ℝ ∞ phi)
+    (N : ℝ) (i j : Fin 4) : coordPartial i (coordPartial j (waveExp phi N)) =
+      fun x ↦ ((N : ℂ) * coordPartial i (coordPartial j phi) x +
+        (N : ℂ) ^ 2 * coordPartial i phi x * coordPartial j phi x) * waveExp phi N x := by
+  rw [coordPartial_waveExp hp N j,
+    coordPartial_mul (contDiff_const.mul (coordPartial_contDiff hp j)) (waveExp_contDiff hp N) i,
+    coordPartial_const_mul (coordPartial_contDiff hp j), coordPartial_waveExp hp N i]
+  ext x
+  ring
+
+theorem smoothSecondOrder_waveExp (g : Fin 4 → Fin 4 → Coord → ℂ)
+    (b : Fin 4 → Coord → ℂ) {phi : Coord → ℂ} (hp : ContDiff ℝ ∞ phi)
+    (N : ℝ) (x : Coord) :
+    smoothSecondOrder g b (waveExp phi N) x =
+      ((N : ℂ) * smoothSecondOrder g b phi x +
+        (N : ℂ) ^ 2 * (smoothEikonal g phi x - 4)) * waveExp phi N x := by
+  simp only [smoothSecondOrder]
+  simp_rw [coordPartial_twice_waveExp hp N]
+  simp only [coordPartial_waveExp hp N, smoothEikonal, add_sub_cancel_right,
+    mul_add, add_mul, Finset.mul_sum, Finset.sum_mul, Finset.sum_add_distrib]
+  simp only [mul_assoc, mul_left_comm]
+  ring
+
+lemma symmetric_cross_sum (G : Fin 4 → Fin 4 → ℂ) (hG : ∀ i j, G i j = G j i)
+    (a q : Fin 4 → ℂ) :
+    (∑ i, ∑ j, G i j * (a i * q j + a j * q i)) =
+      ∑ i, (2 * ∑ j, G i j * q j) * a i := by
+  have hs : (∑ i, ∑ j, G i j * a j * q i) = ∑ i, ∑ j, G i j * a i * q j := by
+    rw [Finset.sum_comm]
+    apply Finset.sum_congr rfl
+    intro i _
+    apply Finset.sum_congr rfl
+    intro j _
+    rw [hG j i]
+  calc
+    _ = (∑ i, ∑ j, G i j * a i * q j) + (∑ i, ∑ j, G i j * a j * q i) := by
+      simp only [mul_add, ← mul_assoc, Finset.sum_add_distrib]
+    _ = 2 * (∑ i, ∑ j, G i j * a i * q j) := by rw [hs]; ring
+    _ = _ := by
+      simp only [Finset.mul_sum, Finset.sum_mul]
+      apply Finset.sum_congr rfl
+      intro i _
+      apply Finset.sum_congr rfl
+      intro j _
+      ring
+
+theorem smooth_wave_residual_identity (g : Fin 4 → Fin 4 → Coord → ℂ)
+    (b : Fin 4 → Coord → ℂ) (hg : ∀ i j x, g i j x = g j i x)
+    {phi a : Coord → ℂ} (hp : ContDiff ℝ ∞ phi) (ha : ContDiff ℝ ∞ a)
+    (N : ℝ) (x : Coord) :
+    smoothSecondOrder g b (fun z ↦ a z * waveExp phi N z) x +
+        ((4 : ℂ) * (N : ℂ) ^ 2 + 6 * (N : ℂ)) * (a x * waveExp phi N x) =
+      waveExp phi N x *
+        ((N : ℂ) ^ 2 * smoothEikonal g phi x * a x +
+          (N : ℂ) * smoothTransport (smoothBeamVector g phi) (smoothBeamScalar g b phi)
+            (fun _ ↦ 0) a x + smoothSecondOrder g b a x) := by
+  rw [smoothSecondOrder_mul g b ha (waveExp_contDiff hp N), smoothSecondOrder_waveExp g b hp,
+    symmetric_cross_sum (fun i j ↦ g i j x) (fun i j ↦ hg i j x)]
+  simp only [coordPartial_waveExp hp N, smoothTransport, smoothBeamVector, smoothBeamScalar,
+    sub_zero]
+  have he : (∑ i, (2 * ∑ j, g i j x * ((N : ℂ) * coordPartial j phi x * waveExp phi N x)) *
+      coordPartial i a x) =
+      (N : ℂ) * waveExp phi N x *
+        (∑ i, (2 * ∑ j, g i j x * coordPartial j phi x) * coordPartial i a x) := by
+    simp only [Finset.mul_sum, Finset.sum_mul]
+    apply Finset.sum_congr rfl
+    intro i _
+    apply Finset.sum_congr rfl
+    intro j _
+    ring
+  rw [he]
+  ring
+
+end
+end Yau.Jets
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/TripleSourceWaves.lean b/lean/OAI/Geometry/NodalSets/Waves/TripleSourceWaves.lean
new file mode 100644
index 0000000000000000000000000000000000000000..ef8f31c83335f0cd6f1be0072292a247ca730e23
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/TripleSourceWaves.lean
@@ -0,0 +1,99 @@
+import OAI.Geometry.NodalSets.Waves.TripleWaveParameters
+import OAI.Geometry.NodalSets.Waves.WitnessedEuclideanWaves
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Filter Set
+open scoped ContDiff Topology
+noncomputable section
+attribute [local instance] clmTopology clmAdd clmModule
+variable {T : Type*} [TopologicalSpace T] [CompactSpace T]
+
+abbrev SourceFrameTriple (g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ) (S : Coord → ℝ)
+    (y : T → Coord) := AdmissibleFrameTriple (g ∘ y) (sourceHessian g S ∘ y) (metricGradient g S ∘ y)
+
+def sourceTripleAlpha (g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ) (S : Coord → ℝ)
+    (y : T → Coord) (z : T × Fin 3) : ℝ :=
+  Real.sqrt (g (y z.1) (metricGradient g S (y z.1)) (metricGradient g S (y z.1)))
+
+def sourceTripleBeta (g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ) (S : Coord → ℝ)
+    (y : T → Coord) (z : T × Fin 3) : ℝ :=
+  Real.sqrt (g (y z.1) (metricGradient g S (y z.1)) (metricGradient g S (y z.1))+4)
+
+structure TripleSourceWaveData
+    (g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ) (w S : Coord → ℝ)
+    (y : T → Coord) (d : SourceFrameTriple g S y) (m J K k0 : ℕ) where
+  beta : ContDiffBump (0:Coord)
+  phi : T × Fin 3 → CPoly
+  A : ℕ → T × Fin 3 → CPoly
+  F : T × Fin 3 → OpenPartialHomeomorph Coord Coord
+  c : ℝ
+  Cw : ℝ
+  Cr : ℝ
+  R : ℝ
+  c_pos : 0 < c
+  Cw_pos : 0 < Cw
+  Cr_pos : 0 < Cr
+  R_pos : 0 < R
+  sourceRadius : ℝ
+  targetRadius : ℝ
+  sourceRadius_pos : 0 < sourceRadius
+  targetRadius_pos : 0 < targetRadius
+  jets : MetricWaveJetFamily (extendedPrincipal g (fun z ↦ y z.1) d.frame beta)
+    (extendedDrift g w (fun z ↦ y z.1) d.frame beta)
+    (fun z ↦ S (y z.1)) (sourceTripleAlpha g S y) (sourceTripleBeta g S y)
+    (fun z ↦ envelopeHessian (S ∘ actualMetricChart g (y z.1) (d.frame z))) m J phi A
+  chart_eq : ∀ z, (F z : Coord → Coord) = actualMetricChart g (y z.1) (d.frame z)
+  chart_domain : ∀ z, (F z).source = Metric.ball 0 sourceRadius ∧
+    Metric.closedBall (y z.1) targetRadius ⊆ (F z).target ∧
+    ContDiffOn ℝ ∞ (F z).symm (F z).target
+  estimates : ∀ᶠ n : ℕ in atTop, ∀ t,
+    let V := chartPushforward (F t) (coordinateWave phi A J (n:ℝ) t)
+    ContDiff ℝ ∞ V ∧ HasCompactSupport V ∧
+    tsupport V ⊆ {z | sourceEuclideanNorm (z-y t.1) ≤ R*(n:ℝ)^(-1/3:ℝ)} ∧
+    ∀ z : Coord,
+      (∀ k : Fin (k0+1), ‖iteratedFDeriv ℝ k.val V z‖ ≤
+        Cw*(n:ℝ)^k.val*Real.exp ((n:ℝ)*S z-c*(n:ℝ)*(sourceEuclideanNorm (z-y t.1))^2)) ∧
+      DerivativeBound k0 (fun z ↦ sourceWeightedOperator g w V z +
+        ((4:ℂ)*(n:ℂ)^2+6*(n:ℂ))*V z) z
+          (Cr*(n:ℝ)^(-(K:ℝ))*Real.exp ((n:ℝ)*S z))
+
+theorem construct_triple_source_waves
+    (g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ) (hg : ContDiff ℝ ∞ g)
+    (hs : ∀ x u v, g x u v = g x v u) (hpos : ∀ x v, v ≠ 0 → 0 < g x v v)
+    (w : Coord → ℝ) (hw : ContDiff ℝ ∞ w) (S : Coord → ℝ) (hS : ContDiff ℝ ∞ S)
+    (y : T → Coord) (hy : Continuous y) (hp0 : ∀ t, metricGradient g S (y t) ≠ 0)
+    (d : SourceFrameTriple g S y) {U : Set Coord} (hU : IsOpen U)
+    (hyU : ∀ t, y t ∈ U) (hwpos : ∀ x ∈ U, 0 < w x)
+    (m J K k0 : ℕ) (hm : 3*K+4*k0+6 < m+1) (hJ : K+k0+1 ≤ J) :
+    Nonempty (TripleSourceWaveData g w S y d m J K k0) := by
+  have hp : Continuous (fun z : T × Fin 3 ↦ metricGradient g S (y z.1)) := (metricGradient_continuous g hg hpos S hS).comp (hy.comp continuous_fst)
+  have hlen : Continuous (fun z : T × Fin 3 ↦
+      g (y z.1) (metricGradient g S (y z.1)) (metricGradient g S (y z.1))) :=
+    (((hg.continuous.comp (hy.comp continuous_fst)).clm_apply hp).clm_apply hp)
+  have hapos (z : T × Fin 3) : 0 < sourceTripleAlpha g S y z :=
+    Real.sqrt_pos.mpr (hpos (y z.1) _ (hp0 z.1))
+  have hbpos (z : T × Fin 3) : 0 < sourceTripleBeta g S y z := by
+    apply Real.sqrt_pos.mpr
+    have h := hpos (y z.1) _ (hp0 z.1)
+    linarith
+  have hab (z : T × Fin 3) : (sourceTripleBeta g S y z)^2 = (sourceTripleAlpha g S y z)^2+4 := by
+    dsimp [sourceTripleAlpha,sourceTripleBeta]
+    rw [Real.sq_sqrt (hpos (y z.1) _ (hp0 z.1)).le,Real.sq_sqrt]
+    have h := hpos (y z.1) _ (hp0 z.1)
+    linarith
+  obtain ⟨beta,phi,A,F,c,Cw,Cr,R,r,delta,hc,hCw,hCr,hR,hr,hdelta,hjets,hF,hdom,hest⟩ :=
+    witnessed_source_wave_estimates_euclidean g hg hs hpos w hw S hS
+      (fun z : T × Fin 3 ↦ y z.1) (hy.comp continuous_fst) d.frame d.continuous_frame
+      d.frame_orthonormal hU (fun z ↦ hyU z.1) hwpos (fun z ↦ d.q z.1 z.2)
+      (sourceTripleAlpha g S y) (sourceTripleBeta g S y) hlen.sqrt (hlen.add continuous_const).sqrt
+      (fun z ↦ ne_of_gt (hapos z)) (fun z ↦ ne_of_gt (hbpos z)) hab
+      (d.frame_first (fun t ↦ hpos (y t)) hp0) (d.frame_second (fun t ↦ hpos (y t)) hp0)
+      (fun z ↦ d.strict z.1 z.2) m J K k0 hm hJ
+  exact ⟨⟨beta,phi,A,F,c,Cw,Cr,R,hc,hCw,hCr,hR,r,delta,hr,hdelta,hjets,hF,hdom,hest⟩⟩
+
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/TripleWaveParameters.lean b/lean/OAI/Geometry/NodalSets/Waves/TripleWaveParameters.lean
new file mode 100644
index 0000000000000000000000000000000000000000..3f04eddcfbef2197a5a41190d0ae5183c937fa06
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/TripleWaveParameters.lean
@@ -0,0 +1,46 @@
+import OAI.Geometry.NodalSets.Charts.LocalFrameTriple
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets
+noncomputable section
+attribute [local instance] clmTopology clmAdd clmModule
+variable {T : Type*} [TopologicalSpace T]
+variable {g H : T → Coord →L[ℝ] Coord →L[ℝ] ℝ} {p : T → Coord}
+namespace AdmissibleFrameTriple
+variable (d : AdmissibleFrameTriple g H p)
+def frame (z : T × Fin 3) : Coord ≃L[ℝ] Coord :=
+  frameEquiv (g z.1) (d.e z.1 z.2) (d.orthonormal z.1 z.2)
+def alpha (t : T) : ℝ := Real.sqrt (g t (p t) (p t))
+def beta (t : T) : ℝ := Real.sqrt (g t (p t) (p t)+4)
+
+lemma continuous_frame : Continuous (fun z ↦ (d.frame z).toContinuousLinearMap) := by
+  apply continuous_prod_of_discrete_right.mpr
+  intro j
+  exact frameEquiv_continuous g (fun t ↦ d.e t j) (d.continuous_e j) (fun t ↦ d.orthonormal t j)
+
+lemma frame_orthonormal (z : T × Fin 3) (i j : Fin 4) :
+    g z.1 (d.frame z (Pi.single i 1)) (d.frame z (Pi.single j 1)) = if i=j then 1 else 0 := by
+  change g z.1 (frameMap (d.e z.1 z.2) (Pi.single i 1))
+    (frameMap (d.e z.1 z.2) (Pi.single j 1)) = _
+  simp only [frameMap_axis,d.orthonormal]
+
+lemma frame_first (hpos : ∀ t v, v ≠ 0 → 0 < g t v v) (hp0 : ∀ t, p t ≠ 0)
+    (z : T × Fin 3) : alpha (g := g) (p := p) z.1 • d.frame z (Pi.single 0 1) = p z.1 := by
+  change alpha z.1 • frameMap (d.e z.1 z.2) (Pi.single 0 1) = _
+  rw [frameMap_axis,d.first]
+  simp [alpha,metricNormalize,smul_smul,ne_of_gt (Real.sqrt_pos.mpr (hpos z.1 _ (hp0 z.1)))]
+
+lemma frame_second (hpos : ∀ t v, v ≠ 0 → 0 < g t v v) (hp0 : ∀ t, p t ≠ 0)
+    (z : T × Fin 3) : beta (g := g) (p := p) z.1 • d.frame z (Pi.single 1 1) = d.q z.1 z.2 := by
+  have hb : 0 < g z.1 (p z.1) (p z.1)+4 := by have h := hpos z.1 _ (hp0 z.1); linarith
+  change beta z.1 • frameMap (d.e z.1 z.2) (Pi.single 1 1) = _
+  rw [frameMap_axis,d.second]
+  simp [beta,metricNormalize,d.length,smul_smul,ne_of_gt (Real.sqrt_pos.mpr hb)]
+end AdmissibleFrameTriple
+
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/TwoValueGaussianMap.lean b/lean/OAI/Geometry/NodalSets/Waves/TwoValueGaussianMap.lean
new file mode 100644
index 0000000000000000000000000000000000000000..8350b6413bd8c889683245230506f946ff06b967
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/TwoValueGaussianMap.lean
@@ -0,0 +1,42 @@
+import OAI.Geometry.NodalSets.Coefficients.TwoPointCoefficientEnergy
+import OAI.Geometry.NodalSets.Waves.FiniteGaussianMap
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Probability MeasureTheory ProbabilityTheory
+open scoped RealInnerProductSpace
+noncomputable section
+variable {ι : Type*} [Fintype ι]
+
+lemma pairJetMap_two_inner (z w : ι → ℂ) (v : EuclideanSpace ℝ (Fin 2))
+    (a : ι × Fin 2 → ℝ) :
+    ⟪v,pairJetMap ![z,w] (WithLp.toLp 2 a)⟫ =
+      pairLinearSum (fun i ↦ (v 0:ℂ)*z i+(v 1:ℂ)*w i) a := by
+  rw [EuclideanSpace.inner_eq_star_dotProduct]
+  simp only [dotProduct,star_trivial,Fin.sum_univ_two,pairJetMap_apply,
+    Matrix.cons_val_zero,Matrix.cons_val_one,pairLinearSum_real_combination]
+  ring
+
+lemma pairJetMap_two_energy (z w : ι → ℂ) (v : EuclideanSpace ℝ (Fin 2)) :
+    ‖(pairJetMap ![z,w]).adjoint v‖^2 =
+      ∑ i, ‖(v 0:ℂ)*z i+(v 1:ℂ)*w i‖^2 := by
+  rw [← variance_gaussianMap_inner]
+  simp_rw [pairJetMap_two_inner]
+  exact variance_pairLinearSum _
+
+lemma two_point_energy_upper (z w : ι → ℂ) (M : ℝ)
+    (hz : ∑ i, ‖z i‖^2 ≤ M) (hw : ∑ i, ‖w i‖^2 ≤ M) (u v : ℝ) :
+    (∑ i, ‖(u:ℂ)*z i+(v:ℂ)*w i‖^2) ≤ 2*M*(u^2+v^2) := by
+  have hh := Finset.sum_le_sum (fun i (_ : i ∈ (Finset.univ : Finset ι)) ↦
+    complex_norm_add_sq_bound ((u:ℂ)*z i) ((v:ℂ)*w i))
+  simp only [norm_mul,mul_pow,Complex.norm_real,Real.norm_eq_abs,sq_abs,
+    Finset.sum_add_distrib,← Finset.mul_sum] at hh
+  have hz' := mul_le_mul_of_nonneg_left hz (sq_nonneg u)
+  have hw' := mul_le_mul_of_nonneg_left hw (sq_nonneg v)
+  nlinarith
+
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/TwoValueSignProbability.lean b/lean/OAI/Geometry/NodalSets/Waves/TwoValueSignProbability.lean
new file mode 100644
index 0000000000000000000000000000000000000000..6fcd0532846d66d5d076ca8fdb85e7c20c852716
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/TwoValueSignProbability.lean
@@ -0,0 +1,60 @@
+import OAI.Geometry.NodalSets.Waves.GaussianFactorTwoSided
+import OAI.Geometry.NodalSets.Waves.GaussianUniformBall
+import OAI.Geometry.NodalSets.Waves.TwoValueGaussianMap
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Probability MeasureTheory ProbabilityTheory Set Metric
+open scoped ENNReal RealInnerProductSpace
+noncomputable section
+local notation "E" => EuclideanSpace ℝ (Fin 2)
+
+lemma two_value_sign_probability (d M : ℝ) (hd : 0 < d) (hM : 0 ≤ M) :
+    ∃ p : ℝ≥0∞, 0 < p ∧ ∀ {ι : Type} [Fintype ι] (z w : ι → ℂ) (mean : E),
+      ‖mean‖ ≤ M → (∑ i, ‖z i‖^2) ≤ 4 → (∑ i, ‖w i‖^2) ≤ 4 →
+      (∀ u v : ℝ, d*(u^2+v^2) ≤ ∑ i, ‖(u:ℂ)*z i+(v:ℂ)*w i‖^2) →
+      p ≤ gaussianPairs {a | 1 ≤ mean 0+pairLinearSum z a ∧ mean 1+pairLinearSum w a ≤ -1} := by
+  let target : E := WithLp.toLp 2 ![2,-2]
+  obtain ⟨p,hp,hball⟩ := uniform_affine_gaussian_ball (ι := Fin 2) (Real.sqrt d)⁻¹ 4 M 1
+    (by positivity) (by norm_num) hM (by norm_num) target
+  refine ⟨p,hp,?_⟩
+  intro ι hfin z w mean hm hz hw hgap
+  let T := pairJetMap ![z,w]
+  have hlo (v : E) : d*‖v‖^2 ≤ ‖T.adjoint v‖^2 := by
+    rw [pairJetMap_two_energy]
+    simpa only [EuclideanSpace.real_norm_sq_eq,Fin.sum_univ_two] using hgap (v 0) (v 1)
+  have hhi (v : E) : ‖T.adjoint v‖^2 ≤ (4:ℝ)^2*‖v‖^2 := by
+    rw [pairJetMap_two_energy]
+    have hh := two_point_energy_upper z w 4 hz hw (v 0) (v 1)
+    have hn : ‖v‖^2 = (v 0)^2+(v 1)^2 := by rw [EuclideanSpace.real_norm_sq_eq,Fin.sum_univ_two]
+    rw [hn]
+    nlinarith [sq_nonneg (v 0),sq_nonneg (v 1)]
+  obtain ⟨L,hLaw,hLi,hL⟩ := exists_gaussian_factor_two_sided T d 4 hd (by norm_num) hlo hhi
+  have hmap : gaussianPairs.map (fun a ↦ mean+T (WithLp.toLp 2 a)) =
+      (stdGaussian E).map (fun v ↦ mean+L v) := by
+    have hh := congrArg (fun μ : Measure E ↦ μ.map (fun v ↦ mean+v)) (pairJetMap_law ![z,w])
+    rw [hLaw] at hh
+    rw [Measure.map_map (by fun_prop) (by fun_prop),
+      Measure.map_map (by fun_prop) (by fun_prop)] at hh
+    exact hh
+  have hb := hball L mean hLi hL hm
+  rw [← hmap,Measure.map_apply (by fun_prop) isOpen_ball.measurableSet] at hb
+  apply hb.trans (measure_mono ?_)
+  intro a ha
+  let y := mean+T (WithLp.toLp 2 a)
+  have hn : ‖y-target‖ < 1 := by simpa only [mem_preimage,mem_ball,dist_eq_norm] using ha
+  have h0 := (PiLp.norm_apply_le (y-target) 0).trans_lt hn
+  have h1 := (PiLp.norm_apply_le (y-target) 1).trans_lt hn
+  change |y 0-2| < 1 at h0
+  change |y 1-(-2)| < 1 at h1
+  have hy0 : y 0 = mean 0+pairLinearSum z a := by simp [y,T,pairJetMap_apply]
+  have hy1 : y 1 = mean 1+pairLinearSum w a := by simp [y,T,pairJetMap_apply]
+  change 1 ≤ mean 0+pairLinearSum z a ∧ mean 1+pairLinearSum w a ≤ -1
+  rw [← hy0,← hy1]
+  exact ⟨by linarith [(abs_lt.mp h0).1],by linarith [(abs_lt.mp h1).2]⟩
+
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/UniformActualWaveSize.lean b/lean/OAI/Geometry/NodalSets/Waves/UniformActualWaveSize.lean
new file mode 100644
index 0000000000000000000000000000000000000000..239494ac367f2d7035a2bf75a2a94ca3f9a67216
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/UniformActualWaveSize.lean
@@ -0,0 +1,64 @@
+import OAI.Geometry.NodalSets.Coefficients.MetricWaveJets
+
+namespace OAI
+
+namespace Yau.Jets
+open Filter Set
+open scoped ContDiff Topology
+noncomputable section
+variable {T : Type*} [TopologicalSpace T]
+
+lemma cutoff_wave_derivative_zero_outside (u : Coord → ℂ) {N : ℝ} (hN : 0 < N)
+    (x : Coord) (k : ℕ) (hx : 2*N^(-1/3:ℝ) < ‖x‖) :
+    iteratedFDeriv ℝ k (fun z ↦ Yau.Waves.scaledCutoff N (0:Coord) z • u z) x = 0 := by
+  have hnear : ∀ᶠ z in 𝓝 x, 2*N^(-1/3:ℝ) < ‖z‖ :=
+    (isOpen_lt continuous_const continuous_norm).mem_nhds hx
+  have he : (fun z ↦ Yau.Waves.scaledCutoff N (0:Coord) z • u z) =ᶠ[𝓝 x] fun _ ↦ (0:ℂ) := by
+    filter_upwards [hnear] with z hz
+    have hb : Yau.Waves.scaledCutoff N (0:Coord) z = 0 := by
+      by_contra hn
+      have hm : z ∈ Function.support (Yau.Waves.scaledCutoff N (0:Coord)) := hn
+      rw [Yau.Waves.scaledCutoff_support hN] at hm
+      have hlt : ‖z‖ < 2*N^(-1/3:ℝ) := by simpa [Metric.mem_ball,dist_eq_norm] using hm
+      linarith
+    simp [hb]
+  simpa using (he.iteratedFDeriv ℝ k).self_of_nhds
+
+theorem uniform_cutoff_wave_size {s : Set T} (hs : IsCompact s)
+    (phi : T → CPoly) (A : ℕ → T → CPoly)
+    (hp : ContinuousPolyFamily phi) (hA : ∀ j, ContinuousPolyFamily (A j))
+    (J k : ℕ) (S : T → Coord → ℝ) (c : ℝ) (hc : 0 < c)
+    (hgap : ∀ᶠ N : ℝ in atTop, ∀ t ∈ s, ∀ x : Coord, ‖x‖ ≤ 2*N^(-1/3:ℝ) →
+      (reval (phi t) x).re-S t x ≤ -c*‖x‖^2) :
+    ∃ C > 0, ∀ᶠ N : ℝ in atTop, ∀ t ∈ s, ∀ x : Coord,
+      ‖iteratedFDeriv ℝ k (fun z ↦ Yau.Waves.scaledCutoff N (0:Coord) z •
+        (finiteAmplitude (fun j ↦ A j t) J N z * waveExp (reval (phi t)) N z)) x‖ ≤
+        C*N^k*Real.exp (N*S t x-c*N*‖x‖^2) := by
+  obtain ⟨Ca,hCa,hamp⟩ := finiteAmplitude_all_parameter_derivatives A hA J k s hs 2
+  obtain ⟨Cp,hCp,hphase⟩ := (hp.uniformSmoothBounded s hs 2).2 k
+  obtain ⟨B,hB,hsize⟩ := cutoff_wave_size k
+  let D := max Cp (1+c)
+  have hD : 1 ≤ D := (le_add_of_nonneg_right hc.le).trans (le_max_right _ _)
+  refine ⟨B*Ca*(k.factorial:ℝ)*D^k,by positivity,?_⟩
+  filter_upwards [hgap,eventually_ge_atTop (1:ℝ)] with N hgN hN
+  intro t ht x
+  by_cases hx : ‖x‖ ≤ 2*N^(-1/3:ℝ)
+  · have hx2 : ‖x‖ ≤ 2 := hx.trans (by
+      have hh := Real.rpow_le_one_of_one_le_of_nonpos hN (show (-1/3:ℝ) ≤ 0 by norm_num)
+      linarith)
+    simpa only [sub_zero,mul_assoc] using hsize (reval (phi t))
+      (finiteAmplitude (fun j ↦ A j t) J N) (reval_contDiff _) (finiteAmplitude_contDiff _ _ _)
+      0 x N Ca D (S t x) c hN hCa.le hD (hamp t ht N hN x hx2)
+      (fun j _ hj ↦ (hphase t ht x hx2 j hj).trans (le_max_left _ _))
+      (by simpa only [sub_zero] using hgN t ht x hx)
+  · rw [cutoff_wave_derivative_zero_outside _ (by linarith) x k (lt_of_not_ge hx),norm_zero]
+    positivity
+
+theorem eventually_nat_frequency {P : ℝ → Prop} (h : ∀ᶠ N : ℝ in atTop, P N) :
+    ∀ᶠ n : ℕ in atTop, P (n:ℝ) :=
+  tendsto_natCast_atTop_atTop.eventually h
+
+end
+end Yau.Jets
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/UniformChartWaveTransfer.lean b/lean/OAI/Geometry/NodalSets/Waves/UniformChartWaveTransfer.lean
new file mode 100644
index 0000000000000000000000000000000000000000..2e0ef5c3318fbedf0b2c72bc270a02374f365b58
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/UniformChartWaveTransfer.lean
@@ -0,0 +1,159 @@
+import OAI.Geometry.NodalSets.Charts.ChartDerivativeTransfer
+import OAI.Geometry.NodalSets.Charts.ChartNormComparison
+import OAI.Geometry.NodalSets.Charts.UniformInverseChartBounds
+import OAI.Geometry.NodalSets.Coefficients.LocalCoordinateResidual
+import OAI.Geometry.NodalSets.Elliptic.OperatorLocality
+import OAI.Geometry.NodalSets.Waves.ConstructedCoordinateWaves
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Filter Set Metric
+open scoped ContDiff Topology
+noncomputable section
+variable {T : Type*} [TopologicalSpace T] [CompactSpace T]
+
+theorem uniform_chart_wave_transfer
+    (g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ) (hg : ContDiff ℝ ∞ g)
+    (hp : ∀ x v, v ≠ 0 → 0 < g x v v) (w : Coord → ℝ) (hw : ContDiff ℝ ∞ w)
+    (p : T → QuadParam Coord) (hpc : Continuous p)
+    (F : T → OpenPartialHomeomorph Coord Coord)
+    (hF : ∀ t, (F t : Coord → Coord) = rawQuadratic (p t))
+    (r : ℝ) (hr : 0 < r) (hsource : ∀ t, (F t).source = ball 0 r)
+    (hInv : ∀ t, ContDiffOn ℝ ∞ (F t).symm (F t).target)
+    (hJ : ∀ t, ∀ x ∈ (F t).source, ∃ J : Coord ≃L[ℝ] Coord,
+      fderiv ℝ (rawQuadratic (p t)) x = J.toContinuousLinearMap)
+    (hwp : ∀ t, ∀ z ∈ (F t).target, 0 < w z)
+    (S : Coord → ℝ) (u : ℕ → T → Coord → ℂ)
+    (hu : ∀ n t, ContDiff ℝ ∞ (u n t))
+    (hsupport : ∀ n, 0 < n → ∀ t, tsupport (u n t) ⊆ closedBall 0 (2*(n:ℝ)^(-1/3:ℝ)))
+    (k0 K : ℕ) (c Cr : ℝ) (Cs : Fin (k0+1) → ℝ)
+    (hc : 0 < c) (hCr : 0 < Cr) (hCs : ∀ k, 0 < Cs k)
+    (hest : ∀ᶠ n : ℕ in atTop, ∀ t x,
+      (∀ k : Fin (k0+1), ‖iteratedFDeriv ℝ k.val (u n t) x‖ ≤
+        Cs k*(n:ℝ)^k.val*Real.exp ((n:ℝ)*S (rawQuadratic (p t) x)-c*(n:ℝ)*‖x‖^2)) ∧
+      DerivativeBound k0 (fun z ↦ smoothSecondOrder (fun i j ↦ complexPrincipal g i j (p t))
+        (fun j ↦ complexDrift g w j (p t)) (u n t) z +
+          ((4:ℂ)*(n:ℂ)^2+6*(n:ℂ))*u n t z) x
+        (Cr*(n:ℝ)^(-(K:ℝ))*Real.exp ((n:ℝ)*S (rawQuadratic (p t) x)))) :
+    ∃ c' Cw Cr' R : ℝ, 0 < c' ∧ 0 < Cw ∧ 0 < Cr' ∧ 0 < R ∧
+      ∀ᶠ n : ℕ in atTop, ∀ t,
+        ContDiff ℝ ∞ (chartPushforward (F t) (u n t)) ∧
+        HasCompactSupport (chartPushforward (F t) (u n t)) ∧
+        tsupport (chartPushforward (F t) (u n t)) ⊆ closedBall (p t).1 (R*(n:ℝ)^(-1/3:ℝ)) ∧
+        ∀ z : Coord,
+          (∀ k : Fin (k0+1), ‖iteratedFDeriv ℝ k.val (chartPushforward (F t) (u n t)) z‖ ≤
+            Cw*(n:ℝ)^k.val*Real.exp ((n:ℝ)*S z-c'*(n:ℝ)*‖z-(p t).1‖^2)) ∧
+          DerivativeBound k0 (fun z ↦ sourceWeightedOperator g w (chartPushforward (F t) (u n t)) z +
+            ((4:ℂ)*(n:ℂ)^2+6*(n:ℂ))*chartPushforward (F t) (u n t) z) z
+              (Cr'*(n:ℝ)^(-(K:ℝ))*Real.exp ((n:ℝ)*S z)) := by
+  have hR (t : T) : closedBall (0:Coord) (r/2) ⊆ (F t).source := by
+    rw [hsource t]
+    exact closedBall_subset_ball (by linarith)
+  obtain ⟨D,hD,hDb⟩ := compact_inverse_derivative_bounds isCompact_univ p hpc F
+    (fun t _ ↦ hF t) (fun t _ ↦ hInv t) (fun t _ ↦ hJ t) (r/2) (fun t _ ↦ hR t) k0
+  obtain ⟨L,hL,hLb⟩ := compact_chart_norm_bound isCompact_univ p hpc (r/2) (by positivity)
+  let Ct : ℝ := ∑ k, Cs k
+  have hCt : 0 < Ct := Finset.sum_pos (fun k _ ↦ hCs k) Finset.univ_nonempty
+  have hCi (k : Fin (k0+1)) : Cs k ≤ Ct :=
+    Finset.single_le_sum (fun j _ ↦ (hCs j).le) (Finset.mem_univ k)
+  let Q : ℝ := (k0.factorial : ℝ)*D^k0
+  have hQ : 0 < Q := by dsimp [Q]; positivity
+  have hQk (k : ℕ) (hk : k ≤ k0) : (k.factorial:ℝ)*D^k ≤ Q := by
+    have hf : (k.factorial:ℝ) ≤ k0.factorial := by exact_mod_cast Nat.factorial_le hk
+    have hd := pow_le_pow_right₀ hD hk
+    dsimp [Q]
+    gcongr
+  refine ⟨c/L^2,Q*Ct,Q*Cr,2*L,by positivity,mul_pos hQ hCt,mul_pos hQ hCr,by positivity,?_⟩
+  have hsmall : ∀ᶠ n : ℕ in atTop, 2*(n:ℝ)^(-1/3:ℝ) < r/2 := by
+    apply eventually_nat_frequency (P := fun N : ℝ ↦ 2*N^(-1/3:ℝ) < r/2)
+    have ht : Tendsto (fun N : ℝ ↦ 2*N^(-1/3:ℝ)) atTop (𝓝 0) := by
+      simpa only [neg_div,mul_zero] using
+        (tendsto_rpow_neg_atTop (by norm_num : (0:ℝ) < 1/3)).const_mul 2
+    exact ht.eventually (gt_mem_nhds (by positivity))
+  filter_upwards [hest,hsmall,eventually_ge_atTop (1:ℕ)] with n hn hsmall hn1
+  have hnpos : 0 < n := by omega
+  have hN : (1:ℝ) ≤ n := by exact_mod_cast hn1
+  intro t
+  have hus := hsupport n hnpos t
+  have huR : tsupport (u n t) ⊆ closedBall (0:Coord) (r/2) :=
+    hus.trans (closedBall_subset_closedBall hsmall.le)
+  have huSource := huR.trans (hR t)
+  have huComp : HasCompactSupport (u n t) :=
+    (isCompact_closedBall (0:Coord) (r/2)).of_isClosed_subset isClosed_closure huR
+  obtain ⟨hVs,hVc,hVsup⟩ := chartPushforward_smooth (F t) (hInv t) (u n t) (hu n t) huComp huSource
+  refine ⟨hVs,hVc,?_,?_⟩
+  · intro z hz
+    obtain ⟨x,hx,rfl⟩ := hVsup hz
+    have hnorm := hLb t (mem_univ t) x (huR hx)
+    have hxN : ‖x‖ ≤ 2*(n:ℝ)^(-1/3:ℝ) := by simpa using hus hx
+    rw [mem_closedBall,dist_eq_norm,hF t]
+    calc
+      _ ≤ L*‖x‖ := hnorm
+      _ ≤ L*(2*(n:ℝ)^(-1/3:ℝ)) := mul_le_mul_of_nonneg_left hxN (by positivity)
+      _ = _ := by ring
+  · intro z
+    by_cases hz : z ∈ tsupport (chartPushforward (F t) (u n t))
+    · obtain ⟨x,hx,rfl⟩ := hVsup hz
+      have hxS := huSource hx
+      have hxt := (F t).map_source hxS
+      have hinv : (F t).symm (F t x) = x := (F t).left_inv hxS
+      have hFx : rawQuadratic (p t) x = F t x := congrFun (hF t).symm x
+      have hdi (i : ℕ) (hi : 1 ≤ i) (hik : i ≤ k0) := hDb t (mem_univ t) x (huR hx) i hi hik
+      have hnorm : ‖F t x-(p t).1‖ ≤ L*‖x‖ := by rw [hF t]; exact hLb t (mem_univ t) x (huR hx)
+      have hgauss : Real.exp ((n:ℝ)*S (rawQuadratic (p t) x)-c*(n:ℝ)*‖x‖^2) ≤
+          Real.exp ((n:ℝ)*S (F t x)-(c/L^2)*(n:ℝ)*‖F t x-(p t).1‖^2) := by
+        apply Real.exp_le_exp.mpr
+        rw [hFx]
+        have hh := gaussian_chart_norm_bound c (n:ℝ) L ‖x‖ ‖F t x-(p t).1‖ hc.le
+          (by positivity) (by positivity) (norm_nonneg _) (norm_nonneg _) hnorm
+        linarith
+      refine ⟨?_,?_⟩
+      · intro k
+        have hk : k.val ≤ k0 := by omega
+        have hb := chart_wave_size_transfer (F t) (hInv t) (u n t) (hu n t) (F t x) hxt k.val
+          Ct D (Real.exp ((n:ℝ)*S (rawQuadratic (p t) x)-c*(n:ℝ)*‖x‖^2)) (n:ℝ)
+          hCt.le (by positivity) hN (fun i hi ↦ by
+            rw [hinv]
+            have hh := (hn t x).1 ⟨i,by omega⟩
+            exact hh.trans (by gcongr; exact hCi ⟨i,by omega⟩))
+          (fun i hi hik ↦ hdi i hi (hik.trans hk))
+        apply hb.trans
+        have hq := hQk k.val hk
+        calc
+          _ = ((k.val.factorial:ℝ)*D^k.val)*Ct*(n:ℝ)^k.val*
+              Real.exp ((n:ℝ)*S (rawQuadratic (p t) x)-c*(n:ℝ)*‖x‖^2) := by ring
+          _ ≤ _ := by gcongr
+      · intro k hk
+        rw [transported_residual_derivative g hg hp w hw (p t) (F t) (hF t) (hInv t) (hJ t)
+          (hwp t) (u n t) (hu n t) huComp huSource _ (F t x) hxt k]
+        have hrs := actual_coordinate_residual_smoothOn g hg hp w hw (p t) (F t).source (hJ t)
+          (fun x hx ↦ by rw [← hF t]; exact hwp t (F t x) ((F t).map_source hx)) (u n t) (hu n t)
+          ((4:ℂ)*(n:ℂ)^2+6*(n:ℂ))
+        have hb := chart_residual_bound_transfer (F t) (hInv t) _ hrs (F t x) hxt k
+          Cr D (Real.exp ((n:ℝ)*S (rawQuadratic (p t) x))) (n:ℝ) (K:ℝ)
+          (fun i hi ↦ by rw [hinv]; exact (hn t x).2 i (hi.trans hk))
+          (fun i hi hik ↦ hdi i hi (hik.trans hk))
+        apply hb.trans
+        rw [hFx]
+        have hq := hQk k hk
+        calc
+          _ = ((k.factorial:ℝ)*D^k)*Cr*(n:ℝ)^(-(K:ℝ))*Real.exp ((n:ℝ)*S (F t x)) := by ring
+          _ ≤ _ := by gcongr
+    · have he : chartPushforward (F t) (u n t) =ᶠ[𝓝 z] (fun _ ↦ 0) :=
+        notMem_tsupport_iff_eventuallyEq.mp hz
+      refine ⟨?_,?_⟩
+      · intro k
+        rw [(he.iteratedFDeriv ℝ k.val).self_of_nhds]
+        have hzder : iteratedFDeriv ℝ k.val (fun _ : Coord ↦ (0:ℂ)) z = 0 := by simp
+        rw [hzder,norm_zero]
+        positivity
+      · intro k _
+        rw [source_residual_derivative_zero g w _ _ z hz k]
+        simp only [norm_zero]
+        positivity
+
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/UniformWaveJet.lean b/lean/OAI/Geometry/NodalSets/Waves/UniformWaveJet.lean
new file mode 100644
index 0000000000000000000000000000000000000000..4f25b8770948d9a4297c635e9ffeed569b37f9fd
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/UniformWaveJet.lean
@@ -0,0 +1,179 @@
+import OAI.Geometry.NodalSets.Elliptic.SmoothBeamJet
+import OAI.Geometry.NodalSets.Elliptic.UniformEikonal
+import OAI.Geometry.NodalSets.Elliptic.UniformTransportPolynomial
+
+namespace OAI
+
+namespace Yau.Jets
+open MvPolynomial
+open scoped ContDiff
+noncomputable section
+variable {T : Type*} [TopologicalSpace T]
+
+lemma beamVector_family (g : T → Fin 4 → Fin 4 → CPoly) (phi : T → CPoly)
+    (hg : ∀ i j, ContinuousPolyFamily (fun t ↦ g t i j))
+    (hp : ContinuousPolyFamily phi) (i : Fin 4) :
+    ContinuousPolyFamily (fun t ↦ beamVector (g t) (phi t) i) := by
+  apply ContinuousPolyFamily.mul (.const _)
+  exact ContinuousPolyFamily.sum _ (fun j _ ↦ (hg i j).mul (hp.pderiv j))
+
+lemma secondOrder_family (g : T → Fin 4 → Fin 4 → CPoly)
+    (b : T → Fin 4 → CPoly) (phi : T → CPoly)
+    (hg : ∀ i j, ContinuousPolyFamily (fun t ↦ g t i j))
+    (hb : ∀ i, ContinuousPolyFamily (fun t ↦ b t i))
+    (hp : ContinuousPolyFamily phi) :
+    ContinuousPolyFamily (fun t ↦ polynomialSecondOrder (g t) (b t) (phi t)) := by
+  apply ContinuousPolyFamily.add
+  · apply ContinuousPolyFamily.sum
+    intro i _
+    exact ContinuousPolyFamily.sum _ (fun j _ ↦ (hg i j).mul ((hp.pderiv j).pderiv i))
+  · exact ContinuousPolyFamily.sum _ (fun i _ ↦ (hb i).mul (hp.pderiv i))
+
+lemma beamScalar_family (g : T → Fin 4 → Fin 4 → CPoly)
+    (b : T → Fin 4 → CPoly) (phi : T → CPoly)
+    (hg : ∀ i j, ContinuousPolyFamily (fun t ↦ g t i j))
+    (hb : ∀ i, ContinuousPolyFamily (fun t ↦ b t i))
+    (hp : ContinuousPolyFamily phi) :
+    ContinuousPolyFamily (fun t ↦ beamScalar (g t) (b t) (phi t)) :=
+  (secondOrder_family g b phi hg hb hp).add (.const _)
+
+theorem polynomial_wave_jets_continuous
+    (v : T → Fin 4 → ℂ) (hv : ∀ j, Continuous (fun t ↦ v t j))
+    (i : Fin 4) (hvi : ∀ t, v t i ≠ 0)
+    (g : T → Fin 4 → Fin 4 → CPoly) (b : T → Fin 4 → CPoly)
+    (hg : ∀ i j, ContinuousPolyFamily (fun t ↦ g t i j))
+    (hb : ∀ i, ContinuousPolyFamily (fun t ↦ b t i))
+    (hmetric : ∀ t i j, homogeneousComponent 0 (g t i j) = if i = j then 1 else 0)
+    (initial : T → Jet) (hi : ∀ t k, (initial t k).IsHomogeneous k)
+    (hic : ∀ k, ContinuousPolyFamily (fun t ↦ initial t k))
+    (hfirst : ∀ t i, pderiv i (initial t 1) = C (v t i))
+    (hcenter : ∀ t, (∑ i, v t i * v t i) + 4 = 0)
+    (hsecond : ∀ t, eikonalCoefficient (v t)
+      (fun r i j ↦ homogeneousComponent r (g t i j)) 0 (initial t) = 0) (m J : ℕ) :
+    ∃ (phi : T → CPoly) (A : ℕ → T → CPoly),
+      ContinuousPolyFamily phi ∧ (∀ j, ContinuousPolyFamily (A j)) ∧
+      (∀ t k, k ≤ 2 → homogeneousComponent k (phi t) = initial t k) ∧
+      (∀ t d, m + 2 * J + 3 < d → homogeneousComponent d (phi t) = 0) ∧
+      (∀ t d, d ≤ m + 2 * J + 2 →
+        homogeneousComponent d (polynomialEikonal (g t) (phi t)) = 0) ∧
+      (∀ j t, homogeneousComponent 0 (A j t) = if j = 0 then 1 else 0) ∧
+      (∀ j t k, m + 1 + 2 * (J - j) < k → homogeneousComponent k (A j t) = 0) ∧
+      (∀ t n, n < m + 1 + 2 * J → homogeneousComponent n
+        (polynomialTransport (beamVector (g t) (phi t)) (beamScalar (g t) (b t) (phi t))
+          0 (A 0 t)) = 0) ∧
+      ∀ j, j < J → ∀ t n, n < m + 1 + 2 * (J - (j + 1)) → homogeneousComponent n
+        (polynomialTransport (beamVector (g t) (phi t)) (beamScalar (g t) (b t) (phi t))
+          (-polynomialSecondOrder (g t) (b t) (A j t)) (A (j + 1) t)) = 0 := by
+  obtain ⟨phi, hpc, hkeep, hdegree, hsolve, _⟩ := finite_eikonal_polynomial_continuous
+    v hv i hvi (fun t r i j ↦ homogeneousComponent r (g t i j))
+    (fun r i j ↦ (hg i j).homogeneousComponent r) initial hi hic hsecond (m + 2 * J + 1)
+  have hf : ∀ t i, pderiv i (homogeneousComponent 1 (phi t)) = C (v t i) := by
+    intro t i
+    rw [show homogeneousComponent 1 (phi t) = initial t 1 from hkeep t 1 (by omega), hfirst]
+  have he : ∀ t d, d ≤ m + 2 * J + 2 →
+      homogeneousComponent d (polynomialEikonal (g t) (phi t)) = 0 := by
+    intro t d hd
+    cases d with
+    | zero =>
+      unfold polynomialEikonal
+      have hc0 (a b : CPoly) : homogeneousComponent 0 (a * b) =
+          homogeneousComponent 0 a * homogeneousComponent 0 b := by
+        simpa using homogeneousComponent_mul a b 0
+      simp only [map_add, map_sum, hc0, homogeneousComponent_pderiv, Nat.zero_add, hf, hmetric]
+      simp only [ite_mul, one_mul, zero_mul, Finset.sum_ite_eq, Finset.mem_univ, ite_true]
+      rw [homogeneousComponent_eq_self (isHomogeneous_C (Fin 4) (4 : ℂ))]
+      simpa only [map_add, map_sum, map_mul, map_zero] using congrArg C (hcenter t)
+    | succ n =>
+      rw [← eikonalCoefficient_eq (v t) (g t) (hmetric t) (phi t) (hf t) n]
+      exact hsolve t n (by omega)
+  have hW := beamVector_family g phi hg hpc
+  have hc := beamScalar_family g b phi hg hb hpc
+  obtain ⟨A, hAc, hA0, hAd, hAfirst, hAnext⟩ := finite_transport_polynomials_continuous
+    (fun t j ↦ 2 * v t j) (fun j ↦ continuous_const.mul (hv j))
+    i (fun t ↦ mul_ne_zero (by norm_num) (hvi t))
+    (fun t r j ↦ homogeneousComponent r (beamVector (g t) (phi t) j))
+    (fun t ↦ polynomialJet (beamScalar (g t) (b t) (phi t)))
+    (fun t r i j ↦ homogeneousComponent r (g t i j))
+    (fun t r i ↦ homogeneousComponent r (b t i))
+    (fun r j ↦ (hW j).homogeneousComponent r) (fun r ↦ hc.homogeneousComponent r)
+    (fun r i j ↦ (hg i j).homogeneousComponent r)
+    (fun r i ↦ (hb i).homogeneousComponent r) m J
+  refine ⟨phi, A, hpc, hAc, hkeep, ?_, he, hA0, hAd, ?_, ?_⟩
+  · simpa [Nat.add_assoc, polynomialJet] using hdegree
+  · intro t n hn
+    rw [← transportCoefficient_eq (fun j ↦ 2 * v t j) _
+      (beamVector_center (v t) (g t) (hmetric t) (phi t) (hf t)) _ 0 (A 0 t) n]
+    have hz : polynomialJet (0 : CPoly) = (0 : Jet) := funext (fun k ↦ map_zero _)
+    rw [hz]
+    exact hAfirst t n hn
+  · intro j hj t n hn
+    rw [← transportCoefficient_eq (fun j ↦ 2 * v t j) _
+      (beamVector_center (v t) (g t) (hmetric t) (phi t) (hf t)) _
+      (-polynomialSecondOrder (g t) (b t) (A j t)) (A (j + 1) t) n]
+    have hforce : polynomialJet (-polynomialSecondOrder (g t) (b t) (A j t)) =
+        fun k ↦ -secondOrderCoefficient (fun r i j ↦ homogeneousComponent r (g t i j))
+          (fun r i ↦ homogeneousComponent r (b t i)) (polynomialJet (A j t)) k := by
+      funext k
+      rw [secondOrderCoefficient_eq]
+      exact map_neg (homogeneousComponent k) _
+    rw [hforce]
+    exact hAnext j hj t n hn
+
+theorem smooth_wave_jets_continuous
+    (v : T → Fin 4 → ℂ) (hv : ∀ j, Continuous (fun t ↦ v t j))
+    (i : Fin 4) (hvi : ∀ t, v t i ≠ 0)
+    (g : T → Fin 4 → Fin 4 → Coord → ℂ) (b : T → Fin 4 → Coord → ℂ)
+    (G : T → Fin 4 → Fin 4 → CPoly) (B : T → Fin 4 → CPoly)
+    (hG : ∀ i j, ContinuousPolyFamily (fun t ↦ G t i j))
+    (hB : ∀ i, ContinuousPolyFamily (fun t ↦ B t i))
+    (hg : ∀ t i j, ContDiff ℝ ∞ (g t i j)) (hb : ∀ t i, ContDiff ℝ ∞ (b t i))
+    (hmetric : ∀ t i j, homogeneousComponent 0 (G t i j) = if i = j then 1 else 0)
+    (initial : T → Jet) (hi : ∀ t k, (initial t k).IsHomogeneous k)
+    (hic : ∀ k, ContinuousPolyFamily (fun t ↦ initial t k))
+    (hfirst : ∀ t i, pderiv i (initial t 1) = C (v t i))
+    (hcenter : ∀ t, (∑ i, v t i * v t i) + 4 = 0)
+    (hsecond : ∀ t, eikonalCoefficient (v t)
+      (fun r i j ↦ homogeneousComponent r (G t i j)) 0 (initial t) = 0) (m J : ℕ)
+    (hgj : ∀ t i j, FlatAt m (fun x ↦ g t i j x - reval (G t i j) x) 0)
+    (hbj : ∀ t i, FlatAt m (fun x ↦ b t i x - reval (B t i) x) 0) :
+    ∃ (phi : T → CPoly) (A : ℕ → T → CPoly),
+      ContinuousPolyFamily phi ∧ (∀ j, ContinuousPolyFamily (A j)) ∧
+      (∀ t k, k ≤ 2 → homogeneousComponent k (phi t) = initial t k) ∧
+      (∀ t d, m + 2 * J + 3 < d → homogeneousComponent d (phi t) = 0) ∧
+      (∀ j t, homogeneousComponent 0 (A j t) = if j = 0 then 1 else 0) ∧
+      (∀ j t k, m + 1 + 2 * (J - j) < k → homogeneousComponent k (A j t) = 0) ∧
+      (∀ t, FlatAt m (smoothEikonal (g t) (reval (phi t))) 0) ∧
+      (∀ t, FlatAt m (smoothTransport (smoothBeamVector (g t) (reval (phi t)))
+        (smoothBeamScalar (g t) (b t) (reval (phi t))) (fun _ ↦ 0) (reval (A 0 t))) 0) ∧
+      ∀ j, j < J → ∀ t, FlatAt m
+        (smoothTransport (smoothBeamVector (g t) (reval (phi t)))
+          (smoothBeamScalar (g t) (b t) (reval (phi t)))
+          (fun x ↦ -smoothSecondOrder (g t) (b t) (reval (A j t)) x)
+          (reval (A (j + 1) t))) 0 := by
+  obtain ⟨phi, A, hpc, hAc, hkeep, hdeg, he, hA0, hAd, ht0, htj⟩ :=
+    polynomial_wave_jets_continuous v hv i hvi G B hG hB hmetric initial hi hic
+      hfirst hcenter hsecond m J
+  refine ⟨phi, A, hpc, hAc, hkeep, hdeg, hA0, hAd, ?_, ?_, ?_⟩
+  · intro t
+    exact smooth_eikonal_flat (G t) (phi t) (hg t) (hgj t) (fun k hk ↦ he t k (by omega))
+  · intro t
+    have hp : ∀ k, k ≤ m → homogeneousComponent k
+        (polynomialTransport (beamVector (G t) (phi t)) (beamScalar (G t) (B t) (phi t))
+          (-polynomialSecondOrder (G t) (B t) 0) (A 0 t)) = 0 := by
+      intro k hk
+      simpa [polynomialSecondOrder] using ht0 t k (by omega)
+    have h := smooth_beam_transport_flat (G t) (B t) (phi t) 0 (A 0 t)
+      (hg t) (hb t) (hgj t) (hbj t) hp
+    have hz : (fun x ↦ -smoothSecondOrder (g t) (b t) (reval 0) x) = fun _ ↦ 0 := by
+      ext x
+      simp [smoothSecondOrder, coordPartial_reval, reval]
+    rw [hz] at h
+    exact h
+  · intro j hj t
+    exact smooth_beam_transport_flat (G t) (B t) (phi t) (A j t) (A (j + 1) t)
+      (hg t) (hb t) (hgj t) (hbj t) (fun k hk ↦ htj j hj t k (by omega))
+
+end
+end Yau.Jets
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/WaveCenterJets.lean b/lean/OAI/Geometry/NodalSets/Waves/WaveCenterJets.lean
new file mode 100644
index 0000000000000000000000000000000000000000..8543a0d524cb6c17bb11fed45945d9952f35a010
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/WaveCenterJets.lean
@@ -0,0 +1,114 @@
+import OAI.Geometry.NodalSets.Elliptic.PhaseActualJet
+import OAI.Geometry.NodalSets.Waves.TripleSourceWaves
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Set Filter
+open scoped ContDiff Topology
+noncomputable section
+attribute [local instance] clmTopology clmAdd clmModule
+variable {T : Type*} [TopologicalSpace T]
+variable {g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ} {w S : Coord → ℝ}
+variable {y : T → Coord} {d : SourceFrameTriple g S y} {m J K k0 : ℕ}
+namespace TripleSourceWaveData
+variable (b : TripleSourceWaveData g w S y d m J K k0)
+
+def phase (t : T × Fin 3) : Coord → ℂ := reval (b.phi t) ∘ (b.F t).symm
+
+def amplitude (N : ℝ) (t : T × Fin 3) : Coord → ℂ :=
+  finiteAmplitude (fun j ↦ b.A j t) J N ∘ (b.F t).symm
+
+def wave (N : ℝ) (t : T × Fin 3) : Coord → ℂ :=
+  chartPushforward (b.F t) (coordinateWave b.phi b.A J N t)
+
+lemma zero_mem_source (t : T × Fin 3) : (0:Coord) ∈ (b.F t).source := by
+  rw [(b.chart_domain t).1]
+  simpa using b.sourceRadius_pos
+
+lemma chart_zero (t : T × Fin 3) : b.F t 0 = y t.1 := by
+  rw [b.chart_eq]
+  exact quadraticChartMap_zero _ _ _
+
+lemma center_mem_target (t : T × Fin 3) : y t.1 ∈ (b.F t).target := by
+  rw [← b.chart_zero t]
+  exact (b.F t).map_source (b.zero_mem_source t)
+
+lemma inverse_center (t : T × Fin 3) : (b.F t).symm (y t.1) = 0 := by
+  rw [← b.chart_zero t]
+  exact (b.F t).left_inv (b.zero_mem_source t)
+
+lemma inverse_derivative (t : T × Fin 3) :
+    HasFDerivAt ((b.F t).symm : Coord → Coord) (d.frame t).symm.toContinuousLinearMap (y t.1) := by
+  apply (b.F t).hasFDerivAt_symm (b.center_mem_target t)
+  rw [b.inverse_center,b.chart_eq]
+  exact quadraticChartMap_deriv_zero _ _ _
+
+lemma phase_smooth_at_center (t : T × Fin 3) : ContDiffAt ℝ ∞ (b.phase t) (y t.1) :=
+  (reval_contDiff _).contDiffAt.comp _ ((b.chart_domain t).2.2.contDiffAt
+    ((b.F t).open_target.mem_nhds (b.center_mem_target t)))
+
+lemma phase_center (t : T × Fin 3) : b.phase t (y t.1) = (S (y t.1):ℂ) := by
+  unfold phase
+  simp only [Function.comp_apply,b.inverse_center]
+  exact retained_phase_value _ _ _ _ (b.jets.phase_retained t 0 (by omega))
+
+lemma amplitude_center (N : ℝ) (t : T × Fin 3) : b.amplitude N t (y t.1) = 1 := by
+  unfold amplitude
+  simp only [Function.comp_apply,b.inverse_center]
+  exact finiteAmplitude_center _ _ _ (fun j ↦ b.jets.amplitude_value j t)
+
+lemma wave_center (N : ℝ) (hN : 0 < N) (t : T × Fin 3) :
+    b.wave N t (y t.1) = Complex.exp ((N:ℂ)*(S (y t.1):ℂ)) := by
+  rw [wave,← b.chart_zero t,chartPushforward_pullback _ _ _ (b.zero_mem_source t)]
+  have hc : Yau.Waves.scaledCutoff N (0:Coord) 0 = 1 :=
+    Yau.Waves.scaledCutoff_eq_one hN 0 0 (by simp; positivity)
+  have hphi := retained_phase_value _ _ _ _ (b.jets.phase_retained t 0 (by omega))
+  simp only [coordinateWave,hc,one_smul,waveExp,hphi,
+    finiteAmplitude_center _ J N (fun j ↦ b.jets.amplitude_value j t),one_mul]
+  rw [b.chart_zero]
+
+lemma wave_ne_zero (N : ℝ) (hN : 0 < N) (t : T × Fin 3) : b.wave N t ≠ 0 := by
+  intro h
+  have hh := congrFun h (y t.1)
+  rw [b.wave_center N hN t] at hh
+  exact Complex.exp_ne_zero _ hh
+
+lemma phase_first (hpos : ∀ x v, v ≠ 0 → 0 < g x v v)
+    (hp0 : ∀ t, metricGradient g S (y t) ≠ 0) (t : T × Fin 3) (v : Coord) :
+    fderiv ℝ (b.phase t) (y t.1) v =
+      (g (y t.1) (metricGradient g S (y t.1)) v:ℂ) +
+        Complex.I * (g (y t.1) (d.q t.1 t.2) v:ℂ) := by
+  have hder := ((reval_contDiff (b.phi t)).differentiable (by simp) 0).hasFDerivAt
+  have he := (b.inverse_derivative t)
+  have hf : fderiv ℝ (b.phase t) (y t.1) v =
+      fderiv ℝ (reval (b.phi t)) 0 ((d.frame t).symm v) := by
+    unfold phase
+    rw [fderiv_comp _ (by rw [b.inverse_center]; exact hder.differentiableAt) he.differentiableAt,
+      he.fderiv,b.inverse_center]
+    rfl
+  rw [hf,fderiv_reval]
+  simp_rw [retained_phase_first _ _ _ _ (b.jets.phase_retained t 1 (by omega))]
+  have hpair0 : g (y t.1) (metricGradient g S (y t.1)) v =
+      sourceTripleAlpha g S y t * (d.frame t).symm v 0 := by
+    have hframe : sourceTripleAlpha g S y t • d.frame t (Pi.single 0 1) = metricGradient g S (y t.1) :=
+      d.frame_first (fun t ↦ hpos (y t)) hp0 t
+    rw [← hframe]
+    simp only [map_smul,smul_apply,smul_eq_mul]
+    rw [frame_pair_coordinate (g (y t.1)) (d.frame t) (d.frame_orthonormal t)]
+  have hpair1 : g (y t.1) (d.q t.1 t.2) v =
+      sourceTripleBeta g S y t * (d.frame t).symm v 1 := by
+    have hframe : sourceTripleBeta g S y t • d.frame t (Pi.single 1 1) = d.q t.1 t.2 :=
+      d.frame_second (fun t ↦ hpos (y t)) hp0 t
+    rw [← hframe]
+    simp only [map_smul,smul_apply,smul_eq_mul]
+    rw [frame_pair_coordinate (g (y t.1)) (d.frame t) (d.frame_orthonormal t)]
+  rw [hpair0,hpair1]
+  simp [normalFrameVector,Fin.sum_univ_succ]
+  ring
+
+end TripleSourceWaveData
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/WaveEstimates.lean b/lean/OAI/Geometry/NodalSets/Waves/WaveEstimates.lean
new file mode 100644
index 0000000000000000000000000000000000000000..daa753576e4b25553c9d94afebf4993b342c7dbf
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/WaveEstimates.lean
@@ -0,0 +1,202 @@
+import Mathlib.Analysis.Calculus.ContDiff.Operations
+import Mathlib.Analysis.Calculus.Taylor
+import Mathlib.Analysis.Complex.Trigonometric
+import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.Ring
+
+namespace OAI
+
+namespace Yau.Waves
+open Set Filter
+open scoped Topology ContDiff
+noncomputable section
+universe u
+variable {E F : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  [NormedAddCommGroup F] [NormedSpace ℝ F]
+
+lemma iteratedDeriv_line {f : E → F} {n k : ℕ} (hf : ContDiff ℝ n f)
+    (hk : k ≤ n) (y z : E) (t : ℝ) :
+    iteratedDeriv k (fun s : ℝ ↦ f (y + s • z)) t =
+      iteratedFDeriv ℝ k f (y + t • z) (fun _ ↦ z) := by
+  let L : ℝ →L[ℝ] E := (ContinuousLinearMap.id ℝ ℝ).smulRight z
+  have hshift : ContDiff ℝ n (fun x ↦ f (y + x)) :=
+    hf.comp (contDiff_const.add contDiff_id)
+  rw [iteratedDeriv_eq_iteratedFDeriv]
+  change iteratedFDeriv ℝ k ((fun x ↦ f (y + x)) ∘ L) t (fun _ ↦ 1) = _
+  rw [L.iteratedFDeriv_comp_right hshift t (by exact_mod_cast hk)]
+  simp only [ContinuousMultilinearMap.compContinuousLinearMap_apply]
+  rw [iteratedFDeriv_comp_add_left]
+  simp [L]
+
+theorem vanishing_jet_taylor_bound {f : E → F} {n : ℕ} (hf : ContDiff ℝ (n + 1) f)
+    (y z : E) (C : ℝ)
+    (hzero : ∀ k, k ≤ n → iteratedFDeriv ℝ k f y = 0)
+    (hbound : ∀ t ∈ Icc (0 : ℝ) 1, ‖iteratedFDeriv ℝ (n + 1) f (y + t • z)‖ ≤ C) :
+    ‖f (y + z)‖ ≤ C * ‖z‖ ^ (n + 1) / (n.factorial : ℝ) := by
+  let g : ℝ → F := fun t ↦ f (y + t • z)
+  have hg : ContDiff ℝ (n + 1) g :=
+    hf.comp (contDiff_const.add (contDiff_id.smul contDiff_const))
+  have hi (k : ℕ) (hk : k ≤ n + 1) (t : ℝ) (ht : t ∈ Icc (0 : ℝ) 1) :
+      iteratedDerivWithin k g (Icc (0 : ℝ) 1) t =
+        iteratedFDeriv ℝ k f (y + t • z) (fun _ ↦ z) := by
+    rw [iteratedDerivWithin_eq_iteratedDeriv (uniqueDiffOn_Icc (by norm_num : (0 : ℝ) < 1))
+      (hg.of_le (by exact_mod_cast hk)).contDiffAt ht]
+    exact iteratedDeriv_line hf hk y z t
+  have htaylor : taylorWithinEval g n (Icc (0 : ℝ) 1) 0 1 = 0 := by
+    rw [taylor_within_apply]
+    apply Finset.sum_eq_zero
+    intro k hk
+    rw [hi k (by have := Finset.mem_range.mp hk; omega) 0 (by constructor <;> norm_num)]
+    simp [hzero k (by have := Finset.mem_range.mp hk; omega)]
+  have hderiv : ∀ t ∈ Icc (0 : ℝ) 1,
+      ‖iteratedDerivWithin (n + 1) g (Icc (0 : ℝ) 1) t‖ ≤ C * ‖z‖ ^ (n + 1) := by
+    intro t ht
+    rw [hi (n + 1) le_rfl t ht]
+    calc
+      _ ≤ ‖iteratedFDeriv ℝ (n + 1) f (y + t • z)‖ * ∏ _ : Fin (n + 1), ‖z‖ :=
+        ContinuousMultilinearMap.le_opNorm _ _
+      _ ≤ C * ‖z‖ ^ (n + 1) := by
+        simp only [Finset.prod_const, Finset.card_univ, Fintype.card_fin]
+        exact mul_le_mul_of_nonneg_right (hbound t ht) (pow_nonneg (norm_nonneg _) _)
+  have h := taylor_mean_remainder_bound (by norm_num : (0 : ℝ) ≤ 1)
+    hg.contDiffOn (by constructor <;> norm_num : (1 : ℝ) ∈ Icc (0 : ℝ) 1) hderiv
+  simpa [htaylor, g] using h
+
+theorem vanishing_jet_derivative_bound {f : E → F} {m l : ℕ}
+    (hf : ContDiff ℝ ∞ f) (hl : l ≤ m) (y z : E) (C : ℝ)
+    (hzero : ∀ k, k ≤ m → iteratedFDeriv ℝ k f y = 0)
+    (hbound : ∀ t ∈ Icc (0 : ℝ) 1,
+      ‖iteratedFDeriv ℝ (m + 1) f (y + t • z)‖ ≤ C) :
+    ‖iteratedFDeriv ℝ l f (y + z)‖ ≤
+      C * ‖z‖ ^ (m + 1 - l) / ((m - l).factorial : ℝ) := by
+  induction l generalizing F m with
+  | zero =>
+    have hf0 : ContDiff ℝ (m + 1 : ℕ) f := hf.of_le (by
+      exact_mod_cast (show (m + 1 : ℕ∞) ≤ ⊤ from le_top))
+    simpa [norm_iteratedFDeriv_zero] using
+      vanishing_jet_taylor_bound (n := m) hf0 y z C hzero hbound
+  | succ l ih =>
+    cases m with
+    | zero => omega
+    | succ m =>
+      have hf' : ContDiff ℝ ∞ (fderiv ℝ f) := hf.fderiv_right (by simp)
+      have hz' : ∀ k, k ≤ m → iteratedFDeriv ℝ k (fderiv ℝ f) y = 0 := by
+        intro k hk
+        apply norm_eq_zero.mp
+        rw [norm_iteratedFDeriv_fderiv, hzero (k + 1) (by omega), norm_zero]
+      have hb' : ∀ t ∈ Icc (0 : ℝ) 1,
+          ‖iteratedFDeriv ℝ (m + 1) (fderiv ℝ f) (y + t • z)‖ ≤ C := by
+        intro t ht
+        rw [norm_iteratedFDeriv_fderiv]
+        exact hbound t ht
+      have h := ih hf' (by omega : l ≤ m) hz' hb'
+      simpa [norm_iteratedFDeriv_fderiv] using h
+
+theorem quadratic_gap_of_cubic_remainder {eta C r gap : ℝ}
+    (hr : 0 ≤ r) (hsmall : C * r ≤ eta / 4)
+    (hgap : gap ≤ -(eta / 2) * r ^ 2 + C * r ^ 3) :
+    gap ≤ -(eta / 4) * r ^ 2 := by
+  have := mul_le_mul_of_nonneg_right hsmall (mul_nonneg hr hr)
+  nlinarith
+
+theorem complex_exponential_gaussian_bound {phi : ℂ} {S c N r : ℝ}
+    (hN : 0 ≤ N) (hgap : phi.re - S ≤ -c * r ^ 2) :
+    ‖Complex.exp ((N : ℂ) * phi)‖ ≤ Real.exp (N * S - c * N * r ^ 2) := by
+  rw [Complex.norm_exp]
+  simp only [Complex.mul_re, Complex.ofReal_re, Complex.ofReal_im, zero_mul, sub_zero]
+  apply Real.exp_le_exp.mpr
+  have := mul_le_mul_of_nonneg_left hgap hN
+  nlinarith
+
+theorem gaussian_absorbs_power (c : ℝ) (hc : 0 < c) (p K : ℝ) :
+    ∀ᶠ N : ℝ in atTop,
+      N ^ p * Real.exp (-c * N ^ (1 / 3 : ℝ)) ≤ N ^ (-K) := by
+  have ht := (tendsto_rpow_mul_exp_neg_mul_atTop_nhds_zero
+    (3 * (p + K)) c hc).comp (tendsto_rpow_atTop (by norm_num : (0 : ℝ) < 1 / 3))
+  have he : ∀ᶠ N : ℝ in atTop,
+      (N ^ (1 / 3 : ℝ)) ^ (3 * (p + K)) * Real.exp (-c * N ^ (1 / 3 : ℝ)) ≤ 1 :=
+    ht.eventually (eventually_le_nhds (by norm_num : (0 : ℝ) < 1))
+  filter_upwards [he, eventually_gt_atTop (0 : ℝ)] with N hN hpos
+  have heq : (N ^ (1 / 3 : ℝ)) ^ (3 * (p + K)) = N ^ (p + K) := by
+    rw [← Real.rpow_mul hpos.le]
+    congr 1
+    ring
+  rw [heq] at hN
+  rw [Real.rpow_neg hpos.le, ← one_div, le_div_iff₀ (Real.rpow_pos_of_pos hpos K)]
+  simpa [Real.rpow_add hpos, mul_assoc, mul_left_comm, mul_comm] using hN
+
+theorem gaussian_cutoff_bound (c b : ℝ) (hc : 0 < c) (hb : 0 < b) (p K : ℝ) :
+    ∀ᶠ N : ℝ in atTop, ∀ S r A : ℝ, 0 ≤ A →
+      b * N ^ (1 / 3 : ℝ) ≤ N * r ^ 2 →
+      A * N ^ p * Real.exp (N * S - c * N * r ^ 2) ≤
+        A * N ^ (-K) * Real.exp (N * S) := by
+  filter_upwards [gaussian_absorbs_power (c * b) (mul_pos hc hb) p K,
+    eventually_gt_atTop (0 : ℝ)] with N hN hpos
+  intro S r A hA hann
+  have hexp : Real.exp (-c * N * r ^ 2) ≤ Real.exp (-(c * b) * N ^ (1 / 3 : ℝ)) := by
+    apply Real.exp_le_exp.mpr
+    have := mul_le_mul_of_nonneg_left hann hc.le
+    nlinarith
+  have hcost : N ^ p * Real.exp (-c * N * r ^ 2) ≤ N ^ (-K) :=
+    (mul_le_mul_of_nonneg_left hexp (Real.rpow_nonneg hpos.le _)).trans hN
+  have h := mul_le_mul_of_nonneg_left hcost hA
+  have h' := mul_le_mul_of_nonneg_right h (Real.exp_pos (N * S)).le
+  rw [show N * S - c * N * r ^ 2 = N * S + (-c * N * r ^ 2) by ring,
+    Real.exp_add]
+  nlinarith [h']
+
+theorem residual_exponent_budgets {m J K k k0 : ℕ}
+    (hm : 3 * K + 4 * k0 + 6 < m + 1) (hJ : K + k0 + 1 ≤ J) (hk : k ≤ k0) :
+    (k : ℝ) + 2 - ((m : ℝ) + 1 - k) / 3 < -(K : ℝ) ∧
+    (k : ℤ) - J ≤ -(K : ℤ) - 1 := by
+  constructor
+  · have hm' : (3 : ℝ) * K + 4 * k0 + 6 < m + 1 := by exact_mod_cast hm
+    have hk' : (k : ℝ) ≤ k0 := by exact_mod_cast hk
+    linarith
+  · omega
+
+theorem scaled_taylor_defect_bound {f : E → F} {m K k k0 : ℕ}
+    (hf : ContDiff ℝ ∞ f) (hm : 3 * K + 4 * k0 + 6 < m + 1) (hk : k ≤ k0)
+    (y z : E) {C R N : ℝ} (hC : 0 ≤ C) (hR : 0 ≤ R) (hN : 1 ≤ N)
+    (hzero : ∀ j, j ≤ m → iteratedFDeriv ℝ j f y = 0)
+    (hbound : ∀ t ∈ Icc (0 : ℝ) 1,
+      ‖iteratedFDeriv ℝ (m + 1) f (y + t • z)‖ ≤ C)
+    (hradius : ‖z‖ ≤ R * N ^ (-1 / 3 : ℝ)) :
+    N ^ ((k : ℝ) + 2) * ‖iteratedFDeriv ℝ k f (y + z)‖ ≤
+      C * R ^ (m + 1 - k) * N ^ (-(K : ℝ)) := by
+  have hkm : k ≤ m := by omega
+  have hpos : 0 < N := lt_of_lt_of_le zero_lt_one hN
+  let q : ℕ := m + 1 - k
+  have ht := vanishing_jet_derivative_bound hf hkm y z C hzero hbound
+  have hfact : (1 : ℝ) ≤ ((m - k).factorial : ℝ) := by
+    exact_mod_cast Nat.succ_le_of_lt (Nat.factorial_pos (m - k))
+  have hsmall : ‖iteratedFDeriv ℝ k f (y + z)‖ ≤ C * (R * N ^ (-1 / 3 : ℝ)) ^ q := by
+    calc
+      _ ≤ C * ‖z‖ ^ q := ht.trans (div_le_self (mul_nonneg hC (pow_nonneg (norm_nonneg _) _)) hfact)
+      _ ≤ _ := mul_le_mul_of_nonneg_left (pow_le_pow_left₀ (norm_nonneg _) hradius q) hC
+  have hp : (N ^ (-1 / 3 : ℝ)) ^ q = N ^ (-(q : ℝ) / 3) := by
+    rw [← Real.rpow_natCast, ← Real.rpow_mul hpos.le]
+    congr 1
+    ring
+  have he : (k : ℝ) + 2 + (-(q : ℝ) / 3) ≤ -(K : ℝ) := by
+    have hq : (q : ℝ) = (m : ℝ) + 1 - k := by
+      dsimp [q]
+      rw [Nat.cast_sub (by omega), Nat.cast_add, Nat.cast_one]
+    rw [hq]
+    have hm' : (3 : ℝ) * K + 4 * k0 + 6 < m + 1 := by exact_mod_cast hm
+    have hk' : (k : ℝ) ≤ k0 := by exact_mod_cast hk
+    linarith
+  calc
+    _ ≤ N ^ ((k : ℝ) + 2) * (C * (R * N ^ (-1 / 3 : ℝ)) ^ q) :=
+      mul_le_mul_of_nonneg_left hsmall (Real.rpow_nonneg hpos.le _)
+    _ = C * R ^ q * N ^ ((k : ℝ) + 2 + (-(q : ℝ) / 3)) := by
+      rw [mul_pow, hp, Real.rpow_add hpos ((k : ℝ) + 2) (-(q : ℝ) / 3)]
+      ring
+    _ ≤ _ := mul_le_mul_of_nonneg_left (Real.rpow_le_rpow_of_exponent_le hN he)
+      (mul_nonneg hC (pow_nonneg hR _))
+
+end
+end Yau.Waves
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/WaveGerm.lean b/lean/OAI/Geometry/NodalSets/Waves/WaveGerm.lean
new file mode 100644
index 0000000000000000000000000000000000000000..bc1ad7a0fd307018ed19819c1b0e75feb4233dda
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/WaveGerm.lean
@@ -0,0 +1,37 @@
+import OAI.Geometry.NodalSets.Waves.WaveCenterJets
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Set Filter
+open scoped ContDiff Topology
+noncomputable section
+variable {T : Type*} [TopologicalSpace T]
+variable {g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ} {w S : Coord → ℝ}
+variable {y : T → Coord} {d : SourceFrameTriple g S y} {m J K k0 : ℕ}
+namespace TripleSourceWaveData
+variable (b : TripleSourceWaveData g w S y d m J K k0)
+
+lemma wave_germ (N : ℝ) (hN : 0 < N) (t : T × Fin 3) :
+    b.wave N t =ᶠ[𝓝 (y t.1)]
+      (fun x ↦ b.amplitude N t x * Complex.exp ((N:ℂ)*b.phase t x)) := by
+  have hcut : Yau.Waves.scaledCutoff N (0:Coord) =ᶠ[𝓝 (0:Coord)] (fun _ ↦ 1) := by
+    filter_upwards [Metric.ball_mem_nhds (0:Coord) (Real.rpow_pos_of_pos hN (-1/3:ℝ))] with z hz
+    exact Yau.Waves.scaledCutoff_eq_one hN 0 z (by simpa only [Metric.mem_ball,dist_eq_norm] using hz.le)
+  have hInv : Tendsto ((b.F t).symm : Coord → Coord) (𝓝 (y t.1)) (𝓝 0) := by
+    have hh := (b.inverse_derivative t).continuousAt
+    simpa only [ContinuousAt,b.inverse_center] using hh
+  filter_upwards [chartPushforward_eventually (b.F t) (coordinateWave b.phi b.A J N t)
+    (y t.1) (b.center_mem_target t),hcut.comp_tendsto hInv] with z hz hc
+  change chartPushforward (b.F t) (coordinateWave b.phi b.A J N t) z = _
+  rw [hz]
+  change Yau.Waves.scaledCutoff N 0 ((b.F t).symm z) • _ = _
+  change Yau.Waves.scaledCutoff N 0 ((b.F t).symm z) = 1 at hc
+  rw [hc,one_smul]
+  rfl
+
+end TripleSourceWaveData
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/WaveHessian.lean b/lean/OAI/Geometry/NodalSets/Waves/WaveHessian.lean
new file mode 100644
index 0000000000000000000000000000000000000000..8dbec7db3d3eadb3e88e54ef2e489babf1494b4b
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/WaveHessian.lean
@@ -0,0 +1,50 @@
+import OAI.Geometry.NodalSets.Waves.WaveSecondJet
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets
+open scoped ContDiff
+noncomputable section
+attribute [local instance] clmTopology clmAdd clmModule
+
+lemma sourceHessian_symmetric_at
+    (g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ) (hg : ContDiff ℝ ∞ g)
+    (hs : ∀ x u v, g x u v = g x v u) (hp : ∀ x v, v ≠ 0 → 0 < g x v v)
+    (f : Coord → ℝ) (x : Coord) (hf : ContDiffAt ℝ ∞ f x) (u v : Coord) :
+    sourceHessian g f x u v = sourceHessian g f x v u := by
+  simp only [sourceHessian_apply]
+  rw [hf.isSymmSndFDerivAt
+    (by simp) u v,
+    actual_connection_symmetric g hg hs hp x u v]
+
+variable {T : Type*} [TopologicalSpace T] [CompactSpace T]
+variable {g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ} {w S : Coord → ℝ}
+variable {y : T → Coord} {d : SourceFrameTriple g S y} {m J K k0 : ℕ}
+namespace TripleSourceWaveData
+variable (b : TripleSourceWaveData g w S y d m J K k0)
+
+omit [CompactSpace T] in
+theorem phase_hessian
+    (hg : ContDiff ℝ ∞ g) (hs : ∀ x u v, g x u v = g x v u)
+    (hp : ∀ x v, v ≠ 0 → 0 < g x v v) (hS : ContDiff ℝ ∞ S)
+    (hp0 : ∀ t, metricGradient g S (y t) ≠ 0) (t : T × Fin 3) (u v : Coord) :
+    sourceHessian g (fun z ↦ (b.phase t z).re) (y t.1) u v =
+      sourceHessian g S (y t.1) u v -
+        sourceDirectionEta (g (y t.1)) (sourceHessian g S (y t.1))
+          (metricGradient g S (y t.1)) (d.q t.1 t.2) * g (y t.1) u v := by
+  have hu := b.phase_second hg hs hp hS hp0 t u
+  have hv := b.phase_second hg hs hp hS hp0 t v
+  have huv := b.phase_second hg hs hp hS hp0 t (u+v)
+  have hsym := sourceHessian_symmetric_at g hg hs hp (fun z ↦ (b.phase t z).re) (y t.1)
+    (Complex.reCLM.contDiff.contDiffAt.comp _ (b.phase_smooth_at_center t)) v u
+  have hSsym := sourceHessian_symmetric_at g hg hs hp S (y t.1) hS.contDiffAt v u
+  simp only [map_add,add_apply] at huv
+  rw [hsym,hSsym,hs (y t.1) v u] at huv
+  nlinarith
+
+end TripleSourceWaveData
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/WaveJetAssembly.lean b/lean/OAI/Geometry/NodalSets/Waves/WaveJetAssembly.lean
new file mode 100644
index 0000000000000000000000000000000000000000..5c4068e0b2c91b87e27ad7c158af5e5ce068a4be
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/WaveJetAssembly.lean
@@ -0,0 +1,96 @@
+import OAI.Geometry.NodalSets.Elliptic.EikonalPolynomial
+
+namespace OAI
+
+namespace Yau.Jets
+open MvPolynomial
+noncomputable section
+
+def beamVector (g : Fin 4 → Fin 4 → CPoly) (phi : CPoly) (i : Fin 4) : CPoly :=
+  C 2 * ∑ j, g i j * pderiv j phi
+
+def beamScalar (g : Fin 4 → Fin 4 → CPoly) (b : Fin 4 → CPoly) (phi : CPoly) : CPoly :=
+  polynomialSecondOrder g b phi + C 6
+
+lemma beamVector_center (v : Fin 4 → ℂ) (g : Fin 4 → Fin 4 → CPoly)
+    (hmetric : ∀ i j, homogeneousComponent 0 (g i j) = if i = j then 1 else 0)
+    (phi : CPoly) (hfirst : ∀ i, pderiv i (homogeneousComponent 1 phi) = C (v i)) :
+    ∀ i, homogeneousComponent 0 (beamVector g phi i) = C (2 * v i) := by
+  intro i
+  have hc0 (a b : CPoly) : homogeneousComponent 0 (a * b) =
+      homogeneousComponent 0 a * homogeneousComponent 0 b := by
+    simpa using homogeneousComponent_mul a b 0
+  unfold beamVector
+  rw [homogeneousComponent_C_mul, map_sum]
+  simp only [hc0, homogeneousComponent_pderiv, Nat.zero_add, hfirst, hmetric,
+    ite_mul, one_mul, zero_mul, Finset.sum_ite_eq, Finset.mem_univ, ite_true, map_mul]
+
+theorem polynomial_transport_hierarchy (w : Fin 4 → ℂ) (hw : w ≠ 0)
+    (W : Fin 4 → CPoly) (hW : ∀ i, homogeneousComponent 0 (W i) = C (w i))
+    (c : CPoly) (g : Fin 4 → Fin 4 → CPoly) (b : Fin 4 → CPoly) (m J : ℕ) :
+    ∃ P : ℕ → CPoly,
+      (∀ j, homogeneousComponent 0 (P j) = if j = 0 then 1 else 0) ∧
+      (∀ j k, m + 1 + 2 * (J - j) < k → homogeneousComponent k (P j) = 0) ∧
+      (∀ n, n < m + 1 + 2 * J → homogeneousComponent n (polynomialTransport W c 0 (P 0)) = 0) ∧
+      ∀ j, j < J → ∀ n, n < m + 1 + 2 * (J - (j + 1)) →
+        homogeneousComponent n (polynomialTransport W c (-polynomialSecondOrder g b (P j))
+          (P (j + 1))) = 0 := by
+  obtain ⟨P, hzero, hdeg, hfirst, hnext⟩ := finite_transport_polynomials w hw
+    (fun r i ↦ homogeneousComponent r (W i)) (polynomialJet c)
+    (fun r i j ↦ homogeneousComponent r (g i j)) (fun r i ↦ homogeneousComponent r (b i)) m J
+  refine ⟨P, hzero, hdeg, ?_, ?_⟩
+  · intro n hn
+    rw [← transportCoefficient_eq w W hW c 0 (P 0) n]
+    have hz : polynomialJet (0 : CPoly) = (0 : Jet) := funext (fun k ↦ map_zero _)
+    rw [hz]
+    exact hfirst n hn
+  · intro j hj n hn
+    rw [← transportCoefficient_eq w W hW c (-polynomialSecondOrder g b (P j)) (P (j + 1)) n]
+    have hforce : polynomialJet (-polynomialSecondOrder g b (P j)) =
+        fun k ↦ -secondOrderCoefficient (fun r i j ↦ homogeneousComponent r (g i j))
+          (fun r i ↦ homogeneousComponent r (b i)) (polynomialJet (P j)) k := by
+      funext k
+      rw [secondOrderCoefficient_eq]
+      exact map_neg (homogeneousComponent k) _
+    rw [hforce]
+    exact hnext j hj n hn
+
+theorem polynomial_wave_jets (v : Fin 4 → ℂ) (hv : v ≠ 0)
+    (g : Fin 4 → Fin 4 → CPoly) (b : Fin 4 → CPoly)
+    (hmetric : ∀ i j, homogeneousComponent 0 (g i j) = if i = j then 1 else 0)
+    (initial : Jet) (hi : ∀ k, (initial k).IsHomogeneous k)
+    (hfirst : ∀ i, pderiv i (initial 1) = C (v i))
+    (hcenter : (∑ i, v i * v i) + 4 = 0)
+    (hsecond : eikonalCoefficient v (fun r i j ↦ homogeneousComponent r (g i j)) 0 initial = 0)
+    (m J : ℕ) :
+    ∃ (phi : CPoly) (A : ℕ → CPoly),
+      (∀ k, k ≤ 2 → homogeneousComponent k phi = initial k) ∧
+      (∀ d, m + 2 * J + 3 < d → homogeneousComponent d phi = 0) ∧
+      (∀ d, d ≤ m + 2 * J + 2 → homogeneousComponent d (polynomialEikonal g phi) = 0) ∧
+      (∀ j, homogeneousComponent 0 (A j) = if j = 0 then 1 else 0) ∧
+      (∀ j k, m + 1 + 2 * (J - j) < k → homogeneousComponent k (A j) = 0) ∧
+      (∀ n, n < m + 1 + 2 * J →
+        homogeneousComponent n (polynomialTransport (beamVector g phi) (beamScalar g b phi) 0 (A 0)) = 0) ∧
+      ∀ j, j < J → ∀ n, n < m + 1 + 2 * (J - (j + 1)) →
+        homogeneousComponent n (polynomialTransport (beamVector g phi) (beamScalar g b phi)
+          (-polynomialSecondOrder g b (A j)) (A (j + 1))) = 0 := by
+  obtain ⟨phi, hkeep, hdegree, heik⟩ := polynomial_eikonal_jets v hv g hmetric initial hi hfirst
+    hcenter hsecond (m + 2 * J + 1)
+  have hf : ∀ i, pderiv i (homogeneousComponent 1 phi) = C (v i) := by
+    intro i
+    rw [show homogeneousComponent 1 phi = initial 1 from hkeep 1 (by omega), hfirst]
+  have hv2 : (fun i ↦ 2 * v i) ≠ (0 : Fin 4 → ℂ) := by
+    intro h
+    apply hv
+    funext i
+    have := congrFun h i
+    simpa using this
+  obtain ⟨A, hA0, hAd, hAfirst, hAnext⟩ := polynomial_transport_hierarchy (fun i ↦ 2 * v i) hv2
+    (beamVector g phi) (beamVector_center v g hmetric phi hf) (beamScalar g b phi) g b m J
+  exact ⟨phi, A, hkeep, by simpa [Nat.add_assoc] using hdegree,
+    by simpa [Nat.add_assoc] using heik, hA0, hAd, hAfirst, hAnext⟩
+
+end
+end Yau.Jets
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/WaveSecondJet.lean b/lean/OAI/Geometry/NodalSets/Waves/WaveSecondJet.lean
new file mode 100644
index 0000000000000000000000000000000000000000..61dfd1eee7411f3be931080c0d8885912a5c18cc
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/WaveSecondJet.lean
@@ -0,0 +1,95 @@
+import OAI.Geometry.NodalSets.Charts.ContactGeometry
+import OAI.Geometry.NodalSets.Coefficients.CoefficientLocality
+import OAI.Geometry.NodalSets.Elliptic.CommonLocalExtension
+import OAI.Geometry.NodalSets.Waves.WaveCenterJets
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Set Filter
+open scoped ContDiff Topology
+noncomputable section
+attribute [local instance] clmTopology clmAdd clmModule
+variable {T : Type*} [TopologicalSpace T] [CompactSpace T]
+variable {g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ} {w S : Coord → ℝ}
+variable {y : T → Coord} {d : SourceFrameTriple g S y} {m J K k0 : ℕ}
+namespace TripleSourceWaveData
+variable (b : TripleSourceWaveData g w S y d m J K k0)
+
+omit [CompactSpace T] in
+theorem phase_second
+    (hg : ContDiff ℝ ∞ g) (hs : ∀ x u v, g x u v = g x v u)
+    (hp : ∀ x v, v ≠ 0 → 0 < g x v v) (hS : ContDiff ℝ ∞ S)
+    (hp0 : ∀ t, metricGradient g S (y t) ≠ 0) (t : T × Fin 3) (v : Coord) :
+    sourceHessian g (fun z ↦ (b.phase t z).re) (y t.1) v v =
+      sourceHessian g S (y t.1) v v -
+        sourceDirectionEta (g (y t.1)) (sourceHessian g S (y t.1))
+          (metricGradient g S (y t.1)) (d.q t.1 t.2) * g (y t.1) v v := by
+  let f : Coord → ℝ := fun z ↦ (b.phase t z).re
+  have hf : ContDiffOn ℝ ∞ f (b.F t).target := by
+    intro z hz
+    exact (Complex.reCLM.contDiff.contDiffAt.comp z
+      ((reval_contDiff _).contDiffAt.comp z
+        ((b.chart_domain t).2.2.contDiffAt ((b.F t).open_target.mem_nhds hz)))).contDiffWithinAt
+  obtain ⟨r,A,hr,hA,_,_,_,_,hAe⟩ := common_local_smooth_extension
+    (b.F t).open_target f hf (b.center_mem_target t)
+  have hAf : A =ᶠ[𝓝 (y t.1)] f := hAe _ (Metric.mem_ball_self hr)
+  let Q : Coord → Coord := actualMetricChart g (y t.1) (d.frame t)
+  have hQ : ContDiff ℝ ∞ Q := quadraticChartMap_smooth _ _ _
+  have hQ0 : Q 0 = y t.1 := quadraticChartMap_zero _ _ _
+  have heF : (b.F t : Coord → Coord) = Q := b.chart_eq t
+  have hAQ : (A ∘ Q) =ᶠ[𝓝 (0:Coord)] (fun z ↦ (reval (b.phi t) z).re) := by
+    have hmap : Tendsto Q (𝓝 0) (𝓝 (y t.1)) := by simpa only [ContinuousAt,hQ0] using (hQ.continuous.continuousAt (x := 0))
+    filter_upwards [hAf.comp_tendsto hmap,(b.F t).open_source.mem_nhds (b.zero_mem_source t)] with z hz hzs
+    change A (Q z) = _
+    change A (Q z) = f (Q z) at hz
+    rw [hz]
+    change (reval (b.phi t) ((b.F t).symm (Q z))).re = _
+    rw [← heF,(b.F t).left_inv hzs]
+  have hHess : sourceHessian g A (y t.1) = sourceHessian g f (y t.1) :=
+    (sourceHessian_eventuallyEq (Filter.EventuallyEq.refl (𝓝 (y t.1)) g) hAf).self_of_nhds
+  let a := sourceTripleAlpha g S y t
+  let bb := sourceTripleBeta g S y t
+  let H := envelopeHessian (S ∘ Q)
+  have ha : a ≠ 0 := ne_of_gt (Real.sqrt_pos.mpr (hp (y t.1) (metricGradient g S (y t.1)) (hp0 t.1)))
+  have hb : bb ≠ 0 := by
+    apply ne_of_gt
+    apply Real.sqrt_pos.mpr
+    have hh := hp (y t.1) (metricGradient g S (y t.1)) (hp0 t.1)
+    linarith
+  have hSQ : ContDiff ℝ ∞ (S ∘ Q) := hS.comp hQ
+  have hj := retained_normal_phase_derivatives (b.phi t) (S (y t.1)) a bb H ha hb
+    (envelopeHessian_symmetric _ hSQ) (b.jets.phase_retained t)
+  have heta : Yau.contactEta a bb (H 0 0) (H 1 1) =
+      sourceDirectionEta (g (y t.1)) (sourceHessian g S (y t.1))
+        (metricGradient g S (y t.1)) (d.q t.1 t.2) := by
+    dsimp [H,Q]
+    rw [charted_envelope_hessian g hg hs hp S hS,charted_envelope_hessian g hg hs hp S hS]
+    exact source_eta_of_adapted_frame _ _ _ (d.frame_orthonormal t) _ _ a bb
+      (d.frame_first (fun t ↦ hp (y t)) hp0 t) (d.frame_second (fun t ↦ hp (y t)) hp0 t)
+  let u := (d.frame t).symm v
+  have hsum : (∑ i, ∑ j, H i j*u i*u j) = sourceHessian g S (y t.1) v v := by
+    rw [← envelopeHessian_diagonal]
+    simpa only [u,Q,ContinuousLinearEquiv.apply_symm_apply] using
+      scalar_chart_covariant_diagonal g hg hs hp S hS (y t.1) (d.frame t) u
+  have hnorm : (∑ i, (u i)^2) = g (y t.1) v v := by
+    simpa only [u,Q,ContinuousLinearEquiv.apply_symm_apply] using
+      (orthonormal_frame_norm_sq (g (y t.1)) (d.frame t) (d.frame_orthonormal t) u).symm
+  calc
+    _ = sourceHessian g A (y t.1) v v := by rw [hHess]
+    _ = iteratedFDeriv ℝ 2 (A ∘ Q) 0 (fun _ ↦ u) := by
+      simpa only [u,Q,ContinuousLinearEquiv.apply_symm_apply] using
+        (scalar_chart_covariant_diagonal g hg hs hp A hA (y t.1) (d.frame t) u).symm
+    _ = iteratedFDeriv ℝ 2 (fun z ↦ (reval (b.phi t) z).re) 0 (fun _ ↦ u) := by
+      rw [(hAQ.iteratedFDeriv ℝ 2).self_of_nhds]
+    _ = (iteratedFDeriv ℝ 2 (reval (b.phi t)) 0 (fun _ ↦ u)).re := by
+      exact congrArg (fun L : Coord [×2]→L[ℝ] ℝ ↦ L (fun _ ↦ u))
+        (Complex.reCLM.iteratedFDeriv_comp_left (reval_contDiff _).contDiffAt
+          (by change (↑(2:ℕ∞):ℕ∞ω) ≤ ↑(⊤:ℕ∞); exact WithTop.coe_le_coe.mpr le_top))
+    _ = _ := by rw [hj.2.2 u,hsum,hnorm,heta]
+
+end TripleSourceWaveData
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/WaveSize.lean b/lean/OAI/Geometry/NodalSets/Waves/WaveSize.lean
new file mode 100644
index 0000000000000000000000000000000000000000..c8a77db4a0bb69a7e6a2b71a06b29760c4bdd63f
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/WaveSize.lean
@@ -0,0 +1,80 @@
+import OAI.Geometry.NodalSets.Elliptic.CutoffProduct
+import OAI.Geometry.NodalSets.Waves.ExponentialWaveBounds
+
+namespace OAI
+
+namespace Yau.Jets
+open scoped ContDiff
+noncomputable section
+
+lemma mul_derivative_power_bound {f u : Coord → ℂ}
+    (hf : ContDiff ℝ ∞ f) (hu : ContDiff ℝ ∞ u) (k : ℕ) (x : Coord)
+    {A B N w : ℝ} (hA : 0 ≤ A) (hN : 0 ≤ N)
+    (hfa : ∀ j, j ≤ k → ‖iteratedFDeriv ℝ j f x‖ ≤ A * N ^ j)
+    (hub : ∀ j, j ≤ k → ‖iteratedFDeriv ℝ j u x‖ ≤ B * N ^ j * w) :
+    ‖iteratedFDeriv ℝ k (fun z ↦ f z * u z) x‖ ≤ (2 : ℝ) ^ k * A * B * N ^ k * w := by
+  have h := norm_iteratedFDeriv_mul_le hf hu x (n := k)
+    (by exact_mod_cast (show (k : ℕ∞) ≤ ⊤ from le_top))
+  calc
+    _ ≤ _ := h
+    _ ≤ ∑ j ∈ Finset.range (k + 1), (k.choose j : ℝ) * (A * B * N ^ k * w) := by
+      apply Finset.sum_le_sum
+      intro j hj
+      have hjk : j ≤ k := by have := Finset.mem_range.mp hj; omega
+      have hm := mul_le_mul (hfa j hjk) (hub (k - j) (Nat.sub_le _ _))
+        (norm_nonneg _) (mul_nonneg hA (pow_nonneg hN _))
+      have he : (A * N ^ j) * (B * N ^ (k - j) * w) = A * B * N ^ k * w := by
+        rw [show (A * N ^ j) * (B * N ^ (k - j) * w) =
+          A * B * (N ^ j * N ^ (k - j)) * w by ring, ← pow_add, Nat.add_sub_of_le hjk]
+      rw [he] at hm
+      simpa only [mul_assoc] using mul_le_mul_of_nonneg_left hm (Nat.cast_nonneg (k.choose j))
+    _ = _ := by
+      rw [← Finset.sum_mul]
+      have he : (∑ j ∈ Finset.range (k + 1), (k.choose j : ℝ)) = (2 : ℝ) ^ k := by
+        exact_mod_cast Nat.sum_range_choose k
+      rw [he]
+      ring
+
+theorem cutoff_wave_size (k : ℕ) : ∃ B > 0, ∀ (phi a : Coord → ℂ),
+    ContDiff ℝ ∞ phi → ContDiff ℝ ∞ a → ∀ (y x : Coord) (N A D S c : ℝ),
+    1 ≤ N → 0 ≤ A → 1 ≤ D →
+    (∀ j, j ≤ k → ‖iteratedFDeriv ℝ j a x‖ ≤ A) →
+    (∀ j, 1 ≤ j → j ≤ k → ‖iteratedFDeriv ℝ j phi x‖ ≤ D) →
+    (phi x).re - S ≤ -c * ‖x - y‖ ^ 2 →
+    ‖iteratedFDeriv ℝ k (fun z ↦ Yau.Waves.scaledCutoff N y z •
+      (a z * waveExp phi N z)) x‖ ≤
+      B * A * (k.factorial : ℝ) * D ^ k * N ^ k *
+        Real.exp (N * S - c * N * ‖x - y‖ ^ 2) := by
+  obtain ⟨B, hB, hcut⟩ := Yau.Waves.cutoff_preserves_derivative_exponent (E := Coord) (F := ℂ) k
+  refine ⟨(2 : ℝ) ^ k * B * (2 : ℝ) ^ k, by positivity, ?_⟩
+  intro phi a hp ha y x N A D S c hN hA hD hamp hphase hgap
+  let w := Real.exp (N * S - c * N * ‖x - y‖ ^ 2)
+  let Q := (k.factorial : ℝ) * D ^ k
+  have hQ : 0 ≤ Q := by dsimp [Q]; positivity
+  have hw : 0 ≤ w := (Real.exp_pos _).le
+  have he (j : ℕ) (hj : j ≤ k) : ‖iteratedFDeriv ℝ j (waveExp phi N) x‖ ≤ Q * N ^ j * w := by
+    apply (waveExp_gaussian_derivative_bound hp j x hD hN
+      (fun i hi hij ↦ hphase i hi (hij.trans hj)) hgap).trans
+    have hfact : (j.factorial : ℝ) ≤ k.factorial := by exact_mod_cast Nat.factorial_le hj
+    have hpow : D ^ j ≤ D ^ k := pow_le_pow_right₀ hD hj
+    dsimp [Q, w]
+    gcongr
+  have hau (j : ℕ) (hj : j ≤ k) :
+      ‖iteratedFDeriv ℝ j (fun z ↦ a z * waveExp phi N z) x‖ ≤
+        ((2 : ℝ) ^ k * A * Q) * N ^ j * w := by
+    apply (mul_derivative_power_bound ha (waveExp_contDiff hp N) j x hA
+      (le_trans zero_le_one hN) _ (fun i hi ↦ he i (hi.trans hj))).trans
+    · have hp2 : (2 : ℝ) ^ j ≤ 2 ^ k := pow_le_pow_right₀ (by norm_num) hj
+      gcongr
+    · intro i hi
+      exact (hamp i (hi.trans hj)).trans (le_mul_of_one_le_right hA (one_le_pow₀ hN))
+  have h := hcut N hN y x (fun z ↦ a z * waveExp phi N z) (ha.mul (waveExp_contDiff hp N))
+    ((2 : ℝ) ^ k * A * Q) w (by positivity) hw hau
+  convert h using 1
+  dsimp [Q, w]
+  ring
+
+end
+end Yau.Jets
+
+end OAI
diff --git a/lean/OAI/Geometry/NodalSets/Waves/WitnessedEuclideanWaves.lean b/lean/OAI/Geometry/NodalSets/Waves/WitnessedEuclideanWaves.lean
new file mode 100644
index 0000000000000000000000000000000000000000..7cb89c745405a65341a59eb34fb6b1cbf292d9af
--- /dev/null
+++ b/lean/OAI/Geometry/NodalSets/Waves/WitnessedEuclideanWaves.lean
@@ -0,0 +1,69 @@
+import OAI.Geometry.NodalSets.Elliptic.CoordinateNorm
+import OAI.Geometry.NodalSets.Waves.ConstructedSourceWavesLemmas
+
+namespace OAI
+
+namespace Yau.Geometry
+open Yau.Jets Filter Set
+open scoped ContDiff Topology
+noncomputable section
+variable {T : Type*} [TopologicalSpace T] [CompactSpace T]
+
+theorem witnessed_source_wave_estimates_euclidean
+    (g : Coord → Coord →L[ℝ] Coord →L[ℝ] ℝ) (hg : ContDiff ℝ ∞ g)
+    (hs : ∀ x u v, g x u v = g x v u) (hpos : ∀ x v, v ≠ 0 → 0 < g x v v)
+    (w : Coord → ℝ) (hw : ContDiff ℝ ∞ w) (S : Coord → ℝ) (hS : ContDiff ℝ ∞ S)
+    (y : T → Coord) (hy : Continuous y) (e : T → Coord ≃L[ℝ] Coord)
+    (he : Continuous (fun t ↦ (e t).toContinuousLinearMap))
+    (ho : ∀ t i j, g (y t) (e t (Pi.single i 1)) (e t (Pi.single j 1)) = if i=j then 1 else 0)
+    {U : Set Coord} (hU : IsOpen U) (hyU : ∀ t, y t ∈ U) (hwpos : ∀ x ∈ U, 0 < w x)
+    (q : T → Coord) (a bb : T → ℝ) (ha : Continuous a) (hbb : Continuous bb)
+    (ha0 : ∀ t, a t ≠ 0) (hb0 : ∀ t, bb t ≠ 0) (hlen : ∀ t, bb t^2 = a t^2+4)
+    (hap : ∀ t, a t • e t (Pi.single 0 1) = metricGradient g S (y t))
+    (hbq : ∀ t, bb t • e t (Pi.single 1 1) = q t)
+    (hstrict : ∀ t, 0 < sourceHessian g S (y t) (metricGradient g S (y t)) (metricGradient g S (y t)) +
+      sourceHessian g S (y t) (q t) (q t)) (m J : ℕ) (K k0 : ℕ) (hm : 3*K+4*k0+6 < m+1) (hJ : K+k0+1 ≤ J) :
+    ∃ (beta : ContDiffBump (0:Coord)) (phi : T → CPoly) (A : ℕ → T → CPoly)
+      (F : T → OpenPartialHomeomorph Coord Coord) (c Cw Cr R r d : ℝ),
+      0 < c ∧ 0 < Cw ∧ 0 < Cr ∧ 0 < R ∧ 0 < r ∧ 0 < d ∧
+      MetricWaveJetFamily (extendedPrincipal g y e beta) (extendedDrift g w y e beta)
+        (fun t ↦ S (y t)) a bb
+        (fun t ↦ envelopeHessian (S ∘ actualMetricChart g (y t) (e t))) m J phi A ∧
+      (∀ t, (F t : Coord → Coord) = actualMetricChart g (y t) (e t)) ∧
+      (∀ t, (F t).source = Metric.ball 0 r ∧ Metric.closedBall (y t) d ⊆ (F t).target ∧
+        ContDiffOn ℝ ∞ (F t).symm (F t).target) ∧
+      ∀ᶠ n : ℕ in atTop, ∀ t,
+        let V := chartPushforward (F t) (coordinateWave phi A J (n:ℝ) t)
+        ContDiff ℝ ∞ V ∧ HasCompactSupport V ∧
+        tsupport V ⊆ {z | sourceEuclideanNorm (z-y t) ≤ R*(n:ℝ)^(-1/3:ℝ)} ∧
+        ∀ z : Coord,
+          (∀ k : Fin (k0+1), ‖iteratedFDeriv ℝ k.val V z‖ ≤
+            Cw*(n:ℝ)^k.val*Real.exp ((n:ℝ)*S z-c*(n:ℝ)*(sourceEuclideanNorm (z-y t))^2)) ∧
+          DerivativeBound k0 (fun z ↦ sourceWeightedOperator g w V z +
+            ((4:ℂ)*(n:ℂ)^2+6*(n:ℂ))*V z) z
+              (Cr*(n:ℝ)^(-(K:ℝ))*Real.exp ((n:ℝ)*S z)) := by
+  obtain ⟨beta,phi,A,F,c,Cw,Cr,R,r,d,hc,hCw,hCr,hR,hr,hd,hjets,hFe,hdom,hest⟩ := witnessed_source_wave_estimates
+    g hg hs hpos w hw S hS y hy e he ho hU hyU hwpos q a bb ha hbb ha0 hb0 hlen hap hbq hstrict
+    m J K k0 hm hJ
+  refine ⟨beta,phi,A,F,c/4,Cw,Cr,2*R,r,d,by positivity,hCw,hCr,by positivity,hr,hd,hjets,hFe,hdom,?_⟩
+  filter_upwards [hest] with n hn
+  intro t
+  obtain ⟨hsm,hcomp,hsup,hbounds⟩ := hn t
+  refine ⟨hsm,hcomp,?_,?_⟩
+  · intro z hz
+    have hzsup : ‖z-y t‖ ≤ R*(n:ℝ)^(-1/3:ℝ) := by simpa only [Metric.mem_closedBall,dist_eq_norm] using hsup hz
+    calc
+      _ ≤ 2*‖z-y t‖ := sourceEuclideanNorm_le _
+      _ ≤ 2*(R*(n:ℝ)^(-1/3:ℝ)) := mul_le_mul_of_nonneg_left hzsup (by norm_num)
+      _ = _ := by ring
+  · intro z
+    refine ⟨?_,(hbounds z).2⟩
+    intro k
+    apply ((hbounds z).1 k).trans
+    have he := gaussian_source_euclidean (z-y t) c (n:ℝ) (S z) hc.le (by positivity)
+    exact mul_le_mul_of_nonneg_left he (by positivity)
+
+end
+end Yau.Geometry
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/ActualFiberFirstMoment.lean b/lean/OAI/Geometry/Perimeter/Regularity/ActualFiberFirstMoment.lean
new file mode 100644
index 0000000000000000000000000000000000000000..49f6a019771e190d29e5831cc3d8f7d2f12ff160
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/ActualFiberFirstMoment.lean
@@ -0,0 +1,135 @@
+import OAI.Geometry.Perimeter.Regularity.PhaseMomentLimit
+import OAI.Geometry.Perimeter.Regularity.ActualFluxWindowLimit
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff Convolution InnerProductSpace
+namespace RPISingularHarnack
+open RPI.Boundary ContinuousLinearMap
+variable {n : ℕ}
+
+theorem fiber_flux_first_moment {q : ℝ → ℝ} (hq : ContDiff ℝ 1 q)
+    {l u : ℝ} (hl : q l=1) (hu : q u=0) :
+    (∫ s in l..u, s*(-deriv q s)) = l+∫ s in l..u, q s := by
+  have h := intervalIntegral.integral_deriv_mul_eq_sub (a := l) (b := u)
+    (fun s _ ↦ hasDerivAt_id s) (fun s _ ↦ (hq.differentiable (by simp) s).hasDerivAt)
+    (continuous_const.intervalIntegrable _ _) (hq.continuous_deriv le_rfl |>.intervalIntegrable _ _)
+  simp only [one_mul,hu,hl,mul_zero,mul_one,zero_sub,id_eq] at h
+  have hi : IntervalIntegrable (fun s ↦ s*deriv q s) volume l u :=
+    (continuous_id.mul (hq.continuous_deriv le_rfl)).intervalIntegrable _ _
+  rw [intervalIntegral.integral_add (hq.continuous.intervalIntegrable _ _) hi] at h
+  simp_rw [mul_neg,intervalIntegral.integral_neg]
+  linarith
+
+theorem mollifiedNormalProfile_cap_values {E : Set (Euc n)}
+    {v : Euc n} (hv : ‖v‖ = 1) {k : Euc n → ℝ} (hu : ∫ x, k x=1)
+    {a b ε R : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) (hab : a+ε < b)
+    (hs : Function.support k ⊆ Metric.ball 0 ε)
+    (hphase : ∀ᵐ x ∂volume, x ∈ Metric.ball 0 R → a < |inner ℝ v x| →
+      (x ∈ E ↔ inner ℝ v x < 0))
+    (y : normalPlane v) (hy : ‖y‖+b+ε ≤ R) :
+    mollifiedNormalProfile E k v y (-b)=1 ∧ mollifiedNormalProfile E k v y b=0 := by
+  constructor
+  · apply phaseMollification_lower_cap hv hu ha hs hphase
+    · have h := norm_add_le (y:Euc n) ((-b)•v)
+      rw [norm_smul,hv,mul_one,Real.norm_eq_abs,abs_neg,abs_of_nonneg hb] at h
+      change ‖(y:Euc n)‖+b+ε ≤ R at hy
+      linarith
+    · rw [normal_fiber_inner hv]; linarith
+  · apply phaseMollification_upper_cap hv ha hs hphase
+    · have h := norm_add_le (y:Euc n) (b•v)
+      rw [norm_smul,hv,mul_one,Real.norm_eq_abs,abs_of_nonneg hb] at h
+      change ‖(y:Euc n)‖+b+ε ≤ R at hy
+      linarith
+    · rw [normal_fiber_inner hv]; exact hab
+
+theorem mollified_fiber_weighted_first_moment {E : Set (Euc n)} (hE : MeasurableSet E)
+    {v : Euc n} (hv : ‖v‖ = 1) {φ : normalPlane v → ℝ}
+    (hφ : Continuous φ) (hcφ : HasCompactSupport φ)
+    {k : Euc n → ℝ} (hk : ContDiff ℝ ∞ k) (hc : HasCompactSupport k)
+    (hu : ∫ x, k x = 1) {a b ε R : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) (hab : a+ε < b)
+    (hs : Function.support k ⊆ Metric.ball 0 ε)
+    (hphase : ∀ᵐ x ∂volume, x ∈ Metric.ball 0 R → a < |inner ℝ v x| →
+      (x ∈ E ↔ inner ℝ v x < 0))
+    (hKR : ∀ y ∈ tsupport φ, ‖y‖+b+ε ≤ R) :
+    (∫ x, normalWindowTest v φ b 1 x * (-fderiv ℝ (phaseMollification E k) x v)) =
+      ∫ y in tsupport φ, φ y*(-b+∫ s in (-b)..b, phaseMollification E k ((y:Euc n)+s•v)) := by
+  have hd := ((phaseMollification_contDiff hE hk hc).continuous_fderiv (by simp)).clm_apply
+    (g := fun _ ↦ v) continuous_const
+  have hi := normalWindowTest_integral hv hφ hcφ hb 1 hd.neg
+  dsimp only [Pi.neg_apply] at hi
+  rw [hi]
+  apply setIntegral_congr_fun hcφ.measurableSet
+  intro y hy
+  simp only [pow_one]
+  simp_rw [← mollifiedNormalProfile_deriv hE hk hc v]
+  have hcap := mollifiedNormalProfile_cap_values hv hu ha hb hab hs hphase y (hKR y hy)
+  rw [fiber_flux_first_moment ((mollifiedNormalProfile_contDiff hE hk hc v y).of_le (by simp))
+    hcap.1 hcap.2]
+  rfl
+
+theorem mollified_fiber_weighted_height_tendsto {E : Set (Euc n)} (hE : MeasurableSet E)
+    {v : Euc n} (hv : ‖v‖ = 1) {φ : normalPlane v → ℝ}
+    (hφ : Continuous φ) (hcφ : HasCompactSupport φ) {b : ℝ} (hb : 0 ≤ b) :
+    Tendsto (fun j ↦ ∫ y in tsupport φ, φ y*
+      (-b+∫ s in (-b)..b, phaseMollification E ((c1Mollifier j).normed volume)
+        ((y:Euc n)+s•v))) atTop
+      (𝓝 (∫ y in tsupport φ, φ y*exactFiberHeight E v b y)) := by
+  apply tendsto_integral_of_dominated_convergence (fun y ↦ ‖φ y‖*b)
+  · intro j
+    exact (hφ.mul (continuous_const.add (continuous_profile_integral (v := v) _
+      (phaseMollification_contDiff hE (c1Mollifier j).contDiff_normed
+        (c1Mollifier j).hasCompactSupport_normed).continuous (-b) b))).aestronglyMeasurable
+  · exact (hφ.continuousOn.integrableOn_compact hcφ).norm.mul_const _
+  · intro j
+    apply ae_of_all
+    intro y
+    rw [norm_mul]
+    apply mul_le_mul_of_nonneg_left _ (norm_nonneg _)
+    have h := interval_height_abs_le
+      (((phaseMollification_contDiff hE (c1Mollifier j).contDiff_normed
+        (c1Mollifier j).hasCompactSupport_normed).continuous.comp
+          (continuous_const.add (continuous_id.smul continuous_const))).measurable)
+      hb (fun s ↦ phaseMollification_c1_mem_Icc hE j ((y:Euc n)+s•v))
+    exact h
+  · filter_upwards [ae_restrict_of_ae (mollified_fiber_height_ae_tendsto hE hv hb)] with y hy
+    exact tendsto_const_nhds.mul hy
+
+theorem actual_fiber_signed_first_moment (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    {v : Euc n} (hv : ‖v‖ = 1) {φ : normalPlane v → ℝ}
+    (hφ : Continuous φ) (hcφ : HasCompactSupport φ)
+    {a b δ R : ℝ} (ha : 0 ≤ a) (hδ : 0 < δ) (hab : a+δ < b) (hR : 0 < R)
+    (hKR : ∀ y ∈ tsupport φ, ‖y‖+b+δ ≤ R)
+    (hh : ∀ x ∈ perimeterSupport E ∩ Metric.ball 0 R, |inner ℝ v x| ≤ a)
+    (hphase : ∀ᵐ x ∂volume, x ∈ Metric.ball 0 R → a < |inner ℝ v x| →
+      (x ∈ E ↔ inner ℝ v x < 0)) :
+    (∫ x in Metric.ball 0 R, normalWindowTest v φ b 1 x *
+      inner ℝ (approximateNormal E x) v ∂perimeterMeasure E) =
+      ∫ y in tsupport φ, φ y*exactFiberHeight E v b y := by
+  have hb : 0 ≤ b := by linarith
+  have hh' : ∀ x ∈ perimeterSupport E ∩ Metric.ball 0 R, |inner ℝ v x| < b :=
+    fun x hx ↦ (hh x hx).trans_lt (by linarith)
+  have hA := actual_normal_flux_window_tendsto hn hmin hv hφ hcφ hb hδ hR 1 hKR hh'
+  have hB := mollified_fiber_weighted_height_tendsto hmin.1 hv hφ hcφ hb
+  apply tendsto_nhds_unique hA
+  apply hB.congr'
+  have he : ∀ᶠ j in atTop, (c1Mollifier (n := n) j).rOut < δ :=
+    c1Mollifier_radius_tendsto.eventually (gt_mem_nhds hδ)
+  filter_upwards [he] with j hj
+  have hs : Function.support ((c1Mollifier (n := n) j).normed volume) ⊆ Metric.ball 0 δ := by
+    intro x hx
+    have hx' := subset_tsupport _ hx
+    rw [(c1Mollifier j).tsupport_normed_eq] at hx'
+    exact Metric.closedBall_subset_ball hj hx'
+  exact (mollified_fiber_weighted_first_moment hmin.1 hv hφ hcφ
+    (c1Mollifier j).contDiff_normed (c1Mollifier j).hasCompactSupport_normed
+    (c1Mollifier j).integral_normed ha hb hab hs hphase hKR).symm
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/ActualFiberMoments.lean b/lean/OAI/Geometry/Perimeter/Regularity/ActualFiberMoments.lean
new file mode 100644
index 0000000000000000000000000000000000000000..5c88007b90d6a8aa7f9f016238e8a4f4ac5295ce
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/ActualFiberMoments.lean
@@ -0,0 +1,86 @@
+import OAI.Geometry.Perimeter.Regularity.ActualFiberSecondMoment
+import OAI.Geometry.Perimeter.Regularity.NormalFluxError
+import OAI.Geometry.Perimeter.Regularity.ActualFiberFirstMoment
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff Convolution InnerProductSpace
+namespace RPISingularHarnack
+open RPI.Boundary ContinuousLinearMap
+variable {n : ℕ}
+
+theorem actual_fiber_first_moment_error (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    {v : Euc n} (hv : ‖v‖ = 1) {φ : normalPlane v → ℝ}
+    (hφ : Continuous φ) (hcφ : HasCompactSupport φ)
+    {M a b δ R : ℝ} (hM : ∀ y, ‖φ y‖ ≤ M)
+    (ha : 0 ≤ a) (hδ : 0 < δ) (hab : a+δ < b) (hR : 0 < R)
+    (hKR : ∀ y ∈ tsupport φ, ‖y‖+b+δ ≤ R)
+    (hh : ∀ x ∈ perimeterSupport E ∩ Metric.ball 0 R, |inner ℝ v x| ≤ a)
+    (hphase : ∀ᵐ x ∂volume, x ∈ Metric.ball 0 R → a < |inner ℝ v x| →
+      (x ∈ E ↔ inner ℝ v x < 0)) :
+    |(∫ x in Metric.ball 0 R, normalWindowTest v φ b 1 x ∂perimeterMeasure E) -
+      (∫ y in tsupport φ, φ y*exactFiberHeight E v b y)| ≤
+        (M*b/2)*orientedTilt E v (Metric.ball 0 R) := by
+  rw [← actual_fiber_signed_first_moment hn hmin hv hφ hcφ ha hδ hab hR hKR hh hphase]
+  have h := normal_flux_error_le_tilt hn hmin hv hR
+    (normalWindowTest_measurable hφ b 1)
+    (normalWindowTest_bound hM (by linarith : 0 ≤ b) 1)
+  simpa only [pow_one,div_mul_eq_mul_div] using h
+
+theorem actual_fiber_second_moment_error (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    {v : Euc n} (hv : ‖v‖ = 1) {φ : normalPlane v → ℝ}
+    (hφ : Continuous φ) (hcφ : HasCompactSupport φ) (hpφ : ∀ y, 0 ≤ φ y)
+    {M a b δ R : ℝ} (hM : ∀ y, ‖φ y‖ ≤ M)
+    (ha : 0 ≤ a) (hδ : 0 < δ) (hab : a+δ < b) (hR : 0 < R)
+    (hKR : ∀ y ∈ tsupport φ, ‖y‖+b+δ ≤ R)
+    (hh : ∀ x ∈ perimeterSupport E ∩ Metric.ball 0 R, |inner ℝ v x| ≤ a)
+    (hphase : ∀ᵐ x ∂volume, x ∈ Metric.ball 0 R → a < |inner ℝ v x| →
+      (x ∈ E ↔ inner ℝ v x < 0)) :
+    |(∫ x in Metric.ball 0 R, normalWindowTest v φ b 2 x ∂perimeterMeasure E) -
+      (∫ y in tsupport φ, φ y*(exactFiberHeight E v b y)^2)| ≤
+        (9/2*b^2*M)*orientedTilt E v (Metric.ball 0 R) := by
+  have hf := normal_flux_error_le_tilt hn hmin hv hR
+    (normalWindowTest_measurable hφ b 2)
+    (normalWindowTest_bound hM (by linarith : 0 ≤ b) 2)
+  have hs := actual_fiber_signed_second_moment hn hmin hv hφ hcφ hpφ hM
+    ha hδ hab hR hKR hh hphase
+  dsimp only at hs
+  have habs : |(∫ x in Metric.ball 0 R, normalWindowTest v φ b 2 x *
+      inner ℝ (approximateNormal E x) v ∂perimeterMeasure E) -
+      (∫ y in tsupport φ, φ y*(exactFiberHeight E v b y)^2)| ≤
+      4*b^2*M*orientedTilt E v (Metric.ball 0 R) := by
+    rw [abs_of_nonneg hs.1]; exact hs.2
+  calc
+    _ ≤ |(∫ x in Metric.ball 0 R, normalWindowTest v φ b 2 x ∂perimeterMeasure E) -
+        (∫ x in Metric.ball 0 R, normalWindowTest v φ b 2 x *
+          inner ℝ (approximateNormal E x) v ∂perimeterMeasure E)| +
+      |(∫ x in Metric.ball 0 R, normalWindowTest v φ b 2 x *
+          inner ℝ (approximateNormal E x) v ∂perimeterMeasure E) -
+        (∫ y in tsupport φ, φ y*(exactFiberHeight E v b y)^2)| := abs_sub_le _ _ _
+    _ ≤ (M*b^2/2)*orientedTilt E v (Metric.ball 0 R) +
+        4*b^2*M*orientedTilt E v (Metric.ball 0 R) := add_le_add (by simpa only [div_mul_eq_mul_div] using hf) habs
+    _ = _ := by ring
+
+theorem normalized_second_moment_defect_tendsto_zero
+    (A B a e : ℕ → ℝ) {C : ℝ} (hC : 0 ≤ C)
+    (ha : ∀ i, 0 < a i) (he : Tendsto e atTop (𝓝 0))
+    (hbound : ∀ i, |A i-B i| ≤ C*(a i)^2*e i) :
+    Tendsto (fun i ↦ (A i-B i)/(a i)^2) atTop (𝓝 0) := by
+  apply squeeze_zero_norm (fun i ↦ ?_) (by simpa using he.norm.const_mul C)
+  rw [Real.norm_eq_abs,abs_div,abs_of_pos (sq_pos_of_pos (ha i))]
+  apply (div_le_iff₀ (sq_pos_of_pos (ha i))).mpr
+  calc
+    |A i-B i| ≤ C*(a i)^2*e i := hbound i
+    _ ≤ C*(a i)^2*‖e i‖ :=
+      mul_le_mul_of_nonneg_left (le_abs_self _) (mul_nonneg hC (sq_nonneg _))
+    _ = C*‖e i‖*(a i)^2 := by ring
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/ActualFiberSecondMoment.lean b/lean/OAI/Geometry/Perimeter/Regularity/ActualFiberSecondMoment.lean
new file mode 100644
index 0000000000000000000000000000000000000000..bc96a524ba625c8fa022272e2d58db40486d7bfc
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/ActualFiberSecondMoment.lean
@@ -0,0 +1,67 @@
+import OAI.Geometry.Perimeter.Regularity.WeightedFiberMoment
+import OAI.Geometry.Perimeter.Regularity.WeightedWrongWayFlux
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff Convolution InnerProductSpace
+namespace RPISingularHarnack
+open RPI.Boundary ContinuousLinearMap
+variable {n : ℕ}
+
+theorem actual_fiber_signed_second_moment (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    {v : Euc n} (hv : ‖v‖ = 1) {φ : normalPlane v → ℝ}
+    (hφ : Continuous φ) (hcφ : HasCompactSupport φ) (hpφ : ∀ y, 0 ≤ φ y)
+    {M a b δ R : ℝ} (hM : ∀ y, ‖φ y‖ ≤ M)
+    (ha : 0 ≤ a) (hδ : 0 < δ) (hab : a+δ < b) (hR : 0 < R)
+    (hKR : ∀ y ∈ tsupport φ, ‖y‖+b+δ ≤ R)
+    (hh : ∀ x ∈ perimeterSupport E ∩ Metric.ball 0 R, |inner ℝ v x| ≤ a)
+    (hphase : ∀ᵐ x ∂volume, x ∈ Metric.ball 0 R → a < |inner ℝ v x| →
+      (x ∈ E ↔ inner ℝ v x < 0)) :
+    let A := ∫ x in Metric.ball 0 R, normalWindowTest v φ b 2 x *
+      inner ℝ (approximateNormal E x) v ∂perimeterMeasure E
+    let B := ∫ y in tsupport φ, φ y*(exactFiberHeight E v b y)^2
+    0 ≤ A-B ∧ A-B ≤ 4*b^2*M*orientedTilt E v (Metric.ball 0 R) := by
+  dsimp only
+  have hb : 0 ≤ b := by linarith
+  have hh' : ∀ x ∈ perimeterSupport E ∩ Metric.ball 0 R, |inner ℝ v x| < b :=
+    fun x hx ↦ (hh x hx).trans_lt (by linarith)
+  have hA := actual_normal_flux_window_tendsto hn hmin hv hφ hcφ hb hδ hR 2 hKR hh'
+  have hB := mollified_fiber_weighted_height_square_tendsto hmin.1 hv hφ hcφ hb
+  have hD := normalWindowTest_diffuse_tendsto_zero hmin.1 hv hφ hcφ b
+  have he : ∀ᶠ j in atTop, (c1Mollifier (n := n) j).rOut < δ :=
+    c1Mollifier_radius_tendsto.eventually (gt_mem_nhds hδ)
+  have hdef : ∀ᶠ j in atTop,
+      let q := phaseMollification E ((c1Mollifier j).normed volume)
+      let A := ∫ x, normalWindowTest v φ b 2 x * (-fderiv ℝ q x v)
+      let B := ∫ y in tsupport φ, φ y*(-b+∫ s in (-b)..b, q ((y:Euc n)+s•v))^2
+      0 ≤ A-B ∧ A-B ≤
+        4*b*(∫ x, normalWindowTest v φ b 0 x * (q x*(1-q x))) +
+        4*b^2*M*orientedTilt E v (Metric.ball 0 R) := by
+    filter_upwards [he] with j hj
+    have hs : Function.support ((c1Mollifier (n := n) j).normed volume) ⊆ Metric.ball 0 δ := by
+      intro x hx
+      have hx' := subset_tsupport _ hx
+      rw [(c1Mollifier j).tsupport_normed_eq] at hx'
+      exact Metric.closedBall_subset_ball hj hx'
+    have ht := mollified_fiber_weighted_defect hmin.1 hv hφ hcφ hpφ
+      (c1Mollifier j).contDiff_normed (c1Mollifier j).hasCompactSupport_normed
+      (c1Mollifier j).nonneg_normed (c1Mollifier j).integral_normed ha hb hab hs hphase hKR
+    have hp := weighted_positive_derivative_le_tilt hn hmin hv hφ hcφ hpφ hM hb hR hKR hj
+    dsimp only at ht ⊢
+    refine ⟨ht.1,ht.2.trans ?_⟩
+    have hm := mul_le_mul_of_nonneg_left hp (by positivity : 0 ≤ 8*b^2)
+    nlinarith
+  refine ⟨ge_of_tendsto (hA.sub hB) (hdef.mono fun j hj ↦ hj.1),?_⟩
+  have ht := le_of_tendsto_of_tendsto (hA.sub hB)
+    ((hD.const_mul (4*b)).add_const (4*b^2*M*orientedTilt E v (Metric.ball 0 R)))
+    (hdef.mono fun j hj ↦ hj.2)
+  simpa only [mul_zero,zero_add] using ht
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/ActualFiberZerothMoment.lean b/lean/OAI/Geometry/Perimeter/Regularity/ActualFiberZerothMoment.lean
new file mode 100644
index 0000000000000000000000000000000000000000..70c5ad50ea3c49c9e9791184bd42ac30e9e0306c
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/ActualFiberZerothMoment.lean
@@ -0,0 +1,90 @@
+import OAI.Geometry.Perimeter.Regularity.NormalFluxError
+import OAI.Geometry.Perimeter.Regularity.ActualFiberFirstMoment
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+open RPI.Boundary ContinuousLinearMap
+variable {n : ℕ}
+
+theorem mollified_fiber_weighted_zeroth_moment {E : Set (Euc n)} (hE : MeasurableSet E)
+    {v : Euc n} (hv : ‖v‖=1) {φ : normalPlane v → ℝ}
+    (hφ : Continuous φ) (hcφ : HasCompactSupport φ)
+    {k : Euc n → ℝ} (hk : ContDiff ℝ ∞ k) (hc : HasCompactSupport k)
+    (hu : ∫ x, k x=1) {a b ε R : ℝ} (ha : 0≤a) (hb : 0≤b) (hab : a+ε<b)
+    (hs : Function.support k ⊆ Metric.ball 0 ε)
+    (hphase : ∀ᵐ x ∂volume, x ∈ Metric.ball 0 R → a < |inner ℝ v x| → (x∈E ↔ inner ℝ v x<0))
+    (hKR : ∀ y ∈ tsupport φ, ‖y‖+b+ε≤R) :
+    (∫ x, normalWindowTest v φ b 0 x*(-fderiv ℝ (phaseMollification E k) x v)) =
+      ∫ y in tsupport φ, φ y := by
+  have hd := ((phaseMollification_contDiff hE hk hc).continuous_fderiv (by simp)).clm_apply
+    (g := fun _ ↦ v) continuous_const
+  have hi := normalWindowTest_integral hv hφ hcφ hb 0 hd.neg
+  dsimp only [Pi.neg_apply] at hi
+  rw [hi]
+  apply setIntegral_congr_fun hcφ.measurableSet
+  intro y hy
+  simp only [pow_zero,one_mul]
+  simp_rw [← mollifiedNormalProfile_deriv hE hk hc v]
+  have hq : ContDiff ℝ 1 (mollifiedNormalProfile E k v y) :=
+    (mollifiedNormalProfile_contDiff hE hk hc v y).of_le (by simp)
+  have hcap := mollifiedNormalProfile_cap_values hv hu ha hb hab hs hphase y (hKR y hy)
+  rw [intervalIntegral.integral_neg,intervalIntegral.integral_eq_sub_of_hasDerivAt
+    (fun s _ ↦ (hq.differentiable (by simp) s).hasDerivAt)
+    ((hq.continuous_deriv le_rfl).intervalIntegrable _ _),hcap.1,hcap.2]
+  ring
+
+theorem actual_fiber_signed_zeroth_moment (hn : 1<n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    {v : Euc n} (hv : ‖v‖=1) {φ : normalPlane v → ℝ}
+    (hφ : Continuous φ) (hcφ : HasCompactSupport φ)
+    {a b δ R : ℝ} (ha : 0≤a) (hδ : 0<δ) (hab : a+δ<b) (hR : 0<R)
+    (hKR : ∀ y ∈ tsupport φ, ‖y‖+b+δ≤R)
+    (hh : ∀ x ∈ perimeterSupport E ∩ Metric.ball 0 R, |inner ℝ v x|≤a)
+    (hphase : ∀ᵐ x ∂volume, x ∈ Metric.ball 0 R → a < |inner ℝ v x| →
+      (x∈E ↔ inner ℝ v x<0)) :
+    (∫ x in Metric.ball 0 R, normalWindowTest v φ b 0 x*
+      inner ℝ (approximateNormal E x) v ∂perimeterMeasure E) = ∫ y in tsupport φ, φ y := by
+  have hb : 0≤b := by linarith
+  have hh' : ∀ x ∈ perimeterSupport E ∩ Metric.ball 0 R, |inner ℝ v x|<b :=
+    fun x hx ↦ (hh x hx).trans_lt (by linarith)
+  have hA := actual_normal_flux_window_tendsto hn hmin hv hφ hcφ hb hδ hR 0 hKR hh'
+  apply tendsto_nhds_unique hA
+  apply tendsto_const_nhds.congr'
+  have he : ∀ᶠ j in atTop, (c1Mollifier (n := n) j).rOut<δ :=
+    c1Mollifier_radius_tendsto.eventually (gt_mem_nhds hδ)
+  filter_upwards [he] with j hj
+  have hs : Function.support ((c1Mollifier (n := n) j).normed volume) ⊆ Metric.ball 0 δ := by
+    intro x hx
+    have hx' := subset_tsupport _ hx
+    rw [(c1Mollifier j).tsupport_normed_eq] at hx'
+    exact Metric.closedBall_subset_ball hj hx'
+  exact (mollified_fiber_weighted_zeroth_moment hmin.1 hv hφ hcφ
+    (c1Mollifier j).contDiff_normed (c1Mollifier j).hasCompactSupport_normed
+    (c1Mollifier j).integral_normed ha hb hab hs hphase hKR).symm
+
+theorem actual_fiber_zeroth_moment_error (hn : 1<n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    {v : Euc n} (hv : ‖v‖=1) {φ : normalPlane v → ℝ}
+    (hφ : Continuous φ) (hcφ : HasCompactSupport φ)
+    {M a b δ R : ℝ} (hM : ∀ y, ‖φ y‖≤M)
+    (ha : 0≤a) (hδ : 0<δ) (hab : a+δ<b) (hR : 0<R)
+    (hKR : ∀ y ∈ tsupport φ, ‖y‖+b+δ≤R)
+    (hh : ∀ x ∈ perimeterSupport E ∩ Metric.ball 0 R, |inner ℝ v x|≤a)
+    (hphase : ∀ᵐ x ∂volume, x ∈ Metric.ball 0 R → a < |inner ℝ v x| →
+      (x∈E ↔ inner ℝ v x<0)) :
+    |(∫ x in Metric.ball 0 R, normalWindowTest v φ b 0 x ∂perimeterMeasure E)-
+      (∫ y in tsupport φ, φ y)| ≤ M/2*orientedTilt E v (Metric.ball 0 R) := by
+  rw [← actual_fiber_signed_zeroth_moment hn hmin hv hφ hcφ ha hδ hab hR hKR hh hphase]
+  have h := normal_flux_error_le_tilt hn hmin hv hR (normalWindowTest_measurable hφ b 0)
+    (normalWindowTest_bound hM (by linarith : 0≤b) 0)
+  simpa only [pow_zero,mul_one,div_mul_eq_mul_div] using h
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/ActualFluxWindowLimit.lean b/lean/OAI/Geometry/Perimeter/Regularity/ActualFluxWindowLimit.lean
new file mode 100644
index 0000000000000000000000000000000000000000..18d9a73d26409548a7d799022c4e8b9cad4768a4
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/ActualFluxWindowLimit.lean
@@ -0,0 +1,89 @@
+import OAI.Geometry.Perimeter.Regularity.PerimeterMomentLimit
+import OAI.Geometry.Perimeter.Regularity.NormalWindowTest
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff Convolution InnerProductSpace
+namespace RPISingularHarnack
+open RPI.Boundary ContinuousLinearMap
+variable {n : ℕ}
+
+theorem c1Mollifier_translated_support {j : ℕ} {x : Euc n} {R : ℝ}
+    (hx : ‖x‖+(c1Mollifier (n := n) j).rOut < R) :
+    tsupport (fun y ↦ (c1Mollifier j).normed volume (x-y)) ⊆ Metric.ball 0 R := by
+  intro y hy
+  have hs : x-y ∈ tsupport ((c1Mollifier j).normed volume) :=
+    tsupport_comp_subset_preimage ((c1Mollifier j).normed volume)
+      (continuous_const.sub continuous_id) hy
+  rw [(c1Mollifier j).tsupport_normed_eq] at hs
+  have hnorm : ‖x-y‖ ≤ (c1Mollifier (n := n) j).rOut := by
+    simpa only [Metric.mem_closedBall,dist_zero_right] using hs
+  rw [Metric.mem_ball,dist_zero_right]
+  have h := norm_sub_le x (x-y)
+  rw [sub_sub_cancel] at h
+  linarith
+
+theorem normalWindowTest_ae_continuousAt {E : Set (Euc n)}
+    (hmin : IsLocallyPerimeterMinimizing E) {v : Euc n}
+    {φ : normalPlane v → ℝ} (hφ : Continuous φ) {b R : ℝ} (m : ℕ)
+    (hh : ∀ x ∈ perimeterSupport E ∩ Metric.ball 0 R, |inner ℝ v x| < b) :
+    ∀ᵐ x ∂(perimeterMeasure E).restrict (Metric.ball 0 R),
+      ContinuousAt (normalWindowTest v φ b m) x := by
+  let := perimeterMeasure_locallyFinite E hmin.2.1
+  have hs : ∀ᵐ x ∂perimeterMeasure E, x ∈ perimeterSupport E := by
+    rw [← perimeterMeasure_support]
+    exact (perimeterMeasure E).support_mem_ae
+  filter_upwards [ae_restrict_of_ae hs,ae_restrict_mem measurableSet_ball] with x hx hxR
+  exact normalWindowTest_continuousAt hφ m (hh x ⟨hx,hxR⟩)
+
+theorem actual_normal_flux_window_tendsto (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    {v : Euc n} (hv : ‖v‖ = 1) {φ : normalPlane v → ℝ}
+    (hφ : Continuous φ) (hcφ : HasCompactSupport φ)
+    {b δ R : ℝ} (hb : 0 ≤ b) (hδ : 0 < δ) (hR : 0 < R) (m : ℕ)
+    (hKR : ∀ y ∈ tsupport φ, ‖y‖+b+δ ≤ R)
+    (hh : ∀ x ∈ perimeterSupport E ∩ Metric.ball 0 R, |inner ℝ v x| < b) :
+    Tendsto (fun j ↦ ∫ x, normalWindowTest v φ b m x *
+      (-fderiv ℝ (phaseMollification E ((c1Mollifier j).normed volume)) x v)) atTop
+      (𝓝 (∫ x in Metric.ball 0 R, normalWindowTest v φ b m x *
+        inner ℝ (approximateNormal E x) v ∂perimeterMeasure E)) := by
+  let μ := (perimeterMeasure E).restrict (Metric.ball 0 R)
+  have : IsFiniteMeasure μ := ⟨by
+    rw [Measure.restrict_apply_univ,perimeterMeasure_open E Metric.isOpen_ball]
+    exact hmin.2.1 0 R hR⟩
+  let f : Euc n → ℝ := fun x ↦ inner ℝ (approximateNormal E x) v
+  have hf : Integrable f μ := (integrable_const (1:ℝ)).mono'
+    (((measurable_approximateNormal hmin.1).inner measurable_const).aestronglyMeasurable)
+    (ae_of_all _ fun x ↦ by
+      change ‖inner ℝ (approximateNormal E x) v‖ ≤ 1
+      exact (norm_inner_le_norm (𝕜 := ℝ) _ _).trans (by
+        rw [hv,mul_one]; exact approximateNormal_norm_le_one E x))
+  obtain ⟨M,hM⟩ := hcφ.exists_bound_of_continuous hφ
+  have ht := weak_c1_convolution_tendsto_of_ae_continuous μ hf
+    (normalWindowTest_measurable hφ b m) (normalWindowTest_bound hM hb m)
+    (normalWindowTest_ae_continuousAt hmin hφ m hh)
+  apply ht.congr'
+  have he : ∀ᶠ j in atTop, (c1Mollifier (n := n) j).rOut < δ :=
+    c1Mollifier_radius_tendsto.eventually (gt_mem_nhds hδ)
+  filter_upwards [he] with j hj
+  apply integral_congr_ae
+  apply ae_of_all
+  intro x
+  dsimp only
+  by_cases hx : normalWindowTest v φ b m x=0
+  · simp only [hx,zero_mul]
+  · have hs := c1Mollifier_translated_support (j := j) (x := x)
+      (R := R) (by have h := normalWindowTest_support_norm hv φ b δ R m hKR hx; linarith)
+    rw [phaseMollification_fderiv hn hmin (c1Mollifier j).contDiff_normed
+      (c1Mollifier j).hasCompactSupport_normed x v hR hs,neg_neg]
+    congr 1
+    apply integral_congr_ae
+    exact ae_of_all _ fun y ↦ mul_comm _ _
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/AnnularCutoffTestAdmissible.lean b/lean/OAI/Geometry/Perimeter/Regularity/AnnularCutoffTestAdmissible.lean
new file mode 100644
index 0000000000000000000000000000000000000000..c3614f6333d125f6dc4c09cd7ff0c20e774fb4cf
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/AnnularCutoffTestAdmissible.lean
@@ -0,0 +1,524 @@
+import OAI.Geometry.Perimeter.Regularity.PerimeterEqOfAe
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+
+theorem annularCutoff_test_admissible {n : ℕ} {U : Set (Euc n)} {V : Euc n → Euc n}
+    (hV : PerimeterAdmissible U V) {c : Euc n} {r a b : ℝ}
+    (ha : 0 < a) (hra : r < a) (hab : a < b) :
+    PerimeterAdmissible (U \ Metric.closedBall c r)
+      (fun x ↦ annularCutoff c a b x • V x) := by
+  have hc : HasCompactSupport V := hV.2.1
+  have hcη : HasCompactSupport ((annularCutoff c a b) • V) :=
+    HasCompactSupport.smul_left (f := annularCutoff c a b) hc
+  refine ⟨(annularCutoff_contDiff c a b).smul hV.1, hcη, ?_, ?_⟩
+  · intro x hx
+    have hsV : x ∈ tsupport V := tsupport_smul_subset_right (annularCutoff c a b) V hx
+    have hsη : x ∈ tsupport (annularCutoff c a b) := tsupport_smul_subset_left (annularCutoff c a b) V hx
+    refine ⟨hV.2.2.1 hsV, ?_⟩
+    have hax := tsupport_annularCutoff_subset ha hab hsη
+    exact not_le.mpr (hra.trans_le hax)
+  · intro x
+    rw [norm_smul, Real.norm_eq_abs, abs_of_nonneg (annularCutoff_mem_Icc c a b x).1]
+    exact (mul_le_of_le_one_left (norm_nonneg _)
+      (annularCutoff_mem_Icc c a b x).2).trans (hV.2.2.2 x)
+
+theorem integral_annularCutoff_divergence_le {n : ℕ} {E U : Set (Euc n)}
+    {V : Euc n → Euc n} (hV : PerimeterAdmissible U V) {c : Euc n} {r a b L : ℝ}
+    (ha : 0 < a) (hra : r < a) (hab : a < b) (hL : 0 < L)
+    (hbound : ∀ s, |deriv Real.smoothTransition s| ≤ L)
+    (hfin : perimeter E (U \ Metric.closedBall c r) ≠ ⊤) :
+    (∫ x in E, annularCutoff c a b x * divergence V x) ≤
+      (perimeter E (U \ Metric.closedBall c r)).toReal +
+        2 * L * n * b ^ (n - 1) * volume.real (Metric.ball (0 : Euc n) 1) := by
+  have hη := annularCutoff_test_admissible (c := c) hV ha hra hab
+  have hd := integrable_divergence hV.1 hV.2.1
+  have hcd : HasCompactSupport (divergence V) :=
+    hV.2.1.of_isClosed_subset (isClosed_tsupport _) (tsupport_divergence_subset V)
+  have hct : HasCompactSupport ((annularCutoff c a b) * (divergence V)) :=
+    HasCompactSupport.mul_left (f := annularCutoff c a b) hcd
+  have ht : Integrable (fun x ↦ annularCutoff c a b x * divergence V x) volume :=
+    ((annularCutoff_contDiff c a b).continuous.mul (continuous_divergence hV.1)).integrable_of_hasCompactSupport
+      hct
+  have hg : Integrable (fun x ↦ ‖gradient (annularCutoff c a b) x‖) volume :=
+    ((gradient_annularCutoff_continuous c a b).integrable_of_hasCompactSupport
+      (gradient_annularCutoff_compact c ha hab)).norm
+  have hinner : ∀ x, |inner ℝ (gradient (annularCutoff c a b) x) (V x)| ≤
+      ‖gradient (annularCutoff c a b) x‖ := by
+    intro x
+    exact (abs_real_inner_le_norm _ _).trans
+      (mul_le_of_le_one_right (norm_nonneg _) (hV.2.2.2 x))
+  have hi : Integrable (fun x ↦ inner ℝ (gradient (annularCutoff c a b) x) (V x)) volume :=
+    hg.mono' ((gradient_annularCutoff_continuous c a b).inner hV.1.continuous).aestronglyMeasurable
+      (Eventually.of_forall hinner)
+  have hvar : (∫ x in E, divergence (fun y ↦ annularCutoff c a b y • V y) x) ≤
+      (perimeter E (U \ Metric.closedBall c r)).toReal := by
+    apply (ENNReal.ofReal_le_iff_le_toReal hfin).mp
+    rw [perimeter_eq_iSup_tests]
+    exact le_iSup_of_le _ (le_iSup_of_le hη le_rfl)
+  have heq : (∫ x in E, divergence (fun y ↦ annularCutoff c a b y • V y) x) =
+      (∫ x in E, annularCutoff c a b x * divergence V x) +
+      (∫ x in E, inner ℝ (gradient (annularCutoff c a b) x) (V x)) := by
+    simp_rw [divergence_smul ((annularCutoff_contDiff c a b).differentiable (by norm_num))
+      (hV.1.differentiable (by norm_num))]
+    exact integral_add ht.integrableOn hi.integrableOn
+  have herr : -(∫ x in E, inner ℝ (gradient (annularCutoff c a b) x) (V x)) ≤
+      ∫ x, ‖gradient (annularCutoff c a b) x‖ := by
+    rw [← integral_neg]
+    exact (integral_mono hi.integrableOn.neg hg.integrableOn
+      (fun x ↦ (neg_le_abs _).trans (hinner x))).trans
+      (setIntegral_le_integral hg (Eventually.of_forall fun x ↦ norm_nonneg _))
+  have henergy := integral_gradient_annularCutoff_le c ha hab hL hbound
+  linarith
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+
+theorem annularCutoff_tendsto {n : ℕ} (c x : Euc n) {r : ℝ} (hr : 0 < r) :
+    Tendsto (fun δ : ℝ ↦ annularCutoff c (r + δ) (r + 2 * δ) x)
+      (𝓝[>] (0 : ℝ)) (𝓝 ((Metric.closedBall c r)ᶜ.indicator (fun _ ↦ (1 : ℝ)) x)) := by
+  by_cases hx : x ∈ Metric.closedBall c r
+  · rw [indicator_of_notMem (by simpa using hx)]
+    apply tendsto_const_nhds.congr'
+    filter_upwards [self_mem_nhdsWithin] with δ hδ
+    change 0 < δ at hδ
+    symm
+    exact annularCutoff_zero (by simpa using add_pos hr hδ) (by linarith [show 0 < δ from hδ])
+      (hx.trans (by linarith : r ≤ r + δ))
+  · rw [indicator_of_mem (show x ∈ (Metric.closedBall c r)ᶜ from hx)]
+    have hd : 0 < (dist x c - r) / 2 := by
+      have hh : r < dist x c := lt_of_not_ge hx
+      linarith
+    apply tendsto_const_nhds.congr'
+    filter_upwards [self_mem_nhdsWithin,
+      (nhdsWithin_le_nhds (gt_mem_nhds hd))] with δ hδ hsmall
+    change 0 < δ at hδ
+    symm
+    exact annularCutoff_one (by simpa using add_pos hr hδ) (by linarith [show 0 < δ from hδ])
+      (by change δ < (dist x c - r) / 2 at hsmall; linarith)
+
+theorem integral_divergence_outside_le {n : ℕ} {E U : Set (Euc n)}
+    {V : Euc n → Euc n} (hV : PerimeterAdmissible U V) {c : Euc n} {r L : ℝ}
+    (hr : 0 < r) (hL : 0 < L) (hbound : ∀ s, |deriv Real.smoothTransition s| ≤ L)
+    (hfin : perimeter E (U \ Metric.closedBall c r) ≠ ⊤) :
+    (∫ x in E \ Metric.closedBall c r, divergence V x) ≤
+      (perimeter E (U \ Metric.closedBall c r)).toReal +
+        2 * L * n * r ^ (n - 1) * volume.real (Metric.ball (0 : Euc n) 1) := by
+  have hd := integrable_divergence hV.1 hV.2.1
+  have hlim : Tendsto (fun δ : ℝ ↦ ∫ x in E,
+      annularCutoff c (r + δ) (r + 2 * δ) x * divergence V x)
+      (𝓝[>] (0 : ℝ)) (𝓝 (∫ x in E \ Metric.closedBall c r, divergence V x)) := by
+    have ht := tendsto_integral_filter_of_dominated_convergence
+      (μ := volume.restrict E) (l := 𝓝[>] (0 : ℝ))
+      (F := fun δ x ↦ annularCutoff c (r + δ) (r + 2 * δ) x * divergence V x)
+      (f := (Metric.closedBall c r)ᶜ.indicator (divergence V)) (fun x ↦ ‖divergence V x‖)
+      (Eventually.of_forall fun δ ↦
+        ((annularCutoff_contDiff c (r + δ) (r + 2 * δ)).continuous.mul
+          (continuous_divergence hV.1)).aestronglyMeasurable)
+      (Eventually.of_forall fun δ ↦ Eventually.of_forall fun x ↦ by
+        rw [norm_mul, Real.norm_eq_abs, abs_of_nonneg (annularCutoff_mem_Icc c _ _ x).1]
+        exact mul_le_of_le_one_left (norm_nonneg _) (annularCutoff_mem_Icc c _ _ x).2)
+      hd.norm.integrableOn (Eventually.of_forall fun x ↦ ?_)
+    · simpa only [integral_indicator measurableSet_closedBall.compl,
+        Measure.restrict_restrict measurableSet_closedBall.compl, inter_comm, ← Set.sdiff_eq] using ht
+    · convert (annularCutoff_tendsto c x hr).mul
+        (tendsto_const_nhds (x := divergence V x)) using 1
+      by_cases hx : x ∈ (Metric.closedBall c r)ᶜ <;> simp [hx]
+  have hpow : Tendsto (fun δ : ℝ ↦
+      (perimeter E (U \ Metric.closedBall c r)).toReal +
+        2 * L * n * (r + 2 * δ) ^ (n - 1) * volume.real (Metric.ball (0 : Euc n) 1))
+      (𝓝[>] (0 : ℝ)) (𝓝 ((perimeter E (U \ Metric.closedBall c r)).toReal +
+        2 * L * n * r ^ (n - 1) * volume.real (Metric.ball (0 : Euc n) 1))) := by
+    have hp : ContinuousAt (fun δ : ℝ ↦
+      (perimeter E (U \ Metric.closedBall c r)).toReal +
+        2 * L * n * (r + 2 * δ) ^ (n - 1) * volume.real (Metric.ball (0 : Euc n) 1)) 0 := by
+      fun_prop
+    simpa using hp.tendsto.mono_left nhdsWithin_le_nhds
+  apply le_of_tendsto_of_tendsto hlim hpow
+  filter_upwards [self_mem_nhdsWithin] with δ hδ
+  change 0 < δ at hδ
+  exact integral_annularCutoff_divergence_le hV (by simpa using add_pos hr hδ)
+    (by linarith [show 0 < δ from hδ]) (by linarith [show 0 < δ from hδ]) hL hbound hfin
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+
+theorem perimeter_remove_closedBall_le {n : ℕ} {E U : Set (Euc n)}
+    {c : Euc n} {r L : ℝ} (hr : 0 < r) (hL : 0 < L)
+    (hbound : ∀ s, |deriv Real.smoothTransition s| ≤ L)
+    (hfin : perimeter E (U \ Metric.closedBall c r) ≠ ⊤) :
+    perimeter (E \ Metric.closedBall c r) U ≤
+      perimeter E (U \ Metric.closedBall c r) +
+        ENNReal.ofReal (2 * L * n * r ^ (n - 1) * volume.real (Metric.ball (0 : Euc n) 1)) := by
+  rw [perimeter_eq_iSup_tests]
+  apply iSup_le
+  intro V
+  apply iSup_le
+  intro hV
+  have h := ENNReal.ofReal_le_ofReal (integral_divergence_outside_le hV hr hL hbound hfin)
+  rwa [ENNReal.ofReal_add ENNReal.toReal_nonneg (by positivity), ENNReal.ofReal_toReal hfin] at h
+
+theorem locallyMinimizing_perimeter_growth {n : ℕ} {E : Set (Euc n)}
+    (hmin : IsLocallyPerimeterMinimizing E) {L : ℝ} (hL : 0 < L)
+    (hbound : ∀ s, |deriv Real.smoothTransition s| ≤ L) (c : Euc n) {r : ℝ} (hr : 0 < r) :
+    perimeter E (Metric.ball c r) ≤
+      ENNReal.ofReal (2 * L * n * r ^ (n - 1) * volume.real (Metric.ball (0 : Euc n) 1)) := by
+  let U := Metric.ball c (r + 1)
+  let F := E \ Metric.closedBall c r
+  have hout : perimeter E (U \ Metric.closedBall c r) ≠ ⊤ :=
+    ne_top_of_le_ne_top (hmin.2.1 c (r + 1) (by linarith)).ne
+      (raw_perimeter_mono_region E sdiff_subset)
+  have hcl : closure (symmDiff E F) ⊆ Metric.closedBall c r := by
+    apply closure_minimal _ Metric.isClosed_closedBall
+    intro x hx
+    by_contra hc
+    rcases hx with hx | hx
+    · exact hx.2 ⟨hx.1, hc⟩
+    · exact hx.2 hx.1.1
+  have hcp : IsCompact (closure (symmDiff E F)) :=
+    (isCompact_closedBall c r).of_isClosed_subset isClosed_closure hcl
+  have hcmp : perimeter E U ≤ perimeter F U :=
+    hmin.2.2 c (r + 1) (by linarith) F
+      (hmin.1.diff measurableSet_closedBall) hcp
+      (hcl.trans (Metric.closedBall_subset_ball (by linarith)))
+  have hdisj : Disjoint (Metric.ball c r) (U \ Metric.closedBall c r) := by
+    apply Set.disjoint_left.mpr
+    intro x hx hy
+    exact hy.2 (Metric.ball_subset_closedBall hx)
+  have hsuper := perimeter_disjoint_superadditive E
+    (show Metric.ball c r ⊆ U from Metric.ball_subset_ball (by linarith)) sdiff_subset hdisj
+  have hremove := perimeter_remove_closedBall_le (E := E) (U := U) (c := c) hr hL hbound hout
+  have htotal := hsuper.trans (hcmp.trans hremove)
+  rw [add_comm (perimeter E (Metric.ball c r))] at htotal
+  exact (ENNReal.add_le_add_iff_left hout).mp htotal
+
+theorem exists_locallyMinimizing_perimeter_growth_constant (n : ℕ) (hn : 0 < n) :
+    ∃ C : ℝ, 0 < C ∧ ∀ E : Set (Euc n), IsLocallyPerimeterMinimizing E →
+      ∀ (c : Euc n) (r : ℝ), 0 < r →
+        perimeter E (Metric.ball c r) ≤ ENNReal.ofReal (C * r ^ (n - 1)) := by
+  obtain ⟨L, hL, hbound⟩ := exists_smoothTransition_deriv_bound
+  let C := 2 * L * n * volume.real (Metric.ball (0 : Euc n) 1)
+  refine ⟨C, ?_, ?_⟩
+  · unfold C
+    have hv : 0 < volume.real (Metric.ball (0 : Euc n) 1) := by
+      rw [measureReal_def, ENNReal.toReal_pos_iff]
+      exact ⟨Metric.measure_ball_pos volume _ (by norm_num),
+        (measure_mono Metric.ball_subset_closedBall).trans_lt
+          (isCompact_closedBall (0 : Euc n) 1).measure_lt_top⟩
+    positivity
+  · intro E hmin c r hr
+    convert locallyMinimizing_perimeter_growth hmin hL hbound c hr using 1
+    unfold C
+    congr 1
+    ring
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+
+def radialNormalTest {n : ℕ} (v x : Euc n) : Euc n :=
+  (1 - Real.smoothTransition (4 * ‖x‖ ^ 2)) • v
+
+def radialFlux {n : ℕ} (v x : Euc n) : ℝ :=
+  (8 * deriv Real.smoothTransition (4 * ‖x‖ ^ 2)) * max (-inner ℝ x v) 0
+
+theorem radialNormalTest_contDiff {n : ℕ} (v : Euc n) :
+    ContDiff ℝ ∞ (radialNormalTest v) := by
+  have hq : ContDiff ℝ ∞ (fun x : Euc n ↦ 4 * ‖x‖ ^ 2) :=
+    contDiff_const.mul (contDiff_id.norm_sq ℝ)
+  exact (contDiff_const.sub (Real.smoothTransition.contDiff.comp hq)).smul contDiff_const
+
+theorem radialNormalTest_support {n : ℕ} (v : Euc n) :
+    tsupport (radialNormalTest v) ⊆ Metric.closedBall 0 (1 / 2) := by
+  apply closure_minimal
+  · intro x hx
+    by_contra h
+    have hn : 1 / 2 < ‖x‖ := by simpa [Metric.mem_closedBall, dist_zero_right] using h
+    have hT := Real.smoothTransition.one_of_one_le (show 1 ≤ 4 * ‖x‖ ^ 2 by nlinarith)
+    exact hx (by simp [radialNormalTest, hT])
+  · exact Metric.isClosed_closedBall
+
+theorem radialNormalTest_admissible {n : ℕ} {v : Euc n} (hv : ‖v‖ = 1) :
+    PerimeterAdmissible (Metric.ball (0 : Euc n) 1) (radialNormalTest v) := by
+  refine ⟨radialNormalTest_contDiff v,
+    (isCompact_closedBall (0 : Euc n) (1 / 2)).of_isClosed_subset (isClosed_tsupport _)
+      (radialNormalTest_support v),
+    (radialNormalTest_support v).trans (Metric.closedBall_subset_ball (by norm_num)), ?_⟩
+  intro x
+  rw [radialNormalTest, norm_smul, hv, mul_one, Real.norm_eq_abs,
+    abs_of_nonneg (sub_nonneg.mpr (Real.smoothTransition.le_one _))]
+  linarith [Real.smoothTransition.nonneg (4 * ‖x‖ ^ 2)]
+
+theorem divergence_radialNormalTest {n : ℕ} (v x : Euc n) :
+    divergence (radialNormalTest v) x =
+      -(8 * deriv Real.smoothTransition (4 * ‖x‖ ^ 2)) * inner ℝ x v := by
+  have hq : HasFDerivAt (fun x : Euc n ↦ 4 * ‖x‖ ^ 2) ((8 : ℝ) • innerSL ℝ x) x := by
+    have h0 : HasFDerivAt (fun x : Euc n ↦ ‖x‖ ^ 2) (2 • innerSL ℝ x) x := by
+      simpa using (hasFDerivAt_id (𝕜 := ℝ) x).norm_sq
+    simpa only [two_smul, smul_add, ← add_smul, show (4 : ℝ) + 4 = 8 by norm_num] using h0.const_mul 4
+  have hd := ((Real.smoothTransition.contDiff : ContDiff ℝ ∞ _).differentiable
+    (by norm_num) _).hasDerivAt.comp_hasFDerivAt x hq
+  have hV := ((hasFDerivAt_const (1 : ℝ) x).sub hd).smul_const v
+  change HasFDerivAt (radialNormalTest v) _ x at hV
+  simp only [divergence, hV.fderiv,
+    ContinuousLinearMap.smulRight_apply, smul_apply,
+    zero_sub, inner_smul_right, innerSL_apply_apply,
+    neg_apply, smul_eq_mul]
+  calc
+    _ = -(8 * deriv Real.smoothTransition (4 * ‖x‖ ^ 2)) *
+        ∑ i : Fin n, inner ℝ x ((EuclideanSpace.basisFun (Fin n) ℝ) i) *
+          inner ℝ ((EuclideanSpace.basisFun (Fin n) ℝ) i) v := by
+      rw [Finset.mul_sum]
+      apply Finset.sum_congr rfl
+      intro i hi
+      ring
+    _ = _ := by rw [(EuclideanSpace.basisFun (Fin n) ℝ).sum_inner_mul_inner]
+
+theorem radialFlux_eq_indicator {n : ℕ} (v x : Euc n) :
+    radialFlux v x = {y | inner ℝ y v < 0}.indicator
+      (divergence (radialNormalTest v)) x := by
+  by_cases hx : inner ℝ x v < 0
+  · rw [Set.indicator_of_mem (show x ∈ {y | inner ℝ y v < 0} from hx), divergence_radialNormalTest]
+    simp only [radialFlux, max_eq_left (by linarith : 0 ≤ -inner ℝ x v)]
+    ring
+  · rw [Set.indicator_of_notMem (show x ∉ {y | inner ℝ y v < 0} from hx)]
+    simp [radialFlux, max_eq_right (by linarith : -inner ℝ x v ≤ 0)]
+
+theorem radialFlux_nonneg {n : ℕ} (v x : Euc n) : 0 ≤ radialFlux v x := by
+  exact mul_nonneg (mul_nonneg (by norm_num) Real.smoothTransition.monotone.deriv_nonneg)
+    (le_max_right _ _)
+
+theorem radialFlux_continuous {n : ℕ} :
+    Continuous (fun p : Euc n × Euc n ↦ radialFlux p.1 p.2) := by
+  have hT : Continuous (deriv Real.smoothTransition) :=
+    (Real.smoothTransition.contDiff : ContDiff ℝ ∞ _).continuous_deriv (by norm_num)
+  exact (hT.comp ((continuous_snd.norm.pow 2).const_mul 4)).const_mul 8 |>.mul
+    (((continuous_snd.inner continuous_fst).neg).max continuous_const)
+
+theorem radialFlux_support {n : ℕ} (v : Euc n) :
+    tsupport (radialFlux v) ⊆ Metric.closedBall 0 (1 / 2) := by
+  apply closure_minimal
+  · intro x hx
+    by_contra h
+    have hn : 1 / 2 < ‖x‖ := by simpa [Metric.mem_closedBall, dist_zero_right] using h
+    have hT := smoothTransition_deriv_zero_of_one_le (show 1 ≤ 4 * ‖x‖ ^ 2 by nlinarith)
+    exact hx (by simp [radialFlux, hT])
+  · exact Metric.isClosed_closedBall
+
+theorem radialFlux_integrable {n : ℕ} (v : Euc n) : Integrable (radialFlux v) := by
+  apply (radialFlux_continuous.comp (continuous_const.prodMk continuous_id)).integrable_of_hasCompactSupport
+  exact (isCompact_closedBall (0 : Euc n) (1 / 2)).of_isClosed_subset (isClosed_tsupport _)
+    (radialFlux_support v)
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+
+theorem integral_radialFlux_pos {n : ℕ} {v : Euc n} (hv : ‖v‖ = 1) :
+    0 < ∫ x, radialFlux v x := by
+  obtain ⟨t, ht, hd⟩ := exists_deriv_eq_slope Real.smoothTransition (by norm_num : (0 : ℝ) < 1)
+    Real.smoothTransition.continuous.continuousOn
+    ((Real.smoothTransition.contDiff : ContDiff ℝ ∞ _).differentiable (by norm_num)).differentiableOn
+  have hd1 : deriv Real.smoothTransition t = 1 := by simpa using hd
+  let x : Euc n := (-(Real.sqrt t / 2)) • v
+  have hr : 0 < Real.sqrt t := Real.sqrt_pos.mpr ht.1
+  have hnorm : ‖x‖ = Real.sqrt t / 2 := by
+    simp [x, norm_smul, hv, Real.norm_eq_abs, abs_of_nonneg (le_of_lt hr)]
+  have hq : 4 * ‖x‖ ^ 2 = t := by
+    rw [hnorm]; nlinarith [Real.sq_sqrt ht.1.le]
+  have hi : inner ℝ x v = -(Real.sqrt t / 2) := by
+    simp [x, inner_smul_left, hv]
+  apply (radialFlux_continuous.comp (continuous_const.prodMk continuous_id)).integral_pos_of_hasCompactSupport_nonneg_nonzero
+    ((isCompact_closedBall (0 : Euc n) (1 / 2)).of_isClosed_subset (isClosed_tsupport _)
+      (radialFlux_support v)) (radialFlux_nonneg v)
+  change radialFlux v x ≠ 0
+  rw [radialFlux, hq, hd1, hi]
+  have hm : max (- -(Real.sqrt t / 2)) 0 = Real.sqrt t / 2 := by
+    simp [max_eq_left (show 0 ≤ Real.sqrt t / 2 by positivity)]
+  rw [hm]
+  positivity
+
+theorem continuous_integral_radialFlux {n : ℕ} :
+    Continuous (fun v : Euc n ↦ ∫ x, radialFlux v x) := by
+  have heq : (fun v : Euc n ↦ ∫ x, radialFlux v x) =
+      (fun v ↦ ∫ x in Metric.closedBall 0 (1 / 2), radialFlux v x) := by
+    funext v
+    exact (setIntegral_eq_integral_of_forall_compl_eq_zero (fun x hx ↦
+      image_eq_zero_of_notMem_tsupport (fun h ↦ hx (radialFlux_support v h)))).symm
+  rw [heq]
+  exact continuous_parametric_integral_of_continuous radialFlux_continuous
+    (isCompact_closedBall (0 : Euc n) (1 / 2))
+
+theorem exists_uniform_radialFlux_lower (n : ℕ) (hn : 0 < n) :
+    ∃ c : ℝ, 0 < c ∧ ∀ v : Euc n, ‖v‖ = 1 → c ≤ ∫ x, radialFlux v x := by
+  have hne : (Metric.sphere (0 : Euc n) 1).Nonempty := by
+    refine ⟨(EuclideanSpace.basisFun (Fin n) ℝ) ⟨0, hn⟩, ?_⟩
+    simp
+  obtain ⟨v, hv, hmin⟩ := (isCompact_sphere (0 : Euc n) 1).exists_isMinOn hne
+    (continuous_integral_radialFlux (n := n)).continuousOn
+  have hv1 : ‖v‖ = 1 := by simpa [Metric.mem_sphere, dist_zero_right] using hv
+  refine ⟨∫ x, radialFlux v x, integral_radialFlux_pos hv1, fun w hw ↦ hmin ?_⟩
+  simpa [Metric.mem_sphere, dist_zero_right] using hw
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+
+def ballNormalTest {n : ℕ} (c : Euc n) (r : ℝ) (v : Euc n) (y : Euc n) : Euc n :=
+  radialNormalTest v (r⁻¹ • (y - c))
+
+theorem ballNormalTest_contDiff {n : ℕ} (c : Euc n) (r : ℝ) (v : Euc n) :
+    ContDiff ℝ ∞ (ballNormalTest c r v) :=
+  (radialNormalTest_contDiff v).comp ((contDiff_id.sub contDiff_const).const_smul r⁻¹)
+
+theorem ballNormalTest_support {n : ℕ} (c : Euc n) {r : ℝ} (hr : 0 < r) (v : Euc n) :
+    tsupport (ballNormalTest c r v) ⊆ Metric.closedBall c (r / 2) := by
+  have hs := tsupport_comp_subset_preimage (f := fun y : Euc n ↦ r⁻¹ • (y - c)) (radialNormalTest v)
+    ((continuous_id.sub continuous_const).const_smul r⁻¹)
+  intro y hy
+  have hmem := radialNormalTest_support v (hs hy)
+  simp only [Metric.mem_closedBall, dist_zero_right, norm_smul, Real.norm_eq_abs,
+    abs_of_pos (inv_pos.mpr hr)] at hmem
+  change dist y c ≤ r / 2
+  rw [dist_eq_norm]
+  have hm := mul_le_mul_of_nonneg_left hmem hr.le
+  simpa [← mul_assoc, hr.ne', div_eq_mul_inv] using hm
+
+theorem ballNormalTest_admissible {n : ℕ} (c : Euc n) {r : ℝ} (hr : 0 < r)
+    {v : Euc n} (hv : ‖v‖ = 1) :
+    PerimeterAdmissible (Metric.ball c r) (ballNormalTest c r v) := by
+  refine ⟨ballNormalTest_contDiff c r v,
+    (isCompact_closedBall c (r / 2)).of_isClosed_subset (isClosed_tsupport _)
+      (ballNormalTest_support c hr v),
+    (ballNormalTest_support c hr v).trans (Metric.closedBall_subset_ball (by linarith)), ?_⟩
+  intro y
+  exact (radialNormalTest_admissible hv).2.2.2 _
+
+theorem divergence_ballNormalTest {n : ℕ} (c : Euc n) (r : ℝ) (v y : Euc n) :
+    divergence (ballNormalTest c r v) y =
+      r⁻¹ * divergence (radialNormalTest v) (r⁻¹ • (y - c)) := by
+  have hq : HasFDerivAt (fun y : Euc n ↦ r⁻¹ • (y - c))
+      (r⁻¹ • ContinuousLinearMap.id ℝ (Euc n)) y :=
+    ((hasFDerivAt_id (𝕜 := ℝ) y).sub_const c).const_smul r⁻¹
+  have hV := ((radialNormalTest_contDiff v).differentiable (by norm_num)
+    (r⁻¹ • (y - c))).hasFDerivAt.comp y hq
+  change HasFDerivAt (ballNormalTest c r v) _ y at hV
+  simp only [divergence, hV.fderiv, ContinuousLinearMap.comp_apply,
+    smul_apply, ContinuousLinearMap.id_apply, map_smul,
+    inner_smul_right, ← Finset.mul_sum]
+
+theorem ballNormalTest_allIntegral {n : ℕ} (E : Set (Euc n)) (hE : MeasurableSet E) (c : Euc n)
+    {r : ℝ} (hr : 0 < r) (v : Euc n) :
+    (∫ y in E, divergence (ballNormalTest c r v) y) =
+      ∫ y in Metric.ball c r, E.indicator (divergence (ballNormalTest c r v)) y := by
+  rw [← integral_indicator (s := E)]
+  · apply (setIntegral_eq_integral_of_forall_compl_eq_zero (fun y hy ↦ ?_)).symm
+    have hns : y ∉ tsupport (divergence (ballNormalTest c r v)) := fun h ↦ hy
+      (((tsupport_divergence_subset _).trans (ballNormalTest_support c hr v)).trans
+        (Metric.closedBall_subset_ball (show r / 2 < r by linarith)) h)
+    by_cases he : y ∈ E
+    · rw [Set.indicator_of_mem he, image_eq_zero_of_notMem_tsupport hns]
+    · exact Set.indicator_of_notMem he _
+  · exact hE
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+
+theorem ballNormalTest_normalized_integral {n : ℕ} (E : Set (Euc n)) (hE : MeasurableSet E)
+    (c : Euc n) {r : ℝ} (hr : 0 < r) (v : Euc n) :
+    (r ^ n)⁻¹ * r * (∫ y in E, divergence (ballNormalTest c r v) y) =
+      ∫ z in Metric.ball (0 : Euc n) 1,
+        E.indicator (fun _ ↦ (1 : ℝ)) (c + r • z) * divergence (radialNormalTest v) z := by
+  rw [ballNormalTest_allIntegral E hE c hr v]
+  calc
+    _ = r * ((r ^ n)⁻¹ * ∫ y in Metric.ball c r,
+        E.indicator (divergence (ballNormalTest c r v)) y) := by ring
+    _ = r * ∫ z in Metric.ball (0 : Euc n) 1,
+        E.indicator (divergence (ballNormalTest c r v)) (c + r • z) := by
+      rw [normalized_ball_integral _ _ hr]
+    _ = _ := by
+      rw [← integral_const_mul]
+      apply setIntegral_congr_fun measurableSet_ball
+      intro z hz
+      by_cases he : c + r • z ∈ E
+      · simp only [Set.indicator_of_mem he, one_mul, divergence_ballNormalTest,
+          add_sub_cancel_left, inv_smul_smul₀ hr.ne', ← mul_assoc, mul_inv_cancel₀ hr.ne']
+      · simp only [Set.indicator_of_notMem he, mul_zero, zero_mul]
+
+theorem ballNormalTest_integral {n : ℕ} (hn : 0 < n) (E : Set (Euc n))
+    (hE : MeasurableSet E) (c : Euc n) {r : ℝ} (hr : 0 < r) (v : Euc n) :
+    (∫ y in E, divergence (ballNormalTest c r v) y) = r ^ (n - 1) *
+      ∫ z in Metric.ball (0 : Euc n) 1,
+        E.indicator (fun _ ↦ (1 : ℝ)) (c + r • z) * divergence (radialNormalTest v) z := by
+  have hc := ballNormalTest_normalized_integral E hE c hr v
+  have hp : r ^ n = r ^ (n - 1) * r := by
+    rw [← pow_succ, Nat.sub_add_cancel hn]
+  rw [hp] at hc
+  calc
+    _ = r ^ (n - 1) * ((r ^ (n - 1) * r)⁻¹ * r *
+        ∫ y in E, divergence (ballNormalTest c r v) y) := by field_simp
+    _ = _ := by rw [hc]
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/BlowupHolderGraph.lean b/lean/OAI/Geometry/Perimeter/Regularity/BlowupHolderGraph.lean
new file mode 100644
index 0000000000000000000000000000000000000000..e489108a39cb13cb481e1bdfe9968fe6a18af9c7
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/BlowupHolderGraph.lean
@@ -0,0 +1,79 @@
+import OAI.Geometry.Perimeter.Regularity.SlabContraction
+import OAI.Geometry.Perimeter.Regularity.InitialSlab
+import OAI.Geometry.Perimeter.Regularity.HolderGraph
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+theorem HasHalfspaceBlowup.exists_holder_graphs (hn : 1<n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    {p v : Euc n} (hp : p∈perimeterSupport E) (hbl : HasHalfspaceBlowup E p v) :
+    ∃ η : ℝ, 0<η ∧ η≤1/2 ∧ ∀ ε : ℝ, 0<ε →
+    ∃ r : ℝ, 0<r ∧ ∃ f : normalPlane v → ℝ,
+      f 0=0 ∧ ContDiffOn ℝ 1 f (Metric.ball 0 (1/4)) ∧
+      (∀ y, ‖y‖<1/4 → |f y|<1/4 ∧
+        (y:Euc n)+f y•v∈perimeterSupport (blownUpSet E p r) ∩ Metric.ball 0 1) ∧
+      (∀ y z, ‖y‖<1/4 → ‖z‖<1/4 → |f y-f z|≤‖y-z‖) ∧
+      (∀ (y : normalPlane v) (t : ℝ), ‖y‖<1/4 → ‖(y:Euc n)+t•v‖<1 →
+        ((y:Euc n)+t•v∈perimeterSupport (blownUpSet E p r) ↔ t=f y)) ∧
+      (∃ a : ℝ, 0<a ∧ a<1/4 ∧ FullSlab (blownUpSet E p r) v 8 a) ∧
+      (∀ x∈Metric.ball (0:normalPlane v) (1/4), GeometricHolderAt η (fderiv ℝ f) x) ∧
+      (∀ x∈Metric.ball (0:normalPlane v) (1/4), ‖fderiv ℝ f x‖<ε) := by
+  let s : ℕ → ℝ := fun i ↦ 1/((i:ℝ)+1)
+  have hs : ∀ i, 0<s i := fun i ↦ by dsimp [s]; positivity
+  have hst : Tendsto s atTop (𝓝 0) := tendsto_one_div_add_atTop_nhds_zero_nat
+  have hHmeas : MeasurableSet {x : Euc n | inner ℝ x v<0} :=
+    (isOpen_lt (continuous_id.inner continuous_const) continuous_const).measurableSet
+  have hH := locally_minimizing_of_local_L1_limit (by omega : 0<n)
+    (fun i ↦ locally_minimizing_blownUpSet hmin p (hs i)) hHmeas (hbl.local_L1 hmin.1 s hs hst)
+  obtain ⟨q,M,δ,hq,hq64,hM,hδ,_,hstep⟩ := exists_slab_contraction hn hbl.1 hH
+  refine ⟨q/2,by positivity,by linarith,?_⟩
+  intro ε hε
+  let b := min (δ/8) (min (1/8) (min (1/(64*(M+1))) (1/(4*(16*M+1/q+1)))))
+  have hb : 0<b := lt_min (by positivity) (lt_min (by norm_num)
+    (lt_min (by positivity) (by positivity)))
+  let a := min b (ε/(32*(M+1)))
+  have ha : 0<a := lt_min hb (by positivity)
+  have hab : a≤b := min_le_left _ _
+  have haε : 16*M*a<ε := by
+    have hh : a≤ε/(32*(M+1)) := min_le_right _ _
+    have hb' := (le_div_iff₀ (by positivity : 0<32*(M+1))).mp hh
+    nlinarith
+  have haδ : 4*a<δ := by
+    have hh : a≤δ/8 := hab.trans (min_le_left _ _)
+    linarith
+  have ha4 : a<1/4 := by
+    have hh : a≤1/8 := hab.trans ((min_le_right _ _).trans (min_le_left _ _))
+    linarith
+  have haM : 8*M*a≤1/2 := by
+    have haB : a≤1/(64*(M+1)) :=
+      hab.trans ((min_le_right _ _).trans ((min_le_right _ _).trans (min_le_left _ _)))
+    have hh := (le_div_iff₀ (by positivity : 0<64*(M+1))).mp haB
+    nlinarith
+  have hachord : a*(16*M+1/q)≤1/2 := by
+    have haB : a≤1/(4*(16*M+1/q+1)) :=
+      hab.trans ((min_le_right _ _).trans ((min_le_right _ _).trans (min_le_right _ _)))
+    have hh := (le_div_iff₀ (by positivity : 0<4*(16*M+1/q+1))).mp haB
+    nlinarith
+  obtain ⟨i,hi⟩ := (hbl.eventually_fullSlab hn hmin s hs hst ha (by linarith : a<1)).exists
+  have hm := locally_minimizing_blownUpSet hmin p (hs i)
+  obtain ⟨f,hf,hfs,hlip,hiff,hhold,hslope⟩ := holder_graph_from_contraction hn hq (by linarith : q<1)
+    hM hstep hm hbl.1 ha haδ ha4 haM hachord hi
+  have hzero : (0:Euc n)∈perimeterSupport (blownUpSet E p (s i)) := by
+    rw [perimeterSupport_blownUpSet hmin.1 p (hs i)]
+    simpa using hp
+  have hf0 : f 0=0 := by
+    have hh := (hiff 0 0 (by simp) (by simp)).mp (by simpa using hzero)
+    exact hh.symm
+  exact ⟨s i,hs i,f,hf0,hf,fun y hy ↦ ⟨((hfs y hy).1.trans_lt ha4),(hfs y hy).2⟩,hlip,hiff,⟨a,ha,ha4,hi⟩,hhold,fun x hx ↦ (hslope x hx).trans_lt haε⟩
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/BlowupHomeomorph.lean b/lean/OAI/Geometry/Perimeter/Regularity/BlowupHomeomorph.lean
new file mode 100644
index 0000000000000000000000000000000000000000..9d1160ce30b4a9ec7d91ade7ad49ecab5f9ec7f4
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/BlowupHomeomorph.lean
@@ -0,0 +1,373 @@
+import OAI.Geometry.Perimeter.Regularity.SmoothFlatOrder
+import OAI.Geometry.Perimeter.Regularity.SetIntegralSubLeSymmDiff
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+open scoped ENNReal Topology
+open scoped ENNReal Topology ContDiff
+open scoped Topology ENNReal
+namespace RPISingularHarnack
+
+def blowupHomeomorph {n : ℕ} (c : Euc n) {r : ℝ} (hr : 0 < r) : Euc n ≃ₜ Euc n :=
+  (Homeomorph.smulOfNeZero r hr.ne').trans (Homeomorph.addLeft c)
+
+@[simp] theorem blowupHomeomorph_apply {n : ℕ} (c : Euc n) {r : ℝ} (hr : 0 < r)
+    (x : Euc n) : blowupHomeomorph c hr x = c + r • x := rfl
+
+@[simp] theorem blowupHomeomorph_symm_apply {n : ℕ} (c : Euc n) {r : ℝ} (hr : 0 < r)
+    (x : Euc n) : (blowupHomeomorph c hr).symm x = r⁻¹ • (x-c) := by
+  change r⁻¹ • (-c + x) = r⁻¹ • (x-c)
+  congr 1
+  abel
+
+theorem admissible_comp_blowup {n : ℕ} {U : Set (Euc n)} {V : Euc n → Euc n}
+    (hV : PerimeterAdmissible U V) (c : Euc n) {r : ℝ} (hr : 0 < r) :
+    PerimeterAdmissible (blownUpSet U c r) (fun z ↦ V (c+r • z)) := by
+  refine ⟨hV.1.comp (contDiff_const.add (contDiff_id.const_smul r)),
+    hV.2.1.comp_homeomorph (blowupHomeomorph c hr), ?_, fun x ↦ hV.2.2.2 _⟩
+  intro x hx
+  exact hV.2.2.1 (tsupport_comp_subset_preimage V (continuous_const.add (continuous_id.const_smul r)) hx)
+
+theorem admissible_comp_blowdown {n : ℕ} {U : Set (Euc n)} {V : Euc n → Euc n}
+    {c : Euc n} {r : ℝ} (hV : PerimeterAdmissible (blownUpSet U c r) V)
+    (hr : 0 < r) : PerimeterAdmissible U (fun y ↦ V (r⁻¹ • (y-c))) := by
+  refine ⟨hV.1.comp ((contDiff_id.sub contDiff_const).const_smul r⁻¹),
+    ?_, ?_, fun x ↦ hV.2.2.2 _⟩
+  · simpa only [Function.comp_def, blowupHomeomorph_symm_apply] using
+      hV.2.1.comp_homeomorph (blowupHomeomorph c hr).symm
+  · intro y hy
+    have hh := hV.2.2.1 (tsupport_comp_subset_preimage V
+      ((continuous_id.sub continuous_const).const_smul r⁻¹) hy)
+    change c + r • (r⁻¹ • (y-c)) ∈ U at hh
+    simpa [smul_inv_smul₀ hr.ne'] using hh
+
+theorem divergence_comp_blowup {n : ℕ} {V : Euc n → Euc n}
+    (hV : Differentiable ℝ V) (c : Euc n) (r : ℝ) (y : Euc n) :
+    divergence (fun z ↦ V (c+r • z)) y = r * divergence V (c+r • y) := by
+  have hd := ((hV _).hasFDerivAt.comp y
+    (((hasFDerivAt_id y).const_smul r).const_add c)).fderiv
+  simp only [Function.comp_def, id_eq, Pi.smul_apply] at hd
+  simp only [divergence, hd, ContinuousLinearMap.comp_apply,
+    smul_apply, ContinuousLinearMap.id_apply, map_smul,
+    inner_smul_right, ← Finset.mul_sum]
+
+theorem integral_divergence_comp_blowup {n : ℕ} {E : Set (Euc n)}
+    (hE : MeasurableSet E) {V : Euc n → Euc n} (hV : Differentiable ℝ V)
+    (c : Euc n) {r : ℝ} (hr : 0 < r) :
+    (∫ z in blownUpSet E c r, divergence (fun y ↦ V (c+r • y)) z) =
+      (r * (r^n)⁻¹) * ∫ y in E, divergence V y := by
+  simp_rw [divergence_comp_blowup hV]
+  rw [integral_const_mul, integral_blownUpSet hE _ c hr]
+  ring
+
+theorem integral_divergence_comp_blowdown {n : ℕ} {E : Set (Euc n)}
+    (hE : MeasurableSet E) {V : Euc n → Euc n} (hV : Differentiable ℝ V)
+    (c : Euc n) {r : ℝ} (hr : 0 < r) :
+    (∫ z in blownUpSet E c r, divergence V z) =
+      (r * (r^n)⁻¹) * ∫ y in E, divergence (fun z ↦ V (r⁻¹ • (z-c))) y := by
+  let W : Euc n → Euc n := fun y ↦ V (r⁻¹ • (y-c))
+  have he (z : Euc n) : divergence W (c+r • z) = r⁻¹ * divergence V z := by
+    dsimp [W]
+    rw [divergence_comp_blowdown hV]
+    simp [inv_smul_smul₀ hr.ne']
+  have hh := integral_blownUpSet hE (divergence W) c hr
+  simp_rw [he] at hh
+  rw [integral_const_mul] at hh
+  have heq := congrArg (fun a : ℝ ↦ r*a) hh
+  simpa only [← mul_assoc, mul_inv_cancel₀ hr.ne', one_mul] using heq
+
+theorem perimeter_blownUpSet_eq {n : ℕ} {E U : Set (Euc n)}
+    (hE : MeasurableSet E) (c : Euc n) {r : ℝ} (hr : 0 < r) :
+    perimeter (blownUpSet E c r) (blownUpSet U c r) =
+      ENNReal.ofReal (r * (r^n)⁻¹) * perimeter E U := by
+  have hp : 0 ≤ r * (r^n)⁻¹ := by positivity
+  apply le_antisymm
+  · rw [perimeter_eq_iSup_tests]
+    refine iSup_le fun V ↦ iSup_le fun hV ↦ ?_
+    rw [integral_divergence_comp_blowdown hE (hV.1.differentiable (by simp)) c hr,
+      ENNReal.ofReal_mul hp]
+    apply mul_le_mul' le_rfl
+    rw [perimeter_eq_iSup_tests]
+    exact le_iSup_of_le _ (le_iSup_of_le (admissible_comp_blowdown hV hr) le_rfl)
+  · rw [perimeter_eq_iSup_tests E U]
+    simp_rw [ENNReal.mul_iSup]
+    refine iSup_le fun V ↦ iSup_le fun hV ↦ ?_
+    rw [← ENNReal.ofReal_mul hp,
+      ← integral_divergence_comp_blowup hE (hV.1.differentiable (by simp)) c hr,
+      perimeter_eq_iSup_tests]
+    exact le_iSup_of_le _ (le_iSup_of_le (admissible_comp_blowup hV c hr) le_rfl)
+
+theorem blownUpSet_ball {n : ℕ} (c a : Euc n) {r : ℝ} (hr : 0 < r) (R : ℝ) :
+    blownUpSet (Metric.ball (c+r • a) (r*R)) c r = Metric.ball a R := by
+  ext x
+  change dist (c+r • x) (c+r • a) < r*R ↔ dist x a < R
+  rw [dist_add_left, dist_smul₀, Real.norm_of_nonneg hr.le, mul_lt_mul_iff_right₀ hr]
+
+theorem locally_minimizing_blownUpSet {n : ℕ} {E : Set (Euc n)}
+    (hmin : IsLocallyPerimeterMinimizing E) (c : Euc n) {r : ℝ} (hr : 0 < r) :
+    IsLocallyPerimeterMinimizing (blownUpSet E c r) := by
+  have hm := measurableSet_blownUpSet hmin.1 c r
+  refine ⟨hm, ?_, ?_⟩
+  · intro a R hR
+    rw [← blownUpSet_ball c a hr R, perimeter_blownUpSet_eq hmin.1 c hr]
+    exact ENNReal.mul_lt_top ENNReal.ofReal_lt_top (hmin.2.1 _ _ (mul_pos hr hR))
+  · intro a R hR G hG hcG hsG
+    let A := blowupHomeomorph c hr
+    let H := A.symm ⁻¹' G
+    have hH : MeasurableSet H := hG.preimage A.symm.continuous.measurable
+    have hEA : A.symm ⁻¹' blownUpSet E c r = E := by
+      ext y
+      change A (A.symm y) ∈ E ↔ y ∈ E
+      rw [A.apply_symm_apply]
+    have hHA : blownUpSet H c r = G := by
+      ext y
+      change A.symm (A y) ∈ G ↔ y ∈ G
+      rw [A.symm_apply_apply]
+    have hdiff : symmDiff E H = A.symm ⁻¹' symmDiff (blownUpSet E c r) G := by
+      rw [preimage_symmDiff, hEA]
+    have hcl : closure (symmDiff E H) = A.symm ⁻¹' closure (symmDiff (blownUpSet E c r) G) := by
+      rw [hdiff, ← A.symm.preimage_closure]
+    have hcH : IsCompact (closure (symmDiff E H)) := by
+      rw [hcl]
+      exact A.symm.isCompact_preimage.mpr hcG
+    have hsH : closure (symmDiff E H) ⊆ Metric.ball (c+r • a) (r*R) := by
+      intro y hy
+      rw [hcl] at hy
+      have hh := hsG hy
+      rw [← blownUpSet_ball c a hr R] at hh
+      change A (A.symm y) ∈ Metric.ball (c+r • a) (r*R) at hh
+      simpa only [A.apply_symm_apply] using hh
+    have hcmp := hmin.2.2 (c+r • a) (r*R) (mul_pos hr hR) H hH hcH hsH
+    calc
+      perimeter (blownUpSet E c r) (Metric.ball a R) =
+          ENNReal.ofReal (r * (r^n)⁻¹) * perimeter E (Metric.ball (c+r • a) (r*R)) := by
+        rw [← blownUpSet_ball c a hr R, perimeter_blownUpSet_eq hmin.1 c hr]
+      _ ≤ ENNReal.ofReal (r * (r^n)⁻¹) * perimeter H (Metric.ball (c+r • a) (r*R)) :=
+        mul_le_mul' le_rfl hcmp
+      _ = perimeter G (Metric.ball a R) := by
+        rw [← perimeter_blownUpSet_eq hH c hr, hHA, blownUpSet_ball c a hr R]
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+open scoped ENNReal Topology
+open scoped ENNReal Topology ContDiff
+open scoped Topology ENNReal
+namespace RPISingularHarnack
+
+theorem exists_uniform_minimizing_exhaustion_cut {n : ℕ} (hn : 0 < n) (m : ℕ) :
+    ∃ C : ℝ≥0∞, C ≠ ⊤ ∧ ∀ E : Set (Euc n), IsLocallyPerimeterMinimizing E →
+      ∃ A : Set (Euc n), MeasurableSet A ∧ A ⊆ Metric.closedBall 0 ((m:ℝ)+2) ∧
+        A ∩ Metric.closedBall 0 ((m:ℝ)+1) = E ∩ Metric.closedBall 0 ((m:ℝ)+1) ∧
+        perimeter A univ ≤ C := by
+  obtain ⟨L, hL, hbound⟩ := exists_smoothTransition_deriv_bound
+  obtain ⟨C, hC, hgrowth⟩ := exists_locallyMinimizing_perimeter_growth_constant n hn
+  let D : ℝ := volume.real (Metric.closedBall (0:Euc n) ((m:ℝ)+2)) + 1
+  have hD : 0 < D := by dsimp [D]; positivity
+  refine ⟨ENNReal.ofReal (C*((m:ℝ)+3)^(n-1)) + ENNReal.ofReal (8*L*D), by finiteness, ?_⟩
+  intro E hmin
+  have hv : volume.real (E ∩ Metric.closedBall (0:Euc n) ((m:ℝ)+2)) <
+      D * (((m:ℝ)+2)-((m:ℝ)+1)) := by
+    have hh := measureReal_mono (μ := (volume : Measure (Euc n)))
+      (s₁ := E ∩ Metric.closedBall 0 ((m:ℝ)+2)) inter_subset_right
+      (isCompact_closedBall (0:Euc n) ((m:ℝ)+2)).measure_ne_top
+    dsimp [D]
+    linarith
+  obtain ⟨r, hr, hc⟩ := exists_controlled_cut_radius E (0:Euc n)
+    (by positivity : (0:ℝ) < (m:ℝ)+1) (by linarith : (m:ℝ)+1 < (m:ℝ)+2) hv
+  refine ⟨E ∩ Metric.closedBall 0 r, hmin.1.inter measurableSet_closedBall,
+    inter_subset_right.trans (Metric.closedBall_subset_closedBall hr.2.le), ?_, ?_⟩
+  · rw [inter_assoc, inter_eq_right.mpr (Metric.closedBall_subset_closedBall hr.1)]
+  · have hh := controlled_radius_perimeter_cut hmin.1 (0:Euc n)
+      (by linarith [hr.1] : 0 < r) hr.2
+      (Metric.closedBall_subset_ball (by linarith : (m:ℝ)+2 < (m:ℝ)+3))
+      hc hD.le hL hbound (hmin.2.1 0 ((m:ℝ)+3) (by positivity)).ne
+    exact (le_add_right le_rfl).trans (hh.trans
+      (add_le_add (hgrowth E hmin 0 ((m:ℝ)+3) (by positivity)) le_rfl))
+
+theorem exists_set_of_indicator_Lp {n : ℕ} (K : Set (Euc n)) (hK : IsCompact K)
+    (g : Lp ℝ 1 (volume.comap (Subtype.val : K → Euc n)))
+    (hg : ∀ᵐ x ∂(volume.comap (Subtype.val : K → Euc n)), g x = 0 ∨ g x = 1) :
+    ∃ F : Set (Euc n), ∃ hF : MeasurableSet F, F ⊆ K ∧ compactIndicatorLp K hK F hF = g := by
+  let A : Set K := {x | g x = 1}
+  have hA : MeasurableSet A := measurableSet_eq_fun (Lp.stronglyMeasurable g).measurable measurable_const
+  let F : Set (Euc n) := Subtype.val '' A
+  have hF : MeasurableSet F :=
+    (MeasurableEmbedding.subtype_coe hK.measurableSet).measurableSet_image.mpr hA
+  refine ⟨F, hF, Subtype.coe_image_subset K A, ?_⟩
+  apply Lp.ext
+  filter_upwards [compactIndicatorLp_ae K hK F hF, hg] with x hx hx01
+  rw [hx]
+  by_cases hx1 : g x = 1
+  · have hxF : (x:Euc n) ∈ F := ⟨x, hx1, rfl⟩
+    simp only [indicator_of_mem hxF, hx1]
+  · have hxF : (x:Euc n) ∉ F := by
+      rintro ⟨y, hy, hyx⟩
+      have : y = x := Subtype.ext hyx
+      subst y
+      exact hx1 hy
+    simp only [indicator_of_notMem hxF, hx01.resolve_right hx1]
+
+theorem exists_simultaneous_indicator_L1_subsequence {n : ℕ}
+    (K : ℕ → Set (Euc n)) (hK : ∀ m, IsCompact (K m))
+    (C : ℕ → ℝ≥0∞) (hC : ∀ m, C m ≠ ⊤)
+    (A : ℕ → ℕ → Set (Euc n)) (hA : ∀ m i, MeasurableSet (A m i))
+    (hAK : ∀ m i, A m i ⊆ K m) (hP : ∀ m i, perimeter (A m i) univ ≤ C m) :
+    ∃ F : ℕ → Set (Euc n), (∀ m, MeasurableSet (F m)) ∧ (∀ m, F m ⊆ K m) ∧
+      ∃ σ : ℕ → ℕ, StrictMono σ ∧ ∀ m,
+        Tendsto (fun i ↦ volume (symmDiff (A m (σ i)) (F m))) atTop (𝓝 0) := by
+  let μ (m : ℕ) : Measure (K m) := volume.comap (Subtype.val : K m → Euc n)
+  let S (m : ℕ) : Set (Lp ℝ 1 (μ m)) := range (fun H : {H : Set (Euc n) //
+    MeasurableSet H ∧ H ⊆ K m ∧ perimeter H univ ≤ C m} ↦
+      compactIndicatorLp (K m) (hK m) H H.property.1)
+  let f (i : ℕ) (m : ℕ) : Lp ℝ 1 (μ m) := compactIndicatorLp (K m) (hK m) (A m i) (hA m i)
+  have hc (m : ℕ) : IsCompact (closure (S m)) :=
+    (bounded_perimeter_indicator_totallyBounded (K m) (hK m) (C m) (hC m)).closure.isCompact_of_isClosed
+      isClosed_closure
+  have hf (i : ℕ) : f i ∈ {x | ∀ m, x m ∈ closure (S m)} := by
+    intro m
+    exact subset_closure ⟨⟨A m i, hA m i, hAK m i, hP m i⟩, rfl⟩
+  obtain ⟨g, hg, σ, hσ, ht⟩ := (isCompact_pi_infinite hc).isSeqCompact hf
+  have hg01 (m : ℕ) : ∀ᵐ x ∂(μ m), g m x = 0 ∨ g m x = 1 := by
+    have hclosed : IsClosed {u : Lp ℝ 1 (μ m) | ∀ᵐ x ∂(μ m), u x ∈ ({0,1}:Set ℝ)} :=
+      isClosed_lp_ae_mem (μ m) _ (isClosed_singleton.union isClosed_singleton)
+    have hs : S m ⊆ {u : Lp ℝ 1 (μ m) | ∀ᵐ x ∂(μ m), u x ∈ ({0,1}:Set ℝ)} := by
+      rintro _ ⟨H, rfl⟩
+      filter_upwards [compactIndicatorLp_ae (K m) (hK m) H H.property.1] with x hx
+      rw [hx]
+      by_cases hxH : (x:Euc n) ∈ (H:Set (Euc n)) <;> simp [hxH]
+    simpa only [mem_ofPred_eq, mem_insert_iff, mem_singleton_iff] using
+      closure_minimal hs hclosed (hg m)
+  choose F hF hFK hFg using fun m ↦ exists_set_of_indicator_Lp (K m) (hK m) (g m) (hg01 m)
+  refine ⟨F, hF, hFK, σ, hσ, ?_⟩
+  intro m
+  have htm := tendsto_pi_nhds.mp ht m
+  have he : Tendsto (fun i ↦ edist (f (σ i) m) (g m)) atTop (𝓝 0) := by
+    simpa only [Function.comp_def, edist_self] using
+      htm.edist (tendsto_const_nhds (x := g m))
+  rw [← hFg m] at he
+  simp only [f, compactIndicatorLp_edist (K m) (hK m) _ _ _ _ (hAK m _) (hFK m)] at he
+  have hvol (i : ℕ) : (∫⁻ x, ‖(A m (σ i)).indicator (fun _ ↦ (1:ℝ)) x -
+      (F m).indicator (fun _ ↦ (1:ℝ)) x‖ₑ) = volume (symmDiff (A m (σ i)) (F m)) := by
+    simpa using lintegral_indicator_difference (U := univ) (hA m (σ i)) (hF m)
+  simpa only [hvol] using he
+
+theorem exists_global_set_of_local_L1_limits {n : ℕ}
+    (E F : ℕ → Set (Euc n)) (hF : ∀ m, MeasurableSet (F m))
+    (hconv : ∀ m, Tendsto (fun i ↦ volume (symmDiff (E i) (F m) ∩
+      Metric.closedBall 0 ((m:ℝ)+1))) atTop (𝓝 0)) :
+    ∃ G : Set (Euc n), MeasurableSet G ∧ ∀ (c : Euc n) (R : ℝ), 0 < R →
+      Tendsto (fun i ↦ volume.real (symmDiff (E i) G ∩ Metric.closedBall c R)) atTop (𝓝 0) := by
+  let B (m : ℕ) : Set (Euc n) := Metric.closedBall 0 ((m:ℝ)+1)
+  have hBmono : Monotone B := fun j m hjm ↦
+    Metric.closedBall_subset_closedBall (by exact_mod_cast Nat.add_le_add_right hjm 1)
+  have hrestrict (j m : ℕ) (hjm : j ≤ m) :
+      Tendsto (fun i ↦ volume (symmDiff (E i) (F m) ∩ B j)) atTop (𝓝 0) :=
+    tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds (hconv m)
+      (fun _ ↦ bot_le) (fun i ↦ measure_mono (inter_subset_inter_right _ (hBmono hjm)))
+  have hcompat (m l : ℕ) : (F m ∩ B (min m l) : Set (Euc n)) =ᵐ[volume] (F l ∩ B (min m l) : Set (Euc n)) := by
+    apply measure_symmDiff_eq_zero_iff.mp
+    apply le_antisymm _ bot_le
+    have ht := (hrestrict (min m l) m (min_le_left _ _)).add
+      (hrestrict (min m l) l (min_le_right _ _))
+    simp only [zero_add] at ht
+    apply ge_of_tendsto ht
+    exact Eventually.of_forall fun i ↦ by
+      simpa only [← inter_symmDiff_distrib_right, symmDiff_comm (F m) (E i)] using
+        (measure_symmDiff_le (μ := (volume : Measure (Euc n)))
+          (F m ∩ B (min m l)) (E i ∩ B (min m l)) (F l ∩ B (min m l)))
+  let G : Set (Euc n) := ⋃ m, F m ∩ B m
+  have hG : MeasurableSet G := MeasurableSet.iUnion fun m ↦
+    (hF m).inter measurableSet_closedBall
+  have hagree (m : ℕ) : (G ∩ B m : Set (Euc n)) =ᵐ[volume] (F m ∩ B m : Set (Euc n)) := by
+    filter_upwards [ae_all_iff.mpr (fun l ↦ hcompat m l)] with x hx
+    apply propext
+    constructor
+    · rintro ⟨hxG, hxm⟩
+      obtain ⟨l, hxl, hxBl⟩ := mem_iUnion.mp hxG
+      have hxB : x ∈ B (min m l) := by
+        rcases le_total m l with h | h
+        · simpa only [min_eq_left h] using hxm
+        · simpa only [min_eq_right h] using hxBl
+      have hh : x ∈ F m ∩ B (min m l) := (eq_iff_iff.mp (hx l)).mpr ⟨hxl, hxB⟩
+      exact ⟨hh.1, hxm⟩
+    · rintro ⟨hxm, hxB⟩
+      exact ⟨mem_iUnion.mpr ⟨m, hxm, hxB⟩, hxB⟩
+  have hconvG (m : ℕ) : Tendsto (fun i ↦ volume (symmDiff (E i) G ∩ B m)) atTop (𝓝 0) := by
+    convert hconv m using 1
+    funext i
+    apply measure_congr
+    filter_upwards [hagree m] with x hx
+    have hh : (x ∈ G ∧ x ∈ B m) ↔ (x ∈ F m ∧ x ∈ B m) := eq_iff_iff.mp hx
+    change (x ∈ symmDiff (E i) G ∩ B m) =
+      (x ∈ symmDiff (E i) (F m) ∩ B m)
+    apply propext
+    simp only [mem_inter_iff, mem_symmDiff]
+    tauto
+  refine ⟨G, hG, ?_⟩
+  intro c R hR
+  obtain ⟨m, hm⟩ := exists_nat_gt (dist c 0 + R)
+  have hs : Metric.closedBall c R ⊆ B m := by
+    intro x hx
+    apply Metric.mem_closedBall.mpr
+    exact (dist_triangle x c 0).trans (by
+      have hh := Metric.mem_closedBall.mp hx
+      linarith)
+  have ht : Tendsto (fun i ↦ volume (symmDiff (E i) G ∩ Metric.closedBall c R)) atTop (𝓝 0) :=
+    tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds (hconvG m)
+      (fun _ ↦ bot_le) (fun i ↦ measure_mono (inter_subset_inter_right _ hs))
+  simpa only [measureReal_def, ENNReal.toReal_zero, Function.comp_def] using
+    (ENNReal.tendsto_toReal ENNReal.zero_ne_top).comp ht
+
+theorem minimizing_global_compactness {n : ℕ} (hn : 0 < n)
+    (E : ℕ → Set (Euc n)) (hE : ∀ i, IsLocallyPerimeterMinimizing (E i)) :
+    ∃ F : Set (Euc n), IsLocallyPerimeterMinimizing F ∧
+      ∃ σ : ℕ → ℕ, StrictMono σ ∧
+        (∀ (c : Euc n) (R : ℝ), 0 < R →
+          Tendsto (fun i ↦ volume.real (symmDiff (E (σ i)) F ∩ Metric.closedBall c R))
+            atTop (𝓝 0)) ∧
+        (∀ (c : Euc n) (r : ℝ), 0 < r → perimeterMeasure F (Metric.sphere c r) = 0 →
+          Tendsto (fun i ↦ perimeter (E (σ i)) (Metric.ball c r)) atTop
+            (𝓝 (perimeter F (Metric.ball c r)))) := by
+  choose C hC hcut using exists_uniform_minimizing_exhaustion_cut hn
+  choose A hA hAK hAE hP using fun m i ↦ hcut m (E i) (hE i)
+  obtain ⟨F, hF, hFK, σ, hσ, ht⟩ := exists_simultaneous_indicator_L1_subsequence
+    (fun m ↦ Metric.closedBall 0 ((m:ℝ)+2)) (fun m ↦ isCompact_closedBall _ _)
+    C hC A hA hAK hP
+  have hlocal (m : ℕ) : Tendsto (fun i ↦ volume (symmDiff (E (σ i)) (F m) ∩
+      Metric.closedBall 0 ((m:ℝ)+1))) atTop (𝓝 0) := by
+    apply tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds (ht m)
+      (fun _ ↦ bot_le) (fun i ↦ ?_)
+    apply measure_mono
+    intro x hx
+    have hh : x ∈ E (σ i) ↔ x ∈ A m (σ i) := by
+      have he := Set.ext_iff.mp (hAE m (σ i)) x
+      simp only [mem_inter_iff] at he
+      exact ⟨fun h ↦ (he.mpr ⟨h, hx.2⟩).1, fun h ↦ (he.mp ⟨h, hx.2⟩).1⟩
+    simpa only [mem_symmDiff, hh] using hx.1
+  obtain ⟨G, hG, hconv⟩ := exists_global_set_of_local_L1_limits (fun i ↦ E (σ i)) F hF hlocal
+  refine ⟨G, locally_minimizing_of_local_L1_limit hn (fun i ↦ hE (σ i)) hG hconv,
+    σ, hσ, hconv, ?_⟩
+  intro c r hr hnull
+  exact tendsto_minimizing_perimeter_ball_of_local_L1 hn (fun i ↦ hE (σ i)) hG hconv c hr hnull
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/C1Graph.lean b/lean/OAI/Geometry/Perimeter/Regularity/C1Graph.lean
new file mode 100644
index 0000000000000000000000000000000000000000..afc33d64b0a948cb081b10d36ca2a93f0351d777
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/C1Graph.lean
@@ -0,0 +1,70 @@
+import OAI.Geometry.Perimeter.Regularity.NeighborhoodPlanes
+import OAI.Geometry.Perimeter.Regularity.GraphDerivative
+import OAI.Geometry.Perimeter.Regularity.UniformDerivativeContinuity
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+theorem c1_graph_from_contraction (hn : 1<n) {q M δ : ℝ}
+    (hq : 0<q) (hq1 : q<1) (hM : 0≤M) (hstep : SlabContractionProperty n q M δ)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    {v : Euc n} (hv : ‖v‖=1) {a : ℝ} (ha : 0<a) (haδ : 4*a<δ)
+    (ha4 : a<1/4) (haM : 8*M*a≤1/2) (hachord : a*(16*M+1/q)≤1/2)
+    (hs : FullSlab E v 8 a) :
+    ∃ f : normalPlane v → ℝ,
+      ContDiffOn ℝ 1 f (Metric.ball 0 (1/4)) ∧
+      (∀ y, ‖y‖<1/4 → |f y|≤a ∧ (y:Euc n)+f y•v∈perimeterSupport E ∩ Metric.ball 0 1) ∧
+      (∀ y z, ‖y‖<1/4 → ‖z‖<1/4 → |f y-f z|≤‖y-z‖) ∧
+      (∀ (y : normalPlane v) (t : ℝ), ‖y‖<1/4 → ‖(y:Euc n)+t•v‖<1 →
+        ((y:Euc n)+t•v∈perimeterSupport E ↔ t=f y)) := by
+  have hchord : ∀ p∈perimeterSupport E ∩ Metric.ball 0 1,
+      ∀ x∈perimeterSupport E ∩ Metric.ball 0 1,
+        |inner ℝ v (x-p)|≤(1/2)*‖x-p‖ := by
+    intro p hp x hx
+    exact (nearby_chord_bound hq hq1 hM hstep hmin hv ha haδ hs hp hx).trans
+      (mul_le_mul_of_nonneg_right hachord (norm_nonneg _))
+  obtain ⟨f,hf,hlip,hiff⟩ := single_graph_from_slab hn hmin hv ha.le ha4 hs hchord
+  have hwex (y : normalPlane v) : ∃ w : Euc n, ‖y‖<1/4 →
+      ‖w-v‖≤8*M*a ∧ ∀ j x, x∈perimeterSupport E → ‖x-((y:Euc n)+f y•v)‖≤2*q^j →
+      |inner ℝ w (x-((y:Euc n)+f y•v))|≤4*a*(2*M+1/(4*q))*(1/2)^j*‖x-((y:Euc n)+f y•v)‖ := by
+    by_cases hy : ‖y‖<1/4
+    · obtain ⟨w,_,hwd,htan⟩ := nearby_full_support_planes hq hq1 hM hstep hmin hv ha haδ hs (hf y hy).2
+      exact ⟨w,fun _ ↦ ⟨hwd,htan⟩⟩
+    · exact ⟨v,fun h ↦ (hy h).elim⟩
+  choose w hw using hwex
+  let D : normalPlane v → normalPlane v →L[ℝ] ℝ := fun y ↦ graphTangentLinear v (w y)
+  let C : ℝ := 16*a*(2*M+1/(4*q))
+  have hrem : ∀ j y, y∈Metric.ball (0:normalPlane v) (1/4) →
+      ∀ z, z∈Metric.ball (0:normalPlane v) (1/4) → ‖z-y‖≤q^j →
+      |f z-f y-D y (z-y)|≤C*(1/2)^j*‖z-y‖ := by
+    intro j y hy z hz hzy
+    have hy' : ‖y‖<1/4 := by simpa only [Metric.mem_ball,dist_zero_right] using hy
+    have hz' : ‖z‖<1/4 := by simpa only [Metric.mem_ball,dist_zero_right] using hz
+    have hh := graph_tangent_remainder hv ((hw y hy').1.trans haM) (hf z hz').2.1
+      (hlip z y hz' hy') hzy ((hw y hy').2 j) (by positivity : 0≤4*a*(2*M+1/(4*q)))
+    convert hh using 1
+    dsimp [C,D]
+    ring
+  have hD : ∀ y∈Metric.ball (0:normalPlane v) (1/4), HasFDerivAt f (D y) y := by
+    intro y hy
+    exact hasFDerivAt_of_geometric_remainder Metric.isOpen_ball hy hq (fun j z hz hzy ↦ hrem j y hy z hz hzy)
+  have hDC := continuous_derivative_of_geometric_remainder Metric.isOpen_ball hq
+    (fun x hx y hy ↦ hlip x y
+      (by simpa only [Metric.mem_ball,dist_zero_right] using hx)
+      (by simpa only [Metric.mem_ball,dist_zero_right] using hy)) hrem
+  refine ⟨f,?_,hf,hlip,hiff⟩
+  change ContDiffOn ℝ (0+1) f (Metric.ball 0 (1/4))
+  rw [contDiffOn_succ_iff_hasFDerivWithinAt_of_uniqueDiffOn Metric.isOpen_ball.uniqueDiffOn]
+  refine ⟨by simp, D,?_,fun y hy ↦ (hD y hy).hasFDerivWithinAt⟩
+  exact contDiffOn_zero.mpr (fun y hy ↦ (hDC y hy).continuousWithinAt)
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/C1MinRegularTest.lean b/lean/OAI/Geometry/Perimeter/Regularity/C1MinRegularTest.lean
new file mode 100644
index 0000000000000000000000000000000000000000..fc91c140d5a46e1b6f7f2ba2e1cdbab9d84147e0
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/C1MinRegularTest.lean
@@ -0,0 +1,460 @@
+import OAI.Geometry.Perimeter.Regularity.RegularSupersolutionC1Test
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPI.Boundary
+section TruncationRegularLimit
+variable {n : ℕ} {S Z : Set (Space n)}
+variable (hS : IsClosed S) (hZ : IsClosed Z)
+variable (hfinite : ∀ R : ℝ, 0 < R → (surfaceMeasure S) (Metric.ball 0 R) < ⊤)
+variable (hgrad : ∀ f : Space n → ℝ, ContDiff ℝ 1 f →
+  AEStronglyMeasurable (surfaceGradient S f) (surfaceMeasure S))
+variable {W : Space n → ℝ} {D : Space n → Space n}
+variable (htrace : ∀ x ∈ S \ Z, HasC1TraceJetAt S W D x)
+variable (hsup : ∀ φ : Space n → ℝ, ContDiff ℝ ∞ φ → HasCompactSupport φ →
+  Disjoint (tsupport φ) Z → (∀ x, 0 ≤ φ x) →
+  0 ≤ ∫ x, inner ℝ (D x) (surfaceGradient S φ x) ∂surfaceMeasure S)
+include hS hZ hfinite hgrad htrace hsup
+
+theorem c1_min_regular_test {φ : Space n → ℝ} (hφ : ContDiff ℝ 1 φ)
+    (hcφ : HasCompactSupport φ) (hφZ : Disjoint (tsupport φ) Z) (hpφ : ∀ x, 0 ≤ φ x) :
+    0 ≤ ∫ x, inner ℝ (truncatedGradient W D x) (surfaceGradient S φ x) ∂surfaceMeasure S := by
+  let b : Space n → ℝ := fun x ↦ inner ℝ (D x) (surfaceGradient S φ x)
+  have hb : Integrable b (surfaceMeasure S) :=
+    c1_integrable_regular_test_pair hS hZ hfinite hgrad htrace hφ hcφ hφZ
+  let q : ℕ → Space n → ℝ := fun j x ↦
+    inner ℝ (truncationSlope (truncationParameter j) (W x) • D x) (surfaceGradient S φ x)
+  have hq : ∀ j, AEStronglyMeasurable (q j) (surfaceMeasure S) := by
+    intro j
+    have htt : ∀ x ∈ S \ Z, HasC1TraceJetAt S (smoothTruncation (truncationParameter j) ∘ W)
+        (fun y ↦ truncationSlope (truncationParameter j) (W y) • D y) x := by
+      intro x hx
+      simpa only [smoothTruncation_deriv] using
+        (htrace x hx).comp hx.1 ((smoothTruncation_contDiff (truncationParameter j)).of_le (by norm_num))
+    exact (c1_integrable_regular_test_pair hS hZ hfinite hgrad htt hφ hcφ hφZ).aestronglyMeasurable
+  have hlim : Tendsto (fun j ↦ ∫ x, q j x ∂surfaceMeasure S) atTop
+      (𝓝 (∫ x, inner ℝ (truncatedGradient W D x) (surfaceGradient S φ x) ∂surfaceMeasure S)) := by
+    apply tendsto_integral_of_dominated_convergence (fun x ↦ ‖b x‖) hq hb.norm
+    · intro j
+      apply ae_of_all
+      intro x
+      simp only [q, real_inner_smul_left, norm_mul]
+      exact mul_le_of_le_one_left (norm_nonneg _) (truncationSlope_norm_le _ _)
+    · apply ae_of_all
+      intro x
+      have ht := (truncationSlope_tendsto (W x)).mul_const (b x)
+      by_cases hx : W x < 1
+      · simpa only [q, b, truncatedGradient, ite_eq_left hx, one_mul, real_inner_smul_left] using ht
+      · simpa only [q, b, truncatedGradient, ite_eq_right hx, zero_mul, inner_zero_left,
+          real_inner_smul_left] using ht
+  apply ge_of_tendsto hlim
+  apply Eventually.of_forall
+  intro j
+  exact c1_smoothTruncation_regular_test hS hZ hfinite hgrad htrace hsup
+    (truncationParameter_pos j) hφ hcφ hφZ hpφ
+
+theorem c1_min_regular_energy (hW0 : ∀ x ∈ S \ Z, 0 ≤ W x)
+    {η : Space n → ℝ} (hη : ContDiff ℝ 1 η) (hcη : HasCompactSupport η)
+    (hηZ : Disjoint (tsupport η) Z) :
+    MemLp (fun x ↦ η x • truncatedGradient W D x) 2 (surfaceMeasure S) ∧
+      vectorEnergy (surfaceMeasure S) (fun x ↦ η x • truncatedGradient W D x) ≤
+        4 * vectorEnergy (surfaceMeasure S) (surfaceGradient S η) := by
+  let A : ℕ → Space n → Space n := fun j x ↦ η x • (truncationSlope (truncationParameter j) (W x) • D x)
+  let H : Space n → Space n := fun x ↦ η x • truncatedGradient W D x
+  have hA : ∀ j, MemLp (A j) 2 (surfaceMeasure S) := by
+    intro j
+    have htt : ∀ x ∈ S \ Z, HasC1TraceJetAt S (smoothTruncation (truncationParameter j) ∘ W)
+        (fun y ↦ truncationSlope (truncationParameter j) (W y) • D y) x := by
+      intro x hx
+      simpa only [smoothTruncation_deriv] using
+        (htrace x hx).comp hx.1 ((smoothTruncation_contDiff (truncationParameter j)).of_le (by norm_num))
+    exact c1_regular_gradient_memLp_test hS hZ hfinite hgrad htt hη hcη hηZ
+  have hB : MemLp (surfaceGradient S η) 2 (surfaceMeasure S) :=
+    c1_surfaceGradient_memLp_of_compact hfinite hgrad hη hcη
+  have he : ∀ j, vectorEnergy (surfaceMeasure S) (A j) ≤
+      4 * vectorEnergy (surfaceMeasure S) (surfaceGradient S η) := by
+    intro j
+    have h := ENNReal.ofReal_le_ofReal (c1_smoothTruncation_regular_energy hS hZ hfinite hgrad
+      htrace hW0 hsup (truncationParameter_pos j) hη hcη hηZ)
+    rw [ENNReal.ofReal_mul (by norm_num : (0 : ℝ) ≤ 4)] at h
+    simpa only [A, vectorEnergy, ofReal_integral_eq_lintegral_ofReal
+      ((memLp_two_iff_integrable_sq_norm (hA j).aestronglyMeasurable).mp (hA j))
+      (ae_of_all _ (fun x ↦ sq_nonneg ‖A j x‖)),
+      ofReal_integral_eq_lintegral_ofReal
+      ((memLp_two_iff_integrable_sq_norm hB.aestronglyMeasurable).mp hB)
+      (ae_of_all _ (fun x ↦ sq_nonneg ‖surfaceGradient S η x‖)), ENNReal.ofReal_ofNat] using h
+  have hlim : ∀ᵐ x ∂surfaceMeasure S, Tendsto (fun j ↦ A j x) atTop (𝓝 (H x)) := by
+    apply ae_of_all
+    intro x
+    have ht := (tendsto_const_nhds (x := η x)).smul
+      ((truncationSlope_tendsto (W x)).smul_const (D x))
+    by_cases hx : W x < 1
+    · simpa only [A, H, truncatedGradient, ite_eq_left hx, one_smul] using ht
+    · simpa only [A, H, truncatedGradient, ite_eq_right hx, zero_smul] using ht
+  have hHm : AEStronglyMeasurable H (surfaceMeasure S) :=
+    aestronglyMeasurable_of_tendsto_ae atTop (fun j ↦ (hA j).aestronglyMeasurable) hlim
+  have hHE := vectorEnergy_le_of_ae_tendsto (fun j ↦ (hA j).aestronglyMeasurable) hlim
+    (Eventually.of_forall he) tendsto_const_nhds
+  refine ⟨(memLp_two_iff_vectorEnergy hHm).mpr (hHE.trans_lt ?_), hHE⟩
+  exact ENNReal.mul_lt_top (by norm_num) ((memLp_two_iff_vectorEnergy hB.aestronglyMeasurable).mp hB)
+
+end TruncationRegularLimit
+end RPI.Boundary
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPI.Boundary
+
+theorem c1_min_local_energy {n : ℕ} (hn : 8 ≤ n)
+    {S Z : Set (Space n)} (hS : IsClosed S) (hZ : IsClosed Z)
+    (hZd : dimH Z ≤ (n - 8 : ℕ))
+    (hfinite : ∀ R : ℝ, 0 < R → (surfaceMeasure S) (Metric.ball 0 R) < ⊤)
+    (hgrad : ∀ f : Space n → ℝ, ContDiff ℝ 1 f →
+      AEStronglyMeasurable (surfaceGradient S f) (surfaceMeasure S))
+    (hgrowth : ∀ R : ℝ, 0 < R → ∃ A r₀ : ℝ, 0 < A ∧ 0 < r₀ ∧
+      ∀ c ∈ Z ∩ Metric.closedBall 0 R, ∀ r : ℝ, 0 < r → r < r₀ →
+        (surfaceMeasure S).restrict (Metric.ball 0 R) (Metric.ball c r) ≤
+          ENNReal.ofReal (A * r ^ (n - 1)))
+    {W : Space n → ℝ} {D : Space n → Space n}
+    (htrace : ∀ x ∈ S \ Z, HasC1TraceJetAt S W D x)
+    (hW0 : ∀ x ∈ S \ Z, 0 ≤ W x)
+    (hsup : ∀ φ : Space n → ℝ, ContDiff ℝ ∞ φ → HasCompactSupport φ →
+      Disjoint (tsupport φ) Z → (∀ x, 0 ≤ φ x) →
+      0 ≤ ∫ x, inner ℝ (D x) (surfaceGradient S φ x) ∂surfaceMeasure S)
+    {R : ℝ} (hR : 0 < R) :
+    MemLp (truncatedGradient W D) 2 ((surfaceMeasure S).restrict (Metric.ball 0 R)) := by
+  let b : ContDiffBump (0 : Space n) := ⟨R + 1, R + 2, by linarith, by linarith⟩
+  let μ := (surfaceMeasure S).restrict (Metric.ball 0 (R + 3))
+  have hR3 : 0 < R + 3 := by linarith
+  have : IsFiniteMeasure μ := ⟨by simpa only [μ, Measure.restrict_apply_univ] using hfinite _ hR3⟩
+  let K := Z ∩ tsupport b
+  have hKc : IsCompact K := b.hasCompactSupport.of_isClosed_subset
+    (hZ.inter (isClosed_tsupport b)) inter_subset_right
+  have hKd : dimH K ≤ (n - 8 : ℕ) := (dimH_mono inter_subset_left).trans hZd
+  have hbs : tsupport b ⊆ Metric.ball (0 : Space n) (R + 3) := by
+    rw [b.tsupport_eq]
+    exact Metric.closedBall_subset_ball (by dsimp [b]; linarith)
+  obtain ⟨A, r₀, hA, hr₀, hgr⟩ := hgrowth _ hR3
+  obtain ⟨χ, hχ, hχlim, hχE⟩ := exists_memLp_capacity_sequence hn S
+    (Metric.ball 0 (R + 3)) K hKc hKd
+    (fun f hf ↦ (hgrad f (hf.of_le (by norm_num))).mono_measure Measure.restrict_le_self) hA hr₀
+    (fun c hc s hs hsr ↦ hgr c ⟨hc.1, Metric.ball_subset_closedBall (hbs hc.2)⟩ s hs hsr)
+  have hb : MemLp (fun x ↦ b x) 2 μ := MemLp.of_bound b.continuous.aestronglyMeasurable 1
+    (ae_of_all _ (fun x ↦ by simpa only [Real.norm_eq_abs, abs_of_nonneg b.nonneg] using b.le_one))
+  have hDb : MemLp (surfaceGradient S b) 2 μ := surfaceGradient_memLp_on_ball b.contDiff
+    ((hgrad b b.contDiff).mono_measure Measure.restrict_le_self)
+  obtain ⟨_, hB, hBt⟩ := capacity_product_pair_tendsto hb hDb χ
+    (fun i ↦ (hχ i).1.continuous.aestronglyMeasurable)
+    (fun i ↦ ae_of_all _ (hχ i).2.1) hχlim zero_le_one
+    (ae_of_all _ (fun x ↦ by simpa only [Real.norm_eq_abs, abs_of_nonneg b.nonneg] using b.le_one))
+    (fun i ↦ surfaceGradient S (χ i)) (fun i ↦ (hχ i).2.2.2) hχE
+  let η : ℕ → Space n → ℝ := fun i x ↦ b x * χ i x
+  let F : ℕ → Space n → Space n := fun i x ↦ η i x • truncatedGradient W D x
+  let H : Space n → Space n := fun x ↦ b x • truncatedGradient W D x
+  have hη : ∀ i, ContDiff ℝ ∞ (η i) := fun i ↦ b.contDiff.mul (hχ i).1
+  have hcη : ∀ i, HasCompactSupport (η i) := fun _ ↦ b.hasCompactSupport.mul_right
+  have hηZ : ∀ i, Disjoint (tsupport (η i)) Z := fun i ↦
+    tsupport_mul_disjoint_Singular (hχ i).2.2.1
+  have hFE : ∀ i, MemLp (F i) 2 (surfaceMeasure S) ∧
+      vectorEnergy (surfaceMeasure S) (F i) ≤
+        4 * vectorEnergy (surfaceMeasure S) (surfaceGradient S (η i)) := by
+    intro i
+    exact c1_min_regular_energy hS hZ hfinite hgrad htrace hsup hW0 ((hη i).of_le (by norm_num)) (hcη i) (hηZ i)
+  have hFm : ∀ i, MemLp (F i) 2 μ := fun i ↦ (hFE i).1.mono_measure Measure.restrict_le_self
+  have hB_eq : ∀ i, (fun x ↦ χ i x • surfaceGradient S b x + b x • surfaceGradient S (χ i) x) =
+      surfaceGradient S (η i) := by
+    intro i
+    funext x
+    change _ = surfaceGradient S (fun y ↦ b y * χ i y) x
+    have hbDiff : ContDiff ℝ ∞ (fun y ↦ b y) := b.contDiff
+    rw [surfaceGradient_mul (hbDiff.differentiable (by simp) x)
+      ((hχ i).1.differentiable (by simp) x), add_comm]
+  have hBE := vectorEnergy_tendsto_of_toLp_tendsto hB hDb hBt.snd_nhds
+  simp_rw [hB_eq] at hBE
+  have hbound : ∀ i, vectorEnergy μ (F i) ≤ 4 * vectorEnergy μ (surfaceGradient S (η i)) := by
+    intro i
+    rw [vectorEnergy_restrict_of_vanish measurableSet_ball, vectorEnergy_restrict_of_vanish measurableSet_ball]
+    · exact (hFE i).2
+    · intro x hx
+      exact surfaceGradient_eq_zero_of_notMem_tsupport S
+        (fun h ↦ hx (hbs (tsupport_mul_subset_left h)))
+    · intro x hx
+      have hb0 : b x = 0 := image_eq_zero_of_notMem_tsupport (fun h ↦ hx (hbs h))
+      simp only [F, η, hb0, zero_mul, zero_smul]
+  have hlim : ∀ᵐ x ∂μ, Tendsto (fun i ↦ F i x) atTop (𝓝 (H x)) := by
+    filter_upwards [hχlim] with x hx
+    simpa only [F, H, η, mul_one] using
+      (tendsto_const_nhds.mul hx).smul_const (truncatedGradient W D x)
+  have hHm : AEStronglyMeasurable H μ :=
+    aestronglyMeasurable_of_tendsto_ae atTop (fun i ↦ (hFm i).aestronglyMeasurable) hlim
+  have hHE := vectorEnergy_le_of_ae_tendsto (fun i ↦ (hFm i).aestronglyMeasurable) hlim
+    (Eventually.of_forall hbound) (ENNReal.Tendsto.const_mul hBE (Or.inr (by norm_num)))
+  have hH : MemLp H 2 μ := (memLp_two_iff_vectorEnergy hHm).mpr (hHE.trans_lt
+    (ENNReal.mul_lt_top (by norm_num) ((memLp_two_iff_vectorEnergy hDb.aestronglyMeasurable).mp hDb)))
+  have hHR : MemLp H 2 ((surfaceMeasure S).restrict (Metric.ball 0 R)) :=
+    hH.mono_measure (Measure.restrict_mono_set _ (Metric.ball_subset_ball (by linarith)))
+  apply hHR.ae_eq
+  filter_upwards [ae_restrict_mem measurableSet_ball] with x hx
+  have hbx : b x = 1 := b.one_of_mem_closedBall (Metric.mem_closedBall.mpr
+    ((Metric.mem_ball.mp hx).le.trans (by dsimp [b]; linarith)))
+  simp only [H, hbx, one_smul]
+
+end RPI.Boundary
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+open RPI RPI.Boundary
+
+theorem c1_min_wholeBoundarySobolev_and_supersolution_of_regular_geometry
+    {n : ℕ} (hn : 8 ≤ n) {E Z : Set (Euc n)} (hE : MeasurableSet E)
+    (hZ : IsClosed Z) (hZd : dimH Z ≤ (n - 8 : ℕ))
+    (hfinite : ∀ R : ℝ, 0 < R → surfaceMeasure (perimeterSupport E) (Metric.ball 0 R) < ⊤)
+    (hgrowth : ∀ R : ℝ, 0 < R → ∃ A r₀ : ℝ, 0 < A ∧ 0 < r₀ ∧
+      ∀ c ∈ Z ∩ Metric.closedBall 0 R, ∀ r : ℝ, 0 < r → r < r₀ →
+        (surfaceMeasure (perimeterSupport E)).restrict (Metric.ball 0 R) (Metric.ball c r) ≤
+          ENNReal.ofReal (A * r ^ (n - 1)))
+    (hreg : ∀ x ∈ perimeterSupport E \ Z, ∃ U : Set (Euc n), IsOpen U ∧ x ∈ U ∧
+      ∃ f : Euc n → ℝ, ContDiffOn ℝ ∞ f U ∧
+        (∀ y ∈ U, fderiv ℝ f y ≠ 0) ∧
+        (∀ y ∈ U, y ∈ perimeterSupport E ↔ f y = 0) ∧
+        (∀ y ∈ U, (f y < 0 → y ∈ E) ∧ (y ∈ E → f y ≤ 0)))
+    {W : Euc n → ℝ} {D : Euc n → Euc n}
+    (htrace : ∀ x ∈ perimeterSupport E \ Z, HasC1TraceJetAt (perimeterSupport E) W D x)
+    (hW0 : ∀ x ∈ perimeterSupport E \ Z, 0 ≤ W x)
+    (hsup : ∀ φ : Euc n → ℝ, ContDiff ℝ ∞ φ → HasCompactSupport φ →
+      Disjoint (tsupport φ) Z → (∀ x, 0 ≤ φ x) →
+      0 ≤ ∫ x, inner ℝ (D x) (surfaceGradient (perimeterSupport E) φ x)
+        ∂surfaceMeasure (perimeterSupport E)) :
+    WholeBoundarySobolev E (fun x ↦ min (W x) 1) (truncatedGradient W D) ∧
+      WholeBoundarySupersolution E (truncatedGradient W D) := by
+  have hS := isClosed_perimeterSupport E
+  have hlevel : ∀ x ∈ perimeterSupport E \ Z, ∃ U : Set (Euc n), IsOpen U ∧ x ∈ U ∧
+      ∃ f : Euc n → ℝ, ContDiffOn ℝ 1 f U ∧
+        (∀ y ∈ U, fderiv ℝ f y ≠ 0) ∧
+        (∀ y ∈ U, y ∈ perimeterSupport E ↔ f y = 0) := by
+    intro x hx
+    obtain ⟨U, hU, hxU, f, hf, hdf, hs, _⟩ := hreg x hx
+    exact ⟨U, hU, hxU, f, hf.of_le (by norm_num), hdf, hs⟩
+  have hfilled : ∀ x ∈ perimeterSupport E \ Z, ∃ f : Euc n → ℝ,
+      ContDiffAt ℝ 1 f x ∧ f x = 0 ∧ fderiv ℝ f x ≠ 0 ∧
+      (∀ᶠ y in 𝓝 x, y ∈ perimeterSupport E ↔ f y = 0) ∧
+      (∀ᶠ y in 𝓝 x, (f y < 0 → y ∈ E) ∧ (y ∈ E → f y ≤ 0)) := by
+    intro x hx
+    obtain ⟨U, hU, hxU, f, hf, hdf, hs, he⟩ := hreg x hx
+    refine ⟨f, (hf.of_le (by norm_num)).contDiffAt (hU.mem_nhds hxU),
+      (hs x hxU).mp hx.1, hdf x hxU, ?_, ?_⟩
+    · filter_upwards [hU.mem_nhds hxU] with y hy using hs y hy
+    · filter_upwards [hU.mem_nhds hxU] with y hy using he y hy
+  have hgrad : ∀ f : Euc n → ℝ, ContDiff ℝ 1 f →
+      AEStronglyMeasurable (surfaceGradient (perimeterSupport E) f)
+        (surfaceMeasure (perimeterSupport E)) := by
+    intro f hf
+    exact surfaceGradient_aestronglyMeasurable_of_regular_locus hn hS.measurableSet
+      hZ.measurableSet hZd hlevel hf
+  have hloc : ∀ R : ℝ, 0 < R → MemLp (truncatedGradient W D) 2
+      ((surfaceMeasure (perimeterSupport E)).restrict (Metric.ball 0 R)) := by
+    intro R hR
+    exact c1_min_local_energy hn hS hZ hZd hfinite hgrad hgrowth htrace hW0 hsup hR
+  have haereg : ∀ᵐ x ∂surfaceMeasure (perimeterSupport E), x ∈ perimeterSupport E \ Z := by
+    have hz : ∀ᵐ x ∂surfaceMeasure (perimeterSupport E), x ∉ Z := by
+      simpa only [ae_iff, not_not, Set.ofPred_mem_eq] using
+        surfaceMeasure_zero_of_singular_dim hn (perimeterSupport E) Z hZd
+    filter_upwards [ae_restrict_mem hS.measurableSet, hz] with x hx hzx
+    exact ⟨hx, hzx⟩
+  constructor
+  · apply wholeBoundarySobolev_of_local_sobolevGraph_filled hn hE hS.measurableSet hZd hfilled
+    intro R hR
+    have : IsFiniteMeasure ((surfaceMeasure (perimeterSupport E)).restrict (Metric.ball 0 R)) :=
+      ⟨by simpa only [Measure.restrict_apply_univ] using hfinite R hR⟩
+    have hw : ∀ᵐ x ∂(surfaceMeasure (perimeterSupport E)).restrict (Metric.ball 0 R), 0 ≤ W x := by
+      filter_upwards [haereg.filter_mono (ae_mono Measure.restrict_le_self)] with x hx
+      exact hW0 x hx
+    obtain ⟨A, r₀, hA, hr₀, hgr⟩ := hgrowth (R + 2) (by linarith)
+    obtain ⟨hh, hpair⟩ := c1_min_one_mem_sobolevGraph_ball hn hS hZ hZd hR htrace hw (hloc R hR)
+      (fun f hf ↦ (hgrad f (hf.of_le (by norm_num))).mono_measure Measure.restrict_le_self) hA hr₀ (by
+        intro c hc r hr hrr
+        apply le_trans _ (hgr c hc r hr hrr)
+        exact Measure.le_iff'.mp (Measure.restrict_mono_set _
+          (Metric.ball_subset_ball (by linarith : R ≤ R + 2))) _)
+    exact ⟨hh, hloc R hR, hpair⟩
+  · intro φ hφ hcφ hpφ
+    have hh := full_test_of_regular_tests hn hZ hZd hloc hfinite (fun f hf ↦ hgrad f (hf.of_le (by norm_num))) hgrowth
+      (fun ψ hψ hcψ hψZ hpψ ↦ c1_min_regular_test hS hZ hfinite hgrad htrace hsup (hψ.of_le (by norm_num)) hcψ hψZ hpψ)
+      hφ hcφ hpφ
+    have heq := surfaceGradient_ae_eq_tangentialGradient_filled hn hE hS.measurableSet hZd hfilled φ
+    rw [boundaryArea_eq_smul_surfaceMeasure, integral_smul_measure]
+    have hi : (∫ x, inner ℝ (truncatedGradient W D x) (tangentialGradient E φ x)
+        ∂surfaceMeasure (perimeterSupport E)) =
+        ∫ x, inner ℝ (truncatedGradient W D x) (surfaceGradient (perimeterSupport E) φ x)
+          ∂surfaceMeasure (perimeterSupport E) := by
+      apply integral_congr_ae
+      filter_upwards [heq] with x hx
+      rw [hx]
+    rw [hi]
+    exact mul_nonneg ENNReal.toReal_nonneg hh
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+open RPI RPI.Boundary
+
+theorem c1_min_wholeBoundarySobolev_and_supersolution_of_minimizing_regular_geometry
+    {n : ℕ} (hn : 8 ≤ n) {E Z : Set (Euc n)}
+    (hmin : IsLocallyPerimeterMinimizing E)
+    (hZ : IsClosed Z) (hZd : dimH Z ≤ (n - 8 : ℕ))
+    (hreg : ∀ x ∈ perimeterSupport E \ Z, ∃ U : Set (Euc n), IsOpen U ∧ x ∈ U ∧
+      ∃ f : Euc n → ℝ, ContDiffOn ℝ ∞ f U ∧
+        (∀ y ∈ U, fderiv ℝ f y ≠ 0) ∧
+        (∀ y ∈ U, y ∈ perimeterSupport E ↔ f y = 0) ∧
+        (∀ y ∈ U, (f y < 0 → y ∈ E) ∧ (y ∈ E → f y ≤ 0)))
+    {W : Euc n → ℝ} {D : Euc n → Euc n}
+    (htrace : ∀ x ∈ perimeterSupport E \ Z, HasC1TraceJetAt (perimeterSupport E) W D x)
+    (hW0 : ∀ x ∈ perimeterSupport E \ Z, 0 ≤ W x)
+    (hsup : ∀ φ : Euc n → ℝ, ContDiff ℝ ∞ φ → HasCompactSupport φ →
+      Disjoint (tsupport φ) Z → (∀ x, 0 ≤ φ x) →
+      0 ≤ ∫ x, inner ℝ (D x) (surfaceGradient (perimeterSupport E) φ x)
+        ∂surfaceMeasure (perimeterSupport E)) :
+    WholeBoundarySobolev E (fun x ↦ min (W x) 1) (truncatedGradient W D) ∧
+      WholeBoundarySupersolution E (truncatedGradient W D) := by
+  have hhalf : ∀ x ∈ perimeterSupport E \ Z, ∃ v, HasHalfspaceBlowup E x v := by
+    intro x hx
+    obtain ⟨U, hU, hxU, f, hf, hdf, hs, he⟩ := hreg x hx
+    refine ⟨_, hasHalfspaceBlowup_gradient_filled hmin.1
+      ((hf.contDiffAt (hU.mem_nhds hxU)).differentiableAt (by norm_num))
+      ((hs x hxU).mp hx.1) (hdf x hxU) ?_⟩
+    filter_upwards [hU.mem_nhds hxU] with y hy using he y hy
+  obtain ⟨A, hA, harea⟩ := exists_minimizing_surfaceMeasure_growth_of_halfspace hn hmin hZd hhalf
+  apply c1_min_wholeBoundarySobolev_and_supersolution_of_regular_geometry hn hmin.1 hZ hZd
+    (fun R hR ↦ (harea 0 R hR).trans_lt ENNReal.ofReal_lt_top) _ hreg htrace hW0 hsup
+  intro R hR
+  refine ⟨A, 1, hA, by norm_num, ?_⟩
+  intro c hc r hr hrr
+  exact (Measure.restrict_apply_le (Metric.ball 0 R) (Metric.ball c r)).trans (harea c r hr)
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+open RPI.Boundary
+open ContinuousLinearMap
+open scoped Convolution
+
+theorem divergence_comp_add_right {n : ℕ} {V : Euc n → Euc n}
+    (hV : Differentiable ℝ V) (t x : Euc n) :
+    divergence (fun y ↦ V (y + t)) x = divergence V (x + t) := by
+  have hd : fderiv ℝ (fun y ↦ V (y + t)) x = fderiv ℝ V (x + t) := by
+    simpa only [Function.comp_def, id_eq, ContinuousLinearMap.comp_id] using ((hV _).hasFDerivAt.comp x ((hasFDerivAt_id x).add_const t)).fderiv
+  simp only [divergence, hd]
+
+theorem integral_divergence_le_perimeter_real {n : ℕ} (E U : Set (Euc n))
+    (hP : perimeter E U ≠ ⊤) {V : Euc n → Euc n} (hV : PerimeterAdmissible U V) :
+    ∫ x in E, divergence V x ≤ (perimeter E U).toReal := by
+  apply (ENNReal.ofReal_le_iff_le_toReal hP).mp
+  rw [perimeter_eq_iSup_tests]
+  exact le_iSup_of_le V (le_iSup_of_le hV le_rfl)
+
+theorem indicator_convolution_variation_le {n : ℕ} {E : Set (Euc n)}
+    (hE : MeasurableSet E) (hP : perimeter E univ ≠ ⊤)
+    {k : Euc n → ℝ} (hkc : Continuous k) (hk : Integrable k volume)
+    (hk0 : ∀ x, 0 ≤ k x) (hk1 : ∫ x, k x = 1)
+    {V : Euc n → Euc n} (hV : PerimeterAdmissible univ V) :
+    (∫ x, (k ⋆[lsmul ℝ ℝ, volume] E.indicator (fun _ ↦ (1 : ℝ))) x * divergence V x) ≤
+      (perimeter E univ).toReal := by
+  let χ : Euc n → ℝ := E.indicator (fun _ ↦ 1)
+  have hmχ : Measurable χ := measurable_const.indicator hE
+  have hbχ : ∀ x, ‖χ x‖ ≤ 1 := by
+    intro x
+    by_cases hx : x ∈ E <;> simp [χ, hx]
+  have hd := integrable_divergence hV.1 hV.2.1
+  let F : Euc n × Euc n → ℝ := fun p ↦ k p.1 * χ (p.2 - p.1) * divergence V p.2
+  have hFi : Integrable F (volume.prod volume) := by
+    apply (hk.norm.mul_prod hd.norm).mono'
+    · exact ((hkc.measurable.comp measurable_fst).mul
+        (hmχ.comp (measurable_snd.sub measurable_fst))).mul
+        ((continuous_divergence hV.1).measurable.comp measurable_snd) |>.aestronglyMeasurable
+    · apply ae_of_all
+      intro p
+      simp only [F, norm_mul]
+      exact mul_le_mul_of_nonneg_right
+        (by nlinarith [hbχ (p.2 - p.1), norm_nonneg (k p.1)]) (norm_nonneg _)
+  have hinner : ∀ t, (∫ x, F (t, x)) = k t * (∫ y in E, divergence V (y + t)) := by
+    intro t
+    simp only [F, mul_assoc, integral_const_mul]
+    congr 1
+    calc
+      (∫ x, χ (x - t) * divergence V x) =
+          ∫ y, χ y * divergence V (y + t) := by
+        simpa only [add_sub_cancel_right] using
+          (integral_add_right_eq_self (fun x ↦ χ (x - t) * divergence V x) t).symm
+      _ = ∫ y in E, divergence V (y + t) := by
+        rw [← integral_indicator hE]
+        apply integral_congr_ae
+        apply ae_of_all
+        intro y
+        by_cases hy : y ∈ E <;> simp [χ, hy]
+  have hInt : Integrable (fun t ↦ k t * (∫ y in E, divergence V (y + t))) volume := by
+    simpa only [hinner] using hFi.integral_prod_left
+  have heq : (∫ x, (k ⋆[lsmul ℝ ℝ, volume] χ) x * divergence V x) =
+      ∫ t, k t * (∫ y in E, divergence V (y + t)) := by
+    calc
+      _ = ∫ x, ∫ t, F (t, x) := by
+        apply integral_congr_ae
+        apply ae_of_all
+        intro x
+        simp only [convolution_def, lsmul_apply, smul_eq_mul, F, integral_mul_const]
+      _ = ∫ t, ∫ x, F (t, x) := integral_integral_swap hFi.swap
+      _ = _ := by simp only [hinner]
+  have hbound : ∀ t, ∫ y in E, divergence V (y + t) ≤ (perimeter E univ).toReal := by
+    intro t
+    let Vt : Euc n → Euc n := fun y ↦ V (y + t)
+    have hVt : PerimeterAdmissible univ Vt := ⟨hV.1.comp (contDiff_id.add contDiff_const),
+      hV.2.1.comp_homeomorph (Homeomorph.addRight t), subset_univ _, fun y ↦ hV.2.2.2 _⟩
+    simpa only [Vt, divergence_comp_add_right (hV.1.differentiable (by simp))] using
+      integral_divergence_le_perimeter_real E univ hP hVt
+  rw [heq]
+  calc
+    (∫ t, k t * (∫ y in E, divergence V (y + t))) ≤
+        ∫ t, k t * (perimeter E univ).toReal := by
+      exact integral_mono hInt (hk.mul_const _) fun t ↦
+        mul_le_mul_of_nonneg_left (hbound t) (hk0 t)
+    _ = (perimeter E univ).toReal := by rw [integral_mul_const, hk1, one_mul]
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/ClosedFilledLineInvariantOfFrontier.lean b/lean/OAI/Geometry/Perimeter/Regularity/ClosedFilledLineInvariantOfFrontier.lean
new file mode 100644
index 0000000000000000000000000000000000000000..e7def427f64a6f7495da4cdb1021c1d7dc1d1fdf
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/ClosedFilledLineInvariantOfFrontier.lean
@@ -0,0 +1,69 @@
+import OAI.Geometry.Perimeter.Regularity.ExistsMinimizingDensityLimit
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+open scoped ENNReal Topology
+open scoped ENNReal Topology ContDiff
+open scoped Topology ENNReal
+namespace RPISingularHarnack
+
+theorem closed_filled_line_invariant_of_frontier {n : ℕ} {F : Set (Euc n)}
+    (hF : IsClosed F) (c : Euc n)
+    (hline : ∀ (t : ℝ) (x : Euc n), x ∈ frontier F → x+t • c ∈ frontier F) :
+    ∀ (t : ℝ) (x : Euc n), x ∈ F → x+t • c ∈ F := by
+  intro t x hx
+  by_cases hxfr : x ∈ frontier F
+  · exact hF.closure_eq ▸ frontier_subset_closure (hline t x hxfr)
+  have hxint : x ∈ interior F := by
+    by_contra hxi
+    apply hxfr
+    rw [frontier, hF.closure_eq]
+    exact ⟨hx, hxi⟩
+  let L : Set (Euc n) := Set.range (fun s : ℝ ↦ x+s • c)
+  have hLfr : Disjoint L (frontier F) := by
+    apply Set.disjoint_left.mpr
+    rintro y ⟨s, rfl⟩ hy
+    apply hxfr
+    have hh := hline (-s) (x+s • c) hy
+    convert hh using 1
+    module
+  have hLsub : L ⊆ interior F ∪ Fᶜ := by
+    intro y hy
+    by_cases hyF : y ∈ F
+    · apply Or.inl
+      by_contra hyi
+      exact (Set.disjoint_left.mp hLfr hy) (by rw [frontier, hF.closure_eq]; exact ⟨hyF, hyi⟩)
+    · exact Or.inr hyF
+  have hLc : IsPreconnected L := by
+    exact isPreconnected_range (by fun_prop : Continuous (fun s : ℝ ↦ x+s • c))
+  have hLInt : L ⊆ interior F := hLc.subset_left_of_subset_union isOpen_interior hF.isOpen_compl
+    (Set.disjoint_left.mpr fun y hy hyc ↦ hyc (interior_subset hy)) hLsub
+    ⟨x, ⟨0, by simp⟩, hxint⟩
+  exact interior_subset (hLInt ⟨t, rfl⟩)
+
+theorem minimizing_cone_tangent_filled_line_invariant {n : ℕ} (hn : 1 < n)
+    {E F : Set (Euc n)} (hE : IsLocallyPerimeterMinimizing E) (hcone : IsCone E)
+    (hF : IsLocallyPerimeterMinimizing F) (hFc : IsClosed F)
+    (hFfront : frontier F = perimeterSupport F) (c : Euc n)
+    (r : ℕ → ℝ) (hr : ∀ i, 0 < r i) (htr : Tendsto r atTop (𝓝 0))
+    (hconv : ∀ (x : Euc n) (R : ℝ), 0 < R →
+      Tendsto (fun i ↦ volume.real (symmDiff (blownUpSet E c (r i)) F ∩ Metric.closedBall x R))
+        atTop (𝓝 0)) :
+    ∀ (t : ℝ) (x : Euc n), x ∈ F → x + t • c ∈ F := by
+  apply closed_filled_line_invariant_of_frontier hFc c
+  intro t x hx
+  rw [hFfront] at hx ⊢
+  exact minimizing_cone_tangent_support_line_invariant hn hE hcone hF c r hr htr hconv t x hx
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/CompactLpExtension.lean b/lean/OAI/Geometry/Perimeter/Regularity/CompactLpExtension.lean
new file mode 100644
index 0000000000000000000000000000000000000000..f139493605be17bf39ad6cba06f8f1c67ffee38d
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/CompactLpExtension.lean
@@ -0,0 +1,84 @@
+import OAI.Geometry.Perimeter.Regularity.HeightCompactness
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal
+namespace RPISingularHarnack
+open RPI.Boundary
+variable {n : ℕ}
+
+def compactLpExtension (K : Set (Euc n))
+    (g : Lp ℝ 1 (volume.comap (Subtype.val : K → Euc n))) : Euc n → ℝ :=
+  Function.extend (Subtype.val : K → Euc n) g (fun _ ↦ 0)
+
+@[simp] theorem compactLpExtension_coe (K : Set (Euc n))
+    (g : Lp ℝ 1 (volume.comap (Subtype.val : K → Euc n))) (x : K) :
+    compactLpExtension K g x = g x :=
+  Subtype.val_injective.extend_apply _ _ x
+
+theorem compactLpExtension_zero (K : Set (Euc n))
+    (g : Lp ℝ 1 (volume.comap (Subtype.val : K → Euc n))) {x : Euc n} (hx : x ∉ K) :
+    compactLpExtension K g x = 0 := by
+  apply Function.extend_apply'
+  rintro ⟨y,rfl⟩; exact hx y.property
+
+theorem compactLpExtension_integrable (K : Set (Euc n)) (hK : IsCompact K)
+    (g : Lp ℝ 1 (volume.comap (Subtype.val : K → Euc n))) :
+    Integrable (compactLpExtension K g) volume := by
+  have hi : IntegrableOn (compactLpExtension K g) K volume := by
+    rw [integrableOn_iff_comap_subtypeVal hK.measurableSet]
+    simpa only [Function.comp_def,compactLpExtension_coe] using
+      (memLp_one_iff_integrable.mp (Lp.memLp g))
+  exact hi.integrable_of_forall_notMem_eq_zero (fun _ hx ↦ compactLpExtension_zero K g hx)
+
+theorem compactLpExtension_error (K : Set (Euc n)) (hK : IsCompact K)
+    (g : Lp ℝ 1 (volume.comap (Subtype.val : K → Euc n)))
+    (f : Euc n → ℝ) (hi : Integrable f volume) (hs : Function.support f ⊆ K) :
+    (∫⁻ x, ‖f x-compactLpExtension K g x‖ₑ) = edist (compactFunctionLp K hK f hi) g := by
+  rw [Lp.edist_def,eLpNorm_one_eq_lintegral_enorm
+    ((Lp.aestronglyMeasurable (compactFunctionLp K hK f hi)).sub (Lp.aestronglyMeasurable g))]
+  have hz : Function.support (fun x ↦ ‖f x-compactLpExtension K g x‖ₑ) ⊆ K := by
+    intro x hx
+    by_contra hnot
+    have hf : f x=0 := Function.notMem_support.mp (fun h ↦ hnot (hs h))
+    simp [hf,compactLpExtension_zero K g hnot] at hx
+  rw [← setLIntegral_eq_of_support_subset hz,← lintegral_subtype_comap hK.measurableSet]
+  apply lintegral_congr_ae
+  filter_upwards [compactFunctionLp_ae K hK f hi] with x hx
+  simp only [compactLpExtension_coe,Pi.sub_apply,hx]
+
+theorem compactLpExtension_tendsto (K : Set (Euc n)) (hK : IsCompact K)
+    (f : ℕ → Euc n → ℝ) (hi : ∀ i, Integrable (f i) volume)
+    (hs : ∀ i, Function.support (f i) ⊆ K)
+    (g : Lp ℝ 1 (volume.comap (Subtype.val : K → Euc n)))
+    (hlim : Tendsto (fun i ↦ compactFunctionLp K hK (f i) (hi i)) atTop (𝓝 g)) :
+    Tendsto (fun i ↦ ∫⁻ x, ‖f i x-compactLpExtension K g x‖ₑ) atTop (𝓝 0) := by
+  have he (i : ℕ) := compactLpExtension_error K hK g (f i) (hi i) (hs i)
+  simp_rw [he]
+  simpa only [edist_self] using hlim.edist (tendsto_const_nhds : Tendsto (fun _ : ℕ ↦ g) atTop (𝓝 g))
+
+theorem compact_support_L1_subsequence
+    (K : Set (Euc n)) (hK : IsCompact K) (f : ℕ → Euc n → ℝ)
+    (hm : ∀ i, Measurable (f i)) (hi : ∀ i, Integrable (f i) volume)
+    (hs : ∀ i, Function.support (f i) ⊆ K)
+    (A : ℝ) (hA : 0 ≤ A) (hb : ∀ i, (∫ x, ‖f i x‖) ≤ A)
+    (D : ℝ≥0∞) (hD : D ≠ ⊤)
+    (ht : ∀ i w, (∫⁻ x, ‖f i (x+w)-f i x‖ₑ) ≤ ‖w‖ₑ * D) :
+    ∃ g : Euc n → ℝ, Integrable g volume ∧ Function.support g ⊆ K ∧
+      ∃ σ : ℕ → ℕ, StrictMono σ ∧
+        Tendsto (fun i ↦ ∫⁻ x, ‖f (σ i) x-g x‖ₑ) atTop (𝓝 0) := by
+  obtain ⟨g,σ,hσ,hg⟩ := compactFunctionLp_subsequence_of_translation K hK f hm hi A hA hb D hD ht
+  refine ⟨compactLpExtension K g,compactLpExtension_integrable K hK g,?_,σ,hσ,?_⟩
+  · intro x hx
+    by_contra hnot
+    exact hx (compactLpExtension_zero K g hnot)
+  · exact compactLpExtension_tendsto K hK (fun i ↦ f (σ i)) (fun i ↦ hi (σ i))
+      (fun i ↦ hs (σ i)) g hg
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/ConeLinkIntersection.lean b/lean/OAI/Geometry/Perimeter/Regularity/ConeLinkIntersection.lean
new file mode 100644
index 0000000000000000000000000000000000000000..ad199e9ad871eb56fde0c154faa0d419f59fc5ae
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/ConeLinkIntersection.lean
@@ -0,0 +1,83 @@
+import OAI.Geometry.Perimeter.Regularity.CoupledConeLink
+import OAI.Geometry.Perimeter.Regularity.ConeLinkSupport
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+namespace RPISingularHarnack
+namespace SphericalFrankel
+
+def fullLink {n : ℕ} (E : Set (Euc n)) : Set (Euc n) :=
+  perimeterSupport E ∩ Metric.sphere 0 1
+
+lemma fullLink_compact {n : ℕ} (E : Set (Euc n)) : IsCompact (fullLink E) :=
+  (isCompact_sphere (0 : Euc n) 1).inter_left (isClosed_perimeterSupport E)
+
+lemma fullLink_nonempty {n : ℕ} (hn : 1 < n) {E : Set (Euc n)}
+    (hE : IsLocallyPerimeterMinimizing E) (hc : IsCone E)
+    (hv : (0 : Euc n) ∈ perimeterSupport E) : (fullLink E).Nonempty := by
+  obtain ⟨x,hx,hx0⟩ := minimizing_support_nonzero_point hn hE hv
+  refine ⟨radialUnit x, (isCone_perimeterSupport hE.1 hc _
+    (inv_pos.mpr (norm_pos_iff.mpr hx0)) x).mpr hx, ?_⟩
+  simpa [Metric.mem_sphere] using radialUnit_norm hx0
+
+lemma product_ne_zero_of_ne_zero {n : ℕ} {μ ν : Measure (Frame n)}
+    [IsFiniteMeasure μ] [IsFiniteMeasure ν] (hμ : μ ≠ 0) (hν : ν ≠ 0) : μ.prod ν ≠ 0 := by
+  intro hz
+  have hp : μ Set.univ * ν Set.univ = 0 := by
+    rw [← Measure.prod_prod, Set.univ_prod_univ, hz]
+    simp
+  rcases mul_eq_zero.mp hp with h | h
+  · exact hμ (Measure.measure_univ_eq_zero.mp h)
+  · exact hν (Measure.measure_univ_eq_zero.mp h)
+
+theorem minimizing_cone_links_intersect {n : ℕ} (hn : 2 < n) {E F : Set (Euc n)}
+    (hE : IsLocallyPerimeterMinimizing E) (hF : IsLocallyPerimeterMinimizing F)
+    (hcE : IsCone E) (hcF : IsCone F)
+    (hvE : (0 : Euc n) ∈ perimeterSupport E)
+    (hvF : (0 : Euc n) ∈ perimeterSupport F) :
+    (fullLink E ∩ fullLink F).Nonempty := by
+  have hn1 : 1 < n := by omega
+  let μ := annularLink E
+  let ν := annularLink F
+  let : IsFiniteMeasure μ := annularLink_finite hE
+  let : IsFiniteMeasure ν := annularLink_finite hF
+  have hμ0 : μ ≠ 0 := annularLink_ne_zero hn1 hE hcE hvE
+  have hν0 : ν ≠ 0 := annularLink_ne_zero hn1 hF hcF hvF
+  have hμ := annularLink_frames hn1 hE hcE
+  have hν := annularLink_frames hn1 hF hcF
+  have hS := minimizing_cones_coupledStationarity hn1 hE hF hcE hcF
+  have hC : Continuous (fun p : Euc n × Euc n ↦ inner ℝ p.1 p.2) :=
+    continuous_fst.inner continuous_snd
+  obtain ⟨p, hp, hmax⟩ := ((fullLink_compact E).prod (fullLink_compact F)).exists_isMaxOn
+    ((fullLink_nonempty hn1 hE hcE hvE).prod (fullLink_nonempty hn1 hF hcF hvF))
+    hC.continuousOn
+  have hpn : ‖p.1‖ = 1 := by simpa [Metric.mem_sphere] using hp.1.2
+  have hqn : ‖p.2‖ = 1 := by simpa [Metric.mem_sphere] using hp.2.2
+  have hle : inner ℝ p.1 p.2 ≤ 1 := by
+    exact le_trans (le_abs_self _) (by simpa [hpn,hqn] using abs_real_inner_le_norm p.1 p.2)
+  by_cases heq : p.1 = p.2
+  · exact ⟨p.1,hp.1,heq ▸ hp.2⟩
+  · have hlt : inner ℝ p.1 p.2 < 1 := lt_of_le_of_ne hle (by
+      intro he
+      exact heq ((inner_eq_one_iff_of_norm_eq_one hpn hqn).mp he))
+    have hbound : ∀ᵐ q ∂μ.prod ν, angle q.1 q.2 ≤ max 0 (inner ℝ p.1 p.2) := by
+      apply (Measure.ae_prod_iff_ae_ae
+        (isClosed_le continuous_angle continuous_const).measurableSet).mpr
+      filter_upwards [annularLink_radial_support hn1 hE hcE] with e he
+      filter_upwards [annularLink_radial_support hn1 hF hcF] with f hf
+      apply le_trans _ (le_max_right _ _)
+      change inner ℝ e.1 f.1 ≤ inner ℝ p.1 p.2
+      exact hmax (a := (e.1,f.1)) ⟨⟨he.1, by simpa [Metric.mem_sphere] using he.2⟩,
+        ⟨hf.1, by simpa [Metric.mem_sphere] using hf.2⟩⟩
+    exact (no_positive_angle_gap hn (product_ne_zero_of_ne_zero hμ0 hν0)
+      (product_frames hμ hν) hS (le_max_left _ _) (max_lt (by norm_num) hlt) hbound).elim
+
+end SphericalFrankel
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/ConeLinkSupport.lean b/lean/OAI/Geometry/Perimeter/Regularity/ConeLinkSupport.lean
new file mode 100644
index 0000000000000000000000000000000000000000..e139f3f4b974c7704f1d9e6715f5a8263d0fc257
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/ConeLinkSupport.lean
@@ -0,0 +1,112 @@
+import OAI.Geometry.Perimeter.Regularity.RawConeLink
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+namespace RPISingularHarnack
+namespace SphericalFrankel
+
+lemma minimizing_perimeter_singleton {n : ℕ} (hn : 1 < n) {E : Set (Euc n)}
+    (hE : IsLocallyPerimeterMinimizing E) (x : Euc n) : perimeterMeasure E {x} = 0 := by
+  obtain ⟨C, D, hC0, hCt, hD0, hDt, hCD⟩ := minimizing_perimeter_hausdorff_comparison hn
+  have hd : (0 : ℝ) < (n - 1 : ℕ) := by exact_mod_cast (by omega : 0 < n-1)
+  let := Measure.nullSingletonClass_hausdorff (Euc n) hd
+  apply le_antisymm _ zero_le
+  calc
+    perimeterMeasure E {x} ≤ (C • (Measure.hausdorffMeasure (n-1 : ℕ)).restrict
+        (perimeterSupport E)) {x} := (hCD E hE).1 {x}
+    _ = 0 := by simp
+
+lemma minimizing_support_nonzero_point {n : ℕ} (hn : 1 < n) {E : Set (Euc n)}
+    (hE : IsLocallyPerimeterMinimizing E) (hv : (0 : Euc n) ∈ perimeterSupport E) :
+    ∃ x ∈ perimeterSupport E, x ≠ 0 := by
+  have hv' : (0 : Euc n) ∈ (perimeterMeasure E).support := by
+    rwa [perimeterMeasure_support]
+  have hp := Metric.nhds_basis_ball.mem_measureSupport.mp hv' 1 (by norm_num : (0 : ℝ)<1)
+  have hdiff : 0 < perimeterMeasure E (Metric.ball 0 1 \ {(0 : Euc n)}) := by
+    rwa [measure_sdiff_null (minimizing_perimeter_singleton hn hE 0)]
+  obtain ⟨x, hx, hs⟩ := (perimeterMeasure E).nonempty_inter_support_of_pos hdiff
+  exact ⟨x, by rwa [perimeterMeasure_support] at hs, hx.2⟩
+
+lemma annularWeight_pos {n : ℕ} {x : Euc n} (h1 : 1 < ‖x‖) (h2 : ‖x‖ < 2) :
+    0 < annularWeight x := by
+  unfold annularWeight annularCut annularProfile
+  apply mul_pos
+  · apply mul_pos <;> apply expNegInvGlue.pos_of_pos
+    · nlinarith [norm_nonneg x]
+    · nlinarith [norm_nonneg x]
+  · exact inv_pos.mpr (lt_trans (by norm_num) h1)
+
+lemma cone_support_annular_point {n : ℕ} (hn : 1 < n) {E : Set (Euc n)}
+    (hE : IsLocallyPerimeterMinimizing E) (hc : IsCone E)
+    (hv : (0 : Euc n) ∈ perimeterSupport E) :
+    ∃ y ∈ perimeterSupport E, ‖y‖ = (3 : ℝ)/2 := by
+  obtain ⟨x, hx, hx0⟩ := minimizing_support_nonzero_point hn hE hv
+  have hn0 : 0 < ‖x‖ := norm_pos_iff.mpr hx0
+  refine ⟨((3:ℝ)/2/‖x‖) • x,
+    (isCone_perimeterSupport hE.1 hc _ (div_pos (by norm_num) hn0) x).mpr hx, ?_⟩
+  rw [norm_smul, Real.norm_of_nonneg (div_nonneg (by norm_num) hn0.le), div_mul_cancel₀ _ hn0.ne']
+
+lemma annularFrameMeasure_ne_zero {n : ℕ} (hn : 1 < n) {E : Set (Euc n)}
+    (hE : IsLocallyPerimeterMinimizing E) (hc : IsCone E)
+    (hv : (0 : Euc n) ∈ perimeterSupport E) : annularFrameMeasure E ≠ 0 := by
+  obtain ⟨y, hy, hyn⟩ := cone_support_annular_point hn hE hc hv
+  have hyp : 0 < annularWeight y := annularWeight_pos (by rw [hyn]; norm_num) (by rw [hyn]; norm_num)
+  have hyb : y ∈ Metric.ball (0 : Euc n) 3 := by simp [Metric.mem_ball, hyn]
+  let U := {x : Euc n | 0 < annularWeight x}
+  have hU : IsOpen U := isOpen_lt continuous_const annularWeight_continuous
+  have hy' : y ∈ (perimeterMeasure E).support := by rwa [perimeterMeasure_support]
+  have hp : 0 < perimeterMeasure E (U ∩ Metric.ball 0 3) :=
+    (Measure.mem_support_iff_forall y).mp hy' _ ((hU.inter Metric.isOpen_ball).mem_nhds ⟨hyp,hyb⟩)
+  intro hz
+  have hzae : (fun x : Euc n ↦ (annularWeightNN x : ℝ≥0∞)) =ᵐ[
+      (perimeterMeasure E).restrict (Metric.ball 0 3)] 0 :=
+    (withDensity_eq_zero_iff annularWeightNN_measurable.coe_nnreal_ennreal.aemeasurable).mp hz
+  have hU0 : (perimeterMeasure E).restrict (Metric.ball 0 3) U = 0 := by
+    apply measure_eq_zero_iff_ae_notMem.mpr
+    filter_upwards [hzae] with x hx
+    intro hxp
+    have hw0 : annularWeight x = 0 := by
+      have hh : annularWeightNN x = 0 := ENNReal.coe_eq_zero.mp hx
+      exact congrArg (fun z : ℝ≥0 ↦ (z : ℝ)) hh
+    exact (ne_of_gt hxp) hw0
+  rw [Measure.restrict_apply hU.measurableSet] at hU0
+  exact hp.ne' hU0
+
+lemma annularLink_ne_zero {n : ℕ} (hn : 1 < n) {E : Set (Euc n)}
+    (hE : IsLocallyPerimeterMinimizing E) (hc : IsCone E)
+    (hv : (0 : Euc n) ∈ perimeterSupport E) : annularLink E ≠ 0 := by
+  intro hz
+  apply annularFrameMeasure_ne_zero hn hE hc hv
+  exact (Measure.map_eq_zero_iff (frameOf_measurable hE.1).aemeasurable).mp hz
+
+lemma annularLink_radial_support {n : ℕ} (hn : 1 < n) {E : Set (Euc n)}
+    (hE : IsLocallyPerimeterMinimizing E) (hc : IsCone E) :
+    ∀ᵐ e ∂annularLink E, e.1 ∈ perimeterSupport E ∧ ‖e.1‖ = 1 := by
+  have hs : ∀ᵐ x ∂perimeterMeasure E, x ∈ perimeterSupport E := by
+    rw [← perimeterMeasure_support]
+    exact (perimeterMeasure E).support_mem_ae
+  have hf := annularLink_frames hn hE hc
+  have hr : ∀ᵐ e ∂annularLink E, e.1 ∈ perimeterSupport E := by
+    rw [annularLink, ae_map_iff (frameOf_measurable hE.1).aemeasurable
+      ((isClosed_perimeterSupport E).preimage continuous_fst).measurableSet,
+      annularFrameMeasure, ae_withDensity_iff annularWeightNN_measurable.coe_nnreal_ennreal]
+    filter_upwards [ae_restrict_of_ae hs] with x hx hρ
+    have hx0 : x ≠ 0 := by
+      intro hh
+      apply hρ
+      apply ENNReal.coe_eq_zero.mpr
+      apply Subtype.ext
+      simp [hh, annularWeightNN, annularWeight]
+    exact (isCone_perimeterSupport hE.1 hc _ (inv_pos.mpr (norm_pos_iff.mpr hx0)) x).mpr hx
+  filter_upwards [hr,hf] with e he hframe
+  exact ⟨he,hframe.1⟩
+
+end SphericalFrankel
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/ConeRadialCap.lean b/lean/OAI/Geometry/Perimeter/Regularity/ConeRadialCap.lean
new file mode 100644
index 0000000000000000000000000000000000000000..f3a9668e6d2f9485b2b5e28c75fbff88bef607a3
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/ConeRadialCap.lean
@@ -0,0 +1,449 @@
+import OAI.Geometry.Perimeter.Regularity.SmulMemPhaseSupport
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+open scoped ENNReal Topology
+open scoped ENNReal Topology ContDiff
+open scoped Topology ENNReal
+namespace RPISingularHarnack
+
+def coneRadialCap {n : ℕ} (R : ℝ) (x : Euc n) : ℝ :=
+  expNegInvGlue (R ^ 2 - ‖x‖ ^ 2)
+
+theorem coneRadialCap_contDiff {n : ℕ} (R : ℝ) :
+    ContDiff ℝ ∞ (coneRadialCap (n := n) R) :=
+  expNegInvGlue.contDiff.comp (contDiff_const.sub (contDiff_id.norm_sq ℝ))
+
+theorem coneRadialCap_tsupport {n : ℕ} {R : ℝ} (hR : 0 < R) :
+    tsupport (coneRadialCap (n := n) R) ⊆ Metric.closedBall (0 : Euc n) R := by
+  apply closure_minimal _ Metric.isClosed_closedBall
+  intro x hx
+  have hh : 0 < R ^ 2 - ‖x‖ ^ 2 := by
+    by_contra hh
+    exact hx (expNegInvGlue.zero_of_nonpos (le_of_not_gt hh))
+  rw [Metric.mem_closedBall, dist_zero_right]
+  nlinarith [norm_nonneg x]
+
+theorem gradient_coneRadialCap {n : ℕ} (R : ℝ) (x : Euc n) :
+    gradient (coneRadialCap R) x =
+      (-2 * ((R ^ 2 - ‖x‖ ^ 2)⁻¹ ^ 2 * expNegInvGlue (R ^ 2 - ‖x‖ ^ 2))) • x := by
+  have hg (s : ℝ) : HasDerivAt expNegInvGlue (s⁻¹ ^ 2 * expNegInvGlue s) s := by
+    simpa using expNegInvGlue.hasDerivAt_polynomial_eval_inv_mul 1 s
+  have hq := ((hasFDerivAt_id (𝕜 := ℝ) x).norm_sq).const_sub (R ^ 2)
+  have hh := (hg (R ^ 2 - ‖x‖ ^ 2)).comp_hasFDerivAt x hq
+  change HasFDerivAt (coneRadialCap R) _ x at hh
+  apply (InnerProductSpace.toDual ℝ (Euc n)).injective
+  rw [toDual_gradient, hh.fderiv]
+  ext y
+  simp only [smul_apply, neg_apply,
+    ContinuousLinearMap.comp_apply, ContinuousLinearMap.id_apply, id_eq,
+    innerSL_apply_apply, map_smul, InnerProductSpace.toDual_apply_apply,
+    smul_eq_mul]
+  ring
+
+theorem minimizing_cone_support_in_halfspace_plane {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E) (hcone : IsCone E)
+    (v : Euc n) (hside : ∀ x ∈ perimeterSupport E, 0 ≤ inner ℝ x v) :
+    ∀ x ∈ perimeterSupport E, inner ℝ x v = 0 := by
+  have hs : ∀ᵐ x ∂perimeterMeasure E, x ∈ perimeterSupport E := by
+    rw [← perimeterMeasure_support]
+    exact (perimeterMeasure E).support_mem_ae
+  have h0 : ∀ᵐ x ∂perimeterMeasure E, 0 ≤ inner ℝ x v := hs.mono fun x hx ↦ hside x hx
+  have hball (R : ℝ) (hR : 0 < R) :
+      ∀ᵐ x ∂(perimeterMeasure E).restrict (Metric.ball 0 R), inner ℝ x v = 0 := by
+    let φ : Euc n → ℝ := coneRadialCap R
+    let b : Euc n → ℝ := fun x ↦ (R ^ 2 - ‖x‖ ^ 2)⁻¹ ^ 2 * expNegInvGlue (R ^ 2 - ‖x‖ ^ 2)
+    let V : Euc n → Euc n := fun x ↦ φ x • v
+    have hφ : ContDiff ℝ ∞ φ := coneRadialCap_contDiff R
+    have htφ : HasCompactSupport φ := (isCompact_closedBall (0 : Euc n) R).of_isClosed_subset
+      (isClosed_tsupport _) (coneRadialCap_tsupport hR)
+    have hV : ContDiff ℝ ∞ V := hφ.smul contDiff_const
+    have htV : HasCompactSupport V := htφ.smul_right
+    have hVs : tsupport V ⊆ Metric.ball (0 : Euc n) (R + 1) :=
+      (tsupport_smul_subset_left _ _).trans
+        ((coneRadialCap_tsupport hR).trans (Metric.closedBall_subset_ball (by linarith)))
+    have hdiv : ∀ᵐ x ∂perimeterMeasure E,
+        tangentialDivergence E V x = -2 * (b x * inner ℝ x v) := by
+      filter_upwards [minimizing_cone_radial_tangent_ae hn hmin hcone] with x hx
+      rw [tangentialDivergence_smul E (hφ.differentiable (by simp)) (differentiable_const v)]
+      have hc : tangentialDivergence E (fun _ : Euc n ↦ v) x = 0 := by
+        simp [tangentialDivergence, divergence]
+      rw [hc, mul_zero, zero_add, tangentialGradient, gradient_coneRadialCap]
+      simp only [real_inner_smul_left, hx, mul_zero, zero_smul, sub_zero]
+      dsimp [b]
+      ring
+    have hdf : Integrable (fun x ↦ b x * inner ℝ x v)
+        ((perimeterMeasure E).restrict (Metric.ball 0 (R + 1))) := by
+      have hh := (tangentialDivergence_integrable_ball_perimeter hmin 0 (by linarith : 0 < R + 1) hV).const_mul (-1/2 : ℝ)
+      apply hh.congr
+      filter_upwards [ae_restrict_of_ae hdiv] with x hx
+      rw [hx]
+      ring
+    have hnn : ∀ᵐ x ∂(perimeterMeasure E).restrict (Metric.ball 0 (R + 1)),
+        0 ≤ b x * inner ℝ x v := by
+      filter_upwards [ae_restrict_of_ae h0] with x hx
+      exact mul_nonneg (mul_nonneg (sq_nonneg _) (expNegInvGlue.nonneg _)) hx
+    have hint : (∫ x in Metric.ball (0 : Euc n) (R + 1), b x * inner ℝ x v ∂perimeterMeasure E) = 0 := by
+      have hh := minimizing_tangentialDivergence_integral_eq_zero hn hmin 0
+        (by linarith : 0 < R + 1) hV htV hVs
+      rw [integral_congr_ae (ae_restrict_of_ae hdiv), integral_const_mul] at hh
+      linarith
+    have he := (integral_eq_zero_iff_of_nonneg_ae hnn hdf).mp hint
+    have her : ∀ᵐ x ∂(perimeterMeasure E).restrict (Metric.ball 0 R), b x * inner ℝ x v = 0 :=
+      ae_mono (Measure.restrict_mono (Metric.ball_subset_ball (by linarith)) le_rfl) he
+    filter_upwards [her, ae_restrict_mem measurableSet_ball] with x hx hxr
+    have hp : 0 < R ^ 2 - ‖x‖ ^ 2 := by
+      have hh : ‖x‖ < R := by simpa using hxr
+      nlinarith [norm_nonneg x]
+    have hb : 0 < b x := mul_pos (sq_pos_of_pos (inv_pos.mpr hp)) (expNegInvGlue.pos_of_pos hp)
+    exact (mul_eq_zero.mp hx).resolve_left hb.ne'
+  have hae : ∀ᵐ x ∂perimeterMeasure E, inner ℝ x v = 0 := by
+    let B (j : ℕ) := Metric.ball (0 : Euc n) (j + 1 : ℝ)
+    have hcover : (⋃ j, B j) = univ := by
+      apply eq_univ_of_forall
+      intro x
+      obtain ⟨j, hj⟩ := exists_nat_gt ‖x‖
+      exact mem_iUnion.mpr ⟨j, by simpa [B] using (show ‖x‖ < (j : ℝ) + 1 by linarith)⟩
+    have hh := (ae_restrict_iUnion_iff B (fun x ↦ inner ℝ x v = 0)).mpr
+      (fun j ↦ hball (j + 1) (by positivity))
+    simpa only [hcover, Measure.restrict_univ] using hh
+  have hsup := Measure.support_subset_of_isClosed
+    (isClosed_eq (continuous_id.inner continuous_const) continuous_const) hae
+  rw [perimeterMeasure_support] at hsup
+  exact hsup
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+open scoped ENNReal Topology
+open scoped ENNReal Topology ContDiff
+open scoped Topology ENNReal
+namespace RPISingularHarnack
+
+theorem minimizing_cone_containing_halfspace_ae_eq {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E) (hcone : IsCone E)
+    (hvertex : (0 : Euc n) ∈ perimeterSupport E)
+    {v : Euc n} (hv : ‖v‖ = 1) (hinside : {x | inner ℝ x v < 0} ⊆ E) :
+    E =ᵐ[volume] {x | inner ℝ x v < 0} := by
+  let K := (volume.restrict E).support
+  have hK : IsClosed K := Measure.isClosed_support
+  have hP : IsOpen {x : Euc n | inner ℝ x v < 0} :=
+    isOpen_lt (continuous_id.inner continuous_const) continuous_const
+  have hPK : {x : Euc n | inner ℝ x v < 0} ⊆ K := by
+    intro x hx
+    apply Measure.interior_inter_support
+    refine ⟨?_, ?_⟩
+    · exact (interior_mono hinside) (hP.interior_eq.symm ▸ hx)
+    · rw [Measure.support_eq_univ]; trivial
+  have hPint : {x : Euc n | inner ℝ x v < 0} ⊆ interior K := by
+    exact hP.interior_eq ▸ interior_mono hPK
+  have hfK : frontier K = perimeterSupport E :=
+    minimizing_frontier_phaseSupport
+      (Real.HolderConjugate.conjExponent (by exact_mod_cast hn : 1 < (n : ℝ))) hmin
+  have hside : ∀ x ∈ perimeterSupport E, 0 ≤ inner ℝ x v := by
+    intro x hx
+    by_contra hh
+    have hi := hPint (lt_of_not_ge hh)
+    rw [← hfK, frontier] at hx
+    exact hx.2 hi
+  have hplane := minimizing_cone_support_in_halfspace_plane hn hmin hcone v hside
+  have hcover : {x : Euc n | 0 < inner ℝ x v} ⊆ interior K ∪ Kᶜ := by
+    intro x hx
+    by_cases hk : x ∈ K
+    · left
+      by_contra hi
+      have hs : x ∈ perimeterSupport E := by
+        rw [← hfK, frontier, hK.closure_eq]
+        exact ⟨hk, hi⟩
+      exact (ne_of_gt hx) (hplane x hs)
+    · exact Or.inr hk
+  have hc : IsPreconnected {x : Euc n | 0 < inner ℝ x v} := by
+    have hh := (convex_halfSpace_gt (innerSL ℝ v).toLinearMap.isLinear (0 : ℝ)).isPreconnected
+    change IsPreconnected {x : Euc n | 0 < inner ℝ v x} at hh
+    simpa only [real_inner_comm v] using hh
+  have hor := hc.subset_or_subset isOpen_interior hK.isOpen_compl
+    (disjoint_compl_right.mono_left interior_subset) hcover
+  have hEK : E =ᵐ[volume] K := minimizing_ae_eq_phaseSupport
+    (Real.HolderConjugate.conjExponent (by exact_mod_cast hn : 1 < (n : ℝ))) hmin
+  rcases hor with hall | hnone
+  · have hae : ∀ᵐ x ∂(volume : Measure (Euc n)), x ∈ K := by
+      filter_upwards [ae_inner_ne_value hv 0] with x hx
+      rcases lt_or_gt_of_ne hx with hn | hp
+      · exact hPK hn
+      · exact interior_subset (hall hp)
+    have hKU : K = univ := by
+      have hh := Measure.support_subset_of_isClosed hK hae
+      rw [Measure.support_eq_univ] at hh
+      exact eq_univ_iff_forall.mpr fun x ↦ hh trivial
+    rw [← hfK, hKU, frontier_univ] at hvertex
+    exact False.elim hvertex
+  · apply hEK.trans
+    apply Filter.eventuallyEqSet_iff.mpr
+    filter_upwards [ae_inner_ne_value hv 0] with x hx
+    rcases lt_or_gt_of_ne hx with hn | hp
+    · exact iff_of_true (hPK hn) hn
+    · exact iff_of_false (hnone hp) (not_lt.mpr hp.le)
+
+theorem minimizing_cone_containing_halfspace_support_eq {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E) (hcone : IsCone E)
+    (hvertex : (0 : Euc n) ∈ perimeterSupport E)
+    {v : Euc n} (hv : ‖v‖ = 1) (hinside : {x | inner ℝ x v < 0} ⊆ E) :
+    perimeterSupport E = perimeterSupport {x | inner ℝ x v < 0} :=
+  perimeterSupport_eq_of_ae
+    (minimizing_cone_containing_halfspace_ae_eq hn hmin hcone hvertex hv hinside)
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+open scoped ENNReal Topology
+open scoped ENNReal Topology ContDiff
+open scoped Topology ENNReal
+namespace RPISingularHarnack
+
+theorem measure_eq_of_Iio_finite {μ ν : Measure ℝ}
+    (hfin : ∀ a : ℝ, μ (Iio a) ≠ ⊤) (heq : ∀ a : ℝ, μ (Iio a) = ν (Iio a)) :
+    μ = ν := by
+  apply Measure.ext_of_Ico' μ ν
+  · intro a b _
+    exact ne_top_of_le_ne_top (hfin b) (measure_mono Ico_subset_Iio_self)
+  · intro a b hab
+    rw [← Iio_sdiff_Iio, measure_sdiff (Iio_subset_Iio hab.le)
+      measurableSet_Iio.nullMeasurableSet (hfin a),
+      measure_sdiff (Iio_subset_Iio hab.le) measurableSet_Iio.nullMeasurableSet
+        (by rw [← heq a]; exact hfin a), heq a, heq b]
+
+noncomputable def radialScaledMeasure {n : ℕ} (μ : Measure (Euc n)) (k : ℕ) (t : ℝ) : Measure ℝ :=
+  ENNReal.ofReal (t^k) • μ.map (fun x ↦ t * ‖x‖)
+
+theorem radialScaledMeasure_Iio {n k : ℕ} (μ : Measure (Euc n))
+    {a t : ℝ} (ht : 0 < t) :
+    radialScaledMeasure μ k t (Iio a) =
+      ENNReal.ofReal (t^k) * μ (Metric.ball 0 (a/t)) := by
+  have hm : Measurable (fun x : Euc n ↦ t * ‖x‖) := by fun_prop
+  rw [radialScaledMeasure, Measure.smul_apply, smul_eq_mul,
+    Measure.map_apply hm measurableSet_Iio]
+  congr 2
+  ext x
+  simp only [mem_preimage, mem_Iio, Metric.mem_ball, dist_zero_right]
+  rw [lt_div_iff₀ ht, mul_comm]
+
+theorem radialScaledMeasure_eq_of_constant_density {n k : ℕ}
+    (μ : Measure (Euc n)) (hfin : ∀ r : ℝ, 0 < r → μ (Metric.ball 0 r) ≠ ⊤)
+    {a : ℝ} (ha : 0 ≤ a) (hdensity : ∀ r : ℝ, 0 < r → μ.real (Metric.ball 0 r) = a * r^k)
+    {s t : ℝ} (hs : 0 < s) (ht : 0 < t) :
+    radialScaledMeasure μ k s = radialScaledMeasure μ k t := by
+  have hval (t : ℝ) (ht : 0 < t) (r : ℝ) :
+      radialScaledMeasure μ k t (Iio r) = ENNReal.ofReal (a * (max r 0)^k) *
+        (if 0 < r then 1 else 0) := by
+    rw [radialScaledMeasure_Iio μ ht]
+    by_cases hr : 0 < r
+    · rw [ite_eq_left hr, mul_one, max_eq_left hr.le]
+      have hrt : 0 < r/t := div_pos hr ht
+      have hp := hdensity (r/t) hrt
+      have he : μ (Metric.ball 0 (r/t)) = ENNReal.ofReal (a * (r/t)^k) := by
+        rw [← hp, Measure.real, ENNReal.ofReal_toReal (hfin _ hrt)]
+      rw [he, ← ENNReal.ofReal_mul' (mul_nonneg ha (pow_nonneg hrt.le _))]
+      congr 1
+      rw [div_pow]
+      field_simp
+    · rw [ite_eq_right hr, mul_zero, Metric.ball_eq_empty.mpr (div_nonpos_of_nonpos_of_nonneg
+        (le_of_not_gt hr) ht.le), measure_empty, mul_zero]
+  apply measure_eq_of_Iio_finite
+  · intro r
+    rw [hval s hs r]
+    split_ifs <;> finiteness
+  · intro r
+    rw [hval s hs r, hval t ht r]
+
+theorem radialDensity_integral_eq_pushforward {n k : ℕ} (μ : Measure (Euc n))
+    {ψ : ℝ → ℝ} (hψ : Continuous ψ) {t : ℝ} (ht : 0 < t) :
+    (∫ x, radialDensityIntegrand 0 k ψ t x ∂μ) =
+      ∫ s, ψ (s^2) ∂radialScaledMeasure μ k t := by
+  have hm : Measurable (fun x : Euc n ↦ t * ‖x‖) := by fun_prop
+  have hc : Continuous (fun s : ℝ ↦ ψ (s^2)) := by fun_prop
+  rw [radialScaledMeasure, integral_smul_measure,
+    integral_map_of_stronglyMeasurable hm hc.stronglyMeasurable,
+    ENNReal.toReal_ofReal (pow_nonneg ht.le _)]
+  simp only [radialDensityIntegrand, sub_zero, smul_eq_mul, mul_pow, integral_const_mul]
+
+theorem radialDensity_integral_ball_eq {n k : ℕ} (μ : Measure (Euc n))
+    {ψ : ℝ → ℝ} (hψz : ∀ s, 1 ≤ s → ψ s = 0)
+    {M t : ℝ} (hM : 0 < M) (ht : 1/M < t) :
+    (∫ x in Metric.ball (0 : Euc n) M, radialDensityIntegrand 0 k ψ t x ∂μ) =
+      ∫ x, radialDensityIntegrand 0 k ψ t x ∂μ := by
+  apply setIntegral_eq_integral_of_forall_compl_eq_zero
+  intro x hx
+  have ht0 : 0 < t := (one_div_pos.mpr hM).trans ht
+  have htM : 1 < t*M := (div_lt_iff₀ hM).mp ht
+  have hxM : M ≤ ‖x‖ := by simpa only [Metric.mem_ball, dist_zero_right, not_lt] using hx
+  have h1 : 1 ≤ t*‖x‖ := by nlinarith
+  have h2 := pow_le_pow_left₀ (by norm_num : (0:ℝ) ≤ 1) h1 2
+  simp only [one_pow, mul_pow] at h2
+  simp only [radialDensityIntegrand, sub_zero, hψz _ h2, mul_zero]
+
+theorem minimizing_radialDensityDerivative_eq_zero_of_constant_density
+    {n : ℕ} {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    {a : ℝ} (ha : 0 ≤ a)
+    (hdensity : ∀ r : ℝ, 0 < r → (perimeterMeasure E).real (Metric.ball 0 r) = a*r^(n-1))
+    {M : ℝ} (hM : 0 < M) {ψ : ℝ → ℝ} (hψ : ContDiff ℝ ∞ ψ)
+    (hψz : ∀ s, 1 ≤ s → ψ s = 0) {t : ℝ} (ht : 1/M < t) :
+    (∫ x in Metric.ball 0 M, radialDensityDerivative 0 (n-1) ψ t x ∂perimeterMeasure E) = 0 := by
+  let μ := (perimeterMeasure E).restrict (Metric.ball 0 M)
+  have : IsFiniteMeasure μ := ⟨by
+    rw [Measure.restrict_apply_univ, perimeterMeasure_open E Metric.isOpen_ball]
+    exact hmin.2.1 0 M hM⟩
+  have hμ : ∀ᵐ x ∂μ, x ∈ Metric.closedBall 0 M :=
+    (ae_restrict_mem measurableSet_ball).mono fun _ hx ↦ Metric.ball_subset_closedBall hx
+  have hd := (radialDensity_integral_hasDerivAt 0 M (n-1) hψ μ hμ t).2
+  have ht0 : 0 < t := (one_div_pos.mpr hM).trans ht
+  have hfin (r : ℝ) (hr : 0 < r) : perimeterMeasure E (Metric.ball 0 r) ≠ ⊤ := by
+    rw [perimeterMeasure_open E Metric.isOpen_ball]
+    exact (hmin.2.1 0 r hr).ne
+  have he (s : ℝ) (hs : 1/M < s) :
+      (∫ x, radialDensityIntegrand 0 (n-1) ψ s x ∂μ) =
+      ∫ x, radialDensityIntegrand 0 (n-1) ψ t x ∂μ := by
+    have hs0 := (one_div_pos.mpr hM).trans hs
+    change (∫ x in Metric.ball 0 M, _ ∂perimeterMeasure E) =
+      ∫ x in Metric.ball 0 M, _ ∂perimeterMeasure E
+    rw [radialDensity_integral_ball_eq _ hψz hM hs,
+      radialDensity_integral_ball_eq _ hψz hM ht,
+      radialDensity_integral_eq_pushforward _ hψ.continuous hs0,
+      radialDensity_integral_eq_pushforward _ hψ.continuous ht0,
+      radialScaledMeasure_eq_of_constant_density _ hfin ha hdensity hs0 ht0]
+  have hdz : HasDerivAt (fun s ↦ ∫ x, radialDensityIntegrand 0 (n-1) ψ s x ∂μ) 0 t := by
+    apply (hasDerivAt_const t (∫ x, radialDensityIntegrand 0 (n-1) ψ t x ∂μ)).congr_of_eventuallyEq
+    filter_upwards [lt_mem_nhds ht] with s hs
+    exact he s hs
+  exact hd.unique hdz
+
+theorem minimizing_radial_tangent_of_constant_density {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    {a : ℝ} (ha : 0 ≤ a)
+    (hdensity : ∀ r : ℝ, 0 < r → (perimeterMeasure E).real (Metric.ball 0 r) = a*r^(n-1)) :
+    ∀ᵐ x ∂perimeterMeasure E, inner ℝ (approximateNormal E x) x = 0 := by
+  let ψ : ℝ → ℝ := fun s ↦ expNegInvGlue (1-s)
+  have hψ : ContDiff ℝ ∞ ψ := expNegInvGlue.contDiff.comp (contDiff_const.sub contDiff_id)
+  have hψz (s : ℝ) (hs : 1 ≤ s) : ψ s = 0 :=
+    expNegInvGlue.zero_of_nonpos (by linarith)
+  have hψd (s : ℝ) : deriv ψ s = -((1-s)⁻¹^2 * expNegInvGlue (1-s)) := by
+    have hg : HasDerivAt expNegInvGlue ((1-s)⁻¹^2 * expNegInvGlue (1-s)) (1-s) := by
+      simpa using expNegInvGlue.hasDerivAt_polynomial_eval_inv_mul 1 (1-s)
+    simpa [ψ, Function.comp_def] using (hg.comp s ((hasDerivAt_id s).const_sub 1)).deriv
+  have hlocal (R : ℝ) (hR : 0 < R) :
+      ∀ᵐ x ∂(perimeterMeasure E).restrict (Metric.ball 0 R),
+        inner ℝ (approximateNormal E x) x = 0 := by
+    let M := 2*R
+    let t := 1/R
+    have hM : 0 < M := by dsimp [M]; positivity
+    have ht0 : 0 < t := one_div_pos.mpr hR
+    have ht : 1/M < t := one_div_lt_one_div_of_lt hR (by dsimp [M]; linarith)
+    have htM : 1/t < M := by dsimp [t, M]; rw [one_div_one_div]; linarith
+    let μ := (perimeterMeasure E).restrict (Metric.ball 0 M)
+    have : IsFiniteMeasure μ := ⟨by
+      rw [Measure.restrict_apply_univ, perimeterMeasure_open E Metric.isOpen_ball]
+      exact hmin.2.1 0 M hM⟩
+    have hμ : ∀ᵐ x ∂μ, x ∈ Metric.closedBall 0 M :=
+      (ae_restrict_mem measurableSet_ball).mono fun _ hx ↦ Metric.ball_subset_closedBall hx
+    let V : Euc n → Euc n := fun x ↦ ψ (t^2*‖x‖^2) • x
+    have hV : ContDiff ℝ ∞ V :=
+      (hψ.comp (contDiff_const.mul (contDiff_id.norm_sq (𝕜 := ℝ)))).smul contDiff_id
+    have hp : tsupport (fun x : Euc n ↦ ψ (t^2*‖x‖^2)) ⊆ Metric.closedBall 0 (1/t) := by
+      simpa only [sub_zero] using (radialProfile_tsupport (n := n) 0 hψz ht0)
+    have hs : tsupport V ⊆ Metric.closedBall 0 (1/t) :=
+      (tsupport_smul_subset_left (fun x : Euc n ↦ ψ (t^2*‖x‖^2))
+        (id : Euc n → Euc n)).trans hp
+    have hc : HasCompactSupport V :=
+      (isCompact_closedBall (0 : Euc n) (1/t)).of_isClosed_subset (isClosed_tsupport _) hs
+    have hdiv := minimizing_tangentialDivergence_integral_eq_zero hn hmin 0 hM hV hc
+      (hs.trans (Metric.closedBall_subset_ball htM))
+    have hD0 := minimizing_radialDensityDerivative_eq_zero_of_constant_density
+      hmin ha hdensity hM hψ hψz ht
+    have hDi := (radialDensity_integral_hasDerivAt 0 M (n-1) hψ μ hμ t).1
+    have hVi := (tangentialDivergence_integrable_ball_perimeter hmin 0 hM hV).const_mul
+      (t^((n-1)-1))
+    let q : Euc n → ℝ := fun x ↦ t^((n-1)-1)*tangentialDivergence E V x -
+      radialDensityDerivative 0 (n-1) ψ t x
+    have hqi : Integrable q μ := hVi.sub hDi
+    have hqformula : ∀ᵐ x ∂μ, q x =
+        (2*t^((n-1)-1)*t^2*(inner ℝ (approximateNormal E x) x)^2) *
+          ((1-t^2*‖x‖^2)⁻¹^2 * expNegInvGlue (1-t^2*‖x‖^2)) := by
+      filter_upwards [ae_restrict_of_ae (minimizing_halfspaceBlowup_ae hn hmin)] with x hx
+      have hpow : t^(n-1) = t^((n-1)-1)*t := by
+        rw [← pow_succ, Nat.sub_add_cancel (by omega : 1 ≤ n-1)]
+      have hvf := tangentialDivergence_radialProfile (by omega : 0<n) E 0 hψ t x hx.1
+      simp only [sub_zero] at hvf
+      dsimp [q, radialDensityDerivative]
+      change t^((n-1)-1)*tangentialDivergence E V x - _ = _
+      rw [show tangentialDivergence E V x = _ from hvf, hpow]
+      simp only [sub_zero, hψd]
+      ring
+    have hqnonneg : ∀ᵐ x ∂μ, 0 ≤ q x := hqformula.mono fun x hx ↦ by
+      rw [hx]
+      exact mul_nonneg (by positivity)
+        (mul_nonneg (sq_nonneg _) (expNegInvGlue.nonneg _))
+    have hqint : (∫ x, q x ∂μ) = 0 := by
+      rw [integral_sub hVi hDi, integral_const_mul]
+      change _ * (∫ x in Metric.ball 0 M, tangentialDivergence E V x ∂perimeterMeasure E) - _ = 0
+      rw [hdiv, hD0]
+      ring
+    have hqzero := (integral_eq_zero_iff_of_nonneg_ae hqnonneg hqi).mp hqint
+    have hqz := (ae_restrict_iff' measurableSet_ball).mp hqzero
+    have hqf := (ae_restrict_iff' measurableSet_ball).mp hqformula
+    apply (ae_restrict_iff' measurableSet_ball).mpr
+    filter_upwards [hqz, hqf] with x hxz hxf
+    intro hx
+    have hxM : x ∈ Metric.ball 0 M := Metric.ball_subset_ball (by dsimp [M]; linarith) hx
+    have htx : t*‖x‖ < 1 := by
+      have hxR : ‖x‖ < R := by simpa only [Metric.mem_ball, dist_zero_right] using hx
+      simpa only [t, one_div, div_eq_mul_inv, mul_comm, mul_one, one_mul] using (div_lt_one hR).mpr hxR
+    have htx0 : 0 ≤ t*‖x‖ := mul_nonneg ht0.le (norm_nonneg x)
+    have hspos : 0 < 1-t^2*‖x‖^2 := by
+      nlinarith [mul_pos (sub_pos.mpr htx) (show 0 < 1+t*‖x‖ by linarith)]
+    have hbpos : 0 < (1-t^2*‖x‖^2)⁻¹^2 * expNegInvGlue (1-t^2*‖x‖^2) :=
+      mul_pos (sq_pos_of_pos (inv_pos.mpr hspos)) (expNegInvGlue.pos_of_pos hspos)
+    have he := hxf hxM
+    rw [hxz hxM] at he
+    have hz : (inner ℝ (approximateNormal E x) x)^2 = 0 := by
+      have hp : 0 < 2*t^((n-1)-1)*t^2 := by positivity
+      exact (mul_eq_zero.mp ((mul_eq_zero.mp he.symm).resolve_right hbpos.ne')).resolve_left hp.ne'
+    exact sq_eq_zero_iff.mp hz
+  let U : ℕ → Set (Euc n) := fun j ↦ Metric.ball 0 (j+1 : ℝ)
+  have hcover : (⋃ j, U j) = univ := by
+    apply eq_univ_of_forall
+    intro x
+    obtain ⟨j, hj⟩ := exists_nat_gt ‖x‖
+    exact mem_iUnion.mpr ⟨j, by simpa [U, Metric.mem_ball, dist_zero_right] using
+      (show ‖x‖ < (j : ℝ)+1 by linarith)⟩
+  have hall := (ae_restrict_iUnion_iff U
+    (fun x ↦ inner ℝ (approximateNormal E x) x = 0)).mpr
+      (fun j ↦ hlocal (j+1) (by positivity))
+  simpa only [hcover, Measure.restrict_univ] using hall
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/ContinuousWithinAtSurfaceGradientOfLocalLevel.lean b/lean/OAI/Geometry/Perimeter/Regularity/ContinuousWithinAtSurfaceGradientOfLocalLevel.lean
new file mode 100644
index 0000000000000000000000000000000000000000..ffa2557ef20e92449ebc25d7cfaeb4ad1f345b5b
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/ContinuousWithinAtSurfaceGradientOfLocalLevel.lean
@@ -0,0 +1,512 @@
+import OAI.Geometry.Perimeter.Regularity.ExistsDiameterCoverOfHausdorffZero
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+namespace RPI.Boundary
+
+theorem continuousWithinAt_surfaceGradient_of_local_level {n : ℕ}
+    {S U : Set (Space n)} {f g : Space n → ℝ} {x : Space n}
+    (hU : IsOpen U) (hxU : x ∈ U) (hxS : x ∈ S)
+    (hf : ContDiffOn ℝ 1 f U) (hdf : ∀ y ∈ U, fderiv ℝ f y ≠ 0)
+    (hlevel : ∀ y ∈ U, y ∈ S ↔ f y = 0)
+    (hg : ContDiffAt ℝ 1 g x) :
+    ContinuousWithinAt (surfaceGradient S g) S x := by
+  let F : Space n → Space n := fun y ↦ gradient g y -
+    (inner ℝ (gradient f y) (gradient g y) / ‖gradient f y‖ ^ 2) • gradient f y
+  have hformula : ∀ y ∈ S ∩ U, surfaceGradient S g y = F y := by
+    intro y hy
+    apply surfaceGradient_eq_level_formula (hf.contDiffAt (hU.mem_nhds hy.2)) (hdf y hy.2)
+    filter_upwards [hU.mem_nhds hy.2] with z hz
+    rw [(hlevel y hy.2).mp hy.1]
+    exact hlevel z hz
+  have hf' : ContinuousAt (gradient f) x :=
+    continuousAt_gradient_of_contDiffAt (hf.contDiffAt (hU.mem_nhds hxU))
+  have hg' : ContinuousAt (gradient g) x := continuousAt_gradient_of_contDiffAt hg
+  have hgn : gradient f x ≠ 0 := by
+    intro he
+    apply hdf x hxU
+    rw [← toDual_gradient, he, map_zero]
+  have hF : ContinuousAt F x :=
+    hg'.sub (((hf'.inner hg').div (hf'.norm.pow 2)
+      (pow_ne_zero _ (norm_ne_zero_iff.mpr hgn))).smul hf')
+  apply hF.continuousWithinAt.congr_of_eventuallyEq _ (hformula x ⟨hxS, hxU⟩)
+  filter_upwards [self_mem_nhdsWithin,
+    mem_nhdsWithin_of_mem_nhds (hU.mem_nhds hxU)] with y hyS hyU
+  exact hformula y ⟨hyS, hyU⟩
+
+end RPI.Boundary
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+namespace RPI.Boundary
+
+theorem surfaceGradient_aestronglyMeasurable_of_regular_locus {n : ℕ} (hn : 8 ≤ n)
+    {S Z : Set (Space n)} (hS : MeasurableSet S) (hZ : MeasurableSet Z)
+    (hdim : dimH Z ≤ (n - 8 : ℕ))
+    (hreg : ∀ x ∈ S \ Z, ∃ U : Set (Space n), IsOpen U ∧ x ∈ U ∧
+      ∃ f : Space n → ℝ, ContDiffOn ℝ 1 f U ∧
+        (∀ y ∈ U, fderiv ℝ f y ≠ 0) ∧
+        (∀ y ∈ U, y ∈ S ↔ f y = 0))
+    {g : Space n → ℝ} (hg : ContDiff ℝ 1 g) :
+    AEStronglyMeasurable (surfaceGradient S g) (surfaceMeasure S) := by
+  have hc : ContinuousOn (surfaceGradient S g) (S \ Z) := by
+    intro x hx
+    obtain ⟨U, hU, hxU, f, hf, hdf, hlevel⟩ := hreg x hx
+    exact (continuousWithinAt_surfaceGradient_of_local_level hU hxU hx.1 hf hdf
+      hlevel hg.contDiffAt).mono sdiff_subset
+  have hs : ∀ᵐ x ∂surfaceMeasure S, x ∈ S := ae_restrict_mem hS
+  have hz : ∀ᵐ x ∂surfaceMeasure S, x ∉ Z := by
+    simpa only [ae_iff, not_not, Set.ofPred_mem_eq] using
+      surfaceMeasure_zero_of_singular_dim hn S Z hdim
+  have hregular : ∀ᵐ x ∂surfaceMeasure S, x ∈ S \ Z := by
+    filter_upwards [hs, hz] with x hx hzx
+    exact ⟨hx, hzx⟩
+  have hmeas := hc.aestronglyMeasurable (μ := surfaceMeasure S) (hS.diff hZ)
+  rwa [Measure.restrict_eq_self_of_ae_mem hregular] at hmeas
+
+end RPI.Boundary
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+namespace RPI.Boundary
+
+theorem exists_geometric_capacity_sequence {n : ℕ} (hn : 8 ≤ n)
+    {S Z : Set (Space n)} (hS : IsClosed S) (V : Set (Space n))
+    (hZc : IsCompact Z) (hZd : dimH Z ≤ (n - 8 : ℕ))
+    (hreg : ∀ x ∈ S \ Z, ∃ U : Set (Space n), IsOpen U ∧ x ∈ U ∧
+      ∃ f : Space n → ℝ, ContDiffOn ℝ 1 f U ∧
+        (∀ y ∈ U, fderiv ℝ f y ≠ 0) ∧
+        (∀ y ∈ U, y ∈ S ↔ f y = 0))
+    {A r₀ : ℝ} (hA : 0 < A) (hr₀ : 0 < r₀)
+    (hgrowth : ∀ c ∈ Z, ∀ r : ℝ, 0 < r → r < r₀ →
+      (surfaceMeasure S).restrict V (Metric.ball c r) ≤
+        ENNReal.ofReal (A * r ^ (n - 1))) :
+    ∃ χ : ℕ → Space n → ℝ,
+      (∀ i, ContDiff ℝ ∞ (χ i) ∧ (∀ x, χ i x ∈ Icc (0 : ℝ) 1) ∧
+        (∃ U : Set (Space n), IsOpen U ∧ Z ⊆ U ∧ ∀ x ∈ U, χ i x = 0) ∧
+        MemLp (surfaceGradient S (χ i)) 2 ((surfaceMeasure S).restrict V)) ∧
+      (∀ᵐ x ∂(surfaceMeasure S).restrict V,
+        Tendsto (fun i ↦ χ i x) atTop (𝓝 1)) ∧
+      Tendsto (fun i ↦ vectorEnergy ((surfaceMeasure S).restrict V)
+        (surfaceGradient S (χ i))) atTop (𝓝 0) := by
+  apply exists_memLp_capacity_sequence hn S V Z hZc hZd _ hA hr₀ hgrowth
+  intro f hf
+  exact (surfaceGradient_aestronglyMeasurable_of_regular_locus hn hS.measurableSet
+    hZc.isClosed.measurableSet hZd hreg (hf.of_le (by norm_num))).mono_measure
+      Measure.restrict_le_self
+
+end RPI.Boundary
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+namespace RPI
+
+theorem vectorEnergy_sub_tendsto_zero_of_toLp_tendsto
+    {X V : Type*} [MeasurableSpace X] [NormedAddCommGroup V] [InnerProductSpace ℝ V]
+    {μ : Measure X} {F : ℕ → X → V} {G : X → V}
+    (hF : ∀ i, MemLp (F i) 2 μ) (hG : MemLp G 2 μ)
+    (ht : Tendsto (fun i ↦ (hF i).toLp (F i)) atTop (𝓝 (hG.toLp G))) :
+    Tendsto (fun i ↦ vectorEnergy μ (fun x ↦ F i x - G x)) atTop (𝓝 0) := by
+  have he : (fun i ↦ vectorEnergy μ (fun x ↦ F i x - G x)) =
+      (fun i ↦ ‖(hF i).toLp (F i) - hG.toLp G‖ₑ ^ 2) := by
+    funext i
+    exact vectorEnergy_eq_enorm_toLp_sq ((hF i).sub hG)
+  rw [he]
+  simpa only [sub_self, enorm_zero, zero_pow (by decide : 2 ≠ 0)] using
+    (ENNReal.Tendsto.pow (ht.sub (tendsto_const_nhds (x := hG.toLp G))).enorm (n := 2))
+
+namespace Boundary
+
+theorem exists_compact_energy_approximants_of_mem_sobolevGraph {n : ℕ}
+    (S V : Set (Space n)) (hV : Bornology.IsBounded V) (hVm : MeasurableSet V)
+    {h : Space n → ℝ} {G : Space n → Space n}
+    (hh : MemLp h 2 ((surfaceMeasure S).restrict V))
+    (hG : MemLp G 2 ((surfaceMeasure S).restrict V))
+    (hp : (hh.toLp h, hG.toLp G) ∈ closure (smoothGradientGraph S V)) :
+    ∃ φ : ℕ → Space n → ℝ,
+      (∀ j, ContDiff ℝ ∞ (φ j) ∧ HasCompactSupport (φ j)) ∧
+      Tendsto (fun j ↦ ∫⁻ x, ENNReal.ofReal (|φ j x - h x| ^ 2)
+        ∂(surfaceMeasure S).restrict V) atTop (𝓝 0) ∧
+      Tendsto (fun j ↦ ∫⁻ x, ENNReal.ofReal
+        (‖surfaceGradient S (φ j) x - G x‖ ^ 2)
+        ∂(surfaceMeasure S).restrict V) atTop (𝓝 0) := by
+  obtain ⟨φ, hφs, hφ, hDφ, hlim⟩ :=
+    exists_compact_approximants_of_mem_sobolevGraph S V hV hVm hh hG hp
+  refine ⟨φ, hφs, ?_, ?_⟩
+  · simpa only [vectorEnergy, Real.norm_eq_abs] using
+      vectorEnergy_sub_tendsto_zero_of_toLp_tendsto hφ hh hlim.fst_nhds
+  · exact vectorEnergy_sub_tendsto_zero_of_toLp_tendsto hDφ hG hlim.snd_nhds
+
+end Boundary
+end RPI
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+namespace RPISingularHarnack
+abbrev Euc (n : ℕ) := EuclideanSpace ℝ (Fin n)
+
+def HasHalfspaceBlowup {n : ℕ} (E : Set (Euc n)) (x v : Euc n) : Prop :=
+  ‖v‖ = 1 ∧
+  Tendsto (fun r : ℝ =>
+    (r ^ n)⁻¹ * ∫ y in Metric.ball x r,
+      |E.indicator (fun _ => (1 : ℝ)) y -
+        {z | inner ℝ (z - x) v < 0}.indicator (fun _ => (1 : ℝ)) y|)
+    (𝓝[>] (0 : ℝ)) (𝓝 0)
+
+def approximateNormal {n : ℕ} (E : Set (Euc n)) (x : Euc n) : Euc n := by
+  classical
+  exact if h : ∃ v, HasHalfspaceBlowup E x v then Classical.choose h else 0
+
+def tangentialGradient {n : ℕ} (E : Set (Euc n)) (φ : Euc n → ℝ)
+    (x : Euc n) : Euc n :=
+  gradient φ x - (inner ℝ (gradient φ x) (approximateNormal E x)) •
+    approximateNormal E x
+
+theorem eventually_sublevel_membership {n : ℕ} {E : Set (Euc n)}
+    {f : Euc n → ℝ} {x z : Euc n} {L : Euc n →L[ℝ] ℝ}
+    (hd : HasFDerivAt f L x) (hx : f x = 0)
+    (hlocal : ∀ᶠ y in 𝓝 x, y ∈ E ↔ f y < 0) (hz : L z ≠ 0) :
+    ∀ᶠ r : ℝ in 𝓝[>] 0, x + r • z ∈ E ↔ L z < 0 := by
+  have hp : HasDerivAt (fun r : ℝ ↦ x + r • z) z 0 := by
+    simpa using ((hasDerivAt_id (0 : ℝ)).smul_const z).const_add x
+  have hc : HasDerivAt (fun r : ℝ ↦ f (x + r • z)) (L z) 0 := by
+    exact (show HasFDerivAt f L (x + (0 : ℝ) • z) from by simpa using hd).comp_hasDerivAt 0 hp
+  have ht : Tendsto (fun r : ℝ ↦ r⁻¹ * f (x + r • z)) (𝓝[>] 0) (𝓝 (L z)) := by
+    simpa [hx] using hc.tendsto_slope_zero_right
+  have hm : ∀ᶠ r : ℝ in 𝓝[>] 0, x + r • z ∈ E ↔ f (x + r • z) < 0 :=
+    (show Tendsto (fun r : ℝ ↦ x + r • z) (𝓝[>] 0) (𝓝 x) from
+      (show Tendsto (fun r : ℝ ↦ x + r • z) (𝓝 0) (𝓝 x) from by
+        simpa [ContinuousAt] using hp.continuousAt).mono_left nhdsWithin_le_nhds).eventually hlocal
+  rcases lt_or_gt_of_ne hz with hneg | hpos
+  · filter_upwards [hm, ht.eventually (gt_mem_nhds hneg), self_mem_nhdsWithin] with r hm hr hpr
+    have hfr : f (x + r • z) < 0 := by
+      have hi : 0 < r⁻¹ := inv_pos.mpr hpr
+      nlinarith
+    simp only [hm, hfr, hneg]
+  · filter_upwards [hm, ht.eventually (lt_mem_nhds hpos), self_mem_nhdsWithin] with r hm hr hpr
+    have hfr : 0 < f (x + r • z) := by
+      have hi : 0 < r⁻¹ := inv_pos.mpr hpr
+      nlinarith
+    simp only [hm, not_lt.mpr hfr.le, not_lt.mpr hpos.le]
+
+theorem normalized_ball_integral {n : ℕ} (g : Euc n → ℝ) (x : Euc n)
+    {r : ℝ} (hr : 0 < r) :
+    (r ^ n)⁻¹ * ∫ y in Metric.ball x r, g y =
+      ∫ z in Metric.ball (0 : Euc n) 1, g (x + r • z) := by
+  have hc := Measure.integral_comp_smul_of_nonneg (volume : Measure (Euc n))
+    (fun y ↦ (Metric.ball x r).indicator g (x + y)) r (hR := hr.le)
+  simp only [finrank_euclideanSpace, Fintype.card_fin, smul_eq_mul,
+    integral_add_left_eq_self, integral_indicator measurableSet_ball] at hc
+  rw [← hc, ← integral_indicator measurableSet_ball]
+  apply integral_congr_ae
+  filter_upwards [] with z
+  have hm : x + r • z ∈ Metric.ball x r ↔ z ∈ Metric.ball (0 : Euc n) 1 := by
+    simp only [Metric.mem_ball, dist_eq_norm, add_sub_cancel_left,
+      sub_zero, norm_smul, Real.norm_eq_abs, abs_of_pos hr]
+    constructor <;> intro h <;> nlinarith
+  by_cases hz : z ∈ Metric.ball (0 : Euc n) 1
+  · simp [hz, hm.mpr hz]
+  · have hn : x + r • z ∉ Metric.ball x r := fun h ↦ hz (hm.mp h)
+    rw [Set.indicator_of_notMem hn, Set.indicator_of_notMem hz]
+
+theorem ae_linear_ne_zero {n : ℕ} {L : Euc n →L[ℝ] ℝ} (hL : L ≠ 0) :
+    ∀ᵐ z ∂(volume : Measure (Euc n)), L z ≠ 0 := by
+  have hk : L.ker ≠ ⊤ := by
+    intro hk
+    apply hL
+    ext z
+    have hz : z ∈ L.ker := hk ▸ Submodule.mem_top
+    exact hz
+  have hn : (volume : Measure (Euc n)) {z | L z = 0} = 0 :=
+    Measure.addHaar_submodule (volume : Measure (Euc n)) L.ker hk
+  simpa only [ae_iff, not_not] using hn
+
+theorem halfspace_indicator_rescale {n : ℕ} (x v z : Euc n) {r : ℝ} (hr : 0 < r) :
+    {y | inner ℝ (y - x) v < 0}.indicator (fun _ ↦ (1 : ℝ)) (x + r • z) =
+      {y | inner ℝ y v < 0}.indicator (fun _ ↦ (1 : ℝ)) z := by
+  have hm : inner ℝ ((x + r • z) - x) v < 0 ↔ inner ℝ z v < 0 := by
+    simp only [add_sub_cancel_left, real_inner_smul_left]
+    constructor <;> intro h <;> nlinarith
+  simp only [Set.indicator, Set.mem_ofPred_eq, hm]
+
+theorem hasHalfspaceBlowup_of_regular_sublevel {n : ℕ} {E : Set (Euc n)}
+    (hE : MeasurableSet E) {f : Euc n → ℝ} {x v : Euc n} {L : Euc n →L[ℝ] ℝ}
+    (hd : HasFDerivAt f L x) (hx : f x = 0)
+    (hlocal : ∀ᶠ y in 𝓝 x, y ∈ E ↔ f y < 0)
+    (hv : ‖v‖ = 1) {a : ℝ} (ha : 0 < a)
+    (hL : ∀ z, L z = a * inner ℝ z v) : HasHalfspaceBlowup E x v := by
+  have hLn : L ≠ 0 := by
+    intro he
+    have hval := hL v
+    simp only [he, zero_apply, real_inner_self_eq_norm_sq, hv,
+      one_pow, mul_one] at hval
+    linarith
+  let H : Set (Euc n) := {z | inner ℝ z v < 0}
+  have hH : MeasurableSet H :=
+    (isOpen_lt (continuous_id.inner continuous_const) continuous_const).measurableSet
+  let F : ℝ → Euc n → ℝ := fun r z ↦
+    |E.indicator (fun _ ↦ (1 : ℝ)) (x + r • z) - H.indicator (fun _ ↦ (1 : ℝ)) z|
+  have hsign : ∀ z, L z < 0 ↔ inner ℝ z v < 0 := by
+    intro z
+    rw [hL]
+    constructor <;> intro h <;> nlinarith
+  have hmeas : ∀ r, AEStronglyMeasurable (F r)
+      (volume.restrict (Metric.ball (0 : Euc n) 1)) := by
+    intro r
+    have he : Measurable (E.indicator (fun _ : Euc n ↦ (1 : ℝ))) :=
+      measurable_const.indicator hE
+    have hh : Measurable (H.indicator (fun _ : Euc n ↦ (1 : ℝ))) :=
+      measurable_const.indicator hH
+    have hm : Measurable (fun z : Euc n ↦ x + r • z) :=
+      measurable_const.add (measurable_id.const_smul r)
+    exact ((he.comp hm).sub hh).abs.aestronglyMeasurable
+  have hb : ∀ r, ∀ᵐ z ∂volume.restrict (Metric.ball (0 : Euc n) 1), ‖F r z‖ ≤ 1 := by
+    intro r
+    filter_upwards [] with z
+    by_cases he : x + r • z ∈ E <;> by_cases hh : z ∈ H <;> simp [F, he, hh]
+  have ht : ∀ᵐ z ∂volume.restrict (Metric.ball (0 : Euc n) 1),
+      Tendsto (fun r ↦ F r z) (𝓝[>] (0 : ℝ)) (𝓝 0) := by
+    filter_upwards [ae_restrict_of_ae (ae_linear_ne_zero hLn)] with z hz
+    apply tendsto_const_nhds.congr'
+    filter_upwards [eventually_sublevel_membership hd hx hlocal hz] with r hr
+    have hmem : x + r • z ∈ E ↔ z ∈ H := hr.trans (hsign z)
+    by_cases he : x + r • z ∈ E
+    · simp [F, he, hmem.mp he]
+    · have hhn : z ∉ H := fun hh ↦ he (hmem.mpr hh)
+      simp only [F, Set.indicator_of_notMem he, Set.indicator_of_notMem hhn,
+        sub_self, abs_zero]
+  have hi : Integrable (fun _ : Euc n ↦ (1 : ℝ))
+      (volume.restrict (Metric.ball 0 1)) := integrableOn_const measure_ball_lt_top.ne
+  have hconv := tendsto_integral_filter_of_dominated_convergence
+    (fun _ : Euc n ↦ (1 : ℝ)) (Eventually.of_forall hmeas) (Eventually.of_forall hb) hi ht
+  refine ⟨hv, ?_⟩
+  have heq : ∀ᶠ r : ℝ in 𝓝[>] 0,
+      (r ^ n)⁻¹ * (∫ y in Metric.ball x r,
+        |E.indicator (fun _ ↦ (1 : ℝ)) y -
+          {z | inner ℝ (z - x) v < 0}.indicator (fun _ ↦ (1 : ℝ)) y|) =
+        ∫ z in Metric.ball (0 : Euc n) 1, F r z := by
+    filter_upwards [self_mem_nhdsWithin] with r hr
+    rw [normalized_ball_integral _ x hr]
+    apply integral_congr_ae
+    filter_upwards [] with z
+    simp only [halfspace_indicator_rescale x v z hr, F, H]
+  exact (show Tendsto (fun r ↦ ∫ z in Metric.ball (0 : Euc n) 1, F r z)
+    (𝓝[>] (0 : ℝ)) (𝓝 0) from by simpa only [integral_zero] using hconv).congr' (Filter.EventuallyEq.symm heq)
+
+theorem hasHalfspaceBlowup_gradient {n : ℕ} {E : Set (Euc n)}
+    (hE : MeasurableSet E) {f : Euc n → ℝ} {x : Euc n}
+    (hd : DifferentiableAt ℝ f x) (hx : f x = 0) (hdf : fderiv ℝ f x ≠ 0)
+    (hlocal : ∀ᶠ y in 𝓝 x, y ∈ E ↔ f y < 0) :
+    HasHalfspaceBlowup E x (‖gradient f x‖⁻¹ • gradient f x) := by
+  have hg : gradient f x ≠ 0 := by
+    intro he
+    apply hdf
+    ext z
+    rw [← inner_gradient_left, he, inner_zero_left]
+    rfl
+  have hng : 0 < ‖gradient f x‖ := norm_pos_iff.mpr hg
+  apply hasHalfspaceBlowup_of_regular_sublevel hE hd.hasFDerivAt hx hlocal
+    (a := ‖gradient f x‖)
+  · rw [norm_smul, Real.norm_of_nonneg (inv_nonneg.mpr hng.le), inv_mul_cancel₀ hng.ne']
+  · exact hng
+  · intro z
+    rw [real_inner_smul_right, ← inner_gradient_left, real_inner_comm]
+    field_simp
+
+theorem halfspace_wedge_nonempty {n : ℕ} {v w : Euc n}
+    (hv : ‖v‖ = 1) (hw : ‖w‖ = 1) (hne : v ≠ w) :
+    (Metric.ball (0 : Euc n) 1 ∩
+      {z | inner ℝ z v < 0} ∩ {z | 0 < inner ℝ z w}).Nonempty := by
+  have hdiff : 0 < ‖w - v‖ := norm_pos_iff.mpr (sub_ne_zero.mpr hne.symm)
+  have hinner : inner ℝ w v < 1 := by
+    have hs := norm_sub_sq_real w v
+    rw [hv, hw] at hs
+    nlinarith [sq_pos_of_pos hdiff]
+  refine ⟨(1 / 4 : ℝ) • (w - v), ⟨?_, ?_⟩, ?_⟩
+  · rw [Metric.mem_ball, dist_zero_right, norm_smul]
+    have hn : ‖w - v‖ ≤ 2 := (norm_sub_le w v).trans (by rw [hw, hv]; norm_num)
+    norm_num
+    linarith
+  · simp only [Set.mem_ofPred_eq, real_inner_smul_left, inner_sub_left,
+      real_inner_self_eq_norm_sq, hv]
+    nlinarith
+  · simp only [Set.mem_ofPred_eq, real_inner_smul_left, inner_sub_left,
+      real_inner_self_eq_norm_sq, hw, real_inner_comm w v]
+    nlinarith
+
+theorem eq_of_halfspace_indicator_ae {n : ℕ} {v w : Euc n}
+    (hv : ‖v‖ = 1) (hw : ‖w‖ = 1)
+    (he : {z | inner ℝ z v < 0}.indicator (fun _ ↦ (1 : ℝ)) =ᵐ[
+      volume.restrict (Metric.ball (0 : Euc n) 1)]
+      {z | inner ℝ z w < 0}.indicator (fun _ ↦ (1 : ℝ))) : v = w := by
+  by_contra hne
+  let U : Set (Euc n) := Metric.ball 0 1 ∩ {z | inner ℝ z v < 0} ∩ {z | 0 < inner ℝ z w}
+  have hUo : IsOpen U := (Metric.isOpen_ball.inter
+    (isOpen_lt (continuous_id.inner continuous_const) continuous_const)).inter
+      (isOpen_lt continuous_const (continuous_id.inner continuous_const))
+  have hUp : 0 < (volume : Measure (Euc n)) U :=
+    hUo.measure_pos _ (halfspace_wedge_nonempty hv hw hne)
+  have hUs : U ⊆ Metric.ball 0 1 := fun _ hz ↦ hz.1.1
+  have hUr : 0 < volume.restrict (Metric.ball (0 : Euc n) 1) U := by
+    rwa [Measure.restrict_apply hUo.measurableSet, inter_eq_left.mpr hUs]
+  obtain ⟨z, hz, hze⟩ := Measure.exists_mem_of_measure_ne_zero_of_ae hUr.ne' (ae_restrict_of_ae he)
+  have hzv : z ∈ {y | inner ℝ y v < 0} := hz.1.2
+  have hzwp : 0 < inner ℝ z w := hz.2
+  have hzw : z ∉ {y | inner ℝ y w < 0} := not_lt.mpr hzwp.le
+  simp only [Set.indicator_of_mem hzv, Set.indicator_of_notMem hzw, one_ne_zero] at hze
+
+theorem integrable_rescaled_indicator {n : ℕ} {E : Set (Euc n)} (hE : MeasurableSet E)
+    (x : Euc n) (r : ℝ) :
+    Integrable (fun z ↦ E.indicator (fun _ ↦ (1 : ℝ)) (x + r • z))
+      (volume.restrict (Metric.ball (0 : Euc n) 1)) := by
+  have hAff : Measurable (fun z : Euc n ↦ x + r • z) :=
+    measurable_const.add (measurable_id.const_smul r)
+  have he : Measurable (E.indicator (fun _ : Euc n ↦ (1 : ℝ))) :=
+    measurable_const.indicator hE
+  apply (integrableOn_const measure_ball_lt_top.ne (C := (1 : ℝ))).mono'
+    (he.comp hAff).aestronglyMeasurable
+  filter_upwards [] with z
+  by_cases hz : x + r • z ∈ E <;> simp [hz]
+
+theorem HasHalfspaceBlowup.tendstoInMeasure {n : ℕ} {E : Set (Euc n)}
+    (hE : MeasurableSet E) {x v : Euc n} (hv : HasHalfspaceBlowup E x v) :
+    TendstoInMeasure (volume.restrict (Metric.ball (0 : Euc n) 1))
+      (fun r z ↦ E.indicator (fun _ ↦ (1 : ℝ)) (x + r • z)) (𝓝[>] (0 : ℝ))
+      ({z | inner ℝ z v < 0}.indicator (fun _ ↦ (1 : ℝ))) := by
+  let μ := (volume : Measure (Euc n)).restrict (Metric.ball 0 1)
+  let χ : ℝ → Euc n → ℝ := fun r z ↦ E.indicator (fun _ ↦ (1 : ℝ)) (x + r • z)
+  let H : Set (Euc n) := {z | inner ℝ z v < 0}
+  let ψ : Euc n → ℝ := H.indicator (fun _ ↦ (1 : ℝ))
+  have hH : MeasurableSet H :=
+    (isOpen_lt (continuous_id.inner continuous_const) continuous_const).measurableSet
+  have hχ : ∀ r, Integrable (χ r) μ := integrable_rescaled_indicator hE x
+  have hψ : Integrable ψ μ :=
+    (integrableOn_const measure_ball_lt_top.ne (C := (1 : ℝ))).indicator hH
+  have hconv : Tendsto (fun r ↦ ∫ z, |χ r z - ψ z| ∂μ) (𝓝[>] (0 : ℝ)) (𝓝 0) := by
+    apply hv.2.congr'
+    filter_upwards [self_mem_nhdsWithin] with r hr
+    rw [normalized_ball_integral _ x hr]
+    apply integral_congr_ae
+    filter_upwards [] with z
+    simp only [χ, ψ, H, halfspace_indicator_rescale x v z hr]
+  apply tendstoInMeasure_of_tendsto_eLpNorm (p := 1) one_ne_zero
+  have heq : (fun r ↦ eLpNorm (χ r - ψ) 1 μ) =
+      (fun r ↦ ENNReal.ofReal (∫ z, |χ r z - ψ z| ∂μ)) := by
+    funext r
+    rw [eLpNorm_one_eq_lintegral_enorm ((hχ r).sub hψ).aestronglyMeasurable,
+      ← ofReal_integral_norm_eq_lintegral_enorm ((hχ r).sub hψ)]
+    rfl
+  change Tendsto (fun r ↦ eLpNorm (χ r - ψ) 1 μ) _ _
+  rw [heq]
+  simpa only [Function.comp_def, ENNReal.ofReal_zero] using
+    ENNReal.continuous_ofReal.continuousAt.tendsto.comp hconv
+
+theorem HasHalfspaceBlowup.unique {n : ℕ} {E : Set (Euc n)} (hE : MeasurableSet E)
+    {x v w : Euc n} (hv : HasHalfspaceBlowup E x v) (hw : HasHalfspaceBlowup E x w) :
+    v = w := by
+  apply eq_of_halfspace_indicator_ae hv.1 hw.1
+  exact tendstoInMeasure_ae_unique (hv.tendstoInMeasure hE) (hw.tendstoInMeasure hE)
+
+theorem approximateNormal_eq_of_blowup {n : ℕ} {E : Set (Euc n)} (hE : MeasurableSet E)
+    {x v : Euc n} (hv : HasHalfspaceBlowup E x v) : approximateNormal E x = v := by
+  have hex : ∃ w, HasHalfspaceBlowup E x w := ⟨v, hv⟩
+  rw [approximateNormal, dite_eq_left hex]
+  exact (Classical.choose_spec hex).unique hE hv
+
+theorem approximateNormal_eq_gradient {n : ℕ} {E : Set (Euc n)}
+    (hE : MeasurableSet E) {f : Euc n → ℝ} {x : Euc n}
+    (hd : DifferentiableAt ℝ f x) (hx : f x = 0) (hdf : fderiv ℝ f x ≠ 0)
+    (hlocal : ∀ᶠ y in 𝓝 x, y ∈ E ↔ f y < 0) :
+    approximateNormal E x = ‖gradient f x‖⁻¹ • gradient f x :=
+  approximateNormal_eq_of_blowup hE (hasHalfspaceBlowup_gradient hE hd hx hdf hlocal)
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+namespace RPI.Boundary
+
+theorem surfaceGradient_eq_tangentialGradient {n : ℕ} {E S : Set (Space n)}
+    (hE : MeasurableSet E) {f g : Space n → ℝ} {x : Space n}
+    (hf : ContDiffAt ℝ 1 f x) (hfx : f x = 0) (hdf : fderiv ℝ f x ≠ 0)
+    (hlevel : ∀ᶠ y in 𝓝 x, y ∈ S ↔ f y = 0)
+    (hfilled : ∀ᶠ y in 𝓝 x, y ∈ E ↔ f y < 0) :
+    surfaceGradient S g x = RPISingularHarnack.tangentialGradient E g x := by
+  have hgrad : gradient f x ≠ 0 := by
+    intro he
+    apply hdf
+    rw [← toDual_gradient, he, map_zero]
+  have hn : ‖gradient f x‖ ≠ 0 := norm_ne_zero_iff.mpr hgrad
+  rw [surfaceGradient_eq_level_formula hf hdf (by simpa only [hfx] using hlevel),
+    RPISingularHarnack.tangentialGradient,
+    RPISingularHarnack.approximateNormal_eq_gradient hE
+      (hf.differentiableAt (by norm_num)) hfx hdf hfilled,
+    real_inner_smul_right, smul_smul, real_inner_comm (gradient f x) (gradient g x)]
+  congr 2
+  field_simp
+
+end RPI.Boundary
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+namespace RPISingularHarnack
+
+def divergence {n : ℕ} (V : Euc n → Euc n) (x : Euc n) : ℝ :=
+  ∑ i : Fin n, inner ℝ ((EuclideanSpace.basisFun (Fin n) ℝ) i)
+    (fderiv ℝ V x ((EuclideanSpace.basisFun (Fin n) ℝ) i))
+
+def perimeter {n : ℕ} (E U : Set (Euc n)) : ℝ≥0∞ :=
+  ⨆ (V : Euc n → Euc n) (_ : ContDiff ℝ ∞ V) (_ : HasCompactSupport V)
+    (_ : tsupport V ⊆ U) (_ : ∀ x, ‖V x‖ ≤ 1),
+    ENNReal.ofReal (∫ x in E, divergence V x)
+
+def perimeterSupport {n : ℕ} (E : Set (Euc n)) : Set (Euc n) :=
+  {x | ∀ r : ℝ, 0 < r → 0 < perimeter E (Metric.ball x r)}
+
+def IsLocallyPerimeterMinimizing {n : ℕ} (E : Set (Euc n)) : Prop :=
+  MeasurableSet E ∧
+  (∀ x (r : ℝ), 0 < r → perimeter E (Metric.ball x r) < ⊤) ∧
+  ∀ x (r : ℝ), 0 < r → ∀ F : Set (Euc n), MeasurableSet F →
+    IsCompact (closure (symmDiff E F)) →
+    closure (symmDiff E F) ⊆ Metric.ball x r →
+    perimeter E (Metric.ball x r) ≤ perimeter F (Metric.ball x r)
+
+def IsCone {n : ℕ} (E : Set (Euc n)) : Prop :=
+  ∀ r : ℝ, 0 < r → ∀ x : Euc n, r • x ∈ E ↔ x ∈ E
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/CoupledConeLink.lean b/lean/OAI/Geometry/Perimeter/Regularity/CoupledConeLink.lean
new file mode 100644
index 0000000000000000000000000000000000000000..c202ea92be70b032c37e4005965be514f0305231
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/CoupledConeLink.lean
@@ -0,0 +1,168 @@
+import OAI.Geometry.Perimeter.Regularity.RawConeLink
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+namespace RPISingularHarnack
+namespace SphericalFrankel
+
+lemma angle_swap {n : ℕ} (e f : Frame n) : angle f e = angle e f := real_inner_comm _ _
+
+lemma cross_swap {n : ℕ} (e f : Frame n) : cross f e = cross e f := real_inner_comm _ _
+
+lemma traceProduct_swap {n : ℕ} (e f : Frame n) : traceProduct f e = traceProduct e f := by
+  simp only [traceProduct, angle_swap, real_inner_comm f.1 e.2,
+    real_inner_comm f.2 e.1, real_inner_comm f.2 e.2]
+  ring
+
+lemma continuous_defect_reverse {n : ℕ} :
+    Continuous (fun p : Frame n × Frame n ↦ defect p.2 p.1) := by
+  unfold defect proj
+  fun_prop
+
+lemma product_frames {n : ℕ} {μ ν : Measure (Frame n)} [IsFiniteMeasure μ] [IsFiniteMeasure ν]
+    (hμ : ∀ᵐ e ∂μ, IsFrame e) (hν : ∀ᵐ e ∂ν, IsFrame e) :
+    ∀ᵐ p ∂μ.prod ν, IsFrame p.1 ∧ IsFrame p.2 := by
+  apply (Measure.ae_prod_iff_ae_ae
+    ((isClosed_frames n).prod (isClosed_frames n)).measurableSet).mpr
+  filter_upwards [hμ] with e he
+  filter_upwards [hν] with f hf
+  exact ⟨he, hf⟩
+
+lemma stationary_product_integral {n : ℕ} {μ ν : Measure (Frame n)}
+    [IsFiniteMeasure μ] [IsFiniteMeasure ν]
+    (hμ : ∀ᵐ e ∂μ, IsFrame e) (hν : ∀ᵐ e ∂ν, IsFrame e)
+    (hS : LinkStationary μ) (W : Frame n → Euc n → Euc n)
+    (hW : ∀ f, ContDiff ℝ ∞ (W f))
+    (G : Frame n × Frame n → ℝ) (hG : Continuous G)
+    (heq : ∀ e f, IsFrame e → IsFrame f → G (e,f) = linkDiv e (W f)) :
+    (∫ p, G p ∂μ.prod ν) = 0 := by
+  rw [integral_prod_symm G (integrable_pairFrameFunction (product_frames hμ hν) hG)]
+  calc
+    (∫ f, ∫ e, G (e, f) ∂μ ∂ν) = ∫ f, (0 : ℝ) ∂ν := by
+      apply integral_congr_ae
+      filter_upwards [hν] with f hf
+      calc
+        (∫ e, G (e,f) ∂μ) = ∫ e, linkDiv e (W f) ∂μ := by
+          apply integral_congr_ae
+          filter_upwards [hμ] with e he
+          exact heq e f he hf
+        _ = 0 := hS.firstVariation (W f) (hW f)
+    _ = 0 := by simp
+
+lemma stationary_gradient_one {n : ℕ} {μ ν : Measure (Frame n)}
+    [IsFiniteMeasure μ] [IsFiniteMeasure ν]
+    (hμ : ∀ᵐ e ∂μ, IsFrame e) (hν : ∀ᵐ e ∂ν, IsFrame e)
+    (hS : LinkStationary μ) {φ : ℝ → ℝ} (hφ : ContDiff ℝ ∞ φ) :
+    (∫ p, deriv φ (angle p.1 p.2) * defect p.1 p.2 ∂μ.prod ν) =
+      ((n : ℝ)-2) * ∫ p, φ (angle p.1 p.2) * angle p.1 p.2 ∂μ.prod ν := by
+  have hd := hφ.continuous_deriv (by simp)
+  have h0 := stationary_product_integral hμ hν hS
+    (fun f z ↦ φ (inner ℝ z f.1) • f.1)
+    (fun f ↦ (hφ.comp (contDiff_id.inner ℝ contDiff_const)).smul contDiff_const)
+    (fun p ↦ deriv φ (angle p.1 p.2) * defect p.1 p.2 -
+      ((n : ℝ)-2) * (φ (angle p.1 p.2) * angle p.1 p.2))
+    (((hd.comp continuous_angle).mul continuous_defect).sub
+      (continuous_const.mul ((hφ.continuous.comp continuous_angle).mul continuous_angle)))
+    (fun e f he hf ↦ by
+      rw [linkDiv_scalar_constant he (hφ.differentiable (by simp))]
+      ring)
+  have hi1 : Integrable (fun p : Frame n × Frame n ↦ deriv φ (angle p.1 p.2) * defect p.1 p.2) (μ.prod ν) :=
+    integrable_pairFrameFunction (product_frames hμ hν)
+      ((hd.comp continuous_angle).mul continuous_defect)
+  have hi2 : Integrable (fun p : Frame n × Frame n ↦ φ (angle p.1 p.2) * angle p.1 p.2) (μ.prod ν) :=
+    integrable_pairFrameFunction (product_frames hμ hν)
+      ((hφ.continuous.comp continuous_angle).mul continuous_angle)
+  rw [integral_sub hi1 (hi2.const_mul ((n : ℝ)-2)), integral_const_mul] at h0
+  exact sub_eq_zero.mp h0
+
+lemma stationary_mixed_one {n : ℕ} {μ ν : Measure (Frame n)}
+    [IsFiniteMeasure μ] [IsFiniteMeasure ν]
+    (hμ : ∀ᵐ e ∂μ, IsFrame e) (hν : ∀ᵐ e ∂ν, IsFrame e)
+    (hS : LinkStationary μ) {φ : ℝ → ℝ} (hφ : ContDiff ℝ ∞ φ) :
+    (∫ p, deriv φ (angle p.1 p.2) * cross p.1 p.2 ∂μ.prod ν) +
+      (∫ p, φ (angle p.1 p.2) * traceProduct p.1 p.2 ∂μ.prod ν) -
+      ((n : ℝ)-2) * ∫ p, φ (angle p.1 p.2) * defect p.2 p.1 ∂μ.prod ν = 0 := by
+  have hd := hφ.continuous_deriv (by simp)
+  have hswap : Continuous (fun p : Frame n × Frame n ↦ defect p.2 p.1) :=
+    continuous_defect_reverse
+  have hC : Continuous (fun p : Frame n × Frame n ↦ deriv φ (angle p.1 p.2) * cross p.1 p.2) :=
+    (hd.comp continuous_angle).mul continuous_cross
+  have hT : Continuous (fun p : Frame n × Frame n ↦ φ (angle p.1 p.2) * traceProduct p.1 p.2) :=
+    (hφ.continuous.comp continuous_angle).mul continuous_traceProduct
+  have hD : Continuous (fun p : Frame n × Frame n ↦ φ (angle p.1 p.2) * defect p.2 p.1) :=
+    (hφ.continuous.comp continuous_angle).mul hswap
+  have h0 := stationary_product_integral hμ hν hS
+    (fun f z ↦ φ (inner ℝ z f.1) • projMap f z)
+    (fun f ↦ (hφ.comp (contDiff_id.inner ℝ contDiff_const)).smul (projMap f).contDiff)
+    (fun p ↦ deriv φ (angle p.1 p.2) * cross p.1 p.2 +
+      φ (angle p.1 p.2) * traceProduct p.1 p.2 -
+      ((n : ℝ)-2) * (φ (angle p.1 p.2) * defect p.2 p.1))
+    ((hC.add hT).sub (continuous_const.mul hD))
+    (fun e f he hf ↦ by
+      rw [linkDiv_scalar_proj he hf (hφ.differentiable (by simp))]
+      ring)
+  have hiC := integrable_pairFrameFunction (product_frames hμ hν) hC
+  have hiT := integrable_pairFrameFunction (product_frames hμ hν) hT
+  have hiD := integrable_pairFrameFunction (product_frames hμ hν) hD
+  have hiCT : Integrable (fun p : Frame n × Frame n ↦
+      deriv φ (angle p.1 p.2) * cross p.1 p.2 +
+      φ (angle p.1 p.2) * traceProduct p.1 p.2) (μ.prod ν) := hiC.add hiT
+  rwa [integral_sub hiCT (hiD.const_mul ((n : ℝ)-2)), integral_add hiC hiT,
+    integral_const_mul] at h0
+
+lemma coupledStationarity_product {n : ℕ} {μ ν : Measure (Frame n)}
+    [IsFiniteMeasure μ] [IsFiniteMeasure ν]
+    (hμ : ∀ᵐ e ∂μ, IsFrame e) (hν : ∀ᵐ e ∂ν, IsFrame e)
+    (hSμ : LinkStationary μ) (hSν : LinkStationary ν) : CoupledStationarity (μ.prod ν) := by
+  constructor
+  · intro φ hφ
+    have hd := hφ.continuous_deriv (by simp)
+    have hi1 : Integrable (fun p : Frame n × Frame n ↦ deriv φ (angle p.1 p.2) * defect p.1 p.2) (μ.prod ν) :=
+      integrable_pairFrameFunction (product_frames hμ hν)
+        ((hd.comp continuous_angle).mul continuous_defect)
+    have hi2 : Integrable (fun p : Frame n × Frame n ↦ deriv φ (angle p.1 p.2) * defect p.2 p.1) (μ.prod ν) :=
+      integrable_pairFrameFunction (product_frames hμ hν)
+        ((hd.comp continuous_angle).mul (continuous_defect_reverse))
+    have h1 := stationary_gradient_one hμ hν hSμ hφ
+    have h2 := stationary_gradient_one hν hμ hSν hφ
+    rw [← integral_prod_swap (fun p : Frame n × Frame n ↦ deriv φ (angle p.1 p.2) * defect p.1 p.2),
+      ← integral_prod_swap (fun p : Frame n × Frame n ↦ φ (angle p.1 p.2) * angle p.1 p.2)] at h2
+    simp only [Prod.fst_swap, Prod.snd_swap, angle_swap] at h2
+    simp_rw [mul_add]
+    rw [integral_add hi1 hi2, h1, h2]
+    ring
+  · intro φ hφ
+    have h1 := stationary_mixed_one hμ hν hSμ hφ
+    have h2 := stationary_mixed_one hν hμ hSν hφ
+    rw [← integral_prod_swap (fun p : Frame n × Frame n ↦ deriv φ (angle p.1 p.2) * cross p.1 p.2),
+      ← integral_prod_swap (fun p : Frame n × Frame n ↦ φ (angle p.1 p.2) * traceProduct p.1 p.2),
+      ← integral_prod_swap (fun p : Frame n × Frame n ↦ φ (angle p.1 p.2) * defect p.2 p.1)] at h2
+    simp only [Prod.fst_swap, Prod.snd_swap, angle_swap, cross_swap, traceProduct_swap] at h2
+    have hi1 : Integrable (fun p : Frame n × Frame n ↦ φ (angle p.1 p.2) * defect p.1 p.2) (μ.prod ν) :=
+      integrable_pairFrameFunction (product_frames hμ hν)
+        ((hφ.continuous.comp continuous_angle).mul continuous_defect)
+    have hi2 : Integrable (fun p : Frame n × Frame n ↦ φ (angle p.1 p.2) * defect p.2 p.1) (μ.prod ν) :=
+      integrable_pairFrameFunction (product_frames hμ hν)
+        ((hφ.continuous.comp continuous_angle).mul (continuous_defect_reverse))
+    simp_rw [mul_add]
+    rw [integral_add hi1 hi2]
+    linarith
+
+lemma minimizing_cones_coupledStationarity {n : ℕ} (hn : 1 < n) {E F : Set (Euc n)}
+    (hE : IsLocallyPerimeterMinimizing E) (hF : IsLocallyPerimeterMinimizing F)
+    (hcE : IsCone E) (hcF : IsCone F) :
+    CoupledStationarity ((annularLink E).prod (annularLink F)) := by
+  let := annularLink_finite hE
+  let := annularLink_finite hF
+  exact coupledStationarity_product (annularLink_frames hn hE hcE) (annularLink_frames hn hF hcF)
+    (annularLink_stationary hn hE hcE) (annularLink_stationary hn hF hcF)
+
+end SphericalFrankel
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/DilatedVolumeTest.lean b/lean/OAI/Geometry/Perimeter/Regularity/DilatedVolumeTest.lean
new file mode 100644
index 0000000000000000000000000000000000000000..bb808f135abf1f0e8a97dcc48826b6347aa85249
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/DilatedVolumeTest.lean
@@ -0,0 +1,321 @@
+import OAI.Geometry.Perimeter.Regularity.ConeRadialCap
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+open scoped ENNReal Topology
+open scoped ENNReal Topology ContDiff
+open scoped Topology ENNReal
+namespace RPISingularHarnack
+
+def dilatedVolumeTest {n : ℕ} (φ : Euc n → ℝ) (t : ℝ) (x : Euc n) : ℝ :=
+  t^n * φ (t • x)
+
+def dilatedVolumeTestDerivative {n : ℕ} (φ : Euc n → ℝ) (t : ℝ) (x : Euc n) : ℝ :=
+  (n : ℝ)*t^(n-1)*φ (t • x) + t^n * fderiv ℝ φ (t • x) x
+
+theorem dilatedVolumeTest_hasDerivAt {n : ℕ} {φ : Euc n → ℝ}
+    (hφ : ContDiff ℝ ∞ φ) (t : ℝ) (x : Euc n) :
+    HasDerivAt (fun s ↦ dilatedVolumeTest φ s x) (dilatedVolumeTestDerivative φ t x) t := by
+  have hp : HasDerivAt (fun s : ℝ ↦ s • x) x t := by
+    simpa using (hasDerivAt_id t).smul_const x
+  have hc := ((hφ.differentiable (by simp)) (t • x)).hasFDerivAt.comp_hasDerivAt t hp
+  convert (hasDerivAt_pow n t).mul hc using 1 <;> rfl
+
+theorem dilatedVolumeTest_continuous {n : ℕ} {φ : Euc n → ℝ}
+    (hφ : ContDiff ℝ ∞ φ) :
+    Continuous (fun p : ℝ × Euc n ↦ dilatedVolumeTest φ p.1 p.2) := by
+  unfold dilatedVolumeTest
+  fun_prop
+
+theorem dilatedVolumeTestDerivative_continuous {n : ℕ} {φ : Euc n → ℝ}
+    (hφ : ContDiff ℝ ∞ φ) :
+    Continuous (fun p : ℝ × Euc n ↦ dilatedVolumeTestDerivative φ p.1 p.2) := by
+  have hd := hφ.continuous_fderiv (by simp)
+  unfold dilatedVolumeTestDerivative
+  fun_prop
+
+theorem dilatedVolumeTest_integral_hasDerivAt {n : ℕ} {φ : Euc n → ℝ}
+    (hφ : ContDiff ℝ ∞ φ) (M : ℝ) (μ : Measure (Euc n)) [IsFiniteMeasure μ]
+    (hμ : ∀ᵐ x ∂μ, x ∈ Metric.closedBall 0 M) (t : ℝ) :
+    Integrable (dilatedVolumeTestDerivative φ t) μ ∧
+    HasDerivAt (fun s ↦ ∫ x, dilatedVolumeTest φ s x ∂μ)
+      (∫ x, dilatedVolumeTestDerivative φ t x ∂μ) t := by
+  have hcF (s : ℝ) : Continuous (dilatedVolumeTest φ s) :=
+    (dilatedVolumeTest_continuous hφ).comp (continuous_const.prodMk continuous_id)
+  have hcD (s : ℝ) : Continuous (dilatedVolumeTestDerivative φ s) :=
+    (dilatedVolumeTestDerivative_continuous hφ).comp (continuous_const.prodMk continuous_id)
+  obtain ⟨C, hC⟩ := (isCompact_closedBall (0:Euc n) M).exists_bound_of_continuousOn (hcF t).continuousOn
+  have hFi : Integrable (dilatedVolumeTest φ t) μ :=
+    Integrable.of_bound (hcF t).aestronglyMeasurable C (hμ.mono (fun x hx ↦ hC x hx))
+  have hcB : ContinuousOn (fun p : ℝ × Euc n ↦ dilatedVolumeTestDerivative φ p.1 p.2)
+      (Icc (t-1) (t+1) ×ˢ Metric.closedBall 0 M) :=
+    (dilatedVolumeTestDerivative_continuous hφ).continuousOn
+  obtain ⟨B, hB⟩ := ((isCompact_Icc : IsCompact (Icc (t-1) (t+1))).prod
+    (isCompact_closedBall (0:Euc n) M)).exists_bound_of_continuousOn hcB
+  apply hasDerivAt_integral_of_dominated_loc_of_deriv_le
+    (s := Ioo (t-1) (t+1)) (bound := fun _ ↦ B) (Ioo_mem_nhds (by linarith) (by linarith))
+    (Eventually.of_forall fun s ↦ (hcF s).aestronglyMeasurable) hFi
+    (hcD t).aestronglyMeasurable ?_ (integrable_const B) (Eventually.of_forall ?_)
+  · filter_upwards [hμ] with x hx
+    intro s hs
+    exact hB (s,x) ⟨⟨hs.1.le,hs.2.le⟩,hx⟩
+  · intro x s _
+    exact dilatedVolumeTest_hasDerivAt hφ s x
+
+theorem dilatedVolumeTestDerivative_eq_divergence {n : ℕ} (hn : 0 < n)
+    {φ : Euc n → ℝ} (hφ : ContDiff ℝ ∞ φ) (t : ℝ) (x : Euc n) :
+    dilatedVolumeTestDerivative φ t x =
+      t^(n-1) * divergence (fun y ↦ φ (t • y) • y) x := by
+  have hη : Differentiable ℝ (fun y : Euc n ↦ φ (t • y)) :=
+    (hφ.differentiable (by simp)).comp (differentiable_id.const_smul t)
+  have hp := divergence_smul hη (differentiable_id (𝕜 := ℝ) (E := Euc n)) x
+  change divergence (fun y ↦ φ (t • y) • y) x = _ at hp
+  rw [hp, inner_gradient_left]
+  have hid : divergence (id : Euc n → Euc n) x = n := by
+    simp [divergence, fderiv_id]
+  have hc := (((hφ.differentiable (by simp)) (t • x)).hasFDerivAt.comp x
+    ((hasFDerivAt_id x).const_smul t)).fderiv
+  simp only [Function.comp_def] at hc
+  rw [hid]
+  simp only [id_eq]
+  have hDx : fderiv ℝ (fun y ↦ φ (t • y)) x x = t * fderiv ℝ φ (t • x) x := by
+    rw [hc]
+    simp only [ContinuousLinearMap.comp_apply, smul_apply,
+      ContinuousLinearMap.id_apply, map_smul, smul_eq_mul]
+  rw [hDx]
+  unfold dilatedVolumeTestDerivative
+  have hpow : t^n = t^(n-1)*t := by rw [← pow_succ, Nat.sub_add_cancel hn]
+  rw [hpow]
+  ring
+
+theorem minimizing_radial_divergence_eq_zero {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    (hrad : ∀ᵐ x ∂perimeterMeasure E, inner ℝ (approximateNormal E x) x = 0)
+    {φ : Euc n → ℝ} (hφ : ContDiff ℝ ∞ φ) (hcφ : HasCompactSupport φ) :
+    (∫ x in E, divergence (fun y ↦ φ y • y) x) = 0 := by
+  obtain ⟨r, hr⟩ := (Metric.isBounded_iff_subset_ball (0 : Euc n)).mp hcφ.isBounded
+  let R := max r 1
+  have hR : 0 < R := lt_of_lt_of_le zero_lt_one (le_max_right _ _)
+  have hsφ : tsupport φ ⊆ Metric.ball (0:Euc n) R :=
+    hr.trans (Metric.ball_subset_ball (le_max_left _ _))
+  let V : Euc n → Euc n := fun y ↦ φ y • y
+  have hV : ContDiff ℝ ∞ V := hφ.smul contDiff_id
+  have hs : tsupport V ⊆ tsupport φ := tsupport_smul_subset_left _ _
+  have hc : HasCompactSupport V := hcφ.of_isClosed_subset (isClosed_tsupport _) hs
+  rw [minimizing_gaussGreen_local hn hmin (Metric.ball 0 R) Metric.isOpen_ball
+    (hmin.2.1 0 R hR).ne V hV hc (hs.trans hsφ)]
+  apply integral_eq_zero_of_ae
+  filter_upwards [ae_restrict_of_ae hrad] with x hx
+  simp only [V, real_inner_smul_right, hx, mul_zero, Pi.zero_apply]
+
+theorem tsupport_dilation_subset_ball {n : ℕ} {φ : Euc n → ℝ} {R : ℝ}
+    (hs : tsupport φ ⊆ Metric.ball (0:Euc n) R) {t : ℝ} (ht : 0 < t) :
+    tsupport (fun x ↦ φ (t • x)) ⊆ Metric.ball (0:Euc n) (R/t) := by
+  intro x hx
+  have hh := hs (tsupport_comp_subset_preimage φ (continuous_id.const_smul t) hx)
+  change t • x ∈ Metric.ball 0 R at hh
+  simp only [Metric.mem_ball, dist_zero_right, norm_smul, Real.norm_of_nonneg ht.le] at hh ⊢
+  exact (lt_div_iff₀ ht).mpr (by simpa only [mul_comm] using hh)
+
+theorem dilatedVolumeTest_zero_outside {n : ℕ} {φ : Euc n → ℝ} {R M t : ℝ}
+    (hs : tsupport φ ⊆ Metric.ball (0:Euc n) R) (ht : 0 < t) (hM : R/t ≤ M)
+    {x : Euc n} (hx : x ∉ Metric.ball 0 M) :
+    dilatedVolumeTest φ t x = 0 ∧ dilatedVolumeTestDerivative φ t x = 0 := by
+  have htx : t • x ∉ tsupport φ := by
+    intro h
+    apply hx
+    have hn := hs h
+    have hn' : x ∈ Metric.ball (0:Euc n) (R/t) := by
+      simp only [Metric.mem_ball, dist_zero_right, norm_smul, Real.norm_of_nonneg ht.le] at hn ⊢
+      exact (lt_div_iff₀ ht).mpr (by simpa only [mul_comm] using hn)
+    exact Metric.ball_subset_ball hM hn'
+  have hz := image_eq_zero_of_notMem_tsupport htx
+  have hd := fderiv_of_notMem_tsupport ℝ htx
+  simp only [dilatedVolumeTest, dilatedVolumeTestDerivative, hz, hd,
+    zero_apply, mul_zero, add_zero, and_self]
+
+theorem minimizing_dilatedVolumeTestDerivative_integral_eq_zero {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    (hrad : ∀ᵐ x ∂perimeterMeasure E, inner ℝ (approximateNormal E x) x = 0)
+    {φ : Euc n → ℝ} (hφ : ContDiff ℝ ∞ φ) (hcφ : HasCompactSupport φ)
+    {t : ℝ} (ht : 0 < t) :
+    (∫ x in E, dilatedVolumeTestDerivative φ t x) = 0 := by
+  have hφt : ContDiff ℝ ∞ (fun x ↦ φ (t • x)) := hφ.comp (contDiff_id.const_smul t)
+  obtain ⟨r, hr⟩ := (Metric.isBounded_iff_subset_ball (0:Euc n)).mp hcφ.isBounded
+  have hs := tsupport_dilation_subset_ball hr ht
+  have hct : HasCompactSupport (fun x ↦ φ (t • x)) :=
+    (isCompact_closedBall (0:Euc n) (r/t)).of_isClosed_subset (isClosed_tsupport _)
+      (hs.trans Metric.ball_subset_closedBall)
+  simp_rw [dilatedVolumeTestDerivative_eq_divergence (by omega : 0<n) hφ,
+    integral_const_mul]
+  rw [minimizing_radial_divergence_eq_zero hn hmin hrad hφt hct, mul_zero]
+
+theorem minimizing_dilatedVolumeTest_integral_hasDerivAt_zero {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    (hrad : ∀ᵐ x ∂perimeterMeasure E, inner ℝ (approximateNormal E x) x = 0)
+    {φ : Euc n → ℝ} (hφ : ContDiff ℝ ∞ φ) (hcφ : HasCompactSupport φ)
+    {t : ℝ} (ht : 0 < t) :
+    HasDerivAt (fun s ↦ ∫ x in E, dilatedVolumeTest φ s x) 0 t := by
+  obtain ⟨r, hr⟩ := (Metric.isBounded_iff_subset_ball (0:Euc n)).mp hcφ.isBounded
+  let R := max r 1
+  have hR : 0 < R := lt_of_lt_of_le zero_lt_one (le_max_right _ _)
+  have hsφ : tsupport φ ⊆ Metric.ball (0:Euc n) R :=
+    hr.trans (Metric.ball_subset_ball (le_max_left _ _))
+  let M := 2*R/t
+  have hbound (s : ℝ) (hs : t/2 < s) : R/s ≤ M := by
+    have hh := div_le_div_of_nonneg_left hR.le (half_pos ht) hs.le
+    have he : R/(t/2) = M := by dsimp [M]; field_simp
+    exact hh.trans_eq he
+  let μ := (volume.restrict E).restrict (Metric.ball (0:Euc n) M)
+  have : IsFiniteMeasure μ := ⟨by
+    rw [Measure.restrict_apply_univ, Measure.restrict_apply measurableSet_ball]
+    exact lt_of_le_of_lt (measure_mono inter_subset_left) measure_ball_lt_top⟩
+  have hμ : ∀ᵐ x ∂μ, x ∈ Metric.closedBall 0 M :=
+    (ae_restrict_mem measurableSet_ball).mono fun _ hx ↦ Metric.ball_subset_closedBall hx
+  have hd := (dilatedVolumeTest_integral_hasDerivAt hφ M μ hμ t).2
+  have he (s : ℝ) (hs : t/2 < s) :
+      (∫ x, dilatedVolumeTest φ s x ∂μ) = ∫ x in E, dilatedVolumeTest φ s x := by
+    apply setIntegral_eq_integral_of_forall_compl_eq_zero
+    intro x hx
+    exact (dilatedVolumeTest_zero_outside hsφ ((half_pos ht).trans hs) (hbound s hs) hx).1
+  have hdE : (∫ x, dilatedVolumeTestDerivative φ t x ∂μ) =
+      ∫ x in E, dilatedVolumeTestDerivative φ t x := by
+    apply setIntegral_eq_integral_of_forall_compl_eq_zero
+    intro x hx
+    exact (dilatedVolumeTest_zero_outside hsφ ht (hbound t (by linarith)) hx).2
+  rw [hdE, minimizing_dilatedVolumeTestDerivative_integral_eq_zero hn hmin hrad hφ hcφ ht] at hd
+  apply hd.congr_of_eventuallyEq
+  filter_upwards [lt_mem_nhds (show t/2 < t by linarith)] with s hs
+  exact (he s hs).symm
+
+theorem minimizing_volume_test_dilation_invariant {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    (hrad : ∀ᵐ x ∂perimeterMeasure E, inner ℝ (approximateNormal E x) x = 0)
+    {φ : Euc n → ℝ} (hφ : ContDiff ℝ ∞ φ) (hcφ : HasCompactSupport φ)
+    {t : ℝ} (ht : 0 < t) :
+    t^n * (∫ x in E, φ (t • x)) = ∫ x in E, φ x := by
+  have hd (s : ℝ) (hs : s ∈ Ioi (0:ℝ)) :=
+    minimizing_dilatedVolumeTest_integral_hasDerivAt_zero hn hmin hrad hφ hcφ hs
+  have hh := isOpen_Ioi.is_const_of_deriv_eq_zero (convex_Ioi (0:ℝ)).isPreconnected
+    (fun s hs ↦ (hd s hs).differentiableAt.differentiableWithinAt)
+    (fun s hs ↦ (hd s hs).deriv) (show t ∈ Ioi (0:ℝ) from ht)
+      (show (1:ℝ) ∈ Ioi (0:ℝ) by norm_num)
+  simpa only [dilatedVolumeTest, one_pow, one_smul, one_mul, integral_const_mul] using hh
+
+theorem minimizing_dilation_ae_eq {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    (hrad : ∀ᵐ x ∂perimeterMeasure E, inner ℝ (approximateNormal E x) x = 0)
+    {t : ℝ} (ht : 0 < t) :
+    {x : Euc n | t⁻¹ • x ∈ E} =ᵐ[volume] E := by
+  let F : Set (Euc n) := {x | t⁻¹ • x ∈ E}
+  have hF : MeasurableSet F := hmin.1.preimage (continuous_id.const_smul t⁻¹).measurable
+  have he : ∀ᵐ x ∂volume, F.indicator (fun _ ↦ (1:ℝ)) x =
+      E.indicator (fun _ ↦ (1:ℝ)) x := by
+    apply ae_eq_of_integral_contDiff_smul_eq
+      (indicator_one_locallyIntegrable hF) (indicator_one_locallyIntegrable hmin.1)
+    intro φ hφ hcφ
+    have hscale := Measure.integral_comp_smul_of_nonneg volume (F.indicator φ) t (hR := ht.le)
+    have hi (x : Euc n) : (F.indicator φ) (t • x) =
+        E.indicator (fun y ↦ φ (t • y)) x := by
+      have hm : t • x ∈ F ↔ x ∈ E := by simp [F, inv_smul_smul₀ ht.ne']
+      by_cases hx : x ∈ E
+      · simp only [Set.indicator_of_mem hx, Set.indicator_of_mem (hm.mpr hx)]
+      · simp only [Set.indicator_of_notMem hx, Set.indicator_of_notMem (mt hm.mp hx)]
+    simp_rw [hi] at hscale
+    simp only [Euc, finrank_euclideanSpace, Fintype.card_fin, smul_eq_mul,
+      integral_indicator hmin.1, integral_indicator hF] at hscale
+    have htest := minimizing_volume_test_dilation_invariant hn hmin hrad hφ hcφ ht
+    have hfint : (∫ x in F, φ x) = ∫ x in E, φ x := by
+      rw [hscale] at htest
+      simpa only [← mul_assoc, mul_inv_cancel₀ (pow_ne_zero _ ht.ne'), one_mul] using htest
+    have hprod (A : Set (Euc n)) (x : Euc n) :
+        φ x • A.indicator (fun _ ↦ (1:ℝ)) x = A.indicator φ x := by
+      by_cases hx : x ∈ A <;> simp [hx]
+    simp_rw [hprod, integral_indicator hF, integral_indicator hmin.1]
+    exact hfint
+  apply Filter.eventuallyEqSet_iff.mpr
+  filter_upwards [he] with x hx
+  change x ∈ F ↔ x ∈ E
+  by_cases hxF : x ∈ F <;> by_cases hxE : x ∈ E <;> simp_all
+
+theorem minimizing_isCone_phaseSupport_of_radial_tangent {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    (hrad : ∀ᵐ x ∂perimeterMeasure E, inner ℝ (approximateNormal E x) x = 0) :
+    IsCone (volume.restrict E).support := by
+  have hforward {x : Euc n} (hx : x ∈ (volume.restrict E).support)
+      {r : ℝ} (hr : 0 < r) : r • x ∈ (volume.restrict E).support := by
+    let F : Set (Euc n) := {y | r⁻¹ • y ∈ E}
+    have hFE : F =ᵐ[volume] E := minimizing_dilation_ae_eq hn hmin hrad hr
+    apply Metric.nhds_basis_ball.mem_measureSupport.mpr
+    intro R hR
+    rw [Measure.restrict_apply measurableSet_ball]
+    have hp := Metric.nhds_basis_ball.mem_measureSupport.mp hx (R/r) (div_pos hR hr)
+    rw [Measure.restrict_apply measurableSet_ball] at hp
+    have hs : r • (Metric.ball x (R/r) ∩ E) ⊆ Metric.ball (r • x) R ∩ F := by
+      rintro y ⟨z, hz, rfl⟩
+      refine ⟨?_, ?_⟩
+      · change dist (r • z) (r • x) < R
+        rw [dist_smul₀, Real.norm_of_nonneg hr.le]
+        simpa only [mul_comm] using (lt_div_iff₀ hr).mp hz.1
+      · simpa only [F, mem_ofPred_eq, inv_smul_smul₀ hr.ne'] using hz.2
+    have hnull : volume (Metric.ball (r • x) R ∩ F) =
+        volume (Metric.ball (r • x) R ∩ E) := by
+      apply measure_congr
+      apply Filter.eventuallyEqSet_iff.mpr
+      filter_upwards [hFE.mem_iff] with y hy
+      exact and_congr_right (fun _ ↦ hy)
+    rw [← hnull]
+    apply lt_of_lt_of_le _ (measure_mono (μ := volume) hs)
+    rw [Measure.addHaar_smul_of_nonneg volume hr.le]
+    exact ENNReal.mul_pos (ENNReal.ofReal_pos.mpr (pow_pos hr _)).ne' hp.ne'
+  intro r hr x
+  constructor
+  · intro hx
+    have hh := hforward hx (inv_pos.mpr hr)
+    simpa only [inv_smul_smul₀ hr.ne'] using hh
+  · exact fun hx ↦ hforward hx hr
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+open scoped ENNReal Topology
+open scoped ENNReal Topology ContDiff
+open scoped Topology ENNReal
+namespace RPISingularHarnack
+
+theorem minimizing_constant_density_conical_representative {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    {a : ℝ} (ha : 0 ≤ a)
+    (hdensity : ∀ r : ℝ, 0 < r → (perimeterMeasure E).real (Metric.ball 0 r) = a*r^(n-1)) :
+    let K := (volume.restrict E).support
+    IsClosed K ∧ E =ᵐ[volume] K ∧ IsLocallyPerimeterMinimizing K ∧ IsCone K ∧
+      frontier K = perimeterSupport E := by
+  let K := (volume.restrict E).support
+  have hK : IsClosed K := Measure.isClosed_support
+  have hp := Real.HolderConjugate.conjExponent (by exact_mod_cast hn : 1 < (n : ℝ))
+  have hEK : E =ᵐ[volume] K := minimizing_ae_eq_phaseSupport hp hmin
+  have hrad := minimizing_radial_tangent_of_constant_density hn hmin ha hdensity
+  exact ⟨hK, hEK, hmin.of_ae hEK hK.measurableSet,
+    minimizing_isCone_phaseSupport_of_radial_tangent hn hmin hrad,
+    minimizing_frontier_phaseSupport hp hmin⟩
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/EllipticC1Limit.lean b/lean/OAI/Geometry/Perimeter/Regularity/EllipticC1Limit.lean
new file mode 100644
index 0000000000000000000000000000000000000000..ec151b1007a7705fdcbea0864c7ab97e32a9e1e5
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/EllipticC1Limit.lean
@@ -0,0 +1,68 @@
+import OAI.Geometry.Perimeter.Regularity.EllipticContraction
+import OAI.Geometry.Perimeter.Regularity.EllipticCompactGradient
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+ theorem C1_limit_of_uniform_gradients {f : ℕ → Euc n → ℝ} {g : Euc n → ℝ}
+    (hf : ∀ i, ContDiff ℝ 1 (f i)) {p : Euc n} {r : ℝ}
+    {W : Euc n → Euc n} (hW : ContinuousOn W (Metric.closedBall p r))
+    (hgrad : TendstoUniformlyOn (fun i ↦ gradient (f i)) W atTop (Metric.closedBall p r))
+    (hlim : ∀ x∈Metric.ball p r, Tendsto (fun i ↦ f i x) atTop (𝓝 (g x))) :
+    ContDiffOn ℝ 1 g (Metric.ball p r) ∧ ∀ x∈Metric.ball p r, gradient g x=W x := by
+  let D : Euc n → Euc n →L[ℝ] ℝ := fun x ↦ InnerProductSpace.toDual ℝ (Euc n) (W x)
+  have hDlim : TendstoUniformlyOn (fun i ↦ fderiv ℝ (f i)) D atTop (Metric.ball p r) := by
+    rw [Metric.tendstoUniformlyOn_iff]
+    intro ε hε
+    filter_upwards [(Metric.tendstoUniformlyOn_iff.mp hgrad) ε hε] with i hi x hx
+    have hh := hi x (Metric.ball_subset_closedBall hx)
+    change dist (InnerProductSpace.toDual ℝ (Euc n) (W x)) (fderiv ℝ (f i) x)<ε
+    rw [show fderiv ℝ (f i) x=InnerProductSpace.toDual ℝ (Euc n) (gradient (f i) x) from
+      (LinearIsometryEquiv.apply_symm_apply _ _).symm,
+      (InnerProductSpace.toDual ℝ (Euc n)).isometry.dist_eq]
+    exact hh
+  have hD (x : Euc n) (hx : x∈Metric.ball p r) : HasFDerivAt g (D x) x :=
+    hasFDerivAt_of_tendstoUniformlyOn Metric.isOpen_ball hDlim
+      (fun i x _ ↦ (hf i).differentiable (by simp) x |>.hasFDerivAt) hlim hx
+  constructor
+  · change ContDiffOn ℝ (0+1) g (Metric.ball p r)
+    rw [contDiffOn_succ_iff_hasFDerivWithinAt_of_uniqueDiffOn Metric.isOpen_ball.uniqueDiffOn]
+    refine ⟨by simp,D,?_,fun x hx ↦ (hD x hx).hasFDerivWithinAt⟩
+    apply contDiffOn_zero.mpr
+    exact (InnerProductSpace.toDual ℝ (Euc n)).continuous.comp_continuousOn
+      (hW.mono Metric.ball_subset_closedBall)
+  · intro x hx
+    rw [gradient,(hD x hx).fderiv]
+    exact LinearIsometryEquiv.symm_apply_apply _ _
+
+ theorem C1_limit_of_geometric_gradient_estimates {f : ℕ → Euc n → ℝ} {g : Euc n → ℝ}
+    (hf : ∀ i, ContDiff ℝ 1 (f i)) {p : Euc n} {r θ R C B : ℝ}
+    (hθ : 0<θ) (hR : 0<R)
+    (hb : ∀ i x, x∈Metric.closedBall p r → ‖gradient (f i) x‖≤B)
+    (hmod : ∀ i j x, x∈Metric.closedBall p r → ∀ y, y∈Metric.closedBall p r →
+      dist x y≤θ^j*R → ‖gradient (f i) x-gradient (f i) y‖≤C*(1/2)^j)
+    (hlim : ∀ x∈Metric.ball p r, Tendsto (fun i ↦ f i x) atTop (𝓝 (g x))) :
+    ContDiffOn ℝ 1 g (Metric.ball p r) ∧
+      ∀ j x, x∈Metric.ball p r → ∀ y, y∈Metric.ball p r → dist x y≤θ^j*R →
+        ‖gradient g x-gradient g y‖≤C*(1/2)^j := by
+  obtain ⟨W,hW,σ,hσ,hconv⟩ := compact_field_subsequence (isCompact_closedBall p r)
+    (fun i ↦ (gradient_continuous_C1 (hf i)).continuousOn) hθ hR hb hmod
+  obtain ⟨hg,hgW⟩ := C1_limit_of_uniform_gradients (fun i ↦ hf (σ i)) hW hconv
+    (fun x hx ↦ (hlim x hx).comp hσ.tendsto_atTop)
+  refine ⟨hg,fun j x hx y hy hxy ↦ ?_⟩
+  rw [hgW x hx,hgW y hy]
+  have ht := ((hconv.tendsto_at (Metric.ball_subset_closedBall hx)).sub
+    (hconv.tendsto_at (Metric.ball_subset_closedBall hy))).norm
+  exact le_of_tendsto ht (Eventually.of_forall (fun i ↦
+    hmod (σ i) j x (Metric.ball_subset_closedBall hx) y (Metric.ball_subset_closedBall hy) hxy))
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/EllipticC1Parts.lean b/lean/OAI/Geometry/Perimeter/Regularity/EllipticC1Parts.lean
new file mode 100644
index 0000000000000000000000000000000000000000..43519f69b86af7df62a9dd92e25661e41e9e3ab8
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/EllipticC1Parts.lean
@@ -0,0 +1,83 @@
+import OAI.Geometry.Perimeter.Regularity.EllipticCurl
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+theorem scalar_compact_integration_parts_C1 {n : ℕ} {u v : Euc n → ℝ}
+    (hu : ContDiff ℝ 1 u) (hv : ContDiff ℝ 1 v) (hcv : HasCompactSupport v)
+    (a : Euc n) :
+    Integrable (fun x ↦ u x * fderiv ℝ v x a) volume ∧
+    Integrable (fun x ↦ fderiv ℝ u x a * v x) volume ∧
+    (∫ x, u x * fderiv ℝ v x a) = -(∫ x, fderiv ℝ u x a * v x) := by
+  have hdu : Continuous (fun x ↦ fderiv ℝ u x a) :=
+    (hu.continuous_fderiv (by simp)).clm_apply continuous_const
+  have hdv : Continuous (fun x ↦ fderiv ℝ v x a) :=
+    (hv.continuous_fderiv (by simp)).clm_apply continuous_const
+  have hcdv : HasCompactSupport (fun x ↦ fderiv ℝ v x a) :=
+    hcv.fderiv_apply ℝ a
+  have h1 : Integrable (fun x ↦ u x * fderiv ℝ v x a) volume :=
+    (hu.continuous.mul hdv).integrable_of_hasCompactSupport hcdv.mul_left
+  have h2 : Integrable (fun x ↦ fderiv ℝ u x a * v x) volume :=
+    (hdu.mul hv.continuous).integrable_of_hasCompactSupport hcv.mul_left
+  refine ⟨h1, h2, ?_⟩
+  exact integral_mul_fderiv_eq_neg_fderiv_mul_of_integrable h2 h1
+    ((hu.continuous.mul hv.continuous).integrable_of_hasCompactSupport hcv.mul_left)
+    (fun x _ ↦ hu.differentiable (by simp) x)
+    (fun x _ ↦ hv.differentiable (by simp) x)
+
+theorem integral_mul_divergence_eq_neg_inner_gradient_C1 {n : ℕ} {u : Euc n → ℝ}
+    {V : Euc n → Euc n} (hu : ContDiff ℝ 1 u)
+    (hV : ContDiff ℝ 1 V) (hcV : HasCompactSupport V) :
+    (∫ x, u x * divergence V x) = - ∫ x, inner ℝ (gradient u x) (V x) := by
+  let b := EuclideanSpace.basisFun (Fin n) ℝ
+  let v : Fin n → Euc n → ℝ := fun i x ↦ inner ℝ (b i) (V x)
+  have hv : ∀ i, ContDiff ℝ 1 (v i) := fun i ↦ (innerSL ℝ (b i)).contDiff.comp hV
+  have hcv : ∀ i, HasCompactSupport (v i) := fun i ↦ hcV.comp_left (by simp)
+  have hdv : ∀ i x, fderiv ℝ (v i) x (b i) =
+      inner ℝ (b i) (fderiv ℝ V x (b i)) := by
+    intro i x
+    have h := ((innerSL ℝ (b i)).hasFDerivAt.comp x
+      (hV.differentiable (by simp) x).hasFDerivAt).fderiv
+    exact congrArg (fun L : Euc n →L[ℝ] ℝ ↦ L (b i)) h
+  have hp := fun i ↦ scalar_compact_integration_parts_C1 hu (hv i) (hcv i) (b i)
+  have hInt1 := fun i ↦ (hp i).2.1
+  have hInt2 := fun i ↦ (hp i).1
+  have hparts := fun i ↦ (hp i).2.2
+  calc
+    (∫ x, u x * divergence V x) = ∫ x, ∑ i, u x * fderiv ℝ (v i) x (b i) := by
+      congr 1
+      funext x
+      simp only [divergence, hdv, Finset.mul_sum, b]
+    _ = ∑ i, ∫ x, u x * fderiv ℝ (v i) x (b i) := integral_finsetSum _ (fun i _ ↦ hInt2 i)
+    _ = -(∑ i, ∫ x, fderiv ℝ u x (b i) * v i x) := by simp only [hparts, Finset.sum_neg_distrib]
+    _ = -(∫ x, ∑ i, fderiv ℝ u x (b i) * v i x) := by
+      rw [integral_finsetSum _ (fun i _ ↦ hInt1 i)]
+    _ = -(∫ x, inner ℝ (gradient u x) (V x)) := by
+      congr 1
+      apply integral_congr_ae
+      apply ae_of_all
+      intro x
+      simp only [← inner_gradient_left, v]
+      exact b.sum_inner_mul_inner _ _
+
+theorem gradient_continuous_C1 {n : ℕ} {f : Euc n → ℝ} (hf : ContDiff ℝ 1 f) :
+    Continuous (gradient f) :=
+  (InnerProductSpace.toDual ℝ (Euc n)).symm.continuous.comp (hf.continuous_fderiv (by norm_num))
+
+theorem C1_gradient_curl_pairing_zero {n : ℕ} {f : Euc n → ℝ} (hf : ContDiff ℝ 1 f)
+    {U : Set (Euc n)} (φ : SmoothLocalTest U) (a b : Euc n) :
+    (∫ x, inner ℝ (gradient f x) (curlTest φ.val a b x))=0 := by
+  have hi := integral_mul_divergence_eq_neg_inner_gradient_C1 hf
+    ((curlTest_contDiff φ.property.1 a b).of_le (by simp))
+    (φ.property.2.1.of_isClosed_subset (isClosed_tsupport _) (curlTest_tsupport _ _ _))
+  rw [curlTest_divergence φ.property.1] at hi
+  simpa only [Pi.zero_apply,mul_zero,integral_zero,eq_neg_iff_add_eq_zero,zero_add] using hi
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/EllipticC1Test.lean b/lean/OAI/Geometry/Perimeter/Regularity/EllipticC1Test.lean
new file mode 100644
index 0000000000000000000000000000000000000000..f8b4df836312d72bdf533324151b14eccf0e8118
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/EllipticC1Test.lean
@@ -0,0 +1,93 @@
+import OAI.Geometry.Perimeter.Regularity.EllipticLinearStep
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+ theorem gradient_tsupport_subset (f : Euc n → ℝ) : tsupport (gradient f) ⊆ tsupport f := by
+  apply closure_minimal _ (isClosed_tsupport _)
+  intro x hx
+  by_contra h
+  have hd := fderiv_of_notMem_tsupport ℝ h
+  have hg : gradient f x=0 := by simp only [gradient,hd,map_zero]
+  exact hx hg
+
+ theorem gradient_norm_eq_fderiv (f : Euc n → ℝ) (x : Euc n) : ‖gradient f x‖=‖fderiv ℝ f x‖ :=
+  (InnerProductSpace.toDual ℝ (Euc n)).symm.norm_map _
+
+ theorem integral_inner_gradient_eq_setIntegral {G : Euc n → Euc n} {φ : Euc n → ℝ}
+    {U : Set (Euc n)} (hU : MeasurableSet U) (hs : tsupport φ ⊆ U) :
+    (∫ x, inner ℝ (G x) (gradient φ x)) = ∫ x in U, inner ℝ (G x) (gradient φ x) := by
+  rw [←integral_indicator hU]
+  congr 1; funext x
+  by_cases hx : x∈U
+  · simp [hx]
+  · have hg : gradient φ x=0 := image_eq_zero_of_notMem_tsupport (fun h ↦ hx (hs (gradient_tsupport_subset φ h)))
+    simp [hx,hg]
+
+ theorem mollified_gradient_pairing_tendsto {G : Euc n → Euc n} (hG : Continuous G)
+    (p : Euc n) (r : ℝ)
+    {φ : Euc n → ℝ} (hφ : ContDiff ℝ 1 φ) (hcφ : HasCompactSupport φ)
+    : Tendsto (fun j ↦ ∫ x in Metric.ball p r, inner ℝ (G x) (gradient (RPI.Boundary.c1Mollify φ j) x)) atTop
+      (𝓝 (∫ x in Metric.ball p r, inner ℝ (G x) (gradient φ x))) := by
+  let φj := RPI.Boundary.c1Mollify φ
+  obtain ⟨M,hM,hb⟩ := RPI.Boundary.c1Mollify_uniform_bound hφ hcφ
+  obtain ⟨C,hC⟩ := (isCompact_closedBall p r).exists_bound_of_continuousOn hG.continuousOn
+  have : IsFiniteMeasure (volume.restrict (Metric.ball p r)) := ⟨by
+    rw [Measure.restrict_apply_univ]; exact measure_ball_lt_top⟩
+  apply tendsto_integral_of_dominated_convergence (fun _ ↦ |C| * M)
+  · intro j
+    exact (hG.inner (gradient_continuous_C1 ((RPI.Boundary.c1Mollify_contDiff hφ.continuous j).of_le (by simp)))).aestronglyMeasurable
+  · exact integrable_const _
+  · intro j
+    filter_upwards [ae_restrict_mem measurableSet_ball] with x hx
+    calc
+      ‖inner ℝ (G x) (gradient (φj j) x)‖ ≤ ‖G x‖*‖gradient (φj j) x‖ := norm_inner_le_norm _ _
+      _ ≤ |C| * M := mul_le_mul ((hC x (Metric.ball_subset_closedBall hx)).trans (le_abs_self C))
+        (by rw [gradient_norm_eq_fderiv]; exact (hb j x).2) (norm_nonneg _) (abs_nonneg _)
+  · apply ae_of_all
+    intro x
+    have hg : Tendsto (fun j ↦ gradient (φj j) x) atTop (𝓝 (gradient φ x)) := by
+      exact (InnerProductSpace.toDual ℝ (Euc n)).symm.continuous.tendsto (fderiv ℝ φ x) |>.comp
+        (RPI.Boundary.c1Mollify_fderiv_tendsto hφ hcφ x)
+    exact (tendsto_const_nhds : Tendsto (fun _ : ℕ ↦ G x) atTop (𝓝 (G x))).inner hg
+
+ theorem continuous_div_free_C1_test {G : Euc n → Euc n} (hG : Continuous G)
+    (p : Euc n) (r : ℝ)
+    (h : ∀ φ : SmoothLocalTest (Metric.ball p r), (∫ x, inner ℝ (G x) (gradient φ.val x))=0)
+    {φ : Euc n → ℝ} (hφ : ContDiff ℝ 1 φ) (hcφ : HasCompactSupport φ)
+    (hsφ : tsupport φ ⊆ Metric.ball p r) :
+    (∫ x, inner ℝ (G x) (gradient φ x))=0 := by
+  let φj := RPI.Boundary.c1Mollify φ
+  have hs := RPI.Boundary.c1Mollify_eventually_tsupport_subset hcφ Metric.isOpen_ball hsφ
+  have hlim := mollified_gradient_pairing_tendsto hG p r hφ hcφ
+  have hz : ∀ᶠ j in atTop, (∫ x in Metric.ball p r, inner ℝ (G x) (gradient (φj j) x))=0 := by
+    filter_upwards [hs] with j hj
+    rw [←integral_inner_gradient_eq_setIntegral measurableSet_ball hj]
+    exact h ⟨φj j,RPI.Boundary.c1Mollify_contDiff hφ.continuous j,
+      RPI.Boundary.c1Mollify_hasCompactSupport hcφ j,hj⟩
+  rw [integral_inner_gradient_eq_setIntegral measurableSet_ball hsφ]
+  exact tendsto_nhds_unique hlim (tendsto_const_nhds.congr' (hz.mono (fun _ hj ↦ hj.symm)))
+
+ theorem WeakLinearEquation.test_C1 {A : Euc n → Euc n →L[ℝ] Euc n}
+    {f : Euc n → ℝ} {F : Euc n → Euc n} (hA : Continuous A) (hf : ContDiff ℝ 1 f)
+    (hF : Continuous F) {p : Euc n} {r : ℝ} (h : WeakLinearEquation A f F (Metric.ball p r))
+    {φ : Euc n → ℝ} (hφ : ContDiff ℝ 1 φ) (hcφ : HasCompactSupport φ)
+    (hsφ : tsupport φ ⊆ Metric.ball p r) :
+    (∫ x, inner ℝ (A x (gradient f x)-F x) (gradient φ x))=0 := by
+  apply continuous_div_free_C1_test ((hA.clm_apply (gradient_continuous_C1 hf)).sub hF) p r _ hφ hcφ hsφ
+  intro ψ
+  change (∫ x, inner ℝ (A x (gradient f x)-F x) (gradient ψ.val x))=0
+  simp_rw [inner_sub_left]
+  erw [integral_sub (inner_gradient_pairing_integrable (hA.clm_apply (gradient_continuous_C1 hf)) ψ)
+    (inner_gradient_pairing_integrable hF ψ),h ψ,sub_self]
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/EllipticCaccioppoli.lean b/lean/OAI/Geometry/Perimeter/Regularity/EllipticCaccioppoli.lean
new file mode 100644
index 0000000000000000000000000000000000000000..b154c015df702e43be4865c06b5f4eb05a07899f
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/EllipticCaccioppoli.lean
@@ -0,0 +1,88 @@
+import OAI.Geometry.Perimeter.Regularity.EllipticC1Test
+import OAI.Geometry.Perimeter.Regularity.EllipticCaccioppoliAlgebra
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+ theorem continuous_inner_gradient_integrable {G : Euc n → Euc n} {f : Euc n → ℝ}
+    (hG : Continuous G) (hf : ContDiff ℝ 1 f) (hcf : HasCompactSupport f) :
+    Integrable (fun x ↦ inner ℝ (G x) (gradient f x)) volume := by
+  apply (hG.inner (gradient_continuous_C1 hf)).integrable_of_hasCompactSupport
+  apply (gradient_hasCompactSupport hcf).of_isClosed_subset (isClosed_tsupport _)
+  apply closure_mono
+  intro x hx
+  exact fun hz ↦ hx (by simp [hz])
+
+ theorem square_weight_integrable {u : Euc n → ℝ} {G : Euc n → Euc n}
+    (hu : Continuous u) (hcu : HasCompactSupport u) (hG : Continuous G) :
+    Integrable (fun x ↦ (u x)^2*‖G x‖^2) volume := by
+  apply ((hu.pow 2).mul (hG.norm.pow 2)).integrable_of_hasCompactSupport
+  apply hcu.of_isClosed_subset (isClosed_tsupport _)
+  apply closure_mono
+  intro x hx
+  exact fun hz ↦ hx (by simp [hz])
+
+ theorem square_compact_field_integrable {u : Euc n → ℝ} {G : Euc n → Euc n}
+    (hu : Continuous u) (hG : Continuous G) (hcG : HasCompactSupport G) :
+    Integrable (fun x ↦ (u x)^2*‖G x‖^2) volume := by
+  apply ((hu.pow 2).mul (hG.norm.pow 2)).integrable_of_hasCompactSupport
+  apply hcG.of_isClosed_subset (isClosed_tsupport _)
+  apply closure_mono
+  intro x hx
+  exact fun hz ↦ hx (by simp [hz])
+
+ theorem caccioppoli_test_bound {A : Euc n → Euc n →L[ℝ] Euc n}
+    {f : Euc n → ℝ} {F : Euc n → Euc n} (hf : ContDiff ℝ 1 f)
+    {p : Euc n} {r : ℝ}
+    (hnear : ∀ x∈Metric.ball p r, ‖ContinuousLinearMap.id ℝ (Euc n)-A x‖≤1/2)
+    {χ : Euc n → ℝ} (hχ : ContDiff ℝ 1 χ)
+    (hsχ : tsupport χ ⊆ Metric.ball p r) (x : Euc n) : (χ x)^2*‖gradient f x‖^2 ≤
+    4*inner ℝ (A x (gradient f x)-F x) (gradient (fun y ↦ χ y*χ y*f y) x)+256*((f x)^2*‖gradient χ x‖^2+(χ x)^2*‖F x‖^2) := by
+  by_cases hx : x∈tsupport χ
+  · rw [gradient_square_test (hf.differentiable (by simp)) (hχ.differentiable (by simp))]
+    exact caccioppoli_pointwise (A x) (hnear x (hsχ hx)) (gradient f x) (gradient χ x) (F x) (χ x) (f x)
+  · have hχ0 := image_eq_zero_of_notMem_tsupport hx
+    have hD0 : gradient χ x=0 := image_eq_zero_of_notMem_tsupport (fun hh ↦ hx (gradient_tsupport_subset χ hh))
+    rw [gradient_square_test (hf.differentiable (by simp)) (hχ.differentiable (by simp)),hχ0,hD0]
+    simp
+
+ theorem WeakLinearEquation.caccioppoli {A : Euc n → Euc n →L[ℝ] Euc n}
+    {f : Euc n → ℝ} {F : Euc n → Euc n} (hA : Continuous A) (hf : ContDiff ℝ 1 f)
+    (hF : Continuous F) {p : Euc n} {r : ℝ} (h : WeakLinearEquation A f F (Metric.ball p r))
+    (hnear : ∀ x∈Metric.ball p r, ‖ContinuousLinearMap.id ℝ (Euc n)-A x‖≤1/2)
+    {χ : Euc n → ℝ} (hχ : ContDiff ℝ 1 χ) (hcχ : HasCompactSupport χ)
+    (hsχ : tsupport χ ⊆ Metric.ball p r) :
+    (∫ x, (χ x)^2*‖gradient f x‖^2) ≤
+      256*((∫ x, (f x)^2*‖gradient χ x‖^2)+(∫ x, (χ x)^2*‖F x‖^2)) := by
+  let φ := fun x ↦ χ x*χ x*f x
+  have hφ : ContDiff ℝ 1 φ := hχ.mul hχ |>.mul hf
+  have hcφ : HasCompactSupport φ := hcχ.mul_right.mul_right
+  have hsφ : tsupport φ ⊆ Metric.ball p r :=
+    tsupport_mul_subset_left.trans (tsupport_mul_subset_left.trans hsχ)
+  have hz := h.test_C1 hA hf hF hφ hcφ hsφ
+  let J := fun x ↦ inner ℝ (A x (gradient f x)-F x) (gradient φ x)
+  have hJ : Integrable J volume := continuous_inner_gradient_integrable
+    ((hA.clm_apply (gradient_continuous_C1 hf)).sub hF) hφ hcφ
+  have hI := square_weight_integrable hχ.continuous hcχ (gradient_continuous_C1 hf)
+  have hI₁ := square_compact_field_integrable hf.continuous (gradient_continuous_C1 hχ)
+    (gradient_hasCompactSupport hcχ)
+  have hI₂ := square_weight_integrable hχ.continuous hcχ hF
+  have hpt := caccioppoli_test_bound (F := F) hf hnear hχ hsχ
+  have hh := integral_mono hI ((hJ.const_mul 4).add ((hI₁.add hI₂).const_mul 256)) hpt
+  simp only [Pi.add_apply] at hh
+  erw [integral_add (hJ.const_mul 4) ((hI₁.add hI₂).const_mul 256),integral_const_mul,
+    integral_const_mul,integral_add hI₁ hI₂] at hh
+  change (∫ x,J x)=0 at hz
+  rw [hz,mul_zero,zero_add] at hh
+  exact hh
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/EllipticCaccioppoliAlgebra.lean b/lean/OAI/Geometry/Perimeter/Regularity/EllipticCaccioppoliAlgebra.lean
new file mode 100644
index 0000000000000000000000000000000000000000..71936033e18e42e952abe52ebe4a93272b924b57
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/EllipticCaccioppoliAlgebra.lean
@@ -0,0 +1,83 @@
+import OAI.Geometry.Perimeter.Regularity.ContinuousWithinAtSurfaceGradientOfLocalLevel
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+ theorem near_identity_bounds (A : Euc n →L[ℝ] Euc n)
+    (hA : ‖ContinuousLinearMap.id ℝ (Euc n)-A‖≤1/2) (ξ : Euc n) :
+    ‖A ξ‖≤2*‖ξ‖ ∧ ‖ξ‖^2/2≤ inner ℝ (A ξ) ξ := by
+  have he := (ContinuousLinearMap.id ℝ (Euc n)-A).le_opNorm ξ
+  simp only [sub_apply,ContinuousLinearMap.id_apply] at he
+  have he' : ‖ξ-A ξ‖≤(1/2)*‖ξ‖ := he.trans (mul_le_mul_of_nonneg_right hA (norm_nonneg ξ))
+  have hn := norm_sub_le ξ (ξ-A ξ)
+  simp only [sub_sub_cancel] at hn
+  have hi := abs_real_inner_le_norm (ξ-A ξ) ξ
+  rw [inner_sub_left,real_inner_self_eq_norm_sq] at hi
+  have hi' := (le_abs_self (‖ξ‖^2-inner ℝ (A ξ) ξ)).trans hi
+  have he'' := mul_le_mul_of_nonneg_right he' (norm_nonneg ξ)
+  constructor <;> nlinarith [norm_nonneg ξ]
+
+ theorem caccioppoli_scalar {X Y Z J : ℝ}
+    (hJ : X^2/2-4*X*Y-X*Z-2*Y*Z≤J) :
+    X^2≤4*J+256*(Y^2+Z^2) := by
+  nlinarith [sq_nonneg (X-16*Y),sq_nonneg (X-4*Z),sq_nonneg (Y-Z),sq_nonneg Y,sq_nonneg Z]
+
+ theorem caccioppoli_pointwise (A : Euc n →L[ℝ] Euc n)
+    (hA : ‖ContinuousLinearMap.id ℝ (Euc n)-A‖≤1/2) (ξ η ζ : Euc n) (c u : ℝ) :
+    c^2*‖ξ‖^2 ≤ 4*inner ℝ (A ξ-ζ) ((c^2)•ξ+(2*c*u)•η)+
+      256*(u^2*‖η‖^2+c^2*‖ζ‖^2) := by
+  obtain ⟨hAn,hAi⟩ := near_identity_bounds A hA ξ
+  have h1 : |(2*c*u)*inner ℝ (A ξ) η|≤4*(|c| * ‖ξ‖)*(|u| * ‖η‖) := by
+    rw [abs_mul]
+    have hh := mul_le_mul_of_nonneg_left (abs_real_inner_le_norm (A ξ) η) (abs_nonneg (2*c*u))
+    have hh' := mul_le_mul_of_nonneg_left (mul_le_mul_of_nonneg_right hAn (norm_nonneg η)) (abs_nonneg (2*c*u))
+    simp only [abs_mul,abs_of_nonneg (by norm_num : (0:ℝ)≤2)] at hh hh' ⊢
+    convert hh.trans hh' using 1
+    ring
+  have h2 : |c^2*inner ℝ ζ ξ|≤(|c| * ‖ξ‖)*(|c| * ‖ζ‖) := by
+    rw [abs_mul,abs_of_nonneg (sq_nonneg c)]
+    have hh := mul_le_mul_of_nonneg_left (abs_real_inner_le_norm ζ ξ) (sq_nonneg c)
+    convert hh using 1
+    rw [←sq_abs c]
+    ring
+  have h3 : |(2*c*u)*inner ℝ ζ η|≤2*(|u| * ‖η‖)*(|c| * ‖ζ‖) := by
+    rw [abs_mul]
+    have hh := mul_le_mul_of_nonneg_left (abs_real_inner_le_norm ζ η) (abs_nonneg (2*c*u))
+    simp only [abs_mul,abs_of_nonneg (by norm_num : (0:ℝ)≤2)] at hh ⊢
+    convert hh using 1
+    ring
+  have hbase := mul_le_mul_of_nonneg_left hAi (sq_nonneg c)
+  have hlo : (|c| * ‖ξ‖)^2/2-4*(|c| * ‖ξ‖)*(|u| * ‖η‖)-
+      (|c| * ‖ξ‖)*(|c| * ‖ζ‖)-2*(|u| * ‖η‖)*(|c| * ‖ζ‖) ≤
+      inner ℝ (A ξ-ζ) ((c^2)•ξ+(2*c*u)•η) := by
+    simp only [inner_add_right,inner_sub_left,real_inner_smul_right]
+    have hh1 := neg_abs_le ((2*c*u)*inner ℝ (A ξ) η)
+    have hh2 := le_abs_self (c^2*inner ℝ ζ ξ)
+    have hh3 := le_abs_self ((2*c*u)*inner ℝ ζ η)
+    simp only [mul_pow,sq_abs]
+    nlinarith
+  have hh := caccioppoli_scalar hlo
+  simpa only [mul_pow,sq_abs] using hh
+
+ theorem gradient_mul_C1 {f g : Euc n → ℝ} (hf : Differentiable ℝ f) (hg : Differentiable ℝ g) (x : Euc n) :
+    gradient (fun y ↦ f y*g y) x=f x•gradient g x+g x•gradient f x := by
+  apply (InnerProductSpace.toDual ℝ (Euc n)).injective
+  simp only [gradient,map_add,map_smul,LinearIsometryEquiv.apply_symm_apply]
+  exact fderiv_mul (hf x) (hg x)
+
+ theorem gradient_square_test {f χ : Euc n → ℝ} (hf : Differentiable ℝ f) (hχ : Differentiable ℝ χ)
+    (x : Euc n) : gradient (fun y ↦ χ y*χ y*f y) x=(χ x)^2•gradient f x+(2*χ x*f x)•gradient χ x := by
+  rw [gradient_mul_C1 (f := fun y ↦ χ y*χ y) (g := f) (hχ.mul hχ) hf,gradient_mul_C1 hχ hχ]
+  simp only [smul_add,smul_smul,pow_two]
+  module
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/EllipticCompactGradient.lean b/lean/OAI/Geometry/Perimeter/Regularity/EllipticCompactGradient.lean
new file mode 100644
index 0000000000000000000000000000000000000000..3668cd346836db8583ccf0a5a0fabe519514779b
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/EllipticCompactGradient.lean
@@ -0,0 +1,70 @@
+import OAI.Geometry.Perimeter.Regularity.ContinuousWithinAtSurfaceGradientOfLocalLevel
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace BoundedContinuousFunction
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+ theorem geometric_equicontinuous {X V : Type*} [PseudoMetricSpace X] [PseudoMetricSpace V]
+    {G : ℕ → X → V} {θ R C : ℝ} (hθ : 0<θ) (hR : 0<R)
+    (hmod : ∀ i j x y, dist x y≤θ^j*R → dist (G i x) (G i y)≤C*(1/2)^j) :
+    Equicontinuous G := by
+  intro x
+  rw [Metric.equicontinuousAt_iff]
+  intro ε hε
+  have ht : Tendsto (fun j : ℕ ↦ C*(1/2:ℝ)^j) atTop (𝓝 0) := by
+    simpa only [mul_zero] using
+      (tendsto_pow_atTop_nhds_zero_of_lt_one (by norm_num : (0:ℝ)≤1/2) (by norm_num)).const_mul C
+  obtain ⟨j,hj⟩ := (ht.eventually (gt_mem_nhds hε)).exists
+  refine ⟨θ^j*R,by positivity,fun y hy i ↦ ?_⟩
+  exact (hmod i j x y (by simpa only [dist_comm] using hy.le)).trans_lt hj
+
+ theorem compact_field_subsequence {V : Type*} [NormedAddCommGroup V] [ProperSpace V]
+    {K : Set (Euc n)} (hK : IsCompact K) {G : ℕ → Euc n → V}
+    (hG : ∀ i, ContinuousOn (G i) K) {θ R C B : ℝ} (hθ : 0<θ) (hR : 0<R)
+    (hb : ∀ i x, x∈K → ‖G i x‖≤B)
+    (hmod : ∀ i j x, x∈K → ∀ y, y∈K → dist x y≤θ^j*R → ‖G i x-G i y‖≤C*(1/2)^j) :
+    ∃ W : Euc n → V, ContinuousOn W K ∧ ∃ σ : ℕ → ℕ, StrictMono σ ∧
+      TendstoUniformlyOn (fun i ↦ G (σ i)) W atTop K := by
+  classical
+  let : CompactSpace K := isCompact_iff_compactSpace.mp hK
+  let F : ℕ → K →ᵇ V := fun i ↦ BoundedContinuousFunction.mkOfCompact
+    ⟨fun x ↦ G i x,continuousOn_iff_continuous_domRestrict.mp (hG i)⟩
+  have hFeq (i : ℕ) (x : K) : F i x=G i x := rfl
+  have heq : Equicontinuous (fun i (x : K) ↦ F i x) := by
+    apply geometric_equicontinuous (C := C) hθ hR
+    intro i j x y hxy
+    simpa only [hFeq,dist_eq_norm] using hmod i j x x.property y y.property hxy
+  have hc : IsCompact (closure (range F)) := by
+    apply BoundedContinuousFunction.arzela_ascoli (Metric.closedBall (0:V) B) (isCompact_closedBall _ _)
+    · rintro g x ⟨i,rfl⟩
+      exact Metric.mem_closedBall.mpr (by simpa only [dist_zero_right,hFeq] using hb i x x.property)
+    · intro x
+      rw [Metric.equicontinuousAt_iff]
+      intro ε hε
+      obtain ⟨δ,hδ,hh⟩ := Metric.equicontinuousAt_iff.mp (heq x) ε hε
+      refine ⟨δ,hδ,fun y hy g ↦ ?_⟩
+      obtain ⟨i,hi⟩ := g.property
+      simpa only [←hi] using hh y hy i
+  obtain ⟨W,hW,σ,hσ,hlim⟩ := hc.tendsto_subseq (fun i ↦ subset_closure (mem_range_self i))
+  let W' : Euc n → V := fun x ↦ if hx : x∈K then W ⟨x,hx⟩ else 0
+  have hW'eq (x : K) : W' x=W x := by simp [W',x.property]
+  have hW' : ContinuousOn W' K := by
+    rw [continuousOn_iff_continuous_domRestrict]
+    exact W.continuous.congr (fun x ↦ (hW'eq x).symm)
+  refine ⟨W',hW',σ,hσ,?_⟩
+  have hu := BoundedContinuousFunction.tendsto_iff_tendstoUniformly.mp hlim
+  rw [Metric.tendstoUniformlyOn_iff]
+  intro ε hε
+  filter_upwards [(Metric.tendstoUniformly_iff.mp hu) ε hε] with i hi x hx
+  have hh : dist (W ⟨x,hx⟩) (G (σ i) x)<ε := hi ⟨x,hx⟩
+  simpa only [←hW'eq ⟨x,hx⟩] using hh
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/EllipticContinuousComparison.lean b/lean/OAI/Geometry/Perimeter/Regularity/EllipticContinuousComparison.lean
new file mode 100644
index 0000000000000000000000000000000000000000..090d1e4d3eb96add9bbc5148ca0802fc648a7b4c
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/EllipticContinuousComparison.lean
@@ -0,0 +1,79 @@
+import OAI.Geometry.Perimeter.Regularity.EllipticCutField
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+def gradientEnergy (f : Euc n → ℝ) (p : Euc n) (r : ℝ) (q : Euc n) : ℝ :=
+  ∫ x in Metric.ball p r, ‖gradient f x-q‖^2
+
+theorem constant_gradient_pairing_zero {U : Set (Euc n)} (φ : SmoothLocalTest U) (q : Euc n) :
+    (∫ x, inner ℝ q (gradient φ.val x))=0 := by
+  have h := (scalar_compact_integration_parts_C1 (u := fun _ : Euc n ↦ (1:ℝ))
+    contDiff_const (φ.property.1.of_le (by simp)) φ.property.2.1 q).2.2
+  simpa [inner_gradient_right] using h
+
+theorem inner_gradient_pairing_integrable {W : Euc n → Euc n} (hW : Continuous W)
+    {U : Set (Euc n)} (φ : SmoothLocalTest U) :
+    Integrable (fun x ↦ inner ℝ (W x) (gradient φ.val x)) volume :=
+  (hW.inner (gradient_contDiff φ.property.1).continuous).integrable_of_hasCompactSupport
+    ((gradient_hasCompactSupport φ.property.2.1).of_isClosed_subset (isClosed_tsupport _) (by
+      apply closure_minimal _ (isClosed_tsupport _)
+      intro x hx
+      by_contra hh
+      exact hx (by simp [image_eq_zero_of_notMem_tsupport hh])))
+
+theorem gradient_sub_linear {f : Euc n → ℝ} (hf : Differentiable ℝ f) (q : Euc n) (x : Euc n) :
+    gradient (fun y ↦ f y-inner ℝ q y) x=gradient f x-q := by
+  apply ext_inner_right ℝ
+  intro y
+  simp only [inner_gradient_left,inner_sub_left]
+  have h := ((hf x).hasFDerivAt.sub (innerSL ℝ q).hasFDerivAt).fderiv
+  exact congrArg (fun L : Euc n →L[ℝ] ℝ ↦ L y) h
+
+theorem continuous_gradient_one_scale (hn : 0<n) :
+    ∃ C : ℝ, 0≤C ∧ ∀ (f : Euc n → ℝ), ContDiff ℝ 1 f →
+      ∀ (e : Euc n → Euc n), Continuous e →
+      (∀ φ : SmoothLocalTest (Metric.ball 0 1),
+        (∫ x, inner ℝ (gradient f x) (gradient φ.val x))=∫ x, inner ℝ (e x) (gradient φ.val x)) →
+      ∃ q : Euc n, ‖q‖^2≤C^2*gradientEnergy f 0 1 0 ∧ ∀ θ : ℝ, θ≤1/2 →
+        gradientEnergy f 0 θ q ≤ 2*(∫ x in Metric.ball 0 1, ‖e x‖^2)+
+          2*C^2*gradientEnergy f 0 1 0*θ^2*volume.real (Metric.ball (0:Euc n) θ) := by
+  obtain ⟨C,hC,hB⟩ := weak_field_one_scale hn
+  refine ⟨C,hC,fun f hf e he h ↦ ?_⟩
+  let u := cutFieldLp (gradient f) (gradient_continuous_C1 hf) 0 1
+  let w := cutFieldLp e he 0 1
+  have heq : ∀ φ : SmoothLocalTest (Metric.ball (0:Euc n) 1),
+      (∫ x, inner ℝ (u x) (gradient φ.val x))=∫ x, inner ℝ (w x) (gradient φ.val x) := by
+    intro φ
+    rw [cutFieldLp_pairing _ _ _ _ ((tsupport_gradient_subset _).trans φ.property.2.2),
+      cutFieldLp_pairing _ _ _ _ ((tsupport_gradient_subset _).trans φ.property.2.2)]
+    exact h φ
+  obtain ⟨q,hq,hqE⟩ := hB u w (cutGradient_curlFree hf 0 1) heq
+  have huE : ‖u‖^2=gradientEnergy f 0 1 0 := by
+    simpa only [gradientEnergy,sub_zero] using cutFieldLp_norm_sq (gradient f) (gradient_continuous_C1 hf) 0 1
+  refine ⟨q,?_,fun θ hθ ↦ ?_⟩
+  · calc
+      ‖q‖^2 ≤ (C*‖u‖)^2 := pow_le_pow_left₀ (norm_nonneg _) hq 2
+      _ = C^2*gradientEnergy f 0 1 0 := by rw [mul_pow,huE]
+  · have hE := hqE θ hθ
+    have heU : (∫ x in Metric.ball 0 θ, ‖u x-q‖^2)=gradientEnergy f 0 θ q := by
+      apply integral_congr_ae
+      filter_upwards [ae_restrict_of_ae (cutFieldLp_ae (gradient f) (gradient_continuous_C1 hf) 0 1),
+        ae_restrict_mem measurableSet_ball] with x hx hxB
+      have hx1 : x∈Metric.ball (0:Euc n) 1 := Metric.ball_subset_ball (by linarith) hxB
+      rw [hx,Set.indicator_of_mem hx1]
+    rw [heU,cutFieldLp_norm_sq] at hE
+    convert hE using 1
+    rw [mul_pow,mul_pow,huE]
+    ring
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/EllipticContraction.lean b/lean/OAI/Geometry/Perimeter/Regularity/EllipticContraction.lean
new file mode 100644
index 0000000000000000000000000000000000000000..38fca31ffff2608877f905cc4e504bd391ae1f18
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/EllipticContraction.lean
@@ -0,0 +1,70 @@
+import OAI.Geometry.Perimeter.Regularity.EllipticScaleStep
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+def LinearContractionProperty (n : ℕ) (θ δ K : ℝ) : Prop :=
+  ∀ (A : Euc n → Euc n →L[ℝ] Euc n), Continuous A →
+  ∀ (f : Euc n → ℝ), ContDiff ℝ 1 f → ∀ F : Euc n → Euc n, Continuous F →
+  ∀ (p : Euc n) (r : ℝ), 0 < r → WeakLinearEquation A f F (Metric.ball p r) →
+  ∀ (A₀ : Euc n →L[ℝ] Euc n) (F₀ q : Euc n) (α β : ℝ), 0 ≤ α → 0 ≤ β →
+    (∀ x∈Metric.ball p r, ‖ContinuousLinearMap.id ℝ (Euc n)-A x‖ ≤ δ ∧
+      ‖A₀-A x‖ ≤ α ∧ ‖F x-F₀‖ ≤ β) →
+    ∃ q' : Euc n, ‖q'-q‖^2 ≤ K*scaledGradientEnergy f p r q ∧
+      scaledGradientEnergy f p (θ*r) q' ≤ scaledGradientEnergy f p r q/16+K*(α*‖q‖+β)^2
+
+theorem exists_linear_contraction (hn : 0 < n) {ρ : ℝ} (hρ : 0 < ρ) :
+    ∃ θ δ K : ℝ, 0 < θ ∧ θ < min ρ (1/2) ∧ 0 < δ ∧ 1 ≤ K ∧ LinearContractionProperty n θ δ K := by
+  obtain ⟨C,hC,hB⟩ := linear_gradient_scaled_step hn
+  let V := volume.real (Metric.ball (0:Euc n) 1)
+  have hV : 0 ≤ V := measureReal_nonneg
+  have htend : Tendsto (fun t : ℝ ↦ 2*C^2*t^2*V) (𝓝 0) (𝓝 0) := by
+    have hh : ContinuousAt (fun t : ℝ ↦ 2*C^2*t^2*V) 0 := by fun_prop
+    simpa only [zero_pow (by norm_num : 2≠0),mul_zero,zero_mul] using hh.tendsto
+  obtain ⟨s,hs,hsb⟩ := Metric.tendsto_nhds_nhds.mp htend (1/32) (by norm_num)
+  let θ := min (s/2) (min (ρ/2) (1/4))
+  have hθ : 0 < θ := lt_min (by positivity) (lt_min (by positivity) (by norm_num))
+  have hθs : θ < s := (min_le_left _ _).trans_lt (by linarith)
+  have hθρ : θ < ρ := ((min_le_right _ _).trans (min_le_left _ _)).trans_lt (by linarith)
+  have hθhalf : θ < 1/2 := ((min_le_right _ _).trans (min_le_right _ _)).trans_lt (by norm_num)
+  have hsmall : 2*C^2*θ^2*V ≤ 1/32 := by
+    have hh := hsb (x := θ) (by simpa only [dist_zero_right,Real.norm_eq_abs,abs_of_pos hθ] using hθs)
+    rw [dist_zero_right,Real.norm_eq_abs,abs_of_nonneg (by positivity)] at hh
+    exact hh.le
+  let δ := Real.sqrt (θ^n/128)
+  have hδ : 0 < δ := Real.sqrt_pos.mpr (by positivity)
+  have hδsq : δ^2=θ^n/128 := Real.sq_sqrt (by positivity)
+  have hdpart : 4*δ^2*(θ^n)⁻¹=1/32 := by rw [hδsq]; field_simp; norm_num
+  let K := max 1 (max (C^2) (4*(θ^n)⁻¹*V))
+  have hK : 1 ≤ K := le_max_left _ _
+  have hKC : C^2 ≤ K := (le_max_left _ _).trans (le_max_right _ _)
+  have hKV : 4*(θ^n)⁻¹*V ≤ K := (le_max_right _ _).trans (le_max_right _ _)
+  refine ⟨θ,δ,K,hθ,lt_min hθρ hθhalf,hδ,hK,?_⟩
+  intro A hA f hf F hF p r hr h A₀ F₀ q α β hα hβ hb
+  obtain ⟨q',hqp,hqE⟩ := hB A hA f hf F hF p r hr h A₀ F₀ q δ α β hδ.le hα hβ hb
+  have hE : 0  ≤  scaledGradientEnergy f p r q := scaledGradientEnergy_nonneg f p hr.le q
+  refine ⟨q',hqp.trans (mul_le_mul_of_nonneg_right hKC hE),?_⟩
+  have hqE' := hqE θ hθ hθhalf.le
+  rw [hdpart] at hqE'
+  have h1 := mul_le_mul_of_nonneg_right hsmall hE
+  have h2 := mul_le_mul_of_nonneg_right hKV (sq_nonneg (α*‖q‖+β))
+  change _ ≤ _ at hqE'
+  dsimp only [V] at h1 h2
+  nlinarith
+
+theorem WeakLinearEquation.mono {A : Euc n → Euc n →L[ℝ] Euc n}
+    {f : Euc n → ℝ} {F : Euc n → Euc n} {U W : Set (Euc n)}
+    (h : WeakLinearEquation A f F U) (hWU : W⊆U) : WeakLinearEquation A f F W := by
+  intro φ
+  exact h ⟨φ.val,φ.property.1,φ.property.2.1,φ.property.2.2.trans hWU⟩
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/EllipticCurl.lean b/lean/OAI/Geometry/Perimeter/Regularity/EllipticCurl.lean
new file mode 100644
index 0000000000000000000000000000000000000000..7a63529eb329db71d620e9173bbcb99f34a85f83
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/EllipticCurl.lean
@@ -0,0 +1,120 @@
+import OAI.Geometry.Perimeter.Regularity.SmoothFlatOrder
+import OAI.Geometry.Perimeter.Regularity.EllipticTestSpace
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+def scalarDirection (φ : Euc n → ℝ) (a : Euc n) (x : Euc n) := fderiv ℝ φ x a
+
+theorem scalarDirection_contDiff {φ : Euc n → ℝ} (hφ : ContDiff ℝ ∞ φ) (a : Euc n) :
+    ContDiff ℝ ∞ (scalarDirection φ a) :=
+  (hφ.fderiv_right (by simp)).clm_apply contDiff_const
+
+theorem scalarDirection_tsupport (φ : Euc n → ℝ) (a : Euc n) :
+    tsupport (scalarDirection φ a) ⊆ tsupport φ := by
+  apply closure_minimal _ (isClosed_tsupport _)
+  intro x hx
+  by_contra hh
+  exact hx (by simp [scalarDirection,fderiv_of_notMem_tsupport ℝ hh])
+
+theorem scalarDirection_fderiv {φ : Euc n → ℝ} (hφ : ContDiff ℝ ∞ φ)
+    (a x b : Euc n) :
+    fderiv ℝ (scalarDirection φ a) x b=fderiv ℝ (fderiv ℝ φ) x b a := by
+  have hd : ContDiff ℝ ∞ (fderiv ℝ φ) := hφ.fderiv_right (by simp)
+  have h := ((ContinuousLinearMap.apply ℝ ℝ a).hasFDerivAt.comp x
+    ((hd.differentiable (by simp)) x).hasFDerivAt).fderiv
+  exact congrArg (fun L : Euc n →L[ℝ] ℝ ↦ L b) h
+
+def directionLocalTest {U : Set (Euc n)} (φ : SmoothLocalTest U) (a : Euc n) : SmoothLocalTest U :=
+  ⟨scalarDirection φ.val a,scalarDirection_contDiff φ.property.1 a,
+    φ.property.2.1.of_isClosed_subset (isClosed_tsupport _) (scalarDirection_tsupport _ _),
+    (scalarDirection_tsupport _ _).trans φ.property.2.2⟩
+
+def curlTest (φ : Euc n → ℝ) (a b : Euc n) (x : Euc n) : Euc n :=
+  scalarDirection φ a x • b - scalarDirection φ b x • a
+
+theorem curlTest_contDiff {φ : Euc n → ℝ} (hφ : ContDiff ℝ ∞ φ) (a b : Euc n) :
+    ContDiff ℝ ∞ (curlTest φ a b) :=
+  ((scalarDirection_contDiff hφ a).smul contDiff_const).sub
+    ((scalarDirection_contDiff hφ b).smul contDiff_const)
+
+theorem curlTest_tsupport (φ : Euc n → ℝ) (a b : Euc n) :
+    tsupport (curlTest φ a b) ⊆ tsupport φ := by
+  apply closure_minimal _ (isClosed_tsupport _)
+  intro x hx
+  by_contra hh
+  exact hx (by simp [curlTest,scalarDirection,fderiv_of_notMem_tsupport ℝ hh])
+
+theorem curlTest_divergence {φ : Euc n → ℝ} (hφ : ContDiff ℝ ∞ φ) (a b : Euc n) :
+    divergence (curlTest φ a b)=0 := by
+  have ha := (scalarDirection_contDiff hφ a).differentiable (by simp)
+  have hb := (scalarDirection_contDiff hφ b).differentiable (by simp)
+  rw [show curlTest φ a b=(fun x ↦ scalarDirection φ a x • b)-(fun x ↦ scalarDirection φ b x • a) from rfl,
+    divergence_sub (ha.smul_const b) (hb.smul_const a),divergence_scalar_direction ha,
+    divergence_scalar_direction hb]
+  funext x
+  simp only [Pi.sub_apply,Pi.zero_apply,scalarDirection_fderiv hφ]
+  exact sub_eq_zero.mpr ((hφ.contDiffAt.isSymmSndFDerivAt (by simp only [minSmoothness_of_isRCLikeNormedField]; exact WithTop.coe_le_coe.mpr le_top)) b a)
+
+def curlTestLp {U : Set (Euc n)} (φ : SmoothLocalTest U) (a b : Euc n) : VectorL2 n :=
+  ((curlTest_contDiff φ.property.1 a b).continuous.memLp_of_hasCompactSupport
+    (φ.property.2.1.of_isClosed_subset (isClosed_tsupport _) (curlTest_tsupport _ _ _))).toLp
+      (curlTest φ.val a b)
+
+theorem curlTestLp_ae {U : Set (Euc n)} (φ : SmoothLocalTest U) (a b : Euc n) :
+    (curlTestLp φ a b : Euc n → Euc n)=ᵐ[volume] curlTest φ.val a b := MemLp.coeFn_toLp _
+
+def CurlFreeLpOn (u : VectorL2 n) (U : Set (Euc n)) : Prop :=
+  ∀ (φ : SmoothLocalTest U) (a b : Euc n), inner ℝ u (curlTestLp φ a b)=0
+
+theorem testGradient_curlTest_inner {U V : Set (Euc n)} (ψ : SmoothLocalTest V)
+    (φ : SmoothLocalTest U) (a b : Euc n) :
+    inner ℝ (testGradientLp ψ) (curlTestLp φ a b)=0 := by
+  rw [MeasureTheory.L2.inner_def]
+  have he : (∫ x, inner ℝ (testGradientLp ψ x) (curlTestLp φ a b x))=
+      ∫ x, inner ℝ (gradient ψ.val x) (curlTest φ.val a b x) := by
+    apply integral_congr_ae
+    filter_upwards [testGradientLp_ae ψ,curlTestLp_ae φ a b] with x hx hy
+    rw [hx,hy]
+  rw [he]
+  have hi := integral_mul_divergence_eq_neg_inner_gradient_local ψ.property.1
+    (curlTest_contDiff φ.property.1 a b)
+    (φ.property.2.1.of_isClosed_subset (isClosed_tsupport _) (curlTest_tsupport _ _ _))
+  rw [curlTest_divergence φ.property.1] at hi
+  simpa only [Pi.zero_apply,mul_zero,integral_zero,eq_neg_iff_add_eq_zero,zero_add] using hi
+
+theorem testGradientSpace_curlFree {U V : Set (Euc n)} {u : VectorL2 n}
+    (hu : u∈testGradientSpace V) : CurlFreeLpOn u U := by
+  intro φ a b
+  have hz := inner_zero_on_closed_span (Set.range (testGradientLp (U := V)))
+    (u := curlTestLp φ a b) (by
+      rintro _ ⟨ψ,rfl⟩
+      rw [real_inner_comm]
+      exact testGradient_curlTest_inner ψ φ a b) u hu
+  rwa [real_inner_comm] at hz
+
+theorem CurlFreeLpOn.sub {U : Set (Euc n)} {u v : VectorL2 n}
+    (hu : CurlFreeLpOn u U) (hv : CurlFreeLpOn v U) : CurlFreeLpOn (u-v) U := by
+  intro φ a b
+  rw [inner_sub_left,hu φ a b,hv φ a b,sub_self]
+
+theorem weak_curl_harmonic_comparison {U : Set (Euc n)} {u e : VectorL2 n}
+    (hu : CurlFreeLpOn u U)
+    (h : ∀ φ : SmoothLocalTest U, (∫ x, inner ℝ (u x) (gradient φ.val x))=
+      ∫ x, inner ℝ (e x) (gradient φ.val x)) :
+    ∃ g : VectorL2 n, DivFreeLpOn g U ∧ CurlFreeLpOn g U ∧ ‖u-g‖≤‖e‖ := by
+  obtain ⟨g,hg,hug,he⟩ := weak_harmonic_gradient_comparison h
+  have hcf := hu.sub (testGradientSpace_curlFree hug)
+  rw [sub_sub_cancel] at hcf
+  exact ⟨g,hg,hcf,he⟩
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/EllipticCutField.lean b/lean/OAI/Geometry/Perimeter/Regularity/EllipticCutField.lean
new file mode 100644
index 0000000000000000000000000000000000000000..8f25049bfd99cc9d6c50b8c8a18219f9e41818a7
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/EllipticCutField.lean
@@ -0,0 +1,65 @@
+import OAI.Geometry.Perimeter.Regularity.EllipticOneScale
+import OAI.Geometry.Perimeter.Regularity.EllipticC1Parts
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+theorem continuous_field_ball_memLp {W : Euc n → Euc n} (hW : Continuous W)
+    (p : Euc n) (r : ℝ) : MemLp ((Metric.ball p r).indicator W) 2 volume := by
+  rw [memLp_indicator_iff_restrict measurableSet_ball]
+  apply (memLp_two_iff_integrable_sq_norm hW.aestronglyMeasurable).mpr
+  exact ((hW.norm.pow 2).continuousOn.integrableOn_compact (isCompact_closedBall p r)).mono_set
+    Metric.ball_subset_closedBall
+
+def cutFieldLp (W : Euc n → Euc n) (hW : Continuous W) (p : Euc n) (r : ℝ) : VectorL2 n :=
+  (continuous_field_ball_memLp hW p r).toLp ((Metric.ball p r).indicator W)
+
+theorem cutFieldLp_ae (W : Euc n → Euc n) (hW : Continuous W) (p : Euc n) (r : ℝ) :
+    (cutFieldLp W hW p r : Euc n → Euc n)=ᵐ[volume] (Metric.ball p r).indicator W :=
+  MemLp.coeFn_toLp _
+
+theorem cutFieldLp_norm_sq (W : Euc n → Euc n) (hW : Continuous W) (p : Euc n) (r : ℝ) :
+    ‖cutFieldLp W hW p r‖^2 = ∫ x in Metric.ball p r, ‖W x‖^2 := by
+  rw [←vectorLp_integral_sq]
+  calc
+    (∫ x, ‖cutFieldLp W hW p r x‖^2) = ∫ x, (Metric.ball p r).indicator (fun x ↦ ‖W x‖^2) x := by
+      apply integral_congr_ae
+      filter_upwards [cutFieldLp_ae W hW p r] with x hx
+      rw [hx]
+      by_cases hb : x∈Metric.ball p r <;> simp [hb]
+    _ = _ := integral_indicator measurableSet_ball
+
+theorem cutFieldLp_pairing (W : Euc n → Euc n) (hW : Continuous W) (p : Euc n) (r : ℝ)
+    {g : Euc n → Euc n} (hg : tsupport g ⊆ Metric.ball p r) :
+    (∫ x, inner ℝ (cutFieldLp W hW p r x) (g x)) = ∫ x, inner ℝ (W x) (g x) := by
+  apply integral_congr_ae
+  filter_upwards [cutFieldLp_ae W hW p r] with x hx
+  rw [hx]
+  by_cases hb : x∈Metric.ball p r
+  · simp [hb]
+  · have hz : g x=0 := image_eq_zero_of_notMem_tsupport (fun h ↦ hb (hg h))
+    simp [hz]
+
+theorem cutGradient_curlFree {f : Euc n → ℝ} (hf : ContDiff ℝ 1 f) (p : Euc n) (r : ℝ) :
+    CurlFreeLpOn (cutFieldLp (gradient f) (gradient_continuous_C1 hf) p r) (Metric.ball p r) := by
+  intro φ a b
+  rw [MeasureTheory.L2.inner_def]
+  calc
+    (∫ x, inner ℝ (cutFieldLp (gradient f) (gradient_continuous_C1 hf) p r x) (curlTestLp φ a b x)) =
+        ∫ x, inner ℝ (cutFieldLp (gradient f) (gradient_continuous_C1 hf) p r x) (curlTest φ.val a b x) := by
+      apply integral_congr_ae
+      filter_upwards [curlTestLp_ae φ a b] with x hx
+      rw [hx]
+    _ = _ := cutFieldLp_pairing _ _ _ _ ((curlTest_tsupport _ _ _).trans φ.property.2.2)
+    _ = 0 := C1_gradient_curl_pairing_zero hf φ a b
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/EllipticCutoffEnergy.lean b/lean/OAI/Geometry/Perimeter/Regularity/EllipticCutoffEnergy.lean
new file mode 100644
index 0000000000000000000000000000000000000000..5c39da3a263ed88a7112585f958dbcd762067d74
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/EllipticCutoffEnergy.lean
@@ -0,0 +1,64 @@
+import OAI.Geometry.Perimeter.Regularity.EllipticCaccioppoli
+import OAI.Geometry.Perimeter.Regularity.EllipticScaleStep
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+ theorem cutoff_energy_bound {A : Euc n → Euc n →L[ℝ] Euc n}
+    {f : Euc n → ℝ} {F : Euc n → Euc n} (hA : Continuous A) (hf : ContDiff ℝ 1 f)
+    (hF : Continuous F) {p : Euc n} {r : ℝ} (h : WeakLinearEquation A f F (Metric.ball p r))
+    (hnear : ∀ x∈Metric.ball p r, ‖ContinuousLinearMap.id ℝ (Euc n)-A x‖≤1/2)
+    {χ : Euc n → ℝ} (hχ : ContDiff ℝ 1 χ) (hcχ : HasCompactSupport χ)
+    (hsχ : tsupport χ ⊆ Metric.ball p r) {B N : ℝ}
+    (hb : ∀ x, |f x|≤B) (hN : ∀ x, ‖F x‖≤N) :
+    (∫ x, (χ x)^2*‖gradient f x‖^2) ≤
+      256*(B^2*(∫ x, ‖gradient χ x‖^2)+N^2*(∫ x, (χ x)^2)) := by
+  have hi₁ : Integrable (fun x ↦ ‖gradient χ x‖^2) volume := by
+    simpa only [one_pow,one_mul] using square_compact_field_integrable
+      (continuous_const : Continuous (fun _ : Euc n ↦ (1:ℝ))) (gradient_continuous_C1 hχ)
+      (gradient_hasCompactSupport hcχ)
+  have hi₂ : Integrable (fun x ↦ (χ x)^2) volume := by
+    apply (hχ.continuous.pow 2).integrable_of_hasCompactSupport
+    apply hcχ.of_isClosed_subset (isClosed_tsupport _)
+    apply closure_mono
+    intro x hx
+    exact fun hz ↦ hx (by simp [hz])
+  have h1 : (∫ x, (f x)^2*‖gradient χ x‖^2)≤B^2*(∫ x, ‖gradient χ x‖^2) := by
+    rw [←integral_const_mul]
+    apply integral_mono (square_compact_field_integrable hf.continuous (gradient_continuous_C1 hχ)
+      (gradient_hasCompactSupport hcχ)) (hi₁.const_mul _)
+    intro x
+    exact mul_le_mul_of_nonneg_right (by nlinarith [hb x,sq_abs (f x),abs_nonneg (f x)]) (sq_nonneg _)
+  have h2 : (∫ x, (χ x)^2*‖F x‖^2)≤N^2*(∫ x, (χ x)^2) := by
+    rw [←integral_const_mul]
+    apply integral_mono (square_weight_integrable hχ.continuous hcχ hF) (hi₂.const_mul _)
+    intro x
+    have ht := mul_le_mul_of_nonneg_left (pow_le_pow_left₀ (norm_nonneg (F x)) (hN x) 2) (sq_nonneg (χ x))
+    simpa only [mul_comm] using ht
+  exact (h.caccioppoli hA hf hF hnear hχ hcχ hsχ).trans (by linarith)
+
+ theorem scaled_energy_le_cutoff {f χ : Euc n → ℝ} (hf : ContDiff ℝ 1 f)
+    (hχ : ContDiff ℝ 1 χ) (hcχ : HasCompactSupport χ) {p : Euc n} {r : ℝ} (hr : 0<r)
+    (hone : ∀ x∈Metric.ball p r, χ x=1) :
+    scaledGradientEnergy f p r 0≤r^(-(n:ℤ))*(∫ x, (χ x)^2*‖gradient f x‖^2) := by
+  have hi := square_weight_integrable hχ.continuous hcχ (gradient_continuous_C1 hf)
+  have he : gradientEnergy f p r 0=(∫ x in Metric.ball p r, (χ x)^2*‖gradient f x‖^2) := by
+    apply setIntegral_congr_fun measurableSet_ball
+    intro x hx
+    simp only [hone x hx,one_pow,one_mul,sub_zero]
+  have hb : gradientEnergy f p r 0≤∫ x, (χ x)^2*‖gradient f x‖^2 := by
+    rw [he]
+    exact setIntegral_le_integral hi (ae_of_all _ (fun x ↦ mul_nonneg (sq_nonneg _) (sq_nonneg _)))
+  have hh := mul_le_mul_of_nonneg_left hb (by positivity : 0≤(r^n)⁻¹)
+  simpa only [scaledGradientEnergy,zpow_neg,zpow_natCast] using hh
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/EllipticDifference.lean b/lean/OAI/Geometry/Perimeter/Regularity/EllipticDifference.lean
new file mode 100644
index 0000000000000000000000000000000000000000..53fe104a0a0bc363ba5fa36605c0ab20d37fb526
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/EllipticDifference.lean
@@ -0,0 +1,117 @@
+import OAI.Geometry.Perimeter.Regularity.EllipticLinearStep
+import OAI.Geometry.Perimeter.Regularity.EllipticSecant
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+def WeakFluxEquation (V : Euc n → Euc n) (U : Set (Euc n)) : Prop :=
+  ∀ φ : SmoothLocalTest U, (∫ x, inner ℝ (V x) (gradient φ.val x))=0
+
+def translateLocalTest {U W : Set (Euc n)} (z : Euc n)
+    (hz : ∀ x∈W, x+z∈U) (φ : SmoothLocalTest W) : SmoothLocalTest U := by
+  let ψ : Euc n → ℝ := fun y ↦ φ.val (y-z)
+  refine ⟨ψ,φ.property.1.comp (contDiff_id.sub contDiff_const),
+    φ.property.2.1.comp_homeomorph (Homeomorph.subRight z),?_⟩
+  intro y hy
+  have hh := φ.property.2.2 (tsupport_comp_subset_preimage φ.val
+    (continuous_id.sub continuous_const) hy)
+  simpa only [sub_add_cancel] using hz (y-z) hh
+
+ theorem gradient_translate {f : Euc n → ℝ} (hf : Differentiable ℝ f)
+    (z x : Euc n) : gradient (fun y ↦ f (y+z)) x=gradient f (x+z) := by
+  apply ext_inner_right ℝ
+  intro y
+  simp only [inner_gradient_left]
+  have hh := ((hf (x+z)).hasFDerivAt.comp x ((hasFDerivAt_id x).add_const z)).fderiv
+  simpa only [ContinuousLinearMap.comp_id,Function.comp_def,id_eq] using congrArg (fun L : Euc n →L[ℝ] ℝ ↦ L y) hh
+
+ theorem WeakFluxEquation.translate {V : Euc n → Euc n} {U W : Set (Euc n)}
+    (h : WeakFluxEquation V U) (z : Euc n) (hz : ∀ x∈W, x+z∈U) :
+    WeakFluxEquation (fun x ↦ V (x+z)) W := by
+  intro φ
+  have hh := h (translateLocalTest z hz φ)
+  have hg (x : Euc n) : gradient (translateLocalTest z hz φ).val (x+z)=gradient φ.val x := by
+    have he : (translateLocalTest z hz φ).val=(fun y ↦ φ.val (y+(-z))) := by
+      funext y; rfl
+    rw [he,gradient_translate (φ.property.1.differentiable (by simp))]
+    simp only [add_neg_cancel_right]
+  rw [←integral_add_right_eq_self (fun x ↦ inner ℝ (V x)
+    (gradient (translateLocalTest z hz φ).val x)) z] at hh
+  simpa only [hg] using hh
+
+ theorem WeakFluxEquation.sub {V W : Euc n → Euc n} {U : Set (Euc n)}
+    (hV : Continuous V) (hW : Continuous W) (h : WeakFluxEquation V U)
+    (h' : WeakFluxEquation W U) : WeakFluxEquation (fun x ↦ V x-W x) U := by
+  intro φ
+  simp_rw [inner_sub_left]
+  rw [integral_sub (inner_gradient_pairing_integrable hV φ)
+    (inner_gradient_pairing_integrable hW φ),h φ,h' φ,sub_self]
+
+ theorem WeakFluxEquation.const_smul {V : Euc n → Euc n} {U : Set (Euc n)}
+    (h : WeakFluxEquation V U) (c : ℝ) : WeakFluxEquation (fun x ↦ c•V x) U := by
+  intro φ
+  simp only [inner_smul_left,conj_trivial,integral_const_mul,h φ,mul_zero]
+
+ def differenceQuotient (f : Euc n → ℝ) (e : Euc n) (h : ℝ) (x : Euc n) :=
+  h⁻¹*(f (x+h•e)-f x)
+
+ theorem differenceQuotient_contDiff {f : Euc n → ℝ} {k : ℕ∞}
+    (hf : ContDiff ℝ k f) (e : Euc n) (h : ℝ) : ContDiff ℝ k (differenceQuotient f e h) :=
+  contDiff_const.mul ((hf.comp (contDiff_id.add contDiff_const)).sub hf)
+
+ theorem differenceQuotient_gradient {f : Euc n → ℝ} (hf : Differentiable ℝ f)
+    (e : Euc n) (h : ℝ) (x : Euc n) :
+    gradient (differenceQuotient f e h) x=h⁻¹•(gradient f (x+h•e)-gradient f x) := by
+  apply ext_inner_right ℝ
+  intro y
+  simp only [inner_gradient_left,inner_smul_left,inner_sub_left,conj_trivial]
+  have hd := ((((hf (x+h•e)).hasFDerivAt.comp x ((hasFDerivAt_id x).add_const (h•e))).sub
+    (hf x).hasFDerivAt).const_mul h⁻¹).fderiv
+  change fderiv ℝ (fun y ↦ h⁻¹*(f (y+h•e)-f y)) x y = _
+  simpa only [differenceQuotient,smul_apply,sub_apply,
+    ContinuousLinearMap.comp_id,smul_eq_mul,Function.comp_def,id_eq,Pi.sub_apply] using congrArg (fun L : Euc n →L[ℝ] ℝ ↦ L y) hd
+
+ theorem differenceQuotient_tendsto {f : Euc n → ℝ} (hf : Differentiable ℝ f)
+    (e x : Euc n) : Tendsto (fun h : ℝ ↦ differenceQuotient f e h x) (𝓝[≠] 0)
+      (𝓝 (fderiv ℝ f x e)) := by
+  have ht : HasDerivAt (fun h : ℝ ↦ x+h•e) e 0 := by
+    simpa using ((hasDerivAt_id (0:ℝ)).smul_const e |>.const_add x)
+  have hd : HasDerivAt (fun h : ℝ ↦ f (x+h•e)) (fderiv ℝ f x e) 0 := by
+    have hf' : HasFDerivAt f (fderiv ℝ f x) (x+(0:ℝ)•e) := by simpa using (hf x).hasFDerivAt
+    have hh := hf'.comp_hasDerivAt (0:ℝ) ht
+    simpa only [zero_smul,add_zero,Function.comp_def] using hh
+  have he : (fun h : ℝ ↦ differenceQuotient f e h x) = _root_.slope (fun h : ℝ ↦ f (x+h•e)) 0 := by
+    funext h
+    simp only [_root_.slope,differenceQuotient,vsub_eq_sub,sub_zero,zero_smul,add_zero,smul_eq_mul]
+  rw [he]
+  exact hd.tendsto_slope
+
+ theorem weak_nonlinear_difference {G : Euc n → Euc n} (hG : ContDiff ℝ 1 G)
+    {f : Euc n → ℝ} (hf : ContDiff ℝ 1 f) {U W : Set (Euc n)}
+    (h : WeakFluxEquation (fun x ↦ G (gradient f x)) U) (hWU : W⊆U)
+    (e : Euc n) (s : ℝ) (hs : ∀ x∈W, x+s•e∈U) :
+    WeakLinearEquation (fun x ↦ derivativeSecant G (gradient f x) (gradient f (x+s•e)))
+      (differenceQuotient f e s) (fun _ ↦ 0) W := by
+  have hbase : WeakFluxEquation (fun x ↦ G (gradient f x)) W := by
+    intro φ
+    exact h ⟨φ.val,φ.property.1,φ.property.2.1,φ.property.2.2.trans hWU⟩
+  have hcont := hG.continuous.comp (gradient_continuous_C1 hf)
+  have hd := ((h.translate (s•e) hs).sub
+    (hcont.comp (continuous_id.add continuous_const)) hcont hbase).const_smul s⁻¹
+  intro φ
+  have he (x : Euc n) : derivativeSecant G (gradient f x) (gradient f (x+s•e))
+      (gradient (differenceQuotient f e s) x)=s⁻¹•(G (gradient f (x+s•e))-G (gradient f x)) := by
+    rw [differenceQuotient_gradient (hf.differentiable (by simp)),map_smul,derivativeSecant_apply hG]
+  simp only [he,inner_zero_left,integral_zero]
+  exact hd φ
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/EllipticExcessStep.lean b/lean/OAI/Geometry/Perimeter/Regularity/EllipticExcessStep.lean
new file mode 100644
index 0000000000000000000000000000000000000000..eb74e6e57bb14510ee72e365387462541eaf1ae0
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/EllipticExcessStep.lean
@@ -0,0 +1,50 @@
+import OAI.Geometry.Perimeter.Regularity.EllipticLinearStep
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+theorem linear_gradient_excess_step (hn : 0<n) :
+    ∃ C : ℝ, 0≤C ∧ ∀ (A : Euc n → Euc n →L[ℝ] Euc n), Continuous A →
+      ∀ (f : Euc n → ℝ), ContDiff ℝ 1 f → ∀ (F : Euc n → Euc n), Continuous F →
+      WeakLinearEquation A f F (Metric.ball 0 1) →
+      ∀ (A₀ : Euc n →L[ℝ] Euc n) (F₀ q : Euc n) (δ α β : ℝ),
+      0≤δ → 0≤α → 0≤β →
+      (∀ x∈Metric.ball (0:Euc n) 1, ‖ContinuousLinearMap.id ℝ (Euc n)-A x‖≤δ ∧
+        ‖A₀-A x‖≤α ∧ ‖F x-F₀‖≤β) →
+      ∃ q' : Euc n, ‖q'-q‖^2≤C^2*gradientEnergy f 0 1 q ∧
+        ∀ θ : ℝ, θ≤1/2 → gradientEnergy f 0 θ q' ≤
+          (4*δ^2+2*C^2*θ^2*volume.real (Metric.ball (0:Euc n) θ))*gradientEnergy f 0 1 q+
+          4*(α*‖q‖+β)^2*volume.real (Metric.ball (0:Euc n) 1) := by
+  obtain ⟨C,hC,hB⟩ := continuous_gradient_one_scale hn
+  refine ⟨C,hC,fun A hA f hf F hF h A₀ F₀ q δ α β hδ hα hβ hb ↦ ?_⟩
+  let g : Euc n → ℝ := fun x ↦ f x-inner ℝ q x
+  have hg : ContDiff ℝ 1 g := hf.sub (innerSL ℝ q).contDiff
+  let e := linearResidual A f F A₀ F₀ q
+  have he : Continuous e := linearResidual_continuous hA hf hF _ _ _
+  obtain ⟨z,hz,hzE⟩ := hB g hg e he (linearResidual_pairing hA hf hF h A₀ F₀ q)
+  have hgrad (x : Euc n) : gradient g x=gradient f x-q := gradient_sub_linear (hf.differentiable (by simp)) q x
+  have hE (r : ℝ) (w : Euc n) : gradientEnergy g 0 r w=gradientEnergy f 0 r (q+w) := by
+    unfold gradientEnergy
+    simp_rw [hgrad,sub_sub]
+  have hBnd := continuous_residual_energy_bound he ((gradient_continuous_C1 hf).sub continuous_const)
+    hδ (add_nonneg (mul_nonneg hα (norm_nonneg _)) hβ) (fun x hx ↦ by
+      have hx' := hb x hx
+      simpa only [add_assoc,Pi.sub_apply,e] using linearResidual_bound (f := f) A₀ F₀ q x hx'.1 hx'.2.1 hx'.2.2)
+  change (∫ x in Metric.ball (0:Euc n) 1, ‖e x‖^2)≤
+    2*δ^2*gradientEnergy f 0 1 q+2*(α*‖q‖+β)^2*volume.real (Metric.ball (0:Euc n) 1) at hBnd
+  refine ⟨q+z,?_,fun θ hθ ↦ ?_⟩
+  · simpa only [add_sub_cancel_left,hE,add_zero] using hz
+  · have hh := hzE θ hθ
+    rw [hE,hE,add_zero] at hh
+    nlinarith
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/EllipticFamilyEstimate.lean b/lean/OAI/Geometry/Perimeter/Regularity/EllipticFamilyEstimate.lean
new file mode 100644
index 0000000000000000000000000000000000000000..16694f2000c893890ab64d0a382bfec316ec345a
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/EllipticFamilyEstimate.lean
@@ -0,0 +1,113 @@
+import OAI.Geometry.Perimeter.Regularity.EllipticCutoffEnergy
+import OAI.Geometry.Perimeter.Regularity.EllipticUniformGradient
+import OAI.Geometry.Perimeter.Regularity.EllipticC1Limit
+import OAI.Geometry.Perimeter.Regularity.GeometricHolder
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+ def ellipticGradientConstant (n : ℕ) (K : ℝ) : ℝ :=
+  4*K+Real.sqrt (2^(n+2)/volume.real (Metric.ball (0:Euc n) 1))
+
+ theorem linear_uniform_gradient_explicit {θ δ K a B H R : ℝ}
+    (hθ : 0<θ) (hθ1 : θ<1) (hK : 1 ≤ K)
+    (hc : LinearContractionProperty n θ δ K)
+    (ha : 0 ≤ a) (hB : 0 ≤ B) (hH : 0 ≤ H) (hR : 0<R)
+    (hsmall : 32*K^2*a ≤ 1) (hforce : 16*K*B ≤ H)
+    {A : Euc n → Euc n →L[ℝ] Euc n} (hA : Continuous A)
+    {f : Euc n → ℝ} (hf : ContDiff ℝ 1 f) {F : Euc n → Euc n} (hF : Continuous F)
+    {U : Set (Euc n)}
+    (h : ∀ p∈U, WeakLinearEquation A f F (Metric.ball p R))
+    (hnear : ∀ p∈U, ∀ x∈Metric.ball p R, ‖ContinuousLinearMap.id ℝ (Euc n)-A x‖ ≤ δ)
+    (hosc : ∀ p∈U, ∀ j x, x∈Metric.ball p (θ^j*R) →
+      ‖A p-A x‖ ≤ a*(1/2)^j ∧ ‖F x-F p‖ ≤ B*(1/2)^j)
+    (hinit : ∀ p∈U, scaledGradientEnergy f p R 0 ≤ H^2) :
+    (∀ p∈U, ‖gradient f p‖ ≤ 2*K*H) ∧
+      ∀ j p, p∈U → ∀ p', p'∈U → dist p p' ≤ θ^j*R/2 →
+        ‖gradient f p-gradient f p'‖ ≤ ellipticGradientConstant n K*H*(1/2)^j := by
+  have hex (p : Euc n) : ∃ q : ℕ → Euc n, p∈U →
+      (∀ j, ‖q j-gradient f p‖ ≤ 2*K*H*(1/2)^j) ∧
+      (∀ j, scaledGradientEnergy f p (θ^j*R) (q j) ≤ H^2*((1/2)^j)^2) ∧
+      ‖gradient f p‖ ≤ 2*K*H := by
+    by_cases hp : p∈U
+    · obtain ⟨q,hq⟩ := linear_pointwise_iteration hθ hθ1 hK hc ha hB hH hR hsmall hforce
+        hA hf hF p (h p hp) (hnear p hp) (hosc p hp) (hinit p hp)
+      exact ⟨q,fun _ ↦ hq⟩
+    · exact ⟨fun _ ↦ 0,fun hh ↦ (hp hh).elim⟩
+  choose q hq using hex
+  have hV : 0<volume.real (Metric.ball (0:Euc n) 1) :=
+    ENNReal.toReal_pos (Metric.measure_ball_pos volume 0 (by norm_num)).ne' measure_ball_lt_top.ne
+  let L := Real.sqrt (2^(n+2)/volume.real (Metric.ball (0:Euc n) 1))
+  have hL : 0 ≤ L := Real.sqrt_nonneg _
+  have hLbig : 2^(n+2) ≤ volume.real (Metric.ball (0:Euc n) 1)*L^2 := by
+    dsimp [L]
+    rw [Real.sq_sqrt (by positivity),mul_div_cancel₀ _ hV.ne']
+  exact ⟨fun p hp ↦ (hq p hp).2.2,gradient_modulus_from_slopes hθ hR hH hL hLbig hf
+    (fun p hp ↦ (hq p hp).1) (fun p hp ↦ (hq p hp).2.1)⟩
+
+ theorem linear_family_C1_limit {θ δ K a B r B₀ N : ℝ}
+    (hθ : 0<θ) (hθ1 : θ<1) (hK : 1 ≤ K)
+    (hc : LinearContractionProperty n θ δ K)
+    (ha : 0 ≤ a) (hB : 0 ≤ B) (hr : 0<r) (hsmall : 32*K^2*a ≤ 1)
+    {p : Euc n} {A : ℕ → Euc n → Euc n →L[ℝ] Euc n}
+    {f : ℕ → Euc n → ℝ} {F : ℕ → Euc n → Euc n}
+    (hA : ∀ i, Continuous (A i)) (hf : ∀ i, ContDiff ℝ 1 (f i)) (hF : ∀ i, Continuous (F i))
+    (h : ∀ i, WeakLinearEquation (A i) (f i) (F i) (Metric.ball p (4*r)))
+    (hnear : ∀ i x, x∈Metric.ball p (4*r) →
+      ‖ContinuousLinearMap.id ℝ (Euc n)-A i x‖ ≤ min δ (1/2))
+    (hosc : ∀ i x, x∈Metric.closedBall p r → ∀ j y, y∈Metric.ball x (θ^j*r) →
+      ‖A i x-A i y‖ ≤ a*(1/2)^j ∧ ‖F i y-F i x‖ ≤ B*(1/2)^j)
+    (hb : ∀ i x, |f i x| ≤ B₀) (hN : ∀ i x, ‖F i x‖ ≤ N)
+    {g : Euc n → ℝ}
+    (hlim : ∀ x∈Metric.ball p r, Tendsto (fun i ↦ f i x) atTop (𝓝 (g x))) :
+    ContDiffOn ℝ 1 g (Metric.ball p r) ∧ GeometricHolderAt θ (gradient g) p := by
+  let χ : ContDiffBump p := ⟨2*r,3*r,by positivity,by linarith⟩
+  have hsχ : tsupport χ⊆Metric.ball p (4*r) := by
+    rw [χ.tsupport_eq]
+    exact Metric.closedBall_subset_ball (by dsimp [χ]; linarith)
+  let S := 256*(B₀^2*(∫ x, ‖gradient χ x‖^2)+N^2*(∫ x, (χ x)^2))
+  have hS : 0 ≤ S := by
+    dsimp [S]
+    positivity
+  have henergy (i : ℕ) : (∫ x, (χ x)^2*‖gradient (f i) x‖^2) ≤ S :=
+    cutoff_energy_bound (hA i) (hf i) (hF i) (h i)
+      (fun x hx ↦ (hnear i x hx).trans (min_le_right _ _)) χ.contDiff χ.hasCompactSupport hsχ (hb i) (hN i)
+  let H := max (16*K*B) (Real.sqrt (r^(-(n:ℤ))*S))
+  have hH : 0 ≤ H := (Real.sqrt_nonneg _).trans (le_max_right _ _)
+  have hHe : r^(-(n:ℤ))*S ≤ H^2 := by
+    have ht := sq_le_sq₀ (Real.sqrt_nonneg (r^(-(n:ℤ))*S)) hH |>.mpr (le_max_right _ _)
+    rwa [Real.sq_sqrt (mul_nonneg (by positivity) hS)] at ht
+  have hball (x : Euc n) (hx : x∈Metric.closedBall p r) : Metric.ball x r⊆Metric.ball p (2*r) := by
+    intro y hy
+    have hh := dist_triangle y x p
+    rw [Metric.mem_closedBall] at hx
+    rw [Metric.mem_ball] at hy ⊢
+    linarith
+  have huniform (i : ℕ) := linear_uniform_gradient_explicit hθ hθ1 hK hc ha hB hH hr hsmall
+    (le_max_left _ _) (hA i) (hf i) (hF i)
+    (fun x hx ↦ (h i).mono ((hball x hx).trans (Metric.ball_subset_ball (by linarith))))
+    (fun x hx y hy ↦ (hnear i y ((hball x hx).trans (Metric.ball_subset_ball (by linarith)) hy)).trans (min_le_left _ _))
+    (hosc i) (fun x hx ↦ (scaled_energy_le_cutoff (hf i) χ.contDiff χ.hasCompactSupport hr
+      (fun y hy ↦ χ.one_of_mem_closedBall (Metric.ball_subset_closedBall (hball x hx hy)))).trans
+      ((mul_le_mul_of_nonneg_left (henergy i) (by positivity)).trans hHe))
+  obtain ⟨hg,hmod⟩ := C1_limit_of_geometric_gradient_estimates hf hθ (by positivity : 0<r/2)
+    (fun i x hx ↦ (huniform i).1 x hx)
+    (fun i j x hx y hy hxy ↦ (huniform i).2 j x hx y hy (by nlinarith)) hlim
+  refine ⟨hg,r/2,ellipticGradientConstant n K*H,by positivity,?_,?_⟩
+  · dsimp [ellipticGradientConstant]
+    positivity
+  · intro j x hx y hy hxy
+    rw [dist_eq_norm]
+    exact hmod j x (Metric.ball_subset_ball (by linarith) hx) y
+      (Metric.ball_subset_ball (by linarith) hy) hxy
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/EllipticGradientLimit.lean b/lean/OAI/Geometry/Perimeter/Regularity/EllipticGradientLimit.lean
new file mode 100644
index 0000000000000000000000000000000000000000..a8776325d6d4fa564e901a7c53fa37733705c7a7
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/EllipticGradientLimit.lean
@@ -0,0 +1,70 @@
+import OAI.Geometry.Perimeter.Regularity.EllipticScaleStep
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+ theorem real_volume_ball_center (p : Euc n) (r : ℝ) :
+    volume.real (Metric.ball p r)=volume.real (Metric.ball (0:Euc n) r) := by
+  exact Measure.addHaar_real_ball_center volume p r
+
+ theorem scaledGradientEnergy_lower {f : Euc n → ℝ} (hf : ContDiff ℝ 1 f)
+    (p : Euc n) {r b : ℝ} (hr : 0<r) (q : Euc n)
+    (hb : ∀ x∈Metric.ball p r, b≤‖gradient f x-q‖^2) :
+    b*volume.real (Metric.ball (0:Euc n) 1) ≤ scaledGradientEnergy f p r q := by
+  have hI : IntegrableOn (fun x ↦ ‖gradient f x-q‖^2) (Metric.ball p r) volume :=
+    ((((gradient_continuous_C1 hf).sub continuous_const).norm.pow 2).continuousOn.integrableOn_compact
+      (isCompact_closedBall p r)).mono_set Metric.ball_subset_closedBall
+  have : IsFiniteMeasure (volume.restrict (Metric.ball p r)) := ⟨by
+    rw [Measure.restrict_apply_univ]; exact measure_ball_lt_top⟩
+  have hh : (∫ x in Metric.ball p r, b) ≤ gradientEnergy f p r q := by
+    apply integral_mono_ae (integrable_const _) hI
+    filter_upwards [ae_restrict_mem measurableSet_ball] with x hx
+    exact hb x hx
+  simp only [integral_const,measureReal_restrict_apply_univ,smul_eq_mul] at hh
+  rw [real_volume_ball_center,real_volume_ball_scale hr] at hh
+  have hh' := mul_le_mul_of_nonneg_left hh (inv_nonneg.mpr (pow_nonneg hr.le n))
+  change _ ≤ (r^n)⁻¹*gradientEnergy f p r q
+  convert hh' using 1
+  field_simp [hr.ne']
+
+ theorem gradient_eq_of_scaled_energy {f : Euc n → ℝ} (hf : ContDiff ℝ 1 f)
+    (p : Euc n) {r ε : ℕ → ℝ} {q : ℕ → Euc n} {w : Euc n}
+    (hr : ∀ j,0<r j) (hrlim : Tendsto r atTop (𝓝 0))
+    (hε : Tendsto ε atTop (𝓝 0)) (hq : Tendsto q atTop (𝓝 w))
+    (hE : ∀ j, scaledGradientEnergy f p (r j) (q j) ≤ ε j) : gradient f p=w := by
+  by_contra hne
+  let d := ‖gradient f p-w‖
+  have hd : 0<d := norm_pos_iff.mpr (sub_ne_zero.mpr hne)
+  obtain ⟨s,hs,hgs⟩ := Metric.continuousAt_iff.mp (gradient_continuous_C1 hf).continuousAt (d/4) (by positivity)
+  have hv : 0<volume.real (Metric.ball (0:Euc n) 1) := by
+    exact ENNReal.toReal_pos (by exact ne_of_gt (Metric.measure_ball_pos volume _ (by norm_num))) measure_ball_lt_top.ne
+  have hsmall := hε.eventually (gt_mem_nhds (show 0<(d/2)^2*volume.real (Metric.ball (0:Euc n) 1) by positivity))
+  have hrsmall := hrlim.eventually (gt_mem_nhds hs)
+  have hqsmall := (Metric.tendsto_atTop.mp hq) (d/4) (by positivity)
+  obtain ⟨N,hN⟩ := hqsmall
+  obtain ⟨j,hjE,hjr,hjq⟩ := (hsmall.and (hrsmall.and (eventually_ge_atTop N))).exists
+  have hlo := scaledGradientEnergy_lower hf p (hr j) (q j) (b := (d/2)^2) (fun x hx ↦ by
+    have hg : ‖gradient f x-gradient f p‖<d/4 := by
+      rw [←dist_eq_norm]
+      exact hgs (by exact (Metric.mem_ball.mp hx).trans hjr)
+    have hq' : ‖q j-w‖<d/4 := by simpa only [dist_eq_norm] using hN j hjq
+    have htri : d≤‖gradient f x-gradient f p‖+‖gradient f x-q j‖+‖q j-w‖ := by
+      have hh := norm_add_le (gradient f p-gradient f x) (gradient f x-q j)
+      have hh' := norm_add_le (gradient f p-q j) (q j-w)
+      simp only [sub_add_sub_cancel,norm_sub_rev (gradient f p) (gradient f x)] at hh hh'
+      dsimp [d]
+      linarith
+    have hn : d/2≤‖gradient f x-q j‖ := by linarith
+    exact pow_le_pow_left₀ (by positivity) hn 2)
+  linarith [hE j]
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/EllipticHarmonicBounds.lean b/lean/OAI/Geometry/Perimeter/Regularity/EllipticHarmonicBounds.lean
new file mode 100644
index 0000000000000000000000000000000000000000..ea683cbb33b2d9a5a13418f47bf26a668b1385ec
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/EllipticHarmonicBounds.lean
@@ -0,0 +1,116 @@
+import OAI.Geometry.Perimeter.Regularity.EllipticLocalNorm
+import OAI.Geometry.Perimeter.Regularity.HarmonicTaylor
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace Convolution
+namespace RPISingularHarnack
+open ContinuousLinearMap
+variable {n : ℕ}
+
+theorem compact_kernel_convolution_bounds {k : Euc n → ℝ}
+    (hk : ContDiff ℝ ∞ k) (hc : HasCompactSupport k) :
+    ∃ C : ℝ, 0≤C ∧ ∀ u : Euc n → ℝ, Integrable u volume →
+      (∀ x, ‖(k ⋆[lsmul ℝ ℝ,volume] u) x‖≤C*(∫ z, ‖u z‖)) ∧
+      ∀ x y, ‖(k ⋆[lsmul ℝ ℝ,volume] u) x-(k ⋆[lsmul ℝ ℝ,volume] u) y‖≤
+        C*(∫ z, ‖u z‖)*‖x-y‖ := by
+  obtain ⟨A,hA⟩ := hc.exists_bound_of_continuous hk.continuous
+  obtain ⟨D,hD0,hD⟩ := compactSupport_bounded_derivative (hk.continuous_fderiv (by simp)) hc
+  let C := max A D
+  have hC0 : 0≤C := hD0.trans (le_max_right _ _)
+  have hB (x : Euc n) : ‖k x‖≤C := (hA x).trans (le_max_left _ _)
+  have hLip (x y : Euc n) : ‖k x-k y‖≤C*‖x-y‖ :=
+    Convex.norm_image_sub_le_of_norm_fderiv_le (𝕜 := ℝ)
+      (fun z _ ↦ hk.differentiable (by simp) z)
+      (fun z _ ↦ (hD z).trans (le_max_right _ _)) convex_univ (mem_univ y) (mem_univ x)
+  refine ⟨C,hC0,fun u hu ↦ ?_⟩
+  have hi (x : Euc n) : Integrable (fun z ↦ u z*k (x-z)) volume :=
+    hu.mul_bdd ((hk.continuous.comp (continuous_const.sub continuous_id)).aestronglyMeasurable)
+      (ae_of_all _ fun z ↦ hB (x-z))
+  constructor
+  · intro x
+    rw [convolution_eq_swap]
+    change ‖∫ z, k (x-z)*u z‖≤_
+    calc
+      _ ≤ ∫ z, C*‖u z‖ := norm_integral_le_of_norm_le (hu.norm.const_mul C)
+        (ae_of_all _ fun z ↦ by rw [norm_mul]; exact mul_le_mul_of_nonneg_right (hB _) (norm_nonneg _))
+      _ = _ := integral_const_mul _ _
+  · intro x y
+    rw [convolution_eq_swap,convolution_eq_swap]
+    change ‖(∫ z, k (x-z)*u z)-(∫ z, k (y-z)*u z)‖≤_
+    simp_rw [mul_comm (k _)]
+    rw [←integral_sub (hi x) (hi y)]
+    have h := norm_integral_le_of_norm_le (hu.norm.const_mul (C*‖x-y‖))
+      (ae_of_all volume fun z ↦ show ‖u z*k (x-z)-u z*k (y-z)‖≤C*‖x-y‖*‖u z‖ from by
+        rw [←mul_sub,norm_mul,mul_comm ‖u z‖]
+        apply mul_le_mul_of_nonneg_right _ (norm_nonneg _)
+        simpa only [sub_sub_sub_cancel_right] using hLip (x-z) (y-z))
+    rw [integral_const_mul] at h
+    convert h using 1
+    ring
+
+theorem harmonic_field_representative_bounds (hn : 0<n) :
+    ∃ C : ℝ, 0≤C ∧ ∀ u : VectorL2 n,
+      DivFreeLpOn u (Metric.ball 0 1) → CurlFreeLpOn u (Metric.ball 0 1) →
+      ∃ G : Euc n → Euc n, ContDiff ℝ ∞ G ∧
+        (u : Euc n → Euc n)=ᵐ[volume.restrict (Metric.ball 0 (1/2))] G ∧
+        (∀ x, ‖G x‖≤C*‖u‖) ∧ ∀ x y, ‖G x-G y‖≤C*‖u‖*‖x-y‖ := by
+  obtain ⟨M,hM,hB⟩ := compact_kernel_convolution_bounds (scaledHarmonicKernel_contDiff (n := n) 4)
+    ((isCompact_closedBall 0 (1/(4:ℝ))).of_isClosed_subset (isClosed_tsupport _)
+      (scaledHarmonicKernel_tsupport (by norm_num)))
+  let C := (n:ℝ)*M*ballL2Mass n 1
+  refine ⟨C,mul_nonneg (mul_nonneg (Nat.cast_nonneg _) hM) (norm_nonneg _),fun u hd hc ↦ ?_⟩
+  let b := EuclideanSpace.basisFun (Fin n) ℝ
+  let w (j : Fin n) := localScalarComponent u (b j)
+  let f (j : Fin n) := harmonicRegularization (w j) 4
+  have hI (j : Fin n) : Integrable (w j) volume := localScalarComponent_integrable u (b j)
+  have hW (j : Fin n) : WeakHarmonicOn (w j) (Metric.ball 0 1) :=
+    localScalarComponent_weakHarmonic hd hc (b j)
+  have hf (j : Fin n) : ContDiff ℝ ∞ (f j) := harmonicRegularization_contDiff (hI j).locallyIntegrable (by norm_num)
+  have he (j : Fin n) : (fun x ↦ inner ℝ (b j) (u x))=ᵐ[volume.restrict (Metric.ball 0 (1/2))] f j := by
+    filter_upwards [ae_restrict_of_ae (weakHarmonic_ae_eq_regularization (hI j).locallyIntegrable (hW j) hn),
+      ae_restrict_mem measurableSet_ball] with x hx hxB
+    have hx1 : x∈Metric.ball (0:Euc n) 1 := Metric.ball_subset_ball (by norm_num) hxB
+    have heq : w j x=inner ℝ (b j) (u x) := Set.indicator_of_mem hx1 _
+    rw [←heq]
+    apply hx (1/2) (by norm_num) _ 4 (by norm_num)
+    intro y hy
+    have ht := dist_triangle y x 0
+    rw [Metric.mem_ball] at hy hxB ⊢
+    linarith
+  have hMass (j : Fin n) : (∫ x, ‖w j x‖)≤ballL2Mass n 1*‖u‖ := by
+    simpa only [b.orthonormal.norm_eq_one j,one_mul] using localScalarComponent_L1 u (b j)
+  have hfb (j : Fin n) (x : Euc n) : ‖f j x‖≤M*ballL2Mass n 1*‖u‖ := by
+    exact ((hB (w j) (hI j)).1 x).trans (by rw [mul_assoc]; exact mul_le_mul_of_nonneg_left (hMass j) hM)
+  have hfl (j : Fin n) (x y : Euc n) : ‖f j x-f j y‖≤M*ballL2Mass n 1*‖u‖*‖x-y‖ := by
+    exact ((hB (w j) (hI j)).2 x y).trans (mul_le_mul_of_nonneg_right
+      (by rw [mul_assoc]; exact mul_le_mul_of_nonneg_left (hMass j) hM) (norm_nonneg _))
+  let G (x : Euc n) := ∑ j : Fin n, f j x • b j
+  refine ⟨G,ContDiff.sum (fun j _ ↦ (hf j).smul contDiff_const),?_,?_,?_⟩
+  · filter_upwards [ae_all_iff.mpr he] with x hx
+    calc
+      u x = ∑ j : Fin n, inner ℝ (b j) (u x) • b j := by
+        simpa only [OrthonormalBasis.repr_apply_apply] using (b.sum_repr (u x)).symm
+      _ = G x := Finset.sum_congr rfl (fun j _ ↦ by rw [hx j])
+  · intro x
+    calc
+      ‖G x‖ ≤ ∑ j : Fin n, ‖f j x • b j‖ := norm_sum_le _ _
+      _ ≤ ∑ j : Fin n, M*ballL2Mass n 1*‖u‖ := Finset.sum_le_sum (fun j _ ↦ by
+        simpa only [norm_smul,b.orthonormal.norm_eq_one j,mul_one] using hfb j x)
+      _ = C*‖u‖ := by simp only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul,C]; ring
+  · intro x y
+    change ‖(∑ j : Fin n, f j x • b j)-(∑ j : Fin n, f j y • b j)‖≤_
+    rw [←Finset.sum_sub_distrib]
+    calc
+      _ ≤ ∑ j : Fin n, ‖f j x • b j-f j y • b j‖ := norm_sum_le _ _
+      _ ≤ ∑ j : Fin n, M*ballL2Mass n 1*‖u‖*‖x-y‖ := Finset.sum_le_sum (fun j _ ↦ by
+        simpa only [←sub_smul,norm_smul,b.orthonormal.norm_eq_one j,mul_one] using hfl j x y)
+      _ = C*‖u‖*‖x-y‖ := by simp only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul,C]; ring
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/EllipticHilbert.lean b/lean/OAI/Geometry/Perimeter/Regularity/EllipticHilbert.lean
new file mode 100644
index 0000000000000000000000000000000000000000..56975b142960015638d8ee7eaf2d2e0309bac0e8
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/EllipticHilbert.lean
@@ -0,0 +1,40 @@
+import Mathlib
+
+namespace OAI
+
+noncomputable section
+open Set
+open scoped InnerProductSpace
+namespace RPISingularHarnack
+
+theorem projection_eq_of_pairing_eq {H : Type*} [NormedAddCommGroup H]
+    [InnerProductSpace ℝ H] (K : Submodule ℝ H) [K.HasOrthogonalProjection]
+    {u e : H} (h : ∀ w∈K, inner ℝ u w=inner ℝ e w) :
+    K.orthogonalProjectionOnto u=K.orthogonalProjectionOnto e := by
+  apply ext_inner_right ℝ
+  intro w
+  simp only [K.inner_orthogonalProjectionOnto_eq_of_mem_right]
+  exact h w w.property
+
+theorem harmonic_projection_error_le {H : Type*} [NormedAddCommGroup H]
+    [InnerProductSpace ℝ H] (K : Submodule ℝ H) [K.HasOrthogonalProjection]
+    {u e : H} (h : ∀ w∈K, inner ℝ u w=inner ℝ e w) :
+    ‖u-(u-K.starProjection u)‖≤‖e‖ := by
+  rw [sub_sub_cancel,Submodule.starProjection_apply,
+    projection_eq_of_pairing_eq K h,Submodule.norm_coe]
+  exact K.norm_orthogonalProjectionOnto_apply_le e
+
+theorem harmonic_projection_exists {H : Type*} [NormedAddCommGroup H]
+    [InnerProductSpace ℝ H] (K : Submodule ℝ H) [K.HasOrthogonalProjection]
+    {u e : H} (h : ∀ w∈K, inner ℝ u w=inner ℝ e w) :
+    ∃ g : H, g∈Kᗮ ∧ u-g∈K ∧ ‖u-g‖≤‖e‖ := by
+  refine ⟨u-K.starProjection u,K.sub_starProjection_mem_orthogonal u,?_,
+    harmonic_projection_error_le K h⟩
+  rw [sub_sub_cancel]
+  exact (K.orthogonalProjectionOnto u).property
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/EllipticHodge.lean b/lean/OAI/Geometry/Perimeter/Regularity/EllipticHodge.lean
new file mode 100644
index 0000000000000000000000000000000000000000..45566674ca55699ba906a271a20384fcec32e990
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/EllipticHodge.lean
@@ -0,0 +1,80 @@
+import OAI.Geometry.Perimeter.Regularity.WeakHarmonicSmooth
+import OAI.Geometry.Perimeter.Regularity.EllipticHodgeIdentity
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+theorem curlTest_inner (u : VectorL2 n) {U : Set (Euc n)}
+    (φ : SmoothLocalTest U) (a b : Euc n) :
+    inner ℝ u (curlTestLp φ a b)=∫ x, inner ℝ (u x) (curlTest φ.val a b x) := by
+  rw [MeasureTheory.L2.inner_def]
+  apply integral_congr_ae
+  filter_upwards [curlTestLp_ae φ a b] with x hx
+  rw [hx]
+
+theorem curlTest_integrable_pairing (u : VectorL2 n) {U : Set (Euc n)}
+    (φ : SmoothLocalTest U) (a b : Euc n) :
+    Integrable (fun x ↦ inner ℝ (u x) (curlTest φ.val a b x)) volume := by
+  apply (MeasureTheory.L2.integrable_inner (𝕜 := ℝ) u (curlTestLp φ a b)).congr
+  filter_upwards [curlTestLp_ae φ a b] with x hx
+  rw [hx]
+
+theorem div_curl_weak_harmonic {u : VectorL2 n} {U : Set (Euc n)}
+    (hd : DivFreeLpOn u U) (hc : CurlFreeLpOn u U) (a : Euc n) :
+    WeakHarmonicOn (fun x ↦ inner ℝ a (u x)) U := by
+  intro ψ hψ hcψ hsψ
+  let φ : SmoothLocalTest U := ⟨ψ,hψ,hcψ,hsψ⟩
+  let b := EuclideanSpace.basisFun (Fin n) ℝ
+  have hi : Integrable (fun x ↦ inner ℝ (u x) (gradient (scalarDirection ψ a) x)) volume :=
+    testGradient_integrable_pairing u (directionLocalTest φ a)
+  have hi' (j : Fin n) : Integrable (fun x ↦ inner ℝ (u x)
+      (curlTest (scalarDirection ψ (b j)) (b j) a x)) volume :=
+    curlTest_integrable_pairing u (directionLocalTest φ (b j)) (b j) a
+  calc
+    (∫ x, inner ℝ a (u x)*divergence (gradient ψ) x)=
+      ∫ x, inner ℝ (u x) (gradient (scalarDirection ψ a) x)+
+        ∑ j : Fin n, inner ℝ (u x) (curlTest (scalarDirection ψ (b j)) (b j) a x) := by
+      congr 1
+      funext x
+      rw [real_inner_comm,mul_comm,←real_inner_smul_right,div_curl_test_identity hψ]
+      simp only [inner_add_right,inner_sum,b]
+    _ = (∫ x, inner ℝ (u x) (gradient (scalarDirection ψ a) x))+
+        ∑ j : Fin n, ∫ x, inner ℝ (u x) (curlTest (scalarDirection ψ (b j)) (b j) a x) := by
+      rw [integral_add hi (integrable_finsetSum Finset.univ (fun j _ ↦ hi' j)),
+        integral_finsetSum Finset.univ (fun j _ ↦ hi' j)]
+    _ = 0 := by
+      rw [show (∫ x, inner ℝ (u x) (gradient (scalarDirection ψ a) x))=0 from hd (directionLocalTest φ a)]
+      simp only [zero_add]
+      apply Finset.sum_eq_zero
+      intro j _
+      exact (curlTest_inner u (directionLocalTest φ (b j)) (b j) a).symm.trans (hc _ _ _)
+
+theorem vectorLp_component_locallyIntegrable (u : VectorL2 n) (a : Euc n) :
+    LocallyIntegrable (fun x ↦ inner ℝ a (u x)) volume :=
+  ((innerSL ℝ a).comp_memLp u).locallyIntegrable (by norm_num)
+
+theorem div_curl_smooth_representative {u : VectorL2 n} {U : Set (Euc n)}
+    (hd : DivFreeLpOn u U) (hc : CurlFreeLpOn u U) (hn : 0<n)
+    (c : Euc n) {r : ℝ} (hr : 0<r) (hU : Metric.ball c (2*r) ⊆ U) :
+    ∃ G : Euc n → Euc n, ContDiff ℝ ∞ G ∧
+      (u : Euc n → Euc n)=ᵐ[volume.restrict (Metric.ball c r)] G := by
+  let b := EuclideanSpace.basisFun (Fin n) ℝ
+  have he (j : Fin n) := weakHarmonic_smooth_representative_ball
+    (vectorLp_component_locallyIntegrable u (b j)) (div_curl_weak_harmonic hd hc (b j)) hn c hr hU
+  choose f hf hfe using he
+  refine ⟨fun x ↦ ∑ j : Fin n, f j x • b j,ContDiff.sum (fun j _ ↦ (hf j).smul contDiff_const),?_⟩
+  filter_upwards [ae_all_iff.mpr hfe] with x hx
+  calc
+    u x = ∑ j : Fin n, inner ℝ (b j) (u x) • b j := by simpa only [OrthonormalBasis.repr_apply_apply] using (b.sum_repr (u x)).symm
+    _ = _ := Finset.sum_congr rfl (fun j _ ↦ by rw [hx j])
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/EllipticHodgeIdentity.lean b/lean/OAI/Geometry/Perimeter/Regularity/EllipticHodgeIdentity.lean
new file mode 100644
index 0000000000000000000000000000000000000000..80560ed7de9d674e295f799bf6561bd9cdb9f347
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/EllipticHodgeIdentity.lean
@@ -0,0 +1,73 @@
+import OAI.Geometry.Perimeter.Regularity.EllipticCurl
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+theorem inner_fderiv_gradient {φ : Euc n → ℝ} (hφ : ContDiff ℝ ∞ φ)
+    (x a b : Euc n) :
+    inner ℝ (fderiv ℝ (gradient φ) x a) b=fderiv ℝ (fderiv ℝ φ) x a b := by
+  have hd : ContDiff ℝ ∞ (fderiv ℝ φ) := hφ.fderiv_right (by simp)
+  have he : fderiv ℝ (gradient φ) x=
+      (InnerProductSpace.toDual ℝ (Euc n)).symm.toContinuousLinearMap.comp
+        (fderiv ℝ (fderiv ℝ φ) x) := by
+    exact (((InnerProductSpace.toDual ℝ (Euc n)).symm.toContinuousLinearMap).hasFDerivAt.comp x
+      (hd.differentiable (by simp) x).hasFDerivAt).fderiv
+  rw [he]
+  change inner ℝ ((InnerProductSpace.toDual ℝ (Euc n)).symm (fderiv ℝ (fderiv ℝ φ) x a)) b=_
+  exact InnerProductSpace.toDual_symm_apply
+
+theorem gradient_scalarDirection {φ : Euc n → ℝ} (hφ : ContDiff ℝ ∞ φ)
+    (x a : Euc n) : gradient (scalarDirection φ a) x=fderiv ℝ (gradient φ) x a := by
+  apply ext_inner_right ℝ
+  intro b
+  rw [inner_gradient_left,scalarDirection_fderiv hφ,inner_fderiv_gradient hφ]
+  exact (hφ.contDiffAt.isSymmSndFDerivAt (by
+    simp only [minSmoothness_of_isRCLikeNormedField]; exact WithTop.coe_le_coe.mpr le_top)) b a
+
+theorem laplacian_scalarDirection_sum {φ : Euc n → ℝ} (hφ : ContDiff ℝ ∞ φ)
+    (x : Euc n) :
+    divergence (gradient φ) x=∑ j : Fin n, scalarDirection (scalarDirection φ
+      ((EuclideanSpace.basisFun (Fin n) ℝ) j)) ((EuclideanSpace.basisFun (Fin n) ℝ) j) x := by
+  unfold divergence
+  apply Finset.sum_congr rfl
+  intro j _
+  rw [real_inner_comm,inner_fderiv_gradient hφ]
+  symm
+  exact scalarDirection_fderiv hφ _ _ _
+
+theorem div_curl_test_identity {φ : Euc n → ℝ} (hφ : ContDiff ℝ ∞ φ)
+    (a x : Euc n) :
+    divergence (gradient φ) x • a=gradient (scalarDirection φ a) x+
+      ∑ j : Fin n, curlTest (scalarDirection φ ((EuclideanSpace.basisFun (Fin n) ℝ) j))
+        ((EuclideanSpace.basisFun (Fin n) ℝ) j) a x := by
+  let b := EuclideanSpace.basisFun (Fin n) ℝ
+  have hs : (∑ j : Fin n, scalarDirection (scalarDirection φ (b j)) a x • b j)=
+      gradient (scalarDirection φ a) x := by
+    calc
+      _ = ∑ j : Fin n, inner ℝ (b j) (gradient (scalarDirection φ a) x) • b j := by
+        apply Finset.sum_congr rfl
+        intro j _
+        congr 1
+        rw [real_inner_comm,inner_gradient_left]
+        change fderiv ℝ (scalarDirection φ (b j)) x a=fderiv ℝ (scalarDirection φ a) x (b j)
+        rw [scalarDirection_fderiv hφ,scalarDirection_fderiv hφ]
+        exact (hφ.contDiffAt.isSymmSndFDerivAt (by
+          simp only [minSmoothness_of_isRCLikeNormedField]; exact WithTop.coe_le_coe.mpr le_top)) a (b j)
+      _ = _ := by simpa only [OrthonormalBasis.repr_apply_apply] using b.sum_repr (gradient (scalarDirection φ a) x)
+  change divergence (gradient φ) x • a=gradient (scalarDirection φ a) x+
+    ∑ j : Fin n, curlTest (scalarDirection φ (b j)) (b j) a x
+  simp only [curlTest,Finset.sum_sub_distrib,←Finset.sum_smul,hs]
+  rw [show (∑ j : Fin n, scalarDirection (scalarDirection φ (b j)) (b j) x)=
+    divergence (gradient φ) x from (laplacian_scalarDirection_sum hφ x).symm]
+  abel
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/EllipticIteration.lean b/lean/OAI/Geometry/Perimeter/Regularity/EllipticIteration.lean
new file mode 100644
index 0000000000000000000000000000000000000000..7d895c58344e8922a793974f2c81e3a0b7c82572
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/EllipticIteration.lean
@@ -0,0 +1,106 @@
+import OAI.Geometry.Perimeter.Regularity.EllipticContraction
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+theorem linear_iteration_scalar {K H a B s Q E E' d : ℝ}
+    (hK : 1 ≤ K) (hH : 0 ≤ H) (ha : 0 ≤ a) (hB : 0 ≤ B) (hs : 0 ≤ s)
+    (hsmall : 32*K^2*a ≤ 1) (hforce : 16*K*B ≤ H)
+    (hQ0 : 0 ≤ Q) (hQ : Q ≤ 2*K*H) (hE : E ≤ H^2*s^2)
+    (hd0 : 0 ≤ d) (hd : d^2 ≤ K*E)
+    (hnext : E' ≤ E/16+K*(a*s*Q+B*s)^2) :
+    d ≤ K*H*s ∧ E' ≤ H^2*(s/2)^2 := by
+  have hK0 : 0 ≤ K := by linarith
+  have hKH : 0 ≤ K*H*s := by positivity
+  have hdE := mul_le_mul_of_nonneg_left hE hK0
+  have hKK : K ≤ K^2 := by nlinarith
+  have hdE' := mul_le_mul_of_nonneg_right hKK (mul_nonneg (sq_nonneg H) (sq_nonneg s))
+  have hd' : d ≤ K*H*s := (sq_le_sq₀ hd0 hKH).mp (by nlinarith)
+  have hsmallH := mul_le_mul_of_nonneg_right hsmall hH
+  have hAQ := mul_le_mul_of_nonneg_left hQ (mul_nonneg hK0 ha)
+  have hbase : K*(a*Q+B) ≤ H/8 := by nlinarith
+  have hb : K*(a*s*Q+B*s) ≤ H*s/8 := by
+    have hh := mul_le_mul_of_nonneg_right hbase hs
+    nlinarith
+  have hb0 : 0 ≤ K*(a*s*Q+B*s) := by positivity
+  have hb2 := pow_le_pow_left₀ hb0 hb 2
+  have hb3 := mul_le_mul_of_nonneg_right hKK (sq_nonneg (a*s*Q+B*s))
+  refine ⟨hd',?_⟩
+  nlinarith [sq_nonneg (H*s)]
+
+theorem linear_energy_iteration {θ δ K a B H R : ℝ}
+    (hθ : 0 < θ) (hθ1 : θ < 1) (hK : 1 ≤ K)
+    (hc : LinearContractionProperty n θ δ K)
+    (ha : 0 ≤ a) (hB : 0 ≤ B) (hH : 0 ≤ H) (hR : 0 < R)
+    (hsmall : 32*K^2*a ≤ 1) (hforce : 16*K*B ≤ H)
+    {A : Euc n → Euc n →L[ℝ] Euc n} (hA : Continuous A)
+    {f : Euc n → ℝ} (hf : ContDiff ℝ 1 f) {F : Euc n → Euc n} (hF : Continuous F)
+    (p : Euc n) (h : WeakLinearEquation A f F (Metric.ball p R))
+    (hnear : ∀ x∈Metric.ball p R, ‖ContinuousLinearMap.id ℝ (Euc n)-A x‖ ≤ δ)
+    (hosc : ∀ j x, x∈Metric.ball p (θ^j*R) →
+      ‖A p-A x‖ ≤ a*(1/2)^j ∧ ‖F x-F p‖ ≤ B*(1/2)^j)
+    (hinit : scaledGradientEnergy f p R 0 ≤ H^2) :
+    ∃ q : ℕ → Euc n, q 0=0 ∧
+      (∀ j, ‖q j‖ ≤ 2*K*H*(1-(1/2)^j)) ∧
+      (∀ j, scaledGradientEnergy f p (θ^j*R) (q j) ≤ H^2*((1/2)^j)^2) ∧
+      (∀ j, ‖q (j+1)-q j‖ ≤ K*H*(1/2)^j) := by
+  have hball (j : ℕ) : Metric.ball p (θ^j*R) ⊆ Metric.ball p R :=
+    Metric.ball_subset_ball (by nlinarith [pow_le_one₀ (n := j) hθ.le hθ1.le])
+  have hstep (j : ℕ) (q : Euc n) : ∃ q' : Euc n,
+      ‖q'-q‖^2 ≤ K*scaledGradientEnergy f p (θ^j*R) q ∧
+      scaledGradientEnergy f p (θ^(j+1)*R) q' ≤ scaledGradientEnergy f p (θ^j*R) q/16+
+        K*(a*(1/2)^j*‖q‖+B*(1/2)^j)^2 := by
+    have hh := hc A hA f hf F hF p (θ^j*R) (by positivity) (h.mono (hball j))
+      (A p) (F p) q (a*(1/2)^j) (B*(1/2)^j) (by positivity) (by positivity)
+      (fun x hx ↦ ⟨hnear x (hball j hx),(hosc j x hx).1,(hosc j x hx).2⟩)
+    simpa only [pow_succ, mul_assoc,mul_left_comm,mul_comm] using hh
+  choose next hnext using hstep
+  let q : ℕ → Euc n := fun j ↦ Nat.rec 0 (fun j z ↦ next j z) j
+  have hq0 : q 0=0 := rfl
+  have hqnext (j : ℕ) : q (j+1)=next j (q j) := rfl
+  have hbounds : ∀ j, ‖q j‖ ≤ 2*K*H*(1-(1/2)^j) ∧
+      scaledGradientEnergy f p (θ^j*R) (q j) ≤ H^2*((1/2)^j)^2 ∧
+      ‖q (j+1)-q j‖ ≤ K*H*(1/2)^j := by
+    intro j
+    induction j with
+    | zero =>
+      have hinit' : scaledGradientEnergy f p (θ^0*R) (q 0) ≤ H^2*((1/2)^0)^2 := by
+        simpa only [hq0,pow_zero,one_pow,one_mul,mul_one] using hinit
+      obtain ⟨hd,_⟩ := linear_iteration_scalar hK hH ha hB (by positivity : 0 ≤ (1/2:ℝ)^0)
+        hsmall hforce (norm_nonneg (q 0)) (by simp only [hq0,norm_zero]; positivity) hinit'
+        (norm_nonneg (next 0 (q 0)-q 0)) (hnext 0 (q 0)).1 (hnext 0 (q 0)).2
+      exact ⟨by simp only [hq0,norm_zero,pow_zero,sub_self,mul_zero,le_refl],hinit',by simpa only [hqnext] using hd⟩
+    | succ j ih =>
+      have hqnorm : ‖q j‖ ≤ 2*K*H := ih.1.trans (by
+        have hh : 0 ≤ 2*K*H*(1/2)^j := by positivity
+        nlinarith)
+      obtain ⟨_,hE⟩ := linear_iteration_scalar hK hH ha hB (by positivity : 0 ≤ (1/2:ℝ)^j)
+        hsmall hforce (norm_nonneg (q j)) hqnorm ih.2.1
+        (norm_nonneg (next j (q j)-q j)) (hnext j (q j)).1 (hnext j (q j)).2
+      have hE' : scaledGradientEnergy f p (θ^(j+1)*R) (q (j+1)) ≤ H^2*((1/2)^(j+1))^2 := by
+        simpa only [hqnext,pow_succ,div_eq_mul_inv,one_div,one_mul] using hE
+      have hn : ‖q (j+1)‖ ≤ 2*K*H*(1-(1/2)^(j+1)) := by
+        have htri := norm_add_le (q j) (q (j+1)-q j)
+        have he : q j + (q (j+1)-q j) = q (j+1) := by abel
+        rw [he] at htri
+        rw [pow_succ]
+        nlinarith [ih.1,ih.2.2]
+      have hn' : ‖q (j+1)‖ ≤ 2*K*H := hn.trans (by
+        have hh : 0 ≤ 2*K*H*(1/2)^(j+1) := by positivity
+        nlinarith)
+      obtain ⟨hd,_⟩ := linear_iteration_scalar hK hH ha hB (by positivity : 0 ≤ (1/2:ℝ)^(j+1))
+        hsmall hforce (norm_nonneg (q (j+1))) hn' hE'
+        (norm_nonneg (next (j+1) (q (j+1))-q (j+1))) (hnext (j+1) (q (j+1))).1 (hnext (j+1) (q (j+1))).2
+      exact ⟨hn,hE',by simpa only [hqnext] using hd⟩
+  exact ⟨q,hq0,fun j ↦ (hbounds j).1,fun j ↦ (hbounds j).2.1,fun j ↦ (hbounds j).2.2⟩
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/EllipticLinearStep.lean b/lean/OAI/Geometry/Perimeter/Regularity/EllipticLinearStep.lean
new file mode 100644
index 0000000000000000000000000000000000000000..8540db75b2415562c9f88b84de3f16790aac5d1c
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/EllipticLinearStep.lean
@@ -0,0 +1,91 @@
+import OAI.Geometry.Perimeter.Regularity.EllipticContinuousComparison
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+def WeakLinearEquation (A : Euc n → Euc n →L[ℝ] Euc n) (f : Euc n → ℝ)
+    (F : Euc n → Euc n) (U : Set (Euc n)) : Prop :=
+  ∀ φ : SmoothLocalTest U,
+    (∫ x, inner ℝ (A x (gradient f x)) (gradient φ.val x)) =
+      ∫ x, inner ℝ (F x) (gradient φ.val x)
+
+def linearResidual (A : Euc n → Euc n →L[ℝ] Euc n) (f : Euc n → ℝ)
+    (F : Euc n → Euc n) (A₀ : Euc n →L[ℝ] Euc n) (F₀ q : Euc n) (x : Euc n) :=
+  (gradient f x-q) - A x (gradient f x) + F x + (A₀ q-F₀)
+
+theorem linearResidual_continuous {A : Euc n → Euc n →L[ℝ] Euc n}
+    {f : Euc n → ℝ} {F : Euc n → Euc n} (hA : Continuous A) (hf : ContDiff ℝ 1 f)
+    (hF : Continuous F) (A₀ : Euc n →L[ℝ] Euc n) (F₀ q : Euc n) :
+    Continuous (linearResidual A f F A₀ F₀ q) :=
+  ((((gradient_continuous_C1 hf).sub continuous_const).sub
+    (hA.clm_apply (gradient_continuous_C1 hf))).add hF).add continuous_const
+
+theorem linearResidual_pairing {A : Euc n → Euc n →L[ℝ] Euc n}
+    {f : Euc n → ℝ} {F : Euc n → Euc n} {U : Set (Euc n)}
+    (hA : Continuous A) (hf : ContDiff ℝ 1 f) (hF : Continuous F)
+    (h : WeakLinearEquation A f F U) (A₀ : Euc n →L[ℝ] Euc n) (F₀ q : Euc n)
+    (φ : SmoothLocalTest U) :
+    (∫ x, inner ℝ (gradient (fun y ↦ f y-inner ℝ q y) x) (gradient φ.val x)) =
+      ∫ x, inner ℝ (linearResidual A f F A₀ F₀ q x) (gradient φ.val x) := by
+  have hg := inner_gradient_pairing_integrable (gradient_continuous_C1 hf) φ
+  have hq := inner_gradient_pairing_integrable (continuous_const : Continuous (fun _ : Euc n ↦ q)) φ
+  have hAf := inner_gradient_pairing_integrable (hA.clm_apply (gradient_continuous_C1 hf)) φ
+  have hFi := inner_gradient_pairing_integrable hF φ
+  have hc := inner_gradient_pairing_integrable
+    (continuous_const : Continuous (fun _ : Euc n ↦ A₀ q-F₀)) φ
+  simp_rw [gradient_sub_linear (hf.differentiable (by simp)),linearResidual,inner_add_left,inner_sub_left]
+  simp_rw [← inner_sub_left (A₀ q) F₀]
+  erw [integral_add ((hg.sub hq).sub hAf |>.add hFi) hc,
+    integral_add ((hg.sub hq).sub hAf) hFi,integral_sub (hg.sub hq) hAf,
+    constant_gradient_pairing_zero φ (A₀ q-F₀),h φ]
+  ring_nf
+  rfl
+
+theorem linearResidual_bound {A : Euc n → Euc n →L[ℝ] Euc n}
+    {f : Euc n → ℝ} {F : Euc n → Euc n} (A₀ : Euc n →L[ℝ] Euc n) (F₀ q x : Euc n)
+    {δ α β : ℝ} (hδ : ‖ContinuousLinearMap.id ℝ (Euc n)-A x‖≤δ)
+    (hα : ‖A₀-A x‖≤α) (hβ : ‖F x-F₀‖≤β) :
+    ‖linearResidual A f F A₀ F₀ q x‖≤δ*‖gradient f x-q‖+α*‖q‖+β := by
+  have he : linearResidual A f F A₀ F₀ q x =
+      (ContinuousLinearMap.id ℝ (Euc n)-A x) (gradient f x-q)+(A₀-A x) q+(F x-F₀) := by
+    simp only [linearResidual,sub_apply,ContinuousLinearMap.id_apply,map_sub]
+    abel
+  rw [he]
+  have h1 := (ContinuousLinearMap.id ℝ (Euc n)-A x).le_opNorm (gradient f x-q)
+  have h2 := (A₀-A x).le_opNorm q
+  have hh := norm_add_le ((ContinuousLinearMap.id ℝ (Euc n)-A x) (gradient f x-q)+(A₀-A x) q) (F x-F₀)
+  have hh' := norm_add_le ((ContinuousLinearMap.id ℝ (Euc n)-A x) (gradient f x-q)) ((A₀-A x) q)
+  have h1' := mul_le_mul_of_nonneg_right hδ (norm_nonneg (gradient f x-q))
+  have h2' := mul_le_mul_of_nonneg_right hα (norm_nonneg q)
+  linarith
+
+theorem continuous_residual_energy_bound {e W : Euc n → Euc n} (he : Continuous e)
+    (hW : Continuous W) {δ b : ℝ} (hδ : 0≤δ) (hb : 0≤b)
+    (h : ∀ x∈Metric.ball (0:Euc n) 1, ‖e x‖≤δ*‖W x‖+b) :
+    (∫ x in Metric.ball (0:Euc n) 1, ‖e x‖^2) ≤
+      2*δ^2*(∫ x in Metric.ball (0:Euc n) 1, ‖W x‖^2)+2*b^2*volume.real (Metric.ball (0:Euc n) 1) := by
+  have hI (G : Euc n → Euc n) (hG : Continuous G) :
+      IntegrableOn (fun x ↦ ‖G x‖^2) (Metric.ball (0:Euc n) 1) volume :=
+    ((hG.norm.pow 2).continuousOn.integrableOn_compact (isCompact_closedBall 0 1)).mono_set Metric.ball_subset_closedBall
+  have : IsFiniteMeasure (volume.restrict (Metric.ball (0:Euc n) 1)) := ⟨by
+    rw [Measure.restrict_apply_univ]; exact measure_ball_lt_top⟩
+  calc
+    _ ≤ ∫ x in Metric.ball (0:Euc n) 1, 2*δ^2*‖W x‖^2+2*b^2 := by
+      apply integral_mono_ae (hI e he) (((hI W hW).const_mul _).add (integrable_const _))
+      filter_upwards [ae_restrict_mem measurableSet_ball] with x hx
+      have hsq := (sq_le_sq₀ (norm_nonneg (e x))
+        (add_nonneg (mul_nonneg hδ (norm_nonneg (W x))) hb)).2 (h x hx)
+      change ‖e x‖^2≤2*δ^2*‖W x‖^2+2*b^2
+      nlinarith [sq_nonneg (δ*‖W x‖-b)]
+    _ = _ := by rw [integral_add ((hI W hW).const_mul _) (integrable_const _),integral_const_mul,integral_const]; simp [smul_eq_mul,mul_comm]
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/EllipticLocalExtension.lean b/lean/OAI/Geometry/Perimeter/Regularity/EllipticLocalExtension.lean
new file mode 100644
index 0000000000000000000000000000000000000000..852a718faa501dd7bc40991c79413c7391bfde96
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/EllipticLocalExtension.lean
@@ -0,0 +1,80 @@
+import OAI.Geometry.Perimeter.Regularity.EllipticDifference
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+ theorem contDiff_cutoff {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V]
+    {f : Euc n → V} {U : Set (Euc n)} (hU : IsOpen U) {k : ℕ∞}
+    (hf : ContDiffOn ℝ k f U) {χ : Euc n → ℝ} (hχ : ContDiff ℝ k χ)
+    (hsχ : tsupport χ⊆U) : ContDiff ℝ k (fun x ↦ χ x•f x) := by
+  rw [contDiff_iff_contDiffAt]
+  intro x
+  by_cases hx : x∈tsupport χ
+  · exact hχ.contDiffAt.smul (hf.contDiffAt (hU.mem_nhds (hsχ hx)))
+  · apply (contDiffAt_const (c := (0:V))).congr_of_eventuallyEq
+    filter_upwards [(isClosed_tsupport χ).isOpen_compl.mem_nhds hx] with y hy
+    simp only [image_eq_zero_of_notMem_tsupport hy,zero_smul]
+
+ theorem local_compact_extension {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V]
+    {f : Euc n → V} {U : Set (Euc n)} (hU : IsOpen U) {p : Euc n} (hp : p∈U)
+    {k : ℕ∞} (hf : ContDiffOn ℝ k f U) :
+    ∃ r : ℝ, 0<r ∧ Metric.closedBall p (4*r)⊆U ∧ ∃ g : Euc n → V,
+      ContDiff ℝ k g ∧ HasCompactSupport g ∧ ∀ x∈Metric.ball p (2*r), g x=f x := by
+  obtain ⟨s,hs,hsU⟩ := Metric.isOpen_iff.mp hU p hp
+  let r := s/8
+  let χ : ContDiffBump p := ⟨s/4,s/2,by positivity,by linarith⟩
+  have hsc : tsupport χ⊆U := by
+    rw [χ.tsupport_eq]
+    exact (Metric.closedBall_subset_ball (by dsimp [χ]; linarith)).trans hsU
+  refine ⟨r,by dsimp [r]; positivity,?_,fun x ↦ χ x•f x,
+    contDiff_cutoff hU hf χ.contDiff hsc,χ.hasCompactSupport.smul_right,?_⟩
+  · exact (Metric.closedBall_subset_ball (by dsimp [r]; linarith)).trans hsU
+  · intro x hx
+    have hh : x∈Metric.closedBall p χ.rIn := by
+      rw [Metric.mem_closedBall]
+      have hd := Metric.mem_ball.mp hx
+      dsimp [χ,r] at *
+      linarith
+    change χ x•f x=f x
+    rw [χ.one_of_mem_closedBall hh,one_smul]
+
+ theorem compact_C1_gradient_bound {f : Euc n → ℝ} (hf : ContDiff ℝ 1 f)
+    (hc : HasCompactSupport f) : ∃ B : ℝ, 0≤B ∧ ∀ x, ‖gradient f x‖≤B := by
+  have hg := gradient_continuous_C1 hf
+  have hgc := gradient_hasCompactSupport hc
+  obtain ⟨B,hB⟩ := hgc.exists_bound_of_continuousOn hg.continuousOn
+  refine ⟨max 0 B,le_max_left _ _,fun x ↦ ?_⟩
+  by_cases hx : x∈tsupport (gradient f)
+  · exact (hB x hx).trans (le_max_right _ _)
+  · rw [image_eq_zero_of_notMem_tsupport hx,norm_zero]
+    exact le_max_left _ _
+
+ theorem differenceQuotient_bound {f : Euc n → ℝ} (hf : Differentiable ℝ f)
+    {B : ℝ} (hB : 0≤B) (hb : ∀ x, ‖gradient f x‖≤B) (e : Euc n) (s : ℝ) (x : Euc n) :
+    |differenceQuotient f e s x|≤B*‖e‖ := by
+  have hnorm (y : Euc n) : ‖fderiv ℝ f y‖≤B := by
+    have he : ‖fderiv ℝ f y‖=‖gradient f y‖ := by
+      rw [gradient,LinearIsometryEquiv.norm_map]
+    rw [he]
+    exact hb y
+  have hh := (convex_univ : Convex ℝ (univ : Set (Euc n))).norm_image_sub_le_of_norm_fderiv_le
+    (fun y _ ↦ hf y) (fun y _ ↦ hnorm y) (mem_univ x) (mem_univ (x+s•e))
+  by_cases hs : s=0
+  · simp only [differenceQuotient,hs,inv_zero,zero_mul,abs_zero]
+    positivity
+  · rw [differenceQuotient,abs_mul,abs_inv]
+    rw [Real.norm_eq_abs,add_sub_cancel_left,norm_smul,Real.norm_eq_abs] at hh
+    have h := mul_le_mul_of_nonneg_left hh (inv_nonneg.mpr (abs_nonneg s))
+    have he : |s|⁻¹*(B*(|s| *‖e‖))=B*‖e‖ := by field_simp [abs_ne_zero.mpr hs]
+    exact h.trans_eq he
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/EllipticLocalNorm.lean b/lean/OAI/Geometry/Perimeter/Regularity/EllipticLocalNorm.lean
new file mode 100644
index 0000000000000000000000000000000000000000..85aa599963169cc5455651b9327f5299938a5c29
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/EllipticLocalNorm.lean
@@ -0,0 +1,88 @@
+import OAI.Geometry.Perimeter.Regularity.EllipticHodge
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+def scalarNormLp (u : VectorL2 n) : Lp ℝ 2 (volume : Measure (Euc n)) :=
+  (Lp.memLp u).norm.toLp (fun x ↦ ‖u x‖)
+
+theorem scalarNormLp_ae (u : VectorL2 n) :
+    (scalarNormLp u : Euc n → ℝ)=ᵐ[volume] fun x ↦ ‖u x‖ := MemLp.coeFn_toLp _
+
+theorem scalarNormLp_norm (u : VectorL2 n) : ‖scalarNormLp u‖=‖u‖ := by
+  rw [scalarNormLp,Lp.norm_toLp,eLpNorm_norm,Lp.norm_def]
+  exact (Lp.memLp u).aestronglyMeasurable
+
+def ballL2Mass (n : ℕ) (r : ℝ) : ℝ :=
+  ‖indicatorConstLp (μ := (volume : Measure (Euc n))) 2 (measurableSet_ball : MeasurableSet (Metric.ball (0:Euc n) r))
+    measure_ball_lt_top.ne (1:ℝ)‖
+
+theorem localL1_le_L2 (u : VectorL2 n) (r : ℝ) :
+    (∫ x in Metric.ball (0:Euc n) r, ‖u x‖) ≤ ballL2Mass n r * ‖u‖ := by
+  rw [←scalarNormLp_norm u]
+  have he : (∫ x in Metric.ball (0:Euc n) r, ‖u x‖)=
+      ∫ x in Metric.ball (0:Euc n) r, scalarNormLp u x :=
+    integral_congr_ae (ae_restrict_of_ae (scalarNormLp_ae u).symm)
+  rw [he,←MeasureTheory.L2.inner_indicatorConstLp_one measurableSet_ball measure_ball_lt_top.ne]
+  exact real_inner_le_norm _ _
+
+theorem WeakHarmonicOn.congr_on {u w : Euc n → ℝ} {U : Set (Euc n)}
+    (hu : WeakHarmonicOn u U) (hw : EqOn u w U) : WeakHarmonicOn w U := by
+  intro φ hφ hcφ hsφ
+  have he : (fun x ↦ w x*divergence (gradient φ) x)=
+      (fun x ↦ u x*divergence (gradient φ) x) := by
+    funext x
+    by_cases hx : x∈U
+    · rw [hw hx]
+    · have hn : x∉tsupport φ := fun hh ↦ hx (hsφ hh)
+      have hz : fderiv ℝ (gradient φ) x=0 :=
+        fderiv_of_notMem_tsupport ℝ (fun hh ↦ hn (tsupport_gradient_subset φ hh))
+      simp [divergence,hz]
+  rw [he]
+  exact hu φ hφ hcφ hsφ
+
+def localScalarComponent (u : VectorL2 n) (a : Euc n) : Euc n → ℝ :=
+  (Metric.ball 0 1).indicator (fun x ↦ inner ℝ a (u x))
+
+theorem localScalarComponent_integrable (u : VectorL2 n) (a : Euc n) :
+    Integrable (localScalarComponent u a) volume := by
+  apply IntegrableOn.integrable_indicator _ measurableSet_ball
+  exact ((((innerSL ℝ a).comp_memLp u).locallyIntegrable (by norm_num)).integrableOn_isCompact
+    (isCompact_closedBall (0:Euc n) 1)).mono_set Metric.ball_subset_closedBall
+
+theorem localScalarComponent_L1 (u : VectorL2 n) (a : Euc n) :
+    (∫ x, ‖localScalarComponent u a x‖) ≤ ‖a‖*ballL2Mass n 1*‖u‖ := by
+  have he : (fun x ↦ ‖localScalarComponent u a x‖)=
+      (Metric.ball 0 1).indicator (fun x ↦ ‖inner ℝ a (u x)‖) := by
+    funext x
+    simp only [localScalarComponent]
+    by_cases hx : x∈Metric.ball (0:Euc n) 1 <;> simp [hx]
+  rw [he,integral_indicator measurableSet_ball]
+  have hI : IntegrableOn (fun x ↦ u x) (Metric.ball (0:Euc n) 1) volume :=
+    integrableOn_Lp_of_measure_ne_top u (by norm_num) measure_ball_lt_top.ne
+  calc
+    _ ≤ ∫ x in Metric.ball (0:Euc n) 1, ‖a‖*‖u x‖ := integral_mono_ae
+      (((((innerSL ℝ a).comp_memLp u).locallyIntegrable (by norm_num)).integrableOn_isCompact
+    (isCompact_closedBall (0:Euc n) 1)).mono_set Metric.ball_subset_closedBall).norm
+      (hI.norm.const_mul _) (ae_of_all _ fun x ↦ norm_inner_le_norm a (u x))
+    _ = ‖a‖*(∫ x in Metric.ball (0:Euc n) 1, ‖u x‖) := integral_const_mul _ _
+    _ ≤ _ := by rw [mul_assoc]; exact mul_le_mul_of_nonneg_left (localL1_le_L2 u 1) (norm_nonneg a)
+
+theorem localScalarComponent_weakHarmonic {u : VectorL2 n}
+    (hd : DivFreeLpOn u (Metric.ball 0 1)) (hc : CurlFreeLpOn u (Metric.ball 0 1))
+    (a : Euc n) : WeakHarmonicOn (localScalarComponent u a) (Metric.ball 0 1) := by
+  apply (div_curl_weak_harmonic hd hc a).congr_on
+  intro x hx
+  change inner ℝ a (u x)=(Metric.ball (0:Euc n) 1).indicator (fun z ↦ inner ℝ a (u z)) x
+  rw [Set.indicator_of_mem hx]
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/EllipticLocality.lean b/lean/OAI/Geometry/Perimeter/Regularity/EllipticLocality.lean
new file mode 100644
index 0000000000000000000000000000000000000000..404ee4805a9793c49b81e47ba704b3896d790f1d
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/EllipticLocality.lean
@@ -0,0 +1,86 @@
+import OAI.Geometry.Perimeter.Regularity.GeoComposition
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+ theorem gradient_eqOn {f g : Euc n → ℝ} {U : Set (Euc n)} (hU : IsOpen U)
+    (h : EqOn f g U) : EqOn (gradient f) (gradient g) U := by
+  intro x hx
+  unfold gradient
+  congr 1
+  exact Filter.EventuallyEq.fderiv_eq (Filter.eventually_of_mem (hU.mem_nhds hx) (fun y hy ↦ h hy))
+
+ theorem fderiv_eqOn {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V]
+    {f g : Euc n → V} {U : Set (Euc n)} (hU : IsOpen U)
+    (h : EqOn f g U) : EqOn (fderiv ℝ f) (fderiv ℝ g) U := by
+  intro x hx
+  exact Filter.EventuallyEq.fderiv_eq (Filter.eventually_of_mem (hU.mem_nhds hx) (fun y hy ↦ h hy))
+
+ theorem flux_pairing_eq {V W : Euc n → Euc n} {U : Set (Euc n)}
+    (h : EqOn V W U) (φ : SmoothLocalTest U) :
+    (fun x ↦ inner ℝ (V x) (gradient φ.val x)) = fun x ↦ inner ℝ (W x) (gradient φ.val x) := by
+  funext x
+  by_cases hx : x∈U
+  · rw [h hx]
+  · have hx' : x∉tsupport φ.val := fun hh ↦ hx (φ.property.2.2 hh)
+    simp only [gradient,fderiv_of_notMem_tsupport ℝ hx',map_zero,inner_zero_right]
+
+ theorem WeakFluxEquation.congr {V W : Euc n → Euc n} {U : Set (Euc n)}
+    (h : WeakFluxEquation V U) (he : EqOn V W U) : WeakFluxEquation W U := by
+  intro φ
+  rw [←flux_pairing_eq he φ]
+  exact h φ
+
+ theorem WeakLinearEquation.congr {A B : Euc n → Euc n →L[ℝ] Euc n}
+    {f g : Euc n → ℝ} {F G : Euc n → Euc n} {U : Set (Euc n)} (hU : IsOpen U)
+    (h : WeakLinearEquation A f F U) (hA : EqOn A B U) (hf : EqOn f g U)
+    (hF : EqOn F G U) : WeakLinearEquation B g G U := by
+  have hAf : EqOn (fun x ↦ A x (gradient f x)) (fun x ↦ B x (gradient g x)) U :=
+    fun x hx ↦ by dsimp only; rw [hA hx,gradient_eqOn hU hf hx]
+  intro φ
+  change (∫ x, inner ℝ (B x (gradient g x)) (gradient φ.val x)) = _
+  rw [←flux_pairing_eq hAf φ,←flux_pairing_eq hF φ]
+  exact h φ
+
+ theorem compact_continuous_bound {V : Type*} [NormedAddCommGroup V]
+    {f : Euc n → V} (hf : Continuous f) (hc : HasCompactSupport f) :
+    ∃ B : ℝ, 0 ≤ B ∧ ∀ x, ‖f x‖ ≤ B := by
+  obtain ⟨B,hB⟩ := hc.exists_bound_of_continuousOn hf.continuousOn
+  refine ⟨max 0 B,le_max_left _ _,fun x ↦ ?_⟩
+  by_cases hx : x∈tsupport f
+  · exact (hB x hx).trans (le_max_right _ _)
+  · rw [image_eq_zero_of_notMem_tsupport hx,norm_zero]
+    exact le_max_left _ _
+
+ theorem local_compact_triple {A : Euc n → Euc n →L[ℝ] Euc n}
+    {f : Euc n → ℝ} {F : Euc n → Euc n} {U : Set (Euc n)} {k l : ℕ∞}
+    (hU : IsOpen U) {p : Euc n} (hp : p∈U)
+    (hA : ContDiffOn ℝ k A U) (hf : ContDiffOn ℝ l f U) (hF : ContDiffOn ℝ k F U) :
+    ∃ W : Set (Euc n), IsOpen W ∧ p∈W ∧ W⊆U ∧
+      ∃ B g G, ContDiff ℝ k B ∧ HasCompactSupport B ∧ ContDiff ℝ l g ∧ HasCompactSupport g ∧
+        ContDiff ℝ k G ∧ HasCompactSupport G ∧ EqOn B A W ∧ EqOn g f W ∧ EqOn G F W := by
+  obtain ⟨a,ha,haU,B,hB,hBc,heB⟩ := local_compact_extension hU hp hA
+  obtain ⟨b,hb,hbU,g,hg,hgc,heg⟩ := local_compact_extension hU hp hf
+  obtain ⟨c,hc,hcU,G,hG,hGc,heG⟩ := local_compact_extension hU hp hF
+  let r := min a (min b c)
+  have hr : 0<r := lt_min ha (lt_min hb hc)
+  have hra : r ≤ 2*a := (min_le_left a (min b c)).trans (by linarith)
+  have hrb : r ≤ 2*b := ((min_le_right a (min b c)).trans (min_le_left b c)).trans (by linarith)
+  have hrc : r ≤ 2*c := ((min_le_right a (min b c)).trans (min_le_right b c)).trans (by linarith)
+  refine ⟨Metric.ball p r,Metric.isOpen_ball,Metric.mem_ball_self hr,
+    (Metric.ball_subset_closedBall.trans ?_),B,g,G,hB,hBc,hg,hgc,hG,hGc,?_,?_,?_⟩
+  · exact (Metric.closedBall_subset_closedBall (by linarith : r ≤ 4*a)).trans haU
+  · exact fun x hx ↦ heB x (Metric.ball_subset_ball hra hx)
+  · exact fun x hx ↦ heg x (Metric.ball_subset_ball hrb hx)
+  · exact fun x hx ↦ heG x (Metric.ball_subset_ball hrc hx)
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/EllipticOneScale.lean b/lean/OAI/Geometry/Perimeter/Regularity/EllipticOneScale.lean
new file mode 100644
index 0000000000000000000000000000000000000000..99aa3bcf2a8a9e27774c56304a64becd810990f7
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/EllipticOneScale.lean
@@ -0,0 +1,110 @@
+import OAI.Geometry.Perimeter.Regularity.EllipticHarmonicBounds
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+theorem weak_curl_harmonic_comparison_norm {U : Set (Euc n)} {u e : VectorL2 n}
+    (hu : CurlFreeLpOn u U)
+    (h : ∀ φ : SmoothLocalTest U, (∫ x, inner ℝ (u x) (gradient φ.val x))=
+      ∫ x, inner ℝ (e x) (gradient φ.val x)) :
+    ∃ g : VectorL2 n, DivFreeLpOn g U ∧ CurlFreeLpOn g U ∧ ‖u-g‖≤‖e‖ ∧ ‖g‖≤‖u‖ := by
+  let K := testGradientSpace U
+  let g := u-K.starProjection u
+  have hmem : u-g∈K := by
+    dsimp [g]
+    rw [sub_sub_cancel]
+    exact (K.orthogonalProjectionOnto u).property
+  have hcf := hu.sub (testGradientSpace_curlFree hmem)
+  refine ⟨g,orthogonal_testGradient_divFree (K.sub_starProjection_mem_orthogonal u),?_,?_,?_⟩
+  · simpa only [sub_sub_cancel] using hcf
+  · exact harmonic_projection_error_le K
+      (testGradient_pairing_ext (fun φ ↦ by simpa only [testGradient_inner] using h φ))
+  · rw [show g=Kᗮ.starProjection u from (K.starProjection_orthogonal_val u).symm]
+    exact Kᗮ.norm_starProjection_apply_le u
+
+theorem vectorLp_integral_sq (u : VectorL2 n) : (∫ x, ‖u x‖^2)=‖u‖^2 := by
+  rw [←real_inner_self_eq_norm_sq,MeasureTheory.L2.inner_def]
+  simp only [real_inner_self_eq_norm_sq]
+
+theorem vectorLp_integrable_sq (u : VectorL2 n) : Integrable (fun x ↦ ‖u x‖^2) volume := by
+  simpa only [real_inner_self_eq_norm_sq] using (MeasureTheory.L2.integrable_inner (𝕜 := ℝ) u u)
+
+theorem vectorLp_setIntegral_sq_le (u : VectorL2 n) (s : Set (Euc n)) :
+    (∫ x in s, ‖u x‖^2)≤‖u‖^2 := by
+  rw [←vectorLp_integral_sq]
+  exact setIntegral_le_integral (vectorLp_integrable_sq u) (ae_of_all _ fun _ ↦ sq_nonneg _)
+
+theorem norm_sub_sq_le_two {V : Type*} [SeminormedAddCommGroup V] (x y z : V) :
+    ‖x-z‖^2≤2*‖x-y‖^2+2*‖y-z‖^2 := by
+  have h := norm_add_le (x-y) (y-z)
+  rw [sub_add_sub_cancel] at h
+  have h' : ‖x-z‖^2≤(‖x-y‖+‖y-z‖)^2 := pow_le_pow_left₀ (norm_nonneg _) h 2
+  nlinarith [sq_nonneg (‖x-y‖-‖y-z‖)]
+
+theorem weak_field_one_scale (hn : 0<n) :
+    ∃ C : ℝ, 0≤C ∧ ∀ u e : VectorL2 n,
+      CurlFreeLpOn u (Metric.ball 0 1) →
+      (∀ φ : SmoothLocalTest (Metric.ball 0 1),
+        (∫ x, inner ℝ (u x) (gradient φ.val x))=∫ x, inner ℝ (e x) (gradient φ.val x)) →
+      ∃ q : Euc n, ‖q‖≤C*‖u‖ ∧ ∀ θ : ℝ, θ≤1/2 →
+        (∫ x in Metric.ball 0 θ, ‖u x-q‖^2) ≤
+          2*‖e‖^2+2*(C*‖u‖*θ)^2*volume.real (Metric.ball (0:Euc n) θ) := by
+  obtain ⟨C,hC,hB⟩ := harmonic_field_representative_bounds hn
+  refine ⟨C,hC,fun u e hu h ↦ ?_⟩
+  obtain ⟨g,hd,hc,he,hgn⟩ := weak_curl_harmonic_comparison_norm hu h
+  obtain ⟨G,hG,hGe,hG0,hGL⟩ := hB g hd hc
+  refine ⟨G 0,(hG0 0).trans (mul_le_mul_of_nonneg_left hgn hC),fun θ hθ ↦ ?_⟩
+  by_cases hθ0 : 0<θ
+  swap
+  · have hz : Metric.ball (0:Euc n) θ=∅ := Metric.ball_eq_empty.mpr (le_of_not_gt hθ0)
+    simp only [hz,Measure.restrict_empty,integral_zero_measure,measureReal_empty,mul_zero,add_zero]
+    positivity
+  have : IsFiniteMeasure (volume.restrict (Metric.ball (0:Euc n) θ)) := ⟨by
+    rw [Measure.restrict_apply_univ]; exact measure_ball_lt_top⟩
+  have hiug : Integrable (fun x ↦ ‖u x-g x‖^2) volume :=
+    (vectorLp_integrable_sq (u-g)).congr (by filter_upwards [Lp.coeFn_sub u g] with x hx; rw [hx]; rfl)
+  have hiu : IntegrableOn (fun x ↦ ‖u x-G 0‖^2) (Metric.ball 0 θ) volume := by
+    have hcst : MemLp (fun _ : Euc n ↦ G 0) 2 (volume.restrict (Metric.ball 0 θ)) := memLp_const _
+    exact (((Lp.memLp u).mono_measure Measure.restrict_le_self).sub hcst).integrable_norm_pow (by norm_num)
+  have hib : IntegrableOn (fun x ↦ 2*‖u x-g x‖^2+2*(C*‖u‖*θ)^2) (Metric.ball 0 θ) volume :=
+    (hiug.integrableOn.const_mul 2).add (integrable_const _)
+  have hb : ∀ᵐ x ∂volume.restrict (Metric.ball (0:Euc n) θ),
+      ‖u x-G 0‖^2≤2*‖u x-g x‖^2+2*(C*‖u‖*θ)^2 := by
+    filter_upwards [ae_restrict_of_ae (ae_restrict_iff' measurableSet_ball |>.mp hGe),
+      ae_restrict_mem measurableSet_ball] with x hx hxB
+    have hxH : x∈Metric.ball (0:Euc n) (1/2) := Metric.ball_subset_ball hθ hxB
+    have hnorm : ‖g x-G 0‖≤C*‖u‖*θ := by
+      rw [hx hxH]
+      have hb' := hGL x 0
+      rw [sub_zero] at hb'
+      exact hb'.trans (mul_le_mul (mul_le_mul_of_nonneg_left hgn hC)
+        (by exact (show ‖x‖<θ by simpa only [Metric.mem_ball,dist_zero_right] using hxB).le)
+        (norm_nonneg _) (mul_nonneg hC (norm_nonneg _)))
+    exact (norm_sub_sq_le_two (u x) (g x) (G 0)).trans (by
+      have hh := pow_le_pow_left₀ (norm_nonneg _) hnorm 2
+      linarith)
+  calc
+    _ ≤ ∫ x in Metric.ball 0 θ, 2*‖u x-g x‖^2+2*(C*‖u‖*θ)^2 := integral_mono_ae hiu hib hb
+    _ = 2*(∫ x in Metric.ball 0 θ, ‖u x-g x‖^2)+
+        2*(C*‖u‖*θ)^2*volume.real (Metric.ball (0:Euc n) θ) := by
+      rw [integral_add (hiug.integrableOn.const_mul 2) (integrable_const _),integral_const_mul,integral_const]
+      simp only [smul_eq_mul,measureReal_def,Measure.restrict_apply_univ]; ring
+    _ ≤ _ := by
+      have hsq : (∫ x in Metric.ball 0 θ, ‖u x-g x‖^2)≤‖u-g‖^2 := by
+        have heq : (∫ x in Metric.ball 0 θ, ‖u x-g x‖^2)=∫ x in Metric.ball 0 θ, ‖(u-g) x‖^2 :=
+          integral_congr_ae (ae_restrict_of_ae (by filter_upwards [Lp.coeFn_sub u g] with x hx; rw [hx]; rfl))
+        rw [heq]
+        exact vectorLp_setIntegral_sq_le _ _
+      have hsq' := pow_le_pow_left₀ (norm_nonneg _) he 2
+      linarith
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/EllipticPointwise.lean b/lean/OAI/Geometry/Perimeter/Regularity/EllipticPointwise.lean
new file mode 100644
index 0000000000000000000000000000000000000000..f7c80e932ec5bf5107a43c202a42aeef3cb7c489
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/EllipticPointwise.lean
@@ -0,0 +1,52 @@
+import OAI.Geometry.Perimeter.Regularity.EllipticIteration
+import OAI.Geometry.Perimeter.Regularity.EllipticGradientLimit
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+ theorem linear_pointwise_iteration {θ δ K a B H R : ℝ}
+    (hθ : 0<θ) (hθ1 : θ<1) (hK : 1≤K)
+    (hc : LinearContractionProperty n θ δ K)
+    (ha : 0≤a) (hB : 0≤B) (hH : 0≤H) (hR : 0<R)
+    (hsmall : 32*K^2*a≤1) (hforce : 16*K*B≤H)
+    {A : Euc n → Euc n →L[ℝ] Euc n} (hA : Continuous A)
+    {f : Euc n → ℝ} (hf : ContDiff ℝ 1 f) {F : Euc n → Euc n} (hF : Continuous F)
+    (p : Euc n) (h : WeakLinearEquation A f F (Metric.ball p R))
+    (hnear : ∀ x∈Metric.ball p R, ‖ContinuousLinearMap.id ℝ (Euc n)-A x‖≤δ)
+    (hosc : ∀ j x, x∈Metric.ball p (θ^j*R) →
+      ‖A p-A x‖≤a*(1/2)^j ∧ ‖F x-F p‖≤B*(1/2)^j)
+    (hinit : scaledGradientEnergy f p R 0≤H^2) :
+    ∃ q : ℕ → Euc n, (∀ j, ‖q j-gradient f p‖≤2*K*H*(1/2)^j) ∧
+      (∀ j, scaledGradientEnergy f p (θ^j*R) (q j)≤H^2*((1/2)^j)^2) ∧
+      ‖gradient f p‖≤2*K*H := by
+  obtain ⟨q,hq0,hqb,hqE,hqinc⟩ := linear_energy_iteration hθ hθ1 hK hc ha hB hH hR
+    hsmall hforce hA hf hF p h hnear hosc hinit
+  have hdist (j : ℕ) : dist (q j) (q (j+1))≤(K*H)*(1/2:ℝ)^j := by
+    rw [dist_eq_norm,norm_sub_rev]; exact hqinc j
+  obtain ⟨w,hw⟩ := cauchySeq_tendsto_of_complete
+    (cauchySeq_of_le_geometric (1/2) (K*H) (by norm_num) hdist)
+  have hwf : gradient f p=w := gradient_eq_of_scaled_energy hf p
+    (fun j ↦ by positivity)
+    (by simpa only [zero_mul] using (tendsto_pow_atTop_nhds_zero_of_lt_one hθ.le hθ1).mul_const R)
+    (by simpa only [zero_pow (by norm_num : (2:ℕ)≠0),mul_zero] using
+      ((tendsto_pow_atTop_nhds_zero_of_lt_one (by norm_num : (0:ℝ)≤1/2) (by norm_num)).pow 2).const_mul (H^2))
+    hw hqE
+  have herror (j : ℕ) : ‖q j-gradient f p‖≤2*K*H*(1/2)^j := by
+    have hh := dist_le_of_le_geometric_of_tendsto (1/2) (K*H) (by norm_num) hdist hw j
+    rw [dist_eq_norm,←hwf] at hh
+    convert hh using 1
+    ring
+  refine ⟨q,herror,hqE,?_⟩
+  have hh := herror 0
+  simpa only [hq0,zero_sub,norm_neg,pow_zero,mul_one] using hh
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/EllipticRescale.lean b/lean/OAI/Geometry/Perimeter/Regularity/EllipticRescale.lean
new file mode 100644
index 0000000000000000000000000000000000000000..509abebe509ee5f7da4c70b13588bfccf474aebd
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/EllipticRescale.lean
@@ -0,0 +1,95 @@
+import OAI.Geometry.Perimeter.Regularity.EllipticLinearStep
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+def ellipticRescale (f : Euc n → ℝ) (p : Euc n) (r : ℝ) (x : Euc n) := r⁻¹*f (p+r•x)
+
+theorem ellipticRescale_contDiff {f : Euc n → ℝ} {k : ℕ∞} (hf : ContDiff ℝ k f)
+    (p : Euc n) (r : ℝ) : ContDiff ℝ k (ellipticRescale f p r) :=
+  contDiff_const.mul (hf.comp (contDiff_const.add (contDiff_id.const_smul r)))
+
+theorem ellipticRescale_gradient {f : Euc n → ℝ} (hf : Differentiable ℝ f)
+    (p : Euc n) {r : ℝ} (hr : r≠0) (x : Euc n) :
+    gradient (ellipticRescale f p r) x=gradient f (p+r•x) := by
+  apply ext_inner_right ℝ
+  intro y
+  simp only [inner_gradient_left]
+  have h := ((((hf (p+r•x)).hasFDerivAt.comp x
+    ((hasFDerivAt_id x).const_smul r |>.const_add p))).const_mul r⁻¹).fderiv
+  have he := congrArg (fun L : Euc n →L[ℝ] ℝ ↦ L y) h
+  change fderiv ℝ (fun y ↦ r⁻¹*f (p+r•y)) x y = _
+  simpa [ContinuousLinearMap.comp_apply,hr,mul_assoc] using he
+
+theorem integral_affine (g : Euc n → ℝ) (p : Euc n) {r : ℝ} (hr : 0<r) :
+    (∫ x, g (p+r•x))=(r^n)⁻¹*∫ x, g x := by
+  have h := Measure.integral_comp_smul_of_nonneg (volume : Measure (Euc n))
+    (fun x ↦ g (p+x)) r (hR := hr.le)
+  simpa only [finrank_euclideanSpace,Fintype.card_fin,smul_eq_mul,integral_add_left_eq_self] using h
+
+theorem blownUpSet_ball_self (p : Euc n) {r : ℝ} (hr : 0<r) (R : ℝ) :
+    blownUpSet (Metric.ball p (r*R)) p r=Metric.ball (0:Euc n) R := by
+  ext x
+  simp only [blownUpSet,mem_preimage,Metric.mem_ball,dist_eq_norm,add_sub_cancel_left,
+    norm_smul,Real.norm_of_nonneg hr.le,sub_zero]
+  exact mul_lt_mul_iff_right₀ hr
+
+theorem gradientEnergy_rescale {f : Euc n → ℝ} (hf : Differentiable ℝ f)
+    (p : Euc n) {r : ℝ} (hr : 0<r) (R : ℝ) (q : Euc n) :
+    gradientEnergy (ellipticRescale f p r) 0 R q=(r^n)⁻¹*gradientEnergy f p (r*R) q := by
+  unfold gradientEnergy
+  simp_rw [ellipticRescale_gradient hf p hr.ne']
+  rw [←blownUpSet_ball_self p hr R]
+  exact integral_blownUpSet (E := Metric.ball p (r*R)) measurableSet_ball (fun x ↦ ‖gradient f x-q‖^2) p hr
+
+def affineLocalTest {U : Set (Euc n)} (p : Euc n) {r : ℝ} (hr : 0<r)
+    (φ : SmoothLocalTest (blownUpSet U p r)) : SmoothLocalTest U := by
+  let ψ : Euc n → ℝ := fun y ↦ r*φ.val (r⁻¹•(y-p))
+  refine ⟨ψ,?_,?_,?_⟩
+  · exact contDiff_const.mul (φ.property.1.comp ((contDiff_id.sub contDiff_const).const_smul r⁻¹))
+  · exact (φ.property.2.1.comp_homeomorph
+      ((Homeomorph.subRight p).trans (Homeomorph.smulOfNeZero r⁻¹ (inv_ne_zero hr.ne')))).mul_left
+  · have hs : tsupport ψ ⊆ tsupport (fun y ↦ φ.val (r⁻¹•(y-p))) := tsupport_mul_subset_right
+    intro y hy
+    have hh := φ.property.2.2 (tsupport_comp_subset_preimage φ.val
+      ((continuous_id.sub continuous_const).const_smul r⁻¹) (hs hy))
+    simpa only [blownUpSet,mem_preimage,Pi.smul_apply,Pi.sub_apply,id_eq,smul_inv_smul₀ hr.ne',add_sub_cancel] using hh
+
+theorem affineLocalTest_gradient {U : Set (Euc n)} (p : Euc n) {r : ℝ} (hr : 0<r)
+    (φ : SmoothLocalTest (blownUpSet U p r)) (x : Euc n) :
+    gradient (affineLocalTest p hr φ).val (p+r•x)=gradient φ.val x := by
+  have hh := ellipticRescale_gradient (φ.property.1.differentiable (by simp))
+    (-(r⁻¹•p)) (inv_ne_zero hr.ne') (p+r•x)
+  have he : (affineLocalTest p hr φ).val=ellipticRescale φ.val (-(r⁻¹•p)) r⁻¹ := by
+    funext y
+    simp only [affineLocalTest,ellipticRescale,inv_inv,sub_eq_add_neg,smul_add,smul_neg,add_comm]
+  rw [he,hh]
+  congr 1
+  simp [smul_add,inv_smul_smul₀ hr.ne',add_comm]
+
+theorem WeakLinearEquation.rescale {A : Euc n → Euc n →L[ℝ] Euc n}
+    {f : Euc n → ℝ} {F : Euc n → Euc n} {U : Set (Euc n)}
+    (h : WeakLinearEquation A f F U) (hf : Differentiable ℝ f)
+    (p : Euc n) {r : ℝ} (hr : 0<r) :
+    WeakLinearEquation (fun x ↦ A (p+r•x)) (ellipticRescale f p r)
+      (fun x ↦ F (p+r•x)) (blownUpSet U p r) := by
+  intro φ
+  have he := h (affineLocalTest p hr φ)
+  have h1 := integral_affine (fun y ↦ inner ℝ (A y (gradient f y))
+    (gradient (affineLocalTest p hr φ).val y)) p hr
+  have h2 := integral_affine (fun y ↦ inner ℝ (F y)
+    (gradient (affineLocalTest p hr φ).val y)) p hr
+  simp_rw [affineLocalTest_gradient p hr φ] at h1 h2
+  simp_rw [ellipticRescale_gradient hf p hr.ne']
+  rw [h1,h2,he]
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/EllipticScaleStep.lean b/lean/OAI/Geometry/Perimeter/Regularity/EllipticScaleStep.lean
new file mode 100644
index 0000000000000000000000000000000000000000..2780c34804e8f59af1872984064ee139a40fb414
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/EllipticScaleStep.lean
@@ -0,0 +1,78 @@
+import OAI.Geometry.Perimeter.Regularity.EllipticExcessStep
+import OAI.Geometry.Perimeter.Regularity.EllipticRescale
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+def scaledGradientEnergy (f : Euc n → ℝ) (p : Euc n) (r : ℝ) (q : Euc n) :=
+  (r^n)⁻¹*gradientEnergy f p r q
+
+theorem scaledGradientEnergy_nonneg (f : Euc n → ℝ) (p : Euc n) {r : ℝ} (hr : 0≤r) (q : Euc n) :
+    0 ≤ scaledGradientEnergy f p r q :=
+  mul_nonneg (inv_nonneg.mpr (pow_nonneg hr _)) (integral_nonneg (fun _ ↦ sq_nonneg _))
+
+theorem real_volume_ball_scale {r : ℝ} (hr : 0<r) :
+    volume.real (Metric.ball (0:Euc n) r)=r^n*volume.real (Metric.ball (0:Euc n) 1) := by
+  have h := integral_blownUpSet (E := Metric.ball (0:Euc n) (r*1)) measurableSet_ball (fun _ ↦ (1:ℝ)) 0 hr
+  rw [blownUpSet_ball_self 0 hr 1] at h
+  simp only [integral_const,measureReal_restrict_apply_univ,smul_eq_mul,mul_one] at h
+  have hh := congrArg (fun a : ℝ ↦ r^n*a) h
+  simpa only [←mul_assoc,mul_inv_cancel₀ (pow_ne_zero _ hr.ne'),one_mul] using hh.symm
+
+theorem linear_gradient_scaled_step (hn : 0<n) :
+    ∃ C : ℝ, 0≤C ∧ ∀ (A : Euc n → Euc n →L[ℝ] Euc n), Continuous A →
+      ∀ (f : Euc n → ℝ), ContDiff ℝ 1 f → ∀ (F : Euc n → Euc n), Continuous F →
+      ∀ (p : Euc n) (r : ℝ), 0<r → WeakLinearEquation A f F (Metric.ball p r) →
+      ∀ (A₀ : Euc n →L[ℝ] Euc n) (F₀ q : Euc n) (δ α β : ℝ),
+      0≤δ → 0≤α → 0≤β →
+      (∀ x∈Metric.ball p r, ‖ContinuousLinearMap.id ℝ (Euc n)-A x‖≤δ ∧
+        ‖A₀-A x‖≤α ∧ ‖F x-F₀‖≤β) →
+      ∃ q' : Euc n, ‖q'-q‖^2≤C^2*scaledGradientEnergy f p r q ∧
+        ∀ θ : ℝ, 0<θ → θ≤1/2 → scaledGradientEnergy f p (θ*r) q' ≤
+          (4*δ^2*(θ^n)⁻¹+2*C^2*θ^2*volume.real (Metric.ball (0:Euc n) 1))*scaledGradientEnergy f p r q+
+          4*(α*‖q‖+β)^2*(θ^n)⁻¹*volume.real (Metric.ball (0:Euc n) 1) := by
+  obtain ⟨C,hC,hB⟩ := linear_gradient_excess_step hn
+  refine ⟨C,hC,fun A hA f hf F hF p r hr h A₀ F₀ q δ α β hδ hα hβ hb ↦ ?_⟩
+  let a := fun x : Euc n ↦ A (p+r•x)
+  let g := ellipticRescale f p r
+  let B := fun x : Euc n ↦ F (p+r•x)
+  have ha : Continuous a := hA.comp (continuous_const.add (continuous_id.const_smul r))
+  have hg : ContDiff ℝ 1 g := ellipticRescale_contDiff hf p r
+  have hB' : Continuous B := hF.comp (continuous_const.add (continuous_id.const_smul r))
+  have hh : WeakLinearEquation a g B (Metric.ball 0 1) := by
+    have hh := h.rescale (hf.differentiable (by simp)) p hr
+    have hset : blownUpSet (Metric.ball p r) p r=Metric.ball (0:Euc n) 1 := by
+      simpa only [mul_one] using blownUpSet_ball_self p hr 1
+    rwa [hset] at hh
+  have hb' : ∀ x∈Metric.ball (0:Euc n) 1, ‖ContinuousLinearMap.id ℝ (Euc n)-a x‖≤δ ∧
+      ‖A₀-a x‖≤α ∧ ‖B x-F₀‖≤β := by
+    intro x hx
+    apply hb
+    have hset := blownUpSet_ball_self p hr 1
+    have hx' : x∈blownUpSet (Metric.ball p (r*1)) p r := by rw [hset]; exact hx
+    simpa only [blownUpSet,mem_preimage,mul_one] using hx'
+  obtain ⟨q',hq',hE⟩ := hB a ha g hg B hB' hh A₀ F₀ q δ α β hδ hα hβ hb'
+  have hunit : gradientEnergy g 0 1 q=scaledGradientEnergy f p r q := by
+    simpa only [mul_one,scaledGradientEnergy] using gradientEnergy_rescale (hf.differentiable (by simp)) p hr 1 q
+  refine ⟨q',by simpa only [hunit] using hq',fun θ hθ hθ' ↦ ?_⟩
+  have hi : scaledGradientEnergy f p (θ*r) q'=(θ^n)⁻¹*gradientEnergy g 0 θ q' := by
+    rw [gradientEnergy_rescale (hf.differentiable (by simp)) p hr θ q']
+    simp only [scaledGradientEnergy,mul_pow,mul_inv_rev,mul_comm θ r]
+    ring
+  rw [hi]
+  have hnθ : θ^n≠0 := pow_ne_zero _ hθ.ne'
+  have hbnd := mul_le_mul_of_nonneg_left (hE θ hθ') (inv_nonneg.mpr (pow_nonneg hθ.le n))
+  rw [hunit,real_volume_ball_scale hθ] at hbnd
+  convert hbnd using 1
+  field_simp
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/EllipticSecant.lean b/lean/OAI/Geometry/Perimeter/Regularity/EllipticSecant.lean
new file mode 100644
index 0000000000000000000000000000000000000000..b2d08d86b941901dcb22026b119418aa10525f93
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/EllipticSecant.lean
@@ -0,0 +1,69 @@
+import OAI.Geometry.Perimeter.Regularity.EllipticCurl
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+def derivativeSecant (G : Euc n → Euc n) (a b : Euc n) : Euc n →L[ℝ] Euc n :=
+  ∫ t in (0:ℝ)..1, fderiv ℝ G (a+t•(b-a))
+
+ theorem derivativeSecant_continuous {G : Euc n → Euc n} (hG : ContDiff ℝ 1 G) :
+    Continuous (fun z : Euc n × Euc n ↦ derivativeSecant G z.1 z.2) := by
+  apply intervalIntegral.continuous_parametric_intervalIntegral_of_continuous'
+  exact (hG.continuous_fderiv (by simp)).comp
+    (continuous_fst.fst.add (continuous_snd.smul (continuous_fst.snd.sub continuous_fst.fst)))
+
+ theorem derivativeSecant_apply {G : Euc n → Euc n} (hG : ContDiff ℝ 1 G) (a b : Euc n) :
+    derivativeSecant G a b (b-a)=G b-G a := by
+  have hc : Continuous (fun t : ℝ ↦ fderiv ℝ G (a+t•(b-a))) :=
+    (hG.continuous_fderiv (by simp)).comp (continuous_const.add (continuous_id.smul continuous_const))
+  rw [derivativeSecant,ContinuousLinearMap.intervalIntegral_apply (hc.intervalIntegrable 0 1)]
+  have hd (t : ℝ) : HasDerivAt (fun s : ℝ ↦ G (a+s•(b-a)))
+      (fderiv ℝ G (a+t•(b-a)) (b-a)) t := by
+    apply (hG.differentiable (by simp) _).hasFDerivAt.comp_hasDerivAt t
+    simpa using ((hasDerivAt_id t).smul_const (b-a) |>.const_add a)
+  simpa only [zero_smul,add_zero,one_smul,add_sub_cancel] using
+    intervalIntegral.integral_eq_sub_of_hasDerivAt (fun t _ ↦ hd t)
+      ((hc.clm_apply continuous_const).intervalIntegrable 0 1)
+
+ theorem derivativeSecant_near {G : Euc n → Euc n} (hG : ContDiff ℝ 1 G)
+    (a b : Euc n) (A : Euc n →L[ℝ] Euc n) (δ : ℝ)
+    (hb : ∀ t∈Icc (0:ℝ) 1, ‖A-fderiv ℝ G (a+t•(b-a))‖≤δ) :
+    ‖A-derivativeSecant G a b‖≤δ := by
+  have hc : Continuous (fun t : ℝ ↦ fderiv ℝ G (a+t•(b-a))) :=
+    (hG.continuous_fderiv (by simp)).comp (continuous_const.add (continuous_id.smul continuous_const))
+  have he : A-derivativeSecant G a b=∫ t in (0:ℝ)..1, A-fderiv ℝ G (a+t•(b-a)) := by
+    let : CompleteSpace (Euc n →L[ℝ] Euc n) := ContinuousLinearMap.instCompleteSpace
+    rw [intervalIntegral.integral_sub intervalIntegrable_const (hc.intervalIntegrable 0 1)]
+    simp only [derivativeSecant, intervalIntegral.integral_const, sub_zero, one_smul]
+  rw [he]
+  have hh := intervalIntegral.norm_integral_le_of_norm_le_const (fun t ht ↦
+    hb t (by
+      rw [Set.uIoc_of_le (by norm_num : (0:ℝ)≤1)] at ht
+      exact Set.Ioc_subset_Icc_self ht))
+  simpa using hh
+
+ theorem derivativeSecant_difference {G : Euc n → Euc n} (hG : ContDiff ℝ 1 G)
+    (a b a' b' : Euc n) (C : ℝ)
+    (hb : ∀ t∈Icc (0:ℝ) 1,
+      ‖fderiv ℝ G (a+t•(b-a))-fderiv ℝ G (a'+t•(b'-a'))‖≤C) :
+    ‖derivativeSecant G a b-derivativeSecant G a' b'‖≤C := by
+  have hc (a b : Euc n) : Continuous (fun t : ℝ ↦ fderiv ℝ G (a+t•(b-a))) :=
+    (hG.continuous_fderiv (by simp)).comp (continuous_const.add (continuous_id.smul continuous_const))
+  rw [derivativeSecant,derivativeSecant,←intervalIntegral.integral_sub
+    ((hc a b).intervalIntegrable 0 1) ((hc a' b').intervalIntegrable 0 1)]
+  have hh := intervalIntegral.norm_integral_le_of_norm_le_const (fun t ht ↦
+    hb t (by
+      rw [Set.uIoc_of_le (by norm_num : (0:ℝ)≤1)] at ht
+      exact Set.Ioc_subset_Icc_self ht))
+  simpa using hh
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/EllipticSecantBounds.lean b/lean/OAI/Geometry/Perimeter/Regularity/EllipticSecantBounds.lean
new file mode 100644
index 0000000000000000000000000000000000000000..b3f593db88df73ecb522548a4055b832c97c7b8c
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/EllipticSecantBounds.lean
@@ -0,0 +1,56 @@
+import OAI.Geometry.Perimeter.Regularity.EllipticSecant
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+ theorem segment_in_ball {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V]
+    {p a b : V} {r t : ℝ} (ha : a∈Metric.ball p r) (hb : b∈Metric.ball p r)
+    (ht : t∈Icc (0:ℝ) 1) : a+t•(b-a)∈Metric.ball p r := by
+  have hh := (convex_ball p r) ha hb (sub_nonneg.mpr ht.2) ht.1 (by ring : 1-t+t=1)
+  convert hh using 1
+  module
+
+ theorem segment_difference_bound {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V]
+    (a b a' b' : V) {t : ℝ} (ht : t∈Icc (0:ℝ) 1) :
+    ‖(a+t•(b-a))-(a'+t•(b'-a'))‖ ≤ ‖a-a'‖+‖b-b'‖ := by
+  have he : (a+t•(b-a))-(a'+t•(b'-a'))=(1-t)•(a-a')+t•(b-b') := by module
+  rw [he]
+  have hh := norm_add_le ((1-t)•(a-a')) (t•(b-b'))
+  rw [norm_smul,norm_smul,Real.norm_of_nonneg (sub_nonneg.mpr ht.2),Real.norm_of_nonneg ht.1] at hh
+  nlinarith [norm_nonneg (a-a'),norm_nonneg (b-b'),mul_nonneg ht.1 (norm_nonneg (a-a')),
+    mul_nonneg (sub_nonneg.mpr ht.2) (norm_nonneg (b-b'))]
+
+ theorem derivativeSecant_lipschitz_ball {G : Euc n → Euc n} (hG : ContDiff ℝ 1 G)
+    {p a b a' b' : Euc n} {r L : ℝ} (hL : 0 ≤ L)
+    (ha : a∈Metric.ball p r) (hb : b∈Metric.ball p r)
+    (ha' : a'∈Metric.ball p r) (hb' : b'∈Metric.ball p r)
+    (hLip : ∀ x∈Metric.ball p r, ∀ y∈Metric.ball p r,
+      ‖fderiv ℝ G x-fderiv ℝ G y‖ ≤ L*‖x-y‖) :
+    ‖derivativeSecant G a b-derivativeSecant G a' b'‖ ≤ L*(‖a-a'‖+‖b-b'‖) := by
+  apply derivativeSecant_difference hG
+  intro t ht
+  exact (hLip _ (segment_in_ball ha hb ht) _ (segment_in_ball ha' hb' ht)).trans
+    (mul_le_mul_of_nonneg_left (segment_difference_bound a b a' b' ht) hL)
+
+ theorem contDiff_local_lipschitz {V W : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V]
+    [NormedAddCommGroup W] [NormedSpace ℝ W] {f : V → W} (hf : ContDiff ℝ 1 f) (p : V) :
+    ∃ r L : ℝ, 0<r ∧ 0≤L ∧ ∀ x∈Metric.ball p r, ∀ y∈Metric.ball p r,
+      ‖f x-f y‖≤L*‖x-y‖ := by
+  have hc := ((hf.continuous_fderiv (by simp)).continuousAt (x := p)).norm
+  obtain ⟨r,hr,hh⟩ := Metric.eventually_nhds_iff.mp
+    (hc.eventually (gt_mem_nhds (lt_add_one ‖fderiv ℝ f p‖)))
+  refine ⟨r,‖fderiv ℝ f p‖+1,hr,by positivity,fun x hx y hy ↦ ?_⟩
+  exact (convex_ball p r).norm_image_sub_le_of_norm_fderiv_le
+    (fun z _ ↦ hf.differentiable (by simp) z)
+    (fun z hz ↦ (hh (Metric.mem_ball.mp hz)).le) hy hx
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/EllipticSlopeComparison.lean b/lean/OAI/Geometry/Perimeter/Regularity/EllipticSlopeComparison.lean
new file mode 100644
index 0000000000000000000000000000000000000000..e47b90f0623fedc18b413a2f1d53eb3175a2d5fb
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/EllipticSlopeComparison.lean
@@ -0,0 +1,68 @@
+import OAI.Geometry.Perimeter.Regularity.EllipticGradientLimit
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+ theorem gradientEnergy_integrable {f : Euc n → ℝ} (hf : ContDiff ℝ 1 f)
+    (p : Euc n) (r : ℝ) (q : Euc n) :
+    IntegrableOn (fun x ↦ ‖gradient f x-q‖^2) (Metric.ball p r) volume :=
+  ((((gradient_continuous_C1 hf).sub continuous_const).norm.pow 2).continuousOn.integrableOn_compact
+    (isCompact_closedBall p r)).mono_set Metric.ball_subset_closedBall
+
+ theorem gradientEnergy_mono {f : Euc n → ℝ} (hf : ContDiff ℝ 1 f)
+    {p p' : Euc n} {r r' : ℝ} (q : Euc n) (h : Metric.ball p r ⊆ Metric.ball p' r') :
+    gradientEnergy f p r q ≤ gradientEnergy f p' r' q :=
+  setIntegral_mono_set (gradientEnergy_integrable hf p' r' q)
+    (ae_of_all _ (fun _ ↦ sq_nonneg _)) (ae_of_all _ (fun x ↦ show x∈Metric.ball p r → x∈Metric.ball p' r' from fun hx ↦ h hx))
+
+ theorem constant_slopes_energy_bound {f : Euc n → ℝ} (hf : ContDiff ℝ 1 f)
+    (p : Euc n) {r : ℝ} (hr : 0<r) (q q' : Euc n) :
+    ‖q-q'‖^2*volume.real (Metric.ball p r) ≤
+      2*gradientEnergy f p r q+2*gradientEnergy f p r q' := by
+  have : IsFiniteMeasure (volume.restrict (Metric.ball p r)) := ⟨by
+    rw [Measure.restrict_apply_univ]; exact measure_ball_lt_top⟩
+  calc
+    _ = ∫ x in Metric.ball p r, ‖q-q'‖^2 := by
+      simp [real_volume_ball_center, real_volume_ball_scale hr, mul_comm]
+    _ ≤ ∫ x in Metric.ball p r, 2*‖gradient f x-q‖^2+2*‖gradient f x-q'‖^2 := by
+      apply integral_mono (integrable_const _) (((gradientEnergy_integrable hf p r q).const_mul 2).add
+        ((gradientEnergy_integrable hf p r q').const_mul 2))
+      intro x
+      have hh := norm_sub_le (q-gradient f x) (q'-gradient f x)
+      have he : (q-gradient f x)-(q'-gradient f x)=q-q' := by abel
+      rw [he,norm_sub_rev q (gradient f x),norm_sub_rev q' (gradient f x)] at hh
+      have hh' := pow_le_pow_left₀ (norm_nonneg (q-q')) hh 2
+      change ‖q-q'‖^2 ≤ 2*‖gradient f x-q‖^2+2*‖gradient f x-q'‖^2
+      nlinarith [sq_nonneg (‖gradient f x-q‖-‖gradient f x-q'‖)]
+    _ = _ := by rw [integral_add ((gradientEnergy_integrable hf p r q).const_mul 2)
+        ((gradientEnergy_integrable hf p r q').const_mul 2),integral_const_mul,integral_const_mul]; rfl
+
+ theorem scaled_slope_comparison {f : Euc n → ℝ} (hf : ContDiff ℝ 1 f)
+    (p p' : Euc n) {r : ℝ} (hr : 0<r) (hd : dist p p'≤r/2) (q q' : Euc n) :
+    volume.real (Metric.ball (0:Euc n) 1)*‖q-q'‖^2 ≤
+      2^(n+1)*(scaledGradientEnergy f p r q+scaledGradientEnergy f p' r q') := by
+  have hb : Metric.ball p' (r/2) ⊆ Metric.ball p r := by
+    intro x hx
+    have hh := dist_triangle x p' p
+    rw [dist_comm p' p] at hh
+    exact Metric.mem_ball.mpr (by linarith [Metric.mem_ball.mp hx])
+  have hb' : Metric.ball p' (r/2) ⊆ Metric.ball p' r := Metric.ball_subset_ball (by linarith)
+  have hh := constant_slopes_energy_bound hf p' (by positivity : 0<r/2) q q'
+  have h1 := gradientEnergy_mono hf q hb
+  have h2 := gradientEnergy_mono hf q' hb'
+  have hh' : ‖q-q'‖^2*volume.real (Metric.ball p' (r/2)) ≤
+      2*gradientEnergy f p r q+2*gradientEnergy f p' r q' := by linarith
+  rw [real_volume_ball_center,real_volume_ball_scale (by positivity : 0<r/2)] at hh'
+  have hm := mul_le_mul_of_nonneg_left hh' (by positivity : 0≤((r/2)^n)⁻¹)
+  convert hm using 1 <;> first | rfl | (simp only [scaledGradientEnergy,div_pow,pow_succ,inv_div]; field_simp [hr.ne'])
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/EllipticTestSpace.lean b/lean/OAI/Geometry/Perimeter/Regularity/EllipticTestSpace.lean
new file mode 100644
index 0000000000000000000000000000000000000000..29c47838d4c15ddbe3409c5fbf92dcaec732f17d
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/EllipticTestSpace.lean
@@ -0,0 +1,86 @@
+import OAI.Geometry.Perimeter.Regularity.EllipticHilbert
+import OAI.Geometry.Perimeter.Regularity.GradientContDiff
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+abbrev VectorL2 (n : ℕ) := Lp (Euc n) 2 (volume : Measure (Euc n))
+
+def SmoothLocalTest (U : Set (Euc n)) :=
+  {φ : Euc n → ℝ // ContDiff ℝ ∞ φ ∧ HasCompactSupport φ ∧ tsupport φ ⊆ U}
+
+def testGradientLp {U : Set (Euc n)} (φ : SmoothLocalTest U) : VectorL2 n :=
+  ((gradient_contDiff φ.property.1).continuous.memLp_of_hasCompactSupport
+    (gradient_hasCompactSupport φ.property.2.1)).toLp (gradient φ.val)
+
+theorem testGradientLp_ae {U : Set (Euc n)} (φ : SmoothLocalTest U) :
+    (testGradientLp φ : Euc n → Euc n)=ᵐ[volume] gradient φ.val :=
+  MemLp.coeFn_toLp _
+
+def testGradientSpace (U : Set (Euc n)) : Submodule ℝ (VectorL2 n) :=
+  (Submodule.span ℝ (Set.range (testGradientLp (U := U)))).topologicalClosure
+
+instance testGradientSpace_complete (U : Set (Euc n)) : CompleteSpace (testGradientSpace U) :=
+  (Submodule.isClosed_topologicalClosure _).completeSpace_coe
+
+theorem testGradient_mem {U : Set (Euc n)} (φ : SmoothLocalTest U) :
+    testGradientLp φ∈testGradientSpace U :=
+  Submodule.le_topologicalClosure _ (Submodule.subset_span ⟨φ,rfl⟩)
+
+theorem inner_zero_on_closed_span {H : Type*} [NormedAddCommGroup H]
+    [InnerProductSpace ℝ H] (s : Set H) {u : H} (h : ∀ w∈s, inner ℝ u w=0) :
+    ∀ w∈(Submodule.span ℝ s).topologicalClosure, inner ℝ u w=0 := by
+  have hs : Submodule.span ℝ s ≤ (innerSL ℝ u).ker := by
+    apply Submodule.span_le.mpr
+    exact h
+  exact (Submodule.span ℝ s).topologicalClosure_minimal hs (innerSL ℝ u).isClosed_ker
+
+theorem testGradient_pairing_ext {U : Set (Euc n)} {u e : VectorL2 n}
+    (h : ∀ φ : SmoothLocalTest U, inner ℝ u (testGradientLp φ)=inner ℝ e (testGradientLp φ)) :
+    ∀ w∈testGradientSpace U, inner ℝ u w=inner ℝ e w := by
+  have hh := inner_zero_on_closed_span (Set.range (testGradientLp (U := U))) (u := u-e)
+    (by rintro _ ⟨φ,rfl⟩; rw [inner_sub_left,h φ,sub_self])
+  intro w hw
+  have hz := hh w hw
+  simpa only [inner_sub_left,sub_eq_zero] using hz
+
+theorem testGradient_inner {U : Set (Euc n)} (u : VectorL2 n) (φ : SmoothLocalTest U) :
+    inner ℝ u (testGradientLp φ)=∫ x, inner ℝ (u x) (gradient φ.val x) := by
+  rw [MeasureTheory.L2.inner_def]
+  apply integral_congr_ae
+  filter_upwards [testGradientLp_ae φ] with x hx
+  rw [hx]
+
+theorem testGradient_integrable_pairing (u : VectorL2 n) {U : Set (Euc n)}
+    (φ : SmoothLocalTest U) : Integrable (fun x ↦ inner ℝ (u x) (gradient φ.val x)) volume := by
+  apply (MeasureTheory.L2.integrable_inner (𝕜 := ℝ) u (testGradientLp φ)).congr
+  filter_upwards [testGradientLp_ae φ] with x hx
+  rw [hx]
+
+def DivFreeLpOn (u : VectorL2 n) (U : Set (Euc n)) : Prop :=
+  ∀ φ : SmoothLocalTest U, (∫ x, inner ℝ (u x) (gradient φ.val x))=0
+
+theorem orthogonal_testGradient_divFree {U : Set (Euc n)} {u : VectorL2 n}
+    (hu : u∈(testGradientSpace U)ᗮ) : DivFreeLpOn u U := by
+  intro φ
+  rw [←testGradient_inner u φ]
+  exact Submodule.inner_left_of_mem_orthogonal (testGradient_mem φ) hu
+
+theorem weak_harmonic_gradient_comparison {U : Set (Euc n)} {u e : VectorL2 n}
+    (h : ∀ φ : SmoothLocalTest U, (∫ x, inner ℝ (u x) (gradient φ.val x))=
+      ∫ x, inner ℝ (e x) (gradient φ.val x)) :
+    ∃ g : VectorL2 n, DivFreeLpOn g U ∧ u-g∈testGradientSpace U ∧ ‖u-g‖≤‖e‖ := by
+  obtain ⟨g,hg,hug,he⟩ := harmonic_projection_exists (testGradientSpace U)
+    (testGradient_pairing_ext (fun φ ↦ by simpa only [testGradient_inner] using h φ))
+  exact ⟨g,orthogonal_testGradient_divFree hg,hug,he⟩
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/EllipticUniformGradient.lean b/lean/OAI/Geometry/Perimeter/Regularity/EllipticUniformGradient.lean
new file mode 100644
index 0000000000000000000000000000000000000000..5d9abbacfbd02e7b26b8a77b6dc0c7adfdbaafd5
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/EllipticUniformGradient.lean
@@ -0,0 +1,87 @@
+import OAI.Geometry.Perimeter.Regularity.EllipticPointwise
+import OAI.Geometry.Perimeter.Regularity.EllipticSlopeComparison
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+ theorem gradient_modulus_from_slopes {θ R K H L : ℝ}
+    (hθ : 0<θ) (hR : 0<R) (hH : 0≤H) (hL : 0≤L)
+    (hLbig : 2^(n+2)≤volume.real (Metric.ball (0:Euc n) 1)*L^2)
+    {f : Euc n → ℝ} (hf : ContDiff ℝ 1 f)
+    {U : Set (Euc n)} {q : Euc n → ℕ → Euc n}
+    (he : ∀ p∈U, ∀ j, ‖q p j-gradient f p‖≤2*K*H*(1/2)^j)
+    (hE : ∀ p∈U, ∀ j, scaledGradientEnergy f p (θ^j*R) (q p j)≤H^2*((1/2)^j)^2) :
+    ∀ j p, p∈U → ∀ p', p'∈U → dist p p'≤θ^j*R/2 →
+       ‖gradient f p-gradient f p'‖≤(4*K+L)*H*(1/2)^j := by
+  intro j p hp p' hp' hd
+  have hV : 0<volume.real (Metric.ball (0:Euc n) 1) :=
+    ENNReal.toReal_pos (Metric.measure_ball_pos volume 0 (by norm_num)).ne' measure_ball_lt_top.ne
+  have hh := scaled_slope_comparison hf p p' (by positivity : 0<θ^j*R) hd (q p j) (q p' j)
+  have hEp := hE p hp j
+  have hEp' := hE p' hp' j
+  have hbound : ‖q p j-q p' j‖≤L*H*(1/2)^j := by
+    have h1 := mul_le_mul_of_nonneg_left (add_le_add hEp hEp') (by positivity : 0≤(2:ℝ)^(n+1))
+    have h2 := mul_le_mul_of_nonneg_right hLbig (by positivity : 0≤H^2*((1/2:ℝ)^j)^2)
+    have heq : (2:ℝ)^(n+2)* (H^2*((1/2:ℝ)^j)^2)=
+        2^(n+1)*(H^2*((1/2:ℝ)^j)^2+H^2*((1/2:ℝ)^j)^2) := by
+      rw [show n+2=(n+1)+1 by omega,pow_succ]; ring
+    rw [heq] at h2
+    have hs : ‖q p j-q p' j‖^2≤(L*H*(1/2)^j)^2 := by
+      apply (mul_le_mul_iff_right₀ hV).mp
+      nlinarith [hh.trans (h1.trans h2)]
+    nlinarith [norm_nonneg (q p j-q p' j),show 0≤L*H*(1/2:ℝ)^j by positivity]
+  have ht := norm_sub_le_norm_sub_add_norm_sub (gradient f p) (q p j) (gradient f p')
+  have ht' := norm_sub_le_norm_sub_add_norm_sub (q p j) (q p' j) (gradient f p')
+  rw [norm_sub_rev (gradient f p) (q p j)] at ht
+  have hep := he p hp j
+  have hep' := he p' hp' j
+  nlinarith
+
+ theorem linear_uniform_gradient {θ δ K a B H R : ℝ}
+    (hθ : 0<θ) (hθ1 : θ<1) (hK : 1≤K)
+    (hc : LinearContractionProperty n θ δ K)
+    (ha : 0≤a) (hB : 0≤B) (hH : 0≤H) (hR : 0<R)
+    (hsmall : 32*K^2*a≤1) (hforce : 16*K*B≤H)
+    {A : Euc n → Euc n →L[ℝ] Euc n} (hA : Continuous A)
+    {f : Euc n → ℝ} (hf : ContDiff ℝ 1 f) {F : Euc n → Euc n} (hF : Continuous F)
+    {U : Set (Euc n)}
+    (h : ∀ p∈U, WeakLinearEquation A f F (Metric.ball p R))
+    (hnear : ∀ p∈U, ∀ x∈Metric.ball p R, ‖ContinuousLinearMap.id ℝ (Euc n)-A x‖≤δ)
+    (hosc : ∀ p∈U, ∀ j x, x∈Metric.ball p (θ^j*R) →
+      ‖A p-A x‖≤a*(1/2)^j ∧ ‖F x-F p‖≤B*(1/2)^j)
+    (hinit : ∀ p∈U, scaledGradientEnergy f p R 0≤H^2) :
+    ∃ C : ℝ, 0<C ∧ (∀ p∈U, ‖gradient f p‖≤2*K*H) ∧
+      ∀ j p, p∈U → ∀ p', p'∈U → dist p p'≤θ^j*R/2 →
+        ‖gradient f p-gradient f p'‖≤C*H*(1/2)^j := by
+  have hex (p : Euc n) : ∃ q : ℕ → Euc n, p∈U →
+      (∀ j, ‖q j-gradient f p‖≤2*K*H*(1/2)^j) ∧
+      (∀ j, scaledGradientEnergy f p (θ^j*R) (q j)≤H^2*((1/2)^j)^2) ∧
+      ‖gradient f p‖≤2*K*H := by
+    by_cases hp : p∈U
+    · obtain ⟨q,hq⟩ := linear_pointwise_iteration hθ hθ1 hK hc ha hB hH hR hsmall hforce
+        hA hf hF p (h p hp) (hnear p hp) (hosc p hp) (hinit p hp)
+      exact ⟨q,fun _ ↦ hq⟩
+    · exact ⟨fun _ ↦ 0,fun hh ↦ (hp hh).elim⟩
+  choose q hq using hex
+  have hV : 0<volume.real (Metric.ball (0:Euc n) 1) :=
+    ENNReal.toReal_pos (Metric.measure_ball_pos volume 0 (by norm_num)).ne' measure_ball_lt_top.ne
+  let L := Real.sqrt (2^(n+2)/volume.real (Metric.ball (0:Euc n) 1))
+  have hL : 0≤L := Real.sqrt_nonneg _
+  have hLbig : 2^(n+2)≤volume.real (Metric.ball (0:Euc n) 1)*L^2 := by
+    dsimp [L]
+    rw [Real.sq_sqrt (by positivity)]
+    rw [mul_div_cancel₀ _ hV.ne']
+  refine ⟨4*K+L,by linarith,fun p hp ↦ (hq p hp).2.2,?_⟩
+  exact gradient_modulus_from_slopes hθ hR hH hL hLbig hf
+    (fun p hp ↦ (hq p hp).1) (fun p hp ↦ (hq p hp).2.1)
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/ExistsCompactTraceOnBounded.lean b/lean/OAI/Geometry/Perimeter/Regularity/ExistsCompactTraceOnBounded.lean
new file mode 100644
index 0000000000000000000000000000000000000000..546539517a60bbca5d38983cbb55c7d302340d8a
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/ExistsCompactTraceOnBounded.lean
@@ -0,0 +1,414 @@
+import OAI.Geometry.Perimeter.Regularity.Space
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+namespace RPI.Boundary
+
+theorem exists_compact_trace_on_bounded {n : ℕ} (S V : Set (Space n))
+    (hV : Bornology.IsBounded V) (f : Space n → ℝ) (hf : ContDiff ℝ ∞ f) :
+    ∃ F : Space n → ℝ, ContDiff ℝ ∞ F ∧ HasCompactSupport F ∧
+      EqOn F f V ∧ EqOn (surfaceGradient S F) (surfaceGradient S f) V := by
+  obtain ⟨r, hr⟩ := Metric.isBounded_iff_subset_ball (0 : Space n) |>.mp hV
+  let b : ContDiffBump (0 : Space n) :=
+    ⟨|r| + 1, |r| + 2, by positivity, by linarith⟩
+  have hb : V ⊆ Metric.ball (0 : Space n) b.rIn := by
+    intro x hx
+    have hx' := Metric.mem_ball.mp (hr hx)
+    change dist x 0 < |r| + 1
+    linarith [le_abs_self r]
+  refine ⟨fun x ↦ b x * f x, b.contDiff.mul hf, b.hasCompactSupport.mul_right, ?_, ?_⟩
+  · intro x hx
+    change b x * f x = f x
+    rw [b.one_of_mem_closedBall (Metric.ball_subset_closedBall (hb hx)), one_mul]
+  · intro x hx
+    apply surfaceGradient_congr_trace
+      ((b.contDiff.mul hf).differentiable (by norm_num) x)
+      (hf.differentiable (by norm_num) x)
+    filter_upwards [b.eventuallyEq_one_of_mem_ball (hb hx)] with y hy
+    intro _
+    simp only [hy, Pi.one_apply, one_mul]
+
+theorem exists_compact_approximants_of_mem_sobolevGraph {n : ℕ} (S V : Set (Space n))
+    (hV : Bornology.IsBounded V) (hVm : MeasurableSet V)
+    {h : Space n → ℝ} {G : Space n → Space n}
+    (hh : MemLp h 2 ((surfaceMeasure S).restrict V))
+    (hG : MemLp G 2 ((surfaceMeasure S).restrict V))
+    (hp : (hh.toLp h, hG.toLp G) ∈ closure (smoothGradientGraph S V)) :
+    ∃ φ : ℕ → Space n → ℝ, (∀ j, ContDiff ℝ ∞ (φ j) ∧ HasCompactSupport (φ j)) ∧
+      ∃ (hφ : ∀ j, MemLp (φ j) 2 ((surfaceMeasure S).restrict V))
+        (hDφ : ∀ j, MemLp (surfaceGradient S (φ j)) 2 ((surfaceMeasure S).restrict V)),
+        Tendsto (fun j ↦ ((hφ j).toLp (φ j),
+          (hDφ j).toLp (surfaceGradient S (φ j)))) atTop (𝓝 (hh.toLp h, hG.toLp G)) := by
+  classical
+  obtain ⟨z, hz, hzt⟩ := mem_closure_iff_seq_limit.mp hp
+  have hw : ∀ j, ∃ φ : Space n → ℝ, ContDiff ℝ ∞ φ ∧ HasCompactSupport φ ∧
+      ∃ (hφ : MemLp φ 2 ((surfaceMeasure S).restrict V))
+        (hDφ : MemLp (surfaceGradient S φ) 2 ((surfaceMeasure S).restrict V)),
+        z j = (hφ.toLp φ, hDφ.toLp (surfaceGradient S φ)) := by
+    intro j
+    obtain ⟨f, hf, hfmem, hDfmem, hzj⟩ := hz j
+    obtain ⟨F, hF, hFc, hFe, hFDe⟩ := exists_compact_trace_on_bounded S V hV f hf
+    have he : F =ᵐ[(surfaceMeasure S).restrict V] f :=
+      (ae_restrict_mem hVm).mono (fun x hx ↦ hFe hx)
+    have hDe : surfaceGradient S F =ᵐ[(surfaceMeasure S).restrict V] surfaceGradient S f :=
+      (ae_restrict_mem hVm).mono (fun x hx ↦ hFDe hx)
+    let hFm := MemLp.ae_eq he.symm hfmem
+    let hDFm := MemLp.ae_eq hDe.symm hDfmem
+    refine ⟨F, hF, hFc, hFm, hDFm, hzj.trans ?_⟩
+    exact Prod.ext (hfmem.toLp_congr hFm he.symm) (hDfmem.toLp_congr hDFm hDe.symm)
+  choose φ hφsmooth hφcompact hφ hDφ hzφ using hw
+  refine ⟨φ, fun j ↦ ⟨hφsmooth j, hφcompact j⟩, hφ, hDφ, ?_⟩
+  have hzfun : z = (fun j ↦ ((hφ j).toLp (φ j),
+      (hDφ j).toLp (surfaceGradient S (φ j)))) := funext hzφ
+  rw [← hzfun]
+  exact hzt
+
+end RPI.Boundary
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+namespace RPI
+
+section BallCutoffs
+variable (V : Type*) [NormedAddCommGroup V] [InnerProductSpace ℝ V]
+  [FiniteDimensional ℝ V]
+
+noncomputable def modelCapacityBump : ContDiffBump (0 : V) :=
+  ⟨1, 2, zero_lt_one, one_lt_two⟩
+
+noncomputable def ballCapacityCutoff (c : V) (r : ℝ) (x : V) : ℝ :=
+  1 - modelCapacityBump V (r⁻¹ • (x - c))
+
+theorem ballCapacityCutoff_contDiff (c : V) (r : ℝ) :
+    ContDiff ℝ ∞ (ballCapacityCutoff V c r) := by
+  apply contDiff_const.sub
+  apply (modelCapacityBump V).contDiff.comp
+  fun_prop
+
+theorem ballCapacityCutoff_mem_Icc (c : V) (r : ℝ) (x : V) :
+    ballCapacityCutoff V c r x ∈ Icc (0 : ℝ) 1 := by
+  constructor
+  · exact sub_nonneg.mpr (modelCapacityBump V).le_one
+  · exact sub_le_self _ (modelCapacityBump V).nonneg
+
+theorem ballCapacityCutoff_zero {c x : V} {r : ℝ} (hr : 0 < r)
+    (hx : dist x c ≤ r) : ballCapacityCutoff V c r x = 0 := by
+  unfold ballCapacityCutoff
+  rw [(modelCapacityBump V).one_of_mem_closedBall, sub_self]
+  change dist (r⁻¹ • (x - c)) 0 ≤ 1
+  rw [dist_zero_right, norm_smul, Real.norm_eq_abs, abs_inv, abs_of_pos hr,
+    ← div_eq_inv_mul, ← dist_eq_norm]
+  exact (div_le_one hr).mpr hx
+
+theorem ballCapacityCutoff_one {c x : V} {r : ℝ} (hr : 0 < r)
+    (hx : 2 * r ≤ dist x c) : ballCapacityCutoff V c r x = 1 := by
+  unfold ballCapacityCutoff
+  rw [(modelCapacityBump V).zero_of_le_dist, sub_zero]
+  change 2 ≤ dist (r⁻¹ • (x - c)) 0
+  rw [dist_zero_right, norm_smul, Real.norm_eq_abs, abs_inv, abs_of_pos hr,
+    ← div_eq_inv_mul, ← dist_eq_norm]
+  exact (le_div_iff₀ hr).mpr hx
+
+theorem exists_ballCapacityCutoff_gradient_bound :
+    ∃ C : ℝ, 0 < C ∧ ∀ (c : V) (r : ℝ), 0 < r →
+      ∀ x : V, ‖gradient (ballCapacityCutoff V c r) x‖ ≤ C / r := by
+  obtain ⟨L, hL⟩ := ContDiff.lipschitzWith_of_hasCompactSupport
+    (modelCapacityBump V).hasCompactSupport
+    (show ContDiff ℝ 1 (modelCapacityBump V) from (modelCapacityBump V).contDiff) one_ne_zero
+  let C : ℝ := (L : ℝ) + 1
+  have hC : 0 < C := by dsimp [C]; positivity
+  refine ⟨C, hC, ?_⟩
+  intro c r hr x
+  have hlip : LipschitzWith ⟨C / r, div_nonneg hC.le hr.le⟩ (ballCapacityCutoff V c r) := by
+    apply LipschitzWith.of_dist_le_mul
+    intro y z
+    have hd : dist (r⁻¹ • (y - c)) (r⁻¹ • (z - c)) = r⁻¹ * dist y z := by
+      simp only [dist_eq_norm, ← smul_sub, sub_sub_sub_cancel_right, norm_smul,
+        Real.norm_eq_abs, abs_inv, abs_of_pos hr]
+    have hbound := hL.dist_le_mul (r⁻¹ • (y - c)) (r⁻¹ • (z - c))
+    rw [hd] at hbound
+    have heq : dist (ballCapacityCutoff V c r y) (ballCapacityCutoff V c r z) =
+        dist (modelCapacityBump V (r⁻¹ • (y - c)))
+          (modelCapacityBump V (r⁻¹ • (z - c))) := by
+      simp only [ballCapacityCutoff, dist_eq_norm, sub_sub_sub_cancel_left, norm_sub_rev]
+    rw [heq]
+    change _ ≤ (C / r) * dist y z
+    calc
+      _ ≤ (L : ℝ) * (r⁻¹ * dist y z) := hbound
+      _ = ((L : ℝ) / r) * dist y z := by ring
+      _ ≤ (C / r) * dist y z := by
+        gcongr
+        dsimp [C]
+        linarith
+  have hb : ‖fderiv ℝ (ballCapacityCutoff V c r) x‖ ≤ C / r :=
+    norm_fderiv_le_of_lipschitz ℝ (x₀ := x) hlip
+  simpa only [gradient, LinearIsometryEquiv.norm_map] using hb
+
+theorem ballCapacityCutoff_gradient_zero {c x : V} {r : ℝ} (hr : 0 < r)
+    (hx : 2 * r ≤ dist x c) : gradient (ballCapacityCutoff V c r) x = 0 := by
+  have hmax : IsLocalMax (ballCapacityCutoff V c r) x := by
+    apply Filter.Eventually.of_forall
+    intro y
+    rw [ballCapacityCutoff_one V hr hx]
+    exact (ballCapacityCutoff_mem_Icc V c r y).2
+  simp only [gradient, hmax.fderiv_eq_zero, map_zero]
+
+end BallCutoffs
+
+section SmoothAggregate
+variable {ι V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V]
+  [FiniteDimensional ℝ V]
+
+noncomputable def capacityGate : ContDiffBump (0 : ℝ) :=
+  ⟨1 / 2, 1, by norm_num, by norm_num⟩
+
+noncomputable def bumpScore (s : Finset ι) (b : ι → V → ℝ) (x : V) : ℝ :=
+  ∑ j ∈ s, (b j x) ^ 2
+
+noncomputable def smoothCutoffAggregate (s : Finset ι) (b : ι → V → ℝ) (x : V) : ℝ :=
+  capacityGate (bumpScore s b x)
+
+omit [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] in
+theorem bumpScore_nonneg [NormedAddCommGroup V] [InnerProductSpace ℝ V]
+    [FiniteDimensional ℝ V] (s : Finset ι) (b : ι → V → ℝ) (x : V) :
+    0 ≤ bumpScore s b x := Finset.sum_nonneg (fun _ _ ↦ sq_nonneg _)
+
+omit [FiniteDimensional ℝ V] in
+theorem smoothCutoffAggregate_contDiff [FiniteDimensional ℝ V]
+    (s : Finset ι) (b : ι → V → ℝ)
+    (hb : ∀ j ∈ s, ContDiff ℝ ∞ (b j)) :
+    ContDiff ℝ ∞ (smoothCutoffAggregate s b) := by
+  apply capacityGate.contDiff.comp
+  exact ContDiff.sum (fun j hj ↦ (hb j hj).pow 2)
+
+omit [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] in
+theorem smoothCutoffAggregate_mem_Icc [NormedAddCommGroup V] [InnerProductSpace ℝ V]
+    [FiniteDimensional ℝ V] (s : Finset ι) (b : ι → V → ℝ) (x : V) :
+    smoothCutoffAggregate s b x ∈ Icc (0 : ℝ) 1 :=
+  ⟨capacityGate.nonneg, capacityGate.le_one⟩
+
+theorem smoothCutoffAggregate_zero {s : Finset ι} {b : ι → V → ℝ} {x : V}
+    (hx : 1 ≤ bumpScore s b x) : smoothCutoffAggregate s b x = 0 := by
+  apply capacityGate.zero_of_le_dist
+  change 1 ≤ dist (bumpScore s b x) 0
+  simpa only [Real.dist_eq, sub_zero, abs_of_nonneg (bumpScore_nonneg s b x)] using hx
+
+theorem smoothCutoffAggregate_zero_of_one {s : Finset ι} {b : ι → V → ℝ} {x : V}
+    {j : ι} (hj : j ∈ s) (hbx : b j x = 1) : smoothCutoffAggregate s b x = 0 := by
+  apply smoothCutoffAggregate_zero
+  have ht := Finset.single_le_sum (s := s) (f := fun j ↦ b j x ^ 2)
+    (fun i hi ↦ sq_nonneg _) hj
+  change 1 ≤ ∑ j ∈ s, b j x ^ 2
+  simpa only [hbx, one_pow] using ht
+
+omit [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] in
+theorem smoothCutoffAggregate_one [NormedAddCommGroup V] [InnerProductSpace ℝ V]
+    [FiniteDimensional ℝ V] {s : Finset ι} {b : ι → V → ℝ} {x : V}
+    (hx : ∀ j ∈ s, b j x = 0) : smoothCutoffAggregate s b x = 1 := by
+  have hscore : bumpScore s b x = 0 := by
+    apply Finset.sum_eq_zero
+    intro j hj
+    simp [hx j hj]
+  unfold smoothCutoffAggregate
+  rw [hscore]
+  exact capacityGate.one_of_mem_closedBall (by norm_num [capacityGate])
+
+theorem norm_sum_smul_sq_le {W : Type*} [NormedAddCommGroup W] [NormedSpace ℝ W]
+    (s : Finset ι) (a : ι → ℝ) (v : ι → W) :
+    ‖∑ j ∈ s, a j • v j‖ ^ 2 ≤
+      (∑ j ∈ s, (a j) ^ 2) * (∑ j ∈ s, ‖v j‖ ^ 2) := by
+  have ht : ‖∑ j ∈ s, a j • v j‖ ≤ ∑ j ∈ s, |a j| * ‖v j‖ := by
+    simpa only [norm_smul, Real.norm_eq_abs] using norm_sum_le s (fun j ↦ a j • v j)
+  calc
+    _ ≤ (∑ j ∈ s, |a j| * ‖v j‖) ^ 2 := by
+      exact (sq_le_sq₀ (norm_nonneg _) (Finset.sum_nonneg
+        (fun _ _ ↦ mul_nonneg (abs_nonneg _) (norm_nonneg _)))).mpr ht
+    _ ≤ (∑ j ∈ s, |a j| ^ 2) * (∑ j ∈ s, ‖v j‖ ^ 2) :=
+      Finset.sum_mul_sq_le_sq_mul_sq s (fun j ↦ |a j|) (fun j ↦ ‖v j‖)
+    _ = _ := by simp only [sq_abs]
+
+omit [FiniteDimensional ℝ V] in
+theorem bumpScore_fderiv [FiniteDimensional ℝ V] (s : Finset ι) (b : ι → V → ℝ) {x : V}
+    (hb : ∀ j ∈ s, DifferentiableAt ℝ (b j) x) :
+    fderiv ℝ (bumpScore s b) x = ∑ j ∈ s, (2 * b j x) • fderiv ℝ (b j) x := by
+  have hd : HasFDerivAt (bumpScore s b)
+      (∑ j ∈ s, (2 * b j x) • fderiv ℝ (b j) x) x := by
+    have h := HasFDerivAt.sum (fun j hj ↦ (hb j hj).hasFDerivAt.pow 2)
+    convert! h using 1
+    · ext y; simp only [bumpScore, Finset.sum_apply]
+    · simp only [Nat.reduceSub, pow_one, nsmul_eq_mul, Nat.cast_ofNat]
+  exact hd.fderiv
+
+theorem bumpScore_norm_fderiv_sq_le (s : Finset ι) (b : ι → V → ℝ) {x : V}
+    (hb : ∀ j ∈ s, DifferentiableAt ℝ (b j) x) :
+    ‖fderiv ℝ (bumpScore s b) x‖ ^ 2 ≤
+      4 * bumpScore s b x * ∑ j ∈ s, ‖fderiv ℝ (b j) x‖ ^ 2 := by
+  rw [bumpScore_fderiv s b hb]
+  have h := norm_sum_smul_sq_le s (fun j ↦ 2 * b j x) (fun j ↦ fderiv ℝ (b j) x)
+  convert! h using 1
+  congr 1
+  simp only [bumpScore, mul_pow, ← Finset.mul_sum]
+  norm_num
+
+theorem exists_smoothCutoffAggregate_gradient_bound :
+    ∃ B : ℝ, 0 < B ∧ ∀ (s : Finset ι) (b : ι → V → ℝ),
+      (∀ j ∈ s, ContDiff ℝ 1 (b j)) → ∀ x : V,
+      ‖gradient (smoothCutoffAggregate s b) x‖ ^ 2 ≤
+        B * ∑ j ∈ s, ‖gradient (b j) x‖ ^ 2 := by
+  obtain ⟨L, hL⟩ := ContDiff.lipschitzWith_of_hasCompactSupport
+    capacityGate.hasCompactSupport (show ContDiff ℝ 1 capacityGate from capacityGate.contDiff)
+    one_ne_zero
+  let B : ℝ := 4 * ((L : ℝ) + 1) ^ 2
+  have hB : 0 < B := by dsimp [B]; positivity
+  refine ⟨B, hB, ?_⟩
+  intro s b hb x
+  have hgrad : ∀ (f : V → ℝ), ‖gradient f x‖ = ‖fderiv ℝ f x‖ := fun _ ↦ by
+    simp only [gradient, LinearIsometryEquiv.norm_map]
+  simp only [hgrad]
+  by_cases hs : 1 ≤ bumpScore s b x
+  · have hmin : IsLocalMin (smoothCutoffAggregate s b) x := by
+      apply Filter.Eventually.of_forall
+      intro y
+      rw [smoothCutoffAggregate_zero hs]
+      exact (smoothCutoffAggregate_mem_Icc s b y).1
+    rw [hmin.fderiv_eq_zero, norm_zero, zero_pow (by decide)]
+    exact mul_nonneg hB.le (Finset.sum_nonneg (fun _ _ ↦ sq_nonneg _))
+  · have hdiff : ∀ j ∈ s, DifferentiableAt ℝ (b j) x := fun j hj ↦
+      (hb j hj).differentiable one_ne_zero |>.differentiableAt
+    have hsdiff : DifferentiableAt ℝ (bumpScore s b) x := by
+      convert! DifferentiableAt.sum (fun j hj ↦ (hdiff j hj).pow 2) using 1
+      ext y; simp only [bumpScore, Finset.sum_apply, Pi.pow_apply]
+    have hcomp : fderiv ℝ (smoothCutoffAggregate s b) x =
+        (fderiv ℝ capacityGate (bumpScore s b x)).comp (fderiv ℝ (bumpScore s b) x) :=
+      fderiv_comp x ((show ContDiff ℝ 1 capacityGate from capacityGate.contDiff).differentiable
+        one_ne_zero).differentiableAt hsdiff
+    have hdL : ‖fderiv ℝ capacityGate (bumpScore s b x)‖ ≤ (L : ℝ) :=
+      norm_fderiv_le_of_lipschitz ℝ hL
+    have hd : ‖fderiv ℝ (smoothCutoffAggregate s b) x‖ ≤
+        (L : ℝ) * ‖fderiv ℝ (bumpScore s b) x‖ := by
+      rw [hcomp]
+      exact (ContinuousLinearMap.opNorm_comp_le _ _).trans
+        (mul_le_mul_of_nonneg_right hdL (norm_nonneg _))
+    have hdsq := bumpScore_norm_fderiv_sq_le s b hdiff
+    have hnnsum : 0 ≤ ∑ j ∈ s, ‖fderiv ℝ (b j) x‖ ^ 2 :=
+      Finset.sum_nonneg (fun _ _ ↦ sq_nonneg _)
+    calc
+      _ ≤ ((L : ℝ) * ‖fderiv ℝ (bumpScore s b) x‖) ^ 2 := by
+        exact (sq_le_sq₀ (norm_nonneg _) (mul_nonneg L.coe_nonneg (norm_nonneg _))).mpr hd
+      _ = (L : ℝ) ^ 2 * ‖fderiv ℝ (bumpScore s b) x‖ ^ 2 := mul_pow _ _ _
+      _ ≤ (L : ℝ) ^ 2 * (4 * bumpScore s b x * ∑ j ∈ s, ‖fderiv ℝ (b j) x‖ ^ 2) := by
+        gcongr
+      _ ≤ B * ∑ j ∈ s, ‖fderiv ℝ (b j) x‖ ^ 2 := by
+        dsimp [B]
+        have hscore : bumpScore s b x ≤ 1 := le_of_not_ge hs
+        have hLpos := L.coe_nonneg
+        nlinarith [mul_nonneg (sq_nonneg (L : ℝ)) hnnsum,
+          mul_le_mul_of_nonneg_left hscore (sq_nonneg (L : ℝ))]
+
+end SmoothAggregate
+
+section Energy
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V]
+  [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V]
+
+noncomputable def smoothDirichletEnergy (μ : Measure V) (f : V → ℝ) : ℝ≥0∞ :=
+  ∫⁻ x, ENNReal.ofReal (‖gradient f x‖ ^ 2) ∂μ
+
+theorem smoothDirichletEnergy_le_ball (μ : Measure V) (f : V → ℝ)
+    (c : V) (r L : ℝ) (hL : 0 ≤ L)
+    (hbound : ∀ x, ‖gradient f x‖ ≤ L)
+    (hzero : ∀ x, r ≤ dist x c → gradient f x = 0) :
+    smoothDirichletEnergy μ f ≤ ENNReal.ofReal (L ^ 2) * μ (Metric.ball c r) := by
+  calc
+    _ ≤ ∫⁻ x, (Metric.ball c r).indicator (fun _ ↦ ENNReal.ofReal (L ^ 2)) x ∂μ := by
+      apply lintegral_mono
+      intro x
+      by_cases hx : x ∈ Metric.ball c r
+      · rw [Set.indicator_of_mem hx]
+        exact ENNReal.ofReal_le_ofReal ((sq_le_sq₀ (norm_nonneg _) hL).mpr (hbound x))
+      · rw [Set.indicator_of_notMem hx]
+        simp only [Metric.mem_ball, not_lt] at hx
+        simp [hzero x hx]
+    _ = _ := by
+      rw [lintegral_indicator Metric.isOpen_ball.measurableSet, lintegral_const,
+        Measure.restrict_apply_univ]
+
+noncomputable def innerCapacityBump (c : V) (r : ℝ) (x : V) : ℝ :=
+  1 - ballCapacityCutoff V c r x
+
+omit [MeasurableSpace V] [BorelSpace V] in
+theorem innerCapacityBump_contDiff [MeasurableSpace V] [BorelSpace V] (c : V) (r : ℝ) :
+    ContDiff ℝ ∞ (innerCapacityBump c r) :=
+  contDiff_const.sub (ballCapacityCutoff_contDiff V c r)
+
+omit [MeasurableSpace V] [BorelSpace V] in
+theorem innerCapacityBump_gradient [MeasurableSpace V] [BorelSpace V] (c : V) (r : ℝ) (x : V) :
+    gradient (innerCapacityBump c r) x = -gradient (ballCapacityCutoff V c r) x := by
+  change gradient (fun y ↦ 1 - ballCapacityCutoff V c r y) x =
+    -gradient (ballCapacityCutoff V c r) x
+  simp only [gradient, fderiv_const_sub, map_neg]
+
+omit [MeasurableSpace V] [BorelSpace V] in
+theorem innerCapacityBump_one [MeasurableSpace V] [BorelSpace V] {c x : V} {r : ℝ} (hr : 0 < r)
+    (hx : dist x c ≤ r) : innerCapacityBump c r x = 1 := by
+  simp only [innerCapacityBump, ballCapacityCutoff_zero V hr hx, sub_zero]
+
+omit [MeasurableSpace V] [BorelSpace V] in
+theorem innerCapacityBump_zero [MeasurableSpace V] [BorelSpace V] {c x : V} {r : ℝ} (hr : 0 < r)
+    (hx : 2 * r ≤ dist x c) : innerCapacityBump c r x = 0 := by
+  simp only [innerCapacityBump, ballCapacityCutoff_one V hr hx, sub_self]
+
+theorem innerCapacityBump_energy_le (μ : Measure V) {C : ℝ} (hC : 0 < C)
+    (hbound : ∀ (c : V) (r : ℝ), 0 < r → ∀ x : V,
+      ‖gradient (ballCapacityCutoff V c r) x‖ ≤ C / r)
+    (c : V) (r : ℝ) (hr : 0 < r) :
+    smoothDirichletEnergy μ (innerCapacityBump c r) ≤
+      ENNReal.ofReal ((C / r) ^ 2) * μ (Metric.ball c (2 * r)) := by
+  apply smoothDirichletEnergy_le_ball μ _ c (2 * r) (C / r) (by positivity)
+  · intro x
+    simpa only [innerCapacityBump_gradient, norm_neg] using hbound c r hr x
+  · intro x hx
+    simp only [innerCapacityBump_gradient, ballCapacityCutoff_gradient_zero V hr hx, neg_zero]
+
+theorem smoothCutoffAggregate_energy_le {ι : Type*} (μ : Measure V)
+    {B : ℝ} (hB : 0 ≤ B) (s : Finset ι) (b : ι → V → ℝ)
+    (hb : ∀ j ∈ s, ContDiff ℝ 1 (b j))
+    (hbound : ∀ x : V, ‖gradient (smoothCutoffAggregate s b) x‖ ^ 2 ≤
+      B * ∑ j ∈ s, ‖gradient (b j) x‖ ^ 2) :
+    smoothDirichletEnergy μ (smoothCutoffAggregate s b) ≤
+      ENNReal.ofReal B * ∑ j ∈ s, smoothDirichletEnergy μ (b j) := by
+  have hmeas : ∀ j ∈ s, Measurable (fun x ↦ ENNReal.ofReal (‖gradient (b j) x‖ ^ 2)) := by
+    intro j hj
+    have hg : Continuous (gradient (b j)) :=
+      (InnerProductSpace.toDual ℝ V).symm.continuous.comp
+        ((hb j hj).continuous_fderiv one_ne_zero)
+    exact (hg.norm.pow 2).measurable.ennreal_ofReal
+  calc
+    _ ≤ ∫⁻ x, ENNReal.ofReal B * ∑ j ∈ s, ENNReal.ofReal (‖gradient (b j) x‖ ^ 2) ∂μ := by
+      apply lintegral_mono
+      intro x
+      change ENNReal.ofReal (‖gradient (smoothCutoffAggregate s b) x‖ ^ 2) ≤
+        ENNReal.ofReal B * ∑ j ∈ s, ENNReal.ofReal (‖gradient (b j) x‖ ^ 2)
+      rw [← ENNReal.ofReal_sum_of_nonneg (s := s) (f := fun j ↦ ‖gradient (b j) x‖ ^ 2)
+        (fun j hj ↦ sq_nonneg _),
+        ← ENNReal.ofReal_mul hB]
+      exact ENNReal.ofReal_le_ofReal (hbound x)
+    _ = _ := by
+      rw [lintegral_const_mul (ENNReal.ofReal B) (Finset.measurable_sum s hmeas),
+        lintegral_finsetSum s hmeas]
+      rfl
+
+end Energy
+end RPI
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/ExistsDiameterCoverOfHausdorffZero.lean b/lean/OAI/Geometry/Perimeter/Regularity/ExistsDiameterCoverOfHausdorffZero.lean
new file mode 100644
index 0000000000000000000000000000000000000000..f6d969e889a1ec6318f5d810a7e1863a479e190d
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/ExistsDiameterCoverOfHausdorffZero.lean
@@ -0,0 +1,496 @@
+import OAI.Geometry.Perimeter.Regularity.VectorEnergy
+import OAI.Geometry.Perimeter.Regularity.ExistsCompactTraceOnBounded
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+namespace RPI
+
+section Covers
+variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X]
+
+theorem exists_diameter_cover_of_hausdorff_zero (S : Set X) (m : ℕ) (hm : 0 < m)
+    (hS : Measure.hausdorffMeasure (m : ℝ) S = 0)
+    {δ ε : ℝ≥0∞} (hδ : 0 < δ) (hε : 0 < ε) :
+    ∃ t : ℕ → Set X, S ⊆ ⋃ j, t j ∧ (∀ j, Metric.ediam (t j) ≤ δ) ∧
+      (∑' j, Metric.ediam (t j) ^ m) < ε := by
+  have hi : (⨅ (t : ℕ → Set X) (_ : S ⊆ ⋃ j, t j)
+      (_ : ∀ j, Metric.ediam (t j) ≤ δ),
+      ∑' j, ⨆ _ : (t j).Nonempty, Metric.ediam (t j) ^ (m : ℝ)) < ε := by
+    apply lt_of_le_of_lt _ (show Measure.hausdorffMeasure (m : ℝ) S < ε by simpa [hS])
+    rw [Measure.hausdorffMeasure_apply]
+    exact le_iSup_of_le δ (le_iSup_of_le hδ le_rfl)
+  obtain ⟨t, ht⟩ := iInf_lt_iff.mp hi
+  obtain ⟨hcover, ht⟩ := iInf_lt_iff.mp ht
+  obtain ⟨hdiam, ht⟩ := iInf_lt_iff.mp ht
+  refine ⟨t, hcover, hdiam, ?_⟩
+  convert ht using 1
+  congr 1
+  funext j
+  by_cases hj : (t j).Nonempty
+  · simp only [ENNReal.rpow_natCast, iSup_pos hj]
+  · have he : t j = ∅ := Set.not_nonempty_iff_eq_empty.mp hj
+    simp [he, hm.ne', ENNReal.zero_rpow_of_pos (show 0 < (m : ℝ) by exact_mod_cast hm)]
+
+theorem exists_finite_ball_cover_of_hausdorff_zero [Nonempty X] (S : Set X) (hcompact : IsCompact S)
+    (m : ℕ) (hm : 0 < m) (hS : Measure.hausdorffMeasure (m : ℝ) S = 0)
+    {δ ε : ℝ} (hδ : 0 < δ) (hε : 0 < ε) :
+    ∃ (s : Finset ℕ) (c : ℕ → X) (r : ℕ → ℝ),
+      (∀ j ∈ s, c j ∈ S ∧ 0 < r j ∧ r j < δ) ∧
+      S ⊆ ⋃ j ∈ s, Metric.ball (c j) (r j) ∧
+      ∑ j ∈ s, (r j) ^ m < ε := by
+  classical
+  by_cases hSne : S.Nonempty
+  · let K : ℝ := 2 ^ (m - 1)
+    have hK : 0 < K := by dsimp [K]; positivity
+    let ε' : ℝ := ε / (4 * K)
+    have hε' : 0 < ε' := by dsimp [ε']; positivity
+    let a : ℝ := min (δ / 4) (min 1 (ε / (8 * K)))
+    have ha : 0 < a := by dsimp [a]; positivity
+    have haδ : a ≤ δ / 4 := min_le_left _ _
+    have ha1 : a ≤ 1 := (min_le_right _ _).trans (min_le_left _ _)
+    have haε : a ≤ ε / (8 * K) := (min_le_right _ _).trans (min_le_right _ _)
+    obtain ⟨t, hcover, hdiam, hcost⟩ := exists_diameter_cover_of_hausdorff_zero S m hm hS
+      (δ := ENNReal.ofReal (δ / 4)) (ε := ENNReal.ofReal ε')
+      (ENNReal.ofReal_pos.mpr (by positivity)) (ENNReal.ofReal_pos.mpr hε')
+    have htfin : ∀ j, Metric.ediam (t j) ≠ ⊤ := fun j ↦
+      ne_top_of_le_ne_top ENNReal.ofReal_ne_top (hdiam j)
+    let d : ℕ → ℝ := fun j ↦ (Metric.ediam (t j)).toReal
+    have hd0 : ∀ j, 0 ≤ d j := fun _ ↦ ENNReal.toReal_nonneg
+    have hdδ : ∀ j, d j ≤ δ / 4 := fun j ↦ by
+      exact (ENNReal.toReal_mono ENNReal.ofReal_ne_top (hdiam j)).trans_eq
+        (ENNReal.toReal_ofReal (by positivity))
+    let b : ℕ → ℝ := fun j ↦ a * (1 / 2 : ℝ) ^ j
+    have hbpos : ∀ j, 0 < b j := fun j ↦ by dsimp [b]; positivity
+    have hble : ∀ j, b j ≤ a := fun j ↦ by
+      dsimp [b]
+      have hp : (1 / 2 : ℝ) ^ j ≤ 1 := pow_le_one₀ (by norm_num) (by norm_num)
+      nlinarith
+    have hbs : Summable b := (summable_geometric_of_abs_lt_one (by norm_num : |(1 / 2 : ℝ)| < 1)).mul_left a
+    have hbtsum : ∑' j, b j = 2 * a := by
+      dsimp [b]
+      rw [tsum_mul_left, tsum_geometric_of_lt_one (by norm_num) (by norm_num)]
+      ring
+    have hc : ∀ j, ∃ x ∈ S, (S ∩ t j).Nonempty → x ∈ t j := by
+      intro j
+      by_cases hj : (S ∩ t j).Nonempty
+      · obtain ⟨x, hx⟩ := hj
+        exact ⟨x, hx.1, fun _ ↦ hx.2⟩
+      · obtain ⟨x, hx⟩ := hSne
+        exact ⟨x, hx, fun hn ↦ (hj hn).elim⟩
+    choose c hcS hct using hc
+    let r : ℕ → ℝ := fun j ↦ d j + b j
+    have hrpos : ∀ j, 0 < r j := fun j ↦ by dsimp [r]; linarith [hd0 j, hbpos j]
+    have hrδ : ∀ j, r j < δ := fun j ↦ by
+      dsimp [r]
+      linarith [hdδ j, (hble j).trans haδ]
+    have hballcover : S ⊆ ⋃ j, Metric.ball (c j) (r j) := by
+      intro x hx
+      obtain ⟨j, hj⟩ := Set.mem_iUnion.mp (hcover hx)
+      apply Set.mem_iUnion.mpr
+      refine ⟨j, ?_⟩
+      have hdxc : dist x (c j) ≤ d j :=
+        Metric.dist_le_diam_of_mem' (htfin j) hj (hct j ⟨x, hx, hj⟩)
+      change dist x (c j) < d j + b j
+      linarith [hbpos j]
+    obtain ⟨s, hscover⟩ := hcompact.elim_finite_subcover
+      (fun j ↦ Metric.ball (c j) (r j)) (fun _ ↦ Metric.isOpen_ball) hballcover
+    refine ⟨s, c, r, fun j _ ↦ ⟨hcS j, hrpos j, hrδ j⟩, hscover, ?_⟩
+    have hdscost : ∑ j ∈ s, d j ^ m < ε' := by
+      have hst := (ENNReal.sum_le_tsum s).trans_lt hcost
+      have heq : (∑ j ∈ s, Metric.ediam (t j) ^ m).toReal = ∑ j ∈ s, d j ^ m := by
+        rw [ENNReal.toReal_sum (fun j hj ↦ ENNReal.pow_ne_top (htfin j))]
+        simp only [ENNReal.toReal_pow, d]
+      have hh := ENNReal.toReal_lt_toReal
+        (ne_top_of_lt hst) ENNReal.ofReal_ne_top |>.mpr hst
+      simpa only [heq, ENNReal.toReal_ofReal hε'.le] using hh
+    have hbspow : ∑ j ∈ s, b j ^ m ≤ 2 * a := by
+      calc
+        _ ≤ ∑ j ∈ s, b j := by
+          apply Finset.sum_le_sum
+          intro j hj
+          exact pow_le_of_le_one (hbpos j).le ((hble j).trans ha1) hm.ne'
+        _ ≤ ∑' j, b j := hbs.sum_le_tsum s (fun _ _ ↦ (hbpos _).le)
+        _ = _ := hbtsum
+    calc
+      _ ≤ ∑ j ∈ s, K * (d j ^ m + b j ^ m) := by
+        apply Finset.sum_le_sum
+        intro j hj
+        exact add_pow_le (hd0 j) (hbpos j).le m
+      _ = K * ((∑ j ∈ s, d j ^ m) + ∑ j ∈ s, b j ^ m) := by
+        rw [← Finset.mul_sum, Finset.sum_add_distrib]
+      _ < K * (ε' + 2 * a) :=
+        mul_lt_mul_of_pos_left (add_lt_add_of_lt_of_le hdscost hbspow) hK
+      _ < ε := by
+        dsimp [ε']
+        have haux : a * (8 * K) ≤ ε := (le_div_iff₀ (by positivity)).mp haε
+        field_simp
+        nlinarith
+  · have hSe : S = ∅ := Set.not_nonempty_iff_eq_empty.mp hSne
+    exact ⟨∅, fun _ ↦ Classical.choice (inferInstance : Nonempty X), fun _ ↦ 0,
+      by simp, by simp [hSe], by simpa using hε⟩
+
+end Covers
+end RPI
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+namespace RPI
+section CapacityConstruction
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V]
+  [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V]
+
+theorem innerCapacityBump_energy_le_radius_power (μ : Measure V)
+    {C A : ℝ} (hC : 0 < C) (hA : 0 < A)
+    (hbound : ∀ (c : V) (r : ℝ), 0 < r → ∀ x : V,
+      ‖gradient (ballCapacityCutoff V c r) x‖ ≤ C / r)
+    (m : ℕ) (c : V) (r : ℝ) (hr : 0 < r)
+    (hgrowth : μ (Metric.ball c (2 * r)) ≤ ENNReal.ofReal (A * (2 * r) ^ (m + 2))) :
+    smoothDirichletEnergy μ (innerCapacityBump c r) ≤
+      ENNReal.ofReal (C ^ 2 * A * 2 ^ (m + 2) * r ^ m) := by
+  calc
+    _ ≤ ENNReal.ofReal ((C / r) ^ 2) * μ (Metric.ball c (2 * r)) :=
+      innerCapacityBump_energy_le μ hC hbound c r hr
+    _ ≤ ENNReal.ofReal ((C / r) ^ 2) * ENNReal.ofReal (A * (2 * r) ^ (m + 2)) := by
+      gcongr
+    _ = _ := by
+      rw [← ENNReal.ofReal_mul (sq_nonneg _)]
+      congr 1
+      rw [mul_pow, pow_add r m 2]
+      field_simp
+
+theorem exists_smooth_capacity_cutoff (μ : Measure V) (S : Set V)
+    (hcompact : IsCompact S) (m : ℕ) (hm : 0 < m)
+    (hS : Measure.hausdorffMeasure (m : ℝ) S = 0)
+    {A r₀ : ℝ} (hA : 0 < A) (hr₀ : 0 < r₀)
+    (hgrowth : ∀ c ∈ S, ∀ r : ℝ, 0 < r → r < r₀ →
+      μ (Metric.ball c r) ≤ ENNReal.ofReal (A * r ^ (m + 2)))
+    {δ ε : ℝ} (hδ : 0 < δ) (hε : 0 < ε) :
+    ∃ χ : V → ℝ, ContDiff ℝ ∞ χ ∧ (∀ x, χ x ∈ Icc (0 : ℝ) 1) ∧
+      (∃ U : Set V, IsOpen U ∧ S ⊆ U ∧ ∀ x ∈ U, χ x = 0) ∧
+      (∀ x, (∀ c ∈ S, 2 * δ ≤ dist x c) → χ x = 1) ∧
+      smoothDirichletEnergy μ χ < ENNReal.ofReal ε := by
+  classical
+  obtain ⟨C, hC, hCbound⟩ := exists_ballCapacityCutoff_gradient_bound V
+  obtain ⟨B, hB, hBbound⟩ := exists_smoothCutoffAggregate_gradient_bound (ι := ℕ) (V := V)
+  let D : ℝ := C ^ 2 * A * 2 ^ (m + 2)
+  have hD : 0 < D := by dsimp [D]; positivity
+  let E : ℝ := B * D
+  have hE : 0 < E := mul_pos hB hD
+  let δ' : ℝ := min δ (r₀ / 4)
+  have hδ' : 0 < δ' := lt_min hδ (by positivity)
+  obtain ⟨s, c, r, hcsr, hcover, hcost⟩ :=
+    exists_finite_ball_cover_of_hausdorff_zero S hcompact m hm hS
+      (δ := δ') (ε := ε / E) hδ' (by positivity)
+  let b : ℕ → V → ℝ := fun j ↦ innerCapacityBump (c j) (r j)
+  have hb : ∀ j, ContDiff ℝ ∞ (b j) := fun j ↦ innerCapacityBump_contDiff (c j) (r j)
+  have hb₁ : ∀ j ∈ s, ContDiff ℝ 1 (b j) := fun j _ ↦ (hb j).of_le (by norm_num)
+  let χ : V → ℝ := smoothCutoffAggregate s b
+  refine ⟨χ, smoothCutoffAggregate_contDiff s b (fun j _ ↦ hb j),
+    smoothCutoffAggregate_mem_Icc s b, ?_, ?_, ?_⟩
+  · refine ⟨⋃ j ∈ s, Metric.ball (c j) (r j), isOpen_iUnion (fun _ ↦ isOpen_iUnion
+        (fun _ ↦ Metric.isOpen_ball)), hcover, ?_⟩
+    intro x hx
+    obtain ⟨j, hxj⟩ := Set.mem_iUnion.mp hx
+    obtain ⟨hj, hxj⟩ := Set.mem_iUnion.mp hxj
+    apply smoothCutoffAggregate_zero_of_one hj
+    exact innerCapacityBump_one (hcsr j hj).2.1 (Metric.mem_ball.mp hxj).le
+  · intro x hx
+    apply smoothCutoffAggregate_one
+    intro j hj
+    apply innerCapacityBump_zero (hcsr j hj).2.1
+    have hrδ := (hcsr j hj).2.2.trans_le (min_le_left δ (r₀ / 4))
+    linarith [hx (c j) (hcsr j hj).1]
+  · have hej : ∀ j ∈ s, smoothDirichletEnergy μ (b j) ≤ ENNReal.ofReal (D * (r j) ^ m) := by
+      intro j hj
+      apply innerCapacityBump_energy_le_radius_power μ hC hA hCbound m (c j) (r j) (hcsr j hj).2.1
+      apply hgrowth (c j) (hcsr j hj).1 (2 * r j)
+        (mul_pos (by norm_num) (hcsr j hj).2.1)
+      have hrr := (hcsr j hj).2.2.trans_le (min_le_right δ (r₀ / 4))
+      linarith
+    have hagg := smoothCutoffAggregate_energy_le μ hB.le s b hb₁ (hBbound s b hb₁)
+    have hsum0 : 0 ≤ ∑ j ∈ s, r j ^ m :=
+      Finset.sum_nonneg (fun j hj ↦ pow_nonneg (hcsr j hj).2.1.le _)
+    calc
+      _ ≤ ENNReal.ofReal B * ∑ j ∈ s, smoothDirichletEnergy μ (b j) := hagg
+      _ ≤ ENNReal.ofReal B * ∑ j ∈ s, ENNReal.ofReal (D * (r j) ^ m) := by
+        gcongr with j hj
+        exact hej j hj
+      _ = ENNReal.ofReal (E * ∑ j ∈ s, r j ^ m) := by
+        rw [← ENNReal.ofReal_sum_of_nonneg (s := s) (f := fun j ↦ D * r j ^ m)
+          (fun j hj ↦ mul_nonneg hD.le (pow_nonneg (hcsr j hj).2.1.le _)),
+          ← ENNReal.ofReal_mul hB.le, ← Finset.mul_sum]
+        congr 1
+        dsimp [E, D]
+        ring
+      _ < ENNReal.ofReal ε := by
+        apply ENNReal.ofReal_lt_ofReal_iff hε |>.mpr
+        have h := (lt_div_iff₀ hE).mp hcost
+        simpa only [mul_comm] using h
+
+theorem exists_smooth_capacity_sequence (μ : Measure V) (S : Set V)
+    (hcompact : IsCompact S) (m : ℕ) (hm : 0 < m)
+    (hS : Measure.hausdorffMeasure (m : ℝ) S = 0)
+    {A r₀ : ℝ} (hA : 0 < A) (hr₀ : 0 < r₀)
+    (hgrowth : ∀ c ∈ S, ∀ r : ℝ, 0 < r → r < r₀ →
+      μ (Metric.ball c r) ≤ ENNReal.ofReal (A * r ^ (m + 2))) :
+    ∃ χ : ℕ → V → ℝ,
+      (∀ i, ContDiff ℝ ∞ (χ i) ∧ (∀ x, χ i x ∈ Icc (0 : ℝ) 1) ∧
+        ∃ U : Set V, IsOpen U ∧ S ⊆ U ∧ ∀ x ∈ U, χ i x = 0) ∧
+      (∀ x ∉ S, ∀ᶠ i in atTop, χ i x = 1) ∧
+      Tendsto (fun i ↦ smoothDirichletEnergy μ (χ i)) atTop (𝓝 0) := by
+  classical
+  let a : ℕ → ℝ := fun i ↦ 1 / ((i : ℝ) + 1)
+  have ha : ∀ i, 0 < a i := fun i ↦ by dsimp [a]; positivity
+  have ha0 : Tendsto a atTop (𝓝 0) := tendsto_one_div_add_atTop_nhds_zero_nat
+  have hχ : ∀ i : ℕ, ∃ χ : V → ℝ, ContDiff ℝ ∞ χ ∧
+      (∀ x, χ x ∈ Icc (0 : ℝ) 1) ∧
+      (∃ U : Set V, IsOpen U ∧ S ⊆ U ∧ ∀ x ∈ U, χ x = 0) ∧
+      (∀ x, (∀ c ∈ S, 2 * a i ≤ dist x c) → χ x = 1) ∧
+      smoothDirichletEnergy μ χ < ENNReal.ofReal (a i) := fun i ↦
+    exists_smooth_capacity_cutoff μ S hcompact m hm hS hA hr₀ hgrowth (ha i) (ha i)
+  choose χ hχsmooth hχbound hχzero hχone hχenergy using hχ
+  refine ⟨χ, fun i ↦ ⟨hχsmooth i, hχbound i, hχzero i⟩, ?_, ?_⟩
+  · intro x hx
+    obtain ⟨d, hd, hball⟩ := Metric.isOpen_iff.mp hcompact.isClosed.isOpen_compl x hx
+    have hfar : ∀ c ∈ S, d ≤ dist x c := by
+      intro c hc
+      by_contra hcclose
+      have hmem : c ∈ Metric.ball x d := by
+        simpa only [Metric.mem_ball, dist_comm] using lt_of_not_ge hcclose
+      exact hball hmem hc
+    have h2a : Tendsto (fun i ↦ 2 * a i) atTop (𝓝 0) := by
+      simpa only [mul_zero] using ha0.const_mul 2
+    filter_upwards [h2a.eventually (gt_mem_nhds hd)] with i hi
+    exact hχone i x (fun c hc ↦ hi.le.trans (hfar c hc))
+  · have hea : Tendsto (fun i ↦ ENNReal.ofReal (a i)) atTop (𝓝 0) := by
+      simpa only [ENNReal.ofReal_zero] using ENNReal.tendsto_ofReal ha0
+    exact tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds hea
+      (fun _ ↦ zero_le) (fun i ↦ (hχenergy i).le)
+
+omit [BorelSpace V] in
+theorem projected_smooth_energy_le (μ : Measure V) (P : V → Submodule ℝ V) (f : V → ℝ) :
+    (∫⁻ x, ENNReal.ofReal
+      (‖((P x).orthogonalProjectionOnto (gradient f x) : V)‖ ^ 2) ∂μ) ≤
+        smoothDirichletEnergy μ f := by
+  apply lintegral_mono
+  intro x
+  apply ENNReal.ofReal_le_ofReal
+  have hp : ‖((P x).orthogonalProjectionOnto (gradient f x) : V)‖ ≤ ‖gradient f x‖ := by
+    calc
+      _ ≤ ‖(P x).orthogonalProjectionOnto‖ * ‖gradient f x‖ :=
+        (P x).orthogonalProjectionOnto.le_opNorm _
+      _ ≤ 1 * ‖gradient f x‖ := mul_le_mul_of_nonneg_right
+        (P x).orthogonalProjectionOnto_norm_le (norm_nonneg _)
+      _ = _ := one_mul _
+  exact (sq_le_sq₀ (norm_nonneg _) (norm_nonneg _)).mpr hp
+
+omit [BorelSpace V] in
+theorem projected_smooth_energy_tendsto_zero (μ : Measure V) (P : V → Submodule ℝ V)
+    (χ : ℕ → V → ℝ)
+    (hχ : Tendsto (fun i ↦ smoothDirichletEnergy μ (χ i)) atTop (𝓝 0)) :
+    Tendsto (fun i ↦ ∫⁻ x, ENNReal.ofReal
+      (‖((P x).orthogonalProjectionOnto (gradient (χ i) x) : V)‖ ^ 2) ∂μ) atTop (𝓝 0) :=
+  tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds hχ
+    (fun _ ↦ zero_le) (fun i ↦ projected_smooth_energy_le μ P (χ i))
+
+omit [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] in
+theorem hausdorff_capacity_dimension {n : ℕ} (hn : 8 ≤ n) (S : Set V)
+    (hdim : dimH S ≤ (n - 8 : ℕ)) :
+    0 < n - 3 ∧ Measure.hausdorffMeasure ((n - 3 : ℕ) : ℝ) S = 0 := by
+  refine ⟨by omega, ?_⟩
+  have hlt : dimH S < (n - 3 : ℕ) := hdim.trans_lt (by
+    exact_mod_cast (show n - 8 < n - 3 by omega))
+  simpa only [NNReal.coe_natCast] using hausdorffMeasure_of_dimH_lt (d := (n - 3 : ℕ)) hlt
+
+end CapacityConstruction
+end RPI
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+namespace RPI.Boundary
+
+theorem exists_memLp_capacity_sequence {n : ℕ} (hn : 8 ≤ n)
+    (S V Z : Set (Space n)) (hZc : IsCompact Z) (hZd : dimH Z ≤ (n - 8 : ℕ))
+    (hgrad : ∀ f : Space n → ℝ, ContDiff ℝ ∞ f →
+      AEStronglyMeasurable (surfaceGradient S f) ((surfaceMeasure S).restrict V))
+    {A r₀ : ℝ} (hA : 0 < A) (hr₀ : 0 < r₀)
+    (hgrowth : ∀ c ∈ Z, ∀ r : ℝ, 0 < r → r < r₀ →
+      (surfaceMeasure S).restrict V (Metric.ball c r) ≤
+        ENNReal.ofReal (A * r ^ (n - 1))) :
+    ∃ χ : ℕ → Space n → ℝ,
+      (∀ i, ContDiff ℝ ∞ (χ i) ∧ (∀ x, χ i x ∈ Icc (0 : ℝ) 1) ∧
+        (∃ U : Set (Space n), IsOpen U ∧ Z ⊆ U ∧ ∀ x ∈ U, χ i x = 0) ∧
+        MemLp (surfaceGradient S (χ i)) 2 ((surfaceMeasure S).restrict V)) ∧
+      (∀ᵐ x ∂(surfaceMeasure S).restrict V,
+        Tendsto (fun i ↦ χ i x) atTop (𝓝 1)) ∧
+      Tendsto (fun i ↦ vectorEnergy ((surfaceMeasure S).restrict V)
+        (surfaceGradient S (χ i))) atTop (𝓝 0) := by
+  obtain ⟨hm, hZd'⟩ := hausdorff_capacity_dimension hn Z hZd
+  have hpow : n - 3 + 2 = n - 1 := by omega
+  obtain ⟨χ, hχ, hχone, hχenergy⟩ := exists_smooth_capacity_sequence
+    ((surfaceMeasure S).restrict V) Z hZc (n - 3) hm hZd' hA hr₀
+    (by simpa only [hpow] using hgrowth)
+  have hE : Tendsto (fun i ↦ vectorEnergy ((surfaceMeasure S).restrict V)
+      (surfaceGradient S (χ i))) atTop (𝓝 0) := by
+    exact projected_smooth_energy_tendsto_zero ((surfaceMeasure S).restrict V)
+      (tangentPlane S) χ hχenergy
+  have hfinite : ∀ᶠ i in atTop, vectorEnergy ((surfaceMeasure S).restrict V)
+      (surfaceGradient S (χ i)) < 1 :=
+    hE.eventually (gt_mem_nhds (by norm_num : (0 : ℝ≥0∞) < 1))
+  obtain ⟨N, hN⟩ := eventually_atTop.mp hfinite
+  have hshift : Tendsto (fun i : ℕ ↦ i + N) atTop atTop := tendsto_add_atTop_nat N
+  refine ⟨fun i ↦ χ (i + N), ?_, ?_, hE.comp hshift⟩
+  · intro i
+    refine ⟨(hχ (i + N)).1, (hχ (i + N)).2.1, (hχ (i + N)).2.2, ?_⟩
+    apply (memLp_two_iff_vectorEnergy (hgrad _ (hχ (i + N)).1)).mpr
+    exact (hN (i + N) (Nat.le_add_left N i)).trans (by norm_num)
+  · have hnull : (surfaceMeasure S).restrict V Z = 0 :=
+      le_antisymm ((Measure.restrict_apply_le V Z).trans_eq
+        (surfaceMeasure_zero_of_singular_dim hn S Z hZd)) zero_le
+    have hae : ∀ᵐ x ∂(surfaceMeasure S).restrict V, x ∉ Z := by
+      simpa only [ae_iff, not_not, Set.ofPred_mem_eq] using hnull
+    filter_upwards [hae] with x hx
+    exact ((tendsto_congr' (hχone x hx)).2 tendsto_const_nhds).comp hshift
+
+theorem hausdorff_trace_mem_sobolevGraph {n : ℕ} (hn : 8 ≤ n)
+    {S Z : Set (Space n)} (hS : IsClosed S) (V : Set (Space n))
+    (hZc : IsCompact Z) (hZd : dimH Z ≤ (n - 8 : ℕ))
+    {h : Space n → ℝ} {G : Space n → Space n}
+    (hreg : ∀ x ∈ S \ Z, HasSmoothTraceJetAt S h G x)
+    (hh : MemLp h 2 ((surfaceMeasure S).restrict V))
+    (hG : MemLp G 2 ((surfaceMeasure S).restrict V))
+    {M : ℝ} (hM : 0 ≤ M)
+    (hbound : ∀ᵐ x ∂(surfaceMeasure S).restrict V, ‖h x‖ ≤ M)
+    (hgrad : ∀ f : Space n → ℝ, ContDiff ℝ ∞ f →
+      AEStronglyMeasurable (surfaceGradient S f) ((surfaceMeasure S).restrict V))
+    {A r₀ : ℝ} (hA : 0 < A) (hr₀ : 0 < r₀)
+    (hgrowth : ∀ c ∈ Z, ∀ r : ℝ, 0 < r → r < r₀ →
+      (surfaceMeasure S).restrict V (Metric.ball c r) ≤
+        ENNReal.ofReal (A * r ^ (n - 1))) :
+    (hh.toLp h, hG.toLp G) ∈ closure (smoothGradientGraph S V) := by
+  obtain ⟨χ, hχ, hχlim, hχE⟩ := exists_memLp_capacity_sequence hn S V Z hZc hZd
+    hgrad hA hr₀ hgrowth
+  exact capacity_trace_mem_sobolevGraph hS V hreg hh hG hM hbound χ
+    (fun i ↦ (hχ i).1) (fun i ↦ (hχ i).2.2.1)
+    (fun i ↦ ae_of_all _ (hχ i).2.1) hχlim (fun i ↦ (hχ i).2.2.2) hχE
+
+end RPI.Boundary
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+namespace RPI.Boundary
+
+theorem tangentPlane_eq_ker_of_local_level {n : ℕ} {S : Set (Space n)}
+    {f : Space n → ℝ} {x : Space n}
+    (hf : ContDiffAt ℝ 1 f x) (hdf : fderiv ℝ f x ≠ 0)
+    (hlevel : ∀ᶠ y in 𝓝 x, y ∈ S ↔ f y = f x) :
+    tangentPlane S x = (fderiv ℝ f x).ker := by
+  have hder := hf.hasStrictFDerivAt (by norm_num)
+  have hrange : (fderiv ℝ f x).range = ⊤ := by
+    apply LinearMap.range_eq_top.mpr
+    have hv : ∃ v, fderiv ℝ f x v ≠ 0 := by
+      by_contra! hv
+      apply hdf
+      ext v
+      exact hv v
+    obtain ⟨v, hv⟩ := hv
+    intro a
+    refine ⟨(a / fderiv ℝ f x v) • v, ?_⟩
+    simp only [map_smul, smul_eq_mul]
+    exact div_mul_cancel₀ a hv
+  apply le_antisymm
+  · apply Submodule.span_le.mpr
+    rintro v ⟨γ, hγx, hγd, hγS⟩
+    have ht : Tendsto γ (𝓝 (0 : ℝ)) (𝓝 x) := hγx ▸ hγd.continuousAt
+    have heq : (f ∘ γ) =ᶠ[𝓝 (0 : ℝ)] fun _ ↦ f x := by
+      filter_upwards [ht.eventually hlevel, hγS] with t he ht
+      exact he.mp ht
+    have hd : HasDerivAt (f ∘ γ) (fderiv ℝ f x v) 0 := by
+      exact (show HasFDerivAt f (fderiv ℝ f x) (γ 0) by
+        simpa only [hγx] using hder.hasFDerivAt).comp_hasDerivAt 0 hγd
+    exact hd.unique ((hasDerivAt_const 0 (f x)).congr_of_eventuallyEq heq)
+  · intro v hv
+    let v' : (fderiv ℝ f x).ker := ⟨v, hv⟩
+    let γ : ℝ → Space n := fun t ↦ hder.implicitFunction f (fderiv ℝ f x)
+      hrange (f x) (t • v')
+    have hγx : γ 0 = x := by simp [γ]
+    have hline : HasDerivAt (fun t : ℝ ↦ t • v') v' 0 := by
+      simpa using (hasDerivAt_id (0 : ℝ)).smul_const v'
+    have hγd : HasDerivAt γ v 0 := by
+      have hd' : HasFDerivAt (hder.implicitFunction f (fderiv ℝ f x)
+          hrange (f x)) (fderiv ℝ f x).ker.subtypeL ((0 : ℝ) • v') := by
+        simpa only [zero_smul] using (hder.to_implicitFunction hrange).hasFDerivAt
+      exact hd'.comp_hasDerivAt 0 hline
+    have ht : Tendsto γ (𝓝 (0 : ℝ)) (𝓝 x) := hγx ▸ hγd.continuousAt
+    have hparam : Tendsto (fun t : ℝ ↦ (f x, t • v')) (𝓝 0)
+        (𝓝 (f x, 0)) := by
+      apply tendsto_const_nhds.prodMk_nhds
+      simpa only [ContinuousAt, zero_smul] using hline.continuousAt
+    have hγS : ∀ᶠ t in 𝓝 (0 : ℝ), γ t ∈ S := by
+      filter_upwards [ht.eventually hlevel,
+        hparam.eventually (hder.map_implicitFunction_eq hrange)] with t he hval
+      exact he.mpr hval
+    exact Submodule.subset_span ⟨γ, hγx, hγd, hγS⟩
+
+end RPI.Boundary
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+namespace RPI.Boundary
+
+theorem surfaceGradient_eq_level_formula {n : ℕ} {S : Set (Space n)}
+    {f g : Space n → ℝ} {x : Space n}
+    (hf : ContDiffAt ℝ 1 f x) (hdf : fderiv ℝ f x ≠ 0)
+    (hlevel : ∀ᶠ y in 𝓝 x, y ∈ S ↔ f y = f x) :
+    surfaceGradient S g x = gradient g x -
+      (inner ℝ (gradient f x) (gradient g x) / ‖gradient f x‖ ^ 2) • gradient f x := by
+  have hgn : gradient f x ≠ 0 := by
+    intro he
+    apply hdf
+    rw [← toDual_gradient, he, map_zero]
+  have hnorm : ‖gradient f x‖ ^ 2 ≠ 0 := pow_ne_zero _ (norm_ne_zero_iff.mpr hgn)
+  have htan := tangentPlane_eq_ker_of_local_level hf hdf hlevel
+  change (tangentPlane S x).starProjection (gradient g x) = _
+  apply Submodule.eq_starProjection_of_mem_of_inner_eq_zero
+  · rw [htan]
+    change fderiv ℝ f x (_ - _) = 0
+    rw [map_sub, map_smul]
+    rw [← inner_gradient_left, ← inner_gradient_left, real_inner_self_eq_norm_sq]
+    simp only [smul_eq_mul]
+    rw [div_mul_cancel₀ _ hnorm, sub_self]
+  · intro w hw
+    rw [htan] at hw
+    rw [sub_sub_cancel, real_inner_smul_left, inner_gradient_left]
+    change fderiv ℝ f x w = 0 at hw
+    simp only [inner_gradient_left, hw, mul_zero]
+
+theorem continuousAt_gradient_of_contDiffAt {n : ℕ} {f : Space n → ℝ} {x : Space n}
+    (hf : ContDiffAt ℝ 1 f x) : ContinuousAt (gradient f) x := by
+  exact (InnerProductSpace.toDual ℝ (Space n)).symm.continuous.continuousAt.comp
+    (hf.continuousAt_fderiv (by norm_num))
+
+end RPI.Boundary
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/ExistsLocalPolarEqApproximateNormal.lean b/lean/OAI/Geometry/Perimeter/Regularity/ExistsLocalPolarEqApproximateNormal.lean
new file mode 100644
index 0000000000000000000000000000000000000000..963b433899ee81242f2103df6818f28e619fb765
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/ExistsLocalPolarEqApproximateNormal.lean
@@ -0,0 +1,402 @@
+import OAI.Geometry.Perimeter.Regularity.TsupportBlowdownSubsetBall
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+namespace RPISingularHarnack
+
+theorem exists_local_polar_eq_approximateNormal {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    (U : Set (Euc n)) (hU : IsOpen U) (hP : perimeter E U ≠ ⊤) :
+    ∃ G : Euc n → Euc n,
+      StronglyMeasurable G ∧ MemLp G 2 ((perimeterMeasure E).restrict U) ∧
+      (∀ᵐ x ∂(perimeterMeasure E).restrict U, HasHalfspaceBlowup E x (G x)) ∧
+      (∀ᵐ x ∂(perimeterMeasure E).restrict U, approximateNormal E x = G x) ∧
+      ∀ V : Euc n → Euc n, ContDiff ℝ ∞ V → HasCompactSupport V → tsupport V ⊆ U →
+        (∫ x in E, divergence V x) = ∫ x in U, inner ℝ (G x) (V x) ∂perimeterMeasure E := by
+  let μ := (perimeterMeasure E).restrict U
+  let : IsFiniteMeasure μ := ⟨by
+    rw [Measure.restrict_apply_univ, perimeterMeasure_open E hU]; exact hP.lt_top⟩
+  obtain ⟨G, hG, hGp, hGn, hdiff, hrep⟩ :=
+    exists_differentiating_unit_polar (lt_trans (by norm_num) hn) hmin U hU hP
+  have hsupp : ∀ᵐ x ∂μ, x ∈ perimeterSupport E := by
+    have hs : ∀ᵐ x ∂perimeterMeasure E, x ∈ perimeterSupport E := by
+      rw [← perimeterMeasure_support E]
+      exact (perimeterMeasure E).support_mem_ae
+    exact ae_restrict_of_ae hs
+  have hUae : ∀ᵐ x ∂μ, x ∈ U := ae_restrict_mem hU.measurableSet
+  have hblow : ∀ᵐ x ∂μ, HasHalfspaceBlowup E x (G x) := by
+    filter_upwards [hGn, hdiff, hsupp, hUae] with x hx hdx hsx hux
+    exact hasHalfspaceBlowup_of_differentiating_polar hn hmin hU μ
+      (MemLp.integrable (by norm_num) hGp) hrep x (G x) hux hsx hx hdx
+  exact ⟨G, hG, hGp, hblow,
+    hblow.mono (fun x hx ↦ approximateNormal_eq_of_blowup hmin.1 hx), hrep⟩
+
+theorem minimizing_gaussGreen_local {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    (U : Set (Euc n)) (hU : IsOpen U) (hP : perimeter E U ≠ ⊤)
+    (V : Euc n → Euc n) (hV : ContDiff ℝ ∞ V) (hVc : HasCompactSupport V)
+    (hVU : tsupport V ⊆ U) :
+    (∫ x in E, divergence V x) =
+      ∫ x in U, inner ℝ (approximateNormal E x) (V x) ∂perimeterMeasure E := by
+  obtain ⟨G, _, _, _, heq, hrep⟩ := exists_local_polar_eq_approximateNormal hn hmin U hU hP
+  rw [hrep V hV hVc hVU]
+  apply integral_congr_ae
+  exact heq.mono fun x hx ↦ by
+    change inner ℝ (G x) (V x) = inner ℝ (approximateNormal E x) (V x)
+    rw [hx]
+
+theorem minimizing_halfspaceBlowup_ae {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E) :
+    ∀ᵐ x ∂perimeterMeasure E, HasHalfspaceBlowup E x (approximateNormal E x) := by
+  let U : ℕ → Set (Euc n) := fun j ↦ Metric.ball 0 (j + 1 : ℝ)
+  have hcover : (⋃ j, U j) = univ := by
+    apply eq_univ_of_forall
+    intro x
+    obtain ⟨j, hj⟩ := exists_nat_gt ‖x‖
+    exact mem_iUnion.mpr ⟨j, by simpa [U, Metric.mem_ball, dist_zero_right] using
+      (show ‖x‖ < (j : ℝ) + 1 by linarith)⟩
+  have hall : ∀ j, ∀ᵐ x ∂(perimeterMeasure E).restrict (U j),
+      HasHalfspaceBlowup E x (approximateNormal E x) := by
+    intro j
+    obtain ⟨G, _, _, hG, heq, _⟩ := exists_local_polar_eq_approximateNormal hn hmin
+      (U j) Metric.isOpen_ball (ne_of_lt (hmin.2.1 0 (j + 1) (by positivity)))
+    filter_upwards [hG, heq] with x hx he
+    rwa [he]
+  have hh := (ae_restrict_iUnion_iff U
+    (fun x ↦ HasHalfspaceBlowup E x (approximateNormal E x))).mpr hall
+  simpa only [hcover, Measure.restrict_univ] using hh
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+namespace RPI
+section BoundedCoefficientLimits
+variable {X V : Type*} [MeasurableSpace X] [NormedAddCommGroup V]
+  [NormedSpace ℝ V] {μ : Measure X}
+
+theorem vectorEnergy_bounded_coefficient_tendsto {G : X → V} (hG : MemLp G 2 μ)
+    (a : ℕ → X → ℝ) (hm : ∀ j, AEStronglyMeasurable (a j) μ)
+    {M : ℝ} (hM : 0 ≤ M)
+    (hb : ∀ j, ∀ᵐ x ∂μ, ‖a j x‖ ≤ M)
+    (ht : ∀ᵐ x ∂μ, Tendsto (fun j ↦ a j x) atTop (𝓝 0)) :
+    Tendsto (fun j ↦ vectorEnergy μ (fun x ↦ a j x • G x)) atTop (𝓝 0) := by
+  have hfin : (∫⁻ x, ENNReal.ofReal (M ^ 2) * ENNReal.ofReal (‖G x‖ ^ 2) ∂μ) ≠ ⊤ := by
+    rw [lintegral_const_mul' (ENNReal.ofReal (M ^ 2)) (fun x ↦ ENNReal.ofReal (‖G x‖ ^ 2)) ENNReal.ofReal_ne_top]
+    exact ENNReal.mul_ne_top ENNReal.ofReal_ne_top
+      ((memLp_two_iff_vectorEnergy hG.aestronglyMeasurable).mp hG).ne
+  have hlim := tendsto_lintegral_of_dominated_convergence'
+    (fun x ↦ ENNReal.ofReal (M ^ 2) * ENNReal.ofReal (‖G x‖ ^ 2))
+    (f := fun _ ↦ (0 : ℝ≥0∞))
+    (F := fun j x ↦ ENNReal.ofReal (‖a j x • G x‖ ^ 2))
+    (fun j ↦ (((hm j).smul hG.aestronglyMeasurable).norm.pow 2).aemeasurable.ennreal_ofReal) ?_ hfin ?_
+  · simpa only [vectorEnergy, lintegral_zero] using hlim
+  · intro j
+    filter_upwards [hb j] with x hx
+    rw [← ENNReal.ofReal_mul (sq_nonneg M)]
+    apply ENNReal.ofReal_le_ofReal
+    rw [norm_smul, mul_pow]
+    exact mul_le_mul_of_nonneg_right ((sq_le_sq₀ (norm_nonneg _) hM).mpr hx) (sq_nonneg _)
+  · filter_upwards [ht] with x hx
+    have hv : Tendsto (fun j ↦ a j x • G x) atTop (𝓝 (0 : V)) := by
+      simpa only [zero_smul] using hx.smul_const (G x)
+    simpa only [norm_zero, zero_pow (by norm_num : (2 : ℕ) ≠ 0), ENNReal.ofReal_zero] using
+      ENNReal.tendsto_ofReal (hv.norm.pow 2)
+
+theorem vectorEnergy_varying_bounded_smul_zero {F : ℕ → X → V}
+    (a : ℕ → X → ℝ) {M : ℝ} (hM : 0 ≤ M)
+    (ha : ∀ j, ∀ᵐ x ∂μ, ‖a j x‖ ≤ M)
+    (hF : Tendsto (fun j ↦ vectorEnergy μ (F j)) atTop (𝓝 0)) :
+    Tendsto (fun j ↦ vectorEnergy μ (fun x ↦ a j x • F j x)) atTop (𝓝 0) := by
+  have hh : Tendsto (fun j ↦ ENNReal.ofReal (M ^ 2) * vectorEnergy μ (F j))
+      atTop (𝓝 0) := by
+    simpa only [mul_zero] using ENNReal.Tendsto.const_mul hF (Or.inr ENNReal.ofReal_ne_top)
+  exact tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds hh
+    (fun _ ↦ bot_le) (fun j ↦ vectorEnergy_bounded_smul hM (ha j))
+
+theorem vectorEnergy_composed_gradient_tendsto
+    {u : ℕ → X → ℝ} {h : X → ℝ} {F : ℕ → X → V} {G : X → V}
+    (hu : ∀ j, AEStronglyMeasurable (u j) μ) (hh : AEStronglyMeasurable h μ)
+    (hF : ∀ j, MemLp (F j) 2 μ) (hG : MemLp G 2 μ)
+    (hFG : Tendsto (fun j ↦ vectorEnergy μ (fun x ↦ F j x - G x)) atTop (𝓝 0))
+    (huh : ∀ᵐ x ∂μ, Tendsto (fun j ↦ u j x) atTop (𝓝 (h x)))
+    (b : ℝ → ℝ) (hb : Continuous b) {M : ℝ} (hM : 0 ≤ M)
+    (hbM : ∀ t, ‖b t‖ ≤ M) :
+    Tendsto (fun j ↦ vectorEnergy μ
+      (fun x ↦ b (u j x) • F j x - b (h x) • G x)) atTop (𝓝 0) := by
+  let A : ℕ → X → V := fun j x ↦ b (u j x) • (F j x - G x)
+  let B : ℕ → X → V := fun j x ↦ (b (u j x) - b (h x)) • G x
+  have hAm : ∀ j, AEStronglyMeasurable (A j) μ := fun j ↦
+    (hb.comp_aestronglyMeasurable (hu j)).smul
+      ((hF j).aestronglyMeasurable.sub hG.aestronglyMeasurable)
+  have hBm : ∀ j, AEStronglyMeasurable (B j) μ := fun j ↦
+    ((hb.comp_aestronglyMeasurable (hu j)).sub
+      (hb.comp_aestronglyMeasurable hh)).smul hG.aestronglyMeasurable
+  have hA0 : Tendsto (fun j ↦ vectorEnergy μ (A j)) atTop (𝓝 0) :=
+    vectorEnergy_varying_bounded_smul_zero (fun j x ↦ b (u j x)) hM
+      (fun j ↦ ae_of_all _ fun x ↦ hbM _) hFG
+  have hB0 : Tendsto (fun j ↦ vectorEnergy μ (B j)) atTop (𝓝 0) := by
+    apply vectorEnergy_bounded_coefficient_tendsto hG
+      (fun j x ↦ b (u j x) - b (h x))
+      (fun j ↦ (hb.comp_aestronglyMeasurable (hu j)).sub (hb.comp_aestronglyMeasurable hh))
+      (M := 2 * M) (by positivity)
+    · intro j
+      exact ae_of_all _ fun x ↦ (norm_sub_le _ _).trans (by linarith [hbM (u j x), hbM (h x)])
+    · filter_upwards [huh] with x hx
+      simpa only [sub_self, Function.comp_def] using (hb.continuousAt.tendsto.comp hx).sub_const (b (h x))
+  have hbound : ∀ j, vectorEnergy μ (fun x ↦ b (u j x) • F j x - b (h x) • G x) ≤
+      2 * vectorEnergy μ (A j) + 2 * vectorEnergy μ (B j) := by
+    intro j
+    have heq : (fun x ↦ b (u j x) • F j x - b (h x) • G x) =
+        fun x ↦ A j x + B j x := by funext x; dsimp [A, B]; module
+    rw [heq]
+    exact vectorEnergy_add_le (hAm j) (hBm j)
+  have hlim : Tendsto (fun j ↦ 2 * vectorEnergy μ (A j) + 2 * vectorEnergy μ (B j))
+      atTop (𝓝 0) := by
+    simpa only [mul_zero, add_zero] using
+      (ENNReal.Tendsto.const_mul hA0 (Or.inr (by norm_num : (2 : ℝ≥0∞) ≠ ⊤))).add
+        (ENNReal.Tendsto.const_mul hB0 (Or.inr (by norm_num : (2 : ℝ≥0∞) ≠ ⊤)))
+  exact tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds hlim
+    (fun _ ↦ bot_le) hbound
+
+end BoundedCoefficientLimits
+end RPI
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+namespace RPI
+section NonlinearTests
+variable {X V : Type*} [MeasurableSpace X] [NormedAddCommGroup V]
+  [NormedSpace ℝ V] {μ : Measure X}
+
+theorem vectorEnergy_smul_bounded_field_le {a : X → ℝ} {D : X → V} {M : ℝ}
+    (hM : 0 ≤ M) (hD : ∀ᵐ x ∂μ, ‖D x‖ ≤ M) :
+    vectorEnergy μ (fun x ↦ a x • D x) ≤ ENNReal.ofReal (M ^ 2) * vectorEnergy μ a := by
+  unfold vectorEnergy
+  rw [← lintegral_const_mul' _ _ ENNReal.ofReal_ne_top]
+  · apply lintegral_mono_ae
+    filter_upwards [hD] with x hx
+    rw [← ENNReal.ofReal_mul (sq_nonneg M)]
+    apply ENNReal.ofReal_le_ofReal
+    rw [norm_smul, mul_pow]
+    nlinarith [mul_le_mul_of_nonneg_left ((sq_le_sq₀ (norm_nonneg _) hM).mpr hx) (sq_nonneg ‖a x‖)]
+
+theorem vectorEnergy_lipschitz_comp_zero {u : ℕ → X → ℝ} {h : X → ℝ}
+    {ψ : ℝ → ℝ} {K : ℝ≥0} (hψ : LipschitzWith K ψ)
+    (hu : Tendsto (fun j ↦ vectorEnergy μ (fun x ↦ u j x - h x)) atTop (𝓝 0)) :
+    Tendsto (fun j ↦ vectorEnergy μ (fun x ↦ ψ (u j x) - ψ (h x))) atTop (𝓝 0) := by
+  have hb : ∀ j, vectorEnergy μ (fun x ↦ ψ (u j x) - ψ (h x)) ≤
+      ENNReal.ofReal ((K : ℝ) ^ 2) * vectorEnergy μ (fun x ↦ u j x - h x) := by
+    intro j
+    unfold vectorEnergy
+    rw [← lintegral_const_mul' _ _ ENNReal.ofReal_ne_top]
+    · apply lintegral_mono
+      intro x
+      dsimp only
+      rw [← ENNReal.ofReal_mul (sq_nonneg (K : ℝ))]
+      apply ENNReal.ofReal_le_ofReal
+      have hh := hψ.norm_sub_le (u j x) (h x)
+      simpa only [mul_pow] using (sq_le_sq₀ (norm_nonneg _) (by positivity)).mpr hh
+  have hlim : Tendsto (fun j ↦ ENNReal.ofReal ((K : ℝ) ^ 2) *
+      vectorEnergy μ (fun x ↦ u j x - h x)) atTop (𝓝 0) := by
+    simpa only [mul_zero] using ENNReal.Tendsto.const_mul hu (Or.inr ENNReal.ofReal_ne_top)
+  exact tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds hlim
+    (fun _ ↦ bot_le) hb
+
+theorem vectorEnergy_nonlinear_test_tendsto
+    {u : ℕ → X → ℝ} {h η : X → ℝ} {F : ℕ → X → V} {G D : X → V}
+    (hu : ∀ j, AEStronglyMeasurable (u j) μ) (hh : AEStronglyMeasurable h μ)
+    (hη : AEStronglyMeasurable η μ) (hD : AEStronglyMeasurable D μ)
+    (hF : ∀ j, MemLp (F j) 2 μ) (hG : MemLp G 2 μ)
+    {M N : ℝ} (hM : 0 ≤ M) (hN : 0 ≤ N)
+    (hηM : ∀ᵐ x ∂μ, ‖η x‖ ≤ M) (hDN : ∀ᵐ x ∂μ, ‖D x‖ ≤ N)
+    (huh : Tendsto (fun j ↦ vectorEnergy μ (fun x ↦ u j x - h x)) atTop (𝓝 0))
+    (hFG : Tendsto (fun j ↦ vectorEnergy μ (fun x ↦ F j x - G x)) atTop (𝓝 0))
+    (hpoint : ∀ᵐ x ∂μ, Tendsto (fun j ↦ u j x) atTop (𝓝 (h x)))
+    {ψ : ℝ → ℝ} {K : ℝ≥0} (hψ : LipschitzWith K ψ)
+    (b : ℝ → ℝ) (hb : Continuous b) {L : ℝ} (hL : 0 ≤ L) (hbL : ∀ t, ‖b t‖ ≤ L) :
+    Tendsto (fun j ↦ vectorEnergy μ (fun x ↦
+      ((2 * η x * ψ (u j x)) • D x + (η x ^ 2 * b (u j x)) • F j x) -
+      ((2 * η x * ψ (h x)) • D x + (η x ^ 2 * b (h x)) • G x))) atTop (𝓝 0) := by
+  let A : ℕ → X → V := fun j x ↦ (2 * η x) • ((ψ (u j x) - ψ (h x)) • D x)
+  let B : ℕ → X → V := fun j x ↦ η x ^ 2 • (b (u j x) • F j x - b (h x) • G x)
+  have hψm := hψ.continuous
+  have hAm : ∀ j, AEStronglyMeasurable (A j) μ := fun j ↦
+    (aestronglyMeasurable_const.mul hη).smul
+      (((hψm.comp_aestronglyMeasurable (hu j)).sub
+        (hψm.comp_aestronglyMeasurable hh)).smul hD)
+  have hBm : ∀ j, AEStronglyMeasurable (B j) μ := fun j ↦
+    (hη.pow 2).smul
+      (((hb.comp_aestronglyMeasurable (hu j)).smul (hF j).aestronglyMeasurable).sub
+        ((hb.comp_aestronglyMeasurable hh).smul hG.aestronglyMeasurable))
+  have hc : Tendsto (fun j ↦ vectorEnergy μ (fun x ↦ (ψ (u j x) - ψ (h x)) • D x))
+      atTop (𝓝 0) := by
+    have hh0 := vectorEnergy_lipschitz_comp_zero hψ huh
+    have hlim : Tendsto (fun j ↦ ENNReal.ofReal (N ^ 2) *
+        vectorEnergy μ (fun x ↦ ψ (u j x) - ψ (h x))) atTop (𝓝 0) := by
+      simpa only [mul_zero] using ENNReal.Tendsto.const_mul hh0 (Or.inr ENNReal.ofReal_ne_top)
+    exact tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds hlim
+      (fun _ ↦ bot_le) (fun _ ↦ vectorEnergy_smul_bounded_field_le hN hDN)
+  have hA0 : Tendsto (fun j ↦ vectorEnergy μ (A j)) atTop (𝓝 0) := by
+    apply vectorEnergy_varying_bounded_smul_zero (fun _ x ↦ 2 * η x) (M := 2 * M) (by positivity) _ hc
+    intro j
+    filter_upwards [hηM] with x hx
+    simpa only [norm_mul, Real.norm_ofNat] using mul_le_mul_of_nonneg_left hx (by norm_num : (0 : ℝ) ≤ 2)
+  have hB0 : Tendsto (fun j ↦ vectorEnergy μ (B j)) atTop (𝓝 0) := by
+    apply vectorEnergy_varying_bounded_smul_zero (fun _ x ↦ η x ^ 2) (M := M ^ 2) (sq_nonneg M) _
+      (vectorEnergy_composed_gradient_tendsto hu hh hF hG hFG hpoint b hb hL hbL)
+    intro j
+    filter_upwards [hηM] with x hx
+    simpa only [norm_pow] using (sq_le_sq₀ (norm_nonneg _) hM).mpr hx
+  have hlim : Tendsto (fun j ↦ 2 * vectorEnergy μ (A j) + 2 * vectorEnergy μ (B j))
+      atTop (𝓝 0) := by
+    simpa only [mul_zero, add_zero] using
+      (ENNReal.Tendsto.const_mul hA0 (Or.inr (by norm_num : (2 : ℝ≥0∞) ≠ ⊤))).add
+        (ENNReal.Tendsto.const_mul hB0 (Or.inr (by norm_num : (2 : ℝ≥0∞) ≠ ⊤)))
+  apply tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds hlim (fun _ ↦ bot_le)
+  intro j
+  have heq : (fun x ↦
+      ((2 * η x * ψ (u j x)) • D x + (η x ^ 2 * b (u j x)) • F j x) -
+      ((2 * η x * ψ (h x)) • D x + (η x ^ 2 * b (h x)) • G x)) =
+      fun x ↦ A j x + B j x := by funext x; dsimp [A, B]; module
+  dsimp only
+  rw [heq]
+  exact vectorEnergy_add_le (hAm j) (hBm j)
+
+end NonlinearTests
+end RPI
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+namespace RPISingularHarnack
+
+def logTest (ε t : ℝ) : ℝ := (Real.sqrt (t ^ 2 + ε ^ 2) - t) / ε ^ 2
+
+def logSlope (ε t : ℝ) : ℝ := (t / Real.sqrt (t ^ 2 + ε ^ 2) - 1) / ε ^ 2
+
+lemma logTest_root_pos {ε : ℝ} (hε : 0 < ε) (t : ℝ) :
+    0 < Real.sqrt (t ^ 2 + ε ^ 2) := Real.sqrt_pos.2 (by positivity)
+
+lemma logTest_root_sq {ε : ℝ} (hε : 0 < ε) (t : ℝ) :
+    Real.sqrt (t ^ 2 + ε ^ 2) ^ 2 = t ^ 2 + ε ^ 2 :=
+  Real.sq_sqrt (by positivity)
+
+lemma logTest_abs_lt_root {ε : ℝ} (hε : 0 < ε) (t : ℝ) :
+    |t| < Real.sqrt (t ^ 2 + ε ^ 2) := by
+  have hs := logTest_root_sq hε t
+  have hr := logTest_root_pos hε t
+  nlinarith [sq_abs t, sq_pos_of_pos hε]
+
+theorem logTest_contDiff {ε : ℝ} (hε : 0 < ε) : ContDiff ℝ ∞ (logTest ε) := by
+  unfold logTest
+  exact ((((contDiff_id.pow 2).add contDiff_const).sqrt
+    (fun t ↦ ne_of_gt (by positivity : 0 < t ^ 2 + ε ^ 2))).sub contDiff_id).div_const _
+
+theorem logTest_hasDerivAt {ε : ℝ} (hε : 0 < ε) (t : ℝ) :
+    HasDerivAt (logTest ε) (logSlope ε t) t := by
+  have hh := (((hasDerivAt_id t).pow 2).add_const (ε ^ 2)).sqrt
+    (ne_of_gt (by positivity : 0 < t ^ 2 + ε ^ 2))
+  have h := (hh.sub (hasDerivAt_id t)).div_const (ε ^ 2)
+  apply h.congr_deriv
+  dsimp [logSlope]
+  ring
+
+theorem logSlope_continuous {ε : ℝ} (hε : 0 < ε) : Continuous (logSlope ε) := by
+  unfold logSlope
+  exact ((continuous_id.div (Real.continuous_sqrt.comp
+    ((continuous_id.pow 2).add continuous_const))
+    (fun t ↦ (logTest_root_pos hε t).ne')).sub continuous_const).div_const _
+
+theorem logTest_pos {ε : ℝ} (hε : 0 < ε) (t : ℝ) : 0 < logTest ε t := by
+  unfold logTest
+  exact div_pos (sub_pos.mpr ((le_abs_self t).trans_lt (logTest_abs_lt_root hε t)))
+    (sq_pos_of_pos hε)
+
+theorem logTest_reciprocal {ε : ℝ} (hε : 0 < ε) (t : ℝ) :
+    logTest ε t = 1 / (Real.sqrt (t ^ 2 + ε ^ 2) + t) := by
+  have hr : 0 < Real.sqrt (t ^ 2 + ε ^ 2) + t := by
+    have hh := (neg_le_abs t).trans_lt (logTest_abs_lt_root hε t)
+    linarith
+  unfold logTest
+  apply (div_eq_div_iff (pow_ne_zero _ hε.ne') hr.ne').2
+  nlinarith [logTest_root_sq hε t]
+
+theorem logSlope_eq {ε : ℝ} (hε : 0 < ε) (t : ℝ) :
+    logSlope ε t = -logTest ε t / Real.sqrt (t ^ 2 + ε ^ 2) := by
+  unfold logSlope logTest
+  field_simp [(logTest_root_pos hε t).ne', hε.ne']
+  ring
+
+theorem logSlope_neg {ε : ℝ} (hε : 0 < ε) (t : ℝ) : logSlope ε t < 0 := by
+  rw [logSlope_eq hε]
+  exact div_neg_of_neg_of_pos (neg_neg_of_pos (logTest_pos hε t)) (logTest_root_pos hε t)
+
+theorem logSlope_bound {ε : ℝ} (hε : 0 < ε) (t : ℝ) :
+    ‖logSlope ε t‖ ≤ 2 / ε ^ 2 := by
+  have hr := logTest_root_pos hε t
+  have ht : |t / Real.sqrt (t ^ 2 + ε ^ 2)| ≤ 1 := by
+    rw [abs_div, abs_of_pos hr]
+    exact (div_le_one hr).2 (logTest_abs_lt_root hε t).le
+  unfold logSlope
+  rw [norm_div, Real.norm_eq_abs, Real.norm_eq_abs, abs_of_pos (sq_pos_of_pos hε)]
+  apply div_le_div_of_nonneg_right _ (sq_nonneg ε)
+  exact (abs_sub (t / Real.sqrt (t ^ 2 + ε ^ 2)) 1).trans (by
+    simpa only [abs_one, add_comm, one_add_one_eq_two] using add_le_add_right ht 1)
+
+theorem logTest_sq_le_neg_slope {ε t : ℝ} (hε : 0 < ε) (ht : 0 ≤ t) :
+    logTest ε t ^ 2 ≤ -logSlope ε t := by
+  have hr := logTest_root_pos hε t
+  have hp := logTest_pos hε t
+  rw [logSlope_eq hε, neg_div, neg_neg]
+  have hmul : logTest ε t * Real.sqrt (t ^ 2 + ε ^ 2) ≤ 1 := by
+    rw [logTest_reciprocal hε, one_div, inv_mul_eq_div]
+    exact (div_le_one (by linarith : 0 < Real.sqrt (t ^ 2 + ε ^ 2) + t)).2 (by linarith)
+  apply (le_div_iff₀ hr).2
+  nlinarith [mul_le_mul_of_nonneg_left hmul hp.le]
+
+theorem logSlope_controls_log_weight {ε t : ℝ} (hε : 0 < ε) (ht : 0 ≤ t) :
+    1 / (t + ε) ^ 2 ≤ 2 * (-logSlope ε t) := by
+  have hr := logTest_root_pos hε t
+  have hte : 0 < t + ε := by positivity
+  have hroot : Real.sqrt (t ^ 2 + ε ^ 2) ≤ t + ε := by
+    nlinarith [logTest_root_sq hε t, mul_nonneg ht hε.le]
+  have hplus : 0 < Real.sqrt (t ^ 2 + ε ^ 2) + t := by positivity
+  rw [logSlope_eq hε, neg_div, neg_neg, logTest_reciprocal hε]
+  apply (div_le_iff₀ (sq_pos_of_pos hte)).2
+  field_simp [hr.ne', hplus.ne']
+  nlinarith [mul_le_mul hroot (show Real.sqrt (t ^ 2 + ε ^ 2) + t ≤ 2 * (t + ε) by linarith)
+    hplus.le hte.le]
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/ExistsLocalVariationExtension.lean b/lean/OAI/Geometry/Perimeter/Regularity/ExistsLocalVariationExtension.lean
new file mode 100644
index 0000000000000000000000000000000000000000..ed6de1b71ad7d7ec43903303d3e46b530633f5e4
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/ExistsLocalVariationExtension.lean
@@ -0,0 +1,544 @@
+import OAI.Geometry.Perimeter.Regularity.SmoothLayerSumBounds
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+
+theorem exists_local_variation_extension {n : ℕ} (E U : Set (Euc n))
+    (hU : IsOpen U) (hP : perimeter E U ≠ ⊤) :
+    ∃ F : Lp (Euc n) 2 ((perimeterMeasure E).restrict U) →ₗ[ℝ] ℝ,
+      (∀ u : smoothVariationDomain U ((perimeterMeasure E).restrict U),
+        F (u : Lp (Euc n) 2 ((perimeterMeasure E).restrict U)) = localVariationFunctional E U hU hP u) ∧
+      ∀ u, |F u| ≤ ∫ x in U, ‖u x‖ ∂perimeterMeasure E := by
+  let μ := (perimeterMeasure E).restrict U
+  let : IsFiniteMeasure μ := ⟨by
+    rw [Measure.restrict_apply_univ, perimeterMeasure_open E hU]; exact hP.lt_top⟩
+  let p : Lp (Euc n) 2 μ →ₗ.[ℝ] ℝ :=
+    ⟨smoothVariationDomain U μ, localVariationFunctional E U hU hP⟩
+  let N : Lp (Euc n) 2 μ → ℝ := fun u ↦ ∫ x, ‖u x‖ ∂μ
+  have hi : ∀ u : Lp (Euc n) 2 μ, Integrable (fun x ↦ ‖u x‖) μ :=
+    fun u ↦ (memLp_one_iff_integrable.mp ((Lp.memLp u).mono_exponent (by norm_num))).norm
+  have hhom : ∀ c : ℝ, 0 < c → ∀ u, N (c • u) = c * N u := by
+    intro c hc u
+    dsimp [N]
+    calc
+      _ = ∫ x, c * ‖u x‖ ∂μ := by
+        apply integral_congr_ae
+        filter_upwards [Lp.coeFn_smul c u] with x hx
+        rw [hx, Pi.smul_apply, norm_smul, Real.norm_of_nonneg hc.le]
+      _ = _ := integral_const_mul _ _
+  have hadd : ∀ u v, N (u + v) ≤ N u + N v := by
+    intro u v
+    dsimp [N]
+    rw [← integral_add (hi u) (hi v)]
+    apply integral_mono_ae (hi (u + v)) ((hi u).add (hi v))
+    filter_upwards [Lp.coeFn_add u v] with x hx
+    rw [hx, Pi.add_apply]
+    exact norm_add_le _ _
+  obtain ⟨F, hF, hFn⟩ := exists_extension_of_le_sublinear p N hhom hadd
+    (fun u ↦ (le_abs_self _).trans (localVariationFunctional_le E U hU hP u))
+  refine ⟨F, hF, fun u ↦ abs_le.mpr ⟨?_, hFn u⟩⟩
+  have hh := hFn (-u)
+  have he : N (-u) = N u := by
+    apply integral_congr_ae
+    filter_upwards [Lp.coeFn_neg u] with x hx
+    rw [hx, Pi.neg_apply, norm_neg]
+  rw [map_neg, he] at hh
+  linarith
+
+theorem local_l2_integral_norm_le {n : ℕ} (μ : Measure (Euc n)) [IsFiniteMeasure μ]
+    (u : Lp (Euc n) 2 μ) :
+    (∫ x, ‖u x‖ ∂μ) ≤ (μ univ ^ (1 / 2 : ℝ)).toReal * ‖u‖ := by
+  have hh := eLpNorm_le_eLpNorm_mul_rpow_measure_univ (f := (u : Euc n → Euc n))
+    (p := 1) (q := 2) (by norm_num) (Lp.aestronglyMeasurable u)
+  norm_num only [ENNReal.toReal_one, ENNReal.toReal_ofNat, div_one,
+    show (1 : ℝ) - 1 / 2 = 1 / 2 by norm_num] at hh
+  have ht : eLpNorm (u : Euc n → Euc n) 2 μ * μ univ ^ (1 / 2 : ℝ) ≠ ⊤ :=
+    ENNReal.mul_ne_top (Lp.eLpNorm_ne_top u)
+      (ENNReal.rpow_ne_top_of_nonneg (by norm_num) (measure_ne_top μ univ))
+  have hr := ENNReal.toReal_mono ht hh
+  rw [eLpNorm_one_eq_lintegral_enorm (Lp.aestronglyMeasurable u), ENNReal.toReal_mul,
+    ← integral_norm_eq_lintegral_enorm (Lp.aestronglyMeasurable u), ← Lp.norm_def] at hr
+  simpa [mul_comm] using hr
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+
+theorem l2_polar_norm_le_one {n : ℕ} (μ : Measure (Euc n)) [IsFiniteMeasure μ]
+    (G : Lp (Euc n) 2 μ)
+    (hG : ∀ u : Lp (Euc n) 2 μ,
+      inner ℝ G u ≤ ∫ x, ‖u x‖ ∂μ) :
+    ∀ᵐ x ∂μ, ‖G x‖ ≤ 1 := by
+  let s := {x | 1 < ‖G x‖}
+  have hs : MeasurableSet s := measurableSet_lt measurable_const (Lp.stronglyMeasurable G).measurable.norm
+  have hu : MemLp (s.indicator (G : Euc n → Euc n)) 2 μ := (Lp.memLp G).indicator hs
+  let u : Lp (Euc n) 2 μ := hu.toLp _
+  have he : (u : Euc n → Euc n) =ᵐ[μ] s.indicator (G : Euc n → Euc n) := hu.coeFn_toLp
+  have hsq : Integrable (fun x ↦ ‖G x‖ ^ 2) μ := by
+    simpa only [real_inner_self_eq_norm_sq] using (L2.integrable_inner (𝕜 := ℝ) G G)
+  have hn : Integrable (fun x ↦ ‖G x‖) μ :=
+    (memLp_one_iff_integrable.mp ((Lp.memLp G).mono_exponent (by norm_num))).norm
+  let a : Euc n → ℝ := s.indicator (fun x ↦ ‖G x‖ ^ 2)
+  let b : Euc n → ℝ := s.indicator (fun x ↦ ‖G x‖)
+  have hai : Integrable a μ := hsq.indicator hs
+  have hbi : Integrable b μ := hn.indicator hs
+  have hea : inner ℝ G u = ∫ x, a x ∂μ := by
+    rw [L2.inner_def]
+    apply integral_congr_ae
+    filter_upwards [he] with x hx
+    rw [hx]
+    by_cases hsx : x ∈ s
+    · simp [a, hsx]
+    · simp [a, hsx]
+  have heb : (∫ x, ‖u x‖ ∂μ) = ∫ x, b x ∂μ := by
+    apply integral_congr_ae
+    filter_upwards [he] with x hx
+    rw [hx]
+    by_cases hsx : x ∈ s <;> simp [b, hsx]
+  have hab := hG u
+  rw [hea, heb] at hab
+  have hnonneg : ∀ x, 0 ≤ a x - b x := by
+    intro x
+    by_cases hx : x ∈ s
+    · have hh : 1 < ‖G x‖ := hx
+      simp only [a, b, indicator_of_mem hx]
+      nlinarith
+    · simp [a, b, hx]
+  have hz : (∫ x, a x - b x ∂μ) = 0 := by
+    apply le_antisymm
+    · rw [integral_sub hai hbi]; linarith
+    · exact integral_nonneg hnonneg
+  have hae : ∀ᵐ x ∂μ, a x - b x = 0 :=
+    (integral_eq_zero_iff_of_nonneg_ae (ae_of_all _ hnonneg) (hai.sub hbi)).mp hz
+  filter_upwards [hae] with x hx
+  by_contra h
+  have hlt : 1 < ‖G x‖ := lt_of_not_ge h
+  have hsx : x ∈ s := hlt
+  simp only [a, b, indicator_of_mem hsx] at hx
+  nlinarith
+
+theorem exists_local_variation_polar {n : ℕ} (E U : Set (Euc n))
+    (hU : IsOpen U) (hP : perimeter E U ≠ ⊤) :
+    ∃ G : Euc n → Euc n,
+      StronglyMeasurable G ∧
+      MemLp G 2 ((perimeterMeasure E).restrict U) ∧
+      (∀ᵐ x ∂(perimeterMeasure E).restrict U, ‖G x‖ ≤ 1) ∧
+      ∀ V : Euc n → Euc n, ContDiff ℝ ∞ V → HasCompactSupport V → tsupport V ⊆ U →
+        (∫ x in E, divergence V x) = ∫ x in U, inner ℝ (G x) (V x) ∂perimeterMeasure E := by
+  let μ := (perimeterMeasure E).restrict U
+  let : IsFiniteMeasure μ := ⟨by
+    rw [Measure.restrict_apply_univ, perimeterMeasure_open E hU]; exact hP.lt_top⟩
+  obtain ⟨F, hF, hFb⟩ := exists_local_variation_extension E U hU hP
+  let C : ℝ := (μ univ ^ (1 / 2 : ℝ)).toReal
+  have hFc : ∀ u, ‖F u‖ ≤ C * ‖u‖ := fun u ↦
+    (hFb u).trans (local_l2_integral_norm_le μ u)
+  let Fc : Lp (Euc n) 2 μ →L[ℝ] ℝ := F.mkContinuous C hFc
+  let G : Lp (Euc n) 2 μ := (InnerProductSpace.toDual ℝ (Lp (Euc n) 2 μ)).symm Fc
+  have hrep : ∀ u, inner ℝ G u = F u := fun u ↦ InnerProductSpace.toDual_symm_apply
+  have hnorm : ∀ᵐ x ∂μ, ‖G x‖ ≤ 1 := l2_polar_norm_le_one μ G
+    (fun u ↦ by rw [hrep]; exact (le_abs_self _).trans (hFb u))
+  refine ⟨G, Lp.stronglyMeasurable G, Lp.memLp G, hnorm, ?_⟩
+  intro V hV hc hs
+  obtain ⟨B, hB⟩ := (hc.isCompact_range hV.continuous).isBounded.exists_norm_le
+  have hVm : MemLp V 2 μ := MemLp.of_bound hV.continuous.aestronglyMeasurable B
+    (ae_of_all _ (fun x ↦ hB _ (mem_range_self x)))
+  let u : Lp (Euc n) 2 μ := hVm.toLp V
+  have heu : (u : Euc n → Euc n) =ᵐ[μ] V := hVm.coeFn_toLp
+  let v : smoothVariationDomain U μ := ⟨u, V, hV, hc, hs, heu⟩
+  obtain ⟨hR, hcR, hsR, heR⟩ := smoothVariationRepresentative_spec v
+  have hval : localVariationFunctional E U hU hP v = ∫ x in E, divergence V x :=
+    integral_divergence_eq_of_variation_ae E U hU hP hR hcR hsR hV hc hs (heR.symm.trans heu)
+  calc
+    (∫ x in E, divergence V x) = F u := hval.symm.trans (hF v).symm
+    _ = inner ℝ G u := (hrep u).symm
+    _ = ∫ x, inner ℝ (G x) (V x) ∂μ := by
+      rw [L2.inner_def]
+      apply integral_congr_ae
+      filter_upwards [heu] with x hx
+      rw [hx]
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+
+theorem exists_local_unit_variation_polar {n : ℕ} (E U : Set (Euc n))
+    (hU : IsOpen U) (hP : perimeter E U ≠ ⊤) :
+    ∃ G : Euc n → Euc n,
+      StronglyMeasurable G ∧
+      MemLp G 2 ((perimeterMeasure E).restrict U) ∧
+      (∀ᵐ x ∂(perimeterMeasure E).restrict U, ‖G x‖ = 1) ∧
+      ∀ V : Euc n → Euc n, ContDiff ℝ ∞ V → HasCompactSupport V → tsupport V ⊆ U →
+        (∫ x in E, divergence V x) = ∫ x in U, inner ℝ (G x) (V x) ∂perimeterMeasure E := by
+  let μ := (perimeterMeasure E).restrict U
+  have hμu : μ univ = perimeter E U := by
+    rw [Measure.restrict_apply_univ, perimeterMeasure_open E hU]
+  let : IsFiniteMeasure μ := ⟨by rw [hμu]; exact hP.lt_top⟩
+  obtain ⟨G, hG, hGp, hGn, hrep⟩ := exists_local_variation_polar E U hU hP
+  have hGi : Integrable (fun x ↦ ‖G x‖) μ :=
+    (memLp_one_iff_integrable.mp (hGp.mono_exponent (by norm_num))).norm
+  have hvar : perimeter E U ≤ ENNReal.ofReal (∫ x, ‖G x‖ ∂μ) := by
+    rw [perimeter_eq_iSup_tests]
+    apply iSup_le
+    intro V
+    apply iSup_le
+    intro hV
+    apply ENNReal.ofReal_le_ofReal
+    rw [hrep V hV.1 hV.2.1 hV.2.2.1]
+    have hb : ∀ᵐ x ∂μ, ‖inner ℝ (G x) (V x)‖ ≤ 1 := by
+      filter_upwards [hGn] with x hx
+      exact (norm_inner_le_norm _ _).trans
+        ((mul_le_of_le_one_left (norm_nonneg _) hx).trans (hV.2.2.2 x))
+    have hi : Integrable (fun x ↦ inner ℝ (G x) (V x)) μ :=
+      Integrable.of_bound (hG.aestronglyMeasurable.inner hV.1.continuous.aestronglyMeasurable) 1 hb
+    apply integral_mono hi hGi
+    intro x
+    exact (real_inner_le_norm _ _).trans
+      ((mul_le_mul_of_nonneg_left (hV.2.2.2 x) (norm_nonneg _)).trans_eq (mul_one _))
+  have hlow : μ.real univ ≤ ∫ x, ‖G x‖ ∂μ := by
+    have hh := ENNReal.toReal_mono ENNReal.ofReal_ne_top hvar
+    rw [ENNReal.toReal_ofReal (integral_nonneg (fun x ↦ norm_nonneg _))] at hh
+    simpa only [measureReal_def, hμu] using hh
+  have hnneg : ∀ᵐ x ∂μ, 0 ≤ (1 : ℝ) - ‖G x‖ := hGn.mono fun x hx ↦ sub_nonneg.mpr hx
+  have hz : (∫ x, (1 : ℝ) - ‖G x‖ ∂μ) = 0 := by
+    apply le_antisymm
+    · rw [integral_sub (integrable_const 1) hGi, integral_const]
+      simp only [smul_eq_mul, mul_one]
+      linarith
+    · exact integral_nonneg_of_ae hnneg
+  have he : ∀ᵐ x ∂μ, (1 : ℝ) - ‖G x‖ = 0 :=
+    (integral_eq_zero_iff_of_nonneg_ae hnneg ((integrable_const 1).sub hGi)).mp hz
+  exact ⟨G, hG, hGp, he.mono (fun x hx ↦ by linarith), hrep⟩
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+
+theorem smooth_translation_lintegral_le {n : ℕ} {u : Euc n → ℝ}
+    (hu : ContDiff ℝ 1 u) (a : Euc n) :
+    (∫⁻ x, ‖u (x + a) - u x‖ₑ) ≤ ‖a‖ₑ * ∫⁻ x, ‖fderiv ℝ u x‖ₑ := by
+  have hfd := hu.continuous_fderiv one_ne_zero
+  have hpoint (x : Euc n) :
+      ‖u (x + a) - u x‖ₑ ≤
+        ∫⁻ t in Ioc (0 : ℝ) 1, ‖a‖ₑ * ‖fderiv ℝ u (x + t • a)‖ₑ := by
+    have hd (t : ℝ) : HasDerivAt (fun s : ℝ ↦ u (x + s • a))
+        (fderiv ℝ u (x + t • a) a) t := by
+      simpa only [Function.comp_def, id_eq, one_smul] using
+        (hu.differentiable one_ne_zero (x + t • a)).hasFDerivAt.comp_hasDerivAt t
+          (HasDerivAt.const_add x ((hasDerivAt_id t).smul_const a))
+    have hi : IntervalIntegrable (fun t : ℝ ↦ fderiv ℝ u (x + t • a) a) volume 0 1 :=
+      ((hfd.comp (continuous_const.add (continuous_id.smul continuous_const))).clm_apply
+        continuous_const).intervalIntegrable _ _
+    have he := intervalIntegral.integral_eq_sub_of_hasDerivAt (fun t _ ↦ hd t) hi
+    simp only [one_smul, zero_smul, add_zero] at he
+    rw [← he, intervalIntegral.integral_of_le (by norm_num : (0 : ℝ) ≤ 1)]
+    apply (enorm_integral_le_lintegral_enorm _).trans
+    apply lintegral_mono
+    intro t
+    simp only [← ofReal_norm]
+    rw [← ENNReal.ofReal_mul (norm_nonneg _)]
+    exact ENNReal.ofReal_le_ofReal ((ContinuousLinearMap.le_opNorm _ _).trans_eq (mul_comm _ _))
+  have hm : Measurable (fun p : Euc n × ℝ ↦ ‖a‖ₑ * ‖fderiv ℝ u (p.1 + p.2 • a)‖ₑ) :=
+    measurable_const.mul ((hfd.comp (continuous_fst.add
+      (continuous_snd.smul continuous_const))).enorm.measurable)
+  calc
+    (∫⁻ x, ‖u (x + a) - u x‖ₑ) ≤
+        ∫⁻ x, ∫⁻ t in Ioc (0 : ℝ) 1, ‖a‖ₑ * ‖fderiv ℝ u (x + t • a)‖ₑ :=
+      lintegral_mono hpoint
+    _ = ∫⁻ t in Ioc (0 : ℝ) 1, ∫⁻ x, ‖a‖ₑ * ‖fderiv ℝ u (x + t • a)‖ₑ :=
+      lintegral_lintegral_swap hm.aemeasurable
+    _ = ∫⁻ t in Ioc (0 : ℝ) 1, ‖a‖ₑ * ∫⁻ x, ‖fderiv ℝ u x‖ₑ := by
+      apply lintegral_congr
+      intro t
+      have hmt : Measurable (fun x : Euc n ↦ ‖fderiv ℝ u (x + t • a)‖ₑ) :=
+        (hfd.comp (continuous_id.add continuous_const)).enorm.measurable
+      rw [lintegral_const_mul _ hmt,
+        lintegral_add_right_eq_self (fun x : Euc n ↦ ‖fderiv ℝ u x‖ₑ) (t • a)]
+    _ = ‖a‖ₑ * ∫⁻ x, ‖fderiv ℝ u x‖ₑ := by simp
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+open RPI.Boundary
+open scoped Convolution
+
+theorem indicator_translation_lintegral_le {n : ℕ} {E : Set (Euc n)}
+    (hE : MeasurableSet E) (hb : Bornology.IsBounded E) (hP : perimeter E univ ≠ ⊤)
+    (a : Euc n) :
+    (∫⁻ x, ‖E.indicator (fun _ ↦ (1 : ℝ)) (x + a) -
+      E.indicator (fun _ ↦ (1 : ℝ)) x‖ₑ) ≤ ‖a‖ₑ * perimeter E univ := by
+  let χ : Euc n → ℝ := E.indicator (fun _ ↦ 1)
+  let u : ℕ → Euc n → ℝ := c1Mollify χ
+  obtain ⟨hχ, hcχ⟩ := indicator_one_integrable_compact hE hb
+  have hu : ∀ j, ContDiff ℝ ∞ (u j) := fun j ↦
+    (c1Mollifier j).hasCompactSupport_normed.contDiff_convolution_left _
+      (c1Mollifier j).contDiff_normed hχ.locallyIntegrable
+  have hlim : ∀ᵐ x ∂volume, Tendsto (fun j ↦ u j x) atTop (𝓝 (χ x)) := by
+    apply ContDiffBump.ae_convolution_tendsto_right_of_locallyIntegrable
+      c1Mollifier_radius_tendsto (K := 2) _ hχ.locallyIntegrable
+    apply Eventually.of_forall
+    intro j
+    dsimp [c1Mollifier]
+    linarith
+  have hlimadd : ∀ᵐ x ∂volume, Tendsto (fun j ↦ u j (x + a)) atTop (𝓝 (χ (x + a))) :=
+    (measurePreserving_add_right volume a).quasiMeasurePreserving.ae hlim
+  have hlimnorm : ∀ᵐ x ∂volume,
+      Tendsto (fun j ↦ ‖u j (x + a) - u j x‖ₑ) atTop (𝓝 (‖χ (x + a) - χ x‖ₑ)) := by
+    filter_upwards [hlimadd, hlim] with x hx hy
+    exact (hx.sub hy).enorm
+  calc
+    (∫⁻ x, ‖χ (x + a) - χ x‖ₑ) =
+        ∫⁻ x, liminf (fun j ↦ ‖u j (x + a) - u j x‖ₑ) atTop := by
+      apply lintegral_congr_ae
+      filter_upwards [hlimnorm] with x hx using hx.liminf_eq.symm
+    _ ≤ liminf (fun j ↦ ∫⁻ x, ‖u j (x + a) - u j x‖ₑ) atTop :=
+      lintegral_liminf_le' (fun j ↦ (((hu j).continuous.comp
+        (continuous_id.add continuous_const)).sub (hu j).continuous).enorm.measurable.aemeasurable)
+    _ ≤ ‖a‖ₑ * perimeter E univ := by
+      have hb' : ∀ j, (∫⁻ x, ‖u j (x + a) - u j x‖ₑ) ≤ ‖a‖ₑ * perimeter E univ := by
+        intro j
+        apply (smooth_translation_lintegral_le ((hu j).of_le (by simp)) a).trans
+        gcongr
+        exact c1Mollify_indicator_gradient_le_perimeter hE hb hP j
+      exact liminf_le_of_frequently_le (Eventually.of_forall hb').frequently
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+open RPI.Boundary ContinuousLinearMap
+open scoped Convolution
+
+theorem c1Mollify_indicator_lintegral_error_le {n : ℕ} {E : Set (Euc n)}
+    (hE : MeasurableSet E) (hb : Bornology.IsBounded E) (hP : perimeter E univ ≠ ⊤)
+    (j : ℕ) :
+    (∫⁻ x, ‖c1Mollify (E.indicator (fun _ ↦ (1 : ℝ))) j x -
+      E.indicator (fun _ ↦ (1 : ℝ)) x‖ₑ) ≤
+      ENNReal.ofReal (c1Mollifier (n := n) j).rOut * perimeter E univ := by
+  let χ : Euc n → ℝ := E.indicator (fun _ ↦ 1)
+  let k : Euc n → ℝ := (c1Mollifier j).normed volume
+  have hmχ : Measurable χ := measurable_const.indicator hE
+  have hbχ : ∀ x, ‖χ x‖ ≤ 1 := by
+    intro x
+    by_cases hx : x ∈ E <;> simp [χ, hx]
+  have hkc : Continuous k := (c1Mollifier j).continuous_normed
+  have hk : Integrable k volume := hkc.integrable_of_hasCompactSupport
+    (c1Mollifier j).hasCompactSupport_normed
+  have hk1 : ∫ x, k x = 1 := (c1Mollifier j).integral_normed
+  have hp (x : Euc n) : ‖c1Mollify χ j x - χ x‖ₑ ≤
+      ∫⁻ y, ‖k y‖ₑ * ‖χ (x - y) - χ x‖ₑ := by
+    have hi : Integrable (fun y ↦ k y * χ (x - y)) volume := by
+      apply hk.norm.mono'
+      · exact (hkc.measurable.mul (hmχ.comp (measurable_const.sub measurable_id))).aestronglyMeasurable
+      · apply ae_of_all
+        intro y
+        rw [norm_mul]
+        exact (mul_le_mul_of_nonneg_left (hbχ (x - y)) (norm_nonneg _)).trans_eq (mul_one _)
+    have he : c1Mollify χ j x - χ x = ∫ y, k y * (χ (x - y) - χ x) := by
+      simp only [c1Mollify, convolution_def, lsmul_apply, smul_eq_mul]
+      simp_rw [mul_sub]
+      rw [integral_sub hi (hk.mul_const _), integral_mul_const, hk1, one_mul]
+    rw [he]
+    apply (enorm_integral_le_lintegral_enorm _).trans_eq
+    apply lintegral_congr
+    intro y
+    exact enorm_mul _ _
+  have hm : Measurable (fun p : Euc n × Euc n ↦
+      ‖k p.2‖ₑ * ‖χ (p.1 - p.2) - χ p.1‖ₑ) :=
+    (hkc.measurable.comp measurable_snd).enorm.mul
+      ((hmχ.comp (measurable_fst.sub measurable_snd)).sub
+        (hmχ.comp measurable_fst)).enorm
+  calc
+    (∫⁻ x, ‖c1Mollify χ j x - χ x‖ₑ) ≤
+        ∫⁻ x, ∫⁻ y, ‖k y‖ₑ * ‖χ (x - y) - χ x‖ₑ := lintegral_mono hp
+    _ = ∫⁻ y, ∫⁻ x, ‖k y‖ₑ * ‖χ (x - y) - χ x‖ₑ :=
+      lintegral_lintegral_swap hm.aemeasurable
+    _ = ∫⁻ y, ‖k y‖ₑ * ∫⁻ x, ‖χ (x - y) - χ x‖ₑ := by
+      apply lintegral_congr
+      intro y
+      apply lintegral_const_mul
+      exact ((hmχ.comp (measurable_id.sub measurable_const)).sub hmχ).enorm
+    _ ≤ ∫⁻ y, ‖k y‖ₑ * (‖y‖ₑ * perimeter E univ) := by
+      apply lintegral_mono
+      intro y
+      have hy : (∫⁻ x, ‖χ (x - y) - χ x‖ₑ) ≤ ‖y‖ₑ * perimeter E univ := by
+        simpa only [sub_eq_add_neg, enorm_neg] using indicator_translation_lintegral_le hE hb hP (-y)
+      exact mul_le_mul_right hy _
+    _ ≤ ∫⁻ y, ‖k y‖ₑ * (ENNReal.ofReal (c1Mollifier (n := n) j).rOut * perimeter E univ) := by
+      apply lintegral_mono
+      intro y
+      by_cases hy : k y = 0
+      · simp only [hy, enorm_zero, zero_mul, le_refl]
+      · have hys : y ∈ Function.support k := hy
+        rw [(c1Mollifier j).support_normed_eq] at hys
+        have hyb : ‖y‖ₑ ≤ ENNReal.ofReal (c1Mollifier (n := n) j).rOut := by
+          rw [← ofReal_norm]
+          exact ENNReal.ofReal_le_ofReal (by
+            have hylt : ‖y‖ < (c1Mollifier (n := n) j).rOut := by
+              simpa only [Metric.mem_ball, dist_zero_right] using hys
+            exact hylt.le)
+        exact mul_le_mul_right (mul_le_mul_left hyb _) _
+    _ = ENNReal.ofReal (c1Mollifier (n := n) j).rOut * perimeter E univ := by
+      rw [lintegral_mul_const _ hkc.measurable.enorm]
+      have hki : (∫⁻ y, ‖k y‖ₑ) = 1 := by
+        have heq : ∀ y, ‖k y‖ₑ = ENNReal.ofReal (k y) := by
+          intro y
+          rw [← ofReal_norm, Real.norm_eq_abs, abs_of_nonneg ((c1Mollifier j).nonneg_normed y)]
+        simp_rw [heq]
+        rw [← ofReal_integral_eq_lintegral_ofReal hk (ae_of_all _ (c1Mollifier j).nonneg_normed), hk1]
+        norm_num
+      rw [hki, one_mul]
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+open RPI.Boundary ContinuousLinearMap
+open scoped Convolution NNReal
+
+theorem c1Mollify_indicator_mem_Icc {n : ℕ} {E : Set (Euc n)}
+    (hE : MeasurableSet E) (j : ℕ) (x : Euc n) :
+    c1Mollify (E.indicator (fun _ ↦ (1 : ℝ))) j x ∈ Icc (0 : ℝ) 1 := by
+  let χ : Euc n → ℝ := E.indicator (fun _ ↦ 1)
+  let k : Euc n → ℝ := (c1Mollifier j).normed volume
+  have hk0 : ∀ y, 0 ≤ k y := (c1Mollifier j).nonneg_normed
+  have hk : Integrable k volume := (c1Mollifier j).continuous_normed.integrable_of_hasCompactSupport
+    (c1Mollifier j).hasCompactSupport_normed
+  have hχb : ∀ y, χ y ∈ Icc (0 : ℝ) 1 := by
+    intro y
+    by_cases hy : y ∈ E <;> simp [χ, hy]
+  change (∫ y, k y * χ (x - y)) ∈ Icc (0 : ℝ) 1
+  refine ⟨integral_nonneg (fun y ↦ mul_nonneg (hk0 y) (hχb _).1), ?_⟩
+  calc
+    (∫ y, k y * χ (x - y)) ≤ ∫ y, k y := by
+      have hnonneg : ∀ᵐ y ∂volume, 0 ≤ k y * χ (x - y) :=
+        ae_of_all _ fun y ↦ mul_nonneg (hk0 y) (hχb _).1
+      have hbound : ∀ᵐ y ∂volume, k y * χ (x - y) ≤ k y := by
+        apply ae_of_all
+        intro y
+        exact (mul_le_mul_of_nonneg_left (hχb _).2 (hk0 y)).trans_eq (mul_one _)
+      have hm : AEStronglyMeasurable (fun y ↦ k y * χ (x - y)) volume :=
+        ((c1Mollifier j).continuous_normed.measurable.mul
+          ((measurable_const.indicator hE).comp
+            (measurable_const.sub measurable_id))).aestronglyMeasurable
+      exact integral_mono_ae (hk.mono_nonneg hm hnonneg hbound) hk hbound
+    _ = 1 := (c1Mollifier j).integral_normed
+
+theorem c1Mollify_indicator_uniform_lipschitz {n : ℕ} (K : Set (Euc n))
+    (hK : Bornology.IsBounded K) (j : ℕ) :
+    ∃ L : ℝ≥0, ∀ E : Set (Euc n), MeasurableSet E → E ⊆ K →
+      LipschitzWith L (c1Mollify (E.indicator (fun _ ↦ (1 : ℝ))) j) := by
+  let k : Euc n → ℝ := (c1Mollifier j).normed volume
+  obtain ⟨C, hC⟩ := ContDiff.lipschitzWith_of_hasCompactSupport
+    (n := ∞) (f := k) (c1Mollifier j).hasCompactSupport_normed
+    (c1Mollifier j).contDiff_normed (by simp)
+  refine ⟨C * (volume K).toNNReal, ?_⟩
+  intro E hE hEK
+  let χ : Euc n → ℝ := E.indicator (fun _ ↦ 1)
+  obtain ⟨hiχ, hcχ⟩ := indicator_one_integrable_compact hE (hK.subset hEK)
+  have hχ0 : ∀ z, 0 ≤ χ z := fun z ↦ indicator_nonneg (fun _ _ ↦ zero_le_one) z
+  have hχ1 : ∫ z, χ z ≤ volume.real K := by
+    rw [show χ = E.indicator (fun _ ↦ (1 : ℝ)) from rfl, integral_indicator hE]
+    simp only [setIntegral_const, smul_eq_mul, mul_one]
+    exact ENNReal.toReal_mono hK.measure_lt_top.ne (measure_mono hEK)
+  have hflip : (lsmul ℝ ℝ : ℝ →L[ℝ] ℝ →L[ℝ] ℝ).flip = lsmul ℝ ℝ := by
+    apply ContinuousLinearMap.ext
+    intro a
+    apply ContinuousLinearMap.ext
+    intro b
+    exact mul_comm b a
+  have hf : c1Mollify χ j = χ ⋆[lsmul ℝ ℝ, volume] k := by
+    change k ⋆[lsmul ℝ ℝ, volume] χ = _
+    rw [← convolution_flip, hflip]
+  have hib (x : Euc n) : Integrable (fun z ↦ χ z * k (x - z)) volume := by
+    obtain ⟨B, hB⟩ := ((c1Mollifier (n := n) j).hasCompactSupport_normed (μ := volume)).exists_bound_of_continuous
+      (c1Mollifier j).continuous_normed
+    apply (hiχ.norm.mul_const B).mono'
+    · exact ((measurable_const.indicator hE).mul
+        ((c1Mollifier j).continuous_normed.measurable.comp
+          (measurable_const.sub measurable_id))).aestronglyMeasurable
+    · apply ae_of_all
+      intro z
+      rw [norm_mul]
+      exact mul_le_mul_of_nonneg_left (hB _) (norm_nonneg _)
+  apply LipschitzWith.of_dist_le_mul
+  intro x y
+  rw [dist_eq_norm, hf]
+  change ‖(∫ z, χ z * k (x - z)) - ∫ z, χ z * k (y - z)‖ ≤ _
+  rw [← integral_sub (hib x) (hib y)]
+  calc
+    ‖∫ z, χ z * k (x - z) - χ z * k (y - z)‖ ≤
+        ∫ z, ‖χ z * k (x - z) - χ z * k (y - z)‖ := norm_integral_le_integral_norm _
+    _ ≤ ∫ z, χ z * ((C : ℝ) * dist x y) := by
+      apply integral_mono ((hib x).sub (hib y)).norm (hiχ.mul_const _)
+      intro z
+      change ‖χ z * k (x - z) - χ z * k (y - z)‖ ≤
+        χ z * ((C : ℝ) * dist x y)
+      rw [← mul_sub, norm_mul, Real.norm_eq_abs, abs_of_nonneg (hχ0 z)]
+      apply mul_le_mul_of_nonneg_left _ (hχ0 z)
+      simpa only [← dist_eq_norm, dist_sub_right] using hC.dist_le_mul (x - z) (y - z)
+    _ = (∫ z, χ z) * ((C : ℝ) * dist x y) := integral_mul_const _ _
+    _ ≤ volume.real K * ((C : ℝ) * dist x y) :=
+      mul_le_mul_of_nonneg_right hχ1 (mul_nonneg C.coe_nonneg dist_nonneg)
+    _ = ((C * (volume K).toNNReal : ℝ≥0) : ℝ) * dist x y := by
+      change volume.real K * ((C : ℝ) * dist x y) =
+        ((C : ℝ) * volume.real K) * dist x y
+      ring
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/ExistsMinimizingBoundaryAreaGrowth.lean b/lean/OAI/Geometry/Perimeter/Regularity/ExistsMinimizingBoundaryAreaGrowth.lean
new file mode 100644
index 0000000000000000000000000000000000000000..7ffe852491ea69058abef264e0db6863d24ac357
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/ExistsMinimizingBoundaryAreaGrowth.lean
@@ -0,0 +1,453 @@
+import OAI.Geometry.Perimeter.Regularity.IndicatorConvolutionVariationLocalLe
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+open RPI
+
+theorem exists_minimizing_boundaryArea_growth {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E) :
+    ∃ A : ℝ, 0 < A ∧ ∀ (c : Euc n) (r : ℝ), 0 < r →
+      boundaryArea E (Metric.ball c r) ≤ ENNReal.ofReal (A * r ^ (n - 1)) := by
+  obtain ⟨A, hA, harea⟩ := exists_minimizing_surfaceMeasure_growth hn hmin
+  let b := hausdorffNormalizationRatio (n - 1)
+  let D := (b * ENNReal.ofReal A).toReal + 1
+  have hD : 0 < D := by dsimp [D]; positivity
+  have hb : b ≠ ⊤ := hausdorffNormalizationRatio_ne_top _
+  have hbA : b * ENNReal.ofReal A ≤ ENNReal.ofReal D := by
+    rw [← ENNReal.ofReal_toReal (ENNReal.mul_ne_top hb ENNReal.ofReal_ne_top)]
+    apply ENNReal.ofReal_le_ofReal
+    dsimp [D]
+    linarith
+  refine ⟨D, hD, ?_⟩
+  intro c r hr
+  rw [boundaryArea_eq_smul_surfaceMeasure]
+  change b * _ ≤ _
+  calc
+    _ ≤ b * ENNReal.ofReal (A * r ^ (n - 1)) := mul_le_mul le_rfl (harea c r hr) (by positivity) (by positivity)
+    _ = (b * ENNReal.ofReal A) * ENNReal.ofReal r ^ (n - 1) := by
+      rw [ENNReal.ofReal_mul hA.le, ENNReal.ofReal_pow hr.le, mul_assoc]
+    _ ≤ ENNReal.ofReal D * ENNReal.ofReal r ^ (n - 1) := mul_le_mul hbA le_rfl (by positivity) (by positivity)
+    _ = _ := by rw [ENNReal.ofReal_mul hD.le, ENNReal.ofReal_pow hr.le]
+
+theorem approximateNormal_norm_le_one {n : ℕ} (E : Set (Euc n)) (x : Euc n) :
+    ‖approximateNormal E x‖ ≤ 1 := by
+  classical
+  unfold approximateNormal
+  split_ifs with h
+  · exact (Classical.choose_spec h).1.le
+  · simp
+
+theorem tangentialGradient_norm_le_twice {n : ℕ} (E : Set (Euc n))
+    (φ : Euc n → ℝ) (x : Euc n) :
+    ‖tangentialGradient E φ x‖ ≤ 2 * ‖gradient φ x‖ := by
+  have hv := approximateNormal_norm_le_one E x
+  have hinner : |inner ℝ (gradient φ x) (approximateNormal E x)| ≤ ‖gradient φ x‖ := by
+    exact (abs_real_inner_le_norm _ _).trans (by nlinarith [norm_nonneg (gradient φ x)])
+  calc
+    _ ≤ ‖gradient φ x‖ + ‖(inner ℝ (gradient φ x) (approximateNormal E x)) •
+        approximateNormal E x‖ := norm_sub_le _ _
+    _ ≤ ‖gradient φ x‖ + ‖gradient φ x‖ := by
+      rw [norm_smul, Real.norm_eq_abs]
+      gcongr
+      exact le_trans (mul_le_mul_of_nonneg_left hv (abs_nonneg _)) (by simpa using hinner)
+    _ = _ := by ring
+
+theorem exists_raw_minimizing_frontier_capacity_sequence {n : ℕ} (hn : 8 ≤ n)
+    {E Z : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    (hZc : IsCompact Z) (hZd : dimH Z ≤ (n - 8 : ℕ)) :
+    ∃ χ : ℕ → Euc n → ℝ,
+      (∀ i, ContDiff ℝ ∞ (χ i) ∧ (∀ x, χ i x ∈ Icc (0 : ℝ) 1) ∧
+        ∃ U : Set (Euc n), IsOpen U ∧ Z ⊆ U ∧ ∀ x ∈ U, χ i x = 0) ∧
+      (∀ x ∉ Z, ∀ᶠ i in atTop, χ i x = 1) ∧
+      Tendsto (fun i ↦ smoothDirichletEnergy (boundaryArea E) (χ i)) atTop (𝓝 0) ∧
+      Tendsto (fun i ↦ vectorEnergy (boundaryArea E) (tangentialGradient E (χ i)))
+        atTop (𝓝 0) := by
+  obtain ⟨A, hA, harea⟩ := exists_minimizing_boundaryArea_growth (by omega : 1 < n) hmin
+  obtain ⟨hm, hnull⟩ := hausdorff_capacity_dimension hn Z hZd
+  have heq : n - 3 + 2 = n - 1 := by omega
+  obtain ⟨χ, hχ, hχone, hχenergy⟩ := exists_smooth_capacity_sequence (boundaryArea E)
+    Z hZc (n - 3) hm hnull hA (by norm_num : (0 : ℝ) < 1)
+    (fun c _ r hr _ ↦ by simpa only [heq] using harea c r hr)
+  refine ⟨χ, hχ, hχone, hχenergy, ?_⟩
+  have hb : ∀ i, vectorEnergy (boundaryArea E) (tangentialGradient E (χ i)) ≤
+      (4 : ℝ≥0∞) * smoothDirichletEnergy (boundaryArea E) (χ i) := by
+    intro i
+    rw [vectorEnergy, smoothDirichletEnergy, ← lintegral_const_mul' _ _ ENNReal.ofNat_ne_top]
+    apply lintegral_mono
+    intro x
+    have hnorm := tangentialGradient_norm_le_twice E (χ i) x
+    calc
+      _ ≤ ENNReal.ofReal (4 * ‖gradient (χ i) x‖ ^ 2) := by
+        apply ENNReal.ofReal_le_ofReal
+        nlinarith [norm_nonneg (tangentialGradient E (χ i) x), norm_nonneg (gradient (χ i) x)]
+      _ = _ := by rw [ENNReal.ofReal_mul (by norm_num : (0 : ℝ) ≤ 4)]; norm_num
+  exact tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds
+    (by simpa only [mul_zero] using ENNReal.Tendsto.const_mul hχenergy (Or.inr (by norm_num : (4 : ℝ≥0∞) ≠ ⊤)))
+    (fun _ ↦ zero_le) hb
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+
+theorem tendsto_normalized_monotone_of_dyadic {a : ℝ → ℝ} (n : ℕ)
+    (ha : ∀ r, 0 < r → 0 ≤ a r)
+    (ham : ∀ r s, 0 < r → r ≤ s → a r ≤ a s)
+    (ht : Tendsto (fun j : ℕ ↦ a ((1 / 2 : ℝ) ^ j) / ((1 / 2 : ℝ) ^ j) ^ n)
+      atTop (𝓝 0)) :
+    Tendsto (fun r : ℝ ↦ a r / r ^ n) (𝓝[>] 0) (𝓝 0) := by
+  apply Metric.tendsto_nhds.mpr
+  intro ε hε
+  have htwo : (0 : ℝ) < 2 ^ n := by positivity
+  have hev := (Metric.tendsto_nhds.mp ht) (ε / (2 : ℝ) ^ n) (div_pos hε htwo)
+  obtain ⟨N, hN⟩ := eventually_atTop.mp hev
+  have hpos : (0 : ℝ) < (1 / 2 : ℝ) ^ N := by positivity
+  have hsmall : ∀ᶠ r : ℝ in 𝓝[>] 0, r < (1 / 2 : ℝ) ^ N :=
+    (gt_mem_nhds hpos).filter_mono nhdsWithin_le_nhds
+  filter_upwards [hsmall, self_mem_nhdsWithin] with r hr hrp
+  have hunit : r ≤ 1 := hr.le.trans (pow_le_one₀ (by norm_num) (by norm_num))
+  obtain ⟨j, hjlo, hjhi⟩ := exists_nat_pow_near_of_lt_one hrp hunit
+    (by norm_num : (0 : ℝ) < 1 / 2) (by norm_num : (1 / 2 : ℝ) < 1)
+  have hjN : N ≤ j := by
+    by_contra h
+    have hNj : j + 1 ≤ N := by omega
+    have hpow : (1 / 2 : ℝ) ^ N ≤ (1 / 2 : ℝ) ^ (j + 1) :=
+      pow_le_pow_of_le_one (by norm_num) (by norm_num) hNj
+    linarith
+  have hspos : (0 : ℝ) < (1 / 2 : ℝ) ^ j := by positivity
+  have hscale : (1 / 2 : ℝ) ^ j ≤ 2 * r := by
+    rw [pow_succ] at hjlo
+    linarith
+  have hpower : ((1 / 2 : ℝ) ^ j) ^ n ≤ (2 : ℝ) ^ n * r ^ n := by
+    simpa only [mul_pow] using pow_le_pow_left₀ hspos.le hscale n
+  have hab := ham r ((1 / 2 : ℝ) ^ j) hrp hjhi
+  have hbound : a r / r ^ n ≤ (2 : ℝ) ^ n *
+      (a ((1 / 2 : ℝ) ^ j) / ((1 / 2 : ℝ) ^ j) ^ n) := by
+    apply le_trans (div_le_div_of_nonneg_right hab (pow_nonneg hrp.le _))
+    rw [← mul_div_assoc, div_le_div_iff₀ (pow_pos hrp _) (pow_pos hspos _)]
+    nlinarith [mul_le_mul_of_nonneg_left hpower (ha _ hspos)]
+  have hj := hN j hjN
+  rw [dist_zero_right, Real.norm_eq_abs, abs_of_nonneg (div_nonneg (ha _ hspos)
+    (pow_nonneg hspos.le _))] at hj
+  rw [dist_zero_right, Real.norm_eq_abs, abs_of_nonneg (div_nonneg (ha _ hrp)
+    (pow_nonneg hrp.le _))]
+  exact hbound.trans_lt ((mul_lt_mul_of_pos_left hj htwo).trans_eq (by field_simp))
+
+def rescaledHalfspaceError {n : ℕ} (E : Set (Euc n)) (r : ℝ)
+    (p : Euc n × Euc n) : ℝ :=
+  ∫ z in Metric.ball (0 : Euc n) 1,
+    |E.indicator (fun _ ↦ (1 : ℝ)) (p.1 + r • z) -
+      {y | inner ℝ y p.2 < 0}.indicator (fun _ ↦ (1 : ℝ)) z|
+
+theorem rescaledHalfspaceError_eq {n : ℕ} (E : Set (Euc n))
+    (x v : Euc n) {r : ℝ} (hr : 0 < r) :
+    rescaledHalfspaceError E r (x, v) =
+      (r ^ n)⁻¹ * ∫ y in Metric.ball x r,
+        |E.indicator (fun _ ↦ (1 : ℝ)) y -
+          {z | inner ℝ (z - x) v < 0}.indicator (fun _ ↦ (1 : ℝ)) y| := by
+  rw [normalized_ball_integral _ x hr]
+  unfold rescaledHalfspaceError
+  apply integral_congr_ae
+  exact ae_of_all _ (fun z ↦ by dsimp only; rw [halfspace_indicator_rescale x v z hr])
+
+theorem hasHalfspaceBlowup_iff_dyadic {n : ℕ} {E : Set (Euc n)}
+    (hE : MeasurableSet E) (x v : Euc n) :
+    HasHalfspaceBlowup E x v ↔ ‖v‖ = 1 ∧
+      Tendsto (fun j : ℕ ↦ rescaledHalfspaceError E ((1 / 2 : ℝ) ^ j) (x, v))
+        atTop (𝓝 0) := by
+  have hpow := tendsto_pow_atTop_nhdsWithin_zero_of_lt_one
+    (by norm_num : (0 : ℝ) < 1 / 2) (by norm_num : (1 / 2 : ℝ) < 1)
+  constructor
+  · rintro ⟨hv, ht⟩
+    refine ⟨hv, ?_⟩
+    convert ht.comp hpow using 1
+    ext j
+    exact rescaledHalfspaceError_eq E x v (by positivity)
+  · rintro ⟨hv, ht⟩
+    refine ⟨hv, ?_⟩
+    let f : Euc n → ℝ := fun y ↦ |E.indicator (fun _ ↦ (1 : ℝ)) y -
+      {z | inner ℝ (z - x) v < 0}.indicator (fun _ ↦ (1 : ℝ)) y|
+    have hf : Measurable f := by
+      apply Measurable.abs
+      exact (measurable_const.indicator hE).sub
+        (measurable_const.indicator (isOpen_lt
+          ((continuous_id.sub continuous_const).inner continuous_const) continuous_const).measurableSet)
+    have hfi : ∀ s : ℝ, IntegrableOn f (Metric.ball x s) volume := by
+      intro s
+      apply (integrableOn_const measure_ball_lt_top.ne (C := (1 : ℝ))).mono' hf.aestronglyMeasurable
+      apply ae_of_all
+      intro y
+      dsimp [f]
+      by_cases he : y ∈ E <;> by_cases hh : inner ℝ (y - x) v < 0 <;> simp [he, hh]
+    have hnon : ∀ s, 0 < s → 0 ≤ ∫ y in Metric.ball x s, f y :=
+      fun s _ ↦ integral_nonneg (fun y ↦ abs_nonneg _)
+    have hmono : ∀ r s, 0 < r → r ≤ s →
+        (∫ y in Metric.ball x r, f y) ≤ ∫ y in Metric.ball x s, f y := by
+      intro r s _ hrs
+      exact setIntegral_mono_set (hfi s) (ae_of_all _ (fun y ↦ abs_nonneg _))
+        (Eventually.of_forall (Metric.ball_subset_ball hrs))
+    have hdy : Tendsto (fun j : ℕ ↦
+        (∫ y in Metric.ball x ((1 / 2 : ℝ) ^ j), f y) / ((1 / 2 : ℝ) ^ j) ^ n)
+        atTop (𝓝 0) := by
+      convert ht using 1
+      ext j
+      rw [rescaledHalfspaceError_eq E x v (by positivity)]
+      simp only [div_eq_mul_inv, mul_comm, f]
+    simpa only [div_eq_mul_inv, mul_comm, f] using
+      tendsto_normalized_monotone_of_dyadic n hnon hmono hdy
+
+theorem measurable_rescaledHalfspaceError {n : ℕ} {E : Set (Euc n)}
+    (hE : MeasurableSet E) (r : ℝ) : Measurable (rescaledHalfspaceError E r) := by
+  have hχ : Measurable (E.indicator (fun _ ↦ (1 : ℝ))) := measurable_const.indicator hE
+  have hH : MeasurableSet {p : (Euc n × Euc n) × Euc n | inner ℝ p.2 p.1.2 < 0} :=
+    (isOpen_lt (continuous_snd.inner (continuous_snd.comp continuous_fst)) continuous_const).measurableSet
+  have hf : Measurable (fun p : (Euc n × Euc n) × Euc n ↦
+      |E.indicator (fun _ ↦ (1 : ℝ)) (p.1.1 + r • p.2) -
+        {y | inner ℝ y p.1.2 < 0}.indicator (fun _ ↦ (1 : ℝ)) p.2|) := by
+    have he : Measurable (fun p : (Euc n × Euc n) × Euc n ↦
+        E.indicator (fun _ ↦ (1 : ℝ)) (p.1.1 + r • p.2)) :=
+      hχ.comp ((measurable_fst.comp measurable_fst).add (measurable_snd.const_smul r))
+    have hh : Measurable (fun p : (Euc n × Euc n) × Euc n ↦
+        {y | inner ℝ y p.1.2 < 0}.indicator (fun _ ↦ (1 : ℝ)) p.2) := by
+      have heq : (fun p : (Euc n × Euc n) × Euc n ↦
+          {y | inner ℝ y p.1.2 < 0}.indicator (fun _ ↦ (1 : ℝ)) p.2) =
+          {p : (Euc n × Euc n) × Euc n | inner ℝ p.2 p.1.2 < 0}.indicator
+            (fun _ ↦ (1 : ℝ)) := by funext p; rfl
+      rw [heq]
+      exact measurable_const.indicator hH
+    exact (he.sub hh).abs
+  exact (hf.stronglyMeasurable.integral_prod_right').measurable
+
+theorem measurableSet_halfspaceBlowup {n : ℕ} {E : Set (Euc n)}
+    (hE : MeasurableSet E) :
+    MeasurableSet {p : Euc n × Euc n | HasHalfspaceBlowup E p.1 p.2} := by
+  have hnorm : MeasurableSet {p : Euc n × Euc n | ‖p.2‖ = 1} :=
+    measurableSet_eq_fun measurable_snd.norm measurable_const
+  have ht := measurableSet_tendsto (l := (atTop : Filter ℕ)) (𝓝 (0 : ℝ))
+    (fun j : ℕ ↦ measurable_rescaledHalfspaceError hE ((1 / 2 : ℝ) ^ j))
+  simpa only [hasHalfspaceBlowup_iff_dyadic hE, Set.ofPred_and] using hnorm.inter ht
+
+theorem measurable_approximateNormal {n : ℕ} {E : Set (Euc n)}
+    (hE : MeasurableSet E) : Measurable (approximateNormal E) := by
+  classical
+  let Γ : Set (Euc n × Euc n) := {p | HasHalfspaceBlowup E p.1 p.2}
+  have hΓ : MeasurableSet Γ := measurableSet_halfspaceBlowup hE
+  have hinj : Set.InjOn Prod.fst Γ := by
+    intro p hp q hq heq
+    apply Prod.ext heq
+    exact (show HasHalfspaceBlowup E p.1 p.2 from hp).unique hE
+      (by simpa only [Γ, Set.mem_ofPred_eq, heq] using hq)
+  let D : Set (Euc n) := Prod.fst '' Γ
+  have hD : MeasurableSet D := hΓ.image_of_measurable_injOn measurable_fst hinj
+  have hDm : ∀ x, x ∈ D ↔ ∃ v, HasHalfspaceBlowup E x v := by
+    intro x
+    constructor
+    · rintro ⟨⟨y, v⟩, hv, hxy⟩
+      change y = x at hxy
+      subst y
+      exact ⟨v, hv⟩
+    · rintro ⟨v, hv⟩
+      exact ⟨(x, v), hv, rfl⟩
+  intro s hs
+  let B : Set (Euc n) := Prod.fst '' (Γ ∩ Prod.snd ⁻¹' s)
+  have hB : MeasurableSet B := (hΓ.inter (hs.preimage measurable_snd)).image_of_measurable_injOn
+    measurable_fst (hinj.mono Set.inter_subset_left)
+  have hBm : ∀ x, x ∈ B ↔ ∃ v, HasHalfspaceBlowup E x v ∧ v ∈ s := by
+    intro x
+    constructor
+    · rintro ⟨⟨y, v⟩, ⟨hv, hvs⟩, hxy⟩
+      change y = x at hxy
+      subst y
+      exact ⟨v, hv, hvs⟩
+    · rintro ⟨v, hv, hvs⟩
+      exact ⟨(x, v), ⟨hv, hvs⟩, rfl⟩
+  have hm : ∀ x, x ∈ D → (approximateNormal E x ∈ s ↔ x ∈ B) := by
+    intro x hx
+    obtain ⟨v, hv⟩ := (hDm x).mp hx
+    rw [approximateNormal_eq_of_blowup hE hv, hBm]
+    constructor
+    · exact fun hvs ↦ ⟨v, hv, hvs⟩
+    · rintro ⟨w, hw, hws⟩
+      rwa [hv.unique hE hw]
+  have hz : ∀ x, x ∉ D → approximateNormal E x = 0 := by
+    intro x hx
+    have hno : ¬ ∃ v, HasHalfspaceBlowup E x v := by simpa only [hDm] using hx
+    simp only [approximateNormal, dite_eq_right hno]
+  have hBD : B ⊆ D := by
+    intro x hx
+    obtain ⟨v, hv, _⟩ := (hBm x).mp hx
+    exact (hDm x).mpr ⟨v, hv⟩
+  by_cases hzero : (0 : Euc n) ∈ s
+  · have heq : approximateNormal E ⁻¹' s = B ∪ Dᶜ := by
+      ext x
+      by_cases hx : x ∈ D
+      · simp only [Set.mem_preimage, Set.mem_union, Set.mem_compl_iff, hx, not_true_eq_false,
+          or_false, hm x hx]
+      · simp only [Set.mem_preimage, Set.mem_union, Set.mem_compl_iff, hx, not_false_eq_true,
+          or_true, hz x hx, hzero]
+    rw [heq]
+    exact hB.union hD.compl
+  · have heq : approximateNormal E ⁻¹' s = B := by
+      ext x
+      by_cases hx : x ∈ D
+      · exact hm x hx
+      · have hb : x ∉ B := fun hb ↦ hx (hBD hb)
+        simp only [Set.mem_preimage, hz x hx, hzero, hb]
+    rw [heq]
+    exact hB
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+open RPI RPI.Boundary
+
+theorem measurable_tangentialGradient {n : ℕ} {E : Set (Euc n)}
+    (hE : MeasurableSet E) {φ : Euc n → ℝ} (hφ : ContDiff ℝ 1 φ) :
+    Measurable (tangentialGradient E φ) := by
+  have hg : Continuous (gradient φ) := continuous_iff_continuousAt.mpr (fun x ↦
+    continuousAt_gradient_of_contDiffAt hφ.contDiffAt)
+  have hv := measurable_approximateNormal hE
+  exact hg.measurable.sub ((hg.measurable.inner (𝕜 := ℝ) hv).smul hv)
+
+theorem minimizing_boundaryArea_ball_lt_top {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    (c : Euc n) {R : ℝ} (hR : 0 < R) : boundaryArea E (Metric.ball c R) < ⊤ := by
+  obtain ⟨A, _, hA⟩ := exists_minimizing_boundaryArea_growth hn hmin
+  exact (hA c R hR).trans_lt ENNReal.ofReal_lt_top
+
+theorem tangentialGradient_memLp_on_ball {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    {R : ℝ} (hR : 0 < R) {φ : Euc n → ℝ} (hφ : ContDiff ℝ 1 φ) :
+    MemLp (tangentialGradient E φ) 2 ((boundaryArea E).restrict (Metric.ball 0 R)) := by
+  let μ := (boundaryArea E).restrict (Metric.ball 0 R)
+  have : IsFiniteMeasure μ := ⟨by
+    simpa only [μ, Measure.restrict_apply_univ] using minimizing_boundaryArea_ball_lt_top hn hmin 0 hR⟩
+  have hg : Continuous (gradient φ) := continuous_iff_continuousAt.mpr (fun x ↦
+    continuousAt_gradient_of_contDiffAt hφ.contDiffAt)
+  obtain ⟨M, hM⟩ := (isCompact_closedBall (0 : Euc n) R).exists_bound_of_continuousOn hg.continuousOn
+  apply MemLp.of_bound (measurable_tangentialGradient hmin.1 hφ).aestronglyMeasurable (2 * M)
+  filter_upwards [ae_restrict_mem measurableSet_ball] with x hx
+  exact (tangentialGradient_norm_le_twice E φ x).trans (by
+    nlinarith [hM x (Metric.ball_subset_closedBall hx)])
+
+theorem tangentialGradient_mul {n : ℕ} (E : Set (Euc n))
+    {f g : Euc n → ℝ} {x : Euc n}
+    (hf : DifferentiableAt ℝ f x) (hg : DifferentiableAt ℝ g x) :
+    tangentialGradient E (fun y ↦ f y * g y) x =
+      f x • tangentialGradient E g x + g x • tangentialGradient E f x := by
+  have hgrad : gradient (fun y ↦ f y * g y) x = f x • gradient g x + g x • gradient f x := by
+    apply (InnerProductSpace.toDual ℝ (Euc n)).injective
+    simp only [map_add, map_smul, toDual_gradient]
+    exact fderiv_mul hf hg
+  simp only [tangentialGradient, hgrad, inner_add_left, real_inner_smul_left]
+  module
+
+theorem tangentialGradient_eq_zero_of_notMem_tsupport {n : ℕ} (E : Set (Euc n))
+    {φ : Euc n → ℝ} {x : Euc n} (hx : x ∉ tsupport φ) : tangentialGradient E φ x = 0 := by
+  simp only [tangentialGradient, gradient, fderiv_of_notMem_tsupport ℝ hx, map_zero,
+    inner_zero_left, zero_smul, sub_self]
+
+theorem boundaryArea_zero_of_singular_dim {n : ℕ} (hn : 8 ≤ n)
+    (E Z : Set (Euc n)) (hZd : dimH Z ≤ (n - 8 : ℕ)) : boundaryArea E Z = 0 := by
+  rw [boundaryArea_eq_smul_surfaceMeasure, Measure.smul_apply,
+    surfaceMeasure_zero_of_singular_dim hn _ _ hZd]
+  exact mul_zero _
+
+theorem wholeBoundarySupersolution_of_regular_tests {n : ℕ} (hn : 8 ≤ n)
+    {E Z : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    (hZ : IsClosed Z) (hZd : dimH Z ≤ (n - 8 : ℕ)) {G : Euc n → Euc n}
+    (hloc : ∀ R : ℝ, 0 < R → MemLp G 2 ((boundaryArea E).restrict (Metric.ball 0 R)))
+    (hreg : ∀ ψ : Euc n → ℝ, ContDiff ℝ ∞ ψ → HasCompactSupport ψ →
+      Disjoint (tsupport ψ) Z → (∀ x, 0 ≤ ψ x) →
+      0 ≤ ∫ x, inner ℝ (G x) (tangentialGradient E ψ x) ∂boundaryArea E) :
+    WholeBoundarySupersolution E G := by
+  intro φ hφ hcφ hpφ
+  obtain ⟨r, hr⟩ := Metric.isBounded_iff_subset_ball (0 : Euc n) |>.mp hcφ.isBounded
+  let R := max r 1
+  have hR : 0 < R := lt_of_lt_of_le zero_lt_one (le_max_right _ _)
+  have hsupp : tsupport φ ⊆ Metric.ball (0 : Euc n) R :=
+    hr.trans (Metric.ball_subset_ball (le_max_left _ _))
+  let μ := (boundaryArea E).restrict (Metric.ball 0 R)
+  have hn1 : 1 < n := by omega
+  have : IsFiniteMeasure μ := ⟨by
+    simpa only [μ, Measure.restrict_apply_univ] using minimizing_boundaryArea_ball_lt_top hn1 hmin 0 hR⟩
+  let K := Z ∩ tsupport φ
+  have hKc : IsCompact K := hcφ.of_isClosed_subset
+    (hZ.inter (isClosed_tsupport φ)) inter_subset_right
+  have hKd : dimH K ≤ (n - 8 : ℕ) := (dimH_mono inter_subset_left).trans hZd
+  obtain ⟨χ, hχ, hχlim, _, hχE⟩ := exists_raw_minimizing_frontier_capacity_sequence hn hmin hKc hKd
+  have hzero : ∀ᵐ x ∂μ, x ∉ K := by
+    have hKzero : μ K = 0 := le_antisymm
+      (Measure.restrict_apply_le _ _ |>.trans_eq (boundaryArea_zero_of_singular_dim hn E K hKd)) zero_le
+    simpa only [ae_iff, not_not, Set.ofPred_mem_eq] using hKzero
+  have hχlim' : ∀ᵐ x ∂μ, Tendsto (fun i ↦ χ i x) atTop (𝓝 1) := by
+    filter_upwards [hzero] with x hx
+    apply tendsto_const_nhds.congr'
+    filter_upwards [hχlim x hx] with i hi
+    exact hi.symm
+  have hχE' : Tendsto (fun i ↦ vectorEnergy μ (tangentialGradient E (χ i))) atTop (𝓝 0) := by
+    apply tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds hχE (fun _ ↦ zero_le)
+    intro i
+    exact lintegral_mono' Measure.restrict_le_self le_rfl
+  have hB : MemLp (tangentialGradient E φ) 2 μ :=
+    tangentialGradient_memLp_on_ball hn1 hmin hR (hφ.of_le (by simp))
+  have hC : ∀ i, MemLp (tangentialGradient E (χ i)) 2 μ := fun i ↦
+    tangentialGradient_memLp_on_ball hn1 hmin hR ((hχ i).1.of_le (by simp))
+  have hC0 : Tendsto (fun i ↦ (eLpNorm (tangentialGradient E (χ i)) 2 μ).toReal)
+      atTop (𝓝 0) := tendsto_eLpNorm_two_toReal_of_vectorEnergy_zero_of_aestronglyMeasurable
+        (fun i ↦ (hC i).aestronglyMeasurable) hχE'
+  have hχnorm : ∀ i, ∀ᵐ x ∂μ, ‖χ i x‖ ≤ 1 := fun i ↦ ae_of_all _ (fun x ↦ by
+    rw [Real.norm_eq_abs, abs_of_nonneg ((hχ i).2.1 x).1]
+    exact ((hχ i).2.1 x).2)
+  obtain ⟨M, hM⟩ := hcφ.exists_bound_of_continuous hφ.continuous
+  have hpair : 0 ≤ ∫ x, inner ℝ (G x) (tangentialGradient E φ x) ∂μ := by
+    apply energy_inequality_of_capacity_cutoffs (hloc R hR) hB χ
+      (fun i ↦ (hχ i).1.continuous.aestronglyMeasurable) hχnorm hχlim'
+      (fun i ↦ tangentialGradient E (χ i)) hC hC0 φ hφ.continuous.aestronglyMeasurable
+      (M := M) (ae_of_all _ hM)
+    intro i
+    have heq : (fun x ↦ χ i x • tangentialGradient E φ x + φ x • tangentialGradient E (χ i) x) =
+        tangentialGradient E (fun x ↦ φ x * χ i x) := by
+      funext x
+      rw [tangentialGradient_mul E (hφ.differentiable (by simp)).differentiableAt
+        ((hχ i).1.differentiable (by simp)).differentiableAt, add_comm]
+    simp_rw [congrFun heq]
+    rw [setIntegral_eq_integral_of_forall_compl_eq_zero]
+    · apply hreg _ (hφ.mul (hχ i).1) hcφ.mul_right
+      · exact tsupport_mul_disjoint_Singular (hχ i).2.2
+      · exact fun x ↦ mul_nonneg (hpφ x) ((hχ i).2.1 x).1
+    · intro x hx
+      rw [tangentialGradient_eq_zero_of_notMem_tsupport E
+        (fun hx' ↦ hx (hsupp (tsupport_mul_subset_left hx'))), inner_zero_right]
+  dsimp only [μ] at hpair
+  rwa [setIntegral_eq_integral_of_forall_compl_eq_zero (fun x hx ↦ by
+    rw [tangentialGradient_eq_zero_of_notMem_tsupport E (fun hx' ↦ hx (hsupp hx')),
+      inner_zero_right])] at hpair
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/ExistsMinimizingDensityLimit.lean b/lean/OAI/Geometry/Perimeter/Regularity/ExistsMinimizingDensityLimit.lean
new file mode 100644
index 0000000000000000000000000000000000000000..a81b4f84c30be6f60d969092f9e11d88007bc8f9
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/ExistsMinimizingDensityLimit.lean
@@ -0,0 +1,530 @@
+import OAI.Geometry.Perimeter.Regularity.DilatedVolumeTest
+import OAI.Geometry.Perimeter.Regularity.BlowupHomeomorph
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+open scoped ENNReal Topology
+open scoped ENNReal Topology ContDiff
+open scoped Topology ENNReal
+namespace RPISingularHarnack
+
+theorem exists_minimizing_density_limit {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hE : IsLocallyPerimeterMinimizing E) (c : Euc n) :
+    ∃ a : ℝ, 0 ≤ a ∧ Tendsto
+      (fun r : ℝ ↦ (perimeterMeasure E).real (Metric.ball c r) / r^(n-1))
+      (𝓝[>] 0) (𝓝 a) := by
+  let f (r : ℝ) := (perimeterMeasure E).real (Metric.ball c r) / r^(n-1)
+  let S : Set ℝ := f '' Ioi 0
+  have hS : S.Nonempty := ⟨f 1, 1, (by norm_num : (1:ℝ) ∈ Ioi 0), rfl⟩
+  have hnonneg : ∀ y ∈ S, 0 ≤ y := by
+    rintro y ⟨r, hr, rfl⟩
+    exact div_nonneg ENNReal.toReal_nonneg (pow_nonneg hr.le _)
+  have hb : BddBelow S := ⟨0, hnonneg⟩
+  have hmono := minimizing_perimeter_density_monotone hn hE c
+  refine ⟨sInf S, le_csInf hS hnonneg, tendsto_order.mpr ⟨?_, ?_⟩⟩
+  · intro b hb'
+    filter_upwards [self_mem_nhdsWithin] with r hr
+    exact hb'.trans_le (csInf_le hb ⟨r, hr, rfl⟩)
+  · intro b hb'
+    obtain ⟨y, ⟨R, hR, rfl⟩, hy⟩ := exists_lt_of_csInf_lt hS hb'
+    filter_upwards [self_mem_nhdsWithin,
+      (gt_mem_nhds hR : ∀ᶠ r in 𝓝 (0:ℝ), r < R).filter_mono nhdsWithin_le_nhds] with r hr hrR
+    exact (hmono hr hR hrR.le).trans_lt hy
+
+theorem minimizing_blowup_ball_real {n : ℕ} (hn : 0 < n)
+    {E : Set (Euc n)} (hE : MeasurableSet E) (c : Euc n)
+    {r R : ℝ} (hr : 0 < r) (hR : 0 < R) :
+    (perimeterMeasure (blownUpSet E c r)).real (Metric.ball 0 R) =
+      ((perimeterMeasure E).real (Metric.ball c (r*R)) / (r*R)^(n-1)) * R^(n-1) := by
+  have hs := blownUpSet_ball c (0:Euc n) hr R
+  simp only [smul_zero, add_zero] at hs
+  rw [measureReal_def, perimeterMeasure_open _ Metric.isOpen_ball,
+    ← hs, perimeter_blownUpSet_eq hE c hr, ENNReal.toReal_mul,
+    ENNReal.toReal_ofReal (by positivity), measureReal_def,
+    perimeterMeasure_open _ Metric.isOpen_ball]
+  have hn' : n = (n-1)+1 := by omega
+  have hpow : r^n = r^(n-1)*r := by conv_lhs => rw [hn', pow_succ]
+  rw [hpow]
+  field_simp [hr.ne', hR.ne']
+  rw [mul_pow]
+  ring
+
+theorem minimizing_dense_null_spheres {n : ℕ} {E : Set (Euc n)}
+    (hE : IsLocallyPerimeterMinimizing E) (c : Euc n) :
+    Dense {r : ℝ | perimeterMeasure E (Metric.sphere c r) = 0} := by
+  let := perimeterMeasure_locallyFinite E hE.2.1
+  have hc := Measure.countable_meas_level_set_pos
+    (μ := perimeterMeasure E) (g := fun x : Euc n ↦ dist x c)
+    (continuous_id.dist continuous_const).measurable
+  have hd := hc.dense_compl ℝ
+  convert hd using 1
+  ext r
+  simp only [mem_compl_iff, mem_ofPred_eq, not_lt, nonpos_iff_eq_zero, Metric.sphere]
+
+theorem minimizing_density_eq_of_null_spheres {n : ℕ} {E : Set (Euc n)}
+    (hE : IsLocallyPerimeterMinimizing E) {a : ℝ} (ha : 0 ≤ a) (k : ℕ)
+    (heq : ∀ r : ℝ, 0 < r → perimeterMeasure E (Metric.sphere 0 r) = 0 →
+      (perimeterMeasure E).real (Metric.ball 0 r) = a*r^k) :
+    ∀ r : ℝ, 0 < r → (perimeterMeasure E).real (Metric.ball 0 r) = a*r^k := by
+  let := perimeterMeasure_finiteOnCompacts E hE.2.1
+  have hd := minimizing_dense_null_spheres hE (0:Euc n)
+  have hm : Monotone (fun r : ℝ ↦ (perimeterMeasure E).real (Metric.ball 0 r)) := by
+    intro r R hrR
+    exact measureReal_mono (Metric.ball_subset_ball hrR)
+      (measure_ne_top_of_subset Metric.ball_subset_closedBall (isCompact_closedBall (0:Euc n) R).measure_ne_top)
+  have ht (r : ℝ) : Tendsto (fun R : ℝ ↦ a*R^k) (𝓝 r) (𝓝 (a*r^k)) := (continuous_const.mul (continuous_id.pow k)).tendsto r
+  intro r hr
+  apply le_antisymm
+  · apply ge_of_tendsto ((ht r).mono_left nhdsWithin_le_nhds :
+      Tendsto (fun R : ℝ ↦ a*R^k) (𝓝[>] r) (𝓝 (a*r^k)))
+    filter_upwards [self_mem_nhdsWithin] with R hR
+    obtain ⟨s, hs, hrs, hsR⟩ := hd.exists_between hR
+    have hs0 := hr.trans hrs
+    calc
+      (perimeterMeasure E).real (Metric.ball 0 r) ≤
+          (perimeterMeasure E).real (Metric.ball 0 s) := hm hrs.le
+      _ = a*s^k := heq s hs0 hs
+      _ ≤ a*R^k := mul_le_mul_of_nonneg_left (pow_le_pow_left₀ hs0.le hsR.le k) ha
+  · apply le_of_tendsto ((ht r).mono_left nhdsWithin_le_nhds :
+      Tendsto (fun R : ℝ ↦ a*R^k) (𝓝[<] r) (𝓝 (a*r^k)))
+    filter_upwards [self_mem_nhdsWithin,
+      (lt_mem_nhds hr : ∀ᶠ R in 𝓝 r, 0 < R).filter_mono nhdsWithin_le_nhds] with R hR hR0
+    obtain ⟨s, hs, hRs, hsr⟩ := hd.exists_between hR
+    calc
+      a*R^k ≤ a*s^k := mul_le_mul_of_nonneg_left (pow_le_pow_left₀ hR0.le hRs.le k) ha
+      _ = (perimeterMeasure E).real (Metric.ball 0 s) := (heq s (hR0.trans hRs) hs).symm
+      _ ≤ (perimeterMeasure E).real (Metric.ball 0 r) := hm hsr.le
+
+theorem minimizing_density_limit_pos {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hE : IsLocallyPerimeterMinimizing E)
+    {c : Euc n} (hc : c ∈ perimeterSupport E) {a : ℝ}
+    (ht : Tendsto (fun r : ℝ ↦ (perimeterMeasure E).real (Metric.ball c r) / r^(n-1))
+      (𝓝[>] 0) (𝓝 a)) : 0 < a := by
+  obtain ⟨b, hb, hbound⟩ := exists_minimizing_perimeter_lower_density n hn
+  apply hb.trans_le
+  apply ge_of_tendsto ht
+  filter_upwards [self_mem_nhdsWithin] with r hr
+  apply (le_div_iff₀ (pow_pos hr _)).mpr
+  rw [measureReal_def, perimeterMeasure_open E Metric.isOpen_ball]
+  exact (ENNReal.ofReal_le_iff_le_toReal (hE.2.1 c r hr).ne).mp (hbound E hE c hc r hr)
+
+theorem minimizing_blowup_limit_constant_density {n : ℕ} (hn : 1 < n)
+    {E F : Set (Euc n)} (hE : IsLocallyPerimeterMinimizing E)
+    (hF : IsLocallyPerimeterMinimizing F) (c : Euc n)
+    (r : ℕ → ℝ) (hr : ∀ i, 0 < r i) (htr : Tendsto r atTop (𝓝 0))
+    (hconv : ∀ (x : Euc n) (R : ℝ), 0 < R →
+      Tendsto (fun i ↦ volume.real (symmDiff (blownUpSet E c (r i)) F ∩ Metric.closedBall x R))
+        atTop (𝓝 0))
+    {a : ℝ} (ha : 0 ≤ a)
+    (hden : Tendsto (fun R : ℝ ↦ (perimeterMeasure E).real (Metric.ball c R) / R^(n-1))
+      (𝓝[>] 0) (𝓝 a)) :
+    ∀ R : ℝ, 0 < R → (perimeterMeasure F).real (Metric.ball 0 R) = a*R^(n-1) := by
+  apply minimizing_density_eq_of_null_spheres hF ha (n-1)
+  intro R hR hnull
+  have htR : Tendsto (fun i ↦ r i * R) atTop (𝓝[>] (0:ℝ)) := by
+    apply tendsto_nhdsWithin_iff.mpr
+    exact ⟨by simpa using htr.mul_const R,
+      Eventually.of_forall fun i ↦ mul_pos (hr i) hR⟩
+  have htden : Tendsto (fun i ↦ (perimeterMeasure (blownUpSet E c (r i))).real (Metric.ball 0 R))
+      atTop (𝓝 (a*R^(n-1))) := by
+    simpa only [minimizing_blowup_ball_real (by omega : 0 < n) hE.1 c (hr _) hR, Function.comp_def] using
+      (hden.comp htR).mul_const (R^(n-1))
+  have htp := tendsto_minimizing_perimeter_ball_of_local_L1 (by omega : 0 < n)
+    (fun i ↦ locally_minimizing_blownUpSet hE c (hr i)) hF.1 hconv (0:Euc n) hR hnull
+  have htF := (ENNReal.tendsto_toReal (hF.2.1 0 R hR).ne).comp htp
+  have htF' : Tendsto (fun i ↦ (perimeterMeasure (blownUpSet E c (r i))).real (Metric.ball 0 R))
+      atTop (𝓝 ((perimeterMeasure F).real (Metric.ball 0 R))) := by
+    simpa only [measureReal_def, perimeterMeasure_open _ Metric.isOpen_ball, Function.comp_def] using htF
+  exact tendsto_nhds_unique htF' htden
+
+theorem minimizing_tangent_cone_subsequence {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hE : IsLocallyPerimeterMinimizing E)
+    {c : Euc n} (hc : c ∈ perimeterSupport E)
+    (r : ℕ → ℝ) (hr : ∀ i, 0 < r i) (htr : Tendsto r atTop (𝓝 0)) :
+    ∃ K : Set (Euc n), IsClosed K ∧ IsLocallyPerimeterMinimizing K ∧ IsCone K ∧
+      K = (volume.restrict K).support ∧ frontier K = perimeterSupport K ∧ (0:Euc n) ∈ perimeterSupport K ∧
+      ∃ σ : ℕ → ℕ, StrictMono σ ∧
+        (∀ (x : Euc n) (R : ℝ), 0 < R →
+          Tendsto (fun i ↦ volume.real (symmDiff (blownUpSet E c (r (σ i))) K ∩ Metric.closedBall x R))
+            atTop (𝓝 0)) ∧
+        (∀ R : ℝ, 0 < R → perimeterMeasure K (Metric.sphere 0 R) = 0 →
+          Tendsto (fun i ↦ perimeter (blownUpSet E c (r (σ i))) (Metric.ball 0 R)) atTop
+            (𝓝 (perimeter K (Metric.ball 0 R)))) := by
+  obtain ⟨F, hF, σ, hσ, hconv, hP⟩ := minimizing_global_compactness (by omega : 0 < n)
+    (fun i ↦ blownUpSet E c (r i)) (fun i ↦ locally_minimizing_blownUpSet hE c (hr i))
+  obtain ⟨a, ha, hden⟩ := exists_minimizing_density_limit hn hE c
+  have ha0 := minimizing_density_limit_pos hn hE hc hden
+  have hdensity := minimizing_blowup_limit_constant_density hn hE hF c
+    (fun i ↦ r (σ i)) (fun i ↦ hr (σ i)) (htr.comp hσ.tendsto_atTop) hconv ha hden
+  let K := (volume.restrict F).support
+  obtain ⟨hK, hFK, hminK, hconeK, hfrontK⟩ :=
+    minimizing_constant_density_conical_representative hn hF ha hdensity
+  have hlocal : ∀ (x : Euc n) (R : ℝ), 0 < R →
+      Tendsto (fun i ↦ volume.real (symmDiff (blownUpSet E c (r (σ i))) K ∩ Metric.closedBall x R))
+        atTop (𝓝 0) := by
+    intro x R hR
+    convert hconv x R hR using 1
+    funext i
+    rw [measureReal_def, measureReal_def]
+    congr 1
+    apply measure_congr
+    filter_upwards [hFK] with y hy
+    simp only [mem_inter_iff, mem_symmDiff]
+    change (y ∈ F) = (y ∈ K) at hy
+    rw [hy]
+  have hcanonical : K = (volume.restrict K).support := by
+    change (volume.restrict F).support = (volume.restrict K).support
+    rw [Measure.restrict_congr_set hFK]
+  refine ⟨K, hK, hminK, hconeK, hcanonical, hfrontK.trans (perimeterSupport_eq_of_ae hFK), ?_,
+    σ, hσ, hlocal, ?_⟩
+  · intro R hR
+    rw [← perimeter_eq_of_ae hFK]
+    have hh : 0 < (perimeter F (Metric.ball 0 R)).toReal := by
+      rw [← perimeterMeasure_open F Metric.isOpen_ball]
+      change 0 < (perimeterMeasure F).real (Metric.ball 0 R)
+      rw [hdensity R hR]
+      positivity
+    exact (ENNReal.toReal_pos_iff.mp hh).1
+  · intro R hR hnull
+    exact tendsto_minimizing_perimeter_ball_of_local_L1 (by omega : 0 < n)
+      (fun i ↦ locally_minimizing_blownUpSet hE c (hr (σ i))) hminK.1 hlocal (0:Euc n) hR hnull
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+open scoped ENNReal Topology
+open scoped ENNReal Topology ContDiff
+open scoped Topology ENNReal
+namespace RPISingularHarnack
+
+theorem ae_subset_of_local_L1_limits {n : ℕ}
+    (E F : ℕ → Set (Euc n)) (K L : Set (Euc n)) (hsub : ∀ i, E i ⊆ F i)
+    (hE : ∀ (x : Euc n) (R : ℝ), 0 < R →
+      Tendsto (fun i ↦ volume.real (symmDiff (E i) K ∩ Metric.closedBall x R)) atTop (𝓝 0))
+    (hF : ∀ (x : Euc n) (R : ℝ), 0 < R →
+      Tendsto (fun i ↦ volume.real (symmDiff (F i) L ∩ Metric.closedBall x R)) atTop (𝓝 0)) :
+    K ≤ᵐ[volume] L := by
+  have hENN (A : ℕ → Set (Euc n)) (B : Set (Euc n)) (x : Euc n) (R : ℝ)
+      (ht : Tendsto (fun i ↦ volume.real (symmDiff (A i) B ∩ Metric.closedBall x R)) atTop (𝓝 0)) :
+      Tendsto (fun i ↦ volume (symmDiff (A i) B ∩ Metric.closedBall x R)) atTop (𝓝 0) := by
+    have h := ENNReal.continuous_ofReal.tendsto (0:ℝ) |>.comp ht
+    simp only [Function.comp_def, measureReal_def, ENNReal.ofReal_zero,
+      ENNReal.ofReal_toReal (measure_ne_top_of_subset inter_subset_right
+        (isCompact_closedBall x R).measure_ne_top)] at h
+    exact h
+  have hz (x : Euc n) (R : ℝ) (hR : 0 < R) : volume ((K \ L) ∩ Metric.closedBall x R) = 0 := by
+    apply le_antisymm _ bot_le
+    have ht := (hENN E K x R (hE x R hR)).add (hENN F L x R (hF x R hR))
+    simp only [zero_add] at ht
+    apply ge_of_tendsto ht
+    exact Eventually.of_forall fun i ↦ by
+      apply (measure_mono (show (K \ L) ∩ Metric.closedBall x R ⊆
+        (symmDiff (E i) K ∩ Metric.closedBall x R) ∪
+          (symmDiff (F i) L ∩ Metric.closedBall x R) from ?_)).trans
+          (measure_union_le _ _)
+      intro y hy
+      by_cases hyE : y ∈ E i
+      · exact Or.inr ⟨Or.inl ⟨hsub i hyE, hy.1.2⟩, hy.2⟩
+      · exact Or.inl ⟨Or.inr ⟨hy.1.1, hyE⟩, hy.2⟩
+  apply ae_le_set.mpr
+  have hs : K \ L ⊆ ⋃ m : ℕ, (K \ L) ∩ Metric.closedBall 0 ((m:ℝ)+1) := by
+    intro x hx
+    obtain ⟨m, hm⟩ := exists_nat_gt (dist x (0:Euc n))
+    exact mem_iUnion.mpr ⟨m, hx, Metric.mem_closedBall.mpr (by linarith)⟩
+  exact le_antisymm ((measure_mono hs).trans (le_of_eq (measure_iUnion_null fun m ↦
+    hz 0 ((m:ℝ)+1) (by positivity)))) bot_le
+
+theorem canonical_subset_of_ae {n : ℕ} {K L : Set (Euc n)}
+    (hK : K = (volume.restrict K).support) (hL : L = (volume.restrict L).support)
+    (hKL : K ≤ᵐ[volume] L) : K ⊆ L := by
+  rw [hK, hL]
+  exact Measure.support_mono (Measure.restrict_mono_ae hKL)
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+open scoped ENNReal Topology
+open scoped ENNReal Topology ContDiff
+open scoped Topology ENNReal
+namespace RPISingularHarnack
+
+theorem nested_minimizing_tangent_cones {n : ℕ} (hn : 1 < n)
+    {E F : Set (Euc n)} (hE : IsLocallyPerimeterMinimizing E)
+    (hF : IsLocallyPerimeterMinimizing F) (hEF : E ⊆ F)
+    {c : Euc n} (hcE : c ∈ perimeterSupport E) (hcF : c ∈ perimeterSupport F)
+    (r : ℕ → ℝ) (hr : ∀ i, 0 < r i) (htr : Tendsto r atTop (𝓝 0)) :
+    ∃ K L : Set (Euc n),
+      IsClosed K ∧ IsClosed L ∧ IsLocallyPerimeterMinimizing K ∧
+      IsLocallyPerimeterMinimizing L ∧ IsCone K ∧ IsCone L ∧ K ⊆ L ∧
+      K = (volume.restrict K).support ∧ L = (volume.restrict L).support ∧
+      frontier K = perimeterSupport K ∧ frontier L = perimeterSupport L ∧
+      (0:Euc n) ∈ perimeterSupport K ∧ (0:Euc n) ∈ perimeterSupport L ∧
+      ∃ σ : ℕ → ℕ, StrictMono σ ∧
+        (∀ (x : Euc n) (R : ℝ), 0 < R →
+          Tendsto (fun i ↦ volume.real (symmDiff (blownUpSet E c (r (σ i))) K ∩ Metric.closedBall x R))
+            atTop (𝓝 0)) ∧
+        (∀ (x : Euc n) (R : ℝ), 0 < R →
+          Tendsto (fun i ↦ volume.real (symmDiff (blownUpSet F c (r (σ i))) L ∩ Metric.closedBall x R))
+            atTop (𝓝 0)) ∧
+        (∀ R : ℝ, 0 < R → perimeterMeasure K (Metric.sphere 0 R) = 0 →
+          Tendsto (fun i ↦ perimeter (blownUpSet E c (r (σ i))) (Metric.ball 0 R)) atTop
+            (𝓝 (perimeter K (Metric.ball 0 R)))) ∧
+        (∀ R : ℝ, 0 < R → perimeterMeasure L (Metric.sphere 0 R) = 0 →
+          Tendsto (fun i ↦ perimeter (blownUpSet F c (r (σ i))) (Metric.ball 0 R)) atTop
+            (𝓝 (perimeter L (Metric.ball 0 R)))) := by
+  obtain ⟨K, hKc, hKm, hKcone, hKcan, hKf, hK0, σ, hσ, hEK, hPK⟩ :=
+    minimizing_tangent_cone_subsequence hn hE hcE r hr htr
+  obtain ⟨L, hLc, hLm, hLcone, hLcan, hLf, hL0, τ, hτ, hFL, hPL⟩ :=
+    minimizing_tangent_cone_subsequence hn hF hcF (fun i ↦ r (σ i))
+      (fun i ↦ hr (σ i)) (htr.comp hσ.tendsto_atTop)
+  have hEK' : ∀ (x : Euc n) (R : ℝ), 0 < R →
+      Tendsto (fun i ↦ volume.real (symmDiff (blownUpSet E c (r (σ (τ i)))) K ∩ Metric.closedBall x R))
+        atTop (𝓝 0) := by
+    intro x R hR
+    exact (hEK x R hR).comp hτ.tendsto_atTop
+  have hKL : K ⊆ L := canonical_subset_of_ae hKcan hLcan
+    (ae_subset_of_local_L1_limits
+      (fun i ↦ blownUpSet E c (r (σ (τ i))))
+      (fun i ↦ blownUpSet F c (r (σ (τ i)))) K L
+      (fun i x hx ↦ hEF hx) hEK' hFL)
+  refine ⟨K, L, hKc, hLc, hKm, hLm, hKcone, hLcone, hKL, hKcan, hLcan,
+    hKf, hLf, hK0, hL0, σ ∘ τ, hσ.comp hτ, hEK', hFL, ?_, hPL⟩
+  intro R hR hnull
+  exact (hPK R hR hnull).comp hτ.tendsto_atTop
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+open scoped ENNReal Topology
+open scoped ENNReal Topology ContDiff
+open scoped Topology ENNReal
+namespace RPISingularHarnack
+
+theorem minimizing_support_limit_of_local_L1 {n : ℕ} (hn : 1 < n)
+    (E : ℕ → Set (Euc n)) {F : Set (Euc n)}
+    (hE : ∀ i, IsLocallyPerimeterMinimizing (E i)) (hF : IsLocallyPerimeterMinimizing F)
+    (hconv : ∀ (x : Euc n) (R : ℝ), 0 < R →
+      Tendsto (fun i ↦ volume.real (symmDiff (E i) F ∩ Metric.closedBall x R)) atTop (𝓝 0))
+    (x : ℕ → Euc n) {y : Euc n} (hx : ∀ i, x i ∈ perimeterSupport (E i))
+    (htx : Tendsto x atTop (𝓝 y)) : y ∈ perimeterSupport F := by
+  by_contra hy
+  change ¬ ∀ R : ℝ, 0 < R → 0 < perimeter F (Metric.ball y R) at hy
+  push Not at hy
+  obtain ⟨R, hR, hzero⟩ := hy
+  have hzero' : perimeter F (Metric.ball y R) = 0 := bot_unique hzero
+  obtain ⟨s, hs, hs0, hsR⟩ := (minimizing_dense_null_spheres hF y).exists_between hR
+  have hsP : perimeter F (Metric.ball y s) = 0 := by
+    apply bot_unique
+    exact (perimeter_mono_region F (Metric.ball_subset_ball hsR.le)).trans hzero
+  have htP := tendsto_minimizing_perimeter_ball_of_local_L1 (by omega : 0 < n)
+    hE hF.1 hconv y hs0 hs
+  rw [hsP] at htP
+  obtain ⟨b, hb, hbound⟩ := exists_minimizing_perimeter_lower_density n hn
+  have hevent : ∀ᶠ i in atTop, ENNReal.ofReal (b * (s/2)^(n-1)) ≤ perimeter (E i) (Metric.ball y s) := by
+    filter_upwards [htx.eventually (Metric.ball_mem_nhds y (by positivity : 0 < s/2))] with i hi
+    apply (hbound (E i) (hE i) (x i) (hx i) (s/2) (by positivity)).trans
+    apply perimeter_mono_region
+    intro z hz
+    apply Metric.mem_ball.mpr
+    have htri := dist_triangle z (x i) y
+    have hz' := Metric.mem_ball.mp hz
+    have hi' := Metric.mem_ball.mp hi
+    linarith
+  have hle := ge_of_tendsto htP hevent
+  have hpos : 0 < ENNReal.ofReal (b * (s/2)^(n-1)) := ENNReal.ofReal_pos.mpr (by positivity)
+  exact (not_le_of_gt hpos) hle
+
+theorem minimizing_support_eventually_hits_of_local_L1 {n : ℕ} (hn : 0 < n)
+    (E : ℕ → Set (Euc n)) {F : Set (Euc n)}
+    (hE : ∀ i, IsLocallyPerimeterMinimizing (E i)) (hF : IsLocallyPerimeterMinimizing F)
+    (hconv : ∀ (x : Euc n) (R : ℝ), 0 < R →
+      Tendsto (fun i ↦ volume.real (symmDiff (E i) F ∩ Metric.closedBall x R)) atTop (𝓝 0))
+    {x : Euc n} (hx : x ∈ perimeterSupport F) {ε : ℝ} (hε : 0 < ε) :
+    ∀ᶠ i in atTop, ∃ y ∈ perimeterSupport (E i), dist y x < ε := by
+  obtain ⟨s, hs, hs0, hsε⟩ := (minimizing_dense_null_spheres hF x).exists_between hε
+  have htP := tendsto_minimizing_perimeter_ball_of_local_L1 hn hE hF.1 hconv x hs0 hs
+  filter_upwards [htP.eventually (lt_mem_nhds (hx s hs0))] with i hi
+  have hi' : perimeterMeasure (E i) (Metric.ball x s) ≠ 0 := by
+    rw [perimeterMeasure_open _ Metric.isOpen_ball]
+    exact hi.ne'
+  have hsupport : ∀ᵐ y ∂perimeterMeasure (E i), y ∈ perimeterSupport (E i) := by
+    rw [← perimeterMeasure_support]; exact (perimeterMeasure (E i)).support_mem_ae
+  obtain ⟨y, hy, hyS⟩ := Measure.exists_mem_of_measure_ne_zero_of_ae hi' (ae_restrict_of_ae hsupport)
+  exact ⟨y, hyS, (Metric.mem_ball.mp hy).trans hsε⟩
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+open scoped ENNReal Topology
+open scoped ENNReal Topology ContDiff
+open scoped Topology ENNReal
+namespace RPISingularHarnack
+
+theorem minimizing_supports_locally_converge {n : ℕ} (hn : 1 < n)
+    (E : ℕ → Set (Euc n)) {F : Set (Euc n)}
+    (hE : ∀ i, IsLocallyPerimeterMinimizing (E i)) (hF : IsLocallyPerimeterMinimizing F)
+    (hconv : ∀ (x : Euc n) (R : ℝ), 0 < R →
+      Tendsto (fun i ↦ volume.real (symmDiff (E i) F ∩ Metric.closedBall x R)) atTop (𝓝 0))
+    {C : Set (Euc n)} (hC : IsCompact C) {ε : ℝ} (hε : 0 < ε) :
+    ∀ᶠ i in atTop,
+      (∀ x ∈ perimeterSupport (E i) ∩ C, ∃ y ∈ perimeterSupport F, dist x y < ε) ∧
+      (∀ y ∈ perimeterSupport F ∩ C, ∃ x ∈ perimeterSupport (E i), dist x y < ε) := by
+  classical
+  have hupper : ∀ᶠ i in atTop,
+      ∀ x ∈ perimeterSupport (E i) ∩ C, ∃ y ∈ perimeterSupport F, dist x y < ε := by
+    by_contra h
+    have hf : ∃ᶠ i in atTop, ∃ x ∈ perimeterSupport (E i) ∩ C,
+        ∀ y ∈ perimeterSupport F, ε ≤ dist x y := by
+      simpa only [not_eventually, not_forall, not_exists, not_and, not_lt, exists_prop] using h
+    obtain ⟨σ, hσ, hσbad⟩ := extraction_of_frequently_atTop hf
+    choose x hx hbad using hσbad
+    obtain ⟨y, hyC, τ, hτ, hy⟩ := hC.tendsto_subseq (fun i ↦ (hx i).2)
+    have hyS : y ∈ perimeterSupport F := minimizing_support_limit_of_local_L1 hn
+      (fun i ↦ E (σ (τ i))) (fun i ↦ hE (σ (τ i))) hF
+      (fun z R hR ↦ (hconv z R hR).comp (hσ.comp hτ).tendsto_atTop)
+      (fun i ↦ x (τ i)) (fun i ↦ (hx (τ i)).1) hy
+    have ht : ∀ᶠ i in atTop, dist (x (τ i)) y < ε :=
+      hy.eventually (Metric.ball_mem_nhds y hε)
+    obtain ⟨i, hi⟩ := ht.exists
+    exact (not_lt_of_ge (hbad (τ i) y hyS)) hi
+  have hlower : ∀ᶠ i in atTop,
+      ∀ y ∈ perimeterSupport F ∩ C, ∃ x ∈ perimeterSupport (E i), dist x y < ε := by
+    by_contra h
+    have hf : ∃ᶠ i in atTop, ∃ y ∈ perimeterSupport F ∩ C,
+        ∀ x ∈ perimeterSupport (E i), ε ≤ dist x y := by
+      simpa only [not_eventually, not_forall, not_exists, not_and, not_lt, exists_prop] using h
+    obtain ⟨σ, hσ, hσbad⟩ := extraction_of_frequently_atTop hf
+    choose y hy hbad using hσbad
+    obtain ⟨z, hzC, τ, hτ, hz⟩ := hC.tendsto_subseq (fun i ↦ (hy i).2)
+    have hzS : z ∈ perimeterSupport F := (isClosed_perimeterSupport F).mem_of_tendsto hz
+      (Eventually.of_forall fun i ↦ (hy (τ i)).1)
+    have hhits := minimizing_support_eventually_hits_of_local_L1 (by omega : 0 < n)
+      (fun i ↦ E (σ (τ i))) (fun i ↦ hE (σ (τ i))) hF
+      (fun x R hR ↦ (hconv x R hR).comp (hσ.comp hτ).tendsto_atTop) hzS (by positivity : 0 < ε/2)
+    have hnear : ∀ᶠ i in atTop, dist (y (τ i)) z < ε/2 :=
+      hz.eventually (Metric.ball_mem_nhds z (by positivity))
+    obtain ⟨i, hi, hni⟩ := (hhits.and hnear).exists
+    obtain ⟨x, hxS, hxz⟩ := hi
+    have htri := dist_triangle x z (y (τ i))
+    rw [dist_comm z (y (τ i))] at htri
+    have hb := hbad (τ i) x hxS
+    linarith
+  exact hupper.and hlower
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+open scoped ENNReal Topology
+open scoped ENNReal Topology ContDiff
+open scoped Topology ENNReal
+namespace RPISingularHarnack
+
+theorem perimeterSupport_blownUpSet {n : ℕ} {E : Set (Euc n)}
+    (hE : MeasurableSet E) (c : Euc n) {r : ℝ} (hr : 0 < r) (x : Euc n) :
+    x ∈ perimeterSupport (blownUpSet E c r) ↔ c + r • x ∈ perimeterSupport E := by
+  have hpos : 0 < ENNReal.ofReal (r * (r^n)⁻¹) := ENNReal.ofReal_pos.mpr (by positivity)
+  constructor
+  · intro hx R hR
+    have h := hx (R/r) (div_pos hR hr)
+    rw [← blownUpSet_ball c x hr, perimeter_blownUpSet_eq hE c hr] at h
+    simp only [mul_div_cancel₀ _ hr.ne'] at h
+    exact (ENNReal.mul_pos_iff.mp h).2
+  · intro hx R hR
+    rw [← blownUpSet_ball c x hr, perimeter_blownUpSet_eq hE c hr]
+    exact ENNReal.mul_pos hpos.ne' (hx (r*R) (mul_pos hr hR)).ne'
+
+theorem cone_blowup_support_affine_mem {n : ℕ} {E : Set (Euc n)}
+    (hE : MeasurableSet E) (hcone : IsCone E) (c : Euc n) {r t : ℝ}
+    (hr : 0 < r) (ht : 0 < 1+r*t) {x : Euc n}
+    (hx : x ∈ perimeterSupport (blownUpSet E c r)) :
+    (1+r*t) • x + t • c ∈ perimeterSupport (blownUpSet E c r) := by
+  rw [perimeterSupport_blownUpSet hE c hr] at hx ⊢
+  have h := (isCone_perimeterSupport hE hcone (1+r*t) ht (c+r • x)).mpr hx
+  convert h using 1
+  module
+
+theorem minimizing_cone_tangent_support_line_invariant {n : ℕ} (hn : 1 < n)
+    {E F : Set (Euc n)} (hE : IsLocallyPerimeterMinimizing E)
+    (hcone : IsCone E) (hF : IsLocallyPerimeterMinimizing F) (c : Euc n)
+    (r : ℕ → ℝ) (hr : ∀ i, 0 < r i) (htr : Tendsto r atTop (𝓝 0))
+    (hconv : ∀ (x : Euc n) (R : ℝ), 0 < R →
+      Tendsto (fun i ↦ volume.real (symmDiff (blownUpSet E c (r i)) F ∩ Metric.closedBall x R))
+        atTop (𝓝 0)) :
+    ∀ (t : ℝ) (x : Euc n), x ∈ perimeterSupport F → x + t • c ∈ perimeterSupport F := by
+  intro t x hx
+  have hmin (i : ℕ) := locally_minimizing_blownUpSet hE c (hr i)
+  have hpositive : ∀ᶠ i in atTop, 0 < 1+r i*t := by
+    have ht : Tendsto (fun i ↦ 1+r i*t) atTop (𝓝 (1:ℝ)) := by
+      simpa using tendsto_const_nhds.add (htr.mul_const t)
+    exact ht.eventually (lt_mem_nhds zero_lt_one)
+  have hhits (j : ℕ) : ∀ᶠ i in atTop,
+      (∃ y ∈ perimeterSupport (blownUpSet E c (r i)), dist y x < (1/2:ℝ)^j) ∧ 0 < 1+r i*t :=
+    (minimizing_support_eventually_hits_of_local_L1 (by omega : 0 < n)
+      (fun i ↦ blownUpSet E c (r i)) hmin hF hconv hx (by positivity)).and hpositive
+  obtain ⟨σ, hσ, hσhit⟩ := extraction_forall_of_eventually hhits
+  choose y hyS hynear using fun i ↦ (hσhit i).1
+  have hty : Tendsto y atTop (𝓝 x) := by
+    apply tendsto_iff_dist_tendsto_zero.mpr
+    apply squeeze_zero (fun _ ↦ dist_nonneg) (fun i ↦ (hynear i).le)
+    exact tendsto_pow_atTop_nhds_zero_of_lt_one (by norm_num) (by norm_num)
+  have htr' : Tendsto (fun i ↦ 1+r (σ i)*t) atTop (𝓝 (1:ℝ)) := by
+    simpa using tendsto_const_nhds.add ((htr.comp hσ.tendsto_atTop).mul_const t)
+  have hty' : Tendsto (fun i ↦ (1+r (σ i)*t) • y i + t • c) atTop (𝓝 (x+t • c)) := by
+    simpa using (htr'.smul hty).add_const (t • c)
+  exact minimizing_support_limit_of_local_L1 hn (fun i ↦ blownUpSet E c (r (σ i)))
+    (fun i ↦ hmin (σ i)) hF (fun z R hR ↦ (hconv z R hR).comp hσ.tendsto_atTop)
+    (fun i ↦ (1+r (σ i)*t) • y i + t • c)
+    (fun i ↦ cone_blowup_support_affine_mem hE.1 hcone c (hr (σ i)) (hσhit i).2 (hyS i)) hty'
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/ExistsTangentialGradientBallBound.lean b/lean/OAI/Geometry/Perimeter/Regularity/ExistsTangentialGradientBallBound.lean
new file mode 100644
index 0000000000000000000000000000000000000000..99175a1a9f53b34a67d9d894b0afeb9a3a3c212f
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/ExistsTangentialGradientBallBound.lean
@@ -0,0 +1,511 @@
+import OAI.Geometry.Perimeter.Regularity.RankOneNormalError
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+open scoped ENNReal Topology
+namespace RPISingularHarnack
+open RPI.Boundary
+
+theorem exists_tangentialGradient_ball_bound {n : ℕ} (E : Set (Euc n))
+    (c : Euc n) (R : ℝ) {f : Euc n → ℝ} (hf : ContDiff ℝ 1 f) :
+    ∃ B : ℝ, ∀ x ∈ Metric.ball c R, ‖tangentialGradient E f x‖ ≤ B := by
+  have hg : Continuous (gradient f) := continuous_iff_continuousAt.mpr (fun x ↦
+    continuousAt_gradient_of_contDiffAt hf.contDiffAt)
+  obtain ⟨M, hM⟩ := (isCompact_closedBall c R).exists_bound_of_continuousOn hg.continuousOn
+  exact ⟨2 * M, fun x hx ↦ (tangentialGradient_norm_le_twice E f x).trans
+    (by nlinarith [hM x (Metric.ball_subset_closedBall hx)])⟩
+
+theorem tangentialDivergence_integrable_ball_perimeter {n : ℕ}
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    (c : Euc n) {R : ℝ} (hR : 0 < R) {V : Euc n → Euc n}
+    (hV : ContDiff ℝ ∞ V) :
+    Integrable (tangentialDivergence E V) ((perimeterMeasure E).restrict (Metric.ball c R)) := by
+  change Integrable (fun x ↦ tangentialDivergence E V x) _
+  simp_rw [tangentialDivergence_eq_sum E (hV.differentiable (by simp))]
+  apply integrable_finsetSum
+  intro i _
+  exact (tangentialGradient_integrable_ball_perimeter hmin c hR
+    ((contDiff_const.inner ℝ hV).of_le (by simp))).inner_const _
+
+theorem minimizing_radial_integral_bound {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    (c a : Euc n) {R ε : ℝ} (hR : 0 < R) (hε : 0 < ε)
+    {f : Euc n → ℝ} (hf : ContDiff ℝ ∞ f) (hcf : HasCompactSupport f)
+    (hsf : tsupport f ⊆ Metric.ball c R) (hf0 : ∀ x, 0 ≤ f x) :
+    (∫ x in Metric.ball c R, f x * (n - 1 : ℕ) *
+      (‖x - a‖ ^ 2 + ε ^ 2) ^ (-((n - 1 : ℕ) : ℝ) / 2 - 1) * ε ^ 2 ∂perimeterMeasure E) ≤
+    ∫ x in Metric.ball c R, ‖tangentialGradient E f x‖ *
+      (radialKernel a ε (n - 1 : ℕ) x * ‖x - a‖) ∂perimeterMeasure E := by
+  let μ := (perimeterMeasure E).restrict (Metric.ball c R)
+  let W : Euc n → Euc n := fun x ↦ radialKernel a ε (n - 1 : ℕ) x • (x - a)
+  have hW : ContDiff ℝ ∞ W := (radialKernel_contDiff a hε _).smul (contDiff_id.sub contDiff_const)
+  have hV : ContDiff ℝ ∞ (fun x ↦ f x • W x) := hf.smul hW
+  have hcV : HasCompactSupport (fun x ↦ f x • W x) := hcf.smul_right
+  have hsV : tsupport (fun x ↦ f x • W x) ⊆ Metric.ball c R :=
+    (tsupport_smul_subset_left _ _).trans hsf
+  have hz := minimizing_tangentialDivergence_integral_eq_zero hn hmin c hR hV hcV hsV
+  have := perimeterMeasure_finiteOnCompacts E hmin.2.1
+  have : IsFiniteMeasure μ := ⟨by
+    rw [Measure.restrict_apply_univ, perimeterMeasure_open E Metric.isOpen_ball]
+    exact hmin.2.1 c R hR⟩
+  have hiW : Integrable W μ := hW.continuous.continuousOn.integrableOn_of_subset_isCompact
+    (isCompact_closedBall c R) measurableSet_ball Metric.ball_subset_closedBall
+    (by rw [perimeterMeasure_open E Metric.isOpen_ball]; exact (hmin.2.1 c R hR).ne)
+  have hmG := (measurable_tangentialGradient hmin.1 (hf.of_le (by simp))).aestronglyMeasurable (μ := μ)
+  obtain ⟨B, hB⟩ := exists_tangentialGradient_ball_bound E c R (hf.of_le (by simp))
+  have hGi : Integrable (fun x ↦ inner ℝ (tangentialGradient E f x) (W x)) μ := by
+    apply Integrable.mono' (hiW.norm.const_mul B) (hmG.inner hW.continuous.aestronglyMeasurable)
+    filter_upwards [ae_restrict_mem measurableSet_ball] with x hx
+    exact (norm_inner_le_norm _ _).trans (mul_le_mul_of_nonneg_right (hB x hx) (norm_nonneg _))
+  obtain ⟨D, hD⟩ := (isCompact_closedBall c R).exists_bound_of_continuousOn hf.continuous.continuousOn
+  have hDi := tangentialDivergence_integrable_ball_perimeter hmin c hR hW
+  have hfi : Integrable (fun x ↦ f x * tangentialDivergence E W x) μ :=
+    hDi.bdd_mul hf.continuous.aestronglyMeasurable
+      ((ae_restrict_mem measurableSet_ball).mono fun x hx ↦ hD x (Metric.ball_subset_closedBall hx))
+  have heq : (∫ x, f x * tangentialDivergence E W x ∂μ) +
+      (∫ x, inner ℝ (tangentialGradient E f x) (W x) ∂μ) = 0 := by
+    rw [← integral_add hfi hGi]
+    simpa only [μ, ← tangentialDivergence_smul E (hf.differentiable (by simp)) (hW.differentiable (by simp))] using hz
+  have hlci : Integrable (fun x ↦ f x * (n - 1 : ℕ) *
+      (‖x - a‖ ^ 2 + ε ^ 2) ^ (-((n - 1 : ℕ) : ℝ) / 2 - 1) * ε ^ 2) μ := by
+    apply ContinuousOn.integrableOn_of_subset_isCompact (s := Metric.ball c R)
+      (K := Metric.closedBall c R) _ (isCompact_closedBall c R) measurableSet_ball Metric.ball_subset_closedBall
+      (by rw [perimeterMeasure_open E Metric.isOpen_ball]; exact (hmin.2.1 c R hR).ne)
+    apply Continuous.continuousOn
+    apply Continuous.mul _ continuous_const
+    apply Continuous.mul (hf.continuous.mul continuous_const)
+    exact (((continuous_id.sub continuous_const).norm.pow 2).add continuous_const).rpow_const
+      (fun x ↦ Or.inl (by change ‖x - a‖ ^ 2 + ε ^ 2 ≠ 0; positivity))
+  have hlow := integral_mono_ae hlci hfi
+    ((ae_restrict_of_ae (minimizing_halfspaceBlowup_ae hn hmin)).mono fun x hx ↦ by
+      dsimp [W]
+      rw [tangentialDivergence_radial E (by omega) a x hε hx.1]
+      have hpos : 0 ≤ (‖x - a‖ ^ 2 + ε ^ 2) ^ (-((n - 1 : ℕ) : ℝ) / 2 - 1) := by positivity
+      nlinarith [mul_nonneg (mul_nonneg (hf0 x) (Nat.cast_nonneg (n - 1))) hpos,
+        sq_nonneg (inner ℝ (x - a) (approximateNormal E x))])
+  have hRhi : Integrable (fun x ↦ ‖tangentialGradient E f x‖ * ‖W x‖) μ :=
+    hiW.norm.bdd_mul hmG.norm ((ae_restrict_mem measurableSet_ball).mono fun x hx ↦ by
+      simpa only [norm_norm] using hB x hx)
+  have hu := integral_mono hGi.neg hRhi (fun x ↦ (neg_le_abs _).trans (abs_real_inner_le_norm _ _))
+  have hconcl : (∫ x, f x * (n - 1 : ℕ) *
+      (‖x - a‖ ^ 2 + ε ^ 2) ^ (-((n - 1 : ℕ) : ℝ) / 2 - 1) * ε ^ 2 ∂μ) ≤
+      ∫ x, ‖tangentialGradient E f x‖ * ‖W x‖ ∂μ := by
+    simp only [Pi.neg_apply, integral_neg] at hu
+    linarith
+  simpa only [W, norm_smul, Real.norm_of_nonneg (radialKernel_pos a _ hε _).le] using hconcl
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+open scoped ENNReal Topology
+namespace RPISingularHarnack
+
+theorem radial_integral_lower_of_density {n : ℕ} (μ : Measure (Euc n)) [IsFiniteMeasure μ]
+    (a : Euc n) {ε k c : ℝ} (hε : 0 < ε) (hk : 0 ≤ k) (hc : 0 ≤ c)
+    (hdensity : c * ε ^ k ≤ μ.real (Metric.ball a ε))
+    {f : Euc n → ℝ} (hf0 : ∀ x, 0 ≤ f x)
+    (hflower : ∀ x ∈ Metric.ball a ε, f a / 2 ≤ f x)
+    (hi : Integrable (fun x ↦ f x * k * (‖x - a‖ ^ 2 + ε ^ 2) ^ (-k / 2 - 1) * ε ^ 2) μ) :
+    (c * k * (2 : ℝ) ^ (-k / 2 - 1) / 2) * f a ≤
+      ∫ x, f x * k * (‖x - a‖ ^ 2 + ε ^ 2) ^ (-k / 2 - 1) * ε ^ 2 ∂μ := by
+  let B : ℝ := f a / 2 * k * (2 * ε ^ 2) ^ (-k / 2 - 1) * ε ^ 2
+  have hB : 0 ≤ B := by dsimp [B]; positivity [hf0 a]
+  have hdist (x : Euc n) (hx : x ∈ Metric.ball a ε) : ‖x - a‖ ^ 2 + ε ^ 2 ≤ 2 * ε ^ 2 := by
+    have hh : ‖x - a‖ < ε := by simpa only [Metric.mem_ball, dist_eq_norm] using hx
+    nlinarith [norm_nonneg (x - a)]
+  have hp (x : Euc n) (hx : x ∈ Metric.ball a ε) :
+      (2 * ε ^ 2) ^ (-k / 2 - 1) ≤ (‖x - a‖ ^ 2 + ε ^ 2) ^ (-k / 2 - 1) :=
+    Real.rpow_le_rpow_of_nonpos (by positivity) (hdist x hx) (by linarith)
+  have hle (x : Euc n) : (Metric.ball a ε).indicator (fun _ ↦ B) x ≤
+      f x * k * (‖x - a‖ ^ 2 + ε ^ 2) ^ (-k / 2 - 1) * ε ^ 2 := by
+    by_cases hx : x ∈ Metric.ball a ε
+    · rw [indicator_of_mem hx]
+      dsimp [B]
+      exact mul_le_mul_of_nonneg_right
+        (mul_le_mul (mul_le_mul_of_nonneg_right (hflower x hx) hk) (hp x hx)
+          (by positivity) (mul_nonneg (hf0 x) hk)) (sq_nonneg ε)
+    · rw [indicator_of_notMem hx]
+      positivity [hf0 x]
+  have hileft : Integrable ((Metric.ball a ε).indicator (fun _ ↦ B)) μ :=
+    (integrable_const B).indicator measurableSet_ball
+  have hh := integral_mono hileft hi hle
+  rw [integral_indicator measurableSet_ball, integral_const, smul_eq_mul,
+    Measure.real, Measure.restrict_apply_univ] at hh
+  have hlow := mul_le_mul hdensity (le_refl B) hB
+    ((mul_nonneg hc (Real.rpow_nonneg hε.le k)).trans hdensity)
+  have he : (c * ε ^ k) * B = (c * k * (2 : ℝ) ^ (-k / 2 - 1) / 2) * f a := by
+    calc
+      _ = (c * k * f a / 2) * (ε ^ k * ε ^ 2 * (2 * ε ^ 2) ^ (-k / 2 - 1)) := by dsimp [B]; ring
+      _ = _ := by rw [radialKernel_mass_scaling hε]; ring
+  exact he ▸ hlow.trans hh
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+open scoped ENNReal Topology
+namespace RPISingularHarnack
+open RPI.Boundary
+
+theorem exists_minimizing_riesz_potential_constant (n : ℕ) (hn : 2 < n) :
+    ∃ S : ℝ, 0 < S ∧ ∀ E : Set (Euc n), IsLocallyPerimeterMinimizing E →
+      ∀ (c : Euc n) (R : ℝ), 0 < R → ∀ f : Euc n → ℝ,
+        ContDiff ℝ ∞ f → HasCompactSupport f → tsupport f ⊆ Metric.ball c R →
+        (∀ x, 0 ≤ f x) → ∀ a ∈ perimeterSupport E,
+        S * f a ≤ ∫ x in Metric.ball c R,
+          ‖tangentialGradient E f x‖ * ‖x - a‖ ^ (-(((n - 1 : ℕ) : ℝ) - 1)) ∂perimeterMeasure E := by
+  have hn1 : 1 < n := by omega
+  have hk : (1 : ℝ) < (n - 1 : ℕ) := by exact_mod_cast (show 1 < n - 1 by omega)
+  obtain ⟨c₀, hc₀, hdensity⟩ := exists_minimizing_perimeter_lower_density n hn1
+  obtain ⟨A, hA, hgrowth⟩ := exists_locallyMinimizing_perimeter_growth_constant n (by omega)
+  let S : ℝ := c₀ * (n - 1 : ℕ) * (2 : ℝ) ^ (-((n - 1 : ℕ) : ℝ) / 2 - 1) / 2
+  have hS : 0 < S := by dsimp [S]; positivity
+  refine ⟨S, hS, ?_⟩
+  intro E hmin c R hR f hf hcf hsf hf0 a ha
+  let μ := (perimeterMeasure E).restrict (Metric.ball c R)
+  have : IsFiniteMeasure μ := ⟨by
+    rw [Measure.restrict_apply_univ, perimeterMeasure_open E Metric.isOpen_ball]
+    exact hmin.2.1 c R hR⟩
+  have hg (b : Euc n) (r : ℝ) (hr : 0 < r) : μ (Metric.ball b r) ≤
+      ENNReal.ofReal (A * r ^ ((n - 1 : ℕ) : ℝ)) := by
+    calc
+      _ ≤ perimeterMeasure E (Metric.ball b r) := Measure.restrict_apply_le _ _
+      _ = perimeter E (Metric.ball b r) := perimeterMeasure_open E Metric.isOpen_ball
+      _ ≤ _ := by simpa only [Real.rpow_natCast] using hgrowth E hmin b r hr
+  have hiK := RPIKernel.integrable_kernel_of_growth μ (by linarith : 0 < ((n - 1 : ℕ) : ℝ) - 1)
+    (by linarith : ((n - 1 : ℕ) : ℝ) - 1 < ((n - 1 : ℕ) : ℝ)) hA.le hg a
+  have hmG := (measurable_tangentialGradient hmin.1 (hf.of_le (by simp))).aestronglyMeasurable (μ := μ)
+  obtain ⟨B, hB⟩ := exists_tangentialGradient_ball_bound E c R (hf.of_le (by simp))
+  have hiGK : Integrable (fun x ↦ ‖tangentialGradient E f x‖ *
+      ‖x - a‖ ^ (-(((n - 1 : ℕ) : ℝ) - 1))) μ := by
+    simpa only [dist_eq_norm] using hiK.bdd_mul hmG.norm
+      ((ae_restrict_mem measurableSet_ball).mono fun x hx ↦ by simpa only [norm_norm] using hB x hx)
+  by_cases hfa : f a = 0
+  · rw [hfa, mul_zero]
+    exact integral_nonneg fun x ↦ by positivity
+  have hfa0 : 0 < f a := lt_of_le_of_ne (hf0 a) (Ne.symm hfa)
+  have haU : a ∈ Metric.ball c R := hsf (subset_tsupport f (by exact hfa))
+  have hnhood : {x | f a / 2 < f x} ∈ 𝓝 a :=
+    hf.continuous.continuousAt.preimage_mem_nhds (Ioi_mem_nhds (by linarith))
+  obtain ⟨ε, hε, hεsub⟩ := Metric.mem_nhds_iff.mp
+    (inter_mem hnhood (Metric.isOpen_ball.mem_nhds haU))
+  have hεU : Metric.ball a ε ⊆ Metric.ball c R := fun x hx ↦ (hεsub hx).2
+  have hμball : μ (Metric.ball a ε) = perimeter E (Metric.ball a ε) := by
+    rw [Measure.restrict_apply measurableSet_ball, inter_eq_left.mpr hεU,
+      perimeterMeasure_open E Metric.isOpen_ball]
+  have hdens : c₀ * ε ^ ((n - 1 : ℕ) : ℝ) ≤ μ.real (Metric.ball a ε) := by
+    have ht := ENNReal.toReal_mono (hmin.2.1 a ε hε).ne (hdensity E hmin a ha ε hε)
+    rw [ENNReal.toReal_ofReal (by positivity)] at ht
+    simpa only [Measure.real, hμball, Real.rpow_natCast] using ht
+  have hlci : Integrable (fun x ↦ f x * (n - 1 : ℕ) *
+      (‖x - a‖ ^ 2 + ε ^ 2) ^ (-((n - 1 : ℕ) : ℝ) / 2 - 1) * ε ^ 2) μ := by
+    apply ContinuousOn.integrableOn_of_subset_isCompact (s := Metric.ball c R)
+      (K := Metric.closedBall c R) _ (isCompact_closedBall c R) measurableSet_ball Metric.ball_subset_closedBall
+      (by rw [perimeterMeasure_open E Metric.isOpen_ball]; exact (hmin.2.1 c R hR).ne)
+    apply Continuous.continuousOn
+    apply Continuous.mul _ continuous_const
+    apply Continuous.mul (hf.continuous.mul continuous_const)
+    exact (((continuous_id.sub continuous_const).norm.pow 2).add continuous_const).rpow_const
+      (fun x ↦ Or.inl (by change ‖x - a‖ ^ 2 + ε ^ 2 ≠ 0; positivity))
+  have hlow := radial_integral_lower_of_density μ a hε (by linarith : 0 ≤ ((n - 1 : ℕ) : ℝ))
+    hc₀.le hdens hf0 (fun x hx ↦ (hεsub hx).1.le) hlci
+  have hmid := minimizing_radial_integral_bound hn1 hmin c a hR hε hf hcf hsf hf0
+  have hmRG : AEStronglyMeasurable (fun x ↦ ‖tangentialGradient E f x‖ *
+      (radialKernel a ε (n - 1 : ℕ) x * ‖x - a‖)) μ :=
+    hmG.norm.mul (((radialKernel_contDiff a hε _).continuous.mul
+      (continuous_id.sub continuous_const).norm).aestronglyMeasurable)
+  have hpoint (x : Euc n) : ‖tangentialGradient E f x‖ *
+      (radialKernel a ε (n - 1 : ℕ) x * ‖x - a‖) ≤
+      ‖tangentialGradient E f x‖ * ‖x - a‖ ^ (-(((n - 1 : ℕ) : ℝ) - 1)) :=
+    mul_le_mul_of_nonneg_left (radialKernel_mul_dist_le a x hε hk) (norm_nonneg _)
+  have hiRG : Integrable (fun x ↦ ‖tangentialGradient E f x‖ *
+      (radialKernel a ε (n - 1 : ℕ) x * ‖x - a‖)) μ :=
+    hiGK.mono' hmRG (ae_of_all μ fun x ↦ by
+      rw [Real.norm_of_nonneg (by positivity [(radialKernel_pos a x hε ((n - 1 : ℕ) : ℝ)).le])]
+      exact hpoint x)
+  exact hlow.trans (hmid.trans (integral_mono hiRG hiGK hpoint))
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+open scoped ENNReal Topology
+namespace RPIKernel
+
+theorem lp_average_bound {X : Type*} [MeasurableSpace X] (μ : Measure X)
+    {p : ℝ} (hp : 1 < p) {f : X → ℝ≥0∞} (hf : AEMeasurable f μ) :
+    (∫⁻ x, f x ∂μ) ^ p ≤ (∫⁻ x, f x ^ p ∂μ) * (μ univ) ^ (p - 1) := by
+  have hc := Real.HolderConjugate.conjExponent hp
+  have hh := ENNReal.lintegral_mul_le_Lp_mul_Lq μ hc hf
+    (aemeasurable_const (b := (1 : ℝ≥0∞)))
+  simp only [Pi.mul_apply, mul_one, ENNReal.one_rpow, lintegral_const, one_mul] at hh
+  have h := ENNReal.rpow_le_rpow hh (by linarith : 0 ≤ p)
+  rw [ENNReal.mul_rpow_of_nonneg _ _ (by linarith : 0 ≤ p), ← ENNReal.rpow_mul,
+    ← ENNReal.rpow_mul, div_mul_cancel₀ _ (ne_of_gt (lt_trans zero_lt_one hp)), ENNReal.rpow_one] at h
+  have he : 1 / p.conjExponent * p = p - 1 := by
+    calc
+      _ = p / p.conjExponent := by ring
+      _ = p - 1 := (div_eq_iff (ne_of_gt hc.symm.pos)).mpr hc.sub_one_mul_conj.symm
+  rwa [he] at h
+
+theorem kernel_lp_bound {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y]
+    (μ : Measure X) (ν : Measure Y) [SFinite μ] [SFinite ν]
+    {p : ℝ} (hp : 1 < p) (K : Y → X → ℝ≥0∞)
+    (hK : Measurable (Function.uncurry K)) (g : X → ℝ≥0∞) (hg : Measurable g)
+    {C : ℝ≥0∞} (hcol : ∀ x, ∫⁻ y, K y x ^ p ∂ν ≤ C) :
+    ∫⁻ y, (∫⁻ x, K y x * g x ∂μ) ^ p ∂ν ≤
+      C * (∫⁻ x, g x ∂μ) ^ p := by
+  have hrow (y : Y) : Measurable (K y) := hK.comp ((measurable_const (a := y)).prodMk measurable_id)
+  have hcolumn (x : X) : Measurable (fun y ↦ K y x) := hK.comp (measurable_id.prodMk (measurable_const (a := x)))
+  have hm : Measurable (Function.uncurry (fun y x ↦ K y x ^ p * g x)) := (hK.pow_const p).mul (hg.comp measurable_snd)
+  have hmain (y : Y) := lp_average_bound (μ.withDensity g) hp ((hrow y).aemeasurable)
+  have hw : (μ.withDensity g) univ = ∫⁻ x, g x ∂μ := by simp
+  have hpt (y : Y) : (∫⁻ x, K y x * g x ∂μ) ^ p ≤
+      (∫⁻ x, K y x ^ p * g x ∂μ) * (∫⁻ x, g x ∂μ) ^ (p - 1) := by
+    have ht := hmain y
+    rw [hw, lintegral_withDensity_eq_lintegral_mul μ hg (hrow y),
+      lintegral_withDensity_eq_lintegral_mul μ hg ((hrow y).pow_const p)] at ht
+    simpa only [Pi.mul_apply, mul_comm (g _)] using ht
+  calc
+    _ ≤ ∫⁻ y, (∫⁻ x, K y x ^ p * g x ∂μ) * (∫⁻ x, g x ∂μ) ^ (p - 1) ∂ν :=
+      lintegral_mono hpt
+    _ = (∫⁻ x, (∫⁻ y, K y x ^ p ∂ν) * g x ∂μ) * (∫⁻ x, g x ∂μ) ^ (p - 1) := by
+      rw [lintegral_mul_const _ hm.lintegral_prod_right]
+      congr 1
+      rw [lintegral_lintegral_swap hm.aemeasurable]
+      apply lintegral_congr
+      intro x
+      rw [lintegral_mul_const _ ((hcolumn x).pow_const p)]
+    _ ≤ (∫⁻ x, C * g x ∂μ) * (∫⁻ x, g x ∂μ) ^ (p - 1) := by
+      gcongr with x
+      exact hcol x
+    _ = _ := by
+      rw [lintegral_const_mul _ hg]
+      rw [mul_assoc]
+      congr 1
+      calc
+        _ = (∫⁻ x, g x ∂μ) ^ (1 : ℝ) * (∫⁻ x, g x ∂μ) ^ (p - 1) := by rw [ENNReal.rpow_one]
+        _ = (∫⁻ x, g x ∂μ) ^ (1 + (p - 1)) := (ENNReal.rpow_add_of_nonneg _ _ (by norm_num) (by linarith)).symm
+        _ = _ := by congr 1; ring
+
+end RPIKernel
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+open scoped ENNReal Topology
+namespace RPIKernel
+open Set Filter MeasureTheory
+open scoped ENNReal Topology
+variable {X : Type*}
+
+lemma ofReal_integral_le_lintegral [MetricSpace X] [MeasurableSpace X] [BorelSpace X]
+    {f : X → ℝ} {μ : Measure X}
+    (hf : ∀ᵐ x ∂μ, 0 ≤ f x) :
+    ENNReal.ofReal (∫ x, f x ∂μ) ≤ ∫⁻ x, ENNReal.ofReal (f x) ∂μ := by
+  by_cases hi : Integrable f μ
+  · exact (ofReal_integral_eq_lintegral_ofReal hi hf).le
+  · simp [integral_undef hi]
+
+variable [MetricSpace X] [MeasurableSpace X] [BorelSpace X]
+
+theorem exists_potential_lp_bound [SecondCountableTopology X] (μ : Measure X) [IsFiniteMeasure μ]
+    {d s p A S : ℝ} (hs : 0 < s) (hp : 1 < p) (hsp : s * p < d)
+    (hA : 0 ≤ A) (hS : 0 < S)
+    (hgrowth : ∀ a r, 0 < r → μ (Metric.ball a r) ≤ ENNReal.ofReal (A * r ^ d)) :
+    ∃ C : ℝ≥0∞, C ≠ ⊤ ∧ ∀ f g : X → ℝ,
+      Measurable g → (∀ x, 0 ≤ g x) →
+      (∀ᵐ a ∂μ, S * f a ≤ ∫ x, g x * dist x a ^ (-s) ∂μ) →
+      (∫⁻ a, ENNReal.ofReal (f a) ^ p ∂μ) ≤ C * (∫⁻ x, ENNReal.ofReal (g x) ∂μ) ^ p := by
+  have hp0 : 0 < p := lt_trans zero_lt_one hp
+  obtain ⟨B, hB, hcol⟩ := exists_kernel_integral_bound μ (mul_pos hs hp0) hsp hA hgrowth
+  let K : X → X → ℝ≥0∞ := fun a x ↦ ENNReal.ofReal (dist x a ^ (-s))
+  have hmK : Measurable (Function.uncurry K) :=
+    ENNReal.measurable_ofReal.comp ((continuous_snd.dist continuous_fst).measurable.pow_const _)
+  have hcol' (x : X) : ∫⁻ a, K a x ^ p ∂μ ≤ B := by
+    calc
+      _ = ∫⁻ a, ENNReal.ofReal (dist a x ^ (-(s * p))) ∂μ := by
+        apply lintegral_congr
+        intro a
+        dsimp [K]
+        rw [ENNReal.ofReal_rpow_of_nonneg (Real.rpow_nonneg dist_nonneg _) hp0.le,
+          ← Real.rpow_mul dist_nonneg, dist_comm]
+        congr 2
+        ring
+      _ ≤ B := hcol x
+  let T : ℝ≥0∞ := (ENNReal.ofReal S)⁻¹ ^ p
+  refine ⟨T * B, ENNReal.mul_ne_top
+    (ENNReal.rpow_ne_top_of_nonneg hp0.le (ENNReal.inv_ne_top.mpr (ENNReal.ofReal_ne_zero_iff.mpr hS))) hB, ?_⟩
+  intro f g hg hg0 hpot
+  have hrow (a : X) : Measurable (K a) := hmK.comp (measurable_const.prodMk measurable_id)
+  have hpot' : ∀ᵐ a ∂μ, ENNReal.ofReal (f a) ≤ (ENNReal.ofReal S)⁻¹ *
+      ∫⁻ x, K a x * ENNReal.ofReal (g x) ∂μ := by
+    filter_upwards [hpot] with a ha
+    apply (ENNReal.mul_le_iff_le_inv (ENNReal.ofReal_ne_zero_iff.mpr hS) ENNReal.ofReal_ne_top).mp
+    calc
+      _ = ENNReal.ofReal (S * f a) := (ENNReal.ofReal_mul hS.le).symm
+      _ ≤ ENNReal.ofReal (∫ x, g x * dist x a ^ (-s) ∂μ) := ENNReal.ofReal_le_ofReal ha
+      _ ≤ ∫⁻ x, ENNReal.ofReal (g x * dist x a ^ (-s)) ∂μ :=
+        ofReal_integral_le_lintegral (ae_of_all _ fun x ↦ mul_nonneg (hg0 x) (Real.rpow_nonneg dist_nonneg _))
+      _ = _ := by
+        apply lintegral_congr
+        intro x
+        rw [ENNReal.ofReal_mul (hg0 x), mul_comm]
+  have hi := kernel_lp_bound μ μ hp K hmK (fun x ↦ ENNReal.ofReal (g x)) hg.ennreal_ofReal hcol'
+  calc
+    _ ≤ ∫⁻ a, T * (∫⁻ x, K a x * ENNReal.ofReal (g x) ∂μ) ^ p ∂μ := by
+      apply lintegral_mono_ae
+      filter_upwards [hpot'] with a ha
+      exact (ENNReal.rpow_le_rpow ha hp0.le).trans_eq
+        (ENNReal.mul_rpow_of_nonneg _ _ hp0.le)
+    _ = T * ∫⁻ a, (∫⁻ x, K a x * ENNReal.ofReal (g x) ∂μ) ^ p ∂μ := by
+      rw [lintegral_const_mul _ (((hmK.mul (hg.ennreal_ofReal.comp measurable_snd)).lintegral_prod_right).pow_const p)]
+    _ ≤ T * (B * (∫⁻ x, ENNReal.ofReal (g x) ∂μ) ^ p) := by gcongr
+    _ = _ := by rw [mul_assoc]
+
+end RPIKernel
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+open scoped ENNReal Topology
+namespace RPISingularHarnack
+
+theorem minimizing_perimeter_sobolev {n : ℕ} (hn : 2 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    (c : Euc n) {R p : ℝ} (hR : 0 < R) (hp : 1 < p)
+    (hpk : (((n - 1 : ℕ) : ℝ) - 1) * p < (n - 1 : ℕ)) :
+    ∃ C : ℝ≥0∞, C ≠ ⊤ ∧ ∀ f : Euc n → ℝ,
+      ContDiff ℝ ∞ f → HasCompactSupport f → tsupport f ⊆ Metric.ball c R →
+      (∀ x, 0 ≤ f x) →
+      (∫⁻ x in Metric.ball c R, ENNReal.ofReal (f x) ^ p ∂perimeterMeasure E) ≤
+        C * (∫⁻ x in Metric.ball c R, ENNReal.ofReal ‖tangentialGradient E f x‖ ∂perimeterMeasure E) ^ p := by
+  obtain ⟨S, hS, hpot⟩ := exists_minimizing_riesz_potential_constant n hn
+  obtain ⟨A, hA, hg⟩ := exists_locallyMinimizing_perimeter_growth_constant n (by omega)
+  let μ := (perimeterMeasure E).restrict (Metric.ball c R)
+  let : IsFiniteMeasure μ := ⟨by
+    rw [Measure.restrict_apply_univ, perimeterMeasure_open E Metric.isOpen_ball]
+    exact hmin.2.1 c R hR⟩
+  have hgrowth (a : Euc n) (r : ℝ) (hr : 0 < r) : μ (Metric.ball a r) ≤
+      ENNReal.ofReal (A * r ^ ((n - 1 : ℕ) : ℝ)) := by
+    calc
+      _ ≤ perimeterMeasure E (Metric.ball a r) := Measure.restrict_apply_le _ _
+      _ = perimeter E (Metric.ball a r) := perimeterMeasure_open E Metric.isOpen_ball
+      _ ≤ _ := by simpa only [Real.rpow_natCast] using hg E hmin a r hr
+  have hk : (1 : ℝ) < (n - 1 : ℕ) := by exact_mod_cast (show 1 < n - 1 by omega)
+  obtain ⟨C, hC, hbound⟩ := RPIKernel.exists_potential_lp_bound μ
+    (by linarith : 0 < ((n - 1 : ℕ) : ℝ) - 1) hp hpk hA.le hS hgrowth
+  refine ⟨C, hC, ?_⟩
+  intro f hf hcf hsf hf0
+  apply hbound f (fun x ↦ ‖tangentialGradient E f x‖)
+    (measurable_tangentialGradient hmin.1 (hf.of_le (by simp))).norm (fun _ ↦ norm_nonneg _)
+  have hs : ∀ᵐ x ∂perimeterMeasure E, x ∈ perimeterSupport E := by
+    rw [← perimeterMeasure_support]; exact (perimeterMeasure E).support_mem_ae
+  filter_upwards [ae_restrict_of_ae hs] with a ha
+  simpa only [dist_eq_norm] using hpot E hmin c R hR f hf hcf hsf hf0 a ha
+
+theorem minimizing_perimeter_boundaryArea_comparison {n : ℕ} (hn : 1 < n) :
+    ∃ C D : ℝ≥0∞, C ≠ 0 ∧ C ≠ ⊤ ∧ D ≠ 0 ∧ D ≠ ⊤ ∧
+      ∀ E : Set (Euc n), IsLocallyPerimeterMinimizing E →
+        perimeterMeasure E ≤ C • boundaryArea E ∧ boundaryArea E ≤ D • perimeterMeasure E := by
+  obtain ⟨C, D, hC0, hCt, hD0, hDt, hb⟩ := minimizing_perimeter_hausdorff_comparison hn
+  let b : ℝ≥0∞ := volume (Metric.ball (0 : Euc (n - 1)) 1) / 2 ^ (n - 1)
+  have hb0 : b ≠ 0 := ENNReal.div_ne_zero.mpr ⟨
+    (Metric.measure_ball_pos volume 0 (by norm_num : (0 : ℝ) < 1)).ne', ENNReal.pow_ne_top (by norm_num)⟩
+  have hbt : b ≠ ⊤ := ENNReal.div_ne_top measure_ball_ne_top (pow_ne_zero _ (by norm_num))
+  refine ⟨C * b⁻¹, b * D, mul_ne_zero hC0 (ENNReal.inv_ne_zero.mpr hbt),
+    ENNReal.mul_ne_top hCt (ENNReal.inv_ne_top.mpr hb0), mul_ne_zero hb0 hD0,
+    ENNReal.mul_ne_top hbt hDt, ?_⟩
+  intro E hmin
+  have he : boundaryArea E = b • ((Measure.hausdorffMeasure (n - 1 : ℕ)).restrict (perimeterSupport E)) := by
+    rw [boundaryArea, geometricHausdorff, Measure.restrict_smul]
+  constructor
+  · rw [he, smul_smul, mul_assoc, ENNReal.inv_mul_cancel hb0 hbt, mul_one]
+    exact (hb E hmin).1
+  · rw [he]
+    apply Measure.le_iff.mpr
+    intro s hs
+    simp only [Measure.smul_apply, smul_eq_mul]
+    calc
+      _ ≤ b * (D * perimeterMeasure E s) := by
+        gcongr
+        exact Measure.le_iff.mp (hb E hmin).2 s hs
+      _ = _ := by rw [mul_assoc]
+
+theorem minimizing_boundaryArea_sobolev {n : ℕ} (hn : 2 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    (c : Euc n) {R p : ℝ} (hR : 0 < R) (hp : 1 < p)
+    (hpk : (((n - 1 : ℕ) : ℝ) - 1) * p < (n - 1 : ℕ)) :
+    ∃ C : ℝ≥0∞, C ≠ ⊤ ∧ ∀ f : Euc n → ℝ,
+      ContDiff ℝ ∞ f → HasCompactSupport f → tsupport f ⊆ Metric.ball c R →
+      (∀ x, 0 ≤ f x) →
+      (∫⁻ x in Metric.ball c R, ENNReal.ofReal (f x) ^ p ∂boundaryArea E) ≤
+        C * (∫⁻ x in Metric.ball c R, ENNReal.ofReal ‖tangentialGradient E f x‖ ∂boundaryArea E) ^ p := by
+  obtain ⟨B, hB, hb⟩ := minimizing_perimeter_sobolev hn hmin c hR hp hpk
+  obtain ⟨D₁, D₂, _, hD₁, _, hD₂, hd⟩ := minimizing_perimeter_boundaryArea_comparison (by omega : 1 < n)
+  have hμ₁ := Measure.restrict_mono_measure (hd E hmin).1 (Metric.ball c R)
+  have hμ₂ := Measure.restrict_mono_measure (hd E hmin).2 (Metric.ball c R)
+  rw [Measure.restrict_smul] at hμ₁ hμ₂
+  have hp0 : 0 ≤ p := (lt_trans zero_lt_one hp).le
+  refine ⟨D₂ * B * D₁ ^ p,
+    ENNReal.mul_ne_top (ENNReal.mul_ne_top hD₂ hB) (ENNReal.rpow_ne_top_of_nonneg hp0 hD₁), ?_⟩
+  intro f hf hcf hsf hf0
+  have hgrad : (∫⁻ x in Metric.ball c R, ENNReal.ofReal ‖tangentialGradient E f x‖ ∂perimeterMeasure E) ≤
+      D₁ * (∫⁻ x in Metric.ball c R, ENNReal.ofReal ‖tangentialGradient E f x‖ ∂boundaryArea E) := by
+    simpa only [lintegral_smul_measure, smul_eq_mul] using lintegral_mono' hμ₁ (le_refl (fun x ↦ ENNReal.ofReal ‖tangentialGradient E f x‖))
+  calc
+    _ ≤ D₂ * (∫⁻ x in Metric.ball c R, ENNReal.ofReal (f x) ^ p ∂perimeterMeasure E) := by
+      simpa only [lintegral_smul_measure, smul_eq_mul] using lintegral_mono' hμ₂ (le_refl (fun x ↦ ENNReal.ofReal (f x) ^ p))
+    _ ≤ D₂ * (B * (∫⁻ x in Metric.ball c R, ENNReal.ofReal ‖tangentialGradient E f x‖ ∂perimeterMeasure E) ^ p) := by
+      gcongr
+      exact hb f hf hcf hsf hf0
+    _ ≤ D₂ * (B * (D₁ * (∫⁻ x in Metric.ball c R, ENNReal.ofReal ‖tangentialGradient E f x‖ ∂boundaryArea E)) ^ p) := by
+      gcongr
+    _ = _ := by rw [ENNReal.mul_rpow_of_nonneg _ _ hp0]; ac_rfl
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/FiberAveraging.lean b/lean/OAI/Geometry/Perimeter/Regularity/FiberAveraging.lean
new file mode 100644
index 0000000000000000000000000000000000000000..570eb17f6ed4b53808eacd20c71fcc0871428435
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/FiberAveraging.lean
@@ -0,0 +1,125 @@
+import OAI.Geometry.Perimeter.Regularity.EllipticLocality
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff Convolution
+namespace RPISingularHarnack
+open RPI.Boundary
+
+def normalFiberAverage {n : ℕ} (v : Euc n) (f : Euc n → ℝ) (x : Euc n) : ℝ :=
+  ∫ s : ℝ, f (x+s•v)
+
+theorem normalFiberAverage_stronglyMeasurable {n : ℕ} {f : Euc n → ℝ}
+    (hf : Measurable f) (v : Euc n) : StronglyMeasurable (normalFiberAverage v f) := by
+  apply StronglyMeasurable.integral_prod_right
+  exact (hf.comp (continuous_fst.add (continuous_snd.smul continuous_const)).measurable).stronglyMeasurable
+
+theorem normalFiberAverage_add_smul {n : ℕ} (v : Euc n) (f : Euc n → ℝ)
+    (x : Euc n) (t : ℝ) :
+    normalFiberAverage v f (x+t•v) = normalFiberAverage v f x := by
+  unfold normalFiberAverage
+  have heq : (fun s : ℝ ↦ f (x+t•v+s•v)) = (fun s : ℝ ↦ f (x+(s+t)•v)) := by
+    funext s
+    congr 1
+    rw [add_smul]
+    abel
+  rw [heq]
+  exact integral_add_right_eq_self (fun s : ℝ ↦ f (x+s•v)) t
+
+theorem fiber_support_subset_interval {n : ℕ} {f : Euc n → ℝ}
+    {v : Euc n} (hv : ‖v‖ = 1) {R : ℝ}
+    (hs : Function.support f ⊆ Metric.ball 0 R) (x : Euc n) :
+    Function.support (fun s : ℝ ↦ f (x+s•v)) ⊆ Icc (-(R+‖x‖)) (R+‖x‖) := by
+  intro s hs'
+  have hx := hs hs'
+  rw [Metric.mem_ball, dist_zero_right] at hx
+  have he : |s| = ‖s•v‖ := by rw [norm_smul, hv, mul_one, Real.norm_eq_abs]
+  have hl : |s| ≤ R+‖x‖ := by
+    rw [he]
+    calc
+      ‖s•v‖ = ‖(x+s•v)-x‖ := by congr 1; abel
+      _ ≤ ‖x+s•v‖+‖x‖ := norm_sub_le _ _
+      _ ≤ R+‖x‖ := by linarith
+  exact abs_le.mp hl
+
+theorem integrable_fiber_of_bounded {n : ℕ} {f : Euc n → ℝ}
+    (hf : Measurable f) {v : Euc n} (hv : ‖v‖ = 1) {R : ℝ}
+    (hs : Function.support f ⊆ Metric.ball 0 R) {C : ℝ}
+    (hb : ∀ x, ‖f x‖ ≤ C) (x : Euc n) :
+    Integrable (fun s : ℝ ↦ f (x+s•v)) volume := by
+  let I := Icc (-(R+‖x‖)) (R+‖x‖)
+  have hI : Integrable (I.indicator (fun _ ↦ C)) volume :=
+    continuous_const.integrableOn_Icc.integrable_indicator measurableSet_Icc
+  apply hI.mono' (hf.comp (continuous_const.add (continuous_id.smul continuous_const)).measurable).aestronglyMeasurable
+  apply ae_of_all
+  intro s
+  by_cases hsI : s ∈ I
+  · simpa [hsI] using hb (x+s•v)
+  · rw [indicator_of_notMem hsI]
+    have hz : f (x+s•v) = 0 := by
+      by_contra hz
+      exact hsI (fiber_support_subset_interval hv hs x hz)
+    simp [hz]
+
+theorem normalFiberAverage_translation_le {n : ℕ} {f : Euc n → ℝ}
+    (hf : Measurable f) {v : Euc n} (hv : ‖v‖ = 1) {R : ℝ} (hR : 0 < R)
+    (hs : Function.support f ⊆ Metric.ball 0 R) {C : ℝ}
+    (hb : ∀ x, ‖f x‖ ≤ C) {S : ℝ} (hS : 0 < S) (w : Euc n) :
+    (∫⁻ x in Metric.ball (0:Euc n) S,
+      ‖normalFiberAverage v f (x+w)-normalFiberAverage v f x‖ₑ) ≤
+        ENNReal.ofReal (2*(R+S+‖w‖)) * ∫⁻ x, ‖f (x+w)-f x‖ₑ := by
+  let T := R+S+‖w‖
+  have hT : 0 < T := add_pos_of_pos_of_nonneg (add_pos hR hS) (norm_nonneg w)
+  let I := Icc (-T) T
+  let B := Metric.ball (0:Euc n) S
+  have hxI (x : Euc n) (hx : x ∈ B) :
+      Function.support (fun s : ℝ ↦ ‖f (x+w+s•v)-f (x+s•v)‖ₑ) ⊆ I := by
+    intro s hs'
+    have hx' : ‖x‖ < S := by simpa only [B, Metric.mem_ball, dist_zero_right] using hx
+    have hzero (y : Euc n) (hy : ‖y‖ ≤ S+‖w‖) (hns : s ∉ I) : f (y+s•v) = 0 := by
+      by_contra hz
+      have ht := fiber_support_subset_interval hv hs y hz
+      exact hns (Icc_subset_Icc (by dsimp [T]; linarith) (by dsimp [T]; linarith) ht)
+    by_contra hns
+    have h₁ := hzero (x+w) ((norm_add_le x w).trans (by linarith)) hns
+    have h₂ := hzero x (hx'.le.trans (le_add_of_nonneg_right (norm_nonneg w))) hns
+    simp only [Function.mem_support, ne_eq, h₁, h₂, sub_self, enorm_zero, not_true_eq_false] at hs'
+  have hm : Measurable (fun p : Euc n × ℝ ↦ ‖f (p.1+w+p.2•v)-f (p.1+p.2•v)‖ₑ) :=
+    ((hf.comp ((continuous_fst.add continuous_const).add
+       (continuous_snd.smul continuous_const)).measurable).sub
+      (hf.comp (continuous_fst.add (continuous_snd.smul continuous_const)).measurable)).enorm
+  calc
+    _ ≤ ∫⁻ x in B, ∫⁻ s in I, ‖f (x+w+s•v)-f (x+s•v)‖ₑ := by
+      apply lintegral_mono_ae
+      filter_upwards [ae_restrict_mem measurableSet_ball] with x hx
+      rw [normalFiberAverage, normalFiberAverage,
+        ← integral_sub (integrable_fiber_of_bounded hf hv hs hb (x+w))
+          (integrable_fiber_of_bounded hf hv hs hb x)]
+      rw [setLIntegral_eq_of_support_subset (hxI x hx)]
+      exact enorm_integral_le_lintegral_enorm _
+    _ ≤ ∫⁻ x, ∫⁻ s in I, ‖f (x+w+s•v)-f (x+s•v)‖ₑ :=
+      setLIntegral_le_lintegral _ _
+    _ = ∫⁻ s in I, ∫⁻ x, ‖f (x+w+s•v)-f (x+s•v)‖ₑ :=
+      lintegral_lintegral_swap hm.aemeasurable
+    _ = ENNReal.ofReal (2*T) * ∫⁻ x, ‖f (x+w)-f x‖ₑ := by
+      have he (s : ℝ) : (∫⁻ x, ‖f (x+w+s•v)-f (x+s•v)‖ₑ) =
+          ∫⁻ x, ‖f (x+w)-f x‖ₑ := by
+        have heq : (fun x ↦ ‖f (x+w+s•v)-f (x+s•v)‖ₑ) =
+            (fun x ↦ ‖f ((x+s•v)+w)-f (x+s•v)‖ₑ) := by
+          funext x
+          congr 3
+          abel
+        rw [heq]
+        exact lintegral_add_right_eq_self (fun x : Euc n ↦ ‖f (x+w)-f x‖ₑ) (s•v)
+      simp_rw [he]
+      simp only [lintegral_const, Measure.restrict_apply_univ, I]
+      rw [← uIcc_of_le (show -T ≤ T by linarith [hT]), Real.volume_interval,
+        sub_neg_eq_add, ← two_mul, abs_of_pos (mul_pos (by norm_num) hT), mul_comm]
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/FiberCompactness.lean b/lean/OAI/Geometry/Perimeter/Regularity/FiberCompactness.lean
new file mode 100644
index 0000000000000000000000000000000000000000..f1bbc548a30a3532942a34f4094075a23396c96e
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/FiberCompactness.lean
@@ -0,0 +1,411 @@
+import OAI.Geometry.Perimeter.Regularity.FlatExcess
+import OAI.Geometry.Perimeter.Regularity.FiberHeight
+import OAI.Geometry.Perimeter.Regularity.HalfspaceHorizontal
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff Convolution
+namespace RPISingularHarnack
+open RPI.Boundary ContinuousLinearMap
+
+theorem integral_phase_error_mul {n : ℕ} {E H : Set (Euc n)}
+    (hE : MeasurableSet E) (hH : MeasurableSet H) {g : Euc n → ℝ}
+    (hg : Integrable g volume) :
+    (∫ x, (E.indicator (fun _ ↦ (1:ℝ)) x - H.indicator (fun _ ↦ (1:ℝ)) x) * g x) =
+      (∫ x in E, g x) - ∫ x in H, g x := by
+  have heq : (fun x ↦ (E.indicator (fun _ ↦ (1:ℝ)) x -
+      H.indicator (fun _ ↦ (1:ℝ)) x) * g x) = E.indicator g - H.indicator g := by
+    funext x
+    by_cases hxE : x ∈ E <;> by_cases hxH : x ∈ H <;> simp [hxE,hxH]
+  rw [heq]
+  change (∫ x, E.indicator g x - H.indicator g x) = _
+  rw [integral_sub (hg.indicator hE) (hg.indicator hH),
+    integral_indicator hE, integral_indicator hH]
+
+theorem integrable_phase_error_mul {n : ℕ} {E H : Set (Euc n)}
+    (hE : MeasurableSet E) (hH : MeasurableSet H) {g : Euc n → ℝ}
+    (hg : Integrable g volume) :
+    Integrable (fun x ↦ (E.indicator (fun _ ↦ (1:ℝ)) x -
+      H.indicator (fun _ ↦ (1:ℝ)) x) * g x) volume := by
+  convert (hg.indicator hE).sub (hg.indicator hH) using 1
+  funext x
+  by_cases hxE : x ∈ E <;> by_cases hxH : x ∈ H <;> simp [hxE,hxH]
+
+theorem cutoffPhaseError_distribution {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    {v w : Euc n} (_hv : ‖v‖ = 1) (hw : inner ℝ v w = 0)
+    {η φ : Euc n → ℝ} (hη : ContDiff ℝ ∞ η) (hc : HasCompactSupport η)
+    (hφ : ContDiff ℝ ∞ φ) {R : ℝ} (hR : 0 < R)
+    (hs : tsupport η ⊆ Metric.ball 0 R) :
+    -(∫ x, cutoffPhaseError E v η x * fderiv ℝ φ x w) =
+      (∫ x, (E.indicator (fun _ ↦ (1:ℝ)) x -
+        {z | inner ℝ v z < 0}.indicator (fun _ ↦ (1:ℝ)) x) *
+          (φ x * fderiv ℝ η x w)) -
+      ∫ x in Metric.ball 0 R, φ x * (η x * inner ℝ (approximateNormal E x) w)
+        ∂perimeterMeasure E := by
+  let H : Set (Euc n) := {x | inner ℝ v x < 0}
+  have heH : H = {x | inner ℝ x v < 0} := by ext x; simp only [H, real_inner_comm]
+  have hHm : MeasurableSet H := (isOpen_lt (continuous_const.inner continuous_id)
+    continuous_const).measurableSet
+  have hηd := hη.differentiable (by simp)
+  have hφd := hφ.differentiable (by simp)
+  have hdi : Integrable (fun x ↦ η x * fderiv ℝ φ x w) volume :=
+    (hη.continuous.mul ((hφ.continuous_fderiv (by simp)).clm_apply continuous_const)).integrable_of_hasCompactSupport
+      hc.mul_right
+  have hdi' : Integrable (fun x ↦ φ x * fderiv ℝ η x w) volume :=
+    (hφ.continuous.mul ((hη.continuous_fderiv (by simp)).clm_apply continuous_const)).integrable_of_hasCompactSupport
+      (hc.fderiv_apply ℝ w).mul_left
+  have hprod (x : Euc n) : fderiv ℝ (fun y ↦ η y * φ y) x w =
+      η x * fderiv ℝ φ x w + φ x * fderiv ℝ η x w := by
+    rw [fderiv_fun_mul (hηd x) (hφd x)]
+    rfl
+  have hgg : (∫ x in E, fderiv ℝ (fun y ↦ η y * φ y) x w) =
+      ∫ x in Metric.ball 0 R, φ x * (η x * inner ℝ (approximateNormal E x) w)
+        ∂perimeterMeasure E := by
+    rw [← divergence_scalar_direction ((hη.mul hφ).differentiable (by simp)) w,
+      minimizing_gaussGreen_local hn hmin (Metric.ball 0 R) Metric.isOpen_ball
+        (hmin.2.1 0 R hR).ne (fun y ↦ (η y * φ y) • w) ((hη.mul hφ).smul contDiff_const)
+        hc.mul_right.smul_right
+        (tsupport_smul_subset_left _ _ |>.trans (tsupport_mul_subset_left.trans hs))]
+    apply integral_congr_ae
+    exact ae_of_all _ fun x ↦ by simp only [real_inner_smul_right]; ring
+  have hzero : (∫ x in H, fderiv ℝ (fun y ↦ η y * φ y) x w) = 0 := by
+    rw [heH]
+    exact halfspace_horizontal_integral hw (hη.mul hφ) hc.mul_right
+  simp_rw [hprod] at hgg hzero
+  rw [integral_add hdi.integrableOn hdi'.integrableOn] at hgg hzero
+  have hleft : (∫ x, cutoffPhaseError E v η x * fderiv ℝ φ x w) =
+      (∫ x in E, η x * fderiv ℝ φ x w) - ∫ x in H, η x * fderiv ℝ φ x w := by
+    rw [← integral_phase_error_mul hmin.1 hHm hdi]
+    apply integral_congr_ae
+    exact ae_of_all _ fun x ↦ by unfold cutoffPhaseError; dsimp [H]; ring
+  rw [hleft, integral_phase_error_mul hmin.1 hHm hdi']
+  linarith
+
+theorem cutoffPhaseError_convolution_fderiv {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    {v w : Euc n} (hv : ‖v‖ = 1) (hw : inner ℝ v w = 0)
+    {η k : Euc n → ℝ} (hη : ContDiff ℝ ∞ η) (hc : HasCompactSupport η)
+    (hk : ContDiff ℝ ∞ k) (hck : HasCompactSupport k) {R : ℝ} (hR : 0 < R)
+    (hs : tsupport η ⊆ Metric.ball 0 R) (x : Euc n) :
+    fderiv ℝ (cutoffPhaseError E v η ⋆[lsmul ℝ ℝ, volume] k) x w =
+      ((fun y ↦ (E.indicator (fun _ ↦ (1:ℝ)) y -
+        {z | inner ℝ v z < 0}.indicator (fun _ ↦ (1:ℝ)) y) * fderiv ℝ η y w)
+          ⋆[lsmul ℝ ℝ, volume] k) x -
+      ((fun y ↦ η y * inner ℝ (approximateNormal E y) w)
+        ⋆[lsmul ℝ ℝ, (perimeterMeasure E).restrict (Metric.ball 0 R)] k) x := by
+  let f := cutoffPhaseError E v η
+  have hfi : Integrable f volume := by
+    have hi := integrable_phase_error_mul (H := {z | inner ℝ v z < 0}) hmin.1
+      ((isOpen_lt (continuous_const.inner continuous_id) continuous_const).measurableSet)
+      (hη.continuous.integrable_of_hasCompactSupport hc)
+    change Integrable (fun y ↦ η y * (E.indicator (fun _ ↦ (1:ℝ)) y -
+      {z | inner ℝ v z < 0}.indicator (fun _ ↦ (1:ℝ)) y)) volume
+    simpa only [mul_comm] using hi
+  let φ : Euc n → ℝ := fun y ↦ k (x-y)
+  have hφ : ContDiff ℝ ∞ φ := hk.comp (contDiff_const.sub contDiff_id)
+  have hd (y : Euc n) : fderiv ℝ φ y w = -fderiv ℝ k (x-y) w := by
+    have h := ((hk.differentiable (by simp) (x-y)).hasFDerivAt.comp y
+      ((hasFDerivAt_const x y).sub (hasFDerivAt_id y))).fderiv
+    simpa only [φ, Function.comp_def, sub_apply, zero_apply, id_apply,
+      zero_sub, ContinuousLinearMap.comp_apply, neg_apply, map_neg] using
+      congrArg (fun L : Euc n →L[ℝ] ℝ ↦ L w) h
+  have hdconv : fderiv ℝ (f ⋆[lsmul ℝ ℝ, volume] k) x w =
+      -(∫ y, f y * fderiv ℝ φ y w) := by
+    rw [(hck.hasFDerivAt_convolution_right (lsmul ℝ ℝ) hfi.locallyIntegrable
+      (hk.of_le (by simp)) x).fderiv,
+      convolution_precompR_apply (lsmul ℝ ℝ) hfi.locallyIntegrable (hck.fderiv ℝ)
+        (hk.continuous_fderiv (by simp))]
+    simp_rw [hd, mul_neg]
+    rw [integral_neg, neg_neg]
+    rfl
+  rw [hdconv, cutoffPhaseError_distribution hn hmin hv hw hη hc hφ hR hs]
+  congr 1 <;> apply integral_congr_ae <;> exact ae_of_all _ fun y ↦ by
+    change _ = _ * k (x-y)
+    dsimp [φ]
+    ring
+
+theorem integral_abs_convolution_le {n : ℕ} (μ : Measure (Euc n)) [SFinite μ]
+    {f k : Euc n → ℝ} (hf : Integrable f μ) (hk : Integrable k volume) :
+    (∫ x, |(f ⋆[lsmul ℝ ℝ, μ] k) x|) ≤
+      (∫ y, |f y| ∂μ) * ∫ x, |k x| := by
+  have hi := (hf.convolution_integrand (lsmul ℝ ℝ) hk).integral_prod_left
+  have hi' := (hf.abs.convolution_integrand (lsmul ℝ ℝ) hk.abs).integral_prod_left
+  calc
+    _ ≤ ∫ x, ((fun y ↦ |f y|) ⋆[lsmul ℝ ℝ, μ] (fun y ↦ |k y|)) x := by
+      apply integral_mono hi.abs hi'
+      intro x
+      change |∫ y, f y * k (x-y) ∂μ| ≤ ∫ y, |f y| * |k (x-y)| ∂μ
+      simpa only [abs_mul] using (abs_integral_le_integral_abs (f := fun y ↦ f y * k (x-y)) (μ := μ))
+    _ = _ := integral_convolution (lsmul ℝ ℝ) hf.abs hk.abs
+
+theorem smooth_directional_translation_lintegral_le {n : ℕ} {u : Euc n → ℝ}
+    (hu : ContDiff ℝ 1 u) (a : Euc n) :
+    (∫⁻ x, ‖u (x+a)-u x‖ₑ) ≤ ∫⁻ x, ‖fderiv ℝ u x a‖ₑ := by
+  have hfd := hu.continuous_fderiv one_ne_zero
+  have hpoint (x : Euc n) : ‖u (x+a)-u x‖ₑ ≤
+      ∫⁻ t in Ioc (0:ℝ) 1, ‖fderiv ℝ u (x+t•a) a‖ₑ := by
+    have hd (t : ℝ) : HasDerivAt (fun s : ℝ ↦ u (x+s•a))
+        (fderiv ℝ u (x+t•a) a) t := by
+      simpa only [Function.comp_def, id_eq, one_smul] using
+        (hu.differentiable one_ne_zero (x+t•a)).hasFDerivAt.comp_hasDerivAt t
+          (HasDerivAt.const_add x ((hasDerivAt_id t).smul_const a))
+    have hi : IntervalIntegrable (fun t : ℝ ↦ fderiv ℝ u (x+t•a) a) volume 0 1 :=
+      ((hfd.comp (continuous_const.add (continuous_id.smul continuous_const))).clm_apply
+        continuous_const).intervalIntegrable _ _
+    have he := intervalIntegral.integral_eq_sub_of_hasDerivAt (fun t _ ↦ hd t) hi
+    simp only [one_smul, zero_smul, add_zero] at he
+    rw [← he, intervalIntegral.integral_of_le (by norm_num : (0:ℝ) ≤ 1)]
+    exact enorm_integral_le_lintegral_enorm _
+  have hm : Measurable (fun p : Euc n × ℝ ↦ ‖fderiv ℝ u (p.1+p.2•a) a‖ₑ) :=
+    ((hfd.comp (continuous_fst.add (continuous_snd.smul continuous_const))).clm_apply
+      continuous_const).enorm.measurable
+  calc
+    _ ≤ ∫⁻ x, ∫⁻ t in Ioc (0:ℝ) 1, ‖fderiv ℝ u (x+t•a) a‖ₑ := lintegral_mono hpoint
+    _ = ∫⁻ t in Ioc (0:ℝ) 1, ∫⁻ x, ‖fderiv ℝ u (x+t•a) a‖ₑ :=
+      lintegral_lintegral_swap hm.aemeasurable
+    _ = ∫⁻ t in Ioc (0:ℝ) 1, ∫⁻ x, ‖fderiv ℝ u x a‖ₑ := by
+      apply lintegral_congr
+      intro t
+      exact lintegral_add_right_eq_self (fun x : Euc n ↦ ‖fderiv ℝ u x a‖ₑ) (t•a)
+    _ = _ := by simp
+
+theorem minimizing_normal_scalar_integrable {n : ℕ} {E : Set (Euc n)}
+    (hmin : IsLocallyPerimeterMinimizing E) {η : Euc n → ℝ}
+    (hη : Continuous η) (hc : HasCompactSupport η) (w : Euc n)
+    {c : Euc n} {R : ℝ} (hR : 0 < R) :
+    Integrable (fun x ↦ η x * inner ℝ (approximateNormal E x) w)
+      ((perimeterMeasure E).restrict (Metric.ball c R)) := by
+  let μ := (perimeterMeasure E).restrict (Metric.ball c R)
+  have : IsFiniteMeasure μ := ⟨by
+    rw [Measure.restrict_apply_univ, perimeterMeasure_open E Metric.isOpen_ball]
+    exact hmin.2.1 c R hR⟩
+  obtain ⟨M,hM⟩ := hc.exists_bound_of_continuous hη
+  have hM0 : 0 ≤ M := (norm_nonneg (η c)).trans (hM c)
+  apply Integrable.of_bound
+    (hη.aestronglyMeasurable.mul
+      ((measurable_approximateNormal hmin.1).inner measurable_const).aestronglyMeasurable)
+    (M*‖w‖)
+  apply ae_of_all μ
+  intro x
+  change ‖η x * inner ℝ (approximateNormal E x) w‖ ≤ _
+  rw [norm_mul]
+  have hb : ‖inner ℝ (approximateNormal E x) w‖ ≤ ‖w‖ :=
+    (norm_inner_le_norm (𝕜 := ℝ) _ _).trans (by
+      simpa only [one_mul] using mul_le_mul_of_nonneg_right
+        (approximateNormal_norm_le_one E x) (norm_nonneg w))
+  exact mul_le_mul (hM x) hb (norm_nonneg _) hM0
+
+theorem cutoffPhaseError_convolution_variation_le {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    {v w : Euc n} (hv : ‖v‖ = 1) (hw : inner ℝ v w = 0)
+    {η k : Euc n → ℝ} (hη : ContDiff ℝ ∞ η) (hc : HasCompactSupport η)
+    (hk : ContDiff ℝ ∞ k) (hck : HasCompactSupport k) {R : ℝ} (hR : 0 < R)
+    (hs : tsupport η ⊆ Metric.ball 0 R) :
+    (∫ x, |fderiv ℝ (cutoffPhaseError E v η ⋆[lsmul ℝ ℝ, volume] k) x w|) ≤
+      ((∫ x, |(E.indicator (fun _ ↦ (1:ℝ)) x -
+        {z | inner ℝ v z < 0}.indicator (fun _ ↦ (1:ℝ)) x) * fderiv ℝ η x w|) +
+      ∫ x in Metric.ball 0 R, |η x * inner ℝ (approximateNormal E x) w|
+        ∂perimeterMeasure E) * ∫ x, |k x| := by
+  let μ := (perimeterMeasure E).restrict (Metric.ball 0 R)
+  have : IsFiniteMeasure μ := ⟨by
+    rw [Measure.restrict_apply_univ, perimeterMeasure_open E Metric.isOpen_ball]
+    exact hmin.2.1 0 R hR⟩
+  let g : Euc n → ℝ := fun x ↦ (E.indicator (fun _ ↦ (1:ℝ)) x -
+    {z | inner ℝ v z < 0}.indicator (fun _ ↦ (1:ℝ)) x) * fderiv ℝ η x w
+  let q : Euc n → ℝ := fun x ↦ η x * inner ℝ (approximateNormal E x) w
+  have hg : Integrable g volume := integrable_phase_error_mul hmin.1
+    (isOpen_lt (continuous_const.inner continuous_id) continuous_const).measurableSet
+    (((hη.continuous_fderiv (by simp)).clm_apply continuous_const).integrable_of_hasCompactSupport
+      (hc.fderiv_apply ℝ w))
+  have hq : Integrable q μ := minimizing_normal_scalar_integrable hmin hη.continuous hc w hR
+  have hki : Integrable k volume := hk.continuous.integrable_of_hasCompactSupport hck
+  have hgk : Integrable (g ⋆[lsmul ℝ ℝ, volume] k) volume :=
+    (hg.convolution_integrand (lsmul ℝ ℝ) hki).integral_prod_left
+  have hqk : Integrable (q ⋆[lsmul ℝ ℝ, μ] k) volume :=
+    (hq.convolution_integrand (lsmul ℝ ℝ) hki).integral_prod_left
+  simp_rw [cutoffPhaseError_convolution_fderiv hn hmin hv hw hη hc hk hck hR hs]
+  calc
+    _ ≤ (∫ x, |(g ⋆[lsmul ℝ ℝ, volume] k) x|) +
+        ∫ x, |(q ⋆[lsmul ℝ ℝ, μ] k) x| := by
+      rw [← integral_add hgk.abs hqk.abs]
+      apply integral_mono (hgk.sub hqk).abs (hgk.abs.add hqk.abs)
+      intro x
+      exact abs_sub _ _
+    _ ≤ (∫ x, |g x|) * (∫ x, |k x|) + (∫ x, |q x| ∂μ) * ∫ x, |k x| :=
+      add_le_add (integral_abs_convolution_le volume hg hki) (integral_abs_convolution_le μ hq hki)
+    _ = _ := by rw [← add_mul]
+
+theorem localizedPhaseError_integrable {n : ℕ} {E : Set (Euc n)}
+    (hE : MeasurableSet E) (v : Euc n) (R : ℝ) :
+    Integrable (localizedPhaseError E v R) volume := by
+  rw [localizedPhaseError, integrable_indicator_iff measurableSet_ball]
+  apply (integrableOn_const measure_ball_lt_top.ne (C := (1:ℝ))).mono'
+    (((measurable_const.indicator hE).sub
+      (measurable_const.indicator (isOpen_lt (continuous_const.inner continuous_id)
+        continuous_const).measurableSet)).aestronglyMeasurable.restrict)
+  exact ae_of_all _ fun x ↦ by
+    by_cases hxE : x ∈ E <;> by_cases hxH : inner ℝ v x < 0 <;> simp [hxE,hxH]
+
+theorem phase_error_cutoff_derivative_le {n : ℕ} {E : Set (Euc n)}
+    (hE : MeasurableSet E) (v w : Euc n)
+    {η : Euc n → ℝ} (hη : ContDiff ℝ ∞ η) (hc : HasCompactSupport η)
+    {R L : ℝ} (hs : tsupport η ⊆ Metric.ball 0 R) (hL : 0 ≤ L)
+    (hgrad : ∀ x ∈ Metric.ball 0 R, ‖gradient η x‖ ≤ L) :
+    (∫ x, |(E.indicator (fun _ ↦ (1:ℝ)) x -
+      {z | inner ℝ v z < 0}.indicator (fun _ ↦ (1:ℝ)) x) * fderiv ℝ η x w|) ≤
+      (L * ‖w‖) * ∫ x, |localizedPhaseError E v R x| := by
+  have hi := integrable_phase_error_mul (H := {z | inner ℝ v z < 0}) hE
+    (isOpen_lt (continuous_const.inner continuous_id) continuous_const).measurableSet
+    (((hη.continuous_fderiv (by simp)).clm_apply continuous_const).integrable_of_hasCompactSupport
+      (hc.fderiv_apply ℝ w))
+  rw [← integral_const_mul]
+  apply integral_mono hi.abs ((localizedPhaseError_integrable hE v R).abs.const_mul _)
+  intro x
+  dsimp only
+  by_cases hx : x ∈ Metric.ball (0:Euc n) R
+  · have hd : |fderiv ℝ η x w| ≤ L * ‖w‖ := by
+      have he : ‖gradient η x‖ = ‖fderiv ℝ η x‖ :=
+        (InnerProductSpace.toDual ℝ (Euc n)).symm.norm_map _
+      exact (ContinuousLinearMap.le_opNorm _ _).trans
+        (mul_le_mul_of_nonneg_right (he ▸ hgrad x hx) (norm_nonneg _))
+    rw [abs_mul]
+    simpa only [localizedPhaseError, indicator_of_mem hx, mul_comm] using
+      mul_le_mul_of_nonneg_left hd
+        (abs_nonneg (E.indicator (fun _ ↦ (1:ℝ)) x -
+          {z | inner ℝ v z < 0}.indicator (fun _ ↦ (1:ℝ)) x))
+  · have hn : x ∉ tsupport η := fun h ↦ hx (hs h)
+    rw [fderiv_of_notMem_tsupport ℝ hn]
+    simp only [zero_apply, mul_zero, abs_zero]
+    positivity
+
+theorem cutoffPhaseError_convolution_height_variation {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    {v w : Euc n} (hv : ‖v‖ = 1) (hw : inner ℝ v w = 0)
+    {η k : Euc n → ℝ} (hη : ContDiff ℝ ∞ η) (hc : HasCompactSupport η)
+    (hk : ContDiff ℝ ∞ k) (hck : HasCompactSupport k) {R : ℝ} (hR : 0 < R)
+    (hs : tsupport η ⊆ Metric.ball 0 R)
+    {a L : ℝ} (ha : 0 ≤ a) (hL : 0 ≤ L)
+    (hheight : ∀ x ∈ perimeterSupport E ∩ Metric.ball 0 R, |inner ℝ v x| ≤ a)
+    (hgrad : ∀ x ∈ Metric.ball 0 R, ‖gradient η x‖ ≤ L)
+    (hphase : ∀ᵐ x ∂volume, x ∈ Metric.ball 0 R → a < |inner ℝ v x| →
+      (x ∈ E ↔ inner ℝ v x < 0)) :
+    (∫ x, |fderiv ℝ (cutoffPhaseError E v η ⋆[lsmul ℝ ℝ, volume] k) x w|) ≤
+      (‖w‖ * a * L * (volume.real (Metric.ball (0:Euc n) (2*R)) / R +
+        2 * (perimeter E (Metric.ball 0 R)).toReal)) * ∫ x, |k x| := by
+  have herr : (∫ x, |localizedPhaseError E v R x|) ≤
+      a * (volume.real (Metric.ball (0:Euc n) (2*R)) / R) := by
+    have hi := slab_integral_bound (localizedPhaseError_measurable hmin.1 v R) hv ha hR
+      (show Function.support (localizedPhaseError E v R) ⊆ Metric.ball 0 R from
+        support_indicator_subset)
+      (localizedPhaseError_slab_bound E v R a hphase)
+    simp only [Real.norm_eq_abs] at hi
+    have ht : 0 < 2*R := by positivity
+    rw [← mul_div_assoc]
+    apply (le_div_iff₀ hR).mpr
+    nlinarith
+  have hn' := minimizing_horizontal_normal_integral_le hn hmin 0 hv hw 0 hR hη hc hs
+    ha hL (by simpa only [sub_zero] using hheight) hgrad
+  have hg := phase_error_cutoff_derivative_le hmin.1 v w hη hc hs hL hgrad
+  have hg' := hg.trans (mul_le_mul_of_nonneg_left herr (by positivity : 0 ≤ L*‖w‖))
+  have hb := add_le_add hg' hn'
+  have hb' : ((∫ x, |(E.indicator (fun _ ↦ (1:ℝ)) x -
+      {z | inner ℝ v z < 0}.indicator (fun _ ↦ (1:ℝ)) x) * fderiv ℝ η x w|) +
+      ∫ x in Metric.ball 0 R, |η x * inner ℝ (approximateNormal E x) w| ∂perimeterMeasure E) ≤
+      ‖w‖ * a * L * (volume.real (Metric.ball (0:Euc n) (2*R)) / R +
+        2 * (perimeter E (Metric.ball 0 R)).toReal) := by
+    exact hb.trans_eq (by ring)
+  exact (cutoffPhaseError_convolution_variation_le hn hmin hv hw hη hc hk hck hR hs).trans
+    (mul_le_mul_of_nonneg_right hb' (integral_nonneg fun x ↦ abs_nonneg _))
+
+theorem translation_lintegral_le_of_mollifier_direction_bound {n : ℕ}
+    {f : Euc n → ℝ} (hf : LocallyIntegrable f volume) (hc : HasCompactSupport f)
+    (a : Euc n) (B : ℝ)
+    (hb : ∀ j, (∫ x, |fderiv ℝ (c1Mollify f j) x a|) ≤ B) :
+    (∫⁻ x, ‖f (x+a)-f x‖ₑ) ≤ ENNReal.ofReal B := by
+  let u : ℕ → Euc n → ℝ := c1Mollify f
+  have hu : ∀ j, ContDiff ℝ ∞ (u j) := fun j ↦
+    (c1Mollifier j).hasCompactSupport_normed.contDiff_convolution_left _
+      (c1Mollifier j).contDiff_normed hf
+  have hcu : ∀ j, HasCompactSupport (u j) := fun j ↦ c1Mollify_hasCompactSupport hc j
+  have hlim : ∀ᵐ x ∂volume, Tendsto (fun j ↦ u j x) atTop (𝓝 (f x)) := by
+    apply ContDiffBump.ae_convolution_tendsto_right_of_locallyIntegrable
+      c1Mollifier_radius_tendsto (K := 2) _ hf
+    apply Eventually.of_forall
+    intro j
+    dsimp [c1Mollifier]
+    linarith
+  have hlimadd : ∀ᵐ x ∂volume, Tendsto (fun j ↦ u j (x+a)) atTop (𝓝 (f (x+a))) :=
+    (measurePreserving_add_right volume a).quasiMeasurePreserving.ae hlim
+  have hlimnorm : ∀ᵐ x ∂volume, Tendsto (fun j ↦ ‖u j (x+a)-u j x‖ₑ) atTop
+      (𝓝 (‖f (x+a)-f x‖ₑ)) := by
+    filter_upwards [hlimadd,hlim] with x hx hy
+    exact (hx.sub hy).enorm
+  calc
+    _ = ∫⁻ x, liminf (fun j ↦ ‖u j (x+a)-u j x‖ₑ) atTop := by
+      apply lintegral_congr_ae
+      filter_upwards [hlimnorm] with x hx using hx.liminf_eq.symm
+    _ ≤ liminf (fun j ↦ ∫⁻ x, ‖u j (x+a)-u j x‖ₑ) atTop :=
+      lintegral_liminf_le' (fun j ↦ (((hu j).continuous.comp
+        (continuous_id.add continuous_const)).sub (hu j).continuous).enorm.measurable.aemeasurable)
+    _ ≤ _ := by
+      have hj : ∀ j, (∫⁻ x, ‖u j (x+a)-u j x‖ₑ) ≤ ENNReal.ofReal B := by
+        intro j
+        apply (smooth_directional_translation_lintegral_le ((hu j).of_le (by simp)) a).trans
+        have hid : Integrable (fun x ↦ fderiv ℝ (u j) x a) volume :=
+          (((hu j).continuous_fderiv (by simp)).clm_apply continuous_const).integrable_of_hasCompactSupport
+            ((hcu j).fderiv_apply ℝ a)
+        rw [← ofReal_integral_norm_eq_lintegral_enorm hid]
+        apply ENNReal.ofReal_le_ofReal
+        simpa only [Real.norm_eq_abs] using hb j
+      exact liminf_le_of_frequently_le (Eventually.of_forall hj).frequently
+
+theorem cutoffPhaseError_horizontal_translation_le {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    {v w : Euc n} (hv : ‖v‖ = 1) (hw : inner ℝ v w = 0)
+    {η : Euc n → ℝ} (hη : ContDiff ℝ ∞ η) (hc : HasCompactSupport η)
+    {R : ℝ} (hR : 0 < R) (hs : tsupport η ⊆ Metric.ball 0 R)
+    {a L : ℝ} (ha : 0 ≤ a) (hL : 0 ≤ L)
+    (hheight : ∀ x ∈ perimeterSupport E ∩ Metric.ball 0 R, |inner ℝ v x| ≤ a)
+    (hgrad : ∀ x ∈ Metric.ball 0 R, ‖gradient η x‖ ≤ L)
+    (hphase : ∀ᵐ x ∂volume, x ∈ Metric.ball 0 R → a < |inner ℝ v x| →
+      (x ∈ E ↔ inner ℝ v x < 0)) :
+    (∫⁻ x, ‖cutoffPhaseError E v η (x+w)-cutoffPhaseError E v η x‖ₑ) ≤
+      ENNReal.ofReal (‖w‖ * a * L * (volume.real (Metric.ball (0:Euc n) (2*R)) / R +
+        2 * (perimeter E (Metric.ball 0 R)).toReal)) := by
+  let f := cutoffPhaseError E v η
+  have hfi : Integrable f volume := by
+    have hi := integrable_phase_error_mul (H := {z | inner ℝ v z < 0}) hmin.1
+      ((isOpen_lt (continuous_const.inner continuous_id) continuous_const).measurableSet)
+      (hη.continuous.integrable_of_hasCompactSupport hc)
+    change Integrable (fun y ↦ η y * (E.indicator (fun _ ↦ (1:ℝ)) y -
+      {z | inner ℝ v z < 0}.indicator (fun _ ↦ (1:ℝ)) y)) volume
+    simpa only [mul_comm] using hi
+  have hcf : HasCompactSupport f := hc.mul_right
+  apply translation_lintegral_le_of_mollifier_direction_bound hfi.locallyIntegrable hcf
+  intro j
+  have hflip : (lsmul ℝ ℝ : ℝ →L[ℝ] ℝ →L[ℝ] ℝ).flip = lsmul ℝ ℝ := by
+    apply ContinuousLinearMap.ext
+    intro a
+    apply ContinuousLinearMap.ext
+    intro b
+    exact mul_comm b a
+  have heq : c1Mollify f j = f ⋆[lsmul ℝ ℝ, volume] (c1Mollifier j).normed volume := by
+    change (c1Mollifier j).normed volume ⋆[lsmul ℝ ℝ, volume] f = _
+    rw [← convolution_flip, hflip]
+  rw [heq]
+  have hk1 : (∫ x, |(c1Mollifier (n := n) j).normed volume x|) = 1 := by
+    simp_rw [abs_of_nonneg ((c1Mollifier j).nonneg_normed _)]
+    exact (c1Mollifier j).integral_normed
+  have hh := cutoffPhaseError_convolution_height_variation (k := (c1Mollifier j).normed volume) hn hmin hv hw hη hc
+    (c1Mollifier j).contDiff_normed (c1Mollifier j).hasCompactSupport_normed hR hs
+    ha hL hheight hgrad hphase
+  simpa only [hk1, mul_one] using hh
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/FiberCrossing.lean b/lean/OAI/Geometry/Perimeter/Regularity/FiberCrossing.lean
new file mode 100644
index 0000000000000000000000000000000000000000..c0b9d589bd9766f25d4973936f6d2c7ecbcb12bc
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/FiberCrossing.lean
@@ -0,0 +1,180 @@
+import OAI.Geometry.Perimeter.Regularity.GeoAlgebra
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal
+namespace RPISingularHarnack
+
+def fiberPrimitive (q : ℝ → ℝ) (l s : ℝ) : ℝ := ∫ t in l..s, q t
+
+theorem fiberPrimitive_hasDerivAt {q : ℝ → ℝ} (hq : Continuous q) (l s : ℝ) :
+    HasDerivAt (fiberPrimitive q l) (q s) s :=
+  intervalIntegral.integral_hasDerivAt_right (hq.intervalIntegrable _ _)
+    hq.stronglyMeasurable.stronglyMeasurableAtFilter hq.continuousAt
+
+theorem fiberPrimitive_continuous {q : ℝ → ℝ} (hq : Continuous q) (l : ℝ) :
+    Continuous (fiberPrimitive q l) :=
+  continuous_iff_continuousAt.mpr (fun s ↦ (fiberPrimitive_hasDerivAt hq l s).continuousAt)
+
+theorem fiber_crossing_identity {q : ℝ → ℝ} (hq : Continuous q) (l u : ℝ) :
+    2 * (∫ s in l..u, (s-l)*q s) - (∫ s in l..u, q s)^2 =
+      2 * ∫ s in l..u, q s * ((s-l)-fiberPrimitive q l s) := by
+  have hF := fiberPrimitive_continuous hq l
+  have hprod : (∫ s in l..u, q s*fiberPrimitive q l s + fiberPrimitive q l s*q s) =
+      (fiberPrimitive q l u)^2 := by
+    have h := intervalIntegral.integral_deriv_mul_eq_sub (a := l) (b := u)
+      (fun s _ ↦ fiberPrimitive_hasDerivAt hq l s)
+      (fun s _ ↦ fiberPrimitive_hasDerivAt hq l s)
+      (hq.intervalIntegrable _ _) (hq.intervalIntegrable _ _)
+    simpa only [fiberPrimitive,intervalIntegral.integral_same,mul_zero,sub_zero,pow_two] using h
+  have hsq : 2*(∫ s in l..u, q s*fiberPrimitive q l s) = (∫ s in l..u, q s)^2 := by
+    have he : (fun s ↦ q s*fiberPrimitive q l s + fiberPrimitive q l s*q s) =
+        (fun s ↦ (2:ℝ)*(q s*fiberPrimitive q l s)) := by funext s; ring
+    rw [he,intervalIntegral.integral_const_mul] at hprod
+    exact hprod
+  have he : (fun s ↦ q s * ((s-l)-fiberPrimitive q l s)) =
+      (fun s ↦ (s-l)*q s-q s*fiberPrimitive q l s) := by funext s; ring
+  rw [he]
+  have hi₁ : IntervalIntegrable (fun s ↦ (s-l)*q s) volume l u :=
+    ((continuous_id.sub continuous_const).mul hq).intervalIntegrable _ _
+  have hi₂ : IntervalIntegrable (fun s ↦ q s*fiberPrimitive q l s) volume l u :=
+    (hq.mul hF).intervalIntegrable _ _
+  rw [intervalIntegral.integral_sub hi₁ hi₂]
+  linarith
+
+theorem fiber_crossing_bound_of_order_defect {q : ℝ → ℝ} (hq : Continuous q)
+    {l u D : ℝ} (hlu : l ≤ u) (hD : 0 ≤ D)
+    (hb : ∀ s ∈ Icc l u, 0 ≤ q s ∧ q s ≤ 1)
+    (horder : ∀ t ∈ Icc l u, ∀ s ∈ Icc t u, q s-q t ≤ D) :
+    0 ≤ 2*(∫ s in l..u, (s-l)*q s)-(∫ s in l..u, q s)^2 ∧
+    2*(∫ s in l..u, (s-l)*q s)-(∫ s in l..u, q s)^2 ≤
+      2*(u-l)*(∫ s in l..u, q s*(1-q s))+2*(u-l)^2*D := by
+  rw [fiber_crossing_identity hq l u]
+  have hF := fiberPrimitive_continuous hq l
+  have hi : IntervalIntegrable (fun s ↦ q s*((s-l)-fiberPrimitive q l s)) volume l u :=
+    (hq.mul ((continuous_id.sub continuous_const).sub hF)).intervalIntegrable _ _
+  have hlen (s : ℝ) : (s-l)-fiberPrimitive q l s = ∫ t in l..s, 1-q t := by
+    rw [intervalIntegral.integral_sub (continuous_const.intervalIntegrable _ _) (hq.intervalIntegrable _ _)]
+    simp [fiberPrimitive]
+  have hpoint (s : ℝ) (hs : s ∈ Icc l u) :
+      0 ≤ q s*((s-l)-fiberPrimitive q l s) ∧
+      q s*((s-l)-fiberPrimitive q l s) ≤
+        (u-l)*(q s*(1-q s))+(u-l)*D := by
+    have hqs := hb s hs
+    have hnon : 0 ≤ (s-l)-fiberPrimitive q l s := by
+      rw [hlen]
+      apply intervalIntegral.integral_nonneg hs.1
+      intro t ht
+      exact sub_nonneg.mpr (hb t ⟨ht.1,ht.2.trans hs.2⟩).2
+    refine ⟨mul_nonneg hqs.1 hnon, ?_⟩
+    have hbound : (s-l)-fiberPrimitive q l s ≤ (s-l)*(1-q s+D) := by
+      rw [hlen]
+      calc
+        _ ≤ ∫ t in l..s, (1-q s+D) := by
+          apply intervalIntegral.integral_mono_on hs.1
+            ((continuous_const.sub hq).intervalIntegrable _ _) (continuous_const.intervalIntegrable _ _)
+          intro t ht
+          change 1-q t ≤ 1-q s+D
+          have h := horder t ⟨ht.1,ht.2.trans hs.2⟩ s ⟨ht.2,hs.2⟩
+          linarith
+        _ = _ := by simp; ring
+    have h₁ := mul_le_mul_of_nonneg_left hbound hqs.1
+    have h₂ : (s-l)*(q s*(1-q s)) ≤ (u-l)*(q s*(1-q s)) := by
+      exact mul_le_mul_of_nonneg_right (by linarith [hs.2]) (mul_nonneg hqs.1 (sub_nonneg.mpr hqs.2))
+    have h₃ : (s-l)*q s*D ≤ (u-l)*D := by
+      have hsq : (s-l)*q s ≤ u-l := by
+        calc
+          _ ≤ (s-l)*1 := mul_le_mul_of_nonneg_left hqs.2 (sub_nonneg.mpr hs.1)
+          _ ≤ _ := by linarith [hs.2]
+      exact mul_le_mul_of_nonneg_right hsq hD
+    nlinarith
+  constructor
+  · exact mul_nonneg (by norm_num) (intervalIntegral.integral_nonneg hlu (fun s hs ↦ (hpoint s hs).1))
+  · have h := intervalIntegral.integral_mono_on hlu hi
+      ((continuous_const.mul (hq.mul (continuous_const.sub hq)) |>.add continuous_const).intervalIntegrable _ _)
+      (fun s hs ↦ (hpoint s hs).2)
+    change (∫ s in l..u, q s*((s-l)-fiberPrimitive q l s)) ≤
+      ∫ s in l..u, (u-l)*(q s*(1-q s))+(u-l)*D at h
+    have hi₁ : IntervalIntegrable (fun s ↦ (u-l)*(q s*(1-q s))) volume l u :=
+      (continuous_const.mul (hq.mul (continuous_const.sub hq))).intervalIntegrable _ _
+    have hi₂ : IntervalIntegrable (fun _ : ℝ ↦ (u-l)*D) volume l u :=
+      continuous_const.intervalIntegrable _ _
+    rw [intervalIntegral.integral_add hi₁ hi₂,intervalIntegral.integral_const_mul] at h
+    simp only [intervalIntegral.integral_const,smul_eq_mul] at h
+    nlinarith
+
+theorem fiber_order_defect_of_positive_derivative {q : ℝ → ℝ}
+    (hq : ContDiff ℝ 1 q) {l u t s : ℝ}
+    (ht : t ∈ Icc l u) (hs : s ∈ Icc t u) :
+    q s-q t ≤ ∫ r in l..u, max (deriv q r) 0 := by
+  have hd : Continuous (deriv q) := hq.continuous_deriv (by simp)
+  rw [← intervalIntegral.integral_deriv_eq_sub
+    (fun x _ ↦ hq.differentiable (by simp) x) (hd.intervalIntegrable _ _)]
+  calc
+    _ ≤ ∫ r in t..s, max (deriv q r) 0 :=
+      intervalIntegral.integral_mono hs.1 (hd.intervalIntegrable _ _)
+        ((hd.max continuous_const).intervalIntegrable _ _) (fun _ ↦ le_max_left _ _)
+    _ ≤ _ := intervalIntegral.integral_mono_interval ht.1 hs.1 hs.2
+      (ae_of_all _ (fun _ ↦ le_max_right _ _)) ((hd.max continuous_const).intervalIntegrable _ _)
+
+theorem fiber_crossing_bound {q : ℝ → ℝ} (hq : ContDiff ℝ 1 q)
+    {l u : ℝ} (hlu : l ≤ u) (hb : ∀ s ∈ Icc l u, 0 ≤ q s ∧ q s ≤ 1) :
+    0 ≤ 2*(∫ s in l..u, (s-l)*q s)-(∫ s in l..u, q s)^2 ∧
+    2*(∫ s in l..u, (s-l)*q s)-(∫ s in l..u, q s)^2 ≤
+      2*(u-l)*(∫ s in l..u, q s*(1-q s))+
+        2*(u-l)^2*(∫ s in l..u, max (deriv q s) 0) := by
+  apply fiber_crossing_bound_of_order_defect hq.continuous hlu
+    (intervalIntegral.integral_nonneg hlu (fun _ _ ↦ le_max_right _ _)) hb
+  exact fun _ ht _ hs ↦ fiber_order_defect_of_positive_derivative hq ht hs
+
+theorem fiber_flux_height_identity {q : ℝ → ℝ} (hq : ContDiff ℝ 1 q)
+    {l u : ℝ} (hl : q l = 1) (hu : q u = 0) :
+    (∫ s in l..u, s^2 * (-deriv q s)) - (l+∫ s in l..u, q s)^2 =
+      2*(∫ s in l..u, (s-l)*q s)-(∫ s in l..u, q s)^2 := by
+  have hd : Continuous (deriv q) := hq.continuous_deriv (by simp)
+  have hpow (s : ℝ) : HasDerivAt (fun t : ℝ ↦ t^2) (2*s) s := by
+    simpa using (hasDerivAt_pow 2 s)
+  have hprod := intervalIntegral.integral_deriv_mul_eq_sub (a := l) (b := u)
+    (fun s _ ↦ hpow s) (fun s _ ↦ (hq.differentiable (by simp) s).hasDerivAt)
+    ((continuous_const.mul continuous_id).intervalIntegrable l u)
+    (hd.intervalIntegrable l u)
+  have hi₁ : IntervalIntegrable (fun s ↦ 2*s*q s) volume l u :=
+    ((continuous_const.mul continuous_id).mul hq.continuous).intervalIntegrable _ _
+  have hi₂ : IntervalIntegrable (fun s ↦ s^2*deriv q s) volume l u :=
+    ((continuous_id.pow 2).mul hd).intervalIntegrable _ _
+  rw [intervalIntegral.integral_add hi₁ hi₂,hl,hu,mul_zero,mul_one,zero_sub] at hprod
+  have he₁ : (∫ s in l..u, s^2*(-deriv q s)) = -(∫ s in l..u, s^2*deriv q s) := by
+    simp_rw [mul_neg]
+    exact intervalIntegral.integral_neg
+  have he₂ : (∫ s in l..u, (s-l)*q s) =
+      (∫ s in l..u, s*q s)-l*(∫ s in l..u, q s) := by
+    simp_rw [sub_mul]
+    have hi₁ : IntervalIntegrable (fun s ↦ s*q s) volume l u :=
+      (continuous_id.mul hq.continuous).intervalIntegrable _ _
+    have hi₂ : IntervalIntegrable (fun s ↦ l*q s) volume l u :=
+      (continuous_const.mul hq.continuous).intervalIntegrable _ _
+    rw [intervalIntegral.integral_sub hi₁ hi₂, intervalIntegral.integral_const_mul]
+  have he₃ : (∫ s in l..u, 2*s*q s) = 2*(∫ s in l..u, s*q s) := by
+    simp_rw [mul_assoc]
+    exact intervalIntegral.integral_const_mul _ _
+  rw [he₃] at hprod
+  rw [he₁,he₂]
+  nlinarith
+
+theorem fiber_flux_height_defect {q : ℝ → ℝ} (hq : ContDiff ℝ 1 q)
+    {l u : ℝ} (hlu : l ≤ u) (hl : q l = 1) (hu : q u = 0)
+    (hb : ∀ s ∈ Icc l u, 0 ≤ q s ∧ q s ≤ 1) :
+    0 ≤ (∫ s in l..u, s^2*(-deriv q s))-(l+∫ s in l..u, q s)^2 ∧
+    (∫ s in l..u, s^2*(-deriv q s))-(l+∫ s in l..u, q s)^2 ≤
+      2*(u-l)*(∫ s in l..u, q s*(1-q s))+
+        2*(u-l)^2*(∫ s in l..u, max (deriv q s) 0) := by
+  rw [fiber_flux_height_identity hq hl hu]
+  exact fiber_crossing_bound hq hlu hb
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/FiberHeight.lean b/lean/OAI/Geometry/Perimeter/Regularity/FiberHeight.lean
new file mode 100644
index 0000000000000000000000000000000000000000..0d1024eabe01b7d404025c1925463b2e285f794f
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/FiberHeight.lean
@@ -0,0 +1,350 @@
+import OAI.Geometry.Perimeter.Regularity.EllipticLocality
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff Pointwise
+namespace RPISingularHarnack
+open RPI.Boundary
+
+theorem minimizing_phase_constant_off_support {n : ℕ} (hn : 1 < n)
+    {E U : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    (hU : IsPreconnected U) (hdis : Disjoint U (perimeterSupport E)) :
+    (∀ᵐ x ∂volume, x ∈ U → x ∈ E) ∨ (∀ᵐ x ∂volume, x ∈ U → x ∉ E) := by
+  let K := (volume.restrict E).support
+  have hK : IsClosed K := Measure.isClosed_support
+  have hp : Real.HolderConjugate (n : ℝ) (Real.conjExponent (n : ℝ)) :=
+    Real.HolderConjugate.conjExponent (by exact_mod_cast hn)
+  have heK : E =ᵐ[volume] K := minimizing_ae_eq_phaseSupport hp hmin
+  have hf : frontier K = perimeterSupport E := minimizing_frontier_phaseSupport hp hmin
+  have hcover : U ⊆ interior K ∪ Kᶜ := by
+    intro x hx
+    by_cases hxK : x ∈ K
+    · left
+      by_contra hxi
+      have hxf : x ∈ frontier K := by
+        rw [frontier, hK.closure_eq]
+        exact ⟨hxK,hxi⟩
+      exact Set.disjoint_left.mp hdis hx (hf ▸ hxf)
+    · exact Or.inr hxK
+  rcases hU.subset_or_subset isOpen_interior hK.isOpen_compl
+    (Set.disjoint_left.mpr fun x hx hx' ↦ hx' (interior_subset hx)) hcover with h | h
+  · left
+    filter_upwards [heK.mem_iff] with x hx
+    intro hxU
+    exact hx.mpr (interior_subset (h hxU))
+  · right
+    filter_upwards [heK.mem_iff] with x hx
+    intro hxU hxE
+    exact h hxU (hx.mp hxE)
+
+theorem minimizing_phase_absent_of_anchor {n : ℕ} (hn : 1 < n)
+    {E U A : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    (hU : IsPreconnected U) (hdis : Disjoint U (perimeterSupport E))
+    (hAU : A ⊆ U) (hanchor : 0 < volume (A \ E)) :
+    ∀ᵐ x ∂volume, x ∈ U → x ∉ E := by
+  rcases minimizing_phase_constant_off_support hn hmin hU hdis with h | h
+  · have hz : volume (A \ E) = 0 := by
+      apply measure_mono_null (fun x hx ↦ show x ∈ {x | ¬ (x ∈ U → x ∈ E)} from
+        fun hh ↦ hx.2 (hh (hAU hx.1)))
+      exact ae_iff.mp h
+    exact (hanchor.ne' hz).elim
+  · exact h
+
+theorem minimizing_phase_present_of_anchor {n : ℕ} (hn : 1 < n)
+    {E U A : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    (hU : IsPreconnected U) (hdis : Disjoint U (perimeterSupport E))
+    (hAU : A ⊆ U) (hanchor : 0 < volume (A ∩ E)) :
+    ∀ᵐ x ∂volume, x ∈ U → x ∈ E := by
+  rcases minimizing_phase_constant_off_support hn hmin hU hdis with h | h
+  · exact h
+  · have hz : volume (A ∩ E) = 0 := by
+      apply measure_mono_null (fun x hx ↦ show x ∈ {x | ¬ (x ∈ U → x ∉ E)} from
+        fun hh ↦ hh (hAU hx.1) hx.2)
+      exact ae_iff.mp h
+    exact (hanchor.ne' hz).elim
+
+theorem minimizing_slab_phase_agreement {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    (v : Euc n) {R a : ℝ} (ha : 0 ≤ a)
+    (hheight : ∀ x ∈ perimeterSupport E ∩ Metric.ball 0 R, |inner ℝ v x| ≤ a)
+    {A B : Set (Euc n)}
+    (hA : A ⊆ Metric.ball 0 R ∩ {x | a < inner ℝ v x})
+    (hB : B ⊆ Metric.ball 0 R ∩ {x | inner ℝ v x < -a})
+    (hanchorA : 0 < volume (A \ E)) (hanchorB : 0 < volume (B ∩ E)) :
+    ∀ᵐ x ∂volume, x ∈ Metric.ball 0 R → a < |inner ℝ v x| →
+      (x ∈ E ↔ inner ℝ v x < 0) := by
+  have hu : IsPreconnected (Metric.ball (0 : Euc n) R ∩ {x | a < inner ℝ v x}) :=
+    ((convex_ball _ _).inter ((convex_Ioi a).linear_preimage (innerSL ℝ v).toLinearMap)).isPreconnected
+  have hl : IsPreconnected (Metric.ball (0 : Euc n) R ∩ {x | inner ℝ v x < -a}) :=
+    ((convex_ball _ _).inter ((convex_Iio (-a)).linear_preimage (innerSL ℝ v).toLinearMap)).isPreconnected
+  have hdu : Disjoint (Metric.ball (0 : Euc n) R ∩ {x | a < inner ℝ v x})
+      (perimeterSupport E) := by
+    apply Set.disjoint_left.mpr
+    intro x hx hs
+    exact (hx.2.trans_le ((le_abs_self _).trans (hheight x ⟨hs,hx.1⟩))).false
+  have hdl : Disjoint (Metric.ball (0 : Euc n) R ∩ {x | inner ℝ v x < -a})
+      (perimeterSupport E) := by
+    apply Set.disjoint_left.mpr
+    intro x hx hs
+    have hh := (abs_le.mp (hheight x ⟨hs,hx.1⟩)).1
+    exact (hx.2.trans_le hh).false
+  have hupper := minimizing_phase_absent_of_anchor hn hmin hu hdu hA hanchorA
+  have hlower := minimizing_phase_present_of_anchor hn hmin hl hdl hB hanchorB
+  filter_upwards [hupper,hlower] with x hx hy
+  intro hxball hxa
+  rcases lt_abs.mp hxa with hpos | hneg
+  · have hnot := hx ⟨hxball,hpos⟩
+    simp only [hnot, false_iff, not_lt]
+    exact ha.trans hpos.le
+  · have hyes := hy ⟨hxball,by change inner ℝ v x < -a; linarith⟩
+    simp only [hyes, true_iff]
+    linarith
+
+def localizedPhaseError {n : ℕ} (E : Set (Euc n)) (v : Euc n) (R : ℝ) : Euc n → ℝ :=
+  (Metric.ball 0 R).indicator (fun x ↦ E.indicator (fun _ ↦ (1:ℝ)) x -
+    {z | inner ℝ v z < 0}.indicator (fun _ ↦ (1:ℝ)) x)
+
+def fiberHeight {n : ℕ} (E : Set (Euc n)) (v : Euc n) (R : ℝ) (x : Euc n) : ℝ :=
+  ∫ s : ℝ, localizedPhaseError E v R (x + s • v)
+
+theorem localizedPhaseError_measurable {n : ℕ} {E : Set (Euc n)}
+    (hE : MeasurableSet E) (v : Euc n) (R : ℝ) :
+    Measurable (localizedPhaseError E v R) := by
+  exact ((measurable_const.indicator hE).sub (measurable_const.indicator
+    (isOpen_lt (continuous_const.inner continuous_id) continuous_const).measurableSet)).indicator
+      measurableSet_ball
+
+theorem localizedPhaseError_norm_le_one {n : ℕ} (E : Set (Euc n))
+    (v : Euc n) (R : ℝ) (x : Euc n) : ‖localizedPhaseError E v R x‖ ≤ 1 := by
+  classical
+  by_cases hx : x ∈ Metric.ball (0 : Euc n) R
+  · by_cases hE : x ∈ E <;> by_cases hH : inner ℝ v x < 0 <;>
+      simp [localizedPhaseError,hx,hE,hH]
+  · simp [localizedPhaseError,hx]
+
+theorem fiberHeight_add_smul {n : ℕ} (E : Set (Euc n)) (v : Euc n) (R t : ℝ)
+    (x : Euc n) : fiberHeight E v R (x+t•v) = fiberHeight E v R x := by
+  unfold fiberHeight
+  have heq : (fun s : ℝ ↦ localizedPhaseError E v R (x+t•v+s•v)) =
+      (fun s : ℝ ↦ localizedPhaseError E v R (x+(s+t)•v)) := by
+    funext s
+    congr 1
+    rw [add_smul]
+    abel
+  rw [heq]
+  exact integral_add_right_eq_self (fun s : ℝ ↦ localizedPhaseError E v R (x+s•v)) t
+
+theorem fiberHeight_stronglyMeasurable {n : ℕ} {E : Set (Euc n)}
+    (hE : MeasurableSet E) (v : Euc n) (R : ℝ) :
+    StronglyMeasurable (fiberHeight E v R) := by
+  apply StronglyMeasurable.integral_prod_right
+  exact ((localizedPhaseError_measurable hE v R).comp
+    (continuous_fst.add (continuous_snd.smul continuous_const)).measurable).stronglyMeasurable
+
+theorem unit_inner_add_smul {n : ℕ} {v : Euc n} (hv : ‖v‖ = 1)
+    (x : Euc n) (s : ℝ) : inner ℝ v (x+s•v) = inner ℝ v x+s := by
+  simp only [inner_add_right, real_inner_smul_right, real_inner_self_eq_norm_sq, hv,
+    one_pow, mul_one]
+
+theorem ae_fiber_norm_bound {n : ℕ} {f : Euc n → ℝ} (hf : Measurable f)
+    {v : Euc n} (hv : ‖v‖ = 1) {a : ℝ}
+    (hbound : ∀ᵐ x ∂volume, ‖f x‖ ≤ (Icc (-a) a).indicator (fun _ ↦ (1:ℝ)) (inner ℝ v x)) :
+    ∀ᵐ x ∂volume, ∀ᵐ s : ℝ ∂volume,
+      ‖f (x+s•v)‖ ≤ (Icc (-a-inner ℝ v x) (a-inner ℝ v x)).indicator (fun _ ↦ (1:ℝ)) s := by
+  have hall : ∀ᵐ s : ℝ ∂volume, ∀ᵐ x ∂volume,
+      ‖f (x+s•v)‖ ≤ (Icc (-a) a).indicator (fun _ ↦ (1:ℝ)) (inner ℝ v (x+s•v)) := by
+    exact ae_of_all _ fun s ↦ (measurePreserving_add_right volume (s•v)).quasiMeasurePreserving.ae hbound
+  have hm : MeasurableSet {p : ℝ × Euc n |
+      ‖f (p.2+p.1•v)‖ ≤ (Icc (-a) a).indicator (fun _ ↦ (1:ℝ)) (inner ℝ v (p.2+p.1•v))} := by
+    apply measurableSet_le
+    · exact (hf.comp (continuous_snd.add (continuous_fst.smul continuous_const)).measurable).norm
+    · exact (measurable_const.indicator measurableSet_Icc).comp
+        ((continuous_const.inner (continuous_snd.add (continuous_fst.smul continuous_const))).measurable)
+  have hh := (Measure.ae_ae_comm hm).mp hall
+  filter_upwards [hh] with x hx
+  filter_upwards [hx] with s hs
+  have he : inner ℝ v (x+s•v) ∈ Icc (-a) a ↔
+      s ∈ Icc (-a-inner ℝ v x) (a-inner ℝ v x) := by
+    rw [unit_inner_add_smul hv]
+    simp only [mem_Icc]
+    constructor <;> rintro ⟨h₁,h₂⟩ <;> constructor <;> linarith
+  simpa only [indicator_apply, he] using hs
+
+theorem norm_fiber_integral_le {n : ℕ} {f : Euc n → ℝ} (hf : Measurable f)
+    {v : Euc n} (hv : ‖v‖ = 1) {a : ℝ} (ha : 0 ≤ a)
+    (hbound : ∀ᵐ x ∂volume, ‖f x‖ ≤ (Icc (-a) a).indicator (fun _ ↦ (1:ℝ)) (inner ℝ v x)) :
+    ∀ᵐ x ∂volume, ‖∫ s : ℝ, f (x+s•v)‖ ≤ 2*a := by
+  filter_upwards [ae_fiber_norm_bound hf hv hbound] with x hx
+  let I := Icc (-a-inner ℝ v x) (a-inner ℝ v x)
+  have hI : Integrable (I.indicator (fun _ ↦ (1:ℝ))) volume := by
+    exact ((continuous_const.integrableOn_Icc).integrable_indicator measurableSet_Icc)
+  have hfx : Integrable (fun s : ℝ ↦ f (x+s•v)) volume :=
+    hI.mono' (hf.comp (continuous_const.add (continuous_id.smul continuous_const)).measurable).aestronglyMeasurable hx
+  calc
+    ‖∫ s : ℝ, f (x+s•v)‖ ≤ ∫ s : ℝ, ‖f (x+s•v)‖ := norm_integral_le_integral_norm _
+    _ ≤ ∫ s : ℝ, I.indicator (fun _ ↦ (1:ℝ)) s := integral_mono_ae hfx.norm hI hx
+    _ = 2*a := by
+      rw [integral_indicator measurableSet_Icc, integral_const]
+      simp only [smul_eq_mul, mul_one, Measure.real, Measure.restrict_apply_univ, Real.volume_Icc]
+      rw [show a-inner ℝ v x-(-a-inner ℝ v x) = 2*a by ring, ENNReal.toReal_ofReal (by positivity)]
+
+theorem localizedPhaseError_slab_bound {n : ℕ} (E : Set (Euc n)) (v : Euc n)
+    (R a : ℝ)
+    (hphase : ∀ᵐ x ∂volume, x ∈ Metric.ball 0 R → a < |inner ℝ v x| →
+      (x ∈ E ↔ inner ℝ v x < 0)) :
+    ∀ᵐ x ∂volume, ‖localizedPhaseError E v R x‖ ≤
+      (Icc (-a) a).indicator (fun _ ↦ (1:ℝ)) (inner ℝ v x) := by
+  classical
+  filter_upwards [hphase] with x hx
+  by_cases hc : inner ℝ v x ∈ Icc (-a) a
+  · simpa only [indicator_of_mem hc] using localizedPhaseError_norm_le_one E v R x
+  · rw [indicator_of_notMem hc]
+    have hez : localizedPhaseError E v R x = 0 := by
+      by_cases hxball : x ∈ Metric.ball (0 : Euc n) R
+      · have hh : a < |inner ℝ v x| := lt_of_not_ge (fun h ↦ hc (abs_le.mp h))
+        have he := hx hxball hh
+        by_cases hxE : x ∈ E
+        · simp [localizedPhaseError,hxball,hxE,he.mp hxE]
+        · have hxH : ¬inner ℝ v x < 0 := fun hh ↦ hxE (he.mpr hh)
+          simp [localizedPhaseError,hxball,hxE,hxH]
+      · simp [localizedPhaseError,hxball]
+    simp only [hez,norm_zero,le_refl]
+
+theorem fiberHeight_slab_bound {n : ℕ} {E : Set (Euc n)} (hE : MeasurableSet E)
+    {v : Euc n} (hv : ‖v‖ = 1) (R : ℝ) {a : ℝ} (ha : 0 ≤ a)
+    (hphase : ∀ᵐ x ∂volume, x ∈ Metric.ball 0 R → a < |inner ℝ v x| →
+      (x ∈ E ↔ inner ℝ v x < 0)) :
+    ∀ᵐ x ∂volume, ‖fiberHeight E v R x‖ ≤ 2*a :=
+  norm_fiber_integral_le (localizedPhaseError_measurable hE v R) hv ha
+    (localizedPhaseError_slab_bound E v R a hphase)
+
+theorem lintegral_fiber_bound {n : ℕ} {f : Euc n → ℝ} (hf : Measurable f)
+    {v : Euc n} (hv : ‖v‖ = 1) {a : ℝ} (ha : 0 ≤ a)
+    (hbound : ∀ᵐ x ∂volume, ‖f x‖ ≤ (Icc (-a) a).indicator (fun _ ↦ (1:ℝ)) (inner ℝ v x)) :
+    ∀ᵐ x ∂volume, (∫⁻ s : ℝ, ‖f (x+s•v)‖ₑ) ≤ ENNReal.ofReal (2*a) := by
+  filter_upwards [ae_fiber_norm_bound hf hv hbound] with x hx
+  calc
+    _ ≤ ∫⁻ s : ℝ, ENNReal.ofReal
+        ((Icc (-a-inner ℝ v x) (a-inner ℝ v x)).indicator (fun _ ↦ (1:ℝ)) s) := by
+      apply lintegral_mono_ae
+      filter_upwards [hx] with s hs
+      rw [← ofReal_norm]
+      exact ENNReal.ofReal_le_ofReal hs
+    _ = ENNReal.ofReal (2*a) := by
+      have he : (fun s : ℝ ↦ ENNReal.ofReal
+          ((Icc (-a-inner ℝ v x) (a-inner ℝ v x)).indicator (fun _ ↦ (1:ℝ)) s)) =
+          (Icc (-a-inner ℝ v x) (a-inner ℝ v x)).indicator (fun _ ↦ (1:ℝ≥0∞)) := by
+        funext s
+        by_cases hs : s ∈ Icc (-a-inner ℝ v x) (a-inner ℝ v x) <;> simp [hs]
+      rw [he]
+      rw [lintegral_indicator measurableSet_Icc]
+      simp only [lintegral_const, Measure.restrict_apply_univ, one_mul]
+      rw [← uIcc_of_le (show -a-inner ℝ v x ≤ a-inner ℝ v x by linarith [ha]),
+        Real.volume_interval, abs_of_nonneg (by linarith [ha])]
+      congr 1
+      ring
+
+theorem slab_lintegral_bound {n : ℕ} {f : Euc n → ℝ} (hf : Measurable f)
+    {v : Euc n} (hv : ‖v‖ = 1) {a : ℝ} (ha : 0 ≤ a) {R : ℝ} (hR : 0 < R)
+    (hs : Function.support f ⊆ Metric.ball 0 R)
+    (hbound : ∀ᵐ x ∂volume, ‖f x‖ ≤ (Icc (-a) a).indicator (fun _ ↦ (1:ℝ)) (inner ℝ v x)) :
+    ENNReal.ofReal (2*R) * (∫⁻ x, ‖f x‖ₑ) ≤
+      ENNReal.ofReal (2*a) * volume (Metric.ball (0:Euc n) (2*R)) := by
+  let I := Icc (-R) R
+  let B := Metric.ball (0:Euc n) (2*R)
+  have hm : Measurable (fun p : Euc n × ℝ ↦ ‖f (p.1+p.2•v)‖ₑ) :=
+    (hf.comp (continuous_fst.add (continuous_snd.smul continuous_const)).measurable).enorm
+  have hswap : ENNReal.ofReal (2*R) * (∫⁻ x, ‖f x‖ₑ) =
+      ∫⁻ x, ∫⁻ s in I, ‖f (x+s•v)‖ₑ := by
+    rw [lintegral_lintegral_swap hm.aemeasurable]
+    simp_rw [lintegral_add_right_eq_self (fun x : Euc n ↦ ‖f x‖ₑ)]
+    simp only [lintegral_const, Measure.restrict_apply_univ, I]
+    rw [← uIcc_of_le (show -R ≤ R by linarith [hR]), Real.volume_interval,
+      sub_neg_eq_add, ← two_mul, abs_of_pos (mul_pos (by norm_num) hR), mul_comm]
+  rw [hswap]
+  calc
+    _ ≤ ∫⁻ x, B.indicator (fun _ ↦ ENNReal.ofReal (2*a)) x := by
+      apply lintegral_mono_ae
+      filter_upwards [lintegral_fiber_bound hf hv ha hbound] with x hx
+      by_cases hxB : x ∈ B
+      · rw [indicator_of_mem hxB]
+        exact (lintegral_mono' Measure.restrict_le_self le_rfl).trans hx
+      · rw [indicator_of_notMem hxB]
+        apply le_of_eq
+        apply (lintegral_eq_zero_iff' (hf.comp (continuous_const.add (continuous_id.smul continuous_const)).measurable).enorm.aemeasurable).mpr
+        filter_upwards [ae_restrict_mem measurableSet_Icc] with s hsI
+        have hz : f (x+s•v) = 0 := by
+          by_contra hnz
+          have hy := hs hnz
+          have hsn : ‖s•v‖ ≤ R := by
+            rw [norm_smul, hv, mul_one, Real.norm_eq_abs]
+            exact abs_le.mpr hsI
+          have hxb : ‖x‖ < 2*R := by
+            have hxy : x = (x+s•v) - s•v := by abel
+            calc
+              ‖x‖ = ‖(x+s•v) - s•v‖ := congrArg norm hxy
+              _ ≤ ‖x+s•v‖ + ‖s•v‖ := norm_sub_le _ _
+              _ < R+R := add_lt_add_of_lt_of_le (by simpa only [Metric.mem_ball, dist_zero_right] using hy) hsn
+              _ = 2*R := by ring
+          exact hxB (Metric.mem_ball.mpr (by simpa only [dist_zero_right] using hxb))
+        change ‖f (x+s•v)‖ₑ = 0
+        simp only [hz, enorm_zero]
+    _ = _ := by rw [lintegral_indicator measurableSet_ball]; simp [mul_comm]
+
+theorem slab_integral_bound {n : ℕ} {f : Euc n → ℝ} (hf : Measurable f)
+    {v : Euc n} (hv : ‖v‖ = 1) {a : ℝ} (ha : 0 ≤ a) {R : ℝ} (hR : 0 < R)
+    (hs : Function.support f ⊆ Metric.ball 0 R)
+    (hbound : ∀ᵐ x ∂volume, ‖f x‖ ≤ (Icc (-a) a).indicator (fun _ ↦ (1:ℝ)) (inner ℝ v x)) :
+    (2*R) * (∫ x, ‖f x‖) ≤ (2*a) * volume.real (Metric.ball (0:Euc n) (2*R)) := by
+  have hfinite : volume (Metric.ball (0:Euc n) (2*R)) ≠ ⊤ :=
+    ne_top_of_le_ne_top (isCompact_closedBall (0:Euc n) (2*R)).measure_ne_top
+      (measure_mono Metric.ball_subset_closedBall)
+  have hi := ENNReal.toReal_mono (ENNReal.mul_ne_top ENNReal.ofReal_ne_top hfinite)
+    (slab_lintegral_bound hf hv ha hR hs hbound)
+  simpa only [ENNReal.toReal_mul, ENNReal.toReal_ofReal (by positivity : 0 ≤ 2*R),
+    ENNReal.toReal_ofReal (by positivity : 0 ≤ 2*a), Measure.real,
+    ← integral_norm_eq_lintegral_enorm hf.aestronglyMeasurable] using hi
+
+def cutoffPhaseError {n : ℕ} (E : Set (Euc n)) (v : Euc n) (η : Euc n → ℝ)
+    (x : Euc n) : ℝ := η x * (E.indicator (fun _ ↦ (1:ℝ)) x -
+      {z | inner ℝ v z < 0}.indicator (fun _ ↦ (1:ℝ)) x)
+
+theorem cutoffPhaseError_measurable {n : ℕ} {E : Set (Euc n)}
+    (hE : MeasurableSet E) (v : Euc n) {η : Euc n → ℝ} (hη : Measurable η) :
+    Measurable (cutoffPhaseError E v η) :=
+  hη.mul ((measurable_const.indicator hE).sub (measurable_const.indicator
+    (isOpen_lt (continuous_const.inner continuous_id) continuous_const).measurableSet))
+
+theorem cutoffPhaseError_support {n : ℕ} (E : Set (Euc n)) (v : Euc n)
+    (η : Euc n → ℝ) : Function.support (cutoffPhaseError E v η) ⊆ Function.support η := by
+  intro x hx hz
+  exact hx (by simp [cutoffPhaseError, hz])
+
+theorem cutoffPhaseError_eq_mul_localized {n : ℕ} (E : Set (Euc n)) (v : Euc n)
+    (η : Euc n → ℝ) {R : ℝ} (hs : Function.support η ⊆ Metric.ball 0 R) :
+    cutoffPhaseError E v η = fun x ↦ η x * localizedPhaseError E v R x := by
+  funext x
+  by_cases hx : x ∈ Metric.ball (0:Euc n) R
+  · simp [cutoffPhaseError, localizedPhaseError, hx]
+  · have hz : η x = 0 := Function.notMem_support.mp (fun hn ↦ hx (hs hn))
+    simp [cutoffPhaseError, hz]
+
+theorem cutoffPhaseError_slab_bound {n : ℕ} (E : Set (Euc n)) (v : Euc n)
+    (η : Euc n → ℝ) {R a : ℝ} (hs : Function.support η ⊆ Metric.ball 0 R)
+    (hη : ∀ x, ‖η x‖ ≤ 1)
+    (hphase : ∀ᵐ x ∂volume, x ∈ Metric.ball 0 R → a < |inner ℝ v x| →
+      (x ∈ E ↔ inner ℝ v x < 0)) :
+    ∀ᵐ x ∂volume, ‖cutoffPhaseError E v η x‖ ≤
+      (Icc (-a) a).indicator (fun _ ↦ (1:ℝ)) (inner ℝ v x) := by
+  rw [cutoffPhaseError_eq_mul_localized E v η hs]
+  filter_upwards [localizedPhaseError_slab_bound E v R a hphase] with x hx
+  rw [norm_mul]
+  exact ((mul_le_mul_of_nonneg_right (hη x) (norm_nonneg _)).trans_eq (one_mul _)).trans hx
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/FiberLocalization.lean b/lean/OAI/Geometry/Perimeter/Regularity/FiberLocalization.lean
new file mode 100644
index 0000000000000000000000000000000000000000..36f8cee4eaa75d69f69d9f9a9d47c0f1b5c10ac6
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/FiberLocalization.lean
@@ -0,0 +1,118 @@
+import OAI.Geometry.Perimeter.Regularity.FiberHeight
+import OAI.Geometry.Perimeter.Regularity.PlaneFiberCompactness
+import OAI.Geometry.Perimeter.Regularity.PhaseMomentLimit
+import OAI.Geometry.Perimeter.Regularity.NormalWindowTest
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal InnerProductSpace ContDiff
+namespace RPISingularHarnack
+open RPI.Boundary
+variable {n m : ℕ}
+
+theorem interval_negative_phase (b : ℝ) (hb : 0 ≤ b) :
+    (∫ s in (-b)..b, {t : ℝ | t<0}.indicator (fun _ ↦ (1:ℝ)) s)=b := by
+  have he : (fun s : ℝ ↦ {t : ℝ | t<0}.indicator (fun _ ↦ (1:ℝ)) s) =ᵐ[volume]
+      (fun s : ℝ ↦ {t : ℝ | t≤0}.indicator (fun _ ↦ (1:ℝ)) s) := by
+    filter_upwards [volume.ae_ne (0:ℝ)] with s hs
+    have hh : s<0 ↔ s≤0 := lt_iff_le_and_ne.trans (and_iff_left hs)
+    simp only [Set.indicator_apply,Set.mem_ofPred_eq,hh]
+  rw [intervalIntegral.integral_congr_ae (he.mono (fun _ h _ ↦ h)),
+    intervalIntegral.integral_indicator (show (0:ℝ) ∈ Icc (-b) b by constructor <;> linarith)]
+  simp
+
+theorem exactFiberHeight_eq_phase_integral {E : Set (Euc n)} (hE : MeasurableSet E)
+    {v : Euc n} (hv : ‖v‖=1) {b : ℝ} (hb : 0 ≤ b) (y : normalPlane v) :
+    exactFiberHeight E v b y = ∫ s in (-b)..b,
+      (E.indicator (fun _ ↦ (1:ℝ)) ((y:Euc n)+s•v)-
+        {z : Euc n | inner ℝ v z<0}.indicator (fun _ ↦ (1:ℝ)) ((y:Euc n)+s•v)) := by
+  have hi : IntervalIntegrable (fun s : ℝ ↦ E.indicator (fun _ ↦ (1:ℝ))
+      ((y:Euc n)+s•v)) volume (-b) b := by
+    apply (intervalIntegrable_const (c := (1:ℝ))).mono_fun
+      (((measurable_const.indicator hE).comp
+        (continuous_const.add (continuous_id.smul continuous_const)).measurable).aestronglyMeasurable)
+    exact ae_of_all _ fun s ↦ by by_cases h : (y:Euc n)+s•v ∈ E <;> simp [h]
+  have hj : IntervalIntegrable (fun s : ℝ ↦ {t : ℝ | t<0}.indicator (fun _ ↦ (1:ℝ)) s)
+      volume (-b) b := by
+    apply (intervalIntegrable_const (c := (1:ℝ))).mono_fun
+      ((measurable_const.indicator measurableSet_Iio).aestronglyMeasurable)
+    exact ae_of_all _ fun s ↦ by by_cases h : s<0 <;> simp [h]
+  have he (s : ℝ) : {z : Euc n | inner ℝ v z<0}.indicator (fun _ ↦ (1:ℝ))
+      ((y:Euc n)+s•v) = {t : ℝ | t<0}.indicator (fun _ ↦ (1:ℝ)) s := by
+    simp only [Set.indicator_apply,Set.mem_ofPred_eq,normal_fiber_inner hv]
+  simp_rw [he]
+  rw [intervalIntegral.integral_sub hi hj,interval_negative_phase b hb,exactFiberHeight]
+  ring
+
+theorem planeFiber_cutoff_eq_exact_ae {E : Set (Euc n)} (hE : MeasurableSet E)
+    {v : Euc n} (hv : ‖v‖=1) {η : Euc n → ℝ} {a b R : ℝ}
+    (ha : 0 ≤ a) (hab : a<b) (hηs : Function.support η ⊆ Metric.ball 0 R)
+    (K : Set (normalPlane v))
+    (hηone : ∀ y ∈ K, ∀ s : ℝ, |s|≤b → η ((y:Euc n)+s•v)=1)
+    (hphase : ∀ᵐ x ∂volume, x ∈ Metric.ball 0 R → a < |inner ℝ v x| →
+      (x ∈ E ↔ inner ℝ v x<0)) :
+    ∀ᵐ y : normalPlane v ∂volume, y ∈ K →
+      planeFiber v (cutoffPhaseError E v η) y = exactFiberHeight E v b y := by
+  have hp := (normalCoordinates_measurePreserving v hv).quasiMeasurePreserving.ae hphase
+  change ∀ᵐ z : normalPlane v × ℝ ∂volume.prod volume, _ at hp
+  have hpf := Measure.ae_ae_of_ae_prod hp
+  filter_upwards [hpf] with y hy
+  intro hyK
+  rw [exactFiberHeight_eq_phase_integral hE hv (ha.trans hab.le) y,
+    intervalIntegral.integral_of_le (by linarith : -b≤b),← integral_indicator measurableSet_Ioc]
+  apply integral_congr_ae
+  filter_upwards [hy] with s hs
+  change ( (y:Euc n)+s•v ∈ Metric.ball 0 R → a < |inner ℝ v ((y:Euc n)+s•v)| →
+      ((y:Euc n)+s•v ∈ E ↔ inner ℝ v ((y:Euc n)+s•v)<0)) at hs
+  rw [normal_fiber_inner hv] at hs
+  by_cases his : s ∈ Ioc (-b) b
+  · have hsabs : |s|≤b := abs_le.mpr ⟨his.1.le,his.2⟩
+    simp only [Set.indicator_of_mem his,cutoffPhaseError,hηone y hyK s hsabs,one_mul]
+  · rw [Set.indicator_of_notMem his]
+    by_cases hη : η ((y:Euc n)+s•v)=0
+    · simp [cutoffPhaseError,hη]
+    · have hxB := hηs hη
+      have hsabs : a < |s| := by
+        have hh : s≤-b ∨ b<s := by simpa only [Set.mem_Ioc,not_and_or,not_lt,not_le] using his
+        rcases hh with hh | hh
+        · exact hab.trans_le ((neg_le_abs s).trans' (by linarith))
+        · exact hab.trans (hh.trans_le (le_abs_self s))
+      have hq := hs hxB hsabs
+      have hq' : ((y:Euc n)+s•v ∈ E ↔ (y:Euc n)+s•v ∈ {z | inner ℝ v z<0}) := by
+        simpa only [Set.mem_ofPred_eq,normal_fiber_inner hv] using hq
+      dsimp only [cutoffPhaseError]
+      by_cases he : (y:Euc n)+s•v ∈ E
+      · rw [Set.indicator_of_mem he,Set.indicator_of_mem (hq'.mp he),sub_self,mul_zero]
+      · rw [Set.indicator_of_notMem he,Set.indicator_of_notMem (fun h ↦ he (hq'.mpr h)),
+          sub_self,mul_zero]
+
+theorem chartFiber_test_eq_exact {E : Set (Euc n)} (hE : MeasurableSet E)
+    {v : Euc n} (hv : ‖v‖=1) (e : Euc m ≃ₗᵢ[ℝ] normalPlane v)
+    {η : Euc n → ℝ} {a b R A : ℝ} (ha : 0 ≤ a) (hab : a<b)
+    (hηs : Function.support η ⊆ Metric.ball 0 R) (φ : normalPlane v → ℝ)
+    (hηone : ∀ y ∈ tsupport φ, ∀ s : ℝ, |s|≤b → η ((y:Euc n)+s•v)=1)
+    (hphase : ∀ᵐ x ∂volume, x ∈ Metric.ball 0 R → a < |inner ℝ v x| →
+      (x ∈ E ↔ inner ℝ v x<0)) :
+    (∫ y, chartFiber v e A (cutoffPhaseError E v η) y * φ (e y)) =
+      A⁻¹ * ∫ y in tsupport φ, φ y*exactFiberHeight E v b y := by
+  simp only [chartFiber,mul_assoc]
+  rw [integral_const_mul]
+  congr 1
+  rw [e.measurePreserving.integral_comp e.toMeasurableEquiv.measurableEmbedding
+    (fun y ↦ planeFiber v (cutoffPhaseError E v η) y * φ y)]
+  rw [← integral_indicator (isClosed_tsupport φ).measurableSet]
+  apply integral_congr_ae
+  filter_upwards [planeFiber_cutoff_eq_exact_ae hE hv ha hab hηs (tsupport φ) hηone hphase]
+    with y hy
+  by_cases hs : y ∈ tsupport φ
+  · rw [Set.indicator_of_mem hs,hy hs,mul_comm]
+  · rw [Set.indicator_of_notMem hs,
+      show φ y=0 from Function.notMem_support.mp (fun hh ↦ hs (subset_closure hh)),mul_zero]
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/FiberMomentIntegration.lean b/lean/OAI/Geometry/Perimeter/Regularity/FiberMomentIntegration.lean
new file mode 100644
index 0000000000000000000000000000000000000000..83795a9abd81abc99e789c4f425a464bec80a180
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/FiberMomentIntegration.lean
@@ -0,0 +1,125 @@
+import OAI.Geometry.Perimeter.Regularity.MollifiedFiberMoment
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff Convolution InnerProductSpace
+namespace RPISingularHarnack
+open RPI.Boundary
+
+variable {n : ℕ}
+
+def normalCylinder (v : Euc n) (hv : ‖v‖ = 1) (K : Set (normalPlane v))
+    (b : ℝ) : Set (Euc n) :=
+  normalCoordinates v hv '' (K ×ˢ Icc (-b) b)
+
+theorem isCompact_normalCylinder {v : Euc n} (hv : ‖v‖ = 1)
+    {K : Set (normalPlane v)} (hK : IsCompact K) (b : ℝ) :
+    IsCompact (normalCylinder v hv K b) :=
+  (hK.prod isCompact_Icc).image (by
+    change Continuous (fun x : normalPlane v × ℝ ↦ (x.1:Euc n)+x.2•v)
+    exact (continuous_subtype_val.comp continuous_fst).add (continuous_snd.smul continuous_const))
+
+theorem integral_normalCylinder {v : Euc n} (hv : ‖v‖ = 1)
+    {K : Set (normalPlane v)} (hK : IsCompact K) {b : ℝ} (hb : 0 ≤ b)
+    {f : Euc n → ℝ} (hf : Continuous f) :
+    (∫ x in normalCylinder v hv K b, f x) =
+      ∫ y in K, ∫ s in (-b)..b, f ((y:Euc n)+s•v) := by
+  rw [normalCylinder, (normalCoordinates_measurePreserving v hv).setIntegral_image_emb
+    (normalCoordinates v hv).measurableEmbedding]
+  have hc : Continuous (fun x : normalPlane v × ℝ ↦ f ((x.1:Euc n)+x.2•v)) :=
+    hf.comp ((continuous_subtype_val.comp continuous_fst).add
+      (continuous_snd.smul continuous_const))
+  have hi : IntegrableOn (fun x : normalPlane v × ℝ ↦ f ((x.1:Euc n)+x.2•v))
+      (K ×ˢ Icc (-b) b) (volume.prod volume) := hc.continuousOn.integrableOn_compact (hK.prod (isCompact_Icc (a := -b) (b := b)))
+  change (∫ x in K ×ˢ Icc (-b) b, f ((x.1:Euc n)+x.2•v) ∂volume.prod volume) = _
+  rw [setIntegral_prod _ hi]
+  apply setIntegral_congr_fun hK.measurableSet
+  intro y _
+  dsimp only
+  rw [intervalIntegral.integral_of_le (by linarith : -b ≤ b), integral_Icc_eq_integral_Ioc]
+
+theorem continuous_profile_integral {v : Euc n} (f : Euc n → ℝ)
+    (hf : Continuous f) (l u : ℝ) :
+    Continuous (fun y : normalPlane v ↦ ∫ s in l..u, f ((y:Euc n)+s•v)) := by
+  apply intervalIntegral.continuous_parametric_intervalIntegral_of_continuous'
+  exact hf.comp ((continuous_subtype_val.comp continuous_fst).add
+    (continuous_snd.smul continuous_const))
+
+theorem mollified_fiber_integrated_defect {E : Set (Euc n)} (hE : MeasurableSet E)
+    {v : Euc n} (hv : ‖v‖ = 1) {k : Euc n → ℝ}
+    (hk : ContDiff ℝ ∞ k) (hc : HasCompactSupport k)
+    (hp : ∀ x, 0 ≤ k x) (hu : ∫ x, k x = 1)
+    {a b ε R : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) (hab : a+ε < b)
+    (hs : Function.support k ⊆ Metric.ball 0 ε)
+    (hphase : ∀ᵐ x ∂volume, x ∈ Metric.ball 0 R → a < |inner ℝ v x| →
+      (x ∈ E ↔ inner ℝ v x < 0))
+    {K : Set (normalPlane v)} (hK : IsCompact K) (hKR : ∀ y ∈ K, ‖y‖+b+ε ≤ R) :
+    let q := phaseMollification E k
+    let H := fun y : normalPlane v ↦ -b+∫ s in (-b)..b, q ((y:Euc n)+s•v)
+    let M := ∫ x in normalCylinder v hv K b,
+      (inner ℝ v x)^2 * (-fderiv ℝ q x v)
+    0 ≤ M - ∫ y in K, (H y)^2 ∧
+    M - ∫ y in K, (H y)^2 ≤
+      4*b*(∫ x in normalCylinder v hv K b, q x*(1-q x)) +
+      8*b^2*(∫ x in normalCylinder v hv K b, max (fderiv ℝ q x v) 0) := by
+  dsimp only
+  let q := phaseMollification E k
+  have hq : Continuous q := (phaseMollification_contDiff hE hk hc).continuous
+  have hd : Continuous (fun x ↦ fderiv ℝ q x v) :=
+    ((phaseMollification_contDiff hE hk hc).continuous_fderiv (by simp)).clm_apply continuous_const
+  have hm : Continuous (fun x ↦ (inner ℝ v x)^2*(-fderiv ℝ q x v)) :=
+    (continuous_const.inner continuous_id).pow 2 |>.mul hd.neg
+  have hiM := integral_normalCylinder hv hK hb hm
+  have hiD := integral_normalCylinder hv hK hb (hq.mul ((continuous_const (y := (1:ℝ))).sub hq))
+  have hiP := integral_normalCylinder hv hK hb (hd.max (continuous_const (y := (0:ℝ))))
+  have he (y : normalPlane v) (s : ℝ) : inner ℝ v ((y:Euc n)+s•v) = s := by
+    rw [inner_add_right, real_inner_smul_right,
+      Submodule.mem_orthogonal_singleton_iff_inner_right.mp y.property,
+      real_inner_self_eq_norm_sq,hv]
+    ring
+  dsimp only [q,Pi.mul_apply,Pi.sub_apply] at hiD
+  dsimp only [q] at hiM hiP
+  simp_rw [he, ← mollifiedNormalProfile_deriv hE hk hc v] at hiM
+  simp_rw [← mollifiedNormalProfile_deriv hE hk hc v] at hiP
+  rw [hiM,hiD,hiP]
+  let A := fun y : normalPlane v ↦ ∫ s in (-b)..b,
+    s^2*(-deriv (mollifiedNormalProfile E k v y) s)
+  let B := fun y : normalPlane v ↦ (-b+∫ s in (-b)..b, q ((y:Euc n)+s•v))^2
+  let D := fun y : normalPlane v ↦ ∫ s in (-b)..b, q ((y:Euc n)+s•v)*(1-q ((y:Euc n)+s•v))
+  let P := fun y : normalPlane v ↦ ∫ s in (-b)..b, max (deriv (mollifiedNormalProfile E k v y) s) 0
+  have hA : Continuous A := by
+    have h := continuous_profile_integral (v := v) _ hm (-b) b
+    dsimp only [q] at h
+    simp_rw [he, ← mollifiedNormalProfile_deriv hE hk hc v] at h
+    exact h
+  have hB : Continuous B :=
+    (continuous_const.add (continuous_profile_integral (v := v) q hq (-b) b)).pow 2
+  have hD : Continuous D := continuous_profile_integral (v := v) _
+    (hq.mul ((continuous_const (y := (1:ℝ))).sub hq)) (-b) b
+  have hP : Continuous P := by
+    have h := continuous_profile_integral (v := v) _ (hd.max (continuous_const (y := (0:ℝ)))) (-b) b
+    dsimp only [q] at h
+    simp_rw [← mollifiedNormalProfile_deriv hE hk hc v] at h
+    exact h
+  have hAi : IntegrableOn A K volume := hA.continuousOn.integrableOn_compact hK
+  have hBi : IntegrableOn B K volume := hB.continuousOn.integrableOn_compact hK
+  have hDi : IntegrableOn D K volume := hD.continuousOn.integrableOn_compact hK
+  have hPi : IntegrableOn P K volume := hP.continuousOn.integrableOn_compact hK
+  change 0 ≤ (∫ y in K, A y)-(∫ y in K, B y) ∧
+    (∫ y in K, A y)-(∫ y in K, B y) ≤ 4*b*(∫ y in K, D y)+8*b^2*(∫ y in K, P y)
+  rw [← integral_sub hAi hBi, ← integral_const_mul, ← integral_const_mul,
+    ← integral_add (hDi.const_mul _) (hPi.const_mul _)]
+  have hpoint (y : normalPlane v) (hy : y ∈ K) :=
+    mollifiedNormalProfile_flux_defect hE hv hk hc hp hu ha hb hab hs hphase y (hKR y hy)
+  constructor
+  · exact setIntegral_nonneg hK.measurableSet fun y hy ↦ (hpoint y hy).1
+  · exact setIntegral_mono_on (hAi.sub hBi) ((hDi.const_mul _).add (hPi.const_mul _))
+      hK.measurableSet fun y hy ↦ (hpoint y hy).2
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/FiberQuadraticLimit.lean b/lean/OAI/Geometry/Perimeter/Regularity/FiberQuadraticLimit.lean
new file mode 100644
index 0000000000000000000000000000000000000000..7e2c409ab0ef33326b5909ddb22ef6aa7bb494b8
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/FiberQuadraticLimit.lean
@@ -0,0 +1,151 @@
+import OAI.Geometry.Perimeter.Regularity.FiberLocalization
+import OAI.Geometry.Perimeter.Regularity.QuadraticError
+import OAI.Geometry.Perimeter.Regularity.WeightedSquareLimit
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+open RPI.Boundary ContinuousLinearMap
+variable {n m : ℕ}
+
+theorem chartFiber_cutoff_exact_ae {E : Set (Euc n)} (hE : MeasurableSet E)
+    {v : Euc n} (hv : ‖v‖=1) (e : Euc m ≃ₗᵢ[ℝ] normalPlane v)
+    {η : Euc n → ℝ} {a b R A : ℝ} (ha : 0≤a) (hab : a<b)
+    (hηs : Function.support η ⊆ Metric.ball 0 R) (K : Set (normalPlane v))
+    (hηone : ∀ y ∈ K, ∀ s : ℝ, |s|≤b → η ((y:Euc n)+s•v)=1)
+    (hphase : ∀ᵐ x ∂volume, x ∈ Metric.ball 0 R → a < |inner ℝ v x| →
+      (x∈E ↔ inner ℝ v x<0)) :
+    ∀ᵐ y : Euc m ∂volume, e y∈K → chartFiber v e A (cutoffPhaseError E v η) y =
+      exactFiberHeight E v b (e y)/A := by
+  have h := e.measurePreserving.quasiMeasurePreserving.ae
+    (planeFiber_cutoff_eq_exact_ae hE hv ha hab hηs K hηone hphase)
+  filter_upwards [h] with y hy
+  intro hyK
+  change A⁻¹*planeFiber v (cutoffPhaseError E v η) (e y) = _
+  rw [hy hyK]
+  ring
+
+theorem chartFiber_quadratic_eq_exact {E : Set (Euc n)} (hE : MeasurableSet E)
+    {v : Euc n} (hv : ‖v‖=1) (e : Euc m ≃ₗᵢ[ℝ] normalPlane v)
+    {η : Euc n → ℝ} {a b R A : ℝ} (ha : 0≤a) (hab : a<b)
+    (hηs : Function.support η ⊆ Metric.ball 0 R) (φ w : normalPlane v → ℝ)
+    (hηone : ∀ y ∈ tsupport φ, ∀ s : ℝ, |s|≤b → η ((y:Euc n)+s•v)=1)
+    (hphase : ∀ᵐ x ∂volume, x ∈ Metric.ball 0 R → a < |inner ℝ v x| →
+      (x∈E ↔ inner ℝ v x<0)) :
+    (∫ y, φ (e y)*(chartFiber v e A (cutoffPhaseError E v η) y-w (e y))^2) =
+      ∫ y, φ y*(exactFiberHeight E v b y/A-w y)^2 := by
+  rw [← e.measurePreserving.integral_comp e.toMeasurableEquiv.measurableEmbedding
+    (fun y ↦ φ y*(exactFiberHeight E v b y/A-w y)^2)]
+  apply integral_congr_ae
+  filter_upwards [chartFiber_cutoff_exact_ae hE hv e ha hab hηs (tsupport φ) hηone hphase]
+    with y hy
+  by_cases hφ : φ (e y)=0
+  · simp only [hφ,zero_mul]
+  · rw [hy (subset_closure hφ)]
+
+theorem chartFiber_normalized_local_bound {E : Set (Euc n)} (hE : MeasurableSet E)
+    {v : Euc n} (hv : ‖v‖=1) (e : Euc m ≃ₗᵢ[ℝ] normalPlane v)
+    {η : Euc n → ℝ} {a R : ℝ} (ha : 0<a)
+    (hηs : Function.support η ⊆ Metric.ball 0 R) (K : Set (normalPlane v))
+    (hηone : ∀ y ∈ K, ∀ s : ℝ, |s|≤2*a → η ((y:Euc n)+s•v)=1)
+    (hphase : ∀ᵐ x ∂volume, x ∈ Metric.ball 0 R → a < |inner ℝ v x| →
+      (x∈E ↔ inner ℝ v x<0)) :
+    ∀ᵐ y : Euc m ∂volume, e y∈K → ‖chartFiber v e a (cutoffPhaseError E v η) y‖≤2 := by
+  filter_upwards [chartFiber_cutoff_exact_ae hE hv e ha.le (by linarith : a<2*a)
+    hηs K hηone hphase] with y hy
+  intro hyK
+  rw [hy hyK,norm_div,Real.norm_of_nonneg ha.le]
+  exact (div_le_iff₀ ha).mpr (by simpa only [mul_comm] using exactFiberHeight_norm_le hE v (by positivity : 0≤2*a) (e y))
+
+theorem cutoffPhaseError_integrable {E : Set (Euc n)} (hE : MeasurableSet E)
+    (v : Euc n) {η : Euc n → ℝ} (hη : Continuous η) (hcη : HasCompactSupport η) :
+    Integrable (cutoffPhaseError E v η) volume := by
+  apply Integrable.mono' (hη.integrable_of_hasCompactSupport hcη).norm
+    (cutoffPhaseError_measurable hE v hη.measurable).aestronglyMeasurable
+  apply ae_of_all
+  intro x
+  dsimp only [cutoffPhaseError]
+  rw [norm_mul]
+  have hb : ‖E.indicator (fun _ ↦ (1:ℝ)) x - {z | inner ℝ v z<0}.indicator (fun _ ↦ (1:ℝ)) x‖≤1 := by
+    by_cases he : x∈E <;> by_cases hh : inner ℝ v x<0 <;> simp [he,hh]
+  simpa using mul_le_mul_of_nonneg_left hb (norm_nonneg (η x))
+
+theorem minimizing_quadratic_height_trace_tendsto_zero (hn : 1<n)
+    (E : ℕ → Set (Euc n)) (hmin : ∀ i, IsLocallyPerimeterMinimizing (E i))
+    {v : Euc n} (hv : ‖v‖=1) (e : Euc m ≃ₗᵢ[ℝ] normalPlane v)
+    {η : Euc n → ℝ} (hη : Continuous η) (hcη : HasCompactSupport η)
+    (hηs : Function.support η ⊆ Metric.ball 0 4)
+    (hηone : ∀ x ∈ Metric.ball (0:Euc n) 3, η x=1)
+    (a : ℕ → ℝ) (ha : ∀ i, 0<a i) (hat : Tendsto a atTop (𝓝 0))
+    (hh : ∀ i x, x ∈ perimeterSupport (E i) ∩ Metric.ball 0 4 → |inner ℝ v x|≤a i)
+    (hphase : ∀ i, ∀ᵐ x ∂volume, x ∈ Metric.ball 0 4 → a i < |inner ℝ v x| →
+      (x∈E i ↔ inner ℝ v x<0))
+    (htilt : Tendsto (fun i ↦ orientedTilt (E i) v (Metric.ball 0 4)) atTop (𝓝 0))
+    {g : Euc m → ℝ} (hg : Integrable g volume)
+    (hL : Tendsto (fun i ↦ ∫⁻ y, ‖chartFiber v e (a i) (cutoffPhaseError (E i) v η) y-g y‖ₑ)
+      atTop (𝓝 0))
+    {φ w : normalPlane v → ℝ} (hφ : Continuous φ) (hcφ : HasCompactSupport φ)
+    (hpφ : ∀ y, 0≤φ y) (hsφ : tsupport φ ⊆ Metric.ball 0 1)
+    (hw : Continuous w) (hwg : ∀ᵐ y : Euc m ∂volume, φ (e y)≠0 → w (e y)=g y) :
+    Tendsto (fun i ↦ ∫ x in Metric.ball 0 4,
+      normalWindowQuadratic v φ w (a i) (2*a i) x ∂perimeterMeasure (E i)) atTop (𝓝 0) := by
+  obtain ⟨M₂,hM₂⟩ := hcφ.exists_bound_of_continuous hφ
+  obtain ⟨M₁,hM₁⟩ := hcφ.mul_right.exists_bound_of_continuous (hφ.mul hw)
+  obtain ⟨M₀,hM₀⟩ := hcφ.mul_right.exists_bound_of_continuous (hφ.mul (hw.pow 2))
+  obtain ⟨B,hB⟩ := hcφ.exists_bound_of_continuousOn hw.continuousOn
+  let B' := max B 0
+  have hB' : 0≤B' := le_max_right _ _
+  have haev : ∀ᶠ i in atTop, a i<1/2 := hat.eventually (gt_mem_nhds (by norm_num : (0:ℝ)<1/2))
+  have hone (i) (hai : a i<1/2) : ∀ y ∈ tsupport φ, ∀ s : ℝ, |s|≤2*a i →
+      η ((y:Euc n)+s•v)=1 := by
+    intro y hy s hs
+    apply hηone
+    have hyb : ‖y‖<1 := by simpa only [Metric.mem_ball,dist_zero_right] using hsφ hy
+    rw [Metric.mem_ball,dist_zero_right]
+    have h := norm_add_le (y:Euc n) (s•v)
+    rw [norm_smul,hv,mul_one,Real.norm_eq_abs] at h
+    change ‖(y:Euc n)+s•v‖≤‖y‖+|s| at h
+    linarith
+  have hbf : ∀ᶠ i in atTop, ∀ᵐ y : Euc m ∂volume, φ (e y)≠0 →
+      ‖chartFiber v e (a i) (cutoffPhaseError (E i) v η) y‖≤2 := by
+    filter_upwards [haev] with i hai
+    filter_upwards [chartFiber_normalized_local_bound (hmin i).1 hv e (ha i) hηs
+      (tsupport φ) (hone i hai) (hphase i)] with y hy
+    exact fun hh ↦ hy (subset_closure hh)
+  have hplane := weighted_square_L1_tendsto_zero
+    (fun i ↦ chartFiber v e (a i) (cutoffPhaseError (E i) v η))
+    (fun i ↦ chartFiber_integrable (cutoffPhaseError_integrable (hmin i).1 v hη hcη) hv e (a i))
+    hg (hw.comp e.continuous).aestronglyMeasurable (hφ.comp e.continuous).aestronglyMeasurable
+    ((norm_nonneg _).trans (hM₂ 0)) (by norm_num : (0:ℝ)≤2) hB'
+    (ae_of_all _ fun y ↦ hM₂ (e y)) hwg
+    (ae_of_all _ fun y hy ↦ (hB (e y) (subset_closure hy)).trans (le_max_left _ _)) hbf hL
+  have hplane' : Tendsto (fun i ↦ ∫ y, φ y*(exactFiberHeight (E i) v (2*a i) y/a i-w y)^2)
+      atTop (𝓝 0) := by
+    apply hplane.congr'
+    filter_upwards [haev] with i hai
+    exact chartFiber_quadratic_eq_exact (hmin i).1 hv e (ha i).le (by linarith [ha i])
+      hηs φ w (hone i hai) (hphase i)
+  have hdef : Tendsto (fun i ↦ (∫ x in Metric.ball 0 4,
+      normalWindowQuadratic v φ w (a i) (2*a i) x ∂perimeterMeasure (E i))-
+      (∫ y, φ y*(exactFiberHeight (E i) v (2*a i) y/a i-w y)^2)) atTop (𝓝 0) := by
+    rw [tendsto_zero_iff_norm_tendsto_zero]
+    apply squeeze_zero' (Eventually.of_forall (fun _ ↦ norm_nonneg _)) _
+      (by simpa only [mul_zero] using htilt.const_mul (18*M₂+2*M₁+M₀/2))
+    filter_upwards [haev] with i hai
+    rw [Real.norm_eq_abs]
+    apply actual_fiber_quadratic_error hn (hmin i) hv hφ hcφ hpφ hw hM₂ hM₁ hM₀
+      (ha i) (show 0<a i/2 from div_pos (ha i) (by norm_num)) (by linarith [ha i]) (by norm_num)
+      _ (hh i) (hphase i)
+    intro y hy
+    have hyb : ‖y‖<1 := by simpa only [Metric.mem_ball,dist_zero_right] using hsφ hy
+    linarith
+  simpa only [sub_add_cancel,add_zero] using hdef.add hplane'
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/FlatExcess.lean b/lean/OAI/Geometry/Perimeter/Regularity/FlatExcess.lean
new file mode 100644
index 0000000000000000000000000000000000000000..f1575bc1447ee4d846b41c1e137470af99642b1f
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/FlatExcess.lean
@@ -0,0 +1,889 @@
+import OAI.Geometry.Perimeter.Regularity.TendstoLintegralNormOfL2
+import OAI.Geometry.Perimeter.Regularity.ExistsMinimizingDensityLimit
+import OAI.Geometry.Perimeter.Regularity.LinearSchauder
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff Pointwise
+namespace RPISingularHarnack
+
+theorem integral_abs_indicator_sub_eq_symmDiff {n : ℕ} {E F U : Set (Euc n)}
+    (hE : MeasurableSet E) (hF : MeasurableSet F) :
+    (∫ x in U, |E.indicator (fun _ ↦ (1:ℝ)) x -
+      F.indicator (fun _ ↦ (1:ℝ)) x|) = volume.real (symmDiff E F ∩ U) := by
+  have hm : AEStronglyMeasurable (fun x ↦ E.indicator (fun _ ↦ (1:ℝ)) x -
+      F.indicator (fun _ ↦ (1:ℝ)) x) (volume.restrict U) :=
+    ((measurable_const.indicator hE).sub
+      (measurable_const.indicator hF)).aestronglyMeasurable.restrict
+  rw [measureReal_def, ← lintegral_indicator_difference hE hF,
+    ← integral_norm_eq_lintegral_enorm hm]
+  simp only [Real.norm_eq_abs]
+
+theorem halfspace_perimeterSupport_subset_plane {n : ℕ} (v : Euc n) :
+    perimeterSupport {x | inner ℝ x v < 0} ⊆ {x | inner ℝ x v = 0} := by
+  let H : Set (Euc n) := {x | inner ℝ x v < 0}
+  have hH : IsOpen H := isOpen_lt (continuous_id.inner continuous_const) continuous_const
+  intro x hx
+  have hx' := (Set.ext_iff.mp (perimeterSupport_eq_phaseSupport_inter hH.measurableSet) x).mp hx
+  have hm : inner ℝ x v ≤ 0 := by
+    have hc : closure H ⊆ {y | inner ℝ y v ≤ 0} :=
+      closure_minimal (fun y hy ↦ show inner ℝ y v ≤ 0 from le_of_lt hy)
+        (isClosed_le (continuous_id.inner continuous_const) continuous_const)
+    exact hc (Measure.support_restrict_subset hx'.1).1
+  have hp : 0 ≤ inner ℝ x v := by
+    have hc : closure Hᶜ = Hᶜ := hH.isClosed_compl.closure_eq
+    have hxC := (Measure.support_restrict_subset hx'.2).1
+    rw [hc] at hxC
+    exact not_lt.mp hxC
+  exact le_antisymm hm hp
+
+theorem halfspace_hasHalfspaceBlowup {n : ℕ} {v x : Euc n}
+    (hv : ‖v‖ = 1) (hx : inner ℝ x v = 0) :
+    HasHalfspaceBlowup {y | inner ℝ y v < 0} x v := by
+  refine ⟨hv, ?_⟩
+  have heq : {z | inner ℝ (z-x) v < 0} = {y | inner ℝ y v < 0} := by
+    ext z
+    simp only [mem_ofPred_eq, inner_sub_left, hx, sub_zero]
+  simp only [heq, sub_self, abs_zero, integral_zero, mul_zero]
+  exact tendsto_const_nhds
+
+theorem HasHalfspaceBlowup.local_L1 {n : ℕ} {E : Set (Euc n)}
+    (hE : MeasurableSet E) {p v : Euc n} (hv : HasHalfspaceBlowup E p v)
+    (r : ℕ → ℝ) (hr : ∀ i, 0 < r i) (htr : Tendsto r atTop (𝓝 0)) :
+    ∀ (x : Euc n) (R : ℝ), 0 < R →
+      Tendsto (fun i ↦ volume.real (symmDiff (blownUpSet E p (r i))
+        {z | inner ℝ z v < 0} ∩ Metric.closedBall x R)) atTop (𝓝 0) := by
+  let H : Set (Euc n) := {z | inner ℝ z v < 0}
+  let Hp : Set (Euc n) := {z | inner ℝ (z-p) v < 0}
+  have hHp : MeasurableSet Hp :=
+    (isOpen_lt ((continuous_id.sub continuous_const).inner continuous_const)
+      continuous_const).measurableSet
+  have he (i : ℕ) : blownUpSet (symmDiff E Hp) p (r i) =
+      symmDiff (blownUpSet E p (r i)) H := by
+    ext z
+    have hz : p + r i • z ∈ Hp ↔ z ∈ H := by
+      change inner ℝ (p + r i • z - p) v < 0 ↔ inner ℝ z v < 0
+      rw [add_sub_cancel_left, real_inner_smul_left]
+      constructor
+      · intro hz
+        exact (mul_neg_iff.mp hz).resolve_right (fun h ↦ (not_lt_of_ge (hr i).le) h.1) |>.2
+      · exact mul_neg_of_pos_of_neg (hr i)
+    simp only [blownUpSet, mem_preimage, mem_symmDiff, hz]
+  have ht : Tendsto (fun t : ℝ ↦ (t^n)⁻¹ *
+      volume.real (symmDiff E Hp ∩ Metric.ball p t)) (𝓝[>] 0) (𝓝 0) := by
+    have hh := hv.2
+    change Tendsto (fun t : ℝ ↦ (t^n)⁻¹ *
+      ∫ x in Metric.ball p t, |E.indicator (fun _ ↦ (1:ℝ)) x -
+        Hp.indicator (fun _ ↦ (1:ℝ)) x|) (𝓝[>] 0) (𝓝 0) at hh
+    simpa only [integral_abs_indicator_sub_eq_symmDiff hE hHp] using hh
+  have hcentral (S : ℝ) (hS : 0 < S) :
+      Tendsto (fun i ↦ volume.real (symmDiff (blownUpSet E p (r i)) H ∩
+        Metric.closedBall (0 : Euc n) S)) atTop (𝓝 0) := by
+    have hscale : Tendsto (fun i ↦ r i * (S+1)) atTop (𝓝[>] (0 : ℝ)) := by
+      apply tendsto_nhdsWithin_iff.2
+      refine ⟨?_, Eventually.of_forall fun i ↦ ?_⟩
+      · simpa using htr.mul_const (S+1)
+      · exact mul_pos (hr i) (by linarith)
+    have ht' := (ht.comp hscale).const_mul ((S+1)^n)
+    simp only [mul_zero] at ht'
+    apply squeeze_zero (fun _ ↦ measureReal_nonneg) (fun i ↦ ?_) ht'
+    rw [← he i, volumeReal_blownUpSet_closedBall (hE.symmDiff hHp) p (hr i)]
+    calc
+      (r i ^ n)⁻¹ * volume.real (symmDiff E Hp ∩ Metric.closedBall p (r i*S)) ≤
+          (r i ^ n)⁻¹ * volume.real (symmDiff E Hp ∩ Metric.ball p (r i*(S+1))) := by
+        apply mul_le_mul_of_nonneg_left _ (inv_nonneg.mpr (pow_nonneg (hr i).le _))
+        apply measureReal_mono
+          (inter_subset_inter_right _ (Metric.closedBall_subset_ball (by nlinarith [hr i])))
+          (measure_ne_top_of_subset inter_subset_right measure_ball_lt_top.ne)
+      _ = (S+1)^n * ((r i*(S+1))^n)⁻¹ *
+          volume.real (symmDiff E Hp ∩ Metric.ball p (r i*(S+1))) := by
+        have hs1 : S+1 ≠ 0 := ne_of_gt (by linarith)
+        rw [mul_pow, mul_inv]
+        field_simp [pow_ne_zero n (hr i).ne', pow_ne_zero n hs1]
+      _ = _ := by dsimp only [Function.comp_def]; ring
+  intro x R hR
+  have hS : 0 < ‖x‖ + R := by positivity
+  have hsub : Metric.closedBall x R ⊆ Metric.closedBall (0 : Euc n) (‖x‖+R) := by
+    intro z hz
+    have htri := dist_triangle z x (0 : Euc n)
+    have hz' := Metric.mem_closedBall.mp hz
+    simp only [dist_zero_right] at htri ⊢
+    exact Metric.mem_closedBall.mpr (by simpa only [dist_zero_right] using
+      (htri.trans (by linarith : dist z x + ‖x‖ ≤ ‖x‖ + R)))
+  exact squeeze_zero (fun _ ↦ measureReal_nonneg)
+    (fun i ↦ measureReal_mono (inter_subset_inter_right _ hsub)
+      (measure_ne_top_of_subset inter_subset_right
+        (isCompact_closedBall (0 : Euc n) (‖x‖+R)).measure_ne_top))
+    (hcentral (‖x‖+R) hS)
+
+theorem HasHalfspaceBlowup.support_height {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    {p v : Euc n} (hv : HasHalfspaceBlowup E p v)
+    (r : ℕ → ℝ) (hr : ∀ i, 0 < r i) (htr : Tendsto r atTop (𝓝 0))
+    {C : Set (Euc n)} (hC : IsCompact C) {ε : ℝ} (hε : 0 < ε) :
+    ∀ᶠ i in atTop, ∀ z ∈ perimeterSupport (blownUpSet E p (r i)) ∩ C,
+      |inner ℝ z v| < ε := by
+  let H : Set (Euc n) := {z | inner ℝ z v < 0}
+  have hH : MeasurableSet H :=
+    (isOpen_lt (continuous_id.inner continuous_const) continuous_const).measurableSet
+  have hconv := hv.local_L1 hmin.1 r hr htr
+  have hEm (i : ℕ) := locally_minimizing_blownUpSet hmin p (hr i)
+  have hHm := locally_minimizing_of_local_L1_limit (by omega : 0 < n) hEm hH hconv
+  filter_upwards [minimizing_supports_locally_converge hn _ hEm hHm hconv hC hε] with i hi
+  intro z hz
+  obtain ⟨y, hy, hzy⟩ := hi.1 z hz
+  have hy0 := halfspace_perimeterSupport_subset_plane v hy
+  have heq : inner ℝ z v = inner ℝ (z-y) v := by
+    rw [inner_sub_left, hy0, sub_zero]
+  rw [heq]
+  calc
+    |inner ℝ (z-y) v| ≤ ‖z-y‖ * ‖v‖ := abs_real_inner_le_norm _ _
+    _ = dist z y := by rw [hv.1, mul_one, dist_eq_norm]
+    _ < ε := hzy
+
+def orientedTilt {n : ℕ} (E : Set (Euc n)) (v : Euc n) (U : Set (Euc n)) : ℝ :=
+  ∫ x in U, ‖approximateNormal E x - v‖^2 ∂perimeterMeasure E
+
+theorem orientedTilt_le_calibration_gap {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    {v : Euc n} (hv : ‖v‖ = 1) {a b : ℝ} (ha : 0 < a) (hab : a < b)
+    {η : Euc n → ℝ} (hη : ContDiff ℝ ∞ η) (hc : HasCompactSupport η)
+    (hs : tsupport η ⊆ Metric.ball 0 b) (hb : ∀ x, η x ∈ Icc (0:ℝ) 1)
+    (hone : ∀ x ∈ Metric.ball 0 a, η x = 1) :
+    orientedTilt E v (Metric.ball 0 a) ≤
+      2 * ((perimeter E (Metric.ball 0 b)).toReal -
+        ∫ x in E, divergence (fun y ↦ η y • v) x) := by
+  let μ := (perimeterMeasure E).restrict (Metric.ball 0 b)
+  have hbf : 0 < b := ha.trans hab
+  have : IsFiniteMeasure μ := ⟨by
+    rw [Measure.restrict_apply_univ, perimeterMeasure_open E Metric.isOpen_ball]
+    exact hmin.2.1 0 b hbf⟩
+  have hnb : ∀ x, |inner ℝ (approximateNormal E x) v| ≤ 1 := by
+    intro x
+    exact (abs_real_inner_le_norm _ _).trans (by
+      rw [hv, mul_one]; exact approximateNormal_norm_le_one E x)
+  have hp : ∀ x, |inner ℝ (approximateNormal E x) (η x • v)| ≤ 1 := by
+    intro x
+    rw [real_inner_smul_right, abs_mul, abs_of_nonneg (hb x).1]
+    exact (mul_le_mul_of_nonneg_left (hnb x) (hb x).1).trans (by simpa using (hb x).2)
+  have hnormal := measurable_approximateNormal hmin.1
+  have hflux : Integrable (fun x ↦ inner ℝ (approximateNormal E x) (η x • v)) μ :=
+    (integrable_const (1:ℝ)).mono'
+      (hnormal.inner (hη.continuous.measurable.smul measurable_const)).aestronglyMeasurable
+      (ae_of_all μ fun x ↦ by simpa only [Real.norm_eq_abs] using hp x)
+  have htilt : Integrable (fun x ↦ ‖approximateNormal E x - v‖^2) μ := by
+    apply (integrable_const (4:ℝ)).mono'
+      ((hnormal.sub measurable_const).norm.pow_const 2).aestronglyMeasurable
+    apply ae_of_all μ
+    intro x
+    rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)]
+    have hbound : ‖approximateNormal E x-v‖ ≤ 2 :=
+      (norm_sub_le _ _).trans (by rw [hv]; linarith [approximateNormal_norm_le_one E x])
+    change ‖approximateNormal E x-v‖^2 ≤ 4
+    nlinarith [norm_nonneg (approximateNormal E x-v)]
+  have hgg := minimizing_gaussGreen_local hn hmin (Metric.ball 0 b) Metric.isOpen_ball
+    (hmin.2.1 0 b hbf).ne (fun x ↦ η x • v) (hη.smul contDiff_const)
+    hc.smul_right (tsupport_smul_subset_left _ _ |>.trans hs)
+  have hle : ∀ x, (Metric.ball (0:Euc n) a).indicator
+      (fun x ↦ ‖approximateNormal E x-v‖^2) x ≤
+        2 * (1 - inner ℝ (approximateNormal E x) (η x • v)) := by
+    intro x
+    by_cases hx : x ∈ Metric.ball (0:Euc n) a
+    · rw [indicator_of_mem hx, hone x hx, one_smul, norm_sub_sq_real, hv]
+      have hh := approximateNormal_norm_le_one E x
+      have hpos := norm_nonneg (approximateNormal E x)
+      nlinarith
+    · rw [indicator_of_notMem hx]
+      have hh := (le_abs_self _).trans (hp x)
+      linarith
+  have hh := integral_mono (htilt.indicator measurableSet_ball)
+    (((integrable_const (1:ℝ)).sub hflux).const_mul 2) hle
+  rw [integral_indicator measurableSet_ball, Measure.restrict_restrict measurableSet_ball,
+    inter_eq_left.mpr (Metric.ball_subset_ball hab.le)] at hh
+  rw [integral_const_mul] at hh
+  simp only [Pi.sub_apply] at hh
+  rw [integral_sub (integrable_const _) hflux, integral_const] at hh
+  change orientedTilt E v (Metric.ball 0 a) ≤ 2 * (μ.real univ * 1 - _) at hh
+  have hmass : μ.real univ = (perimeter E (Metric.ball 0 b)).toReal := by
+    simp [μ, measureReal_def]
+  rw [hmass, mul_one] at hh
+  exact hgg ▸ hh
+
+theorem exists_halfspace_calibration {n : ℕ} (hn : 1 < n) {v : Euc n}
+    (hv : ‖v‖ = 1)
+    (hH : IsLocallyPerimeterMinimizing {x : Euc n | inner ℝ x v < 0})
+    {a b : ℝ} (ha : 0 < a) (hab : a < b) {ε : ℝ} (hε : 0 < ε) :
+    ∃ η : Euc n → ℝ, ContDiff ℝ ∞ η ∧ HasCompactSupport η ∧
+      tsupport η ⊆ Metric.ball 0 b ∧ (∀ x, η x ∈ Icc (0:ℝ) 1) ∧
+      (∀ x ∈ Metric.ball 0 a, η x = 1) ∧
+      (perimeter {x : Euc n | inner ℝ x v < 0} (Metric.ball 0 b)).toReal - ε <
+        ∫ x in {x : Euc n | inner ℝ x v < 0}, divergence (fun y ↦ η y • v) x := by
+  let H : Set (Euc n) := {x | inner ℝ x v < 0}
+  let μ := (perimeterMeasure H).restrict (Metric.ball 0 b)
+  have hb : 0 < b := ha.trans hab
+  have : IsFiniteMeasure μ := ⟨by
+    rw [Measure.restrict_apply_univ, perimeterMeasure_open H Metric.isOpen_ball]
+    exact hH.2.1 0 b hb⟩
+  let s : ℕ → ℝ := fun j ↦ (1-(1/2:ℝ)^(j+1))*b
+  have hs0 (j : ℕ) : 0 < s j := by
+    have hh : (1/2:ℝ)^(j+1) < 1 := pow_lt_one₀ (by norm_num) (by norm_num) (by omega)
+    exact mul_pos (sub_pos.mpr hh) hb
+  have hsb (j : ℕ) : s j < b := by
+    have hh : 0 < (1/2:ℝ)^(j+1) := pow_pos (by norm_num) _
+    dsimp [s]
+    nlinarith
+  have ht : Tendsto s atTop (𝓝 b) := by
+    have hpow := (tendsto_pow_atTop_nhds_zero_of_lt_one (by norm_num : (0:ℝ) ≤ 1/2)
+      (by norm_num : (1/2:ℝ) < 1)).mul_const (1/2:ℝ)
+    simpa only [s, pow_succ, zero_mul, sub_zero, one_mul] using
+      (tendsto_const_nhds (x := (1:ℝ))).sub hpow |>.mul_const b
+  obtain ⟨L, hL, hcut⟩ := exists_interior_cutoff_constant
+  choose η hη hc hsupport hbound hone hgrad using
+    (fun j ↦ hcut n H (s j) b (hs0 j) (hsb j))
+  have hnH : ∀ᵐ x ∂μ, approximateNormal H x = v := by
+    have hsupp : ∀ᵐ x ∂perimeterMeasure H, x ∈ perimeterSupport H := by
+      rw [← perimeterMeasure_support H]
+      exact (perimeterMeasure H).support_mem_ae
+    filter_upwards [ae_restrict_of_ae hsupp] with x hx
+    exact approximateNormal_eq_of_blowup hH.1
+      (halfspace_hasHalfspaceBlowup hv (halfspace_perimeterSupport_subset_plane v hx))
+  have heq (j : ℕ) : (∫ x in H, divergence (fun y ↦ η j y • v) x) = ∫ x, η j x ∂μ := by
+    rw [minimizing_gaussGreen_local hn hH (Metric.ball 0 b) Metric.isOpen_ball
+      (hH.2.1 0 b hb).ne (fun x ↦ η j x • v) ((hη j).smul contDiff_const)
+      (hc j).smul_right (tsupport_smul_subset_left _ _ |>.trans (hsupport j))]
+    apply integral_congr_ae
+    filter_upwards [hnH] with x hx
+    rw [hx, real_inner_smul_right, real_inner_self_eq_norm_sq, hv]
+    ring
+  have hlim : Tendsto (fun j ↦ ∫ x, η j x ∂μ) atTop
+      (𝓝 ((perimeter H (Metric.ball 0 b)).toReal)) := by
+    have hpoint : ∀ᵐ x ∂μ, Tendsto (fun j ↦ η j x) atTop (𝓝 (1:ℝ)) := by
+      filter_upwards [ae_restrict_mem measurableSet_ball] with x hx
+      apply tendsto_const_nhds.congr'
+      filter_upwards [ht.eventually (lt_mem_nhds (Metric.mem_ball.mp hx))] with j hj
+      exact (hone j x (Metric.mem_ball.mpr hj)).symm
+    have hh := tendsto_integral_of_dominated_convergence (μ := μ) (fun _ ↦ (1:ℝ))
+      (fun j ↦ (hη j).continuous.aestronglyMeasurable) (integrable_const _)
+      (fun j ↦ ae_of_all μ fun x ↦ by
+        rw [Real.norm_eq_abs, abs_of_nonneg (hbound j x).1]
+        exact (hbound j x).2) hpoint
+    simpa [μ, integral_const, measureReal_def] using hh
+  have hnear := hlim.eventually
+    (lt_mem_nhds (sub_lt_self _ hε))
+  obtain ⟨j, hj, haj⟩ := (hnear.and (ht.eventually (lt_mem_nhds hab))).exists
+  refine ⟨η j, hη j, hc j, hsupport j, hbound j, ?_, ?_⟩
+  · intro x hx
+    exact hone j x (Metric.ball_subset_ball haj.le hx)
+  · rwa [heq j]
+
+theorem HasHalfspaceBlowup.orientedTilt_tendsto {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    {p v : Euc n} (hv : HasHalfspaceBlowup E p v)
+    (r : ℕ → ℝ) (hr : ∀ i, 0 < r i) (htr : Tendsto r atTop (𝓝 0))
+    {R : ℝ} (hR : 0 < R) :
+    Tendsto (fun i ↦ orientedTilt (blownUpSet E p (r i)) v (Metric.ball 0 R))
+      atTop (𝓝 0) := by
+  let H : Set (Euc n) := {x | inner ℝ x v < 0}
+  have hH : MeasurableSet H :=
+    (isOpen_lt (continuous_id.inner continuous_const) continuous_const).measurableSet
+  have hconv := hv.local_L1 hmin.1 r hr htr
+  have hEm (i : ℕ) := locally_minimizing_blownUpSet hmin p (hr i)
+  have hHm := locally_minimizing_of_local_L1_limit (by omega : 0 < n) hEm hH hconv
+  obtain ⟨b, hnull, hRb, hbR⟩ := (minimizing_dense_null_spheres hHm (0:Euc n)).exists_between
+    (show R < R+1 by linarith)
+  have hb : 0 < b := hR.trans hRb
+  have hmass := (ENNReal.tendsto_toReal (hHm.2.1 0 b hb).ne).comp
+    (tendsto_minimizing_perimeter_ball_of_local_L1 (by omega : 0 < n) hEm hH hconv 0 hb hnull)
+  apply tendsto_order.2
+  constructor
+  · intro A hA
+    exact Eventually.of_forall fun i ↦ hA.trans_le (integral_nonneg fun _ ↦ sq_nonneg _)
+  · intro ε hε
+    obtain ⟨η, hη, hcη, hsη, hbη, honeη, hcal⟩ :=
+      exists_halfspace_calibration hn hv.1 hHm hR hRb (by positivity : 0 < ε/4)
+    have htflux := tendsto_setIntegral_of_local_symmDiff (fun i ↦ (hEm i).1) hH hconv
+      (continuous_divergence (hη.smul contDiff_const))
+      (hcη.smul_right.of_isClosed_subset (isClosed_tsupport _)
+        (tsupport_divergence_subset (fun y ↦ η y • v)))
+    have htgap := (hmass.sub htflux).const_mul 2
+    have hgap : 2 *((perimeter H (Metric.ball 0 b)).toReal -
+        ∫ x in H, divergence (fun y ↦ η y • v) x) < ε := by linarith
+    filter_upwards [htgap.eventually (gt_mem_nhds hgap)] with i hi
+    exact (orientedTilt_le_calibration_gap hn (hEm i) hv.1 hR hRb
+      hη hcη hsη hbη honeη).trans_lt hi
+
+theorem HasHalfspaceBlowup.orientedTilt_tendsto_all_scales {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    {p v : Euc n} (hv : HasHalfspaceBlowup E p v) {R : ℝ} (hR : 0 < R) :
+    Tendsto (fun r : ℝ ↦ orientedTilt (blownUpSet E p r) v (Metric.ball 0 R))
+      (𝓝[>] (0:ℝ)) (𝓝 0) := by
+  apply Filter.tendsto_of_seq_tendsto
+  intro r ht
+  let s : ℕ → ℝ := fun i ↦ if 0 < r i then r i else 1
+  have hs : ∀ i, 0 < s i := by
+    intro i
+    dsimp [s]
+    split_ifs with h
+    · exact h
+    · norm_num
+  have heq : s =ᶠ[atTop] r := by
+    filter_upwards [ht.eventually self_mem_nhdsWithin] with i hi
+    simp [s, hi]
+  have hs0 : Tendsto s atTop (𝓝 (0:ℝ)) :=
+    (tendsto_nhds_of_tendsto_nhdsWithin ht).congr' heq.symm
+  have hh := hv.orientedTilt_tendsto hn hmin s hs hs0 hR
+  apply hh.congr'
+  exact heq.fun_comp (fun a ↦ orientedTilt (blownUpSet E p a) v (Metric.ball 0 R))
+
+theorem HasHalfspaceBlowup.support_height_all_scales {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    {p v : Euc n} (hv : HasHalfspaceBlowup E p v)
+    {C : Set (Euc n)} (hC : IsCompact C) {ε : ℝ} (hε : 0 < ε) :
+    ∀ᶠ r : ℝ in 𝓝[>] 0, ∀ z ∈ perimeterSupport (blownUpSet E p r) ∩ C,
+      |inner ℝ z v| < ε := by
+  apply Filter.eventually_iff_seq_eventually.2
+  intro r ht
+  let s : ℕ → ℝ := fun i ↦ if 0 < r i then r i else 1
+  have hs : ∀ i, 0 < s i := by
+    intro i
+    dsimp [s]
+    split_ifs with h
+    · exact h
+    · norm_num
+  have heq : s =ᶠ[atTop] r := by
+    filter_upwards [ht.eventually self_mem_nhdsWithin] with i hi
+    simp [s, hi]
+  have hs0 : Tendsto s atTop (𝓝 (0:ℝ)) :=
+    (tendsto_nhds_of_tendsto_nhdsWithin ht).congr' heq.symm
+  filter_upwards [hv.support_height hn hmin s hs hs0 hC hε, heq] with i hi he
+  rwa [he] at hi
+
+theorem HasHalfspaceBlowup.small_height_and_orientedTilt {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    {p v : Euc n} (hv : HasHalfspaceBlowup E p v)
+    {R ε : ℝ} (hR : 0 < R) (hε : 0 < ε) :
+    ∀ᶠ r : ℝ in 𝓝[>] 0,
+      (∀ z ∈ perimeterSupport (blownUpSet E p r) ∩ Metric.closedBall 0 R,
+        |inner ℝ z v| < ε) ∧
+      orientedTilt (blownUpSet E p r) v (Metric.ball 0 R) < ε := by
+  exact (hv.support_height_all_scales hn hmin (isCompact_closedBall 0 R) hε).and
+    ((hv.orientedTilt_tendsto_all_scales hn hmin hR).eventually (gt_mem_nhds hε))
+
+theorem minimizing_height_caccioppoli {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    (c v : Euc n) (a : ℝ) {R : ℝ} (hR : 0 < R)
+    {η : Euc n → ℝ} (hη : ContDiff ℝ ∞ η) (hc : HasCompactSupport η)
+    (hs : tsupport η ⊆ Metric.ball c R) :
+    (∫ x in Metric.ball c R,
+      η x ^ 2 * ‖tangentialGradient E (fun y ↦ inner ℝ v y - a) x‖ ^ 2
+      ∂perimeterMeasure E) ≤
+    4 * ∫ x in Metric.ball c R, (inner ℝ v x - a)^2 * ‖gradient η x‖^2
+      ∂perimeterMeasure E := by
+  let H : Euc n → ℝ := fun x ↦ inner ℝ v x - a
+  have hH : ContDiff ℝ ∞ H := (contDiff_const.inner ℝ contDiff_id).sub contDiff_const
+  have hgH (x : Euc n) : gradient H x = v := by
+    apply HasGradientAt.gradient
+    change HasFDerivAt H ((InnerProductSpace.toDual ℝ (Euc n)) v) x
+    exact (innerSL ℝ v).hasFDerivAt.sub_const a
+  let φ : Euc n → ℝ := fun x ↦ (η x * η x) * H x
+  have hφ : ContDiff ℝ ∞ φ := (hη.mul hη).mul hH
+  have hcφ : HasCompactSupport φ := (hc.mul_right).mul_right
+  have hsφ : tsupport φ ⊆ Metric.ball c R :=
+    tsupport_mul_subset_left.trans (tsupport_mul_subset_left.trans hs)
+  let μ := (perimeterMeasure E).restrict (Metric.ball c R)
+  let := perimeterMeasure_locallyFinite E hmin.2.1
+  have : IsFiniteMeasure μ := ⟨by
+    simp only [μ, Measure.restrict_apply_univ, perimeterMeasure_open E Metric.isOpen_ball]
+    exact hmin.2.1 c R hR⟩
+  have hgd : Continuous (gradient η) := continuous_iff_continuousAt.mpr fun x ↦
+    RPI.Boundary.continuousAt_gradient_of_contDiffAt (hη.contDiffAt.of_le (by simp))
+  obtain ⟨M, hM⟩ := hc.exists_bound_of_continuous hη.continuous
+  have hM0 : 0 ≤ M := (norm_nonneg (η c)).trans (hM c)
+  have hA : MemLp (fun x ↦ η x • tangentialGradient E H x) (2:ℝ≥0∞) μ := by
+    apply MemLp.of_bound
+      (hη.continuous.aestronglyMeasurable.smul
+        (measurable_tangentialGradient hmin.1 (hH.of_le (by simp))).aestronglyMeasurable)
+      (M * (2 * ‖v‖))
+    apply ae_of_all μ
+    intro x
+    change ‖η x • tangentialGradient E H x‖ ≤ _
+    rw [norm_smul]
+    exact mul_le_mul (hM x)
+      ((tangentialGradient_norm_le_twice E H x).trans_eq (by rw [hgH]))
+      (norm_nonneg _) hM0
+  have hIA : Integrable (fun x ↦ η x^2 * ‖tangentialGradient E H x‖^2) μ := by
+    simpa only [norm_smul, mul_pow, Real.norm_eq_abs, sq_abs] using
+      hA.integrable_norm_pow (by norm_num : (2:ℕ) ≠ 0)
+  have hIB : Integrable (fun x ↦ H x^2 * ‖gradient η x‖^2) μ := by
+    apply IntegrableOn.mono_set
+      (ContinuousOn.integrableOn_compact (isCompact_closedBall c R)
+        ((hH.continuous.pow 2).mul (hgd.norm.pow 2)).continuousOn)
+      Metric.ball_subset_closedBall
+  have hIJ : Integrable (fun x ↦ inner ℝ (tangentialGradient E φ x) v) μ :=
+    (tangentialGradient_integrable_ball_perimeter hmin c hR
+      (hφ.of_le (by simp))).inner_const v
+  have hz := minimizing_tangentialGradient_integral_eq_zero_local hn hmin c hR hφ hcφ hsφ v
+  have hp : ∀ᵐ x ∂μ,
+      η x^2 * ‖tangentialGradient E H x‖^2 ≤
+      2 * inner ℝ (tangentialGradient E φ x) v + 4 * (H x^2 * ‖gradient η x‖^2) := by
+    filter_upwards [ae_restrict_of_ae (minimizing_halfspaceBlowup_ae hn hmin)] with x hx
+    have hW : inner ℝ (tangentialGradient E H x) v =
+        ‖tangentialGradient E H x‖^2 := by
+      simp only [tangentialGradient, hgH, inner_sub_left, real_inner_smul_left,
+        norm_sub_sq_real, norm_smul, Real.norm_eq_abs, sq_abs,
+        real_inner_smul_right, hx.1, mul_one, real_inner_self_eq_norm_sq]
+      rw [real_inner_comm (approximateNormal E x) v]
+      ring
+    have hG : inner ℝ (tangentialGradient E η x) v =
+        inner ℝ (gradient η x) (tangentialGradient E H x) := by
+      simp only [tangentialGradient, hgH, inner_sub_left, real_inner_smul_left,
+        inner_sub_right, real_inner_smul_right]
+      rw [real_inner_comm (approximateNormal E x) v]
+      ring
+    have hD : inner ℝ (tangentialGradient E φ x) v =
+        η x^2 * ‖tangentialGradient E H x‖^2 +
+        2 * η x * H x * inner ℝ (gradient η x) (tangentialGradient E H x) := by
+      dsimp only [φ]
+      rw [tangentialGradient_mul E ((hη.differentiable (by simp) x).fun_mul
+        (hη.differentiable (by simp) x)) (hH.differentiable (by simp) x),
+        tangentialGradient_mul E (hη.differentiable (by simp) x)
+          (hη.differentiable (by simp) x)]
+      simp only [inner_add_left, real_inner_smul_left, hW, hG]
+      ring
+    have hnn := sq_nonneg ‖η x • tangentialGradient E H x + (2 * H x) • gradient η x‖
+    rw [norm_add_sq_real] at hnn
+    simp only [norm_smul, Real.norm_eq_abs, mul_pow, sq_abs,
+      real_inner_smul_left, real_inner_smul_right] at hnn
+    have hiComm := real_inner_comm (tangentialGradient E H x) (gradient η x)
+    rw [← hiComm] at hnn
+    rw [hD]
+    nlinarith only [hnn]
+  have hi := integral_mono_ae hIA ((hIJ.const_mul 2).add (hIB.const_mul 4)) hp
+  simp only [Pi.add_apply] at hi
+  rw [integral_add (hIJ.const_mul 2) (hIB.const_mul 4), integral_const_mul,
+    integral_const_mul] at hi
+  change (∫ x, η x^2 * ‖tangentialGradient E H x‖^2 ∂μ) ≤
+    2 * (∫ x in Metric.ball c R, inner ℝ (tangentialGradient E φ x) v ∂perimeterMeasure E) + _ at hi
+  rw [hz, mul_zero, zero_add] at hi
+  exact hi
+
+theorem minimizing_height_distribution {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    (c : Euc n) {R : ℝ} (hR : 0 < R) (v : Euc n) (a : ℝ)
+    {φ : Euc n → ℝ} (hφ : ContDiff ℝ ∞ φ) (hcφ : HasCompactSupport φ)
+    (hsφ : tsupport φ ⊆ Metric.ball c R) :
+    (∫ x in Metric.ball c R, (inner ℝ v x - a) *
+      tangentialDivergence E (gradient φ) x ∂perimeterMeasure E) = 0 := by
+  let H : Euc n → ℝ := fun x ↦ inner ℝ v x - a
+  have hH : ContDiff ℝ ∞ H := (contDiff_const.inner ℝ contDiff_id).sub contDiff_const
+  have hgH (x : Euc n) : gradient H x = v := by
+    apply HasGradientAt.gradient
+    change HasFDerivAt H ((InnerProductSpace.toDual ℝ (Euc n)) v) x
+    exact (innerSL ℝ v).hasFDerivAt.sub_const a
+  have hG := gradient_contDiff hφ
+  have hcG := gradient_hasCompactSupport hcφ
+  have hsG : tsupport (gradient φ) ⊆ Metric.ball c R := by
+    exact (tsupport_comp_subset (g := (InnerProductSpace.toDual ℝ (Euc n)).symm)
+      (map_zero _) (fderiv ℝ φ)).trans ((tsupport_fderiv_subset ℝ).trans hsφ)
+  let V : Euc n → Euc n := fun x ↦ H x • gradient φ x
+  have hV : ContDiff ℝ ∞ V := hH.smul (𝕜' := ℝ) hG
+  have hcV : HasCompactSupport V := hcG.smul_left (f := H)
+  have hsV : tsupport V ⊆ Metric.ball c R :=
+    (tsupport_smul_subset_right H (gradient φ)).trans hsG
+  have heq (x : Euc n) :
+      H x * tangentialDivergence E (gradient φ) x =
+        tangentialDivergence E V x - inner ℝ (tangentialGradient E φ x) v := by
+    rw [show V = fun y ↦ H y • gradient φ y from rfl,
+      tangentialDivergence_smul E (hH.differentiable (by simp))
+        (hG.differentiable (by simp))]
+    simp only [tangentialGradient, hgH, inner_sub_left, real_inner_smul_left]
+    rw [real_inner_comm v (gradient φ x),
+      real_inner_comm (approximateNormal E x) (gradient φ x),
+      real_inner_comm (approximateNormal E x) v]
+    ring
+  change (∫ x in Metric.ball c R, H x * tangentialDivergence E (gradient φ) x ∂perimeterMeasure E) = 0
+  simp_rw [heq]
+  rw [integral_sub (tangentialDivergence_integrable_ball_perimeter hmin c hR hV)
+    ((tangentialGradient_integrable_ball_perimeter hmin c hR
+      (hφ.of_le (by simp))).inner_const v),
+    minimizing_tangentialDivergence_integral_eq_zero hn hmin c hR hV hcV hsV,
+    minimizing_tangentialGradient_integral_eq_zero_local hn hmin c hR hφ hcφ hsφ v,
+    sub_self]
+
+def planarDivergence {n : ℕ} (v : Euc n) (V : Euc n → Euc n) (x : Euc n) : ℝ :=
+  divergence V x - inner ℝ v (fderiv ℝ V x v)
+
+theorem normal_quadratic_difference_le {n : ℕ} (L : Euc n →L[ℝ] Euc n)
+    {u v : Euc n} (hu : ‖u‖ ≤ 1) (hv : ‖v‖ ≤ 1) :
+    |inner ℝ u (L u) - inner ℝ v (L v)| ≤ 2 * ‖L‖ * ‖u-v‖ := by
+  have he : inner ℝ u (L u) - inner ℝ v (L v) =
+      inner ℝ (u-v) (L u) + inner ℝ v (L (u-v)) := by
+    rw [map_sub, inner_sub_left, inner_sub_right]
+    ring
+  rw [he]
+  calc
+    _ ≤ |inner ℝ (u-v) (L u)| + |inner ℝ v (L (u-v))| := abs_add_le _ _
+    _ ≤ ‖u-v‖ * (‖L‖ * ‖u‖) + ‖v‖ * (‖L‖ * ‖u-v‖) := by
+      apply add_le_add
+      · simpa only [Real.norm_eq_abs] using
+          (norm_inner_le_norm (𝕜 := ℝ) (u-v) (L u)).trans
+            (mul_le_mul_of_nonneg_left (L.le_opNorm u) (norm_nonneg _))
+      · simpa only [Real.norm_eq_abs] using
+          (norm_inner_le_norm (𝕜 := ℝ) v (L (u-v))).trans
+            (mul_le_mul_of_nonneg_left (L.le_opNorm (u-v)) (norm_nonneg _))
+    _ ≤ ‖u-v‖ * (‖L‖ * 1) + 1 * (‖L‖ * ‖u-v‖) := by
+      apply add_le_add
+      · exact mul_le_mul_of_nonneg_left
+          (mul_le_mul_of_nonneg_left hu (norm_nonneg _)) (norm_nonneg _)
+      · exact mul_le_mul_of_nonneg_right hv (mul_nonneg (norm_nonneg _) (norm_nonneg _))
+    _ = _ := by ring
+
+theorem normal_error_integral_le {n : ℕ} {E : Set (Euc n)}
+    (hmin : IsLocallyPerimeterMinimizing E) (c : Euc n) {R : ℝ} (hR : 0 < R)
+    (v : Euc n) :
+    (∫ x in Metric.ball c R, ‖approximateNormal E x - v‖ ∂perimeterMeasure E) ≤
+      Real.sqrt (orientedTilt E v (Metric.ball c R)) *
+        Real.sqrt ((perimeter E (Metric.ball c R)).toReal) := by
+  let μ := (perimeterMeasure E).restrict (Metric.ball c R)
+  have : IsFiniteMeasure μ := ⟨by
+    rw [Measure.restrict_apply_univ, perimeterMeasure_open E Metric.isOpen_ball]
+    exact hmin.2.1 c R hR⟩
+  have hmem : MemLp (fun x ↦ ‖approximateNormal E x - v‖) (ENNReal.ofReal (2:ℝ)) μ := by
+    apply MemLp.of_bound
+      (((measurable_approximateNormal hmin.1).sub measurable_const).norm.aestronglyMeasurable)
+      (1+‖v‖)
+    exact ae_of_all μ fun x ↦ by
+      rw [Real.norm_of_nonneg (norm_nonneg _)]
+      exact (norm_sub_le _ _).trans (add_le_add (approximateNormal_norm_le_one E x) le_rfl)
+  have hh := integral_mul_le_Lp_mul_Lq_of_nonneg
+    (show (2:ℝ).HolderConjugate 2 from Real.holderConjugate_iff.mpr ⟨by norm_num, by norm_num⟩)
+    (ae_of_all μ fun _ ↦ norm_nonneg _) (ae_of_all μ fun _ ↦ (zero_le_one : (0:ℝ) ≤ 1))
+    hmem (memLp_const (1:ℝ))
+  simpa only [mul_one, Real.rpow_two, one_pow, integral_const, smul_eq_mul, mul_one,
+    μ, Measure.real, Measure.restrict_apply_univ, perimeterMeasure_open E Metric.isOpen_ball,
+    ← Real.sqrt_eq_rpow, orientedTilt] using hh
+
+theorem minimizing_height_planar_residual {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    (c : Euc n) {R : ℝ} (hR : 0 < R) {v : Euc n} (hv : ‖v‖ = 1) (a : ℝ)
+    {φ : Euc n → ℝ} (hφ : ContDiff ℝ ∞ φ) (hcφ : HasCompactSupport φ)
+    (hsφ : tsupport φ ⊆ Metric.ball c R)
+    {A K : ℝ} (hA : 0 ≤ A) (hK : 0 ≤ K)
+    (hheight : ∀ x ∈ perimeterSupport E ∩ Metric.ball c R, |inner ℝ v x - a| ≤ A)
+    (hderiv : ∀ x ∈ Metric.ball c R, ‖fderiv ℝ (gradient φ) x‖ ≤ K) :
+    |∫ x in Metric.ball c R, (inner ℝ v x - a) *
+      planarDivergence v (gradient φ) x ∂perimeterMeasure E| ≤
+      2 * A * K * (Real.sqrt (orientedTilt E v (Metric.ball c R)) *
+        Real.sqrt ((perimeter E (Metric.ball c R)).toReal)) := by
+  let μ := (perimeterMeasure E).restrict (Metric.ball c R)
+  let H : Euc n → ℝ := fun x ↦ inner ℝ v x - a
+  let G := gradient φ
+  have hg := gradient_contDiff hφ
+  have hH : Continuous H := (continuous_const.inner continuous_id).sub continuous_const
+  have := perimeterMeasure_locallyFinite E hmin.2.1
+  have : IsFiniteMeasure μ := ⟨by
+    rw [Measure.restrict_apply_univ, perimeterMeasure_open E Metric.isOpen_ball]
+    exact hmin.2.1 c R hR⟩
+  have hbH : ∀ᵐ x ∂μ, |H x| ≤ A := by
+    have hmm : ∀ᵐ x ∂perimeterMeasure E, x ∈ perimeterSupport E := by
+      rw [← perimeterMeasure_support]; exact (perimeterMeasure E).support_mem_ae
+    filter_upwards [ae_restrict_of_ae hmm, ae_restrict_mem measurableSet_ball] with x hx hx'
+    exact hheight x ⟨hx, hx'⟩
+  have hpcont : Continuous (planarDivergence v G) :=
+    (continuous_divergence hg).sub
+      (continuous_const.inner ((hg.continuous_fderiv (by simp)).clm_apply continuous_const))
+  have hpint : Integrable (fun x ↦ H x * planarDivergence v G x) μ :=
+    ((hH.mul hpcont).continuousOn.integrableOn_compact (isCompact_closedBall c R)).mono_set
+      Metric.ball_subset_closedBall
+  have htint : Integrable (fun x ↦ H x * tangentialDivergence E G x) μ :=
+    (tangentialDivergence_integrable_ball_perimeter hmin c hR hg).bdd_mul
+      hH.aestronglyMeasurable (by simpa only [Real.norm_eq_abs] using hbH)
+  have hiN : Integrable (fun x ↦ ‖approximateNormal E x-v‖) μ := by
+    apply Integrable.of_bound
+      (((measurable_approximateNormal hmin.1).sub measurable_const).norm.aestronglyMeasurable)
+      2
+    apply ae_of_all μ
+    intro x
+    rw [Real.norm_of_nonneg (norm_nonneg _)]
+    exact (norm_sub_le _ _).trans (by
+      simp only [hv]; linarith [approximateNormal_norm_le_one E x])
+  have heq : (∫ x, H x * planarDivergence v G x ∂μ) =
+      ∫ x, H x * planarDivergence v G x - H x * tangentialDivergence E G x ∂μ := by
+    rw [integral_sub hpint htint]
+    have hz := minimizing_height_distribution hn hmin c hR v a hφ hcφ hsφ
+    change (∫ x, H x * tangentialDivergence E G x ∂μ) = 0 at hz
+    rw [hz, sub_zero]
+  have hb : ∀ᵐ x ∂μ,
+      |H x * planarDivergence v G x - H x * tangentialDivergence E G x| ≤
+        (2*A*K) * ‖approximateNormal E x-v‖ := by
+    filter_upwards [hbH, ae_restrict_mem measurableSet_ball] with x hx hxB
+    have hq := normal_quadratic_difference_le (fderiv ℝ G x)
+      (approximateNormal_norm_le_one E x) hv.le
+    have haeq : H x * planarDivergence v G x - H x * tangentialDivergence E G x =
+        H x * (inner ℝ (approximateNormal E x) (fderiv ℝ G x (approximateNormal E x)) -
+          inner ℝ v (fderiv ℝ G x v)) := by
+      dsimp [planarDivergence, tangentialDivergence]
+      ring
+    rw [haeq, abs_mul]
+    calc
+      _ ≤ A * (2 * ‖fderiv ℝ G x‖ * ‖approximateNormal E x-v‖) :=
+        mul_le_mul hx hq (abs_nonneg _) hA
+      _ ≤ A * (2 * K * ‖approximateNormal E x-v‖) := by
+        apply mul_le_mul_of_nonneg_left _ hA
+        exact mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left
+          (hderiv x hxB) (by norm_num)) (norm_nonneg _)
+      _ = _ := by ring
+  change |∫ x, H x * planarDivergence v G x ∂μ| ≤ _
+  rw [heq]
+  calc
+    _ ≤ ∫ x, |H x * planarDivergence v G x - H x * tangentialDivergence E G x| ∂μ :=
+      abs_integral_le_integral_abs
+    _ ≤ ∫ x, (2*A*K) * ‖approximateNormal E x-v‖ ∂μ :=
+      integral_mono_ae (hpint.sub htint).abs (hiN.const_mul _) hb
+    _ = (2*A*K) * ∫ x, ‖approximateNormal E x-v‖ ∂μ := integral_const_mul _ _
+    _ ≤ _ := mul_le_mul_of_nonneg_left (normal_error_integral_le hmin c hR v)
+      (mul_nonneg (mul_nonneg (by norm_num) hA) hK)
+
+theorem horizontal_normal_le_height_projection {n : ℕ} {N v w : Euc n}
+    (hN : ‖N‖ = 1) (hv : ‖v‖ = 1) (hw : inner ℝ v w = 0) :
+    |inner ℝ N w| ≤ ‖v - (inner ℝ N v) • N‖ * ‖w‖ := by
+  have he : inner ℝ (N - (inner ℝ N v) • v) w = inner ℝ N w := by
+    simp [inner_sub_left, real_inner_smul_left, hw]
+  have hs : ‖N - (inner ℝ N v) • v‖ ^ 2 =
+      ‖v - (inner ℝ N v) • N‖ ^ 2 := by
+    simp only [norm_sub_sq_real, norm_smul, Real.norm_eq_abs, sq_abs,
+      real_inner_smul_right, hN, hv, one_pow, mul_one]
+    rw [real_inner_comm v N]
+  have hn : ‖N - (inner ℝ N v) • v‖ = ‖v - (inner ℝ N v) • N‖ := by
+    nlinarith [norm_nonneg (N - (inner ℝ N v) • v),
+      norm_nonneg (v - (inner ℝ N v) • N)]
+  rw [← he]
+  simpa only [Real.norm_eq_abs, hn] using
+    norm_inner_le_norm (𝕜 := ℝ) (N - (inner ℝ N v) • v) w
+
+theorem minimizing_horizontal_normal_square_le {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    (c : Euc n) {v w : Euc n} (hv : ‖v‖ = 1) (hw : inner ℝ v w = 0)
+    (a : ℝ) {R : ℝ} (hR : 0 < R)
+    {η : Euc n → ℝ} (hη : ContDiff ℝ ∞ η) (hc : HasCompactSupport η)
+    (hs : tsupport η ⊆ Metric.ball c R) :
+    (∫ x in Metric.ball c R, (η x * inner ℝ (approximateNormal E x) w)^2
+      ∂perimeterMeasure E) ≤
+    4 * ‖w‖^2 * ∫ x in Metric.ball c R, (inner ℝ v x - a)^2 * ‖gradient η x‖^2
+      ∂perimeterMeasure E := by
+  let μ := (perimeterMeasure E).restrict (Metric.ball c R)
+  let H : Euc n → ℝ := fun x ↦ inner ℝ v x-a
+  have hH : ContDiff ℝ ∞ H := (contDiff_const.inner ℝ contDiff_id).sub contDiff_const
+  have hgH (x : Euc n) : gradient H x = v := by
+    apply HasGradientAt.gradient
+    change HasFDerivAt H ((InnerProductSpace.toDual ℝ (Euc n)) v) x
+    exact (innerSL ℝ v).hasFDerivAt.sub_const a
+  have : IsFiniteMeasure μ := ⟨by
+    rw [Measure.restrict_apply_univ, perimeterMeasure_open E Metric.isOpen_ball]
+    exact hmin.2.1 c R hR⟩
+  obtain ⟨M, hM⟩ := hc.exists_bound_of_continuous hη.continuous
+  have hM0 : 0 ≤ M := (norm_nonneg (η c)).trans (hM c)
+  have hLI : Integrable (fun x ↦ (η x * inner ℝ (approximateNormal E x) w)^2) μ := by
+    have hh : MemLp (fun x ↦ η x * inner ℝ (approximateNormal E x) w) 2 μ := by
+      apply MemLp.of_bound
+        (hη.continuous.aestronglyMeasurable.mul
+          ((measurable_approximateNormal hmin.1).inner measurable_const).aestronglyMeasurable)
+        (M * ‖w‖)
+      apply ae_of_all μ
+      intro x
+      change ‖η x * inner ℝ (approximateNormal E x) w‖ ≤ _
+      rw [norm_mul]
+      have hb : ‖inner ℝ (approximateNormal E x) w‖ ≤ ‖w‖ :=
+        (norm_inner_le_norm (𝕜 := ℝ) _ _).trans (by
+          simpa only [one_mul] using mul_le_mul_of_nonneg_right
+            (approximateNormal_norm_le_one E x) (norm_nonneg w))
+      exact mul_le_mul (hM x) hb (norm_nonneg _) hM0
+    simpa only [Real.norm_eq_abs, sq_abs] using hh.integrable_norm_pow
+      (by norm_num : (2:ℕ) ≠ 0)
+  have hRI : Integrable (fun x ↦ η x^2 * ‖tangentialGradient E H x‖^2) μ := by
+    have hh : MemLp (fun x ↦ η x • tangentialGradient E H x) 2 μ := by
+      apply MemLp.of_bound
+        (hη.continuous.aestronglyMeasurable.smul
+          (measurable_tangentialGradient hmin.1 (hH.of_le (by simp))).aestronglyMeasurable)
+        (M * (2*‖v‖))
+      apply ae_of_all μ
+      intro x
+      change ‖η x • tangentialGradient E H x‖ ≤ _
+      rw [norm_smul]
+      exact mul_le_mul (hM x) ((tangentialGradient_norm_le_twice E H x).trans_eq
+        (by rw [hgH])) (norm_nonneg _) hM0
+    simpa only [norm_smul, mul_pow, Real.norm_eq_abs, sq_abs] using
+      hh.integrable_norm_pow (by norm_num : (2:ℕ) ≠ 0)
+  have hp : ∀ᵐ x ∂μ, (η x * inner ℝ (approximateNormal E x) w)^2 ≤
+      ‖w‖^2 * (η x^2 * ‖tangentialGradient E H x‖^2) := by
+    filter_upwards [ae_restrict_of_ae (minimizing_halfspaceBlowup_ae hn hmin)] with x hx
+    have hh := horizontal_normal_le_height_projection hx.1 hv hw
+    change |inner ℝ (approximateNormal E x) w| ≤
+      ‖v - (inner ℝ (approximateNormal E x) v) • approximateNormal E x‖ * ‖w‖ at hh
+    have hsq := sq_le_sq₀ (abs_nonneg _) (mul_nonneg (norm_nonneg _) (norm_nonneg _)) |>.mpr hh
+    rw [sq_abs, mul_pow] at hsq
+    have hi := mul_le_mul_of_nonneg_left hsq (sq_nonneg (η x))
+    rw [← real_inner_comm (approximateNormal E x) v] at hi
+    simpa only [mul_pow, tangentialGradient, hgH, mul_assoc, mul_left_comm, mul_comm] using hi
+  have hi := integral_mono_ae hLI (hRI.const_mul _) hp
+  rw [integral_const_mul] at hi
+  exact hi.trans (by
+    have hh := mul_le_mul_of_nonneg_left
+      (minimizing_height_caccioppoli hn hmin c v a hR hη hc hs) (sq_nonneg ‖w‖)
+    convert hh using 1
+    ring)
+
+theorem integral_abs_le_of_quadratic_mean_bound {α : Type*} [MeasurableSpace α]
+    (μ : Measure α) [IsFiniteMeasure μ] {f : α → ℝ}
+    (hf : MemLp f 2 μ) {L : ℝ} (hL : 0 ≤ L)
+    (hq : (∫ x, f x^2 ∂μ) ≤ L^2 * (μ univ).toReal) :
+    (∫ x, |f x| ∂μ) ≤ L * (μ univ).toReal := by
+  have hpq : Real.HolderConjugate 2 2 :=
+    Real.holderConjugate_iff.mpr ⟨by norm_num, by norm_num⟩
+  have hm : MemLp (fun x ↦ |f x|) (ENNReal.ofReal (2:ℝ)) μ := by
+    simpa only [ENNReal.ofReal_ofNat, Real.norm_eq_abs] using hf.norm
+  have hh := integral_mul_le_Lp_mul_Lq_of_nonneg hpq
+    (ae_of_all μ fun x ↦ abs_nonneg (f x))
+    (ae_of_all μ fun _ ↦ (zero_le_one : (0:ℝ) ≤ 1)) hm (memLp_const (1:ℝ))
+  have hs : Real.sqrt (∫ x, f x^2 ∂μ) ≤ L * Real.sqrt (μ univ).toReal := by
+    calc
+      _ ≤ Real.sqrt (L^2 * (μ univ).toReal) := Real.sqrt_le_sqrt hq
+      _ = _ := by rw [Real.sqrt_mul (sq_nonneg L), Real.sqrt_sq hL]
+  have hh' : (∫ x, |f x| ∂μ) ≤ Real.sqrt (∫ x, f x^2 ∂μ) * Real.sqrt (μ univ).toReal := by
+    simpa only [mul_one, Real.rpow_two, sq_abs, one_pow, integral_const, smul_eq_mul,
+      mul_one, Measure.real, ← Real.sqrt_eq_rpow] using hh
+  calc
+    _ ≤ (L * Real.sqrt (μ univ).toReal) * Real.sqrt (μ univ).toReal :=
+      hh'.trans (mul_le_mul_of_nonneg_right hs (Real.sqrt_nonneg _))
+    _ = _ := by rw [mul_assoc, Real.mul_self_sqrt ENNReal.toReal_nonneg]
+
+theorem minimizing_horizontal_normal_integral_le {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    (c : Euc n) {v w : Euc n} (hv : ‖v‖ = 1) (hw : inner ℝ v w = 0)
+    (a : ℝ) {R : ℝ} (hR : 0 < R)
+    {η : Euc n → ℝ} (hη : ContDiff ℝ ∞ η) (hc : HasCompactSupport η)
+    (hs : tsupport η ⊆ Metric.ball c R)
+    {A L : ℝ} (hA : 0 ≤ A) (hL : 0 ≤ L)
+    (hheight : ∀ x ∈ perimeterSupport E ∩ Metric.ball c R, |inner ℝ v x-a| ≤ A)
+    (hgrad : ∀ x ∈ Metric.ball c R, ‖gradient η x‖ ≤ L) :
+    (∫ x in Metric.ball c R, |η x * inner ℝ (approximateNormal E x) w|
+      ∂perimeterMeasure E) ≤
+      (2 * ‖w‖ * A * L) * (perimeter E (Metric.ball c R)).toReal := by
+  let μ := (perimeterMeasure E).restrict (Metric.ball c R)
+  have := perimeterMeasure_locallyFinite E hmin.2.1
+  have : IsFiniteMeasure μ := ⟨by
+    rw [Measure.restrict_apply_univ, perimeterMeasure_open E Metric.isOpen_ball]
+    exact hmin.2.1 c R hR⟩
+  obtain ⟨M, hM⟩ := hc.exists_bound_of_continuous hη.continuous
+  have hM0 : 0 ≤ M := (norm_nonneg (η c)).trans (hM c)
+  have hf : MemLp (fun x ↦ η x * inner ℝ (approximateNormal E x) w) 2 μ := by
+    apply MemLp.of_bound
+      (hη.continuous.aestronglyMeasurable.mul
+        ((measurable_approximateNormal hmin.1).inner measurable_const).aestronglyMeasurable)
+      (M * ‖w‖)
+    apply ae_of_all μ
+    intro x
+    change ‖η x * inner ℝ (approximateNormal E x) w‖ ≤ _
+    rw [norm_mul]
+    have hb : ‖inner ℝ (approximateNormal E x) w‖ ≤ ‖w‖ :=
+      (norm_inner_le_norm (𝕜 := ℝ) _ _).trans (by
+        simpa only [one_mul] using mul_le_mul_of_nonneg_right
+          (approximateNormal_norm_le_one E x) (norm_nonneg w))
+    exact mul_le_mul (hM x) hb (norm_nonneg _) hM0
+  have hfi : Integrable (fun x ↦ (inner ℝ v x-a)^2 * ‖gradient η x‖^2) μ :=
+    (((continuous_const.inner continuous_id).sub continuous_const).pow 2 |>.mul
+      ((gradient_contDiff hη).continuous.norm.pow 2)).continuousOn.integrableOn_compact
+      (isCompact_closedBall c R) |>.mono_set Metric.ball_subset_closedBall
+  have hG : (∫ x, (inner ℝ v x-a)^2 * ‖gradient η x‖^2 ∂μ) ≤
+      A^2 * L^2 * (μ univ).toReal := by
+    have hsupp : ∀ᵐ x ∂μ, x ∈ perimeterSupport E := by
+      apply ae_restrict_of_ae
+      rw [← perimeterMeasure_support]
+      exact (perimeterMeasure E).support_mem_ae
+    have hb : ∀ᵐ x ∂μ, (inner ℝ v x-a)^2 * ‖gradient η x‖^2 ≤ A^2 * L^2 := by
+      filter_upwards [hsupp, ae_restrict_mem measurableSet_ball] with x hx hxB
+      have h₁ := (sq_le_sq₀ (abs_nonneg _) hA).mpr (hheight x ⟨hx,hxB⟩)
+      rw [sq_abs] at h₁
+      have h₂ := (sq_le_sq₀ (norm_nonneg _) hL).mpr (hgrad x hxB)
+      exact mul_le_mul h₁ h₂ (sq_nonneg _) (sq_nonneg _)
+    have hi := integral_mono_ae hfi (integrable_const (A^2*L^2)) hb
+    simpa only [integral_const, smul_eq_mul, mul_comm, Measure.real] using hi
+  have hq : (∫ x, (η x * inner ℝ (approximateNormal E x) w)^2 ∂μ) ≤
+      (2*‖w‖*A*L)^2 * (μ univ).toReal := by
+    have hh := minimizing_horizontal_normal_square_le hn hmin c hv hw a hR hη hc hs
+    change (∫ x, (η x * inner ℝ (approximateNormal E x) w)^2 ∂μ) ≤
+      4*‖w‖^2 * _ at hh
+    apply hh.trans
+    have hi := mul_le_mul_of_nonneg_left hG (by positivity : 0 ≤ 4*‖w‖^2)
+    convert hi using 1
+    ring
+  have hh := integral_abs_le_of_quadratic_mean_bound μ hf
+    (show 0 ≤ 2*‖w‖*A*L by positivity) hq
+  simpa only [μ, Measure.restrict_apply_univ, perimeterMeasure_open E Metric.isOpen_ball] using hh
+
+theorem minimizing_horizontal_test_le {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    (c : Euc n) {v w : Euc n} (hv : ‖v‖ = 1) (hw : inner ℝ v w = 0)
+    (a : ℝ) {R : ℝ} (hR : 0 < R)
+    {η φ : Euc n → ℝ} (hη : ContDiff ℝ ∞ η) (hc : HasCompactSupport η)
+    (hs : tsupport η ⊆ Metric.ball c R) (hφ : ContDiff ℝ ∞ φ)
+    (hbφ : ∀ x, |φ x| ≤ 1)
+    {A L : ℝ} (hA : 0 ≤ A) (hL : 0 ≤ L)
+    (hheight : ∀ x ∈ perimeterSupport E ∩ Metric.ball c R, |inner ℝ v x-a| ≤ A)
+    (hgrad : ∀ x ∈ Metric.ball c R, ‖gradient η x‖ ≤ L) :
+    |∫ x in E, fderiv ℝ (fun y ↦ η y * φ y) x w| ≤
+      (2 * ‖w‖ * A * L) * (perimeter E (Metric.ball c R)).toReal := by
+  let μ := (perimeterMeasure E).restrict (Metric.ball c R)
+  have : IsFiniteMeasure μ := ⟨by
+    rw [Measure.restrict_apply_univ, perimeterMeasure_open E Metric.isOpen_ball]
+    exact hmin.2.1 c R hR⟩
+  have htest : (∫ x in E, fderiv ℝ (fun y ↦ η y * φ y) x w) =
+      ∫ x, φ x * (η x * inner ℝ (approximateNormal E x) w) ∂μ := by
+    rw [← divergence_scalar_direction ((hη.mul hφ).differentiable (by simp)) w,
+      minimizing_gaussGreen_local hn hmin (Metric.ball c R) Metric.isOpen_ball
+        (hmin.2.1 c R hR).ne (fun y ↦ (η y * φ y) • w) ((hη.mul hφ).smul contDiff_const)
+        hc.mul_right.smul_right
+        (tsupport_smul_subset_left _ _ |>.trans (tsupport_mul_subset_left.trans hs))]
+    apply integral_congr_ae
+    apply ae_of_all μ
+    intro x
+    simp only [real_inner_smul_right]
+    ring
+  obtain ⟨M,hM⟩ := hc.exists_bound_of_continuous hη.continuous
+  have hM0 : 0 ≤ M := (norm_nonneg (η c)).trans (hM c)
+  have hg : Integrable (fun x ↦ η x * inner ℝ (approximateNormal E x) w) μ := by
+    apply Integrable.of_bound
+      (hη.continuous.aestronglyMeasurable.mul
+        ((measurable_approximateNormal hmin.1).inner measurable_const).aestronglyMeasurable)
+      (M*‖w‖)
+    apply ae_of_all μ
+    intro x
+    change ‖η x * inner ℝ (approximateNormal E x) w‖ ≤ _
+    rw [norm_mul]
+    have hb : ‖inner ℝ (approximateNormal E x) w‖ ≤ ‖w‖ :=
+      (norm_inner_le_norm (𝕜 := ℝ) _ _).trans (by
+        simpa only [one_mul] using mul_le_mul_of_nonneg_right
+          (approximateNormal_norm_le_one E x) (norm_nonneg w))
+    exact mul_le_mul (hM x) hb (norm_nonneg _) hM0
+  have hφg : Integrable (fun x ↦ φ x * (η x * inner ℝ (approximateNormal E x) w)) μ :=
+    hg.bdd_mul hφ.continuous.aestronglyMeasurable (ae_of_all μ fun x ↦ by
+      simpa only [Real.norm_eq_abs] using hbφ x)
+  rw [htest]
+  calc
+    _ ≤ ∫ x, |φ x * (η x * inner ℝ (approximateNormal E x) w)| ∂μ := abs_integral_le_integral_abs
+    _ ≤ ∫ x, |η x * inner ℝ (approximateNormal E x) w| ∂μ := by
+      apply integral_mono_ae hφg.abs hg.abs
+      exact ae_of_all μ fun x ↦ by
+        change |φ x * (η x * inner ℝ (approximateNormal E x) w)| ≤ |η x * inner ℝ (approximateNormal E x) w|
+        rw [abs_mul]
+        simpa only [one_mul] using mul_le_mul_of_nonneg_right (hbφ x) (abs_nonneg _)
+    _ ≤ _ := minimizing_horizontal_normal_integral_le hn hmin c hv hw a hR hη hc hs
+      hA hL hheight hgrad
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/FlatnessImprovement.lean b/lean/OAI/Geometry/Perimeter/Regularity/FlatnessImprovement.lean
new file mode 100644
index 0000000000000000000000000000000000000000..791ccb7fac3fb685e762a1455d25f51b82282a13
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/FlatnessImprovement.lean
@@ -0,0 +1,64 @@
+import OAI.Geometry.Perimeter.Regularity.LocalFlatExcess
+import OAI.Geometry.Perimeter.Regularity.SequentialFlatness
+import OAI.Geometry.Perimeter.Regularity.NormalAdjustment
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+def FullSlab (E : Set (Euc n)) (v : Euc n) (R a : ℝ) : Prop :=
+  (∀ x∈perimeterSupport E ∩ Metric.ball 0 R, |inner ℝ v x|≤a) ∧
+  (∀ᵐ x ∂volume, x∈Metric.ball 0 R → a < |inner ℝ v x| → (x∈E ↔ inner ℝ v x<0))
+
+theorem FullSlab.mono_radius {E : Set (Euc n)} {v : Euc n} {R S a : ℝ}
+    (h : FullSlab E v R a) (hSR : S≤R) : FullSlab E v S a := by
+  refine ⟨fun x hx ↦ h.1 x ⟨hx.1,Metric.ball_subset_ball hSR hx.2⟩,?_⟩
+  filter_upwards [h.2] with x hx
+  exact fun hxS ↦ hx (Metric.ball_subset_ball hSR hxS)
+
+theorem exists_fixed_normal_improvement_constants (n : ℕ) (hn : 1<n) :
+    ∃ K B : ℝ, 0<K ∧ 0≤B ∧ ∀ (v : Euc n), ‖v‖=1 →
+      IsLocallyPerimeterMinimizing {x : Euc n | inner ℝ x v<0} →
+      ∀ r : ℝ, 0<r → r<1/8 → ∃ δ : ℝ, 0<δ ∧
+        ∀ (E : Set (Euc n)), IsLocallyPerimeterMinimizing E → (0:Euc n)∈perimeterSupport E →
+          ∀ a : ℝ, 0<a → a<δ → FullSlab E v 8 a →
+            ∃ L : Euc n →L[ℝ] ℝ, L v=0 ∧ ‖L‖≤B ∧
+              ∀ x∈perimeterSupport E ∩ Metric.ball 0 r,
+                |inner ℝ v x-a*L x|≤a*K*r^2 := by
+  obtain ⟨K,B,hK,hB,hseq⟩ := exists_sequential_flatness_constants n (n-1) hn (by omega)
+  refine ⟨K,B,hK,hB,fun v hv hH r hr hr8 ↦ ?_⟩
+  by_contra! hno
+  have hex (i : ℕ) := hno (1/((i:ℝ)+1)) (by positivity)
+  choose E hmin hzero a ha haδ hslab hfail using hex
+  have hat : Tendsto a atTop (𝓝 0) := by
+    apply squeeze_zero (fun i ↦ (ha i).le) (fun i ↦ (haδ i).le)
+    exact tendsto_one_div_add_atTop_nhds_zero_nat
+  have htilt := orientedTilt_tendsto_of_local_halfspace_L1 hn hmin hv hH
+    (tendsto_phase_slab_error (fun i ↦ (hmin i).1) hv a hat 8 (fun i ↦ (hslab i).2))
+  have hsmall (i) := (hslab i).mono_radius (by norm_num : (4:ℝ)≤8)
+  obtain ⟨σ,hσ,L,hLv,hLB,hflat⟩ := hseq E hmin hzero v hv (normalPlaneEuclideanChart hv)
+    a ha hat (fun i ↦ (hsmall i).1) (fun i ↦ (hsmall i).2) htilt
+  obtain ⟨i,hi⟩ := (hflat r hr hr8).exists
+  have hh : ∀ x∈perimeterSupport (E (σ i)) ∩ Metric.ball 0 r,
+      |inner ℝ v x-a (σ i)*L x|≤a (σ i)*K*r^2 := by
+    intro x hx
+    have hb := mul_le_mul_of_nonneg_left (hi x hx) (ha (σ i)).le
+    have he : a (σ i)*|inner ℝ v x/a (σ i)-L x|=|inner ℝ v x-a (σ i)*L x| := by
+      calc
+        a (σ i)*|inner ℝ v x/a (σ i)-L x| = |a (σ i)*(inner ℝ v x/a (σ i)-L x)| := by
+          rw [abs_mul, abs_of_pos (ha (σ i))]
+        _ = _ := by congr 1; field_simp [(ha (σ i)).ne']
+    rw [he] at hb
+    simpa only [mul_assoc] using hb
+  obtain ⟨x,hx,hbad⟩ := hfail (σ i) L hLv hLB
+  exact (not_lt_of_ge (hh x hx)) hbad
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/GeoAlgebra.lean b/lean/OAI/Geometry/Perimeter/Regularity/GeoAlgebra.lean
new file mode 100644
index 0000000000000000000000000000000000000000..a7313d9af4551286e826d286a00f746a51c48c1f
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/GeoAlgebra.lean
@@ -0,0 +1,100 @@
+import OAI.Geometry.Perimeter.Regularity.GeometricHolderAlgebra
+import OAI.Geometry.Perimeter.Regularity.GeoRegularity
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+variable {V W Z : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V]
+  [NormedAddCommGroup W] [NormedSpace ℝ W] [NormedAddCommGroup Z] [NormedSpace ℝ Z]
+
+ theorem GeoBase.linear {θ : ℝ} (hθ : 0 ≤ θ) {f : Euc n → V} {U : Set (Euc n)}
+    (hU : IsOpen U) (hf : GeoBase θ f U) (L : V →L[ℝ] W) :
+    GeoBase θ (fun x ↦ L (f x)) U :=
+  ⟨L.continuous.comp_continuousOn hf.1,fun p hp ↦ (hf.2 p hp).comp_C1 hθ
+    (hf.1.continuousAt (hU.mem_nhds hp)) L.contDiff⟩
+
+ theorem GeoBase.prodMk {θ : ℝ} (hθ : 0 ≤ θ) {f : Euc n → V} {g : Euc n → W}
+    {U : Set (Euc n)} (hf : GeoBase θ f U) (hg : GeoBase θ g U) :
+    GeoBase θ (fun x ↦ (f x,g x)) U :=
+  ⟨hf.1.prodMk hg.1,fun p hp ↦ (hf.2 p hp).prodMk hθ (hg.2 p hp)⟩
+
+ theorem GeoC.linear {θ : ℝ} (hθ : 0 ≤ θ) {k : ℕ} {f : Euc n → V} {U : Set (Euc n)}
+    (hU : IsOpen U) (hf : GeoC θ k f U) (L : V →L[ℝ] W) :
+    GeoC θ k (fun x ↦ L (f x)) U := by
+  induction k generalizing f with
+  | zero => exact GeoBase.linear hθ hU hf L
+  | succ k ih =>
+    refine ⟨L.contDiff.comp_contDiffOn hf.1,(GeoBase.linear hθ hU hf.2.1 L),fun e ↦ ?_⟩
+    apply (ih (hf.2.2 e)).congr hθ hU
+    intro x hx
+    have hh := L.hasFDerivAt.comp x (((hf.1.contDiffAt (hU.mem_nhds hx)).differentiableAt
+      (by simp)).hasFDerivAt)
+    exact (congrArg (fun A : Euc n →L[ℝ] W ↦ A e) hh.fderiv).symm
+
+ theorem GeoC.prodMk {θ : ℝ} (hθ : 0 ≤ θ) {k : ℕ}
+    {f : Euc n → V} {g : Euc n → W} {U : Set (Euc n)}
+    (hU : IsOpen U) (hf : GeoC θ k f U) (hg : GeoC θ k g U) :
+    GeoC θ k (fun x ↦ (f x,g x)) U := by
+  induction k generalizing f g with
+  | zero => exact GeoBase.prodMk hθ hf hg
+  | succ k ih =>
+    refine ⟨hf.1.prodMk hg.1,GeoBase.prodMk hθ hf.2.1 hg.2.1,fun e ↦ ?_⟩
+    apply (ih (hf.2.2 e) (hg.2.2 e)).congr hθ hU
+    intro x hx
+    have hdf := ((hf.1.contDiffAt (hU.mem_nhds hx)).differentiableAt (by simp)).hasFDerivAt
+    have hdg := ((hg.1.contDiffAt (hU.mem_nhds hx)).differentiableAt (by simp)).hasFDerivAt
+    exact (congrArg (fun A : Euc n →L[ℝ] (V×W) ↦ A e) (hdf.prodMk hdg).fderiv).symm
+
+ theorem GeoC.add {θ : ℝ} (hθ : 0 ≤ θ) {k : ℕ}
+    {f g : Euc n → V} {U : Set (Euc n)} (hU : IsOpen U)
+    (hf : GeoC θ k f U) (hg : GeoC θ k g U) : GeoC θ k (fun x ↦ f x+g x) U :=
+  (hf.prodMk hθ hU hg).linear hθ hU (ContinuousLinearMap.fst ℝ V V+ContinuousLinearMap.snd ℝ V V)
+
+ theorem GeoC.neg {θ : ℝ} (hθ : 0 ≤ θ) {k : ℕ}
+    {f : Euc n → V} {U : Set (Euc n)} (hU : IsOpen U)
+    (hf : GeoC θ k f U) : GeoC θ k (fun x ↦ -f x) U :=
+  hf.linear hθ hU (-ContinuousLinearMap.id ℝ V)
+
+ theorem GeoC.sub {θ : ℝ} (hθ : 0 ≤ θ) {k : ℕ}
+    {f g : Euc n → V} {U : Set (Euc n)} (hU : IsOpen U)
+    (hf : GeoC θ k f U) (hg : GeoC θ k g U) : GeoC θ k (fun x ↦ f x-g x) U := by
+  simpa only [sub_eq_add_neg] using hf.add hθ hU (hg.neg hθ hU)
+
+ theorem GeoC.bilinear {θ : ℝ} (hθ : 0 ≤ θ) {k : ℕ}
+    {f : Euc n → V} {g : Euc n → W} {U : Set (Euc n)} (hU : IsOpen U)
+    (hf : GeoC θ k f U) (hg : GeoC θ k g U) (B : V →L[ℝ] W →L[ℝ] Z) :
+    GeoC θ k (fun x ↦ B (f x) (g x)) U := by
+  have hb : ContDiff ℝ 1 (fun z : V×W ↦ B z.1 z.2) :=
+    (B.contDiff.comp contDiff_fst).clm_apply contDiff_snd
+  have hbase {f : Euc n → V} {g : Euc n → W} (hf : GeoBase θ f U) (hg : GeoBase θ g U) :
+      GeoBase θ (fun x ↦ B (f x) (g x)) U :=
+    ⟨(B.continuous.comp_continuousOn hf.1).clm_apply hg.1,fun p hp ↦
+      ((hf.2 p hp).prodMk hθ (hg.2 p hp)).comp_C1 hθ
+        ((hf.1.continuousAt (hU.mem_nhds hp)).prodMk (hg.1.continuousAt (hU.mem_nhds hp))) hb⟩
+  induction k generalizing f g with
+  | zero => exact hbase hf hg
+  | succ k ih =>
+    refine ⟨(B.contDiff.comp_contDiffOn hf.1).clm_apply hg.1,hbase hf.2.1 hg.2.1,fun e ↦ ?_⟩
+    have h₁ := ih hf.mono_order (hg.2.2 e)
+    have h₂ := ih (hf.2.2 e) hg.mono_order
+    apply (h₁.add hθ hU h₂).congr hθ hU
+    intro x hx
+    have hdf := ((hf.1.contDiffAt (hU.mem_nhds hx)).differentiableAt (by simp)).hasFDerivAt
+    have hdg := ((hg.1.contDiffAt (hU.mem_nhds hx)).differentiableAt (by simp)).hasFDerivAt
+    have hh := ((B.hasFDerivAt.comp x hdf).clm_apply hdg).fderiv
+    exact (congrArg (fun A : Euc n →L[ℝ] Z ↦ A e) hh).symm
+
+ theorem GeoC.clm_apply {θ : ℝ} (hθ : 0 ≤ θ) {k : ℕ}
+    {f : Euc n → V →L[ℝ] W} {g : Euc n → V} {U : Set (Euc n)} (hU : IsOpen U)
+    (hf : GeoC θ k f U) (hg : GeoC θ k g U) : GeoC θ k (fun x ↦ f x (g x)) U :=
+  hf.bilinear hθ hU hg (ContinuousLinearMap.apply ℝ W).flip
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/GeoComposition.lean b/lean/OAI/Geometry/Perimeter/Regularity/GeoComposition.lean
new file mode 100644
index 0000000000000000000000000000000000000000..d1e2ee3936aa3cda3c2206c20b8257e4d2767da6
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/GeoComposition.lean
@@ -0,0 +1,79 @@
+import OAI.Geometry.Perimeter.Regularity.GeoAlgebra
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+ theorem GeoC.const {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V]
+    (θ : ℝ) (k : ℕ) (c : V) (U : Set (Euc n)) : GeoC θ k (fun _ ↦ c) U := by
+  induction k generalizing c with
+  | zero => exact ⟨continuousOn_const,fun p _ ↦ GeometricHolderAt.const θ p c⟩
+  | succ k ih =>
+    refine ⟨contDiffOn_const,⟨continuousOn_const,fun p _ ↦ GeometricHolderAt.const θ p c⟩,fun e ↦ ?_⟩
+    simpa only [fderiv_const_apply,zero_apply] using ih (0:V)
+
+ theorem GeoC.clm_of_apply {V W : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V]
+    [FiniteDimensional ℝ V] [NormedAddCommGroup W] [NormedSpace ℝ W]
+    {θ : ℝ} (hθ : 0 ≤ θ) {k : ℕ} {U : Set (Euc n)} (hU : IsOpen U)
+    {f : Euc n → V →L[ℝ] W} (hf : ∀ e, GeoC θ k (fun x ↦ f x e) U) :
+    GeoC θ k f U := by
+  have hbase {f : Euc n → V →L[ℝ] W} (hf : ∀ e, GeoC θ 0 (fun x ↦ f x e) U) :
+      GeoC θ 0 f U := by
+    have hc : ContinuousOn f U := continuousOn_clm_apply.mpr (fun e ↦ (hf e).1)
+    exact ⟨hc,fun p hp ↦ (GeometricHolderAt.clm_apply_iff hθ
+      (hc.continuousAt (hU.mem_nhds hp))).mpr (fun e ↦ (hf e).2 p hp)⟩
+  induction k generalizing f with
+  | zero => exact hbase hf
+  | succ k ih =>
+    have hc : ContDiffOn ℝ 1 f U := contDiffOn_clm_apply.mpr (fun e ↦ (hf e).1)
+    refine ⟨hc,hbase (fun e ↦ (hf e).2.1),fun e ↦ ih (fun v ↦ ?_)⟩
+    apply ((hf v).2.2 e).congr hθ hU
+    intro x hx
+    have hd := ((hc.contDiffAt (hU.mem_nhds hx)).differentiableAt (by simp)).hasFDerivAt
+    exact congrArg (fun L : Euc n →L[ℝ] W ↦ L e)
+      (((ContinuousLinearMap.apply ℝ W v).hasFDerivAt.comp x hd).fderiv)
+
+ theorem GeoC.fderiv {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V]
+    {θ : ℝ} (hθ : 0 ≤ θ) {k : ℕ} {U : Set (Euc n)} (hU : IsOpen U)
+    {f : Euc n → V} (hf : GeoC θ (k+1) f U) : GeoC θ k (fderiv ℝ f) U :=
+  GeoC.clm_of_apply hθ hU hf.2.2
+
+ theorem GeoC.gradient {θ : ℝ} (hθ : 0 ≤ θ) {k : ℕ} {U : Set (Euc n)} (hU : IsOpen U)
+    {f : Euc n → ℝ} (hf : GeoC θ (k+1) f U) : GeoC θ k (gradient f) U :=
+  (hf.fderiv hθ hU).linear hθ hU (InnerProductSpace.toDual ℝ (Euc n)).symm.toContinuousLinearMap
+
+ theorem geoC_comp (k : ℕ) : ∀ {V W : Type} [NormedAddCommGroup V] [NormedSpace ℝ V]
+    [NormedAddCommGroup W] [NormedSpace ℝ W] {θ : ℝ}, 0 ≤ θ →
+    ∀ {U : Set (Euc n)}, IsOpen U → ∀ {f : Euc n → V}, GeoC θ k f U →
+    ∀ {G : V → W}, ContDiff ℝ ∞ G → GeoC θ k (fun x ↦ G (f x)) U := by
+  induction k with
+  | zero =>
+    intro V W _ _ _ _ θ hθ U hU f hf G hG
+    exact ⟨hG.continuous.comp_continuousOn hf.1,fun p hp ↦ (hf.2 p hp).comp_C1 hθ
+      (hf.1.continuousAt (hU.mem_nhds hp)) (hG.of_le (by simp))⟩
+  | succ k ih =>
+    intro V W _ _ _ _ θ hθ U hU f hf G hG
+    have hcf : ContDiffOn ℝ 1 f U := hf.1
+    have hcG : ContDiff ℝ 1 G := hG.of_le (by simp)
+    have hDG : ContDiff ℝ ∞ (fderiv ℝ G) := hG.fderiv_right (by simp)
+    have hb : GeoC θ 0 (fun x ↦ G (f x)) U :=
+      ⟨hG.continuous.comp_continuousOn hf.2.1.1,fun p hp ↦ (hf.2.1.2 p hp).comp_C1 hθ
+        (hcf.continuousOn.continuousAt (hU.mem_nhds hp)) hcG⟩
+    refine ⟨hcG.comp_contDiffOn hcf,hb,fun e ↦ ?_⟩
+    have h₁ := ih hθ hU hf.mono_order hDG
+    have hh := h₁.clm_apply hθ hU (hf.2.2 e)
+    apply hh.congr hθ hU
+    intro x hx
+    have hd := ((hcf.contDiffAt (hU.mem_nhds hx)).differentiableAt (by simp)).hasFDerivAt
+    exact (congrArg (fun L : Euc n →L[ℝ] W ↦ L e)
+      (((hcG.differentiable (by simp) (f x)).hasFDerivAt.comp x hd).fderiv)).symm
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/GeoRegularity.lean b/lean/OAI/Geometry/Perimeter/Regularity/GeoRegularity.lean
new file mode 100644
index 0000000000000000000000000000000000000000..4aa01e7cd3b679662a091658ba2e826e36bfc812
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/GeoRegularity.lean
@@ -0,0 +1,117 @@
+import OAI.Geometry.Perimeter.Regularity.EllipticLocalExtension
+import OAI.Geometry.Perimeter.Regularity.GeometricHolder
+import OAI.Geometry.Perimeter.Regularity.EllipticSecantBounds
+import OAI.Geometry.Perimeter.Regularity.LinearDifference
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+ abbrev GeoBase {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V]
+    (θ : ℝ) (f : Euc n → V) (U : Set (Euc n)) : Prop :=
+  ContinuousOn f U ∧ ∀ p∈U, GeometricHolderAt θ f p
+
+ def GeoC {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V]
+    (θ : ℝ) : ℕ → (Euc n → V) → Set (Euc n) → Prop
+  | 0,f,U => GeoBase θ f U
+  | k+1,f,U => ContDiffOn ℝ 1 f U ∧ GeoBase θ f U ∧
+      ∀ e, GeoC θ k (fun x ↦ fderiv ℝ f x e) U
+
+ theorem GeoC.contDiff {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V]
+    {θ : ℝ} {k : ℕ} {f : Euc n → V} {U : Set (Euc n)} (hU : IsOpen U)
+    (h : GeoC θ k f U) : ContDiffOn ℝ k f U := by
+  induction k generalizing f with
+  | zero => exact contDiffOn_zero.mpr h.1
+  | succ k ih =>
+    rw [Nat.cast_add,Nat.cast_one,contDiffOn_succ_iff_fderiv_of_isOpen hU]
+    refine ⟨h.1.differentiableOn one_ne_zero,by simp,contDiffOn_clm_apply.mpr ?_⟩
+    intro e
+    exact ih (h.2.2 e)
+
+ theorem GeoC.base {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V]
+    {θ : ℝ} {k : ℕ} {f : Euc n → V} {U : Set (Euc n)}
+    (h : GeoC θ k f U) : GeoC θ 0 f U := by cases k with | zero => exact h | succ k => exact h.2.1
+
+ theorem GeoC.mono_order {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V]
+    {θ : ℝ} {k : ℕ} {f : Euc n → V} {U : Set (Euc n)}
+    (h : GeoC θ (k+1) f U) : GeoC θ k f U := by
+  induction k generalizing f with
+  | zero => exact h.2.1
+  | succ k ih => exact ⟨h.1,h.2.1,fun e ↦ ih (h.2.2 e)⟩
+
+ theorem GeoC.mono {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V]
+    {θ : ℝ} {k : ℕ} {f : Euc n → V} {U W : Set (Euc n)}
+    (h : GeoC θ k f U) (hWU : W⊆U) : GeoC θ k f W := by
+  induction k generalizing f with
+  | zero => exact ⟨h.1.mono hWU,fun p hp ↦ h.2 p (hWU hp)⟩
+  | succ k ih => exact ⟨h.1.mono hWU,⟨h.2.1.1.mono hWU,fun p hp ↦ h.2.1.2 p (hWU hp)⟩,
+      fun e ↦ ih (h.2.2 e)⟩
+
+ theorem GeoC.congr {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V]
+    {θ : ℝ} (hθ : 0 ≤ θ) {k : ℕ} {f g : Euc n → V} {U : Set (Euc n)}
+    (hU : IsOpen U) (h : GeoC θ k f U) (he : EqOn f g U) : GeoC θ k g U := by
+  induction k generalizing f g with
+  | zero =>
+    refine ⟨h.1.congr he.symm,fun p hp ↦ (h.2 p hp).congr hθ ?_⟩
+    exact Filter.eventually_of_mem (hU.mem_nhds hp) (fun x hx ↦ he hx)
+  | succ k ih =>
+    refine ⟨h.1.congr he.symm,?_,fun e ↦ ih (h.2.2 e) ?_⟩
+    · exact ⟨h.2.1.1.congr he.symm,fun p hp ↦ (h.2.1.2 p hp).congr hθ
+        (Filter.eventually_of_mem (hU.mem_nhds hp) (fun x hx ↦ he hx))⟩
+    · intro x hx
+      exact congrArg (fun L : Euc n →L[ℝ] V ↦ L e)
+        ((Filter.EventuallyEq.fderiv_eq (Filter.eventually_of_mem (hU.mem_nhds hx) (fun y hy ↦ he hy))))
+
+ theorem GeoC.local {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V]
+    {θ : ℝ} {k : ℕ} {f : Euc n → V} {U : Set (Euc n)}
+    (h : ∀ p∈U, ∃ W : Set (Euc n), IsOpen W ∧ p∈W ∧ GeoC θ k f W) :
+    GeoC θ k f U := by
+  induction k generalizing f with
+  | zero =>
+    constructor
+    · intro p hp
+      obtain ⟨W,hW,hpW,hh⟩ := h p hp
+      exact (hh.1.continuousAt (hW.mem_nhds hpW)).continuousWithinAt
+    · intro p hp
+      obtain ⟨W,hW,hpW,hh⟩ := h p hp
+      exact hh.2 p hpW
+  | succ k ih =>
+    refine ⟨?_,?_,fun e ↦ ih (fun p hp ↦ ?_)⟩
+    · intro p hp
+      obtain ⟨W,hW,hpW,hh⟩ := h p hp
+      exact (hh.1.contDiffAt (hW.mem_nhds hpW)).contDiffWithinAt
+    · constructor
+      · intro p hp
+        obtain ⟨W,hW,hpW,hh⟩ := h p hp
+        exact (hh.1.continuousOn.continuousAt (hW.mem_nhds hpW)).continuousWithinAt
+      · intro p hp
+        obtain ⟨W,hW,hpW,hh⟩ := h p hp
+        exact hh.2.1.2 p hpW
+    · obtain ⟨W,hW,hpW,hh⟩ := h p hp
+      exact ⟨W,hW,hpW,hh.2.2 e⟩
+
+ theorem geometricHolderAt_of_C1_on {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V]
+    {θ : ℝ} (hθ : 0 ≤ θ) (hθ' : θ ≤ 1/2) {f : Euc n → V}
+    {U : Set (Euc n)} (hU : IsOpen U) (hf : ContDiffOn ℝ 1 f U) {p : Euc n} (hp : p∈U) :
+    GeometricHolderAt θ f p := by
+  obtain ⟨r,hr,hrU,g,hg,hgc,he⟩ := local_compact_extension hU hp hf
+  obtain ⟨s,L,hs,hL,hLip⟩ := contDiff_local_lipschitz hg p
+  have hh : GeometricHolderAt θ g p := geometricHolderAt_of_local_lipschitz hθ hθ' hs hL
+    (fun x hx y hy ↦ by simpa only [dist_eq_norm] using hLip x hx y hy)
+  exact hh.congr hθ (Filter.eventually_of_mem (Metric.ball_mem_nhds p (by positivity : 0<2*r))
+    (fun x hx ↦ he x hx))
+
+ theorem geoC_zero_of_C1 {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V]
+    {θ : ℝ} (hθ : 0 ≤ θ) (hθ' : θ ≤ 1/2) {f : Euc n → V}
+    {U : Set (Euc n)} (hU : IsOpen U) (hf : ContDiffOn ℝ 1 f U) : GeoC θ 0 f U :=
+  ⟨hf.continuousOn,fun _ hp ↦ geometricHolderAt_of_C1_on hθ hθ' hU hf hp⟩
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/GeometricDerivativeModulus.lean b/lean/OAI/Geometry/Perimeter/Regularity/GeometricDerivativeModulus.lean
new file mode 100644
index 0000000000000000000000000000000000000000..055832ccd542673933bb40cacba75c08d28dc496
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/GeometricDerivativeModulus.lean
@@ -0,0 +1,73 @@
+import OAI.Geometry.Perimeter.Regularity.UniformDerivativeContinuity
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+
+ theorem geometricHolder_derivative_of_remainder {V : Type*}
+    [NormedAddCommGroup V] [NormedSpace ℝ V]
+    {f : V → ℝ} {D : V → V →L[ℝ] ℝ} {U : Set V} (hU : IsOpen U)
+    {q C : ℝ} (hq : 0<q) (hq1 : q ≤ 1) (hC : 0 ≤ C)
+    (hlip : ∀ x∈U, ∀ y∈U, |f x-f y|≤‖x-y‖)
+    (hrem : ∀ j x, x∈U → ∀ z, z∈U → ‖z-x‖≤q^j →
+      |f z-f x-D x (z-x)|≤C*(1/2)^j*‖z-x‖)
+    {p : V} (hp : p∈U) : GeometricHolderAt (q/2) D p := by
+  obtain ⟨s,hs,hsU⟩ := Metric.isOpen_iff.mp hU p hp
+  let r := min (s/4) 1
+  have hr : 0<r := lt_min (by positivity) (by norm_num)
+  have hrs : r ≤ s/4 := min_le_left _ _
+  have hr1 : r ≤ 1 := min_le_right _ _
+  refine ⟨r,2+2*C,hr,by positivity,fun j x hx y hy hxy ↦ ?_⟩
+  let t := q^j*r
+  have ht : 0<t := by dsimp [t]; positivity
+  have htq : t ≤ q^j := by dsimp [t]; nlinarith [pow_pos hq j]
+  have htr : t ≤ r := by dsimp [t]; nlinarith [pow_le_one₀ (n := j) hq.le hq1]
+  have hxU : x∈U := hsU (Metric.ball_subset_ball (by linarith) hx)
+  have hyU : y∈U := hsU (Metric.ball_subset_ball (by linarith) hy)
+  have hshift (z : V) (hz : z∈Metric.ball p r) (u : V) (hu : ‖u‖=1) : z+t•u∈U := by
+    apply hsU
+    rw [Metric.mem_ball,dist_eq_norm]
+    have he : z+t•u-p=(z-p)+t•u := by abel
+    rw [he]
+    have hh := norm_add_le (z-p) (t•u)
+    rw [norm_smul,Real.norm_of_nonneg ht.le,hu,mul_one] at hh
+    have hz' := Metric.mem_ball.mp hz
+    rw [dist_eq_norm] at hz'
+    linarith
+  rw [dist_eq_norm]
+  apply ContinuousLinearMap.opNorm_le_of_unit_norm (by positivity)
+  intro u hu
+  have hxu := hshift x hx u hu
+  have hyu := hshift y hy u hu
+  have hnx : ‖(x+t•u)-x‖=t := by simp [norm_smul,Real.norm_of_nonneg ht.le,hu]
+  have hny : ‖(y+t•u)-y‖=t := by simp [norm_smul,Real.norm_of_nonneg ht.le,hu]
+  have hxR := hrem j x hxU (x+t•u) hxu (by rw [hnx]; exact htq)
+  have hyR := hrem j y hyU (y+t•u) hyu (by rw [hny]; exact htq)
+  simp only [add_sub_cancel_left,map_smul,smul_eq_mul] at hxR hyR
+  rw [norm_smul,Real.norm_of_nonneg ht.le,hu,mul_one] at hxR hyR
+  have hsh : ‖(x+t•u)-(y+t•u)‖=‖x-y‖ := by congr 1; abel
+  have hb := four_point_linear_bound ht hxR hyR (hlip x hxU y hyU)
+    (by simpa only [hsh] using hlip (x+t•u) hxu (y+t•u) hyu)
+  change ‖D x u-D y u‖ ≤ _
+  rw [Real.norm_eq_abs]
+  apply hb.trans
+  have hd : ‖x-y‖ ≤ (1/2)^j*t := by
+    rw [←dist_eq_norm]
+    have he : (q/2)^j*r=(1/2:ℝ)^j*t := by
+      dsimp [t]
+      rw [show q/2=(1/2:ℝ)*q by ring,mul_pow]
+      ring
+    rwa [←he]
+  have hd' : 2*‖x-y‖/t ≤ 2*(1/2)^j := by
+    apply (div_le_iff₀ ht).mpr
+    nlinarith
+  nlinarith
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/GeometricHausdorff.lean b/lean/OAI/Geometry/Perimeter/Regularity/GeometricHausdorff.lean
new file mode 100644
index 0000000000000000000000000000000000000000..3f0b92c0c8fccb8d85d02dee92de801eb9612752
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/GeometricHausdorff.lean
@@ -0,0 +1,469 @@
+import OAI.Geometry.Perimeter.Regularity.ContinuousWithinAtSurfaceGradientOfLocalLevel
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+namespace RPISingularHarnack
+
+def geometricHausdorff {n : ℕ} (k : ℕ) : Measure (Euc n) :=
+  ((volume (Metric.ball (0 : Euc k) 1)) / (2 : ℝ≥0∞) ^ k) •
+    Measure.hausdorffMeasure (k : ℝ)
+
+def hausdorffNormalizationRatio (k : ℕ) : ℝ≥0∞ :=
+  ((volume (Metric.ball (0 : Euc k) 1)) / (2 : ℝ≥0∞) ^ k) /
+    (Measure.addHaarScalarFactor (volume : Measure (Euc k))
+      (Measure.hausdorffMeasure (k : ℝ)) : ℝ≥0∞)
+
+theorem hausdorffNormalizationRatio_ne_zero (k : ℕ) :
+    hausdorffNormalizationRatio k ≠ 0 := by
+  apply ENNReal.div_ne_zero.mpr
+  constructor
+  · apply ENNReal.div_ne_zero.mpr
+    constructor
+    · exact (Metric.measure_ball_pos volume 0 (by norm_num : (0 : ℝ) < 1)).ne'
+    · exact ENNReal.pow_ne_top (by norm_num)
+  · exact ENNReal.coe_ne_top
+
+theorem hausdorffNormalizationRatio_ne_top (k : ℕ) :
+    hausdorffNormalizationRatio k ≠ ⊤ := by
+  apply ENNReal.div_ne_top
+  · apply ENNReal.div_ne_top
+    · exact measure_ball_ne_top
+    · exact pow_ne_zero _ (by norm_num)
+  · exact_mod_cast Measure.addHaarScalarFactor_volume_hausdorffMeasure_ne_zero k
+
+theorem geometricHausdorff_eq_smul_euclideanHausdorff {n : ℕ} (k : ℕ) :
+    (geometricHausdorff k : Measure (Euc n)) =
+      hausdorffNormalizationRatio k • Measure.euclideanHausdorffMeasure k := by
+  rw [geometricHausdorff, Measure.euclideanHausdorffMeasure_def,
+    ENNReal.smul_def, smul_smul, hausdorffNormalizationRatio]
+  rw [ENNReal.div_mul_cancel (by exact_mod_cast
+    Measure.addHaarScalarFactor_volume_hausdorffMeasure_ne_zero k) ENNReal.coe_ne_top]
+
+def boundaryArea {n : ℕ} (E : Set (Euc n)) : Measure (Euc n) :=
+  (geometricHausdorff (n - 1)).restrict (perimeterSupport E)
+
+def WholeBoundarySobolev {n : ℕ} (E : Set (Euc n))
+    (h : Euc n → ℝ) (G : Euc n → Euc n) : Prop :=
+  ∀ R : ℝ, 0 < R →
+    let μ := (boundaryArea E).restrict (Metric.ball 0 R)
+    MemLp h 2 μ ∧ MemLp G 2 μ ∧
+    ∃ φ : ℕ → Euc n → ℝ,
+      (∀ j, ContDiff ℝ ∞ (φ j) ∧ HasCompactSupport (φ j)) ∧
+      Tendsto (fun j => ∫⁻ x, ENNReal.ofReal (|φ j x - h x| ^ 2) ∂μ)
+        atTop (𝓝 0) ∧
+      Tendsto (fun j => ∫⁻ x,
+        ENNReal.ofReal (‖tangentialGradient E (φ j) x - G x‖ ^ 2) ∂μ)
+        atTop (𝓝 0)
+
+def WholeBoundarySupersolution {n : ℕ} (E : Set (Euc n))
+    (G : Euc n → Euc n) : Prop :=
+  ∀ φ : Euc n → ℝ, ContDiff ℝ ∞ φ → HasCompactSupport φ →
+    (∀ x, 0 ≤ φ x) →
+    0 ≤ ∫ x, inner ℝ (G x) (tangentialGradient E φ x) ∂boundaryArea E
+
+def ZeroEssentialInfAtVertex {n : ℕ} (E : Set (Euc n)) (h : Euc n → ℝ) : Prop :=
+  ∀ R : ℝ, 0 < R → ∀ ε : ℝ, 0 < ε →
+    0 < boundaryArea E (Metric.ball 0 R ∩ {x | h x < ε})
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+namespace RPI.Boundary
+
+theorem surfaceGradient_ae_eq_tangentialGradient {n : ℕ} (hn : 8 ≤ n)
+    {E S Z : Set (Space n)} (hE : MeasurableSet E) (hS : MeasurableSet S)
+    (hZd : dimH Z ≤ (n - 8 : ℕ))
+    (hreg : ∀ x ∈ S \ Z, ∃ f : Space n → ℝ,
+      ContDiffAt ℝ 1 f x ∧ f x = 0 ∧ fderiv ℝ f x ≠ 0 ∧
+      (∀ᶠ y in 𝓝 x, y ∈ S ↔ f y = 0) ∧
+      (∀ᶠ y in 𝓝 x, y ∈ E ↔ f y < 0)) (g : Space n → ℝ) :
+    surfaceGradient S g =ᵐ[surfaceMeasure S]
+      RPISingularHarnack.tangentialGradient E g := by
+  have hz : ∀ᵐ x ∂surfaceMeasure S, x ∉ Z := by
+    simpa only [ae_iff, not_not, Set.ofPred_mem_eq] using
+      surfaceMeasure_zero_of_singular_dim hn S Z hZd
+  filter_upwards [ae_restrict_mem hS, hz] with x hx hzx
+  obtain ⟨f, hf, hfx, hdf, hlevel, hfilled⟩ := hreg x ⟨hx, hzx⟩
+  exact surfaceGradient_eq_tangentialGradient hE hf hfx hdf hlevel hfilled
+
+end RPI.Boundary
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+namespace RPISingularHarnack
+
+theorem boundaryArea_eq_smul_surfaceMeasure {n : ℕ} (E : Set (Euc n)) :
+    boundaryArea E = hausdorffNormalizationRatio (n - 1) •
+      RPI.Boundary.surfaceMeasure (perimeterSupport E) := by
+  rw [boundaryArea, geometricHausdorff_eq_smul_euclideanHausdorff,
+    Measure.restrict_smul]
+  rfl
+
+theorem wholeBoundarySobolev_of_local_sobolevGraph {n : ℕ} (hn : 8 ≤ n)
+    {E Z : Set (Euc n)} (hE : MeasurableSet E)
+    (hS : MeasurableSet (perimeterSupport E)) (hZd : dimH Z ≤ (n - 8 : ℕ))
+    (hreg : ∀ x ∈ perimeterSupport E \ Z, ∃ f : Euc n → ℝ,
+      ContDiffAt ℝ 1 f x ∧ f x = 0 ∧ fderiv ℝ f x ≠ 0 ∧
+      (∀ᶠ y in 𝓝 x, y ∈ perimeterSupport E ↔ f y = 0) ∧
+      (∀ᶠ y in 𝓝 x, y ∈ E ↔ f y < 0))
+    {h : Euc n → ℝ} {G : Euc n → Euc n}
+    (hgraph : ∀ R : ℝ, 0 < R →
+      ∃ (hh : MemLp h 2 ((RPI.Boundary.surfaceMeasure (perimeterSupport E)).restrict
+          (Metric.ball 0 R)))
+        (hG : MemLp G 2 ((RPI.Boundary.surfaceMeasure (perimeterSupport E)).restrict
+          (Metric.ball 0 R))),
+        (hh.toLp h, hG.toLp G) ∈ closure
+          (RPI.Boundary.smoothGradientGraph (perimeterSupport E) (Metric.ball 0 R))) :
+    WholeBoundarySobolev E h G := by
+  intro R hR
+  obtain ⟨hh, hG, hpair⟩ := hgraph R hR
+  obtain ⟨φ, hφ, hlim, hDlim⟩ :=
+    RPI.Boundary.exists_compact_energy_approximants_of_mem_sobolevGraph
+      (perimeterSupport E) (Metric.ball 0 R) Metric.isBounded_ball
+      measurableSet_ball hh hG hpair
+  dsimp only
+  rw [boundaryArea_eq_smul_surfaceMeasure, Measure.restrict_smul]
+  refine ⟨hh.smul_measure (hausdorffNormalizationRatio_ne_top _),
+    hG.smul_measure (hausdorffNormalizationRatio_ne_top _), φ, hφ, ?_, ?_⟩
+  · simp_rw [lintegral_smul_measure, smul_eq_mul]
+    simpa only [mul_zero] using ENNReal.Tendsto.const_mul hlim
+      (Or.inr (hausdorffNormalizationRatio_ne_top (n - 1)))
+  · have heq : ∀ j, RPI.Boundary.surfaceGradient (perimeterSupport E) (φ j)
+        =ᵐ[(RPI.Boundary.surfaceMeasure (perimeterSupport E)).restrict (Metric.ball 0 R)]
+          tangentialGradient E (φ j) := fun j ↦
+      (RPI.Boundary.surfaceGradient_ae_eq_tangentialGradient hn hE hS hZd hreg (φ j)).filter_mono
+        (ae_mono Measure.restrict_le_self)
+    have hl : Tendsto (fun j ↦ ∫⁻ x,
+        ENNReal.ofReal (‖tangentialGradient E (φ j) x - G x‖ ^ 2)
+        ∂(RPI.Boundary.surfaceMeasure (perimeterSupport E)).restrict (Metric.ball 0 R))
+        atTop (𝓝 0) := by
+      apply hDlim.congr
+      intro j
+      apply lintegral_congr_ae
+      filter_upwards [heq j] with x hx
+      rw [hx]
+    simp_rw [lintegral_smul_measure, smul_eq_mul]
+    simpa only [mul_zero] using ENNReal.Tendsto.const_mul hl
+      (Or.inr (hausdorffNormalizationRatio_ne_top (n - 1)))
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+namespace RPISingularHarnack
+
+theorem eventually_filled_membership {n : ℕ} {E : Set (Euc n)}
+    {f : Euc n → ℝ} {x z : Euc n} {L : Euc n →L[ℝ] ℝ}
+    (hd : HasFDerivAt f L x) (hx : f x = 0)
+    (hlocal : ∀ᶠ y in 𝓝 x, (f y < 0 → y ∈ E) ∧ (y ∈ E → f y ≤ 0)) (hz : L z ≠ 0) :
+    ∀ᶠ r : ℝ in 𝓝[>] 0, x + r • z ∈ E ↔ L z < 0 := by
+  have hp : HasDerivAt (fun r : ℝ ↦ x + r • z) z 0 := by
+    simpa using ((hasDerivAt_id (0 : ℝ)).smul_const z).const_add x
+  have hc : HasDerivAt (fun r : ℝ ↦ f (x + r • z)) (L z) 0 := by
+    exact (show HasFDerivAt f L (x + (0 : ℝ) • z) from by simpa using hd).comp_hasDerivAt 0 hp
+  have ht : Tendsto (fun r : ℝ ↦ r⁻¹ * f (x + r • z)) (𝓝[>] 0) (𝓝 (L z)) := by
+    simpa [hx] using hc.tendsto_slope_zero_right
+  have hm : ∀ᶠ r : ℝ in 𝓝[>] 0,
+      (f (x + r • z) < 0 → x + r • z ∈ E) ∧
+      (x + r • z ∈ E → f (x + r • z) ≤ 0) :=
+    (show Tendsto (fun r : ℝ ↦ x + r • z) (𝓝[>] 0) (𝓝 x) from
+      (show Tendsto (fun r : ℝ ↦ x + r • z) (𝓝 0) (𝓝 x) from by
+        simpa [ContinuousAt] using hp.continuousAt).mono_left nhdsWithin_le_nhds).eventually hlocal
+  rcases lt_or_gt_of_ne hz with hneg | hpos
+  · filter_upwards [hm, ht.eventually (gt_mem_nhds hneg), self_mem_nhdsWithin] with r hm hr hpr
+    have hfr : f (x + r • z) < 0 := by
+      have hi : 0 < r⁻¹ := inv_pos.mpr hpr
+      nlinarith
+    exact iff_of_true (hm.1 hfr) hneg
+  · filter_upwards [hm, ht.eventually (lt_mem_nhds hpos), self_mem_nhdsWithin] with r hm hr hpr
+    have hfr : 0 < f (x + r • z) := by
+      have hi : 0 < r⁻¹ := inv_pos.mpr hpr
+      nlinarith
+    exact iff_of_false (fun he ↦ (not_le.mpr hfr) (hm.2 he)) (not_lt.mpr hpos.le)
+
+theorem hasHalfspaceBlowup_of_regular_filled_chart {n : ℕ} {E : Set (Euc n)}
+    (hE : MeasurableSet E) {f : Euc n → ℝ} {x v : Euc n} {L : Euc n →L[ℝ] ℝ}
+    (hd : HasFDerivAt f L x) (hx : f x = 0)
+    (hlocal : ∀ᶠ y in 𝓝 x, (f y < 0 → y ∈ E) ∧ (y ∈ E → f y ≤ 0))
+    (hv : ‖v‖ = 1) {a : ℝ} (ha : 0 < a)
+    (hL : ∀ z, L z = a * inner ℝ z v) : HasHalfspaceBlowup E x v := by
+  have hLn : L ≠ 0 := by
+    intro he
+    have hval := hL v
+    simp only [he, zero_apply, real_inner_self_eq_norm_sq, hv,
+      one_pow, mul_one] at hval
+    linarith
+  let H : Set (Euc n) := {z | inner ℝ z v < 0}
+  have hH : MeasurableSet H :=
+    (isOpen_lt (continuous_id.inner continuous_const) continuous_const).measurableSet
+  let F : ℝ → Euc n → ℝ := fun r z ↦
+    |E.indicator (fun _ ↦ (1 : ℝ)) (x + r • z) - H.indicator (fun _ ↦ (1 : ℝ)) z|
+  have hsign : ∀ z, L z < 0 ↔ inner ℝ z v < 0 := by
+    intro z
+    rw [hL]
+    constructor <;> intro h <;> nlinarith
+  have hmeas : ∀ r, AEStronglyMeasurable (F r)
+      (volume.restrict (Metric.ball (0 : Euc n) 1)) := by
+    intro r
+    have he : Measurable (E.indicator (fun _ : Euc n ↦ (1 : ℝ))) :=
+      measurable_const.indicator hE
+    have hh : Measurable (H.indicator (fun _ : Euc n ↦ (1 : ℝ))) :=
+      measurable_const.indicator hH
+    have hm : Measurable (fun z : Euc n ↦ x + r • z) :=
+      measurable_const.add (measurable_id.const_smul r)
+    exact ((he.comp hm).sub hh).abs.aestronglyMeasurable
+  have hb : ∀ r, ∀ᵐ z ∂volume.restrict (Metric.ball (0 : Euc n) 1), ‖F r z‖ ≤ 1 := by
+    intro r
+    filter_upwards [] with z
+    by_cases he : x + r • z ∈ E <;> by_cases hh : z ∈ H <;> simp [F, he, hh]
+  have ht : ∀ᵐ z ∂volume.restrict (Metric.ball (0 : Euc n) 1),
+      Tendsto (fun r ↦ F r z) (𝓝[>] (0 : ℝ)) (𝓝 0) := by
+    filter_upwards [ae_restrict_of_ae (ae_linear_ne_zero hLn)] with z hz
+    apply tendsto_const_nhds.congr'
+    filter_upwards [eventually_filled_membership hd hx hlocal hz] with r hr
+    have hmem : x + r • z ∈ E ↔ z ∈ H := hr.trans (hsign z)
+    by_cases he : x + r • z ∈ E
+    · simp [F, he, hmem.mp he]
+    · have hhn : z ∉ H := fun hh ↦ he (hmem.mpr hh)
+      simp only [F, Set.indicator_of_notMem he, Set.indicator_of_notMem hhn,
+        sub_self, abs_zero]
+  have hi : Integrable (fun _ : Euc n ↦ (1 : ℝ))
+      (volume.restrict (Metric.ball 0 1)) := integrableOn_const measure_ball_lt_top.ne
+  have hconv := tendsto_integral_filter_of_dominated_convergence
+    (fun _ : Euc n ↦ (1 : ℝ)) (Eventually.of_forall hmeas) (Eventually.of_forall hb) hi ht
+  refine ⟨hv, ?_⟩
+  have heq : ∀ᶠ r : ℝ in 𝓝[>] 0,
+      (r ^ n)⁻¹ * (∫ y in Metric.ball x r,
+        |E.indicator (fun _ ↦ (1 : ℝ)) y -
+          {z | inner ℝ (z - x) v < 0}.indicator (fun _ ↦ (1 : ℝ)) y|) =
+        ∫ z in Metric.ball (0 : Euc n) 1, F r z := by
+    filter_upwards [self_mem_nhdsWithin] with r hr
+    rw [normalized_ball_integral _ x hr]
+    apply integral_congr_ae
+    filter_upwards [] with z
+    simp only [halfspace_indicator_rescale x v z hr, F, H]
+  exact (show Tendsto (fun r ↦ ∫ z in Metric.ball (0 : Euc n) 1, F r z)
+    (𝓝[>] (0 : ℝ)) (𝓝 0) from by simpa only [integral_zero] using hconv).congr' (Filter.EventuallyEq.symm heq)
+
+theorem hasHalfspaceBlowup_gradient_filled {n : ℕ} {E : Set (Euc n)}
+    (hE : MeasurableSet E) {f : Euc n → ℝ} {x : Euc n}
+    (hd : DifferentiableAt ℝ f x) (hx : f x = 0) (hdf : fderiv ℝ f x ≠ 0)
+    (hlocal : ∀ᶠ y in 𝓝 x, (f y < 0 → y ∈ E) ∧ (y ∈ E → f y ≤ 0)) :
+    HasHalfspaceBlowup E x (‖gradient f x‖⁻¹ • gradient f x) := by
+  have hg : gradient f x ≠ 0 := by
+    intro he
+    apply hdf
+    ext z
+    rw [← inner_gradient_left, he, inner_zero_left]
+    rfl
+  have hng : 0 < ‖gradient f x‖ := norm_pos_iff.mpr hg
+  apply hasHalfspaceBlowup_of_regular_filled_chart hE hd.hasFDerivAt hx hlocal
+    (a := ‖gradient f x‖)
+  · rw [norm_smul, Real.norm_of_nonneg (inv_nonneg.mpr hng.le), inv_mul_cancel₀ hng.ne']
+  · exact hng
+  · intro z
+    rw [real_inner_smul_right, ← inner_gradient_left, real_inner_comm]
+    field_simp
+
+theorem approximateNormal_eq_gradient_filled {n : ℕ} {E : Set (Euc n)}
+    (hE : MeasurableSet E) {f : Euc n → ℝ} {x : Euc n}
+    (hd : DifferentiableAt ℝ f x) (hx : f x = 0) (hdf : fderiv ℝ f x ≠ 0)
+    (hlocal : ∀ᶠ y in 𝓝 x, (f y < 0 → y ∈ E) ∧ (y ∈ E → f y ≤ 0)) :
+    approximateNormal E x = ‖gradient f x‖⁻¹ • gradient f x :=
+  approximateNormal_eq_of_blowup hE
+    (hasHalfspaceBlowup_gradient_filled hE hd hx hdf hlocal)
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPI
+
+def truncationSlope (ε t : ℝ) : ℝ := Real.smoothTransition ((1 - t) / ε)
+
+def smoothTruncation (ε s : ℝ) : ℝ := ∫ t in (0 : ℝ)..s, truncationSlope ε t
+
+theorem truncationSlope_contDiff (ε : ℝ) : ContDiff ℝ ∞ (truncationSlope ε) := by
+  unfold truncationSlope
+  fun_prop
+
+theorem smoothTruncation_hasDerivAt (ε s : ℝ) :
+    HasDerivAt (smoothTruncation ε) (truncationSlope ε s) s := by
+  have hc := (truncationSlope_contDiff ε).continuous
+  exact intervalIntegral.integral_hasDerivAt_right (hc.intervalIntegrable 0 s)
+    hc.stronglyMeasurable.stronglyMeasurableAtFilter hc.continuousAt
+
+@[simp] theorem smoothTruncation_deriv (ε s : ℝ) :
+    deriv (smoothTruncation ε) s = truncationSlope ε s :=
+  (smoothTruncation_hasDerivAt ε s).deriv
+
+theorem smoothTruncation_contDiff (ε : ℝ) : ContDiff ℝ ∞ (smoothTruncation ε) := by
+  rw [contDiff_infty_iff_deriv]
+  exact ⟨fun s ↦ (smoothTruncation_hasDerivAt ε s).differentiableAt,
+    by simpa only [funext (smoothTruncation_deriv ε)] using truncationSlope_contDiff ε⟩
+
+theorem truncationSlope_mem_Icc (ε t : ℝ) : truncationSlope ε t ∈ Icc (0 : ℝ) 1 :=
+  ⟨Real.smoothTransition.nonneg _, Real.smoothTransition.le_one _⟩
+
+theorem truncationSlope_antitone {ε : ℝ} (hε : 0 < ε) : Antitone (truncationSlope ε) := by
+  intro s t hst
+  apply Real.smoothTransition.monotone
+  exact div_le_div_of_nonneg_right (sub_le_sub_left hst 1) hε.le
+
+theorem smoothTruncation_second_deriv_nonpos {ε : ℝ} (hε : 0 < ε) (s : ℝ) :
+    deriv (deriv (smoothTruncation ε)) s ≤ 0 := by
+  rw [show deriv (smoothTruncation ε) = truncationSlope ε from funext (smoothTruncation_deriv ε)]
+  exact (truncationSlope_antitone hε).deriv_nonpos
+
+@[simp] theorem smoothTruncation_zero (ε : ℝ) : smoothTruncation ε 0 = 0 := by
+  simp [smoothTruncation]
+
+theorem smoothTruncation_monotone (ε : ℝ) : Monotone (smoothTruncation ε) := by
+  apply monotone_of_deriv_nonneg
+  · exact fun s ↦ (smoothTruncation_hasDerivAt ε s).differentiableAt
+  · intro s
+    rw [smoothTruncation_deriv]
+    exact (truncationSlope_mem_Icc ε s).1
+
+theorem smoothTruncation_nonneg (ε : ℝ) {s : ℝ} (hs : 0 ≤ s) :
+    0 ≤ smoothTruncation ε s := by
+  simpa only [smoothTruncation_zero] using smoothTruncation_monotone ε hs
+
+theorem smoothTruncation_le_self (ε : ℝ) {s : ℝ} (hs : 0 ≤ s) :
+    smoothTruncation ε s ≤ s := by
+  have h := intervalIntegral.integral_mono hs
+    ((truncationSlope_contDiff ε).continuous.intervalIntegrable 0 s)
+    (intervalIntegrable_const : IntervalIntegrable (fun _ : ℝ ↦ (1 : ℝ)) volume 0 s)
+    (fun t ↦ (truncationSlope_mem_Icc ε t).2)
+  simpa only [smoothTruncation, intervalIntegral.integral_const, sub_zero, smul_eq_mul, mul_one] using h
+
+theorem truncationSlope_eq_zero {ε : ℝ} (hε : 0 < ε) {t : ℝ} (ht : 1 ≤ t) :
+    truncationSlope ε t = 0 := by
+  apply Real.smoothTransition.zero_of_nonpos
+  exact div_nonpos_of_nonpos_of_nonneg (sub_nonpos.mpr ht) hε.le
+
+theorem truncationSlope_eq_one {ε : ℝ} (hε : 0 < ε) {t : ℝ} (ht : t ≤ 1 - ε) :
+    truncationSlope ε t = 1 := by
+  apply Real.smoothTransition.one_of_one_le
+  apply (le_div_iff₀ hε).mpr
+  linarith
+
+theorem smoothTruncation_eq_self {ε s : ℝ} (hε : 0 < ε) (hs : 0 ≤ s)
+    (hsε : s ≤ 1 - ε) : smoothTruncation ε s = s := by
+  unfold smoothTruncation
+  calc
+    _ = ∫ _ in (0 : ℝ)..s, (1 : ℝ) := by
+      apply intervalIntegral.integral_congr
+      intro t ht
+      exact truncationSlope_eq_one hε ((uIcc_of_le hs ▸ ht).2.trans hsε)
+    _ = s := by simp
+
+theorem smoothTruncation_eq_at_one {ε s : ℝ} (hε : 0 < ε) (hs : 1 ≤ s) :
+    smoothTruncation ε s = smoothTruncation ε 1 := by
+  have hc := (truncationSlope_contDiff ε).continuous
+  have he := intervalIntegral.integral_add_adjacent_intervals (μ := volume)
+    (hc.intervalIntegrable 0 1) (hc.intervalIntegrable 1 s)
+  have hz : ∫ t in (1 : ℝ)..s, truncationSlope ε t = 0 := by
+    calc
+      _ = ∫ _ in (1 : ℝ)..s, (0 : ℝ) := by
+        apply intervalIntegral.integral_congr
+        intro t ht
+        exact truncationSlope_eq_zero hε ((uIcc_of_le hs ▸ ht).1)
+      _ = 0 := by simp
+  change _ = _
+  change smoothTruncation ε 1 + (∫ t in (1 : ℝ)..s, truncationSlope ε t) =
+    smoothTruncation ε s at he
+  rw [hz, add_zero] at he
+  exact he.symm
+
+theorem smoothTruncation_le_min {ε s : ℝ} (hε : 0 < ε) (hs : 0 ≤ s) :
+    smoothTruncation ε s ≤ min s 1 := by
+  apply le_min (smoothTruncation_le_self ε hs)
+  by_cases h : s ≤ 1
+  · exact (smoothTruncation_le_self ε hs).trans h
+  · rw [smoothTruncation_eq_at_one hε (le_of_not_ge h)]
+    exact smoothTruncation_le_self ε zero_le_one
+
+theorem min_sub_le_smoothTruncation {ε s : ℝ} (hε : 0 < ε) (hε1 : ε ≤ 1)
+    (hs : 0 ≤ s) : min s 1 - ε ≤ smoothTruncation ε s := by
+  by_cases h : s ≤ 1 - ε
+  · rw [smoothTruncation_eq_self hε hs h]
+    exact (sub_le_self _ hε.le).trans (min_le_left _ _)
+  · have h0 : 0 ≤ 1 - ε := sub_nonneg.mpr hε1
+    have he : smoothTruncation ε (1 - ε) = 1 - ε :=
+      smoothTruncation_eq_self hε h0 le_rfl
+    have hm := smoothTruncation_monotone ε (le_of_not_ge h)
+    rw [he] at hm
+    exact (sub_le_sub_right (min_le_right s 1) ε).trans hm
+
+def truncationParameter (j : ℕ) : ℝ := 1 / (j + 1 : ℝ)
+
+theorem truncationParameter_pos (j : ℕ) : 0 < truncationParameter j := by
+  unfold truncationParameter
+  positivity
+
+theorem truncationParameter_le_one (j : ℕ) : truncationParameter j ≤ 1 := by
+  unfold truncationParameter
+  apply (div_le_one (by positivity)).mpr
+  have hj : (0 : ℝ) ≤ j := Nat.cast_nonneg j
+  linarith
+
+theorem truncationParameter_tendsto : Tendsto truncationParameter atTop (𝓝 (0 : ℝ)) :=
+  tendsto_one_div_add_atTop_nhds_zero_nat
+
+theorem smoothTruncation_tendsto {s : ℝ} (hs : 0 ≤ s) :
+    Tendsto (fun j ↦ smoothTruncation (truncationParameter j) s) atTop (𝓝 (min s 1)) := by
+  have hlow : Tendsto (fun j ↦ min s 1 - truncationParameter j) atTop (𝓝 (min s 1)) := by
+    simpa only [sub_zero] using tendsto_const_nhds.sub truncationParameter_tendsto
+  apply tendsto_of_tendsto_of_tendsto_of_le_of_le hlow tendsto_const_nhds
+  · intro j
+    exact min_sub_le_smoothTruncation (truncationParameter_pos j)
+      (truncationParameter_le_one j) hs
+  · intro j
+    exact smoothTruncation_le_min (truncationParameter_pos j) hs
+
+theorem truncationSlope_tendsto (s : ℝ) :
+    Tendsto (fun j ↦ truncationSlope (truncationParameter j) s) atTop
+      (𝓝 (if s < 1 then (1 : ℝ) else 0)) := by
+  split_ifs with hs
+  · apply tendsto_const_nhds.congr'
+    filter_upwards [truncationParameter_tendsto.eventually (gt_mem_nhds (sub_pos.mpr hs))] with j hj
+    symm
+    exact truncationSlope_eq_one (truncationParameter_pos j) (by linarith)
+  · apply tendsto_const_nhds.congr
+    intro j
+    symm
+    exact truncationSlope_eq_zero (truncationParameter_pos j) (le_of_not_gt hs)
+
+end RPI
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/GeometricHolder.lean b/lean/OAI/Geometry/Perimeter/Regularity/GeometricHolder.lean
new file mode 100644
index 0000000000000000000000000000000000000000..d9973d1514cd57a57fbabf4724c8be8c6d217ca7
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/GeometricHolder.lean
@@ -0,0 +1,96 @@
+import Mathlib
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+
+ def GeometricHolderAt {X Y : Type*} [PseudoMetricSpace X] [PseudoMetricSpace Y]
+    (θ : ℝ) (f : X → Y) (p : X) : Prop :=
+  ∃ r C : ℝ, 0<r ∧ 0 ≤ C ∧ ∀ j x, x∈Metric.ball p r → ∀ y, y∈Metric.ball p r →
+    dist x y ≤ θ^j*r → dist (f x) (f y) ≤ C*(1/2)^j
+
+ theorem GeometricHolderAt.mono_scale {X Y : Type*} [PseudoMetricSpace X] [PseudoMetricSpace Y]
+    {θ η : ℝ} {f : X → Y} {p : X} (h : GeometricHolderAt θ f p)
+    (hη : 0 ≤ η) (he : η ≤ θ) : GeometricHolderAt η f p := by
+  obtain ⟨r,C,hr,hC,h⟩ := h
+  refine ⟨r,C,hr,hC,fun j x hx y hy hxy ↦ h j x hx y hy (hxy.trans ?_)⟩
+  exact mul_le_mul_of_nonneg_right (pow_le_pow_left₀ hη he j) hr.le
+
+ theorem GeometricHolderAt.shrink_amplitude {X Y : Type*} [PseudoMetricSpace X] [PseudoMetricSpace Y]
+    {θ : ℝ} (hθ : 0<θ) {f : X → Y} {p : X} (h : GeometricHolderAt θ f p)
+    {ε R : ℝ} (hε : 0<ε) (hR : 0<R) :
+    ∃ r : ℝ, 0<r ∧ r ≤ R ∧ ∀ j x, x∈Metric.ball p r → ∀ y, y∈Metric.ball p r →
+      dist x y ≤ θ^j*r → dist (f x) (f y) ≤ ε*(1/2)^j := by
+  obtain ⟨s,C,hs,hC,h⟩ := h
+  have ht : Tendsto (fun j : ℕ ↦ C*(1/2:ℝ)^j) atTop (𝓝 0) := by
+    simpa only [mul_zero] using
+      (tendsto_pow_atTop_nhds_zero_of_lt_one (by norm_num : (0:ℝ) ≤ 1/2) (by norm_num)).const_mul C
+  obtain ⟨J,hJ⟩ := (ht.eventually (gt_mem_nhds hε)).exists
+  let r := min R (min s (θ^J*s))
+  have hr : 0<r := lt_min hR (lt_min hs (by positivity))
+  have hrs : r ≤ s := (min_le_right _ _).trans (min_le_left _ _)
+  have hrJ : r ≤ θ^J*s := (min_le_right _ _).trans (min_le_right _ _)
+  refine ⟨r,hr,min_le_left _ _,fun j x hx y hy hxy ↦ ?_⟩
+  have hh := h (j+J) x (Metric.ball_subset_ball hrs hx) y (Metric.ball_subset_ball hrs hy)
+    (hxy.trans (by simpa only [pow_add,mul_assoc] using
+      mul_le_mul_of_nonneg_left hrJ (pow_nonneg hθ.le j)))
+  apply hh.trans
+  rw [pow_add]
+  nlinarith [mul_le_mul_of_nonneg_right hJ.le (by positivity : 0 ≤ (1/2:ℝ)^j)]
+
+ theorem GeometricHolderAt.congr {X Y : Type*} [PseudoMetricSpace X] [PseudoMetricSpace Y]
+    {θ : ℝ} (hθ : 0 ≤ θ) {f g : X → Y} {p : X} (h : GeometricHolderAt θ f p)
+    (he : f=ᶠ[𝓝 p] g) : GeometricHolderAt θ g p := by
+  obtain ⟨r,C,hr,hC,h⟩ := h
+  obtain ⟨s,hs,hsb⟩ := Metric.eventually_nhds_iff.mp he
+  refine ⟨min r s,C,lt_min hr hs,hC,fun j x hx y hy hxy ↦ ?_⟩
+  have hxs : dist x p<s := (Metric.mem_ball.mp hx).trans_le (min_le_right _ _)
+  have hys : dist y p<s := (Metric.mem_ball.mp hy).trans_le (min_le_right _ _)
+  rw [←hsb hxs,←hsb hys]
+  exact h j x (Metric.ball_subset_ball (min_le_left _ _) hx) y
+    (Metric.ball_subset_ball (min_le_left _ _) hy)
+    (hxy.trans (mul_le_mul_of_nonneg_left (min_le_left _ _) (by positivity)))
+
+ theorem GeometricHolderAt.prodMk {X Y Z : Type*} [PseudoMetricSpace X]
+    [PseudoMetricSpace Y] [PseudoMetricSpace Z] {θ : ℝ} (hθ : 0 ≤ θ)
+    {f : X → Y} {g : X → Z} {p : X}
+    (hf : GeometricHolderAt θ f p) (hg : GeometricHolderAt θ g p) :
+    GeometricHolderAt θ (fun x ↦ (f x,g x)) p := by
+  obtain ⟨r,C,hr,hC,hf⟩ := hf
+  obtain ⟨s,D,hs,hD,hg⟩ := hg
+  refine ⟨min r s,max C D,lt_min hr hs,le_trans hC (le_max_left _ _),fun j x hx y hy hxy ↦ ?_⟩
+  rw [Prod.dist_eq]
+  apply max_le
+  · exact (hf j x (Metric.ball_subset_ball (min_le_left _ _) hx) y
+      (Metric.ball_subset_ball (min_le_left _ _) hy)
+      (hxy.trans (mul_le_mul_of_nonneg_left (min_le_left _ _) (pow_nonneg hθ _)))).trans
+      (mul_le_mul_of_nonneg_right (le_max_left _ _) (by positivity))
+  · exact (hg j x (Metric.ball_subset_ball (min_le_right _ _) hx) y
+      (Metric.ball_subset_ball (min_le_right _ _) hy)
+      (hxy.trans (mul_le_mul_of_nonneg_left (min_le_right _ _) (pow_nonneg hθ _)))).trans
+      (mul_le_mul_of_nonneg_right (le_max_right _ _) (by positivity))
+
+ theorem GeometricHolderAt.const {X Y : Type*} [PseudoMetricSpace X] [PseudoMetricSpace Y]
+    (θ : ℝ) (p : X) (c : Y) : GeometricHolderAt θ (fun _ : X ↦ c) p := by
+  refine ⟨1,0,by norm_num,le_rfl,?_⟩
+  intros; simp
+
+ theorem geometricHolderAt_of_local_lipschitz {X Y : Type*}
+    [PseudoMetricSpace X] [PseudoMetricSpace Y] {θ r L : ℝ}
+    (hθ : 0 ≤ θ) (hθ' : θ ≤ 1/2) (hr : 0<r) (hL : 0 ≤ L)
+    {f : X → Y} {p : X}
+    (h : ∀ x∈Metric.ball p r, ∀ y∈Metric.ball p r, dist (f x) (f y) ≤ L*dist x y) :
+    GeometricHolderAt θ f p := by
+  refine ⟨r,L*r,hr,mul_nonneg hL hr.le,fun j x hx y hy hxy ↦ ?_⟩
+  exact (h x hx y hy).trans ((mul_le_mul_of_nonneg_left hxy hL).trans (by
+    have hh := mul_le_mul_of_nonneg_left (pow_le_pow_left₀ hθ hθ' j) (mul_nonneg hL hr.le)
+    nlinarith))
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/GeometricHolderAlgebra.lean b/lean/OAI/Geometry/Perimeter/Regularity/GeometricHolderAlgebra.lean
new file mode 100644
index 0000000000000000000000000000000000000000..3efc55b458cc7de171ec20ca1afb050259bf0fa6
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/GeometricHolderAlgebra.lean
@@ -0,0 +1,86 @@
+import OAI.Geometry.Perimeter.Regularity.GeometricHolder
+import OAI.Geometry.Perimeter.Regularity.EllipticSecantBounds
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+
+ theorem GeometricHolderAt.comp_lipschitz {X V W : Type*} [PseudoMetricSpace X]
+    [PseudoMetricSpace V] [PseudoMetricSpace W] {θ : ℝ} (hθ : 0 ≤ θ)
+    {f : X → V} {g : V → W} {p : X} (hf : GeometricHolderAt θ f p)
+    (hc : ContinuousAt f p) {s L : ℝ} (hs : 0<s) (hL : 0 ≤ L)
+    (hg : ∀ x∈Metric.ball (f p) s, ∀ y∈Metric.ball (f p) s,
+      dist (g x) (g y) ≤ L*dist x y) : GeometricHolderAt θ (fun x ↦ g (f x)) p := by
+  obtain ⟨r,C,hr,hC,hf⟩ := hf
+  obtain ⟨b,hb,hbim⟩ := Metric.continuousAt_iff.mp hc s hs
+  refine ⟨min r b,L*C,lt_min hr hb,mul_nonneg hL hC,fun j x hx y hy hxy ↦ ?_⟩
+  have him {z : X} (hz : z∈Metric.ball p (min r b)) : f z∈Metric.ball (f p) s :=
+    Metric.mem_ball.mpr (hbim ((Metric.mem_ball.mp hz).trans_le (min_le_right _ _)))
+  apply (hg _ (him hx) _ (him hy)).trans
+  have hh := hf j x (Metric.ball_subset_ball (min_le_left _ _) hx) y
+    (Metric.ball_subset_ball (min_le_left _ _) hy)
+    (hxy.trans (mul_le_mul_of_nonneg_left (min_le_left _ _) (pow_nonneg hθ j)))
+  simpa only [mul_assoc] using mul_le_mul_of_nonneg_left hh hL
+
+ theorem GeometricHolderAt.comp_C1 {X V W : Type*} [PseudoMetricSpace X]
+    [NormedAddCommGroup V] [NormedSpace ℝ V] [NormedAddCommGroup W] [NormedSpace ℝ W]
+    {θ : ℝ} (hθ : 0 ≤ θ) {f : X → V} {g : V → W} {p : X}
+    (hf : GeometricHolderAt θ f p) (hc : ContinuousAt f p) (hg : ContDiff ℝ 1 g) :
+    GeometricHolderAt θ (fun x ↦ g (f x)) p := by
+  obtain ⟨r,L,hr,hL,h⟩ := contDiff_local_lipschitz hg (f p)
+  exact hf.comp_lipschitz hθ hc hr hL (fun x hx y hy ↦ by
+    simpa only [dist_eq_norm] using h x hx y hy)
+
+ theorem GeometricHolderAt.pi {ι X V : Type*} [Fintype ι] [PseudoMetricSpace X]
+    [NormedAddCommGroup V] {θ : ℝ} (hθ : 0 ≤ θ) {f : X → ι → V} {p : X}
+    (h : ∀ i, GeometricHolderAt θ (fun x ↦ f x i) p) :
+    GeometricHolderAt θ f p := by
+  classical
+  have hall (s : Finset ι) : ∃ r C : ℝ, 0<r ∧ 0 ≤ C ∧
+      ∀ i∈s, ∀ j x, x∈Metric.ball p r → ∀ y, y∈Metric.ball p r →
+        dist x y ≤ θ^j*r → dist (f x i) (f y i) ≤ C*(1/2)^j := by
+    induction s using Finset.induction_on with
+    | empty => exact ⟨1,0,by norm_num,le_rfl,by simp⟩
+    | @insert i s hi ih =>
+      obtain ⟨r,C,hr,hC,hs⟩ := ih
+      obtain ⟨b,D,hb,hD,hi⟩ := h i
+      refine ⟨min r b,max C D,lt_min hr hb,hC.trans (le_max_left _ _),?_⟩
+      intro k hk j x hx y hy hxy
+      rcases Finset.mem_insert.mp hk with rfl | hk
+      · exact (hi j x (Metric.ball_subset_ball (min_le_right _ _) hx) y
+          (Metric.ball_subset_ball (min_le_right _ _) hy)
+          (hxy.trans (mul_le_mul_of_nonneg_left (min_le_right _ _) (pow_nonneg hθ j)))).trans
+          (mul_le_mul_of_nonneg_right (le_max_right _ _) (by positivity))
+      · exact (hs k hk j x (Metric.ball_subset_ball (min_le_left _ _) hx) y
+          (Metric.ball_subset_ball (min_le_left _ _) hy)
+          (hxy.trans (mul_le_mul_of_nonneg_left (min_le_left _ _) (pow_nonneg hθ j)))).trans
+          (mul_le_mul_of_nonneg_right (le_max_left _ _) (by positivity))
+  obtain ⟨r,C,hr,hC,hh⟩ := hall Finset.univ
+  refine ⟨r,C,hr,hC,fun j x hx y hy hxy ↦ ?_⟩
+  exact (dist_pi_le_iff (by positivity)).mpr (fun i ↦ hh i (Finset.mem_univ _) j x hx y hy hxy)
+
+ theorem GeometricHolderAt.clm_apply_iff {X V W : Type*} [PseudoMetricSpace X]
+    [NormedAddCommGroup V] [NormedSpace ℝ V] [FiniteDimensional ℝ V]
+    [NormedAddCommGroup W] [NormedSpace ℝ W] {θ : ℝ} (hθ : 0 ≤ θ)
+    {f : X → V →L[ℝ] W} {p : X} (hc : ContinuousAt f p) :
+    GeometricHolderAt θ f p ↔ ∀ e, GeometricHolderAt θ (fun x ↦ f x e) p := by
+  constructor
+  · intro h e
+    exact h.comp_C1 hθ hc (ContinuousLinearMap.apply ℝ W e).contDiff
+  · intro h
+    let d := Module.finrank ℝ V
+    have hd : d=Module.finrank ℝ (Fin d → ℝ) := (Module.finrank_fin_fun ℝ).symm
+    let e₁ := ContinuousLinearEquiv.ofFinrankEq hd
+    let e₂ := (e₁.arrowCongr (1 : W ≃L[ℝ] W)).trans (ContinuousLinearEquiv.piRing (Fin d))
+    have hpi : GeometricHolderAt θ (fun x ↦ e₂ (f x)) p := GeometricHolderAt.pi hθ (fun i ↦ h _)
+    have hh := hpi.comp_C1 hθ (e₂.continuous.continuousAt.comp hc) e₂.symm.contDiff
+    simpa only [e₂.symm_apply_apply] using hh
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/GradientContDiff.lean b/lean/OAI/Geometry/Perimeter/Regularity/GradientContDiff.lean
new file mode 100644
index 0000000000000000000000000000000000000000..084a42d33927cb01c3dccc7ebb380da6d0d7a081
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/GradientContDiff.lean
@@ -0,0 +1,445 @@
+import OAI.Geometry.Perimeter.Regularity.C1MinRegularTest
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+open RPI.Boundary
+open ContinuousLinearMap
+
+theorem gradient_contDiff {n : ℕ} {u : Euc n → ℝ} (hu : ContDiff ℝ ∞ u) :
+    ContDiff ℝ ∞ (gradient u) := by
+  exact (InnerProductSpace.toDual ℝ (Euc n)).symm.contDiff.comp
+    (hu.fderiv_right (by simp))
+
+theorem gradient_hasCompactSupport {n : ℕ} {u : Euc n → ℝ}
+    (hu : HasCompactSupport u) : HasCompactSupport (gradient u) := by
+  exact (hu.fderiv ℝ).comp_left (map_zero _)
+
+theorem integral_mul_divergence_eq_neg_inner_gradient {n : ℕ} {u : Euc n → ℝ}
+    {V : Euc n → Euc n} (hu : ContDiff ℝ ∞ u) (hcu : HasCompactSupport u)
+    (hV : ContDiff ℝ ∞ V) (hcV : HasCompactSupport V) :
+    (∫ x, u x * divergence V x) = - ∫ x, inner ℝ (gradient u x) (V x) := by
+  let b := EuclideanSpace.basisFun (Fin n) ℝ
+  let v : Fin n → Euc n → ℝ := fun i x ↦ inner ℝ (b i) (V x)
+  have hv : ∀ i, ContDiff ℝ ∞ (v i) := fun i ↦ (innerSL ℝ (b i)).contDiff.comp hV
+  have hcv : ∀ i, HasCompactSupport (v i) := fun i ↦ hcV.comp_left (by simp)
+  have hdv : ∀ i x, fderiv ℝ (v i) x (b i) =
+      inner ℝ (b i) (fderiv ℝ V x (b i)) := by
+    intro i x
+    have h := ((innerSL ℝ (b i)).hasFDerivAt.comp x
+      (hV.differentiable (by simp) x).hasFDerivAt).fderiv
+    exact congrArg (fun L : Euc n →L[ℝ] ℝ ↦ L (b i)) h
+  have hdi : ∀ i, Continuous (fun x ↦ fderiv ℝ u x (b i)) := fun i ↦
+    (hu.continuous_fderiv (by simp)).clm_apply continuous_const
+  have hdvi : ∀ i, Continuous (fun x ↦ fderiv ℝ (v i) x (b i)) := fun i ↦
+    ((hv i).continuous_fderiv (by simp)).clm_apply continuous_const
+  have hInt1 : ∀ i, Integrable (fun x ↦ fderiv ℝ u x (b i) * v i x) volume := fun i ↦
+    ((hdi i).mul (hv i).continuous).integrable_of_hasCompactSupport (hcv i).mul_left
+  have hInt2 : ∀ i, Integrable (fun x ↦ u x * fderiv ℝ (v i) x (b i)) volume := fun i ↦
+    (hu.continuous.mul (hdvi i)).integrable_of_hasCompactSupport hcu.mul_right
+  have hparts : ∀ i, (∫ x, u x * fderiv ℝ (v i) x (b i)) =
+      -∫ x, fderiv ℝ u x (b i) * v i x := by
+    intro i
+    exact integral_mul_fderiv_eq_neg_fderiv_mul_of_integrable (hInt1 i) (hInt2 i)
+      ((hu.continuous.mul (hv i).continuous).integrable_of_hasCompactSupport hcu.mul_right)
+      (fun x _ ↦ hu.differentiable (by simp) x)
+      (fun x _ ↦ (hv i).differentiable (by simp) x)
+  calc
+    (∫ x, u x * divergence V x) = ∫ x, ∑ i, u x * fderiv ℝ (v i) x (b i) := by
+      congr 1
+      funext x
+      simp only [divergence, hdv, Finset.mul_sum, b]
+    _ = ∑ i, ∫ x, u x * fderiv ℝ (v i) x (b i) := integral_finsetSum _ (fun i _ ↦ hInt2 i)
+    _ = -(∑ i, ∫ x, fderiv ℝ u x (b i) * v i x) := by simp only [hparts, Finset.sum_neg_distrib]
+    _ = -(∫ x, ∑ i, fderiv ℝ u x (b i) * v i x) := by
+      rw [integral_finsetSum _ (fun i _ ↦ hInt1 i)]
+    _ = -(∫ x, inner ℝ (gradient u x) (V x)) := by
+      congr 1
+      apply integral_congr_ae
+      apply ae_of_all
+      intro x
+      simp only [← inner_gradient_left, v]
+      exact b.sum_inner_mul_inner _ _
+
+def gradientDualTest {n : ℕ} (u : Euc n → ℝ) (ε : ℝ) (x : Euc n) : Euc n :=
+  -(Real.sqrt (ε ^ 2 + ‖gradient u x‖ ^ 2))⁻¹ • gradient u x
+
+theorem gradientDualTest_contDiff {n : ℕ} {u : Euc n → ℝ}
+    (hu : ContDiff ℝ ∞ u) {ε : ℝ} (hε : 0 < ε) :
+    ContDiff ℝ ∞ (gradientDualTest u ε) := by
+  have hg := gradient_contDiff hu
+  have hp : ∀ x, 0 < ε ^ 2 + ‖gradient u x‖ ^ 2 := fun x ↦
+    add_pos_of_pos_of_nonneg (sq_pos_of_pos hε) (sq_nonneg _)
+  exact (((contDiff_const.add (hg.norm_sq ℝ)).sqrt (fun x ↦ (hp x).ne')).inv
+    (fun x ↦ (Real.sqrt_pos.2 (hp x)).ne')).neg.smul hg
+
+theorem gradientDualTest_hasCompactSupport {n : ℕ} {u : Euc n → ℝ}
+    (hcu : HasCompactSupport u) (ε : ℝ) : HasCompactSupport (gradientDualTest u ε) := by
+  change HasCompactSupport ((fun x ↦ -(Real.sqrt (ε ^ 2 + ‖gradient u x‖ ^ 2))⁻¹) • gradient u)
+  exact (gradient_hasCompactSupport hcu).smul_left
+
+theorem gradientDualTest_norm_le {n : ℕ} (u : Euc n → ℝ) {ε : ℝ}
+    (hε : 0 < ε) (x : Euc n) : ‖gradientDualTest u ε x‖ ≤ 1 := by
+  have hp : 0 < ε ^ 2 + ‖gradient u x‖ ^ 2 :=
+    add_pos_of_pos_of_nonneg (sq_pos_of_pos hε) (sq_nonneg _)
+  have hs : ‖gradient u x‖ ≤ Real.sqrt (ε ^ 2 + ‖gradient u x‖ ^ 2) := by
+    rw [Real.le_sqrt (norm_nonneg _) hp.le]
+    nlinarith [sq_nonneg ε]
+  simp only [gradientDualTest, norm_smul, Real.norm_eq_abs, abs_neg, abs_inv,
+    abs_of_nonneg (Real.sqrt_nonneg _)]
+  rw [inv_mul_eq_div, div_le_one (Real.sqrt_pos.2 hp)]
+  exact hs
+
+theorem integral_norm_gradient_le_of_test_bound {n : ℕ} {u : Euc n → ℝ}
+    (hu : ContDiff ℝ ∞ u) (hcu : HasCompactSupport u) {P : ℝ}
+    (hbound : ∀ V : Euc n → Euc n, PerimeterAdmissible univ V →
+      (∫ x, u x * divergence V x) ≤ P) :
+    (∫ x, ‖gradient u x‖) ≤ P := by
+  let ε : ℕ → ℝ := fun j ↦ 1 / (j + 1 : ℝ)
+  have hε : ∀ j, 0 < ε j := by intro j; dsimp [ε]; positivity
+  let f : ℕ → Euc n → ℝ := fun j x ↦
+    ‖gradient u x‖ ^ 2 / Real.sqrt (ε j ^ 2 + ‖gradient u x‖ ^ 2)
+  have hg := (gradient_contDiff hu).continuous
+  have hgi : Integrable (fun x ↦ ‖gradient u x‖) volume :=
+    (hg.integrable_of_hasCompactSupport (gradient_hasCompactSupport hcu)).norm
+  have hf0 : ∀ j x, 0 ≤ f j x := by intro j x; exact div_nonneg (sq_nonneg _) (Real.sqrt_nonneg _)
+  have hfle : ∀ j x, f j x ≤ ‖gradient u x‖ := by
+    intro j x
+    have hp : 0 < ε j ^ 2 + ‖gradient u x‖ ^ 2 :=
+      add_pos_of_pos_of_nonneg (sq_pos_of_pos (hε j)) (sq_nonneg _)
+    have hs : ‖gradient u x‖ ≤ Real.sqrt (ε j ^ 2 + ‖gradient u x‖ ^ 2) := by
+      rw [Real.le_sqrt (norm_nonneg _) hp.le]
+      nlinarith [sq_nonneg (ε j)]
+    exact (div_le_iff₀ (Real.sqrt_pos.2 hp)).2 (by nlinarith [norm_nonneg (gradient u x)])
+  have hfcont : ∀ j, Continuous (f j) := by
+    intro j
+    exact (hg.norm.pow 2).div ((continuous_const.add (hg.norm.pow 2)).sqrt)
+      (fun x ↦ (Real.sqrt_pos.2
+        (add_pos_of_pos_of_nonneg (sq_pos_of_pos (hε j)) (sq_nonneg _))).ne')
+  have hflim : ∀ x, Tendsto (fun j ↦ f j x) atTop (𝓝 ‖gradient u x‖) := by
+    intro x
+    have hlim : Tendsto ε atTop (𝓝 0) := by
+      simpa only [ε, Nat.cast_add, Nat.cast_one] using tendsto_one_div_add_atTop_nhds_zero_nat
+    by_cases hx : ‖gradient u x‖ = 0
+    · simp [f, hx]
+    · have hden := ((hlim.pow 2).add_const (‖gradient u x‖ ^ 2)).sqrt
+      have hden' : Tendsto (fun j ↦ Real.sqrt (ε j ^ 2 + ‖gradient u x‖ ^ 2))
+          atTop (𝓝 ‖gradient u x‖) := by
+        simpa [Real.sqrt_sq (norm_nonneg _)] using hden
+      have hh := (tendsto_const_nhds (x := ‖gradient u x‖ ^ 2)).div hden' hx
+      have heq : ‖gradient u x‖ ^ 2 / ‖gradient u x‖ = ‖gradient u x‖ := by
+        rw [pow_two, mul_div_cancel_right₀ _ hx]
+      change Tendsto (fun j ↦ ‖gradient u x‖ ^ 2 / Real.sqrt (ε j ^ 2 + ‖gradient u x‖ ^ 2))
+        atTop (𝓝 (‖gradient u x‖ ^ 2 / ‖gradient u x‖)) at hh
+      rw [heq] at hh
+      exact hh
+  have hlim : Tendsto (fun j ↦ ∫ x, f j x) atTop (𝓝 (∫ x, ‖gradient u x‖)) :=
+    tendsto_integral_of_dominated_convergence (fun x ↦ ‖gradient u x‖)
+      (fun j ↦ (hfcont j).aestronglyMeasurable)
+      hgi (fun j ↦ ae_of_all _ fun x ↦ by simpa only [Real.norm_eq_abs, abs_of_nonneg (hf0 j x)] using hfle j x)
+      (ae_of_all _ hflim)
+  apply le_of_tendsto hlim
+  apply Eventually.of_forall
+  intro j
+  let V := gradientDualTest u (ε j)
+  have hV : PerimeterAdmissible univ V := ⟨gradientDualTest_contDiff hu (hε j),
+    gradientDualTest_hasCompactSupport hcu _, subset_univ _, gradientDualTest_norm_le u (hε j)⟩
+  have heq : (∫ x, u x * divergence V x) = ∫ x, f j x := by
+    rw [integral_mul_divergence_eq_neg_inner_gradient hu hcu hV.1 hV.2.1, ← integral_neg]
+    apply integral_congr_ae
+    apply ae_of_all
+    intro x
+    simp only [V, gradientDualTest, inner_smul_right, real_inner_self_eq_norm_sq,
+      f, mul_neg, neg_neg, div_eq_mul_inv, mul_comm]
+  rw [← heq]
+  exact hbound V hV
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+open RPI.Boundary ContinuousLinearMap
+open scoped Convolution
+
+theorem indicator_one_integrable_compact {n : ℕ} {E : Set (Euc n)}
+    (hE : MeasurableSet E) (hb : Bornology.IsBounded E) :
+    Integrable (E.indicator (fun _ ↦ (1 : ℝ))) volume ∧
+      HasCompactSupport (E.indicator (fun _ ↦ (1 : ℝ))) := by
+  have hfin : volume E ≠ ⊤ := ne_top_of_le_ne_top hb.isCompact_closure.measure_ne_top
+    (measure_mono subset_closure)
+  refine ⟨(integrable_indicator_iff hE).2 (integrableOn_const hfin), ?_⟩
+  apply hb.isCompact_closure.of_isClosed_subset (isClosed_tsupport _)
+  apply closure_mono
+  intro x hx
+  by_contra hn
+  exact hx (indicator_of_notMem hn _)
+
+theorem c1Mollify_indicator_gradient_le_perimeter {n : ℕ} {E : Set (Euc n)}
+    (hE : MeasurableSet E) (hb : Bornology.IsBounded E) (hP : perimeter E univ ≠ ⊤)
+    (j : ℕ) :
+    (∫⁻ x, ‖fderiv ℝ (c1Mollify (E.indicator (fun _ ↦ (1 : ℝ))) j) x‖ₑ) ≤
+      perimeter E univ := by
+  let χ : Euc n → ℝ := E.indicator (fun _ ↦ 1)
+  let u := c1Mollify χ j
+  obtain ⟨hχ, hcχ⟩ := indicator_one_integrable_compact hE hb
+  have hu : ContDiff ℝ ∞ u :=
+    (c1Mollifier j).hasCompactSupport_normed.contDiff_convolution_left _
+      (c1Mollifier j).contDiff_normed hχ.locallyIntegrable
+  have hcu : HasCompactSupport u := c1Mollify_hasCompactSupport hcχ j
+  have hgrad : (∫ x, ‖gradient u x‖) ≤ (perimeter E univ).toReal := by
+    apply integral_norm_gradient_le_of_test_bound hu hcu
+    intro V hV
+    exact indicator_convolution_variation_le hE hP
+      (c1Mollifier j).continuous_normed
+      ((c1Mollifier j).continuous_normed.integrable_of_hasCompactSupport
+        (c1Mollifier j).hasCompactSupport_normed)
+      (c1Mollifier j).nonneg_normed (c1Mollifier j).integral_normed hV
+  have hnorm : ∀ x, ‖gradient u x‖ = ‖fderiv ℝ u x‖ := fun x ↦
+    (InnerProductSpace.toDual ℝ (Euc n)).symm.norm_map _
+  calc
+    (∫⁻ x, ‖fderiv ℝ u x‖ₑ) = ∫⁻ x, ENNReal.ofReal ‖gradient u x‖ := by
+      apply lintegral_congr
+      intro x
+      rw [hnorm, ofReal_norm]
+    _ = ENNReal.ofReal (∫ x, ‖gradient u x‖) :=
+      (ofReal_integral_eq_lintegral_ofReal
+        ((gradient_contDiff hu).continuous.integrable_of_hasCompactSupport
+          (gradient_hasCompactSupport hcu)).norm (ae_of_all _ fun x ↦ norm_nonneg _)).symm
+    _ ≤ ENNReal.ofReal (perimeter E univ).toReal := ENNReal.ofReal_le_ofReal hgrad
+    _ = perimeter E univ := ENNReal.ofReal_toReal hP
+
+theorem raw_perimeter_isoperimetric {n : ℕ} {p : ℝ}
+    (hp : Real.HolderConjugate (n : ℝ) p) {E : Set (Euc n)}
+    (hE : MeasurableSet E) (hb : Bornology.IsBounded E) :
+    volume E ≤ (lintegralPowLePowLIntegralFDerivConst (volume : Measure (Euc n)) p : ℝ≥0∞) *
+      (perimeter E univ) ^ p := by
+  let χ : Euc n → ℝ := E.indicator (fun _ ↦ 1)
+  obtain ⟨hχ, hcχ⟩ := indicator_one_integrable_compact hE hb
+  let u : ℕ → Euc n → ℝ := c1Mollify χ
+  have hu : ∀ j, ContDiff ℝ ∞ (u j) := fun j ↦
+    (c1Mollifier j).hasCompactSupport_normed.contDiff_convolution_left _
+      (c1Mollifier j).contDiff_normed hχ.locallyIntegrable
+  have hcu : ∀ j, HasCompactSupport (u j) := c1Mollify_hasCompactSupport hcχ
+  have hlim : ∀ᵐ x ∂volume, Tendsto (fun j ↦ u j x) atTop (𝓝 (χ x)) := by
+    apply ContDiffBump.ae_convolution_tendsto_right_of_locallyIntegrable
+      c1Mollifier_radius_tendsto (K := 2) _ hχ.locallyIntegrable
+    apply Eventually.of_forall
+    intro j
+    dsimp [c1Mollifier]
+    linarith
+  have hSob : ∀ j, (∫⁻ x, ‖u j x‖ₑ ^ p) ≤
+      (lintegralPowLePowLIntegralFDerivConst (volume : Measure (Euc n)) p : ℝ≥0∞) *
+        (perimeter E univ) ^ p := by
+    intro j
+    have hs := lintegral_pow_le_pow_lintegral_fderiv volume
+      ((hu j).of_le (by simp)) (hcu j) (by simpa using hp)
+    apply hs.trans
+    gcongr
+    · exact hp.symm.pos.le
+    by_cases hP : perimeter E univ = ⊤
+    · rw [hP]; exact le_top
+    · exact c1Mollify_indicator_gradient_le_perimeter hE hb hP j
+  have hlp : ∀ᵐ x ∂volume, Tendsto (fun j ↦ ‖u j x‖ₑ ^ p) atTop (𝓝 (‖χ x‖ₑ ^ p)) := by
+    filter_upwards [hlim] with x hx
+    exact ENNReal.continuous_rpow_const.continuousAt.tendsto.comp hx.enorm
+  calc
+    volume E = ∫⁻ x, ‖χ x‖ₑ ^ p := by
+      have heq : (fun x ↦ ‖χ x‖ₑ ^ p) = E.indicator (fun _ ↦ (1 : ℝ≥0∞)) := by
+        funext x
+        by_cases hx : x ∈ E
+        · simp [χ, hx]
+        · simp [χ, hx, ENNReal.zero_rpow_of_pos hp.symm.pos]
+      rw [heq, lintegral_indicator hE]
+      simp
+    _ = ∫⁻ x, liminf (fun j ↦ ‖u j x‖ₑ ^ p) atTop := by
+      apply lintegral_congr_ae
+      filter_upwards [hlp] with x hx using hx.liminf_eq.symm
+    _ ≤ liminf (fun j ↦ ∫⁻ x, ‖u j x‖ₑ ^ p) atTop :=
+      lintegral_liminf_le' (fun j ↦ ((hu j).continuous.measurable.enorm.pow_const p).aemeasurable)
+    _ ≤ (lintegralPowLePowLIntegralFDerivConst (volume : Measure (Euc n)) p : ℝ≥0∞) *
+        (perimeter E univ) ^ p := liminf_le_of_frequently_le (Eventually.of_forall hSob).frequently
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+
+theorem radial_volume_increment_le {n : ℕ} (E : Set (Euc n)) (c : Euc n)
+    {a b : ℝ} (ha : 0 ≤ a) (hab : a ≤ b) :
+    volume.real (E ∩ Metric.closedBall c b) - volume.real (E ∩ Metric.closedBall c a) ≤
+      (b ^ n - a ^ n) * volume.real (Metric.ball (0 : Euc n) 1) := by
+  have hf : volume (E ∩ Metric.closedBall c a) ≠ ⊤ :=
+    measure_ne_top_of_subset inter_subset_right (isCompact_closedBall c a).measure_ne_top
+  calc
+    _ ≤ volume.real ((E ∩ Metric.closedBall c b) \ (E ∩ Metric.closedBall c a)) :=
+      le_measureReal_sdiff hf
+    _ ≤ volume.real (Metric.closedBall c b \ Metric.closedBall c a) := by
+      apply measureReal_mono
+      · intro x hx
+        exact ⟨hx.1.2, fun h ↦ hx.2 ⟨hx.1.1, h⟩⟩
+      · exact measure_ne_top_of_subset sdiff_subset (isCompact_closedBall c b).measure_ne_top
+    _ = _ := volume_real_annulus c ha hab
+
+theorem radial_volume_abs_sub_le {n : ℕ} (E : Set (Euc n)) (c : Euc n)
+    {a b : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) :
+    |volume.real (E ∩ Metric.closedBall c b) - volume.real (E ∩ Metric.closedBall c a)| ≤
+      |b ^ n - a ^ n| * volume.real (Metric.ball (0 : Euc n) 1) := by
+  rcases le_total a b with hab | hba
+  · rw [abs_of_nonneg (sub_nonneg.mpr (measureReal_mono
+        (inter_subset_inter_right E (Metric.closedBall_subset_closedBall hab))
+        (measure_ne_top_of_subset inter_subset_right (isCompact_closedBall c b).measure_ne_top))),
+      abs_of_nonneg (sub_nonneg.mpr (pow_le_pow_left₀ ha hab n))]
+    exact radial_volume_increment_le E c ha hab
+  · rw [abs_sub_comm (volume.real _), abs_sub_comm (b ^ n)]
+    rw [abs_of_nonneg (sub_nonneg.mpr (measureReal_mono
+        (inter_subset_inter_right E (Metric.closedBall_subset_closedBall hba))
+        (measure_ne_top_of_subset inter_subset_right (isCompact_closedBall c a).measure_ne_top))),
+      abs_of_nonneg (sub_nonneg.mpr (pow_le_pow_left₀ hb hba n))]
+    exact radial_volume_increment_le E c hb hba
+
+theorem radial_volume_continuousAt {n : ℕ} (E : Set (Euc n)) (c : Euc n)
+    {r : ℝ} (hr : 0 < r) :
+    ContinuousAt (fun s : ℝ ↦ volume.real (E ∩ Metric.closedBall c s)) r := by
+  rw [ContinuousAt, ← tendsto_sub_nhds_zero_iff]
+  apply squeeze_zero_norm' (a := fun s : ℝ ↦
+    |s ^ n - r ^ n| * volume.real (Metric.ball (0 : Euc n) 1))
+  · filter_upwards [lt_mem_nhds hr] with s hs
+    simpa only [Real.norm_eq_abs] using radial_volume_abs_sub_le E c hr.le hs.le
+  · have ht : ContinuousAt (fun s : ℝ ↦
+        |s ^ n - r ^ n| * volume.real (Metric.ball (0 : Euc n) 1)) r := by fun_prop
+    simpa using ht.tendsto
+
+theorem exists_controlled_cut_radius {n : ℕ} (E : Set (Euc n)) (c : Euc n)
+    {a b D : ℝ} (ha : 0 < a) (hab : a < b)
+    (hD : volume.real (E ∩ Metric.closedBall c b) < D * (b - a)) :
+    ∃ r ∈ Ico a b, ∀ t ∈ Icc r b,
+      volume.real (E ∩ Metric.closedBall c t) - volume.real (E ∩ Metric.closedBall c r) ≤
+        D * (t - r) := by
+  let F : ℝ → ℝ := fun s ↦ volume.real (E ∩ Metric.closedBall c s)
+  have hcont : ContinuousOn (fun s ↦ F s - D * s) (Icc a b) := by
+    intro s hs
+    exact ((radial_volume_continuousAt E c (ha.trans_le hs.1)).sub
+      (continuousAt_const.mul continuousAt_id)).continuousWithinAt
+  obtain ⟨r, hr, hmax⟩ := isCompact_Icc.exists_isMaxOn (nonempty_Icc.mpr hab.le) hcont
+  have hrb : r < b := by
+    by_contra hh
+    have heq : r = b := le_antisymm hr.2 (not_lt.mp hh)
+    subst r
+    have hc := hmax (show a ∈ Icc a b from ⟨le_rfl, hab.le⟩)
+    have hn : 0 ≤ F a := ENNReal.toReal_nonneg
+    dsimp [F] at hc hn
+    nlinarith
+  refine ⟨r, ⟨hr.1, hrb⟩, ?_⟩
+  intro t ht
+  have hc := hmax (show t ∈ Icc a b from ⟨hr.1.trans ht.1, ht.2⟩)
+  dsimp [F] at hc
+  nlinarith
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+
+theorem integral_gradient_annularCutoff_in_set_le {n : ℕ} (E : Set (Euc n))
+    (c : Euc n) {a b L : ℝ} (ha : 0 < a) (hab : a < b) (hL : 0 < L)
+    (hbound : ∀ s, |deriv Real.smoothTransition s| ≤ L) :
+    (∫ x in E, ‖gradient (annularCutoff c a b) x‖) ≤
+      (2 * L * b / (b ^ 2 - a ^ 2)) *
+        volume.real (E ∩ (Metric.closedBall c b \ Metric.closedBall c a)) := by
+  let A := Metric.closedBall c b \ Metric.closedBall c a
+  have hid : (∫ x in A, ‖gradient (annularCutoff c a b) x‖ ∂volume.restrict E) =
+      ∫ x in E, ‖gradient (annularCutoff c a b) x‖ := by
+    apply setIntegral_eq_integral_of_forall_compl_eq_zero
+    intro x hx
+    rw [gradient_annularCutoff_zero ha hab, norm_zero]
+    by_cases hh : x ∈ Metric.closedBall c a
+    · exact Or.inl hh
+    · exact Or.inr (le_of_lt (lt_of_not_ge (fun h ↦ hx ⟨h, hh⟩)))
+  rw [← hid, Measure.restrict_restrict (measurableSet_closedBall.diff measurableSet_closedBall)]
+  calc
+    _ ≤ ‖∫ x in A ∩ E, ‖gradient (annularCutoff c a b) x‖‖ := le_abs_self _
+    _ ≤ (2 * L * b / (b ^ 2 - a ^ 2)) * volume.real (A ∩ E) := by
+      apply norm_setIntegral_le_of_norm_le_const
+        ((measure_mono (inter_subset_left.trans sdiff_subset)).trans_lt
+          (isCompact_closedBall c b).measure_lt_top)
+      intro x hx
+      simpa only [norm_norm] using gradient_annularCutoff_bound ha hab hL hbound x
+    _ = _ := by rw [inter_comm]
+
+theorem controlled_radius_gradient_bound {n : ℕ} {E : Set (Euc n)}
+    (hE : MeasurableSet E) (c : Euc n) {r B D L δ : ℝ}
+    (hr : 0 < r) (hδ : 0 < δ) (hδB : r + 2 * δ ≤ B)
+    (hcontrol : ∀ t ∈ Icc r B,
+      volume.real (E ∩ Metric.closedBall c t) - volume.real (E ∩ Metric.closedBall c r) ≤
+        D * (t - r))
+    (hD : 0 ≤ D) (hL : 0 < L)
+    (hbound : ∀ s, |deriv Real.smoothTransition s| ≤ L) :
+    (∫ x in E, ‖gradient (annularCutoff c (r + δ) (r + 2 * δ)) x‖) ≤ 4 * L * D := by
+  have ha : 0 < r + δ := by positivity
+  have hab : r + δ < r + 2 * δ := by linarith
+  have hden : 0 < (r + 2 * δ) ^ 2 - (r + δ) ^ 2 := by nlinarith
+  have hv : volume.real (E ∩ (Metric.closedBall c (r + 2 * δ) \ Metric.closedBall c (r + δ))) ≤
+      2 * D * δ := by
+    calc
+      _ ≤ volume.real ((E ∩ Metric.closedBall c (r + 2 * δ)) \ (E ∩ Metric.closedBall c r)) := by
+        apply measureReal_mono
+        · intro x hx
+          refine ⟨⟨hx.1, hx.2.1⟩, ?_⟩
+          intro hh
+          exact hx.2.2 (Metric.closedBall_subset_closedBall (by linarith : r ≤ r + δ) hh.2)
+        · exact measure_ne_top_of_subset (sdiff_subset.trans inter_subset_right)
+            (isCompact_closedBall c (r + 2 * δ)).measure_ne_top
+      _ = volume.real (E ∩ Metric.closedBall c (r + 2 * δ)) -
+          volume.real (E ∩ Metric.closedBall c r) := by
+        exact measureReal_sdiff
+          (inter_subset_inter_right E (Metric.closedBall_subset_closedBall (by linarith)))
+          (hE.inter measurableSet_closedBall)
+          (measure_ne_top_of_subset inter_subset_right
+            (isCompact_closedBall c (r + 2 * δ)).measure_ne_top)
+      _ ≤ D * ((r + 2 * δ) - r) := hcontrol _ ⟨by linarith, hδB⟩
+      _ = _ := by ring
+  have hcoef : 2 * L * (r + 2 * δ) / ((r + 2 * δ) ^ 2 - (r + δ) ^ 2) ≤
+      2 * L / δ := by
+    rw [div_le_div_iff₀ hden hδ]
+    nlinarith [mul_pos hL (mul_pos ha hδ)]
+  calc
+    _ ≤ (2 * L * (r + 2 * δ) / ((r + 2 * δ) ^ 2 - (r + δ) ^ 2)) *
+        volume.real (E ∩ (Metric.closedBall c (r + 2 * δ) \ Metric.closedBall c (r + δ))) :=
+      integral_gradient_annularCutoff_in_set_le E c ha hab hL hbound
+    _ ≤ (2 * L / δ) * (2 * D * δ) :=
+      (mul_le_mul_of_nonneg_left hv (by positivity)).trans
+        (mul_le_mul_of_nonneg_right hcoef (mul_nonneg (mul_nonneg (by norm_num) hD) hδ.le))
+    _ = _ := by field_simp; ring
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/GraphChart.lean b/lean/OAI/Geometry/Perimeter/Regularity/GraphChart.lean
new file mode 100644
index 0000000000000000000000000000000000000000..ed969215e5f3d5d933929628a9a4ffdb4845de4b
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/GraphChart.lean
@@ -0,0 +1,128 @@
+import OAI.Geometry.Perimeter.Regularity.GraphPhase
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+def graphCylinder (v : Euc n) : Set (Euc n) :=
+  {x | ‖normalProjection v x‖<1/4 ∧ |inner ℝ v x|<1/2}
+
+def graphDefining (v : Euc n) (f : normalPlane v → ℝ) (x : Euc n) : ℝ :=
+  inner ℝ v x-f (normalProjection v x)
+
+theorem graphCylinder_open (v : Euc n) : IsOpen (graphCylinder v) :=
+  (isOpen_lt (normalProjection v).continuous.norm continuous_const).inter
+    (isOpen_lt (show Continuous (fun x : Euc n ↦ |inner ℝ v x|) from
+      ((innerSL ℝ v).continuous).abs) continuous_const)
+
+@[simp] theorem graphDefining_fiber {v : Euc n} (hv : ‖v‖=1)
+    (f : normalPlane v → ℝ) (y : normalPlane v) (t : ℝ) :
+    graphDefining v f ((y:Euc n)+t•v)=t-f y := by
+  simp only [graphDefining,normalProjection_fiber,normal_fiber_inner hv]
+
+@[simp] theorem graphCylinder_fiber {v : Euc n} (hv : ‖v‖=1)
+    (y : normalPlane v) (t : ℝ) :
+    (y:Euc n)+t•v∈graphCylinder v ↔ ‖y‖<1/4 ∧ |t|<1/2 := by
+  simp only [graphCylinder,mem_ofPred_eq,normalProjection_fiber,normal_fiber_inner hv]
+
+theorem graphCylinder_norm_lt_one {v : Euc n} (hv : ‖v‖=1)
+    {x : Euc n} (hx : x∈graphCylinder v) : ‖x‖<1 := by
+  obtain ⟨⟨y,t⟩,rfl⟩ := (normalCoordinates v hv).surjective x
+  simp only [normalCoordinates_apply,graphCylinder_fiber hv] at hx ⊢
+  have hh := norm_add_le (y:Euc n) (t•v)
+  rw [norm_smul,Real.norm_eq_abs,hv,mul_one] at hh
+  rw [Submodule.norm_coe] at hh
+  linarith [hx.1,hx.2]
+
+theorem graphDefining_contDiffOn {v : Euc n} {f : normalPlane v → ℝ} {k : ℕ∞ω}
+    (hf : ContDiffOn ℝ k f (Metric.ball 0 (1/4))) :
+    ContDiffOn ℝ k (graphDefining v f) (graphCylinder v) := by
+  apply (contDiff_const.inner ℝ contDiff_id).contDiffOn.sub
+  exact hf.comp (normalProjection v).contDiff.contDiffOn (fun x hx ↦ by
+    simpa only [Metric.mem_ball,dist_zero_right] using hx.1)
+
+theorem graphDefining_hasFDerivAt {v : Euc n} {f : normalPlane v → ℝ}
+    (hf : ContDiffOn ℝ 1 f (Metric.ball 0 (1/4))) {x : Euc n}
+    (hx : x∈graphCylinder v) :
+    HasFDerivAt (graphDefining v f)
+      ((innerSL ℝ v)-(fderiv ℝ f (normalProjection v x)).comp (normalProjection v)) x := by
+  have hy : normalProjection v x∈Metric.ball 0 (1/4) := by
+    simpa only [Metric.mem_ball,dist_zero_right] using hx.1
+  have hfd : DifferentiableAt ℝ f (normalProjection v x) :=
+    (hf.contDiffAt (Metric.isOpen_ball.mem_nhds hy)).differentiableAt (by norm_num)
+  exact (innerSL ℝ v).hasFDerivAt.sub (hfd.hasFDerivAt.comp x (normalProjection v).hasFDerivAt)
+
+theorem graphDefining_fderiv_normal {v : Euc n} (hv : ‖v‖=1)
+    {f : normalPlane v → ℝ} (hf : ContDiffOn ℝ 1 f (Metric.ball 0 (1/4)))
+    {x : Euc n} (hx : x∈graphCylinder v) : fderiv ℝ (graphDefining v f) x v=1 := by
+  rw [(graphDefining_hasFDerivAt hf hx).fderiv]
+  have hvp : normalProjection v v=0 := by
+    simpa only [Submodule.coe_zero,zero_add,one_smul] using normalProjection_fiber v 0 1
+  simp only [sub_apply,ContinuousLinearMap.comp_apply,hvp,map_zero,sub_zero]
+  change inner ℝ v v=1
+  rw [real_inner_self_eq_norm_sq,hv,one_pow]
+
+theorem graphDefining_fderiv_ne_zero {v : Euc n} (hv : ‖v‖=1)
+    {f : normalPlane v → ℝ} (hf : ContDiffOn ℝ 1 f (Metric.ball 0 (1/4)))
+    {x : Euc n} (hx : x∈graphCylinder v) : fderiv ℝ (graphDefining v f) x≠0 := by
+  intro h
+  have hh := graphDefining_fderiv_normal hv hf hx
+  rw [h,zero_apply] at hh
+  norm_num at hh
+
+theorem graphDefining_support_iff {v : Euc n} (hv : ‖v‖=1) {E : Set (Euc n)}
+    {f : normalPlane v → ℝ}
+    (hiff : ∀ (y : normalPlane v) (t : ℝ), ‖y‖<1/4 → ‖(y:Euc n)+t•v‖<1 →
+      ((y:Euc n)+t•v∈perimeterSupport E ↔ t=f y))
+    {x : Euc n} (hx : x∈graphCylinder v) : x∈perimeterSupport E ↔ graphDefining v f x=0 := by
+  obtain ⟨⟨y,t⟩,rfl⟩ := (normalCoordinates v hv).surjective x
+  simp only [normalCoordinates_apply] at hx ⊢
+  have hh := (graphCylinder_fiber hv y t).mp hx
+  rw [graphDefining_fiber hv,sub_eq_zero]
+  exact hiff y t hh.1 (graphCylinder_norm_lt_one hv hx)
+
+theorem graphDefining_phaseSupport_iff (hn : 1<n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    {v : Euc n} (hv : ‖v‖=1) {a : ℝ} (ha : 0≤a) (ha4 : a<1/4)
+    (hs : FullSlab E v 8 a) {f : normalPlane v → ℝ}
+    (hiff : ∀ (y : normalPlane v) (t : ℝ), ‖y‖<1/4 → ‖(y:Euc n)+t•v‖<1 →
+      ((y:Euc n)+t•v∈perimeterSupport E ↔ t=f y))
+    {x : Euc n} (hx : x∈graphCylinder v) :
+    x∈(volume.restrict E).support ↔ graphDefining v f x≤0 := by
+  obtain ⟨⟨y,t⟩,rfl⟩ := (normalCoordinates v hv).surjective x
+  simp only [normalCoordinates_apply] at hx ⊢
+  have hh := (graphCylinder_fiber hv y t).mp hx
+  rw [graphDefining_fiber hv,sub_nonpos]
+  exact full_graph_phaseSupport_iff hn hmin hv ha ha4 hs hiff y hh.1 t hh.2
+
+theorem graphDefining_approximateNormal (hn : 1<n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    {v : Euc n} (hv : ‖v‖=1) {a : ℝ} (ha : 0≤a) (ha4 : a<1/4)
+    (hs : FullSlab E v 8 a) {f : normalPlane v → ℝ}
+    (hf : ContDiffOn ℝ 1 f (Metric.ball 0 (1/4)))
+    (hiff : ∀ (y : normalPlane v) (t : ℝ), ‖y‖<1/4 → ‖(y:Euc n)+t•v‖<1 →
+      ((y:Euc n)+t•v∈perimeterSupport E ↔ t=f y))
+    {x : Euc n} (hx : x∈graphCylinder v) (hxS : x∈perimeterSupport E) :
+    approximateNormal E x=‖gradient (graphDefining v f) x‖⁻¹ • gradient (graphDefining v f) x := by
+  let K := (volume.restrict E).support
+  have hlocal : ∀ᶠ z in 𝓝 x, (graphDefining v f z<0 → z∈K) ∧
+      (z∈K → graphDefining v f z≤0) := by
+    filter_upwards [(graphCylinder_open v).mem_nhds hx] with z hz
+    have hh := graphDefining_phaseSupport_iff hn hmin hv ha ha4 hs hiff hz
+    exact ⟨fun h ↦ hh.mpr h.le,hh.mp⟩
+  have hbl := hasHalfspaceBlowup_gradient_filled Measure.isClosed_support.measurableSet
+    (graphDefining_hasFDerivAt hf hx).differentiableAt
+    ((graphDefining_support_iff hv hiff hx).mp hxS) (graphDefining_fderiv_ne_zero hv hf hx) hlocal
+  have hae := minimizing_ae_eq_phaseSupport
+    (Real.HolderConjugate.conjExponent (by exact_mod_cast hn : 1<(n:ℝ))) hmin
+  exact approximateNormal_eq_of_blowup hmin.1 ((hasHalfspaceBlowup_iff_of_ae hae x _).mpr hbl)
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/GraphCoordinates.lean b/lean/OAI/Geometry/Perimeter/Regularity/GraphCoordinates.lean
new file mode 100644
index 0000000000000000000000000000000000000000..aabbbbe38082f889e6c9fee6f0c100d17d2307be
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/GraphCoordinates.lean
@@ -0,0 +1,76 @@
+import OAI.Geometry.Perimeter.Regularity.EllipticLocality
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+
+section
+variable {V W : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [CompleteSpace V]
+  [NormedAddCommGroup W] [InnerProductSpace ℝ W] [CompleteSpace W]
+
+theorem gradient_comp_isometry (e : V ≃ₗᵢ[ℝ] W) (f : W → ℝ) (x : V) :
+    gradient (f ∘ e) x=e.symm (gradient f (e x)) := by
+  apply ext_inner_right ℝ
+  intro y
+  rw [inner_gradient_left]
+  have he : inner ℝ (e.symm (gradient f (e x))) y=inner ℝ (gradient f (e x)) (e y) := by
+    rw [←e.inner_map_map,e.apply_symm_apply]
+  rw [he,inner_gradient_left]
+  have hd := e.toContinuousLinearEquiv.comp_right_fderiv (f := f) (x := x)
+  exact congrArg (fun L : V →L[ℝ] ℝ ↦ L y) hd
+
+omit [CompleteSpace V] [CompleteSpace W] in
+theorem GeometricHolderAt.precomp_isometry [CompleteSpace V] [CompleteSpace W]
+    {X : Type*} [PseudoMetricSpace X]
+    {θ : ℝ} {f : W → X} {p : V} (e : V ≃ₗᵢ[ℝ] W)
+    (h : GeometricHolderAt θ f (e p)) : GeometricHolderAt θ (f ∘ e) p := by
+  obtain ⟨r,C,hr,hC,hh⟩ := h
+  refine ⟨r,C,hr,hC,fun j x hx y hy hxy ↦ hh j (e x) ?_ (e y) ?_ ?_⟩
+  · simpa only [Metric.mem_ball,e.dist_map] using hx
+  · simpa only [Metric.mem_ball,e.dist_map] using hy
+  · simpa only [e.dist_map] using hxy
+
+end
+
+section Transfer
+variable {m : ℕ} {W : Type*} [NormedAddCommGroup W] [InnerProductSpace ℝ W]
+  [FiniteDimensional ℝ W] [MeasurableSpace W] [BorelSpace W]
+
+theorem minimal_equation_isometry (e : Euc m ≃ₗᵢ[ℝ] W) {f : W → ℝ} {U : Set W}
+    (h : ∀ ψ : W → ℝ, ContDiff ℝ ∞ ψ → HasCompactSupport ψ → tsupport ψ ⊆ U →
+      (∫ y, inner ℝ (gradient f y) (gradient ψ y)/Real.sqrt (1+‖gradient f y‖^2))=0)
+    (φ : SmoothLocalTest (e ⁻¹' U)) :
+    (∫ x, inner ℝ (gradient (f ∘ e) x) (gradient φ.val x)/
+      Real.sqrt (1+‖gradient (f ∘ e) x‖^2))=0 := by
+  let ψ : W → ℝ := φ.val ∘ e.symm
+  have hψ : ContDiff ℝ ∞ ψ := φ.property.1.comp e.symm.contDiff
+  have hcψ : HasCompactSupport ψ := φ.property.2.1.comp_homeomorph e.symm.toHomeomorph
+  have hsψ : tsupport ψ ⊆ U := by
+    intro y hy
+    have hh := φ.property.2.2 (tsupport_comp_subset_preimage φ.val e.symm.continuous hy)
+    simpa only [mem_preimage,e.apply_symm_apply] using hh
+  have hh := h ψ hψ hcψ hsψ
+  have hi := e.measurePreserving.integral_comp e.toHomeomorph.measurableEmbedding
+    (fun y ↦ inner ℝ (gradient f y) (gradient ψ y)/Real.sqrt (1+‖gradient f y‖^2))
+  rw [←hi] at hh
+  convert hh using 1
+  apply integral_congr_ae
+  filter_upwards [] with x
+  rw [gradient_comp_isometry,e.symm.norm_map]
+  have heg : gradient ψ (e x)=e (gradient φ.val x) := by
+    dsimp [ψ]
+    rw [gradient_comp_isometry,e.symm_apply_apply]
+    rfl
+  rw [heg]
+  congr 1
+  rw [←e.inner_map_map,e.apply_symm_apply]
+
+end Transfer
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/GraphDerivative.lean b/lean/OAI/Geometry/Perimeter/Regularity/GraphDerivative.lean
new file mode 100644
index 0000000000000000000000000000000000000000..0a4b33499e92967b1df7558b9a1706bdf4f9fc0b
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/GraphDerivative.lean
@@ -0,0 +1,75 @@
+import OAI.Geometry.Perimeter.Regularity.SingleGraph
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+def graphTangentLinear (v w : Euc n) : normalPlane v →L[ℝ] ℝ :=
+  (-(inner ℝ w v)⁻¹) • ((innerSL ℝ w).comp (normalPlane v).subtypeL)
+
+@[simp] theorem graphTangentLinear_apply (v w : Euc n) (y : normalPlane v) :
+    graphTangentLinear v w y= -(inner ℝ w v)⁻¹ * inner ℝ w (y:Euc n) := rfl
+
+theorem inner_near_unit_lower {v w : Euc n} (hv : ‖v‖=1) (hwv : ‖w-v‖≤1/2) :
+    1/2 ≤ inner ℝ w v := by
+  have hh := abs_real_inner_le_norm (w-v) v
+  rw [inner_sub_left,real_inner_self_eq_norm_sq,hv,mul_one] at hh
+  norm_num only [one_pow] at hh
+  have hlo := (abs_le.mp hh).1
+  linarith
+
+theorem graph_tangent_remainder {E : Set (Euc n)} {v w : Euc n} (hv : ‖v‖=1)
+    {f : normalPlane v → ℝ} {y z : normalPlane v} {C q : ℝ} {j : ℕ}
+    (hwv : ‖w-v‖≤1/2)
+    (hzS : (z:Euc n)+f z•v∈perimeterSupport E)
+    (hlip : |f z-f y|≤‖z-y‖) (hzq : ‖z-y‖≤q^j)
+    (htan : ∀ x, x∈perimeterSupport E →
+      ‖x-((y:Euc n)+f y•v)‖≤2*q^j →
+      |inner ℝ w (x-((y:Euc n)+f y•v))|≤C*(1/2)^j*‖x-((y:Euc n)+f y•v)‖)
+    (hC : 0≤C) :
+    |f z-f y-graphTangentLinear v w (z-y)|≤4*C*(1/2)^j*‖z-y‖ := by
+  let d := inner ℝ w v
+  have hd : 1/2≤d := inner_near_unit_lower hv hwv
+  have hdp : 0<d := by linarith
+  have hnorm := fiber_chord_norm hv z y (f z) (f y)
+  have hn : ‖((z:Euc n)+f z•v)-((y:Euc n)+f y•v)‖≤2*‖z-y‖ := by linarith
+  have hh := htan _ hzS (hn.trans (mul_le_mul_of_nonneg_left hzq (by norm_num)))
+  have he : inner ℝ w (((z:Euc n)+f z•v)-((y:Euc n)+f y•v))=
+      d*(f z-f y-graphTangentLinear v w (z-y)) := by
+    simp only [graphTangentLinear_apply,Submodule.coe_sub,inner_sub_right,inner_add_right,
+      real_inner_smul_right]
+    dsimp [d]
+    field_simp [show inner ℝ w v ≠ 0 from hdp.ne']
+    ring
+  rw [he,abs_mul,abs_of_pos hdp] at hh
+  have hm := mul_le_mul_of_nonneg_left hn (show 0≤C*(1/2:ℝ)^j by positivity)
+  nlinarith [abs_nonneg (f z-f y-graphTangentLinear v w (z-y))]
+
+theorem hasFDerivAt_of_geometric_remainder {V : Type*} [NormedAddCommGroup V]
+    [NormedSpace ℝ V] {f : V → ℝ} {D : V →L[ℝ] ℝ} {U : Set V} (hU : IsOpen U)
+    {y : V} (hy : y∈U) {q C : ℝ} (hq : 0<q)
+    (hrem : ∀ j z, z∈U → ‖z-y‖≤q^j →
+      |f z-f y-D (z-y)|≤C*(1/2)^j*‖z-y‖) : HasFDerivAt f D y := by
+  rw [hasFDerivAt_iff_isLittleO,Asymptotics.isLittleO_iff]
+  intro ε hε
+  have ht : Tendsto (fun j : ℕ ↦ C*(1/2:ℝ)^j) atTop (𝓝 0) := by
+    simpa only [mul_zero] using
+      (tendsto_pow_atTop_nhds_zero_of_lt_one (by norm_num : (0:ℝ)≤1/2) (by norm_num)).const_mul C
+  obtain ⟨j,hj⟩ := (ht.eventually (gt_mem_nhds hε)).exists
+  filter_upwards [hU.mem_nhds hy,Metric.ball_mem_nhds y (pow_pos hq j)] with z hz hzB
+  have hzy : ‖z-y‖≤q^j := by
+    have hh : ‖z-y‖<q^j := by simpa only [Metric.mem_ball,dist_eq_norm] using hzB
+    exact hh.le
+  exact (show ‖f z-f y-D (z-y)‖≤C*(1/2)^j*‖z-y‖ by
+    simpa only [Real.norm_eq_abs] using hrem j z hz hzy).trans
+      (mul_le_mul_of_nonneg_right hj.le (norm_nonneg _))
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/GraphExistence.lean b/lean/OAI/Geometry/Perimeter/Regularity/GraphExistence.lean
new file mode 100644
index 0000000000000000000000000000000000000000..4d001a10b628c183881ab774c65d1ec6956a5622
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/GraphExistence.lean
@@ -0,0 +1,105 @@
+import OAI.Geometry.Perimeter.Regularity.FlatnessImprovement
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+theorem connected_crosses_frontier {X : Type*} [TopologicalSpace X]
+    {S K : Set X} (hS : IsPreconnected S) (hK : IsClosed K)
+    {x y : X} (hx : x∈S) (hxK : x∈K) (hy : y∈S) (hyK : y∉K) :
+    (S ∩ frontier K).Nonempty := by
+  by_contra hn
+  have hsub : S ⊆ interior K ∪ Kᶜ := by
+    intro z hz
+    by_cases hzK : z∈K
+    · left
+      apply (mem_interior_iff_notMem_frontier hzK).mpr
+      intro hzf
+      exact hn ⟨z,hz,hzf⟩
+    · exact Or.inr hzK
+  have hd : Disjoint (interior K) Kᶜ :=
+    Set.disjoint_left.mpr fun z hz hzc ↦ hzc (interior_subset hz)
+  rcases hS.subset_or_subset isOpen_interior hK.isOpen_compl hd hsub with h | h
+  · exact hyK (interior_subset (h hy))
+  · exact h hx hxK
+
+theorem FullSlab.lower_phaseSupport {E : Set (Euc n)} {v : Euc n} {R a : ℝ}
+    (hs : FullSlab E v R a) (ha : 0≤a) {x : Euc n}
+    (hx : x∈Metric.ball 0 R) (hxi : inner ℝ v x < -a) :
+    x∈(volume.restrict E).support := by
+  let U : Set (Euc n) := Metric.ball 0 R ∩ {z | inner ℝ v z < -a}
+  have hU : IsOpen U := Metric.isOpen_ball.inter (isOpen_lt (continuous_const.inner continuous_id) continuous_const)
+  have hUE : U ≤ᵐ[volume] E := by
+    filter_upwards [hs.2] with z hz
+    intro hzU
+    have hi : inner ℝ v z < 0 := lt_of_lt_of_le hzU.2 (neg_nonpos.mpr ha)
+    apply (hz hzU.1 ?_).mpr hi
+    rw [abs_of_neg hi]
+    have hzi : inner ℝ v z < -a := hzU.2
+    linarith
+  apply Measure.support_mono (Measure.restrict_mono_ae hUE)
+  apply Measure.interior_inter_support
+  exact ⟨hU.interior_eq.symm ▸ ⟨hx,hxi⟩,by rw [Measure.support_eq_univ]; trivial⟩
+
+theorem FullSlab.upper_not_phaseSupport {E : Set (Euc n)} {v : Euc n} {R a : ℝ}
+    (hs : FullSlab E v R a) (ha : 0≤a) {x : Euc n}
+    (hx : x∈Metric.ball 0 R) (hxi : a < inner ℝ v x) :
+    x∉(volume.restrict E).support := by
+  let U : Set (Euc n) := Metric.ball 0 R ∩ {z | a < inner ℝ v z}
+  have hU : IsOpen U := Metric.isOpen_ball.inter (isOpen_lt continuous_const (continuous_const.inner continuous_id))
+  have hUE : ∀ᵐ z ∂volume, z∉U∩E := by
+    filter_upwards [hs.2] with z hz
+    rintro ⟨hzU,hzE⟩
+    have hi : 0 < inner ℝ v z := lt_of_le_of_lt ha hzU.2
+    have he := hz hzU.1 (by rw [abs_of_pos hi]; exact hzU.2)
+    exact (not_lt_of_ge hi.le) (he.mp hzE)
+  have hzero : (volume.restrict E) U=0 := by
+    rw [Measure.restrict_apply hU.measurableSet]
+    simpa only [not_not, ofPred_mem_eq] using ae_iff.mp hUE
+  exact Measure.subset_compl_support_of_isOpen hU hzero ⟨hx,hxi⟩
+
+theorem FullSlab.central_fiber_meets_support (hn : 1<n) {E : Set (Euc n)}
+    (hmin : IsLocallyPerimeterMinimizing E) {v : Euc n} (hv : ‖v‖=1)
+    {a : ℝ} (ha : 0≤a) (ha4 : a<1/4) (hs : FullSlab E v 8 a)
+    (y : normalPlane v) (hy : ‖y‖<1/4) :
+    ∃ t : ℝ, |t|≤a ∧ (y:Euc n)+t•v ∈ perimeterSupport E ∩ Metric.ball 0 1 := by
+  let f : ℝ → Euc n := fun t ↦ (y:Euc n)+t•v
+  have hf : Continuous f := continuous_const.add (continuous_id.smul continuous_const)
+  have hnorm (t : ℝ) (ht : t∈Icc (-(1/4)) (1/4)) : ‖f t‖<1 := by
+    have htn : |t|≤1/4 := abs_le.mpr ht
+    have hh := norm_add_le (y:Euc n) (t•v)
+    rw [norm_smul,hv,mul_one,Real.norm_eq_abs] at hh
+    change ‖(y:Euc n)+t•v‖<1
+    change ‖(y:Euc n)‖<1/4 at hy
+    linarith
+  have hball (t : ℝ) (ht : t∈Icc (-(1/4)) (1/4)) : f t∈Metric.ball 0 8 := by
+    rw [Metric.mem_ball,dist_zero_right]
+    linarith [hnorm t ht]
+  have hleft : f (-(1/4))∈(volume.restrict E).support :=
+    hs.lower_phaseSupport ha (hball _ (by norm_num)) (by
+      change inner ℝ v ((y:Euc n)+ (-(1/4):ℝ)•v) < -a
+      rw [normal_fiber_inner hv]; linarith)
+  have hright : f (1/4)∉(volume.restrict E).support :=
+    hs.upper_not_phaseSupport ha (hball _ (by norm_num)) (by
+      change a < inner ℝ v ((y:Euc n)+(1/4:ℝ)•v)
+      rw [normal_fiber_inner hv]; exact ha4)
+  obtain ⟨z,⟨t,ht,rfl⟩,hz⟩ := connected_crosses_frontier
+    (isPreconnected_Icc.image f hf.continuousOn) Measure.isClosed_support
+    (mem_image_of_mem f (by norm_num : (-(1/4):ℝ)∈Icc (-(1/4)) (1/4))) hleft
+    (mem_image_of_mem f (by norm_num : (1/4:ℝ)∈Icc (-(1/4)) (1/4))) hright
+  rw [minimizing_frontier_phaseSupport
+    (Real.HolderConjugate.conjExponent (by exact_mod_cast hn : 1<(n:ℝ))) hmin] at hz
+  have hsmall : f t∈Metric.ball 0 1 := by simpa only [Metric.mem_ball,dist_zero_right] using hnorm t ht
+  refine ⟨t,?_,hz,hsmall⟩
+  have hh := hs.1 (f t) ⟨hz,hball t ht⟩
+  simpa only [f,normal_fiber_inner hv] using hh
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/GraphFluxWeight.lean b/lean/OAI/Geometry/Perimeter/Regularity/GraphFluxWeight.lean
new file mode 100644
index 0000000000000000000000000000000000000000..2422a4f955420308ffcb70606d576b9bcdf7c591
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/GraphFluxWeight.lean
@@ -0,0 +1,67 @@
+import OAI.Geometry.Perimeter.Regularity.GraphNormal
+import OAI.Geometry.Perimeter.Regularity.GraphTestLift
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+def graphFluxWeight (v : Euc n) (f ψ : normalPlane v → ℝ) (y : normalPlane v) : ℝ :=
+  inner ℝ (gradient f y) (gradient ψ y)/graphAreaFactor v f y
+
+theorem graphFluxWeight_zero {v : Euc n} {f ψ : normalPlane v → ℝ}
+    {y : normalPlane v} (hy : y∉tsupport ψ) : graphFluxWeight v f ψ y=0 := by
+  simp [graphFluxWeight,gradient,fderiv_of_notMem_tsupport ℝ hy]
+
+theorem graphFluxWeight_tsupport {v : Euc n} (f ψ : normalPlane v → ℝ) :
+    tsupport (graphFluxWeight v f ψ) ⊆ tsupport ψ := by
+  apply closure_minimal _ (isClosed_tsupport ψ)
+  intro y hy
+  by_contra hn
+  exact hy (graphFluxWeight_zero hn)
+
+theorem graphFluxWeight_continuous {v : Euc n} (hv : ‖v‖=1)
+    {f ψ : normalPlane v → ℝ} (hf : ContDiffOn ℝ 1 f (Metric.ball 0 (1/4)))
+    (hψ : ContDiff ℝ ∞ ψ) (hsψ : tsupport ψ ⊆ Metric.ball 0 (1/4)) :
+    Continuous (graphFluxWeight v f ψ) := by
+  apply continuous_iff_continuousAt.mpr
+  intro y
+  by_cases hy : y∈tsupport ψ
+  · have hgf : ContinuousAt (gradient f) y :=
+      (InnerProductSpace.toDual ℝ (normalPlane v)).symm.continuous.continuousAt.comp
+        ((hf.contDiffAt (Metric.isOpen_ball.mem_nhds (hsψ hy))).continuousAt_fderiv (by norm_num))
+    have hgp : ContinuousAt (gradient ψ) y :=
+      (InnerProductSpace.toDual ℝ (normalPlane v)).symm.continuous.continuousAt.comp
+        (hψ.contDiffAt.continuousAt_fderiv (by simp))
+    have hgq : ContinuousAt (graphAreaFactor v f) y := by
+      exact (continuousAt_const.sub ((normalPlane v).subtypeL.continuous.continuousAt.comp hgf)).norm
+    exact (hgf.inner hgp).div hgq (graphAreaFactor_pos hv f y).ne'
+  · apply continuousAt_const.congr_of_eventuallyEq
+    filter_upwards [(isClosed_tsupport ψ).isOpen_compl.mem_nhds hy] with z hz
+    exact graphFluxWeight_zero hz
+
+theorem graphTest_tangential_flux {v : Euc n} (hv : ‖v‖=1)
+    {E : Set (Euc n)} {f ψ : normalPlane v → ℝ} {χ : ℝ → ℝ}
+    (hψ : Differentiable ℝ ψ) {b : ℝ} (hχone : ∀ s : ℝ, |s|<b → χ s=1)
+    {x : Euc n} (hx : |inner ℝ v x|<b)
+    (hN : approximateNormal E x=(graphAreaFactor v f (normalProjection v x))⁻¹ •
+      graphNormalVector v f (normalProjection v x)) :
+    inner ℝ (tangentialGradient E (normalTestLift v ψ χ) x) v=
+      graphFluxWeight v f ψ (normalProjection v x)*inner ℝ (approximateNormal E x) v := by
+  rw [tangentialGradient,normalTestLift_gradient hψ hχone hx,inner_sub_left,
+    real_inner_smul_left,hN,real_inner_smul_left,graphNormalVector_inner_normal hv,mul_one]
+  have hz : inner ℝ ((gradient ψ (normalProjection v x):normalPlane v):Euc n) v=0 :=
+    Submodule.mem_orthogonal_singleton_iff_inner_left.mp (gradient ψ (normalProjection v x)).property
+  simp only [real_inner_smul_right,graphNormalVector,inner_sub_right,hz,zero_sub,
+    Submodule.coe_inner,graphFluxWeight,div_eq_mul_inv]
+  conv_lhs => rw [real_inner_comm]
+  ring
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/GraphGeoCoordinates.lean b/lean/OAI/Geometry/Perimeter/Regularity/GraphGeoCoordinates.lean
new file mode 100644
index 0000000000000000000000000000000000000000..3654a487ab6d4105d6c79e1e086c0cd10f037f9c
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/GraphGeoCoordinates.lean
@@ -0,0 +1,41 @@
+import OAI.Geometry.Perimeter.Regularity.GraphCoordinates
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {m : ℕ} {W : Type*} [NormedAddCommGroup W] [InnerProductSpace ℝ W]
+  [FiniteDimensional ℝ W]
+
+theorem geoC_one_isometry {θ η : ℝ} (hθ : 0 ≤ θ) (hθh : θ ≤ 1/2) (hθη : θ ≤ η)
+    (e : Euc m ≃ₗᵢ[ℝ] W) {f : W → ℝ} {U : Set W} (hU : IsOpen U)
+    (hf : ContDiffOn ℝ 1 f U)
+    (hh : ∀ p∈U, GeometricHolderAt η (fderiv ℝ f) p) :
+    GeoC θ 1 (f ∘ e) (e ⁻¹' U) := by
+  have hV : IsOpen (e ⁻¹' U) := hU.preimage e.continuous
+  have hc : ContDiffOn ℝ 1 (f ∘ e) (e ⁻¹' U) :=
+    hf.comp e.contDiff.contDiffOn (fun _ hx ↦ hx)
+  have hd : ContinuousOn (fderiv ℝ f) U :=
+    (hf.fderiv_of_isOpen hU (m:=0) (by norm_num)).continuousOn
+  have hdc : ContinuousOn (fderiv ℝ (f ∘ e)) (e ⁻¹' U) :=
+    (hc.fderiv_of_isOpen hV (m:=0) (by norm_num)).continuousOn
+  refine ⟨hc,geoC_zero_of_C1 hθ hθh hV hc,fun u ↦ ⟨hdc.clm_apply continuousOn_const,?_⟩⟩
+  intro p hp
+  have hm := ((hh (e p) hp).mono_scale hθ hθη).precomp_isometry e
+  have hh' := hm.comp_C1 hθ
+    ((hd.continuousAt (hU.mem_nhds hp)).comp e.continuous.continuousAt)
+    (ContinuousLinearMap.apply ℝ ℝ (e u)).contDiff
+  have he : (fun x ↦ fderiv ℝ (f ∘ e) x u)=(fun x ↦ fderiv ℝ f (e x) (e u)) := by
+    funext x
+    exact congrArg (fun L : Euc m →L[ℝ] ℝ ↦ L u)
+      (e.toContinuousLinearEquiv.comp_right_fderiv (f:=f) (x:=x))
+  rw [he]
+  exact hh'
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/GraphNormal.lean b/lean/OAI/Geometry/Perimeter/Regularity/GraphNormal.lean
new file mode 100644
index 0000000000000000000000000000000000000000..c6036dc21e44af92faaac5c21acdd253b1e1e5f1
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/GraphNormal.lean
@@ -0,0 +1,82 @@
+import OAI.Geometry.Perimeter.Regularity.GraphChart
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+def graphNormalVector (v : Euc n) (f : normalPlane v → ℝ) (y : normalPlane v) : Euc n :=
+  v-((gradient f y : normalPlane v):Euc n)
+
+def graphAreaFactor (v : Euc n) (f : normalPlane v → ℝ) (y : normalPlane v) : ℝ :=
+  ‖graphNormalVector v f y‖
+
+theorem graphDefining_gradient {v : Euc n} {f : normalPlane v → ℝ}
+    (hf : ContDiffOn ℝ 1 f (Metric.ball 0 (1/4)))
+    {x : Euc n} (hx : x∈graphCylinder v) :
+    gradient (graphDefining v f) x=graphNormalVector v f (normalProjection v x) := by
+  apply ext_inner_right ℝ
+  intro z
+  rw [inner_gradient_left,(graphDefining_hasFDerivAt hf hx).fderiv]
+  simp only [sub_apply,ContinuousLinearMap.comp_apply]
+  change inner ℝ v z-fderiv ℝ f (normalProjection v x) (normalProjection v z)=
+    inner ℝ (v-((gradient f (normalProjection v x) : normalPlane v):Euc n)) z
+  rw [inner_sub_left,←(normalPlane v).inner_orthogonalProjectionOnto_eq_of_mem_left,inner_gradient_left]
+
+theorem graphNormalVector_inner_normal {v : Euc n} (hv : ‖v‖=1)
+    (f : normalPlane v → ℝ) (y : normalPlane v) : inner ℝ (graphNormalVector v f y) v=1 := by
+  rw [graphNormalVector,inner_sub_left,real_inner_self_eq_norm_sq,hv,one_pow,
+    Submodule.mem_orthogonal_singleton_iff_inner_left.mp (gradient f y).property,sub_zero]
+
+theorem graphAreaFactor_sq {v : Euc n} (hv : ‖v‖=1)
+    (f : normalPlane v → ℝ) (y : normalPlane v) : (graphAreaFactor v f y)^2=1+‖gradient f y‖^2 := by
+  rw [graphAreaFactor,graphNormalVector,norm_sub_sq_real,hv,one_pow,
+    Submodule.mem_orthogonal_singleton_iff_inner_right.mp (gradient f y).property,
+    mul_zero,sub_zero,Submodule.norm_coe]
+
+theorem graphAreaFactor_pos {v : Euc n} (hv : ‖v‖=1)
+    (f : normalPlane v → ℝ) (y : normalPlane v) : 0<graphAreaFactor v f y := by
+  have hh := graphAreaFactor_sq hv f y
+  have hn : 0≤graphAreaFactor v f y := norm_nonneg _
+  nlinarith [sq_nonneg ‖gradient f y‖]
+
+theorem graphAreaFactor_sqrt {v : Euc n} (hv : ‖v‖=1)
+    (f : normalPlane v → ℝ) (y : normalPlane v) :
+    graphAreaFactor v f y=Real.sqrt (1+‖gradient f y‖^2) := by
+  rw [←graphAreaFactor_sq hv f y,Real.sqrt_sq (graphAreaFactor_pos hv f y).le]
+
+theorem graph_approximateNormal (hn : 1<n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    {v : Euc n} (hv : ‖v‖=1) {a : ℝ} (ha : 0≤a) (ha4 : a<1/4)
+    (hs : FullSlab E v 8 a) {f : normalPlane v → ℝ}
+    (hf : ContDiffOn ℝ 1 f (Metric.ball 0 (1/4)))
+    (hiff : ∀ (y : normalPlane v) (t : ℝ), ‖y‖<1/4 → ‖(y:Euc n)+t•v‖<1 →
+      ((y:Euc n)+t•v∈perimeterSupport E ↔ t=f y))
+    {x : Euc n} (hx : x∈graphCylinder v) (hxS : x∈perimeterSupport E) :
+    approximateNormal E x=(graphAreaFactor v f (normalProjection v x))⁻¹ •
+      graphNormalVector v f (normalProjection v x) := by
+  rw [graphDefining_approximateNormal hn hmin hv ha ha4 hs hf hiff hx hxS,
+    graphDefining_gradient hf hx]
+  rfl
+
+theorem graph_normal_flux_positive (hn : 1<n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    {v : Euc n} (hv : ‖v‖=1) {a : ℝ} (ha : 0≤a) (ha4 : a<1/4)
+    (hs : FullSlab E v 8 a) {f : normalPlane v → ℝ}
+    (hf : ContDiffOn ℝ 1 f (Metric.ball 0 (1/4)))
+    (hiff : ∀ (y : normalPlane v) (t : ℝ), ‖y‖<1/4 → ‖(y:Euc n)+t•v‖<1 →
+      ((y:Euc n)+t•v∈perimeterSupport E ↔ t=f y))
+    {x : Euc n} (hx : x∈graphCylinder v) (hxS : x∈perimeterSupport E) :
+    0 < inner ℝ (approximateNormal E x) v := by
+  rw [graph_approximateNormal hn hmin hv ha ha4 hs hf hiff hx hxS,
+    real_inner_smul_left,graphNormalVector_inner_normal hv,mul_one]
+  exact inv_pos.mpr (graphAreaFactor_pos hv f _)
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/GraphPhase.lean b/lean/OAI/Geometry/Perimeter/Regularity/GraphPhase.lean
new file mode 100644
index 0000000000000000000000000000000000000000..59bcb6925cf4ba44b3de440757837df6227660dd
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/GraphPhase.lean
@@ -0,0 +1,67 @@
+import OAI.Geometry.Perimeter.Regularity.GraphExistence
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+theorem full_graph_phaseSupport_iff (hn : 1<n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    {v : Euc n} (hv : ‖v‖=1) {a : ℝ} (ha : 0≤a) (ha4 : a<1/4)
+    (hs : FullSlab E v 8 a) {f : normalPlane v → ℝ}
+    (hiff : ∀ (y : normalPlane v) (t : ℝ), ‖y‖<1/4 → ‖(y:Euc n)+t•v‖<1 →
+      ((y:Euc n)+t•v∈perimeterSupport E ↔ t=f y))
+    (y : normalPlane v) (hy : ‖y‖<1/4) (t : ℝ) (ht : |t|<1/2) :
+    ((y:Euc n)+t•v∈(volume.restrict E).support ↔ t≤f y) := by
+  let g : ℝ → Euc n := fun s ↦ (y:Euc n)+s•v
+  let K := (volume.restrict E).support
+  have hg : Continuous g := continuous_const.add (continuous_id.smul continuous_const)
+  have hnorm (s : ℝ) (hss : |s|≤1/2) : ‖g s‖<1 := by
+    have hh := norm_add_le (y:Euc n) (s•v)
+    rw [norm_smul,Real.norm_eq_abs,hv,mul_one] at hh
+    change ‖(y:Euc n)‖<1/4 at hy
+    dsimp [g]
+    linarith
+  have hball (s : ℝ) (hss : |s|≤1/2) : g s∈Metric.ball 0 8 := by
+    rw [Metric.mem_ball,dist_zero_right]
+    linarith [hnorm s hss]
+  have hleft : g (-(1/2))∈K := hs.lower_phaseSupport ha (hball _ (by norm_num)) (by
+    change inner ℝ v ((y:Euc n)+(-(1/2):ℝ)•v) < -a
+    rw [normal_fiber_inner hv]; linarith)
+  have hright : g (1/2)∉K := hs.upper_not_phaseSupport ha (hball _ (by norm_num)) (by
+    change a < inner ℝ v ((y:Euc n)+(1/2:ℝ)•v)
+    rw [normal_fiber_inner hv]; linarith)
+  have hfront : frontier K=perimeterSupport E := minimizing_frontier_phaseSupport
+    (Real.HolderConjugate.conjExponent (by exact_mod_cast hn : 1<(n:ℝ))) hmin
+  constructor
+  · intro hx
+    by_contra! hf
+    obtain ⟨z,⟨s,hss,rfl⟩,hz⟩ := connected_crosses_frontier
+      (isPreconnected_Icc.image g hg.continuousOn) Measure.isClosed_support
+      (mem_image_of_mem g (show t∈Icc t (1/2) from ⟨le_rfl,(abs_lt.mp ht).2.le⟩)) hx
+      (mem_image_of_mem g (show (1/2:ℝ)∈Icc t (1/2) from ⟨(abs_lt.mp ht).2.le,le_rfl⟩)) hright
+    rw [hfront] at hz
+    have hb : |s|≤1/2 := abs_le.mpr ⟨(abs_lt.mp ht).1.le.trans hss.1,hss.2⟩
+    have he := (hiff y s hy (hnorm s hb)).mp hz
+    linarith [hss.1]
+  · intro htf
+    by_contra hx
+    obtain ⟨z,⟨s,hss,rfl⟩,hz⟩ := connected_crosses_frontier
+      (isPreconnected_Icc.image g hg.continuousOn) Measure.isClosed_support
+      (mem_image_of_mem g (show (-(1/2):ℝ)∈Icc (-(1/2)) t from ⟨le_rfl,(abs_lt.mp ht).1.le⟩)) hleft
+      (mem_image_of_mem g (show t∈Icc (-(1/2)) t from ⟨(abs_lt.mp ht).1.le,le_rfl⟩)) hx
+    have hKz : g s∈K := Measure.isClosed_support.frontier_subset hz
+    rw [hfront] at hz
+    have hb : |s|≤1/2 := abs_le.mpr ⟨hss.1,hss.2.trans (abs_lt.mp ht).2.le⟩
+    have he := (hiff y s hy (hnorm s hb)).mp hz
+    have hst : s=t := by linarith [hss.2]
+    exact hx (hst ▸ hKz)
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/GraphSlope.lean b/lean/OAI/Geometry/Perimeter/Regularity/GraphSlope.lean
new file mode 100644
index 0000000000000000000000000000000000000000..fa1d4b3706aacfc29073c11ca8583c1a0d672ee6
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/GraphSlope.lean
@@ -0,0 +1,38 @@
+import OAI.Geometry.Perimeter.Regularity.GraphDerivative
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+theorem graphTangentLinear_norm_le {v w : Euc n} (hv : ‖v‖=1)
+    (hwv : ‖w-v‖≤1/2) : ‖graphTangentLinear v w‖≤2*‖w-v‖ := by
+  have hd : (1/2:ℝ) ≤ inner ℝ w v := inner_near_unit_lower hv hwv
+  have hd0 : 0 < inner ℝ w v := by linarith
+  apply (graphTangentLinear v w).opNorm_le_bound (mul_nonneg (by norm_num) (norm_nonneg _))
+  intro y
+  rw [Real.norm_eq_abs,graphTangentLinear_apply,abs_mul,abs_neg,
+    abs_of_nonneg (inv_nonneg.mpr hd0.le)]
+  have hy : inner ℝ v (y:Euc n)=0 := by
+    rw [real_inner_comm]
+    exact (Submodule.mem_orthogonal_singleton_iff_inner_left.mp y.property)
+  have he : inner ℝ w (y:Euc n)=inner ℝ (w-v) (y:Euc n) := by rw [inner_sub_left,hy,sub_zero]
+  rw [he]
+  have hi : (inner ℝ w v)⁻¹ ≤ 2 := by
+    rw [inv_eq_one_div]
+    exact (div_le_iff₀ hd0).mpr (by linarith)
+  have hb := abs_real_inner_le_norm (w-v) (y:Euc n)
+  calc
+    (inner ℝ w v)⁻¹*|inner ℝ (w-v) (y:Euc n)| ≤ 2*|inner ℝ (w-v) (y:Euc n)| :=
+      mul_le_mul_of_nonneg_right hi (abs_nonneg _)
+    _ ≤ 2*(‖w-v‖*‖y‖) := mul_le_mul_of_nonneg_left hb (by norm_num)
+    _ = (2*‖w-v‖)*‖y‖ := by ring
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/GraphTestLift.lean b/lean/OAI/Geometry/Perimeter/Regularity/GraphTestLift.lean
new file mode 100644
index 0000000000000000000000000000000000000000..69a65d4410ffcc431bd67256bb7400e175922a07
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/GraphTestLift.lean
@@ -0,0 +1,81 @@
+import OAI.Geometry.Perimeter.Regularity.PlaneWindowLift
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+def normalTestLift (v : Euc n) (ψ : normalPlane v → ℝ) (χ : ℝ → ℝ) (x : Euc n) : ℝ :=
+  ψ (normalProjection v x)*χ (inner ℝ v x)
+
+theorem normalTestLift_contDiff {v : Euc n} {ψ : normalPlane v → ℝ} {χ : ℝ → ℝ}
+    (hψ : ContDiff ℝ ∞ ψ) (hχ : ContDiff ℝ ∞ χ) :
+    ContDiff ℝ ∞ (normalTestLift v ψ χ) :=
+  (hψ.comp (normalProjection v).contDiff).mul
+    (hχ.comp (contDiff_const.inner ℝ contDiff_id))
+
+theorem normalTestLift_tsupport {v : Euc n} (hv : ‖v‖=1)
+    (ψ : normalPlane v → ℝ) (χ : ℝ → ℝ) {r s : ℝ}
+    (hr : Function.support ψ ⊆ Metric.closedBall 0 r)
+    (hs : Function.support χ ⊆ Metric.closedBall 0 s) :
+    tsupport (normalTestLift v ψ χ) ⊆ Metric.closedBall 0 (r+s) := by
+  apply closure_minimal _ Metric.isClosed_closedBall
+  intro x hx
+  have hp : ψ (normalProjection v x)≠0 ∧ χ (inner ℝ v x)≠0 := mul_ne_zero_iff.mp hx
+  have h1 := hr hp.1
+  have h2 := hs hp.2
+  rw [Metric.mem_closedBall,dist_zero_right] at h1 h2 ⊢
+  calc
+    ‖x‖ = ‖(normalProjection v x:Euc n)+inner ℝ v x•v‖ :=
+      congrArg norm (normalProjection_decomposition hv x).symm
+    _ ≤ ‖(normalProjection v x:Euc n)‖+‖inner ℝ v x•v‖ := norm_add_le _ _
+    _ ≤ r+s := by simpa only [Submodule.norm_coe,norm_smul,hv,mul_one] using add_le_add h1 h2
+
+theorem gradient_normalProjection_comp {v : Euc n} {ψ : normalPlane v → ℝ}
+    (hψ : Differentiable ℝ ψ) (x : Euc n) :
+    gradient (fun z ↦ ψ (normalProjection v z)) x=
+      ((gradient ψ (normalProjection v x):normalPlane v):Euc n) := by
+  apply ext_inner_right ℝ
+  intro z
+  rw [inner_gradient_left]
+  have hd : HasFDerivAt (fun z ↦ ψ (normalProjection v z))
+      ((fderiv ℝ ψ (normalProjection v x)).comp (normalProjection v)) x :=
+    (hψ _).hasFDerivAt.comp x (normalProjection v).hasFDerivAt
+  rw [hd.fderiv]
+  rw [←(normalPlane v).inner_orthogonalProjectionOnto_eq_of_mem_left,inner_gradient_left]
+  rfl
+
+theorem normalTestLift_gradient {v : Euc n} {ψ : normalPlane v → ℝ} {χ : ℝ → ℝ}
+    (hψ : Differentiable ℝ ψ) {b : ℝ}
+    (hχone : ∀ s : ℝ, |s|<b → χ s=1) {x : Euc n} (hx : |inner ℝ v x|<b) :
+    gradient (normalTestLift v ψ χ) x=
+      ((gradient ψ (normalProjection v x):normalPlane v):Euc n) := by
+  have he : normalTestLift v ψ χ =ᶠ[𝓝 x] fun z ↦ ψ (normalProjection v z) := by
+    filter_upwards [((continuous_const.inner continuous_id).abs.continuousAt).eventually
+      (gt_mem_nhds hx)] with z hz
+    change |inner ℝ v z|<b at hz
+    dsimp only [normalTestLift]
+    rw [hχone _ hz,mul_one]
+  rw [show gradient (normalTestLift v ψ χ) x=gradient (fun z ↦ ψ (normalProjection v z)) x from
+    congrArg (InnerProductSpace.toDual ℝ (Euc n)).symm (he.fderiv_eq (𝕜 := ℝ))]
+  exact gradient_normalProjection_comp hψ x
+
+theorem normalTestLift_gradient_zero {v : Euc n} {ψ : normalPlane v → ℝ} {χ : ℝ → ℝ}
+    {x : Euc n} (hx : normalProjection v x∉tsupport ψ) :
+    gradient (normalTestLift v ψ χ) x=0 := by
+  have he : normalTestLift v ψ χ =ᶠ[𝓝 x] fun _ ↦ (0:ℝ) := by
+    filter_upwards [(normalProjection v).continuous.continuousAt.eventually
+      ((isClosed_tsupport ψ).isOpen_compl.mem_nhds hx)] with z hz
+    simp only [normalTestLift,image_eq_zero_of_notMem_tsupport hz,zero_mul]
+  rw [show gradient (normalTestLift v ψ χ) x=gradient (fun _ : Euc n ↦ (0:ℝ)) x from
+    congrArg (InnerProductSpace.toDual ℝ (Euc n)).symm (he.fderiv_eq (𝕜 := ℝ))]
+  simp [gradient]
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/HalfspaceHorizontal.lean b/lean/OAI/Geometry/Perimeter/Regularity/HalfspaceHorizontal.lean
new file mode 100644
index 0000000000000000000000000000000000000000..e3f46979176a84da7ff56cc8038b57af67a11016
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/HalfspaceHorizontal.lean
@@ -0,0 +1,62 @@
+import OAI.Geometry.Perimeter.Regularity.GeoComposition
+import OAI.Geometry.Perimeter.Regularity.NormalCoordinates
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology InnerProductSpace ContDiff
+namespace RPISingularHarnack
+open RPI.Boundary ContinuousLinearMap
+
+ theorem halfspace_horizontal_integral_unit {n : ℕ} {v w : Euc n}
+    (hw : ‖w‖ = 1) (hvw : inner ℝ w v = 0)
+    {φ : Euc n → ℝ} (hφ : ContDiff ℝ ∞ φ) (hc : HasCompactSupport φ) :
+    (∫ x in {y | inner ℝ y v < 0}, fderiv ℝ φ x w) = 0 := by
+  let H : Set (Euc n) := {y | inner ℝ y v < 0}
+  have hHm : MeasurableSet H :=
+    (isOpen_lt (continuous_id.inner continuous_const) continuous_const).measurableSet
+  have hi : Integrable φ volume := hφ.continuous.integrable_of_hasCompactSupport hc
+  have hdi : Integrable (fun x ↦ fderiv ℝ φ x w) volume :=
+    ((hφ.continuous_fderiv (by simp)).clm_apply continuous_const).integrable_of_hasCompactSupport
+      (hc.fderiv_apply ℝ w)
+  rw [← integral_indicator hHm]
+  rw [integral_normalCoordinates w hw _ (hdi.indicator hHm)]
+  have hm := normalCoordinates_measurePreserving w hw
+  have hia := ((hm.integrable_comp hi.aestronglyMeasurable).mpr hi).prod_right_ae
+  have hdia := ((hm.integrable_comp hdi.aestronglyMeasurable).mpr hdi).prod_right_ae
+  apply integral_eq_zero_of_ae
+  filter_upwards [hia,hdia] with y hy hdy
+  have hd (s : ℝ) : HasDerivAt (fun t : ℝ ↦ φ ((y:Euc n)+t•w))
+      (fderiv ℝ φ ((y:Euc n)+s•w) w) s := by
+    simpa only [Function.comp_def,id_eq,one_smul] using
+      ((hφ.differentiable (by simp) _).hasFDerivAt).comp_hasDerivAt s
+        (HasDerivAt.const_add (y:Euc n) ((hasDerivAt_id s).smul_const w))
+  have hz := integral_eq_zero_of_hasDerivAt_of_integrable hd hdy hy
+  by_cases hyH : inner ℝ (y:Euc n) v < 0
+  · simpa [H,Set.indicator,inner_add_left,real_inner_smul_left,hvw,hyH] using hz
+  · simp [H,Set.indicator,inner_add_left,real_inner_smul_left,hvw,hyH]
+
+ theorem halfspace_horizontal_integral {n : ℕ} {v w : Euc n}
+    (hvw : inner ℝ v w = 0)
+    {φ : Euc n → ℝ} (hφ : ContDiff ℝ ∞ φ) (hc : HasCompactSupport φ) :
+    (∫ x in {y | inner ℝ y v < 0}, fderiv ℝ φ x w) = 0 := by
+  by_cases hw : w=0
+  · simp [hw]
+  have hn : 0 < ‖w‖ := norm_pos_iff.mpr hw
+  let u : Euc n := ‖w‖⁻¹ • w
+  have hu : ‖u‖=1 := by simp [u,norm_smul,hn.ne']
+  have huv : inner ℝ u v = 0 := by
+    have hvw' : inner ℝ w v=0 := by rw [real_inner_comm]; exact hvw
+    simp only [u,real_inner_smul_left,hvw',mul_zero]
+  have he : w = ‖w‖ • u := by simp [u,smul_smul,hn.ne']
+  have hh := halfspace_horizontal_integral_unit hu huv hφ hc
+  conv_lhs => arg 2; ext x; rw [he,map_smul]
+  simp only [smul_eq_mul]
+  rw [integral_const_mul,hh,mul_zero]
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/HarmonicApproxIdentity.lean b/lean/OAI/Geometry/Perimeter/Regularity/HarmonicApproxIdentity.lean
new file mode 100644
index 0000000000000000000000000000000000000000..9810eff799e77eb5de03c31e0ce65837518c936d
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/HarmonicApproxIdentity.lean
@@ -0,0 +1,54 @@
+import OAI.Geometry.Perimeter.Regularity.HarmonicKernel
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff Convolution InnerProductSpace
+namespace RPISingularHarnack
+open RPI.Boundary ContinuousLinearMap
+variable {n : ℕ}
+
+theorem scaledHarmonicKernel_ball_bound {t : ℝ} (ht : 0 < t) (c x : Euc n) :
+    |scaledHarmonicKernel t (c-x)| ≤
+      (volume.real (Metric.ball (0:Euc n) 1)/harmonicKernelMass n) /
+        volume.real (Metric.closedBall c (1/t)) := by
+  have hω : 0 < volume.real (Metric.ball (0:Euc n) 1) :=
+    ENNReal.toReal_pos (Metric.measure_ball_pos volume (0:Euc n) zero_lt_one).ne'
+      measure_ball_lt_top.ne
+  rw [Measure.addHaar_real_closedBall volume c (one_div_pos.mpr ht).le]
+  simp only [finrank_euclideanSpace,Fintype.card_fin]
+  calc
+    _ ≤ t^n/harmonicKernelMass n := scaledHarmonicKernel_bound ht _
+    _ = _ := by
+      simp only [one_div,inv_pow]
+      field_simp [ht.ne',hω.ne',(harmonicKernelMass_pos n).ne']
+
+theorem scaledHarmonicKernel_ae_tendsto {u : Euc n → ℝ} (hu : LocallyIntegrable u volume) :
+    ∀ᵐ c ∂volume, Tendsto (fun i : ℕ ↦
+      ∫ x, scaledHarmonicKernel ((i:ℝ)+1) (c-x)*u x) atTop (𝓝 (u c)) := by
+  have hd := IsUnifLocDoublingMeasure.ae_tendsto_average_norm_sub volume hu (1:ℝ)
+  filter_upwards [hd] with c hc
+  have hr : Tendsto (fun i : ℕ ↦ 1/((i:ℝ)+1)) atTop (𝓝[>] (0:ℝ)) :=
+    tendsto_nhdsWithin_iff.mpr ⟨tendsto_one_div_add_atTop_nhds_zero_nat,
+      Eventually.of_forall fun i ↦ (show 0 < 1/((i:ℝ)+1) from one_div_pos.mpr (by positivity))⟩
+  have hm := hc (fun _ : ℕ ↦ c) (fun i : ℕ ↦ 1/((i:ℝ)+1)) hr (by
+    apply Eventually.of_forall
+    intro i
+    exact Metric.mem_closedBall_self (by positivity))
+  have h := tendsto_integral_smul_of_tendsto_average_norm_sub
+    (g := fun i : ℕ ↦ fun x ↦ scaledHarmonicKernel ((i:ℝ)+1) (c-x))
+    (volume.real (Metric.ball (0:Euc n) 1)/harmonicKernelMass n) hm
+    (Eventually.of_forall fun i ↦ hu.integrableOn_isCompact (isCompact_closedBall _ _))
+    (show Tendsto (fun i : ℕ ↦ ∫ x, scaledHarmonicKernel ((i:ℝ)+1) (c-x)) atTop (𝓝 1) by
+      exact tendsto_const_nhds.congr (fun i ↦ (shifted_scaledHarmonicKernel_integral
+        (by positivity : 0 < (i:ℝ)+1) c).symm))
+    (Eventually.of_forall fun i ↦ shifted_scaledHarmonicKernel_support (by positivity) c)
+    (Eventually.of_forall fun i x ↦ scaledHarmonicKernel_ball_bound (by positivity) c x)
+  simpa only [smul_eq_mul] using h
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/HarmonicHeightLimit.lean b/lean/OAI/Geometry/Perimeter/Regularity/HarmonicHeightLimit.lean
new file mode 100644
index 0000000000000000000000000000000000000000..a1775a320aca3359e7869e3bb89012c7199bcdc8
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/HarmonicHeightLimit.lean
@@ -0,0 +1,141 @@
+import OAI.Geometry.Perimeter.Regularity.PlaneHeightResidual
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal InnerProductSpace ContDiff
+namespace RPISingularHarnack
+open RPI.Boundary ContinuousLinearMap
+variable {n m : ℕ}
+
+theorem l1_test_pairing_tendsto {f : ℕ → Euc m → ℝ} {g φ : Euc m → ℝ}
+    (hf : ∀ i, Integrable (f i) volume) (hg : Integrable g volume)
+    (hφ : AEStronglyMeasurable φ volume) {M : ℝ} (hb : ∀ x, ‖φ x‖≤M)
+    (hL : Tendsto (fun i ↦ ∫⁻ x, ‖f i x-g x‖ₑ) atTop (𝓝 0)) :
+    Tendsto (fun i ↦ ∫ x, f i x*φ x) atTop (𝓝 (∫ x, g x*φ x)) := by
+  apply tendsto_integral_of_L1 _ (hg.aestronglyMeasurable.mul hφ)
+    (Eventually.of_forall (fun i ↦ (hf i).mul_bdd hφ (ae_of_all _ hb)))
+  have ht : Tendsto (fun i ↦ ENNReal.ofReal M * ∫⁻ x, ‖f i x-g x‖ₑ) atTop (𝓝 0) := by
+    simpa using ENNReal.Tendsto.const_mul hL (Or.inr ENNReal.ofReal_ne_top)
+  apply tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds ht
+    (fun _ ↦ zero_le) (fun i ↦ ?_)
+  rw [← lintegral_const_mul' _ _ ENNReal.ofReal_ne_top]
+  apply lintegral_mono
+  intro x
+  dsimp only [Pi.mul_apply]
+  rw [← sub_mul,enorm_mul,mul_comm]
+  exact mul_le_mul
+    (by simpa only [← ofReal_norm] using ENNReal.ofReal_le_ofReal (hb x)) le_rfl zero_le zero_le
+
+theorem chartLaplacianTest_tsupport {v : Euc n} (e : Euc m ≃ₗᵢ[ℝ] normalPlane v)
+    (ψ : Euc m → ℝ) : tsupport (chartLaplacianTest e ψ) ⊆ e '' tsupport ψ := by
+  intro y hy
+  refine ⟨e.symm y,?_,e.apply_symm_apply y⟩
+  have hs : tsupport (divergence (gradient ψ)) ⊆ tsupport ψ :=
+    (tsupport_divergence_subset _).trans (tsupport_gradient_subset ψ)
+  apply hs
+  change y ∈ tsupport ((divergence (gradient ψ)) ∘ (e.symm.toHomeomorph : _ → _)) at hy
+  rw [tsupport_comp_eq_preimage _ e.symm.toHomeomorph] at hy
+  exact hy
+
+theorem minimizing_plane_height_weakHarmonic (hn : 1<n)
+    (E : ℕ → Set (Euc n)) (hmin : ∀ i, IsLocallyPerimeterMinimizing (E i))
+    {v : Euc n} (hv : ‖v‖=1) (e : Euc m ≃ₗᵢ[ℝ] normalPlane v)
+    {η : Euc n → ℝ} (hη : Continuous η) (hbη : ∀ x, ‖η x‖≤1)
+    (hsη : HasCompactSupport η) (hηs : Function.support η ⊆ Metric.ball 0 4)
+    (hηone : ∀ x ∈ Metric.ball (0:Euc n) 3, η x=1)
+    (a : ℕ → ℝ) (ha : ∀ i, 0<a i) (hat : Tendsto a atTop (𝓝 0))
+    (hh : ∀ i x, x ∈ perimeterSupport (E i) ∩ Metric.ball 0 4 → |inner ℝ v x|≤a i)
+    (hphase : ∀ i, ∀ᵐ x ∂volume, x ∈ Metric.ball 0 4 → a i < |inner ℝ v x| →
+      (x ∈ E i ↔ inner ℝ v x<0))
+    {P : ℝ} (hper : ∀ i, (perimeter (E i) (Metric.ball 0 4)).toReal≤P)
+    (htilt : Tendsto (fun i ↦ orientedTilt (E i) v (Metric.ball 0 4)) atTop (𝓝 0))
+    {g : Euc m → ℝ} (hg : Integrable g volume)
+    (hL : Tendsto (fun i ↦ ∫⁻ y, ‖chartFiber v e (a i) (cutoffPhaseError (E i) v η) y-g y‖ₑ)
+      atTop (𝓝 0)) : WeakHarmonicOn g (Metric.ball 0 1) := by
+  intro ψ hψ hcψ hsψ
+  let χ : ContDiffBump (0:ℝ) := ⟨1/2,1,by norm_num,by norm_num⟩
+  have hχone : ∀ s : ℝ, |s|<1/2 → χ s=1 := by
+    intro s hs
+    exact χ.one_of_mem_closedBall (by simpa only [Metric.mem_closedBall,dist_zero_right,Real.norm_eq_abs] using hs.le)
+  have hcΔ : HasCompactSupport (divergence (gradient ψ)) :=
+    hcψ.of_isClosed_subset (isClosed_tsupport _)
+      ((tsupport_divergence_subset _).trans (tsupport_gradient_subset ψ))
+  obtain ⟨M,hM⟩ := hcΔ.exists_bound_of_continuous (continuous_divergence (gradient_contDiff hψ))
+  have hM0 : 0≤M := (norm_nonneg (divergence (gradient ψ) 0)).trans (hM 0)
+  have hsm := planeWindowLift_contDiff e hψ χ.contDiff
+  obtain ⟨K₀,hK₀⟩ := (isCompact_closedBall (0:Euc n) 4).exists_bound_of_continuousOn
+    ((gradient_contDiff hsm).continuous_fderiv (by simp)).continuousOn
+  let K := max K₀ 0
+  have hK : 0≤K := le_max_right _ _
+  have hd : ∀ x ∈ Metric.ball (0:Euc n) 4,
+      ‖fderiv ℝ (gradient (planeWindowLift e ψ χ)) x‖≤K := by
+    intro x hx
+    exact (hK₀ x (Metric.ball_subset_closedBall hx)).trans (le_max_left _ _)
+  have hs : tsupport (planeWindowLift e ψ χ) ⊆ Metric.ball 0 4 := by
+    apply (planeWindowLift_tsupport hv e ψ χ
+      ((subset_closure.trans hsψ).trans Metric.ball_subset_closedBall) ?_).trans
+      (Metric.closedBall_subset_ball (by norm_num : (1:ℝ)+1<4))
+    simpa only [χ.support_eq] using (Metric.ball_subset_closedBall : Metric.ball (0:ℝ) 1 ⊆ Metric.closedBall 0 1)
+  have hsmall : ∀ᶠ i in atTop, a i<1/8 := hat.eventually (gt_mem_nhds (by norm_num : (0:ℝ)<1/8))
+  have hr : ∀ᶠ i in atTop,
+      |∫ y, chartFiber v e (a i) (cutoffPhaseError (E i) v η) y * divergence (gradient ψ) y| ≤
+        2*K*(Real.sqrt (orientedTilt (E i) v (Metric.ball 0 4))*Real.sqrt P)+
+        M*orientedTilt (E i) v (Metric.ball 0 4) := by
+    filter_upwards [hsmall] with i hi
+    have hai := ha i
+    have hri := minimizing_chart_height_residual hn (hmin i) hv e (ha i)
+      (b := 2*a i) (δ := a i/2) (b₀ := 1/2) (by positivity) (by linarith [ha i])
+      (by linarith) (by norm_num) hηs hψ hcψ χ.contDiff χ.hasCompactSupport hχone
+      (fun y hy s hs ↦ hηone _ (by
+        have hy' := chartLaplacianTest_tsupport e ψ hy
+        obtain ⟨z,hz,rfl⟩ := hy'
+        have hz' := hsψ hz
+        rw [Metric.mem_ball,dist_zero_right] at hz' ⊢
+        have ht := norm_add_le (e z:Euc n) (s•v)
+        rw [norm_smul,hv,mul_one] at ht
+        change ‖(e z:Euc n)+s•v‖≤‖e z‖+‖s‖ at ht
+        rw [e.norm_map,Real.norm_eq_abs] at ht
+        linarith))
+      (fun y hy ↦ by
+        obtain ⟨z,hz,rfl⟩ := chartLaplacianTest_tsupport e ψ hy
+        have hz' := hsψ hz
+        rw [Metric.mem_ball,dist_zero_right] at hz'
+        rw [e.norm_map]
+        linarith) hs hM hK hd (hh i) (hphase i)
+    have hdiv : M*(2*a i)/(2*a i)=M := by field_simp [ne_of_gt hai]
+    rw [hdiv] at hri
+    apply hri.trans
+    gcongr
+    exact hper i
+  have ht : Tendsto (fun i ↦ 2*K*(Real.sqrt (orientedTilt (E i) v (Metric.ball 0 4))*Real.sqrt P)+
+      M*orientedTilt (E i) v (Metric.ball 0 4)) atTop (𝓝 0) := by
+    simpa using (((Real.continuous_sqrt.tendsto 0).comp htilt).mul_const (Real.sqrt P)).const_mul
+      (2*K) |>.add (htilt.const_mul M)
+  have hz : Tendsto (fun i ↦ ∫ y, chartFiber v e (a i) (cutoffPhaseError (E i) v η) y *
+      divergence (gradient ψ) y) atTop (𝓝 0) := by
+    rw [tendsto_zero_iff_norm_tendsto_zero]
+    exact squeeze_zero' (Eventually.of_forall (fun _ ↦ norm_nonneg _)) hr ht
+  have hf (i) : Integrable (cutoffPhaseError (E i) v η) volume := by
+    apply (integrableOn_iff_integrable_of_support_subset
+      ((cutoffPhaseError_support (E i) v η).trans (subset_tsupport η))).mp
+    have : Fact (volume (tsupport η) < (∞ : ℝ≥0∞)) := ⟨hsη.measure_lt_top⟩
+    refine ⟨(cutoffPhaseError_measurable (hmin i).1 v hη.measurable).aestronglyMeasurable.restrict, ?_⟩
+    apply HasFiniteIntegral.of_bounded (C := 1)
+    apply ae_of_all
+    intro x
+    dsimp only [cutoffPhaseError]
+    rw [norm_mul]
+    have hb : ‖(E i).indicator (fun _ ↦ (1:ℝ)) x - {z | inner ℝ v z<0}.indicator (fun _ ↦ (1:ℝ)) x‖≤1 := by
+      by_cases he : x∈E i <;> by_cases hh : inner ℝ v x<0 <;> simp [he,hh]
+    exact (mul_le_mul (hbη x) hb (norm_nonneg _) zero_le_one).trans_eq (one_mul _)
+  have htpair := l1_test_pairing_tendsto (fun i ↦ chartFiber_integrable (hf i) hv e (a i)) hg
+    (continuous_divergence (gradient_contDiff hψ)).aestronglyMeasurable hM hL
+  exact tendsto_nhds_unique htpair hz
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/HarmonicKernel.lean b/lean/OAI/Geometry/Perimeter/Regularity/HarmonicKernel.lean
new file mode 100644
index 0000000000000000000000000000000000000000..c836a439268cc827dac27432adec9cfd5022f954
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/HarmonicKernel.lean
@@ -0,0 +1,106 @@
+import OAI.Geometry.Perimeter.Regularity.SmulMemPhaseSupport
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff Convolution InnerProductSpace
+namespace RPISingularHarnack
+open RPI.Boundary ContinuousLinearMap
+variable {n : ℕ}
+
+def harmonicKernelRaw (x : Euc n) : ℝ := densityProfile 1 (‖x‖^2)
+def harmonicKernelMass (n : ℕ) : ℝ := ∫ x : Euc n, harmonicKernelRaw x
+
+theorem harmonicKernelRaw_contDiff : ContDiff ℝ ∞ (harmonicKernelRaw (n := n)) :=
+  (densityProfile_contDiff 1).comp (contDiff_id.norm_sq (𝕜 := ℝ))
+
+theorem harmonicKernelRaw_tsupport : tsupport (harmonicKernelRaw (n := n)) ⊆
+    Metric.closedBall 0 1 := by
+  have h := radialProfile_tsupport (0:Euc n) (fun s hs ↦ densityProfile_zero 1 hs) zero_lt_one
+  change tsupport (fun x : Euc n ↦ densityProfile 1 (‖x‖^2)) ⊆ _
+  simpa only [one_pow,one_mul,sub_zero,div_one] using h
+
+theorem harmonicKernelRaw_compact : HasCompactSupport (harmonicKernelRaw (n := n)) :=
+  (isCompact_closedBall 0 1).of_isClosed_subset (isClosed_tsupport _) harmonicKernelRaw_tsupport
+
+theorem harmonicKernelMass_pos (n : ℕ) : 0 < harmonicKernelMass n := by
+  apply harmonicKernelRaw_contDiff.continuous.integral_pos_of_hasCompactSupport_nonneg_nonzero
+    harmonicKernelRaw_compact (fun x ↦ (densityProfile_mem_Icc 1 _).1) (x := 0)
+  simp only [harmonicKernelRaw,norm_zero,zero_pow (by decide : 2 ≠ 0),densityProfile]
+  norm_num [Real.smoothTransition.one_of_one_le]
+
+def harmonicKernel (x : Euc n) : ℝ := harmonicKernelRaw x / harmonicKernelMass n
+
+theorem harmonicKernel_contDiff : ContDiff ℝ ∞ (harmonicKernel (n := n)) :=
+  harmonicKernelRaw_contDiff.div_const _
+
+theorem harmonicKernel_tsupport : tsupport (harmonicKernel (n := n)) ⊆ Metric.closedBall 0 1 :=
+  by
+    change tsupport (fun x : Euc n ↦ harmonicKernelRaw x / harmonicKernelMass n) ⊆ _
+    simp only [div_eq_mul_inv]
+    exact tsupport_mul_subset_left.trans harmonicKernelRaw_tsupport
+
+theorem harmonicKernel_compact : HasCompactSupport (harmonicKernel (n := n)) :=
+  (isCompact_closedBall 0 1).of_isClosed_subset (isClosed_tsupport _) harmonicKernel_tsupport
+
+theorem harmonicKernel_integral : (∫ x : Euc n, harmonicKernel x) = 1 := by
+  simp only [harmonicKernel,integral_div]
+  exact div_self (harmonicKernelMass_pos n).ne'
+
+def scaledHarmonicKernel (t : ℝ) (x : Euc n) : ℝ := t^n*harmonicKernel (t•x)
+
+theorem scaledHarmonicKernel_contDiff (t : ℝ) :
+    ContDiff ℝ ∞ (scaledHarmonicKernel (n := n) t) :=
+  contDiff_const.mul (harmonicKernel_contDiff.comp (contDiff_id.const_smul t))
+
+theorem scaledHarmonicKernel_eq {t : ℝ} (ht : 0 < t) (x : Euc n) :
+    scaledHarmonicKernel t x = t^n*densityProfile 1 (t^2*‖x‖^2)/harmonicKernelMass n := by
+  simp only [scaledHarmonicKernel,harmonicKernel,harmonicKernelRaw,norm_smul,
+    Real.norm_eq_abs,abs_of_pos ht,mul_pow]
+  ring
+
+theorem scaledHarmonicKernel_integral {t : ℝ} (ht : 0 < t) :
+    (∫ x : Euc n, scaledHarmonicKernel t x) = 1 := by
+  change (∫ x : Euc n, t^n*harmonicKernel (t•x)) = 1
+  rw [integral_const_mul]
+  have h := Measure.integral_comp_smul_of_nonneg (volume : Measure (Euc n))
+    harmonicKernel t (hR := ht.le)
+  simp only [finrank_euclideanSpace,Fintype.card_fin,smul_eq_mul,harmonicKernel_integral] at h
+  rw [h]
+  simp [ht.ne']
+
+theorem scaledHarmonicKernel_tsupport {t : ℝ} (ht : 0 < t) :
+    tsupport (scaledHarmonicKernel (n := n) t) ⊆ Metric.closedBall 0 (1/t) := by
+  have he : scaledHarmonicKernel (n := n) t = fun x ↦
+      (t^n/harmonicKernelMass n)*densityProfile 1 (t^2*‖x-0‖^2) := by
+    funext x
+    rw [scaledHarmonicKernel_eq ht,sub_zero]; ring
+  rw [he]
+  exact tsupport_mul_subset_right.trans
+    (radialProfile_tsupport 0 (fun s hs ↦ densityProfile_zero 1 hs) ht)
+
+theorem scaledHarmonicKernel_bound {t : ℝ} (ht : 0 < t) (x : Euc n) :
+    |scaledHarmonicKernel t x| ≤ t^n/harmonicKernelMass n := by
+  rw [scaledHarmonicKernel_eq ht,abs_div,abs_of_pos (harmonicKernelMass_pos n),
+    abs_mul,abs_of_pos (pow_pos ht _),abs_of_nonneg (densityProfile_mem_Icc 1 _).1]
+  exact div_le_div_of_nonneg_right
+    (mul_le_of_le_one_right (pow_nonneg ht.le _) (densityProfile_mem_Icc 1 _).2)
+    (harmonicKernelMass_pos n).le
+
+theorem shifted_scaledHarmonicKernel_integral {t : ℝ} (ht : 0 < t) (c : Euc n) :
+    (∫ x, scaledHarmonicKernel t (c-x)) = 1 := by
+  rw [integral_sub_left_eq_self]
+  exact scaledHarmonicKernel_integral ht
+
+theorem shifted_scaledHarmonicKernel_support {t : ℝ} (ht : 0 < t) (c : Euc n) :
+    Function.support (fun x ↦ scaledHarmonicKernel t (c-x)) ⊆ Metric.closedBall c (1/t) := by
+  intro x hx
+  have h := scaledHarmonicKernel_tsupport ht (subset_tsupport _ hx)
+  simpa only [Metric.mem_closedBall,dist_zero_right,dist_eq_norm,sub_zero,norm_sub_rev] using h
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/HarmonicLimitBounds.lean b/lean/OAI/Geometry/Perimeter/Regularity/HarmonicLimitBounds.lean
new file mode 100644
index 0000000000000000000000000000000000000000..4420722dec3d0e492db39be7a63bb0b0721be0b9
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/HarmonicLimitBounds.lean
@@ -0,0 +1,72 @@
+import OAI.Geometry.Perimeter.Regularity.GeoComposition
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff Convolution InnerProductSpace
+namespace RPISingularHarnack
+open RPI.Boundary ContinuousLinearMap
+variable {n : ℕ}
+
+theorem integral_norm_le_of_L1_limit {X : Type*} [MeasurableSpace X]
+    {μ : Measure X} (f : ℕ → X → ℝ) (hf : ∀ i, Integrable (f i) μ)
+    {g : X → ℝ} (hg : Integrable g μ) {V : ℝ}
+    (hb : ∀ i, ∫ x, ‖f i x‖ ∂μ ≤ V)
+    (hL : Tendsto (fun i ↦ ∫⁻ x, ‖f i x-g x‖ₑ ∂μ) atTop (𝓝 0)) :
+    ∫ x, ‖g x‖ ∂μ ≤ V := by
+  have hn : Tendsto (fun i ↦ ∫⁻ x, ‖‖f i x‖-‖g x‖‖ₑ ∂μ) atTop (𝓝 0) := by
+    apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds hL
+      (Eventually.of_forall fun _ ↦ zero_le)
+    exact Eventually.of_forall fun i ↦ lintegral_mono fun x ↦ by
+      rw [← ofReal_norm,← ofReal_norm]
+      exact ENNReal.ofReal_le_ofReal (by simpa only [Real.norm_eq_abs] using abs_norm_sub_norm_le (f i x) (g x))
+  exact le_of_tendsto' (tendsto_integral_of_L1 (fun x ↦ ‖g x‖)
+    hg.norm.aestronglyMeasurable (Eventually.of_forall fun i ↦ (hf i).norm) hn) hb
+
+theorem kernelLinear_uniform_bound {k : Euc n → ℝ}
+    (hk : ContDiff ℝ ∞ k) (hc : HasCompactSupport k) :
+    ∃ M : ℝ, 0≤M ∧ ∀ u : Euc n → ℝ, Integrable u volume →
+      ‖kernelLinear u k‖≤M*(∫ x, ‖u x‖) := by
+  obtain ⟨M,hM⟩ := (hc.fderiv ℝ).exists_bound_of_continuous (hk.continuous_fderiv (by simp))
+  refine ⟨M,(norm_nonneg _).trans (hM 0),fun u hu ↦ ?_⟩
+  apply (norm_integral_le_integral_norm _).trans
+  have h := integral_mono_ae (kernelLinear_integrable hu hk hc).norm (hu.norm.const_mul M)
+    (ae_of_all _ fun x ↦ show ‖u x • fderiv ℝ k (-x)‖≤M*‖u x‖ from by
+      rw [norm_smul,mul_comm M]
+      exact mul_le_mul_of_nonneg_left (hM (-x)) (norm_nonneg _))
+  simpa only [integral_const_mul] using h
+
+theorem harmonicRegularization_uniform_taylor (n : ℕ) :
+    ∃ C : ℝ, 0≤C ∧ ∀ u : Euc n → ℝ, Integrable u volume → ∀ y : Euc n,
+      ‖harmonicRegularization u 4 y-harmonicRegularization u 4 0-
+        kernelLinear u (scaledHarmonicKernel 4) y‖ ≤ C*(∫ x, ‖u x‖)*‖y‖^2 := by
+  exact convolution_uniform_taylor (scaledHarmonicKernel_contDiff 4)
+    ((isCompact_closedBall 0 (1/(4:ℝ))).of_isClosed_subset (isClosed_tsupport _)
+      (scaledHarmonicKernel_tsupport (by norm_num)))
+
+theorem harmonic_height_representative_bounds (hn : 0<n) :
+    ∃ C : ℝ, 0≤C ∧ ∀ g : Euc n → ℝ, Integrable g volume →
+      WeakHarmonicOn g (Metric.ball 0 1) → ∀ V : ℝ, (∫ x, ‖g x‖)≤V →
+      ∃ w : Euc n → ℝ, ∃ ℓ : Euc n →L[ℝ] ℝ,
+        ContDiff ℝ ∞ w ∧ g =ᵐ[volume.restrict (Metric.ball 0 (1/2))] w ∧
+        ∀ y, ‖w y-w 0-ℓ y‖≤C*V*‖y‖^2 := by
+  obtain ⟨C,hC,hTaylor⟩ := harmonicRegularization_uniform_taylor n
+  refine ⟨C,hC,fun g hg hw V hV ↦ ⟨harmonicRegularization g 4,
+    kernelLinear g (scaledHarmonicKernel 4),
+    harmonicRegularization_contDiff hg.locallyIntegrable (by norm_num),?_,?_⟩⟩
+  · filter_upwards [ae_restrict_of_ae (weakHarmonic_ae_eq_regularization hg.locallyIntegrable hw hn),
+      ae_restrict_mem measurableSet_ball] with y hy hyB
+    apply hy (1/2) (by norm_num) _ 4 (by norm_num)
+    intro z hz
+    have hd := dist_triangle z y 0
+    rw [Metric.mem_ball] at hz hyB ⊢
+    linarith
+  · intro y
+    exact (hTaylor g hg y).trans (by gcongr)
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/HarmonicTaylor.lean b/lean/OAI/Geometry/Perimeter/Regularity/HarmonicTaylor.lean
new file mode 100644
index 0000000000000000000000000000000000000000..3b1ed5f2a62f795ef6d48fb0174fe8bd43e8096b
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/HarmonicTaylor.lean
@@ -0,0 +1,107 @@
+import OAI.Geometry.Perimeter.Regularity.EllipticCurl
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff Convolution InnerProductSpace
+namespace RPISingularHarnack
+open RPI.Boundary ContinuousLinearMap
+variable {n : ℕ}
+
+theorem compactSupport_bounded_derivative {V W : Type*}
+    [NormedAddCommGroup V] [NormedSpace ℝ V] [NormedAddCommGroup W] [NormedSpace ℝ W]
+    {f : V → W} (hf : Continuous (fderiv ℝ f)) (hc : HasCompactSupport f) :
+    ∃ C : ℝ, 0≤C ∧ ∀ x, ‖fderiv ℝ f x‖≤C := by
+  obtain ⟨C,hC⟩ := (hc.fderiv ℝ).exists_bound_of_continuous hf
+  exact ⟨C,(norm_nonneg _).trans (hC 0),hC⟩
+
+theorem compact_kernel_bounded_second {k : Euc n → ℝ}
+    (hk : ContDiff ℝ ∞ k) (hc : HasCompactSupport k) :
+    ∃ C : ℝ, 0≤C ∧ ∀ x, ‖fderiv ℝ (fderiv ℝ k) x‖≤C := by
+  have hk' : ContDiff ℝ ∞ (fderiv ℝ k) := hk.fderiv_right (by simp)
+  exact compactSupport_bounded_derivative (hk'.continuous_fderiv (by simp)) (hc.fderiv ℝ)
+
+theorem kernel_derivative_lipschitz {k : Euc n → ℝ}
+    (hk : ContDiff ℝ ∞ k) {C : ℝ}
+    (hC : ∀ x, ‖fderiv ℝ (fderiv ℝ k) x‖≤C) (x z : Euc n) :
+    ‖fderiv ℝ k z-fderiv ℝ k x‖≤C*‖z-x‖ := by
+  have hk' : ContDiff ℝ ∞ (fderiv ℝ k) := hk.fderiv_right (by simp)
+  exact Convex.norm_image_sub_le_of_norm_fderiv_le (𝕜 := ℝ) (f := fderiv ℝ k)
+    (fun z _ ↦ hk'.differentiable (by simp) z) (fun z _ ↦ hC z)
+    convex_univ (mem_univ x) (mem_univ z)
+
+theorem kernel_taylor_of_derivative_lipschitz {k : Euc n → ℝ}
+    (hk : Differentiable ℝ k) {C : ℝ} (hC0 : 0≤C)
+    (hLip : ∀ x z, ‖fderiv ℝ k z-fderiv ℝ k x‖≤C*‖z-x‖)
+    (x y : Euc n) : ‖k (x+y)-k x-fderiv ℝ k x y‖≤C*‖y‖^2 := by
+  have hb : ∀ z ∈ Metric.closedBall x ‖y‖, ‖fderiv ℝ k z-fderiv ℝ k x‖≤C*‖y‖ := by
+    intro z hz
+    exact (hLip x z).trans (mul_le_mul_of_nonneg_left (by simpa only [Metric.mem_closedBall,dist_eq_norm] using hz) hC0)
+  have h := Convex.norm_image_sub_le_of_norm_fderiv_le' (𝕜 := ℝ) (f := k)
+    (fun z (_ : z∈Metric.closedBall x ‖y‖) ↦ hk z) hb
+    (convex_closedBall x ‖y‖) (Metric.mem_closedBall_self (norm_nonneg y))
+    (show x+y ∈ Metric.closedBall x ‖y‖ by simp [Metric.mem_closedBall,dist_eq_norm])
+  simpa only [add_sub_cancel_left,pow_two,mul_assoc] using h
+
+theorem compact_kernel_uniform_taylor {k : Euc n → ℝ}
+    (hk : ContDiff ℝ ∞ k) (hc : HasCompactSupport k) :
+    ∃ C : ℝ, 0≤C ∧ ∀ x y : Euc n,
+      ‖k (x+y)-k x-fderiv ℝ k x y‖≤C*‖y‖^2 := by
+  obtain ⟨C,hC0,hC⟩ := compact_kernel_bounded_second hk hc
+  exact ⟨C,hC0,kernel_taylor_of_derivative_lipschitz (hk.differentiable (by simp)) hC0
+    (kernel_derivative_lipschitz hk hC)⟩
+
+def kernelLinear (u k : Euc n → ℝ) : Euc n →L[ℝ] ℝ :=
+  ∫ x, u x • fderiv ℝ k (-x)
+
+theorem kernelLinear_integrable {u k : Euc n → ℝ} (hu : Integrable u volume)
+    (hk : ContDiff ℝ ∞ k) (hc : HasCompactSupport k) :
+    Integrable (fun x ↦ u x • fderiv ℝ k (-x)) volume := by
+  obtain ⟨M,hM⟩ := (hc.fderiv ℝ).exists_bound_of_continuous (hk.continuous_fderiv (by simp))
+  apply (hu.norm.mul_const M).mono'
+    (hu.aestronglyMeasurable.smul (((hk.continuous_fderiv (by simp)).comp continuous_neg).aestronglyMeasurable))
+  exact ae_of_all _ fun x ↦ by change ‖u x • fderiv ℝ k (-x)‖ ≤ ‖u x‖*M; rw [norm_smul]; exact mul_le_mul_of_nonneg_left (hM (-x)) (norm_nonneg _)
+
+theorem kernelLinear_apply {u k : Euc n → ℝ} (hu : Integrable u volume)
+    (hk : ContDiff ℝ ∞ k) (hc : HasCompactSupport k) (y : Euc n) :
+    kernelLinear u k y = ∫ x, u x*fderiv ℝ k (-x) y := by
+  exact ContinuousLinearMap.integral_apply (kernelLinear_integrable hu hk hc) y
+
+theorem convolution_uniform_taylor {k : Euc n → ℝ}
+    (hk : ContDiff ℝ ∞ k) (hc : HasCompactSupport k) :
+    ∃ C : ℝ, 0≤C ∧ ∀ u : Euc n → ℝ, Integrable u volume → ∀ y : Euc n,
+      ‖(k ⋆[lsmul ℝ ℝ,volume] u) y-(k ⋆[lsmul ℝ ℝ,volume] u) 0-kernelLinear u k y‖ ≤
+        C*(∫ x, ‖u x‖)*‖y‖^2 := by
+  obtain ⟨C,hC,hTaylor⟩ := compact_kernel_uniform_taylor hk hc
+  obtain ⟨M,hM⟩ := hc.exists_bound_of_continuous hk.continuous
+  refine ⟨C,hC,fun u hu y ↦ ?_⟩
+  have hi (z : Euc n) : Integrable (fun x ↦ u x*k (z-x)) volume :=
+    hu.mul_bdd ((hk.continuous.comp (continuous_const.sub continuous_id)).aestronglyMeasurable)
+      (ae_of_all _ fun x ↦ hM (z-x))
+  have hil : Integrable (fun x ↦ u x*fderiv ℝ k (-x) y) volume :=
+    ((ContinuousLinearMap.apply ℝ ℝ y).integrable_comp (kernelLinear_integrable hu hk hc))
+  rw [convolution_eq_swap,convolution_eq_swap,kernelLinear_apply hu hk hc]
+  change ‖(∫ x, k (y-x)*u x)-(∫ x, k (0-x)*u x)-(∫ x, u x*fderiv ℝ k (-x) y)‖ ≤ _
+  simp_rw [mul_comm (k _)]
+  have his : Integrable (fun x ↦ u x*k (y-x)-u x*k (0-x)) volume := (hi y).sub (hi 0)
+  rw [← integral_sub (hi y) (hi 0),← integral_sub his hil]
+  have he : (fun x ↦ u x*k (y-x)-u x*k (0-x)-u x*fderiv ℝ k (-x) y) =
+      fun x ↦ (k ((-x)+y)-k (-x)-fderiv ℝ k (-x) y)*u x := by
+    funext x
+    rw [zero_sub,show y-x=(-x)+y by abel]
+    ring
+  rw [he]
+  have hb : ∀ᵐ x ∂volume,
+      ‖(k ((-x)+y)-k (-x)-fderiv ℝ k (-x) y)*u x‖≤(C*‖y‖^2)*‖u x‖ :=
+    ae_of_all _ fun x ↦ by rw [norm_mul]; exact mul_le_mul_of_nonneg_right (hTaylor (-x) y) (norm_nonneg _)
+  have h := norm_integral_le_of_norm_le (hu.norm.const_mul (C*‖y‖^2)) hb
+  rw [integral_const_mul] at h
+  convert h using 1
+  ring
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/HasC1TraceAt.lean b/lean/OAI/Geometry/Perimeter/Regularity/HasC1TraceAt.lean
new file mode 100644
index 0000000000000000000000000000000000000000..70714c27cce4dc1ddad08b19bfb485e5d46707e4
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/HasC1TraceAt.lean
@@ -0,0 +1,512 @@
+import OAI.Geometry.Perimeter.Regularity.TendstoWeightedHalfspaceBlowup
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPI.Boundary
+
+def HasC1TraceAt {n : ℕ} (S : Set (Space n)) (h : Space n → ℝ)
+    (x : Space n) : Prop :=
+  ∃ U : Set (Space n), IsOpen U ∧ x ∈ U ∧ ∃ f : Space n → ℝ,
+    ContDiffOn ℝ 1 f U ∧ EqOn f h (S ∩ U)
+
+theorem exists_c1_eqOn_of_local_trace {n : ℕ} {S : Set (Space n)}
+    (hS : IsClosed S) (h : Space n → ℝ)
+    (hloc : ∀ x ∈ S, HasC1TraceAt S h x) :
+    ∃ f : Space n → ℝ, ContDiff ℝ 1 f ∧ EqOn f h S := by
+  classical
+  let t : Space n → Set ℝ := fun x ↦ if x ∈ S then {h x} else univ
+  have hc : ∀ x, Convex ℝ (t x) := by
+    intro x
+    dsimp [t]
+    split_ifs
+    · exact convex_singleton _
+    · exact convex_univ
+  have hlocal : ∀ x : Space n, ∃ U ∈ 𝓝 x, ∃ g : Space n → ℝ,
+      ContMDiffOn (modelWithCornersSelf ℝ (Space n)) (modelWithCornersSelf ℝ ℝ) 1 g U ∧
+      ∀ y ∈ U, g y ∈ t y := by
+    intro x
+    by_cases hx : x ∈ S
+    · obtain ⟨U, hU, hxU, f, hf, he⟩ := hloc x hx
+      refine ⟨U, hU.mem_nhds hxU, f, hf.contMDiffOn, ?_⟩
+      intro y hy
+      by_cases hys : y ∈ S
+      · simpa only [t, ite_eq_left hys, mem_singleton_iff] using he ⟨hys, hy⟩
+      · simp only [t, ite_eq_right hys, mem_univ]
+    · refine ⟨Sᶜ, hS.isOpen_compl.mem_nhds hx, fun _ ↦ 0, contMDiffOn_const, ?_⟩
+      intro y hy
+      simp only [t, ite_eq_right hy, mem_univ]
+  obtain ⟨f, hf⟩ := exists_contMDiffMap_forall_mem_convex_of_local
+    (modelWithCornersSelf ℝ (Space n)) (n := (1 : ℕ∞)) hc hlocal
+  refine ⟨f, f.contMDiff.contDiff, ?_⟩
+  intro x hx
+  simpa only [t, ite_eq_left hx, mem_singleton_iff] using hf x
+
+def HasC1TraceJetAt {n : ℕ} (S : Set (Space n)) (h : Space n → ℝ)
+    (G : Space n → Space n) (x : Space n) : Prop :=
+  ∃ U : Set (Space n), IsOpen U ∧ x ∈ U ∧ ∃ f : Space n → ℝ,
+    ContDiffOn ℝ 1 f U ∧ EqOn f h (S ∩ U) ∧ surfaceGradient S f x = G x
+
+lemma HasC1TraceJetAt.trace {n : ℕ} {S : Set (Space n)} {h : Space n → ℝ}
+    {G : Space n → Space n} {x : Space n} (hx : HasC1TraceJetAt S h G x) :
+    HasC1TraceAt S h x := by
+  obtain ⟨U, hU, hxU, f, hf, he, _⟩ := hx
+  exact ⟨U, hU, hxU, f, hf, he⟩
+
+theorem exists_c1_cutoff_trace {n : ℕ} {S Z : Set (Space n)}
+    (hS : IsClosed S) {h : Space n → ℝ} {G : Space n → Space n}
+    (hreg : ∀ x ∈ S \ Z, HasC1TraceJetAt S h G x)
+    (χ : Space n → ℝ) (hχ : ContDiff ℝ 1 χ)
+    (hzero : ∃ U : Set (Space n), IsOpen U ∧ Z ⊆ U ∧ ∀ x ∈ U, χ x = 0) :
+    ∃ F : Space n → ℝ, ContDiff ℝ 1 F ∧ EqOn F (fun x ↦ χ x * h x) S ∧
+      EqOn (surfaceGradient S F)
+        (fun x ↦ χ x • G x + h x • surfaceGradient S χ x) S := by
+  have hj : ∀ x ∈ S, HasC1TraceJetAt S (fun x ↦ χ x * h x)
+      (fun x ↦ χ x • G x + h x • surfaceGradient S χ x) x := by
+    intro x hx
+    by_cases hxZ : x ∈ Z
+    · obtain ⟨U, hU, hZU, he⟩ := hzero
+      have hxU := hZU hxZ
+      refine ⟨U, hU, hxU, fun _ ↦ 0, contDiffOn_const, ?_, ?_⟩
+      · intro y hy
+        simp only [he y hy.2, zero_mul]
+      · have hχgrad : gradient χ x = 0 := by
+          have heq : χ =ᶠ[𝓝 x] (fun _ ↦ (0 : ℝ)) := by
+            filter_upwards [hU.mem_nhds hxU] with y hy using he y hy
+          simpa only [gradient_fun_const] using heq.gradient_eq
+        simp only [he x hxU, zero_smul, surfaceGradient, gradient_fun_const,
+          hχgrad, map_zero, Submodule.coe_zero, smul_zero, add_zero]
+    · obtain ⟨U, hU, hxU, f, hf, he, hG⟩ := hreg x ⟨hx, hxZ⟩
+      refine ⟨U, hU, hxU, fun y ↦ χ y * f y, hχ.contDiffOn.mul hf, ?_, ?_⟩
+      · intro y hy
+        exact congrArg (fun a ↦ χ y * a) (he hy)
+      · rw [surfaceGradient_mul (hχ.differentiable (by simp) x)
+            ((hf.contDiffAt (hU.mem_nhds hxU)).differentiableAt (by simp)),
+          hG, he ⟨hx, hxU⟩]
+  obtain ⟨F, hF, hFe⟩ := exists_c1_eqOn_of_local_trace hS
+    (fun x ↦ χ x * h x) (fun x hx ↦ (hj x hx).trace)
+  refine ⟨F, hF, hFe, ?_⟩
+  intro x hx
+  obtain ⟨U, hU, hxU, f, hf, he, hG⟩ := hj x hx
+  calc
+    surfaceGradient S F x = surfaceGradient S f x := by
+      apply surfaceGradient_congr_trace (hF.differentiable (by simp) x)
+        ((hf.contDiffAt (hU.mem_nhds hxU)).differentiableAt (by simp))
+      filter_upwards [hU.mem_nhds hxU] with y hy
+      intro hys
+      exact (hFe hys).trans (he ⟨hys, hy⟩).symm
+    _ = _ := hG
+
+end RPI.Boundary
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPI.Boundary
+
+lemma surfaceGradient_norm_le_fderiv {n : ℕ} (S : Set (Space n))
+    (f : Space n → ℝ) (x : Space n) : ‖surfaceGradient S f x‖ ≤ ‖fderiv ℝ f x‖ := by
+  calc
+    _ ≤ ‖gradient f x‖ := (tangentPlane S x).norm_orthogonalProjectionOnto_apply_le _
+    _ = _ := (InnerProductSpace.toDual ℝ (Space n)).symm.norm_map _
+
+lemma c1Mollify_surfaceGradient_tendsto {n : ℕ} (S : Set (Space n))
+    {f : Space n → ℝ} (hf : ContDiff ℝ 1 f) (hc : HasCompactSupport f) (x : Space n) :
+    Tendsto (fun j ↦ surfaceGradient S (c1Mollify f j) x) atTop
+      (𝓝 (surfaceGradient S f x)) := by
+  have hcont : Continuous (fun z : Space n →L[ℝ] ℝ ↦
+      ((tangentPlane S x).orthogonalProjectionOnto
+        ((InnerProductSpace.toDual ℝ (Space n)).symm z) : Space n)) := by fun_prop
+  exact hcont.continuousAt.tendsto.comp (c1Mollify_fderiv_tendsto hf hc x)
+
+lemma c1_surfaceGradient_aestronglyMeasurable {n : ℕ} (S : Set (Space n))
+    (μ : Measure (Space n))
+    (hgrad : ∀ f : Space n → ℝ, ContDiff ℝ ∞ f → AEStronglyMeasurable (surfaceGradient S f) μ)
+    {f : Space n → ℝ} (hf : ContDiff ℝ 1 f) (hc : HasCompactSupport f) :
+    AEStronglyMeasurable (surfaceGradient S f) μ :=
+  aestronglyMeasurable_of_tendsto_ae atTop
+    (fun j ↦ hgrad _ (c1Mollify_contDiff hf.continuous j))
+    (ae_of_all _ (c1Mollify_surfaceGradient_tendsto S hf hc))
+
+lemma c1Mollify_pair_bounded {n : ℕ} (S : Set (Space n))
+    {f : Space n → ℝ} (hf : ContDiff ℝ 1 f) (hc : HasCompactSupport f) :
+    ∃ M : ℝ, 0 < M ∧ (∀ j x, ‖c1Mollify f j x‖ ≤ M ∧
+      ‖surfaceGradient S (c1Mollify f j) x‖ ≤ M) ∧
+      ∀ x, ‖f x‖ ≤ M ∧ ‖surfaceGradient S f x‖ ≤ M := by
+  obtain ⟨M, hM, hbound⟩ := c1Mollify_uniform_bound hf hc
+  refine ⟨M, hM, ?_, ?_⟩
+  · intro j x
+    exact ⟨(hbound j x).1, (surfaceGradient_norm_le_fderiv S _ x).trans (hbound j x).2⟩
+  · intro x
+    refine ⟨le_of_tendsto (c1Mollify_tendsto hf.continuous x).norm
+      (Eventually.of_forall fun j ↦ (hbound j x).1), ?_⟩
+    apply le_of_tendsto (c1Mollify_surfaceGradient_tendsto S hf hc x).norm
+    exact Eventually.of_forall fun j ↦ (surfaceGradient_norm_le_fderiv S _ x).trans (hbound j x).2
+
+end RPI.Boundary
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPI
+
+lemma bounded_tendsto_toLp_two {X V : Type*} [MeasurableSpace X]
+    [NormedAddCommGroup V] [InnerProductSpace ℝ V] {μ : Measure X} [IsFiniteMeasure μ]
+    {F : ℕ → X → V} {f : X → V}
+    (hF : ∀ j, MemLp (F j) 2 μ) (hf : MemLp f 2 μ)
+    {M : ℝ} (hM : 0 ≤ M)
+    (hbound : ∀ j, ∀ᵐ x ∂μ, ‖F j x‖ ≤ M)
+    (hfbound : ∀ᵐ x ∂μ, ‖f x‖ ≤ M)
+    (ht : ∀ᵐ x ∂μ, Tendsto (fun j ↦ F j x) atTop (𝓝 (f x))) :
+    Tendsto (fun j ↦ (hF j).toLp (F j)) atTop (𝓝 (hf.toLp f)) := by
+  apply (Lp.tendsto_Lp_iff_tendsto_eLpNorm'' _ hF _ hf).mpr
+  apply tendsto_eLpNorm_two_of_vectorEnergy_zero_of_aestronglyMeasurable
+    (fun index ↦ ((hF index).sub hf).aestronglyMeasurable)
+  have hlim := tendsto_lintegral_of_dominated_convergence'
+    (fun _ : X ↦ ENNReal.ofReal ((2 * M) ^ 2))
+    (f := fun _ ↦ (0 : ℝ≥0∞))
+    (F := fun j x ↦ ENNReal.ofReal (‖F j x - f x‖ ^ 2))
+    (fun j ↦ ((hF j).aestronglyMeasurable.sub hf.aestronglyMeasurable).norm.pow 2
+      |>.aemeasurable.ennreal_ofReal) ?_ ?_ ?_
+  · simpa only [vectorEnergy, Pi.sub_apply, lintegral_zero] using hlim
+  · intro j
+    filter_upwards [hbound j, hfbound] with x hx hy
+    apply ENNReal.ofReal_le_ofReal
+    apply (sq_le_sq₀ (norm_nonneg _) (by positivity)).mpr
+    exact (norm_sub_le _ _).trans (by linarith)
+  · simp only [lintegral_const]
+    exact ENNReal.mul_ne_top ENNReal.ofReal_ne_top (measure_ne_top _ _)
+  · filter_upwards [ht] with x hx
+    simpa only [sub_self, norm_zero, zero_pow (by norm_num : (2 : ℕ) ≠ 0),
+      ENNReal.ofReal_zero] using ENNReal.tendsto_ofReal ((hx.sub_const (f x)).norm.pow 2)
+
+end RPI
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPI.Boundary
+
+theorem c1_compact_mem_sobolevGraph {n : ℕ} (S V : Set (Space n))
+    [IsFiniteMeasure ((surfaceMeasure S).restrict V)]
+    (hgrad : ∀ f : Space n → ℝ, ContDiff ℝ ∞ f →
+      AEStronglyMeasurable (surfaceGradient S f) ((surfaceMeasure S).restrict V))
+    {f : Space n → ℝ} (hf : ContDiff ℝ 1 f) (hc : HasCompactSupport f)
+    (hm : MemLp f 2 ((surfaceMeasure S).restrict V))
+    (hD : MemLp (surfaceGradient S f) 2 ((surfaceMeasure S).restrict V)) :
+    (hm.toLp f, hD.toLp (surfaceGradient S f)) ∈ closure (smoothGradientGraph S V) := by
+  obtain ⟨M, hM, hbound, hfb⟩ := c1Mollify_pair_bounded S hf hc
+  have hfm : ∀ j, MemLp (c1Mollify f j) 2 ((surfaceMeasure S).restrict V) := by
+    intro j
+    exact MemLp.of_bound (c1Mollify_contDiff hf.continuous j).continuous.aestronglyMeasurable M
+      (ae_of_all _ fun x ↦ (hbound j x).1)
+  have hDm : ∀ j, MemLp (surfaceGradient S (c1Mollify f j)) 2
+      ((surfaceMeasure S).restrict V) := by
+    intro j
+    exact MemLp.of_bound (hgrad _ (c1Mollify_contDiff hf.continuous j)) M
+      (ae_of_all _ fun x ↦ (hbound j x).2)
+  have ht := bounded_tendsto_toLp_two hfm hm hM.le
+    (fun j ↦ ae_of_all _ fun x ↦ (hbound j x).1)
+    (ae_of_all _ fun x ↦ (hfb x).1) (ae_of_all _ fun x ↦ c1Mollify_tendsto hf.continuous x)
+  have hDt := bounded_tendsto_toLp_two hDm hD hM.le
+    (fun j ↦ ae_of_all _ fun x ↦ (hbound j x).2)
+    (ae_of_all _ fun x ↦ (hfb x).2) (ae_of_all _ fun x ↦ c1Mollify_surfaceGradient_tendsto S hf hc x)
+  apply mem_closure_of_tendsto (ht.prodMk_nhds hDt)
+  exact Eventually.of_forall fun j ↦ ⟨c1Mollify f j, c1Mollify_contDiff hf.continuous j,
+    hfm j, hDm j, rfl⟩
+
+theorem exists_c1_compact_trace_on_bounded {n : ℕ} (S V : Set (Space n))
+    (hV : Bornology.IsBounded V) (f : Space n → ℝ) (hf : ContDiff ℝ 1 f) :
+    ∃ F : Space n → ℝ, ContDiff ℝ 1 F ∧ HasCompactSupport F ∧
+      EqOn F f V ∧ EqOn (surfaceGradient S F) (surfaceGradient S f) V := by
+  obtain ⟨r, hr⟩ := Metric.isBounded_iff_subset_ball (0 : Space n) |>.mp hV
+  let b : ContDiffBump (0 : Space n) :=
+    ⟨|r| + 1, |r| + 2, by positivity, by linarith⟩
+  have hb : V ⊆ Metric.ball (0 : Space n) b.rIn := by
+    intro x hx
+    have hx' := Metric.mem_ball.mp (hr hx)
+    change dist x 0 < |r| + 1
+    linarith [le_abs_self r]
+  have hb1 : ContDiff ℝ 1 b := b.contDiff
+  refine ⟨fun x ↦ b x * f x, hb1.mul hf, b.hasCompactSupport.mul_right, ?_, ?_⟩
+  · intro x hx
+    change b x * f x = f x
+    rw [b.one_of_mem_closedBall (Metric.ball_subset_closedBall (hb hx)), one_mul]
+  · intro x hx
+    apply surfaceGradient_congr_trace
+      ((hb1.mul hf).differentiable (by norm_num) x)
+      (hf.differentiable (by norm_num) x)
+    filter_upwards [b.eventuallyEq_one_of_mem_ball (hb hx)] with y hy
+    intro _
+    simp only [hy, Pi.one_apply, one_mul]
+
+theorem c1_mem_sobolevGraph_bounded {n : ℕ} (S V : Set (Space n))
+    (hV : Bornology.IsBounded V) (hVm : MeasurableSet V)
+    [IsFiniteMeasure ((surfaceMeasure S).restrict V)]
+    (hgrad : ∀ f : Space n → ℝ, ContDiff ℝ ∞ f →
+      AEStronglyMeasurable (surfaceGradient S f) ((surfaceMeasure S).restrict V))
+    {f : Space n → ℝ} (hf : ContDiff ℝ 1 f)
+    (hm : MemLp f 2 ((surfaceMeasure S).restrict V))
+    (hD : MemLp (surfaceGradient S f) 2 ((surfaceMeasure S).restrict V)) :
+    (hm.toLp f, hD.toLp (surfaceGradient S f)) ∈ closure (smoothGradientGraph S V) := by
+  obtain ⟨F, hF, hcF, he, hDe⟩ := exists_c1_compact_trace_on_bounded S V hV f hf
+  have he' : F =ᵐ[(surfaceMeasure S).restrict V] f := by
+    filter_upwards [ae_restrict_mem hVm] with x hx using he hx
+  have hDe' : surfaceGradient S F =ᵐ[(surfaceMeasure S).restrict V] surfaceGradient S f := by
+    filter_upwards [ae_restrict_mem hVm] with x hx using hDe hx
+  let hmF := hm.ae_eq he'.symm
+  let hDF := hD.ae_eq hDe'.symm
+  have hp := c1_compact_mem_sobolevGraph S V hgrad hF hcF hmF hDF
+  simpa only [hmF.toLp_congr hm he', hDF.toLp_congr hD hDe'] using hp
+
+end RPI.Boundary
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPI.Boundary
+
+theorem c1_capacity_trace_mem_sobolevGraph {n : ℕ} {S Z : Set (Space n)}
+    (hS : IsClosed S) (V : Set (Space n))
+    (hV : Bornology.IsBounded V) (hVm : MeasurableSet V)
+    [IsFiniteMeasure ((surfaceMeasure S).restrict V)]
+    (hgrad : ∀ f : Space n → ℝ, ContDiff ℝ ∞ f →
+      AEStronglyMeasurable (surfaceGradient S f) ((surfaceMeasure S).restrict V))
+    {h : Space n → ℝ} {G : Space n → Space n}
+    (hreg : ∀ x ∈ S \ Z, HasC1TraceJetAt S h G x)
+    (hh : MemLp h 2 ((surfaceMeasure S).restrict V))
+    (hG : MemLp G 2 ((surfaceMeasure S).restrict V))
+    {M : ℝ} (hM : 0 ≤ M)
+    (hbound : ∀ᵐ x ∂(surfaceMeasure S).restrict V, ‖h x‖ ≤ M)
+    (χ : ℕ → Space n → ℝ)
+    (hχ : ∀ i, ContDiff ℝ ∞ (χ i))
+    (hχzero : ∀ i, ∃ U : Set (Space n), IsOpen U ∧ Z ⊆ U ∧ ∀ x ∈ U, χ i x = 0)
+    (hχb : ∀ i, ∀ᵐ x ∂(surfaceMeasure S).restrict V, χ i x ∈ Icc (0 : ℝ) 1)
+    (hχlim : ∀ᵐ x ∂(surfaceMeasure S).restrict V,
+      Tendsto (fun i ↦ χ i x) atTop (𝓝 1))
+    (hC : ∀ i, MemLp (surfaceGradient S (χ i)) 2 ((surfaceMeasure S).restrict V))
+    (hC0 : Tendsto (fun i ↦ vectorEnergy ((surfaceMeasure S).restrict V)
+      (surfaceGradient S (χ i))) atTop (𝓝 0)) :
+    (hh.toLp h, hG.toLp G) ∈ closure (smoothGradientGraph S V) := by
+  obtain ⟨hm, hg, ht⟩ := capacity_product_pair_tendsto hh hG χ
+    (fun i ↦ (hχ i).continuous.aestronglyMeasurable) hχb hχlim hM hbound
+    (fun i ↦ surfaceGradient S (χ i)) hC hC0
+  apply isClosed_closure.mem_of_tendsto ht
+  apply Filter.Eventually.of_forall
+  intro i
+  obtain ⟨F, hF, hFe, hFG⟩ := exists_c1_cutoff_trace hS hreg (χ i) ((hχ i).of_le (by norm_num)) (hχzero i)
+  have hμS : ∀ᵐ x ∂(surfaceMeasure S).restrict V, x ∈ S :=
+    ae_restrict_of_ae (ae_restrict_mem hS.measurableSet)
+  have he : F =ᵐ[(surfaceMeasure S).restrict V] (fun x ↦ χ i x * h x) := by
+    filter_upwards [hμS] with x hx using hFe hx
+  have hge : surfaceGradient S F =ᵐ[(surfaceMeasure S).restrict V]
+      (fun x ↦ χ i x • G x + h x • surfaceGradient S (χ i) x) := by
+    filter_upwards [hμS] with x hx using hFG hx
+  let hfmem := MemLp.ae_eq he.symm (hm i)
+  let hgfmem := MemLp.ae_eq hge.symm (hg i)
+  have hmem := c1_mem_sobolevGraph_bounded S V hV hVm hgrad hF hfmem hgfmem
+  have heLp := (hm i).toLp_congr hfmem he.symm
+  have hgeLp := (hg i).toLp_congr hgfmem hge.symm
+  simpa only [heLp, hgeLp] using hmem
+
+end RPI.Boundary
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPI.Boundary
+theorem HasC1TraceJetAt.comp {n : ℕ} {S : Set (Space n)}
+    {h : Space n → ℝ} {G : Space n → Space n} {x : Space n}
+    (hxS : x ∈ S) (hx : HasC1TraceJetAt S h G x)
+    {T : ℝ → ℝ} (hT : ContDiff ℝ 1 T) :
+    HasC1TraceJetAt S (T ∘ h) (fun y ↦ deriv T (h y) • G y) x := by
+  obtain ⟨U, hU, hxU, f, hf, hfh, hfG⟩ := hx
+  refine ⟨U, hU, hxU, T ∘ f, hT.comp_contDiffOn hf, ?_, ?_⟩
+  · intro y hy
+    exact congrArg T (hfh hy)
+  · rw [surfaceGradient_comp ((hf.contDiffAt (hU.mem_nhds hxU)).differentiableAt (by simp))
+        (hT.differentiable (by simp)).differentiableAt, hfh ⟨hxS, hxU⟩, hfG]
+
+theorem HasC1TraceJetAt.smoothTruncation {n : ℕ} {S : Set (Space n)}
+    {W : Space n → ℝ} {D : Space n → Space n} {x : Space n}
+    (hxS : x ∈ S) (hx : HasC1TraceJetAt S W D x)
+    {ε : ℝ} (hε : 0 < ε) :
+    HasC1TraceJetAt S (fun y ↦ smoothTruncation ε (W y))
+      (fun y ↦ truncationSlope ε (W y) • truncatedGradient W D y) x := by
+  simpa only [Function.comp_def, smoothTruncation_deriv,
+    truncationSlope_smul_truncatedGradient W D hε] using hx.comp hxS ((smoothTruncation_contDiff ε).of_le (by norm_num))
+
+theorem c1_surfaceGradient_memLp_on_ball {n : ℕ} {S : Set (Space n)}
+    {R : ℝ} [IsFiniteMeasure ((surfaceMeasure S).restrict (Metric.ball 0 R))]
+    {φ : Space n → ℝ} (hφ : ContDiff ℝ 1 φ)
+    (hm : AEStronglyMeasurable (surfaceGradient S φ)
+      ((surfaceMeasure S).restrict (Metric.ball 0 R))) :
+    MemLp (surfaceGradient S φ) 2 ((surfaceMeasure S).restrict (Metric.ball 0 R)) := by
+  have hc : Continuous (gradient φ) := continuous_iff_continuousAt.mpr (fun x ↦
+    continuousAt_gradient_of_contDiffAt (hφ.of_le (by norm_num)).contDiffAt)
+  obtain ⟨M, hM⟩ := (isCompact_closedBall (0 : Space n) R).exists_bound_of_continuousOn
+    hc.continuousOn
+  apply MemLp.of_bound hm M
+  filter_upwards [ae_restrict_mem measurableSet_ball] with x hx
+  calc
+    ‖surfaceGradient S φ x‖ ≤ ‖gradient φ x‖ := by
+      calc
+        _ ≤ ‖(tangentPlane S x).orthogonalProjectionOnto‖ * ‖gradient φ x‖ :=
+          (tangentPlane S x).orthogonalProjectionOnto.le_opNorm _
+        _ ≤ 1 * ‖gradient φ x‖ := mul_le_mul_of_nonneg_right
+          (tangentPlane S x).orthogonalProjectionOnto_norm_le (norm_nonneg _)
+        _ = _ := one_mul _
+    _ ≤ M := hM x (Metric.ball_subset_closedBall hx)
+
+end RPI.Boundary
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPI.Boundary
+
+theorem c1_exists_smooth_representative_near_regular_test {n : ℕ} {S Z : Set (Space n)}
+    (hS : IsClosed S) (hZ : IsClosed Z) {W : Space n → ℝ} {D : Space n → Space n}
+    (htrace : ∀ x ∈ S \ Z, HasC1TraceJetAt S W D x)
+    {K : Set (Space n)} (hK : IsClosed K) (hKZ : Disjoint K Z) :
+    ∃ F : Space n → ℝ, ContDiff ℝ 1 F ∧
+      (∀ x ∈ S ∩ K, F x = W x ∧ surfaceGradient S F x = D x) := by
+  obtain ⟨σ, hσ0, hσ1, _⟩ := exists_contMDiffMap_zero_one_nhds_of_isClosed
+    (modelWithCornersSelf ℝ (Space n)) hZ hK hKZ.symm (n := (⊤ : ℕ∞))
+  have hσ : ContDiff ℝ 1 σ := σ.contMDiff.contDiff.of_le (by norm_num)
+  obtain ⟨U, hU, hUZ, hUσ⟩ := mem_nhdsSet_iff_exists.mp hσ0
+  obtain ⟨F, hF, hFW, hFD⟩ := exists_c1_cutoff_trace hS htrace σ hσ
+    ⟨U, hU, hUZ, hUσ⟩
+  refine ⟨F, hF, ?_⟩
+  intro x hx
+  have hσx : σ x = 1 := hσ1.self_of_nhdsSet x hx.2
+  have hσnear : (σ : Space n → ℝ) =ᶠ[𝓝 x] (fun _ ↦ (1 : ℝ)) :=
+    hσ1.filter_mono (nhds_le_nhdsSet hx.2)
+  have hσg : surfaceGradient S σ x = 0 := by
+    have hg : gradient (σ : Space n → ℝ) x = 0 := by
+      simpa only [gradient_fun_const] using hσnear.gradient_eq
+    simp only [surfaceGradient, hg, map_zero, Submodule.coe_zero]
+  exact ⟨by simpa only [hσx, one_mul] using hFW hx.1,
+    by simpa only [hσx, one_smul, hσg, smul_zero, add_zero] using hFD hx.1⟩
+
+end RPI.Boundary
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPI.Boundary
+section RegularTests
+variable {n : ℕ} {S Z : Set (Space n)}
+variable (hS : IsClosed S) (hZ : IsClosed Z)
+variable (hfinite : ∀ R : ℝ, 0 < R → (surfaceMeasure S) (Metric.ball 0 R) < ⊤)
+variable (hgrad : ∀ f : Space n → ℝ, ContDiff ℝ 1 f →
+  AEStronglyMeasurable (surfaceGradient S f) (surfaceMeasure S))
+
+include hfinite hgrad
+
+theorem c1_surfaceGradient_memLp_of_compact {φ : Space n → ℝ}
+    (hφ : ContDiff ℝ 1 φ) (hcφ : HasCompactSupport φ) :
+    MemLp (surfaceGradient S φ) 2 (surfaceMeasure S) := by
+  obtain ⟨r, hr⟩ := Metric.isBounded_iff_subset_ball (0 : Space n) |>.mp hcφ.isBounded
+  let R := max r 1
+  have hR : 0 < R := lt_of_lt_of_le zero_lt_one (le_max_right _ _)
+  have hsupp : tsupport φ ⊆ Metric.ball (0 : Space n) R :=
+    hr.trans (Metric.ball_subset_ball (le_max_left _ _))
+  let : IsFiniteMeasure ((surfaceMeasure S).restrict (Metric.ball 0 R)) :=
+    ⟨by simpa only [Measure.restrict_apply_univ] using hfinite R hR⟩
+  have hloc := c1_surfaceGradient_memLp_on_ball (R := R) hφ
+    ((hgrad φ hφ).mono_measure Measure.restrict_le_self)
+  have hh := (memLp_indicator_iff_restrict measurableSet_ball).mpr hloc
+  apply hh.ae_eq
+  apply ae_of_all
+  intro x
+  by_cases hx : x ∈ Metric.ball (0 : Space n) R
+  · simp only [indicator_of_mem hx]
+  · simp only [indicator_of_notMem hx]
+    exact (surfaceGradient_eq_zero_of_notMem_tsupport S (fun h ↦ hx (hsupp h))).symm
+
+include hS hZ
+
+theorem c1_regular_gradient_memLp_indicator_test {W : Space n → ℝ} {D : Space n → Space n}
+    (htrace : ∀ x ∈ S \ Z, HasC1TraceJetAt S W D x)
+    {φ : Space n → ℝ} (hcφ : HasCompactSupport φ) (hφZ : Disjoint (tsupport φ) Z) :
+    MemLp ((tsupport φ).indicator D) 2 (surfaceMeasure S) := by
+  obtain ⟨F, hF, he⟩ := c1_exists_smooth_representative_near_regular_test hS hZ htrace
+    (isClosed_tsupport φ) hφZ
+  obtain ⟨r, hr⟩ := Metric.isBounded_iff_subset_ball (0 : Space n) |>.mp hcφ.isBounded
+  let R := max r 1
+  have hR : 0 < R := lt_of_lt_of_le zero_lt_one (le_max_right _ _)
+  have hsupp : tsupport φ ⊆ Metric.ball (0 : Space n) R :=
+    hr.trans (Metric.ball_subset_ball (le_max_left _ _))
+  let : IsFiniteMeasure ((surfaceMeasure S).restrict (Metric.ball 0 R)) :=
+    ⟨by simpa only [Measure.restrict_apply_univ] using hfinite R hR⟩
+  have hloc : MemLp (surfaceGradient S F) 2 ((surfaceMeasure S).restrict (tsupport φ)) :=
+    (c1_surfaceGradient_memLp_on_ball (R := R) hF ((hgrad F hF).mono_measure Measure.restrict_le_self)).mono_measure
+      (Measure.restrict_mono_set _ hsupp)
+  apply ((memLp_indicator_iff_restrict (isClosed_tsupport φ).measurableSet).mpr hloc).ae_eq
+  filter_upwards [ae_restrict_mem hS.measurableSet] with x hx
+  by_cases hxt : x ∈ tsupport φ
+  · simp only [indicator_of_mem hxt, (he x ⟨hx, hxt⟩).2]
+  · simp only [indicator_of_notMem hxt]
+
+theorem c1_integrable_regular_test_pair {W : Space n → ℝ} {D : Space n → Space n}
+    (htrace : ∀ x ∈ S \ Z, HasC1TraceJetAt S W D x)
+    {φ : Space n → ℝ} (hφ : ContDiff ℝ 1 φ) (hcφ : HasCompactSupport φ)
+    (hφZ : Disjoint (tsupport φ) Z) :
+    Integrable (fun x ↦ inner ℝ (D x) (surfaceGradient S φ x)) (surfaceMeasure S) := by
+  have hD := c1_regular_gradient_memLp_indicator_test hS hZ hfinite hgrad htrace hcφ hφZ
+  have hB := c1_surfaceGradient_memLp_of_compact hfinite hgrad hφ hcφ
+  apply (integrable_inner_of_memLp hD hB).congr
+  apply ae_of_all
+  intro x
+  by_cases hxt : x ∈ tsupport φ
+  · simp only [indicator_of_mem hxt]
+  · simp only [indicator_of_notMem hxt, surfaceGradient_eq_zero_of_notMem_tsupport S hxt,
+      inner_zero_right]
+
+end RegularTests
+end RPI.Boundary
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/HeightCompactness.lean b/lean/OAI/Geometry/Perimeter/Regularity/HeightCompactness.lean
new file mode 100644
index 0000000000000000000000000000000000000000..4ecd9d25f041622edbc0e231871da1c7c2879cc2
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/HeightCompactness.lean
@@ -0,0 +1,236 @@
+import OAI.Geometry.Perimeter.Regularity.GeoAlgebra
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff Convolution BoundedContinuousFunction
+namespace RPISingularHarnack
+open RPI.Boundary ContinuousLinearMap
+
+theorem c1Mollify_lintegral_error_of_translation {n : ℕ} {f : Euc n → ℝ}
+    (hf : Measurable f) (hi : Integrable f volume) (D : ℝ≥0∞)
+    (ht : ∀ w : Euc n, (∫⁻ x, ‖f (x+w)-f x‖ₑ) ≤ ‖w‖ₑ * D) (j : ℕ) :
+    (∫⁻ x, ‖c1Mollify f j x-f x‖ₑ) ≤
+      ENNReal.ofReal (c1Mollifier (n := n) j).rOut * D := by
+  let k : Euc n → ℝ := (c1Mollifier j).normed volume
+  have hkc : Continuous k := (c1Mollifier j).continuous_normed
+  have hk : Integrable k volume := hkc.integrable_of_hasCompactSupport
+    (c1Mollifier j).hasCompactSupport_normed
+  have hk1 : ∫ x, k x = 1 := (c1Mollifier j).integral_normed
+  obtain ⟨B,hB⟩ := ((c1Mollifier (n := n) j).hasCompactSupport_normed (μ := volume)).exists_bound_of_continuous hkc
+  have hp (x : Euc n) : ‖c1Mollify f j x-f x‖ₑ ≤
+      ∫⁻ y, ‖k y‖ₑ * ‖f (x-y)-f x‖ₑ := by
+    have hif : Integrable (fun y ↦ f (x-y)) volume :=
+      hi.comp_sub_left x
+    have hib : Integrable (fun y ↦ k y * f (x-y)) volume := by
+      apply (hif.norm.const_mul B).mono'
+      · exact (hkc.measurable.mul (hf.comp (measurable_const.sub measurable_id))).aestronglyMeasurable
+      · exact ae_of_all _ fun y ↦ by rw [norm_mul]; exact mul_le_mul_of_nonneg_right (hB y) (norm_nonneg _)
+    have he : c1Mollify f j x-f x = ∫ y, k y * (f (x-y)-f x) := by
+      simp only [c1Mollify, convolution_def, lsmul_apply, smul_eq_mul]
+      simp_rw [mul_sub]
+      rw [integral_sub hib (hk.mul_const _), integral_mul_const, hk1, one_mul]
+    rw [he]
+    apply (enorm_integral_le_lintegral_enorm _).trans_eq
+    exact lintegral_congr (fun _ ↦ enorm_mul _ _)
+  have hm : Measurable (fun p : Euc n × Euc n ↦
+      ‖k p.2‖ₑ * ‖f (p.1-p.2)-f p.1‖ₑ) :=
+    (hkc.measurable.comp measurable_snd).enorm.mul
+      ((hf.comp (measurable_fst.sub measurable_snd)).sub (hf.comp measurable_fst)).enorm
+  calc
+    _ ≤ ∫⁻ x, ∫⁻ y, ‖k y‖ₑ * ‖f (x-y)-f x‖ₑ := lintegral_mono hp
+    _ = ∫⁻ y, ∫⁻ x, ‖k y‖ₑ * ‖f (x-y)-f x‖ₑ := lintegral_lintegral_swap hm.aemeasurable
+    _ = ∫⁻ y, ‖k y‖ₑ * ∫⁻ x, ‖f (x-y)-f x‖ₑ := by
+      apply lintegral_congr
+      intro y
+      exact lintegral_const_mul _ ((hf.comp (measurable_id.sub measurable_const)).sub hf).enorm
+    _ ≤ ∫⁻ y, ‖k y‖ₑ * (‖y‖ₑ * D) := by
+      apply lintegral_mono
+      intro y
+      exact mul_le_mul_right (by simpa only [sub_eq_add_neg, enorm_neg] using ht (-y)) _
+    _ ≤ ∫⁻ y, ‖k y‖ₑ * (ENNReal.ofReal (c1Mollifier (n := n) j).rOut * D) := by
+      apply lintegral_mono
+      intro y
+      by_cases hy : k y = 0
+      · simp only [hy, enorm_zero, zero_mul, le_refl]
+      · have hys : y ∈ Function.support k := hy
+        rw [(c1Mollifier j).support_normed_eq] at hys
+        have hyb : ‖y‖ₑ ≤ ENNReal.ofReal (c1Mollifier (n := n) j).rOut := by
+          rw [← ofReal_norm]
+          exact ENNReal.ofReal_le_ofReal (by
+            have hylt : ‖y‖ < (c1Mollifier (n := n) j).rOut := by
+              simpa only [Metric.mem_ball, dist_zero_right] using hys
+            exact hylt.le)
+        exact mul_le_mul_right (mul_le_mul_left hyb _) _
+    _ = _ := by
+      rw [lintegral_mul_const _ hkc.measurable.enorm]
+      have hki : (∫⁻ y, ‖k y‖ₑ) = 1 := by
+        have heq : ∀ y, ‖k y‖ₑ = ENNReal.ofReal (k y) := by
+          intro y
+          rw [← ofReal_norm, Real.norm_eq_abs, abs_of_nonneg ((c1Mollifier j).nonneg_normed y)]
+        simp_rw [heq]
+        rw [← ofReal_integral_eq_lintegral_ofReal hk (ae_of_all _ (c1Mollifier j).nonneg_normed), hk1, ENNReal.ofReal_one]
+      rw [hki, one_mul]
+
+theorem c1Mollify_uniform_bounds_of_integral_norm {n : ℕ} (A : ℝ) (hA : 0 ≤ A)
+    (j : ℕ) : ∃ L : ℝ≥0, ∃ B : ℝ, ∀ f : Euc n → ℝ,
+      Measurable f → Integrable f volume → (∫ x, ‖f x‖) ≤ A →
+      LipschitzWith L (c1Mollify f j) ∧ ∀ x, ‖c1Mollify f j x‖ ≤ B := by
+  let k : Euc n → ℝ := (c1Mollifier j).normed volume
+  have hc : HasCompactSupport k := (c1Mollifier j).hasCompactSupport_normed
+  have hk : ContDiff ℝ ∞ k := (c1Mollifier j).contDiff_normed
+  obtain ⟨C,hC⟩ := hk.lipschitzWith_of_hasCompactSupport hc (by simp)
+  obtain ⟨B,hB⟩ := hc.exists_bound_of_continuous hk.continuous
+  have hB0 : 0 ≤ B := (norm_nonneg (k 0)).trans (hB 0)
+  refine ⟨C * ⟨A,hA⟩, B*A, ?_⟩
+  intro f hf hi hbound
+  have hflip : (lsmul ℝ ℝ : ℝ →L[ℝ] ℝ →L[ℝ] ℝ).flip = lsmul ℝ ℝ := by
+    apply ContinuousLinearMap.ext
+    intro a
+    apply ContinuousLinearMap.ext
+    intro b
+    exact mul_comm b a
+  have he : c1Mollify f j = f ⋆[lsmul ℝ ℝ,volume] k := by
+    change k ⋆[lsmul ℝ ℝ,volume] f = _
+    rw [← convolution_flip,hflip]
+  have hib (x : Euc n) : Integrable (fun z ↦ f z*k (x-z)) volume := by
+    apply (hi.norm.mul_const B).mono'
+    · exact (hf.mul (hk.continuous.measurable.comp (measurable_const.sub measurable_id))).aestronglyMeasurable
+    · exact ae_of_all _ fun z ↦ by rw [norm_mul]; exact mul_le_mul_of_nonneg_left (hB _) (norm_nonneg _)
+  constructor
+  · apply LipschitzWith.of_dist_le_mul
+    intro x y
+    rw [dist_eq_norm,he]
+    change ‖(∫ z, f z*k (x-z))-(∫ z, f z*k (y-z))‖ ≤ _
+    rw [← integral_sub (hib x) (hib y)]
+    calc
+      _ ≤ ∫ z, ‖f z*k (x-z)-f z*k (y-z)‖ := norm_integral_le_integral_norm _
+      _ ≤ ∫ z, ‖f z‖*((C:ℝ)*dist x y) := by
+        apply integral_mono ((hib x).sub (hib y)).norm (hi.norm.mul_const _)
+        intro z
+        change ‖f z*k (x-z)-f z*k (y-z)‖ ≤ ‖f z‖*((C:ℝ)*dist x y)
+        rw [← mul_sub,norm_mul]
+        apply mul_le_mul_of_nonneg_left _ (norm_nonneg _)
+        simpa only [← dist_eq_norm,dist_sub_right] using hC.dist_le_mul (x-z) (y-z)
+      _ = (∫ z, ‖f z‖)*((C:ℝ)*dist x y) := integral_mul_const _ _
+      _ ≤ A*((C:ℝ)*dist x y) := mul_le_mul_of_nonneg_right hbound (by positivity)
+      _ = _ := by change A*((C:ℝ)*dist x y) = ((C:ℝ)*A)*dist x y; ring
+  · intro x
+    rw [he]
+    change ‖∫ z, f z*k (x-z)‖ ≤ _
+    calc
+      _ ≤ ∫ z, ‖f z*k (x-z)‖ := norm_integral_le_integral_norm _
+      _ ≤ ∫ z, ‖f z‖*B := by
+        apply integral_mono (hib x).norm (hi.norm.mul_const B)
+        intro z
+        change ‖f z*k (x-z)‖ ≤ ‖f z‖*B
+        rw [norm_mul]
+        exact mul_le_mul_of_nonneg_left (hB _) (norm_nonneg _)
+      _ = (∫ z, ‖f z‖)*B := integral_mul_const _ _
+      _ ≤ A*B := mul_le_mul_of_nonneg_right hbound hB0
+      _ = B*A := mul_comm _ _
+
+theorem totallyBounded_toLp_of_lipschitz_norm
+    {α ι : Type*} [MetricSpace α] [CompactSpace α]
+    [MeasurableSpace α] [BorelSpace α] (μ : Measure α) [IsFiniteMeasure μ]
+    (f : ι → α →ᵇ ℝ) (L : NNReal) (B : ℝ)
+    (hf : ∀ i, LipschitzWith L (f i)) (hb : ∀ i x, ‖f i x‖ ≤ B) :
+    TotallyBounded (range (fun i ↦ BoundedContinuousFunction.toLp 1 μ ℝ (f i))) := by
+  have hc : IsCompact (closure (range f)) := by
+    apply BoundedContinuousFunction.arzela_ascoli (Metric.closedBall (0:ℝ) B)
+      (isCompact_closedBall _ _)
+    · rintro g x ⟨i,rfl⟩
+      simpa only [Metric.mem_closedBall,dist_zero_right] using hb i x
+    · apply UniformEquicontinuous.equicontinuous
+      apply LipschitzWith.uniformEquicontinuous _ L
+      intro g
+      obtain ⟨i,hi⟩ := g.property
+      simpa only [← hi] using hf i
+  exact (hc.image (BoundedContinuousFunction.toLp 1 μ ℝ).continuous).totallyBounded.subset
+    (by rintro _ ⟨i,rfl⟩; exact ⟨f i,subset_closure (mem_range_self i),rfl⟩)
+
+noncomputable def compactFunctionLp {n : ℕ} (K : Set (Euc n)) (hK : IsCompact K)
+    (f : Euc n → ℝ) (hi : Integrable f volume) :
+    Lp ℝ 1 (volume.comap (Subtype.val : K → Euc n)) :=
+  (memLp_one_iff_integrable.mpr ((integrableOn_iff_comap_subtypeVal hK.measurableSet).mp hi.integrableOn)).toLp _
+
+theorem compactFunctionLp_ae {n : ℕ} (K : Set (Euc n)) (hK : IsCompact K)
+    (f : Euc n → ℝ) (hi : Integrable f volume) :
+    compactFunctionLp K hK f hi =ᵐ[volume.comap (Subtype.val : K → Euc n)]
+      fun x : K ↦ f x :=
+  by unfold compactFunctionLp; exact MemLp.coeFn_toLp _
+
+theorem compactFunctionLp_totallyBounded_of_translation {n : ℕ} {ι : Type*}
+    (K : Set (Euc n)) (hK : IsCompact K) (f : ι → Euc n → ℝ)
+    (hm : ∀ i, Measurable (f i)) (hi : ∀ i, Integrable (f i) volume)
+    (A : ℝ) (hA : 0 ≤ A) (hb : ∀ i, (∫ x, ‖f i x‖) ≤ A)
+    (D : ℝ≥0∞) (hD : D ≠ ⊤)
+    (ht : ∀ i w, (∫⁻ x, ‖f i (x+w)-f i x‖ₑ) ≤ ‖w‖ₑ * D) :
+    TotallyBounded (range (fun i ↦ compactFunctionLp K hK (f i) (hi i))) := by
+  let μ : Measure K := volume.comap (Subtype.val : K → Euc n)
+  let : IsFiniteMeasure μ := compact_subtype_volume_finite hK
+  let : CompactSpace K := isCompact_iff_compactSpace.mp hK
+  have hmc (i : ι) (j : ℕ) : Continuous (c1Mollify (f i) j) :=
+    ((c1Mollifier j).hasCompactSupport_normed.contDiff_convolution_left (n := (⊤:ℕ∞)) _
+      (c1Mollifier j).contDiff_normed (hi i).locallyIntegrable).continuous
+  let a : ℕ → ι → K →ᵇ ℝ := fun j i ↦ BoundedContinuousFunction.mkOfCompact
+    ⟨fun x ↦ c1Mollify (f i) j x, (hmc i j).comp continuous_subtype_val⟩
+  apply totallyBounded_range_of_uniform_approximation
+    (fun i ↦ compactFunctionLp K hK (f i) (hi i))
+    (fun j i ↦ BoundedContinuousFunction.toLp 1 μ ℝ (a j i))
+  · intro j
+    obtain ⟨L,B,hLB⟩ := c1Mollify_uniform_bounds_of_integral_norm (n := n) A hA j
+    apply totallyBounded_toLp_of_lipschitz_norm μ (a j) L B
+    · intro i
+      apply LipschitzWith.of_dist_le_mul
+      intro x y
+      exact (hLB (f i) (hm i) (hi i) (hb i)).1.dist_le_mul x y
+    · intro i x
+      exact (hLB (f i) (hm i) (hi i) (hb i)).2 x
+  · intro ε hε
+    have htend : Tendsto (fun j ↦ ENNReal.ofReal (c1Mollifier (n := n) j).rOut * D)
+        atTop (𝓝 0) := by
+      simpa only [ENNReal.ofReal_zero,zero_mul] using
+        ENNReal.Tendsto.mul_const (ENNReal.tendsto_ofReal c1Mollifier_radius_tendsto) (Or.inr hD)
+    obtain ⟨j,hj⟩ := (htend.eventually (gt_mem_nhds (ENNReal.ofReal_pos.mpr hε))).exists
+    refine ⟨j,fun i ↦ edist_lt_ofReal.mp ?_⟩
+    apply lt_of_le_of_lt _ hj
+    rw [Lp.edist_def, eLpNorm_one_eq_lintegral_enorm
+      ((Lp.aestronglyMeasurable _).sub (Lp.aestronglyMeasurable _))]
+    calc
+      _ = ∫⁻ x : K, ‖f i x-c1Mollify (f i) j x‖ₑ ∂μ := by
+        apply lintegral_congr_ae
+        filter_upwards [compactFunctionLp_ae K hK (f i) (hi i),
+          BoundedContinuousFunction.coeFn_toLp 1 μ ℝ (a j i)] with x hx hy
+        change ‖(compactFunctionLp K hK (f i) (hi i)) x-
+          (BoundedContinuousFunction.toLp 1 μ ℝ (a j i)) x‖ₑ = _
+        rw [hx,hy]
+        rfl
+      _ = ∫⁻ x in K, ‖f i x-c1Mollify (f i) j x‖ₑ :=
+        lintegral_subtype_comap (μ := volume) hK.measurableSet
+          (fun x : Euc n ↦ ‖f i x-c1Mollify (f i) j x‖ₑ)
+      _ ≤ ∫⁻ x, ‖f i x-c1Mollify (f i) j x‖ₑ := setLIntegral_le_lintegral _ _
+      _ ≤ _ := by
+        simpa only [enorm_sub_rev] using
+          c1Mollify_lintegral_error_of_translation (hm i) (hi i) D (ht i) j
+
+theorem compactFunctionLp_subsequence_of_translation {n : ℕ}
+    (K : Set (Euc n)) (hK : IsCompact K) (f : ℕ → Euc n → ℝ)
+    (hm : ∀ i, Measurable (f i)) (hi : ∀ i, Integrable (f i) volume)
+    (A : ℝ) (hA : 0 ≤ A) (hb : ∀ i, (∫ x, ‖f i x‖) ≤ A)
+    (D : ℝ≥0∞) (hD : D ≠ ⊤)
+    (ht : ∀ i w, (∫⁻ x, ‖f i (x+w)-f i x‖ₑ) ≤ ‖w‖ₑ * D) :
+    ∃ g : Lp ℝ 1 (volume.comap (Subtype.val : K → Euc n)),
+      ∃ σ : ℕ → ℕ, StrictMono σ ∧
+        Tendsto (fun i ↦ compactFunctionLp K hK (f (σ i)) (hi (σ i))) atTop (𝓝 g) := by
+  have hc := (compactFunctionLp_totallyBounded_of_translation K hK f hm hi A hA hb D hD ht).closure.isCompact_of_isClosed isClosed_closure
+  obtain ⟨g,-,σ,hσ,hg⟩ := hc.tendsto_subseq (fun i ↦ subset_closure
+    (mem_range_self (f := fun i ↦ compactFunctionLp K hK (f i) (hi i)) i))
+  exact ⟨g,σ,hσ,hg⟩
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/HeightMeanValue.lean b/lean/OAI/Geometry/Perimeter/Regularity/HeightMeanValue.lean
new file mode 100644
index 0000000000000000000000000000000000000000..6e00647bb4021b3f14cf4c2ddf385ef434e78991
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/HeightMeanValue.lean
@@ -0,0 +1,401 @@
+import OAI.Geometry.Perimeter.Regularity.SmulMemPhaseSupport
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff Pointwise
+namespace RPISingularHarnack
+open RPI.Boundary
+
+private theorem gradient_affine_height {n : ℕ} (v : Euc n) (a : ℝ) (x : Euc n) :
+    gradient (fun y ↦ inner ℝ v y - a) x = v := by
+  apply HasGradientAt.gradient
+  change HasFDerivAt (fun y ↦ inner ℝ v y - a)
+    ((InnerProductSpace.toDual ℝ (Euc n)) v) x
+  exact (innerSL ℝ v).hasFDerivAt.sub_const a
+
+private theorem gradient_radial_primitive {n : ℕ} (c : Euc n) (t : ℝ)
+    {Ψ ψ : ℝ → ℝ} (hΨ : ContDiff ℝ ∞ Ψ) (hdΨ : ∀ s, deriv Ψ s = ψ s)
+    (x : Euc n) :
+    gradient (fun y ↦ Ψ (t^2 * ‖y-c‖^2)) x =
+      (2*t^2*ψ (t^2*‖x-c‖^2)) • (x-c) := by
+  have hi := (((hasFDerivAt_id (𝕜 := ℝ) x).sub_const c).norm_sq).const_mul (t^2)
+  have hh := ((hΨ.differentiable (by simp)) _).hasDerivAt.comp_hasFDerivAt x hi
+  rw [hdΨ] at hh
+  change HasFDerivAt (fun y : Euc n ↦ Ψ (t^2 * ‖y-c‖^2)) _ x at hh
+  apply (InnerProductSpace.toDual ℝ (Euc n)).injective
+  rw [toDual_gradient, hh.fderiv]
+  ext z
+  simp only [smul_apply, ContinuousLinearMap.comp_apply,
+    ContinuousLinearMap.id_apply, id_eq, innerSL_apply_apply, map_smul,
+    InnerProductSpace.toDual_apply_apply, smul_eq_mul]
+  ring
+
+theorem minimizing_height_radialDerivative_nonpos {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    (c v : Euc n) (a : ℝ) {M : ℝ} (hM : 0 < M)
+    {Ψ ψ : ℝ → ℝ} (hΨ : ContDiff ℝ ∞ Ψ) (hψ : ContDiff ℝ ∞ ψ)
+    (hdΨ : ∀ s, deriv Ψ s = ψ s)
+    (hΨz : ∀ s, 1 ≤ s → Ψ s = 0) (hψz : ∀ s, 1 ≤ s → ψ s = 0)
+    (hΨ0 : ∀ s, Ψ s ≤ 0) (hψd : ∀ s, deriv ψ s ≤ 0)
+    {t : ℝ} (ht : 1/M < t) :
+    (∫ x in Metric.ball c M, (inner ℝ v x-a)^2 *
+      radialDensityDerivative c (n-1) ψ t x ∂perimeterMeasure E) ≤ 0 := by
+  let μ := (perimeterMeasure E).restrict (Metric.ball c M)
+  have := perimeterMeasure_locallyFinite E hmin.2.1
+  have : IsFiniteMeasure μ := ⟨by
+    rw [Measure.restrict_apply_univ, perimeterMeasure_open E Metric.isOpen_ball]
+    exact hmin.2.1 c M hM⟩
+  have ht0 : 0 < t := (one_div_pos.mpr hM).trans ht
+  have htM : 1/t < M := (div_lt_iff₀ ht0).mpr (by
+    have := (div_lt_iff₀ hM).mp ht
+    nlinarith)
+  let H : Euc n → ℝ := fun x ↦ inner ℝ v x-a
+  let P : Euc n → ℝ := fun x ↦ Ψ (t^2*‖x-c‖^2)
+  let Q : Euc n → ℝ := fun x ↦ ψ (t^2*‖x-c‖^2)
+  let V : Euc n → Euc n := fun x ↦ Q x • (x-c)
+  let W : Euc n → Euc n := fun x ↦ H x^2 • V x
+  let φ : Euc n → ℝ := fun x ↦ H x * P x
+  have hH : ContDiff ℝ ∞ H := (contDiff_const.inner ℝ contDiff_id).sub contDiff_const
+  have hP : ContDiff ℝ ∞ P :=
+    hΨ.comp (contDiff_const.mul ((contDiff_id.sub contDiff_const).norm_sq (𝕜 := ℝ)))
+  have hQ : ContDiff ℝ ∞ Q :=
+    hψ.comp (contDiff_const.mul ((contDiff_id.sub contDiff_const).norm_sq (𝕜 := ℝ)))
+  have hV : ContDiff ℝ ∞ V := hQ.smul (𝕜' := ℝ) (contDiff_id.sub contDiff_const)
+  have hW : ContDiff ℝ ∞ W := (hH.pow 2).smul (𝕜' := ℝ) hV
+  have hφ : ContDiff ℝ ∞ φ := hH.mul hP
+  have hsV : tsupport V ⊆ Metric.closedBall c (1/t) :=
+    (tsupport_smul_subset_left _ _).trans (radialProfile_tsupport c hψz ht0)
+  have hsW : tsupport W ⊆ Metric.closedBall c (1/t) :=
+    (tsupport_smul_subset_right (fun x ↦ H x^2) V).trans hsV
+  have hsφ : tsupport φ ⊆ Metric.closedBall c (1/t) :=
+    tsupport_mul_subset_right.trans (radialProfile_tsupport c hΨz ht0)
+  have hcW : HasCompactSupport W :=
+    (isCompact_closedBall c (1/t)).of_isClosed_subset (isClosed_tsupport _) hsW
+  have hcφ : HasCompactSupport φ :=
+    (isCompact_closedBall c (1/t)).of_isClosed_subset (isClosed_tsupport _) hsφ
+  have hzW := minimizing_tangentialDivergence_integral_eq_zero hn hmin c hM hW hcW
+    (hsW.trans (Metric.closedBall_subset_ball htM))
+  have hzφ := minimizing_tangentialGradient_integral_eq_zero_local hn hmin c hM hφ hcφ
+    (hsφ.trans (Metric.closedBall_subset_ball htM)) v
+  have hiD : Integrable (fun x ↦ H x^2 * radialDensityDerivative c (n-1) ψ t x) μ := by
+    exact (((hH.continuous.pow 2).mul
+      ((radialDensityDerivative_continuous c (n-1) hψ).comp
+        (continuous_const.prodMk continuous_id))).continuousOn.integrableOn_compact
+        (isCompact_closedBall c M)).mono_set Metric.ball_subset_closedBall
+  have hiW := (tangentialDivergence_integrable_ball_perimeter hmin c hM hW).const_mul
+    (t^((n-1)-1))
+  have hiφ := ((tangentialGradient_integrable_ball_perimeter hmin c hM
+    (hφ.of_le (by simp))).inner_const v).const_mul (t^((n-1)-1)/t^2)
+  have hcmp : ∀ᵐ x ∂μ,
+      H x^2 * radialDensityDerivative c (n-1) ψ t x ≤
+        t^((n-1)-1) * tangentialDivergence E W x -
+          (t^((n-1)-1)/t^2) * inner ℝ (tangentialGradient E φ x) v := by
+    filter_upwards [ae_restrict_of_ae (minimizing_halfspaceBlowup_ae hn hmin)] with x hx
+    have hpow : t^(n-1) = t^((n-1)-1)*t := by
+      rw [← pow_succ, Nat.sub_add_cancel (by omega : 1 ≤ n-1)]
+    have hgH := gradient_affine_height v a x
+    change gradient H x = v at hgH
+    have hGnonneg : 0 ≤ ‖v‖^2 - (inner ℝ v (approximateNormal E x))^2 := by
+      have hh := norm_inner_le_norm (𝕜 := ℝ) v (approximateNormal E x)
+      rw [hx.1, mul_one, Real.norm_eq_abs] at hh
+      have hsq := mul_self_le_mul_self (abs_nonneg _) hh
+      nlinarith [sq_abs (inner ℝ v (approximateNormal E x))]
+    have hf : inner ℝ (tangentialGradient E φ x) v =
+        P x * (‖v‖^2 - (inner ℝ v (approximateNormal E x))^2) +
+        (2*t^2*H x*Q x) * (inner ℝ v (x-c) -
+          inner ℝ v (approximateNormal E x) * inner ℝ (approximateNormal E x) (x-c)) := by
+      dsimp only [φ]
+      rw [tangentialGradient_mul E (hH.differentiable (by simp) x)
+        (hP.differentiable (by simp) x)]
+      simp only [inner_add_left, real_inner_smul_left, tangentialGradient, hgH]
+      rw [show gradient P x = (2*t^2*Q x) • (x-c) from
+        gradient_radial_primitive c t hΨ hdΨ x]
+      simp only [inner_sub_left, real_inner_smul_left,
+        real_inner_self_eq_norm_sq]
+      simp only [inner_sub_right]
+      simp only [real_inner_comm]
+      ring
+    have hgh2 : tangentialGradient E (fun y ↦ H y^2) x =
+        (2*H x) • tangentialGradient E H x := by
+      have he : (fun y ↦ H y^2) = fun y ↦ H y*H y := by funext y; ring
+      rw [he, tangentialGradient_mul E (hH.differentiable (by simp) x)
+        (hH.differentiable (by simp) x)]
+      module
+    have hdiv : tangentialDivergence E W x =
+        H x^2 * ((n-1:ℕ)* Q x + 2*t^2*deriv ψ (t^2*‖x-c‖^2) *
+          (‖x-c‖^2 - (inner ℝ (approximateNormal E x) (x-c))^2)) +
+        (2*H x*Q x) * (inner ℝ v (x-c) -
+          inner ℝ v (approximateNormal E x) * inner ℝ (approximateNormal E x) (x-c)) := by
+      dsimp only [W]
+      rw [tangentialDivergence_smul E ((hH.pow 2).differentiable (by simp))
+        (hV.differentiable (by simp)), hgh2]
+      rw [show tangentialDivergence E V x = _ from
+        tangentialDivergence_radialProfile (by omega) E c hψ t x hx.1]
+      simp only [V, tangentialGradient, hgH, inner_sub_left, real_inner_smul_left,
+        real_inner_smul_right]
+      ring
+    rw [hf, hdiv]
+    have hbad : (t^((n-1)-1)/t^2) * P x *
+        (‖v‖^2 - (inner ℝ v (approximateNormal E x))^2) ≤ 0 :=
+      mul_nonpos_of_nonpos_of_nonneg
+        (mul_nonpos_of_nonneg_of_nonpos (by positivity) (hΨ0 _)) hGnonneg
+    have hrad : 2*t^((n-1)-1)*t^2 * H x^2 *
+        (inner ℝ (approximateNormal E x) (x-c))^2 * deriv ψ (t^2*‖x-c‖^2) ≤ 0 :=
+      mul_nonpos_of_nonneg_of_nonpos (by positivity) (hψd _)
+    have halg : H x^2 * radialDensityDerivative c (n-1) ψ t x =
+        t^((n-1)-1) *
+          (H x^2 * ((n-1:ℕ)*Q x + 2*t^2*deriv ψ (t^2*‖x-c‖^2) *
+            (‖x-c‖^2-(inner ℝ (approximateNormal E x) (x-c))^2)) +
+            (2*H x*Q x)*(inner ℝ v (x-c)-inner ℝ v (approximateNormal E x)*
+              inner ℝ (approximateNormal E x) (x-c))) -
+          (t^((n-1)-1)/t^2) * (P x*(‖v‖^2-(inner ℝ v (approximateNormal E x))^2)+
+            (2*t^2*H x*Q x)*(inner ℝ v (x-c)-inner ℝ v (approximateNormal E x)*
+              inner ℝ (approximateNormal E x) (x-c))) +
+          (t^((n-1)-1)/t^2) * P x*(‖v‖^2-(inner ℝ v (approximateNormal E x))^2) +
+          2*t^((n-1)-1)*t^2*H x^2*(inner ℝ (approximateNormal E x) (x-c))^2*
+            deriv ψ (t^2*‖x-c‖^2) := by
+      dsimp only [radialDensityDerivative, Q]
+      rw [hpow]
+      field_simp
+      ring
+    linarith
+  have hh := integral_mono_ae hiD (hiW.sub hiφ) hcmp
+  simp only [Pi.sub_apply] at hh
+  dsimp only [μ] at hh
+  rw [integral_sub hiW hiφ, integral_const_mul, integral_const_mul] at hh
+  change _ ≤ _ * (∫ x in Metric.ball c M, tangentialDivergence E W x ∂perimeterMeasure E) -
+    _ * (∫ x in Metric.ball c M, inner ℝ (tangentialGradient E φ x) v ∂perimeterMeasure E) at hh
+  simpa only [hzW, hzφ, mul_zero, sub_self] using hh
+
+def densityPrimitive (j : ℕ) (s : ℝ) : ℝ :=
+  ∫ z in (1:ℝ)..s, densityProfile j z
+
+theorem densityPrimitive_hasDerivAt (j : ℕ) (s : ℝ) :
+    HasDerivAt (densityPrimitive j) (densityProfile j s) s := by
+  exact intervalIntegral.integral_hasDerivAt_right
+    ((densityProfile_contDiff j).continuous.intervalIntegrable 1 s)
+    (densityProfile_contDiff j).continuous.stronglyMeasurable.stronglyMeasurableAtFilter
+    (densityProfile_contDiff j).continuous.continuousAt
+
+theorem densityPrimitive_deriv (j : ℕ) (s : ℝ) :
+    deriv (densityPrimitive j) s = densityProfile j s := (densityPrimitive_hasDerivAt j s).deriv
+
+theorem densityPrimitive_contDiff (j : ℕ) : ContDiff ℝ ∞ (densityPrimitive j) := by
+  apply contDiff_infty_iff_deriv.mpr
+  refine ⟨fun s ↦ (densityPrimitive_hasDerivAt j s).differentiableAt, ?_⟩
+  simpa only [funext (densityPrimitive_deriv j)] using densityProfile_contDiff j
+
+theorem densityPrimitive_zero (j : ℕ) {s : ℝ} (hs : 1 ≤ s) :
+    densityPrimitive j s = 0 := by
+  unfold densityPrimitive
+  trans ∫ z in (1:ℝ)..s, (0:ℝ)
+  · apply intervalIntegral.integral_congr
+    intro z hz
+    rw [uIcc_of_le hs] at hz
+    exact densityProfile_zero j hz.1
+  · simp
+
+theorem densityPrimitive_nonpos (j : ℕ) (s : ℝ) : densityPrimitive j s ≤ 0 := by
+  by_cases hs : 1 ≤ s
+  · rw [densityPrimitive_zero j hs]
+  · unfold densityPrimitive
+    rw [intervalIntegral.integral_symm]
+    exact neg_nonpos.mpr (intervalIntegral.integral_nonneg (le_of_not_ge hs)
+      fun z _ ↦ (densityProfile_mem_Icc j z).1)
+
+private theorem integral_hasDerivAt_compact_window {n : ℕ}
+    {F D : ℝ → Euc n → ℝ} (hF : Continuous (Function.uncurry F))
+    (hD : Continuous (Function.uncurry D))
+    (hd : ∀ s x, HasDerivAt (fun t ↦ F t x) (D s x) s)
+    (μ : Measure (Euc n)) [IsFiniteMeasure μ]
+    {K : Set (Euc n)} (hK : IsCompact K) (hμ : ∀ᵐ x ∂μ, x ∈ K) (t : ℝ) :
+    HasDerivAt (fun s ↦ ∫ x, F s x ∂μ) (∫ x, D t x ∂μ) t := by
+  have hFt : Continuous (F t) := hF.comp (continuous_const.prodMk continuous_id)
+  have hDt : Continuous (D t) := hD.comp (continuous_const.prodMk continuous_id)
+  obtain ⟨C, hC⟩ := hK.exists_bound_of_continuousOn hFt.continuousOn
+  have hFi : Integrable (F t) μ :=
+    Integrable.of_bound hFt.aestronglyMeasurable C (hμ.mono fun x hx ↦ hC x hx)
+  obtain ⟨B, hB⟩ := (isCompact_Icc.prod hK).exists_bound_of_continuousOn
+    (hD.continuousOn : ContinuousOn _ (Icc (t-1) (t+1) ×ˢ K))
+  have hbound : ∀ᵐ x ∂μ, ∀ s ∈ Ioo (t-1) (t+1), ‖D s x‖ ≤ B := by
+    filter_upwards [hμ] with x hx
+    intro s hs
+    exact hB (s,x) ⟨⟨hs.1.le, hs.2.le⟩,hx⟩
+  have hderiv : ∀ᵐ x ∂μ, ∀ s ∈ Ioo (t-1) (t+1),
+      HasDerivAt (fun t ↦ F t x) (D s x) s := ae_of_all μ fun x s _ ↦ hd s x
+  exact (hasDerivAt_integral_of_dominated_loc_of_deriv_le
+    (F := F) (F' := D) (s := Ioo (t-1) (t+1)) (bound := fun _ ↦ B)
+    (Ioo_mem_nhds (by linarith) (by linarith))
+    (Eventually.of_forall fun s ↦ (hF.comp ((continuous_const : Continuous (fun _ : Euc n ↦ s)).prodMk continuous_id)).aestronglyMeasurable)
+    hFi hDt.aestronglyMeasurable hbound (integrable_const B) hderiv).2
+
+theorem height_radial_integral_hasDerivAt {n : ℕ}
+    (c v : Euc n) (a M : ℝ) (k : ℕ)
+    {ψ : ℝ → ℝ} (hψ : ContDiff ℝ ∞ ψ) (μ : Measure (Euc n)) [IsFiniteMeasure μ]
+    (hμ : ∀ᵐ x ∂μ, x ∈ Metric.closedBall c M) (t : ℝ) :
+    HasDerivAt (fun s ↦ ∫ x, (inner ℝ v x-a)^2 * radialDensityIntegrand c k ψ s x ∂μ)
+      (∫ x, (inner ℝ v x-a)^2 * radialDensityDerivative c k ψ t x ∂μ) t := by
+  let H : Euc n → ℝ := fun x ↦ (inner ℝ v x-a)^2
+  have hH : Continuous H := ((continuous_const.inner continuous_id).sub continuous_const).pow 2
+  have hcF : Continuous (fun p : ℝ × Euc n ↦ H p.2 * radialDensityIntegrand c k ψ p.1 p.2) :=
+    (hH.comp continuous_snd).mul (radialDensityIntegrand_continuous c k hψ)
+  have hcD : Continuous (fun p : ℝ × Euc n ↦ H p.2 * radialDensityDerivative c k ψ p.1 p.2) :=
+    (hH.comp continuous_snd).mul (radialDensityDerivative_continuous c k hψ)
+  exact integral_hasDerivAt_compact_window hcF hcD
+    (fun s x ↦ (radialDensityIntegrand_hasDerivAt c k hψ s x).const_mul (H x))
+    μ (isCompact_closedBall c M) hμ t
+
+theorem minimizing_height_smoothedDensity_antitoneOn {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    (c v : Euc n) (a : ℝ) {M : ℝ} (hM : 0 < M) (j : ℕ) :
+    AntitoneOn (fun t ↦ ∫ x in Metric.ball c M, (inner ℝ v x-a)^2 *
+      radialDensityIntegrand c (n-1) (densityProfile j) t x ∂perimeterMeasure E) (Ioi (1/M)) := by
+  let μ := (perimeterMeasure E).restrict (Metric.ball c M)
+  have : IsFiniteMeasure μ := ⟨by
+    rw [Measure.restrict_apply_univ, perimeterMeasure_open E Metric.isOpen_ball]
+    exact hmin.2.1 c M hM⟩
+  have hμ : ∀ᵐ x ∂μ, x ∈ Metric.closedBall c M :=
+    (ae_restrict_mem measurableSet_ball).mono fun _ hx ↦ Metric.ball_subset_closedBall hx
+  have hd := fun t ↦ height_radial_integral_hasDerivAt c v a M (n-1)
+    (densityProfile_contDiff j) μ hμ t
+  apply antitoneOn_of_hasDerivWithinAt_nonpos (convex_Ioi (1/M))
+    (fun t _ ↦ (hd t).continuousAt.continuousWithinAt)
+    (fun t _ ↦ (hd t).hasDerivWithinAt)
+  intro t ht
+  exact minimizing_height_radialDerivative_nonpos hn hmin c v a hM
+    (densityPrimitive_contDiff j) (densityProfile_contDiff j) (densityPrimitive_deriv j)
+    (fun s hs ↦ densityPrimitive_zero j hs) (fun s hs ↦ densityProfile_zero j hs)
+    (densityPrimitive_nonpos j) (densityProfile_deriv_nonpos j)
+    (by simpa only [interior_Ioi, mem_Ioi] using ht)
+
+theorem height_radial_integral_tendsto {n : ℕ}
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    (c v : Euc n) (a : ℝ) {M : ℝ} (hM : 0 < M) {t : ℝ} (ht : 1/M < t) :
+    Tendsto (fun j ↦ ∫ x in Metric.ball c M, (inner ℝ v x-a)^2 *
+      radialDensityIntegrand c (n-1) (densityProfile j) t x ∂perimeterMeasure E) atTop
+      (𝓝 (t^(n-1) * ∫ x in Metric.ball c (1/t), (inner ℝ v x-a)^2 ∂perimeterMeasure E)) := by
+  let μ := (perimeterMeasure E).restrict (Metric.ball c M)
+  let H : Euc n → ℝ := fun x ↦ (inner ℝ v x-a)^2
+  have hH : Continuous H := ((continuous_const.inner continuous_id).sub continuous_const).pow 2
+  have := perimeterMeasure_locallyFinite E hmin.2.1
+  have : IsFiniteMeasure μ := ⟨by
+    rw [Measure.restrict_apply_univ, perimeterMeasure_open E Metric.isOpen_ball]
+    exact hmin.2.1 c M hM⟩
+  have ht0 : 0 < t := (one_div_pos.mpr hM).trans ht
+  have htM : 1/t < M := (div_lt_iff₀ ht0).mpr (by
+    have := (div_lt_iff₀ hM).mp ht
+    nlinarith)
+  have hiH : Integrable H μ := (hH.continuousOn.integrableOn_compact
+    (isCompact_closedBall c M)).mono_set Metric.ball_subset_closedBall
+  have hh := tendsto_integral_of_dominated_convergence (μ := μ) (fun x ↦ H x*t^(n-1))
+    (fun j ↦ (hH.mul (((radialDensityIntegrand_continuous c (n-1) (densityProfile_contDiff j)).comp
+      (continuous_const.prodMk continuous_id)))).aestronglyMeasurable)
+    (hiH.mul_const _) (fun j ↦ ae_of_all μ fun x ↦ ?_)
+    (ae_of_all μ fun x ↦ (tendsto_const_nhds (x := H x)).mul
+      (radialDensityIntegrand_tendsto c (n-1) ht0 x))
+  · have heq : (∫ x, H x * (Metric.ball c (1/t)).indicator (fun _ ↦ t^(n-1)) x ∂μ) =
+        t^(n-1) * ∫ x in Metric.ball c (1/t), H x ∂perimeterMeasure E := by
+      have hpt : ∀ x, H x * (Metric.ball c (1/t)).indicator (fun _ ↦ t^(n-1)) x =
+          t^(n-1) * (Metric.ball c (1/t)).indicator H x := by
+        intro x
+        by_cases hx : x ∈ Metric.ball c (1/t)
+        · rw [indicator_of_mem hx, indicator_of_mem hx, mul_comm]
+        · rw [indicator_of_notMem hx, indicator_of_notMem hx, mul_zero, mul_zero]
+      simp_rw [hpt]
+      rw [integral_const_mul, integral_indicator measurableSet_ball]
+      dsimp only [μ]
+      rw [Measure.restrict_restrict measurableSet_ball,
+        inter_eq_left.mpr (Metric.ball_subset_ball htM.le)]
+    rwa [heq] at hh
+  · change |(inner ℝ v x-a)^2 * (t^(n-1)*densityProfile j (t^2*‖x-c‖^2))| ≤
+      (inner ℝ v x-a)^2 * t^(n-1)
+    rw [abs_of_nonneg (mul_nonneg (sq_nonneg _) (mul_nonneg
+      (pow_nonneg ht0.le _) (densityProfile_mem_Icc j _).1))]
+    exact mul_le_mul_of_nonneg_left
+      (mul_le_of_le_one_right (pow_nonneg ht0.le _) (densityProfile_mem_Icc j _).2) (sq_nonneg _)
+
+theorem minimizing_height_density_monotone {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E) (c v : Euc n) (a : ℝ) :
+    MonotoneOn (fun r : ℝ ↦ (∫ x in Metric.ball c r,
+      (inner ℝ v x-a)^2 ∂perimeterMeasure E) / r^(n-1)) (Ioi 0) := by
+  intro r hr R hR hrR
+  have hr0 : 0 < r := hr
+  have hR0 : 0 < R := hR
+  let M := 2*R
+  have hM : 0 < M := by dsimp [M]; positivity
+  have hRM : 1/M < 1/R := one_div_lt_one_div_of_lt hR0 (by dsimp [M]; linarith)
+  have htr : 1/R ≤ 1/r := one_div_le_one_div_of_le hr0 hrR
+  have hrM : 1/M < 1/r := hRM.trans_le htr
+  have hseq (j : ℕ) := minimizing_height_smoothedDensity_antitoneOn hn hmin c v a hM j hRM hrM htr
+  have hh := le_of_tendsto_of_tendsto
+    (height_radial_integral_tendsto hmin c v a hM hrM)
+    (height_radial_integral_tendsto hmin c v a hM hRM) (Eventually.of_forall hseq)
+  simpa [one_div, div_eq_mul_inv, mul_comm] using hh
+
+theorem exists_minimizing_height_mean_value_constant (n : ℕ) (hn : 1 < n) :
+    ∃ C : ℝ, 0 < C ∧ ∀ E : Set (Euc n), IsLocallyPerimeterMinimizing E →
+      ∀ p ∈ perimeterSupport E, ∀ R : ℝ, 0 < R → ∀ (v : Euc n) (a : ℝ),
+        (inner ℝ v p-a)^2 ≤ C/R^(n-1) *
+          ∫ x in Metric.ball p R, (inner ℝ v x-a)^2 ∂perimeterMeasure E := by
+  obtain ⟨c₀, hc₀, hdensity⟩ := exists_minimizing_perimeter_lower_density n hn
+  refine ⟨2/c₀, by positivity, ?_⟩
+  intro E hmin p hp R hR v a
+  let H : Euc n → ℝ := fun x ↦ (inner ℝ v x-a)^2
+  have hH : Continuous H := ((continuous_const.inner continuous_id).sub continuous_const).pow 2
+  have hI0 : 0 ≤ ∫ x in Metric.ball p R, H x ∂perimeterMeasure E :=
+    integral_nonneg fun x ↦ sq_nonneg _
+  by_cases hz : H p = 0
+  · change H p ≤ _
+    rw [hz]
+    exact mul_nonneg (by positivity) hI0
+  have hp0 : 0 < H p := lt_of_le_of_ne (sq_nonneg _) (Ne.symm hz)
+  have ht : {x | H p/2 < H x} ∈ 𝓝 p :=
+    hH.continuousAt.preimage_mem_nhds (Ioi_mem_nhds (by linarith))
+  obtain ⟨r, hr, hsub⟩ := Metric.mem_nhds_iff.mp ht
+  let ρ := min r R
+  have hρ : 0 < ρ := lt_min hr hR
+  have hρr : ρ ≤ r := min_le_left _ _
+  have hρR : ρ ≤ R := min_le_right _ _
+  have := perimeterMeasure_locallyFinite E hmin.2.1
+  have : IsFiniteMeasure ((perimeterMeasure E).restrict (Metric.ball p ρ)) := ⟨by
+    rw [Measure.restrict_apply_univ, perimeterMeasure_open E Metric.isOpen_ball]
+    exact hmin.2.1 p ρ hρ⟩
+  have hiH : IntegrableOn H (Metric.ball p ρ) (perimeterMeasure E) :=
+    (hH.continuousOn.integrableOn_compact (isCompact_closedBall p ρ)).mono_set
+      Metric.ball_subset_closedBall
+  have hlow : (H p/2) * (perimeterMeasure E).real (Metric.ball p ρ) ≤
+      ∫ x in Metric.ball p ρ, H x ∂perimeterMeasure E := by
+    have hh := integral_mono_ae (integrable_const (H p/2)) hiH
+      ((ae_restrict_mem measurableSet_ball).mono fun x hx ↦
+        (hsub (Metric.ball_subset_ball hρr hx)).le)
+    simpa only [integral_const, smul_eq_mul, Measure.real, Measure.restrict_apply_univ,
+      mul_comm] using hh
+  have hd : c₀*ρ^(n-1) ≤ (perimeterMeasure E).real (Metric.ball p ρ) := by
+    have hd₀ := ENNReal.toReal_mono (hmin.2.1 p ρ hρ).ne (hdensity E hmin p hp ρ hρ)
+    rw [ENNReal.toReal_ofReal (by positivity)] at hd₀
+    simpa only [Measure.real, perimeterMeasure_open E Metric.isOpen_ball] using hd₀
+  have hmono := minimizing_height_density_monotone hn hmin p v a hρ hR hρR
+  have hbound : c₀*H p/2 ≤ (∫ x in Metric.ball p R, H x ∂perimeterMeasure E)/R^(n-1) := by
+    calc
+      _ = ((H p/2)*(c₀*ρ^(n-1)))/ρ^(n-1) := by
+        field_simp
+      _ ≤ ((H p/2)*(perimeterMeasure E).real (Metric.ball p ρ))/ρ^(n-1) := by
+        exact div_le_div_of_nonneg_right
+          (mul_le_mul_of_nonneg_left hd (by positivity)) (pow_nonneg hρ.le _)
+      _ ≤ (∫ x in Metric.ball p ρ, H x ∂perimeterMeasure E)/ρ^(n-1) :=
+        div_le_div_of_nonneg_right hlow (pow_nonneg hρ.le _)
+      _ ≤ _ := hmono
+  change H p ≤ (2/c₀)/R^(n-1) * ∫ x in Metric.ball p R, H x ∂perimeterMeasure E
+  calc
+    H p = (2/c₀)*(c₀*H p/2) := by field_simp
+    _ ≤ (2/c₀)*((∫ x in Metric.ball p R, H x ∂perimeterMeasure E)/R^(n-1)) :=
+      mul_le_mul_of_nonneg_left hbound (by positivity)
+    _ = _ := by ring
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/HolderGraph.lean b/lean/OAI/Geometry/Perimeter/Regularity/HolderGraph.lean
new file mode 100644
index 0000000000000000000000000000000000000000..88f8888d6fea48e579b3bfbf5e69b7f0de1fb50a
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/HolderGraph.lean
@@ -0,0 +1,88 @@
+import OAI.Geometry.Perimeter.Regularity.NeighborhoodPlanes
+import OAI.Geometry.Perimeter.Regularity.GraphSlope
+import OAI.Geometry.Perimeter.Regularity.GeometricDerivativeModulus
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+theorem holder_graph_from_contraction (hn : 1<n) {q M δ : ℝ}
+    (hq : 0<q) (hq1 : q<1) (hM : 0≤M) (hstep : SlabContractionProperty n q M δ)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    {v : Euc n} (hv : ‖v‖=1) {a : ℝ} (ha : 0<a) (haδ : 4*a<δ)
+    (ha4 : a<1/4) (haM : 8*M*a≤1/2) (hachord : a*(16*M+1/q)≤1/2)
+    (hs : FullSlab E v 8 a) :
+    ∃ f : normalPlane v → ℝ,
+      ContDiffOn ℝ 1 f (Metric.ball 0 (1/4)) ∧
+      (∀ y, ‖y‖<1/4 → |f y|≤a ∧ (y:Euc n)+f y•v∈perimeterSupport E ∩ Metric.ball 0 1) ∧
+      (∀ y z, ‖y‖<1/4 → ‖z‖<1/4 → |f y-f z|≤‖y-z‖) ∧
+      (∀ (y : normalPlane v) (t : ℝ), ‖y‖<1/4 → ‖(y:Euc n)+t•v‖<1 →
+        ((y:Euc n)+t•v∈perimeterSupport E ↔ t=f y)) ∧
+      (∀ p∈Metric.ball (0:normalPlane v) (1/4), GeometricHolderAt (q/2) (fderiv ℝ f) p) ∧
+      (∀ p∈Metric.ball (0:normalPlane v) (1/4), ‖fderiv ℝ f p‖≤16*M*a) := by
+  have hchord : ∀ p∈perimeterSupport E ∩ Metric.ball 0 1,
+      ∀ x∈perimeterSupport E ∩ Metric.ball 0 1,
+        |inner ℝ v (x-p)|≤(1/2)*‖x-p‖ := by
+    intro p hp x hx
+    exact (nearby_chord_bound hq hq1 hM hstep hmin hv ha haδ hs hp hx).trans
+      (mul_le_mul_of_nonneg_right hachord (norm_nonneg _))
+  obtain ⟨f,hf,hlip,hiff⟩ := single_graph_from_slab hn hmin hv ha.le ha4 hs hchord
+  have hwex (y : normalPlane v) : ∃ w : Euc n, ‖y‖<1/4 →
+      ‖w-v‖≤8*M*a ∧ ∀ j x, x∈perimeterSupport E → ‖x-((y:Euc n)+f y•v)‖≤2*q^j →
+      |inner ℝ w (x-((y:Euc n)+f y•v))|≤4*a*(2*M+1/(4*q))*(1/2)^j*‖x-((y:Euc n)+f y•v)‖ := by
+    by_cases hy : ‖y‖<1/4
+    · obtain ⟨w,_,hwd,htan⟩ := nearby_full_support_planes hq hq1 hM hstep hmin hv ha haδ hs (hf y hy).2
+      exact ⟨w,fun _ ↦ ⟨hwd,htan⟩⟩
+    · exact ⟨v,fun h ↦ (hy h).elim⟩
+  choose w hw using hwex
+  let D : normalPlane v → normalPlane v →L[ℝ] ℝ := fun y ↦ graphTangentLinear v (w y)
+  let C : ℝ := 16*a*(2*M+1/(4*q))
+  have hrem : ∀ j y, y∈Metric.ball (0:normalPlane v) (1/4) →
+      ∀ z, z∈Metric.ball (0:normalPlane v) (1/4) → ‖z-y‖≤q^j →
+      |f z-f y-D y (z-y)|≤C*(1/2)^j*‖z-y‖ := by
+    intro j y hy z hz hzy
+    have hy' : ‖y‖<1/4 := by simpa only [Metric.mem_ball,dist_zero_right] using hy
+    have hz' : ‖z‖<1/4 := by simpa only [Metric.mem_ball,dist_zero_right] using hz
+    have hh := graph_tangent_remainder hv ((hw y hy').1.trans haM) (hf z hz').2.1
+      (hlip z y hz' hy') hzy ((hw y hy').2 j) (by positivity : 0≤4*a*(2*M+1/(4*q)))
+    convert hh using 1
+    dsimp [C,D]
+    ring
+  have hD : ∀ y∈Metric.ball (0:normalPlane v) (1/4), HasFDerivAt f (D y) y := by
+    intro y hy
+    exact hasFDerivAt_of_geometric_remainder Metric.isOpen_ball hy hq (fun j z hz hzy ↦ hrem j y hy z hz hzy)
+  have hDC := continuous_derivative_of_geometric_remainder Metric.isOpen_ball hq
+    (fun x hx y hy ↦ hlip x y
+      (by simpa only [Metric.mem_ball,dist_zero_right] using hx)
+      (by simpa only [Metric.mem_ball,dist_zero_right] using hy)) hrem
+  have hC1 : ContDiffOn ℝ 1 f (Metric.ball 0 (1/4)) := by
+    change ContDiffOn ℝ (0+1) f (Metric.ball 0 (1/4))
+    rw [contDiffOn_succ_iff_hasFDerivWithinAt_of_uniqueDiffOn Metric.isOpen_ball.uniqueDiffOn]
+    refine ⟨by simp, D,?_,fun y hy ↦ (hD y hy).hasFDerivWithinAt⟩
+    exact contDiffOn_zero.mpr (fun y hy ↦ (hDC y hy).continuousWithinAt)
+  refine ⟨f,hC1,hf,hlip,hiff,?_,?_⟩
+  swap
+  · intro p hp
+    rw [(hD p hp).fderiv]
+    have hp' : ‖p‖<1/4 := by simpa only [Metric.mem_ball,dist_zero_right] using hp
+    have hh := graphTangentLinear_norm_le hv ((hw p hp').1.trans haM)
+    change ‖graphTangentLinear v (w p)‖ ≤ _
+    nlinarith [(hw p hp').1]
+  · intro p hp
+    have hm := geometricHolder_derivative_of_remainder Metric.isOpen_ball hq hq1.le
+      (by dsimp [C]; positivity : 0 ≤ C)
+      (fun x hx y hy ↦ hlip x y
+        (by simpa only [Metric.mem_ball,dist_zero_right] using hx)
+        (by simpa only [Metric.mem_ball,dist_zero_right] using hy)) hrem hp
+    exact hm.congr (by positivity) (Filter.eventually_of_mem (Metric.isOpen_ball.mem_nhds hp)
+      (fun y hy ↦ (hD y hy).fderiv.symm))
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/IndicatorConvolutionVariationLocalLe.lean b/lean/OAI/Geometry/Perimeter/Regularity/IndicatorConvolutionVariationLocalLe.lean
new file mode 100644
index 0000000000000000000000000000000000000000..2d2397797ff16e17d69547dde3ff1556f6765b63
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/IndicatorConvolutionVariationLocalLe.lean
@@ -0,0 +1,517 @@
+import OAI.Geometry.Perimeter.Regularity.IntegralDivergenceEqZeroOfPerimeterEqZero
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+open RPI.Boundary
+open ContinuousLinearMap
+open scoped Convolution
+
+theorem indicator_convolution_variation_local_le {n : ℕ} {E : Set (Euc n)}
+    (hE : MeasurableSet E) (U W : Set (Euc n)) (hP : perimeter E W ≠ ⊤)
+    {k : Euc n → ℝ} (hkc : Continuous k) (hk : Integrable k volume)
+    (hk0 : ∀ x, 0 ≤ k x) (hk1 : ∫ x, k x = 1)
+    (hshift : ∀ t ∈ tsupport k, ∀ y, y + t ∈ U → y ∈ W)
+    {V : Euc n → Euc n} (hV : PerimeterAdmissible U V) :
+    (∫ x, (k ⋆[lsmul ℝ ℝ, volume] E.indicator (fun _ ↦ (1 : ℝ))) x * divergence V x) ≤
+      (perimeter E W).toReal := by
+  let χ : Euc n → ℝ := E.indicator (fun _ ↦ 1)
+  have hmχ : Measurable χ := measurable_const.indicator hE
+  have hbχ : ∀ x, ‖χ x‖ ≤ 1 := by
+    intro x
+    by_cases hx : x ∈ E <;> simp [χ, hx]
+  have hd := integrable_divergence hV.1 hV.2.1
+  let F : Euc n × Euc n → ℝ := fun p ↦ k p.1 * χ (p.2 - p.1) * divergence V p.2
+  have hFi : Integrable F (volume.prod volume) := by
+    apply (hk.norm.mul_prod hd.norm).mono'
+    · exact ((hkc.measurable.comp measurable_fst).mul
+        (hmχ.comp (measurable_snd.sub measurable_fst))).mul
+        ((continuous_divergence hV.1).measurable.comp measurable_snd) |>.aestronglyMeasurable
+    · apply ae_of_all
+      intro p
+      simp only [F, norm_mul]
+      exact mul_le_mul_of_nonneg_right
+        (by nlinarith [hbχ (p.2 - p.1), norm_nonneg (k p.1)]) (norm_nonneg _)
+  have hinner : ∀ t, (∫ x, F (t, x)) = k t * (∫ y in E, divergence V (y + t)) := by
+    intro t
+    simp only [F, mul_assoc, integral_const_mul]
+    congr 1
+    calc
+      (∫ x, χ (x - t) * divergence V x) =
+          ∫ y, χ y * divergence V (y + t) := by
+        simpa only [add_sub_cancel_right] using
+          (integral_add_right_eq_self (fun x ↦ χ (x - t) * divergence V x) t).symm
+      _ = ∫ y in E, divergence V (y + t) := by
+        rw [← integral_indicator hE]
+        apply integral_congr_ae
+        apply ae_of_all
+        intro y
+        by_cases hy : y ∈ E <;> simp [χ, hy]
+  have hInt : Integrable (fun t ↦ k t * (∫ y in E, divergence V (y + t))) volume := by
+    simpa only [hinner] using hFi.integral_prod_left
+  have heq : (∫ x, (k ⋆[lsmul ℝ ℝ, volume] χ) x * divergence V x) =
+      ∫ t, k t * (∫ y in E, divergence V (y + t)) := by
+    calc
+      _ = ∫ x, ∫ t, F (t, x) := by
+        apply integral_congr_ae
+        apply ae_of_all
+        intro x
+        simp only [convolution_def, lsmul_apply, smul_eq_mul, F, integral_mul_const]
+      _ = ∫ t, ∫ x, F (t, x) := integral_integral_swap hFi.swap
+      _ = _ := by simp only [hinner]
+  have hbound : ∀ t, k t * (∫ y in E, divergence V (y + t)) ≤ k t * (perimeter E W).toReal := by
+    intro t
+    by_cases ht : k t = 0
+    · simp only [ht, zero_mul, le_refl]
+    have htk : t ∈ tsupport k := subset_closure ht
+    let Vt : Euc n → Euc n := fun y ↦ V (y + t)
+    have hVs : tsupport Vt ⊆ W := by
+      intro y hy
+      apply hshift t htk y
+      exact hV.2.2.1 ((tsupport_comp_subset_preimage V (continuous_id.add continuous_const)) hy)
+    have hVt : PerimeterAdmissible W Vt := ⟨hV.1.comp (contDiff_id.add contDiff_const),
+      hV.2.1.comp_homeomorph (Homeomorph.addRight t), hVs, fun y ↦ hV.2.2.2 _⟩
+    have hh : (∫ y in E, divergence V (y + t)) ≤ (perimeter E W).toReal := by
+      simpa only [Vt, divergence_comp_add_right (hV.1.differentiable (by simp))] using
+        integral_divergence_le_perimeter_real E W hP hVt
+    exact mul_le_mul_of_nonneg_left hh (hk0 t)
+  rw [heq]
+  calc
+    (∫ t, k t * (∫ y in E, divergence V (y + t))) ≤
+        ∫ t, k t * (perimeter E W).toReal := by
+      exact integral_mono hInt (hk.mul_const _) hbound
+    _ = (perimeter E W).toReal := by rw [integral_mul_const, hk1, one_mul]
+
+theorem scalar_compact_integration_parts {n : ℕ} {u v : Euc n → ℝ}
+    (hu : ContDiff ℝ ∞ u) (hv : ContDiff ℝ ∞ v) (hcv : HasCompactSupport v)
+    (a : Euc n) :
+    Integrable (fun x ↦ u x * fderiv ℝ v x a) volume ∧
+    Integrable (fun x ↦ fderiv ℝ u x a * v x) volume ∧
+    (∫ x, u x * fderiv ℝ v x a) = -(∫ x, fderiv ℝ u x a * v x) := by
+  have hdu : Continuous (fun x ↦ fderiv ℝ u x a) :=
+    (hu.continuous_fderiv (by simp)).clm_apply continuous_const
+  have hdv : Continuous (fun x ↦ fderiv ℝ v x a) :=
+    (hv.continuous_fderiv (by simp)).clm_apply continuous_const
+  have hcdv : HasCompactSupport (fun x ↦ fderiv ℝ v x a) :=
+    hcv.fderiv_apply ℝ a
+  have h1 : Integrable (fun x ↦ u x * fderiv ℝ v x a) volume :=
+    (hu.continuous.mul hdv).integrable_of_hasCompactSupport hcdv.mul_left
+  have h2 : Integrable (fun x ↦ fderiv ℝ u x a * v x) volume :=
+    (hdu.mul hv.continuous).integrable_of_hasCompactSupport hcv.mul_left
+  refine ⟨h1, h2, ?_⟩
+  exact integral_mul_fderiv_eq_neg_fderiv_mul_of_integrable h2 h1
+    ((hu.continuous.mul hv.continuous).integrable_of_hasCompactSupport hcv.mul_left)
+    (fun x _ ↦ hu.differentiable (by simp) x)
+    (fun x _ ↦ hv.differentiable (by simp) x)
+
+theorem integral_mul_divergence_eq_neg_inner_gradient_local {n : ℕ} {u : Euc n → ℝ}
+    {V : Euc n → Euc n} (hu : ContDiff ℝ ∞ u)
+    (hV : ContDiff ℝ ∞ V) (hcV : HasCompactSupport V) :
+    (∫ x, u x * divergence V x) = - ∫ x, inner ℝ (gradient u x) (V x) := by
+  let b := EuclideanSpace.basisFun (Fin n) ℝ
+  let v : Fin n → Euc n → ℝ := fun i x ↦ inner ℝ (b i) (V x)
+  have hv : ∀ i, ContDiff ℝ ∞ (v i) := fun i ↦ (innerSL ℝ (b i)).contDiff.comp hV
+  have hcv : ∀ i, HasCompactSupport (v i) := fun i ↦ hcV.comp_left (by simp)
+  have hdv : ∀ i x, fderiv ℝ (v i) x (b i) =
+      inner ℝ (b i) (fderiv ℝ V x (b i)) := by
+    intro i x
+    have h := ((innerSL ℝ (b i)).hasFDerivAt.comp x
+      (hV.differentiable (by simp) x).hasFDerivAt).fderiv
+    exact congrArg (fun L : Euc n →L[ℝ] ℝ ↦ L (b i)) h
+  have hp := fun i ↦ scalar_compact_integration_parts hu (hv i) (hcv i) (b i)
+  have hInt1 := fun i ↦ (hp i).2.1
+  have hInt2 := fun i ↦ (hp i).1
+  have hparts := fun i ↦ (hp i).2.2
+  calc
+    (∫ x, u x * divergence V x) = ∫ x, ∑ i, u x * fderiv ℝ (v i) x (b i) := by
+      congr 1
+      funext x
+      simp only [divergence, hdv, Finset.mul_sum, b]
+    _ = ∑ i, ∫ x, u x * fderiv ℝ (v i) x (b i) := integral_finsetSum _ (fun i _ ↦ hInt2 i)
+    _ = -(∑ i, ∫ x, fderiv ℝ u x (b i) * v i x) := by simp only [hparts, Finset.sum_neg_distrib]
+    _ = -(∫ x, ∑ i, fderiv ℝ u x (b i) * v i x) := by
+      rw [integral_finsetSum _ (fun i _ ↦ hInt1 i)]
+    _ = -(∫ x, inner ℝ (gradient u x) (V x)) := by
+      congr 1
+      apply integral_congr_ae
+      apply ae_of_all
+      intro x
+      simp only [← inner_gradient_left, v]
+      exact b.sum_inner_mul_inner _ _
+
+theorem integral_weighted_norm_gradient_le_of_test_bound {n : ℕ} {u : Euc n → ℝ}
+    (hu : ContDiff ℝ ∞ u) {φ : Euc n → ℝ} (hφ : ContDiff ℝ ∞ φ)
+    (hcφ : HasCompactSupport φ) (hφ0 : ∀ x, 0 ≤ φ x) (hφ1 : ∀ x, φ x ≤ 1)
+    (U : Set (Euc n)) (hsφ : tsupport φ ⊆ U) {P : ℝ}
+    (hbound : ∀ V : Euc n → Euc n, PerimeterAdmissible U V →
+      (∫ x, u x * divergence V x) ≤ P) :
+    (∫ x, φ x * ‖gradient u x‖) ≤ P := by
+  let ε : ℕ → ℝ := fun j ↦ 1 / (j + 1 : ℝ)
+  have hε : ∀ j, 0 < ε j := by intro j; dsimp [ε]; positivity
+  let f : ℕ → Euc n → ℝ := fun j x ↦
+    ‖gradient u x‖ ^ 2 / Real.sqrt (ε j ^ 2 + ‖gradient u x‖ ^ 2)
+  have hg := (gradient_contDiff hu).continuous
+  have hgi : Integrable (fun x ↦ φ x * ‖gradient u x‖) volume :=
+    (hφ.continuous.mul hg.norm).integrable_of_hasCompactSupport hcφ.mul_right
+  have hf0 : ∀ j x, 0 ≤ f j x := by intro j x; exact div_nonneg (sq_nonneg _) (Real.sqrt_nonneg _)
+  have hfle : ∀ j x, f j x ≤ ‖gradient u x‖ := by
+    intro j x
+    have hp : 0 < ε j ^ 2 + ‖gradient u x‖ ^ 2 :=
+      add_pos_of_pos_of_nonneg (sq_pos_of_pos (hε j)) (sq_nonneg _)
+    have hs : ‖gradient u x‖ ≤ Real.sqrt (ε j ^ 2 + ‖gradient u x‖ ^ 2) := by
+      rw [Real.le_sqrt (norm_nonneg _) hp.le]
+      nlinarith [sq_nonneg (ε j)]
+    exact (div_le_iff₀ (Real.sqrt_pos.2 hp)).2 (by nlinarith [norm_nonneg (gradient u x)])
+  have hfcont : ∀ j, Continuous (f j) := by
+    intro j
+    exact (hg.norm.pow 2).div ((continuous_const.add (hg.norm.pow 2)).sqrt)
+      (fun x ↦ (Real.sqrt_pos.2
+        (add_pos_of_pos_of_nonneg (sq_pos_of_pos (hε j)) (sq_nonneg _))).ne')
+  have hflim : ∀ x, Tendsto (fun j ↦ f j x) atTop (𝓝 ‖gradient u x‖) := by
+    intro x
+    have hlim : Tendsto ε atTop (𝓝 0) := by
+      simpa only [ε, Nat.cast_add, Nat.cast_one] using tendsto_one_div_add_atTop_nhds_zero_nat
+    by_cases hx : ‖gradient u x‖ = 0
+    · simp [f, hx]
+    · have hden := ((hlim.pow 2).add_const (‖gradient u x‖ ^ 2)).sqrt
+      have hden' : Tendsto (fun j ↦ Real.sqrt (ε j ^ 2 + ‖gradient u x‖ ^ 2))
+          atTop (𝓝 ‖gradient u x‖) := by
+        simpa [Real.sqrt_sq (norm_nonneg _)] using hden
+      have hh := (tendsto_const_nhds (x := ‖gradient u x‖ ^ 2)).div hden' hx
+      have heq : ‖gradient u x‖ ^ 2 / ‖gradient u x‖ = ‖gradient u x‖ := by
+        rw [pow_two, mul_div_cancel_right₀ _ hx]
+      change Tendsto (fun j ↦ ‖gradient u x‖ ^ 2 / Real.sqrt (ε j ^ 2 + ‖gradient u x‖ ^ 2))
+        atTop (𝓝 (‖gradient u x‖ ^ 2 / ‖gradient u x‖)) at hh
+      rw [heq] at hh
+      exact hh
+  have hlim : Tendsto (fun j ↦ ∫ x, φ x * f j x) atTop
+      (𝓝 (∫ x, φ x * ‖gradient u x‖)) :=
+    tendsto_integral_of_dominated_convergence (fun x ↦ φ x * ‖gradient u x‖)
+      (fun j ↦ (hφ.continuous.mul (hfcont j)).aestronglyMeasurable)
+      hgi (fun j ↦ ae_of_all _ fun x ↦ by
+        rw [Real.norm_eq_abs, abs_of_nonneg (mul_nonneg (hφ0 x) (hf0 j x))]
+        exact mul_le_mul_of_nonneg_left (hfle j x) (hφ0 x))
+      (ae_of_all _ fun x ↦ (hflim x).const_mul (φ x))
+  apply le_of_tendsto hlim
+  apply Eventually.of_forall
+  intro j
+  let V : Euc n → Euc n := fun x ↦ φ x • gradientDualTest u (ε j) x
+  have hV : PerimeterAdmissible U V := by
+    refine ⟨hφ.smul (gradientDualTest_contDiff hu (hε j)), hcφ.smul_right,
+      (tsupport_smul_subset_left φ _).trans hsφ, ?_⟩
+    intro x
+    simp only [V, norm_smul, Real.norm_eq_abs, abs_of_nonneg (hφ0 x)]
+    calc
+      φ x * ‖gradientDualTest u (ε j) x‖ ≤ φ x * 1 :=
+        mul_le_mul_of_nonneg_left (gradientDualTest_norm_le u (hε j) x) (hφ0 x)
+      _ ≤ 1 := by simpa only [mul_one] using hφ1 x
+  have heq : (∫ x, u x * divergence V x) = ∫ x, φ x * f j x := by
+    rw [integral_mul_divergence_eq_neg_inner_gradient_local hu hV.1 hV.2.1, ← integral_neg]
+    apply integral_congr_ae
+    apply ae_of_all
+    intro x
+    simp only [V, gradientDualTest, inner_smul_right, real_inner_self_eq_norm_sq,
+      f, neg_mul, mul_neg, neg_neg, div_eq_mul_inv]
+    ring
+  rw [← heq]
+  exact hbound V hV
+
+theorem c1Mollify_indicator_local_gradient_le {n : ℕ} {E : Set (Euc n)}
+    (hE : MeasurableSet E) (c : Euc n) {R : ℝ} (hR : 0 < R)
+    (hP : perimeter E (Metric.ball c (4 * R)) ≠ ⊤) (j : ℕ)
+    (hj : (c1Mollifier (n := n) j).rOut < R) :
+    (∫ x in Metric.closedBall c R,
+      ‖fderiv ℝ (c1Mollify (E.indicator (fun _ ↦ (1 : ℝ))) j) x‖) ≤
+      (perimeter E (Metric.ball c (4 * R))).toReal := by
+  let χ : Euc n → ℝ := E.indicator (fun _ ↦ 1)
+  let u := c1Mollify χ j
+  let φ : ContDiffBump c := ⟨R, 2 * R, hR, by linarith⟩
+  have hu : ContDiff ℝ ∞ u :=
+    (c1Mollifier j).hasCompactSupport_normed.contDiff_convolution_left _
+      (c1Mollifier j).contDiff_normed (indicator_one_locallyIntegrable hE)
+  have hsφ : tsupport φ ⊆ Metric.ball c (3 * R) := by
+    rw [φ.tsupport_eq]
+    exact Metric.closedBall_subset_ball (by dsimp [φ]; linarith)
+  have hgrad : (∫ x, φ x * ‖gradient u x‖) ≤
+      (perimeter E (Metric.ball c (4 * R))).toReal := by
+    apply integral_weighted_norm_gradient_le_of_test_bound hu φ.contDiff
+      φ.hasCompactSupport (fun _ ↦ φ.nonneg) (fun _ ↦ φ.le_one) (Metric.ball c (3 * R)) hsφ
+    intro V hV
+    apply indicator_convolution_variation_local_le hE
+      (Metric.ball c (3 * R)) (Metric.ball c (4 * R)) hP
+      (c1Mollifier j).continuous_normed
+      ((c1Mollifier j).continuous_normed.integrable_of_hasCompactSupport
+        (c1Mollifier j).hasCompactSupport_normed)
+      (c1Mollifier j).nonneg_normed (c1Mollifier j).integral_normed _ hV
+    intro t ht y hy
+    rw [(c1Mollifier j).tsupport_normed_eq] at ht
+    have ht' : ‖t‖ ≤ (c1Mollifier (n := n) j).rOut := by
+      simpa only [Metric.mem_closedBall, dist_zero_right] using ht
+    have hdist : dist y (y + t) = ‖t‖ := by simp [dist_eq_norm]
+    exact (dist_triangle y (y + t) c).trans_lt (by
+      rw [hdist]
+      linarith [Metric.mem_ball.mp hy])
+  have hnorm : ∀ x, ‖gradient u x‖ = ‖fderiv ℝ u x‖ := fun x ↦
+    (InnerProductSpace.toDual ℝ (Euc n)).symm.norm_map _
+  calc
+    _ = ∫ x in Metric.closedBall c R, φ x * ‖gradient u x‖ := by
+      apply setIntegral_congr_fun measurableSet_closedBall
+      intro x hx
+      change ‖fderiv ℝ u x‖ = φ x * ‖gradient u x‖
+      rw [φ.one_of_mem_closedBall hx, one_mul, hnorm]
+    _ ≤ ∫ x, φ x * ‖gradient u x‖ :=
+      setIntegral_le_integral ((φ.continuous.mul (gradient_contDiff hu).continuous.norm).integrable_of_hasCompactSupport φ.hasCompactSupport.mul_right)
+        (ae_of_all _ fun x ↦ mul_nonneg (φ.nonneg (x := x)) (norm_nonneg _))
+    _ ≤ _ := hgrad
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+open RPI.Boundary
+open ContinuousLinearMap
+open scoped Convolution
+
+theorem c1Mollify_indicator_norm_le_one {n : ℕ} {E : Set (Euc n)}
+    (hE : MeasurableSet E) (j : ℕ) (x : Euc n) :
+    ‖c1Mollify (E.indicator (fun _ ↦ (1 : ℝ))) j x‖ ≤ 1 := by
+  have hh := dist_convolution_le (μ := (volume : Measure (Euc n)))
+    (g := E.indicator (fun _ ↦ (1 : ℝ))) (x₀ := x) (z₀ := 0) (by norm_num : (0 : ℝ) ≤ 1)
+    (c1Mollifier j).support_normed_eq.subset (c1Mollifier j).nonneg_normed
+    (c1Mollifier j).integral_normed (measurable_const.indicator hE).aestronglyMeasurable
+    (fun y _ ↦ by by_cases hy : y ∈ E <;> simp [hy])
+  simpa only [c1Mollify, dist_zero_right] using hh
+
+theorem raw_ball_poincare {n : ℕ} {E : Set (Euc n)} (hE : MeasurableSet E)
+    (c : Euc n) {R : ℝ} (hR : 0 < R)
+    (hP : perimeter E (Metric.ball c (4 * R)) ≠ ⊤) :
+    volume.real (E ∩ Metric.closedBall c R) *
+        volume.real (Eᶜ ∩ Metric.closedBall c R) ≤
+      4 * R * volume.real (Metric.closedBall c R) * 2 ^ n *
+        (perimeter E (Metric.ball c (4 * R))).toReal := by
+  let B := Metric.closedBall c R
+  let μ : Measure (Euc n) := volume.restrict B
+  have : IsFiniteMeasure μ := ⟨by
+    simpa only [μ, Measure.restrict_apply_univ, B] using
+      (isCompact_closedBall c R).measure_lt_top (μ := volume)⟩
+  let χ : Euc n → ℝ := E.indicator (fun _ ↦ 1)
+  let ψ : Euc n → ℝ := Eᶜ.indicator (fun _ ↦ 1)
+  let u : ℕ → Euc n → ℝ := c1Mollify χ
+  have hχ : LocallyIntegrable χ volume := indicator_one_locallyIntegrable hE
+  have hψ : LocallyIntegrable ψ volume := indicator_one_locallyIntegrable hE.compl
+  have hχi : Integrable χ μ := hχ.integrableOn_isCompact (isCompact_closedBall c R)
+  have hψi : Integrable ψ μ := hψ.integrableOn_isCompact (isCompact_closedBall c R)
+  have hu : ∀ j, ContDiff ℝ ∞ (u j) := fun j ↦
+    (c1Mollifier j).hasCompactSupport_normed.contDiff_convolution_left _
+      (c1Mollifier j).contDiff_normed hχ
+  have hub : ∀ j x, ‖u j x‖ ≤ 1 := c1Mollify_indicator_norm_le_one hE
+  have hlim : ∀ᵐ x ∂μ, Tendsto (fun j ↦ u j x) atTop (𝓝 (χ x)) := by
+    apply Filter.Eventually.filter_mono (ae_mono Measure.restrict_le_self)
+    apply ContDiffBump.ae_convolution_tendsto_right_of_locallyIntegrable
+      c1Mollifier_radius_tendsto (K := 2) _ hχ
+    apply Eventually.of_forall
+    intro j
+    dsimp [c1Mollifier]
+    linarith
+  have hlimp : ∀ᵐ p ∂μ.prod μ,
+      Tendsto (fun j ↦ ‖u j p.1 - u j p.2‖) atTop (𝓝 ‖χ p.1 - χ p.2‖) := by
+    filter_upwards [Measure.quasiMeasurePreserving_fst.ae hlim,
+      Measure.quasiMeasurePreserving_snd.ae hlim] with p hp hq
+    exact (hp.sub hq).norm
+  have hDlim : Tendsto (fun j ↦ ∫ p, ‖u j p.1 - u j p.2‖ ∂μ.prod μ) atTop
+      (𝓝 (∫ p, ‖χ p.1 - χ p.2‖ ∂μ.prod μ)) := by
+    apply tendsto_integral_of_dominated_convergence (fun _ ↦ (2 : ℝ))
+    · intro j
+      exact (((hu j).continuous.comp continuous_fst).sub
+        ((hu j).continuous.comp continuous_snd)).norm.aestronglyMeasurable
+    · exact integrable_const _
+    · intro j
+      apply ae_of_all
+      intro p
+      simp only [norm_norm]
+      exact (norm_sub_le _ _).trans (by linarith [hub j p.1, hub j p.2])
+    · exact hlimp
+  have hD : (∫ p, ‖χ p.1 - χ p.2‖ ∂μ.prod μ) ≤
+      4 * R * volume.real B * 2 ^ n * (perimeter E (Metric.ball c (4 * R))).toReal := by
+    apply le_of_tendsto hDlim
+    have hsmall : ∀ᶠ j in atTop, (c1Mollifier (n := n) j).rOut < R :=
+      c1Mollifier_radius_tendsto.eventually (gt_mem_nhds hR)
+    filter_upwards [hsmall] with j hj
+    have hh := smooth_ball_poincare ((hu j).of_le (by simp)) c hR
+    apply hh.trans
+    have hgrad := c1Mollify_indicator_local_gradient_le hE c hR hP j hj
+    have hconst : 0 ≤ 4 * R * volume.real B * (2 : ℝ) ^ n := by positivity
+    calc
+      _ = (4 * R * volume.real B * (2 : ℝ) ^ n) *
+          ∫ x in B, ‖fderiv ℝ (u j) x‖ := by ring
+      _ ≤ _ := mul_le_mul_of_nonneg_left hgrad hconst
+  have hDi : Integrable (fun p : Euc n × Euc n ↦ ‖χ p.1 - χ p.2‖) (μ.prod μ) :=
+    ((hχi.comp_fst μ).sub (hχi.comp_snd μ)).norm
+  calc
+    _ = (∫ x, χ x ∂μ) * ∫ y, ψ y ∂μ := by
+      simp only [χ, ψ, μ, integral_indicator hE, integral_indicator hE.compl,
+        Measure.restrict_restrict hE, Measure.restrict_restrict hE.compl,
+        integral_const, measureReal_restrict_apply_univ, smul_eq_mul, mul_one, B]
+    _ = ∫ p, χ p.1 * ψ p.2 ∂μ.prod μ := (integral_prod_mul χ ψ).symm
+    _ ≤ ∫ p, ‖χ p.1 - χ p.2‖ ∂μ.prod μ := by
+      apply integral_mono (hχi.mul_prod hψi) hDi
+      intro p
+      by_cases hp : p.1 ∈ E <;> by_cases hq : p.2 ∈ E <;> simp [χ, ψ, hp, hq]
+    _ ≤ _ := hD
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+
+theorem exists_minimizing_perimeter_lower_density (n : ℕ) (hn : 1 < n) :
+    ∃ c₀ : ℝ, 0 < c₀ ∧ ∀ E : Set (Euc n), IsLocallyPerimeterMinimizing E →
+      ∀ c ∈ perimeterSupport E, ∀ r : ℝ, 0 < r →
+        ENNReal.ofReal (c₀ * r ^ (n - 1)) ≤ perimeter E (Metric.ball c r) := by
+  have hnr : (1 : ℝ) < n := by exact_mod_cast hn
+  obtain ⟨ε, hε, hphase⟩ := minimizing_scaled_two_phase_volume_lower
+    (Real.HolderConjugate.conjExponent hnr)
+  let V := volume.real (Metric.ball (0 : Euc n) 1)
+  have hV : 0 < V := by
+    change 0 < volume.real (Metric.ball (0 : Euc n) 1)
+    rw [measureReal_def, ENNReal.toReal_pos_iff]
+    exact ⟨Metric.measure_ball_pos volume _ (by norm_num),
+      (measure_mono Metric.ball_subset_closedBall).trans_lt
+        (isCompact_closedBall (0 : Euc n) 1).measure_lt_top⟩
+  let K : ℝ := 4 * V * 2 ^ n
+  have hK : 0 < K := by dsimp [K]; positivity
+  let c₀ : ℝ := ε ^ 2 / (K * 4 ^ (n - 1))
+  have hc₀ : 0 < c₀ := by dsimp [c₀]; positivity
+  refine ⟨c₀, hc₀, ?_⟩
+  intro E hmin c hc r hr
+  let R := r / 4
+  have hR : 0 < R := by dsimp [R]; positivity
+  have h4R : 4 * R = r := by dsimp [R]; ring
+  have hP : perimeter E (Metric.ball c (4 * R)) ≠ ⊤ :=
+    (hmin.2.1 c (4 * R) (by positivity)).ne
+  have hpoin := raw_ball_poincare hmin.1 c hR hP
+  have hvol : volume.real (Metric.closedBall c R) = R ^ n * V := by
+    simpa only [Euc, finrank_euclideanSpace, Fintype.card_fin, V] using
+      (Measure.addHaar_real_closedBall (volume : Measure (Euc n)) c hR.le)
+  rw [hvol] at hpoin
+  obtain ⟨hleft, hright⟩ := hphase E hmin c hc R hR
+  have hprod : (ε * R ^ n) * (ε * R ^ n) ≤
+      volume.real (E ∩ Metric.closedBall c R) *
+        volume.real (Eᶜ ∩ Metric.closedBall c R) :=
+    mul_le_mul hleft.le hright.le (by positivity) measureReal_nonneg
+  have hpow : R ^ n = R ^ (n - 1) * R := by
+    rw [← pow_succ]
+    congr 1
+    omega
+  have hsimple : ε ^ 2 * R ^ (n - 1) ≤ K *
+      (perimeter E (Metric.ball c (4 * R))).toReal := by
+    apply (mul_le_mul_iff_right₀ (show 0 < R ^ n * R by positivity)).mp
+    calc
+      _ = (ε * R ^ n) * (ε * R ^ n) := by rw [hpow]; ring
+      _ ≤ _ := hprod.trans hpoin
+      _ = _ := by dsimp [K]; ring
+  have hfinal : c₀ * r ^ (n - 1) ≤ (perimeter E (Metric.ball c r)).toReal := by
+    apply (mul_le_mul_iff_right₀ hK).mp
+    calc
+      K * (c₀ * r ^ (n - 1)) = ε ^ 2 * R ^ (n - 1) := by
+        rw [← h4R, mul_pow]
+        dsimp [c₀]
+        field_simp
+      _ ≤ _ := by simpa only [h4R] using hsimple
+  exact (ENNReal.ofReal_le_ofReal hfinal).trans_eq (ENNReal.ofReal_toReal
+    (hmin.2.1 c r hr).ne)
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+
+theorem exists_minimizing_surfaceMeasure_growth {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E) :
+    ∃ A : ℝ, 0 < A ∧ ∀ (c : Euc n) (r : ℝ), 0 < r →
+      RPI.Boundary.surfaceMeasure (perimeterSupport E) (Metric.ball c r) ≤
+        ENNReal.ofReal (A * r ^ (n - 1)) := by
+  obtain ⟨c₀, hc₀, hdensity₀⟩ := exists_minimizing_perimeter_lower_density n hn
+  obtain ⟨C, hC, hperim⟩ := exists_locallyMinimizing_perimeter_growth_constant n (by omega)
+  have hdensity : ∀ x ∈ perimeterSupport E, ∀ᶠ r : ℝ in 𝓝[>] (0 : ℝ),
+      ENNReal.ofReal (c₀ * r ^ (n - 1)) ≤ perimeter E (Metric.ball x r) := by
+    intro x hx
+    filter_upwards [self_mem_nhdsWithin] with r hr
+    exact hdensity₀ E hmin x hx r hr
+  let K : ℝ≥0∞ := (8 : ℝ≥0∞) ^ (n - 1) * (ENNReal.ofReal c₀)⁻¹ *
+    ENNReal.ofReal C * (2 : ℝ≥0∞) ^ (n - 1)
+  have hK : K ≠ ⊤ := by
+    unfold K
+    exact ENNReal.mul_ne_top (ENNReal.mul_ne_top
+      (ENNReal.mul_ne_top (by finiteness)
+        (ENNReal.inv_ne_top.mpr (ENNReal.ofReal_ne_zero_iff.mpr hc₀)))
+      ENNReal.ofReal_ne_top) (by finiteness)
+  have hraw : ∀ (c : Euc n) (r : ℝ), 0 < r →
+      Measure.hausdorffMeasure (n - 1 : ℕ) (Metric.ball c r ∩ perimeterSupport E) ≤
+        K * ENNReal.ofReal r ^ (n - 1) := by
+    intro c r hr
+    rw [inter_comm]
+    calc
+      _ ≤ (8 : ℝ≥0∞) ^ (n - 1) * (ENNReal.ofReal c₀)⁻¹ *
+          perimeter E (Metric.ball c (2 * r)) :=
+        hausdorffMeasure_le_perimeter_of_lower_density E (perimeterSupport E) hc₀ hdensity c hr
+      _ ≤ (8 : ℝ≥0∞) ^ (n - 1) * (ENNReal.ofReal c₀)⁻¹ *
+          ENNReal.ofReal (C * (2 * r) ^ (n - 1)) := by
+        exact mul_le_mul le_rfl (hperim E hmin c (2 * r) (by positivity)) (by positivity) (by positivity)
+      _ = _ := by
+        rw [ENNReal.ofReal_mul hC.le, ENNReal.ofReal_pow (by positivity),
+          ENNReal.ofReal_mul (by norm_num : (0 : ℝ) ≤ 2)]
+        norm_num only [ENNReal.ofReal_ofNat]
+        simp only [K, mul_pow, mul_assoc]
+  let a : ℝ≥0∞ := (Measure.addHaarScalarFactor (volume : Measure (Euc (n - 1)))
+    (Measure.hausdorffMeasure ((n - 1 : ℕ) : ℝ)) : ℝ≥0∞)
+  let A : ℝ := (a * K).toReal + 1
+  have hA : 0 < A := by dsimp [A]; positivity
+  refine ⟨A, hA, ?_⟩
+  intro c r hr
+  have haK : a * K ≤ ENNReal.ofReal A := by
+    rw [← ENNReal.ofReal_toReal (ENNReal.mul_ne_top ENNReal.coe_ne_top hK)]
+    exact ENNReal.ofReal_le_ofReal (by dsimp [A]; linarith)
+  rw [RPI.Boundary.surfaceMeasure, Measure.restrict_apply measurableSet_ball,
+    Measure.euclideanHausdorffMeasure_def, Measure.smul_apply, ENNReal.smul_def]
+  change a * Measure.hausdorffMeasure (n - 1 : ℕ) (Metric.ball c r ∩ perimeterSupport E) ≤ _
+  calc
+    _ ≤ a * (K * ENNReal.ofReal r ^ (n - 1)) :=
+      mul_le_mul le_rfl (hraw c r hr) (by positivity) (by positivity)
+    _ ≤ ENNReal.ofReal A * ENNReal.ofReal r ^ (n - 1) := by
+      rw [← mul_assoc]
+      exact mul_le_mul haK le_rfl (by positivity) (by positivity)
+    _ = _ := by rw [ENNReal.ofReal_mul hA.le, ENNReal.ofReal_pow hr.le]
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/InitialSlab.lean b/lean/OAI/Geometry/Perimeter/Regularity/InitialSlab.lean
new file mode 100644
index 0000000000000000000000000000000000000000..37c2694548240c9cee06fcfc90f48a4160bbef1b
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/InitialSlab.lean
@@ -0,0 +1,107 @@
+import OAI.Geometry.Perimeter.Regularity.PhaseRestoration
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+theorem phase_anchors_of_small_symmDiff {E H A B U : Set (Euc n)}
+    (hE : MeasurableSet E) (hU : volume U ≠ (∞ : ℝ≥0∞))
+    (hA : A ⊆ U ∩ Hᶜ) (hB : B ⊆ U ∩ H)
+    (hsmallA : volume.real (symmDiff E H ∩ U) < volume.real A)
+    (hsmallB : volume.real (symmDiff E H ∩ U) < volume.real B) :
+    0 < volume (A\E) ∧ 0 < volume (B∩E) := by
+  have hfin : volume (symmDiff E H ∩ U) ≠ (∞ : ℝ≥0∞) := measure_ne_top_of_subset inter_subset_right hU
+  constructor
+  · by_contra! hz
+    have heq : volume A = volume (A∩E) := by
+      have hh := measure_sdiff_add_inter (μ := volume) A hE
+      rw [le_antisymm hz bot_le,zero_add] at hh
+      exact hh.symm
+    have hle : volume (A∩E) ≤ volume (symmDiff E H ∩ U) := by
+      apply measure_mono
+      intro x hx
+      exact ⟨Or.inl ⟨hx.2,(hA hx.1).2⟩,(hA hx.1).1⟩
+    have hh := ENNReal.toReal_mono hfin (heq ▸ hle)
+    exact (not_lt_of_ge hh) hsmallA
+  · by_contra! hz
+    have heq : volume B = volume (B\E) := by
+      have hh := measure_sdiff_add_inter (μ := volume) B hE
+      rw [le_antisymm hz bot_le,add_zero] at hh
+      exact hh.symm
+    have hle : volume (B\E) ≤ volume (symmDiff E H ∩ U) := by
+      apply measure_mono
+      intro x hx
+      exact ⟨Or.inr ⟨(hB hx.1).2,hx.2⟩,(hB hx.1).1⟩
+    have hh := ENNReal.toReal_mono hfin (heq ▸ hle)
+    exact (not_lt_of_ge hh) hsmallB
+
+theorem HasHalfspaceBlowup.eventually_fullSlab (hn : 1<n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    {p v : Euc n} (hbl : HasHalfspaceBlowup E p v)
+    (r : ℕ → ℝ) (hr : ∀ i, 0<r i) (htr : Tendsto r atTop (𝓝 0))
+    {a : ℝ} (ha : 0<a) (ha1 : a<1) :
+    ∀ᶠ i in atTop, FullSlab (blownUpSet E p (r i)) v 8 a := by
+  let H : Set (Euc n) := {x | inner ℝ x v<0}
+  let A : Set (Euc n) := Metric.ball ((4:ℝ)•v) 1
+  let B : Set (Euc n) := Metric.ball ((-4:ℝ)•v) 1
+  have hAcap : A ⊆ Metric.ball 0 8 ∩ {x | a < inner ℝ v x} := by
+    intro x hx
+    have hh := positive_cap_ball hbl.1 (by norm_num : (0:ℝ)<8)
+      (by simp : ‖v-v‖≤1/4) x (by simpa only [show (8:ℝ)/2=4 by norm_num,show (8:ℝ)/8=1 by norm_num] using (show x∈Metric.ball ((4:ℝ)•v) 1 from hx))
+    exact ⟨hh.1,ha1.trans (by simpa using hh.2.1)⟩
+  have hBcap : B ⊆ Metric.ball 0 8 ∩ {x | inner ℝ v x < -a} := by
+    intro x hx
+    have hxA : -x∈A := by
+      change dist (-x) ((4:ℝ)•v)<1
+      have he : dist (-x) ((4:ℝ)•v)=dist x ((-4:ℝ)•v) := by
+        simpa only [neg_smul,neg_neg] using dist_neg_neg x (-((4:ℝ)•v))
+      rw [he]
+      exact hx
+    have hh := hAcap hxA
+    refine ⟨by simpa only [Metric.mem_ball,dist_zero_right,norm_neg] using hh.1,?_⟩
+    have hi := hh.2
+    simp only [mem_ofPred_eq,inner_neg_right] at hi
+    change inner ℝ v x < -a
+    linarith
+  have hApos : 0<volume.real A := ENNReal.toReal_pos
+    (Metric.measure_ball_pos volume _ (by norm_num : (0:ℝ)<1)).ne'
+    (measure_ball_lt_top.ne)
+  have hBpos : 0<volume.real B := ENNReal.toReal_pos
+    (Metric.measure_ball_pos volume _ (by norm_num : (0:ℝ)<1)).ne'
+    (measure_ball_lt_top.ne)
+  have ht := hbl.local_L1 hmin.1 r hr htr (0:Euc n) 8 (by norm_num)
+  have hheight := hbl.support_height hn hmin r hr htr (isCompact_closedBall (0:Euc n) 8) ha
+  filter_upwards [ht.eventually (gt_mem_nhds hApos),ht.eventually (gt_mem_nhds hBpos),hheight] with i hAi hBi hhi
+  have hm := locally_minimizing_blownUpSet hmin p (hr i)
+  have hh : ∀ x∈perimeterSupport (blownUpSet E p (r i)) ∩ Metric.ball 0 8,
+      |inner ℝ v x|≤a := by
+    intro x hx
+    rw [real_inner_comm]
+    exact (hhi x ⟨hx.1,Metric.ball_subset_closedBall hx.2⟩).le
+  have hAU : A ⊆ Metric.closedBall 0 8 ∩ Hᶜ := by
+    intro x hx
+    have hhA := hAcap hx
+    refine ⟨Metric.ball_subset_closedBall hhA.1,?_⟩
+    change ¬inner ℝ x v<0
+    rw [real_inner_comm]
+    exact not_lt_of_ge (ha.le.trans hhA.2.le)
+  have hBU : B ⊆ Metric.closedBall 0 8 ∩ H := by
+    intro x hx
+    have hhB := hBcap hx
+    refine ⟨Metric.ball_subset_closedBall hhB.1,?_⟩
+    change inner ℝ x v<0
+    rw [real_inner_comm]
+    exact hhB.2.trans (neg_neg_of_pos ha)
+  obtain ⟨hvolA,hvolB⟩ := phase_anchors_of_small_symmDiff hm.1
+    (isCompact_closedBall (0:Euc n) 8).measure_ne_top hAU hBU hAi hBi
+  exact ⟨hh,minimizing_slab_phase_agreement hn hm v ha.le hh hAcap hBcap hvolA hvolB⟩
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/IntegralDivergenceEqZero.lean b/lean/OAI/Geometry/Perimeter/Regularity/IntegralDivergenceEqZero.lean
new file mode 100644
index 0000000000000000000000000000000000000000..6908d0b46e4695833a90348fa1505ffa69ee25b7
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/IntegralDivergenceEqZero.lean
@@ -0,0 +1,329 @@
+import OAI.Geometry.Perimeter.Regularity.PairedCutTest
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+
+theorem integral_divergence_eq_zero {n : ℕ} {V : Euc n → Euc n}
+    (hV : ContDiff ℝ ∞ V) (hcV : HasCompactSupport V) :
+    (∫ x, divergence V x) = 0 := by
+  let b := EuclideanSpace.basisFun (Fin n) ℝ
+  let v : Fin n → Euc n → ℝ := fun i x ↦ inner ℝ (b i) (V x)
+  have hv : ∀ i, ContDiff ℝ ∞ (v i) := fun i ↦ (innerSL ℝ (b i)).contDiff.comp hV
+  have hcv : ∀ i, HasCompactSupport (v i) := fun i ↦ hcV.comp_left (by simp)
+  have hdv : ∀ i x, fderiv ℝ (v i) x (b i) =
+      inner ℝ (b i) (fderiv ℝ V x (b i)) := by
+    intro i x
+    have h := ((innerSL ℝ (b i)).hasFDerivAt.comp x
+      (hV.differentiable (by simp) x).hasFDerivAt).fderiv
+    exact congrArg (fun L : Euc n →L[ℝ] ℝ ↦ L (b i)) h
+  have hInt : ∀ i, Integrable (fun x ↦ fderiv ℝ (v i) x (b i)) volume := by
+    intro i
+    exact (((hv i).continuous_fderiv (by simp)).clm_apply continuous_const).integrable_of_hasCompactSupport
+      ((hcv i).fderiv_apply ℝ (b i))
+  have hz : ∀ i, (∫ x, fderiv ℝ (v i) x (b i)) = 0 := by
+    intro i
+    have h := integral_mul_fderiv_eq_neg_fderiv_mul_of_integrable
+      (f := fun _ : Euc n ↦ (1 : ℝ)) (g := v i) (v := b i) (μ := volume)
+      (by simp)
+      (by simpa using hInt i)
+      (by simpa using (hv i).continuous.integrable_of_hasCompactSupport (hcv i))
+      (fun _ _ ↦ differentiableAt_const _) (fun x _ ↦ (hv i).differentiable (by simp) x)
+    simpa using h
+  calc
+    (∫ x, divergence V x) = ∫ x, ∑ i, fderiv ℝ (v i) x (b i) := by
+      congr 1
+      funext x
+      simp only [divergence, hdv, b]
+    _ = ∑ i, ∫ x, fderiv ℝ (v i) x (b i) := integral_finsetSum _ (fun i _ ↦ hInt i)
+    _ = 0 := by simp only [hz, Finset.sum_const_zero]
+
+theorem divergence_neg {n : ℕ} {V : Euc n → Euc n}
+    (hV : Differentiable ℝ V) : divergence (-V) = -divergence V := by
+  funext x
+  simp only [divergence, (hV x).hasFDerivAt.neg.fderiv, neg_apply,
+    inner_neg_right, Finset.sum_neg_distrib, Pi.neg_apply]
+
+theorem perimeterAdmissible_neg {n : ℕ} {U : Set (Euc n)} {V : Euc n → Euc n}
+    (hV : PerimeterAdmissible U V) : PerimeterAdmissible U (-V) := by
+  exact ⟨hV.1.neg, hV.2.1.neg, by simpa using hV.2.2.1,
+    fun x ↦ by simpa using hV.2.2.2 x⟩
+
+theorem raw_perimeter_compl_le {n : ℕ} {E : Set (Euc n)} (hE : MeasurableSet E)
+    (U : Set (Euc n)) : perimeter Eᶜ U ≤ perimeter E U := by
+  rw [perimeter_eq_iSup_tests]
+  refine iSup_le fun V ↦ iSup_le fun hV ↦ ?_
+  have hsplit := integral_add_compl hE (integrable_divergence hV.1 hV.2.1)
+  rw [integral_divergence_eq_zero hV.1 hV.2.1] at hsplit
+  have heq : (∫ x in Eᶜ, divergence V x) = ∫ x in E, divergence (-V) x := by
+    simp only [divergence_neg (hV.1.differentiable (by simp)), Pi.neg_apply, integral_neg]
+    linarith
+  rw [heq, perimeter_eq_iSup_tests]
+  exact le_iSup_of_le (-V) (le_iSup_of_le (perimeterAdmissible_neg hV) le_rfl)
+
+theorem raw_perimeter_compl {n : ℕ} {E : Set (Euc n)} (hE : MeasurableSet E)
+    (U : Set (Euc n)) : perimeter Eᶜ U = perimeter E U := by
+  apply le_antisymm (raw_perimeter_compl_le hE U)
+  simpa only [compl_compl] using raw_perimeter_compl_le hE.compl U
+
+theorem perimeterSupport_compl {n : ℕ} {E : Set (Euc n)} (hE : MeasurableSet E) :
+    perimeterSupport Eᶜ = perimeterSupport E := by
+  ext x
+  simp only [perimeterSupport, mem_ofPred_eq, raw_perimeter_compl hE]
+
+theorem isLocallyPerimeterMinimizing_compl {n : ℕ} {E : Set (Euc n)}
+    (hmin : IsLocallyPerimeterMinimizing E) : IsLocallyPerimeterMinimizing Eᶜ := by
+  refine ⟨hmin.1.compl, ?_, ?_⟩
+  · intro x r hr
+    rw [raw_perimeter_compl hmin.1]
+    exact hmin.2.1 x r hr
+  · intro x r hr F hF hc hs
+    have heq : symmDiff E Fᶜ = symmDiff Eᶜ F := by
+      ext y
+      simp only [mem_symmDiff, mem_compl_iff]
+      tauto
+    have h := hmin.2.2 x r hr Fᶜ hF.compl (by simpa only [heq] using hc)
+      (by simpa only [heq] using hs)
+    simpa only [raw_perimeter_compl hmin.1, raw_perimeter_compl hF] using h
+
+theorem raw_perimeter_eq_zero_of_volume_inter_eq_zero {n : ℕ} {E U : Set (Euc n)}
+    (hU : MeasurableSet U) (hnull : volume (E ∩ U) = 0) : perimeter E U = 0 := by
+  apply le_antisymm _ bot_le
+  rw [perimeter_eq_iSup_tests]
+  refine iSup_le fun V ↦ iSup_le fun hV ↦ ?_
+  have hzero : divergence V =ᵐ[volume.restrict E] 0 := by
+    have hnonmem : ∀ᵐ x ∂volume.restrict E, x ∉ U := by
+      rw [ae_iff]
+      simp only [not_not]
+      change (volume.restrict E) U = 0
+      rw [Measure.restrict_apply hU, inter_comm]
+      exact hnull
+    filter_upwards [hnonmem] with x hx
+    exact image_eq_zero_of_notMem_tsupport
+      (fun h ↦ hx (hV.2.2.1 (tsupport_divergence_subset V h)))
+  rw [integral_congr_ae hzero]
+  simp
+
+theorem minimizing_two_phase_volume_lower {n : ℕ} {p : ℝ}
+    (hp : Real.HolderConjugate (n : ℝ) p) {R : ℝ} (hR : 0 < R) :
+    ∃ ε : ℝ, 0 < ε ∧ ∀ E : Set (Euc n), IsLocallyPerimeterMinimizing E →
+      ∀ c ∈ perimeterSupport E,
+        ε < volume.real (E ∩ Metric.closedBall c R) ∧
+        ε < volume.real (Eᶜ ∩ Metric.closedBall c R) := by
+  obtain ⟨ε, hε, hclear⟩ := minimizing_clearing_out hp hR
+  have hphase : ∀ E : Set (Euc n), IsLocallyPerimeterMinimizing E →
+      ∀ c ∈ perimeterSupport E, ε < volume.real (E ∩ Metric.closedBall c R) := by
+    intro E hmin c hc
+    by_contra h
+    have hz := hclear E hmin c (le_of_not_gt h)
+    have hpz := raw_perimeter_eq_zero_of_volume_inter_eq_zero measurableSet_ball
+      (measure_mono_null (inter_subset_inter_right E Metric.ball_subset_closedBall) hz)
+    have hpos := hc (R / 2) (by positivity)
+    rw [hpz] at hpos
+    exact (lt_irrefl _ hpos)
+  refine ⟨ε, hε, ?_⟩
+  intro E hmin c hc
+  exact ⟨hphase E hmin c hc, hphase Eᶜ (isLocallyPerimeterMinimizing_compl hmin) c
+    (by simpa only [perimeterSupport_compl hmin.1] using hc)⟩
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+
+theorem holder_volume_scale {n : ℕ} {p R : ℝ}
+    (hp : Real.HolderConjugate (n : ℝ) p) (hR : 0 < R) :
+    (R ^ (n : ℝ) / R) ^ p = R ^ (n : ℝ) := by
+  rw [← Real.rpow_sub_one hR.ne', ← Real.rpow_mul hR.le, hp.sub_one_mul_conj]
+
+theorem minimizing_scaled_clearing_out {n : ℕ} {p : ℝ}
+    (hp : Real.HolderConjugate (n : ℝ) p) :
+    ∃ ε : ℝ, 0 < ε ∧ ∀ E : Set (Euc n), IsLocallyPerimeterMinimizing E →
+      ∀ c : Euc n, ∀ R : ℝ, 0 < R →
+      volume.real (E ∩ Metric.closedBall c R) ≤ ε * R ^ n →
+        volume (E ∩ Metric.closedBall c (R / 2)) = 0 := by
+  obtain ⟨L, hL, hbound⟩ := exists_smoothTransition_deriv_bound
+  let S : ℝ := lintegralPowLePowLIntegralFDerivConst (volume : Measure (Euc n)) p
+  let K : ℝ := 64 * L
+  let C : ℝ := (S + 1) * K ^ p
+  let B : ℝ := (2 : ℝ) ^ p
+  have hS : 0 ≤ S := NNReal.coe_nonneg _
+  have hK : 0 < K := by dsimp [K]; positivity
+  have hC : 0 < C := by dsimp [C]; positivity
+  have hp1 : 1 < p := hp.symm.lt
+  have hB : 1 < B := Real.one_lt_rpow (by norm_num) (by linarith)
+  obtain ⟨ε, hε, hiteration⟩ := nonlinear_iteration_tendsto_zero hp1 hC hB
+  refine ⟨ε, hε, ?_⟩
+  intro E hmin c R hR hsmall
+  let Q : ℝ := R ^ n
+  have hQ : 0 < Q := pow_pos hR n
+  have hscale : (Q / R) ^ p = Q := by
+    simpa only [Q, Real.rpow_natCast] using holder_volume_scale hp hR
+  let u : ℕ → ℝ := fun i ↦ volume.real (E ∩ Metric.closedBall c (clearingRadius R i)) / Q
+  have hu : ∀ i, 0 ≤ u i := fun _ ↦ div_nonneg ENNReal.toReal_nonneg hQ.le
+  have hstart : u 0 ≤ ε := by
+    dsimp [u]
+    rw [clearingRadius_zero]
+    exact (div_le_iff₀ hQ).mpr hsmall
+  have hstep : ∀ i, u (i + 1) ≤ C * B ^ i * (u i) ^ p := by
+    intro i
+    have hm := minimizing_volume_recursion hp hmin c (clearingRadius_pos hR (i + 1))
+      (clearingRadius_succ_lt hR i) hL hbound
+    have heq : 16 * L * volume.real (E ∩ Metric.closedBall c (clearingRadius R i)) /
+        (clearingRadius R i - clearingRadius R (i + 1)) =
+        (K * (2 : ℝ) ^ i * u i) * (Q / R) := by
+      rw [clearingRadius_sub_succ]
+      dsimp [K, u]
+      field_simp
+      ring
+    change volume.real (E ∩ Metric.closedBall c (clearingRadius R (i + 1))) ≤ S * _ at hm
+    rw [heq, Real.mul_rpow (by positivity) (by positivity), hscale] at hm
+    have hpow : ((2 : ℝ) ^ i) ^ p = ((2 : ℝ) ^ p) ^ i := by
+      rw [← Real.rpow_natCast_mul (by norm_num : (0 : ℝ) ≤ 2),
+        mul_comm (i : ℝ), Real.rpow_mul_natCast (by norm_num : (0 : ℝ) ≤ 2)]
+    calc
+      u (i + 1) ≤ S * (K * (2 : ℝ) ^ i * u i) ^ p := by
+        exact (div_le_iff₀ hQ).mpr (by simpa only [mul_assoc] using hm)
+      _ = (S * K ^ p) * B ^ i * (u i) ^ p := by
+        rw [Real.mul_rpow (by positivity) (hu i),
+          Real.mul_rpow hK.le (by positivity), hpow]
+        dsimp [B]
+        ring
+      _ ≤ C * B ^ i * (u i) ^ p := by
+        unfold C
+        gcongr
+        linarith
+  have ht := hiteration u hu hstart hstep
+  have hzero : volume.real (E ∩ Metric.closedBall c (R / 2)) / Q = 0 := by
+    apply le_antisymm _ (div_nonneg ENNReal.toReal_nonneg hQ.le)
+    apply ge_of_tendsto ht
+    apply Eventually.of_forall
+    intro i
+    exact div_le_div_of_nonneg_right (measureReal_mono
+      (inter_subset_inter_right E (Metric.closedBall_subset_closedBall (clearingRadius_half_le hR.le i)))
+      (measure_ne_top_of_subset inter_subset_right (isCompact_closedBall c (clearingRadius R i)).measure_ne_top)) hQ.le
+  have hvol : volume.real (E ∩ Metric.closedBall c (R / 2)) = 0 := (div_eq_zero_iff).mp hzero |>.resolve_right hQ.ne'
+  exact ((ENNReal.toReal_eq_zero_iff _).mp hvol).resolve_right
+    (measure_ne_top_of_subset inter_subset_right (isCompact_closedBall c (R / 2)).measure_ne_top)
+
+theorem minimizing_scaled_two_phase_volume_lower {n : ℕ} {p : ℝ}
+    (hp : Real.HolderConjugate (n : ℝ) p) :
+    ∃ ε : ℝ, 0 < ε ∧ ∀ E : Set (Euc n), IsLocallyPerimeterMinimizing E →
+      ∀ c ∈ perimeterSupport E, ∀ R : ℝ, 0 < R →
+        ε * R ^ n < volume.real (E ∩ Metric.closedBall c R) ∧
+        ε * R ^ n < volume.real (Eᶜ ∩ Metric.closedBall c R) := by
+  obtain ⟨ε, hε, hclear⟩ := minimizing_scaled_clearing_out hp
+  have hphase : ∀ E : Set (Euc n), IsLocallyPerimeterMinimizing E →
+      ∀ c ∈ perimeterSupport E, ∀ R : ℝ, 0 < R →
+        ε * R ^ n < volume.real (E ∩ Metric.closedBall c R) := by
+    intro E hmin c hc R hR
+    by_contra h
+    have hz := hclear E hmin c R hR (le_of_not_gt h)
+    have hpz := raw_perimeter_eq_zero_of_volume_inter_eq_zero measurableSet_ball
+      (measure_mono_null (inter_subset_inter_right E Metric.ball_subset_closedBall) hz)
+    have hpos := hc (R / 2) (by positivity)
+    rw [hpz] at hpos
+    exact (lt_irrefl _ hpos)
+  refine ⟨ε, hε, ?_⟩
+  intro E hmin c hc R hR
+  exact ⟨hphase E hmin c hc R hR, hphase Eᶜ (isLocallyPerimeterMinimizing_compl hmin) c
+    (by simpa only [perimeterSupport_compl hmin.1] using hc) R hR⟩
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+
+theorem indicator_one_locallyIntegrable {n : ℕ} {E : Set (Euc n)}
+    (hE : MeasurableSet E) : LocallyIntegrable (E.indicator (fun _ ↦ (1 : ℝ))) volume := by
+  apply (locallyIntegrable_const (μ := (volume : Measure (Euc n))) (1 : ℝ)).mono
+    (measurable_const.indicator hE).aestronglyMeasurable
+  exact ae_of_all _ fun x ↦ by
+    by_cases hx : x ∈ E <;> simp [hx]
+
+theorem average_indicator_one_closedBall {n : ℕ} {E : Set (Euc n)}
+    (hE : MeasurableSet E) (c : Euc n) (R : ℝ) :
+    (⨍ x in Metric.closedBall c R, E.indicator (fun _ ↦ (1 : ℝ)) x) =
+      volume.real (E ∩ Metric.closedBall c R) / volume.real (Metric.closedBall c R) := by
+  rw [setAverage_eq, integral_indicator hE, Measure.restrict_restrict hE,
+    integral_const, smul_eq_mul, smul_eq_mul]
+  simp only [measureReal_restrict_apply_univ, one_mul, div_eq_mul_inv, mul_comm]
+
+theorem minimizing_phaseSupport_volume_lower {n : ℕ} {p : ℝ}
+    (hp : Real.HolderConjugate (n : ℝ) p) :
+    ∃ ε : ℝ, 0 < ε ∧ ∀ E : Set (Euc n), IsLocallyPerimeterMinimizing E →
+      ∀ c ∈ (volume.restrict E).support, ∀ R : ℝ, 0 < R →
+        ε * R ^ n < volume.real (E ∩ Metric.closedBall c R) := by
+  obtain ⟨ε, hε, hclear⟩ := minimizing_scaled_clearing_out hp
+  refine ⟨ε, hε, ?_⟩
+  intro E hmin c hc R hR
+  by_contra h
+  have hz := hclear E hmin c R hR (le_of_not_gt h)
+  have hpos := (Metric.nhds_basis_ball.mem_measureSupport (μ := volume.restrict E)).mp hc
+    (R / 2) (by positivity)
+  rw [Measure.restrict_apply measurableSet_ball, inter_comm] at hpos
+  have hnull := measure_mono_null (inter_subset_inter_right E Metric.ball_subset_closedBall) hz
+  exact hpos.ne' hnull
+
+theorem minimizing_ae_eq_phaseSupport {n : ℕ} {p : ℝ}
+    (hp : Real.HolderConjugate (n : ℝ) p) {E : Set (Euc n)}
+    (hmin : IsLocallyPerimeterMinimizing E) : E =ᵐ[volume] (volume.restrict E).support := by
+  obtain ⟨ε, hε, hlower⟩ := minimizing_phaseSupport_volume_lower hp
+  let omegaVol : ℝ := volume.real (Metric.ball (0 : Euc n) 1)
+  have homegaVol : 0 < omegaVol := by
+    exact ENNReal.toReal_pos (Metric.measure_ball_pos volume (0 : Euc n) (by norm_num)).ne'
+      measure_ball_lt_top.ne
+  have hmean : ∀ c ∈ (volume.restrict E).support, ∀ R : ℝ, 0 < R →
+      ε / omegaVol ≤ ⨍ x in Metric.closedBall c R, E.indicator (fun _ ↦ (1 : ℝ)) x := by
+    intro c hc R hR
+    rw [average_indicator_one_closedBall hmin.1,
+      Measure.addHaar_real_closedBall volume c hR.le]
+    simp only [finrank_euclideanSpace, Fintype.card_fin]
+    calc
+      ε / omegaVol = (ε * R ^ n) / (R ^ n * omegaVol) := by field_simp [hR.ne', homegaVol.ne']
+      _ ≤ volume.real (E ∩ Metric.closedBall c R) / (R ^ n * omegaVol) :=
+        div_le_div_of_nonneg_right (hlower E hmin c hc R hR).le (by positivity)
+  have hmem : ∀ᵐ x ∂volume, x ∈ E → x ∈ (volume.restrict E).support :=
+    (ae_restrict_iff' hmin.1).mp (volume.restrict E).support_mem_ae
+  have hdiff := IsUnifLocDoublingMeasure.ae_tendsto_average volume
+    (indicator_one_locallyIntegrable hmin.1) (1 : ℝ)
+  filter_upwards [hmem, hdiff] with x hmem hdiff
+  apply propext
+  constructor
+  · exact hmem
+  · intro hx
+    by_contra hxE
+    have ht := hdiff (fun _ : ℝ ↦ x) (id : ℝ → ℝ) tendsto_id
+      (by filter_upwards [self_mem_nhdsWithin] with R hR
+          exact Metric.mem_closedBall_self (by simpa only [one_mul] using hR.le))
+    simp only [indicator_of_notMem hxE] at ht
+    have hle : ε / omegaVol ≤ (0 : ℝ) := ge_of_tendsto ht (by
+      filter_upwards [self_mem_nhdsWithin] with R hR
+      exact hmean x hx R hR)
+    exact (div_pos hε homegaVol).not_ge hle
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/IntegralDivergenceEqZeroOfPerimeterEqZero.lean b/lean/OAI/Geometry/Perimeter/Regularity/IntegralDivergenceEqZeroOfPerimeterEqZero.lean
new file mode 100644
index 0000000000000000000000000000000000000000..e43bcb706f310f4523f2bbff7bf509f65c9d949d
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/IntegralDivergenceEqZeroOfPerimeterEqZero.lean
@@ -0,0 +1,380 @@
+import OAI.Geometry.Perimeter.Regularity.IntegralDivergenceEqZero
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+open RPI.Boundary ContinuousLinearMap
+open scoped Convolution
+
+theorem integral_divergence_eq_zero_of_perimeter_eq_zero {n : ℕ}
+    {E U : Set (Euc n)} (hP : perimeter E U = 0)
+    {V : Euc n → Euc n} (hV : ContDiff ℝ ∞ V) (hc : HasCompactSupport V)
+    (hs : tsupport V ⊆ U) : (∫ x in E, divergence V x) = 0 := by
+  obtain ⟨M, hM⟩ := hc.exists_bound_of_continuous hV.continuous
+  let a : ℝ := (|M| + 1)⁻¹
+  have ha : 0 < a := by dsimp [a]; positivity
+  let W : Euc n → Euc n := fun x ↦ a • V x
+  have hW : PerimeterAdmissible U W := by
+    have hsmul : ContDiff ℝ ∞ (fun z : Euc n ↦ a • z) :=
+      (contDiff_const (c := a)).smul contDiff_id
+    refine ⟨hsmul.comp hV, hc.comp_left (g := fun z : Euc n ↦ a • z) (smul_zero a), ?_, ?_⟩
+    · exact (tsupport_smul_subset_right (fun _ ↦ a) V).trans hs
+    · intro x
+      simp only [W, norm_smul, Real.norm_of_nonneg ha.le]
+      calc
+        a * ‖V x‖ ≤ a * |M| := mul_le_mul_of_nonneg_left ((hM x).trans (le_abs_self M)) ha.le
+        _ ≤ a * (|M| + 1) := mul_le_mul_of_nonneg_left (by linarith) ha.le
+        _ = 1 := inv_mul_cancel₀ (by positivity)
+  have hupper := integral_divergence_le_perimeter_real E U (by rw [hP]; simp) hW
+  have hlower := integral_divergence_le_perimeter_real E U (by rw [hP]; simp)
+    (perimeterAdmissible_neg hW)
+  have hd : divergence W = fun x ↦ a * divergence V x := by
+    funext x
+    rw [divergence_smul (differentiable_const a) (hV.differentiable (by simp))]
+    simp
+  rw [hd, integral_const_mul, hP, ENNReal.toReal_zero] at hupper
+  rw [divergence_neg (hW.1.differentiable (by simp))] at hlower
+  simp only [Pi.neg_apply] at hlower
+  rw [integral_neg, hd,
+    integral_const_mul, hP, ENNReal.toReal_zero] at hlower
+  nlinarith
+
+theorem convolution_fderiv_eq_zero_of_perimeter_eq_zero {n : ℕ}
+    {E U : Set (Euc n)} (hE : MeasurableSet E) (hP : perimeter E U = 0)
+    {k : Euc n → ℝ} (hk : ContDiff ℝ ∞ k) (hck : HasCompactSupport k)
+    (x : Euc n) (hs : ∀ y, x - y ∈ tsupport k → y ∈ U) :
+    fderiv ℝ (E.indicator (fun _ ↦ (1 : ℝ)) ⋆[lsmul ℝ ℝ, volume] k) x = 0 := by
+  let χ : Euc n → ℝ := E.indicator (fun _ ↦ 1)
+  have hχ : LocallyIntegrable χ volume := indicator_one_locallyIntegrable hE
+  have hk1 : ContDiff ℝ 1 k := hk.of_le (by simp)
+  rw [(hck.hasFDerivAt_convolution_right (lsmul ℝ ℝ) hχ hk1 x).fderiv]
+  ext v
+  rw [convolution_precompR_apply (lsmul ℝ ℝ) hχ (hck.fderiv ℝ)
+    (hk.continuous_fderiv (by simp))]
+  let a : Euc n → ℝ := fun y ↦ k (x - y)
+  let V : Euc n → Euc n := fun y ↦ a y • v
+  have ha : ContDiff ℝ ∞ a := hk.comp (contDiff_const.sub contDiff_id)
+  have hca : HasCompactSupport a := hck.comp_homeomorph (Homeomorph.subLeft x)
+  have hsV : tsupport V ⊆ U := by
+    apply (tsupport_smul_subset_left a (fun _ ↦ v)).trans
+    intro y hy
+    apply hs y
+    exact (tsupport_comp_subset_preimage k (continuous_const.sub continuous_id)) hy
+  have hd : divergence V = fun y ↦ -fderiv ℝ k (x - y) v := by
+    funext y
+    rw [divergence_smul (ha.differentiable (by simp)) (differentiable_const v)]
+    simp only [divergence, fderiv_fun_const, Pi.zero_apply, zero_apply, inner_zero_right, Finset.sum_const_zero,
+      mul_zero, zero_add, inner_gradient_left]
+    have h := ((hk.differentiable (by simp) (x - y)).hasFDerivAt.comp y
+      ((hasFDerivAt_const x y).sub (hasFDerivAt_id y))).fderiv
+    simpa only [a, Function.comp_def, sub_apply, zero_apply, id_apply, zero_sub,
+      ContinuousLinearMap.comp_apply, neg_apply, map_neg] using congrArg (fun L : Euc n →L[ℝ] ℝ ↦ L v) h
+  have hz := integral_divergence_eq_zero_of_perimeter_eq_zero hP
+    (ha.smul (contDiff_const (c := v))) hca.smul_right hsV
+  change (∫ x in E, divergence V x) = 0 at hz
+  rw [hd, integral_neg, neg_eq_zero] at hz
+  change (∫ t, χ t * fderiv ℝ k (x - t) v) = 0
+  have heq : (fun t ↦ χ t * fderiv ℝ k (x - t) v) =
+      E.indicator (fun t ↦ fderiv ℝ k (x - t) v) := by
+    funext t
+    by_cases ht : t ∈ E <;> simp [χ, ht]
+  rw [heq, integral_indicator hE]
+  exact hz
+
+theorem volume_inter_eq_zero_or_compl_of_perimeter_eq_zero {n : ℕ}
+    {E : Set (Euc n)} (hE : MeasurableSet E) (c : Euc n) {R : ℝ} (hR : 0 < R)
+    (hP : perimeter E (Metric.ball c R) = 0) :
+    volume (E ∩ Metric.ball c (R / 2)) = 0 ∨
+      volume (Eᶜ ∩ Metric.ball c (R / 2)) = 0 := by
+  let χ : Euc n → ℝ := E.indicator (fun _ ↦ 1)
+  let u : ℕ → Euc n → ℝ := c1Mollify χ
+  have hχ : LocallyIntegrable χ volume := indicator_one_locallyIntegrable hE
+  have hflip : (lsmul ℝ ℝ : ℝ →L[ℝ] ℝ →L[ℝ] ℝ).flip = lsmul ℝ ℝ := by
+    apply ContinuousLinearMap.ext
+    intro a
+    apply ContinuousLinearMap.ext
+    intro b
+    exact mul_comm b a
+  have hu_eq : ∀ j, u j = χ ⋆[lsmul ℝ ℝ, volume] (c1Mollifier j).normed volume := by
+    intro j
+    change (c1Mollifier j).normed volume ⋆[lsmul ℝ ℝ, volume] χ = _
+    rw [← convolution_flip, hflip]
+  have hu : ∀ j, ContDiff ℝ ∞ (u j) := fun j ↦
+    (c1Mollifier j).hasCompactSupport_normed.contDiff_convolution_left _
+      (c1Mollifier j).contDiff_normed hχ
+  have hconst : ∀ᶠ j in atTop, ∀ x ∈ Metric.ball c (R / 2),
+      ∀ y ∈ Metric.ball c (R / 2), u j x = u j y := by
+    have hsmall : ∀ᶠ j in atTop, (c1Mollifier (n := n) j).rOut < R / 2 :=
+      c1Mollifier_radius_tendsto.eventually (gt_mem_nhds (by positivity))
+    filter_upwards [hsmall] with j hj
+    have hzero : (Metric.ball c (R / 2)).EqOn (fderiv ℝ (u j)) 0 := by
+      intro x hx
+      rw [hu_eq j]
+      apply convolution_fderiv_eq_zero_of_perimeter_eq_zero hE hP
+        (c1Mollifier j).contDiff_normed (c1Mollifier j).hasCompactSupport_normed x
+      intro y hy
+      rw [(c1Mollifier j).tsupport_normed_eq] at hy
+      have hxy : dist y x ≤ (c1Mollifier (n := n) j).rOut := by
+        simpa only [Metric.mem_closedBall, dist_zero_right, dist_eq_norm, sub_zero, norm_sub_rev] using hy
+      exact (dist_triangle y x c).trans_lt (by linarith [Metric.mem_ball.mp hx])
+    intro x hx y hy
+    exact Metric.isOpen_ball.is_const_of_fderiv_eq_zero (convex_ball c (R / 2)).isPreconnected
+      ((hu j).differentiable (by simp)).differentiableOn hzero hx hy
+  have hlim : ∀ᵐ x ∂volume, Tendsto (fun j ↦ u j x) atTop (𝓝 (χ x)) := by
+    apply ContDiffBump.ae_convolution_tendsto_right_of_locallyIntegrable
+      c1Mollifier_radius_tendsto (K := 2) _ hχ
+    apply Eventually.of_forall
+    intro j
+    dsimp [c1Mollifier]
+    linarith
+  by_contra hn
+  push Not at hn
+  obtain ⟨x, hx, hlimx⟩ := Measure.exists_mem_of_measure_ne_zero_of_ae hn.1
+    (hlim.filter_mono (ae_mono Measure.restrict_le_self))
+  obtain ⟨y, hy, hlimy⟩ := Measure.exists_mem_of_measure_ne_zero_of_ae hn.2
+    (hlim.filter_mono (ae_mono Measure.restrict_le_self))
+  have hxy : (fun j ↦ u j x) =ᶠ[atTop] (fun j ↦ u j y) :=
+    hconst.mono fun j hj ↦ hj x hx.2 y hy.2
+  have hlimxy := tendsto_nhds_unique hlimx (hlimy.congr' hxy.symm)
+  have : (1 : ℝ) = 0 := by
+    simpa only [χ, indicator_of_mem hx.1, indicator_of_notMem hy.1] using hlimxy
+  exact one_ne_zero this
+
+theorem perimeterSupport_eq_phaseSupport_inter {n : ℕ} {E : Set (Euc n)}
+    (hE : MeasurableSet E) :
+    perimeterSupport E = (volume.restrict E).support ∩ (volume.restrict Eᶜ).support := by
+  ext x
+  constructor
+  · intro hx
+    have hphase : ∀ F : Set (Euc n), MeasurableSet F → x ∈ perimeterSupport F →
+        x ∈ (volume.restrict F).support := by
+      intro F hF hxF
+      apply (Metric.nhds_basis_ball.mem_measureSupport (μ := volume.restrict F)).mpr
+      intro r hr
+      apply pos_iff_ne_zero.mpr
+      intro hz
+      rw [Measure.restrict_apply measurableSet_ball, inter_comm] at hz
+      have hpz := raw_perimeter_eq_zero_of_volume_inter_eq_zero measurableSet_ball hz
+      exact (hxF r hr).ne' hpz
+    exact ⟨hphase E hE hx, hphase Eᶜ hE.compl
+      (by simpa only [perimeterSupport_compl hE] using hx)⟩
+  · rintro ⟨hxE, hxEc⟩ r hr
+    apply pos_iff_ne_zero.mpr
+    intro hpz
+    rcases volume_inter_eq_zero_or_compl_of_perimeter_eq_zero hE x hr hpz with hz | hz
+    · have hpos := (Metric.nhds_basis_ball.mem_measureSupport (μ := volume.restrict E)).mp hxE
+        (r / 2) (by positivity)
+      rw [Measure.restrict_apply measurableSet_ball, inter_comm] at hpos
+      exact hpos.ne' hz
+    · have hpos := (Metric.nhds_basis_ball.mem_measureSupport (μ := volume.restrict Eᶜ)).mp hxEc
+        (r / 2) (by positivity)
+      rw [Measure.restrict_apply measurableSet_ball, inter_comm] at hpos
+      exact hpos.ne' hz
+
+theorem minimizing_frontier_phaseSupport {n : ℕ} {p : ℝ}
+    (hp : Real.HolderConjugate (n : ℝ) p) {E : Set (Euc n)}
+    (hmin : IsLocallyPerimeterMinimizing E) :
+    frontier (volume.restrict E).support = perimeterSupport E := by
+  let K : Set (Euc n) := (volume.restrict E).support
+  have hK : IsClosed K := Measure.isClosed_support
+  have hEK : E =ᵐ[volume] K := minimizing_ae_eq_phaseSupport hp hmin
+  have hEcKc : Eᶜ =ᵐ[volume] Kᶜ := by
+    apply Filter.eventuallyEqSet_iff.mpr
+    filter_upwards [hEK.mem_iff] with x hx
+    exact not_congr hx
+  have hKc : (volume.restrict Kᶜ).support = closure Kᶜ := by
+    apply le_antisymm (Measure.support_restrict_subset.trans inter_subset_left)
+    apply closure_minimal _ Measure.isClosed_support
+    intro x hx
+    apply Measure.interior_inter_support
+    exact ⟨(hK.isOpen_compl.interior_eq).symm ▸ hx, by rw [Measure.support_eq_univ]; trivial⟩
+  have hμ : (volume : Measure (Euc n)).restrict Eᶜ = volume.restrict Kᶜ :=
+    Measure.restrict_congr_set hEcKc
+  rw [perimeterSupport_eq_phaseSupport_inter hmin.1, hμ, hKc,
+    frontier_eq_closure_inter_closure, hK.closure_eq]
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+open ContinuousLinearMap
+
+theorem chord_half_norm_sub_le {n : ℕ} {f : Euc n → ℝ}
+    (hf : ContDiff ℝ 1 f) (c x y : Euc n) {R : ℝ}
+    (hx : x ∈ Metric.closedBall c R) (hy : y ∈ Metric.closedBall c R) :
+    ‖f x - f ((1 / 2 : ℝ) • x + (1 / 2 : ℝ) • y)‖ ≤
+      2 * R * ∫ t in (1 / 2 : ℝ)..1, ‖fderiv ℝ f (t • x + (1 - t) • y)‖ := by
+  let z : ℝ → Euc n := fun t ↦ t • x + (1 - t) • y
+  have hz : ∀ t, HasDerivAt z (x - y) t := by
+    intro t
+    simpa [z, sub_eq_add_neg, Pi.add_def] using
+      ((hasDerivAt_id t).smul_const x).add
+        (((hasDerivAt_const t (1 : ℝ)).sub (hasDerivAt_id t)).smul_const y)
+  have hdf : ∀ t, HasDerivAt (fun t ↦ f (z t)) (fderiv ℝ f (z t) (x - y)) t := fun t ↦
+    (hf.differentiable one_ne_zero (z t)).hasFDerivAt.comp_hasDerivAt t (hz t)
+  have hxy : ‖x - y‖ ≤ 2 * R := by
+    have hh := (dist_triangle x c y).trans (add_le_add (Metric.mem_closedBall.mp hx)
+      (by simpa only [dist_comm] using Metric.mem_closedBall.mp hy))
+    simpa only [dist_eq_norm, two_mul] using hh
+  have hD : Continuous (fun t ↦ ‖fderiv ℝ f (z t)‖) :=
+    ((hf.continuous_fderiv one_ne_zero).comp (by fun_prop)).norm
+  have hi := norm_sub_le_integral_of_norm_deriv_le_of_le (a := (1 / 2 : ℝ))
+    (b := 1) (f := fun t ↦ f (z t)) (B := fun t ↦ 2 * R * ‖fderiv ℝ f (z t)‖) (by norm_num)
+    (hf.continuous.comp (by fun_prop)).continuousOn
+    (fun t _ ↦ (hdf t).differentiableAt.differentiableWithinAt)
+    (ae_of_all _ fun t _ ↦ by
+      rw [(hdf t).deriv]
+      exact ((fderiv ℝ f (z t)).le_opNorm (x - y)).trans
+        (by nlinarith [norm_nonneg (fderiv ℝ f (z t))]))
+    ((continuous_const.mul hD).intervalIntegrable _ _)
+  simpa only [z, Function.comp_def, one_smul, sub_self, zero_smul, add_zero, show (1 : ℝ) - 1 / 2 = 1 / 2 by norm_num,
+    intervalIntegral.integral_const_mul] using hi
+
+theorem chord_half_integral_le {n : ℕ} {G : Euc n → ℝ}
+    (hG : Continuous G) (hG0 : ∀ x, 0 ≤ G x) (c y : Euc n) {R t : ℝ}
+    (hy : y ∈ Metric.closedBall c R) (ht : t ∈ Icc (1 / 2 : ℝ) 1) :
+    (∫ x in Metric.closedBall c R, G (t • x + (1 - t) • y)) ≤
+      2 ^ n * ∫ x in Metric.closedBall c R, G x := by
+  let B := Metric.closedBall c R
+  let g : Euc n → ℝ := B.indicator G
+  have hg : Integrable g volume :=
+    (integrable_indicator_iff measurableSet_closedBall).2
+      (hG.continuousOn.integrableOn_compact (isCompact_closedBall c R))
+  have hg0 : ∀ x, 0 ≤ g x := fun x ↦ indicator_nonneg (fun x _ ↦ hG0 x) x
+  have ht0 : 0 < t := lt_of_lt_of_le (by norm_num) ht.1
+  have hga : Integrable (fun x ↦ g (t • x + (1 - t) • y)) volume :=
+    (hg.comp_add_right ((1 - t) • y)).comp_smul ht0.ne'
+  have heq : ∀ x ∈ B, G (t • x + (1 - t) • y) = g (t • x + (1 - t) • y) := by
+    intro x hx
+    have hz : t • x + (1 - t) • y ∈ B :=
+      (convex_closedBall c R) hx hy ht0.le (sub_nonneg.mpr ht.2) (by ring)
+    exact (indicator_of_mem hz G).symm
+  calc
+    (∫ x in B, G (t • x + (1 - t) • y)) =
+        ∫ x in B, g (t • x + (1 - t) • y) :=
+      setIntegral_congr_fun measurableSet_closedBall heq
+    _ ≤ ∫ x, g (t • x + (1 - t) • y) :=
+      setIntegral_le_integral hga (ae_of_all _ fun x ↦ hg0 _)
+    _ = (t ^ n)⁻¹ * ∫ x in B, G x := by
+      rw [Measure.integral_comp_smul_of_nonneg volume
+        (fun z ↦ g (z + (1 - t) • y)) t (hR := ht0.le)]
+      rw [integral_add_right_eq_self, integral_indicator measurableSet_closedBall]
+      simp only [Euc, finrank_euclideanSpace, Fintype.card_fin, smul_eq_mul, B]
+    _ ≤ 2 ^ n * ∫ x in B, G x := by
+      apply mul_le_mul_of_nonneg_right _ (integral_nonneg fun x ↦ hG0 x)
+      have hh : (t ^ n)⁻¹ ≤ ((1 / 2 : ℝ) ^ n)⁻¹ :=
+        inv_anti₀ (pow_pos (by norm_num : (0 : ℝ) < 1 / 2) _) (pow_le_pow_left₀ (by norm_num) ht.1 _)
+      simpa only [inv_pow, one_div, inv_inv] using hh
+
+theorem integrated_chord_half_le {n : ℕ} {f : Euc n → ℝ}
+    (hf : ContDiff ℝ 1 f) (c y : Euc n) {R : ℝ}
+    (hy : y ∈ Metric.closedBall c R) :
+    (∫ x in Metric.closedBall c R, ‖f x - f ((1 / 2 : ℝ) • x + (1 / 2 : ℝ) • y)‖) ≤
+      (2 * R) * (2 ^ n * ∫ x in Metric.closedBall c R, ‖fderiv ℝ f x‖) := by
+  let B := Metric.closedBall c R
+  let I := Icc (1 / 2 : ℝ) 1
+  let G : Euc n → ℝ := fun x ↦ ‖fderiv ℝ f x‖
+  let F : Euc n × ℝ → ℝ := fun p ↦ G (p.2 • p.1 + (1 - p.2) • y)
+  have hG : Continuous G := (hf.continuous_fderiv one_ne_zero).norm
+  have hF : Continuous F := hG.comp (by fun_prop)
+  have hiF : Integrable F ((volume.restrict B).prod (volume.restrict I)) := by
+    rw [Measure.prod_restrict]
+    exact hF.continuousOn.integrableOn_compact ((isCompact_closedBall c R).prod isCompact_Icc)
+  have hR : 0 ≤ 2 * R := by nlinarith [Metric.mem_closedBall.mp hy, dist_nonneg (x := y) (y := c)]
+  have hp : IntegrableOn
+      (fun x ↦ ‖f x - f ((1 / 2 : ℝ) • x + (1 / 2 : ℝ) • y)‖) B volume :=
+    (hf.continuous.sub (hf.continuous.comp (by fun_prop))).norm.continuousOn.integrableOn_compact
+      (isCompact_closedBall c R)
+  calc
+    _ ≤ ∫ x in B, (2 * R) * ∫ t in I, F (x, t) := by
+      apply setIntegral_mono_on hp (hiF.integral_prod_left.const_mul _) measurableSet_closedBall
+      intro x hx
+      have hh := chord_half_norm_sub_le hf c x y hx hy
+      rw [intervalIntegral.integral_of_le (by norm_num : (1 / 2 : ℝ) ≤ 1),
+        ← integral_Icc_eq_integral_Ioc] at hh
+      exact hh
+    _ = (2 * R) * ∫ t in I, ∫ x in B, F (x, t) := by
+      rw [integral_const_mul, integral_integral_swap hiF]
+    _ ≤ (2 * R) * (2 ^ n * ∫ x in B, G x) := by
+      apply mul_le_mul_of_nonneg_left _ hR
+      have hh : (∫ t in I, ∫ x in B, F (x, t)) ≤
+          ∫ _t in I, (2 : ℝ) ^ n * ∫ x in B, G x := by
+        apply setIntegral_mono_on hiF.integral_prod_right continuous_const.integrableOn_Icc measurableSet_Icc
+        intro t ht
+        exact chord_half_integral_le hG (fun x ↦ norm_nonneg _) c y hy ht
+      calc
+        _ ≤ _ := hh
+        _ ≤ (2 : ℝ) ^ n * ∫ x in B, G x := by
+          rw [integral_const, measureReal_restrict_apply_univ, smul_eq_mul]
+          have hvol : (volume I).toReal ≤ 1 := by
+            norm_num [I, Real.volume_Icc]
+          change (volume I).toReal * _ ≤ _
+          nlinarith [mul_nonneg (pow_nonneg (by norm_num : (0 : ℝ) ≤ 2) n)
+            (integral_nonneg (μ := volume.restrict B) (fun x ↦ norm_nonneg (fderiv ℝ f x))), measureReal_nonneg (μ := volume) (s := I)]
+
+theorem smooth_ball_poincare {n : ℕ} {f : Euc n → ℝ}
+    (hf : ContDiff ℝ 1 f) (c : Euc n) {R : ℝ} (hR : 0 < R) :
+    (∫ p, ‖f p.1 - f p.2‖ ∂((volume.restrict (Metric.closedBall c R)).prod
+      (volume.restrict (Metric.closedBall c R)))) ≤
+      4 * R * volume.real (Metric.closedBall c R) *
+        (2 ^ n * ∫ x in Metric.closedBall c R, ‖fderiv ℝ f x‖) := by
+  let B := Metric.closedBall c R
+  let H : Euc n × Euc n → ℝ := fun p ↦
+    ‖f p.1 - f ((1 / 2 : ℝ) • p.1 + (1 / 2 : ℝ) • p.2)‖
+  have hHc : Continuous H := (hf.continuous.comp continuous_fst |>.sub
+    (hf.continuous.comp (by fun_prop))).norm
+  have hHi : Integrable H ((volume.restrict B).prod (volume.restrict B)) := by
+    rw [Measure.prod_restrict]
+    exact hHc.continuousOn.integrableOn_compact
+      ((isCompact_closedBall c R).prod (isCompact_closedBall c R))
+  have hDi : Integrable (fun p : Euc n × Euc n ↦ ‖f p.1 - f p.2‖)
+      ((volume.restrict B).prod (volume.restrict B)) := by
+    rw [Measure.prod_restrict]
+    exact ((hf.continuous.comp continuous_fst).sub
+      (hf.continuous.comp continuous_snd)).norm.continuousOn.integrableOn_compact
+        ((isCompact_closedBall c R).prod (isCompact_closedBall c R))
+  have hbound : (∫ p, H p ∂((volume.restrict B).prod (volume.restrict B))) ≤
+      volume.real B * ((2 * R) * (2 ^ n * ∫ x in B, ‖fderiv ℝ f x‖)) := by
+    rw [integral_prod_symm H hHi]
+    calc
+      _ ≤ ∫ _y in B, (2 * R) * (2 ^ n * ∫ x in B, ‖fderiv ℝ f x‖) := by
+        apply setIntegral_mono_on hHi.integral_prod_right
+          (integrableOn_const (isCompact_closedBall c R).measure_ne_top) measurableSet_closedBall
+        exact fun y hy ↦ integrated_chord_half_le hf c y hy
+      _ = _ := by rw [integral_const, measureReal_restrict_apply_univ, smul_eq_mul]
+  have htri : ∀ p : Euc n × Euc n, ‖f p.1 - f p.2‖ ≤ H p + H p.swap := by
+    intro p
+    simpa only [H, Prod.fst_swap, Prod.snd_swap, add_comm ((1 / 2 : ℝ) • p.2), norm_sub_rev] using
+      (norm_sub_le_norm_sub_add_norm_sub (f p.1) (f ((1 / 2 : ℝ) • p.1 + (1 / 2 : ℝ) • p.2)) (f p.2))
+  calc
+    _ ≤ ∫ p, H p + H p.swap ∂((volume.restrict B).prod (volume.restrict B)) :=
+      integral_mono hDi (hHi.add hHi.swap) htri
+    _ = 2 * ∫ p, H p ∂((volume.restrict B).prod (volume.restrict B)) := by
+      calc
+        _ = (∫ p, H p ∂((volume.restrict B).prod (volume.restrict B))) +
+            ∫ p, (H ∘ Prod.swap) p ∂((volume.restrict B).prod (volume.restrict B)) :=
+          integral_add hHi hHi.swap
+        _ = _ := by simp only [Function.comp_def, integral_prod_swap]; ring
+    _ ≤ 2 * (volume.real B * ((2 * R) * (2 ^ n * ∫ x in B, ‖fderiv ℝ f x‖))) := by
+      apply (mul_le_mul_iff_right₀ hR).mp
+      nlinarith [mul_le_mul_of_nonneg_right hbound hR.le]
+    _ = _ := by ring
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/IntegralInnerEqInnerToLp.lean b/lean/OAI/Geometry/Perimeter/Regularity/IntegralInnerEqInnerToLp.lean
new file mode 100644
index 0000000000000000000000000000000000000000..99f9d816b27028e24f98968e519c9847abb4cdb0
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/IntegralInnerEqInnerToLp.lean
@@ -0,0 +1,385 @@
+import OAI.Geometry.Perimeter.Regularity.SurfaceGradientComp
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPI
+
+section CutoffTests
+variable {X V : Type*} [MeasurableSpace X]
+variable [NormedAddCommGroup V] [InnerProductSpace ℝ V]
+variable {μ : Measure X}
+
+theorem integral_inner_eq_inner_toLp {G B : X → V}
+    (hG : MemLp G 2 μ) (hB : MemLp B 2 μ) :
+    (∫ x, inner ℝ (G x) (B x) ∂μ) = inner ℝ (hG.toLp G) (hB.toLp B) := by
+  rw [L2.inner_def]
+  apply integral_congr_ae
+  filter_upwards [hG.coeFn_toLp, hB.coeFn_toLp] with x hx hy
+  rw [hx, hy]
+
+theorem integrable_inner_of_memLp {G B : X → V}
+    (hG : MemLp G 2 μ) (hB : MemLp B 2 μ) :
+    Integrable (fun x ↦ inner ℝ (G x) (B x)) μ := by
+  apply (L2.integrable_inner (𝕜 := ℝ) (hG.toLp G) (hB.toLp B)).congr
+  filter_upwards [hG.coeFn_toLp, hB.coeFn_toLp] with x hx hy
+  rw [hx, hy]
+
+theorem tendsto_integral_inner_of_L2_zero {G : X → V} (hG : MemLp G 2 μ)
+    (C : ℕ → X → V) (hC : ∀ i, MemLp (C i) 2 μ)
+    (hC0 : Tendsto (fun i ↦ (eLpNorm (C i) 2 μ).toReal) atTop (𝓝 0)) :
+    Tendsto (fun i ↦ ∫ x, inner ℝ (G x) (C i x) ∂μ) atTop (𝓝 0) := by
+  have hClim : Tendsto (fun i ↦ (hC i).toLp (C i)) atTop
+      (𝓝 (0 : Lp V 2 μ)) := by
+    apply tendsto_iff_norm_sub_tendsto_zero.mpr
+    simpa only [sub_zero, Lp.norm_toLp] using hC0
+  have hlim := Filter.Tendsto.inner (𝕜 := ℝ) (tendsto_const_nhds (x := hG.toLp G)) hClim
+  simp only [inner_zero_right] at hlim
+  convert hlim using 1
+  ext i
+  exact integral_inner_eq_inner_toLp hG (hC i)
+
+theorem energy_inequality_of_capacity_cutoffs
+    {G B : X → V} (hG : MemLp G 2 μ) (hB : MemLp B 2 μ)
+    (χ : ℕ → X → ℝ) (hχ : ∀ i, AEStronglyMeasurable (χ i) μ)
+    (hχbound : ∀ i, ∀ᵐ x ∂μ, ‖χ i x‖ ≤ 1)
+    (hχlim : ∀ᵐ x ∂μ, Tendsto (fun i ↦ χ i x) atTop (𝓝 1))
+    (C : ℕ → X → V) (hC : ∀ i, MemLp (C i) 2 μ)
+    (hC0 : Tendsto (fun i ↦ (eLpNorm (C i) 2 μ).toReal) atTop (𝓝 0))
+    (φ : X → ℝ) (hφ : AEStronglyMeasurable φ μ)
+    {M : ℝ} (hφbound : ∀ᵐ x ∂μ, ‖φ x‖ ≤ M)
+    (htest : ∀ i, 0 ≤ ∫ x, inner ℝ (G x) (χ i x • B x + φ x • C i x) ∂μ) :
+    0 ≤ ∫ x, inner ℝ (G x) (B x) ∂μ := by
+  let b : X → ℝ := fun x ↦ inner ℝ (G x) (B x)
+  have hb : Integrable b μ := integrable_inner_of_memLp hG hB
+  have hmainlim : Tendsto (fun i ↦ ∫ x, χ i x * b x ∂μ) atTop (𝓝 (∫ x, b x ∂μ)) := by
+    apply tendsto_integral_of_dominated_convergence (fun x ↦ ‖b x‖)
+    · intro i
+      exact (hχ i).mul hb.aestronglyMeasurable
+    · exact hb.norm
+    · intro i
+      filter_upwards [hχbound i] with x hx
+      rw [norm_mul]
+      exact mul_le_of_le_one_left (norm_nonneg _) hx
+    · filter_upwards [hχlim] with x hx
+      simpa only [one_mul] using hx.mul_const (b x)
+  have hφG : MemLp (fun x ↦ φ x • G x) 2 μ := by
+    apply hG.of_le_mul (c := M) (hφ.smul hG.aestronglyMeasurable)
+    filter_upwards [hφbound] with x hx
+    change ‖φ x • G x‖ ≤ M * ‖G x‖
+    rw [norm_smul]
+    exact mul_le_mul_of_nonneg_right hx (norm_nonneg _)
+  have herrlim : Tendsto (fun i ↦ ∫ x, φ x * inner ℝ (G x) (C i x) ∂μ)
+      atTop (𝓝 0) := by
+    simpa only [real_inner_smul_left] using tendsto_integral_inner_of_L2_zero hφG C hC hC0
+  have hsumlim := hmainlim.add herrlim
+  rw [add_zero] at hsumlim
+  apply ge_of_tendsto hsumlim
+  apply Filter.Eventually.of_forall
+  intro i
+  have hm : Integrable (fun x ↦ χ i x * b x) μ := hb.bdd_mul (hχ i) (hχbound i)
+  have he : Integrable (fun x ↦ φ x * inner ℝ (G x) (C i x)) μ :=
+    (integrable_inner_of_memLp hG (hC i)).bdd_mul hφ hφbound
+  rw [← integral_add hm he]
+  simpa only [inner_add_right, real_inner_smul_right, b] using htest i
+
+end CutoffTests
+end RPI
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPI.Boundary
+
+theorem surfaceGradient_memLp_on_ball {n : ℕ} {S : Set (Space n)}
+    {R : ℝ} [IsFiniteMeasure ((surfaceMeasure S).restrict (Metric.ball 0 R))]
+    {φ : Space n → ℝ} (hφ : ContDiff ℝ ∞ φ)
+    (hm : AEStronglyMeasurable (surfaceGradient S φ)
+      ((surfaceMeasure S).restrict (Metric.ball 0 R))) :
+    MemLp (surfaceGradient S φ) 2 ((surfaceMeasure S).restrict (Metric.ball 0 R)) := by
+  have hc : Continuous (gradient φ) := continuous_iff_continuousAt.mpr (fun x ↦
+    continuousAt_gradient_of_contDiffAt (hφ.of_le (by norm_num)).contDiffAt)
+  obtain ⟨M, hM⟩ := (isCompact_closedBall (0 : Space n) R).exists_bound_of_continuousOn
+    hc.continuousOn
+  apply MemLp.of_bound hm M
+  filter_upwards [ae_restrict_mem measurableSet_ball] with x hx
+  calc
+    ‖surfaceGradient S φ x‖ ≤ ‖gradient φ x‖ := by
+      calc
+        _ ≤ ‖(tangentPlane S x).orthogonalProjectionOnto‖ * ‖gradient φ x‖ :=
+          (tangentPlane S x).orthogonalProjectionOnto.le_opNorm _
+        _ ≤ 1 * ‖gradient φ x‖ := mul_le_mul_of_nonneg_right
+          (tangentPlane S x).orthogonalProjectionOnto_norm_le (norm_nonneg _)
+        _ = _ := one_mul _
+    _ ≤ M := hM x (Metric.ball_subset_closedBall hx)
+
+theorem surfaceGradient_eq_zero_of_notMem_tsupport {n : ℕ} (S : Set (Space n))
+    {φ : Space n → ℝ} {x : Space n} (hx : x ∉ tsupport φ) :
+    surfaceGradient S φ x = 0 := by
+  simp only [surfaceGradient, gradient, fderiv_of_notMem_tsupport ℝ hx, map_zero,
+    Submodule.coe_zero]
+
+theorem tsupport_mul_disjoint_Singular {n : ℕ} {Z : Set (Space n)}
+    {φ χ : Space n → ℝ}
+    (hzero : ∃ U : Set (Space n), IsOpen U ∧ Z ∩ tsupport φ ⊆ U ∧ ∀ x ∈ U, χ x = 0) :
+    Disjoint (tsupport (fun x ↦ φ x * χ x)) Z := by
+  obtain ⟨U, hU, hZU, he⟩ := hzero
+  apply Set.disjoint_left.mpr
+  intro x hx hxZ
+  have hxφ : x ∈ tsupport φ := tsupport_mul_subset_left hx
+  have heq : (fun x ↦ φ x * χ x) =ᶠ[𝓝 x] (0 : Space n → ℝ) := by
+    filter_upwards [hU.mem_nhds (hZU ⟨hxZ, hxφ⟩)] with y hy
+    simp only [he y hy, mul_zero, Pi.zero_apply]
+  exact (notMem_tsupport_iff_eventuallyEq.mpr heq) hx
+
+theorem full_test_of_regular_tests {n : ℕ} (hn : 8 ≤ n)
+    {S Z : Set (Space n)} (hZ : IsClosed Z) (hZd : dimH Z ≤ (n - 8 : ℕ))
+    {G : Space n → Space n}
+    (hloc : ∀ R : ℝ, 0 < R → MemLp G 2 ((surfaceMeasure S).restrict (Metric.ball 0 R)))
+    (hfinite : ∀ R : ℝ, 0 < R → (surfaceMeasure S) (Metric.ball 0 R) < ⊤)
+    (hgrad : ∀ φ : Space n → ℝ, ContDiff ℝ ∞ φ →
+      AEStronglyMeasurable (surfaceGradient S φ) (surfaceMeasure S))
+    (hgrowth : ∀ R : ℝ, 0 < R → ∃ A r₀ : ℝ, 0 < A ∧ 0 < r₀ ∧
+      ∀ c ∈ Z ∩ Metric.closedBall 0 R, ∀ r : ℝ, 0 < r → r < r₀ →
+        (surfaceMeasure S).restrict (Metric.ball 0 R) (Metric.ball c r) ≤
+          ENNReal.ofReal (A * r ^ (n - 1)))
+    (hreg : ∀ ψ : Space n → ℝ, ContDiff ℝ ∞ ψ → HasCompactSupport ψ →
+      Disjoint (tsupport ψ) Z → (∀ x, 0 ≤ ψ x) →
+      0 ≤ ∫ x, inner ℝ (G x) (surfaceGradient S ψ x) ∂surfaceMeasure S)
+    {φ : Space n → ℝ} (hφ : ContDiff ℝ ∞ φ) (hcφ : HasCompactSupport φ)
+    (hpφ : ∀ x, 0 ≤ φ x) :
+    0 ≤ ∫ x, inner ℝ (G x) (surfaceGradient S φ x) ∂surfaceMeasure S := by
+  obtain ⟨r, hr⟩ := Metric.isBounded_iff_subset_ball (0 : Space n) |>.mp hcφ.isBounded
+  let R := max r 1
+  have hR : 0 < R := lt_of_lt_of_le zero_lt_one (le_max_right _ _)
+  have hsupp : tsupport φ ⊆ Metric.ball (0 : Space n) R :=
+    hr.trans (Metric.ball_subset_ball (le_max_left _ _))
+  let μ := (surfaceMeasure S).restrict (Metric.ball 0 R)
+  let : IsFiniteMeasure μ := ⟨by simpa only [μ, Measure.restrict_apply_univ] using hfinite R hR⟩
+  let K := Z ∩ tsupport φ
+  have hKc : IsCompact K := hcφ.of_isClosed_subset
+    (hZ.inter (isClosed_tsupport φ)) inter_subset_right
+  have hKd : dimH K ≤ (n - 8 : ℕ) := (dimH_mono inter_subset_left).trans hZd
+  obtain ⟨A, r₀, hA, hr₀, hgr⟩ := hgrowth R hR
+  obtain ⟨χ, hχ, hχlim, hχE⟩ := exists_memLp_capacity_sequence hn S (Metric.ball 0 R) K
+    hKc hKd (fun f hf ↦ (hgrad f hf).mono_measure Measure.restrict_le_self) hA hr₀
+    (fun c hc s hs hsr ↦ hgr c ⟨hc.1, Metric.ball_subset_closedBall (hsupp hc.2)⟩ s hs hsr)
+  have hB : MemLp (surfaceGradient S φ) 2 μ := surfaceGradient_memLp_on_ball hφ
+    ((hgrad φ hφ).mono_measure Measure.restrict_le_self)
+  have hC0 : Tendsto (fun i ↦ (eLpNorm (surfaceGradient S (χ i)) 2 μ).toReal)
+      atTop (𝓝 0) := tendsto_eLpNorm_two_toReal_of_vectorEnergy_zero_of_aestronglyMeasurable
+        (fun i ↦ (hχ i).2.2.2.aestronglyMeasurable) hχE
+  obtain ⟨M, hM⟩ := hcφ.exists_bound_of_continuous hφ.continuous
+  have htesteq : ∀ i, (fun x ↦ χ i x • surfaceGradient S φ x + φ x • surfaceGradient S (χ i) x) =
+      surfaceGradient S (fun x ↦ φ x * χ i x) := by
+    intro i
+    ext x
+    rw [surfaceGradient_mul (hφ.differentiable (by simp) x)
+      ((hχ i).1.differentiable (by simp) x), add_comm]
+  have hnonsupp : ∀ x, x ∉ Metric.ball (0 : Space n) R → surfaceGradient S φ x = 0 :=
+    fun x hx ↦ surfaceGradient_eq_zero_of_notMem_tsupport S (fun h ↦ hx (hsupp h))
+  have hχnorm : ∀ i, ∀ᵐ x ∂μ, ‖χ i x‖ ≤ 1 := by
+    intro i
+    apply ae_of_all
+    intro x
+    rw [Real.norm_eq_abs, abs_of_nonneg ((hχ i).2.1 x).1]
+    exact ((hχ i).2.1 x).2
+  have hpair : 0 ≤ ∫ x, inner ℝ (G x) (surfaceGradient S φ x) ∂μ := by
+    apply energy_inequality_of_capacity_cutoffs (hloc R hR) hB χ
+      (fun i ↦ (hχ i).1.continuous.aestronglyMeasurable) hχnorm hχlim
+      (fun i ↦ surfaceGradient S (χ i)) (fun i ↦ (hχ i).2.2.2) hC0
+      φ hφ.continuous.aestronglyMeasurable (M := M) (ae_of_all _ hM)
+    intro i
+    simp_rw [congrFun (htesteq i)]
+    rw [setIntegral_eq_integral_of_forall_compl_eq_zero]
+    · exact hreg _ (hφ.mul (hχ i).1) hcφ.mul_right
+        (tsupport_mul_disjoint_Singular (hχ i).2.2.1)
+        (fun x ↦ mul_nonneg (hpφ x) ((hχ i).2.1 x).1)
+    · intro x hx
+      rw [surfaceGradient_eq_zero_of_notMem_tsupport S
+        (fun hx' ↦ hx (hsupp (tsupport_mul_subset_left hx'))), inner_zero_right]
+  dsimp only [μ] at hpair
+  rwa [setIntegral_eq_integral_of_forall_compl_eq_zero
+    (fun x hx ↦ by rw [hnonsupp x hx, inner_zero_right])] at hpair
+
+end RPI.Boundary
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPI.Boundary
+
+theorem exists_smooth_representative_near_regular_test {n : ℕ} {S Z : Set (Space n)}
+    (hS : IsClosed S) (hZ : IsClosed Z) {W : Space n → ℝ} {D : Space n → Space n}
+    (htrace : ∀ x ∈ S \ Z, HasSmoothTraceJetAt S W D x)
+    {K : Set (Space n)} (hK : IsClosed K) (hKZ : Disjoint K Z) :
+    ∃ F : Space n → ℝ, ContDiff ℝ ∞ F ∧
+      (∀ x ∈ S ∩ K, F x = W x ∧ surfaceGradient S F x = D x) := by
+  obtain ⟨σ, hσ0, hσ1, _⟩ := exists_contMDiffMap_zero_one_nhds_of_isClosed
+    (modelWithCornersSelf ℝ (Space n)) hZ hK hKZ.symm (n := (⊤ : ℕ∞))
+  have hσ : ContDiff ℝ ∞ σ := σ.contMDiff.contDiff
+  obtain ⟨U, hU, hUZ, hUσ⟩ := mem_nhdsSet_iff_exists.mp hσ0
+  obtain ⟨F, hF, hFW, hFD⟩ := exists_smooth_cutoff_trace hS htrace σ hσ
+    ⟨U, hU, hUZ, hUσ⟩
+  refine ⟨F, hF, ?_⟩
+  intro x hx
+  have hσx : σ x = 1 := hσ1.self_of_nhdsSet x hx.2
+  have hσnear : (σ : Space n → ℝ) =ᶠ[𝓝 x] (fun _ ↦ (1 : ℝ)) :=
+    hσ1.filter_mono (nhds_le_nhdsSet hx.2)
+  have hσg : surfaceGradient S σ x = 0 := by
+    have hg : gradient (σ : Space n → ℝ) x = 0 := by
+      simpa only [gradient_fun_const] using hσnear.gradient_eq
+    simp only [surfaceGradient, hg, map_zero, Submodule.coe_zero]
+  exact ⟨by simpa only [hσx, one_mul] using hFW hx.1,
+    by simpa only [hσx, one_smul, hσg, smul_zero, add_zero] using hFD hx.1⟩
+
+end RPI.Boundary
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPI.Boundary
+section RegularTests
+variable {n : ℕ} {S Z : Set (Space n)}
+variable (hS : IsClosed S) (hZ : IsClosed Z)
+variable (hfinite : ∀ R : ℝ, 0 < R → (surfaceMeasure S) (Metric.ball 0 R) < ⊤)
+variable (hgrad : ∀ f : Space n → ℝ, ContDiff ℝ ∞ f →
+  AEStronglyMeasurable (surfaceGradient S f) (surfaceMeasure S))
+
+include hfinite hgrad
+
+theorem surfaceGradient_memLp_of_compact {φ : Space n → ℝ}
+    (hφ : ContDiff ℝ ∞ φ) (hcφ : HasCompactSupport φ) :
+    MemLp (surfaceGradient S φ) 2 (surfaceMeasure S) := by
+  obtain ⟨r, hr⟩ := Metric.isBounded_iff_subset_ball (0 : Space n) |>.mp hcφ.isBounded
+  let R := max r 1
+  have hR : 0 < R := lt_of_lt_of_le zero_lt_one (le_max_right _ _)
+  have hsupp : tsupport φ ⊆ Metric.ball (0 : Space n) R :=
+    hr.trans (Metric.ball_subset_ball (le_max_left _ _))
+  let : IsFiniteMeasure ((surfaceMeasure S).restrict (Metric.ball 0 R)) :=
+    ⟨by simpa only [Measure.restrict_apply_univ] using hfinite R hR⟩
+  have hloc := surfaceGradient_memLp_on_ball (R := R) hφ
+    ((hgrad φ hφ).mono_measure Measure.restrict_le_self)
+  have hh := (memLp_indicator_iff_restrict measurableSet_ball).mpr hloc
+  apply hh.ae_eq
+  apply ae_of_all
+  intro x
+  by_cases hx : x ∈ Metric.ball (0 : Space n) R
+  · simp only [indicator_of_mem hx]
+  · simp only [indicator_of_notMem hx]
+    exact (surfaceGradient_eq_zero_of_notMem_tsupport S (fun h ↦ hx (hsupp h))).symm
+
+include hS hZ
+
+theorem regular_gradient_memLp_indicator_test {W : Space n → ℝ} {D : Space n → Space n}
+    (htrace : ∀ x ∈ S \ Z, HasSmoothTraceJetAt S W D x)
+    {φ : Space n → ℝ} (hcφ : HasCompactSupport φ) (hφZ : Disjoint (tsupport φ) Z) :
+    MemLp ((tsupport φ).indicator D) 2 (surfaceMeasure S) := by
+  obtain ⟨F, hF, he⟩ := exists_smooth_representative_near_regular_test hS hZ htrace
+    (isClosed_tsupport φ) hφZ
+  obtain ⟨r, hr⟩ := Metric.isBounded_iff_subset_ball (0 : Space n) |>.mp hcφ.isBounded
+  let R := max r 1
+  have hR : 0 < R := lt_of_lt_of_le zero_lt_one (le_max_right _ _)
+  have hsupp : tsupport φ ⊆ Metric.ball (0 : Space n) R :=
+    hr.trans (Metric.ball_subset_ball (le_max_left _ _))
+  let : IsFiniteMeasure ((surfaceMeasure S).restrict (Metric.ball 0 R)) :=
+    ⟨by simpa only [Measure.restrict_apply_univ] using hfinite R hR⟩
+  have hloc : MemLp (surfaceGradient S F) 2 ((surfaceMeasure S).restrict (tsupport φ)) :=
+    (surfaceGradient_memLp_on_ball (R := R) hF ((hgrad F hF).mono_measure Measure.restrict_le_self)).mono_measure
+      (Measure.restrict_mono_set _ hsupp)
+  apply ((memLp_indicator_iff_restrict (isClosed_tsupport φ).measurableSet).mpr hloc).ae_eq
+  filter_upwards [ae_restrict_mem hS.measurableSet] with x hx
+  by_cases hxt : x ∈ tsupport φ
+  · simp only [indicator_of_mem hxt, (he x ⟨hx, hxt⟩).2]
+  · simp only [indicator_of_notMem hxt]
+
+theorem integrable_regular_test_pair {W : Space n → ℝ} {D : Space n → Space n}
+    (htrace : ∀ x ∈ S \ Z, HasSmoothTraceJetAt S W D x)
+    {φ : Space n → ℝ} (hφ : ContDiff ℝ ∞ φ) (hcφ : HasCompactSupport φ)
+    (hφZ : Disjoint (tsupport φ) Z) :
+    Integrable (fun x ↦ inner ℝ (D x) (surfaceGradient S φ x)) (surfaceMeasure S) := by
+  have hD := regular_gradient_memLp_indicator_test hS hZ hfinite hgrad htrace hcφ hφZ
+  have hB := surfaceGradient_memLp_of_compact hfinite hgrad hφ hcφ
+  apply (integrable_inner_of_memLp hD hB).congr
+  apply ae_of_all
+  intro x
+  by_cases hxt : x ∈ tsupport φ
+  · simp only [indicator_of_mem hxt]
+  · simp only [indicator_of_notMem hxt, surfaceGradient_eq_zero_of_notMem_tsupport S hxt,
+      inner_zero_right]
+
+theorem smoothTruncation_regular_test {W : Space n → ℝ} {D : Space n → Space n}
+    (htrace : ∀ x ∈ S \ Z, HasSmoothTraceJetAt S W D x)
+    (hsup : ∀ φ : Space n → ℝ, ContDiff ℝ ∞ φ → HasCompactSupport φ →
+      Disjoint (tsupport φ) Z → (∀ x, 0 ≤ φ x) →
+      0 ≤ ∫ x, inner ℝ (D x) (surfaceGradient S φ x) ∂surfaceMeasure S)
+    {ε : ℝ} (hε : 0 < ε) {φ : Space n → ℝ}
+    (hφ : ContDiff ℝ ∞ φ) (hcφ : HasCompactSupport φ)
+    (hφZ : Disjoint (tsupport φ) Z) (hpφ : ∀ x, 0 ≤ φ x) :
+    0 ≤ ∫ x, inner ℝ (truncationSlope ε (W x) • D x)
+      (surfaceGradient S φ x) ∂surfaceMeasure S := by
+  obtain ⟨F, hF, he⟩ := exists_smooth_representative_near_regular_test hS hZ htrace
+    (isClosed_tsupport φ) hφZ
+  let ψ : Space n → ℝ := fun x ↦ φ x * truncationSlope ε (F x)
+  have hψ : ContDiff ℝ ∞ ψ := hφ.mul ((truncationSlope_contDiff ε).comp hF)
+  have hcψ : HasCompactSupport ψ := hcφ.mul_right
+  have hψZ : Disjoint (tsupport ψ) Z := hφZ.mono_left tsupport_mul_subset_left
+  have htt : ∀ x ∈ S \ Z, HasSmoothTraceJetAt S (smoothTruncation ε ∘ W)
+      (fun y ↦ truncationSlope ε (W y) • D y) x := by
+    intro x hx
+    simpa only [smoothTruncation_deriv] using
+      (htrace x hx).comp hx.1 (smoothTruncation_contDiff ε)
+  have hi := integrable_regular_test_pair hS hZ hfinite hgrad htrace hψ hcψ hψZ
+  have hj := integrable_regular_test_pair hS hZ hfinite hgrad htt hφ hcφ hφZ
+  apply (hsup ψ hψ hcψ hψZ (fun x ↦ mul_nonneg (hpφ x)
+    (truncationSlope_mem_Icc ε (F x)).1)).trans
+  apply integral_mono_ae hi hj
+  filter_upwards [ae_restrict_mem hS.measurableSet] with x hx
+  by_cases hxt : x ∈ tsupport φ
+  · have hFx := he x ⟨hx, hxt⟩
+    have hDp : deriv (truncationSlope ε) (W x) ≤ 0 :=
+      (truncationSlope_antitone hε).deriv_nonpos
+    have heq : surfaceGradient S ψ x =
+        φ x • (deriv (truncationSlope ε) (W x) • D x) +
+          truncationSlope ε (W x) • surfaceGradient S φ x := by
+      dsimp only [ψ]
+      rw [surfaceGradient_mul (S := S) (g := fun y ↦ truncationSlope ε (F y)) (hφ.differentiable (by simp) x)
+        (((truncationSlope_contDiff ε).comp hF).differentiable (by simp) x)]
+      rw [show (fun y ↦ truncationSlope ε (F y)) = truncationSlope ε ∘ F from rfl,
+        surfaceGradient_comp (hF.differentiable (by simp) x)
+          ((truncationSlope_contDiff ε).differentiable (by simp) (F x)), hFx.1, hFx.2]
+    rw [heq, inner_add_right, real_inner_smul_right, real_inner_smul_right,
+      real_inner_smul_right, real_inner_smul_left, real_inner_self_eq_norm_sq]
+    have hneg : φ x * (deriv (truncationSlope ε) (W x) * ‖D x‖ ^ 2) ≤ 0 :=
+      mul_nonpos_of_nonneg_of_nonpos (hpφ x) (mul_nonpos_of_nonpos_of_nonneg hDp (sq_nonneg _))
+    linarith
+  · have hxψ : x ∉ tsupport ψ := fun h ↦ hxt (tsupport_mul_subset_left h)
+    simp only [surfaceGradient_eq_zero_of_notMem_tsupport S hxt,
+      surfaceGradient_eq_zero_of_notMem_tsupport S hxψ, inner_zero_right, le_refl]
+
+end RegularTests
+end RPI.Boundary
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/IsConeMemPosTangentCone.lean b/lean/OAI/Geometry/Perimeter/Regularity/IsConeMemPosTangentCone.lean
new file mode 100644
index 0000000000000000000000000000000000000000..7020f89c60a3e04e711342d60fd12b2d0f24f0f5
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/IsConeMemPosTangentCone.lean
@@ -0,0 +1,60 @@
+import OAI.Geometry.Perimeter.Regularity.ConeRadialCap
+import OAI.Geometry.Perimeter.Regularity.GeoComposition
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+namespace RPISingularHarnack
+
+theorem isCone_mem_posTangentCone {n : ℕ} {S : Set (Euc n)}
+    (hcone : IsCone S) {x : Euc n} (hx : x ∈ S) :
+    x ∈ posTangentConeAt S (0 : Euc n) := by
+  apply mem_posTangentConeAt_of_frequently_mem
+  apply Filter.Eventually.frequently
+  filter_upwards [self_mem_nhdsWithin] with r hr
+  simpa only [zero_add] using (hcone r hr x).mpr hx
+
+theorem minimizing_cone_supporting_fderiv_eq_zero {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    (hcone : IsCone E) {f : Euc n → ℝ}
+    (hf : DifferentiableAt ℝ f (0 : Euc n))
+    (hmax : IsLocalMaxOn f (perimeterSupport E) (0 : Euc n)) :
+    ∀ x ∈ perimeterSupport E, fderiv ℝ f (0 : Euc n) x = 0 := by
+  have hc := isCone_perimeterSupport hmin.1 hcone
+  have hneg (x : Euc n) (hx : x ∈ perimeterSupport E) :
+      fderiv ℝ f (0 : Euc n) x ≤ 0 :=
+    hmax.hasFDerivWithinAt_nonpos hf.hasFDerivAt.hasFDerivWithinAt
+      (isCone_mem_posTangentCone hc hx)
+  have hside : ∀ x ∈ perimeterSupport E,
+      0 ≤ inner ℝ x (-gradient f (0 : Euc n)) := by
+    intro x hx
+    rw [inner_neg_right, real_inner_comm, inner_gradient_left]
+    linarith [hneg x hx]
+  have hzero := minimizing_cone_support_in_halfspace_plane hn hmin hcone
+    (-gradient f (0 : Euc n)) hside
+  intro x hx
+  have hh := hzero x hx
+  rw [inner_neg_right, real_inner_comm, inner_gradient_left] at hh
+  linarith
+
+theorem minimizing_cone_supporting_fderiv_zero_of_span {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    (hcone : IsCone E) (hspan : Submodule.span ℝ (perimeterSupport E) = ⊤)
+    {f : Euc n → ℝ} (hf : DifferentiableAt ℝ f (0 : Euc n))
+    (hmax : IsLocalMaxOn f (perimeterSupport E) (0 : Euc n)) :
+    fderiv ℝ f (0 : Euc n) = 0 := by
+  have hz := minimizing_cone_supporting_fderiv_eq_zero hn hmin hcone hf hmax
+  have hker : Submodule.span ℝ (perimeterSupport E) ≤
+      LinearMap.ker (fderiv ℝ f (0 : Euc n)).toLinearMap :=
+    Submodule.span_le.mpr hz
+  rw [hspan] at hker
+  ext x
+  exact hker (by trivial)
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/IsometricMinimizer.lean b/lean/OAI/Geometry/Perimeter/Regularity/IsometricMinimizer.lean
new file mode 100644
index 0000000000000000000000000000000000000000..e57ee10f1d46e3ee94b452a13f5d0373cd67b2f7
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/IsometricMinimizer.lean
@@ -0,0 +1,119 @@
+import OAI.Geometry.Perimeter.Regularity.PlaneLaplacian
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+open RPI.Boundary ContinuousLinearMap
+variable {n : ℕ}
+
+def isometricPullbackField (e : Euc n ≃ₗᵢ[ℝ] Euc n) (V : Euc n → Euc n) : Euc n → Euc n :=
+  fun x ↦ e.symm (V (e x))
+
+theorem divergence_isometricPullback (e : Euc n ≃ₗᵢ[ℝ] Euc n)
+    {V : Euc n → Euc n} (hV : Differentiable ℝ V) (x : Euc n) :
+    divergence (isometricPullbackField e V) x = divergence V (e x) := by
+  have hd : fderiv ℝ (isometricPullbackField e V) x =
+      e.symm.toContinuousLinearEquiv.toContinuousLinearMap.comp
+        ((fderiv ℝ V (e x)).comp e.toContinuousLinearEquiv.toContinuousLinearMap) :=
+    (e.symm.hasFDerivAt.comp x ((hV (e x)).hasFDerivAt.comp x e.hasFDerivAt)).fderiv
+  rw [divergence_eq_trace,hd,divergence_eq_trace]
+  change LinearMap.trace ℝ (Euc n) (e.symm.toLinearMap.comp
+    ((fderiv ℝ V (e x)).toLinearMap.comp e.toLinearMap)) = _
+  rw [LinearMap.trace_comp_comm',LinearMap.comp_assoc]
+  have he : e.toLinearMap.comp e.symm.toLinearMap = LinearMap.id := by
+    apply LinearMap.ext
+    intro z
+    exact e.apply_symm_apply z
+  rw [he,LinearMap.comp_id]
+
+theorem admissible_isometricPullback (e : Euc n ≃ₗᵢ[ℝ] Euc n)
+    {V : Euc n → Euc n} {U : Set (Euc n)} (hV : PerimeterAdmissible U V) :
+    PerimeterAdmissible (e ⁻¹' U) (isometricPullbackField e V) := by
+  refine ⟨e.symm.contDiff.comp (hV.1.comp e.contDiff),
+    (hV.2.1.comp_homeomorph e.toHomeomorph).comp_left (map_zero e.symm), ?_, ?_⟩
+  · have hs : tsupport (isometricPullbackField e V) ⊆ tsupport (V ∘ e) := by
+      apply closure_mono
+      intro x hx
+      contrapose! hx
+      simp only [Function.mem_support, not_not] at hx ⊢
+      change e.symm (V (e x))=0
+      rw [show V (e x)=0 from hx,map_zero]
+    exact hs.trans ((tsupport_comp_subset_preimage V e.continuous).trans
+      (preimage_mono hV.2.2.1))
+  · intro x
+    change ‖e.symm (V (e x))‖≤1
+    rw [e.symm.norm_map]
+    exact hV.2.2.2 _
+
+theorem integral_divergence_isometricPullback (e : Euc n ≃ₗᵢ[ℝ] Euc n)
+    (E : Set (Euc n)) {V : Euc n → Euc n} (hV : Differentiable ℝ V) :
+    (∫ x in e ⁻¹' E, divergence (isometricPullbackField e V) x) =
+      ∫ x in E, divergence V x := by
+  simp_rw [divergence_isometricPullback e hV]
+  exact e.measurePreserving.setIntegral_preimage_emb e.toHomeomorph.measurableEmbedding _ _
+
+theorem perimeter_isometric_preimage (e : Euc n ≃ₗᵢ[ℝ] Euc n) (E U : Set (Euc n)) :
+    perimeter (e ⁻¹' E) (e ⁻¹' U)=perimeter E U := by
+  have hle (d : Euc n ≃ₗᵢ[ℝ] Euc n) (A B : Set (Euc n)) :
+      perimeter A B ≤ perimeter (d ⁻¹' A) (d ⁻¹' B) := by
+    rw [perimeter_eq_iSup_tests]
+    apply iSup_le; intro V
+    apply iSup_le; intro hV
+    rw [← integral_divergence_isometricPullback d A (hV.1.differentiable (by simp))]
+    rw [perimeter_eq_iSup_tests]
+    exact le_iSup_of_le (isometricPullbackField d V)
+      (le_iSup_of_le (admissible_isometricPullback d hV) le_rfl)
+  apply le_antisymm
+  · simpa only [← preimage_comp,Function.comp_def,e.apply_symm_apply,preimage_id'] using
+      hle e.symm (e ⁻¹' E) (e ⁻¹' U)
+  · exact hle e E U
+
+theorem perimeterSupport_isometric_preimage (e : Euc n ≃ₗᵢ[ℝ] Euc n) (E : Set (Euc n)) :
+    perimeterSupport (e ⁻¹' E)=e ⁻¹' perimeterSupport E := by
+  ext x
+  change (∀ r, 0<r → 0<perimeter (e ⁻¹' E) (Metric.ball x r)) ↔
+    ∀ r, 0<r → 0<perimeter E (Metric.ball (e x) r)
+  have he (r : ℝ) : e ⁻¹' Metric.ball (e x) r=Metric.ball x r := by
+    rw [e.preimage_ball,e.symm_apply_apply]
+  simp_rw [← he,perimeter_isometric_preimage]
+
+theorem locally_minimizing_isometric_preimage (e : Euc n ≃ₗᵢ[ℝ] Euc n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E) :
+    IsLocallyPerimeterMinimizing (e ⁻¹' E) := by
+  have hball (x : Euc n) (r : ℝ) : e ⁻¹' Metric.ball (e x) r=Metric.ball x r := by
+    rw [e.preimage_ball,e.symm_apply_apply]
+  refine ⟨hmin.1.preimage e.continuous.measurable, ?_, ?_⟩
+  · intro x r hr
+    rw [← hball,perimeter_isometric_preimage]
+    exact hmin.2.1 _ r hr
+  · intro x r hr F hF hcF hsF
+    let G := e.symm ⁻¹' F
+    have hG : MeasurableSet G := hF.preimage e.symm.continuous.measurable
+    have he : e.symm ⁻¹' (e ⁻¹' E)=E := by ext y; simp
+    have hg : e ⁻¹' G=F := by ext y; simp [G]
+    have hcl : closure (symmDiff E G)=e.symm ⁻¹' closure (symmDiff (e ⁻¹' E) F) := by
+      change _ = e.symm.toHomeomorph ⁻¹' _
+      rw [e.symm.toHomeomorph.preimage_closure,preimage_symmDiff]
+      change _ = closure (symmDiff (e.symm ⁻¹' (e ⁻¹' E)) G)
+      rw [he]
+    have hcG : IsCompact (closure (symmDiff E G)) := by
+      rw [hcl]
+      exact e.symm.toHomeomorph.isCompact_preimage.mpr hcF
+    have hsG : closure (symmDiff E G) ⊆ Metric.ball (e x) r := by
+      intro y hy
+      rw [hcl] at hy
+      have hh := hsF hy
+      rw [← hball] at hh
+      simpa only [mem_preimage,e.apply_symm_apply] using hh
+    have hcmp := hmin.2.2 (e x) r hr G hG hcG hsG
+    rw [← hball,perimeter_isometric_preimage,← hg,perimeter_isometric_preimage]
+    exact hcmp
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/LinearBaseGain.lean b/lean/OAI/Geometry/Perimeter/Regularity/LinearBaseGain.lean
new file mode 100644
index 0000000000000000000000000000000000000000..75a000cab4d652789594a68e74925e28a0558ef6
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/LinearBaseGain.lean
@@ -0,0 +1,103 @@
+import OAI.Geometry.Perimeter.Regularity.EllipticFamilyEstimate
+import OAI.Geometry.Perimeter.Regularity.EllipticLocality
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+ theorem linear_gradient_holder_global {θ δ K : ℝ}
+    (hθ : 0<θ) (hθ' : θ ≤ 1/2) (hK : 1 ≤ K)
+    (hc : LinearContractionProperty n θ δ K)
+    {A : Euc n → Euc n →L[ℝ] Euc n} (hA : Continuous A)
+    {F : Euc n → Euc n} (hF : Continuous F) (hFc : HasCompactSupport F)
+    {f : Euc n → ℝ} (hf : ContDiff ℝ 1 f) (hfc : HasCompactSupport f)
+    {U : Set (Euc n)} (hU : IsOpen U) {p : Euc n} (hp : p∈U)
+    (heq : WeakLinearEquation A f F U)
+    (hAm : GeometricHolderAt θ A p) (hFm : GeometricHolderAt θ F p)
+    (hnear : ‖ContinuousLinearMap.id ℝ (Euc n)-A p‖ < min δ (1/2)) :
+    GeometricHolderAt θ (gradient f) p := by
+  have hθ1 : θ<1 := by linarith
+  have hcn : ContinuousAt (fun x ↦ ‖ContinuousLinearMap.id ℝ (Euc n)-A x‖) p :=
+    (continuousAt_const.sub hA.continuousAt).norm
+  obtain ⟨τ,hτ,hτn⟩ := Metric.eventually_nhds_iff.mp (hcn.eventually (gt_mem_nhds hnear))
+  obtain ⟨S,hS,hSU⟩ := Metric.isOpen_iff.mp hU p hp
+  let a := 1/(32*K^2)
+  have ha : 0<a := by dsimp [a]; positivity
+  obtain ⟨b,hb,hbs,hbm⟩ := (hAm.prodMk hθ.le hFm).shrink_amplitude hθ ha (lt_min hτ hS)
+  let r := b/8
+  have hr : 0<r := by dsimp [r]; positivity
+  have hrb : 4*r<b := by dsimp [r]; linarith
+  have hbS : b ≤ S := hbs.trans (min_le_right _ _)
+  have hbτ : b ≤ τ := hbs.trans (min_le_left _ _)
+  obtain ⟨B,hB,hBb⟩ := compact_continuous_bound hf.continuous hfc
+  obtain ⟨N,hN,hNb⟩ := compact_continuous_bound hF hFc
+  have hsmall : 32*K^2*a ≤ 1 := by dsimp [a]; rw [mul_one_div_cancel (by positivity : 32*K^2≠0)]
+  have hh := linear_family_C1_limit hθ hθ1 hK hc ha.le ha.le hr hsmall
+    (p := p) (A := fun _ ↦ A) (f := fun _ ↦ f) (F := fun _ ↦ F)
+    (fun _ ↦ hA) (fun _ ↦ hf) (fun _ ↦ hF)
+    (fun _ ↦ heq.mono ((Metric.ball_subset_ball (hrb.le.trans hbS)).trans hSU))
+    (fun _ x hx ↦ (hτn ((Metric.mem_ball.mp hx).trans_le (hrb.le.trans hbτ))).le)
+    (fun _ x hx j y hy ↦ ?_) (fun _ x ↦ by simpa only [Real.norm_eq_abs] using hBb x)
+    (fun _ x ↦ hNb x) (g := f) (fun _ _ ↦ tendsto_const_nhds)
+  · exact hh.2
+  · have hθj : θ^j ≤ 1 := pow_le_one₀ hθ.le hθ1.le
+    have hxb : x∈Metric.ball p b := Metric.closedBall_subset_ball (by linarith) hx
+    have hyb : y∈Metric.ball p b := by
+      have ht := dist_triangle y x p
+      have hhx := Metric.mem_closedBall.mp hx
+      have hhy := Metric.mem_ball.mp hy
+      rw [Metric.mem_ball]
+      nlinarith [mul_le_mul_of_nonneg_right hθj hr.le]
+    have hxy : dist x y ≤ θ^j*b := by
+      rw [dist_comm]
+      exact (Metric.mem_ball.mp hy).le.trans (mul_le_mul_of_nonneg_left (by linarith : r≤b) (pow_nonneg hθ.le _))
+    have hm := hbm j x hxb y hyb hxy
+    simp only [Prod.dist_eq,max_le_iff] at hm
+    exact ⟨(dist_eq_norm (A x) (A y)) ▸ hm.1,by simpa only [dist_eq_norm,norm_sub_rev] using hm.2⟩
+
+ theorem linear_gradient_holder {θ δ K : ℝ}
+    (hθ : 0<θ) (hθ' : θ ≤ 1/2) (hK : 1 ≤ K)
+    (hc : LinearContractionProperty n θ δ K)
+    {A : Euc n → Euc n →L[ℝ] Euc n} {F : Euc n → Euc n}
+    {f : Euc n → ℝ} {U : Set (Euc n)} (hU : IsOpen U) {p : Euc n} (hp : p∈U)
+    (hA : GeoC θ 0 A U) (hF : GeoC θ 0 F U) (hf : ContDiffOn ℝ 1 f U)
+    (heq : WeakLinearEquation A f F U)
+    (hnear : ‖ContinuousLinearMap.id ℝ (Euc n)-A p‖ < min δ (1/2)) :
+    GeometricHolderAt θ (gradient f) p := by
+  obtain ⟨W,hW,hpW,hWU,B,g,G,hB,hBc,hg,hgc,hG,hGc,heB,heg,heG⟩ :=
+    local_compact_triple hU hp (contDiffOn_zero.mpr hA.1) hf (contDiffOn_zero.mpr hF.1)
+  have he := (heq.mono hWU).congr hW heB.symm heg.symm heG.symm
+  have hBm := (hA.2 p hp).congr hθ.le (Filter.eventually_of_mem (hW.mem_nhds hpW) (fun x hx ↦ (heB hx).symm))
+  have hGm := (hF.2 p hp).congr hθ.le (Filter.eventually_of_mem (hW.mem_nhds hpW) (fun x hx ↦ (heG hx).symm))
+  have hm := linear_gradient_holder_global hθ hθ' hK hc hB.continuous hG.continuous hGc hg hgc
+    hW hpW he hBm hGm (by rwa [heB hpW])
+  exact hm.congr hθ.le (Filter.eventually_of_mem (hW.mem_nhds hpW) (fun x hx ↦ gradient_eqOn hW heg hx))
+
+ theorem linear_geoC_one {θ δ K : ℝ} (hθ : 0<θ) (hθ' : θ ≤ 1/2) (hK : 1 ≤ K)
+    (hc : LinearContractionProperty n θ δ K)
+    {A : Euc n → Euc n →L[ℝ] Euc n} {F : Euc n → Euc n}
+    {f : Euc n → ℝ} {U : Set (Euc n)} (hU : IsOpen U)
+    (hA : GeoC θ 0 A U) (hF : GeoC θ 0 F U) (hf : ContDiffOn ℝ 1 f U)
+    (heq : WeakLinearEquation A f F U)
+    (hnear : ∀ p∈U, ‖ContinuousLinearMap.id ℝ (Euc n)-A p‖ < min δ (1/2)) :
+    GeoC θ 1 f U := by
+  have hg : GeoC θ 0 (gradient f) U := ⟨
+    (InnerProductSpace.toDual ℝ (Euc n)).symm.continuous.comp_continuousOn
+      (hf.continuousOn_fderiv_of_isOpen hU (by norm_num)),
+    fun p hp ↦ linear_gradient_holder hθ hθ' hK hc hU hp hA hF hf heq (hnear p hp)⟩
+  refine ⟨hf,geoC_zero_of_C1 hθ.le hθ' hU hf,fun e ↦ ?_⟩
+  have hh := hg.linear hθ.le hU (innerSL ℝ e)
+  apply hh.congr hθ.le hU
+  intro x hx
+  dsimp only
+  rw [innerSL_apply_apply,real_inner_comm,inner_gradient_left]
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/LinearDerivativeGain.lean b/lean/OAI/Geometry/Perimeter/Regularity/LinearDerivativeGain.lean
new file mode 100644
index 0000000000000000000000000000000000000000..23d695218f5577228bd88ad49aeebe2534ec7f53
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/LinearDerivativeGain.lean
@@ -0,0 +1,154 @@
+import OAI.Geometry.Perimeter.Regularity.NonlinearDerivativeGain
+import OAI.Geometry.Perimeter.Regularity.GeoComposition
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+ theorem norm_apply_difference (A B : Euc n →L[ℝ] Euc n) (v w : Euc n) :
+    ‖A v-B w‖ ≤ ‖A‖*‖v-w‖+‖A-B‖*‖w‖ := by
+  have he : A v-B w=A (v-w)+(A-B) w := by simp only [map_sub,sub_apply]; abel
+  rw [he]
+  exact (norm_add_le _ _).trans (add_le_add (A.le_opNorm _) ((A-B).le_opNorm _))
+
+ theorem linear_directional_gain {θ δ K : ℝ}
+    (hθ : 0<θ) (hθ' : θ ≤ 1/2) (hK : 1 ≤ K)
+    (hc : LinearContractionProperty n θ δ K)
+    {A : Euc n → Euc n →L[ℝ] Euc n} (hA : ContDiff ℝ 1 A) (hAc : HasCompactSupport A)
+    {F : Euc n → Euc n} (hF : ContDiff ℝ 1 F) (hFc : HasCompactSupport F)
+    {f : Euc n → ℝ} (hf : ContDiff ℝ 1 f) (hfc : HasCompactSupport f)
+    {U : Set (Euc n)} (hU : IsOpen U) {p : Euc n} (hp : p∈U)
+    (heq : WeakLinearEquation A f F U)
+    (hmod : GeometricHolderAt θ (gradient f) p)
+    (hDA : GeometricHolderAt θ (fderiv ℝ A) p)
+    (hDF : GeometricHolderAt θ (fderiv ℝ F) p)
+    (hnear : ‖ContinuousLinearMap.id ℝ (Euc n)-A p‖ < min δ (1/2))
+    (e : Euc n) :
+    ∃ r : ℝ, 0<r ∧ Metric.ball p r⊆U ∧
+      ContDiffOn ℝ 1 (fun x ↦ fderiv ℝ f x e) (Metric.ball p r) ∧
+      GeometricHolderAt θ (gradient (fun x ↦ fderiv ℝ f x e)) p := by
+  have hθ1 : θ<1 := by linarith
+  have hAm := geometricHolderAt_of_C1_on hθ.le hθ' isOpen_univ hA.contDiffOn (mem_univ p)
+  have hJ := hAm.prodMk hθ.le (hDA.prodMk hθ.le (hDF.prodMk hθ.le hmod))
+  have hcn : ContinuousAt (fun x ↦ ‖ContinuousLinearMap.id ℝ (Euc n)-A x‖) p :=
+    (continuousAt_const.sub hA.continuous.continuousAt).norm
+  obtain ⟨τ,hτ,hτn⟩ := Metric.eventually_nhds_iff.mp (hcn.eventually (gt_mem_nhds hnear))
+  obtain ⟨S,hS,hSU⟩ := Metric.isOpen_iff.mp hU p hp
+  let a := 1/(32*K^2)
+  have ha : 0<a := by dsimp [a]; positivity
+  obtain ⟨b,hb,hbs,hbm⟩ := hJ.shrink_amplitude hθ ha (lt_min hτ hS)
+  let r := b/8
+  have hr : 0<r := by dsimp [r]; positivity
+  have hrb : 5*r<b := by dsimp [r]; linarith
+  have hbS : b ≤ S := hbs.trans (min_le_right _ _)
+  have hbτ : b ≤ τ := hbs.trans (min_le_left _ _)
+  have hshift (i : ℕ) (R : ℝ) (x : Euc n) (hx : x∈Metric.ball p R)
+      (t : ℝ) (ht : t∈Icc (0:ℝ) 1) : x+t•(quotientSteps r e i•e)∈Metric.ball p (R+r) := by
+    rw [Metric.mem_ball,dist_eq_norm]
+    have he : x+t•(quotientSteps r e i•e)-p=(x-p)+t•(quotientSteps r e i•e) := by abel
+    rw [he]
+    have hnorm : ‖t•(quotientSteps r e i•e)‖ ≤ ‖quotientSteps r e i•e‖ := by
+      rw [norm_smul,Real.norm_of_nonneg ht.1]
+      exact mul_le_of_le_one_left (norm_nonneg _) ht.2
+    have hh := norm_add_le (x-p) (t•(quotientSteps r e i•e))
+    have hd := Metric.mem_ball.mp hx
+    rw [dist_eq_norm] at hd
+    linarith [quotientSteps_shift_lt hr e i]
+  have hm (j : ℕ) (x : Euc n) (hx : x∈Metric.ball p b) (y : Euc n)
+      (hy : y∈Metric.ball p b) (hxy : dist x y ≤ θ^j*b) :
+      ‖A x-A y‖ ≤ a*(1/2)^j ∧ ‖fderiv ℝ A x-fderiv ℝ A y‖ ≤ a*(1/2)^j ∧
+      ‖fderiv ℝ F x-fderiv ℝ F y‖ ≤ a*(1/2)^j ∧ ‖gradient f x-gradient f y‖ ≤ a*(1/2)^j := by
+    have hh := hbm j x hx y hy hxy
+    simp only [Prod.dist_eq,max_le_iff] at hh
+    exact ⟨(dist_eq_norm (A x) (A y)) ▸ hh.1,
+      (dist_eq_norm (fderiv ℝ A x) (fderiv ℝ A y)) ▸ hh.2.1,
+      (dist_eq_norm (fderiv ℝ F x) (fderiv ℝ F y)) ▸ hh.2.2.1,
+      (dist_eq_norm (gradient f x) (gradient f y)) ▸ hh.2.2.2⟩
+  obtain ⟨Bg,hBg,hBgall⟩ := compact_C1_gradient_bound hf hfc
+  have : CompleteSpace (Euc n →L[ℝ] Euc n) := ContinuousLinearMap.instCompleteSpace
+  obtain ⟨LA,hLA,hLAall⟩ := compact_C1_fderiv_bound hA hAc
+  obtain ⟨LF,hLF,hLFall⟩ := compact_C1_fderiv_bound hF hFc
+  let v : ℕ → Euc n → ℝ := fun i ↦ differenceQuotient f e (quotientSteps r e i)
+  let H : ℕ → Euc n → Euc n := fun i x ↦ vectorQuotient F e (quotientSteps r e i) x-
+    vectorQuotient A e (quotientSteps r e i) x (gradient f (x+quotientSteps r e i•e))
+  have hH (i : ℕ) : Continuous (H i) := (vectorQuotient_continuous hF.continuous e _).sub
+    ((vectorQuotient_continuous hA.continuous e _).clm_apply
+      ((gradient_continuous_C1 hf).comp (continuous_id.add continuous_const)))
+  have hv (i : ℕ) : ContDiff ℝ 1 (v i) := differenceQuotient_contDiff hf e _
+  have h4b : Metric.ball p (4*r)⊆Metric.ball p b := Metric.ball_subset_ball (by linarith)
+  have h4U : Metric.ball p (4*r)⊆U := h4b.trans ((Metric.ball_subset_ball hbS).trans hSU)
+  have hE (i : ℕ) : WeakLinearEquation A (v i) (H i) (Metric.ball p (4*r)) :=
+    weak_linear_difference hA.continuous hf hF.continuous heq h4U e _ (fun x hx ↦ by
+      have hh := hshift i (4*r) x hx 1 (by norm_num)
+      simp only [one_smul] at hh
+      exact hSU (Metric.ball_subset_ball (by linarith : 4*r+r ≤ S) hh))
+  have hN (i : ℕ) (x : Euc n) (hx : x∈Metric.ball p (4*r)) :
+      ‖ContinuousLinearMap.id ℝ (Euc n)-A x‖ ≤ min δ (1/2) :=
+    (hτn ((Metric.mem_ball.mp (h4b hx)).trans_le hbτ)).le
+  let B := a*‖e‖*(1+LA+Bg)
+  have hB : 0 ≤ B := by dsimp [B]; positivity
+  have hosc (i : ℕ) (x : Euc n) (hx : x∈Metric.closedBall p r) (j : ℕ)
+      (y : Euc n) (hy : y∈Metric.ball x (θ^j*r)) :
+      ‖A x-A y‖ ≤ a*(1/2)^j ∧ ‖H i y-H i x‖ ≤ B*(1/2)^j := by
+    have hj : θ^j ≤ 1 := pow_le_one₀ hθ.le hθ1.le
+    have hx2 : x∈Metric.ball p (2*r) := Metric.mem_ball.mpr
+      ((Metric.mem_closedBall.mp hx).trans_lt (by linarith))
+    have hy2 : y∈Metric.ball p (2*r) := by
+      have hd := dist_triangle y x p
+      have hdx := Metric.mem_closedBall.mp hx
+      have hdy := Metric.mem_ball.mp hy
+      have hrj := mul_le_mul_of_nonneg_right hj hr.le
+      rw [Metric.mem_ball]
+      linarith
+    have h2b : Metric.ball p (2*r)⊆Metric.ball p b := Metric.ball_subset_ball (by linarith)
+    have hxy : dist x y ≤ θ^j*b := by
+      rw [dist_comm]
+      exact (Metric.mem_ball.mp hy).le.trans (mul_le_mul_of_nonneg_left (by dsimp [r]; linarith)
+        (pow_nonneg hθ.le j))
+    have hsmod (t : ℝ) (ht : t∈Icc (0:ℝ) 1) := hm j
+      _ (Metric.ball_subset_ball (by linarith : 2*r+r ≤ b) (hshift i (2*r) x hx2 t ht))
+      _ (Metric.ball_subset_ball (by linarith : 2*r+r ≤ b) (hshift i (2*r) y hy2 t ht))
+      (by simpa only [dist_add_right] using hxy)
+    have hqA := vectorQuotient_difference hA e (quotientSteps_pos hr e i).ne' x y
+      (fun t ht ↦ (hsmod t ht).2.1)
+    have hqF := vectorQuotient_difference hF e (quotientSteps_pos hr e i).ne' x y
+      (fun t ht ↦ (hsmod t ht).2.2.1)
+    have hg : ‖gradient f (x+quotientSteps r e i•e)-gradient f (y+quotientSteps r e i•e)‖ ≤ a*(1/2)^j := by
+      simpa only [one_smul] using (hsmod 1 (by norm_num)).2.2.2
+    have haB := vectorQuotient_bound hA hLA hLAall e (quotientSteps r e i) x
+    have hfg := hBgall (y+quotientSteps r e i•e)
+    have hap := norm_apply_difference (vectorQuotient A e (quotientSteps r e i) x)
+      (vectorQuotient A e (quotientSteps r e i) y) (gradient f (x+quotientSteps r e i•e))
+      (gradient f (y+quotientSteps r e i•e))
+    have hap' : ‖vectorQuotient A e (quotientSteps r e i) x (gradient f (x+quotientSteps r e i•e))-
+        vectorQuotient A e (quotientSteps r e i) y (gradient f (y+quotientSteps r e i•e))‖ ≤
+        LA*‖e‖*(a*(1/2)^j)+(a*(1/2)^j*‖e‖)*Bg :=
+      hap.trans (add_le_add (mul_le_mul haB hg (norm_nonneg _) (by positivity))
+        (mul_le_mul hqA hfg (norm_nonneg _) (by positivity)))
+    refine ⟨(hm j x (h2b hx2) y (h2b hy2) hxy).1,?_⟩
+    rw [norm_sub_rev]
+    have he : H i x-H i y=(vectorQuotient F e (quotientSteps r e i) x-vectorQuotient F e (quotientSteps r e i) y)-
+        (vectorQuotient A e (quotientSteps r e i) x (gradient f (x+quotientSteps r e i•e))-
+        vectorQuotient A e (quotientSteps r e i) y (gradient f (y+quotientSteps r e i•e))) := by dsimp [H]; abel
+    rw [he]
+    exact (norm_sub_le _ _).trans ((add_le_add hqF hap').trans_eq (by dsimp [B]; ring))
+  have hglobal (i : ℕ) (x : Euc n) : ‖H i x‖ ≤ LF*‖e‖+LA*‖e‖*Bg := by
+    apply (norm_sub_le _ _).trans
+    apply add_le_add (vectorQuotient_bound hF hLF hLFall e _ x)
+    exact (ContinuousLinearMap.le_opNorm _ _).trans
+      (mul_le_mul (vectorQuotient_bound hA hLA hLAall e _ x) (hBgall _) (norm_nonneg _) (by positivity))
+  obtain ⟨hg,hgmod⟩ := linear_family_C1_limit hθ hθ1 hK hc ha.le hB hr
+    (by dsimp [a]; field_simp; norm_num) (fun _ ↦ hA.continuous) hv hH hE hN hosc
+    (fun i x ↦ differenceQuotient_bound (hf.differentiable (by simp)) hBg hBgall e _ x) hglobal
+    (fun x _ ↦ (differenceQuotient_tendsto (hf.differentiable (by simp)) e x).comp (quotientSteps_tendsto hr e))
+  exact ⟨r,hr,(Metric.ball_subset_ball (by linarith : r ≤ 4*r)).trans h4U,hg,hgmod⟩
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/LinearDifference.lean b/lean/OAI/Geometry/Perimeter/Regularity/LinearDifference.lean
new file mode 100644
index 0000000000000000000000000000000000000000..b0591346879d550f3d43baa47d661d40e73c1349
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/LinearDifference.lean
@@ -0,0 +1,64 @@
+import OAI.Geometry.Perimeter.Regularity.EllipticContraction
+import OAI.Geometry.Perimeter.Regularity.VectorQuotient
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+ theorem WeakFluxEquation.mono {V : Euc n → Euc n} {U W : Set (Euc n)}
+    (h : WeakFluxEquation V U) (hWU : W⊆U) : WeakFluxEquation V W := by
+  intro φ
+  exact h ⟨φ.val,φ.property.1,φ.property.2.1,φ.property.2.2.trans hWU⟩
+
+ theorem weakLinear_iff_flux {A : Euc n → Euc n →L[ℝ] Euc n}
+    {f : Euc n → ℝ} {F : Euc n → Euc n} {U : Set (Euc n)}
+    (hA : Continuous A) (hf : ContDiff ℝ 1 f) (hF : Continuous F) :
+    WeakLinearEquation A f F U ↔ WeakFluxEquation (fun x ↦ A x (gradient f x)-F x) U := by
+  have hc := hA.clm_apply (gradient_continuous_C1 hf)
+  constructor <;> intro h φ
+  · have hh := h φ
+    simp only [inner_sub_left]
+    rw [integral_sub (inner_gradient_pairing_integrable hc φ) (inner_gradient_pairing_integrable hF φ)]
+    exact sub_eq_zero.mpr hh
+  · have hh := h φ
+    simp only [inner_sub_left] at hh
+    rw [integral_sub (inner_gradient_pairing_integrable hc φ) (inner_gradient_pairing_integrable hF φ)] at hh
+    exact sub_eq_zero.mp hh
+
+ theorem weak_linear_difference {A : Euc n → Euc n →L[ℝ] Euc n}
+    {f : Euc n → ℝ} {F : Euc n → Euc n} {U W : Set (Euc n)}
+    (hA : Continuous A) (hf : ContDiff ℝ 1 f) (hF : Continuous F)
+    (h : WeakLinearEquation A f F U) (hWU : W⊆U)
+    (e : Euc n) (s : ℝ) (hs : ∀ x∈W, x+s•e∈U) :
+    WeakLinearEquation A (differenceQuotient f e s)
+      (fun x ↦ vectorQuotient F e s x-vectorQuotient A e s x (gradient f (x+s•e))) W := by
+  let V := fun x ↦ A x (gradient f x)-F x
+  have hV : Continuous V := (hA.clm_apply (gradient_continuous_C1 hf)).sub hF
+  have hflux := (weakLinear_iff_flux hA hf hF).mp h
+  have hq := ((hflux.translate (s•e) hs).sub
+    (hV.comp (continuous_id.add continuous_const)) hV (hflux.mono hWU)).const_smul s⁻¹
+  have hqA : Continuous (vectorQuotient A e s) := by
+    unfold vectorQuotient
+    exact ((hA.comp (continuous_id.add continuous_const)).sub hA).const_smul s⁻¹
+  have hqF : Continuous (fun x ↦ vectorQuotient F e s x-
+      vectorQuotient A e s x (gradient f (x+s•e))) :=
+    (vectorQuotient_continuous hF e s).sub (hqA.clm_apply
+      ((gradient_continuous_C1 hf).comp (continuous_id.add continuous_const)))
+  apply (weakLinear_iff_flux hA (differenceQuotient_contDiff hf e s) hqF).mpr
+  have he (x : Euc n) : A x (gradient (differenceQuotient f e s) x)-
+      (vectorQuotient F e s x-vectorQuotient A e s x (gradient f (x+s•e)))=
+      s⁻¹•(V (x+s•e)-V x) := by
+    rw [differenceQuotient_gradient (hf.differentiable (by simp))]
+    simp only [vectorQuotient,smul_apply,sub_apply,map_smul,map_sub,V]
+    module
+  simpa only [he,Function.comp_def,Pi.add_apply,id_eq] using hq
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/LinearLocalGain.lean b/lean/OAI/Geometry/Perimeter/Regularity/LinearLocalGain.lean
new file mode 100644
index 0000000000000000000000000000000000000000..b344b477d429336c63407e8329784e5e4e1a8d7d
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/LinearLocalGain.lean
@@ -0,0 +1,62 @@
+import OAI.Geometry.Perimeter.Regularity.EllipticLocality
+import OAI.Geometry.Perimeter.Regularity.LinearDerivativeGain
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+ theorem linear_directional_gain_local {θ δ K : ℝ}
+    (hθ : 0<θ) (hθ' : θ ≤ 1/2) (hK : 1 ≤ K)
+    (hc : LinearContractionProperty n θ δ K)
+    {A : Euc n → Euc n →L[ℝ] Euc n} {F : Euc n → Euc n}
+    {f : Euc n → ℝ} {U : Set (Euc n)} (hU : IsOpen U) {p : Euc n} (hp : p∈U)
+    (hA : GeoC θ 1 A U) (hF : GeoC θ 1 F U) (hf : GeoC θ 1 f U)
+    (heq : WeakLinearEquation A f F U)
+    (hnear : ‖ContinuousLinearMap.id ℝ (Euc n)-A p‖ < min δ (1/2)) (e : Euc n) :
+    ∃ r : ℝ, 0<r ∧ Metric.ball p r⊆U ∧
+      ContDiffOn ℝ 1 (fun x ↦ fderiv ℝ f x e) (Metric.ball p r) ∧
+      GeometricHolderAt θ (gradient (fun x ↦ fderiv ℝ f x e)) p := by
+  obtain ⟨W,hW,hpW,hWU,B,g,G,hB,hBc,hg,hgc,hG,hGc,heB,heg,heG⟩ :=
+    local_compact_triple hU hp hA.1 hf.1 hF.1
+  have he := (heq.mono hWU).congr hW heB.symm heg.symm heG.symm
+  have hmf := (hf.gradient hθ.le hU).2 p hp
+  have hmA := (hA.fderiv hθ.le hU).2 p hp
+  have hmF := (hF.fderiv hθ.le hU).2 p hp
+  have hmg := hmf.congr hθ.le (Filter.eventually_of_mem (hW.mem_nhds hpW)
+    (fun x hx ↦ (gradient_eqOn hW heg hx).symm))
+  have hmB := hmA.congr hθ.le (Filter.eventually_of_mem (hW.mem_nhds hpW)
+    (fun x hx ↦ (fderiv_eqOn hW heB hx).symm))
+  have hmG := hmF.congr hθ.le (Filter.eventually_of_mem (hW.mem_nhds hpW)
+    (fun x hx ↦ (fderiv_eqOn hW heG hx).symm))
+  obtain ⟨r,hr,hrW,hdr,hdm⟩ := linear_directional_gain hθ hθ' hK hc hB hBc hG hGc hg hgc
+    hW hpW he hmg hmB hmG (by rwa [heB hpW]) e
+  have hdEq : EqOn (fun x ↦ fderiv ℝ g x e) (fun x ↦ fderiv ℝ f x e) W :=
+    fun x hx ↦ congrArg (fun L : Euc n →L[ℝ] ℝ ↦ L e) (fderiv_eqOn hW heg hx)
+  refine ⟨r,hr,hrW.trans hWU,hdr.congr (fun x hx ↦ (hdEq (hrW hx)).symm),?_⟩
+  exact hdm.congr hθ.le (Filter.eventually_of_mem (hW.mem_nhds hpW)
+    (fun x hx ↦ gradient_eqOn hW hdEq hx))
+
+ theorem linear_C2_gain {θ δ K : ℝ}
+    (hθ : 0<θ) (hθ' : θ ≤ 1/2) (hK : 1 ≤ K)
+    (hc : LinearContractionProperty n θ δ K)
+    {A : Euc n → Euc n →L[ℝ] Euc n} {F : Euc n → Euc n}
+    {f : Euc n → ℝ} {U : Set (Euc n)} (hU : IsOpen U)
+    (hA : GeoC θ 1 A U) (hF : GeoC θ 1 F U) (hf : GeoC θ 1 f U)
+    (heq : WeakLinearEquation A f F U)
+    (hnear : ∀ p∈U, ‖ContinuousLinearMap.id ℝ (Euc n)-A p‖ < min δ (1/2)) :
+    ContDiffOn ℝ 2 f U := by
+  change ContDiffOn ℝ (1+1) f U
+  rw [contDiffOn_succ_iff_fderiv_of_isOpen hU]
+  refine ⟨hf.1.differentiableOn one_ne_zero,by simp,contDiffOn_clm_apply.mpr (fun e p hp ↦ ?_)⟩
+  obtain ⟨r,hr,hrU,hdr,hdm⟩ := linear_directional_gain_local hθ hθ' hK hc hU hp hA hF hf heq (hnear p hp) e
+  exact (hdr.contDiffAt (Metric.ball_mem_nhds p hr)).contDiffWithinAt
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/LinearSchauder.lean b/lean/OAI/Geometry/Perimeter/Regularity/LinearSchauder.lean
new file mode 100644
index 0000000000000000000000000000000000000000..1dc7115b7518fb013251ff87fe73a95662823f87
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/LinearSchauder.lean
@@ -0,0 +1,53 @@
+import OAI.Geometry.Perimeter.Regularity.LinearBaseGain
+import OAI.Geometry.Perimeter.Regularity.LinearLocalGain
+import OAI.Geometry.Perimeter.Regularity.LinearWeakDerivative
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+ theorem GeoC.of_le {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V]
+    {θ : ℝ} {k l : ℕ} {f : Euc n → V} {U : Set (Euc n)}
+    (h : GeoC θ k f U) (hl : l ≤ k) : GeoC θ l f U := by
+  induction hl with
+  | refl => exact h
+  | @step k hk ih => exact ih h.mono_order
+
+ theorem linear_schauder (k : ℕ) {θ δ K : ℝ}
+    (hθ : 0<θ) (hθ' : θ ≤ 1/2) (hK : 1 ≤ K)
+    (hc : LinearContractionProperty n θ δ K)
+    {A : Euc n → Euc n →L[ℝ] Euc n} {F : Euc n → Euc n}
+    {f : Euc n → ℝ} {U : Set (Euc n)} (hU : IsOpen U)
+    (hA : GeoC θ k A U) (hF : GeoC θ k F U) (hf : ContDiffOn ℝ 1 f U)
+    (heq : WeakLinearEquation A f F U)
+    (hnear : ∀ p∈U, ‖ContinuousLinearMap.id ℝ (Euc n)-A p‖ < min δ (1/2)) :
+    GeoC θ (k+1) f U := by
+  induction k generalizing A F f U with
+  | zero => exact linear_geoC_one hθ hθ' hK hc hU hA hF hf heq hnear
+  | succ k ih =>
+    have hff := ih hU hA.mono_order hF.mono_order hf heq hnear
+    have hA1 : GeoC θ 1 A U := hA.of_le (by omega)
+    have hF1 : GeoC θ 1 F U := hF.of_le (by omega)
+    have hf1 : GeoC θ 1 f U := hff.of_le (by omega)
+    have hf2 := linear_C2_gain hθ hθ' hK hc hU hA1 hF1 hf1 heq hnear
+    have hgrad := hff.gradient hθ.le hU
+    refine ⟨hf,geoC_zero_of_C1 hθ.le hθ' hU hf,fun e ↦ ?_⟩
+    have hforce : GeoC θ k (fun x ↦ fderiv ℝ F x e-fderiv ℝ A x e (gradient f x)) U :=
+      (hF.2.2 e).sub hθ.le hU ((hA.2.2 e).clm_apply hθ.le hU hgrad)
+    have hdir : ContDiffOn ℝ 1 (fun x ↦ fderiv ℝ f x e) U :=
+      (hf2.fderiv_of_isOpen hU (by norm_num)).clm_apply contDiffOn_const
+    apply GeoC.local
+    intro p hp
+    obtain ⟨W,hW,hpW,hWU,hd⟩ := weak_linear_derivative_local hU hp hA1.1 hF1.1 hf2 heq e
+    exact ⟨W,hW,hpW,ih hW (hA.mono_order.mono hWU) (hforce.mono hWU) (hdir.mono hWU)
+      hd (fun x hx ↦ hnear x (hWU hx))⟩
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/LinearWeakDerivative.lean b/lean/OAI/Geometry/Perimeter/Regularity/LinearWeakDerivative.lean
new file mode 100644
index 0000000000000000000000000000000000000000..21305f015b297da6298d0cf174838b927704fbd5
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/LinearWeakDerivative.lean
@@ -0,0 +1,64 @@
+import OAI.Geometry.Perimeter.Regularity.EllipticLocality
+import OAI.Geometry.Perimeter.Regularity.WeakDerivative
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+ theorem weak_linear_derivative {A : Euc n → Euc n →L[ℝ] Euc n}
+    {F : Euc n → Euc n} {f : Euc n → ℝ} {U : Set (Euc n)}
+    (hA : ContDiff ℝ 1 A) (hF : ContDiff ℝ 1 F) (hf : ContDiff ℝ 2 f)
+    (h : WeakLinearEquation A f F U) (e : Euc n) :
+    WeakLinearEquation A (fun x ↦ fderiv ℝ f x e)
+      (fun x ↦ fderiv ℝ F x e-fderiv ℝ A x e (gradient f x)) U := by
+  have hdf : ContDiff ℝ 1 (fderiv ℝ f) := hf.fderiv_right (by norm_num)
+  have hg : ContDiff ℝ 1 (gradient f) :=
+    (InnerProductSpace.toDual ℝ (Euc n)).symm.toContinuousLinearMap.contDiff.comp hdf
+  have hV : ContDiff ℝ 1 (fun x ↦ A x (gradient f x)-F x) := (hA.clm_apply hg).sub hF
+  have hd := ((weakLinear_iff_flux hA.continuous (hf.of_le (by norm_num)) hF.continuous).mp h).derivative hV e
+  have hdir : ContDiff ℝ 1 (fun x ↦ fderiv ℝ f x e) := hdf.clm_apply contDiff_const
+  have hH : Continuous (fun x ↦ fderiv ℝ F x e-fderiv ℝ A x e (gradient f x)) :=
+    ((hF.continuous_fderiv (by norm_num)).clm_apply continuous_const).sub
+      (((hA.continuous_fderiv (by norm_num)).clm_apply continuous_const).clm_apply hg.continuous)
+  apply (weakLinear_iff_flux hA.continuous hdir hH).mpr
+  apply hd.congr
+  intro x hx
+  have hr := (((hA.differentiable (by simp) x).hasFDerivAt.clm_apply
+    (hg.differentiable (by simp) x).hasFDerivAt).sub (hF.differentiable (by simp) x).hasFDerivAt).fderiv
+  have he := congrArg (fun L : Euc n →L[ℝ] Euc n ↦ L e) hr
+  dsimp only at *
+  change fderiv ℝ (fun y ↦ A y (gradient f y)-F y) x e = _ at he
+  rw [he,gradient_direction_swap hf]
+  simp only [sub_apply,add_apply,ContinuousLinearMap.comp_apply,
+    ContinuousLinearMap.flip_apply]
+  module
+
+ theorem weak_linear_derivative_local {A : Euc n → Euc n →L[ℝ] Euc n}
+    {F : Euc n → Euc n} {f : Euc n → ℝ} {U : Set (Euc n)}
+    (hU : IsOpen U) {p : Euc n} (hp : p∈U)
+    (hA : ContDiffOn ℝ 1 A U) (hF : ContDiffOn ℝ 1 F U) (hf : ContDiffOn ℝ 2 f U)
+    (h : WeakLinearEquation A f F U) (e : Euc n) :
+    ∃ W : Set (Euc n), IsOpen W ∧ p∈W ∧ W⊆U ∧
+      WeakLinearEquation A (fun x ↦ fderiv ℝ f x e)
+        (fun x ↦ fderiv ℝ F x e-fderiv ℝ A x e (gradient f x)) W := by
+  obtain ⟨W,hW,hpW,hWU,B,g,G,hB,hBc,hg,hgc,hG,hGc,heB,heg,heG⟩ :=
+    local_compact_triple hU hp hA hf hF
+  have he := (h.mono hWU).congr hW heB.symm heg.symm heG.symm
+  have hd := weak_linear_derivative hB hG hg he e
+  refine ⟨W,hW,hpW,hWU,hd.congr hW heB ?_ ?_⟩
+  · intro x hx
+    dsimp only
+    rw [fderiv_eqOn hW heg hx]
+  · intro x hx
+    dsimp only
+    rw [fderiv_eqOn hW heG hx,fderiv_eqOn hW heB hx,gradient_eqOn hW heg hx]
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/LocalFlatExcess.lean b/lean/OAI/Geometry/Perimeter/Regularity/LocalFlatExcess.lean
new file mode 100644
index 0000000000000000000000000000000000000000..dcec2725c58e11e624bd655352508b0e1aa6c05e
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/LocalFlatExcess.lean
@@ -0,0 +1,119 @@
+import OAI.Geometry.Perimeter.Regularity.FlatExcess
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+theorem tendsto_phase_slab_error {E : ℕ → Set (Euc n)}
+    (hE : ∀ i, MeasurableSet (E i)) {v : Euc n} (hv : ‖v‖=1)
+    (a : ℕ → ℝ) (hat : Tendsto a atTop (𝓝 0)) (R : ℝ)
+    (hphase : ∀ i, ∀ᵐ x ∂volume, x∈Metric.ball 0 R → a i < |inner ℝ v x| →
+      (x∈E i ↔ inner ℝ v x<0)) :
+    Tendsto (fun i ↦ volume.real (symmDiff (E i) {x | inner ℝ x v<0} ∩ Metric.ball 0 R))
+      atTop (𝓝 0) := by
+  let H : Set (Euc n) := {x | inner ℝ x v<0}
+  have hH : MeasurableSet H := (isOpen_lt (continuous_id.inner continuous_const) continuous_const).measurableSet
+  let f := fun i x ↦ |(E i).indicator (fun _ ↦ (1:ℝ)) x-H.indicator (fun _ ↦ (1:ℝ)) x|
+  have hmeas (i) : AEStronglyMeasurable (f i) (volume.restrict (Metric.ball 0 R)) :=
+    (((measurable_const.indicator (hE i)).sub (measurable_const.indicator hH)).abs).aestronglyMeasurable.restrict
+  have hb (i) : ∀ᵐ x ∂volume.restrict (Metric.ball (0:Euc n) R), ‖f i x‖≤1 := by
+    apply ae_of_all
+    intro x
+    by_cases hx : x∈E i <;> by_cases hy : x∈H <;> simp [f,hx,hy]
+  have hn : (innerSL ℝ v : Euc n →L[ℝ] ℝ) ≠ 0 := by
+    intro hz
+    have he := congrArg (fun L : Euc n →L[ℝ] ℝ ↦ L v) hz
+    change inner ℝ v v=0 at he
+    rw [real_inner_self_eq_norm_sq,hv] at he
+    norm_num at he
+  have hall := ae_all_iff.mpr hphase
+  have ht : ∀ᵐ x ∂volume.restrict (Metric.ball (0:Euc n) R), Tendsto (fun i ↦ f i x) atTop (𝓝 0) := by
+    filter_upwards [ae_restrict_of_ae hall,ae_restrict_of_ae (ae_linear_ne_zero hn),
+      ae_restrict_mem measurableSet_ball] with x hx hnz hxB
+    apply tendsto_const_nhds.congr'
+    have hpos : 0 < |inner ℝ v x| := abs_pos.mpr hnz
+    filter_upwards [hat.eventually (gt_mem_nhds hpos)] with i hi
+    have hm := hx i hxB hi
+    have hm' : x∈E i ↔ x∈H := by simpa only [H,mem_ofPred_eq,real_inner_comm x v] using hm
+    by_cases he : x∈E i
+    · simp [f,he,hm'.mp he]
+    · simp [f,he,mt hm'.mpr he]
+  have ht' := tendsto_integral_of_dominated_convergence (fun _ : Euc n ↦ (1:ℝ)) hmeas
+    (integrableOn_const measure_ball_lt_top.ne) hb ht
+  simp only [integral_zero] at ht'
+  convert ht' using 1
+  ext i
+  exact (integral_abs_indicator_sub_eq_symmDiff (hE i) hH).symm
+
+theorem tendsto_setIntegral_of_ball_symmDiff {E : ℕ → Set (Euc n)} {F : Set (Euc n)}
+    (hE : ∀ i, MeasurableSet (E i)) (hF : MeasurableSet F) {R : ℝ}
+    (hconv : Tendsto (fun i ↦ volume.real (symmDiff (E i) F ∩ Metric.ball 0 R)) atTop (𝓝 0))
+    {f : Euc n → ℝ} (hf : Continuous f) (hc : HasCompactSupport f)
+    (hs : tsupport f ⊆ Metric.ball 0 R) :
+    Tendsto (fun i ↦ ∫ x in E i, f x) atTop (𝓝 (∫ x in F, f x)) := by
+  obtain ⟨C,hC⟩ := hc.exists_bound_of_continuous hf
+  have hC0 : 0≤C := (norm_nonneg (f 0)).trans (hC 0)
+  have hb (i) := abs_setIntegral_sub_le_symmDiff_volume (hE i) hF measurableSet_ball
+    measure_ball_lt_top.ne (hf.integrable_of_hasCompactSupport hc)
+    (fun x hx ↦ image_eq_zero_of_notMem_tsupport (fun h ↦ hx (hs h))) hC0
+    (fun x ↦ by simpa only [Real.norm_eq_abs] using hC x)
+  apply tendsto_iff_norm_sub_tendsto_zero.mpr
+  apply squeeze_zero (fun i ↦ norm_nonneg _)
+    (fun i ↦ by simpa only [Real.norm_eq_abs] using hb i)
+  simpa using hconv.const_mul C
+
+theorem orientedTilt_tendsto_of_local_halfspace_L1 (hn : 1<n)
+    {E : ℕ → Set (Euc n)} (hmin : ∀ i, IsLocallyPerimeterMinimizing (E i))
+    {v : Euc n} (hv : ‖v‖=1)
+    (hH : IsLocallyPerimeterMinimizing {x : Euc n | inner ℝ x v<0})
+    (hconv : Tendsto (fun i ↦ volume.real
+      (symmDiff (E i) {x | inner ℝ x v<0} ∩ Metric.ball 0 8)) atTop (𝓝 0)) :
+    Tendsto (fun i ↦ orientedTilt (E i) v (Metric.ball 0 4)) atTop (𝓝 0) := by
+  let H : Set (Euc n) := {x | inner ℝ x v<0}
+  apply tendsto_order.2
+  constructor
+  · intro A hA
+    exact Eventually.of_forall fun i ↦ hA.trans_le (integral_nonneg fun _ ↦ sq_nonneg _)
+  · intro ε hε
+    obtain ⟨η,hη,hcη,hsη,hbη,honeη,hcal⟩ := exists_halfspace_calibration hn hv hH
+      (by norm_num : (0:ℝ)<4) (by norm_num : (4:ℝ)<6) (by positivity : 0<ε/8)
+    obtain ⟨a₀,ha₀,hs₀⟩ := exists_lt_subset_ball (isClosed_tsupport η) hsη
+    let b := max a₀ 5
+    have h4b : 4<b := lt_of_lt_of_le (by norm_num) (le_max_right _ _)
+    have hb6 : b<6 := max_lt ha₀ (by norm_num)
+    have hb : 0<b := lt_trans (by norm_num) h4b
+    have hsb : tsupport η ⊆ Metric.ball 0 b := hs₀.trans (Metric.ball_subset_ball (le_max_left _ _))
+    have hann : Tendsto (fun i ↦ volume.real (symmDiff (E i) H ∩
+        (Metric.closedBall 0 6 \ Metric.closedBall 0 b))) atTop (𝓝 0) := by
+      apply squeeze_zero (fun _ ↦ ENNReal.toReal_nonneg) (fun i ↦ ?_) hconv
+      apply measureReal_mono _ (measure_ne_top_of_subset inter_subset_right measure_ball_lt_top.ne)
+      exact inter_subset_inter_right _ (sdiff_subset.trans (Metric.closedBall_subset_ball (by norm_num : (6:ℝ)<8)))
+    have hmass := eventually_minimizing_perimeter_le_of_annular_L1 hmin hH.1 0 hb hb6 hann
+      (by positivity : 0<ε/8)
+    have hflux := tendsto_setIntegral_of_ball_symmDiff (f := divergence (fun y ↦ η y • v)) (fun i ↦ (hmin i).1) hH.1 hconv
+      (continuous_divergence (hη.smul contDiff_const))
+      (hcη.smul_right.of_isClosed_subset (isClosed_tsupport _) (tsupport_divergence_subset _))
+      ((tsupport_divergence_subset _).trans ((tsupport_smul_subset_left _ _).trans
+        (hsη.trans (Metric.ball_subset_ball (by norm_num : (6:ℝ)≤8)))))
+    have hfl := hflux.eventually (lt_mem_nhds (sub_lt_self _ (by positivity : 0<ε/8)))
+    filter_upwards [hmass,hfl] with i hi hfi
+    have hm : (perimeter (E i) (Metric.ball 0 b)).toReal ≤
+        (perimeter H (Metric.ball 0 6)).toReal+ε/8 := by
+      have ht := ENNReal.toReal_mono (ENNReal.add_ne_top.mpr ⟨(hH.2.1 0 6 (by norm_num)).ne,
+        ENNReal.ofReal_ne_top⟩) hi
+      rwa [ENNReal.toReal_add (hH.2.1 0 6 (by norm_num)).ne ENNReal.ofReal_ne_top,
+        ENNReal.toReal_ofReal (by positivity : 0≤ε/8)] at ht
+    have hgap := orientedTilt_le_calibration_gap hn (hmin i) hv (by norm_num : (0:ℝ)<4)
+      h4b hη hcη hsb hbη honeη
+    change (perimeter H (Metric.ball 0 6)).toReal-ε/8 < _ at hcal
+    linarith
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/Main.lean b/lean/OAI/Geometry/Perimeter/Regularity/Main.lean
new file mode 100644
index 0000000000000000000000000000000000000000..f4cf3f07e51ca123a35e67e0a761b606fa29a146
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/Main.lean
@@ -0,0 +1,7 @@
+import OAI.Geometry.Perimeter.Regularity.C1Graph
+import OAI.Geometry.Perimeter.Regularity.ClosedFilledLineInvariantOfFrontier
+import OAI.Geometry.Perimeter.Regularity.ConeLinkIntersection
+import OAI.Geometry.Perimeter.Regularity.IsConeMemPosTangentCone
+import OAI.Geometry.Perimeter.Regularity.PhaseSupportBlownUpSet
+import OAI.Geometry.Perimeter.Regularity.SmoothSheet
+import OAI.Geometry.Perimeter.Regularity.WholeBoundarySublevelMeasureStep
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/MinimalFlux.lean b/lean/OAI/Geometry/Perimeter/Regularity/MinimalFlux.lean
new file mode 100644
index 0000000000000000000000000000000000000000..4a4a93e02260f7f104bf1e8b9bc2a14695cbdec7
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/MinimalFlux.lean
@@ -0,0 +1,43 @@
+import OAI.Geometry.Perimeter.Regularity.GeoComposition
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+def minimalFlux (z : Euc n) : Euc n := (Real.sqrt (1+‖z‖^2))⁻¹ • z
+
+theorem minimalFlux_scalar_smooth : ContDiff ℝ ∞ (fun z : Euc n ↦ (Real.sqrt (1+‖z‖^2))⁻¹) := by
+  apply ContDiff.inv
+  · exact (contDiff_const.add (contDiff_id.norm_sq ℝ)).sqrt (fun z ↦ by positivity)
+  · intro z; positivity
+
+theorem minimalFlux_smooth : ContDiff ℝ ∞ (minimalFlux (n:=n)) :=
+  minimalFlux_scalar_smooth.smul contDiff_id
+
+theorem minimalFlux_fderiv_zero : fderiv ℝ (minimalFlux (n:=n)) 0 = ContinuousLinearMap.id ℝ (Euc n) := by
+  unfold minimalFlux
+  have hc : DifferentiableAt ℝ (fun z : Euc n ↦ (Real.sqrt (1+‖z‖^2))⁻¹) 0 :=
+    (minimalFlux_scalar_smooth.differentiable (by simp)).differentiableAt
+  have he := fderiv_fun_smul hc
+    (differentiableAt_id : DifferentiableAt ℝ (fun z : Euc n ↦ z) 0)
+  convert he using 1 <;> simp
+
+theorem minimalFlux_exists_small {δ : ℝ} (hδ : 0<δ) : ∃ ε : ℝ, 0<ε ∧
+    ∀ z : Euc n, ‖z‖<ε → ‖ContinuousLinearMap.id ℝ (Euc n)-fderiv ℝ minimalFlux z‖< min δ (1/2) := by
+  have hD : ContDiff ℝ ∞ (fderiv ℝ (minimalFlux (n:=n))) :=
+    minimalFlux_smooth.fderiv_right (by simp)
+  have hc : ContinuousAt (fderiv ℝ (minimalFlux (n:=n))) 0 := hD.continuous.continuousAt
+  obtain ⟨ε,hε,hh⟩ := Metric.continuousAt_iff.mp hc (min δ (1/2)) (lt_min hδ (by norm_num))
+  refine ⟨ε,hε,fun z hz ↦ ?_⟩
+  have hv := hh (by simpa only [dist_zero_right] using hz)
+  rwa [minimalFlux_fderiv_zero,dist_eq_norm,norm_sub_rev] at hv
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/MinimalGraphEquation.lean b/lean/OAI/Geometry/Perimeter/Regularity/MinimalGraphEquation.lean
new file mode 100644
index 0000000000000000000000000000000000000000..17a80ffa5248074bce2a3fa0645a587c6bfbddab
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/MinimalGraphEquation.lean
@@ -0,0 +1,71 @@
+import OAI.Geometry.Perimeter.Regularity.GraphFluxWeight
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+theorem full_graph_weak_minimal_equation (hn : 1<n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    {v : Euc n} (hv : ‖v‖=1) {a : ℝ} (ha : 0≤a) (ha4 : a<1/4)
+    (hs : FullSlab E v 8 a) {f : normalPlane v → ℝ}
+    (hf : ContDiffOn ℝ 1 f (Metric.ball 0 (1/4)))
+    (hiff : ∀ (y : normalPlane v) (t : ℝ), ‖y‖<1/4 → ‖(y:Euc n)+t•v‖<1 →
+      ((y:Euc n)+t•v∈perimeterSupport E ↔ t=f y))
+    {ψ : normalPlane v → ℝ} (hψ : ContDiff ℝ ∞ ψ) (hcψ : HasCompactSupport ψ)
+    (hsψ : tsupport ψ ⊆ Metric.ball 0 (1/4)) :
+    (∫ y, inner ℝ (gradient f y) (gradient ψ y)/Real.sqrt (1+‖gradient f y‖^2))=0 := by
+  let φ := graphFluxWeight v f ψ
+  have hφ : Continuous φ := graphFluxWeight_continuous hv hf hψ hsψ
+  have hcφ : HasCompactSupport φ := hcψ.of_isClosed_subset (isClosed_tsupport _)
+    (graphFluxWeight_tsupport f ψ)
+  have hsφ : tsupport φ ⊆ Metric.ball 0 (1/4) := (graphFluxWeight_tsupport f ψ).trans hsψ
+  let χ : ContDiffBump (0:ℝ) := ⟨1/3,1/2,by norm_num,by norm_num⟩
+  have hχone : ∀ t : ℝ, |t|<1/3 → χ t=1 := by
+    intro t ht
+    exact χ.one_of_mem_closedBall (by simpa only [Metric.mem_closedBall,dist_zero_right,Real.norm_eq_abs] using ht.le)
+  let Ψ := normalTestLift v ψ χ
+  have hcΨ : tsupport Ψ ⊆ Metric.closedBall 0 (1/4+1/2) :=
+    normalTestLift_tsupport hv ψ χ (subset_closure.trans (hsψ.trans Metric.ball_subset_closedBall))
+      (by simpa only [χ.support_eq] using (Metric.ball_subset_closedBall : Metric.ball (0:ℝ) (1/2) ⊆ Metric.closedBall 0 (1/2)))
+  have hΨ := normalTestLift_contDiff hψ χ.contDiff
+  have hzero := minimizing_tangentialGradient_integral_eq_zero_local hn hmin 0 (by norm_num : (0:ℝ)<1)
+    hΨ ((isCompact_closedBall 0 (1/4+1/2)).of_isClosed_subset (isClosed_tsupport _) hcΨ)
+    (hcΨ.trans (Metric.closedBall_subset_ball (by norm_num))) v
+  have hslab := hs.mono_radius (by norm_num : (1:ℝ)≤8)
+  have hdegree := actual_fiber_signed_zeroth_moment hn hmin hv hφ hcφ ha
+    (by norm_num : (0:ℝ)<1/8) (by linarith : a+1/8<1/2) (by norm_num : (0:ℝ)<1)
+    (fun y hy ↦ by
+      have hy' : ‖y‖<1/4 := by simpa only [Metric.mem_ball,dist_zero_right] using hsφ hy
+      linarith : ∀ y∈tsupport φ, ‖y‖+1/2+1/8≤1) hslab.1 hslab.2
+  have heq : (∫ x in Metric.ball 0 1, inner ℝ (tangentialGradient E Ψ x) v ∂perimeterMeasure E)=
+      ∫ x in Metric.ball 0 1, normalWindowTest v φ (1/2) 0 x*
+        inner ℝ (approximateNormal E x) v ∂perimeterMeasure E := by
+    apply integral_congr_ae
+    have hsupport : ∀ᵐ x ∂perimeterMeasure E, x∈perimeterSupport E := by
+      rw [←perimeterMeasure_support]; exact (perimeterMeasure E).support_mem_ae
+    filter_upwards [ae_restrict_of_ae hsupport,ae_restrict_mem measurableSet_ball] with x hxS hxB
+    have hxheight := hslab.1 x ⟨hxS,hxB⟩
+    have hwindow : normalWindowTest v φ (1/2) 0 x=φ (normalProjection v x) := by
+      simp only [normalWindowTest,Set.indicator_of_mem (show x∈{z | |inner ℝ v z|≤1/2} from by
+        change |inner ℝ v x|≤1/2; linarith),pow_zero,mul_one]
+    rw [hwindow]
+    by_cases hxP : normalProjection v x∈tsupport ψ
+    · have hxC : x∈graphCylinder v := ⟨by
+        simpa only [Metric.mem_ball,dist_zero_right] using hsψ hxP,by linarith⟩
+      exact graphTest_tangential_flux hv (hψ.differentiable (by simp)) hχone (by linarith)
+        (graph_approximateNormal hn hmin hv ha ha4 hs hf hiff hxC hxS)
+    · change inner ℝ (gradient Ψ x - (inner ℝ (gradient Ψ x) (approximateNormal E x)) • approximateNormal E x) v = _
+      rw [show gradient Ψ x=0 from normalTestLift_gradient_zero hxP]
+      simp [φ,graphFluxWeight_zero hxP]
+  rw [heq,hdegree,setIntegral_tsupport] at hzero
+  simpa only [φ,graphFluxWeight,graphAreaFactor_sqrt hv] using hzero
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/MinimalGraphSmooth.lean b/lean/OAI/Geometry/Perimeter/Regularity/MinimalGraphSmooth.lean
new file mode 100644
index 0000000000000000000000000000000000000000..ac63420c18691e324255ee2f78e6947e63f65eaa
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/MinimalGraphSmooth.lean
@@ -0,0 +1,64 @@
+import OAI.Geometry.Perimeter.Regularity.NonlinearSmooth
+import OAI.Geometry.Perimeter.Regularity.MinimalFlux
+import OAI.Geometry.Perimeter.Regularity.NormalAdjustment
+import OAI.Geometry.Perimeter.Regularity.GraphGeoCoordinates
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+theorem normal_graph_smooth_threshold (hn : 1<n) {v : Euc n} (hv : ‖v‖=1)
+    {η : ℝ} (hη : 0<η) : ∃ ε : ℝ, 0<ε ∧
+    ∀ f : normalPlane v → ℝ,
+    ContDiffOn ℝ 1 f (Metric.ball 0 (1/4)) →
+    (∀ x∈Metric.ball (0:normalPlane v) (1/4), GeometricHolderAt η (fderiv ℝ f) x) →
+    (∀ x∈Metric.ball (0:normalPlane v) (1/4), ‖fderiv ℝ f x‖<ε) →
+    (∀ ψ : normalPlane v → ℝ, ContDiff ℝ ∞ ψ → HasCompactSupport ψ →
+      tsupport ψ ⊆ Metric.ball 0 (1/4) →
+      (∫ y, inner ℝ (gradient f y) (gradient ψ y)/Real.sqrt (1+‖gradient f y‖^2))=0) →
+    ContDiffOn ℝ ∞ f (Metric.ball 0 (1/4)) := by
+  obtain ⟨θ,δ,K,hθ,hθb,hδ,hK,hc⟩ := exists_linear_contraction (by omega : 0<n-1) hη
+  obtain ⟨ε,hε,hsmall⟩ := minimalFlux_exists_small (n:=n-1) hδ
+  refine ⟨ε,hε,fun f hf hh hs heq ↦ ?_⟩
+  let e := normalPlaneEuclideanChart hv
+  let U : Set (normalPlane v) := Metric.ball 0 (1/4)
+  let V := e ⁻¹' U
+  have hU : IsOpen U := Metric.isOpen_ball
+  have hV : IsOpen V := hU.preimage e.continuous
+  have hgeo := geoC_one_isometry hθ.le (hθb.trans_le (min_le_right _ _)).le
+    (hθb.trans_le (min_le_left _ _)).le e hU hf hh
+  have hweak : WeakFluxEquation (fun x ↦ minimalFlux (gradient (f ∘ e) x)) V := by
+    intro φ
+    have hz := minimal_equation_isometry e heq φ
+    change (∫ x, inner ℝ (minimalFlux (gradient (f ∘ e) x)) (gradient φ.val x))=0
+    convert hz using 1
+    apply integral_congr_ae
+    filter_upwards [] with x
+    rw [minimalFlux,real_inner_smul_left,div_eq_mul_inv,mul_comm]
+  have hnear : ∀ x∈V, ‖ContinuousLinearMap.id ℝ (Euc (n-1))-
+      fderiv ℝ minimalFlux (gradient (f ∘ e) x)‖ < min δ (1/2) := by
+    intro x hx
+    apply hsmall
+    rw [gradient_comp_isometry,e.symm.norm_map]
+    have hnorm : ‖gradient f (e x)‖=‖fderiv ℝ f (e x)‖ :=
+      (InnerProductSpace.toDual ℝ (normalPlane v)).symm.norm_map _
+    rw [hnorm]
+    exact hs (e x) hx
+  have hsm := nonlinear_smooth hθ (hθb.trans_le (min_le_right _ _)).le hK hc
+    minimalFlux_smooth hV hgeo hweak hnear
+  have him : MapsTo e.symm U V := by
+    intro x hx
+    change e (e.symm x)∈U
+    simpa only [e.apply_symm_apply] using hx
+  have hout := hsm.comp e.symm.contDiff.contDiffOn him
+  simpa only [Function.comp_def,e.apply_symm_apply] using hout
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/MollifiedFiberMoment.lean b/lean/OAI/Geometry/Perimeter/Regularity/MollifiedFiberMoment.lean
new file mode 100644
index 0000000000000000000000000000000000000000..adc7dfea6f5f54d879a208f4bc6ddef45d89b532
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/MollifiedFiberMoment.lean
@@ -0,0 +1,154 @@
+import OAI.Geometry.Perimeter.Regularity.NormalCoordinates
+import OAI.Geometry.Perimeter.Regularity.FiberCrossing
+import OAI.Geometry.Perimeter.Regularity.NormalPhaseMollification
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff Convolution InnerProductSpace
+namespace RPISingularHarnack
+open RPI.Boundary ContinuousLinearMap
+
+theorem phaseMollification_eq_one_of_ae {n : ℕ} {E : Set (Euc n)}
+    {k : Euc n → ℝ} (hu : ∫ x, k x = 1) (x : Euc n)
+    (he : ∀ᵐ y ∂volume, k (x-y) ≠ 0 → y ∈ E) :
+    phaseMollification E k x = 1 := by
+  change (∫ y, E.indicator (fun _ ↦ (1:ℝ)) y*k (x-y)) = 1
+  calc
+    _ = ∫ y, k (x-y) := by
+      apply integral_congr_ae
+      filter_upwards [he] with y hy
+      by_cases hk : k (x-y) = 0
+      · simp [hk]
+      · simp [hy hk]
+    _ = 1 := by rw [integral_sub_left_eq_self,hu]
+
+theorem phaseMollification_eq_zero_of_ae {n : ℕ} {E : Set (Euc n)}
+    {k : Euc n → ℝ} (x : Euc n)
+    (he : ∀ᵐ y ∂volume, k (x-y) ≠ 0 → y ∉ E) :
+    phaseMollification E k x = 0 := by
+  change (∫ y, E.indicator (fun _ ↦ (1:ℝ)) y*k (x-y)) = 0
+  apply integral_eq_zero_of_ae
+  filter_upwards [he] with y hy
+  by_cases hk : k (x-y) = 0
+  · simp [hk]
+  · simp [hy hk]
+
+theorem phaseMollification_lower_cap {n : ℕ} {E : Set (Euc n)} {v : Euc n}
+    (hv : ‖v‖ = 1) {k : Euc n → ℝ} (hu : ∫ x, k x = 1)
+    {a ε R : ℝ} (ha : 0 ≤ a) (hs : Function.support k ⊆ Metric.ball 0 ε)
+    (hphase : ∀ᵐ y ∂volume, y ∈ Metric.ball 0 R → a < |inner ℝ v y| →
+      (y ∈ E ↔ inner ℝ v y < 0))
+    (x : Euc n) (hx : ‖x‖+ε ≤ R) (hl : inner ℝ v x < -a-ε) :
+    phaseMollification E k x = 1 := by
+  apply phaseMollification_eq_one_of_ae hu x
+  filter_upwards [hphase] with y hy
+  intro hk
+  have hxy : ‖x-y‖ < ε := by simpa only [Metric.mem_ball,dist_zero_right] using hs hk
+  have hyn : y ∈ Metric.ball (0:Euc n) R := by
+    rw [Metric.mem_ball, dist_zero_right]
+    have h := norm_sub_le x (x-y)
+    rw [sub_sub_cancel] at h
+    linarith
+  have hiv : |inner ℝ v (x-y)| ≤ ‖x-y‖ := by
+    simpa only [hv,one_mul] using abs_real_inner_le_norm v (x-y)
+  have hey : inner ℝ v y < -a := by
+    rw [inner_sub_right] at hiv
+    have he := (neg_le_abs (inner ℝ v x-inner ℝ v y)).trans hiv
+    linarith
+  exact (hy hyn (by rw [abs_of_neg (by linarith : inner ℝ v y < 0)]; linarith)).mpr
+    (by linarith)
+
+theorem phaseMollification_upper_cap {n : ℕ} {E : Set (Euc n)} {v : Euc n}
+    (hv : ‖v‖ = 1) {k : Euc n → ℝ}
+    {a ε R : ℝ} (ha : 0 ≤ a) (hs : Function.support k ⊆ Metric.ball 0 ε)
+    (hphase : ∀ᵐ y ∂volume, y ∈ Metric.ball 0 R → a < |inner ℝ v y| →
+      (y ∈ E ↔ inner ℝ v y < 0))
+    (x : Euc n) (hx : ‖x‖+ε ≤ R) (hl : a+ε < inner ℝ v x) :
+    phaseMollification E k x = 0 := by
+  apply phaseMollification_eq_zero_of_ae x
+  filter_upwards [hphase] with y hy
+  intro hk
+  have hxy : ‖x-y‖ < ε := by simpa only [Metric.mem_ball,dist_zero_right] using hs hk
+  have hyn : y ∈ Metric.ball (0:Euc n) R := by
+    rw [Metric.mem_ball, dist_zero_right]
+    have h := norm_sub_le x (x-y)
+    rw [sub_sub_cancel] at h
+    linarith
+  have hiv : |inner ℝ v (x-y)| ≤ ‖x-y‖ := by
+    simpa only [hv,one_mul] using abs_real_inner_le_norm v (x-y)
+  have hey : a < inner ℝ v y := by
+    rw [inner_sub_right] at hiv
+    have he := (le_abs_self (inner ℝ v x-inner ℝ v y)).trans hiv
+    linarith
+  intro he
+  have hn := (hy hyn (by rw [abs_of_pos (by linarith : 0 < inner ℝ v y)]; exact hey)).mp he
+  linarith
+
+def mollifiedNormalProfile {n : ℕ} (E : Set (Euc n)) (k : Euc n → ℝ)
+    (v : Euc n) (y : normalPlane v) (s : ℝ) : ℝ :=
+  phaseMollification E k ((y:Euc n)+s•v)
+
+theorem mollifiedNormalProfile_contDiff {n : ℕ} {E : Set (Euc n)}
+    (hE : MeasurableSet E) {k : Euc n → ℝ} (hk : ContDiff ℝ ∞ k)
+    (hc : HasCompactSupport k) (v : Euc n) (y : normalPlane v) :
+    ContDiff ℝ ∞ (mollifiedNormalProfile E k v y) :=
+  (phaseMollification_contDiff hE hk hc).comp (contDiff_const.add (contDiff_id.smul contDiff_const))
+
+theorem mollifiedNormalProfile_deriv {n : ℕ} {E : Set (Euc n)}
+    (hE : MeasurableSet E) {k : Euc n → ℝ} (hk : ContDiff ℝ ∞ k)
+    (hc : HasCompactSupport k) (v : Euc n) (y : normalPlane v) (s : ℝ) :
+    deriv (mollifiedNormalProfile E k v y) s =
+      fderiv ℝ (phaseMollification E k) ((y:Euc n)+s•v) v := by
+  have hd := ((phaseMollification_contDiff hE hk hc).differentiable (by simp)
+    ((y:Euc n)+s•v)).hasFDerivAt.comp_hasDerivAt s
+      (HasDerivAt.const_add (y:Euc n) ((hasDerivAt_id s).smul_const v))
+  change deriv (fun t ↦ phaseMollification E k ((y:Euc n)+t•v)) s = _
+  simpa only [Function.comp_def,id_eq,one_smul] using hd.deriv
+
+theorem mollifiedNormalProfile_flux_defect {n : ℕ} {E : Set (Euc n)}
+    (hE : MeasurableSet E) {v : Euc n} (hv : ‖v‖ = 1)
+    {k : Euc n → ℝ} (hk : ContDiff ℝ ∞ k) (hc : HasCompactSupport k)
+    (hp : ∀ x, 0 ≤ k x) (hu : ∫ x, k x = 1)
+    {a b ε R : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) (hab : a+ε < b)
+    (hs : Function.support k ⊆ Metric.ball 0 ε)
+    (hphase : ∀ᵐ x ∂volume, x ∈ Metric.ball 0 R → a < |inner ℝ v x| →
+      (x ∈ E ↔ inner ℝ v x < 0))
+    (y : normalPlane v) (hy : ‖y‖+b+ε ≤ R) :
+    let q := mollifiedNormalProfile E k v y
+    0 ≤ (∫ s in (-b)..b, s^2*(-deriv q s))-(-b+∫ s in (-b)..b, q s)^2 ∧
+    (∫ s in (-b)..b, s^2*(-deriv q s))-(-b+∫ s in (-b)..b, q s)^2 ≤
+      4*b*(∫ s in (-b)..b, q s*(1-q s))+
+        8*b^2*(∫ s in (-b)..b, max (deriv q s) 0) := by
+  dsimp only
+  have hself : inner ℝ v v = 1 := by rw [real_inner_self_eq_norm_sq,hv]; norm_num
+  have ho : inner ℝ v (y:Euc n) = 0 :=
+    Submodule.mem_orthogonal_singleton_iff_inner_right.mp y.property
+  have hl : mollifiedNormalProfile E k v y (-b) = 1 := by
+    apply phaseMollification_lower_cap hv hu ha hs hphase
+    · have h := norm_add_le (y:Euc n) ((-b)•v)
+      rw [norm_smul,hv,mul_one,Real.norm_eq_abs,abs_neg,abs_of_nonneg hb] at h
+      change ‖(y:Euc n)‖+b+ε ≤ R at hy
+      linarith
+    · rw [inner_add_right,real_inner_smul_right,ho,hself,mul_one,zero_add]
+      linarith
+  have hr : mollifiedNormalProfile E k v y b = 0 := by
+    apply phaseMollification_upper_cap hv ha hs hphase
+    · have h := norm_add_le (y:Euc n) (b•v)
+      rw [norm_smul,hv,mul_one,Real.norm_eq_abs,abs_of_nonneg hb] at h
+      change ‖(y:Euc n)‖+b+ε ≤ R at hy
+      linarith
+    · rw [inner_add_right,real_inner_smul_right,ho,hself,mul_one,zero_add]
+      exact hab
+  have he := fiber_flux_height_defect
+    ((mollifiedNormalProfile_contDiff hE hk hc v y).of_le (by simp))
+    (by linarith : -b ≤ b) hl hr (fun s _ ↦
+      phaseMollification_mem_Icc hE (hk.continuous.integrable_of_hasCompactSupport hc) hp hu _)
+  simpa only [show 2*(b- -b) = 4*b by ring, show 2*(b- -b)^2 = 8*b^2 by ring] using he
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/NeighborhoodPlanes.lean b/lean/OAI/Geometry/Perimeter/Regularity/NeighborhoodPlanes.lean
new file mode 100644
index 0000000000000000000000000000000000000000..e48ad799cc9520c2d25aa2fcb2eee5e51a62b41f
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/NeighborhoodPlanes.lean
@@ -0,0 +1,93 @@
+import OAI.Geometry.Perimeter.Regularity.SlabScaling
+import OAI.Geometry.Perimeter.Regularity.TangentPlanes
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+theorem nearby_full_support_planes {q M δ : ℝ} (hq : 0 < q) (hq1 : q < 1)
+    (hM : 0≤M) (hstep : SlabContractionProperty n q M δ)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    {v : Euc n} (hv : ‖v‖=1) {a : ℝ} (ha : 0 < a) (haδ : 4*a < δ)
+    (hs : FullSlab E v 8 a) {p : Euc n}
+    (hp : p∈perimeterSupport E ∩ Metric.ball 0 1) :
+    ∃ w : Euc n, ‖w‖=1 ∧ ‖w-v‖≤8*M*a ∧
+      ∀ j x, x∈perimeterSupport E → ‖x-p‖≤2*q^j →
+        |inner ℝ w (x-p)|≤4*a*(2*M+1/(4*q))*(1/2)^j*‖x-p‖ := by
+  have hminF := locally_minimizing_blownUpSet hmin p (by norm_num : (0:ℝ)<1/2)
+  have hzeroF : (0:Euc n)∈perimeterSupport (blownUpSet E p (1/2)) := by
+    rw [perimeterSupport_blownUpSet hmin.1 p (by norm_num),smul_zero,add_zero]
+    exact hp.1
+  obtain ⟨w,hw,hwd,htan⟩ := exists_full_support_tangent_plane hq hq1 hM hstep
+    hminF hzeroF hv (by positivity : 0 < 4*a) haδ (hs.recenter_half hmin.1 ha.le hp)
+  refine ⟨w,hw,by nlinarith [hwd],fun j x hx hxp ↦ ?_⟩
+  let y := (2:ℝ)•(x-p)
+  have hxy : p+(1/2:ℝ)•y=x := by dsimp [y]; module
+  have hyS : y∈perimeterSupport (blownUpSet E p (1/2)) := by
+    rw [perimeterSupport_blownUpSet hmin.1 p (by norm_num),hxy]
+    exact hx
+  have hyn : ‖y‖=2*‖x-p‖ := by rw [norm_smul,Real.norm_of_nonneg (by norm_num)]
+  have hyB : ‖y‖≤4*q^j := by rw [hyn]; linarith
+  have hh := htan j y hyS hyB
+  change |inner ℝ w ((2:ℝ)•(x-p))|≤_ at hh
+  rw [real_inner_smul_right,abs_mul,abs_of_pos (by norm_num : (0:ℝ)<2),hyn] at hh
+  nlinarith
+
+theorem nearby_chord_bound {q M δ : ℝ} (hq : 0 < q) (hq1 : q < 1)
+    (hM : 0≤M) (hstep : SlabContractionProperty n q M δ)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    {v : Euc n} (hv : ‖v‖=1) {a : ℝ} (ha : 0 < a) (haδ : 4*a < δ)
+    (hs : FullSlab E v 8 a) {p x : Euc n}
+    (hp : p∈perimeterSupport E ∩ Metric.ball 0 1)
+    (hx : x∈perimeterSupport E ∩ Metric.ball 0 1) :
+    |inner ℝ v (x-p)|≤a*(16*M+1/q)*‖x-p‖ := by
+  obtain ⟨w,hw,hwd,htan⟩ := nearby_full_support_planes hq hq1 hM hstep hmin hv ha haδ hs hp
+  have hxp : ‖x-p‖≤2*q^0 := by
+    have hh := norm_sub_le x p
+    have hnx : ‖x‖<1 := by simpa only [Metric.mem_ball,dist_zero_right] using hx.2
+    have hnp : ‖p‖<1 := by simpa only [Metric.mem_ball,dist_zero_right] using hp.2
+    simp only [pow_zero,mul_one]
+    linarith
+  have ht := htan 0 x hx.1 hxp
+  simp only [pow_zero,mul_one] at ht
+  have hi : inner ℝ v (x-p)=inner ℝ (v-w) (x-p)+inner ℝ w (x-p) := by rw [inner_sub_left]; ring
+  calc
+    |inner ℝ v (x-p)| ≤ ‖v-w‖*‖x-p‖+|inner ℝ w (x-p)| := by
+      rw [hi]
+      exact (abs_add_le _ _).trans (add_le_add (abs_real_inner_le_norm _ _) le_rfl)
+    _ ≤ (8*M*a)*‖x-p‖+4*a*(2*M+1/(4*q))*‖x-p‖ :=
+      add_le_add (mul_le_mul_of_nonneg_right (by simpa only [norm_sub_rev] using hwd)
+        (norm_nonneg _)) ht
+    _ = a*(16*M+1/q)*‖x-p‖ := by field_simp; ring
+
+theorem projection_injective_on_support {E : Set (Euc n)} {v : Euc n}
+    (hchord : ∀ p∈perimeterSupport E ∩ Metric.ball 0 1,
+      ∀ x∈perimeterSupport E ∩ Metric.ball 0 1,
+        |inner ℝ v (x-p)|≤(1/2)*‖x-p‖) (hv : ‖v‖=1) :
+    Set.InjOn (fun x ↦ x-inner ℝ v x•v) (perimeterSupport E ∩ Metric.ball 0 1) := by
+  intro x hx p hp he
+  have hh := hchord p hp x hx
+  have hex : x-p=inner ℝ v (x-p)•v := by
+    rw [inner_sub_right,sub_smul]
+    change x-inner ℝ v x•v=p-inner ℝ v p•v at he
+    calc
+      x-p = (x-inner ℝ v x•v)-(p-inner ℝ v p•v) +
+        (inner ℝ v x•v-inner ℝ v p•v) := by abel
+      _ = inner ℝ v x•v-inner ℝ v p•v := by rw [he,sub_self,zero_add]
+  have hn : ‖x-p‖=|inner ℝ v (x-p)| := by
+    calc
+      ‖x-p‖ = ‖inner ℝ v (x-p)•v‖ := congrArg norm hex
+      _ = _ := by rw [norm_smul,hv,mul_one,Real.norm_eq_abs]
+  rw [← hn] at hh
+  have hz : ‖x-p‖=0 := by linarith [norm_nonneg (x-p)]
+  exact sub_eq_zero.mp (norm_eq_zero.mp hz)
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/NonlinearDerivativeGain.lean b/lean/OAI/Geometry/Perimeter/Regularity/NonlinearDerivativeGain.lean
new file mode 100644
index 0000000000000000000000000000000000000000..a6f3e4d0014e4d420a27d557e8f2802eb0684419
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/NonlinearDerivativeGain.lean
@@ -0,0 +1,154 @@
+import OAI.Geometry.Perimeter.Regularity.EllipticLocalExtension
+import OAI.Geometry.Perimeter.Regularity.EllipticFamilyEstimate
+import OAI.Geometry.Perimeter.Regularity.EllipticSecantBounds
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+ def quotientSteps (r : ℝ) (e : Euc n) (i : ℕ) : ℝ :=
+  (r/(‖e‖+1))*(1/((i:ℝ)+1))
+
+ theorem quotientSteps_pos {r : ℝ} (hr : 0<r) (e : Euc n) (i : ℕ) :
+    0<quotientSteps r e i := by unfold quotientSteps; positivity
+
+ theorem quotientSteps_shift_lt {r : ℝ} (hr : 0<r) (e : Euc n) (i : ℕ) :
+    ‖quotientSteps r e i•e‖<r := by
+  rw [norm_smul,Real.norm_of_nonneg (quotientSteps_pos hr e i).le]
+  have hi : (1:ℝ)/((i:ℝ)+1) ≤ 1 := by
+    apply (div_le_iff₀ (by positivity)).mpr
+    simp
+  have hh : quotientSteps r e i ≤ r/(‖e‖+1) := by
+    unfold quotientSteps
+    simpa only [mul_one] using mul_le_mul_of_nonneg_left hi (by positivity : 0 ≤ r/(‖e‖+1))
+  apply (mul_le_mul_of_nonneg_right hh (norm_nonneg _)).trans_lt
+  have he : r/(‖e‖+1)*‖e‖<r := by
+    rw [div_mul_eq_mul_div,div_lt_iff₀ (by positivity : 0<‖e‖+1)]
+    nlinarith
+  exact he
+
+ theorem quotientSteps_tendsto {r : ℝ} (hr : 0<r) (e : Euc n) :
+    Tendsto (quotientSteps r e) atTop (𝓝[≠] 0) := by
+  apply tendsto_nhdsWithin_iff.mpr
+  constructor
+  · change Tendsto (fun i : ℕ ↦ (r/(‖e‖+1))*(1/((i:ℝ)+1))) atTop (𝓝 0)
+    simpa only [mul_zero] using
+      tendsto_one_div_add_atTop_nhds_zero_nat.const_mul (r/(‖e‖+1))
+  · exact Eventually.of_forall (fun i ↦ (quotientSteps_pos hr e i).ne')
+
+ theorem nonlinear_directional_gain {θ δ K : ℝ}
+    (hθ : 0<θ) (hθ1 : θ<1) (hK : 1 ≤ K)
+    (hc : LinearContractionProperty n θ δ K)
+    {G : Euc n → Euc n} (hG : ContDiff ℝ 2 G)
+    {f : Euc n → ℝ} (hf : ContDiff ℝ 1 f) (hfc : HasCompactSupport f)
+    {U : Set (Euc n)} (hU : IsOpen U) {p : Euc n} (hp : p∈U)
+    (heq : WeakFluxEquation (fun x ↦ G (gradient f x)) U)
+    (hmod : GeometricHolderAt θ (gradient f) p)
+    (hnear : ‖ContinuousLinearMap.id ℝ (Euc n)-fderiv ℝ G (gradient f p)‖ < min δ (1/2))
+    (e : Euc n) :
+    ∃ r : ℝ, 0<r ∧ Metric.ball p r⊆U ∧
+      ContDiffOn ℝ 1 (fun x ↦ fderiv ℝ f x e) (Metric.ball p r) ∧
+      GeometricHolderAt θ (gradient (fun x ↦ fderiv ℝ f x e)) p := by
+  have hG1 : ContDiff ℝ 1 G := hG.of_le (by norm_num)
+  have hDG : ContDiff ℝ 1 (fderiv ℝ G) := hG.fderiv_right (by norm_num)
+  obtain ⟨ρ,L,hρ,hL,hLip⟩ := contDiff_local_lipschitz hDG (gradient f p)
+  have hcn : ContinuousAt (fun z ↦ ‖ContinuousLinearMap.id ℝ (Euc n)-fderiv ℝ G z‖)
+      (gradient f p) := (continuousAt_const.sub hDG.continuous.continuousAt).norm
+  obtain ⟨τ,hτ,hτn⟩ := Metric.eventually_nhds_iff.mp (hcn.eventually (gt_mem_nhds hnear))
+  let κ := min ρ τ
+  have hκ : 0<κ := lt_min hρ hτ
+  have hgc := gradient_continuous_C1 hf
+  obtain ⟨s,hs,hsg⟩ := Metric.continuousAt_iff.mp hgc.continuousAt κ hκ
+  obtain ⟨S,hS,hSU⟩ := Metric.isOpen_iff.mp hU p hp
+  let a := 1/(32*K^2)
+  have ha : 0<a := by dsimp [a]; positivity
+  let ε := a/(2*(L+1))
+  have hε : 0<ε := by dsimp [ε]; positivity
+  obtain ⟨b,hb,hbs,hbm⟩ := hmod.shrink_amplitude hθ hε (lt_min hs hS)
+  let r := b/8
+  have hr : 0<r := by dsimp [r]; positivity
+  have hrb : 5*r<b := by dsimp [r]; linarith
+  have hbS : b ≤ S := hbs.trans (min_le_right _ _)
+  have hbs' : b ≤ s := hbs.trans (min_le_left _ _)
+  have him (x : Euc n) (hx : x∈Metric.ball p b) : gradient f x∈Metric.ball (gradient f p) κ :=
+    Metric.mem_ball.mpr (hsg ((Metric.mem_ball.mp hx).trans_le hbs'))
+  have hshift (i : ℕ) (R : ℝ) (x : Euc n) (hx : x∈Metric.ball p R) :
+      x+quotientSteps r e i•e∈Metric.ball p (R+r) := by
+    rw [Metric.mem_ball,dist_eq_norm]
+    have he : x+quotientSteps r e i•e-p=(x-p)+quotientSteps r e i•e := by abel
+    rw [he]
+    have hh := norm_add_le (x-p) (quotientSteps r e i•e)
+    have hd := Metric.mem_ball.mp hx
+    rw [dist_eq_norm] at hd
+    linarith [quotientSteps_shift_lt hr e i]
+  let A : ℕ → Euc n → Euc n →L[ℝ] Euc n := fun i x ↦
+    derivativeSecant G (gradient f x) (gradient f (x+quotientSteps r e i•e))
+  let v : ℕ → Euc n → ℝ := fun i ↦ differenceQuotient f e (quotientSteps r e i)
+  have hA (i : ℕ) : Continuous (A i) := (derivativeSecant_continuous hG1).comp
+    (hgc.prodMk (hgc.comp (continuous_id.add continuous_const)))
+  have hv (i : ℕ) : ContDiff ℝ 1 (v i) := differenceQuotient_contDiff hf e _
+  have h4b : Metric.ball p (4*r)⊆Metric.ball p b := Metric.ball_subset_ball (by linarith)
+  have h4U : Metric.ball p (4*r)⊆U := h4b.trans ((Metric.ball_subset_ball hbS).trans hSU)
+  have hE (i : ℕ) : WeakLinearEquation (A i) (v i) (fun _ ↦ 0) (Metric.ball p (4*r)) :=
+    weak_nonlinear_difference hG1 hf heq h4U e _ (fun x hx ↦
+      hSU (Metric.ball_subset_ball (by linarith : 4*r+r ≤ S) (hshift i (4*r) x hx)))
+  have hN (i : ℕ) (x : Euc n) (hx : x∈Metric.ball p (4*r)) :
+      ‖ContinuousLinearMap.id ℝ (Euc n)-A i x‖ ≤ min δ (1/2) := by
+    apply derivativeSecant_near hG1
+    intro t ht
+    have hh := segment_in_ball (him x (h4b hx))
+      (him _ (Metric.ball_subset_ball (by linarith : 4*r+r ≤ b) (hshift i (4*r) x hx))) ht
+    exact (hτn ((Metric.mem_ball.mp hh).trans_le (min_le_right _ _))).le
+  have hosc (i : ℕ) (x : Euc n) (hx : x∈Metric.closedBall p r) (j : ℕ)
+      (y : Euc n) (hy : y∈Metric.ball x (θ^j*r)) :
+      ‖A i x-A i y‖ ≤ a*(1/2)^j ∧ ‖(0:Euc n)-0‖ ≤ (0:ℝ)*(1/2)^j := by
+    have hj : θ^j ≤ 1 := pow_le_one₀ hθ.le hθ1.le
+    have hx2 : x∈Metric.ball p (2*r) := Metric.mem_ball.mpr
+      ((Metric.mem_closedBall.mp hx).trans_lt (by linarith))
+    have hy2 : y∈Metric.ball p (2*r) := by
+      have hd := dist_triangle y x p
+      have hdx := Metric.mem_closedBall.mp hx
+      have hdy := Metric.mem_ball.mp hy
+      have hrj := mul_le_mul_of_nonneg_right hj hr.le
+      rw [Metric.mem_ball]
+      linarith
+    have h2b : Metric.ball p (2*r)⊆Metric.ball p b := Metric.ball_subset_ball (by linarith)
+    have hx' : x+quotientSteps r e i•e∈Metric.ball p b :=
+      Metric.ball_subset_ball (by linarith : 2*r+r ≤ b) (hshift i (2*r) x hx2)
+    have hy' : y+quotientSteps r e i•e∈Metric.ball p b :=
+      Metric.ball_subset_ball (by linarith : 2*r+r ≤ b) (hshift i (2*r) y hy2)
+    have hd : dist x y ≤ θ^j*b := by
+      rw [dist_comm]
+      exact (Metric.mem_ball.mp hy).le.trans (mul_le_mul_of_nonneg_left (by dsimp [r]; linarith) (pow_nonneg hθ.le _))
+    have hg₁ := hbm j x (h2b hx2) y (h2b hy2) hd
+    have hg₂ := hbm j _ hx' _ hy' (by simpa only [dist_add_right] using hd)
+    rw [dist_eq_norm] at hg₁ hg₂
+    have he := derivativeSecant_lipschitz_ball hG1 hL
+      (Metric.ball_subset_ball (min_le_left _ _) (him x (h2b hx2)))
+      (Metric.ball_subset_ball (min_le_left _ _) (him _ hx'))
+      (Metric.ball_subset_ball (min_le_left _ _) (him y (h2b hy2)))
+      (Metric.ball_subset_ball (min_le_left _ _) (him _ hy')) hLip
+    have hεeq : 2*(L+1)*ε=a := by dsimp [ε]; field_simp
+    constructor
+    · apply he.trans
+      have hh := mul_le_mul_of_nonneg_left (add_le_add hg₁ hg₂) hL
+      have hpos : 0 ≤ (1/2:ℝ)^j*ε := mul_nonneg (by positivity) hε.le
+      nlinarith
+    · simp
+  obtain ⟨B₀,hB₀,hbound⟩ := compact_C1_gradient_bound hf hfc
+  obtain ⟨hg,hgmod⟩ := linear_family_C1_limit hθ hθ1 hK hc ha.le (by norm_num : (0:ℝ) ≤ 0)
+    hr (by dsimp [a]; field_simp; norm_num) hA hv (fun _ ↦ continuous_const) hE hN hosc
+    (fun i x ↦ differenceQuotient_bound (hf.differentiable (by simp)) hB₀ hbound e _ x)
+    (by intro i x; simp : ∀ i x, ‖(fun _ : Euc n ↦ (0:Euc n)) x‖ ≤ 0)
+    (fun x _ ↦ (differenceQuotient_tendsto (hf.differentiable (by simp)) e x).comp (quotientSteps_tendsto hr e))
+  exact ⟨r,hr,(Metric.ball_subset_ball (by linarith : r ≤ 4*r)).trans h4U,hg,hgmod⟩
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/NonlinearLocalGain.lean b/lean/OAI/Geometry/Perimeter/Regularity/NonlinearLocalGain.lean
new file mode 100644
index 0000000000000000000000000000000000000000..36837ebecfe05d6835f335852805ea98f7b8f102
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/NonlinearLocalGain.lean
@@ -0,0 +1,88 @@
+import OAI.Geometry.Perimeter.Regularity.NonlinearDerivativeGain
+import OAI.Geometry.Perimeter.Regularity.EllipticLocality
+import OAI.Geometry.Perimeter.Regularity.WeakDerivative
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+ theorem nonlinear_C2_gain {θ δ K : ℝ}
+    (hθ : 0<θ) (hθ' : θ ≤ 1/2) (hK : 1 ≤ K)
+    (hc : LinearContractionProperty n θ δ K)
+    {G : Euc n → Euc n} (hG : ContDiff ℝ 2 G)
+    {f : Euc n → ℝ} {U : Set (Euc n)} (hU : IsOpen U)
+    (hf : GeoC θ 1 f U) (heq : WeakFluxEquation (fun x ↦ G (gradient f x)) U)
+    (hnear : ∀ p∈U, ‖ContinuousLinearMap.id ℝ (Euc n)-fderiv ℝ G (gradient f p)‖ < min δ (1/2)) :
+    ContDiffOn ℝ 2 f U := by
+  have hθ1 : θ<1 := by linarith
+  change ContDiffOn ℝ (1+1) f U
+  rw [contDiffOn_succ_iff_fderiv_of_isOpen hU]
+  refine ⟨hf.1.differentiableOn one_ne_zero,by simp,contDiffOn_clm_apply.mpr (fun e p hp ↦ ?_)⟩
+  obtain ⟨a,ha,haU,g,hg,hgc,heg⟩ := local_compact_extension hU hp hf.1
+  let W := Metric.ball p (2*a)
+  have hW : IsOpen W := Metric.isOpen_ball
+  have hpW : p∈W := Metric.mem_ball_self (by positivity)
+  have hWU : W⊆U := Metric.ball_subset_closedBall.trans ((Metric.closedBall_subset_closedBall (by linarith)).trans haU)
+  have heGrad : EqOn (gradient g) (gradient f) W := gradient_eqOn hW heg
+  have he := (heq.mono hWU).congr (fun x hx ↦ congrArg G (heGrad hx).symm)
+  have hm := ((hf.gradient hθ.le hU).2 p hp).congr hθ.le
+    (Filter.eventually_of_mem (hW.mem_nhds hpW) (fun x hx ↦ (heGrad hx).symm))
+  obtain ⟨r,hr,hrW,hdr,hdm⟩ := nonlinear_directional_gain hθ hθ1 hK hc hG hg hgc hW hpW he hm
+    (by rw [heGrad hpW]; exact hnear p hp) e
+  have hdEq : EqOn (fun x ↦ fderiv ℝ g x e) (fun x ↦ fderiv ℝ f x e) W :=
+    fun x hx ↦ congrArg (fun L : Euc n →L[ℝ] ℝ ↦ L e) (fderiv_eqOn hW heg hx)
+  exact ((hdr.congr (fun x hx ↦ (hdEq (hrW hx)).symm)).contDiffAt
+    (Metric.ball_mem_nhds p hr)).contDiffWithinAt
+
+ theorem weak_nonlinear_derivative {G : Euc n → Euc n} (hG : ContDiff ℝ 1 G)
+    {f : Euc n → ℝ} (hf : ContDiff ℝ 2 f) {U : Set (Euc n)}
+    (h : WeakFluxEquation (fun x ↦ G (gradient f x)) U) (e : Euc n) :
+    WeakLinearEquation (fun x ↦ fderiv ℝ G (gradient f x))
+      (fun x ↦ fderiv ℝ f x e) (fun _ ↦ 0) U := by
+  have hdf : ContDiff ℝ 1 (fderiv ℝ f) := hf.fderiv_right (by norm_num)
+  have hg : ContDiff ℝ 1 (gradient f) :=
+    (InnerProductSpace.toDual ℝ (Euc n)).symm.toContinuousLinearMap.contDiff.comp hdf
+  have hd := h.derivative (hG.comp hg) e
+  have he : EqOn (fun x ↦ fderiv ℝ (fun y ↦ G (gradient f y)) x e)
+      (fun x ↦ fderiv ℝ G (gradient f x) (gradient (fun y ↦ fderiv ℝ f y e) x)) U := by
+    intro x hx
+    dsimp only
+    rw [gradient_direction_swap hf]
+    exact congrArg (fun L : Euc n →L[ℝ] Euc n ↦ L e)
+      (((hG.differentiable (by simp) (gradient f x)).hasFDerivAt.comp x
+        (hg.differentiable (by simp) x).hasFDerivAt).fderiv)
+  intro φ
+  simpa only [inner_zero_left,integral_zero] using hd.congr he φ
+
+ theorem weak_nonlinear_derivative_local {G : Euc n → Euc n} (hG : ContDiff ℝ 1 G)
+    {f : Euc n → ℝ} {U : Set (Euc n)} (hU : IsOpen U) {p : Euc n} (hp : p∈U)
+    (hf : ContDiffOn ℝ 2 f U)
+    (h : WeakFluxEquation (fun x ↦ G (gradient f x)) U) (e : Euc n) :
+    ∃ W : Set (Euc n), IsOpen W ∧ p∈W ∧ W⊆U ∧
+      WeakLinearEquation (fun x ↦ fderiv ℝ G (gradient f x))
+        (fun x ↦ fderiv ℝ f x e) (fun _ ↦ 0) W := by
+  obtain ⟨a,ha,haU,g,hg,hgc,heg⟩ := local_compact_extension hU hp hf
+  let W := Metric.ball p (2*a)
+  have hW : IsOpen W := Metric.isOpen_ball
+  have hpW : p∈W := Metric.mem_ball_self (by positivity)
+  have hWU : W⊆U := Metric.ball_subset_closedBall.trans ((Metric.closedBall_subset_closedBall (by linarith)).trans haU)
+  have heGrad : EqOn (gradient g) (gradient f) W := gradient_eqOn hW heg
+  have he := (h.mono hWU).congr (fun x hx ↦ congrArg G (heGrad hx).symm)
+  have hd := weak_nonlinear_derivative hG hg he e
+  refine ⟨W,hW,hpW,hWU,hd.congr hW ?_ ?_ (fun _ _ ↦ rfl)⟩
+  · intro x hx
+    dsimp only
+    rw [heGrad hx]
+  · intro x hx
+    dsimp only
+    rw [fderiv_eqOn hW heg hx]
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/NonlinearSmooth.lean b/lean/OAI/Geometry/Perimeter/Regularity/NonlinearSmooth.lean
new file mode 100644
index 0000000000000000000000000000000000000000..e05e631e07134575cb3c6ede643fa296e14dc6c1
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/NonlinearSmooth.lean
@@ -0,0 +1,53 @@
+import OAI.Geometry.Perimeter.Regularity.LinearSchauder
+import OAI.Geometry.Perimeter.Regularity.NonlinearLocalGain
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+ theorem nonlinear_geoC_all {θ δ K : ℝ}
+    (hθ : 0<θ) (hθ' : θ ≤ 1/2) (hK : 1 ≤ K)
+    (hc : LinearContractionProperty n θ δ K)
+    {G : Euc n → Euc n} (hG : ContDiff ℝ ∞ G)
+    {f : Euc n → ℝ} {U : Set (Euc n)} (hU : IsOpen U)
+    (hf : GeoC θ 1 f U) (heq : WeakFluxEquation (fun x ↦ G (gradient f x)) U)
+    (hnear : ∀ p∈U, ‖ContinuousLinearMap.id ℝ (Euc n)-fderiv ℝ G (gradient f p)‖ < min δ (1/2)) :
+    ∀ k : ℕ, GeoC θ (k+1) f U := by
+  have hf2 := nonlinear_C2_gain hθ hθ' hK hc (hG.of_le (by exact ENat.natCast_le_of_coe_top_le_withTop le_rfl 2)) hU hf heq hnear
+  have hDG : ContDiff ℝ ∞ (fderiv ℝ G) := hG.fderiv_right (by simp)
+  intro k
+  induction k with
+  | zero => exact hf
+  | succ k ih =>
+    have hgrad := ih.gradient hθ.le hU
+    have hA := geoC_comp k hθ.le hU hgrad hDG
+    refine ⟨hf.1,hf.2.1,fun e ↦ ?_⟩
+    have hdir : ContDiffOn ℝ 1 (fun x ↦ fderiv ℝ f x e) U :=
+      (hf2.fderiv_of_isOpen hU (by norm_num)).clm_apply contDiffOn_const
+    apply GeoC.local
+    intro p hp
+    obtain ⟨W,hW,hpW,hWU,hd⟩ := weak_nonlinear_derivative_local (hG.of_le (by simp)) hU hp hf2 heq e
+    exact ⟨W,hW,hpW,linear_schauder k hθ hθ' hK hc hW (hA.mono hWU)
+      (GeoC.const θ k 0 W) (hdir.mono hWU) hd (fun x hx ↦ hnear x (hWU hx))⟩
+
+ theorem nonlinear_smooth {θ δ K : ℝ}
+    (hθ : 0<θ) (hθ' : θ ≤ 1/2) (hK : 1 ≤ K)
+    (hc : LinearContractionProperty n θ δ K)
+    {G : Euc n → Euc n} (hG : ContDiff ℝ ∞ G)
+    {f : Euc n → ℝ} {U : Set (Euc n)} (hU : IsOpen U)
+    (hf : GeoC θ 1 f U) (heq : WeakFluxEquation (fun x ↦ G (gradient f x)) U)
+    (hnear : ∀ p∈U, ‖ContinuousLinearMap.id ℝ (Euc n)-fderiv ℝ G (gradient f p)‖ < min δ (1/2)) :
+    ContDiffOn ℝ ∞ f U := by
+  rw [contDiffOn_infty]
+  intro k
+  exact ((nonlinear_geoC_all hθ hθ' hK hc hG hU hf heq hnear k).mono_order).contDiff hU
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/NormalAdjustment.lean b/lean/OAI/Geometry/Perimeter/Regularity/NormalAdjustment.lean
new file mode 100644
index 0000000000000000000000000000000000000000..6ac9cadde1437b8058d617500e813fd18c6b1c8a
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/NormalAdjustment.lean
@@ -0,0 +1,81 @@
+import OAI.Geometry.Perimeter.Regularity.NormalCoordinates
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+open RPI.Boundary
+variable {n : ℕ}
+
+theorem normalPlane_finrank {v : Euc n} (hv : ‖v‖=1) :
+    Module.finrank ℝ (normalPlane v)=n-1 := by
+  have hn : v≠0 := by intro hzero; simp [hzero] at hv
+  have h := (ℝ ∙ v).finrank_add_finrank_orthogonal
+  rw [finrank_span_singleton hn,finrank_euclideanSpace,Fintype.card_fin] at h
+  dsimp [normalPlane]
+  omega
+
+def normalPlaneEuclideanChart {v : Euc n} (hv : ‖v‖=1) : Euc (n-1) ≃ₗᵢ[ℝ] normalPlane v :=
+  ((stdOrthonormalBasis ℝ (normalPlane v)).reindex (finCongr (normalPlane_finrank hv))).repr.symm
+
+theorem exists_normal_rotation {v w : Euc n} (h : ‖v‖=‖w‖) :
+    ∃ e : Euc n ≃ₗᵢ[ℝ] Euc n, e v=w :=
+  ⟨(ℝ ∙ (v-w))ᗮ.reflection,Submodule.reflection_sub h⟩
+
+theorem dual_of_normal_linear {v : Euc n} {L : Euc n →L[ℝ] ℝ} (hL : L v=0) :
+    let u := (InnerProductSpace.toDual ℝ (Euc n)).symm L
+    ‖u‖=‖L‖ ∧ inner ℝ v u=0 ∧ ∀ x, inner ℝ u x=L x := by
+  constructor
+  · exact (InnerProductSpace.toDual ℝ (Euc n)).symm.norm_map L
+  constructor
+  · rw [real_inner_comm,InnerProductSpace.toDual_symm_apply,hL]
+  · intro x; exact InnerProductSpace.toDual_symm_apply
+
+theorem exists_adjusted_unit_normal {v : Euc n} (hv : ‖v‖=1)
+    (L : Euc n →L[ℝ] ℝ) (hLv : L v=0) {a : ℝ} (ha : 0≤a) :
+    ∃ w : Euc n, ‖w‖=1 ∧ ‖w-v‖≤2*a*‖L‖ ∧
+      ∀ x, |inner ℝ w x|≤|inner ℝ v x-a*L x| := by
+  let u := (InnerProductSpace.toDual ℝ (Euc n)).symm L
+  have hu := dual_of_normal_linear hLv
+  change ‖u‖=‖L‖ ∧ inner ℝ v u=0 ∧ ∀ x, inner ℝ u x=L x at hu
+  let z := v-a•u
+  have hzsq : ‖z‖^2=1+a^2*‖u‖^2 := by
+    dsimp [z]
+    rw [norm_sub_sq_real,hv,real_inner_smul_right,hu.2.1,mul_zero,mul_zero,sub_zero,
+      norm_smul,Real.norm_of_nonneg ha]
+    ring
+  have hzn : 1≤‖z‖ := by nlinarith [norm_nonneg z,sq_nonneg a,sq_nonneg ‖u‖]
+  have hz : 0<‖z‖ := lt_of_lt_of_le zero_lt_one hzn
+  let w := ‖z‖⁻¹•z
+  have hwn : ‖w‖=1 := by rw [norm_smul,Real.norm_of_nonneg (inv_nonneg.mpr hz.le),inv_mul_cancel₀ hz.ne']
+  have hdist : ‖w-z‖=‖z‖-1 := by
+    have he : w-z=(‖z‖⁻¹-1)•z := by dsimp [w]; rw [sub_smul,one_smul]
+    rw [he,norm_smul,Real.norm_eq_abs,abs_of_nonpos (sub_nonpos.mpr ((inv_le_one₀ hz).mpr hzn))]
+    field_simp
+    ring
+  have hzv : ‖z-v‖=a*‖L‖ := by
+    have he : z-v=-(a•u) := by dsimp [z]; abel
+    rw [he,norm_neg,norm_smul,Real.norm_of_nonneg ha,hu.1]
+  refine ⟨w,hwn,?_,?_⟩
+  · have hrev := norm_sub_norm_le z v
+    rw [hv,hzv] at hrev
+    have htri := norm_sub_le w z
+    have hh := norm_sub_le w v
+    calc
+      ‖w-v‖ ≤ ‖w-z‖+‖z-v‖ := norm_sub_le_norm_sub_add_norm_sub _ _ _
+      _ = ‖z‖-1+a*‖L‖ := by rw [hdist,hzv]
+      _ ≤ 2*a*‖L‖ := by linarith
+  · intro x
+    have he : inner ℝ w x=‖z‖⁻¹*(inner ℝ v x-a*L x) := by
+      dsimp [w,z]
+      rw [real_inner_smul_left,inner_sub_left,real_inner_smul_left,hu.2.2]
+    rw [he,abs_mul,abs_of_nonneg (inv_nonneg.mpr hz.le)]
+    exact mul_le_of_le_one_left (abs_nonneg _) ((inv_le_one₀ hz).mpr hzn)
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/NormalCoordinates.lean b/lean/OAI/Geometry/Perimeter/Regularity/NormalCoordinates.lean
new file mode 100644
index 0000000000000000000000000000000000000000..2c8f50da14bcc7febf8954f1052fcbf56da0b25f
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/NormalCoordinates.lean
@@ -0,0 +1,85 @@
+import OAI.Geometry.Perimeter.Regularity.HasC1TraceAt
+
+namespace OAI
+
+noncomputable section
+open Set MeasureTheory
+open scoped InnerProductSpace ENNReal
+namespace RPISingularHarnack
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V]
+
+abbrev normalPlane (v : V) := (ℝ ∙ v)ᗮ
+
+def normalCoordinateIsometry (v : V) (hv : ‖v‖ = 1) :
+    WithLp 2 (normalPlane v × ℝ) →ₗᵢ[ℝ] V where
+  toLinearMap :=
+    { toFun := fun x ↦ (x.fst : V) + x.snd • v
+      map_add' := by intro x y; simp [add_smul]; abel
+      map_smul' := by intro t x; simp [smul_add,smul_smul] }
+  norm_map' := by
+    intro x
+    have ho : inner ℝ (x.fst : V) v = 0 :=
+      Submodule.mem_orthogonal_singleton_iff_inner_left.mp x.fst.property
+    apply (sq_eq_sq₀ (norm_nonneg _) (norm_nonneg _)).mp
+    change ‖(x.fst : V)+x.snd•v‖^2 = ‖x‖^2
+    rw [norm_add_sq_real, real_inner_smul_right, ho, mul_zero, mul_zero, add_zero,
+      norm_smul, hv, mul_one, WithLp.prod_norm_sq_eq_of_L2]
+    rfl
+
+theorem normalCoordinateIsometry_surjective (v : V) (hv : ‖v‖ = 1) :
+    Function.Surjective (normalCoordinateIsometry v hv) := by
+  have hself : inner ℝ v v = 1 := by rw [real_inner_self_eq_norm_sq,hv]; norm_num
+  intro x
+  let y : normalPlane v := ⟨x-inner ℝ v x•v, by
+    rw [Submodule.mem_orthogonal_singleton_iff_inner_right,inner_sub_right,
+      real_inner_smul_right,hself,mul_one,sub_self]⟩
+  refine ⟨WithLp.toLp 2 (y,inner ℝ v x), ?_⟩
+  change x-inner ℝ v x•v+inner ℝ v x•v=x
+  exact sub_add_cancel _ _
+
+def normalCoordinateEquiv (v : V) (hv : ‖v‖ = 1) :
+    WithLp 2 (normalPlane v × ℝ) ≃ₗᵢ[ℝ] V :=
+  LinearIsometryEquiv.ofSurjective (normalCoordinateIsometry v hv)
+    (normalCoordinateIsometry_surjective v hv)
+
+@[simp] theorem normalCoordinateEquiv_apply (v : V) (hv : ‖v‖ = 1)
+    (x : WithLp 2 (normalPlane v × ℝ)) :
+    normalCoordinateEquiv v hv x = (x.fst : V)+x.snd•v := rfl
+
+variable [MeasurableSpace V] [BorelSpace V]
+
+def normalCoordinates (v : V) (hv : ‖v‖ = 1) : (normalPlane v × ℝ) ≃ᵐ V :=
+  (MeasurableEquiv.toLp 2 (normalPlane v × ℝ)).trans
+    (normalCoordinateEquiv v hv).toMeasurableEquiv
+
+@[simp] theorem normalCoordinates_apply (v : V) (hv : ‖v‖ = 1)
+    (x : normalPlane v × ℝ) : normalCoordinates v hv x = (x.1 : V)+x.2•v := rfl
+
+variable [FiniteDimensional ℝ V]
+
+theorem normalCoordinates_measurePreserving (v : V) (hv : ‖v‖ = 1) :
+    MeasurePreserving (normalCoordinates v hv) volume volume :=
+  (normalCoordinateEquiv v hv).measurePreserving.comp
+    (WithLp.volume_preserving_toLp (normalPlane v) ℝ)
+
+theorem integral_normalCoordinates (v : V) (hv : ‖v‖ = 1) (f : V → ℝ)
+    (hf : Integrable f volume) :
+    (∫ x, f x) = ∫ y : normalPlane v, ∫ s : ℝ, f ((y:V)+s•v) := by
+  have hm := normalCoordinates_measurePreserving v hv
+  have he := hm.integral_comp (normalCoordinates v hv).measurableEmbedding f
+  rw [← he]
+  apply integral_prod
+  exact (hm.integrable_comp hf.aestronglyMeasurable).mpr hf
+
+theorem lintegral_normalCoordinates (v : V) (hv : ‖v‖ = 1) (f : V → ℝ≥0∞)
+    (hf : Measurable f) :
+    (∫⁻ x, f x) = ∫⁻ y : normalPlane v, ∫⁻ s : ℝ, f ((y:V)+s•v) := by
+  rw [← (normalCoordinates_measurePreserving v hv).lintegral_comp hf]
+  apply lintegral_prod
+  exact (hf.comp (normalCoordinates v hv).measurable).aemeasurable
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/NormalFluxError.lean b/lean/OAI/Geometry/Perimeter/Regularity/NormalFluxError.lean
new file mode 100644
index 0000000000000000000000000000000000000000..b66135f59586c6f282323095f8eddf303776a053
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/NormalFluxError.lean
@@ -0,0 +1,70 @@
+import OAI.Geometry.Perimeter.Regularity.FlatExcess
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff Convolution InnerProductSpace
+namespace RPISingularHarnack
+open RPI.Boundary ContinuousLinearMap
+variable {n : ℕ}
+
+theorem normal_flux_error_le_tilt (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    {v : Euc n} (hv : ‖v‖ = 1) {R : ℝ} (hR : 0 < R)
+    {ψ : Euc n → ℝ} (hψ : Measurable ψ) {M : ℝ} (hM : ∀ x, ‖ψ x‖ ≤ M) :
+    |(∫ x in Metric.ball 0 R, ψ x ∂perimeterMeasure E) -
+      (∫ x in Metric.ball 0 R, ψ x*inner ℝ (approximateNormal E x) v ∂perimeterMeasure E)| ≤
+      M * orientedTilt E v (Metric.ball 0 R) / 2 := by
+  let μ := (perimeterMeasure E).restrict (Metric.ball 0 R)
+  have : IsFiniteMeasure μ := ⟨by
+    rw [Measure.restrict_apply_univ,perimeterMeasure_open E Metric.isOpen_ball]
+    exact hmin.2.1 0 R hR⟩
+  have hM0 : 0 ≤ M := (norm_nonneg (ψ 0)).trans (hM 0)
+  have hiψ : Integrable ψ μ := Integrable.of_bound hψ.aestronglyMeasurable M (ae_of_all _ hM)
+  have hN : ∀ x, |inner ℝ (approximateNormal E x) v| ≤ 1 := by
+    intro x
+    exact (abs_real_inner_le_norm _ _).trans (by rw [hv,mul_one]; exact approximateNormal_norm_le_one E x)
+  have hif : Integrable (fun x ↦ ψ x*inner ℝ (approximateNormal E x) v) μ := by
+    apply Integrable.of_bound (hψ.mul
+      ((measurable_approximateNormal hmin.1).inner measurable_const)).aestronglyMeasurable M
+    apply ae_of_all _
+    intro x
+    change ‖ψ x*inner ℝ (approximateNormal E x) v‖ ≤ M
+    rw [norm_mul]
+    exact (mul_le_mul (hM x) (hN x) (abs_nonneg _) hM0).trans_eq (mul_one _)
+  have htilt : Integrable (fun x ↦ ‖approximateNormal E x-v‖^2) μ := by
+    apply Integrable.of_bound
+      (((measurable_approximateNormal hmin.1).sub measurable_const).norm.pow_const 2).aestronglyMeasurable
+      4
+    apply ae_of_all _
+    intro x
+    rw [Real.norm_eq_abs,abs_of_nonneg (sq_nonneg _)]
+    have h := (norm_sub_le (approximateNormal E x) v).trans
+      (show ‖approximateNormal E x‖+‖v‖ ≤ 2 by rw [hv]; linarith [approximateNormal_norm_le_one E x])
+    change ‖approximateNormal E x-v‖^2 ≤ 4
+    nlinarith [norm_nonneg (approximateNormal E x-v)]
+  have hdiff : ∀ᵐ x ∂μ, |ψ x-ψ x*inner ℝ (approximateNormal E x) v| ≤
+      (M/2)*‖approximateNormal E x-v‖^2 := by
+    filter_upwards [ae_restrict_of_ae (minimizing_halfspaceBlowup_ae hn hmin)] with x hx
+    have hid : 1-inner ℝ (approximateNormal E x) v = ‖approximateNormal E x-v‖^2/2 := by
+      rw [norm_sub_sq_real,hx.1,hv]
+      ring
+    rw [show ψ x-ψ x*inner ℝ (approximateNormal E x) v =
+      ψ x*(1-inner ℝ (approximateNormal E x) v) by ring,
+      abs_mul,hid,abs_of_nonneg (show 0 ≤ ‖approximateNormal E x-v‖^2/2 by positivity)]
+    have h := mul_le_mul_of_nonneg_right (hM x) (by positivity : 0 ≤ ‖approximateNormal E x-v‖^2/2)
+    simpa only [Real.norm_eq_abs,mul_div_assoc,div_mul_eq_mul_div] using h
+  change |(∫ x, ψ x ∂μ)-(∫ x, ψ x*inner ℝ (approximateNormal E x) v ∂μ)| ≤ _
+  rw [← integral_sub hiψ hif]
+  calc
+    _ ≤ ∫ x, |ψ x-ψ x*inner ℝ (approximateNormal E x) v| ∂μ := abs_integral_le_integral_abs
+    _ ≤ ∫ x, (M/2)*‖approximateNormal E x-v‖^2 ∂μ :=
+      integral_mono_ae (hiψ.sub hif).abs (htilt.const_mul _) hdiff
+    _ = _ := by rw [integral_const_mul]; dsimp only [orientedTilt,μ]; ring
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/NormalPhaseMollification.lean b/lean/OAI/Geometry/Perimeter/Regularity/NormalPhaseMollification.lean
new file mode 100644
index 0000000000000000000000000000000000000000..be030f758fb8859e7fabc08ffe9bf894c9acc170
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/NormalPhaseMollification.lean
@@ -0,0 +1,177 @@
+import OAI.Geometry.Perimeter.Regularity.FlatExcess
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff Convolution
+namespace RPISingularHarnack
+open RPI.Boundary ContinuousLinearMap
+
+def phaseMollification {n : ℕ} (E : Set (Euc n)) (k : Euc n → ℝ) : Euc n → ℝ :=
+  E.indicator (fun _ ↦ (1:ℝ)) ⋆[lsmul ℝ ℝ, volume] k
+
+theorem phaseMollification_contDiff {n : ℕ} {E : Set (Euc n)}
+    (hE : MeasurableSet E) {k : Euc n → ℝ} (hk : ContDiff ℝ ∞ k)
+    (hc : HasCompactSupport k) : ContDiff ℝ ∞ (phaseMollification E k) :=
+  hc.contDiff_convolution_right _ ((locallyIntegrable_const (1:ℝ)).indicator hE) hk
+
+theorem phaseMollification_eq_setIntegral {n : ℕ} (E : Set (Euc n))
+    (hE : MeasurableSet E) (k : Euc n → ℝ) (x : Euc n) :
+    phaseMollification E k x = ∫ y in E, k (x-y) := by
+  change (∫ y, E.indicator (fun _ ↦ (1:ℝ)) y*k (x-y)) = _
+  have he : (fun y ↦ E.indicator (fun _ ↦ (1:ℝ)) y*k (x-y)) =
+      E.indicator (fun y ↦ k (x-y)) := by
+    funext y
+    by_cases hy : y ∈ E <;> simp [hy]
+  rw [he,integral_indicator hE]
+
+theorem phaseMollification_mem_Icc {n : ℕ} {E : Set (Euc n)}
+    (hE : MeasurableSet E) {k : Euc n → ℝ} (hk : Integrable k volume)
+    (hpos : ∀ x, 0 ≤ k x) (hunit : ∫ x, k x = 1) (x : Euc n) :
+    phaseMollification E k x ∈ Icc (0:ℝ) 1 := by
+  rw [phaseMollification_eq_setIntegral E hE]
+  constructor
+  · exact integral_nonneg fun y ↦ hpos _
+  · calc
+      _ ≤ ∫ y, k (x-y) := setIntegral_le_integral (hk.comp_sub_left x)
+        (ae_of_all _ (fun y ↦ hpos _))
+      _ = 1 := by rw [integral_sub_left_eq_self,hunit]
+
+theorem negative_normal_flux_le_tilt {n : ℕ} (N : Euc n) {v : Euc n}
+    (hv : ‖v‖ = 1) : 2*max (-inner ℝ N v) 0 ≤ ‖N-v‖^2 := by
+  rw [norm_sub_sq_real,hv]
+  rcases le_total (-inner ℝ N v) 0 with h | h
+  · rw [max_eq_right h]
+    nlinarith [sq_nonneg ‖N-v‖,norm_sub_sq_real N v]
+  · rw [max_eq_left h]
+    nlinarith [sq_nonneg ‖N‖]
+
+theorem phaseMollification_fderiv {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    {k : Euc n → ℝ} (hk : ContDiff ℝ ∞ k) (hc : HasCompactSupport k)
+    (x v : Euc n) {R : ℝ} (hR : 0 < R)
+    (hs : tsupport (fun y ↦ k (x-y)) ⊆ Metric.ball 0 R) :
+    fderiv ℝ (phaseMollification E k) x v =
+      -(∫ y in Metric.ball 0 R, k (x-y)*inner ℝ (approximateNormal E y) v
+        ∂perimeterMeasure E) := by
+  let φ : Euc n → ℝ := fun y ↦ k (x-y)
+  have hφ : ContDiff ℝ ∞ φ := hk.comp (contDiff_const.sub contDiff_id)
+  have hcφ : HasCompactSupport φ := hc.comp_homeomorph (Homeomorph.subLeft x)
+  have hd (y : Euc n) : fderiv ℝ φ y v = -fderiv ℝ k (x-y) v := by
+    have h := ((hk.differentiable (by simp) (x-y)).hasFDerivAt.comp y
+      ((hasFDerivAt_const x y).sub (hasFDerivAt_id y))).fderiv
+    simpa only [φ,Function.comp_def,sub_apply,zero_apply,id_apply,
+      zero_sub,ContinuousLinearMap.comp_apply,neg_apply,map_neg] using
+      congrArg (fun L : Euc n →L[ℝ] ℝ ↦ L v) h
+  have hfi : LocallyIntegrable (E.indicator (fun _ ↦ (1:ℝ))) volume :=
+    (locallyIntegrable_const (1:ℝ)).indicator hmin.1
+  have hdiff : fderiv ℝ (phaseMollification E k) x v = -(∫ y in E, fderiv ℝ φ y v) := by
+    rw [phaseMollification,(hc.hasFDerivAt_convolution_right (lsmul ℝ ℝ) hfi
+      (hk.of_le (by simp)) x).fderiv,
+      convolution_precompR_apply (lsmul ℝ ℝ) hfi (hc.fderiv ℝ)
+        (hk.continuous_fderiv (by simp))]
+    change (∫ y, E.indicator (fun _ ↦ (1:ℝ)) y*fderiv ℝ k (x-y) v) = _
+    simp_rw [hd,integral_neg,neg_neg]
+    have he : (fun y ↦ E.indicator (fun _ ↦ (1:ℝ)) y*fderiv ℝ k (x-y) v) =
+        E.indicator (fun y ↦ fderiv ℝ k (x-y) v) := by
+      funext y
+      by_cases hy : y ∈ E <;> simp [hy]
+    rw [he,integral_indicator hmin.1]
+  rw [hdiff,← divergence_scalar_direction (hφ.differentiable (by simp)) v,
+    minimizing_gaussGreen_local hn hmin (Metric.ball 0 R) Metric.isOpen_ball
+      (hmin.2.1 0 R hR).ne (fun y ↦ φ y • v) (hφ.smul contDiff_const)
+      hcφ.smul_right (tsupport_smul_subset_left _ _ |>.trans hs)]
+  congr 1
+  apply integral_congr_ae
+  exact ae_of_all _ fun y ↦ by simp only [φ,real_inner_smul_right]
+
+theorem positive_neg_convolution_le {n : ℕ} (μ : Measure (Euc n))
+    {f k : Euc n → ℝ} (hf : Integrable f μ) (hk : Continuous k)
+    (hc : HasCompactSupport k) (hp : ∀ x, 0 ≤ k x) (x : Euc n) :
+    max (-(f ⋆[lsmul ℝ ℝ, μ] k) x) 0 ≤
+      ((fun y ↦ max (-f y) 0) ⋆[lsmul ℝ ℝ, μ] k) x := by
+  have hq : Integrable (fun y ↦ max (-f y) 0) μ := hf.neg_part
+  have hi := hc.convolutionExists_right (lsmul ℝ ℝ) hf.locallyIntegrable hk x
+  have hj := hc.convolutionExists_right (lsmul ℝ ℝ) hq.locallyIntegrable hk x
+  change max (-(∫ y, f y*k (x-y) ∂μ)) 0 ≤ ∫ y, max (-f y) 0*k (x-y) ∂μ
+  apply max_le
+  · rw [← integral_neg]
+    apply integral_mono hi.neg hj
+    intro y
+    change -(f y*k (x-y)) ≤ max (-f y) 0*k (x-y)
+    simpa only [neg_mul] using mul_le_mul_of_nonneg_right (le_max_left (-f y) 0) (hp _)
+  · exact integral_nonneg fun y ↦ mul_nonneg (le_max_right _ _) (hp _)
+
+theorem phaseMollification_positive_derivative_le_tilt {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E) {v : Euc n}
+    (hv : ‖v‖ = 1) {k : Euc n → ℝ} (hk : ContDiff ℝ ∞ k)
+    (hc : HasCompactSupport k) (hp : ∀ x, 0 ≤ k x) (hu : ∫ x, k x = 1)
+    {R : ℝ} (hR : 0 < R) (c : Euc n) (r : ℝ)
+    (hs : ∀ x ∈ Metric.closedBall c r, tsupport (fun y ↦ k (x-y)) ⊆ Metric.ball 0 R) :
+    (∫ x in Metric.closedBall c r,
+      max (fderiv ℝ (phaseMollification E k) x v) 0) ≤
+      orientedTilt E v (Metric.ball 0 R) / 2 := by
+  let μ := (perimeterMeasure E).restrict (Metric.ball 0 R)
+  have : IsFiniteMeasure μ := ⟨by
+    rw [Measure.restrict_apply_univ, perimeterMeasure_open E Metric.isOpen_ball]
+    exact hmin.2.1 0 R hR⟩
+  let f : Euc n → ℝ := fun y ↦ inner ℝ (approximateNormal E y) v
+  let q : Euc n → ℝ := fun y ↦ max (-f y) 0
+  have hm : Measurable f := (measurable_approximateNormal hmin.1).inner measurable_const
+  have hf : Integrable f μ := (integrable_const (1:ℝ)).mono' hm.aestronglyMeasurable
+    (ae_of_all _ fun y ↦ by
+      change ‖inner ℝ (approximateNormal E y) v‖ ≤ 1
+      exact (norm_inner_le_norm (𝕜 := ℝ) _ _).trans (by
+        rw [hv,mul_one]; exact approximateNormal_norm_le_one E y))
+  have hq : Integrable q μ := hf.neg_part
+  have hki : Integrable k volume := hk.continuous.integrable_of_hasCompactSupport hc
+  have hconv : Integrable (q ⋆[lsmul ℝ ℝ, μ] k) volume :=
+    (hq.convolution_integrand (lsmul ℝ ℝ) hki).integral_prod_left
+  have hpos : ∀ x, 0 ≤ (q ⋆[lsmul ℝ ℝ, μ] k) x := by
+    intro x
+    exact integral_nonneg fun y ↦ mul_nonneg (le_max_right _ _) (hp _)
+  have hdiff := (phaseMollification_contDiff hmin.1 hk hc).continuous_fderiv (by simp)
+  have hleft : IntegrableOn (fun x ↦ max (fderiv ℝ (phaseMollification E k) x v) 0)
+      (Metric.closedBall c r) volume :=
+    ((hdiff.clm_apply continuous_const).max continuous_const).continuousOn.integrableOn_compact
+      (isCompact_closedBall c r)
+  have hpoint : ∀ x ∈ Metric.closedBall c r,
+      max (fderiv ℝ (phaseMollification E k) x v) 0 ≤
+        (q ⋆[lsmul ℝ ℝ, μ] k) x := by
+    intro x hx
+    rw [phaseMollification_fderiv hn hmin hk hc x v hR (hs x hx)]
+    have h := positive_neg_convolution_le μ hf hk.continuous hc hp x
+    convert h using 1
+    congr 2
+    apply integral_congr_ae
+    exact ae_of_all _ fun y ↦ mul_comm _ _
+  have htilt : Integrable (fun y ↦ ‖approximateNormal E y-v‖^2) μ := by
+    apply (integrable_const (4:ℝ)).mono'
+      (((measurable_approximateNormal hmin.1).sub measurable_const).norm.pow_const 2).aestronglyMeasurable
+    apply ae_of_all μ
+    intro y
+    rw [Real.norm_eq_abs,abs_of_nonneg (sq_nonneg _)]
+    have hb : ‖approximateNormal E y-v‖ ≤ 2 :=
+      (norm_sub_le _ _).trans (by rw [hv]; linarith [approximateNormal_norm_le_one E y])
+    change ‖approximateNormal E y-v‖^2 ≤ 4
+    nlinarith [norm_nonneg (approximateNormal E y-v)]
+  calc
+    _ ≤ ∫ x in Metric.closedBall c r, (q ⋆[lsmul ℝ ℝ, μ] k) x :=
+      setIntegral_mono_on hleft hconv.integrableOn measurableSet_closedBall hpoint
+    _ ≤ ∫ x, (q ⋆[lsmul ℝ ℝ, μ] k) x :=
+      setIntegral_le_integral hconv (ae_of_all _ hpos)
+    _ = ∫ y, q y ∂μ := by rw [integral_convolution (lsmul ℝ ℝ) hq hki,hu]; simp
+    _ ≤ _ := by
+      dsimp [orientedTilt]
+      rw [le_div_iff₀ (by norm_num : (0:ℝ) < 2)]
+      rw [← integral_mul_const]
+      apply integral_mono (hq.mul_const 2) htilt
+      intro y
+      simpa only [q,f,mul_comm] using negative_normal_flux_le_tilt (approximateNormal E y) hv
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/NormalWindowTest.lean b/lean/OAI/Geometry/Perimeter/Regularity/NormalWindowTest.lean
new file mode 100644
index 0000000000000000000000000000000000000000..b2b25c573ca55c067cd1a0154cac249b6ae598c8
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/NormalWindowTest.lean
@@ -0,0 +1,175 @@
+import OAI.Geometry.Perimeter.Regularity.FiberMomentIntegration
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff Convolution InnerProductSpace
+namespace RPISingularHarnack
+open RPI.Boundary ContinuousLinearMap
+variable {n : ℕ}
+
+abbrev normalProjection (v : Euc n) : Euc n →L[ℝ] normalPlane v :=
+  (normalPlane v).orthogonalProjectionOnto
+
+@[simp] theorem normalProjection_fiber (v : Euc n) (y : normalPlane v) (s : ℝ) :
+    normalProjection v ((y:Euc n)+s•v) = y := by
+  have hv : normalProjection v v = 0 :=
+    Submodule.orthogonalProjectionOnto_eq_zero_iff.mpr
+      ((ℝ ∙ v).le_orthogonal_orthogonal (Submodule.mem_span_singleton_self v))
+  rw [map_add,map_smul,hv,smul_zero,add_zero]
+  exact Submodule.orthogonalProjectionOnto_mem_subspace_eq_self y
+
+@[simp] theorem normal_fiber_inner {v : Euc n} (hv : ‖v‖ = 1)
+    (y : normalPlane v) (s : ℝ) : inner ℝ v ((y:Euc n)+s•v) = s := by
+  rw [inner_add_right,real_inner_smul_right,
+    Submodule.mem_orthogonal_singleton_iff_inner_right.mp y.property,
+    real_inner_self_eq_norm_sq,hv]
+  ring
+
+def normalWindowTest (v : Euc n) (φ : normalPlane v → ℝ) (b : ℝ) (m : ℕ)
+    (x : Euc n) : ℝ :=
+  {z : Euc n | |inner ℝ v z| ≤ b}.indicator
+    (fun z ↦ φ (normalProjection v z)*(inner ℝ v z)^m) x
+
+theorem normalWindowTest_measurable {v : Euc n} {φ : normalPlane v → ℝ}
+    (hφ : Continuous φ) (b : ℝ) (m : ℕ) : Measurable (normalWindowTest v φ b m) :=
+  ((hφ.comp (normalProjection v).continuous).measurable.mul
+    ((continuous_const.inner continuous_id).pow m).measurable).indicator
+      (isClosed_le (continuous_const.inner continuous_id).abs continuous_const).measurableSet
+
+theorem normalWindowTest_bound {v : Euc n} {φ : normalPlane v → ℝ}
+    {M b : ℝ} (hM : ∀ y, ‖φ y‖ ≤ M) (hb : 0 ≤ b) (m : ℕ) (x : Euc n) :
+    ‖normalWindowTest v φ b m x‖ ≤ M*b^m := by
+  have hM0 : 0 ≤ M := (norm_nonneg (φ 0)).trans (hM 0)
+  by_cases hx : |inner ℝ v x| ≤ b
+  · simp only [normalWindowTest,Set.indicator_apply,Set.mem_ofPred_eq,ite_eq_left hx,norm_mul,norm_pow,
+      Real.norm_eq_abs]
+    exact mul_le_mul (hM _) (pow_le_pow_left₀ (abs_nonneg _) hx m) (pow_nonneg (abs_nonneg _) _) hM0
+  · simp only [normalWindowTest,Set.indicator_apply,Set.mem_ofPred_eq,ite_eq_right hx,norm_zero]
+    exact mul_nonneg hM0 (pow_nonneg hb _)
+
+theorem normalWindowTest_continuousAt {v : Euc n} {φ : normalPlane v → ℝ}
+    (hφ : Continuous φ) {b : ℝ} (m : ℕ) {x : Euc n} (hx : |inner ℝ v x| < b) :
+    ContinuousAt (normalWindowTest v φ b m) x := by
+  have he : ∀ᶠ z in 𝓝 x, |inner ℝ v z| < b :=
+    (continuous_const.inner continuous_id).abs.continuousAt.eventually (gt_mem_nhds hx)
+  have hc : Continuous (fun z ↦ φ (normalProjection v z)*(inner ℝ v z)^m) :=
+    (hφ.comp (normalProjection v).continuous).mul ((continuous_const.inner continuous_id).pow m)
+  apply hc.continuousAt.congr
+  filter_upwards [he] with z hz
+  simp only [normalWindowTest,Set.indicator_apply,Set.mem_ofPred_eq,ite_eq_left hz.le]
+
+@[simp] theorem normalCoordinates_mem_normalCylinder {v : Euc n} (hv : ‖v‖ = 1)
+    (K : Set (normalPlane v)) (b : ℝ) (y : normalPlane v) (s : ℝ) :
+    (y:Euc n)+s•v ∈ normalCylinder v hv K b ↔ y ∈ K ∧ s ∈ Icc (-b) b := by
+  change normalCoordinates v hv (y,s) ∈ normalCoordinates v hv '' (K ×ˢ Icc (-b) b) ↔ _
+  constructor
+  · rintro ⟨z,hz,he⟩
+    have he' : z = (y,s) := (normalCoordinates v hv).injective he
+    simpa only [he',Set.mem_prod] using hz
+  · intro h
+    exact ⟨(y,s),h,rfl⟩
+
+theorem normalWindowTest_eq_indicator {v : Euc n} (hv : ‖v‖ = 1)
+    (φ : normalPlane v → ℝ) (b : ℝ) (m : ℕ) :
+    normalWindowTest v φ b m = (normalCylinder v hv (tsupport φ) b).indicator
+      (fun x ↦ φ (normalProjection v x)*(inner ℝ v x)^m) := by
+  funext x
+  obtain ⟨⟨y,s⟩,rfl⟩ := (normalCoordinates v hv).surjective x
+  simp only [normalCoordinates_apply]
+  by_cases hy : y ∈ tsupport φ
+  · by_cases hs : s ∈ Icc (-b) b
+    · have hs' : |s| ≤ b := abs_le.mpr hs
+      simp [normalWindowTest,normal_fiber_inner hv,hs',normalCoordinates_mem_normalCylinder,hy,hs]
+    · have hs' : ¬ |s| ≤ b := fun h ↦ hs (abs_le.mp h)
+      simp [normalWindowTest,normal_fiber_inner hv,hs',normalCoordinates_mem_normalCylinder,hy,hs]
+  · have hz : φ y=0 := image_eq_zero_of_notMem_tsupport hy
+    simp [normalWindowTest,normal_fiber_inner hv,normalCoordinates_mem_normalCylinder,
+      hy,hz,Set.indicator_apply]
+
+theorem normalWindowTest_integrable {v : Euc n} (hv : ‖v‖ = 1)
+    {φ : normalPlane v → ℝ} (hφ : Continuous φ) (hcφ : HasCompactSupport φ)
+    (b : ℝ) (m : ℕ) : Integrable (normalWindowTest v φ b m) volume := by
+  rw [normalWindowTest_eq_indicator hv]
+  have hf : Continuous (fun x ↦ φ (normalProjection v x)*(inner ℝ v x)^m) :=
+    (hφ.comp (normalProjection v).continuous).mul ((continuous_const.inner continuous_id).pow m)
+  exact (hf.continuousOn.integrableOn_compact (isCompact_normalCylinder hv hcφ b)).integrable_indicator
+    (isCompact_normalCylinder hv hcφ b).measurableSet
+
+theorem normalWindowTest_support_subset {v : Euc n} (hv : ‖v‖ = 1)
+    (φ : normalPlane v → ℝ) (b : ℝ) (m : ℕ) :
+    Function.support (normalWindowTest v φ b m) ⊆ normalCylinder v hv (tsupport φ) b := by
+  rw [normalWindowTest_eq_indicator hv]
+  exact Set.support_indicator_subset
+
+theorem normalWindowTest_mul_integrable {v : Euc n} (hv : ‖v‖ = 1)
+    {φ : normalPlane v → ℝ} (hφ : Continuous φ) (hcφ : HasCompactSupport φ)
+    (b : ℝ) (m : ℕ) {g : Euc n → ℝ} (hg : Continuous g) :
+    Integrable (fun x ↦ normalWindowTest v φ b m x*g x) volume := by
+  rw [normalWindowTest_eq_indicator hv]
+  have he : (fun x ↦ (normalCylinder v hv (tsupport φ) b).indicator
+      (fun z ↦ φ (normalProjection v z)*(inner ℝ v z)^m) x*g x) =
+      (normalCylinder v hv (tsupport φ) b).indicator
+        (fun x ↦ φ (normalProjection v x)*(inner ℝ v x)^m*g x) := by
+    funext x
+    by_cases hx : x ∈ normalCylinder v hv (tsupport φ) b <;> simp [hx]
+  rw [he]
+  have hf : Continuous (fun x ↦ φ (normalProjection v x)*(inner ℝ v x)^m*g x) :=
+    ((hφ.comp (normalProjection v).continuous).mul ((continuous_const.inner continuous_id).pow m)).mul hg
+  exact (hf.continuousOn.integrableOn_compact (isCompact_normalCylinder hv hcφ b)).integrable_indicator
+    (isCompact_normalCylinder hv hcφ b).measurableSet
+
+theorem normalWindowTest_support_norm {v : Euc n} (hv : ‖v‖ = 1)
+    (φ : normalPlane v → ℝ) (b δ R : ℝ) (m : ℕ)
+    (hKR : ∀ y ∈ tsupport φ, ‖y‖+b+δ ≤ R) {x : Euc n}
+    (hx : normalWindowTest v φ b m x ≠ 0) : ‖x‖+δ ≤ R := by
+  obtain ⟨⟨y,s⟩,⟨hy,hs⟩,rfl⟩ := normalWindowTest_support_subset hv φ b m hx
+  have h := norm_add_le (y:Euc n) (s•v)
+  rw [norm_smul,hv,mul_one,Real.norm_eq_abs] at h
+  have hs' : |s| ≤ b := abs_le.mpr hs
+  have hyR := hKR y hy
+  change ‖(y:Euc n)‖+b+δ ≤ R at hyR
+  change ‖(y:Euc n)+s•v‖+δ ≤ R
+  linarith
+
+theorem normalWindowTest_integral {v : Euc n} (hv : ‖v‖ = 1)
+    {φ : normalPlane v → ℝ} (hφ : Continuous φ) (hcφ : HasCompactSupport φ)
+    {b : ℝ} (hb : 0 ≤ b) (m : ℕ) {g : Euc n → ℝ} (hg : Continuous g) :
+    (∫ x, normalWindowTest v φ b m x*g x) =
+      ∫ y in tsupport φ, φ y*(∫ s in (-b)..b, s^m*g ((y:Euc n)+s•v)) := by
+  let f : Euc n → ℝ := fun x ↦ φ (normalProjection v x)*(inner ℝ v x)^m*g x
+  have hf : Continuous f := ((hφ.comp (normalProjection v).continuous).mul
+    ((continuous_const.inner continuous_id).pow m)).mul hg
+  have he : (fun x ↦ normalWindowTest v φ b m x*g x) =
+      (normalCylinder v hv (tsupport φ) b).indicator f := by
+    funext x
+    obtain ⟨⟨y,s⟩,rfl⟩ := (normalCoordinates v hv).surjective x
+    simp only [normalCoordinates_apply]
+    by_cases hy : y ∈ tsupport φ
+    · by_cases hs : s ∈ Icc (-b) b
+      · have hs' : |s| ≤ b := abs_le.mpr hs
+        simp [normalWindowTest,normal_fiber_inner hv,hs',normalCoordinates_mem_normalCylinder,
+          hy,hs,f]
+      · have hs' : ¬ |s| ≤ b := fun h ↦ hs (abs_le.mp h)
+        simp [normalWindowTest,normal_fiber_inner hv,hs',normalCoordinates_mem_normalCylinder,
+          hy,hs,f]
+    · have hz : φ y=0 := image_eq_zero_of_notMem_tsupport hy
+      simp [normalWindowTest,normal_fiber_inner hv,normalCoordinates_mem_normalCylinder,
+        hy,f,hz,Set.indicator_apply]
+  rw [he,integral_indicator (isCompact_normalCylinder hv hcφ b).measurableSet,
+    integral_normalCylinder hv hcφ hb hf]
+  apply setIntegral_congr_fun hcφ.measurableSet
+  intro y _
+  dsimp only [f]
+  simp_rw [normalProjection_fiber,normal_fiber_inner hv]
+  rw [← intervalIntegral.integral_const_mul]
+  congr 1
+  funext s
+  ring
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/NormalizedHeight.lean b/lean/OAI/Geometry/Perimeter/Regularity/NormalizedHeight.lean
new file mode 100644
index 0000000000000000000000000000000000000000..a7d5aae9c0d0a882e8656a9194dab3122f26721b
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/NormalizedHeight.lean
@@ -0,0 +1,205 @@
+import OAI.Geometry.Perimeter.Regularity.FiberCompactness
+import OAI.Geometry.Perimeter.Regularity.FiberAveraging
+import OAI.Geometry.Perimeter.Regularity.WindowAverage
+import OAI.Geometry.Perimeter.Regularity.HeightCompactness
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff Convolution
+namespace RPISingularHarnack
+open RPI.Boundary
+
+def normalizedWindowHeight {n : ℕ} (v : Euc n) (T a : ℝ)
+    (f : Euc n → ℝ) (x : Euc n) : ℝ := a⁻¹ * normalWindowAverage v T f x
+
+theorem normalizedWindowHeight_measurable {n : ℕ} {f : Euc n → ℝ}
+    (hf : Measurable f) (v : Euc n) (T a : ℝ) :
+    Measurable (normalizedWindowHeight v T a f) :=
+  (normalWindowAverage_stronglyMeasurable hf v T).measurable.const_mul _
+
+theorem normalizedWindowHeight_lintegral_le {n : ℕ} {f : Euc n → ℝ}
+    (hf : Measurable f) (v : Euc n) {T a V : ℝ} (ha : 0 < a)
+    (hT : 0 ≤ T) (hV : 0 ≤ V)
+    (hm : (∫⁻ x, ‖f x‖ₑ) ≤ ENNReal.ofReal (a*V)) :
+    (∫⁻ x, ‖normalizedWindowHeight v T a f x‖ₑ) ≤ ENNReal.ofReal (2*T*V) := by
+  have he : (∫⁻ x, ‖normalizedWindowHeight v T a f x‖ₑ) =
+      ENNReal.ofReal a⁻¹ * ∫⁻ x, ‖normalWindowAverage v T f x‖ₑ := by
+    simp only [normalizedWindowHeight]
+    simp_rw [enorm_mul]
+    rw [show ‖a⁻¹‖ₑ = ENNReal.ofReal a⁻¹ by
+      rw [← ofReal_norm,Real.norm_eq_abs,abs_of_pos (inv_pos.mpr ha)]]
+    exact lintegral_const_mul _ (normalWindowAverage_stronglyMeasurable hf v T).measurable.enorm
+  rw [he]
+  calc
+    _ ≤ ENNReal.ofReal a⁻¹ * (ENNReal.ofReal (2*T) * ENNReal.ofReal (a*V)) :=
+      mul_le_mul_right ((normalWindowAverage_lintegral_le hf v T).trans
+        (mul_le_mul_right hm _)) _
+    _ = _ := by
+      rw [← ENNReal.ofReal_mul (by positivity),
+        ← ENNReal.ofReal_mul' (mul_nonneg (mul_nonneg (by norm_num) hT) (mul_nonneg ha.le hV))]
+      congr 1
+      field_simp
+
+theorem normalizedWindowHeight_integrable {n : ℕ} {f : Euc n → ℝ}
+    (hf : Measurable f) (v : Euc n) {T a V : ℝ} (ha : 0 < a)
+    (hT : 0 ≤ T) (hV : 0 ≤ V)
+    (hm : (∫⁻ x, ‖f x‖ₑ) ≤ ENNReal.ofReal (a*V)) :
+    Integrable (normalizedWindowHeight v T a f) volume := by
+  refine ⟨(normalizedWindowHeight_measurable hf v T a).aestronglyMeasurable, ?_⟩
+  exact (normalizedWindowHeight_lintegral_le hf v ha hT hV hm).trans_lt ENNReal.ofReal_lt_top
+
+theorem normalizedWindowHeight_translation_le {n : ℕ} {f : Euc n → ℝ}
+    (hf : Measurable f) {C : ℝ} (hb : ∀ x, ‖f x‖ ≤ C)
+    {v : Euc n} (hv : ‖v‖ = 1) {T a H V : ℝ} (ha : 0 < a)
+    (hT : 0 ≤ T) (hH : 0 ≤ H) (hV : 0 ≤ V)
+    (hm : (∫⁻ x, ‖f x‖ₑ) ≤ ENNReal.ofReal (a*V))
+    (hh : ∀ z, inner ℝ v z = 0 →
+      (∫⁻ x, ‖f (x+z)-f x‖ₑ) ≤ ENNReal.ofReal (‖z‖*(a*H))) (w : Euc n) :
+    (∫⁻ x, ‖normalizedWindowHeight v T a f (x+w)-normalizedWindowHeight v T a f x‖ₑ) ≤
+      ‖w‖ₑ * ENNReal.ofReal (4*T*H+4*V) := by
+  have he : (∫⁻ x, ‖normalizedWindowHeight v T a f (x+w)-normalizedWindowHeight v T a f x‖ₑ) =
+      ENNReal.ofReal a⁻¹ * ∫⁻ x, ‖normalWindowAverage v T f (x+w)-normalWindowAverage v T f x‖ₑ := by
+    simp only [normalizedWindowHeight,← mul_sub]
+    simp_rw [enorm_mul]
+    rw [show ‖a⁻¹‖ₑ = ENNReal.ofReal a⁻¹ by
+      rw [← ofReal_norm,Real.norm_eq_abs,abs_of_pos (inv_pos.mpr ha)]]
+    apply lintegral_const_mul
+    exact (((normalWindowAverage_stronglyMeasurable hf v T).measurable.comp
+      (measurable_id.add measurable_const)).sub
+      (normalWindowAverage_stronglyMeasurable hf v T).measurable).enorm
+  rw [he]
+  calc
+    _ ≤ ENNReal.ofReal a⁻¹ * ENNReal.ofReal (‖w‖*(4*T*(a*H)+4*(a*V))) :=
+      mul_le_mul_right (normalWindowAverage_all_translation_le hf hb hv hT
+        (by positivity) (by positivity) hm hh w) _
+    _ = _ := by
+      rw [← ofReal_norm,← ENNReal.ofReal_mul (by positivity),
+        ← ENNReal.ofReal_mul (norm_nonneg _)]
+      congr 1
+      field_simp
+
+theorem normalizedWindowHeight_subsequence {n : ℕ}
+    (K : Set (Euc n)) (hK : IsCompact K) (f : ℕ → Euc n → ℝ)
+    (hm : ∀ i, Measurable (f i)) (C : ℝ) (hb : ∀ i x, ‖f i x‖ ≤ C)
+    {v : Euc n} (hv : ‖v‖ = 1) {T H V : ℝ}
+    (hT : 0 ≤ T) (hH : 0 ≤ H) (hV : 0 ≤ V)
+    (a : ℕ → ℝ) (ha : ∀ i, 0 < a i)
+    (hmass : ∀ i, (∫⁻ x, ‖f i x‖ₑ) ≤ ENNReal.ofReal (a i*V))
+    (hhor : ∀ i z, inner ℝ v z = 0 →
+      (∫⁻ x, ‖f i (x+z)-f i x‖ₑ) ≤ ENNReal.ofReal (‖z‖*(a i*H))) :
+    ∃ hi : ∀ i, Integrable (normalizedWindowHeight v T (a i) (f i)) volume,
+      ∃ g : Lp ℝ 1 (volume.comap (Subtype.val : K → Euc n)),
+        ∃ σ : ℕ → ℕ, StrictMono σ ∧
+          Tendsto (fun i ↦ compactFunctionLp K hK
+            (normalizedWindowHeight v T (a (σ i)) (f (σ i))) (hi (σ i))) atTop (𝓝 g) := by
+  have hi := fun i ↦ normalizedWindowHeight_integrable (hm i) v (ha i) hT hV (hmass i)
+  refine ⟨hi, ?_⟩
+  refine compactFunctionLp_subsequence_of_translation K hK _
+    (fun i ↦ normalizedWindowHeight_measurable (hm i) v T (a i)) hi
+    (2*T*V) (by positivity) ?_ (ENNReal.ofReal (4*T*H+4*V)) ENNReal.ofReal_ne_top ?_
+  · intro i
+    have h := normalizedWindowHeight_lintegral_le (hm i) v (ha i) hT hV (hmass i)
+    rw [← ofReal_integral_norm_eq_lintegral_enorm (hi i)] at h
+    exact (ENNReal.ofReal_le_ofReal_iff (by positivity)).mp h
+  · intro i w
+    exact normalizedWindowHeight_translation_le (hm i) (hb i) hv (ha i) hT hH hV
+      (hmass i) (hhor i) w
+
+theorem normalWindowAverage_eq_normalFiberAverage {n : ℕ}
+    {f : Euc n → ℝ} {v : Euc n} (hv : ‖v‖ = 1) {R T : ℝ}
+    (hs : Function.support f ⊆ Metric.ball 0 R) (x : Euc n)
+    (hT : R+‖x‖ ≤ T) :
+    normalWindowAverage v T f x = normalFiberAverage v f x := by
+  unfold normalWindowAverage normalFiberAverage
+  apply integral_congr_ae
+  apply ae_of_all
+  intro s
+  by_cases ht : s ∈ Icc (-T) T
+  · simp [lineWindow,ht]
+  · have hz : f (x+s•v) = 0 := by
+      by_contra hz
+      have hi := fiber_support_subset_interval hv hs x hz
+      exact ht ((Icc_subset_Icc (by linarith) hT) hi)
+    simp [hz]
+
+theorem cutoffPhaseError_norm_le_one {n : ℕ} (E : Set (Euc n)) (v : Euc n)
+    {η : Euc n → ℝ} (hη : ∀ x, ‖η x‖ ≤ 1) (x : Euc n) :
+    ‖cutoffPhaseError E v η x‖ ≤ 1 := by
+  unfold cutoffPhaseError
+  rw [norm_mul]
+  have h : ‖E.indicator (fun _ ↦ (1:ℝ)) x -
+      {z | inner ℝ v z < 0}.indicator (fun _ ↦ (1:ℝ)) x‖ ≤ 1 := by
+    by_cases he : x ∈ E <;> by_cases hh : inner ℝ v x < 0 <;> simp [he,hh]
+  exact (mul_le_mul (hη x) h (norm_nonneg _) zero_le_one).trans_eq (one_mul _)
+
+theorem cutoffPhaseError_lintegral_height_le {n : ℕ} {E : Set (Euc n)}
+    (hE : MeasurableSet E) {v : Euc n} (hv : ‖v‖ = 1)
+    {η : Euc n → ℝ} (hη : Measurable η) (hb : ∀ x, ‖η x‖ ≤ 1)
+    {R a : ℝ} (hR : 0 < R) (ha : 0 ≤ a)
+    (hs : Function.support η ⊆ Metric.ball 0 R)
+    (hphase : ∀ᵐ x ∂volume, x ∈ Metric.ball 0 R → a < |inner ℝ v x| →
+      (x ∈ E ↔ inner ℝ v x < 0)) :
+    (∫⁻ x, ‖cutoffPhaseError E v η x‖ₑ) ≤
+      ENNReal.ofReal (a*(volume.real (Metric.ball (0:Euc n) (2*R))/R)) := by
+  have hmass := slab_lintegral_bound (cutoffPhaseError_measurable hE v hη) hv ha hR
+    ((cutoffPhaseError_support E v η).trans hs)
+    (cutoffPhaseError_slab_bound E v η hs hb hphase)
+  have hfinite : volume (Metric.ball (0:Euc n) (2*R)) ≠ ⊤ := measure_ball_lt_top.ne
+  have hnorm : ENNReal.ofReal (a*(volume.real (Metric.ball (0:Euc n) (2*R))/R)) =
+      (ENNReal.ofReal (2*a)*volume (Metric.ball (0:Euc n) (2*R))) / ENNReal.ofReal (2*R) := by
+    rw [← ENNReal.ofReal_toReal hfinite]
+    simp only [← ENNReal.ofReal_mul (by positivity : 0 ≤ 2*a),
+      ← ENNReal.ofReal_div_of_pos (by positivity : 0 < 2*R)]
+    congr 1
+    change a*((volume (Metric.ball (0:Euc n) (2*R))).toReal/R) =
+      2*a*(volume (Metric.ball (0:Euc n) (2*R))).toReal/(2*R)
+    field_simp
+  rw [hnorm]
+  exact (ENNReal.le_div_iff_mul_le (Or.inl (ENNReal.ofReal_pos.mpr (by positivity)).ne')
+    (Or.inl ENNReal.ofReal_ne_top)).mpr
+    (by simpa only [mul_comm] using hmass)
+
+theorem minimizing_normalized_height_subsequence {n : ℕ} (hn : 1 < n)
+    (K : Set (Euc n)) (hK : IsCompact K) (E : ℕ → Set (Euc n))
+    (hmin : ∀ i, IsLocallyPerimeterMinimizing (E i))
+    {v : Euc n} (hv : ‖v‖ = 1)
+    {η : Euc n → ℝ} (hη : ContDiff ℝ ∞ η) (hc : HasCompactSupport η)
+    (hb : ∀ x, ‖η x‖ ≤ 1) {R T L P : ℝ} (hR : 0 < R) (hT : 0 ≤ T)
+    (hL : 0 ≤ L) (hP : 0 ≤ P) (hs : tsupport η ⊆ Metric.ball 0 R)
+    (hgrad : ∀ x ∈ Metric.ball 0 R, ‖gradient η x‖ ≤ L)
+    (a : ℕ → ℝ) (ha : ∀ i, 0 < a i)
+    (hheight : ∀ i x, x ∈ perimeterSupport (E i) ∩ Metric.ball 0 R → |inner ℝ v x| ≤ a i)
+    (hper : ∀ i, (perimeter (E i) (Metric.ball 0 R)).toReal ≤ P)
+    (hphase : ∀ i, ∀ᵐ x ∂volume, x ∈ Metric.ball 0 R → a i < |inner ℝ v x| →
+      (x ∈ E i ↔ inner ℝ v x < 0)) :
+    ∃ hi : ∀ i, Integrable (normalizedWindowHeight v T (a i) (cutoffPhaseError (E i) v η)) volume,
+      ∃ g : Lp ℝ 1 (volume.comap (Subtype.val : K → Euc n)),
+        ∃ σ : ℕ → ℕ, StrictMono σ ∧
+          Tendsto (fun i ↦ compactFunctionLp K hK
+            (normalizedWindowHeight v T (a (σ i)) (cutoffPhaseError (E (σ i)) v η))
+              (hi (σ i))) atTop (𝓝 g) := by
+  let V := volume.real (Metric.ball (0:Euc n) (2*R))/R
+  have hV : 0 ≤ V := div_nonneg measureReal_nonneg hR.le
+  apply normalizedWindowHeight_subsequence K hK (fun i ↦ cutoffPhaseError (E i) v η)
+    (fun i ↦ cutoffPhaseError_measurable (hmin i).1 v hη.continuous.measurable)
+    1 (fun i x ↦ cutoffPhaseError_norm_le_one (E i) v hb x) hv hT
+    (H := L*(V+2*P)) (by positivity) hV a ha
+  · intro i
+    exact cutoffPhaseError_lintegral_height_le (hmin i).1 hv hη.continuous.measurable hb
+      hR (ha i).le ((show Function.support η ⊆ tsupport η from subset_closure).trans hs) (hphase i)
+  · intro i z hz
+    apply (cutoffPhaseError_horizontal_translation_le hn (hmin i) hv hz hη hc hR hs
+      (ha i).le hL (hheight i) hgrad (hphase i)).trans
+    apply ENNReal.ofReal_le_ofReal
+    have hai := (ha i).le
+    have hp := mul_le_mul_of_nonneg_left (hper i) (by positivity : 0 ≤ ‖z‖*a i*L*2)
+    dsimp [V]
+    nlinarith
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/PairedCutTest.lean b/lean/OAI/Geometry/Perimeter/Regularity/PairedCutTest.lean
new file mode 100644
index 0000000000000000000000000000000000000000..2cd39f0577bdb000c5ebf7c02926c34366d44af2
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/PairedCutTest.lean
@@ -0,0 +1,524 @@
+import OAI.Geometry.Perimeter.Regularity.GradientContDiff
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+
+def pairedCutTest {n : ℕ} (c : Euc n) (a b : ℝ)
+    (V W : Euc n → Euc n) (x : Euc n) : Euc n :=
+  V x + annularCutoff c a b x • (W x - V x)
+
+theorem pairedCutTest_eq {n : ℕ} (c : Euc n) (a b : ℝ)
+    (V W : Euc n → Euc n) (x : Euc n) :
+    pairedCutTest c a b V W x =
+      (1 - annularCutoff c a b x) • V x + annularCutoff c a b x • W x := by
+  unfold pairedCutTest
+  module
+
+theorem pairedCutTest_admissible {n : ℕ} {U : Set (Euc n)}
+    {V W : Euc n → Euc n} (hV : PerimeterAdmissible univ V)
+    (hW : PerimeterAdmissible U W) {c : Euc n} {a b : ℝ}
+    (ha : 0 < a) (hab : a < b) (hB : Metric.closedBall c b ⊆ U) :
+    PerimeterAdmissible U (pairedCutTest c a b V W) := by
+  have hs : tsupport (pairedCutTest c a b V W) ⊆ Metric.closedBall c b ∪ tsupport W := by
+    apply closure_minimal
+    · intro x hx
+      by_contra hh
+      have hxb : b ≤ dist x c := le_of_lt (lt_of_not_ge (fun hb ↦ hh (Or.inl hb)))
+      have hxW : x ∉ tsupport W := fun hw ↦ hh (Or.inr hw)
+      exact hx (by simp [pairedCutTest, annularCutoff_one ha hab hxb,
+        image_eq_zero_of_notMem_tsupport hxW])
+    · exact Metric.isClosed_closedBall.union (isClosed_tsupport _)
+  refine ⟨hV.1.add ((annularCutoff_contDiff c a b).smul (hW.1.sub hV.1)),
+    ((isCompact_closedBall c b).union hW.2.1).of_isClosed_subset (isClosed_tsupport _) hs,
+    hs.trans (union_subset hB hW.2.2.1), ?_⟩
+  intro x
+  rw [pairedCutTest_eq]
+  have hη := annularCutoff_mem_Icc c a b x
+  calc
+    _ ≤ ‖(1 - annularCutoff c a b x) • V x‖ + ‖annularCutoff c a b x • W x‖ := norm_add_le _ _
+    _ = (1 - annularCutoff c a b x) * ‖V x‖ + annularCutoff c a b x * ‖W x‖ := by
+      simp [norm_smul, Real.norm_eq_abs, abs_of_nonneg hη.1, abs_of_nonneg (sub_nonneg.mpr hη.2)]
+    _ ≤ (1 - annularCutoff c a b x) * 1 + annularCutoff c a b x * 1 :=
+      add_le_add (mul_le_mul_of_nonneg_left (hV.2.2.2 x) (sub_nonneg.mpr hη.2))
+        (mul_le_mul_of_nonneg_left (hW.2.2.2 x) hη.1)
+    _ = 1 := by ring
+
+theorem divergence_sub {n : ℕ} {V W : Euc n → Euc n}
+    (hV : Differentiable ℝ V) (hW : Differentiable ℝ W) :
+    divergence (V - W) = divergence V - divergence W := by
+  funext x
+  simp only [divergence, fderiv_sub (hV x) (hW x), sub_apply,
+    inner_sub_right, Finset.sum_sub_distrib, Pi.sub_apply]
+
+theorem divergence_pairedCutTest {n : ℕ} {c : Euc n} {a b : ℝ}
+    {V W : Euc n → Euc n} (hV : ContDiff ℝ ∞ V) (hW : ContDiff ℝ ∞ W) (x : Euc n) :
+    divergence (pairedCutTest c a b V W) x =
+      divergence V x + annularCutoff c a b x * (divergence W x - divergence V x) +
+        inner ℝ (gradient (annularCutoff c a b) x) (W x - V x) := by
+  have hdV := hV.differentiable (by norm_num)
+  have hdW := hW.differentiable (by norm_num)
+  have hdη := (annularCutoff_contDiff c a b).differentiable (by norm_num)
+  change divergence (V + (fun y ↦ annularCutoff c a b y • (W - V) y)) x = _
+  rw [divergence_add (W := fun y ↦ annularCutoff c a b y • (W - V) y)
+    hdV (hdη.smul (hdW.sub hdV))]
+  change divergence V x + divergence (fun y ↦ annularCutoff c a b y • (W y - V y)) x = _
+  rw [divergence_smul (V := fun y ↦ W y - V y) hdη (hdW.sub hdV)]
+  rw [show divergence (fun y ↦ W y - V y) x = divergence W x - divergence V x from
+    congrFun (divergence_sub hdW hdV) x]
+  ring
+
+theorem integrable_annularCutoff_mul {n : ℕ} {f : Euc n → ℝ}
+    (hf : Integrable f volume) (c : Euc n) (a b : ℝ) :
+    Integrable (fun x ↦ annularCutoff c a b x * f x) volume := by
+  apply hf.norm.mono' (((annularCutoff_contDiff c a b).continuous.aestronglyMeasurable).mul
+    hf.aestronglyMeasurable)
+  filter_upwards [] with x
+  change ‖annularCutoff c a b x * f x‖ ≤ ‖f x‖
+  rw [norm_mul, Real.norm_eq_abs, abs_of_nonneg (annularCutoff_mem_Icc c a b x).1]
+  exact mul_le_of_le_one_left (norm_nonneg _) (annularCutoff_mem_Icc c a b x).2
+
+theorem integral_pairedCut_le {n : ℕ} {E U : Set (Euc n)}
+    {V W : Euc n → Euc n} (hV : PerimeterAdmissible univ V)
+    (hW : PerimeterAdmissible U W) {c : Euc n} {a b : ℝ}
+    (ha : 0 < a) (hab : a < b) (hB : Metric.closedBall c b ⊆ U)
+    (hfin : perimeter E U ≠ ⊤) :
+    (∫ x in E, divergence V x + annularCutoff c a b x * (divergence W x - divergence V x)) ≤
+      (perimeter E U).toReal + 2 * ∫ x in E, ‖gradient (annularCutoff c a b) x‖ := by
+  have hZ := pairedCutTest_admissible hV hW ha hab hB
+  have hdV := integrable_divergence hV.1 hV.2.1
+  have hdW := integrable_divergence hW.1 hW.2.1
+  have hm : Integrable (fun x ↦ divergence V x +
+      annularCutoff c a b x * (divergence W x - divergence V x)) volume :=
+    hdV.add (integrable_annularCutoff_mul (hdW.sub hdV) c a b)
+  have hg : Integrable (fun x ↦ ‖gradient (annularCutoff c a b) x‖) volume :=
+    ((gradient_annularCutoff_continuous c a b).integrable_of_hasCompactSupport
+    (gradient_annularCutoff_compact c ha hab)).norm
+  have hinner : ∀ x, |inner ℝ (gradient (annularCutoff c a b) x) (W x - V x)| ≤
+      2 * ‖gradient (annularCutoff c a b) x‖ := by
+    intro x
+    calc
+      _ ≤ ‖gradient (annularCutoff c a b) x‖ * ‖W x - V x‖ := abs_real_inner_le_norm _ _
+      _ ≤ ‖gradient (annularCutoff c a b) x‖ * 2 := by
+        gcongr
+        exact (norm_sub_le _ _).trans (by linarith [hV.2.2.2 x, hW.2.2.2 x])
+      _ = _ := by ring
+  have hi : Integrable (fun x ↦ inner ℝ (gradient (annularCutoff c a b) x) (W x - V x)) volume :=
+    (hg.const_mul 2).mono'
+      ((gradient_annularCutoff_continuous c a b).inner (hW.1.continuous.sub hV.1.continuous)).aestronglyMeasurable
+      (Eventually.of_forall hinner)
+  have hp : (∫ x in E, divergence (pairedCutTest c a b V W) x) ≤ (perimeter E U).toReal := by
+    apply (ENNReal.ofReal_le_iff_le_toReal hfin).mp
+    rw [perimeter_eq_iSup_tests]
+    exact le_iSup_of_le _ (le_iSup_of_le hZ le_rfl)
+  simp_rw [divergence_pairedCutTest hV.1 hW.1] at hp
+  rw [integral_add hm.integrableOn hi.integrableOn] at hp
+  have herr : -(∫ x in E, inner ℝ (gradient (annularCutoff c a b) x) (W x - V x)) ≤
+      2 * ∫ x in E, ‖gradient (annularCutoff c a b) x‖ := by
+    rw [← integral_neg, ← integral_const_mul]
+    exact integral_mono hi.integrableOn.neg (hg.const_mul 2).integrableOn
+      (fun x ↦ (neg_le_abs _).trans (hinner x))
+  linarith
+
+theorem tendsto_integral_annularCutoff_mul {n : ℕ} {E : Set (Euc n)}
+    {f : Euc n → ℝ} (hf : Integrable f volume) (c : Euc n) {r : ℝ} (hr : 0 < r) :
+    Tendsto (fun δ : ℝ ↦ ∫ x in E, annularCutoff c (r + δ) (r + 2 * δ) x * f x)
+      (𝓝[>] (0 : ℝ)) (𝓝 (∫ x in E \ Metric.closedBall c r, f x)) := by
+  have ht := tendsto_integral_filter_of_dominated_convergence
+    (μ := volume.restrict E) (l := 𝓝[>] (0 : ℝ))
+    (F := fun δ x ↦ annularCutoff c (r + δ) (r + 2 * δ) x * f x)
+    (f := (Metric.closedBall c r)ᶜ.indicator f) (fun x ↦ ‖f x‖)
+    (Eventually.of_forall fun δ ↦ (integrable_annularCutoff_mul hf c _ _).integrableOn.aestronglyMeasurable)
+    (Eventually.of_forall fun δ ↦ Eventually.of_forall fun x ↦ by
+      rw [norm_mul, Real.norm_eq_abs, abs_of_nonneg (annularCutoff_mem_Icc c _ _ x).1]
+      exact mul_le_of_le_one_left (norm_nonneg _) (annularCutoff_mem_Icc c _ _ x).2)
+    hf.norm.integrableOn (Eventually.of_forall fun x ↦ ?_)
+  · simpa only [integral_indicator measurableSet_closedBall.compl,
+      Measure.restrict_restrict measurableSet_closedBall.compl, inter_comm, ← sdiff_eq] using ht
+  · convert (annularCutoff_tendsto c x hr).mul (tendsto_const_nhds (x := f x)) using 1
+    by_cases hx : x ∈ (Metric.closedBall c r)ᶜ <;> simp [hx]
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+
+theorem nonnegative_perimeter_test_replacement {n : ℕ} (E U : Set (Euc n))
+    {V : Euc n → Euc n} (hV : PerimeterAdmissible U V) :
+    ∃ W : Euc n → Euc n, PerimeterAdmissible U W ∧
+      0 ≤ ∫ x in E, divergence W x ∧
+      ENNReal.ofReal (∫ x in E, divergence W x) = ENNReal.ofReal (∫ x in E, divergence V x) := by
+  by_cases h : 0 ≤ ∫ x in E, divergence V x
+  · exact ⟨V, hV, h, rfl⟩
+  · refine ⟨fun _ ↦ 0, perimeterAdmissible_zero U, ?_, ?_⟩
+    · simp [divergence]
+    · rw [ENNReal.ofReal_of_nonpos (le_of_not_ge h)]
+      simp [divergence]
+
+theorem perimeter_add_le_of_pair_integrals {n : ℕ} (E F U V : Set (Euc n)) (K : ℝ)
+    (h : ∀ X Y : Euc n → Euc n, PerimeterAdmissible U X → PerimeterAdmissible V Y →
+      (∫ x in E, divergence X x) + (∫ x in F, divergence Y x) ≤ K) :
+    perimeter E U + perimeter F V ≤ ENNReal.ofReal K := by
+  rw [perimeter_eq_iSup_tests, perimeter_eq_iSup_tests]
+  apply ENNReal.biSup_add_biSup_le' ⟨_, perimeterAdmissible_zero U⟩ ⟨_, perimeterAdmissible_zero V⟩
+  intro X hX Y hY
+  obtain ⟨X', hX', hnX, hvalX⟩ := nonnegative_perimeter_test_replacement E U hX
+  obtain ⟨Y', hY', hnY, hvalY⟩ := nonnegative_perimeter_test_replacement F V hY
+  rw [← hvalX, ← hvalY, ← ENNReal.ofReal_add hnX hnY]
+  exact ENNReal.ofReal_le_ofReal (h X' Y' hX' hY')
+
+theorem controlled_radius_perimeter_cut {n : ℕ} {E U : Set (Euc n)}
+    (hE : MeasurableSet E) (c : Euc n) {r B D L : ℝ}
+    (hr : 0 < r) (hrB : r < B) (hBU : Metric.closedBall c B ⊆ U)
+    (hcontrol : ∀ t ∈ Icc r B,
+      volume.real (E ∩ Metric.closedBall c t) - volume.real (E ∩ Metric.closedBall c r) ≤
+        D * (t - r))
+    (hD : 0 ≤ D) (hL : 0 < L)
+    (hbound : ∀ s, |deriv Real.smoothTransition s| ≤ L)
+    (hfin : perimeter E U ≠ ⊤) :
+    perimeter (E ∩ Metric.closedBall c r) univ + perimeter (E \ Metric.closedBall c r) U ≤
+      perimeter E U + ENNReal.ofReal (8 * L * D) := by
+  rw [← ENNReal.ofReal_toReal hfin, ← ENNReal.ofReal_add ENNReal.toReal_nonneg (by positivity)]
+  apply perimeter_add_le_of_pair_integrals
+  intro V W hV hW
+  have hdV := integrable_divergence hV.1 hV.2.1
+  have hdW := integrable_divergence hW.1 hW.2.1
+  have ht : Tendsto (fun δ : ℝ ↦ ∫ x in E,
+      divergence V x + annularCutoff c (r + δ) (r + 2 * δ) x * (divergence W x - divergence V x))
+      (𝓝[>] (0 : ℝ)) (𝓝 ((∫ x in E ∩ Metric.closedBall c r, divergence V x) +
+        (∫ x in E \ Metric.closedBall c r, divergence W x))) := by
+    have hlim := (tendsto_const_nhds (x := ∫ x in E, divergence V x)).add
+      (tendsto_integral_annularCutoff_mul (E := E) (hdW.sub hdV) c hr)
+    have heq : (∫ x in E, divergence V x) +
+        (∫ x in E \ Metric.closedBall c r, (divergence W - divergence V) x) =
+        (∫ x in E ∩ Metric.closedBall c r, divergence V x) +
+        (∫ x in E \ Metric.closedBall c r, divergence W x) := by
+      simp only [Pi.sub_apply]
+      rw [integral_sub hdW.integrableOn hdV.integrableOn]
+      have hsplit := integral_inter_add_sdiff (s := E) (t := Metric.closedBall c r) measurableSet_closedBall hdV.integrableOn
+      linarith
+    rw [heq] at hlim
+    convert hlim using 1
+    ext δ
+    exact integral_add hdV.integrableOn
+      (integrable_annularCutoff_mul (hdW.sub hdV) c _ _).integrableOn
+  apply le_of_tendsto ht
+  filter_upwards [self_mem_nhdsWithin,
+    nhdsWithin_le_nhds (gt_mem_nhds (show (0 : ℝ) < (B - r) / 2 by linarith))] with δ hδ hsmall
+  change 0 < δ at hδ
+  change δ < (B - r) / 2 at hsmall
+  have hδB : r + 2 * δ ≤ B := by linarith
+  have hp := integral_pairedCut_le hV hW (c := c)
+    (show 0 < r + δ by positivity) (show r + δ < r + 2 * δ by linarith)
+    ((Metric.closedBall_subset_closedBall hδB).trans hBU) hfin
+  have hg := controlled_radius_gradient_bound hE c hr hδ hδB hcontrol hD hL hbound
+  linarith
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+
+theorem minimizing_controlled_radius_perimeter_le {n : ℕ} {E : Set (Euc n)}
+    (hmin : IsLocallyPerimeterMinimizing E) (c : Euc n) {r B D L : ℝ}
+    (hr : 0 < r) (hrB : r < B)
+    (hcontrol : ∀ t ∈ Icc r B,
+      volume.real (E ∩ Metric.closedBall c t) - volume.real (E ∩ Metric.closedBall c r) ≤
+        D * (t - r))
+    (hD : 0 ≤ D) (hL : 0 < L)
+    (hbound : ∀ s, |deriv Real.smoothTransition s| ≤ L) :
+    perimeter (E ∩ Metric.closedBall c r) univ ≤ ENNReal.ofReal (8 * L * D) := by
+  let U := Metric.ball c (B + 1)
+  let F := E \ Metric.closedBall c r
+  have hfin : perimeter E U ≠ ⊤ := (hmin.2.1 c (B + 1) (by linarith)).ne
+  have hcut := controlled_radius_perimeter_cut hmin.1 c hr hrB
+    (show Metric.closedBall c B ⊆ U from Metric.closedBall_subset_ball (by linarith))
+    hcontrol hD hL hbound hfin
+  have hcl : closure (symmDiff E F) ⊆ Metric.closedBall c r := by
+    apply closure_minimal _ Metric.isClosed_closedBall
+    intro x hx
+    by_contra hc
+    rcases hx with hx | hx
+    · exact hx.2 ⟨hx.1, hc⟩
+    · exact hx.2 hx.1.1
+  have hcmp : perimeter E U ≤ perimeter F U :=
+    hmin.2.2 c (B + 1) (by linarith) F (hmin.1.diff measurableSet_closedBall)
+      ((isCompact_closedBall c r).of_isClosed_subset isClosed_closure hcl)
+      (hcl.trans (Metric.closedBall_subset_ball (by linarith)))
+  have hout : perimeter F U ≠ ⊤ := ne_top_of_le_ne_top
+    (ENNReal.add_ne_top.mpr ⟨hfin, ENNReal.ofReal_ne_top⟩)
+    ((le_add_left le_rfl).trans hcut)
+  have hlast := hcut.trans (add_le_add hcmp le_rfl)
+  rw [add_comm (perimeter (E ∩ Metric.closedBall c r) univ)] at hlast
+  exact (ENNReal.add_le_add_iff_left hout).mp hlast
+
+theorem exists_minimizing_volume_cut {n : ℕ} {E : Set (Euc n)}
+    (hmin : IsLocallyPerimeterMinimizing E) (c : Euc n) {a b L : ℝ}
+    (ha : 0 < a) (hab : a < b)
+    (hv : 0 < volume.real (E ∩ Metric.closedBall c b))
+    (hL : 0 < L) (hbound : ∀ s, |deriv Real.smoothTransition s| ≤ L) :
+    ∃ r ∈ Ico a b, perimeter (E ∩ Metric.closedBall c r) univ ≤
+      ENNReal.ofReal (16 * L * volume.real (E ∩ Metric.closedBall c b) / (b - a)) := by
+  let D := 2 * volume.real (E ∩ Metric.closedBall c b) / (b - a)
+  have hD : 0 < D := by dsimp [D]; positivity
+  have hDwidth : D * (b - a) = 2 * volume.real (E ∩ Metric.closedBall c b) := by
+    dsimp [D]
+    exact div_mul_cancel₀ _ (sub_pos.mpr hab).ne'
+  obtain ⟨r, hr, hcontrol⟩ := exists_controlled_cut_radius E c ha hab
+    (show volume.real (E ∩ Metric.closedBall c b) < D * (b - a) by rw [hDwidth]; linarith)
+  refine ⟨r, hr, ?_⟩
+  have h := minimizing_controlled_radius_perimeter_le hmin c (ha.trans_le hr.1) hr.2
+    hcontrol hD.le hL hbound
+  convert h using 1
+  dsimp [D]
+  congr 1
+  ring
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+
+theorem nonlinear_iteration_geometric_bound {u : ℕ → ℝ} {p C B A q : ℝ}
+    (hp : 0 ≤ p) (hC : 0 ≤ C) (hB : 0 ≤ B) (hA : 0 ≤ A) (hq : 0 ≤ q)
+    (hu : ∀ i, 0 ≤ u i) (hstart : u 0 ≤ A)
+    (hstep : ∀ i, u (i + 1) ≤ C * B ^ i * (u i) ^ p)
+    (hAstep : C * A ^ p ≤ A * q) (hqstep : B * q ^ p ≤ q) :
+    ∀ i, u i ≤ A * q ^ i := by
+  intro i
+  induction i with
+  | zero => simpa using hstart
+  | succ i ih =>
+    have hpow : (q ^ i) ^ p = (q ^ p) ^ i := by
+      rw [← Real.rpow_natCast_mul hq, mul_comm (i : ℝ), Real.rpow_mul_natCast hq]
+    calc
+      u (i + 1) ≤ C * B ^ i * (u i) ^ p := hstep i
+      _ ≤ C * B ^ i * (A * q ^ i) ^ p := by gcongr; exact hu i
+      _ = (C * A ^ p) * (B * q ^ p) ^ i := by
+        rw [Real.mul_rpow hA (pow_nonneg hq _), hpow, mul_pow]
+        ring
+      _ ≤ (A * q) * q ^ i := by
+        exact mul_le_mul hAstep (pow_le_pow_left₀ (mul_nonneg hB (Real.rpow_nonneg hq _)) hqstep i)
+          (pow_nonneg (mul_nonneg hB (Real.rpow_nonneg hq _)) _) (mul_nonneg hA hq)
+      _ = A * q ^ (i + 1) := by rw [pow_succ]; ring
+
+theorem exists_nonlinear_iteration_threshold {p C B : ℝ}
+    (hp : 1 < p) (hC : 0 < C) (hB : 1 < B) :
+    ∃ A q : ℝ, 0 < A ∧ 0 < q ∧ q < 1 ∧
+      C * A ^ p ≤ A * q ∧ B * q ^ p ≤ q := by
+  let q := (1 / (2 * B)) ^ (p - 1)⁻¹
+  have he : 0 < p - 1 := by linarith
+  have hbase : 0 < 1 / (2 * B) := by positivity
+  have hq : 0 < q := Real.rpow_pos_of_pos hbase _
+  have hq1 : q < 1 := by
+    apply Real.rpow_lt_one hbase.le _ (inv_pos.mpr he)
+    rw [div_lt_one (by linarith : 0 < 2 * B)]
+    linarith
+  have hqpow : q ^ (p - 1) = 1 / (2 * B) := by
+    dsimp [q]
+    rw [← Real.rpow_mul hbase.le, inv_mul_cancel₀ he.ne', Real.rpow_one]
+  let A := (q / (2 * C)) ^ (p - 1)⁻¹
+  have hbaseA : 0 < q / (2 * C) := by positivity
+  have hA : 0 < A := Real.rpow_pos_of_pos hbaseA _
+  have hApow : A ^ (p - 1) = q / (2 * C) := by
+    dsimp [A]
+    rw [← Real.rpow_mul hbaseA.le, inv_mul_cancel₀ he.ne', Real.rpow_one]
+  have hrewrite : p = (p - 1) + 1 := by ring
+  refine ⟨A, q, hA, hq, hq1, ?_, ?_⟩
+  · rw [hrewrite, Real.rpow_add hA, hApow, Real.rpow_one]
+    have heq : C * (q / (2 * C) * A) = A * q / 2 := by field_simp
+    rw [heq]
+    nlinarith
+  · rw [hrewrite, Real.rpow_add hq, hqpow, Real.rpow_one]
+    have heq : B * (1 / (2 * B) * q) = q / 2 := by field_simp
+    rw [heq]
+    linarith
+
+theorem nonlinear_iteration_tendsto_zero {p C B : ℝ}
+    (hp : 1 < p) (hC : 0 < C) (hB : 1 < B) :
+    ∃ A : ℝ, 0 < A ∧ ∀ u : ℕ → ℝ, (∀ i, 0 ≤ u i) → u 0 ≤ A →
+      (∀ i, u (i + 1) ≤ C * B ^ i * (u i) ^ p) → Tendsto u atTop (𝓝 0) := by
+  obtain ⟨A, q, hA, hq, hq1, hAstep, hqstep⟩ := exists_nonlinear_iteration_threshold hp hC hB
+  refine ⟨A, hA, ?_⟩
+  intro u hu hstart hstep
+  have hmajor := nonlinear_iteration_geometric_bound (by linarith : 0 ≤ p) hC.le
+    (by linarith : 0 ≤ B) hA.le hq.le hu hstart hstep hAstep hqstep
+  apply squeeze_zero hu hmajor
+  simpa using (tendsto_const_nhds (x := A)).mul (tendsto_pow_atTop_nhds_zero_of_lt_one hq.le hq1)
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+
+theorem minimizing_volume_recursion {n : ℕ} {p : ℝ}
+    (hp : Real.HolderConjugate (n : ℝ) p) {E : Set (Euc n)}
+    (hmin : IsLocallyPerimeterMinimizing E) (c : Euc n) {a b L : ℝ}
+    (ha : 0 < a) (hab : a < b) (hL : 0 < L)
+    (hbound : ∀ s, |deriv Real.smoothTransition s| ≤ L) :
+    volume.real (E ∩ Metric.closedBall c a) ≤
+      (lintegralPowLePowLIntegralFDerivConst (volume : Measure (Euc n)) p : ℝ) *
+        (16 * L * volume.real (E ∩ Metric.closedBall c b) / (b - a)) ^ p := by
+  have hv : 0 ≤ volume.real (E ∩ Metric.closedBall c b) := ENNReal.toReal_nonneg
+  by_cases hz : volume.real (E ∩ Metric.closedBall c b) = 0
+  · have hm : volume.real (E ∩ Metric.closedBall c a) ≤ 0 := by
+      rw [← hz]
+      exact measureReal_mono
+        (inter_subset_inter_right E (Metric.closedBall_subset_closedBall hab.le))
+        (measure_ne_top_of_subset inter_subset_right (isCompact_closedBall c b).measure_ne_top)
+    exact hm.trans (by positivity)
+  obtain ⟨r, hr, hcut⟩ := exists_minimizing_volume_cut hmin c ha hab
+    (lt_of_le_of_ne hv (Ne.symm hz)) hL hbound
+  have hiso := raw_perimeter_isoperimetric hp (hmin.1.inter measurableSet_closedBall)
+    (((isCompact_closedBall c r).isBounded).subset inter_subset_right)
+  have hvol : volume (E ∩ Metric.closedBall c a) ≤
+      (lintegralPowLePowLIntegralFDerivConst (volume : Measure (Euc n)) p : ℝ≥0∞) *
+        (ENNReal.ofReal (16 * L * volume.real (E ∩ Metric.closedBall c b) / (b - a))) ^ p := by
+    calc
+      _ ≤ volume (E ∩ Metric.closedBall c r) := measure_mono
+        (inter_subset_inter_right E (Metric.closedBall_subset_closedBall hr.1))
+      _ ≤ _ := hiso.trans (mul_le_mul_right (ENNReal.rpow_le_rpow hcut hp.symm.pos.le) _)
+  have hQ : 0 ≤ 16 * L * volume.real (E ∩ Metric.closedBall c b) / (b - a) := by positivity
+  have hne : (lintegralPowLePowLIntegralFDerivConst (volume : Measure (Euc n)) p : ℝ≥0∞) *
+      (ENNReal.ofReal (16 * L * volume.real (E ∩ Metric.closedBall c b) / (b - a))) ^ p ≠ ⊤ := by
+    apply ENNReal.mul_ne_top ENNReal.coe_ne_top
+    exact ENNReal.rpow_ne_top_of_nonneg hp.symm.pos.le ENNReal.ofReal_ne_top
+  have hreal := ENNReal.toReal_mono hne hvol
+  rw [ENNReal.toReal_mul, ← ENNReal.toReal_rpow, ENNReal.toReal_ofReal hQ] at hreal
+  simpa only [measureReal_def, ENNReal.coe_toReal] using hreal
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+
+def clearingRadius (R : ℝ) (i : ℕ) : ℝ := R / 2 + R / (2 * (2 : ℝ) ^ i)
+
+theorem clearingRadius_zero (R : ℝ) : clearingRadius R 0 = R := by
+  simp [clearingRadius]
+
+theorem clearingRadius_sub_succ (R : ℝ) (i : ℕ) :
+    clearingRadius R i - clearingRadius R (i + 1) = R / (4 * (2 : ℝ) ^ i) := by
+  unfold clearingRadius
+  rw [pow_succ]
+  field_simp
+  ring
+
+theorem clearingRadius_pos {R : ℝ} (hR : 0 < R) (i : ℕ) : 0 < clearingRadius R i := by
+  unfold clearingRadius
+  positivity
+
+theorem clearingRadius_half_le {R : ℝ} (hR : 0 ≤ R) (i : ℕ) : R / 2 ≤ clearingRadius R i := by
+  unfold clearingRadius
+  exact le_add_of_nonneg_right (div_nonneg hR (by positivity))
+
+theorem clearingRadius_succ_lt {R : ℝ} (hR : 0 < R) (i : ℕ) :
+    clearingRadius R (i + 1) < clearingRadius R i := by
+  have hpos : 0 < clearingRadius R i - clearingRadius R (i + 1) := by
+    rw [clearingRadius_sub_succ]
+    positivity
+  linarith
+
+theorem minimizing_clearing_out {n : ℕ} {p : ℝ}
+    (hp : Real.HolderConjugate (n : ℝ) p) {R : ℝ} (hR : 0 < R) :
+    ∃ ε : ℝ, 0 < ε ∧ ∀ E : Set (Euc n), IsLocallyPerimeterMinimizing E →
+      ∀ c : Euc n, volume.real (E ∩ Metric.closedBall c R) ≤ ε →
+        volume (E ∩ Metric.closedBall c (R / 2)) = 0 := by
+  obtain ⟨L, hL, hbound⟩ := exists_smoothTransition_deriv_bound
+  let S : ℝ := lintegralPowLePowLIntegralFDerivConst (volume : Measure (Euc n)) p
+  let K : ℝ := 64 * L / R
+  let C : ℝ := (S + 1) * K ^ p
+  let B : ℝ := (2 : ℝ) ^ p
+  have hS : 0 ≤ S := NNReal.coe_nonneg _
+  have hK : 0 < K := by dsimp [K]; positivity
+  have hC : 0 < C := by dsimp [C]; positivity
+  have hp1 : 1 < p := hp.symm.lt
+  have hB : 1 < B := Real.one_lt_rpow (by norm_num) (by linarith)
+  obtain ⟨ε, hε, hiteration⟩ := nonlinear_iteration_tendsto_zero hp1 hC hB
+  refine ⟨ε, hε, ?_⟩
+  intro E hmin c hsmall
+  let u : ℕ → ℝ := fun i ↦ volume.real (E ∩ Metric.closedBall c (clearingRadius R i))
+  have hu : ∀ i, 0 ≤ u i := fun _ ↦ ENNReal.toReal_nonneg
+  have hstart : u 0 ≤ ε := by simpa only [u, clearingRadius_zero] using hsmall
+  have hstep : ∀ i, u (i + 1) ≤ C * B ^ i * (u i) ^ p := by
+    intro i
+    have hm := minimizing_volume_recursion hp hmin c (clearingRadius_pos hR (i + 1))
+      (clearingRadius_succ_lt hR i) hL hbound
+    have heq : 16 * L * u i / (clearingRadius R i - clearingRadius R (i + 1)) = K * (2 : ℝ) ^ i * u i := by
+      rw [clearingRadius_sub_succ]
+      dsimp [K]
+      field_simp
+      ring
+    change u (i + 1) ≤ S * (16 * L * u i /
+      (clearingRadius R i - clearingRadius R (i + 1))) ^ p at hm
+    rw [heq] at hm
+    have hpow : ((2 : ℝ) ^ i) ^ p = ((2 : ℝ) ^ p) ^ i := by
+      rw [← Real.rpow_natCast_mul (by norm_num : (0 : ℝ) ≤ 2),
+        mul_comm (i : ℝ), Real.rpow_mul_natCast (by norm_num : (0 : ℝ) ≤ 2)]
+    calc
+      u (i + 1) ≤ S * (K * (2 : ℝ) ^ i * u i) ^ p := hm
+      _ = (S * K ^ p) * B ^ i * (u i) ^ p := by
+        rw [Real.mul_rpow (by positivity) (hu i),
+          Real.mul_rpow hK.le (by positivity), hpow]
+        dsimp [B]
+        ring
+      _ ≤ C * B ^ i * (u i) ^ p := by
+        unfold C
+        gcongr
+        linarith
+  have ht := hiteration u hu hstart hstep
+  have hzero : volume.real (E ∩ Metric.closedBall c (R / 2)) = 0 := by
+    apply le_antisymm _ ENNReal.toReal_nonneg
+    apply ge_of_tendsto ht
+    exact Eventually.of_forall fun i ↦ measureReal_mono
+      (inter_subset_inter_right E (Metric.closedBall_subset_closedBall (clearingRadius_half_le hR.le i)))
+      (measure_ne_top_of_subset inter_subset_right (isCompact_closedBall c (clearingRadius R i)).measure_ne_top)
+  exact ((ENNReal.toReal_eq_zero_iff _).mp hzero).resolve_right
+    (measure_ne_top_of_subset inter_subset_right (isCompact_closedBall c (R / 2)).measure_ne_top)
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/PerimeterAdmissiblePartitionPiece.lean b/lean/OAI/Geometry/Perimeter/Regularity/PerimeterAdmissiblePartitionPiece.lean
new file mode 100644
index 0000000000000000000000000000000000000000..1a752301b78c8f6826d58c8f8d8904e1ec8e8f09
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/PerimeterAdmissiblePartitionPiece.lean
@@ -0,0 +1,535 @@
+import OAI.Geometry.Perimeter.Regularity.IndicatorConvolutionVariationLocalLe
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+open scoped Manifold
+
+theorem perimeterAdmissible_partition_piece {n : ℕ} {ι : Type*}
+    {E U : Set (Euc n)} {V : Euc n → Euc n}
+    (hV : PerimeterAdmissible U V)
+    (ρ : SmoothPartitionOfUnity ι 𝓘(ℝ, Euc n) (Euc n) E)
+    (W : ι → Set (Euc n)) (hρ : ρ.IsSubordinate W) (i : ι) :
+    PerimeterAdmissible (W i) (fun x ↦ ρ i x • V x) := by
+  refine ⟨(ρ i).contMDiff.contDiff.smul hV.1,
+    (hV.2.1.smul_left (f := fun x ↦ ρ i x)), ?_, ?_⟩
+  · exact (tsupport_smul_subset_left _ _).trans (hρ i)
+  · intro x
+    rw [norm_smul, Real.norm_eq_abs, abs_of_nonneg (ρ.nonneg i x)]
+    calc
+      ρ i x * ‖V x‖ ≤ 1 * ‖V x‖ :=
+        mul_le_mul_of_nonneg_right (ρ.le_one i x) (norm_nonneg _)
+      _ ≤ 1 := by simpa only [one_mul] using hV.2.2.2 x
+
+theorem divergence_finset_sum {n : ℕ} {ι : Type*} (s : Finset ι)
+    (V : ι → Euc n → Euc n) (hV : ∀ i ∈ s, Differentiable ℝ (V i)) :
+    divergence (∑ i ∈ s, V i) = ∑ i ∈ s, divergence (V i) := by
+  classical
+  induction s using Finset.induction_on with
+  | empty =>
+    simp only [Finset.sum_empty]
+    funext x
+    simp [divergence]
+  | @insert i s hi ih =>
+    rw [Finset.sum_insert hi, Finset.sum_insert hi,
+      divergence_add (hV i (by simp))
+        (Differentiable.sum (fun j hj ↦ hV j (by simp [hj]))),
+      ih (fun j hj ↦ hV j (by simp [hj]))]
+
+theorem perimeter_iUnion_open_le {n : ℕ} (E : Set (Euc n))
+    {ι : Type*} (U : ι → Set (Euc n)) (hU : ∀ i, IsOpen (U i)) :
+    perimeter E (⋃ i, U i) ≤ ∑' i, perimeter E (U i) := by
+  classical
+  rw [perimeter_eq_iSup_tests]
+  refine iSup_le fun V ↦ iSup_le fun hV ↦ ?_
+  obtain ⟨s, hs⟩ := hV.2.1.elim_finite_subcover U hU hV.2.2.1
+  have hs' : tsupport V ⊆ ⋃ i : s, U i := by
+    simpa only [iUnion_subtype, iUnion_coe_set] using hs
+  obtain ⟨ρ, hρ⟩ := SmoothPartitionOfUnity.exists_isSubordinate
+    𝓘(ℝ, Euc n) (isClosed_tsupport V) (fun i : s ↦ U i) (fun i ↦ hU i) hs'
+  let W : s → Euc n → Euc n := fun i x ↦ ρ i x • V x
+  have hW : ∀ i : s, PerimeterAdmissible (U i) (W i) :=
+    fun i ↦ perimeterAdmissible_partition_piece hV ρ _ hρ i
+  have hsum : V = ∑ i : s, W i := by
+    funext x
+    simp only [Finset.sum_apply, W, ← Finset.sum_smul]
+    by_cases hx : x ∈ tsupport V
+    · have ht := ρ.sum_eq_one hx
+      rw [finsum_eq_sum_of_fintype] at ht
+      rw [ht, one_smul]
+    · rw [image_eq_zero_of_notMem_tsupport hx, smul_zero]
+  calc
+    ENNReal.ofReal (∫ x in E, divergence V x) =
+        ENNReal.ofReal (∑ i : s, ∫ x in E, divergence (W i) x) := by
+      rw [hsum, divergence_finset_sum _ W (fun i _ ↦ (hW i).1.differentiable (by simp))]
+      simp only [Finset.sum_apply]
+      rw [integral_finsetSum _ (fun i _ ↦
+        (integrable_divergence (hW i).1 (hW i).2.1).integrableOn)]
+    _ ≤ ∑ i : s, ENNReal.ofReal (∫ x in E, divergence (W i) x) :=
+      by
+      have h (m : Finset s) :
+          ENNReal.ofReal (∑ i ∈ m, ∫ x in E, divergence (W i) x) ≤
+          ∑ i ∈ m, ENNReal.ofReal (∫ x in E, divergence (W i) x) := by
+        induction m using Finset.induction_on with
+        | empty => simp
+        | @insert i m hi ih =>
+          simp only [Finset.sum_insert hi]
+          exact ENNReal.ofReal_add_le.trans (add_le_add le_rfl ih)
+      exact h Finset.univ
+    _ ≤ ∑ i : s, perimeter E (U i) := by
+      apply Finset.sum_le_sum
+      intro i hi
+      rw [perimeter_eq_iSup_tests]
+      exact le_iSup_of_le (W i) (le_iSup_of_le (hW i) le_rfl)
+    _ ≤ ∑' i, perimeter E (U i) := by
+      exact (Finset.sum_coe_sort s (fun i ↦ perimeter E (U i))).le.trans
+        (ENNReal.sum_le_tsum s)
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+
+@[simp] theorem raw_perimeter_empty_region {n : ℕ} (E : Set (Euc n)) :
+    perimeter E ∅ = 0 := by
+  apply le_antisymm _ zero_le
+  rw [perimeter_eq_iSup_tests]
+  refine iSup_le fun V ↦ iSup_le fun hV ↦ ?_
+  have hzero : V = 0 := by
+    funext x
+    exact image_eq_zero_of_notMem_tsupport (fun hx ↦ hV.2.2.1 hx)
+  simp [hzero, divergence]
+
+noncomputable def openPerimeterCost {n : ℕ} (E U : Set (Euc n)) : ℝ≥0∞ :=
+  by classical exact if IsOpen U then perimeter E U else ⊤
+
+@[simp] theorem openPerimeterCost_empty {n : ℕ} (E : Set (Euc n)) :
+    openPerimeterCost E ∅ = 0 := by
+  simp [openPerimeterCost]
+
+noncomputable def perimeterOuterMeasure {n : ℕ} (E : Set (Euc n)) :
+    OuterMeasure (Euc n) :=
+  OuterMeasure.ofFunction (openPerimeterCost E) (openPerimeterCost_empty E)
+
+theorem perimeterOuterMeasure_le_open {n : ℕ} (E : Set (Euc n))
+    {A U : Set (Euc n)} (hU : IsOpen U) (hAU : A ⊆ U) :
+    perimeterOuterMeasure E A ≤ perimeter E U := by
+  exact ((perimeterOuterMeasure E).mono hAU).trans
+    ((OuterMeasure.ofFunction_le _).trans_eq (by simp [openPerimeterCost, hU]))
+
+theorem perimeterOuterMeasure_eq_iInf_open {n : ℕ} (E A : Set (Euc n)) :
+    perimeterOuterMeasure E A =
+      ⨅ (U : Set (Euc n)) (_ : IsOpen U) (_ : A ⊆ U), perimeter E U := by
+  apply le_antisymm
+  · exact le_iInf fun U ↦ le_iInf fun hU ↦ le_iInf fun hAU ↦
+      perimeterOuterMeasure_le_open E hU hAU
+  · unfold perimeterOuterMeasure
+    rw [OuterMeasure.ofFunction_eq_iInf_mem (openPerimeterCost E)
+      (openPerimeterCost_empty E) (P := IsOpen)
+      (fun U hU ↦ by simp [openPerimeterCost, hU])]
+    refine le_iInf fun U ↦ le_iInf fun hU ↦ le_iInf fun hAU ↦ ?_
+    calc
+      _ ≤ perimeter E (⋃ i, U i) :=
+        iInf_le_of_le (⋃ i, U i) (iInf_le_of_le (isOpen_iUnion hU)
+          (iInf_le_of_le hAU le_rfl))
+      _ ≤ ∑' i, perimeter E (U i) := perimeter_iUnion_open_le E U hU
+      _ = ∑' i, openPerimeterCost E (U i) := by
+        congr 1
+        funext i
+        simp [openPerimeterCost, hU i]
+
+theorem perimeterOuterMeasure_open {n : ℕ} (E : Set (Euc n))
+    {U : Set (Euc n)} (hU : IsOpen U) :
+    perimeterOuterMeasure E U = perimeter E U := by
+  apply le_antisymm (perimeterOuterMeasure_le_open E hU Subset.rfl)
+  rw [perimeterOuterMeasure_eq_iInf_open]
+  exact le_iInf fun W ↦ le_iInf fun _ ↦ le_iInf fun hUW ↦ raw_perimeter_mono_region E hUW
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+
+theorem separated_open_neighborhoods {n : ℕ} {A B : Set (Euc n)}
+    (hsep : Metric.AreSeparated A B) :
+    ∃ U V : Set (Euc n), IsOpen U ∧ IsOpen V ∧ A ⊆ U ∧ B ⊆ V ∧ Disjoint U V := by
+  obtain ⟨r, hr, hsep⟩ := hsep
+  have hc : IsClosed {p : Euc n × Euc n | r ≤ edist p.1 p.2} :=
+    isClosed_le continuous_const continuous_edist
+  have hcl : closure (A ×ˢ B) ⊆ {p : Euc n × Euc n | r ≤ edist p.1 p.2} :=
+    hc.closure_subset_iff.mpr (fun p hp ↦ hsep p.1 hp.1 p.2 hp.2)
+  rw [closure_prod_eq] at hcl
+  have hd : Disjoint (closure A) (closure B) := by
+    apply Set.disjoint_left.mpr
+    intro x hx hy
+    exact hr (le_antisymm (by simpa using hcl (show (x, x) ∈ closure A ×ˢ closure B from ⟨hx, hy⟩)) zero_le)
+  obtain ⟨U, V, hU, hV, hAU, hBV, hUV⟩ :=
+    normal_separation isClosed_closure isClosed_closure hd
+  exact ⟨U, V, hU, hV, subset_closure.trans hAU, subset_closure.trans hBV, hUV⟩
+
+theorem perimeterOuterMeasure_isMetric {n : ℕ} (E : Set (Euc n)) :
+    (perimeterOuterMeasure E).IsMetric := by
+  intro A B hsep
+  apply le_antisymm (measure_union_le (μ := perimeterOuterMeasure E) A B)
+  obtain ⟨U, V, hU, hV, hAU, hBV, hUV⟩ := separated_open_neighborhoods hsep
+  rw [perimeterOuterMeasure_eq_iInf_open E (A ∪ B)]
+  refine le_iInf fun O ↦ le_iInf fun hO ↦ le_iInf fun hABO ↦ ?_
+  calc
+    _ ≤ perimeter E (O ∩ U) + perimeter E (O ∩ V) := by
+      apply add_le_add
+      · exact perimeterOuterMeasure_le_open E (hO.inter hU)
+          (fun x hx ↦ ⟨hABO (Or.inl hx), hAU hx⟩)
+      · exact perimeterOuterMeasure_le_open E (hO.inter hV)
+          (fun x hx ↦ ⟨hABO (Or.inr hx), hBV hx⟩)
+    _ ≤ perimeter E O := perimeter_disjoint_superadditive E
+      inter_subset_left inter_subset_left (hUV.mono inter_subset_right inter_subset_right)
+
+noncomputable def perimeterMeasure {n : ℕ} (E : Set (Euc n)) : Measure (Euc n) :=
+  (perimeterOuterMeasure E).toMeasure (perimeterOuterMeasure_isMetric E).le_caratheodory
+
+@[simp] theorem perimeterMeasure_open {n : ℕ} (E : Set (Euc n))
+    {U : Set (Euc n)} (hU : IsOpen U) : perimeterMeasure E U = perimeter E U := by
+  rw [perimeterMeasure, toMeasure_apply _ _ hU.measurableSet,
+    perimeterOuterMeasure_open E hU]
+
+theorem perimeterMeasure_support {n : ℕ} (E : Set (Euc n)) :
+    (perimeterMeasure E).support = perimeterSupport E := by
+  ext x
+  rw [Measure.mem_support_iff_forall]
+  constructor
+  · intro hx r hr
+    rw [← perimeterMeasure_open E Metric.isOpen_ball]
+    exact hx _ (Metric.ball_mem_nhds x hr)
+  · intro hx U hU
+    obtain ⟨r, hr, hrU⟩ := Metric.mem_nhds_iff.mp hU
+    have hm : 0 < perimeterMeasure E (Metric.ball x r) := by
+      rw [perimeterMeasure_open E Metric.isOpen_ball]
+      exact hx r hr
+    exact hm.trans_le ((perimeterMeasure E).mono hrU)
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+
+theorem perimeterMeasure_locallyFinite {n : ℕ} (E : Set (Euc n))
+    (hfin : ∀ x (r : ℝ), 0 < r → perimeter E (Metric.ball x r) < ⊤) :
+    IsLocallyFiniteMeasure (perimeterMeasure E) := by
+  constructor
+  intro x
+  refine ⟨Metric.ball x 1, Metric.ball_mem_nhds x zero_lt_one, ?_⟩
+  rw [perimeterMeasure_open E Metric.isOpen_ball]
+  exact hfin x 1 zero_lt_one
+
+theorem perimeterMeasure_finiteOnCompacts {n : ℕ} (E : Set (Euc n))
+    (hfin : ∀ x (r : ℝ), 0 < r → perimeter E (Metric.ball x r) < ⊤) :
+    IsFiniteMeasureOnCompacts (perimeterMeasure E) := by
+  let := perimeterMeasure_locallyFinite E hfin
+  infer_instance
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+
+theorem measure_le_hausdorff_of_ball_growth {n k : ℕ} (μ : Measure (Euc n))
+    {C : ℝ} (hC : 0 < C)
+    (hg : ∀ x (r : ℝ), 0 < r → μ (Metric.ball x r) ≤ ENNReal.ofReal (C * r ^ k)) :
+    μ ≤ ENNReal.ofReal C • Measure.hausdorffMeasure (k : ℝ) := by
+  have hbound : ∀ s : Set (Euc n), Metric.ediam s ≤ 1 →
+      μ s ≤ ENNReal.ofReal C * Metric.ediam s ^ k := by
+    intro s hs
+    rcases s.eq_empty_or_nonempty with rfl | ⟨x, hx⟩
+    · simp
+    have hst : Metric.ediam s ≠ ⊤ := ne_top_of_le_ne_top (by simp) hs
+    have hlim : Tendsto (fun r : ℝ ↦ ENNReal.ofReal (C * (Metric.diam s + r) ^ k))
+        (𝓝[>] (0 : ℝ)) (𝓝 (ENNReal.ofReal (C * Metric.diam s ^ k))) := by
+      have hc : Continuous (fun r : ℝ ↦ ENNReal.ofReal (C * (Metric.diam s + r) ^ k)) :=
+        ENNReal.continuous_ofReal.comp (continuous_const.mul
+          ((continuous_const.add continuous_id).pow k))
+      simpa only [add_zero] using (hc.tendsto 0).mono_left nhdsWithin_le_nhds
+    have hh : μ s ≤ ENNReal.ofReal (C * Metric.diam s ^ k) := by
+      apply ge_of_tendsto hlim
+      filter_upwards [self_mem_nhdsWithin] with r hr
+      have hsub : s ⊆ Metric.ball x (Metric.diam s + r) := by
+        intro y hy
+        exact (Metric.dist_le_diam_of_mem' hst hy hx).trans_lt (lt_add_of_pos_right _ hr)
+      exact (measure_mono hsub).trans (hg x _ (add_pos_of_nonneg_of_pos Metric.diam_nonneg hr))
+    rw [ENNReal.ofReal_mul hC.le, ENNReal.ofReal_pow Metric.diam_nonneg] at hh
+    simpa only [Metric.diam, ENNReal.ofReal_toReal hst] using hh
+  have hm : (ENNReal.ofReal C)⁻¹ • μ ≤ Measure.hausdorffMeasure (k : ℝ) := by
+    apply Measure.le_hausdorffMeasure _ _ 1 zero_lt_one
+    intro s hs
+    rw [Measure.smul_apply, smul_eq_mul, ENNReal.rpow_natCast]
+    calc
+      _ ≤ (ENNReal.ofReal C)⁻¹ * (ENNReal.ofReal C * Metric.ediam s ^ k) :=
+        mul_le_mul le_rfl (hbound s hs) zero_le zero_le
+      _ = _ := by rw [← mul_assoc, ENNReal.inv_mul_cancel
+        (ENNReal.ofReal_ne_zero_iff.mpr hC) ENNReal.ofReal_ne_top, one_mul]
+  calc
+    μ = ENNReal.ofReal C • ((ENNReal.ofReal C)⁻¹ • μ) := by
+      rw [smul_smul, ENNReal.mul_inv_cancel (ENNReal.ofReal_ne_zero_iff.mpr hC)
+        ENNReal.ofReal_ne_top, one_smul]
+    _ ≤ ENNReal.ofReal C • Measure.hausdorffMeasure (k : ℝ) := by
+      apply Measure.le_iff.mpr
+      intro s hs
+      simp only [Measure.smul_apply, smul_eq_mul]
+      exact mul_le_mul le_rfl (Measure.le_iff.mp hm s hs) zero_le zero_le
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+
+theorem exists_perimeter_vitali_cover_open {n : ℕ} (E S U : Set (Euc n))
+    (hU : IsOpen U) {c₀ : ℝ} (hc₀ : 0 < c₀)
+    (hdensity : ∀ x ∈ S, ∀ᶠ r : ℝ in 𝓝[>] (0 : ℝ),
+      ENNReal.ofReal (c₀ * r ^ (n - 1)) ≤ perimeter E (Metric.ball x r))
+    {δ : ℝ} (hδ : 0 < δ) :
+    ∃ (u : Set (Euc n)) (r : Euc n → ℝ), u.Countable ∧
+      (∀ x ∈ u, 0 < r x ∧ r x ≤ δ) ∧
+      (S ∩ U ⊆ ⋃ x ∈ u, Metric.ball x (4 * r x)) ∧
+      (∑' x : u, (ENNReal.ofReal (r x)) ^ (n - 1)) ≤
+        (ENNReal.ofReal c₀)⁻¹ * perimeter E U := by
+  classical
+  let t := S ∩ U
+  have hchoose : ∀ x ∈ t, ∃ r : ℝ, 0 < r ∧ r ≤ δ ∧
+      Metric.ball x r ⊆ U ∧
+      ENNReal.ofReal (c₀ * r ^ (n - 1)) ≤ perimeter E (Metric.ball x r) := by
+    intro x hx
+    obtain ⟨ε, hε, hεU⟩ := Metric.mem_nhds_iff.mp (hU.mem_nhds hx.2)
+    have hev := hdensity x hx.1
+    have hsmall : ∀ᶠ r : ℝ in 𝓝[>] (0 : ℝ), r < min δ ε :=
+      (eventually_lt_nhds (lt_min hδ hε)).filter_mono nhdsWithin_le_nhds
+    obtain ⟨r, hr, hden, hsm⟩ :=
+      ((show ∀ᶠ r : ℝ in 𝓝[>] (0 : ℝ), 0 < r from self_mem_nhdsWithin).and
+        (hev.and hsmall)).exists
+    exact ⟨r, hr, hsm.le.trans (min_le_left _ _),
+      (Metric.ball_subset_ball (hsm.le.trans (min_le_right _ _))).trans hεU, hden⟩
+  have htotal : ∀ x : Euc n, ∃ r : ℝ, 0 < r ∧ r ≤ δ ∧
+      (x ∈ t → Metric.ball x r ⊆ U ∧
+        ENNReal.ofReal (c₀ * r ^ (n - 1)) ≤ perimeter E (Metric.ball x r)) := by
+    intro x
+    by_cases hx : x ∈ t
+    · obtain ⟨r, hr, hrδ, hrU, hden⟩ := hchoose x hx
+      exact ⟨r, hr, hrδ, fun _ ↦ ⟨hrU, hden⟩⟩
+    · refine ⟨δ / 2, half_pos hδ, (half_lt_self hδ).le, fun h ↦ ?_⟩
+      exact (hx h).elim
+  choose r hr using htotal
+  obtain ⟨u, hut, hud, hcover⟩ :=
+    Vitali.exists_disjoint_subfamily_covering_enlargement_ball t id r δ
+      (fun x hx ↦ (hr x).2.1) 4 (by norm_num)
+  have huc : u.Countable := hud.countable_of_isOpen (fun _ _ ↦ Metric.isOpen_ball)
+    (fun x hx ↦ ⟨x, Metric.mem_ball_self (hr x).1⟩)
+  refine ⟨u, r, huc, fun x hx ↦ ⟨(hr x).1, (hr x).2.1⟩, ?_, ?_⟩
+  · intro x hx
+    obtain ⟨y, hy, hxy⟩ := hcover x hx
+    exact mem_iUnion.mpr ⟨y, mem_iUnion.mpr ⟨hy, hxy (Metric.mem_ball_self (hr x).1)⟩⟩
+  · have hsum := perimeter_tsum_disjoint_le E
+      (fun x : u ↦ Metric.ball (x : Euc n) (r x)) U
+      (fun x y h ↦ hud x.property y.property (fun hxy ↦ h (Subtype.ext hxy)))
+      (fun x ↦ ((hr x).2.2 (hut x.property)).1)
+    have hweigh : ENNReal.ofReal c₀ * ∑' x : u, ENNReal.ofReal (r x) ^ (n - 1) ≤
+        perimeter E U := by
+      rw [← ENNReal.tsum_mul_left]
+      apply le_trans (ENNReal.tsum_le_tsum (fun x : u ↦ ?_)) hsum
+      rw [← ENNReal.ofReal_pow (hr x).1.le, ← ENNReal.ofReal_mul hc₀.le]
+      exact ((hr x).2.2 (hut x.property)).2
+    calc
+      _ = (ENNReal.ofReal c₀)⁻¹ *
+          (ENNReal.ofReal c₀ * ∑' x : u, ENNReal.ofReal (r x) ^ (n - 1)) := by
+        rw [← mul_assoc, ENNReal.inv_mul_cancel (ENNReal.ofReal_ne_zero_iff.mpr hc₀)
+          ENNReal.ofReal_ne_top, one_mul]
+      _ ≤ _ := mul_le_mul le_rfl hweigh zero_le zero_le
+
+theorem hausdorffMeasure_inter_open_le_perimeter {n : ℕ} (E S U : Set (Euc n))
+    (hU : IsOpen U) {c₀ : ℝ} (hc₀ : 0 < c₀)
+    (hdensity : ∀ x ∈ S, ∀ᶠ r : ℝ in 𝓝[>] (0 : ℝ),
+      ENNReal.ofReal (c₀ * r ^ (n - 1)) ≤ perimeter E (Metric.ball x r)) :
+    Measure.hausdorffMeasure (n - 1 : ℕ) (S ∩ U) ≤
+      (8 : ℝ≥0∞) ^ (n - 1) * (ENNReal.ofReal c₀)⁻¹ * perimeter E U := by
+  classical
+  let δ : ℕ → ℝ := fun j ↦ 1 / (j + 1)
+  have hδ : ∀ j, 0 < δ j := fun j ↦ by dsimp [δ]; positivity
+  choose u r hcount hr hcover hsum using fun j ↦
+    exists_perimeter_vitali_cover_open E S U hU hc₀ hdensity (hδ j)
+  let : ∀ j, Countable (u j) := fun j ↦ (hcount j).to_subtype
+  have hediam : ∀ j (x : u j),
+      Metric.ediam (Metric.ball (x : Euc n) (4 * r j x)) ≤
+        ENNReal.ofReal (8 * r j x) := by
+    intro j x
+    apply Metric.ediam_le
+    intro y hy z hz
+    rw [edist_dist]
+    apply ENNReal.ofReal_le_ofReal
+    simp only [Metric.mem_ball] at hy hz
+    calc dist y z ≤ dist y x + dist z x := dist_triangle_right _ _ _
+      _ ≤ 8 * r j x := by linarith
+  have hδlim : Tendsto (fun j ↦ ENNReal.ofReal (8 * δ j)) atTop (𝓝 0) := by
+    have hh : Tendsto δ atTop (𝓝 0) := tendsto_one_div_add_atTop_nhds_zero_nat
+    simpa only [Function.comp_def, mul_zero, ENNReal.ofReal_zero] using
+      ENNReal.continuous_ofReal.continuousAt.tendsto.comp (hh.const_mul 8)
+  have hh := Measure.hausdorffMeasure_le_liminf_tsum ((n - 1 : ℕ) : ℝ)
+    (S ∩ U) (fun j ↦ ENNReal.ofReal (8 * δ j)) hδlim
+    (fun j (x : u j) ↦ Metric.ball (x : Euc n) (4 * r j x))
+    (Filter.Eventually.of_forall (fun j x ↦
+      (hediam j x).trans (ENNReal.ofReal_le_ofReal (by
+        have h := (hr j x x.property).2; linarith))))
+    (Filter.Eventually.of_forall (fun j ↦ by simpa only [iUnion_subtype] using hcover j))
+  apply hh.trans
+  refine Filter.liminf_le_of_frequently_le ?_ (by isBoundedDefault)
+  apply Filter.Eventually.frequently
+  apply Filter.Eventually.of_forall
+  intro j
+  calc
+    _ ≤ ∑' x : u j, (8 : ℝ≥0∞) ^ (n - 1) *
+        ENNReal.ofReal (r j x) ^ (n - 1) := by
+      apply ENNReal.tsum_le_tsum
+      intro x
+      rw [ENNReal.rpow_natCast]
+      calc
+        _ ≤ (ENNReal.ofReal (8 * r j x)) ^ (n - 1) := pow_le_pow_left' (hediam j x) _
+        _ = _ := by rw [ENNReal.ofReal_mul (by norm_num : (0 : ℝ) ≤ 8), mul_pow]; norm_num
+    _ = (8 : ℝ≥0∞) ^ (n - 1) * ∑' x : u j, ENNReal.ofReal (r j x) ^ (n - 1) :=
+      ENNReal.tsum_mul_left
+    _ ≤ _ := by simpa only [mul_assoc] using mul_le_mul le_rfl (hsum j) zero_le zero_le
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+
+theorem hausdorff_restrict_le_perimeterMeasure {n : ℕ} (E S : Set (Euc n))
+    {c₀ : ℝ} (hc₀ : 0 < c₀)
+    (hdensity : ∀ x ∈ S, ∀ᶠ r : ℝ in 𝓝[>] (0 : ℝ),
+      ENNReal.ofReal (c₀ * r ^ (n - 1)) ≤ perimeter E (Metric.ball x r)) :
+    (Measure.hausdorffMeasure (n - 1 : ℕ)).restrict S ≤
+      ((8 : ℝ≥0∞) ^ (n - 1) * (ENNReal.ofReal c₀)⁻¹) • perimeterMeasure E := by
+  let K : ℝ≥0∞ := 8 ^ (n - 1) * (ENNReal.ofReal c₀)⁻¹
+  have hc0 : ENNReal.ofReal c₀ ≠ 0 := ENNReal.ofReal_ne_zero_iff.mpr hc₀
+  have hK0 : K ≠ 0 := by
+    exact mul_ne_zero (pow_ne_zero _ (by norm_num : (8 : ℝ≥0∞) ≠ 0))
+      (ENNReal.inv_ne_zero.mpr ENNReal.ofReal_ne_top)
+  have hKt : K ≠ ⊤ := by
+    apply ENNReal.mul_ne_top
+    · exact ENNReal.pow_ne_top (by norm_num)
+    · exact ENNReal.inv_ne_top.mpr hc0
+  apply Measure.le_iff.mpr
+  intro A hA
+  rw [Measure.restrict_apply hA, Measure.smul_apply, smul_eq_mul]
+  change Measure.hausdorffMeasure (n - 1 : ℕ) (A ∩ S) ≤ K * perimeterMeasure E A
+  rw [perimeterMeasure, toMeasure_apply _ _ hA, perimeterOuterMeasure_eq_iInf_open,
+    ENNReal.mul_iInf_of_ne hK0 hKt]
+  refine le_iInf fun U ↦ ?_
+  rw [ENNReal.mul_iInf_of_ne hK0 hKt]
+  refine le_iInf fun hU ↦ ?_
+  rw [ENNReal.mul_iInf_of_ne hK0 hKt]
+  refine le_iInf fun hAU ↦ ?_
+  calc
+    _ ≤ Measure.hausdorffMeasure (n - 1 : ℕ) (S ∩ U) :=
+      (Measure.hausdorffMeasure _).mono (fun x hx ↦ ⟨hx.2, hAU hx.1⟩)
+    _ ≤ _ := hausdorffMeasure_inter_open_le_perimeter E S U hU hc₀ hdensity
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+
+theorem minimizing_perimeter_hausdorff_comparison {n : ℕ} (hn : 1 < n) :
+    ∃ C D : ℝ≥0∞, C ≠ 0 ∧ C ≠ ⊤ ∧ D ≠ 0 ∧ D ≠ ⊤ ∧
+      ∀ E : Set (Euc n), IsLocallyPerimeterMinimizing E →
+        perimeterMeasure E ≤ C • ((Measure.hausdorffMeasure (n - 1 : ℕ)).restrict
+          (perimeterSupport E)) ∧
+        (Measure.hausdorffMeasure (n - 1 : ℕ)).restrict (perimeterSupport E) ≤
+          D • perimeterMeasure E := by
+  obtain ⟨C, hC, hg⟩ := exists_locallyMinimizing_perimeter_growth_constant n (by omega)
+  obtain ⟨c₀, hc₀, hd⟩ := exists_minimizing_perimeter_lower_density n hn
+  let D : ℝ≥0∞ := 8 ^ (n - 1) * (ENNReal.ofReal c₀)⁻¹
+  have hD0 : D ≠ 0 := mul_ne_zero (pow_ne_zero _ (by norm_num))
+    (ENNReal.inv_ne_zero.mpr ENNReal.ofReal_ne_top)
+  have hDt : D ≠ ⊤ := ENNReal.mul_ne_top (ENNReal.pow_ne_top (by norm_num))
+    (ENNReal.inv_ne_top.mpr (ENNReal.ofReal_ne_zero_iff.mpr hc₀))
+  refine ⟨ENNReal.ofReal C, D, ENNReal.ofReal_ne_zero_iff.mpr hC,
+    ENNReal.ofReal_ne_top, hD0, hDt, ?_⟩
+  intro E hmin
+  constructor
+  · have hh := measure_le_hausdorff_of_ball_growth (perimeterMeasure E) (k := n - 1) hC
+      (fun x r hr ↦ by rw [perimeterMeasure_open E Metric.isOpen_ball]; exact hg E hmin x r hr)
+    have hs : ∀ᵐ x ∂perimeterMeasure E, x ∈ perimeterSupport E := by
+      rw [← perimeterMeasure_support]
+      exact (perimeterMeasure E).support_mem_ae
+    have hres := Measure.restrict_mono_measure hh (perimeterSupport E)
+    rwa [Measure.restrict_eq_self_of_ae_mem hs, Measure.restrict_smul] at hres
+  · apply hausdorff_restrict_le_perimeterMeasure E (perimeterSupport E) hc₀
+    intro x hx
+    filter_upwards [self_mem_nhdsWithin] with r hr
+    exact hd E hmin x hx r hr
+
+theorem minimizing_perimeter_hausdorff_mutuallyAC {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E) :
+    perimeterMeasure E ≪ (Measure.hausdorffMeasure (n - 1 : ℕ)).restrict (perimeterSupport E) ∧
+    (Measure.hausdorffMeasure (n - 1 : ℕ)).restrict (perimeterSupport E) ≪ perimeterMeasure E := by
+  obtain ⟨C, D, _, _, _, _, hb⟩ := minimizing_perimeter_hausdorff_comparison hn
+  exact ⟨Measure.absolutelyContinuous_of_le_smul (hb E hmin).1,
+    Measure.absolutelyContinuous_of_le_smul (hb E hmin).2⟩
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/PerimeterEqOfAe.lean b/lean/OAI/Geometry/Perimeter/Regularity/PerimeterEqOfAe.lean
new file mode 100644
index 0000000000000000000000000000000000000000..5fd0df53d99b004e786bc24604266a178d1d206f
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/PerimeterEqOfAe.lean
@@ -0,0 +1,469 @@
+import OAI.Geometry.Perimeter.Regularity.SurfaceGradientComp
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+
+theorem perimeter_eq_of_ae {n : ℕ} {E F : Set (Euc n)}
+    (hEF : E =ᵐ[volume] F) (U : Set (Euc n)) : perimeter E U = perimeter F U := by
+  unfold perimeter
+  rw [Measure.restrict_congr_set hEF]
+
+theorem perimeterSupport_eq_of_ae {n : ℕ} {E F : Set (Euc n)}
+    (hEF : E =ᵐ[volume] F) : perimeterSupport E = perimeterSupport F := by
+  ext x
+  simp only [perimeterSupport, mem_ofPred_eq, perimeter_eq_of_ae hEF]
+
+theorem hasHalfspaceBlowup_iff_of_ae {n : ℕ} {E F : Set (Euc n)}
+    (hEF : E =ᵐ[volume] F) (x v : Euc n) :
+    HasHalfspaceBlowup E x v ↔ HasHalfspaceBlowup F x v := by
+  classical
+  have hi : E.indicator (fun _ ↦ (1 : ℝ)) =ᵐ[volume] F.indicator (fun _ ↦ (1 : ℝ)) := by
+    filter_upwards [hEF.mem_iff] with y hy
+    by_cases hx : y ∈ E
+    · simp only [indicator_of_mem hx, indicator_of_mem (hy.mp hx)]
+    · simp only [indicator_of_notMem hx, indicator_of_notMem (mt hy.mpr hx)]
+  have hint : ∀ r : ℝ, (∫ y in Metric.ball x r,
+      |E.indicator (fun _ ↦ (1 : ℝ)) y -
+        {z | inner ℝ (z - x) v < 0}.indicator (fun _ ↦ (1 : ℝ)) y|) =
+      ∫ y in Metric.ball x r,
+      |F.indicator (fun _ ↦ (1 : ℝ)) y -
+        {z | inner ℝ (z - x) v < 0}.indicator (fun _ ↦ (1 : ℝ)) y| := by
+    intro r
+    apply integral_congr_ae
+    filter_upwards [hi.filter_mono (ae_mono Measure.restrict_le_self)] with y hy
+    rw [hy]
+  unfold HasHalfspaceBlowup
+  simp only [hint]
+
+theorem approximateNormal_eq_of_ae {n : ℕ} {E F : Set (Euc n)}
+    (hEF : E =ᵐ[volume] F) (x : Euc n) : approximateNormal E x = approximateNormal F x := by
+  have hp : HasHalfspaceBlowup E x = HasHalfspaceBlowup F x :=
+    funext fun v ↦ propext (hasHalfspaceBlowup_iff_of_ae hEF x v)
+  unfold approximateNormal
+  rw [hp]
+
+theorem tangentialGradient_eq_of_ae {n : ℕ} {E F : Set (Euc n)}
+    (hEF : E =ᵐ[volume] F) (φ : Euc n → ℝ) :
+    tangentialGradient E φ = tangentialGradient F φ := by
+  funext x
+  simp only [tangentialGradient, approximateNormal_eq_of_ae hEF]
+
+theorem boundaryArea_eq_of_ae {n : ℕ} {E F : Set (Euc n)}
+    (hEF : E =ᵐ[volume] F) : boundaryArea E = boundaryArea F := by
+  simp only [boundaryArea, perimeterSupport_eq_of_ae hEF]
+
+theorem wholeBoundarySobolev_iff_of_ae {n : ℕ} {E F : Set (Euc n)}
+    (hEF : E =ᵐ[volume] F) (h : Euc n → ℝ) (G : Euc n → Euc n) :
+    WholeBoundarySobolev E h G ↔ WholeBoundarySobolev F h G := by
+  simp only [WholeBoundarySobolev, boundaryArea_eq_of_ae hEF, tangentialGradient_eq_of_ae hEF]
+
+theorem wholeBoundarySupersolution_iff_of_ae {n : ℕ} {E F : Set (Euc n)}
+    (hEF : E =ᵐ[volume] F) (G : Euc n → Euc n) :
+    WholeBoundarySupersolution E G ↔ WholeBoundarySupersolution F G := by
+  simp only [WholeBoundarySupersolution, boundaryArea_eq_of_ae hEF, tangentialGradient_eq_of_ae hEF]
+
+theorem zeroEssentialInfAtVertex_iff_of_ae {n : ℕ} {E F : Set (Euc n)}
+    (hEF : E =ᵐ[volume] F) (h : Euc n → ℝ) :
+    ZeroEssentialInfAtVertex E h ↔ ZeroEssentialInfAtVertex F h := by
+  simp only [ZeroEssentialInfAtVertex, boundaryArea_eq_of_ae hEF]
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+
+theorem IsLocallyPerimeterMinimizing.of_ae {n : ℕ} {E F : Set (Euc n)}
+    (hmin : IsLocallyPerimeterMinimizing E) (hEF : E =ᵐ[volume] F)
+    (hF : MeasurableSet F) : IsLocallyPerimeterMinimizing F := by
+  obtain ⟨hE, hfin, hcomp⟩ := hmin
+  refine ⟨hF, ?_, ?_⟩
+  · intro x r hr
+    rw [← perimeter_eq_of_ae hEF]
+    exact hfin x r hr
+  · intro x r hr G hG hKc hKB
+    let K := closure (symmDiff F G)
+    let H := (G ∩ K) ∪ (E \ K)
+    have hK : IsClosed K := isClosed_closure
+    have hHm : MeasurableSet H := (hG.inter hK.measurableSet).union (hE.diff hK.measurableSet)
+    have hFGout : ∀ y ∉ K, y ∈ F ↔ y ∈ G := by
+      intro y hy
+      have hdiff : y ∉ symmDiff F G := fun h ↦ hy (subset_closure h)
+      simp only [mem_symmDiff] at hdiff
+      tauto
+    have hHG : H =ᵐ[volume] G := by
+      apply Filter.eventuallyEqSet_iff.mpr
+      filter_upwards [hEF.mem_iff] with y hy
+      by_cases hyK : y ∈ K
+      · simp only [H, mem_union, mem_inter_iff, Set.mem_sdiff, hyK, and_true, not_true_eq_false,
+          and_false, or_false]
+      · have ho := hFGout y hyK
+        simp only [H, mem_union, mem_inter_iff, Set.mem_sdiff, hyK, and_false, not_false_eq_true,
+          and_true, false_or]
+        exact hy.trans ho
+    have hEH : symmDiff E H ⊆ K := by
+      intro y hy
+      by_contra hn
+      have hne : y ∈ H ↔ y ∈ E := by
+        simp only [H, mem_union, mem_inter_iff, Set.mem_sdiff, hn, and_false, not_false_eq_true,
+          and_true, false_or]
+      simp only [mem_symmDiff] at hy
+      tauto
+    have hcl : closure (symmDiff E H) ⊆ K := closure_minimal hEH hK
+    calc
+      perimeter F (Metric.ball x r) = perimeter E (Metric.ball x r) := (perimeter_eq_of_ae hEF _).symm
+      _ ≤ perimeter H (Metric.ball x r) := hcomp x r hr H hHm
+        (hKc.of_isClosed_subset isClosed_closure hcl) (hcl.trans hKB)
+      _ = perimeter G (Metric.ball x r) := perimeter_eq_of_ae hHG _
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+
+theorem tsupport_divergence_subset {n : ℕ} (V : Euc n → Euc n) :
+    tsupport (divergence V) ⊆ tsupport V := by
+  apply closure_minimal
+  · intro x hx
+    by_contra h
+    have hd : fderiv ℝ V x = 0 := fderiv_of_notMem_tsupport ℝ h
+    exact hx (by simp only [divergence, hd, zero_apply, inner_zero_right,
+      Finset.sum_const_zero])
+  · exact isClosed_tsupport V
+
+theorem continuous_divergence {n : ℕ} {V : Euc n → Euc n}
+    (hV : ContDiff ℝ ∞ V) : Continuous (divergence V) := by
+  unfold divergence
+  apply continuous_finsetSum
+  intro i hi
+  exact continuous_const.inner ((hV.continuous_fderiv (by norm_num)).clm_apply continuous_const)
+
+theorem integrable_divergence {n : ℕ} {V : Euc n → Euc n}
+    (hV : ContDiff ℝ ∞ V) (hc : HasCompactSupport V) :
+    Integrable (divergence V) volume :=
+  (continuous_divergence hV).integrable_of_hasCompactSupport
+    (hc.of_isClosed_subset (isClosed_tsupport _) (tsupport_divergence_subset V))
+
+theorem divergence_add {n : ℕ} {V W : Euc n → Euc n}
+    (hV : Differentiable ℝ V) (hW : Differentiable ℝ W) :
+    divergence (V + W) = divergence V + divergence W := by
+  funext x
+  simp only [divergence, fderiv_add (hV x) (hW x), add_apply,
+    inner_add_right, Finset.sum_add_distrib, Pi.add_apply]
+
+theorem raw_perimeter_mono_region {n : ℕ} (E : Set (Euc n)) {U V : Set (Euc n)}
+    (hUV : U ⊆ V) : perimeter E U ≤ perimeter E V := by
+  unfold perimeter
+  refine iSup_le fun f ↦ iSup_le fun hf ↦ iSup_le fun hc ↦ iSup_le fun hs ↦ iSup_le fun hb ↦ ?_
+  exact le_iSup_of_le f (le_iSup_of_le hf (le_iSup_of_le hc
+    (le_iSup_of_le (hs.trans hUV) (le_iSup_of_le hb le_rfl))))
+
+def PerimeterAdmissible {n : ℕ} (U : Set (Euc n)) (V : Euc n → Euc n) : Prop :=
+  ContDiff ℝ ∞ V ∧ HasCompactSupport V ∧ tsupport V ⊆ U ∧ (∀ x, ‖V x‖ ≤ 1)
+
+theorem perimeterAdmissible_zero {n : ℕ} (U : Set (Euc n)) :
+    PerimeterAdmissible U (fun _ ↦ 0) := by
+  exact ⟨contDiff_const, (by simp [HasCompactSupport]), by simp, by simp⟩
+
+theorem perimeter_eq_iSup_tests {n : ℕ} (E U : Set (Euc n)) :
+    perimeter E U = ⨆ V, ⨆ (_ : PerimeterAdmissible U V),
+      ENNReal.ofReal (∫ x in E, divergence V x) := by
+  simp only [perimeter, PerimeterAdmissible, iSup_and]
+
+theorem perimeter_disjoint_superadditive {n : ℕ} (E : Set (Euc n))
+    {U V Z : Set (Euc n)} (hUZ : U ⊆ Z) (hVZ : V ⊆ Z) (hUV : Disjoint U V) :
+    perimeter E U + perimeter E V ≤ perimeter E Z := by
+  rw [perimeter_eq_iSup_tests E U, perimeter_eq_iSup_tests E V]
+  apply ENNReal.biSup_add_biSup_le'
+    ⟨_, perimeterAdmissible_zero U⟩ ⟨_, perimeterAdmissible_zero V⟩
+  rintro F ⟨hF, hcF, hsF, hbF⟩ G ⟨hG, hcG, hsG, hbG⟩
+  by_cases hf : 0 ≤ ∫ x in E, divergence F x
+  · by_cases hg : 0 ≤ ∫ x in E, divergence G x
+    · have hs : tsupport (F + G) ⊆ Z := (tsupport_add F G).trans (union_subset (hsF.trans hUZ) (hsG.trans hVZ))
+      have hb : ∀ x, ‖(F + G) x‖ ≤ 1 := by
+        intro x
+        by_cases hx : x ∈ tsupport F
+        · have hn : x ∉ tsupport G := fun hg ↦ Set.disjoint_left.mp hUV (hsF hx) (hsG hg)
+          simp only [Pi.add_apply, image_eq_zero_of_notMem_tsupport hn, add_zero]
+          exact hbF x
+        · simp only [Pi.add_apply, image_eq_zero_of_notMem_tsupport hx, zero_add]
+          exact hbG x
+      rw [← ENNReal.ofReal_add hf hg]
+      have hi : (∫ x in E, divergence F x) + (∫ x in E, divergence G x) =
+          ∫ x in E, divergence (F + G) x := by
+        rw [divergence_add (hF.differentiable (by norm_num)) (hG.differentiable (by norm_num))]
+        exact (integral_add (integrable_divergence hF hcF).integrableOn
+          (integrable_divergence hG hcG).integrableOn).symm
+      rw [hi]
+      unfold perimeter
+      exact le_iSup_of_le (F + G) (le_iSup_of_le (hF.add hG) (le_iSup_of_le (hcF.add hcG)
+        (le_iSup_of_le hs (le_iSup_of_le hb le_rfl))))
+    · rw [ENNReal.ofReal_of_nonpos (le_of_lt (lt_of_not_ge hg)), add_zero]
+      unfold perimeter
+      exact le_iSup_of_le F (le_iSup_of_le hF (le_iSup_of_le hcF
+        (le_iSup_of_le (hsF.trans hUZ) (le_iSup_of_le hbF le_rfl))))
+  · rw [ENNReal.ofReal_of_nonpos (le_of_lt (lt_of_not_ge hf)), zero_add]
+    unfold perimeter
+    exact le_iSup_of_le G (le_iSup_of_le hG (le_iSup_of_le hcG
+      (le_iSup_of_le (hsG.trans hVZ) (le_iSup_of_le hbG le_rfl))))
+
+theorem divergence_smul {n : ℕ} {η : Euc n → ℝ} {V : Euc n → Euc n}
+    (hη : Differentiable ℝ η) (hV : Differentiable ℝ V) (x : Euc n) :
+    divergence (fun y ↦ η y • V y) x =
+      η x * divergence V x + inner ℝ (gradient η x) (V x) := by
+  simp only [divergence, fderiv_fun_smul (hη x) (hV x),
+    add_apply, smul_apply,
+    ContinuousLinearMap.smulRight_apply, inner_add_right, inner_smul_right,
+    Finset.sum_add_distrib, ← Finset.mul_sum]
+  congr 1
+  simp_rw [← inner_gradient_left]
+  exact (EuclideanSpace.basisFun (Fin n) ℝ).sum_inner_mul_inner _ _
+
+theorem smoothTransition_deriv_zero_of_nonpos {s : ℝ} (hs : s ≤ 0) :
+    deriv Real.smoothTransition s = 0 := by
+  have hm : IsLocalMin Real.smoothTransition s := by
+    filter_upwards [] with x
+    rw [Real.smoothTransition.zero_of_nonpos hs]
+    exact Real.smoothTransition.nonneg x
+  exact hm.deriv_eq_zero
+
+theorem smoothTransition_deriv_zero_of_one_le {s : ℝ} (hs : 1 ≤ s) :
+    deriv Real.smoothTransition s = 0 := by
+  have hm : IsLocalMax Real.smoothTransition s := by
+    filter_upwards [] with x
+    rw [Real.smoothTransition.one_of_one_le hs]
+    exact Real.smoothTransition.le_one x
+  exact hm.deriv_eq_zero
+
+theorem exists_smoothTransition_deriv_bound :
+    ∃ L : ℝ, 0 < L ∧ ∀ s, |deriv Real.smoothTransition s| ≤ L := by
+  have hc : Continuous (deriv Real.smoothTransition) :=
+    (Real.smoothTransition.contDiff : ContDiff ℝ ∞ _).continuous_deriv (by norm_num)
+  obtain ⟨L₀, hL₀⟩ := (isCompact_Icc : IsCompact (Icc (0 : ℝ) 1)).exists_bound_of_continuousOn hc.continuousOn
+  refine ⟨max L₀ 0 + 1, by positivity, ?_⟩
+  intro s
+  by_cases hs : s ∈ Icc (0 : ℝ) 1
+  · have h := hL₀ s hs
+    rw [Real.norm_eq_abs] at h
+    linarith [le_max_left L₀ (0 : ℝ)]
+  · have hh : s ≤ 0 ∨ 1 ≤ s := by simp only [mem_Icc] at hs; by_contra h; push Not at h; exact hs ⟨h.1.le, h.2.le⟩
+    rcases hh with hh | hh
+    · rw [smoothTransition_deriv_zero_of_nonpos hh, abs_zero]; positivity
+    · rw [smoothTransition_deriv_zero_of_one_le hh, abs_zero]; positivity
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+
+def annularCutoff {n : ℕ} (c : Euc n) (a b : ℝ) (x : Euc n) : ℝ :=
+  Real.smoothTransition ((‖x - c‖ ^ 2 - a ^ 2) / (b ^ 2 - a ^ 2))
+
+theorem annularCutoff_contDiff {n : ℕ} (c : Euc n) (a b : ℝ) :
+    ContDiff ℝ ∞ (annularCutoff c a b) := by
+  apply Real.smoothTransition.contDiff.comp
+  exact (((contDiff_id.sub contDiff_const).norm_sq ℝ).sub contDiff_const).div_const _
+
+theorem annularCutoff_mem_Icc {n : ℕ} (c : Euc n) (a b : ℝ) (x : Euc n) :
+    annularCutoff c a b x ∈ Icc (0 : ℝ) 1 :=
+  ⟨Real.smoothTransition.nonneg _, Real.smoothTransition.le_one _⟩
+
+theorem annularCutoff_zero {n : ℕ} {c x : Euc n} {a b : ℝ}
+    (ha : 0 < a) (hab : a < b) (hx : dist x c ≤ a) :
+    annularCutoff c a b x = 0 := by
+  apply Real.smoothTransition.zero_of_nonpos
+  apply div_nonpos_of_nonpos_of_nonneg
+  · rw [dist_eq_norm] at hx
+    nlinarith [norm_nonneg (x - c)]
+  · nlinarith
+
+theorem annularCutoff_one {n : ℕ} {c x : Euc n} {a b : ℝ}
+    (ha : 0 < a) (hab : a < b) (hx : b ≤ dist x c) :
+    annularCutoff c a b x = 1 := by
+  apply Real.smoothTransition.one_of_one_le
+  rw [le_div_iff₀ (by nlinarith : 0 < b ^ 2 - a ^ 2)]
+  rw [dist_eq_norm] at hx
+  nlinarith [norm_nonneg (x - c)]
+
+theorem tsupport_annularCutoff_subset {n : ℕ} {c : Euc n} {a b : ℝ}
+    (ha : 0 < a) (hab : a < b) :
+    tsupport (annularCutoff c a b) ⊆ {x | a ≤ dist x c} := by
+  apply closure_minimal
+  · intro x hx
+    exact (lt_of_not_ge (fun h ↦ hx (annularCutoff_zero ha hab h))).le
+  · exact isClosed_le continuous_const (continuous_id.dist continuous_const)
+
+theorem gradient_annularCutoff {n : ℕ} (c x : Euc n) (a b : ℝ) :
+    gradient (annularCutoff c a b) x =
+      (deriv Real.smoothTransition ((‖x - c‖ ^ 2 - a ^ 2) / (b ^ 2 - a ^ 2)) *
+        (2 / (b ^ 2 - a ^ 2))) • (x - c) := by
+  let q : Euc n → ℝ := fun y ↦ (‖y - c‖ ^ 2 - a ^ 2) / (b ^ 2 - a ^ 2)
+  have h0 : HasFDerivAt (fun y : Euc n ↦ ‖y - c‖ ^ 2 - a ^ 2)
+      (2 • innerSL ℝ (x - c)) x := by
+    simpa using (((hasFDerivAt_id (𝕜 := ℝ) x).sub_const c).norm_sq.sub_const (a ^ 2))
+  have hq : HasFDerivAt q ((2 • innerSL ℝ (x - c)).smulRight
+      ((b ^ 2 - a ^ 2)⁻¹ : ℝ)) x := by
+    simpa only [q, div_eq_mul_inv, smul_eq_mul] using h0.smul_const
+      ((b ^ 2 - a ^ 2)⁻¹ : ℝ)
+  have hT : DifferentiableAt ℝ Real.smoothTransition (q x) :=
+    (Real.smoothTransition.contDiff : ContDiff ℝ ∞ _).differentiable (by norm_num) _
+  apply (InnerProductSpace.toDual ℝ (Euc n)).injective
+  rw [toDual_gradient]
+  have hchain := (hT.hasDerivAt.comp_hasFDerivAt x hq).fderiv
+  change fderiv ℝ (Real.smoothTransition ∘ q) x = _
+  rw [hchain]
+  ext y
+  simp only [map_smul, smul_apply, ContinuousLinearMap.smulRight_apply,
+    InnerProductSpace.toDual_apply_apply, innerSL_apply_apply]
+  simp only [smul_eq_mul]
+  dsimp [q]
+  ring
+
+theorem gradient_annularCutoff_zero {n : ℕ} {c x : Euc n} {a b : ℝ}
+    (ha : 0 < a) (hab : a < b) (hx : dist x c ≤ a ∨ b ≤ dist x c) :
+    gradient (annularCutoff c a b) x = 0 := by
+  rcases hx with hx | hx
+  · have hm : IsLocalMin (annularCutoff c a b) x := by
+      filter_upwards [] with y
+      rw [annularCutoff_zero ha hab hx]
+      exact (annularCutoff_mem_Icc c a b y).1
+    simp only [gradient, hm.fderiv_eq_zero, map_zero]
+  · have hm : IsLocalMax (annularCutoff c a b) x := by
+      filter_upwards [] with y
+      rw [annularCutoff_one ha hab hx]
+      exact (annularCutoff_mem_Icc c a b y).2
+    simp only [gradient, hm.fderiv_eq_zero, map_zero]
+
+theorem gradient_annularCutoff_bound {n : ℕ} {c : Euc n} {a b L : ℝ}
+    (ha : 0 < a) (hab : a < b) (hL : 0 < L)
+    (hbound : ∀ s, |deriv Real.smoothTransition s| ≤ L) (x : Euc n) :
+    ‖gradient (annularCutoff c a b) x‖ ≤ 2 * L * b / (b ^ 2 - a ^ 2) := by
+  have hb : 0 < b := ha.trans hab
+  have hden : 0 < b ^ 2 - a ^ 2 := by nlinarith
+  by_cases hx : b ≤ dist x c
+  · rw [gradient_annularCutoff_zero ha hab (Or.inr hx), norm_zero]
+    exact div_nonneg (mul_nonneg (by positivity) (by linarith)) (by nlinarith)
+  · have hxb : ‖x - c‖ ≤ b := by simpa only [dist_eq_norm] using (lt_of_not_ge hx).le
+    rw [gradient_annularCutoff, norm_smul, Real.norm_eq_abs, abs_mul,
+      abs_div, abs_of_pos (by norm_num : (0 : ℝ) < 2), abs_of_pos (by nlinarith : 0 < b ^ 2 - a ^ 2)]
+    calc
+      _ ≤ (L * (2 / (b ^ 2 - a ^ 2))) * b := by
+        gcongr
+        exact hbound _
+      _ = _ := by ring
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+
+theorem volume_real_annulus {n : ℕ} (c : Euc n) {a b : ℝ}
+    (ha : 0 ≤ a) (hab : a ≤ b) :
+    volume.real (Metric.closedBall c b \ Metric.closedBall c a) =
+      (b ^ n - a ^ n) * volume.real (Metric.ball (0 : Euc n) 1) := by
+  rw [measureReal_sdiff (Metric.closedBall_subset_closedBall hab) measurableSet_closedBall
+      (isCompact_closedBall c b).measure_ne_top,
+    Measure.addHaar_real_closedBall volume c (ha.trans hab),
+    Measure.addHaar_real_closedBall volume c ha]
+  simp only [finrank_euclideanSpace, Fintype.card_fin]
+  ring
+
+theorem gradient_annularCutoff_continuous {n : ℕ} (c : Euc n) (a b : ℝ) :
+    Continuous (gradient (annularCutoff c a b)) := by
+  exact (InnerProductSpace.toDual ℝ (Euc n)).symm.continuous.comp
+    ((annularCutoff_contDiff c a b).continuous_fderiv (by norm_num))
+
+theorem gradient_annularCutoff_compact {n : ℕ} (c : Euc n) {a b : ℝ}
+    (ha : 0 < a) (hab : a < b) :
+    HasCompactSupport (gradient (annularCutoff c a b)) := by
+  apply (isCompact_closedBall c b).of_isClosed_subset (isClosed_tsupport _)
+  apply closure_minimal
+  · intro x hx
+    by_contra hb
+    exact hx (gradient_annularCutoff_zero ha hab (Or.inr (le_of_lt (lt_of_not_ge hb))))
+  · exact Metric.isClosed_closedBall
+
+theorem integral_gradient_annularCutoff_le {n : ℕ} (c : Euc n) {a b L : ℝ}
+    (ha : 0 < a) (hab : a < b) (hL : 0 < L)
+    (hbound : ∀ s, |deriv Real.smoothTransition s| ≤ L) :
+    (∫ x, ‖gradient (annularCutoff c a b) x‖) ≤
+      2 * L * n * b ^ (n - 1) * volume.real (Metric.ball (0 : Euc n) 1) := by
+  have hb : 0 < b := ha.trans hab
+  have hden : 0 < b ^ 2 - a ^ 2 := by nlinarith
+  have hpow : b ^ n - a ^ n ≤ (b - a) * n * b ^ (n - 1) := by
+    have hh := abs_pow_sub_pow_le (a := b) (b := a) (n := n)
+    rw [abs_of_pos hb, abs_of_pos ha, max_eq_left hab.le, abs_of_pos (sub_pos.mpr hab)] at hh
+    exact (le_abs_self _).trans hh
+  let A := Metric.closedBall c b \ Metric.closedBall c a
+  have hid : (∫ x in A, ‖gradient (annularCutoff c a b) x‖) =
+      ∫ x, ‖gradient (annularCutoff c a b) x‖ := by
+    apply setIntegral_eq_integral_of_forall_compl_eq_zero
+    intro x hx
+    rw [gradient_annularCutoff_zero ha hab, norm_zero]
+    by_cases hh : x ∈ Metric.closedBall c a
+    · exact Or.inl hh
+    · exact Or.inr (le_of_lt (lt_of_not_ge (fun h ↦ hx ⟨h, hh⟩)))
+  rw [← hid]
+  calc
+    _ ≤ ‖∫ x in A, ‖gradient (annularCutoff c a b) x‖‖ := le_abs_self _
+    _ ≤ (2 * L * b / (b ^ 2 - a ^ 2)) * volume.real A := by
+      apply norm_setIntegral_le_of_norm_le_const
+        ((measure_mono sdiff_subset).trans_lt (isCompact_closedBall c b).measure_lt_top)
+      intro x hx
+      simpa only [norm_norm] using gradient_annularCutoff_bound ha hab hL hbound x
+    _ = (2 * L * b / (b ^ 2 - a ^ 2)) *
+        ((b ^ n - a ^ n) * volume.real (Metric.ball (0 : Euc n) 1)) := by
+      rw [volume_real_annulus c ha.le hab.le]
+    _ ≤ (2 * L * b / (b ^ 2 - a ^ 2)) *
+        (((b - a) * n * b ^ (n - 1)) * volume.real (Metric.ball (0 : Euc n) 1)) := by
+      gcongr
+    _ ≤ _ := by
+      have heq : (2 * L * b / (b ^ 2 - a ^ 2)) *
+          (((b - a) * n * b ^ (n - 1)) * volume.real (Metric.ball (0 : Euc n) 1)) =
+          (b / (b + a)) * (2 * L * n * b ^ (n - 1) *
+            volume.real (Metric.ball (0 : Euc n) 1)) := by
+        field_simp
+        ring
+      rw [heq]
+      exact mul_le_of_le_one_left (by positivity)
+        ((div_le_one (by linarith : 0 < b + a)).mpr (by linarith))
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/PerimeterMomentLimit.lean b/lean/OAI/Geometry/Perimeter/Regularity/PerimeterMomentLimit.lean
new file mode 100644
index 0000000000000000000000000000000000000000..844f7d0e002d7d2597687b9ae0021f3bd5f31c92
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/PerimeterMomentLimit.lean
@@ -0,0 +1,168 @@
+import OAI.Geometry.Perimeter.Regularity.GeoAlgebra
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff Convolution InnerProductSpace
+namespace RPISingularHarnack
+open RPI.Boundary ContinuousLinearMap
+variable {n : ℕ}
+
+theorem c1Mollify_norm_le_bound {ψ : Euc n → ℝ} (hψ : Continuous ψ)
+    {M : ℝ} (hM : ∀ x, ‖ψ x‖ ≤ M) (j : ℕ) (y : Euc n) :
+    ‖c1Mollify ψ j y‖ ≤ M := by
+  let k : Euc n → ℝ := (c1Mollifier j).normed volume
+  have hk : Integrable k volume := (c1Mollifier j).integrable_normed
+  have hp : ∀ x, 0 ≤ k x := (c1Mollifier j).nonneg_normed
+  have hi : Integrable (fun x ↦ k x*ψ (y-x)) volume :=
+    hk.mul_bdd (hψ.comp (continuous_const.sub continuous_id)).aestronglyMeasurable
+      (ae_of_all _ fun x ↦ hM _)
+  change ‖∫ x, k x*ψ (y-x)‖ ≤ M
+  calc
+    _ ≤ ∫ x, ‖k x*ψ (y-x)‖ := norm_integral_le_integral_norm _
+    _ ≤ ∫ x, k x*M := by
+      apply integral_mono hi.norm (hk.mul_const M)
+      intro x
+      change ‖k x*ψ (y-x)‖ ≤ k x*M
+      rw [norm_mul,Real.norm_eq_abs,abs_of_nonneg (hp x)]
+      exact mul_le_mul_of_nonneg_left (hM _) (hp x)
+    _ = M := by rw [integral_mul_const]; dsimp only [k]; rw [(c1Mollifier j).integral_normed,one_mul]
+
+theorem weak_c1_convolution_identity (μ : Measure (Euc n)) [SFinite μ]
+    {f : Euc n → ℝ} (hf : Integrable f μ) {ψ : Euc n → ℝ}
+    (hψ : Continuous ψ) (hcψ : HasCompactSupport ψ) (j : ℕ) :
+    (∫ x, ψ x*(f ⋆[lsmul ℝ ℝ, μ] (c1Mollifier j).normed volume) x) =
+      ∫ y, f y*c1Mollify ψ j y ∂μ := by
+  let k : Euc n → ℝ := (c1Mollifier j).normed volume
+  obtain ⟨M,hM⟩ := hcψ.exists_bound_of_continuous hψ
+  have hi : Integrable (fun z : Euc n × Euc n ↦ ψ z.1*(f z.2*k (z.1-z.2)))
+      (volume.prod μ) :=
+    (hf.convolution_integrand (lsmul ℝ ℝ) (c1Mollifier j).integrable_normed).bdd_mul
+      (hψ.comp continuous_fst).aestronglyMeasurable (ae_of_all _ fun z ↦ hM _)
+  calc
+    _ = ∫ x, ∫ y, ψ x*(f y*k (x-y)) ∂μ := by
+      apply integral_congr_ae
+      exact ae_of_all _ fun x ↦ (integral_const_mul _ _).symm
+    _ = ∫ y, ∫ x, ψ x*(f y*k (x-y)) ∂volume ∂μ := integral_integral_swap hi
+    _ = ∫ y, f y*c1Mollify ψ j y ∂μ := by
+      apply integral_congr_ae
+      apply ae_of_all
+      intro y
+      dsimp only
+      rw [c1Mollify,convolution_eq_swap,← integral_const_mul]
+      apply integral_congr_ae
+      apply ae_of_all
+      intro x
+      change ψ x*(f y*k (x-y)) = f y*(k (y-x)*ψ x)
+      have he : k (y-x) = k (x-y) := by
+        rw [show y-x=-(x-y) by abel]
+        exact (c1Mollifier j).normed_neg _
+      rw [he]
+      ring
+
+theorem weak_c1_convolution_tendsto (μ : Measure (Euc n)) [SFinite μ]
+    {f : Euc n → ℝ} (hf : Integrable f μ) {ψ : Euc n → ℝ}
+    (hψ : Continuous ψ) (hcψ : HasCompactSupport ψ) :
+    Tendsto (fun j ↦ ∫ x, ψ x*(f ⋆[lsmul ℝ ℝ, μ] (c1Mollifier j).normed volume) x)
+      atTop (𝓝 (∫ y, ψ y*f y ∂μ)) := by
+  simp_rw [weak_c1_convolution_identity μ hf hψ hcψ]
+  obtain ⟨M,hM⟩ := hcψ.exists_bound_of_continuous hψ
+  have h : Tendsto (fun j ↦ ∫ y, f y*c1Mollify ψ j y ∂μ) atTop
+      (𝓝 (∫ y, f y*ψ y ∂μ)) := by
+    apply tendsto_integral_of_dominated_convergence (fun y ↦ ‖f y‖*M)
+    · intro j
+      exact hf.aestronglyMeasurable.mul (c1Mollify_contDiff hψ j).continuous.aestronglyMeasurable
+    · exact hf.norm.mul_const M
+    · intro j
+      apply ae_of_all
+      intro y
+      rw [norm_mul]
+      exact mul_le_mul_of_nonneg_left (c1Mollify_norm_le_bound hψ hM j y) (norm_nonneg _)
+    · exact ae_of_all _ fun y ↦ tendsto_const_nhds.mul (c1Mollify_tendsto hψ y)
+  convert h using 2
+  apply integral_congr_ae
+  exact ae_of_all _ fun y ↦ mul_comm _ _
+
+theorem weak_c1_convolution_tendsto_of_ae_continuous
+    (μ : Measure (Euc n)) [SFinite μ] {f : Euc n → ℝ} (hf : Integrable f μ)
+    {ψ : Euc n → ℝ} (hψ : Measurable ψ) {M : ℝ} (hM : ∀ x, ‖ψ x‖ ≤ M)
+    (hcont : ∀ᵐ y ∂μ, ContinuousAt ψ y) :
+    Tendsto (fun j ↦ ∫ x, ψ x*(f ⋆[lsmul ℝ ℝ, μ] (c1Mollifier j).normed volume) x)
+      atTop (𝓝 (∫ y, ψ y*f y ∂μ)) := by
+  have hM0 : 0 ≤ M := (norm_nonneg (ψ 0)).trans (hM 0)
+  have hloc : LocallyIntegrable ψ volume :=
+    (locallyIntegrable_const M).mono hψ.aestronglyMeasurable
+      (ae_of_all _ fun x ↦ by simpa only [Real.norm_eq_abs,abs_of_nonneg hM0] using hM x)
+  have hc (j : ℕ) : ContDiff ℝ ∞ (c1Mollify ψ j) :=
+    (c1Mollifier j).hasCompactSupport_normed.contDiff_convolution_left _
+      (c1Mollifier j).contDiff_normed hloc
+  have hbound (j : ℕ) (y : Euc n) : ‖c1Mollify ψ j y‖ ≤ M := by
+    let k : Euc n → ℝ := (c1Mollifier j).normed volume
+    have hk : Integrable k volume := (c1Mollifier j).integrable_normed
+    have hp : ∀ x, 0 ≤ k x := (c1Mollifier j).nonneg_normed
+    have hi : Integrable (fun x ↦ k x*ψ (y-x)) volume :=
+      hk.mul_bdd (hψ.comp (measurable_const.sub measurable_id)).aestronglyMeasurable
+        (ae_of_all _ fun x ↦ hM _)
+    change ‖∫ x, k x*ψ (y-x)‖ ≤ M
+    calc
+      _ ≤ ∫ x, ‖k x*ψ (y-x)‖ := norm_integral_le_integral_norm _
+      _ ≤ ∫ x, k x*M := by
+        apply integral_mono hi.norm (hk.mul_const M)
+        intro x
+        change ‖k x*ψ (y-x)‖ ≤ k x*M
+        rw [norm_mul,Real.norm_eq_abs,abs_of_nonneg (hp x)]
+        exact mul_le_mul_of_nonneg_left (hM _) (hp x)
+      _ = M := by rw [integral_mul_const]; dsimp only [k]; rw [(c1Mollifier j).integral_normed,one_mul]
+  have he (j : ℕ) :
+      (∫ x, ψ x*(f ⋆[lsmul ℝ ℝ, μ] (c1Mollifier j).normed volume) x) =
+        ∫ y, f y*c1Mollify ψ j y ∂μ := by
+    let k : Euc n → ℝ := (c1Mollifier j).normed volume
+    have hi : Integrable (fun z : Euc n × Euc n ↦ ψ z.1*(f z.2*k (z.1-z.2)))
+        (volume.prod μ) :=
+      (hf.convolution_integrand (lsmul ℝ ℝ) (c1Mollifier j).integrable_normed).bdd_mul
+        (hψ.comp measurable_fst).aestronglyMeasurable (ae_of_all _ fun z ↦ hM _)
+    calc
+      _ = ∫ x, ∫ y, ψ x*(f y*k (x-y)) ∂μ := by
+        apply integral_congr_ae
+        exact ae_of_all _ fun x ↦ (integral_const_mul _ _).symm
+      _ = ∫ y, ∫ x, ψ x*(f y*k (x-y)) ∂volume ∂μ := integral_integral_swap hi
+      _ = ∫ y, f y*c1Mollify ψ j y ∂μ := by
+        apply integral_congr_ae
+        apply ae_of_all
+        intro y
+        dsimp only
+        rw [c1Mollify,convolution_eq_swap,← integral_const_mul]
+        apply integral_congr_ae
+        apply ae_of_all
+        intro x
+        change ψ x*(f y*k (x-y)) = f y*(k (y-x)*ψ x)
+        have hk : k (y-x) = k (x-y) := by
+          rw [show y-x=-(x-y) by abel]
+          exact (c1Mollifier j).normed_neg _
+        rw [hk]; ring
+  simp_rw [he]
+  have ht : Tendsto (fun j ↦ ∫ y, f y*c1Mollify ψ j y ∂μ) atTop
+      (𝓝 (∫ y, f y*ψ y ∂μ)) := by
+    apply tendsto_integral_of_dominated_convergence (fun y ↦ ‖f y‖*M)
+    · exact fun j ↦ hf.aestronglyMeasurable.mul (hc j).continuous.aestronglyMeasurable
+    · exact hf.norm.mul_const M
+    · intro j
+      apply ae_of_all
+      intro y
+      rw [norm_mul]
+      exact mul_le_mul_of_nonneg_left (hbound j y) (norm_nonneg _)
+    · filter_upwards [hcont] with y hy
+      apply tendsto_const_nhds.mul
+      exact ContDiffBump.convolution_tendsto_right c1Mollifier_radius_tendsto
+        (Eventually.of_forall fun _ ↦ hψ.aestronglyMeasurable)
+        (hy.tendsto.comp tendsto_snd) tendsto_const_nhds
+  convert ht using 2
+  apply integral_congr_ae
+  exact ae_of_all _ fun y ↦ mul_comm _ _
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/PhaseMomentLimit.lean b/lean/OAI/Geometry/Perimeter/Regularity/PhaseMomentLimit.lean
new file mode 100644
index 0000000000000000000000000000000000000000..641bf214f052040025348f8f7fad1be99ffbda59
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/PhaseMomentLimit.lean
@@ -0,0 +1,153 @@
+import OAI.Geometry.Perimeter.Regularity.FiberMomentIntegration
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff Convolution InnerProductSpace
+namespace RPISingularHarnack
+open RPI.Boundary ContinuousLinearMap
+variable {n : ℕ}
+
+theorem phaseMollification_c1_eq (E : Set (Euc n)) (j : ℕ) :
+    phaseMollification E ((c1Mollifier j).normed volume) =
+      c1Mollify (E.indicator (fun _ ↦ (1:ℝ))) j := by
+  funext x
+  rw [phaseMollification,convolution_eq_swap]
+  change (∫ y, E.indicator (fun _ ↦ (1:ℝ)) (x-y) * (c1Mollifier j).normed volume y) = _
+  apply integral_congr_ae
+  exact ae_of_all _ fun y ↦ mul_comm _ _
+
+theorem phaseMollification_ae_tendsto {E : Set (Euc n)} (hE : MeasurableSet E) :
+    ∀ᵐ x ∂volume, Tendsto (fun j ↦ phaseMollification E ((c1Mollifier j).normed volume) x)
+      atTop (𝓝 (E.indicator (fun _ ↦ (1:ℝ)) x)) := by
+  simp_rw [phaseMollification_c1_eq]
+  apply ContDiffBump.ae_convolution_tendsto_right_of_locallyIntegrable
+    c1Mollifier_radius_tendsto (K := 2) _ ((locallyIntegrable_const (1:ℝ)).indicator hE)
+  apply Eventually.of_forall
+  intro j
+  dsimp [c1Mollifier]
+  linarith
+
+theorem phaseMollification_c1_mem_Icc {E : Set (Euc n)} (hE : MeasurableSet E)
+    (j : ℕ) (x : Euc n) :
+    phaseMollification E ((c1Mollifier j).normed volume) x ∈ Icc (0:ℝ) 1 :=
+  phaseMollification_mem_Icc hE (c1Mollifier j).integrable_normed
+    (c1Mollifier j).nonneg_normed (c1Mollifier j).integral_normed x
+
+theorem phaseMollification_diffuse_tendsto_zero {E : Set (Euc n)} (hE : MeasurableSet E)
+    (K : Set (Euc n)) (hK : IsCompact K) :
+    Tendsto (fun j ↦ ∫ x in K,
+      phaseMollification E ((c1Mollifier j).normed volume) x *
+        (1-phaseMollification E ((c1Mollifier j).normed volume) x)) atTop (𝓝 0) := by
+  have : IsFiniteMeasure (volume.restrict K) := ⟨by
+    rw [Measure.restrict_apply_univ]; exact hK.measure_lt_top⟩
+  have hlim : ∀ᵐ x ∂volume.restrict K, Tendsto (fun j ↦
+      phaseMollification E ((c1Mollifier j).normed volume) x *
+        (1-phaseMollification E ((c1Mollifier j).normed volume) x)) atTop (𝓝 (0:ℝ)) := by
+    filter_upwards [ae_restrict_of_ae (phaseMollification_ae_tendsto hE)] with x hx
+    convert hx.mul (tendsto_const_nhds.sub hx) using 1
+    by_cases he : x ∈ E <;> simp [he]
+  have h := tendsto_integral_of_dominated_convergence (μ := volume.restrict K) (fun _ ↦ (1:ℝ))
+    (fun j ↦ ((phaseMollification_contDiff hE (c1Mollifier j).contDiff_normed
+      (c1Mollifier j).hasCompactSupport_normed).continuous.mul
+        (continuous_const.sub (phaseMollification_contDiff hE (c1Mollifier j).contDiff_normed
+          (c1Mollifier j).hasCompactSupport_normed).continuous)).aestronglyMeasurable)
+    (integrable_const _) (fun j ↦ ae_of_all _ fun x ↦ ?_) hlim
+  · simpa only [integral_zero,Pi.mul_apply,Pi.sub_apply] using h
+  · obtain ⟨h0,h1⟩ := phaseMollification_c1_mem_Icc hE j x
+    dsimp only [Pi.mul_apply,Pi.sub_apply]
+    rw [Real.norm_eq_abs,abs_of_nonneg (mul_nonneg h0 (sub_nonneg.mpr h1))]
+    nlinarith
+
+def exactFiberHeight (E : Set (Euc n)) (v : Euc n) (b : ℝ)
+    (y : normalPlane v) : ℝ :=
+  -b+∫ s in (-b)..b, E.indicator (fun _ ↦ (1:ℝ)) ((y:Euc n)+s•v)
+
+theorem phaseMollification_fiber_ae_tendsto {E : Set (Euc n)} (hE : MeasurableSet E)
+    {v : Euc n} (hv : ‖v‖ = 1) :
+    ∀ᵐ y : normalPlane v ∂volume, ∀ᵐ s : ℝ ∂volume,
+      Tendsto (fun j ↦ phaseMollification E ((c1Mollifier j).normed volume)
+        ((y:Euc n)+s•v)) atTop (𝓝 (E.indicator (fun _ ↦ (1:ℝ)) ((y:Euc n)+s•v))) := by
+  have ht := (normalCoordinates_measurePreserving v hv).quasiMeasurePreserving.ae
+    (phaseMollification_ae_tendsto hE)
+  change ∀ᵐ z : normalPlane v × ℝ ∂volume.prod volume, _ at ht
+  exact Measure.ae_ae_of_ae_prod ht
+
+theorem mollified_fiber_height_ae_tendsto {E : Set (Euc n)} (hE : MeasurableSet E)
+    {v : Euc n} (hv : ‖v‖ = 1) {b : ℝ} (hb : 0 ≤ b) :
+    ∀ᵐ y : normalPlane v ∂volume, Tendsto (fun j ↦
+      -b+∫ s in (-b)..b, phaseMollification E ((c1Mollifier j).normed volume)
+        ((y:Euc n)+s•v)) atTop (𝓝 (exactFiberHeight E v b y)) := by
+  filter_upwards [phaseMollification_fiber_ae_tendsto hE hv] with y hy
+  let F : ℕ → ℝ → ℝ := fun j ↦ mollifiedNormalProfile E ((c1Mollifier j).normed volume) v y
+  let f : ℝ → ℝ := fun s ↦ E.indicator (fun _ ↦ (1:ℝ)) ((y:Euc n)+s•v)
+  have hm (j : ℕ) : AEStronglyMeasurable (F j) (volume.restrict (Ioc (-b) b)) :=
+    (mollifiedNormalProfile_contDiff hE (c1Mollifier j).contDiff_normed
+      (c1Mollifier j).hasCompactSupport_normed v y).continuous.aestronglyMeasurable
+  have hbd (j : ℕ) : ∀ᵐ s ∂volume.restrict (Ioc (-b) b), ‖F j s‖ ≤ 1 := by
+    apply ae_of_all
+    intro s
+    obtain ⟨h0,h1⟩ := phaseMollification_c1_mem_Icc hE j ((y:Euc n)+s•v)
+    change ‖phaseMollification E ((c1Mollifier j).normed volume) ((y:Euc n)+s•v)‖ ≤ 1
+    simpa only [Real.norm_eq_abs,abs_of_nonneg h0] using h1
+  have hlim : ∀ᵐ s ∂volume.restrict (Ioc (-b) b), Tendsto (fun j ↦ F j s) atTop (𝓝 (f s)) :=
+    ae_restrict_of_ae hy
+  have h : Tendsto (fun j ↦ ∫ s in Ioc (-b) b, F j s) atTop
+      (𝓝 (∫ s in Ioc (-b) b, f s)) :=
+    tendsto_integral_of_dominated_convergence (fun _ ↦ (1:ℝ)) hm (integrable_const _) hbd hlim
+  have hadd := (tendsto_const_nhds (x := -b)).add h
+  simpa only [exactFiberHeight,intervalIntegral.integral_of_le (by linarith : -b ≤ b),
+    F,f,mollifiedNormalProfile] using hadd
+
+theorem interval_height_abs_le {q : ℝ → ℝ} (hq : Measurable q) {b : ℝ}
+    (hb : 0 ≤ b) (hbound : ∀ s, q s ∈ Icc (0:ℝ) 1) :
+    |-b+∫ s in (-b)..b, q s| ≤ b := by
+  have hi : IntervalIntegrable q volume (-b) b := by
+    apply (intervalIntegrable_const (c := (1:ℝ))).mono_fun hq.aestronglyMeasurable
+    apply ae_of_all
+    intro s
+    simpa only [Real.norm_eq_abs,abs_of_nonneg (hbound s).1,abs_one] using (hbound s).2
+  have h0 : 0 ≤ ∫ s in (-b)..b, q s :=
+    intervalIntegral.integral_nonneg (by linarith) fun s _ ↦ (hbound s).1
+  have h1 : (∫ s in (-b)..b, q s) ≤ 2*b := by
+    calc
+      _ ≤ ∫ s in (-b)..b, (1:ℝ) := intervalIntegral.integral_mono_on
+        (by linarith) hi intervalIntegrable_const (fun s _ ↦ (hbound s).2)
+      _ = _ := by simp; ring
+  rw [abs_le]
+  constructor <;> linarith
+
+theorem mollified_fiber_height_square_tendsto {E : Set (Euc n)} (hE : MeasurableSet E)
+    {v : Euc n} (hv : ‖v‖ = 1) {b : ℝ} (hb : 0 ≤ b)
+    (K : Set (normalPlane v)) (hK : IsCompact K) :
+    Tendsto (fun j ↦ ∫ y in K,
+      (-b+∫ s in (-b)..b, phaseMollification E ((c1Mollifier j).normed volume)
+        ((y:Euc n)+s•v))^2) atTop (𝓝 (∫ y in K, (exactFiberHeight E v b y)^2)) := by
+  have : IsFiniteMeasure (volume.restrict K) := ⟨by
+    rw [Measure.restrict_apply_univ]; exact hK.measure_lt_top⟩
+  apply tendsto_integral_of_dominated_convergence (fun _ ↦ b^2)
+  · intro j
+    exact ((continuous_const.add (continuous_profile_integral (v := v) _
+      (phaseMollification_contDiff hE (c1Mollifier j).contDiff_normed
+        (c1Mollifier j).hasCompactSupport_normed).continuous (-b) b)).pow 2).aestronglyMeasurable
+  · exact integrable_const _
+  · intro j
+    apply ae_of_all
+    intro y
+    have h := interval_height_abs_le
+      (((phaseMollification_contDiff hE (c1Mollifier j).contDiff_normed
+        (c1Mollifier j).hasCompactSupport_normed).continuous.comp
+          (continuous_const.add (continuous_id.smul continuous_const))).measurable)
+      hb (fun s ↦ phaseMollification_c1_mem_Icc hE j ((y:Euc n)+s•v))
+    rw [Real.norm_eq_abs,abs_of_nonneg (sq_nonneg _)]
+    simpa only [sq_abs,Function.comp_def,Pi.add_apply,Pi.smul_apply,Pi.smul_apply',id_eq] using
+      (sq_le_sq₀ (abs_nonneg _) hb).mpr h
+  · filter_upwards [ae_restrict_of_ae (mollified_fiber_height_ae_tendsto hE hv hb)] with y hy
+    exact hy.pow 2
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/PhaseRestoration.lean b/lean/OAI/Geometry/Perimeter/Regularity/PhaseRestoration.lean
new file mode 100644
index 0000000000000000000000000000000000000000..85b29b0e6d13c0798ba748bde50bf145d0a54ba0
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/PhaseRestoration.lean
@@ -0,0 +1,105 @@
+import OAI.Geometry.Perimeter.Regularity.FlatnessImprovement
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+theorem positive_cap_ball {v w : Euc n} (hv : ‖v‖=1)
+    {r : ℝ} (hr : 0 < r) (hvw : ‖w-v‖≤1/4) :
+    ∀ x∈Metric.ball ((r/2)•v) (r/8),
+      x∈Metric.ball 0 r ∧ r/8 < inner ℝ v x ∧ r/8 < inner ℝ w x := by
+  intro x hx
+  have hxc : ‖x-(r/2)•v‖ < r/8 := by simpa only [Metric.mem_ball,dist_eq_norm] using hx
+  have hc : ‖(r/2)•v‖=r/2 := by rw [norm_smul,Real.norm_of_nonneg (by positivity),hv,mul_one]
+  have hnx : ‖x‖<r := by
+    have hh := norm_le_norm_sub_add x ((r/2)•v)
+    rw [hc] at hh
+    linarith
+  have hi : inner ℝ v ((r/2)•v)=r/2 := by
+    rw [real_inner_smul_right,real_inner_self_eq_norm_sq,hv]; ring
+  have hvi := abs_real_inner_le_norm v (x-(r/2)•v)
+  rw [inner_sub_right,hi,hv,one_mul] at hvi
+  have hvx : 3*r/8 < inner ℝ v x := by
+    have hh := (abs_le.mp hvi).1
+    linarith
+  have hwi := abs_real_inner_le_norm (w-v) x
+  rw [inner_sub_left] at hwi
+  have hwb : ‖w-v‖*‖x‖ < r/4 :=
+    (mul_le_mul_of_nonneg_right hvw (norm_nonneg x)).trans_lt (by linarith)
+  have hwx : r/8 < inner ℝ w x := by
+    have hh := (abs_le.mp hwi).1
+    linarith
+  exact ⟨by simpa only [Metric.mem_ball,dist_zero_right] using hnx,by linarith,hwx⟩
+
+theorem FullSlab.restore_phase (hn : 1 < n) {E : Set (Euc n)}
+    (hmin : IsLocallyPerimeterMinimizing E) {v w : Euc n} (hv : ‖v‖=1)
+    {R r a b : ℝ} (hslab : FullSlab E v R a) (hr : 0 < r) (hrR : r≤R)
+    (ha : a<r/8) (hb : 0≤b) (hbr : b<r/8) (hvw : ‖w-v‖≤1/4)
+    (hheight : ∀ x∈perimeterSupport E ∩ Metric.ball 0 r, |inner ℝ w x|≤b) :
+    FullSlab E w r b := by
+  let A : Set (Euc n) := Metric.ball ((r/2)•v) (r/8)
+  let B : Set (Euc n) := {x : Euc n | -x∈A}
+  have hA := positive_cap_ball hv hr hvw
+  have hAcap : A ⊆ Metric.ball 0 r ∩ {x | b < inner ℝ w x} := by
+    intro x hx
+    exact ⟨(hA x hx).1,hbr.trans (hA x hx).2.2⟩
+  have hBcap : B ⊆ Metric.ball 0 r ∩ {x | inner ℝ w x < -b} := by
+    intro x hx
+    have hh := hA (-x) hx
+    refine ⟨by simpa only [Metric.mem_ball,dist_zero_right,norm_neg] using hh.1,?_⟩
+    simp only [inner_neg_right] at hh
+    change inner ℝ w x < -b
+    linarith [hh.2.2]
+  have hAE : ∀ᵐ x ∂volume, x∈A → x∉E := by
+    filter_upwards [hslab.2] with x hx
+    intro hxA hxE
+    have hh := hA x hxA
+    have hpos : 0 < inner ℝ v x := lt_trans (by positivity) hh.2.1
+    have he := hx (Metric.ball_subset_ball hrR hh.1)
+      (by rw [abs_of_pos hpos]; exact ha.trans hh.2.1)
+    exact (not_lt_of_ge hpos.le) (he.mp hxE)
+  have hBE : ∀ᵐ x ∂volume, x∈B → x∈E := by
+    filter_upwards [hslab.2] with x hx
+    intro hxB
+    have hh := hA (-x) hxB
+    have hball : x∈Metric.ball 0 R := Metric.ball_subset_ball hrR
+      (by simpa only [Metric.mem_ball,dist_zero_right,norm_neg] using hh.1)
+    have hneg : inner ℝ v x < 0 := by simp only [inner_neg_right] at hh; linarith [hh.2.1]
+    apply (hx hball ?_).mpr hneg
+    rw [abs_of_neg hneg]
+    simp only [inner_neg_right] at hh
+    exact ha.trans hh.2.1
+  have hvolA : 0 < volume (A\E) := by
+    have he : A\E =ᵐ[volume] A := by
+      filter_upwards [hAE] with x hx
+      exact propext ⟨fun h ↦ h.1,fun h ↦ ⟨h,hx h⟩⟩
+    have heq : volume (A \ E)=volume A := measure_congr he
+    rw [heq]
+    exact Metric.measure_ball_pos volume _ (by positivity)
+  have hvolB : 0 < volume (B∩E) := by
+    have he : (B ∩ E : Set (Euc n)) =ᵐ[volume] B := by
+      filter_upwards [hBE] with x hx
+      exact propext ⟨fun h ↦ h.1,fun h ↦ ⟨h,hx h⟩⟩
+    have heq : volume (B ∩ E)=volume B := measure_congr he
+    rw [heq]
+    have heB : B=Metric.ball (-(r/2)•v) (r/8) := by
+      ext x
+      change dist (-x) ((r/2)•v)<r/8 ↔ dist x (-(r/2)•v)<r/8
+      rw [neg_smul]
+      have hd : dist (-x) ((r/2)•v)=dist x (-((r/2)•v)) := by
+        simpa only [neg_neg] using dist_neg_neg x (-((r/2)•v))
+      rw [hd]
+    rw [heB]
+    exact Metric.measure_ball_pos volume _ (by positivity)
+  exact ⟨hheight,minimizing_slab_phase_agreement hn hmin w hb hheight
+    hAcap hBcap hvolA hvolB⟩
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/PhaseSupportBlownUpSet.lean b/lean/OAI/Geometry/Perimeter/Regularity/PhaseSupportBlownUpSet.lean
new file mode 100644
index 0000000000000000000000000000000000000000..38d77097554d2cc0764bd9ef062bc9474e701889
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/PhaseSupportBlownUpSet.lean
@@ -0,0 +1,110 @@
+import OAI.Geometry.Perimeter.Regularity.MinimalGraphEquation
+import OAI.Geometry.Perimeter.Regularity.MinimalGraphSmooth
+import OAI.Geometry.Perimeter.Regularity.BlowupHolderGraph
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+
+theorem phaseSupport_blownUpSet {n : ℕ} (E : Set (Euc n))
+    (c : Euc n) {r : ℝ} (hr : 0 < r) (x : Euc n) :
+    x ∈ (volume.restrict (blownUpSet E c r)).support ↔
+      c + r • x ∈ (volume.restrict E).support := by
+  let e := blowupHomeomorph c hr
+  have hd (S : Set (Euc n)) :
+      volume (blownUpSet S c r) = ENNReal.ofReal ((r ^ n)⁻¹) * volume S := by
+    change volume ((fun y : Euc n ↦ r • y) ⁻¹' ((fun y ↦ c + y) ⁻¹' S)) = _
+    rw [Measure.addHaar_preimage_smul volume hr.ne', measure_preimage_add]
+    simp [finrank_euclideanSpace, abs_of_pos (inv_pos.mpr (pow_pos hr n))]
+  have hp (S : Set (Euc n)) :
+      0 < volume (blownUpSet S c r) ↔ 0 < volume S := by
+    rw [hd, ENNReal.mul_pos_iff]
+    simp [ENNReal.ofReal_pos.mpr (inv_pos.mpr (pow_pos hr n))]
+  simp only [Measure.support_eq_forall_isOpen, Set.mem_ofPred_eq]
+  constructor
+  · intro hx U hUx hU
+    rw [Measure.restrict_apply hU.measurableSet]
+    apply (hp (U ∩ E)).mp
+    have hOU : IsOpen (e ⁻¹' U) := hU.preimage e.continuous
+    have h := hx (e ⁻¹' U) hUx hOU
+    rw [Measure.restrict_apply hOU.measurableSet] at h
+    exact h
+  · intro hx U hUx hU
+    rw [Measure.restrict_apply hU.measurableSet]
+    have hOU := e.isOpenMap U hU
+    have h := hx (e '' U) ⟨x,hUx,rfl⟩ hOU
+    rw [Measure.restrict_apply hOU.measurableSet] at h
+    have h' := (hp (e '' U ∩ E)).mpr h
+    change 0 < volume (e ⁻¹' (e '' U ∩ E)) at h'
+    change 0 < volume (U ∩ e ⁻¹' E)
+    simpa only [preimage_inter, e.injective.preimage_image] using h'
+
+theorem flat_halfspace_blowup_regular_neighborhood {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    {p v : Euc n} (hp : p ∈ perimeterSupport E) (hbl : HasHalfspaceBlowup E p v) :
+    ∃ U : Set (Euc n), IsOpen U ∧ p ∈ U ∧
+      ∃ f : Euc n → ℝ, ContDiffOn ℝ ∞ f U ∧
+        (∀ y ∈ U, fderiv ℝ f y ≠ 0) ∧
+        (∀ y ∈ U, y ∈ perimeterSupport E ↔ f y = 0) ∧
+        (∀ y ∈ U, (f y < 0 → y ∈ (volume.restrict E).support) ∧
+          (y ∈ (volume.restrict E).support → f y ≤ 0)) := by
+  obtain ⟨η,hη,_,hgraphs⟩ := hbl.exists_holder_graphs hn hmin hp
+  obtain ⟨ε,hε,hsmooth⟩ := normal_graph_smooth_threshold hn hbl.1 hη
+  obtain ⟨r,hr,g,_,hg,_,_,hiff,⟨a,ha,ha4,hs⟩,hh,hslope⟩ := hgraphs ε hε
+  have heq : ∀ ψ : normalPlane v → ℝ, ContDiff ℝ ∞ ψ → HasCompactSupport ψ →
+      tsupport ψ ⊆ Metric.ball 0 (1/4) →
+      (∫ y, inner ℝ (gradient g y) (gradient ψ y) / Real.sqrt (1+‖gradient g y‖^2)) = 0 := by
+    intro ψ hψ hcψ hsψ
+    exact full_graph_weak_minimal_equation hn (locally_minimizing_blownUpSet hmin p hr)
+      hbl.1 ha.le ha4 hs hg hiff hψ hcψ hsψ
+  have hgSmooth := hsmooth g hg hh hslope heq
+  let A : Euc n → Euc n := fun x ↦ r⁻¹ • (x-p)
+  let L : Euc n →L[ℝ] Euc n := r⁻¹ • ContinuousLinearMap.id ℝ (Euc n)
+  have hA : ContDiff ℝ ∞ A := by
+    dsimp [A]
+    exact (contDiff_id.sub contDiff_const).const_smul r⁻¹
+  have hAd (x : Euc n) : HasFDerivAt A L x := by
+    change HasFDerivAt (fun y : Euc n ↦ r⁻¹ • (y-p))
+      (r⁻¹ • ContinuousLinearMap.id ℝ (Euc n)) x
+    convert ((hasFDerivAt_id x).sub_const p).const_smul r⁻¹ using 1
+    rfl
+  have hAp : A p = 0 := by simp [A]
+  have hback (x : Euc n) : p + r • A x = x := by
+    simp only [A, smul_smul, mul_inv_cancel₀ hr.ne', one_smul]
+    abel
+  have hG := graphDefining_contDiffOn hgSmooth
+  refine ⟨A ⁻¹' graphCylinder v, (graphCylinder_open v).preimage hA.continuous,
+    ?_, graphDefining v g ∘ A, hG.comp hA.contDiffOn (fun _ hx ↦ hx), ?_, ?_, ?_⟩
+  · simp [hAp, graphCylinder]
+  · intro x hx hzero
+    have hfd : DifferentiableAt ℝ (graphDefining v g) (A x) :=
+      (hG.contDiffAt ((graphCylinder_open v).mem_nhds hx)).differentiableAt (by simp)
+    have hder : fderiv ℝ (graphDefining v g ∘ A) x =
+        (fderiv ℝ (graphDefining v g) (A x)).comp L :=
+      (hfd.hasFDerivAt.comp x (hAd x)).fderiv
+    apply graphDefining_fderiv_ne_zero hbl.1 hg hx
+    ext w
+    have hz := congrArg (fun T : Euc n →L[ℝ] ℝ ↦ T (r • w)) hzero
+    rw [hder] at hz
+    simpa [ContinuousLinearMap.comp_apply, L, smul_smul, hr.ne'] using hz
+  · intro x hx
+    rw [← hback x, ← perimeterSupport_blownUpSet hmin.1 p hr (A x)]
+    simpa only [hback, Function.comp_apply] using graphDefining_support_iff hbl.1 hiff hx
+  · intro x hx
+    have hphase : x ∈ (volume.restrict E).support ↔ graphDefining v g (A x) ≤ 0 := by
+      have hb : A x ∈ (volume.restrict (blownUpSet E p r)).support ↔
+          x ∈ (volume.restrict E).support := by
+        simpa only [hback] using phaseSupport_blownUpSet E p hr (A x)
+      exact hb.symm.trans (graphDefining_phaseSupport_iff hn
+        (locally_minimizing_blownUpSet hmin p hr) hbl.1 ha.le ha4 hs hiff hx)
+    exact ⟨fun h ↦ hphase.mpr h.le, hphase.mp⟩
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/PlaneFiberCompactness.lean b/lean/OAI/Geometry/Perimeter/Regularity/PlaneFiberCompactness.lean
new file mode 100644
index 0000000000000000000000000000000000000000..2d450cd586f1375c81cc0c636f2f63dd6e5e980f
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/PlaneFiberCompactness.lean
@@ -0,0 +1,100 @@
+import OAI.Geometry.Perimeter.Regularity.EllipticLocality
+import OAI.Geometry.Perimeter.Regularity.NormalCoordinates
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal InnerProductSpace
+namespace RPISingularHarnack
+open RPI.Boundary
+variable {n m : ℕ}
+
+def planeFiber (v : Euc n) (f : Euc n → ℝ) (y : normalPlane v) : ℝ :=
+  ∫ s : ℝ, f ((y:Euc n)+s•v)
+
+theorem planeFiber_measurable {f : Euc n → ℝ} (hf : Measurable f) (v : Euc n) :
+    Measurable (planeFiber v f) := by
+  have h : Measurable (fun p : normalPlane v × ℝ ↦ f ((p.1:Euc n)+p.2•v)) :=
+    hf.comp ((continuous_subtype_val.comp continuous_fst).add
+      (continuous_snd.smul continuous_const)).measurable
+  exact h.stronglyMeasurable.integral_prod_right.measurable
+
+theorem planeFiber_integrable {f : Euc n → ℝ} (hf : Integrable f volume)
+    {v : Euc n} (hv : ‖v‖=1) : Integrable (planeFiber v f) volume := by
+  have hi := ((normalCoordinates_measurePreserving v hv).integrable_comp
+    hf.aestronglyMeasurable).mpr hf
+  exact hi.integral_prod_left
+
+theorem planeFiber_lintegral_le {f : Euc n → ℝ} (hf : Measurable f)
+    {v : Euc n} (hv : ‖v‖=1) :
+    (∫⁻ y : normalPlane v, ‖planeFiber v f y‖ₑ) ≤ ∫⁻ x, ‖f x‖ₑ := by
+  rw [lintegral_normalCoordinates v hv _ hf.enorm]
+  exact lintegral_mono (fun _ ↦ enorm_integral_le_lintegral_enorm _)
+
+theorem planeFiber_translation_le {f : Euc n → ℝ} (hm : Measurable f)
+    (hf : Integrable f volume) {v : Euc n} (hv : ‖v‖=1) (w : normalPlane v) :
+    (∫⁻ y : normalPlane v, ‖planeFiber v f (y+w)-planeFiber v f y‖ₑ) ≤
+      ∫⁻ x, ‖f (x+(w:Euc n))-f x‖ₑ := by
+  have hi := ((normalCoordinates_measurePreserving v hv).integrable_comp
+    hf.aestronglyMeasurable).mpr hf
+  have hit : Integrable (fun x ↦ f (x+(w:Euc n))) volume := hf.comp_add_right _
+  have hit' := ((normalCoordinates_measurePreserving v hv).integrable_comp
+    hit.aestronglyMeasurable).mpr hit
+  calc
+    _ = ∫⁻ y : normalPlane v, ‖planeFiber v (fun x ↦ f (x+(w:Euc n))-f x) y‖ₑ := by
+      apply lintegral_congr_ae
+      filter_upwards [hi.prod_right_ae,hit'.prod_right_ae] with y hy hyt
+      unfold planeFiber
+      have he : (fun s : ℝ ↦ f (((y+w:normalPlane v):Euc n)+s•v)) =
+          (fun s : ℝ ↦ f ((y:Euc n)+s•v+(w:Euc n))) := by
+        funext s; congr 1; change (y:Euc n)+(w:Euc n)+s•v=(y:Euc n)+s•v+(w:Euc n); abel
+      rw [he]
+      dsimp only
+      change Integrable (fun s : ℝ ↦ f ((y:Euc n)+s•v)) volume at hy
+      change Integrable (fun s : ℝ ↦ f ((y:Euc n)+s•v+(w:Euc n))) volume at hyt
+      rw [integral_sub hyt hy]
+    _ ≤ _ := planeFiber_lintegral_le ((hm.comp (measurable_id.add measurable_const)).sub hm) hv
+
+def chartFiber (v : Euc n) (e : Euc m ≃ₗᵢ[ℝ] normalPlane v)
+    (a : ℝ) (f : Euc n → ℝ) (y : Euc m) : ℝ := a⁻¹*planeFiber v f (e y)
+
+theorem chartFiber_measurable {f : Euc n → ℝ} (hf : Measurable f)
+    (v : Euc n) (e : Euc m ≃ₗᵢ[ℝ] normalPlane v) (a : ℝ) :
+    Measurable (chartFiber v e a f) :=
+  measurable_const.mul ((planeFiber_measurable hf v).comp e.continuous.measurable)
+
+theorem chartFiber_integrable {f : Euc n → ℝ} (hf : Integrable f volume)
+    {v : Euc n} (hv : ‖v‖=1) (e : Euc m ≃ₗᵢ[ℝ] normalPlane v) (a : ℝ) :
+    Integrable (chartFiber v e a f) volume := by
+  have hi := planeFiber_integrable hf hv
+  exact ((e.measurePreserving.integrable_comp hi.aestronglyMeasurable).mpr hi).const_mul _
+
+theorem chartFiber_lintegral_le {f : Euc n → ℝ} (hf : Measurable f)
+    {v : Euc n} (hv : ‖v‖=1) (e : Euc m ≃ₗᵢ[ℝ] normalPlane v) (a : ℝ) :
+    (∫⁻ y, ‖chartFiber v e a f y‖ₑ) ≤ ENNReal.ofReal |a⁻¹| * ∫⁻ x, ‖f x‖ₑ := by
+  simp only [chartFiber,enorm_mul]
+  rw [Real.enorm_eq_ofReal_abs (a⁻¹)]
+  rw [lintegral_const_mul' _ _ ENNReal.ofReal_ne_top,
+    e.measurePreserving.lintegral_comp ((planeFiber_measurable hf v).enorm)]
+  exact mul_le_mul le_rfl (planeFiber_lintegral_le hf hv) zero_le zero_le
+
+theorem chartFiber_translation_le {f : Euc n → ℝ} (hm : Measurable f)
+    (hf : Integrable f volume) {v : Euc n} (hv : ‖v‖=1)
+    (e : Euc m ≃ₗᵢ[ℝ] normalPlane v) (a : ℝ) (w : Euc m) :
+    (∫⁻ y, ‖chartFiber v e a f (y+w)-chartFiber v e a f y‖ₑ) ≤
+      ENNReal.ofReal |a⁻¹| * ∫⁻ x, ‖f (x+(e w:Euc n))-f x‖ₑ := by
+  simp only [chartFiber,← mul_sub,enorm_mul,map_add]
+  rw [Real.enorm_eq_ofReal_abs (a⁻¹)]
+  have hfib := planeFiber_measurable hm v
+  have hdiff : Measurable (fun y : normalPlane v ↦ ‖planeFiber v f (y+e w)-planeFiber v f y‖ₑ) :=
+    ((hfib.comp (measurable_id.add measurable_const)).sub hfib).enorm
+  rw [lintegral_const_mul' _ _ ENNReal.ofReal_ne_top,
+    e.measurePreserving.lintegral_comp hdiff]
+  exact mul_le_mul le_rfl (planeFiber_translation_le hm hf hv (e w)) zero_le zero_le
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/PlaneHeightLimit.lean b/lean/OAI/Geometry/Perimeter/Regularity/PlaneHeightLimit.lean
new file mode 100644
index 0000000000000000000000000000000000000000..e1c4673ddc6672abb44a2d338c08e1dedec98ba9
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/PlaneHeightLimit.lean
@@ -0,0 +1,129 @@
+import OAI.Geometry.Perimeter.Regularity.NormalizedHeight
+import OAI.Geometry.Perimeter.Regularity.PlaneFiberCompactness
+import OAI.Geometry.Perimeter.Regularity.CompactLpExtension
+import OAI.Geometry.Perimeter.Regularity.NormalWindowTest
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal InnerProductSpace ContDiff
+namespace RPISingularHarnack
+open RPI.Boundary
+variable {n m : ℕ}
+
+theorem chartFiber_support {f : Euc n → ℝ} {v : Euc n}
+    (e : Euc m ≃ₗᵢ[ℝ] normalPlane v) (a : ℝ) {R : ℝ}
+    (hs : Function.support f ⊆ Metric.ball 0 R) :
+    Function.support (chartFiber v e a f) ⊆ Metric.closedBall 0 R := by
+  intro y hy
+  by_contra hnot
+  have hR : R < ‖y‖ := by simpa only [Metric.mem_closedBall,dist_zero_right,not_le] using hnot
+  have hz (s : ℝ) : f ((e y:Euc n)+s•v)=0 := by
+    apply Function.notMem_support.mp
+    intro hh
+    have hb := hs hh
+    rw [Metric.mem_ball,dist_zero_right] at hb
+    have hp := (normalPlane v).norm_orthogonalProjectionOnto_apply_le ((e y:Euc n)+s•v)
+    change ‖normalProjection v ((e y:Euc n)+s•v)‖ ≤ _ at hp
+    rw [normalProjection_fiber,e.norm_map] at hp
+    linarith
+  exact hy (by simp [chartFiber,planeFiber,hz])
+
+theorem chartFiber_normalized_mass {f : Euc n → ℝ} (hm : Measurable f)
+    {v : Euc n} (hv : ‖v‖=1) (e : Euc m ≃ₗᵢ[ℝ] normalPlane v)
+    {a V : ℝ} (ha : 0 < a)
+    (hV : (∫⁻ x, ‖f x‖ₑ) ≤ ENNReal.ofReal (a*V)) :
+    (∫⁻ y, ‖chartFiber v e a f y‖ₑ) ≤ ENNReal.ofReal V := by
+  apply (chartFiber_lintegral_le hm hv e a).trans
+  calc
+    _ ≤ ENNReal.ofReal |a⁻¹| *ENNReal.ofReal (a*V) := mul_le_mul_right hV _
+    _ = _ := by
+      rw [abs_of_pos (inv_pos.mpr ha),← ENNReal.ofReal_mul (by positivity)]
+      congr 1
+      field_simp
+
+theorem chartFiber_normalized_translation {f : Euc n → ℝ} (hm : Measurable f)
+    (hi : Integrable f volume) {v : Euc n} (hv : ‖v‖=1)
+    (e : Euc m ≃ₗᵢ[ℝ] normalPlane v) {a H : ℝ} (ha : 0 < a)
+    (hhor : ∀ z, inner ℝ v z=0 →
+      (∫⁻ x, ‖f (x+z)-f x‖ₑ) ≤ ENNReal.ofReal (‖z‖*(a*H))) (w : Euc m) :
+    (∫⁻ y, ‖chartFiber v e a f (y+w)-chartFiber v e a f y‖ₑ) ≤
+      ‖w‖ₑ * ENNReal.ofReal H := by
+  apply (chartFiber_translation_le hm hi hv e a w).trans
+  calc
+    _ ≤ ENNReal.ofReal |a⁻¹| * ENNReal.ofReal (‖(e w:Euc n)‖*(a*H)) := by
+      apply mul_le_mul_right
+      exact hhor _ (Submodule.mem_orthogonal_singleton_iff_inner_right.mp (e w).property)
+    _ = _ := by
+      rw [abs_of_pos (inv_pos.mpr ha),← ENNReal.ofReal_mul (by positivity)]
+      change ENNReal.ofReal (a⁻¹*(‖e w‖*(a*H))) = _
+      rw [e.norm_map,← ofReal_norm,← ENNReal.ofReal_mul (norm_nonneg _)]
+      congr 1
+      field_simp
+
+theorem normalized_plane_height_subsequence
+    (f : ℕ → Euc n → ℝ) (hm : ∀ i, Measurable (f i))
+    {v : Euc n} (hv : ‖v‖=1) (e : Euc m ≃ₗᵢ[ℝ] normalPlane v)
+    (a : ℕ → ℝ) (ha : ∀ i, 0 < a i) {R V H : ℝ} (hV : 0 ≤ V)
+    (hs : ∀ i, Function.support (f i) ⊆ Metric.ball 0 R)
+    (hmass : ∀ i, (∫⁻ x, ‖f i x‖ₑ) ≤ ENNReal.ofReal (a i*V))
+    (hhor : ∀ i z, inner ℝ v z=0 →
+      (∫⁻ x, ‖f i (x+z)-f i x‖ₑ) ≤ ENNReal.ofReal (‖z‖*(a i*H))) :
+    ∃ g : Euc m → ℝ, Integrable g volume ∧ Function.support g ⊆ Metric.closedBall 0 R ∧
+      ∃ σ : ℕ → ℕ, StrictMono σ ∧
+        Tendsto (fun i ↦ ∫⁻ y, ‖chartFiber v e (a (σ i)) (f (σ i)) y-g y‖ₑ) atTop (𝓝 0) := by
+  have hi (i) : Integrable (f i) volume :=
+    ⟨(hm i).aestronglyMeasurable,(hmass i).trans_lt ENNReal.ofReal_lt_top⟩
+  have hig (i) := chartFiber_integrable (hi i) hv e (a i)
+  apply compact_support_L1_subsequence (Metric.closedBall 0 R) (isCompact_closedBall _ _)
+    (fun i ↦ chartFiber v e (a i) (f i)) (fun i ↦ chartFiber_measurable (hm i) v e (a i))
+    hig (fun i ↦ chartFiber_support e (a i) (hs i)) V hV _
+    (ENNReal.ofReal H) ENNReal.ofReal_ne_top _
+  · intro i
+    have h := chartFiber_normalized_mass (hm i) hv e (ha i) (hmass i)
+    rw [← ofReal_integral_norm_eq_lintegral_enorm (hig i)] at h
+    exact (ENNReal.ofReal_le_ofReal_iff hV).mp h
+  · intro i w
+    exact chartFiber_normalized_translation (hm i) (hi i) hv e (ha i) (hhor i) w
+
+theorem minimizing_plane_height_subsequence (hn : 1 < n)
+    (E : ℕ → Set (Euc n)) (hmin : ∀ i, IsLocallyPerimeterMinimizing (E i))
+    {v : Euc n} (hv : ‖v‖=1) (e : Euc m ≃ₗᵢ[ℝ] normalPlane v)
+    {η : Euc n → ℝ} (hη : ContDiff ℝ ∞ η) (hc : HasCompactSupport η)
+    (hb : ∀ x, ‖η x‖ ≤ 1) {R L P : ℝ} (hR : 0 < R) (hL : 0 ≤ L)
+    (hs : tsupport η ⊆ Metric.ball 0 R)
+    (hgrad : ∀ x ∈ Metric.ball 0 R, ‖gradient η x‖ ≤ L)
+    (a : ℕ → ℝ) (ha : ∀ i, 0 < a i)
+    (hheight : ∀ i x, x ∈ perimeterSupport (E i) ∩ Metric.ball 0 R → |inner ℝ v x| ≤ a i)
+    (hper : ∀ i, (perimeter (E i) (Metric.ball 0 R)).toReal ≤ P)
+    (hphase : ∀ i, ∀ᵐ x ∂volume, x ∈ Metric.ball 0 R → a i < |inner ℝ v x| →
+      (x ∈ E i ↔ inner ℝ v x < 0)) :
+    ∃ g : Euc m → ℝ, Integrable g volume ∧ Function.support g ⊆ Metric.closedBall 0 R ∧
+      ∃ σ : ℕ → ℕ, StrictMono σ ∧
+        Tendsto (fun i ↦ ∫⁻ y, ‖chartFiber v e (a (σ i))
+          (cutoffPhaseError (E (σ i)) v η) y-g y‖ₑ) atTop (𝓝 0) := by
+  let V := volume.real (Metric.ball (0:Euc n) (2*R))/R
+  have hV : 0 ≤ V := div_nonneg measureReal_nonneg hR.le
+  apply normalized_plane_height_subsequence (fun i ↦ cutoffPhaseError (E i) v η)
+    (fun i ↦ cutoffPhaseError_measurable (hmin i).1 v hη.continuous.measurable)
+    hv e a ha (H := L*(V+2*P)) hV
+  · intro i
+    exact (cutoffPhaseError_support (E i) v η).trans (subset_closure.trans hs)
+  · intro i
+    exact cutoffPhaseError_lintegral_height_le (hmin i).1 hv hη.continuous.measurable hb
+      hR (ha i).le (subset_closure.trans hs) (hphase i)
+  · intro i z hz
+    apply (cutoffPhaseError_horizontal_translation_le hn (hmin i) hv hz hη hc hR hs
+      (ha i).le hL (hheight i) hgrad (hphase i)).trans
+    apply ENNReal.ofReal_le_ofReal
+    have hai := (ha i).le
+    have hp := mul_le_mul_of_nonneg_left (hper i) (by positivity : 0 ≤ ‖z‖*a i*L*2)
+    dsimp only [V]
+    nlinarith
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/PlaneHeightResidual.lean b/lean/OAI/Geometry/Perimeter/Regularity/PlaneHeightResidual.lean
new file mode 100644
index 0000000000000000000000000000000000000000..d2f79dd9c14925ddaf4c7d5d1316ce5e3165b770
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/PlaneHeightResidual.lean
@@ -0,0 +1,100 @@
+import OAI.Geometry.Perimeter.Regularity.ActualFiberMoments
+import OAI.Geometry.Perimeter.Regularity.FiberLocalization
+import OAI.Geometry.Perimeter.Regularity.PlaneWindowLift
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal InnerProductSpace ContDiff
+namespace RPISingularHarnack
+open RPI.Boundary ContinuousLinearMap
+variable {n m : ℕ} {v : Euc n}
+
+theorem planeHeight_tsupport_gradient_subset (ψ : Euc m → ℝ) : tsupport (gradient ψ) ⊆ tsupport ψ := by
+  apply closure_minimal _ (isClosed_tsupport _)
+  intro x hx
+  by_contra hh
+  exact hx (by simp only [gradient,fderiv_of_notMem_tsupport ℝ hh,map_zero])
+
+def chartLaplacianTest (e : Euc m ≃ₗᵢ[ℝ] normalPlane v) (ψ : Euc m → ℝ)
+    (y : normalPlane v) : ℝ := divergence (gradient ψ) (e.symm y)
+
+theorem chartLaplacianTest_continuous (e : Euc m ≃ₗᵢ[ℝ] normalPlane v)
+    {ψ : Euc m → ℝ} (hψ : ContDiff ℝ ∞ ψ) : Continuous (chartLaplacianTest e ψ) :=
+  (continuous_divergence (gradient_contDiff hψ)).comp e.symm.continuous
+
+theorem chartLaplacianTest_hasCompactSupport (e : Euc m ≃ₗᵢ[ℝ] normalPlane v)
+    {ψ : Euc m → ℝ} (hcψ : HasCompactSupport ψ) : HasCompactSupport (chartLaplacianTest e ψ) := by
+  have h : HasCompactSupport (divergence (gradient ψ)) :=
+    hcψ.of_isClosed_subset (isClosed_tsupport _)
+      ((tsupport_divergence_subset _).trans (planeHeight_tsupport_gradient_subset ψ))
+  exact h.comp_homeomorph e.symm.toHomeomorph
+
+theorem minimizing_chart_height_residual (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    (hv : ‖v‖=1) (e : Euc m ≃ₗᵢ[ℝ] normalPlane v)
+    {η : Euc n → ℝ} {a b δ b₀ R : ℝ}
+    (ha : 0<a) (hδ : 0<δ) (hab : a+δ<b) (hab₀ : a<b₀) (hR : 0<R)
+    (hηs : Function.support η ⊆ Metric.ball 0 R)
+    {ψ : Euc m → ℝ} (hψ : ContDiff ℝ ∞ ψ) (hcψ : HasCompactSupport ψ)
+    {χ : ℝ → ℝ} (hχ : ContDiff ℝ ∞ χ) (hcχ : HasCompactSupport χ)
+    (hχone : ∀ s : ℝ, |s|<b₀ → χ s=1)
+    (hηone : ∀ y ∈ tsupport (chartLaplacianTest e ψ), ∀ s : ℝ, |s|≤b →
+      η ((y:Euc n)+s•v)=1)
+    (hKR : ∀ y ∈ tsupport (chartLaplacianTest e ψ), ‖y‖+b+δ≤R)
+    (hs : tsupport (planeWindowLift e ψ χ) ⊆ Metric.ball 0 R)
+    {M K : ℝ} (hM : ∀ y, ‖divergence (gradient ψ) y‖≤M) (hK : 0≤K)
+    (hd : ∀ x ∈ Metric.ball 0 R, ‖fderiv ℝ (gradient (planeWindowLift e ψ χ)) x‖≤K)
+    (hh : ∀ x ∈ perimeterSupport E ∩ Metric.ball 0 R, |inner ℝ v x|≤a)
+    (hphase : ∀ᵐ x ∂volume, x ∈ Metric.ball 0 R → a < |inner ℝ v x| →
+      (x ∈ E ↔ inner ℝ v x<0)) :
+    |∫ y, chartFiber v e a (cutoffPhaseError E v η) y * divergence (gradient ψ) y| ≤
+      2*K*(Real.sqrt (orientedTilt E v (Metric.ball 0 R))*
+        Real.sqrt ((perimeter E (Metric.ball 0 R)).toReal)) +
+      (M*b/(2*a))*orientedTilt E v (Metric.ball 0 R) := by
+  let φ := chartLaplacianTest e ψ
+  have hcφ := chartLaplacianTest_hasCompactSupport e hcψ
+  have hm := actual_fiber_first_moment_error hn hmin hv
+    (chartLaplacianTest_continuous e hψ) hcφ (fun y ↦ hM (e.symm y))
+    ha.le hδ hab hR hKR hh hphase
+  have he := chartFiber_test_eq_exact (A := a) hmin.1 hv e ha.le
+    (by linarith : a<b) hηs φ hηone hphase
+  have hj : (∫ x in Metric.ball 0 R, normalWindowTest v φ b 1 x ∂perimeterMeasure E) =
+      ∫ x in Metric.ball 0 R, (inner ℝ v x-0)*
+        planarDivergence v (gradient (planeWindowLift e ψ χ)) x ∂perimeterMeasure E := by
+    apply integral_congr_ae
+    have hsup : ∀ᵐ x ∂perimeterMeasure E, x ∈ perimeterSupport E := by
+      rw [← perimeterMeasure_support]; exact (perimeterMeasure E).support_mem_ae
+    filter_upwards [ae_restrict_of_ae hsup,ae_restrict_mem measurableSet_ball] with x hx hxB
+    have hxH := hh x ⟨hx,hxB⟩
+    have hxb : |inner ℝ v x|≤b := hxH.trans (by linarith)
+    rw [planeWindowLift_laplacian e hψ hχone (hxH.trans_lt hab₀)]
+    unfold normalWindowTest
+    rw [Set.indicator_of_mem (show x ∈ {z : Euc n | |inner ℝ v z|≤b} from hxb)]
+    simp only [sub_zero,pow_one,φ,chartLaplacianTest,chartProjection,ContinuousLinearMap.comp_apply]
+    exact mul_comm _ _
+  have hr := minimizing_height_planar_residual hn hmin (0:Euc n) hR hv 0
+    (planeWindowLift_contDiff e hψ hχ) (planeWindowLift_hasCompactSupport hv e hcψ hcχ)
+    hs ha.le hK (by simpa only [sub_zero] using hh) hd
+  have hI : |∫ y in tsupport φ, φ y*exactFiberHeight E v b y| ≤
+      2*a*K*(Real.sqrt (orientedTilt E v (Metric.ball 0 R))*
+        Real.sqrt ((perimeter E (Metric.ball 0 R)).toReal)) +
+      (M*b/2)*orientedTilt E v (Metric.ball 0 R) := by
+    have ht := abs_sub_le (∫ y in tsupport φ, φ y*exactFiberHeight E v b y)
+      (∫ x in Metric.ball 0 R, normalWindowTest v φ b 1 x ∂perimeterMeasure E) 0
+    rw [sub_zero,sub_zero,abs_sub_comm] at ht
+    rw [hj] at hm ht
+    linarith
+  have he' : (∫ y, chartFiber v e a (cutoffPhaseError E v η) y * divergence (gradient ψ) y) =
+      a⁻¹ * ∫ y in tsupport φ, φ y*exactFiberHeight E v b y := by
+    simpa only [φ,chartLaplacianTest,e.symm_apply_apply] using he
+  rw [he',abs_mul,abs_inv,abs_of_pos ha]
+  apply (mul_le_mul_of_nonneg_left hI (inv_nonneg.mpr ha.le)).trans_eq
+  field_simp
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/PlaneLaplacian.lean b/lean/OAI/Geometry/Perimeter/Regularity/PlaneLaplacian.lean
new file mode 100644
index 0000000000000000000000000000000000000000..a17ff3be2c5fb06c7ee061c899d9302144dbb75b
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/PlaneLaplacian.lean
@@ -0,0 +1,85 @@
+import OAI.Geometry.Perimeter.Regularity.NormalWindowTest
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology InnerProductSpace ContDiff
+namespace RPISingularHarnack
+open RPI.Boundary ContinuousLinearMap
+variable {n m : ℕ} {v : Euc n}
+
+def chartProjection (e : Euc m ≃ₗᵢ[ℝ] normalPlane v) : Euc n →L[ℝ] Euc m :=
+  e.symm.toContinuousLinearEquiv.toContinuousLinearMap.comp (normalProjection v)
+
+def chartInclusion (e : Euc m ≃ₗᵢ[ℝ] normalPlane v) : Euc m →L[ℝ] Euc n :=
+  (normalPlane v).subtypeL.comp e.toContinuousLinearEquiv.toContinuousLinearMap
+
+@[simp] theorem chartProjection_inclusion (e : Euc m ≃ₗᵢ[ℝ] normalPlane v) (y : Euc m) :
+    chartProjection e (chartInclusion e y)=y := by
+  change e.symm (normalProjection v (e y))=y
+  rw [Submodule.orthogonalProjectionOnto_mem_subspace_eq_self,e.symm_apply_apply]
+
+@[simp] theorem chartProjection_normal (e : Euc m ≃ₗᵢ[ℝ] normalPlane v) :
+    chartProjection e v=0 := by
+  have h : normalProjection v v = 0 :=
+    Submodule.orthogonalProjectionOnto_eq_zero_iff.mpr
+      ((ℝ ∙ v).le_orthogonal_orthogonal (Submodule.mem_span_singleton_self v))
+  change e.symm (normalProjection v v)=0
+  rw [h,map_zero]
+
+theorem chartInclusion_inner (e : Euc m ≃ₗᵢ[ℝ] normalPlane v) (y : Euc m) (x : Euc n) :
+    inner ℝ (chartInclusion e y) x = inner ℝ y (chartProjection e x) := by
+  change inner ℝ (e y:Euc n) x = inner ℝ y (e.symm (normalProjection v x))
+  rw [← (normalPlane v).inner_orthogonalProjectionOnto_eq_of_mem_left (e y) x]
+  simpa only [e.apply_symm_apply] using (e.inner_map_map y (e.symm (normalProjection v x)))
+
+theorem gradient_chart_lift {ψ : Euc m → ℝ} (hψ : Differentiable ℝ ψ)
+    (e : Euc m ≃ₗᵢ[ℝ] normalPlane v) (x : Euc n) :
+    gradient (fun z ↦ ψ (chartProjection e z)) x =
+      chartInclusion e (gradient ψ (chartProjection e x)) := by
+  apply ext_inner_right ℝ
+  intro z
+  rw [inner_gradient_left,chartInclusion_inner,inner_gradient_left]
+  change fderiv ℝ (ψ ∘ chartProjection e) x z = _
+  rw [fderiv_comp x (hψ _) (chartProjection e).differentiableAt,(chartProjection e).fderiv]
+  rfl
+
+theorem divergence_eq_trace {V : Euc n → Euc n} (x : Euc n) :
+    divergence V x = (fderiv ℝ V x).toLinearMap.trace ℝ (Euc n) := by
+  rw [LinearMap.trace_eq_sum_inner _ (EuclideanSpace.basisFun (Fin n) ℝ)]
+  rfl
+
+theorem planar_laplacian_chart_lift {ψ : Euc m → ℝ} (hψ : ContDiff ℝ ∞ ψ)
+    (e : Euc m ≃ₗᵢ[ℝ] normalPlane v) (x : Euc n) :
+    planarDivergence v (gradient (fun z ↦ ψ (chartProjection e z))) x =
+      divergence (gradient ψ) (chartProjection e x) := by
+  have hg := gradient_contDiff hψ
+  have he : gradient (fun z ↦ ψ (chartProjection e z)) =
+      fun z ↦ chartInclusion e (gradient ψ (chartProjection e z)) :=
+    funext (gradient_chart_lift (hψ.differentiable (by simp)) e)
+  have hd : fderiv ℝ (gradient (fun z ↦ ψ (chartProjection e z))) x =
+      (chartInclusion e).comp ((fderiv ℝ (gradient ψ) (chartProjection e x)).comp (chartProjection e)) := by
+    rw [he]
+    exact (((chartInclusion e).hasFDerivAt.comp x
+      (((hg.differentiable (by simp) _).hasFDerivAt).comp x
+        (chartProjection e).hasFDerivAt))).fderiv
+  rw [planarDivergence,divergence_eq_trace,hd,ContinuousLinearMap.comp_apply,
+    ContinuousLinearMap.comp_apply,chartProjection_normal,map_zero,map_zero,inner_zero_right,sub_zero,
+    divergence_eq_trace]
+  change LinearMap.trace ℝ (Euc n)
+    ((chartInclusion e).toLinearMap.comp
+      ((fderiv ℝ (gradient ψ) (chartProjection e x)).toLinearMap.comp
+        (chartProjection e).toLinearMap)) = _
+  rw [LinearMap.trace_comp_comm']
+  have hi : (chartProjection e).toLinearMap.comp (chartInclusion e).toLinearMap = LinearMap.id := by
+    apply LinearMap.ext
+    intro y
+    exact chartProjection_inclusion e y
+  rw [LinearMap.comp_assoc,hi,LinearMap.comp_id]
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/PlaneWindowLift.lean b/lean/OAI/Geometry/Perimeter/Regularity/PlaneWindowLift.lean
new file mode 100644
index 0000000000000000000000000000000000000000..d37adfb141459d8c2e6c4c9c74d782bca93a3a9a
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/PlaneWindowLift.lean
@@ -0,0 +1,85 @@
+import OAI.Geometry.Perimeter.Regularity.PlaneLaplacian
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology InnerProductSpace ContDiff
+namespace RPISingularHarnack
+open RPI.Boundary ContinuousLinearMap
+variable {n m : ℕ} {v : Euc n}
+
+theorem normalProjection_decomposition (hv : ‖v‖=1) (x : Euc n) :
+    (normalProjection v x:Euc n)+inner ℝ v x•v=x := by
+  obtain ⟨⟨y,s⟩,rfl⟩ := (normalCoordinates v hv).surjective x
+  simp only [normalCoordinates_apply,normalProjection_fiber,normal_fiber_inner hv]
+
+@[simp] theorem chartInclusion_projection (e : Euc m ≃ₗᵢ[ℝ] normalPlane v) (x : Euc n) :
+    chartInclusion e (chartProjection e x) = (normalProjection v x:Euc n) := by
+  change ((e (e.symm (normalProjection v x))):Euc n) = _
+  rw [e.apply_symm_apply]
+
+@[simp] theorem norm_chartInclusion (e : Euc m ≃ₗᵢ[ℝ] normalPlane v) (y : Euc m) :
+    ‖chartInclusion e y‖=‖y‖ := e.norm_map y
+
+def planeWindowLift (e : Euc m ≃ₗᵢ[ℝ] normalPlane v) (ψ : Euc m → ℝ)
+    (χ : ℝ → ℝ) (x : Euc n) : ℝ := ψ (chartProjection e x)*χ (inner ℝ v x)
+
+theorem planeWindowLift_contDiff (e : Euc m ≃ₗᵢ[ℝ] normalPlane v)
+    {ψ : Euc m → ℝ} {χ : ℝ → ℝ} (hψ : ContDiff ℝ ∞ ψ) (hχ : ContDiff ℝ ∞ χ) :
+    ContDiff ℝ ∞ (planeWindowLift e ψ χ) :=
+  (hψ.comp (chartProjection e).contDiff).mul
+    (hχ.comp (contDiff_const.inner ℝ contDiff_id))
+
+theorem planeWindowLift_tsupport (hv : ‖v‖=1) (e : Euc m ≃ₗᵢ[ℝ] normalPlane v)
+    (ψ : Euc m → ℝ) (χ : ℝ → ℝ) {r s : ℝ}
+    (hr : Function.support ψ ⊆ Metric.closedBall 0 r)
+    (hs : Function.support χ ⊆ Metric.closedBall 0 s) :
+    tsupport (planeWindowLift e ψ χ) ⊆ Metric.closedBall 0 (r+s) := by
+  apply closure_minimal _ Metric.isClosed_closedBall
+  intro x hx
+  have hprod : ψ (chartProjection e x)≠0 ∧ χ (inner ℝ v x)≠0 := mul_ne_zero_iff.mp hx
+  have h1 := hr hprod.1
+  have h2 := hs hprod.2
+  rw [Metric.mem_closedBall,dist_zero_right] at h1 h2 ⊢
+  have he : chartInclusion e (chartProjection e x)+inner ℝ v x•v=x := by
+    rw [chartInclusion_projection]; exact normalProjection_decomposition hv x
+  calc
+    ‖x‖=‖chartInclusion e (chartProjection e x)+inner ℝ v x•v‖ := congrArg norm he.symm
+    _ ≤ ‖chartInclusion e (chartProjection e x)‖+‖inner ℝ v x•v‖ := norm_add_le _ _
+    _ ≤ r+s := by simpa only [norm_chartInclusion,norm_smul,hv,mul_one] using add_le_add h1 h2
+
+theorem planeWindowLift_hasCompactSupport (hv : ‖v‖=1)
+    (e : Euc m ≃ₗᵢ[ℝ] normalPlane v) {ψ : Euc m → ℝ} {χ : ℝ → ℝ}
+    (hψ : HasCompactSupport ψ) (hχ : HasCompactSupport χ) :
+    HasCompactSupport (planeWindowLift e ψ χ) := by
+  obtain ⟨r,hr⟩ := hψ.isBounded.subset_closedBall (0:Euc m)
+  obtain ⟨s,hs⟩ := hχ.isBounded.subset_closedBall (0:ℝ)
+  exact (isCompact_closedBall 0 (r+s)).of_isClosed_subset (isClosed_tsupport _)
+    (planeWindowLift_tsupport hv e ψ χ (subset_closure.trans hr) (subset_closure.trans hs))
+
+theorem gradient_eventuallyEq {f g : Euc n → ℝ} {x : Euc n}
+    (h : f=ᶠ[𝓝 x] g) : gradient f=ᶠ[𝓝 x] gradient g := by
+  filter_upwards [h.fderiv (𝕜 := ℝ)] with y hy
+  exact congrArg (InnerProductSpace.toDual ℝ (Euc n)).symm hy
+
+theorem planeWindowLift_laplacian (e : Euc m ≃ₗᵢ[ℝ] normalPlane v)
+    {ψ : Euc m → ℝ} {χ : ℝ → ℝ} (hψ : ContDiff ℝ ∞ ψ) {b : ℝ}
+    (hχone : ∀ s : ℝ, |s|<b → χ s=1) {x : Euc n} (hx : |inner ℝ v x|<b) :
+    planarDivergence v (gradient (planeWindowLift e ψ χ)) x =
+      divergence (gradient ψ) (chartProjection e x) := by
+  have he : planeWindowLift e ψ χ =ᶠ[𝓝 x] fun z ↦ ψ (chartProjection e z) := by
+    filter_upwards [((continuous_const.inner continuous_id).abs.continuousAt).eventually
+      (gt_mem_nhds hx)] with z hz
+    change |inner ℝ v z|<b at hz
+    dsimp only [planeWindowLift]
+    rw [hχone _ hz,mul_one]
+  have hd := (gradient_eventuallyEq he).fderiv_eq (𝕜 := ℝ)
+  rw [← planar_laplacian_chart_lift hψ e x]
+  simp only [planarDivergence,divergence,hd]
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/QuadraticError.lean b/lean/OAI/Geometry/Perimeter/Regularity/QuadraticError.lean
new file mode 100644
index 0000000000000000000000000000000000000000..43d46228acbe407d9a3bcd6f31b06ea86542f5aa
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/QuadraticError.lean
@@ -0,0 +1,66 @@
+import OAI.Geometry.Perimeter.Regularity.ActualFiberMoments
+import OAI.Geometry.Perimeter.Regularity.ActualFiberZerothMoment
+import OAI.Geometry.Perimeter.Regularity.QuadraticMoment
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+open RPI.Boundary ContinuousLinearMap
+variable {n : ℕ}
+
+theorem actual_fiber_quadratic_error (hn : 1<n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    {v : Euc n} (hv : ‖v‖=1) {φ w : normalPlane v → ℝ}
+    (hφ : Continuous φ) (hcφ : HasCompactSupport φ) (hpφ : ∀ y, 0≤φ y)
+    (hw : Continuous w) {M₂ M₁ M₀ a δ R : ℝ}
+    (hM₂ : ∀ y, ‖φ y‖≤M₂) (hM₁ : ∀ y, ‖φ y*w y‖≤M₁)
+    (hM₀ : ∀ y, ‖φ y*(w y)^2‖≤M₀)
+    (ha : 0<a) (hδ : 0<δ) (hab : a+δ<2*a) (hR : 0<R)
+    (hKR : ∀ y ∈ tsupport φ, ‖y‖+2*a+δ≤R)
+    (hh : ∀ x ∈ perimeterSupport E ∩ Metric.ball 0 R, |inner ℝ v x|≤a)
+    (hphase : ∀ᵐ x ∂volume, x ∈ Metric.ball 0 R → a < |inner ℝ v x| →
+      (x∈E ↔ inner ℝ v x<0)) :
+    |(∫ x in Metric.ball 0 R, normalWindowQuadratic v φ w a (2*a) x ∂perimeterMeasure E)-
+      (∫ y, φ y*(exactFiberHeight E v (2*a) y/a-w y)^2)| ≤
+        (18*M₂+2*M₁+M₀/2)*orientedTilt E v (Metric.ball 0 R) := by
+  have h2 := actual_fiber_second_moment_error hn hmin hv hφ hcφ hpφ hM₂
+    ha.le hδ hab hR hKR hh hphase
+  have h1 := actual_fiber_first_moment_error hn hmin hv (hφ.mul hw) hcφ.mul_right hM₁
+    ha.le hδ hab hR (fun y hy ↦ hKR y (tsupport_mul_subset_left hy)) hh hphase
+  have h0 := actual_fiber_zeroth_moment_error hn hmin hv (hφ.mul (hw.pow 2)) hcφ.mul_right hM₀
+    ha.le hδ hab hR (fun y hy ↦ hKR y (tsupport_mul_subset_left hy)) hh hphase
+  rw [setIntegral_tsupport_mul] at h2 h1
+  change |(∫ x in Metric.ball 0 R, normalWindowTest v (fun y ↦ φ y*(w y)^2) (2*a) 0 x ∂perimeterMeasure E)-
+    (∫ y in tsupport (fun y ↦ φ y*(w y)^2), φ y*(w y)^2)| ≤ _ at h0
+  have hz : (∫ y in tsupport (fun y ↦ φ y*(w y)^2), φ y*(w y)^2) = ∫ y, φ y*(w y)^2 := by
+    apply setIntegral_eq_integral_of_forall_compl_eq_zero
+    intro y hy
+    exact Function.notMem_support.mp (fun hh ↦ hy (subset_closure hh))
+  rw [hz] at h0
+  rw [normalWindowQuadratic_integral_expand hmin v hφ hcφ hw (by positivity) hR,
+    exactFiberQuadratic_integral_expand hmin.1 v hφ hcφ hw (by positivity)]
+  have htri (A₂ A₁ A₀ B₂ B₁ B₀ : ℝ) :
+      |((a⁻¹)^2*A₂-(2/a)*A₁+A₀)-((a⁻¹)^2*B₂-(2/a)*B₁+B₀)| ≤
+        (a⁻¹)^2*|A₂-B₂|+(2/a)*|A₁-B₁|+|A₀-B₀| := by
+    calc
+      _ = |((a⁻¹)^2*(A₂-B₂)-(2/a)*(A₁-B₁))+(A₀-B₀)| := by congr 1; ring
+      _ ≤ |(a⁻¹)^2*(A₂-B₂)-(2/a)*(A₁-B₁)|+|A₀-B₀| := abs_add_le _ _
+      _ ≤ (|(a⁻¹)^2*(A₂-B₂)|+|(2/a)*(A₁-B₁)|)+|A₀-B₀| :=
+        add_le_add (abs_sub _ _) le_rfl
+      _ = _ := by rw [abs_mul,abs_mul,abs_of_nonneg (sq_nonneg _),abs_of_pos (by positivity : 0<2/a)]
+  refine (htri _ _ _ _ _ _).trans ?_
+  have h := add_le_add (add_le_add (mul_le_mul_of_nonneg_left h2 (sq_nonneg (a⁻¹)))
+    (mul_le_mul_of_nonneg_left h1 (by positivity : 0≤2/a))) h0
+  convert h using 1
+  try rfl
+  field_simp
+  ring
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/QuadraticMoment.lean b/lean/OAI/Geometry/Perimeter/Regularity/QuadraticMoment.lean
new file mode 100644
index 0000000000000000000000000000000000000000..c2c47cf41e6c83d6e4803ea37a7af1d3a38ce5d0
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/QuadraticMoment.lean
@@ -0,0 +1,122 @@
+import OAI.Geometry.Perimeter.Regularity.PhaseMomentLimit
+import OAI.Geometry.Perimeter.Regularity.NormalWindowTest
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+open RPI.Boundary ContinuousLinearMap
+variable {n : ℕ}
+
+theorem exactFiberHeight_measurable {E : Set (Euc n)} (hE : MeasurableSet E)
+    (v : Euc n) {b : ℝ} (hb : 0≤b) : Measurable (exactFiberHeight E v b) := by
+  have hm : Measurable (fun z : normalPlane v × ℝ ↦
+      E.indicator (fun _ ↦ (1:ℝ)) ((z.1:Euc n)+z.2•v)) :=
+    (measurable_const.indicator hE).comp
+      (((normalPlane v).subtypeL.continuous.comp continuous_fst).add
+        (continuous_snd.smul continuous_const)).measurable
+  have hi := hm.stronglyMeasurable.integral_prod_right' (ν := volume.restrict (Ioc (-b) b)) |>.measurable
+  unfold exactFiberHeight
+  simp_rw [intervalIntegral.integral_of_le (by linarith : -b≤b)]
+  exact (measurable_const : Measurable (fun _ : normalPlane v ↦ -b)).add hi
+
+theorem exactFiberHeight_norm_le {E : Set (Euc n)} (hE : MeasurableSet E)
+    (v : Euc n) {b : ℝ} (hb : 0≤b) (y : normalPlane v) : ‖exactFiberHeight E v b y‖≤b := by
+  apply interval_height_abs_le
+    ((measurable_const.indicator hE).comp (continuous_const.add (continuous_id.smul continuous_const)).measurable)
+    hb
+  intro s
+  by_cases hs : (y:Euc n)+s•v∈E <;> simp [hs]
+
+theorem exactFiberMoment_integrable {E : Set (Euc n)} (hE : MeasurableSet E)
+    (v : Euc n) {φ : normalPlane v → ℝ} (hφ : Continuous φ) (hcφ : HasCompactSupport φ)
+    {b : ℝ} (hb : 0≤b) (k : ℕ) : Integrable (fun y ↦ φ y*(exactFiberHeight E v b y)^k) volume := by
+  apply Integrable.mono' ((hφ.integrable_of_hasCompactSupport hcφ).norm.mul_const (b^k))
+    ((hφ.measurable.mul ((exactFiberHeight_measurable hE v hb).pow_const k)).aestronglyMeasurable)
+  exact ae_of_all _ fun y ↦ by
+    dsimp only [Pi.mul_apply]
+    rw [norm_mul,norm_pow]
+    exact mul_le_mul_of_nonneg_left (pow_le_pow_left₀ (norm_nonneg _) (exactFiberHeight_norm_le hE v hb y) k) (norm_nonneg _)
+
+theorem normalWindowTest_integrableOn {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    (v : Euc n) {φ : normalPlane v → ℝ} (hφ : Continuous φ) (hcφ : HasCompactSupport φ)
+    {b R : ℝ} (hb : 0≤b) (hR : 0<R) (k : ℕ) :
+    IntegrableOn (normalWindowTest v φ b k) (Metric.ball 0 R) (perimeterMeasure E) := by
+  have : IsFiniteMeasure ((perimeterMeasure E).restrict (Metric.ball 0 R)) := ⟨by
+    rw [Measure.restrict_apply_univ,perimeterMeasure_open E Metric.isOpen_ball]
+    exact hmin.2.1 0 R hR⟩
+  obtain ⟨M,hM⟩ := hcφ.exists_bound_of_continuous hφ
+  exact (integrable_const (M*b^k)).mono' (normalWindowTest_measurable hφ b k).aestronglyMeasurable
+    (ae_of_all _ (normalWindowTest_bound hM hb k))
+
+theorem setIntegral_tsupport_mul {direction : Euc n} (φ f : normalPlane direction → ℝ) :
+    (∫ y in tsupport φ, φ y*f y) = ∫ y, φ y*f y := by
+  apply setIntegral_eq_integral_of_forall_compl_eq_zero
+  intro y hy
+  rw [Function.notMem_support.mp (fun h ↦ hy (subset_closure h)),zero_mul]
+
+def normalWindowQuadratic (v : Euc n) (φ w : normalPlane v → ℝ) (a b : ℝ)
+    (x : Euc n) : ℝ :=
+  {z:Euc n | |inner ℝ v z|≤b}.indicator
+    (fun z ↦ φ (normalProjection v z)*(inner ℝ v z/a-w (normalProjection v z))^2) x
+
+theorem normalWindowQuadratic_expand (v : Euc n) (φ w : normalPlane v → ℝ) (a b : ℝ) :
+    normalWindowQuadratic v φ w a b = fun x ↦ (a⁻¹)^2*normalWindowTest v φ b 2 x -
+      (2/a)*normalWindowTest v (fun y ↦ φ y*w y) b 1 x +
+      normalWindowTest v (fun y ↦ φ y*(w y)^2) b 0 x := by
+  funext x
+  by_cases hx : |inner ℝ v x|≤b
+  · simp only [normalWindowQuadratic,normalWindowTest,Set.indicator_apply,Set.mem_ofPred_eq,
+      ite_eq_left hx,pow_one,pow_zero,mul_one]
+    ring
+  · simp [normalWindowQuadratic,normalWindowTest,hx]
+
+theorem normalWindowQuadratic_integral_expand {E : Set (Euc n)}
+    (hmin : IsLocallyPerimeterMinimizing E) (v : Euc n) {φ w : normalPlane v → ℝ}
+    (hφ : Continuous φ) (hcφ : HasCompactSupport φ) (hw : Continuous w)
+    {a b R : ℝ} (hb : 0≤b) (hR : 0<R) :
+    (∫ x in Metric.ball 0 R, normalWindowQuadratic v φ w a b x ∂perimeterMeasure E) =
+      (a⁻¹)^2*(∫ x in Metric.ball 0 R, normalWindowTest v φ b 2 x ∂perimeterMeasure E) -
+      (2/a)*(∫ x in Metric.ball 0 R, normalWindowTest v (fun y ↦ φ y*w y) b 1 x ∂perimeterMeasure E) +
+      ∫ x in Metric.ball 0 R, normalWindowTest v (fun y ↦ φ y*(w y)^2) b 0 x ∂perimeterMeasure E := by
+  have hi2 := normalWindowTest_integrableOn hmin v hφ hcφ hb hR 2
+  have hi1 := normalWindowTest_integrableOn hmin v (hφ.mul hw) (hcφ.mul_right) hb hR 1
+  have hi0 := normalWindowTest_integrableOn hmin v (hφ.mul (hw.pow 2)) (hcφ.mul_right) hb hR 0
+  change IntegrableOn (normalWindowTest v (fun y ↦ φ y*w y) b 1) _ _ at hi1
+  change IntegrableOn (normalWindowTest v (fun y ↦ φ y*(w y)^2) b 0) _ _ at hi0
+  simp only [normalWindowQuadratic_expand]
+  have h := integral_add ((hi2.const_mul ((a⁻¹)^2)).sub (hi1.const_mul (2/a))) hi0
+  simp only [Pi.sub_apply] at h
+  rw [integral_sub (hi2.const_mul ((a⁻¹)^2)) (hi1.const_mul (2/a)),
+    integral_const_mul,integral_const_mul] at h
+  exact h
+
+theorem exactFiberQuadratic_integral_expand {E : Set (Euc n)} (hE : MeasurableSet E)
+    (v : Euc n) {φ w : normalPlane v → ℝ}
+    (hφ : Continuous φ) (hcφ : HasCompactSupport φ) (hw : Continuous w)
+    {a b : ℝ} (hb : 0≤b) :
+    (∫ y, φ y*(exactFiberHeight E v b y/a-w y)^2) =
+      (a⁻¹)^2*(∫ y, φ y*(exactFiberHeight E v b y)^2) -
+      (2/a)*(∫ y, (φ y*w y)*exactFiberHeight E v b y) + ∫ y, φ y*(w y)^2 := by
+  have hi2 := exactFiberMoment_integrable hE v hφ hcφ hb 2
+  have hi1 := exactFiberMoment_integrable hE v (hφ.mul hw) (hcφ.mul_right) hb 1
+  simp only [pow_one] at hi1
+  have hi0 := (hφ.mul (hw.pow 2)).integrable_of_hasCompactSupport (μ := volume) (hcφ.mul_right)
+  have he : (fun y ↦ φ y*(exactFiberHeight E v b y/a-w y)^2) = fun y ↦
+      (a⁻¹)^2*(φ y*(exactFiberHeight E v b y)^2) - (2/a)*((φ y*w y)*exactFiberHeight E v b y) +
+        φ y*(w y)^2 := by funext y; ring
+  rw [he]
+  simp only [Pi.mul_apply] at hi1 hi0
+  have h := integral_add ((hi2.const_mul ((a⁻¹)^2)).sub (hi1.const_mul (2/a))) hi0
+  simp only [Pi.sub_apply] at h
+  rw [integral_sub (hi2.const_mul ((a⁻¹)^2)) (hi1.const_mul (2/a)),
+    integral_const_mul,integral_const_mul] at h
+  exact h
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/RankOneNormalError.lean b/lean/OAI/Geometry/Perimeter/Regularity/RankOneNormalError.lean
new file mode 100644
index 0000000000000000000000000000000000000000..7a508cc95fc8a1e38e916619587b6959c367bcc0
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/RankOneNormalError.lean
@@ -0,0 +1,571 @@
+import OAI.Geometry.Perimeter.Regularity.SmoothSublevelPairingBound
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+namespace RPISingularHarnack
+open RPI.Boundary
+
+def rankOneNormalError {n : ℕ} (E : Set (Euc n)) (φ : Euc n → ℝ)
+    (v : Euc n) (x : Euc n) : Euc n :=
+  (fderiv ℝ φ x v) • approximateNormal E x -
+    (inner ℝ (approximateNormal E x) v) • gradient φ x
+
+theorem inner_rankOnePiola {n : ℕ} (E : Set (Euc n)) (φ : Euc n → ℝ)
+    (v : Euc n) (t : ℝ) (W : Euc n → Euc n) (x : Euc n) :
+    inner ℝ (approximateNormal E x) (rankOnePiola φ v t W x) =
+      inner ℝ (approximateNormal E x + t • rankOneNormalError E φ v x)
+        (W (rankOneVariation φ v t x)) := by
+  simp only [rankOnePiola, rankOneNormalError, rankOneJacobian, inner_sub_right,
+    real_inner_smul_right, inner_add_left, real_inner_smul_left, inner_sub_left,
+    inner_gradient_left]
+  ring
+
+theorem rankOneNormalError_norm_le {n : ℕ} (E : Set (Euc n)) {φ : Euc n → ℝ}
+    {L : ℝ≥0} (hL : LipschitzWith L φ) (v x : Euc n) :
+    ‖rankOneNormalError E φ v x‖ ≤ 2 * L * ‖v‖ := by
+  have hD : ‖fderiv ℝ φ x‖ ≤ L := norm_fderiv_le_of_lipschitz ℝ hL
+  have hN := approximateNormal_norm_le_one E x
+  have hgrad : ‖gradient φ x‖ ≤ L := by simpa only [gradient, LinearIsometryEquiv.norm_map] using hD
+  have hDv : ‖fderiv ℝ φ x v‖ ≤ L * ‖v‖ :=
+    ((fderiv ℝ φ x).le_opNorm v).trans (mul_le_mul_of_nonneg_right hD (norm_nonneg v))
+  have hNv : ‖inner ℝ (approximateNormal E x) v‖ ≤ ‖v‖ :=
+    (norm_inner_le_norm _ _).trans (by nlinarith [norm_nonneg v])
+  calc
+    _ ≤ ‖(fderiv ℝ φ x v) • approximateNormal E x‖ +
+        ‖(inner ℝ (approximateNormal E x) v) • gradient φ x‖ := norm_sub_le _ _
+    _ ≤ L * ‖v‖ * 1 + ‖v‖ * L := by
+      simp only [norm_smul]
+      exact add_le_add (mul_le_mul hDv hN (norm_nonneg _) (by positivity))
+        (mul_le_mul hNv hgrad (norm_nonneg _) (norm_nonneg v))
+    _ = _ := by ring
+
+theorem perimeter_image_rankOneVariation_le {n : ℕ} (hn : 1 < n)
+    {E U : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    (hU : IsOpen U) (hP : perimeter E U ≠ ⊤)
+    {φ : Euc n → ℝ} (hφ : ContDiff ℝ ∞ φ) (hφU : tsupport φ ⊆ U)
+    {L : ℝ≥0} (hL : LipschitzWith L φ) (v : Euc n) (t : ℝ)
+    (ht : ‖t‖ * L * ‖v‖ < 1) :
+    perimeter (rankOneVariation φ v t '' E) U ≤
+      ENNReal.ofReal (∫ x in U, ‖approximateNormal E x + t • rankOneNormalError E φ v x‖
+        ∂perimeterMeasure E) := by
+  obtain ⟨e, he⟩ := exists_rankOneVariation_homeomorph (lt_trans (by norm_num) hn) hL v t ht
+  let μ := (perimeterMeasure E).restrict U
+  have : IsFiniteMeasure μ := ⟨by rw [Measure.restrict_apply_univ, perimeterMeasure_open E hU]; exact hP.lt_top⟩
+  have hmN := (measurable_approximateNormal hmin.1).aestronglyMeasurable (μ := μ)
+  have hcgrad : Continuous (gradient φ) := continuous_iff_continuousAt.mpr (fun x ↦
+    continuousAt_gradient_of_contDiffAt (hφ.contDiffAt.of_le (by simp)))
+  have hmQ : AEStronglyMeasurable (rankOneNormalError E φ v) μ := by
+    unfold rankOneNormalError
+    exact (((hφ.fderiv_right (m := ∞) (by simp)).continuous.clm_apply continuous_const).aestronglyMeasurable.smul hmN).sub
+      ((hmN.inner (𝕜 := ℝ) (aestronglyMeasurable_const (b := v))).smul hcgrad.aestronglyMeasurable)
+  have hiQ : Integrable (rankOneNormalError E φ v) μ :=
+    Integrable.of_bound hmQ (2 * L * ‖v‖) (ae_of_all _ (rankOneNormalError_norm_le E hL v))
+  have hiN : Integrable (approximateNormal E) μ :=
+    Integrable.of_bound hmN 1 (ae_of_all _ (approximateNormal_norm_le_one E))
+  have hiA := (hiN.add (hiQ.smul t)).norm
+  unfold perimeter
+  refine iSup_le fun W ↦ iSup_le fun hW ↦ iSup_le fun hcW ↦ iSup_le fun hWU ↦ iSup_le fun hW1 ↦ ?_
+  apply ENNReal.ofReal_le_ofReal
+  have hchange := integral_image_eq_integral_abs_det_fderiv_smul (μ := volume) hmin.1
+    (fun x (_ : x ∈ E) ↦ (rankOneVariation_hasFDerivAt (hφ.differentiable (by simp) x) v t).hasFDerivWithinAt)
+    (show InjOn (rankOneVariation φ v t) E from by rw [← he]; exact e.injective.injOn)
+    (divergence W)
+  have hdiv : (∫ x in rankOneVariation φ v t '' E, divergence W x) =
+      ∫ x in E, divergence (rankOnePiola φ v t W) x := by
+    rw [hchange]
+    apply setIntegral_congr_fun hmin.1
+    intro x _
+    dsimp only
+    rw [det_id_add_smulRight]
+    change |rankOneJacobian φ v t x| * divergence W (rankOneVariation φ v t x) = _
+    rw [abs_of_pos (rankOneJacobian_pos hL v t ht x), divergence_rankOnePiola hφ hW]
+  rw [hdiv, minimizing_gaussGreen_local hn hmin U hU hP _
+    (rankOnePiola_contDiff hφ hW v t) (rankOnePiola_hasCompactSupport hφ v t e he hcW)
+    (rankOnePiola_tsupport_subset_region hφ v t e he hφU hWU)]
+  simp_rw [inner_rankOnePiola]
+  have hiW : Integrable (fun x ↦ inner ℝ (approximateNormal E x + t • rankOneNormalError E φ v x)
+      (W (rankOneVariation φ v t x))) μ := by
+    apply Integrable.mono' hiA
+    · exact (hmN.add (hmQ.const_smul t)).inner (hW.continuous.comp
+        (rankOneVariation_contDiff hφ v t).continuous).aestronglyMeasurable
+    · exact ae_of_all _ fun x ↦ (norm_inner_le_norm _ _).trans
+        (by simpa only [mul_one, Pi.add_apply, Pi.smul_apply] using mul_le_mul_of_nonneg_left (hW1 _) (norm_nonneg _))
+  apply integral_mono_ae hiW hiA
+  exact ae_of_all _ fun x ↦ (real_inner_le_norm _ _).trans
+    (by simpa only [mul_one, Pi.add_apply, Pi.smul_apply] using mul_le_mul_of_nonneg_left (hW1 _) (norm_nonneg _))
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+namespace RPISingularHarnack
+
+theorem norm_unit_add_smul_le_quadratic {V : Type*} [NormedAddCommGroup V]
+    [InnerProductSpace ℝ V] (N Q : V) (hN : ‖N‖ = 1) (t : ℝ) :
+    ‖N + t • Q‖ ≤ 1 + t * inner ℝ N Q + (t ^ 2 / 2) * ‖Q‖ ^ 2 := by
+  have hs := norm_add_sq_real N (t • Q)
+  rw [hN, norm_smul, mul_pow, Real.norm_eq_abs, sq_abs, real_inner_smul_right] at hs
+  nlinarith [sq_nonneg (‖N + t • Q‖ - 1)]
+
+theorem linear_term_eq_zero_of_local_quadratic_nonneg (I J : ℝ)
+    (h : ∀ᶠ t : ℝ in 𝓝 0, 0 ≤ t * I + t ^ 2 * J) : I = 0 := by
+  have hm : IsLocalMin (fun t : ℝ ↦ t * I + t ^ 2 * J) 0 := by
+    simpa only [IsLocalMin, IsMinFilter, zero_mul, zero_pow (by norm_num : (2 : ℕ) ≠ 0),
+      zero_add] using h
+  have hd : HasDerivAt (fun t : ℝ ↦ t * I + t ^ 2 * J) I 0 := by
+    have hh := ((hasDerivAt_id (0 : ℝ)).mul_const I).fun_add
+      (((hasDerivAt_id (0 : ℝ)).pow 2).mul_const J)
+    simpa using hh
+  exact hm.hasDerivAt_eq_zero hd
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+namespace RPISingularHarnack
+open RPI.Boundary
+
+theorem minimizing_tangentialGradient_integral_eq_zero_local {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    (c : Euc n) {R : ℝ} (hR : 0 < R) {φ : Euc n → ℝ}
+    (hφ : ContDiff ℝ ∞ φ) (hcφ : HasCompactSupport φ)
+    (hsφ : tsupport φ ⊆ Metric.ball c R) (v : Euc n) :
+    (∫ x in Metric.ball c R, inner ℝ (tangentialGradient E φ x) v ∂perimeterMeasure E) = 0 := by
+  let U := Metric.ball c R
+  let μ := (perimeterMeasure E).restrict U
+  have hP : perimeter E U ≠ ⊤ := (hmin.2.1 c R hR).ne
+  have : IsFiniteMeasure μ := ⟨by rw [Measure.restrict_apply_univ, perimeterMeasure_open E Metric.isOpen_ball]; exact hP.lt_top⟩
+  obtain ⟨L, hL⟩ := ContDiff.lipschitzWith_of_hasCompactSupport hcφ (hφ.of_le (by simp)) one_ne_zero
+  let N := approximateNormal E
+  let Q := rankOneNormalError E φ v
+  let B : ℝ := 2 * L * ‖v‖
+  have hB : 0 ≤ B := by dsimp [B]; positivity
+  have hmN : AEStronglyMeasurable N μ := (measurable_approximateNormal hmin.1).aestronglyMeasurable
+  have hcgrad : Continuous (gradient φ) := continuous_iff_continuousAt.mpr (fun x ↦
+    continuousAt_gradient_of_contDiffAt (hφ.contDiffAt.of_le (by simp)))
+  have hmQ : AEStronglyMeasurable Q μ := by
+    dsimp [Q, rankOneNormalError]
+    exact (((hφ.continuous_fderiv (by simp)).clm_apply continuous_const).aestronglyMeasurable.smul hmN).sub
+      ((hmN.inner (𝕜 := ℝ) (aestronglyMeasurable_const (b := v))).smul hcgrad.aestronglyMeasurable)
+  have hN1 : ∀ x, ‖N x‖ ≤ 1 := approximateNormal_norm_le_one E
+  have hNu : ∀ᵐ x ∂μ, ‖N x‖ = 1 :=
+    (ae_restrict_of_ae (minimizing_halfspaceBlowup_ae hn hmin)).mono fun _ h ↦ h.1
+  have hQB : ∀ x, ‖Q x‖ ≤ B := rankOneNormalError_norm_le E hL v
+  have hiN : Integrable N μ := Integrable.of_bound hmN 1 (ae_of_all _ hN1)
+  have hiQ : Integrable Q μ := Integrable.of_bound hmQ B (ae_of_all _ hQB)
+  have hiI : Integrable (fun x ↦ inner ℝ (N x) (Q x)) μ :=
+    Integrable.of_bound (hmN.inner hmQ) B (ae_of_all _ fun x ↦
+      (norm_inner_le_norm _ _).trans (by nlinarith [hN1 x, hQB x, norm_nonneg (N x), norm_nonneg (Q x)]))
+  have hiJ : Integrable (fun x ↦ ‖Q x‖ ^ 2) μ :=
+    Integrable.of_bound (hmQ.norm.pow 2) (B ^ 2) (ae_of_all _ fun x ↦ by
+      rw [Real.norm_of_nonneg (sq_nonneg _)]
+      exact (sq_le_sq₀ (norm_nonneg _) hB).2 (hQB x))
+  have hvariation : ∀ t : ℝ, ‖t‖ * L * ‖v‖ < 1 →
+      μ.real univ ≤ ∫ x, ‖N x + t • Q x‖ ∂μ := by
+    intro t ht
+    obtain ⟨e, he⟩ := exists_rankOneVariation_homeomorph (lt_trans (by norm_num) hn) hL v t ht
+    have hmF : MeasurableSet (rankOneVariation φ v t '' E) := by
+      rw [← he]
+      exact e.measurableEmbedding.measurableSet_image.mpr hmin.1
+    have hs := symmDiff_image_rankOneVariation_subset E v t e he
+    have hc : IsCompact (closure (symmDiff E (rankOneVariation φ v t '' E))) :=
+      hcφ.of_isClosed_subset isClosed_closure hs
+    have hlow := hmin.2.2 c R hR _ hmF hc (hs.trans hsφ)
+    have hu := perimeter_image_rankOneVariation_le hn hmin Metric.isOpen_ball hP hφ hsφ hL v t ht
+    have htoreal := ENNReal.toReal_mono ENNReal.ofReal_ne_top (hlow.trans hu)
+    rw [ENNReal.toReal_ofReal (integral_nonneg fun _ ↦ norm_nonneg _)] at htoreal
+    simpa only [μ, U, N, Q, Measure.real, Measure.restrict_apply_univ, perimeterMeasure_open E Metric.isOpen_ball] using htoreal
+  have hevent : ∀ᶠ t : ℝ in 𝓝 0, ‖t‖ * L * ‖v‖ < 1 := by
+    exact (show ContinuousAt (fun t : ℝ ↦ ‖t‖ * L * ‖v‖) 0 by fun_prop).eventually
+      (gt_mem_nhds (by simp : ‖(0 : ℝ)‖ * L * ‖v‖ < 1))
+  have hz : (∫ x, inner ℝ (N x) (Q x) ∂μ) = 0 := by
+    apply linear_term_eq_zero_of_local_quadratic_nonneg _ ((∫ x, ‖Q x‖ ^ 2 ∂μ) / 2)
+    filter_upwards [hevent] with t ht
+    have hiA : Integrable (fun x ↦ ‖N x + t • Q x‖) μ := (hiN.add (hiQ.smul t)).norm
+    have hiB : Integrable (fun x ↦ 1 + t * inner ℝ (N x) (Q x)) μ :=
+      (integrable_const (1 : ℝ)).add (hiI.const_mul t)
+    have hip : Integrable (fun x ↦ 1 + t * inner ℝ (N x) (Q x) +
+        t ^ 2 / 2 * ‖Q x‖ ^ 2) μ := hiB.add (hiJ.const_mul (t ^ 2 / 2))
+    have hupper := integral_mono_ae hiA hip (hNu.mono fun x hx ↦ norm_unit_add_smul_le_quadratic (N x) (Q x) hx t)
+    rw [integral_add hiB (hiJ.const_mul _),
+      integral_add (integrable_const (1 : ℝ)) (hiI.const_mul t), integral_const,
+      integral_const_mul, integral_const_mul, smul_eq_mul, mul_one] at hupper
+    have hlow := hvariation t ht
+    nlinarith
+  have hid : ∀ᵐ x ∂μ, inner ℝ (N x) (Q x) = inner ℝ (tangentialGradient E φ x) v := by
+    filter_upwards [hNu] with x hx
+    dsimp [N, Q, rankOneNormalError, tangentialGradient]
+    rw [inner_sub_right, real_inner_smul_right, real_inner_self_eq_norm_sq, hx,
+      one_pow, mul_one, inner_sub_left, real_inner_smul_left, inner_gradient_left]
+    rw [real_inner_smul_right, real_inner_comm (approximateNormal E x) (gradient φ x)]
+    ring
+  exact (integral_congr_ae hid).symm.trans hz
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+namespace RPISingularHarnack
+
+def tangentialDivergence {n : ℕ} (E : Set (Euc n)) (V : Euc n → Euc n)
+    (x : Euc n) : ℝ :=
+  divergence V x - inner ℝ (approximateNormal E x)
+    (fderiv ℝ V x (approximateNormal E x))
+
+theorem tangentialDivergence_eq_sum {n : ℕ} (E : Set (Euc n))
+    {V : Euc n → Euc n} (hV : Differentiable ℝ V) (x : Euc n) :
+    tangentialDivergence E V x = ∑ i : Fin n,
+      inner ℝ (tangentialGradient E
+        (fun y ↦ inner ℝ ((EuclideanSpace.basisFun (Fin n) ℝ) i) (V y)) x)
+        ((EuclideanSpace.basisFun (Fin n) ℝ) i) := by
+  have hd (i : Fin n) : fderiv ℝ (fun y ↦ inner ℝ ((EuclideanSpace.basisFun (Fin n) ℝ) i) (V y)) x =
+      (innerSL ℝ ((EuclideanSpace.basisFun (Fin n) ℝ) i)).comp (fderiv ℝ V x) :=
+    ((innerSL ℝ ((EuclideanSpace.basisFun (Fin n) ℝ) i)).hasFDerivAt.comp x (hV x).hasFDerivAt).fderiv
+  simp only [tangentialGradient, inner_sub_left, real_inner_smul_left,
+    inner_gradient_left, Finset.sum_sub_distrib]
+  change divergence V x - _ = (∑ i, fderiv ℝ (fun y ↦ inner ℝ ((EuclideanSpace.basisFun (Fin n) ℝ) i) (V y)) x ((EuclideanSpace.basisFun (Fin n) ℝ) i)) - _
+  simp only [hd, ContinuousLinearMap.comp_apply, innerSL_apply_apply]
+  congr 1
+  conv_rhs => arg 2; ext i; rw [mul_comm]
+  exact ((EuclideanSpace.basisFun (Fin n) ℝ).sum_inner_mul_inner _ _).symm
+
+theorem tangentialDivergence_smul {n : ℕ} (E : Set (Euc n))
+    {f : Euc n → ℝ} {V : Euc n → Euc n}
+    (hf : Differentiable ℝ f) (hV : Differentiable ℝ V) (x : Euc n) :
+    tangentialDivergence E (fun y ↦ f y • V y) x =
+      f x * tangentialDivergence E V x +
+        inner ℝ (tangentialGradient E f x) (V x) := by
+  rw [tangentialDivergence, divergence_smul hf hV,
+    fderiv_fun_smul (hf x) (hV x)]
+  simp only [add_apply, smul_apply,
+    ContinuousLinearMap.smulRight_apply, inner_add_right, real_inner_smul_right,
+    tangentialDivergence, tangentialGradient, inner_sub_left, real_inner_smul_left,
+    inner_gradient_left]
+  ring
+
+theorem tangentialDivergence_displacement {n : ℕ} (E : Set (Euc n))
+    (a x : Euc n) (hN : ‖approximateNormal E x‖ = 1) (hn : 0 < n) :
+    tangentialDivergence E (fun y ↦ y - a) x = (n - 1 : ℕ) := by
+  have hd : fderiv ℝ (fun y : Euc n ↦ y - a) x = ContinuousLinearMap.id ℝ (Euc n) := by
+    simpa only [id_eq] using ((hasFDerivAt_id x).sub_const a).fderiv
+  simp only [tangentialDivergence, divergence, hd, ContinuousLinearMap.id_apply,
+    real_inner_self_eq_norm_sq, hN, one_pow]
+  simp only [(EuclideanSpace.basisFun (Fin n) ℝ).norm_eq_one, one_pow,
+    Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul, mul_one]
+  simpa only [Nat.cast_one] using (Nat.cast_sub (R := ℝ) (by omega : 1 ≤ n)).symm
+
+def radialKernel {n : ℕ} (a : Euc n) (ε p : ℝ) (x : Euc n) : ℝ :=
+  (‖x - a‖ ^ 2 + ε ^ 2) ^ (-p / 2)
+
+theorem radialKernel_contDiff {n : ℕ} (a : Euc n) {ε : ℝ} (hε : 0 < ε) (p : ℝ) :
+    ContDiff ℝ ∞ (radialKernel a ε p) := by
+  apply (((show ContDiff ℝ ∞ (fun x : Euc n ↦ x - a) from contDiff_id.sub contDiff_const).norm_sq (𝕜 := ℝ)).add contDiff_const).rpow_const_of_ne
+  intro x
+  positivity
+
+theorem radialKernel_hasFDerivAt {n : ℕ} (a x : Euc n) {ε : ℝ} (hε : 0 < ε) (p : ℝ) :
+    HasFDerivAt (radialKernel a ε p)
+      ((-p * (‖x - a‖ ^ 2 + ε ^ 2) ^ (-p / 2 - 1)) • innerSL ℝ (x - a)) x := by
+  have hq := (((hasFDerivAt_id x).sub_const a).norm_sq).add_const (ε ^ 2)
+  have hh := hq.rpow_const (p := -p / 2) (Or.inl (by positivity : ‖id x - a‖ ^ 2 + ε ^ 2 ≠ 0))
+  have he : (-p / 2 * (‖x - a‖ ^ 2 + ε ^ 2) ^ (-p / 2 - 1)) * 2 =
+      -p * (‖x - a‖ ^ 2 + ε ^ 2) ^ (-p / 2 - 1) := by ring
+  have hmap : (-p / 2 * (‖id x - a‖ ^ 2 + ε ^ 2) ^ (-p / 2 - 1)) •
+      (2 • (innerSL ℝ (id x - a)).comp (ContinuousLinearMap.id ℝ (Euc n))) =
+      (-p * (‖x - a‖ ^ 2 + ε ^ 2) ^ (-p / 2 - 1)) • innerSL ℝ (x - a) := by
+    ext v
+    simp only [smul_apply, ContinuousLinearMap.comp_apply,
+      ContinuousLinearMap.id_apply, innerSL_apply_apply, smul_eq_mul, id_eq]
+    ring
+  rw [hmap] at hh
+  exact hh
+
+theorem tangentialDivergence_radial {n : ℕ} (E : Set (Euc n)) (hn : 0 < n)
+    (a x : Euc n) {ε : ℝ} (hε : 0 < ε) (hN : ‖approximateNormal E x‖ = 1) :
+    tangentialDivergence E (fun y ↦ radialKernel a ε (n - 1 : ℕ) y • (y - a)) x =
+      (n - 1 : ℕ) * (‖x - a‖ ^ 2 + ε ^ 2) ^ (-((n - 1 : ℕ) : ℝ) / 2 - 1) *
+        (ε ^ 2 + (inner ℝ (x - a) (approximateNormal E x)) ^ 2) := by
+  rw [tangentialDivergence_smul E ((radialKernel_contDiff a hε _).differentiable (by simp))
+    (show Differentiable ℝ (fun y : Euc n ↦ y - a) from differentiable_id.sub_const a), tangentialDivergence_displacement E a x hN hn]
+  have hgrad : gradient (radialKernel a ε (n - 1 : ℕ)) x =
+      (-((n - 1 : ℕ) : ℝ) * (‖x - a‖ ^ 2 + ε ^ 2) ^ (-((n - 1 : ℕ) : ℝ) / 2 - 1)) • (x - a) := by
+    apply (InnerProductSpace.toDual ℝ (Euc n)).injective
+    rw [toDual_gradient, (radialKernel_hasFDerivAt a x hε _).fderiv]
+    simp only [map_smul]
+    rfl
+  rw [tangentialGradient, hgrad, inner_sub_left]
+  simp only [real_inner_smul_left, real_inner_self_eq_norm_sq, radialKernel]
+  rw [real_inner_comm (approximateNormal E x) (x - a)]
+  have hp : 0 < ‖x - a‖ ^ 2 + ε ^ 2 := by positivity
+  have hr : (‖x - a‖ ^ 2 + ε ^ 2) ^ (-((n - 1 : ℕ) : ℝ) / 2) =
+      (‖x - a‖ ^ 2 + ε ^ 2) ^ (-((n - 1 : ℕ) : ℝ) / 2 - 1) * (‖x - a‖ ^ 2 + ε ^ 2) := by
+    calc
+      _ = (‖x - a‖ ^ 2 + ε ^ 2) ^ ((-((n - 1 : ℕ) : ℝ) / 2 - 1) + 1) := by congr 1; ring
+      _ = _ := by rw [Real.rpow_add hp, Real.rpow_one]
+  rw [hr]
+  ring
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+namespace RPISingularHarnack
+open RPI.Boundary
+
+theorem tangentialGradient_integrable_ball_perimeter {n : ℕ}
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    (c : Euc n) {R : ℝ} (hR : 0 < R) {φ : Euc n → ℝ}
+    (hφ : ContDiff ℝ 1 φ) :
+    Integrable (tangentialGradient E φ) ((perimeterMeasure E).restrict (Metric.ball c R)) := by
+  let μ := (perimeterMeasure E).restrict (Metric.ball c R)
+  have : IsFiniteMeasure μ := ⟨by
+    rw [Measure.restrict_apply_univ, perimeterMeasure_open E Metric.isOpen_ball]
+    exact hmin.2.1 c R hR⟩
+  have hg : Continuous (gradient φ) := continuous_iff_continuousAt.mpr (fun x ↦
+    continuousAt_gradient_of_contDiffAt hφ.contDiffAt)
+  obtain ⟨M, hM⟩ := (isCompact_closedBall c R).exists_bound_of_continuousOn hg.continuousOn
+  apply Integrable.of_bound (measurable_tangentialGradient hmin.1 hφ).aestronglyMeasurable (2 * M)
+  filter_upwards [ae_restrict_mem measurableSet_ball] with x hx
+  exact (tangentialGradient_norm_le_twice E φ x).trans (by
+    nlinarith [hM x (Metric.ball_subset_closedBall hx)])
+
+theorem minimizing_tangentialDivergence_integral_eq_zero {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    (c : Euc n) {R : ℝ} (hR : 0 < R) {V : Euc n → Euc n}
+    (hV : ContDiff ℝ ∞ V) (hcV : HasCompactSupport V)
+    (hsV : tsupport V ⊆ Metric.ball c R) :
+    (∫ x in Metric.ball c R, tangentialDivergence E V x ∂perimeterMeasure E) = 0 := by
+  let b := EuclideanSpace.basisFun (Fin n) ℝ
+  let f : Fin n → Euc n → ℝ := fun i x ↦ inner ℝ (b i) (V x)
+  have hf (i : Fin n) : ContDiff ℝ ∞ (f i) := contDiff_const.inner ℝ hV
+  have hcf (i : Fin n) : HasCompactSupport (f i) :=
+    hcV.comp_left (show inner ℝ (b i) (0 : Euc n) = 0 from inner_zero_right _)
+  have hsf (i : Fin n) : tsupport (f i) ⊆ Metric.ball c R :=
+    (tsupport_comp_subset (g := fun z : Euc n ↦ inner ℝ (b i) z)
+      (inner_zero_right _) V).trans hsV
+  have hz (i : Fin n) := minimizing_tangentialGradient_integral_eq_zero_local hn hmin c hR
+    (hf i) (hcf i) (hsf i) (b i)
+  have hi (i : Fin n) : Integrable (fun x ↦ inner ℝ (tangentialGradient E (f i) x) (b i))
+      ((perimeterMeasure E).restrict (Metric.ball c R)) :=
+    (tangentialGradient_integrable_ball_perimeter hmin c hR ((hf i).of_le (by simp))).inner_const (b i)
+  simp_rw [tangentialDivergence_eq_sum E (hV.differentiable (by simp))]
+  rw [integral_finsetSum _ (fun i _ ↦ hi i)]
+  exact Finset.sum_eq_zero fun i _ ↦ hz i
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+open scoped ENNReal Topology
+namespace RPIKernel
+
+variable {X : Type*} [MetricSpace X]
+
+theorem kernel_le_dyadic [MeasurableSpace X] [BorelSpace X]
+    (a x : X) {s : ℝ} (hs : 0 < s) :
+    ENNReal.ofReal (dist x a ^ (-s)) ≤ 1 + ∑' j : ℕ,
+      (Metric.ball a (2 * (1 / 2 : ℝ) ^ j)).indicator
+        (fun _ ↦ ENNReal.ofReal (((2 : ℝ) ^ s) ^ (j + 1))) x := by
+  by_cases h0 : dist x a = 0
+  · simp [h0, Real.zero_rpow (ne_of_lt (neg_neg_of_pos hs))]
+  have hr : 0 < dist x a := lt_of_le_of_ne dist_nonneg (Ne.symm h0)
+  by_cases hr1 : 1 ≤ dist x a
+  · have hh : dist x a ^ (-s) ≤ 1 := by
+      simpa using Real.rpow_le_rpow_of_nonpos (by norm_num : (0 : ℝ) < 1) hr1 (by linarith : -s ≤ 0)
+    exact (ENNReal.ofReal_le_one.mpr hh).trans (le_add_of_nonneg_right (by positivity))
+  obtain ⟨j, hj₁, hj₂⟩ := exists_nat_pow_near_of_lt_one hr (le_of_not_ge hr1)
+    (by norm_num : (0 : ℝ) < 1 / 2) (by norm_num : (1 / 2 : ℝ) < 1)
+  have hx : x ∈ Metric.ball a (2 * (1 / 2 : ℝ) ^ j) := by
+    rw [Metric.mem_ball]
+    nlinarith [pow_pos (by norm_num : (0 : ℝ) < 1 / 2) j]
+  have hb : dist x a ^ (-s) ≤ ((2 : ℝ) ^ s) ^ (j + 1) := by
+    refine (Real.rpow_le_rpow_of_nonpos (pow_pos (by norm_num : (0 : ℝ) < 1 / 2) _) hj₁.le (by linarith)).trans_eq ?_
+    rw [← Real.rpow_natCast_mul (by norm_num : (0 : ℝ) ≤ 1 / 2), mul_comm, Real.rpow_mul_natCast (by norm_num : (0 : ℝ) ≤ 1 / 2)]
+    congr 1
+    rw [Real.rpow_neg_eq_inv_rpow]
+    norm_num
+  calc
+    _ ≤ ENNReal.ofReal (((2 : ℝ) ^ s) ^ (j + 1)) := ENNReal.ofReal_le_ofReal hb
+    _ = (Metric.ball a (2 * (1 / 2 : ℝ) ^ j)).indicator
+        (fun _ ↦ ENNReal.ofReal (((2 : ℝ) ^ s) ^ (j + 1))) x := (indicator_of_mem hx (fun _ : X ↦ ENNReal.ofReal (((2 : ℝ) ^ s) ^ (j + 1)))).symm
+    _ ≤ ∑' j : ℕ, (Metric.ball a (2 * (1 / 2 : ℝ) ^ j)).indicator
+        (fun _ ↦ ENNReal.ofReal (((2 : ℝ) ^ s) ^ (j + 1))) x :=
+      ENNReal.le_tsum (f := fun index : ℕ ↦
+        (Metric.ball a (2 * (1 / 2 : ℝ) ^ index)).indicator
+          (fun _ ↦ ENNReal.ofReal (((2 : ℝ) ^ s) ^ (index + 1))) x) j
+    _ ≤ _ := le_add_of_nonneg_left (by positivity)
+
+variable [MeasurableSpace X] [BorelSpace X]
+
+theorem exists_kernel_integral_bound (μ : Measure X) [IsFiniteMeasure μ]
+    {d s A : ℝ} (hs : 0 < s) (hsd : s < d) (hA : 0 ≤ A)
+    (hg : ∀ (a : X) (r : ℝ), 0 < r → μ (Metric.ball a r) ≤ ENNReal.ofReal (A * r ^ d)) :
+    ∃ C : ℝ≥0∞, C ≠ ⊤ ∧ ∀ a : X, ∫⁻ x, ENNReal.ofReal (dist x a ^ (-s)) ∂μ ≤ C := by
+  let q : ℝ := (2 : ℝ) ^ s * (1 / 2 : ℝ) ^ d
+  have hq0 : 0 ≤ q := by dsimp [q]; positivity
+  have hq1 : q < 1 := by
+    have he : q = (2 : ℝ) ^ (s - d) := by
+      dsimp [q]
+      rw [show (1 / 2 : ℝ) = (2 : ℝ)⁻¹ by norm_num, Real.inv_rpow (by norm_num),
+        ← Real.rpow_neg (by norm_num), ← Real.rpow_add (by norm_num)]
+      congr 1
+    rw [he]
+    exact Real.rpow_lt_one_of_one_lt_of_neg (by norm_num) (sub_neg.mpr hsd)
+  let B : ℝ := A * (2 : ℝ) ^ d * (2 : ℝ) ^ s
+  have hB : 0 ≤ B := by dsimp [B]; positivity
+  let C : ℝ≥0∞ := μ univ + ENNReal.ofReal B * ∑' j : ℕ, (ENNReal.ofReal q) ^ j
+  have hqt : ∑' j : ℕ, (ENNReal.ofReal q) ^ j ≠ ⊤ := by
+    rw [ENNReal.tsum_geometric]
+    exact ENNReal.inv_ne_top.mpr (tsub_pos_iff_lt.mpr (ENNReal.ofReal_lt_one.mpr hq1)).ne'
+  refine ⟨C, ENNReal.add_ne_top.mpr ⟨measure_ne_top _ _, ENNReal.mul_ne_top ENNReal.ofReal_ne_top hqt⟩, ?_⟩
+  intro a
+  calc
+    _ ≤ ∫⁻ x, 1 + ∑' j : ℕ, (Metric.ball a (2 * (1 / 2 : ℝ) ^ j)).indicator
+        (fun _ ↦ ENNReal.ofReal (((2 : ℝ) ^ s) ^ (j + 1))) x ∂μ :=
+      lintegral_mono fun x ↦ kernel_le_dyadic a x hs
+    _ = μ univ + ∑' j : ℕ, ENNReal.ofReal (((2 : ℝ) ^ s) ^ (j + 1)) *
+        μ (Metric.ball a (2 * (1 / 2 : ℝ) ^ j)) := by
+      rw [lintegral_add_left measurable_const]
+      rw [lintegral_tsum (fun j ↦ (measurable_const.indicator measurableSet_ball).aemeasurable)]
+      simp only [lintegral_const, one_mul, lintegral_indicator measurableSet_ball,
+        Measure.restrict_apply_univ]
+    _ ≤ μ univ + ∑' j : ℕ, ENNReal.ofReal B * (ENNReal.ofReal q) ^ j := by
+      apply add_le_add le_rfl
+      apply ENNReal.tsum_le_tsum
+      intro j
+      calc
+        _ ≤ ENNReal.ofReal (((2 : ℝ) ^ s) ^ (j + 1)) *
+            ENNReal.ofReal (A * (2 * (1 / 2 : ℝ) ^ j) ^ d) :=
+          by gcongr; exact hg a _ (by positivity)
+        _ = ENNReal.ofReal B * (ENNReal.ofReal q) ^ j := by
+          rw [← ENNReal.ofReal_mul (by positivity), ← ENNReal.ofReal_pow hq0,
+            ← ENNReal.ofReal_mul hB]
+          congr 1
+          rw [Real.mul_rpow (by norm_num) (by positivity),
+            ← Real.rpow_natCast_mul (by norm_num : (0 : ℝ) ≤ 1 / 2),
+            mul_comm (j : ℝ) d, Real.rpow_mul_natCast (by norm_num : (0 : ℝ) ≤ 1 / 2),
+            pow_succ, mul_pow]
+          dsimp [B, q]
+          ring
+    _ = C := by rw [ENNReal.tsum_mul_left]
+
+end RPIKernel
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+open scoped ENNReal Topology
+namespace RPIKernel
+open MeasureTheory
+variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X]
+
+theorem integrable_kernel_of_growth (μ : Measure X) [IsFiniteMeasure μ]
+    {d s A : ℝ} (hs : 0 < s) (hsd : s < d) (hA : 0 ≤ A)
+    (hg : ∀ (a : X) (r : ℝ), 0 < r → μ (Metric.ball a r) ≤ ENNReal.ofReal (A * r ^ d))
+    (a : X) : Integrable (fun x ↦ dist x a ^ (-s)) μ := by
+  obtain ⟨C, hC, hbound⟩ := exists_kernel_integral_bound μ hs hsd hA hg
+  refine ⟨((continuous_id.dist (continuous_const (y := a))).measurable.pow_const _).aestronglyMeasurable, ?_⟩
+  rw [hasFiniteIntegral_iff_ofReal (ae_of_all μ fun _ ↦ Real.rpow_nonneg dist_nonneg _)]
+  exact (hbound a).trans_lt hC.lt_top
+
+end RPIKernel
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+open scoped ENNReal Topology
+namespace RPISingularHarnack
+
+theorem radialKernel_pos {n : ℕ} (a x : Euc n) {ε : ℝ} (hε : 0 < ε) (p : ℝ) :
+    0 < radialKernel a ε p x := by
+  apply Real.rpow_pos_of_pos
+  positivity
+
+theorem radialKernel_mul_dist_le {n : ℕ} (a x : Euc n) {ε p : ℝ}
+    (hε : 0 < ε) (hp : 1 < p) :
+    radialKernel a ε p x * ‖x - a‖ ≤ ‖x - a‖ ^ (-(p - 1)) := by
+  by_cases h0 : ‖x - a‖ = 0
+  · rw [h0, mul_zero]
+    positivity
+  have hr : 0 < ‖x - a‖ := lt_of_le_of_ne (norm_nonneg _) (Ne.symm h0)
+  have hpow := Real.rpow_le_rpow_of_nonpos (sq_pos_of_pos hr)
+    (le_add_of_nonneg_right (sq_pos_of_pos hε).le) (by linarith : -p / 2 ≤ 0)
+  apply (mul_le_mul_of_nonneg_right hpow hr.le).trans_eq
+  calc
+    _ = ‖x - a‖ ^ ((2 : ℝ) * (-p / 2)) * ‖x - a‖ ^ (1 : ℝ) := by
+      rw [Real.rpow_one]
+      congr 1
+      simpa only [Nat.cast_ofNat] using (Real.rpow_natCast_mul hr.le 2 (-p / 2)).symm
+    _ = ‖x - a‖ ^ ((2 : ℝ) * (-p / 2) + 1) := (Real.rpow_add hr _ _).symm
+    _ = _ := by congr 1; ring
+
+theorem radialKernel_mass_scaling {k ε : ℝ} (hε : 0 < ε) :
+    ε ^ k * ε ^ 2 * (2 * ε ^ 2) ^ (-k / 2 - 1) = (2 : ℝ) ^ (-k / 2 - 1) := by
+  rw [Real.mul_rpow (by norm_num) (sq_nonneg ε), ← Real.rpow_natCast_mul hε.le]
+  norm_num only [Nat.cast_ofNat]
+  rw [show ε ^ k * ε ^ 2 * ((2 : ℝ) ^ (-k / 2 - 1) * ε ^ ((2 : ℝ) * (-k / 2 - 1))) =
+      (2 : ℝ) ^ (-k / 2 - 1) * (ε ^ k * ε ^ (2 : ℝ) * ε ^ ((2 : ℝ) * (-k / 2 - 1))) by
+        rw [Real.rpow_two]; ring]
+  rw [← Real.rpow_add hε, ← Real.rpow_add hε]
+  ring_nf
+  simp
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/RawConeLink.lean b/lean/OAI/Geometry/Perimeter/Regularity/RawConeLink.lean
new file mode 100644
index 0000000000000000000000000000000000000000..6140e2924d4102fadf911541742efb9398666ab3
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/RawConeLink.lean
@@ -0,0 +1,513 @@
+import OAI.Geometry.Perimeter.Regularity.EllipticLocality
+import OAI.Geometry.Perimeter.Regularity.SphericalFrankel
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+namespace RPISingularHarnack
+namespace SphericalFrankel
+
+abbrev End (n : ℕ) := Euc n →L[ℝ] Euc n
+
+def linTrace {n : ℕ} (A : End n) : ℝ :=
+  ∑ i : Fin n, inner ℝ ((EuclideanSpace.basisFun (Fin n) ℝ) i)
+    (A ((EuclideanSpace.basisFun (Fin n) ℝ) i))
+
+def linkTrace {n : ℕ} (e : Frame n) (A : End n) : ℝ :=
+  linTrace A - inner ℝ e.1 (A e.1) - inner ℝ e.2 (A e.2)
+
+def projMap {n : ℕ} (e : Frame n) : End n :=
+  ContinuousLinearMap.id ℝ (Euc n) - (innerSL ℝ e.1).smulRight e.1 -
+    (innerSL ℝ e.2).smulRight e.2
+
+lemma projMap_apply {n : ℕ} (e : Frame n) (v : Euc n) :
+    projMap e v = proj e v := rfl
+
+lemma linTrace_sub {n : ℕ} (A B : End n) :
+    linTrace (A - B) = linTrace A - linTrace B := by
+  simp only [linTrace, sub_apply, inner_sub_right,
+    Finset.sum_sub_distrib]
+
+lemma linTrace_add {n : ℕ} (A B : End n) :
+    linTrace (A + B) = linTrace A + linTrace B := by
+  simp only [linTrace, add_apply, inner_add_right,
+    Finset.sum_add_distrib]
+
+lemma linTrace_smul {n : ℕ} (t : ℝ) (A : End n) :
+    linTrace (t • A) = t * linTrace A := by
+  simp only [linTrace, smul_apply, real_inner_smul_right,
+    Finset.mul_sum]
+
+lemma linTrace_id {n : ℕ} : linTrace (ContinuousLinearMap.id ℝ (Euc n)) = n := by
+  simp [linTrace]
+
+lemma linTrace_rank {n : ℕ} (v w : Euc n) :
+    linTrace ((innerSL ℝ v).smulRight w) = inner ℝ v w := by
+  simpa only [linTrace, ContinuousLinearMap.smulRight_apply, innerSL_apply_apply,
+    real_inner_smul_right] using
+    (EuclideanSpace.basisFun (Fin n) ℝ).sum_inner_mul_inner v w
+
+lemma linkTrace_sub {n : ℕ} (e : Frame n) (A B : End n) :
+    linkTrace e (A-B) = linkTrace e A - linkTrace e B := by
+  simp only [linkTrace, linTrace_sub, sub_apply, inner_sub_right]
+  ring
+
+lemma linkTrace_add {n : ℕ} (e : Frame n) (A B : End n) :
+    linkTrace e (A+B) = linkTrace e A + linkTrace e B := by
+  simp only [linkTrace, linTrace_add, add_apply, inner_add_right]
+  ring
+
+lemma linkTrace_smul {n : ℕ} (e : Frame n) (t : ℝ) (A : End n) :
+    linkTrace e (t • A) = t * linkTrace e A := by
+  simp only [linkTrace, linTrace_smul, smul_apply,
+    real_inner_smul_right]
+  ring
+
+lemma linkTrace_rank {n : ℕ} (e : Frame n) (v w : Euc n) :
+    linkTrace e ((innerSL ℝ v).smulRight w) = inner ℝ (proj e v) w := by
+  simp only [linkTrace, linTrace_rank, ContinuousLinearMap.smulRight_apply,
+    innerSL_apply_apply, real_inner_smul_right, proj, inner_sub_left,
+    real_inner_smul_left]
+  rw [real_inner_comm v e.1, real_inner_comm v e.2]
+
+lemma proj_self_left {n : ℕ} {e : Frame n} (he : IsFrame e) : proj e e.1 = 0 := by
+  have h11 : inner ℝ e.1 e.1 = 1 := by simp [he.1]
+  have h21 : inner ℝ e.2 e.1 = 0 := by rw [real_inner_comm, he.2.2]
+  simp only [proj, h11, h21, one_smul, zero_smul, sub_self]
+
+lemma proj_self_right {n : ℕ} {e : Frame n} (he : IsFrame e) : proj e e.2 = 0 := by
+  have h22 : inner ℝ e.2 e.2 = 1 := by simp [he.2.1]
+  simp only [proj, h22, he.2.2, one_smul, zero_smul, sub_zero, sub_self]
+
+lemma inner_proj_swap {n : ℕ} (e : Frame n) (v w : Euc n) :
+    inner ℝ (proj e v) w = inner ℝ v (proj e w) := by
+  simp only [proj, inner_sub_left, inner_sub_right, real_inner_smul_left,
+    real_inner_smul_right]
+  rw [real_inner_comm v e.1, real_inner_comm v e.2]
+  ring
+
+lemma inner_proj_self {n : ℕ} {e : Frame n} (he : IsFrame e) (v : Euc n) :
+    inner ℝ (proj e v) v = ‖proj e v‖ ^ 2 := by
+  rw [← real_inner_self_eq_norm_sq, proj_inner_self he]
+  simp only [proj, inner_sub_left, real_inner_smul_left, real_inner_self_eq_norm_sq]
+  ring
+
+lemma linkTrace_id {n : ℕ} {e : Frame n} (he : IsFrame e) :
+    linkTrace e (ContinuousLinearMap.id ℝ (Euc n)) = (n : ℝ) - 2 := by
+  simp [linkTrace, linTrace_id, he.1, he.2.1]
+  ring
+
+lemma linkTrace_proj {n : ℕ} {e f : Frame n} (he : IsFrame e) (hf : IsFrame f) :
+    linkTrace e (projMap f) = traceProduct e f := by
+  rw [projMap, linkTrace_sub, linkTrace_sub, linkTrace_id he,
+    linkTrace_rank, linkTrace_rank]
+  simp only [proj, inner_sub_left, real_inner_smul_left, real_inner_self_eq_norm_sq,
+    hf.1, hf.2.1, traceProduct, angle]
+  ring
+
+def linkDiv {n : ℕ} (e : Frame n) (W : Euc n → Euc n) : ℝ :=
+  linkTrace e (fderiv ℝ W e.1) - ((n : ℝ)-2) * inner ℝ e.1 (W e.1)
+
+lemma linkDiv_scalar_constant {n : ℕ} {e f : Frame n}
+    (he : IsFrame e) {φ : ℝ → ℝ} (hφ : Differentiable ℝ φ) :
+    linkDiv e (fun z ↦ φ (inner ℝ z f.1) • f.1) =
+      deriv φ (angle e f) * defect e f - ((n : ℝ)-2) * φ (angle e f) * angle e f := by
+  have hi : HasFDerivAt (fun z : Euc n ↦ inner ℝ z f.1) (innerSL ℝ f.1) e.1 := by
+    convert (innerSL ℝ f.1).hasFDerivAt (x := e.1) using 1
+    funext z
+    exact real_inner_comm f.1 z
+  have hd := ((hφ _).hasDerivAt.comp_hasFDerivAt e.1 hi).smul_const f.1
+  have hd' : fderiv ℝ (fun z ↦ φ (inner ℝ z f.1) • f.1) e.1 =
+      deriv φ (angle e f) • (innerSL ℝ f.1).smulRight f.1 := by
+    have hdf := hd.fderiv
+    simp only [Function.comp_def] at hdf
+    rw [hdf]
+    ext v
+    simp [angle, mul_assoc]
+  rw [linkDiv, hd', linkTrace_smul, linkTrace_rank, inner_proj_self he]
+  simp only [defect, real_inner_smul_right, angle]
+  ring
+
+lemma linkDiv_scalar_proj {n : ℕ} {e f : Frame n}
+    (he : IsFrame e) (hf : IsFrame f)
+    {φ : ℝ → ℝ} (hφ : Differentiable ℝ φ) :
+    linkDiv e (fun z ↦ φ (inner ℝ z f.1) • (projMap f z)) =
+      deriv φ (angle e f) * cross e f + φ (angle e f) * traceProduct e f -
+        ((n : ℝ)-2) * φ (angle e f) * defect f e := by
+  have hi : HasFDerivAt (fun z : Euc n ↦ inner ℝ z f.1) (innerSL ℝ f.1) e.1 := by
+    convert (innerSL ℝ f.1).hasFDerivAt (x := e.1) using 1
+    funext z
+    exact real_inner_comm f.1 z
+  have hd := ((hφ _).hasDerivAt.comp_hasFDerivAt e.1 hi).smul (projMap f).hasFDerivAt
+  have hd' : fderiv ℝ (fun z ↦ φ (inner ℝ z f.1) • (projMap f z)) e.1 =
+      deriv φ (angle e f) • (innerSL ℝ f.1).smulRight (proj f e.1) +
+        φ (angle e f) • projMap f := by
+    have hdf := hd.fderiv
+    change fderiv ℝ (fun z ↦ φ (inner ℝ z f.1) • (projMap f z)) e.1 = _ at hdf
+    rw [hdf]
+    ext v
+    simp [angle, projMap_apply, add_comm, mul_assoc]
+  rw [linkDiv, hd', linkTrace_add, linkTrace_smul, linkTrace_smul,
+    linkTrace_rank, linkTrace_proj he hf]
+  simp only [projMap_apply, real_inner_smul_right]
+  have hip : inner ℝ e.1 (proj f e.1) = ‖proj f e.1‖ ^ 2 :=
+    (real_inner_comm _ _).trans (inner_proj_self hf e.1)
+  rw [hip]
+  dsimp [angle, cross, defect]
+  ring
+
+lemma linTrace_rank_general {n : ℕ} (ℓ : Euc n →L[ℝ] ℝ) (w : Euc n) :
+    linTrace (ℓ.smulRight w) = ℓ w := by
+  let b := EuclideanSpace.basisFun (Fin n) ℝ
+  have hw : (∑ i : Fin n, inner ℝ (b i) w • b i) = w := by
+    simpa only [OrthonormalBasis.repr_apply_apply] using b.sum_repr w
+  calc
+    linTrace (ℓ.smulRight w) = ∑ i : Fin n, inner ℝ (b i) w * ℓ (b i) := by
+      simp only [linTrace, ContinuousLinearMap.smulRight_apply, real_inner_smul_right, b]
+      congr 1; ext i; ring
+    _ = ℓ (∑ i : Fin n, inner ℝ (b i) w • b i) := by simp
+    _ = ℓ w := by rw [hw]
+
+def sphereProj {n : ℕ} (z : Euc n) : End n :=
+  ContinuousLinearMap.id ℝ (Euc n) - (innerSL ℝ z).smulRight z
+
+def planeTrace {n : ℕ} (ν : Euc n) (A : End n) : ℝ :=
+  linTrace A - inner ℝ ν (A ν)
+
+lemma planeTrace_add {n : ℕ} (ν : Euc n) (A B : End n) :
+    planeTrace ν (A+B) = planeTrace ν A + planeTrace ν B := by
+  simp only [planeTrace, linTrace_add, add_apply, inner_add_right]
+  ring
+
+lemma planeTrace_sub {n : ℕ} (ν : Euc n) (A B : End n) :
+    planeTrace ν (A-B) = planeTrace ν A - planeTrace ν B := by
+  simp only [planeTrace, linTrace_sub, sub_apply, inner_sub_right]
+  ring
+
+lemma planeTrace_smul {n : ℕ} (ν : Euc n) (a : ℝ) (A : End n) :
+    planeTrace ν (a • A) = a * planeTrace ν A := by
+  simp only [planeTrace, linTrace_smul, smul_apply, real_inner_smul_right]
+  ring
+
+lemma planeTrace_rank {n : ℕ} (ν : Euc n) (ℓ : Euc n →L[ℝ] ℝ) (w : Euc n) :
+    planeTrace ν (ℓ.smulRight w) = ℓ w - ℓ ν * inner ℝ ν w := by
+  simp only [planeTrace, linTrace_rank_general, ContinuousLinearMap.smulRight_apply,
+    real_inner_smul_right]
+
+lemma planeTrace_comp_sphereProj {n : ℕ} {e : Frame n} (he : IsFrame e) (A : End n) :
+    planeTrace e.2 (A.comp (sphereProj e.1)) = linkTrace e A := by
+  have hid : A.comp (sphereProj e.1) = A - (innerSL ℝ e.1).smulRight (A e.1) := by
+    ext v
+    simp [sphereProj]
+  rw [hid, planeTrace_sub, planeTrace_rank]
+  simp only [innerSL_apply_apply, he.2.2, zero_mul, sub_zero, planeTrace, linkTrace]
+  ring
+
+lemma sphereProj_self {n : ℕ} {z : Euc n} (hz : ‖z‖ = 1) : sphereProj z z = 0 := by
+  simp [sphereProj, hz]
+
+lemma planeTrace_sphereProj {n : ℕ} {e : Frame n} (he : IsFrame e) :
+    planeTrace e.2 (sphereProj e.1) = (n : ℝ)-2 := by
+  have hh := planeTrace_comp_sphereProj he (ContinuousLinearMap.id ℝ (Euc n))
+  simpa [linkTrace_id he] using hh
+
+def radialUnit {n : ℕ} (x : Euc n) : Euc n := ‖x‖⁻¹ • x
+
+lemma radialUnit_norm {n : ℕ} {x : Euc n} (hx : x ≠ 0) : ‖radialUnit x‖ = 1 := by
+  simp [radialUnit, norm_smul, norm_ne_zero_iff.mpr hx]
+
+lemma radialUnit_scale {n : ℕ} {x : Euc n} (hx : x ≠ 0) : ‖x‖ • radialUnit x = x := by
+  simp [radialUnit, smul_smul, norm_ne_zero_iff.mpr hx]
+
+lemma radialUnit_contDiffAt {n : ℕ} {x : Euc n} (hx : x ≠ 0) :
+    ContDiffAt ℝ ∞ radialUnit x :=
+  ((contDiffAt_norm ℝ hx).inv (norm_ne_zero_iff.mpr hx)).smul contDiffAt_id
+
+lemma radialUnit_measurable {n : ℕ} : Measurable (radialUnit (n := n)) :=
+  (measurable_norm.inv).smul measurable_id
+
+lemma radialUnit_hasFDerivAt {n : ℕ} {x : Euc n} (hx : x ≠ 0) :
+    HasFDerivAt radialUnit (‖x‖⁻¹ • sphereProj (radialUnit x)) x := by
+  have hr : ‖x‖ ≠ 0 := norm_ne_zero_iff.mpr hx
+  have hnorm : HasFDerivAt (fun y : Euc n ↦ ‖y‖) (‖x‖⁻¹ • innerSL ℝ x) x := by
+    have hh := ((hasFDerivAt_id (𝕜 := ℝ) x).norm_sq).sqrt (pow_ne_zero 2 hr)
+    simp only [Real.sqrt_sq (norm_nonneg _)] at hh
+    apply hh.congr_fderiv
+    ext v
+    simp
+    ring
+  have hinv := (hasDerivAt_inv hr).comp_hasFDerivAt x hnorm
+  have h := hinv.smul (hasFDerivAt_id x)
+  change HasFDerivAt radialUnit _ x at h
+  apply h.congr_fderiv
+  ext v : 1
+  simp only [sphereProj, radialUnit, add_apply,
+    sub_apply, smul_apply,
+    ContinuousLinearMap.smulRight_apply, ContinuousLinearMap.id_apply,
+    innerSL_apply_apply, Function.comp_apply, real_inner_smul_left,
+    smul_eq_mul, smul_smul, id_eq]
+  module
+
+def sphereLift {n : ℕ} (W : Euc n → Euc n) (x : Euc n) : Euc n :=
+  W (radialUnit x) - (inner ℝ (radialUnit x) (W (radialUnit x))) • radialUnit x
+
+lemma inner_radialUnit_sphereLift {n : ℕ} (W : Euc n → Euc n)
+    {x : Euc n} (hx : x ≠ 0) : inner ℝ (radialUnit x) (sphereLift W x) = 0 := by
+  simp [sphereLift, inner_sub_right, real_inner_smul_right, radialUnit_norm hx]
+
+lemma sphereLift_contDiffAt {n : ℕ} {W : Euc n → Euc n}
+    (hW : ContDiff ℝ ∞ W) {x : Euc n} (hx : x ≠ 0) :
+    ContDiffAt ℝ ∞ (sphereLift W) x := by
+  have hq := radialUnit_contDiffAt hx
+  have hWq := hW.contDiffAt.comp x hq
+  exact hWq.sub ((hq.inner ℝ hWq).smul hq)
+
+lemma planeTrace_sphereLift {n : ℕ} {W : Euc n → Euc n}
+    (hW : Differentiable ℝ W) {x ν : Euc n} (hx : x ≠ 0)
+    (he : IsFrame (radialUnit x, ν)) :
+    planeTrace ν (fderiv ℝ (sphereLift W) x) =
+      ‖x‖⁻¹ * linkDiv (radialUnit x, ν) W := by
+  let z := radialUnit x
+  let A := fderiv ℝ W z
+  let B := ‖x‖⁻¹ • sphereProj z
+  let D := A.comp B
+  let ℓ := (fderivInnerCLM ℝ (z, W z)).comp (B.prod D)
+  have hq : HasFDerivAt radialUnit B x := radialUnit_hasFDerivAt hx
+  have hWq : HasFDerivAt (W ∘ radialUnit) D x := (hW z).hasFDerivAt.comp x hq
+  have hd := hWq.sub ((hq.inner ℝ hWq).smul hq)
+  have hdf : fderiv ℝ (sphereLift W) x = D - (inner ℝ z (W z) • B + ℓ.smulRight z) := by
+    exact hd.fderiv
+  have hz : ‖z‖ = 1 := he.1
+  have hBz : B z = 0 := by
+    change ‖x‖⁻¹ • sphereProj z z = 0
+    rw [sphereProj_self hz, smul_zero]
+  have hlz : ℓ z = 0 := by simp [ℓ, D, hBz]
+  have hnz : inner ℝ ν z = 0 := by exact (real_inner_comm _ _).trans he.2.2
+  have hD : planeTrace ν D = ‖x‖⁻¹ * linkTrace (z,ν) A := by
+    have hid : D = ‖x‖⁻¹ • (A.comp (sphereProj z)) := by ext v; simp [D, B]
+    rw [hid, planeTrace_smul, planeTrace_comp_sphereProj he]
+  have hB : planeTrace ν B = ‖x‖⁻¹ * ((n : ℝ)-2) := by
+    change planeTrace ν (‖x‖⁻¹ • sphereProj z) = _
+    rw [planeTrace_smul, planeTrace_sphereProj he]
+  rw [hdf, planeTrace_sub, planeTrace_add, planeTrace_smul, planeTrace_rank,
+    hlz, hnz, mul_zero, sub_zero, hD, hB]
+  dsimp [linkDiv, z, A]
+  ring
+
+def annularProfile (s : ℝ) : ℝ := expNegInvGlue (s - 1) * expNegInvGlue (4 - s)
+
+def annularCut {n : ℕ} (x : Euc n) : ℝ := annularProfile (‖x‖ ^ 2)
+
+lemma annularProfile_contDiff : ContDiff ℝ ∞ annularProfile :=
+  (expNegInvGlue.contDiff.comp (contDiff_id.sub contDiff_const)).mul
+    (expNegInvGlue.contDiff.comp (contDiff_const.sub contDiff_id))
+
+lemma annularCut_contDiff {n : ℕ} : ContDiff ℝ ∞ (annularCut (n := n)) :=
+  annularProfile_contDiff.comp (contDiff_id.norm_sq ℝ)
+
+lemma annularCut_nonneg {n : ℕ} (x : Euc n) : 0 ≤ annularCut x :=
+  mul_nonneg (expNegInvGlue.nonneg _) (expNegInvGlue.nonneg _)
+
+lemma annularCut_eq_zero_inner {n : ℕ} {x : Euc n} (hx : ‖x‖ ≤ 1) :
+    annularCut x = 0 := by
+  have hh : ‖x‖ ^ 2 - 1 ≤ 0 := by nlinarith [norm_nonneg x]
+  simp [annularCut, annularProfile, expNegInvGlue.zero_of_nonpos hh]
+
+lemma annularCut_eq_zero_outer {n : ℕ} {x : Euc n} (hx : 2 ≤ ‖x‖) :
+    annularCut x = 0 := by
+  have hh : 4 - ‖x‖ ^ 2 ≤ 0 := by nlinarith
+  simp [annularCut, annularProfile, expNegInvGlue.zero_of_nonpos hh]
+
+lemma annularCut_pos {n : ℕ} {x : Euc n} (hxi : 1 < ‖x‖) (hxo : ‖x‖ < 2) :
+    0 < annularCut x := by
+  apply mul_pos
+  · exact expNegInvGlue.pos_of_pos (by nlinarith)
+  · exact expNegInvGlue.pos_of_pos (by nlinarith [norm_nonneg x])
+
+lemma annularCut_hasFDerivAt {n : ℕ} (x : Euc n) :
+    HasFDerivAt annularCut
+      ((2 * deriv annularProfile (‖x‖ ^ 2)) • innerSL ℝ x) x := by
+  have hh := ((annularProfile_contDiff.differentiable (by simp)) _).hasDerivAt.comp_hasFDerivAt x
+    ((hasFDerivAt_id x).norm_sq)
+  apply hh.congr_fderiv
+  ext v
+  simp
+  ring
+
+def annularLift {n : ℕ} (W : Euc n → Euc n) (x : Euc n) : Euc n :=
+  annularCut x • sphereLift W x
+
+lemma annularLift_contDiff {n : ℕ} {W : Euc n → Euc n} (hW : ContDiff ℝ ∞ W) :
+    ContDiff ℝ ∞ (annularLift W) := by
+  apply contDiff_iff_contDiffAt.mpr
+  intro x
+  by_cases hx : x = 0
+  · subst x
+    apply (contDiffAt_const (c := (0 : Euc n))).congr_of_eventuallyEq
+    filter_upwards [Metric.ball_mem_nhds (0 : Euc n) (by norm_num : (0 : ℝ)<1)] with y hy
+    have hy' : ‖y‖ ≤ 1 := (by simpa using hy : ‖y‖ < 1).le
+    simp [annularLift, annularCut_eq_zero_inner hy']
+  · exact annularCut_contDiff.contDiffAt.smul (sphereLift_contDiffAt hW hx)
+
+lemma annularLift_tsupport {n : ℕ} (W : Euc n → Euc n) :
+    tsupport (annularLift W) ⊆ Metric.closedBall (0 : Euc n) 2 := by
+  apply closure_minimal _ Metric.isClosed_closedBall
+  intro x hx
+  rw [Metric.mem_closedBall, dist_zero_right]
+  by_contra hn
+  exact hx (by simp [annularLift, annularCut_eq_zero_outer (le_of_not_ge hn)])
+
+lemma planeTrace_annularLift {n : ℕ} {W : Euc n → Euc n}
+    (hW : ContDiff ℝ ∞ W) {x ν : Euc n} (hx : x ≠ 0)
+    (he : IsFrame (radialUnit x,ν)) :
+    planeTrace ν (fderiv ℝ (annularLift W) x) =
+      (annularCut x * ‖x‖⁻¹) * linkDiv (radialUnit x,ν) W := by
+  have hS := (sphereLift_contDiffAt hW hx).differentiableAt (by simp)
+  have hd := (annularCut_hasFDerivAt x).smul hS.hasFDerivAt
+  have hdf : fderiv ℝ (annularLift W) x = annularCut x • fderiv ℝ (sphereLift W) x +
+      ((2 * deriv annularProfile (‖x‖^2)) • innerSL ℝ x).smulRight (sphereLift W x) := hd.fderiv
+  have hxn : inner ℝ x ν = 0 := by
+    rw [← radialUnit_scale hx, real_inner_smul_left, he.2.2, mul_zero]
+  have hxS : inner ℝ x (sphereLift W x) = 0 := by
+    calc
+      inner ℝ x (sphereLift W x) = ‖x‖ * inner ℝ (radialUnit x) (sphereLift W x) := by
+        rw [← real_inner_smul_left, radialUnit_scale hx]
+      _ = 0 := by rw [inner_radialUnit_sphereLift W hx, mul_zero]
+  rw [hdf, planeTrace_add, planeTrace_smul, planeTrace_rank,
+    planeTrace_sphereLift (hW.differentiable (by simp)) hx he]
+  simp only [smul_apply, innerSL_apply_apply, smul_eq_mul, hxn, hxS, mul_zero]
+  ring
+
+lemma annularLift_zero_nhds {n : ℕ} (W : Euc n → Euc n) :
+    annularLift W =ᶠ[𝓝 (0 : Euc n)] (fun _ ↦ 0) := by
+  filter_upwards [Metric.ball_mem_nhds (0 : Euc n) (by norm_num : (0 : ℝ)<1)] with y hy
+  have hy' : ‖y‖ ≤ 1 := (by simpa using hy : ‖y‖ < 1).le
+  simp [annularLift, annularCut_eq_zero_inner hy']
+
+def annularWeight {n : ℕ} (x : Euc n) : ℝ := annularCut x * ‖x‖⁻¹
+
+lemma annularWeight_nonneg {n : ℕ} (x : Euc n) : 0 ≤ annularWeight x :=
+  mul_nonneg (annularCut_nonneg x) (inv_nonneg.mpr (norm_nonneg x))
+
+lemma annularWeight_continuous {n : ℕ} : Continuous (annularWeight (n := n)) := by
+  apply continuous_iff_continuousAt.mpr
+  intro x
+  by_cases hx : x = 0
+  · subst x
+    apply (continuousAt_const (y := (0 : ℝ))).congr_of_eventuallyEq
+    filter_upwards [Metric.ball_mem_nhds (0 : Euc n) (by norm_num : (0 : ℝ)<1)] with y hy
+    have hy' : ‖y‖ ≤ 1 := (by simpa using hy : ‖y‖ < 1).le
+    simp [annularWeight, annularCut_eq_zero_inner hy']
+  · exact annularCut_contDiff.continuous.continuousAt.mul
+      (continuous_norm.continuousAt.inv₀ (norm_ne_zero_iff.mpr hx))
+
+lemma annularWeight_tsupport {n : ℕ} :
+    tsupport (annularWeight (n := n)) ⊆ Metric.closedBall (0 : Euc n) 2 := by
+  apply closure_minimal _ Metric.isClosed_closedBall
+  intro x hx
+  rw [Metric.mem_closedBall, dist_zero_right]
+  by_contra hn
+  exact hx (by simp [annularWeight, annularCut_eq_zero_outer (le_of_not_ge hn)])
+
+def annularWeightNN {n : ℕ} (x : Euc n) : ℝ≥0 := ⟨annularWeight x, annularWeight_nonneg x⟩
+
+lemma annularWeightNN_measurable {n : ℕ} : Measurable (annularWeightNN (n := n)) :=
+  annularWeight_continuous.measurable.nnreal_mk
+
+def annularFrameMeasure {n : ℕ} (E : Set (Euc n)) : Measure (Euc n) :=
+  ((perimeterMeasure E).restrict (Metric.ball 0 3)).withDensity
+    (fun x ↦ (annularWeightNN x : ℝ≥0∞))
+
+def frameOf {n : ℕ} (E : Set (Euc n)) (x : Euc n) : Frame n :=
+  (radialUnit x, approximateNormal E x)
+
+lemma frameOf_measurable {n : ℕ} {E : Set (Euc n)} (hE : MeasurableSet E) :
+    Measurable (frameOf E) := radialUnit_measurable.prodMk (measurable_approximateNormal hE)
+
+def annularLink {n : ℕ} (E : Set (Euc n)) : Measure (Frame n) :=
+  Measure.map (frameOf E) (annularFrameMeasure E)
+
+lemma annularFrameMeasure_finite {n : ℕ} {E : Set (Euc n)}
+    (hE : IsLocallyPerimeterMinimizing E) : IsFiniteMeasure (annularFrameMeasure E) := by
+  let := perimeterMeasure_locallyFinite E hE.2.1
+  have hc : HasCompactSupport (annularWeight (n := n)) :=
+    (isCompact_closedBall (0 : Euc n) 2).of_isClosed_subset isClosed_closure annularWeight_tsupport
+  have hi : Integrable (annularWeight (n := n))
+      ((perimeterMeasure E).restrict (Metric.ball 0 3)) :=
+    annularWeight_continuous.integrable_of_hasCompactSupport hc
+  apply isFiniteMeasure_withDensity
+  have hh := (hasFiniteIntegral_iff_ofReal
+    (Filter.Eventually.of_forall (annularWeight_nonneg (n := n)))).mp hi.2
+  convert hh.ne using 1
+  congr 1
+  funext x
+  exact (ENNReal.ofReal_coe_nnreal (p := annularWeightNN x)).symm
+
+lemma annularLink_finite {n : ℕ} {E : Set (Euc n)}
+    (hE : IsLocallyPerimeterMinimizing E) : IsFiniteMeasure (annularLink E) := by
+  let := annularFrameMeasure_finite hE
+  exact Measure.isFiniteMeasure_map (annularFrameMeasure E) (frameOf E)
+
+lemma annularLink_frames {n : ℕ} (hn : 1 < n) {E : Set (Euc n)}
+    (hE : IsLocallyPerimeterMinimizing E) (hc : IsCone E) :
+    ∀ᵐ e ∂annularLink E, IsFrame e := by
+  rw [annularLink, ae_map_iff (frameOf_measurable hE.1).aemeasurable
+    (isClosed_frames n).measurableSet, annularFrameMeasure,
+    ae_withDensity_iff annularWeightNN_measurable.coe_nnreal_ennreal]
+  filter_upwards [ae_restrict_of_ae (minimizing_halfspaceBlowup_ae hn hE),
+    ae_restrict_of_ae (minimizing_cone_radial_tangent_ae hn hE hc)] with x hx ht hρ
+  have hx0 : x ≠ 0 := by
+    intro hh
+    apply hρ
+    apply ENNReal.coe_eq_zero.mpr
+    apply Subtype.ext
+    simp [hh, annularWeightNN, annularWeight]
+  refine ⟨radialUnit_norm hx0, hx.1, ?_⟩
+  change inner ℝ (radialUnit x) (approximateNormal E x) = 0
+  rw [radialUnit, real_inner_smul_left, ht, mul_zero]
+
+lemma linkDiv_continuous {n : ℕ} {W : Euc n → Euc n} (hW : ContDiff ℝ ∞ W) :
+    Continuous (fun e : Frame n ↦ linkDiv e W) := by
+  have hA := hW.continuous_fderiv (by simp)
+  unfold linkDiv linkTrace linTrace
+  fun_prop
+
+structure LinkStationary {n : ℕ} (μ : Measure (Frame n)) : Prop where
+  firstVariation : ∀ W : Euc n → Euc n, ContDiff ℝ ∞ W → (∫ e, linkDiv e W ∂μ) = 0
+
+lemma annularLink_stationary {n : ℕ} (hn : 1 < n) {E : Set (Euc n)}
+    (hE : IsLocallyPerimeterMinimizing E) (hc : IsCone E) :
+    LinkStationary (annularLink E) := by
+  constructor
+  intro W hW
+  have hV := annularLift_contDiff hW
+  have hs : tsupport (annularLift W) ⊆ Metric.ball (0 : Euc n) 3 :=
+    (annularLift_tsupport W).trans (Metric.closedBall_subset_ball (by norm_num))
+  have hk : HasCompactSupport (annularLift W) :=
+    (isCompact_closedBall (0 : Euc n) 2).of_isClosed_subset isClosed_closure (annularLift_tsupport W)
+  have hstat := minimizing_tangentialDivergence_integral_eq_zero hn hE 0 (by norm_num : (0 : ℝ)<3)
+    hV hk hs
+  rw [annularLink, integral_map (frameOf_measurable hE.1).aemeasurable
+    (linkDiv_continuous hW).measurable.aestronglyMeasurable,
+    annularFrameMeasure, integral_withDensity_eq_integral_smul annularWeightNN_measurable]
+  change (∫ x in Metric.ball 0 3, annularWeight x * linkDiv (frameOf E x) W ∂perimeterMeasure E) = 0
+  apply (integral_congr_ae ?_).trans hstat
+  filter_upwards [ae_restrict_of_ae (minimizing_halfspaceBlowup_ae hn hE),
+    ae_restrict_of_ae (minimizing_cone_radial_tangent_ae hn hE hc)] with x hx ht
+  by_cases hx0 : x = 0
+  · subst x
+    have hd : fderiv ℝ (annularLift W) 0 = 0 :=
+      ((hasFDerivAt_const (0 : Euc n) (0 : Euc n)).congr_of_eventuallyEq (annularLift_zero_nhds W)).fderiv
+    simp [annularWeight, tangentialDivergence, divergence, hd]
+  · have hf : IsFrame (radialUnit x, approximateNormal E x) :=
+      ⟨radialUnit_norm hx0, hx.1, by simp [radialUnit, real_inner_smul_left, ht]⟩
+    exact (planeTrace_annularLift hW hx0 hf).symm
+
+end SphericalFrankel
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/RegularSupersolutionC1Test.lean b/lean/OAI/Geometry/Perimeter/Regularity/RegularSupersolutionC1Test.lean
new file mode 100644
index 0000000000000000000000000000000000000000..3553a0b727f7315d6e6fa4580ce71194649b9a2c
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/RegularSupersolutionC1Test.lean
@@ -0,0 +1,461 @@
+import OAI.Geometry.Perimeter.Regularity.HasC1TraceAt
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPI.Boundary
+
+theorem regular_supersolution_c1_test {n : ℕ} {S Z : Set (Space n)}
+    (hS : IsClosed S) (hZ : IsClosed Z)
+    (hfinite : ∀ R : ℝ, 0 < R → surfaceMeasure S (Metric.ball 0 R) < ⊤)
+    (hgrad : ∀ f : Space n → ℝ, ContDiff ℝ 1 f →
+      AEStronglyMeasurable (surfaceGradient S f) (surfaceMeasure S))
+    {W : Space n → ℝ} {D : Space n → Space n}
+    (htrace : ∀ x ∈ S \ Z, HasC1TraceJetAt S W D x)
+    (hsup : ∀ φ : Space n → ℝ, ContDiff ℝ ∞ φ → HasCompactSupport φ →
+      Disjoint (tsupport φ) Z → (∀ x, 0 ≤ φ x) →
+      0 ≤ ∫ x, inner ℝ (D x) (surfaceGradient S φ x) ∂surfaceMeasure S)
+    {φ : Space n → ℝ} (hφ : ContDiff ℝ 1 φ) (hc : HasCompactSupport φ)
+    (hφZ : Disjoint (tsupport φ) Z) (hpφ : ∀ x, 0 ≤ φ x) :
+    0 ≤ ∫ x, inner ℝ (D x) (surfaceGradient S φ x) ∂surfaceMeasure S := by
+  obtain ⟨r, hr, hKZ⟩ := hc.exists_cthickening_subset_open hZ.isOpen_compl
+    (Set.disjoint_left.mp hφZ)
+  let K := Metric.cthickening r (tsupport φ)
+  have hKc : IsCompact K := hc.cthickening
+  have hφK : tsupport φ ⊆ K := Metric.self_subset_cthickening (tsupport φ)
+  have hKd : Disjoint K Z := Set.disjoint_left.mpr hKZ
+  obtain ⟨F, hF, he⟩ := c1_exists_smooth_representative_near_regular_test hS hZ htrace
+    hKc.isClosed hKd
+  have hFK : D =ᵐ[(surfaceMeasure S).restrict K] surfaceGradient S F := by
+    filter_upwards [ae_restrict_mem hKc.measurableSet,
+      ae_restrict_of_ae (ae_restrict_mem hS.measurableSet)] with x hx hxS
+    exact (he x ⟨hxS, hx⟩).2.symm
+  have hDm : AEStronglyMeasurable D ((surfaceMeasure S).restrict K) :=
+    ((hgrad F hF).mono_measure Measure.restrict_le_self).congr hFK.symm
+  obtain ⟨R, hR⟩ := Metric.isBounded_iff_subset_ball (0 : Space n) |>.mp hKc.isBounded
+  have hKfin : surfaceMeasure S K < ⊤ :=
+    (measure_mono (hR.trans (Metric.ball_subset_ball (le_max_left R 1)))).trans_lt
+      (hfinite (max R 1) (zero_lt_one.trans_le (le_max_right R 1)))
+  let : IsFiniteMeasure ((surfaceMeasure S).restrict K) :=
+    ⟨by simpa only [Measure.restrict_apply_univ] using hKfin⟩
+  obtain ⟨B, hB⟩ := hKc.exists_bound_of_continuousOn
+    (hF.continuous_fderiv (by norm_num)).continuousOn
+  have hDb : ∀ᵐ x ∂(surfaceMeasure S).restrict K, ‖D x‖ ≤ B := by
+    filter_upwards [ae_restrict_mem hKc.measurableSet, hFK] with x hx hFx
+    rw [hFx]
+    exact (surfaceGradient_norm_le_fderiv S F x).trans (hB x hx)
+  obtain ⟨M, hM, hb, _⟩ := c1Mollify_pair_bounded S hφ hc
+  have ht := tendsto_integral_of_dominated_convergence (μ := (surfaceMeasure S).restrict K)
+    (fun _ ↦ B * M)
+    (F := fun j x ↦ inner ℝ (D x) (surfaceGradient S (c1Mollify φ j) x))
+    (f := fun x ↦ inner ℝ (D x) (surfaceGradient S φ x))
+    (fun j ↦ hDm.inner ((hgrad _ ((c1Mollify_contDiff hφ.continuous j).of_le (by norm_num))).mono_measure
+      Measure.restrict_le_self)) (integrable_const _) ?_ ?_
+  · have hsupp : ∀ᶠ j in atTop, tsupport (c1Mollify φ j) ⊆ K := by
+      filter_upwards [c1Mollifier_radius_tendsto.eventually (gt_mem_nhds hr)] with j hj
+      exact (c1Mollify_tsupport_subset φ j).trans (Metric.cthickening_mono hj.le _)
+    have hn : ∀ᶠ j in atTop, 0 ≤ ∫ x in K,
+        inner ℝ (D x) (surfaceGradient S (c1Mollify φ j) x) ∂surfaceMeasure S := by
+      filter_upwards [hsupp] with j hj
+      rw [setIntegral_eq_integral_of_forall_compl_eq_zero (fun x hx ↦ by
+        rw [surfaceGradient_eq_zero_of_notMem_tsupport S (fun h ↦ hx (hj h)), inner_zero_right])]
+      exact hsup _ (c1Mollify_contDiff hφ.continuous j) (c1Mollify_hasCompactSupport hc j)
+        (hKd.mono_left hj) (c1Mollify_nonneg hpφ j)
+    have hle := ge_of_tendsto ht hn
+    rwa [setIntegral_eq_integral_of_forall_compl_eq_zero (fun x hx ↦ by
+      rw [surfaceGradient_eq_zero_of_notMem_tsupport S (fun h ↦ hx (hφK h)), inner_zero_right])] at hle
+  · intro j
+    filter_upwards [hDb] with x hx
+    exact (norm_inner_le_norm _ _).trans (mul_le_mul hx (hb j x).2
+      (norm_nonneg _) (by linarith [norm_nonneg (D x)]))
+  · exact ae_of_all _ fun x ↦ tendsto_const_nhds.inner
+      (c1Mollify_surfaceGradient_tendsto S hφ hc x)
+
+end RPI.Boundary
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPI.Boundary
+
+theorem c1_hausdorff_trace_mem_sobolevGraph {n : ℕ} (hn : 8 ≤ n)
+    {S Z : Set (Space n)} (hS : IsClosed S) (V : Set (Space n))
+    (hV : Bornology.IsBounded V) (hVm : MeasurableSet V)
+    [IsFiniteMeasure ((surfaceMeasure S).restrict V)]
+    (hZc : IsCompact Z) (hZd : dimH Z ≤ (n - 8 : ℕ))
+    {h : Space n → ℝ} {G : Space n → Space n}
+    (hreg : ∀ x ∈ S \ Z, HasC1TraceJetAt S h G x)
+    (hh : MemLp h 2 ((surfaceMeasure S).restrict V))
+    (hG : MemLp G 2 ((surfaceMeasure S).restrict V))
+    {M : ℝ} (hM : 0 ≤ M)
+    (hbound : ∀ᵐ x ∂(surfaceMeasure S).restrict V, ‖h x‖ ≤ M)
+    (hgrad : ∀ f : Space n → ℝ, ContDiff ℝ ∞ f →
+      AEStronglyMeasurable (surfaceGradient S f) ((surfaceMeasure S).restrict V))
+    {A r₀ : ℝ} (hA : 0 < A) (hr₀ : 0 < r₀)
+    (hgrowth : ∀ c ∈ Z, ∀ r : ℝ, 0 < r → r < r₀ →
+      (surfaceMeasure S).restrict V (Metric.ball c r) ≤
+        ENNReal.ofReal (A * r ^ (n - 1))) :
+    (hh.toLp h, hG.toLp G) ∈ closure (smoothGradientGraph S V) := by
+  obtain ⟨χ, hχ, hχlim, hχE⟩ := exists_memLp_capacity_sequence hn S V Z hZc hZd
+    hgrad hA hr₀ hgrowth
+  exact c1_capacity_trace_mem_sobolevGraph hS V hV hVm hgrad hreg hh hG hM hbound χ
+    (fun i ↦ (hχ i).1) (fun i ↦ (hχ i).2.2.1)
+    (fun i ↦ ae_of_all _ (hχ i).2.1) hχlim (fun i ↦ (hχ i).2.2.2) hχE
+
+theorem HasC1TraceJetAt.mul_smooth {n : ℕ} {S : Set (Space n)}
+    {h : Space n → ℝ} {G : Space n → Space n} {x : Space n}
+    (hxS : x ∈ S) (hx : HasC1TraceJetAt S h G x)
+    {ψ : Space n → ℝ} (hψ : ContDiff ℝ ∞ ψ) :
+    HasC1TraceJetAt S (fun y ↦ ψ y * h y)
+      (fun y ↦ ψ y • G y + h y • surfaceGradient S ψ y) x := by
+  obtain ⟨U, hU, hxU, f, hf, he, hG⟩ := hx
+  refine ⟨U, hU, hxU, fun y ↦ ψ y * f y, (hψ.of_le (by norm_num)).contDiffOn.mul hf, ?_, ?_⟩
+  · intro y hy
+    exact congrArg (fun a ↦ ψ y * a) (he hy)
+  · rw [surfaceGradient_mul (hψ.differentiable (by simp) x)
+        ((hf.contDiffAt (hU.mem_nhds hxU)).differentiableAt (by simp)),
+        hG, he ⟨hxS, hxU⟩]
+
+theorem c1_regular_trace_jet_mul_spatial {n : ℕ} {S Z : Set (Space n)}
+    {h : Space n → ℝ} {G : Space n → Space n}
+    (hreg : ∀ x ∈ S \ Z, HasC1TraceJetAt S h G x)
+    {ψ : Space n → ℝ} (hψ : ContDiff ℝ ∞ ψ) :
+    ∀ x ∈ S \ (Z ∩ tsupport ψ), HasC1TraceJetAt S (fun y ↦ ψ y * h y)
+      (fun y ↦ ψ y • G y + h y • surfaceGradient S ψ y) x := by
+  intro x hx
+  by_cases hxZ : x ∈ Z
+  · have hxs : x ∉ tsupport ψ := fun hxψ ↦ hx.2 ⟨hxZ, hxψ⟩
+    have hloc : ψ =ᶠ[𝓝 x] (0 : Space n → ℝ) :=
+      notMem_tsupport_iff_eventuallyEq.mp hxs
+    obtain ⟨U, hUsub, hU, hxU⟩ := mem_nhds_iff.mp hloc
+    have hψgrad : gradient ψ x = 0 := by
+      simpa only [Pi.zero_def, gradient_fun_const] using hloc.gradient_eq
+    have hψx : ψ x = 0 := hloc.self_of_nhds
+    refine ⟨U, hU, hxU, fun _ ↦ 0, contDiffOn_const, ?_, ?_⟩
+    · intro y hy
+      have hy0 : ψ y = 0 := hUsub hy.2
+      simp only [hy0, zero_mul]
+    · simp only [hψx, surfaceGradient, hψgrad, gradient_fun_const,
+        map_zero, Submodule.coe_zero, zero_smul, smul_zero, add_zero]
+  · exact (hreg x ⟨hx.1, hxZ⟩).mul_smooth hx.1 hψ
+
+theorem c1_hausdorff_trace_mem_sobolevGraph_ball {n : ℕ} (hn : 8 ≤ n)
+    {S Z : Set (Space n)} (hS : IsClosed S) (hZ : IsClosed Z)
+    (hZd : dimH Z ≤ (n - 8 : ℕ)) {R : ℝ} (hR : 0 < R)
+    [IsFiniteMeasure ((surfaceMeasure S).restrict (Metric.ball 0 R))]
+    {h : Space n → ℝ} {G : Space n → Space n}
+    (hreg : ∀ x ∈ S \ Z, HasC1TraceJetAt S h G x)
+    (hh : MemLp h 2 ((surfaceMeasure S).restrict (Metric.ball 0 R)))
+    (hG : MemLp G 2 ((surfaceMeasure S).restrict (Metric.ball 0 R)))
+    {M : ℝ} (hM : 0 ≤ M)
+    (hbound : ∀ᵐ x ∂(surfaceMeasure S).restrict (Metric.ball 0 R), ‖h x‖ ≤ M)
+    (hgrad : ∀ f : Space n → ℝ, ContDiff ℝ ∞ f →
+      AEStronglyMeasurable (surfaceGradient S f)
+        ((surfaceMeasure S).restrict (Metric.ball 0 R)))
+    {A r₀ : ℝ} (hA : 0 < A) (hr₀ : 0 < r₀)
+    (hgrowth : ∀ c ∈ Z ∩ Metric.closedBall 0 (R + 2), ∀ r : ℝ, 0 < r → r < r₀ →
+      (surfaceMeasure S).restrict (Metric.ball 0 R) (Metric.ball c r) ≤
+        ENNReal.ofReal (A * r ^ (n - 1))) :
+    (hh.toLp h, hG.toLp G) ∈ closure (smoothGradientGraph S (Metric.ball 0 R)) := by
+  let b : ContDiffBump (0 : Space n) := ⟨R + 1, R + 2, by linarith, by linarith⟩
+  let K : Set (Space n) := Z ∩ tsupport b
+  have hKc : IsCompact K := b.hasCompactSupport.of_isClosed_subset
+    (hZ.inter (isClosed_tsupport b)) inter_subset_right
+  have hKd : dimH K ≤ (n - 8 : ℕ) :=
+    (dimH_mono (show K ⊆ Z from inter_subset_left)).trans hZd
+  have hb : ∀ x ∈ Metric.ball (0 : Space n) R, b =ᶠ[𝓝 x] 1 := by
+    intro x hx
+    apply b.eventuallyEq_one_of_mem_ball
+    exact Metric.mem_ball.mpr ((Metric.mem_ball.mp hx).trans (by dsimp [b]; linarith))
+  have hbG : ∀ x ∈ Metric.ball (0 : Space n) R, surfaceGradient S b x = 0 := by
+    intro x hx
+    have hgr : gradient b x = 0 := by
+      simpa only [Pi.one_def, gradient_fun_const] using (hb x hx).gradient_eq
+    simp only [surfaceGradient, hgr, map_zero, Submodule.coe_zero]
+  let h' := fun x ↦ b x * h x
+  let G' := fun x ↦ b x • G x + h x • surfaceGradient S b x
+  have he : h' =ᵐ[(surfaceMeasure S).restrict (Metric.ball 0 R)] h := by
+    filter_upwards [ae_restrict_mem measurableSet_ball] with x hx
+    dsimp [h']
+    rw [(hb x hx).self_of_nhds]
+    simp
+  have hDe : G' =ᵐ[(surfaceMeasure S).restrict (Metric.ball 0 R)] G := by
+    filter_upwards [ae_restrict_mem measurableSet_ball] with x hx
+    dsimp [G']
+    rw [(hb x hx).self_of_nhds, hbG x hx]
+    simp
+  let hh' := hh.ae_eq he.symm
+  let hG' := hG.ae_eq hDe.symm
+  have hbound' : ∀ᵐ x ∂(surfaceMeasure S).restrict (Metric.ball 0 R), ‖h' x‖ ≤ M := by
+    filter_upwards [he, hbound] with x hx hbx
+    simpa only [hx] using hbx
+  have hmem := c1_hausdorff_trace_mem_sobolevGraph hn hS (Metric.ball 0 R)
+    Metric.isBounded_ball measurableSet_ball hKc hKd
+    (c1_regular_trace_jet_mul_spatial hreg b.contDiff) hh' hG' hM
+    hbound' hgrad hA hr₀ (fun c hc r hr hrr ↦ hgrowth c
+      ⟨hc.1, by simpa only [b.tsupport_eq] using hc.2⟩ r hr hrr)
+  have heLp : hh'.toLp h' = hh.toLp h := hh'.toLp_congr hh he
+  have hDeLp : hG'.toLp G' = hG.toLp G := hG'.toLp_congr hG hDe
+  change (hh'.toLp h', hG'.toLp G') ∈ _ at hmem
+  simpa only [heLp, hDeLp] using hmem
+
+theorem c1_trace_aestronglyMeasurable_of_singular_dim {n : ℕ} (hn : 8 ≤ n)
+    {S Z : Set (Space n)} (hS : MeasurableSet S) (hZ : MeasurableSet Z)
+    (hdim : dimH Z ≤ (n - 8 : ℕ)) {h : Space n → ℝ} {G : Space n → Space n}
+    (hreg : ∀ x ∈ S \ Z, HasC1TraceJetAt S h G x) :
+    AEStronglyMeasurable h (surfaceMeasure S) := by
+  have hc : ContinuousOn h (S \ Z) := by
+    intro x hx
+    obtain ⟨U, hU, hxU, f, hf, he, _⟩ := hreg x hx
+    have hf' : ContinuousWithinAt f (S \ Z) x :=
+      (hf.contDiffAt (hU.mem_nhds hxU)).continuousAt.continuousWithinAt
+    apply hf'.congr_of_eventuallyEq _ (he ⟨hx.1, hxU⟩).symm
+    filter_upwards [self_mem_nhdsWithin,
+      mem_nhdsWithin_of_mem_nhds (hU.mem_nhds hxU)] with y hy hyU
+    exact (he ⟨hy.1, hyU⟩).symm
+  have hs : ∀ᵐ x ∂surfaceMeasure S, x ∈ S := ae_restrict_mem hS
+  have hz : ∀ᵐ x ∂surfaceMeasure S, x ∉ Z := by
+    simpa only [ae_iff, not_not, Set.ofPred_mem_eq] using
+      surfaceMeasure_zero_of_singular_dim hn S Z hdim
+  have hregular : ∀ᵐ x ∂surfaceMeasure S, x ∈ S \ Z := by
+    filter_upwards [hs, hz] with x hx hzx
+    exact ⟨hx, hzx⟩
+  have hmeas := hc.aestronglyMeasurable (μ := surfaceMeasure S) (hS.diff hZ)
+  rwa [Measure.restrict_eq_self_of_ae_mem hregular] at hmeas
+
+theorem c1_min_one_mem_sobolevGraph_ball {n : ℕ} (hn : 8 ≤ n)
+    {S Z : Set (Space n)} (hS : IsClosed S) (hZ : IsClosed Z)
+    (hZd : dimH Z ≤ (n - 8 : ℕ)) {R : ℝ} (hR : 0 < R)
+    [IsFiniteMeasure ((surfaceMeasure S).restrict (Metric.ball 0 R))]
+    {W : Space n → ℝ} {D : Space n → Space n}
+    (htrace : ∀ x ∈ S \ Z, HasC1TraceJetAt S W D x)
+    (hW0 : ∀ᵐ x ∂(surfaceMeasure S).restrict (Metric.ball 0 R), 0 ≤ W x)
+    (hG : MemLp (truncatedGradient W D) 2
+      ((surfaceMeasure S).restrict (Metric.ball 0 R)))
+    (hgrad : ∀ f : Space n → ℝ, ContDiff ℝ ∞ f →
+      AEStronglyMeasurable (surfaceGradient S f)
+        ((surfaceMeasure S).restrict (Metric.ball 0 R)))
+    {A r₀ : ℝ} (hA : 0 < A) (hr₀ : 0 < r₀)
+    (hgrowth : ∀ c ∈ Z ∩ Metric.closedBall 0 (R + 2), ∀ r : ℝ, 0 < r → r < r₀ →
+      (surfaceMeasure S).restrict (Metric.ball 0 R) (Metric.ball c r) ≤
+        ENNReal.ofReal (A * r ^ (n - 1))) :
+    ∃ hh : MemLp (fun x ↦ min (W x) 1) 2
+        ((surfaceMeasure S).restrict (Metric.ball 0 R)),
+      (hh.toLp (fun x ↦ min (W x) 1), hG.toLp (truncatedGradient W D)) ∈
+        closure (smoothGradientGraph S (Metric.ball 0 R)) := by
+  have hW : AEStronglyMeasurable W ((surfaceMeasure S).restrict (Metric.ball 0 R)) :=
+    (c1_trace_aestronglyMeasurable_of_singular_dim hn hS.measurableSet
+    hZ.measurableSet hZd htrace).mono_measure (Measure.restrict_le_self)
+  obtain ⟨hh, hf, hF, hlim⟩ := smoothTruncation_pair_tendsto hW hW0 hG
+  refine ⟨hh, isClosed_closure.mem_of_tendsto hlim (Filter.Eventually.of_forall ?_)⟩
+  intro j
+  apply c1_hausdorff_trace_mem_sobolevGraph_ball hn hS hZ hZd hR
+    (fun x hx ↦ (htrace x hx).smoothTruncation hx.1 (truncationParameter_pos j))
+    (hf j) (hF j) (M := 1) zero_le_one _ hgrad hA hr₀ hgrowth
+  filter_upwards [hW0] with x hx
+  exact smoothTruncation_norm_le (truncationParameter_pos j) hx
+
+end RPI.Boundary
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPI.Boundary
+section C1RegularTests
+variable {n : ℕ} {S Z : Set (Space n)}
+variable (hS : IsClosed S) (hZ : IsClosed Z)
+variable (hfinite : ∀ R : ℝ, 0 < R → (surfaceMeasure S) (Metric.ball 0 R) < ⊤)
+variable (hgrad : ∀ f : Space n → ℝ, ContDiff ℝ 1 f →
+  AEStronglyMeasurable (surfaceGradient S f) (surfaceMeasure S))
+include hS hZ hfinite hgrad
+
+theorem c1_smoothTruncation_regular_test {W : Space n → ℝ} {D : Space n → Space n}
+    (htrace : ∀ x ∈ S \ Z, HasC1TraceJetAt S W D x)
+    (hsup : ∀ φ : Space n → ℝ, ContDiff ℝ ∞ φ → HasCompactSupport φ →
+      Disjoint (tsupport φ) Z → (∀ x, 0 ≤ φ x) →
+      0 ≤ ∫ x, inner ℝ (D x) (surfaceGradient S φ x) ∂surfaceMeasure S)
+    {ε : ℝ} (hε : 0 < ε) {φ : Space n → ℝ}
+    (hφ : ContDiff ℝ 1 φ) (hcφ : HasCompactSupport φ)
+    (hφZ : Disjoint (tsupport φ) Z) (hpφ : ∀ x, 0 ≤ φ x) :
+    0 ≤ ∫ x, inner ℝ (truncationSlope ε (W x) • D x)
+      (surfaceGradient S φ x) ∂surfaceMeasure S := by
+  obtain ⟨F, hF, he⟩ := c1_exists_smooth_representative_near_regular_test hS hZ htrace
+    (isClosed_tsupport φ) hφZ
+  let ψ : Space n → ℝ := fun x ↦ φ x * truncationSlope ε (F x)
+  have hψ : ContDiff ℝ 1 ψ := hφ.mul (((truncationSlope_contDiff ε).of_le (by norm_num)).comp hF)
+  have hcψ : HasCompactSupport ψ := hcφ.mul_right
+  have hψZ : Disjoint (tsupport ψ) Z := hφZ.mono_left tsupport_mul_subset_left
+  have htt : ∀ x ∈ S \ Z, HasC1TraceJetAt S (smoothTruncation ε ∘ W)
+      (fun y ↦ truncationSlope ε (W y) • D y) x := by
+    intro x hx
+    simpa only [smoothTruncation_deriv] using
+      (htrace x hx).comp hx.1 ((smoothTruncation_contDiff ε).of_le (by norm_num))
+  have hi := c1_integrable_regular_test_pair hS hZ hfinite hgrad htrace hψ hcψ hψZ
+  have hj := c1_integrable_regular_test_pair hS hZ hfinite hgrad htt hφ hcφ hφZ
+  apply (regular_supersolution_c1_test hS hZ hfinite hgrad htrace hsup hψ hcψ hψZ (fun x ↦ mul_nonneg (hpφ x)
+    (truncationSlope_mem_Icc ε (F x)).1)).trans
+  apply integral_mono_ae hi hj
+  filter_upwards [ae_restrict_mem hS.measurableSet] with x hx
+  by_cases hxt : x ∈ tsupport φ
+  · have hFx := he x ⟨hx, hxt⟩
+    have hDp : deriv (truncationSlope ε) (W x) ≤ 0 :=
+      (truncationSlope_antitone hε).deriv_nonpos
+    have heq : surfaceGradient S ψ x =
+        φ x • (deriv (truncationSlope ε) (W x) • D x) +
+          truncationSlope ε (W x) • surfaceGradient S φ x := by
+      dsimp only [ψ]
+      rw [surfaceGradient_mul (S := S) (g := fun y ↦ truncationSlope ε (F y)) (hφ.differentiable (by simp) x)
+        ((((truncationSlope_contDiff ε).of_le (by norm_num)).comp hF).differentiable (by simp) x)]
+      rw [show (fun y ↦ truncationSlope ε (F y)) = truncationSlope ε ∘ F from rfl,
+        surfaceGradient_comp (hF.differentiable (by simp) x)
+          ((truncationSlope_contDiff ε).differentiable (by simp) (F x)), hFx.1, hFx.2]
+    rw [heq, inner_add_right, real_inner_smul_right, real_inner_smul_right,
+      real_inner_smul_right, real_inner_smul_left, real_inner_self_eq_norm_sq]
+    have hneg : φ x * (deriv (truncationSlope ε) (W x) * ‖D x‖ ^ 2) ≤ 0 :=
+      mul_nonpos_of_nonneg_of_nonpos (hpφ x) (mul_nonpos_of_nonpos_of_nonneg hDp (sq_nonneg _))
+    linarith
+  · have hxψ : x ∉ tsupport ψ := fun h ↦ hxt (tsupport_mul_subset_left h)
+    simp only [surfaceGradient_eq_zero_of_notMem_tsupport S hxt,
+      surfaceGradient_eq_zero_of_notMem_tsupport S hxψ, inner_zero_right, le_refl]
+
+end C1RegularTests
+end RPI.Boundary
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPI.Boundary
+section RegularEnergy
+variable {n : ℕ} {S Z : Set (Space n)}
+variable (hS : IsClosed S) (hZ : IsClosed Z)
+variable (hfinite : ∀ R : ℝ, 0 < R → (surfaceMeasure S) (Metric.ball 0 R) < ⊤)
+variable (hgrad : ∀ f : Space n → ℝ, ContDiff ℝ 1 f →
+  AEStronglyMeasurable (surfaceGradient S f) (surfaceMeasure S))
+include hS hZ hfinite hgrad
+
+theorem c1_regular_gradient_memLp_test {W : Space n → ℝ} {D : Space n → Space n}
+    (htrace : ∀ x ∈ S \ Z, HasC1TraceJetAt S W D x)
+    {η : Space n → ℝ} (hη : ContDiff ℝ 1 η) (hcη : HasCompactSupport η)
+    (hηZ : Disjoint (tsupport η) Z) :
+    MemLp (fun x ↦ η x • D x) 2 (surfaceMeasure S) := by
+  have hi := c1_regular_gradient_memLp_indicator_test hS hZ hfinite hgrad htrace hcη hηZ
+  obtain ⟨M, hM⟩ := hcη.exists_bound_of_continuous hη.continuous
+  have hh : MemLp (fun x ↦ η x • (tsupport η).indicator D x) 2 (surfaceMeasure S) := by
+    apply hi.of_le_mul (c := M)
+      (hη.continuous.aestronglyMeasurable.smul hi.aestronglyMeasurable)
+    apply ae_of_all
+    intro x
+    change ‖η x • (tsupport η).indicator D x‖ ≤ M * ‖(tsupport η).indicator D x‖
+    rw [norm_smul]
+    exact mul_le_mul_of_nonneg_right (hM x) (norm_nonneg _)
+  apply hh.ae_eq
+  apply ae_of_all
+  intro x
+  by_cases hx : x ∈ tsupport η
+  · simp only [indicator_of_mem hx]
+  · simp only [indicator_of_notMem hx, image_eq_zero_of_notMem_tsupport hx, zero_smul]
+
+theorem c1_smoothTruncation_regular_energy {W : Space n → ℝ} {D : Space n → Space n}
+    (htrace : ∀ x ∈ S \ Z, HasC1TraceJetAt S W D x)
+    (hW0 : ∀ x ∈ S \ Z, 0 ≤ W x)
+    (hsup : ∀ φ : Space n → ℝ, ContDiff ℝ ∞ φ → HasCompactSupport φ →
+      Disjoint (tsupport φ) Z → (∀ x, 0 ≤ φ x) →
+      0 ≤ ∫ x, inner ℝ (D x) (surfaceGradient S φ x) ∂surfaceMeasure S)
+    {ε : ℝ} (hε : 0 < ε) {η : Space n → ℝ}
+    (hη : ContDiff ℝ 1 η) (hcη : HasCompactSupport η)
+    (hηZ : Disjoint (tsupport η) Z) :
+    (∫ x, ‖η x • (truncationSlope ε (W x) • D x)‖ ^ 2 ∂surfaceMeasure S) ≤
+      4 * ∫ x, ‖surfaceGradient S η x‖ ^ 2 ∂surfaceMeasure S := by
+  let T := smoothTruncation ε
+  let G : Space n → Space n := fun x ↦ truncationSlope ε (W x) • D x
+  have htt : ∀ x ∈ S \ Z, HasC1TraceJetAt S (T ∘ W) G x := by
+    intro x hx
+    simpa only [T, G, smoothTruncation_deriv] using
+      (htrace x hx).comp hx.1 ((smoothTruncation_contDiff ε).of_le (by norm_num))
+  obtain ⟨F, hF, he⟩ := c1_exists_smooth_representative_near_regular_test hS hZ htrace
+    (isClosed_tsupport η) hηZ
+  let ψ : Space n → ℝ := fun x ↦ (η x * η x) * (1 - T (F x))
+  have hTF : ContDiff ℝ 1 (T ∘ F) := ((smoothTruncation_contDiff ε).of_le (by norm_num)).comp hF
+  have hψ : ContDiff ℝ 1 ψ := (hη.mul hη).mul (contDiff_const.sub hTF)
+  have hcψ : HasCompactSupport ψ := hcη.mul_right.mul_right
+  have hsub : tsupport ψ ⊆ tsupport η := tsupport_mul_subset_left.trans tsupport_mul_subset_left
+  have hψZ : Disjoint (tsupport ψ) Z := hηZ.mono_left hsub
+  have hψ0 : ∀ x, 0 ≤ ψ x := fun x ↦ mul_nonneg (mul_self_nonneg (η x))
+    (sub_nonneg.mpr (smoothTruncation_le_one_all hε (F x)))
+  have ht : 0 ≤ ∫ x, inner ℝ (G x) (surfaceGradient S ψ x) ∂surfaceMeasure S :=
+    c1_smoothTruncation_regular_test hS hZ hfinite hgrad htrace hsup hε hψ hcψ hψZ hψ0
+  let A : Space n → Space n := fun x ↦ η x • G x
+  let B : Space n → Space n := surfaceGradient S η
+  have hA : MemLp A 2 (surfaceMeasure S) :=
+    c1_regular_gradient_memLp_test hS hZ hfinite hgrad htt hη hcη hηZ
+  have hB : MemLp B 2 (surfaceMeasure S) :=
+    c1_surfaceGradient_memLp_of_compact hfinite hgrad hη hcη
+  have hi := c1_integrable_regular_test_pair hS hZ hfinite hgrad htt hψ hcψ hψZ
+  have hAs := (memLp_two_iff_integrable_sq_norm hA.aestronglyMeasurable).mp hA
+  have hBs := (memLp_two_iff_integrable_sq_norm hB.aestronglyMeasurable).mp hB
+  have hu : Integrable (fun x ↦ 2 * ‖B x‖ ^ 2 - (1 / 2 : ℝ) * ‖A x‖ ^ 2) (surfaceMeasure S) :=
+    (hBs.const_mul 2).sub (hAs.const_mul (1 / 2))
+  have hle : (∫ x, inner ℝ (G x) (surfaceGradient S ψ x) ∂surfaceMeasure S) ≤
+      ∫ x, 2 * ‖B x‖ ^ 2 - (1 / 2 : ℝ) * ‖A x‖ ^ 2 ∂surfaceMeasure S := by
+    apply integral_mono_ae hi hu
+    filter_upwards [ae_restrict_mem hS.measurableSet] with x hx
+    by_cases hxt : x ∈ tsupport η
+    · have hxr : x ∈ S \ Z := ⟨hx, fun hz ↦ Set.disjoint_left.mp hηZ hxt hz⟩
+      have hFx := he x ⟨hx, hxt⟩
+      have hgT : surfaceGradient S (fun y ↦ T (F y)) x = G x := by
+        change surfaceGradient S (T ∘ F) x = G x
+        rw [surfaceGradient_comp (hF.differentiable (by simp) x)
+          ((smoothTruncation_contDiff ε).differentiable (by simp) (F x))]
+        simp only [smoothTruncation_deriv, hFx.1, hFx.2, G]
+      have hgψ : surfaceGradient S ψ x =
+          (η x * η x) • (-G x) + (1 - T (W x)) • (η x • B x + η x • B x) := by
+        dsimp only [ψ]
+        rw [surfaceGradient_mul (S := S) (g := fun y ↦ 1 - T (F y)) ((hη.mul hη).differentiable (by simp) x)
+          ((contDiff_const.sub hTF).differentiable (by simp) x)]
+        rw [surfaceGradient_const_sub S 1, hgT,
+          surfaceGradient_mul (hη.differentiable (by simp) x) (hη.differentiable (by simp) x), hFx.1]
+      have heq : inner ℝ (G x) (surfaceGradient S ψ x) =
+          inner ℝ (A x) ((2 * (1 - T (W x))) • B x - A x) := by
+        rw [hgψ]
+        simp only [A, inner_add_right, real_inner_smul_right, inner_neg_right,
+          real_inner_smul_left, inner_sub_right]
+        ring
+      rw [heq]
+      exact truncation_test_pointwise (A x) (B x)
+        (smoothTruncation_nonneg ε (hW0 x hxr)) (smoothTruncation_le_one_all hε (W x))
+    · have hAx : A x = 0 := by simp only [A, image_eq_zero_of_notMem_tsupport hxt, zero_smul]
+      have hBx : B x = 0 := surfaceGradient_eq_zero_of_notMem_tsupport S hxt
+      rw [surfaceGradient_eq_zero_of_notMem_tsupport S (fun h ↦ hxt (hsub h)),
+        inner_zero_right, hAx, hBx]
+      norm_num
+  rw [integral_sub (hBs.const_mul 2) (hAs.const_mul (1 / 2)),
+    integral_const_mul, integral_const_mul] at hle
+  change (∫ x, ‖A x‖ ^ 2 ∂surfaceMeasure S) ≤ 4 * ∫ x, ‖B x‖ ^ 2 ∂surfaceMeasure S
+  linarith
+
+end RegularEnergy
+end RPI.Boundary
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/SequentialFlatness.lean b/lean/OAI/Geometry/Perimeter/Regularity/SequentialFlatness.lean
new file mode 100644
index 0000000000000000000000000000000000000000..d631e56c218e6493fa1a19f5040bed60af9b40ce
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/SequentialFlatness.lean
@@ -0,0 +1,167 @@
+import OAI.Geometry.Perimeter.Regularity.PlaneHeightLimit
+import OAI.Geometry.Perimeter.Regularity.HarmonicHeightLimit
+import OAI.Geometry.Perimeter.Regularity.FiberQuadraticLimit
+import OAI.Geometry.Perimeter.Regularity.SupportImprovement
+import OAI.Geometry.Perimeter.Regularity.HarmonicLimitBounds
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+open RPI.Boundary ContinuousLinearMap
+variable {n m : ℕ}
+
+theorem minimizing_plane_limit_integral_bound
+    (E : ℕ → Set (Euc n)) (hmin : ∀ i, IsLocallyPerimeterMinimizing (E i))
+    {v : Euc n} (hv : ‖v‖=1) (e : Euc m ≃ₗᵢ[ℝ] normalPlane v)
+    {η : Euc n → ℝ} (hη : Continuous η) (hcη : HasCompactSupport η)
+    (hηs : Function.support η ⊆ Metric.ball 0 4) (hηb : ∀ x, ‖η x‖≤1)
+    (a : ℕ → ℝ) (ha : ∀ i, 0<a i)
+    (hphase : ∀ i, ∀ᵐ x ∂volume, x∈Metric.ball 0 4 → a i < |inner ℝ v x| →
+      (x∈E i ↔ inner ℝ v x<0))
+    {g : Euc m → ℝ} (hg : Integrable g volume)
+    (hL : Tendsto (fun i ↦ ∫⁻ y, ‖chartFiber v e (a i) (cutoffPhaseError (E i) v η) y-g y‖ₑ)
+      atTop (𝓝 0)) :
+    (∫ y, ‖g y‖)≤volume.real (Metric.ball (0:Euc n) 8)/4 := by
+  let V := volume.real (Metric.ball (0:Euc n) 8)/4
+  have hV : 0≤V := by dsimp [V]; positivity
+  have hi (i) := chartFiber_integrable (cutoffPhaseError_integrable (hmin i).1 v hη hcη) hv e (a i)
+  apply integral_norm_le_of_L1_limit _ hi hg _ hL
+  intro i
+  have hb := chartFiber_normalized_mass (cutoffPhaseError_measurable (hmin i).1 v hη.measurable)
+    hv e (ha i) (cutoffPhaseError_lintegral_height_le (hmin i).1 hv hη.measurable hηb
+      (by norm_num : (0:ℝ)<4) (ha i).le hηs (hphase i))
+  rw [← ofReal_integral_norm_eq_lintegral_enorm (hi i)] at hb
+  exact (ENNReal.ofReal_le_ofReal_iff hV).mp (by simpa only [V, show (2:ℝ)*4=8 by norm_num] using hb)
+
+theorem normalPlane_chart_linear {v : Euc n} (e : Euc m ≃ₗᵢ[ℝ] normalPlane v)
+    (L : Euc m →L[ℝ] ℝ) :
+    let ℓ := L.comp (e.symm.toContinuousLinearEquiv.toContinuousLinearMap.comp (normalProjection v))
+    ℓ v=0 ∧ ‖ℓ‖≤‖L‖ := by
+  let ℓ := L.comp (e.symm.toContinuousLinearEquiv.toContinuousLinearMap.comp (normalProjection v))
+  change ℓ v = 0 ∧ ‖ℓ‖ ≤ ‖L‖
+  have hv : normalProjection v v=0 :=
+    Submodule.orthogonalProjectionOnto_eq_zero_iff.mpr
+      ((ℝ ∙ v).le_orthogonal_orthogonal (Submodule.mem_span_singleton_self v))
+  constructor
+  · change L (e.symm (normalProjection v v)) = 0
+    rw [hv, map_zero, map_zero]
+  · refine ContinuousLinearMap.opNorm_le_bound (f := ℓ) (norm_nonneg L) ?_
+    intro x
+    change ‖L (e.symm (normalProjection v x))‖ ≤ ‖L‖ * ‖x‖
+    calc
+      _ ≤ ‖L‖ * ‖e.symm (normalProjection v x)‖ := L.le_opNorm _
+      _ = ‖L‖ * ‖normalProjection v x‖ := by rw [e.symm.norm_map]
+      _ ≤ ‖L‖ * ‖x‖ := mul_le_mul_of_nonneg_left
+        ((normalPlane v).norm_orthogonalProjectionOnto_apply_le x) (norm_nonneg L)
+
+theorem exists_sequential_flatness_constants (n m : ℕ) (hn : 1<n) (hm : 0 < m) :
+    ∃ K B : ℝ, 0<K ∧ 0≤B ∧ ∀ (E : ℕ → Set (Euc n)),
+      (∀ i, IsLocallyPerimeterMinimizing (E i)) →
+      (∀ i, (0:Euc n)∈perimeterSupport (E i)) →
+      ∀ (v : Euc n), ‖v‖=1 → (Euc m ≃ₗᵢ[ℝ] normalPlane v) →
+      ∀ (a : ℕ → ℝ), (∀ i, 0<a i) → Tendsto a atTop (𝓝 0) →
+      (∀ i x, x∈perimeterSupport (E i) ∩ Metric.ball 0 4 → |inner ℝ v x|≤a i) →
+      (∀ i, ∀ᵐ x ∂volume, x∈Metric.ball 0 4 → a i < |inner ℝ v x| →
+        (x∈E i ↔ inner ℝ v x<0)) →
+      Tendsto (fun i ↦ orientedTilt (E i) v (Metric.ball 0 4)) atTop (𝓝 0) →
+      ∃ σ : ℕ → ℕ, StrictMono σ ∧ ∃ ℓ : Euc n →L[ℝ] ℝ,
+        ℓ v=0 ∧ ‖ℓ‖≤B ∧ ∀ r : ℝ, 0<r → r<1/8 → ∀ᶠ i in atTop,
+          ∀ p∈perimeterSupport (E (σ i)) ∩ Metric.ball 0 r,
+            |inner ℝ v p/a (σ i)-ℓ p|≤K*r^2 := by
+  obtain ⟨C,hC,hTaylor⟩ := harmonicRegularization_uniform_taylor m
+  have hcK : HasCompactSupport (scaledHarmonicKernel (n := m) 4) :=
+    (isCompact_closedBall 0 (1/(4:ℝ))).of_isClosed_subset (isClosed_tsupport _)
+      (scaledHarmonicKernel_tsupport (by norm_num))
+  obtain ⟨M,hM,hLin⟩ := kernelLinear_uniform_bound (scaledHarmonicKernel_contDiff 4) hcK
+  let V := volume.real (Metric.ball (0:Euc n) 8)/4
+  have hV : 0≤V := by dsimp [V]; positivity
+  let D := C*V
+  have hD : 0≤D := mul_nonneg hC hV
+  obtain ⟨K,hK,himp⟩ := exists_full_support_improvement_constant n hn D hD
+  refine ⟨K,M*V,hK,mul_nonneg hM hV,fun E hmin hzero v hv e a ha hat hh hphase htilt ↦ ?_⟩
+  let η : ContDiffBump (0:Euc n) := ⟨3,7/2,by norm_num,by norm_num⟩
+  have hηs : tsupport (η:Euc n → ℝ) ⊆ Metric.ball 0 4 := by
+    rw [η.tsupport_eq]; exact Metric.closedBall_subset_ball (by norm_num : (7:ℝ)/2<4)
+  have hηone : ∀ x∈Metric.ball (0:Euc n) 3, η x=1 := fun x hx ↦
+    η.one_of_mem_closedBall (Metric.ball_subset_closedBall hx)
+  have hηb : ∀ x, ‖η x‖≤1 := fun x ↦ by rw [Real.norm_of_nonneg η.nonneg]; exact η.le_one
+  obtain ⟨L₀,hL₀⟩ := (isCompact_closedBall (0:Euc n) 4).exists_bound_of_continuousOn
+    (gradient_contDiff η.contDiff).continuous.continuousOn
+  let L := max L₀ 0
+  have hLg : ∀ x∈Metric.ball (0:Euc n) 4, ‖gradient η x‖≤L :=
+    fun x hx ↦ (hL₀ x (Metric.ball_subset_closedBall hx)).trans (le_max_left _ _)
+  obtain ⟨G,hG,hgrowth⟩ := exists_locallyMinimizing_perimeter_growth_constant n (by omega)
+  have hper (i) : (perimeter (E i) (Metric.ball 0 4)).toReal≤G*4^(n-1) := by
+    have h := ENNReal.toReal_mono ENNReal.ofReal_ne_top (hgrowth (E i) (hmin i) 0 4 (by norm_num))
+    rw [ENNReal.toReal_ofReal (by positivity)] at h
+    exact h
+  obtain ⟨g,hg,hgs,σ,hσ,hL1⟩ := minimizing_plane_height_subsequence hn E hmin hv e
+    η.contDiff η.hasCompactSupport hηb (by norm_num : (0:ℝ)<4) (le_max_right _ _)
+    hηs hLg a ha hh hper hphase
+  have has := hat.comp hσ.tendsto_atTop
+  have ht := htilt.comp hσ.tendsto_atTop
+  have hgv : (∫ y, ‖g y‖)≤V := minimizing_plane_limit_integral_bound
+    (fun i ↦ E (σ i)) (fun i ↦ hmin (σ i)) hv e η.continuous η.hasCompactSupport
+    (subset_closure.trans hηs) hηb (fun i ↦ a (σ i)) (fun i ↦ ha (σ i))
+    (fun i ↦ hphase (σ i)) hg hL1
+  have hgh : WeakHarmonicOn g (Metric.ball 0 1) := minimizing_plane_height_weakHarmonic hn
+    (fun i ↦ E (σ i)) (fun i ↦ hmin (σ i)) hv e η.continuous hηb η.hasCompactSupport
+    (subset_closure.trans hηs) hηone (fun i ↦ a (σ i)) (fun i ↦ ha (σ i)) has
+    (fun i ↦ hh (σ i)) (fun i ↦ hphase (σ i)) (fun i ↦ hper (σ i)) ht hg hL1
+  let f := harmonicRegularization g 4
+  have hf : ContDiff ℝ ∞ f := harmonicRegularization_contDiff hg.locallyIntegrable (by norm_num)
+  have hgf : g =ᵐ[volume.restrict (Metric.ball 0 (1/2))] f := by
+    filter_upwards [ae_restrict_of_ae (weakHarmonic_ae_eq_regularization hg.locallyIntegrable hgh hm),
+      ae_restrict_mem measurableSet_ball] with y hy hyB
+    apply hy (1/2) (by norm_num) _ 4 (by norm_num)
+    intro z hz
+    have hd := dist_triangle z y 0
+    rw [Metric.mem_ball] at hz hyB ⊢
+    linarith
+  let Lh := kernelLinear g (scaledHarmonicKernel 4)
+  let ℓ := Lh.comp (e.symm.toContinuousLinearEquiv.toContinuousLinearMap.comp (normalProjection v))
+  have hℓ := normalPlane_chart_linear e Lh
+  let w : normalPlane v → ℝ := fun y ↦ f (e.symm y)
+  have hw : Continuous w := hf.continuous.comp e.symm.continuous
+  let φ : ContDiffBump (0:normalPlane v) := ⟨1/4,1/3,by norm_num,by norm_num⟩
+  have hφs : tsupport (φ:normalPlane v → ℝ) ⊆ Metric.ball 0 1 := by
+    rw [φ.tsupport_eq]; exact Metric.closedBall_subset_ball (by norm_num : (1:ℝ)/3<1)
+  have hwg : ∀ᵐ y : Euc m ∂volume, φ (e y)≠0 → w (e y)=g y := by
+    have hgf' := (ae_restrict_iff' measurableSet_ball).mp hgf
+    filter_upwards [hgf'] with y hy
+    intro hφy
+    have hyb : y∈Metric.ball (0:Euc m) (1/2) := by
+      have hyB : e y∈Metric.ball 0 (1/3) := by rwa [← φ.support_eq]
+      rw [Metric.mem_ball,dist_zero_right,e.norm_map] at hyB
+      rw [Metric.mem_ball,dist_zero_right]; linarith
+    simpa only [w,e.symm_apply_apply] using (hy hyb).symm
+  have htrace := minimizing_quadratic_height_trace_tendsto_zero hn (fun i ↦ E (σ i))
+    (fun i ↦ hmin (σ i)) hv e η.continuous η.hasCompactSupport (subset_closure.trans hηs)
+    hηone (fun i ↦ a (σ i)) (fun i ↦ ha (σ i)) has (fun i ↦ hh (σ i))
+    (fun i ↦ hphase (σ i)) ht hg hL1 φ.continuous φ.hasCompactSupport (fun _ ↦ φ.nonneg)
+    hφs hw hwg
+  have htaylor : ∀ x∈Metric.ball (0:Euc n) (1/4),
+      |w (normalProjection v x)-f 0-ℓ x|≤D*‖x‖^2 := by
+    intro x _
+    have h := hTaylor g hg (e.symm (normalProjection v x))
+    change ‖w (normalProjection v x)-f 0-ℓ x‖≤C*(∫ y, ‖g y‖)*‖e.symm (normalProjection v x)‖^2 at h
+    rw [Real.norm_eq_abs,e.symm.norm_map] at h
+    apply h.trans
+    dsimp only [D]
+    gcongr
+    exact (normalPlane v).norm_orthogonalProjectionOnto_apply_le x
+  obtain ⟨hb,hflat⟩ := himp (fun i ↦ E (σ i)) (fun i ↦ hmin (σ i)) (fun i ↦ hzero (σ i))
+    v φ w φ.continuous φ.hasCompactSupport (fun _ ↦ φ.nonneg)
+    (fun y hy ↦ φ.one_of_mem_closedBall (Metric.ball_subset_closedBall hy)) hw
+    (fun i ↦ a (σ i)) (fun i ↦ ha (σ i)) (fun i ↦ hh (σ i)) ℓ (f 0) htaylor htrace
+  refine ⟨σ,hσ,ℓ,hℓ.1,?_,hflat⟩
+  exact hℓ.2.trans ((hLin g hg).trans (mul_le_mul_of_nonneg_left hgv hM))
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/SetIntegralSubLeSymmDiff.lean b/lean/OAI/Geometry/Perimeter/Regularity/SetIntegralSubLeSymmDiff.lean
new file mode 100644
index 0000000000000000000000000000000000000000..02edb83b8f10bb78ec4d259a1308e1eb1ab4e457
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/SetIntegralSubLeSymmDiff.lean
@@ -0,0 +1,454 @@
+import OAI.Geometry.Perimeter.Regularity.PerimeterAdmissiblePartitionPiece
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+open scoped ENNReal Topology
+open scoped ENNReal Topology ContDiff
+open scoped Topology ENNReal
+namespace RPISingularHarnack
+
+theorem setIntegral_sub_le_symmDiff {n : ℕ} {E F : Set (Euc n)}
+    (hE : MeasurableSet E) (hF : MeasurableSet F) {f g : Euc n → ℝ}
+    (hf : Integrable f volume) (hg : Integrable g volume)
+    (hfg : ∀ x, |f x| ≤ g x) :
+    (∫ x in F, f x) - (∫ x in E, f x) ≤ ∫ x in symmDiff E F, g x := by
+  rw [← integral_indicator hF, ← integral_indicator hE,
+    ← integral_sub (hf.indicator hF) (hf.indicator hE),
+    ← integral_indicator (hE.symmDiff hF)]
+  apply integral_mono ((hf.indicator hF).sub (hf.indicator hE))
+    (hg.indicator (hE.symmDiff hF))
+  intro x
+  by_cases hxE : x ∈ E <;> by_cases hxF : x ∈ F
+  · simp [hxE, hxF, mem_symmDiff]
+  · simpa [hxE, hxF, mem_symmDiff] using (neg_le_abs (f x)).trans (hfg x)
+  · simpa [hxE, hxF, mem_symmDiff] using (le_abs_self (f x)).trans (hfg x)
+  · simp [hxE, hxF, mem_symmDiff]
+
+theorem integral_smooth_patch_le {n : ℕ} {E F U : Set (Euc n)}
+    (hE : MeasurableSet E) (hF : MeasurableSet F)
+    {V : Euc n → Euc n} (hV : PerimeterAdmissible U V)
+    {c : Euc n} {r a b B : ℝ} (ha : 0 < a) (hra : r < a) (hab : a < b)
+    (hbB : b < B) (hPE : perimeter E (U \ Metric.closedBall c r) ≠ ⊤)
+    (hPF : perimeter F (Metric.ball c B) ≠ ⊤) :
+    (∫ x in F, divergence V x) +
+        (∫ x in E, annularCutoff c a b x * divergence V x) -
+        (∫ x in F, annularCutoff c a b x * divergence V x) ≤
+      (perimeter E (U \ Metric.closedBall c r)).toReal +
+      (perimeter F (Metric.ball c B)).toReal +
+        ∫ x in symmDiff E F, ‖gradient (annularCutoff c a b) x‖ := by
+  have hVE : PerimeterAdmissible (U \ Metric.closedBall c r)
+      (fun x ↦ annularCutoff c a b x • V x) :=
+    annularCutoff_test_admissible hV ha hra hab
+  have hVu : PerimeterAdmissible univ V :=
+    ⟨hV.1, hV.2.1, subset_univ _, hV.2.2.2⟩
+  have hVF := pairedCutTest_admissible hVu (perimeterAdmissible_zero (Metric.ball c B))
+    ha hab (Metric.closedBall_subset_ball hbB)
+  have hd := integrable_divergence hV.1 hV.2.1
+  have hg : Integrable (fun x ↦ ‖gradient (annularCutoff c a b) x‖) volume :=
+    ((gradient_annularCutoff_continuous c a b).integrable_of_hasCompactSupport
+      (gradient_annularCutoff_compact c ha hab)).norm
+  have hinner (x : Euc n) : |inner ℝ (gradient (annularCutoff c a b) x) (V x)| ≤
+      ‖gradient (annularCutoff c a b) x‖ :=
+    (abs_real_inner_le_norm _ _).trans
+      (mul_le_of_le_one_right (norm_nonneg _) (hV.2.2.2 x))
+  have hi : Integrable (fun x ↦ inner ℝ (gradient (annularCutoff c a b) x) (V x)) volume :=
+    hg.mono' ((gradient_annularCutoff_continuous c a b).inner hV.1.continuous).aestronglyMeasurable
+      (Eventually.of_forall hinner)
+  have hmul := integrable_annularCutoff_mul hd c a b
+  have hEi := integral_divergence_le_perimeter_real E _ hPE hVE
+  have hFi := integral_divergence_le_perimeter_real F _ hPF hVF
+  simp_rw [divergence_smul ((annularCutoff_contDiff c a b).differentiable (by simp))
+    (hV.1.differentiable (by simp))] at hEi
+  rw [integral_add hmul.integrableOn hi.integrableOn] at hEi
+  have hformula (x : Euc n) : divergence (pairedCutTest c a b V (fun _ ↦ 0)) x =
+      divergence V x - annularCutoff c a b x * divergence V x -
+        inner ℝ (gradient (annularCutoff c a b) x) (V x) := by
+    rw [divergence_pairedCutTest hV.1 contDiff_const]
+    have hz : divergence (fun _ : Euc n ↦ (0:Euc n)) x = 0 := by simp [divergence]
+    simp only [hz, zero_sub, inner_neg_right]
+    ring
+  simp_rw [hformula] at hFi
+  have hleft : Integrable (fun x ↦ divergence V x - annularCutoff c a b x * divergence V x) volume := hd.sub hmul
+  rw [integral_sub hleft.integrableOn hi.integrableOn,
+    integral_sub hd.integrableOn hmul.integrableOn] at hFi
+  have herr := setIntegral_sub_le_symmDiff hE hF hi hg hinner
+  linarith
+
+def gluedFilledSet {n : ℕ} (E F : Set (Euc n)) (c : Euc n) (r : ℝ) : Set (Euc n) :=
+  (F ∩ Metric.closedBall c r) ∪ (E \ Metric.closedBall c r)
+
+theorem controlled_radius_glued_perimeter_le {n : ℕ} {E F U : Set (Euc n)}
+    (hE : MeasurableSet E) (hF : MeasurableSet F) (c : Euc n) {r B D L : ℝ}
+    (hr : 0 < r) (hrB : r < B)
+    (hcontrol : ∀ t ∈ Icc r B,
+      volume.real (symmDiff E F ∩ Metric.closedBall c t) -
+        volume.real (symmDiff E F ∩ Metric.closedBall c r) ≤ D * (t - r))
+    (hD : 0 ≤ D) (hL : 0 < L)
+    (hbound : ∀ s, |deriv Real.smoothTransition s| ≤ L)
+    (hPE : perimeter E (U \ Metric.closedBall c r) ≠ ⊤)
+    (hPF : perimeter F (Metric.ball c B) ≠ ⊤) :
+    perimeter (gluedFilledSet E F c r) U ≤
+      perimeter E (U \ Metric.closedBall c r) + perimeter F (Metric.ball c B) +
+        ENNReal.ofReal (4*L*D) := by
+  rw [perimeter_eq_iSup_tests]
+  refine iSup_le fun V ↦ iSup_le fun hV ↦ ?_
+  have hd := integrable_divergence hV.1 hV.2.1
+  have hdis : Disjoint (F ∩ Metric.closedBall c r) (E \ Metric.closedBall c r) := by
+    apply disjoint_left.mpr
+    intro x hx hy
+    exact hy.2 hx.2
+  have hsplit := integral_inter_add_sdiff (s := F) (t := Metric.closedBall c r)
+    measurableSet_closedBall hd.integrableOn
+  have hpatch : (∫ x in F, divergence V x) +
+      (∫ x in E \ Metric.closedBall c r, divergence V x) -
+      (∫ x in F \ Metric.closedBall c r, divergence V x) =
+      ∫ x in gluedFilledSet E F c r, divergence V x := by
+    rw [gluedFilledSet, setIntegral_union hdis (hE.diff measurableSet_closedBall)
+      hd.integrableOn hd.integrableOn]
+    linarith
+  have ht := ((tendsto_const_nhds (x := ∫ x in F, divergence V x)).add
+    (tendsto_integral_annularCutoff_mul (E := E) hd c hr)).sub
+      (tendsto_integral_annularCutoff_mul (E := F) hd c hr)
+  rw [hpatch] at ht
+  have hreal : (∫ x in gluedFilledSet E F c r, divergence V x) ≤
+      (perimeter E (U \ Metric.closedBall c r)).toReal +
+      (perimeter F (Metric.ball c B)).toReal + 4*L*D := by
+    apply le_of_tendsto ht
+    filter_upwards [self_mem_nhdsWithin,
+      nhdsWithin_le_nhds (gt_mem_nhds (show (0:ℝ) < (B-r)/2 by linarith))] with δ hδ hsmall
+    change 0 < δ at hδ
+    change δ < (B-r)/2 at hsmall
+    have hh := integral_smooth_patch_le hE hF hV
+      (show 0 < r+δ by positivity) (show r < r+δ by linarith)
+      (show r+δ < r+2*δ by linarith) (show r+2*δ < B by linarith) hPE hPF
+    have hg := controlled_radius_gradient_bound (hE.symmDiff hF) c hr hδ
+      (show r+2*δ ≤ B by linarith) hcontrol hD hL hbound
+    linarith
+  calc
+    _ ≤ ENNReal.ofReal ((perimeter E (U \ Metric.closedBall c r)).toReal +
+        (perimeter F (Metric.ball c B)).toReal + 4*L*D) := ENNReal.ofReal_le_ofReal hreal
+    _ = _ := by
+      rw [ENNReal.ofReal_add (by positivity) (by positivity),
+        ENNReal.ofReal_add ENNReal.toReal_nonneg ENNReal.toReal_nonneg,
+        ENNReal.ofReal_toReal hPE, ENNReal.ofReal_toReal hPF]
+
+theorem minimizing_controlled_replacement_perimeter_le {n : ℕ} {E F : Set (Euc n)}
+    (hmin : IsLocallyPerimeterMinimizing E) (hF : MeasurableSet F)
+    (c : Euc n) {a r B D L : ℝ} (ha : 0 < a) (har : a ≤ r) (hrB : r < B)
+    (hcontrol : ∀ t ∈ Icc r B,
+      volume.real (symmDiff E F ∩ Metric.closedBall c t) -
+        volume.real (symmDiff E F ∩ Metric.closedBall c r) ≤ D*(t-r))
+    (hD : 0 ≤ D) (hL : 0 < L)
+    (hbound : ∀ s, |deriv Real.smoothTransition s| ≤ L)
+    (hPF : perimeter F (Metric.ball c B) ≠ ⊤) :
+    perimeter E (Metric.ball c a) ≤ perimeter F (Metric.ball c B) + ENNReal.ofReal (4*L*D) := by
+  let U := Metric.ball c (B+1)
+  let G := gluedFilledSet E F c r
+  have hr : 0 < r := ha.trans_le har
+  have hUfin : perimeter E U ≠ ⊤ := (hmin.2.1 c (B+1) (by linarith)).ne
+  have hPE : perimeter E (U \ Metric.closedBall c r) ≠ ⊤ :=
+    ne_top_of_le_ne_top hUfin (raw_perimeter_mono_region E sdiff_subset)
+  have hG : MeasurableSet G :=
+    (hF.inter measurableSet_closedBall).union (hmin.1.diff measurableSet_closedBall)
+  have hcl : closure (symmDiff E G) ⊆ Metric.closedBall c r := by
+    apply closure_minimal _ Metric.isClosed_closedBall
+    intro x hx
+    by_contra hb
+    have hmem : x ∈ G ↔ x ∈ E := by simp [G, gluedFilledSet, hb]
+    rcases hx with hx | hx
+    · exact hx.2 (hmem.mpr hx.1)
+    · exact hx.2 (hmem.mp hx.1)
+  have hcmp := hmin.2.2 c (B+1) (by linarith) G hG
+    ((isCompact_closedBall c r).of_isClosed_subset isClosed_closure hcl)
+    (hcl.trans (Metric.closedBall_subset_ball (by linarith)))
+  have hcut := controlled_radius_glued_perimeter_le hmin.1 hF c hr hrB
+    hcontrol hD hL hbound hPE hPF
+  have hdis : Disjoint (Metric.ball c a) (U \ Metric.closedBall c r) := by
+    apply disjoint_left.mpr
+    intro x hx hy
+    exact hy.2 ((Metric.ball_subset_closedBall.trans (Metric.closedBall_subset_closedBall har)) hx)
+  have hadd := perimeter_disjoint_superadditive E
+    (Metric.ball_subset_ball (by linarith : a ≤ B+1)) sdiff_subset hdis
+  have hh : perimeter E (Metric.ball c a) + perimeter E (U \ Metric.closedBall c r) ≤
+      (perimeter F (Metric.ball c B) + ENNReal.ofReal (4*L*D)) +
+        perimeter E (U \ Metric.closedBall c r) := by
+    calc
+      _ ≤ perimeter E U := hadd
+      _ ≤ perimeter G U := hcmp
+      _ ≤ _ := by simpa only [add_assoc, add_comm, add_left_comm] using hcut
+  exact (ENNReal.add_le_add_iff_right hPE).mp hh
+
+theorem exists_controlled_annular_cut_radius {n : ℕ} {T : Set (Euc n)}
+    (hT : MeasurableSet T) (c : Euc n) {a B D : ℝ} (ha : 0 < a) (haB : a < B)
+    (hv : volume.real (T ∩ (Metric.closedBall c B \ Metric.closedBall c a)) < D*(B-a)) :
+    ∃ r ∈ Ico a B, ∀ t ∈ Icc r B,
+      volume.real (T ∩ Metric.closedBall c t) - volume.real (T ∩ Metric.closedBall c r) ≤ D*(t-r) := by
+  let S := T \ Metric.closedBall c a
+  have hvol : volume.real (S ∩ Metric.closedBall c B) < D*(B-a) := by
+    convert hv using 2
+    ext x
+    simp only [S, mem_inter_iff, Set.mem_sdiff]
+    tauto
+  obtain ⟨r, hr, hc⟩ := exists_controlled_cut_radius S c ha haB hvol
+  have he (t : ℝ) (hat : a ≤ t) :
+      volume.real (S ∩ Metric.closedBall c t) =
+      volume.real (T ∩ Metric.closedBall c t) - volume.real (T ∩ Metric.closedBall c a) := by
+    have hset : S ∩ Metric.closedBall c t =
+        (T ∩ Metric.closedBall c t) \ (T ∩ Metric.closedBall c a) := by
+      ext x
+      simp only [S, mem_inter_iff, Set.mem_sdiff]
+      tauto
+    rw [hset]
+    exact measureReal_sdiff (inter_subset_inter_right T (Metric.closedBall_subset_closedBall hat))
+      (hT.inter measurableSet_closedBall)
+      (measure_ne_top_of_subset inter_subset_right (isCompact_closedBall c t).measure_ne_top)
+  refine ⟨r, hr, ?_⟩
+  intro t ht
+  have hh := hc t ht
+  rw [he t (hr.1.trans ht.1), he r hr.1] at hh
+  linarith
+
+theorem minimizing_annular_replacement_perimeter_le {n : ℕ} {E F : Set (Euc n)}
+    (hmin : IsLocallyPerimeterMinimizing E) (hF : MeasurableSet F)
+    (c : Euc n) {a B D L : ℝ} (ha : 0 < a) (haB : a < B)
+    (hv : volume.real (symmDiff E F ∩ (Metric.closedBall c B \ Metric.closedBall c a)) < D*(B-a))
+    (hD : 0 ≤ D) (hL : 0 < L)
+    (hbound : ∀ s, |deriv Real.smoothTransition s| ≤ L) :
+    perimeter E (Metric.ball c a) ≤ perimeter F (Metric.ball c B) + ENNReal.ofReal (4*L*D) := by
+  by_cases hPF : perimeter F (Metric.ball c B) = ⊤
+  · simp [hPF]
+  obtain ⟨r, hr, hc⟩ := exists_controlled_annular_cut_radius (hmin.1.symmDiff hF) c ha haB hv
+  exact minimizing_controlled_replacement_perimeter_le hmin hF c ha hr.1 hr.2
+    hc hD hL hbound hPF
+
+theorem eventually_minimizing_perimeter_le_of_annular_L1 {n : ℕ}
+    {E : ℕ → Set (Euc n)} {F : Set (Euc n)}
+    (hE : ∀ i, IsLocallyPerimeterMinimizing (E i)) (hF : MeasurableSet F)
+    (c : Euc n) {a B : ℝ} (ha : 0 < a) (haB : a < B)
+    (hconv : Tendsto (fun i ↦ volume.real (symmDiff (E i) F ∩
+      (Metric.closedBall c B \ Metric.closedBall c a))) atTop (𝓝 0))
+    {ε : ℝ} (hε : 0 < ε) :
+    ∀ᶠ i in atTop, perimeter (E i) (Metric.ball c a) ≤
+      perimeter F (Metric.ball c B) + ENNReal.ofReal ε := by
+  obtain ⟨L, hL, hbound⟩ := exists_smoothTransition_deriv_bound
+  let D := ε/(4*L)
+  have hD : 0 < D := div_pos hε (by positivity)
+  have hd : 0 < D*(B-a) := mul_pos hD (sub_pos.mpr haB)
+  filter_upwards [hconv.eventually (gt_mem_nhds hd)] with i hi
+  have hh := minimizing_annular_replacement_perimeter_le (hE i) hF c ha haB hi
+    hD.le hL hbound
+  have he : 4*L*D = ε := by dsimp [D]; field_simp
+  simpa only [he] using hh
+
+theorem lintegral_indicator_difference {n : ℕ} {E F U : Set (Euc n)}
+    (hE : MeasurableSet E) (hF : MeasurableSet F) :
+    (∫⁻ x in U, ‖E.indicator (fun _ ↦ (1:ℝ)) x - F.indicator (fun _ ↦ (1:ℝ)) x‖ₑ) =
+      volume (symmDiff E F ∩ U) := by
+  have he (x : Euc n) :
+      ‖E.indicator (fun _ ↦ (1:ℝ)) x - F.indicator (fun _ ↦ (1:ℝ)) x‖ₑ =
+      (symmDiff E F).indicator (fun _ ↦ (1:ℝ≥0∞)) x := by
+    by_cases hxE : x ∈ E <;> by_cases hxF : x ∈ F <;> simp [hxE, hxF, mem_symmDiff]
+  simp_rw [he]
+  rw [lintegral_indicator (hE.symmDiff hF), lintegral_one,
+    Measure.restrict_apply_univ, Measure.restrict_apply (hE.symmDiff hF)]
+
+theorem abs_setIntegral_sub_le_symmDiff_volume {n : ℕ} {E F U : Set (Euc n)}
+    (hE : MeasurableSet E) (hF : MeasurableSet F) (hU : MeasurableSet U)
+    (hUfin : volume U ≠ ⊤) {f : Euc n → ℝ} (hf : Integrable f volume)
+    (hs : ∀ x ∉ U, f x = 0) {C : ℝ} (hC : 0 ≤ C) (hb : ∀ x, |f x| ≤ C) :
+    |(∫ x in E, f x) - (∫ x in F, f x)| ≤
+      C * volume.real (symmDiff E F ∩ U) := by
+  let g := U.indicator (fun _ : Euc n ↦ C)
+  have hg : Integrable g volume := (integrable_indicator_iff hU).2 (integrableOn_const hUfin)
+  have hfg (x : Euc n) : |f x| ≤ g x := by
+    by_cases hx : x ∈ U
+    · simpa [g, hx] using hb x
+    · rw [hs x hx, abs_zero]
+      exact Set.indicator_nonneg (fun _ _ ↦ hC) x
+  have hi : (∫ x in symmDiff E F, g x) = C * volume.real (symmDiff E F ∩ U) := by
+    rw [show g = U.indicator (fun _ : Euc n ↦ C) from rfl, integral_indicator hU,
+      Measure.restrict_restrict hU, integral_const, smul_eq_mul]
+    rw [Set.inter_comm U (symmDiff E F)]
+    simp [Measure.real, mul_comm]
+  have h₁ := setIntegral_sub_le_symmDiff hE hF hf hg hfg
+  have h₂ := setIntegral_sub_le_symmDiff hF hE hf hg hfg
+  rw [symmDiff_comm F E, hi] at h₂
+  rw [hi] at h₁
+  exact abs_le.mpr ⟨by linarith, h₂⟩
+
+theorem tendsto_setIntegral_of_local_symmDiff {n : ℕ}
+    {E : ℕ → Set (Euc n)} {F : Set (Euc n)}
+    (hE : ∀ i, MeasurableSet (E i)) (hF : MeasurableSet F)
+    (hconv : ∀ (c : Euc n) (R : ℝ), 0 < R →
+      Tendsto (fun i ↦ volume.real (symmDiff (E i) F ∩ Metric.closedBall c R)) atTop (𝓝 0))
+    {f : Euc n → ℝ} (hf : Continuous f) (hc : HasCompactSupport f) :
+    Tendsto (fun i ↦ ∫ x in E i, f x) atTop (𝓝 (∫ x in F, f x)) := by
+  obtain ⟨r, hr⟩ := (Metric.isBounded_iff_subset_ball (0:Euc n)).mp hc.isBounded
+  let R := max r 0 + 1
+  have hR : 0 < R := by dsimp [R]; positivity
+  have hs : tsupport f ⊆ Metric.closedBall (0:Euc n) R :=
+    hr.trans (Metric.ball_subset_closedBall.trans
+      (Metric.closedBall_subset_closedBall (by dsimp [R]; linarith [le_max_left r 0])))
+  obtain ⟨C, hC⟩ := hc.exists_bound_of_continuous hf
+  have hC0 : 0 ≤ C := (norm_nonneg (f 0)).trans (hC 0)
+  have hb (i : ℕ) := abs_setIntegral_sub_le_symmDiff_volume (hE i) hF
+    measurableSet_closedBall (isCompact_closedBall (0:Euc n) R).measure_ne_top
+    (hf.integrable_of_hasCompactSupport hc)
+    (fun x hx ↦ image_eq_zero_of_notMem_tsupport (fun h ↦ hx (hs h))) hC0
+    (fun x ↦ by simpa only [Real.norm_eq_abs] using hC x)
+  apply tendsto_iff_norm_sub_tendsto_zero.mpr
+  apply squeeze_zero (fun i ↦ norm_nonneg _) (fun i ↦ by simpa only [Real.norm_eq_abs] using hb i)
+  simpa using (hconv 0 R hR).const_mul C
+
+theorem perimeter_le_of_eventually_le_of_local_L1 {n : ℕ}
+    {E : ℕ → Set (Euc n)} {F U : Set (Euc n)}
+    (hE : ∀ i, MeasurableSet (E i)) (hF : MeasurableSet F)
+    (hconv : ∀ (c : Euc n) (R : ℝ), 0 < R →
+      Tendsto (fun i ↦ volume.real (symmDiff (E i) F ∩ Metric.closedBall c R)) atTop (𝓝 0))
+    {A : ℝ≥0∞} (hb : ∀ᶠ i in atTop, perimeter (E i) U ≤ A) :
+    perimeter F U ≤ A := by
+  by_cases hA : A = ⊤
+  · simp [hA]
+  rw [perimeter_eq_iSup_tests]
+  refine iSup_le fun V ↦ iSup_le fun hV ↦ ?_
+  have hc := hV.2.1.of_isClosed_subset (isClosed_tsupport _) (tsupport_divergence_subset V)
+  have ht := tendsto_setIntegral_of_local_symmDiff hE hF hconv (continuous_divergence hV.1) hc
+  apply (ENNReal.ofReal_le_iff_le_toReal hA).2
+  apply le_of_tendsto ht
+  filter_upwards [hb] with i hi
+  exact (integral_divergence_le_perimeter_real (E i) U
+    (ne_top_of_le_ne_top hA hi) hV).trans (ENNReal.toReal_mono hA hi)
+
+theorem locally_minimizing_of_local_L1_limit {n : ℕ} (hn : 0 < n)
+    {E : ℕ → Set (Euc n)} {F : Set (Euc n)}
+    (hE : ∀ i, IsLocallyPerimeterMinimizing (E i)) (hF : MeasurableSet F)
+    (hconv : ∀ (c : Euc n) (R : ℝ), 0 < R →
+      Tendsto (fun i ↦ volume.real (symmDiff (E i) F ∩ Metric.closedBall c R)) atTop (𝓝 0)) :
+    IsLocallyPerimeterMinimizing F := by
+  obtain ⟨C, hC, hgrowth⟩ := exists_locallyMinimizing_perimeter_growth_constant n hn
+  refine ⟨hF, ?_, ?_⟩
+  · intro c r hr
+    exact (perimeter_le_of_eventually_le_of_local_L1 (fun i ↦ (hE i).1) hF hconv
+      (Eventually.of_forall fun i ↦ hgrowth (E i) (hE i) c r hr)).trans_lt ENNReal.ofReal_lt_top
+  · intro c R hR G hG hcG hsG
+    rw [perimeter_eq_iSup_tests]
+    refine iSup_le fun V ↦ iSup_le fun hV ↦ ?_
+    obtain ⟨a₀, ha₀, hsub⟩ := exists_lt_subset_ball
+      ((isClosed_tsupport V).union isClosed_closure) (union_subset hV.2.2.1 hsG)
+    let a := max a₀ (R/2)
+    have ha : 0 < a := (half_pos hR).trans_le (le_max_right _ _)
+    have haR : a < R := max_lt ha₀ (by linarith)
+    have hasub : tsupport V ∪ closure (symmDiff F G) ⊆ Metric.ball c a :=
+      hsub.trans (Metric.ball_subset_ball (le_max_left _ _))
+    let B := (a+R)/2
+    have haB : a < B := by dsimp [B]; linarith
+    have hBR : B < R := by dsimp [B]; linarith
+    have hB : 0 < B := ha.trans haB
+    have hann : Tendsto (fun i ↦ volume.real (symmDiff (E i) G ∩
+        (Metric.closedBall c B \ Metric.closedBall c a))) atTop (𝓝 0) := by
+      apply squeeze_zero (fun i ↦ ENNReal.toReal_nonneg)
+        (fun i ↦ measureReal_mono (s₂ := symmDiff (E i) F ∩ Metric.closedBall c B) ?_
+          (measure_ne_top_of_subset inter_subset_right (isCompact_closedBall c B).measure_ne_top))
+        (hconv c B hB)
+      intro x hx
+      have hxFG : x ∉ symmDiff F G := by
+        intro hx'
+        exact hx.2.2 (Metric.ball_subset_closedBall
+          (hasub (Or.inr (subset_closure hx'))))
+      refine ⟨?_, hx.2.1⟩
+      simp only [mem_symmDiff] at hxFG ⊢
+      have hh := hx.1
+      simp only [mem_symmDiff] at hh
+      tauto
+    have hle : perimeter F (Metric.ball c a) ≤ perimeter G (Metric.ball c R) := by
+      apply ENNReal.le_of_forall_pos_le_add
+      intro ε hε hfin
+      apply perimeter_le_of_eventually_le_of_local_L1 (fun i ↦ (hE i).1) hF hconv
+      filter_upwards [eventually_minimizing_perimeter_le_of_annular_L1 hE hG c ha haB hann
+        (show 0 < (ε:ℝ) from hε)] with i hi
+      rw [ENNReal.ofReal_coe_nnreal] at hi
+      exact hi.trans (add_le_add (raw_perimeter_mono_region G
+        (Metric.ball_subset_ball hBR.le)) le_rfl)
+    have hvle : ENNReal.ofReal (∫ x in F, divergence V x) ≤ perimeter F (Metric.ball c a) := by
+      rw [perimeter_eq_iSup_tests]
+      exact le_iSup_of_le V (le_iSup_of_le
+        (show PerimeterAdmissible (Metric.ball c a) V from
+          ⟨hV.1, hV.2.1, subset_union_left.trans hasub, hV.2.2.2⟩) le_rfl)
+    exact hvle.trans hle
+
+theorem tendsto_perimeter_ball_right {n : ℕ} {F : Set (Euc n)}
+    (hF : IsLocallyPerimeterMinimizing F) (c : Euc n) {r : ℝ} (hr : 0 < r) :
+    Tendsto (fun R ↦ perimeter F (Metric.ball c R)) (𝓝[>] r)
+      (𝓝 (perimeterMeasure F (Metric.closedBall c r))) := by
+  have ht := tendsto_measure_biInter_gt
+    (μ := perimeterMeasure F) (s := fun R : ℝ ↦ Metric.ball c R) (a := r)
+    (fun R hR ↦ measurableSet_ball.nullMeasurableSet)
+    (fun R S hR hRS ↦ Metric.ball_subset_ball hRS)
+    ⟨r+1, by linarith, by
+      rw [perimeterMeasure_open F Metric.isOpen_ball]
+      exact (hF.2.1 c (r+1) (by linarith)).ne⟩
+  simpa only [Metric.biInter_gt_ball, Function.comp_def,
+    perimeterMeasure_open F Metric.isOpen_ball] using ht
+
+theorem tendsto_minimizing_perimeter_ball_of_local_L1 {n : ℕ} (hn : 0 < n)
+    {E : ℕ → Set (Euc n)} {F : Set (Euc n)}
+    (hE : ∀ i, IsLocallyPerimeterMinimizing (E i)) (hF : MeasurableSet F)
+    (hconv : ∀ (c : Euc n) (R : ℝ), 0 < R →
+      Tendsto (fun i ↦ volume.real (symmDiff (E i) F ∩ Metric.closedBall c R)) atTop (𝓝 0))
+    (c : Euc n) {r : ℝ} (hr : 0 < r)
+    (hnull : perimeterMeasure F (Metric.sphere c r) = 0) :
+    Tendsto (fun i ↦ perimeter (E i) (Metric.ball c r)) atTop
+      (𝓝 (perimeter F (Metric.ball c r))) := by
+  have hmin := locally_minimizing_of_local_L1_limit hn hE hF hconv
+  apply tendsto_order.2
+  constructor
+  · intro A hA
+    rw [perimeter_eq_iSup_tests, lt_iSup_iff] at hA
+    obtain ⟨V, hV⟩ := hA
+    rw [lt_iSup_iff] at hV
+    obtain ⟨hV, hAV⟩ := hV
+    have hc := hV.2.1.of_isClosed_subset (isClosed_tsupport _) (tsupport_divergence_subset V)
+    have ht := ENNReal.continuous_ofReal.tendsto _ |>.comp
+      (tendsto_setIntegral_of_local_symmDiff (fun i ↦ (hE i).1) hF hconv
+        (continuous_divergence hV.1) hc)
+    filter_upwards [ht.eventually (lt_mem_nhds hAV)] with i hi
+    apply hi.trans_le
+    rw [perimeter_eq_iSup_tests]
+    exact le_iSup_of_le V (le_iSup_of_le hV le_rfl)
+  · intro A hA
+    have heq : perimeterMeasure F (Metric.closedBall c r) = perimeter F (Metric.ball c r) := by
+      rw [← perimeterMeasure_open F Metric.isOpen_ball]
+      apply le_antisymm
+      · rw [← Metric.ball_union_sphere]
+        exact (measure_union_le _ _).trans (by rw [hnull, add_zero])
+      · exact measure_mono Metric.ball_subset_closedBall
+    have ht := tendsto_perimeter_ball_right hmin c hr
+    rw [heq] at ht
+    have hh : ∀ᶠ B in 𝓝[>] r, r < B ∧ perimeter F (Metric.ball c B) < A := by
+      filter_upwards [self_mem_nhdsWithin, ht.eventually (gt_mem_nhds hA)] with B hB hBA
+      exact ⟨hB, hBA⟩
+    obtain ⟨B, hBr, hBA⟩ := hh.exists
+    obtain ⟨ε, hε, hεA⟩ := ENNReal.lt_iff_exists_add_pos_lt.mp hBA
+    have hann : Tendsto (fun i ↦ volume.real (symmDiff (E i) F ∩
+        (Metric.closedBall c B \ Metric.closedBall c r))) atTop (𝓝 0) := by
+      exact squeeze_zero (fun _ ↦ ENNReal.toReal_nonneg)
+        (fun i ↦ measureReal_mono (inter_subset_inter_right _ sdiff_subset)
+          (measure_ne_top_of_subset inter_subset_right (isCompact_closedBall c B).measure_ne_top))
+        (hconv c B (hr.trans hBr))
+    filter_upwards [eventually_minimizing_perimeter_le_of_annular_L1 hE hF c hr hBr hann
+      (show 0 < (ε:ℝ) from hε)] with i hi
+    rw [ENNReal.ofReal_coe_nnreal] at hi
+    exact hi.trans_lt hεA
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/SingleGraph.lean b/lean/OAI/Geometry/Perimeter/Regularity/SingleGraph.lean
new file mode 100644
index 0000000000000000000000000000000000000000..60730e2b0901fb7ae6d60e14d0ab9b01aec973b5
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/SingleGraph.lean
@@ -0,0 +1,60 @@
+import OAI.Geometry.Perimeter.Regularity.GraphExistence
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+theorem fiber_chord_norm {v : Euc n} (hv : ‖v‖=1)
+    (y z : normalPlane v) (s t : ℝ) :
+    ‖((y:Euc n)+s•v)-((z:Euc n)+t•v)‖≤‖y-z‖+|s-t| := by
+  have he : ((y:Euc n)+s•v)-((z:Euc n)+t•v)=((y-z:normalPlane v):Euc n)+(s-t)•v := by
+    simp only [Submodule.coe_sub,sub_smul]; abel
+  rw [he]
+  simpa only [norm_smul,Real.norm_eq_abs,hv,mul_one,Submodule.norm_coe] using
+    norm_add_le ((y-z:normalPlane v):Euc n) ((s-t)•v)
+
+theorem single_graph_from_slab (hn : 1<n) {E : Set (Euc n)}
+    (hmin : IsLocallyPerimeterMinimizing E) {v : Euc n} (hv : ‖v‖=1)
+    {a : ℝ} (ha : 0≤a) (ha4 : a<1/4) (hs : FullSlab E v 8 a)
+    (hchord : ∀ p∈perimeterSupport E ∩ Metric.ball 0 1,
+      ∀ x∈perimeterSupport E ∩ Metric.ball 0 1,
+        |inner ℝ v (x-p)|≤(1/2)*‖x-p‖) :
+    ∃ f : normalPlane v → ℝ,
+      (∀ y, ‖y‖<1/4 → |f y|≤a ∧ (y:Euc n)+f y•v∈perimeterSupport E ∩ Metric.ball 0 1) ∧
+      (∀ y z, ‖y‖<1/4 → ‖z‖<1/4 → |f y-f z|≤‖y-z‖) ∧
+      (∀ (y : normalPlane v) (t : ℝ), ‖y‖<1/4 → ‖(y:Euc n)+t•v‖<1 →
+        ((y:Euc n)+t•v∈perimeterSupport E ↔ t=f y)) := by
+  have hex (y : normalPlane v) : ∃ t : ℝ, ‖y‖<1/4 →
+      |t|≤a ∧ (y:Euc n)+t•v∈perimeterSupport E ∩ Metric.ball 0 1 := by
+    by_cases hy : ‖y‖<1/4
+    · obtain ⟨t,ht⟩ := hs.central_fiber_meets_support hn hmin hv ha ha4 y hy
+      exact ⟨t,fun _ ↦ ht⟩
+    · exact ⟨0,fun h ↦ (hy h).elim⟩
+  choose f hf using hex
+  refine ⟨f,hf,?_,?_⟩
+  · intro y z hy hz
+    have hh := hchord _ (hf z hz).2 _ (hf y hy).2
+    rw [inner_sub_right,normal_fiber_inner hv,normal_fiber_inner hv] at hh
+    have hn := fiber_chord_norm hv y z (f y) (f z)
+    linarith
+  · intro y t hy hyt
+    constructor
+    · intro hsT
+      have hh := hchord _ (hf y hy).2 _ ⟨hsT,by simpa only [Metric.mem_ball,dist_zero_right] using hyt⟩
+      rw [inner_sub_right,normal_fiber_inner hv,normal_fiber_inner hv] at hh
+      have hn := fiber_chord_norm hv y y t (f y)
+      simp only [sub_self,norm_zero,zero_add] at hn
+      have hz : |t-f y|=0 := by linarith [abs_nonneg (t-f y)]
+      exact sub_eq_zero.mp (abs_eq_zero.mp hz)
+    · rintro rfl
+      exact (hf y hy).2.1
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/SlabContraction.lean b/lean/OAI/Geometry/Perimeter/Regularity/SlabContraction.lean
new file mode 100644
index 0000000000000000000000000000000000000000..297fafb623bf2532de517d4379f1ce3812fbb12d
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/SlabContraction.lean
@@ -0,0 +1,65 @@
+import OAI.Geometry.Perimeter.Regularity.UniformFlatness
+import OAI.Geometry.Perimeter.Regularity.PhaseRestoration
+import OAI.Geometry.Perimeter.Regularity.SlabScaling
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+theorem exists_slab_contraction (hn : 1 < n) {v₀ : Euc n} (hv₀ : ‖v₀‖=1)
+    (hH : IsLocallyPerimeterMinimizing {x : Euc n | inner ℝ x v₀ < 0}) :
+    ∃ q M δ : ℝ, 0 < q ∧ q < 1/64 ∧ 0≤M ∧ 0 < δ ∧ δ≤1 ∧
+      ∀ (E : Set (Euc n)), IsLocallyPerimeterMinimizing E →
+        (0:Euc n)∈perimeterSupport E → ∀ (v : Euc n), ‖v‖=1 →
+          ∀ a : ℝ, 0 < a → a < δ → FullSlab E v 8 a →
+            ∃ w : Euc n, ‖w‖=1 ∧ ‖w-v‖≤M*a ∧
+              FullSlab (blownUpSet E 0 q) w 8 (a/2) := by
+  obtain ⟨K,B,hK,hB,himp⟩ := exists_uniform_normal_height_improvement hn hv₀ hH
+  let r := min (1/16) (1/(32*(K+1)))
+  have hr : 0 < r := lt_min (by norm_num) (by positivity)
+  have hr16 : r≤1/16 := min_le_left _ _
+  have hr8 : r<1/8 := by linarith
+  have hKr : K*r≤1/16 := by
+    have hh : r≤1/(32*(K+1)) := min_le_right _ _
+    have hp : 0 < 32*(K+1) := by positivity
+    have hb := (le_div_iff₀ hp).mp hh
+    nlinarith
+  obtain ⟨d,hd,hdimp⟩ := himp r hr hr8
+  let δ := min d (min (r/16) (min 1 (1/(16*(B+1)))))
+  have hδ : 0 < δ := lt_min hd (lt_min (by positivity) (lt_min zero_lt_one (by positivity)))
+  have hδd : δ≤d := min_le_left _ _
+  have hδr : δ≤r/16 := (min_le_right _ _).trans (min_le_left _ _)
+  have hδ1 : δ≤1 := (min_le_right _ _).trans ((min_le_right _ _).trans (min_le_left _ _))
+  have hδB : δ≤1/(16*(B+1)) :=
+    (min_le_right _ _).trans ((min_le_right _ _).trans (min_le_right _ _))
+  refine ⟨r/8,2*B,δ,by positivity,by linarith,by positivity,hδ,hδ1,
+    fun E hmin hzero v hv a ha haδ hs ↦ ?_⟩
+  obtain ⟨w,hw,hwd,hheight⟩ := hdimp v hv E hmin hzero a ha (haδ.trans_le hδd) hs
+  have hab : a < r/8 := by linarith [haδ.trans_le hδr]
+  have ha1 : a < 1 := haδ.trans_le hδ1
+  have hdist : ‖w-v‖≤1/4 := by
+    have hh := (lt_div_iff₀ (by positivity : 0 < 16*(B+1))).mp (haδ.trans_le hδB)
+    nlinarith
+  have hb : 0≤a*r/16 := by positivity
+  have hbr : a*r/16<r/8 := by nlinarith
+  have hh : ∀ x∈perimeterSupport E ∩ Metric.ball 0 r, |inner ℝ w x|≤a*r/16 := by
+    intro x hx
+    apply (hheight x hx).trans
+    have hmul := mul_le_mul_of_nonneg_left hKr (mul_nonneg ha.le hr.le)
+    nlinarith
+  have hslab := hs.restore_phase hn hmin hv hr (by linarith) hab hb hbr hdist hh
+  have hscaled := hslab.dilate hmin.1 (by positivity : 0 < r/8)
+  have hR : r/(r/8)=8 := by field_simp
+  have hA : (a*r/16)/(r/8)=a/2 := by field_simp; ring
+  rw [hR,hA] at hscaled
+  exact ⟨w,hw,by nlinarith [hwd],hscaled⟩
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/SlabIteration.lean b/lean/OAI/Geometry/Perimeter/Regularity/SlabIteration.lean
new file mode 100644
index 0000000000000000000000000000000000000000..1107df2b0e8b056a3c6c31de95c2589526a7188d
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/SlabIteration.lean
@@ -0,0 +1,80 @@
+import OAI.Geometry.Perimeter.Regularity.FlatnessImprovement
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+@[simp] theorem blownUpSet_zero_one (E : Set (Euc n)) : blownUpSet E 0 1=E := by
+  ext x; simp [blownUpSet]
+
+theorem blownUpSet_zero_comp (E : Set (Euc n)) (s t : ℝ) :
+    blownUpSet (blownUpSet E 0 s) 0 t=blownUpSet E 0 (s*t) := by
+  ext x; simp [blownUpSet,smul_smul]
+
+def SlabContractionProperty (n : ℕ) (q M δ : ℝ) : Prop :=
+  ∀ (E : Set (Euc n)), IsLocallyPerimeterMinimizing E →
+    (0:Euc n)∈perimeterSupport E → ∀ (v : Euc n), ‖v‖=1 →
+      ∀ a : ℝ, 0 < a → a < δ → FullSlab E v 8 a →
+        ∃ w : Euc n, ‖w‖=1 ∧ ‖w-v‖≤M*a ∧ FullSlab (blownUpSet E 0 q) w 8 (a/2)
+
+theorem slab_normal_sequence {q M δ : ℝ} (hq : 0 < q)
+    (hstep : SlabContractionProperty n q M δ) {E : Set (Euc n)}
+    (hmin : IsLocallyPerimeterMinimizing E) (hzero : (0:Euc n)∈perimeterSupport E)
+    {v₀ : Euc n} (hv₀ : ‖v₀‖=1) {a : ℝ} (ha : 0 < a) (haδ : a < δ)
+    (hs : FullSlab E v₀ 8 a) :
+    ∃ v : ℕ → Euc n, v 0=v₀ ∧ (∀ j, ‖v j‖=1) ∧
+      (∀ j, FullSlab (blownUpSet E 0 (q^j)) (v j) 8 (a*(1/2)^j)) ∧
+      (∀ j, ‖v (j+1)-v j‖≤M*a*(1/2)^j) := by
+  let S (j : ℕ) := {v : Euc n // ‖v‖=1 ∧ FullSlab (blownUpSet E 0 (q^j)) v 8 (a*(1/2)^j)}
+  have hnext (j : ℕ) (s : S j) : ∃ t : S (j+1), ‖t.val-s.val‖≤M*a*(1/2)^j := by
+    have hminj := locally_minimizing_blownUpSet hmin 0 (pow_pos hq j)
+    have hzj : (0:Euc n)∈perimeterSupport (blownUpSet E 0 (q^j)) := by
+      apply (perimeterSupport_blownUpSet hmin.1 0 (pow_pos hq j) 0).mpr
+      simpa only [smul_zero,zero_add] using hzero
+    have haj : 0 < a*(1/2:ℝ)^j := mul_pos ha (pow_pos (by norm_num) _)
+    have hadj : a*(1/2:ℝ)^j < δ :=
+      (mul_le_of_le_one_right ha.le (pow_le_one₀ (by norm_num) (by norm_num))).trans_lt haδ
+    obtain ⟨w,hw,hwd,hws⟩ := hstep _ hminj hzj s.val s.property.1 _ haj hadj s.property.2
+    rw [blownUpSet_zero_comp,← pow_succ] at hws
+    have hpow : a*(1/2:ℝ)^j/2=a*(1/2:ℝ)^(j+1) := by rw [pow_succ]; ring
+    rw [hpow] at hws
+    exact ⟨⟨w,hw,hws⟩,by simpa only [mul_assoc] using hwd⟩
+  choose next hnext using hnext
+  let s₀ : S 0 := ⟨v₀,hv₀,by simpa only [pow_zero,mul_one,blownUpSet_zero_one] using hs⟩
+  let s : (j : ℕ) → S j := fun j ↦ Nat.rec s₀ (fun j t ↦ next j t) j
+  refine ⟨fun j ↦ (s j).val,rfl,fun j ↦ (s j).property.1,fun j ↦ (s j).property.2,?_⟩
+  intro j
+  exact hnext j (s j)
+
+theorem slab_normal_limit {q M δ : ℝ} (hq : 0 < q)
+    (hstep : SlabContractionProperty n q M δ) {E : Set (Euc n)}
+    (hmin : IsLocallyPerimeterMinimizing E) (hzero : (0:Euc n)∈perimeterSupport E)
+    {v₀ : Euc n} (hv₀ : ‖v₀‖=1) {a : ℝ} (ha : 0 < a) (haδ : a < δ)
+    (hs : FullSlab E v₀ 8 a) :
+    ∃ v : ℕ → Euc n, ∃ w : Euc n, v 0=v₀ ∧ (∀ j, ‖v j‖=1) ∧ ‖w‖=1 ∧
+      Tendsto v atTop (𝓝 w) ∧
+      (∀ j, FullSlab (blownUpSet E 0 (q^j)) (v j) 8 (a*(1/2)^j)) ∧
+      (∀ j, ‖v j-w‖≤2*M*a*(1/2)^j) := by
+  obtain ⟨v,hv0,hunit,hslab,hinc⟩ := slab_normal_sequence hq hstep hmin hzero hv₀ ha haδ hs
+  have hdist (j : ℕ) : dist (v j) (v (j+1))≤(M*a)*(1/2:ℝ)^j := by
+    rw [dist_eq_norm,norm_sub_rev]
+    exact hinc j
+  obtain ⟨w,hw⟩ := cauchySeq_tendsto_of_complete
+    (cauchySeq_of_le_geometric (1/2) (M*a) (by norm_num) hdist)
+  have hwn : ‖w‖=1 := tendsto_nhds_unique hw.norm (by simpa only [hunit] using
+    (tendsto_const_nhds : Tendsto (fun _ : ℕ ↦ (1:ℝ)) atTop (𝓝 1)))
+  refine ⟨v,w,hv0,hunit,hwn,hw,hslab,fun j ↦ ?_⟩
+  have hh := dist_le_of_le_geometric_of_tendsto (1/2) (M*a) (by norm_num) hdist hw j
+  rw [dist_eq_norm] at hh
+  convert hh using 1
+  ring
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/SlabScaling.lean b/lean/OAI/Geometry/Perimeter/Regularity/SlabScaling.lean
new file mode 100644
index 0000000000000000000000000000000000000000..1e0348d51f5bcbe244b44204ae8d92612d32f7a8
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/SlabScaling.lean
@@ -0,0 +1,91 @@
+import OAI.Geometry.Perimeter.Regularity.FlatnessImprovement
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+theorem FullSlab.mono_height {E : Set (Euc n)} {v : Euc n} {R a b : ℝ}
+    (h : FullSlab E v R a) (hab : a≤b) : FullSlab E v R b := by
+  refine ⟨fun x hx ↦ (h.1 x hx).trans hab,?_⟩
+  filter_upwards [h.2] with x hx
+  exact fun hball hheight ↦ hx hball (hab.trans_lt hheight)
+
+theorem FullSlab.dilate {E : Set (Euc n)} (hE : MeasurableSet E)
+    {v : Euc n} {R a t : ℝ} (h : FullSlab E v R a) (ht : 0 < t) :
+    FullSlab (blownUpSet E 0 t) v (R/t) (a/t) := by
+  have hball (x : Euc n) (hx : x∈Metric.ball 0 (R/t)) : t•x∈Metric.ball 0 R := by
+    simp only [Metric.mem_ball,dist_zero_right] at hx ⊢
+    rw [norm_smul,Real.norm_of_nonneg ht.le]
+    simpa only [mul_comm] using (lt_div_iff₀ ht).mp hx
+  refine ⟨?_,?_⟩
+  · intro x hx
+    have hs : t•x∈perimeterSupport E := by
+      simpa only [zero_add] using (perimeterSupport_blownUpSet hE 0 ht x).mp hx.1
+    have hh := h.1 (t•x) ⟨hs,hball x hx.2⟩
+    rw [real_inner_smul_right,abs_mul,abs_of_pos ht] at hh
+    exact (le_div_iff₀ ht).mpr (by simpa only [mul_comm] using hh)
+  · have hh := (Measure.quasiMeasurePreserving_smul volume ht.ne').ae h.2
+    filter_upwards [hh] with x hx
+    intro hxb hxa
+    have hb : a < |inner ℝ v (t•x)| := by
+      rw [real_inner_smul_right,abs_mul,abs_of_pos ht]
+      simpa only [mul_comm] using (div_lt_iff₀ ht).mp hxa
+    have he := hx (hball x hxb) hb
+    change (0+t•x∈E ↔ inner ℝ v x < 0)
+    rw [zero_add,he,real_inner_smul_right]
+    constructor
+    · intro hh; nlinarith
+    · exact fun hh ↦ mul_neg_of_pos_of_neg ht hh
+
+theorem FullSlab.recenter_half {E : Set (Euc n)} (hE : MeasurableSet E)
+    {v p : Euc n} {a : ℝ} (ha : 0≤a) (h : FullSlab E v 8 a)
+    (hp : p∈perimeterSupport E ∩ Metric.ball 0 1) :
+    FullSlab (blownUpSet E p (1/2)) v 8 (4*a) := by
+  have hhp : |inner ℝ v p|≤a := h.1 p ⟨hp.1,Metric.ball_subset_ball (by norm_num) hp.2⟩
+  have hball (x : Euc n) (hx : x∈Metric.ball 0 8) : p+(1/2:ℝ)•x∈Metric.ball 0 8 := by
+    have hnp : ‖p‖<1 := by simpa only [Metric.mem_ball,dist_zero_right] using hp.2
+    have hnx : ‖x‖<8 := by simpa only [Metric.mem_ball,dist_zero_right] using hx
+    have hh := norm_add_le p ((1/2:ℝ)•x)
+    rw [norm_smul,Real.norm_of_nonneg (by norm_num : (0:ℝ)≤1/2)] at hh
+    simp only [Metric.mem_ball,dist_zero_right]
+    linarith
+  refine ⟨?_,?_⟩
+  · intro x hx
+    have hs := (perimeterSupport_blownUpSet hE p (by norm_num : (0:ℝ)<1/2) x).mp hx.1
+    have hh := h.1 _ ⟨hs,hball x hx.2⟩
+    rw [inner_add_right,real_inner_smul_right] at hh
+    rcases abs_le.mp hh with ⟨hl,hu⟩
+    rcases abs_le.mp hhp with ⟨hpl,hpu⟩
+    apply abs_le.mpr
+    constructor <;> linarith
+  · have hq : Measure.QuasiMeasurePreserving (fun x : Euc n ↦ p+(1/2:ℝ)•x) volume volume :=
+      (measurePreserving_add_left volume p).quasiMeasurePreserving.comp
+        (Measure.quasiMeasurePreserving_smul volume (by norm_num : (1/2:ℝ)≠0))
+    filter_upwards [hq.ae h.2] with x hx
+    intro hxb hxa
+    rcases abs_le.mp hhp with ⟨hpl,hpu⟩
+    have hi : inner ℝ v (p+(1/2:ℝ)•x)=inner ℝ v p+1/2*inner ℝ v x := by
+      rw [inner_add_right,real_inner_smul_right]
+    have hbig : a < |inner ℝ v (p+(1/2:ℝ)•x)| := by
+      rw [hi]
+      rcases lt_abs.mp hxa with hh | hh
+      · exact (lt_of_lt_of_le (by linarith) (le_abs_self _))
+      · exact (lt_of_lt_of_le (by linarith) (neg_le_abs _))
+    have hsign : inner ℝ v (p+(1/2:ℝ)•x)<0 ↔ inner ℝ v x<0 := by
+      rw [hi]
+      rcases lt_abs.mp hxa with hh | hh
+      · have hpos : 0 < inner ℝ v x := by linarith
+        constructor <;> intro hneg <;> linarith
+      · have hneg : inner ℝ v x < 0 := by linarith
+        constructor <;> intro hneg' <;> linarith
+    exact (hx (hball x hxb) hbig).trans hsign
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/SmoothFlatOrder.lean b/lean/OAI/Geometry/Perimeter/Regularity/SmoothFlatOrder.lean
new file mode 100644
index 0000000000000000000000000000000000000000..66cb84808af78dfffdd69ecc66fab7ebca53e018
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/SmoothFlatOrder.lean
@@ -0,0 +1,540 @@
+import OAI.Geometry.Perimeter.Regularity.TotallyBoundedRangeOfUniformApproximation
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+namespace RPISingularHarnack
+
+theorem smooth_flat_order {n : ℕ} {u : Euc n → ℝ}
+    (hu : ContDiff ℝ 1 u) {v : Euc n} (hv : ‖v‖ = 1)
+    {R : ℝ} (hz : ∀ z ∈ Metric.ball (0 : Euc n) R,
+      ∀ w, inner ℝ w v = 0 → fderiv ℝ u z w = 0)
+    (hneg : ∀ z ∈ Metric.ball (0 : Euc n) R, fderiv ℝ u z v ≤ 0)
+    {x y : Euc n} (hx : x ∈ Metric.ball (0 : Euc n) R)
+    (hy : y ∈ Metric.ball (0 : Euc n) R)
+    (hxy : inner ℝ x v ≤ inner ℝ y v) : u y ≤ u x := by
+  let w := y - x
+  let a := inner ℝ w v
+  have ha : 0 ≤ a := by dsimp [a, w]; rw [inner_sub_left]; linarith
+  have horth : inner ℝ (w - a • v) v = 0 := by
+    simp [inner_sub_left, inner_smul_left, hv, a]
+  have hdir (z : Euc n) (hzball : z ∈ Metric.ball (0 : Euc n) R) :
+      fderiv ℝ u z w ≤ 0 := by
+    have heq : w = (w - a • v) + a • v := by abel
+    rw [heq, map_add, map_smul, hz z hzball (w - a • v) horth, zero_add,
+      smul_eq_mul]
+    exact mul_nonpos_of_nonneg_of_nonpos ha (hneg z hzball)
+  let f : ℝ → ℝ := fun t ↦ u (x + t • w)
+  have hd (t : ℝ) : HasDerivAt f (fderiv ℝ u (x + t • w) w) t := by
+    simpa only [f, Function.comp_def, id_eq, one_smul] using
+      (hu.differentiable one_ne_zero (x + t • w)).hasFDerivAt.comp_hasDerivAt t
+        (HasDerivAt.const_add x ((hasDerivAt_id t).smul_const w))
+  have hm : AntitoneOn f (Icc (0 : ℝ) 1) := by
+    apply antitoneOn_of_hasDerivWithinAt_nonpos (convex_Icc _ _)
+      (hu.continuous.comp (continuous_const.add (continuous_id.smul continuous_const))).continuousOn
+    · intro t ht
+      exact (hd t).hasDerivWithinAt
+    · intro t ht
+      apply hdir
+      exact (convex_ball (0 : Euc n) R).add_smul_sub_mem hx hy (interior_subset ht)
+  have he := hm (show (0 : ℝ) ∈ Icc 0 1 by norm_num)
+    (show (1 : ℝ) ∈ Icc 0 1 by norm_num) (by norm_num)
+  simpa [f, w] using he
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+namespace RPISingularHarnack
+open RPI.Boundary ContinuousLinearMap
+open scoped Convolution
+
+theorem indicator_convolution_fderiv_eq {n : ℕ} {E : Set (Euc n)}
+    (hE : MeasurableSet E) {k : Euc n → ℝ} (hk : ContDiff ℝ ∞ k)
+    (hck : HasCompactSupport k) (x v : Euc n) :
+    fderiv ℝ (E.indicator (fun _ ↦ (1 : ℝ)) ⋆[lsmul ℝ ℝ, volume] k) x v =
+      -(∫ y in E, fderiv ℝ (fun z ↦ k (x - z)) y v) := by
+  let χ : Euc n → ℝ := E.indicator (fun _ ↦ 1)
+  have hχ : LocallyIntegrable χ volume := indicator_one_locallyIntegrable hE
+  rw [(hck.hasFDerivAt_convolution_right (lsmul ℝ ℝ) hχ (hk.of_le (by simp)) x).fderiv]
+  rw [convolution_precompR_apply (lsmul ℝ ℝ) hχ (hck.fderiv ℝ)
+    (hk.continuous_fderiv (by simp))]
+  have hd (y : Euc n) : fderiv ℝ (fun z ↦ k (x - z)) y v =
+      -fderiv ℝ k (x - y) v := by
+    have h := ((hk.differentiable (by simp) (x - y)).hasFDerivAt.comp y
+      ((hasFDerivAt_const x y).sub (hasFDerivAt_id y))).fderiv
+    simpa only [Function.comp_def, sub_apply, zero_apply, id_apply,
+      zero_sub, ContinuousLinearMap.comp_apply, neg_apply, map_neg] using
+      congrArg (fun L : Euc n →L[ℝ] ℝ ↦ L v) h
+  simp_rw [hd]
+  rw [integral_neg, neg_neg]
+  change (∫ t, χ t * fderiv ℝ k (x - t) v) = _
+  have heq : (fun t ↦ χ t * fderiv ℝ k (x - t) v) =
+      E.indicator (fun t ↦ fderiv ℝ k (x - t) v) := by
+    funext t
+    by_cases ht : t ∈ E <;> simp [χ, ht]
+  rw [heq, integral_indicator hE]
+
+theorem halfspace_of_flat_distribution {n : ℕ} {E : Set (Euc n)}
+    (hE : MeasurableSet E) {R : ℝ} (hR : 0 < R) {v : Euc n} (hv : ‖v‖ = 1)
+    (hz : ∀ φ : Euc n → ℝ, ContDiff ℝ ∞ φ → HasCompactSupport φ →
+      tsupport φ ⊆ Metric.ball (0 : Euc n) R →
+      ∀ w, inner ℝ w v = 0 → (∫ x in E, fderiv ℝ φ x w) = 0)
+    (hp : ∀ φ : Euc n → ℝ, ContDiff ℝ ∞ φ → HasCompactSupport φ →
+      tsupport φ ⊆ Metric.ball (0 : Euc n) R → (∀ x, 0 ≤ φ x) →
+      0 ≤ ∫ x in E, fderiv ℝ φ x v) :
+    ∃ t ∈ Icc (-R / 2) (R / 2),
+      ∀ᵐ x ∂volume.restrict (Metric.ball (0 : Euc n) (R / 2)),
+        x ∈ E ↔ inner ℝ x v < t := by
+  let χ : Euc n → ℝ := E.indicator (fun _ ↦ 1)
+  let u : ℕ → Euc n → ℝ := c1Mollify χ
+  have hχ : LocallyIntegrable χ volume := indicator_one_locallyIntegrable hE
+  have hflip : (lsmul ℝ ℝ : ℝ →L[ℝ] ℝ →L[ℝ] ℝ).flip = lsmul ℝ ℝ := by
+    apply ContinuousLinearMap.ext
+    intro a
+    apply ContinuousLinearMap.ext
+    intro b
+    exact mul_comm b a
+  have hu_eq : ∀ j, u j = χ ⋆[lsmul ℝ ℝ, volume] (c1Mollifier j).normed volume := by
+    intro j
+    change (c1Mollifier j).normed volume ⋆[lsmul ℝ ℝ, volume] χ = _
+    rw [← convolution_flip, hflip]
+  have hu : ∀ j, ContDiff ℝ ∞ (u j) := fun j ↦
+    (c1Mollifier j).hasCompactSupport_normed.contDiff_convolution_left _
+      (c1Mollifier j).contDiff_normed hχ
+  have horder : ∀ᶠ j in atTop, ∀ x ∈ Metric.ball (0 : Euc n) (R / 2),
+      ∀ y ∈ Metric.ball (0 : Euc n) (R / 2),
+        inner ℝ x v ≤ inner ℝ y v → u j y ≤ u j x := by
+    have hsmall : ∀ᶠ j in atTop, (c1Mollifier (n := n) j).rOut < R / 2 :=
+      c1Mollifier_radius_tendsto.eventually (gt_mem_nhds (by positivity))
+    filter_upwards [hsmall] with j hj
+    have hs (x : Euc n) (hx : x ∈ Metric.ball (0 : Euc n) (R / 2)) :
+        tsupport (fun z ↦ (c1Mollifier (n := n) j).normed volume (x - z)) ⊆
+          Metric.ball (0 : Euc n) R := by
+      intro y hy
+      have hk := (tsupport_comp_subset_preimage ((c1Mollifier j).normed volume)
+        (continuous_const.sub continuous_id)) hy
+      rw [(c1Mollifier j).tsupport_normed_eq] at hk
+      change x - y ∈ Metric.closedBall (0 : Euc n) (c1Mollifier j).rOut at hk
+      have hxy : dist y x ≤ (c1Mollifier (n := n) j).rOut := by
+        simpa only [Metric.mem_closedBall, dist_zero_right, dist_eq_norm, sub_zero, norm_sub_rev] using hk
+      exact (dist_triangle y x 0).trans_lt (by linarith [Metric.mem_ball.mp hx])
+    have hzero : ∀ z ∈ Metric.ball (0 : Euc n) (R / 2),
+        ∀ w, inner ℝ w v = 0 → fderiv ℝ (u j) z w = 0 := by
+      intro z hzball w hw
+      rw [hu_eq j, indicator_convolution_fderiv_eq hE
+        (c1Mollifier j).contDiff_normed (c1Mollifier j).hasCompactSupport_normed]
+      apply neg_eq_zero.mpr
+      exact hz _ ((c1Mollifier j).contDiff_normed.comp (contDiff_const.sub contDiff_id))
+        ((c1Mollifier j).hasCompactSupport_normed.comp_homeomorph (Homeomorph.subLeft z))
+        (hs z hzball) w hw
+    have hnonpos : ∀ z ∈ Metric.ball (0 : Euc n) (R / 2), fderiv ℝ (u j) z v ≤ 0 := by
+      intro z hzball
+      rw [hu_eq j, indicator_convolution_fderiv_eq hE
+        (c1Mollifier j).contDiff_normed (c1Mollifier j).hasCompactSupport_normed]
+      apply neg_nonpos.mpr
+      exact hp _ ((c1Mollifier j).contDiff_normed.comp (contDiff_const.sub contDiff_id))
+        ((c1Mollifier j).hasCompactSupport_normed.comp_homeomorph (Homeomorph.subLeft z))
+        (hs z hzball) (fun y ↦ (c1Mollifier j).nonneg_normed (z - y))
+    intro x hx y hy hxy
+    exact smooth_flat_order ((hu j).of_le (by simp)) hv hzero hnonpos hx hy hxy
+  have hlim : ∀ᵐ x ∂volume, Tendsto (fun j ↦ u j x) atTop (𝓝 (χ x)) := by
+    apply ContDiffBump.ae_convolution_tendsto_right_of_locallyIntegrable
+      c1Mollifier_radius_tendsto (K := 2) _ hχ
+    apply Eventually.of_forall
+    intro j
+    dsimp [c1Mollifier]
+    linarith
+  let S := {x | Tendsto (fun j ↦ u j x) atTop (𝓝 (χ x))}
+  have hS : ∀ᵐ x ∂volume, x ∈ S := hlim
+  have hd : ∀ x ∈ S ∩ Metric.ball (0 : Euc n) (R / 2),
+      ∀ y ∈ S ∩ Metric.ball (0 : Euc n) (R / 2),
+        inner ℝ x v < inner ℝ y v → y ∈ E → x ∈ E := by
+    intro x hx y hy hxy hyE
+    have hle : χ y ≤ χ x := le_of_tendsto_of_tendsto hy.1 hx.1
+      (horder.mono fun j hj ↦ hj x hx.2 y hy.2 hxy.le)
+    by_contra hxE
+    norm_num [χ, hxE, hyE] at hle
+  simpa only [neg_div] using ordered_indicator_halfspace E S hS hv (by positivity : 0 < R / 2) hd
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+namespace RPISingularHarnack
+
+theorem halfspace_threshold_eq_zero {n : ℕ} {E : Set (Euc n)}
+    {R : ℝ} (hR : 0 < R) {v : Euc n} (hv : ‖v‖ = 1) {t : ℝ}
+    (hhalf : ∀ᵐ x ∂volume.restrict (Metric.ball (0 : Euc n) R),
+      x ∈ E ↔ inner ℝ x v < t)
+    (hphase : ∀ s : ℝ, 0 < s → s ≤ R →
+      0 < volume (E ∩ Metric.ball (0 : Euc n) s) ∧
+      0 < volume (Eᶜ ∩ Metric.ball (0 : Euc n) s)) : t = 0 := by
+  have ha := (ae_restrict_iff' measurableSet_ball).mp hhalf
+  have hinner (x : Euc n) {s : ℝ} (hx : x ∈ Metric.ball (0 : Euc n) s) :
+      |inner ℝ x v| < s := by
+    apply (abs_real_inner_le_norm x v).trans_lt
+    simpa only [hv, mul_one, Metric.mem_ball, dist_zero_right] using hx
+  rcases lt_trichotomy t 0 with ht | ht | ht
+  · let s := min R (-t) / 2
+    have hs : 0 < s := by
+      dsimp [s]
+      exact div_pos (lt_min hR (neg_pos.mpr ht)) (by norm_num)
+    have hsR : s ≤ R := by dsimp [s]; linarith [min_le_left R (-t)]
+    have hst : s < -t := by dsimp [s]; linarith [min_le_right R (-t), lt_min hR (neg_pos.mpr ht)]
+    have hno : ∀ᵐ x ∂volume, x ∉ E ∩ Metric.ball (0 : Euc n) s := by
+      filter_upwards [ha] with x hx
+      intro hxin
+      have hxR := Metric.ball_subset_ball hsR hxin.2
+      have hxt := (hx hxR).mp hxin.1
+      have hb := (abs_lt.mp (hinner x hxin.2)).1
+      linarith
+    have hz : volume (E ∩ Metric.ball (0 : Euc n) s) = 0 := by
+      simpa only [not_not, Set.ofPred_mem_eq] using ae_iff.mp hno
+    exact False.elim (ne_of_gt (hphase s hs hsR).1 hz)
+  · exact ht
+  · let s := min R t / 2
+    have hs : 0 < s := by dsimp [s]; positivity
+    have hsR : s ≤ R := by dsimp [s]; linarith [min_le_left R t]
+    have hst : s < t := by dsimp [s]; linarith [min_le_right R t, lt_min hR ht]
+    have hno : ∀ᵐ x ∂volume, x ∉ Eᶜ ∩ Metric.ball (0 : Euc n) s := by
+      filter_upwards [ha] with x hx
+      intro hxin
+      have hxR := Metric.ball_subset_ball hsR hxin.2
+      have hxt : inner ℝ x v < t := (abs_lt.mp (hinner x hxin.2)).2.trans hst
+      exact hxin.1 ((hx hxR).mpr hxt)
+    have hz : volume (Eᶜ ∩ Metric.ball (0 : Euc n) s) = 0 := by
+      simpa only [not_not, Set.ofPred_mem_eq] using ae_iff.mp hno
+    exact False.elim (ne_of_gt (hphase s hs hsR).2 hz)
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+namespace RPISingularHarnack
+
+def blownUpSet {n : ℕ} (E : Set (Euc n)) (c : Euc n) (r : ℝ) : Set (Euc n) :=
+  (fun z ↦ c + r • z) ⁻¹' E
+
+theorem measurableSet_blownUpSet {n : ℕ} {E : Set (Euc n)} (hE : MeasurableSet E)
+    (c : Euc n) (r : ℝ) : MeasurableSet (blownUpSet E c r) :=
+  hE.preimage (by fun_prop)
+
+theorem integral_blownUpSet {n : ℕ} {E : Set (Euc n)} (hE : MeasurableSet E)
+    (g : Euc n → ℝ) (c : Euc n) {r : ℝ} (hr : 0 < r) :
+    (∫ z in blownUpSet E c r, g (c + r • z)) = (r ^ n)⁻¹ * ∫ y in E, g y := by
+  have hc := Measure.integral_comp_smul_of_nonneg (volume : Measure (Euc n))
+    (fun y ↦ E.indicator g (c + y)) r (hR := hr.le)
+  simp only [finrank_euclideanSpace, Fintype.card_fin, smul_eq_mul,
+    integral_add_left_eq_self, integral_indicator hE] at hc
+  rw [← hc, ← integral_indicator (measurableSet_blownUpSet hE c r)]
+  apply integral_congr_ae
+  exact ae_of_all _ fun z ↦ by
+    by_cases hz : c + r • z ∈ E <;> simp [blownUpSet, hz]
+
+theorem divergence_comp_blowdown {n : ℕ} {V : Euc n → Euc n}
+    (hV : Differentiable ℝ V) (c : Euc n) (r : ℝ) (y : Euc n) :
+    divergence (fun z ↦ V (r⁻¹ • (z - c))) y =
+      r⁻¹ * divergence V (r⁻¹ • (y - c)) := by
+  have hd := ((hV _).hasFDerivAt.comp y
+    (((hasFDerivAt_id y).sub_const c).const_smul r⁻¹)).fderiv
+  simp only [Function.comp_def, id_eq, Pi.smul_apply] at hd
+  simp only [divergence, hd, ContinuousLinearMap.comp_apply,
+    smul_apply, ContinuousLinearMap.id_apply, map_smul,
+    inner_smul_right, ← Finset.mul_sum]
+
+theorem perimeter_blownUpSet_le {n : ℕ} {E : Set (Euc n)} (hE : MeasurableSet E)
+    (c : Euc n) {r R : ℝ} (hr : 0 < r)
+    (hP : perimeter E (Metric.ball c (r * R)) ≠ ⊤) :
+    perimeter (blownUpSet E c r) (Metric.ball (0 : Euc n) R) ≤
+      ENNReal.ofReal (r * (r ^ n)⁻¹ * (perimeter E (Metric.ball c (r * R))).toReal) := by
+  rw [perimeter_eq_iSup_tests]
+  refine iSup_le fun V ↦ iSup_le fun hV ↦ ?_
+  let W : Euc n → Euc n := fun y ↦ V (r⁻¹ • (y - c))
+  have hW : PerimeterAdmissible (Metric.ball c (r * R)) W := by
+    refine ⟨hV.1.comp ((contDiff_id.sub contDiff_const).const_smul r⁻¹), ?_, ?_, ?_⟩
+    · exact hV.2.1.comp_homeomorph
+        ((Homeomorph.subRight c).trans (Homeomorph.smulOfNeZero r⁻¹ (inv_ne_zero hr.ne')))
+    · intro y hy
+      have hys := tsupport_comp_subset_preimage V
+        ((continuous_id.sub continuous_const).const_smul r⁻¹) hy
+      have hyb := hV.2.2.1 hys
+      change r⁻¹ • (y - c) ∈ Metric.ball (0 : Euc n) R at hyb
+      simp only [Metric.mem_ball, dist_zero_right, norm_smul,
+        Real.norm_of_nonneg (inv_nonneg.mpr hr.le)] at hyb
+      change dist y c < r * R
+      rw [dist_eq_norm]
+      have he : r * (r⁻¹ * ‖y - c‖) = ‖y - c‖ := by field_simp
+      linarith [mul_lt_mul_of_pos_left hyb hr]
+    · exact fun y ↦ hV.2.2.2 _
+  have he (z : Euc n) : divergence W (c + r • z) = r⁻¹ * divergence V z := by
+    dsimp [W]
+    rw [divergence_comp_blowdown (hV.1.differentiable (by simp))]
+    simp [inv_smul_smul₀ hr.ne']
+  have hint := integral_blownUpSet hE (divergence W) c hr
+  simp_rw [he] at hint
+  rw [integral_const_mul] at hint
+  have hint' : (∫ z in blownUpSet E c r, divergence V z) =
+      (r * (r ^ n)⁻¹) * ∫ y in E, divergence W y := by
+    have heq := congrArg (fun a : ℝ ↦ r * a) hint
+    have hrinv : r * r⁻¹ = 1 := mul_inv_cancel₀ hr.ne'
+    simpa only [← mul_assoc, hrinv, one_mul] using heq
+  rw [hint']
+  apply ENNReal.ofReal_le_ofReal
+  exact mul_le_mul_of_nonneg_left (integral_divergence_le_perimeter_real E _ hP hW)
+    (by positivity)
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+namespace RPISingularHarnack
+
+theorem exists_fixed_annular_bv_cut {n : ℕ} :
+    ∃ D : ℝ≥0∞, D ≠ ⊤ ∧ ∀ E : Set (Euc n), MeasurableSet E →
+      perimeter E (Metric.ball (0 : Euc n) 4) ≠ ⊤ →
+      ∃ r ∈ Ico (2 : ℝ) 3,
+        perimeter (E ∩ Metric.closedBall (0 : Euc n) r) univ ≤
+          perimeter E (Metric.ball (0 : Euc n) 4) + D := by
+  obtain ⟨L, hL, hbound⟩ := exists_smoothTransition_deriv_bound
+  let D : ℝ := volume.real (Metric.closedBall (0 : Euc n) 3) + 1
+  have hD : 0 < D := by dsimp [D]; positivity
+  refine ⟨ENNReal.ofReal (8 * L * D), ENNReal.ofReal_ne_top, ?_⟩
+  intro E hE hP
+  have hvol : volume.real (E ∩ Metric.closedBall (0 : Euc n) 3) < D * (3 - 2) := by
+    have hm := measureReal_mono (μ := (volume : Measure (Euc n)))
+      (s₁ := E ∩ Metric.closedBall (0 : Euc n) 3) inter_subset_right
+      (isCompact_closedBall (0 : Euc n) 3).measure_ne_top
+    dsimp [D]
+    linarith
+  obtain ⟨r, hr, hc⟩ := exists_controlled_cut_radius E (0 : Euc n)
+    (by norm_num : (0 : ℝ) < 2) (by norm_num : (2 : ℝ) < 3) hvol
+  refine ⟨r, hr, ?_⟩
+  have hcut := controlled_radius_perimeter_cut hE (0 : Euc n)
+    (by linarith [hr.1] : 0 < r) hr.2
+    (Metric.closedBall_subset_ball (by norm_num : (3 : ℝ) < 4))
+    hc hD.le hL hbound hP
+  exact (le_add_right le_rfl).trans hcut
+
+theorem exists_local_indicator_L1_subsequence {n : ℕ} (C : ℝ≥0∞) (hC : C ≠ ⊤)
+    (E : ℕ → Set (Euc n)) (hE : ∀ i, MeasurableSet (E i))
+    (hP : ∀ i, perimeter (E i) (Metric.ball (0 : Euc n) 4) ≤ C) :
+    ∃ F : Set (Euc n), MeasurableSet F ∧ F ⊆ Metric.closedBall (0 : Euc n) 3 ∧
+      ∃ A : ℕ → Set (Euc n), (∀ i, MeasurableSet (A i)) ∧
+        (∀ i, A i ∩ Metric.ball (0 : Euc n) 2 = E i ∩ Metric.ball (0 : Euc n) 2) ∧
+        ∃ σ : ℕ → ℕ, StrictMono σ ∧
+          Tendsto (fun i ↦ ∫⁻ x, ‖(A (σ i)).indicator (fun _ ↦ (1 : ℝ)) x -
+            F.indicator (fun _ ↦ (1 : ℝ)) x‖ₑ) atTop (𝓝 0) := by
+  obtain ⟨D, hD, hcut⟩ := exists_fixed_annular_bv_cut (n := n)
+  choose r hr hpr using fun i ↦ hcut (E i) (hE i) (ne_top_of_le_ne_top hC (hP i))
+  let A : ℕ → Set (Euc n) := fun i ↦ E i ∩ Metric.closedBall 0 (r i)
+  have hA : ∀ i, MeasurableSet (A i) := fun i ↦ (hE i).inter measurableSet_closedBall
+  have hAK : ∀ i, A i ⊆ Metric.closedBall (0 : Euc n) 3 := fun i ↦
+    inter_subset_right.trans (Metric.closedBall_subset_closedBall (hr i).2.le)
+  have hpA : ∀ i, perimeter (A i) univ ≤ C + D := fun i ↦
+    (hpr i).trans (add_le_add (hP i) le_rfl)
+  obtain ⟨F, hF, hFK, σ, hσ, ht⟩ := exists_indicator_L1_subsequence
+    (Metric.closedBall (0 : Euc n) 3) (isCompact_closedBall _ _) (C + D)
+      (ENNReal.add_ne_top.mpr ⟨hC, hD⟩) A hA hAK hpA
+  refine ⟨F, hF, hFK, A, hA, ?_, σ, hσ, ht⟩
+  intro i
+  ext x
+  constructor
+  · exact fun hx ↦ ⟨hx.1.1, hx.2⟩
+  · intro hx
+    exact ⟨⟨hx.1, (Metric.ball_subset_closedBall.trans
+      (Metric.closedBall_subset_closedBall (hr i).1)) hx.2⟩, hx.2⟩
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+namespace RPISingularHarnack
+
+theorem exists_blownUpSet_perimeter_bound {n : ℕ} (hn : 0 < n) :
+    ∃ C : ℝ≥0∞, C ≠ ⊤ ∧ ∀ E : Set (Euc n), IsLocallyPerimeterMinimizing E →
+      ∀ c (r : ℝ), 0 < r →
+        perimeter (blownUpSet E c r) (Metric.ball (0 : Euc n) 4) ≤ C := by
+  obtain ⟨C, hC, hgrowth⟩ := exists_locallyMinimizing_perimeter_growth_constant n hn
+  refine ⟨ENNReal.ofReal (C * 4 ^ (n - 1)), ENNReal.ofReal_ne_top, ?_⟩
+  intro E hmin c r hr
+  have hb := hgrowth E hmin c (r * 4) (by positivity)
+  have hfin : perimeter E (Metric.ball c (r * 4)) ≠ ⊤ := ne_top_of_le_ne_top ENNReal.ofReal_ne_top hb
+  have hreal : (perimeter E (Metric.ball c (r * 4))).toReal ≤ C * (r * 4) ^ (n - 1) := by
+    simpa only [ENNReal.toReal_ofReal (by positivity : 0 ≤ C * (r * 4) ^ (n - 1))] using
+      ENNReal.toReal_mono ENNReal.ofReal_ne_top hb
+  apply (perimeter_blownUpSet_le hmin.1 c hr hfin).trans
+  apply ENNReal.ofReal_le_ofReal
+  calc
+    r * (r ^ n)⁻¹ * (perimeter E (Metric.ball c (r * 4))).toReal ≤
+        r * (r ^ n)⁻¹ * (C * (r * 4) ^ (n - 1)) :=
+      mul_le_mul_of_nonneg_left hreal (by positivity)
+    _ = C * 4 ^ (n - 1) := by
+      have hn' : n = (n - 1) + 1 := (Nat.sub_add_cancel hn).symm
+      have he : r ^ n = r * r ^ (n - 1) := by conv_lhs => rw [hn']; rw [pow_succ'];
+      rw [he, mul_pow]
+      field_simp
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+namespace RPISingularHarnack
+
+theorem divergence_scalar_direction {n : ℕ} {φ : Euc n → ℝ}
+    (hφ : Differentiable ℝ φ) (w : Euc n) :
+    divergence (fun y ↦ φ y • w) = fun y ↦ fderiv ℝ φ y w := by
+  funext y
+  rw [divergence_smul hφ (differentiable_const w)]
+  simp only [divergence, fderiv_fun_const, Pi.zero_apply, zero_apply,
+    inner_zero_right, Finset.sum_const_zero, mul_zero, zero_add, inner_gradient_left]
+
+theorem integral_fderiv_blownUpSet {n : ℕ} {E : Set (Euc n)} (hE : MeasurableSet E)
+    {φ : Euc n → ℝ} (hφ : Differentiable ℝ φ) (c w : Euc n) {r : ℝ} (hr : 0 < r) :
+    (∫ z in blownUpSet E c r, fderiv ℝ φ z w) =
+      (r * (r ^ n)⁻¹) * ∫ y in E, fderiv ℝ (fun z ↦ φ (r⁻¹ • (z - c))) y w := by
+  have hd (y : Euc n) : fderiv ℝ (fun z ↦ φ (r⁻¹ • (z - c))) y w =
+      r⁻¹ * fderiv ℝ φ (r⁻¹ • (y - c)) w := by
+    have h := ((hφ _).hasFDerivAt.comp y
+      (((hasFDerivAt_id y).sub_const c).const_smul r⁻¹)).fderiv
+    simp only [Function.comp_def, id_eq, Pi.smul_apply] at h
+    rw [h, ContinuousLinearMap.comp_apply, smul_apply,
+      ContinuousLinearMap.id_apply, map_smul, smul_eq_mul]
+  have he (z : Euc n) : fderiv ℝ (fun y ↦ φ (r⁻¹ • (y - c))) (c + r • z) w =
+      r⁻¹ * fderiv ℝ φ z w := by rw [hd]; simp [inv_smul_smul₀ hr.ne']
+  have hint := integral_blownUpSet hE
+    (fun y ↦ fderiv ℝ (fun z ↦ φ (r⁻¹ • (z - c))) y w) c hr
+  simp_rw [he] at hint
+  rw [integral_const_mul] at hint
+  have heq := congrArg (fun a : ℝ ↦ r * a) hint
+  simpa only [← mul_assoc, mul_inv_cancel₀ hr.ne', one_mul] using heq
+
+theorem integral_fderiv_blownUpSet_polar {n : ℕ} {E U : Set (Euc n)}
+    (hE : MeasurableSet E) (μ : Measure (Euc n)) (G : Euc n → Euc n)
+    (hrep : ∀ V : Euc n → Euc n, ContDiff ℝ ∞ V → HasCompactSupport V →
+      tsupport V ⊆ U → (∫ y in E, divergence V y) = ∫ y, inner ℝ (G y) (V y) ∂μ)
+    {φ : Euc n → ℝ} (hφ : ContDiff ℝ ∞ φ) (hcφ : HasCompactSupport φ)
+    (c w : Euc n) {r : ℝ} (hr : 0 < r)
+    (hs : tsupport (fun y ↦ φ (r⁻¹ • (y - c))) ⊆ U) :
+    (∫ z in blownUpSet E c r, fderiv ℝ φ z w) =
+      (r * (r ^ n)⁻¹) * ∫ y, φ (r⁻¹ • (y - c)) * inner ℝ (G y) w ∂μ := by
+  let ψ : Euc n → ℝ := fun y ↦ φ (r⁻¹ • (y - c))
+  have hψ : ContDiff ℝ ∞ ψ := hφ.comp ((contDiff_id.sub contDiff_const).const_smul r⁻¹)
+  have hcψ : HasCompactSupport ψ := hcφ.comp_homeomorph
+    ((Homeomorph.subRight c).trans (Homeomorph.smulOfNeZero r⁻¹ (inv_ne_zero hr.ne')))
+  rw [integral_fderiv_blownUpSet hE (hφ.differentiable (by simp)) c w hr]
+  have he := hrep (fun y ↦ ψ y • w) (hψ.smul contDiff_const) hcψ.smul_right
+    ((tsupport_smul_subset_left ψ (fun _ ↦ w)).trans hs)
+  rw [divergence_scalar_direction (hψ.differentiable (by simp))] at he
+  simp only [inner_smul_right] at he
+  rw [he]
+
+theorem polar_test_oscillation_le {n : ℕ} (μ : Measure (Euc n)) [IsFiniteMeasure μ]
+    {G : Euc n → Euc n} (hG : Integrable G μ) (v w : Euc n)
+    {ψ : Euc n → ℝ} (hψ : Continuous ψ) {M : ℝ} (hM : 0 ≤ M)
+    (hb : ∀ y, ‖ψ y‖ ≤ M) (B : Set (Euc n)) (hB : MeasurableSet B)
+    (hs : ∀ y ∉ B, ψ y = 0) :
+    |(∫ y, ψ y * inner ℝ (G y) w ∂μ) -
+      (∫ y, ψ y * inner ℝ v w ∂μ)| ≤
+      M * ‖w‖ * ∫ y in B, ‖G y - v‖ ∂μ := by
+  have hiψ : Integrable ψ μ := Integrable.of_bound hψ.aestronglyMeasurable M (ae_of_all _ hb)
+  have hib : Integrable (fun y ↦ ψ y * inner ℝ (G y - v) w) μ := by
+    apply (hG.sub (integrable_const v)).norm.mul_const (M * ‖w‖) |>.mono'
+    · exact hψ.aestronglyMeasurable.mul
+        ((hG.aestronglyMeasurable.sub stronglyMeasurable_const.aestronglyMeasurable).inner
+          stronglyMeasurable_const.aestronglyMeasurable)
+    · filter_upwards [] with y
+      rw [norm_mul, Real.norm_eq_abs]
+      have hin := abs_real_inner_le_norm (G y - v) w
+      calc
+        ‖ψ y‖ * |inner ℝ (G y - v) w| ≤ M * (‖G y - v‖ * ‖w‖) :=
+          mul_le_mul (hb y) hin (abs_nonneg _) hM
+        _ = ‖G y - v‖ * (M * ‖w‖) := by ring
+  have hiG : Integrable (fun y ↦ ψ y * inner ℝ (G y) w) μ := by
+    have he : (fun y ↦ ψ y * inner ℝ (G y) w) =
+        (fun y ↦ ψ y * inner ℝ (G y - v) w) + (fun y ↦ ψ y * inner ℝ v w) := by
+      funext y
+      simp only [Pi.add_apply, inner_sub_left]
+      ring
+    rw [he]
+    exact hib.add (hiψ.mul_const _)
+  have heq : (∫ y, ψ y * inner ℝ (G y) w ∂μ) - (∫ y, ψ y * inner ℝ v w ∂μ) =
+      ∫ y in B, ψ y * inner ℝ (G y - v) w ∂μ := by
+    rw [← integral_sub hiG (hiψ.mul_const _)]
+    have he : (fun y ↦ ψ y * inner ℝ (G y) w - ψ y * inner ℝ v w) =
+        fun y ↦ ψ y * inner ℝ (G y - v) w := by funext y; rw [inner_sub_left, mul_sub]
+    rw [he]
+    rw [← integral_indicator hB]
+    apply integral_congr_ae
+    exact ae_of_all _ fun point ↦ by
+      by_cases hpoint : point ∈ B
+      · rw [indicator_of_mem hpoint]
+      · simp only [indicator_of_notMem hpoint, hs point hpoint, zero_mul]
+  rw [heq]
+  calc
+    _ ≤ ∫ y in B, |ψ y * inner ℝ (G y - v) w| ∂μ := abs_integral_le_integral_abs
+    _ ≤ ∫ y in B, M * ‖w‖ * ‖G y - v‖ ∂μ := by
+      apply integral_mono hib.integrableOn.norm
+        (((hG.sub (integrable_const v)).norm.const_mul (M * ‖w‖)).integrableOn)
+      intro y
+      change ‖ψ y * inner ℝ (G y - v) w‖ ≤ M * ‖w‖ * ‖G y - v‖
+      rw [norm_mul, Real.norm_eq_abs (inner ℝ (G y - v) w)]
+      calc
+        _ ≤ M * (‖G y - v‖ * ‖w‖) :=
+          mul_le_mul (hb y) (abs_real_inner_le_norm _ _) (abs_nonneg _) hM
+        _ = _ := by ring
+    _ = _ := integral_const_mul _ _
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/SmoothLayerSumBounds.lean b/lean/OAI/Geometry/Perimeter/Regularity/SmoothLayerSumBounds.lean
new file mode 100644
index 0000000000000000000000000000000000000000..a8e122ff3cf4978ca4f7c83a451bf92b10ffa8f0
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/SmoothLayerSumBounds.lean
@@ -0,0 +1,490 @@
+import OAI.Geometry.Perimeter.Regularity.PerimeterAdmissiblePartitionPiece
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+
+theorem smooth_layer_sum_bounds (N : ℕ) {a : ℝ} (ha : 0 ≤ a) (haN : a ≤ N) :
+    a - 1 ≤ ∑ j ∈ Finset.range N, Real.smoothTransition (a - j) ∧
+    (∑ j ∈ Finset.range N, Real.smoothTransition (a - j)) ≤ a + 1 := by
+  induction N generalizing a with
+  | zero => simp only [Nat.cast_zero] at haN; have : a = 0 := le_antisymm haN ha; simp [this]
+  | succ N ih =>
+    rw [Finset.sum_range_succ']
+    by_cases ha1 : 1 ≤ a
+    · have hrec := ih (a := a - 1) (by linarith) (by push_cast at haN; linarith)
+      have he : (∑ j ∈ Finset.range N, Real.smoothTransition (a - (j + 1 : ℕ))) =
+          ∑ j ∈ Finset.range N, Real.smoothTransition ((a - 1) - j) := by
+        apply Finset.sum_congr rfl
+        intro j hj
+        congr 1
+        push_cast
+        ring
+      rw [he, Nat.cast_zero, sub_zero, Real.smoothTransition.one_of_one_le ha1]
+      constructor <;> linarith [hrec.1, hrec.2]
+    · have ha1' : a ≤ 1 := le_of_not_ge ha1
+      have hz : ∑ j ∈ Finset.range N, Real.smoothTransition (a - (j + 1 : ℕ)) = 0 := by
+        apply Finset.sum_eq_zero
+        intro j hj
+        apply Real.smoothTransition.zero_of_nonpos
+        have hj0 : (0 : ℝ) ≤ j := Nat.cast_nonneg j
+        push_cast
+        linarith
+      rw [hz, zero_add, Nat.cast_zero, sub_zero]
+      constructor
+      · linarith [Real.smoothTransition.nonneg a]
+      · linarith [Real.smoothTransition.le_one a]
+
+theorem finite_level_count_le (N : ℕ) {a : ℝ} (ha : 0 ≤ a) :
+    (∑ j ∈ Finset.range N, if (j : ℝ) < a then (1 : ℝ) else 0) ≤ a + 1 := by
+  classical
+  induction N with
+  | zero => simp; linarith
+  | succ N ih =>
+    rw [Finset.sum_range_succ]
+    by_cases hN : (N : ℝ) < a
+    · rw [ite_eq_left hN]
+      have hsum : (∑ j ∈ Finset.range N, if (j : ℝ) < a then (1 : ℝ) else 0) ≤ N := by
+        calc
+          _ ≤ ∑ _j ∈ Finset.range N, (1 : ℝ) := by
+            apply Finset.sum_le_sum
+            intro j hj
+            split_ifs <;> norm_num
+          _ = _ := by simp
+      linarith
+    · simpa [hN] using ih
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+
+theorem abs_integral_divergence_le_bound_perimeter {n : ℕ} (E U : Set (Euc n))
+    (hP : perimeter E U ≠ ⊤) {V : Euc n → Euc n}
+    (hV : ContDiff ℝ ∞ V) (hc : HasCompactSupport V) (hs : tsupport V ⊆ U)
+    {B : ℝ} (hB : 0 < B) (hb : ∀ x, ‖V x‖ ≤ B) :
+    |∫ x in E, divergence V x| ≤ B * (perimeter E U).toReal := by
+  let W : Euc n → Euc n := fun x ↦ B⁻¹ • V x
+  have hW : PerimeterAdmissible U W := by
+    have hsm : ContDiff ℝ ∞ (fun z : Euc n ↦ B⁻¹ • z) :=
+      (contDiff_const (c := B⁻¹)).smul contDiff_id
+    refine ⟨hsm.comp hV, hc.comp_left (g := fun z : Euc n ↦ B⁻¹ • z)
+      (smul_zero _), ?_, ?_⟩
+    · exact (tsupport_smul_subset_right (fun _ ↦ B⁻¹) V).trans hs
+    · intro x
+      simp only [W, norm_smul, Real.norm_of_nonneg (inv_nonneg.mpr hB.le)]
+      exact (mul_le_mul_of_nonneg_left (hb x) (inv_nonneg.mpr hB.le)).trans_eq
+        (inv_mul_cancel₀ hB.ne')
+  have hp := integral_divergence_le_perimeter_real E U hP hW
+  have hn := integral_divergence_le_perimeter_real E U hP (perimeterAdmissible_neg hW)
+  have hd : divergence W = fun x ↦ B⁻¹ * divergence V x := by
+    funext x
+    rw [divergence_smul (differentiable_const _) (hV.differentiable (by simp))]
+    simp
+  rw [divergence_neg (hW.1.differentiable (by simp))] at hn
+  simp only [Pi.neg_apply] at hn
+  rw [integral_neg] at hn
+  rw [hd, integral_const_mul] at hp hn
+  have hab : |B⁻¹ * ∫ x in E, divergence V x| ≤ (perimeter E U).toReal :=
+    abs_le.mpr ⟨by linarith, hp⟩
+  calc
+    _ = B * |B⁻¹ * ∫ x in E, divergence V x| := by
+      rw [abs_mul, abs_inv, abs_of_pos hB, ← mul_assoc, mul_inv_cancel₀ hB.ne', one_mul]
+    _ ≤ _ := mul_le_mul_of_nonneg_left hab hB.le
+
+theorem scalar_layer_admissible {n : ℕ} {U : Set (Euc n)}
+    {Q : Euc n → Euc n} (hQ : PerimeterAdmissible U Q)
+    {f : Euc n → ℝ} (hf : ContDiff ℝ ∞ f) {δ : ℝ} (hδ : 0 < δ) (j : ℕ) :
+    PerimeterAdmissible (U ∩ {x | ((j : ℝ) - 1) * δ < f x})
+      (fun x ↦ Real.smoothTransition (f x / δ - j) • Q x) := by
+  let a : Euc n → ℝ := fun x ↦ Real.smoothTransition (f x / δ - j)
+  have ha : ContDiff ℝ ∞ a :=
+    Real.smoothTransition.contDiff.comp ((hf.div_const δ).sub contDiff_const)
+  have hsa : tsupport a ⊆ {x | (j : ℝ) * δ ≤ f x} := by
+    apply closure_minimal
+    · intro x hx
+      by_contra h
+      have hlt : f x < (j : ℝ) * δ := lt_of_not_ge h
+      have hd : f x / δ - j ≤ 0 := by
+        have := (div_lt_iff₀ hδ).mpr hlt
+        linarith
+      exact hx (Real.smoothTransition.zero_of_nonpos hd)
+    · exact isClosed_le continuous_const hf.continuous
+  refine ⟨ha.smul hQ.1, hQ.2.1.smul_left (f := a), ?_, ?_⟩
+  · intro x hx
+    refine ⟨hQ.2.2.1 (tsupport_smul_subset_right _ _ hx), ?_⟩
+    have h := hsa (tsupport_smul_subset_left _ _ hx)
+    change (j : ℝ) * δ ≤ f x at h
+    change ((j : ℝ) - 1) * δ < f x
+    nlinarith
+  · intro x
+    rw [norm_smul, Real.norm_eq_abs, abs_of_nonneg (Real.smoothTransition.nonneg _)]
+    exact (mul_le_of_le_one_left (norm_nonneg _) (Real.smoothTransition.le_one _)).trans
+      (hQ.2.2.2 x)
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+
+theorem finite_level_integral_le {n : ℕ} (μ : Measure (Euc n)) [IsFiniteMeasure μ]
+    {f : Euc n → ℝ} (hf : Measurable f) (hf0 : ∀ x, 0 ≤ f x) (hfi : Integrable f μ)
+    {δ : ℝ} (hδ : 0 < δ) (N : ℕ) :
+    δ * (∑ j ∈ Finset.range N, μ.real {x | ((j : ℝ) - 1) * δ < f x}) ≤
+      (∫ x, f x ∂μ) + 2 * δ * μ.real univ := by
+  classical
+  let s : ℕ → Set (Euc n) := fun j ↦ {x | ((j : ℝ) - 1) * δ < f x}
+  have hs : ∀ j, MeasurableSet (s j) := fun j ↦ measurableSet_lt measurable_const hf
+  let g : ℕ → Euc n → ℝ := fun j ↦ (s j).indicator (fun _ ↦ 1)
+  have hg : ∀ j, Integrable (g j) μ := fun j ↦ (integrable_const 1).indicator (hs j)
+  have hpoint : ∀ x, δ * ∑ j ∈ Finset.range N, g j x ≤ f x + 2 * δ := by
+    intro x
+    have hcount := finite_level_count_le N (a := f x / δ + 1) (add_nonneg (div_nonneg (hf0 x) hδ.le) zero_le_one)
+    have he : (∑ j ∈ Finset.range N, g j x) =
+        ∑ j ∈ Finset.range N, if (j : ℝ) < f x / δ + 1 then (1 : ℝ) else 0 := by
+      apply Finset.sum_congr rfl
+      intro j hj
+      have hc : ((j : ℝ) - 1) * δ < f x ↔ (j : ℝ) < f x / δ + 1 := by
+        have heq : f x / δ * δ = f x := div_mul_cancel₀ _ hδ.ne'
+        constructor <;> intro h <;> nlinarith
+      simp only [g, Set.indicator_apply, s, Set.mem_ofPred_eq, hc]
+    rw [he]
+    have hh := mul_le_mul_of_nonneg_left hcount hδ.le
+    have heq : δ * (f x / δ) = f x := mul_div_cancel₀ _ hδ.ne'
+    nlinarith
+  calc
+    _ = ∫ x, δ * ∑ j ∈ Finset.range N, g j x ∂μ := by
+      rw [integral_const_mul, integral_finsetSum _ (fun j _ ↦ hg j)]
+      congr 1
+      apply Finset.sum_congr rfl
+      intro j hj
+      simp only [g, integral_indicator (hs j), integral_const, smul_eq_mul, mul_one,
+        measureReal_restrict_apply_univ, s]
+    _ ≤ ∫ x, (f x + 2 * δ) ∂μ := by
+      exact integral_mono ((integrable_finsetSum _ (fun j _ ↦ hg j)).const_mul δ)
+        (hfi.add (integrable_const _)) hpoint
+    _ = _ := by rw [integral_add hfi (integrable_const _), integral_const]; simp [smul_eq_mul, mul_comm]
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+
+theorem integral_divergence_scalar_le {n : ℕ} (E U : Set (Euc n))
+    (hU : IsOpen U) (hP : perimeter E U ≠ ⊤)
+    {Q : Euc n → Euc n} (hQ : PerimeterAdmissible U Q)
+    {f : Euc n → ℝ} (hf : ContDiff ℝ ∞ f) (hf0 : ∀ x, 0 ≤ f x)
+    {M : ℝ} (hfM : ∀ x, f x ≤ M) :
+    (∫ x in E, divergence (fun y ↦ f y • Q y) x) ≤
+      ∫ x in U, f x ∂perimeterMeasure E := by
+  classical
+  let μ := (perimeterMeasure E).restrict U
+  have hμuniv : μ univ = perimeter E U := by
+    rw [Measure.restrict_apply_univ, perimeterMeasure_open E hU]
+  let : IsFiniteMeasure μ := ⟨by rw [hμuniv]; exact hP.lt_top⟩
+  have hμr : μ.real univ = (perimeter E U).toReal := by rw [measureReal_def, hμuniv]
+  have hfi : Integrable f μ := Integrable.of_bound hf.continuous.aestronglyMeasurable M
+    (ae_of_all _ (fun x ↦ by rw [Real.norm_eq_abs, abs_of_nonneg (hf0 x)]; exact hfM x))
+  let V : Euc n → Euc n := fun x ↦ f x • Q x
+  have hVc : ContDiff ℝ ∞ V := hf.smul hQ.1
+  have hVk : HasCompactSupport V := hQ.2.1.smul_left (f := f)
+  have hVs : tsupport V ⊆ U := (tsupport_smul_subset_right _ _).trans hQ.2.2.1
+  have hδbound : ∀ δ : ℝ, 0 < δ →
+      (∫ x in E, divergence V x) ≤ (∫ x, f x ∂μ) + 3 * δ * μ.real univ := by
+    intro δ hδ
+    obtain ⟨N, hN⟩ := exists_nat_ge (M / δ)
+    let s : ℕ → Set (Euc n) := fun j ↦ U ∩ {x | ((j : ℝ) - 1) * δ < f x}
+    have hs : ∀ j, IsOpen (s j) := fun j ↦ hU.inter (isOpen_lt continuous_const hf.continuous)
+    let W : ℕ → Euc n → Euc n := fun j x ↦ Real.smoothTransition (f x / δ - j) • Q x
+    have hW : ∀ j, PerimeterAdmissible (s j) (W j) := scalar_layer_admissible hQ hf hδ
+    let a : Euc n → ℝ := fun x ↦ δ * ∑ j ∈ Finset.range N, Real.smoothTransition (f x / δ - j)
+    have hac : ContDiff ℝ ∞ a := contDiff_const.mul (ContDiff.sum (fun j _ ↦
+      Real.smoothTransition.contDiff.comp ((hf.div_const δ).sub contDiff_const)))
+    let A : Euc n → Euc n := fun x ↦ a x • Q x
+    have hAc : ContDiff ℝ ∞ A := hac.smul hQ.1
+    have hAk : HasCompactSupport A := hQ.2.1.smul_left (f := a)
+    have hAs : tsupport A ⊆ U := (tsupport_smul_subset_right _ _).trans hQ.2.2.1
+    have har : ∀ x, |f x - a x| ≤ δ := by
+      intro x
+      have hh := smooth_layer_sum_bounds N (a := f x / δ) (div_nonneg (hf0 x) hδ.le)
+        ((div_le_div_of_nonneg_right (hfM x) hδ.le).trans hN)
+      have h1 := mul_le_mul_of_nonneg_left hh.1 hδ.le
+      have h2 := mul_le_mul_of_nonneg_left hh.2 hδ.le
+      have heq : δ * (f x / δ) = f x := by field_simp
+      dsimp [a]
+      exact abs_le.mpr ⟨by nlinarith, by nlinarith⟩
+    have hb : ∀ x, ‖(V - A) x‖ ≤ δ := by
+      intro x
+      change ‖f x • Q x - a x • Q x‖ ≤ δ
+      rw [← sub_smul, norm_smul, Real.norm_eq_abs]
+      exact (mul_le_mul (har x) (hQ.2.2.2 x) (norm_nonneg _) hδ.le).trans_eq (mul_one δ)
+    have her := abs_integral_divergence_le_bound_perimeter E U hP
+      (hVc.sub hAc) (hVk.sub hAk)
+      ((tsupport_sub V A).trans (union_subset hVs hAs)) hδ hb
+    change |∫ x in E, divergence (V - A) x| ≤ δ * (perimeter E U).toReal at her
+    rw [divergence_sub (hVc.differentiable (by simp)) (hAc.differentiable (by simp))] at her
+    simp only [Pi.sub_apply] at her
+    rw [integral_sub (integrable_divergence hVc hVk).integrableOn
+      (integrable_divergence hAc hAk).integrableOn] at her
+    have hAid : A = fun x ↦ δ • (∑ j ∈ Finset.range N, W j) x := by
+      funext x
+      simp only [A, a, W, Finset.sum_apply, ← Finset.sum_smul, smul_smul]
+    have hAdiv : divergence A = fun x ↦ δ * ∑ j ∈ Finset.range N, divergence (W j) x := by
+      rw [hAid]
+      funext x
+      rw [divergence_smul (differentiable_const δ)
+        (Differentiable.sum (fun j _ ↦ (hW j).1.differentiable (by simp)))]
+      simp only [gradient_fun_const, inner_zero_left, add_zero]
+      rw [divergence_finset_sum _ W (fun j _ ↦ (hW j).1.differentiable (by simp))]
+      simp only [Finset.sum_apply]
+    have hAint : (∫ x in E, divergence A x) =
+        δ * ∑ j ∈ Finset.range N, ∫ x in E, divergence (W j) x := by
+      rw [hAdiv, integral_const_mul, integral_finsetSum _
+        (fun j _ ↦ (integrable_divergence (hW j).1 (hW j).2.1).integrableOn)]
+    have hWbound : ∀ j, (∫ x in E, divergence (W j) x) ≤
+        μ.real {x | ((j : ℝ) - 1) * δ < f x} := by
+      intro j
+      have hPs : perimeter E (s j) ≠ ⊤ := ne_top_of_le_ne_top hP
+        (raw_perimeter_mono_region E inter_subset_left)
+      have hd := integral_divergence_le_perimeter_real E (s j) hPs (hW j)
+      convert hd using 1
+      rw [measureReal_def, Measure.restrict_apply
+        (measurableSet_lt measurable_const hf.continuous.measurable), inter_comm,
+        perimeterMeasure_open E (hs j)]
+    have hAupper : (∫ x in E, divergence A x) ≤ (∫ x, f x ∂μ) + 2 * δ * μ.real univ := by
+      rw [hAint]
+      exact (mul_le_mul_of_nonneg_left (Finset.sum_le_sum (fun j _ ↦ hWbound j)) hδ.le).trans
+        (finite_level_integral_le μ hf.continuous.measurable hf0 hfi hδ N)
+    rw [← hμr] at her
+    have hh := (le_abs_self _).trans her
+    linarith
+  apply le_of_forall_pos_le_add
+  intro ε hε
+  let δ := ε / (3 * μ.real univ + 1)
+  have hd : 0 < δ := by dsimp [δ]; positivity
+  have hh := hδbound δ hd
+  have hmul : δ * (3 * μ.real univ + 1) = ε := by
+    dsimp [δ]
+    exact div_mul_cancel₀ _ (by positivity)
+  change (∫ x in E, divergence V x) ≤ (∫ x, f x ∂μ) + ε
+  nlinarith
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+
+theorem abs_integral_divergence_le_integral_norm {n : ℕ} (E U : Set (Euc n))
+    (hU : IsOpen U) (hP : perimeter E U ≠ ⊤) {V : Euc n → Euc n}
+    (hV : ContDiff ℝ ∞ V) (hc : HasCompactSupport V) (hs : tsupport V ⊆ U) :
+    |∫ x in E, divergence V x| ≤ ∫ x in U, ‖V x‖ ∂perimeterMeasure E := by
+  classical
+  let μ := (perimeterMeasure E).restrict U
+  let : IsFiniteMeasure μ := ⟨by
+    rw [Measure.restrict_apply_univ, perimeterMeasure_open E hU]; exact hP.lt_top⟩
+  obtain ⟨B, hB⟩ := (hc.isCompact_range hV.continuous).isBounded.exists_norm_le
+  have hb : ∀ x, ‖V x‖ ≤ B := fun x ↦ hB _ (mem_range_self x)
+  have hNi : Integrable (fun x ↦ ‖V x‖) μ :=
+    Integrable.of_bound hV.continuous.norm.aestronglyMeasurable B
+      (ae_of_all _ (fun x ↦ by simpa using hb x))
+  have hbound : ∀ ε : ℝ, 0 < ε →
+      |∫ x in E, divergence V x| ≤ (∫ x, ‖V x‖ ∂μ) + ε * μ.real univ := by
+    intro ε hε
+    let f : Euc n → ℝ := fun x ↦ Real.sqrt (‖V x‖ ^ 2 + ε ^ 2)
+    have hsquare : ContDiff ℝ ∞ (fun x ↦ ‖V x‖ ^ 2 + ε ^ 2) := by
+      simpa only [real_inner_self_eq_norm_sq] using (hV.inner ℝ hV).add (contDiff_const (c := ε^2))
+    have hsq : ∀ x, 0 < ‖V x‖ ^ 2 + ε ^ 2 := fun x ↦ by positivity
+    have hf : ContDiff ℝ ∞ f := hsquare.sqrt (fun x ↦ (hsq x).ne')
+    have hfpos : ∀ x, 0 < f x := fun x ↦ Real.sqrt_pos.2 (hsq x)
+    have hflo : ∀ x, ‖V x‖ ≤ f x := by
+      intro x
+      exact (Real.le_sqrt (norm_nonneg _) (hsq x).le).2 (by nlinarith [sq_nonneg ε])
+    have hfhi : ∀ x, f x ≤ ‖V x‖ + ε := by
+      intro x
+      apply (Real.sqrt_le_iff).2
+      constructor
+      · positivity
+      · nlinarith [norm_nonneg (V x)]
+    let Q : Euc n → Euc n := fun x ↦ (f x)⁻¹ • V x
+    have hQ : PerimeterAdmissible U Q := by
+      refine ⟨(hf.inv (fun x ↦ (hfpos x).ne')).smul hV,
+        hc.smul_left (f := fun x ↦ (f x)⁻¹),
+        (tsupport_smul_subset_right _ _).trans hs, ?_⟩
+      intro x
+      simp only [Q, norm_smul, Real.norm_of_nonneg (inv_nonneg.mpr (hfpos x).le)]
+      exact (mul_le_mul_of_nonneg_left (hflo x) (inv_nonneg.mpr (hfpos x).le)).trans_eq
+        (inv_mul_cancel₀ (hfpos x).ne')
+    have hfM : ∀ x, f x ≤ B + ε := fun x ↦ (hfhi x).trans (add_le_add (hb x) le_rfl)
+    have hfi : Integrable f μ := Integrable.of_bound hf.continuous.aestronglyMeasurable
+      (B + ε) (ae_of_all _ (fun x ↦ by rw [Real.norm_eq_abs, abs_of_pos (hfpos x)]; exact hfM x))
+    have hi := integral_divergence_scalar_le E U hU hP hQ hf (fun x ↦ (hfpos x).le) hfM
+    have hn := integral_divergence_scalar_le E U hU hP (perimeterAdmissible_neg hQ)
+      hf (fun x ↦ (hfpos x).le) hfM
+    have heq : (fun x ↦ f x • Q x) = V := by
+      funext x
+      simp only [Q, smul_smul, mul_inv_cancel₀ (hfpos x).ne', one_smul]
+    have heqn : (fun x ↦ f x • (-Q) x) = -V := by
+      funext x
+      simp only [Pi.neg_apply, smul_neg]
+      rw [congrFun heq]
+    rw [heq] at hi
+    rw [heqn, divergence_neg (hV.differentiable (by simp))] at hn
+    simp only [Pi.neg_apply, integral_neg] at hn
+    have ha : |∫ x in E, divergence V x| ≤ ∫ x, f x ∂μ := abs_le.mpr ⟨by linarith, hi⟩
+    refine ha.trans ?_
+    calc
+      (∫ x, f x ∂μ) ≤ ∫ x, ‖V x‖ + ε ∂μ := integral_mono hfi (hNi.add (integrable_const ε)) hfhi
+      _ = (∫ x, ‖V x‖ ∂μ) + ε * μ.real univ := by
+        rw [integral_add hNi (integrable_const ε), integral_const]; simp [mul_comm]
+  apply le_of_forall_pos_le_add
+  intro η hη
+  let ε := η / (μ.real univ + 1)
+  have hε : 0 < ε := by dsimp [ε]; positivity
+  have hh := hbound ε hε
+  have hm : ε * (μ.real univ + 1) = η := by
+    dsimp [ε]
+    exact div_mul_cancel₀ _ (by positivity)
+  change |∫ x in E, divergence V x| ≤ (∫ x, ‖V x‖ ∂μ) + η
+  nlinarith
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+
+def smoothVariationDomain {n : ℕ} (U : Set (Euc n)) (μ : Measure (Euc n)) :
+    Submodule ℝ (Lp (Euc n) 2 μ) where
+  carrier := {u | ∃ V : Euc n → Euc n, ContDiff ℝ ∞ V ∧ HasCompactSupport V ∧
+    tsupport V ⊆ U ∧ (u : Euc n → Euc n) =ᵐ[μ] V}
+  zero_mem' := by
+    refine ⟨0, contDiff_const, HasCompactSupport.zero, by simp, ?_⟩
+    exact Lp.coeFn_zero _ _ _
+  add_mem' := by
+    rintro u v ⟨V, hV, hcV, hsV, heV⟩ ⟨W, hW, hcW, hsW, heW⟩
+    refine ⟨V + W, hV.add hW, hcV.add hcW,
+      (tsupport_add V W).trans (union_subset hsV hsW), ?_⟩
+    exact (Lp.coeFn_add u v).trans (heV.add heW)
+  smul_mem' := by
+    rintro c u ⟨V, hV, hcV, hsV, heV⟩
+    refine ⟨c • V, hV.const_smul c, hcV.comp_left (g := fun z : Euc n ↦ c • z) (smul_zero _),
+      (tsupport_smul_subset_right (fun _ : Euc n ↦ c) V).trans hsV, ?_⟩
+    exact (Lp.coeFn_smul c u).trans (heV.const_smul c)
+
+noncomputable def smoothVariationRepresentative {n : ℕ} {U : Set (Euc n)}
+    {μ : Measure (Euc n)} (u : smoothVariationDomain U μ) : Euc n → Euc n :=
+  u.property.choose
+
+theorem smoothVariationRepresentative_spec {n : ℕ} {U : Set (Euc n)}
+    {μ : Measure (Euc n)} (u : smoothVariationDomain U μ) :
+    ContDiff ℝ ∞ (smoothVariationRepresentative u) ∧
+    HasCompactSupport (smoothVariationRepresentative u) ∧
+    tsupport (smoothVariationRepresentative u) ⊆ U ∧
+    ((u : Lp (Euc n) 2 μ) : Euc n → Euc n) =ᵐ[μ] smoothVariationRepresentative u :=
+  u.property.choose_spec
+
+theorem integral_divergence_eq_of_variation_ae {n : ℕ} (E U : Set (Euc n))
+    (hU : IsOpen U) (hP : perimeter E U ≠ ⊤) {V W : Euc n → Euc n}
+    (hV : ContDiff ℝ ∞ V) (hcV : HasCompactSupport V) (hsV : tsupport V ⊆ U)
+    (hW : ContDiff ℝ ∞ W) (hcW : HasCompactSupport W) (hsW : tsupport W ⊆ U)
+    (he : V =ᵐ[(perimeterMeasure E).restrict U] W) :
+    (∫ x in E, divergence V x) = ∫ x in E, divergence W x := by
+  have hh := abs_integral_divergence_le_integral_norm E U hU hP
+    (hV.sub hW) (hcV.sub hcW) ((tsupport_sub V W).trans (union_subset hsV hsW))
+  have he0 : (∫ x in U, ‖V x - W x‖ ∂perimeterMeasure E) = 0 := by
+    apply integral_eq_zero_of_ae
+    filter_upwards [he] with x hx
+    simp [hx]
+  change |∫ x in E, divergence (V - W) x| ≤ ∫ x in U, ‖V x - W x‖ ∂perimeterMeasure E at hh
+  rw [he0, abs_nonpos_iff, divergence_sub (hV.differentiable (by simp))
+    (hW.differentiable (by simp))] at hh
+  simp only [Pi.sub_apply] at hh
+  rw [integral_sub (integrable_divergence hV hcV).integrableOn
+    (integrable_divergence hW hcW).integrableOn] at hh
+  exact sub_eq_zero.mp hh
+
+noncomputable def localVariationFunctional {n : ℕ} (E U : Set (Euc n))
+    (hU : IsOpen U) (hP : perimeter E U ≠ ⊤) :
+    smoothVariationDomain U ((perimeterMeasure E).restrict U) →ₗ[ℝ] ℝ where
+  toFun u := ∫ x in E, divergence (smoothVariationRepresentative u) x
+  map_add' u v := by
+    obtain ⟨hVu, hcVu, hsVu, heVu⟩ := smoothVariationRepresentative_spec u
+    obtain ⟨hVv, hcVv, hsVv, heVv⟩ := smoothVariationRepresentative_spec v
+    obtain ⟨hVw, hcVw, hsVw, heVw⟩ := smoothVariationRepresentative_spec (u + v)
+    have he := heVw.symm.trans ((Lp.coeFn_add (u : Lp (Euc n) 2 ((perimeterMeasure E).restrict U)) (v : Lp (Euc n) 2 ((perimeterMeasure E).restrict U))).trans
+      (heVu.add heVv))
+    rw [integral_divergence_eq_of_variation_ae E U hU hP hVw hcVw hsVw
+      (hVu.add hVv) (hcVu.add hcVv)
+      ((tsupport_add (smoothVariationRepresentative u) (smoothVariationRepresentative v)).trans (union_subset hsVu hsVv)) he]
+    change (∫ x in E, divergence (smoothVariationRepresentative u + smoothVariationRepresentative v) x) = _
+    rw [divergence_add (hVu.differentiable (by simp)) (hVv.differentiable (by simp))]
+    exact integral_add (integrable_divergence hVu hcVu).integrableOn
+      (integrable_divergence hVv hcVv).integrableOn
+  map_smul' c u := by
+    obtain ⟨hVu, hcVu, hsVu, heVu⟩ := smoothVariationRepresentative_spec u
+    obtain ⟨hVw, hcVw, hsVw, heVw⟩ := smoothVariationRepresentative_spec (c • u)
+    have he := heVw.symm.trans ((Lp.coeFn_smul c (u : Lp (Euc n) 2 ((perimeterMeasure E).restrict U))).trans (heVu.const_smul c))
+    rw [integral_divergence_eq_of_variation_ae E U hU hP hVw hcVw hsVw
+      (hVu.const_smul c) (hcVu.comp_left (g := fun z : Euc n ↦ c • z) (smul_zero _))
+      ((tsupport_smul_subset_right (fun _ : Euc n ↦ c) (smoothVariationRepresentative u)).trans hsVu) he]
+    change (∫ x in E, divergence (fun y ↦ c • smoothVariationRepresentative u y) x) = _
+    simp_rw [divergence_smul (differentiable_const c) (hVu.differentiable (by simp)),
+      gradient_fun_const, inner_zero_left, add_zero]
+    exact integral_smul c _
+
+theorem localVariationFunctional_le {n : ℕ} (E U : Set (Euc n))
+    (hU : IsOpen U) (hP : perimeter E U ≠ ⊤)
+    (u : smoothVariationDomain U ((perimeterMeasure E).restrict U)) :
+    |localVariationFunctional E U hU hP u| ≤
+      ∫ x in U, ‖(u : Lp (Euc n) 2 ((perimeterMeasure E).restrict U)) x‖ ∂perimeterMeasure E := by
+  obtain ⟨hV, hcV, hsV, heV⟩ := smoothVariationRepresentative_spec u
+  refine (abs_integral_divergence_le_integral_norm E U hU hP hV hcV hsV).trans_eq ?_
+  apply integral_congr_ae
+  filter_upwards [heV] with x hx
+  rw [hx]
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/SmoothSheet.lean b/lean/OAI/Geometry/Perimeter/Regularity/SmoothSheet.lean
new file mode 100644
index 0000000000000000000000000000000000000000..8090c34f818985839620e4a34f9dfc334db0a6ec
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/SmoothSheet.lean
@@ -0,0 +1,88 @@
+import OAI.Geometry.Perimeter.Regularity.MinimalGraphEquation
+import OAI.Geometry.Perimeter.Regularity.MinimalGraphSmooth
+import OAI.Geometry.Perimeter.Regularity.BlowupHolderGraph
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+
+def HasSmoothPerimeterSheetAt {n : ℕ} (E : Set (Euc n)) (p : Euc n) : Prop :=
+  ∃ U : Set (Euc n), IsOpen U ∧ p ∈ U ∧
+    ∃ f : Euc n → ℝ, ContDiffOn ℝ ∞ f U ∧
+      (∀ x ∈ U, x ∈ perimeterSupport E ↔ f x = 0) ∧
+      (∀ x ∈ U, f x = 0 → fderiv ℝ f x ≠ 0)
+
+def FlatBlowupRegularityObligation : Prop :=
+  ∀ (n : ℕ), 1 < n → ∀ (E : Set (Euc n)),
+    IsLocallyPerimeterMinimizing E →
+    ∀ (p v : Euc n), p ∈ perimeterSupport E → HasHalfspaceBlowup E p v →
+      HasSmoothPerimeterSheetAt E p
+
+theorem HasSmoothPerimeterSheetAt.of_blownUpSet {n : ℕ} {E : Set (Euc n)}
+    (hE : MeasurableSet E) {p : Euc n} {r : ℝ} (hr : 0<r)
+    (h : HasSmoothPerimeterSheetAt (blownUpSet E p r) 0) :
+    HasSmoothPerimeterSheetAt E p := by
+  obtain ⟨U,hU,h0,f,hf,hiff,hnz⟩ := h
+  let A : Euc n → Euc n := fun x ↦ r⁻¹ • (x-p)
+  let L : Euc n →L[ℝ] Euc n := r⁻¹ • ContinuousLinearMap.id ℝ (Euc n)
+  have hA : ContDiff ℝ ∞ A := by
+    dsimp [A]
+    exact (contDiff_id.sub contDiff_const).const_smul r⁻¹
+  have hAd (x : Euc n) : HasFDerivAt A L x := by
+    change HasFDerivAt (fun y : Euc n ↦ r⁻¹ • (y-p))
+      (r⁻¹ • ContinuousLinearMap.id ℝ (Euc n)) x
+    convert ((hasFDerivAt_id x).sub_const p).const_smul r⁻¹ using 1
+    rfl
+  have hAp : A p=0 := by simp [A]
+  have hback (x : Euc n) : p+r•A x=x := by
+    simp only [A,smul_smul,mul_inv_cancel₀ hr.ne',one_smul]
+    abel
+  refine ⟨A ⁻¹' U,hU.preimage hA.continuous,by simpa only [mem_preimage,hAp] using h0,
+    f ∘ A,hf.comp hA.contDiffOn (fun _ hx ↦ hx),?_,?_⟩
+  · intro x hx
+    rw [← hback x,← perimeterSupport_blownUpSet hE p hr (A x)]
+    simpa only [hback,Function.comp_apply] using hiff (A x) hx
+  · intro x hx hz hzero
+    have hfd : DifferentiableAt ℝ f (A x) :=
+      (hf.contDiffAt (hU.mem_nhds hx)).differentiableAt (by simp)
+    have hder : fderiv ℝ (f ∘ A) x=(fderiv ℝ f (A x)).comp L :=
+      (hfd.hasFDerivAt.comp x (hAd x)).fderiv
+    apply hnz (A x) hx hz
+    ext w
+    have hh := congrArg (fun T : Euc n →L[ℝ] ℝ ↦ T (r•w)) hzero
+    rw [hder] at hh
+    simpa [ContinuousLinearMap.comp_apply,L,smul_smul,hr.ne'] using hh
+
+theorem smooth_perimeter_sheet_of_halfspace_blowup
+    {n : ℕ} (hn : 1<n) {E : Set (Euc n)}
+    (hmin : IsLocallyPerimeterMinimizing E) {p v : Euc n}
+    (hp : p∈perimeterSupport E) (hbl : HasHalfspaceBlowup E p v) :
+    HasSmoothPerimeterSheetAt E p := by
+  obtain ⟨η,hη,_,hgraphs⟩ := hbl.exists_holder_graphs hn hmin hp
+  obtain ⟨ε,hε,hsmooth⟩ := normal_graph_smooth_threshold hn hbl.1 hη
+  obtain ⟨r,hr,f,_,hf,_,_,hiff,⟨a,ha,ha4,hs⟩,hh,hslope⟩ := hgraphs ε hε
+  have heq : ∀ ψ : normalPlane v → ℝ, ContDiff ℝ ∞ ψ → HasCompactSupport ψ →
+      tsupport ψ ⊆ Metric.ball 0 (1/4) →
+      (∫ y, inner ℝ (gradient f y) (gradient ψ y)/Real.sqrt (1+‖gradient f y‖^2))=0 := by
+    intro ψ hψ hcψ hsψ
+    exact full_graph_weak_minimal_equation hn (locally_minimizing_blownUpSet hmin p hr)
+      hbl.1 ha.le ha4 hs hf hiff hψ hcψ hsψ
+  have hfSmooth := hsmooth f hf hh hslope heq
+  apply HasSmoothPerimeterSheetAt.of_blownUpSet hmin.1 hr
+  refine ⟨graphCylinder v,graphCylinder_open v,?_,graphDefining v f,
+    graphDefining_contDiffOn hfSmooth,?_,?_⟩
+  · simp [graphCylinder]
+  · exact fun x hx ↦ graphDefining_support_iff hbl.1 hiff hx
+  · exact fun x hx _ ↦ graphDefining_fderiv_ne_zero hbl.1 hf hx
+
+theorem flatBlowupRegularity : FlatBlowupRegularityObligation :=
+  fun _ hn _ hmin _ _ hp hbl ↦ smooth_perimeter_sheet_of_halfspace_blowup hn hmin hp hbl
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/SmoothSublevelPairingBound.lean b/lean/OAI/Geometry/Perimeter/Regularity/SmoothSublevelPairingBound.lean
new file mode 100644
index 0000000000000000000000000000000000000000..02668891b4b58ae03d85ab9ce172d1c36e8bdb95
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/SmoothSublevelPairingBound.lean
@@ -0,0 +1,457 @@
+import OAI.Geometry.Perimeter.Regularity.TangentialGradientComp
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+namespace RPISingularHarnack
+open RPI
+
+theorem smooth_sublevel_pairing_bound {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V]
+    (A D : V) (a : ℝ) {ψ χ k : ℝ} (hk : 0 ≤ k) (hψ : 0 ≤ ψ)
+    (hχ : χ ∈ Icc (0 : ℝ) 1) (hψχ : ψ ≤ k * χ) :
+    inner ℝ A ((2 * a * ψ) • D - (a ^ 2 * χ) • A) ≤
+      2 * k ^ 2 * ‖D‖ ^ 2 - (1 / 2 : ℝ) * (a ^ 2 * χ * ‖A‖ ^ 2) := by
+  have hs0 : 0 ≤ Real.sqrt χ := Real.sqrt_nonneg _
+  have hs2 : Real.sqrt χ ^ 2 = χ := Real.sq_sqrt hχ.1
+  have hs1 : Real.sqrt χ ≤ 1 := by nlinarith [hχ.2]
+  have hχs : χ ≤ Real.sqrt χ := by nlinarith [mul_nonneg hs0 (sub_nonneg.mpr hs1)]
+  have hψs : ψ ≤ k * Real.sqrt χ := hψχ.trans (mul_le_mul_of_nonneg_left hχs hk)
+  have hi : a * inner ℝ A D ≤ |a| * (‖A‖ * ‖D‖) := by
+    calc
+      _ ≤ |a * inner ℝ A D| := le_abs_self _
+      _ = |a| * |inner ℝ A D| := abs_mul _ _
+      _ ≤ _ := mul_le_mul_of_nonneg_left (abs_real_inner_le_norm A D) (abs_nonneg a)
+  have hic := mul_le_mul_of_nonneg_left hi (mul_nonneg (show (0 : ℝ) ≤ 2 by norm_num) hψ)
+  have hψb := mul_le_mul_of_nonneg_left hψs
+    (show 0 ≤ 2 * |a| * ‖A‖ * ‖D‖ by positivity)
+  have hY := sq_nonneg (|a| * Real.sqrt χ * ‖A‖ - 2 * k * ‖D‖)
+  have hx : (|a| * Real.sqrt χ * ‖A‖) ^ 2 = a ^ 2 * χ * ‖A‖ ^ 2 := by
+    rw [mul_pow, mul_pow, sq_abs, hs2]
+  rw [inner_sub_right, real_inner_smul_right, real_inner_smul_right, real_inner_self_eq_norm_sq]
+  nlinarith [hx]
+
+theorem sublevel_pairing_bound {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V]
+    (A D : V) (a : ℝ) {ε k t : ℝ} (hε : 0 < ε) (hk : 0 ≤ k) (ht : 0 ≤ t) :
+    inner ℝ A ((2 * a * sublevelTest ε k t) • D -
+      (a ^ 2 * sublevelSlope ε k t) • A) ≤
+        2 * (if t < k then k ^ 2 * ‖D‖ ^ 2 else 0) -
+          (1 / 2 : ℝ) * (a ^ 2 * sublevelSlope ε k t * ‖A‖ ^ 2) := by
+  by_cases htk : t < k
+  · simpa only [ite_eq_left htk, mul_assoc] using smooth_sublevel_pairing_bound A D a hk
+      (sublevelTest_nonneg hε k t) (sublevelSlope_mem_Icc ε k t)
+      (sublevelTest_controls hε hk ht)
+  · simp [htk, sublevelTest_eq_zero hε (le_of_not_gt htk),
+      sublevelSlope_eq_zero hε (le_of_not_gt htk)]
+
+theorem wholeBoundary_sublevel_energy {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    {h : Euc n → ℝ} {G : Euc n → Euc n}
+    (hSob : WholeBoundarySobolev E h G) (hSuper : WholeBoundarySupersolution E G)
+    (hh01 : ∀ᵐ x ∂boundaryArea E, 0 ≤ h x ∧ h x ≤ 1)
+    {R : ℝ} (hR : 0 < R) {η : Euc n → ℝ}
+    (hη : ContDiff ℝ ∞ η) (hcη : HasCompactSupport η)
+    (hsη : tsupport η ⊆ Metric.ball 0 R) {ε k : ℝ} (hε : 0 < ε) (hk : 0 ≤ k) :
+    (∫ x in Metric.ball 0 R, η x ^ 2 * sublevelSlope ε k (h x) * ‖G x‖ ^ 2 ∂boundaryArea E) ≤
+      4 * ∫ x in Metric.ball 0 R,
+        (if h x < k then k ^ 2 * ‖tangentialGradient E η x‖ ^ 2 else 0) ∂boundaryArea E := by
+  let μ := (boundaryArea E).restrict (Metric.ball 0 R)
+  have : IsFiniteMeasure μ := ⟨by
+    simpa only [μ, Measure.restrict_apply_univ] using minimizing_boundaryArea_ball_lt_top hn hmin 0 hR⟩
+  have hh := (hSob R hR).1
+  have hG := (hSob R hR).2.1
+  have hD := tangentialGradient_memLp_on_ball hn hmin hR (hη.of_le (by simp))
+  let D := tangentialGradient E η
+  let χ : Euc n → ℝ := fun x ↦ sublevelSlope ε k (h x)
+  let ψ : Euc n → ℝ := fun x ↦ sublevelTest ε k (h x)
+  have hχc : Continuous (sublevelSlope ε k) := by unfold sublevelSlope; fun_prop
+  have hχm : AEStronglyMeasurable χ μ := hχc.comp_aestronglyMeasurable hh.aestronglyMeasurable
+  have hψm : AEStronglyMeasurable ψ μ :=
+    (sublevelTest_contDiff ε k).continuous.comp_aestronglyMeasurable hh.aestronglyMeasurable
+  obtain ⟨M, hM⟩ := hcη.exists_bound_of_continuous hη.continuous
+  have hM0 : 0 ≤ M := (norm_nonneg (η 0)).trans (hM 0)
+  have hψk : ∀ᵐ x ∂μ, ‖ψ x‖ ≤ k := by
+    filter_upwards [ae_restrict_of_ae hh01] with x hx
+    rw [Real.norm_of_nonneg (sublevelTest_nonneg hε k (h x))]
+    exact (sublevelTest_controls hε hk hx.1).trans
+      (by nlinarith [(sublevelSlope_mem_Icc ε k (h x)).2])
+  have hχb : ∀ x, ‖χ x‖ ≤ 1 := fun x ↦ by
+    rw [Real.norm_of_nonneg (sublevelSlope_mem_Icc ε k (h x)).1]
+    exact (sublevelSlope_mem_Icc ε k (h x)).2
+  have hc₁ : MemLp (fun x ↦ 2 * η x * ψ x) ∞ μ := by
+    apply MemLp.of_bound ((aestronglyMeasurable_const.mul hη.continuous.aestronglyMeasurable).mul hψm)
+      (2 * M * k)
+    filter_upwards [hψk] with x hx
+    simp only [Pi.mul_apply, norm_mul, Real.norm_ofNat]
+    exact mul_le_mul (mul_le_mul_of_nonneg_left (hM x) (by norm_num)) hx (norm_nonneg _) (by positivity)
+  have hc₂ : MemLp (fun x ↦ η x ^ 2 * χ x) ∞ μ := by
+    apply MemLp.of_bound ((hη.continuous.aestronglyMeasurable.pow 2).mul hχm) (M ^ 2)
+    exact ae_of_all _ fun x ↦ by
+      simp only [Pi.mul_apply, Pi.pow_apply, norm_mul, norm_pow]
+      exact (mul_le_mul ((sq_le_sq₀ (norm_nonneg _) hM0).2 (hM x)) (hχb x)
+        (norm_nonneg _) (sq_nonneg M)).trans_eq (mul_one _)
+  let B : Euc n → Euc n := fun x ↦ (2 * η x * ψ x) • D x - (η x ^ 2 * χ x) • G x
+  have hB : MemLp B 2 μ := (hc₁.smul hD).sub (hc₂.smul hG)
+  have hP : Integrable (fun x ↦ inner ℝ (G x) (B x)) μ := integrable_inner_of_memLp hG hB
+  let J : Euc n → ℝ := fun x ↦ η x ^ 2 * χ x * ‖G x‖ ^ 2
+  have hJm : AEStronglyMeasurable J μ :=
+    ((hη.continuous.aestronglyMeasurable.pow 2).mul hχm).mul (hG.aestronglyMeasurable.norm.pow 2)
+  have hGs := (memLp_two_iff_integrable_sq_norm hG.aestronglyMeasurable).mp hG
+  have hJ : Integrable J μ := (hGs.const_mul (M ^ 2)).mono' hJm (by
+    filter_upwards [] with x
+    rw [Real.norm_of_nonneg (by dsimp [J]; positivity [(sublevelSlope_mem_Icc ε k (h x)).1])]
+    apply mul_le_mul_of_nonneg_right _ (sq_nonneg ‖G x‖)
+    have hm : η x ^ 2 ≤ M ^ 2 := by simpa only [Real.norm_eq_abs, sq_abs] using
+      (sq_le_sq₀ (norm_nonneg _) hM0).2 (hM x)
+    exact (mul_le_mul hm (sublevelSlope_mem_Icc ε k (h x)).2
+      (sublevelSlope_mem_Icc ε k (h x)).1 (sq_nonneg M)).trans_eq (mul_one _))
+  let C : Euc n → ℝ := fun x ↦ if h x < k then k ^ 2 * ‖D x‖ ^ 2 else 0
+  have hDs := (memLp_two_iff_integrable_sq_norm hD.aestronglyMeasurable).mp hD
+  have hCm : AEStronglyMeasurable C μ := by
+    exact ((hD.aestronglyMeasurable.norm.pow 2).const_mul (k ^ 2)).indicator₀
+      (hh.aestronglyMeasurable.nullMeasurableSet_lt aestronglyMeasurable_const)
+  have hC : Integrable C μ := (hDs.const_mul (k ^ 2)).mono' hCm (by
+    filter_upwards [] with x
+    change ‖if h x < k then k ^ 2 * ‖D x‖ ^ 2 else 0‖ ≤ _
+    split_ifs
+    · rw [Real.norm_of_nonneg (mul_nonneg (sq_nonneg k) (sq_nonneg ‖D x‖))]
+    · simp only [norm_zero]
+      positivity)
+  have htest := wholeBoundarySupersolution_nonlinear_test hn hmin hSob hSuper hh01
+    hR hη hcη hsη (sublevelTest_contDiff ε k) (sublevelTest_nonneg hε k)
+    (sublevelTest_lipschitz ε k) (show (0 : ℝ) ≤ 1 by norm_num) (fun t ↦ by
+      rw [(sublevelTest_hasDerivAt ε k t).deriv, norm_neg,
+        Real.norm_of_nonneg (sublevelSlope_mem_Icc ε k t).1]
+      exact (sublevelSlope_mem_Icc ε k t).2)
+  have htest' : 0 ≤ ∫ x, inner ℝ (G x) (B x) ∂μ := by
+    convert htest using 1
+    apply integral_congr_ae
+    exact ae_of_all _ fun x ↦ by
+      dsimp only
+      rw [(sublevelTest_hasDerivAt ε k (h x)).deriv]
+      simp only [B, D, ψ, χ, mul_neg, neg_smul, sub_eq_add_neg]
+  have hi := integral_mono_ae hP ((hC.const_mul 2).sub (hJ.const_mul (1 / 2))) (by
+    filter_upwards [ae_restrict_of_ae hh01] with x hx
+    exact sublevel_pairing_bound (G x) (D x) (η x) hε hk hx.1)
+  simp only [Pi.sub_apply] at hi
+  rw [integral_sub (hC.const_mul 2) (hJ.const_mul (1 / 2)), integral_const_mul,
+    integral_const_mul] at hi
+  change (∫ x, J x ∂μ) ≤ 4 * ∫ x, C x ∂μ
+  linarith
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+namespace RPISingularHarnack
+
+def rankOneVariation {n : ℕ} (φ : Euc n → ℝ) (v : Euc n) (t : ℝ) (x : Euc n) : Euc n :=
+  x + (t * φ x) • v
+
+def rankOneJacobian {n : ℕ} (φ : Euc n → ℝ) (v : Euc n) (t : ℝ) (x : Euc n) : ℝ :=
+  1 + t * fderiv ℝ φ x v
+
+def rankOnePiola {n : ℕ} (φ : Euc n → ℝ) (v : Euc n) (t : ℝ)
+    (W : Euc n → Euc n) (x : Euc n) : Euc n :=
+  rankOneJacobian φ v t x • W (rankOneVariation φ v t x) -
+    (t * fderiv ℝ φ x (W (rankOneVariation φ v t x))) • v
+
+theorem rankOneVariation_contDiff {n : ℕ} {φ : Euc n → ℝ}
+    (hφ : ContDiff ℝ ∞ φ) (v : Euc n) (t : ℝ) :
+    ContDiff ℝ ∞ (rankOneVariation φ v t) :=
+  contDiff_id.add ((contDiff_const.mul hφ).smul contDiff_const)
+
+theorem rankOneVariation_hasFDerivAt {n : ℕ} {φ : Euc n → ℝ} {x : Euc n}
+    (hφ : DifferentiableAt ℝ φ x) (v : Euc n) (t : ℝ) :
+    HasFDerivAt (rankOneVariation φ v t)
+      (ContinuousLinearMap.id ℝ (Euc n) + (t • fderiv ℝ φ x).smulRight v) x := by
+  convert (hasFDerivAt_id x).add ((hφ.hasFDerivAt.const_mul t).smul_const v) using 1
+  rfl
+
+theorem divergence_comp_rankOneVariation {n : ℕ} {φ : Euc n → ℝ}
+    {W : Euc n → Euc n} (hφ : Differentiable ℝ φ) (hW : Differentiable ℝ W)
+    (v : Euc n) (t : ℝ) (x : Euc n) :
+    divergence (W ∘ rankOneVariation φ v t) x =
+      divergence W (rankOneVariation φ v t x) +
+        t * fderiv ℝ φ x (fderiv ℝ W (rankOneVariation φ v t x) v) := by
+  rw [divergence]
+  rw [((hW _).hasFDerivAt.comp x (rankOneVariation_hasFDerivAt (hφ x) v t)).fderiv]
+  simp only [ContinuousLinearMap.comp_apply, add_apply,
+    ContinuousLinearMap.id_apply, ContinuousLinearMap.smulRight_apply,
+    smul_apply, smul_eq_mul, map_add, map_smul,
+    inner_add_right, real_inner_smul_right, Finset.sum_add_distrib]
+  change divergence W (rankOneVariation φ v t x) + _ = _
+  congr 1
+  simp_rw [← inner_gradient_left, mul_assoc]
+  rw [← Finset.mul_sum]
+  congr 1
+  exact (EuclideanSpace.basisFun (Fin n) ℝ).sum_inner_mul_inner _ _
+
+theorem rankOneJacobian_hasFDerivAt {n : ℕ} {φ : Euc n → ℝ}
+    (hφ : ContDiff ℝ ∞ φ) (v : Euc n) (t : ℝ) (x : Euc n) :
+    HasFDerivAt (rankOneJacobian φ v t)
+      (t • (ContinuousLinearMap.apply ℝ ℝ v).comp (fderiv ℝ (fderiv ℝ φ) x)) x := by
+  have hd : Differentiable ℝ (fderiv ℝ φ) := (hφ.fderiv_right (m := ∞) (by simp)).differentiable (by simp)
+  convert (((ContinuousLinearMap.apply ℝ ℝ v).hasFDerivAt.comp x
+    (hd x).hasFDerivAt).const_mul t).const_add 1 using 1
+  rfl
+
+theorem rankOneJacobian_contDiff {n : ℕ} {φ : Euc n → ℝ}
+    (hφ : ContDiff ℝ ∞ φ) (v : Euc n) (t : ℝ) :
+    ContDiff ℝ ∞ (rankOneJacobian φ v t) :=
+  contDiff_const.add (contDiff_const.mul ((hφ.fderiv_right (m := ∞) (by simp)).clm_apply contDiff_const))
+
+theorem rankOnePiola_contDiff {n : ℕ} {φ : Euc n → ℝ} {W : Euc n → Euc n}
+    (hφ : ContDiff ℝ ∞ φ) (hW : ContDiff ℝ ∞ W) (v : Euc n) (t : ℝ) :
+    ContDiff ℝ ∞ (rankOnePiola φ v t W) := by
+  have hF := hW.comp (rankOneVariation_contDiff hφ v t)
+  exact ((rankOneJacobian_contDiff hφ v t).smul hF).sub
+    ((contDiff_const.mul ((hφ.fderiv_right (m := ∞) (by simp)).clm_apply hF)).smul contDiff_const)
+
+theorem divergence_rankOnePiola {n : ℕ} {φ : Euc n → ℝ} {W : Euc n → Euc n}
+    (hφ : ContDiff ℝ ∞ φ) (hW : ContDiff ℝ ∞ W) (v : Euc n) (t : ℝ) (x : Euc n) :
+    divergence (rankOnePiola φ v t W) x =
+      rankOneJacobian φ v t x * divergence W (rankOneVariation φ v t x) := by
+  let F := rankOneVariation φ v t
+  let A := fderiv ℝ W (F x)
+  let L := fderiv ℝ φ x
+  let H := fderiv ℝ (fderiv ℝ φ) x
+  have hdφ : Differentiable ℝ φ := hφ.differentiable (by simp)
+  have hdW : Differentiable ℝ W := hW.differentiable (by simp)
+  have hdL : Differentiable ℝ (fderiv ℝ φ) := (hφ.fderiv_right (m := ∞) (by simp)).differentiable (by simp)
+  have hdF : Differentiable ℝ F := (rankOneVariation_contDiff hφ v t).differentiable (by simp)
+  have hdV : Differentiable ℝ (W ∘ F) := hdW.comp hdF
+  have hdJ := (rankOneJacobian_contDiff hφ v t).differentiable (by simp)
+  have hdw : Differentiable ℝ (fun y ↦ t * fderiv ℝ φ y (W (F y))) :=
+    (hdL.clm_apply hdV).const_mul t
+  change divergence ((fun y ↦ rankOneJacobian φ v t y • W (F y)) -
+    (fun y ↦ (t * fderiv ℝ φ y (W (F y))) • v)) x = _
+  have hd1 : Differentiable ℝ (fun y ↦ rankOneJacobian φ v t y • W (F y)) :=
+    hdJ.fun_smul hdV
+  have hd2 : Differentiable ℝ (fun y ↦ (t * fderiv ℝ φ y (W (F y))) • v) :=
+    hdw.smul_const v
+  rw [divergence_sub hd1 hd2, Pi.sub_apply]
+  have hds := divergence_smul hdJ hdV x
+  dsimp only [Function.comp_apply] at hds
+  rw [hds, divergence_smul hdw (differentiable_const v)]
+  have hconst : divergence (fun _ : Euc n ↦ v) x = 0 := by simp [divergence]
+  rw [hconst, mul_zero, zero_add]
+  rw [inner_gradient_left, inner_gradient_left]
+  rw [(rankOneJacobian_hasFDerivAt hφ v t x).fderiv]
+  have hcomp := (hdW (F x)).hasFDerivAt.comp x (rankOneVariation_hasFDerivAt (hdφ x) v t)
+  have happ := (hdL x).hasFDerivAt.clm_apply hcomp
+  have happ' := (happ.const_mul t).fderiv
+  dsimp only [Function.comp_apply] at happ'
+  rw [happ']
+  simp only [smul_apply, ContinuousLinearMap.comp_apply,
+    ContinuousLinearMap.apply_apply, ContinuousLinearMap.flip_apply, smul_eq_mul, add_apply,
+    ContinuousLinearMap.id_apply, ContinuousLinearMap.smulRight_apply]
+  have hsym : H (W (F x)) v = H v (W (F x)) :=
+    second_derivative_symmetric (fun y ↦ (hdφ y).hasFDerivAt) (hdL x).hasFDerivAt _ _
+  have hdiv := divergence_comp_rankOneVariation hdφ hdW v t x
+  change _ = _ at hdiv
+  change rankOneJacobian φ v t x * divergence (W ∘ F) x + t * H (W (F x)) v -
+    t * (L (A (v + (t * L v) • v)) + H v (W (F x))) = _
+  rw [hdiv, hsym]
+  simp only [F, rankOneJacobian, map_add, map_smul, smul_eq_mul, A, L]
+  ring
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+namespace RPISingularHarnack
+
+theorem det_id_add_smulRight {n : ℕ} (L : Euc n →L[ℝ] ℝ) (v : Euc n) :
+    (ContinuousLinearMap.id ℝ (Euc n) + L.smulRight v).det = 1 + L v := by
+  classical
+  let b := (EuclideanSpace.basisFun (Fin n) ℝ).toBasis
+  change LinearMap.det (LinearMap.id + L.toLinearMap.smulRight v) = _
+  rw [← LinearMap.det_toMatrix b, map_add, LinearMap.toMatrix_id,
+    LinearMap.toMatrix_smulRight]
+  have hm : Matrix.vecMulVec (b.repr v) (L.toLinearMap ∘ b) =
+      Matrix.replicateCol Unit (b.repr v) * Matrix.replicateRow Unit (L.toLinearMap ∘ b) := by
+    ext i j
+    simp [Matrix.mul_apply, Matrix.vecMulVec_apply]
+  rw [hm, Matrix.det_one_add_replicateCol_mul_replicateRow]
+  congr 1
+  have hb := b.sum_repr v
+  apply_fun L at hb
+  change (∑ i, L (b i) * b.repr v i) = L v
+  simpa only [map_sum, map_smul, smul_eq_mul, mul_comm] using hb
+
+theorem det_fderiv_rankOneVariation {n : ℕ} {φ : Euc n → ℝ}
+    (hφ : Differentiable ℝ φ) (v : Euc n) (t : ℝ) (x : Euc n) :
+    (fderiv ℝ (rankOneVariation φ v t) x).det = rankOneJacobian φ v t x := by
+  rw [(rankOneVariation_hasFDerivAt (hφ x) v t).fderiv, det_id_add_smulRight]
+  rfl
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+namespace RPISingularHarnack
+
+theorem exists_rankOneVariation_homeomorph {n : ℕ} (hn : 0 < n)
+    {φ : Euc n → ℝ} {L : ℝ≥0} (hL : LipschitzWith L φ)
+    (v : Euc n) (t : ℝ) (ht : ‖t‖ * L * ‖v‖ < 1) :
+    ∃ e : Euc n ≃ₜ Euc n, (e : Euc n → Euc n) = rankOneVariation φ v t := by
+  have : Nonempty (Fin n) := Fin.pos_iff_nonempty.mp hn
+  let c : ℝ≥0 := ‖t‖₊ * L * ‖v‖₊
+  have ha : ApproximatesLinearOn (rankOneVariation φ v t)
+      ((ContinuousLinearEquiv.refl ℝ (Euc n)) : Euc n →L[ℝ] Euc n) univ c := by
+    intro x _ y _
+    have heq : rankOneVariation φ v t x - rankOneVariation φ v t y - (x - y) =
+        (t * (φ x - φ y)) • v := by
+      unfold rankOneVariation
+      rw [mul_sub, sub_smul]
+      abel
+    change ‖rankOneVariation φ v t x - rankOneVariation φ v t y - (x - y)‖ ≤ _
+    rw [heq, norm_smul, norm_mul]
+    have hb := mul_le_mul_of_nonneg_right
+      (mul_le_mul_of_nonneg_left (hL.norm_sub_le x y) (norm_nonneg t)) (norm_nonneg v)
+    simpa only [c, NNReal.coe_mul, coe_nnnorm, mul_assoc, mul_left_comm, mul_comm] using hb
+  have hc : c < ‖((ContinuousLinearEquiv.refl ℝ (Euc n)).symm : Euc n →L[ℝ] Euc n)‖₊⁻¹ := by
+    simpa [c, ← NNReal.coe_lt_coe] using ht
+  exact ⟨ha.toHomeomorph _ (Or.inr hc), rfl⟩
+
+theorem rankOneJacobian_pos {n : ℕ} {φ : Euc n → ℝ} {L : ℝ≥0}
+    (hL : LipschitzWith L φ) (v : Euc n) (t : ℝ)
+    (ht : ‖t‖ * L * ‖v‖ < 1) (x : Euc n) : 0 < rankOneJacobian φ v t x := by
+  have hnorm : ‖t * fderiv ℝ φ x v‖ ≤ ‖t‖ * L * ‖v‖ := by
+    rw [norm_mul]
+    calc
+      _ ≤ ‖t‖ * (‖fderiv ℝ φ x‖ * ‖v‖) :=
+        mul_le_mul_of_nonneg_left ((fderiv ℝ φ x).le_opNorm v) (norm_nonneg t)
+      _ ≤ ‖t‖ * (L * ‖v‖) := mul_le_mul_of_nonneg_left
+        (mul_le_mul_of_nonneg_right (norm_fderiv_le_of_lipschitz ℝ hL) (norm_nonneg v)) (norm_nonneg t)
+      _ = _ := by ring
+  have hb := (neg_abs_le (t * fderiv ℝ φ x v))
+  simp only [Real.norm_eq_abs] at hnorm ht
+  unfold rankOneJacobian
+  linarith
+
+theorem rankOneVariation_eq_self_of_notMem_tsupport {n : ℕ}
+    (φ : Euc n → ℝ) (v : Euc n) (t : ℝ) {x : Euc n} (hx : x ∉ tsupport φ) :
+    rankOneVariation φ v t x = x := by
+  simp [rankOneVariation, image_eq_zero_of_notMem_tsupport hx]
+
+theorem Homeomorph.image_eq_of_eqOn_compl {X : Type*} [TopologicalSpace X]
+    (e : X ≃ₜ X) (U : Set X) (heq : ∀ x ∉ U, e x = x) : e '' U = U := by
+  ext x
+  constructor
+  · rintro ⟨y, hy, rfl⟩
+    by_contra hx
+    have hh := e.injective (heq (e y) hx)
+    exact hx (hh.symm ▸ hy)
+  · intro hx
+    refine ⟨e.symm x, ?_, e.apply_symm_apply x⟩
+    by_contra hn
+    have hh := heq (e.symm x) hn
+    rw [e.apply_symm_apply] at hh
+    exact hn (hh ▸ hx)
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+namespace RPISingularHarnack
+
+theorem rankOnePiola_tsupport_subset {n : ℕ} {φ : Euc n → ℝ} {W : Euc n → Euc n}
+    (hφ : ContDiff ℝ ∞ φ) (v : Euc n) (t : ℝ) :
+    tsupport (rankOnePiola φ v t W) ⊆ (rankOneVariation φ v t) ⁻¹' tsupport W := by
+  apply closure_minimal _ ((isClosed_tsupport W).preimage (rankOneVariation_contDiff hφ v t).continuous)
+  intro x hx
+  by_contra hn
+  have hw : W (rankOneVariation φ v t x) = 0 := image_eq_zero_of_notMem_tsupport hn
+  exact hx (by simp [rankOnePiola, hw])
+
+theorem rankOnePiola_hasCompactSupport {n : ℕ} {φ : Euc n → ℝ} {W : Euc n → Euc n}
+    (hφ : ContDiff ℝ ∞ φ) (v : Euc n) (t : ℝ) (e : Euc n ≃ₜ Euc n)
+    (he : (e : Euc n → Euc n) = rankOneVariation φ v t) (hW : HasCompactSupport W) :
+    HasCompactSupport (rankOnePiola φ v t W) := by
+  have hc : IsCompact ((rankOneVariation φ v t) ⁻¹' tsupport W) := by
+    rw [← he, ← e.image_symm]
+    exact hW.image e.symm.continuous
+  exact hc.of_isClosed_subset (isClosed_tsupport _) (rankOnePiola_tsupport_subset hφ v t)
+
+theorem rankOnePiola_tsupport_subset_region {n : ℕ} {φ : Euc n → ℝ} {W : Euc n → Euc n}
+    {U : Set (Euc n)} (hφ : ContDiff ℝ ∞ φ) (v : Euc n) (t : ℝ)
+    (e : Euc n ≃ₜ Euc n) (he : (e : Euc n → Euc n) = rankOneVariation φ v t)
+    (hφU : tsupport φ ⊆ U) (hWU : tsupport W ⊆ U) :
+    tsupport (rankOnePiola φ v t W) ⊆ U := by
+  intro x hx
+  by_contra hnx
+  have hf : e x = x := by
+    rw [he]
+    exact rankOneVariation_eq_self_of_notMem_tsupport φ v t (fun h ↦ hnx (hφU h))
+  have hW := rankOnePiola_tsupport_subset hφ v t hx
+  change rankOneVariation φ v t x ∈ tsupport W at hW
+  rw [← he, hf] at hW
+  exact hnx (hWU hW)
+
+theorem rankOneVariation_image_region {n : ℕ} {φ : Euc n → ℝ} {U : Set (Euc n)}
+    (v : Euc n) (t : ℝ) (e : Euc n ≃ₜ Euc n)
+    (he : (e : Euc n → Euc n) = rankOneVariation φ v t) (hφU : tsupport φ ⊆ U) :
+    rankOneVariation φ v t '' U = U := by
+  rw [← he]
+  apply Homeomorph.image_eq_of_eqOn_compl
+  intro x hx
+  rw [he]
+  exact rankOneVariation_eq_self_of_notMem_tsupport φ v t (fun h ↦ hx (hφU h))
+
+theorem symmDiff_image_rankOneVariation_subset {n : ℕ} {φ : Euc n → ℝ}
+    (E : Set (Euc n)) (v : Euc n) (t : ℝ) (e : Euc n ≃ₜ Euc n)
+    (he : (e : Euc n → Euc n) = rankOneVariation φ v t) :
+    closure (symmDiff E (rankOneVariation φ v t '' E)) ⊆ tsupport φ := by
+  apply closure_minimal _ (isClosed_tsupport _)
+  intro x hx
+  by_contra hnx
+  have hfix : e x = x := by rw [he]; exact rankOneVariation_eq_self_of_notMem_tsupport φ v t hnx
+  have himem : x ∈ rankOneVariation φ v t '' E ↔ x ∈ E := by
+    rw [← he]
+    constructor
+    · rintro ⟨y, hy, hye⟩
+      exact (e.injective (hye.trans hfix.symm)) ▸ hy
+    · intro h
+      exact ⟨x, h, hfix⟩
+  exact (mem_symmDiff.mp hx).elim (fun h ↦ h.2 (himem.mpr h.1)) (fun h ↦ h.2 (himem.mp h.1))
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/SmulMemPhaseSupport.lean b/lean/OAI/Geometry/Perimeter/Regularity/SmulMemPhaseSupport.lean
new file mode 100644
index 0000000000000000000000000000000000000000..9423765310d5bd894fc81267b7002b972d44280f
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/SmulMemPhaseSupport.lean
@@ -0,0 +1,516 @@
+import OAI.Geometry.Perimeter.Regularity.ExistsTangentialGradientBallBound
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+open scoped ENNReal Topology
+open scoped ENNReal Topology ContDiff
+open scoped Topology ENNReal
+namespace RPISingularHarnack
+open scoped Pointwise
+
+theorem IsCone.smul_mem_phaseSupport {n : ℕ} {E : Set (Euc n)} (hcone : IsCone E)
+    {x : Euc n} (hx : x ∈ (volume.restrict E).support) {r : ℝ} (hr : 0 < r) :
+    r • x ∈ (volume.restrict E).support := by
+  apply Metric.nhds_basis_ball.mem_measureSupport.mpr
+  intro R hR
+  rw [Measure.restrict_apply measurableSet_ball]
+  have hp := Metric.nhds_basis_ball.mem_measureSupport.mp hx (R/r) (div_pos hR hr)
+  rw [Measure.restrict_apply measurableSet_ball] at hp
+  have hs : r • (Metric.ball x (R/r) ∩ E) ⊆ Metric.ball (r • x) R ∩ E := by
+    rintro y ⟨z, hz, rfl⟩
+    refine ⟨?_, (hcone r hr z).mpr hz.2⟩
+    change dist (r • z) (r • x) < R
+    rw [dist_smul₀, Real.norm_of_nonneg hr.le]
+    simpa only [mul_comm] using (lt_div_iff₀ hr).mp hz.1
+  apply lt_of_lt_of_le _ (measure_mono (μ := volume) hs)
+  rw [Measure.addHaar_smul_of_nonneg volume hr.le]
+  exact ENNReal.mul_pos (ENNReal.ofReal_pos.mpr (pow_pos hr _)).ne' hp.ne'
+
+theorem isCone_perimeterSupport {n : ℕ} {E : Set (Euc n)}
+    (hE : MeasurableSet E) (hcone : IsCone E) : IsCone (perimeterSupport E) := by
+  have hcompl : IsCone Eᶜ := fun r hr x ↦ not_congr (hcone r hr x)
+  have hforward (x : Euc n) {r : ℝ} (hr : 0 < r)
+      (hx : x ∈ perimeterSupport E) : r • x ∈ perimeterSupport E := by
+    rw [perimeterSupport_eq_phaseSupport_inter hE] at hx ⊢
+    exact ⟨hcone.smul_mem_phaseSupport hx.1 hr, hcompl.smul_mem_phaseSupport hx.2 hr⟩
+  intro r hr x
+  constructor
+  · intro hx
+    have hh := hforward (r • x) (inv_pos.mpr hr) hx
+    simpa only [inv_smul_smul₀ hr.ne'] using hh
+  · exact hforward x hr
+
+theorem IsCone.isPathConnected_perimeterSupport {n : ℕ} {E : Set (Euc n)}
+    (hE : MeasurableSet E) (hcone : IsCone E)
+    (hvertex : (0 : Euc n) ∈ perimeterSupport E) :
+    IsPathConnected (perimeterSupport E) := by
+  have hc := isCone_perimeterSupport hE hcone
+  have hs : StarConvex ℝ (0 : Euc n) (perimeterSupport E) := by
+    intro x hx a b ha hb hab
+    by_cases hb0 : b = 0
+    · simpa [hb0] using hvertex
+    · simpa only [smul_zero, zero_add] using (hc b (lt_of_le_of_ne hb (Ne.symm hb0)) x).mpr hx
+  exact hs.isPathConnected hvertex
+
+def NestedMinimizingConeMaximumObligation : Prop :=
+  ∀ (n : ℕ), 8 ≤ n → ∀ (E F : Set (Euc n)),
+    IsLocallyPerimeterMinimizing E → IsLocallyPerimeterMinimizing F →
+    IsCone E → IsCone F → E ⊆ F →
+    (0 : Euc n) ∈ perimeterSupport E → (0 : Euc n) ∈ perimeterSupport F →
+    perimeterSupport E = perimeterSupport F
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+open scoped ENNReal Topology
+open scoped ENNReal Topology ContDiff
+open scoped Topology ENNReal
+namespace RPISingularHarnack
+open RPI.Boundary
+
+def radialDensityIntegrand {n : ℕ} (c : Euc n) (k : ℕ) (ψ : ℝ → ℝ)
+    (t : ℝ) (x : Euc n) : ℝ := t ^ k * ψ (t ^ 2 * ‖x-c‖ ^ 2)
+
+def radialDensityDerivative {n : ℕ} (c : Euc n) (k : ℕ) (ψ : ℝ → ℝ)
+    (t : ℝ) (x : Euc n) : ℝ :=
+  (k : ℝ) * t ^ (k-1) * ψ (t ^ 2 * ‖x-c‖ ^ 2) +
+    t ^ k * deriv ψ (t ^ 2 * ‖x-c‖ ^ 2) * (2*t*‖x-c‖ ^ 2)
+
+theorem radialDensityIntegrand_hasDerivAt {n : ℕ} (c : Euc n) (k : ℕ)
+    {ψ : ℝ → ℝ} (hψ : ContDiff ℝ ∞ ψ) (t : ℝ) (x : Euc n) :
+    HasDerivAt (fun s ↦ radialDensityIntegrand c k ψ s x)
+      (radialDensityDerivative c k ψ t x) t := by
+  have hi : HasDerivAt (fun s : ℝ ↦ s^2 * ‖x-c‖^2) (2*t*‖x-c‖^2) t := by
+    simpa using (hasDerivAt_pow 2 t).mul_const (‖x-c‖^2)
+  have hh := (hasDerivAt_pow k t).mul
+    (((hψ.differentiable (by simp)) _).hasDerivAt.comp t hi)
+  convert hh using 1 <;> try rfl
+  · dsimp [radialDensityDerivative]
+    ring
+
+theorem radialDensityIntegrand_continuous {n : ℕ} (c : Euc n) (k : ℕ)
+    {ψ : ℝ → ℝ} (hψ : ContDiff ℝ ∞ ψ) :
+    Continuous (fun p : ℝ × Euc n ↦ radialDensityIntegrand c k ψ p.1 p.2) := by
+  unfold radialDensityIntegrand
+  fun_prop
+
+theorem radialDensityDerivative_continuous {n : ℕ} (c : Euc n) (k : ℕ)
+    {ψ : ℝ → ℝ} (hψ : ContDiff ℝ ∞ ψ) :
+    Continuous (fun p : ℝ × Euc n ↦ radialDensityDerivative c k ψ p.1 p.2) := by
+  have hd := hψ.continuous_deriv (by simp)
+  unfold radialDensityDerivative
+  fun_prop
+
+theorem radialDensity_integral_hasDerivAt {n : ℕ} (c : Euc n) (M : ℝ) (k : ℕ)
+    {ψ : ℝ → ℝ} (hψ : ContDiff ℝ ∞ ψ) (μ : Measure (Euc n)) [IsFiniteMeasure μ]
+    (hμ : ∀ᵐ x ∂μ, x ∈ Metric.closedBall c M) (t : ℝ) :
+    Integrable (radialDensityDerivative c k ψ t) μ ∧
+    HasDerivAt (fun s ↦ ∫ x, radialDensityIntegrand c k ψ s x ∂μ)
+      (∫ x, radialDensityDerivative c k ψ t x ∂μ) t := by
+  have hcF (s : ℝ) : Continuous (radialDensityIntegrand c k ψ s) :=
+    (radialDensityIntegrand_continuous c k hψ).comp (continuous_const.prodMk continuous_id)
+  have hcD (s : ℝ) : Continuous (radialDensityDerivative c k ψ s) :=
+    (radialDensityDerivative_continuous c k hψ).comp (continuous_const.prodMk continuous_id)
+  obtain ⟨C, hC⟩ := (isCompact_closedBall c M).exists_bound_of_continuousOn (hcF t).continuousOn
+  have hFi : Integrable (radialDensityIntegrand c k ψ t) μ :=
+    Integrable.of_bound (hcF t).aestronglyMeasurable C (hμ.mono (fun x hx ↦ hC x hx))
+  obtain ⟨B, hB⟩ := (isCompact_Icc.prod (isCompact_closedBall c M)).exists_bound_of_continuousOn
+    ((radialDensityDerivative_continuous c k hψ).continuousOn :
+      ContinuousOn _ (Icc (t-1) (t+1) ×ˢ Metric.closedBall c M))
+  apply hasDerivAt_integral_of_dominated_loc_of_deriv_le
+    (s := Ioo (t-1) (t+1)) (bound := fun _ ↦ B) (Ioo_mem_nhds (by linarith) (by linarith))
+    (Eventually.of_forall fun s ↦ (hcF s).aestronglyMeasurable) hFi
+    (hcD t).aestronglyMeasurable ?_ (integrable_const B) (Eventually.of_forall ?_)
+  · filter_upwards [hμ] with x hx
+    intro s hs
+    exact hB (s,x) ⟨⟨hs.1.le,hs.2.le⟩,hx⟩
+  · intro x s _
+    exact radialDensityIntegrand_hasDerivAt c k hψ s x
+
+theorem tangentialDivergence_radialProfile {n : ℕ} (hn : 0 < n)
+    (E : Set (Euc n)) (c : Euc n) {ψ : ℝ → ℝ} (hψ : ContDiff ℝ ∞ ψ)
+    (t : ℝ) (x : Euc n) (hN : ‖approximateNormal E x‖ = 1) :
+    tangentialDivergence E (fun y ↦ ψ (t^2 * ‖y-c‖^2) • (y-c)) x =
+      (n-1 : ℕ) * ψ (t^2 * ‖x-c‖^2) +
+        2*t^2*deriv ψ (t^2 * ‖x-c‖^2) *
+          (‖x-c‖^2 - (inner ℝ (approximateNormal E x) (x-c))^2) := by
+  have hf : ContDiff ℝ ∞ (fun y : Euc n ↦ ψ (t^2 * ‖y-c‖^2)) :=
+    hψ.comp (contDiff_const.mul ((contDiff_id.sub contDiff_const).norm_sq (𝕜 := ℝ)))
+  have hV : Differentiable ℝ (fun y : Euc n ↦ y-c) :=
+    differentiable_id.sub (differentiable_const c)
+  rw [tangentialDivergence_smul E (hf.differentiable (by simp)) hV,
+    tangentialDivergence_displacement E c x hN hn]
+  have hd : fderiv ℝ (fun y : Euc n ↦ ψ (t^2 * ‖y-c‖^2)) x =
+      (deriv ψ (t^2 * ‖x-c‖^2) * t^2 * 2) • innerSL ℝ (x-c) := by
+    have hi := (((hasFDerivAt_id x).sub_const c).norm_sq).const_mul (t^2)
+    have hh := ((hψ.differentiable (by simp)) _).hasDerivAt.comp_hasFDerivAt x hi
+    convert hh.fderiv using 1 <;> try rfl
+    ext v
+    simp
+    ring
+  rw [tangentialGradient, inner_sub_left, real_inner_smul_left]
+  simp only [inner_gradient_left, hd, smul_apply,
+    innerSL_apply_apply, smul_eq_mul, real_inner_self_eq_norm_sq]
+  rw [real_inner_comm (x-c) (approximateNormal E x)]
+  ring
+
+theorem radialProfile_tsupport {n : ℕ} (c : Euc n) {ψ : ℝ → ℝ}
+    (hψz : ∀ s, 1 ≤ s → ψ s = 0) {t : ℝ} (ht : 0 < t) :
+    tsupport (fun x : Euc n ↦ ψ (t^2 * ‖x-c‖^2)) ⊆ Metric.closedBall c (1/t) := by
+  apply closure_minimal
+  · intro x hx
+    by_contra hh
+    have hr : 1/t < ‖x-c‖ := by simpa [Metric.mem_closedBall, dist_eq_norm] using hh
+    have htr : 1 < t * ‖x-c‖ := by
+      have := (div_lt_iff₀ ht).mp hr
+      nlinarith
+    have hsq : 1 ≤ t^2 * ‖x-c‖^2 := by nlinarith [sq_nonneg (t * ‖x-c‖ - 1)]
+    exact hx (hψz _ hsq)
+  · exact Metric.isClosed_closedBall
+
+theorem minimizing_radialDensityDerivative_nonpos {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    (c : Euc n) {M : ℝ} (hM : 0 < M) {ψ : ℝ → ℝ} (hψ : ContDiff ℝ ∞ ψ)
+    (hψz : ∀ s, 1 ≤ s → ψ s = 0) (hψd : ∀ s, deriv ψ s ≤ 0)
+    {t : ℝ} (ht : 1/M < t) :
+    (∫ x in Metric.ball c M, radialDensityDerivative c (n-1) ψ t x ∂perimeterMeasure E) ≤ 0 := by
+  let μ := (perimeterMeasure E).restrict (Metric.ball c M)
+  have : IsFiniteMeasure μ := ⟨by
+    rw [Measure.restrict_apply_univ, perimeterMeasure_open E Metric.isOpen_ball]
+    exact hmin.2.1 c M hM⟩
+  have hμ : ∀ᵐ x ∂μ, x ∈ Metric.closedBall c M :=
+    (ae_restrict_mem measurableSet_ball).mono fun _ hx ↦ Metric.ball_subset_closedBall hx
+  have hd := radialDensity_integral_hasDerivAt c M (n-1) hψ μ hμ t
+  have ht0 : 0 < t := (one_div_pos.mpr hM).trans ht
+  have htM : 1/t < M := (div_lt_iff₀ ht0).mpr (by
+    have := (div_lt_iff₀ hM).mp ht
+    nlinarith)
+  let V : Euc n → Euc n := fun x ↦ ψ (t^2 * ‖x-c‖^2) • (x-c)
+  have hf : ContDiff ℝ ∞ (fun x : Euc n ↦ ψ (t^2 * ‖x-c‖^2)) :=
+    hψ.comp (contDiff_const.mul ((contDiff_id.sub contDiff_const).norm_sq (𝕜 := ℝ)))
+  have hV : ContDiff ℝ ∞ V := hf.smul (contDiff_id.sub contDiff_const)
+  have hs : tsupport V ⊆ Metric.closedBall c (1/t) :=
+    (tsupport_smul_subset_left _ _).trans (radialProfile_tsupport c hψz ht0)
+  have hc : HasCompactSupport V :=
+    (isCompact_closedBall c (1/t)).of_isClosed_subset (isClosed_tsupport _) hs
+  have hz := minimizing_tangentialDivergence_integral_eq_zero hn hmin c hM hV hc
+    (hs.trans (Metric.closedBall_subset_ball htM))
+  have hi := (tangentialDivergence_integrable_ball_perimeter hmin c hM hV).const_mul (t^((n-1)-1))
+  have hcmp : ∀ᵐ x ∂μ,
+      radialDensityDerivative c (n-1) ψ t x ≤ t^((n-1)-1) * tangentialDivergence E V x := by
+    filter_upwards [ae_restrict_of_ae (minimizing_halfspaceBlowup_ae hn hmin)] with x hx
+    have hpow : t^(n-1) = t^((n-1)-1)*t := by
+      rw [← pow_succ, Nat.sub_add_cancel (by omega : 1 ≤ n-1)]
+    have heq : radialDensityDerivative c (n-1) ψ t x =
+        t^((n-1)-1) * tangentialDivergence E V x +
+        (2*t^((n-1)-1)*t^2*(inner ℝ (approximateNormal E x) (x-c))^2) *
+          deriv ψ (t^2*‖x-c‖^2) := by
+      dsimp [V, radialDensityDerivative]
+      rw [tangentialDivergence_radialProfile (by omega) E c hψ t x hx.1, hpow]
+      ring
+    rw [heq]
+    exact add_le_of_nonpos_right (mul_nonpos_of_nonneg_of_nonpos (by positivity) (hψd _))
+  have h := integral_mono_ae hd.1 hi hcmp
+  rw [integral_const_mul] at h
+  change _ ≤ _ * (∫ x in Metric.ball c M, tangentialDivergence E V x ∂perimeterMeasure E) at h
+  simpa only [hz, mul_zero] using h
+
+theorem minimizing_smoothedDensity_antitoneOn {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    (c : Euc n) {M : ℝ} (hM : 0 < M) {ψ : ℝ → ℝ} (hψ : ContDiff ℝ ∞ ψ)
+    (hψz : ∀ s, 1 ≤ s → ψ s = 0) (hψd : ∀ s, deriv ψ s ≤ 0) :
+    AntitoneOn (fun t ↦ ∫ x in Metric.ball c M,
+      radialDensityIntegrand c (n-1) ψ t x ∂perimeterMeasure E) (Ioi (1/M)) := by
+  let μ := (perimeterMeasure E).restrict (Metric.ball c M)
+  have : IsFiniteMeasure μ := ⟨by
+    rw [Measure.restrict_apply_univ, perimeterMeasure_open E Metric.isOpen_ball]
+    exact hmin.2.1 c M hM⟩
+  have hμ : ∀ᵐ x ∂μ, x ∈ Metric.closedBall c M :=
+    (ae_restrict_mem measurableSet_ball).mono fun _ hx ↦ Metric.ball_subset_closedBall hx
+  have hd := fun t ↦ (radialDensity_integral_hasDerivAt c M (n-1) hψ μ hμ t).2
+  apply antitoneOn_of_hasDerivWithinAt_nonpos (convex_Ioi (1/M))
+    (fun t _ ↦ (hd t).continuousAt.continuousWithinAt)
+    (fun t _ ↦ (hd t).hasDerivWithinAt)
+  intro t ht
+  exact minimizing_radialDensityDerivative_nonpos hn hmin c hM hψ hψz hψd
+    (by simpa only [interior_Ioi, mem_Ioi] using ht)
+
+def densityProfile (j : ℕ) (s : ℝ) : ℝ :=
+  Real.smoothTransition (((j : ℝ)+1)*(1-s))
+
+theorem densityProfile_contDiff (j : ℕ) : ContDiff ℝ ∞ (densityProfile j) :=
+  Real.smoothTransition.contDiff.comp (contDiff_const.mul (contDiff_const.sub contDiff_id))
+
+theorem densityProfile_mem_Icc (j : ℕ) (s : ℝ) : densityProfile j s ∈ Icc (0:ℝ) 1 :=
+  ⟨Real.smoothTransition.nonneg _, Real.smoothTransition.le_one _⟩
+
+theorem densityProfile_zero (j : ℕ) {s : ℝ} (hs : 1 ≤ s) : densityProfile j s = 0 := by
+  apply Real.smoothTransition.zero_of_nonpos
+  exact mul_nonpos_of_nonneg_of_nonpos (by positivity) (by linarith)
+
+theorem densityProfile_deriv_nonpos (j : ℕ) (s : ℝ) : deriv (densityProfile j) s ≤ 0 := by
+  have hm : Antitone (densityProfile j) := by
+    intro a b hab
+    apply Real.smoothTransition.monotone
+    apply mul_le_mul_of_nonneg_left _ (by positivity)
+    linarith
+  exact hm.deriv_nonpos
+
+theorem densityProfile_eventually_one {s : ℝ} (hs : s < 1) :
+    ∀ᶠ j : ℕ in atTop, densityProfile j s = 1 := by
+  obtain ⟨N, hN⟩ := exists_nat_gt (1/(1-s))
+  filter_upwards [eventually_ge_atTop N] with j hj
+  apply Real.smoothTransition.one_of_one_le
+  have hn : (N : ℝ) ≤ j := by exact_mod_cast hj
+  have hdiv : 1/(1-s) < (j:ℝ)+1 := by linarith
+  exact ((div_lt_iff₀ (by linarith : 0 < 1-s)).mp hdiv).le
+
+theorem radialDensityIntegrand_tendsto {n : ℕ} (c : Euc n) (k : ℕ)
+    {t : ℝ} (ht : 0 < t) (x : Euc n) :
+    Tendsto (fun j ↦ radialDensityIntegrand c k (densityProfile j) t x) atTop
+      (𝓝 ((Metric.ball c (1/t)).indicator (fun _ ↦ t^k) x)) := by
+  by_cases hx : x ∈ Metric.ball c (1/t)
+  · have hr : ‖x-c‖ < 1/t := by simpa [Metric.mem_ball, dist_eq_norm] using hx
+    have htr : t*‖x-c‖ < 1 := by
+      have := (lt_div_iff₀ ht).mp hr
+      nlinarith
+    have hq : t^2*‖x-c‖^2 < 1 := by
+      have := mul_self_lt_mul_self (mul_nonneg ht.le (norm_nonneg (x-c))) htr
+      nlinarith
+    rw [indicator_of_mem hx]
+    apply tendsto_const_nhds.congr'
+    filter_upwards [densityProfile_eventually_one hq] with j hj
+    simp [radialDensityIntegrand, hj]
+  · have hr : 1/t ≤ ‖x-c‖ := by simpa [Metric.mem_ball, dist_eq_norm] using hx
+    have htr : 1 ≤ t*‖x-c‖ := by
+      have := (div_le_iff₀ ht).mp hr
+      nlinarith
+    have hq : 1 ≤ t^2*‖x-c‖^2 := by nlinarith [sq_nonneg (t*‖x-c‖-1)]
+    rw [indicator_of_notMem hx]
+    have heq : (fun j ↦ radialDensityIntegrand c k (densityProfile j) t x) = fun _ ↦ (0:ℝ) := by
+      funext j
+      simp [radialDensityIntegrand, densityProfile_zero j hq]
+    rw [heq]
+    exact tendsto_const_nhds
+
+theorem radialDensity_integral_tendsto {n : ℕ}
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    (c : Euc n) {M : ℝ} (hM : 0 < M) {t : ℝ} (ht : 1/M < t) :
+    Tendsto (fun j ↦ ∫ x in Metric.ball c M,
+      radialDensityIntegrand c (n-1) (densityProfile j) t x ∂perimeterMeasure E) atTop
+      (𝓝 (t^(n-1) * (perimeterMeasure E).real (Metric.ball c (1/t)))) := by
+  let μ := (perimeterMeasure E).restrict (Metric.ball c M)
+  have : IsFiniteMeasure μ := ⟨by
+    rw [Measure.restrict_apply_univ, perimeterMeasure_open E Metric.isOpen_ball]
+    exact hmin.2.1 c M hM⟩
+  have ht0 : 0 < t := (one_div_pos.mpr hM).trans ht
+  have htM : 1/t < M := (div_lt_iff₀ ht0).mpr (by
+    have := (div_lt_iff₀ hM).mp ht
+    nlinarith)
+  have hh := tendsto_integral_of_dominated_convergence (μ := μ) (fun _ ↦ t^(n-1))
+    (fun j ↦ ((radialDensityIntegrand_continuous c (n-1) (densityProfile_contDiff j)).comp
+      (continuous_const.prodMk continuous_id)).aestronglyMeasurable)
+    (integrable_const _) (fun j ↦ ae_of_all μ fun x ↦ ?_)
+    (ae_of_all μ (radialDensityIntegrand_tendsto c (n-1) ht0))
+  · have heq : (∫ x, (Metric.ball c (1/t)).indicator (fun _ ↦ t^(n-1)) x ∂μ) =
+        t^(n-1) * (perimeterMeasure E).real (Metric.ball c (1/t)) := by
+      rw [integral_indicator_const (t^(n-1)) measurableSet_ball, smul_eq_mul, mul_comm]
+      congr 1
+      change (μ (Metric.ball c (1/t))).toReal = _
+      rw [Measure.restrict_apply measurableSet_ball,
+        inter_eq_left.mpr (Metric.ball_subset_ball htM.le)]
+      rfl
+    rwa [heq] at hh
+  · dsimp [radialDensityIntegrand]
+    rw [abs_of_nonneg (mul_nonneg (by positivity) (densityProfile_mem_Icc j _).1)]
+    exact mul_le_of_le_one_right (by positivity) (densityProfile_mem_Icc j _).2
+
+theorem minimizing_perimeter_density_monotone {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E) (c : Euc n) :
+    MonotoneOn (fun r : ℝ ↦ (perimeterMeasure E).real (Metric.ball c r) / r^(n-1)) (Ioi 0) := by
+  intro r hr R hR hrR
+  have hr0 : 0 < r := hr
+  have hR0 : 0 < R := hR
+  let M := 2*R
+  have hM : 0 < M := by dsimp [M]; positivity
+  have hRM : 1/M < 1/R := one_div_lt_one_div_of_lt hR0 (by dsimp [M]; linarith)
+  have htr : 1/R ≤ 1/r := one_div_le_one_div_of_le hr0 hrR
+  have hrM : 1/M < 1/r := hRM.trans_le htr
+  have hseq (j : ℕ) := minimizing_smoothedDensity_antitoneOn hn hmin c hM
+    (densityProfile_contDiff j) (fun s hs ↦ densityProfile_zero j hs)
+    (densityProfile_deriv_nonpos j) hRM hrM htr
+  have hh := le_of_tendsto_of_tendsto
+    (radialDensity_integral_tendsto hmin c hM hrM)
+    (radialDensity_integral_tendsto hmin c hM hRM) (Eventually.of_forall hseq)
+  simpa [one_div, div_eq_mul_inv, mul_comm] using hh
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+open scoped ENNReal Topology
+open scoped ENNReal Topology ContDiff
+open scoped Topology ENNReal
+namespace RPISingularHarnack
+
+theorem support_ray_inner_le_zero {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    {x v w : Euc n} (hv : HasHalfspaceBlowup E x v)
+    (hray : ∀ r : ℝ, 0 < r → r < 1 → x + r • w ∈ perimeterSupport E) :
+    inner ℝ w v ≤ 0 := by
+  by_contra hbad
+  have ha : 0 < inner ℝ w v := lt_of_not_ge hbad
+  let a := inner ℝ w v
+  let δ := a / 4
+  let A := ‖w‖ + δ + 1
+  have hδ : 0 < δ := by dsimp [δ, a]; positivity
+  have hA : 0 < A := by dsimp [A]; positivity
+  let f : Euc n → ℝ := fun y ↦ |E.indicator (fun _ ↦ (1 : ℝ)) y -
+    {z | inner ℝ (z - x) v < 0}.indicator (fun _ ↦ (1 : ℝ)) y|
+  have hfm : Measurable f := by
+    apply Measurable.abs
+    exact (measurable_const.indicator hmin.1).sub
+      (measurable_const.indicator (isOpen_lt
+        ((continuous_id.sub continuous_const).inner continuous_const) continuous_const).measurableSet)
+  have hfi (s : ℝ) : IntegrableOn f (Metric.ball x s) volume := by
+    apply (integrableOn_const measure_ball_lt_top.ne (C := (1 : ℝ))).mono' hfm.aestronglyMeasurable
+    apply ae_of_all
+    intro y
+    dsimp [f]
+    by_cases he : y ∈ E <;> by_cases hh : inner ℝ (y - x) v < 0 <;> simp [he, hh]
+  obtain ⟨ε, hε, hphase⟩ := minimizing_scaled_two_phase_volume_lower
+    (Real.HolderConjugate.conjExponent (by exact_mod_cast hn : 1 < (n : ℝ)))
+  have hlow (r : ℝ) (hr : 0 < r) (hr1 : r < 1) :
+      ε * (r * δ) ^ n ≤ ∫ y in Metric.ball x (r * A), f y := by
+    let B := Metric.closedBall (x + r • w) (r * δ)
+    have hsub : B ⊆ Metric.ball x (r * A) := by
+      intro y hy
+      have ht := dist_triangle y (x + r • w) x
+      have hd : dist (x + r • w) x = r * ‖w‖ := by
+        simp [dist_eq_norm, norm_smul, abs_of_pos hr]
+      rw [hd] at ht
+      have hb : dist y (x + r • w) ≤ r * δ := hy
+      change dist y x < r * A
+      dsimp [A]
+      nlinarith
+    have hpos (y : Euc n) (hy : y ∈ B) : 0 < inner ℝ (y - x) v := by
+      have hi := (abs_le.mp (abs_real_inner_le_norm (y - (x + r • w)) v)).1
+      rw [hv.1, mul_one] at hi
+      have hb : ‖y - (x + r • w)‖ ≤ r * δ := by
+        simpa only [B, Metric.mem_closedBall, dist_eq_norm] using hy
+      have he : inner ℝ (y - x) v = inner ℝ (y - (x + r • w)) v + r * a := by
+        simp [a, inner_sub_left, inner_add_left, real_inner_smul_left]
+        ring
+      dsimp [δ] at hb
+      rw [he]
+      nlinarith
+    have heq : (∫ y in B, f y) = volume.real (E ∩ B) := by
+      calc
+        (∫ y in B, f y) = ∫ y in B, E.indicator (fun _ ↦ (1 : ℝ)) y := by
+          apply setIntegral_congr_fun measurableSet_closedBall
+          intro y hy
+          have hh : ¬ inner ℝ (y - x) v < 0 := not_lt.mpr (hpos y hy).le
+          dsimp [f]
+          by_cases he : y ∈ E <;> simp [he, hh]
+        _ = volume.real (E ∩ B) := by
+          simp only [integral_indicator_const _ hmin.1,
+            measureReal_restrict_apply hmin.1, smul_eq_mul, mul_one]
+    exact le_trans (hphase E hmin _ (hray r hr hr1) _ (mul_pos hr hδ)).1.le
+      (heq ▸ setIntegral_mono_set (hfi _) (ae_of_all _ (fun y ↦ abs_nonneg _))
+        (Eventually.of_forall hsub))
+  have ht : Tendsto (fun r : ℝ ↦ r * A) (𝓝[>] 0) (𝓝[>] 0) := by
+    apply tendsto_nhdsWithin_iff.mpr
+    refine ⟨?_, ?_⟩
+    · simpa using ((tendsto_id : Tendsto (fun r : ℝ ↦ r) (𝓝 0) (𝓝 0)).mono_left nhdsWithin_le_nhds).mul_const A
+    · filter_upwards [self_mem_nhdsWithin] with r hr
+      exact mul_pos hr hA
+  have hlim := hv.2.comp ht
+  change Tendsto (fun r : ℝ ↦ ((r * A) ^ n)⁻¹ * ∫ y in Metric.ball x (r * A), f y)
+    (𝓝[>] 0) (𝓝 0) at hlim
+  have hb : ε * δ ^ n / A ^ n ≤ (0 : ℝ) := by
+    apply ge_of_tendsto hlim
+    filter_upwards [self_mem_nhdsWithin, (eventually_lt_nhds (show (0 : ℝ) < 1 by norm_num)).filter_mono nhdsWithin_le_nhds] with r hr hr1
+    change 0 < r at hr
+    have hl := hlow r hr hr1
+    rw [mul_pow] at hl
+    apply (div_le_iff₀ (pow_pos hA n)).mpr
+    have he : (((r * A) ^ n)⁻¹ * ∫ y in Metric.ball x (r * A), f y) * A ^ n =
+        (r ^ n)⁻¹ * ∫ y in Metric.ball x (r * A), f y := by
+      rw [mul_pow]
+      field_simp
+    rw [he]
+    have hh := mul_le_mul_of_nonneg_left hl (inv_nonneg.mpr (pow_pos hr n).le)
+    convert hh using 1
+    field_simp [ne_of_gt hr]
+  have : 0 < ε * δ ^ n / A ^ n := div_pos (mul_pos hε (pow_pos hδ n)) (pow_pos hA n)
+  linarith
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+open scoped ENNReal Topology
+open scoped ENNReal Topology ContDiff
+open scoped Topology ENNReal
+namespace RPISingularHarnack
+
+theorem minimizing_cone_halfspace_radial_tangent {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E) (hcone : IsCone E)
+    {x v : Euc n} (hx : x ∈ perimeterSupport E) (hv : HasHalfspaceBlowup E x v) :
+    inner ℝ x v = 0 := by
+  have hc := isCone_perimeterSupport hmin.1 hcone
+  have hplus : inner ℝ x v ≤ 0 := support_ray_inner_le_zero hn hmin hv (by
+    intro r hr _
+    have hh := (hc (1 + r) (by linarith) x).mpr hx
+    simpa only [add_smul, one_smul] using hh)
+  have hminus : inner ℝ (-x) v ≤ 0 := support_ray_inner_le_zero hn hmin hv (by
+    intro r _ hr
+    have hh := (hc (1 - r) (by linarith) x).mpr hx
+    rw [sub_smul, one_smul] at hh
+    simpa only [smul_neg, sub_eq_add_neg] using hh)
+  rw [inner_neg_left] at hminus
+  linarith
+
+theorem minimizing_cone_radial_tangent_ae {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E) (hcone : IsCone E) :
+    ∀ᵐ x ∂perimeterMeasure E, inner ℝ x (approximateNormal E x) = 0 := by
+  have hs : ∀ᵐ x ∂perimeterMeasure E, x ∈ perimeterSupport E := by
+    rw [← perimeterMeasure_support]
+    exact (perimeterMeasure E).support_mem_ae
+  filter_upwards [minimizing_halfspaceBlowup_ae hn hmin, hs] with x hx hsx
+  exact minimizing_cone_halfspace_radial_tangent hn hmin hcone hsx hx
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/Space.lean b/lean/OAI/Geometry/Perimeter/Regularity/Space.lean
new file mode 100644
index 0000000000000000000000000000000000000000..63659c686f999729091e6baba71b70d627f047a4
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/Space.lean
@@ -0,0 +1,299 @@
+import Mathlib
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+namespace RPI
+namespace Boundary
+
+abbrev Space (n : ℕ) := EuclideanSpace ℝ (Fin n)
+
+noncomputable def divergence {n : ℕ} (X : Space n → Space n) (x : Space n) : ℝ :=
+  ∑ i : Fin n, (fderiv ℝ X x (EuclideanSpace.single i 1)) i
+
+def PerimeterTest {n : ℕ} (U : Set (Space n)) (X : Space n → Space n) : Prop :=
+  ContDiff ℝ 1 X ∧ HasCompactSupport X ∧ tsupport X ⊆ U ∧ ∀ x, ‖X x‖ ≤ 1
+
+noncomputable def perimeter {n : ℕ} (K U : Set (Space n)) : ℝ≥0∞ :=
+  ⨆ X : Space n → Space n, ⨆ (_ : PerimeterTest U X),
+    ENNReal.ofReal (∫ x in K, divergence X x)
+
+theorem perimeter_mono {n : ℕ} (K : Set (Space n)) {U V : Set (Space n)}
+    (hUV : U ⊆ V) : perimeter K U ≤ perimeter K V := by
+  unfold perimeter
+  refine iSup_le fun X ↦ iSup_le fun hX ↦ ?_
+  apply le_iSup_of_le X
+  apply le_iSup_of_le (show PerimeterTest V X from
+    ⟨hX.1, hX.2.1, hX.2.2.1.trans hUV, hX.2.2.2⟩)
+  exact le_rfl
+
+theorem perimeter_congr_ae {n : ℕ} {K L : Set (Space n)}
+    (hKL : K =ᵐ[(volume : Measure (Space n))] L) (U : Set (Space n)) :
+    perimeter K U = perimeter L U := by
+  simp only [perimeter, Measure.restrict_congr_set hKL]
+
+def LocallyFinitePerimeter {n : ℕ} (K : Set (Space n)) : Prop :=
+  MeasurableSet K ∧ ∀ R : ℝ, 0 < R → perimeter K (Metric.ball 0 R) < ⊤
+
+def CompactVariationIn {n : ℕ} (K L U : Set (Space n)) : Prop :=
+  ∃ F : Set (Space n), IsCompact F ∧ F ⊆ U ∧
+    ∀ᵐ x ∂(volume : Measure (Space n)), x ∉ F → (x ∈ K ↔ x ∈ L)
+
+def LocallyPerimeterMinimizing {n : ℕ} (K : Set (Space n)) : Prop :=
+  LocallyFinitePerimeter K ∧
+    ∀ (L U : Set (Space n)), LocallyFinitePerimeter L → IsOpen U → Bornology.IsBounded U →
+      CompactVariationIn K L U → perimeter K U ≤ perimeter L U
+
+def perimeterSupport {n : ℕ} (K : Set (Space n)) : Set (Space n) :=
+  {x | ∀ r : ℝ, 0 < r → 0 < perimeter K (Metric.ball x r)}
+
+theorem isClosed_perimeterSupport {n : ℕ} (K : Set (Space n)) :
+    IsClosed (perimeterSupport K) := by
+  rw [← isOpen_compl_iff, Metric.isOpen_iff]
+  intro x hx
+  change ¬ ∀ r : ℝ, 0 < r → 0 < perimeter K (Metric.ball x r) at hx
+  push Not at hx
+  obtain ⟨r, hr, hPr⟩ := hx
+  refine ⟨r / 2, half_pos hr, ?_⟩
+  intro y hy
+  change ¬ ∀ s : ℝ, 0 < s → 0 < perimeter K (Metric.ball y s)
+  intro h
+  have hsub : Metric.ball y (r / 2) ⊆ Metric.ball x r := by
+    intro z hz
+    have hzy := Metric.mem_ball.mp hz
+    have hyx := Metric.mem_ball.mp hy
+    have ht := dist_triangle z y x
+    change dist z x < r
+    linarith
+  exact not_lt_of_ge ((perimeter_mono K hsub).trans hPr) (h (r / 2) (half_pos hr))
+
+def IsCone {n : ℕ} (K : Set (Space n)) : Prop :=
+  ∀ r : ℝ, 0 < r → ∀ᵐ x ∂(volume : Measure (Space n)), (r • x ∈ K ↔ x ∈ K)
+
+noncomputable def surfaceMeasure {n : ℕ} (S : Set (Space n)) : Measure (Space n) :=
+  (Measure.euclideanHausdorffMeasure (n - 1)).restrict S
+
+theorem surfaceMeasure_zero_of_dimH_lt {n : ℕ} (S T : Set (Space n))
+    (hT : dimH T < (n - 1 : ℕ)) : surfaceMeasure S T = 0 := by
+  have hraw : Measure.hausdorffMeasure ((n - 1 : ℕ) : ℝ) T = 0 := by
+    simpa only [NNReal.coe_natCast] using
+      hausdorffMeasure_of_dimH_lt (d := (n - 1 : ℕ)) hT
+  have hnorm : (Measure.euclideanHausdorffMeasure (n - 1) : Measure (Space n)) T = 0 := by
+    simp only [Measure.euclideanHausdorffMeasure, Measure.smul_apply, hraw, smul_zero]
+  exact le_antisymm ((Measure.restrict_apply_le S T).trans_eq hnorm) zero_le
+
+theorem surfaceMeasure_zero_of_singular_dim {n : ℕ} (hn : 8 ≤ n)
+    (S T : Set (Space n)) (hT : dimH T ≤ (n - 8 : ℕ)) : surfaceMeasure S T = 0 := by
+  apply surfaceMeasure_zero_of_dimH_lt S T
+  apply hT.trans_lt
+  exact_mod_cast (show n - 8 < n - 1 by omega)
+
+theorem contact_measure_zero {n : ℕ} (S S' Z : Set (Space n))
+    (hS : MeasurableSet S') (hcontact : S ∩ S' ⊆ Z)
+    (hZ : dimH Z < (n - 1 : ℕ)) (U : Set (Space n)) :
+    (surfaceMeasure S).restrict U S' = 0 := by
+  have hbase : surfaceMeasure S S' = 0 := by
+    rw [surfaceMeasure, Measure.restrict_apply hS]
+    have hraw : Measure.hausdorffMeasure ((n - 1 : ℕ) : ℝ) Z = 0 := by
+      simpa only [NNReal.coe_natCast] using
+        hausdorffMeasure_of_dimH_lt (d := (n - 1 : ℕ)) hZ
+    have hnorm : (Measure.euclideanHausdorffMeasure (n - 1) : Measure (Space n)) Z = 0 := by
+      simp only [Measure.euclideanHausdorffMeasure, Measure.smul_apply, hraw, smul_zero]
+    apply le_antisymm _ zero_le
+    apply (measure_mono ?_).trans_eq hnorm
+    exact fun x hx ↦ hcontact ⟨hx.2, hx.1⟩
+  exact le_antisymm ((Measure.restrict_apply_le U S').trans_eq hbase) zero_le
+
+noncomputable def tangentPlane {n : ℕ} (S : Set (Space n)) (x : Space n) :
+    Submodule ℝ (Space n) :=
+  Submodule.span ℝ {v | ∃ γ : ℝ → Space n,
+    γ 0 = x ∧ HasDerivAt γ v 0 ∧ ∀ᶠ t in 𝓝 (0 : ℝ), γ t ∈ S}
+
+noncomputable def surfaceGradient {n : ℕ} (S : Set (Space n))
+    (f : Space n → ℝ) (x : Space n) : Space n :=
+  ((tangentPlane S x).orthogonalProjectionOnto (gradient f x) : Space n)
+
+def smoothGradientGraph {n : ℕ} (S U : Set (Space n)) :
+    Set (Lp ℝ 2 ((surfaceMeasure S).restrict U) ×
+         Lp (Space n) 2 ((surfaceMeasure S).restrict U)) :=
+  {z | ∃ f : Space n → ℝ, ContDiff ℝ ∞ f ∧
+    ∃ hf : MemLp f 2 ((surfaceMeasure S).restrict U),
+    ∃ hDf : MemLp (surfaceGradient S f) 2 ((surfaceMeasure S).restrict U),
+      z = (hf.toLp f, hDf.toLp (surfaceGradient S f))}
+
+def IsLocalSobolevPair {n : ℕ} (S U : Set (Space n))
+    (h : Space n → ℝ) (G : Space n → Space n) : Prop :=
+  ∀ V : Set (Space n), IsOpen V → IsCompact (closure V) → closure V ⊆ U →
+    ∃ hh : MemLp h 2 ((surfaceMeasure S).restrict V),
+    ∃ hG : MemLp G 2 ((surfaceMeasure S).restrict V),
+      (hh.toLp h, hG.toLp G) ∈ closure (smoothGradientGraph S V)
+
+def IsWeakLaplaceSupersolution {n : ℕ} (S U : Set (Space n))
+    (h : Space n → ℝ) (G : Space n → Space n) : Prop :=
+  IsLocalSobolevPair S U h G ∧
+    ∀ φ : Space n → ℝ, ContDiff ℝ ∞ φ → HasCompactSupport φ → tsupport φ ⊆ U →
+      (∀ x, 0 ≤ φ x) →
+      Integrable (fun x ↦ inner ℝ (G x) (surfaceGradient S φ x)) (surfaceMeasure S) ∧
+      0 ≤ ∫ x, inner ℝ (G x) (surfaceGradient S φ x) ∂surfaceMeasure S
+
+end Boundary
+end RPI
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+namespace RPI.Boundary
+
+theorem surfaceGradient_congr_trace {n : ℕ} {S : Set (Space n)}
+    {f g : Space n → ℝ} {x : Space n}
+    (hf : DifferentiableAt ℝ f x) (hg : DifferentiableAt ℝ g x)
+    (heq : ∀ᶠ y in 𝓝 x, y ∈ S → f y = g y) :
+    surfaceGradient S f x = surfaceGradient S g x := by
+  have hker : tangentPlane S x ≤ (fderiv ℝ f x - fderiv ℝ g x).ker := by
+    apply Submodule.span_le.mpr
+    intro v hv
+    obtain ⟨γ, hγx, hγd, hγS⟩ := hv
+    have hfe : HasDerivAt (f ∘ γ) (fderiv ℝ f x v) 0 := by
+      exact (show HasFDerivAt f (fderiv ℝ f x) (γ 0) by
+        simpa only [hγx] using hf.hasFDerivAt).comp_hasDerivAt 0 hγd
+    have hge : HasDerivAt (g ∘ γ) (fderiv ℝ g x v) 0 := by
+      exact (show HasFDerivAt g (fderiv ℝ g x) (γ 0) by
+        simpa only [hγx] using hg.hasFDerivAt).comp_hasDerivAt 0 hγd
+    have hcomp : (f ∘ γ) =ᶠ[𝓝 (0 : ℝ)] (g ∘ γ) := by
+      have ht : Tendsto γ (𝓝 (0 : ℝ)) (𝓝 x) := hγx ▸ hγd.continuousAt
+      filter_upwards [ht.eventually heq, hγS] with t ht hS
+      exact ht hS
+    have hd : fderiv ℝ f x v = fderiv ℝ g x v :=
+      hfe.unique (hge.congr_of_eventuallyEq hcomp)
+    change fderiv ℝ f x v - fderiv ℝ g x v = 0
+    exact sub_eq_zero.mpr hd
+  have ho : gradient f x - gradient g x ∈ (tangentPlane S x)ᗮ := by
+    rw [Submodule.mem_orthogonal]
+    intro v hv
+    have hk := hker hv
+    change fderiv ℝ f x v - fderiv ℝ g x v = 0 at hk
+    simpa only [inner_sub_right, inner_gradient_right, starRingEnd_apply, star_trivial] using hk
+  have hp := (Submodule.orthogonalProjectionOnto_eq_zero_iff).mpr ho
+  have hp' : (tangentPlane S x).orthogonalProjectionOnto (gradient f x) =
+      (tangentPlane S x).orthogonalProjectionOnto (gradient g x) := by
+    simpa only [map_sub, sub_eq_zero] using hp
+  exact congrArg Subtype.val hp'
+
+def HasSmoothTraceAt {n : ℕ} (S : Set (Space n)) (h : Space n → ℝ)
+    (x : Space n) : Prop :=
+  ∃ U : Set (Space n), IsOpen U ∧ x ∈ U ∧ ∃ f : Space n → ℝ,
+    ContDiffOn ℝ ∞ f U ∧ EqOn f h (S ∩ U)
+
+theorem exists_contDiff_eqOn_of_local_trace {n : ℕ} {S : Set (Space n)}
+    (hS : IsClosed S) (h : Space n → ℝ)
+    (hloc : ∀ x ∈ S, HasSmoothTraceAt S h x) :
+    ∃ f : Space n → ℝ, ContDiff ℝ ∞ f ∧ EqOn f h S := by
+  classical
+  let t : Space n → Set ℝ := fun x ↦ if x ∈ S then {h x} else univ
+  have hc : ∀ x, Convex ℝ (t x) := by
+    intro x
+    dsimp [t]
+    split_ifs
+    · exact convex_singleton _
+    · exact convex_univ
+  have hlocal : ∀ x : Space n, ∃ U ∈ 𝓝 x, ∃ g : Space n → ℝ,
+      ContMDiffOn (modelWithCornersSelf ℝ (Space n)) (modelWithCornersSelf ℝ ℝ) ∞ g U ∧
+      ∀ y ∈ U, g y ∈ t y := by
+    intro x
+    by_cases hx : x ∈ S
+    · obtain ⟨U, hU, hxU, f, hf, he⟩ := hloc x hx
+      refine ⟨U, hU.mem_nhds hxU, f, hf.contMDiffOn, ?_⟩
+      intro y hy
+      by_cases hys : y ∈ S
+      · simpa only [t, ite_eq_left hys, mem_singleton_iff] using he ⟨hys, hy⟩
+      · simp only [t, ite_eq_right hys, mem_univ]
+    · refine ⟨Sᶜ, hS.isOpen_compl.mem_nhds hx, fun _ ↦ 0, contMDiffOn_const, ?_⟩
+      intro y hy
+      simp only [t, ite_eq_right hy, mem_univ]
+  obtain ⟨f, hf⟩ := exists_contMDiffMap_forall_mem_convex_of_local
+    (modelWithCornersSelf ℝ (Space n)) (n := (⊤ : ℕ∞)) hc hlocal
+  refine ⟨f, f.contMDiff.contDiff, ?_⟩
+  intro x hx
+  simpa only [t, ite_eq_left hx, mem_singleton_iff] using hf x
+
+def HasSmoothTraceJetAt {n : ℕ} (S : Set (Space n)) (h : Space n → ℝ)
+    (G : Space n → Space n) (x : Space n) : Prop :=
+  ∃ U : Set (Space n), IsOpen U ∧ x ∈ U ∧ ∃ f : Space n → ℝ,
+    ContDiffOn ℝ ∞ f U ∧ EqOn f h (S ∩ U) ∧ surfaceGradient S f x = G x
+
+lemma HasSmoothTraceJetAt.trace {n : ℕ} {S : Set (Space n)} {h : Space n → ℝ}
+    {G : Space n → Space n} {x : Space n} (hx : HasSmoothTraceJetAt S h G x) :
+    HasSmoothTraceAt S h x := by
+  obtain ⟨U, hU, hxU, f, hf, he, _⟩ := hx
+  exact ⟨U, hU, hxU, f, hf, he⟩
+
+theorem surfaceGradient_mul {n : ℕ} {S : Set (Space n)}
+    {f g : Space n → ℝ} {x : Space n}
+    (hf : DifferentiableAt ℝ f x) (hg : DifferentiableAt ℝ g x) :
+    surfaceGradient S (fun y ↦ f y * g y) x =
+      f x • surfaceGradient S g x + g x • surfaceGradient S f x := by
+  have hgrad : gradient (fun y ↦ f y * g y) x =
+      f x • gradient g x + g x • gradient f x := by
+    apply (InnerProductSpace.toDual ℝ (Space n)).injective
+    simp only [map_add, map_smul, toDual_gradient]
+    exact fderiv_mul hf hg
+  simp only [surfaceGradient, hgrad, map_add, map_smul, Submodule.coe_add,
+    Submodule.coe_smul]
+
+@[simp] theorem surfaceGradient_const {n : ℕ} (S : Set (Space n))
+    (c : ℝ) (x : Space n) : surfaceGradient S (fun _ ↦ c) x = 0 := by
+  simp only [surfaceGradient, gradient_fun_const, map_zero, Submodule.coe_zero]
+
+theorem exists_smooth_cutoff_trace {n : ℕ} {S Z : Set (Space n)}
+    (hS : IsClosed S) {h : Space n → ℝ} {G : Space n → Space n}
+    (hreg : ∀ x ∈ S \ Z, HasSmoothTraceJetAt S h G x)
+    (χ : Space n → ℝ) (hχ : ContDiff ℝ ∞ χ)
+    (hzero : ∃ U : Set (Space n), IsOpen U ∧ Z ⊆ U ∧ ∀ x ∈ U, χ x = 0) :
+    ∃ F : Space n → ℝ, ContDiff ℝ ∞ F ∧ EqOn F (fun x ↦ χ x * h x) S ∧
+      EqOn (surfaceGradient S F)
+        (fun x ↦ χ x • G x + h x • surfaceGradient S χ x) S := by
+  have hj : ∀ x ∈ S, HasSmoothTraceJetAt S (fun x ↦ χ x * h x)
+      (fun x ↦ χ x • G x + h x • surfaceGradient S χ x) x := by
+    intro x hx
+    by_cases hxZ : x ∈ Z
+    · obtain ⟨U, hU, hZU, he⟩ := hzero
+      have hxU := hZU hxZ
+      refine ⟨U, hU, hxU, fun _ ↦ 0, contDiffOn_const, ?_, ?_⟩
+      · intro y hy
+        simp only [he y hy.2, zero_mul]
+      · have hχgrad : gradient χ x = 0 := by
+          have heq : χ =ᶠ[𝓝 x] (fun _ ↦ (0 : ℝ)) := by
+            filter_upwards [hU.mem_nhds hxU] with y hy using he y hy
+          simpa only [gradient_fun_const] using heq.gradient_eq
+        simp only [he x hxU, zero_smul, surfaceGradient, gradient_fun_const,
+          hχgrad, map_zero, Submodule.coe_zero, smul_zero, add_zero]
+    · obtain ⟨U, hU, hxU, f, hf, he, hG⟩ := hreg x ⟨hx, hxZ⟩
+      refine ⟨U, hU, hxU, fun y ↦ χ y * f y, hχ.contDiffOn.mul hf, ?_, ?_⟩
+      · intro y hy
+        exact congrArg (fun a ↦ χ y * a) (he hy)
+      · rw [surfaceGradient_mul (hχ.differentiable (by simp) x)
+            ((hf.contDiffAt (hU.mem_nhds hxU)).differentiableAt (by simp)),
+          hG, he ⟨hx, hxU⟩]
+  obtain ⟨F, hF, hFe⟩ := exists_contDiff_eqOn_of_local_trace hS
+    (fun x ↦ χ x * h x) (fun x hx ↦ (hj x hx).trace)
+  refine ⟨F, hF, hFe, ?_⟩
+  intro x hx
+  obtain ⟨U, hU, hxU, f, hf, he, hG⟩ := hj x hx
+  calc
+    surfaceGradient S F x = surfaceGradient S f x := by
+      apply surfaceGradient_congr_trace (hF.differentiable (by simp) x)
+        ((hf.contDiffAt (hU.mem_nhds hxU)).differentiableAt (by simp))
+      filter_upwards [hU.mem_nhds hxU] with y hy
+      intro hys
+      exact (hFe hys).trans (he ⟨hys, hy⟩).symm
+    _ = _ := hG
+
+end RPI.Boundary
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/SphericalFrankel.lean b/lean/OAI/Geometry/Perimeter/Regularity/SphericalFrankel.lean
new file mode 100644
index 0000000000000000000000000000000000000000..10ce2b632a34dfd2eebb2c7ffdcdb70c21462ef0
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/SphericalFrankel.lean
@@ -0,0 +1,449 @@
+import OAI.Geometry.Perimeter.Regularity.GeoComposition
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+namespace RPISingularHarnack
+namespace SphericalFrankel
+
+abbrev Frame (n : ℕ) := Euc n × Euc n
+
+def IsFrame {n : ℕ} (e : Frame n) : Prop :=
+  ‖e.1‖ = 1 ∧ ‖e.2‖ = 1 ∧ inner ℝ e.1 e.2 = 0
+
+def proj {n : ℕ} (e : Frame n) (v : Euc n) : Euc n :=
+  v - (inner ℝ e.1 v) • e.1 - (inner ℝ e.2 v) • e.2
+
+def angle {n : ℕ} (e f : Frame n) : ℝ := inner ℝ e.1 f.1
+
+def defect {n : ℕ} (e f : Frame n) : ℝ := ‖proj e f.1‖ ^ 2
+
+def traceProduct {n : ℕ} (e f : Frame n) : ℝ :=
+  (n : ℝ) - 4 + angle e f ^ 2 + (inner ℝ e.1 f.2) ^ 2 +
+    (inner ℝ e.2 f.1) ^ 2 + (inner ℝ e.2 f.2) ^ 2
+
+def cross {n : ℕ} (e f : Frame n) : ℝ := inner ℝ (proj e f.1) (proj f e.1)
+
+lemma proj_inner_self {n : ℕ} {e : Frame n} (he : IsFrame e) (v : Euc n) :
+    inner ℝ (proj e v) (proj e v) =
+      ‖v‖ ^ 2 - (inner ℝ e.1 v) ^ 2 - (inner ℝ e.2 v) ^ 2 := by
+  have h11 : inner ℝ e.1 e.1 = 1 := by simp [he.1]
+  have h22 : inner ℝ e.2 e.2 = 1 := by simp [he.2.1]
+  have h21 : inner ℝ e.2 e.1 = 0 := by rw [real_inner_comm, he.2.2]
+  simp only [proj, inner_sub_left, inner_sub_right, inner_smul_left, inner_smul_right,
+    conj_trivial, h11, h22, he.2.2, h21, real_inner_self_eq_norm_sq]
+  rw [real_inner_comm v e.1, real_inner_comm v e.2]
+  ring
+
+lemma defect_eq {n : ℕ} {e f : Frame n} (he : IsFrame e) (hf : IsFrame f) :
+    defect e f = 1 - angle e f ^ 2 - (inner ℝ e.2 f.1) ^ 2 := by
+  rw [defect, ← real_inner_self_eq_norm_sq, proj_inner_self he, hf.1]
+  simp [angle]
+
+lemma cross_eq {n : ℕ} {e f : Frame n} (he : IsFrame e) (hf : IsFrame f) :
+    cross e f = angle e f * (-1 + angle e f ^ 2 +
+      (inner ℝ e.2 f.1) ^ 2 + (inner ℝ e.1 f.2) ^ 2) +
+      (inner ℝ e.2 f.1) * (inner ℝ e.1 f.2) * (inner ℝ e.2 f.2) := by
+  have hs : inner ℝ f.1 f.2 = 0 := hf.2.2
+  have ht : inner ℝ e.2 e.1 = 0 := by rw [real_inner_comm, he.2.2]
+  simp only [cross, proj, inner_sub_left, inner_sub_right, inner_smul_left,
+    inner_smul_right, conj_trivial, hs, ht, real_inner_self_eq_norm_sq, he.1, hf.1]
+  dsimp [angle]
+  simp only [real_inner_comm f.1 e.1, real_inner_comm f.2 e.1]
+  ring
+
+lemma cross_abs_le {n : ℕ} (e f : Frame n) :
+    |cross e f| ≤ (defect e f + defect f e) / 2 := by
+  have h := abs_real_inner_le_norm (proj e f.1) (proj f e.1)
+  dsimp [cross, defect]
+  nlinarith [sq_nonneg (‖proj e f.1‖ - ‖proj f e.1‖)]
+
+lemma angle_abs_le_one {n : ℕ} {e f : Frame n} (he : IsFrame e) (hf : IsFrame f) :
+    |angle e f| ≤ 1 := by
+  simpa [angle, he.1, hf.1] using abs_real_inner_le_norm e.1 f.1
+
+lemma traceProduct_eq {n : ℕ} {e f : Frame n} (he : IsFrame e) (hf : IsFrame f) :
+    traceProduct e f = (n : ℝ) - 2 - angle e f ^ 2 +
+      (inner ℝ e.2 f.2) ^ 2 - (defect e f + defect f e) := by
+  rw [traceProduct, defect_eq he hf, defect_eq hf he]
+  rw [real_inner_comm f.2 e.1]
+  have h : angle f e = angle e f := real_inner_comm _ _
+  rw [h]
+  ring
+
+lemma scalar_gap_bound {c f a b d : ℝ}
+    (hc0 : 0 ≤ c) (hc1 : c < 1) (hf0 : 0 ≤ f) (hfc : f ≤ c)
+    (ha : |a| ≤ 1) (hb : |b| ≤ 1) (hd : |d| ≤ 1)
+    (hA : 0 ≤ 1 - f ^ 2 - a ^ 2) (hB : 0 ≤ 1 - f ^ 2 - b ^ 2)
+    (hcross : |f * (-1 + f ^ 2 + a ^ 2 + b ^ 2) + a * b * d| ≤
+      ((1 - f ^ 2 - a ^ 2) + (1 - f ^ 2 - b ^ 2)) / 2) :
+    f ^ 2 - d ^ 2 ≤ 3 / (1 - c ^ 2) ^ 2 *
+      ((1 - f ^ 2 - a ^ 2) + (1 - f ^ 2 - b ^ 2)) := by
+  let s := 1 - f ^ 2
+  let U := (1 - f ^ 2 - a ^ 2) + (1 - f ^ 2 - b ^ 2)
+  have hf1 : f < 1 := hfc.trans_lt hc1
+  have hs : 0 < s := by dsimp [s]; nlinarith
+  have hs1 : s ≤ 1 := by dsimp [s]; nlinarith [sq_nonneg f]
+  have hU : 0 ≤ U := add_nonneg hA hB
+  have ha2 : a ^ 2 ≤ s := by dsimp [s]; linarith
+  have hb2 : b ^ 2 ≤ s := by dsimp [s]; linarith
+  have hab : a ^ 2 * b ^ 2 ≤ s ^ 2 := by
+    simpa [pow_two] using mul_le_mul ha2 hb2 (sq_nonneg b) hs.le
+  have habd : |a * b * d| ≤ 1 := by
+    rw [abs_mul, abs_mul]
+    calc
+      |a| * |b| * |d| ≤ 1 * 1 * 1 :=
+        mul_le_mul (mul_le_mul ha hb (abs_nonneg b) (by norm_num)) hd
+          (abs_nonneg d) (by norm_num)
+      _ = 1 := by ring
+  have hfs : |f * s| ≤ 1 := by
+    rw [abs_of_nonneg (mul_nonneg hf0 hs.le)]
+    nlinarith
+  have hC : |f * s + a * b * d| ≤ (3 / 2 : ℝ) * U := by
+    have hid : f * s + a * b * d =
+        (f * (-1 + f ^ 2 + a ^ 2 + b ^ 2) + a * b * d) + f * U := by
+      dsimp [s, U]; ring
+    rw [hid]
+    calc
+      _ ≤ |f * (-1 + f ^ 2 + a ^ 2 + b ^ 2) + a * b * d| + |f * U| := abs_add_le _ _
+      _ ≤ U / 2 + f * U := by
+        rw [abs_of_nonneg (mul_nonneg hf0 hU)]
+        change _ ≤ ((1 - f ^ 2 - a ^ 2) + (1 - f ^ 2 - b ^ 2)) / 2 + f * U
+        linarith
+      _ ≤ (3 / 2 : ℝ) * U := by nlinarith
+  have hD : |f * s - a * b * d| ≤ 2 :=
+    (abs_sub _ _).trans (by linarith)
+  have hprod : |(f * s) ^ 2 - (a * b * d) ^ 2| ≤ 3 * U := by
+    have hid : (f * s) ^ 2 - (a * b * d) ^ 2 =
+        (f * s + a * b * d) * (f * s - a * b * d) := by ring
+    rw [hid, abs_mul]
+    calc
+      _ ≤ ((3 / 2 : ℝ) * U) * 2 :=
+        mul_le_mul hC hD (abs_nonneg _) (by positivity)
+      _ = 3 * U := by ring
+  have hgap : (f ^ 2 - d ^ 2) * s ^ 2 ≤ 3 * U := by
+    have hmul := mul_le_mul_of_nonneg_right hab (sq_nonneg d)
+    have hsq := (le_abs_self ((f * s) ^ 2 - (a * b * d) ^ 2)).trans hprod
+    nlinarith only [hmul, hsq]
+  have hε : 0 < 1 - c ^ 2 := by nlinarith
+  have hεs : 1 - c ^ 2 ≤ s := by dsimp [s]; nlinarith
+  have hε2 : (1 - c ^ 2) ^ 2 ≤ s ^ 2 := by nlinarith
+  have hεp : 0 < (1 - c ^ 2) ^ 2 := sq_pos_of_pos hε
+  change f ^ 2 - d ^ 2 ≤ 3 / (1 - c ^ 2) ^ 2 * U
+  rw [div_mul_eq_mul_div, le_div_iff₀ hεp]
+  by_cases hsign : 0 ≤ f ^ 2 - d ^ 2
+  · exact (mul_le_mul_of_nonneg_left hε2 hsign).trans hgap
+  · have hg : f ^ 2 - d ^ 2 ≤ 0 := le_of_lt (lt_of_not_ge hsign)
+    exact (mul_nonpos_of_nonpos_of_nonneg hg hεp.le).trans (by positivity)
+
+lemma traceProduct_lower_bound {n : ℕ} {e f : Frame n}
+    (he : IsFrame e) (hf : IsFrame f) {c : ℝ} (hc0 : 0 ≤ c) (hc1 : c < 1)
+    (hangle0 : 0 ≤ angle e f) (hanglec : angle e f ≤ c) :
+    (n : ℝ) - 2 - (1 + 3 / (1 - c ^ 2) ^ 2) * (defect e f + defect f e) ≤
+      traceProduct e f := by
+  have ha : |inner ℝ e.2 f.1| ≤ 1 := by
+    simpa [he.2.1, hf.1] using abs_real_inner_le_norm e.2 f.1
+  have hb : |inner ℝ e.1 f.2| ≤ 1 := by
+    simpa [he.1, hf.2.1] using abs_real_inner_le_norm e.1 f.2
+  have hd : |inner ℝ e.2 f.2| ≤ 1 := by
+    simpa [he.2.1, hf.2.1] using abs_real_inner_le_norm e.2 f.2
+  have hB : defect f e = 1 - angle e f ^ 2 - (inner ℝ e.1 f.2) ^ 2 := by
+    rw [defect_eq hf he, show angle f e = angle e f from real_inner_comm _ _,
+      real_inner_comm f.2 e.1]
+  have hg := scalar_gap_bound hc0 hc1 hangle0 hanglec ha hb hd
+    (show 0 ≤ 1 - angle e f ^ 2 - (inner ℝ e.2 f.1) ^ 2 by
+      rw [← defect_eq he hf]; exact sq_nonneg _)
+    (show 0 ≤ 1 - angle e f ^ 2 - (inner ℝ e.1 f.2) ^ 2 by
+      rw [← hB]; exact sq_nonneg _)
+    (show |angle e f * (-1 + angle e f ^ 2 + (inner ℝ e.2 f.1) ^ 2 +
+        (inner ℝ e.1 f.2) ^ 2) + (inner ℝ e.2 f.1) * (inner ℝ e.1 f.2) *
+        (inner ℝ e.2 f.2)| ≤ _ by
+      rw [← cross_eq he hf, ← defect_eq he hf, ← hB]
+      exact cross_abs_le e f)
+  rw [← defect_eq he hf, ← hB] at hg
+  rw [traceProduct_eq he hf]
+  nlinarith only [hg]
+
+lemma continuous_proj {n : ℕ} : Continuous (fun p : Frame n × Euc n ↦ proj p.1 p.2) := by
+  unfold proj
+  fun_prop
+
+lemma continuous_angle {n : ℕ} : Continuous (fun p : Frame n × Frame n ↦ angle p.1 p.2) := by
+  unfold angle
+  fun_prop
+
+lemma continuous_defect {n : ℕ} : Continuous (fun p : Frame n × Frame n ↦ defect p.1 p.2) := by
+  unfold defect proj
+  fun_prop
+
+lemma continuous_traceProduct {n : ℕ} :
+    Continuous (fun p : Frame n × Frame n ↦ traceProduct p.1 p.2) := by
+  unfold traceProduct angle
+  fun_prop
+
+lemma continuous_cross {n : ℕ} : Continuous (fun p : Frame n × Frame n ↦ cross p.1 p.2) := by
+  unfold cross proj
+  fun_prop
+
+lemma isClosed_frames (n : ℕ) : IsClosed {e : Frame n | IsFrame e} := by
+  have h1 : IsClosed {e : Frame n | ‖e.1‖ = 1} := isClosed_eq (by fun_prop) continuous_const
+  have h2 : IsClosed {e : Frame n | ‖e.2‖ = 1} := isClosed_eq (by fun_prop) continuous_const
+  have h3 : IsClosed {e : Frame n | inner ℝ e.1 e.2 = 0} :=
+    isClosed_eq (by fun_prop) continuous_const
+  exact h1.inter (h2.inter h3)
+
+lemma isCompact_frames (n : ℕ) : IsCompact {e : Frame n | IsFrame e} := by
+  apply (isCompact_closedBall (0 : Frame n) 1).of_isClosed_subset (isClosed_frames n)
+  intro e he
+  simp [Metric.mem_closedBall, dist_zero_right, Prod.norm_def, he.1, he.2.1]
+
+lemma integrable_pairFrameFunction {n : ℕ} {μ : Measure (Frame n × Frame n)}
+    [IsFiniteMeasure μ] (hμ : ∀ᵐ p ∂μ, IsFrame p.1 ∧ IsFrame p.2)
+    {G : Frame n × Frame n → ℝ} (hG : Continuous G) : Integrable G μ := by
+  have hK := (isCompact_frames n).prod (isCompact_frames n)
+  obtain ⟨C, hC⟩ := hK.exists_bound_of_continuousOn hG.continuousOn
+  apply Integrable.of_bound hG.measurable.aestronglyMeasurable C
+  filter_upwards [hμ] with p hp
+  exact hC p hp
+
+def angularWeight (t s : ℝ) : ℝ := Real.exp (t * s) * expNegInvGlue s
+
+lemma angularWeight_contDiff (t : ℝ) : ContDiff ℝ ∞ (angularWeight t) :=
+  ((contDiff_const.mul contDiff_id).exp).mul expNegInvGlue.contDiff
+
+lemma angularWeight_nonneg (t s : ℝ) : 0 ≤ angularWeight t s :=
+  mul_nonneg (Real.exp_pos _).le (expNegInvGlue.nonneg _)
+
+lemma angularWeight_pos_iff (t s : ℝ) : 0 < angularWeight t s ↔ 0 < s := by
+  rw [angularWeight, mul_pos_iff_of_pos_left (Real.exp_pos _)]
+  constructor
+  · intro h
+    by_contra hh
+    rw [expNegInvGlue.zero_of_nonpos (le_of_not_gt hh)] at h
+    exact lt_irrefl 0 h
+  · exact expNegInvGlue.pos_of_pos
+
+lemma angularWeight_zero_of_nonpos (t : ℝ) {s : ℝ} (hs : s ≤ 0) :
+    angularWeight t s = 0 := by simp [angularWeight, expNegInvGlue.zero_of_nonpos hs]
+
+lemma angularWeight_hasDerivAt (t s : ℝ) :
+    HasDerivAt (angularWeight t) ((t + s⁻¹ ^ 2) * angularWeight t s) s := by
+  have hg : HasDerivAt expNegInvGlue (s⁻¹ ^ 2 * expNegInvGlue s) s := by
+    simpa using expNegInvGlue.hasDerivAt_polynomial_eval_inv_mul 1 s
+  have he := (((hasDerivAt_id s).const_mul t).exp).mul hg
+  have hid : (t + s⁻¹ ^ 2) * angularWeight t s =
+      Real.exp (t * id s) * (t * 1) * expNegInvGlue s +
+        Real.exp (t * id s) * (s⁻¹ ^ 2 * expNegInvGlue s) := by
+    dsimp [angularWeight]
+    ring
+  rw [hid]
+  exact he
+
+lemma angularWeight_deriv (t s : ℝ) :
+    deriv (angularWeight t) s = (t + s⁻¹ ^ 2) * angularWeight t s :=
+  (angularWeight_hasDerivAt t s).deriv
+
+lemma angularWeight_deriv_lower (t s : ℝ) :
+    t * angularWeight t s ≤ deriv (angularWeight t) s := by
+  rw [angularWeight_deriv]
+  nlinarith [sq_nonneg s⁻¹, angularWeight_nonneg t s]
+
+lemma angularWeight_deriv_nonneg {t : ℝ} (ht : 0 ≤ t) (s : ℝ) :
+    0 ≤ deriv (angularWeight t) s :=
+  (mul_nonneg ht (angularWeight_nonneg t s)).trans (angularWeight_deriv_lower t s)
+
+structure CoupledStationarity {n : ℕ} (μ : Measure (Frame n × Frame n)) : Prop where
+  gradient : ∀ (φ : ℝ → ℝ), ContDiff ℝ ∞ φ →
+    (∫ p, deriv φ (angle p.1 p.2) * (defect p.1 p.2 + defect p.2 p.1) ∂μ) =
+      2 * ((n : ℝ) - 2) * ∫ p, φ (angle p.1 p.2) * angle p.1 p.2 ∂μ
+  mixed : ∀ (φ : ℝ → ℝ), ContDiff ℝ ∞ φ →
+    2 * (∫ p, φ (angle p.1 p.2) * traceProduct p.1 p.2 ∂μ) -
+      ((n : ℝ) - 2) * (∫ p, φ (angle p.1 p.2) *
+        (defect p.1 p.2 + defect p.2 p.1) ∂μ) +
+      2 * (∫ p, deriv φ (angle p.1 p.2) * cross p.1 p.2 ∂μ) = 0
+
+lemma continuous_defect_sum {n : ℕ} :
+    Continuous (fun p : Frame n × Frame n ↦ defect p.1 p.2 + defect p.2 p.1) :=
+by
+  unfold defect proj
+  fun_prop
+
+lemma angularWeight_integral_pos {n : ℕ} (hn : 2 < n)
+    {μ : Measure (Frame n × Frame n)} [IsFiniteMeasure μ] (hμ0 : μ ≠ 0)
+    (hμ : ∀ᵐ p ∂μ, IsFrame p.1 ∧ IsFrame p.2)
+    (hS : CoupledStationarity μ) (t : ℝ) :
+    0 < ∫ p, angularWeight t (angle p.1 p.2) ∂μ := by
+  have hk : 0 < (n : ℝ) - 2 := by
+    have hh : (2 : ℝ) < n := by exact_mod_cast hn
+    linarith
+  have hiA := integrable_pairFrameFunction hμ (continuous_angle (n := n))
+  have hiD := integrable_pairFrameFunction hμ (continuous_defect_sum (n := n))
+  have hiT := integrable_pairFrameFunction hμ (continuous_traceProduct (n := n))
+  have hiW := integrable_pairFrameFunction hμ
+    ((angularWeight_contDiff t).continuous.comp continuous_angle)
+  have hW : ∀ᵐ p ∂μ, 0 ≤ angularWeight t (angle p.1 p.2) :=
+    Filter.Eventually.of_forall fun p ↦ angularWeight_nonneg t _
+  by_contra hpos
+  have hz : (∫ p, angularWeight t (angle p.1 p.2) ∂μ) = 0 :=
+    le_antisymm (le_of_not_gt hpos) (integral_nonneg_of_ae hW)
+  have heq := (integral_eq_zero_iff_of_nonneg_ae hW hiW).mp hz
+  have hAnonpos : ∀ᵐ p ∂μ, angle p.1 p.2 ≤ 0 := by
+    filter_upwards [heq] with p hp
+    by_contra hh
+    have h := (angularWeight_pos_iff t _).mpr (lt_of_not_ge hh)
+    simp [hp] at h
+  have hA : (∫ p, angle p.1 p.2 ∂μ) = 0 := by
+    have h := hS.gradient (fun _ ↦ 1) contDiff_const
+    simp only [deriv_const, zero_mul, integral_zero, one_mul] at h
+    nlinarith
+  have hAzero : ∀ᵐ p ∂μ, angle p.1 p.2 = 0 := by
+    have hh : (∫ p, -(angle p.1 p.2) ∂μ) = 0 := by rw [integral_neg, hA, neg_zero]
+    have he := (integral_eq_zero_iff_of_nonneg_ae
+      (hAnonpos.mono (fun p hp ↦ neg_nonneg.mpr hp)) hiA.neg).mp hh
+    filter_upwards [he] with p hp
+    exact neg_eq_zero.mp hp
+  have hD : (∫ p, defect p.1 p.2 + defect p.2 p.1 ∂μ) = 0 := by
+    have h := hS.gradient id contDiff_id
+    simp only [id_eq] at h
+    rw [show (∫ p, angle p.1 p.2 * angle p.1 p.2 ∂μ) = 0 by
+      calc
+        _ = ∫ _ : Frame n × Frame n, (0 : ℝ) ∂μ :=
+          integral_congr_ae (hAzero.mono (fun p hp ↦ by simp [hp]))
+        _ = 0 := integral_zero _ _] at h
+    simpa using h
+  have hT : (∫ p, traceProduct p.1 p.2 ∂μ) = 0 := by
+    have h := hS.mixed (fun _ ↦ 1) contDiff_const
+    simp only [deriv_const, zero_mul, integral_zero, one_mul, hD] at h
+    linarith
+  have hlower : ∀ᵐ p ∂μ, (n : ℝ) - 2 ≤
+      traceProduct p.1 p.2 + (defect p.1 p.2 + defect p.2 p.1) := by
+    filter_upwards [hμ, hAzero] with p hp ha
+    rw [traceProduct_eq hp.1 hp.2, ha]
+    nlinarith [sq_nonneg (inner ℝ p.1.2 p.2.2)]
+  have hle := integral_mono_ae (integrable_const ((n : ℝ) - 2)) (hiT.add hiD) hlower
+  change (∫ _ : Frame n × Frame n, (n : ℝ) - 2 ∂μ) ≤
+    ∫ p, traceProduct p.1 p.2 + (defect p.1 p.2 + defect p.2 p.1) ∂μ at hle
+  rw [integral_add hiT hiD, hT, hD, add_zero, integral_const, smul_eq_mul] at hle
+  have hmass : 0 < μ.real univ := by
+    apply ENNReal.toReal_pos _ (measure_ne_top _ _)
+    intro hm
+    exact hμ0 (Measure.measure_univ_eq_zero.mp hm)
+  nlinarith
+
+lemma weighted_trace_lower_bound {n : ℕ} {e f : Frame n}
+    (he : IsFrame e) (hf : IsFrame f) {c : ℝ} (hc0 : 0 ≤ c) (hc1 : c < 1)
+    (hfc : angle e f ≤ c) (t : ℝ) :
+    ((n : ℝ) - 2) * angularWeight t (angle e f) -
+        (1 + 3 / (1 - c ^ 2) ^ 2) *
+          (angularWeight t (angle e f) * (defect e f + defect f e)) ≤
+      angularWeight t (angle e f) * traceProduct e f := by
+  by_cases ha : 0 ≤ angle e f
+  · have h := mul_le_mul_of_nonneg_left
+      (traceProduct_lower_bound he hf hc0 hc1 ha hfc)
+      (angularWeight_nonneg t (angle e f))
+    nlinarith
+  · rw [angularWeight_zero_of_nonpos t (le_of_lt (lt_of_not_ge ha))]
+    ring_nf
+    exact le_rfl
+
+theorem no_positive_angle_gap {n : ℕ} (hn : 2 < n)
+    {μ : Measure (Frame n × Frame n)} [IsFiniteMeasure μ] (hμ0 : μ ≠ 0)
+    (hμ : ∀ᵐ p ∂μ, IsFrame p.1 ∧ IsFrame p.2)
+    (hS : CoupledStationarity μ) {c : ℝ} (hc0 : 0 ≤ c) (hc1 : c < 1)
+    (hc : ∀ᵐ p ∂μ, angle p.1 p.2 ≤ c) : False := by
+  let k : ℝ := n - 2
+  let C : ℝ := 1 + 3 / (1 - c ^ 2) ^ 2
+  have hk : 0 < k := by
+    have hh : (2 : ℝ) < n := by exact_mod_cast hn
+    dsimp [k]; linarith
+  have hC : 0 < C := by dsimp [C]; positivity
+  let t : ℝ := 2 * c * (C + k / 2) / (1 - c) + 1
+  have ht : 0 < t := by dsimp [t]; positivity
+  have hcoef : 2 * k * c * (C + k / 2) < t * k * (1 - c) := by
+    have hden : 1 - c ≠ 0 := ne_of_gt (sub_pos.mpr hc1)
+    dsimp [t]
+    field_simp
+    nlinarith
+  let W : Frame n × Frame n → ℝ := fun p ↦ angularWeight t (angle p.1 p.2)
+  let D : Frame n × Frame n → ℝ := fun p ↦ defect p.1 p.2 + defect p.2 p.1
+  let dW : Frame n × Frame n → ℝ := fun p ↦ deriv (angularWeight t) (angle p.1 p.2)
+  have hW : Continuous W := (angularWeight_contDiff t).continuous.comp continuous_angle
+  have hdW : Continuous dW :=
+    ((angularWeight_contDiff t).continuous_deriv (by simp)).comp continuous_angle
+  have hD : Continuous D := continuous_defect_sum
+  have hiW := integrable_pairFrameFunction hμ hW
+  have hiWD := integrable_pairFrameFunction hμ (hW.mul hD)
+  have hiWA := integrable_pairFrameFunction hμ (hW.mul continuous_angle)
+  have hiWT := integrable_pairFrameFunction hμ (hW.mul continuous_traceProduct)
+  have hidWD := integrable_pairFrameFunction hμ (hdW.mul hD)
+  have hidWC := integrable_pairFrameFunction hμ (hdW.mul continuous_cross)
+  have hWnn (p : Frame n × Frame n) : 0 ≤ W p := angularWeight_nonneg _ _
+  have hDnn (p : Frame n × Frame n) : 0 ≤ D p :=
+    add_nonneg (sq_nonneg _) (sq_nonneg _)
+  have hdWnn (p : Frame n × Frame n) : 0 ≤ dW p := angularWeight_deriv_nonneg ht.le _
+  let J : ℝ := ∫ p, W p ∂μ
+  let B : ℝ := ∫ p, W p * D p ∂μ
+  let A : ℝ := ∫ p, W p * angle p.1 p.2 ∂μ
+  let T : ℝ := ∫ p, W p * traceProduct p.1 p.2 ∂μ
+  let Q : ℝ := ∫ p, dW p * cross p.1 p.2 ∂μ
+  have hJ : 0 < J := angularWeight_integral_pos hn hμ0 hμ hS t
+  have hgrad : (∫ p, dW p * D p ∂μ) = 2 * k * A :=
+    hS.gradient _ (angularWeight_contDiff t)
+  have hmixed : 2 * T - k * B + 2 * Q = 0 :=
+    hS.mixed _ (angularWeight_contDiff t)
+  have hA : A ≤ c * J := by
+    have hp : ∀ᵐ p ∂μ, W p * angle p.1 p.2 ≤ c * W p := by
+      filter_upwards [hc] with p hp
+      nlinarith [mul_le_mul_of_nonneg_left hp (hWnn p)]
+    have h := integral_mono_ae hiWA (hiW.const_mul c) hp
+    simpa only [Pi.mul_apply, integral_const_mul] using h
+  have hB : t * B ≤ 2 * k * A := by
+    have hp : ∀ᵐ p ∂μ, t * (W p * D p) ≤ dW p * D p :=
+      Filter.Eventually.of_forall fun p ↦ by
+        have h := mul_le_mul_of_nonneg_right (angularWeight_deriv_lower t (angle p.1 p.2)) (hDnn p)
+        simpa only [mul_assoc] using h
+    have h := integral_mono_ae (hiWD.const_mul t) hidWD hp
+    simpa only [Pi.mul_apply, integral_const_mul, hgrad] using h
+  have hQ : -(k * A) ≤ Q := by
+    have hp : ∀ᵐ p ∂μ, (-1 / 2 : ℝ) * (dW p * D p) ≤ dW p * cross p.1 p.2 :=
+      Filter.Eventually.of_forall fun p ↦ by
+        have h := (abs_le.mp (cross_abs_le p.1 p.2)).1
+        have hm := mul_le_mul_of_nonneg_left h (hdWnn p)
+        dsimp [D] at *
+        nlinarith
+    have h := integral_mono_ae (hidWD.const_mul (-1 / 2)) hidWC hp
+    simp only [Pi.mul_apply] at h
+    rw [integral_const_mul, hgrad] at h
+    dsimp [Q]
+    linarith
+  have hT : k * J - C * B ≤ T := by
+    have hp : ∀ᵐ p ∂μ, k * W p - C * (W p * D p) ≤ W p * traceProduct p.1 p.2 := by
+      filter_upwards [hμ, hc] with p hp hpc
+      exact weighted_trace_lower_bound hp.1 hp.2 hc0 hc1 hpc t
+    have h := integral_mono_ae ((hiW.const_mul k).sub (hiWD.const_mul C)) hiWT hp
+    simp only [Pi.mul_apply, Pi.sub_apply] at h
+    have hiCB : Integrable (fun p ↦ C * (W p * D p)) μ := by
+      simpa only [Pi.mul_apply] using hiWD.const_mul C
+    rw [integral_sub (hiW.const_mul k) hiCB, integral_const_mul,
+      integral_const_mul] at h
+    exact h
+  have hmain : k * (1 - c) * J ≤ (C + k / 2) * B := by
+    have hu : T ≤ k / 2 * B + k * A := by linarith
+    have hf := mul_le_mul_of_nonneg_left hA hk.le
+    nlinarith
+  have hB' : t * B ≤ 2 * k * c * J := by
+    have h := mul_le_mul_of_nonneg_left hA (by positivity : 0 ≤ 2 * k)
+    nlinarith
+  have h1 := mul_le_mul_of_nonneg_left hmain ht.le
+  have h2 := mul_le_mul_of_nonneg_left hB' (by positivity : 0 ≤ C + k / 2)
+  have h3 := mul_lt_mul_of_pos_right hcoef hJ
+  nlinarith
+
+end SphericalFrankel
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/SupportAffineEstimate.lean b/lean/OAI/Geometry/Perimeter/Regularity/SupportAffineEstimate.lean
new file mode 100644
index 0000000000000000000000000000000000000000..7193dfb63e5622cf08458cebe46c71fe88851c3e
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/SupportAffineEstimate.lean
@@ -0,0 +1,141 @@
+import OAI.Geometry.Perimeter.Regularity.QuadraticMoment
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+open RPI.Boundary ContinuousLinearMap
+variable {n : ℕ}
+
+theorem normalWindowQuadratic_integrableOn {E : Set (Euc n)}
+    (hmin : IsLocallyPerimeterMinimizing E) (v : Euc n) {φ w : normalPlane v → ℝ}
+    (hφ : Continuous φ) (hcφ : HasCompactSupport φ) (hw : Continuous w)
+    {a b R : ℝ} (hb : 0≤b) (hR : 0<R) :
+    IntegrableOn (normalWindowQuadratic v φ w a b) (Metric.ball 0 R) (perimeterMeasure E) := by
+  have hi2 := normalWindowTest_integrableOn hmin v hφ hcφ hb hR 2
+  have hi1 := normalWindowTest_integrableOn hmin v (hφ.mul hw) hcφ.mul_right hb hR 1
+  have hi0 := normalWindowTest_integrableOn hmin v (hφ.mul (hw.pow 2)) hcφ.mul_right hb hR 0
+  rw [normalWindowQuadratic_expand]
+  exact ((hi2.const_mul ((a⁻¹)^2)).sub (hi1.const_mul (2/a))).add hi0
+
+theorem normalWindowQuadratic_nonneg (v : Euc n) {φ w : normalPlane v → ℝ}
+    (hφ : ∀ y, 0≤φ y) (a b : ℝ) (x : Euc n) :
+    0≤normalWindowQuadratic v φ w a b x := by
+  dsimp only [normalWindowQuadratic]
+  exact Set.indicator_nonneg (fun x _ ↦ mul_nonneg (hφ _) (sq_nonneg _)) x
+
+theorem exists_minimizing_linear_mean_value_constant (n : ℕ) (hn : 1<n) :
+    ∃ C : ℝ, 0<C ∧ ∀ E : Set (Euc n), IsLocallyPerimeterMinimizing E →
+      ∀ p ∈ perimeterSupport E, ∀ r : ℝ, 0<r → ∀ (L : Euc n →L[ℝ] ℝ) (b : ℝ),
+        (L p-b)^2≤C/r^(n-1)*∫ x in Metric.ball p r, (L x-b)^2 ∂perimeterMeasure E := by
+  obtain ⟨C,hC,h⟩ := exists_minimizing_height_mean_value_constant n hn
+  refine ⟨C,hC,fun E hE p hp r hr L b ↦ ?_⟩
+  simpa only [InnerProductSpace.toDual_symm_apply] using
+    h E hE p hp r hr ((InnerProductSpace.toDual ℝ (Euc n)).symm L) b
+
+theorem minimizing_affine_error_from_trace {E : Set (Euc n)}
+    (hmin : IsLocallyPerimeterMinimizing E) {C G : ℝ} (hC : 0<C)
+    (hmean : ∀ p ∈ perimeterSupport E, ∀ r : ℝ, 0<r → ∀ (L : Euc n →L[ℝ] ℝ) (b : ℝ),
+      (L p-b)^2≤C/r^(n-1)*∫ x in Metric.ball p r, (L x-b)^2 ∂perimeterMeasure E)
+    (hgrowth : ∀ p r, 0<r → perimeter E (Metric.ball p r)≤ENNReal.ofReal (G*r^(n-1)))
+    (hG : 0≤G) {Q : Euc n → ℝ}
+    (hQ : IntegrableOn Q (Metric.ball 0 4) (perimeterMeasure E))
+    (hQ0 : ∀ x, 0≤Q x) {L : Euc n →L[ℝ] ℝ} {b T r : ℝ}
+    (hr : 0<r) (hrb : r<1) {p : Euc n} (hp : p∈perimeterSupport E)
+    (hpb : p∈Metric.ball 0 r)
+    (hbound : ∀ x∈perimeterSupport E ∩ Metric.ball p r, (L x-b)^2≤2*Q x+T^2) :
+    (L p-b)^2≤C/r^(n-1)*(2*(∫ x in Metric.ball 0 4, Q x ∂perimeterMeasure E)+T^2*(G*r^(n-1))) := by
+  have hsub : Metric.ball p r ⊆ Metric.ball (0:Euc n) 4 := by
+    intro x hx
+    have hd := dist_triangle x p 0
+    rw [Metric.mem_ball] at hx hpb ⊢
+    linarith
+  have := perimeterMeasure_locallyFinite E hmin.2.1
+  have : IsFiniteMeasure ((perimeterMeasure E).restrict (Metric.ball p r)) := ⟨by
+    rw [Measure.restrict_apply_univ,perimeterMeasure_open E Metric.isOpen_ball]
+    exact hmin.2.1 p r hr⟩
+  have hiA : IntegrableOn (fun x ↦ (L x-b)^2) (Metric.ball p r) (perimeterMeasure E) :=
+    (((L.continuous.sub continuous_const).pow 2).continuousOn.integrableOn_compact
+      (isCompact_closedBall p r)).mono_set Metric.ball_subset_closedBall
+  have hiQ := hQ.mono_set hsub
+  have hs : ∀ᵐ x ∂perimeterMeasure E, x∈perimeterSupport E := by
+    rw [← perimeterMeasure_support]; exact (perimeterMeasure E).support_mem_ae
+  have hib : (∫ x in Metric.ball p r, (L x-b)^2 ∂perimeterMeasure E) ≤
+      ∫ x in Metric.ball p r, 2*Q x+T^2 ∂perimeterMeasure E := by
+    apply integral_mono_ae hiA ((hiQ.const_mul 2).add (integrable_const _))
+    filter_upwards [ae_restrict_of_ae hs,ae_restrict_mem measurableSet_ball] with x hx hxB
+    exact hbound x ⟨hx,hxB⟩
+  have hiQle := setIntegral_mono_set hQ (ae_of_all _ hQ0) (ae_of_all _ hsub)
+  have hper : (perimeterMeasure E).real (Metric.ball p r)≤G*r^(n-1) := by
+    have h := ENNReal.toReal_mono ENNReal.ofReal_ne_top (hgrowth p r hr)
+    rw [ENNReal.toReal_ofReal (mul_nonneg hG (pow_nonneg hr.le _))] at h
+    simpa only [Measure.real,perimeterMeasure_open E Metric.isOpen_ball] using h
+  apply (hmean p hp r hr L b).trans
+  apply mul_le_mul_of_nonneg_left _ (by positivity)
+  calc
+    _ ≤ ∫ x in Metric.ball p r, 2*Q x+T^2 ∂perimeterMeasure E := hib
+    _ = 2*(∫ x in Metric.ball p r, Q x ∂perimeterMeasure E)+T^2*(perimeterMeasure E).real (Metric.ball p r) := by
+      rw [integral_add (hiQ.const_mul 2) (integrable_const _),integral_const_mul,integral_const]
+      simp only [smul_eq_mul,Measure.real,Measure.restrict_apply_univ,mul_comm]
+    _ ≤ _ := add_le_add (mul_le_mul_of_nonneg_left hiQle (by norm_num))
+      (mul_le_mul_of_nonneg_left hper (sq_nonneg _))
+
+theorem minimizing_support_taylor_estimate {E : Set (Euc n)}
+    (hmin : IsLocallyPerimeterMinimizing E) {C G : ℝ} (hC : 0<C) (hG : 0≤G)
+    (hmean : ∀ p ∈ perimeterSupport E, ∀ r : ℝ, 0<r → ∀ (L : Euc n →L[ℝ] ℝ) (b : ℝ),
+      (L p-b)^2≤C/r^(n-1)*∫ x in Metric.ball p r, (L x-b)^2 ∂perimeterMeasure E)
+    (hgrowth : ∀ p r, 0<r → perimeter E (Metric.ball p r)≤ENNReal.ofReal (G*r^(n-1)))
+    (v : Euc n) {φ w : normalPlane v → ℝ}
+    (hφ : Continuous φ) (hcφ : HasCompactSupport φ) (hφ0 : ∀ y, 0≤φ y)
+    (hφone : ∀ y∈Metric.ball (0:normalPlane v) (1/4), φ y=1)
+    (hw : Continuous w) {a D r b : ℝ} (ha : 0<a) (hD : 0≤D)
+    (hr : 0<r) (hrb : r<1/8) (ℓ : Euc n →L[ℝ] ℝ)
+    (htaylor : ∀ x∈Metric.ball (0:Euc n) (1/4), |w (normalProjection v x)-b-ℓ x|≤D*‖x‖^2)
+    (hh : ∀ x∈perimeterSupport E ∩ Metric.ball 0 4, |inner ℝ v x|≤a)
+    {p : Euc n} (hp : p∈perimeterSupport E) (hpb : p∈Metric.ball 0 r) :
+    (inner ℝ v p/a-ℓ p-b)^2 ≤ C/r^(n-1)*
+      (2*(∫ x in Metric.ball 0 4, normalWindowQuadratic v φ w a (2*a) x ∂perimeterMeasure E)+
+        (8*D*r^2)^2*(G*r^(n-1))) := by
+  let L : Euc n →L[ℝ] ℝ := a⁻¹ • innerSL ℝ v - ℓ
+  have hL (x : Euc n) : L x=inner ℝ v x/a-ℓ x := by
+    simp only [L,sub_apply,smul_apply,smul_eq_mul]
+    change a⁻¹*inner ℝ v x-ℓ x=inner ℝ v x/a-ℓ x
+    ring
+  rw [← hL]
+  apply minimizing_affine_error_from_trace hmin hC hmean hgrowth hG
+    (normalWindowQuadratic_integrableOn hmin v hφ hcφ hw (by positivity) (by norm_num))
+    (normalWindowQuadratic_nonneg v hφ0 a (2*a)) hr (by linarith) hp hpb
+  intro x hx
+  have hxb : ‖x‖<2*r := by
+    have hd := dist_triangle x p 0
+    have hxB := hx.2
+    rw [Metric.mem_ball] at hxB hpb
+    simpa only [dist_zero_right] using (lt_of_le_of_lt hd (by linarith : dist x p+dist p 0<2*r))
+  have hx4 : x∈Metric.ball (0:Euc n) 4 := by rw [Metric.mem_ball,dist_zero_right]; linarith
+  have hxq : x∈Metric.ball (0:Euc n) (1/4) := by rw [Metric.mem_ball,dist_zero_right]; linarith
+  have hproj : normalProjection v x∈Metric.ball (0:normalPlane v) (1/4) := by
+    rw [Metric.mem_ball,dist_zero_right]
+    exact ((normalPlane v).norm_orthogonalProjectionOnto_apply_le x).trans_lt
+      (by rw [Metric.mem_ball,dist_zero_right] at hxq; exact hxq)
+  have heq : normalWindowQuadratic v φ w a (2*a) x=(inner ℝ v x/a-w (normalProjection v x))^2 := by
+    rw [normalWindowQuadratic,Set.indicator_of_mem (show x∈{z:Euc n | |inner ℝ v z|≤2*a} from
+      (hh x ⟨hx.1,hx4⟩).trans (by linarith)),hφone _ hproj,one_mul]
+  have ht := htaylor x hxq
+  have htx : |w (normalProjection v x)-b-ℓ x|≤4*D*r^2 := by
+    apply ht.trans
+    have hn0 := norm_nonneg x
+    nlinarith [sq_nonneg (2*r-‖x‖),mul_nonneg hD (show 0≤4*r^2-‖x‖^2 by nlinarith)]
+  rw [hL,heq]
+  have hs := sq_le_sq₀ (abs_nonneg (w (normalProjection v x)-b-ℓ x)) (by positivity : 0≤4*D*r^2)
+  have ht2 : (w (normalProjection v x)-b-ℓ x)^2≤(4*D*r^2)^2 := by
+    simpa only [sq_abs] using hs.mpr htx
+  nlinarith [sq_nonneg ((inner ℝ v x/a-w (normalProjection v x))-(w (normalProjection v x)-b-ℓ x)),
+    sq_nonneg (D*r^2)]
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/SupportImprovement.lean b/lean/OAI/Geometry/Perimeter/Regularity/SupportImprovement.lean
new file mode 100644
index 0000000000000000000000000000000000000000..f397615cd70b0b99c9518c7802e9c0444c872032
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/SupportImprovement.lean
@@ -0,0 +1,92 @@
+import OAI.Geometry.Perimeter.Regularity.SupportAffineEstimate
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+open RPI.Boundary ContinuousLinearMap
+variable {n : ℕ}
+
+theorem constant_zero_of_quartic_bound {b M δ : ℝ} (hδ : 0<δ)
+    (hb : ∀ r : ℝ, 0<r → r<δ → b^2≤M*r^4) : b=0 := by
+  have ht : Tendsto (fun r : ℝ ↦ M*r^4) (𝓝[>] 0) (𝓝 0) := by
+    simpa using (((tendsto_id : Tendsto (fun r : ℝ ↦ r) (𝓝 0) (𝓝 0)).pow 4).const_mul M).mono_left nhdsWithin_le_nhds
+  have hzero : b^2≤0 := le_of_tendsto_of_tendsto tendsto_const_nhds ht (by
+    filter_upwards [self_mem_nhdsWithin,nhdsWithin_le_nhds (gt_mem_nhds hδ)] with r hr hrδ
+    exact hb r hr hrδ)
+  nlinarith [sq_nonneg b]
+
+theorem exists_full_support_improvement_constant (n : ℕ) (hn : 1<n)
+    (D : ℝ) (hD : 0≤D) :
+    ∃ K : ℝ, 0<K ∧ ∀ (E : ℕ → Set (Euc n)),
+      (∀ i, IsLocallyPerimeterMinimizing (E i)) →
+      (∀ i, (0:Euc n)∈perimeterSupport (E i)) →
+      ∀ (v : Euc n) (φ w : normalPlane v → ℝ), Continuous φ → HasCompactSupport φ →
+      (∀ y, 0≤φ y) →
+      (∀ y∈Metric.ball (0:normalPlane v) (1/4), φ y=1) → Continuous w →
+      ∀ (a : ℕ → ℝ), (∀ i, 0<a i) →
+      (∀ i x, x∈perimeterSupport (E i) ∩ Metric.ball 0 4 → |inner ℝ v x|≤a i) →
+      ∀ (ℓ : Euc n →L[ℝ] ℝ) (b : ℝ),
+      (∀ x∈Metric.ball (0:Euc n) (1/4), |w (normalProjection v x)-b-ℓ x|≤D*‖x‖^2) →
+      Tendsto (fun i ↦ ∫ x in Metric.ball 0 4,
+        normalWindowQuadratic v φ w (a i) (2*a i) x ∂perimeterMeasure (E i)) atTop (𝓝 0) →
+      b=0 ∧ ∀ r : ℝ, 0<r → r<1/8 → ∀ᶠ i in atTop,
+        ∀ p∈perimeterSupport (E i) ∩ Metric.ball 0 r,
+          |inner ℝ v p/a i-ℓ p|≤K*r^2 := by
+  obtain ⟨C,hC,hmean⟩ := exists_minimizing_linear_mean_value_constant n hn
+  obtain ⟨G,hG,hgrowth⟩ := exists_locallyMinimizing_perimeter_growth_constant n (by omega)
+  let A := C*(2+64*D^2*G)
+  have hA : 0≤A := by dsimp [A]; positivity
+  let K := A+1
+  have hK : 0<K := by dsimp [K]; linarith
+  have hAK : A≤K^2 := by dsimp [K]; nlinarith [sq_nonneg A]
+  refine ⟨K,hK,fun E hmin hzero v φ w hφ hcφ hpφ hφone hw a ha hh ℓ b htaylor htrace ↦ ?_⟩
+  let Q : ℕ → ℝ := fun i ↦ ∫ x in Metric.ball 0 4,
+    normalWindowQuadratic v φ w (a i) (2*a i) x ∂perimeterMeasure (E i)
+  have hest (r : ℝ) (hr : 0<r) (hrb : r<1/8) (i : ℕ)
+      (p : Euc n) (hp : p∈perimeterSupport (E i) ∩ Metric.ball 0 r) :
+      (inner ℝ v p/a i-ℓ p-b)^2≤C/r^(n-1)*(2*Q i+(8*D*r^2)^2*(G*r^(n-1))) :=
+    minimizing_support_taylor_estimate (hmin i) hC hG.le (hmean (E i) (hmin i))
+      (hgrowth (E i) (hmin i)) v hφ hcφ hpφ hφone hw (ha i) hD hr hrb ℓ htaylor
+      (hh i) hp.1 hp.2
+  have hb : b=0 := by
+    apply constant_zero_of_quartic_bound (δ := 1/8) (M := 64*C*D^2*G) (by norm_num)
+    intro r hr hrb
+    have he (i : ℕ) : b^2≤C/r^(n-1)*(2*Q i+(8*D*r^2)^2*(G*r^(n-1))) := by
+      simpa only [inner_zero_right,zero_div,map_zero,sub_zero,zero_sub,neg_sq] using
+        hest r hr hrb i 0 ⟨hzero i,Metric.mem_ball_self hr⟩
+    have ht : Tendsto (fun i ↦ C/r^(n-1)*(2*Q i+(8*D*r^2)^2*(G*r^(n-1))))
+        atTop (𝓝 (C/r^(n-1)*((8*D*r^2)^2*(G*r^(n-1))))) := by
+      simpa only [mul_zero,zero_add] using (htrace.const_mul 2 |>.add_const ((8*D*r^2)^2*(G*r^(n-1)))).const_mul (C/r^(n-1))
+    have h := le_of_tendsto_of_tendsto' tendsto_const_nhds ht he
+    have heq : C/r^(n-1)*((8*D*r^2)^2*(G*r^(n-1)))=(64*C*D^2*G)*r^4 := by
+      field_simp [ne_of_gt hr]
+      ring
+    rw [heq] at h
+    exact h
+  refine ⟨hb,fun r hr hrb ↦ ?_⟩
+  have hsmall : ∀ᶠ i in atTop, Q i<r^(n-1)*r^4 :=
+    htrace.eventually (gt_mem_nhds (by positivity : (0:ℝ)<r^(n-1)*r^4))
+  filter_upwards [hsmall] with i hi
+  intro p hp
+  have h := hest r hr hrb i p hp
+  rw [hb,sub_zero] at h
+  have hsq : (inner ℝ v p/a i-ℓ p)^2≤A*r^4 := by
+    apply h.trans
+    calc
+      _ ≤ C/r^(n-1)*(2*(r^(n-1)*r^4)+(8*D*r^2)^2*(G*r^(n-1))) := by gcongr
+      _ = A*r^4 := by dsimp [A]; field_simp; ring
+  apply (sq_le_sq₀ (abs_nonneg _) (by positivity : 0≤K*r^2)).mp
+  rw [sq_abs]
+  apply hsq.trans
+  calc
+    A*r^4 ≤ K^2*r^4 := mul_le_mul_of_nonneg_right hAK (by positivity)
+    _ = (K*r^2)^2 := by ring
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/SurfaceGradientComp.lean b/lean/OAI/Geometry/Perimeter/Regularity/SurfaceGradientComp.lean
new file mode 100644
index 0000000000000000000000000000000000000000..116011269be1ebb171de8b13953543c86434b96c
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/SurfaceGradientComp.lean
@@ -0,0 +1,499 @@
+import OAI.Geometry.Perimeter.Regularity.GeometricHausdorff
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPI.Boundary
+
+theorem surfaceGradient_comp {n : ℕ} {S : Set (Space n)}
+    {f : Space n → ℝ} {T : ℝ → ℝ} {x : Space n}
+    (hf : DifferentiableAt ℝ f x) (hT : DifferentiableAt ℝ T (f x)) :
+    surfaceGradient S (T ∘ f) x = deriv T (f x) • surfaceGradient S f x := by
+  have hgrad : gradient (T ∘ f) x = deriv T (f x) • gradient f x := by
+    apply (InnerProductSpace.toDual ℝ (Space n)).injective
+    simp only [map_smul, toDual_gradient]
+    exact (hT.hasDerivAt.comp_hasFDerivAt x hf.hasFDerivAt).fderiv
+  simp only [surfaceGradient, hgrad, map_smul, Submodule.coe_smul]
+
+theorem HasSmoothTraceJetAt.comp {n : ℕ} {S : Set (Space n)}
+    {h : Space n → ℝ} {G : Space n → Space n} {x : Space n}
+    (hxS : x ∈ S) (hx : HasSmoothTraceJetAt S h G x)
+    {T : ℝ → ℝ} (hT : ContDiff ℝ ∞ T) :
+    HasSmoothTraceJetAt S (T ∘ h) (fun y ↦ deriv T (h y) • G y) x := by
+  obtain ⟨U, hU, hxU, f, hf, hfh, hfG⟩ := hx
+  refine ⟨U, hU, hxU, T ∘ f, hT.comp_contDiffOn hf, ?_, ?_⟩
+  · intro y hy
+    exact congrArg T (hfh hy)
+  · rw [surfaceGradient_comp ((hf.contDiffAt (hU.mem_nhds hxU)).differentiableAt (by simp))
+        (hT.differentiable (by simp)).differentiableAt, hfh ⟨hxS, hxU⟩, hfG]
+
+def truncatedGradient {n : ℕ} (W : Space n → ℝ) (D : Space n → Space n)
+    (x : Space n) : Space n := if W x < 1 then D x else 0
+
+theorem truncationSlope_smul_truncatedGradient {n : ℕ}
+    (W : Space n → ℝ) (D : Space n → Space n) {ε : ℝ} (hε : 0 < ε) (x : Space n) :
+    truncationSlope ε (W x) • truncatedGradient W D x = truncationSlope ε (W x) • D x := by
+  unfold truncatedGradient
+  split_ifs with h
+  · rfl
+  · simp only [truncationSlope_eq_zero hε (le_of_not_gt h), zero_smul]
+
+theorem HasSmoothTraceJetAt.smoothTruncation {n : ℕ} {S : Set (Space n)}
+    {W : Space n → ℝ} {D : Space n → Space n} {x : Space n}
+    (hxS : x ∈ S) (hx : HasSmoothTraceJetAt S W D x)
+    {ε : ℝ} (hε : 0 < ε) :
+    HasSmoothTraceJetAt S (fun y ↦ smoothTruncation ε (W y))
+      (fun y ↦ truncationSlope ε (W y) • truncatedGradient W D y) x := by
+  simpa only [Function.comp_def, smoothTruncation_deriv,
+    truncationSlope_smul_truncatedGradient W D hε] using hx.comp hxS (smoothTruncation_contDiff ε)
+
+end RPI.Boundary
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPI.Boundary
+
+theorem smoothTruncation_norm_le {ε s : ℝ} (hε : 0 < ε) (hs : 0 ≤ s) :
+    ‖smoothTruncation ε s‖ ≤ 1 := by
+  rw [Real.norm_eq_abs, abs_of_nonneg (smoothTruncation_nonneg _ hs)]
+  exact (smoothTruncation_le_min hε hs).trans (min_le_right _ _)
+
+theorem truncationSlope_norm_le (ε s : ℝ) : ‖truncationSlope ε s‖ ≤ 1 := by
+  rw [Real.norm_eq_abs, abs_of_nonneg (truncationSlope_mem_Icc _ _).1]
+  exact (truncationSlope_mem_Icc _ _).2
+
+theorem min_one_memLp {n : ℕ} {μ : Measure (Space n)} [IsFiniteMeasure μ]
+    {W : Space n → ℝ} (hW : AEStronglyMeasurable W μ) (hW0 : ∀ᵐ x ∂μ, 0 ≤ W x) :
+    MemLp (fun x ↦ min (W x) 1) 2 μ := by
+  apply MemLp.of_bound ((continuous_id.min continuous_const).comp_aestronglyMeasurable hW) 1
+  filter_upwards [hW0] with x hx
+  change ‖min (W x) 1‖ ≤ 1
+  rw [Real.norm_eq_abs, abs_of_nonneg (le_min hx zero_le_one)]
+  exact min_le_right _ _
+
+theorem smoothTruncation_memLp {n : ℕ} {μ : Measure (Space n)} [IsFiniteMeasure μ]
+    {W : Space n → ℝ} (hW : AEStronglyMeasurable W μ) (hW0 : ∀ᵐ x ∂μ, 0 ≤ W x)
+    {ε : ℝ} (hε : 0 < ε) : MemLp (fun x ↦ smoothTruncation ε (W x)) 2 μ := by
+  apply MemLp.of_bound ((smoothTruncation_contDiff ε).continuous.comp_aestronglyMeasurable hW) 1
+  filter_upwards [hW0] with x hx using smoothTruncation_norm_le hε hx
+
+theorem truncationGradient_memLp {n : ℕ} {μ : Measure (Space n)}
+    {W : Space n → ℝ} {D : Space n → Space n}
+    (hW : AEStronglyMeasurable W μ) (hG : MemLp (truncatedGradient W D) 2 μ) (ε : ℝ) :
+    MemLp (fun x ↦ truncationSlope ε (W x) • truncatedGradient W D x) 2 μ := by
+  apply hG.of_le (((truncationSlope_contDiff _).continuous.comp_aestronglyMeasurable hW).smul
+    hG.aestronglyMeasurable)
+  apply ae_of_all
+  intro x
+  change ‖truncationSlope ε (W x) • truncatedGradient W D x‖ ≤ ‖truncatedGradient W D x‖
+  rw [norm_smul]
+  exact mul_le_of_le_one_left (norm_nonneg _) (truncationSlope_norm_le _ _)
+
+theorem smoothTruncation_error_energy_tendsto {n : ℕ} {μ : Measure (Space n)}
+    [IsFiniteMeasure μ] {W : Space n → ℝ}
+    (hW : AEStronglyMeasurable W μ) (hW0 : ∀ᵐ x ∂μ, 0 ≤ W x) :
+    Tendsto (fun j ↦ vectorEnergy μ (fun x ↦ smoothTruncation (truncationParameter j) (W x) -
+      min (W x) 1)) atTop (𝓝 0) := by
+  have hm : AEStronglyMeasurable (fun x ↦ min (W x) 1) μ := (continuous_id.min continuous_const).comp_aestronglyMeasurable hW
+  have hf : ∀ j, AEStronglyMeasurable (fun x ↦ smoothTruncation (truncationParameter j) (W x)) μ :=
+    fun j ↦ (smoothTruncation_contDiff _).continuous.comp_aestronglyMeasurable hW
+  have hmeas : ∀ j, AEMeasurable (fun x ↦ ENNReal.ofReal
+      (‖smoothTruncation (truncationParameter j) (W x) - min (W x) 1‖ ^ 2)) μ :=
+    fun j ↦ (((hf j).sub hm).norm.pow 2).aemeasurable.ennreal_ofReal
+  have hlim := tendsto_lintegral_of_dominated_convergence' (μ := μ)
+    (fun _ : Space n ↦ ENNReal.ofReal ((2 : ℝ) ^ 2)) (f := fun _ ↦ (0 : ℝ≥0∞))
+    (F := fun j x ↦ ENNReal.ofReal (‖smoothTruncation (truncationParameter j) (W x) - min (W x) 1‖ ^ 2))
+    hmeas ?_ ?_ ?_
+  · simpa only [vectorEnergy, lintegral_zero] using hlim
+  · intro j
+    filter_upwards [hW0] with x hx
+    have hm : ‖min (W x) 1‖ ≤ 1 := by
+      rw [Real.norm_eq_abs, abs_of_nonneg (le_min hx zero_le_one)]
+      exact min_le_right _ _
+    apply ENNReal.ofReal_le_ofReal
+    apply (sq_le_sq₀ (norm_nonneg _) (by norm_num)).mpr
+    exact (norm_sub_le _ _).trans (by linarith [smoothTruncation_norm_le (truncationParameter_pos j) hx])
+  · simp only [lintegral_const, ne_eq]
+    exact ENNReal.mul_ne_top ENNReal.ofReal_ne_top (measure_ne_top μ _)
+  · filter_upwards [hW0] with x hx
+    have ht : Tendsto (fun j ↦ smoothTruncation (truncationParameter j) (W x) - min (W x) 1)
+        atTop (𝓝 (0 : ℝ)) := by
+      simpa only [sub_self] using (smoothTruncation_tendsto hx).sub_const (min (W x) 1)
+    simpa only [norm_zero, zero_pow (by norm_num : (2 : ℕ) ≠ 0), ENNReal.ofReal_zero] using
+      ENNReal.tendsto_ofReal (ht.norm.pow 2)
+
+theorem truncationGradient_error_energy_tendsto {n : ℕ} {μ : Measure (Space n)}
+    {W : Space n → ℝ} {D : Space n → Space n}
+    (hW : AEStronglyMeasurable W μ) (hG : MemLp (truncatedGradient W D) 2 μ) :
+    Tendsto (fun j ↦ vectorEnergy μ (fun x ↦
+      truncationSlope (truncationParameter j) (W x) • truncatedGradient W D x -
+        truncatedGradient W D x)) atTop (𝓝 0) := by
+  let G := truncatedGradient W D
+  let F := fun j x ↦ truncationSlope (truncationParameter j) (W x) • G x
+  have hF : ∀ j, MemLp (F j) 2 μ := fun j ↦ truncationGradient_memLp hW hG _
+  have hfin : vectorEnergy μ G ≠ ⊤ :=
+    (memLp_two_iff_vectorEnergy hG.aestronglyMeasurable).mp hG |>.ne
+  have hmeas : ∀ j, AEMeasurable (fun x ↦ ENNReal.ofReal (‖F j x - G x‖ ^ 2)) μ := by
+    intro j
+    exact (((hF j).aestronglyMeasurable.sub hG.aestronglyMeasurable).norm.pow 2).aemeasurable.ennreal_ofReal
+  have hlim := tendsto_lintegral_of_dominated_convergence' (μ := μ)
+    (fun x ↦ ENNReal.ofReal (‖G x‖ ^ 2)) (f := fun _ ↦ (0 : ℝ≥0∞))
+    (F := fun j x ↦ ENNReal.ofReal (‖F j x - G x‖ ^ 2)) hmeas ?_ hfin ?_
+  · simpa only [vectorEnergy, Pi.sub_apply, lintegral_zero] using hlim
+  · intro j
+    apply ae_of_all
+    intro x
+    apply ENNReal.ofReal_le_ofReal
+    apply (sq_le_sq₀ (norm_nonneg _) (norm_nonneg _)).mpr
+    have he : F j x - G x = (truncationSlope (truncationParameter j) (W x) - 1) • G x := by
+      rw [sub_smul, one_smul]
+    rw [he, norm_smul, Real.norm_eq_abs,
+      abs_of_nonpos (sub_nonpos.mpr (truncationSlope_mem_Icc _ _).2)]
+    exact mul_le_of_le_one_left (norm_nonneg _) (by linarith [(truncationSlope_mem_Icc (truncationParameter j) (W x)).1])
+  · apply ae_of_all
+    intro x
+    have ht : Tendsto (fun j ↦ F j x) atTop (𝓝 (G x)) := by
+      have ht := (truncationSlope_tendsto (W x)).smul_const (G x)
+      by_cases hx : W x < 1
+      · simpa only [ite_eq_left hx, one_smul] using ht
+      · simpa only [F, G, truncatedGradient, ite_eq_right hx, smul_zero] using ht
+    have ht' : Tendsto (fun j ↦ F j x - G x) atTop (𝓝 (0 : Space n)) := by
+      simpa only [sub_self] using ht.sub_const (G x)
+    simpa only [norm_zero, zero_pow (by norm_num : (2 : ℕ) ≠ 0), ENNReal.ofReal_zero] using
+      ENNReal.tendsto_ofReal (ht'.norm.pow 2)
+
+theorem smoothTruncation_pair_tendsto {n : ℕ} {μ : Measure (Space n)}
+    [IsFiniteMeasure μ] {W : Space n → ℝ} {D : Space n → Space n}
+    (hW : AEStronglyMeasurable W μ) (hW0 : ∀ᵐ x ∂μ, 0 ≤ W x)
+    (hG : MemLp (truncatedGradient W D) 2 μ) :
+    ∃ (hh : MemLp (fun x ↦ min (W x) 1) 2 μ)
+      (hf : ∀ j, MemLp (fun x ↦ smoothTruncation (truncationParameter j) (W x)) 2 μ)
+      (hF : ∀ j, MemLp (fun x ↦ truncationSlope (truncationParameter j) (W x) •
+        truncatedGradient W D x) 2 μ),
+      Tendsto (fun j ↦ ((hf j).toLp (fun x ↦ smoothTruncation (truncationParameter j) (W x)),
+        (hF j).toLp (fun x ↦ truncationSlope (truncationParameter j) (W x) •
+          truncatedGradient W D x))) atTop
+        (𝓝 (hh.toLp (fun x ↦ min (W x) 1), hG.toLp (truncatedGradient W D))) := by
+  let hh := min_one_memLp hW hW0
+  let hf := fun j ↦ smoothTruncation_memLp hW hW0 (truncationParameter_pos j)
+  let hF := fun j ↦ truncationGradient_memLp hW hG (truncationParameter j)
+  refine ⟨hh, hf, hF, ?_⟩
+  apply Filter.Tendsto.prodMk_nhds
+  · apply (Lp.tendsto_Lp_iff_tendsto_eLpNorm'' _ hf _ hh).mpr
+    exact tendsto_eLpNorm_two_of_vectorEnergy_zero_of_aestronglyMeasurable
+      (fun index ↦ ((hf index).sub hh).aestronglyMeasurable)
+      (smoothTruncation_error_energy_tendsto hW hW0)
+  · apply (Lp.tendsto_Lp_iff_tendsto_eLpNorm'' _ hF _ hG).mpr
+    exact tendsto_eLpNorm_two_of_vectorEnergy_zero_of_aestronglyMeasurable
+      (fun index ↦ ((hF index).sub hG).aestronglyMeasurable)
+      (truncationGradient_error_energy_tendsto hW hG)
+
+end RPI.Boundary
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPI.Boundary
+
+theorem surfaceGradient_eq_tangentialGradient_filled {n : ℕ} {E S : Set (Space n)}
+    (hE : MeasurableSet E) {f g : Space n → ℝ} {x : Space n}
+    (hf : ContDiffAt ℝ 1 f x) (hfx : f x = 0) (hdf : fderiv ℝ f x ≠ 0)
+    (hlevel : ∀ᶠ y in 𝓝 x, y ∈ S ↔ f y = 0)
+    (hfilled : ∀ᶠ y in 𝓝 x, (f y < 0 → y ∈ E) ∧ (y ∈ E → f y ≤ 0)) :
+    surfaceGradient S g x = RPISingularHarnack.tangentialGradient E g x := by
+  have hgrad : gradient f x ≠ 0 := by
+    intro he
+    apply hdf
+    rw [← toDual_gradient, he, map_zero]
+  have hn : ‖gradient f x‖ ≠ 0 := norm_ne_zero_iff.mpr hgrad
+  rw [surfaceGradient_eq_level_formula hf hdf (by simpa only [hfx] using hlevel),
+    RPISingularHarnack.tangentialGradient,
+    RPISingularHarnack.approximateNormal_eq_gradient_filled hE
+      (hf.differentiableAt (by norm_num)) hfx hdf hfilled,
+    real_inner_smul_right, smul_smul, real_inner_comm (gradient f x) (gradient g x)]
+  congr 2
+  field_simp
+
+end RPI.Boundary
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPI.Boundary
+
+theorem surfaceGradient_ae_eq_tangentialGradient_filled {n : ℕ} (hn : 8 ≤ n)
+    {E S Z : Set (Space n)} (hE : MeasurableSet E) (hS : MeasurableSet S)
+    (hZd : dimH Z ≤ (n - 8 : ℕ))
+    (hreg : ∀ x ∈ S \ Z, ∃ f : Space n → ℝ,
+      ContDiffAt ℝ 1 f x ∧ f x = 0 ∧ fderiv ℝ f x ≠ 0 ∧
+      (∀ᶠ y in 𝓝 x, y ∈ S ↔ f y = 0) ∧
+      (∀ᶠ y in 𝓝 x, (f y < 0 → y ∈ E) ∧ (y ∈ E → f y ≤ 0))) (g : Space n → ℝ) :
+    surfaceGradient S g =ᵐ[surfaceMeasure S]
+      RPISingularHarnack.tangentialGradient E g := by
+  have hz : ∀ᵐ x ∂surfaceMeasure S, x ∉ Z := by
+    simpa only [ae_iff, not_not, Set.ofPred_mem_eq] using
+      surfaceMeasure_zero_of_singular_dim hn S Z hZd
+  filter_upwards [ae_restrict_mem hS, hz] with x hx hzx
+  obtain ⟨f, hf, hfx, hdf, hlevel, hfilled⟩ := hreg x ⟨hx, hzx⟩
+  exact surfaceGradient_eq_tangentialGradient_filled hE hf hfx hdf hlevel hfilled
+
+end RPI.Boundary
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+
+theorem wholeBoundarySobolev_of_local_sobolevGraph_filled {n : ℕ} (hn : 8 ≤ n)
+    {E Z : Set (Euc n)} (hE : MeasurableSet E)
+    (hS : MeasurableSet (perimeterSupport E)) (hZd : dimH Z ≤ (n - 8 : ℕ))
+    (hreg : ∀ x ∈ perimeterSupport E \ Z, ∃ f : Euc n → ℝ,
+      ContDiffAt ℝ 1 f x ∧ f x = 0 ∧ fderiv ℝ f x ≠ 0 ∧
+      (∀ᶠ y in 𝓝 x, y ∈ perimeterSupport E ↔ f y = 0) ∧
+      (∀ᶠ y in 𝓝 x, (f y < 0 → y ∈ E) ∧ (y ∈ E → f y ≤ 0)))
+    {h : Euc n → ℝ} {G : Euc n → Euc n}
+    (hgraph : ∀ R : ℝ, 0 < R →
+      ∃ (hh : MemLp h 2 ((RPI.Boundary.surfaceMeasure (perimeterSupport E)).restrict
+          (Metric.ball 0 R)))
+        (hG : MemLp G 2 ((RPI.Boundary.surfaceMeasure (perimeterSupport E)).restrict
+          (Metric.ball 0 R))),
+        (hh.toLp h, hG.toLp G) ∈ closure
+          (RPI.Boundary.smoothGradientGraph (perimeterSupport E) (Metric.ball 0 R))) :
+    WholeBoundarySobolev E h G := by
+  intro R hR
+  obtain ⟨hh, hG, hpair⟩ := hgraph R hR
+  obtain ⟨φ, hφ, hlim, hDlim⟩ :=
+    RPI.Boundary.exists_compact_energy_approximants_of_mem_sobolevGraph
+      (perimeterSupport E) (Metric.ball 0 R) Metric.isBounded_ball
+      measurableSet_ball hh hG hpair
+  dsimp only
+  rw [boundaryArea_eq_smul_surfaceMeasure, Measure.restrict_smul]
+  refine ⟨hh.smul_measure (hausdorffNormalizationRatio_ne_top _),
+    hG.smul_measure (hausdorffNormalizationRatio_ne_top _), φ, hφ, ?_, ?_⟩
+  · simp_rw [lintegral_smul_measure, smul_eq_mul]
+    simpa only [mul_zero] using ENNReal.Tendsto.const_mul hlim
+      (Or.inr (hausdorffNormalizationRatio_ne_top (n - 1)))
+  · have heq : ∀ j, RPI.Boundary.surfaceGradient (perimeterSupport E) (φ j)
+        =ᵐ[(RPI.Boundary.surfaceMeasure (perimeterSupport E)).restrict (Metric.ball 0 R)]
+          tangentialGradient E (φ j) := fun j ↦
+      (RPI.Boundary.surfaceGradient_ae_eq_tangentialGradient_filled hn hE hS hZd hreg (φ j)).filter_mono
+        (ae_mono Measure.restrict_le_self)
+    have hl : Tendsto (fun j ↦ ∫⁻ x,
+        ENNReal.ofReal (‖tangentialGradient E (φ j) x - G x‖ ^ 2)
+        ∂(RPI.Boundary.surfaceMeasure (perimeterSupport E)).restrict (Metric.ball 0 R))
+        atTop (𝓝 0) := by
+      apply hDlim.congr
+      intro j
+      apply lintegral_congr_ae
+      filter_upwards [heq j] with x hx
+      rw [hx]
+    simp_rw [lintegral_smul_measure, smul_eq_mul]
+    simpa only [mul_zero] using ENNReal.Tendsto.const_mul hl
+      (Or.inr (hausdorffNormalizationRatio_ne_top (n - 1)))
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPI.Boundary
+
+theorem HasSmoothTraceJetAt.mul_smooth {n : ℕ} {S : Set (Space n)}
+    {h : Space n → ℝ} {G : Space n → Space n} {x : Space n}
+    (hxS : x ∈ S) (hx : HasSmoothTraceJetAt S h G x)
+    {ψ : Space n → ℝ} (hψ : ContDiff ℝ ∞ ψ) :
+    HasSmoothTraceJetAt S (fun y ↦ ψ y * h y)
+      (fun y ↦ ψ y • G y + h y • surfaceGradient S ψ y) x := by
+  obtain ⟨U, hU, hxU, f, hf, he, hG⟩ := hx
+  refine ⟨U, hU, hxU, fun y ↦ ψ y * f y, hψ.contDiffOn.mul hf, ?_, ?_⟩
+  · intro y hy
+    exact congrArg (fun a ↦ ψ y * a) (he hy)
+  · rw [surfaceGradient_mul (hψ.differentiable (by simp) x)
+        ((hf.contDiffAt (hU.mem_nhds hxU)).differentiableAt (by simp)),
+        hG, he ⟨hxS, hxU⟩]
+
+theorem regular_trace_jet_mul_spatial {n : ℕ} {S Z : Set (Space n)}
+    {h : Space n → ℝ} {G : Space n → Space n}
+    (hreg : ∀ x ∈ S \ Z, HasSmoothTraceJetAt S h G x)
+    {ψ : Space n → ℝ} (hψ : ContDiff ℝ ∞ ψ) :
+    ∀ x ∈ S \ (Z ∩ tsupport ψ), HasSmoothTraceJetAt S (fun y ↦ ψ y * h y)
+      (fun y ↦ ψ y • G y + h y • surfaceGradient S ψ y) x := by
+  intro x hx
+  by_cases hxZ : x ∈ Z
+  · have hxs : x ∉ tsupport ψ := fun hxψ ↦ hx.2 ⟨hxZ, hxψ⟩
+    have hloc : ψ =ᶠ[𝓝 x] (0 : Space n → ℝ) :=
+      notMem_tsupport_iff_eventuallyEq.mp hxs
+    obtain ⟨U, hUsub, hU, hxU⟩ := mem_nhds_iff.mp hloc
+    have hψgrad : gradient ψ x = 0 := by
+      simpa only [Pi.zero_def, gradient_fun_const] using hloc.gradient_eq
+    have hψx : ψ x = 0 := hloc.self_of_nhds
+    refine ⟨U, hU, hxU, fun _ ↦ 0, contDiffOn_const, ?_, ?_⟩
+    · intro y hy
+      have hy0 : ψ y = 0 := hUsub hy.2
+      simp only [hy0, zero_mul]
+    · simp only [hψx, surfaceGradient, hψgrad, gradient_fun_const,
+        map_zero, Submodule.coe_zero, zero_smul, smul_zero, add_zero]
+  · exact (hreg x ⟨hx.1, hxZ⟩).mul_smooth hx.1 hψ
+
+theorem hausdorff_trace_mem_sobolevGraph_ball {n : ℕ} (hn : 8 ≤ n)
+    {S Z : Set (Space n)} (hS : IsClosed S) (hZ : IsClosed Z)
+    (hZd : dimH Z ≤ (n - 8 : ℕ)) {R : ℝ} (hR : 0 < R)
+    {h : Space n → ℝ} {G : Space n → Space n}
+    (hreg : ∀ x ∈ S \ Z, HasSmoothTraceJetAt S h G x)
+    (hh : MemLp h 2 ((surfaceMeasure S).restrict (Metric.ball 0 R)))
+    (hG : MemLp G 2 ((surfaceMeasure S).restrict (Metric.ball 0 R)))
+    {M : ℝ} (hM : 0 ≤ M)
+    (hbound : ∀ᵐ x ∂(surfaceMeasure S).restrict (Metric.ball 0 R), ‖h x‖ ≤ M)
+    (hgrad : ∀ f : Space n → ℝ, ContDiff ℝ ∞ f →
+      AEStronglyMeasurable (surfaceGradient S f)
+        ((surfaceMeasure S).restrict (Metric.ball 0 R)))
+    {A r₀ : ℝ} (hA : 0 < A) (hr₀ : 0 < r₀)
+    (hgrowth : ∀ c ∈ Z ∩ Metric.closedBall 0 (R + 2), ∀ r : ℝ, 0 < r → r < r₀ →
+      (surfaceMeasure S).restrict (Metric.ball 0 R) (Metric.ball c r) ≤
+        ENNReal.ofReal (A * r ^ (n - 1))) :
+    (hh.toLp h, hG.toLp G) ∈ closure (smoothGradientGraph S (Metric.ball 0 R)) := by
+  let b : ContDiffBump (0 : Space n) := ⟨R + 1, R + 2, by linarith, by linarith⟩
+  let K : Set (Space n) := Z ∩ tsupport b
+  have hKc : IsCompact K := b.hasCompactSupport.of_isClosed_subset
+    (hZ.inter (isClosed_tsupport b)) inter_subset_right
+  have hKd : dimH K ≤ (n - 8 : ℕ) :=
+    (dimH_mono (show K ⊆ Z from inter_subset_left)).trans hZd
+  have hb : ∀ x ∈ Metric.ball (0 : Space n) R, b =ᶠ[𝓝 x] 1 := by
+    intro x hx
+    apply b.eventuallyEq_one_of_mem_ball
+    exact Metric.mem_ball.mpr ((Metric.mem_ball.mp hx).trans (by dsimp [b]; linarith))
+  have hbG : ∀ x ∈ Metric.ball (0 : Space n) R, surfaceGradient S b x = 0 := by
+    intro x hx
+    have hgr : gradient b x = 0 := by
+      simpa only [Pi.one_def, gradient_fun_const] using (hb x hx).gradient_eq
+    simp only [surfaceGradient, hgr, map_zero, Submodule.coe_zero]
+  let h' := fun x ↦ b x * h x
+  let G' := fun x ↦ b x • G x + h x • surfaceGradient S b x
+  have he : h' =ᵐ[(surfaceMeasure S).restrict (Metric.ball 0 R)] h := by
+    filter_upwards [ae_restrict_mem measurableSet_ball] with x hx
+    dsimp [h']
+    rw [(hb x hx).self_of_nhds]
+    simp
+  have hDe : G' =ᵐ[(surfaceMeasure S).restrict (Metric.ball 0 R)] G := by
+    filter_upwards [ae_restrict_mem measurableSet_ball] with x hx
+    dsimp [G']
+    rw [(hb x hx).self_of_nhds, hbG x hx]
+    simp
+  let hh' := hh.ae_eq he.symm
+  let hG' := hG.ae_eq hDe.symm
+  have hbound' : ∀ᵐ x ∂(surfaceMeasure S).restrict (Metric.ball 0 R), ‖h' x‖ ≤ M := by
+    filter_upwards [he, hbound] with x hx hbx
+    simpa only [hx] using hbx
+  have hmem := hausdorff_trace_mem_sobolevGraph hn hS (Metric.ball 0 R) hKc hKd
+    (regular_trace_jet_mul_spatial hreg b.contDiff) hh' hG' hM
+    hbound' hgrad hA hr₀ (fun c hc r hr hrr ↦ hgrowth c
+      ⟨hc.1, by simpa only [b.tsupport_eq] using hc.2⟩ r hr hrr)
+  have heLp : hh'.toLp h' = hh.toLp h := hh'.toLp_congr hh he
+  have hDeLp : hG'.toLp G' = hG.toLp G := hG'.toLp_congr hG hDe
+  change (hh'.toLp h', hG'.toLp G') ∈ _ at hmem
+  simpa only [heLp, hDeLp] using hmem
+
+end RPI.Boundary
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPI.Boundary
+
+theorem trace_aestronglyMeasurable_of_singular_dim {n : ℕ} (hn : 8 ≤ n)
+    {S Z : Set (Space n)} (hS : MeasurableSet S) (hZ : MeasurableSet Z)
+    (hdim : dimH Z ≤ (n - 8 : ℕ)) {h : Space n → ℝ} {G : Space n → Space n}
+    (hreg : ∀ x ∈ S \ Z, HasSmoothTraceJetAt S h G x) :
+    AEStronglyMeasurable h (surfaceMeasure S) := by
+  have hc : ContinuousOn h (S \ Z) := by
+    intro x hx
+    obtain ⟨U, hU, hxU, f, hf, he, _⟩ := hreg x hx
+    have hf' : ContinuousWithinAt f (S \ Z) x :=
+      (hf.contDiffAt (hU.mem_nhds hxU)).continuousAt.continuousWithinAt
+    apply hf'.congr_of_eventuallyEq _ (he ⟨hx.1, hxU⟩).symm
+    filter_upwards [self_mem_nhdsWithin,
+      mem_nhdsWithin_of_mem_nhds (hU.mem_nhds hxU)] with y hy hyU
+    exact (he ⟨hy.1, hyU⟩).symm
+  have hs : ∀ᵐ x ∂surfaceMeasure S, x ∈ S := ae_restrict_mem hS
+  have hz : ∀ᵐ x ∂surfaceMeasure S, x ∉ Z := by
+    simpa only [ae_iff, not_not, Set.ofPred_mem_eq] using
+      surfaceMeasure_zero_of_singular_dim hn S Z hdim
+  have hregular : ∀ᵐ x ∂surfaceMeasure S, x ∈ S \ Z := by
+    filter_upwards [hs, hz] with x hx hzx
+    exact ⟨hx, hzx⟩
+  have hmeas := hc.aestronglyMeasurable (μ := surfaceMeasure S) (hS.diff hZ)
+  rwa [Measure.restrict_eq_self_of_ae_mem hregular] at hmeas
+
+theorem min_one_mem_sobolevGraph_ball {n : ℕ} (hn : 8 ≤ n)
+    {S Z : Set (Space n)} (hS : IsClosed S) (hZ : IsClosed Z)
+    (hZd : dimH Z ≤ (n - 8 : ℕ)) {R : ℝ} (hR : 0 < R)
+    [IsFiniteMeasure ((surfaceMeasure S).restrict (Metric.ball 0 R))]
+    {W : Space n → ℝ} {D : Space n → Space n}
+    (htrace : ∀ x ∈ S \ Z, HasSmoothTraceJetAt S W D x)
+    (hW0 : ∀ᵐ x ∂(surfaceMeasure S).restrict (Metric.ball 0 R), 0 ≤ W x)
+    (hG : MemLp (truncatedGradient W D) 2
+      ((surfaceMeasure S).restrict (Metric.ball 0 R)))
+    (hgrad : ∀ f : Space n → ℝ, ContDiff ℝ ∞ f →
+      AEStronglyMeasurable (surfaceGradient S f)
+        ((surfaceMeasure S).restrict (Metric.ball 0 R)))
+    {A r₀ : ℝ} (hA : 0 < A) (hr₀ : 0 < r₀)
+    (hgrowth : ∀ c ∈ Z ∩ Metric.closedBall 0 (R + 2), ∀ r : ℝ, 0 < r → r < r₀ →
+      (surfaceMeasure S).restrict (Metric.ball 0 R) (Metric.ball c r) ≤
+        ENNReal.ofReal (A * r ^ (n - 1))) :
+    ∃ hh : MemLp (fun x ↦ min (W x) 1) 2
+        ((surfaceMeasure S).restrict (Metric.ball 0 R)),
+      (hh.toLp (fun x ↦ min (W x) 1), hG.toLp (truncatedGradient W D)) ∈
+        closure (smoothGradientGraph S (Metric.ball 0 R)) := by
+  have hW : AEStronglyMeasurable W ((surfaceMeasure S).restrict (Metric.ball 0 R)) :=
+    (trace_aestronglyMeasurable_of_singular_dim hn hS.measurableSet
+    hZ.measurableSet hZd htrace).mono_measure (Measure.restrict_le_self)
+  obtain ⟨hh, hf, hF, hlim⟩ := smoothTruncation_pair_tendsto hW hW0 hG
+  refine ⟨hh, isClosed_closure.mem_of_tendsto hlim (Filter.Eventually.of_forall ?_)⟩
+  intro j
+  apply hausdorff_trace_mem_sobolevGraph_ball hn hS hZ hZd hR
+    (fun x hx ↦ (htrace x hx).smoothTruncation hx.1 (truncationParameter_pos j))
+    (hf j) (hF j) (M := 1) zero_le_one _ hgrad hA hr₀ hgrowth
+  filter_upwards [hW0] with x hx
+  exact smoothTruncation_norm_le (truncationParameter_pos j) hx
+
+end RPI.Boundary
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/SurfaceGradientConstSub.lean b/lean/OAI/Geometry/Perimeter/Regularity/SurfaceGradientConstSub.lean
new file mode 100644
index 0000000000000000000000000000000000000000..49199edf755b720a39e12ac800e938342cb6258b
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/SurfaceGradientConstSub.lean
@@ -0,0 +1,510 @@
+import OAI.Geometry.Perimeter.Regularity.IntegralInnerEqInnerToLp
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPI.Boundary
+
+theorem surfaceGradient_const_sub {n : ℕ} (S : Set (Space n)) {f : Space n → ℝ}
+    {x : Space n} (c : ℝ) :
+    surfaceGradient S (fun y ↦ c - f y) x = -surfaceGradient S f x := by
+  have hg : gradient (fun y ↦ c - f y) x = -gradient f x := by
+    apply (InnerProductSpace.toDual ℝ (Space n)).injective
+    simp only [map_neg, toDual_gradient]
+    exact fderiv_const_sub c
+  simp only [surfaceGradient, hg, map_neg, Submodule.coe_neg]
+
+section RegularEnergy
+variable {n : ℕ} {S Z : Set (Space n)}
+variable (hS : IsClosed S) (hZ : IsClosed Z)
+variable (hfinite : ∀ R : ℝ, 0 < R → (surfaceMeasure S) (Metric.ball 0 R) < ⊤)
+variable (hgrad : ∀ f : Space n → ℝ, ContDiff ℝ ∞ f →
+  AEStronglyMeasurable (surfaceGradient S f) (surfaceMeasure S))
+include hS hZ hfinite hgrad
+
+theorem regular_gradient_memLp_test {W : Space n → ℝ} {D : Space n → Space n}
+    (htrace : ∀ x ∈ S \ Z, HasSmoothTraceJetAt S W D x)
+    {η : Space n → ℝ} (hη : ContDiff ℝ ∞ η) (hcη : HasCompactSupport η)
+    (hηZ : Disjoint (tsupport η) Z) :
+    MemLp (fun x ↦ η x • D x) 2 (surfaceMeasure S) := by
+  have hi := regular_gradient_memLp_indicator_test hS hZ hfinite hgrad htrace hcη hηZ
+  obtain ⟨M, hM⟩ := hcη.exists_bound_of_continuous hη.continuous
+  have hh : MemLp (fun x ↦ η x • (tsupport η).indicator D x) 2 (surfaceMeasure S) := by
+    apply hi.of_le_mul (c := M) (hη.continuous.aestronglyMeasurable.smul hi.aestronglyMeasurable)
+    apply ae_of_all
+    intro x
+    change ‖η x • (tsupport η).indicator D x‖ ≤ M * ‖(tsupport η).indicator D x‖
+    rw [norm_smul]
+    exact mul_le_mul_of_nonneg_right (hM x) (norm_nonneg _)
+  apply hh.ae_eq
+  apply ae_of_all
+  intro x
+  by_cases hx : x ∈ tsupport η
+  · simp only [indicator_of_mem hx]
+  · simp only [indicator_of_notMem hx, image_eq_zero_of_notMem_tsupport hx, zero_smul]
+
+omit hS hZ hfinite hgrad in
+theorem smoothTruncation_le_one_all {ε : ℝ} (hε : 0 < ε) (s : ℝ) :
+    smoothTruncation ε s ≤ 1 := by
+  by_cases hs : 0 ≤ s
+  · exact (smoothTruncation_le_min hε hs).trans (min_le_right _ _)
+  · have hh := smoothTruncation_monotone ε (le_of_not_ge hs)
+    have hz : smoothTruncation ε 0 = 0 := by simp [smoothTruncation]
+    exact (hh.trans_eq hz).trans zero_le_one
+
+theorem smoothTruncation_regular_energy {W : Space n → ℝ} {D : Space n → Space n}
+    (htrace : ∀ x ∈ S \ Z, HasSmoothTraceJetAt S W D x)
+    (hW0 : ∀ x ∈ S \ Z, 0 ≤ W x)
+    (hsup : ∀ φ : Space n → ℝ, ContDiff ℝ ∞ φ → HasCompactSupport φ →
+      Disjoint (tsupport φ) Z → (∀ x, 0 ≤ φ x) →
+      0 ≤ ∫ x, inner ℝ (D x) (surfaceGradient S φ x) ∂surfaceMeasure S)
+    {ε : ℝ} (hε : 0 < ε) {η : Space n → ℝ}
+    (hη : ContDiff ℝ ∞ η) (hcη : HasCompactSupport η)
+    (hηZ : Disjoint (tsupport η) Z) :
+    (∫ x, ‖η x • (truncationSlope ε (W x) • D x)‖ ^ 2 ∂surfaceMeasure S) ≤
+      4 * ∫ x, ‖surfaceGradient S η x‖ ^ 2 ∂surfaceMeasure S := by
+  let T := smoothTruncation ε
+  let G : Space n → Space n := fun x ↦ truncationSlope ε (W x) • D x
+  have htt : ∀ x ∈ S \ Z, HasSmoothTraceJetAt S (T ∘ W) G x := by
+    intro x hx
+    simpa only [T, G, smoothTruncation_deriv] using
+      (htrace x hx).comp hx.1 (smoothTruncation_contDiff ε)
+  obtain ⟨F, hF, he⟩ := exists_smooth_representative_near_regular_test hS hZ htrace
+    (isClosed_tsupport η) hηZ
+  let ψ : Space n → ℝ := fun x ↦ (η x * η x) * (1 - T (F x))
+  have hTF : ContDiff ℝ ∞ (T ∘ F) := (smoothTruncation_contDiff ε).comp hF
+  have hψ : ContDiff ℝ ∞ ψ := (hη.mul hη).mul (contDiff_const.sub hTF)
+  have hcψ : HasCompactSupport ψ := hcη.mul_right.mul_right
+  have hsub : tsupport ψ ⊆ tsupport η := tsupport_mul_subset_left.trans tsupport_mul_subset_left
+  have hψZ : Disjoint (tsupport ψ) Z := hηZ.mono_left hsub
+  have hψ0 : ∀ x, 0 ≤ ψ x := fun x ↦ mul_nonneg (mul_self_nonneg (η x))
+    (sub_nonneg.mpr (smoothTruncation_le_one_all hε (F x)))
+  have ht : 0 ≤ ∫ x, inner ℝ (G x) (surfaceGradient S ψ x) ∂surfaceMeasure S :=
+    smoothTruncation_regular_test hS hZ hfinite hgrad htrace hsup hε hψ hcψ hψZ hψ0
+  let A : Space n → Space n := fun x ↦ η x • G x
+  let B : Space n → Space n := surfaceGradient S η
+  have hA : MemLp A 2 (surfaceMeasure S) :=
+    regular_gradient_memLp_test hS hZ hfinite hgrad htt hη hcη hηZ
+  have hB : MemLp B 2 (surfaceMeasure S) :=
+    surfaceGradient_memLp_of_compact hfinite hgrad hη hcη
+  have hi := integrable_regular_test_pair hS hZ hfinite hgrad htt hψ hcψ hψZ
+  have hAs := (memLp_two_iff_integrable_sq_norm hA.aestronglyMeasurable).mp hA
+  have hBs := (memLp_two_iff_integrable_sq_norm hB.aestronglyMeasurable).mp hB
+  have hu : Integrable (fun x ↦ 2 * ‖B x‖ ^ 2 - (1 / 2 : ℝ) * ‖A x‖ ^ 2) (surfaceMeasure S) :=
+    (hBs.const_mul 2).sub (hAs.const_mul (1 / 2))
+  have hle : (∫ x, inner ℝ (G x) (surfaceGradient S ψ x) ∂surfaceMeasure S) ≤
+      ∫ x, 2 * ‖B x‖ ^ 2 - (1 / 2 : ℝ) * ‖A x‖ ^ 2 ∂surfaceMeasure S := by
+    apply integral_mono_ae hi hu
+    filter_upwards [ae_restrict_mem hS.measurableSet] with x hx
+    by_cases hxt : x ∈ tsupport η
+    · have hxr : x ∈ S \ Z := ⟨hx, fun hz ↦ Set.disjoint_left.mp hηZ hxt hz⟩
+      have hFx := he x ⟨hx, hxt⟩
+      have hgT : surfaceGradient S (fun y ↦ T (F y)) x = G x := by
+        change surfaceGradient S (T ∘ F) x = G x
+        rw [surfaceGradient_comp (hF.differentiable (by simp) x)
+          ((smoothTruncation_contDiff ε).differentiable (by simp) (F x))]
+        simp only [smoothTruncation_deriv, hFx.1, hFx.2, G]
+      have hgψ : surfaceGradient S ψ x =
+          (η x * η x) • (-G x) + (1 - T (W x)) • (η x • B x + η x • B x) := by
+        dsimp only [ψ]
+        rw [surfaceGradient_mul (S := S) (g := fun y ↦ 1 - T (F y)) ((hη.mul hη).differentiable (by simp) x)
+          ((contDiff_const.sub hTF).differentiable (by simp) x)]
+        rw [surfaceGradient_const_sub S 1, hgT,
+          surfaceGradient_mul (hη.differentiable (by simp) x) (hη.differentiable (by simp) x), hFx.1]
+      have heq : inner ℝ (G x) (surfaceGradient S ψ x) =
+          inner ℝ (A x) ((2 * (1 - T (W x))) • B x - A x) := by
+        rw [hgψ]
+        simp only [A, inner_add_right, real_inner_smul_right, inner_neg_right,
+          real_inner_smul_left, inner_sub_right]
+        ring
+      rw [heq]
+      exact truncation_test_pointwise (A x) (B x)
+        (smoothTruncation_nonneg ε (hW0 x hxr)) (smoothTruncation_le_one_all hε (W x))
+    · have hAx : A x = 0 := by simp only [A, image_eq_zero_of_notMem_tsupport hxt, zero_smul]
+      have hBx : B x = 0 := surfaceGradient_eq_zero_of_notMem_tsupport S hxt
+      rw [surfaceGradient_eq_zero_of_notMem_tsupport S (fun h ↦ hxt (hsub h)),
+        inner_zero_right, hAx, hBx]
+      norm_num
+  rw [integral_sub (hBs.const_mul 2) (hAs.const_mul (1 / 2)),
+    integral_const_mul, integral_const_mul] at hle
+  change (∫ x, ‖A x‖ ^ 2 ∂surfaceMeasure S) ≤ 4 * ∫ x, ‖B x‖ ^ 2 ∂surfaceMeasure S
+  linarith
+
+end RegularEnergy
+end RPI.Boundary
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPI.Boundary
+section TruncationRegularLimit
+variable {n : ℕ} {S Z : Set (Space n)}
+variable (hS : IsClosed S) (hZ : IsClosed Z)
+variable (hfinite : ∀ R : ℝ, 0 < R → (surfaceMeasure S) (Metric.ball 0 R) < ⊤)
+variable (hgrad : ∀ f : Space n → ℝ, ContDiff ℝ ∞ f →
+  AEStronglyMeasurable (surfaceGradient S f) (surfaceMeasure S))
+variable {W : Space n → ℝ} {D : Space n → Space n}
+variable (htrace : ∀ x ∈ S \ Z, HasSmoothTraceJetAt S W D x)
+variable (hsup : ∀ φ : Space n → ℝ, ContDiff ℝ ∞ φ → HasCompactSupport φ →
+  Disjoint (tsupport φ) Z → (∀ x, 0 ≤ φ x) →
+  0 ≤ ∫ x, inner ℝ (D x) (surfaceGradient S φ x) ∂surfaceMeasure S)
+include hS hZ hfinite hgrad htrace hsup
+
+theorem min_regular_test {φ : Space n → ℝ} (hφ : ContDiff ℝ ∞ φ)
+    (hcφ : HasCompactSupport φ) (hφZ : Disjoint (tsupport φ) Z) (hpφ : ∀ x, 0 ≤ φ x) :
+    0 ≤ ∫ x, inner ℝ (truncatedGradient W D x) (surfaceGradient S φ x) ∂surfaceMeasure S := by
+  let b : Space n → ℝ := fun x ↦ inner ℝ (D x) (surfaceGradient S φ x)
+  have hb : Integrable b (surfaceMeasure S) :=
+    integrable_regular_test_pair hS hZ hfinite hgrad htrace hφ hcφ hφZ
+  let q : ℕ → Space n → ℝ := fun j x ↦
+    inner ℝ (truncationSlope (truncationParameter j) (W x) • D x) (surfaceGradient S φ x)
+  have hq : ∀ j, AEStronglyMeasurable (q j) (surfaceMeasure S) := by
+    intro j
+    have htt : ∀ x ∈ S \ Z, HasSmoothTraceJetAt S (smoothTruncation (truncationParameter j) ∘ W)
+        (fun y ↦ truncationSlope (truncationParameter j) (W y) • D y) x := by
+      intro x hx
+      simpa only [smoothTruncation_deriv] using
+        (htrace x hx).comp hx.1 (smoothTruncation_contDiff (truncationParameter j))
+    exact (integrable_regular_test_pair hS hZ hfinite hgrad htt hφ hcφ hφZ).aestronglyMeasurable
+  have hlim : Tendsto (fun j ↦ ∫ x, q j x ∂surfaceMeasure S) atTop
+      (𝓝 (∫ x, inner ℝ (truncatedGradient W D x) (surfaceGradient S φ x) ∂surfaceMeasure S)) := by
+    apply tendsto_integral_of_dominated_convergence (fun x ↦ ‖b x‖) hq hb.norm
+    · intro j
+      apply ae_of_all
+      intro x
+      simp only [q, real_inner_smul_left, norm_mul]
+      exact mul_le_of_le_one_left (norm_nonneg _) (truncationSlope_norm_le _ _)
+    · apply ae_of_all
+      intro x
+      have ht := (truncationSlope_tendsto (W x)).mul_const (b x)
+      by_cases hx : W x < 1
+      · simpa only [q, b, truncatedGradient, ite_eq_left hx, one_mul, real_inner_smul_left] using ht
+      · simpa only [q, b, truncatedGradient, ite_eq_right hx, zero_mul, inner_zero_left,
+          real_inner_smul_left] using ht
+  apply ge_of_tendsto hlim
+  apply Eventually.of_forall
+  intro j
+  exact smoothTruncation_regular_test hS hZ hfinite hgrad htrace hsup
+    (truncationParameter_pos j) hφ hcφ hφZ hpφ
+
+theorem min_regular_energy (hW0 : ∀ x ∈ S \ Z, 0 ≤ W x)
+    {η : Space n → ℝ} (hη : ContDiff ℝ ∞ η) (hcη : HasCompactSupport η)
+    (hηZ : Disjoint (tsupport η) Z) :
+    MemLp (fun x ↦ η x • truncatedGradient W D x) 2 (surfaceMeasure S) ∧
+      vectorEnergy (surfaceMeasure S) (fun x ↦ η x • truncatedGradient W D x) ≤
+        4 * vectorEnergy (surfaceMeasure S) (surfaceGradient S η) := by
+  let A : ℕ → Space n → Space n := fun j x ↦ η x • (truncationSlope (truncationParameter j) (W x) • D x)
+  let H : Space n → Space n := fun x ↦ η x • truncatedGradient W D x
+  have hA : ∀ j, MemLp (A j) 2 (surfaceMeasure S) := by
+    intro j
+    have htt : ∀ x ∈ S \ Z, HasSmoothTraceJetAt S (smoothTruncation (truncationParameter j) ∘ W)
+        (fun y ↦ truncationSlope (truncationParameter j) (W y) • D y) x := by
+      intro x hx
+      simpa only [smoothTruncation_deriv] using
+        (htrace x hx).comp hx.1 (smoothTruncation_contDiff (truncationParameter j))
+    exact regular_gradient_memLp_test hS hZ hfinite hgrad htt hη hcη hηZ
+  have hB : MemLp (surfaceGradient S η) 2 (surfaceMeasure S) :=
+    surfaceGradient_memLp_of_compact hfinite hgrad hη hcη
+  have he : ∀ j, vectorEnergy (surfaceMeasure S) (A j) ≤
+      4 * vectorEnergy (surfaceMeasure S) (surfaceGradient S η) := by
+    intro j
+    have h := ENNReal.ofReal_le_ofReal (smoothTruncation_regular_energy hS hZ hfinite hgrad
+      htrace hW0 hsup (truncationParameter_pos j) hη hcη hηZ)
+    rw [ENNReal.ofReal_mul (by norm_num : (0 : ℝ) ≤ 4)] at h
+    simpa only [A, vectorEnergy, ofReal_integral_eq_lintegral_ofReal
+      ((memLp_two_iff_integrable_sq_norm (hA j).aestronglyMeasurable).mp (hA j))
+      (ae_of_all _ (fun x ↦ sq_nonneg ‖A j x‖)),
+      ofReal_integral_eq_lintegral_ofReal
+      ((memLp_two_iff_integrable_sq_norm hB.aestronglyMeasurable).mp hB)
+      (ae_of_all _ (fun x ↦ sq_nonneg ‖surfaceGradient S η x‖)), ENNReal.ofReal_ofNat] using h
+  have hlim : ∀ᵐ x ∂surfaceMeasure S, Tendsto (fun j ↦ A j x) atTop (𝓝 (H x)) := by
+    apply ae_of_all
+    intro x
+    have ht := (tendsto_const_nhds (x := η x)).smul
+      ((truncationSlope_tendsto (W x)).smul_const (D x))
+    by_cases hx : W x < 1
+    · simpa only [A, H, truncatedGradient, ite_eq_left hx, one_smul] using ht
+    · simpa only [A, H, truncatedGradient, ite_eq_right hx, zero_smul] using ht
+  have hHm : AEStronglyMeasurable H (surfaceMeasure S) :=
+    aestronglyMeasurable_of_tendsto_ae atTop (fun j ↦ (hA j).aestronglyMeasurable) hlim
+  have hHE := vectorEnergy_le_of_ae_tendsto (fun j ↦ (hA j).aestronglyMeasurable) hlim
+    (Eventually.of_forall he) tendsto_const_nhds
+  refine ⟨(memLp_two_iff_vectorEnergy hHm).mpr (hHE.trans_lt ?_), hHE⟩
+  exact ENNReal.mul_lt_top (by norm_num) ((memLp_two_iff_vectorEnergy hB.aestronglyMeasurable).mp hB)
+
+end TruncationRegularLimit
+end RPI.Boundary
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPI
+
+theorem vectorEnergy_restrict_of_vanish {X V : Type*} [MeasurableSpace X]
+    [NormedAddCommGroup V] {μ : Measure X} {U : Set X} (hU : MeasurableSet U)
+    {F : X → V} (hF : ∀ x ∉ U, F x = 0) :
+    vectorEnergy (μ.restrict U) F = vectorEnergy μ F := by
+  unfold vectorEnergy
+  rw [← lintegral_indicator hU]
+  apply lintegral_congr
+  intro x
+  by_cases hx : x ∈ U
+  · simp only [indicator_of_mem hx]
+  · simp only [indicator_of_notMem hx, hF x hx, norm_zero, ne_eq,
+      OfNat.ofNat_ne_zero, not_false_eq_true, zero_pow, ENNReal.ofReal_zero]
+
+end RPI
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPI.Boundary
+
+theorem min_local_energy {n : ℕ} (hn : 8 ≤ n)
+    {S Z : Set (Space n)} (hS : IsClosed S) (hZ : IsClosed Z)
+    (hZd : dimH Z ≤ (n - 8 : ℕ))
+    (hfinite : ∀ R : ℝ, 0 < R → (surfaceMeasure S) (Metric.ball 0 R) < ⊤)
+    (hgrad : ∀ f : Space n → ℝ, ContDiff ℝ ∞ f →
+      AEStronglyMeasurable (surfaceGradient S f) (surfaceMeasure S))
+    (hgrowth : ∀ R : ℝ, 0 < R → ∃ A r₀ : ℝ, 0 < A ∧ 0 < r₀ ∧
+      ∀ c ∈ Z ∩ Metric.closedBall 0 R, ∀ r : ℝ, 0 < r → r < r₀ →
+        (surfaceMeasure S).restrict (Metric.ball 0 R) (Metric.ball c r) ≤
+          ENNReal.ofReal (A * r ^ (n - 1)))
+    {W : Space n → ℝ} {D : Space n → Space n}
+    (htrace : ∀ x ∈ S \ Z, HasSmoothTraceJetAt S W D x)
+    (hW0 : ∀ x ∈ S \ Z, 0 ≤ W x)
+    (hsup : ∀ φ : Space n → ℝ, ContDiff ℝ ∞ φ → HasCompactSupport φ →
+      Disjoint (tsupport φ) Z → (∀ x, 0 ≤ φ x) →
+      0 ≤ ∫ x, inner ℝ (D x) (surfaceGradient S φ x) ∂surfaceMeasure S)
+    {R : ℝ} (hR : 0 < R) :
+    MemLp (truncatedGradient W D) 2 ((surfaceMeasure S).restrict (Metric.ball 0 R)) := by
+  let b : ContDiffBump (0 : Space n) := ⟨R + 1, R + 2, by linarith, by linarith⟩
+  let μ := (surfaceMeasure S).restrict (Metric.ball 0 (R + 3))
+  have hR3 : 0 < R + 3 := by linarith
+  have : IsFiniteMeasure μ := ⟨by simpa only [μ, Measure.restrict_apply_univ] using hfinite _ hR3⟩
+  let K := Z ∩ tsupport b
+  have hKc : IsCompact K := b.hasCompactSupport.of_isClosed_subset
+    (hZ.inter (isClosed_tsupport b)) inter_subset_right
+  have hKd : dimH K ≤ (n - 8 : ℕ) := (dimH_mono inter_subset_left).trans hZd
+  have hbs : tsupport b ⊆ Metric.ball (0 : Space n) (R + 3) := by
+    rw [b.tsupport_eq]
+    exact Metric.closedBall_subset_ball (by dsimp [b]; linarith)
+  obtain ⟨A, r₀, hA, hr₀, hgr⟩ := hgrowth _ hR3
+  obtain ⟨χ, hχ, hχlim, hχE⟩ := exists_memLp_capacity_sequence hn S
+    (Metric.ball 0 (R + 3)) K hKc hKd
+    (fun f hf ↦ (hgrad f hf).mono_measure Measure.restrict_le_self) hA hr₀
+    (fun c hc s hs hsr ↦ hgr c ⟨hc.1, Metric.ball_subset_closedBall (hbs hc.2)⟩ s hs hsr)
+  have hb : MemLp (fun x ↦ b x) 2 μ := MemLp.of_bound b.continuous.aestronglyMeasurable 1
+    (ae_of_all _ (fun x ↦ by simpa only [Real.norm_eq_abs, abs_of_nonneg b.nonneg] using b.le_one))
+  have hDb : MemLp (surfaceGradient S b) 2 μ := surfaceGradient_memLp_on_ball b.contDiff
+    ((hgrad b b.contDiff).mono_measure Measure.restrict_le_self)
+  obtain ⟨_, hB, hBt⟩ := capacity_product_pair_tendsto hb hDb χ
+    (fun i ↦ (hχ i).1.continuous.aestronglyMeasurable)
+    (fun i ↦ ae_of_all _ (hχ i).2.1) hχlim zero_le_one
+    (ae_of_all _ (fun x ↦ by simpa only [Real.norm_eq_abs, abs_of_nonneg b.nonneg] using b.le_one))
+    (fun i ↦ surfaceGradient S (χ i)) (fun i ↦ (hχ i).2.2.2) hχE
+  let η : ℕ → Space n → ℝ := fun i x ↦ b x * χ i x
+  let F : ℕ → Space n → Space n := fun i x ↦ η i x • truncatedGradient W D x
+  let H : Space n → Space n := fun x ↦ b x • truncatedGradient W D x
+  have hη : ∀ i, ContDiff ℝ ∞ (η i) := fun i ↦ b.contDiff.mul (hχ i).1
+  have hcη : ∀ i, HasCompactSupport (η i) := fun _ ↦ b.hasCompactSupport.mul_right
+  have hηZ : ∀ i, Disjoint (tsupport (η i)) Z := fun i ↦
+    tsupport_mul_disjoint_Singular (hχ i).2.2.1
+  have hFE : ∀ i, MemLp (F i) 2 (surfaceMeasure S) ∧
+      vectorEnergy (surfaceMeasure S) (F i) ≤
+        4 * vectorEnergy (surfaceMeasure S) (surfaceGradient S (η i)) := by
+    intro i
+    exact min_regular_energy hS hZ hfinite hgrad htrace hsup hW0 (hη i) (hcη i) (hηZ i)
+  have hFm : ∀ i, MemLp (F i) 2 μ := fun i ↦ (hFE i).1.mono_measure Measure.restrict_le_self
+  have hB_eq : ∀ i, (fun x ↦ χ i x • surfaceGradient S b x + b x • surfaceGradient S (χ i) x) =
+      surfaceGradient S (η i) := by
+    intro i
+    funext x
+    change _ = surfaceGradient S (fun y ↦ b y * χ i y) x
+    have hbDiff : ContDiff ℝ ∞ (fun y ↦ b y) := b.contDiff
+    rw [surfaceGradient_mul (hbDiff.differentiable (by simp) x)
+      ((hχ i).1.differentiable (by simp) x), add_comm]
+  have hBE := vectorEnergy_tendsto_of_toLp_tendsto hB hDb hBt.snd_nhds
+  simp_rw [hB_eq] at hBE
+  have hbound : ∀ i, vectorEnergy μ (F i) ≤ 4 * vectorEnergy μ (surfaceGradient S (η i)) := by
+    intro i
+    rw [vectorEnergy_restrict_of_vanish measurableSet_ball, vectorEnergy_restrict_of_vanish measurableSet_ball]
+    · exact (hFE i).2
+    · intro x hx
+      exact surfaceGradient_eq_zero_of_notMem_tsupport S
+        (fun h ↦ hx (hbs (tsupport_mul_subset_left h)))
+    · intro x hx
+      have hb0 : b x = 0 := image_eq_zero_of_notMem_tsupport (fun h ↦ hx (hbs h))
+      simp only [F, η, hb0, zero_mul, zero_smul]
+  have hlim : ∀ᵐ x ∂μ, Tendsto (fun i ↦ F i x) atTop (𝓝 (H x)) := by
+    filter_upwards [hχlim] with x hx
+    simpa only [F, H, η, mul_one] using
+      (tendsto_const_nhds.mul hx).smul_const (truncatedGradient W D x)
+  have hHm : AEStronglyMeasurable H μ :=
+    aestronglyMeasurable_of_tendsto_ae atTop (fun i ↦ (hFm i).aestronglyMeasurable) hlim
+  have hHE := vectorEnergy_le_of_ae_tendsto (fun i ↦ (hFm i).aestronglyMeasurable) hlim
+    (Eventually.of_forall hbound) (ENNReal.Tendsto.const_mul hBE (Or.inr (by norm_num)))
+  have hH : MemLp H 2 μ := (memLp_two_iff_vectorEnergy hHm).mpr (hHE.trans_lt
+    (ENNReal.mul_lt_top (by norm_num) ((memLp_two_iff_vectorEnergy hDb.aestronglyMeasurable).mp hDb)))
+  have hHR : MemLp H 2 ((surfaceMeasure S).restrict (Metric.ball 0 R)) :=
+    hH.mono_measure (Measure.restrict_mono_set _ (Metric.ball_subset_ball (by linarith)))
+  apply hHR.ae_eq
+  filter_upwards [ae_restrict_mem measurableSet_ball] with x hx
+  have hbx : b x = 1 := b.one_of_mem_closedBall (Metric.mem_closedBall.mpr
+    ((Metric.mem_ball.mp hx).le.trans (by dsimp [b]; linarith)))
+  simp only [H, hbx, one_smul]
+
+end RPI.Boundary
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+
+theorem perimeter_mono_region {n : ℕ} (E : Set (Euc n)) {U V : Set (Euc n)}
+    (hUV : U ⊆ V) : perimeter E U ≤ perimeter E V := by
+  unfold perimeter
+  refine iSup_le fun f ↦ iSup_le fun hf ↦ iSup_le fun hc ↦ iSup_le fun hs ↦ iSup_le fun hb ↦ ?_
+  exact le_iSup_of_le f (le_iSup_of_le hf (le_iSup_of_le hc
+    (le_iSup_of_le (hs.trans hUV) (le_iSup_of_le hb le_rfl))))
+
+theorem isClosed_perimeterSupport {n : ℕ} (E : Set (Euc n)) :
+    IsClosed (perimeterSupport E) := by
+  rw [← isOpen_compl_iff, Metric.isOpen_iff]
+  intro x hx
+  simp only [mem_compl_iff, perimeterSupport, mem_ofPred_eq] at hx
+  push Not at hx
+  obtain ⟨r, hr, hp⟩ := hx
+  refine ⟨r / 2, half_pos hr, ?_⟩
+  intro y hy
+  change ¬ ∀ s : ℝ, 0 < s → 0 < perimeter E (Metric.ball y s)
+  intro hys
+  have hsub : Metric.ball y (r / 2) ⊆ Metric.ball x r := by
+    intro z hz
+    apply Metric.mem_ball.mpr
+    have h1 := Metric.mem_ball.mp hy
+    have h2 := Metric.mem_ball.mp hz
+    have h3 := dist_triangle z y x
+    linarith
+  exact (not_lt_of_ge ((perimeter_mono_region E hsub).trans hp)) (hys (r / 2) (half_pos hr))
+
+open RPI RPI.Boundary
+
+theorem min_wholeBoundarySobolev_and_supersolution_of_regular_geometry
+    {n : ℕ} (hn : 8 ≤ n) {E Z : Set (Euc n)} (hE : MeasurableSet E)
+    (hZ : IsClosed Z) (hZd : dimH Z ≤ (n - 8 : ℕ))
+    (hfinite : ∀ R : ℝ, 0 < R → surfaceMeasure (perimeterSupport E) (Metric.ball 0 R) < ⊤)
+    (hgrowth : ∀ R : ℝ, 0 < R → ∃ A r₀ : ℝ, 0 < A ∧ 0 < r₀ ∧
+      ∀ c ∈ Z ∩ Metric.closedBall 0 R, ∀ r : ℝ, 0 < r → r < r₀ →
+        (surfaceMeasure (perimeterSupport E)).restrict (Metric.ball 0 R) (Metric.ball c r) ≤
+          ENNReal.ofReal (A * r ^ (n - 1)))
+    (hreg : ∀ x ∈ perimeterSupport E \ Z, ∃ U : Set (Euc n), IsOpen U ∧ x ∈ U ∧
+      ∃ f : Euc n → ℝ, ContDiffOn ℝ ∞ f U ∧
+        (∀ y ∈ U, fderiv ℝ f y ≠ 0) ∧
+        (∀ y ∈ U, y ∈ perimeterSupport E ↔ f y = 0) ∧
+        (∀ y ∈ U, (f y < 0 → y ∈ E) ∧ (y ∈ E → f y ≤ 0)))
+    {W : Euc n → ℝ} {D : Euc n → Euc n}
+    (htrace : ∀ x ∈ perimeterSupport E \ Z, HasSmoothTraceJetAt (perimeterSupport E) W D x)
+    (hW0 : ∀ x ∈ perimeterSupport E \ Z, 0 ≤ W x)
+    (hsup : ∀ φ : Euc n → ℝ, ContDiff ℝ ∞ φ → HasCompactSupport φ →
+      Disjoint (tsupport φ) Z → (∀ x, 0 ≤ φ x) →
+      0 ≤ ∫ x, inner ℝ (D x) (surfaceGradient (perimeterSupport E) φ x)
+        ∂surfaceMeasure (perimeterSupport E)) :
+    WholeBoundarySobolev E (fun x ↦ min (W x) 1) (truncatedGradient W D) ∧
+      WholeBoundarySupersolution E (truncatedGradient W D) := by
+  have hS := isClosed_perimeterSupport E
+  have hlevel : ∀ x ∈ perimeterSupport E \ Z, ∃ U : Set (Euc n), IsOpen U ∧ x ∈ U ∧
+      ∃ f : Euc n → ℝ, ContDiffOn ℝ 1 f U ∧
+        (∀ y ∈ U, fderiv ℝ f y ≠ 0) ∧
+        (∀ y ∈ U, y ∈ perimeterSupport E ↔ f y = 0) := by
+    intro x hx
+    obtain ⟨U, hU, hxU, f, hf, hdf, hs, _⟩ := hreg x hx
+    exact ⟨U, hU, hxU, f, hf.of_le (by norm_num), hdf, hs⟩
+  have hfilled : ∀ x ∈ perimeterSupport E \ Z, ∃ f : Euc n → ℝ,
+      ContDiffAt ℝ 1 f x ∧ f x = 0 ∧ fderiv ℝ f x ≠ 0 ∧
+      (∀ᶠ y in 𝓝 x, y ∈ perimeterSupport E ↔ f y = 0) ∧
+      (∀ᶠ y in 𝓝 x, (f y < 0 → y ∈ E) ∧ (y ∈ E → f y ≤ 0)) := by
+    intro x hx
+    obtain ⟨U, hU, hxU, f, hf, hdf, hs, he⟩ := hreg x hx
+    refine ⟨f, (hf.of_le (by norm_num)).contDiffAt (hU.mem_nhds hxU),
+      (hs x hxU).mp hx.1, hdf x hxU, ?_, ?_⟩
+    · filter_upwards [hU.mem_nhds hxU] with y hy using hs y hy
+    · filter_upwards [hU.mem_nhds hxU] with y hy using he y hy
+  have hgrad : ∀ f : Euc n → ℝ, ContDiff ℝ ∞ f →
+      AEStronglyMeasurable (surfaceGradient (perimeterSupport E) f)
+        (surfaceMeasure (perimeterSupport E)) := by
+    intro f hf
+    exact surfaceGradient_aestronglyMeasurable_of_regular_locus hn hS.measurableSet
+      hZ.measurableSet hZd hlevel (hf.of_le (by norm_num))
+  have hloc : ∀ R : ℝ, 0 < R → MemLp (truncatedGradient W D) 2
+      ((surfaceMeasure (perimeterSupport E)).restrict (Metric.ball 0 R)) := by
+    intro R hR
+    exact min_local_energy hn hS hZ hZd hfinite hgrad hgrowth htrace hW0 hsup hR
+  have haereg : ∀ᵐ x ∂surfaceMeasure (perimeterSupport E), x ∈ perimeterSupport E \ Z := by
+    have hz : ∀ᵐ x ∂surfaceMeasure (perimeterSupport E), x ∉ Z := by
+      simpa only [ae_iff, not_not, Set.ofPred_mem_eq] using
+        surfaceMeasure_zero_of_singular_dim hn (perimeterSupport E) Z hZd
+    filter_upwards [ae_restrict_mem hS.measurableSet, hz] with x hx hzx
+    exact ⟨hx, hzx⟩
+  constructor
+  · apply wholeBoundarySobolev_of_local_sobolevGraph_filled hn hE hS.measurableSet hZd hfilled
+    intro R hR
+    have : IsFiniteMeasure ((surfaceMeasure (perimeterSupport E)).restrict (Metric.ball 0 R)) :=
+      ⟨by simpa only [Measure.restrict_apply_univ] using hfinite R hR⟩
+    have hw : ∀ᵐ x ∂(surfaceMeasure (perimeterSupport E)).restrict (Metric.ball 0 R), 0 ≤ W x := by
+      filter_upwards [haereg.filter_mono (ae_mono Measure.restrict_le_self)] with x hx
+      exact hW0 x hx
+    obtain ⟨A, r₀, hA, hr₀, hgr⟩ := hgrowth (R + 2) (by linarith)
+    obtain ⟨hh, hpair⟩ := min_one_mem_sobolevGraph_ball hn hS hZ hZd hR htrace hw (hloc R hR)
+      (fun f hf ↦ (hgrad f hf).mono_measure Measure.restrict_le_self) hA hr₀ (by
+        intro c hc r hr hrr
+        apply le_trans _ (hgr c hc r hr hrr)
+        exact Measure.le_iff'.mp (Measure.restrict_mono_set _
+          (Metric.ball_subset_ball (by linarith : R ≤ R + 2))) _)
+    exact ⟨hh, hloc R hR, hpair⟩
+  · intro φ hφ hcφ hpφ
+    have hh := full_test_of_regular_tests hn hZ hZd hloc hfinite hgrad hgrowth
+      (fun ψ hψ hcψ hψZ hpψ ↦ min_regular_test hS hZ hfinite hgrad htrace hsup hψ hcψ hψZ hpψ)
+      hφ hcφ hpφ
+    have heq := surfaceGradient_ae_eq_tangentialGradient_filled hn hE hS.measurableSet hZd hfilled φ
+    rw [boundaryArea_eq_smul_surfaceMeasure, integral_smul_measure]
+    have hi : (∫ x, inner ℝ (truncatedGradient W D x) (tangentialGradient E φ x)
+        ∂surfaceMeasure (perimeterSupport E)) =
+        ∫ x, inner ℝ (truncatedGradient W D x) (surfaceGradient (perimeterSupport E) φ x)
+          ∂surfaceMeasure (perimeterSupport E) := by
+      apply integral_congr_ae
+      filter_upwards [heq] with x hx
+      rw [hx]
+    rw [hi]
+    exact mul_nonneg ENNReal.toReal_nonneg hh
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/TangentPlanes.lean b/lean/OAI/Geometry/Perimeter/Regularity/TangentPlanes.lean
new file mode 100644
index 0000000000000000000000000000000000000000..99915d19dce6f6c78f6332c37b595afd2df71aa6
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/TangentPlanes.lean
@@ -0,0 +1,102 @@
+import OAI.Geometry.Perimeter.Regularity.SlabIteration
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+theorem FullSlab.undilate_height {E : Set (Euc n)} (hE : MeasurableSet E)
+    {v : Euc n} {R a t : ℝ} (ht : 0 < t)
+    (h : FullSlab (blownUpSet E 0 t) v R a) :
+    ∀ x∈perimeterSupport E ∩ Metric.ball 0 (t*R), |inner ℝ v x|≤t*a := by
+  intro x hx
+  let y := t⁻¹•x
+  have hxy : t•y=x := smul_inv_smul₀ ht.ne' x
+  have hyS : y∈perimeterSupport (blownUpSet E 0 t) := by
+    rw [perimeterSupport_blownUpSet hE 0 ht,zero_add,hxy]
+    exact hx.1
+  have hyB : y∈Metric.ball 0 R := by
+    have hh : t*‖y‖<t*R := by
+      rw [← show ‖t‖*‖y‖=t*‖y‖ by rw [Real.norm_of_nonneg ht.le],← norm_smul,hxy]
+      simpa only [Metric.mem_ball,dist_zero_right] using hx.2
+    simpa only [Metric.mem_ball,dist_zero_right] using (mul_lt_mul_iff_right₀ ht).mp hh
+  have hh := mul_le_mul_of_nonneg_left (h.1 y ⟨hyS,hyB⟩) ht.le
+  have hi : |inner ℝ v x|=t*|inner ℝ v y| := by
+    rw [← hxy,real_inner_smul_right,abs_mul,abs_of_pos ht]
+  exact hi ▸ hh
+
+theorem relative_tangent_estimate {q M a : ℝ} (hq : 0 < q) (hq1 : q < 1)
+    (hM : 0≤M) (ha : 0≤a) {E : Set (Euc n)} (hE : MeasurableSet E)
+    {v : ℕ → Euc n} {w : Euc n}
+    (hslab : ∀ j, FullSlab (blownUpSet E 0 (q^j)) (v j) 8 (a*(1/2)^j))
+    (hlimit : ∀ j, ‖v j-w‖≤2*M*a*(1/2)^j) :
+    ∀ j x, x∈perimeterSupport E → ‖x‖≤4*q^j →
+      |inner ℝ w x|≤a*(2*M+1/(4*q))*(1/2)^j*‖x‖ := by
+  intro j x hx hnx
+  by_cases hx0 : x=0
+  · simp [hx0]
+  have hxp : 0 < ‖x‖ := norm_pos_iff.mpr hx0
+  have hden : 0 < 4*q^j := by positivity
+  obtain ⟨k,hlo,hhi⟩ := exists_nat_pow_near_of_lt_one
+    (div_pos hxp hden) ((div_le_one hden).mpr hnx) hq hq1
+  let l := j+k
+  have hnlo : 4*q^(l+1)<‖x‖ := by
+    have hh := (lt_div_iff₀ hden).mp hlo
+    dsimp [l]
+    rw [Nat.add_assoc,pow_add q j (k+1)]
+    nlinarith
+  have hnhi : ‖x‖≤4*q^l := by
+    have hh := (div_le_iff₀ hden).mp hhi
+    dsimp [l]
+    rw [pow_add]
+    nlinarith
+  have hxB : x∈Metric.ball 0 (q^l*8) := by
+    simp only [Metric.mem_ball,dist_zero_right]
+    nlinarith [pow_pos hq l]
+  have hheight := (hslab l).undilate_height hE (pow_pos hq l) x ⟨hx,hxB⟩
+  have hqbound : q^l≤‖x‖/(4*q) := by
+    apply (le_div_iff₀ (by positivity)).mpr
+    rw [pow_succ] at hnlo
+    nlinarith
+  have hpow : (1/2:ℝ)^l≤(1/2:ℝ)^j := by
+    dsimp [l]
+    rw [pow_add]
+    exact mul_le_of_le_one_right (by positivity) (pow_le_one₀ (by norm_num) (by norm_num))
+  have hnorm : ‖w-v l‖≤2*M*a*(1/2)^l := by rw [norm_sub_rev]; exact hlimit l
+  have hi : inner ℝ w x=inner ℝ (w-v l) x+inner ℝ (v l) x := by
+    rw [inner_sub_left]; ring
+  calc
+    |inner ℝ w x| ≤ ‖w-v l‖*‖x‖+|inner ℝ (v l) x| := by
+      rw [hi]
+      exact (abs_add_le _ _).trans (add_le_add (abs_real_inner_le_norm (w-v l) x) le_rfl)
+    _ ≤ (2*M*a*(1/2)^l)*‖x‖+q^l*(a*(1/2)^l) :=
+      add_le_add (mul_le_mul_of_nonneg_right hnorm (norm_nonneg x)) hheight
+    _ ≤ (2*M*a*(1/2)^l)*‖x‖+(‖x‖/(4*q))*(a*(1/2)^l) :=
+      add_le_add le_rfl (mul_le_mul_of_nonneg_right hqbound (show 0≤a*(1/2:ℝ)^l by positivity))
+    _ = a*(2*M+1/(4*q))*(1/2)^l*‖x‖ := by ring
+    _ ≤ a*(2*M+1/(4*q))*(1/2)^j*‖x‖ :=
+      mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left hpow (by positivity)) (norm_nonneg x)
+
+theorem exists_full_support_tangent_plane {q M δ : ℝ} (hq : 0 < q) (hq1 : q < 1)
+    (hM : 0≤M) (hstep : SlabContractionProperty n q M δ) {E : Set (Euc n)}
+    (hmin : IsLocallyPerimeterMinimizing E) (hzero : (0:Euc n)∈perimeterSupport E)
+    {v₀ : Euc n} (hv₀ : ‖v₀‖=1) {a : ℝ} (ha : 0 < a) (haδ : a < δ)
+    (hs : FullSlab E v₀ 8 a) :
+    ∃ w : Euc n, ‖w‖=1 ∧ ‖w-v₀‖≤2*M*a ∧
+      ∀ j x, x∈perimeterSupport E → ‖x‖≤4*q^j →
+        |inner ℝ w x|≤a*(2*M+1/(4*q))*(1/2)^j*‖x‖ := by
+  obtain ⟨v,w,hv0,hunit,hwn,hwt,hslab,hlimit⟩ :=
+    slab_normal_limit hq hstep hmin hzero hv₀ ha haδ hs
+  refine ⟨w,hwn,?_,relative_tangent_estimate hq hq1 hM ha.le hmin.1 hslab hlimit⟩
+  have hh := hlimit 0
+  rw [hv0,pow_zero,mul_one,norm_sub_rev] at hh
+  exact hh
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/TangentialGradientComp.lean b/lean/OAI/Geometry/Perimeter/Regularity/TangentialGradientComp.lean
new file mode 100644
index 0000000000000000000000000000000000000000..3b0ae14893663b8a0b17ab0189fa8012079b04f1
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/TangentialGradientComp.lean
@@ -0,0 +1,417 @@
+import OAI.Geometry.Perimeter.Regularity.ExistsLocalPolarEqApproximateNormal
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+namespace RPISingularHarnack
+open RPI RPI.Boundary
+
+theorem tangentialGradient_comp {n : ℕ} (E : Set (Euc n))
+    {f : Euc n → ℝ} {T : ℝ → ℝ} {x : Euc n}
+    (hf : DifferentiableAt ℝ f x) (hT : DifferentiableAt ℝ T (f x)) :
+    tangentialGradient E (T ∘ f) x = deriv T (f x) • tangentialGradient E f x := by
+  have hgrad : gradient (T ∘ f) x = deriv T (f x) • gradient f x := by
+    apply (InnerProductSpace.toDual ℝ (Euc n)).injective
+    simp only [map_smul, toDual_gradient]
+    exact (hT.hasDerivAt.comp_hasFDerivAt x hf.hasFDerivAt).fderiv
+  simp only [tangentialGradient, hgrad, real_inner_smul_left]
+  module
+
+theorem tangentialGradient_nonlinear_test {n : ℕ} (E : Set (Euc n))
+    {η f : Euc n → ℝ} {ψ : ℝ → ℝ} {x : Euc n}
+    (hη : DifferentiableAt ℝ η x) (hf : DifferentiableAt ℝ f x)
+    (hψ : DifferentiableAt ℝ ψ (f x)) :
+    tangentialGradient E (fun y ↦ (η y * η y) * ψ (f y)) x =
+      (2 * η x * ψ (f x)) • tangentialGradient E η x +
+      (η x ^ 2 * deriv ψ (f x)) • tangentialGradient E f x := by
+  have hprod := tangentialGradient_mul E (hη.mul hη) (hψ.comp x hf)
+  simp only [Pi.mul_apply, Function.comp_apply] at hprod
+  rw [hprod]
+  change (η x * η x) • tangentialGradient E (ψ ∘ f) x +
+    ψ (f x) • tangentialGradient E (fun y ↦ η y * η y) x = _
+  rw [tangentialGradient_mul E hη hη]
+  rw [tangentialGradient_comp E hf hψ]
+  module
+
+theorem wholeBoundarySupersolution_nonlinear_test {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    {h : Euc n → ℝ} {G : Euc n → Euc n}
+    (hSob : WholeBoundarySobolev E h G) (hSuper : WholeBoundarySupersolution E G)
+    (hh01 : ∀ᵐ x ∂boundaryArea E, 0 ≤ h x ∧ h x ≤ 1)
+    {R : ℝ} (hR : 0 < R) {η : Euc n → ℝ}
+    (hη : ContDiff ℝ ∞ η) (hcη : HasCompactSupport η)
+    (hsη : tsupport η ⊆ Metric.ball 0 R)
+    {ψ : ℝ → ℝ} (hψ : ContDiff ℝ ∞ ψ) (hψpos : ∀ t, 0 ≤ ψ t)
+    {K : ℝ≥0} (hψLip : LipschitzWith K ψ)
+    {L : ℝ} (hL : 0 ≤ L) (hdψ : ∀ t, ‖deriv ψ t‖ ≤ L) :
+    0 ≤ ∫ x in Metric.ball 0 R, inner ℝ (G x)
+      ((2 * η x * ψ (h x)) • tangentialGradient E η x +
+        (η x ^ 2 * deriv ψ (h x)) • G x) ∂boundaryArea E := by
+  let μ := (boundaryArea E).restrict (Metric.ball 0 R)
+  have : IsFiniteMeasure μ := ⟨by
+    simpa only [μ, Measure.restrict_apply_univ] using minimizing_boundaryArea_ball_lt_top hn hmin 0 hR⟩
+  obtain ⟨hh, hG, u, hu, huh, huG⟩ := hSob R hR
+  have hu0 : Tendsto (fun j ↦ vectorEnergy μ (fun x ↦ u j x - h x)) atTop (𝓝 0) := by
+    simpa only [vectorEnergy, Real.norm_eq_abs] using huh
+  have hum : ∀ j, AEStronglyMeasurable (u j) μ :=
+    fun j ↦ (hu j).1.continuous.aestronglyMeasurable
+  obtain ⟨σ, hσ, hpoint⟩ := (tendstoInMeasure_of_tendsto_eLpNorm
+    (by norm_num : (2 : ℝ≥0∞) ≠ 0)
+    (tendsto_eLpNorm_two_of_vectorEnergy_zero_of_aestronglyMeasurable
+      (fun j ↦ (hum j).sub hh.aestronglyMeasurable) hu0)).exists_seq_tendsto_ae
+  let F : ℕ → Euc n → Euc n := fun j ↦ tangentialGradient E (u (σ j))
+  let D := tangentialGradient E η
+  have hF : ∀ j, MemLp (F j) 2 μ := fun j ↦
+    tangentialGradient_memLp_on_ball hn hmin hR ((hu (σ j)).1.of_le (by simp))
+  have hD : MemLp D 2 μ := tangentialGradient_memLp_on_ball hn hmin hR (hη.of_le (by simp))
+  have hdcont : Continuous (deriv ψ) := hψ.continuous_deriv (by simp)
+  obtain ⟨M₀, hM₀⟩ := hcη.exists_bound_of_continuous hη.continuous
+  let M := max M₀ 0
+  have hM : 0 ≤ M := le_max_right _ _
+  have hηM : ∀ᵐ x ∂μ, ‖η x‖ ≤ M := ae_of_all _ fun x ↦ (hM₀ x).trans (le_max_left _ _)
+  have hgradη : Continuous (gradient η) := continuous_iff_continuousAt.mpr (fun x ↦
+    continuousAt_gradient_of_contDiffAt (hη.of_le (by simp)).contDiffAt)
+  obtain ⟨N₀, hN₀⟩ := (isCompact_closedBall (0 : Euc n) R).exists_bound_of_continuousOn hgradη.continuousOn
+  let N := 2 * max N₀ 0
+  have hN : 0 ≤ N := by positivity
+  have hDN : ∀ᵐ x ∂μ, ‖D x‖ ≤ N := by
+    filter_upwards [ae_restrict_mem measurableSet_ball] with x hx
+    exact (tangentialGradient_norm_le_twice E η x).trans (by
+      dsimp [N]; nlinarith [hN₀ x (Metric.ball_subset_closedBall hx), le_max_left N₀ 0])
+  let B : Euc n → Euc n := fun x ↦
+    (2 * η x * ψ (h x)) • D x + (η x ^ 2 * deriv ψ (h x)) • G x
+  obtain ⟨C₀, hC₀⟩ := (isCompact_Icc : IsCompact (Icc (0 : ℝ) 1)).exists_bound_of_continuousOn
+    hψ.continuous.continuousOn
+  let C := max C₀ 0
+  have hC : 0 ≤ C := le_max_right _ _
+  have hψC : ∀ᵐ x ∂μ, ‖ψ (h x)‖ ≤ C := by
+    filter_upwards [ae_restrict_of_ae hh01] with x hx
+    exact (hC₀ (h x) hx).trans (le_max_left _ _)
+  have hca : MemLp (fun x ↦ 2 * η x * ψ (h x)) ∞ μ := by
+    apply MemLp.of_bound ((aestronglyMeasurable_const.mul hη.continuous.aestronglyMeasurable).mul
+      (hψ.continuous.comp_aestronglyMeasurable hh.aestronglyMeasurable)) (2 * M * C)
+    filter_upwards [hηM, hψC] with x hx hy
+    simpa only [Pi.mul_apply, norm_mul, Real.norm_ofNat] using mul_le_mul
+      (mul_le_mul_of_nonneg_left hx (by norm_num : (0 : ℝ) ≤ 2)) hy (norm_nonneg _) (by positivity)
+  have hcb : MemLp (fun x ↦ η x ^ 2 * deriv ψ (h x)) ∞ μ := by
+    apply MemLp.of_bound ((hη.continuous.aestronglyMeasurable.pow 2).mul
+      (hdcont.comp_aestronglyMeasurable hh.aestronglyMeasurable)) (M ^ 2 * L)
+    filter_upwards [hηM] with x hx
+    rw [Pi.mul_apply, Pi.pow_apply, norm_mul, norm_pow]
+    exact mul_le_mul ((sq_le_sq₀ (norm_nonneg _) hM).mpr hx) (hdψ _) (norm_nonneg _) (sq_nonneg _)
+  have hB : MemLp B 2 μ := (hca.smul hD).add (hcb.smul hG)
+  let ξ : ℕ → Euc n → ℝ := fun j x ↦ (η x * η x) * ψ (u (σ j) x)
+  have hξ : ∀ j, ContDiff ℝ ∞ (ξ j) := fun j ↦ (hη.mul hη).mul (hψ.comp (hu (σ j)).1)
+  have hξc : ∀ j, HasCompactSupport (ξ j) := fun _ ↦ hcη.mul_right.mul_right
+  have hξgrad : ∀ j x, tangentialGradient E (ξ j) x =
+      (2 * η x * ψ (u (σ j) x)) • D x + (η x ^ 2 * deriv ψ (u (σ j) x)) • F j x := by
+    intro j x
+    exact tangentialGradient_nonlinear_test E (hη.differentiable (by simp) x)
+      ((hu (σ j)).1.differentiable (by simp) x) (hψ.differentiable (by simp) _)
+  have henergy : Tendsto (fun j ↦ vectorEnergy μ
+      (fun x ↦ tangentialGradient E (ξ j) x - B x)) atTop (𝓝 0) := by
+    simp_rw [hξgrad]
+    exact vectorEnergy_nonlinear_test_tendsto (fun j ↦ hum (σ j)) hh.aestronglyMeasurable
+      hη.continuous.aestronglyMeasurable hD.aestronglyMeasurable hF hG hM hN hηM hDN
+      (hu0.comp hσ.tendsto_atTop) (huG.comp hσ.tendsto_atTop) hpoint hψLip
+      (deriv ψ) hdcont hL hdψ
+  have hξLp : ∀ j, MemLp (tangentialGradient E (ξ j)) 2 μ := fun j ↦
+    tangentialGradient_memLp_on_ball hn hmin hR ((hξ j).of_le (by simp))
+  have hlim0 := tendsto_integral_inner_of_L2_zero hG
+    (fun j x ↦ tangentialGradient E (ξ j) x - B x) (fun j ↦ (hξLp j).sub hB)
+    (tendsto_eLpNorm_two_toReal_of_vectorEnergy_zero_of_aestronglyMeasurable
+      (fun j ↦ ((hξLp j).sub hB).aestronglyMeasurable) henergy)
+  have hlim : Tendsto (fun j ↦ ∫ x, inner ℝ (G x) (tangentialGradient E (ξ j) x) ∂μ)
+      atTop (𝓝 (∫ x, inner ℝ (G x) (B x) ∂μ)) := by
+    have hj : ∀ j, (∫ x, inner ℝ (G x) (tangentialGradient E (ξ j) x - B x) ∂μ) =
+        (∫ x, inner ℝ (G x) (tangentialGradient E (ξ j) x) ∂μ) -
+        ∫ x, inner ℝ (G x) (B x) ∂μ := fun j ↦ by
+      simp_rw [inner_sub_right]
+      exact integral_sub (integrable_inner_of_memLp hG (hξLp j)) (integrable_inner_of_memLp hG hB)
+    change Tendsto (fun j ↦ ∫ x, inner ℝ (G x) (tangentialGradient E (ξ j) x - B x) ∂μ) atTop (𝓝 0) at hlim0
+    simp_rw [hj] at hlim0
+    simpa only [sub_add_cancel, zero_add] using hlim0.add_const (∫ x, inner ℝ (G x) (B x) ∂μ)
+  apply ge_of_tendsto hlim
+  exact Filter.Eventually.of_forall fun j ↦ by
+    change 0 ≤ ∫ x in Metric.ball 0 R, inner ℝ (G x) (tangentialGradient E (ξ j) x) ∂boundaryArea E
+    rw [setIntegral_eq_integral_of_forall_compl_eq_zero]
+    · exact hSuper (ξ j) (hξ j) (hξc j) (fun x ↦ mul_nonneg (mul_self_nonneg _) (hψpos _))
+    · intro x hx
+      rw [tangentialGradient_eq_zero_of_notMem_tsupport E
+        (fun hx' ↦ hx (hsη (tsupport_mul_subset_left (tsupport_mul_subset_left hx')))), inner_zero_right]
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+namespace RPISingularHarnack
+open RPI
+
+theorem logTest_lipschitz {ε : ℝ} (hε : 0 < ε) :
+    LipschitzWith ⟨2 / ε ^ 2, by positivity⟩ (logTest ε) := by
+  apply lipschitzWith_of_nnnorm_deriv_le ((logTest_contDiff hε).differentiable (by simp))
+  intro t
+  change ‖deriv (logTest ε) t‖ ≤ 2 / ε ^ 2
+  rw [(logTest_hasDerivAt hε t).deriv]
+  exact logSlope_bound hε t
+
+def logRootWeight (ε t : ℝ) : ℝ := Real.sqrt (-logSlope ε t)
+def logTestRatio (ε t : ℝ) : ℝ := logTest ε t / logRootWeight ε t
+
+theorem logRootWeight_pos {ε : ℝ} (hε : 0 < ε) (t : ℝ) :
+    0 < logRootWeight ε t := Real.sqrt_pos.2 (neg_pos.mpr (logSlope_neg hε t))
+
+theorem logRootWeight_sq {ε : ℝ} (hε : 0 < ε) (t : ℝ) :
+    logRootWeight ε t ^ 2 = -logSlope ε t :=
+  Real.sq_sqrt (neg_nonneg.mpr (logSlope_neg hε t).le)
+
+theorem logTestRatio_mem_Icc {ε t : ℝ} (hε : 0 < ε) (ht : 0 ≤ t) :
+    logTestRatio ε t ∈ Icc (0 : ℝ) 1 := by
+  have hw := logRootWeight_pos hε t
+  have hψ := logTest_pos hε t
+  refine ⟨(div_pos hψ hw).le, (div_le_one hw).2 ?_⟩
+  nlinarith [logRootWeight_sq hε t, logTest_sq_le_neg_slope hε ht]
+
+theorem logRootWeight_bound {ε : ℝ} (hε : 0 < ε) (t : ℝ) :
+    logRootWeight ε t ≤ Real.sqrt (2 / ε ^ 2) := by
+  apply Real.sqrt_le_sqrt
+  have hh := logSlope_bound hε t
+  rwa [Real.norm_eq_abs, abs_of_neg (logSlope_neg hε t)] at hh
+
+theorem logTestRatio_continuous {ε : ℝ} (hε : 0 < ε) : Continuous (logTestRatio ε) := by
+  exact (logTest_contDiff hε).continuous.div
+    (Real.continuous_sqrt.comp (logSlope_continuous hε).neg)
+    (fun t ↦ (logRootWeight_pos hε t).ne')
+
+theorem logTest_pairing {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V]
+    {ε : ℝ} (hε : 0 < ε) (t a : ℝ) (G D : V) :
+    inner ℝ ((a * logRootWeight ε t) • G)
+      ((2 * logTestRatio ε t) • D - (a * logRootWeight ε t) • G) =
+      inner ℝ G ((2 * a * logTest ε t) • D + (a ^ 2 * logSlope ε t) • G) := by
+  simp only [inner_sub_right, inner_add_right, real_inner_smul_left, real_inner_smul_right]
+  have heq : logRootWeight ε t * logTestRatio ε t = logTest ε t := by
+    dsimp [logTestRatio]
+    exact mul_div_cancel₀ _ (logRootWeight_pos hε t).ne'
+  calc
+    _ = 2 * a * (logRootWeight ε t * logTestRatio ε t) * inner ℝ G D -
+        a ^ 2 * logRootWeight ε t ^ 2 * inner ℝ G G := by ring
+    _ = _ := by rw [heq, logRootWeight_sq hε]; ring
+
+theorem log_weight_le_energy {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V]
+    {ε t : ℝ} (hε : 0 < ε) (ht : 0 ≤ t) (a : ℝ) (G : V) :
+    a ^ 2 * ‖G‖ ^ 2 / (t + ε) ^ 2 ≤ 2 * ‖(a * logRootWeight ε t) • G‖ ^ 2 := by
+  have hh := mul_le_mul_of_nonneg_left (logSlope_controls_log_weight hε ht)
+    (mul_nonneg (sq_nonneg a) (sq_nonneg ‖G‖))
+  rw [norm_smul, norm_mul, mul_pow, mul_pow, Real.norm_eq_abs, sq_abs,
+    Real.norm_eq_abs, sq_abs, logRootWeight_sq hε]
+  simpa only [div_eq_mul_inv, mul_one, mul_assoc, mul_left_comm, mul_comm] using hh
+
+theorem wholeBoundary_log_caccioppoli {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    {h : Euc n → ℝ} {G : Euc n → Euc n}
+    (hSob : WholeBoundarySobolev E h G) (hSuper : WholeBoundarySupersolution E G)
+    (hh01 : ∀ᵐ x ∂boundaryArea E, 0 ≤ h x ∧ h x ≤ 1)
+    {R : ℝ} (hR : 0 < R) {η : Euc n → ℝ}
+    (hη : ContDiff ℝ ∞ η) (hcη : HasCompactSupport η)
+    (hsη : tsupport η ⊆ Metric.ball 0 R) {ε : ℝ} (hε : 0 < ε) :
+    (∫ x in Metric.ball 0 R, η x ^ 2 * ‖G x‖ ^ 2 / (h x + ε) ^ 2 ∂boundaryArea E) ≤
+      8 * ∫ x in Metric.ball 0 R, ‖tangentialGradient E η x‖ ^ 2 ∂boundaryArea E := by
+  let μ := (boundaryArea E).restrict (Metric.ball 0 R)
+  have hh := (hSob R hR).1
+  have hG := (hSob R hR).2.1
+  have hD := tangentialGradient_memLp_on_ball hn hmin hR (hη.of_le (by simp))
+  let A : Euc n → Euc n := fun x ↦ (η x * logRootWeight ε (h x)) • G x
+  have hwc : Continuous (logRootWeight ε) := Real.continuous_sqrt.comp (logSlope_continuous hε).neg
+  have hAm : AEStronglyMeasurable A μ :=
+    (hη.continuous.aestronglyMeasurable.mul
+      (hwc.comp_aestronglyMeasurable hh.aestronglyMeasurable)).smul hG.aestronglyMeasurable
+  obtain ⟨M, hM⟩ := hcη.exists_bound_of_continuous hη.continuous
+  have hM0 : 0 ≤ M := (norm_nonneg (η 0)).trans (hM 0)
+  have hA : MemLp A 2 μ := by
+    apply hG.of_le_mul (c := M * Real.sqrt (2 / ε ^ 2)) hAm
+    exact ae_of_all _ fun x ↦ by
+      change ‖(η x * logRootWeight ε (h x)) • G x‖ ≤ _
+      rw [norm_smul, norm_mul, Real.norm_of_nonneg (logRootWeight_pos hε _).le]
+      exact mul_le_mul_of_nonneg_right (mul_le_mul (hM x) (logRootWeight_bound hε _)
+        (logRootWeight_pos hε _).le hM0) (norm_nonneg _)
+  have htest0 := wholeBoundarySupersolution_nonlinear_test hn hmin hSob hSuper hh01
+    hR hη hcη hsη (logTest_contDiff hε) (fun t ↦ (logTest_pos hε t).le)
+    (logTest_lipschitz hε) (show 0 ≤ 2 / ε ^ 2 by positivity)
+    (fun t ↦ by rw [(logTest_hasDerivAt hε t).deriv]; exact logSlope_bound hε t)
+  have htest : 0 ≤ ∫ x, inner ℝ (A x)
+      ((2 * (1 - (1 - logTestRatio ε (h x)))) • tangentialGradient E η x - A x) ∂μ := by
+    have heq : (fun x ↦ inner ℝ (A x)
+        ((2 * (1 - (1 - logTestRatio ε (h x)))) • tangentialGradient E η x - A x)) =
+        fun x ↦ inner ℝ (G x)
+          ((2 * η x * logTest ε (h x)) • tangentialGradient E η x +
+            (η x ^ 2 * deriv (logTest ε) (h x)) • G x) := by
+      funext x
+      rw [(logTest_hasDerivAt hε (h x)).deriv]
+      simp only [sub_sub_cancel]
+      exact logTest_pairing hε (h x) (η x) (G x) (tangentialGradient E η x)
+    rwa [heq]
+  have hbound := truncation_test_energy hA hD
+    (aestronglyMeasurable_const.sub
+      ((logTestRatio_continuous hε).comp_aestronglyMeasurable hh.aestronglyMeasurable))
+    (show ∀ᵐ x ∂μ, (1 - logTestRatio ε (h x)) ∈ Icc (0 : ℝ) 1 from by
+      filter_upwards [ae_restrict_of_ae hh01] with x hx
+      have hb := logTestRatio_mem_Icc hε hx.1
+      constructor <;> linarith [hb.1, hb.2]) htest
+  have hAs : Integrable (fun x ↦ ‖A x‖ ^ 2) μ := (memLp_two_iff_integrable_sq_norm hAm).mp hA
+  let W : Euc n → ℝ := fun x ↦ η x ^ 2 * ‖G x‖ ^ 2 / (h x + ε) ^ 2
+  have hWm : AEStronglyMeasurable W μ :=
+    (((hη.continuous.aemeasurable.pow_const 2).mul
+      (hG.aestronglyMeasurable.norm.pow 2).aemeasurable).div
+      ((hh.aestronglyMeasurable.add_const ε).pow 2).aemeasurable).aestronglyMeasurable
+  have hWle : ∀ᵐ x ∂μ, W x ≤ 2 * ‖A x‖ ^ 2 := by
+    filter_upwards [ae_restrict_of_ae hh01] with x hx
+    exact log_weight_le_energy hε hx.1 (η x) (G x)
+  have hW : Integrable W μ := (hAs.const_mul 2).mono' hWm (by
+    filter_upwards [hWle] with x hx
+    rw [Real.norm_of_nonneg (by dsimp [W]; positivity)]
+    exact hx)
+  have hi := integral_mono_ae hW (hAs.const_mul 2) hWle
+  rw [integral_const_mul] at hi
+  change (∫ x, W x ∂μ) ≤ 8 * ∫ x, ‖tangentialGradient E η x‖ ^ 2 ∂μ
+  linarith
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+namespace RPISingularHarnack
+
+def smoothPositive (ε s : ℝ) : ℝ := ∫ t in (0 : ℝ)..s, Real.smoothTransition (t / ε)
+
+theorem smoothPositive_hasDerivAt (ε s : ℝ) :
+    HasDerivAt (smoothPositive ε) (Real.smoothTransition (s / ε)) s := by
+  have hc : Continuous (fun t : ℝ ↦ Real.smoothTransition (t / ε)) := by fun_prop
+  exact intervalIntegral.integral_hasDerivAt_right (hc.intervalIntegrable 0 s)
+    hc.stronglyMeasurable.stronglyMeasurableAtFilter hc.continuousAt
+
+@[simp] theorem smoothPositive_deriv (ε s : ℝ) :
+    deriv (smoothPositive ε) s = Real.smoothTransition (s / ε) :=
+  (smoothPositive_hasDerivAt ε s).deriv
+
+theorem smoothPositive_contDiff (ε : ℝ) : ContDiff ℝ ∞ (smoothPositive ε) := by
+  rw [contDiff_infty_iff_deriv]
+  refine ⟨fun s ↦ (smoothPositive_hasDerivAt ε s).differentiableAt, ?_⟩
+  simp only [funext (smoothPositive_deriv ε)]
+  fun_prop
+
+theorem smoothPositive_of_nonpos {ε s : ℝ} (hε : 0 < ε) (hs : s ≤ 0) :
+    smoothPositive ε s = 0 := by
+  unfold smoothPositive
+  calc
+    _ = ∫ _t in (0 : ℝ)..s, (0 : ℝ) := by
+      apply intervalIntegral.integral_congr
+      intro t ht
+      rw [uIcc_of_ge hs] at ht
+      exact Real.smoothTransition.zero_of_nonpos (div_nonpos_of_nonpos_of_nonneg ht.2 hε.le)
+    _ = 0 := by simp
+
+theorem smoothPositive_nonneg {ε : ℝ} (hε : 0 < ε) (s : ℝ) :
+    0 ≤ smoothPositive ε s := by
+  by_cases hs : 0 ≤ s
+  · exact intervalIntegral.integral_nonneg hs (fun t _ ↦ Real.smoothTransition.nonneg _)
+  · rw [smoothPositive_of_nonpos hε (le_of_not_ge hs)]
+
+theorem smoothPositive_le_mul {ε s : ℝ} (hε : 0 < ε) (hs : 0 ≤ s) :
+    smoothPositive ε s ≤ s * Real.smoothTransition (s / ε) := by
+  have hc : Continuous (fun t : ℝ ↦ Real.smoothTransition (t / ε)) := by fun_prop
+  have hh := intervalIntegral.integral_mono_on hs (hc.intervalIntegrable 0 s)
+    (intervalIntegrable_const : IntervalIntegrable (fun _ : ℝ ↦ Real.smoothTransition (s / ε)) volume 0 s)
+    (fun t ht ↦ Real.smoothTransition.monotone (div_le_div_of_nonneg_right ht.2 hε.le))
+  simpa only [smoothPositive, intervalIntegral.integral_const, sub_zero, smul_eq_mul] using hh
+
+theorem smoothPositive_le_self {ε s : ℝ} (hε : 0 < ε) (hs : 0 ≤ s) :
+    smoothPositive ε s ≤ s := by
+  exact (smoothPositive_le_mul hε hs).trans (by nlinarith [Real.smoothTransition.le_one (s / ε)])
+
+theorem sub_parameter_le_smoothPositive {ε s : ℝ} (hε : 0 < ε) :
+    s - ε ≤ smoothPositive ε s := by
+  by_cases hs : s ≤ ε
+  · exact (sub_nonpos.mpr hs).trans (smoothPositive_nonneg hε s)
+  · have hes : ε ≤ s := le_of_not_ge hs
+    have hc : Continuous (fun t : ℝ ↦ Real.smoothTransition (t / ε)) := by fun_prop
+    unfold smoothPositive
+    rw [← intervalIntegral.integral_add_adjacent_intervals (hc.intervalIntegrable 0 ε)
+      (hc.intervalIntegrable ε s)]
+    have heq : (∫ t in ε..s, Real.smoothTransition (t / ε)) = s - ε := by
+      calc
+        _ = ∫ _t in ε..s, (1 : ℝ) := by
+          apply intervalIntegral.integral_congr
+          intro t ht
+          rw [uIcc_of_le hes] at ht
+          exact Real.smoothTransition.one_of_one_le ((le_div_iff₀ hε).2 (by simpa using ht.1))
+        _ = _ := by simp
+    rw [heq]
+    have hp := intervalIntegral.integral_nonneg (μ := volume) hε.le
+      (fun t (_ : t ∈ Icc (0 : ℝ) ε) ↦ Real.smoothTransition.nonneg (t / ε))
+    linarith
+
+def sublevelTest (ε k t : ℝ) : ℝ := smoothPositive ε (k - t)
+def sublevelSlope (ε k t : ℝ) : ℝ := Real.smoothTransition ((k - t) / ε)
+
+theorem sublevelTest_contDiff (ε k : ℝ) : ContDiff ℝ ∞ (sublevelTest ε k) :=
+  (smoothPositive_contDiff ε).comp (contDiff_const.sub contDiff_id)
+
+theorem sublevelTest_hasDerivAt (ε k t : ℝ) :
+    HasDerivAt (sublevelTest ε k) (-sublevelSlope ε k t) t := by
+  convert (smoothPositive_hasDerivAt ε (k - t)).comp t ((hasDerivAt_id t).const_sub k) using 1 <;>
+    (first | rfl | simp [sublevelSlope])
+
+theorem sublevelTest_nonneg {ε : ℝ} (hε : 0 < ε) (k t : ℝ) :
+    0 ≤ sublevelTest ε k t := smoothPositive_nonneg hε _
+
+theorem sublevelSlope_mem_Icc (ε k t : ℝ) : sublevelSlope ε k t ∈ Icc (0 : ℝ) 1 :=
+  ⟨Real.smoothTransition.nonneg _, Real.smoothTransition.le_one _⟩
+
+theorem sublevelTest_controls {ε k t : ℝ} (hε : 0 < ε) (hk : 0 ≤ k) (ht : 0 ≤ t) :
+    sublevelTest ε k t ≤ k * sublevelSlope ε k t := by
+  by_cases htk : t ≤ k
+  · exact (smoothPositive_le_mul hε (sub_nonneg.mpr htk)).trans
+      (mul_le_mul_of_nonneg_right (by linarith) (sublevelSlope_mem_Icc ε k t).1)
+  · rw [sublevelTest, smoothPositive_of_nonpos hε (by linarith)]
+    exact mul_nonneg hk (sublevelSlope_mem_Icc ε k t).1
+
+theorem sublevelTest_eq_zero {ε k t : ℝ} (hε : 0 < ε) (ht : k ≤ t) :
+    sublevelTest ε k t = 0 := smoothPositive_of_nonpos hε (sub_nonpos.mpr ht)
+
+theorem sublevelSlope_eq_zero {ε k t : ℝ} (hε : 0 < ε) (ht : k ≤ t) :
+    sublevelSlope ε k t = 0 :=
+  Real.smoothTransition.zero_of_nonpos (div_nonpos_of_nonpos_of_nonneg (sub_nonpos.mpr ht) hε.le)
+
+theorem sublevelTest_lipschitz (ε k : ℝ) : LipschitzWith 1 (sublevelTest ε k) := by
+  apply lipschitzWith_of_nnnorm_deriv_le ((sublevelTest_contDiff ε k).differentiable (by simp))
+  intro t
+  change ‖deriv (sublevelTest ε k) t‖ ≤ 1
+  rw [(sublevelTest_hasDerivAt ε k t).deriv, norm_neg,
+    Real.norm_of_nonneg (sublevelSlope_mem_Icc ε k t).1]
+  exact (sublevelSlope_mem_Icc ε k t).2
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/TendstoLintegralNormOfL2.lean b/lean/OAI/Geometry/Perimeter/Regularity/TendstoLintegralNormOfL2.lean
new file mode 100644
index 0000000000000000000000000000000000000000..3e46212c1763dd6486a249832257b7b5bd8493eb
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/TendstoLintegralNormOfL2.lean
@@ -0,0 +1,527 @@
+import OAI.Geometry.Perimeter.Regularity.ExistsTangentialGradientBallBound
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+open scoped ENNReal Topology
+namespace RPIKernel
+variable {X V : Type*} [MeasurableSpace X] [NormedAddCommGroup V]
+
+lemma tendsto_lintegral_norm_of_L2 [NormedSpace ℝ V] (μ : Measure X) [IsFiniteMeasure μ]
+    {F : ℕ → X → V} {G : X → V} (hF : ∀ j, MemLp (F j) 2 μ) (hG : MemLp G 2 μ)
+    (hlim : Tendsto (fun j ↦ eLpNorm (fun x ↦ F j x - G x) 2 μ) atTop (𝓝 0)) :
+    Tendsto (fun j ↦ ∫⁻ x, ENNReal.ofReal ‖F j x‖ ∂μ) atTop
+      (𝓝 (∫⁻ x, ENNReal.ofReal ‖G x‖ ∂μ)) := by
+  have hfinite : μ univ ^ (1 / 2 : ℝ) ≠ ⊤ :=
+    ENNReal.rpow_ne_top_of_nonneg (by norm_num) (measure_ne_top μ univ)
+  have ht : Tendsto (fun j ↦ eLpNorm (fun x ↦ F j x - G x) 2 μ * μ univ ^ (1 / 2 : ℝ)) atTop (𝓝 0) := by
+    simpa only [zero_mul] using ENNReal.Tendsto.mul_const hlim (Or.inr hfinite)
+  have hnorm : Tendsto (fun j ↦ eLpNorm (fun x ↦ ‖F j x‖ - ‖G x‖) 1 μ) atTop (𝓝 0) := by
+    apply tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds ht (fun _ ↦ bot_le)
+    intro j
+    calc
+      _ ≤ eLpNorm (fun x ↦ F j x - G x) 1 μ := eLpNorm_mono_ae
+        ((hF j).aestronglyMeasurable.norm.sub hG.aestronglyMeasurable.norm)
+        (ae_of_all _ fun x ↦ by simpa only [Pi.sub_apply, Real.norm_eq_abs] using abs_norm_sub_norm_le (F j x) (G x))
+      _ ≤ _ := by
+        simpa only [Pi.sub_apply, ENNReal.toReal_one, ENNReal.toReal_ofNat, div_one,
+          show (1 : ℝ) - 1 / 2 = 1 / 2 by norm_num] using
+          eLpNorm_le_eLpNorm_mul_rpow_measure_univ (f := fun x ↦ F j x - G x) (by norm_num : (1 : ℝ≥0∞) ≤ 2) ((hF j).aestronglyMeasurable.sub hG.aestronglyMeasurable)
+  have hiF (j : ℕ) : Integrable (F j) μ := memLp_one_iff_integrable.mp ((hF j).mono_exponent (by norm_num))
+  have hiG : Integrable G μ := memLp_one_iff_integrable.mp (hG.mono_exponent (by norm_num))
+  have hI := tendsto_integral_of_L1' (fun x ↦ ‖G x‖)
+    (Eventually.of_forall fun j ↦ (hiF j).norm) hnorm
+  have H := ENNReal.continuous_ofReal.tendsto (∫ x, ‖G x‖ ∂μ) |>.comp hI
+  change Tendsto (fun j ↦ ENNReal.ofReal (∫ x, ‖F j x‖ ∂μ)) atTop
+    (𝓝 (ENNReal.ofReal (∫ x, ‖G x‖ ∂μ))) at H
+  simp_rw [ofReal_integral_norm_eq_lintegral_enorm (hiF _),
+    ofReal_integral_norm_eq_lintegral_enorm hiG] at H
+  simpa only [ofReal_norm] using H
+
+variable [NormedSpace ℝ V]
+
+theorem sobolev_limit (μ : Measure X) [IsFiniteMeasure μ]
+    {f : ℕ → X → ℝ} {h : X → ℝ} {F : ℕ → X → V} {G : X → V}
+    (hf : ∀ j, AEMeasurable (f j) μ)
+    (hpoint : ∀ᵐ x ∂μ, Tendsto (fun j ↦ f j x) atTop (𝓝 (h x)))
+    (hF : ∀ j, MemLp (F j) 2 μ) (hG : MemLp G 2 μ)
+    (hlim : Tendsto (fun j ↦ eLpNorm (fun x ↦ F j x - G x) 2 μ) atTop (𝓝 0))
+    {p : ℝ} {C : ℝ≥0∞} (hC : C ≠ ⊤)
+    (hbound : ∀ j, (∫⁻ x, ENNReal.ofReal (f j x) ^ p ∂μ) ≤
+      C * (∫⁻ x, ENNReal.ofReal ‖F j x‖ ∂μ) ^ p) :
+    (∫⁻ x, ENNReal.ofReal (h x) ^ p ∂μ) ≤ C * (∫⁻ x, ENNReal.ofReal ‖G x‖ ∂μ) ^ p := by
+  have hI := tendsto_lintegral_norm_of_L2 μ hF hG hlim
+  have hpow := (ENNReal.continuous_rpow_const (y := p)).tendsto (∫⁻ x, ENNReal.ofReal ‖G x‖ ∂μ) |>.comp hI
+  have hR := ENNReal.Tendsto.const_mul hpow (Or.inr hC)
+  calc
+    _ = ∫⁻ x, liminf (fun j ↦ ENNReal.ofReal (f j x) ^ p) atTop ∂μ := by
+      apply lintegral_congr_ae
+      filter_upwards [hpoint] with x hx
+      exact (((ENNReal.continuous_rpow_const (y := p)).comp ENNReal.continuous_ofReal).tendsto (h x) |>.comp hx).liminf_eq.symm
+    _ ≤ liminf (fun j ↦ ∫⁻ x, ENNReal.ofReal (f j x) ^ p ∂μ) atTop :=
+      lintegral_liminf_le' (fun j ↦ ((hf j).ennreal_ofReal).pow_const p)
+    _ ≤ liminf (fun j ↦ C * (∫⁻ x, ENNReal.ofReal ‖F j x‖ ∂μ) ^ p) atTop :=
+      liminf_le_liminf (Eventually.of_forall hbound)
+    _ = _ := hR.liminf_eq
+
+end RPIKernel
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+open scoped ENNReal Topology
+namespace RPISingularHarnack
+open RPI RPI.Boundary
+
+theorem wholeBoundary_nonlinear_approximation {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    {h : Euc n → ℝ} {G : Euc n → Euc n}
+    (hSob : WholeBoundarySobolev E h G)
+    (hh01 : ∀ᵐ x ∂boundaryArea E, 0 ≤ h x ∧ h x ≤ 1)
+    {R : ℝ} (hR : 0 < R) {η : Euc n → ℝ}
+    (hη : ContDiff ℝ ∞ η) (hcη : HasCompactSupport η)
+    (hsη : tsupport η ⊆ Metric.ball 0 R)
+    {ψ : ℝ → ℝ} (hψ : ContDiff ℝ ∞ ψ) (hψpos : ∀ t, 0 ≤ ψ t)
+    {K : ℝ≥0} (hψLip : LipschitzWith K ψ)
+    {L : ℝ} (hL : 0 ≤ L) (hdψ : ∀ t, ‖deriv ψ t‖ ≤ L) :
+    let μ := (boundaryArea E).restrict (Metric.ball 0 R)
+    let B := fun x ↦ (2 * η x * ψ (h x)) • tangentialGradient E η x +
+      (η x ^ 2 * deriv ψ (h x)) • G x
+    MemLp B 2 μ ∧ ∃ ξ : ℕ → Euc n → ℝ,
+      (∀ j, ContDiff ℝ ∞ (ξ j) ∧ HasCompactSupport (ξ j) ∧
+        tsupport (ξ j) ⊆ Metric.ball 0 R ∧ ∀ x, 0 ≤ ξ j x) ∧
+      (∀ᵐ x ∂μ, Tendsto (fun j ↦ ξ j x) atTop (𝓝 (η x ^ 2 * ψ (h x)))) ∧
+      Tendsto (fun j ↦ vectorEnergy μ (fun x ↦ tangentialGradient E (ξ j) x - B x)) atTop (𝓝 0) := by
+  let μ := (boundaryArea E).restrict (Metric.ball 0 R)
+  have : IsFiniteMeasure μ := ⟨by
+    simpa only [μ, Measure.restrict_apply_univ] using minimizing_boundaryArea_ball_lt_top hn hmin 0 hR⟩
+  obtain ⟨hh, hG, u, hu, huh, huG⟩ := hSob R hR
+  have hu0 : Tendsto (fun j ↦ vectorEnergy μ (fun x ↦ u j x - h x)) atTop (𝓝 0) := by
+    simpa only [vectorEnergy, Real.norm_eq_abs] using huh
+  have hum : ∀ j, AEStronglyMeasurable (u j) μ :=
+    fun j ↦ (hu j).1.continuous.aestronglyMeasurable
+  obtain ⟨σ, hσ, hpoint⟩ := (tendstoInMeasure_of_tendsto_eLpNorm
+    (by norm_num : (2 : ℝ≥0∞) ≠ 0)
+    (tendsto_eLpNorm_two_of_vectorEnergy_zero_of_aestronglyMeasurable
+      (fun index ↦ (hum index).sub hh.aestronglyMeasurable) hu0)).exists_seq_tendsto_ae
+  let F : ℕ → Euc n → Euc n := fun j ↦ tangentialGradient E (u (σ j))
+  let D := tangentialGradient E η
+  have hF : ∀ j, MemLp (F j) 2 μ := fun j ↦
+    tangentialGradient_memLp_on_ball hn hmin hR ((hu (σ j)).1.of_le (by simp))
+  have hD : MemLp D 2 μ := tangentialGradient_memLp_on_ball hn hmin hR (hη.of_le (by simp))
+  have hdcont : Continuous (deriv ψ) := hψ.continuous_deriv (by simp)
+  obtain ⟨M₀, hM₀⟩ := hcη.exists_bound_of_continuous hη.continuous
+  let M := max M₀ 0
+  have hM : 0 ≤ M := le_max_right _ _
+  have hηM : ∀ᵐ x ∂μ, ‖η x‖ ≤ M := ae_of_all _ fun x ↦ (hM₀ x).trans (le_max_left _ _)
+  have hgradη : Continuous (gradient η) := continuous_iff_continuousAt.mpr (fun x ↦
+    continuousAt_gradient_of_contDiffAt (hη.of_le (by simp)).contDiffAt)
+  obtain ⟨N₀, hN₀⟩ := (isCompact_closedBall (0 : Euc n) R).exists_bound_of_continuousOn hgradη.continuousOn
+  let N := 2 * max N₀ 0
+  have hN : 0 ≤ N := by positivity
+  have hDN : ∀ᵐ x ∂μ, ‖D x‖ ≤ N := by
+    filter_upwards [ae_restrict_mem measurableSet_ball] with x hx
+    exact (tangentialGradient_norm_le_twice E η x).trans (by
+      dsimp [N]; nlinarith [hN₀ x (Metric.ball_subset_closedBall hx), le_max_left N₀ 0])
+  let B : Euc n → Euc n := fun x ↦
+    (2 * η x * ψ (h x)) • D x + (η x ^ 2 * deriv ψ (h x)) • G x
+  obtain ⟨C₀, hC₀⟩ := (isCompact_Icc : IsCompact (Icc (0 : ℝ) 1)).exists_bound_of_continuousOn
+    hψ.continuous.continuousOn
+  let C := max C₀ 0
+  have hC : 0 ≤ C := le_max_right _ _
+  have hψC : ∀ᵐ x ∂μ, ‖ψ (h x)‖ ≤ C := by
+    filter_upwards [ae_restrict_of_ae hh01] with x hx
+    exact (hC₀ (h x) hx).trans (le_max_left _ _)
+  have hca : MemLp (fun x ↦ 2 * η x * ψ (h x)) ∞ μ := by
+    apply MemLp.of_bound ((aestronglyMeasurable_const.mul hη.continuous.aestronglyMeasurable).mul
+      (hψ.continuous.comp_aestronglyMeasurable hh.aestronglyMeasurable)) (2 * M * C)
+    filter_upwards [hηM, hψC] with x hx hy
+    simpa only [Pi.mul_apply, norm_mul, Real.norm_ofNat] using mul_le_mul
+      (mul_le_mul_of_nonneg_left hx (by norm_num : (0 : ℝ) ≤ 2)) hy (norm_nonneg _) (by positivity)
+  have hcb : MemLp (fun x ↦ η x ^ 2 * deriv ψ (h x)) ∞ μ := by
+    apply MemLp.of_bound ((hη.continuous.aestronglyMeasurable.pow 2).mul
+      (hdcont.comp_aestronglyMeasurable hh.aestronglyMeasurable)) (M ^ 2 * L)
+    filter_upwards [hηM] with x hx
+    rw [Pi.mul_apply, Pi.pow_apply, norm_mul, norm_pow]
+    exact mul_le_mul ((sq_le_sq₀ (norm_nonneg _) hM).mpr hx) (hdψ _) (norm_nonneg _) (sq_nonneg _)
+  have hB : MemLp B 2 μ := (hca.smul hD).add (hcb.smul hG)
+  let ξ : ℕ → Euc n → ℝ := fun j x ↦ (η x * η x) * ψ (u (σ j) x)
+  have hξ : ∀ j, ContDiff ℝ ∞ (ξ j) := fun j ↦ (hη.mul hη).mul (hψ.comp (hu (σ j)).1)
+  have hξc : ∀ j, HasCompactSupport (ξ j) := fun _ ↦ hcη.mul_right.mul_right
+  have hξgrad : ∀ j x, tangentialGradient E (ξ j) x =
+      (2 * η x * ψ (u (σ j) x)) • D x + (η x ^ 2 * deriv ψ (u (σ j) x)) • F j x := by
+    intro j x
+    exact tangentialGradient_nonlinear_test E (hη.differentiable (by simp) x)
+      ((hu (σ j)).1.differentiable (by simp) x) (hψ.differentiable (by simp) _)
+  have henergy : Tendsto (fun j ↦ vectorEnergy μ
+      (fun x ↦ tangentialGradient E (ξ j) x - B x)) atTop (𝓝 0) := by
+    simp_rw [hξgrad]
+    exact vectorEnergy_nonlinear_test_tendsto (fun j ↦ hum (σ j)) hh.aestronglyMeasurable
+      hη.continuous.aestronglyMeasurable hD.aestronglyMeasurable hF hG hM hN hηM hDN
+      (hu0.comp hσ.tendsto_atTop) (huG.comp hσ.tendsto_atTop) hpoint hψLip
+      (deriv ψ) hdcont hL hdψ
+  refine ⟨hB, ξ, ?_, ?_, henergy⟩
+  · intro j
+    refine ⟨hξ j, hξc j, ?_, fun x ↦ mul_nonneg (mul_self_nonneg _) (hψpos _)⟩
+    exact (tsupport_mul_subset_left.trans tsupport_mul_subset_left).trans hsη
+  · filter_upwards [hpoint] with x hx
+    simpa only [ξ, ← sq] using
+      (tendsto_const_nhds.mul ((hψ.continuous.tendsto (h x)).comp hx) :
+        Tendsto (fun j ↦ (η x * η x) * ψ (u (σ j) x)) atTop (𝓝 ((η x * η x) * ψ (h x))))
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+open scoped ENNReal Topology
+namespace RPISingularHarnack
+open RPI
+
+theorem wholeBoundary_sublevel_sobolev {n : ℕ} (hn : 2 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    {h : Euc n → ℝ} {G : Euc n → Euc n}
+    (hSob : WholeBoundarySobolev E h G)
+    (hh01 : ∀ᵐ x ∂boundaryArea E, 0 ≤ h x ∧ h x ≤ 1)
+    {R p : ℝ} (hR : 0 < R) (hp : 1 < p)
+    (hpk : (((n - 1 : ℕ) : ℝ) - 1) * p < (n - 1 : ℕ)) :
+    ∃ C : ℝ≥0∞, C ≠ ⊤ ∧ ∀ η : Euc n → ℝ,
+      ContDiff ℝ ∞ η → HasCompactSupport η → tsupport η ⊆ Metric.ball 0 R →
+      ∀ ε k : ℝ, 0 < ε → 0 ≤ k →
+      (∫⁻ x in Metric.ball 0 R, ENNReal.ofReal (η x ^ 2 * sublevelTest ε k (h x)) ^ p ∂boundaryArea E) ≤
+        C * (∫⁻ x in Metric.ball 0 R, ENNReal.ofReal
+          ‖(2 * η x * sublevelTest ε k (h x)) • tangentialGradient E η x -
+            (η x ^ 2 * sublevelSlope ε k (h x)) • G x‖ ∂boundaryArea E) ^ p := by
+  obtain ⟨C, hC, hb⟩ := minimizing_boundaryArea_sobolev hn hmin 0 hR hp hpk
+  refine ⟨C, hC, ?_⟩
+  intro η hη hcη hsη ε k hε hk
+  let μ := (boundaryArea E).restrict (Metric.ball 0 R)
+  let : IsFiniteMeasure μ := ⟨by
+    simpa only [μ, Measure.restrict_apply_univ] using minimizing_boundaryArea_ball_lt_top (by omega : 1 < n) hmin 0 hR⟩
+  obtain ⟨hB, ξ, hξ, hpoint, henergy⟩ := wholeBoundary_nonlinear_approximation (by omega : 1 < n)
+    hmin hSob hh01 hR hη hcη hsη (sublevelTest_contDiff ε k) (sublevelTest_nonneg hε k)
+    (sublevelTest_lipschitz ε k) (show (0 : ℝ) ≤ 1 by norm_num) (fun t ↦ by
+      rw [(sublevelTest_hasDerivAt ε k t).deriv, norm_neg,
+        Real.norm_of_nonneg (sublevelSlope_mem_Icc ε k t).1]
+      exact (sublevelSlope_mem_Icc ε k t).2)
+  have hF : ∀ j, MemLp (tangentialGradient E (ξ j)) 2 μ := fun j ↦
+    tangentialGradient_memLp_on_ball (by omega : 1 < n) hmin hR ((hξ j).1.of_le (by simp))
+  have H := RPIKernel.sobolev_limit μ (fun j ↦ (hξ j).1.continuous.measurable.aemeasurable)
+    hpoint hF hB (tendsto_eLpNorm_two_of_vectorEnergy_zero_of_aestronglyMeasurable
+      (fun index ↦ (hF index).aestronglyMeasurable.sub hB.aestronglyMeasurable) henergy) hC
+    (fun j ↦ hb (ξ j) (hξ j).1 (hξ j).2.1 (hξ j).2.2.1 (hξ j).2.2.2)
+  simpa only [(sublevelTest_hasDerivAt ε k _).deriv, mul_neg, neg_smul, sub_eq_add_neg] using H
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+open scoped ENNReal Topology
+open scoped ENNReal Topology ContDiff
+namespace RPIKernel
+open Classical
+
+lemma weighted_sublevel_gradient_bound {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V]
+    (G D : V) {a ψ χ k N : ℝ} (ha : a ∈ Icc (0 : ℝ) 1)
+    (hχ : χ ∈ Icc (0 : ℝ) 1) (hψ : 0 ≤ ψ) (hψk : ψ ≤ k * χ)
+    (hk : 0 < k) (hN : 0 < N) (hD : ‖D‖ ≤ N) :
+    ‖(2 * a * ψ) • D - (a ^ 2 * χ) • G‖ ≤
+      (5 / 2 : ℝ) * (k * N) + (a ^ 2 * χ * ‖G‖ ^ 2) / (2 * (k * N)) := by
+  have ha0 := ha.1
+  have ha1 := ha.2
+  have hχ0 := hχ.1
+  have hχ1 := hχ.2
+  have hχa : 0 ≤ a ^ 2 * χ := by positivity
+  have ha2 : a ^ 2 ≤ 1 := by nlinarith [ha.1, ha.2]
+  have hχa1 : a ^ 2 * χ ≤ 1 := by nlinarith [hχ.1, hχ.2, sq_nonneg a]
+  have hL : 0 < k * N := mul_pos hk hN
+  have hψbound : ψ ≤ k := by nlinarith [hχ.2]
+  have hfirst : ‖(2 * a * ψ) • D‖ ≤ 2 * (k * N) := by
+    rw [norm_smul, Real.norm_of_nonneg (by positivity)]
+    calc
+      _ ≤ (2 * 1 * k) * N := by gcongr
+      _ = _ := by ring
+  have hsecond : ‖(a ^ 2 * χ) • G‖ ≤ (1 / 2 : ℝ) * (k * N) +
+      (a ^ 2 * χ * ‖G‖ ^ 2) / (2 * (k * N)) := by
+    rw [norm_smul, Real.norm_of_nonneg hχa]
+    have hh : (a ^ 2 * χ * ‖G‖) ^ 2 ≤ a ^ 2 * χ * ‖G‖ ^ 2 := by
+      nlinarith [mul_nonneg (sub_nonneg.mpr hχa1) (mul_nonneg hχa (sq_nonneg ‖G‖))]
+    have H : a ^ 2 * χ * ‖G‖ * (2 * (k * N)) ≤ (k * N) ^ 2 + a ^ 2 * χ * ‖G‖ ^ 2 := by
+      nlinarith [sq_nonneg (a ^ 2 * χ * ‖G‖ - k * N)]
+    have HD := (le_div_iff₀ (by positivity : 0 < 2 * (k * N))).mpr H
+    calc
+      _ ≤ ((k * N) ^ 2 + a ^ 2 * χ * ‖G‖ ^ 2) / (2 * (k * N)) := HD
+      _ = _ := by field_simp
+  exact (norm_sub_le _ _).trans (by linarith)
+
+lemma integral_norm_sublevel_bound {X V : Type*} [MeasurableSpace X]
+    [NormedAddCommGroup V] [NormedSpace ℝ V]
+    (μ : Measure X) [IsFiniteMeasure μ] {A : Set X} (hA : NullMeasurableSet A μ)
+    {B : X → V} {J : X → ℝ} (hB : Integrable B μ) (hJ : Integrable J μ)
+    {L : ℝ} (hL : 0 < L)
+    (hpoint : ∀ᵐ x ∂μ, ‖B x‖ ≤ (if x ∈ A then (5 / 2 : ℝ) * L else 0) + J x / (2 * L))
+    (henergy : (∫ x, J x ∂μ) ≤ 4 * L ^ 2 * μ.real A) :
+    (∫ x, ‖B x‖ ∂μ) ≤ 5 * L * μ.real A := by
+  classical
+  have hiA : Integrable (fun x ↦ if x ∈ A then (5 / 2 : ℝ) * L else 0) μ :=
+    (integrable_const ((5 / 2 : ℝ) * L)).indicator₀ hA
+  have hmono := integral_mono_ae hB.norm (hiA.add (hJ.div_const _)) hpoint
+  have hIA : (∫ x, (if x ∈ A then (5 / 2 : ℝ) * L else 0) ∂μ) = (5 / 2 : ℝ) * L * μ.real A := by
+    change (∫ x, A.indicator (fun _ ↦ (5 / 2 : ℝ) * L) x ∂μ) = _
+    rw [integral_indicator₀ hA]
+    simp only [setIntegral_const, smul_eq_mul]
+    ring
+  simp only [Pi.add_apply] at hmono
+  rw [integral_add hiA (hJ.div_const _), hIA, integral_div] at hmono
+  have he := div_le_div_of_nonneg_right henergy (by positivity : 0 ≤ 2 * L)
+  have hc : 4 * L ^ 2 * μ.real A / (2 * L) = 2 * L * μ.real A := by field_simp; ring
+  rw [hc] at he
+  nlinarith [(measureReal_nonneg : 0 ≤ μ.real A)]
+
+end RPIKernel
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+open scoped ENNReal Topology
+open scoped ENNReal Topology ContDiff
+namespace RPISingularHarnack
+open RPI RPI.Boundary Classical
+
+theorem wholeBoundary_sublevel_L1 {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    {h : Euc n → ℝ} {G : Euc n → Euc n}
+    (hSob : WholeBoundarySobolev E h G) (hSuper : WholeBoundarySupersolution E G)
+    (hh01 : ∀ᵐ x ∂boundaryArea E, 0 ≤ h x ∧ h x ≤ 1)
+    {R r ε k N : ℝ} (hR : 0 < R) (hrR : r ≤ R) (hε : 0 < ε) (hk : 0 < k) (hN : 0 < N)
+    {η : Euc n → ℝ} (hη : ContDiff ℝ ∞ η) (hcη : HasCompactSupport η)
+    (hsη : tsupport η ⊆ Metric.ball 0 r) (hη01 : ∀ x, η x ∈ Icc (0 : ℝ) 1)
+    (hD : ∀ x, ‖tangentialGradient E η x‖ ≤ N) :
+    (∫⁻ x in Metric.ball 0 R, ENNReal.ofReal
+      ‖(2 * η x * sublevelTest ε k (h x)) • tangentialGradient E η x -
+        (η x ^ 2 * sublevelSlope ε k (h x)) • G x‖ ∂boundaryArea E) ≤
+      ENNReal.ofReal (5 * k * N) * boundaryArea E (Metric.ball 0 r ∩ {x | h x < k}) := by
+  classical
+  let μ := (boundaryArea E).restrict (Metric.ball 0 R)
+  let : IsFiniteMeasure μ := ⟨by
+    simpa only [μ, Measure.restrict_apply_univ] using minimizing_boundaryArea_ball_lt_top hn hmin 0 hR⟩
+  have hsR := hsη.trans (Metric.ball_subset_ball hrR)
+  have hh := (hSob R hR).1
+  have hG := (hSob R hR).2.1
+  let A := Metric.ball (0 : Euc n) r ∩ {x | h x < k}
+  have hAm : NullMeasurableSet A μ := measurableSet_ball.nullMeasurableSet.inter
+    (nullMeasurableSet_lt hh.aestronglyMeasurable.aemeasurable aemeasurable_const)
+  let J := fun x ↦ η x ^ 2 * sublevelSlope ε k (h x) * ‖G x‖ ^ 2
+  let B := fun x ↦ (2 * η x * sublevelTest ε k (h x)) • tangentialGradient E η x -
+    (η x ^ 2 * sublevelSlope ε k (h x)) • G x
+  have hχc : Continuous (sublevelSlope ε k) := by unfold sublevelSlope; fun_prop
+  have hJm : AEStronglyMeasurable J μ :=
+    ((hη.continuous.aestronglyMeasurable.pow 2).mul (hχc.comp_aestronglyMeasurable hh.aestronglyMeasurable)).mul (hG.aestronglyMeasurable.norm.pow 2)
+  have hJ : Integrable J μ := ((memLp_two_iff_integrable_sq_norm hG.aestronglyMeasurable).mp hG).mono' hJm (by
+    filter_upwards [] with x
+    have ha0 := (hη01 x).1
+    have ha1 := (hη01 x).2
+    have hc0 := (sublevelSlope_mem_Icc ε k (h x)).1
+    have hc1 := (sublevelSlope_mem_Icc ε k (h x)).2
+    rw [Real.norm_of_nonneg (by dsimp [J]; positivity)]
+    have hb : η x ^ 2 * sublevelSlope ε k (h x) ≤ 1 := by nlinarith [sq_nonneg (η x)]
+    simpa only [one_mul] using mul_le_mul_of_nonneg_right hb (sq_nonneg ‖G x‖))
+  have hBA : MemLp B 2 μ := by
+    have H := (wholeBoundary_nonlinear_approximation hn hmin hSob hh01 hR hη hcη hsR
+      (sublevelTest_contDiff ε k) (sublevelTest_nonneg hε k) (sublevelTest_lipschitz ε k)
+      (show (0 : ℝ) ≤ 1 by norm_num) (fun t ↦ by
+        rw [(sublevelTest_hasDerivAt ε k t).deriv, norm_neg,
+          Real.norm_of_nonneg (sublevelSlope_mem_Icc ε k t).1]
+        exact (sublevelSlope_mem_Icc ε k t).2)).1
+    simpa only [B, μ, (sublevelTest_hasDerivAt ε k _).deriv, mul_neg, neg_smul, sub_eq_add_neg] using H
+  have hB : Integrable B μ := memLp_one_iff_integrable.mp (hBA.mono_exponent (by norm_num))
+  have hz {x : Euc n} (hx : x ∉ Metric.ball 0 r) : η x = 0 ∧ tangentialGradient E η x = 0 := by
+    have ht : x ∉ tsupport η := fun ht ↦ hx (hsη ht)
+    exact ⟨image_eq_zero_of_notMem_tsupport ht, tangentialGradient_eq_zero_of_notMem_tsupport E ht⟩
+  have hpoint : ∀ᵐ x ∂μ, ‖B x‖ ≤ (if x ∈ A then (5 / 2 : ℝ) * (k * N) else 0) + J x / (2 * (k * N)) := by
+    filter_upwards [ae_restrict_of_ae hh01] with x hx
+    by_cases hxr : x ∈ Metric.ball 0 r
+    · by_cases hxk : h x < k
+      · simp only [A, mem_inter_iff, mem_ofPred_eq, hxr, hxk, and_self, ite_true]
+        exact RPIKernel.weighted_sublevel_gradient_bound (G x) (tangentialGradient E η x)
+          (hη01 x) (sublevelSlope_mem_Icc ε k (h x)) (sublevelTest_nonneg hε k (h x))
+          (sublevelTest_controls hε hk.le hx.1) hk hN (hD x)
+      · simp [B, J, A, hxk, sublevelTest_eq_zero hε (le_of_not_gt hxk),
+          sublevelSlope_eq_zero hε (le_of_not_gt hxk)]
+    · obtain ⟨hηx, hDx⟩ := hz hxr
+      simp [B, J, A, hxr, hηx, hDx]
+  have hi : Integrable (fun x ↦ if h x < k then k ^ 2 * ‖tangentialGradient E η x‖ ^ 2 else 0) μ := by
+    have hm : NullMeasurableSet {x | h x < k} μ := nullMeasurableSet_lt hh.aestronglyMeasurable.aemeasurable aemeasurable_const
+    exact ((memLp_two_iff_integrable_sq_norm
+      (tangentialGradient_memLp_on_ball hn hmin hR (hη.of_le (by simp))).aestronglyMeasurable).mp
+      (tangentialGradient_memLp_on_ball hn hmin hR (hη.of_le (by simp)))).const_mul (k ^ 2) |>.indicator₀ hm
+  have hconst : Integrable (A.indicator (fun _ ↦ (k * N) ^ 2)) μ := (integrable_const _).indicator₀ hAm
+  have hcost : (∫ x, (if h x < k then k ^ 2 * ‖tangentialGradient E η x‖ ^ 2 else 0) ∂μ) ≤
+      (k * N) ^ 2 * μ.real A := by
+    calc
+      _ ≤ ∫ x, A.indicator (fun _ ↦ (k * N) ^ 2) x ∂μ := by
+        apply integral_mono_ae hi hconst
+        exact ae_of_all _ fun x ↦ by
+          by_cases hxr : x ∈ Metric.ball 0 r
+          · by_cases hxk : h x < k
+            · simp only [A, mem_inter_iff, mem_ofPred_eq, hxr, hxk, and_self, ite_true, indicator_of_mem]
+              nlinarith [sq_le_sq₀ (norm_nonneg (tangentialGradient E η x)) hN.le |>.mpr (hD x)]
+            · simp [A, hxk]
+          · simp [A, hxr, (hz hxr).2]
+      _ = _ := by rw [integral_indicator₀ hAm, setIntegral_const]; simp [Measure.real, smul_eq_mul, mul_comm]
+  have henergy := wholeBoundary_sublevel_energy hn hmin hSob hSuper hh01 hR hη hcη hsR hε hk.le
+  have hreal := RPIKernel.integral_norm_sublevel_bound μ hAm hB hJ (mul_pos hk hN) (by
+      filter_upwards [hpoint] with x hx
+      by_cases hxA : x ∈ A <;> simpa only [hxA, ↓reduceIte] using hx)
+    (by simpa only [μ, J, mul_assoc] using
+      henergy.trans (mul_le_mul_of_nonneg_left hcost (by norm_num : (0 : ℝ) ≤ 4)))
+  have hμA : μ A = boundaryArea E A := by
+    change (boundaryArea E).restrict (Metric.ball 0 R) A = _
+    rw [Measure.restrict_apply₀ hAm]
+    congr 1
+    exact inter_eq_left.mpr (fun x hx ↦ Metric.ball_subset_ball hrR hx.1)
+  calc
+    _ = ENNReal.ofReal (∫ x, ‖B x‖ ∂μ) := by
+      symm
+      simpa only [ofReal_norm] using ofReal_integral_norm_eq_lintegral_enorm hB
+    _ ≤ ENNReal.ofReal (5 * (k * N) * μ.real A) := ENNReal.ofReal_le_ofReal hreal
+    _ = _ := by
+      rw [ENNReal.ofReal_mul (by positivity : 0 ≤ 5 * (k * N)), Measure.real,
+        ENNReal.ofReal_toReal (measure_ne_top μ A), hμA, mul_assoc]
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+open scoped ENNReal Topology
+open scoped ENNReal Topology ContDiff
+namespace RPISingularHarnack
+open RPI RPI.Boundary
+
+theorem exists_interior_cutoff_constant : ∃ L : ℝ, 0 < L ∧ ∀ (n : ℕ) (E : Set (Euc n))
+    (s r : ℝ), 0 < s → s < r → ∃ η : Euc n → ℝ,
+    ContDiff ℝ ∞ η ∧ HasCompactSupport η ∧ tsupport η ⊆ Metric.ball 0 r ∧
+    (∀ x, η x ∈ Icc (0 : ℝ) 1) ∧ (∀ x ∈ Metric.ball 0 s, η x = 1) ∧
+    (∀ x, ‖tangentialGradient E η x‖ ≤ L / (r - s)) := by
+  obtain ⟨M, hM, hb⟩ := exists_smoothTransition_deriv_bound
+  refine ⟨8 * M, by positivity, ?_⟩
+  intro n E s r hs hsr
+  let b := (s + r) / 2
+  have hsb : s < b := by dsimp [b]; linarith
+  have hbr : b < r := by dsimp [b]; linarith
+  have hb0 : 0 < b := hs.trans hsb
+  let η := fun x : Euc n ↦ 1 - annularCutoff 0 s b x
+  have hη : ContDiff ℝ ∞ η := contDiff_const.sub (annularCutoff_contDiff 0 s b)
+  have hts : tsupport η ⊆ Metric.closedBall 0 b := by
+    apply closure_minimal
+    · intro x hx
+      by_contra hxb
+      have hdist : b ≤ dist x 0 := le_of_lt (lt_of_not_ge hxb)
+      exact hx (by dsimp [η]; rw [annularCutoff_one hs hsb hdist, sub_self])
+    · exact Metric.isClosed_closedBall
+  have hgrad (x : Euc n) : gradient η x = -gradient (annularCutoff 0 s b) x := by
+    simp only [η, gradient, fderiv_const_sub, map_neg]
+  refine ⟨η, hη, (isCompact_closedBall (0 : Euc n) b).of_isClosed_subset (isClosed_tsupport _) hts,
+    hts.trans (Metric.closedBall_subset_ball hbr), ?_, ?_, ?_⟩
+  · intro x
+    have h := annularCutoff_mem_Icc (0 : Euc n) s b x
+    dsimp [η]
+    exact ⟨by linarith [h.2], by linarith [h.1]⟩
+  · intro x hx
+    dsimp [η]
+    rw [annularCutoff_zero hs hsb (Metric.mem_ball.mp hx).le, sub_zero]
+  · intro x
+    have hden : 0 < b ^ 2 - s ^ 2 := by nlinarith
+    have hwidth : 0 < r - s := sub_pos.mpr hsr
+    calc
+      _ ≤ 2 * ‖gradient η x‖ := tangentialGradient_norm_le_twice E η x
+      _ = 2 * ‖gradient (annularCutoff 0 s b) x‖ := by rw [hgrad, norm_neg]
+      _ ≤ 2 * (2 * M * b / (b ^ 2 - s ^ 2)) :=
+        mul_le_mul_of_nonneg_left (gradient_annularCutoff_bound hs hsb hM hb x) (by norm_num)
+      _ ≤ 8 * M / (r - s) := by
+        rw [← mul_div_assoc, div_le_div_iff₀ hden hwidth]
+        have hbs : b - s = (r - s) / 2 := by dsimp [b]; ring
+        have hp : 0 ≤ M * s * (r - s) := by positivity
+        have heq : 8 * M * (b ^ 2 - s ^ 2) - 2 * (2 * M * b) * (r-s) = 4 * M * s * (r-s) := by
+          dsimp [b]
+          ring
+        nlinarith
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+open scoped ENNReal Topology
+open scoped ENNReal Topology ContDiff
+namespace RPIIteration
+
+lemma sublevel_step_division {a b C d p : ℝ} (hd : 0 < d) (hb : 0 ≤ b)
+    (h : d ^ p * a ≤ C * b ^ p) : a ≤ C * (b / d) ^ p := by
+  have hdp := Real.rpow_pos_of_pos hd p
+  rw [Real.div_rpow hb hd.le, ← mul_div_assoc, le_div_iff₀ hdp]
+  rwa [mul_comm]
+
+lemma sublevel_step_factor {C L k l s r a p : ℝ} (hC : 0 ≤ C) (hL : 0 < L)
+    (hk : 0 < k) (hkl : l < k) (hsr : s < r) (ha : 0 ≤ a) :
+    C * (5 * k * (L / (r-s)) * a / ((k-l) / 2)) ^ p ≤
+      ((C + 1) * (10 * L) ^ p) * (k / ((r-s) * (k-l))) ^ p * a ^ p := by
+  have heq : 5 * k * (L / (r-s)) * a / ((k-l)/2) =
+      (10 * L) * (k / ((r-s) * (k-l))) * a := by field_simp; ring
+  rw [heq, Real.mul_rpow (by positivity) ha,
+    Real.mul_rpow (by positivity) (by positivity)]
+  have hc : C ≤ C + 1 := by linarith
+  calc
+    _ ≤ (C + 1) * ((10 * L) ^ p * (k / ((r-s) * (k-l))) ^ p * a ^ p) := by
+      exact mul_le_mul hc le_rfl (by positivity) (add_nonneg hC zero_le_one)
+    _ = _ := by ring
+
+end RPIIteration
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/TendstoWeightedHalfspaceBlowup.lean b/lean/OAI/Geometry/Perimeter/Regularity/TendstoWeightedHalfspaceBlowup.lean
new file mode 100644
index 0000000000000000000000000000000000000000..a72cb060ccd6e789ddf10ab51e79de5f17912465
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/TendstoWeightedHalfspaceBlowup.lean
@@ -0,0 +1,548 @@
+import OAI.Geometry.Perimeter.Regularity.SurfaceGradientConstSub
+import OAI.Geometry.Perimeter.Regularity.AnnularCutoffTestAdmissible
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+
+theorem tendsto_weighted_halfspaceBlowup {n : ℕ} {E : Set (Euc n)}
+    (hE : MeasurableSet E) {c v : Euc n} (hv : HasHalfspaceBlowup E c v) :
+    Tendsto (fun r : ℝ ↦ ∫ z in Metric.ball (0 : Euc n) 1,
+        E.indicator (fun _ ↦ (1 : ℝ)) (c + r • z) * divergence (radialNormalTest v) z)
+      (𝓝[>] (0 : ℝ)) (𝓝 (∫ z, radialFlux v z)) := by
+  let μ := (volume : Measure (Euc n)).restrict (Metric.ball 0 1)
+  let χ : ℝ → Euc n → ℝ := fun r z ↦ E.indicator (fun _ ↦ (1 : ℝ)) (c + r • z)
+  let H : Set (Euc n) := {z | inner ℝ z v < 0}
+  let ψ : Euc n → ℝ := H.indicator (fun _ ↦ (1 : ℝ))
+  let dV : Euc n → ℝ := divergence (radialNormalTest v)
+  have hH : MeasurableSet H :=
+    (isOpen_lt (continuous_id.inner continuous_const) continuous_const).measurableSet
+  have hχ : ∀ r, Integrable (χ r) μ := integrable_rescaled_indicator hE c
+  have hψ : Integrable ψ μ :=
+    (integrableOn_const measure_ball_lt_top.ne (C := (1 : ℝ))).indicator hH
+  have hdV : Continuous dV := continuous_divergence (radialNormalTest_contDiff v)
+  have hcV : HasCompactSupport dV := (radialNormalTest_admissible hv.1).2.1.of_isClosed_subset
+    (isClosed_tsupport _) (tsupport_divergence_subset _)
+  obtain ⟨D, hD⟩ := hcV.exists_bound_of_continuous hdV
+  have hχV : ∀ r, Integrable (fun z ↦ χ r z * dV z) μ := fun r ↦
+    (hχ r).mul_bdd hdV.aestronglyMeasurable (Eventually.of_forall hD)
+  have hψV : Integrable (fun z ↦ ψ z * dV z) μ :=
+    hψ.mul_bdd hdV.aestronglyMeasurable (Eventually.of_forall hD)
+  have hconv : Tendsto (fun r ↦ ∫ z, |χ r z - ψ z| ∂μ) (𝓝[>] (0 : ℝ)) (𝓝 0) := by
+    apply hv.2.congr'
+    filter_upwards [self_mem_nhdsWithin] with r hr
+    rw [normalized_ball_integral _ c hr]
+    apply integral_congr_ae
+    filter_upwards [] with z
+    simp only [χ, ψ, H, halfspace_indicator_rescale c v z hr]
+  have hbound : ∀ r, ‖(∫ z, χ r z * dV z ∂μ) - ∫ z, ψ z * dV z ∂μ‖ ≤
+      D * ∫ z, |χ r z - ψ z| ∂μ := by
+    intro r
+    calc
+      _ = ‖∫ z, (χ r z - ψ z) * dV z ∂μ‖ := by
+        simp_rw [sub_mul]
+        rw [integral_sub (hχV r) hψV]
+      _ ≤ ∫ z, D * |χ r z - ψ z| ∂μ := by
+        apply norm_integral_le_of_norm_le ((hχ r |>.sub hψ).abs.const_mul D)
+        filter_upwards [] with z
+        rw [norm_mul, Real.norm_eq_abs]
+        exact (mul_le_mul_of_nonneg_left (hD z) (abs_nonneg _)).trans_eq (mul_comm _ _)
+      _ = _ := integral_const_mul _ _
+  have hl : Tendsto (fun r ↦ ∫ z, χ r z * dV z ∂μ) (𝓝[>] (0 : ℝ))
+      (𝓝 (∫ z, ψ z * dV z ∂μ)) := by
+    apply tendsto_iff_norm_sub_tendsto_zero.mpr
+    exact squeeze_zero (fun _ ↦ norm_nonneg _) hbound (by simpa using hconv.const_mul D)
+  have heq : (∫ z, ψ z * dV z ∂μ) = ∫ z, radialFlux v z := by
+    calc
+      _ = ∫ z in Metric.ball (0 : Euc n) 1, radialFlux v z := by
+        apply integral_congr_ae
+        filter_upwards [] with z
+        rw [radialFlux_eq_indicator]
+        by_cases hz : z ∈ H <;> simp [ψ, H, dV, hz]
+      _ = _ := setIntegral_eq_integral_of_forall_compl_eq_zero (fun z hz ↦
+        image_eq_zero_of_notMem_tsupport (fun h ↦ hz
+          ((radialFlux_support v).trans (Metric.closedBall_subset_ball (by norm_num)) h)))
+  rw [heq] at hl
+  exact hl
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+
+theorem exists_halfspace_perimeter_lower_density_constant (n : ℕ) (hn : 0 < n) :
+    ∃ c₀ : ℝ, 0 < c₀ ∧ ∀ E : Set (Euc n), MeasurableSet E →
+      ∀ x v, HasHalfspaceBlowup E x v →
+        ∀ᶠ r : ℝ in 𝓝[>] (0 : ℝ),
+          ENNReal.ofReal (c₀ * r ^ (n - 1)) ≤ perimeter E (Metric.ball x r) := by
+  obtain ⟨c₁, hc₁, hflux⟩ := exists_uniform_radialFlux_lower n hn
+  refine ⟨c₁ / 2, by positivity, ?_⟩
+  intro E hE x v hv
+  have hconv := tendsto_weighted_halfspaceBlowup hE hv
+  have hlt : c₁ / 2 < ∫ z, radialFlux v z := by
+    have hh := hflux v hv.1
+    linarith
+  have hevent := hconv.eventually (Ioi_mem_nhds hlt)
+  filter_upwards [hevent, self_mem_nhdsWithin] with r hr0 hr
+  change 0 < r at hr
+  change c₁ / 2 < _ at hr0
+  calc
+    _ ≤ ENNReal.ofReal (∫ y in E, divergence (ballNormalTest x r v) y) := by
+      apply ENNReal.ofReal_le_ofReal
+      rw [ballNormalTest_integral hn E hE x hr v]
+      exact (mul_le_mul_of_nonneg_right hr0.le (pow_nonneg hr.le _)).trans_eq (mul_comm _ _)
+    _ ≤ perimeter E (Metric.ball x r) := by
+      rw [perimeter_eq_iSup_tests]
+      exact le_iSup_of_le (ballNormalTest x r v)
+        (le_iSup_of_le (ballNormalTest_admissible x hr hv.1) le_rfl)
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+
+theorem perimeter_finset_disjoint_superadditive {n : ℕ} (E : Set (Euc n))
+    {ι : Type*} (s : Finset ι) (U : ι → Set (Euc n))
+    (hU : (s : Set ι).PairwiseDisjoint U) :
+    ∑ i ∈ s, perimeter E (U i) ≤ perimeter E (⋃ i ∈ s, U i) := by
+  classical
+  induction s using Finset.induction_on with
+  | empty => simp only [Finset.sum_empty]; exact bot_le
+  | @insert a s ha ih =>
+    rw [Finset.sum_insert ha]
+    have hs : (s : Set ι).PairwiseDisjoint U := (fun i hi j hj h ↦ hU (by simp [hi]) (by simp [hj]) h)
+    have hd : Disjoint (U a) (⋃ i ∈ s, U i) := by
+      rw [Set.disjoint_iUnion_right]
+      intro i
+      rw [Set.disjoint_iUnion_right]
+      intro hi
+      exact hU (by simp) (by simp [hi]) (by intro h; subst i; exact ha hi)
+    calc
+      _ ≤ perimeter E (U a) + perimeter E (⋃ i ∈ s, U i) := add_le_add le_rfl (ih hs)
+      _ ≤ perimeter E (⋃ i ∈ insert a s, U i) :=
+        perimeter_disjoint_superadditive E
+          (by intro x hx; exact mem_iUnion.mpr ⟨a, mem_iUnion.mpr ⟨by simp, hx⟩⟩)
+          (by intro x hx; simpa only [Finset.mem_insert, iUnion_iUnion_eq_or_left] using
+            (show x ∈ U a ∪ ⋃ i ∈ s, U i from Or.inr hx)) hd
+
+theorem perimeter_tsum_disjoint_le {n : ℕ} (E : Set (Euc n))
+    {ι : Type*} (U : ι → Set (Euc n)) (Z : Set (Euc n))
+    (hU : Pairwise (fun i j ↦ Disjoint (U i) (U j))) (hUZ : ∀ i, U i ⊆ Z) :
+    ∑' i, perimeter E (U i) ≤ perimeter E Z := by
+  apply ENNReal.summable.tsum_le_of_sum_le
+  intro s
+  exact (perimeter_finset_disjoint_superadditive E s U
+    (fun _ _ _ _ h ↦ hU h)).trans
+    (raw_perimeter_mono_region E (by simp only [iUnion_subset_iff]; intro i hi; exact hUZ i))
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+
+theorem exists_perimeter_vitali_cover {n : ℕ} (E S : Set (Euc n))
+    {c₀ : ℝ} (hc₀ : 0 < c₀)
+    (hdensity : ∀ x ∈ S, ∀ᶠ r : ℝ in 𝓝[>] (0 : ℝ),
+      ENNReal.ofReal (c₀ * r ^ (n - 1)) ≤ perimeter E (Metric.ball x r))
+    (c : Euc n) {R δ : ℝ} (hR : 0 < R) (hδ : 0 < δ) :
+    ∃ (u : Set (Euc n)) (r : Euc n → ℝ), u.Countable ∧
+      (∀ x ∈ u, 0 < r x ∧ r x ≤ δ) ∧
+      (S ∩ Metric.ball c R ⊆ ⋃ x ∈ u, Metric.ball x (4 * r x)) ∧
+      (∑' x : u, (ENNReal.ofReal (r x)) ^ (n - 1)) ≤
+        (ENNReal.ofReal c₀)⁻¹ * perimeter E (Metric.ball c (2 * R)) := by
+  classical
+  let t := S ∩ Metric.ball c R
+  have hchoose : ∀ x ∈ t, ∃ r : ℝ, 0 < r ∧ r ≤ min δ R ∧
+      ENNReal.ofReal (c₀ * r ^ (n - 1)) ≤ perimeter E (Metric.ball x r) := by
+    intro x hx
+    have hev := hdensity x hx.1
+    have hsmall : ∀ᶠ r : ℝ in 𝓝[>] (0 : ℝ), r < min δ R :=
+      (eventually_lt_nhds (lt_min hδ hR)).filter_mono nhdsWithin_le_nhds
+    obtain ⟨r, hr, hden, hsm⟩ :=
+      ((show ∀ᶠ r : ℝ in 𝓝[>] (0 : ℝ), 0 < r from self_mem_nhdsWithin).and (hev.and hsmall)).exists
+    exact ⟨r, hr, hsm.le, hden⟩
+  have htotal : ∀ x : Euc n, ∃ r : ℝ, 0 < r ∧ r ≤ min δ R ∧
+      (x ∈ t → ENNReal.ofReal (c₀ * r ^ (n - 1)) ≤ perimeter E (Metric.ball x r)) := by
+    intro x
+    by_cases hx : x ∈ t
+    · obtain ⟨r, hr, hrb, hd⟩ := hchoose x hx
+      exact ⟨r, hr, hrb, fun _ ↦ hd⟩
+    · exact ⟨min δ R / 2, by positivity, by linarith [lt_min hδ hR],
+        (show x ∈ t → ENNReal.ofReal (c₀ * (min δ R / 2) ^ (n - 1)) ≤
+          perimeter E (Metric.ball x (min δ R / 2)) from fun h ↦ False.elim (hx h))⟩
+  choose r hr using htotal
+  obtain ⟨u, hut, hud, hcover⟩ :=
+    Vitali.exists_disjoint_subfamily_covering_enlargement_ball t id r (min δ R)
+      (fun x hx ↦ (hr x).2.1) 4 (by norm_num)
+  have huc : u.Countable := hud.countable_of_isOpen (fun _ _ ↦ Metric.isOpen_ball)
+    (fun x hx ↦ ⟨x, Metric.mem_ball_self (hr x).1⟩)
+  refine ⟨u, r, huc, ?_, ?_, ?_⟩
+  · intro x hx
+    exact ⟨(hr x).1, (hr x).2.1.trans (min_le_left _ _)⟩
+  · intro x hx
+    obtain ⟨y, hy, hxy⟩ := hcover x hx
+    exact mem_iUnion.mpr ⟨y, mem_iUnion.mpr ⟨hy, hxy (Metric.mem_ball_self (hr x).1)⟩⟩
+  · have hballs : ∀ x : u, Metric.ball (x : Euc n) (r x) ⊆ Metric.ball c (2 * R) := by
+      intro x y hy
+      have hx := (hut x.property).2
+      simp only [Metric.mem_ball] at hx hy ⊢
+      have hrR := (hr x).2.1.trans (min_le_right δ R)
+      calc dist y c ≤ dist y x + dist (x : Euc n) c := dist_triangle _ _ _
+        _ < 2 * R := by linarith
+    have hsum := perimeter_tsum_disjoint_le E
+      (fun x : u ↦ Metric.ball (x : Euc n) (r x)) (Metric.ball c (2 * R))
+      (fun x y h ↦ hud x.property y.property (fun hxy ↦ h (Subtype.ext hxy))) hballs
+    have hweigh : ENNReal.ofReal c₀ * ∑' x : u, ENNReal.ofReal (r x) ^ (n - 1) ≤
+        perimeter E (Metric.ball c (2 * R)) := by
+      rw [← ENNReal.tsum_mul_left]
+      apply le_trans (ENNReal.tsum_le_tsum (fun x : u ↦ ?_)) hsum
+      rw [← ENNReal.ofReal_pow (hr x).1.le, ← ENNReal.ofReal_mul hc₀.le]
+      exact (hr x).2.2 (hut x.property)
+    calc
+      _ = (ENNReal.ofReal c₀)⁻¹ *
+          (ENNReal.ofReal c₀ * ∑' x : u, ENNReal.ofReal (r x) ^ (n - 1)) := by
+        rw [← mul_assoc, ENNReal.inv_mul_cancel (ENNReal.ofReal_ne_zero_iff.mpr hc₀)
+          ENNReal.ofReal_ne_top, one_mul]
+      _ ≤ _ := mul_le_mul le_rfl hweigh (by positivity) (by positivity)
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+
+theorem hausdorffMeasure_le_perimeter_of_lower_density {n : ℕ} (E S : Set (Euc n))
+    {c₀ : ℝ} (hc₀ : 0 < c₀)
+    (hdensity : ∀ x ∈ S, ∀ᶠ r : ℝ in 𝓝[>] (0 : ℝ),
+      ENNReal.ofReal (c₀ * r ^ (n - 1)) ≤ perimeter E (Metric.ball x r))
+    (c : Euc n) {R : ℝ} (hR : 0 < R) :
+    Measure.hausdorffMeasure (n - 1 : ℕ) (S ∩ Metric.ball c R) ≤
+      (8 : ℝ≥0∞) ^ (n - 1) * (ENNReal.ofReal c₀)⁻¹ *
+        perimeter E (Metric.ball c (2 * R)) := by
+  classical
+  let δ : ℕ → ℝ := fun j ↦ 1 / (j + 1)
+  have hδ : ∀ j, 0 < δ j := fun j ↦ by dsimp [δ]; positivity
+  choose u r hcount hr hcover hsum using fun j ↦
+    exists_perimeter_vitali_cover E S hc₀ hdensity c hR (hδ j)
+  let : ∀ j, Countable (u j) := fun j ↦ (hcount j).to_subtype
+  have hediam : ∀ j (x : u j),
+      Metric.ediam (Metric.ball (x : Euc n) (4 * r j x)) ≤
+        ENNReal.ofReal (8 * r j x) := by
+    intro j x
+    apply Metric.ediam_le
+    intro y hy z hz
+    rw [edist_dist]
+    apply ENNReal.ofReal_le_ofReal
+    simp only [Metric.mem_ball] at hy hz
+    calc dist y z ≤ dist y x + dist z x := dist_triangle_right _ _ _
+      _ ≤ 8 * r j x := by linarith
+  have hδlim : Tendsto (fun j ↦ ENNReal.ofReal (8 * δ j)) atTop (𝓝 0) := by
+    have hh : Tendsto δ atTop (𝓝 0) := tendsto_one_div_add_atTop_nhds_zero_nat
+    simpa only [Function.comp_def, mul_zero, ENNReal.ofReal_zero] using ENNReal.continuous_ofReal.continuousAt.tendsto.comp (hh.const_mul 8)
+  have hh := Measure.hausdorffMeasure_le_liminf_tsum ((n - 1 : ℕ) : ℝ)
+    (S ∩ Metric.ball c R) (fun j ↦ ENNReal.ofReal (8 * δ j)) hδlim
+    (fun j (x : u j) ↦ Metric.ball (x : Euc n) (4 * r j x))
+    (Filter.Eventually.of_forall (fun j x ↦
+      (hediam j x).trans (ENNReal.ofReal_le_ofReal (by
+        have h := (hr j x x.property).2; linarith))))
+    (Filter.Eventually.of_forall (fun j ↦ by simpa only [iUnion_subtype] using hcover j))
+  apply hh.trans
+  refine Filter.liminf_le_of_frequently_le ?_ (by isBoundedDefault)
+  apply Filter.Eventually.frequently
+  apply Filter.Eventually.of_forall
+  intro j
+  calc
+    _ ≤ ∑' x : u j, (8 : ℝ≥0∞) ^ (n - 1) *
+        ENNReal.ofReal (r j x) ^ (n - 1) := by
+      apply ENNReal.tsum_le_tsum
+      intro x
+      rw [ENNReal.rpow_natCast]
+      calc
+        _ ≤ (ENNReal.ofReal (8 * r j x)) ^ (n - 1) := pow_le_pow_left' (hediam j x) _
+        _ = _ := by rw [ENNReal.ofReal_mul (by norm_num : (0 : ℝ) ≤ 8), mul_pow]; norm_num
+    _ = (8 : ℝ≥0∞) ^ (n - 1) * ∑' x : u j, ENNReal.ofReal (r j x) ^ (n - 1) :=
+      ENNReal.tsum_mul_left
+    _ ≤ _ := by simpa only [mul_assoc] using mul_le_mul le_rfl (hsum j) (by positivity) (by positivity)
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+
+theorem exists_minimizing_surfaceMeasure_growth_of_halfspace {n : ℕ} (hn : 8 ≤ n)
+    {E Z : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    (hZd : dimH Z ≤ (n - 8 : ℕ))
+    (hhalf : ∀ x ∈ perimeterSupport E \ Z, ∃ v, HasHalfspaceBlowup E x v) :
+    ∃ A : ℝ, 0 < A ∧ ∀ (c : Euc n) (r : ℝ), 0 < r →
+      RPI.Boundary.surfaceMeasure (perimeterSupport E) (Metric.ball c r) ≤
+        ENNReal.ofReal (A * r ^ (n - 1)) := by
+  obtain ⟨c₀, hc₀, hdensity₀⟩ := exists_halfspace_perimeter_lower_density_constant n (by omega)
+  obtain ⟨C, hC, hperim⟩ := exists_locallyMinimizing_perimeter_growth_constant n (by omega)
+  have hdensity : ∀ x ∈ perimeterSupport E \ Z, ∀ᶠ r : ℝ in 𝓝[>] (0 : ℝ),
+      ENNReal.ofReal (c₀ * r ^ (n - 1)) ≤ perimeter E (Metric.ball x r) := by
+    intro x hx
+    obtain ⟨v, hv⟩ := hhalf x hx
+    exact hdensity₀ E hmin.1 x v hv
+  have hZnull : Measure.hausdorffMeasure (n - 1 : ℕ) Z = 0 := by
+    apply hausdorffMeasure_of_dimH_lt (d := (n - 1 : ℕ))
+    apply hZd.trans_lt
+    exact_mod_cast (show n - 8 < n - 1 by omega)
+  let K : ℝ≥0∞ := (8 : ℝ≥0∞) ^ (n - 1) * (ENNReal.ofReal c₀)⁻¹ *
+    ENNReal.ofReal C * (2 : ℝ≥0∞) ^ (n - 1)
+  have hK : K ≠ ⊤ := by
+    unfold K
+    exact ENNReal.mul_ne_top (ENNReal.mul_ne_top
+      (ENNReal.mul_ne_top (by finiteness)
+        (ENNReal.inv_ne_top.mpr (ENNReal.ofReal_ne_zero_iff.mpr hc₀)))
+      ENNReal.ofReal_ne_top) (by finiteness)
+  have hraw : ∀ (c : Euc n) (r : ℝ), 0 < r →
+      Measure.hausdorffMeasure (n - 1 : ℕ) (Metric.ball c r ∩ perimeterSupport E) ≤
+        K * ENNReal.ofReal r ^ (n - 1) := by
+    intro c r hr
+    have heq : (Metric.ball c r ∩ perimeterSupport E) \ Z =
+        (perimeterSupport E \ Z) ∩ Metric.ball c r := by ext x; simp only [Set.mem_sdiff, mem_inter_iff]; tauto
+    rw [← measure_sdiff_null hZnull, heq]
+    calc
+      _ ≤ (8 : ℝ≥0∞) ^ (n - 1) * (ENNReal.ofReal c₀)⁻¹ *
+          perimeter E (Metric.ball c (2 * r)) :=
+        hausdorffMeasure_le_perimeter_of_lower_density E (perimeterSupport E \ Z) hc₀ hdensity c hr
+      _ ≤ (8 : ℝ≥0∞) ^ (n - 1) * (ENNReal.ofReal c₀)⁻¹ *
+          ENNReal.ofReal (C * (2 * r) ^ (n - 1)) := by
+        exact mul_le_mul le_rfl (hperim E hmin c (2 * r) (by positivity)) (by positivity) (by positivity)
+      _ = _ := by
+        rw [ENNReal.ofReal_mul hC.le, ENNReal.ofReal_pow (by positivity),
+          ENNReal.ofReal_mul (by norm_num : (0 : ℝ) ≤ 2)]
+        norm_num only [ENNReal.ofReal_ofNat]
+        simp only [K, mul_pow, mul_assoc]
+  let a : ℝ≥0∞ := (Measure.addHaarScalarFactor (volume : Measure (Euc (n - 1)))
+    (Measure.hausdorffMeasure ((n - 1 : ℕ) : ℝ)) : ℝ≥0∞)
+  let A : ℝ := (a * K).toReal + 1
+  have hA : 0 < A := by dsimp [A]; positivity
+  refine ⟨A, hA, ?_⟩
+  intro c r hr
+  have haK : a * K ≤ ENNReal.ofReal A := by
+    rw [← ENNReal.ofReal_toReal (ENNReal.mul_ne_top ENNReal.coe_ne_top hK)]
+    exact ENNReal.ofReal_le_ofReal (by dsimp [A]; linarith)
+  rw [RPI.Boundary.surfaceMeasure, Measure.restrict_apply measurableSet_ball,
+    Measure.euclideanHausdorffMeasure_def, Measure.smul_apply, ENNReal.smul_def]
+  change a * Measure.hausdorffMeasure (n - 1 : ℕ) (Metric.ball c r ∩ perimeterSupport E) ≤ _
+  calc
+    _ ≤ a * (K * ENNReal.ofReal r ^ (n - 1)) :=
+      mul_le_mul le_rfl (hraw c r hr) (by positivity) (by positivity)
+    _ ≤ ENNReal.ofReal A * ENNReal.ofReal r ^ (n - 1) := by
+      rw [← mul_assoc]
+      exact mul_le_mul haK le_rfl (by positivity) (by positivity)
+    _ = _ := by rw [ENNReal.ofReal_mul hA.le, ENNReal.ofReal_pow hr.le]
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPISingularHarnack
+open RPI RPI.Boundary
+
+theorem min_wholeBoundarySobolev_and_supersolution_of_minimizing_regular_geometry
+    {n : ℕ} (hn : 8 ≤ n) {E Z : Set (Euc n)}
+    (hmin : IsLocallyPerimeterMinimizing E)
+    (hZ : IsClosed Z) (hZd : dimH Z ≤ (n - 8 : ℕ))
+    (hreg : ∀ x ∈ perimeterSupport E \ Z, ∃ U : Set (Euc n), IsOpen U ∧ x ∈ U ∧
+      ∃ f : Euc n → ℝ, ContDiffOn ℝ ∞ f U ∧
+        (∀ y ∈ U, fderiv ℝ f y ≠ 0) ∧
+        (∀ y ∈ U, y ∈ perimeterSupport E ↔ f y = 0) ∧
+        (∀ y ∈ U, (f y < 0 → y ∈ E) ∧ (y ∈ E → f y ≤ 0)))
+    {W : Euc n → ℝ} {D : Euc n → Euc n}
+    (htrace : ∀ x ∈ perimeterSupport E \ Z, HasSmoothTraceJetAt (perimeterSupport E) W D x)
+    (hW0 : ∀ x ∈ perimeterSupport E \ Z, 0 ≤ W x)
+    (hsup : ∀ φ : Euc n → ℝ, ContDiff ℝ ∞ φ → HasCompactSupport φ →
+      Disjoint (tsupport φ) Z → (∀ x, 0 ≤ φ x) →
+      0 ≤ ∫ x, inner ℝ (D x) (surfaceGradient (perimeterSupport E) φ x)
+        ∂surfaceMeasure (perimeterSupport E)) :
+    WholeBoundarySobolev E (fun x ↦ min (W x) 1) (truncatedGradient W D) ∧
+      WholeBoundarySupersolution E (truncatedGradient W D) := by
+  have hhalf : ∀ x ∈ perimeterSupport E \ Z, ∃ v, HasHalfspaceBlowup E x v := by
+    intro x hx
+    obtain ⟨U, hU, hxU, f, hf, hdf, hs, he⟩ := hreg x hx
+    refine ⟨_, hasHalfspaceBlowup_gradient_filled hmin.1
+      ((hf.contDiffAt (hU.mem_nhds hxU)).differentiableAt (by norm_num))
+      ((hs x hxU).mp hx.1) (hdf x hxU) ?_⟩
+    filter_upwards [hU.mem_nhds hxU] with y hy using he y hy
+  obtain ⟨A, hA, harea⟩ := exists_minimizing_surfaceMeasure_growth_of_halfspace hn hmin hZd hhalf
+  apply min_wholeBoundarySobolev_and_supersolution_of_regular_geometry hn hmin.1 hZ hZd
+    (fun R hR ↦ (harea 0 R hR).trans_lt ENNReal.ofReal_lt_top) _ hreg htrace hW0 hsup
+  intro R hR
+  refine ⟨A, 1, hA, by norm_num, ?_⟩
+  intro c hc r hr hrr
+  exact (Measure.restrict_apply_le (Metric.ball 0 R) (Metric.ball c r)).trans (harea c r hr)
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+namespace RPI.Boundary
+open scoped Convolution
+open ContinuousLinearMap
+
+noncomputable def c1Mollifier {n : ℕ} (j : ℕ) : ContDiffBump (0 : Space n) :=
+  ⟨((j : ℝ) + 1)⁻¹ / 2, ((j : ℝ) + 1)⁻¹,
+    half_pos (by positivity), half_lt_self (by positivity)⟩
+
+noncomputable def c1Mollify {n : ℕ} (f : Space n → ℝ) (j : ℕ) : Space n → ℝ :=
+  (c1Mollifier j).normed volume ⋆[lsmul ℝ ℝ, volume] f
+
+lemma c1Mollifier_radius_tendsto {n : ℕ} :
+    Tendsto (fun j ↦ (c1Mollifier (n := n) j).rOut) atTop (𝓝 0) := by
+  simpa only [c1Mollifier, one_div] using (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ))
+
+lemma c1Mollify_contDiff {n : ℕ} {f : Space n → ℝ} (hf : Continuous f) (j : ℕ) :
+    ContDiff ℝ ∞ (c1Mollify f j) :=
+  (c1Mollifier j).hasCompactSupport_normed.contDiff_convolution_left _
+    (c1Mollifier j).contDiff_normed hf.locallyIntegrable
+
+lemma c1Mollify_hasCompactSupport {n : ℕ} {f : Space n → ℝ}
+    (hc : HasCompactSupport f) (j : ℕ) : HasCompactSupport (c1Mollify f j) :=
+  (c1Mollifier j).hasCompactSupport_normed.convolution _ hc
+
+lemma c1Mollify_fderiv {n : ℕ} {f : Space n → ℝ} (hf : ContDiff ℝ 1 f)
+    (hc : HasCompactSupport f) (j : ℕ) (x : Space n) :
+    fderiv ℝ (c1Mollify f j) x =
+      ((c1Mollifier j).normed volume ⋆[lsmul ℝ ℝ, volume] fderiv ℝ f) x := by
+  have he : (lsmul ℝ ℝ : ℝ →L[ℝ] ℝ →L[ℝ] ℝ).precompR (Space n) =
+      (lsmul ℝ ℝ : ℝ →L[ℝ] (Space n →L[ℝ] ℝ) →L[ℝ] (Space n →L[ℝ] ℝ)) := by
+    ext scalar linearMap
+    rfl
+  simpa only [c1Mollify, he] using
+    (hc.hasFDerivAt_convolution_right (lsmul ℝ ℝ)
+      (c1Mollifier j).continuous_normed.locallyIntegrable hf x).fderiv
+
+lemma c1Mollify_tendsto {n : ℕ} {f : Space n → ℝ} (hf : Continuous f) (x : Space n) :
+    Tendsto (fun j ↦ c1Mollify f j x) atTop (𝓝 (f x)) :=
+  ContDiffBump.convolution_tendsto_right_of_continuous c1Mollifier_radius_tendsto hf x
+
+lemma c1Mollify_fderiv_tendsto {n : ℕ} {f : Space n → ℝ} (hf : ContDiff ℝ 1 f)
+    (hc : HasCompactSupport f) (x : Space n) :
+    Tendsto (fun j ↦ fderiv ℝ (c1Mollify f j) x) atTop (𝓝 (fderiv ℝ f x)) := by
+  simp_rw [c1Mollify_fderiv hf hc]
+  exact ContDiffBump.convolution_tendsto_right_of_continuous
+    c1Mollifier_radius_tendsto (hf.continuous_fderiv one_ne_zero) x
+
+lemma c1Mollify_nonneg {n : ℕ} {f : Space n → ℝ} (hf : ∀ x, 0 ≤ f x)
+    (j : ℕ) (x : Space n) : 0 ≤ c1Mollify f j x := by
+  apply integral_nonneg
+  intro t
+  exact mul_nonneg ((c1Mollifier j).nonneg_normed t) (hf _)
+
+lemma c1Mollify_uniform_bound {n : ℕ} {f : Space n → ℝ}
+    (hf : ContDiff ℝ 1 f) (hc : HasCompactSupport f) :
+    ∃ M : ℝ, 0 < M ∧ ∀ j x, ‖c1Mollify f j x‖ ≤ M ∧
+      ‖fderiv ℝ (c1Mollify f j) x‖ ≤ M := by
+  obtain ⟨A, hA⟩ := hf.continuous.bounded_above_of_compact_support hc
+  obtain ⟨B, hB⟩ := (hf.continuous_fderiv one_ne_zero).bounded_above_of_compact_support
+    (hc.fderiv ℝ)
+  let M := 3 * (|A| + |B| + 1)
+  refine ⟨M, by dsimp [M]; positivity, ?_⟩
+  intro j x
+  have h1 : dist (c1Mollify f j x) (f x) ≤ 2 * |A| := by
+    apply (c1Mollifier j).dist_normed_convolution_le hf.continuous.aestronglyMeasurable
+    intro y hy
+    calc
+      dist (f y) (f x) ≤ ‖f y‖ + ‖f x‖ := dist_le_norm_add_norm _ _
+      _ ≤ 2 * |A| := by have := hA y; have := hA x; have := le_abs_self A; linarith
+  have h2 : dist (fderiv ℝ (c1Mollify f j) x) (fderiv ℝ f x) ≤ 2 * |B| := by
+    rw [c1Mollify_fderiv hf hc]
+    apply (c1Mollifier j).dist_normed_convolution_le
+      (hf.continuous_fderiv one_ne_zero).aestronglyMeasurable
+    intro y hy
+    calc
+      dist (fderiv ℝ f y) (fderiv ℝ f x) ≤ ‖fderiv ℝ f y‖ + ‖fderiv ℝ f x‖ :=
+        dist_le_norm_add_norm _ _
+      _ ≤ 2 * |B| := by have := hB y; have := hB x; have := le_abs_self B; linarith
+  constructor
+  · have := dist_triangle (c1Mollify f j x) (f x) 0
+    simp only [dist_zero_right] at this
+    have := hA x
+    dsimp [M]
+    have := le_abs_self A
+    have := abs_nonneg B
+    simp only [Real.norm_eq_abs] at *
+    linarith
+  · have := dist_triangle (fderiv ℝ (c1Mollify f j) x) (fderiv ℝ f x) 0
+    simp only [dist_zero_right] at this
+    have := hB x
+    dsimp [M]
+    have := le_abs_self B
+    have := abs_nonneg A
+    linarith
+
+lemma c1Mollify_tsupport_subset {n : ℕ} (f : Space n → ℝ) (j : ℕ) :
+    tsupport (c1Mollify f j) ⊆ Metric.cthickening (c1Mollifier (n := n) j).rOut (tsupport f) := by
+  apply closure_minimal _ Metric.isClosed_cthickening
+  intro x hx
+  obtain ⟨y, hy, z, hz, rfl⟩ := support_convolution_subset_swap (lsmul ℝ ℝ) hx
+  rw [(c1Mollifier j).support_normed_eq] at hz
+  apply Metric.mem_cthickening_of_dist_le _ y _ _ (subset_closure hy)
+  simpa only [dist_eq_norm, add_sub_cancel_left, sub_zero] using (Metric.mem_ball.mp hz).le
+
+lemma c1Mollify_eventually_tsupport_subset {n : ℕ} {f : Space n → ℝ}
+    (hc : HasCompactSupport f) {U : Set (Space n)} (hU : IsOpen U) (hs : tsupport f ⊆ U) :
+    ∀ᶠ j in atTop, tsupport (c1Mollify f j) ⊆ U := by
+  obtain ⟨r, hr, hsub⟩ := hc.exists_cthickening_subset_open hU hs
+  filter_upwards [c1Mollifier_radius_tendsto.eventually (gt_mem_nhds hr)] with j hj
+  exact (c1Mollify_tsupport_subset f j).trans ((Metric.cthickening_mono hj.le _).trans hsub)
+
+end RPI.Boundary
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/TotallyBoundedRangeOfUniformApproximation.lean b/lean/OAI/Geometry/Perimeter/Regularity/TotallyBoundedRangeOfUniformApproximation.lean
new file mode 100644
index 0000000000000000000000000000000000000000..8376683fa353a933bc72bfb7135bd13ea1d5a4b6
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/TotallyBoundedRangeOfUniformApproximation.lean
@@ -0,0 +1,546 @@
+import OAI.Geometry.Perimeter.Regularity.ExistsLocalVariationExtension
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+namespace RPISingularHarnack
+
+theorem totallyBounded_range_of_uniform_approximation
+    {α X : Type*} [PseudoMetricSpace X] (f : α → X) (a : ℕ → α → X)
+    (ha : ∀ j, TotallyBounded (range (a j)))
+    (happrox : ∀ ε : ℝ, 0 < ε → ∃ j, ∀ x, dist (f x) (a j x) < ε) :
+    TotallyBounded (range f) := by
+  apply Metric.totallyBounded_iff.mpr
+  intro ε hε
+  obtain ⟨j, hj⟩ := happrox (ε / 2) (half_pos hε)
+  obtain ⟨t, ht, hcover⟩ := Metric.totallyBounded_iff.mp (ha j) (ε / 2) (half_pos hε)
+  refine ⟨t, ht, ?_⟩
+  rintro _ ⟨x, rfl⟩
+  obtain ⟨y, hy, hxy⟩ := mem_iUnion₂.mp (hcover (mem_range_self x))
+  apply mem_iUnion₂.mpr
+  refine ⟨y, hy, ?_⟩
+  change dist (f x) y < ε
+  calc
+    dist (f x) y ≤ dist (f x) (a j x) + dist (a j x) y := dist_triangle _ _ _
+    _ < ε / 2 + ε / 2 := add_lt_add (hj x) hxy
+    _ = ε := by ring
+
+theorem totallyBounded_toLp_of_lipschitz
+    {α ι : Type*} [MetricSpace α] [CompactSpace α]
+    [MeasurableSpace α] [BorelSpace α] (μ : Measure α) [IsFiniteMeasure μ]
+    (f : ι → α →ᵇ ℝ) (L : NNReal)
+    (hf : ∀ i, LipschitzWith L (f i))
+    (hrange : ∀ i x, f i x ∈ Icc (0 : ℝ) 1) :
+    TotallyBounded (range (fun i ↦ BoundedContinuousFunction.toLp 1 μ ℝ (f i))) := by
+  have hc : IsCompact (closure (range f)) := by
+    apply BoundedContinuousFunction.arzela_ascoli (Icc (0 : ℝ) 1) isCompact_Icc
+    · rintro g x ⟨i, rfl⟩
+      exact hrange i x
+    · apply UniformEquicontinuous.equicontinuous
+      apply LipschitzWith.uniformEquicontinuous _ L
+      intro g
+      obtain ⟨i, hi⟩ := g.property
+      simpa only [← hi] using hf i
+  have hc' := hc.image (BoundedContinuousFunction.toLp 1 μ ℝ).continuous
+  apply hc'.totallyBounded.subset
+  rintro _ ⟨i, rfl⟩
+  exact ⟨f i, subset_closure (mem_range_self i), rfl⟩
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+namespace RPISingularHarnack
+
+theorem isClosed_lp_ae_mem {α : Type*} [MeasurableSpace α] (μ : Measure α)
+    (s : Set ℝ) (hs : IsClosed s) :
+    IsClosed {f : Lp ℝ 1 μ | ∀ᵐ x ∂μ, f x ∈ s} := by
+  apply IsSeqClosed.isClosed
+  intro f g hf hfg
+  obtain ⟨σ, hσ, hconv⟩ := (tendstoInMeasure_of_tendsto_Lp hfg).exists_seq_tendsto_ae
+  filter_upwards [ae_all_iff.mpr hf, hconv] with x hx hc
+  exact hs.mem_of_tendsto hc (Eventually.of_forall (fun j ↦ hx (σ j)))
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+namespace RPISingularHarnack
+open RPI.Boundary
+open scoped BoundedContinuousFunction
+
+theorem compact_subtype_volume_finite {n : ℕ} {K : Set (Euc n)} (hK : IsCompact K) :
+    IsFiniteMeasure (volume.comap (Subtype.val : K → Euc n)) := by
+  constructor
+  rw [(MeasurableEmbedding.subtype_coe hK.measurableSet).comap_apply]
+  simpa only [Set.image_univ, Subtype.range_coe] using hK.measure_lt_top
+
+noncomputable def compactIndicatorLp {n : ℕ} (K : Set (Euc n)) (hK : IsCompact K)
+    (E : Set (Euc n)) (hE : MeasurableSet E) :
+    Lp ℝ 1 (volume.comap (Subtype.val : K → Euc n)) := by
+  letI := compact_subtype_volume_finite hK
+  exact ((memLp_const (1 : ℝ)).indicator (hE.preimage measurable_subtype_coe)).toLp _
+
+theorem compactIndicatorLp_ae {n : ℕ} (K : Set (Euc n)) (hK : IsCompact K)
+    (E : Set (Euc n)) (hE : MeasurableSet E) :
+    compactIndicatorLp K hK E hE =ᵐ[volume.comap (Subtype.val : K → Euc n)]
+      fun x : K ↦ E.indicator (fun _ ↦ (1 : ℝ)) x := by
+  let := compact_subtype_volume_finite hK
+  have hm : MemLp ((Subtype.val ⁻¹' E : Set K).indicator (fun _ ↦ (1 : ℝ))) 1
+      (volume.comap (Subtype.val : K → Euc n)) :=
+    (memLp_const (1 : ℝ)).indicator (hE.preimage measurable_subtype_coe)
+  unfold compactIndicatorLp
+  filter_upwards [hm.coeFn_toLp] with x hx
+  change (hm.toLp _) x = _
+  exact hx.trans (by rfl)
+
+theorem compactIndicatorLp_edist_mollified_le {n : ℕ}
+    (K : Set (Euc n)) (hK : IsCompact K) (E : Set (Euc n))
+    (hE : MeasurableSet E) (hEK : E ⊆ K) (hP : perimeter E univ ≠ ⊤)
+    (j : ℕ) (g : K →ᵇ ℝ)
+    (hg : ∀ x : K, g x = c1Mollify (E.indicator (fun _ ↦ (1 : ℝ))) j x) :
+    letI := compact_subtype_volume_finite hK
+    edist (compactIndicatorLp K hK E hE)
+      (BoundedContinuousFunction.toLp 1
+        (volume.comap (Subtype.val : K → Euc n)) ℝ g) ≤
+      ENNReal.ofReal (c1Mollifier (n := n) j).rOut * perimeter E univ := by
+  let μ : Measure K := volume.comap (Subtype.val : K → Euc n)
+  let : IsFiniteMeasure μ := compact_subtype_volume_finite hK
+  rw [Lp.edist_def, eLpNorm_one_eq_lintegral_enorm
+    ((Lp.aestronglyMeasurable _).sub (Lp.aestronglyMeasurable _))]
+  calc
+    _ = ∫⁻ x : K, ‖E.indicator (fun _ ↦ (1 : ℝ)) x -
+        c1Mollify (E.indicator (fun _ ↦ (1 : ℝ))) j x‖ₑ ∂μ := by
+      apply lintegral_congr_ae
+      filter_upwards [compactIndicatorLp_ae K hK E hE,
+        BoundedContinuousFunction.coeFn_toLp 1 μ ℝ g] with x hx hy
+      change ‖(compactIndicatorLp K hK E hE) x -
+        (BoundedContinuousFunction.toLp 1 μ ℝ g) x‖ₑ = _
+      rw [hx, hy, hg x]
+    _ = ∫⁻ x in K, ‖E.indicator (fun _ ↦ (1 : ℝ)) x -
+        c1Mollify (E.indicator (fun _ ↦ (1 : ℝ))) j x‖ₑ :=
+      lintegral_subtype_comap (μ := volume) hK.measurableSet
+        (fun x : Euc n ↦ ‖E.indicator (fun _ ↦ (1 : ℝ)) x -
+          c1Mollify (E.indicator (fun _ ↦ (1 : ℝ))) j x‖ₑ)
+    _ ≤ ∫⁻ x, ‖E.indicator (fun _ ↦ (1 : ℝ)) x -
+        c1Mollify (E.indicator (fun _ ↦ (1 : ℝ))) j x‖ₑ :=
+      setLIntegral_le_lintegral _ _
+    _ ≤ ENNReal.ofReal (c1Mollifier (n := n) j).rOut * perimeter E univ := by
+      simpa only [enorm_sub_rev] using c1Mollify_indicator_lintegral_error_le hE
+        (hK.isBounded.subset hEK) hP j
+
+theorem bounded_perimeter_indicator_totallyBounded {n : ℕ}
+    (K : Set (Euc n)) (hK : IsCompact K) (C : ℝ≥0∞) (hC : C ≠ ⊤) :
+    TotallyBounded (range (fun E : {E : Set (Euc n) //
+      MeasurableSet E ∧ E ⊆ K ∧ perimeter E univ ≤ C} ↦
+        compactIndicatorLp K hK E E.property.1)) := by
+  let μ : Measure K := volume.comap (Subtype.val : K → Euc n)
+  let : IsFiniteMeasure μ := compact_subtype_volume_finite hK
+  let : CompactSpace K := isCompact_iff_compactSpace.mp hK
+  let ι := {E : Set (Euc n) // MeasurableSet E ∧ E ⊆ K ∧ perimeter E univ ≤ C}
+  let f : ι → Lp ℝ 1 μ := fun E ↦ compactIndicatorLp K hK E E.property.1
+  have hm (E : ι) (j : ℕ) : Continuous
+      (c1Mollify ((E : Set (Euc n)).indicator (fun _ ↦ (1 : ℝ))) j) := by
+    obtain ⟨hi, -⟩ := indicator_one_integrable_compact E.property.1
+      (hK.isBounded.subset E.property.2.1)
+    exact ((c1Mollifier j).hasCompactSupport_normed.contDiff_convolution_left (n := (⊤ : ℕ∞)) _
+      (c1Mollifier j).contDiff_normed hi.locallyIntegrable).continuous
+  let a : ℕ → ι → K →ᵇ ℝ := fun j E ↦ BoundedContinuousFunction.mkOfCompact
+    ⟨fun x ↦ c1Mollify ((E : Set (Euc n)).indicator (fun _ ↦ (1 : ℝ))) j x,
+      (hm E j).comp continuous_subtype_val⟩
+  apply totallyBounded_range_of_uniform_approximation f
+    (fun j E ↦ BoundedContinuousFunction.toLp 1 μ ℝ (a j E))
+  · intro j
+    obtain ⟨L, hL⟩ := c1Mollify_indicator_uniform_lipschitz K hK.isBounded j
+    apply totallyBounded_toLp_of_lipschitz μ (a j) L
+    · intro E
+      apply LipschitzWith.of_dist_le_mul
+      intro x y
+      exact (hL E E.property.1 E.property.2.1).dist_le_mul x y
+    · intro E x
+      exact c1Mollify_indicator_mem_Icc E.property.1 j x
+  · intro ε hε
+    have ht : Tendsto (fun j ↦ ENNReal.ofReal (c1Mollifier (n := n) j).rOut * C)
+        atTop (𝓝 0) := by
+      simpa only [ENNReal.ofReal_zero, zero_mul] using
+        ENNReal.Tendsto.mul_const
+          (ENNReal.tendsto_ofReal c1Mollifier_radius_tendsto) (Or.inr hC)
+    obtain ⟨j, hj⟩ := (ht.eventually (gt_mem_nhds (ENNReal.ofReal_pos.mpr hε))).exists
+    refine ⟨j, fun E ↦ edist_lt_ofReal.mp ?_⟩
+    refine (compactIndicatorLp_edist_mollified_le K hK E E.property.1 E.property.2.1
+      (ne_top_of_le_ne_top hC E.property.2.2) j (a j E) (fun _ ↦ rfl)).trans_lt ?_
+    exact (mul_le_mul' le_rfl E.property.2.2).trans_lt hj
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+namespace RPISingularHarnack
+open RPI.Boundary
+open scoped BoundedContinuousFunction
+
+theorem compactIndicatorLp_edist {n : ℕ} (K : Set (Euc n)) (hK : IsCompact K)
+    (E F : Set (Euc n)) (hE : MeasurableSet E) (hF : MeasurableSet F)
+    (hEK : E ⊆ K) (hFK : F ⊆ K) :
+    edist (compactIndicatorLp K hK E hE) (compactIndicatorLp K hK F hF) =
+      ∫⁻ x, ‖E.indicator (fun _ ↦ (1 : ℝ)) x - F.indicator (fun _ ↦ (1 : ℝ)) x‖ₑ := by
+  let μ : Measure K := volume.comap (Subtype.val : K → Euc n)
+  rw [Lp.edist_def, eLpNorm_one_eq_lintegral_enorm
+    ((Lp.aestronglyMeasurable _).sub (Lp.aestronglyMeasurable _))]
+  calc
+    _ = ∫⁻ x : K, ‖E.indicator (fun _ ↦ (1 : ℝ)) x -
+        F.indicator (fun _ ↦ (1 : ℝ)) x‖ₑ ∂μ := by
+      apply lintegral_congr_ae
+      filter_upwards [compactIndicatorLp_ae K hK E hE,
+        compactIndicatorLp_ae K hK F hF] with x hx hy
+      change ‖(compactIndicatorLp K hK E hE) x - (compactIndicatorLp K hK F hF) x‖ₑ = _
+      rw [hx, hy]
+    _ = ∫⁻ x in K, ‖E.indicator (fun _ ↦ (1 : ℝ)) x -
+        F.indicator (fun _ ↦ (1 : ℝ)) x‖ₑ :=
+      lintegral_subtype_comap (μ := volume) hK.measurableSet
+        (fun x : Euc n ↦ ‖E.indicator (fun _ ↦ (1 : ℝ)) x -
+          F.indicator (fun _ ↦ (1 : ℝ)) x‖ₑ)
+    _ = _ := by
+      rw [← lintegral_indicator hK.measurableSet]
+      apply lintegral_congr
+      intro x
+      by_cases hx : x ∈ K
+      · simp only [indicator_of_mem hx]
+      · have hxE : x ∉ E := fun h ↦ hx (hEK h)
+        have hxF : x ∉ F := fun h ↦ hx (hFK h)
+        simp only [indicator_of_notMem hx, indicator_of_notMem hxE,
+          indicator_of_notMem hxF, sub_self, enorm_zero]
+
+theorem exists_indicator_L1_subsequence {n : ℕ} (K : Set (Euc n)) (hK : IsCompact K)
+    (C : ℝ≥0∞) (hC : C ≠ ⊤) (E : ℕ → Set (Euc n))
+    (hE : ∀ i, MeasurableSet (E i)) (hEK : ∀ i, E i ⊆ K)
+    (hP : ∀ i, perimeter (E i) univ ≤ C) :
+    ∃ F : Set (Euc n), MeasurableSet F ∧ F ⊆ K ∧
+      ∃ σ : ℕ → ℕ, StrictMono σ ∧
+        Tendsto (fun i ↦ ∫⁻ x, ‖(E (σ i)).indicator (fun _ ↦ (1 : ℝ)) x -
+          F.indicator (fun _ ↦ (1 : ℝ)) x‖ₑ) atTop (𝓝 0) := by
+  let μ : Measure K := volume.comap (Subtype.val : K → Euc n)
+  let f : ℕ → Lp ℝ 1 μ := fun i ↦ compactIndicatorLp K hK (E i) (hE i)
+  let S : Set (Lp ℝ 1 μ) := range (fun H : {H : Set (Euc n) //
+    MeasurableSet H ∧ H ⊆ K ∧ perimeter H univ ≤ C} ↦
+      compactIndicatorLp K hK H H.property.1)
+  have hfS (i : ℕ) : f i ∈ S := ⟨⟨E i, hE i, hEK i, hP i⟩, rfl⟩
+  have hc : IsCompact (closure S) :=
+    (bounded_perimeter_indicator_totallyBounded K hK C hC).closure.isCompact_of_isClosed
+      isClosed_closure
+  obtain ⟨g, hg, σ, hσ, hconv⟩ := hc.isSeqCompact (fun i ↦ subset_closure (hfS i))
+  have hclosed : IsClosed {g : Lp ℝ 1 μ | ∀ᵐ x ∂μ, g x ∈ ({0, 1} : Set ℝ)} :=
+    isClosed_lp_ae_mem μ _ (isClosed_singleton.union isClosed_singleton)
+  have hS01 : S ⊆ {g : Lp ℝ 1 μ | ∀ᵐ x ∂μ, g x ∈ ({0, 1} : Set ℝ)} := by
+    rintro _ ⟨H, rfl⟩
+    filter_upwards [compactIndicatorLp_ae K hK H H.property.1] with x hx
+    rw [hx]
+    by_cases hxH : (x : Euc n) ∈ (H : Set (Euc n)) <;> simp [hxH]
+  have hg01 : ∀ᵐ x ∂μ, g x = 0 ∨ g x = 1 := by
+    simpa only [mem_ofPred_eq, mem_insert_iff, mem_singleton_iff] using
+      (closure_minimal hS01 hclosed hg)
+  let A : Set K := {x | g x = 1}
+  have hA : MeasurableSet A := measurableSet_eq_fun (Lp.stronglyMeasurable g).measurable measurable_const
+  let F : Set (Euc n) := Subtype.val '' A
+  have hF : MeasurableSet F :=
+    (MeasurableEmbedding.subtype_coe hK.measurableSet).measurableSet_image.mpr hA
+  have hFK : F ⊆ K := Subtype.coe_image_subset K A
+  have hFg : compactIndicatorLp K hK F hF = g := by
+    apply Lp.ext
+    filter_upwards [compactIndicatorLp_ae K hK F hF, hg01] with x hx hx01
+    rw [hx]
+    by_cases hx1 : g x = 1
+    · have hxF : (x : Euc n) ∈ F := ⟨x, hx1, rfl⟩
+      simp only [indicator_of_mem hxF, hx1]
+    · have hxF : (x : Euc n) ∉ F := by
+        rintro ⟨y, hy, hyx⟩
+        have : y = x := Subtype.ext hyx
+        subst y
+        exact hx1 hy
+      simp only [indicator_of_notMem hxF, hx01.resolve_right hx1]
+  refine ⟨F, hF, hFK, σ, hσ, ?_⟩
+  have ht : Tendsto (fun i ↦ edist (f (σ i)) g) atTop (𝓝 0) := by
+    simpa only [Function.comp_def, edist_self] using
+      hconv.edist (tendsto_const_nhds (x := g))
+  rw [← hFg] at ht
+  simpa only [f, compactIndicatorLp_edist K hK _ F _ hF (hEK _) hFK] using ht
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+namespace RPISingularHarnack
+
+theorem tendsto_setIntegral_of_indicator_L1 {n : ℕ}
+    (E : ℕ → Set (Euc n)) (F : Set (Euc n))
+    (hE : ∀ i, MeasurableSet (E i)) (hF : MeasurableSet F)
+    (hconv : Tendsto (fun i ↦ ∫⁻ x,
+      ‖(E i).indicator (fun _ ↦ (1 : ℝ)) x - F.indicator (fun _ ↦ (1 : ℝ)) x‖ₑ)
+      atTop (𝓝 0))
+    {f : Euc n → ℝ} (hf : Integrable f volume)
+    (hbound : ∃ M : ℝ, ∀ x, ‖f x‖ ≤ M) :
+    Tendsto (fun i ↦ ∫ x in E i, f x) atTop (𝓝 (∫ x in F, f x)) := by
+  obtain ⟨M, hM⟩ := hbound
+  have hM0 : 0 ≤ M := (norm_nonneg (f 0)).trans (hM 0)
+  have hib (i : ℕ) :
+      edist (∫ x in E i, f x) (∫ x in F, f x) ≤
+      ENNReal.ofReal M * ∫⁻ x,
+        ‖(E i).indicator (fun _ ↦ (1 : ℝ)) x - F.indicator (fun _ ↦ (1 : ℝ)) x‖ₑ := by
+    rw [edist_eq_enorm_sub, ← integral_indicator (hE i), ← integral_indicator hF,
+      ← integral_sub (hf.indicator (hE i)) (hf.indicator hF)]
+    apply (enorm_integral_le_lintegral_enorm _).trans
+    rw [← lintegral_const_mul' _ _]
+    · apply lintegral_mono
+      intro x
+      have he : (E i).indicator f x - F.indicator f x =
+          ((E i).indicator (fun _ ↦ (1 : ℝ)) x -
+            F.indicator (fun _ ↦ (1 : ℝ)) x) * f x := by
+        by_cases hx : x ∈ E i <;> by_cases hy : x ∈ F <;> simp [hx,hy]
+      dsimp only
+      rw [he, enorm_mul, mul_comm]
+      exact mul_le_mul' (by simpa only [ofReal_norm] using
+        ENNReal.ofReal_le_ofReal (hM x)) le_rfl
+    · exact ENNReal.ofReal_ne_top
+  apply tendsto_iff_edist_tendsto_0.mpr
+  apply tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds
+    (by simpa only [mul_zero, bot_eq_zero] using
+      ENNReal.Tendsto.const_mul hconv (Or.inr ENNReal.ofReal_ne_top))
+    (fun _ ↦ show (0 : ℝ≥0∞) ≤ _ from bot_le) hib
+
+theorem perimeter_le_of_indicator_L1 {n : ℕ}
+    (E : ℕ → Set (Euc n)) (F U : Set (Euc n)) (C : ℝ≥0∞)
+    (hE : ∀ i, MeasurableSet (E i)) (hF : MeasurableSet F)
+    (hconv : Tendsto (fun i ↦ ∫⁻ x,
+      ‖(E i).indicator (fun _ ↦ (1 : ℝ)) x - F.indicator (fun _ ↦ (1 : ℝ)) x‖ₑ)
+      atTop (𝓝 0))
+    (hP : ∀ i, perimeter (E i) U ≤ C) : perimeter F U ≤ C := by
+  rw [perimeter_eq_iSup_tests]
+  apply iSup_le
+  intro V
+  apply iSup_le
+  intro hV
+  have hd := integrable_divergence hV.1 hV.2.1
+  have hc : HasCompactSupport (divergence V) :=
+    hV.2.1.of_isClosed_subset (isClosed_tsupport _) (tsupport_divergence_subset V)
+  have ht := tendsto_setIntegral_of_indicator_L1 E F hE hF hconv hd
+    (hc.exists_bound_of_continuous (continuous_divergence hV.1))
+  apply le_of_tendsto (ENNReal.continuous_ofReal.tendsto _ |>.comp ht)
+  filter_upwards [] with i
+  have hb : ENNReal.ofReal (∫ x in E i, divergence V x) ≤ perimeter (E i) U := by
+    rw [perimeter_eq_iSup_tests]
+    exact le_iSup_of_le V (le_iSup_of_le hV le_rfl)
+  exact hb.trans (hP i)
+
+theorem exists_bounded_perimeter_L1_subsequence {n : ℕ}
+    (K : Set (Euc n)) (hK : IsCompact K) (C : ℝ≥0∞) (hC : C ≠ ⊤)
+    (E : ℕ → Set (Euc n)) (hE : ∀ i, MeasurableSet (E i))
+    (hEK : ∀ i, E i ⊆ K) (hP : ∀ i, perimeter (E i) univ ≤ C) :
+    ∃ F : Set (Euc n), MeasurableSet F ∧ F ⊆ K ∧ perimeter F univ ≤ C ∧
+      ∃ σ : ℕ → ℕ, StrictMono σ ∧
+        Tendsto (fun i ↦ ∫⁻ x, ‖(E (σ i)).indicator (fun _ ↦ (1 : ℝ)) x -
+          F.indicator (fun _ ↦ (1 : ℝ)) x‖ₑ) atTop (𝓝 0) := by
+  obtain ⟨F, hF, hFK, σ, hσ, ht⟩ := exists_indicator_L1_subsequence K hK C hC E hE hEK hP
+  exact ⟨F, hF, hFK, perimeter_le_of_indicator_L1 (E ∘ σ) F univ C
+    (fun i ↦ hE (σ i)) hF ht (fun i ↦ hP (σ i)), σ, hσ, ht⟩
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+namespace RPISingularHarnack
+
+theorem ae_tendsto_scaled_polar_oscillation {n k : ℕ}
+    (μ : Measure (Euc n)) [IsFiniteMeasure μ] (G : Euc n → Euc n)
+    (hG : Integrable G μ) {C : ℝ} (hC : 0 < C)
+    (hgrowth : ∀ x (r : ℝ), 0 < r →
+      μ (Metric.ball x r) ≤ ENNReal.ofReal (C * r ^ k)) :
+    ∀ᵐ x ∂μ, Tendsto (fun r : ℝ ↦ (r ^ k)⁻¹ *
+      ∫ y in Metric.closedBall x r, ‖G y - G x‖ ∂μ)
+      (𝓝[>] (0 : ℝ)) (𝓝 0) := by
+  filter_upwards [(Besicovitch.vitaliFamily μ).ae_tendsto_average_norm_sub hG.locallyIntegrable]
+    with x hx
+  have ht := hx.comp (Besicovitch.tendsto_filterAt μ x)
+  let A : ℝ → ℝ := fun r ↦ ⨍ y in Metric.closedBall x r, ‖G y - G x‖ ∂μ
+  have ha (r : ℝ) : 0 ≤ A r := integral_nonneg (fun _ ↦ norm_nonneg _)
+  have hb : ∀ᶠ r : ℝ in 𝓝[>] (0 : ℝ),
+      (r ^ k)⁻¹ * ∫ y in Metric.closedBall x r, ‖G y - G x‖ ∂μ ≤
+        (C * 2 ^ k) * A r := by
+    filter_upwards [self_mem_nhdsWithin] with r hr
+    change 0 < r at hr
+    have hμb : μ (Metric.closedBall x r) ≤ ENNReal.ofReal (C * (2 * r) ^ k) :=
+      (measure_mono (Metric.closedBall_subset_ball (by linarith))).trans
+        (hgrowth x (2 * r) (by positivity))
+    have hreal : μ.real (Metric.closedBall x r) ≤ C * (2 * r) ^ k := by
+      simpa only [measureReal_def, ENNReal.toReal_ofReal (by positivity : 0 ≤ C * (2 * r) ^ k)] using
+        ENNReal.toReal_mono ENNReal.ofReal_ne_top hμb
+    have hi := measure_smul_setAverage (μ := μ) (fun y ↦ ‖G y - G x‖)
+      (measure_ne_top μ (Metric.closedBall x r))
+    simp only [smul_eq_mul] at hi
+    rw [← hi]
+    calc
+      _ ≤ (r ^ k)⁻¹ * ((C * (2 * r) ^ k) * A r) := by
+        exact mul_le_mul_of_nonneg_left (mul_le_mul_of_nonneg_right hreal (ha r))
+          (by positivity)
+      _ = _ := by rw [mul_pow]; field_simp [hr.ne']
+  have hnonneg : ∀ᶠ r : ℝ in 𝓝[>] (0 : ℝ), 0 ≤ (r ^ k)⁻¹ *
+      ∫ y in Metric.closedBall x r, ‖G y - G x‖ ∂μ := by
+    filter_upwards [self_mem_nhdsWithin] with r hr
+    exact mul_nonneg (inv_nonneg.mpr (pow_nonneg (le_of_lt hr) k))
+      (integral_nonneg fun _ ↦ norm_nonneg _)
+  refine squeeze_zero' hnonneg hb ?_
+  simpa only [mul_zero, Function.comp_def, A] using ht.const_mul (C * 2 ^ k)
+
+theorem exists_differentiating_unit_polar {n : ℕ} (hn : 0 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    (U : Set (Euc n)) (hU : IsOpen U) (hP : perimeter E U ≠ ⊤) :
+    ∃ G : Euc n → Euc n,
+      StronglyMeasurable G ∧ MemLp G 2 ((perimeterMeasure E).restrict U) ∧
+      (∀ᵐ x ∂(perimeterMeasure E).restrict U, ‖G x‖ = 1) ∧
+      (∀ᵐ x ∂(perimeterMeasure E).restrict U,
+        Tendsto (fun r : ℝ ↦ (r ^ (n - 1))⁻¹ * ∫ y in Metric.closedBall x r,
+          ‖G y - G x‖ ∂(perimeterMeasure E).restrict U) (𝓝[>] (0 : ℝ)) (𝓝 0)) ∧
+      ∀ V : Euc n → Euc n, ContDiff ℝ ∞ V → HasCompactSupport V → tsupport V ⊆ U →
+        (∫ x in E, divergence V x) = ∫ x in U, inner ℝ (G x) (V x) ∂perimeterMeasure E := by
+  let μ := (perimeterMeasure E).restrict U
+  let : IsFiniteMeasure μ := ⟨by
+    rw [Measure.restrict_apply_univ, perimeterMeasure_open E hU]; exact hP.lt_top⟩
+  obtain ⟨G, hG, hGp, hGn, hrep⟩ := exists_local_unit_variation_polar E U hU hP
+  obtain ⟨C, hC, hgrowth⟩ := exists_locallyMinimizing_perimeter_growth_constant n hn
+  have hlocal : ∀ x (r : ℝ), 0 < r → μ (Metric.ball x r) ≤
+      ENNReal.ofReal (C * r ^ (n - 1)) := by
+    intro x r hr
+    exact ((Measure.restrict_le_self (μ := perimeterMeasure E) (s := U)) _).trans
+      (by simpa only [perimeterMeasure_open E Metric.isOpen_ball] using hgrowth E hmin x r hr)
+  exact ⟨G, hG, hGp, hGn,
+    ae_tendsto_scaled_polar_oscillation μ G (MemLp.integrable (by norm_num) hGp) hC hlocal,
+    hrep⟩
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+namespace RPISingularHarnack
+
+theorem ae_inner_ne_value {n : ℕ} {v : Euc n} (hv : ‖v‖ = 1) (t : ℝ) :
+    ∀ᵐ x ∂(volume : Measure (Euc n)), inner ℝ x v ≠ t := by
+  let L : Euc n →L[ℝ] ℝ := innerSL ℝ v
+  have hLv : L v = 1 := by simp [L, hv]
+  have hL : L ≠ 0 := by intro hz; simp [hz] at hLv
+  have hk : L.ker ≠ ⊤ := by
+    intro hk
+    apply hL
+    ext z
+    have hz : z ∈ L.ker := hk ▸ Submodule.mem_top
+    exact hz
+  have hn : (volume : Measure (Euc n)) {z | L z = 0} = 0 :=
+    Measure.addHaar_submodule (volume : Measure (Euc n)) L.ker hk
+  have ha : ∀ᵐ z ∂(volume : Measure (Euc n)), L z ≠ 0 := by
+    simpa only [ae_iff, not_not] using hn
+  have ht := (measurePreserving_add_right (volume : Measure (Euc n)) (-t • v)).quasiMeasurePreserving.ae ha
+  filter_upwards [ht] with x hx
+  change L (x + -t • v) ≠ 0 at hx
+  simp only [map_add, map_smul, hLv, smul_eq_mul, mul_one] at hx
+  intro heq
+  apply hx
+  change inner ℝ v x + -t = 0
+  rw [real_inner_comm, heq]
+  ring
+
+theorem ordered_indicator_halfspace {n : ℕ} (E S : Set (Euc n))
+    (hS : ∀ᵐ x ∂(volume : Measure (Euc n)), x ∈ S)
+    {v : Euc n} (hv : ‖v‖ = 1) {R : ℝ} (hR : 0 < R)
+    (horder : ∀ x ∈ S ∩ Metric.ball (0 : Euc n) R,
+      ∀ y ∈ S ∩ Metric.ball (0 : Euc n) R,
+        inner ℝ x v < inner ℝ y v → y ∈ E → x ∈ E) :
+    ∃ t ∈ Icc (-R) R, ∀ᵐ x ∂volume.restrict (Metric.ball (0 : Euc n) R),
+      x ∈ E ↔ inner ℝ x v < t := by
+  let A : Set ℝ := (fun x : Euc n ↦ inner ℝ x v) '' (E ∩ S ∩ Metric.ball 0 R)
+  have hb (x : Euc n) (hx : x ∈ Metric.ball (0 : Euc n) R) :
+      -R < inner ℝ x v ∧ inner ℝ x v < R := by
+    apply abs_lt.mp
+    apply (abs_real_inner_le_norm x v).trans_lt
+    simpa only [hv, mul_one, Metric.mem_ball, dist_zero_right] using hx
+  have hbA : ∀ a ∈ A, -R < a ∧ a < R := by
+    rintro _ ⟨x, hx, rfl⟩
+    exact hb x hx.2
+  by_cases hA : A.Nonempty
+  · have hAb : BddAbove A := ⟨R, fun a ha ↦ (hbA a ha).2.le⟩
+    let t := sSup A
+    have htl : -R ≤ t := by
+      obtain ⟨a, ha⟩ := hA
+      exact (hbA a ha).1.le.trans (le_csSup hAb ha)
+    have htu : t ≤ R := csSup_le hA (fun a ha ↦ (hbA a ha).2.le)
+    refine ⟨t, ⟨htl, htu⟩, ?_⟩
+    filter_upwards [ae_restrict_mem measurableSet_ball,
+      hS.filter_mono (ae_mono Measure.restrict_le_self),
+      (ae_inner_ne_value hv t).filter_mono (ae_mono Measure.restrict_le_self)] with x hx hxS hxne
+    constructor
+    · intro hxE
+      have ha : inner ℝ x v ∈ A := ⟨x, ⟨⟨hxE, hxS⟩, hx⟩, rfl⟩
+      exact (le_csSup hAb ha).lt_of_ne hxne
+    · intro hxt
+      obtain ⟨a, ha, hxa⟩ := exists_lt_of_lt_csSup hA hxt
+      obtain ⟨y, hy, rfl⟩ := ha
+      exact horder x ⟨hxS, hx⟩ y ⟨hy.1.2, hy.2⟩ hxa hy.1.1
+  · refine ⟨-R, ⟨le_rfl, by linarith⟩, ?_⟩
+    filter_upwards [ae_restrict_mem measurableSet_ball,
+      hS.filter_mono (ae_mono Measure.restrict_le_self)] with x hx hxS
+    have hxE : x ∉ E := by
+      intro hxE
+      exact hA ⟨inner ℝ x v, x, ⟨⟨hxE, hxS⟩, hx⟩, rfl⟩
+    simp only [hxE, false_iff]
+    exact not_lt.mpr (hb x hx).1.le
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/TsupportBlowdownSubsetBall.lean b/lean/OAI/Geometry/Perimeter/Regularity/TsupportBlowdownSubsetBall.lean
new file mode 100644
index 0000000000000000000000000000000000000000..2bdb4de7ae28334982a1df2bb5efd5d2bc122797
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/TsupportBlowdownSubsetBall.lean
@@ -0,0 +1,498 @@
+import OAI.Geometry.Perimeter.Regularity.ExistsMinimizingBoundaryAreaGrowth
+import OAI.Geometry.Perimeter.Regularity.SmoothFlatOrder
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+namespace RPISingularHarnack
+
+theorem tsupport_blowdown_subset_ball {n : ℕ} {φ : Euc n → ℝ} {R : ℝ}
+    (hs : tsupport φ ⊆ Metric.ball (0 : Euc n) R) (c : Euc n) {r : ℝ} (hr : 0 < r) :
+    tsupport (fun y ↦ φ (r⁻¹ • (y - c))) ⊆ Metric.ball c (r * R) := by
+  intro y hy
+  have hyb := hs (tsupport_comp_subset_preimage φ
+    ((continuous_id.sub continuous_const).const_smul r⁻¹) hy)
+  change r⁻¹ • (y - c) ∈ Metric.ball (0 : Euc n) R at hyb
+  simp only [Metric.mem_ball, dist_zero_right, norm_smul,
+    Real.norm_of_nonneg (inv_nonneg.mpr hr.le)] at hyb
+  change dist y c < r * R
+  rw [dist_eq_norm]
+  have he : r * (r⁻¹ * ‖y - c‖) = ‖y - c‖ := by field_simp
+  linarith [mul_lt_mul_of_pos_left hyb hr]
+
+theorem tsupport_directionalDeriv_subset {n : ℕ} (φ : Euc n → ℝ) (w : Euc n) :
+    tsupport (fun y ↦ fderiv ℝ φ y w) ⊆ tsupport φ := by
+  apply closure_minimal
+  · intro y hy
+    by_contra hn
+    exact hy (by dsimp only; rw [fderiv_of_notMem_tsupport ℝ hn]; rfl)
+  · exact isClosed_tsupport φ
+
+theorem setIntegral_eq_of_inter_eq {n : ℕ} {A B U : Set (Euc n)}
+    (hA : MeasurableSet A) (hB : MeasurableSet B) (hAB : A ∩ U = B ∩ U)
+    {f : Euc n → ℝ} (hs : tsupport f ⊆ U) :
+    (∫ x in A, f x) = ∫ x in B, f x := by
+  rw [← integral_indicator hA, ← integral_indicator hB]
+  congr 1
+  funext x
+  by_cases hx : x ∈ U
+  · have hm : x ∈ A ↔ x ∈ B := by
+      have he := Set.ext_iff.mp hAB x
+      simpa only [mem_inter_iff, hx, and_true] using he
+    by_cases ha : x ∈ A
+    · simp [ha, hm.mp ha]
+    · simp [ha, mt hm.mpr ha]
+  · have hz : f x = 0 := image_eq_zero_of_notMem_tsupport (fun hh ↦ hx (hs hh))
+    by_cases ha : x ∈ A <;> by_cases hb : x ∈ B <;> simp [ha, hb, hz]
+
+theorem polar_blowup_test_error_le {n : ℕ} (hn : 0 < n) {E U : Set (Euc n)}
+    (hE : MeasurableSet E) (μ : Measure (Euc n)) [IsFiniteMeasure μ]
+    {G : Euc n → Euc n} (hG : Integrable G μ)
+    (hrep : ∀ V : Euc n → Euc n, ContDiff ℝ ∞ V → HasCompactSupport V →
+      tsupport V ⊆ U → (∫ y in E, divergence V y) = ∫ y, inner ℝ (G y) (V y) ∂μ)
+    {φ : Euc n → ℝ} (hφ : ContDiff ℝ ∞ φ) (hcφ : HasCompactSupport φ)
+    (hsφ : tsupport φ ⊆ Metric.ball (0 : Euc n) 2)
+    (c v w : Euc n) {r : ℝ} (hr : 0 < r) (hrU : Metric.ball c (r * 2) ⊆ U)
+    {M : ℝ} (hM : 0 ≤ M) (hMφ : ∀ y, ‖φ y‖ ≤ M) :
+    |(∫ z in blownUpSet E c r, fderiv ℝ φ z w) -
+      ((r ^ (n - 1))⁻¹ * ∫ y, φ (r⁻¹ • (y - c)) * inner ℝ v w ∂μ)| ≤
+      M * ‖w‖ * ((r ^ (n - 1))⁻¹ * ∫ y in Metric.closedBall c (2 * r), ‖G y - v‖ ∂μ) := by
+  have hs := tsupport_blowdown_subset_ball hsφ c hr
+  rw [integral_fderiv_blownUpSet_polar hE μ G hrep hφ hcφ c w hr (hs.trans hrU)]
+  have hn' : n = (n - 1) + 1 := (Nat.sub_add_cancel hn).symm
+  have he : r * (r ^ n)⁻¹ = (r ^ (n - 1))⁻¹ := by
+    have hp : r ^ n = r * r ^ (n - 1) := by conv_lhs => rw [hn']; rw [pow_succ']
+    rw [hp]; field_simp
+  rw [he, ← mul_sub, abs_mul, abs_of_nonneg (inv_nonneg.mpr (pow_nonneg hr.le _))]
+  have hzero : ∀ y ∉ Metric.closedBall c (2 * r), φ (r⁻¹ • (y - c)) = 0 := by
+    intro y hy
+    have ht : y ∉ tsupport (fun z ↦ φ (r⁻¹ • (z - c))) := fun ht ↦
+      hy (Metric.ball_subset_closedBall (by simpa only [mul_comm] using hs ht))
+    exact image_eq_zero_of_notMem_tsupport (f := fun z ↦ φ (r⁻¹ • (z - c))) ht
+  have hb := polar_test_oscillation_le μ hG v w
+    (hφ.continuous.comp ((continuous_id.sub continuous_const).const_smul r⁻¹))
+    hM (fun y ↦ hMφ _) (Metric.closedBall c (2 * r)) measurableSet_closedBall hzero
+  calc
+    _ ≤ (r ^ (n - 1))⁻¹ * (M * ‖w‖ * ∫ y in Metric.closedBall c (2 * r), ‖G y - v‖ ∂μ) :=
+      mul_le_mul_of_nonneg_left hb (by positivity)
+    _ = _ := by ring
+
+theorem tendsto_doubled_polar_oscillation {n : ℕ} (μ : Measure (Euc n))
+    (G : Euc n → Euc n) (c v : Euc n)
+    (hlim : Tendsto (fun s : ℝ ↦ (s ^ (n - 1))⁻¹ * ∫ y in Metric.closedBall c s,
+      ‖G y - v‖ ∂μ) (𝓝[>] (0 : ℝ)) (𝓝 0))
+    {r : ℕ → ℝ} (hr : ∀ i, 0 < r i) (hr0 : Tendsto r atTop (𝓝 0)) :
+    Tendsto (fun i ↦ ((r i) ^ (n - 1))⁻¹ * ∫ y in Metric.closedBall c (2 * r i),
+      ‖G y - v‖ ∂μ) atTop (𝓝 0) := by
+  have ht : Tendsto (fun i ↦ 2 * r i) atTop (𝓝[>] (0 : ℝ)) := by
+    apply tendsto_nhdsWithin_iff.mpr
+    exact ⟨by simpa only [mul_zero] using hr0.const_mul 2,
+      Eventually.of_forall (fun i ↦ by change 0 < 2 * r i; exact mul_pos (by norm_num) (hr i))⟩
+  have h := (hlim.comp ht).const_mul ((2 : ℝ) ^ (n - 1))
+  simp only [mul_zero] at h
+  convert h using 1
+  funext i
+  dsimp only [Function.comp_def]
+  rw [mul_pow]
+  field_simp [(hr i).ne']
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+namespace RPISingularHarnack
+
+theorem flat_distribution_of_polar_blowup_limit {n : ℕ} (hn : 0 < n)
+    {E U : Set (Euc n)} (hE : MeasurableSet E)
+    (μ : Measure (Euc n)) [IsFiniteMeasure μ]
+    {G : Euc n → Euc n} (hG : Integrable G μ)
+    (hrep : ∀ V : Euc n → Euc n, ContDiff ℝ ∞ V → HasCompactSupport V →
+      tsupport V ⊆ U → (∫ y in E, divergence V y) = ∫ y, inner ℝ (G y) (V y) ∂μ)
+    (c v : Euc n)
+    (hosc : Tendsto (fun s : ℝ ↦ (s ^ (n - 1))⁻¹ * ∫ y in Metric.closedBall c s,
+      ‖G y - v‖ ∂μ) (𝓝[>] (0 : ℝ)) (𝓝 0))
+    {r : ℕ → ℝ} (hr : ∀ i, 0 < r i) (hr0 : Tendsto r atTop (𝓝 0))
+    (hrU : ∀ᶠ i in atTop, Metric.ball c (r i * 2) ⊆ U)
+    (A : ℕ → Set (Euc n)) (hA : ∀ i, MeasurableSet (A i))
+    (hAE : ∀ i, A i ∩ Metric.ball (0 : Euc n) 2 =
+      blownUpSet E c (r i) ∩ Metric.ball (0 : Euc n) 2)
+    (F : Set (Euc n)) (hF : MeasurableSet F)
+    (hconv : Tendsto (fun i ↦ ∫⁻ x,
+      ‖(A i).indicator (fun _ ↦ (1 : ℝ)) x - F.indicator (fun _ ↦ (1 : ℝ)) x‖ₑ)
+      atTop (𝓝 0)) :
+    (∀ φ : Euc n → ℝ, ContDiff ℝ ∞ φ → HasCompactSupport φ →
+      tsupport φ ⊆ Metric.ball (0 : Euc n) 2 →
+      ∀ w, inner ℝ w v = 0 → (∫ x in F, fderiv ℝ φ x w) = 0) ∧
+    (∀ φ : Euc n → ℝ, ContDiff ℝ ∞ φ → HasCompactSupport φ →
+      tsupport φ ⊆ Metric.ball (0 : Euc n) 2 → (∀ x, 0 ≤ φ x) →
+      0 ≤ ∫ x in F, fderiv ℝ φ x v) := by
+  let e : ℕ → ℝ := fun i ↦ ((r i) ^ (n - 1))⁻¹ *
+    ∫ y in Metric.closedBall c (2 * r i), ‖G y - v‖ ∂μ
+  have he : Tendsto e atTop (𝓝 0) :=
+    tendsto_doubled_polar_oscillation μ G c v hosc hr hr0
+  have he0 : ∀ i, 0 ≤ e i := fun i ↦ mul_nonneg
+    (inv_nonneg.mpr (pow_nonneg (hr i).le _)) (integral_nonneg fun _ ↦ norm_nonneg _)
+  have htest (φ : Euc n → ℝ) (hφ : ContDiff ℝ ∞ φ)
+      (hcφ : HasCompactSupport φ) (hsφ : tsupport φ ⊆ Metric.ball (0 : Euc n) 2)
+      (w : Euc n) :
+      Tendsto (fun i ↦ ∫ x in blownUpSet E c (r i), fderiv ℝ φ x w)
+        atTop (𝓝 (∫ x in F, fderiv ℝ φ x w)) := by
+    have hcD : HasCompactSupport (fun x ↦ fderiv ℝ φ x w) :=
+      hcφ.of_isClosed_subset (isClosed_tsupport _) (tsupport_directionalDeriv_subset φ w)
+    have hD : Continuous (fun x ↦ fderiv ℝ φ x w) :=
+      (hφ.continuous_fderiv (by simp)).clm_apply continuous_const
+    have ht := tendsto_setIntegral_of_indicator_L1 A F hA hF hconv
+      (hD.integrable_of_hasCompactSupport hcD) (hcD.exists_bound_of_continuous hD)
+    convert ht using 1
+    funext i
+    exact setIntegral_eq_of_inter_eq (measurableSet_blownUpSet hE c (r i)) (hA i)
+      (hAE i).symm ((tsupport_directionalDeriv_subset φ w).trans hsφ)
+  constructor
+  · intro φ hφ hcφ hsφ w hw
+    obtain ⟨M, hb⟩ := hcφ.exists_bound_of_continuous hφ.continuous
+    have hM : 0 ≤ M := (norm_nonneg (φ 0)).trans (hb 0)
+    have hbe : ∀ᶠ i in atTop, |∫ x in blownUpSet E c (r i), fderiv ℝ φ x w| ≤ M * ‖w‖ * e i := by
+      filter_upwards [hrU] with i hiU
+      have hh := polar_blowup_test_error_le hn hE μ hG hrep hφ hcφ hsφ c v w
+        (hr i) hiU hM hb
+      have hvw : inner ℝ v w = 0 := by rw [real_inner_comm]; exact hw
+      simpa only [hvw, mul_zero, integral_zero, sub_zero] using hh
+    have hz : Tendsto (fun i ↦ ∫ x in blownUpSet E c (r i), fderiv ℝ φ x w)
+        atTop (𝓝 0) := by
+      apply tendsto_zero_iff_norm_tendsto_zero.mpr
+      simpa only [Real.norm_eq_abs] using
+        (squeeze_zero' (Eventually.of_forall fun i ↦ abs_nonneg _) hbe
+          (by simpa only [mul_zero] using he.const_mul (M * ‖w‖)))
+    exact tendsto_nhds_unique (htest φ hφ hcφ hsφ w) hz
+  · intro φ hφ hcφ hsφ hφ0
+    obtain ⟨M, hb⟩ := hcφ.exists_bound_of_continuous hφ.continuous
+    have hM : 0 ≤ M := (norm_nonneg (φ 0)).trans (hb 0)
+    have hbtest : ∀ᶠ i in atTop, -(M * ‖v‖ * e i) ≤
+        ∫ x in blownUpSet E c (r i), fderiv ℝ φ x v := by
+      filter_upwards [hrU] with i hiU
+      have hh := polar_blowup_test_error_le hn hE μ hG hrep hφ hcφ hsφ c v v
+        (hr i) hiU hM hb
+      have hp : 0 ≤ ((r i) ^ (n - 1))⁻¹ *
+          ∫ y, φ ((r i)⁻¹ • (y - c)) * inner ℝ v v ∂μ :=
+        mul_nonneg (inv_nonneg.mpr (pow_nonneg (hr i).le _)) (integral_nonneg fun y ↦
+          mul_nonneg (hφ0 _) (real_inner_self_nonneg))
+      have hlow := (abs_le.mp hh).1
+      change -(M * ‖v‖ * e i) ≤ _ at hlow
+      linarith
+    exact le_of_tendsto_of_tendsto
+      (by simpa only [mul_zero, neg_zero] using (he.const_mul (M * ‖v‖)).neg)
+      (htest φ hφ hcφ hsφ v) hbtest
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+namespace RPISingularHarnack
+
+theorem volumeReal_blownUpSet_closedBall {n : ℕ} {E : Set (Euc n)}
+    (hE : MeasurableSet E) (c : Euc n) {r : ℝ} (hr : 0 < r) (s : ℝ) :
+    volume.real (blownUpSet E c r ∩ Metric.closedBall (0 : Euc n) s) =
+      (r ^ n)⁻¹ * volume.real (E ∩ Metric.closedBall c (r * s)) := by
+  have hi := integral_blownUpSet hE
+    ((Metric.closedBall c (r * s)).indicator (fun _ ↦ (1 : ℝ))) c hr
+  have he : (fun z ↦ (Metric.closedBall c (r * s)).indicator (fun _ ↦ (1 : ℝ)) (c + r • z)) =
+      (Metric.closedBall (0 : Euc n) s).indicator (fun _ ↦ (1 : ℝ)) := by
+    funext z
+    have hm : c + r • z ∈ Metric.closedBall c (r * s) ↔ z ∈ Metric.closedBall (0 : Euc n) s := by
+      simp only [Metric.mem_closedBall, dist_eq_norm, add_sub_cancel_left, sub_zero,
+        norm_smul, Real.norm_of_nonneg hr.le]
+      exact mul_le_mul_iff_right₀ hr
+    by_cases hz : z ∈ Metric.closedBall (0 : Euc n) s
+    · simp [hz, hm.mpr hz]
+    · simp [hz, mt hm.mp hz]
+  rw [he] at hi
+  simpa only [integral_indicator_const _ measurableSet_closedBall,
+    measureReal_restrict_apply measurableSet_closedBall, smul_eq_mul, mul_one, inter_comm] using hi
+
+theorem indicator_L1_compl {n : ℕ} (A F : Set (Euc n)) :
+    (∫⁻ x, ‖Aᶜ.indicator (fun _ ↦ (1 : ℝ)) x - Fᶜ.indicator (fun _ ↦ (1 : ℝ)) x‖ₑ) =
+      ∫⁻ x, ‖A.indicator (fun _ ↦ (1 : ℝ)) x - F.indicator (fun _ ↦ (1 : ℝ)) x‖ₑ := by
+  apply lintegral_congr
+  intro x
+  by_cases hx : x ∈ A <;> by_cases hy : x ∈ F <;> simp [hx, hy]
+
+theorem tendsto_volumeReal_inter_closedBall_of_indicator_L1 {n : ℕ}
+    (A : ℕ → Set (Euc n)) (F : Set (Euc n))
+    (hA : ∀ i, MeasurableSet (A i)) (hF : MeasurableSet F)
+    (hconv : Tendsto (fun i ↦ ∫⁻ x,
+      ‖(A i).indicator (fun _ ↦ (1 : ℝ)) x - F.indicator (fun _ ↦ (1 : ℝ)) x‖ₑ)
+      atTop (𝓝 0)) (c : Euc n) (s : ℝ) :
+    Tendsto (fun i ↦ volume.real (A i ∩ Metric.closedBall c s)) atTop
+      (𝓝 (volume.real (F ∩ Metric.closedBall c s))) := by
+  have hf : Integrable ((Metric.closedBall c s).indicator (fun _ ↦ (1 : ℝ))) volume :=
+    ((continuous_const.locallyIntegrable (μ := (volume : Measure (Euc n)))).integrableOn_isCompact
+      (isCompact_closedBall c s)).integrable_indicator measurableSet_closedBall
+  have hb : ∃ M : ℝ, ∀ x : Euc n, ‖(Metric.closedBall c s).indicator (fun _ ↦ (1 : ℝ)) x‖ ≤ M := by
+    refine ⟨1, ?_⟩
+    intro x
+    by_cases hx : x ∈ Metric.closedBall c s <;> simp [hx]
+  have ht := tendsto_setIntegral_of_indicator_L1 A F hA hF hconv hf hb
+  simpa only [integral_indicator_const _ measurableSet_closedBall,
+    measureReal_restrict_apply measurableSet_closedBall, smul_eq_mul, mul_one, inter_comm] using ht
+
+theorem two_phase_of_indicator_L1 {n : ℕ}
+    (A : ℕ → Set (Euc n)) (F : Set (Euc n))
+    (hA : ∀ i, MeasurableSet (A i)) (hF : MeasurableSet F)
+    (hconv : Tendsto (fun i ↦ ∫⁻ x,
+      ‖(A i).indicator (fun _ ↦ (1 : ℝ)) x - F.indicator (fun _ ↦ (1 : ℝ)) x‖ₑ)
+      atTop (𝓝 0))
+    {ε : ℝ} (hε : 0 < ε)
+    (hphase : ∀ i (s : ℝ), 0 < s → s ≤ 1 →
+      ε * s ^ n ≤ volume.real (A i ∩ Metric.closedBall (0 : Euc n) s) ∧
+      ε * s ^ n ≤ volume.real ((A i)ᶜ ∩ Metric.closedBall (0 : Euc n) s)) :
+    ∀ s : ℝ, 0 < s → s ≤ 1 →
+      0 < volume (F ∩ Metric.ball (0 : Euc n) s) ∧
+      0 < volume (Fᶜ ∩ Metric.ball (0 : Euc n) s) := by
+  intro s hs hs1
+  have hh : 0 < s / 2 := by positivity
+  have hh1 : s / 2 ≤ 1 := by linarith
+  have hl : ε * (s / 2) ^ n ≤ volume.real (F ∩ Metric.closedBall (0 : Euc n) (s / 2)) := by
+    apply ge_of_tendsto (tendsto_volumeReal_inter_closedBall_of_indicator_L1 A F hA hF hconv 0 (s / 2))
+    exact Eventually.of_forall fun i ↦ (hphase i _ hh hh1).1
+  have hconvC : Tendsto (fun i ↦ ∫⁻ x,
+      ‖(A i)ᶜ.indicator (fun _ ↦ (1 : ℝ)) x - Fᶜ.indicator (fun _ ↦ (1 : ℝ)) x‖ₑ)
+      atTop (𝓝 0) := by simpa only [indicator_L1_compl] using hconv
+  have hlc : ε * (s / 2) ^ n ≤ volume.real (Fᶜ ∩ Metric.closedBall (0 : Euc n) (s / 2)) := by
+    apply ge_of_tendsto (tendsto_volumeReal_inter_closedBall_of_indicator_L1
+      (fun i ↦ (A i)ᶜ) Fᶜ (fun i ↦ (hA i).compl) hF.compl hconvC 0 (s / 2))
+    exact Eventually.of_forall fun i ↦ (hphase i _ hh hh1).2
+  have hp : 0 < ε * (s / 2) ^ n := mul_pos hε (pow_pos hh n)
+  have hsub : Metric.closedBall (0 : Euc n) (s / 2) ⊆ Metric.ball (0 : Euc n) s :=
+    Metric.closedBall_subset_ball (by linarith)
+  exact ⟨(ENNReal.toReal_pos_iff.mp (hp.trans_le hl)).1.trans_le
+      (measure_mono (inter_subset_inter_right F hsub)),
+    (ENNReal.toReal_pos_iff.mp (hp.trans_le hlc)).1.trans_le
+      (measure_mono (inter_subset_inter_right Fᶜ hsub))⟩
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+namespace RPISingularHarnack
+
+theorem tendsto_rescaledHalfspaceError_of_indicator_L1 {n : ℕ}
+    (E : Set (Euc n)) (c v : Euc n) (r : ℕ → ℝ)
+    (A : ℕ → Set (Euc n)) (hA : ∀ i, MeasurableSet (A i))
+    (hAE : ∀ i, A i ∩ Metric.ball (0 : Euc n) 2 =
+      blownUpSet E c (r i) ∩ Metric.ball (0 : Euc n) 2)
+    (F : Set (Euc n)) (hF : MeasurableSet F)
+    (hconv : Tendsto (fun i ↦ ∫⁻ x,
+      ‖(A i).indicator (fun _ ↦ (1 : ℝ)) x - F.indicator (fun _ ↦ (1 : ℝ)) x‖ₑ)
+      atTop (𝓝 0))
+    (hhalf : ∀ᵐ x ∂volume.restrict (Metric.ball (0 : Euc n) 1),
+      x ∈ F ↔ inner ℝ x v < 0) :
+    Tendsto (fun i ↦ rescaledHalfspaceError E (r i) (c, v)) atTop (𝓝 0) := by
+  have ht : Tendsto (fun i ↦ ∫⁻ x in Metric.ball (0 : Euc n) 1,
+      ‖(A i).indicator (fun _ ↦ (1 : ℝ)) x - F.indicator (fun _ ↦ (1 : ℝ)) x‖ₑ)
+      atTop (𝓝 0) := by
+    exact tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds hconv
+      (fun _ ↦ bot_le) (fun i ↦ setLIntegral_le_lintegral _ _)
+  have htR := (ENNReal.tendsto_toReal (by norm_num : (0 : ℝ≥0∞) ≠ ⊤)).comp ht
+  simp only [ENNReal.toReal_zero] at htR
+  convert htR using 1
+  funext i
+  dsimp only [Function.comp_def]
+  have hm : AEStronglyMeasurable (fun x ↦
+      (A i).indicator (fun _ ↦ (1 : ℝ)) x - F.indicator (fun _ ↦ (1 : ℝ)) x)
+      (volume.restrict (Metric.ball (0 : Euc n) 1)) :=
+    ((measurable_const.indicator (hA i)).sub
+      (measurable_const.indicator hF)).aestronglyMeasurable.restrict
+  rw [← integral_norm_eq_lintegral_enorm hm]
+  unfold rescaledHalfspaceError
+  apply integral_congr_ae
+  have hzball : ∀ᵐ z ∂volume.restrict (Metric.ball (0 : Euc n) 1),
+      z ∈ Metric.ball (0 : Euc n) 1 := ae_restrict_mem measurableSet_ball
+  filter_upwards [hhalf, hzball] with z hz hz1
+  have hz2 := Metric.ball_subset_ball (by norm_num : (1 : ℝ) ≤ 2) hz1
+  have hAz : z ∈ A i ↔ c + r i • z ∈ E := by
+    have hh := Set.ext_iff.mp (hAE i) z
+    simpa only [blownUpSet, mem_inter_iff, mem_preimage, hz2, and_true] using hh
+  have hχA : (A i).indicator (fun _ ↦ (1 : ℝ)) z =
+      E.indicator (fun _ ↦ (1 : ℝ)) (c + r i • z) := by
+    by_cases ha : z ∈ A i
+    · simp [ha, hAz.mp ha]
+    · simp [ha, mt hAz.mpr ha]
+  have hχF : F.indicator (fun _ ↦ (1 : ℝ)) z =
+      {y | inner ℝ y v < 0}.indicator (fun _ ↦ (1 : ℝ)) z := by
+    by_cases hf : z ∈ F
+    · simp [hf, hz.mp hf]
+    · simp [hf, mt hz.mpr hf]
+  rw [hχA, hχF, Real.norm_eq_abs]
+
+theorem tendsto_of_every_subseq_has_subseq {X : Type*} [TopologicalSpace X]
+    (f : ℕ → X) (x : X)
+    (h : ∀ σ : ℕ → ℕ, StrictMono σ → ∃ τ : ℕ → ℕ, StrictMono τ ∧
+      Tendsto (fun i ↦ f (σ (τ i))) atTop (𝓝 x)) : Tendsto f atTop (𝓝 x) := by
+  by_contra ht
+  rw [tendsto_def] at ht
+  push Not at ht
+  obtain ⟨S, hS, hnot⟩ := ht
+  have hf : ∃ᶠ i in atTop, f i ∉ S := by
+    intro hevent
+    apply hnot
+    filter_upwards [hevent] with i hi
+    exact not_not.mp hi
+  obtain ⟨σ, hσ, hbad⟩ := extraction_of_frequently_atTop hf
+  obtain ⟨τ, hτ, hgood⟩ := h σ hσ
+  obtain ⟨i, hi⟩ := (hgood.eventually hS).exists
+  exact hbad (τ i) hi
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+namespace RPISingularHarnack
+
+theorem inter_eq_on_subdomain {X : Type*} {A B U V : Set X}
+    (hAB : A ∩ U = B ∩ U) (hVU : V ⊆ U) : A ∩ V = B ∩ V := by
+  have hu : U ∩ V = V := inter_eq_right.mpr hVU
+  calc
+    A ∩ V = (A ∩ U) ∩ V := by rw [inter_assoc, hu]
+    _ = (B ∩ U) ∩ V := by rw [hAB]
+    _ = B ∩ V := by rw [inter_assoc, hu]
+
+theorem compl_inter_eq_of_inter_eq {X : Type*} {A B U : Set X}
+    (hAB : A ∩ U = B ∩ U) : Aᶜ ∩ U = Bᶜ ∩ U := by
+  ext x
+  have hh := Set.ext_iff.mp hAB x
+  simp only [mem_inter_iff, mem_compl_iff] at hh ⊢
+  tauto
+
+theorem scaled_two_phase_bound_for_blownUpSet {n : ℕ} {E : Set (Euc n)}
+    (hE : MeasurableSet E) (c : Euc n) {ε : ℝ}
+    (hden : ∀ R : ℝ, 0 < R →
+      ε * R ^ n < volume.real (E ∩ Metric.closedBall c R) ∧
+      ε * R ^ n < volume.real (Eᶜ ∩ Metric.closedBall c R))
+    {r s : ℝ} (hr : 0 < r) (hs : 0 < s) :
+    ε * s ^ n ≤ volume.real (blownUpSet E c r ∩ Metric.closedBall (0 : Euc n) s) ∧
+    ε * s ^ n ≤ volume.real ((blownUpSet E c r)ᶜ ∩ Metric.closedBall (0 : Euc n) s) := by
+  have he : (r ^ n)⁻¹ * (ε * (r * s) ^ n) = ε * s ^ n := by
+    rw [mul_pow]
+    field_simp
+  have hd := hden (r * s) (mul_pos hr hs)
+  constructor
+  · rw [volumeReal_blownUpSet_closedBall hE c hr s, ← he]
+    exact mul_le_mul_of_nonneg_left hd.1.le (by positivity)
+  · change ε * s ^ n ≤ volume.real (blownUpSet Eᶜ c r ∩ Metric.closedBall (0 : Euc n) s)
+    rw [volumeReal_blownUpSet_closedBall hE.compl c hr s, ← he]
+    exact mul_le_mul_of_nonneg_left hd.2.le (by positivity)
+
+theorem exists_halfspace_blowup_subsequence {n : ℕ} (hn : 1 < n)
+    {E U : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E) (hU : IsOpen U)
+    (μ : Measure (Euc n)) [IsFiniteMeasure μ]
+    {G : Euc n → Euc n} (hG : Integrable G μ)
+    (hrep : ∀ V : Euc n → Euc n, ContDiff ℝ ∞ V → HasCompactSupport V →
+      tsupport V ⊆ U → (∫ y in E, divergence V y) = ∫ y, inner ℝ (G y) (V y) ∂μ)
+    (c v : Euc n) (hcU : c ∈ U) (hc : c ∈ perimeterSupport E) (hv : ‖v‖ = 1)
+    (hosc : Tendsto (fun s : ℝ ↦ (s ^ (n - 1))⁻¹ * ∫ y in Metric.closedBall c s,
+      ‖G y - v‖ ∂μ) (𝓝[>] (0 : ℝ)) (𝓝 0))
+    {r : ℕ → ℝ} (hr : ∀ i, 0 < r i) (hr0 : Tendsto r atTop (𝓝 0)) :
+    ∃ σ : ℕ → ℕ, StrictMono σ ∧
+      Tendsto (fun i ↦ rescaledHalfspaceError E (r (σ i)) (c, v)) atTop (𝓝 0) := by
+  have hn0 : 0 < n := lt_trans (by norm_num) hn
+  obtain ⟨C, hC, hbound⟩ := exists_blownUpSet_perimeter_bound hn0
+  obtain ⟨F, hF, hFK, A, hA, hAE, σ, hσ, hconv⟩ :=
+    exists_local_indicator_L1_subsequence C hC (fun i ↦ blownUpSet E c (r i))
+      (fun i ↦ measurableSet_blownUpSet hmin.1 c (r i))
+      (fun i ↦ hbound E hmin c (r i) (hr i))
+  have hnR : (1 : ℝ) < (n : ℝ) := by exact_mod_cast hn
+  obtain ⟨ε, hε, hdensity⟩ := minimizing_scaled_two_phase_volume_lower
+    (Real.HolderConjugate.conjExponent hnR)
+  have hphaseA : ∀ i (s : ℝ), 0 < s → s ≤ 1 →
+      ε * s ^ n ≤ volume.real (A i ∩ Metric.closedBall (0 : Euc n) s) ∧
+      ε * s ^ n ≤ volume.real ((A i)ᶜ ∩ Metric.closedBall (0 : Euc n) s) := by
+    intro i s hs hs1
+    have hsub : Metric.closedBall (0 : Euc n) s ⊆ Metric.ball (0 : Euc n) 2 :=
+      Metric.closedBall_subset_ball (by linarith)
+    rw [inter_eq_on_subdomain (hAE i) hsub,
+      inter_eq_on_subdomain (compl_inter_eq_of_inter_eq (hAE i)) hsub]
+    exact scaled_two_phase_bound_for_blownUpSet hmin.1 c
+      (hdensity E hmin c hc) (hr i) hs
+  have hphase : ∀ s : ℝ, 0 < s → s ≤ 1 →
+      0 < volume (F ∩ Metric.ball (0 : Euc n) s) ∧
+      0 < volume (Fᶜ ∩ Metric.ball (0 : Euc n) s) :=
+    two_phase_of_indicator_L1 (fun i ↦ A (σ i)) F (fun i ↦ hA (σ i)) hF hconv hε
+      (fun i ↦ hphaseA (σ i))
+  have hrU : ∀ᶠ i in atTop, Metric.ball c (r i * 2) ⊆ U := by
+    obtain ⟨δ, hδ, hδU⟩ := Metric.mem_nhds_iff.mp (hU.mem_nhds hcU)
+    have ht : Tendsto (fun i ↦ r i * 2) atTop (𝓝 0) := by
+      simpa only [zero_mul] using hr0.mul_const 2
+    filter_upwards [ht.eventually (gt_mem_nhds hδ)] with i hi
+    exact (Metric.ball_subset_ball hi.le).trans hδU
+  obtain ⟨hflat, hmono⟩ := flat_distribution_of_polar_blowup_limit hn0 hmin.1 μ hG hrep
+    c v hosc (fun i ↦ hr (σ i)) (hr0.comp hσ.tendsto_atTop)
+    (hσ.tendsto_atTop.eventually hrU) (fun i ↦ A (σ i)) (fun i ↦ hA (σ i))
+    (fun i ↦ hAE (σ i)) F hF hconv
+  obtain ⟨t, ht, hhalf⟩ := halfspace_of_flat_distribution hF (by norm_num : (0 : ℝ) < 2)
+    hv hflat hmono
+  have hhalf1 : ∀ᵐ x ∂volume.restrict (Metric.ball (0 : Euc n) 1),
+      x ∈ F ↔ inner ℝ x v < t := by simpa only [show (2 : ℝ) / 2 = 1 by norm_num] using hhalf
+  have ht0 := halfspace_threshold_eq_zero (by norm_num : (0 : ℝ) < 1) hv hhalf1 hphase
+  rw [ht0] at hhalf1
+  exact ⟨σ, hσ, tendsto_rescaledHalfspaceError_of_indicator_L1 E c v (fun i ↦ r (σ i))
+    (fun i ↦ A (σ i)) (fun i ↦ hA (σ i)) (fun i ↦ hAE (σ i)) F hF hconv hhalf1⟩
+
+theorem hasHalfspaceBlowup_of_differentiating_polar {n : ℕ} (hn : 1 < n)
+    {E U : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E) (hU : IsOpen U)
+    (μ : Measure (Euc n)) [IsFiniteMeasure μ]
+    {G : Euc n → Euc n} (hG : Integrable G μ)
+    (hrep : ∀ V : Euc n → Euc n, ContDiff ℝ ∞ V → HasCompactSupport V →
+      tsupport V ⊆ U → (∫ y in E, divergence V y) = ∫ y, inner ℝ (G y) (V y) ∂μ)
+    (c v : Euc n) (hcU : c ∈ U) (hc : c ∈ perimeterSupport E) (hv : ‖v‖ = 1)
+    (hosc : Tendsto (fun s : ℝ ↦ (s ^ (n - 1))⁻¹ * ∫ y in Metric.closedBall c s,
+      ‖G y - v‖ ∂μ) (𝓝[>] (0 : ℝ)) (𝓝 0)) : HasHalfspaceBlowup E c v := by
+  apply (hasHalfspaceBlowup_iff_dyadic hmin.1 c v).mpr
+  refine ⟨hv, tendsto_of_every_subseq_has_subseq
+    (fun i : ℕ ↦ rescaledHalfspaceError E ((1 / 2 : ℝ) ^ i) (c, v)) (0 : ℝ) ?_⟩
+  intro σ hσ
+  have ht : Tendsto (fun i ↦ (1 / 2 : ℝ) ^ σ i) atTop (𝓝 0) :=
+    (tendsto_pow_atTop_nhds_zero_of_lt_one
+      (by norm_num : (0 : ℝ) ≤ 1 / 2) (by norm_num : (1 / 2 : ℝ) < 1)).comp hσ.tendsto_atTop
+  exact exists_halfspace_blowup_subsequence hn hmin hU μ hG hrep c v hcU hc hv hosc
+    (r := fun i : ℕ ↦ (1 / 2 : ℝ) ^ σ i)
+    (fun i ↦ pow_pos (by norm_num : (0 : ℝ) < 1 / 2) (σ i)) ht
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/UniformDerivativeContinuity.lean b/lean/OAI/Geometry/Perimeter/Regularity/UniformDerivativeContinuity.lean
new file mode 100644
index 0000000000000000000000000000000000000000..2aa09945900dbc5a373263a19aa3b5870098a578
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/UniformDerivativeContinuity.lean
@@ -0,0 +1,96 @@
+import OAI.Geometry.Perimeter.Regularity.EllipticLocality
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+
+theorem four_point_linear_bound {t e d A B X X' Y Y' : ℝ} (ht : 0<t)
+    (hx : |X'-X-t*A|≤e*t) (hy : |Y'-Y-t*B|≤e*t)
+    (hxy : |X-Y|≤d) (hxy' : |X'-Y'|≤d) :
+    |A-B|≤2*d/t+2*e := by
+  have he : t*(A-B)= -(X'-X-t*A)+(Y'-Y-t*B)+(X'-Y')-(X-Y) := by ring
+  have hh : t*|A-B|≤e*t+e*t+d+d := by
+    calc
+      t*|A-B| = |t*(A-B)| := by rw [abs_mul,abs_of_pos ht]
+      _ = |-(X'-X-t*A)+(Y'-Y-t*B)+(X'-Y')-(X-Y)| := by rw [he]
+      _ ≤ |X'-X-t*A|+|Y'-Y-t*B|+|X'-Y'|+|X-Y| := by
+        have h1 := abs_add_le (-(X'-X-t*A)) (Y'-Y-t*B)
+        have h2 := abs_add_le (-(X'-X-t*A)+(Y'-Y-t*B)) (X'-Y')
+        have h3 := abs_add_le (-(X'-X-t*A)+(Y'-Y-t*B)+(X'-Y')) (-(X-Y))
+        simp only [abs_neg,← sub_eq_add_neg] at h1 h2 h3
+        linarith
+      _ ≤ e*t+e*t+d+d := by linarith
+  have heq : (2*d/t+2*e)*t=2*d+2*e*t := by field_simp
+  apply (le_of_mul_le_mul_right ?_ ht)
+  rw [heq]
+  nlinarith
+
+theorem continuous_derivative_of_geometric_remainder {V : Type*}
+    [NormedAddCommGroup V] [NormedSpace ℝ V]
+    {f : V → ℝ} {D : V → V →L[ℝ] ℝ} {U : Set V} (hU : IsOpen U)
+    {q C : ℝ} (hq : 0<q)
+    (hlip : ∀ x∈U, ∀ y∈U, |f x-f y|≤‖x-y‖)
+    (hrem : ∀ j x, x∈U → ∀ z, z∈U → ‖z-x‖≤q^j →
+      |f z-f x-D x (z-x)|≤C*(1/2)^j*‖z-x‖) :
+    ∀ y∈U, ContinuousAt D y := by
+  intro y hy
+  rw [Metric.continuousAt_iff]
+  intro ε hε
+  obtain ⟨r,hr,hrU⟩ := Metric.isOpen_iff.mp hU y hy
+  have hp : Tendsto (fun j : ℕ ↦ C*(1/2:ℝ)^j) atTop (𝓝 0) := by
+    simpa only [mul_zero] using
+      (tendsto_pow_atTop_nhds_zero_of_lt_one (by norm_num : (0:ℝ)≤1/2) (by norm_num)).const_mul C
+  obtain ⟨j,hj⟩ := (hp.eventually (gt_mem_nhds (by positivity : 0<ε/8))).exists
+  let t := min (q^j) (r/4)
+  have ht : 0<t := lt_min (pow_pos hq _) (by positivity)
+  have htq : t≤q^j := min_le_left _ _
+  have htr : t≤r/4 := min_le_right _ _
+  let δ := min (r/4) (ε*t/16)
+  have hδ : 0<δ := lt_min (by positivity) (by positivity)
+  refine ⟨δ,hδ,fun x hx ↦ ?_⟩
+  have hxy : ‖x-y‖<δ := by simpa only [dist_eq_norm] using hx
+  have hxyr : ‖x-y‖<r/4 := hxy.trans_le (min_le_left _ _)
+  have hxyε : ‖x-y‖<ε*t/16 := hxy.trans_le (min_le_right _ _)
+  have hxU : x∈U := hrU (by simpa only [Metric.mem_ball,dist_eq_norm] using (by linarith : ‖x-y‖<r))
+  have hnorm : ‖D x-D y‖≤ε/2 := by
+    apply ContinuousLinearMap.opNorm_le_of_unit_norm (by positivity)
+    intro u hu
+    have hyu : y+t•u∈U := by
+      apply hrU
+      simp only [Metric.mem_ball,dist_eq_norm,add_sub_cancel_left,norm_smul,Real.norm_of_nonneg ht.le,hu,mul_one]
+      linarith
+    have hxu : x+t•u∈U := by
+      apply hrU
+      rw [Metric.mem_ball,dist_eq_norm]
+      have he : x+t•u-y=(x-y)+t•u := by abel
+      rw [he]
+      have hh := norm_add_le (x-y) (t•u)
+      rw [norm_smul,Real.norm_of_nonneg ht.le,hu,mul_one] at hh
+      linarith
+    have hnx : ‖(x+t•u)-x‖=t := by simp [norm_smul,Real.norm_of_nonneg ht.le,hu]
+    have hny : ‖(y+t•u)-y‖=t := by simp [norm_smul,Real.norm_of_nonneg ht.le,hu]
+    have hxR := hrem j x hxU (x+t•u) hxu (by rw [hnx]; exact htq)
+    have hyR := hrem j y hy (y+t•u) hyu (by rw [hny]; exact htq)
+    simp only [add_sub_cancel_left,map_smul,smul_eq_mul] at hxR hyR
+    rw [norm_smul,Real.norm_of_nonneg ht.le,hu,mul_one] at hxR hyR
+    have hsh : ‖(x+t•u)-(y+t•u)‖=‖x-y‖ := by congr 1; abel
+    have hb := four_point_linear_bound ht hxR hyR (hlip x hxU y hy)
+      (by simpa only [hsh] using hlip (x+t•u) hxu (y+t•u) hyu)
+    change ‖D x u-D y u‖≤ε/2
+    rw [Real.norm_eq_abs]
+    apply hb.trans
+    have hd : 2*‖x-y‖/t<ε/8 := by
+      apply (div_lt_iff₀ ht).mpr
+      nlinarith
+    linarith
+  rw [dist_eq_norm]
+  exact hnorm.trans_lt (by linarith)
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/UniformFlatness.lean b/lean/OAI/Geometry/Perimeter/Regularity/UniformFlatness.lean
new file mode 100644
index 0000000000000000000000000000000000000000..ceb2fee74102d5eede08ad9213c6b874a15a6797
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/UniformFlatness.lean
@@ -0,0 +1,69 @@
+import OAI.Geometry.Perimeter.Regularity.IsometricMinimizer
+import OAI.Geometry.Perimeter.Regularity.FlatnessImprovement
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+theorem FullSlab.isometric_preimage {E : Set (Euc n)} {v : Euc n} {R a : ℝ}
+    (h : FullSlab E v R a) (e : Euc n ≃ₗᵢ[ℝ] Euc n) :
+    FullSlab (e ⁻¹' E) (e.symm v) R a := by
+  have hi (x : Euc n) : inner ℝ (e.symm v) x=inner ℝ v (e x) := by
+    exact (e.inner_map_map (e.symm v) x).symm.trans (by rw [e.apply_symm_apply])
+  refine ⟨?_,?_⟩
+  · intro x hx
+    rw [perimeterSupport_isometric_preimage] at hx
+    rw [hi]
+    apply h.1 (e x) ⟨hx.1,?_⟩
+    simpa only [Metric.mem_ball,dist_zero_right,e.norm_map] using hx.2
+  · have hh := e.measurePreserving.quasiMeasurePreserving.ae h.2
+    filter_upwards [hh] with x hx
+    intro hxb hxa
+    rw [hi] at hxa ⊢
+    exact hx (by simpa only [Metric.mem_ball,dist_zero_right,e.norm_map] using hxb) hxa
+
+theorem exists_uniform_normal_height_improvement (hn : 1 < n)
+    {v₀ : Euc n} (hv₀ : ‖v₀‖=1)
+    (hH : IsLocallyPerimeterMinimizing {x : Euc n | inner ℝ x v₀ < 0}) :
+    ∃ K B : ℝ, 0 < K ∧ 0 ≤ B ∧ ∀ r : ℝ, 0 < r → r < 1/8 →
+      ∃ δ : ℝ, 0 < δ ∧ ∀ (v : Euc n), ‖v‖=1 →
+        ∀ (E : Set (Euc n)), IsLocallyPerimeterMinimizing E →
+          (0:Euc n)∈perimeterSupport E → ∀ a : ℝ, 0 < a → a < δ → FullSlab E v 8 a →
+            ∃ w : Euc n, ‖w‖=1 ∧ ‖w-v‖≤2*a*B ∧
+              ∀ x∈perimeterSupport E ∩ Metric.ball 0 r, |inner ℝ w x|≤a*K*r^2 := by
+  obtain ⟨K,B,hK,hB,himp⟩ := exists_fixed_normal_improvement_constants n hn
+  refine ⟨K,B,hK,hB,fun r hr hr8 ↦ ?_⟩
+  obtain ⟨δ,hδ,hd⟩ := himp v₀ hv₀ hH r hr hr8
+  refine ⟨δ,hδ,fun v hv E hmin hzero a ha haδ hs ↦ ?_⟩
+  obtain ⟨e,he⟩ := exists_normal_rotation (hv₀.trans hv.symm)
+  have hpre := hs.isometric_preimage e
+  have hve : e.symm v=v₀ := by rw [← he,e.symm_apply_apply]
+  rw [hve] at hpre
+  have hz : (0:Euc n)∈perimeterSupport (e ⁻¹' E) := by
+    rw [perimeterSupport_isometric_preimage]
+    simpa only [mem_preimage,map_zero] using hzero
+  obtain ⟨L,hLv,hLB,hflat⟩ := hd (e ⁻¹' E)
+    (locally_minimizing_isometric_preimage e hmin) hz a ha haδ hpre
+  obtain ⟨w₀,hw₀,hwd,hwflat⟩ := exists_adjusted_unit_normal hv₀ L hLv ha.le
+  refine ⟨e w₀,by simpa only [e.norm_map] using hw₀,?_,?_⟩
+  · rw [← he,← map_sub,e.norm_map]
+    exact hwd.trans (mul_le_mul_of_nonneg_left hLB (by positivity))
+  · intro x hx
+    have hx' : e.symm x∈perimeterSupport (e ⁻¹' E) ∩ Metric.ball 0 r := by
+      rw [perimeterSupport_isometric_preimage]
+      exact ⟨by simpa only [mem_preimage,e.apply_symm_apply] using hx.1,
+        by simpa only [Metric.mem_ball,dist_zero_right,e.symm.norm_map] using hx.2⟩
+    have hh := (hwflat (e.symm x)).trans (hflat (e.symm x) hx')
+    have hi : inner ℝ (e w₀) x=inner ℝ w₀ (e.symm x) := by
+      exact e.inner_map_eq_flip w₀ x
+    simpa only [hi] using hh
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/VectorEnergy.lean b/lean/OAI/Geometry/Perimeter/Regularity/VectorEnergy.lean
new file mode 100644
index 0000000000000000000000000000000000000000..0323cd693e5caa8550af06652b96d3a23a62299c
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/VectorEnergy.lean
@@ -0,0 +1,468 @@
+import OAI.Geometry.Perimeter.Regularity.Space
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+namespace RPI
+section Energy
+variable {X V : Type*} [MeasurableSpace X] [NormedAddCommGroup V]
+variable {μ : Measure X}
+
+noncomputable def vectorEnergy (μ : Measure X) (F : X → V) : ℝ≥0∞ :=
+  ∫⁻ x, ENNReal.ofReal (‖F x‖ ^ 2) ∂μ
+
+theorem memLp_two_iff_vectorEnergy {F : X → V} (hF : AEStronglyMeasurable F μ) :
+    MemLp F 2 μ ↔ vectorEnergy μ F < ⊤ := by
+  rw [memLp_two_iff_integrable_sq_norm hF]
+  have hm : AEStronglyMeasurable (fun x ↦ ‖F x‖ ^ 2) μ := hF.norm.pow 2
+  change (AEStronglyMeasurable (fun x ↦ ‖F x‖ ^ 2) μ ∧ HasFiniteIntegral (fun x ↦ ‖F x‖ ^ 2) μ) ↔ _
+  rw [and_iff_right hm, hasFiniteIntegral_iff_ofReal (ae_of_all μ (fun x ↦ sq_nonneg _))]
+  rfl
+
+theorem eLpNorm_two_eq_vectorEnergy_rpow (F : X → V) :
+    eLpNorm' F 2 μ = vectorEnergy μ F ^ (1 / 2 : ℝ) := by
+  rw [eLpNorm'_eq_lintegral_enorm]
+  congr 1
+  apply lintegral_congr
+  intro x
+  rw [ENNReal.rpow_two, ENNReal.ofReal_pow (norm_nonneg _), ofReal_norm]
+
+theorem eLpNorm_two_eq_vectorEnergy_rpow_of_aestronglyMeasurable {F : X → V}
+    (hF : AEStronglyMeasurable F μ) :
+    eLpNorm F 2 μ = vectorEnergy μ F ^ (1 / 2 : ℝ) := by
+  rw [eLpNorm_eq_eLpNorm' (by norm_num : (2 : ℝ≥0∞) ≠ 0)
+    (by norm_num : (2 : ℝ≥0∞) ≠ ⊤) hF, ENNReal.toReal_ofNat,
+    eLpNorm_two_eq_vectorEnergy_rpow]
+
+theorem vectorEnergy_le_of_ae_tendsto {F : ℕ → X → V} {G : X → V}
+    (hF : ∀ i, AEStronglyMeasurable (F i) μ)
+    (hlim : ∀ᵐ x ∂μ, Tendsto (fun i ↦ F i x) atTop (𝓝 (G x)))
+    {b : ℕ → ℝ≥0∞} {B : ℝ≥0∞}
+    (hb : ∀ᶠ i in atTop, vectorEnergy μ (F i) ≤ b i)
+    (hbB : Tendsto b atTop (𝓝 B)) : vectorEnergy μ G ≤ B := by
+  have ht : ∀ᵐ x ∂μ, Tendsto (fun i ↦ ENNReal.ofReal (‖F i x‖ ^ 2))
+      atTop (𝓝 (ENNReal.ofReal (‖G x‖ ^ 2))) := by
+    filter_upwards [hlim] with x hx
+    exact ENNReal.tendsto_ofReal (hx.norm.pow 2)
+  calc
+    vectorEnergy μ G = ∫⁻ x, liminf (fun i ↦ ENNReal.ofReal (‖F i x‖ ^ 2)) atTop ∂μ := by
+      apply lintegral_congr_ae
+      filter_upwards [ht] with x hx using hx.liminf_eq.symm
+    _ ≤ liminf (fun i ↦ vectorEnergy μ (F i)) atTop :=
+      lintegral_liminf_le' (fun i ↦ ((hF i).norm.pow 2).aemeasurable.ennreal_ofReal)
+    _ ≤ liminf b atTop := liminf_le_liminf hb
+    _ = B := hbB.liminf_eq
+
+theorem tendsto_eLpNorm_two_of_vectorEnergy_zero {F : ℕ → X → V}
+    (hF : Tendsto (fun i ↦ vectorEnergy μ (F i)) atTop (𝓝 0)) :
+    Tendsto (fun i ↦ eLpNorm' (F i) 2 μ) atTop (𝓝 0) := by
+  simp_rw [eLpNorm_two_eq_vectorEnergy_rpow]
+  simpa only [Function.comp_def, ENNReal.zero_rpow_of_pos (by norm_num : (0 : ℝ) < 1 / 2)] using
+    (ENNReal.continuous_rpow_const (y := (1 / 2 : ℝ))).continuousAt.tendsto.comp hF
+
+theorem tendsto_eLpNorm_two_toReal_of_vectorEnergy_zero {F : ℕ → X → V}
+    (hF : Tendsto (fun i ↦ vectorEnergy μ (F i)) atTop (𝓝 0)) :
+    Tendsto (fun i ↦ (eLpNorm' (F i) 2 μ).toReal) atTop (𝓝 0) := by
+  simpa only [Function.comp_def, ENNReal.toReal_zero] using
+    (ENNReal.tendsto_toReal (by simp : (0 : ℝ≥0∞) ≠ ⊤)).comp
+      (tendsto_eLpNorm_two_of_vectorEnergy_zero hF)
+
+theorem tendsto_eLpNorm_two_of_vectorEnergy_zero_of_aestronglyMeasurable {F : ℕ → X → V}
+    (hFm : ∀ i, AEStronglyMeasurable (F i) μ)
+    (hF : Tendsto (fun i ↦ vectorEnergy μ (F i)) atTop (𝓝 0)) :
+    Tendsto (fun i ↦ eLpNorm (F i) 2 μ) atTop (𝓝 0) := by
+  simp_rw [eLpNorm_two_eq_vectorEnergy_rpow_of_aestronglyMeasurable (hFm _)]
+  simpa only [eLpNorm_two_eq_vectorEnergy_rpow] using
+    (tendsto_eLpNorm_two_of_vectorEnergy_zero hF)
+
+theorem tendsto_eLpNorm_two_toReal_of_vectorEnergy_zero_of_aestronglyMeasurable
+    {F : ℕ → X → V} (hFm : ∀ i, AEStronglyMeasurable (F i) μ)
+    (hF : Tendsto (fun i ↦ vectorEnergy μ (F i)) atTop (𝓝 0)) :
+    Tendsto (fun i ↦ (eLpNorm (F i) 2 μ).toReal) atTop (𝓝 0) := by
+  simpa only [Function.comp_def, ENNReal.toReal_zero] using
+    (ENNReal.tendsto_toReal (by simp : (0 : ℝ≥0∞) ≠ ⊤)).comp
+      (tendsto_eLpNorm_two_of_vectorEnergy_zero_of_aestronglyMeasurable hFm hF)
+
+end Energy
+
+section SobolevLimits
+variable {X V : Type*} [MeasurableSpace X]
+  [NormedAddCommGroup V] [NormedSpace ℝ V] {μ : Measure X}
+
+theorem vectorEnergy_bounded_smul {a : X → ℝ} {F : X → V} {M : ℝ}
+    (hM : 0 ≤ M) (ha : ∀ᵐ x ∂μ, ‖a x‖ ≤ M) :
+    vectorEnergy μ (fun x ↦ a x • F x) ≤ ENNReal.ofReal (M ^ 2) * vectorEnergy μ F := by
+  calc
+    _ ≤ ∫⁻ x, ENNReal.ofReal (M ^ 2) * ENNReal.ofReal (‖F x‖ ^ 2) ∂μ := by
+      apply lintegral_mono_ae
+      filter_upwards [ha] with x hx
+      rw [← ENNReal.ofReal_mul (sq_nonneg _)]
+      apply ENNReal.ofReal_le_ofReal
+      rw [norm_smul, mul_pow]
+      exact mul_le_mul_of_nonneg_right
+        ((sq_le_sq₀ (norm_nonneg _) hM).mpr hx) (sq_nonneg _)
+    _ = _ := lintegral_const_mul' _ _ ENNReal.ofReal_ne_top
+
+omit [NormedSpace ℝ V] in
+theorem vectorEnergy_add_le {F G : X → V}
+    (hF : AEStronglyMeasurable F μ) (hG : AEStronglyMeasurable G μ) :
+    vectorEnergy μ (fun x ↦ F x + G x) ≤
+      2 * vectorEnergy μ F + 2 * vectorEnergy μ G := by
+  have hFm : AEMeasurable (fun x ↦ (2 : ℝ≥0∞) * ENNReal.ofReal (‖F x‖ ^ 2)) μ :=
+    aemeasurable_const.mul (hF.norm.pow 2 |>.aemeasurable.ennreal_ofReal)
+  have hGm : AEMeasurable (fun x ↦ (2 : ℝ≥0∞) * ENNReal.ofReal (‖G x‖ ^ 2)) μ :=
+    aemeasurable_const.mul (hG.norm.pow 2 |>.aemeasurable.ennreal_ofReal)
+  calc
+    _ ≤ ∫⁻ x, 2 * ENNReal.ofReal (‖F x‖ ^ 2) + 2 * ENNReal.ofReal (‖G x‖ ^ 2) ∂μ := by
+      apply lintegral_mono
+      intro x
+      have hb : ‖F x + G x‖ ^ 2 ≤ 2 * ‖F x‖ ^ 2 + 2 * ‖G x‖ ^ 2 := by
+        have hsq := (sq_le_sq₀ (norm_nonneg _) (add_nonneg (norm_nonneg _) (norm_nonneg _))).mpr
+          (norm_add_le (F x) (G x))
+        nlinarith [sq_nonneg (‖F x‖ - ‖G x‖)]
+      calc
+        _ ≤ ENNReal.ofReal (2 * ‖F x‖ ^ 2 + 2 * ‖G x‖ ^ 2) := ENNReal.ofReal_le_ofReal hb
+        _ = _ := by
+          rw [ENNReal.ofReal_add (by positivity) (by positivity),
+            ENNReal.ofReal_mul (by norm_num : (0 : ℝ) ≤ 2),
+            ENNReal.ofReal_mul (by norm_num : (0 : ℝ) ≤ 2)]
+          norm_num
+    _ = _ := by
+      rw [lintegral_congr_ae (hFm.ae_eq_mk.fun_add hGm.ae_eq_mk),
+        lintegral_add_aux hFm.measurable_mk hGm.measurable_mk,
+        ← lintegral_congr_ae hFm.ae_eq_mk, ← lintegral_congr_ae hGm.ae_eq_mk,
+        lintegral_const_mul' _ _ (by norm_num : (2 : ℝ≥0∞) ≠ ⊤),
+        lintegral_const_mul' _ _ (by norm_num : (2 : ℝ≥0∞) ≠ ⊤)]
+      rfl
+
+theorem vectorEnergy_omitted_tendsto_zero {G : X → V} (hG : MemLp G 2 μ)
+    (χ : ℕ → X → ℝ) (hχ : ∀ i, AEStronglyMeasurable (χ i) μ)
+    (hχb : ∀ i, ∀ᵐ x ∂μ, χ i x ∈ Icc (0 : ℝ) 1)
+    (hχlim : ∀ᵐ x ∂μ, Tendsto (fun i ↦ χ i x) atTop (𝓝 1)) :
+    Tendsto (fun i ↦ vectorEnergy μ (fun x ↦ (χ i x - 1) • G x)) atTop (𝓝 0) := by
+  have hfin : vectorEnergy μ G ≠ ⊤ :=
+    (memLp_two_iff_vectorEnergy hG.aestronglyMeasurable).mp hG |>.ne
+  have hlim := tendsto_lintegral_of_dominated_convergence'
+    (fun x ↦ ENNReal.ofReal (‖G x‖ ^ 2))
+    (f := fun _ ↦ (0 : ℝ≥0∞))
+    (F := fun i x ↦ ENNReal.ofReal (‖(χ i x - 1) • G x‖ ^ 2))
+    (fun i ↦ (((hχ i).sub aestronglyMeasurable_const).smul hG.aestronglyMeasurable).norm.pow 2
+      |>.aemeasurable.ennreal_ofReal) ?_ hfin ?_
+  · simpa only [vectorEnergy, lintegral_zero] using hlim
+  · intro i
+    filter_upwards [hχb i] with x hx
+    apply ENNReal.ofReal_le_ofReal
+    apply (sq_le_sq₀ (norm_nonneg _) (norm_nonneg _)).mpr
+    rw [norm_smul, Real.norm_eq_abs, abs_of_nonpos (by linarith [hx.2])]
+    exact mul_le_of_le_one_left (norm_nonneg _) (by linarith [hx.1])
+  · filter_upwards [hχlim] with x hx
+    have hv : Tendsto (fun i ↦ (χ i x - 1) • G x) atTop (𝓝 (0 : V)) := by
+      simpa only [sub_self, zero_smul] using (hx.sub_const 1).smul_const (G x)
+    simpa only [norm_zero, zero_pow (by norm_num : (2 : ℕ) ≠ 0), ENNReal.ofReal_zero] using
+      ENNReal.tendsto_ofReal (hv.norm.pow 2)
+
+theorem vectorEnergy_bounded_smul_tendsto_zero {h : X → ℝ} {M : ℝ}
+    (hM : 0 ≤ M) (hh : ∀ᵐ x ∂μ, ‖h x‖ ≤ M)
+    (C : ℕ → X → V)
+    (hC : Tendsto (fun i ↦ vectorEnergy μ (C i)) atTop (𝓝 0)) :
+    Tendsto (fun i ↦ vectorEnergy μ (fun x ↦ h x • C i x)) atTop (𝓝 0) := by
+  have ha : Tendsto (fun i ↦ ENNReal.ofReal (M ^ 2) * vectorEnergy μ (C i)) atTop (𝓝 0) := by
+    simpa only [mul_zero] using ENNReal.Tendsto.const_mul hC (Or.inr ENNReal.ofReal_ne_top)
+  exact tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds ha
+    (fun _ ↦ zero_le) (fun _ ↦ vectorEnergy_bounded_smul hM hh)
+
+theorem memLp_of_capacity_energy_bounds {G : X → V}
+    (χ : ℕ → X → ℝ)
+    (hF : ∀ i, MemLp (fun x ↦ χ i x • G x) 2 μ)
+    (hχlim : ∀ᵐ x ∂μ, Tendsto (fun i ↦ χ i x) atTop (𝓝 1))
+    {a : ℕ → ℝ≥0∞} (ha : Tendsto a atTop (𝓝 0))
+    {A B : ℝ≥0∞} (hA : A ≠ ⊤) (hB : B < ⊤)
+    (hbound : ∀ᶠ i in atTop, vectorEnergy μ (fun x ↦ χ i x • G x) ≤ B + A * a i) :
+    MemLp G 2 μ ∧ vectorEnergy μ G ≤ B := by
+  have hlim : ∀ᵐ x ∂μ, Tendsto (fun i ↦ χ i x • G x) atTop (𝓝 (G x)) := by
+    filter_upwards [hχlim] with x hx
+    simpa only [one_smul] using hx.smul_const (G x)
+  have hG : AEStronglyMeasurable G μ :=
+    aestronglyMeasurable_of_tendsto_ae atTop (fun i ↦ (hF i).aestronglyMeasurable) hlim
+  have hbB : Tendsto (fun i ↦ B + A * a i) atTop (𝓝 B) := by
+    simpa only [mul_zero, add_zero] using
+      tendsto_const_nhds.add (ENNReal.Tendsto.const_mul ha (Or.inr hA))
+  have hE := vectorEnergy_le_of_ae_tendsto (fun i ↦ (hF i).aestronglyMeasurable) hlim hbound hbB
+  exact ⟨(memLp_two_iff_vectorEnergy hG).mpr (hE.trans_lt hB), hE⟩
+
+theorem capacity_productGradient_tendsto {G : X → V} (hG : MemLp G 2 μ)
+    (χ : ℕ → X → ℝ) (hχ : ∀ i, AEStronglyMeasurable (χ i) μ)
+    (hχb : ∀ i, ∀ᵐ x ∂μ, χ i x ∈ Icc (0 : ℝ) 1)
+    (hχlim : ∀ᵐ x ∂μ, Tendsto (fun i ↦ χ i x) atTop (𝓝 1))
+    {h : X → ℝ} (hh : AEStronglyMeasurable h μ) {M : ℝ} (hM : 0 ≤ M)
+    (hhb : ∀ᵐ x ∂μ, ‖h x‖ ≤ M)
+    (C : ℕ → X → V) (hC : ∀ i, MemLp (C i) 2 μ)
+    (hC0 : Tendsto (fun i ↦ vectorEnergy μ (C i)) atTop (𝓝 0)) :
+    Tendsto (fun i ↦ vectorEnergy μ (fun x ↦ χ i x • G x + h x • C i x - G x))
+      atTop (𝓝 0) := by
+  have ho := vectorEnergy_omitted_tendsto_zero hG χ hχ hχb hχlim
+  have he := vectorEnergy_bounded_smul_tendsto_zero hM hhb C hC0
+  have htwo : (2 : ℝ≥0∞) ≠ ⊤ := by norm_num
+  have hsum : Tendsto (fun i ↦
+      2 * vectorEnergy μ (fun x ↦ (χ i x - 1) • G x) +
+      2 * vectorEnergy μ (fun x ↦ h x • C i x)) atTop (𝓝 0) := by
+    simpa only [mul_zero, add_zero] using
+      (ENNReal.Tendsto.const_mul ho (Or.inr htwo)).add
+        (ENNReal.Tendsto.const_mul he (Or.inr htwo))
+  apply tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds hsum (fun _ ↦ zero_le)
+  intro i
+  have h₁ : AEStronglyMeasurable (fun x ↦ (χ i x - 1) • G x) μ :=
+    ((hχ i).sub aestronglyMeasurable_const).smul hG.aestronglyMeasurable
+  have h₂ : AEStronglyMeasurable (fun x ↦ h x • C i x) μ :=
+    hh.smul (hC i).aestronglyMeasurable
+  have hid : (fun x ↦ χ i x • G x + h x • C i x - G x) =
+      (fun x ↦ (χ i x - 1) • G x + h x • C i x) := by
+    funext x
+    simp only [sub_smul, one_smul]
+    abel
+  change vectorEnergy μ (fun x ↦ χ i x • G x + h x • C i x - G x) ≤
+    2 * vectorEnergy μ (fun x ↦ (χ i x - 1) • G x) +
+    2 * vectorEnergy μ (fun x ↦ h x • C i x)
+  rw [hid]
+  exact vectorEnergy_add_le h₁ h₂
+
+theorem capacity_product_pair_tendsto {h : X → ℝ} {G : X → V}
+    (hh : MemLp h 2 μ) (hG : MemLp G 2 μ)
+    (χ : ℕ → X → ℝ) (hχ : ∀ i, AEStronglyMeasurable (χ i) μ)
+    (hχb : ∀ i, ∀ᵐ x ∂μ, χ i x ∈ Icc (0 : ℝ) 1)
+    (hχlim : ∀ᵐ x ∂μ, Tendsto (fun i ↦ χ i x) atTop (𝓝 1))
+    {M : ℝ} (hM : 0 ≤ M) (hhb : ∀ᵐ x ∂μ, ‖h x‖ ≤ M)
+    (C : ℕ → X → V) (hC : ∀ i, MemLp (C i) 2 μ)
+    (hC0 : Tendsto (fun i ↦ vectorEnergy μ (C i)) atTop (𝓝 0)) :
+    ∃ (hm : ∀ i, MemLp (fun x ↦ χ i x * h x) 2 μ)
+      (hg : ∀ i, MemLp (fun x ↦ χ i x • G x + h x • C i x) 2 μ),
+      Tendsto (fun i ↦ ((hm i).toLp (fun x ↦ χ i x * h x),
+        (hg i).toLp (fun x ↦ χ i x • G x + h x • C i x))) atTop
+        (𝓝 (hh.toLp h, hG.toLp G)) := by
+  have hχnorm : ∀ i, ∀ᵐ x ∂μ, ‖χ i x‖ ≤ 1 := by
+    intro i
+    filter_upwards [hχb i] with x hx
+    simpa only [Real.norm_eq_abs, abs_of_nonneg hx.1] using hx.2
+  have hm : ∀ i, MemLp (fun x ↦ χ i x * h x) 2 μ := by
+    intro i
+    apply hh.of_le_mul (c := 1) ((hχ i).mul hh.aestronglyMeasurable)
+    filter_upwards [hχnorm i] with x hx
+    change ‖χ i x * h x‖ ≤ 1 * ‖h x‖
+    rw [norm_mul]
+    exact mul_le_mul_of_nonneg_right hx (norm_nonneg _)
+  have hg : ∀ i, MemLp (fun x ↦ χ i x • G x + h x • C i x) 2 μ := by
+    intro i
+    have hg₁ : MemLp (fun x ↦ χ i x • G x) 2 μ := by
+      apply hG.of_le_mul (c := 1) ((hχ i).smul hG.aestronglyMeasurable)
+      filter_upwards [hχnorm i] with x hx
+      change ‖χ i x • G x‖ ≤ 1 * ‖G x‖
+      rw [norm_smul]
+      exact mul_le_mul_of_nonneg_right hx (norm_nonneg _)
+    have hg₂ : MemLp (fun x ↦ h x • C i x) 2 μ := by
+      apply (hC i).of_le_mul (c := M) (hh.aestronglyMeasurable.smul (hC i).aestronglyMeasurable)
+      filter_upwards [hhb] with x hx
+      change ‖h x • C i x‖ ≤ M * ‖C i x‖
+      rw [norm_smul]
+      exact mul_le_mul_of_nonneg_right hx (norm_nonneg _)
+    exact hg₁.add hg₂
+  refine ⟨hm, hg, ?_⟩
+  apply Filter.Tendsto.prodMk_nhds
+  · apply (Lp.tendsto_Lp_iff_tendsto_eLpNorm'' _ hm h hh).mpr
+    apply tendsto_eLpNorm_two_of_vectorEnergy_zero_of_aestronglyMeasurable
+      (fun i ↦ ((hm i).sub hh).aestronglyMeasurable)
+    change Tendsto (fun i ↦ vectorEnergy μ (fun x ↦ χ i x * h x - h x)) atTop (𝓝 0)
+    have hid : (fun i ↦ vectorEnergy μ (fun x ↦ χ i x * h x - h x)) =
+        (fun i ↦ vectorEnergy μ (fun x ↦ (χ i x - 1) • h x)) := by
+      funext i
+      congr 1
+      funext x
+      change χ i x * h x - h x = (χ i x - 1) * h x
+      ring
+    rw [hid]
+    exact vectorEnergy_omitted_tendsto_zero hh χ hχ hχb hχlim
+  · apply (Lp.tendsto_Lp_iff_tendsto_eLpNorm'' _ hg G hG).mpr
+    apply tendsto_eLpNorm_two_of_vectorEnergy_zero_of_aestronglyMeasurable
+      (fun i ↦ ((hg i).sub hG).aestronglyMeasurable)
+    exact capacity_productGradient_tendsto hG χ hχ hχb hχlim
+      hh.aestronglyMeasurable hM hhb C hC hC0
+
+end SobolevLimits
+
+section Caccioppoli
+variable {X V : Type*} [MeasurableSpace X]
+  [NormedAddCommGroup V] [InnerProductSpace ℝ V] {μ : Measure X}
+
+theorem truncation_test_pointwise (A B : V) {h : ℝ} (h0 : 0 ≤ h) (h1 : h ≤ 1) :
+    inner ℝ A ((2 * (1 - h)) • B - A) ≤
+      2 * ‖B‖ ^ 2 - (1 / 2 : ℝ) * ‖A‖ ^ 2 := by
+  rw [inner_sub_right, real_inner_smul_right, real_inner_self_eq_norm_sq]
+  have hcoef : 0 ≤ 2 * (1 - h) := by positivity
+  have hi := mul_le_mul_of_nonneg_left (real_inner_le_norm A B) hcoef
+  have hn : 0 ≤ ‖A‖ * ‖B‖ := mul_nonneg (norm_nonneg _) (norm_nonneg _)
+  have hh := mul_nonneg h0 hn
+  nlinarith [sq_nonneg (‖A‖ - 2 * ‖B‖)]
+
+theorem truncation_test_energy {A B : X → V}
+    (hA : MemLp A 2 μ) (hB : MemLp B 2 μ)
+    {h : X → ℝ} (hh : AEStronglyMeasurable h μ)
+    (hb : ∀ᵐ x ∂μ, h x ∈ Icc (0 : ℝ) 1)
+    (htest : 0 ≤ ∫ x, inner ℝ (A x) ((2 * (1 - h x)) • B x - A x) ∂μ) :
+    (∫ x, ‖A x‖ ^ 2 ∂μ) ≤ 4 * ∫ x, ‖B x‖ ^ 2 ∂μ := by
+  have hC : MemLp (fun x ↦ (2 * (1 - h x)) • B x) 2 μ := by
+    apply hB.of_le_mul (c := 2) ((aestronglyMeasurable_const.mul
+      (aestronglyMeasurable_const.sub hh)).smul hB.aestronglyMeasurable)
+    filter_upwards [hb] with x hx
+    change ‖(2 * (1 - h x)) • B x‖ ≤ 2 * ‖B x‖
+    rw [norm_smul, Real.norm_eq_abs, abs_of_nonneg (by nlinarith [hx.2])]
+    exact mul_le_mul_of_nonneg_right (by linarith [hx.1]) (norm_nonneg _)
+  have hCsub : MemLp (fun x ↦ (2 * (1 - h x)) • B x - A x) 2 μ := hC.sub hA
+  have hi : Integrable (fun x ↦ inner ℝ (A x)
+      ((2 * (1 - h x)) • B x - A x)) μ := by
+    apply (L2.integrable_inner (𝕜 := ℝ) (hA.toLp A)
+      (hCsub.toLp (fun x ↦ (2 * (1 - h x)) • B x - A x))).congr
+    filter_upwards [hA.coeFn_toLp, hCsub.coeFn_toLp] with x hx hy
+    rw [hx, hy]
+  have hAs := (memLp_two_iff_integrable_sq_norm hA.aestronglyMeasurable).mp hA
+  have hBs := (memLp_two_iff_integrable_sq_norm hB.aestronglyMeasurable).mp hB
+  have hu : Integrable (fun x ↦ 2 * ‖B x‖ ^ 2 - (1 / 2 : ℝ) * ‖A x‖ ^ 2) μ :=
+    (hBs.const_mul 2).sub (hAs.const_mul (1 / 2))
+  have hle := integral_mono_ae hi hu (by
+    filter_upwards [hb] with x hx using truncation_test_pointwise (A x) (B x) hx.1 hx.2)
+  rw [integral_sub (hBs.const_mul 2) (hAs.const_mul (1 / 2)),
+    integral_const_mul, integral_const_mul] at hle
+  linarith
+
+end Caccioppoli
+end RPI
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+namespace RPI.Boundary
+
+theorem capacity_trace_mem_sobolevGraph {n : ℕ} {S Z : Set (Space n)}
+    (hS : IsClosed S) (V : Set (Space n))
+    {h : Space n → ℝ} {G : Space n → Space n}
+    (hreg : ∀ x ∈ S \ Z, HasSmoothTraceJetAt S h G x)
+    (hh : MemLp h 2 ((surfaceMeasure S).restrict V))
+    (hG : MemLp G 2 ((surfaceMeasure S).restrict V))
+    {M : ℝ} (hM : 0 ≤ M)
+    (hbound : ∀ᵐ x ∂(surfaceMeasure S).restrict V, ‖h x‖ ≤ M)
+    (χ : ℕ → Space n → ℝ)
+    (hχ : ∀ i, ContDiff ℝ ∞ (χ i))
+    (hχzero : ∀ i, ∃ U : Set (Space n), IsOpen U ∧ Z ⊆ U ∧ ∀ x ∈ U, χ i x = 0)
+    (hχb : ∀ i, ∀ᵐ x ∂(surfaceMeasure S).restrict V, χ i x ∈ Icc (0 : ℝ) 1)
+    (hχlim : ∀ᵐ x ∂(surfaceMeasure S).restrict V,
+      Tendsto (fun i ↦ χ i x) atTop (𝓝 1))
+    (hC : ∀ i, MemLp (surfaceGradient S (χ i)) 2 ((surfaceMeasure S).restrict V))
+    (hC0 : Tendsto (fun i ↦ vectorEnergy ((surfaceMeasure S).restrict V)
+      (surfaceGradient S (χ i))) atTop (𝓝 0)) :
+    (hh.toLp h, hG.toLp G) ∈ closure (smoothGradientGraph S V) := by
+  obtain ⟨hm, hg, ht⟩ := capacity_product_pair_tendsto hh hG χ
+    (fun i ↦ (hχ i).continuous.aestronglyMeasurable) hχb hχlim hM hbound
+    (fun i ↦ surfaceGradient S (χ i)) hC hC0
+  apply mem_closure_of_tendsto ht
+  apply Filter.Eventually.of_forall
+  intro i
+  obtain ⟨F, hF, hFe, hFG⟩ := exists_smooth_cutoff_trace hS hreg (χ i) (hχ i) (hχzero i)
+  have hμS : ∀ᵐ x ∂(surfaceMeasure S).restrict V, x ∈ S :=
+    ae_restrict_of_ae (ae_restrict_mem hS.measurableSet)
+  have he : F =ᵐ[(surfaceMeasure S).restrict V] (fun x ↦ χ i x * h x) := by
+    filter_upwards [hμS] with x hx using hFe hx
+  have hge : surfaceGradient S F =ᵐ[(surfaceMeasure S).restrict V]
+      (fun x ↦ χ i x • G x + h x • surfaceGradient S (χ i) x) := by
+    filter_upwards [hμS] with x hx using hFG hx
+  let hfmem := MemLp.ae_eq he.symm (hm i)
+  let hgfmem := MemLp.ae_eq hge.symm (hg i)
+  exact ⟨F, hF, hfmem, hgfmem, Prod.ext
+    ((hm i).toLp_congr hfmem he.symm) ((hg i).toLp_congr hgfmem hge.symm)⟩
+
+end RPI.Boundary
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+namespace RPI
+section EnergyClosure
+variable {X V : Type*} [MeasurableSpace X]
+  [NormedAddCommGroup V] {μ : Measure X}
+
+theorem vectorEnergy_eq_enorm_toLp_sq {F : X → V} (hF : MemLp F 2 μ) :
+    vectorEnergy μ F = ‖hF.toLp F‖ₑ ^ 2 := by
+  rw [Lp.enorm_toLp,
+    eLpNorm_two_eq_vectorEnergy_rpow_of_aestronglyMeasurable hF.aestronglyMeasurable,
+    ← ENNReal.rpow_natCast, ← ENNReal.rpow_mul]
+  norm_num
+
+theorem vectorEnergy_tendsto_of_toLp_tendsto {F : ℕ → X → V} {G : X → V}
+    (hF : ∀ i, MemLp (F i) 2 μ) (hG : MemLp G 2 μ)
+    (ht : Tendsto (fun i ↦ (hF i).toLp (F i)) atTop (𝓝 (hG.toLp G))) :
+    Tendsto (fun i ↦ vectorEnergy μ (F i)) atTop (𝓝 (vectorEnergy μ G)) := by
+  have he : (fun i ↦ vectorEnergy μ (F i)) =
+      (fun i ↦ ‖(hF i).toLp (F i)‖ₑ ^ 2) :=
+    funext (fun i ↦ vectorEnergy_eq_enorm_toLp_sq (hF i))
+  rw [he, vectorEnergy_eq_enorm_toLp_sq hG]
+  exact ENNReal.Tendsto.pow ht.enorm
+
+variable [InnerProductSpace ℝ V]
+
+theorem memLp_of_regular_caccioppoli
+    {h η : X → ℝ} {H D : X → V}
+    (hh : AEStronglyMeasurable h μ)
+    (hb : ∀ᵐ x ∂μ, h x ∈ Icc (0 : ℝ) 1)
+    (hη : MemLp η 2 μ) (hD : MemLp D 2 μ)
+    {M : ℝ} (hM : 0 ≤ M) (hηb : ∀ᵐ x ∂μ, ‖η x‖ ≤ M)
+    (χ : ℕ → X → ℝ) (hχ : ∀ i, AEStronglyMeasurable (χ i) μ)
+    (hχb : ∀ i, ∀ᵐ x ∂μ, χ i x ∈ Icc (0 : ℝ) 1)
+    (hχlim : ∀ᵐ x ∂μ, Tendsto (fun i ↦ χ i x) atTop (𝓝 1))
+    (C : ℕ → X → V) (hC : ∀ i, MemLp (C i) 2 μ)
+    (hC0 : Tendsto (fun i ↦ vectorEnergy μ (C i)) atTop (𝓝 0))
+    (hA : ∀ i, MemLp (fun x ↦ χ i x • H x) 2 μ)
+    (htest : ∀ i, 0 ≤ ∫ x, inner ℝ (χ i x • H x)
+      ((2 * (1 - h x)) • (χ i x • D x + η x • C i x) - χ i x • H x) ∂μ) :
+    MemLp H 2 μ ∧ vectorEnergy μ H ≤ 4 * vectorEnergy μ D := by
+  obtain ⟨_, hB, hBt⟩ := capacity_product_pair_tendsto hη hD χ hχ hχb hχlim
+    hM hηb C hC hC0
+  have hBE := vectorEnergy_tendsto_of_toLp_tendsto hB hD hBt.snd_nhds
+  have hbound : ∀ i, vectorEnergy μ (fun x ↦ χ i x • H x) ≤
+      4 * vectorEnergy μ (fun x ↦ χ i x • D x + η x • C i x) := by
+    intro i
+    have he := truncation_test_energy (hA i) (hB i) hh hb (htest i)
+    have he' := ENNReal.ofReal_le_ofReal he
+    rw [ENNReal.ofReal_mul (by norm_num : (0 : ℝ) ≤ 4)] at he'
+    simpa only [vectorEnergy, ofReal_integral_eq_lintegral_ofReal
+      ((memLp_two_iff_integrable_sq_norm (hA i).aestronglyMeasurable).mp (hA i))
+      (ae_of_all μ (fun x ↦ sq_nonneg ‖χ i x • H x‖)),
+      ofReal_integral_eq_lintegral_ofReal
+      ((memLp_two_iff_integrable_sq_norm (hB i).aestronglyMeasurable).mp (hB i))
+      (ae_of_all μ (fun x ↦ sq_nonneg ‖χ i x • D x + η x • C i x‖)),
+      ENNReal.ofReal_ofNat] using he'
+  have hlim : ∀ᵐ x ∂μ, Tendsto (fun i ↦ χ i x • H x) atTop (𝓝 (H x)) := by
+    filter_upwards [hχlim] with x hx
+    simpa only [one_smul] using hx.smul_const (H x)
+  have hH : AEStronglyMeasurable H μ :=
+    aestronglyMeasurable_of_tendsto_ae atTop (fun i ↦ (hA i).aestronglyMeasurable) hlim
+  have hBE' : Tendsto (fun i ↦ 4 * vectorEnergy μ
+      (fun x ↦ χ i x • D x + η x • C i x)) atTop (𝓝 (4 * vectorEnergy μ D)) :=
+    ENNReal.Tendsto.const_mul hBE (Or.inr (by norm_num))
+  have he := vectorEnergy_le_of_ae_tendsto (fun i ↦ (hA i).aestronglyMeasurable) hlim
+    (Eventually.of_forall hbound) hBE'
+  refine ⟨(memLp_two_iff_vectorEnergy hH).mpr (he.trans_lt ?_), he⟩
+  exact ENNReal.mul_lt_top (by norm_num)
+    ((memLp_two_iff_vectorEnergy hD.aestronglyMeasurable).mp hD)
+
+end EnergyClosure
+end RPI
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/VectorQuotient.lean b/lean/OAI/Geometry/Perimeter/Regularity/VectorQuotient.lean
new file mode 100644
index 0000000000000000000000000000000000000000..ae0cfe010a1af18d818e0f2c5a8d5aa90dbc3e48
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/VectorQuotient.lean
@@ -0,0 +1,83 @@
+import OAI.Geometry.Perimeter.Regularity.EllipticDifference
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+variable {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V] [CompleteSpace V]
+
+ def vectorQuotient (G : Euc n → V) (e : Euc n) (s : ℝ) (x : Euc n) : V :=
+  s⁻¹•(G (x+s•e)-G x)
+
+omit [CompleteSpace V] in
+ theorem vectorQuotient_continuous [CompleteSpace V] {G : Euc n → V} (hG : Continuous G) (e : Euc n) (s : ℝ) :
+    Continuous (vectorQuotient G e s) :=
+  continuous_const.smul ((hG.comp (continuous_id.add continuous_const)).sub hG)
+
+ theorem vectorQuotient_integral {G : Euc n → V} (hG : ContDiff ℝ 1 G)
+    (e : Euc n) {s : ℝ} (hs : s≠0) (x : Euc n) :
+    vectorQuotient G e s x = ∫ t in (0:ℝ)..1, fderiv ℝ G (x+t•(s•e)) e := by
+  have hc : Continuous (fun t : ℝ ↦ fderiv ℝ G (x+t•(s•e)) e) :=
+    ((hG.continuous_fderiv (by simp)).comp
+      (continuous_const.add (continuous_id.smul continuous_const))).clm_apply continuous_const
+  have hd (t : ℝ) : HasDerivAt (fun u : ℝ ↦ G (x+u•(s•e)))
+      (s•fderiv ℝ G (x+t•(s•e)) e) t := by
+    have hh := (hG.differentiable (by simp) _).hasFDerivAt.comp_hasDerivAt t
+      ((hasDerivAt_id t).smul_const (s•e) |>.const_add x)
+    simpa only [map_smul,one_smul,id_eq,Function.comp_def] using hh
+  have hh := intervalIntegral.integral_eq_sub_of_hasDerivAt (fun t _ ↦ hd t)
+    ((continuous_const.smul hc).intervalIntegrable 0 1)
+  simp only [intervalIntegral.integral_smul,zero_smul,add_zero,one_smul] at hh
+  rw [vectorQuotient,←hh,smul_smul,inv_mul_cancel₀ hs,one_smul]
+
+ theorem vectorQuotient_bound {G : Euc n → V} (hG : ContDiff ℝ 1 G)
+    {L : ℝ} (hL : 0 ≤ L) (hbound : ∀ x, ‖fderiv ℝ G x‖ ≤ L)
+    (e : Euc n) (s : ℝ) (x : Euc n) : ‖vectorQuotient G e s x‖ ≤ L*‖e‖ := by
+  by_cases hs : s=0
+  · simp only [vectorQuotient,hs,inv_zero,zero_smul,norm_zero]
+    positivity
+  rw [vectorQuotient_integral hG e hs]
+  have hh := intervalIntegral.norm_integral_le_of_norm_le_const (a := 0) (b := 1) (fun t _ ↦
+    ((fderiv ℝ G (x+t•(s•e))).le_opNorm e).trans
+      (mul_le_mul_of_nonneg_right (hbound _) (norm_nonneg _)))
+  simpa using hh
+
+ theorem vectorQuotient_difference {G : Euc n → V} (hG : ContDiff ℝ 1 G)
+    (e : Euc n) {s : ℝ} (hs : s≠0) (x y : Euc n) {C : ℝ}
+    (h : ∀ t∈Icc (0:ℝ) 1,
+      ‖fderiv ℝ G (x+t•(s•e))-fderiv ℝ G (y+t•(s•e))‖ ≤ C) :
+    ‖vectorQuotient G e s x-vectorQuotient G e s y‖ ≤ C*‖e‖ := by
+  have hc (x : Euc n) : Continuous (fun t : ℝ ↦ fderiv ℝ G (x+t•(s•e)) e) :=
+    ((hG.continuous_fderiv (by simp)).comp
+      (continuous_const.add (continuous_id.smul continuous_const))).clm_apply continuous_const
+  rw [vectorQuotient_integral hG e hs,vectorQuotient_integral hG e hs,
+    ←intervalIntegral.integral_sub ((hc x).intervalIntegrable 0 1) ((hc y).intervalIntegrable 0 1)]
+  have hh := intervalIntegral.norm_integral_le_of_norm_le_const (fun t ht ↦ by
+    have ht' : t∈Icc (0:ℝ) 1 := by
+      rw [Set.uIoc_of_le (by norm_num : (0:ℝ) ≤ 1)] at ht
+      exact Ioc_subset_Icc_self ht
+    change ‖(fderiv ℝ G (x+t•(s•e))-fderiv ℝ G (y+t•(s•e))) e‖ ≤ C*‖e‖
+    exact (ContinuousLinearMap.le_opNorm _ e).trans
+      (mul_le_mul_of_nonneg_right (h t ht') (norm_nonneg _)))
+  simpa using hh
+
+omit [CompleteSpace V] in
+ theorem compact_C1_fderiv_bound [CompleteSpace V] {G : Euc n → V} (hG : ContDiff ℝ 1 G)
+    (hgc : HasCompactSupport G) : ∃ L : ℝ, 0 ≤ L ∧ ∀ x, ‖fderiv ℝ G x‖ ≤ L := by
+  have hc := hG.continuous_fderiv (by simp)
+  have hcc := hgc.fderiv ℝ
+  obtain ⟨L,hL⟩ := hcc.exists_bound_of_continuousOn hc.continuousOn
+  refine ⟨max 0 L,le_max_left _ _,fun x ↦ ?_⟩
+  by_cases hx : x∈tsupport (fderiv ℝ G)
+  · exact (hL x hx).trans (le_max_right _ _)
+  · rw [image_eq_zero_of_notMem_tsupport hx,norm_zero]
+    exact le_max_left _ _
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/WeakDerivative.lean b/lean/OAI/Geometry/Perimeter/Regularity/WeakDerivative.lean
new file mode 100644
index 0000000000000000000000000000000000000000..a27c083e02d4860ba122a32b52ffefab139588c1
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/WeakDerivative.lean
@@ -0,0 +1,83 @@
+import OAI.Geometry.Perimeter.Regularity.GeoComposition
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+variable {n : ℕ}
+
+ theorem scalarDirection_swap {φ : Euc n → ℝ} (hφ : ContDiff ℝ ∞ φ)
+    (a b x : Euc n) : fderiv ℝ (scalarDirection φ a) x b =
+      fderiv ℝ (scalarDirection φ b) x a := by
+  rw [scalarDirection_fderiv hφ,scalarDirection_fderiv hφ]
+  exact (hφ.contDiffAt.isSymmSndFDerivAt (by simp only [minSmoothness_of_isRCLikeNormedField]; exact WithTop.coe_le_coe.mpr le_top)) b a
+
+ theorem flux_direction_parts {V : Euc n → Euc n} (hV : ContDiff ℝ 1 V)
+    {U : Set (Euc n)} (φ : SmoothLocalTest U) (e : Euc n) :
+    (∫ x, inner ℝ (fderiv ℝ V x e) (gradient φ.val x)) =
+      -(∫ x, inner ℝ (V x) (gradient (scalarDirection φ.val e) x)) := by
+  let b := EuclideanSpace.basisFun (Fin n) ℝ
+  let u : Fin n → Euc n → ℝ := fun i x ↦ inner ℝ (b i) (V x)
+  have hu (i : Fin n) : ContDiff ℝ 1 (u i) := (innerSL ℝ (b i)).contDiff.comp hV
+  have hdu (i : Fin n) (x : Euc n) : fderiv ℝ (u i) x e =
+      inner ℝ (b i) (fderiv ℝ V x e) := by
+    exact congrArg (fun L : Euc n →L[ℝ] ℝ ↦ L e)
+      (((innerSL ℝ (b i)).hasFDerivAt.comp x (hV.differentiable (by simp) x).hasFDerivAt).fderiv)
+  have hv (i : Fin n) : ContDiff ℝ 1 (scalarDirection φ.val (b i)) :=
+    (scalarDirection_contDiff φ.property.1 (b i)).of_le (by simp)
+  have hvc (i : Fin n) : HasCompactSupport (scalarDirection φ.val (b i)) :=
+    φ.property.2.1.of_isClosed_subset (isClosed_tsupport _) (scalarDirection_tsupport _ _)
+  have hp (i : Fin n) := scalar_compact_integration_parts_C1 (hu i) (hv i) (hvc i) e
+  have hsum (x : Euc n) : inner ℝ (fderiv ℝ V x e) (gradient φ.val x) =
+      ∑ i, fderiv ℝ (u i) x e * scalarDirection φ.val (b i) x := by
+    simp only [hdu,scalarDirection,← inner_gradient_left]
+    simpa only [real_inner_comm] using (b.sum_inner_mul_inner (fderiv ℝ V x e) (gradient φ.val x)).symm
+  have hsum' (x : Euc n) : inner ℝ (V x) (gradient (scalarDirection φ.val e) x) =
+      ∑ i, u i x * fderiv ℝ (scalarDirection φ.val (b i)) x e := by
+    simp only [scalarDirection_swap φ.property.1 (b _) e,u,← inner_gradient_left]
+    simpa only [real_inner_comm] using (b.sum_inner_mul_inner (V x)
+      (gradient (scalarDirection φ.val e) x)).symm
+  simp_rw [hsum,hsum']
+  rw [integral_finsetSum _ (fun i _ ↦ (hp i).2.1),
+    integral_finsetSum _ (fun i _ ↦ (hp i).1)]
+  simp only [(fun i ↦ (hp i).2.2),Finset.sum_neg_distrib,neg_neg]
+
+ theorem WeakFluxEquation.derivative {V : Euc n → Euc n} (hV : ContDiff ℝ 1 V)
+    {U : Set (Euc n)} (h : WeakFluxEquation V U) (e : Euc n) :
+    WeakFluxEquation (fun x ↦ fderiv ℝ V x e) U := by
+  intro φ
+  rw [flux_direction_parts hV φ e]
+  exact neg_eq_zero.mpr (h (directionLocalTest φ e))
+
+ theorem gradient_direction_swap {f : Euc n → ℝ} (hf : ContDiff ℝ 2 f)
+    (e x : Euc n) : gradient (fun y ↦ fderiv ℝ f y e) x =
+      fderiv ℝ (gradient f) x e := by
+  have hdf : ContDiff ℝ 1 (fderiv ℝ f) := hf.fderiv_right (by norm_num)
+  have hg : ContDiff ℝ 1 (gradient f) := (InnerProductSpace.toDual ℝ (Euc n)).symm.toContinuousLinearMap.contDiff.comp hdf
+  apply ext_inner_right ℝ
+  intro a
+  rw [inner_gradient_left]
+  have hs := (hf.contDiffAt (x := x)).isSymmSndFDerivAt (by simp [minSmoothness_of_isRCLikeNormedField]) a e
+  have hd := congrArg (fun L : Euc n →L[ℝ] ℝ ↦ L a)
+    (((ContinuousLinearMap.apply ℝ ℝ e).hasFDerivAt.comp x
+      (hdf.differentiable (by simp) x).hasFDerivAt).fderiv)
+  have hga := congrArg (fun L : Euc n →L[ℝ] ℝ ↦ L e)
+    (((innerSL ℝ a).hasFDerivAt.comp x (hg.differentiable (by simp) x).hasFDerivAt).fderiv)
+  have heq : (fun y ↦ inner ℝ a (gradient f y)) = fun y ↦ fderiv ℝ f y a := by
+    funext y
+    rw [real_inner_comm,inner_gradient_left]
+  change fderiv ℝ (fun y ↦ inner ℝ a (gradient f y)) x e = inner ℝ a (fderiv ℝ (gradient f) x e) at hga
+  rw [heq] at hga
+  have hda := congrArg (fun L : Euc n →L[ℝ] ℝ ↦ L e)
+    (((ContinuousLinearMap.apply ℝ ℝ a).hasFDerivAt.comp x
+      (hdf.differentiable (by simp) x).hasFDerivAt).fderiv)
+  simpa only [ContinuousLinearMap.comp_apply,ContinuousLinearMap.apply_apply,innerSL_apply_apply,
+    real_inner_comm,Function.comp_def] using hd.trans (hs.trans (hda.symm.trans hga))
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/WeakHarmonicRadial.lean b/lean/OAI/Geometry/Perimeter/Regularity/WeakHarmonicRadial.lean
new file mode 100644
index 0000000000000000000000000000000000000000..827c8beb455710a05a1849261c32c4fd18e77e39
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/WeakHarmonicRadial.lean
@@ -0,0 +1,154 @@
+import OAI.Geometry.Perimeter.Regularity.HeightMeanValue
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+open RPI.Boundary ContinuousLinearMap
+variable {n : ℕ}
+
+def WeakHarmonicOn (u : Euc n → ℝ) (U : Set (Euc n)) : Prop :=
+  ∀ φ : Euc n → ℝ, ContDiff ℝ ∞ φ → HasCompactSupport φ → tsupport φ ⊆ U →
+    (∫ x, u x * divergence (gradient φ) x) = 0
+
+theorem divergence_displacement (c x : Euc n) :
+    divergence (fun y ↦ y-c) x = n := by
+  have hd : fderiv ℝ (fun y : Euc n ↦ y-c) x = ContinuousLinearMap.id ℝ (Euc n) :=
+    ((hasFDerivAt_id x).sub_const c).fderiv
+  simp [divergence,hd]
+
+theorem gradient_radialPrimitive (c : Euc n) (t : ℝ)
+    {Ψ ψ : ℝ → ℝ} (hΨ : ContDiff ℝ ∞ Ψ) (hdΨ : ∀ s, deriv Ψ s = ψ s)
+    (x : Euc n) :
+    gradient (fun y ↦ Ψ (t^2*‖y-c‖^2)) x =
+      (2*t^2*ψ (t^2*‖x-c‖^2)) • (x-c) := by
+  have hi := (((hasFDerivAt_id (𝕜 := ℝ) x).sub_const c).norm_sq).const_mul (t^2)
+  have hh := ((hΨ.differentiable (by simp)) _).hasDerivAt.comp_hasFDerivAt x hi
+  rw [hdΨ] at hh
+  change HasFDerivAt (fun y : Euc n ↦ Ψ (t^2*‖y-c‖^2)) _ x at hh
+  apply (InnerProductSpace.toDual ℝ (Euc n)).injective
+  rw [toDual_gradient,hh.fderiv]
+  ext z
+  simp only [smul_apply,ContinuousLinearMap.comp_apply,
+    ContinuousLinearMap.id_apply,id_eq,innerSL_apply_apply,map_smul,
+    InnerProductSpace.toDual_apply_apply,smul_eq_mul]
+  ring
+
+theorem laplacian_radialPrimitive (c : Euc n) (t : ℝ)
+    {Ψ ψ : ℝ → ℝ} (hΨ : ContDiff ℝ ∞ Ψ) (hψ : ContDiff ℝ ∞ ψ)
+    (hdΨ : ∀ s, deriv Ψ s = ψ s) (x : Euc n) :
+    divergence (gradient (fun y ↦ Ψ (t^2*‖y-c‖^2))) x =
+      2*t^2*((n:ℝ)*ψ (t^2*‖x-c‖^2)+
+        2*t^2*‖x-c‖^2*deriv ψ (t^2*‖x-c‖^2)) := by
+  rw [funext (gradient_radialPrimitive c t hΨ hdΨ)]
+  have hq : ContDiff ℝ ∞ (fun y : Euc n ↦ 2*t^2*ψ (t^2*‖y-c‖^2)) :=
+    contDiff_const.mul (hψ.comp (contDiff_const.mul
+      ((contDiff_id.sub contDiff_const).norm_sq (𝕜 := ℝ))))
+  have hV : Differentiable ℝ (fun y : Euc n ↦ y-c) := differentiable_id.sub_const c
+  rw [divergence_smul (hq.differentiable (by simp)) hV,divergence_displacement,inner_gradient_left]
+  have hi := (((hasFDerivAt_id (𝕜 := ℝ) x).sub_const c).norm_sq).const_mul (t^2)
+  have hh := (((hψ.differentiable (by simp)) _).hasDerivAt.comp_hasFDerivAt x hi).const_mul (2*t^2)
+  change HasFDerivAt (fun y : Euc n ↦ 2*t^2*ψ (t^2*‖y-c‖^2)) _ x at hh
+  rw [hh.fderiv]
+  simp only [smul_apply,ContinuousLinearMap.comp_apply,
+    ContinuousLinearMap.id_apply,id_eq,innerSL_apply_apply,smul_eq_mul,
+    real_inner_self_eq_norm_sq]
+  ring
+
+theorem radial_average_hasDerivAt (c : Euc n) (R : ℝ) (k : ℕ)
+    {ψ : ℝ → ℝ} (hψ : ContDiff ℝ ∞ ψ) {u : Euc n → ℝ}
+    (hu : IntegrableOn u (Metric.ball c R) volume) (t : ℝ) :
+    IntegrableOn (fun x ↦ u x*radialDensityDerivative c k ψ t x) (Metric.ball c R) ∧
+    HasDerivAt (fun s ↦ ∫ x in Metric.ball c R, u x*radialDensityIntegrand c k ψ s x)
+      (∫ x in Metric.ball c R, u x*radialDensityDerivative c k ψ t x) t := by
+  let μ := volume.restrict (Metric.ball c R)
+  have hcF (s : ℝ) : Continuous (radialDensityIntegrand c k ψ s) :=
+    (radialDensityIntegrand_continuous c k hψ).comp (continuous_const.prodMk continuous_id)
+  have hcD (s : ℝ) : Continuous (radialDensityDerivative c k ψ s) :=
+    (radialDensityDerivative_continuous c k hψ).comp (continuous_const.prodMk continuous_id)
+  obtain ⟨C,hC⟩ := (isCompact_closedBall c R).exists_bound_of_continuousOn (hcF t).continuousOn
+  have hFi : Integrable (fun x ↦ u x*radialDensityIntegrand c k ψ t x) μ := by
+    apply hu.mul_bdd (hcF t).aestronglyMeasurable
+    filter_upwards [ae_restrict_mem measurableSet_ball] with x hx
+    exact hC x (Metric.ball_subset_closedBall hx)
+  obtain ⟨B,hB⟩ := (isCompact_Icc.prod (isCompact_closedBall c R)).exists_bound_of_continuousOn
+    ((radialDensityDerivative_continuous c k hψ).continuousOn :
+      ContinuousOn _ (Icc (t-1) (t+1) ×ˢ Metric.closedBall c R))
+  apply hasDerivAt_integral_of_dominated_loc_of_deriv_le
+    (s := Ioo (t-1) (t+1)) (bound := fun x ↦ ‖u x‖*B)
+    (Ioo_mem_nhds (by linarith) (by linarith))
+    (Eventually.of_forall fun s ↦ hu.aestronglyMeasurable.mul (hcF s).aestronglyMeasurable)
+    hFi (hu.aestronglyMeasurable.mul (hcD t).aestronglyMeasurable)
+    ?_ (hu.norm.mul_const B) (Eventually.of_forall ?_)
+  · filter_upwards [ae_restrict_mem measurableSet_ball] with x hx
+    intro s hs
+    rw [norm_mul]
+    exact mul_le_mul_of_nonneg_left (hB (s,x) ⟨⟨hs.1.le,hs.2.le⟩,
+      Metric.ball_subset_closedBall hx⟩) (norm_nonneg _)
+  · intro x s _
+    exact (radialDensityIntegrand_hasDerivAt c k hψ s x).const_mul (u x)
+
+theorem tsupport_gradient_subset (φ : Euc n → ℝ) : tsupport (gradient φ) ⊆ tsupport φ := by
+  exact (tsupport_comp_subset (map_zero (InnerProductSpace.toDual ℝ (Euc n)).symm) (fderiv ℝ φ)).trans
+    (tsupport_fderiv_subset ℝ)
+
+theorem weakHarmonic_radial_derivative_zero {u : Euc n → ℝ} {U : Set (Euc n)}
+    (hw : WeakHarmonicOn u U) (hn : 0 < n) (c : Euc n) {R : ℝ} (hR : 0 < R)
+    (hU : Metric.ball c R ⊆ U)
+    {Ψ ψ : ℝ → ℝ} (hΨ : ContDiff ℝ ∞ Ψ) (hψ : ContDiff ℝ ∞ ψ)
+    (hdΨ : ∀ s, deriv Ψ s = ψ s) (hΨz : ∀ s, 1 ≤ s → Ψ s = 0)
+    {t : ℝ} (ht : 1/R < t) :
+    (∫ x in Metric.ball c R, u x*radialDensityDerivative c n ψ t x) = 0 := by
+  have ht0 : 0 < t := (one_div_pos.mpr hR).trans ht
+  have htR : 1/t < R := (div_lt_iff₀ ht0).mpr (by
+    have := (div_lt_iff₀ hR).mp ht
+    nlinarith)
+  let φ : Euc n → ℝ := fun x ↦ Ψ (t^2*‖x-c‖^2)
+  have hφ : ContDiff ℝ ∞ φ :=
+    hΨ.comp (contDiff_const.mul ((contDiff_id.sub contDiff_const).norm_sq (𝕜 := ℝ)))
+  have hsφ : tsupport φ ⊆ Metric.ball c R :=
+    (radialProfile_tsupport c hΨz ht0).trans (Metric.closedBall_subset_ball htR)
+  have hcφ : HasCompactSupport φ := (isCompact_closedBall c R).of_isClosed_subset
+    (isClosed_tsupport φ) (hsφ.trans Metric.ball_subset_closedBall)
+  have hz := hw φ hφ hcφ (hsφ.trans hU)
+  have hsL : Function.support (fun x ↦ u x*divergence (gradient φ) x) ⊆ Metric.ball c R :=
+    (Function.support_mul_subset_right _ _).trans ((subset_tsupport _).trans
+      ((tsupport_divergence_subset _).trans ((tsupport_gradient_subset φ).trans hsφ)))
+  rw [← setIntegral_eq_integral_of_forall_compl_eq_zero (fun x hx ↦
+    Function.notMem_support.mp (fun hh ↦ hx (hsL hh)))] at hz
+  have hp : t^n = t^(n-1)*t := by rw [← pow_succ]; congr 1; omega
+  have hid (x : Euc n) : u x*radialDensityDerivative c n ψ t x =
+      (t^(n-1)/(2*t^2))*(u x*divergence (gradient φ) x) := by
+    dsimp only [φ]
+    rw [laplacian_radialPrimitive c t hΨ hψ hdΨ]
+    dsimp only [radialDensityDerivative]
+    rw [hp]
+    field_simp [ht0.ne']
+  simp_rw [hid]
+  rw [integral_const_mul,hz,mul_zero]
+
+theorem weakHarmonic_radial_average_eq {u : Euc n → ℝ} {U : Set (Euc n)}
+    (hw : WeakHarmonicOn u U) (hn : 0 < n) (c : Euc n) {R : ℝ} (hR : 0 < R)
+    (hu : IntegrableOn u (Metric.ball c R) volume) (hU : Metric.ball c R ⊆ U)
+    (j : ℕ) {s t : ℝ} (hs : 1/R < s) (ht : 1/R < t) :
+    (∫ x in Metric.ball c R, u x*radialDensityIntegrand c n (densityProfile j) s x) =
+      ∫ x in Metric.ball c R, u x*radialDensityIntegrand c n (densityProfile j) t x := by
+  apply isOpen_Ioi.is_const_of_deriv_eq_zero (convex_Ioi (1/R)).isPreconnected
+    (f := fun r ↦ ∫ x in Metric.ball c R, u x*radialDensityIntegrand c n (densityProfile j) r x)
+  · intro r hr
+    exact (radial_average_hasDerivAt c R n (densityProfile_contDiff j) hu r).2.differentiableAt.differentiableWithinAt
+  · intro r hr
+    rw [(radial_average_hasDerivAt c R n (densityProfile_contDiff j) hu r).2.deriv]
+    exact weakHarmonic_radial_derivative_zero hw hn c hR hU
+      (densityPrimitive_contDiff j) (densityProfile_contDiff j) (densityPrimitive_deriv j)
+      (fun z hz ↦ densityPrimitive_zero j hz) hr
+  · exact hs
+  · exact ht
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/WeakHarmonicSmooth.lean b/lean/OAI/Geometry/Perimeter/Regularity/WeakHarmonicSmooth.lean
new file mode 100644
index 0000000000000000000000000000000000000000..46310b69bffad7382e22ea850340e7bfb4d48dba
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/WeakHarmonicSmooth.lean
@@ -0,0 +1,95 @@
+import OAI.Geometry.Perimeter.Regularity.WeakHarmonicRadial
+import OAI.Geometry.Perimeter.Regularity.HarmonicApproxIdentity
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff Convolution InnerProductSpace
+namespace RPISingularHarnack
+open RPI.Boundary ContinuousLinearMap
+variable {n : ℕ}
+
+def harmonicRegularization (u : Euc n → ℝ) (t : ℝ) : Euc n → ℝ :=
+  scaledHarmonicKernel t ⋆[lsmul ℝ ℝ,volume] u
+
+theorem harmonicRegularization_eq_integral (u : Euc n → ℝ) (t : ℝ) (c : Euc n) :
+    harmonicRegularization u t c = ∫ x, scaledHarmonicKernel t (c-x)*u x := by
+  rw [harmonicRegularization,convolution_eq_swap]
+  rfl
+
+theorem harmonicRegularization_contDiff {u : Euc n → ℝ} (hu : LocallyIntegrable u volume)
+    {t : ℝ} (ht : 0 < t) : ContDiff ℝ ∞ (harmonicRegularization u t) := by
+  have hc : HasCompactSupport (scaledHarmonicKernel (n := n) t) :=
+    (isCompact_closedBall 0 (1/t)).of_isClosed_subset (isClosed_tsupport _)
+      (scaledHarmonicKernel_tsupport ht)
+  exact hc.contDiff_convolution_left _ (scaledHarmonicKernel_contDiff t) hu
+
+theorem harmonicRegularization_eq_radial (u : Euc n → ℝ) (c : Euc n)
+    {R t : ℝ} (hR : 0 < R) (ht : 1/R < t) :
+    harmonicRegularization u t c =
+      (harmonicKernelMass n)⁻¹ * ∫ x in Metric.ball c R,
+        u x*radialDensityIntegrand c n (densityProfile 1) t x := by
+  have ht0 : 0 < t := (one_div_pos.mpr hR).trans ht
+  have htR : 1/t < R := (div_lt_iff₀ ht0).mpr (by
+    have := (div_lt_iff₀ hR).mp ht; nlinarith)
+  have hs : Function.support (fun x ↦ scaledHarmonicKernel t (c-x)*u x) ⊆ Metric.ball c R :=
+    (Function.support_mul_subset_left _ _).trans
+      ((shifted_scaledHarmonicKernel_support ht0 c).trans (Metric.closedBall_subset_ball htR))
+  rw [harmonicRegularization_eq_integral,
+    ← setIntegral_eq_integral_of_forall_compl_eq_zero (fun x hx ↦
+      Function.notMem_support.mp (fun hh ↦ hx (hs hh))), ← integral_const_mul]
+  apply setIntegral_congr_fun measurableSet_ball
+  intro x hx
+  dsimp only
+  rw [scaledHarmonicKernel_eq ht0,norm_sub_rev]
+  dsimp only [radialDensityIntegrand]
+  ring
+
+theorem weakHarmonic_regularization_eq {u : Euc n → ℝ} {U : Set (Euc n)}
+    (hu : LocallyIntegrable u volume) (hw : WeakHarmonicOn u U) (hn : 0 < n)
+    (c : Euc n) {R : ℝ} (hR : 0 < R) (hU : Metric.ball c R ⊆ U)
+    {s t : ℝ} (hs : 1/R < s) (ht : 1/R < t) :
+    harmonicRegularization u s c = harmonicRegularization u t c := by
+  rw [harmonicRegularization_eq_radial u c hR hs,harmonicRegularization_eq_radial u c hR ht,
+    weakHarmonic_radial_average_eq hw hn c hR
+      ((hu.integrableOn_isCompact (isCompact_closedBall c R)).mono_set Metric.ball_subset_closedBall)
+      hU 1 hs ht]
+
+theorem weakHarmonic_ae_eq_regularization {u : Euc n → ℝ} {U : Set (Euc n)}
+    (hu : LocallyIntegrable u volume) (hw : WeakHarmonicOn u U) (hn : 0 < n) :
+    ∀ᵐ c ∂volume, ∀ R : ℝ, 0 < R → Metric.ball c R ⊆ U →
+      ∀ t : ℝ, 1/R < t → u c = harmonicRegularization u t c := by
+  filter_upwards [scaledHarmonicKernel_ae_tendsto hu] with c hc
+  intro R hR hU t ht
+  have hl : Tendsto (fun i : ℕ ↦ harmonicRegularization u ((i:ℝ)+1) c) atTop (𝓝 (u c)) := by
+    simpa only [harmonicRegularization_eq_integral] using hc
+  obtain ⟨N,hN⟩ := exists_nat_gt (1/R)
+  have he : ∀ᶠ i : ℕ in atTop,
+      harmonicRegularization u ((i:ℝ)+1) c = harmonicRegularization u t c := by
+    filter_upwards [eventually_ge_atTop N] with i hi
+    apply weakHarmonic_regularization_eq hu hw hn c hR hU _ ht
+    have hni : (N:ℝ) ≤ i := by exact_mod_cast hi
+    linarith
+  exact tendsto_nhds_unique hl (tendsto_const_nhds.congr' (he.mono (fun _ h ↦ h.symm)))
+
+theorem weakHarmonic_smooth_representative_ball {u : Euc n → ℝ} {U : Set (Euc n)}
+    (hu : LocallyIntegrable u volume) (hw : WeakHarmonicOn u U) (hn : 0 < n)
+    (c : Euc n) {r : ℝ} (hr : 0 < r) (hU : Metric.ball c (2*r) ⊆ U) :
+    ∃ f : Euc n → ℝ, ContDiff ℝ ∞ f ∧
+      u =ᵐ[volume.restrict (Metric.ball c r)] f := by
+  refine ⟨harmonicRegularization u (2/r),harmonicRegularization_contDiff hu (by positivity),?_⟩
+  filter_upwards [ae_restrict_of_ae (weakHarmonic_ae_eq_regularization hu hw hn),
+    ae_restrict_mem measurableSet_ball] with x hx hxB
+  apply hx r hr _ (2/r) (by exact div_lt_div_of_pos_right (by norm_num) hr)
+  intro y hy
+  apply hU
+  have hxy := dist_triangle y x c
+  rw [Metric.mem_ball] at hxB hy ⊢
+  linarith
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/WeightedFiberMoment.lean b/lean/OAI/Geometry/Perimeter/Regularity/WeightedFiberMoment.lean
new file mode 100644
index 0000000000000000000000000000000000000000..5ffb59ab8ea02227599476092d3f69029805c972
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/WeightedFiberMoment.lean
@@ -0,0 +1,156 @@
+import OAI.Geometry.Perimeter.Regularity.PhaseMomentLimit
+import OAI.Geometry.Perimeter.Regularity.NormalWindowTest
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff Convolution InnerProductSpace
+namespace RPISingularHarnack
+open RPI.Boundary ContinuousLinearMap
+variable {n : ℕ}
+
+theorem mollified_fiber_weighted_defect {E : Set (Euc n)} (hE : MeasurableSet E)
+    {v : Euc n} (hv : ‖v‖ = 1) {φ : normalPlane v → ℝ}
+    (hφ : Continuous φ) (hcφ : HasCompactSupport φ) (hpφ : ∀ y, 0 ≤ φ y)
+    {k : Euc n → ℝ} (hk : ContDiff ℝ ∞ k) (hc : HasCompactSupport k)
+    (hp : ∀ x, 0 ≤ k x) (hu : ∫ x, k x = 1)
+    {a b ε R : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) (hab : a+ε < b)
+    (hs : Function.support k ⊆ Metric.ball 0 ε)
+    (hphase : ∀ᵐ x ∂volume, x ∈ Metric.ball 0 R → a < |inner ℝ v x| →
+      (x ∈ E ↔ inner ℝ v x < 0))
+    (hKR : ∀ y ∈ tsupport φ, ‖y‖+b+ε ≤ R) :
+    let q := phaseMollification E k
+    let H := fun y : normalPlane v ↦ -b+∫ s in (-b)..b, q ((y:Euc n)+s•v)
+    let M := ∫ x, normalWindowTest v φ b 2 x * (-fderiv ℝ q x v)
+    0 ≤ M - ∫ y in tsupport φ, φ y*(H y)^2 ∧
+    M - ∫ y in tsupport φ, φ y*(H y)^2 ≤
+      4*b*(∫ x, normalWindowTest v φ b 0 x * (q x*(1-q x))) +
+      8*b^2*(∫ x, normalWindowTest v φ b 0 x * max (fderiv ℝ q x v) 0) := by
+  dsimp only
+  let q := phaseMollification E k
+  have hq : Continuous q := (phaseMollification_contDiff hE hk hc).continuous
+  have hd : Continuous (fun x ↦ fderiv ℝ q x v) :=
+    ((phaseMollification_contDiff hE hk hc).continuous_fderiv (by simp)).clm_apply continuous_const
+  have hiM := normalWindowTest_integral hv hφ hcφ hb 2 hd.neg
+  have hiD := normalWindowTest_integral hv hφ hcφ hb 0 (hq.mul ((continuous_const (y := (1:ℝ))).sub hq))
+  have hiP := normalWindowTest_integral hv hφ hcφ hb 0 (hd.max (continuous_const (y := (0:ℝ))))
+  dsimp only [q,Pi.neg_apply,Pi.mul_apply,Pi.sub_apply] at hiM hiD hiP
+  rw [hiM,hiD,hiP]
+  simp only [pow_zero,one_mul]
+  simp_rw [← mollifiedNormalProfile_deriv hE hk hc v]
+  let A := fun y : normalPlane v ↦ ∫ s in (-b)..b,
+    s^2*(-deriv (mollifiedNormalProfile E k v y) s)
+  let B := fun y : normalPlane v ↦ (-b+∫ s in (-b)..b, q ((y:Euc n)+s•v))^2
+  let D := fun y : normalPlane v ↦ ∫ s in (-b)..b, q ((y:Euc n)+s•v)*(1-q ((y:Euc n)+s•v))
+  let P := fun y : normalPlane v ↦ ∫ s in (-b)..b, max (deriv (mollifiedNormalProfile E k v y) s) 0
+  have hA : Continuous A := by
+    have hm : Continuous (fun x ↦ (inner ℝ v x)^2*(-fderiv ℝ q x v)) :=
+      ((continuous_const.inner continuous_id).pow 2).mul hd.neg
+    have h := continuous_profile_integral (v := v) _ hm (-b) b
+    dsimp only [q] at h
+    simp_rw [normal_fiber_inner hv,← mollifiedNormalProfile_deriv hE hk hc v] at h
+    exact h
+  have hB : Continuous B :=
+    (continuous_const.add (continuous_profile_integral (v := v) q hq (-b) b)).pow 2
+  have hD : Continuous D := continuous_profile_integral (v := v) _
+    (hq.mul ((continuous_const (y := (1:ℝ))).sub hq)) (-b) b
+  have hP : Continuous P := by
+    have h := continuous_profile_integral (v := v) _ (hd.max (continuous_const (y := (0:ℝ)))) (-b) b
+    dsimp only [q] at h
+    simp_rw [← mollifiedNormalProfile_deriv hE hk hc v] at h
+    exact h
+  have hAi : IntegrableOn (fun y ↦ φ y*A y) (tsupport φ) volume :=
+    (hφ.mul hA).continuousOn.integrableOn_compact hcφ
+  have hBi : IntegrableOn (fun y ↦ φ y*B y) (tsupport φ) volume :=
+    (hφ.mul hB).continuousOn.integrableOn_compact hcφ
+  have hDi : IntegrableOn (fun y ↦ φ y*D y) (tsupport φ) volume :=
+    (hφ.mul hD).continuousOn.integrableOn_compact hcφ
+  have hPi : IntegrableOn (fun y ↦ φ y*P y) (tsupport φ) volume :=
+    (hφ.mul hP).continuousOn.integrableOn_compact hcφ
+  change 0 ≤ (∫ y in tsupport φ, φ y*A y)-(∫ y in tsupport φ, φ y*B y) ∧
+    (∫ y in tsupport φ, φ y*A y)-(∫ y in tsupport φ, φ y*B y) ≤
+      4*b*(∫ y in tsupport φ, φ y*D y)+8*b^2*(∫ y in tsupport φ, φ y*P y)
+  rw [← integral_sub hAi hBi,← integral_const_mul,← integral_const_mul,
+    ← integral_add (hDi.const_mul _) (hPi.const_mul _)]
+  have hh (y : normalPlane v) (hy : y ∈ tsupport φ) :=
+    mollifiedNormalProfile_flux_defect hE hv hk hc hp hu ha hb hab hs hphase y (hKR y hy)
+  constructor
+  · apply setIntegral_nonneg hcφ.measurableSet
+    intro y hy
+    have h := mul_nonneg (hpφ y) (hh y hy).1
+    change 0 ≤ φ y*A y-φ y*B y
+    change 0 ≤ φ y*(A y-B y) at h
+    nlinarith only [h]
+  · apply setIntegral_mono_on (hAi.sub hBi) ((hDi.const_mul _).add (hPi.const_mul _))
+      hcφ.measurableSet
+    intro y hy
+    have h := mul_le_mul_of_nonneg_left (hh y hy).2 (hpφ y)
+    change φ y*A y-φ y*B y ≤ 4*b*(φ y*D y)+8*b^2*(φ y*P y)
+    change φ y*(A y-B y) ≤ φ y*(4*b*D y+8*b^2*P y) at h
+    nlinarith only [h]
+
+theorem normalWindowTest_diffuse_tendsto_zero {E : Set (Euc n)} (hE : MeasurableSet E)
+    {v : Euc n} (hv : ‖v‖ = 1) {φ : normalPlane v → ℝ}
+    (hφ : Continuous φ) (hcφ : HasCompactSupport φ) (b : ℝ) :
+    Tendsto (fun j ↦ ∫ x, normalWindowTest v φ b 0 x *
+      (phaseMollification E ((c1Mollifier j).normed volume) x *
+        (1-phaseMollification E ((c1Mollifier j).normed volume) x))) atTop (𝓝 0) := by
+  have hi := normalWindowTest_integrable hv hφ hcφ b 0
+  have ht : ∀ᵐ x ∂volume, Tendsto (fun j ↦ normalWindowTest v φ b 0 x *
+      (phaseMollification E ((c1Mollifier j).normed volume) x *
+        (1-phaseMollification E ((c1Mollifier j).normed volume) x))) atTop (𝓝 (0:ℝ)) := by
+    filter_upwards [phaseMollification_ae_tendsto hE] with x hx
+    convert tendsto_const_nhds.mul (hx.mul (tendsto_const_nhds.sub hx)) using 1
+    by_cases he : x ∈ E <;> simp [he]
+  have h := tendsto_integral_of_dominated_convergence (fun x ↦ ‖normalWindowTest v φ b 0 x‖)
+    (fun j ↦ (normalWindowTest_measurable hφ b 0).aestronglyMeasurable.mul
+      (((phaseMollification_contDiff hE (c1Mollifier j).contDiff_normed
+        (c1Mollifier j).hasCompactSupport_normed).continuous.mul
+          (continuous_const.sub (phaseMollification_contDiff hE (c1Mollifier j).contDiff_normed
+            (c1Mollifier j).hasCompactSupport_normed).continuous)).aestronglyMeasurable))
+    hi.norm (fun j ↦ ae_of_all _ fun x ↦ ?_) ht
+  · simpa only [integral_zero,Pi.mul_apply,Pi.sub_apply] using h
+  · obtain ⟨h0,h1⟩ := phaseMollification_c1_mem_Icc hE j x
+    dsimp only [Pi.mul_apply,Pi.sub_apply]
+    rw [norm_mul]
+    have hb : ‖phaseMollification E ((c1Mollifier j).normed volume) x *
+        (1-phaseMollification E ((c1Mollifier j).normed volume) x)‖ ≤ 1 := by
+      rw [Real.norm_eq_abs,abs_of_nonneg (mul_nonneg h0 (sub_nonneg.mpr h1))]
+      nlinarith
+    exact (mul_le_mul_of_nonneg_left hb (norm_nonneg _)).trans_eq (mul_one _)
+
+theorem mollified_fiber_weighted_height_square_tendsto {E : Set (Euc n)} (hE : MeasurableSet E)
+    {v : Euc n} (hv : ‖v‖ = 1) {φ : normalPlane v → ℝ}
+    (hφ : Continuous φ) (hcφ : HasCompactSupport φ) {b : ℝ} (hb : 0 ≤ b) :
+    Tendsto (fun j ↦ ∫ y in tsupport φ, φ y*
+      (-b+∫ s in (-b)..b, phaseMollification E ((c1Mollifier j).normed volume)
+        ((y:Euc n)+s•v))^2) atTop
+      (𝓝 (∫ y in tsupport φ, φ y*(exactFiberHeight E v b y)^2)) := by
+  apply tendsto_integral_of_dominated_convergence (fun y ↦ ‖φ y‖*b^2)
+  · intro j
+    exact (hφ.mul ((continuous_const.add (continuous_profile_integral (v := v) _
+      (phaseMollification_contDiff hE (c1Mollifier j).contDiff_normed
+        (c1Mollifier j).hasCompactSupport_normed).continuous (-b) b)).pow 2)).aestronglyMeasurable
+  · exact (hφ.continuousOn.integrableOn_compact hcφ).norm.mul_const _
+  · intro j
+    apply ae_of_all
+    intro y
+    rw [norm_mul]
+    apply mul_le_mul_of_nonneg_left _ (norm_nonneg _)
+    have h := interval_height_abs_le
+      (((phaseMollification_contDiff hE (c1Mollifier j).contDiff_normed
+        (c1Mollifier j).hasCompactSupport_normed).continuous.comp
+          (continuous_const.add (continuous_id.smul continuous_const))).measurable)
+      hb (fun s ↦ phaseMollification_c1_mem_Icc hE j ((y:Euc n)+s•v))
+    rw [Real.norm_eq_abs,abs_of_nonneg (sq_nonneg _)]
+    simpa only [sq_abs,Function.comp_def,Pi.add_apply,Pi.smul_apply,Pi.smul_apply',id_eq] using
+      (sq_le_sq₀ (abs_nonneg _) hb).mpr h
+  · filter_upwards [ae_restrict_of_ae (mollified_fiber_height_ae_tendsto hE hv hb)] with y hy
+    exact tendsto_const_nhds.mul (hy.pow 2)
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/WeightedSquareLimit.lean b/lean/OAI/Geometry/Perimeter/Regularity/WeightedSquareLimit.lean
new file mode 100644
index 0000000000000000000000000000000000000000..bb01449eb36280798659f104f30728719290c00d
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/WeightedSquareLimit.lean
@@ -0,0 +1,60 @@
+import OAI.Geometry.Perimeter.Regularity.GeoAlgebra
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff InnerProductSpace
+namespace RPISingularHarnack
+open RPI.Boundary ContinuousLinearMap
+variable {n m : ℕ}
+
+theorem weighted_square_L1_tendsto_zero {X : Type*} [MeasurableSpace X]
+    {μ : Measure X} (f : ℕ → X → ℝ) (hf : ∀ i, Integrable (f i) μ)
+    {g w φ : X → ℝ} (hg : Integrable g μ) (hw : AEStronglyMeasurable w μ)
+    (hφ : AEStronglyMeasurable φ μ) {M A B : ℝ} (hM : 0≤M) (hA : 0≤A) (hB : 0≤B)
+    (hbφ : ∀ᵐ x ∂μ, ‖φ x‖≤M)
+    (heq : ∀ᵐ x ∂μ, φ x≠0 → w x=g x)
+    (hbw : ∀ᵐ x ∂μ, φ x≠0 → ‖w x‖≤B)
+    (hbf : ∀ᶠ i in atTop, ∀ᵐ x ∂μ, φ x≠0 → ‖f i x‖≤A)
+    (hL : Tendsto (fun i ↦ ∫⁻ x, ‖f i x-g x‖ₑ ∂μ) atTop (𝓝 0)) :
+    Tendsto (fun i ↦ ∫ x, φ x*(f i x-w x)^2 ∂μ) atTop (𝓝 0) := by
+  have hbound : ∀ᶠ i in atTop, ∀ᵐ x ∂μ,
+      ‖φ x*(f i x-w x)^2‖≤M*(A+B)*‖f i x-g x‖ := by
+    filter_upwards [hbf] with i hi
+    filter_upwards [hi,hbw,heq,hbφ] with x hix hwx hex hφx
+    by_cases hz : φ x=0
+    · simp only [hz,zero_mul,norm_zero]
+      positivity
+    · have hab : ‖f i x-w x‖≤A+B := (norm_sub_le _ _).trans (add_le_add (hix hz) (hwx hz))
+      rw [norm_mul,norm_pow]
+      calc
+        _ ≤ M*(‖f i x-w x‖*‖f i x-w x‖) := by rw [← pow_two]; gcongr
+        _ ≤ M*((A+B)*‖f i x-w x‖) := by gcongr
+        _ = _ := by rw [hex hz]; ring
+  have hi : ∀ᶠ i in atTop, Integrable (fun x ↦ φ x*(f i x-w x)^2) μ := by
+    filter_upwards [hbound] with i hib
+    exact (((hf i).sub hg).norm.const_mul (M*(A+B))).mono'
+      (hφ.mul (((hf i).aestronglyMeasurable.sub hw).pow 2)) hib
+  have ht : Tendsto (fun i ↦ ENNReal.ofReal (M*(A+B)) * ∫⁻ x, ‖f i x-g x‖ₑ ∂μ)
+      atTop (𝓝 0) := by
+    simpa using ENNReal.Tendsto.const_mul hL (Or.inr ENNReal.ofReal_ne_top)
+  have hl : Tendsto (fun i ↦ ∫⁻ x, ‖φ x*(f i x-w x)^2‖ₑ ∂μ) atTop (𝓝 0) := by
+    refine tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds ht
+      (Eventually.of_forall (fun _ ↦ zero_le)) ?_
+    filter_upwards [hbound] with i hib
+    rw [← lintegral_const_mul' _ _ ENNReal.ofReal_ne_top]
+    apply lintegral_mono_ae
+    filter_upwards [hib] with x hx
+    rw [← ofReal_norm,← ofReal_norm,← ENNReal.ofReal_mul (by positivity)]
+    exact ENNReal.ofReal_le_ofReal hx
+  have h := tendsto_integral_of_L1 (fun _ : X ↦ (0:ℝ))
+    (aestronglyMeasurable_zero : AEStronglyMeasurable (fun _ : X ↦ (0:ℝ)) μ) hi
+    (by simpa only [sub_zero] using hl)
+  simpa only [integral_zero] using h
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/WeightedWrongWayFlux.lean b/lean/OAI/Geometry/Perimeter/Regularity/WeightedWrongWayFlux.lean
new file mode 100644
index 0000000000000000000000000000000000000000..c63e3220a7325b605ecd238a7f4d9f676785a89a
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/WeightedWrongWayFlux.lean
@@ -0,0 +1,71 @@
+import OAI.Geometry.Perimeter.Regularity.ActualFluxWindowLimit
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff Convolution InnerProductSpace
+namespace RPISingularHarnack
+open RPI.Boundary ContinuousLinearMap
+variable {n : ℕ}
+
+theorem normalWindowTest_nonneg {v : Euc n} {φ : normalPlane v → ℝ}
+    (hφ : ∀ y, 0 ≤ φ y) (b : ℝ) (x : Euc n) :
+    0 ≤ normalWindowTest v φ b 0 x := by
+  simp only [normalWindowTest,pow_zero,mul_one]
+  exact Set.indicator_nonneg (fun y _ ↦ hφ _) x
+
+theorem weighted_positive_derivative_le_tilt (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    {v : Euc n} (hv : ‖v‖ = 1) {φ : normalPlane v → ℝ}
+    (hφ : Continuous φ) (hcφ : HasCompactSupport φ)
+    (hpφ : ∀ y, 0 ≤ φ y) {M b δ R : ℝ} (hM : ∀ y, ‖φ y‖ ≤ M)
+    (hb : 0 ≤ b) (hR : 0 < R)
+    (hKR : ∀ y ∈ tsupport φ, ‖y‖+b+δ ≤ R)
+    {j : ℕ} (hj : (c1Mollifier (n := n) j).rOut < δ) :
+    (∫ x, normalWindowTest v φ b 0 x *
+      max (fderiv ℝ (phaseMollification E ((c1Mollifier j).normed volume)) x v) 0) ≤
+      M * (orientedTilt E v (Metric.ball 0 R) / 2) := by
+  let q := phaseMollification E ((c1Mollifier j).normed volume)
+  let g : Euc n → ℝ := fun x ↦ max (fderiv ℝ q x v) 0
+  have hg : Continuous g :=
+    (((phaseMollification_contDiff hmin.1 (c1Mollifier j).contDiff_normed
+      (c1Mollifier j).hasCompactSupport_normed).continuous_fderiv (by simp)).clm_apply
+        continuous_const).max continuous_const
+  have hi := normalWindowTest_mul_integrable hv hφ hcφ b 0 hg
+  have hgi : IntegrableOn g (Metric.closedBall 0 (R-δ)) volume :=
+    hg.continuousOn.integrableOn_compact (isCompact_closedBall _ _)
+  have hMt : 0 ≤ M := (norm_nonneg (φ 0)).trans (hM 0)
+  have hdom : ∀ x, normalWindowTest v φ b 0 x * g x ≤
+      (Metric.closedBall 0 (R-δ)).indicator (fun x ↦ M*g x) x := by
+    intro x
+    have hg0 : 0 ≤ g x := le_max_right _ _
+    by_cases hx : normalWindowTest v φ b 0 x=0
+    · rw [hx,zero_mul]
+      exact Set.indicator_nonneg (fun x _ ↦ mul_nonneg hMt (le_max_right _ _)) _
+    · have hxR : x ∈ Metric.closedBall 0 (R-δ) := by
+        rw [Metric.mem_closedBall,dist_zero_right]
+        have := normalWindowTest_support_norm hv φ b δ R 0 hKR hx
+        linarith
+      rw [Set.indicator_of_mem hxR]
+      apply mul_le_mul_of_nonneg_right _ hg0
+      have h := normalWindowTest_bound hM hb 0 x
+      simp only [pow_zero,mul_one] at h
+      rwa [Real.norm_of_nonneg (normalWindowTest_nonneg hpφ b x)] at h
+  have hbound := phaseMollification_positive_derivative_le_tilt hn hmin hv
+    (c1Mollifier j).contDiff_normed (c1Mollifier j).hasCompactSupport_normed
+    (c1Mollifier j).nonneg_normed (c1Mollifier j).integral_normed hR
+    0 (R-δ) (fun x hx ↦ c1Mollifier_translated_support (by
+      rw [Metric.mem_closedBall,dist_zero_right] at hx; linarith))
+  calc
+    _ ≤ ∫ x, (Metric.closedBall 0 (R-δ)).indicator (fun x ↦ M*g x) x :=
+      integral_mono hi (IntegrableOn.integrable_indicator (hgi.const_mul M) measurableSet_closedBall) hdom
+    _ = M * ∫ x in Metric.closedBall 0 (R-δ), g x := by
+      rw [integral_indicator measurableSet_closedBall,integral_const_mul]
+    _ ≤ _ := mul_le_mul_of_nonneg_left hbound hMt
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/WholeBoundarySublevelMeasureStep.lean b/lean/OAI/Geometry/Perimeter/Regularity/WholeBoundarySublevelMeasureStep.lean
new file mode 100644
index 0000000000000000000000000000000000000000..686debd7861c4fdb4c0810cfe123096ad8304c9b
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/WholeBoundarySublevelMeasureStep.lean
@@ -0,0 +1,519 @@
+import OAI.Geometry.Perimeter.Regularity.TendstoLintegralNormOfL2
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+open scoped ENNReal Topology
+open scoped ENNReal Topology ContDiff
+namespace RPISingularHarnack
+open RPI RPI.Boundary Classical
+
+theorem wholeBoundary_sublevel_measure_step {n : ℕ} (hn : 2 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    {h : Euc n → ℝ} {G : Euc n → Euc n}
+    (hSob : WholeBoundarySobolev E h G) (hSuper : WholeBoundarySupersolution E G)
+    (hh01 : ∀ᵐ x ∂boundaryArea E, 0 ≤ h x ∧ h x ≤ 1)
+    {p : ℝ} (hp : 1 < p) (hpk : (((n - 1 : ℕ) : ℝ) - 1) * p < (n - 1 : ℕ)) :
+    ∃ C : ℝ, 0 < C ∧ ∀ s r l k : ℝ, 0 < s → s < r → r ≤ 2 → 0 < l → l < k →
+      (boundaryArea E).real (Metric.ball 0 s ∩ {x | h x < l}) ≤
+        C * (k / ((r-s) * (k-l))) ^ p *
+          ((boundaryArea E).real (Metric.ball 0 r ∩ {x | h x < k})) ^ p := by
+  classical
+  have hp0 : 0 ≤ p := (lt_trans zero_lt_one hp).le
+  have hn1 : 1 < n := by omega
+  obtain ⟨L, hL, hcut⟩ := exists_interior_cutoff_constant
+  obtain ⟨C, hC, hsob⟩ := wholeBoundary_sublevel_sobolev hn hmin hSob hh01 (by norm_num : (0 : ℝ) < 3) hp hpk
+  refine ⟨(C.toReal + 1) * (10 * L) ^ p, by positivity, ?_⟩
+  intro s r l k hs hsr hr2 hl hlk
+  have hk : 0 < k := hl.trans hlk
+  have hd : 0 < (k-l)/2 := by linarith
+  have hN : 0 < L / (r-s) := div_pos hL (sub_pos.mpr hsr)
+  obtain ⟨η, hη, hcη, hsη, hη01, hηone, hηD⟩ := hcut n E s r hs hsr
+  let μ := (boundaryArea E).restrict (Metric.ball 0 3)
+  let : IsFiniteMeasure μ := ⟨by
+    simpa only [μ, Measure.restrict_apply_univ] using minimizing_boundaryArea_ball_lt_top hn1 hmin 0 (by norm_num : (0 : ℝ) < 3)⟩
+  let A := Metric.ball (0 : Euc n) s ∩ {x | h x < l}
+  let B := Metric.ball (0 : Euc n) r ∩ {x | h x < k}
+  have hm := (hSob 3 (by norm_num)).1.aestronglyMeasurable.aemeasurable
+  have hAm : NullMeasurableSet A μ := measurableSet_ball.nullMeasurableSet.inter
+    (nullMeasurableSet_lt hm aemeasurable_const)
+  have hBm : NullMeasurableSet B μ := measurableSet_ball.nullMeasurableSet.inter
+    (nullMeasurableSet_lt hm aemeasurable_const)
+  have hAeq : μ A = boundaryArea E A := by
+    change (boundaryArea E).restrict (Metric.ball 0 3) A = _
+    rw [Measure.restrict_apply₀ hAm, inter_eq_left.mpr]
+    exact fun x hx ↦ Metric.ball_subset_ball (by linarith : s ≤ 3) hx.1
+  have hBeq : μ B = boundaryArea E B := by
+    change (boundaryArea E).restrict (Metric.ball 0 3) B = _
+    rw [Measure.restrict_apply₀ hBm, inter_eq_left.mpr]
+    exact fun x hx ↦ Metric.ball_subset_ball (by linarith : r ≤ 3) hx.1
+  have hAtop : boundaryArea E A ≠ ⊤ := hAeq ▸ measure_ne_top μ A
+  have hBtop : boundaryArea E B ≠ ⊤ := hBeq ▸ measure_ne_top μ B
+  have hlo : ENNReal.ofReal ((k-l)/2) ^ p * boundaryArea E A ≤
+      ∫⁻ x, ENNReal.ofReal (η x ^ 2 * sublevelTest ((k-l)/2) k (h x)) ^ p ∂μ := by
+    rw [← hAeq]
+    calc
+      _ = ∫⁻ x, A.indicator (fun _ ↦ ENNReal.ofReal ((k-l)/2) ^ p) x ∂μ := by
+        rw [lintegral_indicator₀ hAm, lintegral_const, Measure.restrict_apply_univ]
+      _ ≤ _ := lintegral_mono fun x ↦ by
+        by_cases hx : x ∈ A
+        · rw [indicator_of_mem hx, hηone x hx.1, one_pow, one_mul]
+          apply ENNReal.rpow_le_rpow _ hp0
+          apply ENNReal.ofReal_le_ofReal
+          have H := sub_parameter_le_smoothPositive (s := k - h x) hd
+          change k - h x - (k-l)/2 ≤ sublevelTest ((k-l)/2) k (h x) at H
+          have hxl : h x < l := hx.2
+          linarith
+        · rw [indicator_of_notMem hx]
+          exact bot_le
+  have hsR := hsη.trans (Metric.ball_subset_ball (by linarith : r ≤ 3))
+  have hl1 := wholeBoundary_sublevel_L1 hn1 hmin hSob hSuper hh01
+    (by norm_num : (0 : ℝ) < 3) (by linarith : r ≤ 3) hd hk hN hη hcη hsη hη01 hηD
+  have hso := hsob η hη hcη hsR ((k-l)/2) k hd hk.le
+  have H : ENNReal.ofReal ((k-l)/2) ^ p * boundaryArea E A ≤
+      C * (ENNReal.ofReal (5 * k * (L/(r-s))) * boundaryArea E B) ^ p := by
+    exact hlo.trans (hso.trans (mul_le_mul_of_nonneg_left (ENNReal.rpow_le_rpow hl1 hp0) (show (0 : ℝ≥0∞) ≤ C from bot_le)))
+  have hRt : C * (ENNReal.ofReal (5 * k * (L/(r-s))) * boundaryArea E B) ^ p ≠ ⊤ :=
+    ENNReal.mul_ne_top hC (ENNReal.rpow_ne_top_of_nonneg hp0
+      (ENNReal.mul_ne_top ENNReal.ofReal_ne_top hBtop))
+  have Hr := ENNReal.toReal_mono hRt H
+  simp only [ENNReal.toReal_mul, ← ENNReal.toReal_rpow,
+    ENNReal.toReal_ofReal hd.le, ENNReal.toReal_ofReal (show 0 ≤ 5 * k * (L/(r-s)) by positivity)] at Hr
+  exact (RPIIteration.sublevel_step_division hd (by positivity) Hr).trans
+    (RPIIteration.sublevel_step_factor ENNReal.toReal_nonneg hL hk hlk hsr ENNReal.toReal_nonneg)
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+open scoped ENNReal Topology
+open scoped ENNReal Topology ContDiff
+open scoped Topology ENNReal
+namespace RPIIteration
+
+lemma exists_small_power {p A e : ℝ} (hp : 1 < p) (he : 0 < e) :
+    ∃ d : ℝ, 0 < d ∧ d < 1 ∧ A * d ^ p ≤ e * d := by
+  have hq : 0 < p - 1 := sub_pos.mpr hp
+  have ht : Tendsto (fun d : ℝ ↦ A * d ^ (p - 1)) (𝓝 (0 : ℝ)) (𝓝 0) := by
+    simpa only [Real.zero_rpow (ne_of_gt hq), mul_zero] using
+      tendsto_const_nhds.mul ((Real.continuous_rpow_const hq.le).tendsto 0)
+  have hb : ∀ᶠ d : ℝ in 𝓝[>] 0, A * d ^ (p - 1) < e :=
+    (ht.eventually (gt_mem_nhds he)).filter_mono nhdsWithin_le_nhds
+  have h01 : ∀ᶠ d : ℝ in 𝓝[>] 0, 0 < d ∧ d < 1 :=
+    Filter.Eventually.and (show ∀ᶠ d : ℝ in 𝓝[>] 0, 0 < d from self_mem_nhdsWithin)
+      ((gt_mem_nhds (by norm_num : (0 : ℝ) < 1)).filter_mono nhdsWithin_le_nhds)
+  obtain ⟨d, ⟨hd, hd1⟩, hAd⟩ := (h01.and hb).exists
+  refine ⟨d, hd, hd1, ?_⟩
+  have heq : d ^ p = d ^ (p - 1) * d := by
+    calc
+      d ^ p = d ^ ((p - 1) + 1) := by congr 1; ring
+      _ = _ := by rw [Real.rpow_add hd, Real.rpow_one]
+  rw [heq, ← mul_assoc]
+  exact mul_le_mul_of_nonneg_right hAd.le hd.le
+
+theorem superlinear_iteration {p A B : ℝ} (hp : 1 < p) (hA : 0 ≤ A) (hB : 0 ≤ B) :
+    ∃ d : ℝ, 0 < d ∧ ∀ u : ℕ → ℝ, (∀ j, 0 ≤ u j) → u 0 ≤ d →
+      (∀ j, u (j + 1) ≤ A * B ^ j * (u j) ^ p) →
+      Tendsto u atTop (𝓝 0) := by
+  obtain ⟨t, ht, ht1, hBt⟩ := exists_small_power (A := B) hp (by norm_num : (0 : ℝ) < 1)
+  simp only [one_mul] at hBt
+  obtain ⟨d, hd, _, hAd⟩ := exists_small_power (A := A) hp ht
+  refine ⟨d, hd, ?_⟩
+  intro u hu hu0 hstep
+  have hbound : ∀ j, u j ≤ d * t ^ j := by
+    intro j
+    induction j with
+    | zero => simpa using hu0
+    | succ j ih =>
+      calc
+        u (j + 1) ≤ A * B ^ j * (u j) ^ p := hstep j
+        _ ≤ A * B ^ j * (d * t ^ j) ^ p := by gcongr; exact hu j
+        _ = (A * d ^ p) * (B * t ^ p) ^ j := by
+          rw [Real.mul_rpow hd.le (pow_nonneg ht.le _), ← Real.rpow_pow_comm ht.le, mul_pow]
+          ring
+        _ ≤ (t * d) * t ^ j := by
+          apply mul_le_mul hAd (pow_le_pow_left₀ (mul_nonneg hB (Real.rpow_nonneg ht.le _)) hBt j)
+            (pow_nonneg (mul_nonneg hB (Real.rpow_nonneg ht.le _)) _) (by positivity)
+        _ = d * t ^ (j + 1) := by rw [pow_succ]; ring
+  have hgeo := tendsto_pow_atTop_nhds_zero_of_lt_one ht.le ht1
+  have hlim : Tendsto (fun j : ℕ ↦ d * t ^ j) atTop (𝓝 0) := by simpa only [mul_zero] using tendsto_const_nhds.mul hgeo
+  exact squeeze_zero hu hbound hlim
+
+end RPIIteration
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+open scoped ENNReal Topology
+open scoped ENNReal Topology ContDiff
+open scoped Topology ENNReal
+namespace RPIIteration
+
+lemma exists_small_positive_sublevel {X : Type*} [MeasurableSpace X]
+    (μ : Measure X) [IsFiniteMeasure μ] (h : X → ℝ) (hm : AEMeasurable h μ)
+    (hp : ∀ᵐ x ∂μ, 0 < h x) {δ : ℝ} (hδ : 0 < δ) :
+    ∃ k : ℝ, 0 < k ∧ k ≤ 1 ∧ μ.real {x | h x < k} < δ := by
+  let q : ℕ → ℝ := fun j ↦ (1 / 2 : ℝ) ^ j
+  let A : ℕ → Set X := fun j ↦ {x | h x < q j}
+  have hq : Tendsto q atTop (𝓝 0) := tendsto_pow_atTop_nhds_zero_of_lt_one (by norm_num) (by norm_num)
+  have hAm (j : ℕ) : NullMeasurableSet (A j) μ := nullMeasurableSet_lt hm aemeasurable_const
+  have hAa : Antitone A := by
+    intro i j hij x hx
+    change h x < q i
+    change h x < q j at hx
+    exact hx.trans_le (pow_le_pow_of_le_one (by norm_num) (by norm_num) hij)
+  have hzero : μ (⋂ j, A j) = 0 := by
+    apply measure_eq_zero_iff_ae_notMem.mpr
+    filter_upwards [hp] with x hx
+    intro hxA
+    have hle : h x ≤ 0 := ge_of_tendsto hq (Eventually.of_forall fun j ↦ (show h x < q j from (mem_iInter.mp hxA) j).le)
+    exact (not_le_of_gt hx) hle
+  have ht := tendsto_measure_iInter_atTop hAm hAa ⟨0, measure_ne_top μ _⟩
+  rw [hzero] at ht
+  have hreal : Tendsto (fun j ↦ μ.real (A j)) atTop (𝓝 0) := by
+    simpa only [Measure.real, ENNReal.toReal_zero, Function.comp_def] using
+      (ENNReal.tendsto_toReal ENNReal.zero_ne_top).comp ht
+  obtain ⟨j, hj⟩ := (hreal.eventually (gt_mem_nhds hδ)).exists
+  exact ⟨q j, pow_pos (by norm_num) _, pow_le_one₀ (by norm_num) (by norm_num), hj⟩
+
+end RPIIteration
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+open scoped ENNReal Topology
+open scoped ENNReal Topology ContDiff
+open scoped Topology ENNReal
+namespace RPIIteration
+
+def radius (j : ℕ) : ℝ := 1 + (1 / 2 : ℝ) ^ j
+def level (a : ℝ) (j : ℕ) : ℝ := a / 2 * radius j
+
+lemma radius_bounds (j : ℕ) : 1 < radius j ∧ radius j ≤ 2 := by
+  have h0 : 0 < (1/2 : ℝ)^j := pow_pos (by norm_num) _
+  have h1 : (1/2 : ℝ)^j ≤ 1 := pow_le_one₀ (by norm_num) (by norm_num)
+  constructor <;> dsimp [radius] <;> linarith
+
+lemma radius_succ_lt (j : ℕ) : radius (j + 1) < radius j := by
+  dsimp [radius]
+  rw [pow_succ]
+  nlinarith [pow_pos (by norm_num : (0 : ℝ) < 1/2) j]
+
+lemma level_bounds {a : ℝ} (ha : 0 < a) (j : ℕ) :
+    0 < level a j ∧ a/2 < level a j ∧ level a j ≤ a := by
+  have hr := radius_bounds j
+  have hr0 : 0 < radius j := lt_trans zero_lt_one hr.1
+  dsimp [level]
+  constructor
+  · positivity
+  constructor
+  · nlinarith [hr.1]
+  · nlinarith [hr.2]
+
+lemma level_succ_lt {a : ℝ} (ha : 0 < a) (j : ℕ) : level a (j + 1) < level a j := by
+  exact mul_lt_mul_of_pos_left (radius_succ_lt j) (by positivity)
+
+lemma level_zero (a : ℝ) : level a 0 = a := by simp [level, radius]; ring
+
+lemma quotient_bound {a : ℝ} (ha : 0 < a) (j : ℕ) :
+    level a j / ((radius j - radius (j+1)) * (level a j - level a (j+1))) ≤ 8 * (4 : ℝ)^j := by
+  have hden : 0 < (radius j - radius (j+1)) * (level a j - level a (j+1)) :=
+    mul_pos (sub_pos.mpr (radius_succ_lt j)) (sub_pos.mpr (level_succ_lt ha j))
+  rw [div_le_iff₀ hden]
+  have heq : (radius j - radius (j+1)) * (level a j - level a (j+1)) = a/8 * ((1/2 : ℝ)^j)^2 := by
+    simp only [radius, level, pow_succ]
+    ring
+  have hpow : ((1/2 : ℝ)^j)^2 = ((1/2 : ℝ)^2)^j := by
+    rw [← pow_mul, ← pow_mul]
+    congr 1
+    omega
+  have h4 : (4 : ℝ)^j * ((1/2 : ℝ)^j)^2 = 1 := by
+    rw [hpow, ← mul_pow]
+    norm_num
+  rw [heq]
+  calc
+    _ ≤ a := (level_bounds ha j).2.2
+    _ = 8 * (4 : ℝ)^j * (a/8 * ((1/2 : ℝ)^j)^2) := by
+      calc
+        _ = a * ((4 : ℝ)^j * ((1/2 : ℝ)^j)^2) := by rw [h4, mul_one]
+        _ = _ := by ring
+
+lemma power_step {a C p : ℝ} (ha : 0 < a) (hC : 0 ≤ C) (hp : 0 ≤ p)
+    {U V : ℝ} (hU : 0 ≤ U) (j : ℕ)
+    (hstep : V ≤ C * (level a j / ((radius j - radius (j+1)) * (level a j - level a (j+1)))) ^ p * U ^ p) :
+    V ≤ (C * (8 : ℝ)^p) * ((4 : ℝ)^p)^j * U ^ p := by
+  have hpos : 0 ≤ level a j / ((radius j - radius (j+1)) * (level a j - level a (j+1))) := by
+    exact div_nonneg (level_bounds ha j).1.le
+      (mul_nonneg (sub_nonneg.mpr (radius_succ_lt j).le) (sub_nonneg.mpr (level_succ_lt ha j).le))
+  calc
+    _ ≤ C * (level a j / ((radius j - radius (j+1)) * (level a j - level a (j+1)))) ^ p * U ^ p := hstep
+    _ ≤ C * (8 * (4 : ℝ)^j)^p * U ^ p := by
+      gcongr
+      exact quotient_bound ha j
+    _ = _ := by
+      rw [Real.mul_rpow (by norm_num) (by positivity), ← Real.rpow_pow_comm (by norm_num : (0 : ℝ) ≤ 4)]
+      ring
+
+end RPIIteration
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+open scoped ENNReal Topology
+open scoped ENNReal Topology ContDiff
+open scoped Topology ENNReal
+namespace RPISingularHarnack
+open RPI RPI.Boundary
+
+theorem no_positive_bounded_supersolution_with_zero_vertex_inf
+    {n : ℕ} (hn : 8 ≤ n) (E : Set (Euc n))
+    (hmin : IsLocallyPerimeterMinimizing E) :
+    IsCone E → (0 : Euc n) ∈ perimeterSupport E → ∀ (h : Euc n → ℝ) (G : Euc n → Euc n),
+    WholeBoundarySobolev E h G → WholeBoundarySupersolution E G →
+    (∀ᵐ x ∂boundaryArea E, 0 < h x ∧ h x ≤ 1) → ZeroEssentialInfAtVertex E h → False := by
+  intro _ _ h G hSobolev hSuper hpositive hinf
+  classical
+  have hn2 : 2 < n := by omega
+  have hn1 : 1 < n := by omega
+  let m : ℝ := (n-1 : ℕ)
+  have hm : 0 < m := by dsimp [m]; exact_mod_cast (show 0 < n-1 by omega)
+  let p : ℝ := 1 + 1/m
+  have hp : 1 < p := by dsimp [p]; linarith [one_div_pos.mpr hm]
+  have hpk : (((n-1 : ℕ) : ℝ)-1) * p < (n-1 : ℕ) := by
+    have hcan : m * (1/m) = 1 := mul_one_div_cancel (ne_of_gt hm)
+    have hinv : 0 < 1/m := one_div_pos.mpr hm
+    change (m-1) * (1+1/m) < m
+    nlinarith
+  have hh01 : ∀ᵐ x ∂boundaryArea E, 0 ≤ h x ∧ h x ≤ 1 :=
+    hpositive.mono fun _ hx ↦ ⟨hx.1.le, hx.2⟩
+  obtain ⟨C, hC, hstep⟩ := wholeBoundary_sublevel_measure_step hn2 hmin hSobolev hSuper hh01 hp hpk
+  obtain ⟨δ, hδ, hiter⟩ := RPIIteration.superlinear_iteration (p := p)
+    (A := C * (8 : ℝ)^p) (B := (4 : ℝ)^p) hp (by positivity) (by positivity)
+  let μ := (boundaryArea E).restrict (Metric.ball 0 2)
+  let : IsFiniteMeasure μ := ⟨by
+    simpa only [μ, Measure.restrict_apply_univ] using minimizing_boundaryArea_ball_lt_top hn1 hmin 0 (by norm_num : (0 : ℝ) < 2)⟩
+  have hmμ : AEMeasurable h μ := (hSobolev 2 (by norm_num)).1.aestronglyMeasurable.aemeasurable
+  have hpμ : ∀ᵐ x ∂μ, 0 < h x := ae_restrict_of_ae (hpositive.mono fun _ hx ↦ hx.1)
+  obtain ⟨a, ha, ha1, hsmall⟩ := RPIIteration.exists_small_positive_sublevel μ h hmμ hpμ hδ
+  let u : ℕ → ℝ := fun j ↦ (boundaryArea E).real
+    (Metric.ball 0 (RPIIteration.radius j) ∩ {x | h x < RPIIteration.level a j})
+  have hu : ∀ j, 0 ≤ u j := fun _ ↦ measureReal_nonneg
+  have hu0 : u 0 ≤ δ := by
+    have hmA : NullMeasurableSet {x | h x < a} μ := nullMeasurableSet_lt hmμ aemeasurable_const
+    have heq : u 0 = μ.real {x | h x < a} := by
+      dsimp [u, Measure.real]
+      rw [RPIIteration.level_zero]
+      norm_num only [RPIIteration.radius, pow_zero] 
+      change (boundaryArea E (Metric.ball 0 2 ∩ {x | h x < a})).toReal = _
+      rw [Measure.restrict_apply₀ hmA, inter_comm]
+    rw [heq]
+    exact hsmall.le
+  have hrec (j : ℕ) : u (j+1) ≤ (C * (8 : ℝ)^p) * ((4 : ℝ)^p)^j * (u j)^p := by
+    apply RPIIteration.power_step ha hC.le (lt_trans zero_lt_one hp).le (hu j) j
+    exact hstep (RPIIteration.radius (j+1)) (RPIIteration.radius j)
+      (RPIIteration.level a (j+1)) (RPIIteration.level a j)
+      (lt_trans zero_lt_one (RPIIteration.radius_bounds (j+1)).1)
+      (RPIIteration.radius_succ_lt j) (RPIIteration.radius_bounds j).2
+      (RPIIteration.level_bounds ha (j+1)).1 (RPIIteration.level_succ_lt ha j)
+  have hlim := hiter u hu hu0 hrec
+  let A := Metric.ball (0 : Euc n) 1 ∩ {x | h x < a/2}
+  have hApos : 0 < boundaryArea E A := hinf 1 (by norm_num) (a/2) (by positivity)
+  have hAtop : boundaryArea E A ≠ ⊤ := ne_top_of_le_ne_top
+    (ne_of_lt (minimizing_boundaryArea_ball_lt_top hn1 hmin 0 (by norm_num : (0 : ℝ) < 1)))
+    (measure_mono inter_subset_left)
+  have hAr : 0 < (boundaryArea E).real A := ENNReal.toReal_pos hApos.ne' hAtop
+  have hlower (j : ℕ) : (boundaryArea E).real A ≤ u j := by
+    apply ENNReal.toReal_mono
+    · apply ne_top_of_le_ne_top
+        (ne_of_lt (minimizing_boundaryArea_ball_lt_top hn1 hmin 0
+          (lt_trans zero_lt_one (RPIIteration.radius_bounds j).1)))
+      exact measure_mono inter_subset_left
+    · apply measure_mono
+      intro x hx
+      exact ⟨Metric.ball_subset_ball (RPIIteration.radius_bounds j).1.le hx.1,
+        hx.2.trans (RPIIteration.level_bounds ha j).2.1⟩
+  have hzero : (boundaryArea E).real A ≤ 0 := ge_of_tendsto hlim (Eventually.of_forall hlower)
+  exact (not_le_of_gt hAr) hzero
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+open scoped ENNReal Topology
+open scoped ENNReal Topology ContDiff
+open scoped Topology ENNReal
+namespace RPISingularHarnack
+open RPI RPI.Boundary
+
+theorem minimizing_boundaryArea_ball_pos {n : ℕ} (hn : 1 < n)
+    {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    {x : Euc n} (hx : x ∈ perimeterSupport E) {r : ℝ} (hr : 0 < r) :
+    0 < boundaryArea E (Metric.ball x r) := by
+  obtain ⟨C, D, hC0, hCt, hD0, hDt, hc⟩ := minimizing_perimeter_boundaryArea_comparison hn
+  have hh := Measure.le_iff'.mp (hc E hmin).1 (Metric.ball x r)
+  rw [perimeterMeasure_open E Metric.isOpen_ball, Measure.smul_apply, smul_eq_mul] at hh
+  by_contra h
+  have hz : boundaryArea E (Metric.ball x r) = 0 := le_antisymm (le_of_not_gt h) bot_le
+  rw [hz, mul_zero] at hh
+  exact (not_le_of_gt (hx r hr)) hh
+
+theorem regular_ball_nonempty {n : ℕ} (hn : 8 ≤ n)
+    {E Z : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    (hZd : dimH Z ≤ (n - 8 : ℕ))
+    (hvertex : (0 : Euc n) ∈ perimeterSupport E) {r : ℝ} (hr : 0 < r) :
+    ((perimeterSupport E \ Z) ∩ Metric.ball 0 r).Nonempty := by
+  have hp := minimizing_boundaryArea_ball_pos (by omega : 1 < n) hmin hvertex hr
+  have hzs := boundaryArea_zero_of_singular_dim hn E Z hZd
+  by_contra he
+  have hsub : Metric.ball (0 : Euc n) r ∩ perimeterSupport E ⊆ Z := by
+    intro x hx
+    by_contra hxz
+    exact he ⟨x, ⟨⟨hx.2, hxz⟩, hx.1⟩⟩
+  have hb : boundaryArea E (Metric.ball 0 r) =
+      boundaryArea E (Metric.ball 0 r ∩ perimeterSupport E) := by
+    simp only [boundaryArea, Measure.restrict_apply measurableSet_ball,
+      Measure.restrict_apply (measurableSet_ball.inter (isClosed_perimeterSupport E).measurableSet),
+      inter_assoc, inter_self]
+  have hh := measure_mono (μ := boundaryArea E) hsub
+  rw [← hb, hzs] at hh
+  exact (not_le_of_gt hp) hh
+
+theorem zeroEssentialInf_of_median_small_infimum {n : ℕ} (hn : 8 ≤ n)
+    {E Z : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E)
+    (hZd : dimH Z ≤ (n - 8 : ℕ))
+    (hvertex : (0 : Euc n) ∈ perimeterSupport E)
+    {W : Euc n → ℝ} {D : Euc n → Euc n}
+    (htrace : ∀ x ∈ perimeterSupport E \ Z, HasC1TraceJetAt (perimeterSupport E) W D x)
+    (hmedian : ∀ r : ℝ, 0 < r → r ≤ 1 →
+      sInf (W '' ((perimeterSupport E \ Z) ∩ Metric.ball 0 r)) ≤ 3 * r) :
+    ZeroEssentialInfAtVertex E (fun x ↦ min (W x) 1) := by
+  classical
+  intro R hR ε hε
+  obtain ⟨r, hr, hrR, hrε, hr1⟩ : ∃ r : ℝ, 0 < r ∧ r < R ∧ 3 * r < ε ∧ r ≤ 1 := by
+    refine ⟨min (min (R/2) (ε/6)) (1/2), by positivity, ?_, ?_, ?_⟩
+    · have := (min_le_left (min (R/2) (ε/6)) (1/2)).trans (min_le_left (R/2) (ε/6))
+      linarith
+    · have := (min_le_left (min (R/2) (ε/6)) (1/2)).trans (min_le_right (R/2) (ε/6))
+      linarith
+    · exact (min_le_right _ _).trans (by norm_num)
+  obtain ⟨w, hw, hwε⟩ := exists_lt_of_csInf_lt
+    ((regular_ball_nonempty hn hmin hZd hvertex hr).image W) ((hmedian r hr hr1).trans_lt hrε)
+  obtain ⟨x, hx, rfl⟩ := hw
+  obtain ⟨U, hU, hxU, f, hf, he, hD⟩ := htrace x hx.1
+  have hfx : f x < ε := by rw [he ⟨hx.1.1, hxU⟩]; exact hwε
+  have hm : U ∩ Metric.ball (0 : Euc n) R ∩ {y | f y < ε} ∈ 𝓝 x :=
+    inter_mem (inter_mem (hU.mem_nhds hxU)
+      (Metric.isOpen_ball.mem_nhds (Metric.ball_subset_ball hrR.le hx.2)))
+      (((hf.contDiffAt (hU.mem_nhds hxU)).continuousAt).eventually (gt_mem_nhds hfx))
+  obtain ⟨s, hs, hsm⟩ := Metric.mem_nhds_iff.mp hm
+  have hsarea := minimizing_boundaryArea_ball_pos (by omega : 1 < n) hmin hx.1.1 hs
+  have hsub : Metric.ball x s ∩ perimeterSupport E ⊆
+      Metric.ball (0 : Euc n) R ∩ {y | min (W y) 1 < ε} := by
+    intro y hy
+    have hh := hsm hy.1
+    refine ⟨hh.1.2, ?_⟩
+    change min (W y) 1 < ε
+    apply lt_of_le_of_lt (min_le_left _ _)
+    rw [← he ⟨hy.2, hh.1.1⟩]
+    exact hh.2
+  have heq : boundaryArea E (Metric.ball x s ∩ perimeterSupport E) =
+      boundaryArea E (Metric.ball x s) := by
+    simp only [boundaryArea, Measure.restrict_apply measurableSet_ball,
+      Measure.restrict_apply (measurableSet_ball.inter (isClosed_perimeterSupport E).measurableSet),
+      inter_assoc, inter_self]
+  exact hsarea.trans_le (heq ▸ measure_mono (μ := boundaryArea E) hsub)
+
+end RPISingularHarnack
+
+end
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff
+open scoped Topology ENNReal NNReal
+open scoped ENNReal Topology ContDiff Pointwise
+open scoped Topology ENNReal ContDiff
+open scoped BoundedContinuousFunction
+open scoped ENNReal Topology
+open scoped ENNReal Topology ContDiff
+open scoped Topology ENNReal
+namespace RPISingularHarnack
+open RPI RPI.Boundary
+
+theorem no_regular_positive_supersolution_with_median_bound
+    {n : ℕ} (hn : 8 ≤ n) {E Z : Set (Euc n)}
+    (hmin : IsLocallyPerimeterMinimizing E) (hcone : IsCone E)
+    (hvertex : (0 : Euc n) ∈ perimeterSupport E)
+    (hZ : IsClosed Z) (hZd : dimH Z ≤ (n - 8 : ℕ))
+    (hreg : ∀ x ∈ perimeterSupport E \ Z, ∃ U : Set (Euc n), IsOpen U ∧ x ∈ U ∧
+      ∃ f : Euc n → ℝ, ContDiffOn ℝ ∞ f U ∧
+        (∀ y ∈ U, fderiv ℝ f y ≠ 0) ∧
+        (∀ y ∈ U, y ∈ perimeterSupport E ↔ f y = 0) ∧
+        (∀ y ∈ U, (f y < 0 → y ∈ E) ∧ (y ∈ E → f y ≤ 0)))
+    {W : Euc n → ℝ} {D : Euc n → Euc n}
+    (htrace : ∀ x ∈ perimeterSupport E \ Z, HasC1TraceJetAt (perimeterSupport E) W D x)
+    (hpositive : ∀ x ∈ perimeterSupport E \ Z, 0 < W x)
+    (hsup : ∀ φ : Euc n → ℝ, ContDiff ℝ ∞ φ → HasCompactSupport φ →
+      Disjoint (tsupport φ) Z → (∀ x, 0 ≤ φ x) →
+      0 ≤ ∫ x, inner ℝ (D x) (surfaceGradient (perimeterSupport E) φ x)
+        ∂surfaceMeasure (perimeterSupport E))
+    (hmedian : ∀ r : ℝ, 0 < r → r ≤ 1 →
+      sInf (W '' ((perimeterSupport E \ Z) ∩ Metric.ball 0 r)) ≤ 3 * r) : False := by
+  obtain ⟨hSob, hSuper⟩ :=
+    c1_min_wholeBoundarySobolev_and_supersolution_of_minimizing_regular_geometry
+      hn hmin hZ hZd hreg htrace (fun x hx ↦ (hpositive x hx).le) hsup
+  have hinf := zeroEssentialInf_of_median_small_infimum hn hmin hZd hvertex htrace hmedian
+  have hs : ∀ᵐ x ∂boundaryArea E, x ∈ perimeterSupport E :=
+    ae_restrict_mem (isClosed_perimeterSupport E).measurableSet
+  have hz : ∀ᵐ x ∂boundaryArea E, x ∉ Z := by
+    simpa only [ae_iff, not_not, Set.ofPred_mem_eq] using
+      boundaryArea_zero_of_singular_dim hn E Z hZd
+  have hp : ∀ᵐ x ∂boundaryArea E, 0 < min (W x) 1 ∧ min (W x) 1 ≤ 1 := by
+    filter_upwards [hs, hz] with x hx hxz
+    exact ⟨lt_min (hpositive x ⟨hx, hxz⟩) zero_lt_one, min_le_right _ _⟩
+  exact no_positive_bounded_supersolution_with_zero_vertex_inf hn E hmin hcone hvertex
+    (fun x ↦ min (W x) 1) (truncatedGradient W D) hSob hSuper hp hinf
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/Perimeter/Regularity/WindowAverage.lean b/lean/OAI/Geometry/Perimeter/Regularity/WindowAverage.lean
new file mode 100644
index 0000000000000000000000000000000000000000..719c9402b3f0e967bfbce817157f0545644b9cbb
--- /dev/null
+++ b/lean/OAI/Geometry/Perimeter/Regularity/WindowAverage.lean
@@ -0,0 +1,278 @@
+import OAI.Geometry.Perimeter.Regularity.EllipticLocality
+
+namespace OAI
+
+noncomputable section
+open Set Filter MeasureTheory
+open scoped Topology ENNReal NNReal ContDiff Convolution
+namespace RPISingularHarnack
+open RPI.Boundary
+
+def lineWindow (T : ℝ) : ℝ → ℝ := (Icc (-T) T).indicator (fun _ ↦ 1)
+
+theorem lineWindow_measurable (T : ℝ) : Measurable (lineWindow T) :=
+  measurable_const.indicator measurableSet_Icc
+
+theorem lineWindow_integrable (T : ℝ) : Integrable (lineWindow T) volume :=
+  continuous_const.integrableOn_Icc.integrable_indicator measurableSet_Icc
+
+theorem lineWindow_lintegral (T : ℝ) : (∫⁻ s : ℝ, ‖lineWindow T s‖ₑ) =
+    ENNReal.ofReal (2*T) := by
+  have he : (fun s ↦ ‖lineWindow T s‖ₑ) = (Icc (-T) T).indicator (fun _ ↦ (1:ℝ≥0∞)) := by
+    funext s
+    by_cases hs : s ∈ Icc (-T) T <;> simp [lineWindow,hs]
+  rw [he,lintegral_indicator measurableSet_Icc]
+  simp only [lintegral_const,Measure.restrict_apply_univ,one_mul,Real.volume_Icc]
+  congr 1
+  ring
+
+theorem lineWindow_translation_le (T t : ℝ) :
+    (∫⁻ s : ℝ, ‖lineWindow T (s-t)-lineWindow T s‖ₑ) ≤ ENNReal.ofReal (4*|t|) := by
+  let J := Icc (-T-|t|) (-T+|t|)
+  let K := Icc (T-|t|) (T+|t|)
+  have hpoint (s : ℝ) : |lineWindow T (s-t)-lineWindow T s| ≤
+      J.indicator (fun _ ↦ (1:ℝ)) s + K.indicator (fun _ ↦ (1:ℝ)) s := by
+    have hJ0 : 0 ≤ J.indicator (fun _ ↦ (1:ℝ)) s := indicator_nonneg (fun _ _ ↦ zero_le_one) s
+    have hK0 : 0 ≤ K.indicator (fun _ ↦ (1:ℝ)) s := indicator_nonneg (fun _ _ ↦ zero_le_one) s
+    have hab := abs_nonneg t
+    have htl := neg_abs_le t
+    have htu := le_abs_self t
+    by_cases h₁ : s-t ∈ Icc (-T) T
+    · by_cases h₂ : s ∈ Icc (-T) T
+      · simp only [lineWindow,indicator_of_mem h₁,indicator_of_mem h₂,sub_self,abs_zero]
+        exact add_nonneg hJ0 hK0
+      · have hjk : s ∈ J ∨ s ∈ K := by
+          have h₁a := h₁.1
+          have h₁b := h₁.2
+          simp only [mem_Icc,not_and_or,not_le] at h₂
+          rcases h₂ with h₂ | h₂
+          · exact Or.inl ⟨by linarith,by linarith⟩
+          · exact Or.inr ⟨by linarith,by linarith⟩
+        simp only [lineWindow,indicator_of_mem h₁,indicator_of_notMem h₂,sub_zero,abs_one]
+        rcases hjk with hj | hk
+        · rw [indicator_of_mem hj]; linarith
+        · rw [indicator_of_mem hk]; linarith
+    · by_cases h₂ : s ∈ Icc (-T) T
+      · have hjk : s ∈ J ∨ s ∈ K := by
+          have h₂a := h₂.1
+          have h₂b := h₂.2
+          simp only [mem_Icc,not_and_or,not_le] at h₁
+          rcases h₁ with h₁ | h₁
+          · exact Or.inl ⟨by linarith,by linarith⟩
+          · exact Or.inr ⟨by linarith,by linarith⟩
+        simp only [lineWindow,indicator_of_notMem h₁,indicator_of_mem h₂,zero_sub,abs_neg,abs_one]
+        rcases hjk with hj | hk
+        · rw [indicator_of_mem hj]; linarith
+        · rw [indicator_of_mem hk]; linarith
+      · simp only [lineWindow,indicator_of_notMem h₁,indicator_of_notMem h₂,sub_self,abs_zero]
+        exact add_nonneg hJ0 hK0
+  have hd : Integrable (fun s ↦ lineWindow T (s-t)-lineWindow T s) volume :=
+    ((lineWindow_integrable T).comp_sub_right t).sub (lineWindow_integrable T)
+  have hJ : Integrable (J.indicator (fun _ ↦ (1:ℝ))) volume :=
+    continuous_const.integrableOn_Icc.integrable_indicator measurableSet_Icc
+  have hK : Integrable (K.indicator (fun _ ↦ (1:ℝ))) volume :=
+    continuous_const.integrableOn_Icc.integrable_indicator measurableSet_Icc
+  rw [← ofReal_integral_norm_eq_lintegral_enorm hd]
+  apply ENNReal.ofReal_le_ofReal
+  calc
+    _ ≤ ∫ s : ℝ, J.indicator (fun _ ↦ (1:ℝ)) s + K.indicator (fun _ ↦ (1:ℝ)) s :=
+      integral_mono hd.norm (hJ.add hK) hpoint
+    _ = 4*|t| := by
+      rw [integral_add hJ hK,integral_indicator measurableSet_Icc,integral_indicator measurableSet_Icc]
+      simp only [setIntegral_const,smul_eq_mul,mul_one,Measure.real,Real.volume_Icc]
+      rw [show -T+|t|-(-T-|t|) = 2*|t| by ring,
+        show T+|t|-(T-|t|) = 2*|t| by ring,ENNReal.toReal_ofReal (by positivity)]
+      ring
+
+def normalWindowAverage {n : ℕ} (v : Euc n) (T : ℝ) (f : Euc n → ℝ) (x : Euc n) : ℝ :=
+  ∫ s : ℝ, lineWindow T s * f (x+s•v)
+
+theorem normalWindowAverage_stronglyMeasurable {n : ℕ} {f : Euc n → ℝ}
+    (hf : Measurable f) (v : Euc n) (T : ℝ) :
+    StronglyMeasurable (normalWindowAverage v T f) := by
+  apply StronglyMeasurable.integral_prod_right
+  exact (((lineWindow_measurable T).comp measurable_snd).mul
+    (hf.comp (continuous_fst.add (continuous_snd.smul continuous_const)).measurable)).stronglyMeasurable
+
+theorem integrable_window_fiber {n : ℕ} {f : Euc n → ℝ} (hf : Measurable f)
+    {C : ℝ} (hb : ∀ x, ‖f x‖ ≤ C) (v : Euc n) (T : ℝ) (x : Euc n) :
+    Integrable (fun s : ℝ ↦ lineWindow T s * f (x+s•v)) volume := by
+  have hI : Integrable ((Icc (-T) T).indicator (fun _ ↦ C)) volume :=
+    continuous_const.integrableOn_Icc.integrable_indicator measurableSet_Icc
+  apply hI.mono' (((lineWindow_measurable T).mul
+    (hf.comp (continuous_const.add (continuous_id.smul continuous_const)).measurable)).aestronglyMeasurable)
+  apply ae_of_all
+  intro s
+  by_cases hs : s ∈ Icc (-T) T
+  · simpa [lineWindow,hs] using hb (x+s•v)
+  · simp [lineWindow,hs]
+
+theorem normalWindowAverage_lintegral_le {n : ℕ} {f : Euc n → ℝ}
+    (hf : Measurable f) (v : Euc n) (T : ℝ) :
+    (∫⁻ x, ‖normalWindowAverage v T f x‖ₑ) ≤
+      ENNReal.ofReal (2*T) * ∫⁻ x, ‖f x‖ₑ := by
+  have hm : Measurable (fun p : Euc n × ℝ ↦ ‖lineWindow T p.2‖ₑ * ‖f (p.1+p.2•v)‖ₑ) :=
+    ((lineWindow_measurable T).comp measurable_snd).enorm.mul
+      (hf.comp (continuous_fst.add (continuous_snd.smul continuous_const)).measurable).enorm
+  calc
+    _ ≤ ∫⁻ x, ∫⁻ s : ℝ, ‖lineWindow T s‖ₑ * ‖f (x+s•v)‖ₑ := by
+      apply lintegral_mono
+      intro x
+      exact (enorm_integral_le_lintegral_enorm _).trans_eq
+        (lintegral_congr (fun _ ↦ enorm_mul _ _))
+    _ = ∫⁻ s : ℝ, ∫⁻ x, ‖lineWindow T s‖ₑ * ‖f (x+s•v)‖ₑ :=
+      lintegral_lintegral_swap hm.aemeasurable
+    _ = _ := by
+      have he (s : ℝ) : (∫⁻ x, ‖lineWindow T s‖ₑ * ‖f (x+s•v)‖ₑ) =
+          ‖lineWindow T s‖ₑ * ∫⁻ x, ‖f x‖ₑ := by
+        rw [lintegral_const_mul _ (show Measurable (fun x : Euc n ↦ ‖f (x+s•v)‖ₑ) from
+          (hf.comp (measurable_id.add measurable_const)).enorm)]
+        rw [lintegral_add_right_eq_self (fun x : Euc n ↦ ‖f x‖ₑ)]
+      simp_rw [he]
+      rw [lintegral_mul_const _ (lineWindow_measurable T).enorm,lineWindow_lintegral]
+
+theorem normalWindowAverage_translation_le {n : ℕ} {f : Euc n → ℝ}
+    (hf : Measurable f) {C : ℝ} (hb : ∀ x, ‖f x‖ ≤ C)
+    (v : Euc n) (T : ℝ) (w : Euc n) :
+    (∫⁻ x, ‖normalWindowAverage v T f (x+w)-normalWindowAverage v T f x‖ₑ) ≤
+      ENNReal.ofReal (2*T) * ∫⁻ x, ‖f (x+w)-f x‖ₑ := by
+  have he (x : Euc n) : normalWindowAverage v T f (x+w)-normalWindowAverage v T f x =
+      normalWindowAverage v T (fun y ↦ f (y+w)-f y) x := by
+    unfold normalWindowAverage
+    rw [← integral_sub (integrable_window_fiber hf hb v T (x+w))
+      (integrable_window_fiber hf hb v T x)]
+    apply integral_congr_ae
+    apply ae_of_all
+    intro s
+    have he : x+w+s•v = (x+s•v)+w := by abel
+    simp only [he,mul_sub]
+  simp_rw [he]
+  exact normalWindowAverage_lintegral_le ((hf.comp (measurable_id.add measurable_const)).sub hf) v T
+
+theorem normalWindowAverage_add_smul {n : ℕ} (v : Euc n) (T : ℝ)
+    (f : Euc n → ℝ) (x : Euc n) (t : ℝ) :
+    normalWindowAverage v T f (x+t•v) =
+      ∫ s : ℝ, lineWindow T (s-t)*f (x+s•v) := by
+  unfold normalWindowAverage
+  have he : (fun s : ℝ ↦ lineWindow T (s-t)*f (x+s•v)) =
+      (fun s : ℝ ↦ lineWindow T (s-t)*f (x+t•v+(s-t)•v)) := by
+    funext s
+    congr 2
+    rw [sub_smul]
+    abel
+  rw [he]
+  exact (integral_sub_right_eq_self (fun s : ℝ ↦ lineWindow T s*f (x+t•v+s•v)) t).symm
+
+theorem normalWindowAverage_normal_translation_le {n : ℕ} {f : Euc n → ℝ}
+    (hf : Measurable f) {C : ℝ} (hb : ∀ x, ‖f x‖ ≤ C)
+    (v : Euc n) (T t : ℝ) :
+    (∫⁻ x, ‖normalWindowAverage v T f (x+t•v)-normalWindowAverage v T f x‖ₑ) ≤
+      ENNReal.ofReal (4*|t|) * ∫⁻ x, ‖f x‖ₑ := by
+  have hishift (x : Euc n) : Integrable (fun s : ℝ ↦ lineWindow T (s-t)*f (x+s•v)) volume := by
+    have hi := (integrable_window_fiber hf hb v T (x+t•v)).comp_sub_right t
+    convert hi using 1
+    funext s
+    congr 2
+    rw [sub_smul]
+    abel
+  have he (x : Euc n) : normalWindowAverage v T f (x+t•v)-normalWindowAverage v T f x =
+      ∫ s : ℝ, (lineWindow T (s-t)-lineWindow T s)*f (x+s•v) := by
+    rw [normalWindowAverage_add_smul,normalWindowAverage,
+      ← integral_sub (hishift x) (integrable_window_fiber hf hb v T x)]
+    apply integral_congr_ae
+    exact ae_of_all _ fun s ↦ (sub_mul _ _ _).symm
+  let d : ℝ → ℝ := fun s ↦ lineWindow T (s-t)-lineWindow T s
+  have hd : Measurable d := ((lineWindow_measurable T).comp
+    (measurable_id.sub measurable_const)).sub (lineWindow_measurable T)
+  have hm : Measurable (fun p : Euc n × ℝ ↦ ‖d p.2‖ₑ * ‖f (p.1+p.2•v)‖ₑ) :=
+    (hd.comp measurable_snd).enorm.mul
+      (hf.comp (continuous_fst.add (continuous_snd.smul continuous_const)).measurable).enorm
+  calc
+    _ ≤ ∫⁻ x, ∫⁻ s : ℝ, ‖d s‖ₑ * ‖f (x+s•v)‖ₑ := by
+      apply lintegral_mono
+      intro x
+      change ‖normalWindowAverage v T f (x+t•v)-normalWindowAverage v T f x‖ₑ ≤ _
+      rw [he]
+      exact (enorm_integral_le_lintegral_enorm _).trans_eq
+        (lintegral_congr (fun _ ↦ enorm_mul _ _))
+    _ = ∫⁻ s : ℝ, ∫⁻ x, ‖d s‖ₑ * ‖f (x+s•v)‖ₑ :=
+      lintegral_lintegral_swap hm.aemeasurable
+    _ = (∫⁻ s : ℝ, ‖d s‖ₑ) * ∫⁻ x, ‖f x‖ₑ := by
+      have he' (s : ℝ) : (∫⁻ x, ‖d s‖ₑ * ‖f (x+s•v)‖ₑ) =
+          ‖d s‖ₑ * ∫⁻ x, ‖f x‖ₑ := by
+        rw [lintegral_const_mul _ (show Measurable (fun x : Euc n ↦ ‖f (x+s•v)‖ₑ) from
+          (hf.comp (measurable_id.add measurable_const)).enorm)]
+        rw [lintegral_add_right_eq_self (fun x : Euc n ↦ ‖f x‖ₑ)]
+      simp_rw [he']
+      exact lintegral_mul_const _ hd.enorm
+    _ ≤ _ := mul_le_mul_left (lineWindow_translation_le T t) _
+
+theorem translation_lintegral_add_le {n : ℕ} {f : Euc n → ℝ}
+    (hf : Measurable f) (a b : Euc n) :
+    (∫⁻ x, ‖f (x+(a+b))-f x‖ₑ) ≤
+      (∫⁻ x, ‖f (x+a)-f x‖ₑ) + ∫⁻ x, ‖f (x+b)-f x‖ₑ := by
+  calc
+    _ ≤ ∫⁻ x, ‖f ((x+b)+a)-f (x+b)‖ₑ + ‖f (x+b)-f x‖ₑ := by
+      apply lintegral_mono
+      intro x
+      have he : f (x+(a+b))-f x =
+          (f ((x+b)+a)-f (x+b)) + (f (x+b)-f x) := by
+        abel_nf
+      change ‖f (x+(a+b))-f x‖ₑ ≤ _
+      rw [he]
+      exact enorm_add_le _ _
+    _ = _ := by
+      rw [lintegral_add_left (show Measurable (fun x : Euc n ↦ ‖f ((x+b)+a)-f (x+b)‖ₑ) from
+        ((hf.comp ((measurable_id.add measurable_const).add measurable_const)).sub
+          (hf.comp (measurable_id.add measurable_const))).enorm)]
+      rw [lintegral_add_right_eq_self (fun x : Euc n ↦ ‖f (x+a)-f x‖ₑ)]
+
+theorem normalWindowAverage_all_translation_le {n : ℕ} {f : Euc n → ℝ}
+    (hf : Measurable f) {C : ℝ} (hb : ∀ x, ‖f x‖ ≤ C)
+    {v : Euc n} (hv : ‖v‖ = 1) {T H V : ℝ}
+    (hT : 0 ≤ T) (hH : 0 ≤ H) (hV : 0 ≤ V)
+    (hmass : (∫⁻ x, ‖f x‖ₑ) ≤ ENNReal.ofReal V)
+    (hhor : ∀ z, inner ℝ v z = 0 →
+      (∫⁻ x, ‖f (x+z)-f x‖ₑ) ≤ ENNReal.ofReal (‖z‖*H))
+    (w : Euc n) :
+    (∫⁻ x, ‖normalWindowAverage v T f (x+w)-normalWindowAverage v T f x‖ₑ) ≤
+      ENNReal.ofReal (‖w‖*(4*T*H+4*V)) := by
+  let t : ℝ := inner ℝ v w
+  let z : Euc n := w-t•v
+  have ht : |t| ≤ ‖w‖ := by simpa only [hv,one_mul] using abs_real_inner_le_norm v w
+  have hz : inner ℝ v z = 0 := by
+    dsimp [z,t]
+    rw [inner_sub_right,inner_smul_right,real_inner_self_eq_norm_sq,hv]
+    ring
+  have hnz : ‖z‖ ≤ 2*‖w‖ := by
+    calc
+      _ ≤ ‖w‖ + ‖t•v‖ := norm_sub_le _ _
+      _ = ‖w‖+|t| := by rw [norm_smul,Real.norm_eq_abs,hv,mul_one]
+      _ ≤ _ := by linarith
+  have he : w = z+t•v := by dsimp [z]; abel
+  calc
+    _ ≤ (∫⁻ x, ‖normalWindowAverage v T f (x+z)-normalWindowAverage v T f x‖ₑ) +
+        ∫⁻ x, ‖normalWindowAverage v T f (x+t•v)-normalWindowAverage v T f x‖ₑ := by
+      conv_lhs => rw [he]
+      exact translation_lintegral_add_le (normalWindowAverage_stronglyMeasurable hf v T).measurable z (t•v)
+    _ ≤ ENNReal.ofReal (2*T) * ENNReal.ofReal (‖z‖*H) +
+        ENNReal.ofReal (4*|t|)*ENNReal.ofReal V := by
+      exact add_le_add ((normalWindowAverage_translation_le hf hb v T z).trans
+        (mul_le_mul_right (hhor z hz) _))
+        ((normalWindowAverage_normal_translation_le hf hb v T t).trans
+          (mul_le_mul_right hmass _))
+    _ = ENNReal.ofReal ((2*T)*(‖z‖*H)+(4*|t|)*V) := by
+      rw [← ENNReal.ofReal_mul (by positivity),← ENNReal.ofReal_mul (by positivity),
+        ← ENNReal.ofReal_add (by positivity) (by positivity)]
+    _ ≤ _ := by
+      apply ENNReal.ofReal_le_ofReal
+      have h₁ : (2*T)*(‖z‖*H) ≤ (2*T)*(2*‖w‖*H) :=
+        mul_le_mul_of_nonneg_left (mul_le_mul_of_nonneg_right hnz hH) (by positivity)
+      have h₂ : (4*|t|)*V ≤ (4*‖w‖)*V := by gcongr
+      nlinarith
+
+end RPISingularHarnack
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/AETilingCountable.lean b/lean/OAI/Geometry/PeriodicTiling/AETilingCountable.lean
new file mode 100644
index 0000000000000000000000000000000000000000..0b72dcffc687d7b2636e70b1b831743435ffefec
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/AETilingCountable.lean
@@ -0,0 +1,55 @@
+import OAI.Geometry.PeriodicTiling.SeparatedCenters
+import Mathlib.Algebra.Order.Floor.Ring
+import Mathlib.Data.Set.Countable
+
+namespace OAI
+
+noncomputable section
+
+namespace PeriodicTilingThree
+
+open Set MeasureTheory
+
+theorem AETiles.eq_of_voxel_close {d : ℕ} {F : Finset (Lattice d)}
+    {A : Set (Space d)} (h : AETiles (Thickening F) A)
+    {z : Lattice d} (hz : z ∈ F) {a b : A}
+    (hc : ∀ i, |(b : Space d) i - (a : Space d) i| < 1) : a = b := by
+  by_contra hne
+  have hpos := unitVoxel_inter_volume_pos
+    ((a : Space d) + castLattice z) ((b : Space d) + castLattice z) (by
+      intro i
+      simpa only [Pi.add_apply, add_sub_add_right_eq_sub] using hc i)
+  have hsub : ∀ c : A, unitVoxel ((c : Space d) + castLattice z) ⊆
+      {x | x - (c : Space d) ∈ Thickening F} := by
+    intro c x hx
+    refine ⟨z, hz, ?_⟩
+    simpa only [mem_unitVoxel, sub_add_eq_sub_sub] using hx
+  have hnull := (h.aedisjoint hne).mono (hsub a) (hsub b)
+  exact (ne_of_gt hpos) hnull
+
+theorem AETiles.floor_injective {d : ℕ} {F : Finset (Lattice d)}
+    {A : Set (Space d)} (h : AETiles (Thickening F) A) (hF : F.Nonempty) :
+    Function.Injective (fun a : A => (fun i => ⌊(a : Space d) i⌋ : Lattice d)) := by
+  obtain ⟨z, hz⟩ := hF
+  intro a b he
+  apply h.eq_of_voxel_close hz
+  intro i
+  have hi : ⌊(a : Space d) i⌋ = ⌊(b : Space d) i⌋ := congrFun he i
+  have ha := Int.floor_le ((a : Space d) i)
+  have ha' := Int.lt_floor_add_one ((a : Space d) i)
+  have hb := Int.floor_le ((b : Space d) i)
+  have hb' := Int.lt_floor_add_one ((b : Space d) i)
+  rw [hi] at ha ha'
+  apply abs_lt.mpr
+  constructor <;> linarith
+
+theorem AETiles.countable_of_thickening {d : ℕ} {F : Finset (Lattice d)}
+    {A : Set (Space d)} (h : AETiles (Thickening F) A) (hF : F.Nonempty) :
+    A.Countable :=
+  Set.countable_coe_iff.mp (h.floor_injective hF).countable
+
+end PeriodicTilingThree
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/Activation.lean b/lean/OAI/Geometry/PeriodicTiling/Activation.lean
new file mode 100644
index 0000000000000000000000000000000000000000..c183cd0e139b3251a136f9a5b957b4f95b9662d4
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/Activation.lean
@@ -0,0 +1,354 @@
+import OAI.Geometry.PeriodicTiling.ShiftMultiplicity
+import Mathlib.Algebra.BigOperators.Ring.Finset
+import Mathlib.Data.Finset.Max
+import Mathlib.Data.Fintype.BigOperators
+import Mathlib.Tactic.Linarith
+import Lean.Elab.Tactic.Omega
+import Mathlib.Tactic.Ring
+import OAI.Geometry.PeriodicTiling.ActivationTiles
+import OAI.Geometry.PeriodicTiling.Dependence
+import Mathlib.Logic.Equiv.Prod
+
+namespace OAI
+
+universe uα uJ uC uK uZ
+
+noncomputable section
+
+namespace PeriodicTilingThree
+
+open scoped BigOperators
+
+def MixedDifferenceZero {a b : ℕ} (Q : ZMod a × ZMod b → ℤ) : Prop :=
+  ∀ x y, Q (x, y) - Q (x - 1, y) - Q (x, y - 1) + Q (x - 1, y - 1) = 0
+
+private theorem zmod_eq_zero_of_pred_invariant {n : ℕ} [NeZero n] {α : Type uα}
+    (f : ZMod n → α) (hf : ∀ x, f x = f (x - 1)) (x : ZMod n) : f x = f 0 := by
+  have hnat : ∀ m : ℕ, f (m : ZMod n) = f 0 := by
+    intro m
+    induction m with
+    | zero => simp only [Nat.cast_zero]
+    | succ m ih =>
+        calc
+          f ((m + 1 : ℕ) : ZMod n) = f (m : ZMod n) := by
+            simpa only [Nat.cast_add, Nat.cast_one, add_sub_cancel_right] using
+              hf ((m : ZMod n) + 1)
+          _ = f 0 := ih
+  simpa only [ZMod.natCast_zmod_val] using hnat x.val
+
+theorem mixedDifference_rectangle {a b : ℕ} [NeZero a] [NeZero b]
+    (Q : ZMod a × ZMod b → ℤ) (hmixed : MixedDifferenceZero Q)
+    (x x₀ : ZMod a) (y : ZMod b) :
+    Q (x, y) - Q (x₀, y) = Q (x, 0) - Q (x₀, 0) := by
+  have hfirst (z : ZMod a) :
+      Q (z, y) - Q (z - 1, y) = Q (z, 0) - Q (z - 1, 0) := by
+    apply zmod_eq_zero_of_pred_invariant
+      (fun w : ZMod b => Q (z, w) - Q (z - 1, w))
+    intro w
+    have h := hmixed z w
+    linarith only [h]
+  have hrow (z : ZMod a) :
+      Q (z, y) - Q (z, 0) = Q (0, y) - Q (0, 0) := by
+    apply zmod_eq_zero_of_pred_invariant
+      (fun w : ZMod a => Q (w, y) - Q (w, 0))
+    intro w
+    have h := hfirst w
+    linarith only [h]
+  linarith only [hrow x, hrow x₀]
+
+theorem histogram_separable {a b : ℕ} [NeZero a] [NeZero b]
+    (Q : ZMod a × ZMod b → ℕ)
+    (hmixed : MixedDifferenceZero (fun z => (Q z : ℤ))) :
+    ∃ u : ZMod a → ℕ, ∃ v : ZMod b → ℕ, ∀ x y, Q (x, y) = u x + v y := by
+  classical
+  obtain ⟨x₀, _hx₀, hmin⟩ := Finset.exists_min_image
+    (Finset.univ : Finset (ZMod a)) (fun x => Q (x, 0)) Finset.univ_nonempty
+  let u : ZMod a → ℕ := fun x => Q (x, 0) - Q (x₀, 0)
+  let v : ZMod b → ℕ := fun y => Q (x₀, y)
+  refine ⟨u, v, ?_⟩
+  intro x y
+  have hle : Q (x₀, 0) ≤ Q (x, 0) := hmin x (Finset.mem_univ x)
+  have hrectangle := mixedDifference_rectangle
+    (fun z => (Q z : ℤ)) hmixed x x₀ y
+  dsimp [u, v]
+  omega
+
+theorem histogram_mass_eq {a b : ℕ} [NeZero a] [NeZero b]
+    (Q : ZMod a × ZMod b → ℕ) (u : ZMod a → ℕ) (v : ZMod b → ℕ)
+    (hsep : ∀ x y, Q (x, y) = u x + v y) :
+    (∑ z, Q z) = b * (∑ x, u x) + a * (∑ y, v y) := by
+  calc
+    (∑ z, Q z) = ∑ x : ZMod a, ∑ y : ZMod b, (u x + v y) := by
+      rw [Fintype.sum_prod_type]
+      apply Finset.sum_congr rfl
+      intro x _
+      apply Finset.sum_congr rfl
+      intro y _
+      exact hsep x y
+    _ = b * (∑ x, u x) + a * (∑ y, v y) := by
+      simp [Finset.sum_add_distrib, ← Finset.mul_sum]
+
+theorem histogram_mass_dvd {a b D : ℕ} [NeZero a] [NeZero b]
+    (Q : ZMod a × ZMod b → ℕ)
+    (hmixed : MixedDifferenceZero (fun z => (Q z : ℤ)))
+    (hmass : (∑ z, Q z) = D) (hlarge : D < b) : a ∣ D := by
+  obtain ⟨u, v, hsep⟩ := histogram_separable Q hmixed
+  have htotal : D = b * (∑ x, u x) + a * (∑ y, v y) := by
+    rw [← hmass]
+    exact histogram_mass_eq Q u v hsep
+  have hu : (∑ x, u x) = 0 := by
+    by_contra hne
+    have hone : 1 ≤ ∑ x, u x := Nat.one_le_iff_ne_zero.mpr hne
+    have hmul : b ≤ b * (∑ x, u x) := by
+      simpa only [Nat.mul_one] using Nat.mul_le_mul_left b hone
+    omega
+  refine ⟨∑ y, v y, ?_⟩
+  simpa only [hu, Nat.mul_zero, Nat.zero_add] using htotal
+
+private theorem sum_histogram_corner {a b : ℕ} [NeZero a] [NeZero b]
+    (f : ZMod a × ZMod b → ℤ) (x : ZMod a) (y : ZMod b) :
+    (∑ e : ZMod a × ZMod b,
+      ((if e.1 = x then (1 : ℤ) else 0) * (if e.2 = y then 1 else 0)) * f e) =
+        f (x, y) := by
+  classical
+  have hterm (e : ZMod a × ZMod b) :
+      ((if e.1 = x then (1 : ℤ) else 0) * (if e.2 = y then 1 else 0)) * f e =
+        if e = (x, y) then f e else 0 := by
+    rcases e with ⟨x', y'⟩
+    by_cases hx : x' = x <;> by_cases hy : y' = y <;>
+      simp [hx, hy, Prod.mk.injEq]
+  simp_rw [hterm]
+  simp
+
+theorem sum_shiftOmega_mul {a b : ℕ} [NeZero a] [NeZero b]
+    (Q : ZMod a × ZMod b → ℤ) (x : ZMod a) (y : ZMod b) :
+    (∑ e : ZMod a × ZMod b, shiftOmega a b e * Q ((x, y) - e)) =
+      Q (x, y) - Q (x - 1, y) - Q (x, y - 1) + Q (x - 1, y - 1) := by
+  classical
+  have hexpand (e : ZMod a × ZMod b) :
+      shiftOmega a b e * Q ((x, y) - e) =
+        ((if e.1 = 0 then (1 : ℤ) else 0) * (if e.2 = 0 then 1 else 0)) *
+            Q ((x, y) - e) -
+          ((if e.1 = 1 then (1 : ℤ) else 0) * (if e.2 = 0 then 1 else 0)) *
+            Q ((x, y) - e) -
+          ((if e.1 = 0 then (1 : ℤ) else 0) * (if e.2 = 1 then 1 else 0)) *
+            Q ((x, y) - e) +
+          ((if e.1 = 1 then (1 : ℤ) else 0) * (if e.2 = 1 then 1 else 0)) *
+            Q ((x, y) - e) := by
+    unfold shiftOmega
+    ring
+  simp_rw [hexpand]
+  rw [Finset.sum_add_distrib, Finset.sum_sub_distrib, Finset.sum_sub_distrib]
+  simp only [sum_histogram_corner, Prod.mk_sub_mk, sub_zero]
+
+theorem mixedDifferenceZero_of_shift_sum {a b D : ℕ} [NeZero a] [NeZero b]
+    (ha : 2 ≤ a) (hb : 2 ≤ b) (Q : ZMod a × ZMod b → ℕ)
+    (hmass : (∑ z, Q z) = D)
+    (hcoverage : ∀ c : ZMod a × ZMod b,
+      (∑ δ : ShiftIndex a b, Q (c - shift a b δ)) = D) :
+    MixedDifferenceZero (fun z => (Q z : ℤ)) := by
+  classical
+  intro x y
+  have hcount :
+      (∑ δ : ShiftIndex a b, (Q ((x, y) - shift a b δ) : ℤ)) = (D : ℤ) := by
+    exact_mod_cast hcoverage (x, y)
+  rw [sum_shift (fun e : ZMod a × ZMod b => (Q ((x, y) - e) : ℤ))] at hcount
+  simp only [nsmul_eq_mul] at hcount
+  simp_rw [shiftMultiplicity_cast ha hb] at hcount
+  simp only [add_mul, one_mul, Finset.sum_add_distrib] at hcount
+  have hbase : (∑ e : ZMod a × ZMod b, (Q ((x, y) - e) : ℤ)) = (D : ℤ) := by
+    have hinv : Function.Involutive (fun e : ZMod a × ZMod b => (x, y) - e) :=
+      fun e => sub_sub_cancel (x, y) e
+    calc
+      (∑ e : ZMod a × ZMod b, (Q ((x, y) - e) : ℤ)) = ∑ e, (Q e : ℤ) :=
+        hinv.bijective.sum_comp (fun e => (Q e : ℤ))
+      _ = (D : ℤ) := by exact_mod_cast hmass
+  rw [hbase, sum_shiftOmega_mul (fun z => (Q z : ℤ)) x y] at hcount
+  linarith only [hcount]
+
+theorem histogram_mass_dvd_of_shift_sum {a b D : ℕ} [NeZero a] [NeZero b]
+    (ha : 2 ≤ a) (hb : 2 ≤ b) (Q : ZMod a × ZMod b → ℕ)
+    (hmass : (∑ z, Q z) = D) (hlarge : D < b)
+    (hcoverage : ∀ c : ZMod a × ZMod b,
+      (∑ δ : ShiftIndex a b, Q (c - shift a b δ)) = D) : a ∣ D :=
+  histogram_mass_dvd Q (mixedDifferenceZero_of_shift_sum ha hb Q hmass hcoverage)
+    hmass hlarge
+
+theorem sum_firstCoordinate_of_bijective
+    {J : Type uJ} {C : Type uC} {K : Type uK} [Fintype J] [Fintype C] [Fintype K] [DecidableEq C]
+    (f : J → C × K) (hf : Function.Bijective f) (c : C) :
+    (∑ j, if (f j).1 = c then (1 : ℕ) else 0) = Fintype.card K := by
+  calc
+    _ = ∑ z : C × K, if z.1 = c then (1 : ℕ) else 0 :=
+      hf.sum_comp (fun z => if z.1 = c then (1 : ℕ) else 0)
+    _ = _ := by
+      rw [Fintype.sum_prod_type, Finset.sum_comm]
+      dsimp only
+      simp only [Fintype.sum_ite_eq', Finset.sum_const, Finset.card_univ,
+        nsmul_eq_mul, mul_one, Nat.cast_id]
+
+theorem card_firstCoordinate_of_bijective
+    {J : Type uJ} {C : Type uC} {K : Type uK} [Fintype J] [Fintype C] [Fintype K] [DecidableEq C]
+    (f : J → C × K) (hf : Function.Bijective f) (c : C) :
+    Fintype.card {j : J // (f j).1 = c} = Fintype.card K := by
+  simpa only [Fintype.card_subtype, Finset.card_filter] using
+    sum_firstCoordinate_of_bijective f hf c
+
+theorem sum_firstFirstCoordinate_of_bijective
+    {J : Type uJ} {C : Type uC} {K : Type uK} {Z : Type uZ}
+    [Fintype J] [Fintype C] [Fintype K] [Fintype Z]
+    [DecidableEq C] (f : J → (C × K) × Z) (hf : Function.Bijective f) (c : C) :
+    (∑ j, if (f j).1.1 = c then (1 : ℕ) else 0) =
+      Fintype.card K * Fintype.card Z := by
+  have h := sum_firstCoordinate_of_bijective
+    (fun j => Equiv.prodAssoc C K Z (f j))
+    ((Equiv.prodAssoc C K Z).bijective.comp hf) c
+  change (∑ j, if (f j).1.1 = c then (1 : ℕ) else 0) =
+    Fintype.card (K × Z) at h
+  simpa only [Fintype.card_prod] using h
+
+theorem card_firstFirstCoordinate_of_bijective
+    {J : Type uJ} {C : Type uC} {K : Type uK} {Z : Type uZ}
+    [Fintype J] [Fintype C] [Fintype K] [Fintype Z]
+    [DecidableEq C] (f : J → (C × K) × Z) (hf : Function.Bijective f) (c : C) :
+    Fintype.card {j : J // (f j).1.1 = c} =
+      Fintype.card K * Fintype.card Z := by
+  simpa only [Fintype.card_subtype, Finset.card_filter] using
+    sum_firstFirstCoordinate_of_bijective f hf c
+
+def seedHistogram {p : ℕ} [NeZero p] {C : Type uC}
+    (q : Residues p → C) (c : C) : ℕ := by
+  classical
+  exact Fintype.card {s : Residues p // q s = c}
+
+theorem seedHistogram_eq_sum {p : ℕ} [NeZero p] {C : Type uC} [DecidableEq C]
+    (q : Residues p → C) (c : C) :
+    seedHistogram q c = ∑ s : Residues p, if q s = c then (1 : ℕ) else 0 := by
+  simp only [seedHistogram, Fintype.card_subtype, Finset.card_filter]
+
+theorem sum_seedHistogram {p : ℕ} [NeZero p] {C : Type uC} [Fintype C]
+    (q : Residues p → C) : (∑ c, seedHistogram q c) = p ^ 4 := by
+  classical
+  calc
+    (∑ c, seedHistogram q c) =
+        Fintype.card (Σ c : C, {s : Residues p // q s = c}) := by
+      simp only [Fintype.card_sigma, seedHistogram]
+    _ = Fintype.card (Residues p) := Fintype.card_congr (Equiv.sigmaFiberEquiv q)
+    _ = p ^ 4 := by
+      simp only [Residues, Fintype.card_prod, ZMod.card]
+      ring
+
+theorem sum_seedHistogram_translate {p : ℕ} [NeZero p]
+    {C : Type uC} [AddCommGroup C] [DecidableEq C]
+    (q : Residues p → C) (s : Residues p) (e c : C) :
+    (∑ τ : Residues p, if q (s - τ) + e = c then (1 : ℕ) else 0) =
+      seedHistogram q (c - e) := by
+  have hinv : Function.Involutive (fun τ : Residues p => s - τ) :=
+    fun τ => sub_sub_cancel s τ
+  rw [seedHistogram_eq_sum]
+  simpa only [eq_sub_iff_add_eq] using
+    hinv.bijective.sum_comp (fun τ => if q τ + e = c then (1 : ℕ) else 0)
+
+variable {p : ℕ} [NeZero p] (E : EncodingParameters p)
+
+theorem ordinary_active_of_tile (o : Outputs E) (hdep : HasDependence E o)
+    (n : Column p) (htile : Tiles (ordinaryActivationTile E n) (graph o)) :
+    ∀ x, ∃ j, Active E o (.inl n) x j := by
+  classical
+  intro x
+  by_contra hno
+  obtain ⟨c₀, hc⟩ := (no_active_iff_constant E o (.inl n) x).mp
+    (fun j hj => hno ⟨j, hj⟩)
+  have hsource (j : ActivationIndex E (.inl n)) :
+      o.c (ordinarySource E n (x, 0) j) (.inl n) = c₀ := by
+    calc
+      _ = usefulAt E o (.inl n) (ordinarySource E n (x, 0) j).1
+          ((ordinarySource E n (x, 0) j).2 (.inl n)) :=
+        usefulAt_factor E hdep _ _ _
+      _ = usefulAt E o (.inl n) x
+          ((ordinarySource E n (x, 0) j).2 (.inl n)) :=
+        usefulAt_eq_of_lineValue_eq E hdep n (ordinarySource_lineValue E n (x, 0) j) _
+      _ = c₀ := hc _
+  have hmap (j : ActivationIndex E (.inl n)) :
+      (ordinaryActivationMap E n o (x, 0) j).1 =
+        c₀ + shift (E.a (.inl n)) (E.b (.inl n)) j.2 := by
+    change o.c (ordinarySource E n (x, 0) j) (.inl n) + _ = _
+    rw [hsource]
+  have htwo : (∑ δ : ShiftIndex (E.a (.inl n)) (E.b (.inl n)),
+      if shift (E.a (.inl n)) (E.b (.inl n)) δ = 0 then (1 : ℕ) else 0) = 2 := by
+    calc
+      _ = ∑ e : P E (.inl n), shiftMultiplicity (E.a (.inl n)) (E.b (.inl n)) e •
+          (if e = 0 then (1 : ℕ) else 0) := sum_shift _
+      _ = shiftMultiplicity (E.a (.inl n)) (E.b (.inl n)) (0 : P E (.inl n)) := by
+        simp [mul_ite]
+      _ = 2 := by
+        change shiftMultiplicity (E.a (.inl n)) (E.b (.inl n)) (0, 0) = 2
+        exact shiftMultiplicity_zero
+  have hcount := sum_firstCoordinate_of_bijective
+    (ordinaryActivationMap E n o (x, 0))
+    ((ordinaryActivationTile_iff E n o).mp htile (x, 0)) c₀
+  simp_rw [hmap, add_eq_left] at hcount
+  rw [Fintype.sum_prod_type] at hcount
+  have hbad : E.r (.inl n) * 2 = E.r (.inl n) := by
+    simpa only [htwo, Finset.sum_const, Finset.card_univ, nsmul_eq_mul, Nat.cast_id, K, ZMod.card]
+      using hcount
+  have hr := E.r_pos (.inl n)
+  omega
+
+theorem seed_active_somewhere_of_tile (o : Outputs E) (hdep : HasDependence E o)
+    (t : Fin 2) (htile : Tiles (seedActivationTile E t) (graph o)) :
+    ∃ x, Active E o (.inr t) x () := by
+  classical
+  by_contra hno
+  have hinactive (x : Plane) : ¬ Active E o (.inr t) x () :=
+    fun hx => hno ⟨x, hx⟩
+  let q : Residues p → P E (.inr t) := fun s => seedFactor E o t s 0
+  have hconstant (s : Residues p) (w : K E (.inr t)) :
+      seedFactor E o t s w = q s := by
+    obtain ⟨c, hc⟩ := (no_active_iff_constant E o (.inr t)
+      (seedRepresentative p s)).mp (by
+        intro j
+        cases j
+        exact hinactive (seedRepresentative p s))
+    exact (hc w).trans (hc 0).symm
+  have hfactor (x : Plane) (k : Input E) :
+      o.c (x, k) (.inr t) = q (seedResidue p x) :=
+    (seedFactor_spec E hdep t x k).trans (hconstant _ _)
+  let b₀ : Base E := (0, 0)
+  have hmap (j : SeedActivationIndex E t) :
+      (seedActivationMap E t o b₀ j).1.1 =
+        q (seedResidue p b₀.1 - j.2) +
+          shift (E.a (.inr t)) (E.b (.inr t)) j.1.2 := by
+    have hf := hfactor (seedSource E t b₀ j).1 (seedSource E t b₀ j).2
+    change o.c (seedSource E t b₀ j) (.inr t) = _ at hf
+    change o.c (seedSource E t b₀ j) (.inr t) + _ = _
+    rw [hf]
+    have hs : seedResidue p (seedSource E t b₀ j).1 =
+        seedResidue p b₀.1 - j.2 := seedSource_residue E t b₀ j
+    rw [hs]
+  have hcoverage (c : P E (.inr t)) :
+      (∑ δ : ShiftIndex (E.a (.inr t)) (E.b (.inr t)),
+        seedHistogram q (c - shift (E.a (.inr t)) (E.b (.inr t)) δ)) = D p := by
+    have hcount := sum_firstFirstCoordinate_of_bijective
+      (seedActivationMap E t o b₀) ((seedActivationTile_iff E t o).mp htile b₀) c
+    simp_rw [hmap] at hcount
+    rw [Fintype.sum_prod_type, Fintype.sum_prod_type] at hcount
+    simp_rw [sum_seedHistogram_translate] at hcount
+    have hmul : E.r (.inr t) *
+        (∑ δ : ShiftIndex (E.a (.inr t)) (E.b (.inr t)),
+          seedHistogram q (c - shift (E.a (.inr t)) (E.b (.inr t)) δ)) =
+        E.r (.inr t) * D p := by
+      simpa only [Finset.sum_const, Finset.card_univ, nsmul_eq_mul, Nat.cast_id, K, ZMod.card]
+        using hcount
+    exact Nat.eq_of_mul_eq_mul_left (E.r_pos (.inr t)) hmul
+  have hdiv : E.a (.inr t) ∣ p ^ 4 :=
+    histogram_mass_dvd_of_shift_sum (E.a_two_le (.inr t)) (E.b_two_le (.inr t))
+      (seedHistogram q) (sum_seedHistogram q) (E.seed_b_large t) hcoverage
+  exact E.prime_ne_p (.inl (.inr t))
+    (Nat.prime_eq_prime_of_dvd_pow (E.a_prime (.inr t)) E.p_prime hdiv)
+
+end PeriodicTilingThree
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/ActivationOffsets.lean b/lean/OAI/Geometry/PeriodicTiling/ActivationOffsets.lean
new file mode 100644
index 0000000000000000000000000000000000000000..f08db4b1adcde0cd5bde625ef1d45c0881609522
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/ActivationOffsets.lean
@@ -0,0 +1,233 @@
+import OAI.Geometry.PeriodicTiling.CyclicCoordinates
+import OAI.Geometry.PeriodicTiling.ShiftMultiplicity
+import Mathlib.Tactic.FinCases
+import Mathlib.Tactic.Linarith
+import Lean.Elab.Tactic.Omega
+import Mathlib.Tactic.Ring
+
+namespace OAI
+
+universe uJ
+
+namespace PeriodicTilingThree
+
+noncomputable section
+
+def separatedResidueLift {J : Type uJ} [Fintype J] (M : ℕ) [NeZero M]
+    (v : J → ZMod M) (j : J) : ℤ :=
+  ((v j).val : ℤ) + (M : ℤ) * ((Fintype.equivFin J j).val : ℤ)
+
+@[simp] theorem separatedResidueLift_cast {J : Type uJ} [Fintype J]
+    (M : ℕ) [NeZero M] (v : J → ZMod M) (j : J) :
+    (separatedResidueLift M v j : ZMod M) = v j := by
+  simp [separatedResidueLift]
+
+theorem separatedResidueLift_injective {J : Type uJ} [Fintype J]
+    (M : ℕ) [NeZero M] (v : J → ZMod M) :
+    Function.Injective (separatedResidueLift M v) := by
+  intro i j hij
+  change ((v i).val : ℤ) + (M : ℤ) * ((Fintype.equivFin J i).val : ℤ) =
+    ((v j).val : ℤ) + (M : ℤ) * ((Fintype.equivFin J j).val : ℤ) at hij
+  have he : (v i).val + M * (Fintype.equivFin J i).val =
+      (v j).val + M * (Fintype.equivFin J j).val := by
+    exact_mod_cast hij
+  apply (Fintype.equivFin J).injective
+  apply Fin.ext
+  by_contra hne
+  rcases lt_or_gt_of_ne hne with hlt | hgt
+  · have hmul := Nat.mul_le_mul_left M (Nat.succ_le_of_lt hlt)
+    have hi := ZMod.val_lt (v i)
+    nlinarith
+  · have hmul := Nat.mul_le_mul_left M (Nat.succ_le_of_lt hgt)
+    have hj := ZMod.val_lt (v j)
+    nlinarith
+
+variable {p : ℕ} [NeZero p] (E : EncodingParameters p)
+
+abbrev ActivationIndex (i : Channel p) :=
+  K E i × ShiftIndex (E.a i) (E.b i)
+
+abbrev SeedActivationIndex (t : Fin 2) := ActivationIndex E (.inr t) × Residues p
+
+def otherSeed (t : Fin 2) : Fin 2 := ⟨1 - t.val, by omega⟩
+
+@[simp] theorem otherSeed_ne_self (t : Fin 2) : otherSeed t ≠ t := by
+  fin_cases t <;> decide
+
+@[simp] theorem otherSeed_otherSeed (t : Fin 2) : otherSeed (otherSeed t) = t := by
+  fin_cases t <;> decide
+
+def activationModulus (i : Channel p) : ℕ := (E.r i * E.a i) * E.b i
+
+instance activationModulus_neZero (i : Channel p) : NeZero (activationModulus E i) := by
+  unfold activationModulus
+  infer_instance
+
+private theorem activationCoprime (i : Channel p) :
+    Nat.Coprime (E.r i * E.a i) (E.b i) := by
+  rw [Nat.coprime_mul_iff_left]
+  exact ⟨E.r_coprime_b i i, E.a_coprime_b i⟩
+
+def activationCRT (i : Channel p) :
+    ZMod (activationModulus E i) ≃+* ((K E i × ZMod (E.a i)) × ZMod (E.b i)) :=
+  (ZMod.chineseRemainder (activationCoprime E i)).trans
+    (RingEquiv.prodCongr (ZMod.chineseRemainder (E.r_coprime_a i i))
+      (RingEquiv.refl _))
+
+def activationResidue (i : Channel p) (j : ActivationIndex E i) :
+    ZMod (activationModulus E i) :=
+  (activationCRT E i).symm
+    ((j.1, rhoA (E.a_two_le i) (E.b_two_le i) j.2),
+      rhoB (E.a_two_le i) (E.b_two_le i) j.2)
+
+def ordinaryFirst (n : Column p) : ActivationIndex E (.inl n) → ℤ :=
+  separatedResidueLift (activationModulus E (.inl n)) (activationResidue E (.inl n))
+
+def ordinaryOffset (n : Column p) (j : ActivationIndex E (.inl n)) : Plane :=
+  (ordinaryFirst E n j, -(n.val : ℤ) * ordinaryFirst E n j)
+
+theorem ordinaryFirst_residues (n : Column p) (j : ActivationIndex E (.inl n)) :
+    (((ordinaryFirst E n j : K E (.inl n)),
+       (ordinaryFirst E n j : ZMod (E.a (.inl n)))),
+       (ordinaryFirst E n j : ZMod (E.b (.inl n)))) =
+      ((j.1, rhoA (E.a_two_le (.inl n)) (E.b_two_le (.inl n)) j.2),
+        rhoB (E.a_two_le (.inl n)) (E.b_two_le (.inl n)) j.2) := by
+  have h := congrArg (activationCRT E (.inl n))
+    (separatedResidueLift_cast (activationModulus E (.inl n))
+      (activationResidue E (.inl n)) j)
+  rw [activationResidue, RingEquiv.apply_symm_apply] at h
+  refine Prod.ext (Prod.ext ?_ ?_) ?_
+  · simpa only [ordinaryFirst, map_intCast, Prod.fst_intCast] using
+      congrArg (fun z => z.1.1) h
+  · simpa only [ordinaryFirst, map_intCast, Prod.fst_intCast, Prod.snd_intCast] using
+      congrArg (fun z => z.1.2) h
+  · simpa only [ordinaryFirst, map_intCast, Prod.snd_intCast] using
+      congrArg Prod.snd h
+
+@[simp] theorem ordinaryOffset_low (n : Column p) (j : ActivationIndex E (.inl n)) :
+    ((ordinaryOffset E n j).1 : K E (.inl n)) = j.1 :=
+  congrArg (fun z => z.1.1) (ordinaryFirst_residues E n j)
+
+@[simp] theorem ordinaryOffset_rhoA (n : Column p) (j : ActivationIndex E (.inl n)) :
+    ((ordinaryOffset E n j).1 : ZMod (E.a (.inl n))) =
+      rhoA (E.a_two_le (.inl n)) (E.b_two_le (.inl n)) j.2 :=
+  congrArg (fun z => z.1.2) (ordinaryFirst_residues E n j)
+
+@[simp] theorem ordinaryOffset_rhoB (n : Column p) (j : ActivationIndex E (.inl n)) :
+    ((ordinaryOffset E n j).1 : ZMod (E.b (.inl n))) =
+      rhoB (E.a_two_le (.inl n)) (E.b_two_le (.inl n)) j.2 :=
+  congrArg Prod.snd (ordinaryFirst_residues E n j)
+
+@[simp] theorem ordinaryOffset_lineValue (n : Column p) (j : ActivationIndex E (.inl n)) :
+    lineValue n (ordinaryOffset E n j) = 0 := by
+  simp [lineValue, ordinaryOffset]
+
+theorem ordinaryOffset_injective (n : Column p) : Function.Injective (ordinaryOffset E n) := by
+  intro i j hij
+  exact separatedResidueLift_injective _ _ (congrArg Prod.fst hij)
+
+def seedModulus (t : Fin 2) : ℕ :=
+  (activationModulus E (.inr t) * p ^ 2) * E.a (.inr (otherSeed t))
+
+instance seedModulus_neZero (t : Fin 2) : NeZero (seedModulus E t) := by
+  unfold seedModulus
+  infer_instance
+
+private theorem activationCoprime_p (i : Channel p) :
+    Nat.Coprime (activationModulus E i) (p ^ 2) := by
+  rw [Nat.coprime_pow_right_iff (by decide : 0 < 2)]
+  simp only [activationModulus, Nat.coprime_mul_iff_left]
+  exact ⟨⟨E.r_coprime_p i, E.prime_coprime_p (.inl i)⟩,
+    E.prime_coprime_p (.inr (.inl i))⟩
+
+private theorem seedCoprime_other (t : Fin 2) :
+    Nat.Coprime (activationModulus E (.inr t) * p ^ 2)
+      (E.a (.inr (otherSeed t))) := by
+  simp only [activationModulus, Nat.coprime_mul_iff_left]
+  refine ⟨⟨⟨E.r_coprime_a _ _, ?_⟩, ?_⟩, ?_⟩
+  · apply E.prime_coprime
+    simp only [ne_eq, Sum.inl.injEq, Sum.inr.injEq]
+    exact (otherSeed_ne_self t).symm
+  · apply E.prime_coprime
+    simp
+  · exact (E.prime_coprime_p (.inl (.inr (otherSeed t)))).symm.pow_left 2
+
+def seedCRT (t : Fin 2) :
+    ZMod (seedModulus E t) ≃+*
+      ((((K E (.inr t) × ZMod (E.a (.inr t))) × ZMod (E.b (.inr t))) ×
+        ZMod (p ^ 2)) × ZMod (E.a (.inr (otherSeed t)))) :=
+  (ZMod.chineseRemainder (seedCoprime_other E t)).trans
+    (RingEquiv.prodCongr
+      ((ZMod.chineseRemainder (activationCoprime_p E (.inr t))).trans
+        (RingEquiv.prodCongr (activationCRT E (.inr t)) (RingEquiv.refl _)))
+      (RingEquiv.refl _))
+
+def seedActivationResidue (t : Fin 2) (j : SeedActivationIndex E t) :
+    ZMod (seedModulus E t) :=
+  (seedCRT E t).symm
+    ((((j.1.1, rhoA (E.a_two_le (.inr t)) (E.b_two_le (.inr t)) j.1.2),
+      rhoB (E.a_two_le (.inr t)) (E.b_two_le (.inr t)) j.1.2), j.2.1), 0)
+
+def seedFirst (t : Fin 2) : SeedActivationIndex E t → ℤ :=
+  separatedResidueLift (seedModulus E t) (seedActivationResidue E t)
+
+def seedOffset (t : Fin 2) (j : SeedActivationIndex E t) : Plane :=
+  (seedFirst E t j, (j.2.2.val : ℤ))
+
+theorem seedFirst_residues (t : Fin 2) (j : SeedActivationIndex E t) :
+    (((((seedFirst E t j : K E (.inr t)),
+      (seedFirst E t j : ZMod (E.a (.inr t)))),
+      (seedFirst E t j : ZMod (E.b (.inr t)))),
+      (seedFirst E t j : ZMod (p ^ 2))),
+      (seedFirst E t j : ZMod (E.a (.inr (otherSeed t))))) =
+    ((((j.1.1, rhoA (E.a_two_le (.inr t)) (E.b_two_le (.inr t)) j.1.2),
+      rhoB (E.a_two_le (.inr t)) (E.b_two_le (.inr t)) j.1.2), j.2.1), 0) := by
+  have h := congrArg (seedCRT E t)
+    (separatedResidueLift_cast (seedModulus E t) (seedActivationResidue E t) j)
+  rw [seedActivationResidue, RingEquiv.apply_symm_apply] at h
+  refine Prod.ext (Prod.ext (Prod.ext (Prod.ext ?_ ?_) ?_) ?_) ?_
+  · simpa only [seedFirst, map_intCast, Prod.fst_intCast] using
+      congrArg (fun z => z.1.1.1.1) h
+  · simpa only [seedFirst, map_intCast, Prod.fst_intCast, Prod.snd_intCast] using
+      congrArg (fun z => z.1.1.1.2) h
+  · simpa only [seedFirst, map_intCast, Prod.fst_intCast, Prod.snd_intCast] using
+      congrArg (fun z => z.1.1.2) h
+  · simpa only [seedFirst, map_intCast, Prod.fst_intCast, Prod.snd_intCast] using
+      congrArg (fun z => z.1.2) h
+  · simpa only [seedFirst, map_intCast, Prod.snd_intCast] using
+      congrArg Prod.snd h
+
+@[simp] theorem seedOffset_low (t : Fin 2) (j : SeedActivationIndex E t) :
+    ((seedOffset E t j).1 : K E (.inr t)) = j.1.1 :=
+  congrArg (fun z => z.1.1.1.1) (seedFirst_residues E t j)
+
+@[simp] theorem seedOffset_rhoA (t : Fin 2) (j : SeedActivationIndex E t) :
+    ((seedOffset E t j).1 : ZMod (E.a (.inr t))) =
+      rhoA (E.a_two_le (.inr t)) (E.b_two_le (.inr t)) j.1.2 :=
+  congrArg (fun z => z.1.1.1.2) (seedFirst_residues E t j)
+
+@[simp] theorem seedOffset_rhoB (t : Fin 2) (j : SeedActivationIndex E t) :
+    ((seedOffset E t j).1 : ZMod (E.b (.inr t))) =
+      rhoB (E.a_two_le (.inr t)) (E.b_two_le (.inr t)) j.1.2 :=
+  congrArg (fun z => z.1.1.2) (seedFirst_residues E t j)
+
+@[simp] theorem seedOffset_other (t : Fin 2) (j : SeedActivationIndex E t) :
+    ((seedOffset E t j).1 : ZMod (E.a (.inr (otherSeed t)))) = 0 :=
+  congrArg Prod.snd (seedFirst_residues E t j)
+
+@[simp] theorem seedOffset_residue (t : Fin 2) (j : SeedActivationIndex E t) :
+    (((seedOffset E t j).1 : ZMod (p ^ 2)),
+      ((seedOffset E t j).2 : ZMod (p ^ 2))) = j.2 := by
+  apply Prod.ext
+  · exact congrArg (fun z => z.1.2) (seedFirst_residues E t j)
+  · simp [seedOffset]
+
+theorem seedOffset_injective (t : Fin 2) : Function.Injective (seedOffset E t) := by
+  intro i j hij
+  exact separatedResidueLift_injective _ _ (congrArg Prod.fst hij)
+
+end
+
+end PeriodicTilingThree
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/ActivationTiles.lean b/lean/OAI/Geometry/PeriodicTiling/ActivationTiles.lean
new file mode 100644
index 0000000000000000000000000000000000000000..56d6b7cd6dfd39e9b023be198adc816d2b11790d
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/ActivationTiles.lean
@@ -0,0 +1,398 @@
+import OAI.Geometry.PeriodicTiling.ActivationOffsets
+import OAI.Geometry.PeriodicTiling.GraphTests
+import Mathlib.Tactic.Abel
+
+namespace OAI
+
+universe uJ uB
+
+namespace PeriodicTilingThree
+
+noncomputable section
+
+variable {p : ℕ} [NeZero p] (E : EncodingParameters p)
+
+def activationBatch (i : Channel p) (h : Plane) (e : P E i)
+    (freeZ : Bool) : Finset (Ambient E) := by
+  classical
+  exact (Finset.univ.filter fun v : V E =>
+    (v.2 i).1 = (h.1 : ZMod ((E.r i) ^ 2)) ∧ (v.2 i).2 = e ∧
+    (∀ j, j ≠ i → low E j (v.2 j).1 = 0) ∧ (freeZ = true ∨ v.1 = 0)).image
+      (fun v => (h, v))
+
+theorem mem_activationBatch (i : Channel p) (h : Plane) (e : P E i)
+    (freeZ : Bool) (g : Ambient E) :
+    g ∈ activationBatch E i h e freeZ ↔
+      g.1 = h ∧ (g.2.2 i).1 = (h.1 : ZMod ((E.r i) ^ 2)) ∧
+      (g.2.2 i).2 = e ∧ (∀ j, j ≠ i → low E j (g.2.2 j).1 = 0) ∧
+      (freeZ = true ∨ g.2.1 = 0) := by
+  classical
+  simp only [activationBatch, Finset.mem_image, Finset.mem_filter,
+    Finset.mem_univ, true_and]
+  constructor
+  · rintro ⟨v, hv, rfl⟩
+    exact ⟨rfl, hv⟩
+  · rintro ⟨hh, hv⟩
+    exact ⟨g.2, hv, Prod.ext hh.symm rfl⟩
+
+theorem activationBatch_nonempty (i : Channel p) (h : Plane) (e : P E i)
+    (freeZ : Bool) : (activationBatch E i h e freeZ).Nonempty := by
+  classical
+  refine ⟨(h, 0, Function.update 0 i ((h.1 : ZMod ((E.r i) ^ 2)), e)), ?_⟩
+  apply (mem_activationBatch E i h e freeZ _).mpr
+  refine ⟨rfl, ?_, ?_, ?_, Or.inr rfl⟩
+  · simp
+  · simp
+  · intro j hji
+    simp [hji]
+
+theorem activationBatch_q0 (i : Channel p) (h : Plane) (e : P E i)
+    (freeZ : Bool) {g : Ambient E} (hg : g ∈ activationBatch E i h e freeZ) :
+    q0 E g = (h, Function.update (0 : Input E) i (h.1 : K E i)) := by
+  classical
+  obtain ⟨hh, hv, _hc, hother, _hz⟩ := (mem_activationBatch E i h e freeZ g).mp hg
+  apply Prod.ext hh
+  funext j
+  by_cases hji : j = i
+  · subst j
+    change low E i (g.2.2 i).1 = _
+    rw [hv]
+    simp [low]
+  · simpa [hji] using hother j hji
+
+theorem activationBatch_disjoint (i : Channel p) {h h' : Plane}
+    (e e' : P E i) (z z' : Bool) (hh : h ≠ h') :
+    Disjoint (activationBatch E i h e z) (activationBatch E i h' e' z') := by
+  classical
+  apply Finset.disjoint_left.mpr
+  intro g hg hg'
+  exact hh (((mem_activationBatch E i h e z g).mp hg).1.symm.trans
+    ((mem_activationBatch E i h' e' z' g).mp hg').1)
+
+def activationSource (i : Channel p) (b : Base E) (h : Plane) (l : K E i) : Base E :=
+  (b.1 - h, Function.update b.2 i (b.2 i - l))
+
+theorem activationSource_eq_sub (i : Channel p) (b : Base E) (h : Plane) (l : K E i) :
+    activationSource E i b h l = b - (h, Function.update (0 : Input E) i l) := by
+  classical
+  refine Prod.ext rfl ?_
+  funext j
+  by_cases hji : j = i
+  · subst j
+    simp [activationSource]
+  · simp [activationSource, hji]
+
+theorem activationSource_shear (i : Channel p) (b : Base E) (h : Plane) (l β : K E i)
+    (hl : (h.1 : K E i) = l) :
+    shear E i (activationSource E i b h l).1.1
+        ((activationSource E i b h l).2 i) β + (h.1 : ZMod ((E.r i) ^ 2)) =
+      shear E i b.1.1 (b.2 i) β := by
+  simpa [activationSource] using shear_source_add E i b.1.1 h.1 (b.2 i) l β hl
+
+def targetBeta (y : Ambient E) (i : Channel p) : K E i :=
+  (shearEquiv E i y.1.1 ((q0 E y).2 i)).symm ⟨(y.2.2 i).1, rfl⟩
+
+@[simp] theorem shear_targetBeta (y : Ambient E) (i : Channel p) :
+    shear E i y.1.1 ((q0 E y).2 i) (targetBeta E y i) = (y.2.2 i).1 :=
+  congrArg Subtype.val ((shearEquiv E i y.1.1 ((q0 E y).2 i)).apply_symm_apply
+    ⟨(y.2.2 i).1, rfl⟩)
+
+@[simp] theorem targetBeta_point (o : GraphOutputs E) (b : Base E) (i : Channel p) :
+    targetBeta E (point o b) i = o.beta b i := by
+  apply shear_injective E i b.1.1 (b.2 i)
+  simpa only [q0_point, point_horizontal, point_v] using shear_targetBeta E (point o b) i
+
+theorem difference_mem_activationBatch (o : GraphOutputs E) (i : Channel p)
+    (h : Plane) (l : K E i) (hl : (h.1 : K E i) = l) (e : P E i)
+    (freeZ : Bool) (y : Ambient E) (b : Base E) :
+    y - point o b ∈ activationBatch E i h e freeZ ↔
+      activationSource E i (q0 E y) h l = b ∧
+      o.c b i + e = (y.2.2 i).2 ∧ o.beta b i = targetBeta E y i ∧
+      (freeZ = true ∨ o.z b = y.2.1) := by
+  constructor
+  · intro hm
+    have hq := activationBatch_q0 E i h e freeZ hm
+    have hq' : q0 E y - b = (h, Function.update (0 : Input E) i l) := by
+      simpa only [map_sub, q0_point, hl] using hq
+    have hb : activationSource E i (q0 E y) h l = b := by
+      rw [activationSource_eq_sub]
+      have he := congrArg (fun z : Base E => q0 E y - z) hq'
+      simpa using he.symm
+    obtain ⟨_hh, hv, hc, _hother, hz⟩ :=
+      (mem_activationBatch E i h e freeZ (y - point o b)).mp hm
+    refine ⟨hb, ?_, ?_, ?_⟩
+    · change (y.2.2 i).2 - o.c b i = e at hc
+      rw [← hc]
+      abel
+    · apply shear_injective E i y.1.1 ((q0 E y).2 i)
+      rw [shear_targetBeta]
+      have hs := activationSource_shear E i (q0 E y) h l (o.beta b i) hl
+      rw [hb] at hs
+      change (y.2.2 i).1 - shear E i b.1.1 (b.2 i) (o.beta b i) =
+        (h.1 : ZMod ((E.r i) ^ 2)) at hv
+      change shear E i b.1.1 (b.2 i) (o.beta b i) +
+        (h.1 : ZMod ((E.r i) ^ 2)) =
+        shear E i y.1.1 ((q0 E y).2 i) (o.beta b i) at hs
+      rw [← hs, ← hv]
+      abel
+    · rcases hz with hz | hz
+      · exact Or.inl hz
+      · right
+        change y.2.1 - o.z b = 0 at hz
+        exact (sub_eq_zero.mp hz).symm
+  · rintro ⟨hb, hc, hβ, hz⟩
+    apply (mem_activationBatch E i h e freeZ (y - point o b)).mpr
+    have hs := activationSource_shear E i (q0 E y) h l (o.beta b i) hl
+    rw [hb, hβ] at hs
+    change shear E i b.1.1 (b.2 i) (targetBeta E y i) +
+      (h.1 : ZMod ((E.r i) ^ 2)) =
+      shear E i y.1.1 ((q0 E y).2 i) (targetBeta E y i) at hs
+    rw [shear_targetBeta] at hs
+    refine ⟨?_, ?_, ?_, ?_, ?_⟩
+    · change y.1 - b.1 = h
+      rw [← hb]
+      change y.1 - (y.1 - h) = h
+      abel
+    · change (y.2.2 i).1 - shear E i b.1.1 (b.2 i) (o.beta b i) =
+        (h.1 : ZMod ((E.r i) ^ 2))
+      rw [hβ]
+      rw [← hs]
+      abel
+    · change (y.2.2 i).2 - o.c b i = e
+      rw [← hc]
+      abel
+    · intro j hji
+      change low E j ((y.2.2 j).1 - shear E j b.1.1 (b.2 j) (o.beta b j)) = 0
+      rw [map_sub, low_shear, ← hb]
+      simp [activationSource, hji, q0]
+    · rcases hz with hz | hz
+      · exact Or.inl hz
+      · right
+        change y.2.1 - o.z b = 0
+        rw [hz, sub_self]
+
+def ordinaryActivationTile (n : Column p) : Finset (Ambient E) := by
+  classical
+  exact Finset.univ.biUnion fun j : ActivationIndex E (.inl n) =>
+    activationBatch E (.inl n) (ordinaryOffset E n j)
+      (shift (E.a (.inl n)) (E.b (.inl n)) j.2) true
+
+def seedActivationTile (t : Fin 2) : Finset (Ambient E) := by
+  classical
+  exact Finset.univ.biUnion fun j : SeedActivationIndex E t =>
+    activationBatch E (.inr t) (seedOffset E t j)
+      (shift (E.a (.inr t)) (E.b (.inr t)) j.1.2) false
+
+def ordinarySource (n : Column p) (b : Base E) (j : ActivationIndex E (.inl n)) : Base E :=
+  activationSource E (.inl n) b (ordinaryOffset E n j) j.1
+
+def seedSource (t : Fin 2) (b : Base E) (j : SeedActivationIndex E t) : Base E :=
+  activationSource E (.inr t) b (seedOffset E t j) j.1.1
+
+@[simp] theorem ordinarySource_horizontal (n : Column p) (b : Base E)
+    (j : ActivationIndex E (.inl n)) :
+    (ordinarySource E n b j).1 = b.1 - ordinaryOffset E n j := rfl
+
+@[simp] theorem ordinarySource_own (n : Column p) (b : Base E)
+    (j : ActivationIndex E (.inl n)) :
+    (ordinarySource E n b j).2 (.inl n) = b.2 (.inl n) - j.1 := by
+  simp [ordinarySource, activationSource]
+
+@[simp] theorem seedSource_horizontal (t : Fin 2) (b : Base E)
+    (j : SeedActivationIndex E t) :
+    (seedSource E t b j).1 = b.1 - seedOffset E t j := rfl
+
+@[simp] theorem seedSource_own (t : Fin 2) (b : Base E)
+    (j : SeedActivationIndex E t) :
+    (seedSource E t b j).2 (.inr t) = b.2 (.inr t) - j.1.1 := by
+  simp [seedSource, activationSource]
+
+@[simp] theorem ordinarySource_lineValue (n : Column p) (b : Base E)
+    (j : ActivationIndex E (.inl n)) :
+    lineValue n (ordinarySource E n b j).1 = lineValue n b.1 := by
+  simp only [ordinarySource_horizontal, lineValue, ordinaryOffset,
+    Prod.fst_sub, Prod.snd_sub]
+  ring
+
+theorem seedSource_residue (t : Fin 2) (b : Base E) (j : SeedActivationIndex E t) :
+    ((((seedSource E t b j).1.1 : ℤ) : ZMod (p ^ 2)),
+      (((seedSource E t b j).1.2 : ℤ) : ZMod (p ^ 2))) =
+      ((b.1.1 : ZMod (p ^ 2)), (b.1.2 : ZMod (p ^ 2))) - j.2 := by
+  rw [seedSource_horizontal]
+  simpa only [Prod.fst_sub, Prod.snd_sub, Int.cast_sub, Prod.mk_sub_mk] using
+    congrArg (fun z : Residues p =>
+      ((b.1.1 : ZMod (p ^ 2)), (b.1.2 : ZMod (p ^ 2))) - z)
+      (seedOffset_residue E t j)
+
+def ordinaryActivationMap (n : Column p) (o : GraphOutputs E) (b : Base E)
+    (j : ActivationIndex E (.inl n)) : P E (.inl n) × K E (.inl n) :=
+  (o.c (ordinarySource E n b j) (.inl n) + shift (E.a (.inl n)) (E.b (.inl n)) j.2,
+    o.beta (ordinarySource E n b j) (.inl n))
+
+def seedActivationMap (t : Fin 2) (o : GraphOutputs E) (b : Base E)
+    (j : SeedActivationIndex E t) : (P E (.inr t) × K E (.inr t)) × ZMod (D p) :=
+  ((o.c (seedSource E t b j) (.inr t) + shift (E.a (.inr t)) (E.b (.inr t)) j.1.2,
+    o.beta (seedSource E t b j) (.inr t)), o.z (seedSource E t b j))
+
+def ordinaryTarget (n : Column p) (y : Ambient E) : P E (.inl n) × K E (.inl n) :=
+  ((y.2.2 (.inl n)).2, targetBeta E y (.inl n))
+
+def seedTarget (t : Fin 2) (y : Ambient E) :
+    (P E (.inr t) × K E (.inr t)) × ZMod (D p) :=
+  (((y.2.2 (.inr t)).2, targetBeta E y (.inr t)), y.2.1)
+
+theorem ordinarySource_injective (n : Column p) (b : Base E) :
+    Function.Injective (ordinarySource E n b) := by
+  intro j j' hj
+  apply ordinaryOffset_injective E n
+  have hh := congrArg (fun d : Base E => b.1 - d.1) hj
+  simpa [ordinarySource, activationSource] using hh
+
+theorem seedSource_injective (t : Fin 2) (b : Base E) :
+    Function.Injective (seedSource E t b) := by
+  intro j j' hj
+  apply seedOffset_injective E t
+  have hh := congrArg (fun d : Base E => b.1 - d.1) hj
+  simpa [seedSource, activationSource] using hh
+
+theorem difference_mem_ordinaryActivationTile (n : Column p) (o : GraphOutputs E)
+    (y : Ambient E) (b : Base E) :
+    y - point o b ∈ ordinaryActivationTile E n ↔
+      ∃ j, ordinarySource E n (q0 E y) j = b ∧
+        ordinaryActivationMap E n o (q0 E y) j = ordinaryTarget E n y := by
+  classical
+  simp only [ordinaryActivationTile, Finset.mem_biUnion, Finset.mem_univ, true_and]
+  apply exists_congr
+  intro j
+  rw [difference_mem_activationBatch E o (.inl n) (ordinaryOffset E n j) j.1
+    (ordinaryOffset_low E n j)]
+  simp only [true_or, and_true]
+  constructor
+  · rintro ⟨hb, hc, hβ⟩
+    change ordinarySource E n (q0 E y) j = b at hb
+    refine ⟨hb, ?_⟩
+    unfold ordinaryActivationMap ordinaryTarget
+    rw [hb]
+    exact Prod.ext hc hβ
+  · rintro ⟨hb, hmap⟩
+    unfold ordinaryActivationMap ordinaryTarget at hmap
+    rw [hb] at hmap
+    exact ⟨hb, congrArg Prod.fst hmap, congrArg Prod.snd hmap⟩
+
+theorem difference_mem_seedActivationTile (t : Fin 2) (o : GraphOutputs E)
+    (y : Ambient E) (b : Base E) :
+    y - point o b ∈ seedActivationTile E t ↔
+      ∃ j, seedSource E t (q0 E y) j = b ∧
+        seedActivationMap E t o (q0 E y) j = seedTarget E t y := by
+  classical
+  simp only [seedActivationTile, Finset.mem_biUnion, Finset.mem_univ, true_and]
+  apply exists_congr
+  intro j
+  rw [difference_mem_activationBatch E o (.inr t) (seedOffset E t j) j.1.1
+    (seedOffset_low E t j)]
+  simp only [Bool.false_eq_true, false_or]
+  constructor
+  · rintro ⟨hb, hc, hβ, hz⟩
+    change seedSource E t (q0 E y) j = b at hb
+    refine ⟨hb, ?_⟩
+    unfold seedActivationMap seedTarget
+    rw [hb]
+    exact Prod.ext (Prod.ext hc hβ) hz
+  · rintro ⟨hb, hmap⟩
+    unfold seedActivationMap seedTarget at hmap
+    rw [hb] at hmap
+    exact ⟨hb, congrArg (fun z => z.1.1) hmap,
+      congrArg (fun z => z.1.2) hmap, congrArg Prod.snd hmap⟩
+
+private theorem existsUnique_source_iff {J : Type uJ} {B : Type uB} (s : J → B)
+    (hs : Function.Injective s) (Q : J → Prop) :
+    (∃! b, ∃ j, s j = b ∧ Q j) ↔ ∃! j, Q j := by
+  constructor
+  · rintro ⟨b, ⟨j, hj, hQ⟩, huniq⟩
+    refine ⟨j, hQ, ?_⟩
+    intro j' hQ'
+    apply hs
+    exact (huniq (s j') ⟨j', rfl, hQ'⟩).trans hj.symm
+  · rintro ⟨j, hj, huniq⟩
+    refine ⟨s j, ⟨j, rfl, hj⟩, ?_⟩
+    rintro b ⟨j', hj', hQ'⟩
+    rw [← hj', huniq j' hQ']
+
+theorem ordinaryTarget_surjective (n : Column p) (b : Base E)
+    (q : P E (.inl n) × K E (.inl n)) :
+    ∃ y : Ambient E, q0 E y = b ∧ ordinaryTarget E n y = q := by
+  classical
+  let o : GraphOutputs E :=
+    ⟨fun _ => Function.update 0 (.inl n) q.1,
+      fun _ => Function.update 0 (.inl n) q.2, fun _ => 0⟩
+  refine ⟨point o b, q0_point o b, ?_⟩
+  simp [ordinaryTarget, o]
+
+theorem seedTarget_surjective (t : Fin 2) (b : Base E)
+    (q : (P E (.inr t) × K E (.inr t)) × ZMod (D p)) :
+    ∃ y : Ambient E, q0 E y = b ∧ seedTarget E t y = q := by
+  classical
+  let o : GraphOutputs E :=
+    ⟨fun _ => Function.update 0 (.inr t) q.1.1,
+      fun _ => Function.update 0 (.inr t) q.1.2, fun _ => q.2⟩
+  refine ⟨point o b, q0_point o b, ?_⟩
+  simp [seedTarget, o]
+
+theorem ordinaryActivationTile_iff (n : Column p) (o : GraphOutputs E) :
+    Tiles (ordinaryActivationTile E n) (graph o) ↔
+      ∀ b : Base E, Function.Bijective (ordinaryActivationMap E n o b) := by
+  rw [graph, tiles_range_iff_unique_base (point o) (point_injective o)]
+  simp_rw [difference_mem_ordinaryActivationTile,
+    existsUnique_source_iff _ (ordinarySource_injective E n _)]
+  constructor
+  · intro h b
+    apply (Function.bijective_iff_existsUnique _).mpr
+    intro q
+    obtain ⟨y, hy, hq⟩ := ordinaryTarget_surjective E n b q
+    simpa only [hy, hq] using h y
+  · intro h y
+    exact (h (q0 E y)).existsUnique (ordinaryTarget E n y)
+
+theorem seedActivationTile_iff (t : Fin 2) (o : GraphOutputs E) :
+    Tiles (seedActivationTile E t) (graph o) ↔
+      ∀ b : Base E, Function.Bijective (seedActivationMap E t o b) := by
+  rw [graph, tiles_range_iff_unique_base (point o) (point_injective o)]
+  simp_rw [difference_mem_seedActivationTile,
+    existsUnique_source_iff _ (seedSource_injective E t _)]
+  constructor
+  · intro h b
+    apply (Function.bijective_iff_existsUnique _).mpr
+    intro q
+    obtain ⟨y, hy, hq⟩ := seedTarget_surjective E t b q
+    simpa only [hy, hq] using h y
+  · intro h y
+    exact (h (q0 E y)).existsUnique (seedTarget E t y)
+
+theorem ordinaryActivationTile_nonempty (n : Column p) :
+    (ordinaryActivationTile E n).Nonempty := by
+  classical
+  let δ : ShiftIndex (E.a (.inl n)) (E.b (.inl n)) :=
+    ⟨(0, 0), ⟨0, by simp [shiftMultiplicity]⟩⟩
+  let j : ActivationIndex E (.inl n) := (0, δ)
+  obtain ⟨g, hg⟩ := activationBatch_nonempty E (.inl n) (ordinaryOffset E n j)
+    (shift (E.a (.inl n)) (E.b (.inl n)) j.2) true
+  refine ⟨g, ?_⟩
+  simp only [ordinaryActivationTile, Finset.mem_biUnion, Finset.mem_univ, true_and]
+  exact ⟨j, hg⟩
+
+theorem seedActivationTile_nonempty (t : Fin 2) :
+    (seedActivationTile E t).Nonempty := by
+  classical
+  let δ : ShiftIndex (E.a (.inr t)) (E.b (.inr t)) := ⟨(0, 0), ⟨0, by simp [shiftMultiplicity]⟩⟩
+  let j : SeedActivationIndex E t := ((0, δ), 0)
+  obtain ⟨g, hg⟩ := activationBatch_nonempty E (.inr t) (seedOffset E t j)
+    (shift (E.a (.inr t)) (E.b (.inr t)) j.1.2) false
+  refine ⟨g, ?_⟩
+  simp only [seedActivationTile, Finset.mem_biUnion, Finset.mem_univ, true_and]
+  exact ⟨j, hg⟩
+
+end
+
+end PeriodicTilingThree
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/ActivityWord.lean b/lean/OAI/Geometry/PeriodicTiling/ActivityWord.lean
new file mode 100644
index 0000000000000000000000000000000000000000..3eaf2f583de25863a03a5a40667398dd57c89fa0
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/ActivityWord.lean
@@ -0,0 +1,125 @@
+import OAI.Geometry.PeriodicTiling.Dependence
+import Mathlib.GroupTheory.OrderOfElement
+
+namespace OAI
+
+noncomputable section
+
+namespace PeriodicTilingThree
+
+variable {p : ℕ} [NeZero p] (E : EncodingParameters p)
+
+def activeLabels (o : GraphOutputs E) (i : Channel p) (x : Plane) :
+    Finset (Label p i) := by
+  classical
+  exact Finset.univ.filter (Active E o i x)
+
+@[simp] theorem mem_activeLabels (o : GraphOutputs E) (i : Channel p)
+    (x : Plane) (j : Label p i) :
+    j ∈ activeLabels E o i x ↔ Active E o i x j := by
+  classical
+  simp [activeLabels]
+
+def selectSymbol (S : Finset (Symbol p)) : Symbol p := by
+  classical
+  exact if h : S.Nonempty then h.choose else 1
+
+theorem selectSymbol_mem
+    {p : ℕ} [NeZero p]
+    (S : Finset (Symbol p)) (hS : S.Nonempty) :
+    selectSymbol S ∈ S := by
+  classical
+  simp only [selectSymbol, dite_eq_left hS]
+  exact hS.choose_spec
+
+def selectedWord (o : GraphOutputs E) : WordArray p :=
+  fun n m => selectSymbol (activeLabels E o (.inl n) (0, m))
+
+theorem selectedWord_active (o : GraphOutputs E)
+    (hactive : ∀ n x, ∃ j, Active E o (.inl n) x j) (n : Column p) (m : ℤ) :
+    Active E o (.inl n) (0, m) (selectedWord E o n m) := by
+  apply (mem_activeLabels E o (.inl n) (0, m) _).mp
+  apply selectSymbol_mem
+  obtain ⟨j, hj⟩ := hactive n (0, m)
+  exact ⟨j, (mem_activeLabels E o (.inl n) (0, m) j).mpr hj⟩
+
+theorem selectedWord_active_at (o : GraphOutputs E) (hdep : HasDependence E o)
+    (hactive : ∀ n x, ∃ j, Active E o (.inl n) x j) (n : Column p) (x : Plane) :
+    Active E o (.inl n) x (selectedWord E o n (lineValue n x)) := by
+  apply (active_ordinary_iff E hdep n (y := (0, lineValue n x))
+    (by simp) _).mpr
+  exact selectedWord_active E o hactive n _
+
+theorem selectedWord_lineRule (o : GraphOutputs E) (hdep : HasDependence E o)
+    (hactive : ∀ n x, ∃ j, Active E o (.inl n) x j)
+    (hwords : ∀ x (w : Word p), (∀ n, Active E o (.inl n) x (w n)) → Allowed p w) :
+    LineRule p (selectedWord E o) := by
+  intro d e
+  apply hwords (d, e)
+  intro n
+  simpa only [lineValue, mul_comm, add_comm] using
+    selectedWord_active_at E o hdep hactive n (d, e)
+
+theorem selectedWord_nonconstant_of_forced_letters
+    (o : GraphOutputs E) (hdep : HasDependence E o)
+    (hactive : ∀ n x, ∃ j, Active E o (.inl n) x j)
+    (a b : Symbol p) (hab : a ≠ b) (x y : Plane)
+    (hx : ∀ n j, Active E o (.inl n) x j → j = a)
+    (hy : ∀ n j, Active E o (.inl n) y j → j = b) :
+    NonconstantColumns (selectedWord E o) := by
+  intro n
+  refine ⟨lineValue n x, lineValue n y, ?_⟩
+  have ha := hx n _ (selectedWord_active_at E o hdep hactive n x)
+  have hb := hy n _ (selectedWord_active_at E o hdep hactive n y)
+  intro he
+  exact hab (ha.symm.trans (he.trans hb))
+
+theorem graph_period_useful (o : GraphOutputs E) (v : Plane)
+    (hv : Period (graph o) (v, (0 : V E))) (x : Plane) (k : Input E) :
+    o.c (x + v, k) = o.c (x, k) := by
+  have hm : point o (x, k) + (v, (0 : V E)) ∈ graph o :=
+    (hv (point o (x, k))).mpr (point_mem_graph o (x, k))
+  have he := eq_point_of_mem_graph o hm
+  have hq : q0 E (point o (x, k) + (v, (0 : V E))) = (x + v, k) := by
+    rw [map_add, q0_point]
+    refine Prod.ext rfl ?_
+    funext i
+    simp [q0]
+  rw [hq] at he
+  funext i
+  have hi := congrArg (fun z : Ambient E => (z.2.2 i).2) he
+  simpa only [Prod.fst_add, Prod.snd_add, Pi.add_apply, point_c,
+    Prod.fst_zero, Prod.snd_zero, Pi.zero_apply, add_zero] using hi.symm
+
+theorem activeLabels_eq_of_graph_period (o : GraphOutputs E) (v : Plane)
+    (hv : Period (graph o) (v, (0 : V E))) (i : Channel p) (x : Plane) :
+    activeLabels E o i (x + v) = activeLabels E o i x := by
+  classical
+  have hc : usefulAt E o i (x + v) = usefulAt E o i x := by
+    funext w
+    exact congrFun (graph_period_useful E o v hv x (Function.update 0 i w)) i
+  ext j
+  simp only [mem_activeLabels, Active, hc]
+
+theorem selectedWord_period_of_graph_period (o : GraphOutputs E) (M : ℤ)
+    (hv : Period (graph o) ((0, M), (0 : V E))) :
+    VerticalPeriod (selectedWord E o) M := by
+  intro n m
+  have hs := activeLabels_eq_of_graph_period E o (0, M) hv (.inl n) (0, m)
+  have hs' : activeLabels E o (.inl n) (0, m + M) =
+      activeLabels E o (.inl n) (0, m) := by simpa only [Prod.mk_add_mk, zero_add] using hs
+  exact congrArg selectSymbol hs'
+
+theorem exists_positive_vertical_graph_period (o : GraphOutputs E)
+    (hper : FullyPeriodic (graph o)) :
+    ∃ M : ℕ, 0 < M ∧ Period (graph o) ((0, (M : ℤ)), (0 : V E)) := by
+  obtain ⟨Λ, hΛ, hperiods⟩ := hper
+  refine ⟨Λ.index, Nat.pos_of_ne_zero hΛ.index_ne_zero, ?_⟩
+  have h := hperiods _ (Λ.nsmul_index_mem (((0, 1) : Plane), (0 : V E)))
+  simpa only [Prod.smul_mk, smul_zero, nsmul_eq_mul, mul_one] using h
+
+end PeriodicTilingThree
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/AuxiliaryCubes.lean b/lean/OAI/Geometry/PeriodicTiling/AuxiliaryCubes.lean
new file mode 100644
index 0000000000000000000000000000000000000000..dee52e325ccb3e040b5be05fbc6ddfc5fe04fba0
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/AuxiliaryCubes.lean
@@ -0,0 +1,98 @@
+import OAI.Geometry.PeriodicTiling.VoxelMeasure
+import OAI.Geometry.PeriodicTiling.ClosedCoverChains
+import Mathlib.Analysis.Normed.Group.Bounded
+import Mathlib.MeasureTheory.Measure.AEDisjoint
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum
+import Mathlib.Tactic.Positivity
+import Mathlib.Tactic.Ring
+
+namespace OAI
+
+noncomputable section
+
+namespace PeriodicTilingThree
+
+open Set MeasureTheory
+open scoped ENNReal
+
+def auxiliaryCube (m : ℕ) (c : Space 3) : Set (Space 3) :=
+  Metric.closedBall c ((m : ℝ) / 2)
+
+theorem auxiliaryCube_isClosed (m : ℕ) (c : Space 3) :
+    IsClosed (auxiliaryCube m c) := Metric.isClosed_closedBall
+
+theorem auxiliaryCube_measurableSet (m : ℕ) (c : Space 3) :
+    MeasurableSet (auxiliaryCube m c) := (auxiliaryCube_isClosed m c).measurableSet
+
+theorem auxiliaryCube_nonempty (m : ℕ) (c : Space 3) :
+    (auxiliaryCube m c).Nonempty :=
+  ⟨c, Metric.mem_closedBall_self (by positivity)⟩
+
+theorem auxiliaryCube_volume (m : ℕ) (c : Space 3) :
+    volume (auxiliaryCube m c) = ((m ^ 3 : ℕ) : ℝ≥0∞) := by
+  rw [auxiliaryCube, closedBall_volume c (by positivity)]
+  rw [show 2 * ((m : ℝ) / 2) = (m : ℝ) by ring]
+  rw [ENNReal.ofReal_pow (Nat.cast_nonneg m) 3, ENNReal.ofReal_natCast]
+  simp only [Nat.cast_pow]
+
+theorem auxiliaryCube_translate (m : ℕ) (c : Space 3) :
+    {x : Space 3 | x - c ∈ auxiliaryCube m 0} = auxiliaryCube m c := by
+  ext x
+  simp only [mem_ofPred_eq, auxiliaryCube, Metric.mem_closedBall,
+    dist_eq_norm, sub_zero]
+
+theorem auxiliaryCube_aedisjoint {m : ℕ} (hm : 0 < m)
+    {c c' : Space 3} (hsep : (m : ℝ) ≤ ‖c - c'‖) :
+    AEDisjoint volume (auxiliaryCube m c) (auxiliaryCube m c') := by
+  have hmR : (0 : ℝ) < m := by exact_mod_cast hm
+  have hballs : Disjoint (Metric.ball c ((m : ℝ) / 2))
+      (Metric.ball c' ((m : ℝ) / 2)) := by
+    apply Metric.ball_disjoint_ball
+    rw [dist_eq_norm]
+    linarith
+  exact hballs.aedisjoint.congr
+    (closedBall_ae_eq_ball c (by positivity))
+    (closedBall_ae_eq_ball c' (by positivity))
+
+theorem auxiliaryCube_inter_norm_le {m : ℕ} {c c' : Space 3}
+    (h : (auxiliaryCube m c ∩ auxiliaryCube m c').Nonempty) :
+    ‖c' - c‖ ≤ (m : ℝ) := by
+  obtain ⟨x, hx, hx'⟩ := h
+  have hxle : dist x c ≤ (m : ℝ) / 2 := hx
+  have hxle' : dist x c' ≤ (m : ℝ) / 2 := hx'
+  have ht := dist_triangle c' x c
+  rw [dist_comm c' x, dist_eq_norm c' c] at ht
+  linarith
+
+theorem auxiliaryCube_locallyFinite (m : ℕ) (C : Set (Space 3))
+    (hfin : ∀ K : Set (Space 3), Bornology.IsBounded K → (C ∩ K).Finite) :
+    LocallyFinite (fun c : C => auxiliaryCube m c) := by
+  intro x
+  have hindex : {c : C | (c : Space 3) ∈
+      Metric.closedBall x ((m : ℝ) / 2 + 1)}.Finite := by
+    have hf := Set.Finite.preimage
+      (f := fun c : C => (c : Space 3)) Subtype.val_injective.injOn
+      (hfin (Metric.closedBall x ((m : ℝ) / 2 + 1)) Metric.isBounded_closedBall)
+    exact hf.subset fun c hc => ⟨c.property, hc⟩
+  refine ⟨Metric.ball x 1, Metric.ball_mem_nhds x (by norm_num), hindex.subset ?_⟩
+  rintro c ⟨y, hy, hyx⟩
+  have hyc : dist y (c : Space 3) ≤ (m : ℝ) / 2 := hy
+  have hyx' : dist y x < 1 := hyx
+  change dist (c : Space 3) x ≤ (m : ℝ) / 2 + 1
+  have ht := dist_triangle (c : Space 3) y x
+  rw [dist_comm (c : Space 3) y] at ht
+  linarith
+
+theorem auxiliaryCube_cover_of_ae (m : ℕ) (C : Set (Space 3))
+    (hfin : ∀ K : Set (Space 3), Bornology.IsBounded K → (C ∩ K).Finite)
+    (hcover : ∀ᵐ x ∂volume, ∃ c : C, x ∈ auxiliaryCube m c) :
+    ∀ x, ∃ c : C, x ∈ auxiliaryCube m c :=
+  closed_locallyFinite_cover_of_ae volume (fun c : C => auxiliaryCube m c)
+    (fun c => auxiliaryCube_isClosed m c) (auxiliaryCube_locallyFinite m C hfin) hcover
+
+end PeriodicTilingThree
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/BlockActivationInverse.lean b/lean/OAI/Geometry/PeriodicTiling/BlockActivationInverse.lean
new file mode 100644
index 0000000000000000000000000000000000000000..460db7946fd4b027dab59a6ae9799f03e42133d3
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/BlockActivationInverse.lean
@@ -0,0 +1,236 @@
+import OAI.Geometry.PeriodicTiling.LabeledDigitBlocks
+import OAI.Geometry.PeriodicTiling.ShiftMultiplicity
+import Mathlib.Tactic.Abel
+
+namespace OAI
+
+universe uΔ uK uA uB
+
+namespace PeriodicTilingThree
+
+structure BlockShiftData (Δ : Type uΔ) (a b : ℕ) where
+  e : Δ → ZMod a × ZMod b
+  rhoA : Δ → ZMod a
+  rhoB : Δ → ZMod b
+  orderA : Δ ≃ (ZMod b × ZMod a)
+  orderB : Δ ≃ (ZMod a × ZMod b)
+  orderA_apply : ∀ δ, orderA δ = ((e δ).2, (e δ).1 + rhoA δ)
+  orderB_apply : ∀ δ, orderB δ = ((e δ).1, (e δ).2 + rhoB δ)
+
+namespace BlockShiftData
+
+noncomputable def ofMultiplicity {a b : ℕ} [NeZero a] [NeZero b]
+    (ha : 2 ≤ a) (hb : 2 ≤ b) : BlockShiftData (ShiftIndex a b) a b where
+  e := shift a b
+  rhoA := PeriodicTilingThree.rhoA ha hb
+  rhoB := PeriodicTilingThree.rhoB ha hb
+  orderA := shiftOrderA ha hb
+  orderB := shiftOrderB ha hb
+  orderA_apply := shiftOrderA_apply ha hb
+  orderB_apply := shiftOrderB_apply ha hb
+
+variable {K : Type uK} {A : Type uA} {B : Type uB} {Δ : Type uΔ} [AddCommGroup K] {a b : ℕ}
+
+def blockForward (D : LabeledBlocks K A B a b) (S : BlockShiftData Δ a b)
+    (H : K × Δ → ℤ) (x : ℤ) (k : K) (z : K × Δ) :
+    (ZMod a × ZMod b) × K :=
+  (D.C (k - z.1) + S.e z.2, D.T (x - H z) (k - z.1))
+
+def blockInverse (D : LabeledBlocks K A B a b) (S : BlockShiftData Δ a b)
+    (x : ℤ) (k : K) (v : (ZMod a × ZMod b) × K) : K × Δ :=
+  match D.decode v.2 with
+  | .inl (c, q) =>
+      let δ := S.orderA.symm (v.1.2, v.1.1 + (x : ZMod a) - q)
+      (k - D.decode.symm (.inl (c, v.1.1 - (S.e δ).1)), δ)
+  | .inr (c, q) =>
+      let δ := S.orderB.symm (v.1.1, v.1.2 + (x : ZMod b) - q)
+      (k - D.decode.symm (.inr (c, v.1.2 - (S.e δ).2)), δ)
+
+theorem blockInverse_inl (D : LabeledBlocks K A B a b) (S : BlockShiftData Δ a b)
+    (x : ℤ) (k : K) (α : ZMod a) (ξ : ZMod b) (c : A) (q : ZMod a) :
+    blockInverse D S x k ((α, ξ), D.decode.symm (.inl (c, q))) =
+      let δ := S.orderA.symm (ξ, α + (x : ZMod a) - q)
+      (k - D.decode.symm (.inl (c, α - (S.e δ).1)), δ) := by
+  simp only [blockInverse, Equiv.apply_symm_apply]
+
+theorem blockInverse_inr (D : LabeledBlocks K A B a b) (S : BlockShiftData Δ a b)
+    (x : ℤ) (k : K) (α : ZMod a) (ξ : ZMod b) (c : B) (q : ZMod b) :
+    blockInverse D S x k ((α, ξ), D.decode.symm (.inr (c, q))) =
+      let δ := S.orderB.symm (α, ξ + (x : ZMod b) - q)
+      (k - D.decode.symm (.inr (c, ξ - (S.e δ).2)), δ) := by
+  simp only [blockInverse, Equiv.apply_symm_apply]
+
+theorem blockForward_inl (D : LabeledBlocks K A B a b) (S : BlockShiftData Δ a b)
+    (H : K × Δ → ℤ) (hA : ∀ z, (H z : ZMod a) = S.rhoA z.2)
+    (x : ℤ) (k : K) (c : A) (y : ZMod a) (δ : Δ) :
+    blockForward D S H x k (k - D.decode.symm (.inl (c, y)), δ) =
+      ((y + (S.e δ).1, (S.e δ).2),
+        D.decode.symm (.inl (c, y + (x : ZMod a) - S.rhoA δ))) := by
+  simp [blockForward, Int.cast_sub, hA, add_sub_assoc]
+  apply Prod.ext <;> simp
+
+theorem blockForward_inr (D : LabeledBlocks K A B a b) (S : BlockShiftData Δ a b)
+    (H : K × Δ → ℤ) (hB : ∀ z, (H z : ZMod b) = S.rhoB z.2)
+    (x : ℤ) (k : K) (c : B) (y : ZMod b) (δ : Δ) :
+    blockForward D S H x k (k - D.decode.symm (.inr (c, y)), δ) =
+      (((S.e δ).1, y + (S.e δ).2),
+        D.decode.symm (.inr (c, y + (x : ZMod b) - S.rhoB δ))) := by
+  simp [blockForward, Int.cast_sub, hB, add_sub_assoc]
+  apply Prod.ext <;> simp
+
+theorem recoverA (S : BlockShiftData Δ a b) (δ : Δ) (y t : ZMod a) :
+    S.orderA.symm ((S.e δ).2, (y + (S.e δ).1) + t -
+      (y + t - S.rhoA δ)) = δ := by
+  apply S.orderA.injective
+  rw [Equiv.apply_symm_apply, S.orderA_apply]
+  apply Prod.ext
+  · rfl
+  · abel_nf
+
+theorem recoverB (S : BlockShiftData Δ a b) (δ : Δ) (y t : ZMod b) :
+    S.orderB.symm ((S.e δ).1, (y + (S.e δ).2) + t -
+      (y + t - S.rhoB δ)) = δ := by
+  apply S.orderB.injective
+  rw [Equiv.apply_symm_apply, S.orderB_apply]
+  apply Prod.ext
+  · rfl
+  · abel_nf
+
+theorem blockInverse_forward_inl (D : LabeledBlocks K A B a b)
+    (S : BlockShiftData Δ a b) (H : K × Δ → ℤ)
+    (hA : ∀ z, (H z : ZMod a) = S.rhoA z.2)
+    (x : ℤ) (k : K) (c : A) (y : ZMod a) (δ : Δ) :
+    blockInverse D S x k
+      (blockForward D S H x k (k - D.decode.symm (.inl (c, y)), δ)) =
+      (k - D.decode.symm (.inl (c, y)), δ) := by
+  rw [blockForward_inl D S H hA, blockInverse_inl]
+  simp only [recoverA]
+  simp
+
+theorem blockInverse_forward_inr (D : LabeledBlocks K A B a b)
+    (S : BlockShiftData Δ a b) (H : K × Δ → ℤ)
+    (hB : ∀ z, (H z : ZMod b) = S.rhoB z.2)
+    (x : ℤ) (k : K) (c : B) (y : ZMod b) (δ : Δ) :
+    blockInverse D S x k
+      (blockForward D S H x k (k - D.decode.symm (.inr (c, y)), δ)) =
+      (k - D.decode.symm (.inr (c, y)), δ) := by
+  rw [blockForward_inr D S H hB, blockInverse_inr]
+  simp only [recoverB]
+  simp
+
+theorem blockInverse_forward (D : LabeledBlocks K A B a b)
+    (S : BlockShiftData Δ a b) (H : K × Δ → ℤ)
+    (hA : ∀ z, (H z : ZMod a) = S.rhoA z.2)
+    (hB : ∀ z, (H z : ZMod b) = S.rhoB z.2)
+    (x : ℤ) (k : K) :
+    Function.LeftInverse (blockInverse D S x k) (blockForward D S H x k) := by
+  rintro ⟨l, δ⟩
+  cases he : D.decode (k - l) with
+  | inl v =>
+      rcases v with ⟨c, y⟩
+      have hu : D.decode.symm (.inl (c, y)) = k - l := by
+        simpa only [he] using D.decode.symm_apply_apply (k - l)
+      simpa only [hu, sub_sub_cancel] using blockInverse_forward_inl D S H hA x k c y δ
+  | inr v =>
+      rcases v with ⟨c, y⟩
+      have hu : D.decode.symm (.inr (c, y)) = k - l := by
+        simpa only [he] using D.decode.symm_apply_apply (k - l)
+      simpa only [hu, sub_sub_cancel] using blockInverse_forward_inr D S H hB x k c y δ
+
+theorem blockForward_inverse_inl (D : LabeledBlocks K A B a b)
+    (S : BlockShiftData Δ a b) (H : K × Δ → ℤ)
+    (hA : ∀ z, (H z : ZMod a) = S.rhoA z.2)
+    (x : ℤ) (k : K) (α : ZMod a) (ξ : ZMod b) (c : A) (q : ZMod a) :
+    blockForward D S H x k
+      (blockInverse D S x k ((α, ξ), D.decode.symm (.inl (c, q)))) =
+      ((α, ξ), D.decode.symm (.inl (c, q))) := by
+  let δ := S.orderA.symm (ξ, α + (x : ZMod a) - q)
+  have horder : ((S.e δ).2, (S.e δ).1 + S.rhoA δ) =
+      (ξ, α + (x : ZMod a) - q) :=
+    (S.orderA_apply δ).symm.trans (S.orderA.apply_symm_apply _)
+  have he : (S.e δ).2 = ξ := congrArg Prod.fst horder
+  have hr : (S.e δ).1 + S.rhoA δ = α + (x : ZMod a) - q :=
+    congrArg Prod.snd horder
+  have hlabel : (α - (S.e δ).1) + (x : ZMod a) - S.rhoA δ = q := by
+    calc
+      _ = (α + (x : ZMod a)) - ((S.e δ).1 + S.rhoA δ) := by abel
+      _ = (α + (x : ZMod a)) - (α + (x : ZMod a) - q) := by rw [hr]
+      _ = q := sub_sub_cancel _ _
+  rw [blockInverse_inl]
+  change blockForward D S H x k
+    (k - D.decode.symm (.inl (c, α - (S.e δ).1)), δ) = _
+  rw [blockForward_inl D S H hA]
+  simp only [sub_add_cancel, he, hlabel]
+
+theorem blockForward_inverse_inr (D : LabeledBlocks K A B a b)
+    (S : BlockShiftData Δ a b) (H : K × Δ → ℤ)
+    (hB : ∀ z, (H z : ZMod b) = S.rhoB z.2)
+    (x : ℤ) (k : K) (α : ZMod a) (ξ : ZMod b) (c : B) (q : ZMod b) :
+    blockForward D S H x k
+      (blockInverse D S x k ((α, ξ), D.decode.symm (.inr (c, q)))) =
+      ((α, ξ), D.decode.symm (.inr (c, q))) := by
+  let δ := S.orderB.symm (α, ξ + (x : ZMod b) - q)
+  have horder : ((S.e δ).1, (S.e δ).2 + S.rhoB δ) =
+      (α, ξ + (x : ZMod b) - q) :=
+    (S.orderB_apply δ).symm.trans (S.orderB.apply_symm_apply _)
+  have he : (S.e δ).1 = α := congrArg Prod.fst horder
+  have hr : (S.e δ).2 + S.rhoB δ = ξ + (x : ZMod b) - q :=
+    congrArg Prod.snd horder
+  have hlabel : (ξ - (S.e δ).2) + (x : ZMod b) - S.rhoB δ = q := by
+    calc
+      _ = (ξ + (x : ZMod b)) - ((S.e δ).2 + S.rhoB δ) := by abel
+      _ = (ξ + (x : ZMod b)) - (ξ + (x : ZMod b) - q) := by rw [hr]
+      _ = q := sub_sub_cancel _ _
+  rw [blockInverse_inr]
+  change blockForward D S H x k
+    (k - D.decode.symm (.inr (c, ξ - (S.e δ).2)), δ) = _
+  rw [blockForward_inr D S H hB]
+  simp only [sub_add_cancel, he, hlabel]
+
+theorem blockForward_inverse (D : LabeledBlocks K A B a b)
+    (S : BlockShiftData Δ a b) (H : K × Δ → ℤ)
+    (hA : ∀ z, (H z : ZMod a) = S.rhoA z.2)
+    (hB : ∀ z, (H z : ZMod b) = S.rhoB z.2)
+    (x : ℤ) (k : K) :
+    Function.RightInverse (blockInverse D S x k) (blockForward D S H x k) := by
+  rintro ⟨⟨α, ξ⟩, β⟩
+  cases he : D.decode β with
+  | inl v =>
+      rcases v with ⟨c, q⟩
+      have hβ : D.decode.symm (.inl (c, q)) = β := by
+        simpa only [he] using D.decode.symm_apply_apply β
+      simpa only [hβ] using blockForward_inverse_inl D S H hA x k α ξ c q
+  | inr v =>
+      rcases v with ⟨c, q⟩
+      have hβ : D.decode.symm (.inr (c, q)) = β := by
+        simpa only [he] using D.decode.symm_apply_apply β
+      simpa only [hβ] using blockForward_inverse_inr D S H hB x k α ξ c q
+
+def blockInverseEquiv (D : LabeledBlocks K A B a b) (S : BlockShiftData Δ a b)
+    (H : K × Δ → ℤ) (hA : ∀ z, (H z : ZMod a) = S.rhoA z.2)
+    (hB : ∀ z, (H z : ZMod b) = S.rhoB z.2) (x : ℤ) (k : K) :
+    (K × Δ) ≃ ((ZMod a × ZMod b) × K) where
+  toFun := blockForward D S H x k
+  invFun := blockInverse D S x k
+  left_inv := blockInverse_forward D S H hA hB x k
+  right_inv := blockForward_inverse D S H hA hB x k
+
+@[simp] theorem blockInverseEquiv_apply (D : LabeledBlocks K A B a b)
+    (S : BlockShiftData Δ a b) (H : K × Δ → ℤ)
+    (hA : ∀ z, (H z : ZMod a) = S.rhoA z.2)
+    (hB : ∀ z, (H z : ZMod b) = S.rhoB z.2) (x : ℤ) (k : K) (z : K × Δ) :
+    blockInverseEquiv D S H hA hB x k z =
+      (D.C (k - z.1) + S.e z.2, D.T (x - H z) (k - z.1)) := rfl
+
+@[simp] theorem blockInverseEquiv_symm_apply (D : LabeledBlocks K A B a b)
+    (S : BlockShiftData Δ a b) (H : K × Δ → ℤ)
+    (hA : ∀ z, (H z : ZMod a) = S.rhoA z.2)
+    (hB : ∀ z, (H z : ZMod b) = S.rhoB z.2)
+    (x : ℤ) (k : K) (v : (ZMod a × ZMod b) × K) :
+    (blockInverseEquiv D S H hA hB x k).symm v = blockInverse D S x k v := rfl
+
+end BlockShiftData
+
+end PeriodicTilingThree
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/BoundedDifferences.lean b/lean/OAI/Geometry/PeriodicTiling/BoundedDifferences.lean
new file mode 100644
index 0000000000000000000000000000000000000000..e8cb0744630e8dd61730ddf6d04b6ae51a31ab40
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/BoundedDifferences.lean
@@ -0,0 +1,99 @@
+import OAI.Geometry.PeriodicTiling.ConfigurationDifferences
+import OAI.Geometry.PeriodicTiling.ForwardDifference
+import Mathlib.Algebra.Order.Archimedean.Real.Basic
+import Mathlib.Tactic.Linarith
+
+namespace OAI
+
+universe uG uA
+
+namespace PeriodicTilingThree
+
+variable {G : Type uG} [AddCommGroup G]
+
+theorem configDiff_bounded (v : G) (f : G → ℝ)
+    (hf : ∃ B : ℝ, ∀ x, |f x| ≤ B) :
+    ∃ B : ℝ, ∀ x, |configDiff v f x| ≤ B := by
+  obtain ⟨B, hB⟩ := hf
+  refine ⟨B + B, fun x => ?_⟩
+  calc
+    |configDiff v f x| = |f (x + v) + -f x| := rfl
+    _ ≤ |f (x + v)| + |-f x| := abs_add_le _ _
+    _ = |f (x + v)| + |f x| := by rw [abs_neg]
+    _ ≤ B + B := add_le_add (hB _) (hB _)
+
+theorem configDiff_eq_zero_of_second (v : G) (f : G → ℝ)
+    (hf : ∃ B : ℝ, ∀ x, |f x| ≤ B)
+    (hsecond : configDiff v (configDiff v f) = 0) : configDiff v f = 0 := by
+  obtain ⟨B, hB⟩ := hf
+  let g := configDiff v f
+  have hg : Function.Periodic g v := (configDiff_eq_zero_iff v g).mp hsecond
+  ext x
+  have hg_nat : ∀ n : ℕ, g (x + n • v) = g x := by
+    intro n
+    induction n with
+    | zero => simp
+    | succ n ih =>
+        simpa only [succ_nsmul, ← add_assoc] using (hg (x + n • v)).trans ih
+  have hf_nat : ∀ n : ℕ, f (x + n • v) = f x + (n : ℝ) * g x := by
+    intro n
+    induction n with
+    | zero => simp
+    | succ n ih =>
+        have hd := hg_nat n
+        change f (x + n • v + v) - f (x + n • v) = g x at hd
+        simp only [succ_nsmul, ← add_assoc, Nat.cast_add, Nat.cast_one]
+        linarith
+  have hx := abs_le.mp (hB x)
+  have hz : g x = 0 := by
+    rcases lt_trichotomy (g x) 0 with hneg | hzero | hpos
+    · obtain ⟨n, hn⟩ := exists_nat_gt (2 * B / (-g x))
+      have hn' : 2 * B < (n : ℝ) * (-g x) :=
+        (div_lt_iff₀ (neg_pos.mpr hneg)).mp hn
+      have hy := abs_le.mp (hB (x + n • v))
+      have he := hf_nat n
+      nlinarith
+    · exact hzero
+    · obtain ⟨n, hn⟩ := exists_nat_gt (2 * B / g x)
+      have hn' : 2 * B < (n : ℝ) * g x := (div_lt_iff₀ hpos).mp hn
+      have hy := abs_le.mp (hB (x + n • v))
+      have he := hf_nat n
+      nlinarith
+  exact hz
+
+theorem configDiff_eq_zero_of_iterate (v : G) (n : ℕ) (f : G → ℝ)
+    (hf : ∃ B : ℝ, ∀ x, |f x| ≤ B)
+    (hn : 0 < n) (hiter : (configDiff v)^[n] f = 0) :
+    configDiff v f = 0 := by
+  induction n generalizing f with
+  | zero => exact (Nat.not_lt_zero _ hn).elim
+  | succ n ih =>
+      cases n with
+      | zero =>
+          change configDiff v f = 0 at hiter
+          exact hiter
+      | succ n =>
+          have hnext : (configDiff v)^[n + 1] (configDiff v f) = 0 := by
+            simpa only [Function.iterate_succ_apply] using hiter
+          have hsecond : configDiff v (configDiff v f) = 0 :=
+            ih (configDiff v f) (configDiff_bounded v f hf) (Nat.succ_pos _) hnext
+          exact configDiff_eq_zero_of_second v f hf hsecond
+
+theorem periodic_of_bounded_configDiff_iterate (v : G) (n : ℕ) (f : G → ℝ)
+    (hf : ∃ B : ℝ, ∀ x, |f x| ≤ B)
+    (hn : 0 < n) (hiter : (configDiff v)^[n] f = 0) :
+    Function.Periodic f v :=
+  (configDiff_eq_zero_iff v f).mp (configDiff_eq_zero_of_iterate v n f hf hn hiter)
+
+theorem forwardDiff_eq_configDiff {A : Type uA} [AddCommGroup A] :
+    (forwardDiff : (ℤ → A) → ℤ → A) = configDiff (1 : ℤ) := rfl
+
+theorem forwardDiff_eq_zero_of_bounded_iterate (n : ℕ) (f : ℤ → ℝ)
+    (hf : ∃ B : ℝ, ∀ x, |f x| ≤ B)
+    (hn : 0 < n) (hiter : (forwardDiff^[n]) f = 0) : forwardDiff f = 0 := by
+  rw [forwardDiff_eq_configDiff] at hiter ⊢
+  exact configDiff_eq_zero_of_iterate 1 n f hf hn hiter
+
+end PeriodicTilingThree
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/CenterFiniteness.lean b/lean/OAI/Geometry/PeriodicTiling/CenterFiniteness.lean
new file mode 100644
index 0000000000000000000000000000000000000000..99d4fd56a886dd4adfb6b22b2fff612237ab5c1b
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/CenterFiniteness.lean
@@ -0,0 +1,81 @@
+import OAI.Geometry.PeriodicTiling.EuclideanBasic
+import Mathlib.Algebra.Order.Floor.Ring
+import Mathlib.Analysis.Normed.Group.Bounded
+import Mathlib.Data.Int.Interval
+import Mathlib.Data.Pi.Interval
+import Mathlib.Data.Set.Countable
+import Mathlib.Tactic.Linarith
+
+namespace OAI
+
+noncomputable section
+
+namespace PeriodicTilingThree
+
+def centerBoxIndex {n : ℕ} (r : ℝ) (x : Space n) : Lattice n :=
+  fun i => ⌊x i / r⌋
+
+theorem norm_sub_lt_of_centerBoxIndex_eq {n : ℕ} {r : ℝ} (hr : 0 < r)
+    {x y : Space n} (hxy : centerBoxIndex r x = centerBoxIndex r y) :
+    ‖x - y‖ < r := by
+  apply (pi_norm_lt_iff hr).mpr
+  intro i
+  have heq : ⌊x i / r⌋ = ⌊y i / r⌋ := congrFun hxy i
+  have hxl := Int.floor_le (x i / r)
+  have hxu := Int.lt_floor_add_one (x i / r)
+  have hyl := Int.floor_le (y i / r)
+  have hyu := Int.lt_floor_add_one (y i / r)
+  rw [← heq] at hyl hyu
+  have hxl' := (le_div_iff₀ hr).mp hxl
+  have hxu' := (div_lt_iff₀ hr).mp hxu
+  have hyl' := (le_div_iff₀ hr).mp hyl
+  have hyu' := (div_lt_iff₀ hr).mp hyu
+  simp only [Pi.sub_apply, Real.norm_eq_abs]
+  apply abs_lt.mpr
+  constructor <;> nlinarith
+
+theorem centerBoxIndex_injOn {n : ℕ} {C : Set (Space n)} {r : ℝ}
+    (hr : 0 < r) (hsep : C.Pairwise (fun x y => r ≤ ‖x - y‖)) :
+    Set.InjOn (centerBoxIndex r) C := by
+  intro x hx y hy hxy
+  by_contra hne
+  exact (not_lt_of_ge (hsep hx hy hne)) (norm_sub_lt_of_centerBoxIndex_eq hr hxy)
+
+theorem centers_countable {n : ℕ} {C : Set (Space n)} {r : ℝ}
+    (hr : 0 < r) (hsep : C.Pairwise (fun x y => r ≤ ‖x - y‖)) : C.Countable := by
+  apply Set.countable_of_injective_of_countable_image (centerBoxIndex_injOn hr hsep)
+  exact Set.to_countable _
+
+theorem centerBoxIndex_image_finite {n : ℕ} {r : ℝ} (hr : 0 < r)
+    {K : Set (Space n)} (hK : Bornology.IsBounded K) :
+    (centerBoxIndex r '' K).Finite := by
+  obtain ⟨R, hR⟩ := hK.exists_norm_le
+  let lo : Lattice n := fun _ => ⌊-R / r⌋
+  let hi : Lattice n := fun _ => ⌊R / r⌋
+  apply (Set.finite_Icc lo hi).subset
+  rintro z ⟨x, hx, rfl⟩
+  have hcoord (i : Fin n) : |x i| ≤ R := by
+    calc
+      |x i| = ‖x i‖ := by simp only [Real.norm_eq_abs]
+      _ ≤ ‖x‖ := norm_le_pi_norm x i
+      _ ≤ R := hR x hx
+  constructor
+  · intro i
+    change ⌊-R / r⌋ ≤ ⌊x i / r⌋
+    exact Int.floor_mono (div_le_div_of_nonneg_right (abs_le.mp (hcoord i)).1 hr.le)
+  · intro i
+    change ⌊x i / r⌋ ≤ ⌊R / r⌋
+    exact Int.floor_mono (div_le_div_of_nonneg_right (abs_le.mp (hcoord i)).2 hr.le)
+
+theorem centers_finite_on_bounded {n : ℕ} {C K : Set (Space n)} {r : ℝ}
+    (hr : 0 < r) (hsep : C.Pairwise (fun x y => r ≤ ‖x - y‖))
+    (hK : Bornology.IsBounded K) : (C ∩ K).Finite := by
+  have hfin : (centerBoxIndex r '' (C ∩ K)).Finite :=
+    (centerBoxIndex_image_finite hr hK).subset (Set.image_mono Set.inter_subset_right)
+  exact hfin.of_finite_image ((centerBoxIndex_injOn hr hsep).mono Set.inter_subset_left)
+
+end PeriodicTilingThree
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/ChannelDigitBlocks.lean b/lean/OAI/Geometry/PeriodicTiling/ChannelDigitBlocks.lean
new file mode 100644
index 0000000000000000000000000000000000000000..a7e0b2da4128842ee567190d5eea2063effb7c3d
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/ChannelDigitBlocks.lean
@@ -0,0 +1,131 @@
+import OAI.Geometry.PeriodicTiling.LabeledDigitBlocks
+import OAI.Geometry.PeriodicTiling.CyclicCRT
+
+namespace OAI
+
+universe uK uA uB uL uJ uF
+
+noncomputable section
+
+namespace PeriodicTilingThree
+
+variable {p : ℕ} [NeZero p] (E : EncodingParameters p)
+
+abbrev ChannelOtherDigits (i : Channel p) (j : Label p i) :=
+  ∀ q : {q : Label p i // q ≠ j}, ZMod (E.digitPrime i q)
+
+abbrev ChannelBlockA (i : Channel p) (j : Label p i) :=
+  Fin (E.blockA i j) × ChannelOtherDigits E i j
+
+abbrev ChannelBlockB (i : Channel p) (j : Label p i) :=
+  Fin (E.blockB i j) × ChannelOtherDigits E i j
+
+def channelDigitBase (i : Channel p) (j : Label p i) :
+    LabeledBlocks (ZMod (E.digitPrime i j)) (Fin (E.blockA i j))
+      (Fin (E.blockB i j)) (E.a i) (E.b i) :=
+  LabeledBlocks.digit (E.a i) (E.b i) (E.digitPrime i j)
+    (E.blockA i j) (E.blockB i j) (E.digit_eq i j)
+
+def channelDigitBlocks (i : Channel p) (j : Label p i) :
+    LabeledBlocks (ZMod (E.r i)) (ChannelBlockA E i j)
+      (ChannelBlockB E i j) (E.a i) (E.b i) :=
+  (LabeledBlocks.onCoordinate (fun q => ZMod (E.digitPrime i q)) j
+    (channelDigitBase E i j)).transport (digitEquiv E i).toEquiv
+
+@[simp] theorem channelDigitBlocks_C (i : Channel p) (j : Label p i)
+    (w : ZMod (E.r i)) :
+    (channelDigitBlocks E i j).C w =
+      (channelDigitBase E i j).C (digitEquiv E i w j) := by
+  simp [channelDigitBlocks]
+
+theorem channelDigitBlocks_C_eq (i : Channel p) (j : Label p i)
+    {w w' : ZMod (E.r i)} (h : digitEquiv E i w j = digitEquiv E i w' j) :
+    (channelDigitBlocks E i j).C w = (channelDigitBlocks E i j).C w' := by
+  simp only [channelDigitBlocks_C, h]
+
+theorem channelDigitBlocks_active_iff (i : Channel p) (j q : Label p i) :
+    (∃ w w' : ZMod (E.r i), w - w' ∈ digitSubgroup E i q ∧
+      (channelDigitBlocks E i j).C w ≠ (channelDigitBlocks E i j).C w') ↔ q = j := by
+  constructor
+  · rintro ⟨w, w', hdiff, hne⟩
+    by_contra hq
+    apply hne
+    exact channelDigitBlocks_C_eq E i j
+      ((sub_mem_digitSubgroup_iff E i q w w').1 hdiff j (Ne.symm hq))
+  · intro hq
+    subst q
+    let : Fact (1 < E.a i) := ⟨(E.a_prime i).one_lt⟩
+    let : Fact (1 < E.b i) := ⟨(E.b_prime i).one_lt⟩
+    obtain ⟨x, y, hxy, hne⟩ :=
+      LabeledBlocks.C_designated_witness (fun q => ZMod (E.digitPrime i q)) j
+        (channelDigitBase E i j)
+    refine ⟨(digitEquiv E i).symm x, (digitEquiv E i).symm y, ?_, ?_⟩
+    · apply (sub_mem_digitSubgroup_iff E i j _ _).2
+      simpa only [AddEquiv.apply_symm_apply] using hxy
+    · simpa only [channelDigitBlocks_C, AddEquiv.apply_symm_apply,
+        LabeledBlocks.C_onCoordinate] using hne
+
+namespace LabeledBlocks
+
+def castSizes {K : Type uK} {A : Type uA} {B : Type uB} {a b a' b' : ℕ}
+    (D : LabeledBlocks K A B a b)
+    (ha : a = a') (hb : b = b') : LabeledBlocks K A B a' b' := by
+  subst a'
+  subst b'
+  exact D
+
+@[simp] theorem T_transport_apply {K : Type uK} {L : Type uL} {A : Type uA} {B : Type uB} {a b : ℕ}
+    (D : LabeledBlocks K A B a b) (e : L ≃ K) (t : ℤ) (x : L) :
+    (D.transport e).T t x = e.symm (D.T t (e x)) := rfl
+
+theorem T_onCoordinate {J : Type uJ} [DecidableEq J] (F : J → Type uF) (j : J)
+    {A : Type uA} {B : Type uB} {a b : ℕ} (D : LabeledBlocks (F j) A B a b) (t : ℤ) (x : ∀ q, F q) :
+    (onCoordinate F j D).T t x =
+      (Equiv.piSplitAt j F).symm (D.T t (x j), fun q => x q) := by
+  rw [onCoordinate, T_transport_apply, T_copyRight]
+  rfl
+
+theorem T_onCoordinate_other {J : Type uJ} [DecidableEq J] (F : J → Type uF) (j : J)
+    {A : Type uA} {B : Type uB} {a b : ℕ} (D : LabeledBlocks (F j) A B a b) (t : ℤ)
+    (x : ∀ q, F q) (q : J) (hq : q ≠ j) :
+    (onCoordinate F j D).T t x q = x q := by
+  rw [T_onCoordinate]
+  simp [Equiv.piSplitAt, hq]
+
+theorem T_onCoordinate_transport_other {J : Type uJ} {L : Type uL} [DecidableEq J]
+    (F : J → Type uF) (j : J) {A : Type uA} {B : Type uB} {a b : ℕ}
+    (D : LabeledBlocks (F j) A B a b) (e : L ≃ (∀ q, F q)) (t : ℤ)
+    (x : L) (q : J) (hq : q ≠ j) :
+    e (((onCoordinate F j D).transport e).T t x) q = e x q := by
+  rw [T_transport_apply, Equiv.apply_symm_apply]
+  exact T_onCoordinate_other F j D t (e x) q hq
+
+end LabeledBlocks
+
+theorem channelDigitBlocks_T_other (i : Channel p) (j : Label p i) (t : ℤ)
+    (w : ZMod (E.r i)) (q : Label p i) (hq : q ≠ j) :
+    digitEquiv E i ((channelDigitBlocks E i j).T t w) q = digitEquiv E i w q := by
+  simpa only [channelDigitBlocks, AddEquiv.toEquiv_eq_coe, AddEquiv.coe_toEquiv] using
+    (LabeledBlocks.T_onCoordinate_transport_other
+      (fun q : Label p i => ZMod (E.digitPrime i q)) j
+      (channelDigitBase E i j) (digitEquiv E i).toEquiv t w q hq)
+
+def seedOneBlocks :
+    LabeledBlocks (ZMod (E.r (.inr (0 : Fin 2))))
+      (ChannelBlockA E (.inr 0) ()) (ChannelBlockB E (.inr 0) ())
+      2 (E.b (.inr 0)) :=
+  (channelDigitBlocks E (.inr 0) ()).castSizes
+    (by simpa using E.seed_a (0 : Fin 2)) rfl
+
+def seedTwoBlocks :
+    LabeledBlocks (ZMod (E.r (.inr (1 : Fin 2))))
+      (ChannelBlockA E (.inr 1) ()) (ChannelBlockB E (.inr 1) ())
+      3 (E.b (.inr 1)) :=
+  (channelDigitBlocks E (.inr 1) ()).castSizes
+    (by simpa using E.seed_a (1 : Fin 2)) rfl
+
+end PeriodicTilingThree
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/CircleWeylCriterion.lean b/lean/OAI/Geometry/PeriodicTiling/CircleWeylCriterion.lean
new file mode 100644
index 0000000000000000000000000000000000000000..57d70371a35587dfe4289bb6a944fc8db8d26b09
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/CircleWeylCriterion.lean
@@ -0,0 +1,168 @@
+import OAI.Geometry.PeriodicTiling.ScalarAverage
+import Mathlib.Analysis.Fourier.AddCircle
+
+namespace OAI
+
+noncomputable section
+
+namespace PeriodicTilingThree
+
+open Filter MeasureTheory Set
+open scoped Topology
+
+private instance unitPeriodPositive : Fact (0 < (1 : ℝ)) := ⟨zero_lt_one⟩
+
+abbrev UnitAddCircle := AddCircle (1 : ℝ)
+
+def CircleFourierCancellation (u : ℕ → UnitAddCircle) : Prop :=
+  ∀ k : ℤ, k ≠ 0 →
+    Tendsto (complexAverage (fun n => fourier k (u n))) atTop (𝓝 0)
+
+private theorem circle_continuous_integrable (f : C(UnitAddCircle, ℂ)) :
+    Integrable f AddCircle.haarAddCircle :=
+  f.continuous.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace f)
+
+private theorem circle_integral_fourier (k : ℤ) :
+    (∫ x : UnitAddCircle, fourier k x ∂AddCircle.haarAddCircle) =
+      if k = 0 then 1 else 0 := by
+  split_ifs with hk
+  · subst k
+    simp only [fourier_zero, integral_const, probReal_univ, one_smul]
+  · exact integral_eq_zero_of_add_right_eq_neg
+      (μ := AddCircle.haarAddCircle)
+      (fourier_add_half_inv_index hk (show (0 : ℝ) < 1 from zero_lt_one))
+
+private theorem circle_integral_norm_le (f : C(UnitAddCircle, ℂ)) :
+    ‖∫ x, f x ∂AddCircle.haarAddCircle‖ ≤ ‖f‖ := by
+  simpa only [probReal_univ, mul_one] using
+    (norm_integral_le_of_norm_le_const
+      (μ := AddCircle.haarAddCircle)
+      (Filter.Eventually.of_forall fun x => f.norm_coe_le_norm x))
+
+private theorem circle_average_norm_le (u : ℕ → UnitAddCircle)
+    (f : C(UnitAddCircle, ℂ)) (N : ℕ) :
+    ‖complexAverage (fun n => f (u n)) N‖ ≤ ‖f‖ :=
+  norm_complexAverage_le _ _ (norm_nonneg f) (fun n => f.norm_coe_le_norm (u n)) N
+
+private def circleAveragingSubmodule (u : ℕ → UnitAddCircle) :
+    Submodule ℂ C(UnitAddCircle, ℂ) where
+  carrier := {f | Tendsto (complexAverage (fun n => f (u n))) atTop
+    (𝓝 (∫ x, f x ∂AddCircle.haarAddCircle))}
+  zero_mem' := by
+    change Tendsto (complexAverage (fun _ => (0 : ℂ))) atTop
+      (𝓝 (∫ _ : UnitAddCircle, (0 : ℂ) ∂AddCircle.haarAddCircle))
+    have hz : complexAverage (fun _ => (0 : ℂ)) = fun _ => 0 := by
+      funext N
+      simp [complexAverage]
+    rw [hz, integral_zero]
+    exact tendsto_const_nhds
+  add_mem' := by
+    intro f g hf hg
+    change Tendsto (complexAverage (fun n => f (u n) + g (u n))) atTop
+      (𝓝 (∫ x, f x + g x ∂AddCircle.haarAddCircle))
+    have ha : complexAverage (fun n => f (u n) + g (u n)) =
+        fun N => complexAverage (fun n => f (u n)) N +
+          complexAverage (fun n => g (u n)) N := by
+      funext N
+      exact complexAverage_add _ _ N
+    rw [ha, integral_add (circle_continuous_integrable f) (circle_continuous_integrable g)]
+    exact hf.add hg
+  smul_mem' := by
+    intro c f hf
+    change Tendsto (complexAverage (fun n => c * f (u n))) atTop
+      (𝓝 (∫ x, c * f x ∂AddCircle.haarAddCircle))
+    have ha : complexAverage (fun n => c * f (u n)) =
+        fun N => c * complexAverage (fun n => f (u n)) N := by
+      funext N
+      exact complexAverage_const_mul _ _ N
+    rw [ha, integral_const_mul]
+    exact hf.const_mul c
+
+private theorem fourier_mem_circleAveragingSubmodule (u : ℕ → UnitAddCircle)
+    (hu : CircleFourierCancellation u) (k : ℤ) :
+    fourier k ∈ circleAveragingSubmodule u := by
+  change Tendsto _ atTop (𝓝 _)
+  rw [circle_integral_fourier]
+  by_cases hk : k = 0
+  · subst k
+    simp only [ite_true, fourier_zero]
+    apply tendsto_const_nhds.congr'
+    filter_upwards [Filter.eventually_ge_atTop (1 : ℕ)] with N hN
+    exact (complexAverage_const 1 (Nat.ne_of_gt (lt_of_lt_of_le Nat.zero_lt_one hN))).symm
+  · simpa only [ite_eq_right hk] using hu k hk
+
+theorem circle_averages_tendsto_integral (u : ℕ → UnitAddCircle)
+    (hu : CircleFourierCancellation u) (f : C(UnitAddCircle, ℂ)) :
+    Tendsto (complexAverage (fun n => f (u n))) atTop
+      (𝓝 (∫ x, f x ∂AddCircle.haarAddCircle)) := by
+  have hspan : Submodule.span ℂ (Set.range (@fourier (1 : ℝ))) ≤
+      circleAveragingSubmodule u := by
+    apply Submodule.span_le.mpr
+    rintro _ ⟨k, rfl⟩
+    exact fourier_mem_circleAveragingSubmodule u hu k
+  have hdense : Dense (Submodule.span ℂ (Set.range (@fourier (1 : ℝ))) :
+      Set C(UnitAddCircle, ℂ)) :=
+    Submodule.dense_iff_topologicalClosure_eq_top.mpr span_fourier_closure_eq_top
+  apply Metric.tendsto_nhds.mpr
+  intro ε hε
+  obtain ⟨p, hp, hfp⟩ := hdense.exists_dist_lt f (show 0 < ε / 4 by positivity)
+  have hpconv := hspan hp
+  change Tendsto (complexAverage (fun n => p (u n))) atTop
+    (𝓝 (∫ x, p x ∂AddCircle.haarAddCircle)) at hpconv
+  filter_upwards [Metric.tendsto_nhds.mp hpconv (ε / 2) (by positivity)] with N hN
+  have havg : ‖complexAverage (fun n => f (u n)) N -
+      complexAverage (fun n => p (u n)) N‖ ≤ ‖f - p‖ := by
+    rw [← complexAverage_sub]
+    exact circle_average_norm_le u (f - p) N
+  have hint : ‖(∫ x, p x ∂AddCircle.haarAddCircle) -
+      (∫ x, f x ∂AddCircle.haarAddCircle)‖ ≤ ‖f - p‖ := by
+    rw [← integral_sub (circle_continuous_integrable p) (circle_continuous_integrable f)]
+    simpa only [ContinuousMap.sub_apply, norm_sub_rev] using circle_integral_norm_le (p - f)
+  have hbound := norm_add_le
+    (complexAverage (fun n => f (u n)) N - complexAverage (fun n => p (u n)) N)
+    (complexAverage (fun n => p (u n)) N - (∫ x, f x ∂AddCircle.haarAddCircle))
+  have hbound' := norm_add_le
+    (complexAverage (fun n => p (u n)) N - (∫ x, p x ∂AddCircle.haarAddCircle))
+    ((∫ x, p x ∂AddCircle.haarAddCircle) - (∫ x, f x ∂AddCircle.haarAddCircle))
+  simp only [sub_add_sub_cancel] at hbound hbound'
+  rw [dist_eq_norm] at hfp hN ⊢
+  linarith
+
+theorem denseRange_of_circleFourierCancellation (u : ℕ → UnitAddCircle)
+    (hu : CircleFourierCancellation u) : DenseRange u := by
+  rw [Metric.denseRange_iff]
+  intro x r hr
+  by_contra h
+  have haway : ∀ n, r ≤ dist x (u n) :=
+    fun n => le_of_not_gt (fun hn => h ⟨n, hn⟩)
+  let b : C(UnitAddCircle, ℝ) :=
+    ⟨fun y => max (r - dist x y) 0,
+      (continuous_const.sub (continuous_const.dist continuous_id)).max continuous_const⟩
+  have hb0 : ∀ n, b (u n) = 0 := by
+    intro n
+    exact max_eq_right (sub_nonpos.mpr (haway n))
+  have hbx : b x ≠ 0 := by simpa [b, dist_self, max_eq_left hr.le] using hr.ne'
+  have hbnonneg : (0 : UnitAddCircle → ℝ) ≤ b := fun y => le_max_right _ _
+  have hbint : 0 < ∫ y, b y ∂AddCircle.haarAddCircle :=
+    b.continuous.integral_pos_of_hasCompactSupport_nonneg_nonzero
+      (HasCompactSupport.of_compactSpace b) hbnonneg hbx
+  let bc : C(UnitAddCircle, ℂ) := ⟨fun y => (b y : ℂ), Complex.continuous_ofReal.comp b.continuous⟩
+  have hlim := circle_averages_tendsto_integral u hu bc
+  have havgzero : complexAverage (fun n => bc (u n)) = fun _ => 0 := by
+    funext N
+    simp [complexAverage, bc, hb0]
+  rw [havgzero] at hlim
+  have hi : (∫ y, bc y ∂AddCircle.haarAddCircle) = 0 :=
+    tendsto_nhds_unique hlim tendsto_const_nhds
+  have hir : (∫ y, b y ∂AddCircle.haarAddCircle) = 0 := by
+    apply Complex.ofReal_injective
+    change (∫ y, (b y : ℂ) ∂AddCircle.haarAddCircle) = 0 at hi
+    rw [integral_complex_ofReal] at hi
+    simpa only [Complex.ofReal_zero] using hi
+  linarith
+
+end PeriodicTilingThree
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/ClosedBoxRounding.lean b/lean/OAI/Geometry/PeriodicTiling/ClosedBoxRounding.lean
new file mode 100644
index 0000000000000000000000000000000000000000..8349fd90b3b3ce20015845f3ab959405bd34e0f8
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/ClosedBoxRounding.lean
@@ -0,0 +1,116 @@
+import OAI.Geometry.PeriodicTiling.EuclideanBasic
+import Mathlib.Algebra.Order.Floor.Ring
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum
+import Lean.Elab.Tactic.Omega
+
+namespace OAI
+
+noncomputable section
+
+namespace PeriodicTilingThree
+
+private theorem floor_abs_le_nat {m : ℕ} {x : ℝ} (hx : |x| ≤ (m : ℝ)) :
+    |⌊x⌋| ≤ (m : ℤ) := by
+  apply abs_le.mpr
+  constructor
+  · apply Int.le_floor.mpr
+    exact_mod_cast (abs_le.mp hx).1
+  · have hh : (⌊x⌋ : ℝ) ≤ (m : ℝ) := (Int.floor_le x).trans (abs_le.mp hx).2
+    exact_mod_cast hh
+
+private theorem floor_error_lt_one (x : ℝ) : |x - (⌊x⌋ : ℝ)| < 1 := by
+  rw [abs_of_nonneg (sub_nonneg.mpr (Int.floor_le x))]
+  have := Int.lt_floor_add_one x
+  linarith
+
+private theorem floor_successor_error_lt_one {x : ℝ} (hx : x ≠ (⌊x⌋ : ℝ)) :
+    |x - ((⌊x⌋ + 1 : ℤ) : ℝ)| < 1 := by
+  have hlo : (⌊x⌋ : ℝ) < x := lt_of_le_of_ne (Int.floor_le x) hx.symm
+  have hhi := Int.lt_floor_add_one x
+  simp only [Int.cast_add, Int.cast_one]
+  apply abs_lt.mpr
+  constructor <;> linarith
+
+theorem closed_box_rounding_exists {n : ℕ} (m : ℕ) (hm : 2 ≤ m) (δ : Space n)
+    (hδ : ∀ i, |δ i| ≤ (m : ℝ))
+    (hnot : ¬ ∃ z : Lattice n, δ = (m : ℝ) • castLattice z) :
+    ∃ z : Lattice n,
+      (∀ i, |z i| ≤ (m : ℤ)) ∧
+      (∃ i, ¬ (m : ℤ) ∣ z i) ∧
+      (∀ i, |δ i - (z i : ℝ)| < 1) := by
+  classical
+  let a : Lattice n := fun i => ⌊δ i⌋
+  have habs (i : Fin n) : |a i| ≤ (m : ℤ) := floor_abs_le_nat (hδ i)
+  have herr (i : Fin n) : |δ i - (a i : ℝ)| < 1 := floor_error_lt_one (δ i)
+  by_cases hdiv : ∀ i, (m : ℤ) ∣ a i
+  · have hnonint : ∃ j, δ j ≠ (a j : ℝ) := by
+      by_contra h
+      have heq (i : Fin n) : δ i = (a i : ℝ) := by
+        by_contra hi
+        exact h ⟨i, hi⟩
+      let z : Lattice n := fun i => Classical.choose (hdiv i)
+      have hz (i : Fin n) : a i = (m : ℤ) * z i := Classical.choose_spec (hdiv i)
+      apply hnot
+      refine ⟨z, ?_⟩
+      funext i
+      change δ i = (m : ℝ) * (z i : ℝ)
+      rw [heq i]
+      exact_mod_cast hz i
+    obtain ⟨j, hj⟩ := hnonint
+    have hjstrict : (a j : ℝ) < δ j := lt_of_le_of_ne (Int.floor_le (δ j)) hj.symm
+    have hjlt : a j < (m : ℤ) := by
+      have hh : (a j : ℝ) < (m : ℝ) := hjstrict.trans_le (abs_le.mp (hδ j)).2
+      exact_mod_cast hh
+    let z : Lattice n := Function.update a j (a j + 1)
+    refine ⟨z, ?_, ?_, ?_⟩
+    · intro i
+      by_cases hij : i = j
+      · subst i
+        simp only [z, Function.update_self]
+        apply abs_le.mpr
+        have ha := (abs_le.mp (habs j)).1
+        constructor <;> omega
+      · simpa only [z, Function.update_of_ne hij] using habs i
+    · refine ⟨j, ?_⟩
+      simp only [z, Function.update_self]
+      intro hsucc
+      have hone : (m : ℤ) ∣ 1 := by
+        simpa using dvd_sub hsucc (hdiv j)
+      have hmle : (m : ℤ) ≤ 1 := Int.le_of_dvd (by norm_num : (0 : ℤ) < 1) hone
+      have hmtwo : (2 : ℤ) ≤ (m : ℤ) := by exact_mod_cast hm
+      omega
+    · intro i
+      by_cases hij : i = j
+      · subst i
+        simp only [z, Function.update_self]
+        exact floor_successor_error_lt_one hj
+      · simpa only [z, Function.update_of_ne hij] using herr i
+  · exact ⟨a, habs, not_forall.mp hdiv, herr⟩
+
+theorem closed_box_rounding (m : ℕ) (hm : 2 ≤ m) (δ : Space 3)
+    (hδ : ∀ i, |δ i| ≤ (m : ℝ)) (hnot : δ ∉ scaledGrid m) :
+    ∃ z : Lattice 3,
+      (∀ i, |z i| ≤ (m : ℤ)) ∧
+      (∃ i, ¬ (m : ℤ) ∣ z i) ∧
+      (∀ i, |δ i - (z i : ℝ)| < 1) := by
+  apply closed_box_rounding_exists m hm δ hδ
+  rintro ⟨z, hz⟩
+  exact hnot (mem_scaledGrid.mpr ⟨z, hz.symm⟩)
+
+theorem closed_box_rounding_norm (m : ℕ) (hm : 2 ≤ m) (δ : Space 3)
+    (hδ : ∀ i, |δ i| ≤ (m : ℝ)) (hnot : δ ∉ scaledGrid m) :
+    ∃ z : Lattice 3,
+      (∀ i, |z i| ≤ (m : ℤ)) ∧
+      (∃ i, ¬ (m : ℤ) ∣ z i) ∧
+      ‖δ - castLattice z‖ < 1 := by
+  obtain ⟨z, hz, hdiv, herr⟩ := closed_box_rounding m hm δ hδ hnot
+  refine ⟨z, hz, hdiv, (pi_norm_lt_iff (by norm_num : (0 : ℝ) < 1)).mpr ?_⟩
+  intro i
+  simpa only [Pi.sub_apply, castLattice_apply, Real.norm_eq_abs] using herr i
+
+end PeriodicTilingThree
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/ClosedCoverChains.lean b/lean/OAI/Geometry/PeriodicTiling/ClosedCoverChains.lean
new file mode 100644
index 0000000000000000000000000000000000000000..78ba5dfd1889cee836718c2fe410659e86f424b4
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/ClosedCoverChains.lean
@@ -0,0 +1,79 @@
+import Mathlib.Topology.Connected.Basic
+import Mathlib.Topology.LocallyFinite
+import Mathlib.MeasureTheory.Measure.OpenPos
+import Mathlib.Logic.Relation
+
+namespace OAI
+
+universe uX uI
+
+namespace PeriodicTilingThree
+
+open Set MeasureTheory
+
+section Topological
+
+variable {X : Type uX} {I : Type uI} [TopologicalSpace X] [PreconnectedSpace X]
+
+def CoverIntersects (K : I → Set X) (i j : I) : Prop :=
+  (K i ∩ K j).Nonempty
+
+theorem closed_cover_chain_connected (K : I → Set X)
+    (hne : ∀ i, (K i).Nonempty) (hclosed : ∀ i, IsClosed (K i))
+    (hlocal : LocallyFinite K) (hcover : ∀ x : X, ∃ i, x ∈ K i)
+    (i₀ j : I) : Relation.ReflTransGen (CoverIntersects K) i₀ j := by
+  classical
+  let R : I → Prop := Relation.ReflTransGen (CoverIntersects K) i₀
+  let A : Set X := ⋃ i : {i // R i}, K i.val
+  let B : Set X := ⋃ i : {i // ¬ R i}, K i.val
+  have hA : IsClosed A :=
+    (hlocal.comp_injective (g := fun i : {i // R i} => i.val)
+      Subtype.val_injective).isClosed_iUnion (fun i => hclosed i.val)
+  have hB : IsClosed B :=
+    (hlocal.comp_injective (g := fun i : {i // ¬ R i} => i.val)
+      Subtype.val_injective).isClosed_iUnion (fun i => hclosed i.val)
+  have hab : (univ : Set X) ⊆ A ∪ B := by
+    intro x _
+    obtain ⟨i, hxi⟩ := hcover x
+    by_cases hi : R i
+    · exact Or.inl (mem_iUnion.mpr ⟨⟨i, hi⟩, hxi⟩)
+    · exact Or.inr (mem_iUnion.mpr ⟨⟨i, hi⟩, hxi⟩)
+  have hAnonempty : ((univ : Set X) ∩ A).Nonempty := by
+    obtain ⟨x, hx⟩ := hne i₀
+    exact ⟨x, mem_univ x,
+      mem_iUnion.mpr ⟨⟨i₀, Relation.ReflTransGen.refl⟩, hx⟩⟩
+  by_contra hj
+  have hBnonempty : ((univ : Set X) ∩ B).Nonempty := by
+    obtain ⟨x, hx⟩ := hne j
+    exact ⟨x, mem_univ x, mem_iUnion.mpr ⟨⟨j, hj⟩, hx⟩⟩
+  obtain ⟨x, _, hxA, hxB⟩ :=
+    isPreconnected_closed_iff.mp isPreconnected_univ A B hA hB hab hAnonempty hBnonempty
+  obtain ⟨i, hxi⟩ := mem_iUnion.mp hxA
+  obtain ⟨k, hxk⟩ := mem_iUnion.mp hxB
+  exact k.property (i.property.tail ⟨x, hxi, hxk⟩)
+
+end Topological
+
+section Measure
+
+variable {X : Type uX} {I : Type uI} [TopologicalSpace X] [MeasurableSpace X]
+variable (μ : Measure X) [MeasureTheory.Measure.IsOpenPosMeasure μ]
+
+theorem closed_locallyFinite_cover_of_ae (K : I → Set X)
+    (hclosed : ∀ i, IsClosed (K i)) (hlocal : LocallyFinite K)
+    (hcover : ∀ᵐ x ∂μ, ∃ i, x ∈ K i) : ∀ x, ∃ i, x ∈ K i := by
+  have hc : IsClosed (⋃ i, K i) := hlocal.isClosed_iUnion hclosed
+  have hae : ∀ᵐ x ∂μ, x ∈ ⋃ i, K i :=
+    hcover.mono fun _ hx => mem_iUnion.mpr hx
+  have hnull : μ (⋃ i, K i)ᶜ = 0 := mem_ae_iff.mp hae
+  have hempty : (⋃ i, K i)ᶜ = ∅ :=
+    (hc.isOpen_compl.measure_eq_zero_iff μ).mp hnull
+  have huniv : (⋃ i, K i) = univ := Set.compl_empty_iff.mp hempty
+  intro x
+  exact mem_iUnion.mp (by rw [huniv]; exact mem_univ x)
+
+end Measure
+
+end PeriodicTilingThree
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/CommonModel.lean b/lean/OAI/Geometry/PeriodicTiling/CommonModel.lean
new file mode 100644
index 0000000000000000000000000000000000000000..09d34d2c6ada3f4ee5a7d639035e2a203ba36c6b
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/CommonModel.lean
@@ -0,0 +1,100 @@
+import OAI.Geometry.PeriodicTiling.OrdinaryActivation
+import OAI.Geometry.PeriodicTiling.CommonSeedOutputs
+import OAI.Geometry.PeriodicTiling.Dependence
+
+namespace OAI
+
+noncomputable section
+
+namespace PeriodicTilingThree.CommonModel
+
+variable {p : ℕ} [NeZero p] (E : EncodingParameters p)
+
+def outputs : GraphOutputs E where
+  c b i := match i with
+    | .inl n => ordinaryUseful E n b.1 (b.2 (.inl n))
+    | .inr t => seedC E t b.1 b.2
+  beta b i := match i with
+    | .inl n => ordinaryHigh E n b.1 (b.2 (.inl n))
+    | .inr t => seedBeta E t b.1 b.2
+  z b := SharedSeed.commonZ E.p_prime E.p_large b.1 b.2
+
+def commonGraph : Set (Ambient E) := graph (outputs E)
+
+@[simp] theorem outputs_c_ordinary (n : Column p) (x : Plane) (k : Input E) :
+    (outputs E).c (x,k) (.inl n) = ordinaryUseful E n x (k (.inl n)) := rfl
+
+@[simp] theorem outputs_c_seed (t : Fin 2) (x : Plane) (k : Input E) :
+    (outputs E).c (x,k) (.inr t) = seedC E t x k := rfl
+
+@[simp] theorem outputs_beta_ordinary (n : Column p) (x : Plane) (k : Input E) :
+    (outputs E).beta (x,k) (.inl n) = ordinaryHigh E n x (k (.inl n)) := rfl
+
+@[simp] theorem outputs_beta_seed (t : Fin 2) (x : Plane) (k : Input E) :
+    (outputs E).beta (x,k) (.inr t) = seedBeta E t x k := rfl
+
+@[simp] theorem outputs_z (x : Plane) (k : Input E) :
+    (outputs E).z (x,k) = SharedSeed.commonZ E.p_prime E.p_large x k := rfl
+
+@[simp] theorem usefulAt_ordinary (n : Column p) (x : Plane) (w : K E (.inl n)) :
+    usefulAt E (outputs E) (.inl n) x w = ordinaryUseful E n x w := by
+  simp [usefulAt]
+
+@[simp] theorem usefulAt_seed (t : Fin 2) (x : Plane) (w : K E (.inr t)) :
+    usefulAt E (outputs E) (.inr t) x w =
+      seedUseful E t (SharedSeed.residue p x) w := by
+  simp [usefulAt, seedC]
+
+theorem outputs_hasDependence : HasDependence E (outputs E) := by
+  constructor
+  · intro n x k
+    simp only [outputs_c_ordinary, ordinaryFactor, usefulAt_ordinary]
+    apply ordinaryUseful_eq_of_lineValue_eq E n
+    simp
+  · intro t x k
+    simp only [outputs_c_seed, seedFactor, usefulAt_seed, seedC]
+    have hr : SharedSeed.residue p (seedRepresentative p (seedResidue p x)) =
+        SharedSeed.residue p x :=
+      seedResidue_representative (p := p) (seedResidue p x)
+    rw [hr]
+
+theorem active_ordinary_iff (n : Column p) (x : Plane) (j : Symbol p) :
+    Active E (outputs E) (.inl n) x j ↔ j = ordinarySymbol E n x := by
+  simpa only [Active, DigitActive, usefulAt_ordinary] using
+    ordinaryUseful_active_iff E n x j
+
+theorem active_seed_iff (t : Fin 2) (x : Plane) :
+    Active E (outputs E) (.inr t) x () ↔
+      SharedSeed.residue p x ∈ seedActiveRegion p t := by
+  simpa only [Active, DigitActive, usefulAt_seed] using
+    seedUseful_active_iff E t (SharedSeed.residue p x)
+
+theorem active_word_allowed (x : Plane) (w : Word p)
+    (hw : ∀ n, Active E (outputs E) (.inl n) x (w n)) : Allowed p w := by
+  have he : w = ordinaryWord E x :=
+    funext fun n => (active_ordinary_iff E n x (w n)).mp (hw n)
+  rw [he]
+  exact ordinaryWord_allowed E x
+
+theorem active_seed_forces_symbol (t : Fin 2) (n : Column p) (x : Plane)
+    (j : Symbol p) (ht : Active E (outputs E) (.inr t) x ())
+    (hj : Active E (outputs E) (.inl n) x j) :
+    j = lastDigit p E.p_prime ((t.val : ℤ) + 1) := by
+  have hj' := (active_ordinary_iff E n x j).mp hj
+  have hr := (active_seed_iff E t x).mp ht
+  fin_cases t
+  · have hfirst : SharedSeed.residue p x ∈ SharedSeed.firstRegion p := by
+      simpa [seedActiveRegion] using hr
+    simpa using hj'.trans (ordinarySymbol_firstRegion E n x hfirst)
+  · have hsecond : SharedSeed.residue p x ∈ SharedSeed.secondRegion p := by
+      simpa [seedActiveRegion] using hr
+    simpa using hj'.trans (ordinarySymbol_secondRegion E n x hsecond)
+
+theorem commonGraph_tiles_kernel : Tiles (kernelTile E) (commonGraph E) :=
+  kernelTile_graph E (outputs E)
+
+end PeriodicTilingThree.CommonModel
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/CommonSeedOutputs.lean b/lean/OAI/Geometry/PeriodicTiling/CommonSeedOutputs.lean
new file mode 100644
index 0000000000000000000000000000000000000000..264f23bd784a3b4bb96ee5b0d62dfd0fbb0585ac
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/CommonSeedOutputs.lean
@@ -0,0 +1,209 @@
+import OAI.Geometry.PeriodicTiling.SharedSeedSizeTransport
+import OAI.Geometry.PeriodicTiling.ChannelDigitBlocks
+import OAI.Geometry.PeriodicTiling.ActivationOffsets
+
+namespace OAI
+
+noncomputable section
+
+namespace PeriodicTilingThree
+
+open SharedSeed
+
+variable {p : ℕ} [NeZero p] (E : EncodingParameters p)
+
+theorem seedOneSize
+    {p : ℕ} [NeZero p] (E : EncodingParameters p)
+    : E.a (.inr (0 : Fin 2)) = 2 := by
+  simpa using E.seed_a (0 : Fin 2)
+
+theorem seedTwoSize
+    {p : ℕ} [NeZero p] (E : EncodingParameters p)
+    : E.a (.inr (1 : Fin 2)) = 3 := by
+  simpa using E.seed_a (1 : Fin 2)
+
+def seedUseful (t : Fin 2) : Residues p → K E (.inr t) → P E (.inr t) :=
+  Fin.cases
+    (fun s u => usefulOneOfEq (seedOneSize E) (channelDigitBlocks E (.inr 0) ()) u
+      (residueEquiv E.p_prime E.p_large s))
+    (fun j => by
+      have hj : j = 0 := Subsingleton.elim _ _
+      subst j
+      exact fun s u => usefulTwoOfEq (seedTwoSize E) (channelDigitBlocks E (.inr 1) ()) u
+        (residueEquiv E.p_prime E.p_large s)) t
+
+def seedHigh (t : Fin 2) : ℤ → Residues p → K E (.inr t) → K E (.inr t) :=
+  Fin.cases
+    (fun x s u => highOneOfEq (seedOneSize E) (channelDigitBlocks E (.inr 0) ()) x u
+      (residueEquiv E.p_prime E.p_large s))
+    (fun j => by
+      have hj : j = 0 := Subsingleton.elim _ _
+      subst j
+      exact fun x s u => highTwoOfEq (seedTwoSize E) (channelDigitBlocks E (.inr 1) ()) x u
+        (residueEquiv E.p_prime E.p_large s)) t
+
+@[simp] theorem seedUseful_zero (s : Residues p) (u : K E (.inr (0 : Fin 2))) :
+    seedUseful E 0 s u =
+      usefulOneOfEq (seedOneSize E) (channelDigitBlocks E (.inr 0) ()) u
+        (residueEquiv E.p_prime E.p_large s) := rfl
+
+@[simp] theorem seedUseful_one (s : Residues p) (u : K E (.inr (1 : Fin 2))) :
+    seedUseful E 1 s u =
+      usefulTwoOfEq (seedTwoSize E) (channelDigitBlocks E (.inr 1) ()) u
+        (residueEquiv E.p_prime E.p_large s) := rfl
+
+@[simp] theorem seedHigh_zero (x : ℤ) (s : Residues p) (u : K E (.inr (0 : Fin 2))) :
+    seedHigh E 0 x s u =
+      highOneOfEq (seedOneSize E) (channelDigitBlocks E (.inr 0) ()) x u
+        (residueEquiv E.p_prime E.p_large s) := rfl
+
+@[simp] theorem seedHigh_one (x : ℤ) (s : Residues p) (u : K E (.inr (1 : Fin 2))) :
+    seedHigh E 1 x s u =
+      highTwoOfEq (seedTwoSize E) (channelDigitBlocks E (.inr 1) ()) x u
+        (residueEquiv E.p_prime E.p_large s) := rfl
+
+def seedC (t : Fin 2) (x : Plane) (k : Input E) : P E (.inr t) :=
+  seedUseful E t (residue p x) (k (.inr t))
+
+def seedBeta (t : Fin 2) (x : Plane) (k : Input E) : K E (.inr t) :=
+  seedHigh E t x.1 (residue p x) (k (.inr t))
+
+theorem seedC_eq_of_residue_low (t : Fin 2) {x y : Plane} {k l : Input E}
+    (hx : residue p x = residue p y) (hk : k (.inr t) = l (.inr t)) :
+    seedC E t x k = seedC E t y l := by
+  simp only [seedC, hx, hk]
+
+def seedSourceInput (t : Fin 2) (k : Input E) (j : SeedActivationIndex E t) : Input E :=
+  Function.update k (.inr t) (k (.inr t) - j.1.1)
+
+@[simp] theorem seedSourceInput_same (t : Fin 2) (k : Input E)
+    (j : SeedActivationIndex E t) :
+    seedSourceInput E t k j (.inr t) = k (.inr t) - j.1.1 := by
+  simp [seedSourceInput]
+
+theorem seedSourceInput_low (t : Fin 2) (k : Input E) (j : SeedActivationIndex E t) :
+    seedSourceInput E t k j (.inr t) =
+      k (.inr t) - ((seedOffset E t j).1 : K E (.inr t)) := by
+  rw [seedSourceInput_same, seedOffset_low]
+
+theorem seed_source_residue (t : Fin 2) (x : Plane) (j : SeedActivationIndex E t) :
+    residue p (x - seedOffset E t j) = residue p x - j.2 := by
+  have hs : residue p (seedOffset E t j) = j.2 := seedOffset_residue E t j
+  have hfirst : ((seedOffset E t j).1 : ZMod (p ^ 2)) = j.2.1 := congrArg Prod.fst hs
+  have hsecond : ((seedOffset E t j).2 : ZMod (p ^ 2)) = j.2.2 := congrArg Prod.snd hs
+  apply Prod.ext
+  · change ((x.1 - (seedOffset E t j).1 : ℤ) : ZMod (p ^ 2)) =
+      (x.1 : ZMod (p ^ 2)) - j.2.1
+    rw [Int.cast_sub, hfirst]
+  · change ((x.2 - (seedOffset E t j).2 : ℤ) : ZMod (p ^ 2)) =
+      (x.2 : ZMod (p ^ 2)) - j.2.2
+    rw [Int.cast_sub, hsecond]
+
+def seedActualForward (t : Fin 2) (x : Plane) (k : Input E)
+    (j : SeedActivationIndex E t) : (P E (.inr t) × K E (.inr t)) × ZMod (D p) :=
+  let x' := x - seedOffset E t j
+  let k' := seedSourceInput E t k j
+  ((seedC E t x' k' + shift (E.a (.inr t)) (E.b (.inr t)) j.1.2,
+    seedBeta E t x' k'), commonZ E.p_prime E.p_large x' k')
+
+theorem seedActualForward_zero_eq (x : Plane) (k : Input E) :
+    seedActualForward E 0 x k =
+      oneForwardOfEq (seedOneSize E) (channelDigitBlocks E (.inr 0) ())
+        (BlockShiftData.ofMultiplicity (E.a_two_le (.inr 0)) (E.b_two_le (.inr 0)))
+        (residueEquiv E.p_prime E.p_large) (outputEquiv E.p_large)
+        (fun j => (seedOffset E 0 j).1) x.1 (residue p x) (k (.inr 0)) := by
+  funext j
+  simp only [seedActualForward, seedC, seedBeta, seedSourceInput_same,
+    seed_source_residue, seedUseful_zero, seedHigh_zero, commonZ,
+    oneForwardOfEq, BlockShiftData.ofMultiplicity, Prod.fst_sub]
+
+theorem seedActualForward_one_eq (x : Plane) (k : Input E) :
+    seedActualForward E 1 x k =
+      twoForwardOfEq (seedTwoSize E) (channelDigitBlocks E (.inr 1) ())
+        (BlockShiftData.ofMultiplicity (E.a_two_le (.inr 1)) (E.b_two_le (.inr 1)))
+        (residueEquiv E.p_prime E.p_large) (outputEquiv E.p_large)
+        (fun j => (seedOffset E 1 j).1) x.1 (residue p x) (k (.inr 1)) := by
+  funext j
+  simp only [seedActualForward, seedC, seedBeta, seedSourceInput_same,
+    seed_source_residue, seedUseful_one, seedHigh_one, commonZ,
+    twoForwardOfEq, BlockShiftData.ofMultiplicity, Prod.fst_sub]
+
+theorem seedActualForward_bijective (t : Fin 2) (x : Plane) (k : Input E) :
+    Function.Bijective (seedActualForward E t x k) := by
+  fin_cases t
+  · change Function.Bijective (seedActualForward E (0 : Fin 2) x k)
+    rw [seedActualForward_zero_eq]
+    apply oneForwardOfEq_bijective
+    · intro j
+      exact seedOffset_rhoA E 0 j
+    · intro j
+      exact seedOffset_rhoB E 0 j
+    · intro j
+      have h := seedOffset_other E (0 : Fin 2) j
+      have ha : E.a (.inr (otherSeed (0 : Fin 2))) = 3 := by
+        change E.a (.inr (1 : Fin 2)) = 3
+        exact seedTwoSize E
+      exact Eq.mp (congrArg (fun modulus : ℕ =>
+        ((seedOffset E (0 : Fin 2) j).1 : ZMod modulus) = 0) ha) h
+  · change Function.Bijective (seedActualForward E (1 : Fin 2) x k)
+    rw [seedActualForward_one_eq]
+    apply twoForwardOfEq_bijective
+    · intro j
+      exact seedOffset_rhoA E 1 j
+    · intro j
+      exact seedOffset_rhoB E 1 j
+    · intro j
+      have h := seedOffset_other E (1 : Fin 2) j
+      have ha : E.a (.inr (otherSeed (1 : Fin 2))) = 2 := by
+        change E.a (.inr (0 : Fin 2)) = 2
+        exact seedOneSize E
+      exact Eq.mp (congrArg (fun modulus : ℕ =>
+        ((seedOffset E (1 : Fin 2) j).1 : ZMod modulus) = 0) ha) h
+
+theorem both_seed_tests (x : Plane) (k : Input E) :
+    Function.Bijective (seedActualForward E 0 x k) ∧
+      Function.Bijective (seedActualForward E 1 x k) :=
+  ⟨seedActualForward_bijective E 0 x k, seedActualForward_bijective E 1 x k⟩
+
+def seedActiveRegion (p : ℕ) (t : Fin 2) : Set (Residues p) :=
+  if t = 0 then firstRegion p else secondRegion p
+
+theorem seedUseful_active_iff (t : Fin 2) (s : Residues p) :
+    (∃ u v : K E (.inr t), u - v ∈ digitSubgroup E (.inr t) () ∧
+      seedUseful E t s u ≠ seedUseful E t s v) ↔ s ∈ seedActiveRegion p t := by
+  classical
+  fin_cases t
+  · change (∃ u v : K E (.inr (0 : Fin 2)),
+        u - v ∈ digitSubgroup E (.inr (0 : Fin 2)) () ∧
+        seedUseful E 0 s u ≠ seedUseful E 0 s v) ↔
+          s ∈ seedActiveRegion p (0 : Fin 2)
+    have hC := (channelDigitBlocks_active_iff E (.inr (0 : Fin 2)) () ()).mpr rfl
+    have hlabel : ActiveOne (residueEquiv E.p_prime E.p_large s) ↔ s ∈ firstRegion p := by
+      rw [← residueEquiv_first_iff E.p_prime E.p_large s]
+      cases hs : residueEquiv E.p_prime E.p_large s <;> simp [ActiveOne]
+    have h := usefulOneOfEq_activity_iff (seedOneSize E) (channelDigitBlocks E (.inr 0) ())
+      (fun u v => u - v ∈ digitSubgroup E (.inr 0) ()) hC
+      (residueEquiv E.p_prime E.p_large s)
+    rw [seedActiveRegion, ite_eq_left (rfl : (0 : Fin 2) = 0)]
+    simpa only [seedUseful_zero] using h.trans hlabel
+  · change (∃ u v : K E (.inr (1 : Fin 2)),
+        u - v ∈ digitSubgroup E (.inr (1 : Fin 2)) () ∧
+        seedUseful E 1 s u ≠ seedUseful E 1 s v) ↔
+          s ∈ seedActiveRegion p (1 : Fin 2)
+    have hC := (channelDigitBlocks_active_iff E (.inr (1 : Fin 2)) () ()).mpr rfl
+    have hlabel : ActiveTwo (residueEquiv E.p_prime E.p_large s) ↔ s ∈ secondRegion p := by
+      rw [← residueEquiv_second_iff E.p_prime E.p_large s]
+      cases hs : residueEquiv E.p_prime E.p_large s with
+      | inl y => simp [ActiveTwo]
+      | inr u => cases u <;> simp [ActiveTwo]
+    have h := usefulTwoOfEq_activity_iff (seedTwoSize E) (channelDigitBlocks E (.inr 1) ())
+      (fun u v => u - v ∈ digitSubgroup E (.inr 1) ()) hC
+      (residueEquiv E.p_prime E.p_large s)
+    simpa only [seedUseful_one, seedActiveRegion, ite_eq_right (by decide : (1 : Fin 2) ≠ 0)]
+      using h.trans hlabel
+
+end PeriodicTilingThree
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/CommonSolution.lean b/lean/OAI/Geometry/PeriodicTiling/CommonSolution.lean
new file mode 100644
index 0000000000000000000000000000000000000000..3540b0d7dbf421b913e8386a22cfb3540df4647e
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/CommonSolution.lean
@@ -0,0 +1,58 @@
+import OAI.Geometry.PeriodicTiling.CommonModel
+import OAI.Geometry.PeriodicTiling.EncodedSystem
+
+namespace OAI
+
+noncomputable section
+
+namespace PeriodicTilingThree.CommonModel
+
+variable {p : ℕ} [NeZero p] (E : EncodingParameters p)
+
+theorem ordinaryActivationMap_outputs (n : Column p) (b : Base E) :
+    ordinaryActivationMap E n (outputs E) b =
+      ordinaryForward E n b.1 (b.2 (.inl n)) := by
+  funext j
+  simp [ordinaryActivationMap, ordinarySource, activationSource, outputs, ordinaryForward]
+
+theorem seedActivationMap_outputs (t : Fin 2) (b : Base E) :
+    seedActivationMap E t (outputs E) b = seedActualForward E t b.1 b.2 := by
+  rfl
+
+theorem common_solution : Solves E (commonGraph E) := by
+  apply (solves_iff E (commonGraph E)).mpr
+  refine ⟨commonGraph_tiles_kernel E, ?_, ?_, ?_, ?_, ?_⟩
+  · exact (dependenceTiles_iff E (outputs E)).mpr (outputs_hasDependence E)
+  · exact (wordConstraintTiles_iff E (outputs E) (outputs_hasDependence E)).mpr
+      (active_word_allowed E)
+  · apply (seedConstraintTiles_iff E (outputs E) (outputs_hasDependence E)).mpr
+    intro t n x j ht hj
+    exact active_seed_forces_symbol E t n x j ht hj
+  · intro n
+    apply (ordinaryActivationTile_iff E n (outputs E)).mpr
+    intro b
+    rw [ordinaryActivationMap_outputs]
+    exact ordinaryForward_bijective E n b.1 (b.2 (.inl n))
+  · intro t
+    apply (seedActivationTile_iff E t (outputs E)).mpr
+    intro b
+    rw [seedActivationMap_outputs]
+    exact seedActualForward_bijective E t b.1 b.2
+
+theorem commonGraph_tiles_testTile (i : TestIndex E) :
+    Tiles (testTile E i) (commonGraph E) :=
+  (solves_iff_index E (commonGraph E)).mp (common_solution E) i
+
+end PeriodicTilingThree.CommonModel
+
+namespace PeriodicTilingThree
+
+theorem exists_common_solution {p : ℕ} [NeZero p] (E : EncodingParameters p) :
+    ∃ A : Set (Ambient E), Solves E A :=
+  ⟨CommonModel.commonGraph E, CommonModel.common_solution E⟩
+
+end PeriodicTilingThree
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/ComponentTargets.lean b/lean/OAI/Geometry/PeriodicTiling/ComponentTargets.lean
new file mode 100644
index 0000000000000000000000000000000000000000..dcab5d9e39179a8973bb7f62fb4980a342a50497
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/ComponentTargets.lean
@@ -0,0 +1,247 @@
+import OAI.Geometry.PeriodicTiling.FiniteMasks
+import OAI.Geometry.PeriodicTiling.TilesRegion
+import OAI.Geometry.PeriodicTiling.PlaneLattices
+import OAI.Geometry.PeriodicTiling.ConfigurationDifferenceReduction
+import OAI.Geometry.PeriodicTiling.BoundedDifferences
+import Mathlib.Tactic.Choose
+import Lean.Elab.Tactic.Omega
+
+namespace OAI
+
+universe uG uι
+
+noncomputable section
+
+open scoped BigOperators Classical
+
+namespace PeriodicTilingThree
+
+def componentTarget {G : Type uG} [AddCommGroup G]
+    (F : Finset G) (D : Set G) : Set G :=
+  {x | tileCount F D x = 1}
+
+private theorem unique_of_nat_sum_one {ι : Type uι} [Fintype ι]
+    (c : ι → ℕ) (hc : (∑ i, c i) = 1) : ∃! i, c i = 1 := by
+  classical
+  have hle (i : ι) : c i ≤ 1 := by
+    calc
+      c i ≤ ∑ j, c j := Finset.single_le_sum (fun j _ => Nat.zero_le (c j))
+        (Finset.mem_univ i)
+      _ = 1 := hc
+  have hcard : (Finset.univ.filter fun i => c i = 1).card = 1 := by
+    calc
+      (Finset.univ.filter fun i => c i = 1).card =
+          ∑ i, if c i = 1 then (1 : ℕ) else 0 := by simp
+      _ = ∑ i, c i := by
+        apply Finset.sum_congr rfl
+        intro i _
+        have hi := hle i
+        split_ifs <;> omega
+      _ = 1 := hc
+  obtain ⟨i, hi⟩ := Finset.card_eq_one.mp hcard
+  have hmem : i ∈ Finset.univ.filter (fun i => c i = 1) := by
+    rw [hi]
+    exact Finset.mem_singleton_self i
+  refine ⟨i, (Finset.mem_filter.mp hmem).2, ?_⟩
+  intro j hj
+  have hjmem : j ∈ Finset.univ.filter (fun i => c i = 1) := by simp [hj]
+  simpa only [hi, Finset.mem_singleton] using hjmem
+
+theorem component_tileCount_sum {G : Type uG} [AddCommGroup G]
+    {ι : Type uι} [Fintype ι] {F : Finset G} {A : Set G}
+    (parts : ι → Set G) (hTile : Tiles F A)
+    (hpartition : ∀ x, (∑ j, tileIndicator ℝ (parts j) x) = tileIndicator ℝ A x)
+    (x : G) : (∑ j, tileCount F (parts j) x) = 1 := by
+  have hreal : ((∑ j, tileCount F (parts j) x : ℕ) : ℝ) = 1 := by
+    rw [Nat.cast_sum]
+    simp_rw [tileCount_cast]
+    rw [Finset.sum_comm]
+    simp_rw [hpartition]
+    exact hTile.sum_tileIndicator_eq_one ℝ x
+  exact_mod_cast hreal
+
+theorem component_tileCount_le_one {G : Type uG} [AddCommGroup G]
+    {ι : Type uι} [Fintype ι] {F : Finset G} {A : Set G}
+    (parts : ι → Set G) (hTile : Tiles F A)
+    (hpartition : ∀ x, (∑ j, tileIndicator ℝ (parts j) x) = tileIndicator ℝ A x)
+    (j : ι) (x : G) : tileCount F (parts j) x ≤ 1 := by
+  calc
+    tileCount F (parts j) x ≤ ∑ i, tileCount F (parts i) x :=
+      Finset.single_le_sum (f := fun i => tileCount F (parts i) x)
+        (fun i _ => Nat.zero_le _) (Finset.mem_univ j)
+    _ = 1 := component_tileCount_sum parts hTile hpartition x
+
+theorem tilesRegion_componentTarget {G : Type uG} [AddCommGroup G]
+    (F : Finset G) (D : Set G) (hcount : ∀ x, tileCount F D x ≤ 1) :
+    TilesRegion F D (componentTarget F D) := by
+  classical
+  constructor
+  · intro f hf d hd
+    change tileCount F D (f + d) = 1
+    apply Nat.le_antisymm (hcount (f + d))
+    have hterm : tileIndicator ℕ D (f + d - f) = 1 := by
+      simp [tileIndicator, hd]
+    calc
+      1 = tileIndicator ℕ D (f + d - f) := hterm.symm
+      _ ≤ tileCount F D (f + d) :=
+        Finset.single_le_sum (f := fun g => tileIndicator ℕ D (f + d - g))
+          (fun g _ => Nat.zero_le _) hf
+  · intro x hx
+    change tileCount F D x = 1 at hx
+    have hcard : (F.filter fun f => x - f ∈ D).card = 1 := by
+      rw [← tileCount_eq_card_filter]
+      exact hx
+    obtain ⟨f, hf⟩ := Finset.card_eq_one.mp hcard
+    have hm : f ∈ F ∧ x - f ∈ D := by
+      apply Finset.mem_filter.mp
+      rw [hf]
+      exact Finset.mem_singleton_self f
+    refine ⟨⟨f, hm.1⟩, hm.2, ?_⟩
+    intro g hg
+    apply Subtype.ext
+    have hg' : g.val ∈ F.filter (fun f => x - f ∈ D) :=
+      Finset.mem_filter.mpr ⟨g.property, hg⟩
+    simpa only [hf, Finset.mem_singleton] using hg'
+
+theorem component_targets_partition {G : Type uG} [AddCommGroup G]
+    {ι : Type uι} [Fintype ι] {F : Finset G} {A : Set G}
+    (parts : ι → Set G) (hTile : Tiles F A)
+    (hpartition : ∀ x, (∑ j, tileIndicator ℝ (parts j) x) = tileIndicator ℝ A x)
+    (x : G) : ∃! j, x ∈ componentTarget F (parts j) :=
+  unique_of_nat_sum_one (fun j => tileCount F (parts j) x)
+    (component_tileCount_sum parts hTile hpartition x)
+
+private theorem real_tileCount_periodic {G : Type uG} [AddCommGroup G]
+    (F : Finset G) (D : Set G) (v : G) (hD : Period D v) :
+    Function.Periodic (fun x => (tileCount F D x : ℝ)) v := by
+  intro x
+  change (tileCount F D (x + v) : ℝ) = (tileCount F D x : ℝ)
+  rw [tileCount_cast, tileCount_cast]
+  apply Finset.sum_congr rfl
+  intro f _
+  have he : x + v - f = (x - f) + v := by abel
+  simp only [tileIndicator, he, hD (x - f)]
+
+private theorem componentTarget_period_of_count {G : Type uG} [AddCommGroup G]
+    (F : Finset G) (D : Set G) (v : G)
+    (h : Function.Periodic (fun x => (tileCount F D x : ℝ)) v) :
+    Period (componentTarget F D) v := by
+  intro x
+  have he : tileCount F D (x + v) = tileCount F D x := by
+    have hreal : (tileCount F D (x + v) : ℝ) = (tileCount F D x : ℝ) := h x
+    exact_mod_cast hreal
+  change tileCount F D (x + v) = 1 ↔ tileCount F D x = 1
+  rw [he]
+
+private theorem bounded_component_periodic {ι : Type uι} [Fintype ι]
+    (g : ι → Plane → ℝ) (w : ι → Plane) (j : ι) (e : Plane)
+    (hperiod : ∀ i, Function.Periodic (g i) (w i))
+    (hsum : (∑ i, g i) = fun _ => (1 : ℝ))
+    (hbound : ∃ B : ℝ, ∀ x, |g j x| ≤ B)
+    (hother : ∃ i, i ≠ j)
+    (hspan : ∀ i, i ≠ j → ∃ a b : ℤ, e = a • w i + b • w j) :
+    Function.Periodic (g j) e := by
+  classical
+  have hchoice : ∀ i, ∃ a b : ℤ, i ≠ j → e = a • w i + b • w j := by
+    intro i
+    by_cases hij : i = j
+    · exact ⟨0, 0, fun h => (h hij).elim⟩
+    · obtain ⟨a, b, h⟩ := hspan i hij
+      exact ⟨a, b, fun _ => h⟩
+  choose a b hab using hchoice
+  let vs : List Plane := (Finset.univ.erase j).toList.map (fun i => a i • w i)
+  have hmem (i : ι) (hij : i ≠ j) : a i • w i ∈ vs := by
+    apply List.mem_map.mpr
+    refine ⟨i, ?_, rfl⟩
+    simpa only [Finset.mem_toList, Finset.mem_erase, Finset.mem_univ, and_true] using hij
+  have hvs : vs ≠ [] := by
+    obtain ⟨i, hij⟩ := hother
+    intro he
+    have h := hmem i hij
+    rw [he] at h
+    exact List.not_mem_nil h
+  have hkill : ∀ i, i ≠ j → ∃ v ∈ vs, Function.Periodic (g i) v := by
+    intro i hij
+    exact ⟨a i • w i, hmem i hij, (hperiod i).zsmul (a i)⟩
+  have hmods : ∀ u ∈ vs, ∃ k : ℤ, u = e + k • w j := by
+    intro u hu
+    obtain ⟨i, hi, rfl⟩ := List.mem_map.mp hu
+    have hij : i ≠ j := (Finset.mem_erase.mp (Finset.mem_toList.mp hi)).1
+    refine ⟨-b i, ?_⟩
+    rw [hab i hij, neg_smul]
+    abel
+  have hzero := configDiffs_component_zero j vs hvs g 1 hsum hkill
+  rw [configDiffs_eq_iterate_of_period vs e (w j) (g j) (hperiod j) hmods] at hzero
+  exact periodic_of_bounded_configDiff_iterate e vs.length (g j) hbound
+    (List.length_pos_iff.mpr hvs) hzero
+
+theorem componentTarget_fullyPeriodic {m : ℕ} {F : Finset Plane} {A : Set Plane}
+    (parts : Fin m → Set Plane) (w : Fin m → Plane) (hTile : Tiles F A)
+    (hpartition : ∀ x, (∑ j, tileIndicator ℝ (parts j) x) = tileIndicator ℝ A x)
+    (hw : ∀ j, w j ≠ 0) (hperiod : ∀ j, Period (parts j) (w j))
+    (htrans : ∀ i j, i ≠ j → planeDet (w i) (w j) ≠ 0)
+    (j : Fin m) : FullyPeriodic (componentTarget F (parts j)) := by
+  classical
+  let g : Fin m → Plane → ℝ := fun i x => tileCount F (parts i) x
+  have hgperiod : ∀ i, Function.Periodic (g i) (w i) :=
+    fun i => real_tileCount_periodic F (parts i) (w i) (hperiod i)
+  have hsum : (∑ i, g i) = fun _ => (1 : ℝ) := by
+    ext x
+    simp only [Finset.sum_apply, g, ← Nat.cast_sum]
+    rw [component_tileCount_sum parts hTile hpartition x]
+    simp
+  have hbound : ∃ B : ℝ, ∀ x, |g j x| ≤ B := by
+    refine ⟨1, fun x => ?_⟩
+    have hn := component_tileCount_le_one parts hTile hpartition j x
+    have hr : (tileCount F (parts j) x : ℝ) ≤ 1 := by exact_mod_cast hn
+    change |(tileCount F (parts j) x : ℝ)| ≤ 1
+    rw [abs_of_nonneg (Nat.cast_nonneg _)]
+    exact hr
+  by_cases hother : ∃ i, i ≠ j
+  · obtain ⟨e, _he, hdet, hspan⟩ := exists_common_transverse w hw htrans
+    have hge : Function.Periodic (g j) e := bounded_component_periodic g w j e
+      hgperiod hsum hbound hother (fun i hij => hspan i j hij)
+    exact fullyPeriodic_of_independent_periods
+      (componentTarget_period_of_count F (parts j) e hge)
+      (componentTarget_period_of_count F (parts j) (w j) (hgperiod j)) (hdet j)
+  · have hall : ∀ i : Fin m, i = j := by
+      intro i
+      by_contra hij
+      exact hother ⟨i, hij⟩
+    have hone : ∀ x, tileCount F (parts j) x = 1 := by
+      intro x
+      have he := component_tileCount_sum parts hTile hpartition x
+      have hs : (∑ i, tileCount F (parts i) x) = tileCount F (parts j) x := by
+        apply Finset.sum_eq_single j
+        · intro i _ hij
+          exact (hij (hall i)).elim
+        · intro hj
+          exact (hj (Finset.mem_univ j)).elim
+      rwa [hs] at he
+    have hE : componentTarget F (parts j) = Set.univ := by
+      ext x
+      simp only [componentTarget, Set.mem_ofPred_eq, hone x, Set.mem_univ]
+    rw [hE]
+    refine ⟨⊤, ?_, fun v _ x => ?_⟩
+    · exact ⟨by simp⟩
+    · simp
+
+theorem component_targets {m : ℕ} {F : Finset Plane} {A : Set Plane}
+    (parts : Fin m → Set Plane) (w : Fin m → Plane) (hTile : Tiles F A)
+    (hpartition : ∀ x, (∑ j, tileIndicator ℝ (parts j) x) = tileIndicator ℝ A x)
+    (hw : ∀ j, w j ≠ 0) (hperiod : ∀ j, Period (parts j) (w j))
+    (htrans : ∀ i j, i ≠ j → planeDet (w i) (w j) ≠ 0) :
+    (∀ j, TilesRegion F (parts j) (componentTarget F (parts j))) ∧
+      (∀ x, ∃! j, x ∈ componentTarget F (parts j)) ∧
+      ∀ j, FullyPeriodic (componentTarget F (parts j)) := by
+  refine ⟨?_, component_targets_partition parts hTile hpartition, ?_⟩
+  · intro j
+    exact tilesRegion_componentTarget F (parts j)
+      (component_tileCount_le_one parts hTile hpartition j)
+  · exact componentTarget_fullyPeriodic parts w hTile hpartition hw hperiod htrans
+
+end PeriodicTilingThree
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/ConfigurationDifferenceReduction.lean b/lean/OAI/Geometry/PeriodicTiling/ConfigurationDifferenceReduction.lean
new file mode 100644
index 0000000000000000000000000000000000000000..4e7b1a5f72bbc792f16e24c1bf888855c1777d73
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/ConfigurationDifferenceReduction.lean
@@ -0,0 +1,60 @@
+import OAI.Geometry.PeriodicTiling.ConfigurationDifferences
+
+namespace OAI
+
+universe uG uA uB uι
+
+namespace PeriodicTilingThree
+
+variable {G : Type uG} {A : Type uA} {B : Type uB}
+  [AddCommGroup G] [AddCommGroup A] [AddCommGroup B]
+
+theorem configDiffs_eq_iterate_of_period (vs : List G) (v p : G) (f : G → A)
+    (hp : Function.Periodic f p)
+    (hvs : ∀ u ∈ vs, ∃ k : ℤ, u = v + k • p) :
+    configDiffs vs f = (configDiff v)^[vs.length] f := by
+  induction vs with
+  | nil => rfl
+  | cons u vs ih =>
+      obtain ⟨k, hk⟩ := hvs u (List.mem_cons_self ..)
+      have htail : ∀ w ∈ vs, ∃ k : ℤ, w = v + k • p := by
+        intro w hw
+        exact hvs w (List.mem_cons_of_mem _ hw)
+      have hper := (configDiffs_preserves_period vs p f hp).zsmul k
+      rw [configDiffs_cons, hk, configDiff_add_period v (k • p) _ hper,
+        ih htail, List.length_cons, Function.iterate_succ_apply']
+
+theorem configDiffs_component_zero {ι : Type uι} [Fintype ι]
+    (j : ι) (vs : List G) (hvs : vs ≠ []) (f : ι → G → A) (a : A)
+    (hsum : (∑ i, f i) = fun _ => a)
+    (hkill : ∀ i, i ≠ j → ∃ v ∈ vs, Function.Periodic (f i) v) :
+    configDiffs vs (f j) = 0 := by
+  classical
+  have hsum' := congrArg (configDiffs vs) hsum
+  rw [configDiffs_sum, configDiffs_const hvs] at hsum'
+  have hone : (∑ i, configDiffs vs (f i)) = configDiffs vs (f j) := by
+    apply Finset.sum_eq_single j
+    · intro i _hi hij
+      obtain ⟨v, hv, hp⟩ := hkill i hij
+      exact configDiffs_eq_zero_of_mem hv (f i)
+        ((configDiff_eq_zero_iff v (f i)).mpr hp)
+    · intro hj
+      exact (hj (Finset.mem_univ j)).elim
+  rw [hone] at hsum'
+  exact hsum'
+
+theorem map_configDiff (φ : A →+ B) (v : G) (f : G → A) :
+    (fun x => φ (configDiff v f x)) = configDiff v (fun x => φ (f x)) := by
+  ext x
+  exact map_sub φ _ _
+
+theorem map_configDiffs (φ : A →+ B) (vs : List G) (f : G → A) :
+    (fun x => φ (configDiffs vs f x)) = configDiffs vs (fun x => φ (f x)) := by
+  induction vs with
+  | nil => rfl
+  | cons v vs ih =>
+      rw [configDiffs_cons, map_configDiff, ih, configDiffs_cons]
+
+end PeriodicTilingThree
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/ConfigurationDifferences.lean b/lean/OAI/Geometry/PeriodicTiling/ConfigurationDifferences.lean
new file mode 100644
index 0000000000000000000000000000000000000000..0e5c36c0feed5a9ede45bdadfd1a64b086e2a301
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/ConfigurationDifferences.lean
@@ -0,0 +1,154 @@
+import Mathlib.Algebra.BigOperators.Group.Finset.Basic
+import Mathlib.Algebra.Group.Pi.Lemmas
+import Mathlib.Algebra.Ring.Periodic
+import Mathlib.Logic.Function.Iterate
+import Mathlib.Tactic.Abel
+
+namespace OAI
+
+universe uG uA uι
+
+namespace PeriodicTilingThree
+
+variable {G : Type uG} {A : Type uA} [AddCommGroup G] [AddCommGroup A]
+
+def configDiff (v : G) (f : G → A) : G → A :=
+  fun x => f (x + v) - f x
+
+@[simp] theorem configDiff_apply (v : G) (f : G → A) (x : G) :
+    configDiff v f x = f (x + v) - f x := rfl
+
+@[simp] theorem configDiff_zero (v : G) : configDiff v (0 : G → A) = 0 := by
+  ext x
+  simp [configDiff]
+
+@[simp] theorem configDiff_const (v : G) (a : A) :
+    configDiff v (fun _ : G => a) = 0 := by
+  ext x
+  simp [configDiff]
+
+theorem configDiff_add (v : G) (f g : G → A) :
+    configDiff v (f + g) = configDiff v f + configDiff v g := by
+  ext x
+  simp only [configDiff_apply, Pi.add_apply]
+  abel
+
+theorem configDiff_sub (v : G) (f g : G → A) :
+    configDiff v (f - g) = configDiff v f - configDiff v g := by
+  ext x
+  simp only [configDiff_apply, Pi.sub_apply]
+  abel
+
+theorem configDiff_comm (v w : G) (f : G → A) :
+    configDiff v (configDiff w f) = configDiff w (configDiff v f) := by
+  ext x
+  have h : x + v + w = x + w + v := by abel
+  simp only [configDiff_apply, h]
+  abel
+
+theorem configDiff_eq_zero_iff (v : G) (f : G → A) :
+    configDiff v f = 0 ↔ Function.Periodic f v := by
+  constructor
+  · intro h x
+    have hx := congrFun h x
+    exact sub_eq_zero.mp hx
+  · intro h
+    ext x
+    exact sub_eq_zero.mpr (h x)
+
+theorem configDiff_add_period (v p : G) (f : G → A)
+    (hp : Function.Periodic f p) : configDiff (v + p) f = configDiff v f := by
+  ext x
+  simp only [configDiff_apply, ← add_assoc, hp (x + v)]
+
+theorem configDiff_preserves_period (v p : G) (f : G → A)
+    (hp : Function.Periodic f p) : Function.Periodic (configDiff v f) p := by
+  apply (configDiff_eq_zero_iff p _).mp
+  rw [configDiff_comm, (configDiff_eq_zero_iff p f).mpr hp, configDiff_zero]
+
+theorem configDiff_sum {ι : Type uι} (s : Finset ι) (v : G) (f : ι → G → A) :
+    configDiff v (∑ i ∈ s, f i) = ∑ i ∈ s, configDiff v (f i) := by
+  classical
+  induction s using Finset.induction_on with
+  | empty => simp
+  | @insert i s hi ih =>
+      simp only [Finset.sum_insert hi, configDiff_add, ih]
+
+def configDiffs : List G → (G → A) → (G → A)
+  | [], f => f
+  | v :: vs, f => configDiff v (configDiffs vs f)
+
+@[simp] theorem configDiffs_nil (f : G → A) : configDiffs [] f = f := rfl
+
+@[simp] theorem configDiffs_cons (v : G) (vs : List G) (f : G → A) :
+    configDiffs (v :: vs) f = configDiff v (configDiffs vs f) := rfl
+
+@[simp] theorem configDiffs_zero (vs : List G) :
+    configDiffs vs (0 : G → A) = 0 := by
+  induction vs with
+  | nil => rfl
+  | cons v vs ih => simp only [configDiffs_cons, ih, configDiff_zero]
+
+theorem configDiff_configDiffs (v : G) (vs : List G) (f : G → A) :
+    configDiff v (configDiffs vs f) = configDiffs vs (configDiff v f) := by
+  induction vs with
+  | nil => rfl
+  | cons w vs ih =>
+      simp only [configDiffs_cons, configDiff_comm v w, ih]
+
+theorem configDiffs_add (vs : List G) (f g : G → A) :
+    configDiffs vs (f + g) = configDiffs vs f + configDiffs vs g := by
+  induction vs with
+  | nil => rfl
+  | cons v vs ih => simp only [configDiffs_cons, ih, configDiff_add]
+
+theorem configDiffs_sub (vs : List G) (f g : G → A) :
+    configDiffs vs (f - g) = configDiffs vs f - configDiffs vs g := by
+  induction vs with
+  | nil => rfl
+  | cons v vs ih => simp only [configDiffs_cons, ih, configDiff_sub]
+
+theorem configDiffs_sum {ι : Type uι} (s : Finset ι) (vs : List G)
+    (f : ι → G → A) :
+    configDiffs vs (∑ i ∈ s, f i) = ∑ i ∈ s, configDiffs vs (f i) := by
+  classical
+  induction s using Finset.induction_on with
+  | empty => simp
+  | @insert i s hi ih =>
+      simp only [Finset.sum_insert hi, configDiffs_add, ih]
+
+theorem configDiffs_eq_zero_of_mem {v : G} {vs : List G} (hv : v ∈ vs)
+    (f : G → A) (hf : configDiff v f = 0) : configDiffs vs f = 0 := by
+  induction vs with
+  | nil => simp at hv
+  | cons w vs ih =>
+      rcases List.mem_cons.mp hv with h | h
+      · subst w
+        rw [configDiffs_cons, configDiff_configDiffs, hf, configDiffs_zero]
+      · rw [configDiffs_cons, ih h, configDiff_zero]
+
+theorem configDiffs_const {vs : List G} (hvs : vs ≠ []) (a : A) :
+    configDiffs vs (fun _ : G => a) = 0 := by
+  cases vs with
+  | nil => exact (hvs rfl).elim
+  | cons v vs =>
+      rw [configDiffs_cons, configDiff_configDiffs, configDiff_const,
+        configDiffs_zero]
+
+theorem configDiffs_preserves_period (vs : List G) (p : G) (f : G → A)
+    (hp : Function.Periodic f p) : Function.Periodic (configDiffs vs f) p := by
+  apply (configDiff_eq_zero_iff p _).mp
+  rw [configDiff_configDiffs, (configDiff_eq_zero_iff p f).mpr hp,
+    configDiffs_zero]
+
+theorem configDiffs_replicate (n : ℕ) (v : G) (f : G → A) :
+    configDiffs (List.replicate n v) f = (configDiff v)^[n] f := by
+  induction n with
+  | zero => rfl
+  | succ n ih =>
+      simp only [List.replicate_succ, configDiffs_cons, ih,
+        Function.iterate_succ_apply']
+
+end PeriodicTilingThree
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/ConstituentAssembly.lean b/lean/OAI/Geometry/PeriodicTiling/ConstituentAssembly.lean
new file mode 100644
index 0000000000000000000000000000000000000000..153f175059149a582b7a72097f66f6f1c9c6d355
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/ConstituentAssembly.lean
@@ -0,0 +1,200 @@
+import OAI.Geometry.PeriodicTiling.ConstituentGeometry
+import OAI.Geometry.PeriodicTiling.AETilingCountable
+
+namespace OAI
+
+noncomputable section
+
+namespace PeriodicTilingThree.MarkedTile
+
+open Set MeasureTheory Filter
+
+namespace ResidueSection
+
+variable {m : ℕ} [NeZero m] (s : ResidueSection m)
+
+theorem translated_parts_unique_ae {w : Lattice 3} {U : Finset (Lattice 3)}
+    (hw : w ∉ U) (a : Space 3) (u u' : ↥U) :
+    ∀ᵐ x ∂volume,
+      x - center m a u ∈ Thickening (s.constituent w u) →
+      x - center m a u' ∈ Thickening (s.constituent w u') → u = u' := by
+  classical
+  by_cases he : u = u'
+  · exact Eventually.of_forall (fun _ _ _ => he)
+  · have hnull := thickening_inter_measure_zero_of_disjoint (s.parts_disjoint hw he)
+    have havoid : ∀ᵐ x ∂volume,
+        x ∉ Thickening (s.part w u) ∩ Thickening (s.part w u') :=
+      compl_mem_ae_iff.mpr hnull
+    have hshift := (measurePreserving_sub_right volume a).quasiMeasurePreserving.ae havoid
+    filter_upwards [hshift] with x hx
+    intro hu hu'
+    exact False.elim (hx ⟨s.mem_translated_part.mpr hu,
+      s.mem_translated_part.mpr hu'⟩)
+
+theorem indexed_constituents_ae {w : Lattice 3} {U : Finset (Lattice 3)}
+    (h0 : 0 ∈ U) (hw : w ∉ U) {A : Set (Space 3)}
+    (htile : AETiles (Thickening (s.finalTile w U)) A) :
+    ∀ᵐ x ∂volume, ∃! q : A × ↥U,
+      x - center m q.1 q.2 ∈ Thickening (s.constituent w q.2) := by
+  classical
+  have hcount := htile.countable_of_thickening (s.finalTile_nonempty w h0)
+  let : Countable A := hcount.to_subtype
+  have hall : ∀ᵐ x ∂volume, ∀ (a : A) (u u' : ↥U),
+      x - center m a u ∈ Thickening (s.constituent w u) →
+      x - center m a u' ∈ Thickening (s.constituent w u') → u = u' := by
+    apply ae_all_iff.mpr
+    intro a
+    apply ae_all_iff.mpr
+    intro u
+    apply ae_all_iff.mpr
+    intro u'
+    exact s.translated_parts_unique_ae hw a u u'
+  filter_upwards [htile, hall] with x hx hpieces
+  obtain ⟨a, ha, haunique⟩ := hx
+  obtain ⟨u, hu⟩ := s.mem_translated_finalTile.mp ha
+  refine ⟨(a, u), hu, ?_⟩
+  rintro ⟨a', u'⟩ hq
+  have ha' : a' = a := haunique a' (s.mem_translated_finalTile.mpr ⟨u', hq⟩)
+  subst a'
+  exact Prod.ext rfl (hpieces a u' u hq hu)
+
+theorem center_injective {w : Lattice 3} {U : Finset (Lattice 3)}
+    (hm : 2 ≤ m) (h0 : 0 ∈ U) (hw : w ∉ U) {A : Set (Space 3)}
+    (htile : AETiles (Thickening (s.finalTile w U)) A) :
+    Function.Injective (fun q : A × ↥U => center m q.1 q.2) := by
+  classical
+  have hcover := s.indexed_constituents_ae h0 hw htile
+  obtain ⟨t, ht⟩ := s.Tstar_nonempty hm
+  intro q q' he
+  by_contra hne
+  have hnull : volume (unitVoxel (center m q.1 q.2 + castLattice t)) = 0 := by
+    apply measure_mono_null ?_ (ae_iff.mp hcover)
+    intro x hx huniq
+    obtain ⟨r, _hr, hruniq⟩ := huniq
+    have hq : x - center m q.1 q.2 ∈ Thickening (s.constituent w q.2) :=
+      s.common_voxel_subset_constituent w q.2 q.1 ht hx
+    have hq' : x - center m q'.1 q'.2 ∈ Thickening (s.constituent w q'.2) := by
+      apply s.common_voxel_subset_constituent w q'.2 q'.1 ht
+      simpa only [he] using hx
+    exact hne ((hruniq q hq).trans (hruniq q' hq').symm)
+  rw [unitVoxel_volume] at hnull
+  exact one_ne_zero hnull
+
+theorem center_mem_A_iff
+    {m : ℕ} [NeZero m]
+    {U : Finset (Lattice 3)} (h0 : 0 ∈ U)
+    {A : Set (Space 3)}
+    (hinj : Function.Injective (fun q : A × ↥U => center m q.1 q.2))
+    (q : A × ↥U) : center m q.1 q.2 ∈ A ↔ (q.2 : Lattice 3) = 0 := by
+  constructor
+  · intro hc
+    have he : q = ((⟨center m q.1 q.2, hc⟩ : A), (⟨0, h0⟩ : ↥U)) := by
+      apply hinj
+      simp
+    exact congrArg (fun p : A × ↥U => (p.2 : Lattice 3)) he
+  · intro hu
+    simpa only [hu, center_zero] using q.1.property
+
+open scoped Classical in
+theorem constituent_type_eq {w : Lattice 3} {U : Finset (Lattice 3)} (h0 : 0 ∈ U)
+    {A : Set (Space 3)}
+    (hinj : Function.Injective (fun q : A × ↥U => center m q.1 q.2))
+    (q : A × ↥U) :
+    (if center m q.1 q.2 ∈ A then Thickening (s.T1 w) else Thickening s.T0) =
+      Thickening (s.constituent w q.2) := by
+  classical
+  by_cases hu : (q.2 : Lattice 3) = 0
+  · rw [ite_eq_left ((center_mem_A_iff h0 hinj q).mpr hu)]
+    simp only [constituent, ite_eq_left hu]
+  · have hc : center m q.1 q.2 ∉ A := fun h => hu ((center_mem_A_iff h0 hinj q).mp h)
+    simp only [hc, ite_false, constituent, hu]
+
+theorem typed_constituents {w : Lattice 3} {U : Finset (Lattice 3)}
+    (hm : 2 ≤ m) (h0 : 0 ∈ U) (hw : w ∉ U) {A : Set (Space 3)}
+    (htile : AETiles (Thickening (s.finalTile w U)) A) :
+    TypedAETiles (Thickening s.T0) (Thickening (s.T1 w)) (centers m U A) A := by
+  classical
+  have hinj := s.center_injective hm h0 hw htile
+  have hcover := s.indexed_constituents_ae h0 hw htile
+  filter_upwards [hcover] with x hx
+  obtain ⟨q, hq, hqunique⟩ := hx
+  let c : centers m U A := ⟨center m q.1 q.2, ⟨q, rfl⟩⟩
+  refine ⟨c, ?_, ?_⟩
+  · change x - center m q.1 q.2 ∈
+      if center m q.1 q.2 ∈ A then Thickening (s.T1 w) else Thickening s.T0
+    rw [s.constituent_type_eq h0 hinj q]
+    exact hq
+  · intro c' hc'
+    obtain ⟨q', he⟩ := c'.property
+    change center m q'.1 q'.2 = (c' : Space 3) at he
+    have hq' : x - center m q'.1 q'.2 ∈ Thickening (s.constituent w q'.2) := by
+      rw [← s.constituent_type_eq h0 hinj q', he]
+      exact hc'
+    have hqq := hqunique q' hq'
+    apply Subtype.ext
+    change (c' : Space 3) = center m q.1 q.2
+    rw [← he, hqq]
+
+end ResidueSection
+
+theorem period_centers {m : ℕ} {U : Finset (Lattice 3)} {A : Set (Space 3)}
+    {v : Space 3} (hA : Period A v) : Period (centers m U A) v := by
+  intro c
+  constructor
+  · intro hc
+    obtain ⟨a, ha, u, hu, he⟩ := mem_centers.mp hc
+    have hav : a - v ∈ A := (hA (a - v)).mp (by simpa using ha)
+    refine mem_centers.mpr ⟨a - v, hav, u, hu, ?_⟩
+    calc
+      center m (a - v) u = center m a u - v := by unfold center; abel
+      _ = c := by rw [he]; abel
+  · intro hc
+    obtain ⟨a, ha, u, hu, he⟩ := mem_centers.mp hc
+    refine mem_centers.mpr ⟨a + v, (hA a).mpr ha, u, hu, ?_⟩
+    calc
+      center m (a + v) u = center m a u + v := by unfold center; abel
+      _ = c + v := by rw [he]
+
+theorem fullyPeriodic_centers {m : ℕ} {U : Finset (Lattice 3)} {A : Set (Space 3)}
+    (hA : EuclideanFullyPeriodic A) : EuclideanFullyPeriodic (centers m U A) := by
+  obtain ⟨b, hb⟩ := hA
+  exact ⟨b, fun v hv => period_centers (hb v hv)⟩
+
+theorem tiles_of_centers_grid {m : ℕ} {U : Finset (Lattice 3)} {A : Set (Space 3)}
+    {c0 : Space 3}
+    (hinj : Function.Injective (fun q : A × ↥U => center m q.1 q.2))
+    (hgrid : centers m U A = Set.range (fun z : Lattice 3 => c0 + scaledCast m z)) :
+    Tiles U {z : Lattice 3 | c0 + scaledCast m z ∈ A} := by
+  classical
+  rw [tiles_iff_unique_tile]
+  intro z
+  have hz : c0 + scaledCast m z ∈ centers m U A := by
+    rw [hgrid]
+    exact ⟨z, rfl⟩
+  obtain ⟨q, hq⟩ := hz
+  have hd : c0 + scaledCast m (z - (q.2 : Lattice 3)) = (q.1 : Space 3) := by
+    have he : (q.1 : Space 3) + scaledCast m (q.2 : Lattice 3) =
+        c0 + scaledCast m z := hq
+    rw [map_sub]
+    calc
+      c0 + (scaledCast m z - scaledCast m (q.2 : Lattice 3)) =
+          (c0 + scaledCast m z) - scaledCast m (q.2 : Lattice 3) := by abel
+      _ = (q.1 : Space 3) := by rw [← he]; abel
+  refine ⟨q.2, ?_, ?_⟩
+  · change c0 + scaledCast m (z - (q.2 : Lattice 3)) ∈ A
+    rw [hd]
+    exact q.1.property
+  · intro u hu
+    let a : A := ⟨c0 + scaledCast m (z - (u : Lattice 3)), hu⟩
+    have hc : center m a u = c0 + scaledCast m z := by
+      change (c0 + scaledCast m (z - (u : Lattice 3))) + scaledCast m u = _
+      rw [map_sub]
+      abel
+    have he : (a, u) = q := hinj (hc.trans hq.symm)
+    exact congrArg Prod.snd he
+
+end PeriodicTilingThree.MarkedTile
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/ConstituentGeometry.lean b/lean/OAI/Geometry/PeriodicTiling/ConstituentGeometry.lean
new file mode 100644
index 0000000000000000000000000000000000000000..dc982381e65d1c37a4e1411871481a3a32f23236
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/ConstituentGeometry.lean
@@ -0,0 +1,146 @@
+import OAI.Geometry.PeriodicTiling.MarkedTile
+import OAI.Geometry.PeriodicTiling.VoxelMeasure
+import Mathlib.MeasureTheory.Group.Measure
+
+namespace OAI
+
+noncomputable section
+
+namespace PeriodicTilingThree.MarkedTile
+
+open Set MeasureTheory
+
+@[simp] theorem castLattice_scale (m : ℕ) (u : Lattice 3) :
+    castLattice (scale m u) = scaledCast m u := by
+  ext i
+  simp [scale, scaledCast, castLattice]
+
+def center (m : ℕ) (a : Space 3) (u : Lattice 3) : Space 3 :=
+  a + scaledCast m u
+
+@[simp] theorem center_zero (m : ℕ) (a : Space 3) : center m a 0 = a := by
+  simp [center]
+
+def centers (m : ℕ) (U : Finset (Lattice 3)) (A : Set (Space 3)) : Set (Space 3) :=
+  Set.range (fun q : A × ↥U => center m q.1 q.2)
+
+theorem mem_centers {m : ℕ} {U : Finset (Lattice 3)} {A : Set (Space 3)}
+    {c : Space 3} : c ∈ centers m U A ↔ ∃ a ∈ A, ∃ u ∈ U, center m a u = c := by
+  constructor
+  · rintro ⟨⟨a, u⟩, h⟩
+    exact ⟨a, a.property, u, u.property, h⟩
+  · rintro ⟨a, ha, u, hu, h⟩
+    exact ⟨(⟨a, ha⟩, ⟨u, hu⟩), h⟩
+
+theorem subset_centers {m : ℕ} {U : Finset (Lattice 3)} (h0 : 0 ∈ U)
+    (A : Set (Space 3)) : A ⊆ centers m U A := by
+  intro a ha
+  exact mem_centers.mpr ⟨a, ha, 0, h0, center_zero m a⟩
+
+namespace ResidueSection
+
+variable {m : ℕ} [NeZero m] (s : ResidueSection m)
+
+def constituent (w u : Lattice 3) : Finset (Lattice 3) :=
+  if u = 0 then s.T1 w else s.T0
+
+def part (w u : Lattice 3) : Finset (Lattice 3) :=
+  (s.constituent w u).image (fun t => scale m u + t)
+
+theorem mem_part {w u z : Lattice 3} :
+    z ∈ s.part w u ↔ ∃ v, s.point w u v = z := by
+  classical
+  by_cases hu : u = 0
+  · subst u
+    simp [part, constituent, mem_T1]
+  · simp only [part, constituent, hu, ite_false, Finset.mem_image]
+    constructor
+    · rintro ⟨t, ht, he⟩
+      obtain ⟨v, rfl⟩ := s.mem_T0.mp ht
+      exact ⟨v, by simpa only [s.point_of_ne_zero w hu] using he⟩
+    · rintro ⟨v, hv⟩
+      exact ⟨s.t v, s.mem_T0.mpr ⟨v, rfl⟩,
+        by simpa only [s.point_of_ne_zero w hu] using hv⟩
+
+theorem part_subset_finalTile {w : Lattice 3} {U : Finset (Lattice 3)}
+    {u : Lattice 3} (hu : u ∈ U) : s.part w u ⊆ s.finalTile w U := by
+  intro z hz
+  obtain ⟨v, hv⟩ := s.mem_part.mp hz
+  exact s.mem_finalTile.mpr ⟨u, hu, v, hv⟩
+
+theorem parts_disjoint {w : Lattice 3} {U : Finset (Lattice 3)} (hw : w ∉ U)
+    {u u' : ↥U} (hne : u ≠ u') : Disjoint (s.part w u) (s.part w u') := by
+  classical
+  apply Finset.disjoint_left.mpr
+  intro z hz hz'
+  obtain ⟨v, hv⟩ := s.mem_part.mp hz
+  obtain ⟨v', hv'⟩ := s.mem_part.mp hz'
+  have hp : (u, v) = (u', v') := s.point_injective hw (hv.trans hv'.symm)
+  exact hne (congrArg Prod.fst hp)
+
+theorem mem_thickening_part {w u : Lattice 3} {x : Space 3} :
+    x ∈ Thickening (s.part w u) ↔
+      x - scaledCast m u ∈ Thickening (s.constituent w u) := by
+  classical
+  constructor
+  · rintro ⟨z, hz, hx⟩
+    obtain ⟨t, ht, rfl⟩ := Finset.mem_image.mp hz
+    refine ⟨t, ht, ?_⟩
+    simpa only [map_add, castLattice_scale, sub_add_eq_sub_sub] using hx
+  · rintro ⟨t, ht, hx⟩
+    refine ⟨scale m u + t, Finset.mem_image.mpr ⟨t, ht, rfl⟩, ?_⟩
+    simpa only [map_add, castLattice_scale, sub_add_eq_sub_sub] using hx
+
+theorem mem_translated_part {w u : Lattice 3} {a x : Space 3} :
+    x - a ∈ Thickening (s.part w u) ↔
+      x - center m a u ∈ Thickening (s.constituent w u) := by
+  simpa only [center, sub_add_eq_sub_sub] using
+    (s.mem_thickening_part (w := w) (u := u) (x := x - a))
+
+theorem mem_thickening_finalTile {w : Lattice 3} {U : Finset (Lattice 3)}
+    {x : Space 3} :
+    x ∈ Thickening (s.finalTile w U) ↔
+      ∃ u : ↥U, x ∈ Thickening (s.part w u) := by
+  constructor
+  · rintro ⟨z, hz, hx⟩
+    obtain ⟨u, hu, v, hv⟩ := s.mem_finalTile.mp hz
+    exact ⟨⟨u, hu⟩, z, s.mem_part.mpr ⟨v, hv⟩, hx⟩
+  · rintro ⟨u, z, hz, hx⟩
+    exact ⟨z, s.part_subset_finalTile u.property hz, hx⟩
+
+theorem mem_translated_finalTile {w : Lattice 3} {U : Finset (Lattice 3)}
+    {a x : Space 3} :
+    x - a ∈ Thickening (s.finalTile w U) ↔
+      ∃ u : ↥U, x - center m a u ∈ Thickening (s.constituent w u) := by
+  rw [s.mem_thickening_finalTile]
+  exact exists_congr fun u => s.mem_translated_part
+
+theorem Tstar_subset_constituent (w u : Lattice 3) :
+    s.Tstar ⊆ s.constituent w u := by
+  classical
+  intro z hz
+  by_cases hu : u = 0
+  · simp only [constituent, hu, ite_true, s.T1_eq_insert]
+    exact Finset.mem_insert_of_mem hz
+  · simpa only [constituent, hu, ite_false] using (Finset.mem_erase.mp hz).2
+
+theorem Tstar_nonempty (hm : 2 ≤ m) : s.Tstar.Nonempty := by
+  apply Finset.card_pos.mp
+  rw [s.card_Tstar]
+  have hp := Nat.pow_le_pow_left hm 3
+  exact Nat.sub_pos_of_lt ((by decide : 1 < 2 ^ 3).trans_le hp)
+
+theorem common_voxel_subset_constituent (w u : Lattice 3) (a : Space 3)
+    {t : Lattice 3} (ht : t ∈ s.Tstar) :
+    unitVoxel (center m a u + castLattice t) ⊆
+      {x | x - center m a u ∈ Thickening (s.constituent w u)} := by
+  intro x hx
+  refine ⟨t, s.Tstar_subset_constituent w u ht, ?_⟩
+  simpa only [mem_unitVoxel, sub_add_eq_sub_sub] using hx
+
+end ResidueSection
+end PeriodicTilingThree.MarkedTile
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/ConstraintCycles.lean b/lean/OAI/Geometry/PeriodicTiling/ConstraintCycles.lean
new file mode 100644
index 0000000000000000000000000000000000000000..05fa51e167655df9bce0f6e444beabbd4d03ea54
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/ConstraintCycles.lean
@@ -0,0 +1,171 @@
+import OAI.Geometry.PeriodicTiling.CycleTests
+import OAI.Geometry.PeriodicTiling.SeedSymbols
+import Mathlib.Data.Finset.Union
+import Mathlib.Tactic.FinCases
+
+namespace OAI
+
+noncomputable section
+
+namespace PeriodicTilingThree
+
+variable {p : ℕ} [NeZero p]
+
+def ordinaryCycle (p : ℕ) [NeZero p] : ZMod (p ^ 2) ↪ Channel p where
+  toFun j := .inl ((ZMod.finEquiv (p ^ 2)).symm j)
+  inj' _ _ h := (ZMod.finEquiv (p ^ 2)).symm.injective (Sum.inl.inj h)
+
+def ordinaryCycleLabels (w : Word p) (j : ZMod (p ^ 2)) :
+    Label p (ordinaryCycle p j) :=
+  w ((ZMod.finEquiv (p ^ 2)).symm j)
+
+def seedCycle (t : Fin 2) (n : Column p) : ZMod 2 ↪ Channel p where
+  toFun j := Fin.cases (.inr t) (fun _ => .inl n) j
+  inj' := by
+    intro i j h
+    fin_cases i <;> fin_cases j
+    · rfl
+    · change Sum.inr t = Sum.inl n at h
+      cases h
+    · change Sum.inl n = Sum.inr t at h
+      cases h
+    · rfl
+
+@[simp] theorem seedCycle_zero
+    {p : ℕ} [NeZero p]
+    (t : Fin 2) (n : Column p) :
+    seedCycle t n 0 = .inr t := rfl
+
+@[simp] theorem seedCycle_one
+    {p : ℕ} [NeZero p]
+    (t : Fin 2) (n : Column p) :
+    seedCycle t n 1 = .inl n := rfl
+
+def seedCycleLabels (t : Fin 2) (n : Column p) (j : Symbol p) (z : ZMod 2) :
+    Label p (seedCycle t n z) :=
+  Fin.cases (motive := fun z : Fin 2 => Label p (seedCycle t n z))
+    () (fun _ => j) z
+
+@[simp] theorem seedCycleLabels_zero
+    {p : ℕ} [NeZero p]
+    (t : Fin 2) (n : Column p) (j : Symbol p) :
+    seedCycleLabels t n j 0 = () := rfl
+
+@[simp] theorem seedCycleLabels_one
+    {p : ℕ} [NeZero p]
+    (t : Fin 2) (n : Column p) (j : Symbol p) :
+    seedCycleLabels t n j 1 = j := rfl
+
+variable (E : EncodingParameters p)
+
+theorem ordinaryCycle_exclusion_iff (w : Word p) (o : GraphOutputs E)
+    (hdep : HasDependence E o) :
+    (∀ F ∈ cycleTiles E (ordinaryCycle p) (ordinaryCycleLabels w),
+      Tiles F (graph o)) ↔
+      ∀ x, ¬ (∀ n : Column p, Active E o (.inl n) x (w n)) := by
+  classical
+  rw [cycle_exclusion_iff E (ordinaryCycle p) (ordinaryCycleLabels w) o hdep]
+  constructor
+  · intro h x hall
+    obtain ⟨z, hz⟩ := h x
+    exact hz (hall ((ZMod.finEquiv (p ^ 2)).symm z))
+  · intro h x
+    obtain ⟨n, hn⟩ := not_forall.mp (h x)
+    refine ⟨(ZMod.finEquiv (p ^ 2)) n, ?_⟩
+    exact Eq.mpr
+      (congrArg (fun m : Column p => ¬ Active E o (.inl m) x (w m))
+        ((ZMod.finEquiv (p ^ 2)).symm_apply_apply n)) hn
+
+theorem seedCycle_exclusion_iff (t : Fin 2) (n : Column p) (j : Symbol p)
+    (o : GraphOutputs E) (hdep : HasDependence E o) :
+    (∀ F ∈ cycleTiles E (seedCycle t n) (seedCycleLabels t n j),
+      Tiles F (graph o)) ↔
+      ∀ x, ¬ (Active E o (.inr t) x () ∧ Active E o (.inl n) x j) := by
+  classical
+  rw [cycle_exclusion_iff E (seedCycle t n) (seedCycleLabels t n j) o hdep]
+  constructor
+  · intro h x hx
+    obtain ⟨z, hz⟩ := h x
+    fin_cases z
+    · exact hz hx.1
+    · exact hz hx.2
+  · intro h x
+    by_cases hs : Active E o (.inr t) x ()
+    · refine ⟨1, ?_⟩
+      change ¬ Active E o (.inl n) x j
+      intro hj
+      exact h x ⟨hs, hj⟩
+    · exact ⟨0, hs⟩
+
+def wordConstraintTiles : Finset (Finset (Ambient E)) := by
+  classical
+  exact (Finset.univ.filter (fun w : Word p => w ∉ allowedWords p)).biUnion
+    (fun w => cycleTiles E (ordinaryCycle p) (ordinaryCycleLabels w))
+
+def seedConstraintTiles : Finset (Finset (Ambient E)) := by
+  classical
+  exact (Finset.univ.filter (fun q : Fin 2 × Column p × Symbol p =>
+      q.2.2 ≠ seedSymbol E q.1)).biUnion
+    (fun q => cycleTiles E (seedCycle q.1 q.2.1) (seedCycleLabels q.1 q.2.1 q.2.2))
+
+theorem nonempty_of_mem_wordConstraintTiles {F : Finset (Ambient E)}
+    (hF : F ∈ wordConstraintTiles E) : F.Nonempty := by
+  classical
+  obtain ⟨w, _, hF⟩ := Finset.mem_biUnion.mp hF
+  exact nonempty_of_mem_cycleTiles E (ordinaryCycle p) (ordinaryCycleLabels w) hF
+
+theorem nonempty_of_mem_seedConstraintTiles {F : Finset (Ambient E)}
+    (hF : F ∈ seedConstraintTiles E) : F.Nonempty := by
+  classical
+  obtain ⟨q, _, hF⟩ := Finset.mem_biUnion.mp hF
+  exact nonempty_of_mem_cycleTiles E (seedCycle q.1 q.2.1)
+    (seedCycleLabels q.1 q.2.1 q.2.2) hF
+
+theorem wordConstraintTiles_iff (o : GraphOutputs E) (hdep : HasDependence E o) :
+    (∀ F ∈ wordConstraintTiles E, Tiles F (graph o)) ↔
+      ∀ x w, (∀ n : Column p, Active E o (.inl n) x (w n)) → Allowed p w := by
+  classical
+  constructor
+  · intro h x w hw
+    by_contra hnot
+    have hcycle : ∀ F ∈ cycleTiles E (ordinaryCycle p) (ordinaryCycleLabels w),
+        Tiles F (graph o) := by
+      intro F hF
+      apply h F
+      apply Finset.mem_biUnion.mpr
+      exact ⟨w, by simp [hnot], hF⟩
+    exact (ordinaryCycle_exclusion_iff E w o hdep).mp hcycle x hw
+  · intro h F hF
+    obtain ⟨w, hw, hF⟩ := Finset.mem_biUnion.mp hF
+    have hnot : ¬ Allowed p w := by simpa using (Finset.mem_filter.mp hw).2
+    apply (ordinaryCycle_exclusion_iff E w o hdep).mpr ?_ F hF
+    intro x hx
+    exact hnot (h x w hx)
+
+theorem seedConstraintTiles_iff (o : GraphOutputs E) (hdep : HasDependence E o) :
+    (∀ F ∈ seedConstraintTiles E, Tiles F (graph o)) ↔
+      ∀ t n x j, Active E o (.inr t) x () → Active E o (.inl n) x j →
+        j = seedSymbol E t := by
+  classical
+  constructor
+  · intro h t n x j hseed hord
+    by_contra hj
+    have hcycle : ∀ F ∈ cycleTiles E (seedCycle t n) (seedCycleLabels t n j),
+        Tiles F (graph o) := by
+      intro F hF
+      apply h F
+      apply Finset.mem_biUnion.mpr
+      exact ⟨(t, n, j), Finset.mem_filter.mpr ⟨Finset.mem_univ _, hj⟩, hF⟩
+    exact (seedCycle_exclusion_iff E t n j o hdep).mp hcycle x ⟨hseed, hord⟩
+  · intro h F hF
+    obtain ⟨⟨t, n, j⟩, hq, hF⟩ := Finset.mem_biUnion.mp hF
+    have hj : j ≠ seedSymbol E t := (Finset.mem_filter.mp hq).2
+    apply (seedCycle_exclusion_iff E t n j o hdep).mpr ?_ F hF
+    intro x hx
+    exact hj (h t n x j hx.1 hx.2)
+
+end PeriodicTilingThree
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/CosetParts.lean b/lean/OAI/Geometry/PeriodicTiling/CosetParts.lean
new file mode 100644
index 0000000000000000000000000000000000000000..ca5159a62e43bc1961d85b4aedaf5ae79bf2c8fb
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/CosetParts.lean
@@ -0,0 +1,114 @@
+import OAI.Geometry.PeriodicTiling.PlaneLattices
+import OAI.Geometry.PeriodicTiling.FiniteMasks
+import Mathlib.Data.Fintype.EquivFin
+import Mathlib.Algebra.Ring.Periodic
+import Mathlib.Tactic.LinearCombination
+
+namespace OAI
+
+noncomputable section
+
+namespace PeriodicTilingThree
+
+open scoped BigOperators
+
+theorem integerCombination_injective {e H : Plane} (hdet : planeDet e H ≠ 0) :
+    Function.Injective (integerCombination e H) := by
+  rintro ⟨a, b⟩ ⟨c, d⟩ he
+  have hx := congrArg Prod.fst he
+  have hy := congrArg Prod.snd he
+  change a * e.1 + b * H.1 = c * e.1 + d * H.1 at hx
+  change a * e.2 + b * H.2 = c * e.2 + d * H.2 at hy
+  apply Prod.ext
+  · have hz : planeDet e H * (a - c) = 0 := by
+      dsimp [planeDet]
+      linear_combination H.2 * hx - H.1 * hy
+    exact sub_eq_zero.mp ((mul_eq_zero.mp hz).resolve_left hdet)
+  · have hz : planeDet e H * (b - d) = 0 := by
+      dsimp [planeDet]
+      linear_combination e.1 * hy - e.2 * hx
+    exact sub_eq_zero.mp ((mul_eq_zero.mp hz).resolve_left hdet)
+
+def cosetPart (P : AddSubgroup Plane) (A : Set Plane) (c : Plane ⧸ P) : Set Plane :=
+  {x | x ∈ A ∧ (x : Plane ⧸ P) = c}
+
+theorem sum_cosetPart_indicator (P : AddSubgroup Plane) [Fintype (Plane ⧸ P)]
+    (A : Set Plane) (x : Plane) :
+    (∑ c : Plane ⧸ P, tileIndicator ℝ (cosetPart P A c) x) = tileIndicator ℝ A x := by
+  classical
+  by_cases hx : x ∈ A <;> simp [tileIndicator, cosetPart, hx]
+
+private theorem mem_iff_of_tileIndicator_eq (A : Set Plane) (x y : Plane)
+    (h : tileIndicator ℝ A x = tileIndicator ℝ A y) : x ∈ A ↔ y ∈ A := by
+  classical
+  by_cases hx : x ∈ A <;> by_cases hy : y ∈ A <;> simp_all [tileIndicator]
+
+theorem cosetPart_period_of_coordinate_period (A : Set Plane) (e H x v : Plane)
+    (hper : Function.Periodic
+      (fun st : Plane => tileIndicator ℝ A (x + st.1 • e + st.2 • H)) v) :
+    Period (cosetPart (integerSpan2 e H) A (x : Plane ⧸ integerSpan2 e H))
+      (integerCombination e H v) := by
+  classical
+  let P := integerSpan2 e H
+  let L := integerCombination e H
+  have hvP : L v ∈ P := ⟨v, rfl⟩
+  intro y
+  have hclass : ((y + L v : Plane) : Plane ⧸ P) = (y : Plane ⧸ P) :=
+    QuotientAddGroup.mk_add_of_mem y hvP
+  change ((y + L v ∈ A) ∧ ((y + L v : Plane) : Plane ⧸ P) = (x : Plane ⧸ P)) ↔
+    (y ∈ A ∧ (y : Plane ⧸ P) = (x : Plane ⧸ P))
+  rw [hclass]
+  by_cases hyclass : (y : Plane ⧸ P) = (x : Plane ⧸ P)
+  · simp only [hyclass, and_true]
+    have hdiff : y - x ∈ P := QuotientAddGroup.eq_iff_sub_mem.mp hyclass
+    obtain ⟨st, hst⟩ := hdiff
+    have hyrep : y = x + L st := by
+      rw [hst]
+      abel
+    have hyshift : y + L v = x + L (st + v) := by
+      rw [hyrep, map_add]
+      abel
+    apply mem_iff_of_tileIndicator_eq
+    rw [hyshift, hyrep]
+    simpa only [L, integerCombination, AddMonoidHom.coe_mk, ZeroHom.coe_mk, add_assoc] using hper st
+  · simp only [hyclass, and_false]
+
+theorem finite_periodic_parts_of_coset_periods {A : Set Plane}
+    (e H : Plane) (hdet : planeDet e H ≠ 0)
+    (hlocal : ∀ x : Plane, ∃ v : Plane, v ≠ 0 ∧ Function.Periodic
+      (fun st : Plane => tileIndicator ℝ A (x + st.1 • e + st.2 • H)) v) :
+    ∃ m : ℕ, ∃ parts : Fin m → Set Plane, ∃ w : Fin m → Plane,
+      (∀ x, (∑ j, tileIndicator ℝ (parts j) x) = tileIndicator ℝ A x) ∧
+      ∀ j, w j ≠ 0 ∧ Period (parts j) (w j) := by
+  classical
+  let P := integerSpan2 e H
+  let : P.FiniteIndex := integerSpan2_finiteIndex e H hdet
+  let : Fintype (Plane ⧸ P) := P.fintypeQuotientOfFiniteIndex
+  have hex : ∀ c : Plane ⧸ P, ∃ w : Plane, w ≠ 0 ∧ Period (cosetPart P A c) w := by
+    intro c
+    let x : Plane := Quotient.out c
+    have hx : (x : Plane ⧸ P) = c := Quotient.out_eq' c
+    obtain ⟨v, hv, hper⟩ := hlocal x
+    refine ⟨integerCombination e H v, ?_, ?_⟩
+    · intro hz
+      apply hv
+      apply integerCombination_injective hdet
+      simpa only [map_zero] using hz
+    · rw [← hx]
+      exact cosetPart_period_of_coordinate_period A e H x v hper
+  choose w hw0 hwper using hex
+  let eqv : Fin (Fintype.card (Plane ⧸ P)) ≃ (Plane ⧸ P) :=
+    (Fintype.equivFin (Plane ⧸ P)).symm
+  refine ⟨Fintype.card (Plane ⧸ P), (fun j => cosetPart P A (eqv j)),
+    (fun j => w (eqv j)), ?_, ?_⟩
+  · intro x
+    rw [eqv.sum_comp (fun c => tileIndicator ℝ (cosetPart P A c) x)]
+    exact sum_cosetPart_indicator P A x
+  · intro j
+    exact ⟨hw0 (eqv j), hwper (eqv j)⟩
+
+end PeriodicTilingThree
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/CosetPeriodicity.lean b/lean/OAI/Geometry/PeriodicTiling/CosetPeriodicity.lean
new file mode 100644
index 0000000000000000000000000000000000000000..612be1d6640a1055fc1da6e8e5aeea5fe7503234
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/CosetPeriodicity.lean
@@ -0,0 +1,83 @@
+import OAI.Geometry.PeriodicTiling.TilingBasic
+import OAI.Geometry.PeriodicTiling.ScalarPolynomialPeriodicity
+import Mathlib.Algebra.BigOperators.Group.Finset.Piecewise
+import Mathlib.Algebra.Order.BigOperators.Ring.Finset
+import Mathlib.Tactic.Choose
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.Ring
+
+namespace OAI
+
+namespace PeriodicTilingThree
+
+theorem coset_configuration_has_period {m : ℕ}
+    (P : Fin m → ℤ → ℝ) (b : Fin m → ℤ) (a : Plane → ℝ)
+    (hzero : ∀ j n, 0 ≤ P j n)
+    (hpair : ∀ i j, i ≠ j → ∀ n r, P i n + P j r ≤ 1)
+    (hsum : ∀ x : Plane, a x = 1 - ∑ j, P j (x.1 + b j * x.2))
+    (degree : ℕ) (hdegree : 0 < degree)
+    (hpoly : ∀ j, (forwardDiff^[degree])
+      (fun n => (P j n : AddCircle (1 : ℝ))) = 0) :
+    ∃ v : Plane, v ≠ 0 ∧ Function.Periodic a v := by
+  classical
+  by_cases hbounded : ∀ j, ∃ c : ℝ, c < 1 ∧ ∀ n, P j n ≤ c
+  · have hp : ∀ j, ∃ q : ℕ, 0 < q ∧ Function.Periodic (P j) (q : ℤ) := by
+      intro j
+      obtain ⟨c, hc, hPc⟩ := hbounded j
+      exact real_polynomial_periodic_of_bounded_lt_one degree hdegree
+        (P j) c (hzero j) hPc hc (hpoly j)
+    choose p hp hperiod using hp
+    let Q : ℕ := ∏ j, p j
+    have hQ : 0 < Q := Finset.prod_pos fun j _ => hp j
+    refine ⟨(0, (Q : ℤ)), ?_, ?_⟩
+    · intro h
+      have hz : (Q : ℤ) = 0 := congrArg Prod.snd h
+      exact (Nat.cast_ne_zero.mpr (Nat.ne_of_gt hQ)) hz
+    · intro x
+      rw [hsum, hsum]
+      apply congrArg (fun y : ℝ => 1 - y)
+      apply Finset.sum_congr rfl
+      intro j _hj
+      have hd : p j ∣ Q := Finset.dvd_prod_of_mem p (Finset.mem_univ j)
+      obtain ⟨k, hk⟩ := hd
+      have hperiodQ : Function.Periodic (P j) (b j * (Q : ℤ)) := by
+        simpa only [hk, Nat.cast_mul, Int.cast_id, mul_assoc, mul_left_comm, mul_comm] using
+          (hperiod j).int_mul (b j * (k : ℤ))
+      change P j (x.1 + 0 + b j * (x.2 + (Q : ℤ))) =
+        P j (x.1 + b j * x.2)
+      rw [show x.1 + 0 + b j * (x.2 + (Q : ℤ)) =
+        (x.1 + b j * x.2) + b j * (Q : ℤ) by ring]
+      exact hperiodQ _
+  · obtain ⟨j, hj⟩ := not_forall.mp hbounded
+    have hother : ∀ i, i ≠ j → ∀ n, P i n = 0 := by
+      intro i hij n
+      apply le_antisymm _ (hzero i n)
+      apply le_of_not_gt
+      intro hpos
+      apply hj
+      refine ⟨1 - P i n, by linarith, ?_⟩
+      intro r
+      have hcross := hpair j i (Ne.symm hij) r n
+      linarith
+    have hone : ∀ x : Plane, a x = 1 - P j (x.1 + b j * x.2) := by
+      intro x
+      rw [hsum]
+      apply congrArg (fun y : ℝ => 1 - y)
+      apply Finset.sum_eq_single j
+      · intro i _hi hij
+        exact hother i hij _
+      · intro hj'
+        exact (hj' (Finset.mem_univ j)).elim
+    refine ⟨(-b j, 1), ?_, ?_⟩
+    · intro h
+      have hz : (1 : ℤ) = 0 := congrArg Prod.snd h
+      exact one_ne_zero hz
+    · intro x
+      rw [hone, hone]
+      apply congrArg (fun n : ℤ => 1 - P j n)
+      change x.1 + -b j + b j * (x.2 + 1) = x.1 + b j * x.2
+      ring
+
+end PeriodicTilingThree
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/Counterexample.lean b/lean/OAI/Geometry/PeriodicTiling/Counterexample.lean
new file mode 100644
index 0000000000000000000000000000000000000000..7988070a1c82287df771ea98f9b211c95542f573
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/Counterexample.lean
@@ -0,0 +1,35 @@
+import OAI.Geometry.PeriodicTiling.LowerDimensions
+import OAI.Geometry.PeriodicTiling.ThreeDimensions
+
+namespace OAI
+
+namespace PeriodicTilingThree
+
+theorem least_lattice_dimension_three :
+    IsLeast {d : ℕ | HasTileWithoutPeriodicComplement d} 3 := by
+  classical
+  constructor
+  · obtain ⟨T, hT, hTiles, hNoPeriod, _hRealTiles, _hNoRealPeriod⟩ :=
+      exists_three_dimensional_tile
+    exact ⟨T, hT, hTiles, hNoPeriod⟩
+  · intro d hd
+    by_contra hnot
+    have hlt : d < 3 := Nat.lt_of_not_ge hnot
+    obtain ⟨T, _hT, hTiles, hNoPeriod⟩ := hd
+    obtain ⟨A, hA, hPeriod⟩ :=
+      tiles_below_three_have_fullyPeriodic_complements d hlt T hTiles
+    exact hNoPeriod A hA hPeriod
+
+theorem periodic_tiling_counterexample_and_minimality :
+    (∃ T : Finset (Lattice 3), T.Nonempty ∧
+      (∃ A : Set (Lattice 3), Tiles T A) ∧
+      (∀ A : Set (Lattice 3), Tiles T A → ¬ FullyPeriodic A) ∧
+      (∃ A : Set (Space 3), AETiles (Thickening T) A) ∧
+      (∀ A : Set (Space 3),
+        AETiles (Thickening T) A → ¬ EuclideanFullyPeriodic A)) ∧
+    IsLeast {d : ℕ | HasTileWithoutPeriodicComplement d} 3 :=
+  ⟨exists_three_dimensional_tile, least_lattice_dimension_three⟩
+
+end PeriodicTilingThree
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/CycleTests.lean b/lean/OAI/Geometry/PeriodicTiling/CycleTests.lean
new file mode 100644
index 0000000000000000000000000000000000000000..5de21c76db0bbe59dc71aa08105e9b34fbd2e0a3
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/CycleTests.lean
@@ -0,0 +1,434 @@
+import OAI.Geometry.PeriodicTiling.Dependence
+import Mathlib.Algebra.BigOperators.Pi
+import Mathlib.Data.ZMod.Basic
+import Mathlib.Tactic.Abel
+
+namespace OAI
+
+universe uι uM uN uA uB uH uC
+
+noncomputable section
+
+namespace PeriodicTilingThree
+
+open scoped BigOperators
+
+section CycleCombinatorics
+
+variable {ι : Type uι} [Fintype ι]
+variable {M : ι → Type uM} {N : ι → Type uN}
+variable [∀ i, AddCommGroup (M i)] [∀ i, AddCommGroup (N i)]
+variable {t : ℕ} [NeZero t]
+
+def cycleExtend (cycle : ZMod t ↪ ι) (v : ∀ j, M (cycle j)) : ∀ i, M i := by
+  classical
+  exact ∑ j : ZMod t, Pi.single (cycle j) (v j)
+
+@[simp] theorem cycleExtend_apply
+    {ι : Type uι} [Fintype ι]
+    {M : ι → Type uM} [∀ i, AddCommGroup (M i)] {t : ℕ} [NeZero t]
+    (cycle : ZMod t ↪ ι)
+    (v : ∀ j, M (cycle j)) (j : ZMod t) :
+    cycleExtend cycle v (cycle j) = v j := by
+  classical
+  rw [cycleExtend, Finset.sum_apply, Finset.sum_eq_single j]
+  · simp
+  · intro l _ hlj
+    have hne : cycle l ≠ cycle j := fun h => hlj (cycle.injective h)
+    simp [Ne.symm hne]
+  · simp
+
+theorem cycleExtend_eq_zero
+    {ι : Type uι} [Fintype ι]
+    {M : ι → Type uM} [∀ i, AddCommGroup (M i)] {t : ℕ} [NeZero t]
+    (cycle : ZMod t ↪ ι)
+    (v : ∀ j, M (cycle j)) {i : ι} (hi : ∀ j, cycle j ≠ i) :
+    cycleExtend cycle v i = 0 := by
+  classical
+  rw [cycleExtend, Finset.sum_apply]
+  apply Finset.sum_eq_zero
+  intro j _
+  simp [Ne.symm (hi j)]
+
+def cycleShift (cycle : ZMod t ↪ ι) (U : ∀ j, AddSubgroup (M (cycle j)))
+    (d : ∀ j, N (cycle (j - 1)) → U j) (e : ∀ i, N i) : ∀ i, M i :=
+  cycleExtend cycle (fun j => (d j (e (cycle (j - 1))) : M (cycle j)))
+
+@[simp] theorem cycleShift_apply
+    {ι : Type uι} [Fintype ι]
+    {M : ι → Type uM} {N : ι → Type uN}
+    [∀ i, AddCommGroup (M i)] [∀ i, AddCommGroup (N i)] {t : ℕ} [NeZero t]
+    (cycle : ZMod t ↪ ι)
+    (U : ∀ j, AddSubgroup (M (cycle j)))
+    (d : ∀ j, N (cycle (j - 1)) → U j) (e : ∀ i, N i) (j : ZMod t) :
+    cycleShift cycle U d e (cycle j) = (d j (e (cycle (j - 1))) : M (cycle j)) :=
+  cycleExtend_apply cycle _ j
+
+theorem cycleShift_eq_zero
+    {ι : Type uι} [Fintype ι]
+    {M : ι → Type uM} {N : ι → Type uN}
+    [∀ i, AddCommGroup (M i)] [∀ i, AddCommGroup (N i)] {t : ℕ} [NeZero t]
+    (cycle : ZMod t ↪ ι)
+    (U : ∀ j, AddSubgroup (M (cycle j)))
+    (d : ∀ j, N (cycle (j - 1)) → U j) (e : ∀ i, N i)
+    {i : ι} (hi : ∀ j, cycle j ≠ i) : cycleShift cycle U d e i = 0 :=
+  cycleExtend_eq_zero cycle _ hi
+
+def cycleMap (cycle : ZMod t ↪ ι) (U : ∀ j, AddSubgroup (M (cycle j)))
+    (d : ∀ j, N (cycle (j - 1)) → U j)
+    (g : ∀ i, M i → N i) (k : ∀ i, M i) (e : ∀ i, N i) : ∀ i, N i :=
+  fun i => e i + g i (k i - cycleShift cycle U d e i)
+
+@[simp] theorem cycleMap_apply_cycle (cycle : ZMod t ↪ ι)
+    (U : ∀ j, AddSubgroup (M (cycle j)))
+    (d : ∀ j, N (cycle (j - 1)) → U j)
+    (g : ∀ i, M i → N i) (k : ∀ i, M i) (e : ∀ i, N i) (j : ZMod t) :
+    cycleMap cycle U d g k e (cycle j) =
+      e (cycle j) + g (cycle j) (k (cycle j) - d j (e (cycle (j - 1)))) := by
+  simp only [cycleMap, cycleShift_apply]
+
+theorem forall_of_cyclic_step {Q : ZMod t → Prop} (j₀ : ZMod t) (h₀ : Q j₀)
+    (hstep : ∀ j, Q j → Q (j + 1)) : ∀ j, Q j := by
+  have hnat : ∀ n : ℕ, Q (j₀ + (n : ZMod t)) := by
+    intro n
+    induction n with
+    | zero => simpa using h₀
+    | succ n ih => simpa only [Nat.cast_succ, add_assoc] using hstep _ ih
+  intro j
+  have h := hnat (j - j₀).val
+  simpa only [ZMod.natCast_zmod_val, add_sub_cancel] using h
+
+theorem cycleMap_injective_of_inactive (cycle : ZMod t ↪ ι)
+    (U : ∀ j, AddSubgroup (M (cycle j)))
+    (d : ∀ j, N (cycle (j - 1)) → U j)
+    (g : ∀ i, M i → N i) (k : ∀ i, M i)
+    (j₀ : ZMod t) (hinactive : ¬ DigitActive (g (cycle j₀)) (U j₀)) :
+    Function.Injective (cycleMap cycle U d g k) := by
+  classical
+  have hagree {w w' : M (cycle j₀)} (hw : w - w' ∈ U j₀) :
+      g (cycle j₀) w = g (cycle j₀) w' := by
+    by_contra hne
+    exact hinactive ⟨w, w', hw, hne⟩
+  have hconstant (e : ∀ i, N i) :
+      g (cycle j₀) (k (cycle j₀) - d j₀ (e (cycle (j₀ - 1)))) =
+        g (cycle j₀) (k (cycle j₀)) := by
+    apply hagree
+    have heq : (k (cycle j₀) - (d j₀ (e (cycle (j₀ - 1))) : M (cycle j₀))) -
+        k (cycle j₀) = -((d j₀ (e (cycle (j₀ - 1)))) : M (cycle j₀)) := by abel
+    rw [heq]
+    exact (U j₀).neg_mem (d j₀ (e (cycle (j₀ - 1)))).property
+  intro e e' he
+  have hstart : e (cycle j₀) = e' (cycle j₀) := by
+    have h := congrFun he (cycle j₀)
+    rw [cycleMap_apply_cycle, cycleMap_apply_cycle, hconstant, hconstant] at h
+    exact add_right_cancel h
+  have hstep (j : ZMod t) (hj : e (cycle j) = e' (cycle j)) :
+      e (cycle (j + 1)) = e' (cycle (j + 1)) := by
+    have h := congrFun he (cycle (j + 1))
+    simp only [cycleMap_apply_cycle] at h
+    have hprev : e (cycle (j + 1 - 1)) = e' (cycle (j + 1 - 1)) := by
+      exact Eq.mpr (congrArg (fun l : ZMod t => e (cycle l) = e' (cycle l))
+        (add_sub_cancel_right j 1)) hj
+    rw [hprev] at h
+    exact add_right_cancel h
+  have hon := forall_of_cyclic_step j₀ hstart hstep
+  funext i
+  by_cases hi : ∃ j, cycle j = i
+  · obtain ⟨j, rfl⟩ := hi
+    exact hon j
+  · have hoff : ∀ j, cycle j ≠ i := by simpa only [not_exists] using hi
+    have h := congrFun he i
+    simp only [cycleMap, cycleShift_eq_zero cycle U d e hoff,
+      cycleShift_eq_zero cycle U d e' hoff, sub_zero] at h
+    exact add_right_cancel h
+
+theorem cycleMap_bijective_of_inactive [∀ i, Fintype (N i)]
+    (cycle : ZMod t ↪ ι) (U : ∀ j, AddSubgroup (M (cycle j)))
+    (d : ∀ j, N (cycle (j - 1)) → U j)
+    (g : ∀ i, M i → N i) (k : ∀ i, M i)
+    (j₀ : ZMod t) (hinactive : ¬ DigitActive (g (cycle j₀)) (U j₀)) :
+    Function.Bijective (cycleMap cycle U d g k) :=
+  (cycleMap_injective_of_inactive cycle U d g k j₀ hinactive).bijective_of_finite
+
+theorem cycleMap_collision_of_all_active (cycle : ZMod t ↪ ι)
+    (U : ∀ j, AddSubgroup (M (cycle j))) (g : ∀ i, M i → N i)
+    (hactive : ∀ j, DigitActive (g (cycle j)) (U j)) :
+    ∃ (k : ∀ i, M i) (d : ∀ j, N (cycle (j - 1)) → U j)
+      (ε : ∀ i, N i), ε ≠ 0 ∧
+      cycleMap cycle U d g k 0 = cycleMap cycle U d g k ε := by
+  classical
+  choose w w' hmem hne using hactive
+  let ε₀ : ∀ j, N (cycle j) := fun j => g (cycle j) (w j) - g (cycle j) (w' j)
+  have hε₀ (j : ZMod t) : ε₀ j ≠ 0 := sub_ne_zero.mpr (hne j)
+  let ε : ∀ i, N i := cycleExtend cycle ε₀
+  let k : ∀ i, M i := cycleExtend cycle w
+  let d : ∀ j, N (cycle (j - 1)) → U j := fun j v =>
+    if v = ε₀ (j - 1) then ⟨w j - w' j, hmem j⟩ else 0
+  have hd₀ (j : ZMod t) : d j 0 = 0 := by
+    simp only [d, ite_eq_right (Ne.symm (hε₀ (j - 1)))]
+  have hdε (j : ZMod t) : d j (ε (cycle (j - 1))) = ⟨w j - w' j, hmem j⟩ := by
+    simp only [d, ε, cycleExtend_apply, ite_eq_left rfl]
+  refine ⟨k, d, ε, ?_, ?_⟩
+  · intro hz
+    have h := congrFun hz (cycle 0)
+    apply hε₀ 0
+    simpa only [ε, cycleExtend_apply, Pi.zero_apply] using h
+  · funext i
+    by_cases hi : ∃ j, cycle j = i
+    · obtain ⟨j, rfl⟩ := hi
+      calc
+        cycleMap cycle U d g k 0 (cycle j) = g (cycle j) (w j) := by
+          simp only [cycleMap_apply_cycle, Pi.zero_apply, hd₀, ZeroMemClass.coe_zero,
+            sub_zero, zero_add, k, cycleExtend_apply]
+        _ = ε₀ j + g (cycle j) (w' j) := (sub_add_cancel _ _).symm
+        _ = cycleMap cycle U d g k ε (cycle j) := by
+          rw [cycleMap_apply_cycle, hdε]
+          simp only [k, ε, cycleExtend_apply, sub_sub_cancel]
+    · have hoff : ∀ j, cycle j ≠ i := by simpa only [not_exists] using hi
+      simp only [cycleMap, Pi.zero_apply, ε, cycleExtend_eq_zero cycle ε₀ hoff,
+        cycleShift_eq_zero cycle U d 0 hoff,
+        cycleShift_eq_zero cycle U d (cycleExtend cycle ε₀) hoff]
+
+theorem forall_cycleMap_bijective_iff [∀ i, Fintype (N i)]
+    (cycle : ZMod t ↪ ι) (U : ∀ j, AddSubgroup (M (cycle j)))
+    (g : ∀ i, M i → N i) :
+    (∀ (d : ∀ j, N (cycle (j - 1)) → U j) (k : ∀ i, M i),
+      Function.Bijective (cycleMap cycle U d g k)) ↔
+      ∃ j, ¬ DigitActive (g (cycle j)) (U j) := by
+  constructor
+  · intro hall
+    by_contra hnone
+    have hactive : ∀ j, DigitActive (g (cycle j)) (U j) := by
+      simpa only [not_exists, not_not] using hnone
+    obtain ⟨k, d, ε, hε, hcollision⟩ :=
+      cycleMap_collision_of_all_active cycle U g hactive
+    exact hε ((hall d k).1 hcollision).symm
+  · rintro ⟨j, hj⟩ d k
+    exact cycleMap_bijective_of_inactive cycle U d g k j hj
+
+end CycleCombinatorics
+
+theorem existsUnique_fixed_comp_iff {A : Type uA} {B : Type uB} (f : A → B) (g : B → A) :
+    (∃! a, g (f a) = a) ↔ ∃! b, f (g b) = b := by
+  constructor
+  · rintro ⟨a, ha, hu⟩
+    refine ⟨f a, ?_, ?_⟩
+    · exact congrArg f ha
+    · intro b hb
+      have hg : g b = a := hu _ (congrArg g hb)
+      calc
+        b = f (g b) := hb.symm
+        _ = f a := congrArg f hg
+  · rintro ⟨b, hb, hu⟩
+    refine ⟨g b, ?_, ?_⟩
+    · exact congrArg g hb
+    · intro a ha
+      have hf : f a = b := hu _ (congrArg f ha)
+      calc
+        a = g (f a) := ha.symm
+        _ = g b := congrArg g hf
+
+theorem tiles_offset_iff_permutation
+    {H : Type uH} {B : Type uB} {C : Type uC} [AddCommGroup H] [AddCommGroup B] [AddCommGroup C]
+    (q : H →+ B) (ψ : H →+ C) (s : B → H) (hs : ∀ b, q (s b) = b)
+    (hjoint : Function.Surjective (fun y => (q y, ψ y)))
+    (Δ : C → B) (F : Finset H)
+    (hF : ∀ y, y ∈ F ↔ q y = Δ (ψ y)) :
+    Tiles F (Set.range s) ↔
+      ∀ b, Function.Bijective (fun e => e + ψ (s (b - Δ e))) := by
+  have hinj : Function.Injective s := by
+    intro b b' h
+    simpa only [hs] using congrArg q h
+  have hmem (y : H) (b : B) : y - s b ∈ F ↔
+      q y - Δ (ψ y - ψ (s b)) = b := by
+    rw [hF, map_sub, hs, map_sub]
+    simp only [sub_eq_iff_eq_add]
+    exact ⟨fun h => h.trans (add_comm _ _), fun h => h.trans (add_comm _ _)⟩
+  have hfibre (y : H) : (∃! b, y - s b ∈ F) ↔
+      ∃! e, e + ψ (s (q y - Δ e)) = ψ y := by
+    calc
+      _ ↔ ∃! b, q y - Δ (ψ y - ψ (s b)) = b := existsUnique_congr (hmem y)
+      _ ↔ ∃! e, ψ y - ψ (s (q y - Δ e)) = e :=
+        existsUnique_fixed_comp_iff (fun b => ψ y - ψ (s b)) (fun e => q y - Δ e)
+      _ ↔ _ := existsUnique_congr (fun e => by rw [sub_eq_iff_eq_add]; exact eq_comm)
+  rw [tiles_range_iff_unique_base s hinj F]
+  constructor
+  · intro h b
+    apply (Function.bijective_iff_existsUnique _).mpr
+    intro c
+    obtain ⟨y, hy⟩ := hjoint (b, c)
+    have hq : q y = b := congrArg Prod.fst hy
+    have hψ : ψ y = c := congrArg Prod.snd hy
+    simpa only [hq, hψ] using (hfibre y).mp (h y)
+  · intro h y
+    exact (hfibre y).mpr ((h (q y)).existsUnique (ψ y))
+
+section ConcreteTiles
+
+variable {p : ℕ} [NeZero p] (E : EncodingParameters p)
+
+def usefulProjection : Ambient E →+ Useful E where
+  toFun v := fun i => (v.2.2 i).2
+  map_zero' := rfl
+  map_add' _ _ := rfl
+
+@[simp] theorem usefulProjection_apply (v : Ambient E) (i : Channel p) :
+    usefulProjection E v i = (v.2.2 i).2 := rfl
+
+@[simp] theorem usefulProjection_point (o : GraphOutputs E) (b : Base E) :
+    usefulProjection E (point o b) = o.c b := rfl
+
+theorem q0_usefulProjection_surjective :
+    Function.Surjective (fun y : Ambient E => (q0 E y, usefulProjection E y)) := by
+  rintro ⟨b, c⟩
+  let o : GraphOutputs E := ⟨fun _ => c, fun _ => 0, fun _ => 0⟩
+  refine ⟨point o b, ?_⟩
+  exact Prod.ext (q0_point o b) rfl
+
+private theorem offset_horizontal_zero (D : Useful E → Low E)
+    (v : {v : Ambient E // q0 E v = (0, D (usefulProjection E v))}) :
+    (v : Ambient E).1 = 0 := congrArg Prod.fst v.property
+
+noncomputable instance offsetFintype (D : Useful E → Low E) :
+    Fintype {v : Ambient E // q0 E v = (0, D (usefulProjection E v))} :=
+  Fintype.ofInjective (fun v => (v : Ambient E).2) (by
+    intro v w h
+    apply Subtype.ext
+    exact Prod.ext ((offset_horizontal_zero E D v).trans
+      (offset_horizontal_zero E D w).symm) h)
+
+def offsetTile (D : Useful E → Low E) : Finset (Ambient E) :=
+  (Finset.univ : Finset {v : Ambient E // q0 E v = (0, D (usefulProjection E v))}).map
+    ⟨Subtype.val, Subtype.val_injective⟩
+
+@[simp] theorem mem_offsetTile (D : Useful E → Low E) (v : Ambient E) :
+    v ∈ offsetTile E D ↔ q0 E v = (0, D (usefulProjection E v)) := by
+  constructor
+  · intro h
+    obtain ⟨w, _, rfl⟩ := Finset.mem_map.mp h
+    exact w.property
+  · intro h
+    exact Finset.mem_map.mpr ⟨⟨v, h⟩, Finset.mem_univ _, rfl⟩
+
+theorem offsetTile_nonempty (D : Useful E → Low E) : (offsetTile E D).Nonempty := by
+  let o : GraphOutputs E := ⟨fun _ => 0, fun _ => 0, fun _ => 0⟩
+  refine ⟨point o (0, D 0), (mem_offsetTile E D _).mpr ?_⟩
+  exact q0_point o (0, D 0)
+
+theorem offsetTile_iff (D : Useful E → Low E) (o : GraphOutputs E) :
+    Tiles (offsetTile E D) (graph o) ↔
+      ∀ x k, Function.Bijective (fun e : Useful E => e + o.c (x, k - D e)) := by
+  have h := tiles_offset_iff_permutation (q0 E) (usefulProjection E) (point o)
+    (q0_point o) (q0_usefulProjection_surjective E) (fun e => (0, D e))
+    (offsetTile E D) (mem_offsetTile E D)
+  change Tiles (offsetTile E D) (graph o) ↔ _ at h
+  constructor
+  · intro htile x k
+    have hb := h.mp htile (x, k)
+    change Function.Bijective (fun e : Useful E => e + o.c (x - 0, k - D e)) at hb
+    simpa only [sub_zero] using hb
+  · intro hmap
+    apply h.mpr
+    rintro ⟨x, k⟩
+    change Function.Bijective (fun e : Useful E => e + o.c (x - 0, k - D e))
+    simpa only [sub_zero] using hmap x k
+
+variable {t : ℕ} [NeZero t]
+
+def cycleTile (cycle : ZMod t ↪ Channel p)
+    (labels : ∀ j, Label p (cycle j))
+    (d : ∀ j, P E (cycle (j - 1)) → digitSubgroup E (cycle j) (labels j)) :
+    Finset (Ambient E) :=
+  offsetTile E (cycleShift cycle (fun j => digitSubgroup E (cycle j) (labels j)) d)
+
+theorem cycleTile_nonempty (cycle : ZMod t ↪ Channel p)
+    (labels : ∀ j, Label p (cycle j))
+    (d : ∀ j, P E (cycle (j - 1)) → digitSubgroup E (cycle j) (labels j)) :
+    (cycleTile E cycle labels d).Nonempty := offsetTile_nonempty E _
+
+theorem cycleTile_iff (cycle : ZMod t ↪ Channel p)
+    (labels : ∀ j, Label p (cycle j))
+    (d : ∀ j, P E (cycle (j - 1)) → digitSubgroup E (cycle j) (labels j))
+    (o : GraphOutputs E) (g : ∀ i, Plane → K E i → P E i)
+    (hfactor : ∀ x k i, o.c (x, k) i = g i x (k i)) :
+    Tiles (cycleTile E cycle labels d) (graph o) ↔
+      ∀ x k, Function.Bijective
+        (cycleMap cycle (fun j => digitSubgroup E (cycle j) (labels j)) d
+          (fun i => g i x) k) := by
+  let U := fun j => digitSubgroup E (cycle j) (labels j)
+  have hm (x : Plane) (k : Low E) :
+      (fun e : Useful E => e + o.c (x, k - cycleShift cycle U d e)) =
+        cycleMap cycle U d (fun i => g i x) k := by
+    funext e i
+    change e i + o.c (x, k - cycleShift cycle U d e) i =
+      e i + g i x (k i - cycleShift cycle U d e i)
+    exact congrArg (fun v => e i + v) (hfactor x (k - cycleShift cycle U d e) i)
+  simpa only [cycleTile, hm] using offsetTile_iff E (cycleShift cycle U d) o
+
+theorem all_cycleTile_iff (cycle : ZMod t ↪ Channel p)
+    (labels : ∀ j, Label p (cycle j))
+    (o : GraphOutputs E) (g : ∀ i, Plane → K E i → P E i)
+    (hfactor : ∀ x k i, o.c (x, k) i = g i x (k i)) :
+    (∀ d : ∀ j, P E (cycle (j - 1)) → digitSubgroup E (cycle j) (labels j),
+      Tiles (cycleTile E cycle labels d) (graph o)) ↔
+      ∀ x, ∃ j, ¬ DigitActive (g (cycle j) x) (digitSubgroup E (cycle j) (labels j)) := by
+  constructor
+  · intro h x
+    apply (forall_cycleMap_bijective_iff cycle
+      (fun j => digitSubgroup E (cycle j) (labels j)) (fun i => g i x)).mp
+    intro d k
+    exact (cycleTile_iff E cycle labels d o g hfactor).mp (h d) x k
+  · intro h d
+    apply (cycleTile_iff E cycle labels d o g hfactor).mpr
+    intro x k
+    obtain ⟨j, hj⟩ := h x
+    exact cycleMap_bijective_of_inactive cycle
+      (fun j => digitSubgroup E (cycle j) (labels j)) d (fun i => g i x) k j hj
+
+def cycleTiles (cycle : ZMod t ↪ Channel p) (labels : ∀ j, Label p (cycle j)) :
+    Finset (Finset (Ambient E)) := by
+  classical
+  exact Finset.univ.image (cycleTile E cycle labels)
+
+theorem cycleTiles_nonempty (cycle : ZMod t ↪ Channel p)
+    (labels : ∀ j, Label p (cycle j)) : (cycleTiles E cycle labels).Nonempty := by
+  classical
+  exact ⟨cycleTile E cycle labels (fun _ _ => 0),
+    Finset.mem_image.mpr ⟨(fun _ _ => 0), Finset.mem_univ _, rfl⟩⟩
+
+theorem nonempty_of_mem_cycleTiles (cycle : ZMod t ↪ Channel p)
+    (labels : ∀ j, Label p (cycle j)) {F : Finset (Ambient E)}
+    (hF : F ∈ cycleTiles E cycle labels) : F.Nonempty := by
+  classical
+  obtain ⟨d, _, rfl⟩ := Finset.mem_image.mp hF
+  exact cycleTile_nonempty E cycle labels d
+
+theorem forall_mem_cycleTiles_iff (cycle : ZMod t ↪ Channel p)
+    (labels : ∀ j, Label p (cycle j)) (A : Set (Ambient E)) :
+    (∀ F ∈ cycleTiles E cycle labels, Tiles F A) ↔
+      ∀ d : ∀ j, P E (cycle (j - 1)) → digitSubgroup E (cycle j) (labels j),
+        Tiles (cycleTile E cycle labels d) A := by
+  classical
+  simp only [cycleTiles, Finset.mem_image, Finset.mem_univ, true_and]
+  constructor
+  · intro h d
+    exact h _ ⟨d, rfl⟩
+  · intro h F hF
+    obtain ⟨d, rfl⟩ := hF
+    exact h d
+
+theorem cycle_exclusion_iff (cycle : ZMod t ↪ Channel p)
+    (labels : ∀ j, Label p (cycle j)) (o : GraphOutputs E)
+    (hdep : HasDependence E o) :
+    (∀ F ∈ cycleTiles E cycle labels, Tiles F (graph o)) ↔
+      ∀ x, ∃ j, ¬ Active E o (cycle j) x (labels j) := by
+  rw [forall_mem_cycleTiles_iff]
+  exact all_cycleTile_iff E cycle labels o (usefulAt E o) (usefulAt_factor E hdep)
+
+end ConcreteTiles
+
+end PeriodicTilingThree
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/CyclicCRT.lean b/lean/OAI/Geometry/PeriodicTiling/CyclicCRT.lean
new file mode 100644
index 0000000000000000000000000000000000000000..ecc098ec9e9b773fcdb0ba42a5d38d8012f38eb0
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/CyclicCRT.lean
@@ -0,0 +1,78 @@
+import OAI.Geometry.PeriodicTiling.EncodingParameters
+import Mathlib.Data.ZMod.QuotientRing
+import Mathlib.Algebra.Group.Pi.Basic
+
+namespace OAI
+
+noncomputable section
+
+namespace PeriodicTilingThree
+
+open scoped BigOperators
+
+variable {p : ℕ} [NeZero p] (E : EncodingParameters p)
+
+def digitEquivRing (i : Channel p) :
+    ZMod (E.r i) ≃+* (∀ j : Label p i, ZMod (E.digitPrime i j)) :=
+  ZMod.prodEquivPi (E.digitPrime i) (E.digitPrime_pairwise i)
+
+def digitEquiv (i : Channel p) :
+    ZMod (E.r i) ≃+ (∀ j : Label p i, ZMod (E.digitPrime i j)) :=
+  (digitEquivRing E i).toAddEquiv
+
+theorem digitPrime_dvd_r (i : Channel p) (j : Label p i) :
+    E.digitPrime i j ∣ E.r i :=
+  Finset.dvd_prod_of_mem (E.digitPrime i) (Finset.mem_univ j)
+
+@[simp] theorem digitEquiv_intCast (i : Channel p) (z : ℤ) (j : Label p i) :
+    digitEquiv E i (z : ZMod (E.r i)) j = (z : ZMod (E.digitPrime i j)) := by
+  exact congrArg (fun f : ∀ j : Label p i, ZMod (E.digitPrime i j) => f j)
+    (map_intCast (digitEquivRing E i) z)
+
+theorem digitEquiv_apply (i : Channel p) (x : ZMod (E.r i)) (j : Label p i) :
+    digitEquiv E i x j = (ZMod.cast x : ZMod (E.digitPrime i j)) := by
+  obtain ⟨z, rfl⟩ := ZMod.intCast_surjective x
+  rw [digitEquiv_intCast, ZMod.cast_intCast (digitPrime_dvd_r E i j)]
+
+def digitSubgroup (i : Channel p) (j : Label p i) : AddSubgroup (ZMod (E.r i)) where
+  carrier := {x | ∀ k : Label p i, k ≠ j → digitEquiv E i x k = 0}
+  zero_mem' := by
+    intro k _hk
+    simp
+  add_mem' := by
+    intro x y hx hy k hk
+    simp [map_add, hx k hk, hy k hk]
+  neg_mem' := by
+    intro x hx k hk
+    simp [map_neg, hx k hk]
+
+@[simp] theorem mem_digitSubgroup (i : Channel p) (j : Label p i) (x : ZMod (E.r i)) :
+    x ∈ digitSubgroup E i j ↔
+      ∀ k : Label p i, k ≠ j → digitEquiv E i x k = 0 := Iff.rfl
+
+theorem sub_mem_digitSubgroup_iff (i : Channel p) (j : Label p i)
+    (x y : ZMod (E.r i)) :
+    x - y ∈ digitSubgroup E i j ↔
+      ∀ k : Label p i, k ≠ j → digitEquiv E i x k = digitEquiv E i y k := by
+  simp [mem_digitSubgroup, map_sub, sub_eq_zero]
+
+def digitSingle (i : Channel p) (j : Label p i) (x : ZMod (E.digitPrime i j)) :
+    ZMod (E.r i) :=
+  (digitEquiv E i).symm (Pi.single j x)
+
+@[simp] theorem digitEquiv_digitSingle (i : Channel p) (j : Label p i)
+    (x : ZMod (E.digitPrime i j)) :
+    digitEquiv E i (digitSingle E i j x) = Pi.single j x :=
+  (digitEquiv E i).apply_symm_apply _
+
+theorem digitSingle_mem (i : Channel p) (j : Label p i)
+    (x : ZMod (E.digitPrime i j)) : digitSingle E i j x ∈ digitSubgroup E i j := by
+  intro k hk
+  rw [digitEquiv_digitSingle]
+  exact Pi.single_eq_of_ne (M := fun j : Label p i => ZMod (E.digitPrime i j)) hk x
+
+end PeriodicTilingThree
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/CyclicCoordinates.lean b/lean/OAI/Geometry/PeriodicTiling/CyclicCoordinates.lean
new file mode 100644
index 0000000000000000000000000000000000000000..24c2daa53ae082fa2b7a7cab0ddcb3e00d15c767
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/CyclicCoordinates.lean
@@ -0,0 +1,135 @@
+import OAI.Geometry.PeriodicTiling.CyclicTypes
+import OAI.Geometry.PeriodicTiling.CyclicShear
+
+namespace OAI
+
+namespace PeriodicTilingThree
+
+noncomputable section
+
+variable {p : ℕ} [NeZero p] (E : EncodingParameters p)
+
+abbrev low (i : Channel p) : ZMod ((E.r i) ^ 2) →+ K E i :=
+  CyclicShear.low (E.r i)
+
+abbrev highMul (i : Channel p) : K E i →+ ZMod ((E.r i) ^ 2) :=
+  CyclicShear.highMul (E.r i)
+
+abbrev shear (i : Channel p) (x : ℤ) (k b : K E i) : ZMod ((E.r i) ^ 2) :=
+  CyclicShear.shear (E.r i) x k b
+
+abbrev shearEquiv (i : Channel p) (x : ℤ) (k : K E i) :
+    K E i ≃ {v : ZMod ((E.r i) ^ 2) // low E i v = k} :=
+  CyclicShear.shearEquiv (E.r i) x k
+
+@[simp] theorem low_shear (i : Channel p) (x : ℤ) (k b : K E i) :
+    low E i (shear E i x k b) = k := CyclicShear.low_shear _ _ _ _
+
+theorem highMul_injective (i : Channel p) : Function.Injective (highMul E i) :=
+  CyclicShear.highMul_injective _
+
+@[simp] theorem low_highMul (i : Channel p) (b : K E i) : low E i (highMul E i b) = 0 :=
+  CyclicShear.low_highMul _ _
+
+theorem shear_injective (i : Channel p) (x : ℤ) (k : K E i) :
+    Function.Injective (shear E i x k) := CyclicShear.shear_injective _ _ _
+
+theorem shear_add (i : Channel p) (x h : ℤ) (k b : K E i) :
+    shear E i (x + h) (k + (h : K E i)) b =
+      shear E i x k b + (h : ZMod ((E.r i) ^ 2)) :=
+  CyclicShear.shear_add _ _ _ _ _
+
+theorem shear_source_add (i : Channel p) (x h : ℤ) (k l b : K E i)
+    (hl : (h : K E i) = l) :
+    shear E i (x - h) (k - l) b + (h : ZMod ((E.r i) ^ 2)) = shear E i x k b :=
+  CyclicShear.shear_source_add _ _ _ _ _ _ hl
+
+def q0 : Ambient E →+ Base E where
+  toFun g := (g.1, fun i => low E i (g.2.2 i).1)
+  map_zero' := by
+    refine Prod.ext ?_ ?_
+    · rfl
+    · funext i
+      exact map_zero (low E i)
+  map_add' g h := by
+    refine Prod.ext ?_ ?_
+    · rfl
+    · funext i
+      exact map_add (low E i) (g.2.2 i).1 (h.2.2 i).1
+
+@[simp] theorem q0_apply (g : Ambient E) :
+    q0 E g = (g.1, fun i => low E i (g.2.2 i).1) := rfl
+
+variable {E}
+
+def point (o : GraphOutputs E) (b : Base E) : Ambient E :=
+  (b.1, o.z b, fun i => (shear E i b.1.1 (b.2 i) (o.beta b i), o.c b i))
+
+@[simp] theorem point_horizontal (o : GraphOutputs E) (b : Base E) : (point o b).1 = b.1 := rfl
+@[simp] theorem point_z (o : GraphOutputs E) (b : Base E) : (point o b).2.1 = o.z b := rfl
+@[simp] theorem point_c (o : GraphOutputs E) (b : Base E) (i : Channel p) :
+    ((point o b).2.2 i).2 = o.c b i := rfl
+@[simp] theorem point_v (o : GraphOutputs E) (b : Base E) (i : Channel p) :
+    ((point o b).2.2 i).1 = shear E i b.1.1 (b.2 i) (o.beta b i) := rfl
+
+@[simp] theorem q0_point (o : GraphOutputs E) (b : Base E) : q0 E (point o b) = b := by
+  refine Prod.ext ?_ ?_
+  · rfl
+  · funext i
+    exact low_shear E i b.1.1 (b.2 i) (o.beta b i)
+
+theorem point_injective (o : GraphOutputs E) : Function.Injective (point o) := by
+  intro b d h
+  have hq := congrArg (q0 E) h
+  simpa only [q0_point] using hq
+
+def graph (o : GraphOutputs E) : Set (Ambient E) := Set.range (point o)
+
+@[simp] theorem point_mem_graph (o : GraphOutputs E) (b : Base E) : point o b ∈ graph o :=
+  ⟨b, rfl⟩
+
+theorem eq_point_of_mem_graph (o : GraphOutputs E) {g : Ambient E} (hg : g ∈ graph o) :
+    g = point o (q0 E g) := by
+  obtain ⟨b, rfl⟩ := hg
+  rw [q0_point]
+
+def zeroOutputs (E : EncodingParameters p) : GraphOutputs E :=
+  ⟨fun _ _ => 0, fun _ _ => 0, fun _ => 0⟩
+
+theorem q0_surjective (E : EncodingParameters p) : Function.Surjective (q0 E) := by
+  intro b
+  exact ⟨point (zeroOutputs E) b, q0_point _ _⟩
+
+private theorem section_low (s : Base E → Ambient E) (hs : ∀ b, q0 E (s b) = b)
+    (b : Base E) (i : Channel p) : low E i ((s b).2.2 i).1 = b.2 i :=
+  congrArg (fun t : Base E => t.2 i) (hs b)
+
+def outputsOfSection (s : Base E → Ambient E) (hs : ∀ b, q0 E (s b) = b) : GraphOutputs E where
+  c b i := ((s b).2.2 i).2
+  beta b i := (shearEquiv E i b.1.1 (b.2 i)).symm
+    ⟨((s b).2.2 i).1, section_low s hs b i⟩
+  z b := (s b).2.1
+
+@[simp] theorem point_outputsOfSection (s : Base E → Ambient E)
+    (hs : ∀ b, q0 E (s b) = b) (b : Base E) :
+    point (outputsOfSection s hs) b = s b := by
+  refine Prod.ext ?_ ?_
+  · exact (congrArg (fun t : Base E => t.1) (hs b)).symm
+  · refine Prod.ext ?_ ?_
+    · rfl
+    · funext i
+      refine Prod.ext ?_ ?_
+      · exact congrArg Subtype.val ((shearEquiv E i b.1.1 (b.2 i)).apply_symm_apply
+          ⟨((s b).2.2 i).1, section_low s hs b i⟩)
+      · rfl
+
+theorem graph_outputsOfSection (s : Base E → Ambient E) (hs : ∀ b, q0 E (s b) = b) :
+    graph (outputsOfSection s hs) = Set.range s := by
+  exact congrArg (fun f : Base E → Ambient E => Set.range f)
+    (funext (point_outputsOfSection s hs))
+
+end
+
+end PeriodicTilingThree
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/CyclicCounterexample.lean b/lean/OAI/Geometry/PeriodicTiling/CyclicCounterexample.lean
new file mode 100644
index 0000000000000000000000000000000000000000..3a4a84c41759d6d481900031f563d7cae5d097d5
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/CyclicCounterexample.lean
@@ -0,0 +1,35 @@
+import OAI.Geometry.PeriodicTiling.CommonSolution
+import OAI.Geometry.PeriodicTiling.EncodedSystemObstruction
+import OAI.Geometry.PeriodicTiling.EncodingPrimeSelection
+import OAI.Geometry.PeriodicTiling.CyclicFactorCRT
+import OAI.Geometry.PeriodicTiling.CyclicSystemStacking
+
+namespace OAI
+
+namespace PeriodicTilingThree
+
+theorem cyclic_counterexample_of_parameters {p : ℕ} [NeZero p]
+    (E : EncodingParameters p) :
+    ∃ Q : ℕ, 0 < Q ∧
+      ∃ F : Finset (Plane × ZMod Q), F.Nonempty ∧
+        (∃ A, Tiles F A) ∧ ∀ A, Tiles F A → ¬ FullyPeriodic A := by
+  classical
+  let : IsAddCyclic (V E) := isAddCyclic_of_surjective
+    (finiteFactorEquiv E).symm (finiteFactorEquiv E).symm.surjective
+  apply exists_cyclic_stack_of_system (V E) (testTile E)
+  · exact ⟨CommonModel.commonGraph E, CommonModel.commonGraph_tiles_testTile E⟩
+  · intro A hA
+    exact testTile_not_fully_periodic E hA
+
+theorem exists_cyclic_counterexample :
+    ∃ Q : ℕ, 0 < Q ∧
+      ∃ F : Finset (Plane × ZMod Q), F.Nonempty ∧
+        (∃ A, Tiles F A) ∧ ∀ A, Tiles F A → ¬ FullyPeriodic A := by
+  obtain ⟨p, hpbound, hp⟩ := Nat.exists_infinite_primes 201
+  let : NeZero p := ⟨hp.ne_zero⟩
+  let E : EncodingParameters p := encodingParameters hp (by omega)
+  exact cyclic_counterexample_of_parameters E
+
+end PeriodicTilingThree
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/CyclicFactorCRT.lean b/lean/OAI/Geometry/PeriodicTiling/CyclicFactorCRT.lean
new file mode 100644
index 0000000000000000000000000000000000000000..901912b7f894118a3d6c24334c24b01256038d2c
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/CyclicFactorCRT.lean
@@ -0,0 +1,111 @@
+import OAI.Geometry.PeriodicTiling.CyclicTypes
+import Mathlib.Data.ZMod.QuotientRing
+import Mathlib.Algebra.Group.Prod
+import Mathlib.Algebra.Group.Equiv.Basic
+
+namespace OAI
+
+noncomputable section
+
+namespace PeriodicTilingThree
+
+open scoped BigOperators
+
+variable {p : ℕ} [NeZero p]
+
+namespace EncodingParameters
+
+variable (E : EncodingParameters p)
+
+def blockModulus (i : Channel p) : ℕ := (E.r i) ^ 2 * E.a i * E.b i
+def cyclicOrder : ℕ := p ^ 4 * ∏ i : Channel p, E.blockModulus i
+
+theorem blockModulus_pos (i : Channel p) : 0 < E.blockModulus i :=
+  Nat.mul_pos (Nat.mul_pos (pow_pos (E.r_pos i) 2) (E.a_pos i)) (E.b_pos i)
+
+theorem cyclicOrder_pos : 0 < E.cyclicOrder :=
+  Nat.mul_pos (pow_pos E.p_prime.pos 4)
+    (Finset.prod_pos (fun i _ => E.blockModulus_pos i))
+
+instance blockModulus_neZero (i : Channel p) : NeZero (E.blockModulus i) :=
+  ⟨(E.blockModulus_pos i).ne'⟩
+instance cyclicOrder_neZero : NeZero E.cyclicOrder := ⟨E.cyclicOrder_pos.ne'⟩
+
+theorem a_coprime_b_all
+    {p : ℕ} [NeZero p] (E : EncodingParameters p)
+    (i j : Channel p) : Nat.Coprime (E.a i) (E.b j) := by
+  apply E.prime_coprime
+  simp
+
+theorem a_pairwise_coprime
+    {p : ℕ} [NeZero p] (E : EncodingParameters p)
+    : Pairwise (fun i j : Channel p => Nat.Coprime (E.a i) (E.a j)) := by
+  intro i j hij
+  apply E.prime_coprime
+  simpa using hij
+
+theorem b_pairwise_coprime
+    {p : ℕ} [NeZero p] (E : EncodingParameters p)
+    : Pairwise (fun i j : Channel p => Nat.Coprime (E.b i) (E.b j)) := by
+  intro i j hij
+  apply E.prime_coprime
+  simpa using hij
+
+theorem blockModulus_pairwise :
+    Pairwise (fun i j : Channel p => Nat.Coprime (E.blockModulus i) (E.blockModulus j)) := by
+  intro i j hij
+  have hrr := (E.r_pairwise_coprime hij).pow 2 2
+  have hra := (E.r_coprime_a i j).pow_left 2
+  have hrb := (E.r_coprime_b i j).pow_left 2
+  have har := (E.r_coprime_a j i).symm.pow_right 2
+  have haa := E.a_pairwise_coprime hij
+  have hab := E.a_coprime_b_all i j
+  have hbr := (E.r_coprime_b j i).symm.pow_right 2
+  have hba := (E.a_coprime_b_all j i).symm
+  have hbb := E.b_pairwise_coprime hij
+  simp only [blockModulus, Nat.coprime_mul_iff_left, Nat.coprime_mul_iff_right]
+  exact ⟨⟨⟨⟨hrr, har⟩, hbr⟩, ⟨⟨hra, haa⟩, hba⟩⟩, ⟨⟨hrb, hab⟩, hbb⟩⟩
+
+theorem p_coprime_blockModulus (i : Channel p) :
+    Nat.Coprime p (E.blockModulus i) := by
+  have hr := (E.r_coprime_p i).symm.pow_right 2
+  have ha : Nat.Coprime p (E.a i) := (E.prime_coprime_p (.inl i)).symm
+  have hb : Nat.Coprime p (E.b i) := (E.prime_coprime_p (.inr (.inl i))).symm
+  simp only [blockModulus, Nat.coprime_mul_iff_right]
+  exact ⟨⟨hr, ha⟩, hb⟩
+
+theorem p4_coprime_blocks :
+    Nat.Coprime (p ^ 4) (∏ i : Channel p, E.blockModulus i) := by
+  apply Nat.coprime_fintype_prod_right_iff.mpr
+  intro i
+  exact (E.p_coprime_blockModulus i).pow_left 4
+
+end EncodingParameters
+
+variable (E : EncodingParameters p)
+
+def channelCRT (i : Channel p) :
+    ZMod (E.blockModulus i) ≃+ (ZMod ((E.r i) ^ 2) × P E i) := by
+  have hRA : Nat.Coprime ((E.r i) ^ 2) (E.a i) := (E.r_coprime_a i i).pow_left 2
+  have hRAB : Nat.Coprime ((E.r i) ^ 2 * E.a i) (E.b i) :=
+    Nat.coprime_mul_iff_left.mpr ⟨(E.r_coprime_b i i).pow_left 2, E.a_coprime_b i⟩
+  exact ((ZMod.chineseRemainder hRAB).toAddEquiv.trans
+    (AddEquiv.prodCongr (ZMod.chineseRemainder hRA).toAddEquiv
+      (AddEquiv.refl (ZMod (E.b i))))).trans AddEquiv.prodAssoc
+
+def cyclicToFiniteFactor : ZMod E.cyclicOrder ≃+ V E :=
+  (ZMod.chineseRemainder E.p4_coprime_blocks).toAddEquiv.trans
+    (AddEquiv.prodCongr (AddEquiv.refl (ZMod (p ^ 4)))
+      ((ZMod.prodEquivPi E.blockModulus E.blockModulus_pairwise).toAddEquiv.trans
+        (AddEquiv.piCongrRight (fun i => channelCRT E i))))
+
+def finiteFactorEquiv : V E ≃+ ZMod E.cyclicOrder := (cyclicToFiniteFactor E).symm
+
+def ambientCyclicEquiv : Ambient E ≃+ (Plane × ZMod E.cyclicOrder) :=
+  AddEquiv.prodCongr (AddEquiv.refl Plane) (finiteFactorEquiv E)
+
+end PeriodicTilingThree
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/CyclicQuotient.lean b/lean/OAI/Geometry/PeriodicTiling/CyclicQuotient.lean
new file mode 100644
index 0000000000000000000000000000000000000000..c4dbcb4efced01181d9d8d4652b9e0a142b8f71e
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/CyclicQuotient.lean
@@ -0,0 +1,156 @@
+import OAI.Geometry.PeriodicTiling.TilingBasic
+import Mathlib.Data.ZMod.Basic
+import Mathlib.LinearAlgebra.Matrix.Notation
+import Mathlib.Tactic.Abel
+import Mathlib.Tactic.NormNum
+
+namespace OAI
+
+namespace PeriodicTilingThree
+namespace CyclicQuotient
+
+abbrev Group (Q : ℕ) := Plane × ZMod Q
+
+def projection (Q : ℕ) : Lattice 3 →+ Group Q where
+  toFun z := ((z 0, z 1), (z 2 : ZMod Q))
+  map_zero' := by simp
+  map_add' x y := by simp
+
+def kernelStep (Q : ℕ) : Lattice 3 := ![0, 0, (Q : ℤ)]
+
+@[simp] theorem projection_kernelStep (Q : ℕ) :
+    projection Q (kernelStep Q) = 0 := by
+  simp [projection, kernelStep]
+
+theorem kernelStep_ne_zero (Q : ℕ) [NeZero Q] : kernelStep Q ≠ 0 := by
+  intro h
+  have hz : (Q : ℤ) = 0 := by
+    simpa [kernelStep] using congrFun h 2
+  exact (NeZero.ne Q) (Int.ofNat_inj.mp hz)
+
+def lift (Q : ℕ) (g : Group Q) : Lattice 3 :=
+  ![g.1.1, g.1.2, (g.2.val : ℤ)]
+
+@[simp] theorem projection_lift (Q : ℕ) [NeZero Q] (g : Group Q) :
+    projection Q (lift Q g) = g := by
+  rcases g with ⟨⟨x, y⟩, r⟩
+  simp [projection, lift]
+
+@[simp] theorem lift_zero (Q : ℕ) : lift Q 0 = 0 := by
+  ext i
+  fin_cases i <;> simp [lift]
+
+theorem projection_surjective (Q : ℕ) [NeZero Q] :
+    Function.Surjective (projection Q) :=
+  fun g => ⟨lift Q g, projection_lift Q g⟩
+
+theorem lift_injective (Q : ℕ) [NeZero Q] : Function.Injective (lift Q) := by
+  intro g h hgh
+  have he := congrArg (projection Q) hgh
+  simpa only [projection_lift] using he
+
+theorem mem_ker_iff_zsmul_kernelStep (Q : ℕ) (z : Lattice 3) :
+    z ∈ (projection Q).ker ↔ ∃ k : ℤ, z = k • kernelStep Q := by
+  constructor
+  · intro hz
+    have hp : projection Q z = 0 := hz
+    have h0 : z 0 = 0 := congrArg (fun g : Group Q => g.1.1) hp
+    have h1 : z 1 = 0 := congrArg (fun g : Group Q => g.1.2) hp
+    have h2 : (z 2 : ZMod Q) = 0 := congrArg (fun g : Group Q => g.2) hp
+    obtain ⟨k, hk⟩ := (ZMod.intCast_zmod_eq_zero_iff_dvd (z 2) Q).mp h2
+    refine ⟨k, ?_⟩
+    ext i
+    fin_cases i <;> simp [kernelStep, h0, h1, hk, mul_comm]
+  · rintro ⟨k, rfl⟩
+    change projection Q (k • kernelStep Q) = 0
+    simp only [map_zsmul, projection_kernelStep, zsmul_zero]
+
+def representatives (Q : ℕ) (F : Finset (Group Q)) : Finset (Lattice 3) :=
+  F.image (lift Q)
+
+theorem lift_mem_representatives (Q : ℕ) {F : Finset (Group Q)}
+    {f : Group Q} (hf : f ∈ F) : lift Q f ∈ representatives Q F :=
+  Finset.mem_image_of_mem _ hf
+
+theorem projection_mem_of_mem_representatives (Q : ℕ) [NeZero Q]
+    {F : Finset (Group Q)} {u : Lattice 3} (hu : u ∈ representatives Q F) :
+    projection Q u ∈ F := by
+  obtain ⟨f, hf, rfl⟩ := Finset.mem_image.mp hu
+  simpa only [projection_lift] using hf
+
+theorem projection_injOn_representatives (Q : ℕ) [NeZero Q]
+    (F : Finset (Group Q)) :
+    Set.InjOn (projection Q) (representatives Q F : Set (Lattice 3)) := by
+  intro u hu v hv huv
+  obtain ⟨f, hf, rfl⟩ := Finset.mem_image.mp hu
+  obtain ⟨g, hg, rfl⟩ := Finset.mem_image.mp hv
+  have hfg : f = g := by simpa only [projection_lift] using huv
+  exact congrArg (lift Q) hfg
+
+theorem projection_image_representatives (Q : ℕ) [NeZero Q]
+    (F : Finset (Group Q)) :
+    (representatives Q F).image (projection Q) = F := by
+  ext f
+  constructor
+  · intro hf
+    obtain ⟨u, hu, huf⟩ := Finset.mem_image.mp hf
+    rw [← huf]
+    exact projection_mem_of_mem_representatives Q hu
+  · intro hf
+    exact Finset.mem_image.mpr
+      ⟨lift Q f, lift_mem_representatives Q hf, projection_lift Q f⟩
+
+theorem zero_mem_representatives (Q : ℕ) {F : Finset (Group Q)}
+    (hF : 0 ∈ F) : 0 ∈ representatives Q F := by
+  simpa only [lift_zero] using lift_mem_representatives Q hF
+
+theorem eq_zero_of_mem_representatives_mem_ker (Q : ℕ) [NeZero Q]
+    {F : Finset (Group Q)} {u : Lattice 3}
+    (hu : u ∈ representatives Q F) (hk : u ∈ (projection Q).ker) : u = 0 := by
+  obtain ⟨f, hf, rfl⟩ := Finset.mem_image.mp hu
+  have hf0 : f = 0 := by
+    simpa only [AddMonoidHom.mem_ker, projection_lift] using hk
+  rw [hf0, lift_zero]
+
+theorem kernelStep_not_mem_representatives (Q : ℕ) [NeZero Q]
+    (F : Finset (Group Q)) : kernelStep Q ∉ representatives Q F := by
+  intro h
+  exact kernelStep_ne_zero Q
+    (eq_zero_of_mem_representatives_mem_ker Q h (projection_kernelStep Q))
+
+theorem preimage_tiles (Q : ℕ) [NeZero Q]
+    {F : Finset (Group Q)} {A : Set (Group Q)} (h : Tiles F A) :
+    Tiles (representatives Q F) ((projection Q) ⁻¹' A) := by
+  rw [tiles_iff_unique_tile]
+  intro x
+  obtain ⟨f, hf, huniq⟩ := (tiles_iff_unique_tile.mp h) (projection Q x)
+  refine ⟨⟨lift Q f, lift_mem_representatives Q f.property⟩, ?_, ?_⟩
+  · change projection Q (x - lift Q f) ∈ A
+    simpa only [map_sub, projection_lift] using hf
+  · intro u hu
+    have hpu : projection Q u ∈ F :=
+      projection_mem_of_mem_representatives Q u.property
+    have hp : projection Q x - projection Q u ∈ A := by
+      simpa only [Set.mem_preimage, map_sub] using hu
+    have he := huniq ⟨projection Q u, hpu⟩ hp
+    apply Subtype.ext
+    apply projection_injOn_representatives Q F u.property
+      (lift_mem_representatives Q f.property)
+    simpa only [projection_lift] using congrArg Subtype.val he
+
+theorem period_preimage_of_mem_ker (Q : ℕ) (A : Set (Group Q))
+    {k : Lattice 3} (hk : k ∈ (projection Q).ker) :
+    Period ((projection Q) ⁻¹' A) k := by
+  intro x
+  change projection Q (x + k) ∈ A ↔ projection Q x ∈ A
+  have hk' : projection Q k = 0 := hk
+  simp only [map_add, hk', add_zero]
+
+theorem period_preimage_kernelStep (Q : ℕ) (A : Set (Group Q)) :
+    Period ((projection Q) ⁻¹' A) (kernelStep Q) :=
+  period_preimage_of_mem_ker Q A (projection_kernelStep Q)
+
+end CyclicQuotient
+end PeriodicTilingThree
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/CyclicShear.lean b/lean/OAI/Geometry/PeriodicTiling/CyclicShear.lean
new file mode 100644
index 0000000000000000000000000000000000000000..1a2620eb96dac72c4e3c33ca2cfa7e414ff198dd
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/CyclicShear.lean
@@ -0,0 +1,153 @@
+import Mathlib.Data.ZMod.Basic
+import Mathlib.Tactic.Abel
+import Mathlib.Tactic.NormNum
+import Mathlib.Tactic.Positivity
+
+namespace OAI
+
+namespace PeriodicTilingThree.CyclicShear
+
+noncomputable section
+
+def low (r : ℕ) : ZMod (r ^ 2) →+ ZMod r :=
+  (ZMod.castHom (dvd_pow_self r (by decide : 2 ≠ 0)) (ZMod r)).toAddMonoidHom
+
+@[simp] theorem low_natCast (r m : ℕ) : low r (m : ZMod (r ^ 2)) = (m : ZMod r) := by
+  simp [low]
+
+@[simp] theorem low_intCast (r : ℕ) (m : ℤ) :
+    low r (m : ZMod (r ^ 2)) = (m : ZMod r) := by
+  simp [low]
+
+private def highInteger (r : ℕ) : ℤ →+ ZMod (r ^ 2) where
+  toFun z := (r : ZMod (r ^ 2)) * (z : ZMod (r ^ 2))
+  map_zero' := by simp
+  map_add' x y := by simp [mul_add]
+
+private theorem highInteger_modulus (r : ℕ) : highInteger r r = 0 := by
+  dsimp only [highInteger]
+  change (r : ZMod (r ^ 2)) * ((r : ℤ) : ZMod (r ^ 2)) = 0
+  rw [Int.cast_natCast, ← pow_two, ← Nat.cast_pow]
+  exact ZMod.natCast_self (r ^ 2)
+
+def highMul (r : ℕ) : ZMod r →+ ZMod (r ^ 2) :=
+  ZMod.lift r ⟨highInteger r, highInteger_modulus r⟩
+
+@[simp] theorem highMul_intCast (r : ℕ) (m : ℤ) :
+    highMul r (m : ZMod r) = (r : ZMod (r ^ 2)) * (m : ZMod (r ^ 2)) := by
+  exact ZMod.lift_coe r ⟨highInteger r, highInteger_modulus r⟩ m
+
+@[simp] theorem highMul_natCast (r m : ℕ) :
+    highMul r (m : ZMod r) = (r : ZMod (r ^ 2)) * (m : ZMod (r ^ 2)) := by
+  simpa using highMul_intCast r (m : ℤ)
+
+theorem highMul_apply (r : ℕ) [NeZero r] (b : ZMod r) :
+    highMul r b = (r : ZMod (r ^ 2)) * (b.val : ZMod (r ^ 2)) := by
+  simpa using highMul_natCast r b.val
+
+theorem highMul_val (r : ℕ) [NeZero r] (b : ZMod r) :
+    (highMul r b).val = r * b.val := by
+  rw [highMul_apply, ← Nat.cast_mul]
+  apply ZMod.val_natCast_of_lt
+  have hr : 0 < r := Nat.pos_of_ne_zero (NeZero.ne r)
+  simpa [pow_two] using Nat.mul_lt_mul_of_pos_left (ZMod.val_lt b) hr
+
+theorem highMul_injective (r : ℕ) [NeZero r] : Function.Injective (highMul r) := by
+  intro b c h
+  have hv := congrArg ZMod.val h
+  rw [highMul_val, highMul_val] at hv
+  have hv' : b.val = c.val := Nat.eq_of_mul_eq_mul_left
+    (Nat.pos_of_ne_zero (NeZero.ne r)) hv
+  exact ZMod.val_injective r hv'
+
+@[simp] theorem low_highMul (r : ℕ) (b : ZMod r) : low r (highMul r b) = 0 := by
+  obtain ⟨m, rfl⟩ := ZMod.intCast_surjective b
+  rw [highMul_intCast]
+  change (ZMod.castHom (dvd_pow_self r (by decide : 2 ≠ 0)) (ZMod r))
+    ((r : ZMod (r ^ 2)) * (m : ZMod (r ^ 2))) = 0
+  simp only [map_mul, map_natCast, map_intCast, ZMod.natCast_self, zero_mul]
+
+theorem mem_range_highMul_iff (r : ℕ) [NeZero r] (v : ZMod (r ^ 2)) :
+    v ∈ Set.range (highMul r) ↔ low r v = 0 := by
+  constructor
+  · rintro ⟨b, rfl⟩
+    exact low_highMul r b
+  · intro h
+    have hcast : (v.val : ZMod r) = 0 := by
+      have hv := congrArg (low r) (ZMod.natCast_zmod_val v)
+      simpa only [low_natCast, h] using hv
+    obtain ⟨m, hm⟩ := (ZMod.natCast_eq_zero_iff v.val r).mp hcast
+    refine ⟨(m : ZMod r), ?_⟩
+    rw [highMul_natCast, ← Nat.cast_mul, ← hm]
+    exact ZMod.natCast_zmod_val v
+
+theorem range_highMul (r : ℕ) [NeZero r] :
+    Set.range (highMul r) = ((low r).ker : Set (ZMod (r ^ 2))) := by
+  ext v
+  exact mem_range_highMul_iff r v
+
+def highEquiv (r : ℕ) [NeZero r] : ZMod r ≃ (low r).ker :=
+  Equiv.ofBijective (fun b => ⟨highMul r b, low_highMul r b⟩)
+    ⟨fun _ _ h => highMul_injective r (congrArg Subtype.val h), by
+      intro v
+      obtain ⟨b, hb⟩ := (mem_range_highMul_iff r v).mpr v.property
+      exact ⟨b, Subtype.ext hb⟩⟩
+
+@[simp] theorem highEquiv_apply (r : ℕ) [NeZero r] (b : ZMod r) :
+    (highEquiv r b : ZMod (r ^ 2)) = highMul r b := rfl
+
+def shear (r : ℕ) (x : ℤ) (k b : ZMod r) : ZMod (r ^ 2) :=
+  (x : ZMod (r ^ 2)) + ((k - (x : ZMod r)).val : ZMod (r ^ 2)) + highMul r b
+
+@[simp] theorem low_shear (r : ℕ) [NeZero r] (x : ℤ) (k b : ZMod r) :
+    low r (shear r x k b) = k := by
+  simp only [shear, map_add, low_intCast, low_natCast, low_highMul, add_zero,
+    ZMod.natCast_zmod_val]
+  abel
+
+theorem shear_injective (r : ℕ) [NeZero r] (x : ℤ) (k : ZMod r) :
+    Function.Injective (shear r x k) := by
+  intro b c h
+  apply highMul_injective r
+  exact add_left_cancel h
+
+theorem shear_eq_zero_add (r : ℕ) (x : ℤ) (k b : ZMod r) :
+    shear r x k b = shear r x k 0 + highMul r b := by
+  simp [shear]
+
+theorem shear_surjective_fiber (r : ℕ) [NeZero r] (x : ℤ) (k : ZMod r) :
+    Function.Surjective (fun b : ZMod r =>
+      (⟨shear r x k b, low_shear r x k b⟩ : {v : ZMod (r ^ 2) // low r v = k})) := by
+  intro v
+  have hz : low r (v.val - shear r x k 0) = 0 := by
+    rw [map_sub, v.property, low_shear, sub_self]
+  obtain ⟨b, hb⟩ := (mem_range_highMul_iff r _).mpr hz
+  refine ⟨b, Subtype.ext ?_⟩
+  change shear r x k b = v.val
+  rw [shear_eq_zero_add, hb]
+  abel
+
+def shearEquiv (r : ℕ) [NeZero r] (x : ℤ) (k : ZMod r) :
+    ZMod r ≃ {v : ZMod (r ^ 2) // low r v = k} :=
+  Equiv.ofBijective (fun b => ⟨shear r x k b, low_shear r x k b⟩)
+    ⟨fun _ _ h => shear_injective r x k (congrArg Subtype.val h),
+      shear_surjective_fiber r x k⟩
+
+@[simp] theorem shearEquiv_apply (r : ℕ) [NeZero r] (x : ℤ) (k b : ZMod r) :
+    (shearEquiv r x k b : ZMod (r ^ 2)) = shear r x k b := rfl
+
+theorem shear_add (r : ℕ) (x h : ℤ) (k b : ZMod r) :
+    shear r (x + h) (k + (h : ZMod r)) b = shear r x k b + (h : ZMod (r ^ 2)) := by
+  simp only [shear, Int.cast_add]
+  abel_nf
+
+theorem shear_source_add (r : ℕ) (x h : ℤ) (k l b : ZMod r)
+    (hl : (h : ZMod r) = l) :
+    shear r (x - h) (k - l) b + (h : ZMod (r ^ 2)) = shear r x k b := by
+  simpa [hl] using (shear_add r (x - h) h (k - l) b).symm
+
+end
+
+end PeriodicTilingThree.CyclicShear
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/CyclicStack.lean b/lean/OAI/Geometry/PeriodicTiling/CyclicStack.lean
new file mode 100644
index 0000000000000000000000000000000000000000..66b0051cd35e476e6096ce3dfeae28d0b6c2975c
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/CyclicStack.lean
@@ -0,0 +1,149 @@
+import OAI.Geometry.PeriodicTiling.Stacking
+import Mathlib.GroupTheory.SpecificGroups.Cyclic
+
+namespace OAI
+
+universe uV uι
+
+namespace PeriodicTilingThree
+
+theorem not_dvd_card_of_lt (V : Type uV) [AddCommGroup V] [Finite V]
+    {q : ℕ} (hq : Nat.card V < q) : ¬ q ∣ Nat.card V := by
+  intro hdiv
+  exact (Nat.not_le_of_gt hq) (Nat.le_of_dvd Nat.card_pos hdiv)
+
+theorem cyclicStackOrder_pos (V : Type uV) [AddCommGroup V] [Finite V]
+    (q : ℕ) (hq : q.Prime) : 0 < q * Nat.card V :=
+  Nat.mul_pos hq.pos Nat.card_pos
+
+noncomputable def freshCyclicEquiv
+    (V : Type uV) [AddCommGroup V] [Finite V] [IsAddCyclic V]
+    (q : ℕ) (hq : q.Prime) (havoid : ¬ q ∣ Nat.card V) :
+    V × ZMod q ≃+ ZMod (q * Nat.card V) := by
+  let eV : V ≃+ ZMod (Nat.card V) :=
+    (zmodAddCyclicAddEquiv (G := V) inferInstance).symm
+  let hcop : q.Coprime (Nat.card V) :=
+    hq.coprime_iff_not_dvd.mpr havoid
+  exact
+    ((eV.prodCongr (AddEquiv.refl (ZMod q))).trans
+      (AddEquiv.prodComm :
+        ZMod (Nat.card V) × ZMod q ≃+
+          ZMod q × ZMod (Nat.card V))).trans
+      (ZMod.chineseRemainder hcop).toAddEquiv.symm
+
+noncomputable def cyclicStackEquiv
+    (V : Type uV) [AddCommGroup V] [Finite V] [IsAddCyclic V]
+    (q : ℕ) (hq : q.Prime) (havoid : ¬ q ∣ Nat.card V) :
+    ((Plane × V) × ZMod q) ≃+ Plane × ZMod (q * Nat.card V) :=
+  (AddEquiv.prodAssoc :
+    ((Plane × V) × ZMod q) ≃+ Plane × (V × ZMod q)).trans
+      ((AddEquiv.refl Plane).prodCongr (freshCyclicEquiv V q hq havoid))
+
+@[simp]
+theorem cyclicStackEquiv_fst
+    (V : Type uV) [AddCommGroup V] [Finite V] [IsAddCyclic V]
+    (q : ℕ) (hq : q.Prime) (havoid : ¬ q ∣ Nat.card V)
+    (x : (Plane × V) × ZMod q) :
+    (cyclicStackEquiv V q hq havoid x).1 = x.1.1 := rfl
+
+section Transport
+
+variable (V : Type uV) [AddCommGroup V] [Finite V] [IsAddCyclic V]
+variable (q : ℕ) (hq : q.Prime) (havoid : ¬ q ∣ Nat.card V)
+
+noncomputable def cyclicStackComplement (A : Set (Plane × V)) :
+    Set (Plane × ZMod (q * Nat.card V)) :=
+  cyclicStackEquiv V q hq havoid '' (A ×ˢ ({0} : Set (ZMod q)))
+
+noncomputable def cyclicStackProjection
+    (B : Set (Plane × ZMod (q * Nat.card V))) : Set (Plane × V) :=
+  Prod.fst '' (cyclicStackEquiv V q hq havoid ⁻¹' B)
+
+theorem fullyPeriodic_cyclicStackComplement {A : Set (Plane × V)}
+    (hA : FullyPeriodic A) :
+    FullyPeriodic (cyclicStackComplement V q hq havoid A) := by
+  let : NeZero q := ⟨hq.ne_zero⟩
+  exact (fullyPeriodic_image_addEquiv_iff
+    (cyclicStackEquiv V q hq havoid)).mpr hA.prod_singleton_zero
+
+theorem fullyPeriodic_cyclicStackProjection
+    {B : Set (Plane × ZMod (q * Nat.card V))} (hB : FullyPeriodic B) :
+    FullyPeriodic (cyclicStackProjection V q hq havoid B) := by
+  exact FullyPeriodic.image_of_surjective
+    (AddMonoidHom.fst (Plane × V) (ZMod q))
+    (fun x => ⟨(x, 0), rfl⟩)
+    (FullyPeriodic.preimage_addEquiv (cyclicStackEquiv V q hq havoid) hB)
+
+end Transport
+
+section FiniteFamily
+
+variable {ι : Type uι} [Fintype ι]
+variable (V : Type uV) [AddCommGroup V] [Finite V] [IsAddCyclic V]
+variable (q : ℕ) (hq : q.Prime) (havoid : ¬ q ∣ Nat.card V)
+
+noncomputable def cyclicStackTile
+    (F : ι → Finset (Plane × V)) (E : ι → Finset (ZMod q)) :
+    Finset (Plane × ZMod (q * Nat.card V)) := by
+  classical
+  letI : NeZero q := ⟨hq.ne_zero⟩
+  exact (stack F E).image (cyclicStackEquiv V q hq havoid)
+
+theorem tiles_cyclicStackTile
+    {F : ι → Finset (Plane × V)} {E : ι → Finset (ZMod q)}
+    (hpart : IsColorPartition E) {A : Set (Plane × V)}
+    (hA : ∀ ν, Tiles (F ν) A) :
+    Tiles (cyclicStackTile V q hq havoid F E)
+      (cyclicStackComplement V q hq havoid A) := by
+  classical
+  let : NeZero q := ⟨hq.ne_zero⟩
+  exact Tiles.image_addEquiv (cyclicStackEquiv V q hq havoid)
+    (tiles_stack_lift hpart hA)
+
+theorem tiles_stack_of_tiles_cyclicStackTile
+    {F : ι → Finset (Plane × V)} {E : ι → Finset (ZMod q)}
+    {B : Set (Plane × ZMod (q * Nat.card V))}
+    (hB : Tiles (cyclicStackTile V q hq havoid F E) B) :
+    Tiles (stack F E) (cyclicStackEquiv V q hq havoid ⁻¹' B) := by
+  classical
+  let : NeZero q := ⟨hq.ne_zero⟩
+  simpa only [cyclicStackTile, Finset.image_image, Function.comp_def,
+    AddEquiv.symm_apply_apply, Finset.image_id'] using
+    Tiles.preimage_addEquiv (cyclicStackEquiv V q hq havoid) hB
+
+theorem tiles_of_tiles_cyclicStackTile
+    {F : ι → Finset (Plane × V)} {E : ι → Finset (ZMod q)}
+    (hpart : IsColorPartition E) (hfull : HasFullDifferences E)
+    (hE : ∀ ν, (E ν).Nonempty)
+    {B : Set (Plane × ZMod (q * Nat.card V))}
+    (hB : Tiles (cyclicStackTile V q hq havoid F E) B) :
+    ∀ ν, Tiles (F ν) (cyclicStackProjection V q hq havoid B) := by
+  classical
+  let : NeZero q := ⟨hq.ne_zero⟩
+  exact tiles_of_tiles_stack hpart hfull hE
+    (tiles_stack_of_tiles_cyclicStackTile V q hq havoid hB)
+
+theorem cyclicStack_counterexample_of_common_system
+    {F : ι → Finset (Plane × V)} {E : ι → Finset (ZMod q)}
+    (hpart : IsColorPartition E) (hfull : HasFullDifferences E)
+    (hE : ∀ ν, (E ν).Nonempty)
+    (hcommon : ∃ A, ∀ ν, Tiles (F ν) A)
+    (hnone : ∀ A, (∀ ν, Tiles (F ν) A) → ¬ FullyPeriodic A) :
+    (cyclicStackTile V q hq havoid F E).Nonempty ∧
+      (∃ A, Tiles (cyclicStackTile V q hq havoid F E)
+        (cyclicStackComplement V q hq havoid A)) ∧
+      ∀ B, Tiles (cyclicStackTile V q hq havoid F E) B →
+        ¬ FullyPeriodic B := by
+  obtain ⟨A, hA⟩ := hcommon
+  have ht := tiles_cyclicStackTile V q hq havoid hpart hA
+  refine ⟨ht.tile_nonempty, ⟨A, ht⟩, ?_⟩
+  intro B hB hperiod
+  exact hnone (cyclicStackProjection V q hq havoid B)
+    (tiles_of_tiles_cyclicStackTile V q hq havoid hpart hfull hE hB)
+    (fullyPeriodic_cyclicStackProjection V q hq havoid hperiod)
+
+end FiniteFamily
+
+end PeriodicTilingThree
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/CyclicSystemStacking.lean b/lean/OAI/Geometry/PeriodicTiling/CyclicSystemStacking.lean
new file mode 100644
index 0000000000000000000000000000000000000000..8858c0ffa21be34c34afe886abc6bfe8f5d9bc37
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/CyclicSystemStacking.lean
@@ -0,0 +1,44 @@
+import OAI.Geometry.PeriodicTiling.CyclicStack
+import OAI.Geometry.PeriodicTiling.FullDifferencePartition
+
+namespace OAI
+
+universe uι uV
+
+namespace PeriodicTilingThree
+
+theorem exists_cyclic_stack_of_system
+    {ι : Type uι} [Fintype ι] [Nonempty ι]
+    (V : Type uV) [AddCommGroup V] [Finite V] [IsAddCyclic V]
+    (F : ι → Finset (Plane × V))
+    (hcommon : ∃ A, ∀ ν, Tiles (F ν) A)
+    (hnone : ∀ A, (∀ ν, Tiles (F ν) A) → ¬ FullyPeriodic A) :
+    ∃ Q : ℕ, 0 < Q ∧
+      ∃ T : Finset (Plane × ZMod Q), T.Nonempty ∧
+        (∃ B, Tiles T B) ∧ ∀ B, Tiles T B → ¬ FullyPeriodic B := by
+  classical
+  let s := Fintype.card ι
+  have hs : 0 < s := Fintype.card_pos
+  let e : ι ≃ Fin s := Fintype.equivFin ι
+  let F' : Fin s → Finset (Plane × V) := fun ν => F (e.symm ν)
+  have hcommon' : ∃ A, ∀ ν, Tiles (F' ν) A := by
+    obtain ⟨A, hA⟩ := hcommon
+    exact ⟨A, fun ν => hA (e.symm ν)⟩
+  have hnone' : ∀ A, (∀ ν, Tiles (F' ν) A) → ¬ FullyPeriodic A := by
+    intro A hA
+    apply hnone A
+    intro ν
+    simpa only [F', Equiv.symm_apply_apply] using hA (e ν)
+  obtain ⟨q, hq, hcard, _hfresh, E, hE, hpart, hfull⟩ :=
+    exists_fresh_prime_fullDifference_partition s hs ∅ (Nat.card V)
+  let havoid : ¬ q ∣ Nat.card V := not_dvd_card_of_lt V hcard
+  have hc := cyclicStack_counterexample_of_common_system V q hq havoid
+    (F := F') (E := E) hpart hfull hE hcommon' hnone'
+  refine ⟨q * Nat.card V, cyclicStackOrder_pos V q hq,
+    cyclicStackTile V q hq havoid F' E, hc.1, ?_, hc.2.2⟩
+  obtain ⟨A, hA⟩ := hc.2.1
+  exact ⟨cyclicStackComplement V q hq havoid A, hA⟩
+
+end PeriodicTilingThree
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/CyclicTypes.lean b/lean/OAI/Geometry/PeriodicTiling/CyclicTypes.lean
new file mode 100644
index 0000000000000000000000000000000000000000..135ec887e80697618673f96034c8841f95f8aa6a
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/CyclicTypes.lean
@@ -0,0 +1,35 @@
+import OAI.Geometry.PeriodicTiling.EncodingParameters
+
+namespace OAI
+
+namespace PeriodicTilingThree
+
+abbrev D (p : ℕ) := p ^ 4
+abbrev Residues (p : ℕ) := ZMod (p ^ 2) × ZMod (p ^ 2)
+
+variable {p : ℕ} [NeZero p]
+
+abbrev K (E : EncodingParameters p) (i : Channel p) := ZMod (E.r i)
+abbrev P (E : EncodingParameters p) (i : Channel p) := ZMod (E.a i) × ZMod (E.b i)
+abbrev Input (E : EncodingParameters p) := ∀ i : Channel p, K E i
+abbrev Useful (E : EncodingParameters p) := ∀ i : Channel p, P E i
+
+abbrev V (E : EncodingParameters p) :=
+  ZMod (D p) × (∀ i : Channel p, ZMod ((E.r i) ^ 2) × P E i)
+
+abbrev Ambient (E : EncodingParameters p) := Plane × V E
+abbrev Base (E : EncodingParameters p) := Plane × Input E
+
+abbrev Low (E : EncodingParameters p) := Input E
+abbrev G (E : EncodingParameters p) := Ambient E
+
+structure GraphOutputs (E : EncodingParameters p) where
+  c : Base E → Useful E
+  beta : Base E → Input E
+  z : Base E → ZMod (D p)
+
+abbrev Outputs (E : EncodingParameters p) := GraphOutputs E
+
+end PeriodicTilingThree
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/Dependence.lean b/lean/OAI/Geometry/PeriodicTiling/Dependence.lean
new file mode 100644
index 0000000000000000000000000000000000000000..6749d0524a2463c775c221f480452516a62d8ce6
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/Dependence.lean
@@ -0,0 +1,355 @@
+import OAI.Geometry.PeriodicTiling.GraphTests
+import OAI.Geometry.PeriodicTiling.CyclicCRT
+import Mathlib.Algebra.Ring.Periodic
+import Mathlib.Data.Finset.Piecewise
+import Mathlib.Data.Fintype.Sigma
+import Mathlib.Tactic.Ring
+
+namespace OAI
+
+universe uι uA uB
+
+noncomputable section
+
+namespace PeriodicTilingThree
+
+section CoordinateReplacement
+
+variable {ι : Type uι} [Fintype ι] [DecidableEq ι]
+variable {A : ι → Type uA} {B : Type uB}
+
+theorem eq_of_update_invariant (f : (∀ i, A i) → B)
+    (hf : ∀ x i (a : A i), f (Function.update x i a) = f x)
+    (x y : ∀ i, A i) : f x = f y := by
+  have h : ∀ s : Finset ι, f (s.piecewise x y) = f y := by
+    intro s
+    induction s using Finset.induction_on with
+    | empty => simp
+    | insert i s hi ih => rw [Finset.piecewise_insert, hf, ih]
+  simpa using h Finset.univ
+
+theorem eq_of_update_invariant_off (f : (∀ i, A i) → B) (i : ι)
+    (hf : ∀ j, j ≠ i → ∀ x (a : A j), f (Function.update x j a) = f x)
+    (x y : ∀ i, A i) (hxy : x i = y i) : f x = f y := by
+  have h : ∀ s : Finset ι, f (s.piecewise x y) = f y := by
+    intro s
+    induction s using Finset.induction_on with
+    | empty => simp
+    | insert j s hj ih =>
+      rw [Finset.piecewise_insert]
+      by_cases hji : j = i
+      · subst j
+        have hv : s.piecewise x y i = x i := by
+          by_cases hi : i ∈ s <;> simp [Finset.piecewise, hi, hxy]
+        rw [← hv, Function.update_eq_self]
+        exact ih
+      · exact (hf j hji _ _).trans ih
+  simpa using h Finset.univ
+
+end CoordinateReplacement
+
+section Activity
+
+variable {A : Type uA} {B : Type uB} [AddGroup A]
+
+def DigitActive (g : A → B) (U : AddSubgroup A) : Prop :=
+  ∃ w w', w - w' ∈ U ∧ g w ≠ g w'
+
+theorem not_digitActive_iff (g : A → B) (U : AddSubgroup A) :
+    ¬ DigitActive g U ↔ ∀ w w', w - w' ∈ U → g w = g w' := by
+  simp only [DigitActive, not_exists, not_and, not_not]
+
+end Activity
+
+variable {p : ℕ} [NeZero p] (E : EncodingParameters p)
+
+def seedResidue (p : ℕ) (x : Plane) : Residues p :=
+  ((x.1 : ZMod (p ^ 2)), (x.2 : ZMod (p ^ 2)))
+
+def seedRepresentative (p : ℕ) (s : Residues p) : Plane :=
+  ((s.1.val : ℤ), (s.2.val : ℤ))
+
+@[simp] theorem seedResidue_representative (s : Residues p) :
+    seedResidue p (seedRepresentative p s) = s := by
+  apply Prod.ext <;> simp [seedResidue, seedRepresentative]
+
+def usefulAt (o : Outputs E) (i : Channel p) (x : Plane) (w : K E i) : P E i :=
+  o.c (x, Function.update 0 i w) i
+
+def ordinaryFactor (o : Outputs E) (n : Column p) (m : ℤ)
+    (w : K E (.inl n)) : P E (.inl n) :=
+  usefulAt E o (.inl n) (0, m) w
+
+def seedFactor (o : Outputs E) (t : Fin 2) (s : Residues p)
+    (w : K E (.inr t)) : P E (.inr t) :=
+  usefulAt E o (.inr t) (seedRepresentative p s) w
+
+def HasDependence (o : Outputs E) : Prop :=
+  (∀ n x k, o.c (x, k) (.inl n) =
+    ordinaryFactor E o n (lineValue n x) (k (.inl n))) ∧
+  (∀ t x k, o.c (x, k) (.inr t) =
+    seedFactor E o t (seedResidue p x) (k (.inr t)))
+
+theorem ordinaryFactor_spec {o : Outputs E} (h : HasDependence E o)
+    (n : Column p) (x : Plane) (k : Low E) :
+    o.c (x, k) (.inl n) = ordinaryFactor E o n (lineValue n x) (k (.inl n)) :=
+  h.1 n x k
+
+theorem seedFactor_spec {o : Outputs E} (h : HasDependence E o)
+    (t : Fin 2) (x : Plane) (k : Low E) :
+    o.c (x, k) (.inr t) = seedFactor E o t (seedResidue p x) (k (.inr t)) :=
+  h.2 t x k
+
+theorem usefulAt_factor {o : Outputs E} (h : HasDependence E o)
+    (x : Plane) (k : Low E) (i : Channel p) :
+    o.c (x, k) i = usefulAt E o i x (k i) := by
+  cases i with
+  | inl n =>
+    rw [h.1 n x k]
+    symm
+    simpa [usefulAt] using h.1 n x (Function.update 0 (.inl n) (k (.inl n)))
+  | inr t =>
+    rw [h.2 t x k]
+    symm
+    simpa [usefulAt] using h.2 t x (Function.update 0 (.inr t) (k (.inr t)))
+
+theorem usefulAt_eq_of_lineValue_eq {o : Outputs E} (h : HasDependence E o)
+    (n : Column p) {x y : Plane} (hxy : lineValue n x = lineValue n y)
+    (w : K E (.inl n)) :
+    usefulAt E o (.inl n) x w = usefulAt E o (.inl n) y w := by
+  unfold usefulAt
+  rw [h.1 n x _, h.1 n y _, hxy]
+
+theorem usefulAt_eq_of_seedResidue_eq {o : Outputs E} (h : HasDependence E o)
+    (t : Fin 2) {x y : Plane} (hxy : seedResidue p x = seedResidue p y)
+    (w : K E (.inr t)) :
+    usefulAt E o (.inr t) x w = usefulAt E o (.inr t) y w := by
+  unfold usefulAt
+  rw [h.2 t x _, h.2 t y _, hxy]
+
+def Active (o : Outputs E) (i : Channel p) (x : Plane) (j : Label p i) : Prop :=
+  DigitActive (usefulAt E o i x) (digitSubgroup E i j)
+
+theorem digitActive_iff_coordinates (i : Channel p) (g : K E i → P E i)
+    (j : Label p i) :
+    DigitActive g (digitSubgroup E i j) ↔
+      ∃ w w', (∀ q, q ≠ j → digitEquiv E i w q = digitEquiv E i w' q) ∧
+        g w ≠ g w' := by
+  simp only [DigitActive, sub_mem_digitSubgroup_iff]
+
+theorem no_digitActive_iff_constant (i : Channel p) (g : K E i → P E i) :
+    (∀ j, ¬ DigitActive g (digitSubgroup E i j)) ↔ ∃ c, ∀ w, g w = c := by
+  constructor
+  · intro h
+    let e := digitEquiv E i
+    have hu : ∀ z j (a : ZMod (E.digitPrime i j)),
+        g (e.symm (Function.update z j a)) = g (e.symm z) := by
+      intro z j a
+      apply (not_digitActive_iff g (digitSubgroup E i j)).mp (h j)
+      apply (sub_mem_digitSubgroup_iff E i j _ _).mpr
+      intro q hq
+      simp [e, hq]
+    refine ⟨g 0, fun w => ?_⟩
+    simpa [e] using eq_of_update_invariant (fun z => g (e.symm z)) hu (e w) (e 0)
+  · rintro ⟨c, hc⟩ j
+    apply (not_digitActive_iff g (digitSubgroup E i j)).mpr
+    intro w w' _hww'
+    exact (hc w).trans (hc w').symm
+
+theorem no_active_iff_constant (o : Outputs E) (i : Channel p) (x : Plane) :
+    (∀ j, ¬ Active E o i x j) ↔ ∃ c, ∀ w, usefulAt E o i x w = c :=
+  no_digitActive_iff_constant E i (usefulAt E o i x)
+
+theorem active_ordinary_iff {o : Outputs E} (h : HasDependence E o)
+    (n : Column p) {x y : Plane} (hxy : lineValue n x = lineValue n y)
+    (j : Label p (.inl n)) : Active E o (.inl n) x j ↔ Active E o (.inl n) y j := by
+  have he : usefulAt E o (.inl n) x = usefulAt E o (.inl n) y :=
+    funext (usefulAt_eq_of_lineValue_eq E h n hxy)
+  unfold Active
+  rw [he]
+
+theorem active_seed_iff {o : Outputs E} (h : HasDependence E o)
+    (t : Fin 2) {x y : Plane} (hxy : seedResidue p x = seedResidue p y)
+    (j : Label p (.inr t)) : Active E o (.inr t) x j ↔ Active E o (.inr t) y j := by
+  have he : usefulAt E o (.inr t) x = usefulAt E o (.inr t) y :=
+    funext (usefulAt_eq_of_seedResidue_eq E h t hxy)
+  unfold Active
+  rw [he]
+
+abbrev DependenceIndex (p : ℕ) :=
+  (Σ i : Channel p, {j : Channel p // j ≠ i}) ⊕
+    (Column p ⊕ (Fin 2 × Bool))
+
+def lowGenerator (j : Channel p) : Base E :=
+  (0, Function.update 0 j 1)
+
+def seedStep (p : ℕ) (second : Bool) : Plane :=
+  if second then (0, ((p ^ 2 : ℕ) : ℤ)) else (((p ^ 2 : ℕ) : ℤ), 0)
+
+def dependenceChannel : DependenceIndex p → Channel p
+  | .inl ⟨i, _⟩ => i
+  | .inr (.inl n) => .inl n
+  | .inr (.inr (t, _)) => .inr t
+
+def dependenceShift : DependenceIndex p → Base E
+  | .inl ⟨_, j⟩ => lowGenerator E j.1
+  | .inr (.inl n) => (lineKernelStep n, 0)
+  | .inr (.inr (_, axis)) => (seedStep p axis, 0)
+
+def dependenceTile (d : DependenceIndex p) : Finset (G E) :=
+  invarianceTile E (dependenceChannel d) (dependenceShift E d)
+
+theorem dependenceTile_nonempty (d : DependenceIndex p) :
+    (dependenceTile E d).Nonempty :=
+  invarianceTile_nonempty E (dependenceChannel d) (dependenceShift E d)
+
+theorem dependenceTile_card (d : DependenceIndex p) :
+    (dependenceTile E d).card = (kernelTile E).card :=
+  invarianceTile_card E (dependenceChannel d) (dependenceShift E d)
+
+def dependenceTiles : Finset (Finset (G E)) := by
+  classical
+  exact Finset.univ.image (dependenceTile E)
+
+def DependenceTests (o : Outputs E) : Prop :=
+  ∀ d : DependenceIndex p, Tiles (dependenceTile E d) (graph o)
+
+def DependenceInvariance (o : Outputs E) : Prop :=
+  ∀ d : DependenceIndex p, ∀ b : Base E,
+    o.c (b - dependenceShift E d) (dependenceChannel d) = o.c b (dependenceChannel d)
+
+private theorem periodic_of_sub_eq {A : Type uA} {B : Type uB} [AddGroup A]
+    (f : A → B) (v : A) (h : ∀ x, f (x - v) = f x) : Function.Periodic f v := by
+  intro x
+  simpa using (h (x + v)).symm
+
+private theorem useful_update_of_generator (o : Outputs E) (i j : Channel p)
+    (h : ∀ b, o.c (b - lowGenerator E j) i = o.c b i)
+    (x : Plane) (k : Low E) (w : K E j) :
+    o.c (x, Function.update k j w) i = o.c (x, k) i := by
+  obtain ⟨a, ha⟩ := ZMod.intCast_surjective (k j - w)
+  have he : (x, Function.update k j w) = (x, k) - a • lowGenerator E j := by
+    apply Prod.ext
+    · simp [lowGenerator]
+    · funext l
+      by_cases hlj : l = j
+      · subst l
+        simp [lowGenerator, zsmul_eq_mul, ha]
+      · simp [lowGenerator, hlj]
+  rw [he]
+  exact (periodic_of_sub_eq (fun b => o.c b i) (lowGenerator E j) h).sub_zsmul_eq a
+
+private theorem usefulAt_factor_of_generators (o : Outputs E)
+    (h : ∀ i j, j ≠ i → ∀ b, o.c (b - lowGenerator E j) i = o.c b i)
+    (x : Plane) (k : Low E) (i : Channel p) :
+    o.c (x, k) i = usefulAt E o i x (k i) := by
+  unfold usefulAt
+  apply eq_of_update_invariant_off (fun k => o.c (x, k) i) i
+  · intro j hji k' w
+    exact useful_update_of_generator E o i j (h i j hji) x k' w
+  · simp
+
+@[simp] theorem seedResidue_sub_step
+    {p : ℕ} [NeZero p]
+    (x : Plane) (axis : Bool) :
+    seedResidue p (x - seedStep p axis) = seedResidue p x := by
+  have hp2 : (p : ZMod (p ^ 2)) ^ 2 = 0 := by
+    rw [← Nat.cast_pow, ZMod.natCast_self]
+  cases axis <;> apply Prod.ext <;> simp [seedResidue, seedStep, hp2]
+
+theorem dependenceInvariance_iff (o : Outputs E) :
+    DependenceInvariance E o ↔ HasDependence E o := by
+  constructor
+  · intro h
+    have hlow : ∀ i j, j ≠ i → ∀ b, o.c (b - lowGenerator E j) i = o.c b i := by
+      intro i j hji b
+      exact h (.inl ⟨i, ⟨j, hji⟩⟩) b
+    have hfactor := usefulAt_factor_of_generators E o hlow
+    constructor
+    · intro n x k
+      have hstep : ∀ b : Base E,
+          o.c (b - (lineKernelStep n, 0)) (.inl n) = o.c b (.inl n) :=
+        h (.inr (.inl n))
+      have hp := periodic_of_sub_eq (fun b : Base E => o.c b (.inl n))
+        (lineKernelStep n, 0) hstep
+      have he : (x, k) - x.1 • (lineKernelStep n, (0 : Low E)) =
+          ((0, lineValue n x), k) := by
+        apply Prod.ext
+        · apply Prod.ext
+          · simp [lineKernelStep]
+          · simp [lineKernelStep, lineValue]
+            ring
+        · simp
+      have hs := hp.sub_zsmul_eq (x := (x, k)) x.1
+      rw [he] at hs
+      exact hs.symm.trans (hfactor (0, lineValue n x) k (.inl n))
+    · intro t x k
+      have hfirst : ∀ b : Base E,
+          o.c (b - (seedStep p false, 0)) (.inr t) = o.c b (.inr t) :=
+        h (.inr (.inr (t, false)))
+      have hsecond : ∀ b : Base E,
+          o.c (b - (seedStep p true, 0)) (.inr t) = o.c b (.inr t) :=
+        h (.inr (.inr (t, true)))
+      have hp := periodic_of_sub_eq (fun b : Base E => o.c b (.inr t))
+        (seedStep p false, 0) hfirst
+      have hq := periodic_of_sub_eq (fun b : Base E => o.c b (.inr t))
+        (seedStep p true, 0) hsecond
+      obtain ⟨a, ha⟩ := (ZMod.intCast_eq_iff (p ^ 2) x.1 (seedResidue p x).1).mp rfl
+      obtain ⟨b, hb⟩ := (ZMod.intCast_eq_iff (p ^ 2) x.2 (seedResidue p x).2).mp rfl
+      have he : (x, k) - a • (seedStep p false, (0 : Low E)) -
+          b • (seedStep p true, (0 : Low E)) =
+          (seedRepresentative p (seedResidue p x), k) := by
+        apply Prod.ext
+        · apply Prod.ext
+          · simpa [seedStep, seedRepresentative, zsmul_eq_mul,
+              sub_eq_iff_eq_add, mul_comm] using ha
+          · simpa [seedStep, seedRepresentative, zsmul_eq_mul,
+              sub_eq_iff_eq_add, mul_comm] using hb
+        · simp
+      have hs := (hq.sub_zsmul_eq
+        (x := (x, k) - a • (seedStep p false, (0 : Low E))) b).trans
+          (hp.sub_zsmul_eq (x := (x, k)) a)
+      rw [he] at hs
+      exact hs.symm.trans (hfactor (seedRepresentative p (seedResidue p x)) k (.inr t))
+  · intro h d xb
+    rcases xb with ⟨x, k⟩
+    rcases d with ⟨i, j⟩ | d
+    · change o.c ((x, k) - (0, Function.update (0 : Low E) j.1 1)) i = o.c (x, k) i
+      simp only [Prod.mk_sub_mk, sub_zero]
+      rw [usefulAt_factor E h, usefulAt_factor E h]
+      simp [Ne.symm j.property]
+    · rcases d with n | ⟨t, axis⟩
+      · change o.c ((x, k) - (lineKernelStep n, 0)) (.inl n) = o.c (x, k) (.inl n)
+        simp only [Prod.mk_sub_mk, sub_zero]
+        rw [h.1 n (x - lineKernelStep n) k, h.1 n x k]
+        have he : lineValue n (x - lineKernelStep n) = lineValue n x := by
+          simp [lineValue, lineKernelStep] ; ring
+        rw [he]
+      · change o.c ((x, k) - (seedStep p axis, 0)) (.inr t) = o.c (x, k) (.inr t)
+        simp only [Prod.mk_sub_mk, sub_zero]
+        rw [h.2 t (x - seedStep p axis) k, h.2 t x k, seedResidue_sub_step]
+
+theorem dependenceTests_iff (o : Outputs E) :
+    DependenceTests E o ↔ HasDependence E o := by
+  have he : DependenceTests E o ↔ DependenceInvariance E o := by
+    unfold DependenceTests DependenceInvariance dependenceTile
+    simp only [invarianceTile_iff]
+  exact he.trans (dependenceInvariance_iff E o)
+
+theorem dependenceTiles_iff (o : Outputs E) :
+    (∀ F ∈ dependenceTiles E, Tiles F (graph o)) ↔ HasDependence E o := by
+  classical
+  rw [← dependenceTests_iff E o]
+  constructor
+  · intro h d
+    apply h (dependenceTile E d)
+    exact Finset.mem_image.mpr ⟨d, Finset.mem_univ d, rfl⟩
+  · intro h F hF
+    obtain ⟨d, _hd, rfl⟩ := Finset.mem_image.mp hF
+    exact h d
+
+end PeriodicTilingThree
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/DirectionalLimits.lean b/lean/OAI/Geometry/PeriodicTiling/DirectionalLimits.lean
new file mode 100644
index 0000000000000000000000000000000000000000..d572ac2f18dc1b001457c96b5a839c4a4bb86b55
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/DirectionalLimits.lean
@@ -0,0 +1,274 @@
+import OAI.Geometry.PeriodicTiling.TileDilation
+import OAI.Geometry.PeriodicTiling.PlaneLattices
+import Mathlib.Algebra.BigOperators.Field
+import Mathlib.Algebra.Ring.Periodic
+import Mathlib.Analysis.SpecificLimits.Basic
+import Mathlib.Data.Fintype.EquivFin
+import Mathlib.Topology.Bases
+import Mathlib.Topology.Compactness.Compact
+import Mathlib.Topology.Order.Compact
+import Mathlib.Topology.Sequences
+import Mathlib.Tactic.Abel
+import Mathlib.Tactic.FieldSimp
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.Positivity
+import Mathlib.Tactic.Ring
+
+namespace OAI
+
+universe uι
+
+noncomputable section
+
+namespace PeriodicTilingThree
+
+open Set Filter
+open scoped BigOperators Topology
+
+def directionalAverage (a : Plane → ℝ) (q : ℕ) (v : Plane) (N : ℕ) (x : Plane) : ℝ :=
+  (∑ k ∈ Finset.range (N + 1), a (x - (1 + q * (k + 1)) • v)) / (N + 1 : ℕ)
+
+theorem directionalAverage_nonneg {a : Plane → ℝ} (ha : ∀ x, 0 ≤ a x)
+    (q : ℕ) (v : Plane) (N : ℕ) (x : Plane) :
+    0 ≤ directionalAverage a q v N x := by
+  unfold directionalAverage
+  exact div_nonneg (Finset.sum_nonneg fun k _ => ha _) (by positivity)
+
+theorem directionalAverage_le_one {a : Plane → ℝ} (ha : ∀ x, a x ≤ 1)
+    (q : ℕ) (v : Plane) (N : ℕ) (x : Plane) :
+    directionalAverage a q v N x ≤ 1 := by
+  have hN : (0 : ℝ) < (N + 1 : ℕ) := by positivity
+  apply (div_le_iff₀ hN).2
+  calc
+    _ ≤ ∑ _k ∈ Finset.range (N + 1), (1 : ℝ) :=
+      Finset.sum_le_sum fun k _ => ha _
+    _ = _ := by simp
+
+theorem directionalAverage_translate_sub (a : Plane → ℝ) (q : ℕ)
+    (v : Plane) (N : ℕ) (x : Plane) :
+    directionalAverage a q v N (x + q • v) - directionalAverage a q v N x =
+      (a (x - v) - a (x - (1 + q * (N + 1)) • v)) / (N + 1 : ℕ) := by
+  have hshift (k : ℕ) :
+      x + q • v - (1 + q * (k + 1)) • v = x - (1 + q * k) • v := by
+    have hk : 1 + q * (k + 1) = (1 + q * k) + q := by ring
+    rw [hk, add_nsmul]
+    abel
+  unfold directionalAverage
+  simp_rw [hshift]
+  rw [← sub_div, ← Finset.sum_sub_distrib]
+  congr 1
+  simpa only [Nat.mul_zero, Nat.add_zero, one_nsmul] using
+    Finset.sum_range_sub' (fun k => a (x - (1 + q * k) • v)) (N + 1)
+
+theorem directionalAverage_translate_error {a : Plane → ℝ}
+    (ha0 : ∀ x, 0 ≤ a x) (ha1 : ∀ x, a x ≤ 1)
+    (q : ℕ) (v : Plane) (N : ℕ) (x : Plane) :
+    |directionalAverage a q v N (x + q • v) - directionalAverage a q v N x| ≤
+      1 / (N + 1 : ℕ) := by
+  rw [directionalAverage_translate_sub, abs_div,
+    abs_of_pos (show (0 : ℝ) < (N + 1 : ℕ) by positivity)]
+  apply div_le_div_of_nonneg_right _ (by positivity)
+  apply abs_le.mpr
+  constructor <;> linarith [ha0 (x - v), ha1 (x - v),
+    ha0 (x - (1 + q * (N + 1)) • v), ha1 (x - (1 + q * (N + 1)) • v)]
+
+theorem sum_directionalAverage {F : Finset Plane} (h0 : (0 : Plane) ∈ F)
+    {a : Plane → ℝ} {q : ℕ}
+    (hD : ∀ (k : ℕ) (x : Plane), ∑ f ∈ F, a (x - (1 + q * k) • f) = 1)
+    (N : ℕ) (x : Plane) :
+    (∑ f ∈ F.erase 0, directionalAverage a q f N x) = 1 - a x := by
+  classical
+  have hs (k : ℕ) :
+      (∑ f ∈ F.erase 0, a (x - (1 + q * (k + 1)) • f)) = 1 - a x := by
+    have he := Finset.sum_erase_add F
+      (fun f => a (x - (1 + q * (k + 1)) • f)) h0
+    rw [hD (k + 1) x] at he
+    simpa only [smul_zero, sub_zero] using (eq_sub_iff_add_eq.mpr he)
+  unfold directionalAverage
+  rw [← Finset.sum_div, Finset.sum_comm]
+  simp_rw [hs]
+  simp only [Finset.sum_const, Finset.card_range, nsmul_eq_mul]
+  exact mul_div_cancel_left₀ (1 - a x) (by positivity)
+
+theorem exists_common_directional_subsequence {ι : Type uι} [Countable ι]
+    (u : ℕ → ι → Plane → ℝ)
+    (hu0 : ∀ N i x, 0 ≤ u N i x) (hu1 : ∀ N i x, u N i x ≤ 1) :
+    ∃ (φ : ι → Plane → ℝ) (σ : ℕ → ℕ), StrictMono σ ∧
+      (∀ i x, 0 ≤ φ i x ∧ φ i x ≤ 1) ∧
+      ∀ i x, Tendsto (fun N => u (σ N) i x) atTop (𝓝 (φ i x)) := by
+  let u' : ℕ → ((ι × Plane) → Set.Icc (0 : ℝ) 1) :=
+    fun N ix => ⟨u N ix.1 ix.2, hu0 N ix.1 ix.2, hu1 N ix.1 ix.2⟩
+  obtain ⟨l, σ, hσ, hl⟩ := CompactSpace.tendsto_subseq u'
+  refine ⟨fun i x => (l (i, x) : ℝ), σ, hσ, ?_, ?_⟩
+  · intro i x
+    exact (l (i, x)).property
+  · intro i x
+    exact ((continuous_subtype_val.comp (continuous_apply (i, x))).tendsto l).comp hl
+
+theorem exists_ungrouped_directional_limits {F : Finset Plane} (h0 : (0 : Plane) ∈ F)
+    {a : Plane → ℝ} (ha0 : ∀ x, 0 ≤ a x) (ha1 : ∀ x, a x ≤ 1)
+    {q : ℕ}
+    (hD : ∀ (k : ℕ) (x : Plane), ∑ f ∈ F, a (x - (1 + q * k) • f) = 1) :
+    ∃ φ : ↥(F.erase 0) → Plane → ℝ,
+      (∀ f x, 0 ≤ φ f x ∧ φ f x ≤ 1) ∧
+      (∀ x, a x + ∑ f, φ f x = 1) ∧
+      ∀ f, Function.Periodic (φ f) (q • (f : Plane)) := by
+  classical
+  obtain ⟨φ, σ, hσ, hφ, hlim⟩ := exists_common_directional_subsequence
+    (fun N (f : ↥(F.erase 0)) x => directionalAverage a q f N x)
+    (fun N f x => directionalAverage_nonneg ha0 q f N x)
+    (fun N f x => directionalAverage_le_one ha1 q f N x)
+  refine ⟨φ, hφ, ?_, ?_⟩
+  · intro x
+    have hsum := tendsto_finsetSum Finset.univ (fun (f : ↥(F.erase 0)) _ => hlim f x)
+    have havg (N : ℕ) :
+        (∑ f : ↥(F.erase 0), directionalAverage a q f (σ N) x) = 1 - a x := by
+      exact (Finset.sum_coe_sort (F.erase 0)
+        (fun f : Plane => directionalAverage a q f (σ N) x)).trans
+          (sum_directionalAverage h0 hD (σ N) x)
+    have heq : (∑ f, φ f x) = 1 - a x :=
+      tendsto_nhds_unique hsum (by simpa only [havg] using
+        (tendsto_const_nhds : Tendsto (fun _ : ℕ => 1 - a x) atTop (𝓝 (1 - a x))))
+    linarith
+  · intro f x
+    have herr := (hlim f (x + q • (f : Plane))).sub (hlim f x)
+    have hbound : Tendsto (fun N : ℕ => (1 : ℝ) / (σ N + 1 : ℕ)) atTop (𝓝 0) :=
+      (tendsto_one_div_atTop_nhds_zero_nat.comp (tendsto_add_atTop_nat 1)).comp
+        hσ.tendsto_atTop
+    have hz := squeeze_zero (fun N => abs_nonneg
+      (directionalAverage a q f (σ N) (x + q • (f : Plane)) -
+        directionalAverage a q f (σ N) x))
+      (fun N => directionalAverage_translate_error ha0 ha1 q f (σ N) x) hbound
+    have heq : |φ f (x + q • (f : Plane)) - φ f x| = 0 :=
+      tendsto_nhds_unique herr.abs hz
+    exact sub_eq_zero.mp (abs_eq_zero.mp heq)
+
+theorem Tiles.exists_ungrouped_directional_limits {F : Finset Plane} {A : Set Plane}
+    (h : Tiles F A) (h0 : (0 : Plane) ∈ F) :
+    ∃ φ : ↥(F.erase 0) → Plane → ℝ,
+      (∀ f x, 0 ≤ φ f x ∧ φ f x ≤ 1) ∧
+      (∀ x, tileIndicator ℝ A x + ∑ f, φ f x = 1) ∧
+      ∀ f, Function.Periodic (φ f) (F.card.factorial • (f : Plane)) := by
+  exact PeriodicTilingThree.exists_ungrouped_directional_limits h0
+    (fun x => tileIndicator_nonneg A x) (fun x => tileIndicator_le_one A x)
+    (fun k x => h.sum_tileIndicator_dilate_factorial k x)
+
+theorem exists_grouped_directional_functions {ι : Type uι} [Fintype ι]
+    (v : ι → Plane) (hv : ∀ i, v i ≠ 0) (φ : ι → Plane → ℝ)
+    (hφ : ∀ i x, 0 ≤ φ i x) (hp : ∀ i, Function.Periodic (φ i) (v i))
+    {a : Plane → ℝ} (ha : ∀ x, 0 ≤ a x)
+    (hsum : ∀ x, a x + ∑ i, φ i x = 1) :
+    ∃ (m : ℕ) (w : Fin m → Plane) (ψ : Fin m → Plane → ℝ),
+      (∀ j, w j ≠ 0) ∧
+      (∀ i j, i ≠ j → planeDet (w i) (w j) ≠ 0) ∧
+      (∀ j x, 0 ≤ ψ j x ∧ ψ j x ≤ 1) ∧
+      (∀ x, a x + ∑ j, ψ j x = 1) ∧
+      ∀ j, Function.Periodic (ψ j) (w j) := by
+  classical
+  let r : Setoid ι :=
+    { r := fun i j => planeDet (v i) (v j) = 0
+      iseqv := ⟨fun i => planeDet_self (v i),
+        fun {i j} hij => by rw [planeDet_swap, hij, neg_zero],
+        fun {i j k} hij hjk => planeDet_zero_trans (hv j) hij hjk⟩ }
+  let Q := Quotient r
+  let : Fintype Q := Fintype.ofFinite Q
+  let cls : ι → Q := Quotient.mk r
+  let fiber (j : Q) : Finset ι := Finset.univ.filter (fun i => cls i = j)
+  have mem_fiber (i : ι) (j : Q) : i ∈ fiber j ↔ cls i = j := by
+    simp [fiber]
+  have out_mem (j : Q) : j.out ∈ fiber j :=
+    (mem_fiber _ _).2 (Quotient.out_eq j)
+  have hex (j : Q) : ∃ w : Plane, w ≠ 0 ∧ ∀ i ∈ fiber j, ∃ n : ℤ, w = n • v i := by
+    have himage : ((fiber j).image v).Nonempty :=
+      ⟨v j.out, Finset.mem_image_of_mem v (out_mem j)⟩
+    obtain ⟨w, hw, hm⟩ := exists_common_collinear_multiple ((fiber j).image v) himage
+      (by
+        intro z hz
+        obtain ⟨i, _, rfl⟩ := Finset.mem_image.mp hz
+        exact hv i)
+      (by
+        intro z hz z' hz'
+        obtain ⟨i, hi, rfl⟩ := Finset.mem_image.mp hz
+        obtain ⟨i', hi', rfl⟩ := Finset.mem_image.mp hz'
+        exact Quotient.exact (((mem_fiber i j).1 hi).trans ((mem_fiber i' j).1 hi').symm))
+    exact ⟨w, hw, fun i hi => hm (v i) (Finset.mem_image_of_mem v hi)⟩
+  choose w hw hm using hex
+  let ψ : Q → Plane → ℝ := fun j x => ∑ i ∈ fiber j, φ i x
+  have hψ (j : Q) (x : Plane) : 0 ≤ ψ j x :=
+    Finset.sum_nonneg (fun i _ => hφ i x)
+  have htotal (x : Plane) : (∑ j, ψ j x) = ∑ i, φ i x := by
+    simp only [ψ, fiber, Finset.sum_filter]
+    rw [Finset.sum_comm]
+    apply Finset.sum_congr rfl
+    intro i _
+    rw [Finset.sum_eq_single (cls i)]
+    · simp only [ite_true]
+    · intro j _ hji
+      exact ite_eq_right (Ne.symm hji)
+    · intro hi
+      exact False.elim (hi (Finset.mem_univ _))
+  have hψone (j : Q) (x : Plane) : ψ j x ≤ 1 := by
+    have hj : ψ j x ≤ ∑ j', ψ j' x :=
+      Finset.single_le_sum (fun j' _ => hψ j' x) (Finset.mem_univ j)
+    rw [htotal] at hj
+    linarith [hsum x, ha x]
+  have hper (j : Q) : Function.Periodic (ψ j) (w j) := by
+    intro x
+    apply Finset.sum_congr rfl
+    intro i hi
+    obtain ⟨n, hn⟩ := hm j i hi
+    rw [hn]
+    exact (hp i).zsmul n x
+  have hpair (j k : Q) (hjk : j ≠ k) : planeDet (w j) (w k) ≠ 0 := by
+    intro hdet
+    obtain ⟨n, hn⟩ := hm j j.out (out_mem j)
+    obtain ⟨m, hmk⟩ := hm k k.out (out_mem k)
+    have hn0 : n ≠ 0 := by
+      intro hn0
+      exact hw j (by simpa [hn0] using hn)
+    have hm0 : m ≠ 0 := by
+      intro hm0
+      exact hw k (by simpa [hm0] using hmk)
+    rw [hn, hmk, planeDet_zsmul_left, planeDet_zsmul_right] at hdet
+    have hz : planeDet (v j.out) (v k.out) = 0 :=
+      (mul_eq_zero.mp ((mul_eq_zero.mp hdet).resolve_left hn0)).resolve_left hm0
+    apply hjk
+    calc
+      j = cls j.out := (Quotient.out_eq j).symm
+      _ = cls k.out := Quotient.sound hz
+      _ = k := Quotient.out_eq k
+  let e : Fin (Fintype.card Q) ≃ Q := (Fintype.equivFin Q).symm
+  refine ⟨Fintype.card Q, fun j => w (e j), fun j => ψ (e j),
+    fun j => hw (e j), ?_, ?_, ?_, fun j => hper (e j)⟩
+  · intro i j hij
+    exact hpair (e i) (e j) (fun he => hij (e.injective he))
+  · intro j x
+    exact ⟨hψ (e j) x, hψone (e j) x⟩
+  · intro x
+    rw [e.sum_comp (fun j => ψ j x), htotal]
+    exact hsum x
+
+theorem Tiles.exists_directional_decomposition {F : Finset Plane} {A : Set Plane}
+    (h : Tiles F A) (h0 : (0 : Plane) ∈ F) :
+    ∃ (m : ℕ) (v : Fin m → Plane) (φ : Fin m → Plane → ℝ),
+      (∀ j, v j ≠ 0) ∧
+      (∀ i j, i ≠ j → planeDet (v i) (v j) ≠ 0) ∧
+      (∀ j x, 0 ≤ φ j x ∧ φ j x ≤ 1) ∧
+      (∀ x, tileIndicator ℝ A x + ∑ j, φ j x = 1) ∧
+      ∀ j, Function.Periodic (φ j) (v j) := by
+  classical
+  obtain ⟨φ, hφ, hsum, hp⟩ := h.exists_ungrouped_directional_limits h0
+  apply exists_grouped_directional_functions
+    (fun f : ↥(F.erase 0) => F.card.factorial • (f : Plane)) _ φ
+    (fun f x => (hφ f x).1) hp (fun x => tileIndicator_nonneg A x) hsum
+  intro f
+  have hq : (F.card.factorial : ℤ) ≠ 0 := by
+    exact_mod_cast Nat.factorial_ne_zero F.card
+  simpa only [natCast_zsmul] using
+    zsmul_plane_ne_zero hq (Finset.mem_erase.mp f.property).1
+
+end PeriodicTilingThree
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/DirectionalSlices.lean b/lean/OAI/Geometry/PeriodicTiling/DirectionalSlices.lean
new file mode 100644
index 0000000000000000000000000000000000000000..1d3fd8f30e5739a95654e749a44d43436cbb3143
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/DirectionalSlices.lean
@@ -0,0 +1,133 @@
+import OAI.Geometry.PeriodicTiling.ConfigurationDifferenceReduction
+import OAI.Geometry.PeriodicTiling.ForwardDifference
+import OAI.Geometry.PeriodicTiling.FiniteMasks
+import OAI.Geometry.PeriodicTiling.PlaneLattices
+import Mathlib.Topology.Instances.AddCircle.Defs
+import Mathlib.Tactic.Choose
+import Mathlib.Tactic.Linarith
+
+namespace OAI
+
+universe uB
+
+namespace PeriodicTilingThree
+
+theorem forwardDiff_slice {B : Type uB} [AddCommGroup B]
+    (f : Plane → B) (x e : Plane) :
+    forwardDiff (fun n : ℤ => f (x + n • e)) =
+      fun n => configDiff e f (x + n • e) := by
+  ext n
+  simp only [forwardDiff_apply, configDiff_apply, add_zsmul, one_zsmul, add_assoc]
+
+theorem forwardDiff_iterate_slice {B : Type uB} [AddCommGroup B]
+    (k : ℕ) (f : Plane → B) (x e : Plane) :
+    (forwardDiff^[k]) (fun n : ℤ => f (x + n • e)) =
+      fun n => ((configDiff e)^[k] f) (x + n • e) := by
+  induction k generalizing f with
+  | zero => rfl
+  | succ k ih =>
+      rw [Function.iterate_succ_apply, forwardDiff_slice, ih,
+        Function.iterate_succ_apply]
+
+theorem circle_sum_of_indicator_decomposition {m : ℕ} (A : Set Plane)
+    (φ : Fin m → Plane → ℝ)
+    (hsum : ∀ x, tileIndicator ℝ A x + ∑ j, φ j x = 1) :
+    (∑ j, (fun x => (φ j x : AddCircle (1 : ℝ)))) = fun _ => 0 := by
+  classical
+  ext x
+  simp only [Finset.sum_apply]
+  change (∑ j, (φ j x : AddCircle (1 : ℝ))) = 0
+  have hcast : (∑ j, (φ j x : AddCircle (1 : ℝ))) =
+      ((∑ j, φ j x : ℝ) : AddCircle (1 : ℝ)) := by
+    exact (map_sum (QuotientAddGroup.mk' (AddSubgroup.zmultiples (1 : ℝ)))
+      (fun j => φ j x) Finset.univ).symm
+  rw [hcast, show (∑ j, φ j x) = 1 - tileIndicator ℝ A x by linarith [hsum x]]
+  by_cases hx : x ∈ A <;> simp [tileIndicator, hx, AddCircle.coe_period]
+
+theorem directional_circle_slice_polynomial {m : ℕ} (hm : 0 < m)
+    (A : Set Plane) (v : Fin m → Plane) (φ : Fin m → Plane → ℝ)
+    (e : Plane)
+    (hspan : ∀ i j, i ≠ j → ∃ a b : ℤ, e = a • v i + b • v j)
+    (hp : ∀ j, Function.Periodic (φ j) (v j))
+    (hsum : ∀ x, tileIndicator ℝ A x + ∑ j, φ j x = 1)
+    (j : Fin m) (x : Plane) :
+    (forwardDiff^[m]) (fun n : ℤ =>
+      (φ j (x + n • e) : AddCircle (1 : ℝ))) = 0 := by
+  classical
+  let ψ : Fin m → Plane → AddCircle (1 : ℝ) := fun i y => (φ i y : AddCircle (1 : ℝ))
+  have hψ : ∀ i, Function.Periodic (ψ i) (v i) := by
+    intro i y
+    change (φ i (y + v i) : AddCircle (1 : ℝ)) = (φ i y : AddCircle (1 : ℝ))
+    rw [hp i y]
+  have hrep : ∀ i : Fin m, ∃ a b : ℤ, i = j ∨ e = a • v i + b • v j := by
+    intro i
+    by_cases hij : i = j
+    · exact ⟨0, 0, Or.inl hij⟩
+    · obtain ⟨a, b, hab⟩ := hspan i j hij
+      exact ⟨a, b, Or.inr hab⟩
+  choose α β hrep using hrep
+  let u : Fin m → Plane := fun i => if i = j then e else α i • v i
+  let vs : List Plane := Finset.univ.toList.map u
+  have hlen : vs.length = m := by simp [vs]
+  have hne : vs ≠ [] := by
+    intro h
+    have hz : m = 0 := by simpa only [h, List.length_nil] using hlen.symm
+    exact (Nat.ne_of_gt hm) hz
+  have hkill : ∀ i, i ≠ j → ∃ w ∈ vs, Function.Periodic (ψ i) w := by
+    intro i hij
+    refine ⟨u i, ?_, ?_⟩
+    · exact List.mem_map.mpr ⟨i, by simp, rfl⟩
+    · simpa only [u, ite_eq_right hij] using (hψ i).zsmul (α i)
+  have hz : configDiffs vs (ψ j) = 0 :=
+    configDiffs_component_zero j vs hne ψ 0
+      (circle_sum_of_indicator_decomposition A φ hsum) hkill
+  have hreplace : ∀ w ∈ vs, ∃ k : ℤ, w = e + k • v j := by
+    intro w hw
+    obtain ⟨i, _hi, rfl⟩ := List.mem_map.mp hw
+    by_cases hij : i = j
+    · refine ⟨0, ?_⟩
+      simp [u, hij]
+    · refine ⟨-β i, ?_⟩
+      have he := (hrep i).resolve_left hij
+      simp only [u, ite_eq_right hij, he, neg_smul]
+      abel
+  rw [configDiffs_eq_iterate_of_period vs e (v j) (ψ j) (hψ j) hreplace,
+    hlen] at hz
+  change (forwardDiff^[m]) (fun n : ℤ => ψ j (x + n • e)) = 0
+  rw [forwardDiff_iterate_slice, hz]
+  rfl
+
+theorem directional_slice_pair_bound {m : ℕ} (A : Set Plane)
+    (v : Fin m → Plane) (φ : Fin m → Plane → ℝ) (e : Plane)
+    (hspan : ∀ i j, i ≠ j → ∃ a b : ℤ, e = a • v i + b • v j)
+    (hφ : ∀ j x, 0 ≤ φ j x)
+    (hp : ∀ j, Function.Periodic (φ j) (v j))
+    (hsum : ∀ x, tileIndicator ℝ A x + ∑ j, φ j x = 1)
+    (x : Plane) (i j : Fin m) (hij : i ≠ j) (n r : ℤ) :
+    φ i (x + n • e) + φ j (x + r • e) ≤ 1 := by
+  classical
+  obtain ⟨a, b, he⟩ := hspan i j hij
+  let y : Plane := x + n • e + ((r - n) * a) • v i
+  have hy : y = x + r • e + ((n - r) * b) • v j := by
+    dsimp [y]
+    rw [he]
+    simp only [smul_add, mul_smul, sub_smul]
+    abel
+  have hi : φ i y = φ i (x + n • e) :=
+    (hp i).zsmul ((r - n) * a) _
+  have hj : φ j y = φ j (x + r • e) := by
+    rw [hy]
+    exact (hp j).zsmul ((n - r) * b) _
+  have hle : φ i y + φ j y ≤ ∑ k, φ k y := by
+    have h := Finset.sum_le_sum_of_subset_of_nonneg
+      (Finset.subset_univ ({i, j} : Finset (Fin m)))
+      (fun k _hk _hnot => hφ k y)
+    simpa [hij] using h
+  have htotal := hsum y
+  have ha := tileIndicator_nonneg A y
+  rw [hi, hj] at hle
+  linarith
+
+end PeriodicTilingThree
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/DisjointSuccessorPairs.lean b/lean/OAI/Geometry/PeriodicTiling/DisjointSuccessorPairs.lean
new file mode 100644
index 0000000000000000000000000000000000000000..6db3e98600b64517d0447a03a62e6a22a539f51c
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/DisjointSuccessorPairs.lean
@@ -0,0 +1,121 @@
+import Mathlib.Data.Fintype.EquivFin
+import Mathlib.Tactic.FinCases
+
+namespace OAI
+
+universe uV
+
+namespace PeriodicTilingThree
+
+private theorem exists_successor_anchors {V : Type uV} [Fintype V] [DecidableEq V]
+    (τ : Equiv.Perm V) (hτ : ∀ v, τ v ≠ v) (marked : V) (k : ℕ)
+    (hcard : 3 * (k + 1) ≤ Fintype.card V) :
+    ∃ M : Finset V, M.card = k ∧
+      (∀ u ∈ M, u ≠ marked ∧ τ u ≠ marked) ∧
+      ∀ u ∈ M, ∀ v ∈ M, u ≠ τ v := by
+  revert hcard
+  induction k with
+  | zero =>
+      intro _
+      refine ⟨∅, rfl, ?_, ?_⟩ <;> simp
+  | succ k ih =>
+      intro hcard
+      obtain ⟨M, hM, hmarked, hseparated⟩ := ih (by omega)
+      let B : Finset V := insert marked (insert (τ.symm marked)
+        ((M ∪ M.image τ) ∪ M.image τ.symm))
+      have hB : B.card ≤ 3 * k + 2 := by
+        have hi₁ : (M.image τ).card ≤ M.card := Finset.card_image_le
+        have hi₂ : (M.image τ.symm).card ≤ M.card := Finset.card_image_le
+        have hu₁ := Finset.card_union_le M (M.image τ)
+        have hu₂ := Finset.card_union_le (M ∪ M.image τ) (M.image τ.symm)
+        have ha₁ := Finset.card_insert_le (τ.symm marked)
+          ((M ∪ M.image τ) ∪ M.image τ.symm)
+        have ha₂ := Finset.card_insert_le marked (insert (τ.symm marked)
+          ((M ∪ M.image τ) ∪ M.image τ.symm))
+        dsimp only [B]
+        omega
+      have hBlt : B.card < (Finset.univ : Finset V).card := by
+        rw [Finset.card_univ]
+        omega
+      obtain ⟨x, _, hx⟩ := Finset.exists_mem_notMem_of_card_lt_card hBlt
+      have hxM : x ∉ M := by
+        intro h
+        exact hx (by simp [B, h])
+      have hximage : x ∉ M.image τ := by
+        intro h
+        exact hx (by simp [B, h])
+      have hxpreimage : x ∉ M.image τ.symm := by
+        intro h
+        exact hx (by simp [B, h])
+      have hxmarked : x ≠ marked := by
+        intro h
+        exact hx (by simp [B, h])
+      have hτxmarked : τ x ≠ marked := by
+        intro h
+        have he : x = τ.symm marked := by
+          rw [← h]
+          exact (τ.symm_apply_apply x).symm
+        exact hx (by simp [B, he])
+      refine ⟨insert x M, ?_, ?_, ?_⟩
+      · rw [Finset.card_insert_of_notMem hxM, hM]
+      · intro u hu
+        rcases Finset.mem_insert.mp hu with rfl | hu
+        · exact ⟨hxmarked, hτxmarked⟩
+        · exact hmarked u hu
+      · intro u hu v hv
+        rcases Finset.mem_insert.mp hu with hux | huM
+        · subst u
+          rcases Finset.mem_insert.mp hv with hvx | hvM
+          · subst v
+            exact (hτ x).symm
+          · intro he
+            exact hximage (Finset.mem_image.mpr ⟨v, hvM, he.symm⟩)
+        · rcases Finset.mem_insert.mp hv with hvx | hvM
+          · subst v
+            intro he
+            apply hxpreimage
+            refine Finset.mem_image.mpr ⟨u, huM, ?_⟩
+            rw [he, τ.symm_apply_apply]
+          · exact hseparated u huM v hvM
+
+theorem exists_disjoint_successor_pairs {V : Type uV} [Fintype V]
+    (τ : Equiv.Perm V) (hτ : ∀ v, τ v ≠ v) (marked : V) (k : ℕ)
+    (hcard : 3 * (k + 1) ≤ Fintype.card V) :
+    ∃ pair : (Fin k × Fin 2) ↪ V,
+      (∀ j, pair (j, 1) = τ (pair (j, 0))) ∧ ∀ x, pair x ≠ marked := by
+  classical
+  obtain ⟨M, hM, hmarked, hseparated⟩ :=
+    exists_successor_anchors τ hτ marked k hcard
+  let e : Fin k ≃ ↥M := (Finset.equivFinOfCardEq hM).symm
+  let a : Fin k → V := fun j => (e j : V)
+  have ha : Function.Injective a := by
+    intro i j h
+    apply e.injective
+    exact Subtype.ext h
+  have hamem (j : Fin k) : a j ∈ M := (e j).property
+  let f : Fin k × Fin 2 → V :=
+    fun p => if p.2 = 0 then a p.1 else τ (a p.1)
+  have hf : Function.Injective f := by
+    rintro ⟨i, b⟩ ⟨j, c⟩ h
+    fin_cases b <;> fin_cases c
+    · have hij : i = j := ha (by simpa [f] using h)
+      cases hij
+      rfl
+    · exact False.elim (hseparated (a i) (hamem i) (a j) (hamem j)
+        (by simpa [f] using h))
+    · exact False.elim (hseparated (a j) (hamem j) (a i) (hamem i)
+        (by simpa [f] using h.symm))
+    · have hij : i = j := ha (τ.injective (by simpa [f] using h))
+      cases hij
+      rfl
+  refine ⟨⟨f, hf⟩, ?_, ?_⟩
+  · intro j
+    simp [f]
+  · rintro ⟨j, b⟩
+    fin_cases b
+    · simpa [f] using (hmarked (a j) (hamem j)).1
+    · simpa [f] using (hmarked (a j) (hamem j)).2
+
+end PeriodicTilingThree
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/EncodedSystem.lean b/lean/OAI/Geometry/PeriodicTiling/EncodedSystem.lean
new file mode 100644
index 0000000000000000000000000000000000000000..024f6552c6ac579e0cbc0f65a481f9e6e374db34
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/EncodedSystem.lean
@@ -0,0 +1,120 @@
+import OAI.Geometry.PeriodicTiling.ConstraintCycles
+import OAI.Geometry.PeriodicTiling.ActivationTiles
+
+namespace OAI
+
+noncomputable section
+
+namespace PeriodicTilingThree
+
+variable {p : ℕ} [NeZero p] (E : EncodingParameters p)
+
+def testTiles : Finset (Finset (Ambient E)) := by
+  classical
+  exact {kernelTile E} ∪ dependenceTiles E ∪ wordConstraintTiles E ∪
+    seedConstraintTiles E ∪ Finset.univ.image (ordinaryActivationTile E) ∪
+    Finset.univ.image (seedActivationTile E)
+
+theorem mem_testTiles_iff (F : Finset (Ambient E)) :
+    F ∈ testTiles E ↔ F = kernelTile E ∨ F ∈ dependenceTiles E ∨
+      F ∈ wordConstraintTiles E ∨ F ∈ seedConstraintTiles E ∨
+      (∃ n : Column p, ordinaryActivationTile E n = F) ∨
+      (∃ t : Fin 2, seedActivationTile E t = F) := by
+  classical
+  simp only [testTiles, Finset.mem_union, Finset.mem_singleton, Finset.mem_image,
+    Finset.mem_univ, true_and, or_assoc]
+
+theorem kernelTile_mem_testTiles : kernelTile E ∈ testTiles E :=
+  (mem_testTiles_iff E _).mpr (Or.inl rfl)
+
+theorem dependenceTiles_subset_testTiles : dependenceTiles E ⊆ testTiles E := by
+  intro F hF
+  exact (mem_testTiles_iff E F).mpr (Or.inr (Or.inl hF))
+
+theorem wordConstraintTiles_subset_testTiles : wordConstraintTiles E ⊆ testTiles E := by
+  intro F hF
+  exact (mem_testTiles_iff E F).mpr (Or.inr (Or.inr (Or.inl hF)))
+
+theorem seedConstraintTiles_subset_testTiles : seedConstraintTiles E ⊆ testTiles E := by
+  intro F hF
+  exact (mem_testTiles_iff E F).mpr (Or.inr (Or.inr (Or.inr (Or.inl hF))))
+
+theorem ordinaryActivationTile_mem_testTiles (n : Column p) :
+    ordinaryActivationTile E n ∈ testTiles E :=
+  (mem_testTiles_iff E _).mpr
+    (Or.inr (Or.inr (Or.inr (Or.inr (Or.inl ⟨n, rfl⟩)))))
+
+theorem seedActivationTile_mem_testTiles (t : Fin 2) :
+    seedActivationTile E t ∈ testTiles E :=
+  (mem_testTiles_iff E _).mpr
+    (Or.inr (Or.inr (Or.inr (Or.inr (Or.inr ⟨t, rfl⟩)))))
+
+theorem testTiles_nonempty : (testTiles E).Nonempty :=
+  ⟨kernelTile E, kernelTile_mem_testTiles E⟩
+
+theorem nonempty_of_mem_testTiles {F : Finset (Ambient E)}
+    (hF : F ∈ testTiles E) : F.Nonempty := by
+  classical
+  rcases (mem_testTiles_iff E F).mp hF with rfl | hF | hF | hF | ⟨n, rfl⟩ | ⟨t, rfl⟩
+  · exact kernelTile_nonempty E
+  · obtain ⟨d, _hd, rfl⟩ := Finset.mem_image.mp hF
+    exact invarianceTile_nonempty E (dependenceChannel d) (dependenceShift E d)
+  · exact nonempty_of_mem_wordConstraintTiles E hF
+  · exact nonempty_of_mem_seedConstraintTiles E hF
+  · exact ordinaryActivationTile_nonempty E n
+  · exact seedActivationTile_nonempty E t
+
+def Solves (A : Set (Ambient E)) : Prop :=
+  ∀ F ∈ testTiles E, Tiles F A
+
+theorem solves_iff (A : Set (Ambient E)) : Solves E A ↔
+    Tiles (kernelTile E) A ∧
+    (∀ F ∈ dependenceTiles E, Tiles F A) ∧
+    (∀ F ∈ wordConstraintTiles E, Tiles F A) ∧
+    (∀ F ∈ seedConstraintTiles E, Tiles F A) ∧
+    (∀ n, Tiles (ordinaryActivationTile E n) A) ∧
+    (∀ t, Tiles (seedActivationTile E t) A) := by
+  constructor
+  · intro h
+    exact ⟨h _ (kernelTile_mem_testTiles E),
+      fun F hF => h F (dependenceTiles_subset_testTiles E hF),
+      fun F hF => h F (wordConstraintTiles_subset_testTiles E hF),
+      fun F hF => h F (seedConstraintTiles_subset_testTiles E hF),
+      fun n => h _ (ordinaryActivationTile_mem_testTiles E n),
+      fun t => h _ (seedActivationTile_mem_testTiles E t)⟩
+  · rintro ⟨hg, hd, hw, hs, ha, hb⟩ F hF
+    rcases (mem_testTiles_iff E F).mp hF with rfl | hF | hF | hF | ⟨n, rfl⟩ | ⟨t, rfl⟩
+    · exact hg
+    · exact hd F hF
+    · exact hw F hF
+    · exact hs F hF
+    · exact ha n
+    · exact hb t
+
+abbrev TestIndex := ↥(testTiles E)
+
+def testTile (i : TestIndex E) : Finset (Ambient E) := i.val
+
+theorem testTile_nonempty (i : TestIndex E) : (testTile E i).Nonempty :=
+  nonempty_of_mem_testTiles E i.property
+
+instance testIndex_nonempty : Nonempty (TestIndex E) :=
+  ⟨⟨kernelTile E, kernelTile_mem_testTiles E⟩⟩
+
+theorem solves_iff_index (A : Set (Ambient E)) :
+    Solves E A ↔ ∀ i : TestIndex E, Tiles (testTile E i) A := by
+  constructor
+  · intro h i
+    exact h i.val i.property
+  · intro h F hF
+    exact h ⟨F, hF⟩
+
+theorem exists_graph_of_solves {A : Set (Ambient E)} (hA : Solves E A) :
+    ∃ o : GraphOutputs E, graph o = A :=
+  (kernelTile_iff_graph E A).mp (hA _ (kernelTile_mem_testTiles E))
+
+end PeriodicTilingThree
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/EncodedSystemObstruction.lean b/lean/OAI/Geometry/PeriodicTiling/EncodedSystemObstruction.lean
new file mode 100644
index 0000000000000000000000000000000000000000..c0f50d4aa5dcea28b8d9829c0e84399591a39578
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/EncodedSystemObstruction.lean
@@ -0,0 +1,35 @@
+import OAI.Geometry.PeriodicTiling.EncodedSystem
+import OAI.Geometry.PeriodicTiling.Activation
+import OAI.Geometry.PeriodicTiling.SystemObstruction
+
+namespace OAI
+
+namespace PeriodicTilingThree
+
+variable {p : ℕ} [NeZero p] (E : EncodingParameters p)
+
+theorem solves_not_fully_periodic {A : Set (Ambient E)} (hA : Solves E A) :
+    ¬ FullyPeriodic A := by
+  obtain ⟨o, rfl⟩ := exists_graph_of_solves E hA
+  obtain ⟨_hg, hd, hw, hs, ho, ht⟩ := (solves_iff E (graph o)).mp hA
+  have hdep : HasDependence E o := (dependenceTiles_iff E o).mp hd
+  apply graph_not_fully_periodic_of_activity E o hdep
+  · exact (wordConstraintTiles_iff E o hdep).mp hw
+  · exact (seedConstraintTiles_iff E o hdep).mp hs
+  · intro n
+    exact ordinary_active_of_tile E o hdep n (ho n)
+  · intro t
+    exact seed_active_somewhere_of_tile E o hdep t (ht t)
+
+theorem testTile_not_fully_periodic {A : Set (Ambient E)}
+    (hA : ∀ i : TestIndex E, Tiles (testTile E i) A) : ¬ FullyPeriodic A :=
+  solves_not_fully_periodic E ((solves_iff_index E A).mpr hA)
+
+theorem no_periodic_common_solution :
+    ¬ ∃ A : Set (Ambient E), Solves E A ∧ FullyPeriodic A := by
+  rintro ⟨A, hA, hperiod⟩
+  exact solves_not_fully_periodic E hA hperiod
+
+end PeriodicTilingThree
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/EncodingParameters.lean b/lean/OAI/Geometry/PeriodicTiling/EncodingParameters.lean
new file mode 100644
index 0000000000000000000000000000000000000000..4ee85efc0d47a24b18fd56f3f6391a8a5e40c1b0
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/EncodingParameters.lean
@@ -0,0 +1,127 @@
+import OAI.Geometry.PeriodicTiling.WordRule
+import Mathlib.Data.Nat.Prime.Basic
+import Mathlib.Data.Nat.GCD.BigOperators
+import Mathlib.Algebra.Order.BigOperators.Ring.Finset
+
+namespace OAI
+
+namespace PeriodicTilingThree
+
+open scoped BigOperators
+
+abbrev PrimeTag (p : ℕ) :=
+  Channel p ⊕ (Channel p ⊕ (Σ i : Channel p, Label p i))
+
+structure EncodingParameters (p : ℕ) where
+  p_prime : Nat.Prime p
+  p_large : 200 < p
+  primeAt : PrimeTag p → ℕ
+  prime_isPrime : ∀ t, Nat.Prime (primeAt t)
+  prime_injective : Function.Injective primeAt
+  prime_ne_p : ∀ t, primeAt t ≠ p
+  seed_a_eq : ∀ t : Fin 2, primeAt (.inl (.inr t)) = t.val + 2
+  seed_b_large : ∀ t : Fin 2, p ^ 4 < primeAt (.inr (.inl (.inr t)))
+  blockA : ∀ i : Channel p, Label p i → ℕ
+  blockB : ∀ i : Channel p, Label p i → ℕ
+  blockA_pos : ∀ i j, 0 < blockA i j
+  blockB_pos : ∀ i j, 0 < blockB i j
+  digit_representation : ∀ i j,
+    primeAt (.inr (.inr ⟨i, j⟩)) =
+      blockA i j * primeAt (.inl i) + blockB i j * primeAt (.inr (.inl i))
+
+namespace EncodingParameters
+
+variable {p : ℕ} (E : EncodingParameters p)
+
+def a (i : Channel p) : ℕ := E.primeAt (.inl i)
+def b (i : Channel p) : ℕ := E.primeAt (.inr (.inl i))
+def digitPrime (i : Channel p) (j : Label p i) : ℕ :=
+  E.primeAt (.inr (.inr ⟨i, j⟩))
+
+theorem a_prime (i : Channel p) : Nat.Prime (E.a i) := E.prime_isPrime _
+theorem b_prime (i : Channel p) : Nat.Prime (E.b i) := E.prime_isPrime _
+theorem digitPrime_prime (i : Channel p) (j : Label p i) :
+    Nat.Prime (E.digitPrime i j) := E.prime_isPrime _
+
+theorem a_pos (i : Channel p) : 0 < E.a i := (E.a_prime i).pos
+theorem b_pos (i : Channel p) : 0 < E.b i := (E.b_prime i).pos
+theorem digitPrime_pos (i : Channel p) (j : Label p i) :
+    0 < E.digitPrime i j := (E.digitPrime_prime i j).pos
+
+theorem a_two_le (i : Channel p) : 2 ≤ E.a i := (E.a_prime i).two_le
+theorem b_two_le (i : Channel p) : 2 ≤ E.b i := (E.b_prime i).two_le
+
+theorem seed_a (t : Fin 2) : E.a (.inr t) = t.val + 2 := E.seed_a_eq t
+
+theorem digit_eq (i : Channel p) (j : Label p i) :
+    E.digitPrime i j = E.blockA i j * E.a i + E.blockB i j * E.b i :=
+  E.digit_representation i j
+
+theorem prime_coprime {s t : PrimeTag p} (hst : s ≠ t) :
+    Nat.Coprime (E.primeAt s) (E.primeAt t) :=
+  (Nat.coprime_primes (E.prime_isPrime s) (E.prime_isPrime t)).2
+    (fun h => hst (E.prime_injective h))
+
+theorem prime_coprime_p (t : PrimeTag p) : Nat.Coprime (E.primeAt t) p :=
+  (Nat.coprime_primes (E.prime_isPrime t) E.p_prime).2 (E.prime_ne_p t)
+
+theorem a_coprime_b (i : Channel p) : Nat.Coprime (E.a i) (E.b i) := by
+  apply E.prime_coprime
+  simp
+
+theorem digitPrime_pairwise (i : Channel p) :
+    Pairwise (fun j j' : Label p i => Nat.Coprime (E.digitPrime i j) (E.digitPrime i j')) := by
+  intro j j' hj
+  apply E.prime_coprime
+  intro he
+  exact hj (by simpa using he)
+
+section Finite
+
+variable [NeZero p]
+
+def r (i : Channel p) : ℕ := ∏ j : Label p i, E.digitPrime i j
+
+theorem r_pos (i : Channel p) : 0 < E.r i := by
+  exact Finset.prod_pos (fun j _ => E.digitPrime_pos i j)
+
+instance a_neZero (i : Channel p) : NeZero (E.a i) := ⟨(E.a_pos i).ne'⟩
+instance b_neZero (i : Channel p) : NeZero (E.b i) := ⟨(E.b_pos i).ne'⟩
+instance digitPrime_neZero (i : Channel p) (j : Label p i) :
+    NeZero (E.digitPrime i j) := ⟨(E.digitPrime_pos i j).ne'⟩
+instance r_neZero (i : Channel p) : NeZero (E.r i) := ⟨(E.r_pos i).ne'⟩
+
+theorem r_coprime_a (i j : Channel p) : Nat.Coprime (E.r i) (E.a j) := by
+  apply Nat.coprime_fintype_prod_left_iff.mpr
+  intro k
+  apply E.prime_coprime
+  simp
+
+theorem r_coprime_b (i j : Channel p) : Nat.Coprime (E.r i) (E.b j) := by
+  apply Nat.coprime_fintype_prod_left_iff.mpr
+  intro k
+  apply E.prime_coprime
+  simp
+
+theorem r_coprime_p (i : Channel p) : Nat.Coprime (E.r i) p := by
+  apply Nat.coprime_fintype_prod_left_iff.mpr
+  intro k
+  exact E.prime_coprime_p _
+
+theorem r_pairwise_coprime : Pairwise (fun i j : Channel p => Nat.Coprime (E.r i) (E.r j)) := by
+  intro i j hij
+  apply Nat.coprime_fintype_prod_left_iff.mpr
+  intro k
+  apply Nat.coprime_fintype_prod_right_iff.mpr
+  intro l
+  apply E.prime_coprime
+  intro he
+  have hs : (⟨i, k⟩ : Σ i : Channel p, Label p i) = ⟨j, l⟩ :=
+    Sum.inr.inj (Sum.inr.inj he)
+  exact hij (congrArg Sigma.fst hs)
+
+end Finite
+end EncodingParameters
+end PeriodicTilingThree
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/EncodingPrimeSelection.lean b/lean/OAI/Geometry/PeriodicTiling/EncodingPrimeSelection.lean
new file mode 100644
index 0000000000000000000000000000000000000000..0918ca18aec148dade303bfa27a9aa404220a535
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/EncodingPrimeSelection.lean
@@ -0,0 +1,236 @@
+import OAI.Geometry.PeriodicTiling.EncodingParameters
+import Mathlib.Data.Nat.Prime.Infinite
+import Mathlib.Data.Fintype.EquivFin
+import Mathlib.Data.Fintype.Sigma
+import Mathlib.Data.Finset.Lattice.Fold
+import Mathlib.Data.Sum.Basic
+import Mathlib.Logic.Equiv.Sum
+import Mathlib.NumberTheory.FrobeniusNumber
+import Mathlib.Tactic.FinCases
+import Mathlib.Tactic.NormNum
+import Lean.Elab.Tactic.Omega
+import Mathlib.Tactic.Choose
+
+namespace OAI
+
+universe uι
+
+noncomputable section
+
+namespace PeriodicTilingThree
+namespace EncodingPrimeSelection
+
+private def nextPrime (bound : ℕ) : ℕ :=
+  Classical.choose (Nat.exists_infinite_primes (bound + 1))
+
+private theorem nextPrime_prime (bound : ℕ) : Nat.Prime (nextPrime bound) :=
+  (Classical.choose_spec (Nat.exists_infinite_primes (bound + 1))).2
+
+private theorem nextPrime_gt (bound : ℕ) : bound < nextPrime bound := by
+  have h := (Classical.choose_spec (Nat.exists_infinite_primes (bound + 1))).1
+  exact h
+
+private def primeSequence (bound : ℕ) : ℕ → ℕ
+  | 0 => nextPrime bound
+  | n + 1 => nextPrime (primeSequence bound n)
+
+private theorem primeSequence_strictMono (bound : ℕ) :
+    StrictMono (primeSequence bound) := by
+  apply strictMono_nat_of_lt_succ
+  intro n
+  exact nextPrime_gt _
+
+private theorem primeSequence_prime (bound n : ℕ) :
+    Nat.Prime (primeSequence bound n) := by
+  cases n with
+  | zero => exact nextPrime_prime _
+  | succ n => exact nextPrime_prime _
+
+private theorem primeSequence_gt (bound n : ℕ) : bound < primeSequence bound n :=
+  lt_of_lt_of_le (nextPrime_gt bound)
+    ((primeSequence_strictMono bound).monotone (Nat.zero_le n))
+
+theorem exists_prime_family (ι : Type uι) [Fintype ι] (bound : ℕ) :
+    ∃ f : ι → ℕ, Function.Injective f ∧
+      ∀ i, Nat.Prime (f i) ∧ bound < f i := by
+  let e := Fintype.equivFin ι
+  refine ⟨fun i => primeSequence bound (e i).val, ?_, ?_⟩
+  · intro i j hij
+    apply e.injective
+    apply Fin.ext
+    exact (primeSequence_strictMono bound).injective hij
+  · intro i
+    exact ⟨primeSequence_prime _ _, primeSequence_gt _ _⟩
+
+theorem exists_positive_combination {a b r : ℕ} (hab : Nat.Coprime a b)
+    (ha : 1 < a) (hb : 1 < b) (hr : a * b + a + b < r) :
+    ∃ A B : ℕ, 0 < A ∧ 0 < B ∧ r = A * a + B * b := by
+  have hmem : r - a - b ∈ AddSubmonoid.closure ({a, b} : Set ℕ) := by
+    by_contra hnot
+    have hle := (frobeniusNumber_pair hab ha hb).2 hnot
+    have hbound : a * b - a - b ≤ a * b :=
+      (Nat.sub_le (a * b - a) b).trans (Nat.sub_le (a * b) a)
+    omega
+  obtain ⟨A, B, hAB⟩ := (AddSubmonoid.mem_closure_pair a b (r - a - b)).mp hmem
+  simp only [smul_eq_mul] at hAB
+  refine ⟨A + 1, B + 1, by omega, by omega, ?_⟩
+  calc
+    r = (r - a - b) + a + b := by omega
+    _ = (A * a + B * b) + a + b := by rw [hAB]
+    _ = (A + 1) * a + (B + 1) * b := by ring
+
+private abbrev UsefulTag (p : ℕ) := Channel p ⊕ Channel p
+private abbrev FreshUsefulTag (p : ℕ) := Column p ⊕ Channel p
+private abbrev DigitTag (p : ℕ) := Σ i, Label p i
+
+private def usefulTagEquiv (p : ℕ) :
+    UsefulTag p ≃ Fin 2 ⊕ FreshUsefulTag p :=
+  (Equiv.sumCongr (Equiv.sumComm (Column p) (Fin 2))
+    (Equiv.refl (Channel p))).trans (Equiv.sumAssoc (Fin 2) (Column p) (Channel p))
+
+private def usefulPrime {p : ℕ} (f : FreshUsefulTag p → ℕ) : UsefulTag p → ℕ :=
+  Sum.elim (fun t : Fin 2 => t.val + 2) f ∘ usefulTagEquiv p
+
+private theorem usefulPrime_injective {p : ℕ} {f : FreshUsefulTag p → ℕ}
+    (hf : Function.Injective f) (hlarge : ∀ t, 3 < f t) :
+    Function.Injective (usefulPrime f) := by
+  have hseed : Function.Injective (fun t : Fin 2 => t.val + 2) := by
+    intro t u htu
+    apply Fin.ext
+    change t.val + 2 = u.val + 2 at htu
+    exact Nat.add_right_cancel htu
+  have hsep : ∀ t : Fin 2, ∀ i, t.val + 2 ≠ f i := by
+    intro t i heq
+    have ht := t.isLt
+    have hi := hlarge i
+    omega
+  exact (hseed.sumElim hf hsep).comp (usefulTagEquiv p).injective
+
+private theorem usefulPrime_prime {p : ℕ} {f : FreshUsefulTag p → ℕ}
+    (hf : ∀ i, Nat.Prime (f i)) (i : UsefulTag p) :
+    Nat.Prime (usefulPrime f i) := by
+  rcases i with (i | t) | i
+  · exact hf (.inl i)
+  · change Nat.Prime (t.val + 2)
+    fin_cases t
+    · exact Nat.prime_two
+    · exact Nat.prime_three
+  · exact hf (.inr i)
+
+private def attachDigits {p : ℕ} (f : UsefulTag p → ℕ) (g : DigitTag p → ℕ) :
+    PrimeTag p → ℕ :=
+  Sum.elim f g ∘ (Equiv.sumAssoc (Channel p) (Channel p) (DigitTag p)).symm
+
+private theorem attachDigits_injective {p : ℕ}
+    {f : UsefulTag p → ℕ} {g : DigitTag p → ℕ}
+    (hf : Function.Injective f) (hg : Function.Injective g)
+    (hsep : ∀ i j, f i ≠ g j) : Function.Injective (attachDigits f g) :=
+  (hf.sumElim hg hsep).comp
+    (Equiv.sumAssoc (Channel p) (Channel p) (DigitTag p)).symm.injective
+
+theorem exists_parameters {p : ℕ} (hp : Nat.Prime p) (hlarge : 200 < p) :
+    Nonempty (EncodingParameters p) := by
+  classical
+  let : NeZero p := ⟨hp.ne_zero⟩
+  obtain ⟨f, hfinj, hf⟩ := exists_prime_family (FreshUsefulTag p) (p ^ 4 + p + 3)
+  let useful : UsefulTag p → ℕ := usefulPrime f
+  have huseful_prime : ∀ i, Nat.Prime (useful i) :=
+    usefulPrime_prime (fun i => (hf i).1)
+  have huseful_inj : Function.Injective useful := by
+    apply usefulPrime_injective hfinj
+    intro i
+    have := (hf i).2
+    omega
+  have huseful_ne_p : ∀ i, useful i ≠ p := by
+    intro i
+    rcases i with (i | t) | i
+    · change f (.inl i) ≠ p
+      have := (hf (.inl i)).2
+      omega
+    · change t.val + 2 ≠ p
+      have := t.isLt
+      omega
+    · change f (.inr i) ≠ p
+      have := (hf (.inr i)).2
+      omega
+  let threshold : Channel p → ℕ := fun i =>
+    useful (.inl i) * useful (.inr i) + useful (.inl i) + useful (.inr i)
+  let bound := p + Finset.univ.sup useful + Finset.univ.sup threshold
+  obtain ⟨g, hginj, hg⟩ := exists_prime_family (DigitTag p) bound
+  have hseparate : ∀ i j, useful i < g j := by
+    intro i j
+    have hi : useful i ≤ Finset.univ.sup useful := Finset.le_sup (Finset.mem_univ i)
+    have hj := (hg j).2
+    dsimp [bound] at hj
+    omega
+  have hthreshold : ∀ i j, threshold i < g ⟨i, j⟩ := by
+    intro i j
+    have hi : threshold i ≤ Finset.univ.sup threshold :=
+      Finset.le_sup (Finset.mem_univ i)
+    have hj := (hg ⟨i, j⟩).2
+    dsimp [bound] at hj
+    omega
+  have hrepresentation : ∀ i (j : Label p i), ∃ A B : ℕ,
+      0 < A ∧ 0 < B ∧ g ⟨i, j⟩ = A * useful (.inl i) + B * useful (.inr i) := by
+    intro i j
+    have hcop : Nat.Coprime (useful (.inl i)) (useful (.inr i)) := by
+      apply (Nat.coprime_primes (huseful_prime _) (huseful_prime _)).2
+      intro h
+      have htags := huseful_inj h
+      cases htags
+    exact exists_positive_combination hcop
+      (huseful_prime _).one_lt (huseful_prime _).one_lt (hthreshold i j)
+  choose A B hA hB hrepresentation using hrepresentation
+  refine ⟨{
+    p_prime := hp
+    p_large := hlarge
+    primeAt := attachDigits useful g
+    prime_isPrime := ?_
+    prime_injective := attachDigits_injective huseful_inj hginj
+      (fun i j => (hseparate i j).ne)
+    prime_ne_p := ?_
+    seed_a_eq := ?_
+    seed_b_large := ?_
+    blockA := A
+    blockB := B
+    blockA_pos := hA
+    blockB_pos := hB
+    digit_representation := ?_
+  }⟩
+  · intro t
+    rcases t with i | (i | j)
+    · exact huseful_prime (.inl i)
+    · exact huseful_prime (.inr i)
+    · exact (hg j).1
+  · intro t
+    rcases t with i | (i | j)
+    · exact huseful_ne_p (.inl i)
+    · exact huseful_ne_p (.inr i)
+    · change g j ≠ p
+      have hj := (hg j).2
+      dsimp [bound] at hj
+      omega
+  · intro t
+    rfl
+  · intro t
+    change p ^ 4 < f (.inr (.inr t))
+    have ht := (hf (.inr (.inr t))).2
+    omega
+  · intro i j
+    exact hrepresentation i j
+
+end EncodingPrimeSelection
+
+theorem exists_encodingParameters {p : ℕ} (hp : Nat.Prime p) (hlarge : 200 < p) :
+    Nonempty (EncodingParameters p) :=
+  EncodingPrimeSelection.exists_parameters hp hlarge
+
+def encodingParameters {p : ℕ} (hp : Nat.Prime p) (hlarge : 200 < p) :
+    EncodingParameters p :=
+  Classical.choice (exists_encodingParameters hp hlarge)
+
+end PeriodicTilingThree
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/EuclideanBasic.lean b/lean/OAI/Geometry/PeriodicTiling/EuclideanBasic.lean
new file mode 100644
index 0000000000000000000000000000000000000000..d231ea92d41dbe60dce92b3651b4fc4fa888735c
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/EuclideanBasic.lean
@@ -0,0 +1,95 @@
+import OAI.Geometry.PeriodicTiling.TilingBasic
+import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
+import Mathlib.Algebra.Module.ZLattice.Basic
+
+namespace OAI
+
+noncomputable section
+
+namespace PeriodicTilingThree
+
+open MeasureTheory
+
+@[simp] theorem castLattice_apply {d : ℕ} (z : Lattice d) (i : Fin d) :
+    castLattice z i = (z i : ℝ) := rfl
+
+theorem castLattice_injective {d : ℕ} :
+    Function.Injective (castLattice : Lattice d → Space d) := by
+  intro z w h
+  funext i
+  exact Int.cast_injective (congrArg (fun x : Space d => x i) h)
+
+def unitVoxel {d : ℕ} (c : Space d) : Set (Space d) :=
+  {x | x - c ∈ unitCube d}
+
+theorem mem_unitCube {d : ℕ} {x : Space d} :
+    x ∈ unitCube d ↔ ∀ i, 0 ≤ x i ∧ x i ≤ 1 := by
+  constructor
+  · intro h i
+    exact ⟨h.1 i, h.2 i⟩
+  · intro h
+    exact ⟨fun i => (h i).1, fun i => (h i).2⟩
+
+@[simp] theorem mem_unitVoxel {d : ℕ} {c x : Space d} :
+    x ∈ unitVoxel c ↔ x - c ∈ unitCube d := Iff.rfl
+
+theorem unitVoxel_eq_Icc {d : ℕ} (c : Space d) :
+    unitVoxel c = Set.Icc c (c + 1) := by
+  ext x
+  constructor
+  · intro hx
+    have h := mem_unitCube.mp hx
+    constructor
+    · intro i
+      exact sub_nonneg.mp (h i).1
+    · intro i
+      have hi := (sub_le_iff_le_add.mp (h i).2)
+      simpa [add_comm] using hi
+  · intro hx
+    apply mem_unitCube.mpr
+    intro i
+    constructor
+    · exact sub_nonneg.mpr (hx.1 i)
+    · apply sub_le_iff_le_add.mpr
+      simpa [add_comm] using hx.2 i
+
+@[simp] theorem mem_Thickening {d : ℕ} {F : Finset (Lattice d)} {x : Space d} :
+    x ∈ Thickening F ↔ ∃ z ∈ F, x - castLattice z ∈ unitCube d := Iff.rfl
+
+theorem unitVoxel_subset_Thickening {d : ℕ} {F : Finset (Lattice d)}
+    {z : Lattice d} (hz : z ∈ F) : unitVoxel (castLattice z) ⊆ Thickening F := by
+  intro x hx
+  exact ⟨z, hz, hx⟩
+
+def TypedAETiles {d : ℕ} (Ω0 Ω1 C C1 : Set (Space d)) : Prop := by
+  classical
+  exact ∀ᵐ x ∂volume, ∃! c : C,
+    x - (c : Space d) ∈ if (c : Space d) ∈ C1 then Ω1 else Ω0
+
+def scaledCast {d : ℕ} (m : ℕ) : Lattice d →+ Space d where
+  toFun z := (m : ℝ) • castLattice z
+  map_zero' := by simp
+  map_add' z w := by simp [smul_add]
+
+@[simp] theorem scaledCast_apply {d : ℕ} (m : ℕ) (z : Lattice d) :
+    scaledCast m z = (m : ℝ) • castLattice z := rfl
+
+theorem scaledCast_injective {d : ℕ} {m : ℕ} (hm : m ≠ 0) :
+    Function.Injective (scaledCast m : Lattice d → Space d) := by
+  intro z w h
+  funext i
+  have hi := congrArg (fun x : Space d => x i) h
+  change (m : ℝ) * (z i : ℝ) = (m : ℝ) * (w i : ℝ) at hi
+  exact Int.cast_injective (mul_left_cancel₀ (Nat.cast_ne_zero.mpr hm) hi)
+
+def scaledGrid (m : ℕ) : AddSubgroup (Space 3) :=
+  (scaledCast m : Lattice 3 →+ Space 3).range
+
+theorem mem_scaledGrid {m : ℕ} {x : Space 3} :
+    x ∈ scaledGrid m ↔ ∃ z : Lattice 3, (m : ℝ) • castLattice z = x := Iff.rfl
+
+end PeriodicTilingThree
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/FiniteCylinder.lean b/lean/OAI/Geometry/PeriodicTiling/FiniteCylinder.lean
new file mode 100644
index 0000000000000000000000000000000000000000..f79fb392142e9f522cf44dbbefa0fa0079ca9a44
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/FiniteCylinder.lean
@@ -0,0 +1,125 @@
+import Mathlib.Data.Fintype.Pigeonhole
+import Mathlib.Data.ZMod.Basic
+import Mathlib.Tactic.Linarith
+import Lean.Elab.Tactic.Omega
+
+namespace OAI
+
+universe uA
+
+namespace PeriodicTilingThree.FiniteCylinder
+
+def wrap (i L n : ℤ) : ℤ := i + (n - i) % L
+
+theorem wrap_bounds (i n : ℤ) {L : ℤ} (hL : 0 < L) :
+    i ≤ wrap i L n ∧ wrap i L n < i + L := by
+  have hnonneg := Int.emod_nonneg (n - i) (ne_of_gt hL)
+  have hlt := Int.emod_lt_of_pos (n - i) hL
+  dsimp [wrap]
+  omega
+
+theorem wrap_eq_self {i L n : ℤ} (hi : i ≤ n) (hn : n < i + L) :
+    wrap i L n = n := by
+  unfold wrap
+  rw [Int.emod_eq_of_lt (by omega) (by omega)]
+  omega
+
+theorem wrap_add_period (i L n : ℤ) : wrap i L (n + L) = wrap i L n := by
+  unfold wrap
+  rw [show n + L - i = (n - i) + L by omega]
+  simp
+
+theorem wrap_dvd (i L n : ℤ) : L ∣ n - wrap i L n := by
+  refine ⟨(n - i) / L, ?_⟩
+  have h := Int.emod_add_mul_ediv (n - i) L
+  dsimp [wrap]
+  omega
+
+theorem wrap_shift (i L n k : ℤ) :
+    wrap i L (n + k) = wrap i L (wrap i L n + k) := by
+  unfold wrap
+  congr 1
+  rw [show n + k - i = (n - i) + k by omega,
+    show i + (n - i) % L + k - i = (n - i) % L + k by omega]
+  exact (Int.emod_add_emod (n - i) L k).symm
+
+theorem window_at_wrap {A : Type uA} (a : ℤ → A) (H : ℕ) (i L : ℤ)
+    (hL : (H : ℤ) < L)
+    (hrows : ∀ c : Fin H, a (i + (c : ℕ)) = a (i + L + (c : ℕ)))
+    (n k : ℤ) (hk0 : 0 ≤ k) (hkH : k ≤ H) :
+    a (wrap i L (n + k)) = a (wrap i L n + k) := by
+  have hL0 : 0 < L := by omega
+  obtain ⟨hb0, hbL⟩ := wrap_bounds i n hL0
+  rw [wrap_shift]
+  by_cases hcross : wrap i L n + k < i + L
+  · exact congrArg a (wrap_eq_self (by omega) hcross)
+  · let c : ℤ := wrap i L n + k - (i + L)
+    have hc0 : 0 ≤ c := by dsimp [c]; omega
+    have hcH : c < H := by dsimp [c]; omega
+    have hcL : c < L := by omega
+    let cN : Fin H := ⟨c.toNat, by omega⟩
+    have hcN : ((cN : ℕ) : ℤ) = c := by dsimp [cN]; omega
+    have heq : wrap i L n + k = (i + c) + L := by dsimp [c]; omega
+    have h := hrows cN
+    rw [hcN] at h
+    calc
+      a (wrap i L (wrap i L n + k)) = a (wrap i L ((i + c) + L)) :=
+        congrArg (fun z => a (wrap i L z)) heq
+      _ = a (i + c) := congrArg a
+        ((wrap_add_period i L (i + c)).trans (wrap_eq_self (by omega) (by omega)))
+      _ = a (i + L + c) := h
+      _ = a (wrap i L n + k) := congrArg a (by dsimp [c]; omega)
+
+theorem exists_periodic_windows {A : Type uA} [Fintype A]
+    (a : ℤ → A) (q H : ℕ) (hq : 0 < q) :
+    ∃ (b : ℤ → A) (L : ℤ), 0 < L ∧ (q : ℤ) ∣ L ∧
+      (∀ n : ℤ, b (n + L) = b n) ∧
+      (∀ n : ℤ, ∃ t : ℤ, (q : ℤ) ∣ t - n ∧
+        ∀ k : ℤ, 0 ≤ k → k ≤ H → b (n + k) = a (t + k)) := by
+  classical
+  let : NeZero q := ⟨Nat.ne_of_gt hq⟩
+  let State := ZMod q × (Fin H → A)
+  let time (k : ℕ) : ℤ := (k : ℤ) * ((H : ℤ) + 1)
+  let state (k : ℕ) : State :=
+    ((time k : ZMod q), fun c => a (time k + (c : ℕ)))
+  let N := Fintype.card State
+  obtain ⟨x, y, hxy, hs⟩ := Fintype.exists_ne_map_eq_of_card_lt
+    (fun k : Fin (N + 1) => state k.val) (by simp [N])
+  have hvals : x.val ≠ y.val := fun h => hxy (Fin.ext h)
+  have ordered : ∃ x y : ℕ, x < y ∧ state x = state y := by
+    rcases lt_or_gt_of_ne hvals with hlt | hgt
+    · exact ⟨x.val, y.val, hlt, hs⟩
+    · exact ⟨y.val, x.val, hgt, hs.symm⟩
+  obtain ⟨x, y, hxy, hs⟩ := ordered
+  let i := time x
+  let L := time y - time x
+  have hxyZ : (x : ℤ) + 1 ≤ (y : ℤ) := by omega
+  have hL : (H : ℤ) < L := by
+    dsimp [L, time]
+    have hH : (0 : ℤ) ≤ H := by omega
+    nlinarith
+  have hL0 : 0 < L := by omega
+  have hphase : (q : ℤ) ∣ L := by
+    have heq : (time x : ZMod q) = (time y : ZMod q) := congrArg Prod.fst hs
+    exact (ZMod.intCast_eq_intCast_iff_dvd_sub (time x) (time y) q).mp heq
+  have hrows : ∀ c : Fin H, a (i + (c : ℕ)) = a (i + L + (c : ℕ)) := by
+    intro c
+    have heq := congrFun (congrArg Prod.snd hs) c
+    change a (time x + (c : ℕ)) = a (time y + (c : ℕ)) at heq
+    have htime : i + L = time y := by dsimp [i, L]; omega
+    rw [htime]
+    exact heq
+  let b : ℤ → A := fun n => a (wrap i L n)
+  refine ⟨b, L, hL0, hphase, ?_, ?_⟩
+  · intro n
+    exact congrArg a (wrap_add_period i L n)
+  · intro n
+    refine ⟨wrap i L n, ?_, ?_⟩
+    · have hd : (q : ℤ) ∣ n - wrap i L n := hphase.trans (wrap_dvd i L n)
+      simpa only [neg_sub] using (dvd_neg.mpr hd)
+    · intro k hk0 hkH
+      exact window_at_wrap a H i L hL hrows n k hk0 hkH
+
+end PeriodicTilingThree.FiniteCylinder
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/FiniteDifferencePolynomial.lean b/lean/OAI/Geometry/PeriodicTiling/FiniteDifferencePolynomial.lean
new file mode 100644
index 0000000000000000000000000000000000000000..0cae29441746d4bbc72c806352d150f9ed60bc98
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/FiniteDifferencePolynomial.lean
@@ -0,0 +1,142 @@
+import OAI.Geometry.PeriodicTiling.ForwardDifference
+import Mathlib.Algebra.BigOperators.Fin
+import Mathlib.Basic.Real.Basic
+import Mathlib.RingTheory.Binomial
+import Mathlib.Topology.Instances.AddCircle.Defs
+
+namespace OAI
+
+universe uA
+
+noncomputable section
+
+namespace PeriodicTilingThree
+
+open scoped BigOperators
+
+section AddCommGroup
+
+variable {A : Type uA} [AddCommGroup A]
+
+theorem intSequence_eq_of_forwardDiff_eq {f g : ℤ → A}
+    (hd : forwardDiff f = forwardDiff g) (hzero : f 0 = g 0) : f = g := by
+  have hsucc (n : ℤ) (hn : f n = g n) : f (n + 1) = g (n + 1) := by
+    have h := congrFun hd n
+    change f (n + 1) - f n = g (n + 1) - g n at h
+    rw [hn] at h
+    exact sub_left_inj.mp h
+  have hpred (n : ℤ) (hn : f n = g n) : f (n - 1) = g (n - 1) := by
+    have h := congrFun hd (n - 1)
+    change f (n - 1 + 1) - f (n - 1) = g (n - 1 + 1) - g (n - 1) at h
+    rw [sub_add_cancel, hn] at h
+    exact sub_right_inj.mp h
+  funext n
+  refine Int.induction_on n hzero ?_ ?_
+  · intro m hm
+    exact hsucc (m : ℤ) hm
+  · intro m hm
+    exact hpred (-(m : ℤ)) hm
+
+def newtonSequence {k : ℕ} (c : Fin k → A) (n : ℤ) : A :=
+  ∑ i : Fin k, Ring.choose n i.val • c i
+
+theorem newtonSequence_cons {k : ℕ} (a : A) (c : Fin k → A) (n : ℤ) :
+    newtonSequence (Fin.cons a c) n =
+      a + ∑ i : Fin k, Ring.choose n (i.val + 1) • c i := by
+  rw [newtonSequence, Fin.sum_univ_succ]
+  simp only [Fin.val_zero, Ring.choose_zero_right, one_zsmul,
+    Fin.cons_zero, Fin.val_succ, Fin.cons_succ]
+
+@[simp] theorem newtonSequence_cons_zero {k : ℕ} (a : A) (c : Fin k → A) :
+    newtonSequence (Fin.cons a c) 0 = a := by
+  simp [newtonSequence_cons]
+
+theorem forwardDiff_newtonSequence_cons {k : ℕ} (a : A) (c : Fin k → A) :
+    forwardDiff (newtonSequence (Fin.cons a c)) = newtonSequence c := by
+  funext n
+  simp only [forwardDiff_apply, newtonSequence_cons]
+  unfold newtonSequence
+  calc
+    (a + ∑ i : Fin k, Ring.choose (n + 1) (i.val + 1) • c i) -
+        (a + ∑ i : Fin k, Ring.choose n (i.val + 1) • c i) =
+        ∑ i : Fin k,
+          (Ring.choose (n + 1) (i.val + 1) • c i -
+            Ring.choose n (i.val + 1) • c i) := by
+      rw [Finset.sum_sub_distrib]
+      abel
+    _ = ∑ i : Fin k, Ring.choose n i.val • c i := by
+      apply Finset.sum_congr rfl
+      intro i hi
+      rw [Ring.choose_succ_succ, add_zsmul, add_sub_cancel_right]
+
+theorem exists_newtonSequence_of_iter_forwardDiff_eq_zero
+    (k : ℕ) (f : ℤ → A) (hk : (forwardDiff^[k]) f = 0) :
+    ∃ c : Fin k → A, f = newtonSequence c := by
+  induction k generalizing f with
+  | zero =>
+      have hf : f = 0 := by simpa using hk
+      refine ⟨Fin.elim0, ?_⟩
+      subst f
+      funext n
+      simp [newtonSequence]
+  | succ k ih =>
+      have hdiff : (forwardDiff^[k]) (forwardDiff f) = 0 := by
+        simpa only [Function.iterate_succ_apply] using hk
+      obtain ⟨c, hc⟩ := ih (forwardDiff f) hdiff
+      refine ⟨Fin.cons (f 0) c, intSequence_eq_of_forwardDiff_eq ?_ ?_⟩
+      · rw [forwardDiff_newtonSequence_cons]
+        exact hc
+      · simp
+
+end AddCommGroup
+
+def newtonBasisPolynomial (k : ℕ) : Polynomial ℝ :=
+  Polynomial.C ((k.factorial : ℝ)⁻¹) * descPochhammer ℝ k
+
+theorem descPochhammer_eval_int_eq_factorial_mul_choose (k : ℕ) (n : ℤ) :
+    (descPochhammer ℝ k).eval (n : ℝ) =
+      (k.factorial : ℝ) * ((Ring.choose n k : ℤ) : ℝ) := by
+  have h : (descPochhammer ℤ k).eval n = (k.factorial : ℤ) * Ring.choose n k := by
+    simpa only [Polynomial.eval_eq_smeval, nsmul_eq_mul] using
+      (Ring.descPochhammer_eq_factorial_smul_choose n k)
+  rw [← descPochhammer_eval_cast ℝ k n, h, Int.cast_mul, Int.cast_natCast]
+
+@[simp] theorem newtonBasisPolynomial_eval_int (k : ℕ) (n : ℤ) :
+    (newtonBasisPolynomial k).eval (n : ℝ) = ((Ring.choose n k : ℤ) : ℝ) := by
+  have hk : (k.factorial : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr (Nat.factorial_ne_zero k)
+  rw [newtonBasisPolynomial, Polynomial.eval_mul, Polynomial.eval_C,
+    descPochhammer_eval_int_eq_factorial_mul_choose, ← mul_assoc, inv_mul_cancel₀ hk, one_mul]
+
+def newtonPolynomial {k : ℕ} (c : Fin k → ℝ) : Polynomial ℝ :=
+  ∑ i : Fin k, Polynomial.C (c i) * newtonBasisPolynomial i.val
+
+theorem newtonPolynomial_eval_int {k : ℕ} (c : Fin k → ℝ) (n : ℤ) :
+    (newtonPolynomial c).eval (n : ℝ) =
+      ∑ i : Fin k, Ring.choose n i.val • c i := by
+  rw [newtonPolynomial, Polynomial.eval_finsetSum]
+  apply Finset.sum_congr rfl
+  intro i hi
+  simp only [Polynomial.eval_mul, Polynomial.eval_C,
+    newtonBasisPolynomial_eval_int, zsmul_eq_mul]
+  exact mul_comm (c i) ((Ring.choose n i.val : ℤ) : ℝ)
+
+theorem exists_polynomial_lift_of_iter_forwardDiff_eq_zero
+    (f : ℤ → AddCircle (1 : ℝ)) (k : ℕ) (hk : (forwardDiff^[k]) f = 0) :
+    ∃ P : Polynomial ℝ, ∀ n : ℤ, ((P.eval (n : ℝ) : ℝ) : AddCircle (1 : ℝ)) = f n := by
+  obtain ⟨c, hc⟩ := exists_newtonSequence_of_iter_forwardDiff_eq_zero k f hk
+  have hrepr : ∀ i : Fin k, ∃ x : ℝ, (x : AddCircle (1 : ℝ)) = c i := by
+    intro i
+    exact QuotientAddGroup.mk_surjective (c i)
+  choose a ha using hrepr
+  refine ⟨newtonPolynomial a, ?_⟩
+  intro n
+  change QuotientAddGroup.mk' (AddSubgroup.zmultiples (1 : ℝ))
+    ((newtonPolynomial a).eval (n : ℝ)) = f n
+  rw [newtonPolynomial_eval_int, map_sum, hc]
+  simp only [newtonSequence, map_zsmul, QuotientAddGroup.mk'_apply, ha]
+
+end PeriodicTilingThree
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/FiniteEventUnion.lean b/lean/OAI/Geometry/PeriodicTiling/FiniteEventUnion.lean
new file mode 100644
index 0000000000000000000000000000000000000000..b66c569129c7248f11102d09b1dfd18681a10cf0
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/FiniteEventUnion.lean
@@ -0,0 +1,48 @@
+import Mathlib.Algebra.BigOperators.Ring.Finset
+import Mathlib.Algebra.Order.BigOperators.Group.Finset
+import Mathlib.Algebra.Order.Field.Rat
+
+namespace OAI
+
+universe uΩ uI
+
+namespace PeriodicTilingThree
+
+open scoped BigOperators
+
+theorem exists_avoiding_of_sum_card_lt {Ω : Type uΩ} {I : Type uI} [Fintype Ω]
+    (events : Finset I) (bad : I → Finset Ω)
+    (hsmall : (∑ i ∈ events, (bad i).card) < Fintype.card Ω) :
+    ∃ ω, ∀ i ∈ events, ω ∉ bad i := by
+  classical
+  have hUnion : (events.biUnion bad).card < (Finset.univ : Finset Ω).card := by
+    rw [Finset.card_univ]
+    exact lt_of_le_of_lt Finset.card_biUnion_le hsmall
+  obtain ⟨ω, _, hω⟩ := Finset.exists_mem_notMem_of_card_lt_card hUnion
+  refine ⟨ω, ?_⟩
+  intro i hi hbad
+  exact hω (Finset.mem_biUnion.mpr ⟨i, hi, hbad⟩)
+
+theorem exists_avoiding_of_uniform_ratio_lt {Ω : Type uΩ} {I : Type uI} [Fintype Ω]
+    (events : Finset I) (bad : I → Finset Ω) (r : ℚ)
+    (hΩ : 0 < Fintype.card Ω)
+    (hbound : ∀ i ∈ events, ((bad i).card : ℚ) / (Fintype.card Ω : ℚ) ≤ r)
+    (hsmall : (events.card : ℚ) * r < 1) :
+    ∃ ω, ∀ i ∈ events, ω ∉ bad i := by
+  have hpos : (0 : ℚ) < (Fintype.card Ω : ℚ) := Nat.cast_pos.mpr hΩ
+  have hsum : (∑ i ∈ events, ((bad i).card : ℚ)) < (Fintype.card Ω : ℚ) := by
+    calc
+      (∑ i ∈ events, ((bad i).card : ℚ))
+          ≤ ∑ i ∈ events, r * (Fintype.card Ω : ℚ) :=
+        Finset.sum_le_sum fun i hi => (div_le_iff₀ hpos).mp (hbound i hi)
+      _ = (events.card : ℚ) * r * (Fintype.card Ω : ℚ) := by
+        simp only [Finset.sum_const, nsmul_eq_mul, mul_assoc]
+      _ < 1 * (Fintype.card Ω : ℚ) := mul_lt_mul_of_pos_right hsmall hpos
+      _ = (Fintype.card Ω : ℚ) := one_mul _
+  apply exists_avoiding_of_sum_card_lt events bad
+  apply (Nat.cast_lt (α := ℚ)).mp
+  simpa only [Nat.cast_sum] using hsum
+
+end PeriodicTilingThree
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/FiniteMasks.lean b/lean/OAI/Geometry/PeriodicTiling/FiniteMasks.lean
new file mode 100644
index 0000000000000000000000000000000000000000..5e73b3c05068240332ab8a68dde4854b90d7b5fe
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/FiniteMasks.lean
@@ -0,0 +1,220 @@
+import OAI.Geometry.PeriodicTiling.TilingBasic
+import Mathlib.Algebra.MonoidAlgebra.Basic
+import Mathlib.Algebra.BigOperators.Ring.Finset
+import Mathlib.Basic.Real.Basic
+import Mathlib.Tactic.Abel
+
+namespace OAI
+
+universe uG uR uι
+
+noncomputable section
+
+open scoped BigOperators Classical
+
+namespace PeriodicTilingThree
+
+variable {G : Type uG}
+
+def tileIndicator (R : Type uR) [Zero R] [One R] (A : Set G) (x : G) : R :=
+  if x ∈ A then 1 else 0
+
+theorem tileIndicator_nonneg (A : Set G) (x : G) :
+    0 ≤ tileIndicator ℝ A x := by
+  classical
+  unfold tileIndicator
+  split_ifs <;> simp
+
+theorem tileIndicator_le_one (A : Set G) (x : G) :
+    tileIndicator ℝ A x ≤ 1 := by
+  classical
+  unfold tileIndicator
+  split_ifs <;> simp
+
+theorem tileIndicator_nat_le_one (A : Set G) (x : G) :
+    tileIndicator ℕ A x ≤ 1 := by
+  classical
+  unfold tileIndicator
+  split_ifs <;> simp
+
+theorem sum_tileIndicator_cast (R : Type uR) [AddCommMonoidWithOne R]
+    {ι : Type uι} (s : Finset ι) (A : Set G) (v : ι → G) :
+    ((∑ i ∈ s, tileIndicator ℕ A (v i) : ℕ) : R) =
+      ∑ i ∈ s, tileIndicator R A (v i) := by
+  classical
+  simp only [Nat.cast_sum]
+  apply Finset.sum_congr rfl
+  intro i hi
+  by_cases h : v i ∈ A <;> simp [tileIndicator, h]
+
+section Counts
+
+variable [AddCommGroup G]
+
+def tileCount (F : Finset G) (A : Set G) (x : G) : ℕ :=
+  ∑ f ∈ F, tileIndicator ℕ A (x - f)
+
+theorem tileCount_eq_card_filter (F : Finset G) (A : Set G) (x : G) :
+    tileCount F A x = (F.filter fun f => x - f ∈ A).card := by
+  classical
+  simp [tileCount, tileIndicator]
+
+theorem tileCount_le_card (F : Finset G) (A : Set G) (x : G) :
+    tileCount F A x ≤ F.card := by
+  classical
+  rw [tileCount_eq_card_filter]
+  exact Finset.card_filter_le _ _
+
+theorem tileCount_cast (R : Type uR) [AddCommMonoidWithOne R]
+    (F : Finset G) (A : Set G) (x : G) :
+    (tileCount F A x : R) = ∑ f ∈ F, tileIndicator R A (x - f) :=
+  sum_tileIndicator_cast R F A (fun f => x - f)
+
+theorem tiles_iff_tileCount_eq_one {F : Finset G} {A : Set G} :
+    Tiles F A ↔ ∀ x : G, tileCount F A x = 1 := by
+  classical
+  rw [tiles_iff_unique_tile]
+  constructor
+  · intro h x
+    obtain ⟨f, hf, hu⟩ := h x
+    have he : (F.filter fun g => x - g ∈ A) = {f.val} := by
+      ext g
+      simp only [Finset.mem_filter, Finset.mem_singleton]
+      constructor
+      · rintro ⟨hg, hga⟩
+        exact congrArg Subtype.val (hu ⟨g, hg⟩ hga)
+      · intro hg
+        subst g
+        exact ⟨f.property, hf⟩
+    rw [tileCount_eq_card_filter, he, Finset.card_singleton]
+  · intro h x
+    obtain ⟨f, hf⟩ := Finset.card_eq_one.mp ((tileCount_eq_card_filter F A x).symm.trans (h x))
+    have hm : f ∈ F ∧ x - f ∈ A := by
+      apply Finset.mem_filter.mp
+      rw [hf]
+      exact Finset.mem_singleton_self f
+    refine ⟨⟨f, hm.1⟩, hm.2, ?_⟩
+    intro g hg
+    apply Subtype.ext
+    have hg' : g.val ∈ F.filter fun f => x - f ∈ A :=
+      Finset.mem_filter.mpr ⟨g.property, hg⟩
+    simpa only [hf, Finset.mem_singleton] using hg'
+
+theorem Tiles.tileCount_eq_one {F : Finset G} {A : Set G}
+    (h : Tiles F A) (x : G) : tileCount F A x = 1 :=
+  tiles_iff_tileCount_eq_one.mp h x
+
+theorem Tiles.sum_tileIndicator_eq_one {F : Finset G} {A : Set G}
+    (h : Tiles F A) (R : Type uR) [AddCommMonoidWithOne R] (x : G) :
+    (∑ f ∈ F, tileIndicator R A (x - f)) = 1 := by
+  rw [← tileCount_cast, h.tileCount_eq_one x, Nat.cast_one]
+
+end Counts
+
+section MaskAction
+
+variable {R : Type uR} [Semiring R] [AddCommGroup G]
+
+def maskAction (m : AddMonoidAlgebra R G) (a : G → R) (x : G) : R :=
+  m.coeff.sum fun y c => c * a (x - y)
+
+@[simp] theorem maskAction_zero (a : G → R) (x : G) :
+    maskAction (0 : AddMonoidAlgebra R G) a x = 0 := by
+  simp [maskAction]
+
+@[simp] theorem maskAction_single (y : G) (c : R) (a : G → R) (x : G) :
+    maskAction (AddMonoidAlgebra.single y c) a x = c * a (x - y) := by
+  simp [maskAction]
+
+theorem maskAction_add (m n : AddMonoidAlgebra R G) (a : G → R) (x : G) :
+    maskAction (m + n) a x = maskAction m a x + maskAction n a x := by
+  simpa only [maskAction, AddMonoidAlgebra.coeff_add] using
+    (Finsupp.sum_add_index' (f := m.coeff) (g := n.coeff)
+      (h := fun y c => c * a (x - y))
+      (fun _ => zero_mul _) (fun _ _ _ => add_mul _ _ _))
+
+theorem maskAction_sum {ι : Type uι} (s : Finset ι)
+    (m : ι → AddMonoidAlgebra R G) (a : G → R) (x : G) :
+    maskAction (∑ i ∈ s, m i) a x = ∑ i ∈ s, maskAction (m i) a x := by
+  simpa only [maskAction, AddMonoidAlgebra.coeff_sum] using
+    (Finsupp.sum_finsetSum_index (s := s) (g := fun i => (m i).coeff)
+      (h := fun y c => c * a (x - y))
+      (fun _ => zero_mul _) (fun _ _ _ => add_mul _ _ _)).symm
+
+private theorem maskAction_single_mul (y : G) (c : R)
+    (m : AddMonoidAlgebra R G) (a : G → R) (x : G) :
+    maskAction (AddMonoidAlgebra.single y c * m) a x =
+      c * maskAction m a (x - y) := by
+  induction m using AddMonoidAlgebra.induction with
+  | zero => simp
+  | single_add z d m hz hd ih =>
+    rw [mul_add, maskAction_add, maskAction_add, mul_add, ih]
+    rw [AddMonoidAlgebra.single_mul_single, maskAction_single, maskAction_single]
+    rw [mul_assoc, sub_sub]
+
+theorem maskAction_mul (m n : AddMonoidAlgebra R G) (a : G → R) (x : G) :
+    maskAction (m * n) a x = maskAction m (fun y => maskAction n a y) x := by
+  induction m using AddMonoidAlgebra.induction with
+  | zero => simp
+  | single_add y c m hy hc ih =>
+    rw [add_mul, maskAction_add, maskAction_add, ih]
+    rw [maskAction_single_mul, maskAction_single]
+
+@[simp] theorem maskAction_one (a : G → R) (x : G) :
+    maskAction (1 : AddMonoidAlgebra R G) a x = a x := by
+  rw [AddMonoidAlgebra.one_def, maskAction_single]
+  simp
+
+theorem maskAction_translate (m : AddMonoidAlgebra R G) (a : G → R) (x v : G) :
+    maskAction m (fun y => a (y + v)) x = maskAction m a (x + v) := by
+  unfold maskAction
+  apply Finsupp.sum_congr
+  intro y hy
+  congr 2
+  abel_nf
+
+def familyMask (R : Type uR) [Semiring R] {ι : Type uι} (s : Finset ι) (v : ι → G) :
+    AddMonoidAlgebra R G :=
+  ∑ i ∈ s, AddMonoidAlgebra.single (v i) 1
+
+def tileMask (R : Type uR) [Semiring R] (F : Finset G) : AddMonoidAlgebra R G :=
+  familyMask R F id
+
+theorem maskAction_familyMask {ι : Type uι} (s : Finset ι) (v : ι → G)
+    (a : G → R) (x : G) :
+    maskAction (familyMask R s v) a x = ∑ i ∈ s, a (x - v i) := by
+  simp only [familyMask, maskAction_sum, maskAction_single, one_mul]
+
+theorem maskAction_tileMask (F : Finset G) (a : G → R) (x : G) :
+    maskAction (tileMask R F) a x = ∑ f ∈ F, a (x - f) :=
+  maskAction_familyMask F id a x
+
+theorem maskAction_familyMask_const {ι : Type uι} (s : Finset ι) (v : ι → G)
+    (c : R) (x : G) :
+    maskAction (familyMask R s v) (fun _ => c) x = (s.card : R) * c := by
+  simp [maskAction_familyMask, nsmul_eq_mul]
+
+theorem maskAction_familyMask_pow_const {ι : Type uι} (s : Finset ι) (v : ι → G)
+    (n : ℕ) (c : R) (x : G) :
+    maskAction (familyMask R s v ^ n) (fun _ => c) x = (s.card : R) ^ n * c := by
+  induction n generalizing x with
+  | zero => simp
+  | succ n ih =>
+    rw [pow_succ', maskAction_mul]
+    simp_rw [ih]
+    rw [maskAction_familyMask_const, pow_succ', mul_assoc]
+
+theorem maskAction_familyMask_pow_succ {ι : Type uι} (s : Finset ι) (v : ι → G)
+    (a : G → R) (h : ∀ x, (∑ i ∈ s, a (x - v i)) = 1) (n : ℕ) (x : G) :
+    maskAction (familyMask R s v ^ (n + 1)) a x = (s.card : R) ^ n := by
+  rw [pow_succ, maskAction_mul]
+  simp_rw [maskAction_familyMask, h]
+  rw [maskAction_familyMask_pow_const, mul_one]
+
+end MaskAction
+
+end PeriodicTilingThree
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/FinitePairCounting.lean b/lean/OAI/Geometry/PeriodicTiling/FinitePairCounting.lean
new file mode 100644
index 0000000000000000000000000000000000000000..389a9c1da5d629f063cd779fad930b9732386617
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/FinitePairCounting.lean
@@ -0,0 +1,126 @@
+import Mathlib.Data.Fintype.BigOperators
+import Mathlib.Data.Fintype.Card
+import Mathlib.Data.Fintype.Sum
+import Mathlib.Logic.Equiv.Set
+import Mathlib.Logic.Equiv.Prod
+import Mathlib.Algebra.Order.BigOperators.Ring.Finset
+import Mathlib.Tactic.FinCases
+
+namespace OAI
+
+universe uV uC uι uA uB
+
+namespace PeriodicTilingThree
+
+open scoped BigOperators Classical
+
+section
+
+variable {V : Type uV} {C : Type uC} [Fintype V] [Fintype C] {k : ℕ}
+
+noncomputable def allBadAssignments
+    (pair : (Fin k × Fin 2) ↪ V) (Good : Fin k → Finset (C × C)) :
+    Finset (V → C) := by
+  classical
+  exact Finset.univ.filter fun f =>
+    ∀ j, (f (pair (j, 0)), f (pair (j, 1))) ∉ Good j
+
+@[simp] theorem mem_allBadAssignments
+    (pair : (Fin k × Fin 2) ↪ V) (Good : Fin k → Finset (C × C)) (f : V → C) :
+    f ∈ allBadAssignments pair Good ↔
+      ∀ j, (f (pair (j, 0)), f (pair (j, 1))) ∉ Good j := by
+  classical
+  simp [allBadAssignments]
+
+private abbrev Outside (pair : (Fin k × Fin 2) ↪ V) :=
+  {v : V // v ∉ Set.range pair}
+
+private noncomputable def vertexDecomposition (pair : (Fin k × Fin 2) ↪ V) :
+    ((Fin k × Fin 2) ⊕ Outside pair) ≃ V := by
+  classical
+  exact (Equiv.sumCongr (Equiv.ofInjective pair pair.injective) (Equiv.refl _)).trans
+    (Equiv.Set.sumCompl (Set.range pair))
+
+private theorem vertexDecomposition_inl {V : Type uV} [Fintype V] (pair : (Fin k × Fin 2) ↪ V)
+    (x : Fin k × Fin 2) : vertexDecomposition pair (Sum.inl x) = pair x := rfl
+
+private theorem outside_card (pair : (Fin k × Fin 2) ↪ V) :
+    Fintype.card (Outside pair) = Fintype.card V - 2 * k := by
+  classical
+  have h := Fintype.card_congr (vertexDecomposition pair)
+  simp only [Fintype.card_sum, Fintype.card_prod, Fintype.card_fin] at h
+  omega
+
+private def edgeAssignmentEquiv : ((Fin k × Fin 2) → C) ≃ (Fin k → C × C) where
+  toFun f j := (f (j, 0), f (j, 1))
+  invFun f x := if x.2 = 0 then (f x.1).1 else (f x.1).2
+  left_inv f := by
+    funext x
+    rcases x with ⟨j, b⟩
+    fin_cases b <;> simp
+  right_inv f := by
+    funext j
+    simp
+
+private noncomputable def assignmentDecomposition (pair : (Fin k × Fin 2) ↪ V) :
+    (V → C) ≃ (Fin k → C × C) × (Outside pair → C) :=
+  ((Equiv.arrowCongr (vertexDecomposition pair).symm (Equiv.refl C)).trans
+    (Equiv.sumArrowEquivProdArrow _ _ _)).trans
+    (Equiv.prodCongr edgeAssignmentEquiv (Equiv.refl _))
+
+private theorem assignmentDecomposition_apply
+    {V : Type uV} {C : Type uC} [Fintype V] [Fintype C] (pair : (Fin k × Fin 2) ↪ V)
+    (f : V → C) (j : Fin k) :
+    (assignmentDecomposition pair f).1 j = (f (pair (j, 0)), f (pair (j, 1))) := rfl
+
+private def constrainedProductEquiv {ι : Type uι} {A : Type uA} {B : Type uB} (p : ι → A → Prop) :
+    {x : (ι → A) × B // ∀ i, p i (x.1 i)} ≃ (∀ i, {a : A // p i a}) × B where
+  toFun x := (fun i => ⟨x.1.1 i, x.2 i⟩, x.1.2)
+  invFun x := ⟨(fun i => (x.1 i).1, x.2), fun i => (x.1 i).2⟩
+  left_inv x := by cases x; rfl
+  right_inv x := by cases x; rfl
+
+private noncomputable def badAssignmentEquiv
+    (pair : (Fin k × Fin 2) ↪ V) (Good : Fin k → Finset (C × C)) :
+    {f : V → C // ∀ j, (f (pair (j, 0)), f (pair (j, 1))) ∉ Good j} ≃
+      (∀ j : Fin k, {c : C × C // c ∉ Good j}) × (Outside pair → C) :=
+  ((assignmentDecomposition pair).subtypeEquiv (fun _ => Iff.rfl)).trans
+    (constrainedProductEquiv fun j c => c ∉ Good j)
+
+theorem allBadAssignments_card
+    (pair : (Fin k × Fin 2) ↪ V) (Good : Fin k → Finset (C × C)) :
+    (allBadAssignments pair Good).card =
+      (∏ j : Fin k, (Fintype.card C ^ 2 - (Good j).card)) *
+        Fintype.card C ^ (Fintype.card V - 2 * k) := by
+  classical
+  have hbad : (allBadAssignments pair Good).card =
+      Fintype.card {f : V → C // ∀ j, (f (pair (j, 0)), f (pair (j, 1))) ∉ Good j} := by
+    rw [Fintype.card_subtype]
+    rfl
+  have hcolor (j : Fin k) : Fintype.card {c : C × C // c ∉ Good j} =
+      Fintype.card C ^ 2 - (Good j).card := by
+    rw [Fintype.card_subtype_compl]
+    simp [Fintype.card_prod, pow_two]
+  rw [hbad, Fintype.card_congr (badAssignmentEquiv pair Good), Fintype.card_prod,
+    Fintype.card_pi, Fintype.card_fun, outside_card]
+  simp_rw [hcolor]
+
+theorem allBadAssignments_card_le
+    (pair : (Fin k × Fin 2) ↪ V) (Good : Fin k → Finset (C × C)) (g : ℕ)
+    (hgood : ∀ j, g ≤ (Good j).card) :
+    (allBadAssignments pair Good).card ≤
+      (Fintype.card C ^ 2 - g) ^ k * Fintype.card C ^ (Fintype.card V - 2 * k) := by
+  classical
+  rw [allBadAssignments_card]
+  apply Nat.mul_le_mul_right
+  calc
+    (∏ j : Fin k, (Fintype.card C ^ 2 - (Good j).card)) ≤
+        ∏ _j : Fin k, (Fintype.card C ^ 2 - g) :=
+      Finset.prod_le_prod (fun j _ => Nat.sub_le_sub_left (hgood j) _)
+    _ = (Fintype.card C ^ 2 - g) ^ k := by simp
+
+end
+
+end PeriodicTilingThree
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/FinitePairRatio.lean b/lean/OAI/Geometry/PeriodicTiling/FinitePairRatio.lean
new file mode 100644
index 0000000000000000000000000000000000000000..ee4cf599225246675d4d3c241dd4263be7972f9f
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/FinitePairRatio.lean
@@ -0,0 +1,49 @@
+import OAI.Geometry.PeriodicTiling.FinitePairCounting
+import Mathlib.Algebra.Order.Field.Rat
+import Mathlib.Algebra.Order.Field.Power
+
+namespace OAI
+
+universe uV uC
+
+namespace PeriodicTilingThree
+
+open scoped Classical
+
+theorem allBadAssignments_card_ratio_le {V : Type uV} {C : Type uC} [Fintype V] [Fintype C]
+    {k : ℕ} (pair : (Fin k × Fin 2) ↪ V) (Good : Fin k → Finset (C × C))
+    (g : ℕ) (hC : 0 < Fintype.card C) (hgB : g ≤ Fintype.card C ^ 2)
+    (hgood : ∀ j, g ≤ (Good j).card) :
+    ((allBadAssignments pair Good).card : ℚ) / (Fintype.card (V → C) : ℚ) ≤
+      (1 - (g : ℚ) / (Fintype.card C : ℚ) ^ 2) ^ k := by
+  let B : ℚ := Fintype.card C
+  let N : ℕ := Fintype.card V
+  have hBpos : 0 < B := Nat.cast_pos.mpr hC
+  have hBne : B ≠ 0 := ne_of_gt hBpos
+  have hdim : 2 * k ≤ N := by
+    have h := Fintype.card_le_of_embedding pair
+    simpa only [Fintype.card_prod, Fintype.card_fin, mul_comm, N] using h
+  have hcast : ((allBadAssignments pair Good).card : ℚ) ≤
+      (B ^ 2 - (g : ℚ)) ^ k * B ^ (N - 2 * k) := by
+    have h := (Nat.cast_le (α := ℚ)).mpr
+      (allBadAssignments_card_le pair Good g hgood)
+    simpa only [Nat.cast_mul, Nat.cast_pow, Nat.cast_sub hgB, B, N] using h
+  have hsplit : B ^ N = (B ^ 2) ^ k * B ^ (N - 2 * k) := by
+    rw [← pow_mul, ← pow_add, Nat.add_sub_of_le hdim]
+  rw [Fintype.card_fun, Nat.cast_pow]
+  change ((allBadAssignments pair Good).card : ℚ) / B ^ N ≤
+    (1 - (g : ℚ) / B ^ 2) ^ k
+  calc
+    ((allBadAssignments pair Good).card : ℚ) / B ^ N
+        ≤ ((B ^ 2 - (g : ℚ)) ^ k * B ^ (N - 2 * k)) / B ^ N :=
+      div_le_div_of_nonneg_right hcast (pow_pos hBpos N).le
+    _ = (B ^ 2 - (g : ℚ)) ^ k / (B ^ 2) ^ k := by
+      rw [hsplit]
+      exact mul_div_mul_right _ _ (pow_ne_zero _ hBne)
+    _ = ((B ^ 2 - (g : ℚ)) / B ^ 2) ^ k := (div_pow _ _ _).symm
+    _ = (1 - (g : ℚ) / B ^ 2) ^ k := by
+      rw [sub_div, div_self (pow_ne_zero _ hBne)]
+
+end PeriodicTilingThree
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/ForwardDifference.lean b/lean/OAI/Geometry/PeriodicTiling/ForwardDifference.lean
new file mode 100644
index 0000000000000000000000000000000000000000..07aef27f095199292d111ae146968a8287cdb76b
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/ForwardDifference.lean
@@ -0,0 +1,18 @@
+import Mathlib.Algebra.Group.Pi.Basic
+import Mathlib.Logic.Function.Iterate
+
+namespace OAI
+
+universe uA
+
+namespace PeriodicTilingThree
+
+def forwardDiff {A : Type uA} [AddCommGroup A] (f : ℤ → A) (n : ℤ) : A :=
+  f (n + 1) - f n
+
+@[simp] theorem forwardDiff_apply {A : Type uA} [AddCommGroup A]
+    (f : ℤ → A) (n : ℤ) : forwardDiff f n = f (n + 1) - f n := rfl
+
+end PeriodicTilingThree
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/FullDifferenceBound.lean b/lean/OAI/Geometry/PeriodicTiling/FullDifferenceBound.lean
new file mode 100644
index 0000000000000000000000000000000000000000..cf5c849db9f9732a173dd11c2e04770063a39670
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/FullDifferenceBound.lean
@@ -0,0 +1,84 @@
+import Mathlib.Data.Nat.Choose.Sum
+
+namespace OAI
+
+namespace PeriodicTilingThree
+
+open scoped BigOperators
+
+private theorem binomial_two_terms_le (a m : ℕ) (hm : 2 ≤ m) :
+    a ^ m + a ^ (m - 2) * m.choose 2 ≤ (1 + a) ^ m := by
+  have hsub : ({0, 2} : Finset ℕ) ⊆ Finset.range (m + 1) := by
+    intro k hk
+    simp only [Finset.mem_insert, Finset.mem_singleton] at hk
+    rcases hk with rfl | rfl
+    · simp only [Finset.mem_range]
+      omega
+    · simp only [Finset.mem_range]
+      omega
+  calc
+    a ^ m + a ^ (m - 2) * m.choose 2 =
+        ∑ k ∈ ({0, 2} : Finset ℕ), 1 ^ k * a ^ (m - k) * m.choose k := by
+      simp
+    _ ≤ ∑ k ∈ Finset.range (m + 1), 1 ^ k * a ^ (m - k) * m.choose k :=
+      Finset.sum_le_sum_of_subset hsub
+    _ = (1 + a) ^ m := (add_pow 1 a m).symm
+
+theorem fullDifference_bad_bound (s m : ℕ) (hs : 2 ≤ s)
+    (hm : 4 * s * (s ^ 2 - 1) ^ 2 + 1 ≤ m) :
+    s * (2 * m) * (s ^ 2 - 1) ^ m < s ^ (2 * m) := by
+  let a : ℕ := s ^ 2 - 1
+  have hsquare : 4 ≤ s ^ 2 := by
+    simpa using Nat.pow_le_pow_left hs 2
+  have ha : 0 < a := by
+    dsimp [a]
+    omega
+  have ha_succ : 1 + a = s ^ 2 := by
+    dsimp [a]
+    omega
+  change 4 * s * a ^ 2 + 1 ≤ m at hm
+  have hthreshold : 0 < 4 * s * a ^ 2 :=
+    Nat.mul_pos (Nat.mul_pos (by decide) (by omega)) (pow_pos ha 2)
+  have hm_two : 2 ≤ m := by omega
+  have hpred : 4 * s * a ^ 2 ≤ m - 1 := by omega
+  have hchoose : m.choose 2 * 2 = m * (m - 1) := by
+    simpa using Nat.choose_succ_right_eq m 1
+  have hdoubled : (2 * s * m * a ^ 2) * 2 ≤ m.choose 2 * 2 := by
+    calc
+      (2 * s * m * a ^ 2) * 2 = m * (4 * s * a ^ 2) := by ring
+      _ ≤ m * (m - 1) := Nat.mul_le_mul_left m hpred
+      _ = m.choose 2 * 2 := hchoose.symm
+  have hcoefficient : 2 * s * m * a ^ 2 ≤ m.choose 2 := by
+    omega
+  have hpower : a ^ 2 * a ^ (m - 2) = a ^ m := by
+    rw [← pow_add, Nat.add_sub_cancel' hm_two]
+  have hterm : (2 * s * m) * a ^ m ≤ a ^ (m - 2) * m.choose 2 := by
+    calc
+      (2 * s * m) * a ^ m = a ^ (m - 2) * (2 * s * m * a ^ 2) := by
+        rw [← hpower]
+        ring
+      _ ≤ a ^ (m - 2) * m.choose 2 :=
+        Nat.mul_le_mul_left (a ^ (m - 2)) hcoefficient
+  have hpositive : 0 < a ^ m := pow_pos ha m
+  calc
+    s * (2 * m) * (s ^ 2 - 1) ^ m = (2 * s * m) * a ^ m := by
+      dsimp [a]
+      ring
+    _ < a ^ m + a ^ (m - 2) * m.choose 2 := by omega
+    _ ≤ (1 + a) ^ m := binomial_two_terms_le a m hm_two
+    _ = s ^ (2 * m) := by rw [ha_succ, ← pow_mul]
+
+theorem fullDifference_bad_bound_poly (s m : ℕ) (hs : 2 ≤ s)
+    (hm : 4 * s ^ 5 + 1 ≤ m) :
+    s * (2 * m) * (s ^ 2 - 1) ^ m < s ^ (2 * m) := by
+  apply fullDifference_bad_bound s m hs
+  calc
+    4 * s * (s ^ 2 - 1) ^ 2 + 1 ≤ 4 * s * (s ^ 2) ^ 2 + 1 :=
+      Nat.add_le_add_right
+        (Nat.mul_le_mul_left (4 * s) (Nat.pow_le_pow_left (Nat.sub_le _ _) 2)) 1
+    _ = 4 * s ^ 5 + 1 := by ring
+    _ ≤ m := hm
+
+end PeriodicTilingThree
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/FullDifferencePartition.lean b/lean/OAI/Geometry/PeriodicTiling/FullDifferencePartition.lean
new file mode 100644
index 0000000000000000000000000000000000000000..745aad3c43f7b3f821f995dfc79b4462d04cdaeb
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/FullDifferencePartition.lean
@@ -0,0 +1,183 @@
+import OAI.Geometry.PeriodicTiling.FinitePairCounting
+import OAI.Geometry.PeriodicTiling.PrimeDifferencePairs
+import OAI.Geometry.PeriodicTiling.FullDifferenceBound
+import Mathlib.Data.Nat.Prime.Infinite
+import Mathlib.Algebra.BigOperators.Group.Finset.Basic
+
+namespace OAI
+
+noncomputable section
+
+namespace PeriodicTilingThree
+
+open Classical
+open scoped BigOperators
+
+abbrev Coloring (q s : ℕ) := ZMod q → Fin s
+
+def BadDifference {q s : ℕ} (c : Coloring q s) (ν : Fin s) (d : ZMod q) : Prop :=
+  ¬ ∃ x y, c x = ν ∧ c y = ν ∧ x - y = d
+
+def badDifferenceColorings {q s : ℕ} [NeZero q]
+    (ν : Fin s) (d : ZMod q) : Finset (Coloring q s) :=
+  Finset.univ.filter (fun c => BadDifference c ν d)
+
+@[simp]
+theorem mem_badDifferenceColorings {q s : ℕ} [NeZero q]
+    (c : Coloring q s) (ν : Fin s) (d : ZMod q) :
+    c ∈ badDifferenceColorings ν d ↔ BadDifference c ν d := by
+  simp [badDifferenceColorings]
+
+theorem badDifference_card_le {q s m : ℕ} [Fact q.Prime]
+    (hq : q = 2 * m + 1) (ν : Fin s) (d : ZMod q) (hd : d ≠ 0) :
+    (badDifferenceColorings ν d).card ≤ s * (s ^ 2 - 1) ^ m := by
+  let pair := primeDifferencePairs q m hq d hd
+  let good : Fin m → Finset (Fin s × Fin s) := fun _ => {(ν, ν)}
+  have hsub : badDifferenceColorings ν d ⊆ allBadAssignments pair good := by
+    intro c hc
+    apply (mem_allBadAssignments pair good c).mpr
+    intro j hj
+    have heq : (c (pair (j, 0)), c (pair (j, 1))) = (ν, ν) :=
+      Finset.mem_singleton.mp hj
+    apply (mem_badDifferenceColorings c ν d).mp hc
+    refine ⟨pair (j, 1), pair (j, 0), congrArg Prod.snd heq,
+      congrArg Prod.fst heq, ?_⟩
+    exact primeDifferencePairs_step q m hq d hd j
+  have hleft : q - 2 * m = 1 := by omega
+  calc
+    (badDifferenceColorings ν d).card ≤ (allBadAssignments pair good).card :=
+      Finset.card_le_card hsub
+    _ ≤ (Fintype.card (Fin s) ^ 2 - 1) ^ m *
+        Fintype.card (Fin s) ^ (Fintype.card (ZMod q) - 2 * m) :=
+      allBadAssignments_card_le pair good 1 (by intro j; simp [good])
+    _ = s * (s ^ 2 - 1) ^ m := by
+      simp only [Fintype.card_fin, ZMod.card, hleft, pow_one]
+      exact Nat.mul_comm _ _
+
+def badDifferenceLabels (q s : ℕ) [NeZero q] : Finset (Fin s × ZMod q) :=
+  Finset.univ.product ((Finset.univ : Finset (ZMod q)).erase 0)
+
+@[simp]
+theorem badDifferenceLabels_card (q s : ℕ) [NeZero q] :
+    (badDifferenceLabels q s).card = s * (q - 1) := by
+  simp [badDifferenceLabels, ZMod.card]
+
+def someBadDifferenceColorings (q s : ℕ) [NeZero q] : Finset (Coloring q s) :=
+  (badDifferenceLabels q s).biUnion (fun p => badDifferenceColorings p.1 p.2)
+
+theorem someBadDifferenceColorings_card_le {q s m : ℕ} [Fact q.Prime]
+    (hq : q = 2 * m + 1) :
+    (someBadDifferenceColorings q s).card ≤
+      (s * (q - 1)) * (s * (s ^ 2 - 1) ^ m) := by
+  calc
+    (someBadDifferenceColorings q s).card ≤
+        ∑ p ∈ badDifferenceLabels q s, (badDifferenceColorings p.1 p.2).card :=
+      Finset.card_biUnion_le
+    _ ≤ ∑ _p ∈ badDifferenceLabels q s, s * (s ^ 2 - 1) ^ m := by
+      apply Finset.sum_le_sum
+      intro p hp
+      have hd : p.2 ≠ 0 := (Finset.mem_erase.mp (Finset.mem_product.mp hp).2).1
+      exact badDifference_card_le hq p.1 p.2 hd
+    _ = (s * (q - 1)) * (s * (s ^ 2 - 1) ^ m) := by
+      rw [Finset.sum_const, nsmul_eq_mul, badDifferenceLabels_card, Nat.cast_id]
+
+theorem exists_coloring_nonzero_differences {q s m : ℕ} [Fact q.Prime]
+    (hs : 2 ≤ s) (hq : q = 2 * m + 1) (hm : 4 * s ^ 5 + 1 ≤ m) :
+    ∃ c : Coloring q s, ∀ ν d, d ≠ 0 →
+      ∃ x y, c x = ν ∧ c y = ν ∧ x - y = d := by
+  have hspos : 0 < s := by omega
+  have hstrict := Nat.mul_lt_mul_of_pos_left
+    (fullDifference_bad_bound_poly s m hs hm) hspos
+  have hcard : (someBadDifferenceColorings q s).card <
+      (Finset.univ : Finset (Coloring q s)).card := by
+    calc
+      (someBadDifferenceColorings q s).card ≤
+          (s * (q - 1)) * (s * (s ^ 2 - 1) ^ m) :=
+        someBadDifferenceColorings_card_le hq
+      _ = s * (s * (2 * m) * (s ^ 2 - 1) ^ m) := by
+        rw [hq]
+        simp only [Nat.add_sub_cancel]
+        ring
+      _ < s * s ^ (2 * m) := hstrict
+      _ = (Finset.univ : Finset (Coloring q s)).card := by
+        simp only [Finset.card_univ, Fintype.card_fun, Fintype.card_fin, ZMod.card]
+        rw [hq, pow_succ]
+        exact Nat.mul_comm _ _
+  obtain ⟨c, _hc, hgood⟩ := Finset.exists_mem_notMem_of_card_lt_card hcard
+  refine ⟨c, ?_⟩
+  intro ν d hd
+  by_contra hbad
+  apply hgood
+  apply Finset.mem_biUnion.mpr
+  refine ⟨(ν, d), ?_, ?_⟩
+  · exact Finset.mem_product.mpr
+      ⟨Finset.mem_univ _, Finset.mem_erase.mpr ⟨hd, Finset.mem_univ _⟩⟩
+  · exact (mem_badDifferenceColorings c ν d).mpr hbad
+
+theorem exists_partition_of_nonzero_coloring {q s : ℕ} [Fact q.Prime]
+    (c : Coloring q s)
+    (hc : ∀ ν d, d ≠ 0 → ∃ x y, c x = ν ∧ c y = ν ∧ x - y = d) :
+    ∃ E : Fin s → Finset (ZMod q),
+      (∀ ν, (E ν).Nonempty) ∧
+      (∀ x, ∃! ν, x ∈ E ν) ∧
+      (∀ ν d, ∃ x ∈ E ν, ∃ y ∈ E ν, x - y = d) := by
+  let E : Fin s → Finset (ZMod q) :=
+    fun ν => Finset.univ.filter (fun x => c x = ν)
+  have hne : ∀ ν, (E ν).Nonempty := by
+    intro ν
+    obtain ⟨x, _y, hx, _hy, _hxy⟩ := hc ν 1 one_ne_zero
+    exact ⟨x, by simp [E, hx]⟩
+  refine ⟨E, hne, ?_, ?_⟩
+  · intro x
+    refine ⟨c x, by simp [E], ?_⟩
+    intro ν hν
+    exact (Finset.mem_filter.mp hν).2.symm
+  · intro ν d
+    by_cases hd : d = 0
+    · subst d
+      obtain ⟨x, hx⟩ := hne ν
+      exact ⟨x, hx, x, hx, sub_self x⟩
+    · obtain ⟨x, y, hx, hy, hxy⟩ := hc ν d hd
+      exact ⟨x, by simp [E, hx], y, by simp [E, hy], hxy⟩
+
+theorem exists_fresh_prime_fullDifference_partition
+    (s : ℕ) (hs : 0 < s) (forbidden : Finset ℕ) (lower : ℕ) :
+    ∃ q : ℕ, q.Prime ∧ lower < q ∧ q ∉ forbidden ∧
+      ∃ E : Fin s → Finset (ZMod q),
+        (∀ ν, (E ν).Nonempty) ∧
+        (∀ x, ∃! ν, x ∈ E ν) ∧
+        (∀ ν d, ∃ x ∈ E ν, ∃ y ∈ E ν, x - y = d) := by
+  let N := max (8 * s ^ 5 + 3) (max (lower + 1) (forbidden.sup id + 1))
+  obtain ⟨q, hqN, hprime⟩ := Nat.exists_infinite_primes N
+  have hqbig : 8 * s ^ 5 + 3 ≤ q := (le_max_left _ _).trans hqN
+  have hqrest : max (lower + 1) (forbidden.sup id + 1) ≤ q :=
+    (le_max_right _ _).trans hqN
+  have hqlower : lower + 1 ≤ q := (le_max_left _ _).trans hqrest
+  have hqforbidden : forbidden.sup id + 1 ≤ q := (le_max_right _ _).trans hqrest
+  have hfresh : q ∉ forbidden := by
+    intro hmem
+    have hsup : q ≤ forbidden.sup id := Finset.le_sup (f := id) hmem
+    omega
+  have hq2 : 2 < q := by omega
+  let : Fact q.Prime := ⟨hprime⟩
+  refine ⟨q, hprime, by omega, hfresh, ?_⟩
+  by_cases hs1 : s = 1
+  · subst s
+    let c : Coloring q 1 := fun _ => 0
+    apply exists_partition_of_nonzero_coloring c
+    intro ν d _hd
+    have hν : (0 : Fin 1) = ν := Subsingleton.elim _ _
+    exact ⟨d, 0, hν, hν, sub_zero d⟩
+  · have hs2 : 2 ≤ s := by omega
+    have hodd : q % 2 = 1 := hprime.eq_two_or_odd.resolve_left (by omega)
+    let m := q / 2
+    have hqm : q = 2 * m + 1 := by dsimp [m]; omega
+    have hm : 4 * s ^ 5 + 1 ≤ m := by dsimp [m]; omega
+    obtain ⟨c, hc⟩ := exists_coloring_nonzero_differences hs2 hqm hm
+    exact exists_partition_of_nonzero_coloring c hc
+
+end PeriodicTilingThree
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/GraphTests.lean b/lean/OAI/Geometry/PeriodicTiling/GraphTests.lean
new file mode 100644
index 0000000000000000000000000000000000000000..79da27f45651855e422ebdc8ce1be1f27d92a22f
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/GraphTests.lean
@@ -0,0 +1,323 @@
+import OAI.Geometry.PeriodicTiling.CyclicCoordinates
+import Mathlib.Tactic.Abel
+
+namespace OAI
+
+universe uH uB uC
+
+namespace PeriodicTilingThree
+
+section RepresentationFibres
+
+variable {H : Type uH} {B : Type uB} {C : Type uC} [AddCommGroup H]
+
+theorem tiles_range_iff_unique_base (s : B → H) (hs : Function.Injective s)
+    (F : Finset H) :
+    Tiles F (Set.range s) ↔ ∀ y : H, ∃! b : B, y - s b ∈ F := by
+  rw [tiles_iff_unique_tile]
+  constructor
+  · intro h y
+    obtain ⟨f, hf, huniq⟩ := h y
+    obtain ⟨b, hb⟩ := hf
+    have hbf : y - s b = (f : H) := by rw [hb]; simp
+    refine ⟨b, ?_, ?_⟩
+    · change y - s b ∈ F
+      rw [hbf]
+      exact f.property
+    · intro b' hb'
+      let f' : ↥F := ⟨y - s b', hb'⟩
+      have hf' : y - (f' : H) ∈ Set.range s := by
+        refine ⟨b', ?_⟩
+        simp [f']
+      have he : (f' : H) = (f : H) := congrArg Subtype.val (huniq f' hf')
+      apply hs
+      exact sub_right_inj.mp (he.trans hbf.symm)
+  · intro h y
+    obtain ⟨b, hb, huniq⟩ := h y
+    let f : ↥F := ⟨y - s b, hb⟩
+    refine ⟨f, ?_, ?_⟩
+    · refine ⟨b, ?_⟩
+      simp [f]
+    · intro f' hf'
+      obtain ⟨b', hb'⟩ := hf'
+      have hbf : y - s b' = (f' : H) := by rw [hb']; simp
+      have hbb : b' = b := huniq b' (by
+        change y - s b' ∈ F
+        rw [hbf]
+        exact f'.property)
+      apply Subtype.ext
+      change (f' : H) = y - s b
+      rw [← hbf, hbb]
+
+variable [AddCommGroup B] [AddCommGroup C]
+
+theorem tiles_kernel_iff_unique_fibre (q : H →+ B)
+    (hq : Function.Surjective q) (F : Finset H)
+    (hF : ∀ x : H, x ∈ F ↔ q x = 0) (A : Set H) :
+    Tiles F A ↔ ∀ b : B, ∃! a : A, q (a : H) = b := by
+  constructor
+  · intro h b
+    obtain ⟨y, hy⟩ := hq b
+    obtain ⟨⟨f, a⟩, hfa⟩ := h.2 y
+    change (f : H) + (a : H) = y at hfa
+    have hqa : q (a : H) = b := by
+      have he := congrArg q hfa
+      rw [map_add, (hF _).mp f.property, zero_add, hy] at he
+      exact he
+    refine ⟨a, hqa, ?_⟩
+    intro a' ha'
+    have hmem : y - (a' : H) ∈ F := by
+      apply (hF _).mpr
+      rw [map_sub, hy, ha', sub_self]
+    let f' : ↥F := ⟨y - (a' : H), hmem⟩
+    have he : ((f', a') : ↥F × A) = (f, a) := by
+      apply h.1
+      change (y - (a' : H)) + (a' : H) = (f : H) + (a : H)
+      rw [sub_add_cancel, hfa]
+    exact congrArg Prod.snd he
+  · intro h
+    constructor
+    · rintro ⟨f, a⟩ ⟨f', a'⟩ he
+      have hqa : q (a : H) = q (a' : H) := by
+        have he' := congrArg q he
+        simpa only [map_add, (hF _).mp f.property,
+          (hF _).mp f'.property, zero_add] using he'
+      obtain ⟨a₀, _ha₀, huniq⟩ := h (q (a : H))
+      have haa : a = a' := (huniq a rfl).trans (huniq a' hqa.symm).symm
+      have hff : f = f' := by
+        apply Subtype.ext
+        have hv := congrArg Subtype.val haa
+        change (f : H) + (a : H) = (f' : H) + (a' : H) at he
+        rw [hv] at he
+        exact add_right_cancel he
+      exact Prod.ext hff haa
+    · intro y
+      obtain ⟨a, ha, _huniq⟩ := h (q y)
+      have hmem : y - (a : H) ∈ F := by
+        apply (hF _).mpr
+        rw [map_sub, ha, sub_self]
+      refine ⟨(⟨y - (a : H), hmem⟩, a), ?_⟩
+      exact sub_add_cancel y (a : H)
+
+theorem tiles_invariance_iff (q : H →+ B) (ψ : H →+ C)
+    (s : B → H) (hs : ∀ b, q (s b) = b)
+    (σ : B) (F : Finset H)
+    (hF : ∀ v : H, v ∈ F ↔
+      (q v = 0 ∧ ψ v ≠ 0) ∨ (q v = σ ∧ ψ v = 0)) :
+    Tiles F (Set.range s) ↔ ∀ b, ψ (s (b - σ)) = ψ (s b) := by
+  have hinj : Function.Injective s := fun b b' h => by
+    have := congrArg q h
+    simpa only [hs] using this
+  have hmem (y : H) (b : B) : y - s b ∈ F ↔
+      (q y = b ∧ ψ y ≠ ψ (s b)) ∨
+      (q y = σ + b ∧ ψ y = ψ (s b)) := by
+    simpa only [map_sub, hs, sub_eq_zero, sub_ne_zero,
+      sub_eq_iff_eq_add, zero_add] using hF (y - s b)
+  rw [tiles_range_iff_unique_base s hinj F]
+  constructor
+  · intro h b
+    obtain ⟨b', hb', _huniq⟩ := h (s b)
+    rcases (hmem _ _).mp hb' with ⟨hq, hne⟩ | ⟨hq, he⟩
+    · rw [hs] at hq
+      exact False.elim (hne (congrArg (fun b => ψ (s b)) hq))
+    · rw [hs] at hq
+      have hb : b' = b - σ := by rw [hq]; abel
+      rw [hb] at he
+      exact he.symm
+  · intro h y
+    by_cases he : ψ y = ψ (s (q y))
+    · refine ⟨q y - σ, ?_, ?_⟩
+      · apply (hmem _ _).mpr
+        exact Or.inr ⟨by abel, he.trans (h (q y)).symm⟩
+      · intro b hb
+        rcases (hmem _ _).mp hb with ⟨hq, hne⟩ | ⟨hq, _he⟩
+        · exact False.elim (hne (hq ▸ he))
+        · rw [hq]
+          abel
+    · refine ⟨q y, (hmem _ _).mpr (Or.inl ⟨rfl, he⟩), ?_⟩
+      intro b hb
+      rcases (hmem _ _).mp hb with ⟨hq, _hne⟩ | ⟨hq, he'⟩
+      · exact hq.symm
+      · have hb : b = q y - σ := by rw [hq]; abel
+        rw [hb, h (q y)] at he'
+        exact False.elim (he he')
+
+end RepresentationFibres
+
+noncomputable section Concrete
+
+variable {p : ℕ} [NeZero p] (E : EncodingParameters p)
+
+local instance : DecidableEq (Ambient E) := Classical.decEq _
+
+def usefulCoord (i : Channel p) : Ambient E →+ P E i where
+  toFun v := (v.2.2 i).2
+  map_zero' := rfl
+  map_add' _ _ := rfl
+
+@[simp] theorem usefulCoord_apply (i : Channel p) (v : Ambient E) :
+    usefulCoord E i v = (v.2.2 i).2 := rfl
+
+@[simp] theorem usefulCoord_point (i : Channel p) (o : GraphOutputs E) (b : Base E) :
+    usefulCoord E i (point o b) = o.c b i := rfl
+
+private theorem kernel_horizontal_zero (v : (q0 E).ker) :
+    (v : Ambient E).1 = 0 := by
+  have hv := congrArg Prod.fst v.property
+  exact hv
+
+noncomputable instance kernelFintype : Fintype (q0 E).ker :=
+  Fintype.ofInjective (fun v : (q0 E).ker => (v : Ambient E).2) (by
+    intro v w h
+    apply Subtype.ext
+    exact Prod.ext ((kernel_horizontal_zero E v).trans
+      (kernel_horizontal_zero E w).symm) h)
+
+def kernelTile : Finset (Ambient E) :=
+  (Finset.univ : Finset (q0 E).ker).map ⟨Subtype.val, Subtype.val_injective⟩
+
+@[simp] theorem mem_kernelTile (v : Ambient E) :
+    v ∈ kernelTile E ↔ q0 E v = 0 := by
+  constructor
+  · intro hv
+    obtain ⟨w, _hw, rfl⟩ := Finset.mem_map.mp hv
+    exact w.property
+  · intro hv
+    exact Finset.mem_map.mpr ⟨⟨v, hv⟩, Finset.mem_univ _, rfl⟩
+
+theorem kernelTile_nonempty : (kernelTile E).Nonempty :=
+  ⟨0, (mem_kernelTile E 0).mpr (map_zero _)⟩
+
+def baseLift (b : Base E) : Ambient E := point (zeroOutputs E) b
+
+@[simp] theorem q0_baseLift (b : Base E) : q0 E (baseLift E b) = b :=
+  q0_point (zeroOutputs E) b
+
+@[simp] theorem usefulCoord_baseLift (i : Channel p) (b : Base E) :
+    usefulCoord E i (baseLift E b) = 0 := rfl
+
+theorem kernelTile_iff_unique_fibre (A : Set (Ambient E)) :
+    Tiles (kernelTile E) A ↔
+      ∀ b : Base E, ∃! a : A, q0 E (a : Ambient E) = b :=
+  tiles_kernel_iff_unique_fibre (q0 E) (q0_surjective E) (kernelTile E)
+    (mem_kernelTile E) A
+
+theorem kernelTile_graph (o : GraphOutputs E) : Tiles (kernelTile E) (graph o) := by
+  apply (tiles_range_iff_unique_base (point o) (point_injective o) (kernelTile E)).mpr
+  intro y
+  refine ⟨q0 E y, ?_, ?_⟩
+  · change y - point o (q0 E y) ∈ kernelTile E
+    rw [mem_kernelTile, map_sub, q0_point, sub_self]
+  · intro b hb
+    rw [mem_kernelTile, map_sub, q0_point, sub_eq_zero] at hb
+    exact hb.symm
+
+theorem kernelTile_iff_graph (A : Set (Ambient E)) :
+    Tiles (kernelTile E) A ↔ ∃ o : GraphOutputs E, graph o = A := by
+  classical
+  constructor
+  · intro h
+    have hf := (kernelTile_iff_unique_fibre E A).mp h
+    let a : Base E → A := fun b => Classical.choose (hf b)
+    have ha (b : Base E) : q0 E (a b : Ambient E) = b :=
+      (Classical.choose_spec (hf b)).1
+    let s : Base E → Ambient E := fun b => (a b : Ambient E)
+    have hs : ∀ b, q0 E (s b) = b := ha
+    refine ⟨outputsOfSection s hs, ?_⟩
+    rw [graph_outputsOfSection]
+    ext y
+    constructor
+    · rintro ⟨b, rfl⟩
+      exact (a b).property
+    · intro hy
+      refine ⟨q0 E y, ?_⟩
+      have he : (⟨y, hy⟩ : A) = a (q0 E y) :=
+        (Classical.choose_spec (hf (q0 E y))).2 ⟨y, hy⟩ rfl
+      exact (congrArg Subtype.val he).symm
+  · rintro ⟨o, rfl⟩
+    exact kernelTile_graph E o
+
+def invarianceZero (i : Channel p) : Finset (Ambient E) :=
+  (kernelTile E).filter (fun v => usefulCoord E i v = 0)
+
+def invarianceNonzero (i : Channel p) : Finset (Ambient E) :=
+  (kernelTile E).filter (fun v => usefulCoord E i v ≠ 0)
+
+def invarianceTile (i : Channel p) (σ : Base E) : Finset (Ambient E) :=
+  invarianceNonzero E i ∪
+    (invarianceZero E i).image (fun v => baseLift E σ + v)
+
+theorem mem_invarianceTile (i : Channel p) (σ : Base E) (v : Ambient E) :
+    v ∈ invarianceTile E i σ ↔
+      (q0 E v = 0 ∧ usefulCoord E i v ≠ 0) ∨
+      (q0 E v = σ ∧ usefulCoord E i v = 0) := by
+  constructor
+  · intro hv
+    rcases Finset.mem_union.mp hv with hv | hv
+    · exact Or.inl (by simpa only [invarianceNonzero, Finset.mem_filter,
+        mem_kernelTile] using hv)
+    · obtain ⟨w, hw, rfl⟩ := Finset.mem_image.mp hv
+      have hw' : q0 E w = 0 ∧ usefulCoord E i w = 0 := by
+        simpa only [invarianceZero, Finset.mem_filter, mem_kernelTile] using hw
+      exact Or.inr ⟨by rw [map_add, q0_baseLift, hw'.1, add_zero],
+        by rw [map_add, usefulCoord_baseLift, hw'.2, add_zero]⟩
+  · rintro (⟨hq, hc⟩ | ⟨hq, hc⟩)
+    · apply Finset.mem_union.mpr
+      exact Or.inl (by simpa only [invarianceNonzero, Finset.mem_filter,
+        mem_kernelTile] using And.intro hq hc)
+    · apply Finset.mem_union.mpr
+      apply Or.inr
+      apply Finset.mem_image.mpr
+      refine ⟨v - baseLift E σ, ?_, ?_⟩
+      · rw [invarianceZero, Finset.mem_filter, mem_kernelTile]
+        constructor
+        · rw [map_sub, hq, q0_baseLift, sub_self]
+        · rw [map_sub, hc, usefulCoord_baseLift, sub_self]
+      · abel
+
+theorem invarianceTile_nonempty (i : Channel p) (σ : Base E) :
+    (invarianceTile E i σ).Nonempty := by
+  refine ⟨baseLift E σ, (mem_invarianceTile E i σ _).mpr (Or.inr ?_)⟩
+  exact ⟨q0_baseLift E σ, usefulCoord_baseLift E i σ⟩
+
+theorem invarianceTile_disjoint (i : Channel p) (σ : Base E) :
+    Disjoint (invarianceNonzero E i)
+      ((invarianceZero E i).image (fun v => baseLift E σ + v)) := by
+  apply Finset.disjoint_left.mpr
+  intro v hv hv'
+  have hne : usefulCoord E i v ≠ 0 := (Finset.mem_filter.mp hv).2
+  obtain ⟨w, hw, rfl⟩ := Finset.mem_image.mp hv'
+  have hw0 : usefulCoord E i w = 0 := (Finset.mem_filter.mp hw).2
+  apply hne
+  rw [map_add, usefulCoord_baseLift, hw0, add_zero]
+
+theorem invarianceTile_card (i : Channel p) (σ : Base E) :
+    (invarianceTile E i σ).card = (kernelTile E).card := by
+  rw [invarianceTile, Finset.card_union_of_disjoint (invarianceTile_disjoint E i σ),
+    Finset.card_image_of_injective _ (fun _ _ h => add_left_cancel h)]
+  have hpart : invarianceNonzero E i ∪ invarianceZero E i = kernelTile E := by
+    ext v
+    simp only [invarianceNonzero, invarianceZero, Finset.mem_union, Finset.mem_filter]
+    constructor
+    · rintro (⟨hv, _⟩ | ⟨hv, _⟩) <;> exact hv
+    · intro hv
+      by_cases hc : usefulCoord E i v = 0
+      · exact Or.inr ⟨hv, hc⟩
+      · exact Or.inl ⟨hv, hc⟩
+  have hdisj : Disjoint (invarianceNonzero E i) (invarianceZero E i) := by
+    apply Finset.disjoint_left.mpr
+    intro v hv hw
+    exact (Finset.mem_filter.mp hv).2 (Finset.mem_filter.mp hw).2
+  rw [← Finset.card_union_of_disjoint hdisj, hpart]
+
+theorem invarianceTile_iff (i : Channel p) (σ : Base E) (o : GraphOutputs E) :
+    Tiles (invarianceTile E i σ) (graph o) ↔
+      ∀ b, o.c (b - σ) i = o.c b i := by
+  exact tiles_invariance_iff (q0 E) (usefulCoord E i) (point o)
+    (q0_point o) σ (invarianceTile E i σ) (mem_invarianceTile E i σ)
+
+end Concrete
+
+end PeriodicTilingThree
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/GridRigidity.lean b/lean/OAI/Geometry/PeriodicTiling/GridRigidity.lean
new file mode 100644
index 0000000000000000000000000000000000000000..3403668a915a8bb1011bd66d7f06143b5f385764
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/GridRigidity.lean
@@ -0,0 +1,152 @@
+import OAI.Geometry.PeriodicTiling.SeparatedCenters
+import OAI.Geometry.PeriodicTiling.CenterFiniteness
+import OAI.Geometry.PeriodicTiling.AuxiliaryCubes
+import OAI.Geometry.PeriodicTiling.PeriodicVolume
+import Mathlib.Topology.Order.IntermediateValue
+import Mathlib.Tactic.Abel
+
+namespace OAI
+
+noncomputable section
+
+namespace PeriodicTilingThree
+
+open Set MeasureTheory
+
+theorem grid_difference_of_cube_chain {m : ℕ} {C : Set (Space 3)}
+    (hnear : ∀ c c' : C, ‖(c' : Space 3) - (c : Space 3)‖ ≤ (m : ℝ) →
+      (c' : Space 3) - (c : Space 3) ∈ scaledGrid m)
+    {c₀ c : C}
+    (hchain : Relation.ReflTransGen
+      (CoverIntersects (fun a : C => auxiliaryCube m a)) c₀ c) :
+    (c : Space 3) - (c₀ : Space 3) ∈ scaledGrid m := by
+  induction hchain with
+  | refl => simpa only [sub_self] using (scaledGrid m).zero_mem
+  | @tail a b _ hab ih =>
+      have hedge := hnear a b (auxiliaryCube_inter_norm_le hab)
+      have hid : (b : Space 3) - (c₀ : Space 3) =
+          ((b : Space 3) - (a : Space 3)) + ((a : Space 3) - (c₀ : Space 3)) := by
+        abel
+      rw [hid]
+      exact (scaledGrid m).add_mem hedge ih
+
+theorem grid_coset_of_auxiliary_cover {m : ℕ} (hm : 0 < m)
+    (C : Set (Space 3))
+    (hlocal : LocallyFinite (fun c : C => auxiliaryCube m c))
+    (hcover : ∀ x : Space 3, ∃ c : C, x ∈ auxiliaryCube m c)
+    (hnear : ∀ c c' : C, ‖(c' : Space 3) - (c : Space 3)‖ ≤ (m : ℝ) →
+      (c' : Space 3) - (c : Space 3) ∈ scaledGrid m) :
+    ∃ c₀ ∈ C, C = {x : Space 3 | x - c₀ ∈ scaledGrid m} := by
+  obtain ⟨c₀, _hc₀⟩ := hcover 0
+  have hmR : (0 : ℝ) < m := by exact_mod_cast hm
+  have hgrid (c : C) : (c : Space 3) - (c₀ : Space 3) ∈ scaledGrid m := by
+    apply grid_difference_of_cube_chain hnear
+    exact closed_cover_chain_connected (fun a : C => auxiliaryCube m a)
+      (fun a => auxiliaryCube_nonempty m a) (fun a => auxiliaryCube_isClosed m a)
+      hlocal hcover c₀ c
+  refine ⟨c₀, c₀.property, ?_⟩
+  ext x
+  constructor
+  · intro hx
+    exact hgrid ⟨x, hx⟩
+  · intro hx
+    obtain ⟨c, hxc⟩ := hcover x
+    have hdiff : x - (c : Space 3) ∈ scaledGrid m := by
+      have h := (scaledGrid m).sub_mem hx (hgrid c)
+      have hid : (x - (c₀ : Space 3)) - ((c : Space 3) - (c₀ : Space 3)) =
+          x - (c : Space 3) := by abel
+      rwa [hid] at h
+    have hnorm : ‖x - (c : Space 3)‖ < (m : ℝ) := by
+      have hle : dist x (c : Space 3) ≤ (m : ℝ) / 2 := hxc
+      rw [dist_eq_norm] at hle
+      linarith
+    have hxc' : x = (c : Space 3) :=
+      sub_eq_zero.mp (scaledGrid_eq_zero_of_norm_lt hm hdiff hnorm)
+    rw [hxc']
+    exact c.property
+
+theorem periodic_voxel_assembly_grid
+    {m : ℕ} (hm : 2 ≤ m) {S0 S1 S : Finset (Lattice 3)}
+    (h0 : S ⊆ S0) (h1 : S ⊆ S1) (hdiff : RigidDifferences m S)
+    (hcard0 : S0.card = m ^ 3) (hcard1 : S1.card = m ^ 3)
+    {C C1 : Set (Space 3)}
+    (b : Module.Basis (Fin 3) ℝ (Space 3))
+    (hp : ∀ v ∈ latticeOfBasis b, Period C v)
+    (ht : ∀ v ∈ latticeOfBasis b, Period C1 v)
+    (htile : TypedAETiles (Thickening S0) (Thickening S1) C C1) :
+    ∃ c₀ ∈ C, C = {x : Space 3 | x - c₀ ∈ scaledGrid m} := by
+  have hmpos : 0 < m := lt_of_lt_of_le (by decide : 0 < 2) hm
+  have hmR : (0 : ℝ) < m := by exact_mod_cast hmpos
+  have hsep : C.Pairwise (fun x y => (m : ℝ) ≤ ‖x - y‖) := by
+    intro x hx y hy hxy
+    have hne : (⟨y, hy⟩ : C) ≠ ⟨x, hx⟩ := by
+      intro heq
+      exact hxy (congrArg Subtype.val heq).symm
+    exact typed_centers_separated hm h0 h1 hdiff htile hne
+  have hcount := centers_countable hmR hsep
+  have hfin : ∀ K : Set (Space 3), Bornology.IsBounded K → (C ∩ K).Finite :=
+    fun _ hK => centers_finite_on_bounded hmR hsep hK
+  have hnear : ∀ c c' : C, ‖(c' : Space 3) - (c : Space 3)‖ ≤ (m : ℝ) →
+      (c' : Space 3) - (c : Space 3) ∈ scaledGrid m := by
+    intro c c' hcc'
+    by_cases heq : c = c'
+    · subst c'
+      simpa only [sub_self] using (scaledGrid m).zero_mem
+    · exact typed_nearby_centers_mem_scaledGrid hm h0 h1 hdiff htile heq hcc'
+  have hvol : volume (Thickening S1) = volume (Thickening S0) := by
+    rw [thickening_volume, thickening_volume, hcard0, hcard1]
+  have hJvol : volume (auxiliaryCube m 0) = volume (Thickening S0) := by
+    rw [auxiliaryCube_volume, thickening_volume, hcard0]
+  have hpack : Pairwise fun c c' : C => AEDisjoint volume
+      (PeriodicVolume.translate c (auxiliaryCube m 0))
+      (PeriodicVolume.translate c' (auxiliaryCube m 0)) := by
+    intro c c' hne
+    change AEDisjoint volume {x | x - (c : Space 3) ∈ auxiliaryCube m 0}
+      {x | x - (c' : Space 3) ∈ auxiliaryCube m 0}
+    rw [auxiliaryCube_translate, auxiliaryCube_translate]
+    apply auxiliaryCube_aedisjoint hmpos
+    exact hsep c.property c'.property (fun heq => hne (Subtype.ext heq))
+  have hae := PeriodicVolume.replacement_ae_covers b
+    (Thickening S0) (Thickening S1) (auxiliaryCube m 0) C C1
+    hcount hfin (thickening_measurableSet S0) (thickening_measurableSet S1)
+    (auxiliaryCube_measurableSet m 0) hvol hJvol hp ht htile hpack
+  have haecubes : ∀ᵐ x ∂volume, ∃ c : C, x ∈ auxiliaryCube m c := by
+    filter_upwards [hae] with x hx
+    obtain ⟨c, hc⟩ := hx
+    refine ⟨c, ?_⟩
+    rw [← auxiliaryCube_translate]
+    exact hc
+  exact grid_coset_of_auxiliary_cover hmpos C (auxiliaryCube_locallyFinite m C hfin)
+    (auxiliaryCube_cover_of_ae m C hfin haecubes) hnear
+
+theorem periodic_voxel_assembly_grid_range
+    {m : ℕ} (hm : 2 ≤ m) {S0 S1 S : Finset (Lattice 3)}
+    (h0 : S ⊆ S0) (h1 : S ⊆ S1) (hdiff : RigidDifferences m S)
+    (hcard0 : S0.card = m ^ 3) (hcard1 : S1.card = m ^ 3)
+    {C C1 : Set (Space 3)}
+    (b : Module.Basis (Fin 3) ℝ (Space 3))
+    (hp : ∀ v ∈ latticeOfBasis b, Period C v)
+    (ht : ∀ v ∈ latticeOfBasis b, Period C1 v)
+    (htile : TypedAETiles (Thickening S0) (Thickening S1) C C1) :
+    ∃ c₀ ∈ C, C = Set.range (fun z : Lattice 3 => c₀ + (m : ℝ) • castLattice z) := by
+  obtain ⟨c₀, hc₀, heq⟩ := periodic_voxel_assembly_grid hm h0 h1 hdiff
+    hcard0 hcard1 b hp ht htile
+  refine ⟨c₀, hc₀, heq.trans ?_⟩
+  ext x
+  constructor
+  · intro hx
+    obtain ⟨z, hz⟩ := mem_scaledGrid.mp hx
+    refine ⟨z, ?_⟩
+    change c₀ + (m : ℝ) • castLattice z = x
+    rw [hz, add_sub_cancel]
+  · rintro ⟨z, rfl⟩
+    apply mem_scaledGrid.mpr
+    refine ⟨z, ?_⟩
+    change (m : ℝ) • castLattice z = (c₀ + (m : ℝ) • castLattice z) - c₀
+    exact (add_sub_cancel_left c₀ ((m : ℝ) • castLattice z)).symm
+
+end PeriodicTilingThree
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/IntegerMatrixIndex.lean b/lean/OAI/Geometry/PeriodicTiling/IntegerMatrixIndex.lean
new file mode 100644
index 0000000000000000000000000000000000000000..7ac6e19c23f3dccf3dcbccf5fff0b5541b9742bc
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/IntegerMatrixIndex.lean
@@ -0,0 +1,100 @@
+import OAI.Geometry.PeriodicTiling.EuclideanBasic
+import OAI.Geometry.PeriodicTiling.MarkedTile
+import Mathlib.LinearAlgebra.Matrix.Adjugate
+import Mathlib.LinearAlgebra.Determinant
+
+namespace OAI
+
+noncomputable section
+
+namespace PeriodicTilingThree
+
+open Matrix
+open scoped BigOperators
+
+theorem integer_mulVec_mem_of_columns
+    (L : AddSubgroup (Lattice 3)) (M : Matrix (Fin 3) (Fin 3) ℤ)
+    (hM : ∀ j, M.col j ∈ L) (z : Lattice 3) : M *ᵥ z ∈ L := by
+  have hs : (∑ j : Fin 3, z j • M.col j) ∈ L := by
+    apply L.sum_mem
+    intro j _hj
+    exact L.zsmul_mem (hM j) (z j)
+  have he : M *ᵥ z = ∑ j : Fin 3, z j • M.col j := by
+    ext i
+    simp [Matrix.mulVec, dotProduct, Matrix.col, mul_comm]
+  rwa [he]
+
+theorem determinant_smul_mem_of_columns
+    (L : AddSubgroup (Lattice 3)) (M : Matrix (Fin 3) (Fin 3) ℤ)
+    (hM : ∀ j, M.col j ∈ L) (z : Lattice 3) : M.det • z ∈ L := by
+  have h := integer_mulVec_mem_of_columns L M hM (M.adjugate *ᵥ z)
+  simpa only [Matrix.mulVec_mulVec, Matrix.mul_adjugate,
+    Matrix.smul_mulVec, Matrix.one_mulVec] using h
+
+theorem finiteIndex_of_integer_columns
+    (L : AddSubgroup (Lattice 3)) (M : Matrix (Fin 3) (Fin 3) ℤ)
+    (hdet : M.det ≠ 0) (hM : ∀ j, M.col j ∈ L) : L.FiniteIndex := by
+  let d : ℕ := M.det.natAbs
+  have hd : d ≠ 0 := (Int.natAbs_pos.mpr hdet).ne'
+  let : NeZero d := ⟨hd⟩
+  have hker : (MarkedTile.residue d).ker ≤ L := by
+    intro x hx
+    obtain ⟨z, hz⟩ := MarkedTile.exists_scale_of_residue_eq_zero d hx
+    have habs : (d : ℤ) • z ∈ L := by
+      have h := determinant_smul_mem_of_columns L M hM z
+      change (M.det.natAbs : ℤ) • z ∈ L
+      rw [Int.natCast_natAbs]
+      by_cases hsign : 0 ≤ M.det
+      · simpa only [abs_of_nonneg hsign] using h
+      · simpa only [abs_of_nonpos (le_of_not_ge hsign), neg_smul] using L.neg_mem h
+    have he : MarkedTile.scale d z = (d : ℤ) • z := by
+      ext i
+      simp only [MarkedTile.scale_apply, Pi.smul_apply, smul_eq_mul]
+    rw [← hz, he]
+    exact habs
+  let : (MarkedTile.residue d).ker.FiniteIndex := inferInstance
+  exact AddSubgroup.finiteIndex_of_le hker
+
+theorem det_ne_zero_of_scaled_columns_basis (m : ℕ)
+    (b : Module.Basis (Fin 3) ℝ (Space 3)) (z : Fin 3 → Lattice 3)
+    (hz : ∀ j, scaledCast m (z j) = b j) :
+    Matrix.det (fun i j : Fin 3 => z j i) ≠ 0 := by
+  let M : Matrix (Fin 3) (Fin 3) ℤ := fun i j => z j i
+  let e := Pi.basisFun ℝ (Fin 3)
+  have hreal : (e.toMatrix b).det ≠ 0 := (e.isUnit_det b).ne_zero
+  have hmatrix : e.toMatrix b = (m : ℝ) • (M.map fun a => (a : ℝ)) := by
+    ext i j
+    change (e.repr (b j)) i = (m : ℝ) * (z j i : ℝ)
+    rw [Pi.basisFun_repr]
+    exact (congrFun (hz j) i).symm
+  change M.det ≠ 0
+  intro hzero
+  have hcast : (M.map fun a => (a : ℝ)).det = 0 := by
+    rw [← Int.cast_det, hzero, Int.cast_zero]
+  apply hreal
+  rw [hmatrix, Matrix.det_smul, hcast, mul_zero]
+
+theorem lattice_pullback_finiteIndex (m : ℕ)
+    (b : Module.Basis (Fin 3) ℝ (Space 3))
+    (hb : latticeOfBasis b ≤ scaledGrid m) :
+    ((latticeOfBasis b).comap (scaledCast m)).FiniteIndex := by
+  have hbmem : ∀ j : Fin 3, b j ∈ latticeOfBasis b := by
+    intro j
+    exact Submodule.subset_span ⟨j, rfl⟩
+  have hex : ∀ j : Fin 3, ∃ z : Lattice 3, scaledCast m z = b j := by
+    intro j
+    exact hb (hbmem j)
+  choose z hz using hex
+  let M : Matrix (Fin 3) (Fin 3) ℤ := fun i j => z j i
+  apply finiteIndex_of_integer_columns _ M
+    (det_ne_zero_of_scaled_columns_basis m b z hz)
+  intro j
+  change scaledCast m (z j) ∈ latticeOfBasis b
+  rw [hz j]
+  exact hbmem j
+
+end PeriodicTilingThree
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/LabeledDigitBlocks.lean b/lean/OAI/Geometry/PeriodicTiling/LabeledDigitBlocks.lean
new file mode 100644
index 0000000000000000000000000000000000000000..56161e5ba5a8b02e03f12eef1c4281bf483c505b
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/LabeledDigitBlocks.lean
@@ -0,0 +1,251 @@
+import Mathlib.Data.ZMod.Basic
+import Mathlib.Data.Fintype.EquivFin
+import Mathlib.Logic.Equiv.Prod
+import Mathlib.Tactic.FinCases
+
+namespace OAI
+
+universe uK uA uB uL uR uJ uF
+
+namespace PeriodicTilingThree
+
+structure LabeledBlocks (K : Type uK) (A : Type uA) (B : Type uB) (a b : ℕ) where
+  decode : K ≃ ((A × ZMod a) ⊕ (B × ZMod b))
+
+namespace LabeledBlocks
+
+variable {K : Type uK} {A : Type uA} {B : Type uB} {a b : ℕ}
+
+def labelRotation (t : ℤ) : Equiv.Perm ((A × ZMod a) ⊕ (B × ZMod b)) where
+  toFun
+    | .inl (c, y) => .inl (c, y + (t : ZMod a))
+    | .inr (c, y) => .inr (c, y + (t : ZMod b))
+  invFun
+    | .inl (c, y) => .inl (c, y - (t : ZMod a))
+    | .inr (c, y) => .inr (c, y - (t : ZMod b))
+  left_inv := by rintro (⟨c, y⟩ | ⟨c, y⟩) <;> simp
+  right_inv := by rintro (⟨c, y⟩ | ⟨c, y⟩) <;> simp
+
+def C (D : LabeledBlocks K A B a b) (x : K) : ZMod a × ZMod b :=
+  match D.decode x with
+  | .inl (_, y) => (y, 0)
+  | .inr (_, y) => (0, y)
+
+def T (D : LabeledBlocks K A B a b) (t : ℤ) : Equiv.Perm K :=
+  D.decode.trans ((labelRotation t).trans D.decode.symm)
+
+def blockId (D : LabeledBlocks K A B a b) (x : K) : A ⊕ B :=
+  match D.decode x with
+  | .inl (c, _) => .inl c
+  | .inr (c, _) => .inr c
+
+def block (D : LabeledBlocks K A B a b) (c : A ⊕ B) : Set K :=
+  {x | D.blockId x = c}
+
+theorem unique_block (D : LabeledBlocks K A B a b) (x : K) :
+    ∃! c : A ⊕ B, x ∈ D.block c := by
+  exact ⟨D.blockId x, rfl, fun c hc => hc.symm⟩
+
+noncomputable def blockEquivA (D : LabeledBlocks K A B a b) (c : A) :
+    {x : K // x ∈ D.block (.inl c)} ≃ ZMod a :=
+  (Equiv.ofBijective
+    (fun y : ZMod a => (⟨D.decode.symm (.inl (c, y)), by simp [block, blockId]⟩ :
+      {x : K // x ∈ D.block (.inl c)}))
+    (by
+      constructor
+      · intro y z h
+        have hh := congrArg (fun x => D.decode x.val) h
+        simpa using hh
+      · rintro ⟨x, hx⟩
+        cases h : D.decode x with
+        | inl z =>
+            have hz : z.1 = c := by simpa [block, blockId, h] using hx
+            refine ⟨z.2, Subtype.ext ?_⟩
+            apply D.decode.injective
+            rw [Equiv.apply_symm_apply, h]
+            exact congrArg Sum.inl (Prod.ext hz.symm rfl)
+        | inr z => simp [block, blockId, h] at hx)).symm
+
+noncomputable def blockEquivB (D : LabeledBlocks K A B a b) (c : B) :
+    {x : K // x ∈ D.block (.inr c)} ≃ ZMod b :=
+  (Equiv.ofBijective
+    (fun y : ZMod b => (⟨D.decode.symm (.inr (c, y)), by simp [block, blockId]⟩ :
+      {x : K // x ∈ D.block (.inr c)}))
+    (by
+      constructor
+      · intro y z h
+        have hh := congrArg (fun x => D.decode x.val) h
+        simpa using hh
+      · rintro ⟨x, hx⟩
+        cases h : D.decode x with
+        | inl z => simp [block, blockId, h] at hx
+        | inr z =>
+            have hz : z.1 = c := by simpa [block, blockId, h] using hx
+            refine ⟨z.2, Subtype.ext ?_⟩
+            apply D.decode.injective
+            rw [Equiv.apply_symm_apply, h]
+            exact congrArg Sum.inr (Prod.ext hz.symm rfl))).symm
+
+@[simp] theorem C_decode_symm_inl (D : LabeledBlocks K A B a b)
+    (c : A) (y : ZMod a) : D.C (D.decode.symm (.inl (c, y))) = (y, 0) := by
+  simp [C]
+
+@[simp] theorem C_decode_symm_inr (D : LabeledBlocks K A B a b)
+    (c : B) (y : ZMod b) : D.C (D.decode.symm (.inr (c, y))) = (0, y) := by
+  simp [C]
+
+@[simp] theorem decode_T (D : LabeledBlocks K A B a b) (t : ℤ) (x : K) :
+    D.decode (D.T t x) = labelRotation t (D.decode x) := by
+  simp [T]
+
+@[simp] theorem T_decode_symm_inl (D : LabeledBlocks K A B a b)
+    (t : ℤ) (c : A) (y : ZMod a) :
+    D.T t (D.decode.symm (.inl (c, y))) =
+      D.decode.symm (.inl (c, y + (t : ZMod a))) := by
+  simp [T, labelRotation]
+
+@[simp] theorem T_decode_symm_inr (D : LabeledBlocks K A B a b)
+    (t : ℤ) (c : B) (y : ZMod b) :
+    D.T t (D.decode.symm (.inr (c, y))) =
+      D.decode.symm (.inr (c, y + (t : ZMod b))) := by
+  simp [T, labelRotation]
+
+@[simp] theorem blockId_T (D : LabeledBlocks K A B a b) (t : ℤ) (x : K) :
+    D.blockId (D.T t x) = D.blockId x := by
+  unfold blockId
+  rw [decode_T]
+  cases D.decode x <;> rfl
+
+@[simp] theorem T_zero (D : LabeledBlocks K A B a b) (x : K) : D.T 0 x = x := by
+  apply D.decode.injective
+  rw [decode_T]
+  cases h : D.decode x with
+  | inl z => simp [labelRotation]
+  | inr z => simp [labelRotation]
+
+theorem T_add (D : LabeledBlocks K A B a b) (t u : ℤ) (x : K) :
+    D.T (t + u) x = D.T t (D.T u x) := by
+  apply D.decode.injective
+  simp only [decode_T]
+  cases h : D.decode x with
+  | inl z => simp [labelRotation, add_comm, add_left_comm]
+  | inr z => simp [labelRotation, add_comm, add_left_comm]
+
+@[simp] theorem T_neg_T (D : LabeledBlocks K A B a b) (t : ℤ) (x : K) :
+    D.T (-t) (D.T t x) = x := by
+  rw [← T_add]
+  simp
+
+@[simp] theorem T_T_neg (D : LabeledBlocks K A B a b) (t : ℤ) (x : K) :
+    D.T t (D.T (-t) x) = x := by
+  rw [← T_add]
+  simp
+
+def transport {L : Type uL} (D : LabeledBlocks K A B a b) (e : L ≃ K) :
+    LabeledBlocks L A B a b := ⟨e.trans D.decode⟩
+
+@[simp] theorem C_transport {L : Type uL} (D : LabeledBlocks K A B a b)
+    (e : L ≃ K) (x : L) : (D.transport e).C x = D.C (e x) := rfl
+
+def copyRight (D : LabeledBlocks K A B a b) (R : Type uR) :
+    LabeledBlocks (K × R) (A × R) (B × R) a b where
+  decode :=
+    { toFun := fun x => match D.decode x.1 with
+        | .inl (c, y) => .inl ((c, x.2), y)
+        | .inr (c, y) => .inr ((c, x.2), y)
+      invFun := fun x => match x with
+        | .inl ((c, r), y) => (D.decode.symm (.inl (c, y)), r)
+        | .inr ((c, r), y) => (D.decode.symm (.inr (c, y)), r)
+      left_inv := by
+        rintro ⟨x, r⟩
+        cases h : D.decode x with
+        | inl z => simpa [h] using congrArg (fun w => (w, r)) (D.decode.symm_apply_apply x)
+        | inr z => simpa [h] using congrArg (fun w => (w, r)) (D.decode.symm_apply_apply x)
+      right_inv := by
+        rintro (⟨⟨c, r⟩, y⟩ | ⟨⟨c, r⟩, y⟩) <;> simp }
+
+@[simp] theorem C_copyRight (D : LabeledBlocks K A B a b) (R : Type uR) (x : K × R) :
+    (D.copyRight R).C x = D.C x.1 := by
+  rcases x with ⟨x, r⟩
+  cases h : D.decode x <;> simp [C, copyRight, h]
+
+@[simp] theorem T_copyRight (D : LabeledBlocks K A B a b) (R : Type uR) (t : ℤ)
+    (x : K × R) : (D.copyRight R).T t x = (D.T t x.1, x.2) := by
+  rcases x with ⟨x, r⟩
+  cases h : D.decode x with
+  | inl z => simp [T, copyRight, labelRotation, h]
+  | inr z => simp [T, copyRight, labelRotation, h]
+
+noncomputable def digit (a b r A B : ℕ) [NeZero a] [NeZero b] [NeZero r]
+    (h : r = A * a + B * b) : LabeledBlocks (ZMod r) (Fin A) (Fin B) a b where
+  decode := Fintype.equivOfCardEq (by
+    simp only [ZMod.card, Fintype.card_sum, Fintype.card_prod, Fintype.card_fin]
+    exact h)
+
+theorem C_nonconstant (D : LabeledBlocks K A B a b) [Nonempty K]
+    [Nontrivial (ZMod a)] [Nontrivial (ZMod b)] :
+    ∃ x y : K, D.C x ≠ D.C y := by
+  obtain ⟨x⟩ := ‹Nonempty K›
+  cases h : D.decode x with
+  | inl z =>
+      refine ⟨D.decode.symm (.inl (z.1, 0)), D.decode.symm (.inl (z.1, 1)), ?_⟩
+      rw [C_decode_symm_inl, C_decode_symm_inl]
+      intro he
+      exact zero_ne_one (congrArg Prod.fst he)
+  | inr z =>
+      refine ⟨D.decode.symm (.inr (z.1, 0)), D.decode.symm (.inr (z.1, 1)), ?_⟩
+      rw [C_decode_symm_inr, C_decode_symm_inr]
+      intro he
+      exact zero_ne_one (congrArg Prod.snd he)
+
+section Coordinate
+
+variable {J : Type uJ} [DecidableEq J] (F : J → Type uF) (j : J)
+variable {a b : ℕ} {A : Type uA} {B : Type uB}
+
+def onCoordinate (D : LabeledBlocks (F j) A B a b) :
+    LabeledBlocks (∀ q, F q)
+      (A × (∀ q : {q // q ≠ j}, F q))
+      (B × (∀ q : {q // q ≠ j}, F q)) a b :=
+  (D.copyRight (∀ q : {q // q ≠ j}, F q)).transport (Equiv.piSplitAt j F)
+
+@[simp] theorem C_onCoordinate (D : LabeledBlocks (F j) A B a b) (x : ∀ q, F q) :
+    (onCoordinate F j D).C x = D.C (x j) := by
+  simp [onCoordinate]
+
+theorem C_eq_of_coordinate_eq (D : LabeledBlocks (F j) A B a b)
+    {x y : ∀ q, F q} (h : x j = y j) :
+    (onCoordinate F j D).C x = (onCoordinate F j D).C y := by
+  simp only [C_onCoordinate, h]
+
+theorem C_designated_witness (D : LabeledBlocks (F j) A B a b)
+    [∀ q, Nonempty (F q)] [Nontrivial (ZMod a)] [Nontrivial (ZMod b)] :
+    ∃ x y : ∀ q, F q, (∀ q, q ≠ j → x q = y q) ∧
+      (onCoordinate F j D).C x ≠ (onCoordinate F j D).C y := by
+  classical
+  obtain ⟨u, v, huv⟩ := D.C_nonconstant
+  let base : ∀ q, F q := fun q => Classical.choice (inferInstance : Nonempty (F q))
+  refine ⟨Function.update base j u, Function.update base j v, ?_, ?_⟩
+  · intro q hq
+    simp [Function.update_of_ne hq]
+  · simpa only [C_onCoordinate, Function.update_self] using huv
+
+theorem C_active_iff (D : LabeledBlocks (F j) A B a b)
+    [∀ q, Nonempty (F q)] [Nontrivial (ZMod a)] [Nontrivial (ZMod b)] (q : J) :
+    (∃ x y : ∀ r, F r, (∀ r, r ≠ q → x r = y r) ∧
+      (onCoordinate F j D).C x ≠ (onCoordinate F j D).C y) ↔ q = j := by
+  constructor
+  · rintro ⟨x, y, hxy, hne⟩
+    by_contra hq
+    exact hne (C_eq_of_coordinate_eq F j D (hxy j (Ne.symm hq)))
+  · intro hq
+    subst q
+    exact C_designated_witness F j D
+
+end Coordinate
+
+end LabeledBlocks
+
+end PeriodicTilingThree
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/LastDigit.lean b/lean/OAI/Geometry/PeriodicTiling/LastDigit.lean
new file mode 100644
index 0000000000000000000000000000000000000000..8abab296e1e8f3bc84ae1e25dc1337d7b272d2f5
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/LastDigit.lean
@@ -0,0 +1,226 @@
+import OAI.Geometry.PeriodicTiling.WordRule
+import Mathlib.NumberTheory.Padics.PadicVal.Basic
+import Mathlib.Data.Int.GCD
+import Mathlib.Tactic.Ring
+import Lean.Elab.Tactic.Omega
+
+namespace OAI
+
+noncomputable section
+
+namespace PeriodicTilingThree
+
+def pFreeQuotient (p : ℕ) (t : ℤ) : ℤ :=
+  t / (p : ℤ) ^ padicValInt p t
+
+theorem pFreeQuotient_factor (p : ℕ) (_hp : p.Prime) (t : ℤ) :
+    t = (p : ℤ) ^ padicValInt p t * pFreeQuotient p t := by
+  exact (Int.mul_ediv_cancel' (padicValInt_dvd (p := p) t)).symm
+
+theorem pFreeQuotient_not_dvd (p : ℕ) (hp : p.Prime) {t : ℤ}
+    (ht : t ≠ 0) : ¬ (p : ℤ) ∣ pFreeQuotient p t := by
+  let : Fact p.Prime := ⟨hp⟩
+  rintro ⟨u, hu⟩
+  have hpow : (p : ℤ) ^ (padicValInt p t + 1) ∣ t := by
+    refine ⟨u, ?_⟩
+    calc
+      t = (p : ℤ) ^ padicValInt p t * pFreeQuotient p t :=
+        pFreeQuotient_factor p hp t
+      _ = (p : ℤ) ^ padicValInt p t * ((p : ℤ) * u) := by rw [hu]
+      _ = (p : ℤ) ^ (padicValInt p t + 1) * u := by
+        rw [pow_succ, mul_assoc]
+  rcases (padicValInt_dvd_iff (p := p) (padicValInt p t + 1) t).mp hpow with h | h
+  · exact ht h
+  · omega
+
+theorem pFreeQuotient_prime_pow_mul (p : ℕ) (hp : p.Prime)
+    (r : ℕ) (u : ℤ) (hu : ¬ (p : ℤ) ∣ u) :
+    pFreeQuotient p ((p : ℤ) ^ r * u) = u := by
+  let : Fact p.Prime := ⟨hp⟩
+  have hp0 : (p : ℤ) ≠ 0 := Int.natCast_ne_zero.mpr hp.ne_zero
+  have hu0 : u ≠ 0 := by
+    intro h
+    apply hu
+    rw [h]
+    exact dvd_zero _
+  have hvpow : padicValInt p ((p : ℤ) ^ r) = r := by
+    rw [← Int.natCast_pow, padicValInt.of_nat, padicValNat.prime_pow]
+  have hv : padicValInt p ((p : ℤ) ^ r * u) = r := by
+    rw [padicValInt.mul (pow_ne_zero _ hp0) hu0, hvpow,
+      padicValInt.eq_zero_of_not_dvd hu, add_zero]
+  rw [pFreeQuotient, hv]
+  exact Int.mul_ediv_cancel_left u (pow_ne_zero r hp0)
+
+theorem pFreeQuotient_mul_prime_pow (p : ℕ) (hp : p.Prime)
+    (r : ℕ) {t : ℤ} (ht : t ≠ 0) :
+    pFreeQuotient p ((p : ℤ) ^ r * t) = pFreeQuotient p t := by
+  have hfactor : (p : ℤ) ^ r * t =
+      (p : ℤ) ^ (r + padicValInt p t) * pFreeQuotient p t := by
+    calc
+      (p : ℤ) ^ r * t =
+          (p : ℤ) ^ r * ((p : ℤ) ^ padicValInt p t * pFreeQuotient p t) :=
+        congrArg (fun z : ℤ => (p : ℤ) ^ r * z) (pFreeQuotient_factor p hp t)
+      _ = (p : ℤ) ^ (r + padicValInt p t) * pFreeQuotient p t := by
+        rw [pow_add, mul_assoc]
+  rw [hfactor]
+  exact pFreeQuotient_prime_pow_mul p hp _ _ (pFreeQuotient_not_dvd p hp ht)
+
+theorem exists_primitive_pair (p : ℕ) (hp : p.Prime) (d e : ℤ)
+    (hde : d ≠ 0 ∨ e ≠ 0) :
+    ∃ r : ℕ, ∃ a b : ℤ, d = (p : ℤ) ^ r * a ∧
+      e = (p : ℤ) ^ r * b ∧ (¬ (p : ℤ) ∣ a ∨ ¬ (p : ℤ) ∣ b) := by
+  let : Fact p.Prime := ⟨hp⟩
+  let r := padicValInt p (Int.gcd d e : ℤ)
+  have hg : (Int.gcd d e : ℤ) ≠ 0 :=
+    Int.natCast_ne_zero.mpr (ne_of_gt (Int.gcd_pos_iff.mpr hde))
+  have hr : (p : ℤ) ^ r ∣ (Int.gcd d e : ℤ) := padicValInt_dvd _
+  obtain ⟨a, ha⟩ := hr.trans (show (Int.gcd d e : ℤ) ∣ d from Int.gcd_dvd_left d e)
+  obtain ⟨b, hb⟩ := hr.trans (show (Int.gcd d e : ℤ) ∣ e from Int.gcd_dvd_right d e)
+  refine ⟨r, a, b, ha, hb, ?_⟩
+  by_cases hpa : (p : ℤ) ∣ a
+  · right
+    intro hpb
+    have hd' : (p : ℤ) ^ (r + 1) ∣ d := by
+      obtain ⟨u, hu⟩ := hpa
+      refine ⟨u, ?_⟩
+      rw [ha, hu, pow_succ, mul_assoc]
+    have he' : (p : ℤ) ^ (r + 1) ∣ e := by
+      obtain ⟨v, hv⟩ := hpb
+      refine ⟨v, ?_⟩
+      rw [hb, hv, pow_succ, mul_assoc]
+    rcases (padicValInt_dvd_iff (p := p) (r + 1) (Int.gcd d e : ℤ)).mp
+        (Int.dvd_coe_gcd hd' he') with hzero | hle
+    · exact hg hzero
+    · change r + 1 ≤ r at hle
+      omega
+  · exact Or.inl hpa
+
+def lastDigit (p : ℕ) (hp : p.Prime) (t : ℤ) : Symbol p := by
+  letI : Fact p.Prime := ⟨hp⟩
+  exact if ht : t = 0 then 1 else
+    Units.mk0 (pFreeQuotient p t : ZMod p) (by
+      intro h
+      exact pFreeQuotient_not_dvd p hp ht
+        ((ZMod.intCast_zmod_eq_zero_iff_dvd (pFreeQuotient p t) p).mp h))
+
+@[simp] theorem lastDigit_zero (p : ℕ) (hp : p.Prime) :
+    lastDigit p hp 0 = 1 := by
+  simp [lastDigit]
+
+theorem lastDigit_val_of_ne_zero (p : ℕ) (hp : p.Prime) {t : ℤ}
+    (ht : t ≠ 0) :
+    (lastDigit p hp t : ZMod p) = (pFreeQuotient p t : ZMod p) := by
+  simp [lastDigit, ht]
+
+theorem lastDigit_eq_of_not_dvd (p : ℕ) (hp : p.Prime) {t : ℤ}
+    (ht : ¬ (p : ℤ) ∣ t) : (lastDigit p hp t : ZMod p) = (t : ZMod p) := by
+  have ht0 : t ≠ 0 := by
+    intro h
+    apply ht
+    rw [h]
+    exact dvd_zero _
+  rw [lastDigit_val_of_ne_zero p hp ht0]
+  simp only [pFreeQuotient, padicValInt.eq_zero_of_not_dvd ht, pow_zero,
+    Int.ediv_one]
+
+@[simp] theorem lastDigit_one (p : ℕ) (hp : p.Prime) :
+    lastDigit p hp 1 = 1 := by
+  apply Units.ext
+  have hnot : ¬ (p : ℤ) ∣ (1 : ℤ) :=
+    fun h => hp.not_dvd_one (Int.natCast_dvd_natCast.mp h)
+  simpa only [Units.val_one, Int.cast_one] using lastDigit_eq_of_not_dvd p hp hnot
+
+theorem lastDigit_two (p : ℕ) (hp : p.Prime) (hlarge : 2 < p) :
+    (lastDigit p hp 2 : ZMod p) = 2 := by
+  have hnot : ¬ (p : ℤ) ∣ (2 : ℤ) := by
+    intro h
+    exact Nat.not_dvd_of_pos_of_lt (by decide : 0 < 2) hlarge
+      (Int.natCast_dvd_natCast.mp h)
+  simpa only [Int.cast_ofNat] using lastDigit_eq_of_not_dvd p hp hnot
+
+theorem lastDigit_mul_prime_pow (p : ℕ) (hp : p.Prime) (r : ℕ) (t : ℤ) :
+    lastDigit p hp ((p : ℤ) ^ r * t) = lastDigit p hp t := by
+  by_cases ht : t = 0
+  · subst t
+    simp
+  · have hp0 : (p : ℤ) ≠ 0 := Int.natCast_ne_zero.mpr hp.ne_zero
+    have hprod : (p : ℤ) ^ r * t ≠ 0 := mul_ne_zero (pow_ne_zero _ hp0) ht
+    apply Units.ext
+    rw [lastDigit_val_of_ne_zero p hp hprod, lastDigit_val_of_ne_zero p hp ht,
+      pFreeQuotient_mul_prime_pow p hp r ht]
+
+theorem lastDigit_eq_div_of_dvd_not_sq_dvd (p : ℕ) (hp : p.Prime) {t : ℤ}
+    (hdiv : (p : ℤ) ∣ t) (hsq : ¬ (p : ℤ) ^ 2 ∣ t) :
+    (lastDigit p hp t : ZMod p) = ((t / (p : ℤ) : ℤ) : ZMod p) := by
+  have ht : t ≠ 0 := by
+    intro h
+    apply hsq
+    rw [h]
+    exact dvd_zero _
+  have hquot : ¬ (p : ℤ) ∣ t / (p : ℤ) := by
+    rintro ⟨u, hu⟩
+    apply hsq
+    refine ⟨u, ?_⟩
+    calc
+      t = (p : ℤ) * (t / (p : ℤ)) := (Int.mul_ediv_cancel' hdiv).symm
+      _ = (p : ℤ) * ((p : ℤ) * u) := by rw [hu]
+      _ = (p : ℤ) ^ 2 * u := by rw [pow_two, mul_assoc]
+  have hfree := pFreeQuotient_prime_pow_mul p hp 1 (t / (p : ℤ)) hquot
+  rw [pow_one, Int.mul_ediv_cancel' hdiv] at hfree
+  rw [lastDigit_val_of_ne_zero p hp ht, hfree]
+
+theorem lastDigit_affine_allowed (p : ℕ) (hp : p.Prime) (d e : ℤ) :
+    Allowed p (fun n => lastDigit p hp (d * (n.val : ℤ) + e)) := by
+  classical
+  by_cases hzero : d = 0 ∧ e = 0
+  · rcases hzero with ⟨hd, he⟩
+    subst d
+    subst e
+    have hnot : ¬ (p : ℤ) ∣ (1 : ℤ) :=
+      fun h => hp.not_dvd_one (Int.natCast_dvd_natCast.mp h)
+    refine ⟨0, 1, Or.inr hnot, ?_⟩
+    intro n
+    dsimp only
+    constructor
+    · intro _
+      simp only [zero_mul, zero_add, lastDigit_zero, Units.val_one, Int.cast_one]
+    · intro h _
+      exact (hnot (by simpa only [zero_mul, zero_add] using h)).elim
+  · obtain ⟨r, a, b, hd, he, hab⟩ :=
+      exists_primitive_pair p hp d e (not_and_or.mp hzero)
+    refine ⟨a, b, hab, ?_⟩
+    intro n
+    have harg : d * (n.val : ℤ) + e =
+        (p : ℤ) ^ r * (a * (n.val : ℤ) + b) := by
+      rw [hd, he]
+      ring
+    dsimp only
+    rw [harg, lastDigit_mul_prime_pow]
+    exact ⟨fun h => lastDigit_eq_of_not_dvd p hp h,
+      fun h₁ h₂ => lastDigit_eq_div_of_dvd_not_sq_dvd p hp h₁ h₂⟩
+
+theorem lastDigit_lineRule (p : ℕ) (hp : p.Prime) :
+    LineRule p (fun _ m => lastDigit p hp m) := by
+  intro d e
+  exact lastDigit_affine_allowed p hp d e
+
+theorem lastDigit_nonconstantColumns (p : ℕ) (hp : p.Prime) (hlarge : 2 < p) :
+    NonconstantColumns (fun (_ : Column p) m => lastDigit p hp m) := by
+  intro n
+  refine ⟨1, 2, ?_⟩
+  intro h
+  have hcoe : (1 : ZMod p) = 2 := by
+    simpa only [lastDigit_one, Units.val_one, lastDigit_two p hp hlarge] using
+      congrArg (fun u : Symbol p => (u : ZMod p)) h
+  have hnat : 1 % p = 2 % p :=
+    (ZMod.natCast_eq_natCast_iff' 1 2 p).mp
+      (by simpa only [Nat.cast_one, Nat.cast_ofNat] using hcoe)
+  have h1 : 1 < p := lt_trans (by decide : 1 < 2) hlarge
+  rw [Nat.mod_eq_of_lt h1, Nat.mod_eq_of_lt hlarge] at hnat
+  omega
+
+end PeriodicTilingThree
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/LatticeDescent.lean b/lean/OAI/Geometry/PeriodicTiling/LatticeDescent.lean
new file mode 100644
index 0000000000000000000000000000000000000000..41c4ef3f45c513c402675fbb824de8016409315a
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/LatticeDescent.lean
@@ -0,0 +1,91 @@
+import OAI.Geometry.PeriodicTiling.IntegerMatrixIndex
+import Mathlib.MeasureTheory.Group.Measure
+import Mathlib.Tactic.Abel
+
+namespace OAI
+
+noncomputable section
+
+namespace PeriodicTilingThree
+
+open MeasureTheory
+open scoped Classical
+
+def normalizedComplement (m : ℕ) (c₀ : Space 3) (A : Set (Space 3)) :
+    Set (Lattice 3) := {z | c₀ + scaledCast m z ∈ A}
+
+theorem period_lattice_le_scaledGrid (m : ℕ) (c₀ : Space 3)
+    {A : Set (Space 3)} (hne : A.Nonempty)
+    (hgrid : ∀ a ∈ A, ∃ z : Lattice 3, c₀ + scaledCast m z = a)
+    (b : Module.Basis (Fin 3) ℝ (Space 3))
+    (hp : ∀ v ∈ latticeOfBasis b, Period A v) :
+    latticeOfBasis b ≤ scaledGrid m := by
+  obtain ⟨a, ha⟩ := hne
+  obtain ⟨z₀, hz₀⟩ := hgrid a ha
+  intro v hv
+  obtain ⟨z₁, hz₁⟩ := hgrid (a + v) ((hp v hv a).mpr ha)
+  refine ⟨z₁ - z₀, ?_⟩
+  change scaledCast m (z₁ - z₀) = v
+  calc
+    scaledCast m (z₁ - z₀) =
+        (c₀ + scaledCast m z₁) - (c₀ + scaledCast m z₀) := by
+      rw [map_sub]
+      abel
+    _ = (a + v) - a := by rw [hz₀, hz₁]
+    _ = v := by abel
+
+theorem fullyPeriodic_normalizedComplement (m : ℕ) (c₀ : Space 3)
+    {A : Set (Space 3)} (hne : A.Nonempty)
+    (hgrid : ∀ a ∈ A, ∃ z : Lattice 3, c₀ + scaledCast m z = a)
+    (hperiodic : EuclideanFullyPeriodic A) :
+    FullyPeriodic (normalizedComplement m c₀ A) := by
+  obtain ⟨b, hb⟩ := hperiodic
+  have hle := period_lattice_le_scaledGrid m c₀ hne hgrid b hb
+  refine ⟨(latticeOfBasis b).comap (scaledCast m),
+    lattice_pullback_finiteIndex m b hle, ?_⟩
+  intro v hv z
+  change c₀ + scaledCast m (z + v) ∈ A ↔ c₀ + scaledCast m z ∈ A
+  rw [map_add, ← add_assoc]
+  exact hb (scaledCast m v) hv (c₀ + scaledCast m z)
+
+theorem normalized_typedAETiles (m : ℕ) (hm : m ≠ 0) (c₀ : Space 3)
+    {Ω₀ Ω₁ C A : Set (Space 3)}
+    (hgrid : C = Set.range (fun z : Lattice 3 => c₀ + scaledCast m z))
+    (htile : TypedAETiles Ω₀ Ω₁ C A) :
+    ∀ᵐ x ∂volume, ∃! z : Lattice 3,
+      x - scaledCast m z ∈
+        if z ∈ normalizedComplement m c₀ A then Ω₁ else Ω₀ := by
+  classical
+  have hshift : ∀ᵐ x ∂volume, ∃! c : C,
+      (c₀ + x) - (c : Space 3) ∈
+        if (c : Space 3) ∈ A then Ω₁ else Ω₀ :=
+    (eventually_add_left_iff volume c₀).mpr htile
+  filter_upwards [hshift] with x hx
+  obtain ⟨c, hc, huniq⟩ := hx
+  obtain ⟨z, hz⟩ : ∃ z : Lattice 3, c₀ + scaledCast m z = (c : Space 3) := by
+    exact Eq.mp
+      (congrArg (fun S : Set (Space 3) => (c : Space 3) ∈ S) hgrid) c.property
+  refine ⟨z, ?_, ?_⟩
+  · rw [← hz] at hc
+    have he : (c₀ + x) - (c₀ + scaledCast m z) = x - scaledCast m z := by abel
+    by_cases hzA : c₀ + scaledCast m z ∈ A
+    · simpa only [he, normalizedComplement, Set.mem_ofPred_eq, hzA, ite_true] using hc
+    · simpa only [he, normalizedComplement, Set.mem_ofPred_eq, hzA, ite_false] using hc
+  · intro y hy
+    let cy : C := ⟨c₀ + scaledCast m y, by rw [hgrid]; exact ⟨y, rfl⟩⟩
+    have hcy : (c₀ + x) - (cy : Space 3) ∈
+        if (cy : Space 3) ∈ A then Ω₁ else Ω₀ := by
+      change (c₀ + x) - (c₀ + scaledCast m y) ∈
+        if c₀ + scaledCast m y ∈ A then Ω₁ else Ω₀
+      have he : (c₀ + x) - (c₀ + scaledCast m y) = x - scaledCast m y := by abel
+      rw [he]
+      exact hy
+    have he : c₀ + scaledCast m y = c₀ + scaledCast m z := by
+      exact (congrArg Subtype.val (huniq cy hcy)).trans hz.symm
+    exact scaledCast_injective hm (add_left_cancel he)
+
+end PeriodicTilingThree
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/LatticeDimensions.lean b/lean/OAI/Geometry/PeriodicTiling/LatticeDimensions.lean
new file mode 100644
index 0000000000000000000000000000000000000000..694878c3d0fa8f1d6b9925b06fd653527ecee423
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/LatticeDimensions.lean
@@ -0,0 +1,32 @@
+import OAI.Geometry.PeriodicTiling.TilingBasic
+
+namespace OAI
+
+namespace PeriodicTilingThree
+
+def intEquiv : ℤ ≃+ Lattice 1 where
+  toFun z := fun _ => z
+  invFun x := x 0
+  left_inv _ := rfl
+  right_inv x := by
+    funext i
+    have hi : i = 0 := Subsingleton.elim _ _
+    subst i
+    rfl
+  map_add' _ _ := rfl
+
+theorem fullyPeriodic_lattice_zero (A : Set (Lattice 0)) : FullyPeriodic A := by
+  refine ⟨⊤, inferInstance, ?_⟩
+  intro v _hv x
+  have hx : x + v = x := Subsingleton.elim _ _
+  rw [hx]
+
+theorem lattice_zero_tile_has_fullyPeriodic_complement
+    (F : Finset (Lattice 0)) (h : ∃ A, Tiles F A) :
+    ∃ A, Tiles F A ∧ FullyPeriodic A := by
+  obtain ⟨A, hA⟩ := h
+  exact ⟨A, hA, fullyPeriodic_lattice_zero A⟩
+
+end PeriodicTilingThree
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/LatticePeriodicity.lean b/lean/OAI/Geometry/PeriodicTiling/LatticePeriodicity.lean
new file mode 100644
index 0000000000000000000000000000000000000000..b5fd9fa7ca30b695e0025fde1474f93b72cf70f3
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/LatticePeriodicity.lean
@@ -0,0 +1,55 @@
+import OAI.Geometry.PeriodicTiling.EuclideanBasic
+import OAI.Geometry.PeriodicTiling.TilingTransport
+import Mathlib.GroupTheory.OrderOfElement
+import Mathlib.LinearAlgebra.Basis.SMul
+import Mathlib.LinearAlgebra.StdBasis
+
+namespace OAI
+
+noncomputable section
+
+namespace PeriodicTilingThree
+
+def scaledCoordinateBasis (d q : ℕ) (hq : q ≠ 0) : Module.Basis (Fin d) ℝ (Space d) :=
+  (Units.mk0 (q : ℝ) (Nat.cast_ne_zero.mpr hq)) • Pi.basisFun ℝ (Fin d)
+
+theorem scaledCoordinateBasis_eq_cast {d q : ℕ} (hq : q ≠ 0) (i : Fin d) :
+    scaledCoordinateBasis d q hq i =
+      castLattice (q • (Pi.single i 1 : Lattice d)) := by
+  classical
+  ext j
+  by_cases hji : j = i
+  · subst j
+    simp [scaledCoordinateBasis, Module.Basis.smul_apply, Units.smul_def]
+  · simp [scaledCoordinateBasis, Module.Basis.smul_apply,
+      hji]
+
+theorem exists_positive_scalar_sublattice {d : ℕ} (P : AddSubgroup (Lattice d))
+    (hP : P.FiniteIndex) : ∃ q : ℕ, 0 < q ∧ ∀ z : Lattice d, q • z ∈ P := by
+  refine ⟨P.index, Nat.pos_of_ne_zero hP.index_ne_zero, ?_⟩
+  intro z
+  exact P.nsmul_index_mem z
+
+theorem FullyPeriodic.castLattice_image {d : ℕ} {A : Set (Lattice d)}
+    (hA : FullyPeriodic A) : EuclideanFullyPeriodic (castLattice '' A) := by
+  classical
+  obtain ⟨P, hP, hPeriods⟩ := hA
+  obtain ⟨q, hq, hqP⟩ := exists_positive_scalar_sublattice P hP
+  refine ⟨scaledCoordinateBasis d q hq.ne', ?_⟩
+  intro v hv
+  change v ∈ Submodule.span ℤ (Set.range (scaledCoordinateBasis d q hq.ne')) at hv
+  induction hv using Submodule.span_induction with
+  | mem v hv =>
+      obtain ⟨i, rfl⟩ := hv
+      rw [scaledCoordinateBasis_eq_cast]
+      exact (hPeriods _ (hqP (Pi.single i 1))).image castLattice
+  | zero => exact period_zero _
+  | add x y _ _ hx hy => exact hx.add hy
+  | smul n x _ hx =>
+      exact (periodSubgroup (castLattice '' A)).zsmul_mem hx n
+
+end PeriodicTilingThree
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/LatticeThickening.lean b/lean/OAI/Geometry/PeriodicTiling/LatticeThickening.lean
new file mode 100644
index 0000000000000000000000000000000000000000..263cceb4b23d6e8032da79a93c73e6a8004d904a
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/LatticeThickening.lean
@@ -0,0 +1,90 @@
+import OAI.Geometry.PeriodicTiling.LatticePeriodicity
+import Mathlib.MeasureTheory.Constructions.Pi
+import Mathlib.Algebra.Order.Floor.Ring
+import Mathlib.Tactic.Abel
+import Mathlib.Tactic.Linarith
+
+namespace OAI
+
+noncomputable section
+
+namespace PeriodicTilingThree
+
+open MeasureTheory
+
+def floorLattice {d : ℕ} (x : Space d) : Lattice d := fun i => ⌊x i⌋
+
+theorem ae_noninteger_coordinates (d : ℕ) :
+    ∀ᵐ x : Space d ∂volume, ∀ i : Fin d, ∀ z : ℤ, x i ≠ (z : ℝ) := by
+  apply ae_all_iff.mpr
+  intro i
+  apply ae_all_iff.mpr
+  intro z
+  exact Measure.ae_eval_ne (fun _ : Fin d => (volume : Measure ℝ)) i (z : ℝ)
+
+theorem sub_floorLattice_mem_unitCube {d : ℕ} (x : Space d) :
+    x - castLattice (floorLattice x) ∈ unitCube d := by
+  apply mem_unitCube.mpr
+  intro i
+  change 0 ≤ x i - (⌊x i⌋ : ℝ) ∧ x i - (⌊x i⌋ : ℝ) ≤ 1
+  have hl := Int.floor_le (x i)
+  have hu := Int.lt_floor_add_one (x i)
+  constructor <;> linarith
+
+theorem floorLattice_eq_of_mem_unitVoxel {d : ℕ} {x : Space d}
+    (hx : ∀ i : Fin d, ∀ z : ℤ, x i ≠ (z : ℝ)) {z : Lattice d}
+    (hz : x - castLattice z ∈ unitCube d) : floorLattice x = z := by
+  funext i
+  apply Int.floor_eq_iff.mpr
+  have hi := mem_unitCube.mp hz i
+  change 0 ≤ x i - (z i : ℝ) ∧ x i - (z i : ℝ) ≤ 1 at hi
+  constructor
+  · linarith
+  · have hne : x i ≠ (z i : ℝ) + 1 := by
+      simpa using hx i (z i + 1)
+    by_contra hn
+    apply hne
+    linarith
+
+theorem Tiles.aeTiles_thickening {d : ℕ} {F : Finset (Lattice d)}
+    {A : Set (Lattice d)} (h : Tiles F A) :
+    AETiles (Thickening F) (castLattice '' A) := by
+  apply (ae_noninteger_coordinates d).mono
+  intro x hx
+  obtain ⟨⟨f, a⟩, hfa⟩ := h.2 (floorLattice x)
+  change (f : Lattice d) + (a : Lattice d) = floorLattice x at hfa
+  refine ⟨⟨castLattice (a : Lattice d), ⟨(a : Lattice d), a.property, rfl⟩⟩, ?_, ?_⟩
+  · refine ⟨(f : Lattice d), f.property, ?_⟩
+    change x - castLattice (a : Lattice d) - castLattice (f : Lattice d) ∈ unitCube d
+    have heq : x - castLattice (a : Lattice d) - castLattice (f : Lattice d) =
+        x - castLattice (floorLattice x) := by
+      rw [← hfa, map_add]
+      abel
+    rw [heq]
+    exact sub_floorLattice_mem_unitCube x
+  · intro b hb
+    obtain ⟨b', hb', hb_eq⟩ := b.property
+    obtain ⟨f', hf', hf'b⟩ := hb
+    have hvoxel : x - castLattice (f' + b') ∈ unitCube d := by
+      have heq : x - castLattice (f' + b') =
+          x - (b : Space d) - castLattice f' := by
+        rw [map_add, ← hb_eq]
+        abel
+      rw [heq]
+      exact hf'b
+    have hcorner := floorLattice_eq_of_mem_unitVoxel hx hvoxel
+    have hp : ((⟨f', hf'⟩ : ↥F), (⟨b', hb'⟩ : A)) = (f, a) := by
+      apply h.1
+      change f' + b' = (f : Lattice d) + (a : Lattice d)
+      exact hcorner.symm.trans hfa.symm
+    apply Subtype.ext
+    calc
+      (b : Space d) = castLattice b' := hb_eq.symm
+      _ = castLattice (a : Lattice d) :=
+        congrArg castLattice (congrArg (fun p : ↥F × A => (p.2 : Lattice d)) hp)
+
+end PeriodicTilingThree
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/LowerDimensions.lean b/lean/OAI/Geometry/PeriodicTiling/LowerDimensions.lean
new file mode 100644
index 0000000000000000000000000000000000000000..a3bcf64c3d895a1dc9c597a99578f929468700a1
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/LowerDimensions.lean
@@ -0,0 +1,38 @@
+import OAI.Geometry.PeriodicTiling.OneDimensionalLattice
+import OAI.Geometry.PeriodicTiling.PlanarPeriodicity
+import OAI.Geometry.PeriodicTiling.TilingTransport
+import Lean.Elab.Tactic.Omega
+
+namespace OAI
+
+namespace PeriodicTilingThree
+
+theorem lattice_two_tile_has_fullyPeriodic_complement
+    (F : Finset (Lattice 2)) (h : ∃ A, Tiles F A) :
+    ∃ B, Tiles F B ∧ FullyPeriodic B := by
+  classical
+  obtain ⟨A, hA⟩ := h
+  have hPlane := hA.preimage_addEquiv planeEquiv
+  obtain ⟨B, hB, hPeriod⟩ := plane_tile_has_fullyPeriodic_complement
+    (F.image planeEquiv.symm) hPlane.tile_nonempty ⟨_, hPlane⟩
+  refine ⟨planeEquiv '' B, ?_, (fullyPeriodic_image_addEquiv_iff planeEquiv).mpr hPeriod⟩
+  have hImage := hB.image_addEquiv planeEquiv
+  simpa only [Finset.image_image, Function.comp_def, planeEquiv.apply_symm_apply,
+    Finset.image_id'] using hImage
+
+theorem tiles_below_three_have_fullyPeriodic_complements
+    (d : ℕ) (hd : d < 3) (F : Finset (Lattice d)) (h : ∃ A, Tiles F A) :
+    ∃ B, Tiles F B ∧ FullyPeriodic B := by
+  cases d with
+  | zero => exact lattice_zero_tile_has_fullyPeriodic_complement F h
+  | succ d =>
+      cases d with
+      | zero => exact lattice_one_tile_has_fullyPeriodic_complement F h
+      | succ d =>
+          cases d with
+          | zero => exact lattice_two_tile_has_fullyPeriodic_complement F h
+          | succ d => omega
+
+end PeriodicTilingThree
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/Main.lean b/lean/OAI/Geometry/PeriodicTiling/Main.lean
new file mode 100644
index 0000000000000000000000000000000000000000..df1299355309afd1ba5d7f1e93402483584a2e9e
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/Main.lean
@@ -0,0 +1 @@
+import OAI.Geometry.PeriodicTiling.Counterexample
diff --git a/lean/OAI/Geometry/PeriodicTiling/MarkedCell.lean b/lean/OAI/Geometry/PeriodicTiling/MarkedCell.lean
new file mode 100644
index 0000000000000000000000000000000000000000..f13125187a7fe82a6763c86c8ffda62084d2b17d
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/MarkedCell.lean
@@ -0,0 +1,155 @@
+import OAI.Geometry.PeriodicTiling.MarkedTile
+import OAI.Geometry.PeriodicTiling.CyclicQuotient
+import OAI.Geometry.PeriodicTiling.VoxelMeasure
+import Mathlib.Tactic.Abel
+
+namespace OAI
+
+noncomputable section
+
+namespace PeriodicTilingThree.MarkedTile
+
+open MeasureTheory
+open scoped Classical
+
+private theorem cast_scale_eq (m : ℕ) (z : Lattice 3) :
+    castLattice (scale m z) = scaledCast m z := by
+  ext i
+  simp [scaledCast, scale]
+
+namespace ResidueSection
+
+variable {m : ℕ} [NeZero m] (s : ResidueSection m)
+
+theorem Tstar_subset_T0 : s.Tstar ⊆ s.T0 := Finset.erase_subset _ _
+
+theorem Tstar_subset_T1 (w : Lattice 3) : s.Tstar ⊆ s.T1 w := by
+  rw [s.T1_eq_insert w]
+  exact Finset.subset_insert _ _
+
+theorem marked_cell_mem_thickening_iff
+    {m : ℕ} [NeZero m] (s : ResidueSection m)
+    {a g : Lattice 3} {x : Space 3}
+    (hx : x ∈ openUnitVoxel (castLattice (scale m a + s.t0)))
+    (F : Finset (Lattice 3)) :
+    x - scaledCast m g ∈ Thickening F ↔
+      ∃ z ∈ F, scale m g + z = scale m a + s.t0 := by
+  have he (z : Lattice 3) :
+      x - castLattice (scale m g + z) =
+        (x - scaledCast m g) - castLattice z := by
+    rw [map_add, cast_scale_eq]
+    abel
+  constructor
+  · rintro ⟨z, hz, hcell⟩
+    refine ⟨z, hz, (mem_unitVoxel_cast_iff_of_mem_open hx).mp ?_⟩
+    change x - castLattice (scale m g + z) ∈ unitCube 3
+    rwa [he z]
+  · rintro ⟨z, hz, hcorner⟩
+    refine ⟨z, hz, ?_⟩
+    have hcell := (mem_unitVoxel_cast_iff_of_mem_open hx).mpr hcorner
+    change x - castLattice (scale m g + z) ∈ unitCube 3 at hcell
+    rwa [he z] at hcell
+
+theorem marked_cell_T0_iff {a g : Lattice 3} {x : Space 3}
+    (hx : x ∈ openUnitVoxel (castLattice (scale m a + s.t0))) :
+    x - scaledCast m g ∈ Thickening s.T0 ↔ g = a := by
+  rw [s.marked_cell_mem_thickening_iff hx]
+  constructor
+  · rintro ⟨z, hz, he⟩
+    obtain ⟨v, hv⟩ := s.mem_T0.mp hz
+    rw [← hv] at he
+    have hp : (g, v) = (a, (0 : Residue m)) := by
+      apply s.coordinates_injective
+      exact he
+    exact congrArg Prod.fst hp
+  · rintro rfl
+    exact ⟨s.t0, s.t0_mem_T0, rfl⟩
+
+theorem marked_cell_T1_iff (w : Lattice 3) {a g : Lattice 3} {x : Space 3}
+    (hx : x ∈ openUnitVoxel (castLattice (scale m a + s.t0))) :
+    x - scaledCast m g ∈ Thickening (s.T1 w) ↔ g = a - w := by
+  rw [s.marked_cell_mem_thickening_iff hx]
+  constructor
+  · rintro ⟨z, hz, he⟩
+    obtain ⟨v, hv⟩ := s.mem_T1.mp hz
+    rw [← hv] at he
+    have hv0 : v = 0 := by
+      have hr := congrArg (residue m) he
+      simpa only [map_add, residue_scale, s.residue_t1, s.residue_t0,
+        zero_add] using hr
+    subst v
+    have hga : g + w = a := by
+      apply scale_injective m
+      apply add_right_cancel (b := s.t0)
+      simpa only [t1, ite_eq_left rfl, ite_true, map_add, add_assoc] using he
+    exact eq_sub_iff_add_eq.mpr hga
+  · rintro rfl
+    refine ⟨s.t1 w 0, s.mem_T1.mpr ⟨0, rfl⟩, ?_⟩
+    simp only [t1, ite_true]
+    rw [← add_assoc, ← map_add, sub_add_cancel]
+
+theorem marked_cell_sources (w : Lattice 3) (B : Set (Lattice 3))
+    {a g : Lattice 3} {x : Space 3}
+    (hx : x ∈ openUnitVoxel (castLattice (scale m a + s.t0))) :
+    (x - scaledCast m g ∈ if g ∈ B then Thickening (s.T1 w) else Thickening s.T0) ↔
+      (g = a ∧ g ∉ B) ∨ (g = a - w ∧ g ∈ B) := by
+  by_cases hg : g ∈ B
+  · simp only [hg, ite_eq_left, not_true_eq_false, and_false, false_or, and_true,
+      s.marked_cell_T1_iff w hx]
+  · simp only [hg, ite_false, not_false_eq_true, and_true, and_false, or_false,
+      s.marked_cell_T0_iff hx]
+
+theorem period_of_marked_cells (w : Lattice 3) (hw : w ≠ 0)
+    (B : Set (Lattice 3))
+    (hcover : ∀ᵐ x ∂volume, ∃! g : Lattice 3,
+      x - scaledCast m g ∈
+        if g ∈ B then Thickening (s.T1 w) else Thickening s.T0) :
+    Period B w := by
+  have hiff : ∀ a : Lattice 3, a ∈ B ↔ a - w ∈ B := by
+    intro a
+    have hvol : volume (openUnitVoxel (castLattice (scale m a + s.t0))) ≠ 0 := by
+      rw [openUnitVoxel_volume]
+      exact one_ne_zero
+    obtain ⟨x, hx, hgcover⟩ :=
+      Measure.exists_mem_of_measure_ne_zero_of_ae hvol (ae_restrict_of_ae hcover)
+    obtain ⟨g, hg, huniq⟩ := hgcover
+    constructor
+    · intro ha
+      rcases (s.marked_cell_sources w B hx).mp hg with ⟨hga, hgB⟩ | ⟨hga, hgB⟩
+      · exact False.elim (hgB (hga.symm ▸ ha))
+      · exact hga ▸ hgB
+    · intro haw
+      by_contra ha
+      have hpa : x - scaledCast m a ∈
+          if a ∈ B then Thickening (s.T1 w) else Thickening s.T0 :=
+        (s.marked_cell_sources w B hx).mpr (Or.inl ⟨rfl, ha⟩)
+      have hpaw : x - scaledCast m (a - w) ∈
+          if a - w ∈ B then Thickening (s.T1 w) else Thickening s.T0 :=
+        (s.marked_cell_sources w B hx).mpr (Or.inr ⟨rfl, haw⟩)
+      have he : a = a - w := (huniq a hpa).trans (huniq (a - w) hpaw).symm
+      apply hw
+      calc
+        w = a - (a - w) := by abel
+        _ = a - a := by rw [← he]
+        _ = 0 := sub_self a
+  intro z
+  simpa only [add_sub_cancel_right] using hiff (z + w)
+
+theorem kernel_periods_of_marked_cells (Q : ℕ) [NeZero Q]
+    (B : Set (Lattice 3))
+    (hcover : ∀ᵐ x ∂volume, ∃! g : Lattice 3,
+      x - scaledCast m g ∈ if g ∈ B then
+        Thickening (s.T1 (CyclicQuotient.kernelStep Q)) else Thickening s.T0) :
+    ∀ k ∈ (CyclicQuotient.projection Q).ker, Period B k := by
+  have hw := s.period_of_marked_cells (CyclicQuotient.kernelStep Q)
+    (CyclicQuotient.kernelStep_ne_zero Q) B hcover
+  intro k hk
+  obtain ⟨n, rfl⟩ := (CyclicQuotient.mem_ker_iff_zsmul_kernelStep Q k).mp hk
+  exact (periodSubgroup B).zsmul_mem hw n
+
+end ResidueSection
+end PeriodicTilingThree.MarkedTile
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/MarkedTile.lean b/lean/OAI/Geometry/PeriodicTiling/MarkedTile.lean
new file mode 100644
index 0000000000000000000000000000000000000000..39e9c04e37e263db691a6596c29df54db32874d0
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/MarkedTile.lean
@@ -0,0 +1,309 @@
+import OAI.Geometry.PeriodicTiling.TilingBasic
+import Mathlib.Data.ZMod.Basic
+import Mathlib.Data.Fintype.BigOperators
+import Mathlib.Data.Finset.Union
+import Mathlib.Tactic.Abel
+import Mathlib.Tactic.NormNum
+
+namespace OAI
+
+namespace PeriodicTilingThree.MarkedTile
+
+abbrev Residue (m : ℕ) := Fin 3 → ZMod m
+
+def scale (m : ℕ) : Lattice 3 →+ Lattice 3 where
+  toFun x i := (m : ℤ) * x i
+  map_zero' := by ext i; simp
+  map_add' x y := by ext i; simp [mul_add]
+
+@[simp] theorem scale_apply (m : ℕ) (x : Lattice 3) (i : Fin 3) :
+    scale m x i = (m : ℤ) * x i := rfl
+
+theorem scale_injective (m : ℕ) [NeZero m] : Function.Injective (scale m) := by
+  have hm : (m : ℤ) ≠ 0 := by exact_mod_cast (NeZero.ne m)
+  intro x y h
+  funext i
+  exact mul_left_cancel₀ hm (congrFun h i)
+
+def residue (m : ℕ) : Lattice 3 →+ Residue m where
+  toFun x i := (x i : ZMod m)
+  map_zero' := by ext i; simp
+  map_add' x y := by ext i; simp
+
+@[simp] theorem residue_apply (m : ℕ) (x : Lattice 3) (i : Fin 3) :
+    residue m x i = (x i : ZMod m) := rfl
+
+@[simp] theorem residue_scale (m : ℕ) (x : Lattice 3) :
+    residue m (scale m x) = 0 := by
+  ext i
+  simp [scale, residue]
+
+theorem exists_scale_of_residue_eq_zero (m : ℕ) {x : Lattice 3}
+    (h : residue m x = 0) : ∃ a : Lattice 3, scale m a = x := by
+  have hd : ∀ i : Fin 3, ∃ a : ℤ, x i = (m : ℤ) * a := by
+    intro i
+    apply (ZMod.intCast_zmod_eq_zero_iff_dvd (x i) m).mp
+    exact congrFun h i
+  choose a ha using hd
+  exact ⟨a, funext fun i => (ha i).symm⟩
+
+structure ResidueSection (m : ℕ) where
+  t : Residue m → Lattice 3
+  residue_t : ∀ v, residue m (t v) = v
+
+namespace ResidueSection
+
+variable {m : ℕ} (s : ResidueSection m)
+
+@[simp] theorem residue_t_apply (v : Residue m) : residue m (s.t v) = v :=
+  s.residue_t v
+
+theorem t_injective : Function.Injective s.t := by
+  intro v v' h
+  have := congrArg (residue m) h
+  simpa using this
+
+def t0 : Lattice 3 := s.t 0
+
+@[simp] theorem residue_t0 : residue m s.t0 = 0 := s.residue_t 0
+
+theorem coordinates_injective [NeZero m] :
+    Function.Injective (fun p : Lattice 3 × Residue m => scale m p.1 + s.t p.2) := by
+  rintro ⟨a, v⟩ ⟨a', v'⟩ h
+  have hv : v = v' := by
+    have := congrArg (residue m) h
+    simpa using this
+  subst v'
+  have ha : a = a' := scale_injective m (add_right_cancel h)
+  exact Prod.ext ha rfl
+
+theorem coordinates_surjective :
+    Function.Surjective (fun p : Lattice 3 × Residue m => scale m p.1 + s.t p.2) := by
+  intro x
+  have hz : residue m (x - s.t (residue m x)) = 0 := by simp
+  obtain ⟨a, ha⟩ := exists_scale_of_residue_eq_zero m hz
+  refine ⟨(a, residue m x), ?_⟩
+  change scale m a + s.t (residue m x) = x
+  rw [ha, sub_add_cancel]
+
+theorem coordinates_bijective [NeZero m] :
+    Function.Bijective (fun p : Lattice 3 × Residue m => scale m p.1 + s.t p.2) :=
+  ⟨s.coordinates_injective, s.coordinates_surjective⟩
+
+noncomputable section
+
+variable [NeZero m]
+
+def T0 : Finset (Lattice 3) := Finset.univ.image s.t
+
+def Tstar : Finset (Lattice 3) := s.T0.erase s.t0
+
+def t1 (w : Lattice 3) (v : Residue m) : Lattice 3 :=
+  if v = 0 then scale m w + s.t0 else s.t v
+
+def T1 (w : Lattice 3) : Finset (Lattice 3) := Finset.univ.image (s.t1 w)
+
+@[simp] theorem mem_T0 {x : Lattice 3} : x ∈ s.T0 ↔ ∃ v, s.t v = x := by
+  classical
+  simp [T0]
+
+@[simp] theorem t0_mem_T0 : s.t0 ∈ s.T0 := by
+  exact s.mem_T0.mpr ⟨0, rfl⟩
+
+theorem mem_Tstar {x : Lattice 3} :
+    x ∈ s.Tstar ↔ ∃ v : Residue m, v ≠ 0 ∧ s.t v = x := by
+  classical
+  constructor
+  · intro hx
+    obtain ⟨hx0, hxT⟩ := Finset.mem_erase.mp hx
+    obtain ⟨v, hv⟩ := s.mem_T0.mp hxT
+    refine ⟨v, ?_, hv⟩
+    intro h0
+    apply hx0
+    simpa [h0, t0] using hv.symm
+  · rintro ⟨v, hv0, rfl⟩
+    apply Finset.mem_erase.mpr
+    refine ⟨?_, s.mem_T0.mpr ⟨v, rfl⟩⟩
+    intro h
+    exact hv0 (s.t_injective h)
+
+@[simp] theorem residue_t1
+    {m : ℕ} (s : ResidueSection m) [NeZero m]
+    (w : Lattice 3) (v : Residue m) :
+    residue m (s.t1 w v) = v := by
+  classical
+  by_cases hv : v = 0
+  · simp [t1, hv]
+  · simp [t1, hv]
+
+theorem t1_injective (w : Lattice 3) : Function.Injective (s.t1 w) := by
+  intro v v' h
+  have := congrArg (residue m) h
+  simpa using this
+
+@[simp] theorem mem_T1 {w x : Lattice 3} : x ∈ s.T1 w ↔ ∃ v, s.t1 w v = x := by
+  classical
+  simp [T1]
+
+theorem T1_eq_insert (w : Lattice 3) :
+    s.T1 w = insert (scale m w + s.t0) s.Tstar := by
+  classical
+  ext x
+  constructor
+  · intro hx
+    obtain ⟨v, rfl⟩ := s.mem_T1.mp hx
+    by_cases hv : v = 0
+    · simp [t1, hv]
+    · apply Finset.mem_insert_of_mem
+      exact s.mem_Tstar.mpr ⟨v, hv, by simp [t1, hv]⟩
+  · intro hx
+    rcases Finset.mem_insert.mp hx with hx | hx
+    · exact s.mem_T1.mpr ⟨0, by simp [t1, hx]⟩
+    · obtain ⟨v, hv, rfl⟩ := s.mem_Tstar.mp hx
+      exact s.mem_T1.mpr ⟨v, by simp [t1, hv]⟩
+
+theorem card_T0 : s.T0.card = m ^ 3 := by
+  classical
+  rw [T0, Finset.card_image_of_injective _ s.t_injective]
+  simp [Residue]
+
+theorem card_Tstar : s.Tstar.card = m ^ 3 - 1 := by
+  classical
+  rw [Tstar, Finset.card_erase_of_mem s.t0_mem_T0, s.card_T0]
+
+theorem card_T1 (w : Lattice 3) : (s.T1 w).card = m ^ 3 := by
+  classical
+  rw [T1, Finset.card_image_of_injective _ (s.t1_injective w)]
+  simp [Residue]
+
+def point (w u : Lattice 3) (v : Residue m) : Lattice 3 :=
+  if u = 0 ∧ v = 0 then scale m w + s.t0 else scale m u + s.t v
+
+@[simp] theorem point_zero
+    {m : ℕ} (s : ResidueSection m) [NeZero m]
+    (w : Lattice 3) (v : Residue m) :
+    s.point w 0 v = s.t1 w v := by
+  classical
+  by_cases hv : v = 0 <;> simp [point, t1, hv]
+
+theorem point_of_ne_zero
+    {m : ℕ} (s : ResidueSection m) [NeZero m]
+    (w : Lattice 3) {u : Lattice 3} (hu : u ≠ 0)
+    (v : Residue m) : s.point w u v = scale m u + s.t v := by
+  classical
+  simp [point, hu]
+
+@[simp] theorem residue_point
+    {m : ℕ} (s : ResidueSection m) [NeZero m]
+    (w u : Lattice 3) (v : Residue m) :
+    residue m (s.point w u v) = v := by
+  classical
+  by_cases h : u = 0 ∧ v = 0
+  · simp [point, h]
+  · simp [point, h]
+
+theorem point_injective {w : Lattice 3} {U : Finset (Lattice 3)}
+    (hw : w ∉ U) :
+    Function.Injective (fun q : ↥U × Residue m => s.point w q.1 q.2) := by
+  classical
+  rintro ⟨u, v⟩ ⟨u', v'⟩ h
+  have hv : v = v' := by
+    have := congrArg (residue m) h
+    simpa using this
+  subst v'
+  suffices hu : u = u' from congrArg (fun a : ↥U => (a, v)) hu
+  apply Subtype.ext
+  by_cases h0 : (u : Lattice 3) = 0
+  · by_cases h0' : (u' : Lattice 3) = 0
+    · exact h0.trans h0'.symm
+    · by_cases hv0 : v = 0
+      · have he : scale m w = scale m (u' : Lattice 3) := by
+          apply add_right_cancel (b := s.t0)
+          simpa [point, h0, h0', hv0, t0] using h
+        have he' := scale_injective m he
+        exact False.elim (hw (he'.symm ▸ u'.property))
+      · have he : scale m (u : Lattice 3) = scale m (u' : Lattice 3) := by
+          apply add_right_cancel (b := s.t v)
+          simpa [point, hv0] using h
+        exact scale_injective m he
+  · by_cases h0' : (u' : Lattice 3) = 0
+    · by_cases hv0 : v = 0
+      · have he : scale m (u : Lattice 3) = scale m w := by
+          apply add_right_cancel (b := s.t0)
+          simpa [point, h0, h0', hv0, t0] using h
+        have he' := scale_injective m he
+        exact False.elim (hw (he' ▸ u.property))
+      · have he : scale m (u : Lattice 3) = scale m (u' : Lattice 3) := by
+          apply add_right_cancel (b := s.t v)
+          simpa [point, hv0] using h
+        exact scale_injective m he
+    · have he : scale m (u : Lattice 3) = scale m (u' : Lattice 3) := by
+        apply add_right_cancel (b := s.t v)
+        simpa [point, h0, h0'] using h
+      exact scale_injective m he
+
+def finalTile (w : Lattice 3) (U : Finset (Lattice 3)) : Finset (Lattice 3) :=
+  Finset.univ.image (fun q : ↥U × Residue m => s.point w q.1 q.2)
+
+theorem mem_finalTile {w x : Lattice 3} {U : Finset (Lattice 3)} :
+    x ∈ s.finalTile w U ↔ ∃ u ∈ U, ∃ v, s.point w u v = x := by
+  classical
+  simp only [finalTile, Finset.mem_image, Finset.mem_univ, true_and]
+  constructor
+  · rintro ⟨⟨u, v⟩, h⟩
+    exact ⟨u, u.property, v, h⟩
+  · rintro ⟨u, hu, v, h⟩
+    exact ⟨(⟨u, hu⟩, v), h⟩
+
+theorem card_finalTile {w : Lattice 3} {U : Finset (Lattice 3)} (hw : w ∉ U) :
+    (s.finalTile w U).card = U.card * m ^ 3 := by
+  classical
+  rw [finalTile, Finset.card_image_of_injective _ (s.point_injective hw)]
+  simp [Residue]
+
+theorem finalTile_nonempty (w : Lattice 3) {U : Finset (Lattice 3)}
+    (h0 : 0 ∈ U) : (s.finalTile w U).Nonempty := by
+  exact ⟨s.point w 0 0, s.mem_finalTile.mpr ⟨0, h0, 0, rfl⟩⟩
+
+theorem finalTile_eq_union (w : Lattice 3) {U : Finset (Lattice 3)} (h0 : 0 ∈ U) :
+    s.finalTile w U = s.T1 w ∪
+      (U.erase 0).biUnion (fun u => s.T0.image (fun x => scale m u + x)) := by
+  classical
+  ext x
+  constructor
+  · intro hx
+    obtain ⟨u, hu, v, rfl⟩ := s.mem_finalTile.mp hx
+    by_cases hu0 : u = 0
+    · apply Finset.mem_union_left
+      exact s.mem_T1.mpr ⟨v, by simp [hu0]⟩
+    · apply Finset.mem_union_right
+      apply Finset.mem_biUnion.mpr
+      refine ⟨u, Finset.mem_erase.mpr ⟨hu0, hu⟩, ?_⟩
+      apply Finset.mem_image.mpr
+      exact ⟨s.t v, s.mem_T0.mpr ⟨v, rfl⟩, (s.point_of_ne_zero w hu0 v).symm⟩
+  · intro hx
+    rcases Finset.mem_union.mp hx with hx | hx
+    · obtain ⟨v, hv⟩ := s.mem_T1.mp hx
+      exact s.mem_finalTile.mpr ⟨0, h0, v, by simpa using hv⟩
+    · obtain ⟨u, hu, hx⟩ := Finset.mem_biUnion.mp hx
+      obtain ⟨hu0, huU⟩ := Finset.mem_erase.mp hu
+      obtain ⟨t, ht, htx⟩ := Finset.mem_image.mp hx
+      obtain ⟨v, hv⟩ := s.mem_T0.mp ht
+      refine s.mem_finalTile.mpr ⟨u, huU, v, ?_⟩
+      rw [s.point_of_ne_zero w hu0 v, hv, htx]
+
+theorem constituent_disjoint {w : Lattice 3} {U : Finset (Lattice 3)}
+    (hw : w ∉ U) {u u' : ↥U} (hne : u ≠ u') :
+    Disjoint (Set.range (s.point w (u : Lattice 3)))
+      (Set.range (s.point w (u' : Lattice 3))) := by
+  rw [Set.disjoint_left]
+  rintro x ⟨v, hv⟩ ⟨v', hv'⟩
+  apply hne
+  have hp : (u, v) = (u', v') := s.point_injective hw (hv.trans hv'.symm)
+  exact congrArg Prod.fst hp
+
+end
+end ResidueSection
+end PeriodicTilingThree.MarkedTile
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/MarkedTileDescent.lean b/lean/OAI/Geometry/PeriodicTiling/MarkedTileDescent.lean
new file mode 100644
index 0000000000000000000000000000000000000000..e264c40ac29513be69173986d6ce444ac39c3826
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/MarkedTileDescent.lean
@@ -0,0 +1,65 @@
+import OAI.Geometry.PeriodicTiling.ConstituentAssembly
+import OAI.Geometry.PeriodicTiling.MarkedCell
+import OAI.Geometry.PeriodicTiling.GridRigidity
+import OAI.Geometry.PeriodicTiling.LatticeDescent
+import OAI.Geometry.PeriodicTiling.QuotientTiling
+import OAI.Geometry.PeriodicTiling.TilingTransport
+
+namespace OAI
+
+noncomputable section
+
+namespace PeriodicTilingThree.MarkedTile.ResidueSection
+
+open CyclicQuotient
+
+variable {m : ℕ} [NeZero m] (s : ResidueSection m)
+
+theorem no_euclideanFullyPeriodic
+    (hm : 2 ≤ m) (hdiff : RigidDifferences m s.Tstar)
+    (Q : ℕ) [NeZero Q] (F : Finset (CyclicQuotient.Group Q)) (hF0 : 0 ∈ F)
+    (haperiodic : ∀ B, Tiles F B → ¬ FullyPeriodic B)
+    {A : Set (Space 3)}
+    (htile : AETiles
+      (Thickening (s.finalTile (kernelStep Q) (representatives Q F))) A) :
+    ¬ EuclideanFullyPeriodic A := by
+  classical
+  rintro ⟨b, hb⟩
+  let U := representatives Q F
+  let w := kernelStep Q
+  have h0 : 0 ∈ U := zero_mem_representatives Q hF0
+  have hw : w ∉ U := kernelStep_not_mem_representatives Q F
+  have htyped := s.typed_constituents hm h0 hw htile
+  have hinj := s.center_injective hm h0 hw htile
+  obtain ⟨c₀, _hc₀, hgrid'⟩ := periodic_voxel_assembly_grid_range hm
+    s.Tstar_subset_T0 (s.Tstar_subset_T1 w) hdiff s.card_T0 (s.card_T1 w)
+    b (fun v hv => period_centers (hb v hv)) hb htyped
+  have hgrid : centers m U A =
+      Set.range (fun z : Lattice 3 => c₀ + scaledCast m z) := hgrid'
+  let B := normalizedComplement m c₀ A
+  have hBtile : Tiles U B := tiles_of_centers_grid hinj hgrid
+  have hAne : A.Nonempty := by
+    obtain ⟨z, hz⟩ := hBtile.complement_nonempty
+    exact ⟨c₀ + scaledCast m z, hz⟩
+  have hAgrid : ∀ a ∈ A, ∃ z : Lattice 3, c₀ + scaledCast m z = a := by
+    intro a ha
+    have hc := subset_centers (m := m) h0 A ha
+    rw [hgrid] at hc
+    exact hc
+  have hBperiodic : FullyPeriodic B :=
+    fullyPeriodic_normalizedComplement m c₀ hAne hAgrid ⟨b, hb⟩
+  have hnormalized := normalized_typedAETiles m (NeZero.ne m) c₀ hgrid htyped
+  have hkernel : ∀ k ∈ (projection Q).ker, Period B k :=
+    s.kernel_periods_of_marked_cells Q B hnormalized
+  have hquotient : Tiles F ((projection Q) '' B) := by
+    have h := tiles_image_of_kernel_periods (projection Q) (projection_surjective Q)
+      hBtile hkernel
+    simpa only [U, projection_image_representatives] using h
+  exact haperiodic _ hquotient
+    (FullyPeriodic.image_of_surjective (projection Q) (projection_surjective Q) hBperiodic)
+
+end PeriodicTilingThree.MarkedTile.ResidueSection
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/MarkedTileExistence.lean b/lean/OAI/Geometry/PeriodicTiling/MarkedTileExistence.lean
new file mode 100644
index 0000000000000000000000000000000000000000..5bf9a97b627bfd8081f8915432a251f037f0ab95
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/MarkedTileExistence.lean
@@ -0,0 +1,145 @@
+import OAI.Geometry.PeriodicTiling.MarkedTile
+import OAI.Geometry.PeriodicTiling.CyclicQuotient
+
+namespace OAI
+
+namespace PeriodicTilingThree.MarkedTile
+
+noncomputable section
+
+def periodEquiv {B : Set (Lattice 3)} {w : Lattice 3} (hB : Period B w) : B ≃ B where
+  toFun b := ⟨(b : Lattice 3) + w, (hB b).mpr b.property⟩
+  invFun b := ⟨(b : Lattice 3) - w, by
+    have h := (hB ((b : Lattice 3) - w)).mp
+    apply h
+    simp [b.property]⟩
+  left_inv b := by
+    apply Subtype.ext
+    change ((b : Lattice 3) + w) - w = (b : Lattice 3)
+    exact add_sub_cancel_right _ _
+  right_inv b := by
+    apply Subtype.ext
+    change ((b : Lattice 3) - w) + w = (b : Lattice 3)
+    exact sub_add_cancel _ _
+
+@[simp] theorem periodEquiv_apply {B : Set (Lattice 3)} {w : Lattice 3}
+    (hB : Period B w) (b : B) : (periodEquiv hB b : Lattice 3) = (b : Lattice 3) + w := rfl
+
+@[simp] theorem periodEquiv_symm_apply {B : Set (Lattice 3)} {w : Lattice 3}
+    (hB : Period B w) (b : B) : ((periodEquiv hB).symm b : Lattice 3) = (b : Lattice 3) - w := rfl
+
+def relocate {m : ℕ} {U : Finset (Lattice 3)} {B : Set (Lattice 3)}
+    {w : Lattice 3} (hB : Period B w) :
+    ((↥U × B) × Residue m) ≃ ((↥U × B) × Residue m) where
+  toFun q := if (q.1.1 : Lattice 3) = 0 ∧ q.2 = 0 then
+    ((q.1.1, periodEquiv hB q.1.2), q.2) else q
+  invFun q := if (q.1.1 : Lattice 3) = 0 ∧ q.2 = 0 then
+    ((q.1.1, (periodEquiv hB).symm q.1.2), q.2) else q
+  left_inv q := by
+    rcases q with ⟨⟨u, b⟩, v⟩
+    by_cases h : (u : Lattice 3) = 0 ∧ v = 0 <;> simp [h]
+  right_inv q := by
+    rcases q with ⟨⟨u, b⟩, v⟩
+    by_cases h : (u : Lattice 3) = 0 ∧ v = 0 <;> simp [h]
+
+namespace ResidueSection
+
+variable {m : ℕ} [NeZero m] (s : ResidueSection m)
+
+def baseline {U : Finset (Lattice 3)} {B : Set (Lattice 3)}
+    (q : (↥U × B) × Residue m) : Lattice 3 :=
+  scale m ((q.1.1 : Lattice 3) + (q.1.2 : Lattice 3)) + s.t q.2
+
+theorem baseline_bijective {U : Finset (Lattice 3)} {B : Set (Lattice 3)}
+    (h : Tiles U B) : Function.Bijective (s.baseline (U := U) (B := B)) := by
+  constructor
+  · rintro ⟨⟨u, b⟩, v⟩ ⟨⟨u', b'⟩, v'⟩ he
+    have hc : ((u : Lattice 3) + (b : Lattice 3), v) =
+        ((u' : Lattice 3) + (b' : Lattice 3), v') :=
+      s.coordinates_injective he
+    have hi : (u, b) = (u', b') := h.1 (congrArg Prod.fst hc)
+    have hv : v = v' := congrArg (fun p : Lattice 3 × Residue m => p.2) hc
+    exact Prod.ext hi hv
+  · intro x
+    obtain ⟨⟨a, v⟩, hx⟩ := s.coordinates_surjective x
+    obtain ⟨⟨u, b⟩, ha⟩ := h.2 a
+    refine ⟨((u, b), v), ?_⟩
+    change scale m ((u : Lattice 3) + (b : Lattice 3)) + s.t v = x
+    change (u : Lattice 3) + (b : Lattice 3) = a at ha
+    rw [ha]
+    exact hx
+
+def placement {U : Finset (Lattice 3)} {B : Set (Lattice 3)}
+    (w : Lattice 3) (q : (↥U × B) × Residue m) : Lattice 3 :=
+  s.point w q.1.1 q.2 + scale m q.1.2
+
+theorem placement_eq_baseline_relocate
+    {m : ℕ} [NeZero m] (s : ResidueSection m)
+    {U : Finset (Lattice 3)}
+    {B : Set (Lattice 3)} {w : Lattice 3} (hB : Period B w)
+    (q : (↥U × B) × Residue m) :
+    s.placement w q = s.baseline (relocate hB q) := by
+  classical
+  rcases q with ⟨⟨u, b⟩, v⟩
+  by_cases h : (u : Lattice 3) = 0 ∧ v = 0
+  · simp only [placement, point, relocate, Equiv.coe_fn_mk, baseline,
+      h.1, h.2, and_self, ite_true, periodEquiv_apply, zero_add, map_add, t0]
+    abel
+  · simp only [placement, point, h, ite_false, relocate, Equiv.coe_fn_mk, baseline,
+      map_add]
+    abel
+
+theorem placement_bijective {U : Finset (Lattice 3)} {B : Set (Lattice 3)}
+    {w : Lattice 3} (h : Tiles U B) (hB : Period B w) :
+    Function.Bijective (s.placement (U := U) (B := B) w) := by
+  have he : s.placement (U := U) (B := B) w = s.baseline ∘ relocate hB := by
+    funext q
+    exact s.placement_eq_baseline_relocate hB q
+  rw [he]
+  exact (s.baseline_bijective h).comp (relocate hB).bijective
+
+theorem finalTile_tiles {U : Finset (Lattice 3)} {B : Set (Lattice 3)}
+    {w : Lattice 3} (h : Tiles U B) (hB : Period B w) :
+    Tiles (s.finalTile w U) ((scale m) '' B) := by
+  classical
+  have hp := s.placement_bijective h hB
+  constructor
+  · rintro ⟨⟨t, ht⟩, ⟨c, hc⟩⟩ ⟨⟨t', ht'⟩, ⟨c', hc'⟩⟩ he
+    obtain ⟨u, hu, v, hv⟩ := s.mem_finalTile.mp ht
+    obtain ⟨u', hu', v', hv'⟩ := s.mem_finalTile.mp ht'
+    obtain ⟨b, hb, hbc⟩ := hc
+    obtain ⟨b', hb', hbc'⟩ := hc'
+    have hi : (((⟨u, hu⟩ : ↥U), (⟨b, hb⟩ : B)), v) =
+        (((⟨u', hu'⟩ : ↥U), (⟨b', hb'⟩ : B)), v') := by
+      apply hp.1
+      change s.point w u v + scale m b = s.point w u' v' + scale m b'
+      rw [hv, hv', hbc, hbc']
+      exact he
+    have htt : t = t' := by
+      have hh := congrArg
+        (fun q : (↥U × B) × Residue m => s.point w q.1.1 q.2) hi
+      simpa only [hv, hv'] using hh
+    have hcc : c = c' := by
+      have hh := congrArg
+        (fun q : (↥U × B) × Residue m => scale m q.1.2) hi
+      simpa only [hbc, hbc'] using hh
+    exact Prod.ext (Subtype.ext htt) (Subtype.ext hcc)
+  · intro x
+    obtain ⟨⟨⟨u, b⟩, v⟩, hx⟩ := hp.2 x
+    refine ⟨(⟨s.point w u v, s.mem_finalTile.mpr ⟨u, u.property, v, rfl⟩⟩,
+      ⟨scale m b, ⟨b, b.property, rfl⟩⟩), ?_⟩
+    exact hx
+
+theorem cyclic_finalTile_tiles (Q : ℕ) [NeZero Q]
+    {F : Finset (CyclicQuotient.Group Q)} {A : Set (CyclicQuotient.Group Q)}
+    (h : Tiles F A) :
+    Tiles (s.finalTile (CyclicQuotient.kernelStep Q) (CyclicQuotient.representatives Q F))
+      ((scale m) '' ((CyclicQuotient.projection Q) ⁻¹' A)) :=
+  s.finalTile_tiles (CyclicQuotient.preimage_tiles Q h)
+    (CyclicQuotient.period_preimage_kernelStep Q A)
+
+end ResidueSection
+end
+end PeriodicTilingThree.MarkedTile
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/Model.lean b/lean/OAI/Geometry/PeriodicTiling/Model.lean
new file mode 100644
index 0000000000000000000000000000000000000000..f191d50f2e570c9fcf5de5c5e3e6a2d35b65b820
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/Model.lean
@@ -0,0 +1,60 @@
+import Mathlib
+
+namespace OAI
+
+noncomputable section
+
+universe u
+
+namespace PeriodicTilingThree
+
+abbrev Lattice (d : ℕ) := Fin d → ℤ
+
+section AdditiveGroup
+
+variable {G : Type u} [AddCommGroup G]
+
+def Tiles (F : Finset G) (A : Set G) : Prop :=
+  Function.Bijective (fun p : ↥F × A => (p.1 : G) + (p.2 : G))
+
+def Period (A : Set G) (v : G) : Prop :=
+  ∀ x : G, x + v ∈ A ↔ x ∈ A
+
+def FullyPeriodic (A : Set G) : Prop :=
+  ∃ P : AddSubgroup G, P.FiniteIndex ∧ ∀ v ∈ P, Period A v
+
+end AdditiveGroup
+
+open MeasureTheory
+
+abbrev Space (d : ℕ) := Fin d → ℝ
+
+def castLattice {d : ℕ} : Lattice d →+ Space d where
+  toFun z i := (z i : ℝ)
+  map_zero' := by ext i; simp
+  map_add' z w := by ext i; simp
+
+def unitCube (d : ℕ) : Set (Space d) := Set.Icc 0 1
+
+def Thickening {d : ℕ} (F : Finset (Lattice d)) : Set (Space d) :=
+  {x | ∃ z ∈ F, x - castLattice z ∈ unitCube d}
+
+def AETiles {d : ℕ} (Ω A : Set (Space d)) : Prop :=
+  ∀ᵐ x ∂volume, ∃! a : A, x - (a : Space d) ∈ Ω
+
+def latticeOfBasis {d : ℕ} (b : Module.Basis (Fin d) ℝ (Space d)) :
+    AddSubgroup (Space d) := (Submodule.span ℤ (Set.range b)).toAddSubgroup
+
+def EuclideanFullyPeriodic {d : ℕ} (A : Set (Space d)) : Prop :=
+  ∃ b : Module.Basis (Fin d) ℝ (Space d), ∀ v ∈ latticeOfBasis b, Period A v
+
+def HasTileWithoutPeriodicComplement (d : ℕ) : Prop :=
+  ∃ T : Finset (Lattice d), T.Nonempty ∧
+    (∃ A : Set (Lattice d), Tiles T A) ∧
+    ∀ A : Set (Lattice d), Tiles T A → ¬ FullyPeriodic A
+
+end PeriodicTilingThree
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/OneDimensional.lean b/lean/OAI/Geometry/PeriodicTiling/OneDimensional.lean
new file mode 100644
index 0000000000000000000000000000000000000000..09579d0cbdbed8a090ae35da99260c71fa5ca58f
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/OneDimensional.lean
@@ -0,0 +1,286 @@
+import OAI.Geometry.PeriodicTiling.TilingBasic
+import Mathlib.Algebra.Ring.Periodic
+import Mathlib.Data.Finset.Max
+import Mathlib.Data.Fintype.BigOperators
+import Mathlib.Data.Fintype.Pigeonhole
+import Mathlib.Data.ZMod.QuotientGroup
+import Lean.Elab.Tactic.Omega
+
+namespace OAI
+
+noncomputable section
+
+open scoped BigOperators
+
+namespace PeriodicTilingThree
+
+def oneDValue (b : Fin 2) : ℤ := (b.val : ℤ)
+
+private theorem oneDValue_injective : Function.Injective oneDValue := by
+  intro a b h
+  apply Fin.ext
+  unfold oneDValue at h
+  omega
+
+def OneDConv (T : Finset ℤ) (a : ℤ → Fin 2) : Prop :=
+  ∀ z : ℤ, (∑ t ∈ T, oneDValue (a (z - t))) = 1
+
+def oneDWindow (M : ℕ) (a : ℤ → Fin 2) (n : ℤ) : Fin M → Fin 2 :=
+  fun i => a (n + (i.val : ℤ))
+
+def OneDBlockEq (M : ℕ) (a : ℤ → Fin 2) (n m : ℤ) : Prop :=
+  ∀ i : ℤ, 0 ≤ i → i < (M : ℤ) → a (n + i) = a (m + i)
+
+private theorem oneDBlockEq_of_window_eq {M : ℕ} {a : ℤ → Fin 2} {n m : ℤ}
+    (h : oneDWindow M a n = oneDWindow M a m) : OneDBlockEq M a n m := by
+  intro i hi hiM
+  have hcast : (i.toNat : ℤ) = i := Int.toNat_of_nonneg hi
+  have hbound : i.toNat < M := by omega
+  have he := congrFun h (⟨i.toNat, hbound⟩ : Fin M)
+  simpa only [oneDWindow, hcast] using he
+
+def oneDIndicator (A : Set ℤ) (n : ℤ) : Fin 2 := by
+  classical
+  exact if n ∈ A then 1 else 0
+
+theorem oneDConv_of_tiles {T : Finset ℤ} {A : Set ℤ} (hTile : Tiles T A) :
+    OneDConv T (oneDIndicator A) := by
+  classical
+  intro z
+  obtain ⟨t₀, ht₀, huniq⟩ := tiles_iff_unique_tile.mp hTile z
+  calc
+    (∑ t ∈ T, oneDValue (oneDIndicator A (z - t))) =
+        oneDValue (oneDIndicator A (z - t₀.val)) := by
+      apply Finset.sum_eq_single t₀.val
+      · intro t ht hne
+        have hnot : z - t ∉ A := by
+          intro hmem
+          have he := congrArg Subtype.val (huniq (⟨t, ht⟩ : ↥T) hmem)
+          exact hne he
+        simp [oneDIndicator, hnot, oneDValue]
+      · intro hnot
+        exact (hnot t₀.property).elim
+    _ = 1 := by simp [oneDIndicator, ht₀, oneDValue]
+
+theorem oneD_right_recurrence {T : Finset ℤ} {a : ℤ → Fin 2}
+    (lo : ℤ) (M : ℕ) (hlo : lo ∈ T) (hConv : OneDConv T a) (n : ℤ) :
+    oneDValue (a (n + (M : ℤ))) =
+      1 - ∑ t ∈ T.erase lo, oneDValue (a (n + (M : ℤ) + lo - t)) := by
+  have hs := Finset.sum_erase_add T
+    (fun t => oneDValue (a (n + (M : ℤ) + lo - t))) hlo
+  rw [hConv (n + (M : ℤ) + lo)] at hs
+  have harg : n + (M : ℤ) + lo - lo = n + (M : ℤ) := by omega
+  rw [harg] at hs
+  omega
+
+theorem oneD_left_recurrence {T : Finset ℤ} {a : ℤ → Fin 2}
+    (lo : ℤ) (M : ℕ) (hhi : lo + (M : ℤ) ∈ T)
+    (hConv : OneDConv T a) (n : ℤ) :
+    oneDValue (a (n - 1)) =
+      1 - ∑ t ∈ T.erase (lo + (M : ℤ)),
+        oneDValue (a (n + (M : ℤ) + lo - 1 - t)) := by
+  have hs := Finset.sum_erase_add T
+    (fun t => oneDValue (a (n + (M : ℤ) + lo - 1 - t))) hhi
+  rw [hConv (n + (M : ℤ) + lo - 1)] at hs
+  have harg : n + (M : ℤ) + lo - 1 - (lo + (M : ℤ)) = n - 1 := by omega
+  rw [harg] at hs
+  omega
+
+theorem oneD_block_eq_add_one {T : Finset ℤ} {a : ℤ → Fin 2}
+    {lo : ℤ} {M : ℕ} (hlo : lo ∈ T)
+    (hBounds : ∀ t ∈ T, lo ≤ t ∧ t ≤ lo + (M : ℤ))
+    (hConv : OneDConv T a) {n m : ℤ} (hBlock : OneDBlockEq M a n m) :
+    OneDBlockEq M a (n + 1) (m + 1) := by
+  have hsum :
+      (∑ t ∈ T.erase lo, oneDValue (a (n + (M : ℤ) + lo - t))) =
+      ∑ t ∈ T.erase lo, oneDValue (a (m + (M : ℤ) + lo - t)) := by
+    apply Finset.sum_congr rfl
+    intro t ht
+    obtain ⟨hne, hmem⟩ := Finset.mem_erase.mp ht
+    obtain ⟨hlo', hhi'⟩ := hBounds t hmem
+    have he := hBlock ((M : ℤ) + lo - t) (by omega) (by omega)
+    have hn : n + ((M : ℤ) + lo - t) = n + (M : ℤ) + lo - t := by omega
+    have hm : m + ((M : ℤ) + lo - t) = m + (M : ℤ) + lo - t := by omega
+    rw [hn, hm] at he
+    exact congrArg oneDValue he
+  have hnext : a (n + (M : ℤ)) = a (m + (M : ℤ)) := by
+    apply oneDValue_injective
+    rw [oneD_right_recurrence lo M hlo hConv n,
+      oneD_right_recurrence lo M hlo hConv m, hsum]
+  intro i hi hiM
+  by_cases hinner : i + 1 < (M : ℤ)
+  · have he := hBlock (i + 1) (by omega) hinner
+    have hn : n + (i + 1) = n + 1 + i := by omega
+    have hm : m + (i + 1) = m + 1 + i := by omega
+    simpa only [hn, hm] using he
+  · have hieq : i + 1 = (M : ℤ) := by omega
+    have hn : n + 1 + i = n + (M : ℤ) := by omega
+    have hm : m + 1 + i = m + (M : ℤ) := by omega
+    simpa only [hn, hm] using hnext
+
+theorem oneD_block_eq_sub_one {T : Finset ℤ} {a : ℤ → Fin 2}
+    {lo : ℤ} {M : ℕ} (hhi : lo + (M : ℤ) ∈ T)
+    (hBounds : ∀ t ∈ T, lo ≤ t ∧ t ≤ lo + (M : ℤ))
+    (hConv : OneDConv T a) {n m : ℤ} (hBlock : OneDBlockEq M a n m) :
+    OneDBlockEq M a (n - 1) (m - 1) := by
+  have hsum :
+      (∑ t ∈ T.erase (lo + (M : ℤ)),
+        oneDValue (a (n + (M : ℤ) + lo - 1 - t))) =
+      ∑ t ∈ T.erase (lo + (M : ℤ)),
+        oneDValue (a (m + (M : ℤ) + lo - 1 - t)) := by
+    apply Finset.sum_congr rfl
+    intro t ht
+    obtain ⟨hne, hmem⟩ := Finset.mem_erase.mp ht
+    obtain ⟨hlo', hhi'⟩ := hBounds t hmem
+    have he := hBlock ((M : ℤ) + lo - 1 - t) (by omega) (by omega)
+    have hn : n + ((M : ℤ) + lo - 1 - t) = n + (M : ℤ) + lo - 1 - t := by omega
+    have hm : m + ((M : ℤ) + lo - 1 - t) = m + (M : ℤ) + lo - 1 - t := by omega
+    rw [hn, hm] at he
+    exact congrArg oneDValue he
+  have hprev : a (n - 1) = a (m - 1) := by
+    apply oneDValue_injective
+    rw [oneD_left_recurrence lo M hhi hConv n,
+      oneD_left_recurrence lo M hhi hConv m, hsum]
+  intro i hi hiM
+  by_cases hiz : i = 0
+  · simpa only [hiz, add_zero] using hprev
+  · have he := hBlock (i - 1) (by omega) (by omega)
+    have hn : n + (i - 1) = n - 1 + i := by omega
+    have hm : m + (i - 1) = m - 1 + i := by omega
+    simpa only [hn, hm] using he
+
+theorem oneD_block_eq_add_int {T : Finset ℤ} {a : ℤ → Fin 2}
+    {lo : ℤ} {M : ℕ} (hlo : lo ∈ T) (hhi : lo + (M : ℤ) ∈ T)
+    (hBounds : ∀ t ∈ T, lo ≤ t ∧ t ≤ lo + (M : ℤ))
+    (hConv : OneDConv T a) {n m : ℤ} (hBlock : OneDBlockEq M a n m) :
+    ∀ z : ℤ, OneDBlockEq M a (n + z) (m + z) := by
+  have hpos : ∀ r : ℕ, OneDBlockEq M a (n + (r : ℤ)) (m + (r : ℤ)) := by
+    intro r
+    induction r with
+    | zero => simpa only [Nat.cast_zero, add_zero] using hBlock
+    | succ r ih =>
+        have he := oneD_block_eq_add_one hlo hBounds hConv ih
+        simpa only [Nat.cast_add, Nat.cast_one, Nat.succ_eq_add_one, add_assoc] using he
+  have hneg : ∀ r : ℕ, OneDBlockEq M a (n - (r : ℤ)) (m - (r : ℤ)) := by
+    intro r
+    induction r with
+    | zero => simpa only [Nat.cast_zero, sub_zero] using hBlock
+    | succ r ih =>
+        have he := oneD_block_eq_sub_one hhi hBounds hConv ih
+        simpa only [Nat.cast_add, Nat.cast_one, Nat.succ_eq_add_one, sub_sub] using he
+  intro z
+  cases z with
+  | ofNat r => exact hpos r
+  | negSucc r =>
+      have hn : n + Int.negSucc r = n - ((r + 1 : ℕ) : ℤ) := by omega
+      have hm : m + Int.negSucc r = m - ((r + 1 : ℕ) : ℤ) := by omega
+      rw [hn, hm]
+      exact hneg (r + 1)
+
+theorem oneD_period_of_conv
+    (T : Finset ℤ) (a : ℤ → Fin 2) (lo : ℤ) (M : ℕ) (hM : 0 < M)
+    (hlo : lo ∈ T) (hhi : lo + (M : ℤ) ∈ T)
+    (hBounds : ∀ t ∈ T, lo ≤ t ∧ t ≤ lo + (M : ℤ))
+    (hConv : OneDConv T a) :
+    ∃ p : ℕ, 0 < p ∧ p ≤ 2 ^ M ∧ Function.Periodic a (p : ℤ) := by
+  classical
+  have hcard : Fintype.card (Fin M → Fin 2) < Fintype.card (Fin (2 ^ M + 1)) := by
+    simp only [Fintype.card_fun, Fintype.card_fin]
+    omega
+  obtain ⟨i, j, hne, heq⟩ := Fintype.exists_ne_map_eq_of_card_lt
+    (fun i : Fin (2 ^ M + 1) => oneDWindow M a (i.val : ℤ)) hcard
+  have derive : ∀ i j : Fin (2 ^ M + 1), i.val < j.val →
+      oneDWindow M a (i.val : ℤ) = oneDWindow M a (j.val : ℤ) →
+      ∃ p : ℕ, 0 < p ∧ p ≤ 2 ^ M ∧ Function.Periodic a (p : ℤ) := by
+    intro i j hij hstate
+    refine ⟨j.val - i.val, by omega, by have := j.isLt; omega, ?_⟩
+    intro x
+    have hBlock := oneDBlockEq_of_window_eq hstate
+    have hshift := oneD_block_eq_add_int hlo hhi hBounds hConv hBlock (x - (i.val : ℤ))
+    have he := hshift 0 (by omega) (by omega)
+    have hn : (i.val : ℤ) + (x - (i.val : ℤ)) + 0 = x := by omega
+    have hm : (j.val : ℤ) + (x - (i.val : ℤ)) + 0 =
+        x + ((j.val - i.val : ℕ) : ℤ) := by omega
+    rw [hn, hm] at he
+    exact he.symm
+  have hneval : i.val ≠ j.val := fun h => hne (Fin.ext h)
+  rcases lt_or_gt_of_ne hneval with hij | hji
+  · exact derive i j hij heq
+  · exact derive j i hji heq.symm
+
+theorem periodic_of_tiles_int
+    (T : Finset ℤ) (hT : T.Nonempty) (A : Set ℤ) (hTile : Tiles T A) :
+    ∃ p : ℕ, 0 < p ∧ Period A (p : ℤ) := by
+  classical
+  let lo : ℤ := T.min' hT
+  let hi : ℤ := T.max' hT
+  have hlo : lo ∈ T := T.min'_mem hT
+  have hhi : hi ∈ T := T.max'_mem hT
+  have hlohi : lo ≤ hi := T.min'_le hi hhi
+  let M : ℕ := (hi - lo).toNat
+  have hMcast : (M : ℤ) = hi - lo := Int.toNat_of_nonneg (by omega)
+  have hmax : lo + (M : ℤ) ∈ T := by
+    have he : lo + (M : ℤ) = hi := by omega
+    rw [he]
+    exact hhi
+  have hBounds : ∀ t ∈ T, lo ≤ t ∧ t ≤ lo + (M : ℤ) := by
+    intro t ht
+    have hl : lo ≤ t := T.min'_le t ht
+    have hr : t ≤ hi := T.le_max' t ht
+    constructor <;> omega
+  by_cases hzero : M = 0
+  · have hall : ∀ z : ℤ, z ∈ A := by
+      intro z
+      obtain ⟨t, ht, _⟩ := tiles_iff_unique_tile.mp hTile (lo + z)
+      obtain ⟨hl, hr⟩ := hBounds t.val t.property
+      have hteq : t.val = lo := by omega
+      simpa only [hteq, add_sub_cancel_left] using ht
+    refine ⟨1, by omega, ?_⟩
+    intro z
+    exact ⟨fun _ => hall z, fun _ => hall (z + (1 : ℤ))⟩
+  · have hM : 0 < M := Nat.pos_of_ne_zero hzero
+    obtain ⟨p, hp, _, hperiod⟩ := oneD_period_of_conv T (oneDIndicator A) lo M hM
+      hlo hmax hBounds (oneDConv_of_tiles hTile)
+    refine ⟨p, hp, ?_⟩
+    intro z
+    have he := hperiod z
+    constructor
+    · intro hz
+      by_contra hn
+      simp [oneDIndicator, hz, hn] at he
+    · intro hz
+      by_contra hn
+      simp [oneDIndicator, hz, hn] at he
+
+theorem fullyPeriodic_of_nat_period
+    (A : Set ℤ) (p : ℕ) (hp : 0 < p) (hPeriod : Period A (p : ℤ)) :
+    FullyPeriodic A := by
+  refine ⟨AddSubgroup.zmultiples (p : ℤ), ?_, ?_⟩
+  · refine ⟨?_⟩
+    simpa only [Int.index_zmultiples, Int.natAbs_natCast] using (Nat.ne_of_gt hp)
+  · intro v hv
+    obtain ⟨k, rfl⟩ := Int.mem_zmultiples_iff.mp hv
+    have hFun : Function.Periodic (fun z : ℤ => z ∈ A) (p : ℤ) :=
+      fun z => propext (hPeriod z)
+    intro z
+    have he : (z + (p : ℤ) * k ∈ A) = (z ∈ A) := by
+      simpa only [Int.cast_id, id_def, mul_comm] using hFun.int_mul k z
+    exact ⟨fun hz => Eq.mp he hz, fun hz => Eq.mpr he hz⟩
+
+theorem fullyPeriodic_of_tiles_int
+    (T : Finset ℤ) (hT : T.Nonempty) (A : Set ℤ) (hTile : Tiles T A) :
+    FullyPeriodic A := by
+  obtain ⟨p, hp, hPeriod⟩ := periodic_of_tiles_int T hT A hTile
+  exact fullyPeriodic_of_nat_period A p hp hPeriod
+
+theorem exists_fullyPeriodic_tiling_int
+    (T : Finset ℤ) (hT : T.Nonempty) (hTile : ∃ A : Set ℤ, Tiles T A) :
+    ∃ A : Set ℤ, Tiles T A ∧ FullyPeriodic A := by
+  obtain ⟨A, hA⟩ := hTile
+  exact ⟨A, hA, fullyPeriodic_of_tiles_int T hT A hA⟩
+
+end PeriodicTilingThree
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/OneDimensionalLattice.lean b/lean/OAI/Geometry/PeriodicTiling/OneDimensionalLattice.lean
new file mode 100644
index 0000000000000000000000000000000000000000..2fc84cef5b9ce7c923285264bfc4feb081751798
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/OneDimensionalLattice.lean
@@ -0,0 +1,24 @@
+import OAI.Geometry.PeriodicTiling.LatticeDimensions
+import OAI.Geometry.PeriodicTiling.OneDimensional
+import OAI.Geometry.PeriodicTiling.TilingTransport
+
+namespace OAI
+
+namespace PeriodicTilingThree
+
+theorem fullyPeriodic_of_tiles_lattice_one
+    {F : Finset (Lattice 1)} {A : Set (Lattice 1)} (h : Tiles F A) :
+    FullyPeriodic A := by
+  have hInt := h.preimage_addEquiv intEquiv
+  have hPeriod := fullyPeriodic_of_tiles_int _ hInt.tile_nonempty _ hInt
+  exact (fullyPeriodic_preimage_addEquiv_iff intEquiv).mp hPeriod
+
+theorem lattice_one_tile_has_fullyPeriodic_complement
+    (F : Finset (Lattice 1)) (h : ∃ A, Tiles F A) :
+    ∃ A, Tiles F A ∧ FullyPeriodic A := by
+  obtain ⟨A, hA⟩ := h
+  exact ⟨A, hA, fullyPeriodic_of_tiles_lattice_one hA⟩
+
+end PeriodicTilingThree
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/OrdinaryActivation.lean b/lean/OAI/Geometry/PeriodicTiling/OrdinaryActivation.lean
new file mode 100644
index 0000000000000000000000000000000000000000..b26d33625dc83d6bff5b85a96c3f5f8323edd449
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/OrdinaryActivation.lean
@@ -0,0 +1,38 @@
+import OAI.Geometry.PeriodicTiling.OrdinaryModel
+import OAI.Geometry.PeriodicTiling.BlockActivationInverse
+
+namespace OAI
+
+noncomputable section
+
+namespace PeriodicTilingThree.CommonModel
+
+variable {p : ℕ} [NeZero p] (E : EncodingParameters p)
+
+def ordinaryForward (n : Column p) (x : Plane) (k : K E (.inl n))
+    (j : ActivationIndex E (.inl n)) : P E (.inl n) × K E (.inl n) :=
+  (ordinaryUseful E n (x - ordinaryOffset E n j) (k - j.1) +
+    shift (E.a (.inl n)) (E.b (.inl n)) j.2,
+    ordinaryHigh E n (x - ordinaryOffset E n j) (k - j.1))
+
+theorem ordinaryForward_bijective (n : Column p) (x : Plane) (k : K E (.inl n)) :
+    Function.Bijective (ordinaryForward E n x k) := by
+  let D := channelDigitBlocks E (.inl n) (ordinarySymbol E n x)
+  let S := BlockShiftData.ofMultiplicity
+    (E.a_two_le (.inl n)) (E.b_two_le (.inl n))
+  let e := BlockShiftData.blockInverseEquiv D S
+    (fun j => (ordinaryOffset E n j).1)
+    (fun j => ordinaryOffset_rhoA E n j)
+    (fun j => ordinaryOffset_rhoB E n j) x.1 k
+  have he : ordinaryForward E n x k = e := by
+    funext j
+    simp only [ordinaryForward, ordinaryUseful_offset, ordinaryHigh_offset]
+    rfl
+  rw [he]
+  exact e.bijective
+
+end PeriodicTilingThree.CommonModel
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/OrdinaryModel.lean b/lean/OAI/Geometry/PeriodicTiling/OrdinaryModel.lean
new file mode 100644
index 0000000000000000000000000000000000000000..45dd203be448fa9daf0bc83c0abb9e131f0389a0
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/OrdinaryModel.lean
@@ -0,0 +1,125 @@
+import OAI.Geometry.PeriodicTiling.ChannelDigitBlocks
+import OAI.Geometry.PeriodicTiling.LastDigit
+import OAI.Geometry.PeriodicTiling.ActivationOffsets
+import OAI.Geometry.PeriodicTiling.SharedSeedRegions
+import Mathlib.Tactic.Ring
+
+namespace OAI
+
+noncomputable section
+
+namespace PeriodicTilingThree.CommonModel
+
+variable {p : ℕ} [NeZero p] (E : EncodingParameters p)
+
+def ordinarySymbol (n : Column p) (x : Plane) : Symbol p :=
+  lastDigit p E.p_prime (lineValue n x)
+
+def ordinaryUseful (n : Column p) (x : Plane) (w : K E (.inl n)) : P E (.inl n) :=
+  (channelDigitBlocks E (.inl n) (ordinarySymbol E n x)).C w
+
+def ordinaryHigh (n : Column p) (x : Plane) (w : K E (.inl n)) : K E (.inl n) :=
+  (channelDigitBlocks E (.inl n) (ordinarySymbol E n x)).T x.1 w
+
+def ordinaryWord (x : Plane) : Word p := fun n => ordinarySymbol E n x
+
+theorem ordinaryWord_allowed
+    {p : ℕ} [NeZero p] (E : EncodingParameters p)
+    (x : Plane) : Allowed p (ordinaryWord E x) := by
+  have hword : ordinaryWord E x =
+      (fun n : Column p => lastDigit p E.p_prime (x.1 * (n.val : ℤ) + x.2)) := by
+    funext n
+    change lastDigit p E.p_prime (lineValue n x) = _
+    congr 1
+    unfold lineValue
+    ring
+  rw [hword]
+  exact lastDigit_affine_allowed p E.p_prime x.1 x.2
+
+theorem ordinaryUseful_eq_of_lineValue_eq (n : Column p) {x y : Plane}
+    (hxy : lineValue n x = lineValue n y) (w : K E (.inl n)) :
+    ordinaryUseful E n x w = ordinaryUseful E n y w := by
+  exact congrArg
+    (fun t : ℤ => (channelDigitBlocks E (.inl n) (lastDigit p E.p_prime t)).C w) hxy
+
+theorem ordinaryUseful_active_iff (n : Column p) (x : Plane) (j : Symbol p) :
+    (∃ w w' : K E (.inl n), w - w' ∈ digitSubgroup E (.inl n) j ∧
+      ordinaryUseful E n x w ≠ ordinaryUseful E n x w') ↔
+        j = ordinarySymbol E n x :=
+  channelDigitBlocks_active_iff E (.inl n) (ordinarySymbol E n x) j
+
+theorem ordinarySymbol_offset (n : Column p) (x : Plane)
+    (j : ActivationIndex E (.inl n)) :
+    ordinarySymbol E n (x - ordinaryOffset E n j) = ordinarySymbol E n x := by
+  unfold ordinarySymbol
+  congr 1
+  simp only [lineValue, ordinaryOffset, Prod.fst_sub, Prod.snd_sub]
+  ring
+
+theorem ordinaryUseful_offset (n : Column p) (x : Plane)
+    (j : ActivationIndex E (.inl n)) (w : K E (.inl n)) :
+    ordinaryUseful E n (x - ordinaryOffset E n j) w = ordinaryUseful E n x w := by
+  exact congrArg (fun s : Symbol p => (channelDigitBlocks E (.inl n) s).C w)
+    (ordinarySymbol_offset E n x j)
+
+theorem ordinaryHigh_offset (n : Column p) (x : Plane)
+    (j : ActivationIndex E (.inl n)) (w : K E (.inl n)) :
+    ordinaryHigh E n (x - ordinaryOffset E n j) w =
+      (channelDigitBlocks E (.inl n) (ordinarySymbol E n x)).T
+        (x.1 - (ordinaryOffset E n j).1) w := by
+  exact congrArg (fun s : Symbol p => (channelDigitBlocks E (.inl n) s).T
+    (x.1 - (ordinaryOffset E n j).1) w) (ordinarySymbol_offset E n x j)
+
+theorem ordinarySymbol_val_of_residue
+    {p : ℕ} [NeZero p] (E : EncodingParameters p)
+    (n : Column p) (x : Plane)
+    (t : ZMod p) (ht : t ≠ 0)
+    (hx₁ : (x.1 : ZMod p) = 0) (hx₂ : (x.2 : ZMod p) = t) :
+    (ordinarySymbol E n x : ZMod p) = t := by
+  have hv : (lineValue n x : ZMod p) = t := by
+    simp [lineValue, hx₁, hx₂]
+  have hnot : ¬ (p : ℤ) ∣ lineValue n x := by
+    intro h
+    have hz : (lineValue n x : ZMod p) = 0 :=
+      (ZMod.intCast_zmod_eq_zero_iff_dvd _ _).mpr h
+    exact ht (hv.symm.trans hz)
+  exact (lastDigit_eq_of_not_dvd p E.p_prime hnot).trans hv
+
+theorem reduce_residue
+    {p : ℕ} [NeZero p]
+    (x : Plane) :
+    SharedSeed.reduce p (SharedSeed.residue p x) =
+      ((x.1 : ZMod p), (x.2 : ZMod p)) := by
+  simp [SharedSeed.reduce, SharedSeed.residue]
+
+theorem ordinarySymbol_firstRegion (n : Column p) (x : Plane)
+    (hx : SharedSeed.residue p x ∈ SharedSeed.firstRegion p) :
+    ordinarySymbol E n x = 1 := by
+  let : Fact (1 < p) := ⟨E.p_prime.one_lt⟩
+  have h := SharedSeed.reduce_of_firstRegion hx
+  rw [reduce_residue (p := p) x] at h
+  apply Units.ext
+  exact ordinarySymbol_val_of_residue E n x 1 one_ne_zero
+    (congrArg Prod.fst h) (congrArg Prod.snd h)
+
+theorem ordinarySymbol_secondRegion (n : Column p) (x : Plane)
+    (hx : SharedSeed.residue p x ∈ SharedSeed.secondRegion p) :
+    ordinarySymbol E n x = lastDigit p E.p_prime 2 := by
+  have h := SharedSeed.reduce_of_secondRegion hx
+  rw [reduce_residue (p := p) x] at h
+  have htwo : (2 : ZMod p) ≠ 0 := by
+    intro hz
+    have hd : p ∣ 2 := (ZMod.natCast_eq_zero_iff 2 p).mp hz
+    have hle := Nat.le_of_dvd (by decide : 0 < 2) hd
+    have hlarge := E.p_large
+    omega
+  apply Units.ext
+  rw [lastDigit_two p E.p_prime (by have h := E.p_large; omega)]
+  exact ordinarySymbol_val_of_residue E n x 2 htwo
+    (congrArg Prod.fst h) (congrArg Prod.snd h)
+
+end PeriodicTilingThree.CommonModel
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/PeriodicPartGrouping.lean b/lean/OAI/Geometry/PeriodicTiling/PeriodicPartGrouping.lean
new file mode 100644
index 0000000000000000000000000000000000000000..95b917c80d2ab52b46999749770decc064e9bd7d
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/PeriodicPartGrouping.lean
@@ -0,0 +1,152 @@
+import OAI.Geometry.PeriodicTiling.FiniteMasks
+import OAI.Geometry.PeriodicTiling.PlaneLattices
+import Mathlib.Data.Fintype.EquivFin
+import Mathlib.Algebra.Order.BigOperators.Group.Finset
+
+namespace OAI
+
+universe uG uι
+
+noncomputable section
+
+namespace PeriodicTilingThree
+
+open scoped BigOperators
+
+theorem tileIndicator_finite_union_eq_sum {G : Type uG} {ι : Type uι} [Fintype ι]
+    (parts : ι → Set G)
+    (hbound : ∀ x, (∑ i, tileIndicator ℝ (parts i) x) ≤ 1)
+    (s : Finset ι) (x : G) :
+    tileIndicator ℝ {x | ∃ i ∈ s, x ∈ parts i} x =
+      ∑ i ∈ s, tileIndicator ℝ (parts i) x := by
+  classical
+  by_cases hx : ∃ i ∈ s, x ∈ parts i
+  · obtain ⟨i, hi, hix⟩ := hx
+    have hlo : (1 : ℝ) ≤ ∑ j ∈ s, tileIndicator ℝ (parts j) x := by
+      simpa only [tileIndicator, ite_eq_left hix] using
+        (Finset.single_le_sum (fun j _ => tileIndicator_nonneg (parts j) x) hi)
+    have hhi : (∑ j ∈ s, tileIndicator ℝ (parts j) x) ≤ 1 := by
+      calc
+        _ ≤ ∑ j, tileIndicator ℝ (parts j) x :=
+          Finset.sum_le_univ_sum_of_nonneg (fun j => tileIndicator_nonneg (parts j) x)
+        _ ≤ 1 := hbound x
+    have hmem : x ∈ {x | ∃ j ∈ s, x ∈ parts j} := ⟨i, hi, hix⟩
+    rw [tileIndicator, ite_eq_left hmem]
+    exact le_antisymm hlo hhi
+  · have hz : (∑ i ∈ s, tileIndicator ℝ (parts i) x) = 0 := by
+      apply Finset.sum_eq_zero
+      intro i hi
+      have hix : x ∉ parts i := fun h => hx ⟨i, hi, h⟩
+      simp only [tileIndicator, ite_eq_right hix]
+    rw [hz]
+    simp only [tileIndicator, Set.mem_ofPred_eq, ite_eq_right hx]
+
+theorem exists_grouped_periodic_parts {ι : Type uι} [Fintype ι]
+    (parts : ι → Set Plane) (v : ι → Plane)
+    (hv : ∀ i, v i ≠ 0) (hp : ∀ i, Period (parts i) (v i))
+    {A : Set Plane}
+    (hsum : ∀ x, (∑ i, tileIndicator ℝ (parts i) x) = tileIndicator ℝ A x) :
+    ∃ (m : ℕ) (w : Fin m → Plane) (grouped : Fin m → Set Plane),
+      (∀ j, w j ≠ 0) ∧
+      (∀ i j, i ≠ j → planeDet (w i) (w j) ≠ 0) ∧
+      (∀ x, (∑ j, tileIndicator ℝ (grouped j) x) = tileIndicator ℝ A x) ∧
+      ∀ j, Period (grouped j) (w j) := by
+  classical
+  let r : Setoid ι :=
+    { r := fun i j => planeDet (v i) (v j) = 0
+      iseqv := ⟨fun i => planeDet_self (v i),
+        fun {i j} hij => by rw [planeDet_swap, hij, neg_zero],
+        fun {i j k} hij hjk => planeDet_zero_trans (hv j) hij hjk⟩ }
+  let Q := Quotient r
+  let : Fintype Q := Fintype.ofFinite Q
+  let cls : ι → Q := Quotient.mk r
+  let fiber (j : Q) : Finset ι := Finset.univ.filter (fun i => cls i = j)
+  have mem_fiber (i : ι) (j : Q) : i ∈ fiber j ↔ cls i = j := by
+    simp [fiber]
+  have out_mem (j : Q) : j.out ∈ fiber j :=
+    (mem_fiber _ _).2 (Quotient.out_eq j)
+  have hex (j : Q) : ∃ w : Plane, w ≠ 0 ∧
+      ∀ i ∈ fiber j, ∃ n : ℤ, w = n • v i := by
+    have himage : ((fiber j).image v).Nonempty :=
+      ⟨v j.out, Finset.mem_image_of_mem v (out_mem j)⟩
+    obtain ⟨w, hw, hm⟩ := exists_common_collinear_multiple ((fiber j).image v) himage
+      (by
+        intro z hz
+        obtain ⟨i, _, rfl⟩ := Finset.mem_image.mp hz
+        exact hv i)
+      (by
+        intro z hz z' hz'
+        obtain ⟨i, hi, rfl⟩ := Finset.mem_image.mp hz
+        obtain ⟨i', hi', rfl⟩ := Finset.mem_image.mp hz'
+        exact Quotient.exact (((mem_fiber i j).1 hi).trans ((mem_fiber i' j).1 hi').symm))
+    exact ⟨w, hw, fun i hi => hm (v i) (Finset.mem_image_of_mem v hi)⟩
+  choose w hw hm using hex
+  let grouped : Q → Set Plane := fun j => {x | ∃ i ∈ fiber j, x ∈ parts i}
+  have hbound (x : Plane) : (∑ i, tileIndicator ℝ (parts i) x) ≤ 1 := by
+    rw [hsum]
+    exact tileIndicator_le_one A x
+  have hind (j : Q) (x : Plane) : tileIndicator ℝ (grouped j) x =
+      ∑ i ∈ fiber j, tileIndicator ℝ (parts i) x :=
+    tileIndicator_finite_union_eq_sum parts hbound (fiber j) x
+  have htotal (x : Plane) : (∑ j, tileIndicator ℝ (grouped j) x) =
+      tileIndicator ℝ A x := by
+    simp_rw [hind]
+    simp only [fiber, Finset.sum_filter]
+    rw [Finset.sum_comm]
+    calc
+      (∑ i, ∑ j : Q, if cls i = j then tileIndicator ℝ (parts i) x else 0) =
+          ∑ i, tileIndicator ℝ (parts i) x := by
+        apply Finset.sum_congr rfl
+        intro i _
+        simpa only [ite_eq_left rfl, ite_true] using
+          (Finset.sum_eq_single (s := Finset.univ)
+            (f := fun j : Q => if cls i = j then tileIndicator ℝ (parts i) x else 0)
+            (cls i) (fun j _ hji => ite_eq_right (Ne.symm hji))
+            (fun h => (h (Finset.mem_univ (cls i))).elim))
+      _ = tileIndicator ℝ A x := hsum x
+  have hper (j : Q) : Period (grouped j) (w j) := by
+    intro x
+    change (∃ i ∈ fiber j, x + w j ∈ parts i) ↔ ∃ i ∈ fiber j, x ∈ parts i
+    constructor
+    · rintro ⟨i, hi, hx⟩
+      obtain ⟨n, hn⟩ := hm j i hi
+      refine ⟨i, hi, ?_⟩
+      rw [hn] at hx
+      exact ((hp i).zsmul n x).mp hx
+    · rintro ⟨i, hi, hx⟩
+      obtain ⟨n, hn⟩ := hm j i hi
+      refine ⟨i, hi, ?_⟩
+      rw [hn]
+      exact ((hp i).zsmul n x).mpr hx
+  have hpair (j k : Q) (hjk : j ≠ k) : planeDet (w j) (w k) ≠ 0 := by
+    intro hdet
+    obtain ⟨n, hn⟩ := hm j j.out (out_mem j)
+    obtain ⟨m, hmk⟩ := hm k k.out (out_mem k)
+    have hn0 : n ≠ 0 := by
+      intro hn0
+      exact hw j (by simpa [hn0] using hn)
+    have hm0 : m ≠ 0 := by
+      intro hm0
+      exact hw k (by simpa [hm0] using hmk)
+    rw [hn, hmk, planeDet_zsmul_left, planeDet_zsmul_right] at hdet
+    have hz : planeDet (v j.out) (v k.out) = 0 :=
+      (mul_eq_zero.mp ((mul_eq_zero.mp hdet).resolve_left hn0)).resolve_left hm0
+    apply hjk
+    calc
+      j = cls j.out := (Quotient.out_eq j).symm
+      _ = cls k.out := Quotient.sound hz
+      _ = k := Quotient.out_eq k
+  let e : Fin (Fintype.card Q) ≃ Q := (Fintype.equivFin Q).symm
+  refine ⟨Fintype.card Q, fun j => w (e j), fun j => grouped (e j),
+    fun j => hw (e j), ?_, ?_, fun j => hper (e j)⟩
+  · intro i j hij
+    exact hpair (e i) (e j) (fun he => hij (e.injective he))
+  · intro x
+    rw [e.sum_comp (fun j => tileIndicator ℝ (grouped j) x)]
+    exact htotal x
+
+end PeriodicTilingThree
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/PeriodicRegion.lean b/lean/OAI/Geometry/PeriodicTiling/PeriodicRegion.lean
new file mode 100644
index 0000000000000000000000000000000000000000..d8acfe8852565d4a27a3e3df54b6d847ae6cbad8
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/PeriodicRegion.lean
@@ -0,0 +1,141 @@
+import OAI.Geometry.PeriodicTiling.TilesRegion
+import OAI.Geometry.PeriodicTiling.FiniteCylinder
+import OAI.Geometry.PeriodicTiling.PlaneLattices
+import Mathlib.Tactic.NormNum
+
+namespace OAI
+
+namespace PeriodicTilingThree
+
+namespace PeriodicRegion
+
+def residue (m : ℕ) (hm : 0 < m) (x : ℤ) : Fin m :=
+  ⟨(x % (m : ℤ)).toNat, by
+    have hmZ : (0 : ℤ) < m := by exact_mod_cast hm
+    have hlo := Int.emod_nonneg x (ne_of_gt hmZ)
+    have hhi := Int.emod_lt_of_pos x hmZ
+    omega⟩
+
+theorem residue_cast (m : ℕ) (hm : 0 < m) (x : ℤ) :
+    ((residue m hm x : ℕ) : ℤ) = x % (m : ℤ) := by
+  have hmZ : (0 : ℤ) < m := by exact_mod_cast hm
+  have hlo := Int.emod_nonneg x (ne_of_gt hmZ)
+  dsimp [residue]
+  omega
+
+theorem residue_add (m : ℕ) (hm : 0 < m) (x : ℤ) :
+    residue m hm (x + m) = residue m hm x := by
+  apply Fin.ext
+  simp [residue]
+
+private theorem coordinate_sub (T : Plane ≃+ Plane) (x y : ℤ) (f : Plane) :
+    T (x, y) - f = T (x - (T.symm f).1, y - (T.symm f).2) := by
+  calc
+    T (x, y) - f = T (x, y) - T (T.symm f) := by rw [T.apply_symm_apply]
+    _ = T ((x, y) - T.symm f) := (map_sub T (x, y) (T.symm f)).symm
+    _ = T (x - (T.symm f).1, y - (T.symm f).2) := rfl
+
+private theorem target_phase {E : Set Plane} (T : Plane ≃+ Plane) (q : ℕ)
+    (hE : Period E (T (0, (q : ℤ)))) (x y y' : ℤ)
+    (hphase : (q : ℤ) ∣ y' - y) :
+    T (x, y') ∈ E ↔ T (x, y) ∈ E := by
+  obtain ⟨c, hc⟩ := hphase
+  have hv := hE.zsmul c
+  have heq : T (x, y) + c • T (0, (q : ℤ)) = T (x, y') := by
+    rw [← map_zsmul, ← map_add]
+    congr 1
+    apply Prod.ext
+    · simp
+    · simp
+      nlinarith
+  simpa only [heq] using hv (T (x, y))
+
+end PeriodicRegion
+
+theorem periodic_region_replacement {F : Finset Plane} {D E : Set Plane}
+    (htile : TilesRegion F D E)
+    (hperiod : ∃ h : Plane, h ≠ 0 ∧ Period D h)
+    (htarget : FullyPeriodic E) :
+    ∃ D' : Set Plane, TilesRegion F D' E ∧ FullyPeriodic D' := by
+  classical
+  obtain ⟨h, hh, hD⟩ := hperiod
+  obtain ⟨m, hm, T, hT⟩ := exists_primitive_coordinates hh
+  have hDcoord : Period D (T ((m : ℤ), 0)) := hT.symm ▸ hD
+  obtain ⟨q, hq, hEcoord⟩ := htarget.exists_coordinate_period T
+  let row : ℤ → (Fin m → Bool) := fun n r => decide (T ((r : ℕ), n) ∈ D)
+  have row_iff (x n : ℤ) :
+      row n (PeriodicRegion.residue m hm x) = true ↔ T (x, n) ∈ D := by
+    change decide (T (((PeriodicRegion.residue m hm x : ℕ) : ℤ), n) ∈ D) = true ↔
+      T (x, n) ∈ D
+    rw [decide_eq_true_eq, PeriodicRegion.residue_cast]
+    exact (hDcoord.mem_coordinate_mod x n).symm
+  let R : ℕ := F.sup (fun f => (T.symm f).2.natAbs)
+  have offset_bounds (f : ↥F) :
+      0 ≤ (R : ℤ) - (T.symm (f : Plane)).2 ∧
+      (R : ℤ) - (T.symm (f : Plane)).2 ≤ (2 * R : ℕ) := by
+    have habs : (T.symm (f : Plane)).2.natAbs ≤ R :=
+      Finset.le_sup (f := fun f : Plane => (T.symm f).2.natAbs) f.property
+    have habsZ : ((T.symm (f : Plane)).2.natAbs : ℤ) ≤ R := by
+      exact_mod_cast habs
+    have hupper : (T.symm (f : Plane)).2 ≤
+        ((T.symm (f : Plane)).2.natAbs : ℤ) := Int.le_natAbs
+    have hlower : -(T.symm (f : Plane)).2 ≤
+        ((T.symm (f : Plane)).2.natAbs : ℤ) := by
+      simpa only [Int.natAbs_neg] using
+        (Int.le_natAbs (a := -(T.symm (f : Plane)).2))
+    have htwo : ((2 * R : ℕ) : ℤ) = 2 * (R : ℤ) := by simp
+    rw [htwo]
+    omega
+  obtain ⟨b, L, hL, hqL, hbperiod, hwindows⟩ :=
+    FiniteCylinder.exists_periodic_windows row q (2 * R) hq
+  let D' : Set Plane := {z | b (T.symm z).2
+    (PeriodicRegion.residue m hm (T.symm z).1) = true}
+  have mem_new (x y : ℤ) :
+      T (x, y) ∈ D' ↔ b y (PeriodicRegion.residue m hm x) = true := by
+    simp only [D', Set.mem_ofPred_eq, T.symm_apply_apply]
+  have htiles : TilesRegion F D' E := by
+    apply htile.of_local_copy
+    intro z
+    obtain ⟨⟨x, y⟩, rfl⟩ := T.surjective z
+    obtain ⟨t, htphase, hwindow⟩ := hwindows (y - R)
+    refine ⟨T (x, t + R), ?_, ?_⟩
+    · apply PeriodicRegion.target_phase T q hEcoord
+      have heq : t + (R : ℤ) - y = t - (y - (R : ℤ)) := by omega
+      simpa only [heq] using htphase
+    · intro f
+      obtain ⟨hk0, hkH⟩ := offset_bounds f
+      have hrow := hwindow ((R : ℤ) - (T.symm (f : Plane)).2) hk0 hkH
+      have hind : y - (R : ℤ) + ((R : ℤ) - (T.symm (f : Plane)).2) =
+          y - (T.symm (f : Plane)).2 := by omega
+      rw [hind] at hrow
+      rw [PeriodicRegion.coordinate_sub T x y f,
+        PeriodicRegion.coordinate_sub T x (t + R) f, mem_new, hrow]
+      have heq : t + (R : ℤ) - (T.symm (f : Plane)).2 =
+          t + ((R : ℤ) - (T.symm (f : Plane)).2) := by omega
+      rw [heq]
+      exact row_iff (x - (T.symm (f : Plane)).1)
+        (t + ((R : ℤ) - (T.symm (f : Plane)).2))
+  have hx : Period D' (T ((m : ℤ), 0)) := by
+    intro z
+    obtain ⟨⟨x, y⟩, rfl⟩ := T.surjective z
+    have heq : T (x, y) + T ((m : ℤ), 0) = T (x + m, y) := by
+      rw [← map_add]
+      change T (x + (m : ℤ), y + 0) = T (x + (m : ℤ), y)
+      rw [add_zero]
+    rw [heq, mem_new, mem_new, PeriodicRegion.residue_add]
+  have hy : Period D' (T (0, L)) := by
+    intro z
+    obtain ⟨⟨x, y⟩, rfl⟩ := T.surjective z
+    have heq : T (x, y) + T (0, L) = T (x, y + L) := by
+      rw [← map_add]
+      change T (x + 0, y + L) = T (x, y + L)
+      rw [add_zero]
+    rw [heq, mem_new, mem_new, hbperiod]
+  have hLn : 0 < L.toNat := by omega
+  have hLcast : (L.toNat : ℤ) = L := by omega
+  refine ⟨D', htiles, fullyPeriodic_of_coordinate_periods T m L.toNat hm hLn hx ?_⟩
+  simpa only [hLcast] using hy
+
+end PeriodicTilingThree
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/PeriodicVolume.lean b/lean/OAI/Geometry/PeriodicTiling/PeriodicVolume.lean
new file mode 100644
index 0000000000000000000000000000000000000000..f0cac4f89d8f11b65eb31cb0ab48a5cf1267904d
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/PeriodicVolume.lean
@@ -0,0 +1,207 @@
+import OAI.Geometry.PeriodicTiling.PeriodicVolumeMass
+import Mathlib.Tactic.Tauto
+
+namespace OAI
+
+noncomputable section
+
+open Set MeasureTheory Filter
+open scoped Pointwise ENNReal
+
+namespace PeriodicTilingThree.PeriodicVolume
+
+variable {d : ℕ}
+
+def typedPiece (Ω0 Ω1 C1 : Set (Space d)) (c : Space d) : Set (Space d) := by
+  classical
+  exact if c ∈ C1 then Ω1 else Ω0
+
+theorem typedPiece_measurable (Ω0 Ω1 C1 : Set (Space d))
+    (h0 : MeasurableSet Ω0) (h1 : MeasurableSet Ω1) (c : Space d) :
+    MeasurableSet (typedPiece Ω0 Ω1 C1 c) := by
+  classical
+  unfold typedPiece
+  split_ifs <;> assumption
+
+theorem typedPiece_periodic (b : Module.Basis (Fin d) ℝ (Space d))
+    (Ω0 Ω1 C1 : Set (Space d))
+    (ht : ∀ v ∈ latticeOfBasis b, Period C1 v)
+    (g : latticeOfBasis b) (x : Space d) :
+    typedPiece Ω0 Ω1 C1 (x + (g : Space d)) = typedPiece Ω0 Ω1 C1 x := by
+  classical
+  simp only [typedPiece, ht g g.property x]
+
+theorem typedPiece_volume (Ω0 Ω1 C1 : Set (Space d))
+    (hv : volume Ω1 = volume Ω0) (c : Space d) :
+    volume (typedPiece Ω0 Ω1 C1 c) = volume Ω0 := by
+  classical
+  unfold typedPiece
+  split_ifs
+  · exact hv
+  · rfl
+
+theorem typed_aedisjoint (Ω0 Ω1 C C1 : Set (Space d))
+    (htile : TypedAETiles Ω0 Ω1 C C1) :
+    Pairwise fun c c' : C => AEDisjoint volume
+      (translate c (typedPiece Ω0 Ω1 C1 c))
+      (translate c' (typedPiece Ω0 Ω1 C1 c')) := by
+  classical
+  intro c c' hne
+  have ha : ∀ᵐ x ∂volume,
+      x ∉ translate c (typedPiece Ω0 Ω1 C1 c) ∩
+        translate c' (typedPiece Ω0 Ω1 C1 c') := by
+    filter_upwards [htile] with x hx
+    intro hmem
+    exact hne (hx.unique hmem.1 hmem.2)
+  simpa only [AEDisjoint, not_not, Set.ofPred_mem_eq] using (ae_iff.mp ha)
+
+theorem typed_assembly_ae_eq_univ (Ω0 Ω1 C C1 : Set (Space d))
+    (htile : TypedAETiles Ω0 Ω1 C C1) :
+    assembly C (typedPiece Ω0 Ω1 C1) =ᵐ[volume] Set.univ := by
+  classical
+  filter_upwards [htile] with x hx
+  apply propext
+  exact ⟨fun _ => mem_univ x, fun _ => mem_assembly.mpr hx.exists⟩
+
+theorem typed_cell_mass (b : Module.Basis (Fin d) ℝ (Space d))
+    (Ω0 Ω1 C C1 : Set (Space d))
+    (hC : C.Countable)
+    (hfin : ∀ B : Set (Space d), Bornology.IsBounded B → (C ∩ B).Finite)
+    (h0 : MeasurableSet Ω0) (h1 : MeasurableSet Ω1)
+    (hv : volume Ω1 = volume Ω0)
+    (hp : ∀ v ∈ latticeOfBasis b, Period C v)
+    (ht : ∀ v ∈ latticeOfBasis b, Period C1 v)
+    (htile : TypedAETiles Ω0 Ω1 C C1) :
+    volume (ZSpan.fundamentalDomain b) =
+      Nat.card (representatives b C) • volume Ω0 := by
+  classical
+  let := (representatives_finite b C hfin).fintype
+  have hu := ae_eq_set_inter (typed_assembly_ae_eq_univ Ω0 Ω1 C C1 htile)
+    (Filter.EventuallyEq.rfl : ZSpan.fundamentalDomain b =ᵐ[volume]
+      ZSpan.fundamentalDomain b)
+  calc
+    volume (ZSpan.fundamentalDomain b) =
+        volume (assembly C (typedPiece Ω0 Ω1 C1) ∩ ZSpan.fundamentalDomain b) := by
+      simpa only [univ_inter] using (measure_congr hu).symm
+    _ = ∑' r : representatives b C, volume (typedPiece Ω0 Ω1 C1 r) :=
+      packed_assembly_mass b C hC hp (typedPiece Ω0 Ω1 C1)
+        (typedPiece_periodic b Ω0 Ω1 C1 ht)
+        (fun c _ => typedPiece_measurable Ω0 Ω1 C1 h0 h1 c)
+        (typed_aedisjoint Ω0 Ω1 C C1 htile)
+    _ = Nat.card (representatives b C) • volume Ω0 := by
+      simp only [typedPiece_volume Ω0 Ω1 C1 hv, tsum_fintype,
+        Finset.sum_const, Finset.card_univ, Nat.card_eq_fintype_card]
+
+theorem assembly_period (b : Module.Basis (Fin d) ℝ (Space d))
+    (C : Set (Space d)) (hp : ∀ v ∈ latticeOfBasis b, Period C v)
+    (K : Space d → Set (Space d))
+    (hKp : ∀ (g : latticeOfBasis b) x, K (x + (g : Space d)) = K x)
+    (g : latticeOfBasis b) : Period (assembly C K) (g : Space d) := by
+  intro x
+  rw [mem_assembly, mem_assembly]
+  constructor
+  · rintro ⟨c, hc⟩
+    have hcg : (c : Space d) - (g : Space d) ∈ C := by
+      simpa only [sub_eq_add_neg] using
+        (hp _ ((latticeOfBasis b).neg_mem g.property) c).2 c.property
+    refine ⟨⟨(c : Space d) - (g : Space d), hcg⟩, ?_⟩
+    have hKm : K ((c : Space d) - (g : Space d)) = K c := by
+      simpa only [NegMemClass.coe_neg, sub_eq_add_neg] using hKp (-g) c
+    change x - ((c : Space d) - (g : Space d)) ∈ K ((c : Space d) - (g : Space d))
+    rw [hKm]
+    have he : x - ((c : Space d) - (g : Space d)) =
+        (x + (g : Space d)) - (c : Space d) := by abel
+    rw [he]
+    exact hc
+  · rintro ⟨c, hc⟩
+    have hcg : (c : Space d) + (g : Space d) ∈ C :=
+      (hp g g.property c).2 c.property
+    refine ⟨⟨(c : Space d) + (g : Space d), hcg⟩, ?_⟩
+    change (x + (g : Space d)) - ((c : Space d) + (g : Space d)) ∈
+      K ((c : Space d) + (g : Space d))
+    rw [hKp g c, add_sub_add_right_eq_sub]
+    exact hc
+
+theorem lattice_vadd_assembly (b : Module.Basis (Fin d) ℝ (Space d))
+    (C : Set (Space d)) (hp : ∀ v ∈ latticeOfBasis b, Period C v)
+    (K : Space d → Set (Space d))
+    (hKp : ∀ (g : latticeOfBasis b) x, K (x + (g : Space d)) = K x)
+    (g : latticeOfBasis b) : g +ᵥ assembly C K = assembly C K := by
+  ext x
+  rw [mem_vadd_set_iff_neg_vadd_mem]
+  change -(g : Space d) + x ∈ assembly C K ↔ x ∈ assembly C K
+  simpa only [NegMemClass.coe_neg, add_comm] using
+    assembly_period b C hp K hKp (-g) x
+
+theorem ae_covers_of_cell_mass (b : Module.Basis (Fin d) ℝ (Space d))
+    (C : Set (Space d)) (hC : C.Countable)
+    (hp : ∀ v ∈ latticeOfBasis b, Period C v)
+    (K : Space d → Set (Space d))
+    (hKp : ∀ (g : latticeOfBasis b) x, K (x + (g : Space d)) = K x)
+    (hK : ∀ c ∈ C, MeasurableSet (K c))
+    (hmass : volume (assembly C K ∩ ZSpan.fundamentalDomain b) =
+      volume (ZSpan.fundamentalDomain b)) :
+    ∀ᵐ x ∂volume, ∃ c : C, x - (c : Space d) ∈ K c := by
+  let : Countable (Submodule.span ℤ (Set.range b)).toAddSubgroup :=
+    inferInstanceAs (Countable (Submodule.span ℤ (Set.range b)))
+  have hU := measurableSet_assembly C hC K hK
+  have hP := ZSpan.fundamentalDomain_measurableSet b
+  have hPfin : volume (ZSpan.fundamentalDomain b) ≠ ⊤ :=
+    (ZSpan.fundamentalDomain_isBounded b).measure_lt_top.ne
+  have hae : (assembly C K ∩ ZSpan.fundamentalDomain b : Set (Space d)) =ᵐ[volume]
+      ZSpan.fundamentalDomain b :=
+    ae_eq_of_subset_of_measure_ge inter_subset_right hmass.symm.le
+      (hU.inter hP).nullMeasurableSet hPfin
+  have hnull := (ae_eq_set.mp hae).2
+  have hdiff : ZSpan.fundamentalDomain b \
+      (assembly C K ∩ ZSpan.fundamentalDomain b) =
+      (assembly C K)ᶜ ∩ ZSpan.fundamentalDomain b := by
+    ext x
+    simp only [Set.mem_sdiff, mem_inter_iff, mem_compl_iff]
+    tauto
+  rw [hdiff] at hnull
+  have hglobal : volume (assembly C K)ᶜ = 0 :=
+    (ZSpan.isAddFundamentalDomain' b volume).measure_zero_of_invariant
+      ((assembly C K)ᶜ)
+      (fun g => by
+        ext x
+        rw [mem_vadd_set_iff_neg_vadd_mem]
+        change (-(g : Space d) + x ∉ assembly C K) ↔ x ∉ assembly C K
+        simpa only [NegMemClass.coe_neg, add_comm] using
+          not_congr (assembly_period b C hp K hKp (-g) x))
+      hnull
+  have hcover : ∀ᵐ x ∂volume, x ∈ assembly C K := ae_iff.mpr hglobal
+  exact hcover.mono fun _ hx => mem_assembly.mp hx
+
+theorem replacement_ae_covers (b : Module.Basis (Fin d) ℝ (Space d))
+    (Ω0 Ω1 J C C1 : Set (Space d))
+    (hC : C.Countable)
+    (hfin : ∀ B : Set (Space d), Bornology.IsBounded B → (C ∩ B).Finite)
+    (h0 : MeasurableSet Ω0) (h1 : MeasurableSet Ω1) (hJ : MeasurableSet J)
+    (hv : volume Ω1 = volume Ω0) (hJv : volume J = volume Ω0)
+    (hp : ∀ v ∈ latticeOfBasis b, Period C v)
+    (ht : ∀ v ∈ latticeOfBasis b, Period C1 v)
+    (htile : TypedAETiles Ω0 Ω1 C C1)
+    (hpack : Pairwise fun c c' : C =>
+      AEDisjoint volume (translate c J) (translate c' J)) :
+    ∀ᵐ x ∂volume, ∃ c : C, x - (c : Space d) ∈ J := by
+  classical
+  let := (representatives_finite b C hfin).fintype
+  apply ae_covers_of_cell_mass b C hC hp (fun _ => J) (fun _ _ => rfl)
+    (fun _ _ => hJ)
+  calc
+    volume (assembly C (fun _ => J) ∩ ZSpan.fundamentalDomain b) =
+        ∑' _r : representatives b C, volume J :=
+      packed_assembly_mass b C hC hp (fun _ => J) (fun _ _ => rfl)
+        (fun _ _ => hJ) hpack
+    _ = Nat.card (representatives b C) • volume Ω0 := by
+      simp only [hJv, tsum_fintype, Finset.sum_const, Finset.card_univ,
+        Nat.card_eq_fintype_card]
+    _ = volume (ZSpan.fundamentalDomain b) :=
+      (typed_cell_mass b Ω0 Ω1 C C1 hC hfin h0 h1 hv hp ht htile).symm
+
+end PeriodicTilingThree.PeriodicVolume
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/PeriodicVolumeMass.lean b/lean/OAI/Geometry/PeriodicTiling/PeriodicVolumeMass.lean
new file mode 100644
index 0000000000000000000000000000000000000000..68ccf38aa53b53ab346e3fb6b860941dc90e0ac4
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/PeriodicVolumeMass.lean
@@ -0,0 +1,134 @@
+import OAI.Geometry.PeriodicTiling.PeriodicVolumeOrbits
+import Mathlib.MeasureTheory.Measure.Typeclasses.Finite
+import Mathlib.Tactic.Abel
+
+namespace OAI
+
+noncomputable section
+
+open Set MeasureTheory
+open scoped Pointwise ENNReal
+
+namespace PeriodicTilingThree.PeriodicVolume
+
+variable {d : ℕ}
+
+def translate (c : Space d) (K : Set (Space d)) : Set (Space d) :=
+  {x | x - c ∈ K}
+
+@[simp] theorem mem_translate {c x : Space d} {K : Set (Space d)} :
+    x ∈ translate c K ↔ x - c ∈ K := Iff.rfl
+
+theorem translate_eq_vadd (c : Space d) (K : Set (Space d)) :
+    translate c K = c +ᵥ K := by
+  ext x
+  simp only [mem_translate, mem_vadd_set_iff_neg_vadd_mem, vadd_eq_add]
+  rw [sub_eq_add_neg, add_comm x (-c)]
+
+@[simp] theorem measure_translate (c : Space d) (K : Set (Space d)) :
+    volume (translate c K) = volume K := by
+  rw [translate_eq_vadd, measure_vadd]
+
+theorem measurableSet_translate (c : Space d) {K : Set (Space d)}
+    (hK : MeasurableSet K) : MeasurableSet (translate c K) :=
+  hK.preimage (measurable_id.sub measurable_const)
+
+theorem lattice_vadd_translate (b : Module.Basis (Fin d) ℝ (Space d))
+    (g : latticeOfBasis b) (r : Space d) (K : Set (Space d)) :
+    g +ᵥ translate r K = translate (r + (g : Space d)) K := by
+  ext x
+  rw [mem_vadd_set_iff_neg_vadd_mem]
+  change (-(g : Space d) + x) - r ∈ K ↔ x - (r + (g : Space d)) ∈ K
+  have he : (-(g : Space d) + x) - r = x - (r + (g : Space d)) := by abel
+  rw [he]
+
+def assembly (C : Set (Space d)) (K : Space d → Set (Space d)) : Set (Space d) :=
+  ⋃ c : C, translate c (K c)
+
+@[simp] theorem mem_assembly {C : Set (Space d)} {K : Space d → Set (Space d)}
+    {x : Space d} : x ∈ assembly C K ↔ ∃ c : C, x - (c : Space d) ∈ K c := by
+  simp only [assembly, mem_iUnion, mem_translate]
+
+theorem measurableSet_assembly (C : Set (Space d)) (hC : C.Countable)
+    (K : Space d → Set (Space d)) (hK : ∀ c ∈ C, MeasurableSet (K c)) :
+    MeasurableSet (assembly C K) := by
+  let : Countable C := hC.to_subtype
+  exact MeasurableSet.iUnion fun c : C => measurableSet_translate c (hK c c.property)
+
+theorem periodic_piece_mass (b : Module.Basis (Fin d) ℝ (Space d))
+    (C : Set (Space d)) (hp : ∀ v ∈ latticeOfBasis b, Period C v)
+    (K : Space d → Set (Space d))
+    (hKp : ∀ (g : latticeOfBasis b) x, K (x + (g : Space d)) = K x) :
+    (∑' c : C, volume (translate c (K c) ∩ ZSpan.fundamentalDomain b)) =
+      ∑' r : representatives b C, volume (K r) := by
+  let : Countable (Submodule.span ℤ (Set.range b)).toAddSubgroup :=
+    inferInstanceAs (Countable (Submodule.span ℤ (Set.range b)))
+  have hP := ZSpan.isAddFundamentalDomain' b (volume : Measure (Space d))
+  calc
+    (∑' c : C, volume (translate c (K c) ∩ ZSpan.fundamentalDomain b)) =
+        ∑' p : representatives b C × latticeOfBasis b,
+          volume (translate (orbitEquiv b C hp p) (K (orbitEquiv b C hp p)) ∩
+            ZSpan.fundamentalDomain b) :=
+      ((orbitEquiv b C hp).tsum_eq _).symm
+    _ = ∑' r : representatives b C, ∑' g : latticeOfBasis b,
+          volume (translate ((r : Space d) + (g : Space d))
+            (K ((r : Space d) + (g : Space d))) ∩ ZSpan.fundamentalDomain b) :=
+      ENNReal.tsum_prod'
+    _ = ∑' r : representatives b C, volume (K r) := by
+      apply tsum_congr
+      intro r
+      calc
+        (∑' g : latticeOfBasis b,
+            volume (translate ((r : Space d) + (g : Space d))
+              (K ((r : Space d) + (g : Space d))) ∩ ZSpan.fundamentalDomain b)) =
+            ∑' g : latticeOfBasis b,
+              volume ((g +ᵥ translate r (K r)) ∩ ZSpan.fundamentalDomain b) := by
+          apply tsum_congr
+          intro g
+          rw [hKp g r, lattice_vadd_translate]
+        _ = volume (translate r (K r)) := (hP.measure_eq_tsum _).symm
+        _ = volume (K r) := measure_translate _ _
+
+theorem periodic_piece_mass_eq_card (b : Module.Basis (Fin d) ℝ (Space d))
+    (C : Set (Space d)) (hp : ∀ v ∈ latticeOfBasis b, Period C v)
+    (hfin : ∀ B : Set (Space d), Bornology.IsBounded B → (C ∩ B).Finite)
+    (K : Space d → Set (Space d))
+    (hKp : ∀ (g : latticeOfBasis b) x, K (x + (g : Space d)) = K x)
+    (v : ℝ≥0∞) (hv : ∀ c ∈ C, volume (K c) = v) :
+    (∑' c : C, volume (translate c (K c) ∩ ZSpan.fundamentalDomain b)) =
+      Nat.card (representatives b C) • v := by
+  let := (representatives_finite b C hfin).fintype
+  rw [periodic_piece_mass b C hp K hKp]
+  have hvr : ∀ r : representatives b C, volume (K r) = v := fun r => hv r r.property.1
+  simp only [hvr, tsum_fintype, Finset.sum_const,
+    Finset.card_univ, Nat.card_eq_fintype_card]
+
+theorem packed_assembly_mass (b : Module.Basis (Fin d) ℝ (Space d))
+    (C : Set (Space d)) (hC : C.Countable)
+    (hp : ∀ v ∈ latticeOfBasis b, Period C v)
+    (K : Space d → Set (Space d))
+    (hKp : ∀ (g : latticeOfBasis b) x, K (x + (g : Space d)) = K x)
+    (hK : ∀ c ∈ C, MeasurableSet (K c))
+    (hpack : Pairwise fun c c' : C =>
+      AEDisjoint volume (translate c (K c)) (translate c' (K c'))) :
+    volume (assembly C K ∩ ZSpan.fundamentalDomain b) =
+      ∑' r : representatives b C, volume (K r) := by
+  let : Countable C := hC.to_subtype
+  calc
+    volume (assembly C K ∩ ZSpan.fundamentalDomain b) =
+        ∑' c : C, volume (translate c (K c) ∩ ZSpan.fundamentalDomain b) := by
+      rw [assembly, iUnion_inter]
+      apply measure_iUnion₀
+      · intro c c' hne
+        exact (hpack hne).mono inter_subset_left inter_subset_left
+      · intro c
+        exact ((measurableSet_translate c (hK c c.property)).inter
+          (ZSpan.fundamentalDomain_measurableSet b)).nullMeasurableSet
+    _ = ∑' r : representatives b C, volume (K r) :=
+      periodic_piece_mass b C hp K hKp
+
+end PeriodicTilingThree.PeriodicVolume
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/PeriodicVolumeOrbits.lean b/lean/OAI/Geometry/PeriodicTiling/PeriodicVolumeOrbits.lean
new file mode 100644
index 0000000000000000000000000000000000000000..9ec7763e35e2f496a21dd4c6572cee59d4ec26ca
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/PeriodicVolumeOrbits.lean
@@ -0,0 +1,87 @@
+import OAI.Geometry.PeriodicTiling.EuclideanBasic
+import Mathlib.Algebra.Module.ZLattice.Basic
+
+namespace OAI
+
+noncomputable section
+
+open Set MeasureTheory
+open scoped Pointwise ENNReal
+
+namespace PeriodicTilingThree.PeriodicVolume
+
+variable {d : ℕ}
+
+def representatives (b : Module.Basis (Fin d) ℝ (Space d)) (C : Set (Space d)) : Set (Space d) :=
+  C ∩ ZSpan.fundamentalDomain b
+
+theorem representatives_finite (b : Module.Basis (Fin d) ℝ (Space d)) (C : Set (Space d))
+    (hfin : ∀ K : Set (Space d), Bornology.IsBounded K → (C ∩ K).Finite) :
+    (representatives b C).Finite :=
+  hfin _ (ZSpan.fundamentalDomain_isBounded b)
+
+def orbitMap (b : Module.Basis (Fin d) ℝ (Space d)) (C : Set (Space d))
+    (hp : ∀ v ∈ latticeOfBasis b, Period C v)
+    (p : representatives b C × latticeOfBasis b) : C :=
+  ⟨(p.1 : Space d) + (p.2 : Space d),
+    (hp p.2 p.2.property p.1).2 p.1.property.1⟩
+
+theorem orbitMap_injective (b : Module.Basis (Fin d) ℝ (Space d)) (C : Set (Space d))
+    (hp : ∀ v ∈ latticeOfBasis b, Period C v) :
+    Function.Injective (orbitMap b C hp) := by
+  rintro ⟨r, g⟩ ⟨s, h⟩ he
+  have he' : (r : Space d) + (g : Space d) = s + (h : Space d) :=
+    congrArg Subtype.val he
+  have hr : ZSpan.fract b (r : Space d) = r :=
+    ZSpan.fract_eq_self.mpr r.property.2
+  have hs : ZSpan.fract b (s : Space d) = s :=
+    ZSpan.fract_eq_self.mpr s.property.2
+  have hfract := congrArg (ZSpan.fract b) he'
+  rw [ZSpan.fract_add_ZSpan b _ g.property,
+    ZSpan.fract_add_ZSpan b _ h.property, hr, hs] at hfract
+  have hrs : r = s := Subtype.ext hfract
+  subst s
+  have hgh : g = h := Subtype.ext (add_left_cancel he')
+  subst h
+  rfl
+
+theorem orbitMap_surjective (b : Module.Basis (Fin d) ℝ (Space d)) (C : Set (Space d))
+    (hp : ∀ v ∈ latticeOfBasis b, Period C v) :
+    Function.Surjective (orbitMap b C hp) := by
+  intro c
+  let g : latticeOfBasis b := ZSpan.floor b (c : Space d)
+  have hneg : -(g : Space d) ∈ latticeOfBasis b :=
+    (latticeOfBasis b).neg_mem g.property
+  have hc : ZSpan.fract b (c : Space d) ∈ C := by
+    rw [ZSpan.fract_apply, sub_eq_add_neg]
+    exact (hp _ hneg c).2 c.property
+  let r : representatives b C :=
+    ⟨ZSpan.fract b c, hc, ZSpan.fract_mem_fundamentalDomain b c⟩
+  refine ⟨(r, g), Subtype.ext ?_⟩
+  change ZSpan.fract b (c : Space d) + (ZSpan.floor b (c : Space d) : Space d) = c
+  rw [ZSpan.fract_apply, sub_add_cancel]
+
+def orbitEquiv (b : Module.Basis (Fin d) ℝ (Space d)) (C : Set (Space d))
+    (hp : ∀ v ∈ latticeOfBasis b, Period C v) :
+    representatives b C × latticeOfBasis b ≃ C :=
+  Equiv.ofBijective (orbitMap b C hp)
+    ⟨orbitMap_injective b C hp, orbitMap_surjective b C hp⟩
+
+@[simp] theorem orbitEquiv_apply_coe (b : Module.Basis (Fin d) ℝ (Space d))
+    (C : Set (Space d)) (hp : ∀ v ∈ latticeOfBasis b, Period C v)
+    (r : representatives b C) (g : latticeOfBasis b) :
+    ((orbitEquiv b C hp (r, g) : C) : Space d) = (r : Space d) + (g : Space d) :=
+  rfl
+
+theorem type_constant_on_orbit (b : Module.Basis (Fin d) ℝ (Space d))
+    (C C1 : Set (Space d)) (hp : ∀ v ∈ latticeOfBasis b, Period C v)
+    (ht : ∀ v ∈ latticeOfBasis b, Period C1 v)
+    (r : representatives b C) (g : latticeOfBasis b) :
+    ((orbitEquiv b C hp (r, g) : C) : Space d) ∈ C1 ↔ (r : Space d) ∈ C1 :=
+  ht g g.property r
+
+end PeriodicTilingThree.PeriodicVolume
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/PlanarPeriodicity.lean b/lean/OAI/Geometry/PeriodicTiling/PlanarPeriodicity.lean
new file mode 100644
index 0000000000000000000000000000000000000000..515fb966a0f2d51f54c58c1ad403aa5088f372c7
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/PlanarPeriodicity.lean
@@ -0,0 +1,117 @@
+import OAI.Geometry.PeriodicTiling.WeakPeriodicity
+import OAI.Geometry.PeriodicTiling.PeriodicPartGrouping
+import OAI.Geometry.PeriodicTiling.ComponentTargets
+import OAI.Geometry.PeriodicTiling.PeriodicRegion
+
+namespace OAI
+
+universe uG uι
+
+noncomputable section
+
+namespace PeriodicTilingThree
+
+open scoped BigOperators Classical
+
+theorem Tiles.image_add {G : Type uG} [AddCommGroup G]
+    {F : Finset G} {A : Set G} (h : Tiles F A) (u : G) :
+    Tiles (F.image (fun f => f + u)) A := by
+  classical
+  apply tiles_iff_unique_tile.mpr
+  intro x
+  obtain ⟨f, hf, huniq⟩ := tiles_iff_unique_tile.mp h (x - u)
+  let f' : ↥(F.image (fun f => f + u)) :=
+    ⟨(f : G) + u, Finset.mem_image_of_mem _ f.property⟩
+  have hsub (g : G) : x - (g + u) = (x - u) - g := by abel
+  refine ⟨f', ?_, ?_⟩
+  · change x - ((f : G) + u) ∈ A
+    rwa [hsub]
+  · intro g hg
+    obtain ⟨z, hz, hzg⟩ := Finset.mem_image.mp g.property
+    have hzA : (x - u) - z ∈ A := by
+      rw [← hsub, hzg]
+      exact hg
+    have hzf : (⟨z, hz⟩ : ↥F) = f := huniq ⟨z, hz⟩ hzA
+    apply Subtype.ext
+    change (g : G) = (f : G) + u
+    have hzf' : z = (f : G) := congrArg Subtype.val hzf
+    rw [← hzg, hzf']
+
+theorem fullyPeriodic_finite_union {G : Type uG} {ι : Type uι} [AddCommGroup G] [Finite ι]
+    (parts : ι → Set G) (hp : ∀ i, FullyPeriodic (parts i)) :
+    FullyPeriodic {x | ∃ i, x ∈ parts i} := by
+  classical
+  choose P hindex hperiod using hp
+  let common : AddSubgroup G := ⨅ i, P i
+  refine ⟨common, AddSubgroup.finiteIndex_iInf hindex, ?_⟩
+  intro v hv x
+  have hv' (i : ι) : v ∈ P i := (AddSubgroup.mem_iInf.mp hv) i
+  change (∃ i, x + v ∈ parts i) ↔ ∃ i, x ∈ parts i
+  constructor
+  · rintro ⟨i, hi⟩
+    exact ⟨i, (hperiod i v (hv' i) x).mp hi⟩
+  · rintro ⟨i, hi⟩
+    exact ⟨i, (hperiod i v (hv' i) x).mpr hi⟩
+
+theorem tiles_union_of_region_partition {G : Type uG} {ι : Type uι} [AddCommGroup G]
+    (F : Finset G) (parts targets : ι → Set G)
+    (htile : ∀ i, TilesRegion F (parts i) (targets i))
+    (hcover : ∀ x, ∃! i, x ∈ targets i) :
+    Tiles F {x | ∃ i, x ∈ parts i} := by
+  apply tiles_iff_unique_tile.mpr
+  intro x
+  obtain ⟨i, hi, huniq⟩ := hcover x
+  obtain ⟨f, hf, hfuniq⟩ := (htile i).unique x hi
+  refine ⟨f, ⟨i, hf⟩, ?_⟩
+  rintro g ⟨j, hj⟩
+  have hxj : x ∈ targets j := by
+    simpa [add_comm] using (htile j).add_mem g g.property (x - g) hj
+  have hji : j = i := huniq j hxj
+  subst j
+  exact hfuniq g hj
+
+theorem Tiles.fullyPeriodic_of_finite_periodic_partition
+    {ι : Type uι} [Fintype ι] {F : Finset Plane} {A : Set Plane}
+    (h : Tiles F A) (parts : ι → Set Plane) (v : ι → Plane)
+    (hpartition : ∀ x, (∑ i, tileIndicator ℝ (parts i) x) = tileIndicator ℝ A x)
+    (hv : ∀ i, v i ≠ 0) (hperiod : ∀ i, Period (parts i) (v i)) :
+    ∃ B : Set Plane, Tiles F B ∧ FullyPeriodic B := by
+  classical
+  obtain ⟨m, w, grouped, hw, htrans, hsum, hp⟩ :=
+    exists_grouped_periodic_parts parts v hv hperiod hpartition
+  obtain ⟨htiles, hcover, htargets⟩ := component_targets grouped w h hsum hw hp htrans
+  have hex (j : Fin m) : ∃ B : Set Plane,
+      TilesRegion F B (componentTarget F (grouped j)) ∧ FullyPeriodic B :=
+    periodic_region_replacement (htiles j) ⟨w j, hw j, hp j⟩ (htargets j)
+  choose replacements hregion hfull using hex
+  refine ⟨{x | ∃ j, x ∈ replacements j}, ?_, ?_⟩
+  · exact tiles_union_of_region_partition F replacements
+      (fun j => componentTarget F (grouped j)) hregion hcover
+  · exact fullyPeriodic_finite_union replacements hfull
+
+theorem plane_tile_has_fullyPeriodic_complement
+    (F : Finset Plane) (hF : F.Nonempty)
+    (h : ∃ A : Set Plane, Tiles F A) :
+    ∃ B : Set Plane, Tiles F B ∧ FullyPeriodic B := by
+  classical
+  obtain ⟨A, hA⟩ := h
+  obtain ⟨f, hf⟩ := hF
+  let : DecidableEq Plane := fun a b => Classical.propDecidable (a = b)
+  let F₀ := F.image (fun x => x + -f)
+  have hzero : (0 : Plane) ∈ F₀ := by
+    apply Finset.mem_image.mpr
+    exact ⟨f, hf, add_neg_cancel f⟩
+  have hA₀ : Tiles F₀ A := hA.image_add (-f)
+  obtain ⟨m, parts, v, hpartition, hv, hp⟩ := hA₀.exists_weak_periodic_partition hzero
+  obtain ⟨B, hB, hfull⟩ :=
+    hA₀.fullyPeriodic_of_finite_periodic_partition parts v hpartition hv hp
+  refine ⟨B, ?_, hfull⟩
+  have hback := hB.image_add f
+  simpa only [F₀, Finset.image_image, Function.comp_def, add_assoc,
+    neg_add_cancel, add_zero, Finset.image_id'] using hback
+
+end PeriodicTilingThree
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/PlaneLattices.lean b/lean/OAI/Geometry/PeriodicTiling/PlaneLattices.lean
new file mode 100644
index 0000000000000000000000000000000000000000..60bf6ef36906bb4ff7ce1d2395de099f054bf0d6
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/PlaneLattices.lean
@@ -0,0 +1,361 @@
+import OAI.Geometry.PeriodicTiling.TilingBasic
+import Mathlib.Data.Int.GCD
+import Mathlib.Data.ZMod.QuotientGroup
+import Mathlib.Data.Set.Finite.Basic
+import Mathlib.Tactic.Ring
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum
+import Mathlib.Tactic.Abel
+
+namespace OAI
+
+universe uG uι
+
+noncomputable section
+
+namespace PeriodicTilingThree
+
+open scoped BigOperators
+
+section Periods
+
+variable {G : Type uG} [AddCommGroup G] {A : Set G} {v : G}
+
+theorem Period.zsmul (hv : Period A v) (n : ℤ) : Period A (n • v) :=
+  (periodSubgroup A).zsmul_mem hv n
+
+theorem Period.nsmul (hv : Period A v) (n : ℕ) : Period A (n • v) :=
+  (periodSubgroup A).nsmul_mem hv n
+
+theorem exists_pos_zsmul_mem (P : AddSubgroup G) (hP : P.FiniteIndex) (v : G) :
+    ∃ q : ℕ, 0 < q ∧ (q : ℤ) • v ∈ P := by
+  obtain ⟨q, hq, _, hmem⟩ :=
+    AddSubgroup.exists_nsmul_mem_of_index_ne_zero (H := P) hP.index_ne_zero v
+  exact ⟨q, hq, by simpa only [natCast_zsmul] using hmem⟩
+
+end Periods
+
+def bezoutCoordinates (a b s t : ℤ) (h : s * a + t * b = 1) : Plane ≃+ Plane where
+  toFun z := (a * z.1 - t * z.2, b * z.1 + s * z.2)
+  invFun z := (s * z.1 + t * z.2, -b * z.1 + a * z.2)
+  left_inv := by
+    rintro ⟨x, y⟩
+    apply Prod.ext
+    · change s * (a * x - t * y) + t * (b * x + s * y) = x
+      calc
+        _ = (s * a + t * b) * x := by ring
+        _ = x := by rw [h, one_mul]
+    · change -b * (a * x - t * y) + a * (b * x + s * y) = y
+      calc
+        _ = (s * a + t * b) * y := by ring
+        _ = y := by rw [h, one_mul]
+  right_inv := by
+    rintro ⟨x, y⟩
+    apply Prod.ext
+    · change a * (s * x + t * y) - t * (-b * x + a * y) = x
+      calc
+        _ = (s * a + t * b) * x := by ring
+        _ = x := by rw [h, one_mul]
+    · change b * (s * x + t * y) + s * (-b * x + a * y) = y
+      calc
+        _ = (s * a + t * b) * y := by ring
+        _ = y := by rw [h, one_mul]
+  map_add' := by
+    rintro ⟨x, y⟩ ⟨z, w⟩
+    apply Prod.ext <;> dsimp <;> ring
+
+theorem exists_primitive_coordinates {h : Plane} (hh : h ≠ 0) :
+    ∃ m : ℕ, 0 < m ∧ ∃ T : Plane ≃+ Plane, T ((m : ℤ), 0) = h := by
+  have hcoords : h.1 ≠ 0 ∨ h.2 ≠ 0 := by
+    by_cases hx : h.1 = 0
+    · right
+      intro hy
+      exact hh (Prod.ext hx hy)
+    · exact Or.inl hx
+  have hg : 0 < Int.gcd h.1 h.2 := Int.gcd_pos_iff.mpr hcoords
+  obtain ⟨a, b, hab, ha, hb⟩ := Int.exists_gcd_one hg
+  have hbez : Int.gcdA a b * a + Int.gcdB a b * b = 1 := by
+    have he := Int.gcd_eq_gcd_ab a b
+    rw [hab] at he
+    simpa only [Int.natCast_one, mul_comm] using he.symm
+  refine ⟨Int.gcd h.1 h.2, hg, bezoutCoordinates a b _ _ hbez, ?_⟩
+  change (a * (Int.gcd h.1 h.2 : ℤ) - Int.gcdB a b * 0,
+    b * (Int.gcd h.1 h.2 : ℤ) + Int.gcdA a b * 0) = h
+  apply Prod.ext
+  · simpa using ha.symm
+  · simpa using hb.symm
+
+def coordinateSubgroup (T : Plane ≃+ Plane) (m L : ℤ) : AddSubgroup Plane :=
+  ((AddSubgroup.zmultiples m).prod (AddSubgroup.zmultiples L)).map T.toAddMonoidHom
+
+theorem index_coordinateSubgroup (T : Plane ≃+ Plane) (m L : ℤ) :
+    (coordinateSubgroup T m L).index = m.natAbs * L.natAbs := by
+  rw [coordinateSubgroup, AddSubgroup.index_map_of_bijective T.bijective,
+    AddSubgroup.index_prod, Int.index_zmultiples, Int.index_zmultiples]
+
+theorem coordinateSubgroup_le (P : AddSubgroup Plane) (T : Plane ≃+ Plane) (m L : ℤ)
+    (hm : T (m, 0) ∈ P) (hL : T (0, L) ∈ P) : coordinateSubgroup T m L ≤ P := by
+  rintro z ⟨x, hx, rfl⟩
+  obtain ⟨a, ha⟩ := AddSubgroup.mem_zmultiples_iff.mp hx.1
+  obtain ⟨b, hb⟩ := AddSubgroup.mem_zmultiples_iff.mp hx.2
+  have he : a • (m, (0 : ℤ)) + b • ((0 : ℤ), L) = x := by
+    apply Prod.ext
+    · simpa using ha
+    · simpa using hb
+  have hmem := P.add_mem (P.zsmul_mem hm a) (P.zsmul_mem hL b)
+  rwa [← map_zsmul, ← map_zsmul, ← map_add, he] at hmem
+
+theorem finiteIndex_of_coordinate_mem (P : AddSubgroup Plane) (T : Plane ≃+ Plane)
+    (m L : ℤ) (hm : m ≠ 0) (hL : L ≠ 0)
+    (hpm : T (m, 0) ∈ P) (hpL : T (0, L) ∈ P) : P.FiniteIndex := by
+  let : (coordinateSubgroup T m L).FiniteIndex := ⟨by
+    rw [index_coordinateSubgroup]
+    exact mul_ne_zero (fun h => hm (Int.natAbs_eq_zero.mp h))
+      (fun h => hL (Int.natAbs_eq_zero.mp h))⟩
+  exact AddSubgroup.finiteIndex_of_le (coordinateSubgroup_le P T m L hpm hpL)
+
+theorem FullyPeriodic.exists_coordinate_period {E : Set Plane}
+    (hE : FullyPeriodic E) (T : Plane ≃+ Plane) :
+    ∃ Q : ℕ, 0 < Q ∧ Period E (T (0, (Q : ℤ))) := by
+  obtain ⟨P, hP, hperiod⟩ := hE
+  obtain ⟨Q, hQ, hmem⟩ := exists_pos_zsmul_mem P hP (T (0, 1))
+  have he : (Q : ℤ) • T (0, 1) = T (0, (Q : ℤ)) := by
+    rw [← map_zsmul]
+    congr 1
+    ext <;> simp
+  refine ⟨Q, hQ, ?_⟩
+  rw [← he]
+  exact hperiod _ hmem
+
+theorem fullyPeriodic_of_coordinate_periods {E : Set Plane}
+    (T : Plane ≃+ Plane) (m L : ℕ) (hm : 0 < m) (hL : 0 < L)
+    (hx : Period E (T ((m : ℤ), 0))) (hy : Period E (T (0, (L : ℤ)))) :
+    FullyPeriodic E := by
+  refine ⟨periodSubgroup E, ?_, fun _ hv => hv⟩
+  exact finiteIndex_of_coordinate_mem (periodSubgroup E) T m L
+    (by exact_mod_cast (ne_of_gt hm)) (by exact_mod_cast (ne_of_gt hL)) hx hy
+
+theorem Period.mem_coordinate_mod {D : Set Plane} {T : Plane ≃+ Plane} {m : ℕ}
+    (hv : Period D (T ((m : ℤ), 0))) (x y : ℤ) :
+    T (x, y) ∈ D ↔ T (x % (m : ℤ), y) ∈ D := by
+  have he : T (x % (m : ℤ), y) + (x / (m : ℤ)) • T ((m : ℤ), 0) = T (x, y) := by
+    rw [← map_zsmul, ← map_add]
+    congr 1
+    apply Prod.ext
+    · change x % (m : ℤ) + (x / (m : ℤ)) * (m : ℤ) = x
+      simpa only [add_comm, mul_comm] using Int.emod_add_mul_ediv x (m : ℤ)
+    · simp
+  have hp := (hv.zsmul (x / (m : ℤ))) (T (x % (m : ℤ), y))
+  rwa [he] at hp
+
+def planeDet (u v : Plane) : ℤ := u.1 * v.2 - u.2 * v.1
+
+@[simp] theorem planeDet_self (u : Plane) : planeDet u u = 0 := by
+  simp [planeDet, mul_comm]
+
+theorem planeDet_swap (u v : Plane) : planeDet u v = -planeDet v u := by
+  dsimp [planeDet]
+  ring
+
+theorem planeDet_zsmul_left (n : ℤ) (u v : Plane) :
+    planeDet (n • u) v = n * planeDet u v := by
+  change (n * u.1) * v.2 - (n * u.2) * v.1 = n * (u.1 * v.2 - u.2 * v.1)
+  ring
+
+theorem planeDet_zsmul_right (n : ℤ) (u v : Plane) :
+    planeDet u (n • v) = n * planeDet u v := by
+  change u.1 * (n * v.2) - u.2 * (n * v.1) = n * (u.1 * v.2 - u.2 * v.1)
+  ring
+
+theorem zsmul_plane_ne_zero {n : ℤ} {v : Plane} (hn : n ≠ 0) (hv : v ≠ 0) :
+    n • v ≠ 0 := by
+  intro he
+  apply hv
+  apply Prod.ext
+  · have h := congrArg Prod.fst he
+    change n * v.1 = 0 at h
+    exact (mul_eq_zero.mp h).resolve_left hn
+  · have h := congrArg Prod.snd he
+    change n * v.2 = 0 at h
+    exact (mul_eq_zero.mp h).resolve_left hn
+
+def integerCombination (u v : Plane) : Plane →+ Plane where
+  toFun z := z.1 • u + z.2 • v
+  map_zero' := by simp
+  map_add' := by
+    intro x y
+    change (x.1 + y.1) • u + (x.2 + y.2) • v =
+      (x.1 • u + x.2 • v) + (y.1 • u + y.2 • v)
+    rw [add_smul, add_smul]
+    abel
+
+def integerSpan2 (u v : Plane) : AddSubgroup Plane := (integerCombination u v).range
+
+theorem mem_integerSpan2 (u v z : Plane) :
+    z ∈ integerSpan2 u v ↔ ∃ a b : ℤ, a • u + b • v = z := by
+  constructor
+  · rintro ⟨⟨a, b⟩, he⟩
+    exact ⟨a, b, he⟩
+  · rintro ⟨a, b, he⟩
+    exact ⟨(a, b), he⟩
+
+theorem determinant_smul_eq (u v z : Plane) :
+    planeDet u v • z = planeDet z v • u + planeDet u z • v := by
+  apply Prod.ext
+  · change (u.1 * v.2 - u.2 * v.1) * z.1 =
+      (z.1 * v.2 - z.2 * v.1) * u.1 + (u.1 * z.2 - u.2 * z.1) * v.1
+    ring
+  · change (u.1 * v.2 - u.2 * v.1) * z.2 =
+      (z.1 * v.2 - z.2 * v.1) * u.2 + (u.1 * z.2 - u.2 * z.1) * v.2
+    ring
+
+theorem integerSpan2_finiteIndex (u v : Plane) (huv : planeDet u v ≠ 0) :
+    (integerSpan2 u v).FiniteIndex := by
+  have hm (z : Plane) : planeDet u v • z ∈ integerSpan2 u v :=
+    (mem_integerSpan2 u v _).mpr ⟨planeDet z v, planeDet u z,
+      (determinant_smul_eq u v z).symm⟩
+  apply finiteIndex_of_coordinate_mem (integerSpan2 u v) (AddEquiv.refl Plane)
+    (planeDet u v) (planeDet u v) huv huv
+  · simpa using hm (1, 0)
+  · simpa using hm (0, 1)
+
+theorem fullyPeriodic_of_independent_periods {E : Set Plane} {u v : Plane}
+    (hu : Period E u) (hv : Period E v) (huv : planeDet u v ≠ 0) : FullyPeriodic E := by
+  refine ⟨integerSpan2 u v, integerSpan2_finiteIndex u v huv, ?_⟩
+  intro z hz
+  obtain ⟨a, b, rfl⟩ := (mem_integerSpan2 u v z).mp hz
+  exact (hu.zsmul a).add (hv.zsmul b)
+
+theorem exists_smul_eq_of_collinear {u v : Plane}
+    (hu : u ≠ 0) (hv : v ≠ 0) (huv : planeDet u v = 0) :
+    ∃ a b : ℤ, a ≠ 0 ∧ b ≠ 0 ∧ a • u = b • v := by
+  by_cases hu₁ : u.1 = 0
+  · have hu₂ : u.2 ≠ 0 := fun h => hu (Prod.ext hu₁ h)
+    have he : u.2 * v.1 = 0 := by
+      simpa only [planeDet, hu₁, zero_mul, zero_sub, neg_eq_zero] using huv
+    have hv₁ : v.1 = 0 := (mul_eq_zero.mp he).resolve_left hu₂
+    have hv₂ : v.2 ≠ 0 := fun h => hv (Prod.ext hv₁ h)
+    refine ⟨v.2, u.2, hv₂, hu₂, ?_⟩
+    apply Prod.ext
+    · change v.2 * u.1 = u.2 * v.1
+      simp [hu₁, hv₁]
+    · change v.2 * u.2 = u.2 * v.2
+      exact mul_comm _ _
+  · have he : u.1 * v.2 = u.2 * v.1 := sub_eq_zero.mp huv
+    have hv₁ : v.1 ≠ 0 := by
+      intro hv₁
+      have hv₂ : v.2 = 0 := (mul_eq_zero.mp (by simpa [hv₁] using he)).resolve_left hu₁
+      exact hv (Prod.ext hv₁ hv₂)
+    refine ⟨v.1, u.1, hv₁, hu₁, ?_⟩
+    apply Prod.ext
+    · change v.1 * u.1 = u.1 * v.1
+      exact mul_comm _ _
+    · change v.1 * u.2 = u.1 * v.2
+      simpa only [mul_comm] using he.symm
+
+theorem planeDet_zero_trans {u v w : Plane} (hv : v ≠ 0)
+    (huv : planeDet u v = 0) (hvw : planeDet v w = 0) : planeDet u w = 0 := by
+  by_cases hu : u = 0
+  · simp [hu, planeDet]
+  obtain ⟨a, b, ha, _, he⟩ := exists_smul_eq_of_collinear hu hv huv
+  have hd : planeDet (a • u) w = planeDet (b • v) w :=
+    congrArg (fun z => planeDet z w) he
+  rw [planeDet_zsmul_left, planeDet_zsmul_left, hvw, mul_zero] at hd
+  exact (mul_eq_zero.mp hd).resolve_left ha
+
+theorem exists_common_collinear_multiple (s : Finset Plane) (hne : s.Nonempty)
+    (h0 : ∀ v ∈ s, v ≠ 0)
+    (hcol : ∀ v ∈ s, ∀ w ∈ s, planeDet v w = 0) :
+    ∃ u : Plane, u ≠ 0 ∧ ∀ v ∈ s, ∃ n : ℤ, u = n • v := by
+  classical
+  obtain ⟨v₀, hv₀⟩ := hne
+  have hex : ∀ w : ↥s, ∃ a b : ℤ, a ≠ 0 ∧ b ≠ 0 ∧ a • v₀ = b • (w : Plane) :=
+    fun w => exists_smul_eq_of_collinear (h0 v₀ hv₀) (h0 w w.property)
+      (hcol v₀ hv₀ w w.property)
+  choose a b ha hb he using hex
+  let c : ℤ := ∏ w : ↥s, a w
+  have hc : c ≠ 0 := Finset.prod_ne_zero_iff.mpr (fun w _ => ha w)
+  refine ⟨c • v₀, zsmul_plane_ne_zero hc (h0 v₀ hv₀), ?_⟩
+  intro v hv
+  let w : ↥s := ⟨v, hv⟩
+  let d : ℤ := ∏ w' ∈ (Finset.univ.erase w), a w'
+  have hdc : d * a w = c := by
+    exact Finset.prod_erase_mul _ _ (Finset.mem_univ w)
+  refine ⟨d * b w, ?_⟩
+  calc
+    c • v₀ = (d * a w) • v₀ := by rw [hdc]
+    _ = d • (a w • v₀) := mul_smul _ _ _
+    _ = d • (b w • v) := by rw [he w]
+    _ = (d * b w) • v := (mul_smul _ _ _).symm
+
+theorem exists_transverse_vector (s : Finset Plane) (h0 : ∀ v ∈ s, v ≠ 0) :
+    ∃ e : Plane, e ≠ 0 ∧ ∀ v ∈ s, planeDet e v ≠ 0 := by
+  classical
+  obtain ⟨N, hN⟩ := (s.image (fun v => v.2 / v.1)).exists_notMem
+  refine ⟨(1, N), by simp, ?_⟩
+  intro v hv he
+  have he' : v.2 = N * v.1 := by
+    change 1 * v.2 - N * v.1 = 0 at he
+    simpa only [one_mul] using sub_eq_zero.mp he
+  have hv₁ : v.1 ≠ 0 := by
+    intro hv₁
+    have hv₂ : v.2 = 0 := by simpa [hv₁] using he'
+    exact h0 v hv (Prod.ext hv₁ hv₂)
+  apply hN
+  apply Finset.mem_image.mpr
+  refine ⟨v, hv, ?_⟩
+  rw [he']
+  exact Int.mul_ediv_cancel N hv₁
+
+theorem exists_common_transverse {ι : Type uι} [Fintype ι]
+    (v : ι → Plane) (h0 : ∀ i, v i ≠ 0)
+    (hpair : ∀ i j, i ≠ j → planeDet (v i) (v j) ≠ 0) :
+    ∃ e : Plane, e ≠ 0 ∧ (∀ i, planeDet e (v i) ≠ 0) ∧
+      ∀ i j, i ≠ j → ∃ a b : ℤ, e = a • v i + b • v j := by
+  classical
+  obtain ⟨w, hw, htrans⟩ := exists_transverse_vector (Finset.univ.image v) (by
+    intro z hz
+    obtain ⟨i, _, rfl⟩ := Finset.mem_image.mp hz
+    exact h0 i)
+  let J := {ij : ι × ι // ij.1 ≠ ij.2}
+  let P : AddSubgroup Plane := ⨅ ij : J, integerSpan2 (v ij.val.1) (v ij.val.2)
+  have hP : P.FiniteIndex := AddSubgroup.finiteIndex_iInf
+    (fun ij : J => integerSpan2_finiteIndex _ _ (hpair _ _ ij.property))
+  obtain ⟨q, hq, hmem⟩ := exists_pos_zsmul_mem P hP w
+  have hq' : (q : ℤ) ≠ 0 := by exact_mod_cast (ne_of_gt hq)
+  refine ⟨(q : ℤ) • w, zsmul_plane_ne_zero hq' hw, ?_, ?_⟩
+  · intro i
+    rw [planeDet_zsmul_left]
+    exact mul_ne_zero hq' (htrans (v i) (Finset.mem_image_of_mem v (Finset.mem_univ i)))
+  · intro i j hij
+    have hm : (q : ℤ) • w ∈ integerSpan2 (v i) (v j) :=
+      (AddSubgroup.mem_iInf.mp hmem) (⟨(i, j), hij⟩ : J)
+    obtain ⟨a, b, he⟩ := (mem_integerSpan2 _ _ _).mp hm
+    exact ⟨a, b, he.symm⟩
+
+theorem exists_common_second_vector {ι : Type uι} [Fintype ι]
+    (v : ι → Plane) (i₀ : ι) (e : Plane)
+    (htrans : ∀ i, planeDet e (v i) ≠ 0) :
+    ∃ K : ℕ, 0 < K ∧ ∀ i, ∃ a b : ℤ, a ≠ 0 ∧
+      (K : ℤ) • v i₀ = b • e + a • v i := by
+  let P : AddSubgroup Plane := ⨅ i, integerSpan2 e (v i)
+  have hP : P.FiniteIndex := AddSubgroup.finiteIndex_iInf
+    (fun i => integerSpan2_finiteIndex e (v i) (htrans i))
+  obtain ⟨K, hK, hmem⟩ := exists_pos_zsmul_mem P hP (v i₀)
+  have hK' : (K : ℤ) ≠ 0 := by exact_mod_cast (ne_of_gt hK)
+  refine ⟨K, hK, ?_⟩
+  intro i
+  obtain ⟨b, a, he⟩ := (mem_integerSpan2 e (v i) _).mp
+    ((AddSubgroup.mem_iInf.mp hmem) i)
+  refine ⟨a, b, ?_, he.symm⟩
+  intro ha
+  have hd : planeDet e ((K : ℤ) • v i₀) = 0 := by
+    rw [← he, ha, zero_smul, add_zero, planeDet_zsmul_right, planeDet_self, mul_zero]
+  rw [planeDet_zsmul_right] at hd
+  exact mul_ne_zero hK' (htrans i₀) hd
+
+end PeriodicTilingThree
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/PolynomialShiftDifference.lean b/lean/OAI/Geometry/PeriodicTiling/PolynomialShiftDifference.lean
new file mode 100644
index 0000000000000000000000000000000000000000..303dbc43a7a739c48bfedd2f7756f5f9cad5f09a
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/PolynomialShiftDifference.lean
@@ -0,0 +1,104 @@
+import Mathlib.Algebra.Polynomial.Taylor
+import Mathlib.NumberTheory.Real.Irrational
+import Mathlib.Tactic.NormNum
+import Mathlib.Tactic.Ring
+import Lean.Elab.Tactic.Omega
+
+namespace OAI
+
+namespace PeriodicTilingThree
+
+open Polynomial
+
+theorem polynomial_taylor_coeff_top (P : Polynomial ℝ) (r : ℝ) :
+    (taylor r P).coeff P.natDegree = P.leadingCoeff := by
+  have hd : (hasseDeriv P.natDegree P).natDegree < 1 := by
+    have h := natDegree_hasseDeriv_le P P.natDegree
+    omega
+  rw [taylor_coeff, eval_eq_sum_range' hd]
+  simp only [Finset.sum_range_succ, Finset.sum_range_zero, zero_add,
+    pow_zero, mul_one, hasseDeriv_coeff, Nat.choose_self,
+    Nat.cast_one, one_mul, Polynomial.leadingCoeff]
+
+theorem polynomial_taylor_coeff_pred (P : Polynomial ℝ)
+    (hP : 0 < P.natDegree) (r : ℝ) :
+    (taylor r P).coeff (P.natDegree - 1) =
+      P.coeff (P.natDegree - 1) +
+        (P.natDegree : ℝ) * P.leadingCoeff * r := by
+  have hd : (hasseDeriv (P.natDegree - 1) P).natDegree < 2 := by
+    have h := natDegree_hasseDeriv_le P (P.natDegree - 1)
+    omega
+  have hs : 1 + (P.natDegree - 1) = P.natDegree := by omega
+  have hc : P.natDegree.choose (P.natDegree - 1) = P.natDegree := by
+    have hs' : P.natDegree - 1 + 1 = P.natDegree := by omega
+    simpa only [hs'] using Nat.choose_succ_self_right (P.natDegree - 1)
+  rw [taylor_coeff, eval_eq_sum_range' hd]
+  simp only [Finset.sum_range_succ, Finset.sum_range_zero, zero_add,
+    pow_zero, pow_one, mul_one, hasseDeriv_coeff,
+    Nat.choose_self, Nat.cast_one, one_mul, hs, hc, Polynomial.leadingCoeff]
+
+theorem polynomial_shift_difference_coeff_pred (P : Polynomial ℝ)
+    (hP : 0 < P.natDegree) (a b : ℝ) :
+    (taylor a P - taylor b P).coeff (P.natDegree - 1) =
+      (P.natDegree : ℝ) * P.leadingCoeff * (a - b) := by
+  rw [coeff_sub, polynomial_taylor_coeff_pred P hP,
+    polynomial_taylor_coeff_pred P hP]
+  ring
+
+theorem polynomial_shift_difference_natDegree_le (P : Polynomial ℝ)
+    (hP : 0 < P.natDegree) (a b : ℝ) :
+    (taylor a P - taylor b P).natDegree ≤ P.natDegree - 1 := by
+  apply natDegree_le_iff_coeff_eq_zero.mpr
+  intro n hn
+  by_cases heq : n = P.natDegree
+  · subst n
+    rw [coeff_sub, polynomial_taylor_coeff_top, polynomial_taylor_coeff_top,
+      sub_self]
+  · have hlt : P.natDegree < n := by omega
+    have ha : (taylor a P).coeff n = 0 :=
+      coeff_eq_zero_of_natDegree_lt (by simpa only [natDegree_taylor] using hlt)
+    have hb : (taylor b P).coeff n = 0 :=
+      coeff_eq_zero_of_natDegree_lt (by simpa only [natDegree_taylor] using hlt)
+    rw [coeff_sub, ha, hb, sub_self]
+
+theorem polynomial_shift_difference_degree_irrational (P : Polynomial ℝ)
+    (hP : 1 < P.natDegree) (hI : Irrational P.leadingCoeff)
+    (h k : ℕ) (hhk : h ≠ k) :
+    let D := taylor (h : ℝ) P - taylor (k : ℝ) P
+    D.natDegree = P.natDegree - 1 ∧ Irrational D.leadingCoeff := by
+  dsimp only
+  have hp : 0 < P.natDegree := by omega
+  have hdiff : (h : ℤ) - (k : ℤ) ≠ 0 := by
+    apply sub_ne_zero.mpr
+    exact_mod_cast hhk
+  have hprod : Irrational
+      ((P.natDegree : ℝ) * P.leadingCoeff * ((h : ℝ) - (k : ℝ))) := by
+    simpa only [Int.cast_sub, Int.cast_natCast] using
+      (hI.natCast_mul (Nat.ne_of_gt hp)).mul_intCast hdiff
+  have hc := polynomial_shift_difference_coeff_pred P hp (h : ℝ) (k : ℝ)
+  have hcn : (taylor (h : ℝ) P - taylor (k : ℝ) P).coeff
+      (P.natDegree - 1) ≠ 0 := by
+    rw [hc]
+    exact hprod.ne_zero
+  have hd := natDegree_eq_of_le_of_coeff_ne_zero
+    (polynomial_shift_difference_natDegree_le P hp (h : ℝ) (k : ℝ)) hcn
+  refine ⟨hd, ?_⟩
+  rw [Polynomial.leadingCoeff, hd, hc]
+  exact hprod
+
+theorem polynomial_shift_difference_leadingCoeff (P : Polynomial ℝ)
+    (hP : 1 < P.natDegree) (hI : Irrational P.leadingCoeff)
+    (h k : ℕ) (hhk : h ≠ k) :
+    (taylor (h : ℝ) P - taylor (k : ℝ) P).leadingCoeff =
+      (P.natDegree : ℝ) * P.leadingCoeff * ((h : ℝ) - (k : ℝ)) := by
+  have hd := (polynomial_shift_difference_degree_irrational P hP hI h k hhk).1
+  rw [Polynomial.leadingCoeff, hd]
+  exact polynomial_shift_difference_coeff_pred P (by omega) (h : ℝ) (k : ℝ)
+
+theorem polynomial_shift_difference_eval (P : Polynomial ℝ) (a b x : ℝ) :
+    (taylor a P - taylor b P).eval x = P.eval (x + a) - P.eval (x + b) := by
+  rw [eval_sub, taylor_eval, taylor_eval]
+
+end PeriodicTilingThree
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/PolynomialWeyl.lean b/lean/OAI/Geometry/PeriodicTiling/PolynomialWeyl.lean
new file mode 100644
index 0000000000000000000000000000000000000000..78ea3191cd55fc7f8394e4499c4995cb294385d5
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/PolynomialWeyl.lean
@@ -0,0 +1,137 @@
+import OAI.Geometry.PeriodicTiling.CircleWeylCriterion
+import OAI.Geometry.PeriodicTiling.ScalarVanDerCorput
+import OAI.Geometry.PeriodicTiling.PolynomialShiftDifference
+import Mathlib.NumberTheory.Real.Irrational
+import Mathlib.Tactic.Ring
+import Lean.Elab.Tactic.Omega
+
+namespace OAI
+
+noncomputable section
+
+namespace PeriodicTilingThree
+
+open Filter Polynomial
+open scoped Topology ComplexConjugate
+
+def scalarPhase (x : ℝ) : ℂ :=
+  AddCircle.toCircle (x : UnitAddCircle)
+
+@[simp] theorem scalarPhase_zero : scalarPhase 0 = 1 := by
+  simp only [scalarPhase, AddCircle.coe_zero, AddCircle.toCircle_zero, Circle.coe_one]
+
+@[simp] theorem norm_scalarPhase (x : ℝ) : ‖scalarPhase x‖ = 1 :=
+  Circle.norm_coe _
+
+theorem scalarPhase_add (x y : ℝ) :
+    scalarPhase (x + y) = scalarPhase x * scalarPhase y := by
+  simp only [scalarPhase, AddCircle.coe_add, AddCircle.toCircle_add, Circle.coe_mul]
+
+theorem scalarPhase_nat_mul (n : ℕ) (x : ℝ) :
+    scalarPhase ((n : ℝ) * x) = scalarPhase x ^ n := by
+  induction n with
+  | zero => simp
+  | succ n ih =>
+      rw [Nat.cast_succ, add_mul, one_mul, scalarPhase_add, ih, pow_succ]
+
+theorem scalarPhase_sub (x y : ℝ) :
+    scalarPhase (x - y) = scalarPhase x * conj (scalarPhase y) := by
+  have hy : scalarPhase y ≠ 0 := Circle.coe_ne_zero _
+  apply mul_right_cancel₀ hy
+  rw [← scalarPhase_add, sub_add_cancel, mul_assoc, Complex.conj_mul', norm_scalarPhase]
+  norm_num
+
+theorem scalarPhase_ne_one_of_irrational {x : ℝ} (hx : Irrational x) :
+    scalarPhase x ≠ 1 := by
+  intro h
+  have he : AddCircle.toCircle (x : UnitAddCircle) = AddCircle.toCircle (0 : UnitAddCircle) := by
+    apply Subtype.ext
+    simpa only [AddCircle.toCircle_zero, Circle.coe_one, scalarPhase] using h
+  have hz : (x : UnitAddCircle) = 0 :=
+    AddCircle.injective_toCircle (by norm_num : (1 : ℝ) ≠ 0) he
+  obtain ⟨m, hm⟩ := (AddCircle.coe_eq_zero_iff (1 : ℝ)).mp hz
+  exact hx.ne_int m (by simpa only [zsmul_eq_mul, mul_one] using hm.symm)
+
+theorem fourier_eq_scalarPhase (k : ℤ) (x : ℝ) :
+    fourier k (x : UnitAddCircle) = scalarPhase ((k : ℝ) * x) := by
+  simp only [fourier_apply, scalarPhase, ← AddCircle.coe_zsmul, zsmul_eq_mul]
+
+private theorem eval_degree_one (P : Polynomial ℝ) (hP : P.natDegree = 1) (x : ℝ) :
+    P.eval x = P.coeff 0 + P.leadingCoeff * x := by
+  have hc : P.coeff 1 = P.leadingCoeff := by
+    simpa only [hP] using P.coeff_natDegree
+  rw [Polynomial.eval_eq_sum_range' (by omega : P.natDegree < 2)]
+  simp only [Finset.sum_range_succ, Finset.sum_range_zero, zero_add, pow_zero,
+    mul_one, pow_one, hc]
+
+theorem polynomial_phase_average_tendsto_zero (P : Polynomial ℝ)
+    (hP : 0 < P.natDegree) (hI : Irrational P.leadingCoeff) :
+    Tendsto (complexAverage (fun n : ℕ => scalarPhase (P.eval (n : ℝ))))
+      atTop (𝓝 0) := by
+  have hall : ∀ d : ℕ, ∀ Q : Polynomial ℝ, Q.natDegree = d →
+      0 < d → Irrational Q.leadingCoeff →
+      Tendsto (complexAverage (fun n : ℕ => scalarPhase (Q.eval (n : ℝ))))
+        atTop (𝓝 0) := by
+    intro d
+    induction d using Nat.strong_induction_on with
+    | h d ih =>
+      intro Q hQ hd hI
+      by_cases hd1 : d = 1
+      · have heval : ∀ n : ℕ, scalarPhase (Q.eval (n : ℝ)) =
+            scalarPhase (Q.coeff 0) * scalarPhase Q.leadingCoeff ^ n := by
+          intro n
+          rw [eval_degree_one Q (hQ.trans hd1), scalarPhase_add,
+            mul_comm Q.leadingCoeff (n : ℝ), scalarPhase_nat_mul]
+        simpa only [heval] using complexAverage_const_mul_geometric_tendsto
+          (scalarPhase (Q.coeff 0)) (scalarPhase Q.leadingCoeff)
+          (norm_scalarPhase _) (scalarPhase_ne_one_of_irrational hI)
+      · apply complexAverage_vanDerCorput
+        · intro n
+          exact (norm_scalarPhase _).le
+        · intro h k hhk
+          let D := Polynomial.taylor (h : ℝ) Q - Polynomial.taylor (k : ℝ) Q
+          obtain ⟨hDdeg, hDirr⟩ :=
+            polynomial_shift_difference_degree_irrational Q (by omega) hI h k hhk
+          have hDpos : 0 < D.natDegree := by dsimp [D]; rw [hDdeg]; omega
+          have hDlt : D.natDegree < d := by dsimp [D]; rw [hDdeg, hQ]; omega
+          have hDlim := ih D.natDegree hDlt D rfl hDpos hDirr
+          have hseq :
+              (fun n : ℕ => scalarPhase (Q.eval ((n + h : ℕ) : ℝ)) *
+                conj (scalarPhase (Q.eval ((n + k : ℕ) : ℝ)))) =
+              (fun n : ℕ => scalarPhase (D.eval (n : ℝ))) := by
+            funext n
+            rw [← scalarPhase_sub]
+            congr 1
+            simp only [D, Polynomial.eval_sub, Polynomial.taylor_eval, Nat.cast_add]
+          rw [hseq]
+          exact hDlim
+  exact hall P.natDegree P rfl hP hI
+
+theorem denseRange_circlePolynomial_of_irrational_leadingCoeff (P : Polynomial ℝ)
+    (hP : 0 < P.natDegree) (hI : Irrational P.leadingCoeff) :
+    DenseRange (fun n : ℕ => ((P.eval (n : ℝ) : ℝ) : UnitAddCircle)) := by
+  apply denseRange_of_circleFourierCancellation
+  intro k hk
+  let Q : Polynomial ℝ := Polynomial.C (k : ℝ) * P
+  have hkreal : (k : ℝ) ≠ 0 := Int.cast_ne_zero.mpr hk
+  have hQdeg : Q.natDegree = P.natDegree := Polynomial.natDegree_C_mul hkreal
+  have hQirr : Irrational Q.leadingCoeff := by
+    simpa only [Q, Polynomial.leadingCoeff_mul, Polynomial.leadingCoeff_C]
+      using hI.intCast_mul hk
+  have hQlim := polynomial_phase_average_tendsto_zero Q (hQdeg.symm ▸ hP) hQirr
+  have hseq :
+      (fun n : ℕ => fourier k (((P.eval (n : ℝ) : ℝ) : UnitAddCircle))) =
+      (fun n : ℕ => scalarPhase (Q.eval (n : ℝ))) := by
+    funext n
+    rw [fourier_eq_scalarPhase]
+    simp only [Q, Polynomial.eval_mul, Polynomial.eval_C]
+  change Tendsto (complexAverage
+    (fun n : ℕ => fourier k (((P.eval (n : ℝ) : ℝ) : UnitAddCircle)))) atTop (𝓝 0)
+  rw [hseq]
+  exact hQlim
+
+end PeriodicTilingThree
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/PrimeDifferencePairs.lean b/lean/OAI/Geometry/PeriodicTiling/PrimeDifferencePairs.lean
new file mode 100644
index 0000000000000000000000000000000000000000..26373d9ec8a930a01c648da5cb668903fc00bcd7
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/PrimeDifferencePairs.lean
@@ -0,0 +1,49 @@
+import Mathlib.Algebra.Field.ZMod
+
+namespace OAI
+
+namespace PeriodicTilingThree
+
+def primeDifferencePairs (q m : ℕ) [Fact q.Prime]
+    (hq : q = 2 * m + 1) (d : ZMod q) (hd : d ≠ 0) :
+    (Fin m × Fin 2) ↪ ZMod q where
+  toFun p := ((2 * p.1.val + p.2.val : ℕ) : ZMod q) * d
+  inj' := by
+    rintro ⟨j, b⟩ ⟨k, c⟩ h
+    have hj := j.isLt
+    have hk := k.isLt
+    have hb := b.isLt
+    have hc := c.isLt
+    have hjb : 2 * j.val + b.val < q := by omega
+    have hkc : 2 * k.val + c.val < q := by omega
+    have hcast : ((2 * j.val + b.val : ℕ) : ZMod q) =
+        ((2 * k.val + c.val : ℕ) : ZMod q) :=
+      mul_right_cancel₀ hd h
+    have hindices := congrArg ZMod.val hcast
+    rw [ZMod.val_natCast_of_lt hjb, ZMod.val_natCast_of_lt hkc] at hindices
+    apply Prod.ext
+    · apply Fin.ext
+      change j.val = k.val
+      omega
+    · apply Fin.ext
+      change b.val = c.val
+      omega
+
+@[simp]
+theorem primeDifferencePairs_apply (q m : ℕ) [Fact q.Prime]
+    (hq : q = 2 * m + 1) (d : ZMod q) (hd : d ≠ 0)
+    (j : Fin m) (b : Fin 2) :
+    primeDifferencePairs q m hq d hd (j, b) =
+      ((2 * j.val + b.val : ℕ) : ZMod q) * d := rfl
+
+theorem primeDifferencePairs_step (q m : ℕ) [Fact q.Prime]
+    (hq : q = 2 * m + 1) (d : ZMod q) (hd : d ≠ 0) (j : Fin m) :
+    primeDifferencePairs q m hq d hd (j, 1) -
+      primeDifferencePairs q m hq d hd (j, 0) = d := by
+  change ((2 * j.val + 1 : ℕ) : ZMod q) * d -
+    ((2 * j.val : ℕ) : ZMod q) * d = d
+  rw [Nat.cast_add, Nat.cast_one, add_mul, one_mul, add_sub_cancel_left]
+
+end PeriodicTilingThree
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/QuotientTiling.lean b/lean/OAI/Geometry/PeriodicTiling/QuotientTiling.lean
new file mode 100644
index 0000000000000000000000000000000000000000..34259e05b86ca85041e3f83cbdc55dc813bcf8f1
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/QuotientTiling.lean
@@ -0,0 +1,75 @@
+import OAI.Geometry.PeriodicTiling.TilingBasic
+
+namespace OAI
+
+universe uG uH
+
+namespace PeriodicTilingThree
+
+section
+
+variable {G : Type uG} {H : Type uH} [AddCommGroup G] [AddCommGroup H] [DecidableEq H]
+
+theorem tiles_preimage_of_section (φ : G →+ H)
+    {U : Finset G} {F : Finset H} {A : Set H}
+    (himage : U.image φ = F) (hinj : Set.InjOn φ (U : Set G))
+    (h : Tiles F A) : Tiles U (φ ⁻¹' A) := by
+  classical
+  apply tiles_iff_unique_tile.mpr
+  intro g
+  obtain ⟨f, hf, huniq⟩ := tiles_iff_unique_tile.mp h (φ g)
+  have hfimage : (f : H) ∈ U.image φ := by
+    rw [himage]
+    exact f.property
+  obtain ⟨u, hu, huf⟩ := Finset.mem_image.mp hfimage
+  refine ⟨⟨u, hu⟩, ?_, ?_⟩
+  · change φ (g - u) ∈ A
+    simpa only [map_sub, huf] using hf
+  · intro v hv
+    have hvF : φ (v : G) ∈ F := by
+      rw [← himage]
+      exact Finset.mem_image.mpr ⟨v, v.property, rfl⟩
+    have hvA : φ g - φ (v : G) ∈ A := by
+      simpa only [Set.mem_preimage, map_sub] using hv
+    have hvf : (⟨φ (v : G), hvF⟩ : ↥F) = f := huniq _ hvA
+    apply Subtype.ext
+    apply hinj v.property hu
+    exact (congrArg Subtype.val hvf).trans huf.symm
+
+theorem tiles_image_of_kernel_periods (φ : G →+ H)
+    (hφ : Function.Surjective φ) {U : Finset G} {B : Set G}
+    (h : Tiles U B) (hker : ∀ k ∈ φ.ker, Period B k) :
+    Tiles (U.image φ) (φ '' B) := by
+  classical
+  apply tiles_iff_unique_tile.mpr
+  intro y
+  obtain ⟨g, hg⟩ := hφ y
+  obtain ⟨u, hu, huniq⟩ := tiles_iff_unique_tile.mp h g
+  have hfu : φ (u : G) ∈ U.image φ :=
+    Finset.mem_image.mpr ⟨u, u.property, rfl⟩
+  refine ⟨⟨φ (u : G), hfu⟩, ?_, ?_⟩
+  · refine ⟨g - (u : G), hu, ?_⟩
+    simp only [map_sub, hg]
+  · intro f hf
+    obtain ⟨v, hv, hvf⟩ := Finset.mem_image.mp f.property
+    obtain ⟨b, hb, hbf⟩ := hf
+    have hgvb : φ (g - v - b) = 0 := by
+      simp only [map_sub, hg, hvf, hbf, sub_self]
+    have hk : g - v - b ∈ φ.ker := hgvb
+    have hgv : g - v ∈ B := by
+      have hp := (hker (g - v - b) hk b).mpr hb
+      have heq : b + (g - v - b) = g - v := by
+        rw [add_comm, sub_add_cancel]
+      rw [heq] at hp
+      exact hp
+    have hvu : (⟨v, hv⟩ : ↥U) = u := huniq _ hgv
+    apply Subtype.ext
+    calc
+      (f : H) = φ v := hvf.symm
+      _ = φ (u : G) := congrArg φ (congrArg Subtype.val hvu)
+
+end
+
+end PeriodicTilingThree
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/RationalPolynomialPeriod.lean b/lean/OAI/Geometry/PeriodicTiling/RationalPolynomialPeriod.lean
new file mode 100644
index 0000000000000000000000000000000000000000..0f7a5b067924f5d0ba85b80032e16ffec41e7d32
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/RationalPolynomialPeriod.lean
@@ -0,0 +1,169 @@
+import Mathlib.Algebra.BigOperators.Group.Finset.Piecewise
+import Mathlib.Algebra.Ring.GeomSum
+import Mathlib.Algebra.Order.BigOperators.Ring.Finset
+import Mathlib.Algebra.Polynomial.Degree.Lemmas
+import Mathlib.NumberTheory.Real.Irrational
+import Mathlib.Topology.Instances.AddCircle.Defs
+
+namespace OAI
+
+universe uι
+
+noncomputable section
+
+open scoped BigOperators
+
+namespace PeriodicTilingThree
+
+private theorem finite_common_denominator {ι : Type uι} (s : Finset ι) (c : ι → ℝ)
+    (hc : ∀ i ∈ s, ∃ q : ℚ, c i = (q : ℝ)) :
+    ∃ a : ℕ, 0 < a ∧ ∀ i ∈ s, ∃ z : ℤ, c i * (a : ℝ) = (z : ℝ) := by
+  classical
+  let q : s → ℚ := fun i => Classical.choose (hc i.1 i.2)
+  have hq : ∀ i : s, c i.1 = (q i : ℝ) := fun i => Classical.choose_spec (hc i.1 i.2)
+  let a : ℕ := ∏ i : s, (q i).den
+  refine ⟨a, Finset.prod_pos (fun i _ => (q i).den_pos), ?_⟩
+  intro i hi
+  let j : s := ⟨i, hi⟩
+  have hj : (q j).den ∣ a := Finset.dvd_prod_of_mem _ (Finset.mem_univ j)
+  obtain ⟨b, hb⟩ := hj
+  refine ⟨(q j).num * (b : ℤ), ?_⟩
+  have hiq : c i = (q j : ℝ) := hq j
+  have hden : ((q j).den : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr (q j).den_ne_zero
+  rw [hiq, hb, Nat.cast_mul, ← mul_assoc, Rat.cast_def,
+    div_mul_cancel₀ _ hden,
+    Int.cast_mul, Int.cast_natCast]
+
+private theorem circle_eval_eq_sum (P : Polynomial ℝ) (x : ℝ) :
+    ((P.eval x : ℝ) : AddCircle (1 : ℝ)) =
+      ∑ i ∈ P.support, ((P.coeff i * x ^ i : ℝ) : AddCircle (1 : ℝ)) := by
+  change (QuotientAddGroup.mk' (AddSubgroup.zmultiples (1 : ℝ))) (P.eval x) = _
+  rw [Polynomial.eval_eq_sum, Polynomial.sum_def, map_sum]
+  simp only [QuotientAddGroup.mk'_apply]
+
+private theorem circle_monomial_add_period (c : ℝ) (a : ℕ) (z : ℤ)
+    (hc : c * (a : ℝ) = (z : ℝ)) (i : ℕ) (n : ℤ) :
+    ((c * ((n + (a : ℤ) : ℤ) : ℝ) ^ i : ℝ) : AddCircle (1 : ℝ)) =
+      ((c * (n : ℝ) ^ i : ℝ) : AddCircle (1 : ℝ)) := by
+  apply sub_eq_zero.mp
+  rw [← AddCircle.coe_sub]
+  apply (AddCircle.coe_eq_zero_iff (1 : ℝ)).mpr
+  have hdiv : (a : ℤ) ∣ (n + (a : ℤ)) ^ i - n ^ i := by
+    simpa using sub_dvd_pow_sub_pow (n + (a : ℤ)) n i
+  obtain ⟨k, hk⟩ := hdiv
+  refine ⟨z * k, ?_⟩
+  have hk' : (((n + (a : ℤ) : ℤ) : ℝ) ^ i - (n : ℝ) ^ i) =
+      (a : ℝ) * (k : ℝ) := by
+    exact_mod_cast hk
+  rw [← mul_sub, hk', ← mul_assoc, hc]
+  simp only [zsmul_eq_mul, mul_one, Int.cast_mul]
+
+theorem rational_nonconstant_circlePolynomial_periodic (P : Polynomial ℝ)
+    (hP : ∀ i : ℕ, 0 < i → ∃ q : ℚ, P.coeff i = (q : ℝ)) :
+    ∃ q : ℕ, 0 < q ∧ ∀ n : ℤ,
+      ((P.eval ((n + (q : ℤ) : ℤ) : ℝ) : ℝ) : AddCircle (1 : ℝ)) =
+        ((P.eval (n : ℝ) : ℝ) : AddCircle (1 : ℝ)) := by
+  classical
+  obtain ⟨a, ha, hcoeff⟩ := finite_common_denominator (P.support.erase 0) P.coeff
+    (fun i hi => hP i (Nat.pos_of_ne_zero (Finset.mem_erase.mp hi).1))
+  refine ⟨a, ha, ?_⟩
+  intro n
+  rw [circle_eval_eq_sum, circle_eval_eq_sum]
+  apply Finset.sum_congr rfl
+  intro i hi
+  by_cases hi0 : i = 0
+  · simp only [hi0, pow_zero, mul_one]
+  · obtain ⟨z, hz⟩ := hcoeff i (Finset.mem_erase.mpr ⟨hi0, hi⟩)
+    exact circle_monomial_add_period (P.coeff i) a z hz i n
+
+private theorem circle_monomial_scaled_eq_zero (c : ℝ) (a : ℕ) (z : ℤ)
+    (hc : c * (a : ℝ) = (z : ℝ)) (i : ℕ) (hi : 0 < i) (n : ℕ) :
+    ((c * ((a * n : ℕ) : ℝ) ^ i : ℝ) : AddCircle (1 : ℝ)) = 0 := by
+  obtain ⟨k, rfl⟩ := Nat.exists_eq_succ_of_ne_zero hi.ne'
+  apply (AddCircle.coe_eq_zero_iff (1 : ℝ)).mpr
+  refine ⟨z * (a : ℤ) ^ k * (n : ℤ) ^ (k + 1), ?_⟩
+  have hmul : c * ((a * n : ℕ) : ℝ) ^ (k + 1) =
+      (z : ℝ) * (a : ℝ) ^ k * (n : ℝ) ^ (k + 1) := by
+    calc
+      _ = (c * (a : ℝ)) * ((a : ℝ) ^ k * (n : ℝ) ^ (k + 1)) := by
+        rw [Nat.cast_mul, mul_pow, pow_succ]
+        ring
+      _ = _ := by rw [hc]; ring
+  simpa only [zsmul_eq_mul, mul_one, Int.cast_mul, Int.cast_pow, Int.cast_natCast]
+    using hmul.symm
+
+theorem exists_progression_irrational_leadingCoeff (P : Polynomial ℝ)
+    (hP : ∃ i : ℕ, 0 < i ∧ Irrational (P.coeff i)) :
+    ∃ a : ℕ, 0 < a ∧ ∃ Q : Polynomial ℝ,
+      0 < Q.natDegree ∧ Irrational Q.leadingCoeff ∧ ∀ n : ℕ,
+        ((P.eval ((a * n : ℕ) : ℝ) : ℝ) : AddCircle (1 : ℝ)) =
+          ((Q.eval (n : ℝ) : ℝ) : AddCircle (1 : ℝ)) := by
+  classical
+  let s := P.support.filter fun i => 0 < i ∧ Irrational (P.coeff i)
+  have hs : s.Nonempty := by
+    obtain ⟨i, hi, hirr⟩ := hP
+    exact ⟨i, Finset.mem_filter.mpr ⟨Polynomial.mem_support_iff.mpr hirr.ne_zero, hi, hirr⟩⟩
+  let d := s.max' hs
+  have hdmem : d ∈ s := Finset.max'_mem s hs
+  have hdpos : 0 < d := (Finset.mem_filter.mp hdmem).2.1
+  have hdirr : Irrational (P.coeff d) := (Finset.mem_filter.mp hdmem).2.2
+  have hrat : ∀ j : ℕ, d < j → ∃ q : ℚ, P.coeff j = (q : ℝ) := by
+    intro j hj
+    have hnot : ¬ Irrational (P.coeff j) := by
+      intro hirr
+      have hjmem : j ∈ s := Finset.mem_filter.mpr
+        ⟨Polynomial.mem_support_iff.mpr hirr.ne_zero, hdpos.trans hj, hirr⟩
+      exact (not_le_of_gt hj) (Finset.le_max' s j hjmem)
+    simpa only [Irrational, Set.mem_range, not_not, eq_comm] using hnot
+  obtain ⟨a, ha, hcoeff⟩ :=
+    finite_common_denominator (P.support.filter fun i => d < i) P.coeff
+      (fun i hi => hrat i (Finset.mem_filter.mp hi).2)
+  let t := P.support.filter fun i => i ≤ d
+  let Q : Polynomial ℝ :=
+    ∑ i ∈ t, Polynomial.monomial i (P.coeff i * (a : ℝ) ^ i)
+  have hQcoeff : ∀ j : ℕ, Q.coeff j =
+      if j ∈ t then P.coeff j * (a : ℝ) ^ j else 0 := by
+    intro j
+    simp only [Q, Polynomial.finsetSum_coeff, Polynomial.coeff_monomial,
+      Finset.sum_ite_eq']
+  have hdmemt : d ∈ t := Finset.mem_filter.mpr
+    ⟨Polynomial.mem_support_iff.mpr hdirr.ne_zero, le_rfl⟩
+  have hQirr : Irrational (Q.coeff d) := by
+    rw [hQcoeff, ite_eq_left hdmemt]
+    simpa only [Nat.cast_pow] using hdirr.mul_natCast (pow_ne_zero d ha.ne')
+  have hQdeg : Q.natDegree = d := by
+    apply Polynomial.natDegree_eq_of_le_of_coeff_ne_zero _ hQirr.ne_zero
+    apply Polynomial.natDegree_le_iff_coeff_eq_zero.mpr
+    intro j hj
+    rw [hQcoeff, ite_eq_right]
+    intro hjmem
+    exact (not_le_of_gt hj) (Finset.mem_filter.mp hjmem).2
+  refine ⟨a, ha, Q, hQdeg.symm ▸ hdpos, ?_, ?_⟩
+  · simpa only [Polynomial.leadingCoeff, hQdeg] using hQirr
+  · intro n
+    have hQeval : ((Q.eval (n : ℝ) : ℝ) : AddCircle (1 : ℝ)) =
+        ∑ i ∈ t, ((P.coeff i * ((a * n : ℕ) : ℝ) ^ i : ℝ) : AddCircle (1 : ℝ)) := by
+      simp only [Q, Polynomial.eval_finsetSum, Polynomial.eval_monomial]
+      change (QuotientAddGroup.mk' (AddSubgroup.zmultiples (1 : ℝ)))
+        (∑ i ∈ t, (P.coeff i * (a : ℝ) ^ i) * (n : ℝ) ^ i) = _
+      rw [map_sum]
+      simp only [QuotientAddGroup.mk'_apply]
+      apply Finset.sum_congr rfl
+      intro i hi
+      congr 1
+      rw [Nat.cast_mul, mul_pow]
+      ring
+    rw [circle_eval_eq_sum, hQeval]
+    symm
+    apply Finset.sum_subset (Finset.filter_subset _ _)
+    intro i hi hit
+    have hdi : d < i := lt_of_not_ge fun hid =>
+      hit (Finset.mem_filter.mpr ⟨hi, hid⟩)
+    obtain ⟨z, hz⟩ := hcoeff i (Finset.mem_filter.mpr ⟨hi, hdi⟩)
+    exact circle_monomial_scaled_eq_zero (P.coeff i) a z hz i (hdpos.trans hdi) n
+
+end PeriodicTilingThree
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/ResidueCoordinates.lean b/lean/OAI/Geometry/PeriodicTiling/ResidueCoordinates.lean
new file mode 100644
index 0000000000000000000000000000000000000000..0cc8048eaa078ce6fb093f759fae571e5bddc52a
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/ResidueCoordinates.lean
@@ -0,0 +1,145 @@
+import OAI.Geometry.PeriodicTiling.TilingBasic
+import Mathlib.Data.ZMod.Basic
+import Mathlib.Data.Int.Interval
+import Mathlib.Data.Fintype.BigOperators
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.Ring
+
+namespace OAI
+
+namespace PeriodicTilingThree
+
+abbrev Residue3 (m : ℕ) := Fin 3 → ZMod m
+abbrev LiftAlphabet := Fin 3 → Fin 5
+abbrev LiftAssignment (m : ℕ) := Residue3 m → LiftAlphabet
+
+def residueMod (m : ℕ) (d : Lattice 3) : Residue3 m := fun i => (d i : ZMod m)
+
+def can {m : ℕ} (v : Residue3 m) : Lattice 3 := fun i => ((v i).val : ℤ)
+
+def signed (h : LiftAlphabet) : Lattice 3 := fun i => (h i).val - (2 : ℤ)
+
+def representative (m : ℕ) [NeZero m] (ω : LiftAssignment m)
+    (v : Residue3 m) : Lattice 3 :=
+  fun i => can v i + (m : ℤ) * signed (ω v) i
+
+@[simp] theorem residueMod_can {m : ℕ} [NeZero m] (v : Residue3 m) :
+    residueMod m (can v) = v := by
+  funext i
+  simp [residueMod, can]
+
+@[simp] theorem representative_residue (m : ℕ) [NeZero m]
+    (ω : LiftAssignment m) (v : Residue3 m) :
+    residueMod m (representative m ω v) = v := by
+  funext i
+  simp [residueMod, representative, can]
+
+theorem representative_injective (m : ℕ) [NeZero m] (ω : LiftAssignment m) :
+    Function.Injective (representative m ω) := by
+  intro u v huv
+  have h := congrArg (residueMod m) huv
+  simpa only [representative_residue] using h
+
+theorem can_nonneg {m : ℕ} (v : Residue3 m) (i : Fin 3) : 0 ≤ can v i := by
+  dsimp only [can]
+  exact_mod_cast (Nat.zero_le (v i).val)
+
+theorem can_lt {m : ℕ} [NeZero m] (v : Residue3 m) (i : Fin 3) :
+    can v i < (m : ℤ) := by
+  dsimp only [can]
+  exact_mod_cast (v i).val_lt
+
+theorem signed_bounds (h : LiftAlphabet) (i : Fin 3) :
+    -2 ≤ signed h i ∧ signed h i ≤ 2 := by
+  have hi := (h i).isLt
+  dsimp [signed]
+  omega
+
+def Requested (m : ℕ) (d : Lattice 3) : Prop :=
+  (∀ i, |d i| ≤ (m : ℤ)) ∧ ¬ ∀ i, (m : ℤ) ∣ d i
+
+noncomputable def requestedVectors (m : ℕ) : Finset (Lattice 3) := by
+  classical
+  exact (Fintype.piFinset (fun _ : Fin 3 => Finset.Icc (-(m : ℤ)) (m : ℤ))).filter
+    (fun d => ¬ ∀ i, (m : ℤ) ∣ d i)
+
+@[simp] theorem mem_requestedVectors {m : ℕ} {d : Lattice 3} :
+    d ∈ requestedVectors m ↔ Requested m d := by
+  classical
+  simp only [requestedVectors, Finset.mem_filter, Fintype.mem_piFinset,
+    Finset.mem_Icc, Requested, abs_le]
+
+theorem card_requestedVectors_le (m : ℕ) :
+    (requestedVectors m).card ≤ (2 * m + 1) ^ 3 := by
+  classical
+  have hc : (Finset.Icc (-(m : ℤ)) (m : ℤ)).card = 2 * m + 1 := by
+    rw [Int.card_Icc]
+    omega
+  calc
+    (requestedVectors m).card ≤
+        (Fintype.piFinset (fun _ : Fin 3 => Finset.Icc (-(m : ℤ)) (m : ℤ))).card :=
+      Finset.card_filter_le _ _
+    _ = (2 * m + 1) ^ 3 := by rw [Fintype.card_piFinset_const, hc]
+
+theorem requested_residueMod_ne_zero {m : ℕ} {d : Lattice 3}
+    (hd : Requested m d) : residueMod m d ≠ 0 := by
+  intro h
+  apply hd.2
+  intro i
+  apply (ZMod.intCast_zmod_eq_zero_iff_dvd (d i) m).mp
+  exact congrFun h i
+
+theorem requested_successor_ne {m : ℕ} {d : Lattice 3}
+    (hd : Requested m d) (u : Residue3 m) : u + residueMod m d ≠ u := by
+  intro h
+  apply requested_residueMod_ne_zero hd
+  apply add_left_cancel (a := u)
+  simpa using h
+
+theorem card_residue3 (m : ℕ) [NeZero m] : Fintype.card (Residue3 m) = m ^ 3 := by
+  simp [Residue3]
+
+theorem card_liftAlphabet : Fintype.card LiftAlphabet = 125 := by
+  norm_num [LiftAlphabet]
+
+theorem card_liftAlphabet_pairs : Fintype.card (LiftAlphabet × LiftAlphabet) = 15625 := by
+  norm_num [Fintype.card_prod, card_liftAlphabet]
+
+def liftDifference (m : ℕ) (d : Lattice 3) (u v : Residue3 m) : Lattice 3 :=
+  fun i => (d i - (can v i - can u i)) / (m : ℤ)
+
+theorem liftDifference_eq {m : ℕ} [NeZero m] {d : Lattice 3} {u v : Residue3 m}
+    (hv : v = u + residueMod m d) (i : Fin 3) :
+    (m : ℤ) * liftDifference m d u v i = d i - (can v i - can u i) := by
+  apply Int.mul_ediv_cancel'
+  apply (ZMod.intCast_zmod_eq_zero_iff_dvd _ m).mp
+  have hvi := congrFun hv i
+  change v i = u i + (d i : ZMod m) at hvi
+  simp only [Int.cast_sub, can, Int.cast_natCast, ZMod.natCast_zmod_val]
+  rw [hvi]
+  ring
+
+theorem liftDifference_bounds {m : ℕ} [NeZero m] {d : Lattice 3} {u v : Residue3 m}
+    (hd : Requested m d) (hv : v = u + residueMod m d) (i : Fin 3) :
+    -1 ≤ liftDifference m d u v i ∧ liftDifference m d u v i ≤ 1 := by
+  have hm : (0 : ℤ) < m := by exact_mod_cast Nat.pos_of_ne_zero (NeZero.ne m)
+  have hdu := (abs_le.mp (hd.1 i)).1
+  have hdv := (abs_le.mp (hd.1 i)).2
+  have hu0 := can_nonneg u i
+  have hv0 := can_nonneg v i
+  have hum := can_lt u i
+  have hvm := can_lt v i
+  have heq := liftDifference_eq hv i
+  constructor
+  · by_contra h
+    have hz : liftDifference m d u v i ≤ -2 := by omega
+    have hmz := mul_le_mul_of_nonneg_left hz (le_of_lt hm)
+    nlinarith
+  · by_contra h
+    have hz : 2 ≤ liftDifference m d u v i := by omega
+    have hmz := mul_le_mul_of_nonneg_left hz (le_of_lt hm)
+    nlinarith
+
+end PeriodicTilingThree
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/ResidueCountBound.lean b/lean/OAI/Geometry/PeriodicTiling/ResidueCountBound.lean
new file mode 100644
index 0000000000000000000000000000000000000000..6cee7ebf5ca31f3468889d250e0c944c842ef77f
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/ResidueCountBound.lean
@@ -0,0 +1,80 @@
+import Mathlib.Algebra.Order.Field.Rat
+import Mathlib.Algebra.Order.GroupWithZero.Basic
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.Ring
+
+namespace OAI
+
+namespace PeriodicTilingThree
+
+theorem one_sub_pow_mul_linear_le_one (c : ℚ) (_hc0 : 0 ≤ c) (hc1 : c ≤ 1)
+    (r : ℕ) : (1 - c) ^ r * (1 + (r : ℚ) * c) ≤ 1 := by
+  induction r with
+  | zero => simp
+  | succ r ih =>
+      have hstep : (1 - c) * (1 + ((r : ℚ) + 1) * c) ≤ 1 + (r : ℚ) * c := by
+        calc
+          (1 - c) * (1 + ((r : ℚ) + 1) * c) =
+              (1 + (r : ℚ) * c) - ((r : ℚ) + 1) * c ^ 2 := by ring
+          _ ≤ 1 + (r : ℚ) * c :=
+            sub_le_self _ (mul_nonneg (add_nonneg (Nat.cast_nonneg r) zero_le_one)
+              (sq_nonneg c))
+      calc
+        (1 - c) ^ (r + 1) * (1 + ((r + 1 : ℕ) : ℚ) * c) =
+            (1 - c) ^ r * ((1 - c) * (1 + ((r : ℚ) + 1) * c)) := by
+              rw [pow_succ, Nat.cast_add, Nat.cast_one]
+              ring
+        _ ≤ (1 - c) ^ r * (1 + (r : ℚ) * c) :=
+          mul_le_mul_of_nonneg_left hstep (pow_nonneg (sub_nonneg.mpr hc1) r)
+        _ ≤ 1 := ih
+
+private theorem scalar_half_bound (x c : ℚ) (hx : 0 ≤ x)
+    (hc : (1 : ℚ) / 1000 ≤ c) (h : x * (1 + 1000 * c) ≤ 1) : x ≤ 1 / 2 := by
+  have htwo : (2 : ℚ) ≤ 1 + 1000 * c := by linarith only [hc]
+  apply (le_div_iff₀ (by norm_num : (0 : ℚ) < 2)).mpr
+  exact (mul_le_mul_of_nonneg_left htwo hx).trans h
+
+theorem residue_block_bound (c : ℚ) (hc0 : 0 ≤ c) (hc1 : c ≤ 1)
+    (hc : (1 : ℚ) / 1000 ≤ c) : (1 - c) ^ 1000 ≤ 1 / 2 := by
+  apply scalar_half_bound ((1 - c) ^ 1000) c
+    (pow_nonneg (sub_nonneg.mpr hc1) 1000) hc
+  exact one_sub_pow_mul_linear_le_one c hc0 hc1 1000
+
+theorem residue_failure_block_bound : ((15598 : ℚ) / 15625) ^ 1000 ≤ 1 / 2 := by
+  have h := residue_block_bound ((27 : ℚ) / 15625) (by norm_num)
+    (by norm_num) (by norm_num)
+  have heq : (1 : ℚ) - 27 / 15625 = 15598 / 15625 := by norm_num
+  rw [heq] at h
+  exact h
+
+private theorem union_bound_from_block (p : ℚ) (hp : 0 ≤ p)
+    (hblock : p ^ 1000 ≤ 1 / 2) : (129 : ℚ) ^ 3 * p ^ 32000 < 1 := by
+  have hblocks : p ^ 32000 ≤ ((1 : ℚ) / 2) ^ 32 := by
+    rw [show (32000 : ℕ) = 1000 * 32 by norm_num, pow_mul]
+    exact pow_le_pow_left₀ (a := p ^ 1000) (b := (1 : ℚ) / 2)
+      (pow_nonneg hp 1000) hblock 32
+  calc
+    (129 : ℚ) ^ 3 * p ^ 32000 ≤
+        (129 : ℚ) ^ 3 * ((1 : ℚ) / 2) ^ 32 :=
+      mul_le_mul_of_nonneg_left hblocks (by norm_num)
+    _ < 1 := by norm_num
+
+theorem residue_union_bound_32000 :
+    (129 : ℚ) ^ 3 * ((15598 : ℚ) / 15625) ^ 32000 < 1 := by
+  exact union_bound_from_block ((15598 : ℚ) / 15625) (by norm_num)
+    residue_failure_block_bound
+
+theorem residue_union_bound_of_large_exponent {k : ℕ} (hk : 32000 ≤ k) :
+    (129 : ℚ) ^ 3 * ((15598 : ℚ) / 15625) ^ k < 1 := by
+  apply lt_of_le_of_lt _ residue_union_bound_32000
+  exact mul_le_mul_of_nonneg_left
+    (pow_le_pow_of_le_one (by norm_num) (by norm_num) hk) (by norm_num)
+
+theorem residue_cube_64 : (64 : ℕ) ^ 3 = 262144 := by norm_num
+
+theorem residue_matching_budget_64 : 32000 ≤ (64 : ℕ) ^ 3 / 3 - 1 := by
+  norm_num
+
+end PeriodicTilingThree
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/ResidueLiftPairs.lean b/lean/OAI/Geometry/PeriodicTiling/ResidueLiftPairs.lean
new file mode 100644
index 0000000000000000000000000000000000000000..71b4af255932ab0557d0a9029dff1e23b71997c8
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/ResidueLiftPairs.lean
@@ -0,0 +1,111 @@
+import OAI.Geometry.PeriodicTiling.ResidueCoordinates
+
+namespace OAI
+
+universe uA uB
+
+namespace PeriodicTilingThree
+
+noncomputable def PairGood (m : ℕ) (d : Lattice 3) (u v : Residue3 m) :
+    Finset (LiftAlphabet × LiftAlphabet) := by
+  classical
+  exact Finset.univ.filter (fun p => ∀ i,
+    can v i + (m : ℤ) * signed p.2 i - (can u i + (m : ℤ) * signed p.1 i) = d i)
+
+@[simp] theorem mem_PairGood {m : ℕ} {d : Lattice 3} {u v : Residue3 m}
+    {p : LiftAlphabet × LiftAlphabet} :
+    p ∈ PairGood m d u v ↔ ∀ i,
+      can v i + (m : ℤ) * signed p.2 i - (can u i + (m : ℤ) * signed p.1 i) = d i := by
+  classical
+  simp only [PairGood, Finset.mem_filter, Finset.mem_univ, true_and]
+
+theorem pairGood_assignment_iff {m : ℕ} [NeZero m] {d : Lattice 3}
+    (ω : LiftAssignment m) (u v : Residue3 m) :
+    (ω u, ω v) ∈ PairGood m d u v ↔ representative m ω v - representative m ω u = d := by
+  rw [mem_PairGood, funext_iff]
+  rfl
+
+private def coordinateLiftPair (z : ℤ) (hz : -2 ≤ z ∧ z ≤ 2) (j : Fin 3) :
+    Fin 5 × Fin 5 :=
+  (⟨j.val + (-z).toNat, by have hj := j.isLt; omega⟩,
+   ⟨j.val + z.toNat, by have hj := j.isLt; omega⟩)
+
+private theorem coordinateLiftPair_difference (z : ℤ) (hz : -2 ≤ z ∧ z ≤ 2)
+    (j : Fin 3) :
+    ((coordinateLiftPair z hz j).2.val : ℤ) - (coordinateLiftPair z hz j).1.val = z := by
+  dsimp [coordinateLiftPair]
+  omega
+
+private theorem coordinateLiftPair_injective (z : ℤ) (hz : -2 ≤ z ∧ z ≤ 2) :
+    Function.Injective (coordinateLiftPair z hz) := by
+  intro a b hab
+  apply Fin.ext
+  have h := congrArg (fun p : Fin 5 × Fin 5 => p.1.val) hab
+  dsimp [coordinateLiftPair] at h
+  exact Nat.add_right_cancel h
+
+private def liftPairForDifference (z : Lattice 3) (hz : ∀ i, -2 ≤ z i ∧ z i ≤ 2)
+    (a : Fin 3 → Fin 3) : LiftAlphabet × LiftAlphabet :=
+  (fun i => (coordinateLiftPair (z i) (hz i) (a i)).1,
+   fun i => (coordinateLiftPair (z i) (hz i) (a i)).2)
+
+private theorem liftPairForDifference_injective (z : Lattice 3)
+    (hz : ∀ i, -2 ≤ z i ∧ z i ≤ 2) : Function.Injective (liftPairForDifference z hz) := by
+  intro a b hab
+  funext i
+  apply coordinateLiftPair_injective (z i) (hz i)
+  apply Prod.ext
+  · exact congrArg (fun p : LiftAlphabet × LiftAlphabet => p.1 i) hab
+  · exact congrArg (fun p : LiftAlphabet × LiftAlphabet => p.2 i) hab
+
+private theorem liftPairForDifference_signed_sub (z : Lattice 3)
+    (hz : ∀ i, -2 ≤ z i ∧ z i ≤ 2) (a : Fin 3 → Fin 3) (i : Fin 3) :
+    signed (liftPairForDifference z hz a).2 i -
+      signed (liftPairForDifference z hz a).1 i = z i := by
+  have h := coordinateLiftPair_difference (z i) (hz i) (a i)
+  dsimp [signed, liftPairForDifference]
+  omega
+
+private theorem card_le_finset_of_injective {A : Type uA} {B : Type uB} [Fintype A]
+    (s : Finset B) (f : A → B) (hfinj : Function.Injective f)
+    (hmem : ∀ a, f a ∈ s) : Fintype.card A ≤ s.card := by
+  classical
+  let g : A → ↥s := fun a => ⟨f a, hmem a⟩
+  have hginj : Function.Injective g := by
+    intro a b hab
+    exact hfinj (congrArg Subtype.val hab)
+  have hc := Fintype.card_le_of_injective g hginj
+  simpa only [Fintype.card_coe] using hc
+
+theorem card_PairGood_ge {m : ℕ} [NeZero m] {d : Lattice 3} {u v : Residue3 m}
+    (hd : Requested m d) (hv : v = u + residueMod m d) :
+    27 ≤ (PairGood m d u v).card := by
+  classical
+  let z := liftDifference m d u v
+  have hz : ∀ i, -2 ≤ z i ∧ z i ≤ 2 := by
+    intro i
+    have h := liftDifference_bounds hd hv i
+    change -2 ≤ liftDifference m d u v i ∧ liftDifference m d u v i ≤ 2
+    omega
+  let f : (Fin 3 → Fin 3) → LiftAlphabet × LiftAlphabet := liftPairForDifference z hz
+  have hfinj : Function.Injective f := liftPairForDifference_injective z hz
+  have hf : ∀ a, f a ∈ PairGood m d u v := by
+    intro a
+    rw [mem_PairGood]
+    intro i
+    have hdiff := liftPairForDifference_signed_sub z hz a i
+    have hcarry := liftDifference_eq hv i
+    change (m : ℤ) * z i = d i - (can v i - can u i) at hcarry
+    change can v i + (m : ℤ) * signed (liftPairForDifference z hz a).2 i -
+      (can u i + (m : ℤ) * signed (liftPairForDifference z hz a).1 i) = d i
+    rw [← hdiff] at hcarry
+    nlinarith
+  generalize hS : PairGood m d u v = S at hf ⊢
+  have hc : Fintype.card (Fin 3 → Fin 3) ≤ S.card :=
+    card_le_finset_of_injective S f hfinj hf
+  have h27 : Fintype.card (Fin 3 → Fin 3) = 27 := by norm_num
+  rwa [h27] at hc
+
+end PeriodicTilingThree
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/ResidueTransversal.lean b/lean/OAI/Geometry/PeriodicTiling/ResidueTransversal.lean
new file mode 100644
index 0000000000000000000000000000000000000000..72a96bf45cbbf9aa83f6aae6d6c88e5b3d636576
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/ResidueTransversal.lean
@@ -0,0 +1,199 @@
+import OAI.Geometry.PeriodicTiling.ResidueLiftPairs
+import OAI.Geometry.PeriodicTiling.ResidueCountBound
+import OAI.Geometry.PeriodicTiling.DisjointSuccessorPairs
+import OAI.Geometry.PeriodicTiling.FinitePairRatio
+import OAI.Geometry.PeriodicTiling.FiniteEventUnion
+import OAI.Geometry.PeriodicTiling.MarkedTile
+import Mathlib.Tactic.Push
+
+namespace OAI
+
+universe uV uC uI
+
+namespace PeriodicTilingThree
+
+open scoped Classical
+
+private def residueSuccessor (m : ℕ) (d : Lattice 3) : Equiv.Perm (Residue3 m) where
+  toFun u := u + residueMod m d
+  invFun u := u - residueMod m d
+  left_inv u := add_sub_cancel_right u _
+  right_inv u := sub_add_cancel u _
+
+private theorem exists_residue_pairs
+    {D : Finset (Lattice 3)} (hD : ∀ d, d ∈ D ↔ Requested 64 d) (d : ↥D) :
+    ∃ pair : (Fin 32000 × Fin 2) ↪ Residue3 64,
+      (∀ j, pair (j, 1) = pair (j, 0) + residueMod 64 d.1) ∧
+      ∀ x, pair x ≠ 0 := by
+  have hd : Requested 64 d.1 := (hD d.1).mp d.2
+  exact exists_disjoint_successor_pairs (residueSuccessor 64 d.1)
+    (requested_successor_ne hd) 0 32000 (by rw [card_residue3]; norm_num)
+
+private noncomputable def residuePairs
+    {D : Finset (Lattice 3)} (hD : ∀ d, d ∈ D ↔ Requested 64 d) (d : ↥D) :
+    (Fin 32000 × Fin 2) ↪ Residue3 64 :=
+  Classical.choose (exists_residue_pairs hD d)
+
+private theorem residuePairs_step
+    {D : Finset (Lattice 3)} (hD : ∀ d, d ∈ D ↔ Requested 64 d)
+    (d : ↥D) (j : Fin 32000) :
+    residuePairs hD d (j, 1) = residuePairs hD d (j, 0) + residueMod 64 d.1 :=
+  (Classical.choose_spec (exists_residue_pairs hD d)).1 j
+
+private theorem residuePairs_ne_zero
+    {D : Finset (Lattice 3)} (hD : ∀ d, d ∈ D ↔ Requested 64 d)
+    (d : ↥D) (x : Fin 32000 × Fin 2) : residuePairs hD d x ≠ 0 :=
+  (Classical.choose_spec (exists_residue_pairs hD d)).2 x
+
+private noncomputable def residuePairGood
+    {D : Finset (Lattice 3)} (hD : ∀ d, d ∈ D ↔ Requested 64 d)
+    (d : ↥D) (j : Fin 32000) : Finset (LiftAlphabet × LiftAlphabet) :=
+  PairGood 64 d.1 (residuePairs hD d (j, 0)) (residuePairs hD d (j, 1))
+
+private theorem normalized_bad_assignment_ratio
+    {V : Type uV} {C : Type uC} [Fintype V] [Fintype C] {k : ℕ}
+    (pair : (Fin k × Fin 2) ↪ V) (Good : Fin k → Finset (C × C))
+    (Bad : Finset (V → C)) (hBad : Bad = allBadAssignments pair Good)
+    (N : ℕ) (hN : N = Fintype.card C ^ Fintype.card V)
+    (hC : Fintype.card C = 125) (hgood : ∀ j, 27 ≤ (Good j).card) :
+    (Bad.card : ℚ) / (N : ℚ) ≤ ((15598 : ℚ) / 15625) ^ k := by
+  have hCpos : 0 < Fintype.card C := by rw [hC]; norm_num
+  have hg : 27 ≤ Fintype.card C ^ 2 := by rw [hC]; norm_num
+  have heq : (1 : ℚ) - ((27 : ℕ) : ℚ) / (Fintype.card C : ℚ) ^ 2 =
+      15598 / 15625 := by
+    rw [hC]
+    norm_num
+  have h := allBadAssignments_card_ratio_le pair Good 27 hCpos hg hgood
+  rw [Fintype.card_fun, ← hN, ← hBad, heq] at h
+  exact h
+
+private theorem residuePairGood_card
+    {D : Finset (Lattice 3)} (hD : ∀ d, d ∈ D ↔ Requested 64 d)
+    (d : ↥D) (j : Fin 32000) : 27 ≤ (residuePairGood hD d j).card := by
+  have hd : Requested 64 d.1 := (hD d.1).mp d.2
+  exact card_PairGood_ge (m := 64) (d := d.1)
+    (u := residuePairs hD d (j, 0)) (v := residuePairs hD d (j, 1))
+    hd (residuePairs_step hD d j)
+
+private theorem representative_difference_of_pairGood {m : ℕ} [NeZero m]
+    (ω : LiftAssignment m) (d : Lattice 3) (u v : Residue3 m) :
+    (ω u, ω v) ∈ PairGood m d u v →
+      representative m ω v - representative m ω u = d := by
+  have hiff := pairGood_assignment_iff (m := m) (d := d) ω u v
+  generalize hset : PairGood m d u v = S at hiff ⊢
+  exact hiff.mp
+
+private theorem exists_good_pair {V : Type uV} {C : Type uC} [Fintype V] [Fintype C] {k : ℕ}
+    (pair : (Fin k × Fin 2) ↪ V) (Good : Fin k → Finset (C × C)) (ω : V → C)
+    (P : Fin k → Prop)
+    (hP : ∀ j, (ω (pair (j, 0)), ω (pair (j, 1))) ∈ Good j → P j)
+    (hbad : ω ∉ allBadAssignments pair Good) : ∃ j, P j := by
+  rw [mem_allBadAssignments] at hbad
+  push Not at hbad
+  obtain ⟨j, hj⟩ := hbad
+  exact ⟨j, hP j hj⟩
+
+private theorem exists_simultaneous_good_pairs
+    {I : Type uI} {V : Type uV} {C : Type uC} [Fintype V] [Fintype C] {k : ℕ}
+    (events : Finset I) (pair : I → (Fin k × Fin 2) ↪ V)
+    (Good : I → Fin k → Finset (C × C))
+    (hC : Fintype.card C = 125) (hgood : ∀ i j, 27 ≤ (Good i j).card)
+    (P : I → (V → C) → Fin k → Prop)
+    (hP : ∀ i ω j, (ω (pair i (j, 0)), ω (pair i (j, 1))) ∈ Good i j → P i ω j)
+    (hsmall : (events.card : ℚ) * ((15598 : ℚ) / 15625) ^ k < 1) :
+    ∃ ω : V → C, ∀ i ∈ events, ∃ j, P i ω j := by
+  classical
+  have hΩ : 0 < Fintype.card (V → C) := by
+    rw [Fintype.card_fun, hC]
+    exact pow_pos (by norm_num) _
+  have hbound (i : I) :
+      ((allBadAssignments (pair i) (Good i)).card : ℚ) /
+          (Fintype.card (V → C) : ℚ) ≤ ((15598 : ℚ) / 15625) ^ k := by
+    exact normalized_bad_assignment_ratio (pair i) (Good i)
+      (allBadAssignments (pair i) (Good i)) rfl (Fintype.card (V → C))
+      (by rw [Fintype.card_fun]) hC (hgood i)
+  obtain ⟨ω, hω⟩ := exists_avoiding_of_uniform_ratio_lt events
+    (fun i => allBadAssignments (pair i) (Good i))
+    (((15598 : ℚ) / 15625) ^ k) hΩ (fun i _ => hbound i) hsmall
+  refine ⟨ω, ?_⟩
+  intro i hi
+  exact exists_good_pair (pair i) (Good i) ω (P i ω) (hP i ω) (hω i hi)
+
+private theorem exists_rigid_assignment_of_requests
+    (D : Finset (Lattice 3)) (hD : ∀ d, d ∈ D ↔ Requested 64 d)
+    (hcardD : D.card ≤ 129 ^ 3) :
+    ∃ ω : LiftAssignment 64, ∀ d : Lattice 3, Requested 64 d →
+      ∃ u v : Residue3 64, u ≠ 0 ∧ v ≠ 0 ∧
+        representative 64 ω v - representative 64 ω u = d := by
+  classical
+  have hcard : (Finset.univ : Finset ↥D).card ≤ 129 ^ 3 := by
+    rw [Finset.card_univ, Fintype.card_coe]
+    exact hcardD
+  have hsmall : ((Finset.univ : Finset ↥D).card : ℚ) *
+      ((15598 : ℚ) / 15625) ^ 32000 < 1 := by
+    apply lt_of_le_of_lt _ residue_union_bound_32000
+    apply mul_le_mul_of_nonneg_right
+    · exact_mod_cast hcard
+    · exact pow_nonneg (by norm_num) _
+  have himp (e : ↥D) (ω : LiftAssignment 64) (j : Fin 32000) :
+      (ω (residuePairs hD e (j, 0)), ω (residuePairs hD e (j, 1))) ∈
+        residuePairGood hD e j →
+      representative 64 ω (residuePairs hD e (j, 1)) -
+        representative 64 ω (residuePairs hD e (j, 0)) = e.1 := by
+    dsimp only [residuePairGood]
+    exact representative_difference_of_pairGood (m := 64) ω e.1
+      (residuePairs hD e (j, 0)) (residuePairs hD e (j, 1))
+  obtain ⟨ω, hω⟩ := exists_simultaneous_good_pairs
+    (Finset.univ : Finset ↥D) (residuePairs hD) (residuePairGood hD)
+    card_liftAlphabet (residuePairGood_card hD)
+    (fun e ω j => representative 64 ω (residuePairs hD e (j, 1)) -
+      representative 64 ω (residuePairs hD e (j, 0)) = e.1) himp hsmall
+  refine ⟨ω, ?_⟩
+  intro d hd
+  let e : ↥D := ⟨d, (hD d).mpr hd⟩
+  obtain ⟨j, hj⟩ := hω e (Finset.mem_univ _)
+  exact ⟨residuePairs hD e (j, 0), residuePairs hD e (j, 1),
+    residuePairs_ne_zero hD e (j, 0), residuePairs_ne_zero hD e (j, 1), hj⟩
+
+theorem exists_rigid_assignment64 :
+    ∃ ω : LiftAssignment 64, ∀ d : Lattice 3, Requested 64 d →
+      ∃ u v : Residue3 64, u ≠ 0 ∧ v ≠ 0 ∧
+        representative 64 ω v - representative 64 ω u = d := by
+  exact exists_rigid_assignment_of_requests (requestedVectors 64)
+    (fun d => mem_requestedVectors (m := 64) (d := d)) (card_requestedVectors_le 64)
+
+noncomputable def rigidAssignment64 : LiftAssignment 64 :=
+  Classical.choose exists_rigid_assignment64
+
+theorem rigidAssignment64_difference (d : Lattice 3) (hd : Requested 64 d) :
+    ∃ u v : Residue3 64, u ≠ 0 ∧ v ≠ 0 ∧
+      representative 64 rigidAssignment64 v - representative 64 rigidAssignment64 u = d :=
+  Classical.choose_spec exists_rigid_assignment64 d hd
+
+noncomputable def rigidSection : MarkedTile.ResidueSection 64 where
+  t := representative 64 rigidAssignment64
+  residue_t v := representative_residue 64 rigidAssignment64 v
+
+theorem rigidSection_difference (d : Lattice 3) (hd : Requested 64 d) :
+    ∃ u v : MarkedTile.Residue 64, u ≠ 0 ∧ v ≠ 0 ∧ rigidSection.t v - rigidSection.t u = d :=
+  rigidAssignment64_difference d hd
+
+private theorem section_difference_to_Tstar {m : ℕ} [NeZero m]
+    (s : MarkedTile.ResidueSection m) (d : Lattice 3)
+    (h : ∃ u v : MarkedTile.Residue m,
+      u ≠ 0 ∧ v ≠ 0 ∧ s.t v - s.t u = d) :
+    ∃ x ∈ s.Tstar, ∃ y ∈ s.Tstar, x - y = d := by
+  obtain ⟨u, v, hu, hv, hdiff⟩ := h
+  have hvT : s.t v ∈ s.Tstar :=
+    (s.mem_Tstar (x := s.t v)).mpr ⟨v, hv, rfl⟩
+  have huT : s.t u ∈ s.Tstar :=
+    (s.mem_Tstar (x := s.t u)).mpr ⟨u, hu, rfl⟩
+  exact ⟨s.t v, hvT, s.t u, huT, hdiff⟩
+
+theorem rigidSection_Tstar_difference (d : Lattice 3) (hd : Requested 64 d) :
+    ∃ x ∈ rigidSection.Tstar, ∃ y ∈ rigidSection.Tstar, x - y = d := by
+  exact section_difference_to_Tstar rigidSection d (rigidSection_difference d hd)
+
+end PeriodicTilingThree
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/RigidSectionGeometry.lean b/lean/OAI/Geometry/PeriodicTiling/RigidSectionGeometry.lean
new file mode 100644
index 0000000000000000000000000000000000000000..3c08c3bda8c4430fcf6a7107c45b638682d757fe
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/RigidSectionGeometry.lean
@@ -0,0 +1,38 @@
+import OAI.Geometry.PeriodicTiling.ResidueTransversal
+import OAI.Geometry.PeriodicTiling.SeparatedCenters
+import Mathlib.Tactic.NormNum
+
+namespace OAI
+
+namespace PeriodicTilingThree
+
+theorem rigidSection_rigidDifferences : RigidDifferences 64 rigidSection.Tstar := by
+  intro d hbound hdiv
+  apply rigidSection_Tstar_difference d
+  refine ⟨hbound, ?_⟩
+  intro hall
+  obtain ⟨i, hi⟩ := hdiv
+  exact hi (hall i)
+
+theorem rigidSection_Tstar_subset_T0 : rigidSection.Tstar ⊆ rigidSection.T0 :=
+  Finset.erase_subset _ _
+
+theorem rigidSection_Tstar_subset_T1 (w : Lattice 3) :
+    rigidSection.Tstar ⊆ rigidSection.T1 w := by
+  rw [rigidSection.T1_eq_insert w]
+  exact Finset.subset_insert _ _
+
+theorem rigidSection_Tstar_nonempty : rigidSection.Tstar.Nonempty := by
+  apply Finset.card_pos.mp
+  rw [rigidSection.card_Tstar]
+  norm_num
+
+theorem rigidSection_T0_card : rigidSection.T0.card = 64 ^ 3 :=
+  rigidSection.card_T0
+
+theorem rigidSection_T1_card (w : Lattice 3) : (rigidSection.T1 w).card = 64 ^ 3 :=
+  rigidSection.card_T1 w
+
+end PeriodicTilingThree
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/ScalarAverage.lean b/lean/OAI/Geometry/PeriodicTiling/ScalarAverage.lean
new file mode 100644
index 0000000000000000000000000000000000000000..afa7867c53c61f4398fb4bb6e160361a939e0c8b
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/ScalarAverage.lean
@@ -0,0 +1,109 @@
+import Mathlib.Analysis.Complex.Basic
+import Mathlib.Analysis.SpecificLimits.Normed
+import Mathlib.Tactic.FieldSimp
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.Positivity
+
+namespace OAI
+
+universe uE
+
+noncomputable section
+
+namespace PeriodicTilingThree
+
+open Finset Filter
+open scoped Topology
+
+def complexAverage (u : ℕ → ℂ) (N : ℕ) : ℂ :=
+  (N : ℂ)⁻¹ * ∑ n ∈ range N, u n
+
+def vectorAverage {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E]
+    (u : ℕ → E) (N : ℕ) : E :=
+  (N : ℝ)⁻¹ • ∑ n ∈ range N, u n
+
+theorem complexAverage_eq_vectorAverage (u : ℕ → ℂ) (N : ℕ) :
+    complexAverage u N = vectorAverage u N := by
+  simp only [complexAverage, vectorAverage, Complex.real_smul,
+    Complex.ofReal_inv, Complex.ofReal_natCast]
+
+@[simp] theorem complexAverage_zero (u : ℕ → ℂ) : complexAverage u 0 = 0 := by
+  simp [complexAverage]
+
+theorem complexAverage_add (u v : ℕ → ℂ) (N : ℕ) :
+    complexAverage (fun n => u n + v n) N = complexAverage u N + complexAverage v N := by
+  simp only [complexAverage, Finset.sum_add_distrib, mul_add]
+
+theorem complexAverage_sub (u v : ℕ → ℂ) (N : ℕ) :
+    complexAverage (fun n => u n - v n) N = complexAverage u N - complexAverage v N := by
+  simp only [complexAverage, Finset.sum_sub_distrib, mul_sub]
+
+theorem complexAverage_const (c : ℂ) {N : ℕ} (hN : N ≠ 0) :
+    complexAverage (fun _ => c) N = c := by
+  have hNc : (N : ℂ) ≠ 0 := Nat.cast_ne_zero.mpr hN
+  simp only [complexAverage, Finset.sum_const, Finset.card_range, nsmul_eq_mul,
+    ← mul_assoc, inv_mul_cancel₀ hNc, one_mul]
+
+theorem complexAverage_mul_const (u : ℕ → ℂ) (c : ℂ) (N : ℕ) :
+    complexAverage (fun n => u n * c) N = complexAverage u N * c := by
+  simp only [complexAverage, ← Finset.sum_mul, mul_assoc]
+
+theorem complexAverage_const_mul (u : ℕ → ℂ) (c : ℂ) (N : ℕ) :
+    complexAverage (fun n => c * u n) N = c * complexAverage u N := by
+  simp only [complexAverage, ← Finset.mul_sum]
+  ring
+
+theorem norm_complexAverage (u : ℕ → ℂ) (N : ℕ) :
+    ‖complexAverage u N‖ = (N : ℝ)⁻¹ * ‖∑ n ∈ range N, u n‖ := by
+  simp only [complexAverage, norm_mul, norm_inv, Complex.norm_natCast]
+
+theorem norm_complexAverage_le (u : ℕ → ℂ) (C : ℝ) (hC : 0 ≤ C)
+    (hu : ∀ n, ‖u n‖ ≤ C) (N : ℕ) : ‖complexAverage u N‖ ≤ C := by
+  by_cases hN : N = 0
+  · simpa [hN] using hC
+  have hNr : (N : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr hN
+  have hs : ‖∑ n ∈ range N, u n‖ ≤ (N : ℝ) * C := by
+    calc
+      _ ≤ ∑ n ∈ range N, ‖u n‖ := norm_sum_le _ _
+      _ ≤ ∑ _n ∈ range N, C := Finset.sum_le_sum fun n _ => hu n
+      _ = (N : ℝ) * C := by simp
+  rw [norm_complexAverage]
+  calc
+    _ ≤ (N : ℝ)⁻¹ * ((N : ℝ) * C) :=
+      mul_le_mul_of_nonneg_left hs (by positivity)
+    _ = C := by rw [← mul_assoc, inv_mul_cancel₀ hNr, one_mul]
+
+theorem norm_geometric_sum_le (z : ℂ) (hz : ‖z‖ = 1) (hz1 : z ≠ 1) (N : ℕ) :
+    ‖∑ n ∈ range N, z ^ n‖ ≤ 2 / ‖z - 1‖ := by
+  rw [geom_sum_eq hz1, norm_div]
+  apply div_le_div_of_nonneg_right _ (norm_nonneg _)
+  calc
+    ‖z ^ N - 1‖ ≤ ‖z ^ N‖ + ‖(1 : ℂ)‖ := norm_sub_le _ _
+    _ = 2 := by norm_num [norm_pow, hz]
+
+theorem complexAverage_geometric_tendsto (z : ℂ) (hz : ‖z‖ = 1) (hz1 : z ≠ 1) :
+    Tendsto (complexAverage (fun n => z ^ n)) atTop (𝓝 0) := by
+  apply tendsto_zero_iff_norm_tendsto_zero.mpr
+  have hb : Tendsto (fun N : ℕ => (N : ℝ)⁻¹ * (2 / ‖z - 1‖)) atTop (𝓝 0) := by
+    simpa only [zero_mul] using
+      tendsto_inv_atTop_nhds_zero_nat.mul_const (2 / ‖z - 1‖)
+  refine squeeze_zero (fun N => norm_nonneg _) (fun N => ?_) hb
+  rw [norm_complexAverage]
+  exact mul_le_mul_of_nonneg_left (norm_geometric_sum_le z hz hz1 N) (by positivity)
+
+theorem complexAverage_const_mul_geometric_tendsto (c z : ℂ)
+    (hz : ‖z‖ = 1) (hz1 : z ≠ 1) :
+    Tendsto (complexAverage (fun n => c * z ^ n)) atTop (𝓝 0) := by
+  have hfun : complexAverage (fun n => c * z ^ n) =
+      fun N => c * complexAverage (fun n => z ^ n) N := by
+    funext N
+    exact complexAverage_const_mul (fun n => z ^ n) c N
+  rw [hfun]
+  simpa only [mul_zero] using
+    (complexAverage_geometric_tendsto z hz hz1).const_mul c
+
+end PeriodicTilingThree
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/ScalarPolynomialPeriodicity.lean b/lean/OAI/Geometry/PeriodicTiling/ScalarPolynomialPeriodicity.lean
new file mode 100644
index 0000000000000000000000000000000000000000..3fbc8588797636c2ca0a8b53ce8a980ee4d6ca17
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/ScalarPolynomialPeriodicity.lean
@@ -0,0 +1,97 @@
+import OAI.Geometry.PeriodicTiling.FiniteDifferencePolynomial
+import OAI.Geometry.PeriodicTiling.PolynomialWeyl
+import OAI.Geometry.PeriodicTiling.RationalPolynomialPeriod
+
+namespace OAI
+
+noncomputable section
+
+namespace PeriodicTilingThree
+
+open Set Polynomial
+open scoped Topology
+
+theorem denseRange_circlePolynomial_of_irrational_coefficient (P : Polynomial ℝ)
+    (hP : ∃ i : ℕ, 0 < i ∧ Irrational (P.coeff i)) :
+    DenseRange (fun n : ℤ => ((P.eval (n : ℝ) : ℝ) : UnitAddCircle)) := by
+  obtain ⟨a, ha, Q, hQ, hI, hvalues⟩ := exists_progression_irrational_leadingCoeff P hP
+  have hd := denseRange_circlePolynomial_of_irrational_leadingCoeff Q hQ hI
+  apply Dense.mono (s₁ := Set.range (fun n : ℕ => ((Q.eval (n : ℝ) : ℝ) : UnitAddCircle))) _ hd
+  rintro _ ⟨n, rfl⟩
+  refine ⟨((a * n : ℕ) : ℤ), ?_⟩
+  simpa only [Int.cast_natCast] using hvalues n
+
+theorem circle_polynomial_periodic_of_not_dense (f : ℤ → UnitAddCircle)
+    (r : ℕ) (hpoly : (forwardDiff^[r]) f = 0) (hnd : ¬ DenseRange f) :
+    ∃ q : ℕ, 0 < q ∧ Function.Periodic f (q : ℤ) := by
+  obtain ⟨P, hP⟩ := exists_polynomial_lift_of_iter_forwardDiff_eq_zero f r hpoly
+  have hrat : ∀ i : ℕ, 0 < i → ∃ q : ℚ, P.coeff i = (q : ℝ) := by
+    intro i hi
+    by_contra h
+    have hirr : Irrational (P.coeff i) := by
+      rintro ⟨q, hq⟩
+      exact h ⟨q, hq.symm⟩
+    apply hnd
+    have hd := denseRange_circlePolynomial_of_irrational_coefficient P ⟨i, hi, hirr⟩
+    have heq : (fun n : ℤ => ((P.eval (n : ℝ) : ℝ) : UnitAddCircle)) = f := funext hP
+    rwa [heq] at hd
+  obtain ⟨q, hq, hper⟩ := rational_nonconstant_circlePolynomial_periodic P hrat
+  refine ⟨q, hq, ?_⟩
+  intro n
+  rw [← hP (n + q), ← hP n]
+  exact hper n
+
+theorem circle_polynomial_periodic_of_omits_open (f : ℤ → UnitAddCircle)
+    (r : ℕ) (hpoly : (forwardDiff^[r]) f = 0)
+    (U : Set UnitAddCircle) (hU : IsOpen U) (hne : U.Nonempty)
+    (hmiss : ∀ n, f n ∉ U) :
+    ∃ q : ℕ, 0 < q ∧ Function.Periodic f (q : ℤ) := by
+  apply circle_polynomial_periodic_of_not_dense f r hpoly
+  intro hd
+  obtain ⟨n, hn⟩ := hd.exists_mem_open hU hne
+  exact hmiss n hn
+
+private theorem not_denseRange_circle_of_bounded_lt_one (f : ℤ → ℝ) (c : ℝ)
+    (h0 : ∀ n, 0 ≤ f n) (hc : ∀ n, f n ≤ c) (hc1 : c < 1) :
+    ¬ DenseRange (fun n => (f n : UnitAddCircle)) := by
+  intro hd
+  have hc0 : 0 ≤ c := (h0 0).trans (hc 0)
+  let K : Set UnitAddCircle := ((↑) : ℝ → UnitAddCircle) '' Set.Icc 0 c
+  have hK : IsClosed K :=
+    (isCompact_Icc.image (AddCircle.continuous_mk' (1 : ℝ))).isClosed
+  have hsub : Set.range (fun n => (f n : UnitAddCircle)) ⊆ K := by
+    rintro _ ⟨n, rfl⟩
+    exact ⟨f n, ⟨h0 n, hc n⟩, rfl⟩
+  have hm : (((c + 1) / 2 : ℝ) : UnitAddCircle) ∈ K :=
+    (closure_minimal hsub hK) (hd _)
+  obtain ⟨y, hy, heq⟩ := hm
+  have hyI : y ∈ Set.Ico (0 : ℝ) (0 + 1) := ⟨hy.1, by linarith [hy.2]⟩
+  have hmI : (c + 1) / 2 ∈ Set.Ico (0 : ℝ) (0 + 1) := by
+    constructor <;> linarith
+  have hyEq : y = (c + 1) / 2 :=
+    (AddCircle.coe_eq_coe_iff_of_mem_Ico hyI hmI).mp heq
+  rw [hyEq] at hy
+  linarith [hy.2]
+
+theorem real_polynomial_periodic_of_bounded_lt_one
+    (r : ℕ) (_hr : 0 < r) (f : ℤ → ℝ) (c : ℝ)
+    (h0 : ∀ n, 0 ≤ f n) (hc : ∀ n, f n ≤ c) (hc1 : c < 1)
+    (hpoly : (forwardDiff^[r]) (fun n => (f n : AddCircle (1 : ℝ))) = 0) :
+    ∃ q : ℕ, 0 < q ∧ Function.Periodic f (q : ℤ) := by
+  obtain ⟨q, hq, hper⟩ := circle_polynomial_periodic_of_not_dense
+    (fun n => (f n : UnitAddCircle)) r hpoly
+    (not_denseRange_circle_of_bounded_lt_one f c h0 hc hc1)
+  refine ⟨q, hq, ?_⟩
+  intro n
+  apply (AddCircle.coe_eq_coe_iff_of_mem_Ico
+    (show f (n + q) ∈ Set.Ico (0 : ℝ) (0 + 1) from
+      ⟨h0 _, by simpa using (hc _).trans_lt hc1⟩)
+    (show f n ∈ Set.Ico (0 : ℝ) (0 + 1) from
+      ⟨h0 _, by simpa using (hc _).trans_lt hc1⟩)).mp
+  exact hper n
+
+end PeriodicTilingThree
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/ScalarVanDerCorput.lean b/lean/OAI/Geometry/PeriodicTiling/ScalarVanDerCorput.lean
new file mode 100644
index 0000000000000000000000000000000000000000..f39f596eb4da72b6134becc95b5683f0b233e5eb
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/ScalarVanDerCorput.lean
@@ -0,0 +1,325 @@
+import OAI.Geometry.PeriodicTiling.ScalarAverage
+import Mathlib.Analysis.InnerProductSpace.Basic
+import Mathlib.Algebra.Order.Chebyshev
+import Mathlib.Tactic.Abel
+import Mathlib.Tactic.Module
+
+namespace OAI
+
+noncomputable section
+
+universe uE uInner
+
+namespace PeriodicTilingThree
+
+open Finset Filter
+open scoped Topology InnerProductSpace ComplexConjugate
+
+section Normed
+
+variable {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E]
+
+theorem norm_vectorAverage (u : ℕ → E) (N : ℕ) :
+    ‖vectorAverage u N‖ = (N : ℝ)⁻¹ * ‖∑ n ∈ range N, u n‖ := by
+  rw [vectorAverage, norm_smul, Real.norm_eq_abs,
+    abs_of_nonneg (inv_nonneg.mpr (Nat.cast_nonneg N))]
+
+theorem sum_shift_sub_sum {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E]
+    (u : ℕ → E) (N h : ℕ) :
+    (∑ n ∈ range N, u (n + h)) - (∑ n ∈ range N, u n) =
+      (∑ n ∈ range h, u (n + N)) - (∑ n ∈ range h, u n) := by
+  have h₁ : (∑ n ∈ range (N + h), u n) =
+      (∑ n ∈ range N, u n) + (∑ n ∈ range h, u (n + N)) := by
+    simpa only [Nat.add_comm] using sum_range_add u N h
+  have h₂ : (∑ n ∈ range (N + h), u n) =
+      (∑ n ∈ range h, u n) + (∑ n ∈ range N, u (n + h)) := by
+    simpa only [Nat.add_comm] using sum_range_add u h N
+  apply sub_eq_sub_iff_add_eq_add.mpr
+  simpa only [add_comm] using h₂.symm.trans h₁
+
+theorem norm_sum_range_le_length
+    {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E]
+    (u : ℕ → E) (hu : ∀ n, ‖u n‖ ≤ 1) (N : ℕ) :
+    ‖∑ n ∈ range N, u n‖ ≤ (N : ℝ) := by
+  calc
+    _ ≤ ∑ n ∈ range N, ‖u n‖ := norm_sum_le _ _
+    _ ≤ ∑ _n ∈ range N, (1 : ℝ) := sum_le_sum fun n _ => hu n
+    _ = (N : ℝ) := by simp
+
+theorem norm_shift_sum_sub_le (u : ℕ → E) (hu : ∀ n, ‖u n‖ ≤ 1) (N h : ℕ) :
+    ‖(∑ n ∈ range N, u (n + h)) - (∑ n ∈ range N, u n)‖ ≤ 2 * (h : ℝ) := by
+  rw [sum_shift_sub_sum]
+  calc
+    _ ≤ ‖∑ n ∈ range h, u (n + N)‖ + ‖∑ n ∈ range h, u n‖ := norm_sub_le _ _
+    _ ≤ (h : ℝ) + h := add_le_add
+      (norm_sum_range_le_length (fun n => u (n + N)) (fun n => hu (n + N)) h)
+      (norm_sum_range_le_length u hu h)
+    _ = 2 * (h : ℝ) := by ring
+
+theorem norm_shift_average_sub_le (u : ℕ → E) (hu : ∀ n, ‖u n‖ ≤ 1) (N h : ℕ) :
+    ‖vectorAverage (fun n => u (n + h)) N - vectorAverage u N‖ ≤
+      (N : ℝ)⁻¹ * (2 * (h : ℝ)) := by
+  simp only [vectorAverage, ← smul_sub]
+  rw [norm_smul, Real.norm_eq_abs,
+    abs_of_nonneg (inv_nonneg.mpr (Nat.cast_nonneg N))]
+  exact mul_le_mul_of_nonneg_left (norm_shift_sum_sub_le u hu N h) (by positivity)
+
+def smoothedAverage (u : ℕ → E) (H N : ℕ) : E :=
+  (H : ℝ)⁻¹ • vectorAverage (fun n => ∑ h ∈ range H, u (n + h)) N
+
+theorem smoothedAverage_eq_sum (u : ℕ → E) (H N : ℕ) :
+    smoothedAverage u H N =
+      (H : ℝ)⁻¹ • ∑ h ∈ range H, vectorAverage (fun n => u (n + h)) N := by
+  simp only [smoothedAverage, vectorAverage, smul_sum]
+  rw [sum_comm]
+
+theorem smoothedAverage_sub (u : ℕ → E) {H : ℕ} (hH : 0 < H) (N : ℕ) :
+    smoothedAverage u H N - vectorAverage u N =
+      (H : ℝ)⁻¹ • ∑ h ∈ range H,
+        (vectorAverage (fun n => u (n + h)) N - vectorAverage u N) := by
+  have hHr : (H : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr hH.ne'
+  rw [sum_sub_distrib, smul_sub, ← smoothedAverage_eq_sum]
+  congr 1
+  simp only [sum_const, card_range, ← Nat.cast_smul_eq_nsmul ℝ,
+    smul_smul, inv_mul_cancel₀ hHr, one_smul]
+
+theorem norm_smoothedAverage_sub_le (u : ℕ → E) (hu : ∀ n, ‖u n‖ ≤ 1)
+    {H : ℕ} (hH : 0 < H) (N : ℕ) :
+    ‖smoothedAverage u H N - vectorAverage u N‖ ≤
+      (N : ℝ)⁻¹ * (2 * (H : ℝ)) := by
+  have hHr : (H : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr hH.ne'
+  rw [smoothedAverage_sub u hH N, norm_smul, Real.norm_eq_abs,
+    abs_of_nonneg (inv_nonneg.mpr (Nat.cast_nonneg H))]
+  have hsum :
+      ‖∑ h ∈ range H, (vectorAverage (fun n => u (n + h)) N - vectorAverage u N)‖ ≤
+        (H : ℝ) * ((N : ℝ)⁻¹ * (2 * (H : ℝ))) := by
+    calc
+      _ ≤ ∑ h ∈ range H,
+          ‖vectorAverage (fun n => u (n + h)) N - vectorAverage u N‖ := norm_sum_le _ _
+      _ ≤ ∑ _h ∈ range H, ((N : ℝ)⁻¹ * (2 * (H : ℝ))) := by
+        apply sum_le_sum
+        intro h hh
+        refine (norm_shift_average_sub_le u hu N h).trans ?_
+        apply mul_le_mul_of_nonneg_left _ (by positivity)
+        have hh' : (h : ℝ) ≤ H := by exact_mod_cast (mem_range.mp hh).le
+        linarith
+      _ = (H : ℝ) * ((N : ℝ)⁻¹ * (2 * (H : ℝ))) := by simp
+  calc
+    _ ≤ (H : ℝ)⁻¹ * ((H : ℝ) * ((N : ℝ)⁻¹ * (2 * (H : ℝ)))) :=
+      mul_le_mul_of_nonneg_left hsum (by positivity)
+    _ = (N : ℝ)⁻¹ * (2 * (H : ℝ)) := by
+      rw [← mul_assoc, inv_mul_cancel₀ hHr, one_mul]
+
+theorem norm_vectorAverage_sq_le (u : ℕ → E) (N : ℕ) :
+    ‖vectorAverage u N‖ ^ 2 ≤ (N : ℝ)⁻¹ * ∑ n ∈ range N, ‖u n‖ ^ 2 := by
+  by_cases hN : N = 0
+  · simp [hN, vectorAverage]
+  have hNr : (N : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr hN
+  have hsum : ‖∑ n ∈ range N, u n‖ ^ 2 ≤ (∑ n ∈ range N, ‖u n‖) ^ 2 := by
+    have hn : 0 ≤ ∑ n ∈ range N, ‖u n‖ := sum_nonneg fun _ _ => norm_nonneg _
+    exact (sq_le_sq₀ (norm_nonneg _) hn).mpr (norm_sum_le (range N) u)
+  have hc : (∑ n ∈ range N, ‖u n‖) ^ 2 ≤
+      (N : ℝ) * ∑ n ∈ range N, ‖u n‖ ^ 2 := by
+    simpa only [card_range] using
+      (sq_sum_le_card_mul_sum_sq (s := range N) (f := fun n => ‖u n‖))
+  rw [norm_vectorAverage, mul_pow]
+  calc
+    _ ≤ ((N : ℝ)⁻¹) ^ 2 * ((N : ℝ) * ∑ n ∈ range N, ‖u n‖ ^ 2) :=
+      mul_le_mul_of_nonneg_left (hsum.trans hc) (sq_nonneg _)
+    _ = (N : ℝ)⁻¹ * ∑ n ∈ range N, ‖u n‖ ^ 2 := by
+      field_simp [hNr]
+
+end Normed
+
+section InnerProduct
+
+variable {E : Type uInner} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+
+def offDiagonalAverage (u : ℕ → E) (H N : ℕ) : ℝ :=
+  ∑ h ∈ range H, ∑ k ∈ range H,
+    if h = k then 0 else vectorAverage (fun n => ⟪u (n + h), u (n + k)⟫_ℝ) N
+
+theorem block_energy_eq (u : ℕ → E) (H N : ℕ) :
+    (N : ℝ)⁻¹ * ∑ n ∈ range N, ‖∑ h ∈ range H, u (n + h)‖ ^ 2 =
+      ∑ h ∈ range H, ∑ k ∈ range H,
+        vectorAverage (fun n => ⟪u (n + h), u (n + k)⟫_ℝ) N := by
+  simp only [← real_inner_self_eq_norm_sq, sum_inner, inner_sum,
+    vectorAverage, smul_eq_mul, mul_sum]
+  rw [sum_comm]
+  apply sum_congr rfl
+  intro h _hh
+  rw [sum_comm]
+  apply sum_congr rfl
+  intro k _hk
+  apply sum_congr rfl
+  intro n _hn
+  rw [real_inner_comm]
+
+theorem diagonal_average_le_one (u : ℕ → E) (hu : ∀ n, ‖u n‖ ≤ 1) (h N : ℕ) :
+    vectorAverage (fun n => ⟪u (n + h), u (n + h)⟫_ℝ) N ≤ 1 := by
+  by_cases hN : N = 0
+  · simp [hN, vectorAverage]
+  have hNr : (N : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr hN
+  have hb : ∀ n, ⟪u (n + h), u (n + h)⟫_ℝ ≤ 1 := by
+    intro n
+    rw [real_inner_self_eq_norm_sq]
+    simpa only [one_pow] using
+      (sq_le_sq₀ (norm_nonneg (u (n + h))) zero_le_one).mpr (hu (n + h))
+  simp only [vectorAverage, smul_eq_mul]
+  calc
+    _ ≤ (N : ℝ)⁻¹ * (∑ _n ∈ range N, (1 : ℝ)) :=
+      mul_le_mul_of_nonneg_left (sum_le_sum fun n _ => hb n) (by positivity)
+    _ = 1 := by simp [inv_mul_cancel₀ hNr]
+
+theorem block_energy_le (u : ℕ → E) (hu : ∀ n, ‖u n‖ ≤ 1) (H N : ℕ) :
+    (N : ℝ)⁻¹ * ∑ n ∈ range N, ‖∑ h ∈ range H, u (n + h)‖ ^ 2 ≤
+      (H : ℝ) + offDiagonalAverage u H N := by
+  rw [block_energy_eq]
+  calc
+    _ ≤ ∑ h ∈ range H, (1 + ∑ k ∈ range H,
+        if h = k then 0 else vectorAverage (fun n => ⟪u (n + h), u (n + k)⟫_ℝ) N) := by
+      apply sum_le_sum
+      intro h hh
+      calc
+        _ ≤ ∑ k ∈ range H,
+            ((if h = k then (1 : ℝ) else 0) +
+              (if h = k then 0 else vectorAverage (fun n => ⟪u (n + h), u (n + k)⟫_ℝ) N)) := by
+          apply sum_le_sum
+          intro k _hk
+          by_cases he : h = k
+          · subst k
+            simpa using diagonal_average_le_one u hu h N
+          · simp [he]
+        _ = _ := by simp [sum_add_distrib, hh]
+    _ = (H : ℝ) + offDiagonalAverage u H N := by
+      simp [sum_add_distrib, offDiagonalAverage]
+
+theorem norm_smoothedAverage_sq_le (u : ℕ → E) (hu : ∀ n, ‖u n‖ ≤ 1)
+    {H : ℕ} (hH : 0 < H) (N : ℕ) :
+    ‖smoothedAverage u H N‖ ^ 2 ≤
+      (H : ℝ)⁻¹ + ((H : ℝ)⁻¹) ^ 2 * offDiagonalAverage u H N := by
+  have hHr : (H : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr hH.ne'
+  have hb := (norm_vectorAverage_sq_le (fun n => ∑ h ∈ range H, u (n + h)) N).trans
+    (block_energy_le u hu H N)
+  rw [smoothedAverage, norm_smul, Real.norm_eq_abs,
+    abs_of_nonneg (inv_nonneg.mpr (Nat.cast_nonneg H)), mul_pow]
+  calc
+    _ ≤ ((H : ℝ)⁻¹) ^ 2 * ((H : ℝ) + offDiagonalAverage u H N) :=
+      mul_le_mul_of_nonneg_left hb (sq_nonneg _)
+    _ = (H : ℝ)⁻¹ + ((H : ℝ)⁻¹) ^ 2 * offDiagonalAverage u H N := by
+      field_simp [hHr]
+
+theorem finite_vanDerCorput (u : ℕ → E) (hu : ∀ n, ‖u n‖ ≤ 1)
+    {H : ℕ} (hH : 0 < H) (N : ℕ) :
+    ‖vectorAverage u N‖ ^ 2 ≤
+      2 * ((H : ℝ)⁻¹ + ((H : ℝ)⁻¹) ^ 2 * offDiagonalAverage u H N) +
+        2 * ((N : ℝ)⁻¹ * (2 * (H : ℝ))) ^ 2 := by
+  have hs := norm_smoothedAverage_sq_le u hu hH N
+  have he := norm_smoothedAverage_sub_le u hu hH N
+  have ht : ‖vectorAverage u N‖ ≤
+      ‖smoothedAverage u H N‖ + ‖smoothedAverage u H N - vectorAverage u N‖ := by
+    have ht' := norm_sub_le (smoothedAverage u H N)
+      (smoothedAverage u H N - vectorAverage u N)
+    simpa only [sub_sub_cancel] using ht'
+  have he0 : 0 ≤ (N : ℝ)⁻¹ * (2 * (H : ℝ)) := by positivity
+  have hn := norm_nonneg (smoothedAverage u H N)
+  have hn' := norm_nonneg (smoothedAverage u H N - vectorAverage u N)
+  have hn'' := norm_nonneg (vectorAverage u N)
+  have hsq : ‖vectorAverage u N‖ ^ 2 ≤
+      2 * ‖smoothedAverage u H N‖ ^ 2 +
+        2 * ‖smoothedAverage u H N - vectorAverage u N‖ ^ 2 := by
+    have ht₂ := (sq_le_sq₀ hn'' (add_nonneg hn hn')).mpr ht
+    nlinarith [sq_nonneg (‖smoothedAverage u H N‖ -
+      ‖smoothedAverage u H N - vectorAverage u N‖)]
+  have he' : ‖smoothedAverage u H N - vectorAverage u N‖ ^ 2 ≤
+      ((N : ℝ)⁻¹ * (2 * (H : ℝ))) ^ 2 := (sq_le_sq₀ hn' he0).mpr he
+  linarith
+
+theorem offDiagonalAverage_tendsto_zero (u : ℕ → E)
+    (hcor : ∀ h k : ℕ, h ≠ k →
+      Tendsto (vectorAverage (fun n => ⟪u (n + h), u (n + k)⟫_ℝ)) atTop (𝓝 0))
+    (H : ℕ) : Tendsto (offDiagonalAverage u H) atTop (𝓝 0) := by
+  have hk (h k : ℕ) : Tendsto
+      (fun N => if h = k then (0 : ℝ) else
+        vectorAverage (fun n => ⟪u (n + h), u (n + k)⟫_ℝ) N) atTop (𝓝 0) := by
+    by_cases he : h = k
+    · simp only [he, ite_true]
+      exact tendsto_const_nhds
+    · simpa only [he, ite_false] using hcor h k he
+  have hh (h : ℕ) : Tendsto
+      (fun N => ∑ k ∈ range H, if h = k then (0 : ℝ) else
+        vectorAverage (fun n => ⟪u (n + h), u (n + k)⟫_ℝ) N) atTop (𝓝 0) := by
+    simpa using tendsto_finsetSum (range H) (fun k _ => hk h k)
+  unfold offDiagonalAverage
+  simpa only [sum_const_zero] using
+    tendsto_finsetSum (range H) (fun h _ => hh h)
+
+theorem vectorAverage_tendsto_zero_of_shift_inner (u : ℕ → E)
+    (hu : ∀ n, ‖u n‖ ≤ 1)
+    (hcor : ∀ h k : ℕ, h ≠ k →
+      Tendsto (vectorAverage (fun n => ⟪u (n + h), u (n + k)⟫_ℝ)) atTop (𝓝 0)) :
+    Tendsto (vectorAverage u) atTop (𝓝 0) := by
+  have hsq : Tendsto (fun N => ‖vectorAverage u N‖ ^ 2) atTop (𝓝 (0 : ℝ)) := by
+    apply tendsto_order.mpr
+    constructor
+    · intro a ha
+      exact Filter.Eventually.of_forall fun N => lt_of_lt_of_le ha (sq_nonneg _)
+    · intro b hb
+      obtain ⟨H, hH⟩ := exists_nat_gt (2 / b)
+      have hHr : (0 : ℝ) < H := lt_trans (by positivity : (0 : ℝ) < 2 / b) hH
+      have hHn : 0 < H := Nat.cast_pos.mp hHr
+      have hlimit : 2 * (H : ℝ)⁻¹ < b := by
+        have hx : 2 < (H : ℝ) * b := (div_lt_iff₀ hb).mp hH
+        have hy : 2 / (H : ℝ) < b := (div_lt_iff₀ hHr).mpr (by nlinarith)
+        simpa only [div_eq_mul_inv] using hy
+      have hoff := offDiagonalAverage_tendsto_zero u hcor H
+      have hfirst : Tendsto
+          (fun N => (H : ℝ)⁻¹ + ((H : ℝ)⁻¹) ^ 2 * offDiagonalAverage u H N)
+          atTop (𝓝 ((H : ℝ)⁻¹)) := by
+        simpa only [mul_zero, add_zero] using
+          tendsto_const_nhds.add (hoff.const_mul (((H : ℝ)⁻¹) ^ 2))
+      have hsecond : Tendsto (fun N : ℕ => ((N : ℝ)⁻¹ * (2 * (H : ℝ))) ^ 2)
+          atTop (𝓝 (0 : ℝ)) := by
+        simpa only [zero_mul, zero_pow (by decide : (2 : ℕ) ≠ 0)] using
+          (tendsto_inv_atTop_nhds_zero_nat.mul_const (2 * (H : ℝ))).pow 2
+      have hg : Tendsto
+          (fun N => 2 * ((H : ℝ)⁻¹ + ((H : ℝ)⁻¹) ^ 2 * offDiagonalAverage u H N) +
+            2 * ((N : ℝ)⁻¹ * (2 * (H : ℝ))) ^ 2)
+          atTop (𝓝 (2 * (H : ℝ)⁻¹)) := by
+        simpa only [mul_zero, add_zero] using (hfirst.const_mul 2).add (hsecond.const_mul 2)
+      filter_upwards [(tendsto_order.mp hg).2 b hlimit] with N hN
+      exact lt_of_le_of_lt (finite_vanDerCorput u hu hHn N) hN
+  apply tendsto_zero_iff_norm_tendsto_zero.mpr
+  simpa only [Real.sqrt_sq_eq_abs, abs_norm, Real.sqrt_zero] using hsq.sqrt
+
+end InnerProduct
+
+theorem vectorAverage_complex_inner_eq_re (u : ℕ → ℂ) (h k N : ℕ) :
+    vectorAverage (fun n => ⟪u (n + h), u (n + k)⟫_ℝ) N =
+      (complexAverage (fun n => u (n + k) * conj (u (n + h))) N).re := by
+  rw [complexAverage_eq_vectorAverage]
+  simp only [vectorAverage, Complex.inner, Complex.real_smul, Complex.mul_re,
+    Complex.ofReal_re, Complex.ofReal_im, zero_mul, sub_zero, Complex.re_sum, smul_eq_mul]
+
+theorem complexAverage_vanDerCorput (u : ℕ → ℂ)
+    (hu : ∀ n, ‖u n‖ ≤ 1)
+    (hcor : ∀ h k : ℕ, h ≠ k →
+      Tendsto (complexAverage (fun n => u (n + h) * conj (u (n + k)))) atTop (𝓝 0)) :
+    Tendsto (complexAverage u) atTop (𝓝 0) := by
+  have hi : ∀ h k : ℕ, h ≠ k →
+      Tendsto (vectorAverage (fun n => ⟪u (n + h), u (n + k)⟫_ℝ)) atTop (𝓝 0) := by
+    intro h k hne
+    have hr := (Complex.continuous_re.tendsto 0).comp (hcor k h hne.symm)
+    simp only [Complex.zero_re] at hr
+    convert hr using 1
+    · funext N
+      exact vectorAverage_complex_inner_eq_re u h k N
+  convert vectorAverage_tendsto_zero_of_shift_inner u hu hi using 1
+  funext N
+  exact complexAverage_eq_vectorAverage u N
+
+end PeriodicTilingThree
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/SeedSymbols.lean b/lean/OAI/Geometry/PeriodicTiling/SeedSymbols.lean
new file mode 100644
index 0000000000000000000000000000000000000000..a79ba8148f62d81e12949def81074bbc25b423b8
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/SeedSymbols.lean
@@ -0,0 +1,64 @@
+import OAI.Geometry.PeriodicTiling.EncodingParameters
+import OAI.Geometry.PeriodicTiling.LastDigit
+
+namespace OAI
+
+noncomputable section
+
+namespace PeriodicTilingThree
+
+def seedSymbol {p : ℕ} [NeZero p] (E : EncodingParameters p) (t : Fin 2) : Symbol p :=
+  lastDigit p E.p_prime ((t.val : ℤ) + 1)
+
+variable {p : ℕ} [NeZero p] (E : EncodingParameters p)
+
+@[simp] theorem seedSymbol_coe (t : Fin 2) :
+    (seedSymbol E t : ZMod p) = (t.val : ZMod p) + 1 := by
+  have hlt : t.val + 1 < p := by
+    have ht := t.isLt
+    have hp := E.p_large
+    omega
+  have hnot : ¬ (p : ℤ) ∣ (t.val : ℤ) + 1 := by
+    intro h
+    apply Nat.not_dvd_of_pos_of_lt (Nat.succ_pos t.val) hlt
+    apply Int.natCast_dvd_natCast.mp
+    simpa only [Nat.cast_succ] using h
+  simpa only [seedSymbol, Int.cast_add, Int.cast_natCast, Int.cast_one] using
+    lastDigit_eq_of_not_dvd p E.p_prime hnot
+
+theorem seedSymbol_injective : Function.Injective (seedSymbol E) := by
+  intro s t h
+  apply Fin.ext
+  have hcoe : (seedSymbol E s : ZMod p) = (seedSymbol E t : ZMod p) :=
+    congrArg (fun u : Symbol p => (u : ZMod p)) h
+  rw [seedSymbol_coe, seedSymbol_coe] at hcoe
+  have hcast : (s.val : ZMod p) = (t.val : ZMod p) := add_right_cancel hcoe
+  have hmod := (ZMod.natCast_eq_natCast_iff' s.val t.val p).mp hcast
+  have hp : 2 < p := by have := E.p_large; omega
+  rw [Nat.mod_eq_of_lt (lt_trans s.isLt hp),
+    Nat.mod_eq_of_lt (lt_trans t.isLt hp)] at hmod
+  exact hmod
+
+@[simp] theorem seedSymbol_eq_iff {s t : Fin 2} :
+    seedSymbol E s = seedSymbol E t ↔ s = t :=
+  ⟨fun h => seedSymbol_injective E h, fun h => congrArg (seedSymbol E) h⟩
+
+theorem seedSymbol_ne {s t : Fin 2} (h : s ≠ t) :
+    seedSymbol E s ≠ seedSymbol E t :=
+  fun hst => h (seedSymbol_injective E hst)
+
+@[simp] theorem seedSymbol_zero : seedSymbol E 0 = 1 := by
+  simp [seedSymbol]
+
+@[simp] theorem seedSymbol_one_coe : (seedSymbol E 1 : ZMod p) = 2 := by
+  change (lastDigit p E.p_prime 2 : ZMod p) = 2
+  exact lastDigit_two p E.p_prime (by have := E.p_large; omega)
+
+theorem seedSymbol_zero_ne_one : seedSymbol E 0 ≠ seedSymbol E 1 :=
+  seedSymbol_ne E (by decide)
+
+end PeriodicTilingThree
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/SeparatedCenters.lean b/lean/OAI/Geometry/PeriodicTiling/SeparatedCenters.lean
new file mode 100644
index 0000000000000000000000000000000000000000..f60ed12c053a210e647222a34a50499532d7d538
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/SeparatedCenters.lean
@@ -0,0 +1,124 @@
+import OAI.Geometry.PeriodicTiling.EuclideanBasic
+import OAI.Geometry.PeriodicTiling.ClosedBoxRounding
+import OAI.Geometry.PeriodicTiling.VoxelMeasure
+import Mathlib.MeasureTheory.Measure.AEDisjoint
+import Mathlib.Tactic.Abel
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum
+import Mathlib.Tactic.Ring
+
+namespace OAI
+
+noncomputable section
+
+namespace PeriodicTilingThree
+
+open Set MeasureTheory
+open scoped Classical
+
+def RigidDifferences (m : ℕ) (S : Finset (Lattice 3)) : Prop :=
+  ∀ d : Lattice 3, (∀ i, |d i| ≤ (m : ℤ)) → (∃ i, ¬ (m : ℤ) ∣ d i) →
+    ∃ t ∈ S, ∃ t' ∈ S, t - t' = d
+
+theorem TypedAETiles.aedisjoint {d : ℕ} {Ω0 Ω1 C C1 : Set (Space d)}
+    (h : TypedAETiles Ω0 Ω1 C C1) {c c' : C} (hne : c ≠ c') :
+    AEDisjoint volume
+      {x | x - (c : Space d) ∈ if (c : Space d) ∈ C1 then Ω1 else Ω0}
+      {x | x - (c' : Space d) ∈ if (c' : Space d) ∈ C1 then Ω1 else Ω0} := by
+  apply measure_mono_null ?_ (ae_iff.mp h)
+  rintro x ⟨hx, hx'⟩ ⟨a, _ha, huniq⟩
+  exact hne ((huniq c hx).trans (huniq c' hx').symm)
+
+theorem AETiles.aedisjoint {d : ℕ} {Ω A : Set (Space d)}
+    (h : AETiles Ω A) {a a' : A} (hne : a ≠ a') :
+    AEDisjoint volume {x | x - (a : Space d) ∈ Ω}
+      {x | x - (a' : Space d) ∈ Ω} := by
+  apply measure_mono_null ?_ (ae_iff.mp h)
+  rintro x ⟨hx, hx'⟩ ⟨c, _hc, huniq⟩
+  exact hne ((huniq a hx).trans (huniq a' hx').symm)
+
+theorem common_unitVoxel_subset_piece {d : ℕ}
+    {S0 S1 S : Finset (Lattice d)} (h0 : S ⊆ S0) (h1 : S ⊆ S1)
+    (C1 : Set (Space d)) (c : Space d) {t : Lattice d} (ht : t ∈ S) :
+    unitVoxel (c + castLattice t) ⊆
+      {x | x - c ∈ if c ∈ C1 then Thickening S1 else Thickening S0} := by
+  intro x hx
+  have hcube : (x - c) - castLattice t ∈ unitCube d := by
+    simpa only [mem_unitVoxel, sub_add_eq_sub_sub] using hx
+  by_cases hc : c ∈ C1
+  · simp only [hc, ite_eq_left, mem_ofPred_eq]
+    exact ⟨t, h1 ht, hcube⟩
+  · simp only [hc, mem_ofPred_eq]
+    exact ⟨t, h0 ht, hcube⟩
+
+theorem typed_nearby_centers_mem_scaledGrid
+    {m : ℕ} (hm : 2 ≤ m) {S0 S1 S : Finset (Lattice 3)}
+    (h0 : S ⊆ S0) (h1 : S ⊆ S1) (hdiff : RigidDifferences m S)
+    {C C1 : Set (Space 3)} (htile : TypedAETiles (Thickening S0) (Thickening S1) C C1)
+    {c c' : C} (hne : c ≠ c') (hclose : ‖(c' : Space 3) - (c : Space 3)‖ ≤ (m : ℝ)) :
+    (c' : Space 3) - (c : Space 3) ∈ scaledGrid m := by
+  by_contra hnot
+  have hcoords : ∀ i, |((c' : Space 3) - (c : Space 3)) i| ≤ (m : ℝ) := by
+    intro i
+    have hcoord : |((c' : Space 3) - (c : Space 3)) i| ≤
+        ‖(c' : Space 3) - (c : Space 3)‖ := by
+      simpa only [Real.norm_eq_abs] using
+        norm_le_pi_norm ((c' : Space 3) - (c : Space 3)) i
+    exact hcoord.trans hclose
+  obtain ⟨d, hd, hdiv, hround⟩ :=
+    closed_box_rounding m hm ((c' : Space 3) - (c : Space 3)) hcoords hnot
+  obtain ⟨t, ht, t', ht', htd⟩ := hdiff d hd hdiv
+  have hcorners : ∀ i,
+      |((c' : Space 3) + castLattice t') i - ((c : Space 3) + castLattice t) i| < 1 := by
+    intro i
+    have hid : ((c' : Space 3) + castLattice t') i -
+        ((c : Space 3) + castLattice t) i =
+        (((c' : Space 3) - (c : Space 3)) i - (d i : ℝ)) := by
+      rw [← htd]
+      simp only [Pi.add_apply, Pi.sub_apply, castLattice_apply, Int.cast_sub]
+      ring
+    rw [hid]
+    exact hround i
+  have hpos := unitVoxel_inter_volume_pos
+    ((c : Space 3) + castLattice t) ((c' : Space 3) + castLattice t') hcorners
+  have hnull := (htile.aedisjoint hne).mono
+    (common_unitVoxel_subset_piece h0 h1 C1 c ht)
+    (common_unitVoxel_subset_piece h0 h1 C1 c' ht')
+  exact (ne_of_gt hpos) hnull
+
+theorem scaledGrid_eq_zero_of_norm_lt {m : ℕ} (hm : 0 < m)
+    {x : Space 3} (hx : x ∈ scaledGrid m) (hnorm : ‖x‖ < (m : ℝ)) : x = 0 := by
+  obtain ⟨z, rfl⟩ := mem_scaledGrid.mp hx
+  have hmR : (0 : ℝ) < m := by exact_mod_cast hm
+  have hz : z = 0 := by
+    funext i
+    have hi : |(m : ℝ) * (z i : ℝ)| < (m : ℝ) := by
+      have hle := norm_le_pi_norm ((m : ℝ) • castLattice z) i
+      have hle' : |(m : ℝ) * (z i : ℝ)| ≤ ‖(m : ℝ) • castLattice z‖ := by
+        simpa only [Pi.smul_apply, smul_eq_mul, castLattice_apply,
+          Real.norm_eq_abs] using hle
+      exact hle'.trans_lt hnorm
+    rw [abs_mul, abs_of_nonneg hmR.le] at hi
+    have hzi : |(z i : ℝ)| < 1 := by
+      exact (mul_lt_mul_iff_right₀ hmR).mp (by simpa only [mul_one] using hi)
+    rw [← Int.cast_abs, ← Int.cast_one, Int.cast_lt] at hzi
+    exact Int.abs_lt_one_iff.mp hzi
+  simp [hz]
+
+theorem typed_centers_separated
+    {m : ℕ} (hm : 2 ≤ m) {S0 S1 S : Finset (Lattice 3)}
+    (h0 : S ⊆ S0) (h1 : S ⊆ S1) (hdiff : RigidDifferences m S)
+    {C C1 : Set (Space 3)} (htile : TypedAETiles (Thickening S0) (Thickening S1) C C1)
+    {c c' : C} (hne : c ≠ c') : (m : ℝ) ≤ ‖(c' : Space 3) - (c : Space 3)‖ := by
+  by_contra h
+  have hsmall := lt_of_not_ge h
+  have hgrid := typed_nearby_centers_mem_scaledGrid hm h0 h1 hdiff htile hne hsmall.le
+  have hzero := scaledGrid_eq_zero_of_norm_lt
+    (lt_of_lt_of_le (by decide : 0 < 2) hm) hgrid hsmall
+  exact hne (Subtype.ext (sub_eq_zero.mp hzero).symm)
+
+end PeriodicTilingThree
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/SharedSeedInactive.lean b/lean/OAI/Geometry/PeriodicTiling/SharedSeedInactive.lean
new file mode 100644
index 0000000000000000000000000000000000000000..2ac1ec639b5e61628f08301efdb9ca60939841f9
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/SharedSeedInactive.lean
@@ -0,0 +1,69 @@
+import OAI.Geometry.PeriodicTiling.BlockActivationInverse
+import Mathlib.Tactic.Abel
+
+namespace OAI
+
+universe uK uΔ
+
+noncomputable section
+
+namespace PeriodicTilingThree.SharedSeed
+
+variable {K : Type uK} {Δ : Type uΔ} [AddCommGroup K] {a b : ℕ}
+
+def inactiveForward (S : BlockShiftData Δ a b) (x : ZMod a) (k : K)
+    (v : (K × Δ) × ZMod a) : ((ZMod a × ZMod b) × K) × ZMod a :=
+  ((((v.2 + (S.e v.1.2).1), (S.e v.1.2).2), k - v.1.1),
+    v.2 + x - S.rhoA v.1.2)
+
+def inactiveInverse (S : BlockShiftData Δ a b) (x : ZMod a) (k : K)
+    (v : ((ZMod a × ZMod b) × K) × ZMod a) : (K × Δ) × ZMod a :=
+  let δ := S.orderA.symm (v.1.1.2, v.1.1.1 + x - v.2)
+  ((k - v.1.2, δ), v.1.1.1 - (S.e δ).1)
+
+theorem inactiveInverse_forward (S : BlockShiftData Δ a b) (x : ZMod a) (k : K)
+    (v : (K × Δ) × ZMod a) :
+    inactiveInverse S x k (inactiveForward S x k v) = v := by
+  rcases v with ⟨⟨l, δ⟩, y⟩
+  have hd : S.orderA.symm ((S.e δ).2,
+      (y + (S.e δ).1) + x - (y + x - S.rhoA δ)) = δ := by
+    apply S.orderA.injective
+    rw [Equiv.apply_symm_apply, S.orderA_apply]
+    apply Prod.ext
+    · rfl
+    · abel_nf
+  dsimp [inactiveInverse, inactiveForward]
+  rw [hd]
+  simp
+
+theorem inactiveForward_inverse (S : BlockShiftData Δ a b) (x : ZMod a) (k : K)
+    (v : ((ZMod a × ZMod b) × K) × ZMod a) :
+    inactiveForward S x k (inactiveInverse S x k v) = v := by
+  rcases v with ⟨⟨⟨α, ξ⟩, β⟩, q⟩
+  let δ := S.orderA.symm (ξ, α + x - q)
+  have hd : ((S.e δ).2, (S.e δ).1 + S.rhoA δ) = (ξ, α + x - q) := by
+    rw [← S.orderA_apply]
+    exact S.orderA.apply_symm_apply _
+  have hb : (S.e δ).2 = ξ := congrArg Prod.fst hd
+  have ha : (S.e δ).1 + S.rhoA δ = α + x - q := congrArg Prod.snd hd
+  have hq : α - (S.e δ).1 + x - S.rhoA δ = q := by
+    calc
+      α - (S.e δ).1 + x - S.rhoA δ = α + x - ((S.e δ).1 + S.rhoA δ) := by abel
+      _ = α + x - (α + x - q) := by rw [ha]
+      _ = q := by abel
+  change ((((α - (S.e δ).1 + (S.e δ).1), (S.e δ).2), k - (k - β)),
+    α - (S.e δ).1 + x - S.rhoA δ) = (((α, ξ), β), q)
+  simp only [sub_add_cancel, sub_sub_cancel, hb, hq]
+
+def inactiveEquiv (S : BlockShiftData Δ a b) (x : ZMod a) (k : K) :
+    ((K × Δ) × ZMod a) ≃ (((ZMod a × ZMod b) × K) × ZMod a) where
+  toFun := inactiveForward S x k
+  invFun := inactiveInverse S x k
+  left_inv := inactiveInverse_forward S x k
+  right_inv := inactiveForward_inverse S x k
+
+end PeriodicTilingThree.SharedSeed
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/SharedSeedInverse.lean b/lean/OAI/Geometry/PeriodicTiling/SharedSeedInverse.lean
new file mode 100644
index 0000000000000000000000000000000000000000..a25ce54f9d60396c057aa862a0eec8a2db9e63ed
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/SharedSeedInverse.lean
@@ -0,0 +1,197 @@
+import OAI.Geometry.PeriodicTiling.SharedSeedLabelEquiv
+
+namespace OAI
+
+universe uK uΔ uA uB uR
+
+noncomputable section
+
+namespace PeriodicTilingThree.SharedSeed
+
+variable {K : Type uK} {Δ : Type uΔ} {A : Type uA} {B : Type uB} {R : Type uR}
+  [AddCommGroup K] [AddCommGroup R] {m b n : ℕ}
+
+def sourceResidueEquiv (sx : R) (e : R ≃ LabelledResidue m) : R ≃ LabelledResidue m where
+  toFun τ := e (sx - τ)
+  invFun s := sx - e.symm s
+  left_inv τ := by simp
+  right_inv s := by simp
+
+@[simp] theorem sourceResidueEquiv_apply (sx : R) (e : R ≃ LabelledResidue m) (τ : R) :
+    sourceResidueEquiv sx e τ = e (sx - τ) := rfl
+
+@[simp] theorem sourceResidueEquiv_symm_apply (sx : R) (e : R ≃ LabelledResidue m)
+    (s : LabelledResidue m) : (sourceResidueEquiv sx e).symm s = sx - e.symm s := rfl
+
+def usefulOne (D : LabeledBlocks K A B 2 b) (u : K) : LabelledResidue m → ZMod 2 × ZMod b
+  | Sum.inl _ => D.C u
+  | Sum.inr (Sum.inl (_, y)) => (y, 0)
+  | Sum.inr (Sum.inr (_, y, _)) => (y, 0)
+
+def highOne (D : LabeledBlocks K A B 2 b) (t : ℤ) (u : K) : LabelledResidue m → K
+  | Sum.inl _ => D.T t u
+  | Sum.inr _ => u
+
+def usefulTwo (D : LabeledBlocks K A B 3 b) (u : K) : LabelledResidue m → ZMod 3 × ZMod b
+  | Sum.inl y => (y, 0)
+  | Sum.inr (Sum.inl _) => D.C u
+  | Sum.inr (Sum.inr (_, _, y)) => (y, 0)
+
+def highTwo (D : LabeledBlocks K A B 3 b) (t : ℤ) (u : K) : LabelledResidue m → K
+  | Sum.inl _ => u
+  | Sum.inr (Sum.inl _) => D.T t u
+  | Sum.inr (Sum.inr _) => u
+
+def oneForward (D : LabeledBlocks K A B 2 b) (S : BlockShiftData Δ 2 b)
+    (e : R ≃ LabelledResidue m) (code : R ≃ ZMod n)
+    (H : (K × Δ) × R → ℤ) (x : ℤ) (sx : R) (k : K)
+    (v : (K × Δ) × R) : ((ZMod 2 × ZMod b) × K) × ZMod n :=
+  let s := e (sx - v.2)
+  let u := k - v.1.1
+  let t := x - H v
+  ((usefulOne D u s + S.e v.1.2, highOne D t u s),
+    sharedOutput e code t (sx - v.2))
+
+def twoForward (D : LabeledBlocks K A B 3 b) (S : BlockShiftData Δ 3 b)
+    (e : R ≃ LabelledResidue m) (code : R ≃ ZMod n)
+    (H : (K × Δ) × R → ℤ) (x : ℤ) (sx : R) (k : K)
+    (v : (K × Δ) × R) : ((ZMod 3 × ZMod b) × K) × ZMod n :=
+  let s := e (sx - v.2)
+  let u := k - v.1.1
+  let t := x - H v
+  ((usefulTwo D u s + S.e v.1.2, highTwo D t u s),
+    sharedOutput e code t (sx - v.2))
+
+def oneActivationEquiv (D : LabeledBlocks K A B 2 b) (S : BlockShiftData Δ 2 b)
+    (e : R ≃ LabelledResidue m) (code : R ≃ ZMod n)
+    (H : (K × Δ) × R → ℤ)
+    (hA : ∀ v, (H v : ZMod 2) = S.rhoA v.1.2)
+    (hB : ∀ v, (H v : ZMod b) = S.rhoB v.1.2)
+    (x : ℤ) (sx : R) (k : K) :
+    ((K × Δ) × R) ≃ (((ZMod 2 × ZMod b) × K) × ZMod n) :=
+  let E := sourceResidueEquiv sx e
+  let blocks := fun y : ZMod 3 =>
+    BlockShiftData.blockInverseEquiv D S (fun v => H (v, E.symm (Sum.inl y)))
+      (fun v => hA (v, E.symm (Sum.inl y)))
+      (fun v => hB (v, E.symm (Sum.inl y))) x k
+  ((Equiv.prodCongr (Equiv.refl (K × Δ)) E).trans
+    (oneLabelEquiv S x k blocks)).trans
+      (Equiv.prodCongr (Equiv.refl ((ZMod 2 × ZMod b) × K)) (e.symm.trans code))
+
+def twoActivationEquiv (D : LabeledBlocks K A B 3 b) (S : BlockShiftData Δ 3 b)
+    (e : R ≃ LabelledResidue m) (code : R ≃ ZMod n)
+    (H : (K × Δ) × R → ℤ)
+    (hA : ∀ v, (H v : ZMod 3) = S.rhoA v.1.2)
+    (hB : ∀ v, (H v : ZMod b) = S.rhoB v.1.2)
+    (x : ℤ) (sx : R) (k : K) :
+    ((K × Δ) × R) ≃ (((ZMod 3 × ZMod b) × K) × ZMod n) :=
+  let E := sourceResidueEquiv sx e
+  let blocks := fun y : Fin 2 × ZMod 2 =>
+    BlockShiftData.blockInverseEquiv D S (fun v => H (v, E.symm (Sum.inr (Sum.inl y))))
+      (fun v => hA (v, E.symm (Sum.inr (Sum.inl y))))
+      (fun v => hB (v, E.symm (Sum.inr (Sum.inl y)))) x k
+  ((Equiv.prodCongr (Equiv.refl (K × Δ)) E).trans
+    (twoLabelEquiv S x k blocks)).trans
+      (Equiv.prodCongr (Equiv.refl ((ZMod 3 × ZMod b) × K)) (e.symm.trans code))
+
+theorem oneActivationEquiv_apply
+    (D : LabeledBlocks K A B 2 b) (S : BlockShiftData Δ 2 b)
+    (e : R ≃ LabelledResidue m) (code : R ≃ ZMod n)
+    (H : (K × Δ) × R → ℤ)
+    (hA : ∀ v, (H v : ZMod 2) = S.rhoA v.1.2)
+    (hB : ∀ v, (H v : ZMod b) = S.rhoB v.1.2)
+    (hOther : ∀ v, (H v : ZMod 3) = 0)
+    (x : ℤ) (sx : R) (k : K) (v : (K × Δ) × R) :
+    oneActivationEquiv D S e code H hA hB x sx k v =
+      oneForward D S e code H x sx k v := by
+  rcases v with ⟨⟨l, δ⟩, τ⟩
+  have hback : sx - e.symm (e (sx - τ)) = τ := by simp
+  cases hs : e (sx - τ) with
+  | inl y =>
+      rw [hs] at hback
+      simp [oneActivationEquiv, oneLabelEquiv, sourceResidueEquiv,
+        BlockShiftData.blockInverseEquiv, BlockShiftData.blockForward,
+        oneForward, usefulOne, highOne, sharedOutput, rotation, rotate, rotateMap,
+        hs, hback, Int.cast_sub, hOther]
+  | inr s =>
+      cases s with
+      | inl cy =>
+          rcases cy with ⟨c, y⟩
+          simp [oneActivationEquiv, oneLabelEquiv, sourceResidueEquiv,
+            inactiveEquiv, inactiveForward, oneForward, usefulOne, highOne,
+            sharedOutput, rotation, rotate, rotateMap, hs, Int.cast_sub, hA,
+            add_sub_assoc, Prod.add_def]
+      | inr cyz =>
+          rcases cyz with ⟨c, y, z⟩
+          simp [oneActivationEquiv, oneLabelEquiv, sourceResidueEquiv,
+            inactiveEquiv, inactiveForward, oneForward, usefulOne, highOne,
+            sharedOutput, rotation, rotate, rotateMap, hs, Int.cast_sub, hA, hOther,
+            add_sub_assoc, Prod.add_def]
+
+theorem twoActivationEquiv_apply
+    (D : LabeledBlocks K A B 3 b) (S : BlockShiftData Δ 3 b)
+    (e : R ≃ LabelledResidue m) (code : R ≃ ZMod n)
+    (H : (K × Δ) × R → ℤ)
+    (hA : ∀ v, (H v : ZMod 3) = S.rhoA v.1.2)
+    (hB : ∀ v, (H v : ZMod b) = S.rhoB v.1.2)
+    (hOther : ∀ v, (H v : ZMod 2) = 0)
+    (x : ℤ) (sx : R) (k : K) (v : (K × Δ) × R) :
+    twoActivationEquiv D S e code H hA hB x sx k v =
+      twoForward D S e code H x sx k v := by
+  rcases v with ⟨⟨l, δ⟩, τ⟩
+  have hback : sx - e.symm (e (sx - τ)) = τ := by simp
+  cases hs : e (sx - τ) with
+  | inl y =>
+      simp [twoActivationEquiv, twoLabelEquiv, sourceResidueEquiv,
+        inactiveEquiv, inactiveForward, twoForward, usefulTwo, highTwo,
+        sharedOutput, rotation, rotate, rotateMap, hs, Int.cast_sub, hA,
+        add_sub_assoc, Prod.add_def]
+  | inr s =>
+      cases s with
+      | inl cy =>
+          rcases cy with ⟨c, y⟩
+          rw [hs] at hback
+          simp [twoActivationEquiv, twoLabelEquiv, sourceResidueEquiv,
+            BlockShiftData.blockInverseEquiv, BlockShiftData.blockForward,
+            twoForward, usefulTwo, highTwo, sharedOutput, rotation, rotate, rotateMap,
+            hs, hback, Int.cast_sub, hOther]
+      | inr cyz =>
+          rcases cyz with ⟨c, y, z⟩
+          simp [twoActivationEquiv, twoLabelEquiv, sourceResidueEquiv,
+            inactiveEquiv, inactiveForward, twoForward, usefulTwo, highTwo,
+            sharedOutput, rotation, rotate, rotateMap, hs, Int.cast_sub, hA, hOther,
+            add_sub_assoc, Prod.add_def]
+
+theorem oneForward_bijective
+    (D : LabeledBlocks K A B 2 b) (S : BlockShiftData Δ 2 b)
+    (e : R ≃ LabelledResidue m) (code : R ≃ ZMod n)
+    (H : (K × Δ) × R → ℤ)
+    (hA : ∀ v, (H v : ZMod 2) = S.rhoA v.1.2)
+    (hB : ∀ v, (H v : ZMod b) = S.rhoB v.1.2)
+    (hOther : ∀ v, (H v : ZMod 3) = 0) (x : ℤ) (sx : R) (k : K) :
+    Function.Bijective (oneForward D S e code H x sx k) := by
+  have he : oneForward D S e code H x sx k =
+      oneActivationEquiv D S e code H hA hB x sx k :=
+    funext fun v => (oneActivationEquiv_apply D S e code H hA hB hOther x sx k v).symm
+  rw [he]
+  exact (oneActivationEquiv D S e code H hA hB x sx k).bijective
+
+theorem twoForward_bijective
+    (D : LabeledBlocks K A B 3 b) (S : BlockShiftData Δ 3 b)
+    (e : R ≃ LabelledResidue m) (code : R ≃ ZMod n)
+    (H : (K × Δ) × R → ℤ)
+    (hA : ∀ v, (H v : ZMod 3) = S.rhoA v.1.2)
+    (hB : ∀ v, (H v : ZMod b) = S.rhoB v.1.2)
+    (hOther : ∀ v, (H v : ZMod 2) = 0) (x : ℤ) (sx : R) (k : K) :
+    Function.Bijective (twoForward D S e code H x sx k) := by
+  have he : twoForward D S e code H x sx k =
+      twoActivationEquiv D S e code H hA hB x sx k :=
+    funext fun v => (twoActivationEquiv_apply D S e code H hA hB hOther x sx k v).symm
+  rw [he]
+  exact (twoActivationEquiv D S e code H hA hB x sx k).bijective
+
+end PeriodicTilingThree.SharedSeed
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/SharedSeedLabelEquiv.lean b/lean/OAI/Geometry/PeriodicTiling/SharedSeedLabelEquiv.lean
new file mode 100644
index 0000000000000000000000000000000000000000..e01bdfff5ee586c83340cb4fb359729858ab73bc
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/SharedSeedLabelEquiv.lean
@@ -0,0 +1,81 @@
+import OAI.Geometry.PeriodicTiling.SharedSeedLabels
+import OAI.Geometry.PeriodicTiling.SharedSeedInactive
+
+namespace OAI
+
+universe uK uΔ
+
+noncomputable section
+
+namespace PeriodicTilingThree.SharedSeed
+
+variable {K : Type uK} {Δ : Type uΔ} [AddCommGroup K] {m b : ℕ}
+
+def oneLabelEquiv (S : BlockShiftData Δ 2 b) (x : ℤ) (k : K)
+    (B : ZMod 3 → (K × Δ) ≃ ((ZMod 2 × ZMod b) × K)) :
+    ((K × Δ) × LabelledResidue m) ≃ (((ZMod 2 × ZMod b) × K) × LabelledResidue m) where
+  toFun
+    | (v, Sum.inl y) => (B y v, Sum.inl (y + (x : ZMod 3)))
+    | (v, Sum.inr (Sum.inl (c, y))) =>
+        let w := inactiveEquiv S (x : ZMod 2) k (v, y)
+        (w.1, Sum.inr (Sum.inl (c, w.2)))
+    | (v, Sum.inr (Sum.inr (c, y, z))) =>
+        let w := inactiveEquiv S (x : ZMod 2) k (v, y)
+        (w.1, Sum.inr (Sum.inr (c, w.2, z + (x : ZMod 3))))
+  invFun
+    | (v, Sum.inl q) => ((B (q - (x : ZMod 3))).symm v, Sum.inl (q - (x : ZMod 3)))
+    | (v, Sum.inr (Sum.inl (c, q))) =>
+        let w := (inactiveEquiv S (x : ZMod 2) k).symm (v, q)
+        (w.1, Sum.inr (Sum.inl (c, w.2)))
+    | (v, Sum.inr (Sum.inr (c, q, r))) =>
+        let w := (inactiveEquiv S (x : ZMod 2) k).symm (v, q)
+        (w.1, Sum.inr (Sum.inr (c, w.2, r - (x : ZMod 3))))
+  left_inv v := by
+    rcases v with ⟨v, y | (⟨c, y⟩ | ⟨c, y, z⟩)⟩
+    · simp only [add_sub_cancel_right, Equiv.symm_apply_apply]
+    · simp only [Prod.mk.eta, Equiv.symm_apply_apply]
+    · simp only [Prod.mk.eta, Equiv.symm_apply_apply, add_sub_cancel_right]
+  right_inv v := by
+    rcases v with ⟨v, q | (⟨c, q⟩ | ⟨c, q, r⟩)⟩
+    · simp only [Equiv.apply_symm_apply, sub_add_cancel]
+    · simp only [Prod.mk.eta, Equiv.apply_symm_apply]
+    · simp only [Prod.mk.eta, Equiv.apply_symm_apply, sub_add_cancel]
+
+def twoLabelEquiv (S : BlockShiftData Δ 3 b) (x : ℤ) (k : K)
+    (B : (Fin 2 × ZMod 2) → (K × Δ) ≃ ((ZMod 3 × ZMod b) × K)) :
+    ((K × Δ) × LabelledResidue m) ≃ (((ZMod 3 × ZMod b) × K) × LabelledResidue m) where
+  toFun
+    | (v, Sum.inl y) =>
+        let w := inactiveEquiv S (x : ZMod 3) k (v, y)
+        (w.1, Sum.inl w.2)
+    | (v, Sum.inr (Sum.inl (c, y))) =>
+        (B (c, y) v, Sum.inr (Sum.inl (c, y + (x : ZMod 2))))
+    | (v, Sum.inr (Sum.inr (c, y, z))) =>
+        let w := inactiveEquiv S (x : ZMod 3) k (v, z)
+        (w.1, Sum.inr (Sum.inr (c, y + (x : ZMod 2), w.2)))
+  invFun
+    | (v, Sum.inl q) =>
+        let w := (inactiveEquiv S (x : ZMod 3) k).symm (v, q)
+        (w.1, Sum.inl w.2)
+    | (v, Sum.inr (Sum.inl (c, q))) =>
+        ((B (c, q - (x : ZMod 2))).symm v,
+          Sum.inr (Sum.inl (c, q - (x : ZMod 2))))
+    | (v, Sum.inr (Sum.inr (c, q, r))) =>
+        let w := (inactiveEquiv S (x : ZMod 3) k).symm (v, r)
+        (w.1, Sum.inr (Sum.inr (c, q - (x : ZMod 2), w.2)))
+  left_inv v := by
+    rcases v with ⟨v, y | (⟨c, y⟩ | ⟨c, y, z⟩)⟩
+    · simp only [Prod.mk.eta, Equiv.symm_apply_apply]
+    · simp only [add_sub_cancel_right, Equiv.symm_apply_apply]
+    · simp only [Prod.mk.eta, Equiv.symm_apply_apply, add_sub_cancel_right]
+  right_inv v := by
+    rcases v with ⟨v, q | (⟨c, q⟩ | ⟨c, q, r⟩)⟩
+    · simp only [Prod.mk.eta, Equiv.apply_symm_apply]
+    · simp only [Equiv.apply_symm_apply, sub_add_cancel]
+    · simp only [Prod.mk.eta, Equiv.apply_symm_apply, sub_add_cancel]
+
+end PeriodicTilingThree.SharedSeed
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/SharedSeedLabels.lean b/lean/OAI/Geometry/PeriodicTiling/SharedSeedLabels.lean
new file mode 100644
index 0000000000000000000000000000000000000000..5097f3756273388df8cc4baf0fda82fd9d64d863
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/SharedSeedLabels.lean
@@ -0,0 +1,135 @@
+import OAI.Geometry.PeriodicTiling.SharedSeedRegions
+import OAI.Geometry.PeriodicTiling.TilingBasic
+
+namespace OAI
+
+universe uS uK
+
+noncomputable section
+
+namespace PeriodicTilingThree.SharedSeed
+
+def ActiveOne {m : ℕ} : LabelledResidue m → Prop
+  | Sum.inl _ => True
+  | Sum.inr _ => False
+
+def ActiveTwo {m : ℕ} : LabelledResidue m → Prop
+  | Sum.inl _ => False
+  | Sum.inr (Sum.inl _) => True
+  | Sum.inr (Sum.inr _) => False
+
+def firstLabel {m : ℕ} : LabelledResidue m → ZMod 2
+  | Sum.inl _ => 0
+  | Sum.inr (Sum.inl v) => v.2
+  | Sum.inr (Sum.inr v) => v.2.1
+
+def secondLabel {m : ℕ} : LabelledResidue m → ZMod 3
+  | Sum.inl y => y
+  | Sum.inr (Sum.inl _) => 0
+  | Sum.inr (Sum.inr v) => v.2.2
+
+def rotateMap {m : ℕ} (t : ℤ) : LabelledResidue m → LabelledResidue m
+  | Sum.inl y => Sum.inl (y + (t : ZMod 3))
+  | Sum.inr (Sum.inl (c, y)) => Sum.inr (Sum.inl (c, y + (t : ZMod 2)))
+  | Sum.inr (Sum.inr (c, y, z)) =>
+      Sum.inr (Sum.inr (c, y + (t : ZMod 2), z + (t : ZMod 3)))
+
+@[simp] theorem rotateMap_zero {m : ℕ} (s : LabelledResidue m) :
+    rotateMap 0 s = s := by
+  rcases s with y | (⟨c, y⟩ | ⟨c, y, z⟩) <;> simp [rotateMap]
+
+theorem rotateMap_add {m : ℕ} (t u : ℤ) (s : LabelledResidue m) :
+    rotateMap (t + u) s = rotateMap t (rotateMap u s) := by
+  rcases s with y | (⟨c, y⟩ | ⟨c, y, z⟩) <;>
+    simp [rotateMap, Int.cast_add, add_comm, add_left_comm]
+
+def rotate {m : ℕ} (t : ℤ) : Equiv.Perm (LabelledResidue m) where
+  toFun := rotateMap t
+  invFun := rotateMap (-t)
+  left_inv s := by rw [← rotateMap_add, neg_add_cancel, rotateMap_zero]
+  right_inv s := by rw [← rotateMap_add, add_neg_cancel, rotateMap_zero]
+
+@[simp] theorem rotate_apply {m : ℕ} (t : ℤ) (s : LabelledResidue m) :
+    rotate t s = rotateMap t s := rfl
+
+@[simp] theorem rotate_zero {m : ℕ} (s : LabelledResidue m) : rotate 0 s = s :=
+  rotateMap_zero s
+
+theorem rotate_add {m : ℕ} (t u : ℤ) (s : LabelledResidue m) :
+    rotate (t + u) s = rotate t (rotate u s) := rotateMap_add t u s
+
+@[simp] theorem activeOne_rotate {m : ℕ} (t : ℤ) (s : LabelledResidue m) :
+    ActiveOne (rotate t s) ↔ ActiveOne s := by
+  rcases s with y | (⟨c, y⟩ | ⟨c, y, z⟩) <;> rfl
+
+@[simp] theorem activeTwo_rotate {m : ℕ} (t : ℤ) (s : LabelledResidue m) :
+    ActiveTwo (rotate t s) ↔ ActiveTwo s := by
+  rcases s with y | (⟨c, y⟩ | ⟨c, y, z⟩) <;> rfl
+
+theorem firstLabel_rotate {m : ℕ} (t : ℤ) (s : LabelledResidue m)
+    (hs : ¬ ActiveOne s) : firstLabel (rotate t s) = firstLabel s + (t : ZMod 2) := by
+  rcases s with y | (⟨c, y⟩ | ⟨c, y, z⟩)
+  · exact False.elim (hs trivial)
+  · rfl
+  · rfl
+
+theorem secondLabel_rotate {m : ℕ} (t : ℤ) (s : LabelledResidue m)
+    (hs : ¬ ActiveTwo s) : secondLabel (rotate t s) = secondLabel s + (t : ZMod 3) := by
+  rcases s with y | (⟨c, y⟩ | ⟨c, y, z⟩)
+  · rfl
+  · exact False.elim (hs trivial)
+  · rfl
+
+theorem rotate_fixed_on_activeOne {m : ℕ} {t : ℤ} {s : LabelledResidue m}
+    (hs : ActiveOne s) (ht : (t : ZMod 3) = 0) : rotate t s = s := by
+  rcases s with y | (⟨c, y⟩ | ⟨c, y, z⟩)
+  · simp [rotate, rotateMap, ht]
+  · exact False.elim hs
+  · exact False.elim hs
+
+theorem rotate_fixed_on_activeTwo {m : ℕ} {t : ℤ} {s : LabelledResidue m}
+    (hs : ActiveTwo s) (ht : (t : ZMod 2) = 0) : rotate t s = s := by
+  rcases s with y | (⟨c, y⟩ | ⟨c, y, z⟩)
+  · exact False.elim hs
+  · simp [rotate, rotateMap, ht]
+  · exact False.elim hs
+
+def rotation {S : Type uS} {m : ℕ} (e : S ≃ LabelledResidue m) (t : ℤ) : Equiv.Perm S :=
+  (e.trans (rotate t)).trans e.symm
+
+@[simp] theorem rotation_apply {S : Type uS} {m : ℕ}
+    (e : S ≃ LabelledResidue m) (t : ℤ) (s : S) :
+    rotation e t s = e.symm (rotate t (e s)) := rfl
+
+@[simp] theorem label_rotation {S : Type uS} {m : ℕ}
+    (e : S ≃ LabelledResidue m) (t : ℤ) (s : S) :
+    e (rotation e t s) = rotate t (e s) := e.apply_symm_apply _
+
+theorem rotation_add {S : Type uS} {m : ℕ}
+    (e : S ≃ LabelledResidue m) (t u : ℤ) (s : S) :
+    rotation e (t + u) s = rotation e t (rotation e u s) := by
+  simp only [rotation_apply, Equiv.apply_symm_apply, rotate_add]
+
+def sharedOutput {S : Type uS} {m D : ℕ}
+    (e : S ≃ LabelledResidue m) (code : S ≃ ZMod D) (t : ℤ) (s : S) : ZMod D :=
+  code (rotation e t s)
+
+@[simp] theorem decode_sharedOutput {S : Type uS} {m D : ℕ}
+    (e : S ≃ LabelledResidue m) (code : S ≃ ZMod D) (t : ℤ) (s : S) :
+    e (code.symm (sharedOutput e code t s)) = rotate t (e s) := by
+  simp only [sharedOutput, Equiv.symm_apply_apply, label_rotation]
+
+def sharedZ {p m : ℕ} {K : Type uK}
+    (e : Residues p ≃ LabelledResidue m) (code : Residues p ≃ ZMod (p ^ 4))
+    (x : Plane) (_k : K) : ZMod (p ^ 4) :=
+  sharedOutput e code x.1 ((x.1 : ZMod (p ^ 2)), (x.2 : ZMod (p ^ 2)))
+
+def commonZ {p : ℕ} {K : Type uK} (hp : p.Prime) (hlarge : 200 < p)
+    (x : Plane) (_k : K) : ZMod (p ^ 4) :=
+  sharedOutput (residueEquiv hp hlarge) (outputEquiv hlarge) x.1 (residue p x)
+
+end PeriodicTilingThree.SharedSeed
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/SharedSeedRegions.lean b/lean/OAI/Geometry/PeriodicTiling/SharedSeedRegions.lean
new file mode 100644
index 0000000000000000000000000000000000000000..141aa42c0c1eb54d089d194867f0b8af57f5ac87
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/SharedSeedRegions.lean
@@ -0,0 +1,260 @@
+import Mathlib.Data.ZMod.Basic
+import Mathlib.Data.Nat.Prime.Basic
+import Mathlib.Data.Fintype.EquivFin
+import Mathlib.Data.Fintype.Sum
+import Mathlib.Data.Fintype.Prod
+import Mathlib.Logic.Equiv.Set
+import Mathlib.Logic.Equiv.Sum
+import Lean.Elab.Tactic.Omega
+import Mathlib.Tactic.NormNum
+import Mathlib.Tactic.Positivity
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.Ring
+
+namespace OAI
+
+namespace PeriodicTilingThree.SharedSeed
+
+abbrev Residues (p : ℕ) := ZMod (p ^ 2) × ZMod (p ^ 2)
+abbrev ActiveLabels := ZMod 3 ⊕ (Fin 2 × ZMod 2)
+abbrev InactiveLabels (m : ℕ) := Fin m × (ZMod 2 × ZMod 3)
+abbrev LabelledResidue (m : ℕ) :=
+  ZMod 3 ⊕ ((Fin 2 × ZMod 2) ⊕ InactiveLabels m)
+
+def residue (p : ℕ) (x : ℤ × ℤ) : Residues p := (x.1, x.2)
+
+def firstPoint (p : ℕ) (y : ZMod 3) : Residues p :=
+  ((p * y.val : ℕ), 1)
+
+def secondPoint (p : ℕ) (u : Fin 2 × ZMod 2) : Residues p :=
+  ((p * (2 * u.1.val + u.2.val) : ℕ), 2)
+
+def activePoint (p : ℕ) : ActiveLabels → Residues p :=
+  Sum.elim (firstPoint p) (secondPoint p)
+
+def firstRegion (p : ℕ) : Set (Residues p) := Set.range (firstPoint p)
+def secondRegion (p : ℕ) : Set (Residues p) := Set.range (secondPoint p)
+def Rest (p : ℕ) := {s : Residues p // s ∉ Set.range (activePoint p)}
+def copyCount (p : ℕ) : ℕ := (p ^ 4 - 7) / 6
+
+private theorem nat_multiple_cast_injective {p u v : ℕ}
+    (hp : 0 < p) (hu : u < p) (hv : v < p)
+    (h : (p * u : ZMod (p ^ 2)) = (p * v : ZMod (p ^ 2))) : u = v := by
+  have hu' : p * u < p ^ 2 := by
+    simpa [pow_two] using Nat.mul_lt_mul_of_pos_left hu hp
+  have hv' : p * v < p ^ 2 := by
+    simpa [pow_two] using Nat.mul_lt_mul_of_pos_left hv hp
+  have hn : ((p * u : ℕ) : ZMod (p ^ 2)) = ((p * v : ℕ) : ZMod (p ^ 2)) := by
+    simpa only [Nat.cast_mul] using h
+  have hh := congrArg ZMod.val hn
+  rw [ZMod.val_natCast_of_lt hu', ZMod.val_natCast_of_lt hv'] at hh
+  exact Nat.eq_of_mul_eq_mul_left hp hh
+
+theorem firstPoint_injective {p : ℕ} (hlarge : 200 < p) :
+    Function.Injective (firstPoint p) := by
+  intro y y' h
+  apply ZMod.val_injective 3
+  apply nat_multiple_cast_injective (by omega)
+      (lt_trans (ZMod.val_lt y) (by omega : 3 < p))
+      (lt_trans (ZMod.val_lt y') (by omega : 3 < p))
+  simpa only [firstPoint, Nat.cast_mul] using congrArg Prod.fst h
+
+theorem secondPoint_injective {p : ℕ} (hlarge : 200 < p) :
+    Function.Injective (secondPoint p) := by
+  rintro ⟨c,y⟩ ⟨c',y'⟩ h
+  have hc := c.isLt
+  have hc' := c'.isLt
+  have hy := ZMod.val_lt y
+  have hy' := ZMod.val_lt y'
+  have hcode : 2 * c.val + y.val = 2 * c'.val + y'.val := by
+    apply nat_multiple_cast_injective (p := p) (by omega) (by omega) (by omega)
+    simpa only [secondPoint, Nat.cast_mul, Nat.cast_add, Nat.cast_ofNat]
+      using congrArg Prod.fst h
+  have hcc : c.val = c'.val := by omega
+  have hyy : y.val = y'.val := by omega
+  exact Prod.ext (Fin.ext hcc) (ZMod.val_injective 2 hyy)
+
+theorem firstPoint_ne_secondPoint {p : ℕ} (hlarge : 200 < p)
+    (y : ZMod 3) (u : Fin 2 × ZMod 2) : firstPoint p y ≠ secondPoint p u := by
+  intro h
+  have hp2 : 2 < p ^ 2 := lt_of_lt_of_le (by norm_num : 2 < 201 ^ 2)
+    (Nat.pow_le_pow_left (by omega : 201 ≤ p) 2)
+  have hh := congrArg (fun s : Residues p => s.2.val) h
+  change (1 : ZMod (p ^ 2)).val = (2 : ZMod (p ^ 2)).val at hh
+  have h1 : (1 : ZMod (p ^ 2)).val = 1 := by
+    simpa only [Nat.cast_one] using ZMod.val_natCast_of_lt (by omega : 1 < p ^ 2)
+  have h2 : (2 : ZMod (p ^ 2)).val = 2 := by
+    simpa only [Nat.cast_ofNat] using ZMod.val_natCast_of_lt hp2
+  have h12 : (1 : ℕ) = 2 := h1.symm.trans (hh.trans h2)
+  omega
+
+theorem activePoint_injective {p : ℕ} (hlarge : 200 < p) :
+    Function.Injective (activePoint p) := by
+  intro a b h
+  cases a with
+  | inl y =>
+    cases b with
+    | inl y' => exact congrArg Sum.inl (firstPoint_injective hlarge h)
+    | inr u => exact (firstPoint_ne_secondPoint hlarge y u h).elim
+  | inr u =>
+    cases b with
+    | inl y => exact (firstPoint_ne_secondPoint hlarge y u h.symm).elim
+    | inr u' => exact congrArg Sum.inr (secondPoint_injective hlarge h)
+
+theorem first_second_disjoint {p : ℕ} (hlarge : 200 < p) :
+    Disjoint (firstRegion p) (secondRegion p) := by
+  rw [Set.disjoint_left]
+  rintro s ⟨y, rfl⟩ ⟨u, hu⟩
+  exact firstPoint_ne_secondPoint hlarge y u hu.symm
+
+theorem seven_le_pow {p : ℕ} (hlarge : 200 < p) : 7 ≤ p ^ 4 := by
+  exact le_trans (by norm_num : 7 ≤ 201 ^ 4)
+    (Nat.pow_le_pow_left (by omega : 201 ≤ p) 4)
+
+theorem pow_four_mod_six {p : ℕ} (hp : p.Prime) (hlarge : 200 < p) :
+    p ^ 4 % 6 = 1 := by
+  have htwo : p % 2 = 1 := hp.eq_two_or_odd.resolve_left (by omega)
+  have hthree : p % 3 ≠ 0 := by
+    intro h
+    have hd : 3 ∣ p := Nat.dvd_of_mod_eq_zero h
+    have he : 3 = p := (Nat.prime_dvd_prime_iff_eq Nat.prime_three hp).mp hd
+    omega
+  have hsix : p % 6 = 1 ∨ p % 6 = 5 := by omega
+  rw [Nat.pow_mod]
+  rcases hsix with h | h <;> norm_num [h]
+
+theorem six_dvd_remainder {p : ℕ} (hp : p.Prime) (hlarge : 200 < p) :
+    6 ∣ p ^ 4 - 7 := by
+  apply Nat.dvd_of_mod_eq_zero
+  have hmod := pow_four_mod_six hp hlarge
+  have hge := seven_le_pow hlarge
+  omega
+
+theorem copyCount_mul_six {p : ℕ} (hp : p.Prime) (hlarge : 200 < p) :
+    copyCount p * 6 = p ^ 4 - 7 := by
+  exact Nat.div_mul_cancel (six_dvd_remainder hp hlarge)
+
+theorem six_mul_copyCount_add_seven {p : ℕ} (hp : p.Prime)
+    (hlarge : 200 < p) : 6 * copyCount p + 7 = p ^ 4 := by
+  have h := copyCount_mul_six hp hlarge
+  have hge := seven_le_pow hlarge
+  omega
+
+noncomputable def restEquiv {p : ℕ} (hp : p.Prime) (hlarge : 200 < p) :
+    Rest p ≃ InactiveLabels (copyCount p) := by
+  classical
+  letI : NeZero (p ^ 2) := ⟨pow_ne_zero 2 (by omega)⟩
+  letI : Fintype (Rest p) := by unfold Rest; infer_instance
+  apply Fintype.equivOfCardEq
+  have hactive : Fintype.card (Set.range (activePoint p)) = 7 := by
+    rw [← Fintype.card_congr (Equiv.ofInjective _ (activePoint_injective hlarge))]
+    norm_num [ActiveLabels]
+  have hrest : Fintype.card (Rest p) = p ^ 4 - 7 := by
+    change Fintype.card ((Set.range (activePoint p))ᶜ : Set (Residues p)) = _
+    rw [Fintype.card_compl_set, hactive]
+    simp only [Residues, Fintype.card_prod, ZMod.card]
+    congr 1
+    ring
+  rw [hrest]
+  simpa [InactiveLabels, Fintype.card_prod, Nat.mul_assoc]
+    using (copyCount_mul_six hp hlarge).symm
+
+noncomputable def residueAssembly {p : ℕ} (hp : p.Prime) (hlarge : 200 < p) :
+    LabelledResidue (copyCount p) ≃ Residues p := by
+  classical
+  exact (Equiv.sumAssoc (ZMod 3) (Fin 2 × ZMod 2)
+      (InactiveLabels (copyCount p))).symm |>.trans
+    ((Equiv.sumCongr (Equiv.ofInjective (activePoint p)
+      (activePoint_injective hlarge)) (restEquiv hp hlarge).symm).trans
+      (Equiv.Set.sumCompl (Set.range (activePoint p))))
+
+noncomputable def residueEquiv {p : ℕ} (hp : p.Prime) (hlarge : 200 < p) :
+    Residues p ≃ LabelledResidue (copyCount p) :=
+  (residueAssembly hp hlarge).symm
+
+@[simp] theorem residueAssembly_first {p : ℕ} (hp : p.Prime)
+    (hlarge : 200 < p) (y : ZMod 3) :
+    residueAssembly hp hlarge (Sum.inl y) = firstPoint p y := rfl
+
+@[simp] theorem residueAssembly_second {p : ℕ} (hp : p.Prime)
+    (hlarge : 200 < p) (u : Fin 2 × ZMod 2) :
+    residueAssembly hp hlarge (Sum.inr (Sum.inl u)) = secondPoint p u := rfl
+
+@[simp] theorem residueEquiv_firstPoint {p : ℕ} (hp : p.Prime)
+    (hlarge : 200 < p) (y : ZMod 3) :
+    residueEquiv hp hlarge (firstPoint p y) = Sum.inl y :=
+  (residueAssembly hp hlarge).symm_apply_apply (Sum.inl y)
+
+@[simp] theorem residueEquiv_secondPoint {p : ℕ} (hp : p.Prime)
+    (hlarge : 200 < p) (u : Fin 2 × ZMod 2) :
+    residueEquiv hp hlarge (secondPoint p u) = Sum.inr (Sum.inl u) :=
+  (residueAssembly hp hlarge).symm_apply_apply (Sum.inr (Sum.inl u))
+
+theorem residueEquiv_first_iff {p : ℕ} (hp : p.Prime)
+    (hlarge : 200 < p) (s : Residues p) :
+    (∃ y, residueEquiv hp hlarge s = Sum.inl y) ↔ s ∈ firstRegion p := by
+  constructor
+  · rintro ⟨y, hy⟩
+    refine ⟨y, ?_⟩
+    apply (residueEquiv hp hlarge).injective
+    rw [residueEquiv_firstPoint, hy]
+  · rintro ⟨y, rfl⟩
+    exact ⟨y, residueEquiv_firstPoint hp hlarge y⟩
+
+theorem residueEquiv_second_iff {p : ℕ} (hp : p.Prime)
+    (hlarge : 200 < p) (s : Residues p) :
+    (∃ u, residueEquiv hp hlarge s = Sum.inr (Sum.inl u)) ↔
+      s ∈ secondRegion p := by
+  constructor
+  · rintro ⟨u, hu⟩
+    refine ⟨u, ?_⟩
+    apply (residueEquiv hp hlarge).injective
+    rw [residueEquiv_secondPoint, hu]
+  · rintro ⟨u, rfl⟩
+    exact ⟨u, residueEquiv_secondPoint hp hlarge u⟩
+
+def reduce (p : ℕ) (s : Residues p) : ZMod p × ZMod p :=
+  let f := ZMod.castHom (show p ∣ p ^ 2 from ⟨p, by ring⟩) (ZMod p)
+  (f s.1, f s.2)
+
+@[simp] theorem reduce_firstPoint (p : ℕ) (y : ZMod 3) :
+    reduce p (firstPoint p y) = (0, 1) := by
+  let f : ZMod (p ^ 2) →+* ZMod p :=
+    ZMod.castHom (show p ∣ p ^ 2 from ⟨p, by ring⟩) (ZMod p)
+  change (f ((p * y.val : ℕ) : ZMod (p ^ 2)), f 1) = (0, 1)
+  apply Prod.ext
+  · rw [map_natCast]
+    simp [Nat.cast_mul]
+  · exact map_one f
+
+@[simp] theorem reduce_secondPoint (p : ℕ) (u : Fin 2 × ZMod 2) :
+    reduce p (secondPoint p u) = (0, 2) := by
+  let f : ZMod (p ^ 2) →+* ZMod p :=
+    ZMod.castHom (show p ∣ p ^ 2 from ⟨p, by ring⟩) (ZMod p)
+  change (f ((p * (2 * u.1.val + u.2.val) : ℕ) : ZMod (p ^ 2)), f 2) = (0, 2)
+  apply Prod.ext
+  · rw [map_natCast]
+    simp [Nat.cast_mul]
+  · simpa only [Nat.cast_ofNat] using map_natCast f 2
+
+theorem reduce_of_firstRegion {p : ℕ} {s : Residues p}
+    (hs : s ∈ firstRegion p) : reduce p s = (0, 1) := by
+  obtain ⟨y, rfl⟩ := hs
+  exact reduce_firstPoint p y
+
+theorem reduce_of_secondRegion {p : ℕ} {s : Residues p}
+    (hs : s ∈ secondRegion p) : reduce p s = (0, 2) := by
+  obtain ⟨u, rfl⟩ := hs
+  exact reduce_secondPoint p u
+
+noncomputable def outputEquiv {p : ℕ} (hlarge : 200 < p) :
+    Residues p ≃ ZMod (p ^ 4) := by
+  letI : NeZero (p ^ 2) := ⟨pow_ne_zero 2 (by omega)⟩
+  letI : NeZero (p ^ 4) := ⟨pow_ne_zero 4 (by omega)⟩
+  apply Fintype.equivOfCardEq
+  simp only [Residues, Fintype.card_prod, ZMod.card]
+  ring
+
+end PeriodicTilingThree.SharedSeed
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/SharedSeedSizeTransport.lean b/lean/OAI/Geometry/PeriodicTiling/SharedSeedSizeTransport.lean
new file mode 100644
index 0000000000000000000000000000000000000000..2eb97c952a4172d7ec8bda2826f11714ed294ac8
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/SharedSeedSizeTransport.lean
@@ -0,0 +1,150 @@
+import OAI.Geometry.PeriodicTiling.SharedSeedInverse
+
+namespace OAI
+
+universe uK uΔ uA uB uR
+
+noncomputable section
+
+namespace PeriodicTilingThree.SharedSeed
+
+variable {K : Type uK} {Δ : Type uΔ} {A : Type uA} {B : Type uB} {R : Type uR}
+  [AddCommGroup K] [AddCommGroup R] {a b m n : ℕ}
+
+def usefulOneOfEq (ha : a = 2) (D : LabeledBlocks K A B a b)
+    (u : K) (s : LabelledResidue m) : ZMod a × ZMod b := by
+  subst a
+  exact usefulOne D u s
+
+def highOneOfEq (ha : a = 2) (D : LabeledBlocks K A B a b)
+    (t : ℤ) (u : K) (s : LabelledResidue m) : K := by
+  subst a
+  exact highOne D t u s
+
+def usefulTwoOfEq (ha : a = 3) (D : LabeledBlocks K A B a b)
+    (u : K) (s : LabelledResidue m) : ZMod a × ZMod b := by
+  subst a
+  exact usefulTwo D u s
+
+def highTwoOfEq (ha : a = 3) (D : LabeledBlocks K A B a b)
+    (t : ℤ) (u : K) (s : LabelledResidue m) : K := by
+  subst a
+  exact highTwo D t u s
+
+def oneForwardOfEq (ha : a = 2) (D : LabeledBlocks K A B a b) (S : BlockShiftData Δ a b)
+    (e : R ≃ LabelledResidue m) (code : R ≃ ZMod n)
+    (H : (K × Δ) × R → ℤ) (x : ℤ) (sx : R) (k : K)
+    (v : (K × Δ) × R) : ((ZMod a × ZMod b) × K) × ZMod n :=
+  ((usefulOneOfEq ha D (k - v.1.1) (e (sx - v.2)) + S.e v.1.2,
+    highOneOfEq ha D (x - H v) (k - v.1.1) (e (sx - v.2))),
+    sharedOutput e code (x - H v) (sx - v.2))
+
+def twoForwardOfEq (ha : a = 3) (D : LabeledBlocks K A B a b) (S : BlockShiftData Δ a b)
+    (e : R ≃ LabelledResidue m) (code : R ≃ ZMod n)
+    (H : (K × Δ) × R → ℤ) (x : ℤ) (sx : R) (k : K)
+    (v : (K × Δ) × R) : ((ZMod a × ZMod b) × K) × ZMod n :=
+  ((usefulTwoOfEq ha D (k - v.1.1) (e (sx - v.2)) + S.e v.1.2,
+    highTwoOfEq ha D (x - H v) (k - v.1.1) (e (sx - v.2))),
+    sharedOutput e code (x - H v) (sx - v.2))
+
+theorem oneForwardOfEq_bijective
+    (ha : a = 2) (D : LabeledBlocks K A B a b) (S : BlockShiftData Δ a b)
+    (e : R ≃ LabelledResidue m) (code : R ≃ ZMod n)
+    (H : (K × Δ) × R → ℤ)
+    (hA : ∀ v, (H v : ZMod a) = S.rhoA v.1.2)
+    (hB : ∀ v, (H v : ZMod b) = S.rhoB v.1.2)
+    (hOther : ∀ v, (H v : ZMod 3) = 0) (x : ℤ) (sx : R) (k : K) :
+    Function.Bijective (oneForwardOfEq ha D S e code H x sx k) := by
+  subst a
+  exact oneForward_bijective D S e code H hA hB hOther x sx k
+
+theorem twoForwardOfEq_bijective
+    (ha : a = 3) (D : LabeledBlocks K A B a b) (S : BlockShiftData Δ a b)
+    (e : R ≃ LabelledResidue m) (code : R ≃ ZMod n)
+    (H : (K × Δ) × R → ℤ)
+    (hA : ∀ v, (H v : ZMod a) = S.rhoA v.1.2)
+    (hB : ∀ v, (H v : ZMod b) = S.rhoB v.1.2)
+    (hOther : ∀ v, (H v : ZMod 2) = 0) (x : ℤ) (sx : R) (k : K) :
+    Function.Bijective (twoForwardOfEq ha D S e code H x sx k) := by
+  subst a
+  exact twoForward_bijective D S e code H hA hB hOther x sx k
+
+theorem usefulOneOfEq_active
+    {K : Type uK} {A : Type uA} {B : Type uB}
+    [AddCommGroup K] {a b m : ℕ}
+    (ha : a = 2) (D : LabeledBlocks K A B a b)
+    (u : K) {s : LabelledResidue m} (hs : ActiveOne s) : usefulOneOfEq ha D u s = D.C u := by
+  subst a
+  rcases s with y | (⟨c, y⟩ | ⟨c, y, z⟩)
+  · rfl
+  · exact False.elim hs
+  · exact False.elim hs
+
+theorem usefulTwoOfEq_active
+    {K : Type uK} {A : Type uA} {B : Type uB}
+    [AddCommGroup K] {a b m : ℕ}
+    (ha : a = 3) (D : LabeledBlocks K A B a b)
+    (u : K) {s : LabelledResidue m} (hs : ActiveTwo s) : usefulTwoOfEq ha D u s = D.C u := by
+  subst a
+  rcases s with y | (⟨c, y⟩ | ⟨c, y, z⟩)
+  · exact False.elim hs
+  · rfl
+  · exact False.elim hs
+
+theorem usefulOneOfEq_inactive_eq
+    {K : Type uK} {A : Type uA} {B : Type uB}
+    [AddCommGroup K] {a b m : ℕ}
+    (ha : a = 2) (D : LabeledBlocks K A B a b)
+    (u v : K) {s : LabelledResidue m} (hs : ¬ ActiveOne s) :
+    usefulOneOfEq ha D u s = usefulOneOfEq ha D v s := by
+  subst a
+  rcases s with y | (⟨c, y⟩ | ⟨c, y, z⟩)
+  · exact False.elim (hs trivial)
+  · rfl
+  · rfl
+
+theorem usefulTwoOfEq_inactive_eq
+    {K : Type uK} {A : Type uA} {B : Type uB}
+    [AddCommGroup K] {a b m : ℕ}
+    (ha : a = 3) (D : LabeledBlocks K A B a b)
+    (u v : K) {s : LabelledResidue m} (hs : ¬ ActiveTwo s) :
+    usefulTwoOfEq ha D u s = usefulTwoOfEq ha D v s := by
+  subst a
+  rcases s with y | (⟨c, y⟩ | ⟨c, y, z⟩)
+  · rfl
+  · exact False.elim (hs trivial)
+  · rfl
+
+theorem usefulOneOfEq_activity_iff (ha : a = 2) (D : LabeledBlocks K A B a b)
+    (rel : K → K → Prop) (hC : ∃ u v, rel u v ∧ D.C u ≠ D.C v)
+    (s : LabelledResidue m) :
+    (∃ u v, rel u v ∧ usefulOneOfEq ha D u s ≠ usefulOneOfEq ha D v s) ↔ ActiveOne s := by
+  classical
+  constructor
+  · rintro ⟨u, v, _, hne⟩
+    by_contra hs
+    exact hne (usefulOneOfEq_inactive_eq ha D u v hs)
+  · intro hs
+    obtain ⟨u, v, huv, hne⟩ := hC
+    refine ⟨u, v, huv, ?_⟩
+    simpa only [usefulOneOfEq_active ha D u hs, usefulOneOfEq_active ha D v hs] using hne
+
+theorem usefulTwoOfEq_activity_iff (ha : a = 3) (D : LabeledBlocks K A B a b)
+    (rel : K → K → Prop) (hC : ∃ u v, rel u v ∧ D.C u ≠ D.C v)
+    (s : LabelledResidue m) :
+    (∃ u v, rel u v ∧ usefulTwoOfEq ha D u s ≠ usefulTwoOfEq ha D v s) ↔ ActiveTwo s := by
+  classical
+  constructor
+  · rintro ⟨u, v, _, hne⟩
+    by_contra hs
+    exact hne (usefulTwoOfEq_inactive_eq ha D u v hs)
+  · intro hs
+    obtain ⟨u, v, huv, hne⟩ := hC
+    refine ⟨u, v, huv, ?_⟩
+    simpa only [usefulTwoOfEq_active ha D u hs, usefulTwoOfEq_active ha D v hs] using hne
+
+end PeriodicTilingThree.SharedSeed
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/ShiftMultiplicity.lean b/lean/OAI/Geometry/PeriodicTiling/ShiftMultiplicity.lean
new file mode 100644
index 0000000000000000000000000000000000000000..52cb945752a61a556cfe9f93c87e49b3bf96b9cf
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/ShiftMultiplicity.lean
@@ -0,0 +1,255 @@
+import Mathlib.Data.ZMod.Basic
+import Mathlib.Algebra.BigOperators.Ring.Finset
+import Mathlib.Data.Fintype.BigOperators
+import Mathlib.Data.Fintype.EquivFin
+import Mathlib.Logic.Equiv.Prod
+import Mathlib.Tactic.NormNum
+
+namespace OAI
+
+universe uM
+
+namespace PeriodicTilingThree
+
+open scoped BigOperators
+
+def shiftMultiplicity (a b : ℕ) (e : ZMod a × ZMod b) : ℕ :=
+  if e.2 = 0 then
+    if e.1 = 0 then 2 else if e.1 = 1 then 0 else 1
+  else if e.2 = 1 then
+    if e.1 = 0 then 0 else if e.1 = 1 then 2 else 1
+  else 1
+
+def shiftOmega (a b : ℕ) (e : ZMod a × ZMod b) : ℤ :=
+  ((if e.1 = 0 then 1 else 0) - (if e.1 = 1 then 1 else 0)) *
+    ((if e.2 = 0 then 1 else 0) - (if e.2 = 1 then 1 else 0))
+
+abbrev ShiftIndex (a b : ℕ) :=
+  Σ e : ZMod a × ZMod b, Fin (shiftMultiplicity a b e)
+
+def shift (a b : ℕ) (δ : ShiftIndex a b) : ZMod a × ZMod b := δ.1
+
+section Counts
+
+variable {a b : ℕ} [NeZero a] [NeZero b]
+
+theorem shiftMultiplicity_cast {a b : ℕ} [NeZero a] [NeZero b] (ha : 2 ≤ a) (hb : 2 ≤ b)
+    (e : ZMod a × ZMod b) :
+    (shiftMultiplicity a b e : ℤ) = 1 + shiftOmega a b e := by
+  let : Fact (1 < a) := ⟨ha⟩
+  let : Fact (1 < b) := ⟨hb⟩
+  have ha01 : (0 : ZMod a) ≠ 1 := zero_ne_one
+  have hb01 : (0 : ZMod b) ≠ 1 := zero_ne_one
+  by_cases hα0 : e.1 = 0 <;> by_cases hα1 : e.1 = 1 <;>
+    by_cases hξ0 : e.2 = 0 <;> by_cases hξ1 : e.2 = 1 <;>
+    simp_all [shiftMultiplicity, shiftOmega]
+
+@[simp] theorem shiftMultiplicity_zero {a b : ℕ} [NeZero a] [NeZero b] :
+    shiftMultiplicity a b (0, 0) = 2 := by
+  simp [shiftMultiplicity]
+
+theorem sum_shiftMultiplicity_fst (ha : 2 ≤ a) (hb : 2 ≤ b)
+    (ξ : ZMod b) :
+    ∑ α : ZMod a, shiftMultiplicity a b (α, ξ) = a := by
+  have hω : (∑ α : ZMod a, shiftOmega a b (α, ξ)) = 0 := by
+    unfold shiftOmega
+    dsimp only
+    rw [← Finset.sum_mul (Finset.univ : Finset (ZMod a))
+      (fun α : ZMod a =>
+        (if α = 0 then (1 : ℤ) else 0) -
+          (if α = 1 then (1 : ℤ) else 0))
+      ((if ξ = 0 then (1 : ℤ) else 0) -
+        (if ξ = 1 then (1 : ℤ) else 0))]
+    rw [Finset.sum_sub_distrib]
+    simp
+  have h : (∑ α : ZMod a, (shiftMultiplicity a b (α, ξ) : ℤ)) = a := by
+    simp_rw [shiftMultiplicity_cast ha hb]
+    rw [Finset.sum_add_distrib, hω]
+    simp [ZMod.card]
+  exact_mod_cast h
+
+theorem sum_shiftMultiplicity_snd (ha : 2 ≤ a) (hb : 2 ≤ b)
+    (α : ZMod a) :
+    ∑ ξ : ZMod b, shiftMultiplicity a b (α, ξ) = b := by
+  have hω : (∑ ξ : ZMod b, shiftOmega a b (α, ξ)) = 0 := by
+    unfold shiftOmega
+    dsimp only
+    rw [← Finset.mul_sum (Finset.univ : Finset (ZMod b))
+      (fun ξ : ZMod b =>
+        (if ξ = 0 then (1 : ℤ) else 0) -
+          (if ξ = 1 then (1 : ℤ) else 0))
+      ((if α = 0 then (1 : ℤ) else 0) -
+        (if α = 1 then (1 : ℤ) else 0))]
+    rw [Finset.sum_sub_distrib]
+    simp
+  have h : (∑ ξ : ZMod b, (shiftMultiplicity a b (α, ξ) : ℤ)) = b := by
+    simp_rw [shiftMultiplicity_cast ha hb]
+    rw [Finset.sum_add_distrib, hω]
+    simp [ZMod.card]
+  exact_mod_cast h
+
+theorem sum_shiftMultiplicity (ha : 2 ≤ a) (hb : 2 ≤ b) :
+    ∑ e : ZMod a × ZMod b, shiftMultiplicity a b e = a * b := by
+  rw [Fintype.sum_prod_type]
+  simp_rw [sum_shiftMultiplicity_snd ha hb]
+  simp [ZMod.card]
+
+theorem card_shiftIndex (ha : 2 ≤ a) (hb : 2 ≤ b) :
+    Fintype.card (ShiftIndex a b) = a * b := by
+  simpa only [ShiftIndex, Fintype.card_sigma, Fintype.card_fin] using
+    sum_shiftMultiplicity ha hb
+
+theorem sum_shift {M : Type uM} [AddCommMonoid M] (f : ZMod a × ZMod b → M) :
+    (∑ δ : ShiftIndex a b, f (shift a b δ)) =
+      ∑ e : ZMod a × ZMod b, shiftMultiplicity a b e • f e := by
+  simp [ShiftIndex, shift, Fintype.sum_sigma]
+
+private def shiftFibreEquiv (e : ZMod a × ZMod b) :
+    {δ : ShiftIndex a b // shift a b δ = e} ≃ Fin (shiftMultiplicity a b e) where
+  toFun δ := by
+    rcases δ with ⟨⟨e', k⟩, h⟩
+    change e' = e at h
+    subst e'
+    exact k
+  invFun k := ⟨⟨e, k⟩, rfl⟩
+  left_inv := by
+    rintro ⟨⟨e', k⟩, h⟩
+    change e' = e at h
+    subst e'
+    rfl
+  right_inv _ := rfl
+
+theorem card_shift_fibre (e : ZMod a × ZMod b) :
+    Fintype.card {δ : ShiftIndex a b // shift a b δ = e} =
+      shiftMultiplicity a b e := by
+  simpa using Fintype.card_congr (shiftFibreEquiv e)
+
+private def shiftFibreAEquiv (ξ : ZMod b) :
+    {δ : ShiftIndex a b // (shift a b δ).2 = ξ} ≃
+      Σ α : ZMod a, Fin (shiftMultiplicity a b (α, ξ)) where
+  toFun δ := by
+    rcases δ with ⟨⟨⟨α, ξ'⟩, k⟩, h⟩
+    change ξ' = ξ at h
+    subst ξ'
+    exact ⟨α, k⟩
+  invFun δ := ⟨⟨(δ.1, ξ), δ.2⟩, rfl⟩
+  left_inv := by
+    rintro ⟨⟨⟨α, ξ'⟩, k⟩, h⟩
+    change ξ' = ξ at h
+    subst ξ'
+    rfl
+  right_inv _ := rfl
+
+private def shiftFibreBEquiv (α : ZMod a) :
+    {δ : ShiftIndex a b // (shift a b δ).1 = α} ≃
+      Σ ξ : ZMod b, Fin (shiftMultiplicity a b (α, ξ)) where
+  toFun δ := by
+    rcases δ with ⟨⟨⟨α', ξ⟩, k⟩, h⟩
+    change α' = α at h
+    subst α'
+    exact ⟨ξ, k⟩
+  invFun δ := ⟨⟨(α, δ.1), δ.2⟩, rfl⟩
+  left_inv := by
+    rintro ⟨⟨⟨α', ξ⟩, k⟩, h⟩
+    change α' = α at h
+    subst α'
+    rfl
+  right_inv _ := rfl
+
+theorem card_shift_fibreA (ha : 2 ≤ a) (hb : 2 ≤ b) (ξ : ZMod b) :
+    Fintype.card {δ : ShiftIndex a b // (shift a b δ).2 = ξ} = a := by
+  calc
+    _ = Fintype.card (Σ α : ZMod a, Fin (shiftMultiplicity a b (α, ξ))) :=
+      Fintype.card_congr (shiftFibreAEquiv ξ)
+    _ = a := by
+      simpa only [Fintype.card_sigma, Fintype.card_fin] using
+        sum_shiftMultiplicity_fst ha hb ξ
+
+theorem card_shift_fibreB (ha : 2 ≤ a) (hb : 2 ≤ b) (α : ZMod a) :
+    Fintype.card {δ : ShiftIndex a b // (shift a b δ).1 = α} = b := by
+  calc
+    _ = Fintype.card (Σ ξ : ZMod b, Fin (shiftMultiplicity a b (α, ξ))) :=
+      Fintype.card_congr (shiftFibreBEquiv α)
+    _ = b := by
+      simpa only [Fintype.card_sigma, Fintype.card_fin] using
+        sum_shiftMultiplicity_snd ha hb α
+
+end Counts
+
+section Orderings
+
+variable {a b : ℕ} [NeZero a] [NeZero b] (ha : 2 ≤ a) (hb : 2 ≤ b)
+
+noncomputable def fibreOrderA (ξ : ZMod b) :
+    {δ : ShiftIndex a b // (shift a b δ).2 = ξ} ≃ ZMod a :=
+  Fintype.equivOfCardEq ((card_shift_fibreA ha hb ξ).trans (ZMod.card a).symm)
+
+noncomputable def fibreOrderB (α : ZMod a) :
+    {δ : ShiftIndex a b // (shift a b δ).1 = α} ≃ ZMod b :=
+  Fintype.equivOfCardEq ((card_shift_fibreB ha hb α).trans (ZMod.card b).symm)
+
+noncomputable def rhoA (δ : ShiftIndex a b) : ZMod a :=
+  fibreOrderA ha hb (shift a b δ).2 ⟨δ, rfl⟩ - (shift a b δ).1
+
+noncomputable def rhoB (δ : ShiftIndex a b) : ZMod b :=
+  fibreOrderB ha hb (shift a b δ).1 ⟨δ, rfl⟩ - (shift a b δ).2
+
+theorem fibreOrderA_apply (ξ : ZMod b)
+    (δ : {δ : ShiftIndex a b // (shift a b δ).2 = ξ}) :
+    fibreOrderA ha hb ξ δ = (shift a b δ.1).1 + rhoA ha hb δ.1 := by
+  rcases δ with ⟨δ, h⟩
+  subst ξ
+  simp [rhoA]
+
+theorem fibreOrderB_apply (α : ZMod a)
+    (δ : {δ : ShiftIndex a b // (shift a b δ).1 = α}) :
+    fibreOrderB ha hb α δ = (shift a b δ.1).2 + rhoB ha hb δ.1 := by
+  rcases δ with ⟨δ, h⟩
+  subst α
+  simp [rhoB]
+
+noncomputable def shiftOrderA : ShiftIndex a b ≃ ZMod b × ZMod a :=
+  (Equiv.sigmaFiberEquiv (fun δ : ShiftIndex a b => (shift a b δ).2)).symm.trans
+    (Equiv.sigmaEquivProdOfEquiv (fibreOrderA ha hb))
+
+noncomputable def shiftOrderB : ShiftIndex a b ≃ ZMod a × ZMod b :=
+  (Equiv.sigmaFiberEquiv (fun δ : ShiftIndex a b => (shift a b δ).1)).symm.trans
+    (Equiv.sigmaEquivProdOfEquiv (fibreOrderB ha hb))
+
+theorem shiftOrderA_apply (δ : ShiftIndex a b) :
+    shiftOrderA ha hb δ = ((shift a b δ).2, (shift a b δ).1 + rhoA ha hb δ) := by
+  change ((shift a b δ).2, fibreOrderA ha hb (shift a b δ).2 ⟨δ, rfl⟩) = _
+  exact congrArg (fun z : ZMod a => ((shift a b δ).2, z))
+    (fibreOrderA_apply ha hb (shift a b δ).2 ⟨δ, rfl⟩)
+
+theorem shiftOrderB_apply (δ : ShiftIndex a b) :
+    shiftOrderB ha hb δ = ((shift a b δ).1, (shift a b δ).2 + rhoB ha hb δ) := by
+  change ((shift a b δ).1, fibreOrderB ha hb (shift a b δ).1 ⟨δ, rfl⟩) = _
+  exact congrArg (fun z : ZMod b => ((shift a b δ).1, z))
+    (fibreOrderB_apply ha hb (shift a b δ).1 ⟨δ, rfl⟩)
+
+theorem bijective_fibreOrderA (ξ : ZMod b) :
+    Function.Bijective (fun δ : {δ : ShiftIndex a b // (shift a b δ).2 = ξ} =>
+      (shift a b δ.1).1 + rhoA ha hb δ.1) := by
+  have he : (fun δ : {δ : ShiftIndex a b // (shift a b δ).2 = ξ} =>
+      (shift a b δ.1).1 + rhoA ha hb δ.1) = (fibreOrderA ha hb ξ) := by
+    funext δ
+    exact (fibreOrderA_apply ha hb ξ δ).symm
+  rw [he]
+  exact (fibreOrderA ha hb ξ).bijective
+
+theorem bijective_fibreOrderB (α : ZMod a) :
+    Function.Bijective (fun δ : {δ : ShiftIndex a b // (shift a b δ).1 = α} =>
+      (shift a b δ.1).2 + rhoB ha hb δ.1) := by
+  have he : (fun δ : {δ : ShiftIndex a b // (shift a b δ).1 = α} =>
+      (shift a b δ.1).2 + rhoB ha hb δ.1) = (fibreOrderB ha hb α) := by
+    funext δ
+    exact (fibreOrderB_apply ha hb α δ).symm
+  rw [he]
+  exact (fibreOrderB ha hb α).bijective
+
+end Orderings
+
+end PeriodicTilingThree
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/Stacking.lean b/lean/OAI/Geometry/PeriodicTiling/Stacking.lean
new file mode 100644
index 0000000000000000000000000000000000000000..b63a5e2eadaf19d1fe2ea47c2fe1ddb61240d5d4
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/Stacking.lean
@@ -0,0 +1,93 @@
+import OAI.Geometry.PeriodicTiling.StackingContributions
+import OAI.Geometry.PeriodicTiling.TilingTransport
+
+namespace OAI
+
+universe uH uK uι
+
+namespace PeriodicTilingThree
+
+section Stacking
+
+variable {H : Type uH} {K : Type uK} {ι : Type uι} [AddCommGroup H] [AddCommGroup K]
+variable [Fintype K] [Fintype ι]
+
+theorem tiles_stack_lift
+    {H : Type uH} {K : Type uK} {ι : Type uι}
+    [AddCommGroup H] [AddCommGroup K] [Fintype K] [Fintype ι]
+    {F : ι → Finset H} {E : ι → Finset K} {A : Set H}
+    (hpart : IsColorPartition E) (hA : ∀ ν, Tiles (F ν) A) :
+    Tiles (stack F E) (A ×ˢ ({0} : Set K)) := by
+  classical
+  apply tiles_iff_unique_tile.mpr
+  intro x
+  obtain ⟨ν, hν, _⟩ := hpart x.2
+  obtain ⟨f, hf, huniq⟩ := tiles_iff_unique_tile.mp (hA ν) x.1
+  refine ⟨⟨((f : H), x.2), mem_stack.mpr ⟨ν, f.property, hν⟩⟩, ?_, ?_⟩
+  · change x.1 - (f : H) ∈ A ∧ x.2 - x.2 ∈ ({0} : Set K)
+    exact ⟨hf, by simp⟩
+  · intro y hy
+    change x.1 - y.1.1 ∈ A ∧ x.2 - y.1.2 ∈ ({0} : Set K) at hy
+    obtain ⟨μ, hyF, hyE⟩ := mem_stack.mp y.property
+    have hy2 : y.1.2 = x.2 := (sub_eq_zero.mp (Set.mem_singleton_iff.mp hy.2)).symm
+    have hμν : μ = ν := hpart.eq_of_mem (hy2 ▸ hyE) hν
+    cases hμν
+    have hy1 : (⟨y.1.1, hyF⟩ : ↥(F ν)) = f := huniq ⟨y.1.1, hyF⟩ hy.1
+    apply Subtype.ext
+    exact Prod.ext (congrArg Subtype.val hy1) hy2
+
+theorem fullyPeriodic_stack_lift {A : Set H} (hA : FullyPeriodic A) :
+    FullyPeriodic (A ×ˢ ({0} : Set K)) :=
+  hA.prod_singleton_zero
+
+theorem fullyPeriodic_stack_project
+    {H : Type uH} {K : Type uK}
+    [AddCommGroup H] [AddCommGroup K] [Fintype K]
+    {B : Set (H × K)} (hB : FullyPeriodic B) :
+    FullyPeriodic (Prod.fst '' B) :=
+  FullyPeriodic.image_of_surjective (AddMonoidHom.fst H K)
+    (fun x => ⟨(x, 0), rfl⟩) hB
+
+theorem exists_tiles_stack_iff {F : ι → Finset H} {E : ι → Finset K}
+    (hpart : IsColorPartition E) (hfull : HasFullDifferences E)
+    (hE : ∀ ν, (E ν).Nonempty) :
+    (∃ A : Set H, ∀ ν, Tiles (F ν) A) ↔
+      ∃ B : Set (H × K), Tiles (stack F E) B := by
+  constructor
+  · rintro ⟨A, hA⟩
+    exact ⟨A ×ˢ ({0} : Set K), tiles_stack_lift hpart hA⟩
+  · rintro ⟨B, hB⟩
+    exact ⟨Prod.fst '' B, tiles_of_tiles_stack hpart hfull hE hB⟩
+
+theorem exists_fullyPeriodic_tiles_stack_iff
+    {F : ι → Finset H} {E : ι → Finset K}
+    (hpart : IsColorPartition E) (hfull : HasFullDifferences E)
+    (hE : ∀ ν, (E ν).Nonempty) :
+    (∃ A : Set H, (∀ ν, Tiles (F ν) A) ∧ FullyPeriodic A) ↔
+      ∃ B : Set (H × K), Tiles (stack F E) B ∧ FullyPeriodic B := by
+  constructor
+  · rintro ⟨A, hA, hperiod⟩
+    exact ⟨A ×ˢ ({0} : Set K), tiles_stack_lift hpart hA,
+      fullyPeriodic_stack_lift hperiod⟩
+  · rintro ⟨B, hB, hperiod⟩
+    exact ⟨Prod.fst '' B, tiles_of_tiles_stack hpart hfull hE hB,
+      fullyPeriodic_stack_project hperiod⟩
+
+theorem stack_counterexample_of_common_system [Nonempty ι]
+    {F : ι → Finset H} {E : ι → Finset K}
+    (hF : ∀ ν, (F ν).Nonempty) (hE : ∀ ν, (E ν).Nonempty)
+    (hpart : IsColorPartition E) (hfull : HasFullDifferences E)
+    (hcommon : ∃ A : Set H, ∀ ν, Tiles (F ν) A)
+    (hnone : ∀ A : Set H, (∀ ν, Tiles (F ν) A) → ¬ FullyPeriodic A) :
+    (stack F E).Nonempty ∧ (∃ B, Tiles (stack F E) B) ∧
+      ∀ B, Tiles (stack F E) B → ¬ FullyPeriodic B := by
+  refine ⟨stack_nonempty hF hE, (exists_tiles_stack_iff hpart hfull hE).mp hcommon, ?_⟩
+  intro B hB hperiod
+  exact hnone (Prod.fst '' B) (tiles_of_tiles_stack hpart hfull hE hB)
+    (fullyPeriodic_stack_project hperiod)
+
+end Stacking
+
+end PeriodicTilingThree
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/StackingContributions.lean b/lean/OAI/Geometry/PeriodicTiling/StackingContributions.lean
new file mode 100644
index 0000000000000000000000000000000000000000..cfb6a8ae150c88ad90a1b239b0e688a515b6ff5d
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/StackingContributions.lean
@@ -0,0 +1,275 @@
+import OAI.Geometry.PeriodicTiling.TilingBasic
+import Mathlib.Data.Finset.Prod
+import Mathlib.Data.Finset.Union
+import Mathlib.Data.Fintype.BigOperators
+import Mathlib.Algebra.Order.BigOperators.Group.Finset
+import Mathlib.Tactic.Abel
+import Lean.Elab.Tactic.Omega
+
+namespace OAI
+
+universe uH uK uι
+
+namespace PeriodicTilingThree
+
+section FiniteStack
+
+variable {H : Type uH} {K : Type uK} {ι : Type uι} [Fintype ι]
+
+def IsColorPartition (E : ι → Finset K) : Prop :=
+  ∀ t : K, ∃! ν : ι, t ∈ E ν
+
+def HasFullDifferences [AddGroup K] (E : ι → Finset K) : Prop :=
+  ∀ (ν : ι) (d : K), ∃ e ∈ E ν, ∃ e' ∈ E ν, e - e' = d
+
+noncomputable def stack (F : ι → Finset H) (E : ι → Finset K) :
+    Finset (H × K) := by
+  classical
+  exact Finset.univ.biUnion (fun ν => (F ν).product (E ν))
+
+@[simp] theorem mem_stack {F : ι → Finset H} {E : ι → Finset K} {p : H × K} :
+    p ∈ stack F E ↔ ∃ ν, p.1 ∈ F ν ∧ p.2 ∈ E ν := by
+  classical
+  simp [stack]
+
+theorem stack_nonempty [Nonempty ι] {F : ι → Finset H} {E : ι → Finset K}
+    (hF : ∀ ν, (F ν).Nonempty) (hE : ∀ ν, (E ν).Nonempty) :
+    (stack F E).Nonempty := by
+  obtain ⟨ν⟩ := ‹Nonempty ι›
+  obtain ⟨f, hf⟩ := hF ν
+  obtain ⟨e, he⟩ := hE ν
+  exact ⟨(f, e), mem_stack.mpr ⟨ν, hf, he⟩⟩
+
+theorem IsColorPartition.eq_of_mem
+    {K : Type uK} {ι : Type uι} [Fintype ι]
+    {E : ι → Finset K} (hpart : IsColorPartition E)
+    {ν μ : ι} {e : K} (hν : e ∈ E ν) (hμ : e ∈ E μ) : ν = μ := by
+  obtain ⟨i, _, hi⟩ := hpart e
+  exact (hi ν hν).trans (hi μ hμ).symm
+
+theorem HasFullDifferences.nonempty
+    {K : Type uK} {ι : Type uι} [Fintype ι]
+    [AddGroup K] {E : ι → Finset K}
+    (hfull : HasFullDifferences E) (ν : ι) : (E ν).Nonempty := by
+  obtain ⟨e, he, _, _, _⟩ := hfull ν 0
+  exact ⟨e, he⟩
+
+theorem card_stack {F : ι → Finset H} {E : ι → Finset K}
+    (hpart : IsColorPartition E) :
+    (stack F E).card = ∑ ν, (F ν).card * (E ν).card := by
+  classical
+  have hdisj : ((Finset.univ : Finset ι) : Set ι).PairwiseDisjoint
+      (fun ν => (F ν).product (E ν)) := by
+    intro ν _ μ _ hne
+    apply Finset.disjoint_left.mpr
+    intro p hp hp'
+    exact hne (hpart.eq_of_mem (Finset.mem_product.mp hp).2
+      (Finset.mem_product.mp hp').2)
+  simpa only [stack, Finset.product_eq_sprod, Finset.card_product] using
+    Finset.card_biUnion hdisj
+
+theorem IsColorPartition.sum_card [Fintype K] {E : ι → Finset K}
+    (hpart : IsColorPartition E) : ∑ ν, (E ν).card = Fintype.card K := by
+  classical
+  let f : (Σ ν, ↥(E ν)) → K := fun z => z.2.1
+  have hf : Function.Bijective f := by
+    constructor
+    · rintro ⟨ν, e⟩ ⟨μ, e'⟩ heq
+      change (e : K) = (e' : K) at heq
+      have hνμ : ν = μ := hpart.eq_of_mem e.property (heq.symm ▸ e'.property)
+      cases hνμ
+      have he : e = e' := Subtype.ext heq
+      cases he
+      rfl
+    · intro e
+      obtain ⟨ν, hν, _⟩ := hpart e
+      exact ⟨⟨ν, ⟨e, hν⟩⟩, rfl⟩
+  simpa only [Fintype.card_sigma, Fintype.card_coe] using
+    Fintype.card_congr (Equiv.ofBijective f hf)
+
+end FiniteStack
+
+section Contributions
+
+variable {H : Type uH} {K : Type uK} {ι : Type uι} [AddCommGroup H] [AddCommGroup K]
+variable [Fintype K] [Fintype ι]
+
+noncomputable def stackContributions (F : ι → Finset H) (B : Set (H × K))
+    (g : H) (ν : ι) : Finset (H × K) := by
+  classical
+  exact ((F ν).product Finset.univ).filter (fun ft => (g - ft.1, ft.2) ∈ B)
+
+@[simp] theorem mem_stackContributions
+    {H : Type uH} {K : Type uK} {ι : Type uι}
+    [AddCommGroup H] [AddCommGroup K] [Fintype K] [Fintype ι]
+    {F : ι → Finset H} {B : Set (H × K)}
+    {g : H} {ν : ι} {c : H × K} :
+    c ∈ stackContributions F B g ν ↔ c.1 ∈ F ν ∧ (g - c.1, c.2) ∈ B := by
+  classical
+  simp [stackContributions]
+
+theorem stackContributions_eq_of_mem {F : ι → Finset H} {E : ι → Finset K}
+    {B : Set (H × K)} (hfull : HasFullDifferences E) (hB : Tiles (stack F E) B)
+    {g : H} {ν : ι} {c c' : H × K}
+    (hc : c ∈ stackContributions F B g ν)
+    (hc' : c' ∈ stackContributions F B g ν) : c = c' := by
+  obtain ⟨hcf, hcB⟩ := mem_stackContributions.mp hc
+  obtain ⟨hc'f, hc'B⟩ := mem_stackContributions.mp hc'
+  obtain ⟨e, he, e', he', hd⟩ := hfull ν (c'.2 - c.2)
+  have hvertical : c.2 + e = c'.2 + e' := by
+    calc
+      c.2 + e = c.2 + (e - e') + e' := by abel
+      _ = c.2 + (c'.2 - c.2) + e' := by rw [hd]
+      _ = c'.2 + e' := by abel
+  let p : ↥(stack F E) × B :=
+    (⟨(c.1, e), mem_stack.mpr ⟨ν, hcf, he⟩⟩, ⟨(g - c.1, c.2), hcB⟩)
+  let p' : ↥(stack F E) × B :=
+    (⟨(c'.1, e'), mem_stack.mpr ⟨ν, hc'f, he'⟩⟩, ⟨(g - c'.1, c'.2), hc'B⟩)
+  have hp : p = p' := by
+    apply hB.1
+    apply Prod.ext
+    · change c.1 + (g - c.1) = c'.1 + (g - c'.1)
+      simp
+    · change e + c.2 = e' + c'.2
+      simpa only [add_comm] using hvertical
+  apply Prod.ext
+  · exact congrArg (fun z : ↥(stack F E) × B => z.1.1.1) hp
+  · exact congrArg (fun z : ↥(stack F E) × B => z.2.1.2) hp
+
+theorem stackContributions_card_le_one {F : ι → Finset H} {E : ι → Finset K}
+    {B : Set (H × K)} (hfull : HasFullDifferences E) (hB : Tiles (stack F E) B)
+    (g : H) (ν : ι) : (stackContributions F B g ν).card ≤ 1 := by
+  classical
+  exact Finset.card_le_one.mpr fun _ hc _ hc' =>
+    stackContributions_eq_of_mem hfull hB hc hc'
+
+theorem stack_projection_injective [Nonempty ι]
+    {F : ι → Finset H} {E : ι → Finset K} {B : Set (H × K)}
+    (hF : ∀ ν, (F ν).Nonempty) (hfull : HasFullDifferences E)
+    (hB : Tiles (stack F E) B) : Set.InjOn Prod.fst B := by
+  obtain ⟨ν⟩ := ‹Nonempty ι›
+  obtain ⟨f, hf⟩ := hF ν
+  intro b hb b' hb' heq
+  have hc : (f, b.2) ∈ stackContributions F B (b.1 + f) ν := by
+    apply mem_stackContributions.mpr
+    exact ⟨hf, by simpa using hb⟩
+  have hc' : (f, b'.2) ∈ stackContributions F B (b.1 + f) ν := by
+    apply mem_stackContributions.mpr
+    refine ⟨hf, ?_⟩
+    simpa [heq] using hb'
+  have ht := congrArg Prod.snd (stackContributions_eq_of_mem hfull hB hc hc')
+  exact Prod.ext heq ht
+
+def StackFiber (F : ι → Finset H) (E : ι → Finset K)
+    (B : Set (H × K)) (g : H) :=
+  Σ ν : ι, ↥(stackContributions F B g ν) × ↥(E ν)
+
+def stackFiberSum {F : ι → Finset H} {E : ι → Finset K}
+    {B : Set (H × K)} {g : H} (z : StackFiber F E B g) : K :=
+  z.2.1.1.2 + z.2.2.1
+
+theorem stackFiberSum_bijective {F : ι → Finset H} {E : ι → Finset K}
+    {B : Set (H × K)} (hpart : IsColorPartition E) (hB : Tiles (stack F E) B)
+    (g : H) : Function.Bijective (stackFiberSum (F := F) (E := E) (B := B) (g := g)) := by
+  classical
+  constructor
+  · rintro ⟨ν, c, e⟩ ⟨μ, c', e'⟩ heq
+    obtain ⟨hcf, hcB⟩ := mem_stackContributions.mp c.property
+    obtain ⟨hc'f, hc'B⟩ := mem_stackContributions.mp c'.property
+    let p : ↥(stack F E) × B :=
+      (⟨(c.1.1, e.1), mem_stack.mpr ⟨ν, hcf, e.property⟩⟩,
+        ⟨(g - c.1.1, c.1.2), hcB⟩)
+    let p' : ↥(stack F E) × B :=
+      (⟨(c'.1.1, e'.1), mem_stack.mpr ⟨μ, hc'f, e'.property⟩⟩,
+        ⟨(g - c'.1.1, c'.1.2), hc'B⟩)
+    have hp : p = p' := by
+      apply hB.1
+      apply Prod.ext
+      · change c.1.1 + (g - c.1.1) = c'.1.1 + (g - c'.1.1)
+        simp
+      · change e.1 + c.1.2 = e'.1 + c'.1.2
+        simpa only [stackFiberSum, add_comm] using heq
+    have hf : c.1.1 = c'.1.1 :=
+      congrArg (fun z : ↥(stack F E) × B => z.1.1.1) hp
+    have ht : c.1.2 = c'.1.2 :=
+      congrArg (fun z : ↥(stack F E) × B => z.2.1.2) hp
+    have he : (e : K) = (e' : K) :=
+      congrArg (fun z : ↥(stack F E) × B => z.1.1.2) hp
+    have hνμ : ν = μ := hpart.eq_of_mem e.property (he.symm ▸ e'.property)
+    cases hνμ
+    have hc : c = c' := Subtype.ext (Prod.ext hf ht)
+    have he' : e = e' := Subtype.ext he
+    cases hc
+    cases he'
+    rfl
+  · intro u
+    obtain ⟨p, hp⟩ := hB.2 (g, u)
+    obtain ⟨ν, hf, he⟩ := mem_stack.mp p.1.property
+    have hhorizontal : p.1.1.1 + p.2.1.1 = g := congrArg Prod.fst hp
+    have hcenter : g - p.1.1.1 = p.2.1.1 := by
+      rw [← hhorizontal]
+      simp
+    have hc : (p.1.1.1, p.2.1.2) ∈ stackContributions F B g ν := by
+      apply mem_stackContributions.mpr
+      exact ⟨hf, by simpa only [hcenter] using p.2.property⟩
+    refine ⟨⟨ν, ⟨(p.1.1.1, p.2.1.2), hc⟩, ⟨p.1.1.2, he⟩⟩, ?_⟩
+    change p.2.1.2 + p.1.1.2 = u
+    have hvertical : p.1.1.2 + p.2.1.2 = u := congrArg Prod.snd hp
+    simpa only [add_comm] using hvertical
+
+theorem stackContributions_sum_card {F : ι → Finset H} {E : ι → Finset K}
+    {B : Set (H × K)} (hpart : IsColorPartition E) (hB : Tiles (stack F E) B)
+    (g : H) :
+    ∑ ν, (stackContributions F B g ν).card * (E ν).card = Fintype.card K := by
+  classical
+  let e : (Σ ν : ι, ↥(stackContributions F B g ν) × ↥(E ν)) ≃ K :=
+    Equiv.ofBijective (stackFiberSum (F := F) (E := E) (B := B) (g := g))
+      (stackFiberSum_bijective hpart hB g)
+  simpa only [Fintype.card_sigma, Fintype.card_prod, Fintype.card_coe] using
+    Fintype.card_congr e
+
+theorem stackContributions_card_eq_one {F : ι → Finset H} {E : ι → Finset K}
+    {B : Set (H × K)} (hpart : IsColorPartition E) (hfull : HasFullDifferences E)
+    (hE : ∀ ν, (E ν).Nonempty) (hB : Tiles (stack F E) B)
+    (g : H) (ν : ι) : (stackContributions F B g ν).card = 1 := by
+  classical
+  have hle := stackContributions_card_le_one hfull hB g ν
+  by_contra hne
+  have hz : (stackContributions F B g ν).card = 0 := by omega
+  have hstrict :
+      (∑ i, (stackContributions F B g i).card * (E i).card) < ∑ i, (E i).card := by
+    apply Finset.sum_lt_sum
+    · intro i _
+      calc
+        (stackContributions F B g i).card * (E i).card ≤ 1 * (E i).card :=
+          Nat.mul_le_mul_right _ (stackContributions_card_le_one hfull hB g i)
+        _ = (E i).card := one_mul _
+    · exact ⟨ν, Finset.mem_univ ν, by simpa only [hz, zero_mul] using (hE ν).card_pos⟩
+  rw [stackContributions_sum_card hpart hB g, hpart.sum_card] at hstrict
+  exact (lt_irrefl _) hstrict
+
+theorem tiles_of_tiles_stack {F : ι → Finset H} {E : ι → Finset K}
+    {B : Set (H × K)} (hpart : IsColorPartition E) (hfull : HasFullDifferences E)
+    (hE : ∀ ν, (E ν).Nonempty) (hB : Tiles (stack F E) B) :
+    ∀ ν, Tiles (F ν) (Prod.fst '' B) := by
+  classical
+  intro ν
+  apply tiles_iff_unique_tile.mpr
+  intro g
+  have hcard := stackContributions_card_eq_one hpart hfull hE hB g ν
+  obtain ⟨c, hc⟩ := Finset.card_pos.mp (by omega : 0 < (stackContributions F B g ν).card)
+  obtain ⟨hcf, hcB⟩ := mem_stackContributions.mp hc
+  refine ⟨⟨c.1, hcf⟩, ⟨(g - c.1, c.2), hcB, rfl⟩, ?_⟩
+  intro f hf
+  obtain ⟨b, hb, hbf⟩ := hf
+  have hc' : ((f : H), b.2) ∈ stackContributions F B g ν := by
+    apply mem_stackContributions.mpr
+    exact ⟨f.property, by simpa only [← hbf] using hb⟩
+  apply Subtype.ext
+  exact congrArg Prod.fst (stackContributions_eq_of_mem hfull hB hc' hc)
+
+end Contributions
+
+end PeriodicTilingThree
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/SystemObstruction.lean b/lean/OAI/Geometry/PeriodicTiling/SystemObstruction.lean
new file mode 100644
index 0000000000000000000000000000000000000000..fb457c7fd6e4894237fff5c4a28d8321ab55c496
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/SystemObstruction.lean
@@ -0,0 +1,35 @@
+import OAI.Geometry.PeriodicTiling.ActivityWord
+import OAI.Geometry.PeriodicTiling.SeedSymbols
+import OAI.Geometry.PeriodicTiling.WordObstruction
+
+namespace OAI
+
+namespace PeriodicTilingThree
+
+variable {p : ℕ} [NeZero p] (E : EncodingParameters p)
+
+theorem graph_not_fully_periodic_of_activity
+    (o : GraphOutputs E) (hdep : HasDependence E o)
+    (hwords : ∀ x (w : Word p), (∀ n, Active E o (.inl n) x (w n)) → Allowed p w)
+    (hseeds : ∀ (t : Fin 2) (n : Column p) (x : Plane) (j : Symbol p),
+      Active E o (.inr t) x () → Active E o (.inl n) x j → j = seedSymbol E t)
+    (hordinary : ∀ n x, ∃ j, Active E o (.inl n) x j)
+    (hseedactive : ∀ t : Fin 2, ∃ x : Plane, Active E o (.inr t) x ()) :
+    ¬ FullyPeriodic (graph o) := by
+  intro hper
+  obtain ⟨x, hx⟩ := hseedactive 0
+  obtain ⟨y, hy⟩ := hseedactive 1
+  have hline : LineRule p (selectedWord E o) :=
+    selectedWord_lineRule E o hdep hordinary hwords
+  have hcolumns : NonconstantColumns (selectedWord E o) :=
+    selectedWord_nonconstant_of_forced_letters E o hdep hordinary
+      (seedSymbol E 0) (seedSymbol E 1) (seedSymbol_zero_ne_one E) x y
+      (fun n j hj => hseeds 0 n x j hx hj)
+      (fun n j hj => hseeds 1 n y j hy hj)
+  obtain ⟨M, hM, hperiod⟩ := exists_positive_vertical_graph_period E o hper
+  exact no_positive_vertical_period E.p_prime E.p_large hline hcolumns
+    (Int.natCast_pos.mpr hM) (selectedWord_period_of_graph_period E o (M : ℤ) hperiod)
+
+end PeriodicTilingThree
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/ThreeDimensionalTransfer.lean b/lean/OAI/Geometry/PeriodicTiling/ThreeDimensionalTransfer.lean
new file mode 100644
index 0000000000000000000000000000000000000000..c7e4a17cdc9add27e3c6eb472a904885378f5edf
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/ThreeDimensionalTransfer.lean
@@ -0,0 +1,47 @@
+import OAI.Geometry.PeriodicTiling.TilingTranslation
+import OAI.Geometry.PeriodicTiling.RigidSectionGeometry
+import OAI.Geometry.PeriodicTiling.MarkedTileExistence
+import OAI.Geometry.PeriodicTiling.MarkedTileDescent
+import OAI.Geometry.PeriodicTiling.LatticeThickening
+import OAI.Geometry.PeriodicTiling.LatticePeriodicity
+
+namespace OAI
+
+noncomputable section
+
+namespace PeriodicTilingThree
+
+theorem three_dimensional_tile_of_cyclic_counterexample
+    (Q : ℕ) [NeZero Q] (F : Finset (CyclicQuotient.Group Q))
+    (hF : F.Nonempty) (htile : ∃ A, Tiles F A)
+    (haperiodic : ∀ A, Tiles F A → ¬ FullyPeriodic A) :
+    ∃ T : Finset (Lattice 3),
+      T.Nonempty ∧
+      (∃ A : Set (Lattice 3), Tiles T A) ∧
+      (∀ A : Set (Lattice 3), Tiles T A → ¬ FullyPeriodic A) ∧
+      (∃ A : Set (Space 3), AETiles (Thickening T) A) ∧
+      (∀ A : Set (Space 3), AETiles (Thickening T) A →
+        ¬ EuclideanFullyPeriodic A) := by
+  classical
+  obtain ⟨F₀, hF₀, ⟨B, hB⟩, hBaperiodic⟩ :=
+    exists_zero_mem_normalization F hF htile haperiodic
+  let T : Finset (Lattice 3) := rigidSection.finalTile
+    (CyclicQuotient.kernelStep Q) (CyclicQuotient.representatives Q F₀)
+  let C : Set (Lattice 3) :=
+    (MarkedTile.scale 64) '' ((CyclicQuotient.projection Q) ⁻¹' B)
+  have hT : Tiles T C := rigidSection.cyclic_finalTile_tiles Q hB
+  have hreal : ∀ A : Set (Space 3),
+      AETiles (Thickening T) A → ¬ EuclideanFullyPeriodic A := by
+    intro A hA
+    exact rigidSection.no_euclideanFullyPeriodic (by norm_num)
+      rigidSection_rigidDifferences Q F₀ hF₀ hBaperiodic hA
+  refine ⟨T, hT.tile_nonempty, ⟨C, hT⟩, ?_,
+    ⟨castLattice '' C, hT.aeTiles_thickening⟩, hreal⟩
+  intro A hA hperiodic
+  exact hreal (castLattice '' A) hA.aeTiles_thickening hperiodic.castLattice_image
+
+end PeriodicTilingThree
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/ThreeDimensions.lean b/lean/OAI/Geometry/PeriodicTiling/ThreeDimensions.lean
new file mode 100644
index 0000000000000000000000000000000000000000..d1ae7b54308ac067dbb1f938d8c48625b29051cb
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/ThreeDimensions.lean
@@ -0,0 +1,22 @@
+import OAI.Geometry.PeriodicTiling.CyclicCounterexample
+import OAI.Geometry.PeriodicTiling.ThreeDimensionalTransfer
+
+namespace OAI
+
+namespace PeriodicTilingThree
+
+theorem exists_three_dimensional_tile :
+    ∃ T : Finset (Lattice 3),
+      T.Nonempty ∧
+      (∃ A : Set (Lattice 3), Tiles T A) ∧
+      (∀ A : Set (Lattice 3), Tiles T A → ¬ FullyPeriodic A) ∧
+      (∃ A : Set (Space 3), AETiles (Thickening T) A) ∧
+      (∀ A : Set (Space 3), AETiles (Thickening T) A →
+        ¬ EuclideanFullyPeriodic A) := by
+  obtain ⟨Q, hQ, F, hF, htile, haperiodic⟩ := exists_cyclic_counterexample
+  let : NeZero Q := ⟨Nat.ne_of_gt hQ⟩
+  exact three_dimensional_tile_of_cyclic_counterexample Q F hF htile haperiodic
+
+end PeriodicTilingThree
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/TileDilation.lean b/lean/OAI/Geometry/PeriodicTiling/TileDilation.lean
new file mode 100644
index 0000000000000000000000000000000000000000..fe64dd24bda2d8353ff25dc3381a50f69396ea4c
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/TileDilation.lean
@@ -0,0 +1,193 @@
+import OAI.Geometry.PeriodicTiling.FiniteMasks
+import Mathlib.FieldTheory.Finite.Basic
+import Mathlib.Data.Nat.Factorization.Induction
+import Mathlib.Data.Nat.Factorial.Basic
+import Lean.Elab.Tactic.Omega
+
+namespace OAI
+
+universe uG uι
+
+noncomputable section
+
+open scoped BigOperators Classical
+
+namespace PeriodicTilingThree
+
+section ArbitraryGroup
+
+variable {G : Type uG} [AddCommGroup G]
+
+theorem sum_tileIndicator_nat_le_card
+    {G : Type uG} [AddCommGroup G]
+    {ι : Type uι} (s : Finset ι)
+    (A : Set G) (v : ι → G) :
+    (∑ i ∈ s, tileIndicator ℕ A (v i)) ≤ s.card := by
+  calc
+    (∑ i ∈ s, tileIndicator ℕ A (v i)) ≤ ∑ _i ∈ s, (1 : ℕ) :=
+      Finset.sum_le_sum fun i _hi => tileIndicator_nat_le_one A (v i)
+    _ = s.card := by simp
+
+theorem familyMask_pow_prime {ι : Type uι} (s : Finset ι) (v : ι → G)
+    {p : ℕ} (hp : p.Prime) :
+    familyMask (ZMod p) s v ^ p = familyMask (ZMod p) s (fun i => p • v i) := by
+  let : Fact p.Prime := ⟨hp⟩
+  let : CharP (AddMonoidAlgebra (ZMod p) G) p :=
+    ⟨fun n => by
+      rw [AddMonoidAlgebra.natCast_def, AddMonoidAlgebra.single_eq_zero,
+        CharP.cast_eq_zero_iff (ZMod p) p]⟩
+  unfold familyMask
+  rw [sum_pow_char]
+  apply Finset.sum_congr rfl
+  intro i hi
+  rw [AddMonoidAlgebra.single_pow, one_pow]
+
+theorem sum_tileIndicator_dilate_prime {ι : Type uι} (s : Finset ι)
+    (v : ι → G) (A : Set G)
+    (h : ∀ x, (∑ i ∈ s, tileIndicator ℕ A (x - v i)) = 1)
+    {p : ℕ} (hp : p.Prime) (hcard : s.card < p) (x : G) :
+    (∑ i ∈ s, tileIndicator ℕ A (x - p • v i)) = 1 := by
+  let : Fact p.Prime := ⟨hp⟩
+  have hpos : 0 < s.card := by
+    have hb := sum_tileIndicator_nat_le_card s A (fun i => (0 : G) - v i)
+    rw [h 0] at hb
+    exact lt_of_lt_of_le Nat.zero_lt_one hb
+  have hne : (s.card : ZMod p) ≠ 0 := by
+    intro hz
+    have hv := congrArg ZMod.val hz
+    rw [ZMod.val_natCast_of_lt hcard, ZMod.val_zero] at hv
+    exact (Nat.ne_of_gt hpos) hv
+  have hfermat : (s.card : ZMod p) ^ (p - 1) = 1 := by
+    simpa only [ZMod.card] using
+      (FiniteField.pow_card_sub_one_eq_one (s.card : ZMod p) hne)
+  have hmod : ∀ y, (∑ i ∈ s, tileIndicator (ZMod p) A (y - v i)) = 1 := by
+    intro y
+    rw [← sum_tileIndicator_cast, h y, Nat.cast_one]
+  have hpower : maskAction (familyMask (ZMod p) s v ^ p)
+      (tileIndicator (ZMod p) A) x = (s.card : ZMod p) ^ (p - 1) := by
+    simpa only [Nat.sub_add_cancel hp.one_lt.le] using
+      (maskAction_familyMask_pow_succ s v (tileIndicator (ZMod p) A) hmod (p - 1) x)
+  have hmod_dilate :
+      (∑ i ∈ s, tileIndicator (ZMod p) A (x - p • v i)) = 1 := by
+    rw [← maskAction_familyMask, ← familyMask_pow_prime s v hp, hpower, hfermat]
+  have hc :
+      ((∑ i ∈ s, tileIndicator ℕ A (x - p • v i) : ℕ) : ZMod p) = 1 := by
+    rw [sum_tileIndicator_cast]
+    exact hmod_dilate
+  have hv := congrArg ZMod.val hc
+  have hbound : (∑ i ∈ s, tileIndicator ℕ A (x - p • v i)) < p :=
+    lt_of_le_of_lt (sum_tileIndicator_nat_le_card s A _) hcard
+  rwa [ZMod.val_natCast_of_lt hbound, ZMod.val_one_eq_one_mod,
+    Nat.mod_eq_of_lt hp.one_lt] at hv
+
+theorem sum_tileIndicator_dilate_of_prime_factors {ι : Type uι} (s : Finset ι)
+    (v : ι → G) (A : Set G)
+    (h : ∀ x, (∑ i ∈ s, tileIndicator ℕ A (x - v i)) = 1)
+    (r : ℕ) (hr : r ≠ 0)
+    (hlarge : ∀ p : ℕ, p.Prime → p ∣ r → s.card < p) (x : G) :
+    (∑ i ∈ s, tileIndicator ℕ A (x - r • v i)) = 1 := by
+  have hall : ∀ n : ℕ, n ≠ 0 →
+      (∀ p : ℕ, p.Prime → p ∣ n → s.card < p) →
+      ∀ y, (∑ i ∈ s, tileIndicator ℕ A (y - n • v i)) = 1 := by
+    apply induction_on_primes
+    · intro hn
+      exact (hn rfl).elim
+    · intro _hn _hpr y
+      simpa only [one_nsmul] using h y
+    · intro p n hp ih hpn hpr y
+      have hn : n ≠ 0 := by
+        intro hn
+        apply hpn
+        simp [hn]
+      have hprn : ∀ q : ℕ, q.Prime → q ∣ n → s.card < q := by
+        intro q hq hqn
+        exact hpr q hq (dvd_mul_of_dvd_right hqn p)
+      have hpp : s.card < p := hpr p hp (dvd_mul_right p n)
+      simpa only [mul_smul] using
+        (sum_tileIndicator_dilate_prime s (fun i => n • v i) A
+          (ih hn hprn) hp hpp y)
+  exact hall r hr hlarge x
+
+theorem prime_factor_factorial_mul_add_one_gt (N k p : ℕ)
+    (hp : p.Prime) (hd : p ∣ 1 + N.factorial * k) : N < p := by
+  by_contra hnp
+  have hpf : p ∣ N.factorial := Nat.dvd_factorial hp.pos (Nat.le_of_not_gt hnp)
+  have hpm : p ∣ N.factorial * k := dvd_mul_of_dvd_left hpf k
+  have hpone : p ∣ 1 := (Nat.dvd_add_left hpm).mp hd
+  exact hp.ne_one (Nat.dvd_one.mp hpone)
+
+theorem Tiles.sum_tileIndicator_nat_dilate_factorial {F : Finset G} {A : Set G}
+    (h : Tiles F A) (k : ℕ) (x : G) :
+    (∑ f ∈ F, tileIndicator ℕ A (x - (1 + F.card.factorial * k) • f)) = 1 := by
+  apply sum_tileIndicator_dilate_of_prime_factors F id A
+    (fun y => h.tileCount_eq_one y) (1 + F.card.factorial * k)
+  · omega
+  · intro p hp hd
+    exact prime_factor_factorial_mul_add_one_gt F.card k p hp hd
+
+theorem Tiles.sum_tileIndicator_dilate_factorial {F : Finset G} {A : Set G}
+    (h : Tiles F A) (k : ℕ) (x : G) :
+    (∑ f ∈ F, tileIndicator ℝ A (x - (1 + F.card.factorial * k) • f)) = 1 := by
+  rw [← sum_tileIndicator_cast, h.sum_tileIndicator_nat_dilate_factorial k x,
+    Nat.cast_one]
+
+def dilateTile (r : ℕ) (F : Finset G) : Finset G :=
+  F.image (fun f => r • f)
+
+end ArbitraryGroup
+
+theorem plane_nsmul_injective {r : ℕ} (hr : r ≠ 0) :
+    Function.Injective (fun x : Plane => r • x) := by
+  intro x y hxy
+  have hrz : (r : ℤ) ≠ 0 := Int.natCast_ne_zero.mpr hr
+  apply Prod.ext
+  · apply mul_left_cancel₀ hrz
+    have hx : r • x.1 = r • y.1 := congrArg Prod.fst hxy
+    simpa only [nsmul_eq_mul] using hx
+  · apply mul_left_cancel₀ hrz
+    have hy : r • x.2 = r • y.2 := congrArg Prod.snd hxy
+    simpa only [nsmul_eq_mul] using hy
+
+theorem dilateTile_card (r : ℕ) (hr : r ≠ 0) (F : Finset Plane) :
+    (dilateTile r F).card = F.card := by
+  let : DecidableEq Plane := fun a b => Classical.propDecidable (a = b)
+  exact Finset.card_image_of_injective F (plane_nsmul_injective hr)
+
+theorem tileCount_dilateTile (r : ℕ) (hr : r ≠ 0)
+    (F : Finset Plane) (A : Set Plane) (x : Plane) :
+    tileCount (dilateTile r F) A x = ∑ f ∈ F, tileIndicator ℕ A (x - r • f) := by
+  let : DecidableEq Plane := fun a b => Classical.propDecidable (a = b)
+  unfold tileCount dilateTile
+  apply Finset.sum_image
+  intro f hf g hg hfg
+  exact plane_nsmul_injective hr hfg
+
+theorem Tiles.dilate_prime {F : Finset Plane} {A : Set Plane} (h : Tiles F A)
+    {p : ℕ} (hp : p.Prime) (hcard : F.card < p) : Tiles (dilateTile p F) A := by
+  apply tiles_iff_tileCount_eq_one.mpr
+  intro x
+  rw [tileCount_dilateTile p hp.ne_zero]
+  exact sum_tileIndicator_dilate_prime F id A (fun y => h.tileCount_eq_one y) hp hcard x
+
+theorem Tiles.dilate_of_prime_factors {F : Finset Plane} {A : Set Plane}
+    (h : Tiles F A) (r : ℕ) (hr : r ≠ 0)
+    (hlarge : ∀ p : ℕ, p.Prime → p ∣ r → F.card < p) :
+    Tiles (dilateTile r F) A := by
+  apply tiles_iff_tileCount_eq_one.mpr
+  intro x
+  rw [tileCount_dilateTile r hr]
+  exact sum_tileIndicator_dilate_of_prime_factors F id A
+    (fun y => h.tileCount_eq_one y) r hr hlarge x
+
+theorem Tiles.dilate_factorial {F : Finset Plane} {A : Set Plane}
+    (h : Tiles F A) (k : ℕ) : Tiles (dilateTile (1 + F.card.factorial * k) F) A := by
+  apply h.dilate_of_prime_factors (1 + F.card.factorial * k)
+  · omega
+  · intro p hp hd
+    exact prime_factor_factorial_mul_add_one_gt F.card k p hp hd
+
+end PeriodicTilingThree
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/TilesRegion.lean b/lean/OAI/Geometry/PeriodicTiling/TilesRegion.lean
new file mode 100644
index 0000000000000000000000000000000000000000..2648c942f48d32245487c64f98570a5bc8aee7a6
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/TilesRegion.lean
@@ -0,0 +1,59 @@
+import OAI.Geometry.PeriodicTiling.TilingBasic
+
+namespace OAI
+
+universe uG
+
+namespace PeriodicTilingThree
+
+section Region
+
+variable {G : Type uG} [AddCommGroup G]
+
+structure TilesRegion (F : Finset G) (D E : Set G) : Prop where
+  add_mem : ∀ f ∈ F, ∀ d ∈ D, f + d ∈ E
+  unique : ∀ x ∈ E, ∃! f : ↥F, x - (f : G) ∈ D
+
+theorem tilesRegion_univ_iff {F : Finset G} {D : Set G} :
+    TilesRegion F D Set.univ ↔ Tiles F D := by
+  rw [tiles_iff_unique_tile]
+  constructor
+  · intro h x
+    exact h.unique x (Set.mem_univ x)
+  · intro h
+    exact ⟨fun _ _ _ _ => Set.mem_univ _, fun x _ => h x⟩
+
+theorem TilesRegion.mem_iff {F : Finset G} {D E : Set G}
+    (h : TilesRegion F D E) (x : G) :
+    x ∈ E ↔ ∃ f : ↥F, x - (f : G) ∈ D := by
+  constructor
+  · intro hx
+    obtain ⟨f, hf, _⟩ := h.unique x hx
+    exact ⟨f, hf⟩
+  · rintro ⟨f, hf⟩
+    simpa [add_comm] using h.add_mem f f.property (x - f) hf
+
+theorem TilesRegion.of_local_copy {F : Finset G} {D D' E : Set G}
+    (h : TilesRegion F D E)
+    (hcopy : ∀ x : G, ∃ y : G, (y ∈ E ↔ x ∈ E) ∧
+      ∀ f : ↥F, (x - (f : G) ∈ D' ↔ y - (f : G) ∈ D)) :
+    TilesRegion F D' E := by
+  constructor
+  · intro f hf d hd
+    obtain ⟨y, hE, htests⟩ := hcopy (f + d)
+    apply hE.mp
+    apply (h.mem_iff y).mpr
+    refine ⟨⟨f, hf⟩, (htests ⟨f, hf⟩).mp ?_⟩
+    simpa using hd
+  · intro x hx
+    obtain ⟨y, hE, htests⟩ := hcopy x
+    obtain ⟨f, hf, huniq⟩ := h.unique y (hE.mpr hx)
+    refine ⟨f, (htests f).mpr hf, ?_⟩
+    intro g hg
+    exact huniq g ((htests g).mp hg)
+
+end Region
+
+end PeriodicTilingThree
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/TilingBasic.lean b/lean/OAI/Geometry/PeriodicTiling/TilingBasic.lean
new file mode 100644
index 0000000000000000000000000000000000000000..b4d275421c4df218636f4322ab9fceeeaafde9ed
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/TilingBasic.lean
@@ -0,0 +1,95 @@
+import OAI.Geometry.PeriodicTiling.Model
+import Mathlib.GroupTheory.Index
+import Mathlib.Logic.Equiv.Fin.Basic
+import Mathlib.Tactic.FinCases
+
+namespace OAI
+
+universe uG
+
+namespace PeriodicTilingThree
+
+abbrev Plane := ℤ × ℤ
+
+def planeEquiv : Plane ≃+ Lattice 2 where
+  toEquiv := (finTwoArrowEquiv ℤ).symm
+  map_add' p q := by
+    ext i
+    fin_cases i <;> rfl
+
+section AdditiveGroup
+
+variable {G : Type uG} [AddCommGroup G]
+
+theorem tiles_iff_unique_tile {F : Finset G} {A : Set G} :
+    Tiles F A ↔ ∀ x : G, ∃! f : ↥F, x - (f : G) ∈ A := by
+  constructor
+  · intro h x
+    obtain ⟨⟨f, a⟩, hx⟩ := h.2 x
+    have hfa : x - (f : G) = (a : G) := by
+      rw [← hx]
+      simp
+    refine ⟨f, ?_, ?_⟩
+    · change x - (f : G) ∈ A
+      rw [hfa]
+      exact a.property
+    · intro g hg
+      have hp : (g, (⟨x - (g : G), hg⟩ : A)) = (f, a) := by
+        apply h.1
+        change (g : G) + (x - (g : G)) = (f : G) + (a : G)
+        simpa [add_comm] using hx.symm
+      exact congrArg Prod.fst hp
+  · intro h
+    constructor
+    · rintro ⟨f, a⟩ ⟨g, b⟩ he
+      obtain ⟨f₀, _hf₀, huniq⟩ := h ((f : G) + (a : G))
+      have hf : (f : G) + (a : G) - (f : G) ∈ A := by
+        simp [a.property]
+      have hg : (f : G) + (a : G) - (g : G) ∈ A := by
+        change (f : G) + (a : G) = (g : G) + (b : G) at he
+        rw [he]
+        simp [b.property]
+      have hfg : f = g := (huniq f hf).trans (huniq g hg).symm
+      cases hfg
+      have hab : a = b := Subtype.ext (add_left_cancel he)
+      exact Prod.ext rfl hab
+    · intro x
+      obtain ⟨f, hf, _huniq⟩ := h x
+      refine ⟨(f, ⟨x - (f : G), hf⟩), ?_⟩
+      change (f : G) + (x - (f : G)) = x
+      simp
+
+theorem Tiles.tile_nonempty {F : Finset G} {A : Set G} (h : Tiles F A) :
+    F.Nonempty := by
+  obtain ⟨⟨f, _a⟩, _hx⟩ := h.2 0
+  exact ⟨f, f.property⟩
+
+theorem Tiles.complement_nonempty {F : Finset G} {A : Set G} (h : Tiles F A) :
+    A.Nonempty := by
+  obtain ⟨⟨_f, a⟩, _hx⟩ := h.2 0
+  exact ⟨a, a.property⟩
+
+theorem period_zero (A : Set G) : Period A 0 := by
+  intro x
+  simp
+
+theorem Period.add {A : Set G} {v w : G} (hv : Period A v) (hw : Period A w) :
+    Period A (v + w) := by
+  intro x
+  rw [← add_assoc, hw, hv]
+
+theorem Period.neg {A : Set G} {v : G} (hv : Period A v) : Period A (-v) := by
+  intro x
+  simpa [sub_eq_add_neg, add_assoc] using (hv (x - v)).symm
+
+def periodSubgroup (A : Set G) : AddSubgroup G where
+  carrier := {v | Period A v}
+  zero_mem' := period_zero A
+  add_mem' := fun hv hw => hv.add hw
+  neg_mem' := fun hv => hv.neg
+
+end AdditiveGroup
+
+end PeriodicTilingThree
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/TilingTranslation.lean b/lean/OAI/Geometry/PeriodicTiling/TilingTranslation.lean
new file mode 100644
index 0000000000000000000000000000000000000000..371bdf188cf5d844f8ac5c2ff4af0f6e5e44c3d1
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/TilingTranslation.lean
@@ -0,0 +1,59 @@
+import OAI.Geometry.PeriodicTiling.TilingBasic
+import Mathlib.Tactic.Abel
+
+namespace OAI
+
+universe uG
+
+namespace PeriodicTilingThree
+
+variable {G : Type uG} [AddCommGroup G] [DecidableEq G]
+
+theorem Tiles.translate_tile {F : Finset G} {A : Set G}
+    (h : Tiles F A) (v : G) : Tiles (F.image (fun f => f + v)) A := by
+  classical
+  apply tiles_iff_unique_tile.mpr
+  intro x
+  obtain ⟨f, hf, huniq⟩ := tiles_iff_unique_tile.mp h (x - v)
+  refine ⟨⟨(f : G) + v, Finset.mem_image.mpr ⟨f, f.property, rfl⟩⟩, ?_, ?_⟩
+  · change x - ((f : G) + v) ∈ A
+    have heq : x - ((f : G) + v) = x - v - (f : G) := by abel
+    rw [heq]
+    exact hf
+  · intro g hg
+    obtain ⟨f', hf', hfg⟩ := Finset.mem_image.mp g.property
+    have heq : x - (f' + v) = x - v - f' := by abel
+    change x - (g : G) ∈ A at hg
+    rw [← hfg, heq] at hg
+    have hff : (⟨f', hf'⟩ : ↥F) = f := huniq _ hg
+    apply Subtype.ext
+    change (g : G) = (f : G) + v
+    rw [← hfg]
+    exact congrArg (fun z : G => z + v) (congrArg Subtype.val hff)
+
+theorem tiles_translate_tile_iff (F : Finset G) (A : Set G) (v : G) :
+    Tiles (F.image (fun f => f + v)) A ↔ Tiles F A := by
+  constructor
+  · intro h
+    have h' := h.translate_tile (-v)
+    simpa only [Finset.image_image, Function.comp_def, add_neg_cancel_right, Finset.image_id'] using h'
+  · intro h
+    exact h.translate_tile v
+
+theorem exists_zero_mem_normalization (F : Finset G) (hF : F.Nonempty)
+    (htile : ∃ A, Tiles F A)
+    (haperiodic : ∀ A, Tiles F A → ¬ FullyPeriodic A) :
+    ∃ U : Finset G, 0 ∈ U ∧
+      (∃ A, Tiles U A) ∧ (∀ A, Tiles U A → ¬ FullyPeriodic A) := by
+  classical
+  obtain ⟨f0, hf0⟩ := hF
+  refine ⟨F.image (fun f => f + -f0), ?_, ?_, ?_⟩
+  · exact Finset.mem_image.mpr ⟨f0, hf0, add_neg_cancel f0⟩
+  · obtain ⟨A, hA⟩ := htile
+    exact ⟨A, hA.translate_tile (-f0)⟩
+  · intro A hA
+    exact haperiodic A ((tiles_translate_tile_iff F A (-f0)).mp hA)
+
+end PeriodicTilingThree
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/TilingTransport.lean b/lean/OAI/Geometry/PeriodicTiling/TilingTransport.lean
new file mode 100644
index 0000000000000000000000000000000000000000..9887cb86c3f8a18b0f1edb7b3e68fe0910ee4032
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/TilingTransport.lean
@@ -0,0 +1,151 @@
+import OAI.Geometry.PeriodicTiling.TilingBasic
+import Mathlib.Logic.Equiv.Set
+
+namespace OAI
+
+universe uG uH uK
+
+namespace PeriodicTilingThree
+
+section AdditiveEquivalence
+
+variable {G : Type uG} {H : Type uH} [AddCommGroup G] [AddCommGroup H]
+
+theorem Tiles.image_addEquiv [DecidableEq H] {F : Finset G} {A : Set G}
+    (e : G ≃+ H) (h : Tiles F A) : Tiles (F.image e) (e '' A) := by
+  classical
+  apply tiles_iff_unique_tile.mpr
+  intro x
+  obtain ⟨f, hf, huniq⟩ := tiles_iff_unique_tile.mp h (e.symm x)
+  refine ⟨⟨e (f : G), Finset.mem_image.mpr ⟨(f : G), f.property, rfl⟩⟩, ?_, ?_⟩
+  · refine ⟨e.symm x - (f : G), hf, ?_⟩
+    change e (e.symm x - (f : G)) = x - e (f : G)
+    rw [map_sub, e.apply_symm_apply]
+  · intro g hg
+    obtain ⟨y, hyF, hey⟩ := Finset.mem_image.mp g.property
+    obtain ⟨a, ha, hea⟩ := hg
+    have hyA : e.symm x - y ∈ A := by
+      have heq : e.symm x - y = a := by
+        apply e.injective
+        rw [map_sub, e.apply_symm_apply, hey]
+        exact hea.symm
+      rw [heq]
+      exact ha
+    have hyf : (⟨y, hyF⟩ : ↥F) = f := huniq ⟨y, hyF⟩ hyA
+    apply Subtype.ext
+    calc
+      (g : H) = e y := hey.symm
+      _ = e (f : G) := congrArg e (congrArg Subtype.val hyf)
+
+theorem tiles_image_addEquiv_iff [DecidableEq H] (e : G ≃+ H)
+    {F : Finset G} {A : Set G} :
+    Tiles (F.image e) (e '' A) ↔ Tiles F A := by
+  classical
+  constructor
+  · intro h
+    simpa only [Finset.image_image, Set.image_image, Function.comp_def,
+      e.symm_apply_apply, Finset.image_id', Set.image_id']
+      using Tiles.image_addEquiv e.symm h
+  · exact Tiles.image_addEquiv e
+
+theorem addEquiv_image_symm_eq_preimage (e : G ≃+ H) (A : Set H) :
+    e.symm '' A = e ⁻¹' A := by
+  exact e.toEquiv.image_symm_eq_preimage A
+
+theorem Tiles.preimage_addEquiv [DecidableEq G] (e : G ≃+ H)
+    {F : Finset H} {A : Set H} (h : Tiles F A) :
+    Tiles (F.image e.symm) (e ⁻¹' A) := by
+  rw [← addEquiv_image_symm_eq_preimage e A]
+  exact Tiles.image_addEquiv e.symm h
+
+theorem tiles_preimage_addEquiv_iff [DecidableEq G] (e : G ≃+ H)
+    {F : Finset H} {A : Set H} :
+    Tiles (F.image e.symm) (e ⁻¹' A) ↔ Tiles F A := by
+  rw [← addEquiv_image_symm_eq_preimage e A]
+  exact tiles_image_addEquiv_iff e.symm
+
+end AdditiveEquivalence
+
+section PeriodImages
+
+variable {G : Type uG} {H : Type uH} [AddCommGroup G] [AddCommGroup H]
+
+theorem Period.image {A : Set G} {v : G} (hv : Period A v) (f : G →+ H) :
+    Period (f '' A) (f v) := by
+  intro y
+  constructor
+  · rintro ⟨x, hx, hxy⟩
+    refine ⟨x - v, ?_, ?_⟩
+    · apply (hv (x - v)).mp
+      simpa only [sub_add_cancel] using hx
+    · rw [map_sub, hxy, add_sub_cancel_right]
+  · rintro ⟨x, hx, rfl⟩
+    exact ⟨x + v, (hv x).mpr hx, f.map_add x v⟩
+
+theorem finiteIndex_map_of_surjective (P : AddSubgroup G) (hP : P.FiniteIndex)
+    (f : G →+ H) (hf : Function.Surjective f) : (P.map f).FiniteIndex := by
+  constructor
+  intro hz
+  have hd := P.index_map_dvd hf
+  rw [hz] at hd
+  exact hP.index_ne_zero (Nat.eq_zero_of_zero_dvd hd)
+
+theorem FullyPeriodic.image_of_surjective (f : G →+ H)
+    (hf : Function.Surjective f) {A : Set G} (h : FullyPeriodic A) :
+    FullyPeriodic (f '' A) := by
+  obtain ⟨P, hP, hperiod⟩ := h
+  refine ⟨P.map f, finiteIndex_map_of_surjective P hP f hf, ?_⟩
+  intro w hw
+  obtain ⟨v, hv, rfl⟩ := AddSubgroup.mem_map.mp hw
+  exact (hperiod v hv).image f
+
+theorem fullyPeriodic_image_addEquiv_iff (e : G ≃+ H) {A : Set G} :
+    FullyPeriodic (e '' A) ↔ FullyPeriodic A := by
+  constructor
+  · intro h
+    have h' := FullyPeriodic.image_of_surjective e.symm.toAddMonoidHom
+      e.symm.surjective h
+    change FullyPeriodic (e.symm '' (e '' A)) at h'
+    simpa only [Set.image_image, Function.comp_def, e.symm_apply_apply,
+      Set.image_id'] using h'
+  · exact FullyPeriodic.image_of_surjective e.toAddMonoidHom e.surjective
+
+theorem FullyPeriodic.preimage_addEquiv (e : G ≃+ H) {A : Set H}
+    (h : FullyPeriodic A) : FullyPeriodic (e ⁻¹' A) := by
+  rw [← addEquiv_image_symm_eq_preimage e A]
+  exact FullyPeriodic.image_of_surjective e.symm.toAddMonoidHom e.symm.surjective h
+
+theorem fullyPeriodic_preimage_addEquiv_iff (e : G ≃+ H) {A : Set H} :
+    FullyPeriodic (e ⁻¹' A) ↔ FullyPeriodic A := by
+  rw [← addEquiv_image_symm_eq_preimage e A]
+  exact fullyPeriodic_image_addEquiv_iff e.symm
+
+end PeriodImages
+
+section FiniteCoordinate
+
+variable {G : Type uG} {K : Type uK} [AddCommGroup G] [AddCommGroup K] [Finite K]
+
+theorem finiteIndex_prod_bot (P : AddSubgroup G) (hP : P.FiniteIndex) :
+    (P.prod (⊥ : AddSubgroup K)).FiniteIndex := by
+  constructor
+  rw [AddSubgroup.index_prod, AddSubgroup.index_bot]
+  exact mul_ne_zero hP.index_ne_zero (ne_of_gt (Nat.card_pos : 0 < Nat.card K))
+
+theorem FullyPeriodic.prod_singleton_zero {A : Set G} (h : FullyPeriodic A) :
+    FullyPeriodic (A ×ˢ ({0} : Set K)) := by
+  obtain ⟨P, hP, hperiod⟩ := h
+  refine ⟨P.prod (⊥ : AddSubgroup K), finiteIndex_prod_bot P hP, ?_⟩
+  intro v hv x
+  obtain ⟨hp, hz⟩ := AddSubgroup.mem_prod.mp hv
+  have hzero : v.2 = 0 := AddSubgroup.mem_bot.mp hz
+  change (x.1 + v.1 ∈ A ∧ x.2 + v.2 ∈ ({0} : Set K)) ↔
+    (x.1 ∈ A ∧ x.2 ∈ ({0} : Set K))
+  rw [hzero, add_zero]
+  exact and_congr (hperiod v.1 hp x.1) Iff.rfl
+
+end FiniteCoordinate
+
+end PeriodicTilingThree
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/VoxelCubeMeasure.lean b/lean/OAI/Geometry/PeriodicTiling/VoxelCubeMeasure.lean
new file mode 100644
index 0000000000000000000000000000000000000000..a7e3a726a7ee4ce63371c76e24c52684736d237d
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/VoxelCubeMeasure.lean
@@ -0,0 +1,94 @@
+import OAI.Geometry.PeriodicTiling.VoxelGeometry
+import Mathlib.MeasureTheory.Constructions.Pi
+import Mathlib.MeasureTheory.Measure.AEDisjoint
+
+namespace OAI
+
+noncomputable section
+
+namespace PeriodicTilingThree
+
+open Set MeasureTheory Filter
+open scoped ENNReal
+
+theorem unitVoxel_volume {d : ℕ} (c : Space d) :
+    volume (unitVoxel c) = 1 := by
+  rw [unitVoxel_eq_Icc, Real.volume_Icc_pi]
+  simp
+
+theorem unitCube_volume (d : ℕ) : volume (unitCube d) = 1 := by
+  simpa only [unitVoxel, sub_zero, Set.ofPred_mem_eq] using
+    unitVoxel_volume (0 : Space d)
+
+theorem openUnitVoxel_volume {d : ℕ} (c : Space d) :
+    volume (openUnitVoxel c) = 1 := by
+  rw [openUnitVoxel, Real.volume_pi_Ioo]
+  simp
+
+theorem openUnitVoxel_ae_eq_unitVoxel {d : ℕ} (c : Space d) :
+    openUnitVoxel c =ᵐ[volume] unitVoxel c := by
+  rw [unitVoxel_eq_Icc]
+  exact Measure.univ_pi_Ioo_ae_eq_Icc
+
+theorem unitVoxel_frontier_volume_zero {d : ℕ} (c : Space d) :
+    volume (frontier (unitVoxel c)) = 0 := by
+  have hnull : volume (unitVoxel c \ openUnitVoxel c) = 0 :=
+    ae_le_set.mp (openUnitVoxel_ae_eq_unitVoxel c).symm.le
+  apply measure_mono_null _ hnull
+  intro x hx
+  refine ⟨(unitVoxel_isClosed c).frontier_subset hx, ?_⟩
+  intro ho
+  exact hx.2 (interior_maximal (openUnitVoxel_subset c)
+    (isOpen_openUnitVoxel c) ho)
+
+theorem unitVoxel_inter_volume_pos {d : ℕ} (a b : Space d)
+    (h : ∀ i, |b i - a i| < 1) :
+    0 < volume (unitVoxel a ∩ unitVoxel b) := by
+  rw [unitVoxel_inter_eq_Icc, Real.volume_Icc_pi]
+  apply bot_lt_iff_ne_bot.mpr
+  apply Finset.prod_ne_zero_iff.mpr
+  intro i _
+  apply ne_of_gt
+  apply ENNReal.ofReal_pos.mpr
+  apply sub_pos.mpr
+  apply max_lt
+  · apply lt_min
+    · linarith
+    · have hi := (abs_lt.mp (h i)).1
+      linarith
+  · apply lt_min
+    · have hi := (abs_lt.mp (h i)).2
+      linarith
+    · linarith
+
+theorem closedBall_volume {d : ℕ} (c : Space d) {r : ℝ} (hr : 0 ≤ r) :
+    volume (Metric.closedBall c r) = ENNReal.ofReal ((2 * r) ^ d) := by
+  simpa using Real.volume_pi_closedBall c hr
+
+theorem closedBall_ae_eq_ball {d : ℕ} (c : Space d) {r : ℝ} (hr : 0 < r) :
+    Metric.closedBall c r =ᵐ[volume] Metric.ball c r := by
+  rw [closedBall_pi c hr.le, ball_pi c hr]
+  simp only [Real.closedBall_eq_Icc, Real.ball_eq_Ioo]
+  exact Measure.pi_Ioo_ae_eq_pi_Icc.symm
+
+theorem closedBall_frontier_volume_zero {d : ℕ} (c : Space d)
+    {r : ℝ} (hr : 0 < r) :
+    volume (frontier (Metric.closedBall c r)) = 0 := by
+  have hnull : volume (Metric.closedBall c r \ Metric.ball c r) = 0 :=
+    ae_le_set.mp (closedBall_ae_eq_ball c hr).le
+  apply measure_mono_null _ hnull
+  intro x hx
+  refine ⟨Metric.isClosed_closedBall.frontier_subset hx, ?_⟩
+  intro hb
+  exact hx.2 (Metric.ball_subset_interior_closedBall hb)
+
+theorem unitVoxel_aedisjoint_int {d : ℕ} {z z' : Lattice d} (hne : z ≠ z') :
+    AEDisjoint volume (unitVoxel (castLattice z)) (unitVoxel (castLattice z')) := by
+  exact (openUnitVoxel_disjoint_unitVoxel_int hne).aedisjoint.congr
+    (openUnitVoxel_ae_eq_unitVoxel _).symm EventuallyEq.rfl
+
+end PeriodicTilingThree
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/VoxelGeometry.lean b/lean/OAI/Geometry/PeriodicTiling/VoxelGeometry.lean
new file mode 100644
index 0000000000000000000000000000000000000000..0bb56ffe49adfb75d5ad5a6fb8fe36c0dc82d4c4
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/VoxelGeometry.lean
@@ -0,0 +1,103 @@
+import OAI.Geometry.PeriodicTiling.EuclideanBasic
+import Mathlib.Tactic.Linarith
+
+namespace OAI
+
+noncomputable section
+
+namespace PeriodicTilingThree
+
+open Set MeasureTheory
+
+def openUnitVoxel {d : ℕ} (c : Space d) : Set (Space d) :=
+  Set.pi Set.univ (fun i => Set.Ioo (c i) (c i + 1))
+
+theorem mem_openUnitVoxel {d : ℕ} {c x : Space d} :
+    x ∈ openUnitVoxel c ↔ ∀ i, c i < x i ∧ x i < c i + 1 := by
+  simp [openUnitVoxel, Set.mem_pi]
+
+theorem openUnitVoxel_nonempty {d : ℕ} (c : Space d) :
+    (openUnitVoxel c).Nonempty := by
+  refine ⟨fun i => c i + 1 / 2, mem_openUnitVoxel.mpr ?_⟩
+  intro i
+  constructor <;> linarith
+
+theorem isOpen_openUnitVoxel {d : ℕ} (c : Space d) :
+    IsOpen (openUnitVoxel c) :=
+  isOpen_set_pi Set.finite_univ (fun _ _ => isOpen_Ioo)
+
+theorem openUnitVoxel_subset {d : ℕ} (c : Space d) :
+    openUnitVoxel c ⊆ unitVoxel c := by
+  intro x hx
+  rw [unitVoxel_eq_Icc]
+  have h := mem_openUnitVoxel.mp hx
+  exact ⟨fun i => (h i).1.le, fun i => (h i).2.le⟩
+
+theorem unitVoxel_isCompact {d : ℕ} (c : Space d) :
+    IsCompact (unitVoxel c) := by
+  rw [unitVoxel_eq_Icc]
+  exact isCompact_Icc
+
+theorem unitVoxel_isClosed {d : ℕ} (c : Space d) :
+    IsClosed (unitVoxel c) := (unitVoxel_isCompact c).isClosed
+
+theorem unitVoxel_measurableSet {d : ℕ} (c : Space d) :
+    MeasurableSet (unitVoxel c) := (unitVoxel_isClosed c).measurableSet
+
+theorem unitVoxel_inter_eq_Icc {d : ℕ} (a b : Space d) :
+    unitVoxel a ∩ unitVoxel b =
+      Set.Icc (fun i => max (a i) (b i))
+        (fun i => min (a i + 1) (b i + 1)) := by
+  rw [unitVoxel_eq_Icc, unitVoxel_eq_Icc]
+  ext x
+  constructor
+  · rintro ⟨ha, hb⟩
+    exact ⟨fun i => max_le (ha.1 i) (hb.1 i),
+      fun i => le_min (ha.2 i) (hb.2 i)⟩
+  · rintro ⟨hl, hu⟩
+    exact ⟨⟨fun i => (le_max_left _ _).trans (hl i),
+      fun i => (hu i).trans (min_le_left _ _)⟩,
+      ⟨fun i => (le_max_right _ _).trans (hl i),
+      fun i => (hu i).trans (min_le_right _ _)⟩⟩
+
+theorem mem_unitVoxel_cast_iff_of_mem_open {d : ℕ} {x : Space d}
+    {z z' : Lattice d} (hx : x ∈ openUnitVoxel (castLattice z)) :
+    x ∈ unitVoxel (castLattice z') ↔ z' = z := by
+  constructor
+  · intro hx'
+    have ho := mem_openUnitVoxel.mp hx
+    rw [unitVoxel_eq_Icc] at hx'
+    funext i
+    have hi := ho i
+    have hl := hx'.1 i
+    have hu := hx'.2 i
+    change (z i : ℝ) < x i ∧ x i < (z i : ℝ) + 1 at hi
+    change (z' i : ℝ) ≤ x i at hl
+    change x i ≤ (z' i : ℝ) + 1 at hu
+    rcases lt_trichotomy (z' i) (z i) with hlt | heq | hgt
+    · have hh : (z' i : ℝ) + 1 ≤ (z i : ℝ) := by
+        have hh' : ((z' i + 1 : ℤ) : ℝ) ≤ (z i : ℝ) :=
+          Int.cast_le.mpr (Int.add_one_le_of_lt hlt)
+        simpa only [Int.cast_add, Int.cast_one] using hh'
+      linarith
+    · exact heq
+    · have hh : (z i : ℝ) + 1 ≤ (z' i : ℝ) := by
+        have hh' : ((z i + 1 : ℤ) : ℝ) ≤ (z' i : ℝ) :=
+          Int.cast_le.mpr (Int.add_one_le_of_lt hgt)
+        simpa only [Int.cast_add, Int.cast_one] using hh'
+      linarith
+  · rintro rfl
+    exact openUnitVoxel_subset _ hx
+
+theorem openUnitVoxel_disjoint_unitVoxel_int {d : ℕ} {z z' : Lattice d}
+    (hne : z ≠ z') :
+    Disjoint (openUnitVoxel (castLattice z)) (unitVoxel (castLattice z')) := by
+  apply Set.disjoint_left.mpr
+  intro x hx hx'
+  exact hne ((mem_unitVoxel_cast_iff_of_mem_open hx).mp hx').symm
+
+end PeriodicTilingThree
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/VoxelMeasure.lean b/lean/OAI/Geometry/PeriodicTiling/VoxelMeasure.lean
new file mode 100644
index 0000000000000000000000000000000000000000..3972a2e575a26f7c7dae34651a1fa06eec8b60ec
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/VoxelMeasure.lean
@@ -0,0 +1,62 @@
+import OAI.Geometry.PeriodicTiling.VoxelCubeMeasure
+import Mathlib.Topology.MetricSpace.Bounded
+
+namespace OAI
+
+noncomputable section
+
+namespace PeriodicTilingThree
+
+open Set MeasureTheory
+open scoped ENNReal
+
+theorem Thickening_eq_iUnion {d : ℕ} (F : Finset (Lattice d)) :
+    Thickening F = ⋃ z ∈ F, unitVoxel (castLattice z) := by
+  ext x
+  simp only [mem_Thickening, Set.mem_iUnion, mem_unitVoxel, exists_prop]
+
+theorem thickening_isCompact {d : ℕ} (F : Finset (Lattice d)) :
+    IsCompact (Thickening F) := by
+  rw [Thickening_eq_iUnion]
+  exact F.isCompact_biUnion (fun z _ => unitVoxel_isCompact (castLattice z))
+
+theorem thickening_isClosed {d : ℕ} (F : Finset (Lattice d)) :
+    IsClosed (Thickening F) := (thickening_isCompact F).isClosed
+
+theorem thickening_measurableSet {d : ℕ} (F : Finset (Lattice d)) :
+    MeasurableSet (Thickening F) := (thickening_isClosed F).measurableSet
+
+theorem thickening_isBounded {d : ℕ} (F : Finset (Lattice d)) :
+    Bornology.IsBounded (Thickening F) := (thickening_isCompact F).isBounded
+
+theorem thickening_volume {d : ℕ} (F : Finset (Lattice d)) :
+    volume (Thickening F) = (F.card : ℝ≥0∞) := by
+  classical
+  rw [Thickening_eq_iUnion]
+  rw [measure_biUnion_finset₀
+    (fun z _ z' _ hne => unitVoxel_aedisjoint_int hne)
+    (fun z _ => (unitVoxel_measurableSet (castLattice z)).nullMeasurableSet)]
+  simp [unitVoxel_volume]
+
+theorem thickening_aedisjoint_of_disjoint {d : ℕ}
+    {S T : Finset (Lattice d)} (h : Disjoint S T) :
+    AEDisjoint volume (Thickening S) (Thickening T) := by
+  classical
+  simp only [Thickening_eq_iUnion, AEDisjoint.iUnion_left_iff,
+    AEDisjoint.iUnion_right_iff]
+  intro z hz z' hz'
+  apply unitVoxel_aedisjoint_int
+  intro heq
+  subst z'
+  exact Finset.disjoint_left.mp h hz' hz
+
+theorem thickening_inter_measure_zero_of_disjoint {d : ℕ}
+    {S T : Finset (Lattice d)} (h : Disjoint S T) :
+    volume (Thickening S ∩ Thickening T) = 0 :=
+  (thickening_aedisjoint_of_disjoint h).eq
+
+end PeriodicTilingThree
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/WeakPeriodicity.lean b/lean/OAI/Geometry/PeriodicTiling/WeakPeriodicity.lean
new file mode 100644
index 0000000000000000000000000000000000000000..27bcc57f2be596bb7a6f9678669a0c0a103d6849
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/WeakPeriodicity.lean
@@ -0,0 +1,84 @@
+import OAI.Geometry.PeriodicTiling.DirectionalLimits
+import OAI.Geometry.PeriodicTiling.DirectionalSlices
+import OAI.Geometry.PeriodicTiling.CosetPeriodicity
+import OAI.Geometry.PeriodicTiling.CosetParts
+import Mathlib.Tactic.Choose
+import Mathlib.Tactic.Ring
+
+namespace OAI
+
+namespace PeriodicTilingThree
+
+theorem Tiles.exists_weak_periodic_partition {F : Finset Plane} {A : Set Plane}
+    (h : Tiles F A) (h0 : (0 : Plane) ∈ F) :
+    ∃ (m : ℕ) (parts : Fin m → Set Plane) (v : Fin m → Plane),
+      (∀ x, (∑ j, tileIndicator ℝ (parts j) x) = tileIndicator ℝ A x) ∧
+      (∀ j, v j ≠ 0) ∧ ∀ j, Period (parts j) (v j) := by
+  classical
+  obtain ⟨m, v, φ, hv, hpair, hφ, hsum, hp⟩ :=
+    h.exists_directional_decomposition h0
+  by_cases hm0 : m = 0
+  · subst m
+    have hA : A = Set.univ := by
+      apply Set.eq_univ_of_forall
+      intro x
+      have hx := hsum x
+      by_contra hnot
+      simp [tileIndicator, hnot] at hx
+    refine ⟨1, fun _ => A, fun _ => (1, 0), ?_, ?_, ?_⟩
+    · intro x
+      simp
+    · intro j hj
+      have hz : (1 : ℤ) = 0 := congrArg Prod.fst hj
+      exact one_ne_zero hz
+    · intro j x
+      simp [hA]
+  · have hm : 0 < m := Nat.pos_of_ne_zero hm0
+    let j₀ : Fin m := ⟨0, hm⟩
+    obtain ⟨e, he, htrans, hspan⟩ := exists_common_transverse v hv hpair
+    obtain ⟨K, hK, hKrep⟩ := exists_common_second_vector v j₀ e htrans
+    let H : Plane := (K : ℤ) • v j₀
+    choose α β hα hH using hKrep
+    have hdet : planeDet e H ≠ 0 := by
+      dsimp [H]
+      rw [planeDet_zsmul_right]
+      exact mul_ne_zero (Nat.cast_ne_zero.mpr (Nat.ne_of_gt hK)) (htrans j₀)
+    have hlocal : ∀ x : Plane, ∃ w : Plane, w ≠ 0 ∧
+        Function.Periodic
+          (fun st : Plane => tileIndicator ℝ A (x + st.1 • e + st.2 • H)) w := by
+      intro x
+      let P : Fin m → ℤ → ℝ := fun j n => φ j (x + n • e)
+      have hcoord (j : Fin m) (s t : ℤ) :
+          φ j (x + s • e + t • H) = P j (s + β j * t) := by
+        have hpoint : x + s • e + t • H =
+            (x + (s + β j * t) • e) + (t * α j) • v j := by
+          change x + s • e + t • ((K : ℤ) • v j₀) = _
+          rw [hH j]
+          apply Prod.ext <;> simp [smul_add, add_smul, zsmul_eq_mul] <;> ring
+        rw [hpoint]
+        exact (hp j).zsmul (t * α j) _
+      have hlocalSum (st : Plane) :
+          tileIndicator ℝ A (x + st.1 • e + st.2 • H) =
+            1 - ∑ j, P j (st.1 + β j * st.2) := by
+        have hz := hsum (x + st.1 • e + st.2 • H)
+        have heq : (∑ j, φ j (x + st.1 • e + st.2 • H)) =
+            ∑ j, P j (st.1 + β j * st.2) := by
+          apply Finset.sum_congr rfl
+          intro j _
+          exact hcoord j st.1 st.2
+        rw [heq] at hz
+        linarith
+      refine coset_configuration_has_period P β _
+        (fun j n => (hφ j _).1) ?_ hlocalSum m hm ?_
+      · intro i j hij n r
+        exact directional_slice_pair_bound A v φ e hspan
+          (fun j y => (hφ j y).1) hp hsum x i j hij n r
+      · intro j
+        exact directional_circle_slice_polynomial hm A v φ e hspan hp hsum j x
+    obtain ⟨n, parts, w, hparts, hw⟩ :=
+      finite_periodic_parts_of_coset_periods e H hdet hlocal
+    exact ⟨n, parts, w, hparts, fun j => (hw j).1, fun j => (hw j).2⟩
+
+end PeriodicTilingThree
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/WordAffine.lean b/lean/OAI/Geometry/PeriodicTiling/WordAffine.lean
new file mode 100644
index 0000000000000000000000000000000000000000..102e70f0071bd8d69e25cfa2727279dc609a8881
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/WordAffine.lean
@@ -0,0 +1,352 @@
+import OAI.Geometry.PeriodicTiling.WordAffineAlgebra
+import OAI.Geometry.PeriodicTiling.WordAffineCount
+import OAI.Geometry.PeriodicTiling.WordAffineGeometry
+import Mathlib.Tactic.Push
+
+namespace OAI
+
+noncomputable section
+
+namespace PeriodicTilingThree
+
+theorem allowed_affine {p : ℕ} {w : Word p} (hw : Allowed p w) :
+    ∃ a b : ZMod p, (a ≠ 0 ∨ b ≠ 0) ∧
+      ∀ n : Column p, a * (n.val : ZMod p) + b ≠ 0 →
+        (w n : ZMod p) = a * (n.val : ZMod p) + b := by
+  obtain ⟨a, b, hab, hw⟩ := hw
+  refine ⟨(a : ZMod p), (b : ZMod p), ?_, ?_⟩
+  · simpa only [ne_eq, ZMod.intCast_zmod_eq_zero_iff_dvd] using hab
+  · intro n hn
+    have hnd : ¬ (p : ℤ) ∣ a * (n.val : ℤ) + b := by
+      intro hd
+      apply hn
+      have h := (ZMod.intCast_zmod_eq_zero_iff_dvd
+        (a * (n.val : ℤ) + b) p).mpr hd
+      simpa only [Int.cast_add, Int.cast_mul, Int.cast_natCast] using h
+    simpa only [Int.cast_add, Int.cast_mul, Int.cast_natCast] using (hw n).1 hnd
+
+private def intColumn {p : ℕ} (n : ℤ) (hn : 0 ≤ n ∧ n < (p ^ 2 : ℕ)) :
+    Column p := ⟨n.toNat, by omega⟩
+
+private theorem intColumn_val {p : ℕ} (n : ℤ)
+    (hn : 0 ≤ n ∧ n < (p ^ 2 : ℕ)) :
+    ((intColumn n hn).val : ℤ) = n := Int.toNat_of_nonneg hn.1
+
+private theorem intColumn_cast {p : ℕ} (n : ℤ)
+    (hn : 0 ≤ n ∧ n < (p ^ 2 : ℕ)) :
+    ((intColumn n hn).val : ZMod p) = (n : ZMod p) := by
+  simpa only [Int.cast_natCast] using
+    congrArg (fun x : ℤ => (x : ZMod p)) (intColumn_val n hn)
+
+private def rawArray {p : ℕ} (W : WordArray p) (n m : ℤ) : ZMod p :=
+  if hn : 0 ≤ n ∧ n < (p ^ 2 : ℕ) then (W (intColumn n hn) m : ZMod p) else 1
+
+private theorem rawArray_column {p : ℕ} (W : WordArray p) (n : Column p) (m : ℤ) :
+    rawArray W n.val m = (W n m : ZMod p) := by
+  have hn : (0 : ℤ) ≤ n.val ∧ (n.val : ℤ) < (p ^ 2 : ℕ) := by
+    exact ⟨Int.natCast_nonneg _, by exact_mod_cast n.isLt⟩
+  have hc : intColumn (n.val : ℤ) hn = n := by
+    apply Fin.ext
+    simp [intColumn]
+  rw [rawArray, dite_eq_left hn, hc]
+
+private theorem rawArray_ne_zero {p : ℕ} [Fact p.Prime] (W : WordArray p)
+    (n m : ℤ) (hn : 0 ≤ n ∧ n < (p ^ 2 : ℕ)) : rawArray W n m ≠ 0 := by
+  rw [rawArray, dite_eq_left hn]
+  exact Units.ne_zero _
+
+private structure LineAffine {p : ℕ} (W : WordArray p) (d e : ℤ) where
+  a : ZMod p
+  b : ZMod p
+  nonzero : a ≠ 0 ∨ b ≠ 0
+  agrees : ∀ n : Column p, a * (n.val : ZMod p) + b ≠ 0 →
+    (W n (d * (n.val : ℤ) + e) : ZMod p) = a * (n.val : ZMod p) + b
+
+private def chooseLine {p : ℕ} {W : WordArray p} (hW : LineRule p W)
+    (d e : ℤ) : LineAffine W d e := by
+  let h := allowed_affine (hW d e)
+  exact ⟨h.choose, h.choose_spec.choose, h.choose_spec.choose_spec.1,
+    h.choose_spec.choose_spec.2⟩
+
+private theorem LineAffine.raw_agrees {p : ℕ} {W : WordArray p} {d e : ℤ}
+    (L : LineAffine W d e) (n : ℤ) (hn : 0 ≤ n ∧ n < (p ^ 2 : ℕ))
+    (hz : L.a * (n : ZMod p) + L.b ≠ 0) :
+    rawArray W n (d * n + e) = L.a * (n : ZMod p) + L.b := by
+  have h := L.agrees (intColumn n hn) (by simpa only [intColumn_cast] using hz)
+  rw [intColumn_val, intColumn_cast] at h
+  simpa only [rawArray, dite_eq_left hn] using h
+
+private theorem clean_second_difference {p : ℕ} {W : WordArray p}
+    (hW : LineRule p W) (r q : ℤ)
+    (hc : ∀ i j : Fin 4, ∀ d : ℤ, (d = -1 ∨ d = 0 ∨ d = 1) →
+      rawArray W (r + i.val) (q + j.val) =
+        (chooseLine hW d ((q + j.val) - d * (r + i.val))).a *
+          ((r + i.val : ℤ) : ZMod p) +
+        (chooseLine hW d ((q + j.val) - d * (r + i.val))).b)
+    (x₀ y₀ x₁ y₁ x₂ y₂ : Fin 4) (d : ℤ)
+    (hd : d = -1 ∨ d = 0 ∨ d = 1)
+    (h₁ : (y₁.val : ℤ) - y₀.val = d * ((x₁.val : ℤ) - x₀.val))
+    (h₂ : (y₂.val : ℤ) - y₀.val = d * ((x₂.val : ℤ) - x₀.val))
+    (hx : (x₂.val : ℤ) - 2 * x₁.val + x₀.val = 0) :
+    rawArray W (r + x₂.val) (q + y₂.val) -
+      2 * rawArray W (r + x₁.val) (q + y₁.val) +
+      rawArray W (r + x₀.val) (q + y₀.val) = 0 := by
+  let e := (q + y₀.val) - d * (r + x₀.val)
+  have he₁ : (q + y₁.val) - d * (r + x₁.val) = e := by
+    dsimp [e]
+    linear_combination h₁
+  have he₂ : (q + y₂.val) - d * (r + x₂.val) = e := by
+    dsimp [e]
+    linear_combination h₂
+  have h₀' := hc x₀ y₀ d hd
+  have h₁' := hc x₁ y₁ d hd
+  have h₂' := hc x₂ y₂ d hd
+  change rawArray W (r + x₀.val) (q + y₀.val) =
+    (chooseLine hW d e).a * ((r + x₀.val : ℤ) : ZMod p) +
+      (chooseLine hW d e).b at h₀'
+  rw [he₁] at h₁'
+  rw [he₂] at h₂'
+  have hx' : (x₂.val : ZMod p) - 2 * x₁.val + x₀.val = 0 := by
+    simpa only [Int.cast_sub, Int.cast_add, Int.cast_mul, Int.cast_ofNat,
+      Int.cast_natCast, Int.cast_zero] using
+      congrArg (fun z : ℤ => (z : ZMod p)) hx
+  rw [h₀', h₁', h₂']
+  push_cast
+  linear_combination (chooseLine hW d e).a * hx'
+
+private theorem affine_four_line_closure {p : ℕ} (hp : p.Prime) (hlarge : 200 < p)
+    {W : WordArray p} (hW : LineRule p W) (A B C : ZMod p) :
+    FourLineClosure (p ^ 2 : ℕ)
+      (fun n m => A * (n : ZMod p) + B * (m : ZMod p) + C = 0 ∨
+        rawArray W n m = A * (n : ZMod p) + B * (m : ZMod p) + C) := by
+  let : Fact p.Prime := ⟨hp⟩
+  intro d e hd r hr₀ hr₃ hgood n hn₀ hn
+  let a : ZMod p := A + B * (d : ZMod p)
+  let b : ZMod p := B * (e : ZMod p) + C
+  let L := chooseLine hW d e
+  have hrest (k : ℤ) : A * (k : ZMod p) + B * ((d * k + e : ℤ) : ZMod p) + C =
+      a * (k : ZMod p) + b := by
+    dsimp [a, b]
+    push_cast
+    ring
+  by_cases ha : a = 0
+  · by_cases hb : b = 0
+    · left
+      rw [hrest, ha, hb]
+      simp
+    · have hab : a ≠ 0 ∨ b ≠ 0 := Or.inr hb
+      obtain ⟨i, j, hij, hiA, hiL, hjA, hjL⟩ := two_common_nonzero_of_four
+        (fun k : Fin 4 => ((r + k.val : ℤ) : ZMod p))
+        (four_residues_injective (by omega) r) a b L.a L.b hab L.nonzero
+      have heq (k : Fin 4) (hkA : a * ((r + k.val : ℤ) : ZMod p) + b ≠ 0)
+          (hkL : L.a * ((r + k.val : ℤ) : ZMod p) + L.b ≠ 0) :
+          a * ((r + k.val : ℤ) : ZMod p) + b =
+            L.a * ((r + k.val : ℤ) : ZMod p) + L.b := by
+        have hk := hgood k
+        dsimp only at hk
+        rw [hrest] at hk
+        have hk' := hk.resolve_left hkA
+        have hbound : 0 ≤ r + k.val ∧ r + k.val < (p ^ 2 : ℕ) := by
+          have := k.isLt
+          omega
+        exact hk'.symm.trans (L.raw_agrees _ hbound hkL)
+      have hcoef := affine_coefficients_eq_of_two
+        (fun h => hij ((four_residues_injective (by omega) r) h))
+        (heq i hiA hiL) (heq j hjA hjL)
+      by_cases hz : a * (n : ZMod p) + b = 0
+      · left
+        rw [hrest]
+        exact hz
+      · right
+        rw [hrest]
+        rw [hcoef.1, hcoef.2] at hz ⊢
+        exact L.raw_agrees n ⟨hn₀, hn⟩ hz
+  · have hab : a ≠ 0 ∨ b ≠ 0 := Or.inl ha
+    obtain ⟨i, j, hij, hiA, hiL, hjA, hjL⟩ := two_common_nonzero_of_four
+      (fun k : Fin 4 => ((r + k.val : ℤ) : ZMod p))
+      (four_residues_injective (by omega) r) a b L.a L.b hab L.nonzero
+    have heq (k : Fin 4) (hkA : a * ((r + k.val : ℤ) : ZMod p) + b ≠ 0)
+        (hkL : L.a * ((r + k.val : ℤ) : ZMod p) + L.b ≠ 0) :
+        a * ((r + k.val : ℤ) : ZMod p) + b =
+          L.a * ((r + k.val : ℤ) : ZMod p) + L.b := by
+      have hk := hgood k
+      dsimp only at hk
+      rw [hrest] at hk
+      have hk' := hk.resolve_left hkA
+      have hbound : 0 ≤ r + k.val ∧ r + k.val < (p ^ 2 : ℕ) := by
+        have := k.isLt
+        omega
+      exact hk'.symm.trans (L.raw_agrees _ hbound hkL)
+    have hcoef := affine_coefficients_eq_of_two
+      (fun h => hij ((four_residues_injective (by omega) r) h))
+      (heq i hiA hiL) (heq j hjA hjL)
+    by_cases hz : a * (n : ZMod p) + b = 0
+    · left
+      rw [hrest]
+      exact hz
+    · right
+      rw [hrest]
+      rw [hcoef.1, hcoef.2] at hz ⊢
+      exact L.raw_agrees n ⟨hn₀, hn⟩ hz
+
+theorem global_affine_approximation {p : ℕ} (hp : p.Prime) (hlarge : 200 < p)
+    {W : WordArray p} (hW : LineRule p W) :
+    ∃ A B C : ZMod p, AffineApproximation W A B C := by
+  classical
+  let : Fact p.Prime := ⟨hp⟩
+  let bad : ℤ → ℤ → ℤ → Prop := fun d e n =>
+    (chooseLine hW d e).a * (n : ZMod p) + (chooseLine hW d e).b = 0
+  have hbad : ∀ d e, d = -1 ∨ d = 0 ∨ d = 1 →
+      ((Finset.Ico (0 : ℤ) (p ^ 2 : ℕ)).filter (bad d e)).card ≤ p := by
+    intro d e _
+    exact affine_int_roots_card_le hp _ _ (chooseLine hW d e).nonzero
+  obtain ⟨r, q, hr₀, hr₃, _, _, hclean⟩ :=
+    exists_clean_four_square p hlarge bad
+      (by
+        intro d e hd
+        have hfilter := @Finset.filter_congr_decidable ℤ
+          (Finset.Ico (0 : ℤ) (p ^ 2 : ℕ)) (bad d e)
+          (fun n => Classical.propDecidable (bad d e n))
+          (fun n => ZMod.decidableEq p
+            ((chooseLine hW d e).a * (n : ZMod p) + (chooseLine hW d e).b) 0)
+        exact (congrArg (fun s : Finset ℤ => s.card ≤ p) hfilter).mpr (hbad d e hd))
+  have hcell : ∀ i j : Fin 4, ∀ d : ℤ, (d = -1 ∨ d = 0 ∨ d = 1) →
+      rawArray W (r + i.val) (q + j.val) =
+        (chooseLine hW d ((q + j.val) - d * (r + i.val))).a *
+          ((r + i.val : ℤ) : ZMod p) +
+        (chooseLine hW d ((q + j.val) - d * (r + i.val))).b := by
+    intro i j d hd
+    have hn : 0 ≤ r + i.val ∧ r + i.val < (p ^ 2 : ℕ) := by
+      have := i.isLt
+      omega
+    have h := (chooseLine hW d ((q + j.val) - d * (r + i.val))).raw_agrees
+      (r + i.val) hn (hclean i j d hd)
+    have he : d * (r + i.val) + ((q + j.val) - d * (r + i.val)) = q + j.val := by ring
+    simpa only [he] using h
+  let U : Fin 4 → ZMod p := fun j => (chooseLine hW 0 (q + j.val)).a
+  let V : Fin 4 → ZMod p := fun j => U j * (r : ZMod p) +
+    (chooseLine hW 0 (q + j.val)).b
+  have hrow (i j : Fin 4) : rawArray W (r + i.val) (q + j.val) =
+      U j * (i.val : ZMod p) + V j := by
+    have h := hcell i j 0 (Or.inr (Or.inl rfl))
+    have he : (q + j.val) - (0 : ℤ) * (r + i.val) = q + j.val := by ring
+    rw [he] at h
+    rw [h]
+    dsimp [V, U]
+    push_cast
+    ring
+  have hv0 : (0 : Fin 4).val = 0 := rfl
+  have hv1 : (1 : Fin 4).val = 1 := rfl
+  have hv2 : (2 : Fin 4).val = 2 := rfl
+  have hv3 : (3 : Fin 4).val = 3 := rfl
+  have hp₀ := clean_second_difference hW r q hcell 0 0 1 1 2 2 1
+    (Or.inr (Or.inr rfl))
+    (by norm_num [hv0, hv1, hv2, hv3])
+    (by norm_num [hv0, hv1, hv2, hv3])
+    (by norm_num [hv0, hv1, hv2, hv3])
+  have hp₀' := clean_second_difference hW r q hcell 1 0 2 1 3 2 1
+    (Or.inr (Or.inr rfl))
+    (by norm_num [hv0, hv1, hv2, hv3])
+    (by norm_num [hv0, hv1, hv2, hv3])
+    (by norm_num [hv0, hv1, hv2, hv3])
+  have hm₀ := clean_second_difference hW r q hcell 2 0 1 1 0 2 (-1)
+    (Or.inl rfl)
+    (by norm_num [hv0, hv1, hv2, hv3])
+    (by norm_num [hv0, hv1, hv2, hv3])
+    (by norm_num [hv0, hv1, hv2, hv3])
+  have hp₁ := clean_second_difference hW r q hcell 0 1 1 2 2 3 1
+    (Or.inr (Or.inr rfl))
+    (by norm_num [hv0, hv1, hv2, hv3])
+    (by norm_num [hv0, hv1, hv2, hv3])
+    (by norm_num [hv0, hv1, hv2, hv3])
+  have hp₁' := clean_second_difference hW r q hcell 1 1 2 2 3 3 1
+    (Or.inr (Or.inr rfl))
+    (by norm_num [hv0, hv1, hv2, hv3])
+    (by norm_num [hv0, hv1, hv2, hv3])
+    (by norm_num [hv0, hv1, hv2, hv3])
+  have hm₁ := clean_second_difference hW r q hcell 2 1 1 2 0 3 (-1)
+    (Or.inl rfl)
+    (by norm_num [hv0, hv1, hv2, hv3])
+    (by norm_num [hv0, hv1, hv2, hv3])
+    (by norm_num [hv0, hv1, hv2, hv3])
+  have htwo : (2 : ZMod p) ≠ 0 := by
+    intro h
+    have hv := congrArg ZMod.val h
+    have hlt : 2 < p := by omega
+    have hv₂ : (2 : ZMod p).val = 2 := ZMod.val_natCast_of_lt hlt
+    rw [hv₂, ZMod.val_zero] at hv
+    omega
+  have hpatch := four_rows_affine htwo U V
+    (by
+      have h := hp₀
+      simp only [hrow] at h
+      simp only [hv0, hv1, hv2, Nat.cast_zero, Nat.cast_one, Nat.cast_ofNat] at h
+      linear_combination h)
+    (by
+      have h := hp₀'
+      simp only [hrow] at h
+      simp only [hv1, hv2, hv3, Nat.cast_one, Nat.cast_ofNat] at h
+      linear_combination h)
+    (by
+      have h := hm₀
+      simp only [hrow] at h
+      simp only [hv0, hv1, hv2, Nat.cast_zero, Nat.cast_one, Nat.cast_ofNat] at h
+      linear_combination h)
+    (by
+      have h := hp₁
+      simp only [hrow] at h
+      simp only [hv0, hv1, hv2, Nat.cast_zero, Nat.cast_one, Nat.cast_ofNat] at h
+      linear_combination h)
+    (by
+      have h := hp₁'
+      simp only [hrow] at h
+      simp only [hv1, hv2, hv3, Nat.cast_one, Nat.cast_ofNat] at h
+      linear_combination h)
+    (by
+      have h := hm₁
+      simp only [hrow] at h
+      simp only [hv0, hv1, hv2, Nat.cast_zero, Nat.cast_one, Nat.cast_ofNat] at h
+      linear_combination h)
+  let A := U 0
+  let B := V 1 - V 0
+  let C := V 0 - A * (r : ZMod p) - B * (q : ZMod p)
+  have hsquare (i j : Fin 4) : rawArray W (r + i.val) (q + j.val) =
+      A * ((r + i.val : ℤ) : ZMod p) +
+        B * ((q + j.val : ℤ) : ZMod p) + C := by
+    rw [hrow, hpatch]
+    dsimp [A, B, C]
+    push_cast
+    ring
+  have hnonzero : A ≠ 0 ∨ B ≠ 0 ∨ C ≠ 0 := by
+    by_cases hA : A = 0
+    · by_cases hB : B = 0
+      · refine Or.inr (Or.inr ?_)
+        intro hC
+        have hz := hsquare 0 0
+        simp only [Fin.val_zero, Nat.cast_zero, add_zero, hA, hB, hC,
+          zero_mul] at hz
+        exact rawArray_ne_zero W r q ⟨hr₀, by omega⟩ hz
+      · exact Or.inr (Or.inl hB)
+    · exact Or.inl hA
+  have hN : (8 : ℤ) ≤ (p ^ 2 : ℕ) := by
+    have hp' : 201 ≤ p := by omega
+    have hsq : 201 ^ 2 ≤ p ^ 2 := Nat.pow_le_pow_left hp' 2
+    exact_mod_cast (show 8 ≤ p ^ 2 by omega)
+  have hglobal := good_everywhere_of_four_square (p ^ 2 : ℕ) hN
+    (fun n m => A * (n : ZMod p) + B * (m : ZMod p) + C = 0 ∨
+      rawArray W n m = A * (n : ZMod p) + B * (m : ZMod p) + C)
+    (affine_four_line_closure hp hlarge hW A B C) r q hr₀ hr₃
+    (fun i j => Or.inr (hsquare i j))
+  refine ⟨A, B, C, hnonzero, ?_⟩
+  intro n m hn
+  have h := hglobal (n.val : ℤ) m (Int.natCast_nonneg _)
+    (by exact_mod_cast n.isLt)
+  rw [rawArray_column] at h
+  simpa only [affineValue, Int.cast_natCast] using
+    (h.resolve_left (by simpa only [affineValue, Int.cast_natCast] using hn))
+
+end PeriodicTilingThree
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/WordAffineAlgebra.lean b/lean/OAI/Geometry/PeriodicTiling/WordAffineAlgebra.lean
new file mode 100644
index 0000000000000000000000000000000000000000..6519fd0e69d974cc1bf489e784997ef3a59dc307
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/WordAffineAlgebra.lean
@@ -0,0 +1,139 @@
+import OAI.Geometry.PeriodicTiling.WordRule
+import Mathlib.Data.Int.Interval
+import Mathlib.Data.Finset.Card
+import Mathlib.Tactic.LinearCombination
+import Mathlib.Tactic.FinCases
+import Mathlib.Tactic.Ring
+
+namespace OAI
+
+universe uK uα
+
+namespace PeriodicTilingThree
+
+attribute [local instance] Classical.propDecidable
+
+section Field
+
+variable {K : Type uK} [Field K]
+
+theorem affine_coefficients_eq_of_two {a b c d x y : K} (hxy : x ≠ y)
+    (hx : a * x + b = c * x + d) (hy : a * y + b = c * y + d) :
+    a = c ∧ b = d := by
+  have hm : (a - c) * (x - y) = 0 := by linear_combination hx - hy
+  have ha : a = c := sub_eq_zero.mp ((mul_eq_zero.mp hm).resolve_right
+    (sub_ne_zero.mpr hxy))
+  refine ⟨ha, ?_⟩
+  linear_combination hx - x * ha
+
+theorem affine_root_unique {a b x y : K} (hab : a ≠ 0 ∨ b ≠ 0)
+    (hx : a * x + b = 0) (hy : a * y + b = 0) : x = y := by
+  by_contra hxy
+  have h := affine_coefficients_eq_of_two hxy
+    (show a * x + b = 0 * x + 0 by simpa using hx)
+    (show a * y + b = 0 * y + 0 by simpa using hy)
+  exact hab.elim (fun ha => ha h.1) (fun hb => hb h.2)
+
+theorem affine_roots_card_le_one {α : Type uα} [DecidableEq α]
+    (s : Finset α) (x : α → K) (hx : Set.InjOn x s)
+    (a b : K) (hab : a ≠ 0 ∨ b ≠ 0) :
+    (s.filter (fun i => a * x i + b = 0)).card ≤ 1 := by
+  classical
+  apply Finset.card_le_one.mpr
+  intro i hi j hj
+  exact hx (Finset.mem_filter.mp hi).1 (Finset.mem_filter.mp hj).1
+    (affine_root_unique hab (Finset.mem_filter.mp hi).2 (Finset.mem_filter.mp hj).2)
+
+theorem two_common_nonzero_of_four (x : Fin 4 → K) (hx : Function.Injective x)
+    (a b c d : K) (hab : a ≠ 0 ∨ b ≠ 0) (hcd : c ≠ 0 ∨ d ≠ 0) :
+    ∃ i j : Fin 4, i ≠ j ∧ a * x i + b ≠ 0 ∧ c * x i + d ≠ 0 ∧
+      a * x j + b ≠ 0 ∧ c * x j + d ≠ 0 := by
+  classical
+  let U := Finset.univ.filter (fun i => a * x i + b = 0)
+  let V := Finset.univ.filter (fun i => c * x i + d = 0)
+  have hu : U.card ≤ 1 := affine_roots_card_le_one _ _ (fun _ _ _ _ h => hx h) _ _ hab
+  have hv : V.card ≤ 1 := affine_roots_card_le_one _ _ (fun _ _ _ _ h => hx h) _ _ hcd
+  have huv : (U ∪ V).card ≤ 2 := (Finset.card_union_le U V).trans (by omega)
+  have hsub : U ∪ V ⊆ (Finset.univ : Finset (Fin 4)) := Finset.subset_univ _
+  have hcard := Finset.card_sdiff_add_card_eq_card hsub
+  have hfour : (Finset.univ : Finset (Fin 4)).card = 4 := by simp
+  have hgood : 1 < (Finset.univ \ (U ∪ V)).card := by omega
+  obtain ⟨i, hi, j, hj, hij⟩ := Finset.one_lt_card.mp hgood
+  have hi' : a * x i + b ≠ 0 ∧ c * x i + d ≠ 0 := by
+    simpa [U, V] using hi
+  have hj' : a * x j + b ≠ 0 ∧ c * x j + d ≠ 0 := by
+    simpa [U, V] using hj
+  exact ⟨i, j, hij, hi'.1, hi'.2, hj'.1, hj'.2⟩
+
+private theorem three_rows (h2 : (2 : K) ≠ 0)
+    (u₀ u₁ u₂ v₀ v₁ v₂ : K)
+    (hp : 2 * u₂ + v₂ - 2 * (u₁ + v₁) + v₀ = 0)
+    (hp' : 3 * u₂ + v₂ - 2 * (2 * u₁ + v₁) + u₀ + v₀ = 0)
+    (hm : v₂ - 2 * (u₁ + v₁) + 2 * u₀ + v₀ = 0) :
+    u₁ = u₀ ∧ u₂ = u₀ ∧ v₂ - 2 * v₁ + v₀ = 0 := by
+  have hu : 2 * (u₂ - u₀) = 0 := by linear_combination hp - hm
+  have hu₂ : u₂ = u₀ := sub_eq_zero.mp ((mul_eq_zero.mp hu).resolve_left h2)
+  have hu' : 2 * (u₁ - u₀) = 0 := by linear_combination hp - hp' + hu₂
+  have hu₁ : u₁ = u₀ := sub_eq_zero.mp ((mul_eq_zero.mp hu').resolve_left h2)
+  refine ⟨hu₁, hu₂, ?_⟩
+  linear_combination hp - 2 * hu₂ + 2 * hu₁
+
+theorem four_rows_affine (h2 : (2 : K) ≠ 0) (U V : Fin 4 → K)
+    (hp₀ : 2 * U 2 + V 2 - 2 * (U 1 + V 1) + V 0 = 0)
+    (hp₀' : 3 * U 2 + V 2 - 2 * (2 * U 1 + V 1) + U 0 + V 0 = 0)
+    (hm₀ : V 2 - 2 * (U 1 + V 1) + 2 * U 0 + V 0 = 0)
+    (hp₁ : 2 * U 3 + V 3 - 2 * (U 2 + V 2) + V 1 = 0)
+    (hp₁' : 3 * U 3 + V 3 - 2 * (2 * U 2 + V 2) + U 1 + V 1 = 0)
+    (hm₁ : V 3 - 2 * (U 2 + V 2) + 2 * U 1 + V 1 = 0) :
+    ∀ i j : Fin 4, U j * (i.val : K) + V j =
+      U 0 * (i.val : K) + (V 1 - V 0) * (j.val : K) + V 0 := by
+  obtain ⟨hu₁, hu₂, hv₂⟩ := three_rows h2 _ _ _ _ _ _ hp₀ hp₀' hm₀
+  obtain ⟨_, hu₃, hv₃⟩ := three_rows h2 _ _ _ _ _ _ hp₁ hp₁' hm₁
+  have hu₃' : U 3 = U 0 := hu₃.trans hu₁
+  have hv₂' : V 2 = 2 * V 1 - V 0 := by linear_combination hv₂
+  have hv₃' : V 3 = 3 * V 1 - 2 * V 0 := by linear_combination hv₃ + 2 * hv₂
+  intro i j
+  fin_cases j <;> simp only [Fin.reduceFinMk,
+    Nat.cast_zero, Nat.cast_one, Nat.cast_ofNat, hu₁, hu₂, hu₃', hv₂', hv₃'] <;> ring
+
+end Field
+
+theorem four_residues_injective {p : ℕ} (hp : 4 ≤ p) (r : ℤ) :
+    Function.Injective (fun i : Fin 4 => ((r + i.val : ℤ) : ZMod p)) := by
+  intro i j hij
+  have hv : (i.val : ZMod p) = (j.val : ZMod p) := by
+    simpa only [Int.cast_add, Int.cast_natCast, add_right_inj] using hij
+  apply Fin.ext
+  have h := congrArg ZMod.val hv
+  simpa only [ZMod.val_natCast_of_lt (i.isLt.trans_le hp),
+    ZMod.val_natCast_of_lt (j.isLt.trans_le hp)] using h
+
+theorem affine_int_roots_card_le {p : ℕ} (hp : p.Prime) (a b : ZMod p)
+    (hab : a ≠ 0 ∨ b ≠ 0) :
+    ((Finset.Ico (0 : ℤ) (p ^ 2 : ℕ)).filter
+      (fun n : ℤ => a * (n : ZMod p) + b = 0)).card ≤ p := by
+  classical
+  let : Fact p.Prime := ⟨hp⟩
+  have hpz : (0 : ℤ) < p := by exact_mod_cast hp.pos
+  have hc : (Finset.Ico (0 : ℤ) (p : ℤ)).card = p := by simp
+  refine le_trans ?_ (le_of_eq hc)
+  apply Finset.card_le_card_of_injOn (fun n : ℤ => n / p)
+  · intro n hn
+    have hn' := Finset.mem_Ico.mp (Finset.mem_filter.mp hn).1
+    apply Finset.mem_Ico.mpr
+    refine ⟨Int.ediv_nonneg hn'.1 hpz.le, ?_⟩
+    apply Int.ediv_lt_of_lt_mul hpz
+    simpa only [pow_two, Nat.cast_mul] using hn'.2
+  · intro n hn m hm hq
+    have hnm : (n : ZMod p) = (m : ZMod p) := affine_root_unique hab
+      (Finset.mem_filter.mp hn).2 (Finset.mem_filter.mp hm).2
+    have hr := (ZMod.intCast_eq_intCast_iff' n m p).mp hnm
+    have hnq := Int.emod_add_mul_ediv n (p : ℤ)
+    have hmq := Int.emod_add_mul_ediv m (p : ℤ)
+    change n / (p : ℤ) = m / (p : ℤ) at hq
+    rw [hr, hq] at hnq
+    omega
+
+end PeriodicTilingThree
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/WordAffineCount.lean b/lean/OAI/Geometry/PeriodicTiling/WordAffineCount.lean
new file mode 100644
index 0000000000000000000000000000000000000000..6d7e02510719985809ec63f64ee51123f8ede0af
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/WordAffineCount.lean
@@ -0,0 +1,157 @@
+import Mathlib.Data.Int.Interval
+import Mathlib.Data.Finset.Prod
+import Mathlib.Algebra.Order.BigOperators.Group.Finset
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.NormNum
+import Mathlib.Tactic.Ring
+
+namespace OAI
+
+namespace PeriodicTilingThree
+
+noncomputable section
+
+attribute [local instance] Classical.propDecidable
+
+def wordExceptionalCells (N : ℕ) (bad : ℤ → ℤ → ℤ → Prop) : Finset (ℤ × ℤ) :=
+  (Finset.Icc (-1 : ℤ) 1).biUnion fun d =>
+    (Finset.Ico (-(N : ℤ)) (2 * (N : ℤ))).biUnion fun e =>
+      ((Finset.Ico (0 : ℤ) N).filter (bad d e)).image fun n => (n, d * n + e)
+
+theorem card_wordExceptionalCells_le (N p : ℕ) (bad : ℤ → ℤ → ℤ → Prop)
+    (hbad : ∀ d e : ℤ, (d = -1 ∨ d = 0 ∨ d = 1) →
+      ((Finset.Ico (0 : ℤ) N).filter (bad d e)).card ≤ p) :
+    (wordExceptionalCells N bad).card ≤ 9 * N * p := by
+  have hE : (Finset.Ico (-(N : ℤ)) (2 * (N : ℤ))).card = 3 * N := by
+    rw [Int.card_Ico]
+    omega
+  have hD : (Finset.Icc (-1 : ℤ) 1).card = 3 := by
+    norm_num [Int.card_Icc]
+  have hinner : ∀ d ∈ Finset.Icc (-1 : ℤ) 1,
+      ((Finset.Ico (-(N : ℤ)) (2 * (N : ℤ))).biUnion fun e =>
+        ((Finset.Ico (0 : ℤ) N).filter (bad d e)).image
+          fun n => (n, d * n + e)).card ≤ 3 * N * p := by
+    intro d hd
+    have hd' : d = -1 ∨ d = 0 ∨ d = 1 := by
+      have hd'' := Finset.mem_Icc.mp hd
+      omega
+    calc
+      _ ≤ (Finset.Ico (-(N : ℤ)) (2 * (N : ℤ))).card * p := by
+        apply Finset.card_biUnion_le_card_mul
+        intro e _he
+        exact Finset.card_image_le.trans (hbad d e hd')
+      _ = 3 * N * p := by rw [hE]
+  calc
+    (wordExceptionalCells N bad).card ≤
+        (Finset.Icc (-1 : ℤ) 1).card * (3 * N * p) :=
+      Finset.card_biUnion_le_card_mul _ _ _ hinner
+    _ = 9 * N * p := by rw [hD]; ring
+
+theorem mem_wordExceptionalCells (N : ℕ) (bad : ℤ → ℤ → ℤ → Prop)
+    (n m d : ℤ) (hn0 : 0 ≤ n) (hnN : n < N)
+    (hm0 : 0 ≤ m) (hmN : m < N)
+    (hd : d = -1 ∨ d = 0 ∨ d = 1) (hb : bad d (m - d * n) n) :
+    (n, m) ∈ wordExceptionalCells N bad := by
+  have hdmem : d ∈ Finset.Icc (-1 : ℤ) 1 := by
+    rcases hd with rfl | rfl | rfl <;> simp
+  have hemem : m - d * n ∈ Finset.Ico (-(N : ℤ)) (2 * (N : ℤ)) := by
+    apply Finset.mem_Ico.mpr
+    rcases hd with rfl | rfl | rfl <;>
+      simp only [neg_one_mul, zero_mul, one_mul, sub_zero] <;> constructor <;> omega
+  apply Finset.mem_biUnion.mpr
+  refine ⟨d, hdmem, Finset.mem_biUnion.mpr ⟨m - d * n, hemem, ?_⟩⟩
+  apply Finset.mem_image.mpr
+  refine ⟨n, Finset.mem_filter.mpr ⟨Finset.mem_Ico.mpr ⟨hn0, hnN⟩, hb⟩, ?_⟩
+  exact Prod.ext rfl (by omega)
+
+def wordBlockedOrigins (B : Finset (ℤ × ℤ)) : Finset (ℤ × ℤ) :=
+  B.biUnion fun c =>
+    ((Finset.Icc (0 : ℤ) 3).product (Finset.Icc (0 : ℤ) 3)).image
+      fun o => (c.1 - o.1, c.2 - o.2)
+
+theorem card_wordBlockedOrigins_le (B : Finset (ℤ × ℤ)) :
+    (wordBlockedOrigins B).card ≤ B.card * 16 := by
+  apply Finset.card_biUnion_le_card_mul
+  intro c _hc
+  calc
+    _ ≤ ((Finset.Icc (0 : ℤ) 3).product (Finset.Icc (0 : ℤ) 3)).card :=
+      Finset.card_image_le
+    _ = 16 := by norm_num [Finset.card_product, Int.card_Icc]
+
+theorem mem_wordBlockedOrigins (B : Finset (ℤ × ℤ)) (r q : ℤ)
+    (i j : Fin 4) (hc : (r + i.val, q + j.val) ∈ B) :
+    (r, q) ∈ wordBlockedOrigins B := by
+  apply Finset.mem_biUnion.mpr
+  refine ⟨(r + i.val, q + j.val), hc, Finset.mem_image.mpr ?_⟩
+  refine ⟨((i.val : ℤ), (j.val : ℤ)), ?_, ?_⟩
+  · apply Finset.mem_product.mpr
+    constructor <;> apply Finset.mem_Icc.mpr <;> constructor <;> omega
+  · apply Prod.ext <;> simp
+
+private theorem four_square_count_inequality (p : ℕ) (hp : 200 < p) :
+    144 * (p ^ 2) * p < (p ^ 2 - 3) ^ 2 := by
+  have hp0 : 0 < p := by omega
+  have hp201 : 201 ≤ p := by omega
+  have hpred : p - 1 + 1 = p := by omega
+  have hpredsq := congrArg (fun k : ℕ => k * k) hpred
+  have hpredmul := congrArg (fun k : ℕ => p * k) hpred
+  have hsq : 3 ≤ p ^ 2 := by nlinarith
+  have hsub : p ^ 2 - 3 + 3 = p ^ 2 := Nat.sub_add_cancel hsq
+  have hmul : 201 * p ≤ p * p := by nlinarith [Nat.mul_le_mul_right p hp201]
+  have hgap : 144 * p < (p - 1) * (p - 1) := by nlinarith
+  have hsmall : p * (p - 1) ≤ p ^ 2 - 3 := by nlinarith
+  have hlarge := Nat.mul_lt_mul_of_pos_left hgap (pow_pos hp0 2)
+  have hbound := Nat.mul_le_mul hsmall hsmall
+  nlinarith
+
+theorem exists_clean_four_square (p : ℕ) (hp : 200 < p)
+    (bad : ℤ → ℤ → ℤ → Prop)
+    (hbad : ∀ d e : ℤ, (d = -1 ∨ d = 0 ∨ d = 1) →
+      ((Finset.Ico (0 : ℤ) (p ^ 2 : ℕ)).filter (bad d e)).card ≤ p) :
+    ∃ r q : ℤ, 0 ≤ r ∧ r + 3 < (p ^ 2 : ℕ) ∧
+      0 ≤ q ∧ q + 3 < (p ^ 2 : ℕ) ∧
+      ∀ i j : Fin 4, ∀ d : ℤ, (d = -1 ∨ d = 0 ∨ d = 1) →
+        ¬ bad d ((q + j.val) - d * (r + i.val)) (r + i.val) := by
+  let N : ℕ := p ^ 2
+  have hN : 3 ≤ N := by dsimp [N]; nlinarith
+  let C : Finset (ℤ × ℤ) :=
+    (Finset.Ico (0 : ℤ) (N - 3 : ℕ)).product
+      (Finset.Ico (0 : ℤ) (N - 3 : ℕ))
+  let B := wordBlockedOrigins (wordExceptionalCells N bad)
+  have hC : C.card = (N - 3) ^ 2 := by
+    simp [C, Finset.card_product, Int.card_Ico, pow_two]
+  have hB : B.card ≤ 144 * N * p := by
+    calc
+      B.card ≤ (wordExceptionalCells N bad).card * 16 :=
+        card_wordBlockedOrigins_le _
+      _ ≤ (9 * N * p) * 16 :=
+        Nat.mul_le_mul_right 16 (card_wordExceptionalCells_le N p bad hbad)
+      _ = 144 * N * p := by ring
+  have hlt : B.card < C.card := by
+    rw [hC]
+    exact hB.trans_lt (four_square_count_inequality p hp)
+  obtain ⟨⟨r, q⟩, hc, hnblocked⟩ := Finset.exists_mem_notMem_of_card_lt_card hlt
+  obtain ⟨hr, hq⟩ := Finset.mem_product.mp hc
+  have hr' := Finset.mem_Ico.mp hr
+  have hq' := Finset.mem_Ico.mp hq
+  have hr0 : 0 ≤ r := hr'.1
+  have hq0 : 0 ≤ q := hq'.1
+  have hr3 : r + 3 < (N : ℤ) := by omega
+  have hq3 : q + 3 < (N : ℤ) := by omega
+  refine ⟨r, q, hr0, hr3, hq0, hq3, ?_⟩
+  intro i j d hd hb
+  apply hnblocked
+  apply mem_wordBlockedOrigins _ r q i j
+  apply mem_wordExceptionalCells N bad (r + i.val) (q + j.val) d
+  · omega
+  · omega
+  · omega
+  · omega
+  · exact hd
+  · exact hb
+
+end
+
+end PeriodicTilingThree
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/WordAffineGeometry.lean b/lean/OAI/Geometry/PeriodicTiling/WordAffineGeometry.lean
new file mode 100644
index 0000000000000000000000000000000000000000..bb8728e4acc173491173eb32fbac5cbe994e8402
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/WordAffineGeometry.lean
@@ -0,0 +1,120 @@
+import Mathlib.Data.Int.Basic
+import Mathlib.Algebra.Order.Ring.Int
+
+namespace OAI
+
+namespace PeriodicTilingThree
+
+def FourLineClosure (N : ℤ) (Good : ℤ → ℤ → Prop) : Prop :=
+  ∀ d e : ℤ, (d = -1 ∨ d = 0 ∨ d = 1) →
+    ∀ r : ℤ, 0 ≤ r → r + 3 < N →
+      (∀ i : Fin 4, Good (r + i.val) (d * (r + i.val) + e)) →
+      ∀ n : ℤ, 0 ≤ n → n < N → Good n (d * n + e)
+
+def FourGoodRows (N : ℤ) (Good : ℤ → ℤ → Prop) (q : ℤ) : Prop :=
+  ∀ m : ℤ, q ≤ m → m ≤ q + 3 →
+    ∀ n : ℤ, 0 ≤ n → n < N → Good n m
+
+theorem fourGoodRows_row_above (N : ℤ) (hN : 8 ≤ N)
+    (Good : ℤ → ℤ → Prop) (hline : FourLineClosure N Good)
+    (q : ℤ) (hrows : FourGoodRows N Good q) :
+    ∀ n : ℤ, 0 ≤ n → n < N → Good n (q + 4) := by
+  intro n hn0 hnN
+  by_cases hr : n + 4 < N
+  · have hseed : ∀ i : Fin 4,
+        Good (n + 1 + i.val) ((-1) * (n + 1 + i.val) + (n + q + 4)) := by
+      intro i
+      have hi : 0 ≤ (i.val : ℤ) ∧ (i.val : ℤ) < 4 := by omega
+      have hg := hrows (q + 3 - i.val) (by omega) (by omega)
+        (n + 1 + i.val) (by omega) (by omega)
+      convert hg using 1; simp only [neg_one_mul]; omega
+    have h := hline (-1) (n + q + 4) (Or.inl rfl)
+      (n + 1) (by omega) (by omega) hseed n hn0 hnN
+    convert h using 1; simp only [neg_one_mul]; omega
+  · have hl : 4 ≤ n := by omega
+    have hseed : ∀ i : Fin 4,
+        Good (n - 4 + i.val) (1 * (n - 4 + i.val) + (q + 4 - n)) := by
+      intro i
+      have hi : 0 ≤ (i.val : ℤ) ∧ (i.val : ℤ) < 4 := by omega
+      have hg := hrows (q + i.val) (by omega) (by omega)
+        (n - 4 + i.val) (by omega) (by omega)
+      convert hg using 1; simp only [one_mul]; omega
+    have h := hline 1 (q + 4 - n) (Or.inr (Or.inr rfl))
+      (n - 4) (by omega) (by omega) hseed n hn0 hnN
+    convert h using 1; simp only [one_mul]; omega
+
+theorem fourGoodRows_row_below (N : ℤ) (hN : 8 ≤ N)
+    (Good : ℤ → ℤ → Prop) (hline : FourLineClosure N Good)
+    (q : ℤ) (hrows : FourGoodRows N Good q) :
+    ∀ n : ℤ, 0 ≤ n → n < N → Good n (q - 1) := by
+  intro n hn0 hnN
+  by_cases hr : n + 4 < N
+  · have hseed : ∀ i : Fin 4,
+        Good (n + 1 + i.val) (1 * (n + 1 + i.val) + (q - 1 - n)) := by
+      intro i
+      have hi : 0 ≤ (i.val : ℤ) ∧ (i.val : ℤ) < 4 := by omega
+      have hg := hrows (q + i.val) (by omega) (by omega)
+        (n + 1 + i.val) (by omega) (by omega)
+      convert hg using 1; simp only [one_mul]; omega
+    have h := hline 1 (q - 1 - n) (Or.inr (Or.inr rfl))
+      (n + 1) (by omega) (by omega) hseed n hn0 hnN
+    convert h using 1; simp only [one_mul]; omega
+  · have hl : 4 ≤ n := by omega
+    have hseed : ∀ i : Fin 4,
+        Good (n - 4 + i.val) ((-1) * (n - 4 + i.val) + (q - 1 + n)) := by
+      intro i
+      have hi : 0 ≤ (i.val : ℤ) ∧ (i.val : ℤ) < 4 := by omega
+      have hg := hrows (q + 3 - i.val) (by omega) (by omega)
+        (n - 4 + i.val) (by omega) (by omega)
+      convert hg using 1; simp only [neg_one_mul]; omega
+    have h := hline (-1) (q - 1 + n) (Or.inl rfl)
+      (n - 4) (by omega) (by omega) hseed n hn0 hnN
+    convert h using 1; simp only [neg_one_mul]; omega
+
+theorem fourGoodRows_add_one (N : ℤ) (hN : 8 ≤ N)
+    (Good : ℤ → ℤ → Prop) (hline : FourLineClosure N Good)
+    (q : ℤ) (hrows : FourGoodRows N Good q) :
+    FourGoodRows N Good (q + 1) := by
+  intro m hm0 hm3 n hn0 hnN
+  by_cases hm : m ≤ q + 3
+  · exact hrows m (by omega) hm n hn0 hnN
+  · have he : m = q + 4 := by omega
+    rw [he]
+    exact fourGoodRows_row_above N hN Good hline q hrows n hn0 hnN
+
+theorem fourGoodRows_sub_one (N : ℤ) (hN : 8 ≤ N)
+    (Good : ℤ → ℤ → Prop) (hline : FourLineClosure N Good)
+    (q : ℤ) (hrows : FourGoodRows N Good q) :
+    FourGoodRows N Good (q - 1) := by
+  intro m hm0 hm3 n hn0 hnN
+  by_cases hm : q ≤ m
+  · exact hrows m hm (by omega) n hn0 hnN
+  · have he : m = q - 1 := by omega
+    rw [he]
+    exact fourGoodRows_row_below N hN Good hline q hrows n hn0 hnN
+
+theorem good_everywhere_of_four_square (N : ℤ) (hN : 8 ≤ N)
+    (Good : ℤ → ℤ → Prop) (hline : FourLineClosure N Good)
+    (r q : ℤ) (hr0 : 0 ≤ r) (hr3 : r + 3 < N)
+    (hsquare : ∀ i j : Fin 4, Good (r + i.val) (q + j.val)) :
+    ∀ n m : ℤ, 0 ≤ n → n < N → Good n m := by
+  have hbase : FourGoodRows N Good q := by
+    intro m hm0 hm3 n hn0 hnN
+    let j : Fin 4 := ⟨(m - q).toNat, by omega⟩
+    have hj : q + (j.val : ℤ) = m := by dsimp [j]; omega
+    have hseed : ∀ i : Fin 4, Good (r + i.val) (0 * (r + i.val) + m) := by
+      intro i
+      simpa only [zero_mul, zero_add, hj] using hsquare i j
+    have h := hline 0 m (Or.inr (Or.inl rfl)) r hr0 hr3 hseed n hn0 hnN
+    simpa only [zero_mul, zero_add] using h
+  have hall : ∀ t : ℤ, FourGoodRows N Good t := by
+    intro t
+    exact Int.inductionOn' t q hbase
+      (fun k _ hk => fourGoodRows_add_one N hN Good hline k hk)
+      (fun k _ hk => fourGoodRows_sub_one N hN Good hline k hk)
+  intro n m hn0 hnN
+  exact hall m m le_rfl (by omega) n hn0 hnN
+
+end PeriodicTilingThree
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/WordObstruction.lean b/lean/OAI/Geometry/PeriodicTiling/WordObstruction.lean
new file mode 100644
index 0000000000000000000000000000000000000000..45ead822bbe1279221588de20d629a685fb58ab4
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/WordObstruction.lean
@@ -0,0 +1,374 @@
+import OAI.Geometry.PeriodicTiling.WordAffine
+import OAI.Geometry.PeriodicTiling.WordAffineAlgebra
+import Mathlib.Tactic.LinearCombination
+import Mathlib.Tactic.Ring
+import Mathlib.Tactic.Linarith
+import Lean.Elab.Tactic.Omega
+
+namespace OAI
+
+noncomputable section
+
+namespace PeriodicTilingThree
+
+private theorem word_one_ne_two {p : ℕ} (hlarge : 200 < p) :
+    (1 : ZMod p) ≠ 2 := by
+  intro h
+  have hv := congrArg (fun x : ZMod p => x.val) h
+  rw [ZMod.val_one_eq_one_mod, ZMod.val_two_eq_two_mod,
+    Nat.mod_eq_of_lt (by omega : 1 < p),
+    Nat.mod_eq_of_lt (by omega : 2 < p)] at hv
+  omega
+
+private theorem word_two_ne_zero {p : ℕ} (hlarge : 200 < p) :
+    (2 : ZMod p) ≠ 0 := by
+  intro h
+  have hd : p ∣ 2 := (ZMod.natCast_eq_zero_iff 2 p).mp h
+  exact Nat.not_dvd_of_pos_of_lt (by decide : 0 < 2) (by omega : 2 < p) hd
+
+private def smallColumn {p : ℕ} (hlarge : 200 < p) (n : Fin p) : Column p :=
+  ⟨n.val, by have := n.isLt; nlinarith⟩
+
+private def fourColumn {p : ℕ} (hlarge : 200 < p) (i : Fin 4) : Column p :=
+  smallColumn hlarge ⟨i.val, by have := i.isLt; omega⟩
+
+private theorem small_affine_nonzero {p : ℕ} (hp : p.Prime)
+    (A C : ZMod p) (hAC : A ≠ 0 ∨ C ≠ 0) :
+    ∃ n : Fin p, A * (n.val : ZMod p) + C ≠ 0 := by
+  by_cases hC : C = 0
+  · have hA : A ≠ 0 := hAC.resolve_right (fun h => h hC)
+    refine ⟨⟨1, hp.one_lt⟩, ?_⟩
+    simpa only [Nat.cast_one, mul_one, hC, add_zero] using hA
+  · refine ⟨⟨0, hp.pos⟩, ?_⟩
+    simpa only [Nat.cast_zero, mul_zero, zero_add] using hC
+
+private theorem exists_vertical_value {p : ℕ} (hp : p.Prime)
+    (A B C : ZMod p) (hB : B ≠ 0) (n : Column p) (t : ZMod p) :
+    ∃ m : ℤ, affineValue A B C n m = t := by
+  let : Fact p.Prime := ⟨hp⟩
+  obtain ⟨m, hm⟩ := ZMod.intCast_surjective
+    (B⁻¹ * (t - A * (n.val : ZMod p) - C))
+  refine ⟨m, ?_⟩
+  have hBm : B * (m : ZMod p) = t - A * (n.val : ZMod p) - C := by
+    rw [hm, ← mul_assoc, mul_inv_cancel₀ hB, one_mul]
+  unfold affineValue
+  rw [hBm]
+  ring
+
+theorem nonconstantColumns_iff_vertical_coefficient {p : ℕ} (hp : p.Prime)
+    (hlarge : 200 < p) {W : WordArray p} {A B C : ZMod p}
+    (hH : AffineApproximation W A B C) : NonconstantColumns W ↔ B ≠ 0 := by
+  let : Fact p.Prime := ⟨hp⟩
+  constructor
+  · intro hcolumns hB
+    have hAC : A ≠ 0 ∨ C ≠ 0 := by
+      rcases hH.1 with hA | hB' | hC
+      · exact Or.inl hA
+      · exact (hB' hB).elim
+      · exact Or.inr hC
+    obtain ⟨n₀, hn₀⟩ := small_affine_nonzero hp A C hAC
+    let n := smallColumn hlarge n₀
+    obtain ⟨m, m', hmm'⟩ := hcolumns n
+    apply hmm'
+    apply Units.ext
+    have hval (z : ℤ) : (W n z : ZMod p) = A * (n₀.val : ZMod p) + C := by
+      have hn : affineValue A B C n z ≠ 0 := by
+        simpa only [affineValue, n, smallColumn, hB, zero_mul, add_zero] using hn₀
+      simpa only [affineValue, n, smallColumn, hB, zero_mul, add_zero] using hH.2 n z hn
+    exact (hval m).trans (hval m').symm
+  · intro hB n
+    obtain ⟨m, hm⟩ := exists_vertical_value hp A B C hB n 1
+    obtain ⟨m', hm'⟩ := exists_vertical_value hp A B C hB n 2
+    refine ⟨m, m', ?_⟩
+    intro heq
+    have hwm : (W n m : ZMod p) = 1 :=
+      (hH.2 n m (by rw [hm]; exact one_ne_zero)).trans hm
+    have hwm' : (W n m' : ZMod p) = 2 :=
+      (hH.2 n m' (by rw [hm']; exact word_two_ne_zero hlarge)).trans hm'
+    exact word_one_ne_two hlarge
+      (hwm.symm.trans ((congrArg (fun u : Symbol p => (u : ZMod p)) heq).trans hwm'))
+
+theorem prime_dvd_vertical_period {p : ℕ} (hp : p.Prime)
+    (hlarge : 200 < p) {W : WordArray p} {A B C : ZMod p}
+    (hH : AffineApproximation W A B C) (hB : B ≠ 0) {M : ℤ}
+    (hperiod : VerticalPeriod W M) : (p : ℤ) ∣ M := by
+  let : Fact p.Prime := ⟨hp⟩
+  let n : Column p := ⟨0, pow_pos hp.pos 2⟩
+  let x : Fin 4 → ZMod p := fun i => (i.val : ZMod p)
+  have hx : Function.Injective x := by
+    simpa only [zero_add, Int.cast_natCast] using
+      (four_residues_injective (p := p) (by omega : 4 ≤ p) 0)
+  obtain ⟨i, j, hij, hi, hiM, hj, hjM⟩ := two_common_nonzero_of_four x hx
+    B (A * (n.val : ZMod p) + C)
+    B (B * (M : ZMod p) + A * (n.val : ZMod p) + C)
+    (Or.inl hB) (Or.inl hB)
+  let m : ℤ := i.val
+  have hm : affineValue A B C n m ≠ 0 := by
+    have heq : affineValue A B C n m = B * x i + (A * (n.val : ZMod p) + C) := by
+      simp only [affineValue, m, x, Int.cast_natCast]
+      ring
+    rw [heq]
+    exact hi
+  have hmM : affineValue A B C n (m + M) ≠ 0 := by
+    have heq : affineValue A B C n (m + M) =
+        B * x i + (B * (M : ZMod p) + A * (n.val : ZMod p) + C) := by
+      simp only [affineValue, m, x, Int.cast_add, Int.cast_natCast]
+      ring
+    rw [heq]
+    exact hiM
+  have heq : affineValue A B C n (m + M) = affineValue A B C n m :=
+    (hH.2 n (m + M) hmM).symm.trans
+      ((congrArg (fun u : Symbol p => (u : ZMod p)) (hperiod n m)).trans (hH.2 n m hm))
+  have hBM : B * (M : ZMod p) = 0 := by
+    simp only [affineValue, Int.cast_add] at heq
+    linear_combination heq
+  exact (ZMod.intCast_zmod_eq_zero_iff_dvd M p).mp
+    ((mul_eq_zero.mp hBM).resolve_left hB)
+
+def straighten {p : ℕ} (W : WordArray p) (u v : ℤ) : WordArray p :=
+  fun n m => W n (m + u * (n.val : ℤ) + v)
+
+theorem straighten_lineRule {p : ℕ} {W : WordArray p}
+    (hW : LineRule p W) (u v : ℤ) : LineRule p (straighten W u v) := by
+  intro d e
+  have heq : (fun n => straighten W u v n (d * (n.val : ℤ) + e)) =
+      (fun n => W n ((d + u) * (n.val : ℤ) + (e + v))) := by
+    funext n
+    unfold straighten
+    congr 1
+    ring
+  rw [heq]
+  exact hW (d + u) (e + v)
+
+theorem straighten_nonconstantColumns_iff {p : ℕ} (W : WordArray p) (u v : ℤ) :
+    NonconstantColumns (straighten W u v) ↔ NonconstantColumns W := by
+  constructor
+  · intro h n
+    obtain ⟨m, m', hmm'⟩ := h n
+    exact ⟨m + u * (n.val : ℤ) + v, m' + u * (n.val : ℤ) + v, hmm'⟩
+  · intro h n
+    obtain ⟨m, m', hmm'⟩ := h n
+    refine ⟨m - u * (n.val : ℤ) - v, m' - u * (n.val : ℤ) - v, ?_⟩
+    have hm : m - u * (n.val : ℤ) - v + u * (n.val : ℤ) + v = m := by ring
+    have hm' : m' - u * (n.val : ℤ) - v + u * (n.val : ℤ) + v = m' := by ring
+    simpa only [straighten, hm, hm'] using hmm'
+
+theorem straighten_verticalPeriod_iff {p : ℕ} (W : WordArray p)
+    (u v M : ℤ) : VerticalPeriod (straighten W u v) M ↔ VerticalPeriod W M := by
+  constructor
+  · intro h n m
+    have hh := h n (m - u * (n.val : ℤ) - v)
+    change W n ((m - u * (n.val : ℤ) - v + M) + u * (n.val : ℤ) + v) =
+      W n ((m - u * (n.val : ℤ) - v) + u * (n.val : ℤ) + v) at hh
+    convert hh using 1 <;> congr 1 <;> ring
+  · intro h n m
+    change W n ((m + M) + u * (n.val : ℤ) + v) = W n (m + u * (n.val : ℤ) + v)
+    convert h n (m + u * (n.val : ℤ) + v) using 1 ; congr 1 ; ring
+
+theorem exists_straighten_affine {p : ℕ} (hp : p.Prime)
+    {W : WordArray p} {A B C : ZMod p} (hH : AffineApproximation W A B C)
+    (hB : B ≠ 0) : ∃ u v : ℤ, AffineApproximation (straighten W u v) 0 B 0 := by
+  let : Fact p.Prime := ⟨hp⟩
+  obtain ⟨u, hu⟩ := ZMod.intCast_surjective (-(B⁻¹ * A))
+  obtain ⟨v, hv⟩ := ZMod.intCast_surjective (-(B⁻¹ * C))
+  have hbu : A + B * (u : ZMod p) = 0 := by
+    rw [hu, mul_neg, ← mul_assoc, mul_inv_cancel₀ hB, one_mul, add_neg_cancel]
+  have hbv : C + B * (v : ZMod p) = 0 := by
+    rw [hv, mul_neg, ← mul_assoc, mul_inv_cancel₀ hB, one_mul, add_neg_cancel]
+  refine ⟨u, v, Or.inr (Or.inl hB), ?_⟩
+  intro n m hm
+  have hnorm : affineValue A B C n (m + u * (n.val : ℤ) + v) =
+      affineValue 0 B 0 n m := by
+    simp only [affineValue, Int.cast_add, Int.cast_mul, Int.cast_natCast]
+    linear_combination (n.val : ZMod p) * hbu + hbv
+  exact (hH.2 n _ (by rw [hnorm]; exact hm)).trans hnorm
+
+def rescale {p : ℕ} (W : WordArray p) : WordArray p :=
+  fun n m => W n ((p : ℤ) * m)
+
+theorem rescale_lineRule {p : ℕ} {W : WordArray p} (hW : LineRule p W) :
+    LineRule p (rescale W) := by
+  intro d e
+  have heq : (fun n => rescale W n (d * (n.val : ℤ) + e)) =
+      (fun n => W n (((p : ℤ) * d) * (n.val : ℤ) + (p : ℤ) * e)) := by
+    funext n
+    unfold rescale
+    congr 1
+    ring
+  rw [heq]
+  exact hW ((p : ℤ) * d) ((p : ℤ) * e)
+
+private theorem exists_nonzero_affine_ne {p : ℕ} (hp : p.Prime)
+    (hlarge : 200 < p) (a z c : ZMod p) (ha : a ≠ 0) :
+    ∃ k : Fin p, z + a * (k.val : ZMod p) ≠ 0 ∧
+      z + a * (k.val : ZMod p) ≠ c := by
+  let : Fact p.Prime := ⟨hp⟩
+  let : NeZero p := ⟨hp.ne_zero⟩
+  have ht : ∃ t : ZMod p, t ≠ 0 ∧ t ≠ c := by
+    by_cases hc : c = 1
+    · exact ⟨2, word_two_ne_zero hlarge, by
+        rw [hc]
+        exact Ne.symm (word_one_ne_two hlarge)⟩
+    · exact ⟨1, one_ne_zero, Ne.symm hc⟩
+  obtain ⟨t, ht0, htc⟩ := ht
+  let x : ZMod p := a⁻¹ * (t - z)
+  let k : Fin p := ⟨x.val, ZMod.val_lt x⟩
+  have hk : (k.val : ZMod p) = x := ZMod.natCast_zmod_val x
+  have heq : z + a * (k.val : ZMod p) = t := by
+    rw [hk]
+    dsimp only [x]
+    rw [← mul_assoc, mul_inv_cancel₀ ha, one_mul]
+    ring
+  exact ⟨k, by rw [heq]; exact ht0, by rw [heq]; exact htc⟩
+
+private def fiberColumn {p : ℕ} (n₀ k : Fin p) : Column p :=
+  ⟨n₀.val + p * k.val, by
+    calc
+      n₀.val + p * k.val < p + p * k.val := Nat.add_lt_add_right n₀.isLt _
+      _ = p * (k.val + 1) := by ring
+      _ ≤ p * p := Nat.mul_le_mul_left p (Nat.succ_le_of_lt k.isLt)
+      _ = p ^ 2 := by ring⟩
+
+theorem rescale_vertical_coefficient_ne_zero {p : ℕ} (hp : p.Prime)
+    (hlarge : 200 < p) {W : WordArray p} (hW : LineRule p W)
+    {B : ZMod p} (hB : B ≠ 0) (hH : AffineApproximation W 0 B 0)
+    {A₁ B₁ C₁ : ZMod p} (hH₁ : AffineApproximation (rescale W) A₁ B₁ C₁) :
+    B₁ ≠ 0 := by
+  classical
+  let : Fact p.Prime := ⟨hp⟩
+  intro hB₁
+  have hAC : A₁ ≠ 0 ∨ C₁ ≠ 0 := by
+    rcases hH₁.1 with hA | hB' | hC
+    · exact Or.inl hA
+    · exact (hB' hB₁).elim
+    · exact Or.inr hC
+  obtain ⟨n₀, hc⟩ := small_affine_nonzero hp A₁ C₁ hAC
+  let c : ZMod p := A₁ * (n₀.val : ZMod p) + C₁
+  obtain ⟨a, b, hab, hw⟩ := hW 1 (-(n₀.val : ℤ))
+  have hab' : (a : ZMod p) ≠ 0 ∨ (b : ZMod p) ≠ 0 := by
+    rcases hab with ha | hb
+    · exact Or.inl (fun h => ha ((ZMod.intCast_zmod_eq_zero_iff_dvd a p).mp h))
+    · exact Or.inr (fun h => hb ((ZMod.intCast_zmod_eq_zero_iff_dvd b p).mp h))
+  let x : Fin 4 → ZMod p := fun i => (i.val : ZMod p)
+  have hx : Function.Injective x := by
+    simpa only [zero_add, Int.cast_natCast] using
+      (four_residues_injective (p := p) (by omega : 4 ≤ p) 0)
+  obtain ⟨i, j, hij, hai, hbi, haj, hbj⟩ := two_common_nonzero_of_four x hx
+    (a : ZMod p) (b : ZMod p) B (-B * (n₀.val : ZMod p)) hab' (Or.inl hB)
+  have hmatch (s : Fin 4)
+      (has : (a : ZMod p) * x s + (b : ZMod p) ≠ 0)
+      (hbs : B * x s + -B * (n₀.val : ZMod p) ≠ 0) :
+      (a : ZMod p) * x s + (b : ZMod p) =
+        B * x s + -B * (n₀.val : ZMod p) := by
+    let n := fourColumn hlarge s
+    have hnot : ¬ (p : ℤ) ∣ a * (n.val : ℤ) + b := by
+      intro hd
+      apply has
+      simpa only [Int.cast_add, Int.cast_mul, Int.cast_natCast, n, fourColumn, smallColumn, x] using
+        (ZMod.intCast_zmod_eq_zero_iff_dvd (a * (n.val : ℤ) + b) p).mpr hd
+    have harg : affineValue 0 B 0 n (1 * (n.val : ℤ) + -(n₀.val : ℤ)) =
+        B * x s + -B * (n₀.val : ZMod p) := by
+      simp only [affineValue, zero_mul, zero_add, add_zero, one_mul,
+        Int.cast_add, Int.cast_neg, Int.cast_natCast]
+      change B * ((s.val : ZMod p) + -(n₀.val : ZMod p)) =
+        B * (s.val : ZMod p) + -B * (n₀.val : ZMod p)
+      ring
+    have hw' := (hw n).1 hnot
+    dsimp only at hw'
+    have hH' := hH.2 n (1 * (n.val : ℤ) + -(n₀.val : ℤ))
+      (by rw [harg]; exact hbs)
+    calc
+      (a : ZMod p) * x s + (b : ZMod p) =
+          (W n (1 * (n.val : ℤ) + -(n₀.val : ℤ)) : ZMod p) := by
+        simpa only [Int.cast_add, Int.cast_mul, Int.cast_natCast, n, fourColumn, smallColumn, x] using hw'.symm
+      _ = B * x s + -B * (n₀.val : ZMod p) := hH'.trans harg
+  obtain ⟨haB, hbB⟩ := affine_coefficients_eq_of_two (fun h => hij (hx h))
+    (hmatch i hai hbi) (hmatch j haj hbj)
+  have ha0 : (a : ZMod p) ≠ 0 := by rw [haB]; exact hB
+  have hdiv : (p : ℤ) ∣ a * (n₀.val : ℤ) + b := by
+    apply (ZMod.intCast_zmod_eq_zero_iff_dvd _ p).mp
+    simp only [Int.cast_add, Int.cast_mul, Int.cast_natCast]
+    rw [haB, hbB]
+    ring
+  obtain ⟨z, hz⟩ := hdiv
+  obtain ⟨k, hk0, hkc⟩ := exists_nonzero_affine_ne hp hlarge (a : ZMod p) (z : ZMod p) c ha0
+  let n := fiberColumn n₀ k
+  have hnv : (n.val : ℤ) = (n₀.val : ℤ) + (p : ℤ) * (k.val : ℤ) := by
+    simp only [n, fiberColumn, Nat.cast_add, Nat.cast_mul]
+  have hnvmod : (n.val : ZMod p) = (n₀.val : ZMod p) := by
+    simp only [n, fiberColumn, Nat.cast_add, Nat.cast_mul, ZMod.natCast_self,
+      zero_mul, add_zero]
+  have hnew : affineValue A₁ B₁ C₁ n (k.val : ℤ) = c := by
+    simp only [affineValue, hB₁, zero_mul, add_zero, hnvmod, c]
+  have hvalue : (W n ((p : ℤ) * (k.val : ℤ)) : ZMod p) = c :=
+    (hH₁.2 n (k.val : ℤ) (by rw [hnew]; exact hc)).trans hnew
+  have hfactor : a * (n.val : ℤ) + b = (p : ℤ) * (z + a * (k.val : ℤ)) := by
+    rw [hnv]
+    linear_combination hz
+  have hnot : ¬ (p : ℤ) ∣ z + a * (k.val : ℤ) := by
+    intro hd
+    apply hk0
+    simpa only [Int.cast_add, Int.cast_mul, Int.cast_natCast] using
+      (ZMod.intCast_zmod_eq_zero_iff_dvd (z + a * (k.val : ℤ)) p).mpr hd
+  have hp0 : (p : ℤ) ≠ 0 := Int.natCast_ne_zero.mpr hp.ne_zero
+  have hnotSq : ¬ (p : ℤ) ^ 2 ∣ a * (n.val : ℤ) + b := by
+    intro hd
+    apply hnot
+    apply (Int.mul_dvd_mul_iff_left hp0).mp
+    simpa only [pow_two, hfactor] using hd
+  have hdivn : (p : ℤ) ∣ a * (n.val : ℤ) + b := ⟨z + a * (k.val : ℤ), hfactor⟩
+  have hsecond := (hw n).2 hdivn hnotSq
+  dsimp only at hsecond
+  have hline : 1 * (n.val : ℤ) + -(n₀.val : ℤ) = (p : ℤ) * (k.val : ℤ) := by
+    rw [hnv]
+    ring
+  rw [hline, hvalue, hfactor, Int.mul_ediv_cancel_left _ hp0] at hsecond
+  apply hkc
+  simpa only [Int.cast_add, Int.cast_mul, Int.cast_natCast] using hsecond.symm
+
+theorem no_positive_nat_vertical_period {p : ℕ} (hp : p.Prime)
+    (hlarge : 200 < p) :
+    ∀ M : ℕ, ∀ W : WordArray p, LineRule p W → NonconstantColumns W →
+      0 < M → VerticalPeriod W (M : ℤ) → False := by
+  intro M
+  induction M using Nat.strong_induction_on with
+  | h M ih =>
+    intro W hW hcolumns hM hperiod
+    obtain ⟨A, B, C, hH⟩ := global_affine_approximation hp hlarge hW
+    have hB := (nonconstantColumns_iff_vertical_coefficient hp hlarge hH).mp hcolumns
+    have hdivZ := prime_dvd_vertical_period hp hlarge hH hB hperiod
+    have hdiv : p ∣ M := Int.natCast_dvd_natCast.mp hdivZ
+    obtain ⟨u, v, hnorm⟩ := exists_straighten_affine hp hH hB
+    let W₀ := straighten W u v
+    have hW₀ : LineRule p W₀ := straighten_lineRule hW u v
+    have hperiod₀ : VerticalPeriod W₀ (M : ℤ) :=
+      (straighten_verticalPeriod_iff W u v (M : ℤ)).mpr hperiod
+    have hW₁ : LineRule p (rescale W₀) := rescale_lineRule hW₀
+    obtain ⟨A₁, B₁, C₁, hH₁⟩ := global_affine_approximation hp hlarge hW₁
+    have hB₁ := rescale_vertical_coefficient_ne_zero hp hlarge hW₀ hB hnorm hH₁
+    have hcolumns₁ := (nonconstantColumns_iff_vertical_coefficient hp hlarge hH₁).mpr hB₁
+    have hsmall : M / p < M := Nat.div_lt_self hM hp.one_lt
+    have hpos : 0 < M / p := Nat.div_pos (Nat.le_of_dvd hM hdiv) hp.pos
+    have hquot : (p : ℤ) * ((M / p : ℕ) : ℤ) = (M : ℤ) := by
+      exact_mod_cast Nat.mul_div_cancel' hdiv
+    have hperiod₁ : VerticalPeriod (rescale W₀) ((M / p : ℕ) : ℤ) := by
+      intro n m
+      change W₀ n ((p : ℤ) * (m + ((M / p : ℕ) : ℤ))) = W₀ n ((p : ℤ) * m)
+      rw [mul_add, hquot]
+      exact hperiod₀ n ((p : ℤ) * m)
+    exact ih (M / p) hsmall (rescale W₀) hW₁ hcolumns₁ hpos hperiod₁
+
+theorem no_positive_vertical_period {p : ℕ} (hp : p.Prime)
+    (hlarge : 200 < p) {W : WordArray p} (hW : LineRule p W)
+    (hcolumns : NonconstantColumns W) {M : ℤ} (hM : 0 < M)
+    (hperiod : VerticalPeriod W M) : False := by
+  have hpos : 0 < M.toNat := Int.pos_iff_toNat_pos.mp hM
+  have hcast : (M.toNat : ℤ) = M := Int.toNat_of_nonneg hM.le
+  exact no_positive_nat_vertical_period hp hlarge M.toNat W hW hcolumns hpos
+    (by simpa only [hcast] using hperiod)
+
+end PeriodicTilingThree
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PeriodicTiling/WordRule.lean b/lean/OAI/Geometry/PeriodicTiling/WordRule.lean
new file mode 100644
index 0000000000000000000000000000000000000000..b1e1e14558f43a53e2c052593d2ef6f46b9b1d53
--- /dev/null
+++ b/lean/OAI/Geometry/PeriodicTiling/WordRule.lean
@@ -0,0 +1,76 @@
+import OAI.Geometry.PeriodicTiling.TilingBasic
+import Mathlib.Algebra.Field.ZMod
+import Mathlib.Data.Fintype.Pi
+import Mathlib.Data.Fintype.Sum
+
+namespace OAI
+
+namespace PeriodicTilingThree
+
+abbrev Column (p : ℕ) := Fin (p ^ 2)
+abbrev Symbol (p : ℕ) := (ZMod p)ˣ
+abbrev Word (p : ℕ) := Column p → Symbol p
+abbrev WordArray (p : ℕ) := Column p → ℤ → Symbol p
+abbrev Channel (p : ℕ) := Column p ⊕ Fin 2
+
+def Label (p : ℕ) : Channel p → Type
+  | .inl _ => Symbol p
+  | .inr _ => Unit
+
+instance instFintypeLabel (p : ℕ) [NeZero p] (i : Channel p) : Fintype (Label p i) := by
+  cases i <;> dsimp [Label] <;> infer_instance
+
+instance instDecidableEqLabel (p : ℕ) (i : Channel p) : DecidableEq (Label p i) := by
+  cases i <;> dsimp [Label] <;> infer_instance
+
+instance instInhabitedLabel (p : ℕ) (i : Channel p) : Inhabited (Label p i) := by
+  cases i <;> dsimp [Label] <;> infer_instance
+
+def Allowed (p : ℕ) (w : Word p) : Prop :=
+  ∃ a b : ℤ,
+    (¬ (p : ℤ) ∣ a ∨ ¬ (p : ℤ) ∣ b) ∧
+    ∀ n : Column p,
+      let t := a * (n.val : ℤ) + b
+      (¬ (p : ℤ) ∣ t → (w n : ZMod p) = (t : ZMod p)) ∧
+      ((p : ℤ) ∣ t → ¬ (p : ℤ) ^ 2 ∣ t →
+        (w n : ZMod p) = ((t / (p : ℤ) : ℤ) : ZMod p))
+
+def LineRule (p : ℕ) (W : WordArray p) : Prop :=
+  ∀ d e : ℤ, Allowed p (fun n => W n (d * (n.val : ℤ) + e))
+
+def NonconstantColumns {p : ℕ} (W : WordArray p) : Prop :=
+  ∀ n, ∃ m m' : ℤ, W n m ≠ W n m'
+
+def VerticalPeriod {p : ℕ} (W : WordArray p) (M : ℤ) : Prop :=
+  ∀ n m, W n (m + M) = W n m
+
+def affineValue {p : ℕ} (A B C : ZMod p) (n : Column p) (m : ℤ) : ZMod p :=
+  A * (n.val : ZMod p) + B * (m : ZMod p) + C
+
+def AffineApproximation {p : ℕ} (W : WordArray p) (A B C : ZMod p) : Prop :=
+  (A ≠ 0 ∨ B ≠ 0 ∨ C ≠ 0) ∧
+  ∀ n m, affineValue A B C n m ≠ 0 →
+    (W n m : ZMod p) = affineValue A B C n m
+
+def lineValue {p : ℕ} (n : Column p) (x : Plane) : ℤ :=
+  x.2 + (n.val : ℤ) * x.1
+
+def lineKernelStep {p : ℕ} (n : Column p) : Plane :=
+  (1, -(n.val : ℤ))
+
+@[simp] theorem lineValue_vertical {p : ℕ} (n : Column p) (m : ℤ) :
+    lineValue n (0, m) = m := by
+  simp [lineValue]
+
+noncomputable def allowedWords (p : ℕ) [NeZero p] : Finset (Word p) := by
+  classical
+  exact Finset.univ.filter (Allowed p)
+
+@[simp] theorem mem_allowedWords {p : ℕ} [NeZero p] (w : Word p) :
+    w ∈ allowedWords p ↔ Allowed p w := by
+  classical
+  simp [allowedWords]
+
+end PeriodicTilingThree
+
+end OAI
diff --git a/lean/OAI/Geometry/PinnedDistances/ArithmeticContradiction.lean b/lean/OAI/Geometry/PinnedDistances/ArithmeticContradiction.lean
new file mode 100644
index 0000000000000000000000000000000000000000..7147fd9873f2c0bc7803858a867d0cd74470e18f
--- /dev/null
+++ b/lean/OAI/Geometry/PinnedDistances/ArithmeticContradiction.lean
@@ -0,0 +1,69 @@
+import OAI.Geometry.PinnedDistances.BoundedContradiction
+
+namespace OAI
+
+universe u1 u2 u3
+
+section
+noncomputable section
+open Filter
+open scoped Topology
+namespace WeakPinned
+
+lemma sequence_bounded_subsequence (W : ℕ → ℝ) (hW : ¬Tendsto W atTop atTop)
+    (p : ℕ → Prop) (hp : ∀ᶠ n in atTop,p n) :
+    ∃ (φ : ℕ → ℕ) (C : ℝ),Tendsto φ atTop atTop ∧ 0≤C ∧ ∀ n,W (φ n)≤C ∧ p (φ n) := by
+  rw [tendsto_atTop] at hW
+  push Not at hW
+  obtain ⟨C,hC⟩ := hW
+  obtain ⟨φ,hφ,hφC⟩ := exists_seq_forall_of_frequently (hC.and_eventually hp)
+  exact ⟨φ,max C 0,hφ,le_max_right _ _,fun n => ⟨(hφC n).1.le.trans (le_max_left _ _),(hφC n).2⟩⟩
+
+end WeakPinned
+
+end
+end
+section
+noncomputable section
+open Filter
+open scoped Topology
+namespace WeakPinned
+open Global
+
+variable {V : ℕ → Type u1} {I : ℕ → Type u2} {K : ℕ → Type u3} [∀ j,Fintype (V j)] [∀ j,Fintype (I j)] [∀ j,DecidableEq (V j)]
+  [∀ j,Field (K j)] [∀ j,NumberField (K j)]
+
+/-- Both alternatives of the genuine arithmetic sequence have been eliminated. -/
+theorem arithmetic_sequence_impossible
+    (G : ∀ j,Core.Fibers (V j) (I j)) (θ γ c : ℝ) (η : ℕ → ℝ)
+    (Z₁ Z₂ : ∀ j,V j → K j) (h₁ : ∀ j,Function.Injective (Z₁ j)) (h₂ : ∀ j,Function.Injective (Z₂ j))
+    (hθ : 0<θ) (hγ : 0<γ) (hγ1 : γ<1) (hc : 0<c) (hcγ : c<γ)
+    (hn : Tendsto (fun j => (Fintype.card (V j):ℝ)) atTop atTop)
+    (hη : Tendsto η atTop (𝓝 0)) (hη0 : ∀ j,0≤η j) (hηb : ∀ j,η j≤θ/2)
+    (hG : ∀ j,PreparedGraph (V j) (I j) (G j) θ γ c (η j))
+    (t : ∀ j,I j → K j) (ht : ∀ j e y,y∈(G j).targets e →
+      (Z₁ j ((G j).source e)-Z₁ j y)*(Z₂ j ((G j).source e)-Z₂ j y)=t j e) : False := by
+  let W := fun j => overlapScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)
+  let A := fun j => potentialScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)
+  by_cases htop : Tendsto W atTop atTop
+  · exact arithmetic_no_unbounded G θ γ c η Z₁ Z₂ h₁ h₂ hθ hγ hγ1 hc hcγ hn hη hη0 hηb hG t ht htop
+  have hcomp := arithmetic_scale_comparison G θ γ c η Z₁ Z₂ h₁ h₂ hθ hγ hγ1 hcγ hn hη hη0 hηb hG t ht
+  obtain ⟨φ,C,hφ,hC,hbound⟩ := sequence_bounded_subsequence W htop
+    (fun j => A j≤(64+8/θ)*(W j+1)) hcomp
+  apply arithmetic_no_bounded (fun j => G (φ j)) θ γ c (η ∘ φ)
+    (fun j => Z₁ (φ j)) (fun j => Z₂ (φ j)) (fun j => h₁ (φ j)) (fun j => h₂ (φ j))
+    hθ hγ hγ1 hc hcγ (hn.comp hφ) (hη.comp hφ) (fun j => hη0 (φ j)) (fun j => hηb (φ j))
+    (fun j => hG (φ j)) (fun j => t (φ j)) (fun j => ht (φ j))
+    (C+(64+8/θ)*(C+1)) (by positivity)
+  intro j
+  change W (φ j)+A (φ j)≤_
+  have hA := (hbound j).2.trans (mul_le_mul_of_nonneg_left
+    (show W (φ j)+1≤C+1 by linarith [(hbound j).1]) (by positivity : 0≤64+8/θ))
+  linarith [(hbound j).1]
+
+end WeakPinned
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PinnedDistances/ArithmeticGraphs.lean b/lean/OAI/Geometry/PinnedDistances/ArithmeticGraphs.lean
new file mode 100644
index 0000000000000000000000000000000000000000..b3cdae4e8cdfe28f9c92d6a4c22597571766999b
--- /dev/null
+++ b/lean/OAI/Geometry/PinnedDistances/ArithmeticGraphs.lean
@@ -0,0 +1,487 @@
+import OAI.Geometry.PinnedDistances.IncomingFibers
+
+namespace OAI
+
+universe u1 u2 u3 u4 u5 u6 u7 u8 u9 u10 u11 u12
+universe u13 u14 u15
+
+section
+noncomputable section
+open NumberField
+open scoped BigOperators
+noncomputable section
+open MeasureTheory
+open scoped Topology
+noncomputable section
+open MeasureTheory
+open scoped Topology ENNReal
+noncomputable section
+open MeasureTheory NumberField
+open scoped BigOperators
+noncomputable section
+open scoped BigOperators
+open MeasureTheory
+noncomputable section
+open Filter
+open scoped Topology
+noncomputable section
+open scoped BigOperators
+
+namespace WeakPinned
+open scoped BigOperators
+open Variance
+variable {V : Type u1} {I : Type u2} [Fintype V] [Fintype I] [DecidableEq V]
+variable {G : Core.Fibers V I} {θ γ c η : ℝ}
+
+lemma PreparedGraph.edgeCount_pos (h : PreparedGraph V I G θ γ c η) (hη : η<θ) : 0<G.edgeCount := by
+  have hnR : (2:ℝ)≤Fintype.card V := by exact_mod_cast h.card_ge
+  have hp : 0<(θ-η)*(Fintype.card V : ℝ)*((Fintype.card V : ℝ)-1) :=
+    mul_pos (mul_pos (sub_pos.mpr hη) (by linarith)) (by linarith)
+  exact hp.trans_le h.lower
+
+lemma PreparedGraph.edge_domination (h : PreparedGraph V I G θ γ c η) (hη : η<θ) (x y : V) :
+    G.edgeLaw x y ≤ (1/(θ-η))*pairProb Finset.univ x y := by
+  have hnR : (2:ℝ)≤Fintype.card V := by exact_mod_cast h.card_ge
+  have hden : 0<(Fintype.card V : ℝ)*((Fintype.card V : ℝ)-1) := mul_pos (by linarith) (by linarith)
+  by_cases hxy : x=y
+  · subst y; simp [G.edgeLaw_diag,pairProb]
+  · have he : pairProb Finset.univ x y=1/((Fintype.card V : ℝ)*((Fintype.card V : ℝ)-1)) := by simp [pairProb,hxy]
+    rw [he]
+    unfold Core.Fibers.edgeLaw
+    split_ifs
+    · have hh := one_div_le_one_div_of_le (mul_pos (sub_pos.mpr hη) hden) (by simpa only [mul_assoc] using h.lower)
+      calc
+        1 / G.edgeCount ≤ 1 / ((θ-η)*((Fintype.card V : ℝ)*((Fintype.card V : ℝ)-1))) := hh
+        _ = _ := by rw [one_div_mul_one_div]
+    · positivity
+
+lemma PreparedGraph.degree_law (h : PreparedGraph V I G θ γ c η) (hη : η<θ) (hη0 : 0≤η) (x : V) :
+    2*((θ-η)/(θ+η/8))*prob Finset.univ x ≤ (∑ y, G.edgeLaw x y)+(∑ y, G.edgeLaw y x) := by
+  rw [G.degree_marginal]
+  have hnR : (2:ℝ)≤Fintype.card V := by exact_mod_cast h.card_ge
+  have hn0 : (0:ℝ)<Fintype.card V := by linarith
+  have hnm : 0<(Fintype.card V : ℝ)-1 := by linarith
+  have hθ : 0<θ+η/8 := by linarith
+  have hd := h.degree x
+  have hu := h.upper
+  have hh : (2*(θ-η)*((Fintype.card V : ℝ)-1))/((θ+η/8)*(Fintype.card V : ℝ)*((Fintype.card V : ℝ)-1)) ≤
+      G.degree (-1) Finset.univ x/G.edgeCount := by
+    apply div_le_div₀ _ hd (h.edgeCount_pos hη) hu
+    exact (mul_nonneg (mul_nonneg (by norm_num) (sub_nonneg.mpr hη.le)) hnm.le).trans hd
+  have he : (2*(θ-η)*((Fintype.card V : ℝ)-1))/((θ+η/8)*(Fintype.card V : ℝ)*((Fintype.card V : ℝ)-1)) =
+      2*((θ-η)/(θ+η/8))*prob Finset.univ x := by
+    simp only [prob,Finset.mem_univ,ite_true,Finset.card_univ]
+    field_simp
+  rwa [he] at hh
+
+lemma PreparedGraph.heavy_bound (h : PreparedGraph V I G θ γ c η) (hη : η<θ) (hη0 : 0≤η) (D : Finset V) :
+    G.heavyCount (1/Real.log (Fintype.card V : ℝ)) D / G.edgeCount ≤
+      ((θ+η)/(θ-η))*pairCell Finset.univ D := by
+  have hnR : (2:ℝ)≤Fintype.card V := by exact_mod_cast h.card_ge
+  have hden : 0<(Fintype.card V : ℝ)*((Fintype.card V : ℝ)-1) := mul_pos (by linarith) (by linarith)
+  have hD : 0 ≤ (D.card : ℝ)*((D.card : ℝ)-1) := by
+    by_cases hz : D.card=0
+    · simp [hz]
+    · have hh : (1:ℝ)≤D.card := by exact_mod_cast (show 1≤D.card by omega)
+      positivity
+  have hb := h.subset D
+  have hcmp : G.heavyCount (1/Real.log (Fintype.card V : ℝ)) D / G.edgeCount ≤
+      ((θ+η)*(D.card : ℝ)*((D.card : ℝ)-1))/((θ-η)*(Fintype.card V : ℝ)*((Fintype.card V : ℝ)-1)) := by
+    apply div_le_div₀ _ hb (mul_pos (mul_pos (sub_pos.mpr hη) (by linarith)) (by linarith)) h.lower
+    simpa only [mul_assoc] using mul_nonneg (show 0≤θ+η by linarith) hD
+  convert hcmp using 1
+  rw [pairCell_formula]
+  simp only [Finset.univ_inter,Finset.card_univ]
+  field_simp
+
+lemma PreparedGraph.small_heavy_bound (h : PreparedGraph V I G θ γ c η) (hη : η<θ) (hη0 : 0≤η)
+    (D : Finset V) (hDsz : (D.card : ℝ)≤(Fintype.card V : ℝ)^(1-c)) :
+    G.heavyCount (1/Real.log (Fintype.card V : ℝ)) D / G.edgeCount ≤
+      (η/(θ-η))*pairCell Finset.univ D := by
+  have hnR : (2:ℝ)≤Fintype.card V := by exact_mod_cast h.card_ge
+  have hD : 0 ≤ (D.card : ℝ)*((D.card : ℝ)-1) := by
+    by_cases hz : D.card=0
+    · simp [hz]
+    · have hh : (1:ℝ)≤D.card := by exact_mod_cast (show 1≤D.card by omega)
+      positivity
+  have hcmp : G.heavyCount (1/Real.log (Fintype.card V : ℝ)) D / G.edgeCount ≤
+      (η*(D.card : ℝ)*((D.card : ℝ)-1))/((θ-η)*(Fintype.card V : ℝ)*((Fintype.card V : ℝ)-1)) := by
+    apply div_le_div₀ _ (h.small D hDsz) (mul_pos (mul_pos (sub_pos.mpr hη) (by linarith)) (by linarith)) h.lower
+    simpa only [mul_assoc] using mul_nonneg hη0 hD
+  convert hcmp using 1
+  rw [pairCell_formula]
+  simp only [Finset.univ_inter,Finset.card_univ]
+  field_simp
+
+end WeakPinned
+
+namespace WeakPinned.Core.Fibers
+open scoped BigOperators
+open Variance SmallFiber
+variable {V : Type u3} {I : Type u4} [Fintype V] [Fintype I] [DecidableEq V]
+variable (G : Fibers V I)
+
+def qCell (C : Finset V) : ℝ := ∑ x ∈ C, ∑ y ∈ C, G.edgeLaw x y
+
+lemma qCell_eq (C : Finset V) : G.qCell C = SmallFiber.edgeCell G.fiberWeight (fun e => prob (G.targets e)) G.source C := by
+  unfold qCell SmallFiber.edgeCell SmallFiber.edgeLaw
+  simp_rw [G.sampling_fiber]
+
+omit [Fintype V] in
+omit [Fintype I] in
+lemma fiber_mass (e : I) (C : Finset V) : mass (prob (G.targets e)) C=(G.size C e : ℝ)/(G.targets e).card := by
+  rw [mass,sum_prob_cell]
+  rfl
+
+lemma qCell_split (hne : ∀ e, (G.targets e).Nonempty) (lam : ℝ) (C : Finset V) :
+    G.qCell C = slice G.fiberWeight (fun e => prob (G.targets e)) G.source lam C + G.heavyCount lam C/G.edgeCount := by
+  classical
+  rw [G.qCell_eq,edgeCell_formula]
+  unfold slice heavyCount
+  simp only [Finset.sum_div,← Finset.sum_add_distrib]
+  apply Finset.sum_congr rfl
+  intro e _
+  have hk : (0:ℝ)<(G.targets e).card := by exact_mod_cast Finset.card_pos.mpr (hne e)
+  have hm := G.fiber_mass e C
+  have hprod : G.fiberWeight e*mass (prob (G.targets e)) C=(G.size C e : ℝ)/G.edgeCount := by
+    rw [hm,fiberWeight]
+    field_simp
+  have hcmp : lam*((G.targets e).card : ℝ) ≤ (G.size C e : ℝ) ↔ lam ≤ mass (prob (G.targets e)) C := by
+    rw [hm,le_div_iff₀ hk]
+  simp only [hcmp]
+  by_cases hx : G.source e ∈ C <;> by_cases hp : mass (prob (G.targets e)) C<lam
+  · simp [hx,hp,not_le.mpr hp]
+  · simp [hx,hp,le_of_not_gt hp,hprod]
+  · simp [hx]
+  · simp [hx]
+
+omit [Fintype I] in
+lemma qCell_bound {B : ℝ} (hdom : ∀ x y, G.edgeLaw x y ≤ B*pairProb Finset.univ x y) (C : Finset V) :
+    G.qCell C ≤ B*pairCell Finset.univ C := by
+  unfold qCell pairCell
+  simp only [Finset.mul_sum]
+  exact Finset.sum_le_sum (fun x _ => Finset.sum_le_sum (fun y _ => hdom x y))
+
+end WeakPinned.Core.Fibers
+
+namespace WeakPinned.Variance
+open scoped BigOperators
+variable {V : Type u5} [Fintype V] [DecidableEq V]
+
+lemma weighted_pair (m : Finset V → ℝ) (q : V → V → ℝ) :
+    weightedIntegral m (fun C => ∑ x ∈ C, ∑ y ∈ C, q x y)=
+      ∑ x, ∑ y, q x y*weightedOverlap m x y := by
+  have hc (C : Finset V) : (∑ x ∈ C, ∑ y ∈ C, q x y)=
+      ∑ x, ∑ y, q x y*(if x∈C ∧ y∈C then (1:ℝ) else 0) := by
+    simp only [mul_ite,mul_one,mul_zero,ite_and,Finset.sum_ite_irrel,Finset.sum_const_zero]
+    simp
+  simp only [weightedIntegral,hc,Finset.mul_sum]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro x _
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro y _
+  simp only [weightedOverlap,weightedIntegral,Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro C _
+  ring
+
+lemma weighted_additive (m : Finset V → ℝ) (b : ℝ) (S : V → ℝ)
+    (hL : ∀ x y, x≠y → weightedOverlap m x y=b+S x+S y)
+    (q : V → V → ℝ) (hdiag : ∀ x, q x x=0) (h1 : (∑ x, ∑ y, q x y)=1) :
+    weightedIntegral m (fun C => ∑ x ∈ C, ∑ y ∈ C, q x y)=
+      b+∑ x, ((∑ y, q x y)+(∑ y, q y x))*S x := by
+  rw [weighted_pair]
+  have hp (x y : V) : q x y*weightedOverlap m x y=q x y*(b+S x+S y) := by
+    by_cases hxy : x=y
+    · subst y; simp [hdiag]
+    · rw [hL x y hxy]
+  simp_rw [hp,mul_add,Finset.sum_add_distrib]
+  have hb : (∑ x, ∑ y, q x y*b)=b := by
+    simp only [← Finset.sum_mul,h1,one_mul]
+  rw [hb]
+  simp only [← Finset.sum_mul,add_mul,Finset.sum_add_distrib]
+  rw [Finset.sum_comm (f:=fun x y => q x y*S y)]
+  simp only [← Finset.sum_mul]
+  ring
+
+lemma uniform_additive (m : Finset V → ℝ) (b : ℝ) (S : V → ℝ)
+    (hL : ∀ x y, x≠y → weightedOverlap m x y=b+S x+S y) (hn : 2≤Fintype.card V) :
+    weightedIntegral m (pairCell Finset.univ)=b+2*(∑ x, prob Finset.univ x*S x) := by
+  have hsum : (∑ x : V, ∑ y : V, pairProb Finset.univ x y)=1 := by
+    simp_rw [sum_pairProb Finset.univ (by simpa using hn)]
+    exact sum_prob Finset.univ (Finset.card_pos.mp (by simpa using (show 0<Fintype.card V by omega)))
+  have hh := weighted_additive m b S hL (pairProb Finset.univ) (by intro x; simp [pairProb]) hsum
+  change weightedIntegral m (fun C => ∑ x ∈ C, ∑ y ∈ C, pairProb Finset.univ x y) = _
+  rw [hh]
+  simp_rw [pairProb_symm Finset.univ _ _,sum_pairProb Finset.univ (by simpa using hn)]
+  simp only [Finset.mul_sum]
+  congr 1
+  apply Finset.sum_congr rfl
+  intro x _
+  ring
+
+end WeakPinned.Variance
+
+namespace WeakPinned
+open scoped BigOperators
+open Variance SmallFiber
+variable {V : Type u6} {I : Type u7} [Fintype V] [Fintype I] [DecidableEq V]
+variable {G : Core.Fibers V I} {θ γ c η : ℝ}
+
+lemma PreparedGraph.overlap_lower (h : PreparedGraph V I G θ γ c η) (hη : η<θ) (hη0 : 0≤η)
+    (m : Finset V → ℝ) (b : ℝ) (S : V → ℝ) (hS : ∀ x, 0≤S x)
+    (hL : ∀ x y, x≠y → weightedOverlap m x y=b+S x+S y) :
+    weightedIntegral m (pairCell Finset.univ) -
+      (2-2*((θ-η)/(θ+η/8)))*(∑ x, prob Finset.univ x*S x) ≤ weightedIntegral m G.qCell := by
+  change _ ≤ weightedIntegral m (fun C => ∑ x ∈ C, ∑ y ∈ C, G.edgeLaw x y)
+  rw [weighted_additive m b S hL G.edgeLaw G.edgeLaw_diag (G.sum_edgeLaw (h.edgeCount_pos hη)),
+    uniform_additive m b S hL h.card_ge]
+  have hh := Finset.sum_le_sum (s:=Finset.univ) (fun x _ =>
+    mul_le_mul_of_nonneg_right (h.degree_law hη hη0 x) (hS x))
+  have he : (∑ x : V, 2*((θ-η)/(θ+η/8))*prob Finset.univ x*S x)=
+      (2*((θ-η)/(θ+η/8)))*(∑ x : V, prob Finset.univ x*S x) := by
+    simp only [Finset.mul_sum,mul_assoc]
+  rw [he] at hh
+  linarith
+
+/-- A fully quantitative version of small-overlap before either scale branch.
+The slice is the actual original fiber mass, not an assumed error term. -/
+lemma PreparedGraph.small_overlap (h : PreparedGraph V I G θ γ c η) (hη : η<θ) (hη0 : 0≤η)
+    (m : Finset V → ℝ) (hm : ∀ C, 0 ≤ m C) (b : ℝ) (S : V → ℝ) (hS : ∀ x, 0≤S x)
+    (hL : ∀ x y, x≠y → weightedOverlap m x y=b+S x+S y) :
+    (θ/(θ-η))*weightedIntegral m (fun C => if (C.card : ℝ)≤(Fintype.card V : ℝ)^(1-c)
+      then pairCell Finset.univ C else 0) ≤
+      weightedIntegral m (slice G.fiberWeight (fun e => prob (G.targets e)) G.source (1/Real.log (Fintype.card V : ℝ))) +
+      ((θ+η)/(θ-η)-1)*weightedIntegral m (pairCell Finset.univ) +
+      (2-2*((θ-η)/(θ+η/8)))*(∑ x, prob Finset.univ x*S x) := by
+  have hlo := h.overlap_lower hη hη0 m b S hS hL
+  have hne : ∀ e, (G.targets e).Nonempty := by
+    intro e
+    have hn : (0:ℝ)<Fintype.card V := by exact_mod_cast (show 0<Fintype.card V by have := h.card_ge; omega)
+    exact Finset.card_pos.mp (by exact_mod_cast ((Real.rpow_pos_of_pos hn γ).trans (h.fiber e)))
+  have hpoint (C : Finset V) : G.qCell C ≤
+      slice G.fiberWeight (fun e => prob (G.targets e)) G.source (1/Real.log (Fintype.card V : ℝ)) C +
+      ((θ+η)/(θ-η))*pairCell Finset.univ C -
+      (θ/(θ-η))*(if (C.card : ℝ)≤(Fintype.card V : ℝ)^(1-c) then pairCell Finset.univ C else 0) := by
+    rw [G.qCell_split hne (1/Real.log (Fintype.card V : ℝ)) C]
+    by_cases hC : (C.card : ℝ)≤(Fintype.card V : ℝ)^(1-c)
+    · rw [ite_eq_left hC]
+      have hh := h.small_heavy_bound hη hη0 C hC
+      have he : (θ+η)/(θ-η)-θ/(θ-η)=η/(θ-η) := by ring
+      calc
+        _ ≤ slice G.fiberWeight (fun e => prob (G.targets e)) G.source (1/Real.log (Fintype.card V : ℝ)) C + (η/(θ-η))*pairCell Finset.univ C := add_le_add_right hh _
+        _ = _ := by rw [← he]; ring
+    · simp only [ite_eq_right hC,mul_zero,sub_zero]
+      exact add_le_add_right (h.heavy_bound hη hη0 C) _
+  have hint := Finset.sum_le_sum (s:=Finset.univ) (fun C _ => mul_le_mul_of_nonneg_left (hpoint C) (hm C))
+  change weightedIntegral m G.qCell ≤ _ at hint
+  simp only [mul_sub,mul_add,Finset.sum_sub_distrib,Finset.sum_add_distrib] at hint
+  have hcomm (a : ℝ) (f : Finset V → ℝ) : (∑ C, m C*(a*f C))=a*weightedIntegral m f := by
+    simp only [weightedIntegral,Finset.mul_sum]
+    exact Finset.sum_congr rfl (fun C _ => by ring)
+  simp only [hcomm] at hint
+  change weightedIntegral m G.qCell ≤ weightedIntegral m _+_ - _ at hint
+  nlinarith
+
+end WeakPinned
+
+namespace WeakPinned
+open scoped BigOperators
+open Variance SmallFiber Global
+variable {V : Type u8} {I : Type u9} [Fintype V] [Fintype I] [DecidableEq V]
+variable {G : Core.Fibers V I} {θ γ c η : ℝ}
+variable {K : Type u10} [Field K] [NumberField K]
+variable (Z₁ Z₂ : V → K) (h₁ : Function.Injective Z₁) (h₂ : Function.Injective Z₂)
+
+def overlapScale : ℝ := weightedIntegral (totalMass Z₁ Z₂ h₁ h₂) (pairCell Finset.univ)
+def potentialScale : ℝ := ∑ x, prob Finset.univ x * totalPotential Z₁ Z₂ h₁ h₂ x
+
+def smallOverlapScale (c : ℝ) : ℝ := weightedIntegral (totalMass Z₁ Z₂ h₁ h₂)
+  (fun C => if (C.card : ℝ) ≤ (Fintype.card V : ℝ)^(1-c) then pairCell Finset.univ C else 0)
+
+lemma overlapScale_nonneg : 0 ≤ overlapScale Z₁ Z₂ h₁ h₂ := by
+  exact Finset.sum_nonneg (fun C _ => mul_nonneg (totalMass_nonneg Z₁ Z₂ h₁ h₂ C) (pairCell_nonneg _ _))
+
+lemma potentialScale_nonneg : 0 ≤ potentialScale Z₁ Z₂ h₁ h₂ := by
+  apply Finset.sum_nonneg
+  intro x _
+  apply mul_nonneg _ (totalPotential_nonneg Z₁ Z₂ h₁ h₂ x)
+  simp only [prob,Finset.mem_univ,ite_true]
+  positivity
+
+lemma smallOverlapScale_nonneg (c : ℝ) : 0 ≤ smallOverlapScale Z₁ Z₂ h₁ h₂ c := by
+  apply Finset.sum_nonneg
+  intro C _
+  apply mul_nonneg (totalMass_nonneg Z₁ Z₂ h₁ h₂ C)
+  dsimp only
+  split_ifs
+  · exact pairCell_nonneg _ _
+  · exact le_refl _
+
+lemma PreparedGraph.arithmetic_slice (h : PreparedGraph V I G θ γ c η) (hη : η < θ)
+    (t : I → K) (ht : ∀ e y, y ∈ G.targets e → (Z₁ (G.source e)-Z₁ y)*(Z₂ (G.source e)-Z₂ y)=t e)
+    (τ lam H : ℝ) (hlam : 0 ≤ lam) (hlam2 : lam ≤ 1/2)
+    (hτ : 0 < τ) (hτ1 : τ ≤ 1) (hτn : 2 ≤ τ*(Fintype.card V:ℝ)) (hH : 0 ≤ H) :
+    weightedIntegral (totalMass Z₁ Z₂ h₁ h₂) (slice G.fiberWeight (fun e => prob (G.targets e)) G.source lam) ≤
+      (2*lam/τ^2+2*lam/(θ-η))*overlapScale Z₁ Z₂ h₁ h₂ +
+      (2*τ/(θ-η))*potentialScale Z₁ Z₂ h₁ h₂ + 4*lam*H + 32*Real.exp (-H/2) := by
+  have hne : ∀ e, (G.targets e).Nonempty := by
+    intro e
+    have hn : (0:ℝ)<Fintype.card V := by exact_mod_cast (show 0<Fintype.card V by have := h.card_ge; omega)
+    exact Finset.card_pos.mp (by exact_mod_cast ((Real.rpow_pos_of_pos hn γ).trans (h.fiber e)))
+  have hp : ∀ e y, 0 ≤ prob (G.targets e) y := by intro e y; unfold prob; split_ifs <;> positivity
+  have hx : ∀ e, prob (G.targets e) (G.source e)=0 := by intro e; simp [prob,G.no_loop]
+  have ht' : ∀ e y, prob (G.targets e) y ≠ 0 → (Z₁ (G.source e)-Z₁ y)*(Z₂ (G.source e)-Z₂ y)=t e := by
+    intro e y hy
+    apply ht e y
+    by_contra hn
+    exact hy (by simp [prob,hn])
+  have hdom : ∀ a b, a ≠ b → SmallFiber.edgeLaw G.fiberWeight (fun e => prob (G.targets e)) G.source a b ≤
+      (1/(θ-η))/((Fintype.card V:ℝ)*((Fintype.card V:ℝ)-1)) := by
+    intro a b hab
+    have hh := h.edge_domination hη a b
+    simpa [SmallFiber.edgeLaw,G.sampling_fiber,pairProb,hab, div_eq_mul_inv,mul_assoc,mul_comm,mul_left_comm] using hh
+  have hh := arithmetic_small_fiber Z₁ Z₂ h₁ h₂ h.card_ge G.fiberWeight (fun e => prob (G.targets e)) G.source t
+    G.fiberWeight_nonneg (G.sum_fiberWeight (h.edgeCount_pos hη)) hp (fun e => sum_prob _ (hne e)) hx ht'
+    (1/(θ-η)) τ lam H (by positivity) hlam hlam2 hτ hτ1 hτn hH hdom
+  have he : (∑ x : V, totalPotential Z₁ Z₂ h₁ h₂ x)/(Fintype.card V:ℝ) = potentialScale Z₁ Z₂ h₁ h₂ := by
+    simp only [potentialScale,prob,Finset.mem_univ,ite_true,Finset.card_univ,Finset.sum_div]
+    exact Finset.sum_congr rfl (fun x _ => by ring)
+  have ha : 2*(1/(θ-η))*τ/(Fintype.card V:ℝ)*(∑ x, totalPotential Z₁ Z₂ h₁ h₂ x) =
+      (2*τ/(θ-η))*potentialScale Z₁ Z₂ h₁ h₂ := by rw [← he]; ring
+  rw [ha] at hh
+  convert hh using 1; unfold overlapScale; ring
+
+/-- The genuine scale estimate, valid for every prepared arithmetic model.
+All error terms are explicit and can be sent to zero uniformly in the field. -/
+lemma PreparedGraph.arithmetic_small_overlap (h : PreparedGraph V I G θ γ c η) (hη : η < θ) (hη0 : 0 ≤ η)
+    (t : I → K) (ht : ∀ e y, y ∈ G.targets e → (Z₁ (G.source e)-Z₁ y)*(Z₂ (G.source e)-Z₂ y)=t e)
+    (τ H : ℝ) (hlam : 0 ≤ 1/Real.log (Fintype.card V:ℝ)) (hlam2 : 1/Real.log (Fintype.card V:ℝ) ≤ 1/2)
+    (hτ : 0 < τ) (hτ1 : τ ≤ 1) (hτn : 2 ≤ τ*(Fintype.card V:ℝ)) (hH : 0 ≤ H) :
+    smallOverlapScale Z₁ Z₂ h₁ h₂ c ≤
+      (2*(1/Real.log (Fintype.card V:ℝ))/τ^2+2*(1/Real.log (Fintype.card V:ℝ))/(θ-η)+((θ+η)/(θ-η)-1))*overlapScale Z₁ Z₂ h₁ h₂ +
+      (2*τ/(θ-η)+(2-2*((θ-η)/(θ+η/8))))*potentialScale Z₁ Z₂ h₁ h₂ +
+      4*(1/Real.log (Fintype.card V:ℝ))*H + 32*Real.exp (-H/2) := by
+  have hs := h.small_overlap hη hη0 (totalMass Z₁ Z₂ h₁ h₂) (totalMass_nonneg Z₁ Z₂ h₁ h₂)
+    (baseConstant Z₁ Z₂ h₁ h₂) (totalPotential Z₁ Z₂ h₁ h₂) (totalPotential_nonneg Z₁ Z₂ h₁ h₂)
+    (totalMass_overlap Z₁ Z₂ h₁ h₂ h.card_ge)
+  have htS := h.arithmetic_slice Z₁ Z₂ h₁ h₂ hη t ht τ (1/Real.log (Fintype.card V:ℝ)) H hlam hlam2 hτ hτ1 hτn hH
+  have hf : 1 ≤ θ/(θ-η) := (one_le_div (sub_pos.mpr hη)).mpr (by linarith)
+  have he := mul_le_mul_of_nonneg_right hf (smallOverlapScale_nonneg Z₁ Z₂ h₁ h₂ c)
+  change (θ/(θ-η))*smallOverlapScale Z₁ Z₂ h₁ h₂ c ≤ _ at hs
+  change _ ≤ _ + ((θ+η)/(θ-η)-1)*overlapScale Z₁ Z₂ h₁ h₂ +
+    (2-2*((θ-η)/(θ+η/8)))*potentialScale Z₁ Z₂ h₁ h₂ at hs
+  nlinarith
+end WeakPinned
+
+namespace WeakPinned.Variance
+open scoped BigOperators
+variable {P : Type u11} {I : Type u12} [Fintype P] [DecidableEq P] [Fintype I]
+lemma varianceCell_nonneg (w : I → ℝ) (B : I → Finset P) (T : ℝ)
+    (hw : ∀ i, 0≤w i) (C : Finset P) : 0≤varianceCell w B T C := by
+  unfold varianceCell
+  split_ifs
+  · exact Finset.sum_nonneg (fun i _ => mul_nonneg (hw i) (sq_nonneg _))
+  · apply mul_nonneg (by norm_num)
+    apply Finset.sum_nonneg
+    intro i _
+    apply mul_nonneg (hw i)
+    split_ifs <;> positivity
+
+lemma reciprocal_card_sub_one {n k : ℝ} (hn : 2≤n) (hnk : n≤k) :
+    1/(k-1) ≤ 2/n := by
+  apply (div_le_div_iff₀ (by linarith : 0<k-1) (by linarith : 0<n)).mpr
+  nlinarith
+
+end WeakPinned.Variance
+
+namespace WeakPinned
+open scoped BigOperators Topology
+open Filter Variance Global
+variable {V : ℕ → Type u13} {I : ℕ → Type u14} {K : ℕ → Type u15} [∀ j, Fintype (V j)] [∀ j, Fintype (I j)] [∀ j, DecidableEq (V j)]
+  [∀ j, Field (K j)] [∀ j, NumberField (K j)]
+variable (Z₁ Z₂ : ∀ j, V j → K j) (h₁ : ∀ j, Function.Injective (Z₁ j)) (h₂ : ∀ j, Function.Injective (Z₂ j))
+
+/-- Asymptotic arithmetic variance for arbitrary genuine uniform-subset
+families. The domination and size assumptions are finite sampling data.
+The exact arithmetic cancellation and small-overlap rate are already proved. -/
+theorem arithmetic_variance_rate (w : ∀ j, I j → ℝ) (B : ∀ j, I j → Finset (V j))
+    (γ c M₀ : ℝ) (hγ : 0<γ) (hγ1 : γ<1) (hcγ : c<γ) (_hc1 : c<1) (hM : 0≤M₀)
+    (hn : Tendsto (fun j => (Fintype.card (V j):ℝ)) atTop atTop)
+    (hw : ∀ j i, 0≤w j i) (hW : ∀ j, (∑ i, w j i)≤1)
+    (hsize : ∀ᶠ j in atTop, ∀ i, (Fintype.card (V j):ℝ)^γ ≤ ((B j i).card:ℝ))
+    (hdom : ∀ᶠ j in atTop, ∀ x, (∑ i, w j i*prob (B j i) x)≤ M₀*prob Finset.univ x)
+    (hsmall : Tendsto (fun j => smallOverlapScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j) c /
+      (overlapScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)+potentialScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)+1)) atTop (𝓝 0)) :
+    Tendsto (fun j => weightedIntegral (totalMass (Z₁ j) (Z₂ j) (h₁ j) (h₂ j))
+      (varianceCell (w j) (B j) ((Fintype.card (V j):ℝ)^(1-c))) /
+      (overlapScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)+potentialScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)+1)) atTop (𝓝 0) := by
+  let n := fun j => (Fintype.card (V j):ℝ)
+  let U := fun j => overlapScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)
+  let A := fun j => potentialScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)
+  let D := fun j => smallOverlapScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j) c
+  have hU (j) : 0≤U j := overlapScale_nonneg _ _ _ _
+  have hA (j) : 0≤A j := potentialScale_nonneg _ _ _ _
+  have hden (j) : 0<U j+A j+1 := by linarith [hU j,hA j]
+  have hNγ : Tendsto (fun j => n j^γ) atTop atTop := (tendsto_rpow_atTop hγ).comp hn
+  have hcT : Tendsto (fun j => n j^(c-γ)) atTop (𝓝 0) :=
+    by
+    have h := (tendsto_rpow_neg_atTop (by linarith : 0<γ-c)).comp hn
+    convert h using 1; funext j; congr 1; ring
+  have hlim : Tendsto (fun j => 4*(M₀+1)*n j^(c-γ)+4*M₀*(D j/(U j+A j+1))) atTop (𝓝 0) := by
+    convert (hcT.const_mul (4*(M₀+1))).add (hsmall.const_mul (4*M₀)) using 1; simp
+  apply squeeze_zero' _ _ hlim
+  · exact Filter.Eventually.of_forall fun j => div_nonneg
+      (Finset.sum_nonneg fun C _ => mul_nonneg (totalMass_nonneg _ _ _ _ C)
+        (varianceCell_nonneg _ _ _ (hw j) C)) (hden j).le
+  · filter_upwards [hn.eventually (eventually_ge_atTop (2:ℝ)),
+      hNγ.eventually (eventually_ge_atTop (2:ℝ)), hsize,hdom] with j hjn hjγ hjs hjd
+    have hnpos : 0<n j := by linarith
+    have hnγpos : 0<n j^γ := Real.rpow_pos_of_pos hnpos _
+    have hnP : 2≤Fintype.card (V j) := by exact_mod_cast hjn
+    have hbs : ∀ i, 2≤(B j i).card := fun i => by exact_mod_cast (hjγ.trans (hjs i))
+    have hbn : n j^γ ≤ n j := by
+      simpa only [Real.rpow_one] using Real.rpow_le_rpow_of_exponent_le (show 1≤n j by linarith) hγ1.le
+    have hk : ∀ i, 1/(((B j i).card:ℝ)-1)≤2/(n j^γ) := fun i => reciprocal_card_sub_one hjγ (hjs i)
+    have hnerr : 1/(n j-1)≤2/(n j^γ) := reciprocal_card_sub_one hjγ hbn
+    have hT : 1≤n j^(1-c) := Real.one_le_rpow (by linarith) (by linarith)
+    have hτ : 0<n j^(-c) := Real.rpow_pos_of_pos hnpos _
+    have hscale : n j^(-c)*n j≤n j^(1-c) := by
+      calc
+        _ = n j^(-c)*n j^(1:ℝ) := by rw [Real.rpow_one]
+        _ = n j^((-c)+1) := (Real.rpow_add hnpos _ _).symm
+        _ = n j^(1-c) := by congr 1;ring
+        _ ≤ n j^(1-c) := le_rfl
+    have hv := arithmetic_variance (Z₁ j) (Z₂ j) (h₁ j) (h₂ j) hnP (w j) (B j)
+      (n j^(1-c)) M₀ (2/(n j^γ)) (n j^(-c)) (hw j) (hW j) hT hM
+      (by positivity) hτ hscale hbs hk hnerr hjd
+    have he : 2*(2/(n j^γ))*(M₀+1)/(n j^(-c))=4*(M₀+1)*n j^(c-γ) := by
+      rw [Real.rpow_sub hnpos,Real.rpow_neg hnpos.le]
+      field_simp
+      ring
+    rw [he] at hv
+    change _≤4*(M₀+1)*n j^(c-γ)*U j+4*M₀*D j at hv
+    have hratio : U j/(U j+A j+1)≤1 := (div_le_one (hden j)).mpr (by linarith [hA j])
+    have hcoeff : 0≤4*(M₀+1)*n j^(c-γ) := by positivity
+    calc
+      _ ≤ (4*(M₀+1)*n j^(c-γ)*U j+4*M₀*D j)/(U j+A j+1) := div_le_div_of_nonneg_right hv (hden j).le
+      _ = (4*(M₀+1)*n j^(c-γ))*(U j/(U j+A j+1))+4*M₀*(D j/(U j+A j+1)) := by ring
+      _ ≤ _ := by nlinarith [mul_le_mul_of_nonneg_left hratio hcoeff]
+end WeakPinned
+noncomputable section
+open Filter
+open scoped Topology
+
+end
+end
+end
+end
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PinnedDistances/ArithmeticHierarchy.lean b/lean/OAI/Geometry/PinnedDistances/ArithmeticHierarchy.lean
new file mode 100644
index 0000000000000000000000000000000000000000..69eb0e9483af29d12f73282c4259d7577f0fb8b7
--- /dev/null
+++ b/lean/OAI/Geometry/PinnedDistances/ArithmeticHierarchy.lean
@@ -0,0 +1,378 @@
+import OAI.Geometry.PinnedDistances.RandomGrids
+
+namespace OAI
+
+universe u1 u2 u3 u4
+
+section
+noncomputable section
+open NumberField
+open scoped BigOperators
+noncomputable section
+open MeasureTheory
+open scoped Topology
+noncomputable section
+open MeasureTheory
+open scoped Topology ENNReal
+
+noncomputable section
+open MeasureTheory NumberField
+open scoped BigOperators
+namespace WeakPinned.Global
+open Hierarchy
+variable {P : Type u1} [Fintype P] [DecidableEq P]
+variable {K : Type u2} [Field K] [NumberField K]
+variable (Z : P → K) (hZ : Function.Injective Z)
+include hZ
+
+def good (v : InfinitePlace K) (ω : Grid.FullSample) : Prop :=
+  ∀ x y z : P, x ≠ y → y ≠ z → x ≠ z →
+    min (Grid.depth ω (v.embedding (Z x)) (v.embedding (Z y)))
+      (Grid.depth ω (v.embedding (Z y)) (v.embedding (Z z))) ≤
+        Grid.depth ω (v.embedding (Z x)) (v.embedding (Z z))
+
+omit [Fintype P] [DecidableEq P] [NumberField K] in
+lemma embedding_ne (v : InfinitePlace K) {x y : P} (hne : x ≠ y) :
+    v.embedding (Z x) ≠ v.embedding (Z y) :=
+  fun hh => hne (hZ (v.embedding.injective hh))
+
+omit [DecidableEq P] [NumberField K] in
+lemma measurableSet_good (v : InfinitePlace K) : MeasurableSet {ω | good Z v ω} := by
+  simp only [good, Set.ofPred_forall]
+  apply MeasurableSet.iInter
+  intro x
+  apply MeasurableSet.iInter
+  intro y
+  apply MeasurableSet.iInter
+  intro z
+  apply MeasurableSet.iInter
+  intro hxy
+  apply MeasurableSet.iInter
+  intro hyz
+  apply MeasurableSet.iInter
+  intro hxz
+  exact measurableSet_le
+    ((Grid.measurable_depth (embedding_ne Z hZ v hxy)).min
+      (Grid.measurable_depth (embedding_ne Z hZ v hyz)))
+    (Grid.measurable_depth (embedding_ne Z hZ v hxz))
+
+omit [DecidableEq P] [NumberField K] in
+lemma good_ae (v : InfinitePlace K) : ∀ᵐ ω ∂Grid.fullLaw, good Z v ω := by
+  simp only [good, Filter.eventually_all]
+  intro x y z hxy hyz hxz
+  exact Grid.depth_ultra_ae (embedding_ne Z hZ v hxy)
+    (embedding_ne Z hZ v hyz) (embedding_ne Z hZ v hxz)
+
+instance good_decidable (v : InfinitePlace K) (ω : Grid.FullSample) : Decidable (good Z v ω) :=
+  Classical.propDecidable _
+
+def arch (v : InfinitePlace K) (ω : Grid.FullSample) : Depth P where
+  value x y := if good Z v ω ∧ x ≠ y then Grid.depth ω (v.embedding (Z x)) (v.embedding (Z y)) else 0
+  symm x y := by
+    by_cases h : good Z v ω
+    · by_cases hxy : x = y
+      · subst y; rfl
+      · simp [h, hxy, Ne.symm hxy, Grid.depth_symm]
+    · simp [h]
+  ultra x y z hxy hyz hxz := by
+    by_cases h : good Z v ω
+    · simpa [h,hxy,hyz,hxz] using h x y z hxy hyz hxz
+    · simp [h]
+
+omit [NumberField K] in
+lemma arch_value_ae (v : InfinitePlace K) {x y : P} (hxy : x ≠ y) :
+    (fun ω => (arch Z v ω).value x y) =ᵐ[Grid.fullLaw]
+      (fun ω => Grid.depth ω (v.embedding (Z x)) (v.embedding (Z y))) := by
+  filter_upwards [good_ae Z hZ v] with ω h
+  simp [arch,h,hxy]
+
+omit [NumberField K] in
+lemma arch_measurable (v : InfinitePlace K) (x y : P) :
+    Measurable (fun ω => (arch Z v ω).value x y) := by
+  by_cases hxy : x = y
+  · subst y
+    simpa only [arch,ne_eq,not_true_eq_false,and_false,reduceIte] using
+      (measurable_const : Measurable (fun _ : Grid.FullSample => (0:ℝ)))
+  · simpa [arch,hxy] using (Grid.measurable_depth (embedding_ne Z hZ v hxy)).ite
+      (measurableSet_good Z hZ v) (measurable_const : Measurable (fun _ : Grid.FullSample => (0:ℝ)))
+
+omit [NumberField K] in
+lemma arch_integrable (v : InfinitePlace K) (x y : P) :
+    Integrable (fun ω => (arch Z v ω).value x y) Grid.fullLaw := by
+  by_cases hxy : x = y
+  · subst y
+    simpa only [arch,ne_eq,not_true_eq_false,and_false,reduceIte] using
+      (integrable_const (0:ℝ) : Integrable (fun _ : Grid.FullSample => (0:ℝ)) Grid.fullLaw)
+  · have hi := (Grid.error_integrable_full (embedding_ne Z hZ v hxy)).sub
+      (integrable_const (Real.log ‖v.embedding (Z x) - v.embedding (Z y)‖))
+    have he : (fun ω => Grid.error ω (v.embedding (Z x)) (v.embedding (Z y)) -
+        Real.log ‖v.embedding (Z x) - v.embedding (Z y)‖) =
+        (fun ω => Grid.depth ω (v.embedding (Z x)) (v.embedding (Z y))) := by
+      funext ω; simp only [Grid.error,add_sub_cancel_right]
+    change Integrable (fun ω => Grid.error ω (v.embedding (Z x)) (v.embedding (Z y)) -
+        Real.log ‖v.embedding (Z x) - v.embedding (Z y)‖) Grid.fullLaw at hi
+    rw [he] at hi
+    exact hi.congr (arch_value_ae Z hZ v hxy).symm
+
+lemma arch_mean (v : InfinitePlace K) (x y : P) (hxy : x ≠ y) :
+    (∫ ω, (arch Z v ω).value x y ∂Grid.fullLaw) = -Real.log (v (Z x-Z y)) + Grid.kappa := by
+  rw [integral_congr_ae (arch_value_ae Z hZ v hxy)]
+  have hh := Grid.mean_error_full (embedding_ne Z hZ v hxy)
+  have hi := (arch_integrable Z hZ v x y).congr (arch_value_ae Z hZ v hxy)
+  simp only [Grid.error] at hh
+  rw [integral_add hi (integrable_const _)] at hh
+  simp only [integral_const,probReal_univ,one_smul] at hh
+  rw [← map_sub, InfinitePlace.norm_embedding_eq] at hh
+  linarith
+
+def finite (v : FinitePlace K) : Depth P where
+  value x y := if x = y then 0 else -Real.log (v (Z x-Z y))
+  symm x y := by
+    by_cases hxy : x = y
+    · subst y; rfl
+    · simp only [hxy,Ne.symm hxy,reduceIte]
+      rw [← neg_sub (Z x) (Z y), map_neg_eq_map]
+  ultra x y z hxy hyz hxz := by
+    simp only [hxy,hyz,hxz,reduceIte]
+    exact Arithmetic.finite_raw_ultra v (hZ.ne hxy) (hZ.ne hyz) (hZ.ne hxz)
+
+abbrev DiffPair := {p : P × P // p.1 ≠ p.2}
+
+def badPlaces : Finset (FinitePlace K) :=
+  (Arithmetic.finite_nonunit_places (fun p : DiffPair (P := P) => Z p.1.1 - Z p.1.2)
+    (fun p => sub_ne_zero.mpr (hZ.ne p.2))).toFinset
+
+omit [DecidableEq P] in
+lemma badPlaces_mem (v : FinitePlace K) : v ∈ badPlaces Z hZ ↔ ∃ x y, x ≠ y ∧ v (Z x-Z y) ≠ 1 := by
+  simp only [badPlaces, Set.Finite.mem_toFinset]
+  constructor
+  · rintro ⟨⟨⟨x,y⟩,hne⟩,hh⟩
+    exact ⟨x,y,hne,hh⟩
+  · rintro ⟨x,y,hne,hh⟩
+    exact ⟨⟨(x,y),hne⟩,hh⟩
+
+lemma finite_value_sum (x y : P) (hxy : x ≠ y) :
+    (∑ v ∈ badPlaces Z hZ, (finite Z hZ v).value x y) =
+      ∑ᶠ v : FinitePlace K, -Real.log (v (Z x-Z y)) := by
+  simp only [finite,hxy,reduceIte]
+  symm
+  apply finsum_eq_sum_of_support_subset
+  intro v hv
+  apply (badPlaces_mem Z hZ v).mpr
+  refine ⟨x,y,hxy,?_⟩
+  intro hh
+  exact hv (by simp [hh])
+
+def S (x : P) : ℝ :=
+  (∑ v : InfinitePlace K, Arithmetic.archWeight v * ∫ ω, (arch Z v ω).potential x ∂Grid.fullLaw) +
+    (Module.finrank ℚ K : ℝ)⁻¹ * ∑ v ∈ badPlaces Z hZ, (finite Z hZ v).potential x
+
+def L (x y : P) (hxy : x ≠ y) : ℝ :=
+  (∑ v : InfinitePlace K, Arithmetic.archWeight v * ∫ ω, (arch Z v ω).overlapLength x y hxy ∂Grid.fullLaw) +
+    (Module.finrank ℚ K : ℝ)⁻¹ * ∑ v ∈ badPlaces Z hZ, (finite Z hZ v).overlapLength x y hxy
+
+def M : ℝ :=
+  (∑ v : InfinitePlace K, Arithmetic.archWeight v * ∫ ω, (arch Z v ω).M ∂Grid.fullLaw) +
+    (Module.finrank ℚ K : ℝ)⁻¹ * ∑ v ∈ badPlaces Z hZ, (finite Z hZ v).M
+
+lemma coordinate_overlap_identity (hn : 2 ≤ Fintype.card P) (x y : P) (hxy : x ≠ y) :
+    L Z hZ x y hxy = Grid.kappa - M Z hZ + S Z hZ x + S Z hZ y := by
+  have harch : (∑ v : InfinitePlace K, Arithmetic.archWeight v *
+      (∫ ω, (arch Z v ω).M ∂Grid.fullLaw)) +
+      (∑ v : InfinitePlace K, Arithmetic.archWeight v *
+      (∫ ω, (arch Z v ω).overlapLength x y hxy ∂Grid.fullLaw)) -
+      (∑ v : InfinitePlace K, Arithmetic.archWeight v *
+      (∫ ω, (arch Z v ω).potential x ∂Grid.fullLaw)) -
+      (∑ v : InfinitePlace K, Arithmetic.archWeight v *
+      (∫ ω, (arch Z v ω).potential y ∂Grid.fullLaw)) =
+      (∑ v : InfinitePlace K, Arithmetic.archWeight v * (-Real.log (v (Z x-Z y)))) + Grid.kappa := by
+    rw [← Finset.sum_add_distrib, ← Finset.sum_sub_distrib, ← Finset.sum_sub_distrib]
+    have he : ∀ v : InfinitePlace K,
+        Arithmetic.archWeight v * (∫ ω, (arch Z v ω).M ∂Grid.fullLaw) +
+        Arithmetic.archWeight v * (∫ ω, (arch Z v ω).overlapLength x y hxy ∂Grid.fullLaw) -
+        Arithmetic.archWeight v * (∫ ω, (arch Z v ω).potential x ∂Grid.fullLaw) -
+        Arithmetic.archWeight v * (∫ ω, (arch Z v ω).potential y ∂Grid.fullLaw) =
+        Arithmetic.archWeight v * (-Real.log (v (Z x-Z y))) + Arithmetic.archWeight v * Grid.kappa := by
+      intro v
+      have hh := Depth.integral_depth_identity (arch Z v) (arch_measurable Z hZ v)
+        (arch_integrable Z hZ v) hn x y hxy
+      rw [arch_mean Z hZ v x y hxy] at hh
+      linear_combination -Arithmetic.archWeight v * hh
+    simp_rw [he]
+    rw [Finset.sum_add_distrib, ← Finset.sum_mul, Arithmetic.sum_archWeight,one_mul]
+  have hfin : (∑ v ∈ badPlaces Z hZ, (finite Z hZ v).M) +
+      (∑ v ∈ badPlaces Z hZ, (finite Z hZ v).overlapLength x y hxy) -
+      (∑ v ∈ badPlaces Z hZ, (finite Z hZ v).potential x) -
+      (∑ v ∈ badPlaces Z hZ, (finite Z hZ v).potential y) =
+      ∑ᶠ v : FinitePlace K, -Real.log (v (Z x-Z y)) := by
+    rw [← finite_value_sum Z hZ x y hxy, ← Finset.sum_add_distrib,
+      ← Finset.sum_sub_distrib, ← Finset.sum_sub_distrib]
+    apply Finset.sum_congr rfl
+    intro v _
+    exact ((finite Z hZ v).depth_identity hn x y hxy).symm
+  have hprod := Arithmetic.normalized_log_product_formula (Z x-Z y) (sub_ne_zero.mpr (hZ.ne hxy))
+  dsimp only [L,M,S]
+  linear_combination harch + (Module.finrank ℚ K : ℝ)⁻¹ * hfin + hprod
+
+lemma S_nonneg (x : P) : 0 ≤ S Z hZ x := by
+  apply add_nonneg
+  · exact Finset.sum_nonneg (fun v _ => mul_nonneg (Arithmetic.archWeight_nonneg v)
+      (integral_nonneg (fun ω => (arch Z v ω).potential_nonneg x)))
+  · exact mul_nonneg (by positivity) (Finset.sum_nonneg (fun v _ => (finite Z hZ v).potential_nonneg x))
+
+lemma L_nonneg (x y : P) (hxy : x ≠ y) : 0 ≤ L Z hZ x y hxy := by
+  apply add_nonneg
+  · exact Finset.sum_nonneg (fun v _ => mul_nonneg (Arithmetic.archWeight_nonneg v)
+      (integral_nonneg (fun ω => (arch Z v ω).overlapLength_nonneg x y hxy)))
+  · exact mul_nonneg (by positivity) (Finset.sum_nonneg (fun v _ => (finite Z hZ v).overlapLength_nonneg x y hxy))
+
+end WeakPinned.Global
+
+namespace WeakPinned.Hierarchy.Depth
+open MeasureTheory
+variable {P : Type u3} [Fintype P] [DecidableEq P]
+variable (d : Depth P)
+
+def count (r : ℝ) (C : Finset P) : ℝ := (d.transformed r).count C
+
+def mass (C : Finset P) : ℝ := if 2 ≤ C.card then ∫ r, d.count r C else 0
+
+lemma sum_map_count (m : Multiset (Finset P)) (f : Finset P → ℝ) :
+    (m.map f).sum = ∑ C, (m.count C : ℝ) * f C := by
+  induction m using Multiset.induction_on with
+  | empty => simp
+  | @cons a m ih =>
+      simp only [Multiset.map_cons, Multiset.sum_cons, ih, Multiset.count_cons]
+      simp_rw [Nat.cast_add, add_mul]
+      rw [Finset.sum_add_distrib]
+      simp [add_comm]
+
+lemma sumCells_count (f : Finset P → ℝ) (r : ℝ) :
+    d.sumCells f r = ∑ C, d.count r C * f C := sum_map_count _ _
+
+lemma count_nonneg (r : ℝ) (C : Finset P) : 0 ≤ d.count r C := Nat.cast_nonneg _
+
+lemma count_le_overlap (r : ℝ) (C : Finset P) (x y : P) (hx : x ∈ C) (hy : y ∈ C) :
+    d.count r C ≤ d.pairOverlap r x y := by
+  rw [pairOverlap, d.sumCells_count]
+  calc
+    d.count r C = d.count r C * (if x ∈ C ∧ y ∈ C then (1 : ℝ) else 0) := by simp [hx,hy]
+    _ ≤ _ := Finset.single_le_sum (f := fun D => d.count r D * (if x ∈ D ∧ y ∈ D then (1 : ℝ) else 0)) (fun D _ => mul_nonneg (d.count_nonneg r D) (by positivity)) (Finset.mem_univ C)
+
+lemma measurable_count (C : Finset P) : Measurable (fun r => d.count r C) := by
+  let F : Finset (Finset P) × Finset (Finset P) → ℝ := fun st =>
+    (((st.1.filter (fun D => ¬Fintype.card P < 2*D.card)).val +
+       st.2.val.map (fun D => Dᶜ)).count C : ℝ)
+  have hh : Measurable (fun r => F (d.cells r, d.giants r)) :=
+    (measurable_of_finite F).comp (d.measurable_cells.prodMk d.measurable_giants)
+  change Measurable (fun r => F (d.cells r, d.giants r))
+  exact hh
+
+lemma integrable_count (hn : 2 ≤ Fintype.card P) (C : Finset P) (hC : 2 ≤ C.card) :
+    Integrable (fun r => d.count r C) := by
+  obtain ⟨x,hx,y,hy,hxy⟩ := Finset.one_lt_card.mp hC
+  apply (d.integrable_pairOverlap hn x y hxy).mono' (d.measurable_count C).aestronglyMeasurable
+  filter_upwards with r
+  simpa only [Real.norm_eq_abs, abs_of_nonneg (d.count_nonneg r C)] using d.count_le_overlap r C x y hx hy
+
+lemma mass_nonneg (C : Finset P) : 0 ≤ d.mass C := by
+  unfold mass
+  split_ifs
+  · exact integral_nonneg (fun r => d.count_nonneg r C)
+  · rfl
+
+lemma mass_singleton (C : Finset P) (hC : C.card ≤ 1) : d.mass C = 0 := by
+  simp [mass, show ¬2 ≤ C.card by omega]
+
+lemma mass_le_overlap (hn : 2 ≤ Fintype.card P) (C : Finset P) (x y : P)
+    (hxy : x ≠ y) (hx : x ∈ C) (hy : y ∈ C) :
+    d.mass C ≤ d.overlapLength x y hxy := by
+  have hC : 2 ≤ C.card := Finset.one_lt_card.mpr ⟨x,hx,y,hy,hxy⟩
+  rw [mass, ite_eq_left hC, overlapLength]
+  exact integral_mono (d.integrable_count hn C hC) (d.integrable_pairOverlap hn x y hxy)
+    (fun r => d.count_le_overlap r C x y hx hy)
+
+lemma integral_sumCells (hn : 2 ≤ Fintype.card P) (f : Finset P → ℝ)
+    (hf : ∀ C, C.card ≤ 1 → f C = 0) :
+    (∫ r, d.sumCells f r) = ∑ C, d.mass C * f C := by
+  have hi (C : Finset P) : Integrable (fun r => d.count r C * f C) := by
+    by_cases hC : 2 ≤ C.card
+    · exact (d.integrable_count hn C hC).mul_const _
+    · simp [hf C (by omega)]
+  simp_rw [d.sumCells_count]
+  rw [integral_finsetSum _ (fun C _ => hi C)]
+  apply Finset.sum_congr rfl
+  intro C _
+  by_cases hC : 2 ≤ C.card
+  · simp [mass,hC,integral_mul_const]
+  · simp [mass,hC,hf C (by omega)]
+
+lemma mass_overlap (hn : 2 ≤ Fintype.card P) (x y : P) (hxy : x ≠ y) :
+    (∑ C, d.mass C * (if x ∈ C ∧ y ∈ C then (1 : ℝ) else 0)) =
+      d.overlapLength x y hxy := by
+  symm
+  apply d.integral_sumCells hn
+  intro C hC
+  have hh : ¬(x ∈ C ∧ y ∈ C) := by
+    rintro ⟨hx,hy⟩
+    have := Finset.one_lt_card.mpr ⟨x,hx,y,hy,hxy⟩
+    omega
+  simp [hh]
+
+section Param
+variable {Ω : Type u4} [MeasurableSpace Ω] (D : Ω → Depth P)
+    (hD : ∀ x y, Measurable (fun ω => (D ω).value x y))
+include hD
+lemma measurable_count_param (C : Finset P) :
+    Measurable (fun p : Ω × ℝ => (D p.1).count p.2 C) := by
+  let F : Finset (Finset P) × Finset (Finset P) → ℝ := fun st =>
+    (((st.1.filter (fun E => ¬Fintype.card P < 2*E.card)).val +
+       st.2.val.map (fun E => Eᶜ)).count C : ℝ)
+  have hm : Measurable (fun p : Ω×ℝ => ((D p.1).cells p.2, (D p.1).giants p.2)) :=
+    (measurable_cells_param (P:=P) (Ω:=Ω×ℝ) (fun p : Ω×ℝ => D p.1)
+      (fun x y => (hD x y).comp measurable_fst) Prod.snd measurable_snd).prodMk
+    (measurable_giants_param (P:=P) (Ω:=Ω×ℝ) (fun p : Ω×ℝ => D p.1)
+      (fun x y => (hD x y).comp measurable_fst) Prod.snd measurable_snd)
+  have hh : Measurable (fun p : Ω×ℝ => F ((D p.1).cells p.2, (D p.1).giants p.2)) :=
+    (measurable_of_finite F).comp hm
+  change Measurable (fun p : Ω×ℝ => F ((D p.1).cells p.2, (D p.1).giants p.2))
+  exact hh
+
+lemma measurable_mass_param (C : Finset P) : Measurable (fun ω => (D ω).mass C) := by
+  unfold mass
+  split_ifs with hC
+  · exact (measurable_count_param D hD C).stronglyMeasurable.integral_prod_right.measurable
+  · exact measurable_const
+
+lemma integrable_mass_param {μ : Measure Ω}
+    (hi : ∀ x y, Integrable (fun ω => (D ω).value x y) μ)
+    (hn : 2 ≤ Fintype.card P) (C : Finset P) :
+    Integrable (fun ω => (D ω).mass C) μ := by
+  by_cases hC : 2 ≤ C.card
+  · obtain ⟨x,hx,y,hy,hxy⟩ := Finset.one_lt_card.mp hC
+    apply (integrable_overlapLength_param D hD hi hn x y hxy).mono'
+      (measurable_mass_param D hD C).aestronglyMeasurable
+    filter_upwards with ω
+    simpa only [Real.norm_eq_abs, abs_of_nonneg ((D ω).mass_nonneg C)] using
+      (D ω).mass_le_overlap hn C x y hxy hx hy
+  · simp only [mass, hC, reduceIte]
+    exact integrable_zero _ _ _
+
+lemma averaged_mass_overlap {μ : Measure Ω}
+    (hi : ∀ x y, Integrable (fun ω => (D ω).value x y) μ)
+    (hn : 2 ≤ Fintype.card P) (x y : P) (hxy : x ≠ y) :
+    (∑ C, (∫ ω, (D ω).mass C ∂μ) * (if x ∈ C ∧ y ∈ C then (1 : ℝ) else 0)) =
+      ∫ ω, (D ω).overlapLength x y hxy ∂μ := by
+  simp_rw [← integral_mul_const]
+  rw [← integral_finsetSum _ (fun C _ => (integrable_mass_param D hD hi hn C).mul_const _)]
+  exact integral_congr_ae (Filter.Eventually.of_forall (fun ω => (D ω).mass_overlap hn x y hxy))
+end Param
+end WeakPinned.Hierarchy.Depth
+
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PinnedDistances/ArithmeticSequence.lean b/lean/OAI/Geometry/PinnedDistances/ArithmeticSequence.lean
new file mode 100644
index 0000000000000000000000000000000000000000..8a5036e662b7ce3f6b920d68a08db3769e6e4b2e
--- /dev/null
+++ b/lean/OAI/Geometry/PinnedDistances/ArithmeticSequence.lean
@@ -0,0 +1,305 @@
+import OAI.Geometry.PinnedDistances.ArithmeticContradiction
+
+namespace OAI
+
+universe u1 u2 u3 u4 u5 u6 u7 u8 u9 u10 u11
+
+section
+noncomputable section
+namespace WeakPinned
+open Filter
+open scoped Topology BigOperators
+
+/-- Diagonalized extraction of genuine configurations. The errors tend to zero
+and vertex numbers tend to infinity. -/
+theorem graph_sequence_of_failure (hfail : ¬ MainStatement) :
+    ∃ c s γ : ℝ, 0<c ∧ c<γ ∧ γ<s ∧ s<1 ∧ 0<Theta s ∧
+      ∃ (P : ℕ → Finset Plane) (C : ∀ j, Finset (P j)),
+        (∀ j, j<(C j).card) ∧
+        ∀ j, PreparedGraph (C j) ((planarFibers (P j) s).Active (cutoff s (P j).card) (C j))
+          (coreGraph (P j) s (C j)) (Theta s) γ c (Theta s/(j+2)) := by
+  classical
+  obtain ⟨c,s,γ,hc,hc3,hcs,hs,hcγ,hγs,hct,hθ⟩ := extremal_parameters hfail
+  have hcrit := critical_le_one hfail
+  have hsc : (1-c)*criticalExponent<s := (le_max_right _ _).trans_lt hcs
+  have hc1 : c<1 := by linarith
+  have hγ0 : 0<γ := hc.trans hcγ
+  have hex (j : ℕ) := arbitrarily_large_graph hfail hγ0 hγs hc hc1 hsc hct
+    (Theta s/(j+2)) (div_pos hθ (by positivity)) j
+  choose P C hC hG using hex
+  exact ⟨c,s,γ,hc,hcγ,hγs,hs.trans_le hcrit,hθ,P,C,hC,hG⟩
+
+lemma graph_card_tendsto {P : ℕ → Finset Plane} {C : ∀ j, Finset (P j)}
+    (hC : ∀ j, j<(C j).card) : Tendsto (fun j => (C j).card) atTop atTop :=
+  tendsto_atTop_mono (fun j => (hC j).le) tendsto_id
+
+lemma graph_error_tendsto (θ : ℝ) : Tendsto (fun j : ℕ => θ/((j:ℝ)+2)) atTop (𝓝 0) := by
+  have ht : Tendsto (fun j : ℕ => (j:ℝ)+2) atTop atTop := tendsto_atTop_add_const_right _ _ tendsto_natCast_atTop_atTop
+  exact tendsto_const_nhds.div_atTop ht
+
+end WeakPinned
+
+/- Algebraic representation of the finite rational polynomial pattern of a
+   real configuration in a number field. -/
+
+noncomputable section
+
+namespace WeakPinned.Transfer
+open MvPolynomial
+
+/-- A system of rational polynomial equations with a complex solution has an
+    algebraic solution. No finite set assumption on the equations is needed. -/
+theorem equations {ι : Type u1} [Finite ι]
+    (S : Set (MvPolynomial ι ℚ)) (x : ι → ℂ)
+    (hx : ∀ p ∈ S, aeval x p = 0) :
+    ∃ y : ι → AlgebraicClosure ℚ, ∀ p ∈ S, aeval y p = 0 := by
+  let I : Ideal (MvPolynomial ι ℚ) := vanishingIdeal ℚ {x}
+  have hI : I ≠ ⊤ := by
+    intro h
+    have h1 : (1 : MvPolynomial ι ℚ) ∈ I := by rw [h]; trivial
+    have hz := (mem_vanishingIdeal_singleton_iff x 1).mp h1
+    simp at hz
+  obtain ⟨M, hM, hIM⟩ := Ideal.exists_le_maximal I hI
+  obtain ⟨y, hy⟩ := eq_vanishingIdeal_singleton_of_isMaximal (AlgebraicClosure ℚ) hM
+  refine ⟨y, fun p hp => ?_⟩
+  apply (mem_vanishingIdeal_singleton_iff y p).mp
+  rw [← hy]
+  exact hIM ((mem_vanishingIdeal_singleton_iff x p).mpr (hx p hp))
+
+/-- Simultaneous equations and finitely many inequations transfer to algebraic
+    coordinates. The inverse-variable equation enforces every inequation. -/
+theorem equations_and_inequations {ι : Type u2} [Finite ι]
+    (S : Set (MvPolynomial ι ℚ)) (T : Finset (MvPolynomial ι ℚ)) (x : ι → ℂ)
+    (hx : ∀ p ∈ S, aeval x p = 0) (hT : ∀ p ∈ T, aeval x p ≠ 0) :
+    ∃ y : ι → AlgebraicClosure ℚ,
+      (∀ p ∈ S, aeval y p = 0) ∧ (∀ p ∈ T, aeval y p ≠ 0) := by
+  let q : MvPolynomial ι ℚ := ∏ p ∈ T, p
+  let r : MvPolynomial (Option ι) ℚ := X none * rename some q - 1
+  let S' : Set (MvPolynomial (Option ι) ℚ) := (rename some) '' S ∪ {r}
+  let x' : Option ι → ℂ := fun j => j.elim (aeval x q)⁻¹ x
+  have hq : aeval x q ≠ 0 := by
+    simpa [q] using Finset.prod_ne_zero_iff.mpr hT
+  have hl (p : MvPolynomial ι ℚ) : aeval x' (rename some p) = aeval x p := by
+    simp [aeval_rename, Function.comp_def, x']
+  have hx' : ∀ p ∈ S', aeval x' p = 0 := by
+    intro p hp
+    rcases hp with ⟨p, hp, rfl⟩ | rfl
+    · rw [hl]
+      exact hx p hp
+    · simp [r, hl, x', hq]
+  obtain ⟨y', hy'⟩ := equations S' x' hx'
+  let y : ι → AlgebraicClosure ℚ := fun i => y' (some i)
+  have hl' (p : MvPolynomial ι ℚ) : aeval y' (rename some p) = aeval y p := by
+    simp [aeval_rename, Function.comp_def, y]
+  refine ⟨y, ?_, ?_⟩
+  · intro p hp
+    rw [← hl']
+    exact hy' _ (Or.inl ⟨p, hp, rfl⟩)
+  · have hr := hy' r (Or.inr rfl)
+    have hq' : aeval y q ≠ 0 := by
+      intro hz
+      simp [r, hl', hz] at hr
+    have hprod : (∏ p ∈ T, aeval y p) ≠ 0 := by simpa [q] using hq'
+    exact Finset.prod_ne_zero_iff.mp hprod
+
+/-- The algebraic realization lies in a single number field generated by its
+    finitely many coordinates. The field is not fixed in advance. -/
+theorem numberField_pattern {ι : Type u3} [Finite ι]
+    (S : Set (MvPolynomial ι ℚ)) (T : Finset (MvPolynomial ι ℚ)) (x : ι → ℂ)
+    (hx : ∀ p ∈ S, aeval x p = 0) (hT : ∀ p ∈ T, aeval x p ≠ 0) :
+    ∃ K : IntermediateField ℚ (AlgebraicClosure ℚ), NumberField K ∧
+      ∃ y : ι → K, (∀ p ∈ S, aeval y p = 0) ∧ (∀ p ∈ T, aeval y p ≠ 0) := by
+  obtain ⟨y, hy, hTy⟩ := equations_and_inequations S T x hx hT
+  let K := IntermediateField.adjoin ℚ (Set.range y)
+  have : Finite (Set.range y) := (Set.finite_range y).to_subtype
+  have : Algebra.IsAlgebraic ℚ (AlgebraicClosure ℚ) := AlgebraicClosure.isAlgebraic ℚ
+  have : Algebra.IsIntegral ℚ (AlgebraicClosure ℚ) := Algebra.IsAlgebraic.isIntegral
+  have hfd : FiniteDimensional ℚ K :=
+    IntermediateField.finiteDimensional_adjoin (fun z _ => Algebra.IsIntegral.isIntegral z)
+  have hnf : NumberField K := { to_charZero := inferInstance, to_finiteDimensional := hfd }
+  let z : ι → K := fun i => ⟨y i, IntermediateField.subset_adjoin ℚ (Set.range y) ⟨i, rfl⟩⟩
+  have he (p : MvPolynomial ι ℚ) : K.val (aeval z p) = aeval y p :=
+    comp_aeval_apply z K.val p
+  refine ⟨K, hnf, z, ?_, ?_⟩
+  · intro p hp
+    apply K.val.injective
+    change K.val (aeval z p) = K.val 0
+    rw [he, map_zero, hy p hp]
+  · intro p hp hz
+    apply hTy p hp
+    rw [← he, hz, map_zero]
+
+/-- Preservation of an arbitrary finite zero-pattern, simultaneously, in one
+    number field. In particular this preserves all selected equalities exactly. -/
+theorem numberField_zero_pattern {ι : Type u4} [Finite ι]
+    (T : Finset (MvPolynomial ι ℚ)) (x : ι → ℂ) :
+    ∃ K : IntermediateField ℚ (AlgebraicClosure ℚ), NumberField K ∧
+      ∃ y : ι → K, ∀ p ∈ T, aeval y p = 0 ↔ aeval x p = 0 := by
+  classical
+  let S : Set (MvPolynomial ι ℚ) := {p | p ∈ T ∧ aeval x p = 0}
+  let U := T.filter (fun p => aeval x p ≠ 0)
+  obtain ⟨K, hK, y, hy, hUy⟩ := numberField_pattern S U x
+    (fun _ hp => hp.2) (fun p hp => (Finset.mem_filter.mp hp).2)
+  refine ⟨K, hK, y, fun p hp => ⟨?_, ?_⟩⟩
+  · intro h
+    by_contra h'
+    exact hUy p (Finset.mem_filter.mpr ⟨hp, h'⟩) h
+  · intro h
+    exact hy p ⟨hp, h⟩
+
+abbrev Plane := EuclideanSpace ℝ (Fin 2)
+
+/-- Complexified variables; the extra variable forces a square root of -1
+    into the field, so every archimedean place is complex. -/
+abbrev Variables (ι : Type u5) := Option (Bool × ι)
+
+def secondCoord (b : Bool) (z : Plane) : ℂ :=
+  if b then ⟨z 0, -z 1⟩ else ⟨z 0, z 1⟩
+
+def complexAssignment {ι : Type u6} (x : ι → Plane) : Variables ι → ℂ :=
+  fun j => j.elim Complex.I (fun bi => secondCoord bi.1 (x bi.2))
+
+def distancePoly {ι : Type u7} (a b : ι) : MvPolynomial (Variables ι) ℚ :=
+  (X (some (false, a)) - X (some (false, b))) *
+  (X (some (true, a)) - X (some (true, b)))
+
+lemma eval_distancePoly {ι : Type u8} (x : ι → Plane) (a b : ι) :
+    aeval (complexAssignment x) (distancePoly a b) = ((dist (x a) (x b)) ^ 2 : ℝ) := by
+  simp only [distancePoly, map_mul, map_sub, aeval_X, complexAssignment,
+    Option.elim_some, secondCoord, Bool.false_eq_true, reduceIte]
+  rw [EuclideanSpace.dist_sq_eq]
+  simp only [Fin.sum_univ_two, Real.dist_eq, sq_abs]
+  apply Complex.ext <;> simp [-Complex.ofReal_pow] <;> ring
+
+/-- Every finite real planar configuration has a faithful number-field model
+    for BOTH distance factors. The real configuration is not restricted or
+    replaced: the exact equality pattern of ALL its distances is retained.
+    The factorization at every place does not require conjugacy in the new field. -/
+theorem planar_factor_model {ι : Type u9} [Finite ι]
+    (x : ι → Plane) (hx : Function.Injective x) :
+    ∃ K : IntermediateField ℚ (AlgebraicClosure ℚ), NumberField K ∧
+      ∃ (ii : K) (Z₁ Z₂ : ι → K), ii ^ 2 = -1 ∧
+        Function.Injective Z₁ ∧ Function.Injective Z₂ ∧
+        ∀ a b c d, (Z₁ a - Z₁ b) * (Z₂ a - Z₂ b) =
+          (Z₁ c - Z₁ d) * (Z₂ c - Z₂ d) ↔ dist (x a) (x b) = dist (x c) (x d) := by
+  classical
+  let := Fintype.ofFinite ι
+  let D := @distancePoly ι
+  let root : MvPolynomial (Variables ι) ℚ := X none ^ 2 + 1
+  let pairs := Finset.univ.image (fun t : ι × ι => D t.1 t.2)
+  let quadruples := Finset.univ.image (fun t : (ι × ι) × (ι × ι) =>
+    D t.1.1 t.1.2 - D t.2.1 t.2.2)
+  let T := pairs ∪ quadruples ∪ {root}
+  obtain ⟨K, hK, y, hy⟩ := numberField_zero_pattern T (complexAssignment x)
+  let Z₁ : ι → K := fun a => y (some (false, a))
+  let Z₂ : ι → K := fun a => y (some (true, a))
+  have he (a b : ι) : aeval y (D a b) = (Z₁ a - Z₁ b) * (Z₂ a - Z₂ b) := by
+    simp [D, distancePoly, Z₁, Z₂]
+  have hp (a b : ι) : D a b ∈ T := by
+    apply Finset.mem_union.mpr
+    left
+    apply Finset.mem_union.mpr
+    left
+    exact Finset.mem_image.mpr ⟨(a,b), Finset.mem_univ _, rfl⟩
+  have h0 (a b : ι) : (Z₁ a - Z₁ b) * (Z₂ a - Z₂ b) = 0 ↔ a = b := by
+    rw [← he, hy _ (hp a b), eval_distancePoly]
+    simp [hx.eq_iff]
+  have hdist (a b c d : ι) : D a b - D c d ∈ T := by
+    apply Finset.mem_union.mpr
+    left
+    apply Finset.mem_union.mpr
+    right
+    exact Finset.mem_image.mpr ⟨((a,b),(c,d)), Finset.mem_univ _, rfl⟩
+  have hroot : y none ^ 2 + 1 = 0 := by
+    have hm : root ∈ T := Finset.mem_union.mpr (Or.inr (Finset.mem_singleton_self _))
+    have hc : aeval (complexAssignment x) root = 0 := by
+      simp [root, complexAssignment, Complex.I_sq]
+    simpa [root] using (hy root hm).mpr hc
+  refine ⟨K, hK, y none, Z₁, Z₂, eq_neg_of_add_eq_zero_left hroot, ?_, ?_, ?_⟩
+  · intro a b hab
+    apply (h0 a b).mp
+    rw [hab, sub_self, zero_mul]
+  · intro a b hab
+    apply (h0 a b).mp
+    rw [hab, sub_self, mul_zero]
+  · intro a b c d
+    rw [← he, ← he]
+    have h := hy _ (hdist a b c d)
+    rw [map_sub, sub_eq_zero, map_sub, sub_eq_zero, eval_distancePoly,
+      eval_distancePoly, Complex.ofReal_inj] at h
+    exact h.trans (sq_eq_sq₀ dist_nonneg dist_nonneg)
+
+end WeakPinned.Transfer
+
+namespace WeakPinned
+open scoped BigOperators
+
+/-- The exact arithmetic model for each extracted real graph. No point is
+assumed algebraic; the original graph's complete distance-equality pattern
+is preserved by the proved finite specialization theorem. -/
+theorem core_arithmetic_model (P : Finset Plane) (s : ℝ) (C : Finset P)
+    (hne : ∀ e, ((coreGraph P s C).targets e).Nonempty) :
+    ∃ K : IntermediateField ℚ (AlgebraicClosure ℚ), NumberField K ∧
+      ∃ (ii : K) (Z₁ Z₂ : C → K) (t : (planarFibers P s).Active (cutoff s P.card) C → K),
+        ii^2 = -1 ∧ Function.Injective Z₁ ∧ Function.Injective Z₂ ∧
+        (∀ e, t e ≠ 0) ∧
+        (∀ e y, y ∈ (coreGraph P s C).targets e →
+          (Z₁ ((coreGraph P s C).source e)-Z₁ y)*(Z₂ ((coreGraph P s C).source e)-Z₂ y)=t e) ∧
+        (∀ a b c d : C, (Z₁ a-Z₁ b)*(Z₂ a-Z₂ b)=(Z₁ c-Z₁ d)*(Z₂ c-Z₂ d) ↔
+          dist a.val.val b.val.val=dist c.val.val d.val.val) := by
+  classical
+  let x : C → Plane := fun a => a.val.val
+  have hx : Function.Injective x := Subtype.val_injective.comp Subtype.val_injective
+  obtain ⟨K,hK,ii,Z₁,Z₂,hii,h₁,h₂,hpat⟩ := Transfer.planar_factor_model x hx
+  choose y hy using hne
+  let G := coreGraph P s C
+  let t := fun e => (Z₁ (G.source e)-Z₁ (y e))*(Z₂ (G.source e)-Z₂ (y e))
+  refine ⟨K,hK,ii,Z₁,Z₂,t,hii,h₁,h₂,?_,?_,hpat⟩
+  · intro e
+    have hxy : G.source e ≠ y e := by intro he; exact G.no_loop e (he ▸ hy e)
+    exact mul_ne_zero (sub_ne_zero.mpr (h₁.ne hxy)) (sub_ne_zero.mpr (h₂.ne hxy))
+  · intro e z hz
+    apply (hpat _ _ _ _).mpr
+    have hz' := (Finset.mem_filter.mp hz).2
+    have hy' := (Finset.mem_filter.mp (hy e)).2
+    have hzdist := (planar_target_iff P s e.val z.val).mp hz'
+    have hydist := (planar_target_iff P s e.val (y e).val).mp hy'
+    change dist e.val.val.1 z.val.val=dist e.val.val.1 (y e).val.val
+    rw [dist_comm,dist_comm e.val.val.1,hzdist,hydist]
+
+lemma PreparedGraph.targets_nonempty {V : Type u10} {I : Type u11} [Fintype V] [Fintype I] [DecidableEq V]
+    {G : Core.Fibers V I} {θ γ c η : ℝ} (h : PreparedGraph V I G θ γ c η) :
+    ∀ e, (G.targets e).Nonempty := by
+  intro e
+  have hn : (0:ℝ)<Fintype.card V := by exact_mod_cast (show 0<Fintype.card V by have := h.card_ge; omega)
+  have hh : (0:ℝ)<(G.targets e).card := (Real.rpow_pos_of_pos hn γ).trans (h.fiber e)
+  exact Finset.card_pos.mp (by exact_mod_cast hh)
+
+/-- Specialization is available independently for EVERY graph in the genuine
+counterexample sequence, over number fields of unrestricted degree. -/
+theorem sequence_arithmetic_model (hfail : ¬ MainStatement) :
+    ∃ c s γ : ℝ, 0<c ∧ c<γ ∧ γ<s ∧ s<1 ∧ 0<Theta s ∧
+      ∃ (P : ℕ → Finset Plane) (C : ∀ j, Finset (P j)),
+        (∀ j, j<(C j).card) ∧
+        (∀ j, PreparedGraph (C j) ((planarFibers (P j) s).Active (cutoff s (P j).card) (C j))
+          (coreGraph (P j) s (C j)) (Theta s) γ c (Theta s/(j+2))) ∧
+        ∀ j, ∃ K : IntermediateField ℚ (AlgebraicClosure ℚ), NumberField K ∧
+          ∃ (ii : K) (Z₁ Z₂ : C j → K) (t : (planarFibers (P j) s).Active (cutoff s (P j).card) (C j) → K),
+            ii^2 = -1 ∧ Function.Injective Z₁ ∧ Function.Injective Z₂ ∧
+            (∀ e, t e ≠ 0) ∧
+            (∀ e y, y ∈ (coreGraph (P j) s (C j)).targets e →
+              (Z₁ ((coreGraph (P j) s (C j)).source e)-Z₁ y)*(Z₂ ((coreGraph (P j) s (C j)).source e)-Z₂ y)=t e) := by
+  obtain ⟨c,s,γ,hc,hcγ,hγs,hs,hθ,P,C,hC,hG⟩ := graph_sequence_of_failure hfail
+  refine ⟨c,s,γ,hc,hcγ,hγs,hs,hθ,P,C,hC,hG,?_⟩
+  intro j
+  obtain ⟨K,hK,ii,Z₁,Z₂,t,hii,h₁,h₂,ht,hfiber,_⟩ :=
+    core_arithmetic_model (P j) s (C j) (hG j).targets_nonempty
+  exact ⟨K,hK,ii,Z₁,Z₂,t,hii,h₁,h₂,ht,hfiber⟩
+
+end WeakPinned
+
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PinnedDistances/ArithmeticVariance.lean b/lean/OAI/Geometry/PinnedDistances/ArithmeticVariance.lean
new file mode 100644
index 0000000000000000000000000000000000000000..3a2cdb98b0da5ed5d0fc47d7ebc843ee93148f8e
--- /dev/null
+++ b/lean/OAI/Geometry/PinnedDistances/ArithmeticVariance.lean
@@ -0,0 +1,201 @@
+import OAI.Geometry.PinnedDistances.Variance
+
+namespace OAI
+
+universe u1 u2 u3 u4 u5 u6 u7
+
+section
+noncomputable section
+open NumberField
+open scoped BigOperators
+noncomputable section
+open MeasureTheory
+open scoped Topology
+noncomputable section
+open MeasureTheory
+open scoped Topology ENNReal
+noncomputable section
+open MeasureTheory NumberField
+open scoped BigOperators
+noncomputable section
+open scoped BigOperators
+open MeasureTheory
+
+namespace WeakPinned.Global
+open Hierarchy
+variable {P : Type u1} [Fintype P] [DecidableEq P]
+variable {K : Type u2} [Field K] [NumberField K]
+variable (Z : P → K) (hZ : Function.Injective Z)
+
+def cellMass (C : Finset P) : ℝ :=
+  (∑ v : InfinitePlace K, Arithmetic.archWeight v * ∫ ω, (arch Z v ω).mass C ∂Grid.fullLaw) +
+    (Module.finrank ℚ K : ℝ)⁻¹ * ∑ v ∈ badPlaces Z hZ, (finite Z hZ v).mass C
+
+lemma cellMass_nonneg (C : Finset P) : 0 ≤ cellMass Z hZ C := by
+  exact add_nonneg
+    (Finset.sum_nonneg (fun v _ => mul_nonneg (Arithmetic.archWeight_nonneg v)
+      (integral_nonneg (fun ω => (arch Z v ω).mass_nonneg C))))
+    (mul_nonneg (by positivity) (Finset.sum_nonneg (fun v _ => (finite Z hZ v).mass_nonneg C)))
+
+lemma cellMass_singleton (C : Finset P) (hC : C.card ≤ 1) : cellMass Z hZ C = 0 := by
+  simp only [cellMass, Depth.mass_singleton _ C hC, integral_zero, mul_zero, Finset.sum_const_zero, add_zero]
+
+lemma cellMass_overlap (hn : 2 ≤ Fintype.card P) (x y : P) (hxy : x ≠ y) :
+    (∑ C, cellMass Z hZ C * (if x ∈ C ∧ y ∈ C then (1 : ℝ) else 0)) = L Z hZ x y hxy := by
+  simp only [cellMass, add_mul, Finset.sum_add_distrib, Finset.sum_mul,
+    mul_assoc, L]
+  rw [← Finset.mul_sum]
+  congr 1
+  · rw [Finset.sum_comm]
+    apply Finset.sum_congr rfl
+    intro v _
+    rw [← Finset.mul_sum, Depth.averaged_mass_overlap (arch Z v) (arch_measurable Z hZ v)
+      (arch_integrable Z hZ v) hn x y hxy]
+  · congr 1
+    rw [Finset.sum_comm]
+    apply Finset.sum_congr rfl
+    intro v _
+    exact (finite Z hZ v).mass_overlap hn x y hxy
+
+end WeakPinned.Global
+
+namespace WeakPinned.Variance
+variable {P : Type u3} [Fintype P] [DecidableEq P]
+
+/-- Finite linear model of the actual occurrence integral on expressions
+    vanishing on singleton and empty cells. Multiplicities are in m. -/
+def weightedIntegral (m : Finset P → ℝ) (f : Finset P → ℝ) : ℝ := ∑ C, m C * f C
+
+def weightedOverlap (m : Finset P → ℝ) (x y : P) : ℝ :=
+  weightedIntegral m (fun C => if x ∈ C ∧ y ∈ C then 1 else 0)
+
+lemma weighted_cancel (m : Finset P → ℝ) (b : ℝ) (S : P → ℝ)
+    (hL : ∀ x y, x ≠ y → weightedOverlap m x y = b+S x+S y)
+    (B : Finset P) (hB : 2 ≤ B.card) : weightedIntegral m (signedCell B) = 0 := by
+  have he : weightedIntegral m (signedCell B) =
+      ∑ x, ∑ y, signedPair B x y * weightedOverlap m x y := by
+    simp only [weightedIntegral, signedCell_eq]
+    have hc (C : Finset P) : (∑ x ∈ C, ∑ y ∈ C, signedPair B x y) =
+        ∑ x, ∑ y, signedPair B x y * (if x ∈ C ∧ y ∈ C then (1:ℝ) else 0) := by
+      simp only [mul_ite, mul_one, mul_zero, ite_and]
+      simp only [Finset.sum_ite_irrel, Finset.sum_const_zero]
+      simp
+    simp_rw [hc, Finset.mul_sum]
+    rw [Finset.sum_comm]
+    apply Finset.sum_congr rfl
+    intro x _
+    rw [Finset.sum_comm]
+    apply Finset.sum_congr rfl
+    intro y _
+    simp only [weightedOverlap, weightedIntegral, Finset.mul_sum]
+    apply Finset.sum_congr rfl
+    intro C _
+    ring
+  rw [he]
+  exact variance_cancel B hB _ b S hL
+
+lemma weighted_variance {I : Type u4} [Fintype I]
+    (m : Finset P → ℝ) (hm : ∀ C, 0 ≤ m C) (b : ℝ) (S : P → ℝ)
+    (hL : ∀ x y, x ≠ y → weightedOverlap m x y = b+S x+S y)
+    (w : I → ℝ) (B : I → Finset P) (T M ε τ : ℝ)
+    (hw : ∀ i, 0 ≤ w i) (hW : (∑ i, w i) ≤ 1)
+    (hT : 1 ≤ T) (hM : 0 ≤ M) (hε : 0 ≤ ε) (hτ : 0 < τ)
+    (hscale : τ * Fintype.card P ≤ T)
+    (hB : ∀ i, 2 ≤ (B i).card)
+    (hk : ∀ i, 1 / (((B i).card : ℝ)-1) ≤ ε)
+    (hn : 1 / ((Fintype.card P : ℝ)-1) ≤ ε)
+    (hdom : ∀ x, (∑ i, w i * prob (B i) x) ≤ M * prob Finset.univ x) :
+    weightedIntegral m (varianceCell w B T) ≤
+      (2*ε*(M+1)/τ)*weightedIntegral m (pairCell Finset.univ) +
+      4*M*weightedIntegral m (fun C => if (C.card : ℝ) ≤ T then pairCell Finset.univ C else 0) := by
+  have hh := Finset.sum_le_sum (s := Finset.univ) (fun C _ =>
+    mul_le_mul_of_nonneg_left
+      (varianceCell_bound w B T M ε τ hw hW hT hM hε hτ hscale hB hk hn hdom C) (hm C))
+  simp only [mul_add, Finset.sum_add_distrib] at hh
+  have hz : (∑ C, m C * ∑ i, w i * signedCell (B i) C) = 0 := by
+    simp_rw [Finset.mul_sum]
+    rw [Finset.sum_comm]
+    have hi (i : I) : (∑ C, m C * (w i * signedCell (B i) C)) = 0 := by
+      calc
+        _ = w i * weightedIntegral m (signedCell (B i)) := by
+          simp only [weightedIntegral, Finset.mul_sum]
+          apply Finset.sum_congr rfl
+          intro C _
+          ring
+        _ = 0 := by rw [weighted_cancel m b S hL _ (hB i),mul_zero]
+    simp_rw [hi]
+    exact Finset.sum_const_zero
+  rw [hz,zero_add] at hh
+  simp only [← mul_add] at hh
+  change weightedIntegral m (varianceCell w B T) ≤ _ at hh
+  have he₁ : (∑ C, m C * (2*ε*(M+1)/τ * pairCell Finset.univ C)) =
+      (2*ε*(M+1)/τ)*weightedIntegral m (pairCell Finset.univ) := by
+    simp only [weightedIntegral, Finset.mul_sum]
+    apply Finset.sum_congr rfl
+    intro C _
+    ring
+  have he₂ : (∑ C, m C * (4*M*(if (C.card : ℝ) ≤ T then pairCell Finset.univ C else 0))) =
+      4*M*weightedIntegral m (fun C => if (C.card : ℝ) ≤ T then pairCell Finset.univ C else 0) := by
+    simp only [weightedIntegral, Finset.mul_sum]
+    apply Finset.sum_congr rfl
+    intro C _
+    ring
+  rwa [he₁,he₂] at hh
+end WeakPinned.Variance
+
+namespace WeakPinned.Global
+open Hierarchy Variance
+variable {P : Type u5} [Fintype P] [DecidableEq P]
+variable {K : Type u6} [Field K] [NumberField K]
+variable (Z₁ Z₂ : P → K) (h₁ : Function.Injective Z₁) (h₂ : Function.Injective Z₂)
+
+def totalMass (C : Finset P) : ℝ := cellMass Z₁ h₁ C + cellMass Z₂ h₂ C
+
+def totalPotential (x : P) : ℝ := S Z₁ h₁ x + S Z₂ h₂ x
+
+def baseConstant : ℝ := 2*Grid.kappa - M Z₁ h₁ - M Z₂ h₂
+
+lemma totalMass_nonneg (C : Finset P) : 0 ≤ totalMass Z₁ Z₂ h₁ h₂ C :=
+  add_nonneg (cellMass_nonneg Z₁ h₁ C) (cellMass_nonneg Z₂ h₂ C)
+
+lemma totalPotential_nonneg (x : P) : 0 ≤ totalPotential Z₁ Z₂ h₁ h₂ x :=
+  add_nonneg (S_nonneg Z₁ h₁ x) (S_nonneg Z₂ h₂ x)
+
+lemma totalMass_overlap (hn : 2 ≤ Fintype.card P) (x y : P) (hxy : x ≠ y) :
+    weightedOverlap (totalMass Z₁ Z₂ h₁ h₂) x y =
+      baseConstant Z₁ Z₂ h₁ h₂ + totalPotential Z₁ Z₂ h₁ h₂ x + totalPotential Z₁ Z₂ h₁ h₂ y := by
+  simp only [weightedOverlap,weightedIntegral,totalMass,add_mul,Finset.sum_add_distrib]
+  rw [cellMass_overlap Z₁ h₁ hn x y hxy,cellMass_overlap Z₂ h₂ hn x y hxy,
+    coordinate_overlap_identity Z₁ h₁ hn x y hxy,coordinate_overlap_identity Z₂ h₂ hn x y hxy]
+  simp only [baseConstant,totalPotential]
+  ring
+
+/-- Unconditional arithmetic-hierarchy instance of the quantitative variance
+    bound. The overlap identity follows from the product formula, and the
+    family hypotheses are exactly finite sampling data. -/
+theorem arithmetic_variance {I : Type u7} [Fintype I]
+    (hnP : 2 ≤ Fintype.card P)
+    (w : I → ℝ) (B : I → Finset P) (T M₀ ε τ : ℝ)
+    (hw : ∀ i, 0 ≤ w i) (hW : (∑ i, w i) ≤ 1)
+    (hT : 1 ≤ T) (hM : 0 ≤ M₀) (hε : 0 ≤ ε) (hτ : 0 < τ)
+    (hscale : τ * Fintype.card P ≤ T)
+    (hB : ∀ i, 2 ≤ (B i).card)
+    (hk : ∀ i, 1 / (((B i).card : ℝ)-1) ≤ ε)
+    (hn : 1 / ((Fintype.card P : ℝ)-1) ≤ ε)
+    (hdom : ∀ x, (∑ i, w i * prob (B i) x) ≤ M₀ * prob Finset.univ x) :
+    weightedIntegral (totalMass Z₁ Z₂ h₁ h₂) (varianceCell w B T) ≤
+      (2*ε*(M₀+1)/τ)*weightedIntegral (totalMass Z₁ Z₂ h₁ h₂) (pairCell Finset.univ) +
+      4*M₀*weightedIntegral (totalMass Z₁ Z₂ h₁ h₂)
+        (fun C => if (C.card : ℝ) ≤ T then pairCell Finset.univ C else 0) :=
+  weighted_variance _ (totalMass_nonneg Z₁ Z₂ h₁ h₂) _ _
+    (totalMass_overlap Z₁ Z₂ h₁ h₂ hnP) w B T M₀ ε τ hw hW hT hM hε hτ hscale hB hk hn hdom
+end WeakPinned.Global
+
+end
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PinnedDistances/BoundedContradiction.lean b/lean/OAI/Geometry/PinnedDistances/BoundedContradiction.lean
new file mode 100644
index 0000000000000000000000000000000000000000..bacf7f0cab7501b6a7c261022cda87d230dbf92f
--- /dev/null
+++ b/lean/OAI/Geometry/PinnedDistances/BoundedContradiction.lean
@@ -0,0 +1,117 @@
+import OAI.Geometry.PinnedDistances.BoundedRates
+
+namespace OAI
+
+universe u1 u2 u3
+
+section
+noncomputable section
+open Filter MeasureTheory
+open scoped Topology BigOperators
+namespace WeakPinned
+open Bounded Global Variance Hierarchy NumberField
+
+variable {V : ℕ → Type u1} {I : ℕ → Type u2} {K : ℕ → Type u3} [∀ j,Fintype (V j)] [∀ j,Fintype (I j)] [∀ j,DecidableEq (V j)]
+  [∀ j,Field (K j)] [∀ j,NumberField (K j)]
+variable (G : ∀ j,Core.Fibers (V j) (I j)) (θ γ c : ℝ) (η : ℕ → ℝ)
+  (Z₁ Z₂ : ∀ j,V j → K j) (h₁ : ∀ j,Function.Injective (Z₁ j)) (h₂ : ∀ j,Function.Injective (Z₂ j))
+
+/-- The bounded arithmetic regime is impossible as well. The embedding and
+normalization are chosen from the actual global mass and finite graph laws. -/
+theorem arithmetic_no_bounded (hθ : 0<θ) (hγ : 0<γ) (hγ1 : γ<1) (hc : 0<c) (hcγ : c<γ)
+    (hn : Tendsto (fun j => (Fintype.card (V j):ℝ)) atTop atTop)
+    (hη : Tendsto η atTop (𝓝 0)) (hη0 : ∀ j,0≤η j) (hηb : ∀ j,η j≤θ/2)
+    (hG : ∀ j,PreparedGraph (V j) (I j) (G j) θ γ c (η j))
+    (t : ∀ j,I j → K j) (ht : ∀ j e y,y∈(G j).targets e →
+      (Z₁ j ((G j).source e)-Z₁ j y)*(Z₂ j ((G j).source e)-Z₂ j y)=t j e)
+    (C : ℝ) (hC : 0≤C) (hscale : ∀ j,overlapScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)+
+      potentialScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)≤C) : False := by
+  let n := fun j => (Fintype.card (V j):ℝ)
+  let T := fun j => n j^(1-c)
+  let S := fun j => n j^γ
+  let q := fun j => varianceCell (G j).fiberWeight (G j).targets (T j)
+  let r := fun j => varianceCell ((G j).retainedWeight (S j)) ((G j).retainedSet (S j)) (T j)
+  have hq (j D) : 0≤q j D := varianceCell_nonneg _ _ _ (G j).fiberWeight_nonneg D
+  have hr (j D) : 0≤r j D := varianceCell_nonneg _ _ _ ((G j).retainedWeight_nonneg _) D
+  have h2 (j) : 2≤Fintype.card (V j) := (hG j).card_ge
+  have hn2 (j) : (2:ℝ)≤n j := by dsimp only [n]; exact_mod_cast h2 j
+  have hn0 (j) : 0<n j := by linarith [hn2 j]
+  have hpos (j) : 0<(G j).edgeCount := (hG j).edgeCount_pos (by linarith [hηb j])
+  have hdom (j x y) := (hG j).uniform_domination hθ (hηb j) x y
+  have hT (j) : 1≤T j := Real.one_le_rpow (by linarith [hn2 j]) (by linarith)
+  have hS (j) : 1<S j := Real.one_lt_rpow (by linarith [hn2 j]) hγ
+  have hSn (j) : S j≤n j := by
+    simpa only [Real.rpow_one] using Real.rpow_le_rpow_of_exponent_le (show 1≤n j by linarith [hn2 j]) hγ1.le
+  have hB (j e) : 2≤((G j).targets e).card := by
+    have hh := (hS j).trans ((hG j).fiber e)
+    exact_mod_cast hh
+  have hR (j y) : 2≤((G j).retainedSet (S j) y).card := by
+    have hh := (hS j).trans_le ((G j).retainedSet_size (S j) (hSn j) y)
+    exact_mod_cast hh
+  have hvar := sampling_variance_rates G θ γ c η Z₁ Z₂ h₁ h₂ hθ hγ hγ1 hcγ hn hη hη0 hηb hG t ht
+  let W := fun j => overlapScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)+potentialScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)
+  have hW (j) : 0≤W j := add_nonneg (overlapScale_nonneg _ _ _ _) (potentialScale_nonneg _ _ _ _)
+  have hf0 (j) : 0≤fiberVariance G c Z₁ Z₂ h₁ h₂ j :=
+    Finset.sum_nonneg (fun D _ => mul_nonneg (totalMass_nonneg _ _ _ _ D) (hq j D))
+  have hr0 (j) : 0≤ incomingVariance G γ c Z₁ Z₂ h₁ h₂ j :=
+    Finset.sum_nonneg (fun D _ => mul_nonneg (totalMass_nonneg _ _ _ _ D) (hr j D))
+  have hft := bounded_denominator_rate _ W C hf0 hW hscale hvar.1
+  have hrt := bounded_denominator_rate _ W C hr0 hW hscale hvar.2
+  obtain ⟨v,hv,hvf,hvr⟩ := bounded_place_selection Z₁ Z₂ h₁ h₂ h2 q r hq hr C hC hscale hft hrt
+  let z₁ := fun j x => (v j).embedding (Z₁ j x)
+  let z₂ := fun j x => (v j).embedding (Z₂ j x)
+  have hz₁ (j) : Function.Injective (z₁ j) := (v j).embedding.injective.comp (h₁ j)
+  have hz₂ (j) : Function.Injective (z₂ j) := (v j).embedding.injective.comp (h₂ j)
+  let u₁ := fun j => logNormalize (z₁ j)
+  let u₂ := fun j => logNormalize (z₂ j)
+  let D : ℝ := 4*(2*C+1)+8
+  have hD : 0≤D := by dsimp [D]; positivity
+  have hlog₁ (j) : logAverage (u₁ j)≤D := by
+    rw [logNormalize_bound _ (hz₁ j) (h2 j)]
+    exact (arch_log_center_bound (Z₁ j) (Z₂ j) (h₁ j) (h₂ j) (h2 j) (v j)).trans (by dsimp [D]; linarith [hv j])
+  have hlog₂ (j) : logAverage (u₂ j)≤D := by
+    rw [logNormalize_bound _ (hz₂ j) (h2 j)]
+    have hh := arch_log_center_bound (Z₂ j) (Z₁ j) (h₂ j) (h₁ j) (h2 j) (v j)
+    rw [archMean_scale_swap] at hh
+    exact hh.trans (by dsimp [D]; linarith [hv j])
+  choose x₁ hx₁ using (fun j => log_center (u₁ j) D (h2 j) (hlog₁ j))
+  choose x₂ hx₂ using (fun j => log_center (u₂ j) D (h2 j) (hlog₂ j))
+  have hsmall := power_fraction_rate n hn0 hn c hc
+  let m₁ := fun j => pairMean (rawDepth (z₁ j))
+  let m₂ := fun j => pairMean (rawDepth (z₂ j))
+  have hd₁ := sphere_arch_test_rate Z₁ Z₂ h₁ h₂ v m₁ (fun j => -u₁ j (x₁ j))
+    (fun j => (G j).fiberWeight) (fun j => (G j).targets) T (2/θ) h2 hT (by positivity)
+    (fun j => (G j).fiberWeight_nonneg) (fun j => (G j).sum_fiberWeight (hpos j) |>.le) hB
+    (fun j => (G j).fiber_domination _ (hdom j) (h2 j)) hvf hsmall
+  have hvf₂ : Tendsto (fun j => archMean (Z₂ j) (Z₁ j) (massEval (q j)) (v j)) atTop (𝓝 0) := by
+    simpa only [archMean_mass_swap] using hvf
+  have hd₂ := sphere_arch_test_rate Z₂ Z₁ h₂ h₁ v m₂ (fun j => -u₂ j (x₂ j))
+    (fun j => (G j).fiberWeight) (fun j => (G j).targets) T (2/θ) h2 hT (by positivity)
+    (fun j => (G j).fiberWeight_nonneg) (fun j => (G j).sum_fiberWeight (hpos j) |>.le) hB
+    (fun j => (G j).fiber_domination _ (hdom j) (h2 j)) hvf₂ hsmall
+  have hdR := sphere_arch_test_rate Z₁ Z₂ h₁ h₂ v m₁ (fun j => -u₁ j (x₁ j))
+    (fun j => (G j).retainedWeight (S j)) (fun j => (G j).retainedSet (S j)) T (2/θ) h2 hT (by positivity)
+    (fun j => (G j).retainedWeight_nonneg _) (fun j => (G j).sum_retainedWeight _ (hpos j)) hR
+    (fun j => (G j).retained_domination _ _ (hdom j) (h2 j)) hvr hsmall
+  apply normalized_graph_impossible G u₁ u₂ x₁ x₂
+    (fun j => logNormalize_injective _ (hz₁ j)) (fun j => logNormalize_injective _ (hz₂ j))
+    hpos (fun j e => Finset.card_pos.mp (by have := hB j e; omega)) h2 hn D hD hlog₁ hlog₂ hx₁ hx₂
+    (by simpa only [sub_eq_add_neg,u₁,logNormalize,z₁,m₁] using hd₁)
+    (by simpa only [sub_eq_add_neg,u₂,logNormalize,z₂,m₂] using hd₂)
+    S (fun j => Real.rpow_pos_of_pos (hn0 j) γ)
+    (by simpa only [sub_eq_add_neg,u₁,logNormalize,z₁,m₁] using hdR)
+    (omitted_rate G θ γ c η hθ hγ1 hηb hn hG) (2/θ) (by positivity) hdom
+  intro j e y z hy hz
+  have he := (ht j e y hy).trans (ht j e z hz).symm
+  have hem := congrArg (v j).embedding he
+  simp only [map_mul,map_sub] at hem
+  dsimp only [u₁,u₂,logNormalize,z₁,z₂]
+  linear_combination (Real.exp (pairMean (rawDepth (z₁ j))):ℂ)*
+    (Real.exp (pairMean (rawDepth (z₂ j))):ℂ)*hem
+
+end WeakPinned
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PinnedDistances/BoundedRates.lean b/lean/OAI/Geometry/PinnedDistances/BoundedRates.lean
new file mode 100644
index 0000000000000000000000000000000000000000..0451ca253f19afcaa74767ff4012d8858b52e378
--- /dev/null
+++ b/lean/OAI/Geometry/PinnedDistances/BoundedRates.lean
@@ -0,0 +1,133 @@
+import OAI.Geometry.PinnedDistances.NormalizedGraphs
+
+namespace OAI
+
+universe u1 u2 u3 u4
+
+section
+noncomputable section
+open MeasureTheory Filter
+open scoped Topology BigOperators
+namespace WeakPinned.Bounded
+open Hierarchy Global Variance NumberField
+
+lemma bounded_denominator_rate (x w : ℕ → ℝ) (C : ℝ)
+    (hx : ∀ n,0≤x n) (hw : ∀ n,0≤w n) (hC : ∀ n,w n≤C)
+    (ht : Tendsto (fun n => x n/(w n+1)) atTop (𝓝 0)) :
+    Tendsto x atTop (𝓝 0) := by
+  apply squeeze_zero hx _ (by simpa only [mul_zero] using tendsto_const_nhds.mul ht (a:=C+1))
+  intro n
+  have hp : 0<w n+1 := by linarith [hw n]
+  calc
+    x n = (w n+1)*(x n/(w n+1)) := by field_simp
+    _ ≤ (C+1)*(x n/(w n+1)) :=
+      mul_le_mul_of_nonneg_right (by linarith [hC n]) (div_nonneg (hx n) hp.le)
+
+variable {P : Type u1} [Fintype P] [DecidableEq P]
+variable {K : Type u2} [Field K] [NumberField K]
+
+lemma massEval_add (q r : Finset P → ℝ) (d d' : Depth P) :
+    massEval (fun C => q C+r C) d d'=massEval q d d'+massEval r d d' := by
+  simp only [massEval,weightedIntegral,mul_add,Finset.sum_add_distrib]
+  ring
+
+omit [NumberField K] in
+lemma archMean_mass_add (Z₁ Z₂ : P → K) (h₁ : Function.Injective Z₁) (h₂ : Function.Injective Z₂)
+    (hn : 2≤Fintype.card P) (q r : Finset P → ℝ) (v : InfinitePlace K) :
+    archMean Z₁ Z₂ (massEval (fun C => q C+r C)) v=
+      archMean Z₁ Z₂ (massEval q) v+archMean Z₁ Z₂ (massEval r) v := by
+  unfold archMean
+  simp only [massEval_add]
+  exact integral_add (integrable_massEval Z₁ Z₂ h₁ h₂ hn q v)
+    (integrable_massEval Z₁ Z₂ h₁ h₂ hn r v)
+
+omit [NumberField K] in
+lemma archMean_mass_swap (Z₁ Z₂ : P → K) (q : Finset P → ℝ) (v : InfinitePlace K) :
+    archMean Z₂ Z₁ (massEval q) v=archMean Z₁ Z₂ (massEval q) v := by
+  unfold archMean
+  congr 1
+  funext ω
+  exact add_comm _ _
+
+omit [NumberField K] in
+lemma archMean_scale_swap (Z₁ Z₂ : P → K) (v : InfinitePlace K) :
+    archMean Z₂ Z₁ scaleEval v=archMean Z₁ Z₂ scaleEval v := by
+  unfold archMean
+  congr 1
+  funext ω
+  simp only [scaleEval,massEval,potentialEval]
+  congr 1
+  · exact add_comm _ _
+  · apply Finset.sum_congr rfl
+    intro x _
+    rw [add_comm ((arch Z₂ v ω).potential x)]
+
+omit [DecidableEq P] in
+lemma weightedIntegral_add (m q r : Finset P → ℝ) :
+    weightedIntegral m (fun C => q C+r C)=weightedIntegral m q+weightedIntegral m r := by
+  simp only [weightedIntegral,mul_add,Finset.sum_add_distrib]
+
+variable {V : ℕ → Type u3} {L : ℕ → Type u4} [∀ n,Fintype (V n)] [∀ n,DecidableEq (V n)]
+  [∀ n,Field (L n)] [∀ n,NumberField (L n)]
+
+lemma bounded_place_selection (Z₁ Z₂ : ∀ n,V n → L n)
+    (h₁ : ∀ n,Function.Injective (Z₁ n)) (h₂ : ∀ n,Function.Injective (Z₂ n))
+    (hn : ∀ n,2≤Fintype.card (V n)) (q r : ∀ n,Finset (V n) → ℝ)
+    (hq : ∀ n C,0≤q n C) (hr : ∀ n C,0≤r n C)
+    (C : ℝ) (hC : 0≤C)
+    (hscale : ∀ n,overlapScale (Z₁ n) (Z₂ n) (h₁ n) (h₂ n)+
+      potentialScale (Z₁ n) (Z₂ n) (h₁ n) (h₂ n)≤C)
+    (hqt : Tendsto (fun n => weightedIntegral (totalMass (Z₁ n) (Z₂ n) (h₁ n) (h₂ n)) (q n)) atTop (𝓝 0))
+    (hrt : Tendsto (fun n => weightedIntegral (totalMass (Z₁ n) (Z₂ n) (h₁ n) (h₂ n)) (r n)) atTop (𝓝 0)) :
+    ∃ v : ∀ n,InfinitePlace (L n),
+      (∀ n,archMean (Z₁ n) (Z₂ n) scaleEval (v n)≤2*C+1) ∧
+      Tendsto (fun n => archMean (Z₁ n) (Z₂ n) (massEval (q n)) (v n)) atTop (𝓝 0) ∧
+      Tendsto (fun n => archMean (Z₁ n) (Z₂ n) (massEval (r n)) (v n)) atTop (𝓝 0) := by
+  let m := fun n => totalMass (Z₁ n) (Z₂ n) (h₁ n) (h₂ n)
+  let ε := fun n => weightedIntegral (m n) (q n)+weightedIntegral (m n) (r n)+1/((n:ℝ)+1)
+  have hq0 (n) : 0≤weightedIntegral (m n) (q n) :=
+    Finset.sum_nonneg (fun D _ => mul_nonneg (totalMass_nonneg _ _ _ _ _) (hq n D))
+  have hr0 (n) : 0≤weightedIntegral (m n) (r n) :=
+    Finset.sum_nonneg (fun D _ => mul_nonneg (totalMass_nonneg _ _ _ _ _) (hr n D))
+  have he (n) : 0<ε n := by
+    have hh : 0<1/((n:ℝ)+1) := by positivity
+    dsimp only [ε]
+    linarith [hq0 n,hr0 n]
+  have het : Tendsto ε atTop (𝓝 0) := by
+    have hi : Tendsto (fun n : ℕ => 1/((n:ℝ)+1)) atTop (𝓝 0) := by
+      simpa only [one_div,Function.comp_def] using tendsto_inv_atTop_zero.comp (tendsto_atTop_add_const_right _ _ tendsto_natCast_atTop_atTop)
+    simpa only [add_zero] using (hqt.add hrt).add hi
+  have hs (n) := select_arch (Z₁ n) (Z₂ n) (h₁ n) (h₂ n) (hn n)
+    (fun D => q n D+r n D) (fun D => add_nonneg (hq n D) (hr n D)) C (ε n) hC (he n) (hscale n)
+    (by rw [weightedIntegral_add]; dsimp [ε,m]; linarith [show 0≤1/((n:ℝ)+1) by positivity])
+  choose v hvscale hv using hs
+  have hvq (n) : 0≤archMean (Z₁ n) (Z₂ n) (massEval (q n)) (v n) :=
+    archMean_nonneg _ _ _ (massEval_nonneg _ (hq n)) _
+  have hvr (n) : 0≤archMean (Z₁ n) (Z₂ n) (massEval (r n)) (v n) :=
+    archMean_nonneg _ _ _ (massEval_nonneg _ (hr n)) _
+  have hsum (n) := hv n
+  simp only [archMean_mass_add _ _ (h₁ _) (h₂ _) (hn _)] at hsum
+  refine ⟨v,hvscale,?_,?_⟩
+  · apply squeeze_zero hvq _ (by simpa only [mul_zero] using het.const_mul 2)
+    intro n
+    linarith [hsum n,hvr n]
+  · apply squeeze_zero hvr _ (by simpa only [mul_zero] using het.const_mul 2)
+    intro n
+    linarith [hsum n,hvq n]
+
+lemma power_fraction_rate (n : ℕ → ℝ) (hn : ∀ j,0<n j)
+    (ht : Tendsto n atTop atTop) (c : ℝ) (hc : 0<c) :
+    Tendsto (fun j => (n j)^(1-c)/n j) atTop (𝓝 0) := by
+  have hh := (tendsto_rpow_neg_atTop hc).comp ht
+  convert hh using 1
+  funext j
+  rw [show 1-c=1+(-c) by ring,Real.rpow_add (hn j),Real.rpow_one]
+  simp only [Function.comp_apply]
+  exact mul_div_cancel_left₀ _ (hn j).ne'
+
+end WeakPinned.Bounded
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PinnedDistances/ChartLimits.lean b/lean/OAI/Geometry/PinnedDistances/ChartLimits.lean
new file mode 100644
index 0000000000000000000000000000000000000000..fdbd80d4f955fe2ba7a75561051521a2b09c545b
--- /dev/null
+++ b/lean/OAI/Geometry/PinnedDistances/ChartLimits.lean
@@ -0,0 +1,174 @@
+import OAI.Geometry.PinnedDistances.FiniteMixtures
+
+namespace OAI
+
+universe u1 u2 u3 u4 u5 u6 u7 u8 u9 u10 u11 u12
+
+section
+noncomputable section
+open MeasureTheory Filter Set
+open scoped Topology
+namespace WeakPinned.Bounded
+
+variable {X : Type u1} {Y : Type u2} [MeasurableSpace X] [MeasurableSpace Y]
+
+lemma chart_probability (j : X → Y) (hj : MeasurableEmbedding j)
+    (μ : Measure Y) [IsProbabilityMeasure μ] (hfin : ∀ᵐ y ∂  μ,y∈range j) :
+    IsProbabilityMeasure (μ.comap j) := by
+  constructor
+  rw [hj.comap_apply,image_univ]
+  exact measure_eq_measure_of_null_sdiff (subset_univ _) (by
+    convert ae_iff.mp hfin using 2
+    ext y
+    simp) |>.trans measure_univ
+
+lemma chart_map_comap (j : X → Y) (hj : MeasurableEmbedding j)
+    (μ : Measure Y) (hfin : ∀ᵐ y ∂  μ,y∈range j) : (μ.comap j).map j=μ := by
+  rw [hj.map_comap]
+  exact Measure.restrict_eq_self_of_ae_mem hfin
+
+lemma chart_ae_iff (j : X → Y) (hj : MeasurableEmbedding j) (μ : Measure Y)
+    {s : Set X} : (∀ᵐ x ∂  μ.comap j,x∈s) ↔ μ (j '' sᶜ)=0 := by
+  rw [ae_iff,hj.comap_apply]
+  rfl
+
+variable [TopologicalSpace X] [TopologicalSpace Y] [BorelSpace X] [BorelSpace Y]
+
+lemma weak_chart_closed (j : X → Y) (hj : Topology.IsOpenEmbedding j)
+    [HasOuterApproxClosed Y] (α : ℕ → ProbabilityMeasure Y) (β : ProbabilityMeasure Y)
+    (ht : Tendsto α atTop (𝓝 β)) (s : Set X) (hs : IsClosed s)
+    (hα : ∀ n,(α n : Measure Y) (j '' sᶜ)=0) :
+    ∀ᵐ x ∂  (β : Measure Y).comap j,x∈s := by
+  rw [chart_ae_iff j hj.measurableEmbedding]
+  apply le_antisymm _ zero_le
+  apply (ProbabilityMeasure.le_liminf_measure_open_of_tendsto ht
+    (hj.isOpenMap _ hs.isOpen_compl)).trans
+  apply liminf_le_of_frequently_le'
+  exact (Eventually.of_forall (fun n => le_of_eq (hα n))).frequently
+
+end WeakPinned.Bounded
+
+end
+end
+section
+noncomputable section
+open MeasureTheory Filter Set
+open scoped Topology
+namespace WeakPinned.Bounded
+
+lemma map_transport {A : Type u3} {B : Type u4} {X : Type u5} {Y : Type u6} [MeasurableSpace A] [MeasurableSpace B]
+    [MeasurableSpace X] [MeasurableSpace Y] (j : A → X) (k : B → Y)
+    (hj : Measurable j) (hk : MeasurableEmbedding k)
+    (f : A → B) (g : X → Y) (hf : Measurable f) (hg : Measurable g)
+    (μ : Measure A) (ν : Measure B) (hcomm : k ∘ f=g ∘ j)
+    (hm : (μ.map j).map g=ν.map k) : μ.map f=ν := by
+  apply hk.map_injective
+  rw [Measure.map_map hk.measurable hf,hcomm,← Measure.map_map hg hj]
+  exact hm
+
+lemma map_pull_ac {A : Type u7} {B : Type u8} [MeasurableSpace A] [MeasurableSpace B]
+    (j : A → B) (hj : MeasurableEmbedding j) (μ ν : Measure A)
+    (h : μ.map j ≪ ν.map j) : μ≪ν := by
+  intro s hs
+  have hν : ν.map j (j '' s)=0 := by
+    rw [hj.map_apply,hj.injective.preimage_image]
+    exact hs
+  have hμ := h hν
+  rwa [hj.map_apply,hj.injective.preimage_image] at hμ
+
+lemma map_full_range {A : Type u9} {B : Type u10} [MeasurableSpace A] [MeasurableSpace B]
+    (j : A → B) (hj : MeasurableEmbedding j) (μ : Measure A) :
+    ∀ᵐ x ∂ μ.map j,x∈range j := by
+  apply (ae_map_iff hj.measurable.aemeasurable hj.measurableSet_range).mpr
+  exact Eventually.of_forall (fun _ => mem_range_self _)
+
+abbrev pairCoe : ℂ × ℂ → Sphere × Sphere := Prod.map OnePoint.some OnePoint.some
+abbrev quadCoe : Quad ℂ → Quad Sphere := Prod.map OnePoint.some
+  (Prod.map OnePoint.some pairCoe)
+abbrev expandedCoe : Expanded ℂ → Expanded Sphere := Prod.map quadCoe (Prod.map pairCoe pairCoe)
+
+lemma pairCoe_open : Topology.IsOpenEmbedding pairCoe :=
+  OnePoint.isOpenEmbedding_coe.prodMap OnePoint.isOpenEmbedding_coe
+
+lemma quadCoe_open : Topology.IsOpenEmbedding quadCoe :=
+  OnePoint.isOpenEmbedding_coe.prodMap (OnePoint.isOpenEmbedding_coe.prodMap pairCoe_open)
+
+lemma expandedCoe_open : Topology.IsOpenEmbedding expandedCoe :=
+  quadCoe_open.prodMap (pairCoe_open.prodMap pairCoe_open)
+
+end WeakPinned.Bounded
+
+end
+end
+section
+noncomputable section
+open MeasureTheory Filter
+open scoped BigOperators Topology ENNReal
+namespace WeakPinned.Bounded
+open Variance
+variable {V : Type u11} {I : Type u12} [Fintype V] [DecidableEq V] [Fintype I]
+variable (G : Core.Fibers V I)
+local instance : MeasurableSpace Sphere := borel Sphere
+local instance : BorelSpace Sphere := ⟨rfl⟩
+
+omit [Fintype I] in
+lemma graph_edgeWeight_nonneg (r : I × V) :
+    0≤edgeWeight G.fiberWeight (fun e => prob (G.targets e)) r :=
+  mul_nonneg (G.fiberWeight_nonneg _) (Core.Fibers.prob_nn _ _)
+
+omit [Fintype I] in
+lemma graph_tripleWeight_nonneg (r : I × V × V × V) :
+    0≤tripleWeight G.fiberWeight (fun e => prob (G.targets e)) r :=
+  mul_nonneg (mul_nonneg (mul_nonneg (G.fiberWeight_nonneg _) (Core.Fibers.prob_nn _ _))
+    (Core.Fibers.prob_nn _ _)) (Core.Fibers.prob_nn _ _)
+
+def edgeProbability (U₁ U₂ : V → Sphere) (hG : 0<G.edgeCount)
+    (hB : ∀ e,(G.targets e).Nonempty) : ProbabilityMeasure (Quad Sphere) :=
+  finiteProbability (edgeWeight G.fiberWeight (fun e => prob (G.targets e)))
+    (edgeValue G.source U₁ U₂) (graph_edgeWeight_nonneg G)
+      (sum_edgeWeight _ _ (G.sum_fiberWeight hG) (fun e => sum_prob _ (hB e)))
+
+def expandedProbability (U₁ U₂ : V → Sphere) (hG : 0<G.edgeCount)
+    (hB : ∀ e,(G.targets e).Nonempty) : ProbabilityMeasure (Expanded Sphere) :=
+  finiteProbability (tripleWeight G.fiberWeight (fun e => prob (G.targets e)))
+    (tripleValue G.source U₁ U₂) (graph_tripleWeight_nonneg G)
+      (sum_tripleWeight _ _ (G.sum_fiberWeight hG) (fun e => sum_prob _ (hB e)))
+
+lemma expandedProbability_edge (U₁ U₂ : V → Sphere) (hG : 0<G.edgeCount)
+    (hB : ∀ e,(G.targets e).Nonempty) :
+    (expandedProbability G U₁ U₂ hG hB:Measure (Expanded Sphere)).map Prod.fst=
+      (edgeProbability G U₁ U₂ hG hB:Measure (Quad Sphere)) := by
+  change (finiteLaw _ _).map _=finiteLaw _ _
+  rw [finiteLaw_map _ _ _ measurable_fst]
+  exact tripleLaw_edge G.fiberWeight (fun e => prob (G.targets e)) (edgeValue G.source U₁ U₂) G.fiberWeight_nonneg (fun e => Core.Fibers.prob_nn _) (fun e => sum_prob _ (hB e))
+
+lemma edgeProbability_dom₁ (U₁ U₂ : V → Sphere) (hG : 0<G.edgeCount)
+    (hB : ∀ e,(G.targets e).Nonempty) (hn : 2≤Fintype.card V) (M : ℝ) (hM : 0≤M)
+    (hdom : ∀ x y,G.edgeLaw x y≤M*pairProb Finset.univ x y) :
+    (edgeProbability G U₁ U₂ hG hB:Measure (Quad Sphere)).map (fun q => (q.1,q.2.2.1))≤
+      ENNReal.ofReal (2*M) • (uniformLaw U₁).prod (uniformLaw U₁) := by
+  change (finiteLaw (edgeWeight G.fiberWeight (fun e => prob (G.targets e)))
+    (edgeValue G.source U₁ U₂)).map (fun q => (q.1,q.2.2.1))≤_
+  rw [finiteLaw_map _ _ (fun q : Quad Sphere => (q.1,q.2.2.1)) (by fun_prop)]
+  change finiteLaw (edgeWeight G.fiberWeight (fun e => prob (G.targets e))) (fun r => (U₁ (G.source r.1),U₁ r.2))≤_
+  rw [graph_edge_law G (fun x y => (U₁ x,U₁ y)) hG hB]
+  exact graph_domination G U₁ hn M hM hdom
+
+lemma edgeProbability_dom₂ (U₁ U₂ : V → Sphere) (hG : 0<G.edgeCount)
+    (hB : ∀ e,(G.targets e).Nonempty) (hn : 2≤Fintype.card V) (M : ℝ) (hM : 0≤M)
+    (hdom : ∀ x y,G.edgeLaw x y≤M*pairProb Finset.univ x y) :
+    (edgeProbability G U₁ U₂ hG hB:Measure (Quad Sphere)).map (fun q => (q.2.1,q.2.2.2))≤
+      ENNReal.ofReal (2*M) • (uniformLaw U₂).prod (uniformLaw U₂) := by
+  change (finiteLaw (edgeWeight G.fiberWeight (fun e => prob (G.targets e)))
+    (edgeValue G.source U₁ U₂)).map (fun q => (q.2.1,q.2.2.2))≤_
+  rw [finiteLaw_map _ _ (fun q : Quad Sphere => (q.2.1,q.2.2.2)) (by fun_prop)]
+  change finiteLaw (edgeWeight G.fiberWeight (fun e => prob (G.targets e))) (fun r => (U₂ (G.source r.1),U₂ r.2))≤_
+  rw [graph_edge_law G (fun x y => (U₂ x,U₂ y)) hG hB]
+  exact graph_domination G U₂ hn M hM hdom
+
+end WeakPinned.Bounded
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PinnedDistances/ConditionalIntegral.lean b/lean/OAI/Geometry/PinnedDistances/ConditionalIntegral.lean
new file mode 100644
index 0000000000000000000000000000000000000000..7eef3aaf5f7183dc45261c73bdec94a14ebe332c
--- /dev/null
+++ b/lean/OAI/Geometry/PinnedDistances/ConditionalIntegral.lean
@@ -0,0 +1,67 @@
+import OAI.Geometry.PinnedDistances.FiniteLaw
+
+namespace OAI
+
+universe u1 u2 u3 u4 u5
+
+section
+noncomputable section
+open MeasureTheory Filter
+open scoped Topology
+namespace WeakPinned.Bounded
+
+lemma integrable_one_bound {X : Type u1} [MeasurableSpace X] (μ : Measure X) [IsFiniteMeasure μ]
+    (f : X → ℝ) (hf : Measurable f) (hbound : ∀ x,‖f x‖≤1) : Integrable f μ := by
+  exact (integrable_const (1:ℝ)).mono' hf.aestronglyMeasurable (Eventually.of_forall hbound)
+
+variable {Ω : Type u2} {E : Type u3} {X : Type u4} {Y : Type u5} [MeasurableSpace Ω] [MeasurableSpace E]
+  [MeasurableSpace X] [MeasurableSpace Y]
+
+lemma integral_fiber_indicator (Q : Measure E) (μ : Measure X)
+    [IsProbabilityMeasure Q] [IsProbabilityMeasure μ]
+    (f : E × X → ℝ) (hf : Measurable f) (hbound : ∀ p,‖f p‖≤1)
+    (s : Set E) (hs : MeasurableSet s) :
+    (∫ e in s, ∫ x,f (e,x) ∂ μ ∂ Q)=
+      ∫ p,(s ×ˢ Set.univ).indicator f p ∂ Q.prod μ := by
+  have hi := integrable_one_bound (Q.prod μ) f hf hbound
+  rw [integral_prod _ (hi.indicator (hs.prod MeasurableSet.univ))]
+  rw [← integral_indicator hs]
+  apply integral_congr_ae
+  filter_upwards [] with e
+  by_cases he : e∈s
+  · simp [Set.indicator_of_mem,he]
+  · simp [Set.indicator_of_notMem,he]
+
+lemma conditional_integral_eq
+    (R : Measure Ω) (Q : Measure E) (μ : Measure X) (ν : Measure Y)
+    [IsProbabilityMeasure R] [IsProbabilityMeasure Q]
+    [IsProbabilityMeasure μ] [IsProbabilityMeasure ν]
+    (e : Ω → E) (z : Ω → X) (w : Ω → Y)
+    (he : Measurable e) (hz : Measurable z) (hw : Measurable w)
+    (hmz : R.map (fun r => (e r,z r))=Q.prod μ)
+    (hmw : R.map (fun r => (e r,w r))=Q.prod ν)
+    (f : E × X → ℝ) (g : E × Y → ℝ)
+    (hf : Measurable f) (hg : Measurable g)
+    (hfb : ∀ p,‖f p‖≤1) (hgb : ∀ p,‖g p‖≤1)
+    (heq : ∀ᵐ r ∂ R,f (e r,z r)=g (e r,w r)) :
+    ∀ᵐ a ∂ Q,(∫ x,f (a,x) ∂ μ)=(∫ y,g (a,y) ∂ ν) := by
+  apply Integrable.ae_eq_of_forall_setIntegral_eq
+  · exact (integrable_one_bound (Q.prod μ) f hf hfb).integral_prod_left
+  · exact (integrable_one_bound (Q.prod ν) g hg hgb).integral_prod_left
+  · intro s hs _
+    rw [integral_fiber_indicator Q μ f hf hfb s hs,
+      integral_fiber_indicator Q ν g hg hgb s hs,← hmz,← hmw]
+    rw [integral_map_of_stronglyMeasurable (he.prodMk hz)
+      (hf.indicator (hs.prod MeasurableSet.univ)).stronglyMeasurable,
+      integral_map_of_stronglyMeasurable (he.prodMk hw)
+      (hg.indicator (hs.prod MeasurableSet.univ)).stronglyMeasurable]
+    apply integral_congr_ae
+    filter_upwards [heq] with r hr
+    by_cases h : e r∈s <;> simp [h,hr]
+
+end WeakPinned.Bounded
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PinnedDistances/CutApproximation.lean b/lean/OAI/Geometry/PinnedDistances/CutApproximation.lean
new file mode 100644
index 0000000000000000000000000000000000000000..efb3afb2c81bbfc675962b642f012440031b9d20
--- /dev/null
+++ b/lean/OAI/Geometry/PinnedDistances/CutApproximation.lean
@@ -0,0 +1,368 @@
+import OAI.Geometry.PinnedDistances.GlobalPotentials
+
+namespace OAI
+
+universe u1 u2 u3 u4 u5 u6 u7 u8 u9
+
+section
+noncomputable section
+open NumberField
+open scoped BigOperators
+noncomputable section
+open MeasureTheory
+open scoped Topology
+noncomputable section
+open MeasureTheory
+open scoped Topology ENNReal
+noncomputable section
+open MeasureTheory NumberField
+open scoped BigOperators
+noncomputable section
+open scoped BigOperators
+open MeasureTheory
+noncomputable section
+open Filter
+open scoped Topology
+noncomputable section
+open scoped BigOperators
+noncomputable section
+open Filter
+open scoped Topology
+noncomputable section
+open MeasureTheory
+open scoped BigOperators
+noncomputable section
+
+namespace WeakPinned.Cuts
+open scoped BigOperators
+
+def tent (δ u : ℝ) : ℝ := max 0 (min (u-δ) (1-δ-u))
+lemma tent_nonneg (δ u : ℝ) : 0≤tent δ u := le_max_left _ _
+lemma tent_symm (δ u : ℝ) : tent δ (1-u)=tent δ u := by
+  unfold tent
+  rw [show 1-u-δ=1-δ-u by ring,show 1-δ-(1-u)=u-δ by ring,min_comm]
+lemma tent_zero_left {δ u : ℝ} (hu : u≤δ) : tent δ u=0 := by
+  unfold tent
+  exact max_eq_left ((min_le_left _ _).trans (sub_nonpos.mpr hu))
+lemma tent_zero_right {δ u : ℝ} (hu : 1-δ≤u) : tent δ u=0 := by
+  rw [← tent_symm]
+  exact tent_zero_left (by linarith)
+lemma tent_le_one {δ u : ℝ} (hδ : 0≤δ) (hu : u≤1) : tent δ u≤1 := by
+  exact max_le (by norm_num) ((min_le_left _ _).trans (by linarith))
+lemma tent_lipschitz (δ u v : ℝ) : |tent δ u-tent δ v|≤|u-v| := by
+  have hm := abs_max_sub_max_le_max (0:ℝ) (min (u-δ) (1-δ-u)) 0 (min (v-δ) (1-δ-v))
+  have hn := abs_min_sub_min_le_max (u-δ) (1-δ-u) (v-δ) (1-δ-v)
+  have he : 1-δ-u-(1-δ-v)=-(u-v) := by ring
+  have hf : u-δ-(v-δ)=u-v := by ring
+  simp only [he,hf,abs_neg,max_self] at hn
+  exact hm.trans (max_le (by simp [abs_nonneg]) hn)
+lemma tent_positive {δ u : ℝ} (hu : 0<tent δ u) : δ<u ∧ u<1-δ := by
+  have h : 0 < min (u-δ) (1-δ-u) := (lt_max_iff.mp hu).resolve_left (lt_irrefl _)
+  have := lt_min_iff.mp h
+  constructor <;> linarith
+lemma tent_lower {δ u : ℝ} (hu : 2*δ≤u) (hu' : u≤1-2*δ) : δ≤tent δ u := by
+  exact le_max_iff.mpr (Or.inr (le_min (by linarith) (by linarith)))
+
+def cutError (δ p a : ℝ) : ℝ := |tent δ p-tent δ a|+tent δ a*|p-a|
+lemma cutError_nonneg (δ p a : ℝ) : 0≤cutError δ p a :=
+  add_nonneg (abs_nonneg _) (mul_nonneg (tent_nonneg _ _) (abs_nonneg _))
+lemma cutError_symm (δ p a : ℝ) : cutError δ (1-p) (1-a)=cutError δ p a := by
+  simp only [cutError,tent_symm,show 1-p-(1-a)=-(p-a) by ring,abs_neg]
+lemma cutError_le {δ p a : ℝ} (hδ : 0≤δ) (ha : a≤1) : cutError δ p a≤2*|p-a| := by
+  have := tent_lipschitz δ p a
+  have := mul_le_mul_of_nonneg_right (tent_le_one hδ ha) (abs_nonneg (p-a))
+  unfold cutError
+  linarith
+lemma cutError_young {δ p a ε : ℝ} (hδ : 0≤δ) (ha : a≤1) (hε : 0<ε) :
+    cutError δ p a≤ε+(p-a)^2/ε := by
+  apply (cutError_le hδ ha).trans
+  apply (mul_le_mul_iff_left₀ hε).mp
+  have he : (ε+(p-a)^2/ε)*ε=ε^2+(p-a)^2 := by field_simp
+  rw [he]
+  nlinarith [sq_nonneg (|p-a|-ε),sq_abs (p-a)]
+
+/-- Tiny uniform cells have nonzero balanced-cut weight only if their sampled
+mass is separated from zero. The variance quadratic then pays the cut error. -/
+lemma cutError_tiny {δ p a : ℝ} (hδ : 0<δ) (ha : a<δ/2) (hp : p≤1) :
+    cutError δ p a≤(4/δ^2)*(p-a)^2 ∧ cutError δ p a≤(4/δ^2)*(p^2/2) := by
+  have hz : tent δ a=0 := tent_zero_left (by linarith)
+  rw [cutError,hz,sub_zero,abs_of_nonneg (tent_nonneg _ _),zero_mul,add_zero]
+  by_cases hh : tent δ p=0
+  · rw [hh]; constructor <;> positivity
+  · have ht : 0<tent δ p := lt_of_le_of_ne (tent_nonneg _ _) (Ne.symm hh)
+    have hpd := (tent_positive ht).1
+    have ht1 := tent_le_one hδ.le hp
+    have hd2 : 0<δ^2 := sq_pos_of_pos hδ
+    constructor
+    · apply ht1.trans
+      apply (mul_le_mul_iff_left₀ hd2).mp
+      have he : ((4/δ^2)*(p-a)^2)*δ^2=4*(p-a)^2 := by field_simp
+      rw [he]
+      nlinarith [sq_nonneg (p-a-δ/2),mul_nonneg (by linarith : 0≤p-a-δ/2) hδ.le]
+    · apply ht1.trans
+      apply (mul_le_mul_iff_left₀ hd2).mp
+      have he : ((4/δ^2)*(p^2/2))*δ^2=2*p^2 := by field_simp; ring
+      rw [he]
+      nlinarith [sq_nonneg (p-δ),mul_nonneg (by linarith : 0≤p-δ) hδ.le]
+end WeakPinned.Cuts
+
+namespace WeakPinned.Variance
+open scoped BigOperators
+open Cuts
+variable {P : Type u1} {I : Type u2} [Fintype P] [Fintype I] [DecidableEq P]
+
+noncomputable def familyCutError (w : I → ℝ) (B : I → Finset P) (δ : ℝ) (C : Finset P) : ℝ :=
+  ∑ i, w i*cutError δ (cellProb (B i) C) (cellProb Finset.univ C)
+
+lemma familyCutError_bound (w : I → ℝ) (B : I → Finset P) (T δ ε : ℝ)
+    (hw : ∀ i, 0≤w i) (hW : (∑ i, w i)≤1) (hδ : 0<δ) (hε : 0<ε)
+    (hk : ∀ i, 2/((B i).card:ℝ)≤δ) (C : Finset P)
+    (hlarge : δ/2≤cellProb Finset.univ C → T<C.card)
+    (hpair : δ/2≤cellProb Finset.univ C → 1≤(8/δ^2)*pairCell Finset.univ C) :
+    familyCutError w B δ C≤(8*ε/δ^2)*pairCell Finset.univ C+
+      (1/ε+4/δ^2)*varianceCell w B T C := by
+  have hv : 0≤varianceCell w B T C := varianceCell_nonneg w B T hw C
+  by_cases ha : δ/2≤cellProb Finset.univ C
+  · have he := Finset.sum_le_sum (s:=Finset.univ) (fun i _ =>
+      mul_le_mul_of_nonneg_left (cutError_young (p:=cellProb (B i) C) hδ.le (cellProb_le_one (Finset.univ : Finset P) C) hε) (hw i))
+    change familyCutError w B δ C≤_ at he
+    simp only [mul_add,Finset.sum_add_distrib,← Finset.sum_mul,← mul_div_assoc,← Finset.sum_div] at he
+    have hvf : varianceCell w B T C=∑ i, w i*(cellProb (B i) C-cellProb Finset.univ C)^2 := by
+      simp only [varianceCell,ite_eq_left (hlarge ha)]
+    rw [← hvf] at he
+    have he2 : (∑ i, w i)*ε≤ε := by nlinarith
+    have he3 : ε≤(8*ε/δ^2)*pairCell Finset.univ C := by
+      have hh := mul_le_mul_of_nonneg_left (hpair ha) hε.le
+      calc
+        ε = ε*1 := by ring
+        _ ≤ ε*((8/δ^2)*pairCell Finset.univ C) := hh
+        _ = _ := by ring
+    have hx : 0≤(4/δ^2)*varianceCell w B T C := mul_nonneg (by positivity) hv
+    have hh : varianceCell w B T C/ε=(1/ε)*varianceCell w B T C := by ring
+    rw [hh] at he
+    nlinarith
+  · have hat : cellProb Finset.univ C<δ/2 := lt_of_not_ge ha
+    have hb : familyCutError w B δ C≤(4/δ^2)*varianceCell w B T C := by
+      unfold familyCutError varianceCell
+      split_ifs with hl
+      · rw [Finset.mul_sum]
+        apply Finset.sum_le_sum
+        intro i _
+        have hh := (cutError_tiny hδ hat (cellProb_le_one (B i) C)).1
+        have hm := mul_le_mul_of_nonneg_left hh (hw i)
+        convert hm using 1; ring
+      · simp only [Finset.mul_sum]
+        apply Finset.sum_le_sum
+        intro i _
+        by_cases hpk : 2/((B i).card:ℝ)≤cellProb (B i) C
+        · rw [ite_eq_left hpk]
+          have hh := (cutError_tiny hδ hat (cellProb_le_one (B i) C)).2
+          have hm := mul_le_mul_of_nonneg_left hh (hw i)
+          convert hm using 1; ring
+        · rw [ite_eq_right hpk]
+          have hp : cellProb (B i) C≤δ := (le_of_lt (lt_of_not_ge hpk)).trans (hk i)
+          simp only [cutError,tent_zero_left hp,tent_zero_left (show cellProb Finset.univ C≤δ by linarith),
+            sub_self,abs_zero,zero_mul,add_zero,mul_zero]
+          norm_num
+    have hx : 0≤(8*ε/δ^2)*pairCell Finset.univ C := mul_nonneg (by positivity) (pairCell_nonneg _ _)
+    have hy : 0≤(1/ε)*varianceCell w B T C := mul_nonneg (by positivity) hv
+    nlinarith
+end WeakPinned.Variance
+
+namespace WeakPinned.Variance
+open scoped BigOperators
+variable {P : Type u3} {I : Type u4} [Fintype P] [Fintype I] [DecidableEq P]
+
+lemma balanced_cell_size (δ : ℝ) (hδ : 0<δ) (hn : 4≤δ*(Fintype.card P:ℝ))
+    (C : Finset P) (ha : δ/2≤cellProb Finset.univ C) : 2≤C.card := by
+  have hn0 : (0:ℝ)<Fintype.card P := by
+    by_contra! h
+    have := mul_nonpos_of_nonneg_of_nonpos hδ.le h
+    linarith
+  simp only [cellProb,Finset.univ_inter,Finset.card_univ] at ha
+  have hh := (le_div_iff₀ hn0).mp ha
+  exact_mod_cast (show (2:ℝ)≤C.card by nlinarith)
+
+lemma balanced_cell_pair (δ : ℝ) (hδ : 0<δ) (hn : 4≤δ*(Fintype.card P:ℝ))
+    (C : Finset P) (ha : δ/2≤cellProb Finset.univ C) :
+    1≤(8/δ^2)*pairCell Finset.univ C := by
+  have hpair := (uniform_pair_comparison C (balanced_cell_size δ hδ hn C ha)).1
+  have han := cellProb_nonneg (Finset.univ : Finset P) C
+  have hh : δ^2≤8*pairCell Finset.univ C := by nlinarith
+  have he : (8/δ^2)*pairCell Finset.univ C=(8*pairCell Finset.univ C)/δ^2 := by ring
+  rw [he]
+  exact (le_div_iff₀ (sq_pos_of_pos hδ)).mpr (by simpa using hh)
+
+/-- Integrated approximation to balanced tree cuts, quantitatively, with no
+selection of a realizing grid and no hidden count of occurrence multiplicities. -/
+lemma weighted_cut_approximation (m : Finset P → ℝ) (hm : ∀ C, 0 ≤ m C)
+    (w : I → ℝ) (B : I → Finset P) (T δ ε : ℝ)
+    (hw : ∀ i, 0≤w i) (hW : (∑ i, w i)≤1) (hδ : 0<δ) (hε : 0<ε)
+    (hn : 4≤δ*(Fintype.card P:ℝ)) (hT : T<δ*(Fintype.card P:ℝ)/2)
+    (hk : ∀ i, 2/((B i).card:ℝ)≤δ) :
+    weightedIntegral m (familyCutError w B δ)≤
+      (8*ε/δ^2)*weightedIntegral m (pairCell Finset.univ)+
+      (1/ε+4/δ^2)*weightedIntegral m (varianceCell w B T) := by
+  have hn0 : (0:ℝ)<Fintype.card P := by
+    by_contra! h
+    have := mul_nonpos_of_nonneg_of_nonpos hδ.le h
+    linarith
+  have hpoint (C : Finset P) := familyCutError_bound w B T δ ε hw hW hδ hε hk C
+    (fun ha => by
+      simp only [cellProb,Finset.univ_inter,Finset.card_univ] at ha
+      have hh := (le_div_iff₀ hn0).mp ha
+      linarith) (balanced_cell_pair δ hδ hn C)
+  have hh := Finset.sum_le_sum (s:=Finset.univ) (fun C _ =>
+    mul_le_mul_of_nonneg_left (hpoint C) (hm C))
+  change weightedIntegral m (familyCutError w B δ)≤_ at hh
+  convert hh using 1
+  simp only [weightedIntegral,mul_add,Finset.sum_add_distrib,Finset.mul_sum]
+  congr 1 <;> apply Finset.sum_congr rfl <;> intro C _ <;> ring
+
+end WeakPinned.Variance
+
+namespace WeakPinned
+open Filter
+open scoped Topology
+
+/-- A Young estimate for every fixed positive ε implies the required little-o
+cut approximation. D is the actual (positive) overlap normalization. -/
+lemma cut_approximation_rate (E V W D : ℕ → ℝ) (δ K : ℝ) (hδ : 0<δ) (hK : 0<K)
+    (hE : ∀ j, 0≤E j) (hD : ∀ j, 0<D j)
+    (hW : ∀ᶠ j in atTop, W j≤K*D j)
+    (hV : Tendsto (fun j => V j/D j) atTop (𝓝 0))
+    (hb : ∀ ε : ℝ, 0<ε → ∀ᶠ j in atTop,
+      E j≤(8*ε/δ^2)*W j+(1/ε+4/δ^2)*V j) :
+    Tendsto (fun j => E j/D j) atTop (𝓝 0) := by
+  apply Metric.tendsto_atTop.mpr
+  intro ζ hζ
+  let ε : ℝ := ζ*δ^2/(32*K)
+  have hε : 0<ε := by dsimp [ε]; positivity
+  have heq : (8*ε/δ^2)*K=ζ/4 := by dsimp [ε]; field_simp; ring
+  have hc : 0<1/ε+4/δ^2 := by positivity
+  have hh : ∀ᶠ j in atTop, |(1/ε+4/δ^2)*(V j/D j)|<ζ/2 := by
+    have hh0 : Tendsto (fun j => (1/ε+4/δ^2)*(V j/D j)) atTop (𝓝 0) := by
+      simpa only [mul_zero] using hV.const_mul (1/ε+4/δ^2)
+    exact (hh0.eventually (Metric.ball_mem_nhds (0:ℝ) (half_pos hζ))).mono
+      (fun j hj => by simpa only [Metric.mem_ball,Real.dist_eq,sub_zero] using hj)
+  have hh' : ∀ᶠ j in atTop, |E j/D j-0|<ζ := by
+    filter_upwards [hb ε hε,hW,hh] with j hj hjw hjv
+    have hn := div_nonneg (hE j) (hD j).le
+    rw [sub_zero,abs_of_nonneg hn]
+    have hp := mul_le_mul_of_nonneg_left hjw (show 0≤8*ε/δ^2 by positivity)
+    have hle : E j/D j≤(8*ε/δ^2)*K+(1/ε+4/δ^2)*(V j/D j) := by
+      apply (div_le_iff₀ (hD j)).mpr
+      have he : ((8*ε/δ^2)*K+(1/ε+4/δ^2)*(V j/D j))*D j =
+        (8*ε/δ^2)*(K*D j)+(1/ε+4/δ^2)*V j := by field_simp [ne_of_gt (hD j)]
+      rw [he]
+      linarith
+    rw [heq] at hle
+    have := (le_abs_self _).trans_lt hjv
+    linarith
+  obtain ⟨N,hN⟩ := eventually_atTop.mp hh'
+  exact ⟨N,fun j hj => by simpa only [Real.dist_eq] using hN j hj⟩
+end WeakPinned
+
+namespace WeakPinned
+open Filter Variance Global
+open scoped Topology BigOperators
+variable {P : ℕ → Type u5} {I : ℕ → Type u6} [∀ j, Fintype (P j)] [∀ j, Fintype (I j)] [∀ j, DecidableEq (P j)]
+
+lemma variance_to_cut_rate (m : ∀ j, Finset (P j) → ℝ) (hm : ∀ j C, 0 ≤ m j C)
+    (w : ∀ j, I j → ℝ) (B : ∀ j, I j → Finset (P j)) (D : ℕ → ℝ) (γ c δ : ℝ)
+    (hD : ∀ j, 0<D j) (hδ : 0<δ) (hγ : 0<γ) (hc : 0<c)
+    (hn : Tendsto (fun j => (Fintype.card (P j):ℝ)) atTop atTop)
+    (hw : ∀ j i, 0≤w j i) (hW : ∀ j, (∑ i, w j i)≤1)
+    (hsize : ∀ j i, (Fintype.card (P j):ℝ)^γ≤((B j i).card:ℝ))
+    (hWD : ∀ j, weightedIntegral (m j) (pairCell Finset.univ)≤D j)
+    (hV : Tendsto (fun j => weightedIntegral (m j)
+      (varianceCell (w j) (B j) ((Fintype.card (P j):ℝ)^(1-c)))/D j) atTop (𝓝 0)) :
+    Tendsto (fun j => weightedIntegral (m j) (familyCutError (w j) (B j) δ)/D j) atTop (𝓝 0) := by
+  let n := fun j => (Fintype.card (P j):ℝ)
+  have hnγ : Tendsto (fun j => n j^γ) atTop atTop := (tendsto_rpow_atTop hγ).comp hn
+  have hnc : Tendsto (fun j => n j^(-c)) atTop (𝓝 0) := (tendsto_rpow_neg_atTop hc).comp hn
+  apply cut_approximation_rate _ _ (fun j => weightedIntegral (m j) (pairCell Finset.univ)) D δ 1 hδ (by norm_num)
+    (fun j => Finset.sum_nonneg (fun C _ => mul_nonneg (hm j C)
+      (Finset.sum_nonneg (fun i _ => mul_nonneg (hw j i) (Cuts.cutError_nonneg _ _ _))))) hD
+    (Eventually.of_forall (fun j => by simpa using hWD j)) hV
+  intro ε hε
+  filter_upwards [hn.eventually (eventually_gt_atTop (4/δ)),
+    hnγ.eventually (eventually_ge_atTop (2/δ)), hnc.eventually (gt_mem_nhds (half_pos hδ))] with j hj hjγ hjc
+  have hn0 : 0<n j := (by positivity : 0<4/δ).trans hj
+  have hnj : 4≤δ*n j := by
+    have hh := (div_lt_iff₀ hδ).mp hj
+    nlinarith
+  have hT : n j^(1-c)<δ*n j/2 := by
+    rw [sub_eq_add_neg,Real.rpow_add hn0,Real.rpow_one]
+    nlinarith
+  have hk (i) : 2/((B j i).card:ℝ)≤δ := by
+    have hbs : 0<((B j i).card:ℝ) := (Real.rpow_pos_of_pos hn0 γ).trans_le (hsize j i)
+    apply (div_le_iff₀ hbs).mpr
+    have hh := (div_le_iff₀ hδ).mp (hjγ.trans (hsize j i))
+    nlinarith
+  exact weighted_cut_approximation (m j) (hm j) (w j) (B j) (n j^(1-c)) δ ε
+    (hw j) (hW j) hδ hε hnj hT hk
+
+end WeakPinned
+
+namespace WeakPinned
+open Filter Variance Global
+open scoped Topology BigOperators
+variable {V : ℕ → Type u7} {I : ℕ → Type u8} {K : ℕ → Type u9} [∀ j, Fintype (V j)] [∀ j, Fintype (I j)] [∀ j, DecidableEq (V j)]
+  [∀ j, Field (K j)] [∀ j, NumberField (K j)]
+variable (G : ∀ j, Core.Fibers (V j) (I j)) (θ γ c : ℝ) (η : ℕ → ℝ)
+  (Z₁ Z₂ : ∀ j, V j → K j) (h₁ : ∀ j, Function.Injective (Z₁ j)) (h₂ : ∀ j, Function.Injective (Z₂ j))
+
+/-- Cut approximation for both sampling families, normalized by W₂+A+1
+    and hence by W in the unbounded regime. -/
+theorem sampling_cut_approximation_rates (δ : ℝ) (hδ : 0<δ) (hθ : 0<θ)
+    (hγ : 0<γ) (hγ1 : γ<1) (hc : 0<c) (hcγ : c<γ)
+    (hn : Tendsto (fun j => (Fintype.card (V j):ℝ)) atTop atTop)
+    (hη : Tendsto η atTop (𝓝 0)) (hη0 : ∀ j, 0≤η j) (hηb : ∀ j, η j≤θ/2)
+    (hG : ∀ j, PreparedGraph (V j) (I j) (G j) θ γ c (η j))
+    (t : ∀ j, I j → K j) (ht : ∀ j e y, y ∈ (G j).targets e →
+      (Z₁ j ((G j).source e)-Z₁ j y)*(Z₂ j ((G j).source e)-Z₂ j y)=t j e) :
+    Tendsto (fun j => weightedIntegral (totalMass (Z₁ j) (Z₂ j) (h₁ j) (h₂ j))
+        (familyCutError (G j).fiberWeight (G j).targets δ)/
+      (overlapScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)+potentialScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)+1)) atTop (𝓝 0) ∧
+    Tendsto (fun j => weightedIntegral (totalMass (Z₁ j) (Z₂ j) (h₁ j) (h₂ j))
+        (familyCutError ((G j).retainedWeight ((Fintype.card (V j):ℝ)^γ))
+          ((G j).retainedSet ((Fintype.card (V j):ℝ)^γ)) δ)/
+      (overlapScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)+potentialScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)+1)) atTop (𝓝 0) := by
+  have hv := sampling_variance_rates G θ γ c η Z₁ Z₂ h₁ h₂ hθ hγ hγ1 hcγ hn hη hη0 hηb hG t ht
+  let m := fun j => totalMass (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)
+  let D := fun j => overlapScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)+potentialScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)+1
+  have hD (j) : 0<D j := by dsimp [D]; linarith [overlapScale_nonneg (Z₁ j) (Z₂ j) (h₁ j) (h₂ j),potentialScale_nonneg (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)]
+  have hWD (j) : weightedIntegral (m j) (pairCell Finset.univ)≤D j := by
+    change overlapScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)≤_
+    dsimp [D]
+    linarith [potentialScale_nonneg (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)]
+  have hpos (j) : 0<(G j).edgeCount := (hG j).edgeCount_pos (by linarith [hηb j])
+  constructor
+  · exact variance_to_cut_rate m (fun j => totalMass_nonneg _ _ _ _)
+      (fun j => (G j).fiberWeight) (fun j => (G j).targets) D γ c δ hD hδ hγ hc hn
+      (fun j => (G j).fiberWeight_nonneg) (fun j => ((G j).sum_fiberWeight (hpos j)).le)
+      (fun j i => ((hG j).fiber i).le) hWD hv.1
+  · apply variance_to_cut_rate m (fun j => totalMass_nonneg _ _ _ _)
+      (fun j => (G j).retainedWeight ((Fintype.card (V j):ℝ)^γ))
+      (fun j => (G j).retainedSet ((Fintype.card (V j):ℝ)^γ)) D γ c δ hD hδ hγ hc hn
+      (fun j => (G j).retainedWeight_nonneg _) (fun j => (G j).sum_retainedWeight _ (hpos j)) _ hWD hv.2
+    intro j y
+    apply (G j).retainedSet_size
+    have hn1 : (1:ℝ)≤Fintype.card (V j) := by exact_mod_cast (by have := (hG j).card_ge; omega : 1≤Fintype.card (V j))
+    simpa only [Real.rpow_one] using Real.rpow_le_rpow_of_exponent_le hn1 hγ1.le
+end WeakPinned
+
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PinnedDistances/CutGeometry.lean b/lean/OAI/Geometry/PinnedDistances/CutGeometry.lean
new file mode 100644
index 0000000000000000000000000000000000000000..27c59a46087893fc759cea4791201f3d576bced7
--- /dev/null
+++ b/lean/OAI/Geometry/PinnedDistances/CutGeometry.lean
@@ -0,0 +1,625 @@
+import OAI.Geometry.PinnedDistances.TreeTransport
+
+namespace OAI
+
+universe u1 u2 u3 u4 u5 u6 u7 u8 u9
+
+section
+noncomputable section
+open NumberField
+open scoped BigOperators
+noncomputable section
+open MeasureTheory
+open scoped Topology
+noncomputable section
+open MeasureTheory
+open scoped Topology ENNReal
+noncomputable section
+open MeasureTheory NumberField
+open scoped BigOperators
+noncomputable section
+open scoped BigOperators
+open MeasureTheory
+noncomputable section
+open Filter
+open scoped Topology
+noncomputable section
+open scoped BigOperators
+noncomputable section
+open Filter
+open scoped Topology
+noncomputable section
+open MeasureTheory
+open scoped BigOperators
+noncomputable section
+noncomputable section
+open scoped BigOperators
+
+namespace WeakPinned.Hierarchy.Depth
+open MeasureTheory
+variable {P : Type u1} [Fintype P] [DecidableEq P]
+variable (d : Depth P)
+
+def allCuts : Finset (Finset P) := by
+  classical
+  exact Finset.univ.filter (fun C => ∃ r : ℝ, C ∈ d.cells r)
+
+lemma mem_allCuts (C : Finset P) : C ∈ d.allCuts ↔ ∃ r : ℝ, C ∈ d.cells r := by
+  classical
+  simp [allCuts]
+
+lemma cells_subset_allCuts (r : ℝ) : d.cells r ⊆ d.allCuts :=
+  fun C hC => (d.mem_allCuts C).mpr ⟨r,hC⟩
+
+lemma allCuts_laminar : Cuts.Laminar d.allCuts := by
+  intro C hC B hB
+  obtain ⟨r,hr⟩ := (d.mem_allCuts C).mp hC
+  obtain ⟨t,ht⟩ := (d.mem_allCuts B).mp hB
+  by_cases hi : Disjoint C B
+  · exact Or.inr (Or.inr hi)
+  · obtain ⟨x,hxC,hxB⟩ := Finset.not_disjoint_iff.mp hi
+    rw [d.cell_of_mem hr hxC,d.cell_of_mem ht hxB]
+    rcases le_total t r with h | h
+    · exact Or.inl (d.cell_mono h x)
+    · exact Or.inr (Or.inl (d.cell_mono h x))
+
+def cutAt (f : Finset P → ℝ) (C : Finset P) (r : ℝ) : ℝ :=
+  if C ∈ d.cells r then f C else 0
+
+def cutLength (f : Finset P → ℝ) (C : Finset P) : ℝ := ∫ r, d.cutAt f C r
+
+lemma measurable_cutAt (f : Finset P → ℝ) (C : Finset P) : Measurable (d.cutAt f C) :=
+  (measurable_of_finite (fun s : Finset (Finset P) => if C ∈ s then f C else 0)).comp d.measurable_cells
+
+lemma cutAt_bound (f : Finset P → ℝ) (C : Finset P) (r : ℝ) : ‖d.cutAt f C r‖ ≤ |f C| := by
+  unfold cutAt
+  split_ifs <;> simp [Real.norm_eq_abs,abs_nonneg]
+
+lemma cutAt_vanish (f : Finset P → ℝ) (hfu : f Finset.univ=0)
+    (hfs : ∀ x, f {x}=0) (C : Finset P) (r : ℝ) (hr : r ∉ Set.Icc (-d.radius) d.radius) :
+    d.cutAt f C r=0 := by
+  unfold cutAt
+  split_ifs with hc
+  · obtain ⟨x,rfl⟩ := (d.mem_cells r C).mp hc
+    rcases not_and_or.mp hr with hlo | hhi
+    · rw [d.cell_of_low (le_of_lt (lt_of_not_ge hlo)),hfu]
+    · rw [d.cell_of_high (lt_of_not_ge hhi),hfs]
+  · rfl
+
+lemma integrable_cutAt (f : Finset P → ℝ) (hfu : f Finset.univ=0)
+    (hfs : ∀ x, f {x}=0) (C : Finset P) : Integrable (d.cutAt f C) := by
+  exact integrable_supported_Icc (d.measurable_cutAt f C) (d.cutAt_bound f C)
+    (d.cutAt_vanish f hfu hfs C)
+
+lemma cutLength_nonneg (f : Finset P → ℝ) (hf : ∀ C, 0 ≤ f C) (C : Finset P) :
+    0 ≤ d.cutLength f C := by
+  apply integral_nonneg
+  intro r
+  unfold cutAt
+  split_ifs
+  · exact hf C
+  · exact le_rfl
+
+lemma cutLength_eq_zero (f : Finset P → ℝ) (C : Finset P) (hC : f C=0) :
+    d.cutLength f C=0 := by simp [cutLength,cutAt,hC]
+
+lemma cutLength_outside (f : Finset P → ℝ) (C : Finset P) (hC : C ∉ d.allCuts) :
+    d.cutLength f C=0 := by
+  have hh (r : ℝ) : C ∉ d.cells r := fun hc => hC ((d.mem_allCuts C).mpr ⟨r,hc⟩)
+  simp [cutLength,cutAt,hh]
+
+lemma integral_cut_sum (f : Finset P → ℝ) (hfu : f Finset.univ=0)
+    (hfs : ∀ x, f {x}=0) (g : Finset P → ℝ) :
+    (∫ r, ∑ C ∈ d.cells r, f C*g C) = ∑ C ∈ d.allCuts, d.cutLength f C*g C := by
+  have he (r : ℝ) : (∑ C ∈ d.cells r, f C*g C) =
+      ∑ C ∈ d.allCuts, d.cutAt f C r*g C := by
+    simp only [cutAt,ite_mul,zero_mul,← Finset.sum_filter]
+    congr 1
+    ext C
+    simp only [Finset.mem_filter]
+    exact ⟨fun hc => ⟨d.cells_subset_allCuts r hc,hc⟩,fun hc => hc.2⟩
+  simp only [he]
+  rw [integral_finsetSum _ (fun C _ => (d.integrable_cutAt f hfu hfs C).mul_const (g C))]
+  exact Finset.sum_congr rfl (fun C _ => by rw [integral_mul_const]; rfl)
+
+lemma symmetric_integral_cut_sum (hn : 2 ≤ Fintype.card P) (f : Finset P → ℝ)
+    (hfs : ∀ C, C.card ≤ 1 → f C=0) (hsym : ∀ C, f Cᶜ=f C) :
+    (∑ C, d.mass C*f C) = ∑ C ∈ d.allCuts, d.cutLength f C := by
+  have hu : f Finset.univ=0 := by simpa using (hsym ∅).trans (hfs ∅ (by simp))
+  rw [← d.integral_sumCells hn f hfs]
+  have he (r : ℝ) : d.sumCells f r=∑ C ∈ d.cells r, f C := by
+    simp only [d.sumCells_formula,hsym,sub_self,Finset.sum_const_zero,add_zero]
+  simp only [he]
+  simpa using d.integral_cut_sum f hu (fun x => hfs {x} (by simp)) (fun _ => 1)
+
+lemma cutLength_balanced (p : P → ℝ) (w : ℝ → ℝ) (η : ℝ)
+    (hws : ∀ u, ¬(η ≤ u ∧ η ≤ 1-u) → w u=0) (C : Finset P)
+    (hC : 0 < d.cutLength (fun B => w (Cuts.mass p B)) C) :
+    η ≤ Cuts.mass p C ∧ η ≤ 1-Cuts.mass p C := by
+  by_contra hn
+  have hz := d.cutLength_eq_zero (fun B => w (Cuts.mass p B)) C (hws _ hn)
+  linarith
+
+lemma actual_cut_fluctuation (p : P → ℝ) (hp0 : ∀ x, 0 ≤ p x) (hp : ∑ x, p x=1)
+    (w : ℝ → ℝ) (hw : ∀ u, 0 ≤ w u) (η : ℝ) (hη : 0 ≤ η)
+    (hws : ∀ u, ¬(η ≤ u ∧ η ≤ 1-u) → w u=0) :
+    η^3*(∑ C ∈ d.allCuts, d.cutLength (fun B => w (Cuts.mass p B)) C) ≤
+      ∑ y, p y*Cuts.doubleMean d.allCuts (d.cutLength (fun B => w (Cuts.mass p B))) p y := by
+  exact Cuts.fluctuation_lower d.allCuts d.allCuts_laminar _
+    (fun C _ => d.cutLength_nonneg _ (fun C => hw _) C) p hp0 hp η hη
+    (fun C _ hC => d.cutLength_balanced p w η hws C hC)
+
+end WeakPinned.Hierarchy.Depth
+
+namespace WeakPinned.Hierarchy.Depth
+open MeasureTheory
+variable {P : Type u2} [Fintype P] [DecidableEq P]
+variable (d : Depth P)
+
+omit [Fintype P] [DecidableEq P] in
+lemma value_eq_of_lt (x y z : P) (hxy : x ≠ y) (hyz : y ≠ z) (hxz : x ≠ z)
+    (hlt : d.value x y < d.value y z) : d.value x z=d.value x y := by
+  have h1 := d.ultra x y z hxy hyz hxz
+  rw [min_eq_left (le_of_lt hlt)] at h1
+  apply le_antisymm _ h1
+  by_contra! hn
+  have h2 := d.ultra x z y hxz hyz.symm hxy
+  rw [d.symm z y] at h2
+  exact (lt_min hn hlt).not_ge h2
+
+def differenceTail (x y : P) (u : ℝ) : Finset P :=
+  Finset.univ.filter (fun z => z=y ∨ (z ≠ x ∧ u ≤ d.value y z-d.value x z))
+
+lemma differenceTail_pos (x y : P) (hxy : x ≠ y) (u : ℝ) (hu : 0 < u) :
+    d.differenceTail x y u=d.cell (d.value x y+u) y := by
+  ext z
+  simp only [differenceTail,Finset.mem_filter,Finset.mem_univ,true_and,mem_cell,share]
+  by_cases hzy : z=y
+  · simp [hzy]
+  by_cases hzx : z=x
+  · subst z
+    have hlt : ¬d.value x y+u ≤ d.value y x := by rw [d.symm y x]; linarith
+    simp [hxy,hxy.symm,hlt]
+  · simp only [hzx,ne_eq,false_or,not_false_eq_true,true_and,eq_comm (a:=y) (b:=z),hzy]
+    constructor
+    · intro h
+      have hlt : d.value x z < d.value z y := by rw [d.symm z y]; linarith
+      have he := d.value_eq_of_lt x z y (Ne.symm hzx) hzy hxy hlt
+      linarith
+    · intro h
+      have hlt : d.value x y < d.value y z := by linarith
+      rw [d.value_eq_of_lt x y z hxy (Ne.symm hzy) (Ne.symm hzx) hlt]
+      linarith
+
+lemma differenceTail_neg_compl (x y : P) (hxy : x ≠ y) (u : ℝ) (hu : u < 0)
+    (hne : ∀ z, u ≠ d.value x y-d.value x z) :
+    (d.differenceTail x y u)ᶜ=d.cell (d.value x y-u) x := by
+  ext z
+  simp only [Finset.mem_compl,differenceTail,Finset.mem_filter,Finset.mem_univ,true_and,mem_cell,share]
+  by_cases hzx : z=x
+  · subst z
+    simp [hxy]
+  by_cases hzy : z=y
+  · subst z
+    have hlt : ¬d.value x y-u ≤ d.value x y := by linarith
+    simp [hxy,hxy.symm,hlt]
+  · simp only [hzy,false_or,ne_eq,not_false_eq_true,true_and,eq_comm (a:=x) (b:=z),hzx,not_le]
+    constructor
+    · intro h
+      have hlt : d.value y z < d.value z x := by rw [d.symm z x]; linarith
+      have he := d.value_eq_of_lt y z x (Ne.symm hzy) hzx hxy.symm hlt
+      rw [d.symm y x] at he
+      linarith
+    · intro h
+      have hlt : d.value y x < d.value x z := by rw [d.symm y x]; linarith
+      have he := d.value_eq_of_lt y x z hxy.symm (Ne.symm hzx) (Ne.symm hzy) hlt
+      rw [d.symm y x] at he
+      have hn := hne z
+      rw [he]
+      rcases lt_or_eq_of_le h with h | h
+      · linarith
+      · exfalso
+        apply hn
+        linarith
+
+lemma mass_compl (p : P → ℝ) (hp : ∑ x, p x=1) (C : Finset P) :
+    Cuts.mass p Cᶜ=1-Cuts.mass p C := by
+  have hh := Finset.sum_add_sum_compl C p
+  simp only [Cuts.mass]
+  rw [hp] at hh
+  linarith
+
+lemma ae_differenceTail_formula (p : P → ℝ) (hp : ∑ x, p x=1) (w : ℝ → ℝ)
+    (hw : ∀ u, w (1-u)=w u) (x y : P) (hxy : x ≠ y) :
+    ∀ᵐ u : ℝ, w (Cuts.mass p (d.differenceTail x y u)) =
+      (if d.value x y < d.value x y+u then w (Cuts.mass p (d.cell (d.value x y+u) y)) else 0)+
+      (if d.value x y < d.value x y-u then w (Cuts.mass p (d.cell (d.value x y-u) x)) else 0) := by
+  have hn : ∀ᵐ u : ℝ, ∀ z : P, u ≠ d.value x y-d.value x z :=
+    ae_all_iff.mpr (fun z => volume.ae_ne _)
+  filter_upwards [hn,volume.ae_ne (0 : ℝ)] with u hne hu
+  rcases lt_or_gt_of_ne hu with hu | hu
+  · have he := d.differenceTail_neg_compl x y hxy u hu hne
+    have hm := mass_compl p hp (d.differenceTail x y u)
+    rw [he] at hm
+    rw [ite_eq_right (by linarith),ite_eq_left (by linarith),zero_add,hm,hw]
+  · rw [d.differenceTail_pos x y hxy u hu,ite_eq_left (by linarith),ite_eq_right (by linarith),add_zero]
+
+end WeakPinned.Hierarchy.Depth
+
+namespace WeakPinned.Hierarchy.Depth
+open MeasureTheory
+variable {P : Type u3} [Fintype P] [DecidableEq P]
+variable (d : Depth P)
+
+def aboveCell (f : Finset P → ℝ) (a : ℝ) (x : P) (r : ℝ) : ℝ :=
+  if a < r then f (d.cell r x) else 0
+
+omit [DecidableEq P] in
+lemma measurable_aboveCell (f : Finset P → ℝ) (a : ℝ) (x : P) :
+    Measurable (d.aboveCell f a x) := by
+  apply Measurable.ite (measurableSet_lt measurable_const measurable_id)
+  · exact (measurable_of_finite f).comp (d.measurable_cell x)
+  · exact measurable_const
+
+omit [DecidableEq P] in
+lemma integrable_aboveCell (f : Finset P → ℝ) (hfs : ∀ x, f {x}=0)
+    (a : ℝ) (ha : -d.radius ≤ a) (x : P) : Integrable (d.aboveCell f a x) := by
+  apply integrable_supported_Icc (d.measurable_aboveCell f a x)
+      (C:=∑ C : Finset P, |f C|) (a:= -d.radius) (b:=d.radius)
+  · intro r
+    unfold aboveCell
+    split_ifs
+    · simpa only [Real.norm_eq_abs] using (Finset.single_le_sum (s:=Finset.univ) (f:=fun C : Finset P => |f C|)
+        (fun _ _ => abs_nonneg _) (Finset.mem_univ (d.cell r x)))
+    · simpa using Finset.sum_nonneg (fun C (_ : C ∈ (Finset.univ : Finset (Finset P))) => abs_nonneg (f C))
+  · intro r hr
+    rcases not_and_or.mp hr with hlo | hhi
+    · have hh : ¬a < r := by have := lt_of_not_ge hlo; linarith
+      simp [aboveCell,hh]
+    · simp [aboveCell,d.cell_of_high (lt_of_not_ge hhi) x,hfs]
+
+lemma sum_cut_separation (f : Finset P → ℝ) (r : ℝ) (x y : P) (hxy : x ≠ y) :
+    (∑ C ∈ d.cells r, f C*|Cuts.ind C x-Cuts.ind C y|) =
+      d.aboveCell f (d.value x y) x r+d.aboveCell f (d.value x y) y r := by
+  have hmem (z : P) (C : Finset P) (hC : C ∈ d.cells r) : z ∈ C ↔ C=d.cell r z :=
+    ⟨fun hz => d.cell_of_mem hC hz,fun he => by rw [he]; exact d.self_mem_cell r z⟩
+  by_cases hr : r ≤ d.value x y
+  · have hce : d.cell r x=d.cell r y := d.cell_eq_of_share (Or.inr hr)
+    have hz : ∑ C ∈ d.cells r, f C*|Cuts.ind C x-Cuts.ind C y|=0 := by
+      apply Finset.sum_eq_zero
+      intro C hC
+      have hm : x ∈ C ↔ y ∈ C := by rw [hmem x C hC,hmem y C hC,hce]
+      simp only [Cuts.ind,hm,sub_self,abs_zero,mul_zero]
+    simp [hz,aboveCell,not_lt.mpr hr]
+  · have hlt : d.value x y < r := lt_of_not_ge hr
+    have hns : ¬d.share r x y := by simpa [share,hxy] using hr
+    have he (C : Finset P) (hC : C ∈ d.cells r) :
+        f C*|Cuts.ind C x-Cuts.ind C y| =
+          (if C=d.cell r x then f (d.cell r x) else 0)+
+          (if C=d.cell r y then f (d.cell r y) else 0) := by
+      by_cases hx : x ∈ C <;> by_cases hy : y ∈ C
+      · have hc := d.cell_of_mem hC hx
+        have hh : d.share r x y := by rwa [hc,d.mem_cell] at hy
+        exact (hns hh).elim
+      · have hc := (hmem x C hC).mp hx
+        have hn : C ≠ d.cell r y := fun h => hy ((hmem y C hC).mpr h)
+        simp only [Cuts.ind,ite_eq_left hx,ite_eq_right hy,sub_zero,abs_one,mul_one,ite_eq_left hc,ite_eq_right hn,add_zero]
+        rw [hc]
+      · have hc := (hmem y C hC).mp hy
+        have hn : C ≠ d.cell r x := fun h => hx ((hmem x C hC).mpr h)
+        simp only [Cuts.ind,ite_eq_right hx,ite_eq_left hy,zero_sub,abs_neg,abs_one,mul_one,ite_eq_left hc,ite_eq_right hn,zero_add]
+        rw [hc]
+      · have hn1 : C ≠ d.cell r x := fun h => hx ((hmem x C hC).mpr h)
+        have hn2 : C ≠ d.cell r y := fun h => hy ((hmem y C hC).mpr h)
+        simp [Cuts.ind,hx,hy,hn1,hn2]
+    rw [Finset.sum_congr rfl he,Finset.sum_add_distrib]
+    simp [d.cell_mem_cells,aboveCell,hlt]
+
+lemma cut_distance_two_integrals (f : Finset P → ℝ) (hfu : f Finset.univ=0)
+    (hfs : ∀ x, f {x}=0) (x y : P) (hxy : x ≠ y) :
+    Cuts.distance d.allCuts (d.cutLength f) x y =
+      (∫ r, d.aboveCell f (d.value x y) x r)+(∫ r, d.aboveCell f (d.value x y) y r) := by
+  rw [Cuts.distance,← d.integral_cut_sum f hfu hfs (fun C => |Cuts.ind C x-Cuts.ind C y|)]
+  simp only [d.sum_cut_separation f _ x y hxy]
+  exact integral_add (d.integrable_aboveCell f hfs _ (abs_le.mp (d.abs_value_le_radius x y)).1 x)
+    (d.integrable_aboveCell f hfs _ (abs_le.mp (d.abs_value_le_radius x y)).1 y)
+
+end WeakPinned.Hierarchy.Depth
+
+namespace WeakPinned.Hierarchy.Depth
+open MeasureTheory
+variable {P : Type u4} [Fintype P] [DecidableEq P]
+variable (d : Depth P)
+
+/-- Exact projection identity, including the formal infinite values at the
+    two endpoints, represented by their tail-set membership. -/
+lemma cut_projection (p : P → ℝ) (hp : ∑ x, p x=1) (w : ℝ → ℝ)
+    (hwu : w 1=0) (hws : ∀ x, w (p x)=0) (hwsym : ∀ u, w (1-u)=w u)
+    (x y : P) (hxy : x ≠ y) :
+    Cuts.distance d.allCuts (d.cutLength (fun C => w (Cuts.mass p C))) x y =
+      ∫ u, w (Cuts.mass p (d.differenceTail x y u)) := by
+  let f : Finset P → ℝ := fun C => w (Cuts.mass p C)
+  have hfu : f Finset.univ=0 := by simpa [f,Cuts.mass,hp] using hwu
+  have hfs : ∀ z, f {z}=0 := by intro z; simpa [f,Cuts.mass] using hws z
+  have hi (z : P) := d.integrable_aboveCell f hfs (d.value x y)
+    (abs_le.mp (d.abs_value_le_radius x y)).1 z
+  have hiY : Integrable (fun u => d.aboveCell f (d.value x y) y (d.value x y+u)) := by
+    exact ((measurePreserving_add_left volume (d.value x y)).integrable_comp (hi y).aestronglyMeasurable).mpr (hi y)
+  have hiX : Integrable (fun u => d.aboveCell f (d.value x y) x (d.value x y-u)) :=
+    (integrable_comp_sub_left _ _).mpr (hi x)
+  rw [d.cut_distance_two_integrals f hfu hfs x y hxy]
+  have he := integral_congr_ae (d.ae_differenceTail_formula p hp w hwsym x y hxy)
+  change (∫ u, w (Cuts.mass p (d.differenceTail x y u))) =
+    ∫ u, d.aboveCell f (d.value x y) y (d.value x y+u)+d.aboveCell f (d.value x y) x (d.value x y-u) at he
+  rw [he,integral_add hiY hiX,integral_add_left_eq_self,integral_sub_left_eq_self]
+  exact add_comm _ _
+
+end WeakPinned.Hierarchy.Depth
+
+namespace WeakPinned.Tail
+open MeasureTheory
+open scoped NNReal
+variable {P : Type u5} [Fintype P]
+
+def step (a u : ℝ) : ℝ := if u ≤ a then 1 else 0
+
+def law (b : ℝ) (p : P → ℝ) (f : P → ℝ) (u : ℝ) : ℝ := b+∑ z, p z*step (f z) u
+
+lemma measurable_step (a : ℝ) : Measurable (step a) := by
+  exact Measurable.ite (measurableSet_le measurable_id measurable_const) measurable_const measurable_const
+
+lemma measurable_law (b : ℝ) (p f : P → ℝ) : Measurable (law b p f) :=
+  measurable_const.add (Finset.measurable_sum _ (fun _z _ => measurable_const.mul (measurable_step _)))
+
+lemma step_difference (a b u : ℝ) : |step a u-step b u| =
+    if min a b < u ∧ u ≤ max a b then 1 else 0 := by
+  unfold step
+  rcases le_total a b with h | h
+  · rw [min_eq_left h,max_eq_right h]
+    by_cases hua : u ≤ a
+    · have hub : u ≤ b := hua.trans h
+      simp [hua,hub,not_lt.mpr hua]
+    · have hau : a < u := lt_of_not_ge hua
+      by_cases hub : u ≤ b <;> simp [hua,hub,hau]
+  · rw [min_eq_right h,max_eq_left h]
+    by_cases hub : u ≤ b
+    · have hua : u ≤ a := hub.trans h
+      simp [hua,hub,not_lt.mpr hub]
+    · have hbu : b < u := lt_of_not_ge hub
+      by_cases hua : u ≤ a <;> simp [hua,hub,hbu]
+
+lemma step_difference_indicator (a b : ℝ) : (fun u => |step a u-step b u|) =
+    (Set.Ioc (min a b) (max a b)).indicator (fun _ : ℝ => 1) := by
+  funext u
+  simp [step_difference,Set.indicator_apply]
+
+lemma integrable_step_difference (a b : ℝ) : Integrable (fun u => |step a u-step b u|) := by
+  rw [step_difference_indicator,integrable_indicator_iff measurableSet_Ioc]
+  exact integrableOn_const (by rw [Real.volume_Ioc]; exact ENNReal.ofReal_ne_top)
+
+lemma integral_step_difference (a b : ℝ) : (∫ u, |step a u-step b u|) = |a-b| := by
+  rw [step_difference_indicator,integral_indicator measurableSet_Ioc]
+  simp only [integral_const,Measure.restrict_apply_univ,smul_eq_mul,mul_one,measureReal_def,Real.volume_Ioc]
+  rw [ENNReal.toReal_ofReal (sub_nonneg.mpr (min_le_max (a:=a) (b:=b)))]
+  rcases le_total a b with hh | hh
+  · simp [max_eq_right hh,min_eq_left hh,abs_of_nonpos (sub_nonpos.mpr hh)]
+  · simp [max_eq_left hh,min_eq_right hh,abs_of_nonneg (sub_nonneg.mpr hh)]
+
+lemma law_difference_bound (b : ℝ) (p f g : P → ℝ) (hp : ∀ z, 0 ≤ p z) (u : ℝ) :
+    |law b p f u-law b p g u| ≤ ∑ z, p z*|step (f z) u-step (g z) u| := by
+  have he : law b p f u-law b p g u=∑ z, p z*(step (f z) u-step (g z) u) := by
+    simp only [law,mul_sub,Finset.sum_sub_distrib]
+    ring
+  rw [he]
+  calc
+    _ ≤ ∑ z, |p z*(step (f z) u-step (g z) u)| := Finset.abs_sum_le_sum_abs _ _
+    _ = _ := by simp only [abs_mul,abs_of_nonneg (hp _)]
+
+lemma lipschitz_tail_bound (b : ℝ) (p f g : P → ℝ) (hp : ∀ z, 0 ≤ p z)
+    (w : ℝ → ℝ) (Λ : ℝ≥0) (hw : LipschitzWith Λ w) :
+    (∫ u, |w (law b p f u)-w (law b p g u)|) ≤ (Λ:ℝ)*∑ z, p z*|f z-g z| := by
+  let h : ℝ → ℝ := fun u => (Λ:ℝ)*∑ z, p z*|step (f z) u-step (g z) u|
+  have hi : Integrable h := (integrable_finsetSum (s:=Finset.univ)
+    (fun z _ => (integrable_step_difference (f z) (g z)).const_mul (p z))).const_mul _
+  have hb (u : ℝ) : |w (law b p f u)-w (law b p g u)| ≤ h u := by
+    have hh := hw.norm_sub_le (law b p f u) (law b p g u)
+    simp only [Real.norm_eq_abs] at hh
+    exact hh.trans (mul_le_mul_of_nonneg_left (law_difference_bound b p f g hp u) Λ.coe_nonneg)
+  have hm : Measurable (fun u => |w (law b p f u)-w (law b p g u)|) :=
+    ((hw.continuous.measurable.comp (measurable_law b p f)).sub
+      (hw.continuous.measurable.comp (measurable_law b p g))).abs
+  have hi' := hi.mono' hm.aestronglyMeasurable (Filter.Eventually.of_forall (fun u => by
+    simpa only [Real.norm_eq_abs,abs_abs] using hb u))
+  calc
+    _ ≤ ∫ u, h u := integral_mono hi' hi hb
+    _ = _ := by
+      simp only [h,integral_const_mul]
+      rw [integral_finsetSum _ (fun z _ => (integrable_step_difference (f z) (g z)).const_mul (p z))]
+      simp only [integral_const_mul,integral_step_difference]
+
+end WeakPinned.Tail
+
+namespace WeakPinned.Hierarchy.Depth
+open MeasureTheory
+variable {P : Type u6} [Fintype P] [DecidableEq P]
+variable (d : Depth P)
+
+omit [DecidableEq P] in
+lemma integrable_cellWeight (f : Finset P → ℝ) (hfu : f Finset.univ=0)
+    (hfs : ∀ x, f {x}=0) (x : P) : Integrable (fun r => f (d.cell r x)) := by
+  apply integrable_supported_Icc ((measurable_of_finite f).comp (d.measurable_cell x))
+    (C:=∑ C : Finset P, |f C|) (a:= -d.radius) (b:=d.radius)
+  · intro r
+    simpa only [Function.comp_def,Real.norm_eq_abs] using (Finset.single_le_sum (s:=Finset.univ) (f:=fun C : Finset P => |f C|)
+      (fun _ _ => abs_nonneg _) (Finset.mem_univ (d.cell r x)))
+  · intro r hr
+    change f (d.cell r x)=0
+    rcases not_and_or.mp hr with hlo | hhi
+    · rw [d.cell_of_low (le_of_lt (lt_of_not_ge hlo)),hfu]
+    · rw [d.cell_of_high (lt_of_not_ge hhi),hfs]
+
+lemma cut_root_projection (f : Finset P → ℝ) (hfu : f Finset.univ=0)
+    (hfs : ∀ x, f {x}=0) (y : P) :
+    Cuts.rootDistance d.allCuts (d.cutLength f) y = ∫ r, f (d.cell r y) := by
+  rw [Cuts.rootDistance,← d.integral_cut_sum f hfu hfs (fun C => Cuts.ind C y)]
+  apply integral_congr_ae
+  apply Filter.Eventually.of_forall
+  intro r
+  have he (C : Finset P) (hC : C ∈ d.cells r) : f C*Cuts.ind C y =
+      if C=d.cell r y then f (d.cell r y) else 0 := by
+    by_cases hy : y ∈ C
+    · have hc := d.cell_of_mem hC hy
+      subst C
+      simp [Cuts.ind,d.self_mem_cell]
+    · have hn : C ≠ d.cell r y := fun h => hy (h ▸ d.self_mem_cell r y)
+      simp [Cuts.ind,hy,hn]
+  change (∑ C ∈ d.cells r, f C*Cuts.ind C y)=f (d.cell r y)
+  rw [Finset.sum_congr rfl he]
+  simp [d.cell_mem_cells]
+
+lemma integrable_differenceTail (p : P → ℝ) (hp : ∑ x, p x=1) (w : ℝ → ℝ)
+    (hws : ∀ x, w (p x)=0) (hwsym : ∀ u, w (1-u)=w u)
+    (x y : P) (hxy : x ≠ y) : Integrable (fun u => w (Cuts.mass p (d.differenceTail x y u))) := by
+  let f : Finset P → ℝ := fun C => w (Cuts.mass p C)
+  have hfs : ∀ z, f {z}=0 := by intro z; simpa [f,Cuts.mass] using hws z
+  have hi (z : P) := d.integrable_aboveCell f hfs (d.value x y)
+    (abs_le.mp (d.abs_value_le_radius x y)).1 z
+  have hiY : Integrable (fun u => d.aboveCell f (d.value x y) y (d.value x y+u)) :=
+    ((measurePreserving_add_left volume (d.value x y)).integrable_comp (hi y).aestronglyMeasurable).mpr (hi y)
+  have hiX : Integrable (fun u => d.aboveCell f (d.value x y) x (d.value x y-u)) :=
+    (integrable_comp_sub_left _ _).mpr (hi x)
+  apply (hiY.add hiX).congr
+  filter_upwards [d.ae_differenceTail_formula p hp w hwsym x y hxy] with u hu
+  exact hu.symm
+
+end WeakPinned.Hierarchy.Depth
+
+namespace WeakPinned.Tail
+open MeasureTheory Hierarchy
+open scoped NNReal
+variable {P : Type u7} [Fintype P] [DecidableEq P]
+
+def offTwo (p : P → ℝ) (x y z : P) : ℝ := if z=x ∨ z=y then 0 else p z
+
+omit [Fintype P] in
+lemma offTwo_nonneg (p : P → ℝ) (hp : ∀ z, 0 ≤ p z) (x y z : P) : 0 ≤ offTwo p x y z := by
+  unfold offTwo
+  split_ifs <;> simp [hp]
+
+lemma mass_as_sum (p : P → ℝ) (C : Finset P) : Cuts.mass p C=∑ z, if z ∈ C then p z else 0 := by
+  simp [Cuts.mass]
+
+lemma differenceTail_law (d : Depth P) (p : P → ℝ) (x y : P) (hxy : x ≠ y) (u : ℝ) :
+    Cuts.mass p (d.differenceTail x y u)=
+      law (p y) (offTwo p x y) (fun z => d.value y z-d.value x z) u := by
+  rw [mass_as_sum]
+  have he (z : P) : (if z ∈ d.differenceTail x y u then p z else 0)=
+      (if z=y then p y else 0)+offTwo p x y z*step (d.value y z-d.value x z) u := by
+    by_cases hy : z=y
+    · subst z; simp [Depth.differenceTail,offTwo]
+    by_cases hx : z=x
+    · subst z; simp [Depth.differenceTail,offTwo,hxy]
+    · simp [Depth.differenceTail,offTwo,step,hx,hy,mul_ite]
+  simp only [he,Finset.sum_add_distrib]
+  simp [law]
+
+lemma rootTail_law (d : Depth P) (p : P → ℝ) (x y : P) (hx : p x=0) (hxy : x ≠ y) (c u : ℝ) :
+    Cuts.mass p (d.cell (u-c) y)=law (p y) (offTwo p x y) (fun z => d.value y z+c) u := by
+  rw [mass_as_sum]
+  have he (z : P) : (if z ∈ d.cell (u-c) y then p z else 0)=
+      (if z=y then p y else 0)+offTwo p x y z*step (d.value y z+c) u := by
+    by_cases hy : z=y
+    · subst z; simp [offTwo,d.self_mem_cell]
+    by_cases hz : z=x
+    · subst z; simp [offTwo,hx,hxy]
+    · have hh : u-c ≤ d.value y z ↔ u ≤ d.value y z+c := by constructor <;> intro hh <;> linarith
+      simp [Depth.mem_cell,Depth.share,offTwo,step,hz,hy,Ne.symm hy,hh,mul_ite]
+  simp only [he,Finset.sum_add_distrib]
+  simp [law]
+
+/-- Quantitative fiber-tree comparison, before inserting the arithmetic
+    raw-depth equation. The y endpoint is included; x has zero fiber mass. -/
+lemma fiber_cut_error (d d' : Depth P) (p : P → ℝ) (hp0 : ∀ z, 0 ≤ p z) (hp : ∑ z, p z=1)
+    (w : ℝ → ℝ) (Λ : ℝ≥0) (hw : LipschitzWith Λ w)
+    (hwu : w 1=0) (hws : ∀ z, w (p z)=0) (hwsym : ∀ u, w (1-u)=w u)
+    (x y : P) (hpx : p x=0) (hxy : x ≠ y) (c : ℝ) :
+    |Cuts.distance d.allCuts (d.cutLength (fun C => w (Cuts.mass p C))) x y-
+      Cuts.rootDistance d'.allCuts (d'.cutLength (fun C => w (Cuts.mass p C))) y| ≤
+      (Λ:ℝ)*∑ z, offTwo p x y z*|d.value y z-d.value x z-(d'.value y z+c)| := by
+  let f : Finset P → ℝ := fun C => w (Cuts.mass p C)
+  have hfu : f Finset.univ=0 := by simpa [f,Cuts.mass,hp] using hwu
+  have hfs : ∀ z, f {z}=0 := by intro z; simpa [f,Cuts.mass] using hws z
+  let a := law (p y) (offTwo p x y) (fun z => d.value y z-d.value x z)
+  let b := law (p y) (offTwo p x y) (fun z => d'.value y z+c)
+  have hia : Integrable (fun u => w (a u)) := by
+    simpa only [differenceTail_law d p x y hxy] using
+      d.integrable_differenceTail p hp w hws hwsym x y hxy
+  have hib0 := d'.integrable_cellWeight f hfu hfs y
+  have hib : Integrable (fun u => w (b u)) := by
+    have hh := ((measurePreserving_add_right volume (-c)).integrable_comp hib0.aestronglyMeasurable).mpr hib0
+    simpa only [Function.comp_def,f,← sub_eq_add_neg,rootTail_law d' p x y hpx hxy c] using hh
+  have he : (∫ u, w (b u))=Cuts.rootDistance d'.allCuts (d'.cutLength f) y := by
+    rw [d'.cut_root_projection f hfu hfs y]
+    have hh := integral_add_right_eq_self (μ:=volume) (fun r => f (d'.cell r y)) (-c)
+    simpa only [f,← sub_eq_add_neg,rootTail_law d' p x y hpx hxy c] using hh
+  rw [d.cut_projection p hp w hwu hws hwsym x y hxy,← he]
+  simp only [differenceTail_law d p x y hxy]
+  change |(∫ u, w (a u))-(∫ u, w (b u))| ≤ _
+  rw [← integral_sub hia hib]
+  calc
+    _ ≤ ∫ u, |w (a u)-w (b u)| := by simpa only [Real.norm_eq_abs] using norm_integral_le_integral_norm (fun u => w (a u)-w (b u))
+    _ ≤ _ := lipschitz_tail_bound (p y) (offTwo p x y) _ _ (offTwo_nonneg p hp0 x y) w Λ hw
+
+end WeakPinned.Tail
+
+namespace WeakPinned.Arithmetic
+variable {K : Type u8} [Field K]
+
+lemma paired_log (v : AbsoluteValue K ℝ) (a b c d z w t : K)
+    (hc : c ≠ a) (hz : z ≠ a) (hcz : c ≠ z) (ht : t ≠ 0)
+    (hcd : (c-a)*(d-b)=t) (hzw : (z-a)*(w-b)=t) :
+    -Real.log (v (w-d)) = -Real.log (v (c-z))+Real.log (v (z-a))-
+      Real.log (v t)+Real.log (v (c-a)) := by
+  rw [paired_difference a b c d z w t hc hz hcd hzw,
+    map_div₀,map_mul,map_mul,
+    Real.log_div (mul_ne_zero (v.pos ht).ne' (v.pos (sub_ne_zero.mpr hcz)).ne')
+      (mul_ne_zero (v.pos (sub_ne_zero.mpr hz)).ne' (v.pos (sub_ne_zero.mpr hc)).ne'),
+    Real.log_mul (v.pos ht).ne' (v.pos (sub_ne_zero.mpr hcz)).ne',
+    Real.log_mul (v.pos (sub_ne_zero.mpr hz)).ne' (v.pos (sub_ne_zero.mpr hc)).ne']
+  ring
+
+lemma absval_sub_symm (v : AbsoluteValue K ℝ) (a b : K) : v (a-b)=v (b-a) := by
+  rw [← neg_sub b a,map_neg_eq_map]
+
+/-- The raw cross-fiber identity at every place. The three points are distinct
+    and the source is outside its fiber. -/
+lemma raw_cross_fiber {P : Type u9} (Z₁ Z₂ : P → K) (h₁ : Function.Injective Z₁)
+    (h₂ : Function.Injective Z₂) (v : AbsoluteValue K ℝ) (x y z : P)
+    (hxy : x ≠ y) (hxz : x ≠ z) (hyz : y ≠ z) (t : K)
+    (hy : (Z₁ x-Z₁ y)*(Z₂ x-Z₂ y)=t)
+    (hz : (Z₁ x-Z₁ z)*(Z₂ x-Z₂ z)=t) :
+    -Real.log (v (Z₂ y-Z₂ z)) = -Real.log (v (Z₁ y-Z₁ z))-
+      (-Real.log (v (Z₁ x-Z₁ z)))-Real.log (v t)-(-Real.log (v (Z₁ x-Z₁ y))) := by
+  have ht : t ≠ 0 := by rw [← hy]; exact mul_ne_zero (sub_ne_zero.mpr (h₁.ne hxy)) (sub_ne_zero.mpr (h₂.ne hxy))
+  have hh := paired_log v (Z₁ x) (Z₂ x) (Z₁ y) (Z₂ y) (Z₁ z) (Z₂ z) t
+    (h₁.ne hxy.symm) (h₁.ne hxz.symm) (h₁.ne hyz) ht
+    (by convert hy using 1; ring) (by convert hz using 1; ring)
+  rw [absval_sub_symm v (Z₂ z) (Z₂ y),absval_sub_symm v (Z₁ z) (Z₁ x),
+    absval_sub_symm v (Z₁ y) (Z₁ x)] at hh
+  linarith
+end WeakPinned.Arithmetic
+
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PinnedDistances/EmpiricalCompactness.lean b/lean/OAI/Geometry/PinnedDistances/EmpiricalCompactness.lean
new file mode 100644
index 0000000000000000000000000000000000000000..3ad2afb53a71277c634d3b9c05d88c47ec9a91c4
--- /dev/null
+++ b/lean/OAI/Geometry/PinnedDistances/EmpiricalCompactness.lean
@@ -0,0 +1,112 @@
+import OAI.Geometry.PinnedDistances.LogBounds
+
+namespace OAI
+
+universe u1 u2 u3 u4
+
+section
+noncomputable section
+open MeasureTheory Filter Set
+open scoped Topology BigOperators ENNReal
+namespace WeakPinned.Bounded
+attribute [local instance] Classical.propDecidable
+local instance : MeasurableSpace Sphere := borel Sphere
+local instance : BorelSpace Sphere := ⟨rfl⟩
+
+variable {V : Type u1} {X : Type u2} [Fintype V] [MeasurableSpace X] [MeasurableSingletonClass X]
+
+def uniformLaw (U : V → X) : Measure X := finiteLaw (fun _ => (Fintype.card V:ℝ)⁻¹) U
+
+omit [MeasurableSingletonClass X] in
+lemma uniformLaw_probability (U : V → X) (hn : 0<Fintype.card V) : IsProbabilityMeasure (uniformLaw U) := by
+  apply finiteLaw_probability
+  · intro; positivity
+  · simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
+    exact mul_inv_cancel₀ (by exact_mod_cast hn.ne')
+
+def uniformProbability (U : V → X) (hn : 0<Fintype.card V) : ProbabilityMeasure X :=
+  ⟨uniformLaw U,uniformLaw_probability U hn⟩
+
+lemma uniformLaw_apply (U : V → X) (s : Set X) [DecidablePred (fun x => U x∈s)] :
+    uniformLaw U s=ENNReal.ofReal (((Finset.univ.filter (fun x => U x∈s)).card:ℝ)/(Fintype.card V:ℝ)) := by
+  simp only [uniformLaw,finiteLaw,Measure.finsetSum_apply,Measure.smul_apply,
+    smul_eq_mul,Measure.dirac_apply,Set.indicator_apply,Pi.one_apply,mul_ite,mul_one,mul_zero]
+  rw [← Finset.sum_filter,Finset.sum_const,nsmul_eq_mul]
+  rw [div_eq_mul_inv,ENNReal.ofReal_mul (Nat.cast_nonneg _),ENNReal.ofReal_natCast]
+
+omit [MeasurableSingletonClass X] in
+lemma uniformLaw_map {Y : Type u3} [MeasurableSpace Y] (U : V → X) (f : X → Y) (hf : Measurable f) :
+    (uniformLaw U).map f=uniformLaw (f ∘ U) := finiteLaw_map _ _ _ hf
+
+variable {P : ℕ → Type u4} [∀ n,Fintype (P n)] [∀ n,DecidableEq (P n)]
+
+lemma normalized_empirical_nonatomic (u : ∀ n,P n → ℂ) (x : ∀ n,P n)
+    (hu : ∀ n,Function.Injective (u n)) (hn : ∀ n,2≤Fintype.card (P n))
+    (hcard : Tendsto (fun n => (Fintype.card (P n):ℝ)) atTop atTop)
+    (C : ℝ) (hC : 0≤C) (hlog : ∀ n,logAverage (u n)≤C)
+    (hx : ∀ n,(∑ y,logCost (u n) (x n) y)≤C*(Fintype.card (P n):ℝ))
+    (β : ProbabilityMeasure Sphere)
+    (ht : Tendsto (fun n => uniformProbability (fun y => ((u n y-u n (x n)):Sphere))
+      (by have := hn n; omega)) atTop (𝓝 β)) :
+    NullSingletonClass (β:Measure Sphere) := by
+  apply weak_limit_nonatomic ht
+  · intro ε hε
+    let R := Real.exp (C/ε+1)
+    have hR : 1<R := by dsimp [R]; rw [Real.one_lt_exp_iff]; positivity
+    refine ⟨R,Eventually.of_forall (fun n => ?_)⟩
+    change uniformLaw (fun y => ((u n y-u n (x n)):Sphere)) _≤_
+    rw [uniformLaw_apply]
+    apply ENNReal.ofReal_le_ofReal
+    have hh := normalized_tail_card (u n) C (x n) (hx n) (hn n) R hR
+    have hs : (Finset.univ.filter (fun y => (((u n y-u n (x n)):Sphere))∈
+        (OnePoint.some '' Metric.closedBall (0:ℂ) R)ᶜ))=
+        Finset.univ.filter (fun y => R<‖u n y-u n (x n)‖) := by
+      ext y
+      simp [Metric.mem_closedBall,dist_eq_norm]
+    change (((Finset.univ.filter (fun y => (((u n y-u n (x n)):Sphere))∈
+      (OnePoint.some '' Metric.closedBall (0:ℂ) R)ᶜ)).card:ℝ)/(Fintype.card (P n):ℝ))≤ε
+    rw [hs]
+    apply hh.trans
+    rw [show Real.log R=C/ε+1 by simp [R]]
+    apply (div_le_iff₀ (by positivity : 0<C/ε+1)).mpr
+    have he : ε*(C/ε+1)=C+ε := by field_simp
+    rw [he]
+    linarith
+  · intro z ε hε
+    let T := 2*C/ε^2+1
+    let r := Real.exp (-T)/2
+    have hT : 0<T := by dsimp [T]; positivity
+    have hr : 0<r := by dsimp [r]; positivity
+    have hr2 : r<1/2 := by dsimp [r]; have hh := Real.exp_lt_one_iff.mpr (neg_neg_of_pos hT); linarith
+    have hsmall : Tendsto (fun n => 1/(Fintype.card (P n):ℝ)) atTop (𝓝 0) := by simpa only [one_div,Function.comp_def] using (tendsto_inv_atTop_zero.comp hcard)
+    have hsmall' : ∀ᶠ n in atTop,1/(Fintype.card (P n):ℝ)≤ε^2/2 :=
+      hsmall.eventually_le_const (by positivity)
+    refine ⟨r,hr,hsmall'.mono (fun n hn' => ?_)⟩
+    change uniformLaw (fun y => ((u n y-u n (x n)):Sphere)) _≤_
+    rw [uniformLaw_apply]
+    apply ENNReal.ofReal_le_ofReal
+    have hs : (Finset.univ.filter (fun y => (((u n y-u n (x n)):Sphere))∈
+        OnePoint.some '' Metric.ball z r))=
+        Finset.univ.filter (fun y => ‖u n y-(z+u n (x n))‖<r) := by
+      ext y
+      simp only [Finset.mem_filter,Finset.mem_univ,true_and,Set.mem_image,OnePoint.coe_eq_coe,exists_eq_right,
+        Metric.mem_ball,dist_eq_norm]
+      ring_nf
+    rw [hs]
+    have hh := small_ball_card (u n) (hu n) C (hn n) (hlog n) (z+u n (x n)) r hr hr2
+    have hl : -Real.log (2*r)=T := by
+      rw [show 2*r=Real.exp (-T) by dsimp [r]; ring,Real.log_exp,neg_neg]
+    rw [hl] at hh
+    have hCT : C/T≤ε^2/2 := by
+      apply (div_le_iff₀ hT).mpr
+      have he : ε^2/2*T=C+ε^2/2 := by dsimp [T]; field_simp
+      rw [he]
+      linarith [sq_nonneg ε]
+    nlinarith
+
+end WeakPinned.Bounded
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PinnedDistances/EnergyLaw.lean b/lean/OAI/Geometry/PinnedDistances/EnergyLaw.lean
new file mode 100644
index 0000000000000000000000000000000000000000..95b7d3e493e8a17ee3b4aa1bfb9f37866ba3c0f9
--- /dev/null
+++ b/lean/OAI/Geometry/PinnedDistances/EnergyLaw.lean
@@ -0,0 +1,102 @@
+import OAI.Geometry.PinnedDistances.MobiusObstruction
+
+namespace OAI
+
+universe u1 u2
+
+section
+noncomputable section
+open MeasureTheory Filter
+open scoped Topology
+namespace WeakPinned.Bounded
+
+abbrev Quad (X : Type u1) := X × X × X × X
+abbrev Expanded (X : Type u2) := Quad X × ((X × X) × (X × X))
+
+lemma no_energy_law (μ ν : Measure ℂ) [IsProbabilityMeasure μ]
+    [IsProbabilityMeasure ν] [NullSingletonClass μ] [NullSingletonClass ν]
+    (Q : Measure (Quad ℂ)) [IsProbabilityMeasure Q]
+    (hind : Q.map (fun q => (q.1,q.2.2))=μ.prod (Q.map (fun q => q.2.2)))
+    (hsupp : ∀ᵐ q ∂ Q,q.2.2.1∈μ.support)
+    (he : ∀ᵐ q ∂ Q,crossEnergy μ q.1 q.2.2.1=targetEnergy ν q.2.2.2) : False := by
+  let τ : Measure (ℂ × ℂ) := Q.map (fun q => q.2.2)
+  have : IsProbabilityMeasure τ := inferInstance
+  have hp := measurableSet_energy_eq μ ν
+  have he' : ∀ᵐ v ∂ μ.prod τ,crossEnergy μ v.1 v.2.1=targetEnergy ν v.2.2 := by
+    rw [← hind]
+    exact (ae_map_iff (by fun_prop) hp).mpr he
+  have hf : ∀ᵐ cd ∂ τ,∀ᵐ a ∂ μ,crossEnergy μ a cd.1=targetEnergy ν cd.2 :=
+    (Measure.ae_ae_comm hp).mp ((Measure.ae_prod_iff_ae_ae hp).mp he')
+  have hs : ∀ᵐ cd ∂ τ,cd.1∈μ.support :=
+    (ae_map_iff (by fun_prop) (μ.isClosed_support.measurableSet.preimage measurable_fst)).mpr hsupp
+  obtain ⟨cd,hc,hcd⟩ := (hs.and hf).exists
+  exact not_ae_energy_eq μ ν cd.1 cd.2 hc hcd
+
+lemma mobius_of_product (a b c d z v : ℂ) (hz : z≠a)
+    (hp : (z-a)*(v-b)=(c-a)*(d-b)) : v=mobius a b c d z := by
+  unfold mobius
+  have hh : v-b=(c-a)*(d-b)/(z-a) := (eq_div_iff (sub_ne_zero.mpr hz)).mpr (by simpa [mul_comm] using hp)
+  linear_combination hh
+
+lemma expanded_kernel_eq (μ ν : Measure ℂ) [IsProbabilityMeasure μ]
+    [IsProbabilityMeasure ν] [NullSingletonClass μ]
+    (Q : Measure (Quad ℂ)) (R : Measure (Expanded ℂ))
+    [IsProbabilityMeasure Q] [IsProbabilityMeasure R]
+    (he : R.map Prod.fst=Q)
+    (hm : R.map (fun r => (r.1,(r.2.1.1,r.2.2.1)))=Q.prod (μ.prod μ))
+    (hbd : ∀ᵐ q ∂ Q,q.2.1≠q.2.2.2)
+    (hz : ∀ᵐ r ∂ R,(r.2.1.1-r.1.1)*(r.2.1.2-r.1.2.1)=
+      (r.1.2.2.1-r.1.1)*(r.1.2.2.2-r.1.2.1))
+    (hw : ∀ᵐ r ∂ R,(r.2.2.1-r.1.1)*(r.2.2.2-r.1.2.1)=
+      (r.1.2.2.1-r.1.1)*(r.1.2.2.2-r.1.2.1)) :
+    ∀ᵐ r ∂ R,crossKernel r.1.1 r.1.2.2.1 (r.2.1.1,r.2.2.1)=
+      targetKernel r.1.2.2.2 (r.2.1.2,r.2.2.2) := by
+  have hpred : MeasurableSet {p : Quad ℂ × (ℂ × ℂ) |
+      p.2.1≠p.1.1 ∧ p.2.2≠p.1.1 ∧ p.2.2≠p.1.2.2.1} := by
+    measurability
+  have hpair : ∀ᵐ p ∂ Q.prod (μ.prod μ),
+      p.2.1≠p.1.1 ∧ p.2.2≠p.1.1 ∧ p.2.2≠p.1.2.2.1 := by
+    rw [Measure.ae_prod_iff_ae_ae hpred]
+    exact Eventually.of_forall fun e => by
+      filter_upwards [ae_pair_ne μ e.1,ae_pair_ne μ e.2.2.1] with zw ha hc
+      exact ⟨ha.1,ha.2,hc.2⟩
+  rw [← hm] at hpair
+  have hp := (ae_map_iff (by fun_prop) hpred).mp hpair
+  have hd : ∀ᵐ r ∂ R,r.1.2.1≠r.1.2.2.2 := by
+    rw [← he] at hbd
+    exact (ae_map_iff measurable_fst.aemeasurable (by measurability)).mp hbd
+  filter_upwards [hp,hd,hz,hw] with r hp hd hz hw
+  have hz' := mobius_of_product r.1.1 r.1.2.1 r.1.2.2.1 r.1.2.2.2 _ _ hp.1 hz
+  have hw' := mobius_of_product r.1.1 r.1.2.1 r.1.2.2.1 r.1.2.2.2 _ _ hp.2.1 hw
+  have hh := (kernel_mobius r.1.1 r.1.2.1 r.1.2.2.1 r.1.2.2.2
+    (r.2.1.1,r.2.2.1) hd hp.1 hp.2.1 hp.2.2).symm
+  simpa only [← hz',← hw'] using hh
+
+lemma expanded_energy_eq (μ ν : Measure ℂ) [IsProbabilityMeasure μ]
+    [IsProbabilityMeasure ν] (Q : Measure (Quad ℂ)) (R : Measure (Expanded ℂ))
+    [IsProbabilityMeasure Q] [IsProbabilityMeasure R]
+    (hm₁ : R.map (fun r => (r.1,(r.2.1.1,r.2.2.1)))=Q.prod (μ.prod μ))
+    (hm₂ : R.map (fun r => (r.1,(r.2.1.2,r.2.2.2)))=Q.prod (ν.prod ν))
+    (hk : ∀ᵐ r ∂ R,crossKernel r.1.1 r.1.2.2.1 (r.2.1.1,r.2.2.1)=
+      targetKernel r.1.2.2.2 (r.2.1.2,r.2.2.2)) :
+    ∀ᵐ q ∂ Q,crossEnergy μ q.1 q.2.2.1=targetEnergy ν q.2.2.2 := by
+  apply conditional_integral_eq R Q (μ.prod μ) (ν.prod ν) Prod.fst
+    (fun r => (r.2.1.1,r.2.2.1)) (fun r => (r.2.1.2,r.2.2.2))
+    (by fun_prop) (by fun_prop) (by fun_prop) hm₁ hm₂
+    (fun p => crossKernel p.1.1 p.1.2.2.1 p.2)
+    (fun p => targetKernel p.1.2.2.2 p.2) ?_ ?_ ?_ ?_ hk
+  · unfold crossKernel; fun_prop
+  · unfold targetKernel; fun_prop
+  · intro p
+    rw [Real.norm_eq_abs,abs_of_nonneg (crossKernel_nonneg _ _ _)]
+    exact crossKernel_le_one _ _ _
+  · intro p
+    rw [Real.norm_eq_abs,abs_of_nonneg (targetKernel_nonneg _ _)]
+    exact targetKernel_le_one _ _
+
+end WeakPinned.Bounded
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PinnedDistances/ExtremalCore.lean b/lean/OAI/Geometry/PinnedDistances/ExtremalCore.lean
new file mode 100644
index 0000000000000000000000000000000000000000..b59c9d28ca245eb32be66e09b49bbdede2743b45
--- /dev/null
+++ b/lean/OAI/Geometry/PinnedDistances/ExtremalCore.lean
@@ -0,0 +1,647 @@
+import OAI.Geometry.PinnedDistances.ExtremalFractions
+
+namespace OAI
+
+universe u1 u2 u3 u4 u5 u6
+
+section
+noncomputable section
+open NumberField
+open scoped BigOperators
+noncomputable section
+open MeasureTheory
+open scoped Topology
+noncomputable section
+open MeasureTheory
+open scoped Topology ENNReal
+noncomputable section
+open MeasureTheory NumberField
+open scoped BigOperators
+noncomputable section
+open scoped BigOperators
+open MeasureTheory
+noncomputable section
+open Filter
+open scoped Topology
+
+/- Finite extremal core construction. The truncated-fiber score gives the same
+   core as pruning while avoiding an arbitrary pruning order. It uses the
+   original fibers without a generic-position hypothesis. -/
+
+noncomputable section
+open scoped BigOperators
+namespace WeakPinned.Core
+variable {V : Type u1} {I : Type u2} [Fintype V] [Fintype I] [DecidableEq V]
+
+structure Fibers (V : Type u3) (I : Type u4) [DecidableEq V] where
+  source : I → V
+  targets : I → Finset V
+  no_loop : ∀ e, source e ∉ targets e
+  disjoint : ∀ e f, e ≠ f → source e=source f → Disjoint (targets e) (targets f)
+
+namespace Fibers
+variable (G : Fibers V I)
+
+def size (C : Finset V) (e : I) : ℕ := (G.targets e ∩ C).card
+
+def contribution (K : ℝ) (C : Finset V) (e : I) : ℝ :=
+  if G.source e ∈ C then max ((G.size C e : ℝ)-K) 0 else 0
+
+def score (K : ℝ) (C : Finset V) : ℝ := ∑ e, G.contribution K C e
+
+def active (K : ℝ) (C : Finset V) (e : I) : Prop :=
+  G.source e ∈ C ∧ K < (G.size C e : ℝ)
+
+instance (K : ℝ) (C : Finset V) (e : I) : Decidable (G.active K C e) := Classical.propDecidable _
+
+def edge (K : ℝ) (C : Finset V) (x y : V) : Prop :=
+  ∃ e, G.active K C e ∧ G.source e=x ∧ y ∈ G.targets e ∩ C
+
+instance (K : ℝ) (C : Finset V) (x y : V) : Decidable (G.edge K C x y) := Classical.propDecidable _
+
+def degree (K : ℝ) (C : Finset V) (v : V) : ℝ :=
+  (Finset.univ.filter (G.edge K C v)).card + (Finset.univ.filter (fun x => G.edge K C x v)).card
+
+def fiberDegree (K : ℝ) (C : Finset V) (v : V) (e : I) : ℝ :=
+  if G.active K C e then
+    (if G.source e=v then (G.size C e : ℝ) else 0) +
+    (if v ∈ G.targets e ∩ C then 1 else 0)
+  else 0
+
+omit [Fintype V] [Fintype I] in
+lemma size_mono {C D : Finset V} (h : C ⊆ D) (e : I) : G.size C e ≤ G.size D e :=
+  Finset.card_le_card (Finset.inter_subset_inter_left h)
+
+omit [Fintype V] [Fintype I] in
+lemma size_erase (C : Finset V) (v : V) (e : I) :
+    G.size (C.erase v) e = G.size C e - if v ∈ G.targets e ∩ C then 1 else 0 := by
+  have he : G.targets e ∩ C.erase v=(G.targets e ∩ C).erase v := by
+    ext z
+    simp only [Finset.mem_inter,Finset.mem_erase]
+    tauto
+  change (G.targets e ∩ C.erase v).card=_
+  rw [he]
+  by_cases hv : v ∈ G.targets e ∩ C
+  · simp [size,hv,Finset.card_erase_of_mem hv]
+  · simp [size,hv]
+
+omit [Fintype V] [Fintype I] in
+lemma contribution_nonneg (K : ℝ) (C : Finset V) (e : I) : 0 ≤ G.contribution K C e := by
+  unfold contribution
+  split_ifs <;> positivity
+
+omit [Fintype V] [Fintype I] in
+lemma contribution_mono (K : ℝ) {C D : Finset V} (h : C ⊆ D) (e : I) :
+    G.contribution K C e ≤ G.contribution K D e := by
+  unfold contribution
+  by_cases hs : G.source e ∈ C
+  · simp only [ite_eq_left hs,ite_eq_left (h hs)]
+    exact max_le_max (sub_le_sub_right (by exact_mod_cast G.size_mono h e) K) le_rfl
+  · simp only [ite_eq_right hs]
+    split_ifs <;> positivity
+
+omit [Fintype V] [Fintype I] in
+lemma contribution_erase_loss (K : ℝ) (hK : 0 ≤ K) (C : Finset V) (v : V) (e : I) :
+    G.contribution K C e-G.contribution K (C.erase v) e ≤ G.fiberDegree K C v e := by
+  by_cases ha : G.active K C e
+  · obtain ⟨hs,hk⟩ := ha
+    rw [fiberDegree,ite_eq_left ⟨hs,hk⟩]
+    by_cases hv : G.source e=v
+    · have hno : v ∉ G.targets e ∩ C := by
+        intro hh
+        exact G.no_loop e (hv ▸ (Finset.mem_inter.mp hh).1)
+      have he1 : G.contribution K C e=(G.size C e : ℝ)-K := by
+        simp [contribution,hs,max_eq_left (le_of_lt (sub_pos.mpr hk))]
+      have he0 : G.contribution K (C.erase v) e=0 := by simp [contribution,hv]
+      rw [he1,he0,ite_eq_left hv,ite_eq_right hno]
+      linarith
+    · have hse : G.source e ∈ C.erase v := Finset.mem_erase.mpr ⟨hv,hs⟩
+      simp only [contribution,ite_eq_left hs,ite_eq_left hse,ite_eq_right hv,zero_add,
+        max_eq_left (le_of_lt (sub_pos.mpr hk))]
+      by_cases hm : v ∈ G.targets e ∩ C
+      · rw [ite_eq_left hm,G.size_erase,ite_eq_left hm]
+        have hpos : 1 ≤ G.size C e := Finset.one_le_card.mpr ⟨v,hm⟩
+        rw [Nat.cast_sub hpos]
+        have hh := le_max_left ((G.size C e : ℝ)-1-K) 0
+        norm_num only [Nat.cast_one]
+        linarith
+      · rw [ite_eq_right hm,G.size_erase,ite_eq_right hm,Nat.sub_zero,max_eq_left (le_of_lt (sub_pos.mpr hk))]
+        simp
+  · have hz : G.contribution K C e=0 := by
+      unfold contribution
+      by_cases hs : G.source e ∈ C
+      · rw [ite_eq_left hs,max_eq_right]
+        exact sub_nonpos.mpr (le_of_not_gt (fun hk => ha ⟨hs,hk⟩))
+      · rw [ite_eq_right hs]
+    rw [hz,fiberDegree,ite_eq_right ha,zero_sub]
+    exact neg_nonpos.mpr (G.contribution_nonneg K _ e)
+
+omit [Fintype V] [Fintype I] in
+lemma edge_no_loop (K : ℝ) (C : Finset V) (v : V) : ¬G.edge K C v v := by
+  rintro ⟨e,_,he,hv⟩
+  exact G.no_loop e (he ▸ (Finset.mem_inter.mp hv).1)
+
+omit [Fintype V] [Fintype I] in
+lemma edge_unique (_K : ℝ) (_C : Finset V) (x y : V) {e f : I}
+    (he : G.source e=x) (hf : G.source f=x) (hye : y ∈ G.targets e) (hyf : y ∈ G.targets f) : e=f := by
+  by_contra hn
+  exact Finset.disjoint_left.mp (G.disjoint e f hn (he.trans hf.symm)) hye hyf
+
+lemma sum_fiberDegree (K : ℝ) (C : Finset V) (v : V) :
+    (∑ e, G.fiberDegree K C v e)=G.degree K C v := by
+  classical
+  have hout (e : I) : (if G.active K C e then (if G.source e=v then (G.size C e : ℝ) else 0) else 0)=
+      ∑ y : V, if G.active K C e ∧ G.source e=v ∧ y ∈ G.targets e ∩ C then (1:ℝ) else 0 := by
+    by_cases ha : G.active K C e <;> by_cases hs : G.source e=v <;>
+      simp [ha,hs,size]
+    congr 1
+    ext y
+    simp
+  have hin (e : I) : (if G.active K C e then (if v ∈ G.targets e ∩ C then (1:ℝ) else 0) else 0)=
+      ∑ x : V, if G.active K C e ∧ G.source e=x ∧ v ∈ G.targets e ∩ C then (1:ℝ) else 0 := by
+    by_cases ha : G.active K C e <;> by_cases hv : v ∈ G.targets e ∩ C <;> simp [ha,hv]
+  have hone (x y : V) : (∑ e, if G.active K C e ∧ G.source e=x ∧ y ∈ G.targets e ∩ C then (1:ℝ) else 0)=
+      if G.edge K C x y then 1 else 0 := by
+    by_cases hh : G.edge K C x y
+    · obtain ⟨e,he,hs,hy⟩ := hh
+      rw [ite_eq_left ⟨e,he,hs,hy⟩]
+      rw [Finset.sum_eq_single e]
+      · simp [he,hs,hy]
+      · intro f _ hfe
+        rw [ite_eq_right]
+        rintro ⟨_,hfs,hfy⟩
+        exact hfe (G.edge_unique K C x y hfs hs (Finset.mem_inter.mp hfy).1 (Finset.mem_inter.mp hy).1)
+      · simp
+    · have hn : ∀ e, ¬(G.active K C e ∧ G.source e=x ∧ y ∈ G.targets e ∩ C) :=
+        fun e he => hh ⟨e,he⟩
+      simp only [ite_eq_right (hn _),Finset.sum_const_zero,ite_eq_right hh]
+  have hs (e : I) : G.fiberDegree K C v e =
+      (if G.active K C e then (if G.source e=v then (G.size C e : ℝ) else 0) else 0)+
+      (if G.active K C e then (if v ∈ G.targets e ∩ C then (1:ℝ) else 0) else 0) := by
+    unfold fiberDegree
+    split_ifs <;> ring
+  simp_rw [hs,hout,hin]
+  rw [Finset.sum_add_distrib,Finset.sum_comm,Finset.sum_comm (s:=Finset.univ) (t:=Finset.univ) (f:=fun e x => if G.active K C e ∧ G.source e=x ∧ v ∈ G.targets e ∩ C then (1:ℝ) else 0)]
+  simp only [hone,degree,← Finset.sum_filter,Finset.sum_const,nsmul_eq_mul,mul_one]
+
+lemma score_erase_loss (K : ℝ) (hK : 0 ≤ K) (C : Finset V) (v : V) :
+    G.score K C-G.score K (C.erase v) ≤ G.degree K C v := by
+  rw [score,score,← Finset.sum_sub_distrib,← G.sum_fiberDegree K C v]
+  exact Finset.sum_le_sum (fun e _ => G.contribution_erase_loss K hK C v e)
+
+def penalty (a : ℝ) (C : Finset V) : ℝ := a*(C.card : ℝ)*((C.card : ℝ)-1)
+
+def objective (K a : ℝ) (C : Finset V) : ℝ := G.score K C-penalty a C
+
+/-- Constants here are finite numeric bounds on the original extremal scores.
+    The proof is a maximization on subsets, not an assumption of a core. -/
+theorem exists_core (K a U : ℝ) (hK : 0 ≤ K) (m₀ : ℕ) (hm : m₀ < Fintype.card V)
+    (hboundary : ∀ C : Finset V, C.card=m₀ → G.score K C ≤ U*(m₀ : ℝ)*((m₀ : ℝ)-1))
+    (hgap : (U-a)*(m₀ : ℝ)*((m₀ : ℝ)-1) < G.objective K a Finset.univ) :
+    ∃ C : Finset V, m₀ < C.card ∧ G.objective K a Finset.univ ≤ G.objective K a C ∧
+      ∀ v ∈ C, 2*a*((C.card : ℝ)-1) ≤ G.degree K C v := by
+  classical
+  let S : Finset (Finset V) := Finset.univ.filter (fun C => m₀ ≤ C.card)
+  have hu : (Finset.univ : Finset V) ∈ S := by simp [S,le_of_lt hm]
+  obtain ⟨C,hC,hmax⟩ := Finset.exists_max_image S (G.objective K a) ⟨_,hu⟩
+  have hmc : m₀ ≤ C.card := (Finset.mem_filter.mp hC).2
+  have hge := hmax _ hu
+  have hlt : m₀ < C.card := by
+    by_contra hh
+    have he : C.card=m₀ := le_antisymm (le_of_not_gt hh) hmc
+    have hb := hboundary C he
+    unfold objective penalty at hge hgap
+    rw [he] at hge
+    nlinarith
+  refine ⟨C,hlt,hge,?_⟩
+  intro v hv
+  have hce : C.erase v ∈ S := by
+    simp only [S,Finset.mem_filter,Finset.mem_univ,true_and,Finset.card_erase_of_mem hv]
+    omega
+  have hh := hmax _ hce
+  have hl := G.score_erase_loss K hK C v
+  have hn : 1 ≤ C.card := Finset.one_le_card.mpr ⟨v,hv⟩
+  unfold objective penalty at hh
+  rw [Finset.card_erase_of_mem hv,Nat.cast_sub hn] at hh
+  norm_num at hh
+  nlinarith
+
+end Fibers
+end WeakPinned.Core
+
+namespace WeakPinned.Core.Fibers
+open scoped BigOperators
+variable {V : Type u5} {I : Type u6} [Fintype V] [Fintype I] [DecidableEq V]
+variable (G : Fibers V I)
+
+def edgeSet (K : ℝ) (C : Finset V) : Finset (V × V) :=
+  Finset.univ.filter (fun xy => G.edge K C xy.1 xy.2)
+
+omit [Fintype V] in
+lemma edge_indicator (K : ℝ) (C : Finset V) (x y : V) :
+    (∑ e, if G.active K C e ∧ G.source e=x ∧ y ∈ G.targets e ∩ C then (1:ℝ) else 0)=
+      if G.edge K C x y then 1 else 0 := by
+  classical
+  by_cases hh : G.edge K C x y
+  · obtain ⟨e,he,hs,hy⟩ := hh
+    rw [ite_eq_left ⟨e,he,hs,hy⟩,Finset.sum_eq_single e]
+    · simp [he,hs,hy]
+    · intro f _ hfe
+      rw [ite_eq_right]
+      rintro ⟨_,hfs,hfy⟩
+      exact hfe (G.edge_unique K C x y hfs hs (Finset.mem_inter.mp hfy).1 (Finset.mem_inter.mp hy).1)
+    · simp
+  · have hn : ∀ e, ¬(G.active K C e ∧ G.source e=x ∧ y ∈ G.targets e ∩ C) :=
+      fun e he => hh ⟨e,he⟩
+    simp only [ite_eq_right (hn _),Finset.sum_const_zero,ite_eq_right hh]
+
+lemma edgeSet_card (K : ℝ) (C : Finset V) :
+    ((G.edgeSet K C).card : ℝ)=∑ e, if G.active K C e then (G.size C e : ℝ) else 0 := by
+  classical
+  have he (e : I) : (∑ xy : V × V, if G.active K C e ∧ G.source e=xy.1 ∧ xy.2 ∈ G.targets e ∩ C then (1:ℝ) else 0)=
+      if G.active K C e then (G.size C e : ℝ) else 0 := by
+    by_cases ha : G.active K C e
+    · simp only [ha,true_and,Fintype.sum_prod_type]
+      rw [Finset.sum_eq_single (G.source e)]
+      · simp only [true_and]
+        rw [← Finset.sum_filter]
+        simp only [Finset.sum_const,nsmul_eq_mul,mul_one]
+        change ((Finset.univ.filter (fun y => y ∈ G.targets e ∩ C)).card : ℝ) = (G.size C e : ℝ)
+        simp only [size,Finset.mem_inter]
+        congr 2
+        ext y
+        simp
+      · intro x _ hx
+        simp [Ne.symm hx]
+      · simp
+    · simp [ha]
+  change ((Finset.univ.filter (fun xy : V × V => G.edge K C xy.1 xy.2)).card : ℝ)=_
+  rw [← Finset.sum_boole]
+  simp_rw [← G.edge_indicator K C]
+  rw [Finset.sum_comm]
+  exact Finset.sum_congr rfl (fun e _ => he e)
+
+omit [Fintype V] [Fintype I] in
+lemma contribution_formula (K : ℝ) (C : Finset V) (e : I) :
+    G.contribution K C e = if G.active K C e then (G.size C e : ℝ)-K else 0 := by
+  unfold contribution active
+  by_cases hs : G.source e ∈ C
+  · by_cases hk : K < (G.size C e : ℝ)
+    · simp [hs,hk,max_eq_left (le_of_lt (sub_pos.mpr hk))]
+    · simp [hs,hk,max_eq_right (sub_nonpos.mpr (le_of_not_gt hk))]
+  · simp [hs]
+
+lemma score_le_edgeSet (K : ℝ) (hK : 0 ≤ K) (C : Finset V) :
+    G.score K C ≤ (G.edgeSet K C).card := by
+  rw [score,G.edgeSet_card]
+  apply Finset.sum_le_sum
+  intro e _
+  rw [G.contribution_formula]
+  split_ifs <;> linarith
+
+lemma score_univ_lower (K : ℝ) :
+    (∑ e, (G.targets e).card : ℝ)-K*(Fintype.card I : ℝ) ≤ G.score K Finset.univ := by
+  have hh : ∀ e, (G.targets e).card-K ≤ G.contribution K Finset.univ e := by
+    intro e
+    simp [contribution,size]
+  have hs := Finset.sum_le_sum (s:=Finset.univ) (fun e _ => hh e)
+  simpa [Finset.sum_sub_distrib,score,mul_comm] using hs
+
+lemma sum_sizes_le_edgeSet_univ (K : ℝ) (h : ∀ e, K < ((G.targets e).card : ℝ)) :
+    (∑ e, ((G.targets e).card : ℝ))=(G.edgeSet K Finset.univ).card := by
+  rw [G.edgeSet_card]
+  apply Finset.sum_congr rfl
+  intro e _
+  have ha : G.active K Finset.univ e := by simp [active,size,h e]
+  simp [ha,size]
+
+lemma number_fibers (L K : ℝ) (hKL : K < L)
+    (hL : ∀ e, L ≤ ((G.targets e).card : ℝ)) :
+    (Fintype.card I : ℝ)*L ≤ (G.edgeSet K Finset.univ).card := by
+  rw [← G.sum_sizes_le_edgeSet_univ K (fun e => hKL.trans_le (hL e))]
+  simpa using Finset.sum_le_sum (s:=Finset.univ) (fun e _ => hL e)
+
+end WeakPinned.Core.Fibers
+
+namespace WeakPinned
+open WeakPinned.Core
+
+def fiberIndices (P : Finset Plane) (s : ℝ) : Finset (Plane × ℝ) :=
+  (richPairs P s).image (fun xy => (xy.1,dist xy.2 xy.1))
+
+lemma index_source_mem (P : Finset Plane) (s : ℝ) (e : fiberIndices P s) : e.val.1 ∈ P := by
+  obtain ⟨xy,hxy,he⟩ := Finset.mem_image.mp e.property
+  obtain ⟨hP,_,_⟩ := Finset.mem_filter.mp hxy
+  rw [← he]
+  exact (Finset.mem_product.mp hP).1
+
+lemma index_radius_pos (P : Finset Plane) (s : ℝ) (e : fiberIndices P s) : 0 < e.val.2 := by
+  obtain ⟨xy,hxy,he⟩ := Finset.mem_image.mp e.property
+  obtain ⟨_,hne,_⟩ := Finset.mem_filter.mp hxy
+  rw [← he]
+  exact dist_pos.mpr hne.symm
+
+def planarFibers (P : Finset Plane) (s : ℝ) : Core.Fibers P (fiberIndices P s) where
+  source e := ⟨e.val.1,index_source_mem P s e⟩
+  targets e := Finset.univ.filter (fun y : P => dist y.val e.val.1=e.val.2)
+  no_loop e := by
+    simp only [Finset.mem_filter,Finset.mem_univ,true_and,dist_self]
+    exact (index_radius_pos P s e).ne
+  disjoint e f hef hsrc := by
+    apply Finset.disjoint_left.mpr
+    intro y hye hyf
+    have hs : e.val.1=f.val.1 := congrArg Subtype.val hsrc
+    have he := (Finset.mem_filter.mp hye).2
+    have hf := (Finset.mem_filter.mp hyf).2
+    apply hef
+    apply Subtype.ext
+    apply Prod.ext hs
+    rw [← he,← hf,hs]
+
+lemma planar_target_iff (P : Finset Plane) (s : ℝ) (e : fiberIndices P s) (y : P) :
+    y ∈ (planarFibers P s).targets e ↔ dist y.val e.val.1=e.val.2 := by
+  simp [planarFibers]
+
+lemma planar_fiber_card (P : Finset Plane) (s : ℝ) (e : fiberIndices P s) :
+    (P.card : ℝ)^s ≤ ((planarFibers P s).targets e).card := by
+  obtain ⟨xy,hxy,he⟩ := Finset.mem_image.mp e.property
+  obtain ⟨hP,hne,hk⟩ := Finset.mem_filter.mp hxy
+  have hsrc : e.val.1=xy.1 := (congrArg Prod.fst he).symm
+  have hrad : e.val.2=dist xy.2 xy.1 := (congrArg Prod.snd he).symm
+  have hi : Function.Injective (fun y : P => y.val) := Subtype.val_injective
+  have heq : ((planarFibers P s).targets e).image (fun y : P => y.val)=distanceFiber P xy.1 xy.2 := by
+    ext y
+    simp only [Finset.mem_image,distanceFiber,Finset.mem_filter,Finset.mem_erase]
+    constructor
+    · rintro ⟨z,hz,rfl⟩
+      have hz' := (planar_target_iff P s e z).mp hz
+      rw [hsrc,hrad] at hz'
+      refine ⟨⟨?_,z.property⟩,hz'⟩
+      intro hh
+      rw [hh,dist_self] at hz'
+      exact (dist_pos.mpr hne.symm).ne hz'
+    · rintro ⟨⟨_,hy⟩,hd⟩
+      refine ⟨⟨y,hy⟩,?_,rfl⟩
+      apply (planar_target_iff _ _ _ _).mpr
+      simpa [hsrc,hrad] using hd
+  have hc : ((planarFibers P s).targets e).card = k P xy.1 xy.2 := by
+    rw [← Finset.card_image_of_injective _ hi,heq]
+    rfl
+  rw [hc]
+  exact hk
+
+/-- Return from a subset of labelled vertices to the original plane, retaining
+    all distances and its exact cardinality. -/
+def planeSubset (P : Finset Plane) (C : Finset P) : Finset Plane := C.image Subtype.val
+
+lemma planeSubset_card (P : Finset Plane) (C : Finset P) : (planeSubset P C).card=C.card :=
+  Finset.card_image_of_injective _ Subtype.val_injective
+
+lemma restricted_targets_subset (P : Finset Plane) (s : ℝ) (C : Finset P)
+    (e : fiberIndices P s) (y : P) (hy : y ∈ (planarFibers P s).targets e) :
+    (((planarFibers P s).targets e ∩ C).image Subtype.val) ⊆
+      distanceFiber (planeSubset P C) e.val.1 y.val := by
+  intro z hz
+  obtain ⟨w,hw,he⟩ := Finset.mem_image.mp hz
+  subst z
+  obtain ⟨hzt,hzC⟩ := Finset.mem_inter.mp hw
+  apply Finset.mem_filter.mpr
+  refine ⟨Finset.mem_erase.mpr ⟨?_,Finset.mem_image.mpr ⟨w,hzC,rfl⟩⟩,?_⟩
+  · intro hh
+    have hd := (planar_target_iff P s e w).mp hzt
+    rw [hh,dist_self] at hd
+    exact (index_radius_pos P s e).ne hd
+  · rw [(planar_target_iff _ _ _ _).mp hzt,(planar_target_iff _ _ _ _).mp hy]
+
+lemma core_edge_rich (P : Finset Plane) (s K r : ℝ) (C : Finset P)
+    (hK : (C.card : ℝ)^r ≤ K) (x y : P)
+    (hxy : (planarFibers P s).edge K C x y) :
+    (x.val,y.val) ∈ richPairs (planeSubset P C) r := by
+  obtain ⟨e,⟨hs,hsize⟩,hsrc,hy⟩ := hxy
+  have hxC : x ∈ C := hsrc ▸ hs
+  have hyC := (Finset.mem_inter.mp hy).2
+  have hyT := (Finset.mem_inter.mp hy).1
+  have hne : x ≠ y := by
+    intro hh
+    exact (planarFibers P s).no_loop e (hsrc ▸ hh ▸ hyT)
+  have hsrc' : e.val.1=x.val := congrArg Subtype.val hsrc
+  apply Finset.mem_filter.mpr
+  refine ⟨Finset.mem_product.mpr ⟨Finset.mem_image.mpr ⟨x,hxC,rfl⟩,
+    Finset.mem_image.mpr ⟨y,hyC,rfl⟩⟩,?_,?_⟩
+  · exact fun hh => hne (Subtype.ext hh)
+  · rw [planeSubset_card]
+    have hc := Finset.card_le_card (restricted_targets_subset P s C e y hyT)
+    rw [Finset.card_image_of_injective _ Subtype.val_injective,hsrc'] at hc
+    have hc' : ((planarFibers P s).size C e : ℝ) ≤ k (planeSubset P C) x.val y.val := by
+      exact_mod_cast hc
+    exact hK.trans ((le_of_lt hsize).trans hc')
+
+end WeakPinned
+
+namespace WeakPinned
+open scoped BigOperators
+
+lemma subtype_pair_injective (P : Finset Plane) :
+    Function.Injective (fun xy : P × P => (xy.1.val,xy.2.val)) := by
+  intro x y h
+  apply Prod.ext <;> apply Subtype.ext
+  · exact congrArg Prod.fst h
+  · exact congrArg Prod.snd h
+
+lemma core_edge_card (P : Finset Plane) (s K r : ℝ) (C : Finset P)
+    (hK : (C.card : ℝ)^r ≤ K) :
+    ((planarFibers P s).edgeSet K C).card ≤ (richPairs (planeSubset P C) r).card := by
+  rw [← Finset.card_image_of_injective _ (subtype_pair_injective P)]
+  apply Finset.card_le_card
+  intro z hz
+  obtain ⟨xy,he,hval⟩ := Finset.mem_image.mp hz
+  subst z
+  exact core_edge_rich P s K r C hK xy.1 xy.2 (Finset.mem_filter.mp he).2
+
+lemma core_score_bound (P : Finset Plane) (s K r : ℝ) (C : Finset P)
+    (hK0 : 0 ≤ K) (hK : (C.card : ℝ)^r ≤ K) (hn : 2 ≤ C.card) :
+    (planarFibers P s).score K C ≤ F C.card r*(C.card : ℝ)*((C.card : ℝ)-1) := by
+  have hQ := fraction_le_F (planeSubset P C) r (by simpa [planeSubset_card] using hn)
+  have hpos : 0 < (C.card : ℝ)*((C.card : ℝ)-1) := by
+    have hh : (2:ℝ) ≤ C.card := by exact_mod_cast hn
+    nlinarith
+  rw [pairFraction,planeSubset_card,div_le_iff₀ hpos] at hQ
+  calc
+    _ ≤ (((planarFibers P s).edgeSet K C).card : ℝ) := (planarFibers P s).score_le_edgeSet K hK0 C
+    _ ≤ ((richPairs (planeSubset P C) r).card : ℝ) := by exact_mod_cast core_edge_card P s K r C hK
+    _ ≤ _ := by simpa only [mul_assoc] using hQ
+
+lemma initial_edge_image (P : Finset Plane) (s K : ℝ) (hK : K < (P.card : ℝ)^s) :
+    ((planarFibers P s).edgeSet K Finset.univ).image (fun xy : P × P => (xy.1.val,xy.2.val)) =
+      richPairs P s := by
+  ext xy
+  constructor
+  · intro h
+    obtain ⟨⟨x,y⟩,hxy,rfl⟩ := Finset.mem_image.mp h
+    obtain ⟨e,_,hsrc,hy⟩ := (Finset.mem_filter.mp hxy).2
+    have hyT := (Finset.mem_inter.mp hy).1
+    have hsrc' : e.val.1=x.val := congrArg Subtype.val hsrc
+    have hr : dist y.val e.val.1=e.val.2 := (planar_target_iff P s e y).mp hyT
+    have hne : x.val ≠ y.val := by
+      intro hh
+      rw [hsrc',hh,dist_self] at hr
+      exact (index_radius_pos P s e).ne hr
+    apply Finset.mem_filter.mpr
+    refine ⟨Finset.mem_product.mpr ⟨x.property,y.property⟩,hne,?_⟩
+    have hc := planar_fiber_card P s e
+    have heq : ((planarFibers P s).targets e).image Subtype.val=distanceFiber P x.val y.val := by
+      ext z
+      constructor
+      · intro hz
+        obtain ⟨w,hw,rfl⟩ := Finset.mem_image.mp hz
+        have hd := (planar_target_iff P s e w).mp hw
+        rw [hsrc'] at hd hr
+        apply Finset.mem_filter.mpr
+        refine ⟨Finset.mem_erase.mpr ⟨?_,w.property⟩,hd.trans hr.symm⟩
+        intro hh
+        rw [hh,dist_self] at hd
+        exact (index_radius_pos P s e).ne hd
+      · intro hz
+        obtain ⟨hz,hd⟩ := Finset.mem_filter.mp hz
+        have hzP := (Finset.mem_erase.mp hz).2
+        refine Finset.mem_image.mpr ⟨⟨z,hzP⟩,?_,rfl⟩
+        apply (planar_target_iff P s e _).mpr
+        simpa [hsrc'] using hd.trans (by simpa [hsrc'] using hr)
+    have hcard : ((planarFibers P s).targets e).card=k P x.val y.val := by
+      rw [← Finset.card_image_of_injective _ Subtype.val_injective,heq]
+      rfl
+    simpa only [hcard] using hc
+  · intro hxy
+    obtain ⟨hP,hne,hk⟩ := Finset.mem_filter.mp hxy
+    obtain ⟨hx,hy⟩ := Finset.mem_product.mp hP
+    let x : P := ⟨xy.1,hx⟩
+    let y : P := ⟨xy.2,hy⟩
+    let e : fiberIndices P s := ⟨(xy.1,dist xy.2 xy.1),Finset.mem_image.mpr ⟨xy,hxy,rfl⟩⟩
+    refine Finset.mem_image.mpr ⟨(x,y),Finset.mem_filter.mpr ⟨Finset.mem_univ _,?_⟩,rfl⟩
+    refine ⟨e,⟨Finset.mem_univ _,?_⟩,rfl,Finset.mem_inter.mpr ⟨?_,Finset.mem_univ _⟩⟩
+    · simpa only [Core.Fibers.size,Finset.inter_univ] using hK.trans_le (planar_fiber_card P s e)
+    · exact (planar_target_iff P s e y).mpr rfl
+
+lemma initial_edge_card (P : Finset Plane) (s K : ℝ) (hK : K < (P.card : ℝ)^s) :
+    ((planarFibers P s).edgeSet K Finset.univ).card=(richPairs P s).card := by
+  rw [← Finset.card_image_of_injective _ (subtype_pair_injective P),initial_edge_image P s K hK]
+
+lemma initial_score_lower (P : Finset Plane) (s K : ℝ) (hK : K < (P.card : ℝ)^s) :
+    (richPairs P s).card-K*(Fintype.card (fiberIndices P s) : ℝ) ≤
+      (planarFibers P s).score K Finset.univ := by
+  have hh := (planarFibers P s).score_univ_lower K
+  rw [(planarFibers P s).sum_sizes_le_edgeSet_univ K (fun e => hK.trans_le (planar_fiber_card P s e)),
+    initial_edge_card P s K hK] at hh
+  exact hh
+
+lemma index_card_bound (P : Finset Plane) (s K : ℝ) (hK : K < (P.card : ℝ)^s) :
+    (Fintype.card (fiberIndices P s) : ℝ)*(P.card : ℝ)^s ≤ (richPairs P s).card := by
+  have hh := (planarFibers P s).number_fibers ((P.card : ℝ)^s) K hK (planar_fiber_card P s)
+  simpa only [initial_edge_card P s K hK] using hh
+
+end WeakPinned
+
+namespace WeakPinned
+open scoped BigOperators
+
+lemma initial_score_rate (P : Finset Plane) (s K ρ : ℝ)
+    (hρ : 0 ≤ ρ) (hK : K < (P.card : ℝ)^s) (hKL : K ≤ ρ*(P.card : ℝ)^s) :
+    (richPairs P s).card-ρ*((P.card : ℝ)*((P.card : ℝ)-1)) ≤
+      (planarFibers P s).score K Finset.univ := by
+  have hf := index_card_bound P s K hK
+  have hc : ((richPairs P s).card : ℝ) ≤ (P.card : ℝ)*((P.card : ℝ)-1) := by
+    by_cases hn : 1 ≤ P.card
+    · have hh := richPairs_card_le P s
+      have hc' : ((richPairs P s).card : ℝ) ≤ (P.card*(P.card-1) : ℕ) := by exact_mod_cast hh
+      simpa only [Nat.cast_mul,Nat.cast_sub hn,Nat.cast_one] using hc'
+    · have hz : P.card=0 := by omega
+      have hzero : (richPairs P s).card=0 := by
+        have hh := richPairs_card_le P s
+        rw [hz] at hh
+        simpa using hh
+      simp [hz,hzero]
+  have hi0 : 0 ≤ (Fintype.card (fiberIndices P s) : ℝ) := Nat.cast_nonneg _
+  have h1 := mul_le_mul_of_nonneg_right hKL hi0
+  have h2 := mul_le_mul_of_nonneg_left hf hρ
+  have h3 := mul_le_mul_of_nonneg_left hc hρ
+  have h4 := initial_score_lower P s K hK
+  nlinarith
+
+/-- A finite actual planar core. All hypotheses before the conclusion are
+    scalar extremal bounds; the graph, degrees and surviving fibers are
+    constructed, not postulated. -/
+theorem finite_extremal_core (P : Finset Plane) (s r K θ η : ℝ) (m₀ : ℕ)
+    (hn : 2 ≤ P.card) (hm0 : 2 ≤ m₀) (hm : m₀ < P.card)
+    (hη : 0 < η) (hK0 : 0 ≤ K) (hK : K < (P.card : ℝ)^s)
+    (hloss : K ≤ (η/8)*(P.card : ℝ)^s)
+    (hinit : θ-η/8 ≤ pairFraction P s)
+    (hboundary : (m₀ : ℝ)*((m₀ : ℝ)-1) ≤ (P.card : ℝ)*((P.card : ℝ)-1)/2)
+    (hthreshold : ∀ m : ℕ, m₀ ≤ m → m ≤ P.card → (m : ℝ)^r ≤ K)
+    (hext : ∀ m : ℕ, m₀ ≤ m → m ≤ P.card → F m r ≤ θ+η/8) :
+    ∃ C : Finset P, m₀ < C.card ∧
+      (θ-η)*(C.card : ℝ)*((C.card : ℝ)-1) ≤ ((planarFibers P s).edgeSet K C).card ∧
+      (((planarFibers P s).edgeSet K C).card : ℝ) ≤ (θ+η/8)*(C.card : ℝ)*((C.card : ℝ)-1) ∧
+      (∀ x ∈ C, 2*(θ-η)*((C.card : ℝ)-1) ≤ (planarFibers P s).degree K C x) := by
+  let G := planarFibers P s
+  let A : ℝ := (P.card : ℝ)*((P.card : ℝ)-1)
+  have hA : 0 < A := by
+    have hh : (2:ℝ) ≤ P.card := by exact_mod_cast hn
+    dsimp [A]
+    nlinarith
+  have hinit' : (θ-η/8)*A ≤ (richPairs P s).card := by
+    simpa only [pairFraction,A,le_div_iff₀ hA] using hinit
+  have hscore := initial_score_rate P s K (η/8) (by positivity) hK hloss
+  have hscore' : (θ-η/4)*A ≤ G.score K Finset.univ := by
+    change (richPairs P s).card-η/8*A ≤ G.score K Finset.univ at hscore
+    nlinarith
+  have hobj : 3*η/4*A ≤ G.objective K (θ-η) Finset.univ := by
+    simp only [Core.Fibers.objective,Core.Fibers.penalty,Finset.card_univ,Fintype.card_coe]
+    unfold A at hscore' ⊢
+    nlinarith
+  obtain ⟨C,hC,hge,hdeg⟩ := G.exists_core K (θ-η) (θ+η/8) hK0 m₀
+    (by simpa using hm) (fun C hCm => by
+      have hc := core_score_bound P s K r C hK0 (hCm ▸ hthreshold m₀ le_rfl (le_of_lt hm)) (by omega)
+      rw [hCm] at hc
+      have hmR : (1:ℝ) ≤ m₀ := by exact_mod_cast (show 1 ≤ m₀ by omega)
+      have he := mul_le_mul_of_nonneg_right (hext m₀ le_rfl (le_of_lt hm))
+        (mul_nonneg (Nat.cast_nonneg m₀) (sub_nonneg.mpr hmR))
+      dsimp [G]
+      nlinarith) (by
+      have hh := mul_le_mul_of_nonneg_left hboundary (show 0 ≤ 9*η/8 by positivity)
+      change _ < G.objective K (θ-η) Finset.univ
+      unfold A at hobj
+      have hηA := mul_pos hη hA
+      unfold A at hηA
+      nlinarith)
+  have hCn : C.card ≤ P.card := by simpa using Finset.card_le_univ C
+  have hCm : m₀ ≤ C.card := le_of_lt hC
+  have hC1 : (1:ℝ) ≤ C.card := by exact_mod_cast (show 1 ≤ C.card by omega)
+  have hprod : 0 ≤ (C.card : ℝ)*((C.card : ℝ)-1) := mul_nonneg (Nat.cast_nonneg _) (sub_nonneg.mpr hC1)
+  have hscore0 : (θ-η)*(C.card : ℝ)*((C.card : ℝ)-1) ≤ G.score K C := by
+    have hh : 0 ≤ G.objective K (θ-η) Finset.univ := (le_of_lt (by positivity : 0 < 3*η/4*A)).trans hobj
+    unfold Core.Fibers.objective Core.Fibers.penalty at hge hh
+    linarith
+  have hedge := core_edge_card P s K r C (hthreshold C.card hCm hCn)
+  have hQ := fraction_le_F (planeSubset P C) r (by simpa [planeSubset_card] using (show 2 ≤ C.card by omega))
+  rw [pairFraction,planeSubset_card] at hQ
+  have hp : 0 < (C.card : ℝ)*((C.card : ℝ)-1) := by
+    have h2 : (2:ℝ) ≤ C.card := by exact_mod_cast (show 2 ≤ C.card by omega)
+    nlinarith
+  rw [div_le_iff₀ hp] at hQ
+  have he := mul_le_mul_of_nonneg_right (hext C.card hCm hCn) hprod
+  refine ⟨C,hC,hscore0.trans (G.score_le_edgeSet K hK0 C),?_,hdeg⟩
+  have he' : (((planarFibers P s).edgeSet K C).card : ℝ) ≤ (richPairs (planeSubset P C) r).card := by exact_mod_cast hedge
+  nlinarith
+
+end WeakPinned
+
+end
+end
+end
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PinnedDistances/ExtremalFractions.lean b/lean/OAI/Geometry/PinnedDistances/ExtremalFractions.lean
new file mode 100644
index 0000000000000000000000000000000000000000..ff97f4f9235d3e189566162715564bda17a41438
--- /dev/null
+++ b/lean/OAI/Geometry/PinnedDistances/ExtremalFractions.lean
@@ -0,0 +1,176 @@
+import OAI.Geometry.PinnedDistances.GlobalOverlap
+
+namespace OAI
+
+section
+noncomputable section
+open NumberField
+open scoped BigOperators
+noncomputable section
+open MeasureTheory
+open scoped Topology
+noncomputable section
+open MeasureTheory
+open scoped Topology ENNReal
+noncomputable section
+open MeasureTheory NumberField
+open scoped BigOperators
+noncomputable section
+open scoped BigOperators
+open MeasureTheory
+noncomputable section
+open Filter
+open scoped Topology
+
+namespace WeakPinned
+open Filter
+open scoped Topology
+
+lemma richPairs_subset_offDiag (P : Finset Plane) (s : ℝ) : richPairs P s ⊆ P.offDiag := by
+  intro xy h
+  obtain ⟨hP,hh,_⟩ := Finset.mem_filter.mp h
+  exact Finset.mem_offDiag.mpr ⟨(Finset.mem_product.mp hP).1,(Finset.mem_product.mp hP).2,hh⟩
+
+lemma richPairs_card_le (P : Finset Plane) (s : ℝ) : (richPairs P s).card ≤ P.card*(P.card-1) := by
+  calc
+    _ ≤ P.offDiag.card := Finset.card_le_card (richPairs_subset_offDiag P s)
+    _ = _ := by rw [Finset.offDiag_card,Nat.mul_sub_one]
+
+lemma pairFraction_nonneg (P : Finset Plane) (s : ℝ) : 0 ≤ pairFraction P s := by
+  by_cases hn : 1 ≤ P.card
+  · exact div_nonneg (Nat.cast_nonneg _) (mul_nonneg (Nat.cast_nonneg _) (sub_nonneg.mpr (by exact_mod_cast hn)))
+  · have hz : P.card=0 := by omega
+    simp [pairFraction,hz]
+
+lemma pairFraction_le_one (P : Finset Plane) (s : ℝ) : pairFraction P s ≤ 1 := by
+  by_cases hn : 2 ≤ P.card
+  · have hd : 0 < (P.card : ℝ)*((P.card : ℝ)-1) := by
+      have : (2:ℝ) ≤ P.card := by exact_mod_cast hn
+      nlinarith
+    rw [pairFraction,div_le_one hd]
+    have hh := richPairs_card_le P s
+    have h1 : 1 ≤ P.card := by omega
+    exact_mod_cast (show (richPairs P s).card ≤ P.card*(P.card-1) from hh)
+  · have hc : P.card=0 ∨ P.card=1 := by omega
+    rcases hc with hc | hc <;> simp [pairFraction,hc]
+
+lemma fraction_values_nonempty (n : ℕ) (s : ℝ) :
+    ({a : ℝ | ∃ P : Finset Plane, P.card=n ∧ a=pairFraction P s}).Nonempty := by
+  obtain ⟨P,hP⟩ := Finset.exists_card_eq (α:=Plane) n
+  exact ⟨_,P,hP,rfl⟩
+
+lemma fraction_values_bddAbove (n : ℕ) (s : ℝ) :
+    BddAbove {a : ℝ | ∃ P : Finset Plane, P.card=n ∧ a=pairFraction P s} := by
+  refine ⟨1,?_⟩
+  rintro a ⟨P,_,rfl⟩
+  exact pairFraction_le_one P s
+
+lemma fraction_le_F (P : Finset Plane) (s : ℝ) (hn : 2 ≤ P.card) : pairFraction P s ≤ F P.card s := by
+  rw [F,ite_eq_left hn]
+  exact le_csSup (fraction_values_bddAbove _ _) ⟨P,rfl,rfl⟩
+
+lemma F_nonneg (n : ℕ) (s : ℝ) : 0 ≤ F n s := by
+  by_cases hn : 2 ≤ n
+  · obtain ⟨P,hP⟩ := Finset.exists_card_eq (α:=Plane) n
+    rw [← hP] at hn ⊢
+    exact (pairFraction_nonneg P s).trans (fraction_le_F P s hn)
+  · simp [F,hn]
+
+lemma F_le_one (n : ℕ) (s : ℝ) : F n s ≤ 1 := by
+  unfold F
+  split_ifs
+  · exact csSup_le (fraction_values_nonempty _ _) (by rintro a ⟨P,_,rfl⟩; exact pairFraction_le_one P s)
+  · norm_num
+
+lemma F_attained (n : ℕ) (hn : 2 ≤ n) (s : ℝ) :
+    ∃ P : Finset Plane, P.card=n ∧ F n s=pairFraction P s := by
+  classical
+  let A : Finset ℕ := (Finset.range (n*(n-1)+1)).filter (fun k => ∃ P : Finset Plane, P.card=n ∧ (richPairs P s).card=k)
+  obtain ⟨P,hP⟩ := Finset.exists_card_eq (α:=Plane) n
+  have hA : (richPairs P s).card ∈ A := by
+    simp only [A,Finset.mem_filter,Finset.mem_range]
+    refine ⟨?_,P,hP,rfl⟩
+    have hh := richPairs_card_le P s
+    rw [hP] at hh
+    omega
+  obtain ⟨k,hk,hmax⟩ := Finset.exists_max_image A id ⟨_,hA⟩
+  obtain ⟨Q,hQ,hkQ⟩ := (Finset.mem_filter.mp hk).2
+  refine ⟨Q,hQ,le_antisymm ?_ ?_⟩
+  · rw [F,ite_eq_left hn]
+    apply csSup_le (fraction_values_nonempty _ _)
+    rintro a ⟨R,hR,rfl⟩
+    have hm : (richPairs R s).card ∈ A := by
+      simp only [A,Finset.mem_filter,Finset.mem_range]
+      refine ⟨?_,R,hR,rfl⟩
+      have hh := richPairs_card_le R s
+      rw [hR] at hh
+      omega
+    have hh := hmax _ hm
+    change (richPairs R s).card ≤ k at hh
+    simp only [pairFraction,hR,hQ]
+    exact div_le_div_of_nonneg_right (by exact_mod_cast (hkQ ▸ hh)) (by
+      have : (2:ℝ) ≤ n := by exact_mod_cast hn
+      nlinarith)
+  · rw [← hQ]
+    exact fraction_le_F Q s (hQ ▸ hn)
+
+lemma k_lt_card {P : Finset Plane} {x y : Plane} (hx : x ∈ P) : k P x y < P.card := by
+  calc
+    _ ≤ (P.erase x).card := Finset.card_le_card (Finset.filter_subset _ _)
+    _ < P.card := Finset.card_erase_lt_of_mem hx
+
+lemma richPairs_antitone {P : Finset Plane} (hn : 1 ≤ P.card) {s t : ℝ} (hst : s ≤ t) :
+    richPairs P t ⊆ richPairs P s := by
+  intro xy h
+  obtain ⟨hP,hne,hk⟩ := Finset.mem_filter.mp h
+  apply Finset.mem_filter.mpr
+  refine ⟨hP,hne,?_⟩
+  exact (Real.rpow_le_rpow_of_exponent_le (by exact_mod_cast hn) hst).trans hk
+
+lemma fraction_antitone {P : Finset Plane} (hn : 2 ≤ P.card) : Antitone (pairFraction P) := by
+  intro s t hst
+  unfold pairFraction
+  apply div_le_div_of_nonneg_right
+  · exact_mod_cast Finset.card_le_card (richPairs_antitone (by omega) hst)
+  · have : (2:ℝ) ≤ P.card := by exact_mod_cast hn
+    nlinarith
+
+lemma F_antitone (n : ℕ) : Antitone (F n) := by
+  intro s t hst
+  by_cases hn : 2 ≤ n
+  · obtain ⟨P,hP,hF⟩ := F_attained n hn t
+    rw [hF]
+    have hPn : 2 ≤ P.card := hP ▸ hn
+    calc
+      _ ≤ pairFraction P s := fraction_antitone hPn hst
+      _ ≤ F n s := by simpa [hP] using fraction_le_F P s hPn
+  · simp [F,hn]
+
+lemma F_eq_zero_of_one_le (n : ℕ) {s : ℝ} (hs : 1 ≤ s) : F n s=0 := by
+  by_cases hn : 2 ≤ n
+  · obtain ⟨P,hP,hF⟩ := F_attained n hn s
+    have hz : richPairs P s=∅ := by
+      apply Finset.eq_empty_iff_forall_notMem.mpr
+      intro xy hh
+      obtain ⟨hxy,_,hk⟩ := Finset.mem_filter.mp hh
+      have hlt := k_lt_card (x:=xy.1) (y:=xy.2) (Finset.mem_product.mp hxy).1
+      have h1 : (1:ℝ) ≤ P.card := by rw [hP]; exact_mod_cast (show 1 ≤ n by omega)
+      have hr := Real.rpow_le_rpow_of_exponent_le h1 hs
+      rw [Real.rpow_one] at hr
+      have hlt' : (k P xy.1 xy.2 : ℝ) < P.card := by exact_mod_cast hlt
+      linarith
+    rw [hF,pairFraction,hz]
+    simp
+  · simp [F,hn]
+
+end WeakPinned
+
+end
+end
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PinnedDistances/FiberOverlap.lean b/lean/OAI/Geometry/PinnedDistances/FiberOverlap.lean
new file mode 100644
index 0000000000000000000000000000000000000000..7a30a7521d8feab19e6f9fba2b2148a19b432f4b
--- /dev/null
+++ b/lean/OAI/Geometry/PinnedDistances/FiberOverlap.lean
@@ -0,0 +1,431 @@
+import OAI.Geometry.PinnedDistances.SmallCells
+
+namespace OAI
+
+universe u1 u2 u3 u4 u5 u6 u7 u8 u9 u10
+
+section
+noncomputable section
+open NumberField
+open scoped BigOperators
+noncomputable section
+open MeasureTheory
+open scoped Topology
+noncomputable section
+open MeasureTheory
+open scoped Topology ENNReal
+noncomputable section
+open MeasureTheory NumberField
+open scoped BigOperators
+noncomputable section
+open scoped BigOperators
+open MeasureTheory
+
+namespace WeakPinned.SmallFiber
+open MeasureTheory Hierarchy Variance
+variable {P : Type u1} {I : Type u2} [Fintype P] [DecidableEq P] [Fintype I]
+
+def slice (w : I → ℝ) (p : I → P → ℝ) (x : I → P) (lam : ℝ) (C : Finset P) : ℝ :=
+  ∑ i, w i * (if x i ∈ C ∧ mass (p i) C < lam then mass (p i) C else 0)
+
+def edgeCell (w : I → ℝ) (p : I → P → ℝ) (x : I → P) (C : Finset P) : ℝ :=
+  ∑ a ∈ C, ∑ b ∈ C, edgeLaw w p x a b
+
+omit [Fintype P] in
+lemma edgeCell_formula (w : I → ℝ) (p : I → P → ℝ) (x : I → P) (C : Finset P) :
+    edgeCell w p x C = ∑ i, w i * (if x i ∈ C then mass (p i) C else 0) := by
+  classical
+  unfold edgeCell edgeLaw
+  calc
+    _ = ∑ a ∈ C, ∑ i, ∑ b ∈ C, if x i = a then w i * p i b else 0 := by
+      apply Finset.sum_congr rfl
+      intro a _
+      exact Finset.sum_comm
+    _ = ∑ i, ∑ a ∈ C, ∑ b ∈ C, if x i = a then w i * p i b else 0 := Finset.sum_comm
+    _ = _ := by
+      apply Finset.sum_congr rfl
+      intro i _
+      have he (a : P) : (∑ b ∈ C, if x i = a then w i*p i b else 0) =
+          if x i = a then w i*mass (p i) C else 0 := by
+        by_cases h : x i = a <;> simp [h,mass,Finset.mul_sum]
+      simp_rw [he]
+      by_cases h : x i ∈ C <;> simp [h,eq_comm]
+
+omit [Fintype P] in
+lemma slice_nonneg (w : I → ℝ) (p : I → P → ℝ) (x : I → P)
+    (hw : ∀ i, 0 ≤ w i) (hp : ∀ i y, 0 ≤ p i y) (lam : ℝ) (C : Finset P) :
+    0 ≤ slice w p x lam C := by
+  classical
+  unfold slice
+  exact Finset.sum_nonneg (fun i _ => mul_nonneg (hw i) (by split_ifs; exact mass_nonneg (hp i) _; rfl))
+
+omit [Fintype P] in
+lemma slice_le_edgeCell (w : I → ℝ) (p : I → P → ℝ) (x : I → P)
+    (hw : ∀ i, 0 ≤ w i) (hp : ∀ i y, 0 ≤ p i y) (lam : ℝ) (C : Finset P) :
+    slice w p x lam C ≤ edgeCell w p x C := by
+  rw [edgeCell_formula]
+  apply Finset.sum_le_sum
+  intro i _
+  apply mul_le_mul_of_nonneg_left _ (hw i)
+  split_ifs <;> simp_all [mass_nonneg (hp i)]
+
+omit [Fintype P] in
+lemma slice_le_lam (w : I → ℝ) (p : I → P → ℝ) (x : I → P)
+    (hw : ∀ i, 0 ≤ w i) (hW : ∑ i, w i = 1) (lam : ℝ) (hlam : 0 ≤ lam) (C : Finset P) :
+    slice w p x lam C ≤ lam := by
+  calc
+    _ ≤ ∑ i, w i * lam := by
+      apply Finset.sum_le_sum
+      intro i _
+      apply mul_le_mul_of_nonneg_left _ (hw i)
+      split_ifs with h
+      · exact h.2.le
+      · exact hlam
+    _ = _ := by rw [← Finset.sum_mul,hW,one_mul]
+
+omit [Fintype P] in
+lemma edgeLaw_diag (w : I → ℝ) (p : I → P → ℝ) (x : I → P)
+    (hx : ∀ i, p i (x i) = 0) (a : P) : edgeLaw w p x a a = 0 := by
+  classical
+  apply Finset.sum_eq_zero
+  intro i _
+  by_cases h : x i = a
+  · rw [ite_eq_left h,← h,hx i,mul_zero]
+  · simp [h]
+
+lemma edgeCell_dom (w : I → ℝ) (p : I → P → ℝ) (x : I → P)
+    (hx : ∀ i, p i (x i) = 0) (B : ℝ)
+    (hdom : ∀ a b, a ≠ b → edgeLaw w p x a b ≤ B/((Fintype.card P:ℝ)*((Fintype.card P:ℝ)-1)))
+    (C : Finset P) : edgeCell w p x C ≤ B*pairCell Finset.univ C := by
+  classical
+  simp only [pairCell,Finset.mul_sum,edgeCell]
+  apply Finset.sum_le_sum
+  intro a _
+  apply Finset.sum_le_sum
+  intro b _
+  by_cases hab : a = b
+  · subst b; simp [edgeLaw_diag w p x hx,pairProb_diag]
+  · simpa [pairProb,hab,div_eq_mul_inv] using hdom a b hab
+
+omit [Fintype P] in
+lemma slice_singleton (w : I → ℝ) (p : I → P → ℝ) (x : I → P)
+    (hx : ∀ i, p i (x i) = 0) (lam : ℝ) (C : Finset P) (hC : C.card ≤ 1) :
+    slice w p x lam C = 0 := by
+  classical
+  apply Finset.sum_eq_zero
+  intro i _
+  by_cases hi : x i ∈ C
+  · have he : C = {x i} := Finset.eq_singleton_iff_unique_mem.mpr
+      ⟨hi,fun y hy => (Finset.card_le_one.mp hC y hy (x i) hi)⟩
+    simp [he,mass,hx]
+  · simp [hi]
+
+lemma large_slice_bound (w : I → ℝ) (p : I → P → ℝ) (x : I → P)
+    (hw : ∀ i, 0 ≤ w i) (hW : ∑ i, w i = 1) (hn : 2 ≤ Fintype.card P)
+    (lam τ : ℝ) (hlam : 0 ≤ lam) (hτ : 0 < τ)
+    (hτn : 2 ≤ τ*(Fintype.card P:ℝ)) (C : Finset P)
+    (hC : τ*(Fintype.card P:ℝ) ≤ C.card) :
+    slice w p x lam C ≤ (2*lam/τ^2)*pairCell Finset.univ C := by
+  have hn0 : 0 < (Fintype.card P:ℝ) := by exact_mod_cast (show 0 < Fintype.card P by omega)
+  have hm : 2 ≤ C.card := by exact_mod_cast hτn.trans hC
+  have hcomp := (uniform_pair_comparison C hm).1
+  have ha : τ ≤ cellProb Finset.univ C := by
+    simpa [cellProb] using (le_div_iff₀ hn0).mpr hC
+  have hτsq : τ^2 ≤ (cellProb Finset.univ C)^2 := sq_le_sq₀ hτ.le (hτ.le.trans ha) |>.mpr ha
+  have hlow : τ^2 ≤ 2*pairCell Finset.univ C := by linarith
+  have hmul := mul_le_mul_of_nonneg_left hlow hlam
+  have he : lam ≤ (2*lam/τ^2)*pairCell Finset.univ C := by
+    rw [div_mul_eq_mul_div]
+    apply (le_div_iff₀ (sq_pos_of_pos hτ)).mpr
+    nlinarith
+  exact (slice_le_lam w p x hw hW lam hlam C).trans he
+
+lemma small_slice_bound (w : I → ℝ) (p : I → P → ℝ) (x : I → P)
+    (hw : ∀ i, 0 ≤ w i) (hp : ∀ i y, 0 ≤ p i y) (hx : ∀ i, p i (x i) = 0)
+    (B τ lam : ℝ) (hB : 0 ≤ B) (hτ : 0 ≤ τ)
+    (hdom : ∀ a b, a ≠ b → edgeLaw w p x a b ≤ B/((Fintype.card P:ℝ)*((Fintype.card P:ℝ)-1)))
+    (C : Finset P) (hC : (C.card:ℝ) < τ*Fintype.card P) :
+    slice w p x lam C ≤ B*τ*cellProb Finset.univ C := by
+  by_cases hm : 2 ≤ C.card
+  · have hn : 0 < (Fintype.card P:ℝ) := by exact_mod_cast (show 0 < Fintype.card P from lt_of_lt_of_le (by omega) (Finset.card_le_univ C))
+    have ha : cellProb Finset.univ C ≤ τ := by
+      simpa [cellProb] using (div_le_iff₀ hn).mpr hC.le
+    have ha0 : 0 ≤ cellProb (Finset.univ : Finset P) C := cellProb_nonneg _ _
+    calc
+      _ ≤ B*pairCell Finset.univ C := (slice_le_edgeCell w p x hw hp lam C).trans (edgeCell_dom w p x hx B hdom C)
+      _ ≤ B*(cellProb Finset.univ C)^2 := mul_le_mul_of_nonneg_left (uniform_pair_comparison C hm).2 hB
+      _ ≤ B*τ*cellProb Finset.univ C := by nlinarith [mul_nonneg hB (mul_nonneg ha0 (sub_nonneg.mpr ha))]
+  · rw [slice_singleton w p x hx lam C (by omega)]
+    exact mul_nonneg (mul_nonneg hB hτ) (cellProb_nonneg _ _)
+
+end WeakPinned.SmallFiber
+
+namespace WeakPinned.Hierarchy.Depth
+variable {P : Type u3} [Fintype P] [DecidableEq P]
+variable (d : Depth P)
+lemma sum_source_cell (f : Finset P → ℝ) (r : ℝ) (x : P) :
+    (∑ C ∈ d.cells r, if x ∈ C then f C else 0) = f (d.cell r x) := by
+  classical
+  have he : ∀ C ∈ d.cells r, (if x ∈ C then f C else 0) =
+      if C = d.cell r x then f (d.cell r x) else 0 := by
+    intro C hC
+    by_cases hx : x ∈ C
+    · rw [d.cell_of_mem hC hx]; simp [d.share_refl]
+    · have hn : C ≠ d.cell r x := by intro h; apply hx; rw [h]; exact d.self_mem_cell r x
+      simp [hx,hn]
+  rw [Finset.sum_congr rfl he]
+  simp [d.cell_mem_cells r x]
+
+lemma sumCells_split (f : Finset P → ℝ) (r : ℝ) :
+    d.sumCells f r =
+      (∑ C ∈ d.cells r, if ¬Fintype.card P < 2*C.card then f C else 0) +
+      ∑ C ∈ d.giants r, f Cᶜ := by
+  classical
+  simp only [sumCells,transformed,Multiset.map_add,Multiset.sum_add,
+    Multiset.map_map,Function.comp_def]
+  change (∑ C ∈ (d.cells r).filter (fun C => ¬Fintype.card P < 2*C.card), f C) + _ = _
+  rw [Finset.sum_filter]
+  rfl
+
+lemma sum_complement_prob (r : ℝ) :
+    (∑ C ∈ d.giants r, Variance.cellProb Finset.univ Cᶜ) =
+      (∑ y, d.outsideIndicator r y)/(Fintype.card P:ℝ) := by
+  classical
+  by_cases hg : (d.giants r).Nonempty
+  · obtain ⟨G,hG⟩ := hg
+    have hs : d.giants r = {G} := Finset.ext fun C => by
+      simp only [Finset.mem_singleton]
+      exact ⟨fun hC => d.giants_unique hC hG,fun h => h ▸ hG⟩
+    have hne : (d.giants r).Nonempty := ⟨G,hG⟩
+    have hi (y : P) : d.outsideIndicator r y = if y ∈ Gᶜ then 1 else 0 := by
+      simp [outsideIndicator,hne,d.giant_eq hG]
+    simp_rw [hi]
+    rw [Finset.sum_ite_mem]
+    simp only [Finset.univ_inter,Finset.sum_const,nsmul_eq_mul,mul_one]
+    simp [hs,Variance.cellProb]
+  · simp [Finset.not_nonempty_iff_eq_empty.mp hg,outsideIndicator]
+end WeakPinned.Hierarchy.Depth
+
+namespace WeakPinned.SmallFiber
+open MeasureTheory Hierarchy Variance
+variable {P : Type u4} {I : Type u5} [Fintype P] [DecidableEq P] [Fintype I]
+
+lemma nongiant_small_slice (d : Depth P) (w : I → ℝ) (p : I → P → ℝ) (x : I → P)
+    (T lam r : ℝ) :
+    (∑ C ∈ d.cells r, if ¬Fintype.card P < 2*C.card ∧ (C.card:ℝ) < T then slice w p x lam C else 0) =
+      ∑ i, w i * activeMass d (p i) (x i) T lam r := by
+  classical
+  have he (C : Finset P) :
+      (if ¬Fintype.card P < 2*C.card ∧ (C.card:ℝ) < T then slice w p x lam C else 0) =
+      ∑ i, w i * (if x i ∈ C then
+        (if ¬Fintype.card P < 2*C.card ∧ (C.card:ℝ) < T ∧ mass (p i) C < lam then mass (p i) C else 0) else 0) := by
+    by_cases h : ¬Fintype.card P < 2*C.card ∧ (C.card:ℝ) < T
+    · simp only [h,slice]
+      apply Finset.sum_congr rfl
+      intro i _
+      by_cases hx : x i ∈ C <;> simp [hx]
+    · simp only [h,ite_false]
+      symm
+      apply Finset.sum_eq_zero
+      intro i _
+      split_ifs with hx hh
+      · exact False.elim (h ⟨hh.1,hh.2.1⟩)
+      · exact mul_zero _
+      · exact mul_zero _
+  simp_rw [he]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro i _
+  rw [←Finset.mul_sum,d.sum_source_cell]
+  congr 1
+  unfold activeMass Depth.sourceGiant
+  split_ifs <;> rfl
+
+lemma pointwise_split (d : Depth P) (w : I → ℝ) (p : I → P → ℝ) (x : I → P)
+    (hw : ∀ i, 0 ≤ w i) (hW : ∑ i, w i = 1) (hp : ∀ i y, 0 ≤ p i y)
+    (hx : ∀ i, p i (x i) = 0) (hn : 2 ≤ Fintype.card P)
+    (B τ lam r : ℝ) (hB : 0 ≤ B) (hlam : 0 ≤ lam) (hτ : 0 < τ)
+    (hτn : 2 ≤ τ*(Fintype.card P:ℝ))
+    (hdom : ∀ a b, a ≠ b → edgeLaw w p x a b ≤ B/((Fintype.card P:ℝ)*((Fintype.card P:ℝ)-1))) :
+    d.sumCells (slice w p x lam) r ≤
+      (2*lam/τ^2)*d.sumCells (pairCell Finset.univ) r +
+      B*τ/(Fintype.card P:ℝ)*(∑ y, d.outsideIndicator r y) +
+      ∑ i, w i*activeMass d (p i) (x i) (τ*Fintype.card P) lam r := by
+  classical
+  have hcoef : 0 ≤ 2*lam/τ^2 := by positivity
+  have hnng : ∀ C : Finset P, 0 ≤ (2*lam/τ^2)*pairCell Finset.univ C :=
+    fun C => mul_nonneg hcoef (pairCell_nonneg _ _)
+  have hactual :
+      (∑ C ∈ d.cells r, if ¬Fintype.card P < 2*C.card then slice w p x lam C else 0) ≤
+      (2*lam/τ^2)*(∑ C ∈ d.cells r, if ¬Fintype.card P < 2*C.card then pairCell Finset.univ C else 0) +
+      ∑ i, w i*activeMass d (p i) (x i) (τ*Fintype.card P) lam r := by
+    rw [←nongiant_small_slice,Finset.mul_sum,←Finset.sum_add_distrib]
+    apply Finset.sum_le_sum
+    intro C _
+    by_cases hg : ¬Fintype.card P < 2*C.card
+    · by_cases hc : (C.card:ℝ) < τ*Fintype.card P
+      · simp only [hg,hc,not_false_eq_true,true_and,ite_true]
+        linarith [hnng C]
+      · simp only [hg,hc,not_false_eq_true,true_and,ite_true,ite_false,add_zero]
+        exact large_slice_bound w p x hw hW hn lam τ hlam hτ hτn C (le_of_not_gt hc)
+    · simp [hg]
+  have hgiant : (∑ C ∈ d.giants r, slice w p x lam Cᶜ) ≤
+      (2*lam/τ^2)*(∑ C ∈ d.giants r, pairCell Finset.univ Cᶜ) +
+      B*τ*(∑ C ∈ d.giants r, cellProb Finset.univ Cᶜ) := by
+    simp only [Finset.mul_sum,←Finset.sum_add_distrib]
+    apply Finset.sum_le_sum
+    intro C _
+    by_cases hc : (Cᶜ.card:ℝ) < τ*Fintype.card P
+    · exact (small_slice_bound w p x hw hp hx B τ lam hB hτ.le hdom Cᶜ hc).trans (le_add_of_nonneg_left (hnng Cᶜ))
+    · exact (large_slice_bound w p x hw hW hn lam τ hlam hτ hτn Cᶜ (le_of_not_gt hc)).trans
+        (le_add_of_nonneg_right (mul_nonneg (mul_nonneg hB hτ.le) (cellProb_nonneg _ _)))
+  rw [d.sum_complement_prob] at hgiant
+  rw [d.sumCells_split,d.sumCells_split]
+  have he : B*τ*((∑ y, d.outsideIndicator r y)/(Fintype.card P:ℝ)) =
+      B*τ/(Fintype.card P:ℝ)*(∑ y, d.outsideIndicator r y) := by ring
+  rw [he] at hgiant
+  linarith
+
+end WeakPinned.SmallFiber
+
+namespace WeakPinned.Hierarchy.Depth
+open MeasureTheory
+variable {P : Type u6} [Fintype P] [DecidableEq P]
+variable (d : Depth P)
+lemma integrable_sumCells (hn : 2 ≤ Fintype.card P) (f : Finset P → ℝ)
+    (hf : ∀ C, C.card ≤ 1 → f C = 0) : Integrable (d.sumCells f) := by
+  change Integrable (fun r => d.sumCells f r)
+  simp_rw [d.sumCells_count]
+  apply integrable_finsetSum
+  intro C _
+  by_cases hC : 2 ≤ C.card
+  · exact (d.integrable_count hn C hC).mul_const _
+  · simp only [hf C (by omega),mul_zero]
+    exact integrable_zero _ _ _
+end WeakPinned.Hierarchy.Depth
+
+namespace WeakPinned.SmallFiber
+open MeasureTheory Hierarchy Variance
+variable {P : Type u7} {I : Type u8} [Fintype P] [DecidableEq P] [Fintype I]
+
+lemma integral_split (d : Depth P) (w : I → ℝ) (p : I → P → ℝ) (x : I → P)
+    (hw : ∀ i, 0 ≤ w i) (hW : ∑ i, w i = 1) (hp : ∀ i y, 0 ≤ p i y)
+    (hp1 : ∀ i, ∑ y, p i y = 1) (hx : ∀ i, p i (x i) = 0) (hn : 2 ≤ Fintype.card P)
+    (B τ lam : ℝ) (hB : 0 ≤ B) (hlam : 0 ≤ lam) (hτ : 0 < τ)
+    (hτn : 2 ≤ τ*(Fintype.card P:ℝ))
+    (hdom : ∀ a b, a ≠ b → edgeLaw w p x a b ≤ B/((Fintype.card P:ℝ)*((Fintype.card P:ℝ)-1))) :
+    (∑ C, d.mass C * slice w p x lam C) ≤
+      (2*lam/τ^2)*(∑ C, d.mass C*pairCell Finset.univ C) +
+      B*τ/(Fintype.card P:ℝ)*(∑ y, d.potential y) +
+      ∑ i, w i*(∫ r, activeMass d (p i) (x i) (τ*Fintype.card P) lam r) := by
+  classical
+  have hi₁ := (d.integrable_sumCells hn (pairCell Finset.univ) (singleton_pairCell _)).const_mul (2*lam/τ^2)
+  have hi₂ := (integrable_finsetSum (s:=Finset.univ) (fun y _ => d.integrable_outsideIndicator hn y)).const_mul (B*τ/Fintype.card P)
+  have hi₃ := integrable_finsetSum (s:=Finset.univ) (fun i _ =>
+    (activeMass_integrable d hn (p i) (hp i) (hp1 i) (x i) (hx i) (τ*Fintype.card P) lam).const_mul (w i))
+  have hb := integral_mono (d.integrable_sumCells hn _ (slice_singleton w p x hx lam))
+    ((hi₁.add hi₂).add hi₃) (fun r => pointwise_split d w p x hw hW hp hx hn B τ lam r hB hlam hτ hτn hdom)
+  have ha := integral_add (hi₁.add hi₂) hi₃
+  have hb' := integral_add hi₁ hi₂
+  simp only [Pi.add_apply] at hb ha hb'
+  rw [ha,hb',integral_const_mul,integral_const_mul,
+    d.integral_sumCells hn _ (singleton_pairCell _),
+    d.integral_sumCells hn _ (slice_singleton w p x hx lam),
+    integral_finsetSum _ (fun y _ => d.integrable_outsideIndicator hn y),
+    integral_finsetSum _ (fun i _ =>
+      (activeMass_integrable d hn (p i) (hp i) (hp1 i) (x i) (hx i) (τ*Fintype.card P) lam).const_mul (w i))] at hb
+  simpa only [integral_const_mul,Depth.potential] using hb
+
+lemma local_small_fiber (d d' : Depth P) (w : I → ℝ) (p : I → P → ℝ) (x : I → P)
+    (hw : ∀ i, 0 ≤ w i) (hW : ∑ i, w i = 1) (hp : ∀ i y, 0 ≤ p i y)
+    (hp1 : ∀ i, ∑ y, p i y = 1) (hx : ∀ i, p i (x i) = 0) (hn : 2 ≤ Fintype.card P)
+    (h : I → ℝ) (B τ lam H : ℝ) (hB : 0 ≤ B) (hlam : 0 ≤ lam) (hlam2 : lam ≤ 1/2)
+    (hτ : 0 < τ) (hτ1 : τ ≤ 1) (hτn : 2 ≤ τ*(Fintype.card P:ℝ)) (hH : 0 ≤ H)
+    (hdom : ∀ a b, a ≠ b → edgeLaw w p x a b ≤ B/((Fintype.card P:ℝ)*((Fintype.card P:ℝ)-1))) :
+    (∑ C, d.mass C * slice w p x lam C) ≤
+      (2*lam/τ^2)*(∑ C, d.mass C*pairCell Finset.univ C) +
+      B*τ/(Fintype.card P:ℝ)*((∑ y, d.potential y)+(∑ y, d'.potential y)) +
+      2*lam*(∑ i, w i*∑ y, p i y*∫ r, d'.pairOverlap r (x i) y) + 2*lam*H +
+      ∑ i, w i*∑ y, p i y*max (|d.value (x i) y+d'.value (x i) y-h i|-H) 0 := by
+  have hb := integral_split d w p x hw hW hp hp1 hx hn B τ lam hB hlam hτ hτn hdom
+  have ha := Finset.sum_le_sum (s:=Finset.univ) (fun i _ => mul_le_mul_of_nonneg_left
+    (active_integral d d' hn (p i) (hp i) (hp1 i) (x i) (hx i) (h i) H (τ*Fintype.card P) lam hH hlam hlam2) (hw i))
+  have hnbr := fiber_neighbor_bound d d' hn w p x B τ hB hτ hτ1 hdom
+  simp only [mul_add,Finset.sum_add_distrib] at ha
+  have hc : (∑ i, w i*(2*lam*H)) = 2*lam*H := by rw [←Finset.sum_mul,hW,one_mul]
+  have hi : (∑ i, w i*(2*lam*(∑ y, p i y*∫ r, d'.pairOverlap r (x i) y))) =
+      2*lam*(∑ i, w i*(∑ y, p i y*∫ r, d'.pairOverlap r (x i) y)) := by
+    rw [Finset.mul_sum]
+    exact Finset.sum_congr rfl (fun i _ => by ring)
+  rw [hc,hi] at ha
+  linarith
+
+end WeakPinned.SmallFiber
+
+namespace WeakPinned.SmallFiber
+open MeasureTheory Hierarchy Variance
+variable {P : Type u9} {I : Type u10} [Fintype P] [DecidableEq P] [Fintype I]
+
+lemma pair_integral_edgeCell (d : Depth P) (hn : 2 ≤ Fintype.card P)
+    (w : I → ℝ) (p : I → P → ℝ) (x : I → P) (hx : ∀ i, p i (x i) = 0) :
+    (∑ i, w i*∑ y, p i y * ∫ r, d.pairOverlap r (x i) y) =
+      ∑ C, d.mass C * edgeCell w p x C := by
+  classical
+  have hpair (i : I) (y : P) :
+      p i y * (∫ r, d.pairOverlap r (x i) y) =
+      ∑ C, d.mass C * (if x i ∈ C ∧ y ∈ C then p i y else 0) := by
+    by_cases hxy : x i = y
+    · subst y; simp [hx]
+    · have h := d.mass_overlap hn (x i) y hxy
+      dsimp [Depth.overlapLength] at h
+      rw [← h, Finset.mul_sum]
+      apply Finset.sum_congr rfl
+      intro C _
+      split_ifs <;> ring
+  simp_rw [hpair,Finset.mul_sum]
+  conv_lhs => arg 2; ext i; rw [Finset.sum_comm]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro C _
+  rw [edgeCell_formula,Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro i _
+  by_cases hi : x i ∈ C
+  · simp only [hi,true_and,ite_true,mass_as_sum,Finset.mul_sum]
+    apply Finset.sum_congr rfl
+    intro y _
+    split_ifs <;> ring
+  · simp [hi]
+
+lemma local_small_fiber_reduced (d d' : Depth P) (w : I → ℝ) (p : I → P → ℝ) (x : I → P)
+    (hw : ∀ i, 0 ≤ w i) (hW : ∑ i, w i = 1) (hp : ∀ i y, 0 ≤ p i y)
+    (hp1 : ∀ i, ∑ y, p i y = 1) (hx : ∀ i, p i (x i) = 0) (hn : 2 ≤ Fintype.card P)
+    (h : I → ℝ) (B τ lam H : ℝ) (hB : 0 ≤ B) (hlam : 0 ≤ lam) (hlam2 : lam ≤ 1/2)
+    (hτ : 0 < τ) (hτ1 : τ ≤ 1) (hτn : 2 ≤ τ*(Fintype.card P:ℝ)) (hH : 0 ≤ H)
+    (hdom : ∀ a b, a ≠ b → edgeLaw w p x a b ≤ B/((Fintype.card P:ℝ)*((Fintype.card P:ℝ)-1))) :
+    (∑ C, d.mass C * slice w p x lam C) ≤
+      (2*lam/τ^2)*(∑ C, d.mass C*pairCell Finset.univ C) +
+      B*τ/(Fintype.card P:ℝ)*((∑ y, d.potential y)+(∑ y, d'.potential y)) +
+      2*lam*B*(∑ C, d'.mass C*pairCell Finset.univ C) + 2*lam*H +
+      ∑ i, w i*∑ y, p i y*max (|d.value (x i) y+d'.value (x i) y-h i|-H) 0 := by
+  have hb := local_small_fiber d d' w p x hw hW hp hp1 hx hn h B τ lam H hB hlam hlam2 hτ hτ1 hτn hH hdom
+  rw [pair_integral_edgeCell d' hn w p x hx] at hb
+  have hd : (∑ C, d'.mass C*edgeCell w p x C) ≤ B*∑ C, d'.mass C*pairCell Finset.univ C := by
+    rw [Finset.mul_sum]
+    apply Finset.sum_le_sum
+    intro C _
+    calc
+      _ ≤ d'.mass C*(B*pairCell Finset.univ C) := mul_le_mul_of_nonneg_left (edgeCell_dom w p x hx B hdom C) (d'.mass_nonneg C)
+      _ = _ := by ring
+  have hh := mul_le_mul_of_nonneg_left hd (show 0 ≤ 2*lam by positivity)
+  nlinarith
+end WeakPinned.SmallFiber
+
+end
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PinnedDistances/FiberTransport.lean b/lean/OAI/Geometry/PinnedDistances/FiberTransport.lean
new file mode 100644
index 0000000000000000000000000000000000000000..6d2208e5abe6516984569cd2d12252c61ca264b3
--- /dev/null
+++ b/lean/OAI/Geometry/PinnedDistances/FiberTransport.lean
@@ -0,0 +1,503 @@
+import OAI.Geometry.PinnedDistances.CutGeometry
+
+namespace OAI
+
+universe u1 u2 u3 u4 u5 u6 u7 u8 u9 u10
+
+section
+noncomputable section
+open NumberField
+open scoped BigOperators
+noncomputable section
+open MeasureTheory
+open scoped Topology
+noncomputable section
+open MeasureTheory
+open scoped Topology ENNReal
+noncomputable section
+open MeasureTheory NumberField
+open scoped BigOperators
+noncomputable section
+open scoped BigOperators
+open MeasureTheory
+noncomputable section
+open Filter
+open scoped Topology
+noncomputable section
+open scoped BigOperators
+noncomputable section
+open Filter
+open scoped Topology
+noncomputable section
+open MeasureTheory
+open scoped BigOperators
+noncomputable section
+noncomputable section
+open scoped BigOperators
+
+namespace WeakPinned.Hierarchy.Depth
+open MeasureTheory Cuts
+open scoped BigOperators
+variable {P : Type u1} [Fintype P] [DecidableEq P]
+variable (d : Depth P)
+
+lemma cutLength_bound (f : Finset P → ℝ) (hfu : f Finset.univ=0)
+    (hfs : ∀ x, f {x}=0) (C : Finset P) :
+    |d.cutLength f C|≤2*d.radius*|f C| := by
+  rw [← Real.norm_eq_abs,cutLength]
+  apply (norm_integral_le_integral_norm _).trans
+  exact integral_bound_supported d.radius_nonneg (d.integrable_cutAt f hfu hfs C).norm
+    (d.cutAt_bound f C) (fun r hr => by rw [d.cutAt_vanish f hfu hfs C r hr,norm_zero])
+
+lemma sum_cutLength_univ (f : Finset P → ℝ) (g : Finset P → ℝ) :
+    (∑ C ∈ d.allCuts, d.cutLength f C*g C)=∑ C, d.cutLength f C*g C := by
+  exact Finset.sum_subset (Finset.subset_univ _) (fun C _ hC => by rw [d.cutLength_outside f C hC,zero_mul])
+
+lemma cutLength_mul (f g : Finset P → ℝ) (C : Finset P) :
+    d.cutLength (fun B => f B*g B) C=d.cutLength f C*g C := by
+  unfold cutLength
+  rw [← integral_mul_const]
+  congr 1
+  funext r
+  simp only [cutAt,ite_mul,zero_mul]
+
+lemma cutLength_difference_le (f g : Finset P → ℝ)
+    (hfu : f Finset.univ=0) (hfs : ∀ x, f {x}=0)
+    (hgu : g Finset.univ=0) (hgs : ∀ x, g {x}=0) (C : Finset P) :
+    |d.cutLength f C-d.cutLength g C|≤d.cutLength (fun B => |f B-g B|) C := by
+  rw [cutLength,cutLength,← integral_sub (d.integrable_cutAt f hfu hfs C) (d.integrable_cutAt g hgu hgs C)]
+  rw [← Real.norm_eq_abs]
+  apply (norm_integral_le_integral_norm _).trans_eq
+  congr 1
+  funext r
+  simp only [cutAt,Real.norm_eq_abs]
+  split_ifs <;> simp
+
+lemma distance_cut_difference (f g : Finset P → ℝ)
+    (hfu : f Finset.univ=0) (hfs : ∀ x, f {x}=0)
+    (hgu : g Finset.univ=0) (hgs : ∀ x, g {x}=0) (x y : P) :
+    |distance d.allCuts (d.cutLength f) x y-distance d.allCuts (d.cutLength g) x y|≤
+      ∑ C ∈ d.allCuts, d.cutLength (fun B => |f B-g B|) C := by
+  unfold distance
+  rw [← Finset.sum_sub_distrib]
+  apply (Finset.abs_sum_le_sum_abs _ _).trans
+  apply Finset.sum_le_sum
+  intro C hC
+  rw [← sub_mul,abs_mul,abs_abs]
+  have hi : |ind C x-ind C y|≤1 := by unfold ind; split_ifs <;> norm_num
+  have h1 := mul_le_mul_of_nonneg_right (d.cutLength_difference_le f g hfu hfs hgu hgs C) (abs_nonneg (ind C x-ind C y))
+  exact h1.trans (by simpa using mul_le_mul_of_nonneg_left hi (d.cutLength_nonneg _ (fun B => abs_nonneg _) C))
+
+lemma root_cut_difference (f g : Finset P → ℝ)
+    (hfu : f Finset.univ=0) (hfs : ∀ x, f {x}=0)
+    (hgu : g Finset.univ=0) (hgs : ∀ x, g {x}=0) (y : P) :
+    |rootDistance d.allCuts (d.cutLength f) y-rootDistance d.allCuts (d.cutLength g) y|≤
+      ∑ C ∈ d.allCuts, d.cutLength (fun B => |f B-g B|) C := by
+  unfold rootDistance
+  rw [← Finset.sum_sub_distrib]
+  apply (Finset.abs_sum_le_sum_abs _ _).trans
+  apply Finset.sum_le_sum
+  intro C hC
+  rw [← sub_mul,abs_mul]
+  have hi : |ind C y|≤1 := by unfold ind; split_ifs <;> norm_num
+  have h1 := mul_le_mul_of_nonneg_right (d.cutLength_difference_le f g hfu hfs hgu hgs C) (abs_nonneg (ind C y))
+  exact h1.trans (by simpa using mul_le_mul_of_nonneg_left hi (d.cutLength_nonneg _ (fun B => abs_nonneg _) C))
+
+lemma cost_cut_formula (f : Finset P → ℝ) (p a : P → ℝ) :
+    cost d.allCuts (d.cutLength f) p a =
+      ∑ C ∈ d.allCuts, d.cutLength (fun B => f B*|Cuts.mass p B-Cuts.mass a B|) C := by
+  simp only [cost,d.cutLength_mul]
+
+variable {Ω : Type u2} [MeasurableSpace Ω] (D : Ω → Depth P)
+    (hD : ∀ x y, Measurable (fun ω => (D ω).value x y))
+include hD
+lemma measurable_cutLength_param (f : Finset P → ℝ) (C : Finset P) :
+    Measurable (fun ω => (D ω).cutLength f C) := by
+  have hm : Measurable (fun p : Ω×ℝ => (D p.1).cutAt f C p.2) :=
+    (measurable_of_finite (fun s : Finset (Finset P) => if C ∈ s then f C else 0)).comp
+    (measurable_cells_param (P:=P) (Ω:=Ω×ℝ) (fun p : Ω×ℝ => D p.1)
+      (fun x y => (hD x y).comp measurable_fst) Prod.snd measurable_snd)
+  exact hm.stronglyMeasurable.integral_prod_right.measurable
+
+lemma integrable_cutLength_param {μ : Measure Ω}
+    (hi : ∀ x y, Integrable (fun ω => (D ω).value x y) μ)
+    (f : Finset P → ℝ) (hfu : f Finset.univ=0) (hfs : ∀ x, f {x}=0) (C : Finset P) :
+    Integrable (fun ω => (D ω).cutLength f C) μ := by
+  apply (((integrable_radius_param D hi).const_mul 2).mul_const |f C|).mono'
+    (measurable_cutLength_param D hD f C).aestronglyMeasurable
+  exact Filter.Eventually.of_forall (fun ω => (D ω).cutLength_bound f hfu hfs C)
+
+lemma integrable_cutSum_param {μ : Measure Ω}
+    (hi : ∀ x y, Integrable (fun ω => (D ω).value x y) μ)
+    (f g : Finset P → ℝ) (hfu : f Finset.univ=0) (hfs : ∀ x, f {x}=0) :
+    Integrable (fun ω => ∑ C ∈ (D ω).allCuts, (D ω).cutLength f C*g C) μ := by
+  simp only [sum_cutLength_univ]
+  exact integrable_finsetSum _ (fun C _ => (integrable_cutLength_param D hD hi f hfu hfs C).mul_const (g C))
+
+lemma integrable_cutDistance_param {μ : Measure Ω}
+    (hi : ∀ x y, Integrable (fun ω => (D ω).value x y) μ)
+    (f : Finset P → ℝ) (hfu : f Finset.univ=0) (hfs : ∀ x, f {x}=0) (x y : P) :
+    Integrable (fun ω => distance (D ω).allCuts ((D ω).cutLength f) x y) μ :=
+  integrable_cutSum_param D hD hi f (fun B => |ind B x-ind B y|) hfu hfs
+
+lemma integrable_rootDistance_param {μ : Measure Ω}
+    (hi : ∀ x y, Integrable (fun ω => (D ω).value x y) μ)
+    (f : Finset P → ℝ) (hfu : f Finset.univ=0) (hfs : ∀ x, f {x}=0) (y : P) :
+    Integrable (fun ω => rootDistance (D ω).allCuts ((D ω).cutLength f) y) μ :=
+  integrable_cutSum_param D hD hi f (fun B => ind B y) hfu hfs
+end WeakPinned.Hierarchy.Depth
+
+namespace WeakPinned.Cuts
+lemma tent_lipschitzWith (δ : ℝ) : LipschitzWith 1 (tent δ) := by
+  rw [lipschitzWith_iff_dist_le_mul]
+  intro x y
+  simpa only [Real.dist_eq,NNReal.coe_one,one_mul] using tent_lipschitz δ x y
+end WeakPinned.Cuts
+
+namespace WeakPinned.Global
+open Hierarchy MeasureTheory Variance Cuts
+open scoped BigOperators
+variable {P : Type u3} [Fintype P] [DecidableEq P]
+variable {K : Type u4} [Field K] [NumberField K]
+variable (Z₁ Z₂ : P → K) (h₁ : Function.Injective Z₁) (h₂ : Function.Injective Z₂)
+include h₁ h₂
+
+lemma arch_cross_defect (v : NumberField.InfinitePlace K) (x y z : P)
+    (hxy : x≠y) (hxz : x≠z) (hyz : y≠z) (t : K)
+    (hy : (Z₁ x-Z₁ y)*(Z₂ x-Z₂ y)=t) (hz : (Z₁ x-Z₁ z)*(Z₂ x-Z₂ z)=t) :
+    (fun ω => (arch Z₁ v ω).value y z-(arch Z₁ v ω).value x z-
+      ((arch Z₂ v ω).value y z+(Real.log (v t)-Real.log (v (Z₁ x-Z₁ y))))) =ᵐ[Grid.fullLaw]
+    (fun ω => Grid.error ω (v.embedding (Z₁ y)) (v.embedding (Z₁ z))-
+      Grid.error ω (v.embedding (Z₁ x)) (v.embedding (Z₁ z))-
+      Grid.error ω (v.embedding (Z₂ y)) (v.embedding (Z₂ z))) := by
+  have he := Arithmetic.raw_cross_fiber Z₁ Z₂ h₁ h₂ v.val x y z hxy hxz hyz t hy hz
+  filter_upwards [arch_value_ae Z₁ h₁ v hyz,arch_value_ae Z₁ h₁ v hxz,arch_value_ae Z₂ h₂ v hyz] with ω ha hb hc
+  rw [ha,hb,hc]
+  simp only [Grid.error,← map_sub,NumberField.InfinitePlace.norm_embedding_eq]
+  change -Real.log (v (Z₂ y-Z₂ z)) = -Real.log (v (Z₁ y-Z₁ z)) -
+    -Real.log (v (Z₁ x-Z₁ z)) - Real.log (v t) - -Real.log (v (Z₁ x-Z₁ y)) at he
+  linarith
+
+lemma arch_cross_defect_mean (v : NumberField.InfinitePlace K) (x y z : P)
+    (hxy : x≠y) (hxz : x≠z) (hyz : y≠z) (t : K)
+    (hy : (Z₁ x-Z₁ y)*(Z₂ x-Z₂ y)=t) (hz : (Z₁ x-Z₁ z)*(Z₂ x-Z₂ z)=t) :
+    (∫ ω, |(arch Z₁ v ω).value y z-(arch Z₁ v ω).value x z-
+      ((arch Z₂ v ω).value y z+(Real.log (v t)-Real.log (v (Z₁ x-Z₁ y))))| ∂Grid.fullLaw)≤12 := by
+  have hEq := arch_cross_defect Z₁ Z₂ h₁ h₂ v x y z hxy hxz hyz t hy hz
+  have heq := integral_congr_ae (hEq.fun_comp (abs : ℝ → ℝ))
+  dsimp only [Function.comp_def] at heq
+  rw [heq]
+  have ha := Grid.error_integrable_full (embedding_ne Z₁ h₁ v hyz)
+  have hb := Grid.error_integrable_full (embedding_ne Z₁ h₁ v hxz)
+  have hc := Grid.error_integrable_full (embedding_ne Z₂ h₂ v hyz)
+  have hh := integral_mono ((ha.sub hb).sub hc).abs ((ha.abs.add hb.abs).add hc.abs) (fun ω =>
+    (abs_sub _ _).trans (add_le_add (abs_sub _ _) le_rfl))
+  dsimp only [Pi.sub_apply,Pi.add_apply] at hh
+  have hAdd := integral_add (ha.abs.add hb.abs) hc.abs
+  have hAdd2 := integral_add ha.abs hb.abs
+  dsimp only [Pi.add_apply] at hAdd hAdd2
+  rw [hAdd,hAdd2] at hh
+  have h1 := Grid.mean_abs_error_full (embedding_ne Z₁ h₁ v hyz)
+  have h2 := Grid.mean_abs_error_full (embedding_ne Z₁ h₁ v hxz)
+  have h3 := Grid.mean_abs_error_full (embedding_ne Z₂ h₂ v hyz)
+  linarith
+
+/-- Actual local cut projection under an arithmetic fiber probability. The
+atom at y is retained and the source has zero mass. -/
+lemma arch_fiber_cut (v : NumberField.InfinitePlace K) (p : P → ℝ)
+    (hp0 : ∀ z, 0≤p z) (hp : ∑ z, p z=1) (x y : P) (hpx : p x=0) (hpy : p y≠0)
+    (t : K) (ht : ∀ z, p z≠0 → (Z₁ x-Z₁ z)*(Z₂ x-Z₂ z)=t)
+    (δ : ℝ) (hδ : 0≤δ) (hps : ∀ z, p z≤δ) :
+    (∫ ω, |distance (arch Z₁ v ω).allCuts ((arch Z₁ v ω).cutLength (fun C => tent δ (mass p C))) x y-
+      rootDistance (arch Z₂ v ω).allCuts ((arch Z₂ v ω).cutLength (fun C => tent δ (mass p C))) y| ∂Grid.fullLaw)≤12 := by
+  have hxy : x≠y := fun hh => hpy (hh ▸ hpx)
+  let f : Finset P → ℝ := fun C => tent δ (mass p C)
+  have hfu : f Finset.univ=0 := by simp only [f,mass,hp]; exact tent_zero_right (by linarith)
+  have hfs (z) : f {z}=0 := by simp only [f,mass,Finset.sum_singleton]; exact tent_zero_left (hps z)
+  have hl := Depth.integrable_cutDistance_param (arch Z₁ v) (arch_measurable Z₁ h₁ v) (arch_integrable Z₁ h₁ v) f hfu hfs x y
+  have hr := Depth.integrable_rootDistance_param (arch Z₂ v) (arch_measurable Z₂ h₂ v) (arch_integrable Z₂ h₂ v) f hfu hfs y
+  let c := Real.log (v t)-Real.log (v (Z₁ x-Z₁ y))
+  let E : Grid.FullSample → ℝ := fun ω => ∑ z, Tail.offTwo p x y z*
+    |(arch Z₁ v ω).value y z-(arch Z₁ v ω).value x z-((arch Z₂ v ω).value y z+c)|
+  have hEi (z : P) : Integrable (fun ω => Tail.offTwo p x y z*
+    |(arch Z₁ v ω).value y z-(arch Z₁ v ω).value x z-((arch Z₂ v ω).value y z+c)|) Grid.fullLaw :=
+    ((((arch_integrable Z₁ h₁ v y z).sub (arch_integrable Z₁ h₁ v x z)).sub
+      ((arch_integrable Z₂ h₂ v y z).add (integrable_const c))).abs.const_mul _)
+  have hi : Integrable E Grid.fullLaw := integrable_finsetSum _ (fun z _ => hEi z)
+  have hh := integral_mono (hl.sub hr).abs hi (fun ω => by
+    simpa only [f,E,Pi.sub_apply,NNReal.coe_one,one_mul] using Tail.fiber_cut_error
+      (arch Z₁ v ω) (arch Z₂ v ω) p hp0 hp (tent δ) 1 (tent_lipschitzWith δ)
+      (tent_zero_right (by linarith)) (fun z => tent_zero_left (hps z)) (tent_symm δ) x y hpx hxy c)
+  apply hh.trans
+  unfold E
+  rw [integral_finsetSum Finset.univ (fun z _ => hEi z)]
+  simp only [integral_const_mul]
+  apply le_trans (Finset.sum_le_sum (fun z _ => show Tail.offTwo p x y z* _≤12*p z from ?_))
+    (by rw [← Finset.mul_sum,hp,mul_one])
+  by_cases hzx : z=x
+  · subst z; simp [Tail.offTwo,hpx]
+  by_cases hzy : z=y
+  · subst z; simp only [Tail.offTwo,or_true,reduceIte,zero_mul]; exact mul_nonneg (by norm_num) (hp0 y)
+  by_cases hpz : p z=0
+  · simp [Tail.offTwo,hzx,hzy,hpz]
+  · simp only [Tail.offTwo,hzx,hzy,or_self,reduceIte]
+    have he := arch_cross_defect_mean Z₁ Z₂ h₁ h₂ v x y z hxy (Ne.symm hzx) (Ne.symm hzy) t (ht y hpy) (ht z hpz)
+    dsimp [c]
+    simpa [mul_comm] using mul_le_mul_of_nonneg_left he (hp0 z)
+
+end WeakPinned.Global
+noncomputable section
+
+namespace WeakPinned.Global
+open Hierarchy MeasureTheory Variance Cuts
+open scoped BigOperators
+variable {P : Type u5} [Fintype P] [DecidableEq P]
+variable {K : Type u6} [Field K] [NumberField K]
+variable (Z₁ Z₂ : P → K) (h₁ : Function.Injective Z₁) (h₂ : Function.Injective Z₂)
+include h₁ h₂
+
+lemma finite_fiber_cut (v : NumberField.FinitePlace K) (p : P → ℝ)
+    (hp0 : ∀ z, 0≤p z) (hp : ∑ z, p z=1) (x y : P) (hpx : p x=0) (hpy : p y≠0)
+    (t : K) (ht : ∀ z, p z≠0 → (Z₁ x-Z₁ z)*(Z₂ x-Z₂ z)=t)
+    (δ : ℝ) (hδ : 0≤δ) (hps : ∀ z, p z≤δ) :
+    |distance (finite Z₁ h₁ v).allCuts ((finite Z₁ h₁ v).cutLength (fun C => tent δ (mass p C))) x y-
+      rootDistance (finite Z₂ h₂ v).allCuts ((finite Z₂ h₂ v).cutLength (fun C => tent δ (mass p C))) y|≤0 := by
+  have hxy : x≠y := fun hh => hpy (hh ▸ hpx)
+  let c := Real.log (v t)-Real.log (v (Z₁ x-Z₁ y))
+  have hh := Tail.fiber_cut_error (finite Z₁ h₁ v) (finite Z₂ h₂ v) p hp0 hp (tent δ) 1
+    (tent_lipschitzWith δ) (tent_zero_right (by linarith)) (fun z => tent_zero_left (hps z))
+    (tent_symm δ) x y hpx hxy c
+  simp only [NNReal.coe_one,one_mul] at hh
+  apply hh.trans
+  apply le_of_eq
+  apply Finset.sum_eq_zero
+  intro z _
+  by_cases hxz : x=z
+  · subst z; simp [Tail.offTwo]
+  by_cases hyz : y=z
+  · subst z; simp [Tail.offTwo]
+  by_cases hpz : p z=0
+  · simp [Tail.offTwo,hpz]
+  · have he := Arithmetic.raw_cross_fiber Z₁ Z₂ h₁ h₂ v.val x y z hxy hxz hyz t (ht y hpy) (ht z hpz)
+    change -Real.log (v (Z₂ y-Z₂ z)) = -Real.log (v (Z₁ y-Z₁ z)) -
+      -Real.log (v (Z₁ x-Z₁ z)) - Real.log (v t) - -Real.log (v (Z₁ x-Z₁ y)) at he
+    suffices (finite Z₁ h₁ v).value y z-(finite Z₁ h₁ v).value x z-
+      ((finite Z₂ h₂ v).value y z+c)=0 by rw [this,abs_zero,mul_zero]
+    simp only [finite,ite_eq_right hyz,ite_eq_right hxz,c]
+    linarith
+
+omit [NumberField K] in
+lemma integrable_arch_cut_error (v : NumberField.InfinitePlace K) (p : P → ℝ)
+    (δ : ℝ) (hpu : tent δ (mass p Finset.univ)=0) (hps : ∀ x, tent δ (mass p {x})=0)
+    (x y : P) :
+    Integrable (fun ω => |distance (arch Z₁ v ω).allCuts ((arch Z₁ v ω).cutLength (fun C => tent δ (mass p C))) x y-
+      rootDistance (arch Z₂ v ω).allCuts ((arch Z₂ v ω).cutLength (fun C => tent δ (mass p C))) y|) Grid.fullLaw := by
+  exact ((Depth.integrable_cutDistance_param (arch Z₁ v) (arch_measurable Z₁ h₁ v)
+    (arch_integrable Z₁ h₁ v) _ hpu hps x y).sub
+    (Depth.integrable_rootDistance_param (arch Z₂ v) (arch_measurable Z₂ h₂ v)
+      (arch_integrable Z₂ h₂ v) _ hpu hps y)).abs
+
+variable {I : Type u7} [Fintype I]
+omit Z₁ Z₂ h₁ h₂ in
+def fiberCutDiscrepancy (w : I → ℝ) (p : I → P → ℝ) (src : I → P) (δ : ℝ) (d d' : Depth P) : ℝ :=
+  ∑ i, w i*∑ y, p i y*|distance d.allCuts (d.cutLength (fun C => tent δ (mass (p i) C))) (src i) y-
+    rootDistance d'.allCuts (d'.cutLength (fun C => tent δ (mass (p i) C))) y|
+
+omit [NumberField K] in
+lemma fiber_cut_discrepancy_integrable (w : I → ℝ) (p : I → P → ℝ) (src : I → P)
+    (hp : ∀ i, ∑ z, p i z=1) (δ : ℝ) (hδ : 0≤δ) (hps : ∀ i z, p i z≤δ) :
+    AverageIntegrable Z₁ Z₂ (fiberCutDiscrepancy w p src δ) := by
+  intro v
+  apply integrable_finsetSum
+  intro i _
+  apply Integrable.const_mul
+  apply integrable_finsetSum
+  intro y _
+  apply Integrable.const_mul
+  exact integrable_arch_cut_error Z₁ Z₂ h₁ h₂ v (p i) δ
+    (by simp only [mass,hp]; exact tent_zero_right (by linarith))
+    (fun z => by simp only [mass,Finset.sum_singleton]; exact tent_zero_left (hps i z)) (src i) y
+
+/-- A bounded projection estimate with exact fiber atoms and no restriction on
+    degree, finite places or variation of number fields. -/
+lemma arithmetic_fiber_cut (w : I → ℝ) (p : I → P → ℝ) (src : I → P) (t : I → K)
+    (hw : ∀ i, 0≤w i) (hW : ∑ i, w i=1) (hp0 : ∀ i z, 0≤p i z)
+    (hp : ∀ i, ∑ z, p i z=1) (hpx : ∀ i, p i (src i)=0)
+    (ht : ∀ i z, p i z≠0 → (Z₁ (src i)-Z₁ z)*(Z₂ (src i)-Z₂ z)=t i)
+    (δ : ℝ) (hδ : 0≤δ) (hps : ∀ i z, p i z≤δ) :
+    average Z₁ Z₂ h₁ h₂ (fiberCutDiscrepancy w p src δ)≤12 := by
+  classical
+  have hi (v : NumberField.InfinitePlace K) (i : I) (y : P) :=
+    integrable_arch_cut_error Z₁ Z₂ h₁ h₂ v (p i) δ
+      (by simp only [mass,hp]; exact tent_zero_right (by linarith))
+      (fun z => by simp only [mass,Finset.sum_singleton]; exact tent_zero_left (hps i z)) (src i) y
+  have ha (v : NumberField.InfinitePlace K) :
+      (∫ ω, fiberCutDiscrepancy w p src δ (arch Z₁ v ω) (arch Z₂ v ω) ∂Grid.fullLaw)≤12 := by
+    unfold fiberCutDiscrepancy
+    rw [integral_finsetSum _ (fun i _ => (integrable_finsetSum _ (fun y _ => (hi v i y).const_mul (p i y))).const_mul (w i))]
+    simp only [integral_const_mul]
+    have he (i : I) : (∫ ω, ∑ y, p i y*|distance (arch Z₁ v ω).allCuts
+        ((arch Z₁ v ω).cutLength (fun C => tent δ (mass (p i) C))) (src i) y-
+        rootDistance (arch Z₂ v ω).allCuts ((arch Z₂ v ω).cutLength (fun C => tent δ (mass (p i) C))) y| ∂Grid.fullLaw)≤12 := by
+      rw [integral_finsetSum _ (fun y _ => (hi v i y).const_mul (p i y))]
+      simp only [integral_const_mul]
+      calc
+        _ ≤ ∑ y, p i y*12 := Finset.sum_le_sum (fun y _ => by
+          by_cases hy : p i y=0
+          · simp [hy]
+          · exact mul_le_mul_of_nonneg_left (arch_fiber_cut Z₁ Z₂ h₁ h₂ v (p i) (hp0 i) (hp i)
+              (src i) y (hpx i) hy (t i) (ht i) δ hδ (hps i)) (hp0 i y))
+        _ = 12 := by rw [← Finset.sum_mul,hp,one_mul]
+    exact (Finset.sum_le_sum (fun i _ => mul_le_mul_of_nonneg_left (he i) (hw i))).trans_eq
+      (by rw [← Finset.sum_mul,hW,one_mul])
+  have hv (v : NumberField.FinitePlace K) :
+      fiberCutDiscrepancy w p src δ (finite Z₁ h₁ v) (finite Z₂ h₂ v)≤0 := by
+    apply Finset.sum_nonpos
+    intro i _
+    apply mul_nonpos_of_nonneg_of_nonpos (hw i)
+    apply Finset.sum_nonpos
+    intro y _
+    by_cases hy : p i y=0
+    · simp [hy]
+    · exact mul_nonpos_of_nonneg_of_nonpos (hp0 i y) (finite_fiber_cut Z₁ Z₂ h₁ h₂ v
+        (p i) (hp0 i) (hp i) (src i) y (hpx i) hy (t i) (ht i) δ hδ (hps i))
+  change placeAverage (placeUnion Z₁ Z₂ h₁ h₂) _ _≤12
+  have hh := placeAverage_mono (placeUnion Z₁ Z₂ h₁ h₂) ha (fun v _ => hv v)
+  rwa [placeAverage_arch_const] at hh
+end WeakPinned.Global
+noncomputable section
+namespace WeakPinned.Cuts
+open MeasureTheory Hierarchy Variance
+open scoped BigOperators
+variable {P : Type u8} [Fintype P] [DecidableEq P]
+
+lemma mass_complement (p : P → ℝ) (hp : ∑ x, p x=1) (C : Finset P) : mass p Cᶜ=1-mass p C := by
+  have hh := Finset.sum_compl_add_sum C p
+  rw [hp] at hh
+  exact eq_sub_of_add_eq hh
+
+omit [Fintype P] [DecidableEq P] in
+lemma mass_small (p : P → ℝ) (δ : ℝ) (hδ : 0≤δ) (hp : ∀ x, p x≤δ)
+    (C : Finset P) (hC : C.card≤1) : mass p C≤δ := by
+  have hh := Finset.sum_le_sum (s:=C) (fun x _ => hp x)
+  simp only [Finset.sum_const,nsmul_eq_mul] at hh
+  have hc : (C.card:ℝ)≤1 := by exact_mod_cast hC
+  exact hh.trans (by nlinarith)
+
+omit [Fintype P] [DecidableEq P] in
+lemma cutError_small (p a : P → ℝ) (δ : ℝ) (hδ : 0≤δ)
+    (hp : ∀ x, p x≤δ) (ha : ∀ x, a x≤δ) (C : Finset P) (hC : C.card≤1) :
+    cutError δ (mass p C) (mass a C)=0 := by
+  simp only [cutError,tent_zero_left (mass_small p δ hδ hp C hC),
+    tent_zero_left (mass_small a δ hδ ha C hC),sub_self,abs_zero,zero_mul,add_zero]
+
+lemma cutError_complement (p a : P → ℝ) (hp : ∑ x, p x=1) (ha : ∑ x, a x=1)
+    (δ : ℝ) (C : Finset P) :
+    cutError δ (mass p Cᶜ) (mass a Cᶜ)=cutError δ (mass p C) (mass a C) := by
+  rw [mass_complement p hp,mass_complement a ha,cutError_symm]
+
+def charge (d : Depth P) (δ : ℝ) (p a : P → ℝ) : ℝ :=
+  ∑ C ∈ d.allCuts, d.cutLength (fun B => cutError δ (mass p B) (mass a B)) C
+
+lemma charge_nonneg (d : Depth P) (δ : ℝ) (p a : P → ℝ) : 0≤charge d δ p a :=
+  Finset.sum_nonneg (fun C _ => d.cutLength_nonneg _ (fun _ => cutError_nonneg _ _ _) C)
+
+lemma charge_occurrence (d : Depth P) (hn : 2≤Fintype.card P) (δ : ℝ) (hδ : 0≤δ)
+    (p a : P → ℝ) (hp : ∑ x, p x=1) (ha : ∑ x, a x=1)
+    (hps : ∀ x, p x≤δ) (has : ∀ x, a x≤δ) :
+    charge d δ p a=weightedIntegral d.mass (fun C => cutError δ (mass p C) (mass a C)) := by
+  symm
+  exact d.symmetric_integral_cut_sum hn _ (cutError_small p a δ hδ hps has)
+    (cutError_complement p a hp ha δ)
+
+omit [DecidableEq P] in
+lemma tent_mass_univ (p : P → ℝ) (hp : ∑ x, p x=1) (δ : ℝ) (hδ : 0≤δ) :
+    tent δ (mass p Finset.univ)=0 := by
+  simp only [mass,hp]
+  exact tent_zero_right (by linarith)
+omit [Fintype P] [DecidableEq P] in
+lemma tent_mass_singleton (p : P → ℝ) (δ : ℝ) (hp : ∀ x, p x≤δ) (x : P) :
+    tent δ (mass p {x})=0 := by
+  simp only [mass,Finset.sum_singleton]
+  exact tent_zero_left (hp x)
+
+end WeakPinned.Cuts
+namespace WeakPinned.Hierarchy.Depth
+open Cuts MeasureTheory
+variable {P : Type u9} [Fintype P] [DecidableEq P]
+lemma cutLength_mono (d : Depth P) (f g : Finset P → ℝ)
+    (hf : f Finset.univ=0) (hfs : ∀ x, f {x}=0) (hg : g Finset.univ=0) (hgs : ∀ x, g {x}=0)
+    (hfg : ∀ C, f C≤g C) (C : Finset P) : d.cutLength f C≤d.cutLength g C := by
+  apply integral_mono (d.integrable_cutAt f hf hfs C) (d.integrable_cutAt g hg hgs C)
+  intro r
+  simp only [cutAt]
+  split_ifs
+  · exact hfg C
+  · rfl
+end WeakPinned.Hierarchy.Depth
+
+namespace WeakPinned.Cuts
+open MeasureTheory Hierarchy Variance
+variable {P : Type u10} [Fintype P] [DecidableEq P]
+lemma distance_change_charge (d : Depth P) (δ : ℝ) (hδ : 0≤δ) (p a : P → ℝ)
+    (hp : ∑ x, p x=1) (ha : ∑ x, a x=1) (hps : ∀ x, p x≤δ) (has : ∀ x, a x≤δ) (x y : P) :
+    |distance d.allCuts (d.cutLength (fun C => tent δ (mass p C))) x y-
+      distance d.allCuts (d.cutLength (fun C => tent δ (mass a C))) x y|≤charge d δ p a := by
+  apply (d.distance_cut_difference _ _ (tent_mass_univ p hp δ hδ) (tent_mass_singleton p δ hps)
+    (tent_mass_univ a ha δ hδ) (tent_mass_singleton a δ has) x y).trans
+  apply Finset.sum_le_sum
+  intro C _
+  apply d.cutLength_mono
+  · simp only [tent_mass_univ p hp δ hδ,tent_mass_univ a ha δ hδ,sub_self,abs_zero]
+  · intro z; simp only [tent_mass_singleton p δ hps,tent_mass_singleton a δ has,sub_self,abs_zero]
+  · simp only [cutError,tent_mass_univ p hp δ hδ,tent_mass_univ a ha δ hδ,sub_self,abs_zero,zero_mul,add_zero]
+  · intro z; exact cutError_small p a δ hδ hps has _ (by simp)
+  · intro C; exact le_add_of_nonneg_right (mul_nonneg (tent_nonneg _ _) (abs_nonneg _))
+
+lemma root_change_charge (d : Depth P) (δ : ℝ) (hδ : 0≤δ) (p a : P → ℝ)
+    (hp : ∑ x, p x=1) (ha : ∑ x, a x=1) (hps : ∀ x, p x≤δ) (has : ∀ x, a x≤δ) (y : P) :
+    |rootDistance d.allCuts (d.cutLength (fun C => tent δ (mass p C))) y-
+      rootDistance d.allCuts (d.cutLength (fun C => tent δ (mass a C))) y|≤charge d δ p a := by
+  apply (d.root_cut_difference _ _ (tent_mass_univ p hp δ hδ) (tent_mass_singleton p δ hps)
+    (tent_mass_univ a ha δ hδ) (tent_mass_singleton a δ has) y).trans
+  apply Finset.sum_le_sum
+  intro C _
+  apply d.cutLength_mono
+  · simp only [tent_mass_univ p hp δ hδ,tent_mass_univ a ha δ hδ,sub_self,abs_zero]
+  · intro z; simp only [tent_mass_singleton p δ hps,tent_mass_singleton a δ has,sub_self,abs_zero]
+  · simp only [cutError,tent_mass_univ p hp δ hδ,tent_mass_univ a ha δ hδ,sub_self,abs_zero,zero_mul,add_zero]
+  · intro z; exact cutError_small p a δ hδ hps has _ (by simp)
+  · intro C; exact le_add_of_nonneg_right (mul_nonneg (tent_nonneg _ _) (abs_nonneg _))
+
+lemma cost_le_charge (d : Depth P) (δ : ℝ) (hδ : 0≤δ) (p a : P → ℝ)
+    (hp : ∑ x, p x=1) (ha : ∑ x, a x=1) (hps : ∀ x, p x≤δ) (has : ∀ x, a x≤δ) :
+    cost d.allCuts (d.cutLength (fun C => tent δ (mass a C))) p a≤charge d δ p a := by
+  rw [d.cost_cut_formula]
+  apply Finset.sum_le_sum
+  intro C _
+  apply d.cutLength_mono
+  · simp only [tent_mass_univ a ha δ hδ,zero_mul]
+  · intro z; simp only [tent_mass_singleton a δ has,zero_mul]
+  · simp only [cutError,tent_mass_univ p hp δ hδ,tent_mass_univ a ha δ hδ,sub_self,abs_zero,zero_mul,add_zero]
+  · intro z; exact cutError_small p a δ hδ hps has _ (by simp)
+  · intro C; exact le_add_of_nonneg_left (abs_nonneg _)
+end WeakPinned.Cuts
+noncomputable section
+
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PinnedDistances/FiniteLaw.lean b/lean/OAI/Geometry/PinnedDistances/FiniteLaw.lean
new file mode 100644
index 0000000000000000000000000000000000000000..52d6963e0d082dfd484b8378f734638863786470
--- /dev/null
+++ b/lean/OAI/Geometry/PinnedDistances/FiniteLaw.lean
@@ -0,0 +1,58 @@
+import OAI.Geometry.PinnedDistances.Model
+
+namespace OAI
+
+universe u1 u2 u3
+
+section
+noncomputable section
+open scoped BigOperators Topology ENNReal
+open Filter MeasureTheory
+namespace WeakPinned.Bounded
+
+variable {I : Type u1} {X : Type u2} [Fintype I] [MeasurableSpace X] [MeasurableSingletonClass X]
+
+def finiteLaw (w : I → ℝ) (U : I → X) : Measure X :=
+  ∑ i,ENNReal.ofReal (w i) • Measure.dirac (U i)
+
+omit [MeasurableSingletonClass X] in
+lemma finiteLaw_univ (w : I → ℝ) (U : I → X) (hw : ∀ i,0≤ w i) :
+    finiteLaw w U Set.univ=ENNReal.ofReal (∑ i,w i) := by
+  simp [finiteLaw,Measure.finsetSum_apply,ENNReal.ofReal_sum_of_nonneg (fun i _ => hw i)]
+
+omit [MeasurableSingletonClass X] in
+lemma finiteLaw_probability (w : I → ℝ) (U : I → X)
+    (hw : ∀ i,0≤ w i) (hW : ∑ i,w i=1) : IsProbabilityMeasure (finiteLaw w U) := by
+  constructor
+  rw [finiteLaw_univ w U hw,hW,ENNReal.ofReal_one]
+
+lemma integral_finiteLaw (w : I → ℝ) (U : I → X) (hw : ∀ i,0≤ w i) (f : X → ℝ) :
+    ∫ x,f x ∂ finiteLaw w U=∑ i,w i*f (U i) := by
+  rw [finiteLaw,integral_finsetSum_measure]
+  · simp only [integral_smul_measure,integral_dirac,ENNReal.toReal_ofReal (hw _),smul_eq_mul]
+  · intro i _
+    exact (integrable_dirac (by simp)).smul_measure ENNReal.ofReal_ne_top
+
+lemma finiteLaw_ae (w : I → ℝ) (U : I → X) {p : X → Prop}
+    (hp : ∀ i,w i≠0 → p (U i)) : ∀ᵐ x ∂ finiteLaw w U,p x := by
+  rw [finiteLaw,ae_finsetSum_measure_iff]
+  intro i _
+  by_cases h : w i=0
+  · simp [h]
+  · exact Measure.ae_smul_measure (by simpa only [ae_dirac_eq,Filter.eventually_pure] using hp i h) _
+
+omit [MeasurableSingletonClass X] in
+lemma finiteLaw_map {Y : Type u3} [MeasurableSpace Y] (w : I → ℝ) (U : I → X)
+    (f : X → Y) (hf : Measurable f) : (finiteLaw w U).map f=finiteLaw w (f ∘ U) := by
+  simp only [finiteLaw, Measure.map_finset_sum' hf.aemeasurable,
+    Measure.map_smul _ hf.aemeasurable, Measure.map_dirac' hf, Function.comp_def]
+
+def finiteProbability (w : I → ℝ) (U : I → X) (hw : ∀ i,0≤ w i) (hW : ∑ i,w i=1) :
+    ProbabilityMeasure X := ⟨finiteLaw w U,finiteLaw_probability w U hw hW⟩
+
+end WeakPinned.Bounded
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PinnedDistances/FiniteMixtures.lean b/lean/OAI/Geometry/PinnedDistances/FiniteMixtures.lean
new file mode 100644
index 0000000000000000000000000000000000000000..a1ef46549f3877e7810ccc0437d2eb48529d0906
--- /dev/null
+++ b/lean/OAI/Geometry/PinnedDistances/FiniteMixtures.lean
@@ -0,0 +1,139 @@
+import OAI.Geometry.PinnedDistances.EmpiricalCompactness
+
+namespace OAI
+
+universe u1 u2 u3 u4 u5 u6 u7
+
+section
+noncomputable section
+open MeasureTheory Filter
+open scoped BigOperators Topology ENNReal
+namespace WeakPinned.Bounded
+
+variable {I : Type u1} {V : Type u2} {X : Type u3} {Y : Type u4} [Fintype I] [Fintype V]
+  [MeasurableSpace X] [MeasurableSpace Y]
+
+lemma finiteLaw_fst (w : I → ℝ) (p : I → V → ℝ) (U : I → X)
+    (hw : ∀ i,0≤w i) (hp : ∀ i x,0≤p i x) (hpsum : ∀ i,∑ x,p i x=1) :
+    finiteLaw (edgeWeight w p) (fun r => U r.1)=finiteLaw w U := by
+  unfold finiteLaw edgeWeight
+  rw [Fintype.sum_prod_type]
+  apply Finset.sum_congr rfl
+  intro i _
+  dsimp only
+  rw [← Finset.sum_smul,← ENNReal.ofReal_sum_of_nonneg (fun x _ => mul_nonneg (hw i) (hp i x)),
+    ← Finset.mul_sum,hpsum,mul_one]
+
+lemma tripleLaw_edge (w : I → ℝ) (p : I → V → ℝ) (U : I × V → X)
+    (hw : ∀ i,0≤w i) (hp : ∀ i x,0≤p i x) (hpsum : ∀ i,∑ x,p i x=1) :
+    finiteLaw (tripleWeight w p) (fun r => U (r.1,r.2.1))=finiteLaw (edgeWeight w p) U := by
+  unfold finiteLaw tripleWeight edgeWeight
+  simp only [Fintype.sum_prod_type]
+  apply Finset.sum_congr rfl
+  intro i _
+  apply Finset.sum_congr rfl
+  intro x _
+  rw [show (∑ z,∑ t, ENNReal.ofReal (w i*p i x*p i z*p i t) • Measure.dirac (U (i,x))) =
+    ∑ z,ENNReal.ofReal (w i*p i x*p i z) • Measure.dirac (U (i,x)) by
+      apply Finset.sum_congr rfl
+      intro z _
+      rw [← Finset.sum_smul,← ENNReal.ofReal_sum_of_nonneg (fun t _ => mul_nonneg (mul_nonneg (mul_nonneg (hw i) (hp i x)) (hp i z)) (hp i t)),
+        ← Finset.mul_sum,hpsum,mul_one]]
+  rw [← Finset.sum_smul,← ENNReal.ofReal_sum_of_nonneg (fun z _ => mul_nonneg (mul_nonneg (hw i) (hp i x)) (hp i z)),
+    ← Finset.mul_sum,hpsum,mul_one]
+
+lemma finiteLaw_mono (w v : I → ℝ) (U : I → X) (hle : ∀ i,w i≤v i) :
+    finiteLaw w U≤finiteLaw v U := by
+  apply Measure.le_iff.mpr
+  intro s hs
+  simp only [finiteLaw,Measure.finsetSum_apply,Measure.smul_apply,smul_eq_mul]
+  apply Finset.sum_le_sum
+  intro i _
+  gcongr
+  exact hle i
+
+lemma finiteLaw_prod (w : I → ℝ) (v : V → ℝ) (U : I → X) (W : V → Y)
+    (hw : ∀ i,0≤w i) :
+    (finiteLaw w U).prod (finiteLaw v W)=
+      finiteLaw (fun r : I × V => w r.1*v r.2) (fun r => (U r.1,W r.2)) := by
+  simp only [finiteLaw,← Measure.sum_fintype,Measure.prod_sum,
+    Measure.prod_smul_left,Measure.prod_smul_right,Measure.dirac_prod_dirac,
+    smul_smul,ENNReal.ofReal_mul (hw _),mul_comm]
+
+lemma finiteLaw_smul (w : I → ℝ) (U : I → X) (c : ℝ) (hc : 0≤c) :
+    finiteLaw (fun i => c*w i) U=ENNReal.ofReal c •finiteLaw w U := by
+  simp only [finiteLaw,ENNReal.ofReal_mul hc,Finset.smul_sum,smul_smul]
+
+end WeakPinned.Bounded
+
+end
+end
+section
+noncomputable section
+open MeasureTheory Filter
+open scoped BigOperators Topology ENNReal
+namespace WeakPinned.Bounded
+open Variance
+variable {V : Type u5} {I : Type u6} {X : Type u7} [Fintype V] [DecidableEq V] [Fintype I]
+  [MeasurableSpace X] [MeasurableSingletonClass X]
+  [TopologicalSpace X] [BorelSpace X] [HasOuterApproxClosed X]
+variable (G : Core.Fibers V I)
+
+lemma graph_edge_eval (F : V → V → ℝ) :
+    (∑ e,G.fiberWeight e*∑ y,prob (G.targets e) y*F (G.source e) y)=
+      ∑ x,∑ y,G.edgeLaw x y*F x y := by
+  rw [← G.incoming_eval]
+  simp only [Finset.mul_sum,← mul_assoc,G.sampling_conditional]
+  exact Finset.sum_comm
+
+lemma graph_edge_law (U : V → V → X) (hG : 0<G.edgeCount)
+    (hB : ∀ e,(G.targets e).Nonempty) :
+    finiteLaw (edgeWeight G.fiberWeight (fun e => prob (G.targets e)))
+      (fun r => U (G.source r.1) r.2)=
+    finiteLaw (fun r : V × V => G.edgeLaw r.1 r.2) (fun r => U r.1 r.2) := by
+  have hw (r : I × V) : 0≤edgeWeight G.fiberWeight (fun e => prob (G.targets e)) r :=
+    mul_nonneg (G.fiberWeight_nonneg _) (Core.Fibers.prob_nn _ _)
+  let : IsProbabilityMeasure (finiteLaw (edgeWeight G.fiberWeight (fun e => prob (G.targets e)))
+      (fun r => U (G.source r.1) r.2)) := finiteLaw_probability _ _ hw
+        (sum_edgeWeight _ _ (G.sum_fiberWeight hG) (fun e => sum_prob _ (hB e)))
+  let : IsProbabilityMeasure (finiteLaw (fun r : V × V => G.edgeLaw r.1 r.2) (fun r => U r.1 r.2)) :=
+    finiteLaw_probability _ _ (fun r => G.edgeLaw_nonneg _ _)
+      (by simpa only [Fintype.sum_prod_type] using G.sum_edgeLaw hG)
+  apply ext_of_forall_integral_eq_of_IsFiniteMeasure
+  intro f
+  rw [integral_finiteLaw _ _ hw,integral_finiteLaw _ _ (fun r => G.edgeLaw_nonneg _ _)]
+  simp only [edgeWeight,Fintype.sum_prod_type,mul_assoc,← Finset.mul_sum]
+  exact graph_edge_eval G (fun x y => f (U x y))
+
+omit G in
+lemma pairProb_le_uniform (hn : 2≤Fintype.card V) (x y : V) :
+    pairProb Finset.univ x y≤2*(Fintype.card V:ℝ)⁻¹*(Fintype.card V:ℝ)⁻¹ := by
+  have hnR : (2:ℝ)≤Fintype.card V := by exact_mod_cast hn
+  have hn0 : (0:ℝ)<Fintype.card V := by linarith
+  simp only [pairProb,Finset.mem_univ,true_and,Finset.card_univ]
+  split_ifs
+  · field_simp
+    apply (div_le_iff₀ (by linarith : 0<(Fintype.card V:ℝ)-1)).mpr
+    nlinarith
+  · positivity
+
+omit [Fintype I] [MeasurableSingletonClass X] [TopologicalSpace X] [BorelSpace X] [HasOuterApproxClosed X] in
+lemma graph_domination (U : V → X) (hn : 2≤Fintype.card V) (M : ℝ) (hM : 0≤M)
+    (hdom : ∀ x y,G.edgeLaw x y≤M*pairProb Finset.univ x y) :
+    finiteLaw (fun r : V × V => G.edgeLaw r.1 r.2) (fun r => (U r.1,U r.2))≤
+      ENNReal.ofReal (2*M) • (uniformLaw U).prod (uniformLaw U) := by
+  rw [uniformLaw,finiteLaw_prod _ _ _ _ (fun _ => by positivity),← finiteLaw_smul _ _ _ (by positivity)]
+  apply finiteLaw_mono
+  intro r
+  calc
+    _ ≤ M*pairProb Finset.univ r.1 r.2 := hdom _ _
+    _ ≤ M*(2*(Fintype.card V:ℝ)⁻¹*(Fintype.card V:ℝ)⁻¹) :=
+      mul_le_mul_of_nonneg_left (pairProb_le_uniform hn _ _) hM
+    _ = _ := by ring
+
+end WeakPinned.Bounded
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PinnedDistances/FiniteObstruction.lean b/lean/OAI/Geometry/PinnedDistances/FiniteObstruction.lean
new file mode 100644
index 0000000000000000000000000000000000000000..c7bd6ed597165bfb4b57443fa5684374eed64210
--- /dev/null
+++ b/lean/OAI/Geometry/PinnedDistances/FiniteObstruction.lean
@@ -0,0 +1,206 @@
+import OAI.Geometry.PinnedDistances.MeasureLimits
+
+namespace OAI
+
+universe u1 u2
+
+section
+noncomputable section
+open MeasureTheory Filter
+namespace WeakPinned.Bounded
+lemma no_finite_expanded (μ ν : Measure ℂ) [IsProbabilityMeasure μ] [IsProbabilityMeasure ν]
+    [NullSingletonClass μ] [NullSingletonClass ν]
+    (Q : Measure (Quad ℂ)) (R : Measure (Expanded ℂ))
+    [IsProbabilityMeasure Q] [IsProbabilityMeasure R]
+    (he : R.map Prod.fst=Q)
+    (hm₁ : R.map (fun r => (r.1,(r.2.1.1,r.2.2.1)))=Q.prod (μ.prod μ))
+    (hm₂ : R.map (fun r => (r.1,(r.2.1.2,r.2.2.2)))=Q.prod (ν.prod ν))
+    (hind : Q.map (fun q => (q.1,q.2.2))=μ.prod (Q.map (fun q => q.2.2)))
+    (hac₁ : Q.map (fun q => (q.1,q.2.2.1))≪μ.prod μ)
+    (hac₂ : Q.map (fun q => (q.2.1,q.2.2.2))≪ν.prod ν)
+    (hz : ∀ᵐ r ∂  R,(r.2.1.1-r.1.1)*(r.2.1.2-r.1.2.1)=
+      (r.1.2.2.1-r.1.1)*(r.1.2.2.2-r.1.2.1))
+    (hw : ∀ᵐ r ∂  R,(r.2.2.1-r.1.1)*(r.2.2.2-r.1.2.1)=
+      (r.1.2.2.1-r.1.1)*(r.1.2.2.2-r.1.2.1)) : False := by
+  have hbd : ∀ᵐ q ∂  Q,q.2.1≠q.2.2.2 :=
+    (ae_map_iff (μ:=Q) (f:=fun q => (q.2.1,q.2.2.2)) (p:=fun p : ℂ × ℂ => p.1≠p.2) (by fun_prop)
+      (measurableSet_eq_fun measurable_fst measurable_snd).compl).mp (hac₂.ae_le (ae_distinct_pair ν))
+  have hsupp : ∀ᵐ q ∂  Q,q.2.2.1∈μ.support := by
+    have hs : ∀ᵐ p ∂  μ.prod μ,p.2∈μ.support := by
+      rw [Measure.ae_prod_iff_ae_ae (μ.isClosed_support.measurableSet.preimage measurable_snd)]
+      exact Eventually.of_forall (fun _ => μ.support_mem_ae)
+    exact (ae_map_iff (μ:=Q) (f:=fun q => (q.1,q.2.2.1)) (p:=fun p : ℂ × ℂ => p.2∈μ.support) (by fun_prop)
+      (μ.isClosed_support.measurableSet.preimage measurable_snd)).mp (hac₁.ae_le hs)
+  exact no_energy_law μ ν Q hind hsupp
+    (expanded_energy_eq μ ν Q R hm₁ hm₂ (expanded_kernel_eq μ ν Q R he hm₁ hbd hz hw))
+end WeakPinned.Bounded
+
+end
+end
+section
+noncomputable section
+open MeasureTheory Filter Set
+open scoped Topology
+namespace WeakPinned.Bounded
+local instance : MeasurableSpace Sphere := borel Sphere
+local instance : BorelSpace Sphere := ⟨rfl⟩
+
+lemma no_sphere_expanded (β₁ β₂ : ProbabilityMeasure Sphere)
+    [NullSingletonClass (β₁ : Measure Sphere)] [NullSingletonClass (β₂ : Measure Sphere)]
+    (Q : Measure (Quad Sphere)) (R : Measure (Expanded Sphere))
+    [IsProbabilityMeasure Q] [IsProbabilityMeasure R]
+    (he : R.map Prod.fst=Q)
+    (hm₁ : R.map (fun r => (r.1,(r.2.1.1,r.2.2.1)))=Q.prod ((β₁:Measure Sphere).prod β₁))
+    (hm₂ : R.map (fun r => (r.1,(r.2.1.2,r.2.2.2)))=Q.prod ((β₂:Measure Sphere).prod β₂))
+    (hind : Q.map (fun q => (q.1,q.2.2))=(β₁:Measure Sphere).prod (Q.map (fun q => q.2.2)))
+    (hac₁ : Q.map (fun q => (q.1,q.2.2.1))≪(β₁:Measure Sphere).prod β₁)
+    (hac₂ : Q.map (fun q => (q.2.1,q.2.2.2))≪(β₂:Measure Sphere).prod β₂)
+    (hz : ∀ᵐ r ∂ R.comap expandedCoe,(r.2.1.1-r.1.1)*(r.2.1.2-r.1.2.1)=
+      (r.1.2.2.1-r.1.1)*(r.1.2.2.2-r.1.2.1))
+    (hw : ∀ᵐ r ∂ R.comap expandedCoe,(r.2.2.1-r.1.1)*(r.2.2.2-r.1.2.1)=
+      (r.1.2.2.1-r.1.1)*(r.1.2.2.2-r.1.2.1)) : False := by
+  let μ := finitePart β₁
+  let ν := finitePart β₂
+  have hμ : μ.map OnePoint.some=(β₁:Measure Sphere) := finitePart_map β₁ (measure_singleton _)
+  have hν : ν.map OnePoint.some=(β₂:Measure Sphere) := finitePart_map β₂ (measure_singleton _)
+  let : IsProbabilityMeasure μ := finitePart_probability β₁ (measure_singleton _)
+  let : IsProbabilityMeasure ν := finitePart_probability β₂ (measure_singleton _)
+  let : NullSingletonClass μ := finitePart_nonatomic β₁
+  let : NullSingletonClass ν := finitePart_nonatomic β₂
+  have hQfin : ∀ᵐ q ∂ Q,q∈range quadCoe := sphere_quad_finite μ ν Q
+    (by simpa only [hμ] using hac₁) (by simpa only [hν] using hac₂)
+  let Qc := Q.comap quadCoe
+  let : IsProbabilityMeasure Qc := chart_probability _ quadCoe_open.measurableEmbedding Q hQfin
+  have hQ : Qc.map quadCoe=Q := chart_map_comap _ quadCoe_open.measurableEmbedding Q hQfin
+  have hRfin : ∀ᵐ r ∂ R,r∈range expandedCoe := sphere_expanded_finite μ ν Qc R
+    (by simpa only [hQ,hμ] using hm₁) (by simpa only [hQ,hν] using hm₂)
+  let Rc := R.comap expandedCoe
+  let : IsProbabilityMeasure Rc := chart_probability _ expandedCoe_open.measurableEmbedding R hRfin
+  have hR : Rc.map expandedCoe=R := chart_map_comap _ expandedCoe_open.measurableEmbedding R hRfin
+  have hec : Rc.map Prod.fst=Qc := by
+    apply map_transport expandedCoe quadCoe expandedCoe_open.measurableEmbedding.measurable
+      quadCoe_open.measurableEmbedding Prod.fst Prod.fst measurable_fst measurable_fst Rc Qc rfl
+    rw [hR,hQ,he]
+  have hm₁c : Rc.map (fun r => (r.1,(r.2.1.1,r.2.2.1)))=Qc.prod (μ.prod μ) := by
+    apply map_transport expandedCoe (Prod.map quadCoe pairCoe)
+      expandedCoe_open.measurableEmbedding.measurable
+      (quadCoe_open.measurableEmbedding.prodMap pairCoe_open.measurableEmbedding)
+      (fun r => (r.1,(r.2.1.1,r.2.2.1))) (fun r => (r.1,(r.2.1.1,r.2.2.1)))
+      (by fun_prop) (by fun_prop) Rc (Qc.prod (μ.prod μ)) rfl
+    rw [hR,hm₁,← Measure.map_prod_map _ _ quadCoe_open.measurableEmbedding.measurable
+      pairCoe_open.measurableEmbedding.measurable,← Measure.map_prod_map _ _
+      measurableEmbedding_sphere_coe.measurable measurableEmbedding_sphere_coe.measurable,hQ,hμ]
+  have hm₂c : Rc.map (fun r => (r.1,(r.2.1.2,r.2.2.2)))=Qc.prod (ν.prod ν) := by
+    apply map_transport expandedCoe (Prod.map quadCoe pairCoe)
+      expandedCoe_open.measurableEmbedding.measurable
+      (quadCoe_open.measurableEmbedding.prodMap pairCoe_open.measurableEmbedding)
+      (fun r => (r.1,(r.2.1.2,r.2.2.2))) (fun r => (r.1,(r.2.1.2,r.2.2.2)))
+      (by fun_prop) (by fun_prop) Rc (Qc.prod (ν.prod ν)) rfl
+    rw [hR,hm₂,← Measure.map_prod_map _ _ quadCoe_open.measurableEmbedding.measurable
+      pairCoe_open.measurableEmbedding.measurable,← Measure.map_prod_map _ _
+      measurableEmbedding_sphere_coe.measurable measurableEmbedding_sphere_coe.measurable,hQ,hν]
+  have htarget : (Qc.map (fun q => q.2.2)).map pairCoe=Q.map (fun q => q.2.2) := by
+    rw [Measure.map_map pairCoe_open.measurableEmbedding.measurable (by fun_prop)]
+    change Qc.map ((fun q : Quad Sphere => q.2.2) ∘ quadCoe)=_
+    rw [← Measure.map_map (by fun_prop) quadCoe_open.measurableEmbedding.measurable,hQ]
+  have hindc : Qc.map (fun q => (q.1,q.2.2))=μ.prod (Qc.map (fun q => q.2.2)) := by
+    apply map_transport quadCoe (Prod.map OnePoint.some pairCoe)
+      quadCoe_open.measurableEmbedding.measurable
+      (measurableEmbedding_sphere_coe.prodMap pairCoe_open.measurableEmbedding)
+      (fun q => (q.1,q.2.2)) (fun q => (q.1,q.2.2))
+      (by fun_prop) (by fun_prop) Qc (μ.prod (Qc.map (fun q => q.2.2))) rfl
+    rw [hQ,hind,← Measure.map_prod_map _ _ measurableEmbedding_sphere_coe.measurable
+      pairCoe_open.measurableEmbedding.measurable,hμ,htarget]
+  have hac₁c : Qc.map (fun q => (q.1,q.2.2.1))≪μ.prod μ := by
+    apply map_transport_ac quadCoe pairCoe quadCoe_open.measurableEmbedding.measurable
+      pairCoe_open.measurableEmbedding (fun q => (q.1,q.2.2.1)) (fun q => (q.1,q.2.2.1))
+      (by fun_prop) (by fun_prop) Qc (μ.prod μ) rfl
+    rw [hQ,← Measure.map_prod_map _ _ measurableEmbedding_sphere_coe.measurable
+      measurableEmbedding_sphere_coe.measurable,hμ]
+    exact hac₁
+  have hac₂c : Qc.map (fun q => (q.2.1,q.2.2.2))≪ν.prod ν := by
+    apply map_transport_ac quadCoe pairCoe quadCoe_open.measurableEmbedding.measurable
+      pairCoe_open.measurableEmbedding (fun q => (q.2.1,q.2.2.2)) (fun q => (q.2.1,q.2.2.2))
+      (by fun_prop) (by fun_prop) Qc (ν.prod ν) rfl
+    rw [hQ,← Measure.map_prod_map _ _ measurableEmbedding_sphere_coe.measurable
+      measurableEmbedding_sphere_coe.measurable,hν]
+    exact hac₂
+  exact no_finite_expanded μ ν Qc Rc hec hm₁c hm₂c hindc hac₁c hac₂c hz hw
+
+end WeakPinned.Bounded
+
+end
+end
+section
+noncomputable section
+open MeasureTheory Filter Set
+open scoped BigOperators Topology
+namespace WeakPinned.Bounded
+open Variance
+local instance : MeasurableSpace Sphere := borel Sphere
+local instance : BorelSpace Sphere := ⟨rfl⟩
+local instance : MeasurableSingletonClass (Quad Sphere) := inferInstance
+local instance : MeasurableSingletonClass ((Sphere × Sphere) × (Sphere × Sphere)) := inferInstance
+local instance : MeasurableSingletonClass (Expanded Sphere) := Prod.instMeasurableSingletonClass
+variable {V : Type u1} {I : Type u2} [Fintype V] [DecidableEq V] [Fintype I]
+variable (G : Core.Fibers V I)
+
+def zEquation : Set (Expanded ℂ) := {r | (r.2.1.1-r.1.1)*(r.2.1.2-r.1.2.1)=
+    (r.1.2.2.1-r.1.1)*(r.1.2.2.2-r.1.2.1)}
+def wEquation : Set (Expanded ℂ) := {r | (r.2.2.1-r.1.1)*(r.2.2.2-r.1.2.1)=
+    (r.1.2.2.1-r.1.1)*(r.1.2.2.2-r.1.2.1)}
+
+lemma zEquation_closed : IsClosed zEquation := isClosed_eq (by fun_prop) (by fun_prop)
+lemma wEquation_closed : IsClosed wEquation := isClosed_eq (by fun_prop) (by fun_prop)
+
+omit [Fintype I] in
+lemma tripleWeight_mem {r : I × V × V × V}
+    (hr : tripleWeight G.fiberWeight (fun e => prob (G.targets e)) r≠0) :
+    r.2.1∈G.targets r.1 ∧ r.2.2.1∈G.targets r.1 ∧ r.2.2.2∈G.targets r.1 := by
+  refine ⟨?_,?_,?_⟩
+  all_goals
+    by_contra h
+    simp [tripleWeight,prob,h] at hr
+
+lemma expanded_chart_zero (U₁ U₂ : V → ℂ) (hG : 0<G.edgeCount)
+    (hB : ∀ e,(G.targets e).Nonempty)
+    (hrel : ∀ e y z,y∈G.targets e → z∈G.targets e →
+      (U₁ y-U₁ (G.source e))*(U₂ y-U₂ (G.source e))=
+      (U₁ z-U₁ (G.source e))*(U₂ z-U₂ (G.source e))) :
+    (expandedProbability G (fun x => (U₁ x:Sphere)) (fun x => (U₂ x:Sphere)) hG hB :
+      Measure (Expanded Sphere)) (expandedCoe '' zEquationᶜ)=0 ∧
+    (expandedProbability G (fun x => (U₁ x:Sphere)) (fun x => (U₂ x:Sphere)) hG hB :
+      Measure (Expanded Sphere)) (expandedCoe '' wEquationᶜ)=0 := by
+  have hzero (s : Set (Expanded ℂ))
+      (hs : ∀ r,tripleWeight G.fiberWeight (fun e => prob (G.targets e)) r≠0 →
+        tripleValue G.source U₁ U₂ r∈s) :
+      (expandedProbability G (fun x => (U₁ x:Sphere)) (fun x => (U₂ x:Sphere)) hG hB :
+        Measure (Expanded Sphere)) (expandedCoe '' sᶜ)=0 := by
+    have hh : ∀ᵐ r ∂  finiteLaw (tripleWeight G.fiberWeight (fun e => prob (G.targets e)))
+        (tripleValue G.source (fun x => (U₁ x:Sphere)) (fun x => (U₂ x:Sphere))),
+        r∉expandedCoe '' sᶜ := by
+      apply finiteLaw_ae
+      intro r hr
+      rintro ⟨q,hq,hqe⟩
+      have he : q=tripleValue G.source U₁ U₂ r := expandedCoe_open.injective hqe
+      subst q
+      exact hq (hs r hr)
+    change (finiteLaw (tripleWeight G.fiberWeight (fun e => prob (G.targets e)))
+      (tripleValue G.source (fun x => (U₁ x:Sphere)) (fun x => (U₂ x:Sphere)))) (expandedCoe '' sᶜ)=0
+    simpa only [not_not, Set.ofPred_mem_eq] using ae_iff.mp hh
+  constructor
+  · apply hzero
+    intro r hr
+    have hm := tripleWeight_mem G hr
+    exact hrel r.1 r.2.2.1 r.2.1 hm.2.1 hm.1
+  · apply hzero
+    intro r hr
+    have hm := tripleWeight_mem G hr
+    exact hrel r.1 r.2.2.2 r.2.1 hm.2.2 hm.1
+
+end WeakPinned.Bounded
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PinnedDistances/GlobalOverlap.lean b/lean/OAI/Geometry/PinnedDistances/GlobalOverlap.lean
new file mode 100644
index 0000000000000000000000000000000000000000..b001e4eade5a7a24c22d0e43894bf817c28f8dc1
--- /dev/null
+++ b/lean/OAI/Geometry/PinnedDistances/GlobalOverlap.lean
@@ -0,0 +1,414 @@
+import OAI.Geometry.PinnedDistances.FiberOverlap
+
+namespace OAI
+
+universe u1 u2 u3
+
+section
+noncomputable section
+open NumberField
+open scoped BigOperators
+noncomputable section
+open MeasureTheory
+open scoped Topology
+noncomputable section
+open MeasureTheory
+open scoped Topology ENNReal
+noncomputable section
+open MeasureTheory NumberField
+open scoped BigOperators
+noncomputable section
+open scoped BigOperators
+open MeasureTheory
+
+namespace WeakPinned.Grid
+open MeasureTheory
+
+lemma truncated_error_integrable {z w : ℂ} (hne : z ≠ w) (t : ℝ) :
+    Integrable (fun ω => max (|error ω z w|-t) 0) fullLaw :=
+  (((error_integrable_full hne).abs.sub (integrable_const t)).sup (integrable_zero _ _ _))
+
+lemma truncated_error_mean {z w : ℂ} (hne : z ≠ w) {t : ℝ} (ht : 0 ≤ t) :
+    (∫ ω, max (|error ω z w|-t) 0 ∂fullLaw) ≤ 4*Real.exp (-t) := by
+  have hm : Measurable (fun ω => max (|error ω z w|-t) 0) :=
+    ((measurable_error hne).abs.sub_const t).max measurable_const
+  have hn (ω : FullSample) : 0 ≤ max (|error ω z w|-t) 0 := le_max_right _ _
+  have hi := truncated_error_integrable hne t
+  have he : IntegrableOn (fun u : ℝ => (4*Real.exp (-t))*Real.exp (-u)) (Set.Ioi 0) volume := by
+    simpa only [neg_one_mul, IntegrableOn] using
+      (integrableOn_exp_mul_Ioi (by norm_num : (-1:ℝ)<0) 0).const_mul (4*Real.exp (-t))
+  have hb : (∫⁻ ω, ENNReal.ofReal (max (|error ω z w|-t) 0) ∂fullLaw) ≤
+      ENNReal.ofReal (4*Real.exp (-t)) := by
+    rw [lintegral_eq_lintegral_meas_lt fullLaw
+      (Filter.Eventually.of_forall hn) hm.aemeasurable]
+    calc
+      _ ≤ ∫⁻ u : ℝ in Set.Ioi 0, ENNReal.ofReal ((4*Real.exp (-t))*Real.exp (-u)) := by
+        apply lintegral_mono_ae
+        filter_upwards [ae_restrict_mem measurableSet_Ioi] with u hu
+        have hu0 : 0 < u := hu
+        have hs : {ω : FullSample | u < max (|error ω z w|-t) 0} =
+            {ω : FullSample | u+t < |error ω z w|} := by
+          ext ω
+          simp only [Set.mem_ofPred_eq,lt_max_iff]
+          constructor
+          · rintro (h | h) <;> linarith
+          · intro h; exact Or.inl (by linarith)
+        rw [hs, ← ENNReal.ofReal_toReal (measure_ne_top fullLaw _)]
+        have hh := abs_error_tail_full hne (add_nonneg hu.le ht)
+        apply ENNReal.ofReal_le_ofReal
+        calc
+          _ ≤ 4*Real.exp (-(u+t)) := hh
+          _ = _ := by rw [neg_add,Real.exp_add]; ring
+      _ = ENNReal.ofReal (4*Real.exp (-t)) := by
+        rw [← ofReal_integral_eq_lintegral_ofReal he (by filter_upwards with u; positivity)]
+        congr 1
+        rw [integral_const_mul]
+        have hz := integral_exp_mul_Ioi (by norm_num : (-1:ℝ)<0) (0:ℝ)
+        simp only [neg_one_mul,mul_zero,Real.exp_zero,neg_div_neg_eq,div_one] at hz
+        rw [hz,mul_one]
+  rw [← ofReal_integral_eq_lintegral_ofReal hi (Filter.Eventually.of_forall hn)] at hb
+  exact (ENNReal.ofReal_le_ofReal_iff (by positivity)).mp hb
+
+lemma truncated_sum_bound (a b H : ℝ) :
+    max (|a+b|-H) 0 ≤ 2*max (|a|-H/2) 0 + 2*max (|b|-H/2) 0 := by
+  apply max_le
+  · have ht := abs_add_le a b
+    have h₁ := le_max_left (|a|-H/2) 0
+    have h₂ := le_max_left (|b|-H/2) 0
+    have h₃ := le_max_right (|a|-H/2) 0
+    have h₄ := le_max_right (|b|-H/2) 0
+    linarith
+  · positivity
+
+lemma paired_error_mean {z₁ w₁ z₂ w₂ : ℂ} (h₁ : z₁ ≠ w₁) (h₂ : z₂ ≠ w₂)
+    {H : ℝ} (hH : 0 ≤ H) :
+    (∫ ω, max (|error ω z₁ w₁ + error ω z₂ w₂|-H) 0 ∂fullLaw) ≤
+      16*Real.exp (-H/2) := by
+  have hi : Integrable (fun ω => max (|error ω z₁ w₁ + error ω z₂ w₂|-H) 0) fullLaw := by
+    exact Integrable.sup (((error_integrable_full h₁).add (error_integrable_full h₂)).abs.sub
+      (integrable_const H)) (integrable_zero _ _ _)
+  have hia := truncated_error_integrable h₁ (H/2)
+  have hib := truncated_error_integrable h₂ (H/2)
+  have hh := integral_mono hi ((hia.const_mul 2).add (hib.const_mul 2))
+    (fun ω => truncated_sum_bound (error ω z₁ w₁) (error ω z₂ w₂) H)
+  simp only [Pi.add_apply] at hh
+  rw [integral_add (hia.const_mul 2) (hib.const_mul 2)] at hh
+  simp only [integral_const_mul] at hh
+  have hta := truncated_error_mean h₁ (div_nonneg hH (by norm_num : (0:ℝ)≤2))
+  have htb := truncated_error_mean h₂ (div_nonneg hH (by norm_num : (0:ℝ)≤2))
+  rw [neg_div'] at hta htb
+  linarith
+
+end WeakPinned.Grid
+
+namespace WeakPinned.Global
+open Hierarchy Variance SmallFiber
+variable {P : Type u1} {I : Type u2} [Fintype P] [DecidableEq P] [Fintype I]
+variable {K : Type u3} [Field K] [NumberField K]
+variable (Z₁ Z₂ : P → K) (h₁ : Function.Injective Z₁) (h₂ : Function.Injective Z₂)
+include h₁ h₂
+
+lemma arch_fiber_defect (v : InfinitePlace K) (a b : P) (hab : a ≠ b) (t : K)
+    (ht : (Z₁ a-Z₁ b)*(Z₂ a-Z₂ b)=t) :
+    (fun ω => (arch Z₁ v ω).value a b+(arch Z₂ v ω).value a b+Real.log (v t)) =ᵐ[Grid.fullLaw]
+      (fun ω => Grid.error ω (v.embedding (Z₁ a)) (v.embedding (Z₁ b))+
+        Grid.error ω (v.embedding (Z₂ a)) (v.embedding (Z₂ b))) := by
+  have he := Arithmetic.raw_fiber_identity v.val (Z₁ a-Z₁ b) (Z₂ a-Z₂ b)
+    (sub_ne_zero.mpr (h₁.ne hab)) (sub_ne_zero.mpr (h₂.ne hab))
+  rw [ht] at he
+  change -Real.log (v (Z₁ a-Z₁ b)) + -Real.log (v (Z₂ a-Z₂ b)) = -Real.log (v t) at he
+  filter_upwards [arch_value_ae Z₁ h₁ v hab,arch_value_ae Z₂ h₂ v hab] with ω hA hB
+  rw [hA,hB]
+  simp only [Grid.error,← map_sub,InfinitePlace.norm_embedding_eq]
+  linarith
+
+lemma arch_defect_mean (v : InfinitePlace K) (a b : P) (hab : a ≠ b) (t : K)
+    (ht : (Z₁ a-Z₁ b)*(Z₂ a-Z₂ b)=t) (H : ℝ) (hH : 0 ≤ H) :
+    (∫ ω, max (|(arch Z₁ v ω).value a b+(arch Z₂ v ω).value a b+Real.log (v t)|-H) 0 ∂Grid.fullLaw) ≤
+      16*Real.exp (-H/2) := by
+  have he := arch_fiber_defect Z₁ Z₂ h₁ h₂ v a b hab t ht
+  have he' : (fun ω => max (|(arch Z₁ v ω).value a b+(arch Z₂ v ω).value a b+Real.log (v t)|-H) 0) =ᵐ[Grid.fullLaw]
+      (fun ω => max (|Grid.error ω (v.embedding (Z₁ a)) (v.embedding (Z₁ b))+
+        Grid.error ω (v.embedding (Z₂ a)) (v.embedding (Z₂ b))|-H) 0) := by
+    filter_upwards [he] with ω hh
+    rw [hh]
+  rw [integral_congr_ae he']
+  exact Grid.paired_error_mean (embedding_ne Z₁ h₁ v hab) (embedding_ne Z₂ h₂ v hab) hH
+
+lemma arch_weighted_defect (v : InfinitePlace K) (w : I → ℝ) (p : I → P → ℝ)
+    (x : I → P) (t : I → K) (hw : ∀ i, 0 ≤ w i) (hW : ∑ i, w i=1)
+    (hp : ∀ i y, 0 ≤ p i y) (hp1 : ∀ i, ∑ y, p i y=1) (hx : ∀ i, p i (x i)=0)
+    (ht : ∀ i y, p i y ≠ 0 → (Z₁ (x i)-Z₁ y)*(Z₂ (x i)-Z₂ y)=t i)
+    (H : ℝ) (hH : 0 ≤ H) :
+    (∫ ω, ∑ i, w i*∑ y, p i y*max (|(arch Z₁ v ω).value (x i) y+
+      (arch Z₂ v ω).value (x i) y+Real.log (v (t i))|-H) 0 ∂Grid.fullLaw) ≤ 16*Real.exp (-H/2) := by
+  have hi (i : I) (y : P) : Integrable (fun ω => max (|(arch Z₁ v ω).value (x i) y+
+      (arch Z₂ v ω).value (x i) y+Real.log (v (t i))|-H) 0) Grid.fullLaw :=
+    ((((arch_integrable Z₁ h₁ v _ _).add (arch_integrable Z₂ h₂ v _ _)).add (integrable_const _)).abs.sub
+      (integrable_const H)).sup (integrable_zero _ _ _)
+  rw [integral_finsetSum _ (fun i _ => (integrable_finsetSum (s:=Finset.univ)
+    (fun y _ => (hi i y).const_mul _)).const_mul _)]
+  simp_rw [integral_const_mul,integral_finsetSum _ (fun y _ => (hi _ y).const_mul _),integral_const_mul]
+  calc
+    _ ≤ ∑ i, w i * ∑ y, p i y*(16*Real.exp (-H/2)) := by
+      apply Finset.sum_le_sum
+      intro i _
+      apply mul_le_mul_of_nonneg_left _ (hw i)
+      apply Finset.sum_le_sum
+      intro y _
+      by_cases hy : p i y=0
+      · simp [hy]
+      · have hxy : x i ≠ y := by intro he; rw [← he,hx i] at hy; exact hy rfl
+        exact mul_le_mul_of_nonneg_left (arch_defect_mean Z₁ Z₂ h₁ h₂ v _ _ hxy _ (ht i y hy) H hH) (hp i y)
+    _ = _ := by simp only [← Finset.sum_mul,hp1,one_mul,hW]
+
+lemma arch_small_fiber (hn : 2 ≤ Fintype.card P) (v : InfinitePlace K)
+    (w : I → ℝ) (p : I → P → ℝ) (x : I → P) (t : I → K)
+    (hw : ∀ i, 0 ≤ w i) (hW : ∑ i, w i=1) (hp : ∀ i y, 0 ≤ p i y)
+    (hp1 : ∀ i, ∑ y, p i y=1) (hx : ∀ i, p i (x i)=0)
+    (ht : ∀ i y, p i y ≠ 0 → (Z₁ (x i)-Z₁ y)*(Z₂ (x i)-Z₂ y)=t i)
+    (B τ lam H : ℝ) (hB : 0 ≤ B) (hlam : 0 ≤ lam) (hlam2 : lam ≤ 1/2)
+    (hτ : 0 < τ) (hτ1 : τ ≤ 1) (hτn : 2 ≤ τ*(Fintype.card P:ℝ)) (hH : 0 ≤ H)
+    (hdom : ∀ a b, a ≠ b → edgeLaw w p x a b ≤ B/((Fintype.card P:ℝ)*((Fintype.card P:ℝ)-1))) :
+    (∑ C, (∫ ω, (arch Z₁ v ω).mass C ∂Grid.fullLaw)*slice w p x lam C) ≤
+      (2*lam/τ^2)*(∑ C, (∫ ω, (arch Z₁ v ω).mass C ∂Grid.fullLaw)*pairCell Finset.univ C) +
+      B*τ/(Fintype.card P:ℝ)*((∑ y, ∫ ω, (arch Z₁ v ω).potential y ∂Grid.fullLaw)+
+        (∑ y, ∫ ω, (arch Z₂ v ω).potential y ∂Grid.fullLaw)) +
+      2*lam*B*(∑ C, (∫ ω, (arch Z₂ v ω).mass C ∂Grid.fullLaw)*pairCell Finset.univ C) + 2*lam*H +
+      16*Real.exp (-H/2) := by
+  have hiM₁ (C : Finset P) := Depth.integrable_mass_param (arch Z₁ v) (arch_measurable Z₁ h₁ v)
+    (arch_integrable Z₁ h₁ v) hn C
+  have hiM₂ (C : Finset P) := Depth.integrable_mass_param (arch Z₂ v) (arch_measurable Z₂ h₂ v)
+    (arch_integrable Z₂ h₂ v) hn C
+  have hiS₁ (y : P) := Depth.integrable_potential_param (arch Z₁ v) (arch_measurable Z₁ h₁ v)
+    (arch_integrable Z₁ h₁ v) hn y
+  have hiS₂ (y : P) := Depth.integrable_potential_param (arch Z₂ v) (arch_measurable Z₂ h₂ v)
+    (arch_integrable Z₂ h₂ v) hn y
+  have hiE (i : I) (y : P) : Integrable (fun ω => max (|(arch Z₁ v ω).value (x i) y+
+      (arch Z₂ v ω).value (x i) y+Real.log (v (t i))|-H) 0) Grid.fullLaw := by
+    have hd := ((arch_integrable Z₁ h₁ v (x i) y).add (arch_integrable Z₂ h₂ v (x i) y)).add
+      (integrable_const (Real.log (v (t i))))
+    exact (hd.abs.sub (integrable_const H)).sup (integrable_zero _ _ _)
+  have hA := (integrable_finsetSum (s:=Finset.univ) (fun C _ => (hiM₁ C).mul_const (pairCell Finset.univ C))).const_mul (2*lam/τ^2)
+  have hS := ((integrable_finsetSum (s:=Finset.univ) (fun y _ => hiS₁ y)).add
+    (integrable_finsetSum (s:=Finset.univ) (fun y _ => hiS₂ y))).const_mul (B*τ/Fintype.card P)
+  have hA' := (integrable_finsetSum (s:=Finset.univ) (fun C _ => (hiM₂ C).mul_const (pairCell Finset.univ C))).const_mul (2*lam*B)
+  have hE := integrable_finsetSum (s:=Finset.univ) (fun i _ =>
+    (integrable_finsetSum (s:=Finset.univ) (fun y _ => (hiE i y).const_mul (p i y))).const_mul (w i))
+  have hb := integral_mono (integrable_finsetSum (s:=Finset.univ) (fun C _ => (hiM₁ C).mul_const (slice w p x lam C)))
+    ((((hA.add hS).add hA').add (integrable_const (2*lam*H))).add hE)
+    (fun ω => by
+      simpa only [Pi.add_apply,sub_neg_eq_add] using
+        (local_small_fiber_reduced (arch Z₁ v ω) (arch Z₂ v ω) w p x hw hW hp hp1 hx hn
+          (fun i => -Real.log (v (t i))) B τ lam H hB hlam hlam2 hτ hτ1 hτn hH hdom))
+  have hI₁ := integral_add (((hA.add hS).add hA').add (integrable_const (2*lam*H))) hE
+  have hI₂ := integral_add ((hA.add hS).add hA') (integrable_const (2*lam*H))
+  have hI₃ := integral_add (hA.add hS) hA'
+  have hI₄ := integral_add hA hS
+  have hI₅ := integral_add (integrable_finsetSum (s:=Finset.univ) (fun y _ => hiS₁ y))
+    (integrable_finsetSum (s:=Finset.univ) (fun y _ => hiS₂ y))
+  simp only [Pi.add_apply] at hb hI₁ hI₂ hI₃ hI₄ hI₅
+  rw [hI₁,hI₂,hI₃,hI₄] at hb
+  simp only [integral_const,probReal_univ,one_smul,integral_const_mul] at hb
+  rw [hI₅] at hb
+  rw [integral_finsetSum _ (fun C _ => (hiM₁ C).mul_const (slice w p x lam C)),
+    integral_finsetSum _ (fun C _ => (hiM₁ C).mul_const (pairCell Finset.univ C)),
+    integral_finsetSum _ (fun C _ => (hiM₂ C).mul_const (pairCell Finset.univ C)),
+    integral_finsetSum _ (fun y _ => hiS₁ y),integral_finsetSum _ (fun y _ => hiS₂ y)] at hb
+  simp only [integral_mul_const] at hb
+  have he := arch_weighted_defect Z₁ Z₂ h₁ h₂ v w p x t hw hW hp hp1 hx ht H hH
+  linarith
+
+lemma finite_small_fiber (hn : 2 ≤ Fintype.card P) (v : FinitePlace K)
+    (w : I → ℝ) (p : I → P → ℝ) (x : I → P) (t : I → K)
+    (hw : ∀ i, 0 ≤ w i) (hW : ∑ i, w i=1) (hp : ∀ i y, 0 ≤ p i y)
+    (hp1 : ∀ i, ∑ y, p i y=1) (hx : ∀ i, p i (x i)=0)
+    (ht : ∀ i y, p i y ≠ 0 → (Z₁ (x i)-Z₁ y)*(Z₂ (x i)-Z₂ y)=t i)
+    (B τ lam : ℝ) (hB : 0 ≤ B) (hlam : 0 ≤ lam) (hlam2 : lam ≤ 1/2)
+    (hτ : 0 < τ) (hτ1 : τ ≤ 1) (hτn : 2 ≤ τ*(Fintype.card P:ℝ))
+    (hdom : ∀ a b, a ≠ b → edgeLaw w p x a b ≤ B/((Fintype.card P:ℝ)*((Fintype.card P:ℝ)-1))) :
+    (∑ C, (finite Z₁ h₁ v).mass C*slice w p x lam C) ≤
+      (2*lam/τ^2)*(∑ C, (finite Z₁ h₁ v).mass C*pairCell Finset.univ C) +
+      B*τ/(Fintype.card P:ℝ)*((∑ y, (finite Z₁ h₁ v).potential y)+(∑ y, (finite Z₂ h₂ v).potential y)) +
+      2*lam*B*(∑ C, (finite Z₂ h₂ v).mass C*pairCell Finset.univ C) := by
+  have hb := local_small_fiber_reduced (finite Z₁ h₁ v) (finite Z₂ h₂ v) w p x hw hW hp hp1 hx hn
+    (fun i => -Real.log (v (t i))) B τ lam 0 hB hlam hlam2 hτ hτ1 hτn (le_refl _) hdom
+  have he (i : I) (y : P) : p i y * max (|(finite Z₁ h₁ v).value (x i) y+
+      (finite Z₂ h₂ v).value (x i) y- -Real.log (v (t i))|-0) 0=0 := by
+    by_cases hy : p i y=0
+    · simp [hy]
+    · have hxy : x i ≠ y := by intro he; rw [← he,hx i] at hy; exact hy rfl
+      have hh := Arithmetic.raw_fiber_identity v.val (Z₁ (x i)-Z₁ y) (Z₂ (x i)-Z₂ y)
+        (sub_ne_zero.mpr (h₁.ne hxy)) (sub_ne_zero.mpr (h₂.ne hxy))
+      change -Real.log (v (Z₁ (x i)-Z₁ y)) + -Real.log (v (Z₂ (x i)-Z₂ y)) =
+        -Real.log (v ((Z₁ (x i)-Z₁ y)*(Z₂ (x i)-Z₂ y))) at hh
+      rw [ht i y hy] at hh
+      simp only [finite,hxy,reduceIte,hh,sub_self,abs_zero,max_self,mul_zero]
+  simpa only [he,Finset.sum_const_zero,mul_zero,add_zero] using hb
+
+omit h₁ h₂ in
+lemma finite_zero_outside {Z : P → K} (hZ : Function.Injective Z) (v : FinitePlace K)
+    (hv : v ∉ badPlaces Z hZ) : ∀ a b, (finite Z hZ v).value a b=0 := by
+  intro a b
+  by_cases hab : a=b
+  · simp [finite,hab]
+  · have hunit : v (Z a-Z b)=1 := by
+      by_contra h
+      exact hv ((badPlaces_mem Z hZ v).mpr ⟨a,b,hab,h⟩)
+    simp [finite,hab,hunit]
+
+omit h₁ h₂ in
+lemma finite_mass_zero {Z : P → K} (hZ : Function.Injective Z) (hn : 2 ≤ Fintype.card P)
+    (v : FinitePlace K) (hv : v ∉ badPlaces Z hZ) (C : Finset P) : (finite Z hZ v).mass C=0 := by
+  have hz := (finite Z hZ v).zero_values hn (finite_zero_outside hZ v hv)
+  by_cases hC : C.card ≤ 1
+  · exact (finite Z hZ v).mass_singleton C hC
+  · obtain ⟨a,ha,b,hb,hab⟩ := Finset.one_lt_card.mp (by omega : 1<C.card)
+    have hl := (finite Z hZ v).mass_le_overlap hn C a b hab ha hb
+    rw [hz.2.2 a b hab] at hl
+    exact le_antisymm hl ((finite Z hZ v).mass_nonneg C)
+
+omit h₁ h₂ in
+lemma sum_finite_mass {Z : P → K} (hZ : Function.Injective Z) (hn : 2 ≤ Fintype.card P)
+    (V : Finset (FinitePlace K)) (hV : badPlaces Z hZ ⊆ V) (C : Finset P) :
+    (∑ v ∈ V, (finite Z hZ v).mass C) = ∑ v ∈ badPlaces Z hZ, (finite Z hZ v).mass C := by
+  symm
+  exact Finset.sum_subset hV (fun v _ hv => finite_mass_zero hZ hn v hv C)
+
+omit h₁ h₂ in
+lemma sum_finite_potential {Z : P → K} (hZ : Function.Injective Z) (hn : 2 ≤ Fintype.card P)
+    (V : Finset (FinitePlace K)) (hV : badPlaces Z hZ ⊆ V) (a : P) :
+    (∑ v ∈ V, (finite Z hZ v).potential a) = ∑ v ∈ badPlaces Z hZ, (finite Z hZ v).potential a := by
+  symm
+  exact Finset.sum_subset hV (fun v _ hv =>
+    ((finite Z hZ v).zero_values hn (finite_zero_outside hZ v hv)).2.1 a)
+
+omit h₁ h₂ in
+def placeAverage (V : Finset (FinitePlace K)) (A : InfinitePlace K → ℝ) (F : FinitePlace K → ℝ) : ℝ :=
+  (∑ v, Arithmetic.archWeight v * A v) + (Module.finrank ℚ K:ℝ)⁻¹ * ∑ v ∈ V, F v
+
+omit h₁ h₂ in
+lemma placeAverage_mono (V : Finset (FinitePlace K)) {A A' : InfinitePlace K → ℝ} {F F' : FinitePlace K → ℝ}
+    (hA : ∀ v, A v ≤ A' v) (hF : ∀ v ∈ V, F v ≤ F' v) : placeAverage V A F ≤ placeAverage V A' F' := by
+  apply add_le_add
+  · exact Finset.sum_le_sum (fun v _ => mul_le_mul_of_nonneg_left (hA v) (Arithmetic.archWeight_nonneg v))
+  · exact mul_le_mul_of_nonneg_left (Finset.sum_le_sum hF) (by positivity)
+
+omit h₁ h₂ in
+lemma placeAverage_add (V : Finset (FinitePlace K)) (A A' : InfinitePlace K → ℝ) (F F' : FinitePlace K → ℝ) :
+    placeAverage V (fun v => A v+A' v) (fun v => F v+F' v) = placeAverage V A F + placeAverage V A' F' := by
+  simp only [placeAverage,mul_add,Finset.sum_add_distrib]
+  ring
+
+omit h₁ h₂ in
+lemma placeAverage_mul (V : Finset (FinitePlace K)) (a : ℝ) (A : InfinitePlace K → ℝ) (F : FinitePlace K → ℝ) :
+    placeAverage V (fun v => a*A v) (fun v => a*F v) = a*placeAverage V A F := by
+  simp only [placeAverage,Finset.mul_sum,mul_add]
+  congr 1
+  · exact Finset.sum_congr rfl (fun v _ => by ring)
+  · exact Finset.sum_congr rfl (fun v _ => by ring)
+
+omit h₁ h₂ in
+lemma placeAverage_arch_const (V : Finset (FinitePlace K)) (a : ℝ) :
+    placeAverage V (fun _ => a) (fun _ => 0) = a := by
+  simp only [placeAverage,Finset.sum_const_zero,mul_zero,add_zero,← Finset.sum_mul,Arithmetic.sum_archWeight,one_mul]
+
+omit h₁ h₂ in
+lemma cellIntegral_placeAverage {Z : P → K} (hZ : Function.Injective Z) (hn : 2 ≤ Fintype.card P)
+    (V : Finset (FinitePlace K)) (hV : badPlaces Z hZ ⊆ V) (f : Finset P → ℝ) :
+    weightedIntegral (cellMass Z hZ) f = placeAverage V
+      (fun v => ∑ C, (∫ ω, (arch Z v ω).mass C ∂Grid.fullLaw)*f C)
+      (fun v => ∑ C, (finite Z hZ v).mass C*f C) := by
+  simp only [weightedIntegral,cellMass,placeAverage,add_mul,Finset.sum_add_distrib]
+  congr 1
+  · simp_rw [Finset.mul_sum,Finset.sum_mul]
+    rw [Finset.sum_comm]
+    apply Finset.sum_congr rfl
+    intro v _
+    exact Finset.sum_congr rfl (fun C _ => by ring)
+  · rw [Finset.sum_comm (s:=V)]
+    simp_rw [← Finset.sum_mul, sum_finite_mass hZ hn V hV,Finset.mul_sum]
+    exact Finset.sum_congr rfl (fun C _ => by rw [← Finset.mul_sum]; ring)
+
+omit h₁ h₂ in
+lemma potentialIntegral_placeAverage {Z : P → K} (hZ : Function.Injective Z) (hn : 2 ≤ Fintype.card P)
+    (V : Finset (FinitePlace K)) (hV : badPlaces Z hZ ⊆ V) :
+    (∑ y, S Z hZ y) = placeAverage V
+      (fun v => ∑ y, ∫ ω, (arch Z v ω).potential y ∂Grid.fullLaw)
+      (fun v => ∑ y, (finite Z hZ v).potential y) := by
+  simp only [S,placeAverage,Finset.sum_add_distrib]
+  congr 1
+  · simp_rw [Finset.mul_sum]
+    rw [Finset.sum_comm]
+  · rw [Finset.sum_comm (s:=V)]
+    simp_rw [sum_finite_potential hZ hn V hV,Finset.mul_sum]
+
+lemma arithmetic_small_fiber_one (hn : 2 ≤ Fintype.card P)
+    (w : I → ℝ) (p : I → P → ℝ) (x : I → P) (t : I → K)
+    (hw : ∀ i, 0 ≤ w i) (hW : ∑ i, w i=1) (hp : ∀ i y, 0 ≤ p i y)
+    (hp1 : ∀ i, ∑ y, p i y=1) (hx : ∀ i, p i (x i)=0)
+    (ht : ∀ i y, p i y ≠ 0 → (Z₁ (x i)-Z₁ y)*(Z₂ (x i)-Z₂ y)=t i)
+    (B τ lam H : ℝ) (hB : 0 ≤ B) (hlam : 0 ≤ lam) (hlam2 : lam ≤ 1/2)
+    (hτ : 0 < τ) (hτ1 : τ ≤ 1) (hτn : 2 ≤ τ*(Fintype.card P:ℝ)) (hH : 0 ≤ H)
+    (hdom : ∀ a b, a ≠ b → edgeLaw w p x a b ≤ B/((Fintype.card P:ℝ)*((Fintype.card P:ℝ)-1))) :
+    weightedIntegral (cellMass Z₁ h₁) (slice w p x lam) ≤
+      (2*lam/τ^2)*weightedIntegral (cellMass Z₁ h₁) (pairCell Finset.univ) +
+      B*τ/(Fintype.card P:ℝ)*((∑ y, S Z₁ h₁ y)+(∑ y, S Z₂ h₂ y)) +
+      2*lam*B*weightedIntegral (cellMass Z₂ h₂) (pairCell Finset.univ) + 2*lam*H +
+      16*Real.exp (-H/2) := by
+  classical
+  let V := badPlaces Z₁ h₁ ∪ badPlaces Z₂ h₂
+  have hV₁ : badPlaces Z₁ h₁ ⊆ V := Finset.subset_union_left
+  have hV₂ : badPlaces Z₂ h₂ ⊆ V := Finset.subset_union_right
+  have hf (v : FinitePlace K) : (∑ C, (finite Z₁ h₁ v).mass C*slice w p x lam C) ≤
+      (2*lam/τ^2)*(∑ C, (finite Z₁ h₁ v).mass C*pairCell Finset.univ C) +
+      B*τ/(Fintype.card P:ℝ)*((∑ y, (finite Z₁ h₁ v).potential y)+(∑ y, (finite Z₂ h₂ v).potential y)) +
+      2*lam*B*(∑ C, (finite Z₂ h₂ v).mass C*pairCell Finset.univ C) + 0 + 0 := by
+    simpa only [add_zero] using finite_small_fiber Z₁ Z₂ h₁ h₂ hn v w p x t hw hW hp hp1 hx ht B τ lam hB hlam hlam2 hτ hτ1 hτn hdom
+  have hb := placeAverage_mono V
+    (fun v => arch_small_fiber Z₁ Z₂ h₁ h₂ hn v w p x t hw hW hp hp1 hx ht B τ lam H hB hlam hlam2 hτ hτ1 hτn hH hdom)
+    (fun v _ => hf v)
+  simp only [placeAverage_add,placeAverage_mul,placeAverage_arch_const] at hb
+  rw [← cellIntegral_placeAverage h₁ hn V hV₁,← cellIntegral_placeAverage h₁ hn V hV₁,
+    ← cellIntegral_placeAverage h₂ hn V hV₂,← potentialIntegral_placeAverage h₁ hn V hV₁,
+    ← potentialIntegral_placeAverage h₂ hn V hV₂] at hb
+  exact hb
+
+/-- The global small-fiber estimate, uniformly over number fields, places and configurations. -/
+
+lemma arithmetic_small_fiber (hn : 2 ≤ Fintype.card P)
+    (w : I → ℝ) (p : I → P → ℝ) (x : I → P) (t : I → K)
+    (hw : ∀ i, 0 ≤ w i) (hW : ∑ i, w i=1) (hp : ∀ i y, 0 ≤ p i y)
+    (hp1 : ∀ i, ∑ y, p i y=1) (hx : ∀ i, p i (x i)=0)
+    (ht : ∀ i y, p i y ≠ 0 → (Z₁ (x i)-Z₁ y)*(Z₂ (x i)-Z₂ y)=t i)
+    (B τ lam H : ℝ) (hB : 0 ≤ B) (hlam : 0 ≤ lam) (hlam2 : lam ≤ 1/2)
+    (hτ : 0 < τ) (hτ1 : τ ≤ 1) (hτn : 2 ≤ τ*(Fintype.card P:ℝ)) (hH : 0 ≤ H)
+    (hdom : ∀ a b, a ≠ b → edgeLaw w p x a b ≤ B/((Fintype.card P:ℝ)*((Fintype.card P:ℝ)-1))) :
+    weightedIntegral (totalMass Z₁ Z₂ h₁ h₂) (slice w p x lam) ≤
+      (2*lam/τ^2+2*lam*B)*weightedIntegral (totalMass Z₁ Z₂ h₁ h₂) (pairCell Finset.univ) +
+      2*B*τ/(Fintype.card P:ℝ)*(∑ y, totalPotential Z₁ Z₂ h₁ h₂ y) + 4*lam*H +
+      32*Real.exp (-H/2) := by
+  have hA := arithmetic_small_fiber_one Z₁ Z₂ h₁ h₂ hn w p x t hw hW hp hp1 hx ht B τ lam H hB hlam hlam2 hτ hτ1 hτn hH hdom
+  have ht' : ∀ i y, p i y ≠ 0 → (Z₂ (x i)-Z₂ y)*(Z₁ (x i)-Z₁ y)=t i := by
+    intro i y hy
+    rw [mul_comm]
+    exact ht i y hy
+  have hB' := arithmetic_small_fiber_one Z₂ Z₁ h₂ h₁ hn w p x t hw hW hp hp1 hx ht' B τ lam H hB hlam hlam2 hτ hτ1 hτn hH hdom
+  simp only [weightedIntegral,totalMass,totalPotential,add_mul,Finset.sum_add_distrib] at hA hB' ⊢
+  linear_combination hA+hB'
+
+end WeakPinned.Global
+
+noncomputable section
+open Filter
+open scoped Topology
+
+end
+end
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PinnedDistances/GlobalPotentials.lean b/lean/OAI/Geometry/PinnedDistances/GlobalPotentials.lean
new file mode 100644
index 0000000000000000000000000000000000000000..67146c5922d288190c21a39fa11ead737d8dcf7a
--- /dev/null
+++ b/lean/OAI/Geometry/PinnedDistances/GlobalPotentials.lean
@@ -0,0 +1,524 @@
+import OAI.Geometry.PinnedDistances.MinPotentials
+
+namespace OAI
+
+universe u1 u2 u3 u4 u5 u6 u7 u8 u9 u10 u11 u12
+universe u13 u14 u15 u16
+
+section
+noncomputable section
+open NumberField
+open scoped BigOperators
+noncomputable section
+open MeasureTheory
+open scoped Topology
+noncomputable section
+open MeasureTheory
+open scoped Topology ENNReal
+noncomputable section
+open MeasureTheory NumberField
+open scoped BigOperators
+noncomputable section
+open scoped BigOperators
+open MeasureTheory
+noncomputable section
+open Filter
+open scoped Topology
+noncomputable section
+open scoped BigOperators
+noncomputable section
+open Filter
+open scoped Topology
+noncomputable section
+open MeasureTheory
+open scoped BigOperators
+noncomputable section
+
+namespace WeakPinned.Global
+open Hierarchy MeasureTheory Variance
+open scoped BigOperators
+variable {P : Type u1} [Fintype P] [DecidableEq P]
+variable {K : Type u2} [Field K] [NumberField K]
+variable (Z₁ Z₂ : P → K) (h₁ : Function.Injective Z₁) (h₂ : Function.Injective Z₂)
+
+def placeUnion : Finset (NumberField.FinitePlace K)  := by
+  classical
+  exact badPlaces Z₁ h₁ ∪ badPlaces Z₂ h₂
+
+def average (f : Depth P → Depth P → ℝ) : ℝ :=
+  (∑ v : NumberField.InfinitePlace K, Arithmetic.archWeight v *
+    ∫ ω, f (arch Z₁ v ω) (arch Z₂ v ω) ∂Grid.fullLaw) +
+  (Module.finrank ℚ K : ℝ)⁻¹ * ∑ v ∈ placeUnion Z₁ Z₂ h₁ h₂,
+    f (finite Z₁ h₁ v) (finite Z₂ h₂ v)
+
+def AverageIntegrable (f : Depth P → Depth P → ℝ) : Prop :=
+  ∀ v : NumberField.InfinitePlace K,
+    Integrable (fun ω => f (arch Z₁ v ω) (arch Z₂ v ω)) Grid.fullLaw
+
+lemma average_mono {f g : Depth P → Depth P → ℝ}
+    (hi : AverageIntegrable Z₁ Z₂ f) (hj : AverageIntegrable Z₁ Z₂ g)
+    (hfg : ∀ d d', f d d'≤g d d') :
+    average Z₁ Z₂ h₁ h₂ f≤average Z₁ Z₂ h₁ h₂ g  := by
+  classical
+  apply add_le_add
+  · apply Finset.sum_le_sum
+    intro v _
+    exact mul_le_mul_of_nonneg_left (integral_mono (hi v) (hj v) (fun ω => hfg _ _))
+      (Arithmetic.archWeight_nonneg v)
+  · apply mul_le_mul_of_nonneg_left _ (by positivity)
+    exact Finset.sum_le_sum (fun v _ => hfg _ _)
+
+lemma average_add {f g : Depth P → Depth P → ℝ}
+    (hi : AverageIntegrable Z₁ Z₂ f) (hj : AverageIntegrable Z₁ Z₂ g) :
+    average Z₁ Z₂ h₁ h₂ (fun d d' => f d d'+g d d')=
+      average Z₁ Z₂ h₁ h₂ f+average Z₁ Z₂ h₁ h₂ g  := by
+  classical
+  unfold average
+  simp only [integral_add (hi _) (hj _),mul_add,Finset.sum_add_distrib]
+  ring
+
+lemma average_mul (f : Depth P → Depth P → ℝ) (c : ℝ) :
+    average Z₁ Z₂ h₁ h₂ (fun d d' => c*f d d')=c*average Z₁ Z₂ h₁ h₂ f  := by
+  classical
+  unfold average
+  simp only [integral_const_mul,Finset.mul_sum,mul_add]
+  congr 1 <;> apply Finset.sum_congr rfl <;> intro v _ <;> ring
+
+lemma average_sum {J : Type u3} [Fintype J] (f : J → Depth P → Depth P → ℝ)
+    (hi : ∀ j, AverageIntegrable Z₁ Z₂ (f j)) :
+    average Z₁ Z₂ h₁ h₂ (fun d d' => ∑ j, f j d d')=
+      ∑ j, average Z₁ Z₂ h₁ h₂ (f j)  := by
+  classical
+  unfold average
+  simp_rw [integral_finsetSum _ (fun j _ => hi j _),Finset.mul_sum]
+  rw [Finset.sum_comm,Finset.sum_comm (s:=placeUnion Z₁ Z₂ h₁ h₂)]
+  simp only [Finset.sum_add_distrib]
+
+lemma average_potential_left (hn : 2≤Fintype.card P) (x : P) :
+    average Z₁ Z₂ h₁ h₂ (fun d _ => d.potential x)=S Z₁ h₁ x  := by
+  classical
+  unfold average S
+  congr 2
+  exact sum_finite_potential h₁ hn _ Finset.subset_union_left x
+
+lemma average_potential_right (hn : 2≤Fintype.card P) (x : P) :
+    average Z₁ Z₂ h₁ h₂ (fun _ d' => d'.potential x)=S Z₂ h₂ x  := by
+  classical
+  unfold average S
+  congr 2
+  exact sum_finite_potential h₂ hn _ Finset.subset_union_right x
+
+lemma average_mass_left (hn : 2≤Fintype.card P) (C : Finset P) :
+    average Z₁ Z₂ h₁ h₂ (fun d _ => d.mass C)=cellMass Z₁ h₁ C  := by
+  classical
+  unfold average cellMass
+  congr 2
+  exact sum_finite_mass h₁ hn _ Finset.subset_union_left C
+
+lemma average_mass_right (hn : 2≤Fintype.card P) (C : Finset P) :
+    average Z₁ Z₂ h₁ h₂ (fun _ d' => d'.mass C)=cellMass Z₂ h₂ C  := by
+  classical
+  unfold average cellMass
+  congr 2
+  exact sum_finite_mass h₂ hn _ Finset.subset_union_right C
+
+omit [NumberField K] in
+include h₁ in
+lemma integrable_potential_left (hn : 2≤Fintype.card P) (x : P) :
+    AverageIntegrable Z₁ Z₂ (fun d _ => d.potential x) :=
+  fun v => Depth.integrable_potential_param (arch Z₁ v) (arch_measurable Z₁ h₁ v)
+    (arch_integrable Z₁ h₁ v) hn x
+
+omit [NumberField K] in
+include h₂ in
+lemma integrable_potential_right (hn : 2≤Fintype.card P) (x : P) :
+    AverageIntegrable Z₁ Z₂ (fun _ d' => d'.potential x) :=
+  fun v => Depth.integrable_potential_param (arch Z₂ v) (arch_measurable Z₂ h₂ v)
+    (arch_integrable Z₂ h₂ v) hn x
+
+omit [NumberField K] in
+include h₁ in
+lemma integrable_mass_left (hn : 2≤Fintype.card P) (C : Finset P) :
+    AverageIntegrable Z₁ Z₂ (fun d _ => d.mass C) :=
+  fun v => Depth.integrable_mass_param (arch Z₁ v) (arch_measurable Z₁ h₁ v)
+    (arch_integrable Z₁ h₁ v) hn C
+
+omit [NumberField K] in
+include h₂ in
+lemma integrable_mass_right (hn : 2≤Fintype.card P) (C : Finset P) :
+    AverageIntegrable Z₁ Z₂ (fun _ d' => d'.mass C) :=
+  fun v => Depth.integrable_mass_param (arch Z₂ v) (arch_measurable Z₂ h₂ v)
+    (arch_integrable Z₂ h₂ v) hn C
+
+end WeakPinned.Global
+
+namespace WeakPinned.Global
+open Hierarchy MeasureTheory Variance
+open scoped BigOperators
+variable {P : Type u4} [Fintype P] [DecidableEq P]
+variable {K : Type u5} [Field K] [NumberField K]
+variable (Z₁ Z₂ : P → K) (h₁ : Function.Injective Z₁) (h₂ : Function.Injective Z₂)
+
+omit Z₁ Z₂ h₁ h₂ in
+def potentialEval (p : P → ℝ) (d d' : Depth P) : ℝ := ∑ y, p y*(d.potential y+d'.potential y)
+omit Z₁ Z₂ h₁ h₂ in
+def massEval (q : Finset P → ℝ) (d d' : Depth P) : ℝ :=
+  weightedIntegral d.mass q+weightedIntegral d'.mass q
+
+omit [NumberField K] in
+include h₁ h₂ in
+lemma integrable_potentialEval (hn : 2≤Fintype.card P) (p : P → ℝ) :
+    AverageIntegrable Z₁ Z₂ (potentialEval p) := by
+  intro v
+  exact integrable_finsetSum (s:=Finset.univ) (fun y _ =>
+    ((integrable_potential_left Z₁ Z₂ h₁ hn y v).add
+      (integrable_potential_right Z₁ Z₂ h₂ hn y v)).const_mul (p y))
+
+omit [NumberField K] in
+include h₁ h₂ in
+lemma integrable_massEval (hn : 2≤Fintype.card P) (q : Finset P → ℝ) :
+    AverageIntegrable Z₁ Z₂ (massEval q) := by
+  intro v
+  exact (integrable_finsetSum (s:=Finset.univ) (fun C _ =>
+      (integrable_mass_left Z₁ Z₂ h₁ hn C v).mul_const (q C))).add
+    (integrable_finsetSum (s:=Finset.univ) (fun C _ =>
+      (integrable_mass_right Z₁ Z₂ h₂ hn C v).mul_const (q C)))
+
+lemma average_potentialEval (hn : 2≤Fintype.card P) (p : P → ℝ) :
+    average Z₁ Z₂ h₁ h₂ (potentialEval p)=∑ y, p y*totalPotential Z₁ Z₂ h₁ h₂ y := by
+  unfold potentialEval
+  rw [average_sum Z₁ Z₂ h₁ h₂ _ (fun y v =>
+    ((integrable_potential_left Z₁ Z₂ h₁ hn y v).add
+      (integrable_potential_right Z₁ Z₂ h₂ hn y v)).const_mul (p y))]
+  apply Finset.sum_congr rfl
+  intro y _
+  rw [average_mul,average_add Z₁ Z₂ h₁ h₂
+    (integrable_potential_left Z₁ Z₂ h₁ hn y) (integrable_potential_right Z₁ Z₂ h₂ hn y),
+    average_potential_left Z₁ Z₂ h₁ h₂ hn,average_potential_right Z₁ Z₂ h₁ h₂ hn]
+  rfl
+
+lemma average_massEval (hn : 2≤Fintype.card P) (q : Finset P → ℝ) :
+    average Z₁ Z₂ h₁ h₂ (massEval q)=weightedIntegral (totalMass Z₁ Z₂ h₁ h₂) q := by
+  have he : massEval q=(fun d d' => ∑ C, q C*(d.mass C+d'.mass C)) := by
+    funext d d'
+    simp only [massEval,weightedIntegral,mul_add,Finset.sum_add_distrib]
+    congr 1 <;> apply Finset.sum_congr rfl <;> intro C _ <;> ring
+  rw [he,average_sum Z₁ Z₂ h₁ h₂ _ (fun C v =>
+    ((integrable_mass_left Z₁ Z₂ h₁ hn C v).add
+      (integrable_mass_right Z₁ Z₂ h₂ hn C v)).const_mul (q C))]
+  apply Finset.sum_congr rfl
+  intro C _
+  rw [average_mul,average_add Z₁ Z₂ h₁ h₂
+    (integrable_mass_left Z₁ Z₂ h₁ hn C) (integrable_mass_right Z₁ Z₂ h₂ hn C),
+    average_mass_left Z₁ Z₂ h₁ h₂ hn,average_mass_right Z₁ Z₂ h₁ h₂ hn]
+  simp only [totalMass]; ring
+
+omit Z₁ Z₂ h₁ h₂ in
+lemma massEval_edge {I : Type u6} [Fintype I] (w : I → ℝ) (p : I → P → ℝ) (x : I → P)
+    (d d' : Depth P) :
+    massEval (SmallFiber.edgeCell w p x) d d'=
+      ∑ i, w i*∑ y, p i y*(weightedOverlap d.mass (x i) y+weightedOverlap d'.mass (x i) y) := by
+  have he (m : Finset P → ℝ) : weightedIntegral m (SmallFiber.edgeCell w p x)=
+      ∑ i, w i*∑ y, p i y*weightedOverlap m (x i) y := by
+    unfold SmallFiber.edgeCell
+    rw [weighted_pair]
+    unfold SmallFiber.edgeLaw
+    simp only [Finset.sum_mul]
+    calc
+      _ = ∑ a, ∑ i, ∑ y, (if x i=a then w i*p i y else 0)*weightedOverlap m a y := by
+        apply Finset.sum_congr rfl
+        intro a _
+        exact Finset.sum_comm
+      _ = ∑ i, ∑ a, ∑ y, (if x i=a then w i*p i y else 0)*weightedOverlap m a y := Finset.sum_comm
+      _ = _ := by
+        simp only [ite_mul,zero_mul,Finset.sum_ite_irrel,Finset.sum_const_zero]
+        simp [Finset.mul_sum,mul_assoc]
+  -- commute the source sum after inserting the indicator
+  simp only [massEval,he,mul_add,Finset.sum_add_distrib]
+end WeakPinned.Global
+
+namespace WeakPinned.Global
+open Hierarchy MeasureTheory Variance
+open scoped BigOperators
+variable {P : Type u7} {I : Type u8} [Fintype P] [DecidableEq P] [Fintype I]
+variable {K : Type u9} [Field K] [NumberField K]
+variable (Z₁ Z₂ : P → K) (h₁ : Function.Injective Z₁) (h₂ : Function.Injective Z₂)
+
+omit Z₁ Z₂ h₁ h₂ in
+lemma potentialEval_mixture (w : I → ℝ) (p : I → P → ℝ) (d d' : Depth P) :
+    potentialEval (fun y => ∑ i, w i*p i y) d d'=∑ i, w i*∑ y, p i y*(d.potential y+d'.potential y) := by
+  simp only [potentialEval,Finset.sum_mul,Finset.mul_sum,mul_assoc]
+  exact Finset.sum_comm
+
+lemma average_sub {f g : Depth P → Depth P → ℝ}
+    (hi : AverageIntegrable Z₁ Z₂ f) (hj : AverageIntegrable Z₁ Z₂ g) :
+    average Z₁ Z₂ h₁ h₂ (fun d d' => f d d'-g d d')=
+      average Z₁ Z₂ h₁ h₂ f-average Z₁ Z₂ h₁ h₂ g := by
+  classical
+  unfold average
+  simp only [integral_sub (hi _) (hj _),mul_sub,Finset.sum_sub_distrib]
+  ring
+
+/-- The first-moment raw-depth error is archimedean only. Its global constant
+is independent of the degree and all coordinates. -/
+lemma arithmetic_potential_fiber_bound (hn : 2≤Fintype.card P)
+    (w : I → ℝ) (B : I → Finset P) (src : I → P) (t : I → K) (T M κ : ℝ)
+    (hw : ∀ i, 0≤w i) (hW : (∑ i, w i)=1) (hT : 1≤T) (hκ : 0≤κ)
+    (hB : ∀ i, 2≤(B i).card) (hk : ∀ i, 2/((B i).card:ℝ)≤κ)
+    (hsrc : ∀ i, src i∉B i)
+    (hdom : ∀ x, (∑ i, w i*prob (B i) x)≤M*prob Finset.univ x)
+    (ht : ∀ i y, y∈B i → (Z₁ (src i)-Z₁ y)*(Z₂ (src i)-Z₂ y)=t i) :
+    (∑ y, (∑ i, w i*prob (B i) y)*totalPotential Z₁ Z₂ h₁ h₂ y) ≤
+      4*weightedIntegral (totalMass Z₁ Z₂ h₁ h₂) (varianceCell w B T)+
+      8*overlapScale Z₁ Z₂ h₁ h₂+
+      2*κ*M*potentialScale Z₁ Z₂ h₁ h₂+
+      weightedIntegral (totalMass Z₁ Z₂ h₁ h₂) (SmallFiber.edgeCell w (fun i => prob (B i)) src)+16 := by
+  classical
+  let p : I → P → ℝ := fun i => prob (B i)
+  let r : P → ℝ := fun y => ∑ i, w i*p i y
+  let q := SmallFiber.edgeCell w p src
+  let vcell := varianceCell w B T
+  let pc := pairCell (Finset.univ : Finset P)
+  let unif := prob (Finset.univ : Finset P)
+  let R : Depth P → Depth P → ℝ := fun d d' =>
+    4*massEval vcell d d'+8*massEval pc d d'+2*κ*M*potentialEval unif d d'+massEval q d d'
+  let f : Depth P → Depth P → ℝ := fun d d' => potentialEval r d d'-R d d'
+  have hp (i y) : 0≤p i y := by dsimp [p,prob]; split_ifs <;> positivity
+  have hp1 (i) : ∑ y, p i y=1 := sum_prob (B i) (Finset.card_pos.mp (by have := hB i; omega))
+  have hx (i) : p i (src i)=0 := by simp [p,prob,hsrc i]
+  have ht' (i y) (hy : p i y≠0) : (Z₁ (src i)-Z₁ y)*(Z₂ (src i)-Z₂ y)=t i := by
+    apply ht i y
+    by_contra hn'; exact hy (by simp [p,prob,hn'])
+  have hL := integrable_potentialEval Z₁ Z₂ h₁ h₂ hn r
+  have hV := integrable_massEval Z₁ Z₂ h₁ h₂ hn vcell
+  have hP := integrable_massEval Z₁ Z₂ h₁ h₂ hn pc
+  have hA := integrable_potentialEval Z₁ Z₂ h₁ h₂ hn unif
+  have hQ := integrable_massEval Z₁ Z₂ h₁ h₂ hn q
+  have hR : AverageIntegrable Z₁ Z₂ R := fun v =>
+    ((((hV v).const_mul 4).add ((hP v).const_mul 8)).add ((hA v).const_mul (2*κ*M))).add (hQ v)
+  have hf : AverageIntegrable Z₁ Z₂ f := fun v => (hL v).sub (hR v)
+  have hb (d d' : Depth P) (a : I → ℝ) :
+      f d d'≤∑ i, w i*∑ y, p i y*|d.value (src i) y+d'.value (src i) y-a i| := by
+    have hh := d.local_potential_fiber_bound d' hn w B src a T M κ hw hW.le hT hκ hB hk hsrc hdom
+    rw [← potentialEval_mixture,← massEval_edge] at hh
+    change potentialEval r d d'≤R d d'+_ at hh
+    exact sub_le_iff_le_add'.mpr hh
+  have ha (v : NumberField.InfinitePlace K) :
+      (∫ ω, f (arch Z₁ v ω) (arch Z₂ v ω) ∂Grid.fullLaw)≤16 := by
+    let E : Grid.FullSample → ℝ := fun ω => ∑ i, w i*∑ y, p i y*
+      |(arch Z₁ v ω).value (src i) y+(arch Z₂ v ω).value (src i) y+Real.log (v (t i))|
+    have hi : Integrable E Grid.fullLaw := integrable_finsetSum (s:=Finset.univ) (fun i _ =>
+      (integrable_finsetSum (s:=Finset.univ) (fun y _ =>
+        ((((arch_integrable Z₁ h₁ v _ _).add (arch_integrable Z₂ h₂ v _ _)).add
+          (integrable_const _)).abs.const_mul (p i y)))).const_mul (w i))
+    have hu : (∫ ω, E ω ∂Grid.fullLaw)≤16 := by
+      simpa only [E,sub_zero,max_eq_left (abs_nonneg (_ : ℝ)),neg_zero,zero_div,Real.exp_zero,mul_one]
+        using arch_weighted_defect Z₁ Z₂ h₁ h₂ v w p src t hw hW hp hp1 hx ht' 0 (by norm_num)
+    exact (integral_mono (hf v) hi (fun ω => by
+      simpa only [sub_neg_eq_add] using hb (arch Z₁ v ω) (arch Z₂ v ω) (fun i => -Real.log (v (t i))))).trans hu
+  have hv (v : NumberField.FinitePlace K) : f (finite Z₁ h₁ v) (finite Z₂ h₂ v)≤0 := by
+    have he (i y) : p i y*|(finite Z₁ h₁ v).value (src i) y+
+        (finite Z₂ h₂ v).value (src i) y-(-Real.log (v (t i)))|=0 := by
+      by_cases hy : p i y=0
+      · simp [hy]
+      · have hxy : src i≠y := by intro hh; exact hsrc i (hh ▸ (by
+          by_contra hm; exact hy (by simp [p,prob,hm])))
+        have hh := Arithmetic.raw_fiber_identity v.val (Z₁ (src i)-Z₁ y) (Z₂ (src i)-Z₂ y)
+          (sub_ne_zero.mpr (h₁.ne hxy)) (sub_ne_zero.mpr (h₂.ne hxy))
+        rw [ht' i y hy] at hh
+        change -Real.log (v (Z₁ (src i)-Z₁ y)) + -Real.log (v (Z₂ (src i)-Z₂ y)) = -Real.log (v (t i)) at hh
+        simp only [finite,ite_eq_right hxy,hh,sub_self,abs_zero,mul_zero]
+    simpa only [he,Finset.sum_const_zero,mul_zero] using
+      hb (finite Z₁ h₁ v) (finite Z₂ h₂ v) (fun i => -Real.log (v (t i)))
+  have hg : average Z₁ Z₂ h₁ h₂ f≤16 := by
+    change placeAverage (placeUnion Z₁ Z₂ h₁ h₂) _ _≤16
+    have hh := placeAverage_mono (placeUnion Z₁ Z₂ h₁ h₂) ha (fun v _ => hv v)
+    rwa [placeAverage_arch_const] at hh
+  have hRe : average Z₁ Z₂ h₁ h₂ R =
+      4*weightedIntegral (totalMass Z₁ Z₂ h₁ h₂) vcell+
+      8*weightedIntegral (totalMass Z₁ Z₂ h₁ h₂) pc+
+      2*κ*M*(∑ y, unif y*totalPotential Z₁ Z₂ h₁ h₂ y)+
+      weightedIntegral (totalMass Z₁ Z₂ h₁ h₂) q := by
+    unfold R
+    rw [average_add Z₁ Z₂ h₁ h₂ (fun v => (((hV v).const_mul 4).add ((hP v).const_mul 8)).add ((hA v).const_mul (2*κ*M))) hQ,
+      average_add Z₁ Z₂ h₁ h₂ (fun v => ((hV v).const_mul 4).add ((hP v).const_mul 8)) (fun v => (hA v).const_mul (2*κ*M)),
+      average_add Z₁ Z₂ h₁ h₂ (fun v => (hV v).const_mul 4) (fun v => (hP v).const_mul 8)]
+    simp only [average_mul,average_massEval Z₁ Z₂ h₁ h₂ hn,average_potentialEval Z₁ Z₂ h₁ h₂ hn]
+  change average Z₁ Z₂ h₁ h₂ (fun d d' => potentialEval r d d'-R d d')≤16 at hg
+  rw [average_sub Z₁ Z₂ h₁ h₂ hL hR,average_potentialEval Z₁ Z₂ h₁ h₂ hn,hRe] at hg
+  exact sub_le_iff_le_add'.mp hg
+end WeakPinned.Global
+
+namespace WeakPinned.Variance
+open scoped BigOperators
+variable {P : Type u10} [Fintype P] [DecidableEq P]
+
+/-- Nonnegative additive off-diagonal overlap controls the potential at every
+vertex, not merely at vertices typical for the edge marginal. -/
+lemma point_potential_lower (m : Finset P → ℝ) (b : ℝ) (S : P → ℝ)
+    (hm : ∀ C : Finset P, (0 : ℝ) ≤ m C) (hL : ∀ x y, x≠y → weightedOverlap m x y=b+S x+S y)
+    (hn : 3≤Fintype.card P) (x : P) :
+    (∑ y, prob Finset.univ y*S y)-2*weightedIntegral m (pairCell Finset.univ)≤S x := by
+  let n : ℝ := Fintype.card P
+  have hn3 : 3≤n := by dsimp [n]; exact_mod_cast hn
+  have hn0 : n≠0 := by linarith
+  let A : ℝ := (∑ y, prob Finset.univ y*S y)
+  let W : ℝ := weightedIntegral m (pairCell Finset.univ)
+  have hW : 0≤W := Finset.sum_nonneg (fun C _ => mul_nonneg (hm C) (pairCell_nonneg _ _))
+  have hu : W=b+2*A := uniform_additive m b S hL (by omega)
+  have hs : (∑ y, S y)=n*A := by
+    dsimp [A,prob,n]
+    simp only [Finset.mem_univ,ite_true]
+    rw [← Finset.mul_sum]
+    have hn0' : (Fintype.card P : ℝ) ≠ 0 := hn0
+    field_simp
+  have hnn : 0 ≤ ∑ y ∈ Finset.univ.erase x, (b+S x+S y) := by
+    apply Finset.sum_nonneg
+    intro y hy
+    rw [← hL x y (Finset.ne_of_mem_erase hy).symm]
+    exact Finset.sum_nonneg (fun C _ => mul_nonneg (hm C) (by dsimp; split_ifs <;> norm_num))
+  have hsx : (∑ y ∈ Finset.univ.erase x, S y)=(∑ y, S y)-S x := by
+    exact eq_sub_of_add_eq (Finset.sum_erase_add Finset.univ S (Finset.mem_univ x))
+  have hcount : ((Finset.univ.erase x).card:ℝ)=n-1 := by
+    rw [Finset.card_erase_of_mem (Finset.mem_univ x),Finset.card_univ,Nat.cast_sub (by omega),Nat.cast_one]
+  simp only [Finset.sum_add_distrib,Finset.sum_const,nsmul_eq_mul,hcount,hsx,hs] at hnn
+  change A-2*W≤S x
+  nlinarith [mul_nonneg (by linarith : 0≤n-3) hW]
+
+end WeakPinned.Variance
+
+namespace WeakPinned
+open Global Variance
+open scoped BigOperators
+variable {P : Type u11} {I : Type u12} [Fintype P] [Fintype I] [DecidableEq P]
+variable {K : Type u13} [Field K] [NumberField K]
+
+/-- Quantitative scale balance for arithmetic fibers, without an unboundedness assumption. -/
+
+theorem arithmetic_scale_balance (G : Core.Fibers P I)
+    (Z₁ Z₂ : P → K) (h₁ : Function.Injective Z₁) (h₂ : Function.Injective Z₂)
+    (hn : 3≤Fintype.card P) (hG : 0<G.edgeCount)
+    (T M κ : ℝ) (hT : 1≤T) (hκ : 0≤κ)
+    (hB : ∀ i, 2≤(G.targets i).card) (hk : ∀ i, 2/((G.targets i).card:ℝ)≤κ)
+    (hdom : ∀ x y, G.edgeLaw x y≤M*pairProb Finset.univ x y)
+    (t : I → K) (ht : ∀ i y, y∈G.targets i →
+      (Z₁ (G.source i)-Z₁ y)*(Z₂ (G.source i)-Z₂ y)=t i) :
+    (1-2*κ*M)*potentialScale Z₁ Z₂ h₁ h₂ ≤
+      4*weightedIntegral (totalMass Z₁ Z₂ h₁ h₂) (varianceCell G.fiberWeight G.targets T)+
+      (10+M)*overlapScale Z₁ Z₂ h₁ h₂+16 := by
+  have hn2 : 2≤Fintype.card P := by omega
+  have hh := arithmetic_potential_fiber_bound Z₁ Z₂ h₁ h₂ hn2 G.fiberWeight G.targets G.source t T M κ
+    G.fiberWeight_nonneg (G.sum_fiberWeight hG) hT hκ hB hk G.no_loop
+    (G.fiber_domination M hdom hn2) ht
+  let m := totalMass Z₁ Z₂ h₁ h₂
+  let S := totalPotential Z₁ Z₂ h₁ h₂
+  let r : P → ℝ := fun y => ∑ i, G.fiberWeight i*prob (G.targets i) y
+  have hr0 (y) : 0≤r y := by
+    apply Finset.sum_nonneg
+    intro i _
+    apply mul_nonneg (G.fiberWeight_nonneg i)
+    unfold prob; split_ifs <;> positivity
+  have hr1 : ∑ y, r y=1 := by
+    dsimp [r]
+    rw [Finset.sum_comm]
+    have hp1 (i : I) : ∑ y, prob (G.targets i) y=1 :=
+      sum_prob (G.targets i) (Finset.card_pos.mp (by have := hB i; omega))
+    simp_rw [← Finset.mul_sum,hp1,mul_one]
+    exact G.sum_fiberWeight hG
+  have hl : potentialScale Z₁ Z₂ h₁ h₂-2*overlapScale Z₁ Z₂ h₁ h₂≤∑ y, r y*S y := by
+    have hlo (y) := point_potential_lower m (baseConstant Z₁ Z₂ h₁ h₂) S
+      (totalMass_nonneg Z₁ Z₂ h₁ h₂) (totalMass_overlap Z₁ Z₂ h₁ h₂ hn2) hn y
+    have hu := Finset.sum_le_sum (s:=Finset.univ) (fun y _ => mul_le_mul_of_nonneg_left (hlo y) (hr0 y))
+    simpa only [← Finset.sum_mul,hr1,one_mul,potentialScale,overlapScale,m,S] using hu
+  have he : SmallFiber.edgeCell G.fiberWeight (fun i => prob (G.targets i)) G.source=G.qCell := by
+    funext C; exact (G.qCell_eq C).symm
+  have hq : weightedIntegral m G.qCell≤M*overlapScale Z₁ Z₂ h₁ h₂ := by
+    simp only [weightedIntegral,Finset.mul_sum,overlapScale]
+    calc
+      _ ≤ ∑ C, m C*(M*pairCell Finset.univ C) := Finset.sum_le_sum (fun C _ =>
+        mul_le_mul_of_nonneg_left (G.qCell_bound hdom C) (totalMass_nonneg Z₁ Z₂ h₁ h₂ C))
+      _ = _ := by apply Finset.sum_congr rfl; intro C _; dsimp [m]; ring
+  rw [he] at hh
+  change (∑ y, r y*S y)≤_ at hh
+  change weightedIntegral (totalMass Z₁ Z₂ h₁ h₂) G.qCell≤_ at hq
+  nlinarith
+end WeakPinned
+
+namespace WeakPinned
+open Filter
+open scoped Topology
+
+/-- Elementary absorption of the two asymptotically small errors. -/
+lemma scale_balance_rate (W A V δ : ℕ → ℝ) (B : ℝ) (hB : 0≤B)
+    (hW : ∀ j, 0≤W j) (hA : ∀ j, 0≤A j)
+    (hδ : Tendsto δ atTop (𝓝 0))
+    (hV : Tendsto (fun j => V j/(W j+A j+1)) atTop (𝓝 0))
+    (hbal : ∀ᶠ j in atTop, (1-δ j)*A j≤4*V j+(10+B)*W j+16) :
+    ∀ᶠ j in atTop, A j≤(64+4*B)*(W j+1) := by
+  filter_upwards [hbal,hδ.eventually (eventually_lt_nhds (by norm_num : (0:ℝ)<1/4)),
+    hV.eventually (eventually_lt_nhds (by norm_num : (0:ℝ)<1/16))] with j hb hd hv
+  have hden : 0<W j+A j+1 := by linarith [hW j,hA j]
+  have hv' := (div_lt_iff₀ hden).mp hv
+  nlinarith [hW j,hA j,mul_nonneg (hW j) hB]
+end WeakPinned
+
+namespace WeakPinned
+open Global Variance Filter
+open scoped Topology BigOperators
+variable {V : ℕ → Type u14} {I : ℕ → Type u15} {K : ℕ → Type u16} [∀ j, Fintype (V j)] [∀ j, Fintype (I j)] [∀ j, DecidableEq (V j)]
+  [∀ j, Field (K j)] [∀ j, NumberField (K j)]
+variable (G : ∀ j, Core.Fibers (V j) (I j)) (θ γ c : ℝ) (η : ℕ → ℝ)
+  (Z₁ Z₂ : ∀ j, V j → K j) (h₁ : ∀ j, Function.Injective (Z₁ j)) (h₂ : ∀ j, Function.Injective (Z₂ j))
+
+theorem arithmetic_scale_comparison (hθ : 0<θ) (hγ : 0<γ) (hγ1 : γ<1) (hcγ : c<γ)
+    (hn : Tendsto (fun j => (Fintype.card (V j):ℝ)) atTop atTop)
+    (hη : Tendsto η atTop (𝓝 0)) (hη0 : ∀ j, 0≤η j) (hηb : ∀ j, η j≤θ/2)
+    (hG : ∀ j, PreparedGraph (V j) (I j) (G j) θ γ c (η j))
+    (t : ∀ j, I j → K j) (ht : ∀ j e y, y ∈ (G j).targets e →
+      (Z₁ j ((G j).source e)-Z₁ j y)*(Z₂ j ((G j).source e)-Z₂ j y)=t j e) :
+    ∀ᶠ j in atTop, potentialScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j) ≤
+      (64+8/θ)*(overlapScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)+1) := by
+  let n : ℕ → ℝ := fun j => Fintype.card (V j)
+  let κ : ℕ → ℝ := fun j => 2/(n j)^γ
+  have hnγ : Tendsto (fun j => (n j)^γ) atTop atTop := (tendsto_rpow_atTop hγ).comp hn
+  have hκ : Tendsto κ atTop (𝓝 0) := by
+    simpa only [κ,div_eq_mul_inv,mul_zero,Function.comp_def] using tendsto_const_nhds.mul (tendsto_inv_atTop_zero.comp hnγ)
+  have hv := (sampling_variance_rates G θ γ c η Z₁ Z₂ h₁ h₂ hθ hγ hγ1 hcγ hn hη hη0 hηb hG t ht).1
+  have hs := scale_balance_rate
+    (fun j => overlapScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j))
+    (fun j => potentialScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j))
+    (fiberVariance G c Z₁ Z₂ h₁ h₂) (fun j => 2*κ j*(2/θ)) (2/θ) (by positivity)
+    (fun j => overlapScale_nonneg _ _ _ _) (fun j => potentialScale_nonneg _ _ _ _)
+    (by simpa using (tendsto_const_nhds.mul hκ).mul_const (2/θ)) hv
+  have hb : ∀ᶠ j in atTop,
+      (1-2*κ j*(2/θ))*potentialScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j) ≤
+      4*fiberVariance G c Z₁ Z₂ h₁ h₂ j+(10+2/θ)*overlapScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)+16 := by
+    filter_upwards [hn.eventually (eventually_ge_atTop (3:ℝ)),hnγ.eventually (eventually_ge_atTop (2:ℝ))] with j hj hjγ
+    have hn3 : 3≤Fintype.card (V j) := by exact_mod_cast hj
+    have hne : 0<(G j).edgeCount := (hG j).edgeCount_pos (by linarith [hηb j])
+    have hn0 : 0<n j := by change (0:ℝ)<Fintype.card (V j); linarith
+    have hBk (i) : (n j)^γ≤((G j).targets i).card := (hG j).fiber i |>.le
+    have hBk2 (i) : 2≤((G j).targets i).card := by exact_mod_cast hjγ.trans (hBk i)
+    have hk (i) : 2/(((G j).targets i).card:ℝ)≤κ j :=
+      div_le_div_of_nonneg_left (by norm_num) (Real.rpow_pos_of_pos hn0 _) (hBk i)
+    have hT : 1≤(n j)^(1-c) := Real.one_le_rpow (by linarith) (by linarith)
+    exact arithmetic_scale_balance (G j) (Z₁ j) (Z₂ j) (h₁ j) (h₂ j) hn3 hne
+      ((n j)^(1-c)) (2/θ) (κ j) hT (by dsimp [κ]; positivity) hBk2 hk
+      ((hG j).uniform_domination hθ (hηb j)) (t j) (ht j)
+  have hh := hs hb
+  convert hh using 1; ring_nf
+end WeakPinned
+
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PinnedDistances/GridBounds.lean b/lean/OAI/Geometry/PinnedDistances/GridBounds.lean
new file mode 100644
index 0000000000000000000000000000000000000000..8ea052c1f6c4ff34d55477e124a25b853e731d0a
--- /dev/null
+++ b/lean/OAI/Geometry/PinnedDistances/GridBounds.lean
@@ -0,0 +1,231 @@
+import OAI.Geometry.PinnedDistances.ChartLimits
+
+namespace OAI
+
+universe u1 u2 u3
+
+section
+noncomputable section
+open scoped BigOperators Topology
+open Filter MeasureTheory
+namespace WeakPinned.Grid
+
+def level (ω : FullSample) (r : ℝ) : ℤ := ⌊(-r-ω.1.1)/Real.log 2⌋
+
+def label (ω : FullSample) (r : ℝ) (z : ℂ) : ℤ × ℤ :=
+  (bin ((2:ℝ)^level ω r) (shiftZ (level ω r) ω.2.1) (rotatePair ω.1.1 ω.1.2 z).1,
+   bin ((2:ℝ)^level ω r) (shiftZ (level ω r) ω.2.2) (rotatePair ω.1.1 ω.1.2 z).2)
+
+lemma regular_pair_ae (z w : ℂ) :
+    ∀ᵐ ω ∂ fullLaw,
+      (∃ m, shareAt m ω.2 (rotatePair ω.1.1 ω.1.2 z) (rotatePair ω.1.1 ω.1.2 w)) ∧
+      (∀ m, shareAt m ω.2 (rotatePair ω.1.1 ω.1.2 z) (rotatePair ω.1.1 ω.1.2 w) →
+        shareAt (m+1) ω.2 (rotatePair ω.1.1 ω.1.2 z) (rotatePair ω.1.1 ω.1.2 w)) := by
+  have hm (m : ℤ) : MeasurableSet {ω : FullSample | shareAt m ω.2
+      (rotatePair ω.1.1 ω.1.2 z) (rotatePair ω.1.1 ω.1.2 w)} :=
+    measurableSet_shareAt_param measurable_snd
+      ((measurable_rotatePair z).comp measurable_fst) ((measurable_rotatePair w).comp measurable_fst) m
+  have hs : MeasurableSet {ω : FullSample |
+      (∃ m, shareAt m ω.2 (rotatePair ω.1.1 ω.1.2 z) (rotatePair ω.1.1 ω.1.2 w)) ∧
+      (∀ m, shareAt m ω.2 (rotatePair ω.1.1 ω.1.2 z) (rotatePair ω.1.1 ω.1.2 w) →
+        shareAt (m+1) ω.2 (rotatePair ω.1.1 ω.1.2 z) (rotatePair ω.1.1 ω.1.2 w))} := by
+    simp only [Set.ofPred_and,Set.ofPred_exists,Set.ofPred_forall]
+    refine (MeasurableSet.iUnion hm).inter (MeasurableSet.iInter (fun m => ?_))
+    convert (hm m).compl.union (hm (m+1)) using 1
+    ext ω
+    exact imp_iff_not_or
+  apply (Measure.ae_prod_iff_ae_ae hs).mpr
+  filter_upwards with p
+  have hn : ∀ᵐ ω ∂ squareLaw, ∀ m, shareAt m ω (rotatePair p.1 p.2 z) (rotatePair p.1 p.2 w) →
+      shareAt (m+1) ω (rotatePair p.1 p.2 z) (rotatePair p.1 p.2 w) := by
+    exact eventually_countable_forall.mpr (fun m => shareAt_nested_ae m _ _)
+  filter_upwards [share_eventually_exists (rotatePair p.1 p.2 z) (rotatePair p.1 p.2 w),hn] with ω he hn
+  obtain ⟨n,he⟩ := he
+  exact ⟨⟨(n:ℤ),he⟩,hn⟩
+
+lemma label_eq_iff {ω : FullSample} {z w : ℂ} (hzw : z≠w)
+    (he : ∃ m, shareAt m ω.2 (rotatePair ω.1.1 ω.1.2 z) (rotatePair ω.1.1 ω.1.2 w))
+    (hn : ∀ m, shareAt m ω.2 (rotatePair ω.1.1 ω.1.2 z) (rotatePair ω.1.1 ω.1.2 w) →
+      shareAt (m+1) ω.2 (rotatePair ω.1.1 ω.1.2 z) (rotatePair ω.1.1 ω.1.2 w)) (r : ℝ) :
+    label ω r z=label ω r w ↔ r≤depth ω z w := by
+  rw [show label ω r z=label ω r w ↔ shareAt (level ω r) ω.2
+    (rotatePair ω.1.1 ω.1.2 z) (rotatePair ω.1.1 ω.1.2 w) by simp only [label,Prod.mk.injEq,shareAt]]
+  rw [shareAt_iff_minScale_le (rotatePair_ne _ _ hzw) he hn]
+  unfold level depth gridDepth
+  rw [Int.le_floor,le_div_iff₀ (Real.log_pos (by norm_num : (1:ℝ)<2))]
+  constructor <;> intro h <;> linarith
+
+/-- Side length in the original (unrotated) plane, after scaling by exp m. -/
+def side (ω : FullSample) (r m : ℝ) : ℝ := Real.exp (m+ω.1.1)*(2:ℝ)^level ω r
+
+lemma side_pos (ω : FullSample) (r m : ℝ) : 0<side ω r m := by unfold side; positivity
+
+lemma log_side (ω : FullSample) (r m : ℝ) :
+    Real.log (side ω r m)=m+ω.1.1+(level ω r:ℝ)*Real.log 2 := by
+  unfold side
+  rw [Real.log_mul (Real.exp_ne_zero _) (by positivity),Real.log_exp,Real.log_zpow]
+
+lemma side_bounds (ω : FullSample) (r m : ℝ) :
+    Real.exp (m-r)/2 < side ω r m ∧ side ω r m ≤ Real.exp (m-r) := by
+  have hl := Int.floor_le ((-r-ω.1.1)/Real.log 2)
+  have hu := Int.lt_floor_add_one ((-r-ω.1.1)/Real.log 2)
+  have hp : 0<Real.log 2 := Real.log_pos (by norm_num)
+  have hl' := (le_div_iff₀ hp).mp hl
+  have hu' := (div_lt_iff₀ hp).mp hu
+  change (level ω r:ℝ)*Real.log 2≤ -r-ω.1.1 at hl'
+  change -r-ω.1.1<((level ω r:ℝ)+1)*Real.log 2 at hu'
+  constructor
+  · rw [← Real.exp_log (by norm_num : (0:ℝ)<2),← Real.exp_sub]
+    rw [← Real.exp_log (side_pos ω r m),Real.exp_lt_exp,log_side]
+    nlinarith
+  · rw [← Real.exp_log (side_pos ω r m),Real.exp_le_exp,log_side]
+    linarith
+
+lemma label_diameter {ω : FullSample} {r m : ℝ} {z w : ℂ}
+    (h : label ω r z=label ω r w) :
+    ‖(Real.exp m:ℂ)*z-(Real.exp m:ℂ)*w‖ < 2*side ω r m := by
+  have h₁ := shares_gap (by positivity : 0<(2:ℝ)^level ω r) (congrArg Prod.fst h)
+  have h₂ := shares_gap (by positivity : 0<(2:ℝ)^level ω r) (congrArg Prod.snd h)
+  have hh := (rotatePair_gaps ω.1.1 ω.1.2 z w).1
+  have hm := mul_lt_mul_of_pos_left (show Real.exp (-ω.1.1)*‖z-w‖ < 2*(2:ℝ)^level ω r by linarith)
+    (Real.exp_pos (m+ω.1.1))
+  have he : Real.exp (m+ω.1.1)*Real.exp (-ω.1.1)=Real.exp m := by rw [← Real.exp_add]; congr 1; ring
+  rw [← mul_sub,norm_mul,Complex.norm_real,Real.norm_eq_abs,abs_of_pos (Real.exp_pos _)]
+  unfold side
+  rw [← mul_assoc,he] at hm
+  nlinarith
+
+end WeakPinned.Grid
+
+namespace WeakPinned.Global
+open NumberField Hierarchy
+variable {P : Type u1} {K : Type u2} [Fintype P] [DecidableEq P] [Field K] [NumberField K]
+variable (Z : P → K) (hZ : Function.Injective Z) (v : InfinitePlace K)
+
+omit [NumberField K] in
+include hZ in
+lemma arch_label_ae : ∀ᵐ ω ∂ Grid.fullLaw, ∀ r x y,
+    (arch Z v ω).share r x y ↔ Grid.label ω r (v.embedding (Z x))=Grid.label ω r (v.embedding (Z y)) := by
+  have hr : ∀ᵐ ω ∂ Grid.fullLaw, ∀ x y,
+      (∃ m, Grid.shareAt m ω.2 (Grid.rotatePair ω.1.1 ω.1.2 (v.embedding (Z x))) (Grid.rotatePair ω.1.1 ω.1.2 (v.embedding (Z y)))) ∧
+      (∀ m, Grid.shareAt m ω.2 (Grid.rotatePair ω.1.1 ω.1.2 (v.embedding (Z x))) (Grid.rotatePair ω.1.1 ω.1.2 (v.embedding (Z y))) →
+        Grid.shareAt (m+1) ω.2 (Grid.rotatePair ω.1.1 ω.1.2 (v.embedding (Z x))) (Grid.rotatePair ω.1.1 ω.1.2 (v.embedding (Z y)))) := by
+    simp only [Filter.eventually_all]
+    exact fun x y => Grid.regular_pair_ae _ _
+  filter_upwards [good_ae Z hZ v,hr] with ω hg hr
+  intro r x y
+  by_cases hxy : x=y
+  · subst y; simp [Depth.share]
+  · rw [Depth.share,or_iff_right hxy]
+    change (r ≤ if good Z v ω ∧ x≠y then Grid.depth ω (v.embedding (Z x)) (v.embedding (Z y)) else 0) ↔ _
+    rw [ite_eq_left ⟨hg,hxy⟩]
+    exact (Grid.label_eq_iff (embedding_ne Z hZ v hxy) (hr x y).1 (hr x y).2 r).symm
+
+end WeakPinned.Global
+
+end
+end
+section
+noncomputable section
+open scoped BigOperators Topology
+open Filter MeasureTheory
+namespace WeakPinned.Grid
+
+lemma bin_gap {l : ℝ} (hl : 0<l) (s x y : ℝ) :
+    |(bin l s x : ℝ)-(bin l s y : ℝ)| < |x-y|/l+1 := by
+  have hx := Int.floor_le ((x-s)/l)
+  have hy := Int.lt_floor_add_one ((y-s)/l)
+  have hx' := Int.lt_floor_add_one ((x-s)/l)
+  have hy' := Int.floor_le ((y-s)/l)
+  have hh : (x-s)/l-(y-s)/l=(x-y)/l := by ring
+  have hle := div_le_div_of_nonneg_right (le_abs_self (x-y)) hl.le
+  have hge := div_le_div_of_nonneg_right (neg_abs_le (x-y)) hl.le
+  rw [neg_div] at hge
+  unfold bin
+  apply abs_lt.mpr
+  constructor <;> linarith
+
+lemma scaled_ratio (ω : FullSample) (r m : ℝ) (z w : ℂ) :
+    Real.exp (-ω.1.1)*‖z-w‖/(2:ℝ)^level ω r =
+      ‖(Real.exp m:ℂ)*z-(Real.exp m:ℂ)*w‖/side ω r m := by
+  rw [← mul_sub,norm_mul,Complex.norm_real,Real.norm_eq_abs,abs_of_pos (Real.exp_pos _)]
+  unfold side
+  have he : Real.exp (-ω.1.1)*Real.exp (m+ω.1.1)=Real.exp m := by
+    rw [← Real.exp_add]; congr 1; ring
+  field_simp
+  nlinarith [congrArg (fun t : ℝ => ‖z-w‖*t) he]
+
+lemma label_gap (ω : FullSample) (r m : ℝ) (z w : ℂ) :
+    |((label ω r z).1:ℝ)-((label ω r w).1:ℝ)| <
+      ‖(Real.exp m:ℂ)*z-(Real.exp m:ℂ)*w‖/side ω r m+1 ∧
+    |((label ω r z).2:ℝ)-((label ω r w).2:ℝ)| <
+      ‖(Real.exp m:ℂ)*z-(Real.exp m:ℂ)*w‖/side ω r m+1 := by
+  have h₁ : |(rotatePair ω.1.1 ω.1.2 z).1-(rotatePair ω.1.1 ω.1.2 w).1|≤
+      Real.exp (-ω.1.1)*‖z-w‖ := by
+    simpa only [← norm_rotateC ω.1.1 ω.1.2 (z-w),rotateC_sub,rotatePair,Complex.sub_re] using
+      Complex.abs_re_le_norm (rotateC ω.1.1 ω.1.2 z-rotateC ω.1.1 ω.1.2 w)
+  have h₂ : |(rotatePair ω.1.1 ω.1.2 z).2-(rotatePair ω.1.1 ω.1.2 w).2|≤
+      Real.exp (-ω.1.1)*‖z-w‖ := by
+    simpa only [← norm_rotateC ω.1.1 ω.1.2 (z-w),rotateC_sub,rotatePair,Complex.sub_im] using
+      Complex.abs_im_le_norm (rotateC ω.1.1 ω.1.2 z-rotateC ω.1.1 ω.1.2 w)
+  have hp : 0<(2:ℝ)^level ω r := by positivity
+  constructor
+  · exact (bin_gap hp _ _ _).trans_le (add_le_add
+      ((div_le_div_of_nonneg_right h₁ hp.le).trans_eq (scaled_ratio ω r m z w)) le_rfl)
+  · exact (bin_gap hp _ _ _).trans_le (add_le_add
+      ((div_le_div_of_nonneg_right h₂ hp.le).trans_eq (scaled_ratio ω r m z w)) le_rfl)
+
+/-- A uniform bound on the number of occupied squares meeting the disk.
+Only the lower bound on the actual side is needed. -/
+lemma card_labels_bound {P : Type u3} [DecidableEq P] (A : Finset P) (u : P → ℂ)
+    (ω : FullSample) (r m : ℝ) (b : ℂ) (R L : ℝ) (M : ℕ)
+    (hR : 0≤R) (hL : 0<L) (hside : L ≤ side ω r m)
+    (hM : 2*R/L+1≤M)
+    (hA : ∀ x∈A, ‖(Real.exp m:ℂ)*u x+b‖≤R) :
+    (A.image (fun x => label ω r (u x))).card ≤ (2*M+1)^2 := by
+  classical
+  rcases A.eq_empty_or_nonempty with h | ⟨x₀,hx₀⟩
+  · simp [h]
+  let l₀ := label ω r (u x₀)
+  have hd (x : P) (hx : x∈A) :
+      ‖(Real.exp m:ℂ)*u x-(Real.exp m:ℂ)*u x₀‖≤2*R := by
+    rw [show (Real.exp m:ℂ)*u x-(Real.exp m:ℂ)*u x₀=
+      ((Real.exp m:ℂ)*u x+b)-((Real.exp m:ℂ)*u x₀+b) by ring]
+    exact (norm_sub_le _ _).trans (by linarith [hA x hx,hA x₀ hx₀])
+  have hg (x : P) (hx : x∈A) :
+      |((label ω r (u x)).1:ℝ)-(l₀.1:ℝ)|≤M ∧
+      |((label ω r (u x)).2:ℝ)-(l₀.2:ℝ)|≤M := by
+    have hb : ‖(Real.exp m:ℂ)*u x-(Real.exp m:ℂ)*u x₀‖/side ω r m+1≤M := by
+      apply le_trans _ hM
+      apply add_le_add _ le_rfl
+      exact div_le_div₀ (by positivity) (hd x hx) hL hside
+    exact ⟨(label_gap ω r m _ _).1.le.trans hb,(label_gap ω r m _ _).2.le.trans hb⟩
+  have hs : A.image (fun x => label ω r (u x)) ⊆
+      (Finset.Icc (l₀.1-(M:ℤ)) (l₀.1+(M:ℤ))) ×ˢ
+      (Finset.Icc (l₀.2-(M:ℤ)) (l₀.2+(M:ℤ))) := by
+    intro p hp
+    obtain ⟨x,hx,rfl⟩ := Finset.mem_image.mp hp
+    have hh := hg x hx
+    rcases abs_le.mp hh.1 with ⟨ha,hb⟩
+    rcases abs_le.mp hh.2 with ⟨hc,hd⟩
+    simp only [Finset.mem_product,Finset.mem_Icc]
+    constructor
+    · constructor
+      · exact_mod_cast (show (l₀.1:ℝ)-(M:ℝ)≤((label ω r (u x)).1:ℝ) by linarith)
+      · exact_mod_cast (show ((label ω r (u x)).1:ℝ)≤(l₀.1:ℝ)+(M:ℝ) by linarith)
+    · constructor
+      · exact_mod_cast (show (l₀.2:ℝ)-(M:ℝ)≤((label ω r (u x)).2:ℝ) by linarith)
+      · exact_mod_cast (show ((label ω r (u x)).2:ℝ)≤(l₀.2:ℝ)+(M:ℝ) by linarith)
+  calc
+    _ ≤ _ := Finset.card_le_card hs
+    _ = (2*M+1)^2 := by
+      rw [Finset.card_product,Int.card_Icc,Int.card_Icc]
+      have he (z : ℤ) : (z+(M:ℤ)+1-(z-(M:ℤ))).toNat=2*M+1 := by omega
+      rw [he,he,pow_two]
+
+end WeakPinned.Grid
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PinnedDistances/Hierarchy.lean b/lean/OAI/Geometry/PinnedDistances/Hierarchy.lean
new file mode 100644
index 0000000000000000000000000000000000000000..b0d2c9853839b2029ba318ca57aa3cd81834f3e1
--- /dev/null
+++ b/lean/OAI/Geometry/PinnedDistances/Hierarchy.lean
@@ -0,0 +1,644 @@
+import OAI.Geometry.PinnedDistances.ProductFormula
+
+namespace OAI
+
+universe u1 u2 u3
+
+section
+noncomputable section
+open NumberField
+open scoped BigOperators
+
+/- Deterministic giant-removal identity. -/
+
+noncomputable section
+open MeasureTheory
+open scoped Topology
+
+namespace WeakPinned.Hierarchy
+variable {P : Type u1} [Fintype P] [DecidableEq P]
+
+structure Depth (P : Type u2) where
+  value : P → P → ℝ
+  symm : ∀ x y, value x y = value y x
+  ultra : ∀ x y z, x ≠ y → y ≠ z → x ≠ z →
+    min (value x y) (value y z) ≤ value x z
+
+namespace Depth
+
+variable (d : Depth P)
+
+def share (r : ℝ) (x y : P) : Prop := x = y ∨ r ≤ d.value x y
+instance (r : ℝ) (x y : P) : Decidable (d.share r x y) := Classical.propDecidable _
+
+omit [Fintype P] [DecidableEq P] in
+lemma share_refl (r : ℝ) (x : P) : d.share r x x := Or.inl rfl
+omit [Fintype P] [DecidableEq P] in
+lemma share_symm {r : ℝ} {x y : P} (h : d.share r x y) : d.share r y x := by
+  rcases h with h | h
+  · exact Or.inl h.symm
+  · exact Or.inr (by simpa only [d.symm y x] using h)
+omit [Fintype P] in
+lemma share_trans {r : ℝ} {x y z : P} (hxy : d.share r x y) (hyz : d.share r y z) :
+    d.share r x z := by
+  by_cases h₁ : x = y
+  · subst y; exact hyz
+  by_cases h₂ : y = z
+  · subst z; exact hxy
+  by_cases h₃ : x = z
+  · exact Or.inl h₃
+  exact Or.inr ((le_min (hxy.resolve_left h₁) (hyz.resolve_left h₂)).trans
+    (d.ultra x y z h₁ h₂ h₃))
+omit [Fintype P] [DecidableEq P] in
+lemma share_mono {r t : ℝ} (htr : t ≤ r) {x y : P} (h : d.share r x y) :
+    d.share t x y := h.imp_right (htr.trans ·)
+
+def cell (r : ℝ) (x : P) : Finset P := Finset.univ.filter (d.share r x)
+omit [DecidableEq P] in
+@[simp] lemma mem_cell (r : ℝ) (x y : P) : y ∈ d.cell r x ↔ d.share r x y := by
+  simp [cell]
+omit [DecidableEq P] in
+@[simp] lemma self_mem_cell (r : ℝ) (x : P) : x ∈ d.cell r x := by
+  simp [share_refl]
+omit [DecidableEq P] in
+lemma cell_nonempty (r : ℝ) (x : P) : (d.cell r x).Nonempty := ⟨x, d.self_mem_cell r x⟩
+
+lemma cell_eq_of_share {r : ℝ} {x y : P} (h : d.share r x y) : d.cell r x = d.cell r y := by
+  ext z
+  simp only [mem_cell]
+  exact ⟨fun hz => d.share_trans (d.share_symm h) hz, fun hz => d.share_trans h hz⟩
+
+omit [DecidableEq P] in
+lemma cell_mono {r t : ℝ} (htr : t ≤ r) (x : P) : d.cell r x ⊆ d.cell t x := by
+  intro z hz
+  exact (d.mem_cell t x z).mpr (d.share_mono htr ((d.mem_cell r x z).mp hz))
+
+def cells (r : ℝ) : Finset (Finset P) := Finset.univ.image (d.cell r)
+@[simp] lemma mem_cells (r : ℝ) (C : Finset P) : C ∈ d.cells r ↔ ∃ x, d.cell r x = C := by
+  simp [cells]
+lemma cell_mem_cells (r : ℝ) (x : P) : d.cell r x ∈ d.cells r := by simp
+
+lemma cell_of_mem {r : ℝ} {C : Finset P} (hC : C ∈ d.cells r) {x : P} (hx : x ∈ C) :
+    C = d.cell r x := by
+  obtain ⟨z, rfl⟩ := (d.mem_cells r C).mp hC
+  exact d.cell_eq_of_share ((d.mem_cell r z x).mp hx)
+
+lemma cells_disjoint {r : ℝ} {C D : Finset P} (hC : C ∈ d.cells r)
+    (hD : D ∈ d.cells r) (hne : C ≠ D) : Disjoint C D := by
+  apply Finset.disjoint_left.mpr
+  intro x hx hd
+  exact hne ((d.cell_of_mem hC hx).trans (d.cell_of_mem hD hd).symm)
+
+lemma cells_pair_sum (r : ℝ) (x y : P) :
+    (∑ C ∈ d.cells r, (if x ∈ C ∧ y ∈ C then (1 : ℝ) else 0)) =
+      if d.share r x y then 1 else 0 := by
+  have h : ∀ C ∈ d.cells r, (if x ∈ C ∧ y ∈ C then (1 : ℝ) else 0) =
+      if C = d.cell r x then (if d.share r x y then 1 else 0) else 0 := by
+    intro C hC
+    by_cases hx : x ∈ C
+    · rw [d.cell_of_mem hC hx]
+      simp [d.share_refl]
+    · have hne : C ≠ d.cell r x := by intro he; apply hx; simpa [he] using d.self_mem_cell r x
+      simp [hx, hne]
+  rw [Finset.sum_congr rfl h]
+  simp [d.cell_mem_cells r x]
+
+def giants (r : ℝ) : Finset (Finset P) :=
+  (d.cells r).filter (fun C => Fintype.card P < 2*C.card)
+
+@[simp] lemma mem_giants (r : ℝ) (C : Finset P) : C ∈ d.giants r ↔
+    C ∈ d.cells r ∧ Fintype.card P < 2*C.card := by simp [giants]
+
+lemma giants_unique {r : ℝ} {C D : Finset P} (hC : C ∈ d.giants r)
+    (hD : D ∈ d.giants r) : C = D := by
+  rw [d.mem_giants] at hC hD
+  by_contra hne
+  have hd := d.cells_disjoint hC.1 hD.1 hne
+  have hc := Finset.card_le_univ (C ∪ D)
+  rw [Finset.card_union_of_disjoint hd] at hc
+  omega
+
+lemma giants_card_le_one (r : ℝ) : (d.giants r).card ≤ 1 :=
+  Finset.card_le_one.mpr (fun _C hC _D hD => d.giants_unique hC hD)
+
+def giant (r : ℝ) : Finset P := (d.giants r).biUnion id
+
+lemma giant_eq {r : ℝ} {C : Finset P} (hC : C ∈ d.giants r) : d.giant r = C := by
+  have he : d.giants r = {C} := Finset.ext fun D => by
+    simp only [Finset.mem_singleton]
+    exact ⟨fun hd => d.giants_unique hd hC, fun hd => hd ▸ hC⟩
+  simp [giant, he]
+
+def transformed (r : ℝ) : Multiset (Finset P) :=
+  ((d.cells r).filter (fun C => ¬Fintype.card P < 2*C.card)).val +
+  ((d.giants r).val.map (fun C => Cᶜ))
+
+def sumCells (f : Finset P → ℝ) (r : ℝ) : ℝ := ((d.transformed r).map f).sum
+
+def pairOverlap (r : ℝ) (x y : P) : ℝ :=
+  d.sumCells (fun C => if x ∈ C ∧ y ∈ C then 1 else 0) r
+
+def giantIndicator (r : ℝ) : ℝ := if (d.giants r).Nonempty then 1 else 0
+
+def outsideIndicator (r : ℝ) (x : P) : ℝ :=
+  if (d.giants r).Nonempty ∧ x ∉ d.giant r then 1 else 0
+
+lemma sumCells_formula (f : Finset P → ℝ) (r : ℝ) :
+    d.sumCells f r = (∑ C ∈ d.cells r, f C) +
+      ∑ C ∈ d.giants r, (f Cᶜ - f C) := by
+  classical
+  simp only [sumCells, transformed, Multiset.map_add, Multiset.sum_add,
+    Multiset.map_map, Finset.sum, Function.comp_def]
+  change (∑ C ∈ (d.cells r).filter (fun C => ¬Fintype.card P < 2*C.card), f C) +
+    (∑ C ∈ d.giants r, f Cᶜ) = (∑ C ∈ d.cells r, f C) +
+      ∑ C ∈ d.giants r, (f Cᶜ - f C)
+  rw [Finset.sum_sub_distrib]
+  have hh := Finset.sum_filter_add_sum_filter_not (d.cells r)
+    (fun C => Fintype.card P < 2*C.card) f
+  change (∑ C ∈ d.giants r, f C) + _ = _ at hh
+  linarith
+
+lemma pairOverlap_formula (r : ℝ) (x y : P) :
+    d.pairOverlap r x y = (if d.share r x y then 1 else 0) -
+      d.giantIndicator r + d.outsideIndicator r x + d.outsideIndicator r y := by
+  unfold pairOverlap
+  rw [d.sumCells_formula, d.cells_pair_sum]
+  unfold giantIndicator outsideIndicator
+  by_cases hg : (d.giants r).Nonempty
+  · obtain ⟨G,hG⟩ := hg
+    have he : d.giants r = {G} := Finset.ext fun C => by
+      simp only [Finset.mem_singleton]
+      exact ⟨fun hc => d.giants_unique hc hG, fun hc => hc ▸ hG⟩
+    rw [d.giant_eq hG]
+    have hg : (d.giants r).Nonempty := ⟨G,hG⟩
+    simp only [he, Finset.sum_singleton, Finset.mem_compl]
+    by_cases hx : x ∈ G <;> by_cases hy : y ∈ G <;> simp [hx,hy]
+    ring
+  · have he : d.giants r = ∅ := Finset.not_nonempty_iff_eq_empty.mp hg
+    simp [he]
+
+lemma giants_nonempty_iff (r : ℝ) : (d.giants r).Nonempty ↔
+    ∃ x : P, ∃ C : Finset P, Fintype.card P < 2*C.card ∧ ∀ y ∈ C, d.share r x y := by
+  constructor
+  · rintro ⟨C,hC⟩
+    obtain ⟨hC,hc⟩ := (d.mem_giants r C).mp hC
+    obtain ⟨x,rfl⟩ := (d.mem_cells r C).mp hC
+    exact ⟨x,d.cell r x,hc,fun y hy => (d.mem_cell r x y).mp hy⟩
+  · rintro ⟨x,C,hc,hC⟩
+    have hsub : C ⊆ d.cell r x := fun y hy => (d.mem_cell r x y).mpr (hC y hy)
+    refine ⟨d.cell r x, (d.mem_giants r _).mpr ⟨d.cell_mem_cells r x, ?_⟩⟩
+    have := Finset.card_le_card hsub
+    omega
+
+lemma giants_mono {r t : ℝ} (htr : t ≤ r) (hr : (d.giants r).Nonempty) :
+    (d.giants t).Nonempty := by
+  rw [d.giants_nonempty_iff] at hr ⊢
+  obtain ⟨x,C,hc,hC⟩ := hr
+  exact ⟨x,C,hc,fun y hy => d.share_mono htr (hC y hy)⟩
+
+lemma isClosed_giants : IsClosed {r : ℝ | (d.giants r).Nonempty} := by
+  simp_rw [d.giants_nonempty_iff, Set.ofPred_exists]
+  apply isClosed_iUnion_of_finite
+  intro x
+  apply isClosed_iUnion_of_finite
+  intro C
+  simp only [Set.ofPred_and]
+  apply IsClosed.inter
+  · by_cases h : Fintype.card P < 2*C.card <;> simp [h]
+  · simp_rw [Set.ofPred_forall]
+    apply isClosed_iInter
+    intro y
+    apply isClosed_iInter
+    intro hy
+    by_cases hxy : x = y
+    · simp [share, hxy]
+    · simp only [share, hxy, false_or]
+      exact isClosed_le continuous_id continuous_const
+
+def radius : ℝ := ∑ x, ∑ y, |d.value x y|
+
+omit [DecidableEq P] in
+lemma radius_nonneg : 0 ≤ d.radius := by unfold radius; positivity
+
+omit [DecidableEq P] in
+lemma abs_value_le_radius (x y : P) : |d.value x y| ≤ d.radius := by
+  have hx : (∑ y, |d.value x y|) ≤ d.radius :=
+    Finset.single_le_sum (fun z _ => Finset.sum_nonneg (fun w _ => abs_nonneg (d.value z w)))
+      (Finset.mem_univ x)
+  exact (Finset.single_le_sum (fun z _ => abs_nonneg (d.value x z)) (Finset.mem_univ y)).trans hx
+
+omit [DecidableEq P] in
+lemma cell_of_low {r : ℝ} (hr : r ≤ -d.radius) (x : P) : d.cell r x = Finset.univ := by
+  ext y
+  simp only [mem_cell, Finset.mem_univ, iff_true]
+  exact Or.inr (hr.trans (abs_le.mp (d.abs_value_le_radius x y)).1)
+
+omit [DecidableEq P] in
+lemma cell_of_high {r : ℝ} (hr : d.radius < r) (x : P) : d.cell r x = {x} := by
+  ext y
+  simp only [mem_cell, share, Finset.mem_singleton]
+  have hh : ¬r ≤ d.value x y := not_le.mpr ((abs_le.mp (d.abs_value_le_radius x y)).2.trans_lt hr)
+  simp [hh, eq_comm]
+
+lemma giants_of_low (hn : 2 ≤ Fintype.card P) {r : ℝ} (hr : r ≤ -d.radius) :
+    (d.giants r).Nonempty := by
+  have : Nonempty P := Fintype.card_pos_iff.mp (by omega)
+  obtain ⟨x⟩ := ‹Nonempty P›
+  refine ⟨Finset.univ, (d.mem_giants r _).mpr ⟨?_, ?_⟩⟩
+  · exact (d.mem_cells r _).mpr ⟨x, d.cell_of_low hr x⟩
+  · simp only [Finset.card_univ]
+    omega
+
+lemma giants_le_radius (hn : 2 ≤ Fintype.card P) {r : ℝ} (hr : (d.giants r).Nonempty) :
+    r ≤ d.radius := by
+  by_contra! h
+  obtain ⟨C,hC⟩ := hr
+  obtain ⟨hC,hc⟩ := (d.mem_giants r C).mp hC
+  obtain ⟨x,rfl⟩ := (d.mem_cells r C).mp hC
+  rw [d.cell_of_high h x, Finset.card_singleton] at hc
+  omega
+
+/-- The last level containing a giant. -/
+def M : ℝ := sSup {r : ℝ | (d.giants r).Nonempty}
+
+lemma giants_cutoff (hn : 2 ≤ Fintype.card P) (r : ℝ) :
+    (d.giants r).Nonempty ↔ r ≤ d.M := by
+  have hne : Set.Nonempty {t : ℝ | (d.giants t).Nonempty} := ⟨-d.radius,d.giants_of_low hn le_rfl⟩
+  have hbd : BddAbove {t : ℝ | (d.giants t).Nonempty} := ⟨d.radius,fun t ht => d.giants_le_radius hn ht⟩
+  exact ⟨fun hr => le_csSup hbd hr, fun hr =>
+    d.giants_mono hr (d.isClosed_giants.csSup_mem hne hbd)⟩
+
+lemma M_bounds (hn : 2 ≤ Fintype.card P) : -d.radius ≤ d.M ∧ d.M ≤ d.radius := by
+  constructor
+  · exact (d.giants_cutoff hn _).mp (d.giants_of_low hn le_rfl)
+  · exact d.giants_le_radius hn ((d.giants_cutoff hn _).mpr le_rfl)
+
+omit [DecidableEq P] in
+lemma measurable_cell (x : P) : Measurable (fun r : ℝ => d.cell r x) := by
+  apply measurable_finset_iff.mpr
+  intro y
+  simp only [mem_cell, share]
+  exact measurable_const.or (measurableSet_setOfPred.mp (measurableSet_le measurable_id measurable_const))
+
+lemma measurable_cells : Measurable d.cells := by
+  have hm : Measurable (fun r : ℝ => fun x : P => d.cell r x) :=
+    measurable_pi_iff.mpr (d.measurable_cell)
+  exact (measurable_of_finite (fun f : P → Finset P => Finset.univ.image f)).comp hm
+
+lemma measurable_giants : Measurable d.giants :=
+  (measurable_of_finite (fun s : Finset (Finset P) =>
+    s.filter (fun C : Finset P => Fintype.card P < 2*C.card))).comp d.measurable_cells
+
+lemma measurable_giant : Measurable d.giant :=
+  (measurable_of_finite (fun s : Finset (Finset P) => s.biUnion id)).comp d.measurable_giants
+
+lemma measurable_giantIndicator : Measurable d.giantIndicator :=
+  (measurable_of_finite (fun s : Finset (Finset P) => if s.Nonempty then (1 : ℝ) else 0)).comp
+    d.measurable_giants
+
+lemma measurable_outsideIndicator (x : P) : Measurable (fun r => d.outsideIndicator r x) := by
+  have hm : Measurable (fun r : ℝ => (d.giants r, d.giant r)) :=
+    d.measurable_giants.prodMk d.measurable_giant
+  exact (measurable_of_finite
+    (fun st : Finset (Finset P) × Finset P => if st.1.Nonempty ∧ x ∉ st.2 then (1 : ℝ) else 0)).comp hm
+
+lemma measurable_pairOverlap (x y : P) : Measurable (fun r => d.pairOverlap r x y) := by
+  have hm : Measurable (fun r : ℝ => if d.share r x y then (1 : ℝ) else 0) := by
+    apply Measurable.ite _ measurable_const measurable_const
+    apply measurableSet_setOfPred.mpr
+    exact measurable_const.or (measurableSet_setOfPred.mp (measurableSet_le measurable_id measurable_const))
+  have ht : Measurable (fun r =>
+      (if d.share r x y then (1 : ℝ) else 0) - d.giantIndicator r +
+        d.outsideIndicator r x + d.outsideIndicator r y) :=
+    ((hm.sub d.measurable_giantIndicator).add (d.measurable_outsideIndicator x)).add
+      (d.measurable_outsideIndicator y)
+  simpa only [pairOverlap_formula] using ht
+
+lemma outsideIndicator_bounds (r : ℝ) (x : P) :
+    0 ≤ d.outsideIndicator r x ∧ d.outsideIndicator r x ≤ 1 := by
+  unfold outsideIndicator
+  split_ifs <;> norm_num
+
+lemma pairOverlap_bounds (r : ℝ) (x y : P) :
+    0 ≤ d.pairOverlap r x y ∧ d.pairOverlap r x y ≤ 2 := by
+  have lo : 0 ≤ d.pairOverlap r x y := by
+    unfold pairOverlap sumCells
+    apply Multiset.sum_nonneg
+    intro t ht
+    obtain ⟨C,hC,rfl⟩ := Multiset.mem_map.mp ht
+    split_ifs <;> norm_num
+  refine ⟨lo, ?_⟩
+  rw [pairOverlap_formula]
+  have hx := (d.outsideIndicator_bounds r x).2
+  have hy := (d.outsideIndicator_bounds r y).2
+  by_cases hg : (d.giants r).Nonempty
+  · simp only [giantIndicator, hg, reduceIte]
+    split_ifs <;> linarith
+  · simp [giantIndicator, outsideIndicator, hg]
+    split_ifs <;> norm_num
+
+lemma outsideIndicator_zero_low (hn : 2 ≤ Fintype.card P) {r : ℝ} (hr : r ≤ -d.radius)
+    (x : P) : d.outsideIndicator r x = 0 := by
+  have hg : Finset.univ ∈ d.giants r := (d.mem_giants r _).mpr ⟨
+    (d.mem_cells r _).mpr ⟨x,d.cell_of_low hr x⟩, by simp only [Finset.card_univ]; omega⟩
+  simp [outsideIndicator, d.giant_eq hg]
+
+lemma outsideIndicator_zero_high (hn : 2 ≤ Fintype.card P) {r : ℝ} (hr : d.radius < r)
+    (x : P) : d.outsideIndicator r x = 0 := by
+  have hg : ¬(d.giants r).Nonempty := fun hg => not_le.mpr hr (d.giants_le_radius hn hg)
+  simp [outsideIndicator, hg]
+
+lemma pairOverlap_zero_low (hn : 2 ≤ Fintype.card P) {r : ℝ} (hr : r ≤ -d.radius)
+    (x y : P) : d.pairOverlap r x y = 0 := by
+  have hxy : d.share r x y := Or.inr (hr.trans (abs_le.mp (d.abs_value_le_radius x y)).1)
+  simp [pairOverlap_formula, giantIndicator, d.giants_of_low hn hr,
+    d.outsideIndicator_zero_low hn hr, hxy]
+
+lemma pairOverlap_zero_high (hn : 2 ≤ Fintype.card P) {r : ℝ} (hr : d.radius < r)
+    (x y : P) (hxy : x ≠ y) : d.pairOverlap r x y = 0 := by
+  have hg : ¬(d.giants r).Nonempty := fun hg => not_le.mpr hr (d.giants_le_radius hn hg)
+  have hs : ¬d.share r x y := by
+    simp only [share, hxy, false_or]
+    exact not_le.mpr ((abs_le.mp (d.abs_value_le_radius x y)).2.trans_lt hr)
+  simp [pairOverlap_formula, giantIndicator, hg, hs, d.outsideIndicator_zero_high hn hr]
+
+/- The diagonal is deliberately absent from overlapLength's domain: the
+   transformed singleton tails have infinite diagonal overlap. -/
+def potential (x : P) : ℝ := ∫ r : ℝ, d.outsideIndicator r x
+
+def overlapLength (x y : P) (_ : x ≠ y) : ℝ := ∫ r : ℝ, d.pairOverlap r x y
+
+lemma integrable_supported_Icc {f : ℝ → ℝ} {a b C : ℝ}
+    (hm : Measurable f) (hb : ∀ r, ‖f r‖ ≤ C)
+    (hz : ∀ r ∉ Set.Icc a b, f r = 0) : Integrable f := by
+  have hi : IntegrableOn f (Set.Icc a b) :=
+    IntegrableOn.of_bound measure_Icc_lt_top hm.aestronglyMeasurable C
+      (Filter.Eventually.of_forall hb)
+  have he : (Set.Icc a b).indicator f = f := by
+    apply Set.indicator_eq_self.mpr
+    intro r hr
+    by_contra hn
+    exact hr (hz r hn)
+  simpa only [he] using hi.integrable_indicator measurableSet_Icc
+
+lemma outsideIndicator_zero_compl (hn : 2 ≤ Fintype.card P) (x : P) :
+    ∀ r ∉ Set.Icc (-d.radius) d.radius, d.outsideIndicator r x = 0 := by
+  intro r hr
+  by_cases hlo : r ≤ -d.radius
+  · exact d.outsideIndicator_zero_low hn hlo x
+  · exact d.outsideIndicator_zero_high hn
+      (lt_of_not_ge (fun hhi => hr ⟨le_of_not_ge hlo, hhi⟩)) x
+
+lemma pairOverlap_zero_compl (hn : 2 ≤ Fintype.card P) (x y : P) (hxy : x ≠ y) :
+    ∀ r ∉ Set.Icc (-d.radius) d.radius, d.pairOverlap r x y = 0 := by
+  intro r hr
+  by_cases hlo : r ≤ -d.radius
+  · exact d.pairOverlap_zero_low hn hlo x y
+  · exact d.pairOverlap_zero_high hn
+      (lt_of_not_ge (fun hhi => hr ⟨le_of_not_ge hlo, hhi⟩)) x y hxy
+
+lemma integrable_outsideIndicator (hn : 2 ≤ Fintype.card P) (x : P) :
+    Integrable (fun r => d.outsideIndicator r x) := by
+  apply integrable_supported_Icc (d.measurable_outsideIndicator x) (C := 1)
+  · intro r
+    rw [Real.norm_eq_abs, abs_of_nonneg (d.outsideIndicator_bounds r x).1]
+    exact (d.outsideIndicator_bounds r x).2
+  · exact d.outsideIndicator_zero_compl hn x
+
+lemma integrable_pairOverlap (hn : 2 ≤ Fintype.card P) (x y : P) (hxy : x ≠ y) :
+    Integrable (fun r => d.pairOverlap r x y) := by
+  apply integrable_supported_Icc (d.measurable_pairOverlap x y) (C := 2)
+  · intro r
+    rw [Real.norm_eq_abs, abs_of_nonneg (d.pairOverlap_bounds r x y).1]
+    exact (d.pairOverlap_bounds r x y).2
+  · exact d.pairOverlap_zero_compl hn x y hxy
+
+lemma integral_cutoff {R a : ℝ} (hR : 0 ≤ R) (ha : a ∈ Set.Icc (-R) R) :
+    (∫ r : ℝ in Set.Icc (-R) R, if r ≤ a then (1 : ℝ) else 0) = a + R := by
+  rw [integral_Icc_eq_integral_Ioc, ← intervalIntegral.integral_of_le (by linarith : -R ≤ R)]
+  change (∫ r : ℝ in (-R)..R, {r : ℝ | r ≤ a}.indicator (fun _ => (1 : ℝ)) r) = a + R
+  rw [intervalIntegral.integral_indicator ha, intervalIntegral.integral_const]
+  simp
+
+lemma depth_identity (hn : 2 ≤ Fintype.card P) (x y : P) (hxy : x ≠ y) :
+    d.value x y = d.M + d.overlapLength x y hxy - d.potential x - d.potential y := by
+  let I := Set.Icc (-d.radius) d.radius
+  have hoc : IntegrableOn (fun r : ℝ => if r ≤ d.value x y then (1 : ℝ) else 0) I := by
+    apply IntegrableOn.of_bound measure_Icc_lt_top
+      ((measurable_const.ite (measurableSet_le measurable_id measurable_const)
+        measurable_const).aestronglyMeasurable) 1
+    exact Filter.Eventually.of_forall (fun r => by split_ifs <;> norm_num)
+  have hgc : IntegrableOn (fun r : ℝ => if r ≤ d.M then (1 : ℝ) else 0) I := by
+    apply IntegrableOn.of_bound measure_Icc_lt_top
+      ((measurable_const.ite (measurableSet_le measurable_id measurable_const)
+        measurable_const).aestronglyMeasurable) 1
+    exact Filter.Eventually.of_forall (fun r => by split_ifs <;> norm_num)
+  have hs : IntegrableOn (fun r : ℝ => (if r ≤ d.value x y then (1 : ℝ) else 0) -
+      (if r ≤ d.M then (1 : ℝ) else 0)) I := hoc.sub hgc
+  have hx := (d.integrable_outsideIndicator hn x).integrableOn (s := I)
+  have hy := (d.integrable_outsideIndicator hn y).integrableOn (s := I)
+  have ha : IntegrableOn (fun r : ℝ => (if r ≤ d.value x y then (1 : ℝ) else 0) -
+      (if r ≤ d.M then (1 : ℝ) else 0) + d.outsideIndicator r x) I := hs.add hx
+  have he : (∫ r in I, d.pairOverlap r x y) =
+      (∫ r in I, if r ≤ d.value x y then (1 : ℝ) else 0) -
+      (∫ r in I, if r ≤ d.M then (1 : ℝ) else 0) +
+      (∫ r in I, d.outsideIndicator r x) + (∫ r in I, d.outsideIndicator r y) := by
+    calc
+      _ = ∫ r in I, ((if r ≤ d.value x y then (1 : ℝ) else 0) -
+          (if r ≤ d.M then 1 else 0) + d.outsideIndicator r x + d.outsideIndicator r y) := by
+        apply setIntegral_congr_fun measurableSet_Icc
+        intro r _
+        simp only [pairOverlap_formula, share, hxy, false_or, giantIndicator,
+          d.giants_cutoff hn]
+      _ = _ := by rw [integral_add ha hy, integral_add hs hx, integral_sub hoc hgc]
+  rw [setIntegral_eq_integral_of_forall_compl_eq_zero (d.pairOverlap_zero_compl hn x y hxy)] at he
+  rw [setIntegral_eq_integral_of_forall_compl_eq_zero (d.outsideIndicator_zero_compl hn x),
+    setIntegral_eq_integral_of_forall_compl_eq_zero (d.outsideIndicator_zero_compl hn y)] at he
+  change d.overlapLength x y hxy = _ at he
+  dsimp only [I] at he
+  rw [integral_cutoff d.radius_nonneg (abs_le.mp (d.abs_value_le_radius x y)),
+    integral_cutoff d.radius_nonneg (d.M_bounds hn)] at he
+  change d.overlapLength x y hxy = _ - _ + d.potential x + d.potential y at he
+  linarith
+
+lemma potential_nonneg (x : P) : 0 ≤ d.potential x :=
+  integral_nonneg (fun r => (d.outsideIndicator_bounds r x).1)
+
+lemma overlapLength_nonneg (x y : P) (hxy : x ≠ y) : 0 ≤ d.overlapLength x y hxy :=
+  integral_nonneg (fun r => (d.pairOverlap_bounds r x y).1)
+
+lemma integral_bound_supported {f : ℝ → ℝ} {R C : ℝ} (hR : 0 ≤ R)
+    (hi : Integrable f) (hb : ∀ r, f r ≤ C)
+    (hz : ∀ r ∉ Set.Icc (-R) R, f r = 0) : (∫ r, f r) ≤ 2*R*C := by
+  rw [← setIntegral_eq_integral_of_forall_compl_eq_zero hz]
+  have hh := integral_mono (hi.integrableOn (s := Set.Icc (-R) R))
+    (integrableOn_const (s := Set.Icc (-R) R) (μ := volume)
+      (C := C) (measure_Icc_lt_top.ne)) (fun r => hb r)
+  rw [setIntegral_const] at hh
+  simp only [Measure.real, Real.volume_Icc, sub_neg_eq_add,
+    ENNReal.toReal_ofReal (by linarith : 0 ≤ R+R), smul_eq_mul] at hh
+  nlinarith
+
+lemma potential_le (hn : 2 ≤ Fintype.card P) (x : P) : d.potential x ≤ 2*d.radius := by
+  have hh := integral_bound_supported d.radius_nonneg (d.integrable_outsideIndicator hn x)
+    (fun r => (d.outsideIndicator_bounds r x).2) (d.outsideIndicator_zero_compl hn x)
+  change d.potential x ≤ _ at hh
+  simpa only [mul_one] using hh
+
+lemma overlapLength_le (hn : 2 ≤ Fintype.card P) (x y : P) (hxy : x ≠ y) :
+    d.overlapLength x y hxy ≤ 4*d.radius := by
+  have hh := integral_bound_supported d.radius_nonneg (d.integrable_pairOverlap hn x y hxy)
+    (fun r => (d.pairOverlap_bounds r x y).2) (d.pairOverlap_zero_compl hn x y hxy)
+  change d.overlapLength x y hxy ≤ _ at hh
+  linarith
+
+lemma zero_values (hn : 2 ≤ Fintype.card P) (hzero : ∀ x y, d.value x y = 0) :
+    d.M = 0 ∧ (∀ x, d.potential x = 0) ∧
+      (∀ x y (hxy : x ≠ y), d.overlapLength x y hxy = 0) := by
+  have hr : d.radius = 0 := by simp [radius, hzero]
+  refine ⟨?_, ?_, ?_⟩
+  · have := d.M_bounds hn
+    rw [hr] at this
+    linarith [this.1, this.2]
+  · intro x
+    have := d.potential_le hn x
+    rw [hr] at this
+    linarith [d.potential_nonneg x]
+  · intro x y hxy
+    have := d.overlapLength_le hn x y hxy
+    rw [hr] at this
+    linarith [d.overlapLength_nonneg x y hxy]
+
+/- Joint measurability and averaging over a family of realized hierarchies. -/
+section Param
+variable {Ω : Type u3} [MeasurableSpace Ω] (D : Ω → Depth P)
+  (hD : ∀ x y, Measurable (fun ω => (D ω).value x y))
+  (R : Ω → ℝ) (hR : Measurable R)
+include hD hR
+
+omit [DecidableEq P] in
+lemma measurable_cell_param (x : P) : Measurable (fun ω => (D ω).cell (R ω) x) := by
+  apply measurable_finset_iff.mpr
+  intro y
+  simp only [mem_cell, share]
+  exact measurable_const.or (measurableSet_setOfPred.mp (measurableSet_le hR (hD x y)))
+
+lemma measurable_cells_param : Measurable (fun ω => (D ω).cells (R ω)) := by
+  have hm : Measurable (fun ω => fun x : P => (D ω).cell (R ω) x) :=
+    measurable_pi_iff.mpr (measurable_cell_param D hD R hR)
+  exact (measurable_of_finite (fun f : P → Finset P => Finset.univ.image f)).comp hm
+
+lemma measurable_giants_param : Measurable (fun ω => (D ω).giants (R ω)) :=
+  (measurable_of_finite (fun s : Finset (Finset P) =>
+    s.filter (fun C : Finset P => Fintype.card P < 2*C.card))).comp
+      (measurable_cells_param D hD R hR)
+
+lemma measurable_giant_param : Measurable (fun ω => (D ω).giant (R ω)) :=
+  (measurable_of_finite (fun s : Finset (Finset P) => s.biUnion id)).comp
+    (measurable_giants_param D hD R hR)
+
+lemma measurable_giantIndicator_param : Measurable (fun ω => (D ω).giantIndicator (R ω)) :=
+  (measurable_of_finite (fun s : Finset (Finset P) => if s.Nonempty then (1 : ℝ) else 0)).comp
+    (measurable_giants_param D hD R hR)
+
+lemma measurable_outsideIndicator_param (x : P) : Measurable (fun ω => (D ω).outsideIndicator (R ω) x) := by
+  have hm := (measurable_giants_param D hD R hR).prodMk (measurable_giant_param D hD R hR)
+  exact (measurable_of_finite
+    (fun st : Finset (Finset P) × Finset P => if st.1.Nonempty ∧ x ∉ st.2 then (1 : ℝ) else 0)).comp hm
+
+lemma measurable_pairOverlap_param (x y : P) : Measurable (fun ω => (D ω).pairOverlap (R ω) x y) := by
+  have hm : Measurable (fun ω => if (D ω).share (R ω) x y then (1:ℝ) else 0) := by
+    apply Measurable.ite _ measurable_const measurable_const
+    apply measurableSet_setOfPred.mpr
+    exact measurable_const.or (measurableSet_setOfPred.mp (measurableSet_le hR (hD x y)))
+  have hh : Measurable (fun ω => (if (D ω).share (R ω) x y then (1:ℝ) else 0) -
+      (D ω).giantIndicator (R ω) + (D ω).outsideIndicator (R ω) x + (D ω).outsideIndicator (R ω) y) :=
+    ((hm.sub (measurable_giantIndicator_param D hD R hR)).add
+    (measurable_outsideIndicator_param D hD R hR x)).add
+    (measurable_outsideIndicator_param D hD R hR y)
+  simpa only [pairOverlap_formula, Pi.add_apply, Pi.sub_apply] using hh
+
+omit hR
+lemma measurable_potential_param (x : P) : Measurable (fun ω => (D ω).potential x) := by
+  have hm : Measurable (fun p : Ω × ℝ => (D p.1).outsideIndicator p.2 x) :=
+    measurable_outsideIndicator_param (P := P) (Ω := Ω × ℝ) (fun p : Ω × ℝ => D p.1)
+      (fun x y => (hD x y).comp measurable_fst) Prod.snd measurable_snd x
+  simpa only [potential] using hm.stronglyMeasurable.integral_prod_right.measurable
+
+lemma measurable_overlapLength_param (x y : P) (hxy : x ≠ y) :
+    Measurable (fun ω => (D ω).overlapLength x y hxy) := by
+  have hm : Measurable (fun p : Ω × ℝ => (D p.1).pairOverlap p.2 x y) :=
+    measurable_pairOverlap_param (P := P) (Ω := Ω × ℝ) (fun p : Ω × ℝ => D p.1)
+      (fun x y => (hD x y).comp measurable_fst) Prod.snd measurable_snd x y
+  simpa only [overlapLength] using hm.stronglyMeasurable.integral_prod_right.measurable
+
+lemma measurable_M_param (hn : 2 ≤ Fintype.card P) : Measurable (fun ω => (D ω).M) := by
+  obtain ⟨x,hx,y,hy,hxy⟩ := Finset.one_lt_card.mp (show 1 < (Finset.univ : Finset P).card by simpa using (show 1 < Fintype.card P by omega))
+  have hh := (((hD x y).sub (measurable_overlapLength_param D hD x y hxy)).add
+    (measurable_potential_param D hD x)).add (measurable_potential_param D hD y)
+  have he : (fun ω => (D ω).value x y - (D ω).overlapLength x y hxy +
+      (D ω).potential x + (D ω).potential y) = (fun ω => (D ω).M) := by
+    funext ω
+    have := (D ω).depth_identity hn x y hxy
+    linarith
+  change Measurable (fun ω => (D ω).value x y - (D ω).overlapLength x y hxy +
+      (D ω).potential x + (D ω).potential y) at hh
+  rwa [he] at hh
+
+variable {μ : Measure Ω} (hi : ∀ x y, Integrable (fun ω => (D ω).value x y) μ)
+include hi
+
+omit [DecidableEq P] in
+omit hD in
+lemma integrable_radius_param : Integrable (fun ω => (D ω).radius) μ := by
+  unfold radius
+  exact integrable_finsetSum _ (fun x _ => integrable_finsetSum _ (fun y _ => (hi x y).abs))
+
+lemma integrable_potential_param (hn : 2 ≤ Fintype.card P) (x : P) :
+    Integrable (fun ω => (D ω).potential x) μ := by
+  apply Integrable.mono' ((integrable_radius_param D hi).const_mul 2)
+    (measurable_potential_param D hD x).aestronglyMeasurable
+  filter_upwards with ω
+  rw [Real.norm_eq_abs, abs_of_nonneg ((D ω).potential_nonneg x)]
+  exact (D ω).potential_le hn x
+
+lemma integrable_overlapLength_param (hn : 2 ≤ Fintype.card P) (x y : P) (hxy : x ≠ y) :
+    Integrable (fun ω => (D ω).overlapLength x y hxy) μ := by
+  apply Integrable.mono' ((integrable_radius_param D hi).const_mul 4)
+    (measurable_overlapLength_param D hD x y hxy).aestronglyMeasurable
+  filter_upwards with ω
+  rw [Real.norm_eq_abs, abs_of_nonneg ((D ω).overlapLength_nonneg x y hxy)]
+  exact (D ω).overlapLength_le hn x y hxy
+
+lemma integrable_M_param (hn : 2 ≤ Fintype.card P) : Integrable (fun ω => (D ω).M) μ := by
+  apply Integrable.mono' (integrable_radius_param D hi) (measurable_M_param D hD hn).aestronglyMeasurable
+  filter_upwards with ω
+  rw [Real.norm_eq_abs]
+  exact abs_le.mpr ((D ω).M_bounds hn)
+
+lemma integral_depth_identity (hn : 2 ≤ Fintype.card P) (x y : P) (hxy : x ≠ y) :
+    (∫ ω, (D ω).value x y ∂μ) =
+      (∫ ω, (D ω).M ∂μ) + (∫ ω, (D ω).overlapLength x y hxy ∂μ) -
+      (∫ ω, (D ω).potential x ∂μ) - (∫ ω, (D ω).potential y ∂μ) := by
+  have hiM := integrable_M_param D hD hi hn
+  have hiL := integrable_overlapLength_param D hD hi hn x y hxy
+  have hix := integrable_potential_param D hD hi hn x
+  have hiy := integrable_potential_param D hD hi hn y
+  calc
+    _ = ∫ ω, (D ω).M + (D ω).overlapLength x y hxy - (D ω).potential x - (D ω).potential y ∂μ := by
+      apply integral_congr_ae
+      exact Filter.Eventually.of_forall (fun ω => (D ω).depth_identity hn x y hxy)
+    _ = _ := by
+      have h₁ : Integrable (fun ω => (D ω).M + (D ω).overlapLength x y hxy) μ := hiM.add hiL
+      have h₂ : Integrable (fun ω => (D ω).M + (D ω).overlapLength x y hxy - (D ω).potential x) μ := h₁.sub hix
+      rw [integral_sub h₂ hiy, integral_sub h₁ hix, integral_add hiM hiL]
+end Param
+
+end Depth
+end WeakPinned.Hierarchy
+
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PinnedDistances/IncomingFibers.lean b/lean/OAI/Geometry/PinnedDistances/IncomingFibers.lean
new file mode 100644
index 0000000000000000000000000000000000000000..bc050983691b7de48a25ea810fc964df5127699f
--- /dev/null
+++ b/lean/OAI/Geometry/PinnedDistances/IncomingFibers.lean
@@ -0,0 +1,389 @@
+import OAI.Geometry.PinnedDistances.UniformCore
+
+namespace OAI
+
+universe u1 u2 u3 u4 u5 u6 u7 u8
+
+section
+noncomputable section
+open NumberField
+open scoped BigOperators
+noncomputable section
+open MeasureTheory
+open scoped Topology
+noncomputable section
+open MeasureTheory
+open scoped Topology ENNReal
+noncomputable section
+open MeasureTheory NumberField
+open scoped BigOperators
+noncomputable section
+open scoped BigOperators
+open MeasureTheory
+noncomputable section
+open Filter
+open scoped Topology
+noncomputable section
+open scoped BigOperators
+
+namespace WeakPinned.Core.Fibers
+open scoped BigOperators
+open Variance
+variable {V : Type u1} {I : Type u2} [Fintype V] [Fintype I] [DecidableEq V]
+variable (G : Fibers V I)
+
+def edges : Finset (V × V) := G.edgeSet (-1) Finset.univ
+
+def edgeCount : ℝ := G.edges.card
+
+def edgeLaw (x y : V) : ℝ := if (x,y) ∈ G.edges then 1/G.edgeCount else 0
+
+def fiberWeight (e : I) : ℝ := (G.targets e).card / G.edgeCount
+
+def incoming (y : V) : Finset V := Finset.univ.filter (fun x => (x,y) ∈ G.edges)
+
+def incomingWeight (y : V) : ℝ := (G.incoming y).card / G.edgeCount
+
+omit [Fintype I] in
+lemma active_univ_neg_one (e : I) : G.active (-1) Finset.univ e := by
+  refine ⟨Finset.mem_univ _,?_⟩
+  have h := Nat.cast_nonneg (α:=ℝ) (G.size Finset.univ e)
+  linarith
+
+lemma edgeCount_eq : G.edgeCount = ∑ e, ((G.targets e).card : ℝ) := by
+  unfold edgeCount edges
+  rw [edgeSet_card]
+  simp [G.active_univ_neg_one,size]
+
+omit [Fintype I] in
+lemma edgeLaw_nonneg (x y : V) : 0 ≤ G.edgeLaw x y := by
+  unfold edgeLaw edgeCount
+  split_ifs <;> positivity
+
+omit [Fintype I] in
+lemma edgeLaw_diag (x : V) : G.edgeLaw x x=0 := by
+  have hh : (x,x) ∉ G.edges := by
+    intro h
+    exact G.edge_no_loop (-1) Finset.univ x (Finset.mem_filter.mp h).2
+  simp [edgeLaw,hh]
+
+omit [Fintype I] in
+lemma fiberWeight_nonneg (e : I) : 0 ≤ G.fiberWeight e := by
+  unfold fiberWeight edgeCount
+  positivity
+
+lemma sum_fiberWeight (hG : 0 < G.edgeCount) : ∑ e, G.fiberWeight e=1 := by
+  simp only [fiberWeight,← Finset.sum_div,← G.edgeCount_eq,div_self hG.ne']
+
+omit [Fintype I] in
+lemma fiberWeight_prob (e : I) (y : V) :
+    G.fiberWeight e*prob (G.targets e) y = if y ∈ G.targets e then 1/G.edgeCount else 0 := by
+  by_cases hy : y ∈ G.targets e
+  · have hk : ((G.targets e).card : ℝ) ≠ 0 := by
+      exact_mod_cast (Finset.card_ne_zero.mpr ⟨y,hy⟩)
+    simp only [fiberWeight,prob,ite_eq_left hy]
+    field_simp
+  · simp [prob,hy]
+
+lemma sampling_fiber (x y : V) :
+    (∑ e, if G.source e=x then G.fiberWeight e * prob (G.targets e) y else 0) =
+      G.edgeLaw x y := by
+  have he := G.edge_indicator (-1) Finset.univ x y
+  simp only [G.active_univ_neg_one,true_and,Finset.mem_inter,Finset.mem_univ,and_true] at he
+  have hh : (∑ e, if G.source e=x then G.fiberWeight e * prob (G.targets e) y else 0) =
+      (∑ e, if G.source e=x ∧ y ∈ G.targets e then (1:ℝ) else 0)/G.edgeCount := by
+    simp only [Finset.sum_div,G.fiberWeight_prob]
+    apply Finset.sum_congr rfl
+    intro e _
+    by_cases hs : G.source e=x <;> by_cases hy : y ∈ G.targets e <;> simp [hs,hy]
+  rw [hh,he]
+  by_cases he : G.edge (-1) Finset.univ x y <;> simp [edgeLaw,edges,edgeSet,he]
+
+lemma fiber_marginal (y : V) :
+    (∑ e, G.fiberWeight e * prob (G.targets e) y) = ∑ x, G.edgeLaw x y := by
+  simp_rw [← G.sampling_fiber]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro e _
+  simp
+
+omit [Fintype I] in
+lemma incoming_marginal (y : V) : ∑ x, G.edgeLaw x y=G.incomingWeight y := by
+  simp only [edgeLaw,incomingWeight,incoming,← Finset.sum_filter,Finset.sum_const,nsmul_eq_mul]
+  simp only [mul_one_div]
+
+omit [Fintype I] in
+lemma sampling_incoming (x y : V) :
+    G.incomingWeight y*prob (G.incoming y) x=G.edgeLaw x y := by
+  by_cases he : (x,y) ∈ G.edges
+  · have hm : x ∈ G.incoming y := by simp [incoming,he]
+    have hk : ((G.incoming y).card : ℝ) ≠ 0 := by exact_mod_cast (Finset.card_ne_zero.mpr ⟨x,hm⟩)
+    simp only [incomingWeight,prob,ite_eq_left hm,edgeLaw,ite_eq_left he]
+    field_simp
+  · have hm : x ∉ G.incoming y := by simp [incoming,he]
+    simp [prob,hm,edgeLaw,he]
+
+omit [Fintype I] in
+lemma sum_edgeLaw (hG : 0 < G.edgeCount) : (∑ x, ∑ y, G.edgeLaw x y)=1 := by
+  rw [← Fintype.sum_prod_type (f:=fun xy : V × V => G.edgeLaw xy.1 xy.2)]
+  simp only [edgeLaw,← Finset.sum_filter,Finset.sum_const,nsmul_eq_mul]
+  have he : Finset.univ.filter (fun xy : V × V => xy ∈ G.edges)=G.edges := by simp
+  rw [he]
+  change G.edgeCount*(1/G.edgeCount)=1
+  field_simp
+
+omit [Fintype I] in
+lemma sum_incomingWeight (hG : 0 < G.edgeCount) : ∑ y, G.incomingWeight y=1 := by
+  simp_rw [← G.incoming_marginal]
+  rw [Finset.sum_comm,G.sum_edgeLaw hG]
+
+lemma fiber_domination (B : ℝ) (hdom : ∀ x y, G.edgeLaw x y ≤ B*pairProb Finset.univ x y)
+    (hn : 2 ≤ Fintype.card V) (y : V) :
+    (∑ e, G.fiberWeight e*prob (G.targets e) y) ≤ B*prob Finset.univ y := by
+  rw [G.fiber_marginal]
+  have hh := Finset.sum_le_sum (s:=Finset.univ) (fun x _ => hdom x y)
+  simpa only [← Finset.mul_sum,pairProb_symm _ _ y,sum_pairProb Finset.univ (by simpa using hn)] using hh
+
+omit [Fintype I] in
+lemma incoming_domination (B : ℝ) (hdom : ∀ x y, G.edgeLaw x y ≤ B*pairProb Finset.univ x y)
+    (hn : 2 ≤ Fintype.card V) (x : V) :
+    (∑ y, G.incomingWeight y*prob (G.incoming y) x) ≤ B*prob Finset.univ x := by
+  simp_rw [G.sampling_incoming]
+  have hh := Finset.sum_le_sum (s:=Finset.univ) (fun y _ => hdom x y)
+  simpa only [← Finset.mul_sum,sum_pairProb Finset.univ (by simpa using hn)] using hh
+
+omit [Fintype I] in
+/-- The exact bound on omitted small incoming degrees, not just a formal
+    little-o assertion. -/
+lemma small_incoming_mass (T : ℝ) (hT : 0 ≤ T) :
+    (∑ y, if ((G.incoming y).card : ℝ)<T then G.incomingWeight y else 0) ≤
+      (Fintype.card V : ℝ)*T/G.edgeCount := by
+  have hn : 0 ≤ G.edgeCount := by unfold edgeCount; positivity
+  have hh : ∀ y, (if ((G.incoming y).card : ℝ)<T then G.incomingWeight y else 0) ≤ T/G.edgeCount := by
+    intro y
+    split_ifs with hy
+    · exact div_le_div_of_nonneg_right hy.le hn
+    · exact div_nonneg hT hn
+  have hs := Finset.sum_le_sum (s:=Finset.univ) (fun y _ => hh y)
+  simpa only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul,mul_div_assoc] using hs
+
+end WeakPinned.Core.Fibers
+
+namespace WeakPinned.Core.Fibers
+open scoped BigOperators
+variable {V : Type u3} {I : Type u4} [Fintype V] [Fintype I] [DecidableEq V]
+variable (G : Fibers V I)
+
+abbrev Active (K : ℝ) (C : Finset V) := {e : I // G.active K C e}
+
+instance (K : ℝ) (C : Finset V) : Fintype (G.Active K C) := by
+  classical
+  exact Fintype.ofFinite _
+
+def onCore (K : ℝ) (C : Finset V) : Fibers C (G.Active K C) where
+  source e := ⟨G.source e.val,e.property.1⟩
+  targets e := Finset.univ.filter (fun y : C => y.val ∈ G.targets e.val)
+  no_loop e := by simpa using G.no_loop e.val
+  disjoint e f hef hs := by
+    apply Finset.disjoint_left.mpr
+    intro y hye hyf
+    have hne : e.val ≠ f.val := fun h => hef (Subtype.ext h)
+    have hs' : G.source e.val=G.source f.val := congrArg Subtype.val hs
+    exact Finset.disjoint_left.mp (G.disjoint _ _ hne hs')
+      (Finset.mem_filter.mp hye).2 (Finset.mem_filter.mp hyf).2
+
+omit [Fintype V] [Fintype I] in
+lemma onCore_targets (K : ℝ) (C : Finset V) (e : G.Active K C) :
+    ((G.onCore K C).targets e).image Subtype.val=G.targets e.val ∩ C := by
+  ext y
+  simp only [onCore,Finset.mem_image,Finset.mem_filter,Finset.mem_univ,true_and,Finset.mem_inter]
+  constructor
+  · rintro ⟨z,hz,rfl⟩; exact ⟨hz,z.property⟩
+  · rintro ⟨hy,hC⟩; exact ⟨⟨y,hC⟩,hy,rfl⟩
+
+omit [Fintype V] [Fintype I] in
+lemma onCore_size (K : ℝ) (C : Finset V) (e : G.Active K C) :
+    ((G.onCore K C).targets e).card=G.size C e.val := by
+  rw [← Finset.card_image_of_injective _ Subtype.val_injective,G.onCore_targets,size]
+
+omit [Fintype I] in
+omit [Fintype V] in
+lemma onCore_edge (K : ℝ) (C : Finset V) (x y : C) :
+    (G.onCore K C).edge (-1) Finset.univ x y ↔ G.edge K C x.val y.val := by
+  simp only [edge,active_univ_neg_one,true_and,Finset.mem_inter,Finset.mem_univ,and_true]
+  constructor
+  · rintro ⟨e,hs,hy⟩
+    refine ⟨e.val,e.property,congrArg Subtype.val hs,?_⟩
+    exact ⟨(Finset.mem_filter.mp hy).2,y.property⟩
+  · rintro ⟨e,ha,hs,hy⟩
+    exact ⟨⟨e,ha⟩,Subtype.ext hs,Finset.mem_filter.mpr ⟨Finset.mem_univ _,hy.1⟩⟩
+
+omit [Fintype I] in
+lemma onCore_edges_image (K : ℝ) (C : Finset V) :
+    ((G.onCore K C).edges).image (fun xy : C × C => (xy.1.val,xy.2.val))=G.edgeSet K C := by
+  ext xy
+  simp only [edges,edgeSet,Finset.mem_image,Finset.mem_filter,Finset.mem_univ,true_and]
+  constructor
+  · rintro ⟨⟨x,y⟩,hxy,rfl⟩
+    exact (G.onCore_edge K C x y).mp hxy
+  · intro hxy
+    obtain ⟨e,ha,hs,hy⟩ := hxy
+    have hxC : xy.1 ∈ C := hs ▸ ha.1
+    have hyC : xy.2 ∈ C := (Finset.mem_inter.mp hy).2
+    refine ⟨(⟨xy.1,hxC⟩,⟨xy.2,hyC⟩),?_,rfl⟩
+    exact (G.onCore_edge K C _ _).mpr ⟨e,ha,hs,hy⟩
+
+omit [Fintype I] in
+lemma onCore_card (K : ℝ) (C : Finset V) :
+    ((G.onCore K C).edges).card=(G.edgeSet K C).card := by
+  rw [← G.onCore_edges_image]
+  apply (Finset.card_image_of_injective _ _).symm
+  intro a b h
+  exact Prod.ext (Subtype.ext (congrArg Prod.fst h)) (Subtype.ext (congrArg Prod.snd h))
+
+omit [Fintype I] in
+lemma onCore_degree (K : ℝ) (C : Finset V) (x : C) :
+    (G.onCore K C).degree (-1) Finset.univ x=G.degree K C x.val := by
+  simp only [degree,G.onCore_edge]
+  have hsum (f : V → ℝ) (hf : ∀ y, y ∉ C → f y=0) : (∑ y : C, f y.val)=∑ y, f y := by
+    rw [Finset.sum_coe_sort]
+    apply Finset.sum_subset (Finset.subset_univ _) (fun y _ hy => hf y hy)
+  simp only [← Finset.sum_boole,← Finset.sum_add_distrib]
+  apply hsum (fun y => (if G.edge K C x.val y then 1 else 0)+(if G.edge K C y x.val then 1 else 0))
+  intro y hy
+  have h1 : ¬ G.edge K C x.val y := by
+    rintro ⟨e,ha,hs,he⟩
+    exact hy (Finset.mem_inter.mp he).2
+  have h2 : ¬ G.edge K C y x.val := by
+    rintro ⟨e,ha,hs,he⟩
+    exact hy (hs ▸ ha.1)
+  simp [h1,h2]
+
+end WeakPinned.Core.Fibers
+
+namespace WeakPinned.Core.Fibers
+open scoped BigOperators
+variable {V : Type u5} {I : Type u6} [Fintype V] [Fintype I] [DecidableEq V]
+variable (G : Fibers V I)
+
+omit [Fintype V] [Fintype I] in
+lemma onCore_size_subset (K : ℝ) (C : Finset V) (D : Finset C) (e : G.Active K C) :
+    (G.onCore K C).size D e=G.size (D.image Subtype.val) e.val := by
+  unfold size
+  rw [← Finset.card_image_of_injective _ Subtype.val_injective]
+  congr 1
+  ext y
+  simp only [Finset.mem_image,Finset.mem_inter]
+  constructor
+  · rintro ⟨z,⟨hz,hd⟩,rfl⟩
+    exact ⟨(Finset.mem_filter.mp hz).2,⟨z,hd,rfl⟩⟩
+  · rintro ⟨hy,z,hz,rfl⟩
+    exact ⟨z,⟨Finset.mem_filter.mpr ⟨Finset.mem_univ _,hy⟩,hz⟩,rfl⟩
+
+def heavyCount (lam : ℝ) (D : Finset V) : ℝ :=
+  ∑ e, if lam*((G.targets e).card : ℝ) ≤ (G.size D e : ℝ) ∧ G.source e ∈ D
+    then (G.size D e : ℝ) else 0
+
+omit [Fintype V] in
+lemma onCore_heavyCount (K lam : ℝ) (C : Finset V) (D : Finset C) :
+    (G.onCore K C).heavyCount lam D =
+      ∑ e, if G.active K C e ∧ lam*(G.size C e : ℝ) ≤ (G.size (D.image Subtype.val) e : ℝ) ∧
+        G.source e ∈ D.image Subtype.val then (G.size (D.image Subtype.val) e : ℝ) else 0 := by
+  classical
+  unfold heavyCount
+  simp_rw [G.onCore_size_subset,G.onCore_size]
+  have hs (e : G.Active K C) : (G.onCore K C).source e ∈ D ↔ G.source e.val ∈ D.image Subtype.val := by
+    constructor
+    · intro h; exact Finset.mem_image.mpr ⟨_,h,rfl⟩
+    · intro h
+      obtain ⟨y,hy,he⟩ := Finset.mem_image.mp h
+      have hh : y=(G.onCore K C).source e := Subtype.ext he
+      simpa only [hh] using hy
+  simp_rw [hs]
+  rw [← Finset.sum_subtype (p:=G.active K C) (Finset.univ.filter (G.active K C)) (by simp)
+    (fun e => if lam*(G.size C e : ℝ) ≤ (G.size (D.image Subtype.val) e : ℝ) ∧
+      G.source e ∈ D.image Subtype.val then (G.size (D.image Subtype.val) e : ℝ) else 0), Finset.sum_filter]
+  apply Finset.sum_congr rfl
+  intro e _
+  by_cases he : G.active K C e <;> simp [he]
+
+omit [Fintype I] in
+lemma degree_marginal (x : V) : (∑ y, G.edgeLaw x y)+(∑ y, G.edgeLaw y x)=
+    G.degree (-1) Finset.univ x/G.edgeCount := by
+  simp only [edgeLaw,edges,edgeSet,Finset.mem_filter,Finset.mem_univ,true_and,
+    ← Finset.sum_filter,Finset.sum_const,nsmul_eq_mul,degree,add_div,mul_one_div]
+
+end WeakPinned.Core.Fibers
+
+namespace WeakPinned
+open scoped BigOperators
+open WeakPinned.Core
+
+abbrev coreGraph (P : Finset Plane) (s : ℝ) (C : Finset P) :=
+  (planarFibers P s).onCore (cutoff s P.card) C
+
+lemma coreGraph_heavy (P : Finset Plane) (s lam : ℝ) (C : Finset P) (D : Finset C) :
+    (coreGraph P s C).heavyCount lam D = heavyCount P s (cutoff s P.card) lam C (D.image Subtype.val) := by
+  rw [(planarFibers P s).onCore_heavyCount]
+  unfold heavyCount heavyFiber
+  apply Finset.sum_congr rfl
+  intro e _
+  simp only [and_assoc]
+
+structure PreparedGraph (V : Type u7) (I : Type u8) [Fintype V] [Fintype I] [DecidableEq V]
+    (G : Fibers V I) (θ γ c η : ℝ) : Prop where
+  card_ge : 2 ≤ Fintype.card V
+  lower : (θ-η)*(Fintype.card V : ℝ)*((Fintype.card V : ℝ)-1) ≤ G.edgeCount
+  upper : G.edgeCount ≤ (θ+η/8)*(Fintype.card V : ℝ)*((Fintype.card V : ℝ)-1)
+  degree : ∀ x, 2*(θ-η)*((Fintype.card V : ℝ)-1) ≤ G.degree (-1) Finset.univ x
+  fiber : ∀ e, (Fintype.card V : ℝ)^γ < ((G.targets e).card : ℝ)
+  subset : ∀ D, G.heavyCount (1/Real.log (Fintype.card V : ℝ)) D ≤ (θ+η)*(D.card : ℝ)*((D.card : ℝ)-1)
+  small : ∀ D, (D.card : ℝ) ≤ (Fintype.card V : ℝ)^(1-c) →
+    G.heavyCount (1/Real.log (Fintype.card V : ℝ)) D ≤ η*(D.card : ℝ)*((D.card : ℝ)-1)
+
+lemma PreparedCore.graph {s γ c η : ℝ} {P : Finset Plane} {C : Finset P}
+    (h : PreparedCore s γ c η P C) :
+    PreparedGraph C ((planarFibers P s).Active (cutoff s P.card) C) (coreGraph P s C) (Theta s) γ c η := by
+  have hn : Fintype.card C=C.card := Fintype.card_coe C
+  have he : (coreGraph P s C).edgeCount = ((planarFibers P s).edgeSet (cutoff s P.card) C).card := by
+    unfold Core.Fibers.edgeCount
+    rw [(planarFibers P s).onCore_card]
+  refine ⟨by simpa only [hn] using h.card_two,?_,?_,?_,?_,?_,?_⟩
+  · simpa only [he,hn] using h.lower
+  · simpa only [he,hn] using h.upper
+  · intro x
+    simpa only [hn,(planarFibers P s).onCore_degree] using h.degree x.val x.property
+  · intro e
+    rw [(planarFibers P s).onCore_size,hn]
+    exact h.large_fiber.trans_lt e.property.2
+  · intro D
+    rw [hn,coreGraph_heavy]
+    simpa only [Finset.card_image_of_injective _ Subtype.val_injective] using h.subset (D.image Subtype.val)
+  · intro D hD
+    rw [hn,coreGraph_heavy]
+    have hc : (D.image Subtype.val).card=D.card := Finset.card_image_of_injective _ Subtype.val_injective
+    simpa only [hc] using h.small (D.image Subtype.val) (by simpa only [hc,hn] using hD)
+
+/-- Actual arbitrarily large real planar graphs, with all uniform subset
+bounds simultaneously. No graph condition is added to the main hypothesis. -/
+theorem arbitrarily_large_graph (hfail : ¬ MainStatement) {s γ c : ℝ}
+    (hγ : 0<γ) (hγs : γ<s) (hc : 0<c) (hc1 : c<1)
+    (hsc : (1-c)*criticalExponent<s) (hs : ContinuousAt Theta s)
+    (η : ℝ) (hη : 0<η) (M : ℕ) :
+    ∃ (P : Finset Plane) (C : Finset P), M<C.card ∧
+      PreparedGraph C ((planarFibers P s).Active (cutoff s P.card) C) (coreGraph P s C) (Theta s) γ c η := by
+  obtain ⟨P,C,hM,hC⟩ := arbitrarily_large_prepared hfail hγ hγs hc hc1 hsc hs η hη M
+  exact ⟨P,C,hM,hC.graph⟩
+
+end WeakPinned
+
+end
+end
+end
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PinnedDistances/LimitObstruction.lean b/lean/OAI/Geometry/PinnedDistances/LimitObstruction.lean
new file mode 100644
index 0000000000000000000000000000000000000000..aa3536cf213b3b808869ddaa94888d6310a24971
--- /dev/null
+++ b/lean/OAI/Geometry/PinnedDistances/LimitObstruction.lean
@@ -0,0 +1,73 @@
+import OAI.Geometry.PinnedDistances.FiniteObstruction
+
+namespace OAI
+
+universe u1 u2
+
+section
+noncomputable section
+open MeasureTheory Filter Set
+open scoped Topology
+namespace WeakPinned.Bounded
+local instance : MeasurableSpace Sphere := borel Sphere
+local instance : BorelSpace Sphere := ⟨rfl⟩
+variable {V : ℕ → Type u1} {I : ℕ → Type u2} [∀ n,Fintype (V n)] [∀ n,DecidableEq (V n)] [∀ n,Fintype (I n)]
+
+/-- The actual finite expanded graph laws cannot have nonatomic coordinate limits
+and the two test-independence properties supplied by the variance estimates. -/
+theorem graph_limit_impossible
+    (G : ∀ n,Core.Fibers (V n) (I n)) (U₁ U₂ : ∀ n,V n → ℂ)
+    (hG : ∀ n,0<(G n).edgeCount) (hB : ∀ n e,((G n).targets e).Nonempty)
+    (hn : ∀ n,2≤Fintype.card (V n))
+    (β₁ β₂ : ProbabilityMeasure Sphere) (Q : ProbabilityMeasure (Quad Sphere))
+    (R : ProbabilityMeasure (Expanded Sphere))
+    [NullSingletonClass (β₁:Measure Sphere)] [NullSingletonClass (β₂:Measure Sphere)]
+    (hα₁ : Tendsto (fun n => uniformProbability (fun x => (U₁ n x:Sphere)) (by have:=hn n;omega)) atTop (𝓝 β₁))
+    (hα₂ : Tendsto (fun n => uniformProbability (fun x => (U₂ n x:Sphere)) (by have:=hn n;omega)) atTop (𝓝 β₂))
+    (hQ : Tendsto (fun n => edgeProbability (G n) (fun x => (U₁ n x:Sphere))
+      (fun x => (U₂ n x:Sphere)) (hG n) (hB n)) atTop (𝓝 Q))
+    (hR : Tendsto (fun n => expandedProbability (G n) (fun x => (U₁ n x:Sphere))
+      (fun x => (U₂ n x:Sphere)) (hG n) (hB n)) atTop (𝓝 R))
+    (hd₁ : ∀ f : C(Sphere,ℝ),Tendsto (fun n => testDiscrep (G n).fiberWeight (G n).targets
+      (fun x => f (U₁ n x:Sphere))) atTop (𝓝 0))
+    (hd₂ : ∀ f : C(Sphere,ℝ),Tendsto (fun n => testDiscrep (G n).fiberWeight (G n).targets
+      (fun x => f (U₂ n x:Sphere))) atTop (𝓝 0))
+    (T : ℕ → ℝ) (hT : ∀ n,0<T n)
+    (hdR : ∀ f : C(Sphere,ℝ),Tendsto (fun n => testDiscrep ((G n).retainedWeight (T n))
+      ((G n).retainedSet (T n)) (fun x => f (U₁ n x:Sphere))) atTop (𝓝 0))
+    (ho : Tendsto (fun n => (G n).omittedMass (T n)) atTop (𝓝 0))
+    (M : ℝ) (hM : 0≤M) (hdom : ∀ n x y,(G n).edgeLaw x y≤M*Variance.pairProb Finset.univ x y)
+    (hrel : ∀ n e y z,y∈(G n).targets e → z∈(G n).targets e →
+      (U₁ n y-U₁ n ((G n).source e))*(U₂ n y-U₂ n ((G n).source e))=
+      (U₁ n z-U₁ n ((G n).source e))*(U₂ n z-U₂ n ((G n).source e))) : False := by
+  let : TopologicalSpace.PseudoMetrizableSpace Sphere := inferInstance
+  let : TopologicalSpace.PseudoMetrizableSpace (Quad Sphere) := inferInstance
+  let : TopologicalSpace.PseudoMetrizableSpace (Sphere × Sphere) := inferInstance
+  let : TopologicalSpace.PseudoMetrizableSpace (Expanded Sphere) := inferInstance
+  let S₁ := fun n x => (U₁ n x:Sphere)
+  let S₂ := fun n x => (U₂ n x:Sphere)
+  have hm₁ := expanded_marginal_limit G S₁ S₂ hG hB hn Prod.fst continuous_fst β₁ Q R hα₁ hQ hR hd₁
+  have hm₂ := expanded_marginal_limit G S₁ S₂ hG hB hn Prod.snd continuous_snd β₂ Q R hα₂ hQ hR hd₂
+  have he := weak_map_eq
+    (fun n => expandedProbability (G n) (S₁ n) (S₂ n) (hG n) (hB n)) R
+    (fun n => edgeProbability (G n) (S₁ n) (S₂ n) (hG n) (hB n)) Q
+    Prod.fst continuous_fst hR hQ
+    (fun n => expandedProbability_edge (G n) (S₁ n) (S₂ n) (hG n) (hB n))
+  have hi := incoming_limit G S₁ S₂ hG hB hn T hT β₁ Q hα₁ hQ hdR ho
+  have hac := edge_ac_limit G S₁ S₂ hG hB hn β₁ β₂ Q hα₁ hα₂ hQ M hM hdom
+  have hz := weak_chart_closed expandedCoe expandedCoe_open
+    (fun n => expandedProbability (G n) (S₁ n) (S₂ n) (hG n) (hB n)) R hR
+    zEquation zEquation_closed
+    (fun n => (expanded_chart_zero (G n) (U₁ n) (U₂ n) (hG n) (hB n) (hrel n)).1)
+  have hw := weak_chart_closed expandedCoe expandedCoe_open
+    (fun n => expandedProbability (G n) (S₁ n) (S₂ n) (hG n) (hB n)) R hR
+    wEquation wEquation_closed
+    (fun n => (expanded_chart_zero (G n) (U₁ n) (U₂ n) (hG n) (hB n) (hrel n)).2)
+  exact no_sphere_expanded β₁ β₂ Q R he hm₁ hm₂ hi hac.1 hac.2 hz hw
+
+end WeakPinned.Bounded
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PinnedDistances/LogBounds.lean b/lean/OAI/Geometry/PinnedDistances/LogBounds.lean
new file mode 100644
index 0000000000000000000000000000000000000000..724c5686c55b0dd3773138b227ab3462d18811c4
--- /dev/null
+++ b/lean/OAI/Geometry/PinnedDistances/LogBounds.lean
@@ -0,0 +1,406 @@
+import OAI.Geometry.PinnedDistances.WeightedSelection
+
+namespace OAI
+
+universe u1 u2 u3 u4 u5 u6
+
+section
+/- Finite log-distance normalization. The diagonal is excluded in the energy
+   and counted explicitly, with mass 1/n, in the ball estimate. -/
+
+noncomputable section
+open scoped BigOperators
+namespace WeakPinned.Bounded
+variable {P : Type u1} [Fintype P] [DecidableEq P]
+
+def logCost (u : P → ℂ) (x y : P) : ℝ :=
+  if x=y then 0 else |Real.log ‖u x-u y‖|
+
+def logAverage (u : P → ℂ) : ℝ :=
+  (∑ x, ∑ y, logCost u x y)/((Fintype.card P : ℝ)*((Fintype.card P : ℝ)-1))
+
+omit [Fintype P] in
+lemma logCost_nonneg (u : P → ℂ) (x y : P) : 0 ≤ logCost u x y := by
+  unfold logCost; split_ifs <;> positivity
+
+lemma logAverage_sum_bound (u : P → ℂ) (C : ℝ) (hn : 2 ≤ Fintype.card P)
+    (hC : logAverage u ≤ C) :
+    (∑ x, ∑ y, logCost u x y) ≤ C*(Fintype.card P : ℝ)*((Fintype.card P : ℝ)-1) := by
+  have hnR : (2:ℝ) ≤ Fintype.card P := by exact_mod_cast hn
+  have hp : 0 < (Fintype.card P : ℝ)*((Fintype.card P : ℝ)-1) := mul_pos (by linarith) (by linarith)
+  simpa only [mul_assoc] using (div_le_iff₀ hp).mp hC
+
+lemma logAverage_nonneg (u : P → ℂ) (hn : 2 ≤ Fintype.card P) : 0 ≤ logAverage u := by
+  have hnR : (2:ℝ) ≤ Fintype.card P := by exact_mod_cast hn
+  exact div_nonneg (Finset.sum_nonneg (fun _ _ => Finset.sum_nonneg (fun _ _ => logCost_nonneg _ _ _))) (mul_nonneg (by linarith) (by linarith))
+
+/-- One actual vertex serves as a tightness center: a row of the nonnegative
+    distance-log matrix has at most the average row energy. -/
+lemma log_center (u : P → ℂ) (C : ℝ) (hn : 2 ≤ Fintype.card P) (hC : logAverage u ≤ C) :
+    ∃ x : P, (∑ y, logCost u x y) ≤ C*(Fintype.card P : ℝ) := by
+  have hnR : (2:ℝ) ≤ Fintype.card P := by exact_mod_cast hn
+  have hC0 : 0 ≤ C := (logAverage_nonneg u hn).trans hC
+  have hsum := logAverage_sum_bound u C hn hC
+  by_contra hnot
+  push Not at hnot
+  have hpos : Finset.univ.Nonempty (α:=P) := Finset.card_pos.mp (by simpa using (show 0<Fintype.card P by omega))
+  have hs := Finset.sum_lt_sum_of_nonempty hpos (fun x _ => hnot x)
+  simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul] at hs
+  nlinarith
+
+omit [Fintype P] in
+lemma log_cost_large (u : P → ℂ) (R : ℝ) (hR : 1 < R) (x y : P) (hxy : R < ‖u x-u y‖) :
+    Real.log R ≤ logCost u x y := by
+  have hne : x ≠ y := by intro he; subst y; simp at hxy; linarith
+  rw [logCost,ite_eq_right hne]
+  have hl := Real.log_le_log (by linarith : 0<R) hxy.le
+  exact hl.trans (le_abs_self _)
+
+/-- A normalized empirical tail bound from the average logarithmic cost. -/
+lemma normalized_tail_card (u : P → ℂ) (C : ℝ) (x : P)
+    (hx : (∑ y, logCost u x y) ≤ C*(Fintype.card P : ℝ))
+    (hn : 2 ≤ Fintype.card P) (R : ℝ) (hR : 1 < R) :
+    ((Finset.univ.filter (fun y => R < ‖u y-u x‖)).card : ℝ)/(Fintype.card P : ℝ) ≤ C/Real.log R := by
+  let A := Finset.univ.filter (fun y => R < ‖u y-u x‖)
+  have hlog := Real.log_pos hR
+  have hn0 : (0:ℝ)<Fintype.card P := by exact_mod_cast (show 0<Fintype.card P by omega)
+  have hs : (A.card : ℝ)*Real.log R ≤ ∑ y, logCost u x y := by
+    calc
+      (A.card : ℝ)*Real.log R = ∑ y ∈ A, Real.log R := by simp
+      _ ≤ ∑ y ∈ A, logCost u x y := by
+        apply Finset.sum_le_sum
+        intro y hy
+        apply log_cost_large u R hR x y
+        simpa only [norm_sub_rev] using (Finset.mem_filter.mp hy).2
+      _ ≤ ∑ y, logCost u x y := Finset.sum_le_sum_of_subset_of_nonneg
+        (Finset.subset_univ _) (fun y _ _ => logCost_nonneg u x y)
+  apply (div_le_div_iff₀ hn0 hlog).mpr
+  nlinarith
+
+omit [Fintype P] in
+lemma log_cost_close (u : P → ℂ) (hu : Function.Injective u) (z : ℂ) (r : ℝ)
+    (_hr : 0<r) (_hr2 : r<1/2) (x y : P) (hx : ‖u x-z‖<r) (hy : ‖u y-z‖<r) (hxy : x≠y) :
+    -Real.log (2*r) ≤ logCost u x y := by
+  have hne : u x-u y ≠ 0 := sub_ne_zero.mpr (fun h => hxy (hu h))
+  have hp := norm_pos_iff.mpr hne
+  have hd : ‖u x-u y‖<2*r := by
+    have hh := norm_sub_le_norm_sub_add_norm_sub (u x) z (u y)
+    rw [norm_sub_rev z] at hh
+    linarith
+  have hh := Real.log_le_log hp hd.le
+  rw [logCost,ite_eq_right hxy]
+  have habs := neg_le_abs (Real.log ‖u x-u y‖)
+  linarith
+
+/-- The true n^-1 diagonal term in the concentration estimate. The points may
+    be arbitrarily close; injectivity is the only geometric condition. -/
+lemma small_ball_card (u : P → ℂ) (hu : Function.Injective u) (C : ℝ)
+    (hn : 2 ≤ Fintype.card P) (hC : logAverage u ≤ C) (z : ℂ) (r : ℝ)
+    (hr : 0<r) (hr2 : r<1/2) :
+    (((Finset.univ.filter (fun x => ‖u x-z‖<r)).card : ℝ)/(Fintype.card P : ℝ))^2 ≤
+      1/(Fintype.card P : ℝ)+C/(-Real.log (2*r)) := by
+  let A := Finset.univ.filter (fun x => ‖u x-z‖<r)
+  let T := -Real.log (2*r)
+  have hT : 0<T := by dsimp [T]; exact neg_pos.mpr (Real.log_neg (by positivity) (by linarith))
+  have hnR : (2:ℝ) ≤ Fintype.card P := by exact_mod_cast hn
+  have hn0 : (0:ℝ)<Fintype.card P := by linarith
+  have hC0 : 0 ≤ C := (logAverage_nonneg u hn).trans hC
+  have ha : (A.card : ℝ) ≤ Fintype.card P := by exact_mod_cast Finset.card_le_univ A
+  have ha0 : (0:ℝ) ≤ A.card := Nat.cast_nonneg _
+  have hsum := logAverage_sum_bound u C hn hC
+  have hpoint : ∀ x y : P, (if x∈A ∧ y∈A ∧ x≠y then T else 0) ≤ logCost u x y := by
+    intro x y
+    split_ifs with h
+    · exact log_cost_close u hu z r hr hr2 x y (Finset.mem_filter.mp h.1).2
+        (Finset.mem_filter.mp h.2.1).2 h.2.2
+    · exact logCost_nonneg _ _ _
+  have hs := Finset.sum_le_sum (s:=Finset.univ) (fun x _ =>
+    Finset.sum_le_sum (s:=Finset.univ) (fun y _ => hpoint x y))
+  have heq : (∑ x : P, ∑ y : P, if x∈A ∧ y∈A ∧ x≠y then T else 0) =
+      (A.card : ℝ)*((A.card : ℝ)-1)*T := by
+    have hrw (x : P) : (∑ y : P, if x∈A ∧ y∈A ∧ x≠y then T else 0) =
+        if x∈A then ((A.card : ℝ)-1)*T else 0 := by
+      by_cases hx : x∈A
+      · simp only [hx,true_and,ite_eq_left]
+        have hf : Finset.univ.filter (fun y => y∈A ∧ x≠y)=A.erase x := by
+          ext y; simp [ne_comm, and_comm]
+        rw [← Finset.sum_filter,hf,Finset.sum_const,Finset.card_erase_of_mem hx,nsmul_eq_mul]
+        rw [Nat.cast_sub (Finset.one_le_card.mpr ⟨x,hx⟩),Nat.cast_one]
+      · simp [hx]
+    simp_rw [hrw]
+    rw [← Finset.sum_filter]
+    simp only [Finset.filter_mem_eq_inter,Finset.univ_inter,Finset.sum_const,nsmul_eq_mul]
+    ring
+  rw [heq] at hs
+  have hcub : (A.card : ℝ)^2*T ≤ (Fintype.card P : ℝ)*T+C*(Fintype.card P : ℝ)^2 := by
+    have hCT := mul_nonneg hC0 hn0.le
+    have hAT := mul_le_mul_of_nonneg_right ha hT.le
+    nlinarith
+  change ((A.card : ℝ)/(Fintype.card P : ℝ))^2 ≤ 1/(Fintype.card P : ℝ)+C/T
+  rw [div_pow]
+  apply (div_le_iff₀ (sq_pos_of_pos hn0)).mpr
+  apply (mul_le_mul_iff_left₀ hT).mp
+  have he : (1/(Fintype.card P : ℝ)+C/T)*(Fintype.card P : ℝ)^2*T =
+      (Fintype.card P : ℝ)*T+C*(Fintype.card P : ℝ)^2 := by field_simp [ne_of_gt hn0, ne_of_gt hT]
+  rw [he]
+  exact hcub
+
+end WeakPinned.Bounded
+
+namespace WeakPinned.Bounded
+open Hierarchy Variance Global NumberField MeasureTheory
+open scoped BigOperators
+
+lemma weighted_deviation {J : Type u2} [Fintype J] (p f : J → ℝ) (a : ℝ)
+    (hp : ∀ j, 0≤ p j) (hs : ∑ j, p j=1) :
+    (∑ j, p j*|f j-∑ k, p k*f k|) ≤ 2*∑ j, p j*|f j-a| := by
+  have hm : |a-∑ j, p j*f j| ≤ ∑ j, p j*|f j-a| := by
+    have he : a-(∑ j,p j*f j) = ∑ j,p j*(a-f j) := by
+      simp only [mul_sub,Finset.sum_sub_distrib,← Finset.sum_mul,hs,one_mul]
+    rw [he]
+    refine (Finset.abs_sum_le_sum_abs _ _).trans ?_
+    apply Finset.sum_le_sum
+    intro j _
+    rw [abs_mul,abs_of_nonneg (hp j),abs_sub_comm]
+  calc
+    _ ≤ ∑ j, p j*(|f j-a|+|a-∑ k,p k*f k|) := by
+      apply Finset.sum_le_sum
+      intro j _
+      exact mul_le_mul_of_nonneg_left (abs_sub_le (f j) a _) (hp j)
+    _ = (∑ j,p j*|f j-a|)+|a-∑ k,p k*f k| := by
+      simp only [mul_add,Finset.sum_add_distrib,← Finset.sum_mul,hs,one_mul]
+    _ ≤ _ := by linarith
+
+variable {P : Type u3} [Fintype P] [DecidableEq P]
+
+def pairMean (f : P → P → ℝ) : ℝ := ∑ x, ∑ y, pairProb Finset.univ x y * f x y
+
+lemma pairWeight_nonneg (hn : 2≤ Fintype.card P) (x y : P) : 0 ≤ pairProb Finset.univ x y := by
+  have hn' : (2:ℝ)≤ Fintype.card P := by exact_mod_cast hn
+  simp only [pairProb,Finset.mem_univ,true_and,Finset.card_univ]
+  split_ifs
+  · exact div_nonneg zero_le_one (mul_nonneg (by positivity) (by linarith))
+  · exact le_rfl
+
+lemma pairMean_one (hn : 2≤ Fintype.card P) : pairMean (fun _ _ : P => 1)=1 := by
+  simp only [pairMean,mul_one,sum_pairProb Finset.univ (by simpa using hn)]
+  apply sum_prob
+  exact Finset.card_pos.mp (by simpa using (show 0<Fintype.card P by omega))
+
+lemma pairMean_mono (hn : 2≤ Fintype.card P) {f g : P → P → ℝ}
+    (h : ∀ x y, x≠y → f x y≤ g x y) : pairMean f≤ pairMean g := by
+  apply Finset.sum_le_sum
+  intro x _
+  apply Finset.sum_le_sum
+  intro y _
+  by_cases hxy : x=y
+  · subst y; simp [pairProb]
+  · exact mul_le_mul_of_nonneg_left (h x y hxy) (pairWeight_nonneg hn x y)
+
+lemma pairMean_add (f g : P → P → ℝ) : pairMean (fun x y => f x y+g x y)=pairMean f+pairMean g := by
+  simp only [pairMean,mul_add,Finset.sum_add_distrib]
+
+lemma pairMean_mul (c : ℝ) (f : P → P → ℝ) : pairMean (fun x y => c*f x y)=c*pairMean f := by
+  simp only [pairMean,Finset.mul_sum]; congr 1; funext x
+  apply Finset.sum_congr rfl; intro y _; ring
+
+lemma pairMean_left (hn : 2≤ Fintype.card P) (f : P → ℝ) :
+    pairMean (fun x _ => f x)=∑ x, prob Finset.univ x*f x := by
+  simp only [pairMean,← Finset.sum_mul,sum_pairProb Finset.univ (by simpa using hn)]
+
+lemma pairMean_right (hn : 2≤ Fintype.card P) (f : P → ℝ) :
+    pairMean (fun _ y => f y)=∑ y, prob Finset.univ y*f y := by
+  rw [pairMean,Finset.sum_comm]
+  simp_rw [pairProb_symm Finset.univ _ _,← Finset.sum_mul,sum_pairProb Finset.univ (by simpa using hn)]
+
+lemma pairMean_deviation (hn : 2≤ Fintype.card P) (f : P → P → ℝ) (a : ℝ) :
+    pairMean (fun x y => |f x y-pairMean f|)≤2*pairMean (fun x y => |f x y-a|) := by
+  have h := weighted_deviation (fun z : P×P => pairProb Finset.univ z.1 z.2)
+    (fun z => f z.1 z.2) a (fun z => pairWeight_nonneg hn z.1 z.2)
+    (by simpa only [Fintype.sum_prod_type,pairMean,mul_one] using pairMean_one (P:=P) hn)
+  simpa only [Fintype.sum_prod_type,pairMean] using h
+
+lemma pairMean_overlap (d : Depth P) (_hn : 2≤ Fintype.card P) :
+    pairMean (weightedOverlap d.mass)=weightedIntegral d.mass (pairCell Finset.univ) := by
+  exact (weighted_pair d.mass (pairProb Finset.univ)).symm
+
+lemma depth_center_bound (d : Depth P) (hn : 2≤ Fintype.card P) :
+    pairMean (fun x y => |d.value x y-d.M|) ≤ weightedIntegral d.mass (pairCell Finset.univ)+
+      2*(∑ x, prob Finset.univ x*d.potential x) := by
+  calc
+    _ ≤ pairMean (fun x y => weightedOverlap d.mass x y+d.potential x+d.potential y) := by
+      apply pairMean_mono hn
+      intro x y hxy
+      rw [show weightedOverlap d.mass x y=d.overlapLength x y hxy from d.mass_overlap hn x y hxy,
+        d.depth_identity hn x y hxy]
+      have hL := d.overlapLength_nonneg x y hxy
+      have hx := d.potential_nonneg x
+      have hy := d.potential_nonneg y
+      exact abs_le.mpr ⟨by linarith,by linarith⟩
+    _ = _ := by rw [pairMean_add,pairMean_add,pairMean_overlap d hn,pairMean_left hn,pairMean_right hn]; ring
+
+end WeakPinned.Bounded
+
+namespace WeakPinned.Bounded
+open Hierarchy Variance Global NumberField MeasureTheory
+open scoped BigOperators
+variable {P : Type u4} [Fintype P] [DecidableEq P]
+
+def singleScale (d : Depth P) : ℝ :=
+  weightedIntegral d.mass (pairCell Finset.univ)+∑ x, prob Finset.univ x*d.potential x
+
+lemma singleScale_nonneg (d : Depth P) : 0 ≤ singleScale d := by
+  refine add_nonneg (Finset.sum_nonneg (fun C _ => mul_nonneg (d.mass_nonneg C) (pairCell_nonneg _ _))) ?_
+  apply Finset.sum_nonneg
+  intro x _
+  apply mul_nonneg _ (d.potential_nonneg x)
+  simp only [prob,Finset.mem_univ,ite_true]; positivity
+
+lemma scaleEval_eq (d d' : Depth P) : scaleEval d d'=singleScale d+singleScale d' := by
+  simp only [scaleEval,singleScale,massEval,potentialEval,mul_add,Finset.sum_add_distrib]; ring
+
+lemma singleScale_le_left (d d' : Depth P) : singleScale d≤ scaleEval d d' := by
+  rw [scaleEval_eq]; exact le_add_of_nonneg_right (singleScale_nonneg d')
+
+lemma singleScale_le_right (d d' : Depth P) : singleScale d'≤ scaleEval d d' := by
+  rw [scaleEval_eq]; exact le_add_of_nonneg_left (singleScale_nonneg d)
+
+def rawDepth (u : P → ℂ) (x y : P) : ℝ := -Real.log ‖u x-u y‖
+def pairGridError (u : P → ℂ) (ω : Grid.FullSample) (x y : P) : ℝ :=
+  if x=y then 0 else |Grid.error ω (u x) (u y)|
+
+lemma raw_center_grid_bound (u : P → ℂ) (d : Depth P) (ω : Grid.FullSample)
+    (hn : 2≤ Fintype.card P) (hd : ∀ x y, x≠y → d.value x y=Grid.depth ω (u x) (u y)) :
+    pairMean (fun x y => |rawDepth u x y-pairMean (rawDepth u)|) ≤
+      4*singleScale d+2*pairMean (pairGridError u ω) := by
+  have h1 := pairMean_deviation hn (rawDepth u) d.M
+  have h2 : pairMean (fun x y => |rawDepth u x y-d.M|) ≤
+      pairMean (fun x y => |d.value x y-d.M|)+pairMean (pairGridError u ω) := by
+    rw [← pairMean_add]
+    apply pairMean_mono hn
+    intro x y hxy
+    simp only [pairGridError,ite_eq_right hxy]
+    have he : rawDepth u x y-d.M=(d.value x y-d.M)-Grid.error ω (u x) (u y) := by
+      rw [hd x y hxy]; simp only [rawDepth,Grid.error]; ring
+    rw [he]
+    exact abs_sub _ _
+  have h3 := depth_center_bound d hn
+  have hL : 0≤ weightedIntegral d.mass (pairCell Finset.univ) :=
+    Finset.sum_nonneg (fun C _ => mul_nonneg (d.mass_nonneg C) (pairCell_nonneg _ _))
+  unfold singleScale
+  linarith
+
+lemma pairGridError_integrable (u : P → ℂ) (hu : Function.Injective u) :
+    Integrable (fun ω => pairMean (pairGridError u ω)) Grid.fullLaw := by
+  apply integrable_finsetSum
+  intro x _
+  apply integrable_finsetSum
+  intro y _
+  by_cases hxy : x=y
+  · simp only [pairGridError,hxy,ite_true,mul_zero]; exact integrable_zero _ _ _
+  · simpa only [pairGridError,ite_eq_right hxy] using
+      ((Grid.error_integrable_full (hu.ne hxy)).abs.const_mul (pairProb Finset.univ x y))
+
+lemma mean_pairGridError (u : P → ℂ) (hu : Function.Injective u) (hn : 2≤ Fintype.card P) :
+    (∫ ω, pairMean (pairGridError u ω) ∂ Grid.fullLaw)≤4 := by
+  have hi (x y : P) : Integrable (fun ω => pairProb Finset.univ x y*pairGridError u ω x y) Grid.fullLaw := by
+    by_cases hxy : x=y
+    · simp only [pairGridError,hxy,ite_true,mul_zero]; exact integrable_zero _ _ _
+    · simpa only [pairGridError,ite_eq_right hxy] using
+        ((Grid.error_integrable_full (hu.ne hxy)).abs.const_mul (pairProb Finset.univ x y))
+  simp only [pairMean]
+  rw [integral_finsetSum _ (fun x _ => integrable_finsetSum _ (fun y _ => hi x y))]
+  simp_rw [integral_finsetSum _ (fun y _ => hi _ y),integral_const_mul]
+  calc
+    _ ≤ ∑ x,∑ y,pairProb Finset.univ x y*4 := by
+      apply Finset.sum_le_sum; intro x _
+      apply Finset.sum_le_sum; intro y _
+      apply mul_le_mul_of_nonneg_left _ (pairWeight_nonneg hn x y)
+      by_cases hxy : x=y
+      · simp only [pairGridError,hxy,ite_true,integral_zero]; norm_num
+      · simpa only [pairGridError,ite_eq_right hxy] using Grid.mean_abs_error_full (hu.ne hxy)
+    _ = 4 := by
+      have h := pairMean_one (P:=P) hn
+      simp only [pairMean,mul_one] at h
+      simp only [← Finset.sum_mul,h,one_mul]
+
+variable {K : Type u5} [Field K] [NumberField K]
+variable (Z₁ Z₂ : P → K) (h₁ : Function.Injective Z₁) (h₂ : Function.Injective Z₂)
+
+omit [NumberField K] in
+include h₁ h₂ in
+lemma arch_log_center_bound (hn : 2≤ Fintype.card P) (v : InfinitePlace K) :
+    pairMean (fun x y => |rawDepth (fun x => v.embedding (Z₁ x)) x y-
+      pairMean (rawDepth (fun x => v.embedding (Z₁ x)))|) ≤ 4*archMean Z₁ Z₂ scaleEval v+8 := by
+  let u : P → ℂ := fun x => v.embedding (Z₁ x)
+  let L := pairMean (fun x y => |rawDepth u x y-pairMean (rawDepth u)|)
+  have hu : Function.Injective u := v.embedding.injective.comp h₁
+  have he := mean_pairGridError u hu hn
+  have hbound : ∀ᵐ ω ∂ Grid.fullLaw, L≤4*scaleEval (arch Z₁ v ω) (arch Z₂ v ω)+2*pairMean (pairGridError u ω) := by
+    filter_upwards [good_ae Z₁ h₁ v] with ω hgood
+    have h := raw_center_grid_bound u (arch Z₁ v ω) ω hn (fun x y hxy => by simp [arch,hgood,hxy,u])
+    have hs := singleScale_le_left (arch Z₁ v ω) (arch Z₂ v ω)
+    dsimp only [L]
+    linarith
+  have hh := integral_mono_ae (integrable_const L)
+    (((integrable_scaleEval Z₁ Z₂ h₁ h₂ hn v).const_mul 4).add ((pairGridError_integrable u hu).const_mul 2)) hbound
+  simp only [Pi.add_apply] at hh
+  rw [integral_add ((integrable_scaleEval Z₁ Z₂ h₁ h₂ hn v).const_mul 4)
+    ((pairGridError_integrable u hu).const_mul 2),integral_const_mul,integral_const_mul] at hh
+  simp only [integral_const,MeasureTheory.probReal_univ,one_smul] at hh
+  change L≤4*archMean Z₁ Z₂ scaleEval v+8
+  change L≤4*archMean Z₁ Z₂ scaleEval v+2*(∫ ω,pairMean (pairGridError u ω) ∂ Grid.fullLaw) at hh
+  linarith
+
+end WeakPinned.Bounded
+
+namespace WeakPinned.Bounded
+open Variance
+variable {P : Type u6} [Fintype P] [DecidableEq P]
+
+def logNormalize (u : P → ℂ) (x : P) : ℂ := (Real.exp (pairMean (rawDepth u)) : ℂ)*u x
+
+lemma logNormalize_injective (u : P → ℂ) (hu : Function.Injective u) :
+    Function.Injective (logNormalize u) := by
+  intro x y h
+  apply hu
+  exact mul_left_cancel₀ (by exact_mod_cast (Real.exp_ne_zero (pairMean (rawDepth u)))) h
+
+lemma logAverage_pairMean (u : P → ℂ) :
+    logAverage u=pairMean (fun x y => |Real.log ‖u x-u y‖|) := by
+  unfold logAverage pairMean
+  simp only [Finset.sum_div]
+  apply Finset.sum_congr rfl; intro x _
+  apply Finset.sum_congr rfl; intro y _
+  by_cases hxy : x=y
+  · simp [pairProb,logCost,hxy]
+  · simp [pairProb,logCost,hxy,div_eq_mul_inv,mul_comm]
+
+lemma logNormalize_log (u : P → ℂ) (hu : Function.Injective u) (x y : P) (hxy : x≠y) :
+    Real.log ‖logNormalize u x-logNormalize u y‖=pairMean (rawDepth u)+Real.log ‖u x-u y‖ := by
+  have hz : ‖u x-u y‖≠0 := norm_ne_zero_iff.mpr (sub_ne_zero.mpr (hu.ne hxy))
+  simp only [logNormalize,← mul_sub,norm_mul,Complex.norm_real,Real.norm_eq_abs,
+    abs_of_pos (Real.exp_pos _)]
+  rw [Real.log_mul (Real.exp_ne_zero _) hz,Real.log_exp]
+
+lemma logNormalize_bound (u : P → ℂ) (hu : Function.Injective u) (hn : 2≤Fintype.card P) :
+    logAverage (logNormalize u)=pairMean (fun x y => |rawDepth u x y-pairMean (rawDepth u)|) := by
+  rw [logAverage_pairMean]
+  apply le_antisymm
+  all_goals
+    apply pairMean_mono hn
+    intro x y hxy
+    rw [logNormalize_log u hu x y hxy]
+    have he : |pairMean (rawDepth u)+Real.log ‖u x-u y‖|=|rawDepth u x y-pairMean (rawDepth u)| := by
+      rw [← abs_neg]; congr 1; unfold rawDepth; ring
+    rw [he]
+
+lemma logAverage_translate (u : P → ℂ) (b : ℂ) : logAverage (fun x => u x+b)=logAverage u := by
+  simp only [logAverage,logCost,add_sub_add_right_eq_sub]
+
+end WeakPinned.Bounded
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PinnedDistances/Main.lean b/lean/OAI/Geometry/PinnedDistances/Main.lean
new file mode 100644
index 0000000000000000000000000000000000000000..c81398d15d7539560d4ef2fee00f61ec022cc46c
--- /dev/null
+++ b/lean/OAI/Geometry/PinnedDistances/Main.lean
@@ -0,0 +1,37 @@
+import OAI.Geometry.PinnedDistances.ArithmeticSequence
+
+namespace OAI
+
+section
+noncomputable section
+open Filter
+open scoped Topology
+namespace WeakPinned
+
+/-- Uniform weak pinned planar distances for arbitrary real configurations. -/
+theorem mainStatement : MainStatement := by
+  classical
+  by_contra hfail
+  obtain ⟨c,s,γ,hc,hcγ,hγs,hs,hθ,P,C,hcard,hG,hmodels⟩ := sequence_arithmetic_model hfail
+  choose L hL ii Z₁ Z₂ t hii h₁ h₂ ht0 ht using hmodels
+  let K := fun j => L j
+  let (j : ℕ) : NumberField (K j) := hL j
+  have hn : Tendsto (fun j => (Fintype.card (C j):ℝ)) atTop atTop := by
+    simpa only [Fintype.card_coe, Function.comp_def] using tendsto_natCast_atTop_atTop.comp (graph_card_tendsto hcard)
+  apply arithmetic_sequence_impossible (fun j => coreGraph (P j) s (C j)) (Theta s) γ c
+    (fun j => Theta s/((j:ℝ)+2)) Z₁ Z₂ h₁ h₂ hθ (hc.trans hcγ) (hγs.trans hs) hc hcγ hn
+    (graph_error_tendsto (Theta s)) (fun j => by positivity) _ hG t ht
+  intro j
+  apply div_le_div_of_nonneg_left hθ.le (by norm_num : (0:ℝ)<2)
+  have : 0 ≤ (j : ℝ) := Nat.cast_nonneg j
+  linarith
+
+theorem main (s : ℝ) (hs : 0<s) : Tendsto (fun n : ℕ => F n s) atTop (𝓝 0) :=
+  mainStatement s hs
+
+end WeakPinned
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PinnedDistances/MeasureLimits.lean b/lean/OAI/Geometry/PinnedDistances/MeasureLimits.lean
new file mode 100644
index 0000000000000000000000000000000000000000..a439dc5d77613344d0b9a34d98e739a573793534
--- /dev/null
+++ b/lean/OAI/Geometry/PinnedDistances/MeasureLimits.lean
@@ -0,0 +1,180 @@
+import OAI.Geometry.PinnedDistances.ProductLimits
+
+namespace OAI
+
+universe u1 u2 u3 u4 u5 u6
+
+section
+noncomputable section
+open MeasureTheory Filter
+open scoped BigOperators Topology BoundedContinuousFunction NNReal ENNReal
+namespace WeakPinned.Bounded
+local instance : MeasurableSpace Sphere := borel Sphere
+local instance : BorelSpace Sphere := ⟨rfl⟩
+variable {V : ℕ → Type u1} {I : ℕ → Type u2} [∀ n,Fintype (V n)] [∀ n,DecidableEq (V n)] [∀ n,Fintype (I n)]
+variable (G : ∀ n,Core.Fibers (V n) (I n)) (U₁ U₂ : ∀ n,V n → Sphere)
+  (hG : ∀ n,0<(G n).edgeCount) (hB : ∀ n e,((G n).targets e).Nonempty)
+  (hn : ∀ n,2≤Fintype.card (V n))
+
+lemma expanded_marginal_limit (π : Sphere × Sphere → Sphere) (hπ : Continuous π)
+    (β : ProbabilityMeasure Sphere) (Q : ProbabilityMeasure (Quad Sphere))
+    (R : ProbabilityMeasure (Expanded Sphere))
+    (hα : Tendsto (fun n => uniformProbability (fun x => π (U₁ n x,U₂ n x)) (by have:=hn n;omega)) atTop (𝓝 β))
+    (hQ : Tendsto (fun n => edgeProbability (G n) (U₁ n) (U₂ n) (hG n) (hB n)) atTop (𝓝 Q))
+    (hR : Tendsto (fun n => expandedProbability (G n) (U₁ n) (U₂ n) (hG n) (hB n)) atTop (𝓝 R))
+    (hd : ∀ f : C(Sphere,ℝ),Tendsto (fun n => testDiscrep (G n).fiberWeight (G n).targets
+      (fun x => f (π (U₁ n x,U₂ n x)))) atTop (𝓝 0)) :
+    (R:Measure (Expanded Sphere)).map (fun r => (r.1,(π r.2.1,π r.2.2)))=
+      (Q:Measure (Quad Sphere)).prod ((β:Measure Sphere).prod β) := by
+  apply weak_double_product_limit (fun n => expandedProbability (G n) (U₁ n) (U₂ n) (hG n) (hB n)) R
+    (fun n => edgeProbability (G n) (U₁ n) (U₂ n) (hG n) (hB n)) Q
+    (fun n => uniformProbability (fun x => π (U₁ n x,U₂ n x)) (by have:=hn n;omega)) β
+    Prod.fst (fun r => π r.2.1) (fun r => π r.2.2) continuous_fst
+    (hπ.comp (continuous_fst.comp continuous_snd)) (hπ.comp (continuous_snd.comp continuous_snd)) hR hQ hα
+  exact expanded_test_rate G U₁ U₂ hG hB hn π hd
+
+lemma incoming_limit (T : ℕ → ℝ) (hT : ∀ n,0<T n)
+    (β : ProbabilityMeasure Sphere) (Q : ProbabilityMeasure (Quad Sphere))
+    (hα : Tendsto (fun n => uniformProbability (U₁ n) (by have:=hn n;omega)) atTop (𝓝 β))
+    (hQ : Tendsto (fun n => edgeProbability (G n) (U₁ n) (U₂ n) (hG n) (hB n)) atTop (𝓝 Q))
+    (hd : ∀ f : C(Sphere,ℝ),Tendsto (fun n => testDiscrep ((G n).retainedWeight (T n))
+      ((G n).retainedSet (T n)) (fun x => f (U₁ n x))) atTop (𝓝 0))
+    (ho : Tendsto (fun n => (G n).omittedMass (T n)) atTop (𝓝 0)) :
+    (Q:Measure (Quad Sphere)).map (fun q => (q.1,q.2.2))=
+      (β:Measure Sphere).prod ((Q:Measure (Quad Sphere)).map (fun q => q.2.2)) := by
+  let Qn := fun n => edgeProbability (G n) (U₁ n) (U₂ n) (hG n) (hB n)
+  have hc : Continuous (fun q : Quad Sphere => (q.1,q.2.2)) := by fun_prop
+  have ht : Continuous (fun q : Quad Sphere => q.2.2) := by fun_prop
+  apply weak_pair_product_limit (fun n => (Qn n).map (fun q => (q.1,q.2.2)))
+    (Q.map (fun q => (q.1,q.2.2)))
+    (fun n => uniformProbability (U₁ n) (by have:=hn n;omega)) β
+    (fun n => (Qn n).map (fun q => q.2.2)) (Q.map (fun q => q.2.2))
+    (ProbabilityMeasure.tendsto_map_of_tendsto_of_continuous Qn Q hQ hc) hα
+    (ProbabilityMeasure.tendsto_map_of_tendsto_of_continuous Qn Q hQ ht)
+  intro f g
+  have hh := incoming_law_test_rate G U₁ U₂ hG hB hn T hT hd ho f g
+  convert hh using 1
+  funext n
+  change (∫ q,f q.1*g q.2 ∂ (Qn n:Measure (Quad Sphere)).map (fun q => (q.1,q.2.2)))-
+    (∫ z,f z ∂ uniformLaw (U₁ n))*(∫ p,g p ∂ (Qn n:Measure (Quad Sphere)).map (fun q => q.2.2))=_
+  rw [integral_map_of_stronglyMeasurable hc.measurable (by fun_prop),
+    integral_map_of_stronglyMeasurable ht.measurable g.continuous.stronglyMeasurable]
+
+lemma edge_ac_limit (β₁ β₂ : ProbabilityMeasure Sphere) (Q : ProbabilityMeasure (Quad Sphere))
+    (hα₁ : Tendsto (fun n => uniformProbability (U₁ n) (by have:=hn n;omega)) atTop (𝓝 β₁))
+    (hα₂ : Tendsto (fun n => uniformProbability (U₂ n) (by have:=hn n;omega)) atTop (𝓝 β₂))
+    (hQ : Tendsto (fun n => edgeProbability (G n) (U₁ n) (U₂ n) (hG n) (hB n)) atTop (𝓝 Q))
+    (M : ℝ) (hM : 0≤M) (hdom : ∀ n x y,(G n).edgeLaw x y≤M*Variance.pairProb Finset.univ x y) :
+    (Q:Measure (Quad Sphere)).map (fun q => (q.1,q.2.2.1))≪(β₁:Measure Sphere).prod β₁ ∧
+    (Q:Measure (Quad Sphere)).map (fun q => (q.2.1,q.2.2.2))≪(β₂:Measure Sphere).prod β₂ := by
+  let c : ℝ≥0 := ⟨2*M,by positivity⟩
+  have hc : (c:ℝ≥0∞)=ENNReal.ofReal (2*M) := by
+    rw [ENNReal.ofReal_eq_coe_nnreal (by positivity)]
+    rfl
+  constructor
+  · apply weak_map_domination
+      (fun n => edgeProbability (G n) (U₁ n) (U₂ n) (hG n) (hB n)) Q
+      (fun n => (uniformProbability (U₁ n) (by have:=hn n;omega)).prod
+        (uniformProbability (U₁ n) (by have:=hn n;omega))) (β₁.prod β₁)
+      (fun q => (q.1,q.2.2.1)) (by fun_prop) hQ
+      (ProbabilityMeasure.continuous_prod.tendsto (β₁,β₁) |>.comp (hα₁.prodMk_nhds hα₁)) c
+    intro n
+    rw [hc]
+    exact edgeProbability_dom₁ (G n) (U₁ n) (U₂ n) (hG n) (hB n) (hn n) M hM (hdom n)
+  · apply weak_map_domination
+      (fun n => edgeProbability (G n) (U₁ n) (U₂ n) (hG n) (hB n)) Q
+      (fun n => (uniformProbability (U₂ n) (by have:=hn n;omega)).prod
+        (uniformProbability (U₂ n) (by have:=hn n;omega))) (β₂.prod β₂)
+      (fun q => (q.2.1,q.2.2.2)) (by fun_prop) hQ
+      (ProbabilityMeasure.continuous_prod.tendsto (β₂,β₂) |>.comp (hα₂.prodMk_nhds hα₂)) c
+    intro n
+    rw [hc]
+    exact edgeProbability_dom₂ (G n) (U₁ n) (U₂ n) (hG n) (hB n) (hn n) M hM (hdom n)
+
+end WeakPinned.Bounded
+
+end
+end
+section
+noncomputable section
+open MeasureTheory Filter Set
+open scoped Topology
+namespace WeakPinned.Bounded
+
+local instance : MeasurableSpace Sphere := borel Sphere
+local instance : BorelSpace Sphere := ⟨rfl⟩
+
+lemma quadCoe_range (q : Quad Sphere)
+    (h₁ : (q.1,q.2.2.1)∈range pairCoe) (h₂ : (q.2.1,q.2.2.2)∈range pairCoe) :
+    q∈range quadCoe := by
+  obtain ⟨⟨a,c⟩,ha⟩ := h₁
+  obtain ⟨⟨b,d⟩,hb⟩ := h₂
+  exact ⟨(a,b,c,d),by ext <;> simp_all [pairCoe,quadCoe,Prod.mk.injEq]⟩
+
+lemma expandedCoe_range (r : Expanded Sphere)
+    (h₁ : (r.1,(r.2.1.1,r.2.2.1))∈range (Prod.map quadCoe pairCoe))
+    (h₂ : (r.1,(r.2.1.2,r.2.2.2))∈range (Prod.map quadCoe pairCoe)) :
+    r∈range expandedCoe := by
+  obtain ⟨⟨q,z,w⟩,hz⟩ := h₁
+  obtain ⟨⟨q',v,t⟩,hv⟩ := h₂
+  refine ⟨(q,((z,v),(w,t))),?_⟩
+  have he : quadCoe q=r.1 := congrArg Prod.fst hz
+  have hz' := congrArg Prod.snd hz
+  have hv' := congrArg Prod.snd hv
+  apply Prod.ext he
+  dsimp [expandedCoe]
+  ext <;> simp_all [pairCoe,Prod.mk.injEq]
+
+lemma map_transport_ac {A : Type u3} {B : Type u4} {X : Type u5} {Y : Type u6} [MeasurableSpace A] [MeasurableSpace B]
+    [MeasurableSpace X] [MeasurableSpace Y] (j : A → X) (k : B → Y)
+    (hj : Measurable j) (hk : MeasurableEmbedding k)
+    (f : A → B) (g : X → Y) (hf : Measurable f) (hg : Measurable g)
+    (μ : Measure A) (ν : Measure B) (hcomm : k ∘ f=g ∘ j)
+    (hm : (μ.map j).map g≪ν.map k) : μ.map f≪ν := by
+  apply map_pull_ac k hk
+  rw [Measure.map_map hk.measurable hf,hcomm,← Measure.map_map hg hj]
+  exact hm
+
+lemma sphere_quad_finite (μ ν : Measure ℂ) [IsProbabilityMeasure μ] [IsProbabilityMeasure ν]
+    (Q : Measure (Quad Sphere))
+    (h₁ : Q.map (fun q => (q.1,q.2.2.1))≪(μ.map OnePoint.some).prod (μ.map OnePoint.some))
+    (h₂ : Q.map (fun q => (q.2.1,q.2.2.2))≪(ν.map OnePoint.some).prod (ν.map OnePoint.some)) :
+    ∀ᵐ q ∂ Q,q∈range quadCoe := by
+  have hpμ := map_full_range pairCoe pairCoe_open.measurableEmbedding (μ.prod μ)
+  have hpν := map_full_range pairCoe pairCoe_open.measurableEmbedding (ν.prod ν)
+  rw [← Measure.map_prod_map _ _ measurableEmbedding_sphere_coe.measurable
+    measurableEmbedding_sphere_coe.measurable] at hpμ hpν
+  have hq₁ := (ae_map_iff (by fun_prop) pairCoe_open.measurableEmbedding.measurableSet_range).mp
+    (h₁.ae_le hpμ)
+  have hq₂ := (ae_map_iff (by fun_prop) pairCoe_open.measurableEmbedding.measurableSet_range).mp
+    (h₂.ae_le hpν)
+  exact (hq₁.and hq₂).mono (fun q h => quadCoe_range q h.1 h.2)
+
+lemma sphere_expanded_finite (μ ν : Measure ℂ) [IsProbabilityMeasure μ] [IsProbabilityMeasure ν]
+    (Q : Measure (Quad ℂ)) (R : Measure (Expanded Sphere)) [IsProbabilityMeasure Q]
+    (h₁ : R.map (fun r => (r.1,(r.2.1.1,r.2.2.1)))=
+      (Q.map quadCoe).prod ((μ.map OnePoint.some).prod (μ.map OnePoint.some)))
+    (h₂ : R.map (fun r => (r.1,(r.2.1.2,r.2.2.2)))=
+      (Q.map quadCoe).prod ((ν.map OnePoint.some).prod (ν.map OnePoint.some))) :
+    ∀ᵐ r ∂ R,r∈range expandedCoe := by
+  have hj := quadCoe_open.measurableEmbedding.prodMap pairCoe_open.measurableEmbedding
+  have hpμ := map_full_range _ hj (Q.prod (μ.prod μ))
+  have hpν := map_full_range _ hj (Q.prod (ν.prod ν))
+  rw [← Measure.map_prod_map _ _ quadCoe_open.measurableEmbedding.measurable
+    pairCoe_open.measurableEmbedding.measurable,
+    ← Measure.map_prod_map _ _ measurableEmbedding_sphere_coe.measurable
+    measurableEmbedding_sphere_coe.measurable,← h₁] at hpμ
+  rw [← Measure.map_prod_map _ _ quadCoe_open.measurableEmbedding.measurable
+    pairCoe_open.measurableEmbedding.measurable,
+    ← Measure.map_prod_map _ _ measurableEmbedding_sphere_coe.measurable
+    measurableEmbedding_sphere_coe.measurable,← h₂] at hpν
+  have hq₁ := (ae_map_iff (by fun_prop) hj.measurableSet_range).mp hpμ
+  have hq₂ := (ae_map_iff (by fun_prop) hj.measurableSet_range).mp hpν
+  exact (hq₁.and hq₂).mono (fun r h => expandedCoe_range r h.1 h.2)
+
+end WeakPinned.Bounded
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PinnedDistances/MinPotentials.lean b/lean/OAI/Geometry/PinnedDistances/MinPotentials.lean
new file mode 100644
index 0000000000000000000000000000000000000000..a2544e09ffb8278a44e2d8ee9cee0ded5dc4f71b
--- /dev/null
+++ b/lean/OAI/Geometry/PinnedDistances/MinPotentials.lean
@@ -0,0 +1,563 @@
+import OAI.Geometry.PinnedDistances.OverlapRates
+
+namespace OAI
+
+universe u1 u2 u3 u4 u5 u6 u7 u8 u9 u10
+
+section
+noncomputable section
+open NumberField
+open scoped BigOperators
+noncomputable section
+open MeasureTheory
+open scoped Topology
+noncomputable section
+open MeasureTheory
+open scoped Topology ENNReal
+noncomputable section
+open MeasureTheory NumberField
+open scoped BigOperators
+noncomputable section
+open scoped BigOperators
+open MeasureTheory
+noncomputable section
+open Filter
+open scoped Topology
+noncomputable section
+open scoped BigOperators
+noncomputable section
+open Filter
+open scoped Topology
+
+namespace WeakPinned.Variance
+open scoped BigOperators
+variable {P : Type u1} {I : Type u2} [Fintype P] [DecidableEq P] [Fintype I]
+
+/-- The giant-complement second moment estimate needed to compare W and W₂.
+The linear error, unlike an absolute singleton error, can be integrated. -/
+lemma cell_second_moment (w : I → ℝ) (B : I → Finset P) (T M κ : ℝ)
+    (hw : ∀ i, 0≤w i) (hW : (∑ i, w i)≤1) (hT : 1≤T) (hκ : 0≤κ)
+    (hk : ∀ i, 2/((B i).card:ℝ)≤κ)
+    (hdom : ∀ x, (∑ i, w i*prob (B i) x)≤M*prob Finset.univ x) (C : Finset P) :
+    (∑ i, w i*(cellProb (B i) C)^2) ≤
+      2*varianceCell w B T C + 4*pairCell Finset.univ C + κ*M*cellProb Finset.univ C := by
+  have hmean := mean_cell_bound w B C M hdom
+  have hm : 0≤M*cellProb Finset.univ C := (Finset.sum_nonneg (fun i _ => mul_nonneg (hw i) (cellProb_nonneg _ _))).trans hmean
+  unfold varianceCell
+  split_ifs with hlarge
+  · have hmC : 2≤C.card := by have : (1:ℝ)<C.card := hT.trans_lt hlarge; exact_mod_cast this
+    have hcomp := (uniform_pair_comparison C hmC).1
+    have hb : (∑ i, w i*(cellProb Finset.univ C)^2) ≤ (cellProb Finset.univ C)^2 := by
+      rw [← Finset.sum_mul]
+      nlinarith [sq_nonneg (cellProb Finset.univ C)]
+    have hh : (∑ i, w i*(cellProb (B i) C)^2) ≤
+        2*(∑ i, w i*(cellProb (B i) C-cellProb Finset.univ C)^2) +
+        2*(∑ i, w i*(cellProb Finset.univ C)^2) := by
+      simp only [Finset.mul_sum,← Finset.sum_add_distrib]
+      apply Finset.sum_le_sum
+      intro i _
+      have h := mul_le_mul_of_nonneg_left (show (cellProb (B i) C)^2 ≤
+        2*(cellProb (B i) C-cellProb Finset.univ C)^2+2*(cellProb Finset.univ C)^2 by
+          nlinarith [sq_nonneg (cellProb (B i) C-2*cellProb Finset.univ C)]) (hw i)
+      nlinarith
+    nlinarith [mul_nonneg hκ hm]
+  · have hh : (∑ i, w i*(cellProb (B i) C)^2) ≤
+        (∑ i, w i*(if 2/((B i).card:ℝ)≤cellProb (B i) C then (cellProb (B i) C)^2 else 0)) +
+        κ*(∑ i, w i*cellProb (B i) C) := by
+      simp only [Finset.mul_sum,← Finset.sum_add_distrib]
+      apply Finset.sum_le_sum
+      intro i _
+      split_ifs with hi
+      · nlinarith [mul_nonneg hκ (mul_nonneg (hw i) (cellProb_nonneg (B i) C))]
+      · have hp : cellProb (B i) C≤κ := (le_of_lt (lt_of_not_ge hi)).trans (hk i)
+        have h := mul_le_mul_of_nonneg_left hp (mul_nonneg (hw i) (cellProb_nonneg (B i) C))
+        nlinarith
+    have hκmean := mul_le_mul_of_nonneg_left hmean hκ
+    nlinarith [pairCell_nonneg Finset.univ C]
+
+/-- Small positive-mass cut contributions are slice contributions or variance. -/
+lemma small_mass_qCell {w : I → ℝ} {B : I → Finset P} (src : I → P) (T δ : ℝ)
+    (hw : ∀ i, 0≤w i) (hδ : 0<δ) (hk : ∀ i, 2/((B i).card:ℝ)≤2*δ)
+    (C : Finset P) (hC : cellProb Finset.univ C≤δ) :
+    (∑ i, w i*(if src i ∈ C then cellProb (B i) C else 0)) ≤
+      (∑ i, w i*(if src i ∈ C ∧ cellProb (B i) C<2*δ then cellProb (B i) C else 0)) +
+      (2/δ)*varianceCell w B T C := by
+  let err := fun i => if T<C.card then (cellProb (B i) C-cellProb Finset.univ C)^2
+    else (1/2)*(if 2/((B i).card:ℝ)≤cellProb (B i) C then (cellProb (B i) C)^2 else 0)
+  have he : varianceCell w B T C=∑ i, w i*err i := by
+    unfold varianceCell err
+    split_ifs <;> simp only [Finset.mul_sum]
+    apply Finset.sum_congr rfl
+    intro i _
+    ring
+  have hnon (i) : 0≤err i := by dsimp [err]; split_ifs <;> positivity
+  have hpoint (i) : (if src i ∈ C then cellProb (B i) C else 0) ≤
+      (if src i ∈ C ∧ cellProb (B i) C<2*δ then cellProb (B i) C else 0)+(2/δ)*err i := by
+    by_cases hs : src i ∈ C
+    · by_cases hsmall : cellProb (B i) C<2*δ
+      · simp only [hs,hsmall,true_and,ite_true]
+        exact le_add_of_nonneg_right (mul_nonneg (by positivity) (hnon i))
+      · have hp2 : 2*δ≤cellProb (B i) C := le_of_not_gt hsmall
+        simp only [hs,hsmall,true_and,ite_true,ite_false,zero_add]
+        rw [show (2/δ)*err i=2*err i/δ by ring]
+        apply (le_div_iff₀ hδ).mpr
+        dsimp [err]
+        split_ifs with hl hp
+        · have hd : δ≤cellProb (B i) C-cellProb Finset.univ C := by linarith
+          have hp := cellProb_nonneg (B i) C
+          nlinarith [mul_nonneg (by linarith : 0≤cellProb (B i) C-2*cellProb Finset.univ C)
+            (by linarith : 0≤cellProb (B i) C-cellProb Finset.univ C),
+            mul_nonneg hp (sub_nonneg.mpr hd)]
+        · nlinarith [mul_nonneg (cellProb_nonneg (B i) C)
+            (by linarith : 0≤cellProb (B i) C-δ)]
+        · exact False.elim (hp ((hk i).trans hp2))
+    · simp only [hs,false_and,ite_false,zero_add]
+      exact mul_nonneg (by positivity) (hnon i)
+  rw [he,Finset.mul_sum,← Finset.sum_add_distrib]
+  apply Finset.sum_le_sum
+  intro i _
+  have hh := mul_le_mul_of_nonneg_left (hpoint i) (hw i)
+  convert hh using 1; ring
+
+end WeakPinned.Variance
+
+namespace WeakPinned.Hierarchy.Depth
+open MeasureTheory
+open scoped BigOperators
+variable {P : Type u3} [Fintype P] [DecidableEq P] (d : Depth P)
+
+lemma mem_giant_iff_cell (r : ℝ) (x : P) :
+    x ∈ d.giant r ↔ Fintype.card P < 2*(d.cell r x).card := by
+  constructor
+  · intro hx
+    obtain ⟨C,hC,hx⟩ := Finset.mem_biUnion.mp hx
+    have hc := (d.mem_giants r C).mp hC
+    simpa only [d.cell_of_mem hc.1 hx] using hc.2
+  · intro hx
+    have hc : d.cell r x ∈ d.giants r := (d.mem_giants r _).mpr ⟨d.cell_mem_cells _ _,hx⟩
+    rw [d.giant_eq hc]
+    exact d.self_mem_cell r x
+
+lemma mem_giant_exists (r : ℝ) (x : P) :
+    x ∈ d.giant r ↔ ∃ C : Finset P, Fintype.card P < 2*C.card ∧ ∀ y ∈ C, d.share r x y := by
+  rw [d.mem_giant_iff_cell]
+  constructor
+  · intro hc
+    exact ⟨d.cell r x,hc,fun y hy => (d.mem_cell r x y).mp hy⟩
+  · rintro ⟨C,hc,hs⟩
+    have hsub : C ⊆ d.cell r x := fun y hy => (d.mem_cell r x y).mpr (hs y hy)
+    have := Finset.card_le_card hsub
+    omega
+
+lemma mem_giant_mono {r t : ℝ} (htr : t ≤ r) {x : P} (hx : x ∈ d.giant r) : x ∈ d.giant t := by
+  rw [d.mem_giant_iff_cell] at hx ⊢
+  have := Finset.card_le_card (d.cell_mono htr x)
+  omega
+
+lemma isClosed_mem_giant (x : P) : IsClosed {r : ℝ | x ∈ d.giant r} := by
+  simp_rw [d.mem_giant_exists,Set.ofPred_exists]
+  apply isClosed_iUnion_of_finite
+  intro C
+  simp only [Set.ofPred_and]
+  apply IsClosed.inter
+  · by_cases h : Fintype.card P < 2*C.card <;> simp [h]
+  · simp_rw [Set.ofPred_forall]
+    apply isClosed_iInter
+    intro y
+    apply isClosed_iInter
+    intro hy
+    by_cases hxy : x=y
+    · simp [share,hxy]
+    · simp only [share,hxy,false_or]
+      exact isClosed_le continuous_id continuous_const
+
+lemma mem_giant_low (hn : 2 ≤ Fintype.card P) (x : P) : x ∈ d.giant (-d.radius) := by
+  rw [d.mem_giant_iff_cell,d.cell_of_low le_rfl,Finset.card_univ]
+  omega
+
+lemma mem_giant_le_M (hn : 2 ≤ Fintype.card P) {r : ℝ} {x : P} (hx : x ∈ d.giant r) : r ≤ d.M := by
+  obtain ⟨C,hC,_⟩ := Finset.mem_biUnion.mp hx
+  exact (d.giants_cutoff hn r).mp ⟨C,hC⟩
+
+/-- The last level at which the vertex belongs to the nested giant. -/
+def departure (x : P) : ℝ := sSup {r : ℝ | x ∈ d.giant r}
+
+lemma departure_cutoff (hn : 2 ≤ Fintype.card P) (x : P) (r : ℝ) :
+    x ∈ d.giant r ↔ r ≤ d.departure x := by
+  have hne : Set.Nonempty {t : ℝ | x ∈ d.giant t} := ⟨-d.radius,d.mem_giant_low hn x⟩
+  have hbd : BddAbove {t : ℝ | x ∈ d.giant t} := ⟨d.M,fun t ht => d.mem_giant_le_M hn ht⟩
+  exact ⟨fun hx => le_csSup hbd hx,fun hr => d.mem_giant_mono hr ((d.isClosed_mem_giant x).csSup_mem hne hbd)⟩
+
+lemma departure_le_M (hn : 2 ≤ Fintype.card P) (x : P) : d.departure x ≤ d.M :=
+  d.mem_giant_le_M hn ((d.departure_cutoff hn x _).mpr le_rfl)
+
+lemma outside_departure (hn : 2 ≤ Fintype.card P) (x : P) :
+    (fun r => d.outsideIndicator r x) = (Set.Ioc (d.departure x) d.M).indicator (fun _ => (1:ℝ)) := by
+  funext r
+  simp only [outsideIndicator,d.giants_cutoff hn,d.departure_cutoff hn,not_le,Set.indicator,Set.mem_Ioc]
+  congr 1
+  exact propext and_comm
+
+lemma integral_one_Ioc {a b : ℝ} (hab : a ≤ b) :
+    (∫ r : ℝ, (Set.Ioc a b).indicator (fun _ => (1:ℝ)) r)=b-a := by
+  rw [integral_indicator measurableSet_Ioc,integral_const]
+  simp [Measure.real,Real.volume_Ioc,ENNReal.toReal_ofReal (sub_nonneg.mpr hab)]
+
+lemma potential_departure (hn : 2 ≤ Fintype.card P) (x : P) : d.potential x = d.M-d.departure x := by
+  unfold potential
+  rw [d.outside_departure hn x,integral_one_Ioc (d.departure_le_M hn x)]
+
+lemma outside_product (hn : 2 ≤ Fintype.card P) (x y : P) :
+    (fun r => d.outsideIndicator r x*d.outsideIndicator r y) =
+      (Set.Ioc (max (d.departure x) (d.departure y)) d.M).indicator (fun _ => (1:ℝ)) := by
+  funext r
+  rw [congrFun (d.outside_departure hn x) r,congrFun (d.outside_departure hn y) r]
+  simp only [Set.indicator,Set.mem_Ioc,max_lt_iff]
+  split_ifs <;> simp_all
+
+/-- Exact layer-cake identity, including the finite set of threshold ties. -/
+lemma min_potential (hn : 2 ≤ Fintype.card P) (x y : P) :
+    min (d.potential x) (d.potential y) = ∫ r, d.outsideIndicator r x*d.outsideIndicator r y := by
+  rw [d.outside_product hn x y,integral_one_Ioc (max_le (d.departure_le_M hn x) (d.departure_le_M hn y)),
+    d.potential_departure hn x,d.potential_departure hn y]
+  exact min_sub_sub_left d.M (d.departure x) (d.departure y)
+
+end WeakPinned.Hierarchy.Depth
+
+noncomputable section
+open MeasureTheory
+open scoped BigOperators
+namespace WeakPinned.MinKernel
+variable {P : Type u4} [Fintype P]
+
+def step (a : ℝ) : ℝ → ℝ := (Set.Ioc 0 a).indicator (fun _ => 1)
+
+lemma integrable_step (a : ℝ) : Integrable (step a) := by
+  rw [step,integrable_indicator_iff measurableSet_Ioc]
+  exact integrableOn_const (by rw [Real.volume_Ioc]; exact ENNReal.ofReal_ne_top)
+
+lemma step_mul (a b : ℝ) : (fun r => step a r*step b r)=step (min a b) := by
+  funext r
+  simp only [step,Set.indicator,Set.mem_Ioc,le_min_iff]
+  split_ifs <;> simp_all
+
+lemma integral_step (a : ℝ) (ha : 0≤a) : (∫ r, step a r)=a := by
+  rw [step,integral_indicator measurableSet_Ioc]
+  simp [integral_const,Measure.real,Real.volume_Ioc,ENNReal.toReal_ofReal ha]
+
+def tail (p a : P → ℝ) (r : ℝ) : ℝ := ∑ x, p x*step (a x) r
+
+def energy (p a b : P → ℝ) : ℝ := ∑ x, ∑ y, p x*p y*min (a x) (b y)
+
+lemma tail_product (p a b : P → ℝ) :
+    (fun r => tail p a r*tail p b r) = fun r => ∑ x, ∑ y, (p x*p y)*step (min (a x) (b y)) r := by
+  funext r
+  simp only [tail]
+  rw [Finset.sum_mul]
+  simp only [Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro x _
+  apply Finset.sum_congr rfl
+  intro y _
+  rw [← congrFun (step_mul (a x) (b y)) r]
+  ring
+
+lemma integrable_tail_product (p a b : P → ℝ) : Integrable (fun r => tail p a r*tail p b r) := by
+  rw [tail_product]
+  exact integrable_finsetSum _ fun x _ => integrable_finsetSum _ fun y _ =>
+    (integrable_step _).const_mul _
+
+lemma energy_integral (p a b : P → ℝ) (ha : ∀ x, 0≤a x) (hb : ∀ x, 0≤b x) :
+    energy p a b = ∫ r, tail p a r*tail p b r := by
+  rw [tail_product,integral_finsetSum _ (fun x _ => integrable_finsetSum _ fun y _ => (integrable_step _).const_mul _)]
+  simp_rw [integral_finsetSum _ (fun y _ => (integrable_step _).const_mul _),integral_const_mul]
+  simp only [integral_step _ (le_min (ha _) (hb _)),energy]
+
+lemma energy_cross (p a b : P → ℝ) (ha : ∀ x, 0≤a x) (hb : ∀ x, 0≤b x) :
+    2*energy p a b ≤ energy p a a+energy p b b := by
+  rw [energy_integral p a b ha hb,energy_integral p a a ha ha,energy_integral p b b hb hb]
+  rw [← integral_const_mul,← integral_add (integrable_tail_product p a a) (integrable_tail_product p b b)]
+  apply integral_mono ((integrable_tail_product p a b).const_mul 2)
+    ((integrable_tail_product p a a).add (integrable_tail_product p b b))
+  intro r
+  simp only [Pi.add_apply]
+  nlinarith [sq_nonneg (tail p a r-tail p b r)]
+
+lemma min_add_four (a b c d : ℝ) (ha : 0≤a) (hb : 0≤b) (hc : 0≤c) (hd : 0≤d) :
+    min (a+b) (c+d) ≤ min a c+min a d+min b c+min b d := by
+  simp only [min_def]
+  split_ifs <;> linarith
+
+lemma energy_sum (p a b : P → ℝ) (hp : ∀ x, 0≤p x) (ha : ∀ x, 0≤a x) (hb : ∀ x, 0≤b x) :
+    energy p (a+b) (a+b) ≤ 2*(energy p a a+energy p b b) := by
+  have hh : energy p (a+b) (a+b) ≤ energy p a a+energy p a b+energy p b a+energy p b b := by
+    unfold energy
+    simp only [← Finset.sum_add_distrib]
+    apply Finset.sum_le_sum
+    intro x _
+    apply Finset.sum_le_sum
+    intro y _
+    have h := mul_le_mul_of_nonneg_left (min_add_four (a x) (b x) (a y) (b y) (ha x) (hb x) (ha y) (hb y))
+      (mul_nonneg (hp x) (hp y))
+    simpa only [Pi.add_apply,mul_add] using h
+  nlinarith [energy_cross p a b ha hb,energy_cross p b a hb ha]
+end WeakPinned.MinKernel
+
+namespace WeakPinned.Hierarchy.Depth
+open MeasureTheory
+open scoped BigOperators
+variable {P : Type u5} [Fintype P] [DecidableEq P]
+variable (d : Depth P)
+
+def outsideMass (p : P → ℝ) (r : ℝ) : ℝ := ∑ x, p x*d.outsideIndicator r x
+
+lemma integrable_outside_product (hn : 2 ≤ Fintype.card P) (x y : P) :
+    Integrable (fun r => d.outsideIndicator r x*d.outsideIndicator r y) := by
+  rw [d.outside_product hn x y,integrable_indicator_iff measurableSet_Ioc]
+  exact integrableOn_const (by rw [Real.volume_Ioc]; exact ENNReal.ofReal_ne_top)
+
+lemma outsideMass_square (p : P → ℝ) (r : ℝ) :
+    (d.outsideMass p r)^2 = ∑ x, ∑ y, p x*p y*(d.outsideIndicator r x*d.outsideIndicator r y) := by
+  unfold outsideMass
+  rw [pow_two,Finset.sum_mul]
+  simp only [Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro x _
+  apply Finset.sum_congr rfl
+  intro y _
+  ring
+
+lemma iid_min_potential (hn : 2 ≤ Fintype.card P) (p : P → ℝ) :
+    MinKernel.energy p d.potential d.potential = ∫ r, (d.outsideMass p r)^2 := by
+  simp only [d.outsideMass_square]
+  rw [integral_finsetSum _ (fun x _ => integrable_finsetSum _ fun y _ => (d.integrable_outside_product hn x y).const_mul _)]
+  simp_rw [integral_finsetSum _ (fun y _ => (d.integrable_outside_product hn _ y).const_mul _),integral_const_mul]
+  simp only [MinKernel.energy,d.min_potential hn]
+
+/-- The source min-potential upper bound before averaging places and grids. -/
+
+theorem iid_min_two_potentials (d' : Depth P) (hn : 2 ≤ Fintype.card P)
+    (p : P → ℝ) (hp : ∀ x, 0≤p x) :
+    (∑ x, ∑ y, p x*p y*min (d.potential x+d'.potential x) (d.potential y+d'.potential y)) ≤
+      2*((∫ r, (d.outsideMass p r)^2)+(∫ r, (d'.outsideMass p r)^2)) := by
+  rw [← d.iid_min_potential hn p,← d'.iid_min_potential hn p]
+  exact MinKernel.energy_sum p d.potential d'.potential hp d.potential_nonneg d'.potential_nonneg
+end WeakPinned.Hierarchy.Depth
+
+namespace WeakPinned.Hierarchy.Depth
+open MeasureTheory Variance
+open scoped BigOperators
+variable {P : Type u6} {I : Type u7} [Fintype P] [DecidableEq P] [Fintype I]
+variable (d : Depth P)
+
+lemma outsideMass_prob (B : Finset P) {r : ℝ} {G : Finset P} (hG : G∈d.giants r) :
+    d.outsideMass (prob B) r=cellProb B Gᶜ := by
+  have hne : (d.giants r).Nonempty := ⟨G,hG⟩
+  have he (x) : prob B x*d.outsideIndicator r x=if x∈Gᶜ then prob B x else 0 := by
+    simp only [outsideIndicator,hne,d.giant_eq hG,true_and]
+    by_cases hx : x∈G <;> simp [hx]
+  simp only [outsideMass,he,Finset.sum_ite_mem,Finset.univ_inter,sum_prob_cell]
+
+lemma outsideMass_zero (p : P → ℝ) {r : ℝ} (hG : ¬(d.giants r).Nonempty) : d.outsideMass p r=0 := by
+  simp [outsideMass,outsideIndicator,hG]
+
+lemma sumCells_nonneg (f : Finset P → ℝ) (hf : ∀ C, 0≤f C) (r : ℝ) : 0≤d.sumCells f r := by
+  rw [d.sumCells_count]
+  exact Finset.sum_nonneg (fun C _ => mul_nonneg (by unfold count; positivity) (hf C))
+
+lemma complement_le_sumCells (f : Finset P → ℝ) (hf : ∀ C, 0≤f C) {r : ℝ} {G : Finset P}
+    (hG : G∈d.giants r) : f Gᶜ≤d.sumCells f r := by
+  have hs : d.giants r={G} := Finset.ext fun C => by
+    simp only [Finset.mem_singleton]
+    exact ⟨fun hC => d.giants_unique hC hG,fun h => h ▸ hG⟩
+  rw [d.sumCells_split,hs,Finset.sum_singleton]
+  exact le_add_of_nonneg_left (Finset.sum_nonneg (fun C _ => by split_ifs <;> simp_all only [le_refl]))
+
+lemma outside_square_bound (w : I → ℝ) (B : I → Finset P) (T M κ : ℝ)
+    (hw : ∀ i, 0≤w i) (hW : (∑ i, w i)≤1) (hT : 1≤T) (hκ : 0≤κ)
+    (hk : ∀ i, 2/((B i).card:ℝ)≤κ)
+    (hdom : ∀ x, (∑ i, w i*prob (B i) x)≤M*prob Finset.univ x) (r : ℝ) :
+    (∑ i, w i*(d.outsideMass (prob (B i)) r)^2) ≤
+      d.sumCells (fun C => 2*varianceCell w B T C+4*pairCell Finset.univ C) r +
+      κ*M*d.outsideMass (prob Finset.univ) r := by
+  have hf (C : Finset P) : 0≤2*varianceCell w B T C+4*pairCell Finset.univ C :=
+    add_nonneg (mul_nonneg (by norm_num) (varianceCell_nonneg w B T hw C))
+      (mul_nonneg (by norm_num) (pairCell_nonneg _ _))
+  by_cases hg : (d.giants r).Nonempty
+  · obtain ⟨G,hG⟩ := hg
+    simp_rw [d.outsideMass_prob _ hG]
+    exact (cell_second_moment w B T M κ hw hW hT hκ hk hdom Gᶜ).trans
+      (add_le_add (d.complement_le_sumCells _ hf hG) le_rfl)
+  · simp only [d.outsideMass_zero _ hg,zero_pow (by norm_num : 2≠0),mul_zero,Finset.sum_const_zero,add_zero]
+    exact d.sumCells_nonneg _ hf r
+
+lemma integrable_outsideMass_square (hn : 2≤Fintype.card P) (p : P → ℝ) :
+    Integrable (fun r => (d.outsideMass p r)^2) := by
+  simp only [d.outsideMass_square]
+  exact integrable_finsetSum _ fun x _ => integrable_finsetSum _ fun y _ =>
+    (d.integrable_outside_product hn x y).const_mul _
+
+lemma integral_outsideMass (hn : 2≤Fintype.card P) (p : P → ℝ) :
+    (∫ r, d.outsideMass p r)=∑ x, p x*d.potential x := by
+  simp only [outsideMass]
+  rw [integral_finsetSum _ (fun x _ => (d.integrable_outsideIndicator hn x).const_mul _)]
+  simp only [integral_const_mul,potential]
+
+/-- The integrable complement second moment estimate. All singleton terms
+are charged to the finite potential, not integrated as constant errors. -/
+theorem integrated_outside_square (hn : 2≤Fintype.card P)
+    (w : I → ℝ) (B : I → Finset P) (T M κ : ℝ)
+    (hw : ∀ i, 0≤w i) (hW : (∑ i, w i)≤1) (hT : 1≤T) (hκ : 0≤κ)
+    (hB : ∀ i, 2≤(B i).card) (hk : ∀ i, 2/((B i).card:ℝ)≤κ)
+    (hdom : ∀ x, (∑ i, w i*prob (B i) x)≤M*prob Finset.univ x) :
+    (∑ i, w i*(∫ r, (d.outsideMass (prob (B i)) r)^2)) ≤
+      2*weightedIntegral d.mass (varianceCell w B T) +
+      4*weightedIntegral d.mass (pairCell Finset.univ) +
+      κ*M*(∑ x, prob Finset.univ x*d.potential x) := by
+  let f := fun C => 2*varianceCell w B T C+4*pairCell Finset.univ C
+  have hf : ∀ C, C.card≤1 → f C=0 := by
+    intro C hC
+    simp [f,varianceCell_singleton w B T hT hB C hC,singleton_pairCell Finset.univ C hC]
+  have hi₁ : Integrable (fun r => ∑ i, w i*(d.outsideMass (prob (B i)) r)^2) :=
+    integrable_finsetSum _ (fun i _ => (d.integrable_outsideMass_square hn _).const_mul _)
+  have hi₂ : Integrable (fun r => d.sumCells f r) := d.integrable_sumCells hn f hf
+  have hi₃ : Integrable (fun r => κ*M*d.outsideMass (prob Finset.univ) r) :=
+    (integrable_finsetSum _ (fun x _ => (d.integrable_outsideIndicator hn x).const_mul _)).const_mul _
+  have hh := integral_mono hi₁ (hi₂.add hi₃) (d.outside_square_bound w B T M κ hw hW hT hκ hk hdom)
+  simp only [Pi.add_apply] at hh
+  rw [integral_finsetSum _ (fun i _ => (d.integrable_outsideMass_square hn _).const_mul _),
+    integral_add hi₂ hi₃,integral_const_mul,d.integral_outsideMass hn _,d.integral_sumCells hn f hf] at hh
+  try simp only [integral_const_mul] at hh
+  have he : (∑ C, d.mass C*f C)=
+      2*weightedIntegral d.mass (varianceCell w B T)+4*weightedIntegral d.mass (pairCell Finset.univ) := by
+    simp only [weightedIntegral,f,mul_add,Finset.sum_add_distrib,Finset.mul_sum]
+    apply congrArg₂ (·+·)
+    · apply Finset.sum_congr rfl; intro C _; ring
+    · apply Finset.sum_congr rfl; intro C _; ring
+  rwa [he] at hh
+
+end WeakPinned.Hierarchy.Depth
+
+namespace WeakPinned.MinKernel
+open scoped BigOperators
+variable {P : Type u8} [Fintype P]
+
+/-- A fiber whose potential is a constant plus nonnegative overlap and a
+small signed grid error has a large iid minimum. This preserves self-pairs
+in the sampling, exactly as in the unbounded-scale argument. -/
+lemma energy_lower (p S L E : P → ℝ) (c : ℝ)
+    (hp : ∀ x, 0≤p x) (hsum : (∑ x, p x)=1)
+    (hL : ∀ x, 0≤L x)
+    (hS : ∀ x, p x≠0 → S x=c+L x-E x) :
+    (∑ x, p x*S x)-(∑ x, p x*L x)-(∑ x, p x*|E x|)≤energy p S S := by
+  have hpair (x y : P) :
+      p x*p y*((S x+S y)/2-(L x+L y+|E x|+|E y|)/2) ≤
+        p x*p y*min (S x) (S y) := by
+    by_cases hx : p x=0
+    · simp [hx]
+    by_cases hy : p y=0
+    · simp [hy]
+    apply mul_le_mul_of_nonneg_left _ (mul_nonneg (hp x) (hp y))
+    have hxS := hS x hx
+    have hyS := hS y hy
+    apply le_min <;> linarith [hL x,hL y,le_abs_self (E x),le_abs_self (E y),neg_abs_le (E x),neg_abs_le (E y)]
+  have hh := Finset.sum_le_sum (s:=Finset.univ) (fun x _ =>
+    Finset.sum_le_sum (s:=Finset.univ) (fun y _ => hpair x y))
+  change _≤energy p S S at hh
+  have he : (∑ x, ∑ y, p x*p y*((S x+S y)/2-(L x+L y+|E x|+|E y|)/2))=
+      (∑ x, p x*S x)-(∑ x, p x*L x)-(∑ x, p x*|E x|) := by
+    have hform (x y : P) : p x*p y*((S x+S y)/2-(L x+L y+|E x|+|E y|)/2)=
+        (p x*S x)*p y/2+p x*(p y*S y)/2-
+        ((p x*L x)*p y/2+p x*(p y*L y)/2+(p x*|E x|)*p y/2+p x*(p y*|E y|)/2) := by ring
+    simp_rw [hform,Finset.sum_sub_distrib,Finset.sum_add_distrib,← Finset.sum_div,
+      ← Finset.mul_sum,← Finset.sum_mul,hsum,mul_one,one_mul]
+    ring
+  rwa [he] at hh
+end WeakPinned.MinKernel
+
+namespace WeakPinned.Hierarchy.Depth
+open MeasureTheory Variance
+open scoped BigOperators
+variable {P : Type u9} {I : Type u10} [Fintype P] [DecidableEq P] [Fintype I]
+
+lemma min_upper_family (d d' : Depth P) (hn : 2≤Fintype.card P)
+    (w : I → ℝ) (B : I → Finset P) (T M κ : ℝ)
+    (hw : ∀ i, 0≤w i) (hW : (∑ i, w i)≤1) (hT : 1≤T) (hκ : 0≤κ)
+    (hB : ∀ i, 2≤(B i).card) (hk : ∀ i, 2/((B i).card:ℝ)≤κ)
+    (hdom : ∀ x, (∑ i, w i*prob (B i) x)≤M*prob Finset.univ x) :
+    (∑ i, w i*MinKernel.energy (prob (B i)) (d.potential+d'.potential) (d.potential+d'.potential)) ≤
+      4*(weightedIntegral d.mass (varianceCell w B T)+weightedIntegral d'.mass (varianceCell w B T))+
+      8*(weightedIntegral d.mass (pairCell Finset.univ)+weightedIntegral d'.mass (pairCell Finset.univ))+
+      2*κ*M*(∑ x, prob Finset.univ x*(d.potential x+d'.potential x)) := by
+  have hm (i) := d.iid_min_two_potentials d' hn (prob (B i))
+    (fun x => by unfold prob; split_ifs <;> positivity)
+  have hsum := Finset.sum_le_sum (s:=Finset.univ) (fun i _ => mul_le_mul_of_nonneg_left (hm i) (hw i))
+  change (∑ i, w i*MinKernel.energy (prob (B i)) (d.potential+d'.potential) (d.potential+d'.potential))≤_ at hsum
+  have h₁ := d.integrated_outside_square hn w B T M κ hw hW hT hκ hB hk hdom
+  have h₂ := d'.integrated_outside_square hn w B T M κ hw hW hT hκ hB hk hdom
+  have hr : (∑ i, w i*(2*((∫ r, (d.outsideMass (prob (B i)) r)^2)+
+      (∫ r, (d'.outsideMass (prob (B i)) r)^2))))=
+      2*((∑ i, w i*(∫ r, (d.outsideMass (prob (B i)) r)^2))+
+      (∑ i, w i*(∫ r, (d'.outsideMass (prob (B i)) r)^2))) := by
+    simp only [Finset.mul_sum,← Finset.sum_add_distrib]
+    apply Finset.sum_congr rfl; intro i _; ring
+  rw [hr] at hsum
+  simp only [mul_add,Finset.sum_add_distrib]
+  linarith
+
+lemma local_potential_fiber_bound (d d' : Depth P) (hn : 2≤Fintype.card P)
+    (w : I → ℝ) (B : I → Finset P) (src : I → P) (t : I → ℝ) (T M κ : ℝ)
+    (hw : ∀ i, 0≤w i) (hW : (∑ i, w i)≤1) (hT : 1≤T) (hκ : 0≤κ)
+    (hB : ∀ i, 2≤(B i).card) (hk : ∀ i, 2/((B i).card:ℝ)≤κ)
+    (hsrc : ∀ i, src i∉B i)
+    (hdom : ∀ x, (∑ i, w i*prob (B i) x)≤M*prob Finset.univ x) :
+    (∑ i, w i*∑ y, prob (B i) y*(d.potential y+d'.potential y)) ≤
+      4*(weightedIntegral d.mass (varianceCell w B T)+weightedIntegral d'.mass (varianceCell w B T))+
+      8*(weightedIntegral d.mass (pairCell Finset.univ)+weightedIntegral d'.mass (pairCell Finset.univ))+
+      2*κ*M*(∑ x, prob Finset.univ x*(d.potential x+d'.potential x))+
+      (∑ i, w i*∑ y, prob (B i) y*(weightedOverlap d.mass (src i) y+weightedOverlap d'.mass (src i) y))+
+      (∑ i, w i*∑ y, prob (B i) y*|d.value (src i) y+d'.value (src i) y-t i|) := by
+  have hp (i : I) (y : P) : 0≤prob (B i) y := by unfold prob; split_ifs <;> positivity
+  have hL (dd : Depth P) (x y : P) : 0≤weightedOverlap dd.mass x y := by
+    apply Finset.sum_nonneg; intro C _
+    exact mul_nonneg (dd.mass_nonneg C) (by dsimp; split_ifs <;> norm_num)
+  have hs (i : I) (y : P) (hy : prob (B i) y≠0) : y∈B i := by
+    by_contra hh; exact hy (by simp [prob,hh])
+  have hv (dd : Depth P) (i : I) (y : P) (hy : prob (B i) y≠0) :
+      weightedOverlap dd.mass (src i) y=dd.overlapLength (src i) y (fun he => hsrc i (he ▸ hs i y hy)) :=
+    dd.mass_overlap hn _ _ _
+  have he (i : I) := MinKernel.energy_lower (prob (B i))
+    (d.potential+d'.potential)
+    (fun y => weightedOverlap d.mass (src i) y+weightedOverlap d'.mass (src i) y)
+    (fun y => d.value (src i) y+d'.value (src i) y-t i)
+    (d.M+d'.M-d.potential (src i)-d'.potential (src i)-t i)
+    (hp i) (sum_prob (B i) (Finset.card_pos.mp (by have := hB i; omega)))
+    (fun y => add_nonneg (hL d _ _) (hL d' _ _)) (by
+      intro y hy
+      have hxy : src i≠y := fun h => hsrc i (h ▸ hs i y hy)
+      have hx := d.depth_identity hn (src i) y hxy
+      have hx' := d'.depth_identity hn (src i) y hxy
+      rw [hv d i y hy,hv d' i y hy]
+      simp only [Pi.add_apply]
+      linarith)
+  have hh := Finset.sum_le_sum (s:=Finset.univ) (fun i _ => mul_le_mul_of_nonneg_left (he i) (hw i))
+  have hu := min_upper_family d d' hn w B T M κ hw hW hT hκ hB hk hdom
+  simp only [Pi.add_apply,mul_sub,Finset.sum_sub_distrib] at hh
+  linarith
+
+end WeakPinned.Hierarchy.Depth
+noncomputable section
+
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PinnedDistances/MixtureLaw.lean b/lean/OAI/Geometry/PinnedDistances/MixtureLaw.lean
new file mode 100644
index 0000000000000000000000000000000000000000..78acca028b3757c9257fbe72f60301730421af97
--- /dev/null
+++ b/lean/OAI/Geometry/PinnedDistances/MixtureLaw.lean
@@ -0,0 +1,110 @@
+import OAI.Geometry.PinnedDistances.EnergyLaw
+
+namespace OAI
+
+universe u1 u2 u3
+
+section
+noncomputable section
+open MeasureTheory Filter
+open scoped Topology BigOperators
+namespace WeakPinned.Bounded
+
+variable {I : Type u1} {V : Type u2} {X : Type u3} [Fintype I] [Fintype V]
+
+def edgeWeight (w : I → ℝ) (p : I → V → ℝ) (r : I × V) : ℝ := w r.1*p r.1 r.2
+
+def tripleWeight (w : I → ℝ) (p : I → V → ℝ) (r : I × V × V × V) : ℝ :=
+  w r.1*p r.1 r.2.1*p r.1 r.2.2.1*p r.1 r.2.2.2
+
+lemma sum_edgeWeight (w : I → ℝ) (p : I → V → ℝ)
+    (hw : ∑ i,w i=1) (hp : ∀ i,∑ x,p i x=1) : ∑ r,edgeWeight w p r=1 := by
+  simp only [edgeWeight,Fintype.sum_prod_type,← Finset.mul_sum,hp,mul_one,hw]
+
+lemma sum_tripleWeight (w : I → ℝ) (p : I → V → ℝ)
+    (hw : ∑ i,w i=1) (hp : ∀ i,∑ x,p i x=1) : ∑ r,tripleWeight w p r=1 := by
+  simp only [tripleWeight,Fintype.sum_prod_type,← Finset.mul_sum,hp,mul_one,hw]
+
+def edgeValue (s : I → V) (U₁ U₂ : V → X) (r : I × V) : Quad X :=
+  (U₁ (s r.1),U₂ (s r.1),U₁ r.2,U₂ r.2)
+
+def tripleValue (s : I → V) (U₁ U₂ : V → X) (r : I × V × V × V) : Expanded X :=
+  (edgeValue s U₁ U₂ (r.1,r.2.1),((U₁ r.2.2.1,U₂ r.2.2.1),(U₁ r.2.2.2,U₂ r.2.2.2)))
+
+lemma triple_integral (w : I → ℝ) (p : I → V → ℝ) (F : I → V → ℝ) (g h : V → ℝ) :
+    (∑ r,tripleWeight w p r*(F r.1 r.2.1*g r.2.2.1*h r.2.2.2))=
+      ∑ i,w i*(∑ y,p i y*F i y)*(∑ z,p i z*g z)*(∑ z,p i z*h z) := by
+  simp only [tripleWeight,Fintype.sum_prod_type]
+  apply Finset.sum_congr rfl
+  intro i _
+  calc
+    _ = ∑ y,∑ z,∑ t,(w i*p i y*F i y)*(p i z*g z)*(p i t*h t) := by
+      apply Finset.sum_congr rfl; intro y _
+      apply Finset.sum_congr rfl; intro z _
+      apply Finset.sum_congr rfl; intro t _
+      ring
+    _ = (∑ y,w i*(p i y*F i y))*(∑ z,p i z*g z)*(∑ t,p i t*h t) := by
+      simp only [← Finset.mul_sum,← Finset.sum_mul]
+      simp only [mul_assoc]
+      rw [← Finset.mul_sum]
+      ring
+    _ = _ := by rw [← Finset.mul_sum]
+
+lemma avg_abs_bound (p : V → ℝ) (f : V → ℝ) (H : ℝ)
+    (hp : ∀ x,0≤p x) (hpsum : ∑ x,p x=1) (hf : ∀ x,|f x|≤H) :
+    |∑ x,p x*f x|≤H := by
+  calc
+    _ ≤ ∑ x,|p x*f x| := Finset.abs_sum_le_sum_abs _ _
+    _ = ∑ x,p x*|f x| := by simp only [abs_mul,abs_of_nonneg (hp _)]
+    _ ≤ ∑ x,p x*H := Finset.sum_le_sum (fun x _ => mul_le_mul_of_nonneg_left (hf x) (hp x))
+    _ = H := by rw [← Finset.sum_mul,hpsum,one_mul]
+
+lemma pair_test_error (m₁ m₂ A₁ A₂ H₁ H₂ : ℝ) (hm₂ : |m₂|≤H₂) (hA₁ : |A₁|≤H₁) :
+    |m₁*m₂-A₁*A₂|≤H₂*|m₁-A₁|+H₁*|m₂-A₂| := by
+  calc
+    _ = |(m₁-A₁)*m₂+A₁*(m₂-A₂)| := by congr 1; ring
+    _ ≤ |(m₁-A₁)*m₂|+|A₁*(m₂-A₂)| := abs_add_le _ _
+    _ ≤ _ := by rw [abs_mul,abs_mul]; nlinarith [abs_nonneg (m₁-A₁),abs_nonneg (m₂-A₂)]
+
+lemma mixture_pair_test_bound (w : I → ℝ) (p : I → V → ℝ) (F : I → V → ℝ)
+    (g h : V → ℝ) (A₁ A₂ C H₁ H₂ : ℝ)
+    (hw : ∀ i,0≤w i) (hp : ∀ i x,0≤p i x) (hpsum : ∀ i,∑ x,p i x=1)
+    (hC : 0≤C) (hF : ∀ i x,|F i x|≤C) (hh : ∀ x,|h x|≤H₂)
+    (hA₁ : |A₁|≤H₁) :
+    |(∑ r,tripleWeight w p r*(F r.1 r.2.1*g r.2.2.1*h r.2.2.2))-
+      (∑ r,edgeWeight w p r*F r.1 r.2)*A₁*A₂|≤
+      C*(H₂*(∑ i,w i*|(∑ z,p i z*g z)-A₁|)+H₁*(∑ i,w i*|(∑ z,p i z*h z)-A₂|)) := by
+  let M := fun i => ∑ y,p i y*F i y
+  let G := fun i => ∑ z,p i z*g z
+  let H := fun i => ∑ z,p i z*h z
+  have hMC (i) : |M i|≤C := avg_abs_bound _ _ _ (hp i) (hpsum i) (hF i)
+  rw [triple_integral]
+  have he : (∑ r,edgeWeight w p r*F r.1 r.2)=∑ i,w i*M i := by
+    simp only [edgeWeight,Fintype.sum_prod_type,M,Finset.mul_sum,mul_assoc]
+  rw [he]
+  change |(∑ i,w i*M i*G i*H i)-(∑ i,w i*M i)*A₁*A₂|≤_
+  simp only [Finset.sum_mul,← Finset.sum_sub_distrib]
+  calc
+    _ ≤ ∑ i,|w i*M i*G i*H i-w i*M i*A₁*A₂| := Finset.abs_sum_le_sum_abs _ _
+    _ = ∑ i,w i*|M i| *|G i*H i-A₁*A₂| := by
+      apply Finset.sum_congr rfl; intro i _
+      rw [show w i*M i*G i*H i-w i*M i*A₁*A₂ = w i*M i*(G i*H i-A₁*A₂) by ring]
+      rw [abs_mul,abs_mul,abs_of_nonneg (hw i)]
+    _ ≤ ∑ i,w i*C*(H₂*|G i-A₁|+H₁*|H i-A₂|) := by
+      apply Finset.sum_le_sum
+      intro i _
+      apply mul_le_mul
+      · exact mul_le_mul_of_nonneg_left (hMC i) (hw i)
+      · exact pair_test_error _ _ _ _ _ _ (avg_abs_bound _ _ _ (hp i) (hpsum i) hh) hA₁
+      · exact abs_nonneg _
+      · exact mul_nonneg (hw i) hC
+    _ = _ := by simp only [Finset.mul_sum,← Finset.sum_add_distrib,G,H]; apply Finset.sum_congr rfl; intros; ring
+
+end WeakPinned.Bounded
+
+end
+end
+section
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PinnedDistances/MobiusObstruction.lean b/lean/OAI/Geometry/PinnedDistances/MobiusObstruction.lean
new file mode 100644
index 0000000000000000000000000000000000000000..b187a171dae44bf96bba7aa2e78b56e946d06d5e
--- /dev/null
+++ b/lean/OAI/Geometry/PinnedDistances/MobiusObstruction.lean
@@ -0,0 +1,431 @@
+import OAI.Geometry.PinnedDistances.ConditionalIntegral
+
+namespace OAI
+
+section
+/- A non-atomic obstruction for bounded cross-ratio energy. -/
+
+noncomputable section
+open Filter MeasureTheory
+open scoped Topology ENNReal
+
+namespace WeakPinned.Bounded
+
+def crossKernel (a c : ℂ) (zw : ℂ × ℂ) : ℝ :=
+  min 1 ‖(c - a) * (zw.2 - zw.1) / ((zw.1 - a) * (c - zw.2))‖
+
+def targetKernel (d : ℂ) (zw : ℂ × ℂ) : ℝ :=
+  min 1 ‖(zw.1 - zw.2) / (zw.2 - d)‖
+
+def crossEnergy (μ : Measure ℂ) (a c : ℂ) : ℝ :=
+  ∫ zw, crossKernel a c zw ∂ μ.prod μ
+
+def targetEnergy (ν : Measure ℂ) (d : ℂ) : ℝ :=
+  ∫ zw, targetKernel d zw ∂ ν.prod ν
+
+lemma measurable_crossKernel (a c : ℂ) : Measurable (crossKernel a c) := by
+  unfold crossKernel
+  fun_prop
+
+lemma measurable_targetKernel (d : ℂ) : Measurable (targetKernel d) := by
+  unfold targetKernel
+  fun_prop
+
+lemma crossKernel_nonneg (a c : ℂ) (zw : ℂ × ℂ) : 0 ≤ crossKernel a c zw := by
+  exact le_min zero_le_one (norm_nonneg _)
+
+lemma crossKernel_le_one (a c : ℂ) (zw : ℂ × ℂ) : crossKernel a c zw ≤ 1 :=
+  min_le_left _ _
+
+lemma targetKernel_nonneg (d : ℂ) (zw : ℂ × ℂ) : 0 ≤ targetKernel d zw := by
+  exact le_min zero_le_one (norm_nonneg _)
+
+lemma targetKernel_le_one (d : ℂ) (zw : ℂ × ℂ) : targetKernel d zw ≤ 1 :=
+  min_le_left _ _
+
+lemma integrable_crossKernel (μ : Measure ℂ) [IsFiniteMeasure μ] (a c : ℂ) :
+    Integrable (crossKernel a c) (μ.prod μ) := by
+  apply Integrable.mono' (integrable_const (1 : ℝ))
+    (measurable_crossKernel a c).aestronglyMeasurable
+  exact Filter.Eventually.of_forall fun zw => by
+    rw [Real.norm_eq_abs, abs_of_nonneg (crossKernel_nonneg a c zw)]
+    exact crossKernel_le_one a c zw
+
+lemma integrable_targetKernel (ν : Measure ℂ) [IsFiniteMeasure ν] (d : ℂ) :
+    Integrable (targetKernel d) (ν.prod ν) := by
+  apply Integrable.mono' (integrable_const (1 : ℝ))
+    (measurable_targetKernel d).aestronglyMeasurable
+  exact Filter.Eventually.of_forall fun zw => by
+    rw [Real.norm_eq_abs, abs_of_nonneg (targetKernel_nonneg d zw)]
+    exact targetKernel_le_one d zw
+
+lemma crossKernel_tendsto (c : ℂ) (zw : ℂ × ℂ)
+    (hz : zw.1 ≠ c) (hw : zw.2 ≠ c) :
+    Tendsto (fun a => crossKernel a c zw) (nhds c) (nhds 0) := by
+  have hd : (zw.1 - c) * (c - zw.2) ≠ 0 :=
+    mul_ne_zero (sub_ne_zero.mpr hz) (sub_ne_zero.mpr hw.symm)
+  have hc : ContinuousAt (fun a : ℂ => crossKernel a c zw) c := by
+    unfold crossKernel
+    fun_prop
+  simpa [crossKernel] using hc.tendsto
+
+lemma ae_pair_ne (μ : Measure ℂ) [NullSingletonClass μ] [SFinite μ] (c : ℂ) :
+    ∀ᵐ zw ∂ μ.prod μ, zw.1 ≠ c ∧ zw.2 ≠ c := by
+  rw [Measure.ae_prod_iff_ae_ae]
+  · filter_upwards [μ.ae_ne c] with z hz
+    filter_upwards [μ.ae_ne c] with w hw
+    exact ⟨hz, hw⟩
+  · exact ((measurableSet_eq_fun measurable_fst measurable_const).compl).inter
+      ((measurableSet_eq_fun measurable_snd measurable_const).compl)
+
+lemma crossEnergy_tendsto (μ : Measure ℂ) [IsFiniteMeasure μ] [NullSingletonClass μ] (c : ℂ) :
+    Tendsto (fun a => crossEnergy μ a c) (nhds c) (nhds 0) := by
+  have h := tendsto_integral_filter_of_dominated_convergence
+    (μ := μ.prod μ) (l := nhds c) (F := fun a zw => crossKernel a c zw)
+    (f := fun _ => (0 : ℝ)) (fun _ => (1 : ℝ))
+    (Filter.Eventually.of_forall fun a => (measurable_crossKernel a c).aestronglyMeasurable)
+    (Filter.Eventually.of_forall fun a => Filter.Eventually.of_forall fun zw => by
+      rw [Real.norm_eq_abs, abs_of_nonneg (crossKernel_nonneg a c zw)]
+      exact crossKernel_le_one a c zw)
+    (integrable_const (1 : ℝ))
+    ((ae_pair_ne μ c).mono fun zw hzw => crossKernel_tendsto c zw hzw.1 hzw.2)
+  simpa [crossEnergy] using h
+
+lemma ae_distinct_pair (μ : Measure ℂ) [NullSingletonClass μ] [SFinite μ] :
+    ∀ᵐ zw ∂ μ.prod μ, zw.1 ≠ zw.2 := by
+  apply (Measure.ae_prod_iff_ae_ae (p := fun zw : ℂ × ℂ => zw.1 ≠ zw.2)
+    (measurableSet_eq_fun measurable_fst measurable_snd).compl).mpr
+  exact Filter.Eventually.of_forall fun z =>
+    (μ.ae_ne z).mono fun w hw => hw.symm
+
+lemma targetEnergy_pos (ν : Measure ℂ) [IsProbabilityMeasure ν] [NullSingletonClass ν]
+    (d : ℂ) : 0 < targetEnergy ν d := by
+  have hp : ∀ᵐ zw ∂ ν.prod ν, 0 < targetKernel d zw := by
+    filter_upwards [ae_distinct_pair ν, ae_pair_ne ν d] with zw hzw hd
+    exact lt_min zero_lt_one (norm_pos_iff.mpr
+      (div_ne_zero (sub_ne_zero.mpr hzw) (sub_ne_zero.mpr hd.2)))
+  have hn : 0 ≤ targetEnergy ν d := integral_nonneg (targetKernel_nonneg d)
+  apply lt_of_le_of_ne hn
+  intro he
+  have hz : targetKernel d =ᵐ[ν.prod ν] 0 :=
+    (integral_eq_zero_iff_of_nonneg (targetKernel_nonneg d)
+      (integrable_targetKernel ν d)).mp he.symm
+  have hf : ∀ᵐ zw ∂ ν.prod ν, False := hp.and hz |>.mono fun zw h => by
+    simpa [h.2] using h.1
+  simpa using (ae_iff.mp hf)
+
+lemma not_ae_energy_eq (μ ν : Measure ℂ) [IsProbabilityMeasure μ]
+    [IsProbabilityMeasure ν] [NullSingletonClass μ] [NullSingletonClass ν]
+    (c d : ℂ) (hc : c ∈ μ.support) :
+    ¬ (∀ᵐ a ∂ μ, crossEnergy μ a c = targetEnergy ν d) := by
+  intro he
+  have ht := targetEnergy_pos ν d
+  have hn : {a | crossEnergy μ a c < targetEnergy ν d} ∈ nhds c :=
+    (crossEnergy_tendsto μ c).eventually (eventually_lt_nhds ht)
+  have hpos := (Measure.mem_support_iff_forall c).mp hc _ hn
+  have hzero : μ {a | crossEnergy μ a c < targetEnergy ν d} = 0 := by
+    have hnot : ∀ᵐ a ∂ μ, ¬ crossEnergy μ a c < targetEnergy ν d :=
+      he.mono fun a ha => by simp [ha]
+    simpa only [ae_iff, not_not, Set.ofPred_mem_eq] using hnot
+  exact (ne_of_gt hpos) hzero
+
+lemma measurable_crossEnergy (μ : Measure ℂ) [SFinite μ] :
+    Measurable (fun ac : ℂ × ℂ => crossEnergy μ ac.1 ac.2) := by
+  have hm : Measurable (fun v : (ℂ × ℂ) × (ℂ × ℂ) =>
+      crossKernel v.1.1 v.1.2 v.2) := by
+    unfold crossKernel
+    fun_prop
+  exact hm.stronglyMeasurable.integral_prod_right'.measurable
+
+lemma measurable_targetEnergy (ν : Measure ℂ) [SFinite ν] :
+    Measurable (targetEnergy ν) := by
+  have hm : Measurable (fun v : ℂ × (ℂ × ℂ) => targetKernel v.1 v.2) := by
+    unfold targetKernel
+    fun_prop
+  exact hm.stronglyMeasurable.integral_prod_right'.measurable
+
+def mobius (a b c d z : ℂ) : ℂ := b + (c - a) * (d - b) / (z - a)
+
+lemma measurable_mobius (a b c d : ℂ) : Measurable (mobius a b c d) := by
+  unfold mobius
+  fun_prop
+
+lemma mobius_sub_target (a b c d z : ℂ) (hz : z ≠ a) :
+    mobius a b c d z - d = (d - b) * (c - z) / (z - a) := by
+  unfold mobius
+  field_simp
+  ring
+
+lemma mobius_ne_target (a b c d w : ℂ) (hbd : b ≠ d)
+    (hwa : w ≠ a) (hwc : w ≠ c) : mobius a b c d w ≠ d := by
+  apply sub_ne_zero.mp
+  rw [mobius_sub_target a b c d w hwa]
+  exact div_ne_zero (mul_ne_zero (sub_ne_zero.mpr hbd.symm)
+    (sub_ne_zero.mpr hwc.symm)) (sub_ne_zero.mpr hwa)
+
+lemma kernel_mobius (a b c d : ℂ) (zw : ℂ × ℂ) (hbd : b ≠ d)
+    (hz : zw.1 ≠ a) (hw : zw.2 ≠ a) (hwc : zw.2 ≠ c) :
+    targetKernel d (mobius a b c d zw.1, mobius a b c d zw.2) =
+      crossKernel a c zw := by
+  unfold targetKernel crossKernel
+  congr 2
+  have hma := sub_ne_zero.mpr hz
+  have hwa := sub_ne_zero.mpr hw
+  have hc := sub_ne_zero.mpr hwc.symm
+  have hdb := sub_ne_zero.mpr hbd.symm
+  rw [mobius_sub_target a b c d zw.2 hw]
+  unfold mobius
+  field_simp
+  ring
+
+lemma energy_of_mobius_map (μ ν : Measure ℂ) [IsProbabilityMeasure μ]
+    [NullSingletonClass μ] (a b c d : ℂ) (hbd : b ≠ d)
+    (hmap : Measure.map (mobius a b c d) μ = ν) :
+    crossEnergy μ a c = targetEnergy ν d := by
+  unfold targetEnergy
+  rw [← hmap, Measure.map_prod_map μ μ (measurable_mobius a b c d)
+    (measurable_mobius a b c d)]
+  rw [integral_map_of_stronglyMeasurable
+    ((measurable_mobius a b c d).prodMap (measurable_mobius a b c d))
+    (measurable_targetKernel d).stronglyMeasurable]
+  apply integral_congr_ae
+  filter_upwards [ae_pair_ne μ a, ae_pair_ne μ c] with zw ha hc
+  exact (kernel_mobius a b c d zw hbd ha.1 ha.2 hc.2).symm
+
+attribute [local irreducible] crossEnergy targetEnergy
+
+lemma measurableSet_energy_eq (μ ν : Measure ℂ) [SFinite μ] [SFinite ν] :
+    MeasurableSet {v : ℂ × (ℂ × ℂ) |
+      crossEnergy μ v.1 v.2.1 = targetEnergy ν v.2.2} := by
+  have h₁ : Measurable (fun v : ℂ × (ℂ × ℂ) => crossEnergy μ v.1 v.2.1) :=
+    (measurable_crossEnergy μ).comp
+      (show Measurable (fun v : ℂ × (ℂ × ℂ) => (v.1, v.2.1)) from by fun_prop)
+  have h₂ : Measurable (fun v : ℂ × (ℂ × ℂ) => targetEnergy ν v.2.2) :=
+    (measurable_targetEnergy ν).comp
+      (show Measurable (fun v : ℂ × (ℂ × ℂ) => v.2.2) from by fun_prop)
+  exact measurableSet_eq_fun h₁ h₂
+
+/-- The finite-plane non-atomic obstruction. -/
+
+theorem no_finite_law (μ ν : Measure ℂ) [IsProbabilityMeasure μ]
+    [IsProbabilityMeasure ν] [NullSingletonClass μ] [NullSingletonClass ν]
+    (Q : Measure (ℂ × ℂ × ℂ × ℂ)) [IsProbabilityMeasure Q]
+    (hlaw : Q.map (fun q => q.1) = μ)
+    (hind : ProbabilityTheory.IndepFun (fun q => q.1) (fun q => q.2.2) Q)
+    (hsupp : ∀ᵐ q ∂ Q, q.2.2.1 ∈ μ.support)
+    (hbd : ∀ᵐ q ∂ Q, q.2.1 ≠ q.2.2.2)
+    (hmap : ∀ᵐ q ∂ Q, μ.map (mobius q.1 q.2.1 q.2.2.1 q.2.2.2) = ν) : False := by
+  let τ : Measure (ℂ × ℂ) := Q.map (fun q => q.2.2)
+  have : IsProbabilityMeasure τ := inferInstance
+  have hp : MeasurableSet {v : ℂ × (ℂ × ℂ) |
+      crossEnergy μ v.1 v.2.1 = targetEnergy ν v.2.2} :=
+    measurableSet_energy_eq μ ν
+  have he : ∀ᵐ q ∂ Q,
+      crossEnergy μ q.1 q.2.2.1 = targetEnergy ν q.2.2.2 := by
+    filter_upwards [hbd, hmap] with q hd hm
+    exact energy_of_mobius_map μ ν q.1 q.2.1 q.2.2.1 q.2.2.2 hd hm
+  have hj : Q.map (fun q => (q.1, q.2.2)) = μ.prod τ := by
+    rw [hind.map_prod_eq_prod_map_map (by fun_prop) (by fun_prop), hlaw]
+  have he' : ∀ᵐ v ∂ μ.prod τ, crossEnergy μ v.1 v.2.1 = targetEnergy ν v.2.2 := by
+    rw [← hj]
+    exact (ae_map_iff (by fun_prop) hp).mpr he
+  have hf : ∀ᵐ cd ∂ τ, ∀ᵐ a ∂ μ, crossEnergy μ a cd.1 = targetEnergy ν cd.2 :=
+    (Measure.ae_ae_comm hp).mp ((Measure.ae_prod_iff_ae_ae hp).mp he')
+  have hs : ∀ᵐ cd ∂ τ, cd.1 ∈ μ.support := by
+    exact (ae_map_iff (by fun_prop)
+      (μ.isClosed_support.measurableSet.preimage measurable_fst)).mpr hsupp
+  obtain ⟨cd, hc, hcd⟩ := (hs.and hf).exists
+  exact not_ae_energy_eq μ ν cd.1 cd.2 hc hcd
+
+/-- The Riemann sphere as the one-point compactification of the complex plane. -/
+abbrev Sphere := OnePoint ℂ
+local instance : MeasurableSpace Sphere := borel Sphere
+local instance : BorelSpace Sphere := ⟨rfl⟩
+
+lemma measurableEmbedding_sphere_coe : MeasurableEmbedding ((↑) : ℂ → Sphere) :=
+  OnePoint.isOpenEmbedding_coe.measurableEmbedding
+
+/-- The sphere Möbius map has its pole at infinity and its value at infinity is b;
+    no finite-valued division convention is used there. -/
+def sphereMobius (a b c d : ℂ) : Sphere → Sphere :=
+  OnePoint.rec (b : Sphere) (fun z => if z = a then OnePoint.infty
+    else (mobius a b c d z : Sphere))
+
+@[simp] lemma sphereMobius_coe (a b c d z : ℂ) (hz : z ≠ a) :
+    sphereMobius a b c d (z : Sphere) = (mobius a b c d z : Sphere) := by
+  simp [sphereMobius, OnePoint.rec, hz]
+
+@[simp] lemma sphereMobius_pole (a b c d : ℂ) :
+    sphereMobius a b c d (a : Sphere) = OnePoint.infty := by
+  simp [sphereMobius, OnePoint.rec]
+
+@[simp] lemma sphereMobius_infty (a b c d : ℂ) :
+    sphereMobius a b c d OnePoint.infty = (b : Sphere) := rfl
+
+lemma map_sphereMobius (μ : Measure ℂ) [NullSingletonClass μ] (a b c d : ℂ) :
+    (μ.map ((↑) : ℂ → Sphere)).map (sphereMobius a b c d) =
+      (μ.map (mobius a b c d)).map ((↑) : ℂ → Sphere) := by
+  have he : ((fun z : ℂ => (mobius a b c d z : Sphere))) =ᵐ[μ]
+      (sphereMobius a b c d ∘ ((↑) : ℂ → Sphere)) := by
+    filter_upwards [μ.ae_ne a] with z hz
+    exact (sphereMobius_coe a b c d z hz).symm
+  have hm0 : Measurable (fun z : ℂ => (mobius a b c d z : Sphere)) :=
+    measurableEmbedding_sphere_coe.measurable.comp (measurable_mobius a b c d)
+  have hm : AEMeasurable (sphereMobius a b c d ∘ ((↑) : ℂ → Sphere)) μ :=
+    hm0.aemeasurable.congr he
+  rw [AEMeasurable.map_map_of_aemeasurable (measurableEmbedding_sphere_coe.aemeasurable_map_iff.mpr hm)
+    measurableEmbedding_sphere_coe.measurable.aemeasurable]
+  rw [Measure.map_map measurableEmbedding_sphere_coe.measurable (measurable_mobius a b c d)]
+  exact Measure.map_congr he.symm
+
+lemma finite_map_of_sphere_map (μ ν : Measure ℂ) [NullSingletonClass μ] (a b c d : ℂ)
+    (hmap : (μ.map ((↑) : ℂ → Sphere)).map (sphereMobius a b c d) =
+      ν.map ((↑) : ℂ → Sphere)) : μ.map (mobius a b c d) = ν := by
+  apply measurableEmbedding_sphere_coe.map_injective
+  rw [← map_sphereMobius μ a b c d]
+  exact hmap
+
+/-- The non-atomic Möbius obstruction with its pushforward identity on the Riemann sphere. -/
+
+theorem bounded_mobius (μ ν : Measure ℂ) [IsProbabilityMeasure μ]
+    [IsProbabilityMeasure ν] [NullSingletonClass μ] [NullSingletonClass ν]
+    (Q : Measure (ℂ × ℂ × ℂ × ℂ)) [IsProbabilityMeasure Q]
+    (hsep : ∀ᵐ q ∂ Q, q.1 ≠ q.2.2.1 ∧ q.2.1 ≠ q.2.2.2)
+    (hsupp : ∀ᵐ q ∂ Q, q.2.2.1 ∈ μ.support ∧ q.2.2.2 ∈ ν.support)
+    (hlaw : Q.map (fun q => q.1) = μ)
+    (hind : ProbabilityTheory.IndepFun (fun q => q.1) (fun q => q.2.2) Q)
+    (hmap : ∀ᵐ q ∂ Q, (μ.map ((↑) : ℂ → Sphere)).map
+      (sphereMobius q.1 q.2.1 q.2.2.1 q.2.2.2) = ν.map ((↑) : ℂ → Sphere)) : False := by
+  apply no_finite_law μ ν Q hlaw hind (hsupp.mono fun _ h => h.1)
+    (hsep.mono fun _ h => h.2)
+  exact hmap.mono fun q h => finite_map_of_sphere_map μ ν q.1 q.2.1 q.2.2.1 q.2.2.2 h
+
+end WeakPinned.Bounded
+
+namespace WeakPinned.Bounded
+open TopologicalSpace
+local instance : MeasurableSpace Sphere := borel Sphere
+local instance : BorelSpace Sphere := ⟨rfl⟩
+
+/-- The one-point compactification of the complex plane has a countable basis. -/
+instance sphereSecondCountable : SecondCountableTopology Sphere := by
+  let B : Set (Set Sphere) :=
+    (fun U : Set ℂ => OnePoint.some '' U) '' countableBasis ℂ ∪
+      Set.range (fun n : ℕ => (OnePoint.some '' Metric.closedBall (0:ℂ) (n:ℝ))ᶜ)
+  have hcount : B.Countable := (countable_countableBasis ℂ).image _ |>.union (Set.countable_range _)
+  have hbase : IsTopologicalBasis B := by
+    apply isTopologicalBasis_of_isOpen_of_nhds
+    · intro U hU
+      rcases hU with ⟨V,hV,rfl⟩ | ⟨n,rfl⟩
+      · exact OnePoint.isOpen_image_coe.mpr ((isBasis_countableBasis ℂ).isOpen hV)
+      · exact OnePoint.isOpen_compl_image_coe.mpr ⟨Metric.isClosed_closedBall,isCompact_closedBall _ _⟩
+    · intro a U ha hU
+      induction a using OnePoint.rec with
+      | coe a =>
+        have hp : IsOpen (OnePoint.some ⁻¹' U : Set ℂ) := hU.preimage OnePoint.continuous_coe
+        obtain ⟨V,hV,haV,hVU⟩ := (isBasis_countableBasis ℂ).mem_nhds_iff.mp (hp.mem_nhds ha)
+        refine ⟨OnePoint.some '' V,Or.inl ⟨V,hV,rfl⟩,⟨a,haV,rfl⟩,?_⟩
+        rintro z ⟨w,hw,rfl⟩
+        exact hVU hw
+      | infty =>
+        have hc := ((OnePoint.isOpen_iff_of_mem ha).mp hU).2
+        obtain ⟨R,hR⟩ := hc.isBounded.subset_closedBall (0:ℂ)
+        obtain ⟨n,hn⟩ := exists_nat_gt R
+        refine ⟨(OnePoint.some '' Metric.closedBall (0:ℂ) (n:ℝ))ᶜ,Or.inr ⟨n,rfl⟩,?_,?_⟩
+        · exact OnePoint.infty_notMem_image_coe
+        · intro z hz
+          induction z using OnePoint.rec with
+          | infty => exact ha
+          | coe z =>
+            by_contra hnot
+            have hzR := hR hnot
+            have hzn := Metric.closedBall_subset_closedBall hn.le hzR
+            exact hz ⟨z,hzn,rfl⟩
+  exact hbase.secondCountableTopology hcount
+
+lemma sphere_pair_subsequence (α : ℕ → ProbabilityMeasure Sphere × ProbabilityMeasure Sphere) :
+    ∃ β : ProbabilityMeasure Sphere × ProbabilityMeasure Sphere,
+      ∃ φ : ℕ → ℕ, StrictMono φ ∧ Tendsto (α ∘ φ) atTop (𝓝 β) := by
+  obtain ⟨β,_,φ,hφ,hT⟩ := isCompact_univ.tendsto_subseq (fun n => Set.mem_univ (α n))
+  exact ⟨β,φ,hφ,hT⟩
+
+/-- A useful Portmanteau consequence with an eventual bound; measures, not
+    arbitrary test function samples, are passed to the limit. -/
+lemma weak_open_bound {α : ℕ → ProbabilityMeasure Sphere} {β : ProbabilityMeasure Sphere}
+    (hT : Tendsto α atTop (𝓝 β)) {U : Set Sphere} (hU : IsOpen U) (b : ℝ≥0∞)
+    (hbound : ∀ᶠ n in atTop, (α n : Measure Sphere) U ≤ b) :
+    (β : Measure Sphere) U ≤ b := by
+  have hp := ProbabilityMeasure.le_liminf_measure_open_of_tendsto hT hU
+  exact hp.trans (liminf_le_of_frequently_le hbound.frequently)
+
+/-- Neighborhood control is enough to remove a possible atom in a weak limit. -/
+lemma weak_singleton_zero {α : ℕ → ProbabilityMeasure Sphere} {β : ProbabilityMeasure Sphere}
+    (hT : Tendsto α atTop (𝓝 β)) (z : Sphere)
+    (hsmall : ∀ ε : ℝ, 0 < ε → ∃ U : Set Sphere, IsOpen U ∧ z ∈ U ∧
+      ∀ᶠ n in atTop, (α n : Measure Sphere) U ≤ ENNReal.ofReal ε) :
+    (β : Measure Sphere) {z}=0 := by
+  apply le_antisymm _ zero_le
+  apply ENNReal.le_of_forall_pos_le_add
+  intro ε hε hεtop
+  have hεR : (0:ℝ) < ε := by exact_mod_cast hε
+  obtain ⟨U,hU,hz,hB⟩ := hsmall (ε:ℝ) hεR
+  have hb := weak_open_bound hT hU (ENNReal.ofReal (ε:ℝ)) hB
+  rw [ENNReal.ofReal_coe_nnreal] at hb
+  simpa only [zero_add] using (measure_mono (Set.singleton_subset_iff.mpr hz)).trans hb
+
+/-- Finiteness and nonatomicity on the sphere follow from open-tail and disk
+    bounds through the Portmanteau inequality. -/
+theorem weak_limit_nonatomic {α : ℕ → ProbabilityMeasure Sphere} {β : ProbabilityMeasure Sphere}
+    (hT : Tendsto α atTop (𝓝 β))
+    (htail : ∀ ε : ℝ, 0 < ε → ∃ R : ℝ, ∀ᶠ n in atTop,
+      (α n : Measure Sphere) (OnePoint.some '' Metric.closedBall (0:ℂ) R)ᶜ ≤ ENNReal.ofReal ε)
+    (hball : ∀ z : ℂ, ∀ ε : ℝ, 0 < ε → ∃ r : ℝ, 0 < r ∧ ∀ᶠ n in atTop,
+      (α n : Measure Sphere) (OnePoint.some '' Metric.ball z r) ≤ ENNReal.ofReal ε) :
+    NullSingletonClass (β : Measure Sphere) := by
+  constructor
+  intro z
+  apply weak_singleton_zero hT z
+  intro ε hε
+  induction z using OnePoint.rec with
+  | infty =>
+    obtain ⟨R,hR⟩ := htail ε hε
+    exact ⟨_,OnePoint.isOpen_compl_image_coe.mpr ⟨Metric.isClosed_closedBall,isCompact_closedBall _ _⟩,
+      OnePoint.infty_notMem_image_coe,hR⟩
+  | coe z =>
+    obtain ⟨r,hr,hR⟩ := hball z ε hε
+    exact ⟨_,OnePoint.isOpen_image_coe.mpr Metric.isOpen_ball,⟨z,Metric.mem_ball_self hr,rfl⟩,hR⟩
+
+/-- Restriction of a sphere probability measure to the complex plane.
+    This is a probability measure when there is no mass at infinity. -/
+def finitePart (β : ProbabilityMeasure Sphere) : Measure ℂ :=
+  Measure.comap ((↑) : ℂ → Sphere) (β : Measure Sphere)
+
+lemma finitePart_probability (β : ProbabilityMeasure Sphere)
+    (hinfty : (β : Measure Sphere) {OnePoint.infty}=0) : IsProbabilityMeasure (finitePart β) := by
+  constructor
+  rw [finitePart,measurableEmbedding_sphere_coe.comap_apply,Set.image_univ]
+  have hh : (β : Measure Sphere) (Set.range ((↑) : ℂ → Sphere))ᶜ=0 := by
+    simpa only [OnePoint.compl_range_coe] using hinfty
+  exact measure_eq_measure_of_null_sdiff (Set.subset_univ _) (by simpa only [← Set.compl_eq_univ_sdiff] using hh) |>.trans (measure_univ)
+
+lemma finitePart_nonatomic (β : ProbabilityMeasure Sphere) [NullSingletonClass (β : Measure Sphere)] :
+    NullSingletonClass (finitePart β) := by
+  constructor
+  intro z
+  rw [finitePart,measurableEmbedding_sphere_coe.comap_apply,Set.image_singleton,measure_singleton]
+
+lemma finitePart_map (β : ProbabilityMeasure Sphere)
+    (hinfty : (β : Measure Sphere) {OnePoint.infty}=0) :
+    (finitePart β).map ((↑) : ℂ → Sphere)=(β : Measure Sphere) := by
+  rw [finitePart,measurableEmbedding_sphere_coe.map_comap]
+  apply Measure.restrict_eq_self_of_ae_mem
+  rw [ae_iff]
+  change (β : Measure Sphere) (Set.range ((↑) : ℂ → Sphere))ᶜ=0
+  simpa only [OnePoint.compl_range_coe] using hinfty
+
+end WeakPinned.Bounded
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PinnedDistances/Model.lean b/lean/OAI/Geometry/PinnedDistances/Model.lean
new file mode 100644
index 0000000000000000000000000000000000000000..4ff245fa9d2626d1f320b070536e2f35259631d4
--- /dev/null
+++ b/lean/OAI/Geometry/PinnedDistances/Model.lean
@@ -0,0 +1,53 @@
+import Mathlib
+
+namespace OAI
+
+noncomputable section
+open Filter
+open scoped Topology
+
+namespace WeakPinned
+abbrev Plane := EuclideanSpace ℝ (Fin 2)
+
+def distanceFiber (P : Finset Plane) (x y : Plane) : Finset Plane :=
+  (P.erase x).filter (fun z => dist z x = dist y x)
+
+def k (P : Finset Plane) (x y : Plane) : ℕ :=
+  (distanceFiber P x y).card
+
+def richPairs (P : Finset Plane) (s : ℝ) : Finset (Plane × Plane) :=
+  (P ×ˢ P).filter (fun xy => xy.1 ≠ xy.2 ∧
+    (P.card : ℝ) ^ s ≤ (k P xy.1 xy.2 : ℝ))
+
+def pairFraction (P : Finset Plane) (s : ℝ) : ℝ :=
+  (richPairs P s).card / ((P.card : ℝ) * ((P.card : ℝ) - 1))
+
+def F (n : ℕ) (s : ℝ) : ℝ :=
+  if 2 ≤ n then sSup {a : ℝ | ∃ P : Finset Plane, P.card = n ∧ a = pairFraction P s}
+  else 0
+
+def distances (P : Finset Plane) (x : Plane) : Finset ℝ :=
+  (P.erase x).image (fun y => dist y x)
+
+def badPins (P : Finset Plane) (ε : ℝ) : Finset Plane :=
+  P.filter (fun x => ((distances P x).card : ℝ) < (P.card : ℝ) ^ (1 - ε))
+
+def badPinFraction (P : Finset Plane) (ε : ℝ) : ℝ :=
+  (badPins P ε).card / (P.card : ℝ)
+
+def B (n : ℕ) (ε : ℝ) : ℝ :=
+  if 0 < n then sSup {a : ℝ | ∃ P : Finset Plane, P.card = n ∧ a = badPinFraction P ε}
+  else 0
+
+/- Uniform multiplicity and pinned-distance convergence statements. -/
+def MainStatement : Prop :=
+  ∀ s : ℝ, 0 < s → Tendsto (fun n : ℕ => F n s) atTop (nhds 0)
+
+def PinsStatement : Prop :=
+  ∀ ε : ℝ, 0 < ε → Tendsto (fun n : ℕ => B n ε) atTop (nhds 0)
+
+end WeakPinned
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PinnedDistances/NormalizedGraphs.lean b/lean/OAI/Geometry/PinnedDistances/NormalizedGraphs.lean
new file mode 100644
index 0000000000000000000000000000000000000000..0f4da28f66c4397baa55c764d9773b59da796ebf
--- /dev/null
+++ b/lean/OAI/Geometry/PinnedDistances/NormalizedGraphs.lean
@@ -0,0 +1,88 @@
+import OAI.Geometry.PinnedDistances.LimitObstruction
+
+namespace OAI
+
+universe u1 u2
+
+section
+noncomputable section
+open MeasureTheory Filter Set
+open scoped Topology BigOperators
+namespace WeakPinned.Bounded
+local instance : MeasurableSpace Sphere := borel Sphere
+local instance : BorelSpace Sphere := ⟨rfl⟩
+variable {V : ℕ → Type u1} {I : ℕ → Type u2} [∀ n,Fintype (V n)] [∀ n,DecidableEq (V n)] [∀ n,Fintype (I n)]
+
+/-- Compactness is applied to the four genuine empirical graph laws together.
+The only asymptotic hypotheses are the proved test-discrepancy estimates. -/
+theorem normalized_graph_impossible
+    (G : ∀ n,Core.Fibers (V n) (I n)) (u₁ u₂ : ∀ n,V n → ℂ)
+    (x₁ x₂ : ∀ n,V n) (hu₁ : ∀ n,Function.Injective (u₁ n)) (hu₂ : ∀ n,Function.Injective (u₂ n))
+    (hG : ∀ n,0<(G n).edgeCount) (hB : ∀ n e,((G n).targets e).Nonempty)
+    (hn : ∀ n,2≤Fintype.card (V n))
+    (hcard : Tendsto (fun n => (Fintype.card (V n):ℝ)) atTop atTop)
+    (C : ℝ) (hC : 0≤C) (hlog₁ : ∀ n,logAverage (u₁ n)≤C) (hlog₂ : ∀ n,logAverage (u₂ n)≤C)
+    (hx₁ : ∀ n,(∑ y,logCost (u₁ n) (x₁ n) y)≤C*(Fintype.card (V n):ℝ))
+    (hx₂ : ∀ n,(∑ y,logCost (u₂ n) (x₂ n) y)≤C*(Fintype.card (V n):ℝ))
+    (hd₁ : ∀ f : C(Sphere,ℝ),Tendsto (fun n => testDiscrep (G n).fiberWeight (G n).targets
+      (fun x => f ((u₁ n x-u₁ n (x₁ n)):Sphere))) atTop (𝓝 0))
+    (hd₂ : ∀ f : C(Sphere,ℝ),Tendsto (fun n => testDiscrep (G n).fiberWeight (G n).targets
+      (fun x => f ((u₂ n x-u₂ n (x₂ n)):Sphere))) atTop (𝓝 0))
+    (T : ℕ → ℝ) (hT : ∀ n,0<T n)
+    (hdR : ∀ f : C(Sphere,ℝ),Tendsto (fun n => testDiscrep ((G n).retainedWeight (T n))
+      ((G n).retainedSet (T n)) (fun x => f ((u₁ n x-u₁ n (x₁ n)):Sphere))) atTop (𝓝 0))
+    (ho : Tendsto (fun n => (G n).omittedMass (T n)) atTop (𝓝 0))
+    (M : ℝ) (hM : 0≤M) (hdom : ∀ n x y,(G n).edgeLaw x y≤M*Variance.pairProb Finset.univ x y)
+    (hrel : ∀ n e y z,y∈(G n).targets e → z∈(G n).targets e →
+      (u₁ n y-u₁ n ((G n).source e))*(u₂ n y-u₂ n ((G n).source e))=
+      (u₁ n z-u₁ n ((G n).source e))*(u₂ n z-u₂ n ((G n).source e))) : False := by
+  let : TopologicalSpace.PseudoMetrizableSpace Sphere := inferInstance
+  let : TopologicalSpace.PseudoMetrizableSpace (Quad Sphere) := inferInstance
+  let : TopologicalSpace.PseudoMetrizableSpace (Sphere × Sphere) := inferInstance
+  let : TopologicalSpace.PseudoMetrizableSpace (Expanded Sphere) := inferInstance
+  let : TopologicalSpace (ProbabilityMeasure Sphere) := inferInstance
+  let : TopologicalSpace (ProbabilityMeasure (Quad Sphere)) := inferInstance
+  let : TopologicalSpace (ProbabilityMeasure (Expanded Sphere)) := inferInstance
+  let : CompactSpace (Quad Sphere) := inferInstance
+  let : CompactSpace (Expanded Sphere) := inferInstance
+  let : CompactSpace (ProbabilityMeasure Sphere) := inferInstance
+  let : CompactSpace (ProbabilityMeasure (Quad Sphere)) := inferInstance
+  let : CompactSpace (ProbabilityMeasure (Expanded Sphere)) := inferInstance
+  let U₁ := fun n x => u₁ n x-u₁ n (x₁ n)
+  let U₂ := fun n x => u₂ n x-u₂ n (x₂ n)
+  let α₁ := fun n => uniformProbability (fun x => (U₁ n x:Sphere)) (by have:=hn n;omega)
+  let α₂ := fun n => uniformProbability (fun x => (U₂ n x:Sphere)) (by have:=hn n;omega)
+  let Qn := fun n => edgeProbability (G n) (fun x => (U₁ n x:Sphere))
+    (fun x => (U₂ n x:Sphere)) (hG n) (hB n)
+  let Rn := fun n => expandedProbability (G n) (fun x => (U₁ n x:Sphere))
+    (fun x => (U₂ n x:Sphere)) (hG n) (hB n)
+  let L := fun n => (α₁ n,α₂ n,Qn n,Rn n)
+  obtain ⟨l,_,φ,hφ,hl⟩ := isCompact_univ.tendsto_subseq (fun n => mem_univ (L n))
+  have ha₁ : Tendsto (α₁ ∘ φ) atTop (𝓝 l.1) := continuous_fst.tendsto l |>.comp hl
+  have ha₂ : Tendsto (α₂ ∘ φ) atTop (𝓝 l.2.1) :=
+    (continuous_fst.comp continuous_snd).tendsto l |>.comp hl
+  have hq : Tendsto (Qn ∘ φ) atTop (𝓝 l.2.2.1) :=
+    (continuous_fst.comp (continuous_snd.comp continuous_snd)).tendsto l |>.comp hl
+  have hr : Tendsto (Rn ∘ φ) atTop (𝓝 l.2.2.2) :=
+    (continuous_snd.comp (continuous_snd.comp continuous_snd)).tendsto l |>.comp hl
+  let : NullSingletonClass (l.1:Measure Sphere) := normalized_empirical_nonatomic
+    (fun n => u₁ (φ n)) (fun n => x₁ (φ n)) (fun n => hu₁ (φ n)) (fun n => hn (φ n))
+    (hcard.comp hφ.tendsto_atTop) C hC (fun n => hlog₁ (φ n)) (fun n => hx₁ (φ n)) l.1 ha₁
+  let : NullSingletonClass (l.2.1:Measure Sphere) := normalized_empirical_nonatomic
+    (fun n => u₂ (φ n)) (fun n => x₂ (φ n)) (fun n => hu₂ (φ n)) (fun n => hn (φ n))
+    (hcard.comp hφ.tendsto_atTop) C hC (fun n => hlog₂ (φ n)) (fun n => hx₂ (φ n)) l.2.1 ha₂
+  apply graph_limit_impossible (fun n => G (φ n)) (fun n => U₁ (φ n)) (fun n => U₂ (φ n))
+    (fun n => hG (φ n)) (fun n => hB (φ n)) (fun n => hn (φ n))
+    l.1 l.2.1 l.2.2.1 l.2.2.2 ha₁ ha₂ hq hr
+    (fun f => (hd₁ f).comp hφ.tendsto_atTop) (fun f => (hd₂ f).comp hφ.tendsto_atTop)
+    (T ∘ φ) (fun n => hT (φ n)) (fun f => (hdR f).comp hφ.tendsto_atTop)
+    (ho.comp hφ.tendsto_atTop) M hM (fun n => hdom (φ n))
+  intro n e y z hy hz
+  simpa only [U₁,U₂,sub_sub_sub_cancel_right] using hrel (φ n) e y z hy hz
+
+end WeakPinned.Bounded
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PinnedDistances/OverlapRates.lean b/lean/OAI/Geometry/PinnedDistances/OverlapRates.lean
new file mode 100644
index 0000000000000000000000000000000000000000..9826b73fab52939696032c4573dee4fa9160eab3
--- /dev/null
+++ b/lean/OAI/Geometry/PinnedDistances/OverlapRates.lean
@@ -0,0 +1,276 @@
+import OAI.Geometry.PinnedDistances.ArithmeticGraphs
+
+namespace OAI
+
+universe u1 u2 u3 u4 u5 u6 u7 u8 u9 u10
+
+section
+noncomputable section
+open NumberField
+open scoped BigOperators
+noncomputable section
+open MeasureTheory
+open scoped Topology
+noncomputable section
+open MeasureTheory
+open scoped Topology ENNReal
+noncomputable section
+open MeasureTheory NumberField
+open scoped BigOperators
+noncomputable section
+open scoped BigOperators
+open MeasureTheory
+noncomputable section
+open Filter
+open scoped Topology
+noncomputable section
+open scoped BigOperators
+noncomputable section
+open Filter
+open scoped Topology
+
+namespace WeakPinned
+
+/-- The scalar limiting step for small overlap, with fixed positive τ and H. -/
+
+theorem small_overlap_rate (n η U A D : ℕ → ℝ) (θ : ℝ) (hθ : 0 < θ)
+    (hn : Tendsto n atTop atTop) (hη : Tendsto η atTop (𝓝 0))
+    (hη0 : ∀ j, 0 ≤ η j) (hU : ∀ j, 0 ≤ U j) (hA : ∀ j, 0 ≤ A j) (hD : ∀ j, 0 ≤ D j)
+    (hbound : ∀ τ H : ℝ, 0 < τ → τ ≤ 1 → 0 ≤ H → ∀ᶠ j in atTop,
+      D j ≤ (2*(1/Real.log (n j))/τ^2+2*(1/Real.log (n j))/(θ-η j)+((θ+η j)/(θ-η j)-1))*U j +
+        (2*τ/(θ-η j)+(2-2*((θ-η j)/(θ+η j/8))))*A j +
+        4*(1/Real.log (n j))*H + 32*Real.exp (-H/2)) :
+    Tendsto (fun j => D j/(U j+A j+1)) atTop (𝓝 0) := by
+  have hlamT : Tendsto (fun j => 1/Real.log (n j)) atTop (𝓝 0) :=
+    tendsto_const_nhds.div_atTop (Real.tendsto_log_atTop.comp hn)
+  apply tendsto_order.mpr
+  constructor
+  · intro a ha
+    exact Filter.Eventually.of_forall fun j => ha.trans_le (div_nonneg (hD j) (by linarith [hU j,hA j]))
+  · intro ε hε
+    let τ := min 1 (ε*θ/16)
+    have hτ : 0 < τ := lt_min (by norm_num) (by positivity)
+    have hτ1 : τ ≤ 1 := min_le_left _ _
+    have hτb : 2*τ/θ ≤ ε/8 := by
+      apply (div_le_iff₀ hθ).mpr
+      have := min_le_right (1:ℝ) (ε*θ/16)
+      dsimp [τ] at ⊢
+      linarith
+    have htExp : Tendsto (fun H : ℝ => 32*Real.exp (-H/2)) atTop (𝓝 0) := by
+      have hd : Tendsto (fun H : ℝ => H/2) atTop atTop := by
+        simpa only [div_eq_mul_inv, mul_comm, id_eq, one_mul] using
+          (tendsto_const_mul_atTop_of_pos (by norm_num : (0:ℝ)<1/2)).mpr tendsto_id
+      simpa only [Function.comp_def,neg_div,mul_zero] using
+        (Real.tendsto_exp_neg_atTop_nhds_zero.comp hd).const_mul 32
+    obtain ⟨H,hH,hHb⟩ := (eventually_ge_atTop (0:ℝ) |>.and (htExp.eventually (eventually_lt_nhds (show (0:ℝ)<ε/8 by linarith)))).exists
+    let a := fun j => 2*(1/Real.log (n j))/τ^2+2*(1/Real.log (n j))/(θ-η j)+((θ+η j)/(θ-η j)-1)
+    let b := fun j => 2*τ/(θ-η j)+(2-2*((θ-η j)/(θ+η j/8)))
+    let e := fun j => 4*(1/Real.log (n j))*H+32*Real.exp (-H/2)
+    have hθm : Tendsto (fun j => θ-η j) atTop (𝓝 θ) := by
+      simpa using (tendsto_const_nhds (x:=θ)).sub hη
+    have hθp : Tendsto (fun j => θ+η j) atTop (𝓝 θ) := by
+      simpa using (tendsto_const_nhds (x:=θ)).add hη
+    have hθp8 : Tendsto (fun j => θ+η j/8) atTop (𝓝 θ) := by
+      simpa using (tendsto_const_nhds (x:=θ)).add (hη.div_const 8)
+    have ha : Tendsto a atTop (𝓝 0) := by
+      have ha1 : Tendsto (fun j => 2*(1/Real.log (n j))/τ^2) atTop (𝓝 0) := by
+        simpa using (hlamT.const_mul 2).div_const (τ^2)
+      have ha2 : Tendsto (fun j => 2*(1/Real.log (n j))/(θ-η j)) atTop (𝓝 0) := by
+        convert (hlamT.const_mul 2).div hθm hθ.ne' using 1
+        simp
+      have ha3 : Tendsto (fun j => (θ+η j)/(θ-η j)-1) atTop (𝓝 0) := by
+        simpa [hθ.ne',Pi.div_apply] using (hθp.div hθm hθ.ne').sub (tendsto_const_nhds (x:=1))
+      simpa [a] using (ha1.add ha2).add ha3
+    have hb : Tendsto b atTop (𝓝 (2*τ/θ)) := by
+      have hb1 := (tendsto_const_nhds (x:=2*τ)).div hθm hθ.ne'
+      have hb2 : Tendsto (fun j => 2-2*((θ-η j)/(θ+η j/8))) atTop (𝓝 0) := by
+        simpa [hθ.ne',Pi.div_apply] using (tendsto_const_nhds (x:= (2:ℝ))).sub ((hθm.div hθp8 hθ.ne').const_mul 2)
+      simpa [b,Pi.div_apply] using hb1.add hb2
+    have he : Tendsto e atTop (𝓝 (32*Real.exp (-H/2))) := by
+      simpa [e] using ((hlamT.const_mul 4).mul_const H).add (tendsto_const_nhds (x:=32*Real.exp (-H/2)))
+    have hlt : 2*τ/θ+32*Real.exp (-H/2)<ε := by linarith
+    have hsum := (ha.add hb |>.add he).eventually (eventually_lt_nhds (by simpa using hlt))
+    filter_upwards [hbound τ H hτ hτ1 hH,hsum,hη.eventually (eventually_lt_nhds hθ),
+      (Real.tendsto_log_atTop.comp hn).eventually (eventually_gt_atTop (0:ℝ))] with j hj hjlt hjη hjlog
+    have ha0 : 0 ≤ a j := by
+      have hr : 1 ≤ (θ+η j)/(θ-η j) := (one_le_div (sub_pos.mpr hjη)).mpr (by linarith [hη0 j])
+      dsimp [a]
+      positivity
+    have hb0 : 0 ≤ b j := by
+      have hr : (θ-η j)/(θ+η j/8) ≤ 1 := (div_le_one (by linarith [hη0 j] : 0<θ+η j/8)).mpr (by linarith [hη0 j])
+      have hr0 : 0 ≤ 2-2*((θ-η j)/(θ+η j/8)) := by linarith
+      dsimp [b]
+      positivity
+    have he0 : 0 ≤ e j := by dsimp [e]; positivity
+    have hden : 0 < U j+A j+1 := by linarith [hU j,hA j]
+    have hupper : D j ≤ (a j+b j+e j)*(U j+A j+1) := by
+      have hj' : D j ≤ a j*U j+b j*A j+e j := by
+        simpa only [a,b,e,add_assoc] using hj
+      nlinarith only [hj',hU j,hA j,ha0,hb0,he0,mul_nonneg ha0 (hA j),mul_nonneg hb0 (hU j),mul_nonneg he0 (hU j),mul_nonneg he0 (hA j)]
+    exact (div_le_iff₀ hden).mpr hupper |>.trans_lt hjlt
+end WeakPinned
+
+namespace WeakPinned
+open Filter
+open scoped Topology
+open Global
+variable (V : ℕ → Type u1) (I : ℕ → Type u2) (K : ℕ → Type u3) [∀ j, Fintype (V j)] [∀ j, Fintype (I j)] [∀ j, DecidableEq (V j)]
+  [∀ j, Field (K j)] [∀ j, NumberField (K j)]
+variable (G : ∀ j, Core.Fibers (V j) (I j)) (θ γ c : ℝ) (η : ℕ → ℝ)
+  (Z₁ Z₂ : ∀ j, V j → K j) (h₁ : ∀ j, Function.Injective (Z₁ j)) (h₂ : ∀ j, Function.Injective (Z₂ j))
+
+/-- Application of the scalar rate to the genuinely prepared arithmetic graphs,
+with no degree bound on the sequence of number fields. -/
+theorem arithmetic_small_overlap_rate (hθ : 0<θ)
+    (hn : Tendsto (fun j => (Fintype.card (V j) : ℝ)) atTop atTop)
+    (hη : Tendsto η atTop (𝓝 0)) (hη0 : ∀ j, 0≤η j)
+    (hG : ∀ j, PreparedGraph (V j) (I j) (G j) θ γ c (η j))
+    (t : ∀ j, I j → K j)
+    (ht : ∀ j e y, y ∈ (G j).targets e →
+      (Z₁ j ((G j).source e)-Z₁ j y)*(Z₂ j ((G j).source e)-Z₂ j y)=t j e) :
+    Tendsto (fun j => smallOverlapScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j) c /
+      (overlapScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)+potentialScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)+1))
+      atTop (𝓝 0) := by
+  apply small_overlap_rate (fun j => (Fintype.card (V j):ℝ)) η _ _ _ θ hθ hn hη hη0
+    (fun j => overlapScale_nonneg _ _ _ _) (fun j => potentialScale_nonneg _ _ _ _)
+    (fun j => smallOverlapScale_nonneg _ _ _ _ c)
+  intro τ H hτ hτ1 hH
+  filter_upwards [hη.eventually (eventually_lt_nhds hθ),
+    (Real.tendsto_log_atTop.comp hn).eventually (eventually_ge_atTop (2:ℝ)),
+    hn.eventually (eventually_ge_atTop (2/τ))] with j hjη hjlog hjn
+  have hlog : 0 ≤ 1/Real.log (Fintype.card (V j):ℝ) := by positivity
+  have hlog2 : 1/Real.log (Fintype.card (V j):ℝ) ≤ 1/2 := by
+    exact one_div_le_one_div_of_le (by norm_num) hjlog
+  have hτn : 2≤τ*(Fintype.card (V j):ℝ) := by
+    have := (div_le_iff₀ hτ).mp hjn
+    nlinarith
+  exact (hG j).arithmetic_small_overlap (Z₁ j) (Z₂ j) (h₁ j) (h₂ j) hjη (hη0 j)
+    (t j) (ht j) τ H hlog hlog2 hτ hτ1 hτn hH
+
+end WeakPinned
+
+namespace WeakPinned.Core.Fibers
+open scoped BigOperators
+open Variance
+variable {V : Type u4} {I : Type u5} [Fintype V] [Fintype I] [DecidableEq V]
+variable (G : Fibers V I)
+
+def retainedWeight (T : ℝ) (y : V) : ℝ := if T≤((G.incoming y).card:ℝ) then G.incomingWeight y else 0
+
+def retainedSet (T : ℝ) (y : V) : Finset V := if T≤((G.incoming y).card:ℝ) then G.incoming y else Finset.univ
+
+omit [Fintype I] in
+lemma retainedWeight_nonneg (T : ℝ) (y : V) : 0≤G.retainedWeight T y := by
+  unfold retainedWeight incomingWeight edgeCount
+  split_ifs <;> positivity
+
+omit [Fintype I] in
+lemma retainedWeight_le (T : ℝ) (y : V) : G.retainedWeight T y≤G.incomingWeight y := by
+  unfold retainedWeight
+  split_ifs
+  · exact le_rfl
+  · unfold incomingWeight edgeCount; positivity
+
+omit [Fintype I] in
+lemma sum_retainedWeight (T : ℝ) (hG : 0<G.edgeCount) : (∑ y, G.retainedWeight T y)≤1 := by
+  rw [← G.sum_incomingWeight hG]
+  exact Finset.sum_le_sum (fun y _ => G.retainedWeight_le T y)
+
+omit [Fintype I] in
+lemma retainedSet_size (T : ℝ) (hT : T≤Fintype.card V) (y : V) : T≤((G.retainedSet T y).card:ℝ) := by
+  unfold retainedSet
+  split_ifs with h
+  · exact h
+  · simpa using hT
+
+omit [Fintype I] in
+lemma retained_domination (T M : ℝ) (hdom : ∀ x y, G.edgeLaw x y≤M*pairProb Finset.univ x y)
+    (hn : 2≤Fintype.card V) (x : V) :
+    (∑ y, G.retainedWeight T y*prob (G.retainedSet T y) x)≤M*prob Finset.univ x := by
+  refine le_trans (Finset.sum_le_sum (fun y _ => ?_)) (G.incoming_domination M hdom hn x)
+  unfold retainedWeight retainedSet
+  split_ifs
+  · exact le_rfl
+  · simp only [zero_mul]
+    unfold incomingWeight edgeCount prob
+    split_ifs <;> positivity
+end WeakPinned.Core.Fibers
+
+namespace WeakPinned
+open Filter Variance Global
+open scoped Topology BigOperators
+variable {V : ℕ → Type u6} {I : ℕ → Type u7} {K : ℕ → Type u8} [∀ j, Fintype (V j)] [∀ j, Fintype (I j)] [∀ j, DecidableEq (V j)]
+  [∀ j, Field (K j)] [∀ j, NumberField (K j)]
+variable (G : ∀ j, Core.Fibers (V j) (I j)) (θ γ c : ℝ) (η : ℕ → ℝ)
+  (Z₁ Z₂ : ∀ j, V j → K j) (h₁ : ∀ j, Function.Injective (Z₁ j)) (h₂ : ∀ j, Function.Injective (Z₂ j))
+
+lemma PreparedGraph.uniform_domination {V : Type u9} {I : Type u10} [Fintype V] [Fintype I] [DecidableEq V]
+    {G : Core.Fibers V I} {θ γ c η : ℝ} (h : PreparedGraph V I G θ γ c η) (hθ : 0<θ)
+    (hη : η≤θ/2) (x y : V) : G.edgeLaw x y≤(2/θ)*pairProb Finset.univ x y := by
+  have hh : η<θ := by linarith
+  refine (h.edge_domination hh x y).trans (mul_le_mul_of_nonneg_right ?_ (by
+    unfold pairProb
+    split_ifs
+    · have hn : (2:ℝ)≤Fintype.card V := by exact_mod_cast h.card_ge
+      simp only [Finset.card_univ]
+      exact div_nonneg (by norm_num) (mul_nonneg (by positivity) (by linarith))
+    · exact le_rfl))
+  apply (div_le_div_iff₀ (by linarith : 0<θ-η) hθ).mpr
+  linarith
+
+def fiberVariance (j : ℕ) : ℝ := weightedIntegral (totalMass (Z₁ j) (Z₂ j) (h₁ j) (h₂ j))
+  (varianceCell (G j).fiberWeight (G j).targets ((Fintype.card (V j):ℝ)^(1-c)))
+
+def incomingVariance (j : ℕ) : ℝ := weightedIntegral (totalMass (Z₁ j) (Z₂ j) (h₁ j) (h₂ j))
+  (varianceCell ((G j).retainedWeight ((Fintype.card (V j):ℝ)^γ))
+    ((G j).retainedSet ((Fintype.card (V j):ℝ)^γ)) ((Fintype.card (V j):ℝ)^(1-c)))
+
+/-- Both source sampling families satisfy the proved vanishing arithmetic
+variance estimate, on every sequence of prepared models. No uniform degree
+bound on the number fields is used. -/
+theorem sampling_variance_rates (hθ : 0<θ) (hγ : 0<γ) (hγ1 : γ<1) (hcγ : c<γ)
+    (hn : Tendsto (fun j => (Fintype.card (V j):ℝ)) atTop atTop)
+    (hη : Tendsto η atTop (𝓝 0)) (hη0 : ∀ j, 0≤η j) (hηb : ∀ j, η j≤θ/2)
+    (hG : ∀ j, PreparedGraph (V j) (I j) (G j) θ γ c (η j))
+    (t : ∀ j, I j → K j) (ht : ∀ j e y, y ∈ (G j).targets e →
+      (Z₁ j ((G j).source e)-Z₁ j y)*(Z₂ j ((G j).source e)-Z₂ j y)=t j e) :
+    Tendsto (fun j => fiberVariance G c Z₁ Z₂ h₁ h₂ j /
+      (overlapScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)+potentialScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)+1)) atTop (𝓝 0) ∧
+    Tendsto (fun j => incomingVariance G γ c Z₁ Z₂ h₁ h₂ j /
+      (overlapScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)+potentialScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)+1)) atTop (𝓝 0) := by
+  have hsmall := arithmetic_small_overlap_rate V I K G θ γ c η Z₁ Z₂ h₁ h₂ hθ hn hη hη0 hG t ht
+  have hηlt (j) : η j<θ := by linarith [hηb j]
+  have hpos (j) : 0<(G j).edgeCount := (hG j).edgeCount_pos (hηlt j)
+  have hdom (j x y) := (hG j).uniform_domination hθ (hηb j) x y
+  constructor
+  · exact arithmetic_variance_rate Z₁ Z₂ h₁ h₂ (fun j => (G j).fiberWeight) (fun j => (G j).targets)
+      γ c (2/θ) hγ hγ1 hcγ (hcγ.trans hγ1) (by positivity) hn
+      (fun j => (G j).fiberWeight_nonneg) (fun j => (G j).sum_fiberWeight (hpos j) |>.le)
+      (Eventually.of_forall fun j e => (hG j).fiber e |>.le)
+      (Eventually.of_forall fun j x => (G j).fiber_domination _ (hdom j) (hG j).card_ge x) hsmall
+  · apply arithmetic_variance_rate Z₁ Z₂ h₁ h₂
+      (fun j => (G j).retainedWeight ((Fintype.card (V j):ℝ)^γ))
+      (fun j => (G j).retainedSet ((Fintype.card (V j):ℝ)^γ)) γ c (2/θ)
+      hγ hγ1 hcγ (hcγ.trans hγ1) (by positivity) hn
+      (fun j y => (G j).retainedWeight_nonneg _ y)
+      (fun j => (G j).sum_retainedWeight _ (hpos j)) _ _ hsmall
+    · apply Eventually.of_forall
+      intro j y
+      apply (G j).retainedSet_size
+      have hn1 : (1:ℝ)≤Fintype.card (V j) := by exact_mod_cast (by have := (hG j).card_ge; omega : 1≤Fintype.card (V j))
+      simpa only [Real.rpow_one] using Real.rpow_le_rpow_of_exponent_le hn1 hγ1.le
+    · exact Eventually.of_forall fun j x => (G j).retained_domination _ _ (hdom j) (hG j).card_ge x
+
+end WeakPinned
+
+end
+end
+end
+end
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PinnedDistances/PartitionIntegrals.lean b/lean/OAI/Geometry/PinnedDistances/PartitionIntegrals.lean
new file mode 100644
index 0000000000000000000000000000000000000000..57cada3cc01f01a7bcad1301b1b30b70adc4fddc
--- /dev/null
+++ b/lean/OAI/Geometry/PinnedDistances/PartitionIntegrals.lean
@@ -0,0 +1,460 @@
+import OAI.Geometry.PinnedDistances.GridBounds
+
+namespace OAI
+
+universe u1 u2 u3 u4 u5 u6 u7 u8 u9 u10
+
+section
+noncomputable section
+open scoped BigOperators Topology
+open Filter MeasureTheory
+namespace WeakPinned.Hierarchy.Depth
+variable {P : Type u1} [Fintype P] [DecidableEq P]
+variable (d : Depth P)
+
+lemma sum_by_cells (r : ℝ) (f : P → ℝ) :
+    (∑ C ∈ d.cells r, ∑ x∈C, f x)=∑ x,f x := by
+  have hh (x : P) : (∑ C ∈ d.cells r, if x∈C then f x else 0)=f x := by
+    rw [Finset.sum_eq_single (d.cell r x)]
+    · simp [d.share_refl]
+    · intro C hC hne
+      rw [ite_eq_right]
+      exact fun hx => hne (d.cell_of_mem hC hx)
+    · simp
+  calc
+    _ = ∑ C ∈ d.cells r, ∑ x, if x∈C then f x else 0 := by simp
+    _ = ∑ x, ∑ C ∈ d.cells r, if x∈C then f x else 0 := Finset.sum_comm
+    _ = _ := by simp_rw [hh]
+
+lemma sum_cell_value (r : ℝ) (p : P → ℝ) (v : Finset P → ℝ) :
+    (∑ x,p x*v (d.cell r x))=∑ C∈d.cells r, v C*(∑ x∈C,p x) := by
+  rw [← d.sum_by_cells r (fun x => p x*v (d.cell r x))]
+  apply Finset.sum_congr rfl
+  intro C hC
+  rw [Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro x hx
+  rw [← d.cell_of_mem hC hx]
+  ring
+
+end WeakPinned.Hierarchy.Depth
+
+namespace WeakPinned.Bounded
+open Hierarchy Variance
+variable {P : Type u2} [Fintype P] [DecidableEq P]
+
+omit [DecidableEq P] in
+lemma weighted_test_bound (p f g : P → ℝ) (hp : ∀ x,0 ≤ p x) (h1 : ∑ x,p x=1)
+    (η : ℝ) (hfg : ∀ x,|f x-g x| ≤ η) :
+    |(∑ x,p x*f x)-(∑ x,p x*g x)| ≤ η := by
+  rw [← Finset.sum_sub_distrib]
+  calc
+    _  ≤  ∑ x, |p x*f x-p x*g x| := Finset.abs_sum_le_sum_abs _ _
+    _ = ∑ x,p x*|f x-g x| := by simp_rw [← mul_sub,abs_mul,abs_of_nonneg (hp _)]
+    _  ≤  ∑ x,p x*η := Finset.sum_le_sum (fun x _ => mul_le_mul_of_nonneg_left (hfg x) (hp x))
+    _ = η := by rw [← Finset.sum_mul,h1,one_mul]
+
+lemma partition_test_bound (d : Depth P) (r : ℝ) (p a f : P → ℝ)
+    (hp : ∀ x,0 ≤ p x) (ha : ∀ x,0 ≤ a x) (hp1 : ∑ x,p x=1) (ha1 : ∑ x,a x=1)
+    (A : Finset (Finset P)) (hA : A⊆d.cells r) (v : Finset P → ℝ) (η H : ℝ)
+    (hf : ∀ x,|f x-v (d.cell r x)| ≤ η)
+    (hv : ∀ C∈d.cells r, |v C| ≤ if C∈A then H else 0) :
+    |(∑ x,p x*f x)-(∑ x,a x*f x)|  ≤
+      2*η+H*(∑ C∈A, |(∑ x∈C,p x)-(∑ x∈C,a x)|) := by
+  have hpa : |(∑ x,p x*v (d.cell r x))-(∑ x,a x*v (d.cell r x))| ≤
+      H*(∑ C∈A,|(∑ x∈C,p x)-(∑ x∈C,a x)|) := by
+    rw [d.sum_cell_value,d.sum_cell_value,← Finset.sum_sub_distrib]
+    calc
+      _  ≤  ∑ C∈d.cells r, |v C*(∑ x∈C,p x)-v C*(∑ x∈C,a x)| := Finset.abs_sum_le_sum_abs _ _
+      _  ≤  ∑ C∈d.cells r, if C∈A then H*|(∑ x∈C,p x)-(∑ x∈C,a x)| else 0 := by
+        apply Finset.sum_le_sum
+        intro C hC
+        rw [← mul_sub,abs_mul]
+        simpa only [ite_mul,zero_mul] using mul_le_mul_of_nonneg_right (hv C hC) (abs_nonneg _)
+      _ = _ := by rw [Finset.sum_ite_mem,Finset.inter_eq_right.mpr hA,Finset.mul_sum]
+  have h₁ := weighted_test_bound p f (fun x => v (d.cell r x)) hp hp1 η hf
+  have h₂ := weighted_test_bound a f (fun x => v (d.cell r x)) ha ha1 η hf
+  have hh := abs_add_three ((∑ x,p x*f x)-(∑ x,p x*v (d.cell r x)))
+    ((∑ x,p x*v (d.cell r x))-(∑ x,a x*v (d.cell r x)))
+    ((∑ x,a x*v (d.cell r x))-(∑ x,a x*f x))
+  rw [abs_sub_comm (∑ x,a x*v (d.cell r x))] at hh
+  convert le_trans hh (by linarith : _ ≤ 2*η+H*(∑ C∈A, |(∑ x∈C,p x)-(∑ x∈C,a x)|)) using 1
+  congr 1
+  ring
+
+lemma cellProb_compl (B C : Finset P) (hB : B.Nonempty) :
+    cellProb B Cᶜ=1-cellProb B C := by
+  rw [← sum_prob_cell,← sum_prob_cell,← sum_prob B hB]
+  have hh := Finset.sum_compl_add_sum C (prob B)
+  linarith
+
+variable {I : Type u3} [Fintype I]
+
+def meanCellDiscrep (w : I → ℝ) (B : I → Finset P) (C : Finset P) : ℝ :=
+  ∑ i,w i*|cellProb (B i) C-cellProb Finset.univ C|
+
+lemma meanCellDiscrep_nonneg (w : I → ℝ) (B : I → Finset P) (hw : ∀ i,0 ≤ w i) (C : Finset P) :
+    0 ≤ meanCellDiscrep w B C := Finset.sum_nonneg (fun i _ => mul_nonneg (hw i) (abs_nonneg _))
+
+lemma meanCellDiscrep_compl (w : I → ℝ) (B : I → Finset P) (hB : ∀ i,(B i).Nonempty)
+    (hn : 0<Fintype.card P) (C : Finset P) :
+    meanCellDiscrep w B Cᶜ=meanCellDiscrep w B C := by
+  unfold meanCellDiscrep
+  apply Finset.sum_congr rfl
+  intro i _
+  rw [cellProb_compl _ _ (hB i),cellProb_compl _ _ (Finset.univ_nonempty_iff.mpr (Fintype.card_pos_iff.mp hn)),
+    sub_sub_sub_cancel_left,abs_sub_comm]
+
+lemma meanCellDiscrep_small (w : I → ℝ) (B : I → Finset P) (M : ℝ)
+    (hw : ∀ i,0 ≤ w i) (hW : ∑ i,w i ≤ 1)
+    (hdom : ∀ x,(∑ i,w i*prob (B i) x) ≤ M*prob Finset.univ x) (C : Finset P) :
+    meanCellDiscrep w B C ≤ (M+1)*cellProb Finset.univ C := by
+  have hmean := mean_cell_bound w B C M hdom
+  calc
+    _  ≤  ∑ i,w i*(cellProb (B i) C+cellProb Finset.univ C) := by
+      apply Finset.sum_le_sum
+      intro i _
+      exact mul_le_mul_of_nonneg_left (abs_sub_le_iff.mpr ⟨by linarith [cellProb_nonneg (Finset.univ : Finset P) C],by linarith [cellProb_nonneg (B i) C]⟩) (hw i)
+    _ = (∑ i,w i*cellProb (B i) C)+(∑ i,w i)*cellProb Finset.univ C := by simp [mul_add,Finset.sum_add_distrib,Finset.sum_mul]
+    _  ≤  (M+1)*cellProb Finset.univ C := by nlinarith [mul_le_mul_of_nonneg_right hW (cellProb_nonneg (Finset.univ : Finset P) C)]
+
+lemma meanCellDiscrep_young (w : I → ℝ) (B : I → Finset P) (M T δ : ℝ)
+    (hw : ∀ i,0 ≤ w i) (hW : ∑ i,w i ≤ 1) (hM : 0 ≤ M) (hT : 0 ≤ T) (hδ : 0<δ)
+    (hdom : ∀ x,(∑ i,w i*prob (B i) x) ≤ M*prob Finset.univ x)
+    (C : Finset P) :
+    meanCellDiscrep w B C  ≤  varianceCell w B T C/δ+δ+(M+1)*T/Fintype.card P := by
+  by_cases hC : T<(C.card:ℝ)
+  · have hy (t : ℝ) : |t| ≤ t^2/δ+δ := by
+      have hh : |t| *δ ≤ t^2+δ*δ := by nlinarith [sq_nonneg (|t| -δ),sq_abs t]
+      calc
+        |t| = (|t| *δ)/δ := by field_simp
+        _ ≤ (t^2+δ*δ)/δ := div_le_div_of_nonneg_right hh hδ.le
+        _ = _ := by field_simp
+    have hh : meanCellDiscrep w B C ≤ varianceCell w B T C/δ+δ := by
+      rw [varianceCell,ite_eq_left hC]
+      calc
+        _  ≤  ∑ i,w i*((cellProb (B i) C-cellProb Finset.univ C)^2/δ+δ) :=
+          Finset.sum_le_sum (fun i _ => mul_le_mul_of_nonneg_left (hy _) (hw i))
+        _ = (∑ i,w i*(cellProb (B i) C-cellProb Finset.univ C)^2)/δ+(∑ i,w i)*δ := by
+          simp only [mul_add,← mul_div_assoc,Finset.sum_add_distrib,← Finset.sum_div,← Finset.sum_mul]
+        _  ≤  _ := by nlinarith [mul_le_mul_of_nonneg_right hW hδ.le]
+    exact hh.trans (le_add_of_nonneg_right (by positivity))
+  · have hh := meanCellDiscrep_small w B M hw hW hdom C
+    have hc : cellProb (Finset.univ : Finset P) C ≤ T/Fintype.card P := by
+      simp only [cellProb,Finset.univ_inter,Finset.card_univ]
+      exact div_le_div_of_nonneg_right (le_of_not_gt hC) (by positivity)
+    have hv := varianceCell_nonneg w B T hw C
+    have hm := mul_le_mul_of_nonneg_left hc (show 0 ≤ M+1 by linarith)
+    rw [← mul_div_assoc] at hm
+    linarith [div_nonneg hv hδ.le]
+
+end WeakPinned.Bounded
+
+end
+end
+section
+noncomputable section
+open scoped BigOperators Topology
+open Filter MeasureTheory
+namespace WeakPinned.Hierarchy.Depth
+variable {P : Type u4} [Fintype P] [DecidableEq P]
+variable (d : Depth P)
+
+def transformCell (C : Finset P) : Finset P := if Fintype.card P<2*C.card then Cᶜ else C
+
+lemma sum_transformCell (r : ℝ) (f : Finset P → ℝ) :
+    (∑ C∈d.cells r,f (transformCell C))=d.sumCells f r := by
+  classical
+  simp only [sumCells,transformed,Multiset.map_add,Multiset.sum_add,Multiset.map_map,
+    Function.comp_def,transformCell]
+  simp_rw [apply_ite f]
+  rw [Finset.sum_ite]
+  change (∑ C∈(d.cells r).filter (fun C => Fintype.card P<2*C.card),f Cᶜ)+
+    (∑ C∈(d.cells r).filter (fun C => ¬Fintype.card P<2*C.card),f C) =
+    (∑ C∈(d.cells r).filter (fun C => ¬Fintype.card P<2*C.card),f C)+
+    (∑ C∈(d.cells r).filter (fun C => Fintype.card P<2*C.card),f Cᶜ)
+  ring
+
+end WeakPinned.Hierarchy.Depth
+
+namespace WeakPinned.Bounded
+open Hierarchy Variance
+variable {P : Type u5} {I : Type u6} [Fintype P] [DecidableEq P] [Fintype I]
+
+lemma meanCellDiscrep_transform (w : I → ℝ) (B : I → Finset P) (hB : ∀ i,(B i).Nonempty)
+    (hn : 0<Fintype.card P) (C : Finset P) :
+    meanCellDiscrep w B (Depth.transformCell C)=meanCellDiscrep w B C := by
+  unfold Depth.transformCell
+  split_ifs
+  · exact meanCellDiscrep_compl w B hB hn C
+  · rfl
+
+lemma active_meanDiscrep_bound (d : Depth P) (r : ℝ) (A : Finset (Finset P))
+    (N M T δ : ℝ) (w : I → ℝ) (B : I → Finset P)
+    (hn : 0<Fintype.card P) (hA : A⊆d.cells r) (hN : (A.card:ℝ) ≤ N)
+    (hw : ∀ i,0 ≤ w i) (hW : ∑ i,w i ≤ 1) (hB : ∀ i,(B i).Nonempty)
+    (hM : 0 ≤ M) (hT : 0 ≤ T) (hδ : 0<δ)
+    (hdom : ∀ x,(∑ i,w i*prob (B i) x) ≤ M*prob Finset.univ x) :
+    (∑ C∈A,meanCellDiscrep w B C) ≤
+      d.sumCells (varianceCell w B T) r/δ+N*(δ+(M+1)*T/Fintype.card P) := by
+  have hv (C) : 0 ≤ varianceCell w B T C := varianceCell_nonneg w B T hw C
+  calc
+    _ = ∑ C∈A,meanCellDiscrep w B (Depth.transformCell C) := by
+      simp_rw [meanCellDiscrep_transform w B hB hn]
+    _ ≤ ∑ C∈A,(varianceCell w B T (Depth.transformCell C)/δ+(δ+(M+1)*T/Fintype.card P)) := by
+      apply Finset.sum_le_sum
+      intro C _
+      simpa only [add_assoc] using meanCellDiscrep_young w B M T δ hw hW hM hT hδ hdom (Depth.transformCell C)
+    _ = (∑ C∈A,varianceCell w B T (Depth.transformCell C))/δ+
+        (A.card:ℝ)*(δ+(M+1)*T/Fintype.card P) := by
+      simp [Finset.sum_add_distrib,Finset.sum_div]
+      ring
+    _ ≤ (∑ C∈d.cells r,varianceCell w B T (Depth.transformCell C))/δ+
+        N*(δ+(M+1)*T/Fintype.card P) := by
+      apply add_le_add
+      · exact div_le_div_of_nonneg_right (Finset.sum_le_sum_of_subset_of_nonneg hA (fun C _ _ => hv _)) hδ.le
+      · apply mul_le_mul_of_nonneg_right hN
+        positivity
+    _ = _ := by rw [d.sum_transformCell]
+
+def testDiscrep (w : I → ℝ) (B : I → Finset P) (f : P → ℝ) : ℝ :=
+  ∑ i,w i*|(∑ x,prob (B i) x*f x)-(∑ x,prob Finset.univ x*f x)|
+
+lemma testDiscrep_nonneg (w : I → ℝ) (B : I → Finset P) (f : P → ℝ) (hw : ∀ i,0 ≤ w i) :
+    0 ≤ testDiscrep w B f := Finset.sum_nonneg (fun i _ => mul_nonneg (hw i) (abs_nonneg _))
+
+lemma family_partition_test_bound (d : Depth P) (r : ℝ) (A : Finset (Finset P))
+    (N M T δ η H : ℝ) (w : I → ℝ) (B : I → Finset P)
+    (hn : 0<Fintype.card P) (hA : A⊆d.cells r) (hN : (A.card:ℝ) ≤ N)
+    (hw : ∀ i,0 ≤ w i) (hW : ∑ i,w i ≤ 1) (hB : ∀ i,(B i).Nonempty)
+    (hM : 0 ≤ M) (hT : 0 ≤ T) (hδ : 0<δ) (hη : 0 ≤ η) (hH : 0 ≤ H)
+    (hdom : ∀ x,(∑ i,w i*prob (B i) x) ≤ M*prob Finset.univ x)
+    (f : P → ℝ) (v : Finset P → ℝ)
+    (hf : ∀ x,|f x-v (d.cell r x)| ≤ η)
+    (hv : ∀ C∈d.cells r,|v C| ≤ if C∈A then H else 0) :
+    testDiscrep w B f ≤ 2*η+H*(d.sumCells (varianceCell w B T) r/δ+
+      N*(δ+(M+1)*T/Fintype.card P)) := by
+  have hp (S : Finset P) (x : P) : 0 ≤ prob S x := by unfold prob; split_ifs <;> positivity
+  have hh (i : I) := partition_test_bound d r (prob (B i)) (prob Finset.univ) f
+    (hp _) (hp _) (sum_prob _ (hB i))
+    (sum_prob _ (Finset.univ_nonempty_iff.mpr (Fintype.card_pos_iff.mp hn))) A hA v η H hf hv
+  simp only [sum_prob_cell] at hh
+  have hmean := active_meanDiscrep_bound d r A N M T δ w B hn hA hN hw hW hB hM hT hδ hdom
+  calc
+    _ ≤ ∑ i,w i*(2*η+H*(∑ C∈A,|cellProb (B i) C-cellProb Finset.univ C|)) :=
+      Finset.sum_le_sum (fun i _ => mul_le_mul_of_nonneg_left (hh i) (hw i))
+    _ = (∑ i,w i)*(2*η)+H*(∑ C∈A,meanCellDiscrep w B C) := by
+      simp only [mul_add,Finset.sum_add_distrib,← Finset.sum_mul]
+      congr 1
+      simp only [meanCellDiscrep,Finset.mul_sum]
+      rw [Finset.sum_comm]
+      apply Finset.sum_congr rfl
+      intro C _
+      apply Finset.sum_congr rfl
+      intro i _
+      ring
+    _ ≤ _ := add_le_add (by simpa only [one_mul] using mul_le_mul_of_nonneg_right hW (show 0≤2*η by positivity))
+      (mul_le_mul_of_nonneg_left hmean hH)
+
+end WeakPinned.Bounded
+
+end
+end
+section
+noncomputable section
+open scoped BigOperators Topology
+open Filter MeasureTheory
+namespace WeakPinned.Bounded
+open Hierarchy
+variable {P : Type u7} [Fintype P] [DecidableEq P]
+
+lemma grid_test_data (d : Depth P) (u : P → ℂ) (ω : Grid.FullSample) (r m : ℝ) (b : ℂ)
+    (hn : 0<Fintype.card P)
+    (hshare : ∀ x y,d.share r x y ↔ Grid.label ω r (u x)=Grid.label ω r (u y))
+    (R L η H ε : ℝ) (M : ℕ) (hR : 0≤R) (hL : 0<L)
+    (hside : L ≤ Grid.side ω r m) (hupper : Grid.side ω r m ≤ ε)
+    (hM : 2*R/L+1≤M)
+    (f : ℂ → ℝ) (hbound : ∀ z,|f z|≤H)
+    (hcont : ∀ z w,‖z-w‖<2*ε → |f z-f w|≤η)
+    (hout : ∀ z,R<‖z‖ → |f z|≤η) :
+    ∃ A : Finset (Finset P), ∃ v : Finset P → ℝ,
+      A⊆d.cells r ∧ A.card≤(2*M+1)^2 ∧
+      (∀ x,|f ((Real.exp m:ℂ)*u x+b)-v (d.cell r x)|≤η) ∧
+      (∀ C∈d.cells r,|v C|≤ if C∈A then H else 0) := by
+  classical
+  let : Nonempty P := Fintype.card_pos_iff.mp hn
+  let D : Finset P := Finset.univ.filter (fun x => ‖(Real.exp m:ℂ)*u x+b‖≤R)
+  let A := D.image (d.cell r)
+  let rep (C : Finset P) : P := if h : C∈A then (Finset.mem_image.mp h).choose else Classical.arbitrary P
+  have hr (C : Finset P) (hC : C∈A) : rep C∈D ∧ d.cell r (rep C)=C := by
+    simpa only [rep,dite_eq_left hC] using (Finset.mem_image.mp hC).choose_spec
+  have hAc : A⊆d.cells r := Finset.image_subset_image (Finset.filter_subset _ _)
+  have hcount : A.card≤(2*M+1)^2 := by
+    apply le_trans (Finset.card_le_card_of_injOn (fun C => Grid.label ω r (u (rep C)))
+      (t:=D.image (fun x => Grid.label ω r (u x))) ?_ ?_)
+    · apply Grid.card_labels_bound D u ω r m b R L M hR hL hside hM
+      intro x hx
+      exact (Finset.mem_filter.mp hx).2
+    · intro C hC
+      exact Finset.mem_image.mpr ⟨rep C,(hr C hC).1,rfl⟩
+    · intro C hC E hE he
+      rw [← (hr C hC).2,← (hr E hE).2]
+      exact d.cell_eq_of_share ((hshare _ _).mpr he)
+  let v (C : Finset P) := if C∈A then f ((Real.exp m:ℂ)*u (rep C)+b) else 0
+  refine ⟨A,v,hAc,hcount,?_,?_⟩
+  · intro x
+    by_cases hx : d.cell r x∈A
+    · simp only [v,ite_eq_left hx]
+      apply hcont
+      have he : d.share r x (rep (d.cell r x)) := by
+        rw [← d.mem_cell]
+        have hmem := d.self_mem_cell r (rep (d.cell r x))
+        rwa [(hr _ hx).2] at hmem
+      have hd := Grid.label_diameter (m:=m) ((hshare _ _).mp he)
+      have heq : ((Real.exp m:ℂ)*u x+b)-((Real.exp m:ℂ)*u (rep (d.cell r x))+b)=
+          (Real.exp m:ℂ)*u x-(Real.exp m:ℂ)*u (rep (d.cell r x)) := by ring
+      rw [heq]
+      exact hd.trans_le (by linarith)
+    · simp only [v,ite_eq_right hx,sub_zero]
+      apply hout
+      by_contra hh
+      have hxD : x∈D := Finset.mem_filter.mpr ⟨Finset.mem_univ x,le_of_not_gt hh⟩
+      exact hx (Finset.mem_image.mpr ⟨x,hxD,rfl⟩)
+  · intro C _
+    by_cases hC : C∈A
+    · simpa only [v,ite_eq_left hC] using hbound ((Real.exp m:ℂ)*u (rep C)+b)
+    · simp only [v,ite_eq_right hC,abs_zero,le_refl]
+
+end WeakPinned.Bounded
+
+end
+end
+section
+noncomputable section
+open scoped BigOperators Topology
+open Filter MeasureTheory
+namespace WeakPinned.Bounded
+
+lemma le_unit_integral (g : ℝ → ℝ) (hg : Integrable g) (hg0 : ∀ r,0≤g r)
+    (t D a b : ℝ) (hb : 0≤b) (h : ∀ r∈Set.Icc t (t+1), D≤a+b*g r) :
+    D≤a+b*(∫ r,g r) := by
+  have hi (c : ℝ) : IntegrableOn (fun _ : ℝ => c) (Set.Icc t (t+1)) :=
+    integrableOn_const isCompact_Icc.measure_lt_top.ne
+  have hv : (volume (Set.Icc t (t+1))).toReal=1 := by simp [Real.volume_Icc]
+  have hc (c : ℝ) : (∫ r in Set.Icc t (t+1),c)=c := by
+    rw [integral_const]
+    simp only [Measure.real,Measure.restrict_apply_univ,hv,one_smul]
+  have hh := setIntegral_mono_on (hi D) ((hi a).add (hg.integrableOn.const_mul b)) measurableSet_Icc h
+  simp only [Pi.add_apply] at hh
+  rw [integral_add (hi a) (hg.integrableOn.const_mul b),hc D,hc a,integral_const_mul] at hh
+  exact hh.trans (add_le_add le_rfl (mul_le_mul_of_nonneg_left
+    (setIntegral_le_integral hg (ae_of_all _ hg0)) hb))
+
+end WeakPinned.Bounded
+
+namespace WeakPinned.Grid
+lemma side_unit_bounds (ω : FullSample) (m ε r : ℝ) (hε : 0<ε)
+    (hr : r∈Set.Icc (m-Real.log ε) (m-Real.log ε+1)) :
+    ε/(2*Real.exp 1) ≤ side ω r m ∧ side ω r m ≤ ε := by
+  have he : Real.exp (Real.log ε-1)/2=ε/(2*Real.exp 1) := by
+    rw [Real.exp_sub,Real.exp_log hε]
+    ring
+  constructor
+  · rw [← he]
+    exact (div_le_div_of_nonneg_right (Real.exp_le_exp.mpr (by linarith [hr.2])) (by norm_num)).trans
+      (side_bounds ω r m).1.le
+  · exact (side_bounds ω r m).2.trans (by rw [← Real.exp_log hε]; exact Real.exp_le_exp.mpr (by linarith [hr.1]))
+end WeakPinned.Grid
+
+end
+end
+section
+noncomputable section
+open scoped BigOperators Topology
+open Filter MeasureTheory
+namespace WeakPinned.Bounded
+open Hierarchy Variance Global NumberField
+variable {P : Type u8} {I : Type u9} [Fintype P] [DecidableEq P] [Fintype I]
+
+lemma grid_test_integrated (d : Depth P) (u : P → ℂ) (ω : Grid.FullSample) (m : ℝ) (b : ℂ)
+    (hn : 2≤ Fintype.card P)
+    (hshare : ∀ r x y,d.share r x y ↔ Grid.label ω r (u x)=Grid.label ω r (u y))
+    (R ε η H M T δ : ℝ) (N : ℕ) (hR : 0≤ R) (hε : 0<ε)
+    (hN : 2*R/(ε/(2*Real.exp 1))+1≤ N)
+    (hη : 0≤ η) (hH : 0≤ H) (hM : 0≤ M) (hT : 1≤ T) (hδ : 0<δ)
+    (w : I → ℝ) (B : I → Finset P)
+    (hw : ∀ i,0≤ w i) (hW : ∑ i,w i≤ 1) (hB : ∀ i,2≤ (B i).card)
+    (hdom : ∀ x, (∑ i,w i*prob (B i) x)≤ M*prob Finset.univ x)
+    (f : ℂ → ℝ) (hbound : ∀ z,|f z|≤ H)
+    (hcont : ∀ z z',‖z-z'‖<2*ε → |f z-f z'|≤ η)
+    (hout : ∀ z,R<‖z‖ → |f z|≤ η) :
+    testDiscrep w B (fun x => f ((Real.exp m:ℂ)*u x+b)) ≤
+      2*η+H*(weightedIntegral d.mass (varianceCell w B T)/δ+
+        ((2*N+1)^2:ℕ)*(δ+(M+1)*T/Fintype.card P)) := by
+  have hq (C) : 0≤ varianceCell w B T C := varianceCell_nonneg w B T hw C
+  have hi := d.integrable_sumCells hn _ (varianceCell_singleton w B T hT hB)
+  have hsum (r) : 0≤ d.sumCells (varianceCell w B T) r := by
+    rw [d.sumCells_count]
+    exact Finset.sum_nonneg (fun C _ => mul_nonneg (d.count_nonneg r C) (hq C))
+  let a := 2*η+H*((((2*N+1)^2:ℕ):ℝ)*(δ+(M+1)*T/Fintype.card P))
+  have hh := le_unit_integral _ hi hsum (m-Real.log ε)
+    (testDiscrep w B (fun x => f ((Real.exp m:ℂ)*u x+b))) a (H/δ) (by positivity) ?_
+  · rw [d.integral_sumCells hn _ (varianceCell_singleton w B T hT hB)] at hh
+    convert hh using 1; simp only [a,weightedIntegral]; ring
+  · intro r hr
+    have hs := Grid.side_unit_bounds ω m ε r hε hr
+    obtain ⟨A,v,hA,hAc,hf,hv⟩ := grid_test_data d u ω r m b (by omega) (hshare r)
+      R (ε/(2*Real.exp 1)) η H ε N hR (by positivity) hs.1 hs.2 hN f hbound hcont hout
+    have h := family_partition_test_bound d r A (((2*N+1)^2:ℕ):ℝ) M T δ η H w B
+      (by omega) hA (by exact_mod_cast hAc) hw hW
+      (fun i => Finset.card_pos.mp (by have := hB i; omega)) hM (by linarith) hδ hη hH hdom _ v hf hv
+    convert h using 1; simp only [a]; ring
+
+variable {K : Type u10} [Field K] [NumberField K]
+
+omit [NumberField K] in
+lemma arch_test_bound (Z₁ Z₂ : P → K) (h₁ : Function.Injective Z₁) (h₂ : Function.Injective Z₂)
+    (v : InfinitePlace K) (m : ℝ) (b : ℂ) (hn : 2≤ Fintype.card P)
+    (R ε η H M T δ : ℝ) (N : ℕ) (hR : 0≤ R) (hε : 0<ε)
+    (hN : 2*R/(ε/(2*Real.exp 1))+1≤ N)
+    (hη : 0≤ η) (hH : 0≤ H) (hM : 0≤ M) (hT : 1≤ T) (hδ : 0<δ)
+    (w : I → ℝ) (B : I → Finset P)
+    (hw : ∀ i,0≤ w i) (hW : ∑ i,w i≤ 1) (hB : ∀ i,2≤ (B i).card)
+    (hdom : ∀ x, (∑ i,w i*prob (B i) x)≤ M*prob Finset.univ x)
+    (f : ℂ → ℝ) (hbound : ∀ z,|f z|≤ H)
+    (hcont : ∀ z z',‖z-z'‖<2*ε → |f z-f z'|≤ η)
+    (hout : ∀ z,R<‖z‖ → |f z|≤ η) :
+    testDiscrep w B (fun x => f ((Real.exp m:ℂ)*v.embedding (Z₁ x)+b)) ≤
+      2*η+H*(archMean Z₁ Z₂ (massEval (varianceCell w B T)) v/δ+
+        ((2*N+1)^2:ℕ)*(δ+(M+1)*T/Fintype.card P)) := by
+  let D := testDiscrep w B (fun x => f ((Real.exp m:ℂ)*v.embedding (Z₁ x)+b))
+  let a := 2*η+H*((((2*N+1)^2:ℕ):ℝ)*(δ+(M+1)*T/Fintype.card P))
+  let q := varianceCell w B T
+  have hq (C) : 0≤ q C := varianceCell_nonneg w B T hw C
+  have hi := integrable_massEval Z₁ Z₂ h₁ h₂ hn q v
+  have hh : ∀ᵐ ω ∂ Grid.fullLaw,D≤ a+(H/δ)*massEval q (arch Z₁ v ω) (arch Z₂ v ω) := by
+    filter_upwards [arch_label_ae Z₁ h₁ v] with ω hω
+    have hh := grid_test_integrated (arch Z₁ v ω) (fun x => v.embedding (Z₁ x)) ω m b hn hω
+      R ε η H M T δ N hR hε hN hη hH hM hT hδ w B hw hW hB hdom f hbound hcont hout
+    have hmass : weightedIntegral (arch Z₁ v ω).mass q≤ massEval q (arch Z₁ v ω) (arch Z₂ v ω) := by
+      apply le_add_of_nonneg_right
+      exact Finset.sum_nonneg (fun C _ => mul_nonneg ((arch Z₂ v ω).mass_nonneg C) (hq C))
+    have he : 2*η+H*(weightedIntegral (arch Z₁ v ω).mass q/δ+
+        ((2*N+1)^2:ℕ)*(δ+(M+1)*T/Fintype.card P))=a+(H/δ)*weightedIntegral (arch Z₁ v ω).mass q := by
+      simp only [a]; ring
+    rw [show varianceCell w B T=q from rfl,he] at hh
+    exact hh.trans (add_le_add le_rfl (mul_le_mul_of_nonneg_left hmass (by positivity)))
+  have hh' := integral_mono_ae (integrable_const D) ((integrable_const a).add (hi.const_mul (H/δ))) hh
+  simp only [Pi.add_apply] at hh'
+  rw [integral_add (integrable_const a) (hi.const_mul (H/δ)),integral_const_mul] at hh'
+  simp only [integral_const,probReal_univ,one_smul] at hh'
+  calc
+    _ ≤ a+(H/δ)*archMean Z₁ Z₂ (massEval q) v := hh'
+    _ = _ := by dsimp only [a,q]; ring
+
+end WeakPinned.Bounded
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PinnedDistances/Pins.lean b/lean/OAI/Geometry/PinnedDistances/Pins.lean
new file mode 100644
index 0000000000000000000000000000000000000000..2afd4b351f13646c0869679f0daa4216bd9c9d49
--- /dev/null
+++ b/lean/OAI/Geometry/PinnedDistances/Pins.lean
@@ -0,0 +1,222 @@
+import OAI.Geometry.PinnedDistances.Main
+
+namespace OAI
+
+section
+noncomputable section
+open Filter
+open scoped Topology BigOperators
+namespace WeakPinned
+
+def lightTargets (P : Finset Plane) (s : ℝ) (x : Plane) : Finset Plane :=
+  (P.erase x).filter (fun y => (k P x y : ℝ) < (P.card : ℝ)^s)
+
+def richTargets (P : Finset Plane) (s : ℝ) (x : Plane) : Finset Plane :=
+  (P.erase x).filter (fun y => (P.card : ℝ)^s ≤ (k P x y : ℝ))
+
+lemma light_card_le (P : Finset Plane) (s : ℝ) (x : Plane) :
+    ((lightTargets P s x).card : ℝ) ≤ (distances P x).card * (P.card : ℝ)^s := by
+  classical
+  have hmap : (↑(lightTargets P s x) : Set Plane).MapsTo (fun y => dist y x) (distances P x) := by
+    intro y hy
+    exact Finset.mem_image_of_mem _ (Finset.mem_filter.mp hy).1
+  have hcard := Finset.card_eq_sum_card_fiberwise hmap
+  rw [hcard, Nat.cast_sum]
+  calc
+    _ ≤ ∑ r ∈ distances P x, (P.card : ℝ)^s := by
+      apply Finset.sum_le_sum
+      intro r _
+      let D := (lightTargets P s x).filter (fun y => dist y x = r)
+      by_cases hD : D.Nonempty
+      · obtain ⟨y,hy⟩ := hD
+        have hsub : D ⊆ distanceFiber P x y := by
+          intro z hz
+          obtain ⟨hzL,hzr⟩ := Finset.mem_filter.mp hz
+          obtain ⟨hyL,hyr⟩ := Finset.mem_filter.mp hy
+          exact Finset.mem_filter.mpr ⟨(Finset.mem_filter.mp hzL).1, hzr.trans hyr.symm⟩
+        have hylt : (k P x y : ℝ) < (P.card : ℝ)^s :=
+          (Finset.mem_filter.mp (Finset.mem_filter.mp hy).1).2
+        exact (show (D.card : ℝ) ≤ (k P x y : ℝ) by exact_mod_cast Finset.card_le_card hsub).trans hylt.le
+      · change (D.card : ℝ) ≤ (P.card : ℝ)^s
+        rw [Finset.not_nonempty_iff_eq_empty.mp hD]
+        simp only [Finset.card_empty, Nat.cast_zero]
+        positivity
+    _ = _ := by simp
+
+lemma row_card_split (P : Finset Plane) (s : ℝ) (x : Plane) :
+    (lightTargets P s x).card + (richTargets P s x).card = (P.erase x).card := by
+  classical
+  convert Finset.card_filter_add_card_filter_not (s := P.erase x)
+    (fun y => (k P x y : ℝ) < (P.card : ℝ)^s) using 1
+  simp [lightTargets,richTargets]
+
+lemma rich_card_sum (P : Finset Plane) (s : ℝ) :
+    ∑ x ∈ P, (richTargets P s x).card = (richPairs P s).card := by
+  classical
+  have hr (x : Plane) : richTargets P s x = P.filter (fun y => x ≠ y ∧ (P.card : ℝ)^s ≤ (k P x y : ℝ)) := by
+    ext y
+    simp only [richTargets, Finset.mem_filter, Finset.mem_erase]
+    tauto
+  simp only [hr, richPairs, Finset.card_eq_sum_ones, Finset.sum_filter, Finset.sum_product]
+
+lemma bad_fraction_bound (P : Finset Plane) (ε : ℝ) (hn : 2 ≤ P.card) :
+    badPinFraction P ε ≤ 2*pairFraction P (ε/2) +
+      2*((P.card : ℝ)^(1-ε/2)/(P.card : ℝ)) := by
+  classical
+  have hn2 : (2 : ℝ) ≤ P.card := by exact_mod_cast hn
+  have hn0 : (0 : ℝ) < P.card := by linarith
+  have hm0 : (0 : ℝ) < (P.card : ℝ)-1 := by linarith
+  have h1 : 1 ≤ P.card := by omega
+  have hlight (x) (hx : x ∈ badPins P ε) :
+      ((lightTargets P (ε/2) x).card : ℝ) ≤ (P.card : ℝ)^(1-ε/2) := by
+    have hd := (Finset.mem_filter.mp hx).2
+    calc
+      _ ≤ (distances P x).card * (P.card : ℝ)^(ε/2) := light_card_le _ _ _
+      _ ≤ (P.card : ℝ)^(1-ε)*(P.card : ℝ)^(ε/2) :=
+        mul_le_mul_of_nonneg_right hd.le (Real.rpow_nonneg (by positivity) _)
+      _ = _ := by rw [← Real.rpow_add hn0]; congr 1; ring
+  have hrow (x) (hx : x ∈ badPins P ε) :
+      (P.card : ℝ)-1 ≤ (richTargets P (ε/2) x).card+(P.card : ℝ)^(1-ε/2) := by
+    have hxP : x ∈ P := (Finset.mem_filter.mp hx).1
+    have he : ((lightTargets P (ε/2) x).card : ℝ)+(richTargets P (ε/2) x).card=(P.card : ℝ)-1 := by
+      rw [← Nat.cast_add, row_card_split, Finset.card_erase_of_mem hxP, Nat.cast_sub h1]
+      norm_num
+    linarith [hlight x hx]
+  have hsum : ((badPins P ε).card : ℝ)*((P.card : ℝ)-1) ≤
+      (∑ x ∈ badPins P ε, ((richTargets P (ε/2) x).card : ℝ)) +
+      (badPins P ε).card*(P.card : ℝ)^(1-ε/2) := by
+    simpa only [Finset.sum_add_distrib, Finset.sum_const, nsmul_eq_mul] using
+      Finset.sum_le_sum hrow
+  have hRsum : (∑ x ∈ badPins P ε, ((richTargets P (ε/2) x).card : ℝ)) ≤
+      ((richPairs P (ε/2)).card : ℝ) := by
+    calc
+      _ ≤ ∑ x ∈ P, ((richTargets P (ε/2) x).card : ℝ) :=
+        Finset.sum_le_sum_of_subset_of_nonneg (Finset.filter_subset _ _) (by intros; positivity)
+      _ = _ := by rw [← Nat.cast_sum, rich_card_sum]
+  have hb : ((badPins P ε).card : ℝ) ≤ P.card := by
+    exact_mod_cast Finset.card_le_card (Finset.filter_subset (fun x => ((distances P x).card : ℝ)<(P.card : ℝ)^(1-ε)) P)
+  have hl0 : (0 : ℝ) ≤ (P.card : ℝ)^(1-ε/2) := Real.rpow_nonneg (by positivity) _
+  have htotal : ((badPins P ε).card : ℝ)*((P.card : ℝ)-1) ≤
+      (richPairs P (ε/2)).card+(P.card : ℝ)*(P.card : ℝ)^(1-ε/2) := by
+    linarith [mul_le_mul_of_nonneg_right hb hl0]
+  have hlratio : (P.card : ℝ)^(1-ε/2)/((P.card : ℝ)-1) ≤
+      2*((P.card : ℝ)^(1-ε/2)/(P.card : ℝ)) := by
+    rw [← mul_div_assoc, div_le_div_iff₀ hm0 hn0]
+    nlinarith
+  calc
+    badPinFraction P ε = ((badPins P ε).card : ℝ)*((P.card : ℝ)-1)/
+      ((P.card : ℝ)*((P.card : ℝ)-1)) := by unfold badPinFraction; field_simp
+    _ ≤ ((richPairs P (ε/2)).card+(P.card : ℝ)*(P.card : ℝ)^(1-ε/2))/
+      ((P.card : ℝ)*((P.card : ℝ)-1)) :=
+      div_le_div_of_nonneg_right htotal (mul_pos hn0 hm0).le
+    _ = pairFraction P (ε/2)+(P.card : ℝ)^(1-ε/2)/((P.card : ℝ)-1) := by
+      unfold pairFraction
+      field_simp
+    _ ≤ 2*pairFraction P (ε/2)+2*((P.card : ℝ)^(1-ε/2)/(P.card : ℝ)) := by
+      linarith [pairFraction_nonneg P (ε/2)]
+
+lemma badPinFraction_nonneg (P : Finset Plane) (ε : ℝ) : 0 ≤ badPinFraction P ε := by
+  unfold badPinFraction
+  positivity
+
+lemma badPinFraction_le_one (P : Finset Plane) (ε : ℝ) : badPinFraction P ε ≤ 1 := by
+  have hb : ((badPins P ε).card : ℝ) ≤ P.card := by
+    exact_mod_cast Finset.card_le_card (Finset.filter_subset (fun x => ((distances P x).card : ℝ)<(P.card : ℝ)^(1-ε)) P)
+  by_cases hn : P.card=0
+  · simp [badPinFraction,hn]
+  · rw [badPinFraction,div_le_one (by exact_mod_cast Nat.pos_of_ne_zero hn)]
+    exact hb
+
+lemma bad_values_nonempty (n : ℕ) (ε : ℝ) :
+    ({a : ℝ | ∃ P : Finset Plane, P.card=n ∧ a=badPinFraction P ε}).Nonempty := by
+  obtain ⟨P,hP⟩ := Finset.exists_card_eq (α:=Plane) n
+  exact ⟨_,P,hP,rfl⟩
+
+lemma bad_values_bddAbove (n : ℕ) (ε : ℝ) :
+    BddAbove {a : ℝ | ∃ P : Finset Plane, P.card=n ∧ a=badPinFraction P ε} := by
+  refine ⟨1,?_⟩
+  rintro a ⟨P,_,rfl⟩
+  exact badPinFraction_le_one P ε
+
+lemma badPinFraction_le_B (P : Finset Plane) (ε : ℝ) (hn : 0<P.card) :
+    badPinFraction P ε ≤ B P.card ε := by
+  rw [B,ite_eq_left hn]
+  exact le_csSup (bad_values_bddAbove _ _) ⟨P,rfl,rfl⟩
+
+lemma B_nonneg (n : ℕ) (ε : ℝ) : 0 ≤ B n ε := by
+  by_cases hn : 0<n
+  · obtain ⟨P,hP⟩ := Finset.exists_card_eq (α:=Plane) n
+    rw [← hP] at hn ⊢
+    exact (badPinFraction_nonneg P ε).trans (badPinFraction_le_B P ε hn)
+  · simp [B,hn]
+
+lemma B_bound (n : ℕ) (hn : 2≤n) (ε : ℝ) :
+    B n ε ≤ 2*F n (ε/2)+2*((n : ℝ)^(1-ε/2)/(n : ℝ)) := by
+  rw [B,ite_eq_left (by omega : 0<n)]
+  apply csSup_le (bad_values_nonempty _ _)
+  rintro a ⟨P,hP,rfl⟩
+  have hPn : 2≤P.card := by omega
+  have hf := fraction_le_F P (ε/2) hPn
+  have hb := bad_fraction_bound P ε hPn
+  rw [hP] at hf hb
+  linarith
+
+/-- All but a uniformly vanishing fraction of pins determine at least
+`n^(1-ε)` distinct nonzero distances. -/
+theorem pins (ε : ℝ) (hε : 0<ε) : Tendsto (fun n : ℕ => B n ε) atTop (𝓝 0) := by
+  have hpower : Tendsto (fun n : ℕ => (n : ℝ)^(1-ε/2)/(n : ℝ)) atTop (𝓝 0) := by
+    have ht := (tendsto_rpow_neg_atTop (half_pos hε)).comp
+      (tendsto_natCast_atTop_atTop (R:=ℝ))
+    apply ht.congr'
+    filter_upwards [eventually_ge_atTop 1] with n hn
+    have hn0 : (0 : ℝ)<n := by exact_mod_cast (show 0<n by omega)
+    simp only [Function.comp_apply]
+    rw [show 1-ε/2=1+(-(ε/2)) by ring, Real.rpow_add hn0, Real.rpow_one]
+    exact (mul_div_cancel_left₀ _ hn0.ne').symm
+  apply squeeze_zero' (Eventually.of_forall (fun n => B_nonneg n ε))
+    (eventually_atTop.2 ⟨2,fun n hn => B_bound n hn ε⟩)
+  simpa only [mul_zero,add_zero] using ((main (ε/2) (half_pos hε)).const_mul 2).add (hpower.const_mul 2)
+
+theorem pinsStatement : PinsStatement := pins
+
+end WeakPinned
+
+end
+end
+section
+noncomputable section
+open Filter
+open scoped Topology
+namespace WeakPinned
+
+lemma distances_pos {P : Finset Plane} {x : Plane} {r : ℝ} (hr : r ∈ distances P x) :
+    0 < r := by
+  obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hr
+  exact dist_pos.mpr (Finset.mem_erase.mp hy).1
+
+/-- Every sufficiently large configuration has a pin with at least
+    the prescribed power of its cardinality many distinct nonzero distances. -/
+theorem exists_pin_eventually (ε : ℝ) (hε : 0 < ε) :
+    ∀ᶠ n : ℕ in atTop, ∀ P : Finset Plane, P.card = n →
+      ∃ x ∈ P, (P.card : ℝ) ^ (1-ε) ≤ ((distances P x).card : ℝ) := by
+  have hb : ∀ᶠ n : ℕ in atTop, B n ε < 1 :=
+    (pins ε hε).eventually (gt_mem_nhds (by norm_num : (0 : ℝ) < 1))
+  filter_upwards [hb, eventually_ge_atTop 1] with n hn hn1 P hP
+  have hP0 : 0 < P.card := by omega
+  by_contra hf
+  have hbad : badPins P ε = P := by
+    apply Finset.filter_eq_self.mpr
+    intro x hx
+    exact lt_of_not_ge (fun h => hf ⟨x,hx,h⟩)
+  have hfrac : badPinFraction P ε = 1 := by
+    rw [badPinFraction, hbad, div_self (by exact_mod_cast Nat.ne_of_gt hP0)]
+  have he := badPinFraction_le_B P ε hP0
+  rw [hfrac,hP] at he
+  exact (not_lt_of_ge he) hn
+
+end WeakPinned
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PinnedDistances/ProductFormula.lean b/lean/OAI/Geometry/PinnedDistances/ProductFormula.lean
new file mode 100644
index 0000000000000000000000000000000000000000..5d815fcf807447ffc12edc59c29c80656fe16f7b
--- /dev/null
+++ b/lean/OAI/Geometry/PinnedDistances/ProductFormula.lean
@@ -0,0 +1,126 @@
+import OAI.Geometry.PinnedDistances.MixtureLaw
+
+namespace OAI
+
+universe u1 u2 u3 u4
+
+section
+
+/- The finite-place absolute values are norm-normalized. Dividing their logarithms
+   by [K:ℚ] gives the weighted contribution of locally normalized absolute values. -/
+
+noncomputable section
+open NumberField
+open scoped BigOperators
+
+namespace WeakPinned.Arithmetic
+
+variable {K : Type u1} [Field K]
+
+lemma totallyComplex_of_sq_neg_one (ii : K) (hii : ii ^ 2 = -1) :
+    NumberField.IsTotallyComplex K := by
+  constructor
+  intro v
+  rw [← InfinitePlace.not_isReal_iff_isComplex]
+  intro hv
+  let φ := InfinitePlace.embedding_of_isReal hv
+  have h := congrArg φ hii
+  simp only [map_pow, map_neg, map_one] at h
+  nlinarith [sq_nonneg (φ ii)]
+
+variable [NumberField K]
+
+lemma log_product_formula (z : K) (hz : z ≠ 0) :
+    (∑ v : InfinitePlace K, (v.mult : ℝ) * Real.log (v z)) +
+      ∑ᶠ v : FinitePlace K, Real.log (v z) = 0 := by
+  have hi : ∀ v : InfinitePlace K, 0 < v z := fun _ => InfinitePlace.pos_iff.mpr hz
+  have hf : ∀ v : FinitePlace K, 0 < v z := fun _ => FinitePlace.pos_iff.mpr hz
+  have h := congrArg Real.log (NumberField.prod_abs_eq_one hz)
+  rw [Real.log_mul (Finset.prod_ne_zero_iff.mpr fun v _ => pow_ne_zero _ (hi v).ne')
+    (finprod_ne_zero fun v => (hf v).ne'), Real.log_prod (fun v _ => pow_ne_zero _ (hi v).ne'),
+    Real.log_finprod hf, Real.log_one] at h
+  simpa only [Real.log_pow] using h
+
+def archWeight (v : InfinitePlace K) : ℝ :=
+  (v.mult : ℝ) / Module.finrank ℚ K
+
+lemma sum_archWeight : ∑ v : InfinitePlace K, archWeight v = 1 := by
+  classical
+  unfold archWeight
+  rw [← Finset.sum_div, ← Nat.cast_sum, InfinitePlace.sum_mult_eq, div_self]
+  exact_mod_cast (Module.finrank_pos (R := ℚ) (M := K)).ne'
+
+lemma archWeight_nonneg (v : InfinitePlace K) : 0 ≤ archWeight v := by
+  unfold archWeight
+  positivity
+
+/-- The finitely many nonunit places for a given finite list of differences. -/
+lemma finite_nonunit_places {ι : Type u2} [Finite ι] (z : ι → K)
+    (hz : ∀ i, z i ≠ 0) :
+    {v : FinitePlace K | ∃ i, v (z i) ≠ 1}.Finite := by
+  have he : {v : FinitePlace K | ∃ i, v (z i) ≠ 1} =
+      ⋃ i, Function.mulSupport (fun v : FinitePlace K => v (z i)) := by
+    ext v
+    simp
+  rw [he]
+  exact Set.finite_iUnion (fun i => FinitePlace.hasFiniteMulSupport (hz i))
+
+omit [NumberField K] in
+/-- Multiplicativity gives the constant depth sum on each nonzero fiber at
+    every absolute value; neither the embeddings nor the field are fixed. -/
+lemma raw_fiber_identity (v : AbsoluteValue K ℝ) (a b : K) (ha : a ≠ 0) (hb : b ≠ 0) :
+    -Real.log (v a) + -Real.log (v b) = -Real.log (v (a * b)) := by
+  rw [map_mul, Real.log_mul (v.pos ha).ne' (v.pos hb).ne']
+  ring
+
+/-- The algebraic inversion identity used both by the unbounded tree argument
+    and the bounded Mobius argument. This is valid in any field. -/
+lemma paired_coordinate {F : Type u3} [Field F] (a b c d t : F)
+    (ha : c ≠ a) (h : (c-a)*(d-b)=t) : d = b + t/(c-a) := by
+  have h' : d-b = t/(c-a) := (eq_div_iff (sub_ne_zero.mpr ha)).mpr
+    (by simpa [mul_comm] using h)
+  rw [← h']
+  ring
+
+lemma paired_difference {F : Type u4} [Field F] (a b c d z w t : F)
+    (hc : c ≠ a) (hz : z ≠ a)
+    (hcd : (c-a)*(d-b)=t) (hzw : (z-a)*(w-b)=t) :
+    w-d = t * (c-z) / ((z-a)*(c-a)) := by
+  rw [paired_coordinate a b c d t hc hcd, paired_coordinate a b z w t hz hzw]
+  field_simp
+  ring
+
+lemma finite_raw_ultra (v : FinitePlace K) {x y z : K}
+    (_hxy : x ≠ y) (_hyz : y ≠ z) (hxz : x ≠ z) :
+    min (-Real.log (v (x-y))) (-Real.log (v (y-z))) ≤ -Real.log (v (x-z)) := by
+  have h := v.add_le (x-y) (y-z)
+  rw [sub_add_sub_cancel] at h
+  have hp : 0 < v (x-z) := FinitePlace.pos_iff.mpr (sub_ne_zero.mpr hxz)
+  have hl := Real.log_le_log hp h
+  by_cases hc : v (x-y) ≤ v (y-z)
+  · rw [max_eq_right hc] at hl
+    exact (min_le_right _ _).trans (neg_le_neg hl)
+  · rw [max_eq_left (le_of_not_ge hc)] at hl
+    exact (min_le_left _ _).trans (neg_le_neg hl)
+
+lemma normalized_log_product_formula (z : K) (hz : z ≠ 0) :
+    (∑ v : InfinitePlace K, archWeight v * (-Real.log (v z))) +
+      (Module.finrank ℚ K : ℝ)⁻¹ * ∑ᶠ v : FinitePlace K, -Real.log (v z) = 0 := by
+  have hh := log_product_formula z hz
+  simp only [archWeight, div_eq_mul_inv]
+  rw [finsum_neg_distrib]
+  have he : (∑ v : InfinitePlace K, (v.mult:ℝ) * (Module.finrank ℚ K : ℝ)⁻¹ * (-Real.log (v z))) =
+      -(Module.finrank ℚ K : ℝ)⁻¹ * ∑ v : InfinitePlace K, (v.mult:ℝ) * Real.log (v z) := by
+    rw [Finset.mul_sum]
+    apply Finset.sum_congr rfl
+    intro v _
+    ring
+  rw [he]
+  linear_combination -(Module.finrank ℚ K : ℝ)⁻¹ * hh
+
+end WeakPinned.Arithmetic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PinnedDistances/ProductLimits.lean b/lean/OAI/Geometry/PinnedDistances/ProductLimits.lean
new file mode 100644
index 0000000000000000000000000000000000000000..923c3e5ef0326fdf794981fc6a06cc02e18a24f9
--- /dev/null
+++ b/lean/OAI/Geometry/PinnedDistances/ProductLimits.lean
@@ -0,0 +1,222 @@
+import OAI.Geometry.PinnedDistances.VarianceRates
+
+namespace OAI
+
+universe u1 u2 u3 u4 u5 u6 u7 u8 u9 u10 u11 u12
+universe u13 u14
+
+section
+noncomputable section
+open MeasureTheory Filter
+open scoped Topology BigOperators BoundedContinuousFunction
+namespace WeakPinned.Bounded
+
+variable {E : Type u1} {X : Type u2} {J : Type u3} [MeasurableSpace E] [MeasurableSpace X]
+  [TopologicalSpace E] [TopologicalSpace X] [BorelSpace E] [BorelSpace X]
+  [SecondCountableTopology E] [SecondCountableTopology X]
+  [HasOuterApproxClosed E] [HasOuterApproxClosed X] [Fintype J]
+
+omit [SecondCountableTopology E] in
+lemma weak_product_limit
+    (Rn : ℕ → ProbabilityMeasure (E × (J → X))) (R : ProbabilityMeasure (E × (J → X)))
+    (Qn : ℕ → ProbabilityMeasure E) (Q : ProbabilityMeasure E)
+    (αn : ℕ → J → ProbabilityMeasure X) (β : J → ProbabilityMeasure X)
+    (hR : Tendsto Rn atTop (𝓝 R)) (hQ : Tendsto Qn atTop (𝓝 Q))
+    (hα : ∀ j,Tendsto (fun n => αn n j) atTop (𝓝 (β j)))
+    (herr : ∀ (f : E →ᵇ ℝ) (g : J → X →ᵇ ℝ),
+      Tendsto (fun n => (∫ p,f p.1*(∏ j,g j (p.2 j)) ∂ (Rn n : Measure (E × (J → X))))-
+        (∫ e,f e ∂ (Qn n : Measure E))*(∏ j,∫ x,g j x ∂ (αn n j : Measure X))) atTop (𝓝 0)) :
+    (R : Measure (E × (J → X)))=(Q : Measure E).prod (Measure.pi (fun j => (β j : Measure X))) := by
+  apply Measure.eq_prod_of_integral_mul_prod_boundedContinuousFunction
+  intro f g
+  let F : E × (J → X) →ᵇ ℝ :=
+    f.compContinuous ⟨Prod.fst,continuous_fst⟩ *
+      ∏ j,(g j).compContinuous ⟨fun p : E × (J → X) => p.2 j,by fun_prop⟩
+  have hRf := ProbabilityMeasure.tendsto_iff_forall_integral_tendsto.mp hR F
+  have hQf := ProbabilityMeasure.tendsto_iff_forall_integral_tendsto.mp hQ f
+  have hβg (j) := ProbabilityMeasure.tendsto_iff_forall_integral_tendsto.mp (hα j) (g j)
+  have ht := hRf.sub (hQf.mul (tendsto_finsetProd Finset.univ (fun j _ => hβg j)))
+  have hF (p : E × (J → X)) : F p=f p.1 * ∏ j,g j (p.2 j) := by simp [F]
+  simp only [hF] at ht
+  have he := tendsto_nhds_unique ht (herr f g)
+  rw [integral_fintype_prod_eq_prod]
+  exact sub_eq_zero.mp he
+
+end WeakPinned.Bounded
+
+end
+end
+section
+noncomputable section
+open MeasureTheory Filter
+open scoped Topology BigOperators BoundedContinuousFunction
+namespace WeakPinned.Bounded
+
+lemma weak_pair_product_limit {X : Type u4} {Y : Type u5} [MeasurableSpace X] [MeasurableSpace Y]
+    [TopologicalSpace X] [TopologicalSpace Y] [BorelSpace X] [BorelSpace Y]
+    [HasOuterApproxClosed X] [HasOuterApproxClosed Y]
+    [SecondCountableTopology X] [SecondCountableTopology Y]
+    (Rn : ℕ → ProbabilityMeasure (X × Y)) (R : ProbabilityMeasure (X × Y))
+    (μn : ℕ → ProbabilityMeasure X) (μ : ProbabilityMeasure X)
+    (νn : ℕ → ProbabilityMeasure Y) (ν : ProbabilityMeasure Y)
+    (hR : Tendsto Rn atTop (𝓝 R)) (hμ : Tendsto μn atTop (𝓝 μ)) (hν : Tendsto νn atTop (𝓝 ν))
+    (herr : ∀ (f : X →ᵇ ℝ) (g : Y →ᵇ ℝ),
+      Tendsto (fun n => (∫ p,f p.1*g p.2 ∂ (Rn n : Measure (X × Y)))-
+        (∫ x,f x ∂ (μn n : Measure X))*(∫ y,g y ∂ (νn n : Measure Y))) atTop (𝓝 0)) :
+    (R : Measure (X × Y))=(μ : Measure X).prod (ν : Measure Y) := by
+  apply Measure.eq_prod_of_integral_mul_boundedContinuousFunction
+  intro f g
+  let F : X × Y →ᵇ ℝ := f.compContinuous ⟨Prod.fst,continuous_fst⟩ *
+    g.compContinuous ⟨Prod.snd,continuous_snd⟩
+  have ht := (ProbabilityMeasure.tendsto_iff_forall_integral_tendsto.mp hR F).sub
+    ((ProbabilityMeasure.tendsto_iff_forall_integral_tendsto.mp hμ f).mul
+      (ProbabilityMeasure.tendsto_iff_forall_integral_tendsto.mp hν g))
+  exact sub_eq_zero.mp (tendsto_nhds_unique ht (herr f g))
+
+lemma pair_of_pi_map {T : Type u6} {E : Type u7} {X : Type u8} [MeasurableSpace T] [MeasurableSpace E] [MeasurableSpace X]
+    (R : Measure T) (Q : Measure E) (μ : Measure X) [SFinite Q] [IsFiniteMeasure μ]
+    (e : T → E) (z w : T → X) (he : Measurable e) (hz : Measurable z) (hw : Measurable w)
+    (hm : R.map (fun r => (e r,![z r,w r]))=Q.prod (Measure.pi (fun _ : Fin 2 => μ))) :
+    R.map (fun r => (e r,(z r,w r)))=Q.prod (μ.prod μ) := by
+  have hp : Measurable (fun r => (e r,![z r,w r])) := by
+    apply he.prodMk
+    apply Measurable.of_eval
+    intro j
+    fin_cases j
+    · exact hz
+    · exact hw
+  let hme := MeasurableEquiv.finTwoArrow (α:=X)
+  have hm2 : Measurable (Prod.map (id : E → E) hme) := measurable_id.prodMap hme.measurable
+  have hh := congrArg (fun m : Measure (E × (Fin 2 → X)) => m.map (Prod.map id hme)) hm
+  rw [Measure.map_map hm2 hp,← Measure.map_prod_map _ _ measurable_id hme.measurable,
+    Measure.map_id,(measurePreserving_finTwoArrow μ).map_eq] at hh
+  exact hh
+
+end WeakPinned.Bounded
+
+end
+end
+section
+noncomputable section
+open MeasureTheory Filter
+open scoped Topology BigOperators BoundedContinuousFunction
+namespace WeakPinned.Bounded
+
+variable {E : Type u9} {X : Type u10} {T : Type u11} [MeasurableSpace E] [MeasurableSpace X] [MeasurableSpace T]
+  [TopologicalSpace E] [TopologicalSpace X] [TopologicalSpace T]
+  [BorelSpace E] [BorelSpace X] [BorelSpace T]
+  [SecondCountableTopology E] [SecondCountableTopology X]
+  [HasOuterApproxClosed E] [HasOuterApproxClosed X]
+
+lemma weak_double_product_limit (Rn : ℕ → ProbabilityMeasure T) (R : ProbabilityMeasure T)
+    (Qn : ℕ → ProbabilityMeasure E) (Q : ProbabilityMeasure E)
+    (αn : ℕ → ProbabilityMeasure X) (β : ProbabilityMeasure X)
+    (e : T → E) (z w : T → X) (he : Continuous e) (hz : Continuous z) (hw : Continuous w)
+    (hR : Tendsto Rn atTop (𝓝 R)) (hQ : Tendsto Qn atTop (𝓝 Q)) (hα : Tendsto αn atTop (𝓝 β))
+    (herr : ∀ (f : E →ᵇ ℝ) (g h : X →ᵇ ℝ),
+      Tendsto (fun n => (∫ r,f (e r)*g (z r)*h (w r) ∂ (Rn n : Measure T))-
+        (∫ q,f q ∂ (Qn n : Measure E))*(∫ x,g x ∂ (αn n : Measure X))*
+          (∫ x,h x ∂ (αn n : Measure X))) atTop (𝓝 0)) :
+    (R:Measure T).map (fun r => (e r,(z r,w r)))=(Q:Measure E).prod ((β:Measure X).prod β) := by
+  let k : T → E × (Fin 2 → X) := fun r => (e r,![z r,w r])
+  have hk : Continuous k := by
+    apply he.prodMk
+    apply continuous_pi
+    intro i
+    fin_cases i
+    · exact hz
+    · exact hw
+  have hRm := ProbabilityMeasure.tendsto_map_of_tendsto_of_continuous Rn R hR hk
+  have hh := weak_product_limit (fun n => (Rn n).map k)
+    (R.map k) Qn Q (fun n _ => αn n) (fun _ : Fin 2 => β) hRm hQ (fun _ => hα) ?_
+  · exact pair_of_pi_map (R:Measure T) (Q:Measure E) (β:Measure X) e z w he.measurable hz.measurable hw.measurable hh
+  · intro f g
+    have hl := herr f (g 0) (g 1)
+    convert hl using 1
+    funext n
+    rw [show ((Rn n).map k : Measure (E × (Fin 2 → X)))=(Rn n:Measure T).map k from rfl]
+    rw [integral_map_of_stronglyMeasurable hk.measurable (by fun_prop)]
+    simp only [Fin.prod_univ_two,k,Matrix.cons_val_zero,Matrix.cons_val_one,Matrix.cons_val_fin_one,mul_assoc]
+
+end WeakPinned.Bounded
+
+end
+end
+section
+noncomputable section
+open MeasureTheory Filter
+open scoped Topology ENNReal NNReal BoundedContinuousFunction
+namespace WeakPinned.Bounded
+
+variable {X : Type u12} [MeasurableSpace X] [TopologicalSpace X] [BorelSpace X]
+  [TopologicalSpace.PseudoMetrizableSpace X]
+
+lemma measure_le_of_continuous_lintegral (μ ν : Measure X) [IsFiniteMeasure μ] [IsFiniteMeasure ν]
+    (h : ∀ f : X →ᵇ ℝ≥0,(∫⁻ x,f x ∂ μ)≤∫⁻ x,f x ∂ ν) : μ≤ν := by
+  have hclosed (K : Set X) (hK : IsClosed K) : μ K≤ν K := by
+    exact le_of_tendsto_of_tendsto
+      (HasOuterApproxClosed.tendsto_lintegral_apprSeq hK μ)
+      (HasOuterApproxClosed.tendsto_lintegral_apprSeq hK ν)
+      (Eventually.of_forall (fun n => h (hK.apprSeq n)))
+  apply Measure.le_iff.mpr
+  intro s hs
+  rw [hs.measure_eq_iSup_isClosed_of_ne_top (measure_ne_top μ s)]
+  exact iSup_le (fun K => iSup_le (fun hKs => iSup_le (fun hK => (hclosed K hK).trans (measure_mono hKs))))
+
+lemma weak_domination (μn νn : ℕ → ProbabilityMeasure X) (μ ν : ProbabilityMeasure X)
+    (hμ : Tendsto μn atTop (𝓝 μ)) (hν : Tendsto νn atTop (𝓝 ν)) (c : ℝ≥0)
+    (hle : ∀ n,(μn n : Measure X)≤(c:ℝ≥0∞) • (νn n : Measure X)) :
+    (μ : Measure X)≤(c:ℝ≥0∞) • (ν : Measure X) := by
+  have : IsFiniteMeasure ((c:ℝ≥0∞) • (ν:Measure X)) := (ν:Measure X).smul_finite ENNReal.coe_ne_top
+  apply measure_le_of_continuous_lintegral
+  intro f
+  have hμf := ProbabilityMeasure.tendsto_iff_forall_lintegral_tendsto.mp hμ f
+  have hνf := ProbabilityMeasure.tendsto_iff_forall_lintegral_tendsto.mp hν f
+  rw [lintegral_smul_measure]
+  apply le_of_tendsto_of_tendsto hμf
+    (ENNReal.Tendsto.const_mul hνf (Or.inr ENNReal.coe_ne_top))
+  filter_upwards [] with n
+  simpa only [lintegral_smul_measure,smul_eq_mul] using lintegral_mono' (hle n) (le_refl (fun x => (f x:ℝ≥0∞)))
+
+end WeakPinned.Bounded
+
+end
+end
+section
+noncomputable section
+open MeasureTheory Filter
+open scoped Topology NNReal ENNReal
+namespace WeakPinned.Bounded
+variable {X : Type u13} {Y : Type u14} [MeasurableSpace X] [MeasurableSpace Y]
+  [TopologicalSpace X] [TopologicalSpace Y] [BorelSpace X] [BorelSpace Y]
+  [HasOuterApproxClosed Y]
+
+lemma weak_map_eq (αn : ℕ → ProbabilityMeasure X) (α : ProbabilityMeasure X)
+    (βn : ℕ → ProbabilityMeasure Y) (β : ProbabilityMeasure Y)
+    (f : X → Y) (hf : Continuous f) (hα : Tendsto αn atTop (𝓝 α))
+    (hβ : Tendsto βn atTop (𝓝 β)) (hm : ∀ n,(αn n:Measure X).map f=(βn n:Measure Y)) :
+    (α:Measure X).map f=(β:Measure Y) := by
+  have he (n) : (αn n).map f=βn n := ProbabilityMeasure.toMeasure_injective (hm n)
+  have hl := ProbabilityMeasure.tendsto_map_of_tendsto_of_continuous αn α hα hf
+  simp only [he] at hl
+  exact congrArg ProbabilityMeasure.toMeasure (tendsto_nhds_unique hl hβ)
+
+omit [HasOuterApproxClosed Y] in
+lemma weak_map_domination [TopologicalSpace.PseudoMetrizableSpace Y]
+    (αn : ℕ → ProbabilityMeasure X) (α : ProbabilityMeasure X)
+    (βn : ℕ → ProbabilityMeasure Y) (β : ProbabilityMeasure Y)
+    (f : X → Y) (hf : Continuous f) (hα : Tendsto αn atTop (𝓝 α))
+    (hβ : Tendsto βn atTop (𝓝 β)) (c : ℝ≥0)
+    (hm : ∀ n,(αn n:Measure X).map f≤(c:ℝ≥0∞) • (βn n:Measure Y)) :
+    (α:Measure X).map f≪(β:Measure Y) := by
+  have hl := weak_domination (fun n => (αn n).map f)
+    βn (α.map f) β
+    (ProbabilityMeasure.tendsto_map_of_tendsto_of_continuous αn α hα hf) hβ c hm
+  exact Measure.absolutelyContinuous_of_le_smul hl
+
+end WeakPinned.Bounded
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PinnedDistances/RandomGrids.lean b/lean/OAI/Geometry/PinnedDistances/RandomGrids.lean
new file mode 100644
index 0000000000000000000000000000000000000000..c16682733d0206c59e476104284509a6f5076e19
--- /dev/null
+++ b/lean/OAI/Geometry/PinnedDistances/RandomGrids.lean
@@ -0,0 +1,1174 @@
+import OAI.Geometry.PinnedDistances.Hierarchy
+
+namespace OAI
+
+universe u1 u2 u3
+
+section
+noncomputable section
+open NumberField
+open scoped BigOperators
+noncomputable section
+open MeasureTheory
+open scoped Topology
+
+/- Square-grid construction and the finite-shift sharing law. -/
+
+noncomputable section
+open MeasureTheory
+open scoped Topology ENNReal
+namespace WeakPinned.Grid
+
+def bin (l s x : ℝ) : ℤ := ⌊(x-s)/l⌋
+def shares (l s x y : ℝ) : ℝ := if bin l s x = bin l s y then 1 else 0
+
+lemma bin_period {l : ℝ} (hl : 0 < l) (s x : ℝ) :
+    bin l (s+l) x = bin l s x - 1 := by
+  unfold bin
+  rw [show (x-(s+l))/l = (x-s)/l - 1 by field_simp; ring]
+  exact Int.floor_sub_one _
+
+lemma shares_period {l : ℝ} (hl : 0 < l) (x y : ℝ) :
+    Function.Periodic (fun s => shares l s x y) l := by
+  intro s
+  simp [shares, bin_period hl]
+
+lemma shares_symm (l s x y : ℝ) : shares l s x y = shares l s y x := by
+  simp only [shares, eq_comm]
+
+lemma measurable_shares (l x y : ℝ) : Measurable (fun s => shares l s x y) := by
+  apply Measurable.ite _ measurable_const measurable_const
+  exact measurableSet_eq_fun
+    (((measurable_const.sub measurable_id).div_const l).floor)
+    (((measurable_const.sub measurable_id).div_const l).floor)
+
+lemma bin_small_shift {l x s : ℝ} (hl : 0 < l) (hs : s ∈ Set.Ioc x (x+l)) :
+    bin l s x = -1 := by
+  apply Int.floor_eq_iff.mpr
+  constructor
+  · apply (le_div_iff₀ hl).mpr
+    norm_num
+    linarith [hs.2]
+  · apply (div_lt_iff₀ hl).mpr
+    norm_num
+    linarith [hs.1]
+
+lemma shares_small_shift {l x y s : ℝ} (hl : 0 < l) (hxy : x ≤ y)
+    (hs : s ∈ Set.Ioc x (x+l)) : shares l s x y = if y < s then 1 else 0 := by
+  have he : bin l s y = -1 ↔ y < s := by
+    simp only [bin, Int.floor_eq_iff]
+    constructor
+    · intro h
+      have := (div_lt_iff₀ hl).mp h.2
+      norm_num at this
+      linarith
+    · intro h
+      constructor
+      · apply (le_div_iff₀ hl).mpr
+        norm_num
+        linarith [hs.2]
+      · apply (div_lt_iff₀ hl).mpr
+        norm_num
+        linarith
+  simp only [shares, bin_small_shift hl hs, eq_comm (a := (-1 : ℤ))]
+  simp only [he]
+
+lemma shares_integral {l : ℝ} (hl : 0 < l) (x y : ℝ) :
+    (∫ s : ℝ in Set.Ioc 0 l, shares l s x y) = max 0 (l-|x-y|) := by
+  wlog hxy : x ≤ y
+  · simpa only [shares_symm l _ x y, abs_sub_comm x y] using this hl y x (le_of_not_ge hxy)
+  have hp := (shares_period hl x y).intervalIntegral_add_eq 0 x
+  simp only [zero_add] at hp
+  rw [intervalIntegral.integral_of_le hl.le,
+    intervalIntegral.integral_of_le (by linarith : x ≤ x+l)] at hp
+  rw [hp]
+  have he : (∫ s : ℝ in Set.Ioc x (x+l), shares l s x y) =
+      ∫ s : ℝ in Set.Ioc x (x+l), if y < s then (1 : ℝ) else 0 := by
+    apply setIntegral_congr_fun measurableSet_Ioc
+    intro s hs
+    exact shares_small_shift hl hxy hs
+  rw [he]
+  change (∫ s : ℝ in Set.Ioc x (x+l), (Set.Ioi y).indicator (fun _ => (1 : ℝ)) s) = _
+  rw [setIntegral_indicator measurableSet_Ioi]
+  rw [show Set.Ioc x (x+l) ∩ Set.Ioi y = Set.Ioc y (x+l) by
+    ext s; simp only [Set.mem_inter_iff, Set.mem_Ioc, Set.mem_Ioi]; constructor
+    · rintro ⟨⟨h₁,h₂⟩,h₃⟩; exact ⟨h₃,h₂⟩
+    · intro h; exact ⟨⟨hxy.trans_lt h.1,h.2⟩,h.1⟩]
+  rw [setIntegral_const]
+  simp only [Measure.real, Real.volume_Ioc, smul_eq_mul, mul_one, ENNReal.toReal_ofReal']
+  rw [abs_of_nonpos (sub_nonpos.mpr hxy)]
+  rw [max_comm]
+  congr 1
+  ring
+
+lemma shares_integrableOn (l x y a b : ℝ) :
+    IntegrableOn (fun s => shares l s x y) (Set.Ioc a b) := by
+  apply IntegrableOn.of_bound measure_Ioc_lt_top (measurable_shares l x y).aestronglyMeasurable 1
+  exact Filter.Eventually.of_forall (fun s => by unfold shares; split_ifs <;> norm_num)
+
+/-- Independent uniform shifts in the two axes give a product of overlap lengths.
+    This integral is unnormalized planar Lebesgue mass. -/
+lemma square_shares_integral {l : ℝ} (hl : 0 < l) (x y : ℝ × ℝ) :
+    (∫ s₁ : ℝ in Set.Ioc 0 l, ∫ s₂ : ℝ in Set.Ioc 0 l,
+      shares l s₁ x.1 y.1 * shares l s₂ x.2 y.2) =
+      max 0 (l-|x.1-y.1|) * max 0 (l-|x.2-y.2|) := by
+  simp_rw [integral_const_mul]
+  rw [integral_mul_const, shares_integral hl, shares_integral hl]
+
+lemma shares_gap {l : ℝ} (hl : 0 < l) {s x y : ℝ}
+    (h : bin l s x = bin l s y) : |x-y| < l := by
+  have hx := Int.floor_le ((x-s)/l)
+  have hy := Int.lt_floor_add_one ((y-s)/l)
+  have hx' := Int.floor_le ((y-s)/l)
+  have hy' := Int.lt_floor_add_one ((x-s)/l)
+  change (bin l s x : ℝ) ≤ (x-s)/l at hx
+  change (y-s)/l < (bin l s y : ℝ) + 1 at hy
+  change (bin l s y : ℝ) ≤ (y-s)/l at hx'
+  change (x-s)/l < (bin l s x : ℝ) + 1 at hy'
+  rw [h] at hx hy'
+  rw [abs_lt]
+  constructor
+  · have : (y-s)/l < (x-s)/l+1 := by linarith
+    have hh := (div_lt_iff₀ hl).mp this
+    field_simp at hh
+    linarith
+  · have : (x-s)/l < (y-s)/l+1 := by linarith
+    have hh := (div_lt_iff₀ hl).mp this
+    field_simp at hh
+    linarith
+
+lemma bin_double {l : ℝ} (hl : 0 < l) (s x : ℝ) (b : ℤ) :
+    bin (2*l) (s+b*l) x = (bin l s x-b)/2 := by
+  unfold bin
+  rw [show (x-(s+b*l))/(2*l) = ((x-s)/l-b)/2 by field_simp; ring]
+  have hh := Int.floor_div_natCast (((x-s)/l)-(b:ℝ)) 2
+  norm_num only [Nat.cast_ofNat, Int.floor_sub_intCast] at hh
+  exact hh
+
+lemma nested_bins {l : ℝ} (hl : 0 < l) (s x y : ℝ) (b : ℤ)
+    (h : bin l s x = bin l s y) : bin (2*l) (s+b*l) x = bin (2*l) (s+b*l) y := by
+  rw [bin_double hl, bin_double hl, h]
+
+/-- Normalized Lebesgue measure on a half-open interval of positive length. -/
+def uniformInterval (l : ℝ) : Measure ℝ :=
+  (ENNReal.ofReal l)⁻¹ • volume.restrict (Set.Ioc 0 l)
+
+lemma uniformInterval_probability {l : ℝ} (hl : 0 < l) :
+    IsProbabilityMeasure (uniformInterval l) := by
+  constructor
+  simp only [uniformInterval, Measure.smul_apply, Measure.restrict_apply MeasurableSet.univ,
+    Set.univ_inter, Real.volume_Ioc, sub_zero, smul_eq_mul]
+  exact ENNReal.inv_mul_cancel (ne_of_gt (ENNReal.ofReal_pos.mpr hl)) ENNReal.ofReal_ne_top
+
+instance uniformInterval_finite (l : ℝ) : IsFiniteMeasure (uniformInterval l) := by
+  by_cases hl : 0 < l
+  · let := uniformInterval_probability hl
+    infer_instance
+  · have hz : uniformInterval l = 0 := by
+      simp [uniformInterval, Set.Ioc_eq_empty_of_le (not_lt.mp hl)]
+    rw [hz]
+    infer_instance
+
+lemma translate_restrict_Ioc (l : ℝ) :
+    (volume.restrict (Set.Ioc 0 l)).map (fun x => x+l) =
+      volume.restrict (Set.Ioc l (l+l)) := by
+  have hh := (measurePreserving_add_right volume l).restrict_preimage
+    (measurableSet_Ioc (a := l) (b := l+l))
+  have he : (fun x : ℝ => x+l) ⁻¹' Set.Ioc l (l+l) = Set.Ioc 0 l := by
+    ext x
+    simp only [Set.mem_preimage, Set.mem_Ioc]
+    constructor <;> intro h <;> constructor <;> linarith [h.1,h.2]
+  rw [he] at hh
+  exact hh.map_eq
+
+open ProbabilityTheory in
+/-- A fair binary step of length `l`. -/
+def binaryStep (l : ℝ) : Measure ℝ := bernoulliMeasure 0 l ⟨1/2, by norm_num⟩
+
+instance binaryStep_probability (l : ℝ) : IsProbabilityMeasure (binaryStep l) := by
+  unfold binaryStep
+  infer_instance
+
+lemma binaryStep_eq (l : ℝ) :
+    binaryStep l = (2 : ℝ≥0∞)⁻¹ • Measure.dirac 0 + (2 : ℝ≥0∞)⁻¹ • Measure.dirac l := by
+  simp only [binaryStep, ProbabilityTheory.bernoulliMeasure_def]
+  rw [← Measure.coe_nnreal_smul, ← Measure.coe_nnreal_smul]
+  congr 2
+  · rw [← ENNReal.ofReal_coe_nnreal]
+    change ENNReal.ofReal ((1:ℝ)/2) = _
+    rw [ENNReal.ofReal_div_of_pos (by norm_num)]
+    norm_num
+  · rw [← ENNReal.ofReal_coe_nnreal]
+    change ENNReal.ofReal (1-(1:ℝ)/2) = _
+    norm_num only [show (1-(1:ℝ)/2) = 1/2 by norm_num]
+    rw [ENNReal.ofReal_div_of_pos (by norm_num)]
+    norm_num
+
+lemma uniform_double {l : ℝ} (hl : 0 < l) :
+    (uniformInterval l).conv (binaryStep l) = uniformInterval (2*l) := by
+  rw [binaryStep_eq, Measure.conv_add, Measure.conv_smul_right,
+    Measure.conv_smul_right, Measure.conv_dirac_zero, Measure.conv_dirac]
+  rw [uniformInterval, Measure.map_smul _ (by fun_prop), translate_restrict_Ioc]
+  have hu : Set.Ioc 0 l ∪ Set.Ioc l (l+l) = Set.Ioc 0 (2*l) := by
+    rw [Set.Ioc_union_Ioc_eq_Ioc hl.le (by linarith)]
+    congr 1; linarith
+  have hd : Disjoint (Set.Ioc 0 l) (Set.Ioc l (l+l)) :=
+    Set.disjoint_left.mpr (by intro x h₁ h₂; exact (not_lt_of_ge h₁.2) h₂.1)
+  rw [smul_smul, smul_smul, ← smul_add, ← Measure.restrict_union hd measurableSet_Ioc, hu]
+  congr 1
+  rw [ENNReal.ofReal_mul (by norm_num : (0:ℝ) ≤ 2)]
+  norm_num [ENNReal.mul_inv]
+
+/-- One initial uniform shift followed by independent fair coarsening bits,
+    before introducing phase and orientation. -/
+def incrementLaw : ℕ → Measure ℝ
+  | 0 => uniformInterval 1
+  | n+1 => binaryStep (2^n)
+
+instance incrementLaw_probability (n : ℕ) : IsProbabilityMeasure (incrementLaw n) := by
+  cases n with
+  | zero => exact uniformInterval_probability zero_lt_one
+  | succ n => exact binaryStep_probability _
+
+def axisLaw : Measure (ℕ → ℝ) := Measure.infinitePi incrementLaw
+instance axisLaw_probability : IsProbabilityMeasure axisLaw := by
+  unfold axisLaw
+  infer_instance
+
+def shift (n : ℕ) (ω : ℕ → ℝ) : ℝ := ∑ i ∈ Finset.range (n+1), ω i
+
+lemma measurable_shift (n : ℕ) : Measurable (shift n) := by
+  exact Finset.measurable_sum _ (fun i _ => measurable_pi_apply i)
+
+lemma shift_zero (ω : ℕ → ℝ) : shift 0 ω = ω 0 := by simp [shift]
+lemma shift_succ (n : ℕ) (ω : ℕ → ℝ) :
+    shift (n+1) ω = shift n ω + ω (n+1) := by simp [shift, Finset.sum_range_succ]
+
+lemma map_shift (n : ℕ) : axisLaw.map (shift n) = uniformInterval (2^n) := by
+  induction n with
+  | zero =>
+      simp only [pow_zero]
+      rw [show shift 0 = (fun ω => ω 0) by funext ω; exact shift_zero ω]
+      exact Measure.infinitePi_map_eval incrementLaw 0
+  | succ n ih =>
+      have hind := ProbabilityTheory.iIndepFun_infinitePi
+        (P := incrementLaw) (X := fun _ => id) (fun _ => measurable_id)
+      have hi := hind.indepFun_finsetSum_of_notMem
+        (fun i => measurable_pi_apply i) (s := Finset.range (n+1)) (i := n+1) (by simp)
+      have he : (∑ j ∈ Finset.range (n+1), (fun ω : ℕ → ℝ => id (ω j))) = shift n := by
+        funext ω
+        simp [shift]
+      rw [he] at hi
+      have hm := hi.map_add_eq_map_conv_map₀ (measurable_shift n).aemeasurable
+        (measurable_pi_apply (n+1)).aemeasurable
+      change axisLaw.map (fun ω => shift n ω + ω (n+1)) = _ at hm
+      simp only [← shift_succ] at hm
+      change axisLaw.map (shift (n+1)) = (axisLaw.map (shift n)).conv ((Measure.infinitePi incrementLaw).map (fun f => f (n+1))) at hm
+      rw [ih, Measure.infinitePi_map_eval incrementLaw (n+1)] at hm
+      rw [incrementLaw, uniform_double (by positivity)] at hm
+      simpa only [pow_succ, mul_comm (2^n) (2:ℝ)] using hm
+
+lemma integral_uniform_shares {l : ℝ} (hl : 0 < l) (x y : ℝ) :
+    (∫ s, shares l s x y ∂uniformInterval l) = max 0 (1-|x-y|/l) := by
+  rw [uniformInterval, integral_smul_measure, shares_integral hl]
+  simp only [ENNReal.toReal_inv, ENNReal.toReal_ofReal hl.le, smul_eq_mul]
+  rw [mul_max_of_nonneg _ _ (inv_nonneg.mpr hl.le), mul_zero]
+  congr 1
+  field_simp
+
+lemma integral_shift_shares (n : ℕ) (x y : ℝ) :
+    (∫ ω, shares (2^n) (shift n ω) x y ∂axisLaw) =
+      max 0 (1-|x-y|/(2^n)) := by
+  rw [← integral_map (measurable_shift n).aemeasurable
+    (measurable_shares (2^n) x y).aestronglyMeasurable, map_shift]
+  exact integral_uniform_shares (by positivity) x y
+
+lemma integral_shares_nat_periods {l : ℝ} (hl : 0 < l) (k : ℕ) (x y : ℝ) :
+    (∫ s in Set.Ioc 0 (k*l), shares l s x y) = (k:ℝ)*max 0 (l-|x-y|) := by
+  have hh := (shares_period hl x y).intervalIntegral_add_zsmul_eq (k:ℤ) 0
+    (fun a b => ⟨shares_integrableOn l x y a b, shares_integrableOn l x y b a⟩)
+  simp only [zero_add, zsmul_eq_mul, Int.cast_natCast,
+    intervalIntegral.integral_of_le (by positivity : (0:ℝ) ≤ k*l),
+    intervalIntegral.integral_of_le hl.le, shares_integral hl] at hh
+  exact hh
+
+/-- The fine-scale shift is the original unit shift. Its sharing law is still
+    the correct uniform-modulo-side law, not a fresh independent grid. -/
+lemma integral_fine_shares (n : ℕ) (x y : ℝ) :
+    (∫ ω, shares ((2:ℝ)^(- (n:ℤ))) (shift 0 ω) x y ∂axisLaw) =
+      max 0 (1-|x-y|/((2:ℝ)^(- (n:ℤ)))) := by
+  rw [← integral_map (measurable_shift 0).aemeasurable
+    (measurable_shares _ x y).aestronglyMeasurable, map_shift]
+  simp only [pow_zero, uniformInterval, ENNReal.ofReal_one, inv_one, one_smul]
+  have hl : 0 < (2:ℝ)^(-(n:ℤ)) := by positivity
+  have he : ((2^n:ℕ):ℝ)*(2:ℝ)^(-(n:ℤ)) = 1 := by
+    simp [zpow_neg, zpow_natCast]
+  have hh := integral_shares_nat_periods hl (2^n) x y
+  rw [he] at hh
+  rw [hh, mul_max_of_nonneg _ _ (by positivity : (0:ℝ) ≤ ((2^n:ℕ):ℝ)), mul_zero]
+  congr 1
+  simp only [Nat.cast_pow, Nat.cast_ofNat, zpow_neg, zpow_natCast, div_inv_eq_mul]
+  field_simp
+
+lemma binaryStep_ae (l : ℝ) : ∀ᵐ x ∂binaryStep l, x = 0 ∨ x = l := by
+  rw [binaryStep_eq]
+  simp only [ae_add_measure_iff]
+  constructor
+  · apply Measure.ae_smul_measure
+    exact (ae_dirac_iff (by measurability)).mpr (Or.inl rfl)
+  · apply Measure.ae_smul_measure
+    exact (ae_dirac_iff (by measurability)).mpr (Or.inr rfl)
+
+lemma axis_increment_ae (n : ℕ) :
+    ∀ᵐ ω ∂axisLaw, ω (n+1) = 0 ∨ ω (n+1) = (2:ℝ)^n := by
+  have hh : ∀ᵐ x ∂incrementLaw (n+1), x = 0 ∨ x = (2:ℝ)^n := binaryStep_ae _
+  apply ae_of_ae_map (μ := axisLaw) (f := fun ω : ℕ → ℝ => ω (n+1))
+    (p := fun x : ℝ => x = 0 ∨ x = (2:ℝ)^n) (measurable_pi_apply (n+1)).aemeasurable
+  simpa only [axisLaw, Measure.infinitePi_map_eval] using hh
+
+lemma bins_nested_ae (n : ℕ) (x y : ℝ) :
+    ∀ᵐ ω ∂axisLaw, bin (2^n) (shift n ω) x = bin (2^n) (shift n ω) y →
+      bin (2^(n+1)) (shift (n+1) ω) x = bin (2^(n+1)) (shift (n+1) ω) y := by
+  filter_upwards [axis_increment_ae n] with ω hω
+  intro h
+  rcases hω with hω | hω
+  · simpa only [shift_succ, hω, pow_succ, mul_comm (2^n) (2:ℝ),
+      Int.cast_zero, zero_mul, add_zero] using nested_bins (by positivity) (shift n ω) x y 0 h
+  · simpa only [shift_succ, hω, pow_succ, mul_comm (2^n) (2:ℝ),
+      Int.cast_one, one_mul] using nested_bins (by positivity) (shift n ω) x y 1 h
+
+/-- Both independent axis constructions, before phase and rotation. -/
+abbrev GridSample := (ℕ → ℝ) × (ℕ → ℝ)
+def squareLaw : Measure GridSample := axisLaw.prod axisLaw
+instance squareLaw_probability : IsProbabilityMeasure squareLaw := by
+  unfold squareLaw
+  infer_instance
+
+def shiftZ (m : ℤ) (ω : ℕ → ℝ) : ℝ := shift m.toNat ω
+
+def shareAt (m : ℤ) (ω : GridSample) (x y : ℝ × ℝ) : Prop :=
+  bin ((2:ℝ)^m) (shiftZ m ω.1) x.1 = bin ((2:ℝ)^m) (shiftZ m ω.1) y.1 ∧
+  bin ((2:ℝ)^m) (shiftZ m ω.2) x.2 = bin ((2:ℝ)^m) (shiftZ m ω.2) y.2
+
+lemma measurableSet_shareAt (m : ℤ) (x y : ℝ × ℝ) :
+    MeasurableSet {ω | shareAt m ω x y} := by
+  unfold shareAt bin shiftZ
+  apply MeasurableSet.inter
+  · exact measurableSet_eq_fun
+      (((measurable_const.sub ((measurable_shift _).comp measurable_fst)).div_const _).floor)
+      (((measurable_const.sub ((measurable_shift _).comp measurable_fst)).div_const _).floor)
+  · exact measurableSet_eq_fun
+      (((measurable_const.sub ((measurable_shift _).comp measurable_snd)).div_const _).floor)
+      (((measurable_const.sub ((measurable_shift _).comp measurable_snd)).div_const _).floor)
+
+lemma integral_shiftZ_shares (m : ℤ) (x y : ℝ) :
+    (∫ ω, shares ((2:ℝ)^m) (shiftZ m ω) x y ∂axisLaw) =
+      max 0 (1-|x-y|/((2:ℝ)^m)) := by
+  cases m with
+  | ofNat n =>
+      change (∫ ω, shares ((2:ℝ)^(n:ℤ)) (shift n ω) x y ∂axisLaw) =
+        max 0 (1-|x-y|/((2:ℝ)^(n:ℤ)))
+      simpa only [zpow_natCast] using integral_shift_shares n x y
+  | negSucc n =>
+      simpa [shiftZ, Int.negSucc_eq, Int.toNat_of_nonpos] using
+        integral_fine_shares (n+1) x y
+
+lemma sharing_probability (m : ℤ) (x y : ℝ × ℝ) :
+    squareLaw.real {ω | shareAt m ω x y} =
+      max 0 (1-|x.1-y.1|/((2:ℝ)^m)) * max 0 (1-|x.2-y.2|/((2:ℝ)^m)) := by
+  rw [← integral_indicator_one (measurableSet_shareAt m x y)]
+  have he : (fun ω : GridSample => Set.indicator {ω | shareAt m ω x y} 1 ω) =
+      (fun ω => shares ((2:ℝ)^m) (shiftZ m ω.1) x.1 y.1 *
+        shares ((2:ℝ)^m) (shiftZ m ω.2) x.2 y.2) := by
+    funext ω
+    simp only [Set.indicator, Set.mem_ofPred_eq, shareAt, shares, Pi.one_apply]
+    by_cases h₁ : bin ((2:ℝ)^m) (shiftZ m ω.1) x.1 = bin ((2:ℝ)^m) (shiftZ m ω.1) y.1 <;>
+      by_cases h₂ : bin ((2:ℝ)^m) (shiftZ m ω.2) x.2 = bin ((2:ℝ)^m) (shiftZ m ω.2) y.2 <;>
+      simp [h₁,h₂]
+  rw [he, squareLaw, integral_prod_mul (fun ω : ℕ → ℝ => shares ((2:ℝ)^m) (shiftZ m ω) x.1 y.1)
+    (fun ω : ℕ → ℝ => shares ((2:ℝ)^m) (shiftZ m ω) x.2 y.2),
+    integral_shiftZ_shares, integral_shiftZ_shares]
+
+lemma max_sharing_bound {a b : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) :
+    1 - max 0 (1-a) * max 0 (1-b) ≤ a+b := by
+  by_cases ha' : a ≤ 1
+  · by_cases hb' : b ≤ 1
+    · rw [max_eq_right (by linarith), max_eq_right (by linarith)]
+      nlinarith [mul_nonneg ha hb]
+    · rw [max_eq_left (show 1-b ≤ 0 by linarith)]
+      nlinarith
+  · rw [max_eq_left (by linarith)]
+    nlinarith
+
+lemma separation_probability_le (m : ℤ) (x y : ℝ × ℝ) :
+    squareLaw.real {ω | ¬ shareAt m ω x y} ≤
+      (|x.1-y.1|+|x.2-y.2|)/((2:ℝ)^m) := by
+  change squareLaw.real {ω | shareAt m ω x y}ᶜ ≤ _
+  rw [measureReal_compl (measurableSet_shareAt m x y), probReal_univ,
+    sharing_probability, add_div]
+  exact max_sharing_bound (by positivity) (by positivity)
+
+lemma share_eventually_exists (x y : ℝ × ℝ) :
+    ∀ᵐ ω ∂squareLaw, ∃ n : ℕ, shareAt n ω x y := by
+  rw [ae_iff]
+  apply (measureReal_eq_zero_iff).mp
+  apply le_antisymm _ (measureReal_nonneg)
+  have hb (n : ℕ) : squareLaw.real {ω | ¬ ∃ n : ℕ, shareAt n ω x y} ≤
+      (|x.1-y.1|+|x.2-y.2|)/((2:ℝ)^n) := by
+    refine (measureReal_mono (s₂ := {ω | ¬ shareAt (n:ℤ) ω x y})
+      (by intro ω h hn; exact h ⟨n, hn⟩)).trans ?_
+    simpa only [zpow_natCast] using separation_probability_le (n:ℤ) x y
+  exact le_of_tendsto_of_tendsto' tendsto_const_nhds
+    (tendsto_const_nhds.div_atTop (tendsto_pow_atTop_atTop_of_one_lt (by norm_num : (1:ℝ)<2))) hb
+
+lemma axis_nestedZ_ae (m : ℤ) (x y : ℝ) :
+    ∀ᵐ ω ∂axisLaw, bin ((2:ℝ)^m) (shiftZ m ω) x = bin ((2:ℝ)^m) (shiftZ m ω) y →
+      bin ((2:ℝ)^(m+1)) (shiftZ (m+1) ω) x = bin ((2:ℝ)^(m+1)) (shiftZ (m+1) ω) y := by
+  by_cases hm : 0 ≤ m
+  · obtain ⟨n,rfl⟩ := Int.eq_ofNat_of_zero_le hm
+    rw [show (n:ℤ)+1 = ((n+1:ℕ):ℤ) by omega]
+    simpa only [shiftZ, Int.toNat_natCast, zpow_natCast] using bins_nested_ae n x y
+  · have hm' : m+1 ≤ 0 := by omega
+    filter_upwards with ω
+    intro h
+    simp only [shiftZ, Int.toNat_of_nonpos (by omega : m ≤ 0)] at h
+    have hh := nested_bins (by positivity : 0 < (2:ℝ)^m) (shift 0 ω) x y 0
+    simpa only [shiftZ, Int.toNat_of_nonpos (by omega : m ≤ 0),
+      Int.toNat_of_nonpos hm', zpow_add₀ (by norm_num : (2:ℝ)≠0), zpow_one,
+      mul_comm ((2:ℝ)^m) (2:ℝ), Int.cast_zero, zero_mul, add_zero] using hh h
+
+lemma shareAt_nested_ae (m : ℤ) (x y : ℝ × ℝ) :
+    ∀ᵐ ω ∂squareLaw, shareAt m ω x y → shareAt (m+1) ω x y := by
+  filter_upwards [Measure.quasiMeasurePreserving_fst.ae (axis_nestedZ_ae m x.1 y.1),
+    Measure.quasiMeasurePreserving_snd.ae (axis_nestedZ_ae m x.2 y.2)] with ω hx hy
+  exact fun h => ⟨hx h.1, hy h.2⟩
+
+lemma shareAt_symm (m : ℤ) (ω : GridSample) (x y : ℝ × ℝ) :
+    shareAt m ω x y ↔ shareAt m ω y x := by simp [shareAt, eq_comm]
+lemma shareAt_trans (m : ℤ) (ω : GridSample) (x y z : ℝ × ℝ)
+    (hxy : shareAt m ω x y) (hyz : shareAt m ω y z) : shareAt m ω x z :=
+  ⟨hxy.1.trans hyz.1, hxy.2.trans hyz.2⟩
+
+lemma shareAt_bddBelow (ω : GridSample) {x y : ℝ × ℝ} (hxy : x ≠ y) :
+    BddBelow {m : ℤ | shareAt m ω x y} := by
+  have bound {a b : ℝ} (h : a ≠ b) (f : ℤ → ℝ) :
+      BddBelow {m : ℤ | bin ((2:ℝ)^m) (f m) a = bin ((2:ℝ)^m) (f m) b} := by
+    obtain ⟨k,hk⟩ := exists_mem_Ico_zpow (abs_pos.mpr (sub_ne_zero.mpr h)) (by norm_num : (1:ℝ)<2)
+    refine ⟨k, fun m hm => ?_⟩
+    have hh := shares_gap (by positivity : 0 < (2:ℝ)^m) hm
+    exact (zpow_le_zpow_iff_right₀ (by norm_num : (1:ℝ)<2)).mp (hk.1.trans hh.le)
+  by_cases h₁ : x.1 = y.1
+  · have h₂ : x.2 ≠ y.2 := fun h₂ => hxy (Prod.ext h₁ h₂)
+    exact (bound h₂ (fun m => shiftZ m ω.2)).mono (fun _ h => h.2)
+  · exact (bound h₁ (fun m => shiftZ m ω.1)).mono (fun _ h => h.1)
+
+def minScale (ω : GridSample) (x y : ℝ × ℝ) : ℤ := sInf {m | shareAt m ω x y}
+
+def gridDepth (ω : GridSample) (x y : ℝ × ℝ) : ℝ :=
+  -(minScale ω x y : ℝ) * Real.log 2
+
+lemma minScale_symm (ω : GridSample) (x y : ℝ × ℝ) :
+    minScale ω x y = minScale ω y x := by
+  unfold minScale
+  congr 1
+  ext m
+  exact shareAt_symm m ω x y
+
+lemma minScale_mem {ω : GridSample} {x y : ℝ × ℝ} (hxy : x ≠ y)
+    (he : ∃ m, shareAt m ω x y) : shareAt (minScale ω x y) ω x y :=
+  Int.csInf_mem he (shareAt_bddBelow ω hxy)
+
+lemma minScale_le {ω : GridSample} {x y : ℝ × ℝ} (hxy : x ≠ y)
+    {m : ℤ} (hm : shareAt m ω x y) : minScale ω x y ≤ m :=
+  csInf_le (shareAt_bddBelow ω hxy) hm
+
+lemma shareAt_iff_minScale_le {ω : GridSample} {x y : ℝ × ℝ} (hxy : x ≠ y)
+    (he : ∃ m, shareAt m ω x y)
+    (hn : ∀ m, shareAt m ω x y → shareAt (m+1) ω x y) (m : ℤ) :
+    shareAt m ω x y ↔ minScale ω x y ≤ m := by
+  constructor
+  · exact minScale_le hxy
+  · intro hm
+    have hmono : Monotone (fun k => shareAt k ω x y) := monotone_int_of_le_succ hn
+    exact hmono hm (minScale_mem hxy he)
+
+lemma minScale_eq_iff {ω : GridSample} {x y : ℝ × ℝ} (hxy : x ≠ y) (m : ℤ) :
+    minScale ω x y = m ↔
+      (shareAt m ω x y ∧ ∀ k : ℤ, k < m → ¬ shareAt k ω x y) ∨
+      (m = 0 ∧ ¬ ∃ k : ℤ, shareAt k ω x y) := by
+  by_cases he : ∃ k, shareAt k ω x y
+  · have hs := minScale_mem hxy he
+    constructor
+    · intro h
+      left
+      rw [h] at hs
+      refine ⟨hs, fun k hk hks => ?_⟩
+      have hh := minScale_le hxy hks
+      omega
+    · rintro (⟨hm,hmin⟩ | ⟨_,hn⟩)
+      · apply le_antisymm (minScale_le hxy hm)
+        by_contra h
+        exact hmin _ (by omega) hs
+      · exact False.elim (hn he)
+  · have hz : minScale ω x y = 0 := by
+      unfold minScale
+      rw [show {m | shareAt m ω x y} = ∅ by ext m; simp only [Set.mem_ofPred_eq, Set.mem_empty_iff_false]; exact iff_false_intro (fun h => he ⟨m,h⟩)]
+      exact Int.csInf_empty
+    simp only [hz]
+    constructor
+    · intro hm
+      exact Or.inr ⟨hm.symm,he⟩
+    · rintro (⟨hm,_⟩ | ⟨rfl,_⟩)
+      · exact False.elim (he ⟨m,hm⟩)
+      · rfl
+
+lemma measurable_minScale {x y : ℝ × ℝ} (hxy : x ≠ y) :
+    Measurable (fun ω => minScale ω x y) := by
+  apply measurable_to_countable'
+  intro m
+  change MeasurableSet {ω | minScale ω x y = m}
+  have he : {ω : GridSample | minScale ω x y = m} =
+      ({ω | shareAt m ω x y} ∩ {ω | ∀ k < m, ¬shareAt k ω x y}) ∪
+      ({ω | m = 0} ∩ {ω | ¬∃ k, shareAt k ω x y}) := by
+    ext ω
+    exact minScale_eq_iff hxy m
+  rw [he]
+  apply MeasurableSet.union
+  · apply (measurableSet_shareAt m x y).inter
+    have he : {ω : GridSample | ∀ k < m, ¬shareAt k ω x y} =
+        ⋂ k : ℤ, ⋂ (_ : k < m), {ω : GridSample | shareAt k ω x y}ᶜ := by
+      ext ω
+      simp
+    rw [he]
+    exact MeasurableSet.iInter fun k =>
+      MeasurableSet.iInter fun (_ : k < m) => (measurableSet_shareAt k x y).compl
+  · apply (MeasurableSet.const (m = 0)).inter
+    have he : {ω : GridSample | ¬∃ k, shareAt k ω x y} =
+        (⋃ k : ℤ, {ω : GridSample | shareAt k ω x y})ᶜ := by
+      ext ω
+      simp
+    rw [he]
+    exact (MeasurableSet.iUnion fun k => measurableSet_shareAt k x y).compl
+
+lemma measurable_gridDepth {x y : ℝ × ℝ} (hxy : x ≠ y) :
+    Measurable (fun ω => gridDepth ω x y) := by
+  exact (((measurable_of_countable (fun n : ℤ => (n:ℝ))).comp (measurable_minScale hxy)).neg).mul_const _
+
+lemma gridDepth_ultra_ae {x y z : ℝ × ℝ} (hxy : x ≠ y) (hyz : y ≠ z) (hxz : x ≠ z) :
+    ∀ᵐ ω ∂squareLaw, min (gridDepth ω x y) (gridDepth ω y z) ≤ gridDepth ω x z := by
+  have hn : ∀ᵐ ω ∂squareLaw, ∀ m : ℤ,
+      (shareAt m ω x y → shareAt (m+1) ω x y) ∧
+      (shareAt m ω y z → shareAt (m+1) ω y z) := by
+    rw [ae_all_iff]
+    intro m
+    filter_upwards [shareAt_nested_ae m x y, shareAt_nested_ae m y z] with ω h₁ h₂ using ⟨h₁,h₂⟩
+  filter_upwards [share_eventually_exists x y, share_eventually_exists y z, hn] with ω h₁ h₂ hn
+  have e₁ : ∃ m : ℤ, shareAt m ω x y := by obtain ⟨n,hn⟩ := h₁; exact ⟨n,hn⟩
+  have e₂ : ∃ m : ℤ, shareAt m ω y z := by obtain ⟨n,hn⟩ := h₂; exact ⟨n,hn⟩
+  have hm : minScale ω x z ≤ max (minScale ω x y) (minScale ω y z) := by
+    apply minScale_le hxz
+    apply shareAt_trans _ ω x y z
+    · exact (shareAt_iff_minScale_le hxy e₁ (fun m => (hn m).1) _).mpr (le_max_left _ _)
+    · exact (shareAt_iff_minScale_le hyz e₂ (fun m => (hn m).2) _).mpr (le_max_right _ _)
+  have hlog : 0 < Real.log 2 := Real.log_pos (by norm_num)
+  unfold gridDepth
+  have hm' : (minScale ω x z : ℝ) ≤ max (minScale ω x y : ℝ) (minScale ω y z : ℝ) := by exact_mod_cast hm
+  by_cases hl : (minScale ω x y : ℝ) ≤ minScale ω y z
+  · rw [max_eq_right hl] at hm'
+    apply (min_le_right _ _).trans
+    nlinarith
+  · rw [max_eq_left (le_of_not_ge hl)] at hm'
+    apply (min_le_left _ _).trans
+    nlinarith
+
+/-- Euclidean norm can subsequently be used for `R`: only its elementary
+    comparison with the two coordinate gaps is needed for the tail bound. -/
+lemma error_upper_ae {x y : ℝ × ℝ} (hxy : x ≠ y) {R : ℝ} (hR : 0 < R)
+    (hgap : R ≤ |x.1-y.1|+|x.2-y.2|) :
+    ∀ᵐ ω ∂squareLaw, gridDepth ω x y + Real.log R ≤ Real.log 2 := by
+  filter_upwards [share_eventually_exists x y] with ω he
+  obtain ⟨n,hn⟩ := he
+  have hs := minScale_mem hxy ⟨n,hn⟩
+  have h₁ := shares_gap (by positivity : 0 < (2:ℝ)^(minScale ω x y)) hs.1
+  have h₂ := shares_gap (by positivity : 0 < (2:ℝ)^(minScale ω x y)) hs.2
+  have hr : R ≤ 2*(2:ℝ)^(minScale ω x y) := by linarith
+  have hh := Real.log_le_log hR hr
+  rw [Real.log_mul (by norm_num : (2:ℝ)≠0) (by positivity), Real.log_zpow] at hh
+  unfold gridDepth
+  linarith
+
+lemma error_lower_tail {x y : ℝ × ℝ} (hxy : x ≠ y) {R : ℝ} (hR : 0 < R)
+    (hgap : |x.1-y.1|+|x.2-y.2| ≤ 2*R) (u : ℝ) :
+    squareLaw.real {ω | gridDepth ω x y + Real.log R < -u} ≤ 4*Real.exp (-u) := by
+  obtain ⟨m,hm⟩ := exists_mem_Ico_zpow (mul_pos hR (Real.exp_pos u))
+    (by norm_num : (1:ℝ)<2)
+  have hl : 0 < Real.log 2 := Real.log_pos (by norm_num)
+  have hm₁ : (m:ℝ)*Real.log 2 ≤ Real.log R + u := by
+    have hh := Real.log_le_log (by positivity : 0 < (2:ℝ)^m) hm.1
+    rw [Real.log_zpow, Real.log_mul hR.ne' (Real.exp_ne_zero _), Real.log_exp] at hh
+    exact hh
+  have hsub : {ω | gridDepth ω x y + Real.log R < -u} ⊆
+      {ω | ¬shareAt m ω x y} := by
+    intro ω hω hs
+    have hk : (minScale ω x y : ℝ) ≤ (m:ℝ) := by exact_mod_cast minScale_le hxy hs
+    have := mul_le_mul_of_nonneg_right hk hl.le
+    change gridDepth ω x y + Real.log R < -u at hω
+    unfold gridDepth at hω
+    linarith
+  refine (measureReal_mono hsub).trans ((separation_probability_le m x y).trans ?_)
+  have hpow : 0 < (2:ℝ)^m := by positivity
+  have hm₂ : R*Real.exp u < 2*(2:ℝ)^m := by
+    simpa only [zpow_add₀ (by norm_num : (2:ℝ)≠0), zpow_one, mul_comm] using hm.2
+  apply (div_le_iff₀ hpow).mpr
+  have hex : Real.exp u * Real.exp (-u) = 1 := by rw [← Real.exp_add]; simp
+  have hh := mul_le_mul_of_nonneg_right hm₂.le (Real.exp_nonneg (-u))
+  nlinarith
+
+lemma error_abs_tail {x y : ℝ × ℝ} (hxy : x ≠ y) {R : ℝ} (hR : 0 < R)
+    (hgap₁ : R ≤ |x.1-y.1|+|x.2-y.2|)
+    (hgap₂ : |x.1-y.1|+|x.2-y.2| ≤ 2*R) {u : ℝ} (_hu : 0 ≤ u) :
+    squareLaw.real {ω | u < |gridDepth ω x y + Real.log R|} ≤ 4*Real.exp (-u) := by
+  by_cases hsmall : u ≤ Real.log 2
+  · have hp : squareLaw.real {ω | u < |gridDepth ω x y + Real.log R|} ≤ 1 :=
+      measureReal_le_one
+    have hex : (1:ℝ)/2 ≤ Real.exp (-u) := by
+      have := Real.exp_le_exp.mpr (neg_le_neg hsmall)
+      rw [Real.exp_neg, Real.exp_log (by norm_num : (0:ℝ)<2)] at this
+      simpa only [one_div] using this
+    linarith
+  · have hle : squareLaw.real {ω | u < |gridDepth ω x y + Real.log R|} ≤
+        squareLaw.real {ω | gridDepth ω x y + Real.log R < -u} := by
+      apply ENNReal.toReal_mono (measure_ne_top _ _)
+      apply measure_mono_ae
+      filter_upwards [error_upper_ae hxy hR hgap₁] with ω hω
+      intro hh
+      rcases lt_abs.mp hh with hh | hh
+      · exact False.elim (hsmall (hh.le.trans hω))
+      · linarith
+    exact hle.trans (error_lower_tail hxy hR hgap₂ u)
+
+lemma error_abs_lintegral_le {x y : ℝ × ℝ} (hxy : x ≠ y) {R : ℝ} (hR : 0 < R)
+    (hgap₁ : R ≤ |x.1-y.1|+|x.2-y.2|)
+    (hgap₂ : |x.1-y.1|+|x.2-y.2| ≤ 2*R) :
+    (∫⁻ ω, ENNReal.ofReal |gridDepth ω x y + Real.log R| ∂squareLaw) ≤ ENNReal.ofReal 4 := by
+  rw [lintegral_eq_lintegral_meas_lt squareLaw
+    (by filter_upwards with ω; exact abs_nonneg _)
+    (((measurable_gridDepth hxy).add_const _).abs.aemeasurable)]
+  have he : IntegrableOn (fun u : ℝ => 4 * Real.exp (-u)) (Set.Ioi 0) volume := by
+    change Integrable (fun u : ℝ => 4 * Real.exp (-u)) (volume.restrict (Set.Ioi 0))
+    simpa only [neg_one_mul] using (integrableOn_exp_mul_Ioi (by norm_num : (-1:ℝ)<0) 0).const_mul 4
+  calc
+    _ ≤ ∫⁻ u : ℝ in Set.Ioi 0, ENNReal.ofReal (4*Real.exp (-u)) := by
+      apply lintegral_mono_ae
+      filter_upwards [ae_restrict_mem measurableSet_Ioi] with u hu
+      rw [← ENNReal.ofReal_toReal (measure_ne_top squareLaw _)]
+      exact ENNReal.ofReal_le_ofReal (error_abs_tail hxy hR hgap₁ hgap₂ hu.le)
+    _ = ENNReal.ofReal 4 := by
+      rw [← ofReal_integral_eq_lintegral_ofReal he (by filter_upwards with u; positivity)]
+      congr 1
+      rw [integral_const_mul]
+      have hi := integral_exp_mul_Ioi (by norm_num : (-1:ℝ)<0) (0:ℝ)
+      simp only [neg_one_mul, mul_zero, Real.exp_zero, neg_div_neg_eq, div_one] at hi
+      rw [hi]
+      norm_num
+
+lemma error_integrable {x y : ℝ × ℝ} (hxy : x ≠ y) {R : ℝ} (hR : 0 < R)
+    (hgap₁ : R ≤ |x.1-y.1|+|x.2-y.2|)
+    (hgap₂ : |x.1-y.1|+|x.2-y.2| ≤ 2*R) :
+    Integrable (fun ω => gridDepth ω x y + Real.log R) squareLaw := by
+  have hm : AEStronglyMeasurable (fun ω => gridDepth ω x y + Real.log R) squareLaw :=
+    ((measurable_gridDepth hxy).add_const (Real.log R)).aestronglyMeasurable
+  apply (integrable_norm_iff hm).mp
+  refine ⟨hm.norm, ?_⟩
+  apply (hasFiniteIntegral_iff_ofReal (by filter_upwards with ω; exact norm_nonneg _)).mpr
+  simpa only [Real.norm_eq_abs] using
+    (error_abs_lintegral_le hxy hR hgap₁ hgap₂).trans_lt ENNReal.ofReal_lt_top
+
+lemma mean_abs_error_le {x y : ℝ × ℝ} (hxy : x ≠ y) {R : ℝ} (hR : 0 < R)
+    (hgap₁ : R ≤ |x.1-y.1|+|x.2-y.2|)
+    (hgap₂ : |x.1-y.1|+|x.2-y.2| ≤ 2*R) :
+    (∫ ω, |gridDepth ω x y + Real.log R| ∂squareLaw) ≤ 4 := by
+  have hi := (error_integrable hxy hR hgap₁ hgap₂).abs
+  have hh := error_abs_lintegral_le hxy hR hgap₁ hgap₂
+  rw [← ofReal_integral_eq_lintegral_ofReal hi (by filter_upwards with ω; exact abs_nonneg _)] at hh
+  have hh' := ENNReal.toReal_mono (by simp : ENNReal.ofReal (4:ℝ) ≠ ⊤) hh
+  simpa only [ENNReal.toReal_ofReal (integral_nonneg fun ω => abs_nonneg _),
+    ENNReal.toReal_ofReal (by norm_num : (0:ℝ)≤4)] using hh'
+
+lemma minScale_cdf {x y : ℝ × ℝ} (hxy : x ≠ y) (m : ℤ) :
+    squareLaw.real {ω | minScale ω x y ≤ m} =
+      max 0 (1-|x.1-y.1|/((2:ℝ)^m)) * max 0 (1-|x.2-y.2|/((2:ℝ)^m)) := by
+  have hn : ∀ᵐ ω ∂squareLaw, ∀ k : ℤ, shareAt k ω x y → shareAt (k+1) ω x y := by
+    rw [ae_all_iff]
+    exact fun k => shareAt_nested_ae k x y
+  have he : {ω | minScale ω x y ≤ m} =ᵐ[squareLaw] {ω | shareAt m ω x y} := by
+    filter_upwards [share_eventually_exists x y, hn] with ω he hn
+    apply propext
+    obtain ⟨n,hn'⟩ := he
+    exact (shareAt_iff_minScale_le hxy ⟨n,hn'⟩ hn m).symm
+  rw [measureReal_def, measure_congr he]
+  exact sharing_probability m x y
+
+lemma minScale_law_same_gaps {x y x' y' : ℝ × ℝ} (hxy : x ≠ y) (hxy' : x' ≠ y')
+    (h₁ : |x.1-y.1| = |x'.1-y'.1|) (h₂ : |x.2-y.2| = |x'.2-y'.2|) :
+    Measure.map (fun ω => minScale ω x y) squareLaw =
+      Measure.map (fun ω => minScale ω x' y') squareLaw := by
+  apply Measure.ext_of_Iic
+  intro m
+  rw [Measure.map_apply (measurable_minScale hxy) measurableSet_Iic,
+    Measure.map_apply (measurable_minScale hxy') measurableSet_Iic]
+  apply (ENNReal.toReal_eq_toReal_iff' (measure_ne_top _ _) (measure_ne_top _ _)).mp
+  change squareLaw.real {ω | minScale ω x y ≤ m} = squareLaw.real {ω | minScale ω x' y' ≤ m}
+  rw [minScale_cdf hxy, minScale_cdf hxy', h₁, h₂]
+
+lemma mean_depth_same_gaps {x y x' y' : ℝ × ℝ} (hxy : x ≠ y) (hxy' : x' ≠ y')
+    (h₁ : |x.1-y.1| = |x'.1-y'.1|) (h₂ : |x.2-y.2| = |x'.2-y'.2|) :
+    (∫ ω, gridDepth ω x y ∂squareLaw) = ∫ ω, gridDepth ω x' y' ∂squareLaw := by
+  have hf : Measurable (fun n : ℤ => -(n:ℝ)*Real.log 2) := measurable_of_countable _
+  unfold gridDepth
+  rw [← integral_map (measurable_minScale hxy).aemeasurable hf.aestronglyMeasurable,
+      ← integral_map (measurable_minScale hxy').aemeasurable hf.aestronglyMeasurable,
+      minScale_law_same_gaps hxy hxy' h₁ h₂]
+
+lemma minScale_law_double {x y : ℝ × ℝ} (hxy : x ≠ y) :
+    Measure.map (fun ω => minScale ω (2*x.1,2*x.2) (2*y.1,2*y.2)) squareLaw =
+      Measure.map (fun ω => minScale ω x y + 1) squareLaw := by
+  have hd : (2*x.1,2*x.2) ≠ (2*y.1,2*y.2) := by
+    intro h
+    apply hxy
+    have h₁ := congrArg Prod.fst h
+    have h₂ := congrArg Prod.snd h
+    apply Prod.ext <;> dsimp at h₁ h₂ ⊢ <;> linarith
+  apply Measure.ext_of_Iic
+  intro m
+  rw [Measure.map_apply (measurable_minScale hd) measurableSet_Iic,
+    Measure.map_apply ((measurable_minScale hxy).add_const 1) measurableSet_Iic]
+  apply (ENNReal.toReal_eq_toReal_iff' (measure_ne_top _ _) (measure_ne_top _ _)).mp
+  change squareLaw.real {ω | minScale ω (2*x.1,2*x.2) (2*y.1,2*y.2) ≤ m} =
+    squareLaw.real {ω | minScale ω x y + 1 ≤ m}
+  rw [show {ω | minScale ω x y + 1 ≤ m} = {ω | minScale ω x y ≤ m-1} by ext ω; simp only [Set.mem_ofPred_eq]; omega,
+    minScale_cdf hd, minScale_cdf hxy]
+  have hpow : (2:ℝ)^(m-1) = (2:ℝ)^m / 2 := by rw [zpow_sub₀ (by norm_num), zpow_one]
+  have hq (a b : ℝ) : |2*a-2*b| / (2:ℝ)^m = |a-b| / ((2:ℝ)^m/2) := by
+    rw [← mul_sub, abs_mul]
+    norm_num
+    ring
+  rw [hpow, hq x.1 y.1, hq x.2 y.2]
+
+lemma gridDepth_integrable {x y : ℝ × ℝ} (hxy : x ≠ y) :
+    Integrable (fun ω => gridDepth ω x y) squareLaw := by
+  let R := |x.1-y.1|+|x.2-y.2|
+  have hR : 0 < R := by
+    by_cases h₁ : x.1 = y.1
+    · have h₂ : x.2 ≠ y.2 := fun h₂ => hxy (Prod.ext h₁ h₂)
+      exact add_pos_of_nonneg_of_pos (abs_nonneg _) (abs_pos.mpr (sub_ne_zero.mpr h₂))
+    · exact add_pos_of_pos_of_nonneg (abs_pos.mpr (sub_ne_zero.mpr h₁)) (abs_nonneg _)
+  have hi := error_integrable hxy hR (le_refl R) (by linarith : R ≤ 2*R)
+  have hh := hi.sub (integrable_const (Real.log R))
+  change Integrable (fun ω => (gridDepth ω x y + Real.log R) - Real.log R) squareLaw at hh
+  simpa only [add_sub_cancel_right] using hh
+
+lemma mean_depth_double {x y : ℝ × ℝ} (hxy : x ≠ y) :
+    (∫ ω, gridDepth ω (2*x.1,2*x.2) (2*y.1,2*y.2) ∂squareLaw) =
+      (∫ ω, gridDepth ω x y ∂squareLaw) - Real.log 2 := by
+  have hd : (2*x.1,2*x.2) ≠ (2*y.1,2*y.2) := by
+    intro h
+    apply hxy
+    have h₁ := congrArg Prod.fst h
+    have h₂ := congrArg Prod.snd h
+    apply Prod.ext <;> dsimp at h₁ h₂ ⊢ <;> linarith
+  have hid : ProbabilityTheory.IdentDistrib
+      (fun ω => minScale ω (2*x.1,2*x.2) (2*y.1,2*y.2))
+      (fun ω => minScale ω x y + 1) squareLaw squareLaw :=
+    ⟨(measurable_minScale hd).aemeasurable,
+      ((measurable_minScale hxy).add_const 1).aemeasurable, minScale_law_double hxy⟩
+  have hh := (hid.comp (measurable_of_countable (fun n : ℤ => -(n:ℝ)*Real.log 2))).integral_eq
+  change (∫ ω, gridDepth ω (2*x.1,2*x.2) (2*y.1,2*y.2) ∂squareLaw) = _ at hh
+  rw [hh]
+  calc
+    _ = ∫ ω, gridDepth ω x y - Real.log 2 ∂squareLaw := by
+      apply integral_congr_ae
+      filter_upwards with ω
+      simp [gridDepth]
+      ring
+    _ = _ := by rw [integral_sub (gridDepth_integrable hxy) (integrable_const _)]; simp
+
+/-- Parameter measurability, needed before integrating orientation and phase. -/
+lemma measurableSet_shareAt_param {T : Type u1} [MeasurableSpace T]
+    {W : T → GridSample} {X Y : T → ℝ × ℝ}
+    (hW : Measurable W) (hX : Measurable X) (hY : Measurable Y) (m : ℤ) :
+    MeasurableSet {t | shareAt m (W t) (X t) (Y t)} := by
+  unfold shareAt bin shiftZ
+  apply MeasurableSet.inter
+  · exact measurableSet_eq_fun
+      ((hX.fst.sub ((measurable_shift _).comp hW.fst)).div_const _).floor
+      ((hY.fst.sub ((measurable_shift _).comp hW.fst)).div_const _).floor
+  · exact measurableSet_eq_fun
+      ((hX.snd.sub ((measurable_shift _).comp hW.snd)).div_const _).floor
+      ((hY.snd.sub ((measurable_shift _).comp hW.snd)).div_const _).floor
+
+lemma measurable_minScale_param {T : Type u2} [MeasurableSpace T]
+    {W : T → GridSample} {X Y : T → ℝ × ℝ}
+    (hW : Measurable W) (hX : Measurable X) (hY : Measurable Y) (hne : ∀ t, X t ≠ Y t) :
+    Measurable (fun t => minScale (W t) (X t) (Y t)) := by
+  apply measurable_to_countable'
+  intro m
+  change MeasurableSet {t | minScale (W t) (X t) (Y t) = m}
+  have he : {t | minScale (W t) (X t) (Y t) = m} =
+      ({t | shareAt m (W t) (X t) (Y t)} ∩
+        ⋂ k : ℤ, ⋂ (_ : k < m), {t | shareAt k (W t) (X t) (Y t)}ᶜ) ∪
+      ({t | m = 0} ∩ (⋃ k : ℤ, {t | shareAt k (W t) (X t) (Y t)})ᶜ) := by
+    ext t
+    simp only [Set.mem_ofPred_eq, Set.mem_union, Set.mem_inter_iff, Set.mem_iInter,
+      Set.mem_compl_iff, Set.mem_iUnion]
+    exact minScale_eq_iff (hne t) m
+  rw [he]
+  exact ((measurableSet_shareAt_param hW hX hY m).inter
+    (MeasurableSet.iInter fun k => MeasurableSet.iInter fun (_ : k < m) =>
+      (measurableSet_shareAt_param hW hX hY k).compl)).union
+    ((MeasurableSet.const (m = 0)).inter (MeasurableSet.iUnion fun k =>
+      measurableSet_shareAt_param hW hX hY k).compl)
+
+lemma measurable_gridDepth_param {T : Type u3} [MeasurableSpace T]
+    {W : T → GridSample} {X Y : T → ℝ × ℝ}
+    (hW : Measurable W) (hX : Measurable X) (hY : Measurable Y) (hne : ∀ t, X t ≠ Y t) :
+    Measurable (fun t => gridDepth (W t) (X t) (Y t)) := by
+  exact (((measurable_of_countable (fun n : ℤ => (n:ℝ))).comp
+    (measurable_minScale_param hW hX hY hne)).neg).mul_const _
+
+def polarPair (t θ : ℝ) : ℝ × ℝ := (Real.exp t * Real.cos θ, Real.exp t * Real.sin θ)
+
+def polarMean (t θ : ℝ) : ℝ :=
+  (∫ ω, gridDepth ω (0,0) (polarPair t θ) ∂squareLaw) + t
+
+lemma zero_ne_polarPair (t θ : ℝ) : (0,0) ≠ polarPair t θ := by
+  intro h
+  have he : Real.exp t ≠ 0 := Real.exp_ne_zero t
+  have h₁ : Real.cos θ = 0 := (mul_eq_zero.mp (congrArg Prod.fst h).symm).resolve_left he
+  have h₂ : Real.sin θ = 0 := (mul_eq_zero.mp (congrArg Prod.snd h).symm).resolve_left he
+  have hh := Real.sin_sq_add_cos_sq θ
+  rw [h₁,h₂] at hh
+  norm_num at hh
+
+lemma polarPair_gaps (t θ : ℝ) :
+    Real.exp t ≤ |(polarPair t θ).1|+|(polarPair t θ).2| ∧
+    |(polarPair t θ).1|+|(polarPair t θ).2| ≤ 2*Real.exp t := by
+  have hsum : 1 ≤ |Real.cos θ|+|Real.sin θ| := by
+    have hsq := Real.sin_sq_add_cos_sq θ
+    have hpos := mul_nonneg (abs_nonneg (Real.cos θ)) (abs_nonneg (Real.sin θ))
+    have hc := abs_nonneg (Real.cos θ)
+    have hs := abs_nonneg (Real.sin θ)
+    nlinarith [sq_abs (Real.cos θ), sq_abs (Real.sin θ)]
+  have hsum' : |Real.cos θ|+|Real.sin θ| ≤ 2 := by
+    linarith [Real.abs_cos_le_one θ, Real.abs_sin_le_one θ]
+  simp only [polarPair, abs_mul, abs_of_pos (Real.exp_pos t)]
+  constructor
+  · nlinarith [mul_le_mul_of_nonneg_left hsum (Real.exp_nonneg t)]
+  · nlinarith [mul_le_mul_of_nonneg_left hsum' (Real.exp_nonneg t)]
+
+lemma polarMean_integral (t θ : ℝ) :
+    polarMean t θ = ∫ ω, gridDepth ω (0,0) (polarPair t θ) + t ∂squareLaw := by
+  rw [integral_add (gridDepth_integrable (zero_ne_polarPair t θ)) (integrable_const _)]
+  simp [polarMean]
+
+lemma polarMean_bound (t θ : ℝ) : |polarMean t θ| ≤ 4 := by
+  rw [polarMean_integral]
+  have hg := polarPair_gaps t θ
+  have hh := mean_abs_error_le (zero_ne_polarPair t θ) (Real.exp_pos t)
+    (by simpa only [zero_sub, abs_neg] using hg.1)
+    (by simpa only [zero_sub, abs_neg] using hg.2)
+  rw [Real.log_exp] at hh
+  have hb : |∫ ω, gridDepth ω (0,0) (polarPair t θ) + t ∂squareLaw| ≤
+      ∫ ω, |gridDepth ω (0,0) (polarPair t θ) + t| ∂squareLaw := by
+    simpa only [Real.norm_eq_abs] using (norm_integral_le_integral_norm
+      (f := fun ω => gridDepth ω (0,0) (polarPair t θ) + t) (μ := squareLaw))
+  exact hb.trans hh
+
+lemma measurable_polarPair : Measurable (fun p : ℝ × ℝ => polarPair p.1 p.2) := by
+  exact (measurable_fst.exp.mul measurable_snd.cos).prodMk
+    (measurable_fst.exp.mul measurable_snd.sin)
+
+lemma measurable_polarMean : Measurable (fun p : ℝ × ℝ => polarMean p.1 p.2) := by
+  have hm : Measurable (fun p : (ℝ × ℝ) × GridSample =>
+      gridDepth p.2 (0,0) (polarPair p.1.1 p.1.2)) :=
+    measurable_gridDepth_param measurable_snd measurable_const
+      (measurable_polarPair.comp measurable_fst) (fun p => zero_ne_polarPair _ _)
+  have hi : StronglyMeasurable (fun p : ℝ × ℝ => ∫ ω,
+      gridDepth ω (0,0) (polarPair p.1 p.2) ∂squareLaw) :=
+    hm.stronglyMeasurable.integral_prod_right
+  exact hi.measurable.add measurable_fst
+
+lemma polarPair_double (t θ : ℝ) : polarPair (t+Real.log 2) θ =
+    (2*(polarPair t θ).1,2*(polarPair t θ).2) := by
+  simp only [polarPair,Real.exp_add,Real.exp_log (by norm_num : (0:ℝ)<2)]
+  congr 1 <;> ring
+
+lemma polarMean_period_phase (θ : ℝ) : Function.Periodic (fun t => polarMean t θ) (Real.log 2) := by
+  intro t
+  have hm := mean_depth_double (zero_ne_polarPair t θ)
+  simp only [mul_zero] at hm
+  simp only [polarMean,polarPair_double,hm]
+  ring
+
+lemma polarMean_period_angle (t : ℝ) : Function.Periodic (polarMean t) (2*Real.pi) := by
+  intro θ
+  simp only [polarMean,polarPair,Real.cos_periodic θ,Real.sin_periodic θ]
+
+lemma uniformInterval_integral {l : ℝ} (hl : 0 < l) (f : ℝ → ℝ) :
+    (∫ x, f x ∂uniformInterval l) = l⁻¹ * ∫ x in (0:ℝ)..l, f x := by
+  rw [uniformInterval, integral_smul_measure, ENNReal.toReal_inv,
+    ENNReal.toReal_ofReal hl.le, smul_eq_mul, intervalIntegral.integral_of_le hl.le]
+
+lemma uniformInterval_period_shift {l : ℝ} (hl : 0 < l) {f : ℝ → ℝ}
+    (hf : Function.Periodic f l) (a : ℝ) :
+    (∫ x, f (x+a) ∂uniformInterval l) = ∫ x, f x ∂uniformInterval l := by
+  rw [uniformInterval_integral hl, uniformInterval_integral hl,
+    intervalIntegral.integral_comp_add_right]
+  congr 1
+  simpa only [zero_add, add_comm l a] using hf.intervalIntegral_add_eq a 0
+
+lemma uniformInterval_period_sub {l : ℝ} (hl : 0 < l) {f : ℝ → ℝ}
+    (hf : Function.Periodic f l) (a : ℝ) :
+    (∫ x, f (a-x) ∂uniformInterval l) = ∫ x, f (-x) ∂uniformInterval l := by
+  have hp : Function.Periodic (fun x => f (-x)) l := by
+    intro x
+    simpa only [neg_add, sub_eq_add_neg] using hf.sub_eq (-x)
+  have hh := uniformInterval_period_shift hl hp (-a)
+  simpa only [neg_add, neg_neg, sub_eq_add_neg, add_comm] using hh
+
+/-- The universal archimedean error mean, before any configuration is chosen. -/
+def kappa : ℝ := ∫ ξ, ∫ θ, polarMean (-ξ) θ
+  ∂uniformInterval (2*Real.pi) ∂uniformInterval (Real.log 2)
+
+lemma polar_average_constant (t φ : ℝ) :
+    (∫ ξ, ∫ θ, polarMean (t-ξ) (θ+φ)
+      ∂uniformInterval (2*Real.pi) ∂uniformInterval (Real.log 2)) = kappa := by
+  simp_rw [uniformInterval_period_shift (by positivity : 0 < 2*Real.pi)
+    (polarMean_period_angle _) φ]
+  have hp : Function.Periodic
+      (fun t => ∫ θ, polarMean t θ ∂uniformInterval (2*Real.pi)) (Real.log 2) := by
+    intro x
+    apply integral_congr_ae
+    filter_upwards with θ
+    exact polarMean_period_phase θ x
+  exact uniformInterval_period_sub (Real.log_pos (by norm_num)) hp t
+
+def rotateC (ξ θ : ℝ) (z : ℂ) : ℂ :=
+  (Real.exp (-ξ) : ℂ) * Complex.exp ((θ : ℂ)*Complex.I) * z
+
+def rotatePair (ξ θ : ℝ) (z : ℂ) : ℝ × ℝ :=
+  ((rotateC ξ θ z).re, (rotateC ξ θ z).im)
+
+lemma rotateC_sub (ξ θ : ℝ) (z w : ℂ) :
+    rotateC ξ θ (z-w) = rotateC ξ θ z - rotateC ξ θ w := by
+  exact mul_sub _ _ _
+
+lemma norm_rotateC (ξ θ : ℝ) (z : ℂ) :
+    ‖rotateC ξ θ z‖ = Real.exp (-ξ) * ‖z‖ := by
+  simp only [rotateC, norm_mul, Complex.norm_real, Real.norm_eq_abs,
+    abs_of_pos (Real.exp_pos _), Complex.norm_exp_ofReal_mul_I, mul_one]
+
+lemma rotatePair_ne (ξ θ : ℝ) {z w : ℂ} (hne : z ≠ w) :
+    rotatePair ξ θ z ≠ rotatePair ξ θ w := by
+  intro he
+  have h : rotateC ξ θ z = rotateC ξ θ w := Complex.ext
+    (congrArg Prod.fst he) (congrArg Prod.snd he)
+  have hn := norm_rotateC ξ θ (z-w)
+  rw [rotateC_sub, h, sub_self, norm_zero] at hn
+  have hz : ‖z-w‖ ≠ 0 := norm_ne_zero_iff.mpr (sub_ne_zero.mpr hne)
+  exact (mul_ne_zero (Real.exp_ne_zero _) hz) hn.symm
+
+lemma rotatePair_gaps (ξ θ : ℝ) (z w : ℂ) :
+    Real.exp (-ξ)*‖z-w‖ ≤
+      |(rotatePair ξ θ z).1-(rotatePair ξ θ w).1| +
+      |(rotatePair ξ θ z).2-(rotatePair ξ θ w).2| ∧
+    |(rotatePair ξ θ z).1-(rotatePair ξ θ w).1| +
+      |(rotatePair ξ θ z).2-(rotatePair ξ θ w).2| ≤ 2*(Real.exp (-ξ)*‖z-w‖) := by
+  have hh := Complex.norm_le_abs_re_add_abs_im (rotateC ξ θ (z-w))
+  have hr := Complex.abs_re_le_norm (rotateC ξ θ (z-w))
+  have hi := Complex.abs_im_le_norm (rotateC ξ θ (z-w))
+  rw [norm_rotateC, rotateC_sub] at hh hr hi
+  simp only [Complex.sub_re, Complex.sub_im] at hh hr hi
+  exact ⟨hh, by dsimp only [rotatePair]; linarith⟩
+
+lemma rotateC_polar (ξ θ : ℝ) {z : ℂ} (hz : z ≠ 0) :
+    rotateC ξ θ z = (Real.exp (Real.log ‖z‖-ξ) : ℂ) *
+      Complex.exp (((θ+z.arg : ℝ) : ℂ)*Complex.I) := by
+  rw [Complex.ofReal_add, add_mul, Complex.exp_add,
+    Real.exp_sub, Real.exp_log (norm_pos_iff.mpr hz),
+    div_eq_mul_inv, ← Real.exp_neg, Complex.ofReal_mul]
+  calc
+    _ = (Real.exp (-ξ) : ℂ) * Complex.exp ((θ:ℂ)*Complex.I) *
+        ((‖z‖:ℂ) * Complex.exp ((z.arg:ℂ)*Complex.I)) := by
+      rw [Complex.norm_mul_exp_arg_mul_I]
+      rfl
+    _ = _ := by ring
+
+lemma exp_polarPair (t θ : ℝ) :
+    ((Real.exp t : ℂ) * Complex.exp ((θ:ℂ)*Complex.I)).re = (polarPair t θ).1 ∧
+    ((Real.exp t : ℂ) * Complex.exp ((θ:ℂ)*Complex.I)).im = (polarPair t θ).2 := by
+  simp [polarPair, Complex.mul_re, Complex.mul_im, Complex.exp_re, Complex.exp_im]
+
+lemma rotated_mean (ξ θ : ℝ) {z w : ℂ} (hne : z ≠ w) :
+    (∫ ω, gridDepth ω (rotatePair ξ θ z) (rotatePair ξ θ w) ∂squareLaw)
+      - ξ + Real.log ‖z-w‖ = polarMean (Real.log ‖z-w‖-ξ) (θ+(z-w).arg) := by
+  have hp := exp_polarPair (Real.log ‖z-w‖-ξ) (θ+(z-w).arg)
+  rw [← rotateC_polar ξ θ (sub_ne_zero.mpr hne), rotateC_sub] at hp
+  have hm := mean_depth_same_gaps (rotatePair_ne ξ θ hne)
+    (zero_ne_polarPair (Real.log ‖z-w‖-ξ) (θ+(z-w).arg))
+    (by dsimp only [rotatePair]; simp only [zero_sub, abs_neg, ← hp.1]; rfl)
+    (by dsimp only [rotatePair]; simp only [zero_sub, abs_neg, ← hp.2]; rfl)
+  rw [hm, polarMean]
+  ring
+
+abbrev FullSample := (ℝ × ℝ) × GridSample
+
+def phaseLaw : Measure (ℝ × ℝ) :=
+  (uniformInterval (Real.log 2)).prod (uniformInterval (2*Real.pi))
+
+def fullLaw : Measure FullSample := phaseLaw.prod squareLaw
+
+instance phaseLaw_probability : IsProbabilityMeasure phaseLaw := by
+  let := uniformInterval_probability (Real.log_pos (by norm_num : (1:ℝ) < 2))
+  let := uniformInterval_probability (by positivity : 0 < 2*Real.pi)
+  exact Measure.prod.instIsProbabilityMeasure _ _
+
+instance fullLaw_probability : IsProbabilityMeasure fullLaw := by
+  exact Measure.prod.instIsProbabilityMeasure _ _
+
+/-- Actual jointly defined hierarchy depth in the original complex plane. -/
+def depth (ω : FullSample) (z w : ℂ) : ℝ :=
+  gridDepth ω.2 (rotatePair ω.1.1 ω.1.2 z) (rotatePair ω.1.1 ω.1.2 w) - ω.1.1
+
+def error (ω : FullSample) (z w : ℂ) : ℝ := depth ω z w + Real.log ‖z-w‖
+
+lemma measurable_rotatePair (z : ℂ) : Measurable (fun p : ℝ × ℝ => rotatePair p.1 p.2 z) := by
+  unfold rotatePair rotateC
+  fun_prop
+
+lemma measurable_depth {z w : ℂ} (hne : z ≠ w) : Measurable (fun ω => depth ω z w) := by
+  unfold depth
+  exact (measurable_gridDepth_param measurable_snd
+    ((measurable_rotatePair z).comp measurable_fst)
+    ((measurable_rotatePair w).comp measurable_fst)
+    (fun p => rotatePair_ne _ _ hne)).sub measurable_fst.fst
+
+lemma measurable_error {z w : ℂ} (hne : z ≠ w) : Measurable (fun ω => error ω z w) :=
+  (measurable_depth hne).add_const _
+
+lemma error_fixed_phase {z w : ℂ} (hne : z ≠ w) (p : ℝ × ℝ) (ω : GridSample) :
+    error (p,ω) z w = gridDepth ω (rotatePair p.1 p.2 z) (rotatePair p.1 p.2 w) +
+      Real.log (Real.exp (-p.1)*‖z-w‖) := by
+  rw [Real.log_mul (Real.exp_ne_zero _) (norm_ne_zero_iff.mpr (sub_ne_zero.mpr hne)), Real.log_exp]
+  simp only [error, depth]
+  ring
+
+lemma error_integrable_fixed {z w : ℂ} (hne : z ≠ w) (p : ℝ × ℝ) :
+    Integrable (fun ω => error (p,ω) z w) squareLaw := by
+  simp_rw [error_fixed_phase hne p]
+  exact error_integrable (rotatePair_ne _ _ hne)
+    (mul_pos (Real.exp_pos _) (norm_pos_iff.mpr (sub_ne_zero.mpr hne)))
+    (rotatePair_gaps _ _ _ _).1 (rotatePair_gaps _ _ _ _).2
+
+lemma mean_abs_error_fixed {z w : ℂ} (hne : z ≠ w) (p : ℝ × ℝ) :
+    (∫ ω, |error (p,ω) z w| ∂squareLaw) ≤ 4 := by
+  simp_rw [error_fixed_phase hne p]
+  exact mean_abs_error_le (rotatePair_ne _ _ hne)
+    (mul_pos (Real.exp_pos _) (norm_pos_iff.mpr (sub_ne_zero.mpr hne)))
+    (rotatePair_gaps _ _ _ _).1 (rotatePair_gaps _ _ _ _).2
+
+lemma error_integrable_full {z w : ℂ} (hne : z ≠ w) :
+    Integrable (fun ω => error ω z w) fullLaw := by
+  apply (integrable_prod_iff (measurable_error hne).aestronglyMeasurable).mpr
+  constructor
+  · exact Filter.Eventually.of_forall (error_integrable_fixed hne)
+  · apply Integrable.of_bound ((measurable_error hne).norm.stronglyMeasurable.integral_prod_right.aestronglyMeasurable) 4
+    filter_upwards with p
+    rw [Real.norm_eq_abs, abs_of_nonneg (integral_nonneg (fun _ => norm_nonneg _))]
+    simpa only [Real.norm_eq_abs] using mean_abs_error_fixed hne p
+
+lemma mean_error_fixed {z w : ℂ} (hne : z ≠ w) (p : ℝ × ℝ) :
+    (∫ ω, error (p,ω) z w ∂squareLaw) =
+      polarMean (Real.log ‖z-w‖-p.1) (p.2+(z-w).arg) := by
+  have hi : Integrable (fun ω => gridDepth ω (rotatePair p.1 p.2 z) (rotatePair p.1 p.2 w)) squareLaw :=
+    gridDepth_integrable (rotatePair_ne _ _ hne)
+  have his : Integrable (fun ω => gridDepth ω (rotatePair p.1 p.2 z) (rotatePair p.1 p.2 w) - p.1) squareLaw :=
+    hi.sub (integrable_const p.1)
+  simp only [error, depth]
+  rw [integral_add his (integrable_const (Real.log ‖z-w‖)), integral_sub hi (integrable_const p.1)]
+  simp only [integral_const, probReal_univ, one_smul]
+  exact rotated_mean _ _ hne
+
+lemma measurable_shifted_polarMean (t φ : ℝ) :
+    Measurable (fun p : ℝ × ℝ => polarMean (t-p.1) (p.2+φ)) := by
+  have h₁ : Measurable (fun p : ℝ × ℝ => t-p.1) := measurable_const.sub measurable_fst
+  have h₂ : Measurable (fun p : ℝ × ℝ => p.2+φ) := measurable_snd.add_const φ
+  change Measurable ((fun p : ℝ × ℝ => polarMean p.1 p.2) ∘
+    (fun p : ℝ × ℝ => (t-p.1,p.2+φ)))
+  apply Measurable.comp measurable_polarMean
+  exact h₁.prodMk h₂
+
+lemma mean_error_full {z w : ℂ} (hne : z ≠ w) :
+    (∫ ω, error ω z w ∂fullLaw) = kappa := by
+  rw [fullLaw, integral_prod _ (error_integrable_full hne)]
+  simp_rw [mean_error_fixed hne]
+  have hm := measurable_shifted_polarMean (Real.log ‖z-w‖) (z-w).arg
+  have hi := Integrable.of_bound (μ := phaseLaw) hm.aestronglyMeasurable 4
+    (Filter.Eventually.of_forall (fun p => by simpa only [Real.norm_eq_abs] using polarMean_bound _ _))
+  rw [phaseLaw, integral_prod _ hi]
+  exact polar_average_constant _ _
+
+lemma abs_error_tail_full {z w : ℂ} (hne : z ≠ w) {u : ℝ} (hu : 0 ≤ u) :
+    fullLaw.real {ω | u < |error ω z w|} ≤ 4*Real.exp (-u) := by
+  have hs : MeasurableSet {ω : FullSample | u < |error ω z w|} :=
+    measurableSet_lt measurable_const (measurable_error hne).abs
+  have hb : fullLaw {ω | u < |error ω z w|} ≤ ENNReal.ofReal (4*Real.exp (-u)) := by
+    rw [fullLaw, Measure.prod_apply hs]
+    calc
+      _ ≤ ∫⁻ _p, ENNReal.ofReal (4*Real.exp (-u)) ∂phaseLaw := by
+        apply lintegral_mono
+        intro p
+        have he : squareLaw.real {ω | u < |error (p,ω) z w|} ≤ 4*Real.exp (-u) := by
+          simp_rw [error_fixed_phase hne p]
+          exact error_abs_tail (rotatePair_ne _ _ hne)
+            (mul_pos (Real.exp_pos _) (norm_pos_iff.mpr (sub_ne_zero.mpr hne)))
+            (rotatePair_gaps _ _ _ _).1 (rotatePair_gaps _ _ _ _).2 hu
+        have hh := ENNReal.ofReal_le_ofReal he
+        rw [measureReal_def, ENNReal.ofReal_toReal (measure_ne_top _ _)] at hh
+        exact hh
+      _ = _ := by simp
+  have hh := ENNReal.toReal_mono ENNReal.ofReal_ne_top hb
+  rw [measureReal_def]
+  simpa only [ENNReal.toReal_ofReal (by positivity : 0 ≤ 4*Real.exp (-u))] using hh
+
+lemma mean_abs_error_full {z w : ℂ} (hne : z ≠ w) :
+    (∫ ω, |error ω z w| ∂fullLaw) ≤ 4 := by
+  rw [fullLaw, integral_prod _ (error_integrable_full hne).abs]
+  calc
+    _ ≤ ∫ _p, (4:ℝ) ∂phaseLaw := integral_mono
+      (error_integrable_full hne).abs.integral_prod_left (integrable_const _)
+      (mean_abs_error_fixed hne)
+    _ = _ := by simp
+
+lemma depth_symm (ω : FullSample) (z w : ℂ) : depth ω z w = depth ω w z := by
+  simp only [depth,gridDepth,minScale_symm ω.2 (rotatePair ω.1.1 ω.1.2 z)
+    (rotatePair ω.1.1 ω.1.2 w)]
+
+lemma depth_ultra_ae {x y z : ℂ} (hxy : x ≠ y) (hyz : y ≠ z) (hxz : x ≠ z) :
+    ∀ᵐ ω ∂fullLaw, min (depth ω x y) (depth ω y z) ≤ depth ω x z := by
+  have hs : MeasurableSet {ω : FullSample | min (depth ω x y) (depth ω y z) ≤ depth ω x z} :=
+    measurableSet_le ((measurable_depth hxy).min (measurable_depth hyz)) (measurable_depth hxz)
+  apply (Measure.ae_prod_iff_ae_ae hs).mpr
+  filter_upwards with p
+  filter_upwards [gridDepth_ultra_ae (rotatePair_ne p.1 p.2 hxy)
+    (rotatePair_ne p.1 p.2 hyz) (rotatePair_ne p.1 p.2 hxz)] with ω hh
+  simpa only [depth,min_sub_sub_right,sub_le_sub_iff_right] using hh
+
+end WeakPinned.Grid
+
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PinnedDistances/SmallCells.lean b/lean/OAI/Geometry/PinnedDistances/SmallCells.lean
new file mode 100644
index 0000000000000000000000000000000000000000..cb4fbbfa4bd44206d138285e345ae61fa6ce8c4e
--- /dev/null
+++ b/lean/OAI/Geometry/PinnedDistances/SmallCells.lean
@@ -0,0 +1,826 @@
+import OAI.Geometry.PinnedDistances.ArithmeticVariance
+
+namespace OAI
+
+universe u1 u2 u3 u4 u5 u6 u7 u8 u9
+
+section
+noncomputable section
+open NumberField
+open scoped BigOperators
+noncomputable section
+open MeasureTheory
+open scoped Topology
+noncomputable section
+open MeasureTheory
+open scoped Topology ENNReal
+noncomputable section
+open MeasureTheory NumberField
+open scoped BigOperators
+noncomputable section
+open scoped BigOperators
+open MeasureTheory
+
+namespace WeakPinned.Hierarchy.Depth
+open MeasureTheory
+variable {P : Type u1} [Fintype P] [DecidableEq P]
+variable (d : Depth P)
+
+def sourceGiant (x : P) (r : ℝ) : Prop := Fintype.card P < 2*(d.cell r x).card
+instance (x : P) (r : ℝ) : Decidable (d.sourceGiant x r) := Classical.propDecidable _
+
+omit [DecidableEq P] in
+lemma sourceGiant_mono (x : P) {r t : ℝ} (htr : t ≤ r) (hr : d.sourceGiant x r) :
+    d.sourceGiant x t := by
+  classical
+  have hc := Finset.card_le_card (d.cell_mono htr x)
+  unfold sourceGiant at *
+  omega
+
+omit [DecidableEq P] in
+lemma sourceGiant_iff (x : P) (r : ℝ) : d.sourceGiant x r ↔
+    ∃ C : Finset P, Fintype.card P < 2*C.card ∧ ∀ y ∈ C, d.share r x y := by
+  classical
+  constructor
+  · intro h; exact ⟨d.cell r x,h,fun y hy => (d.mem_cell r x y).mp hy⟩
+  · rintro ⟨C,hc,hC⟩
+    have hsub : C ⊆ d.cell r x := fun y hy => (d.mem_cell r x y).mpr (hC y hy)
+    have := Finset.card_le_card hsub
+    unfold sourceGiant
+    omega
+
+lemma isClosed_sourceGiant (x : P) : IsClosed {r : ℝ | d.sourceGiant x r} := by
+  classical
+  simp_rw [d.sourceGiant_iff, Set.ofPred_exists]
+  apply isClosed_iUnion_of_finite
+  intro C
+  simp only [Set.ofPred_and]
+  apply IsClosed.inter
+  · by_cases h : Fintype.card P < 2*C.card <;> simp [h]
+  · simp_rw [Set.ofPred_forall]
+    apply isClosed_iInter
+    intro y
+    apply isClosed_iInter
+    intro hy
+    by_cases hxy : x = y
+    · simp [share,hxy]
+    · simp only [share,hxy,false_or]
+      exact isClosed_le continuous_id continuous_const
+
+def sourceM (x : P) : ℝ := sSup {r : ℝ | d.sourceGiant x r}
+
+lemma sourceGiant_cutoff (hn : 2 ≤ Fintype.card P) (x : P) (r : ℝ) :
+    d.sourceGiant x r ↔ r ≤ d.sourceM x := by
+  classical
+  have hlo : d.sourceGiant x (-d.radius) := by
+    classical
+    unfold sourceGiant
+    rw [d.cell_of_low le_rfl,Finset.card_univ]
+    omega
+  have hle : ∀ t, d.sourceGiant x t → t ≤ d.radius := by
+    classical
+    intro t ht
+    exact d.giants_le_radius hn ⟨d.cell t x, (d.mem_giants t _).mpr ⟨d.cell_mem_cells t x,ht⟩⟩
+  have hne : Set.Nonempty {t : ℝ | d.sourceGiant x t} := ⟨-d.radius,hlo⟩
+  have hbd : BddAbove {t : ℝ | d.sourceGiant x t} := ⟨d.radius, fun t ht => hle t ht⟩
+  have hmem : d.sourceGiant x (d.sourceM x) := (d.isClosed_sourceGiant x).csSup_mem hne hbd
+  exact ⟨fun hr => le_csSup hbd hr, fun hr => d.sourceGiant_mono x hr hmem⟩
+
+lemma sourceM_le_M (hn : 2 ≤ Fintype.card P) (x : P) : d.sourceM x ≤ d.M := by
+  classical
+  apply (d.giants_cutoff hn _).mp
+  exact ⟨d.cell (d.sourceM x) x, (d.mem_giants _ _).mpr
+    ⟨d.cell_mem_cells _ x,(d.sourceGiant_cutoff hn x _).mpr le_rfl⟩⟩
+
+lemma sourceGiant_mem_giant (x : P) (r : ℝ) :
+    d.sourceGiant x r ↔ (d.giants r).Nonempty ∧ x ∈ d.giant r := by
+  classical
+  constructor
+  · intro hs
+    have hc : d.cell r x ∈ d.giants r := (d.mem_giants r _).mpr ⟨d.cell_mem_cells r x,hs⟩
+    exact ⟨⟨_,hc⟩,by rw [d.giant_eq hc]; exact d.self_mem_cell r x⟩
+  · rintro ⟨⟨G,hG⟩,hx⟩
+    rw [d.giant_eq hG] at hx
+    obtain ⟨hGc,hGm⟩ := (d.mem_giants r G).mp hG
+    rwa [d.cell_of_mem hGc hx] at hGm
+
+lemma outsideIndicator_source (hn : 2 ≤ Fintype.card P) (x : P) (r : ℝ) :
+    d.outsideIndicator r x = if d.sourceM x < r ∧ r ≤ d.M then 1 else 0 := by
+  classical
+  have he : ((d.giants r).Nonempty ∧ x ∉ d.giant r) ↔ d.sourceM x < r ∧ r ≤ d.M := by
+    classical
+    rw [← not_le, ← d.sourceGiant_cutoff hn x r, d.sourceGiant_mem_giant,
+      ← d.giants_cutoff hn r]
+    tauto
+  simp only [outsideIndicator,he]
+
+lemma potential_source (hn : 2 ≤ Fintype.card P) (x : P) :
+    d.potential x = d.M - d.sourceM x := by
+  classical
+  have he : (fun r => d.outsideIndicator r x) =
+      (Set.Ioc (d.sourceM x) d.M).indicator (fun _ : ℝ => (1:ℝ)) := by
+    classical
+    funext r
+    simp only [d.outsideIndicator_source hn, Set.indicator_apply, Set.mem_Ioc]
+  rw [potential,he,integral_indicator measurableSet_Ioc]
+  rw [← intervalIntegral.integral_of_le (d.sourceM_le_M hn x), intervalIntegral.integral_const]
+  simp
+
+/-- The level set is an interval of length 2H, regardless of the fiber center-level h. -/
+
+lemma straddle_integral (hn : 2 ≤ Fintype.card P) (x : P) (h H : ℝ) (hH : 0 ≤ H) :
+    (∫ r : ℝ, if d.sourceGiant x (h-r-H) ∧ ¬d.sourceGiant x (h-r+H) then (1:ℝ) else 0) = 2*H := by
+  classical
+  have he : (fun r : ℝ => if d.sourceGiant x (h-r-H) ∧ ¬d.sourceGiant x (h-r+H) then (1:ℝ) else 0) =
+      (Set.Ico (h-H-d.sourceM x) (h+H-d.sourceM x)).indicator (fun _ => (1:ℝ)) := by
+    classical
+    funext r
+    simp only [d.sourceGiant_cutoff hn, not_le,Set.indicator_apply,Set.mem_Ico]
+    congr 1
+    apply propext
+    constructor <;> rintro ⟨h₁,h₂⟩ <;> constructor <;> linarith
+  rw [he,integral_indicator measurableSet_Ico]
+  rw [integral_Ico_eq_integral_Ioc, ← intervalIntegral.integral_of_le (by linarith : h-H-d.sourceM x ≤ h+H-d.sourceM x), intervalIntegral.integral_const]
+  simp
+  ring
+
+def smallCells (y : P) (T : ℝ) : Finset (Finset P) := by
+  classical
+  exact Finset.univ.filter (fun C => ∃ r : ℝ, C = d.cell r y ∧ (C.card : ℝ) < T)
+
+def smallNeighbors (y : P) (T : ℝ) : Finset P := (d.smallCells y T).biUnion id
+
+omit [DecidableEq P] in
+lemma smallCells_chain (y : P) (T : ℝ) {C E : Finset P}
+    (hC : C ∈ d.smallCells y T) (hE : E ∈ d.smallCells y T) : C ⊆ E ∨ E ⊆ C := by
+  classical
+  obtain ⟨r,rfl,hr⟩ := (Finset.mem_filter.mp hC).2
+  obtain ⟨t,rfl,ht⟩ := (Finset.mem_filter.mp hE).2
+  rcases le_total t r with h | h
+  · exact Or.inl (d.cell_mono h y)
+  · exact Or.inr (d.cell_mono h y)
+
+lemma smallNeighbors_card (y : P) (T : ℝ) (hT : 0 < T) :
+    ((d.smallNeighbors y T).card : ℝ) < T := by
+  classical
+  by_cases hne : (d.smallCells y T).Nonempty
+  · obtain ⟨C,hC,hmax⟩ := Finset.exists_max_image (d.smallCells y T) Finset.card hne
+    have he : d.smallNeighbors y T = C := by
+      classical
+      apply Finset.Subset.antisymm
+      · intro x hx
+        obtain ⟨E,hE,hx⟩ := Finset.mem_biUnion.mp hx
+        change x ∈ E at hx
+        rcases d.smallCells_chain y T hE hC with h | h
+        · exact h hx
+        · have he : C = E := Finset.eq_of_subset_of_card_le h (hmax E hE)
+          simpa only [he] using hx
+      · intro x hx
+        exact Finset.mem_biUnion.mpr ⟨C,hC,hx⟩
+    rw [he]
+    obtain ⟨r,hr,ht⟩ := (Finset.mem_filter.mp hC).2
+    exact ht
+  · rw [smallNeighbors,Finset.not_nonempty_iff_eq_empty.mp hne]
+    simpa using hT
+
+lemma source_mem_smallNeighbors (y x : P) (r T : ℝ)
+    (hxy : y ∈ d.cell r x) (hT : ((d.cell r x).card : ℝ) < T) :
+    x ∈ d.smallNeighbors y T := by
+  classical
+  have he := d.cell_eq_of_share ((d.mem_cell r x y).mp hxy)
+  apply Finset.mem_biUnion.mpr
+  refine ⟨d.cell r y, ?_, (d.mem_cell r y x).mpr (d.share_symm ((d.mem_cell r x y).mp hxy))⟩
+  apply Finset.mem_filter.mpr
+  exact ⟨Finset.mem_univ _,r,rfl,by rwa [← he]⟩
+end WeakPinned.Hierarchy.Depth
+
+namespace WeakPinned.Hierarchy.Depth
+open MeasureTheory
+variable {P : Type u2} [Fintype P] [DecidableEq P]
+variable (d : Depth P)
+
+lemma smallNeighbors_self (y : P) (T : ℝ) (hne : (d.smallNeighbors y T).Nonempty) :
+    y ∈ d.smallNeighbors y T := by
+  classical
+  obtain ⟨x,hx⟩ := hne
+  obtain ⟨C,hC,hx⟩ := Finset.mem_biUnion.mp hx
+  obtain ⟨r,rfl,hr⟩ := (Finset.mem_filter.mp hC).2
+  apply Finset.mem_biUnion.mpr
+  exact ⟨d.cell r y,hC,d.self_mem_cell r y⟩
+
+lemma smallNeighbors_erase_bound (hn : 2 ≤ Fintype.card P) (y : P)
+    (τ : ℝ) (hτ : 0 < τ) (hτ1 : τ ≤ 1) :
+    ((d.smallNeighbors y (τ*Fintype.card P)).erase y).card ≤ τ*((Fintype.card P : ℝ)-1) := by
+  classical
+  have hn0 : 0 < (Fintype.card P : ℝ) := by exact_mod_cast (show 0 < Fintype.card P by omega)
+  have hn1 : 0 ≤ (Fintype.card P : ℝ)-1 := by
+    have : (2:ℝ) ≤ Fintype.card P := by exact_mod_cast hn
+    linarith
+  by_cases hne : (d.smallNeighbors y (τ*Fintype.card P)).Nonempty
+  · have hy := d.smallNeighbors_self y _ hne
+    have hc := d.smallNeighbors_card y (τ*Fintype.card P) (mul_pos hτ hn0)
+    rw [Finset.card_erase_of_mem hy,Nat.cast_sub (Finset.one_le_card.mpr hne),Nat.cast_one]
+    nlinarith
+  · rw [Finset.not_nonempty_iff_eq_empty.mp hne]
+    simp only [Finset.erase_empty,Finset.card_empty,Nat.cast_zero]
+    exact mul_nonneg hτ.le hn1
+
+/-- All candidate sources for each target lie in one small neighborhood,
+    allowing a level-dependent fiber reparametrization. -/
+
+lemma smallNeighbors_pair_bound (hn : 2 ≤ Fintype.card P)
+    (q : P → P → ℝ) (B τ : ℝ) (hB : 0 ≤ B) (hτ : 0 < τ) (hτ1 : τ ≤ 1)
+    (hq : ∀ x y, x ≠ y → q x y ≤ B / ((Fintype.card P : ℝ)*((Fintype.card P : ℝ)-1)))
+    (C : Finset P) :
+    (∑ y ∈ C, ∑ x ∈ (d.smallNeighbors y (τ*Fintype.card P)).erase y, q x y) ≤
+      B*τ*((C.card : ℝ)/Fintype.card P) := by
+  classical
+  have hn0 : 0 < (Fintype.card P : ℝ) := by exact_mod_cast (show 0 < Fintype.card P by omega)
+  have hn1 : 0 < (Fintype.card P : ℝ)-1 := by
+    have : (2:ℝ) ≤ Fintype.card P := by exact_mod_cast hn
+    linarith
+  have hden : 0 ≤ B / ((Fintype.card P : ℝ)*((Fintype.card P : ℝ)-1)) :=
+    div_nonneg hB (mul_pos hn0 hn1).le
+  have hy (y : P) : (∑ x ∈ (d.smallNeighbors y (τ*Fintype.card P)).erase y, q x y) ≤
+      B*τ/Fintype.card P := by
+    classical
+    calc
+      _ ≤ ∑ x ∈ (d.smallNeighbors y (τ*Fintype.card P)).erase y,
+          B / ((Fintype.card P : ℝ)*((Fintype.card P : ℝ)-1)) :=
+        Finset.sum_le_sum (fun x hx => hq x y (Finset.ne_of_mem_erase hx))
+      _ = (((d.smallNeighbors y (τ*Fintype.card P)).erase y).card : ℝ) *
+          (B / ((Fintype.card P : ℝ)*((Fintype.card P : ℝ)-1))) := by simp
+      _ ≤ (τ*((Fintype.card P : ℝ)-1)) *
+          (B / ((Fintype.card P : ℝ)*((Fintype.card P : ℝ)-1))) :=
+        mul_le_mul_of_nonneg_right (d.smallNeighbors_erase_bound hn y τ hτ hτ1) hden
+      _ = B*τ/Fintype.card P := by field_simp
+  calc
+    _ ≤ ∑ y ∈ C, B*τ/Fintype.card P := Finset.sum_le_sum (fun y _ => hy y)
+    _ = _ := by simp; ring
+
+end WeakPinned.Hierarchy.Depth
+
+namespace WeakPinned.SmallFiber
+open MeasureTheory
+variable {P : Type u3} [Fintype P] [DecidableEq P]
+
+/-- Probability of a finite subset; the fiber weights used below will be the
+    exact uniform probabilities on the original retained distance fibers. -/
+def mass (p : P → ℝ) (C : Finset P) : ℝ := ∑ y ∈ C, p y
+
+omit [Fintype P] [DecidableEq P] in
+lemma mass_nonneg {p : P → ℝ} (hp : ∀ y, 0 ≤ p y) (C : Finset P) : 0 ≤ mass p C :=
+  Finset.sum_nonneg (fun y _ => hp y)
+
+omit [Fintype P] [DecidableEq P] in
+lemma mass_mono {p : P → ℝ} (hp : ∀ y, 0 ≤ p y) {C D : Finset P} (h : C ⊆ D) :
+    mass p C ≤ mass p D := Finset.sum_le_sum_of_subset_of_nonneg h (fun y _ _ => hp y)
+
+omit [DecidableEq P] in
+lemma mass_le_one {p : P → ℝ} (hp : ∀ y, 0 ≤ p y) (h1 : ∑ y, p y = 1) (C : Finset P) :
+    mass p C ≤ 1 := by rw [← h1]; exact mass_mono hp (Finset.subset_univ C)
+
+omit [Fintype P] in
+lemma mass_diff_inter (p : P → ℝ) (C D : Finset P) :
+    mass p C = mass p (C \ D) + mass p (C ∩ D) := by
+  classical
+  symm
+  have hh := Finset.sum_sdiff (f:=p) (Finset.inter_subset_left (s₁:=C) (s₂:=D))
+  have he : C \ (C ∩ D) = C \ D := by ext y; simp
+  rw [he] at hh
+  exact hh
+
+lemma mass_cover (p : P → ℝ) (hp : ∀ y, 0 ≤ p y) (h1 : ∑ y, p y = 1)
+    (C D : Finset P) : 1 ≤ mass p C + mass p D + mass p (C ∪ D)ᶜ := by
+  classical
+  rw [← h1]
+  have he := Finset.sum_compl_add_sum (C ∪ D) p
+  have hu := (Finset.sum_union_inter (s₁:=C) (s₂:=D) (f:=p))
+  have hn := mass_nonneg hp (C ∩ D)
+  unfold mass at *
+  linarith
+
+lemma small_case_one (p : P → ℝ) (hp : ∀ y, 0 ≤ p y) (h1 : ∑ y, p y = 1)
+    (C D : Finset P) (lam : ℝ) (hlam : 0 ≤ lam) (hlam2 : lam ≤ 1/2)
+    (hC : mass p C ≤ lam) :
+    mass p C ≤ 2*lam*mass p D + mass p (C ∪ D)ᶜ := by
+  classical
+  have hc := mass_cover p hp h1 C D
+  have hnC := mass_nonneg hp C
+  have hnD := mass_nonneg hp D
+  have hnE := mass_nonneg hp (C∪D)ᶜ
+  have hp2 : mass p C ≤ 1/2 := hC.trans hlam2
+  nlinarith
+
+end WeakPinned.SmallFiber
+
+namespace WeakPinned.Hierarchy.Depth
+open MeasureTheory
+variable {P : Type u4} [Fintype P] [DecidableEq P]
+variable (d : Depth P)
+
+lemma nongiant_share_overlap {r : ℝ} {x y : P} (hg : ¬ d.sourceGiant x r)
+    (hy : y ∈ d.cell r x) : 1 ≤ d.pairOverlap r x y := by
+  classical
+  have hs := (d.mem_cell r x y).mp hy
+  rw [d.pairOverlap_formula, ite_eq_left hs]
+  by_cases he : (d.giants r).Nonempty
+  · have hx : x ∉ d.giant r := by
+      intro hx
+      exact hg ((d.sourceGiant_mem_giant x r).mpr ⟨he,hx⟩)
+    have hy' : y ∉ d.giant r := by
+      intro hy'
+      have hgy := (d.sourceGiant_mem_giant y r).mpr ⟨he,hy'⟩
+      exact hg (by simpa only [sourceGiant, d.cell_eq_of_share hs] using hgy)
+    simp [giantIndicator,outsideIndicator,he,hx,hy']
+  · simp [giantIndicator,outsideIndicator,he]
+
+end WeakPinned.Hierarchy.Depth
+
+namespace WeakPinned.SmallFiber
+open MeasureTheory Hierarchy
+variable {P : Type u5} [Fintype P] [DecidableEq P]
+
+lemma mass_as_sum (p : P → ℝ) (C : Finset P) :
+    mass p C = ∑ y, if y ∈ C then p y else 0 := by
+  classical
+  simp [mass]
+
+lemma mass_as_sum_off (p : P → ℝ) (x : P) (hx : p x = 0) (C : Finset P) :
+    mass p C = ∑ y, if x ≠ y ∧ y ∈ C then p y else 0 := by
+  classical
+  rw [mass_as_sum]
+  apply Finset.sum_congr rfl
+  intro y _
+  by_cases hxy : x = y
+  · subst y; simp [hx]
+  · simp [hxy]
+
+lemma nongiant_mass_bound (d : Depth P) (p : P → ℝ) (hp : ∀ y, 0 ≤ p y)
+    (x : P) (r : ℝ) :
+    (if ¬d.sourceGiant x r then mass p (d.cell r x) else 0) ≤
+      ∑ y, p y * d.pairOverlap r x y := by
+  classical
+  by_cases hg : d.sourceGiant x r
+  · simp only [hg,not_true_eq_false,ite_false]
+    exact Finset.sum_nonneg (fun y _ => mul_nonneg (hp y) (d.pairOverlap_bounds r x y).1)
+  · simp only [hg,not_false_eq_true,ite_true]
+    rw [mass_as_sum]
+    apply Finset.sum_le_sum
+    intro y _
+    by_cases hy : y ∈ d.cell r x
+    · simp only [hy,ite_true]
+      nlinarith [d.nongiant_share_overlap hg hy, hp y]
+    · simpa only [hy,ite_false] using mul_nonneg (hp y) (d.pairOverlap_bounds r x y).1
+
+lemma integral_open_interval (a b : ℝ) :
+    (∫ r : ℝ, if a < r ∧ r < b then (1:ℝ) else 0) = max (b-a) 0 := by
+  have he : (fun r : ℝ => if a < r ∧ r < b then (1:ℝ) else 0) =
+      (Set.Ioo a b).indicator (fun _ : ℝ => (1:ℝ)) := by
+    funext r; simp [Set.indicator_apply]
+  rw [he, integral_indicator measurableSet_Ioo]
+  simp [integral_const,measureReal_def,Real.volume_Ioo,ENNReal.toReal_ofReal']
+
+lemma integral_closed_interval (a b : ℝ) :
+    (∫ r : ℝ, if a ≤ r ∧ r ≤ b then (1:ℝ) else 0) = max (b-a) 0 := by
+  have he : (fun r : ℝ => if a ≤ r ∧ r ≤ b then (1:ℝ) else 0) =
+      (Set.Icc a b).indicator (fun _ : ℝ => (1:ℝ)) := by
+    funext r; simp [Set.indicator_apply]
+  rw [he, integral_indicator measurableSet_Icc]
+  simp [integral_const,measureReal_def,Real.volume_Icc,ENNReal.toReal_ofReal']
+
+lemma integrable_open_interval (a b : ℝ) :
+    Integrable (fun r : ℝ => if a < r ∧ r < b then (1:ℝ) else 0) := by
+  have he : (fun r : ℝ => if a < r ∧ r < b then (1:ℝ) else 0) =
+      (Set.Ioo a b).indicator (fun _ : ℝ => (1:ℝ)) := by
+    funext r; simp [Set.indicator_apply]
+  rw [he, integrable_indicator_iff measurableSet_Ioo]
+  exact integrableOn_const (by rw [Real.volume_Ioo]; exact ENNReal.ofReal_ne_top)
+
+lemma integrable_closed_interval (a b : ℝ) :
+    Integrable (fun r : ℝ => if a ≤ r ∧ r ≤ b then (1:ℝ) else 0) := by
+  have he : (fun r : ℝ => if a ≤ r ∧ r ≤ b then (1:ℝ) else 0) =
+      (Set.Icc a b).indicator (fun _ : ℝ => (1:ℝ)) := by
+    funext r; simp [Set.indicator_apply]
+  rw [he, integrable_indicator_iff measurableSet_Icc]
+  exact integrableOn_const (by rw [Real.volume_Icc]; exact ENNReal.ofReal_ne_top)
+
+def minusError (d d' : Depth P) (p : P → ℝ) (x : P) (h H r : ℝ) : ℝ :=
+  mass p (d.cell r x ∪ d'.cell (h-r-H) x)ᶜ
+
+def plusError (d d' : Depth P) (p : P → ℝ) (x : P) (h H r : ℝ) : ℝ :=
+  mass p (d.cell r x ∩ d'.cell (h-r+H) x)
+
+lemma minusError_formula (d d' : Depth P) (p : P → ℝ) (x : P) (hx : p x = 0)
+    (h H r : ℝ) :
+    minusError d d' p x h H r = ∑ y, p y *
+      (if d.value x y < r ∧ r < h-H-d'.value x y then (1:ℝ) else 0) := by
+  classical
+  rw [minusError,mass_as_sum]
+  apply Finset.sum_congr rfl
+  intro y _
+  by_cases hxy : x = y
+  · subst y; simp [hx]
+  · simp only [Finset.mem_compl,Finset.mem_union,Depth.mem_cell,Depth.share,hxy,false_or,not_or,not_le]
+    have he : (d.value x y < r ∧ d'.value x y < h-r-H) ↔
+        d.value x y < r ∧ r < h-H-d'.value x y := by constructor <;> rintro ⟨a,b⟩ <;> constructor <;> linarith
+    simp only [he]
+    split_ifs <;> simp
+
+lemma plusError_formula (d d' : Depth P) (p : P → ℝ) (x : P) (hx : p x = 0)
+    (h H r : ℝ) :
+    plusError d d' p x h H r = ∑ y, p y *
+      (if h+H-d'.value x y ≤ r ∧ r ≤ d.value x y then (1:ℝ) else 0) := by
+  classical
+  rw [plusError,mass_as_sum]
+  apply Finset.sum_congr rfl
+  intro y _
+  by_cases hxy : x = y
+  · subst y; simp [hx]
+  · simp only [Finset.mem_inter,Depth.mem_cell,Depth.share,hxy,false_or]
+    have he : (r ≤ d.value x y ∧ h-r+H ≤ d'.value x y) ↔
+        h+H-d'.value x y ≤ r ∧ r ≤ d.value x y := by constructor <;> rintro ⟨a,b⟩ <;> constructor <;> linarith
+    simp only [he]
+    split_ifs <;> simp
+
+lemma minusError_integrable (d d' : Depth P) (p : P → ℝ) (x : P) (hx : p x = 0)
+    (h H : ℝ) : Integrable (minusError d d' p x h H) := by
+  change Integrable (fun r => minusError d d' p x h H r)
+  simp_rw [minusError_formula d d' p x hx]
+  exact integrable_finsetSum _ (fun y _ => (integrable_open_interval _ _).const_mul (p y))
+
+lemma plusError_integrable (d d' : Depth P) (p : P → ℝ) (x : P) (hx : p x = 0)
+    (h H : ℝ) : Integrable (plusError d d' p x h H) := by
+  change Integrable (fun r => plusError d d' p x h H r)
+  simp_rw [plusError_formula d d' p x hx]
+  exact integrable_finsetSum _ (fun y _ => (integrable_closed_interval _ _).const_mul (p y))
+
+lemma minusError_integral (d d' : Depth P) (p : P → ℝ) (x : P) (hx : p x = 0)
+    (h H : ℝ) : (∫ r, minusError d d' p x h H r) =
+      ∑ y, p y * max (h-d.value x y-d'.value x y-H) 0 := by
+  simp_rw [minusError_formula d d' p x hx]
+  rw [integral_finsetSum _ (fun y _ => (integrable_open_interval _ _).const_mul (p y))]
+  simp_rw [integral_const_mul,integral_open_interval]
+  apply Finset.sum_congr rfl
+  intro y _
+  congr 2; ring
+
+lemma plusError_integral (d d' : Depth P) (p : P → ℝ) (x : P) (hx : p x = 0)
+    (h H : ℝ) : (∫ r, plusError d d' p x h H r) =
+      ∑ y, p y * max (d.value x y+d'.value x y-h-H) 0 := by
+  simp_rw [plusError_formula d d' p x hx]
+  rw [integral_finsetSum _ (fun y _ => (integrable_closed_interval _ _).const_mul (p y))]
+  simp_rw [integral_const_mul,integral_closed_interval]
+  apply Finset.sum_congr rfl
+  intro y _
+  congr 2; ring
+
+lemma two_truncations (a H : ℝ) (hH : 0 ≤ H) :
+    max (-a-H) 0 + max (a-H) 0 = max (|a|-H) 0 := by
+  rcases le_total 0 a with ha | ha
+  · rw [abs_of_nonneg ha,max_eq_right (by linarith : -a-H ≤ 0),zero_add]
+  · rw [abs_of_nonpos ha,max_eq_right (by linarith : a-H ≤ 0),add_zero]
+
+lemma errors_integral (d d' : Depth P) (p : P → ℝ) (x : P) (hx : p x = 0)
+    (h H : ℝ) (hH : 0 ≤ H) :
+    (∫ r, minusError d d' p x h H r + plusError d d' p x h H r) =
+      ∑ y, p y * max (|d.value x y+d'.value x y-h|-H) 0 := by
+  rw [integral_add (minusError_integrable d d' p x hx h H) (plusError_integrable d d' p x hx h H),
+    minusError_integral d d' p x hx, plusError_integral d d' p x hx,← Finset.sum_add_distrib]
+  apply Finset.sum_congr rfl
+  intro y _
+  have he : h-d.value x y-d'.value x y-H = -(d.value x y+d'.value x y-h)-H := by ring
+  rw [he,← mul_add,two_truncations _ H hH]
+
+end WeakPinned.SmallFiber
+
+namespace WeakPinned.SmallFiber
+open MeasureTheory Hierarchy
+variable {P : Type u6} [Fintype P] [DecidableEq P]
+
+/-- Actual source cell, not a giant-complement occurrence. -/
+def activeMass (d : Depth P) (p : P → ℝ) (x : P) (T lam r : ℝ) : ℝ := by
+  classical
+  exact if ¬d.sourceGiant x r ∧ ((d.cell r x).card : ℝ) < T ∧ mass p (d.cell r x) < lam
+    then mass p (d.cell r x) else 0
+
+def nongiantMass (d : Depth P) (p : P → ℝ) (x : P) (r : ℝ) : ℝ :=
+  if ¬d.sourceGiant x r then mass p (d.cell r x) else 0
+
+def straddle (d : Depth P) (x : P) (h H r : ℝ) : ℝ :=
+  if d.sourceGiant x (h-r-H) ∧ ¬d.sourceGiant x (h-r+H) then 1 else 0
+
+def neighborMass (d d' : Depth P) (p : P → ℝ) (x : P) (T t : ℝ) : ℝ := by
+  classical
+  exact ∑ y, p y * (if x ∈ (d.smallNeighbors y T).erase y then d'.outsideIndicator t y else 0)
+
+lemma neighborMass_nonneg (d d' : Depth P) (p : P → ℝ) (hp : ∀ y, 0 ≤ p y)
+    (x : P) (T t : ℝ) : 0 ≤ neighborMass d d' p x T t := by
+  classical
+  exact Finset.sum_nonneg (fun y _ => mul_nonneg (hp y) (by split_ifs; exact (d'.outsideIndicator_bounds t y).1; rfl))
+
+omit [DecidableEq P] in
+lemma nongiantMass_nonneg (d : Depth P) (p : P → ℝ) (hp : ∀ y, 0 ≤ p y)
+    (x : P) (r : ℝ) : 0 ≤ nongiantMass d p x r := by
+  unfold nongiantMass
+  split_ifs <;> first | exact mass_nonneg hp _ | rfl
+
+lemma case_three_bound (d d' : Depth P) (p : P → ℝ) (hp : ∀ y, 0 ≤ p y)
+    (x : P) (hx : p x = 0) (r t T : ℝ)
+    (hT : ((d.cell r x).card : ℝ) < T) (hg : d'.sourceGiant x t) :
+    mass p (d.cell r x \ d'.cell t x) ≤ neighborMass d d' p x T t := by
+  classical
+  rw [mass_as_sum_off p x hx]
+  unfold neighborMass
+  apply Finset.sum_le_sum
+  intro y _
+  by_cases hy : x ≠ y ∧ y ∈ d.cell r x \ d'.cell t x
+  · obtain ⟨hxy,hyD⟩ := hy
+    obtain ⟨hyU,hyV⟩ := Finset.mem_sdiff.mp hyD
+    have hn : x ∈ (d.smallNeighbors y T).erase y :=
+      Finset.mem_erase.mpr ⟨hxy, d.source_mem_smallNeighbors y x r T hyU hT⟩
+    have hG : d'.cell t x ∈ d'.giants t := (d'.mem_giants t _).mpr ⟨d'.cell_mem_cells t x,hg⟩
+    have he : (d'.giants t).Nonempty := ⟨_,hG⟩
+    have hyo : y ∉ d'.giant t := by rwa [d'.giant_eq hG]
+    simp [hxy,hyD,hn,Depth.outsideIndicator,he,hyo]
+  · simp only [hy,ite_false]
+    exact mul_nonneg (hp y) (by split_ifs; exact (d'.outsideIndicator_bounds t y).1; rfl)
+
+/-- The pointwise small-fiber bound. The only independent error is the exact
+    pair of intervals already evaluated. -/
+lemma active_pointwise (d d' : Depth P) (p : P → ℝ) (hp : ∀ y, 0 ≤ p y)
+    (h1 : ∑ y, p y = 1) (x : P) (hx : p x = 0) (h H T lam r : ℝ)
+    (hlam : 0 ≤ lam) (hlam2 : lam ≤ 1/2) :
+    activeMass d p x T lam r ≤
+      2*lam*nongiantMass d' p x (h-r-H) + lam*straddle d' x h H r +
+      minusError d d' p x h H r + plusError d d' p x h H r +
+      neighborMass d d' p x T (h-r+H) := by
+  classical
+  have hminus : 0 ≤ minusError d d' p x h H r := mass_nonneg hp _
+  have hplus : 0 ≤ plusError d d' p x h H r := mass_nonneg hp _
+  have hnbr := neighborMass_nonneg d d' p hp x T (h-r+H)
+  have hng := nongiantMass_nonneg d' p hp x (h-r-H)
+  have hstr : 0 ≤ straddle d' x h H r := by unfold straddle; split_ifs <;> norm_num
+  unfold activeMass
+  split_ifs with ha
+  · obtain ⟨haG,haT,haP⟩ := ha
+    by_cases hg : d'.sourceGiant x (h-r-H)
+    · by_cases hg' : d'.sourceGiant x (h-r+H)
+      · have hthree := case_three_bound d d' p hp x hx r (h-r+H) T haT hg'
+        have he := mass_diff_inter p (d.cell r x) (d'.cell (h-r+H) x)
+        change mass p (d.cell r x) = _ + plusError d d' p x h H r at he
+        nlinarith
+      · have hs : straddle d' x h H r = 1 := by simp [straddle,hg,hg']
+        rw [hs]
+        nlinarith
+    · have hone := small_case_one p hp h1 (d.cell r x) (d'.cell (h-r-H) x) lam hlam hlam2 haP.le
+      have hnge : nongiantMass d' p x (h-r-H) = mass p (d'.cell (h-r-H) x) := by
+        simp [nongiantMass,hg]
+      rw [hnge]
+      change mass p (d.cell r x) ≤ _ + minusError d d' p x h H r at hone
+      nlinarith
+  · positivity
+
+omit [DecidableEq P] in
+lemma nongiantMass_measurable (d : Depth P) (p : P → ℝ) (x : P) :
+    Measurable (nongiantMass d p x) := by
+  classical
+  let f : Finset P → ℝ := fun C => if ¬Fintype.card P < 2*C.card then mass p C else 0
+  have h := (measurable_of_finite f).comp (d.measurable_cell x)
+  have he : nongiantMass d p x = f ∘ (fun r => d.cell r x) := by
+    funext r
+    by_cases hg : Fintype.card P < 2*(d.cell r x).card
+    · simp [nongiantMass,f,Depth.sourceGiant,hg]
+    · simp [nongiantMass,f,Depth.sourceGiant,hg]
+  rw [he]
+  exact h
+
+omit [DecidableEq P] in
+lemma activeMass_measurable (d : Depth P) (p : P → ℝ) (x : P) (T lam : ℝ) :
+    Measurable (activeMass d p x T lam) := by
+  classical
+  let f : Finset P → ℝ := fun C => if ¬Fintype.card P < 2*C.card ∧ (C.card : ℝ) < T ∧ mass p C < lam
+      then mass p C else 0
+  have h := (measurable_of_finite f).comp (d.measurable_cell x)
+  have he : activeMass d p x T lam = f ∘ (fun r => d.cell r x) := by
+    funext r
+    by_cases hg : Fintype.card P < 2*(d.cell r x).card
+    · simp [activeMass,f,Depth.sourceGiant,hg]
+    · simp [activeMass,f,Depth.sourceGiant,hg,Nat.not_lt.mp hg]
+  rw [he]
+  exact h
+
+omit [DecidableEq P] in
+lemma nongiantMass_integrable (d : Depth P) (hn : 2 ≤ Fintype.card P)
+    (p : P → ℝ) (hp : ∀ y, 0 ≤ p y) (h1 : ∑ y, p y = 1) (x : P) (hx : p x = 0) :
+    Integrable (nongiantMass d p x) := by
+  classical
+  apply Depth.integrable_supported_Icc (nongiantMass_measurable d p x) (a:=-d.radius) (b:=d.radius) (C:=1)
+  · intro r
+    rw [Real.norm_eq_abs,abs_of_nonneg (nongiantMass_nonneg d p hp x r)]
+    unfold nongiantMass
+    split_ifs <;> first | exact mass_le_one hp h1 _ | norm_num
+  · intro r hr
+    by_cases hlo : r ≤ -d.radius
+    · have hG : d.sourceGiant x r := by
+        unfold Depth.sourceGiant
+        rw [d.cell_of_low hlo x,Finset.card_univ]
+        omega
+      simp [nongiantMass,hG]
+    · have hhi : d.radius < r := by
+        by_contra! hhi
+        exact hr ⟨(not_le.mp hlo).le,hhi⟩
+      simp only [nongiantMass]
+      rw [d.cell_of_high hhi x]
+      have he : mass p {x} = 0 := by simp [mass,hx]
+      split_ifs <;> simp [he]
+
+omit [DecidableEq P] in
+lemma activeMass_integrable (d : Depth P) (hn : 2 ≤ Fintype.card P)
+    (p : P → ℝ) (hp : ∀ y, 0 ≤ p y) (h1 : ∑ y, p y = 1) (x : P) (hx : p x = 0)
+    (T lam : ℝ) : Integrable (activeMass d p x T lam) := by
+  classical
+  apply (nongiantMass_integrable d hn p hp h1 x hx).mono'
+    (activeMass_measurable d p x T lam).aestronglyMeasurable
+  filter_upwards with r
+  unfold activeMass nongiantMass
+  split_ifs <;> simp_all [abs_of_nonneg (mass_nonneg hp (d.cell r x)),mass_nonneg hp]
+
+lemma straddle_integrable (d : Depth P) (hn : 2 ≤ Fintype.card P) (x : P) (h H : ℝ) :
+    Integrable (straddle d x h H) := by
+  classical
+  have he : straddle d x h H =
+      (Set.Ico (h-H-d.sourceM x) (h+H-d.sourceM x)).indicator (fun _ : ℝ => (1:ℝ)) := by
+    funext r
+    simp only [straddle,d.sourceGiant_cutoff hn,not_le,Set.indicator_apply,Set.mem_Ico]
+    congr 1
+    apply propext
+    constructor <;> rintro ⟨a,b⟩ <;> constructor <;> linarith
+  rw [he,integrable_indicator_iff measurableSet_Ico]
+  exact integrableOn_const (by rw [Real.volume_Ico]; exact ENNReal.ofReal_ne_top)
+
+lemma neighborMass_integrable (d d' : Depth P) (hn : 2 ≤ Fintype.card P)
+    (p : P → ℝ) (x : P) (T : ℝ) : Integrable (neighborMass d d' p x T) := by
+  classical
+  unfold neighborMass
+  apply integrable_finsetSum
+  intro y _
+  apply Integrable.const_mul
+  by_cases h : x ∈ (d.smallNeighbors y T).erase y
+  · simpa only [h,ite_true] using d'.integrable_outsideIndicator hn y
+  · simp only [h,ite_false]
+    exact integrable_zero _ _ _
+
+lemma neighborMass_integral (d d' : Depth P) (hn : 2 ≤ Fintype.card P)
+    (p : P → ℝ) (x : P) (T : ℝ) :
+    (∫ r, neighborMass d d' p x T r) =
+      ∑ y, p y * (if x ∈ (d.smallNeighbors y T).erase y then d'.potential y else 0) := by
+  classical
+  unfold neighborMass
+  rw [integral_finsetSum]
+  · apply Finset.sum_congr rfl
+    intro y _
+    rw [integral_const_mul]
+    by_cases hh : x ∈ (d.smallNeighbors y T).erase y
+    · simp only [hh,ite_true,Depth.potential]
+    · simp only [hh,ite_false,integral_zero]
+  · intro y _
+    apply Integrable.const_mul
+    by_cases hh : x ∈ (d.smallNeighbors y T).erase y
+    · simpa only [hh,ite_true] using d'.integrable_outsideIndicator hn y
+    · simp only [hh,ite_false]; exact integrable_zero _ _ _
+
+end WeakPinned.SmallFiber
+
+namespace WeakPinned.SmallFiber
+open MeasureTheory Hierarchy
+variable {P : Type u7} [Fintype P] [DecidableEq P]
+
+lemma pairMass_integrable (d : Depth P) (hn : 2 ≤ Fintype.card P)
+    (p : P → ℝ) (x : P) (hx : p x = 0) :
+    Integrable (fun r => ∑ y, p y * d.pairOverlap r x y) := by
+  apply integrable_finsetSum
+  intro y _
+  by_cases hxy : x = y
+  · subst y; simp only [hx,zero_mul]; exact integrable_zero _ _ _
+  · exact (d.integrable_pairOverlap hn x y hxy).const_mul _
+
+lemma integral_nongiantMass_bound (d : Depth P) (hn : 2 ≤ Fintype.card P)
+    (p : P → ℝ) (hp : ∀ y, 0 ≤ p y) (h1 : ∑ y, p y = 1) (x : P) (hx : p x = 0) :
+    (∫ r, nongiantMass d p x r) ≤ ∑ y, p y * ∫ r, d.pairOverlap r x y := by
+  calc
+    _ ≤ ∫ r : ℝ, ∑ y, p y * d.pairOverlap r x y :=
+      integral_mono (nongiantMass_integrable d hn p hp h1 x hx) (pairMass_integrable d hn p x hx)
+        (fun r => nongiant_mass_bound d p hp x r)
+    _ = _ := by
+      rw [integral_finsetSum]
+      · exact Finset.sum_congr rfl (fun y _ => integral_const_mul (p y) _)
+      · intro y _
+        by_cases hxy : x = y
+        · subst y; simp only [hx,zero_mul]; exact integrable_zero _ _ _
+        · exact (d.integrable_pairOverlap hn x y hxy).const_mul _
+
+/-- The local small-fiber bound for two entire realized hierarchies. This
+    includes the exact signed-depth error; it will vanish at finite places
+    and have an exponentially small grid mean at archimedean places. -/
+lemma active_integral (d d' : Depth P) (hn : 2 ≤ Fintype.card P)
+    (p : P → ℝ) (hp : ∀ y, 0 ≤ p y) (h1 : ∑ y, p y = 1)
+    (x : P) (hx : p x = 0) (h H T lam : ℝ)
+    (hH : 0 ≤ H) (hlam : 0 ≤ lam) (hlam2 : lam ≤ 1/2) :
+    (∫ r, activeMass d p x T lam r) ≤
+      2*lam*(∑ y, p y * ∫ r, d'.pairOverlap r x y) + 2*lam*H +
+      (∑ y, p y * max (|d.value x y+d'.value x y-h|-H) 0) +
+      ∑ y, p y * (if x ∈ (d.smallNeighbors y T).erase y then d'.potential y else 0) := by
+  classical
+  have hshift₁ (r : ℝ) : h-r-H = (h-H)-r := by ring
+  have hshift₂ (r : ℝ) : h-r+H = (h+H)-r := by ring
+  have hi₁ : Integrable (fun r => 2*lam*nongiantMass d' p x (h-r-H)) := by
+    simp_rw [hshift₁]
+    exact ((integrable_comp_sub_left _ (h-H)).mpr (nongiantMass_integrable d' hn p hp h1 x hx)).const_mul _
+  have hi₂ : Integrable (fun r => lam*straddle d' x h H r) :=
+    (straddle_integrable d' hn x h H).const_mul _
+  have hi₃ := (minusError_integrable d d' p x hx h H).add (plusError_integrable d d' p x hx h H)
+  have hi₄ : Integrable (fun r => neighborMass d d' p x T (h-r+H)) := by
+    simp_rw [hshift₂]
+    exact (integrable_comp_sub_left _ (h+H)).mpr (neighborMass_integrable d d' hn p x T)
+  have hb := integral_mono (activeMass_integrable d hn p hp h1 x hx T lam)
+    (((hi₁.add hi₂).add hi₃).add hi₄) (fun r => by
+      have hh := active_pointwise d d' p hp h1 x hx h H T lam r hlam hlam2
+      dsimp only [Pi.add_apply]
+      linarith)
+  simp only [Pi.add_apply] at hb
+  have he₁ := integral_add ((hi₁.add hi₂).add hi₃) hi₄
+  have he₂ := integral_add (hi₁.add hi₂) hi₃
+  have he₃ := integral_add hi₁ hi₂
+  simp only [Pi.add_apply] at he₁ he₂ he₃
+  rw [he₁,he₂,he₃] at hb
+  rw [integral_const_mul,integral_const_mul] at hb
+  simp_rw [hshift₁,hshift₂] at hb
+  rw [integral_sub_left_eq_self,integral_sub_left_eq_self,
+    neighborMass_integral d d' hn p x T,errors_integral d d' p x hx h H hH] at hb
+  have hs : (∫ r, straddle d' x h H r) = 2*H := d'.straddle_integral hn x h H hH
+  rw [hs] at hb
+  have hng := mul_le_mul_of_nonneg_left (integral_nongiantMass_bound d' hn p hp h1 x hx)
+    (show 0 ≤ 2*lam by positivity)
+  linarith
+
+/-- Exact edge law induced by fiber sampling; no independence of fiber
+    labels or levels is postulated. -/
+def edgeLaw {I : Type u8} [Fintype I] (w : I → ℝ) (p : I → P → ℝ) (x : I → P) (a b : P) : ℝ :=
+  ∑ i, if x i = a then w i * p i b else 0
+
+lemma fiber_neighbor_bound {I : Type u9} [Fintype I]
+    (d d' : Depth P) (hn : 2 ≤ Fintype.card P)
+    (w : I → ℝ) (p : I → P → ℝ) (x : I → P)
+    (B τ : ℝ) (hB : 0 ≤ B) (hτ : 0 < τ) (hτ1 : τ ≤ 1)
+    (hdom : ∀ a b, a ≠ b → edgeLaw w p x a b ≤ B/((Fintype.card P:ℝ)*((Fintype.card P:ℝ)-1))) :
+    (∑ i, w i * ∑ y, p i y *
+      (if x i ∈ (d.smallNeighbors y (τ*Fintype.card P)).erase y then d'.potential y else 0)) ≤
+      B*τ/Fintype.card P * ∑ y, d'.potential y := by
+  classical
+  have hn0 : 0 < (Fintype.card P:ℝ) := by exact_mod_cast (show 0 < Fintype.card P by omega)
+  have hn1 : 0 < (Fintype.card P:ℝ)-1 := by
+    have : (2:ℝ) ≤ Fintype.card P := by exact_mod_cast hn
+    linarith
+  have hid (y : P) :
+      (∑ i, w i * (p i y * (if x i ∈ (d.smallNeighbors y (τ*Fintype.card P)).erase y then d'.potential y else 0))) =
+      (∑ a ∈ (d.smallNeighbors y (τ*Fintype.card P)).erase y, edgeLaw w p x a y) * d'.potential y := by
+    rw [Finset.sum_mul]
+    simp only [edgeLaw,Finset.sum_mul]
+    rw [Finset.sum_comm]
+    apply Finset.sum_congr rfl
+    intro i _
+    rw [← Finset.sum_mul]
+    have he : (∑ a ∈ (d.smallNeighbors y (τ*Fintype.card P)).erase y,
+        if x i = a then w i*p i y else 0) =
+        if x i ∈ (d.smallNeighbors y (τ*Fintype.card P)).erase y then w i*p i y else 0 := by
+      simp [eq_comm]
+    rw [he]
+    split_ifs <;> ring
+  simp_rw [Finset.mul_sum]
+  rw [Finset.sum_comm]
+  simp_rw [hid]
+  have hb (y : P) : (∑ a ∈ (d.smallNeighbors y (τ*Fintype.card P)).erase y, edgeLaw w p x a y) ≤ B*τ/Fintype.card P := by
+    calc
+      _ ≤ ((d.smallNeighbors y (τ*Fintype.card P)).erase y).card *
+          (B/((Fintype.card P:ℝ)*((Fintype.card P:ℝ)-1))) := by
+        calc
+          _ ≤ ∑ a ∈ (d.smallNeighbors y (τ*Fintype.card P)).erase y,
+              B/((Fintype.card P:ℝ)*((Fintype.card P:ℝ)-1)) :=
+            Finset.sum_le_sum (fun a ha => hdom a y (Finset.ne_of_mem_erase ha))
+          _ = _ := by simp
+      _ ≤ (τ*((Fintype.card P:ℝ)-1)) * (B/((Fintype.card P:ℝ)*((Fintype.card P:ℝ)-1))) :=
+        mul_le_mul_of_nonneg_right (d.smallNeighbors_erase_bound hn y τ hτ hτ1) (by positivity)
+      _ = _ := by field_simp
+  exact Finset.sum_le_sum (fun y _ => mul_le_mul_of_nonneg_right (hb y) (d'.potential_nonneg y))
+
+end WeakPinned.SmallFiber
+
+end
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PinnedDistances/TreeContradiction.lean b/lean/OAI/Geometry/PinnedDistances/TreeContradiction.lean
new file mode 100644
index 0000000000000000000000000000000000000000..1a3b9a3be6a956c4dcf2c0ded33b062f3fb02923
--- /dev/null
+++ b/lean/OAI/Geometry/PinnedDistances/TreeContradiction.lean
@@ -0,0 +1,300 @@
+import OAI.Geometry.PinnedDistances.FiberTransport
+
+namespace OAI
+
+universe u1 u2 u3 u4
+
+section
+noncomputable section
+open NumberField
+open scoped BigOperators
+noncomputable section
+open MeasureTheory
+open scoped Topology
+noncomputable section
+open MeasureTheory
+open scoped Topology ENNReal
+noncomputable section
+open MeasureTheory NumberField
+open scoped BigOperators
+noncomputable section
+open scoped BigOperators
+open MeasureTheory
+noncomputable section
+open Filter
+open scoped Topology
+noncomputable section
+open scoped BigOperators
+noncomputable section
+open Filter
+open scoped Topology
+noncomputable section
+open MeasureTheory
+open scoped BigOperators
+noncomputable section
+noncomputable section
+open scoped BigOperators
+noncomputable section
+noncomputable section
+noncomputable section
+
+namespace WeakPinned.Cuts
+open MeasureTheory Hierarchy Variance
+open scoped BigOperators
+variable {P : Type u1} {I : Type u2} [Fintype P] [DecidableEq P] [Fintype I]
+
+omit [Fintype P] [DecidableEq P] in
+lemma cost_mixture (s : Finset (Finset P)) (l : Finset P → ℝ) (hl : ∀ C ∈ s, 0≤l C)
+    (w : I → ℝ) (p : I → P → ℝ) (a : P → ℝ) (hw : ∀ i, 0≤w i) (hW : ∑ i, w i=1) :
+    cost s l (fun y => ∑ i, w i*p i y) a≤∑ i, w i*cost s l (p i) a := by
+  have hm (C : Finset P) : mass (fun y => ∑ i, w i*p i y) C-mass a C=
+      ∑ i, w i*(mass (p i) C-mass a C) := by
+    simp only [mass,mul_sub,Finset.sum_sub_distrib,← Finset.sum_mul,hW,one_mul]
+    rw [Finset.sum_comm]
+    simp only [Finset.mul_sum]
+  unfold cost
+  simp only [hm]
+  calc
+    _ ≤ ∑ C ∈ s, l C*∑ i, w i*|mass (p i) C-mass a C| := by
+      apply Finset.sum_le_sum
+      intro C hC
+      apply mul_le_mul_of_nonneg_left _ (hl C hC)
+      convert Finset.abs_sum_le_sum_abs (s:=Finset.univ) (fun i => w i*(mass (p i) C-mass a C)) using 1
+      simp only [abs_mul,abs_of_nonneg (hw _)]
+    _ = _ := by
+      simp only [Finset.mul_sum]
+      rw [Finset.sum_comm]
+      exact Finset.sum_congr rfl (fun i _ => Finset.sum_congr rfl (fun C _ => by ring))
+
+omit [DecidableEq P] in
+lemma mass_upper (p : P → ℝ) (hp0 : ∀ x, 0≤p x) (hp : ∑ x, p x=1) (C : Finset P) : mass p C≤1 :=
+  (Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ _) (fun x _ _ => hp0 x)).trans_eq hp
+
+omit [DecidableEq P] in
+lemma cost_bound (s : Finset (Finset P)) (l : Finset P → ℝ) (hl : ∀ C ∈ s, 0≤l C)
+    (p a : P → ℝ) (hp0 : ∀ x, 0≤p x) (ha0 : ∀ x, 0≤a x)
+    (hp : ∑ x, p x=1) (ha : ∑ x, a x=1) : cost s l p a≤∑ C ∈ s, l C := by
+  apply Finset.sum_le_sum
+  intro C hC
+  have hh : |mass p C-mass a C|≤1 := abs_le.mpr ⟨by linarith [mass_nonneg p hp0 C,mass_upper a ha0 ha C],
+    by linarith [mass_nonneg a ha0 C,mass_upper p hp0 hp C]⟩
+  simpa using mul_le_mul_of_nonneg_left hh (hl C hC)
+
+lemma projection_replacement (d d' : Depth P) (δ : ℝ) (hδ : 0≤δ) (p a : P → ℝ)
+    (hp : ∑ x, p x=1) (ha : ∑ x, a x=1) (hps : ∀ x, p x≤δ) (has : ∀ x, a x≤δ) (x y : P) :
+    |distance d.allCuts (d.cutLength (fun C => tent δ (mass a C))) x y-
+        rootDistance d'.allCuts (d'.cutLength (fun C => tent δ (mass a C))) y|≤
+      |distance d.allCuts (d.cutLength (fun C => tent δ (mass p C))) x y-
+        rootDistance d'.allCuts (d'.cutLength (fun C => tent δ (mass p C))) y|+
+      charge d δ p a+charge d' δ p a := by
+  have h1 := distance_change_charge d δ hδ p a hp ha hps has x y
+  have h2 := root_change_charge d' δ hδ p a hp ha hps has y
+  let A := distance d.allCuts (d.cutLength (fun C => tent δ (mass a C))) x y
+  let B := rootDistance d'.allCuts (d'.cutLength (fun C => tent δ (mass a C))) y
+  let C := distance d.allCuts (d.cutLength (fun C => tent δ (mass p C))) x y
+  let D := rootDistance d'.allCuts (d'.cutLength (fun C => tent δ (mass p C))) y
+  change |C-A|≤charge d δ p a at h1
+  change |D-B|≤charge d' δ p a at h2
+  change |A-B|≤|C-D|+charge d δ p a+charge d' δ p a
+  have hh1 : |A-B|≤|A-C|+|C-B| := abs_sub_le A C B
+  have hh2 : |C-B|≤|C-D|+|D-B| := abs_sub_le C D B
+  have hh3 : |A-C|=|C-A| := abs_sub_comm A C
+  rw [hh3] at hh1
+  linarith only [hh1,hh2,h1,h2]
+
+lemma family_charge_occurrence (d : Depth P) (hn : 2≤Fintype.card P) (w : I → ℝ)
+    (B : I → Finset P) (hB : ∀ i, (B i).Nonempty) (δ : ℝ) (hδ : 0≤δ)
+    (hp : ∀ i x, prob (B i) x≤δ) (ha : ∀ x, prob (Finset.univ : Finset P) x≤δ) :
+    (∑ i, w i*charge d δ (prob (B i)) (prob Finset.univ))=
+      weightedIntegral d.mass (familyCutError w B δ) := by
+  have ha1 : ∑ x, prob (Finset.univ : Finset P) x=1 := sum_prob _ (by
+    apply Finset.univ_nonempty_iff.mpr
+    exact Fintype.card_pos_iff.mp (by omega))
+  simp only [charge_occurrence d hn δ hδ _ _ (sum_prob _ (hB _)) ha1 (hp _) ha]
+  simp only [weightedIntegral,Finset.mul_sum,familyCutError]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro C _
+  apply Finset.sum_congr rfl
+  intro i _
+  simp only [mass,sum_prob_cell]
+  ring
+
+/-- One coordinate of the finite tree contradiction, before arithmetic
+averaging. The incoming laws are arbitrary probabilities representing the
+actual conditional source law, not an independence hypothesis. -/
+lemma length_upper (d d' : Depth P) (δ : ℝ) (hδ : 0≤δ)
+    (a t : P → ℝ) (ρ : P → P → ℝ) (w : I → ℝ) (p : I → P → ℝ) (src : I → P)
+    (ha0 : ∀ x, 0≤a x) (ht0 : ∀ y, 0≤t y) (hρ0 : ∀ y x, 0≤ρ y x) (hw : ∀ i, 0≤w i)
+    (ha : ∑ x, a x=1) (ht : ∑ y, t y=1) (hρ : ∀ y, ∑ x, ρ y x=1) (hW : ∑ i, w i=1)
+    (hp : ∀ i, ∑ y, p i y=1) (hp0 : ∀ i y, 0≤p i y)
+    (has : ∀ x, a x≤δ) (hps : ∀ i x, p i x≤δ)
+    (htm : t=fun y => ∑ i, w i*p i y)
+    (hedge : ∀ F : P → P → ℝ, (∑ y, t y*∑ x, ρ y x*F x y)=∑ i, w i*∑ y, p i y*F (src i) y) :
+    δ^3*(∑ C ∈ d.allCuts, d.cutLength (fun B => tent δ (mass a B)) C)≤
+      2*Global.fiberCutDiscrepancy w p src δ d d'+
+      4*(∑ i, w i*charge d δ (p i) a)+2*(∑ i, w i*charge d' δ (p i) a)+
+      2*(∑ y, t y*cost d.allCuts (d.cutLength (fun B => tent δ (mass a B))) (ρ y) a) := by
+  let l := d.cutLength (fun B => tent δ (mass a B))
+  have hl (C) (_ : C∈d.allCuts) : 0≤l C := d.cutLength_nonneg _ (fun C => tent_nonneg _ _) C
+  have hlow := d.actual_cut_fluctuation a ha0 ha (tent δ) (tent_nonneg δ) δ hδ (fun u hn => by
+    apply le_antisymm _ (tent_nonneg δ u)
+    by_contra hh
+    have hu := tent_positive (lt_of_not_ge hh)
+    exact hn ⟨hu.1.le,by linarith [hu.2]⟩)
+  have hupper := iid_transfer d.allCuts d.allCuts_laminar l hl a t ρ ha0 ht0 hρ0 ha ht hρ
+    (rootDistance d'.allCuts (d'.cutLength (fun B => tent δ (mass a B))))
+  rw [hedge] at hupper
+  have hproj : (∑ i, w i*∑ y, p i y*
+      |distance d.allCuts l (src i) y-rootDistance d'.allCuts (d'.cutLength (fun B => tent δ (mass a B))) y|)≤
+      Global.fiberCutDiscrepancy w p src δ d d'+∑ i, w i*charge d δ (p i) a+∑ i, w i*charge d' δ (p i) a := by
+    have hh := Finset.sum_le_sum (s:=Finset.univ) (fun i _ => mul_le_mul_of_nonneg_left
+      (Finset.sum_le_sum (s:=Finset.univ) (fun y _ => mul_le_mul_of_nonneg_left
+        (projection_replacement d d' δ hδ (p i) a (hp i) ha (hps i) has (src i) y) (hp0 i y))) (hw i))
+    simpa only [Global.fiberCutDiscrepancy,mul_add,Finset.sum_add_distrib,← Finset.sum_mul,hp,one_mul,l] using hh
+  have hcost : cost d.allCuts l t a≤∑ i, w i*charge d δ (p i) a := by
+    rw [htm]
+    exact (cost_mixture d.allCuts l hl w p a hw hW).trans (Finset.sum_le_sum (fun i _ =>
+      mul_le_mul_of_nonneg_left (cost_le_charge d δ hδ (p i) a (hp i) ha (hps i) has) (hw i)))
+  dsimp only [l] at hupper hproj hcost
+  linarith
+end WeakPinned.Cuts
+noncomputable section
+namespace WeakPinned.Core.Fibers
+open MeasureTheory Hierarchy Variance Cuts
+open scoped BigOperators
+variable {V : Type u3} {I : Type u4} [Fintype V] [Fintype I] [DecidableEq V]
+variable (G : Fibers V I)
+
+omit [Fintype V] in
+lemma prob_nn (B : Finset V) (x : V) : 0≤prob B x := by
+  unfold prob
+  split_ifs <;> positivity
+
+def conditionalSource (y : V) : V → ℝ :=
+  if (G.incoming y).Nonempty then prob (G.incoming y) else prob Finset.univ
+
+omit [Fintype I] in
+lemma conditionalSource_nonneg (y x : V) : 0≤G.conditionalSource y x := by
+  unfold conditionalSource
+  split_ifs <;> exact prob_nn _ _
+
+omit [Fintype I] in
+lemma conditionalSource_sum (hn : 2≤Fintype.card V) (y : V) : ∑ x, G.conditionalSource y x=1 := by
+  unfold conditionalSource
+  split_ifs with h
+  · exact sum_prob _ h
+  · exact sum_prob _ (Finset.univ_nonempty_iff.mpr (Fintype.card_pos_iff.mp (by omega)))
+
+omit [Fintype I] in
+lemma incomingWeight_nonneg (y : V) : 0≤G.incomingWeight y := by unfold incomingWeight edgeCount; positivity
+
+omit [Fintype I] in
+lemma sampling_conditional (y x : V) : G.incomingWeight y*G.conditionalSource y x=G.edgeLaw x y := by
+  by_cases h : (G.incoming y).Nonempty
+  · simpa only [conditionalSource,ite_eq_left h] using G.sampling_incoming x y
+  · have he : G.incoming y=∅ := Finset.not_nonempty_iff_eq_empty.mp h
+    have hh := G.sampling_incoming x y
+    have hw : G.incomingWeight y=0 := by simp [incomingWeight,he]
+    simp only [hw,zero_mul] at hh ⊢
+    exact hh
+
+lemma incoming_eval (F : V → V → ℝ) :
+    (∑ y, G.incomingWeight y*∑ x, G.conditionalSource y x*F x y)=
+      ∑ e, G.fiberWeight e*∑ y, prob (G.targets e) y*F (G.source e) y := by
+  simp only [Finset.mul_sum,← mul_assoc,G.sampling_conditional,← G.sampling_fiber,Finset.sum_mul]
+  conv_lhs =>
+    arg 2
+    ext y
+    rw [Finset.sum_comm]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro e _
+  apply Finset.sum_congr rfl
+  intro y _
+  simp only [ite_mul,zero_mul]
+  simp
+
+lemma incoming_repr : G.incomingWeight=fun y => ∑ e, G.fiberWeight e*prob (G.targets e) y := by
+  funext y
+  rw [G.fiber_marginal,G.incoming_marginal]
+
+def omittedMass (T : ℝ) : ℝ := ∑ y, if ((G.incoming y).card:ℝ)<T then G.incomingWeight y else 0
+
+omit [Fintype I] in
+lemma omittedMass_nonneg (T : ℝ) : 0≤G.omittedMass T := by
+  apply Finset.sum_nonneg
+  intro y _
+  split_ifs
+  · exact G.incomingWeight_nonneg y
+  · rfl
+
+omit [Fintype I] in
+lemma retained_cost (d : Depth V) (δ T : ℝ) (hδ : 0≤δ) (hT : 0<T)
+    (has : ∀ x, prob (Finset.univ : Finset V) x≤δ)
+    (hrs : ∀ y x, prob (G.retainedSet T y) x≤δ)
+    (hrn : ∀ y, (G.retainedSet T y).Nonempty) (hn : 2≤Fintype.card V) :
+    (∑ y, G.incomingWeight y*cost d.allCuts (d.cutLength (fun B => tent δ (mass (prob Finset.univ) B)))
+      (G.conditionalSource y) (prob Finset.univ))≤
+      (∑ y, G.retainedWeight T y*charge d δ (prob (G.retainedSet T y)) (prob Finset.univ))+
+      G.omittedMass T*(∑ C ∈ d.allCuts, d.cutLength (fun B => tent δ (mass (prob Finset.univ) B)) C) := by
+  have ha : ∑ x, prob (Finset.univ : Finset V) x=1 :=
+    sum_prob _ (Finset.univ_nonempty_iff.mpr (Fintype.card_pos_iff.mp (by omega)))
+  unfold omittedMass
+  rw [Finset.sum_mul,← Finset.sum_add_distrib]
+  apply Finset.sum_le_sum
+  intro y _
+  by_cases hs : T≤((G.incoming y).card:ℝ)
+  · have hy : (G.incoming y).Nonempty := Finset.card_pos.mp (by exact_mod_cast hT.trans_le hs)
+    have he : G.retainedSet T y=G.incoming y := by simp only [retainedSet,ite_eq_left hs]
+    have hh := cost_le_charge d δ hδ (prob (G.retainedSet T y)) (prob Finset.univ)
+      (sum_prob _ (hrn y)) ha (hrs y) has
+    simp only [he] at hh
+    simpa only [retainedWeight,ite_eq_left hs,conditionalSource,ite_eq_left hy,not_lt_of_ge hs,ite_false,zero_mul,add_zero,he]
+      using mul_le_mul_of_nonneg_left hh (G.incomingWeight_nonneg y)
+  · have hh := cost_bound d.allCuts (d.cutLength (fun B => tent δ (mass (prob Finset.univ) B)))
+      (fun C _ => d.cutLength_nonneg _ (fun B => tent_nonneg _ _) C)
+      (G.conditionalSource y) (prob Finset.univ) (G.conditionalSource_nonneg y) (prob_nn _)
+      (G.conditionalSource_sum hn y) ha
+    simpa only [retainedWeight,ite_eq_right hs,lt_of_not_ge hs,ite_true,zero_mul,zero_add] using
+      mul_le_mul_of_nonneg_left hh (G.incomingWeight_nonneg y)
+
+lemma graph_length_upper (d d' : Depth V) (δ T : ℝ) (hδ : 0≤δ) (hT : 0<T)
+    (hn : 2≤Fintype.card V) (hG : 0<G.edgeCount)
+    (has : ∀ x, prob (Finset.univ : Finset V) x≤δ)
+    (hps : ∀ e x, prob (G.targets e) x≤δ) (hpn : ∀ e, (G.targets e).Nonempty)
+    (hrs : ∀ y x, prob (G.retainedSet T y) x≤δ) (hrn : ∀ y, (G.retainedSet T y).Nonempty) :
+    δ^3*(∑ C ∈ d.allCuts, d.cutLength (fun B => tent δ (mass (prob Finset.univ) B)) C)≤
+      2*Global.fiberCutDiscrepancy G.fiberWeight (fun e => prob (G.targets e)) G.source δ d d'+
+      4*(∑ e, G.fiberWeight e*charge d δ (prob (G.targets e)) (prob Finset.univ))+
+      2*(∑ e, G.fiberWeight e*charge d' δ (prob (G.targets e)) (prob Finset.univ))+
+      2*(∑ y, G.retainedWeight T y*charge d δ (prob (G.retainedSet T y)) (prob Finset.univ))+
+      2*G.omittedMass T*(∑ C ∈ d.allCuts, d.cutLength (fun B => tent δ (mass (prob Finset.univ) B)) C) := by
+  have ha : ∑ x, prob (Finset.univ : Finset V) x=1 :=
+    sum_prob _ (Finset.univ_nonempty_iff.mpr (Fintype.card_pos_iff.mp (by omega)))
+  have hh := length_upper d d' δ hδ (prob Finset.univ) G.incomingWeight G.conditionalSource G.fiberWeight
+    (fun e => prob (G.targets e)) G.source (prob_nn _) G.incomingWeight_nonneg G.conditionalSource_nonneg
+    G.fiberWeight_nonneg ha (G.sum_incomingWeight hG) (G.conditionalSource_sum hn) (G.sum_fiberWeight hG)
+    (fun e => sum_prob _ (hpn e)) (fun e => prob_nn _) has hps G.incoming_repr G.incoming_eval
+  have hc := G.retained_cost d δ T hδ hT has hrs hrn hn
+  linarith
+end WeakPinned.Core.Fibers
+
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PinnedDistances/TreeTransport.lean b/lean/OAI/Geometry/PinnedDistances/TreeTransport.lean
new file mode 100644
index 0000000000000000000000000000000000000000..c8d2db00b464e45271fd4cd816f1b8c798893be4
--- /dev/null
+++ b/lean/OAI/Geometry/PinnedDistances/TreeTransport.lean
@@ -0,0 +1,458 @@
+import OAI.Geometry.PinnedDistances.CutApproximation
+
+namespace OAI
+
+universe u1
+
+section
+noncomputable section
+open NumberField
+open scoped BigOperators
+noncomputable section
+open MeasureTheory
+open scoped Topology
+noncomputable section
+open MeasureTheory
+open scoped Topology ENNReal
+noncomputable section
+open MeasureTheory NumberField
+open scoped BigOperators
+noncomputable section
+open scoped BigOperators
+open MeasureTheory
+noncomputable section
+open Filter
+open scoped Topology
+noncomputable section
+open scoped BigOperators
+noncomputable section
+open Filter
+open scoped Topology
+noncomputable section
+open MeasureTheory
+open scoped BigOperators
+noncomputable section
+
+/- Finite tree transport by laminar-cut telescoping, without optimal-coupling selection. -/
+
+noncomputable section
+open scoped BigOperators
+namespace WeakPinned.Cuts
+variable {P : Type u1} [Fintype P] [DecidableEq P]
+
+def Laminar (s : Finset (Finset P)) : Prop :=
+  ∀ A ∈ s, ∀ B ∈ s, A ⊆ B ∨ B ⊆ A ∨ Disjoint A B
+
+def value (s : Finset (Finset P)) (d : Finset P → ℝ) (a : ℝ) (x : P) : ℝ :=
+  a + ∑ C ∈ s, if x ∈ C then d C else 0
+
+def mass (b : P → ℝ) (C : Finset P) : ℝ := ∑ x ∈ C, b x
+
+omit [Fintype P] in
+lemma laminar_erase {s : Finset (Finset P)} (hs : Laminar s) (C : Finset P) : Laminar (s.erase C) :=
+  fun _ hA _ hB => hs _ (Finset.mem_of_mem_erase hA) _ (Finset.mem_of_mem_erase hB)
+
+omit [Fintype P] in
+lemma minimal_constant {s : Finset (Finset P)} (hs : Laminar s)
+    (C : Finset P) (hC : C ∈ s) (hmin : ∀ D ∈ s, C.card ≤ D.card)
+    (d : Finset P → ℝ) (a : ℝ) :
+    ∀ x ∈ C, value (s.erase C) d a x = a+∑ D ∈ s.erase C, if C ⊆ D then d D else 0 := by
+  intro x hx
+  unfold value
+  congr 1
+  apply Finset.sum_congr rfl
+  intro D hD
+  have hDs := Finset.mem_of_mem_erase hD
+  have hDC : D ≠ C := (Finset.mem_erase.mp hD).1
+  have hnot : ¬D ⊆ C := by
+    intro hh
+    exact hDC (Finset.eq_of_subset_of_card_le hh (hmin D hDs))
+  by_cases hCD : C ⊆ D
+  · simp [hCD,hCD hx]
+  · have hd : Disjoint C D := (hs C hC D hDs).resolve_left hCD |>.resolve_left hnot
+    have hxD : x ∉ D := fun hxD => Finset.disjoint_left.mp hd hx hxD
+    simp [hCD,hxD]
+
+omit [Fintype P] in
+lemma value_erase {s : Finset (Finset P)} (C : Finset P) (hC : C ∈ s)
+    (d : Finset P → ℝ) (a : ℝ) (x : P) :
+    value s d a x = value (s.erase C) d a x + (if x ∈ C then d C else 0) := by
+  unfold value
+  rw [← Finset.add_sum_erase _ _ hC]
+  ring
+
+/-- Dual tree transport with completely explicit cut cost. The signed measure
+    b has total mass zero. Neither nonnegativity nor coupling existence is needed. -/
+lemma signed_lipschitz_bound (s : Finset (Finset P)) (hs : Laminar s)
+    (b : P → ℝ) (hb : ∑ x, b x=0) (d : Finset P → ℝ) (a : ℝ)
+    (g : ℝ → ℝ) (hg : ∀ u v, |g u-g v| ≤ |u-v|) :
+    |∑ x, b x*g (value s d a x)| ≤ ∑ C ∈ s, |d C| * |mass b C| := by
+  induction s using Finset.strongInductionOn with
+  | _ s ih =>
+    by_cases hsn : s.Nonempty
+    · obtain ⟨C,hC,hmin⟩ := Finset.exists_min_image s Finset.card hsn
+      let q := a+∑ D ∈ s.erase C, if C ⊆ D then d D else 0
+      let Δ := g (q+d C)-g q
+      have hΔ : |Δ| ≤ |d C| := by
+        simpa only [Δ,add_sub_cancel_left] using hg (q+d C) q
+      have he (x : P) : g (value s d a x) =
+          g (value (s.erase C) d a x)+(if x ∈ C then Δ else 0) := by
+        rw [value_erase C hC]
+        by_cases hx : x ∈ C
+        · rw [ite_eq_left hx,ite_eq_left hx,minimal_constant hs C hC hmin d a x hx]
+          dsimp only [Δ,q]
+          ring
+        · simp [hx]
+      have hi := ih (s.erase C) (Finset.erase_ssubset hC) (laminar_erase hs C)
+      have he' : (∑ x, b x*g (value s d a x)) =
+          (∑ x, b x*g (value (s.erase C) d a x))+Δ*mass b C := by
+        simp_rw [he,mul_add,Finset.sum_add_distrib]
+        congr 1
+        simp only [mul_ite,mul_zero,mass,Finset.mul_sum]
+        simp only [← Finset.sum_filter,Finset.filter_mem_eq_inter,Finset.univ_inter]
+        exact Finset.sum_congr rfl (fun x _ => by ring)
+      rw [he',← Finset.add_sum_erase _ _ hC]
+      calc
+        _ ≤ |∑ x, b x*g (value (s.erase C) d a x)|+|Δ*mass b C| := abs_add_le _ _
+        _ ≤ (∑ D ∈ s.erase C, |d D| * |mass b D|)+|d C| * |mass b C| := by
+          apply add_le_add hi
+          rw [abs_mul]
+          exact mul_le_mul_of_nonneg_right hΔ (abs_nonneg _)
+        _ = _ := by ring
+    · have hs0 := Finset.not_nonempty_iff_eq_empty.mp hsn
+      simp only [hs0,value,Finset.sum_empty,add_zero,← Finset.sum_mul,hb,zero_mul,abs_zero,le_refl]
+
+def ind (C : Finset P) (x : P) : ℝ := if x ∈ C then 1 else 0
+
+def distance (s : Finset (Finset P)) (l : Finset P → ℝ) (x y : P) : ℝ :=
+  ∑ C ∈ s, l C*|ind C x-ind C y|
+
+def rootDistance (s : Finset (Finset P)) (l : Finset P → ℝ) (y : P) : ℝ :=
+  ∑ C ∈ s, l C*ind C y
+
+def coeff (l : Finset P → ℝ) (y : P) (C : Finset P) : ℝ :=
+  if y ∈ C then -l C else l C
+
+def cost (s : Finset (Finset P)) (l : Finset P → ℝ) (p q : P → ℝ) : ℝ :=
+  ∑ C ∈ s, l C*|mass p C-mass q C|
+
+omit [Fintype P] in
+lemma distance_symm (s : Finset (Finset P)) (l : Finset P → ℝ) (x y : P) :
+    distance s l x y=distance s l y x := by
+  simp only [distance,abs_sub_comm]
+
+omit [Fintype P] in
+lemma distance_nonneg (s : Finset (Finset P)) (l : Finset P → ℝ) (hl : ∀ C ∈ s, 0 ≤ l C)
+    (x y : P) : 0 ≤ distance s l x y :=
+  Finset.sum_nonneg (fun C hC => mul_nonneg (hl C hC) (abs_nonneg _))
+
+omit [Fintype P] in
+lemma distance_as_value (s : Finset (Finset P)) (l : Finset P → ℝ) (x y : P) :
+    distance s l x y=value s (coeff l y) (rootDistance s l y) x := by
+  simp only [distance,value,rootDistance,← Finset.sum_add_distrib]
+  apply Finset.sum_congr rfl
+  intro C _
+  by_cases hx : x ∈ C <;> by_cases hy : y ∈ C <;> simp [ind,coeff,hx,hy]
+
+omit [Fintype P] in
+lemma abs_coeff (l : Finset P → ℝ) (y : P) (C : Finset P) (hC : 0 ≤ l C) :
+    |coeff l y C|=l C := by
+  unfold coeff
+  split_ifs <;> simp [abs_of_nonneg hC]
+
+lemma function_transfer (s : Finset (Finset P)) (hs : Laminar s)
+    (l : Finset P → ℝ) (hl : ∀ C ∈ s, 0 ≤ l C) (p q : P → ℝ)
+    (hpq : ∑ x, p x=∑ x, q x) (y : P) (g : ℝ → ℝ)
+    (hg : ∀ u v, |g u-g v| ≤ |u-v|) :
+    |(∑ x, p x*g (distance s l x y))-(∑ x, q x*g (distance s l x y))| ≤ cost s l p q := by
+  have hb : (∑ x, (p x-q x))=0 := by rw [Finset.sum_sub_distrib,hpq,sub_self]
+  have hh := signed_lipschitz_bound s hs (fun x => p x-q x) hb (coeff l y) (rootDistance s l y) g hg
+  simp only [← distance_as_value,sub_mul,Finset.sum_sub_distrib,mass] at hh
+  simp only [cost,mass]
+  convert hh using 1
+  exact Finset.sum_congr rfl (fun C hC => by rw [abs_coeff l y C (hl C hC)])
+
+lemma first_fluctuation_transfer (s : Finset (Finset P)) (hs : Laminar s)
+    (l : Finset P → ℝ) (hl : ∀ C ∈ s, 0 ≤ l C) (p q : P → ℝ)
+    (hpq : ∑ x, p x=∑ x, q x) (x' y : P) :
+    |(∑ x, p x*|distance s l x y-distance s l x' y|)-
+      (∑ x, q x*|distance s l x y-distance s l x' y|)| ≤ cost s l p q := by
+  apply function_transfer s hs l hl p q hpq y (fun u => |u-distance s l x' y|)
+  intro u v
+  simpa only [sub_sub_sub_cancel_right] using abs_abs_sub_abs_le_abs_sub (u-distance s l x' y) (v-distance s l x' y)
+
+omit [Fintype P] in
+lemma value_sub (s : Finset (Finset P)) (d e : Finset P → ℝ) (a b : ℝ) (x : P) :
+    value s d a x-value s e b x=value s (fun C => d C-e C) (a-b) x := by
+  simp only [value]
+  have he : (∑ C ∈ s, if x ∈ C then d C-e C else 0) =
+      (∑ C ∈ s, if x ∈ C then d C else 0)-(∑ C ∈ s, if x ∈ C then e C else 0) := by
+    rw [← Finset.sum_sub_distrib]
+    exact Finset.sum_congr rfl (fun C _ => by split_ifs <;> ring)
+  rw [he]
+  ring
+
+lemma last_fluctuation_transfer (s : Finset (Finset P)) (hs : Laminar s)
+    (l : Finset P → ℝ) (hl : ∀ C ∈ s, 0 ≤ l C) (p q : P → ℝ)
+    (hpq : ∑ x, p x=∑ x, q x) (x x' : P) :
+    |(∑ y, p y*|distance s l x y-distance s l x' y|)-
+      (∑ y, q y*|distance s l x y-distance s l x' y|)| ≤ 2*cost s l p q := by
+  have hb : (∑ y, (p y-q y))=0 := by rw [Finset.sum_sub_distrib,hpq,sub_self]
+  have hh := signed_lipschitz_bound s hs (fun y => p y-q y) hb
+    (fun C => coeff l x C-coeff l x' C) (rootDistance s l x-rootDistance s l x') abs abs_abs_sub_abs_le_abs_sub
+  have he (y : P) : value s (fun C => coeff l x C-coeff l x' C) (rootDistance s l x-rootDistance s l x') y =
+      distance s l x y-distance s l x' y := by
+    rw [← value_sub,← distance_as_value,← distance_as_value,distance_symm s l y x,distance_symm s l y x']
+  simp only [he,sub_mul,Finset.sum_sub_distrib,mass] at hh
+  refine hh.trans ?_
+  simp only [cost,mass,Finset.mul_sum]
+  apply Finset.sum_le_sum
+  intro C hC
+  have hc : |coeff l x C-coeff l x' C| ≤ 2*l C := by
+    calc
+      _ ≤ |coeff l x C|+|coeff l x' C| := abs_sub _ _
+      _ = _ := by rw [abs_coeff l x C (hl C hC),abs_coeff l x' C (hl C hC)]; ring
+  calc
+    _ ≤ (2*l C)*|∑ y ∈ C, p y-∑ y ∈ C, q y| := mul_le_mul_of_nonneg_right hc (abs_nonneg _)
+    _ = _ := by ring
+
+omit [Fintype P] [DecidableEq P] in
+lemma mass_nonneg (p : P → ℝ) (hp : ∀ x, 0 ≤ p x) (C : Finset P) : 0 ≤ mass p C :=
+  Finset.sum_nonneg (fun x _ => hp x)
+
+lemma expected_cut (p : P → ℝ) (hp : ∑ x, p x=1) (C : Finset P) (y : P) :
+    (∑ x, p x*|ind C x-ind C y|) = if y ∈ C then 1-mass p C else mass p C := by
+  by_cases hy : y ∈ C
+  · have he (x : P) : p x*|ind C x-ind C y|=p x-(if x ∈ C then p x else 0) := by
+      by_cases hx : x ∈ C <;> simp [ind,hx,hy]
+    simp_rw [he]
+    simp only [Finset.sum_sub_distrib,hp,← Finset.sum_filter,
+      Finset.filter_mem_eq_inter,Finset.univ_inter,ite_eq_left hy,mass]
+  · have he (x : P) : p x*|ind C x-ind C y|=if x ∈ C then p x else 0 := by
+      by_cases hx : x ∈ C <;> simp [ind,hx,hy]
+    simp_rw [he]
+    simp only [← Finset.sum_filter,Finset.filter_mem_eq_inter,Finset.univ_inter,ite_eq_right hy,mass]
+
+lemma expected_distance_ge (s : Finset (Finset P)) (l : Finset P → ℝ)
+    (hl : ∀ C ∈ s, 0 ≤ l C) (p : P → ℝ) (hp : ∑ x, p x=1) (η : ℝ)
+    (hbal : ∀ C ∈ s, 0 < l C → η ≤ mass p C ∧ η ≤ 1-mass p C) (y : P) :
+    η*(∑ C ∈ s, l C) ≤ ∑ x, p x*distance s l x y := by
+  simp only [distance,Finset.mul_sum]
+  rw [Finset.sum_comm]
+  apply Finset.sum_le_sum
+  intro C hC
+  have he : (∑ x, p x*(l C*|ind C x-ind C y|)) =
+      l C*(∑ x, p x*|ind C x-ind C y|) := by
+    rw [Finset.mul_sum]
+    exact Finset.sum_congr rfl (fun x _ => by ring)
+  rw [he,expected_cut p hp]
+  by_cases hpos : 0 < l C
+  · rcases hbal C hC hpos with ⟨h1,h2⟩
+    have hh : η ≤ if y ∈ C then 1-mass p C else mass p C := by split_ifs <;> assumption
+    simpa [mul_comm] using mul_le_mul_of_nonneg_left hh (hl C hC)
+  · have hz : l C=0 := le_antisymm (le_of_not_gt hpos) (hl C hC)
+    simp [hz]
+
+omit [Fintype P] in
+lemma exists_zero_cell (s : Finset (Finset P)) (hs : Laminar s)
+    (l : Finset P → ℝ) (hl : ∀ C ∈ s, 0 ≤ l C)
+    (hn : (s.filter (fun C => 0 < l C)).Nonempty) :
+    ∃ C ∈ s, 0 < l C ∧ ∀ x ∈ C, ∀ y ∈ C, distance s l x y=0 := by
+  obtain ⟨C,hC,hmin⟩ := Finset.exists_min_image (s.filter (fun C => 0 < l C)) Finset.card hn
+  rcases Finset.mem_filter.mp hC with ⟨hCs,hCl⟩
+  refine ⟨C,hCs,hCl,?_⟩
+  intro x hx y hy
+  unfold distance
+  apply Finset.sum_eq_zero
+  intro D hD
+  by_cases hd : 0 < l D
+  · have hcD : C.card ≤ D.card := hmin D (Finset.mem_filter.mpr ⟨hD,hd⟩)
+    rcases hs C hCs D hD with hCD | hDC | hdis
+    · simp [ind,hCD hx,hCD hy]
+    · have he : C=D := (Finset.eq_of_subset_of_card_le hDC hcD).symm
+      simp [ind,← he,hx,hy]
+    · have hxD : x ∉ D := fun h => Finset.disjoint_left.mp hdis hx h
+      have hyD : y ∉ D := fun h => Finset.disjoint_left.mp hdis hy h
+      simp [ind,hxD,hyD]
+  · have hz : l D=0 := le_antisymm (le_of_not_gt hd) (hl D hD)
+    simp [hz]
+
+/-- A minimal positive cut collapses a set of mass at least η to a single
+    pseudometric point. This gives a direct fluctuation bound without
+    constructing a tree graph. -/
+lemma fluctuation_lower (s : Finset (Finset P)) (hs : Laminar s)
+    (l : Finset P → ℝ) (hl : ∀ C ∈ s, 0 ≤ l C) (p : P → ℝ)
+    (hp0 : ∀ x, 0 ≤ p x) (hp : ∑ x, p x=1) (η : ℝ) (hη : 0 ≤ η)
+    (hbal : ∀ C ∈ s, 0 < l C → η ≤ mass p C ∧ η ≤ 1-mass p C) :
+    η^3*(∑ C ∈ s, l C) ≤
+      ∑ y, p y*∑ x', p x'*∑ x, p x*|distance s l x y-distance s l x' y| := by
+  let L := ∑ C ∈ s, l C
+  have hL : 0 ≤ L := Finset.sum_nonneg hl
+  have hf0 (y x' : P) : 0 ≤ ∑ x, p x*|distance s l x y-distance s l x' y| :=
+    Finset.sum_nonneg (fun x _ => mul_nonneg (hp0 x) (abs_nonneg _))
+  by_cases hn : (s.filter (fun C => 0 < l C)).Nonempty
+  · obtain ⟨C,hC,hCl,hzero⟩ := exists_zero_cell s hs l hl hn
+    have hm : η ≤ mass p C := (hbal C hC hCl).1
+    have hc0 := mass_nonneg p hp0 C
+    have hy (y : P) (hy : y ∈ C) : mass p C*(η*L) ≤
+        ∑ x', p x'*∑ x, p x*|distance s l x y-distance s l x' y| := by
+      calc
+        _ = ∑ x' ∈ C, p x'*(η*L) := by simp only [mass,Finset.sum_mul]
+        _ ≤ ∑ x' ∈ C, p x'*∑ x, p x*|distance s l x y-distance s l x' y| := by
+          apply Finset.sum_le_sum
+          intro x' hx'
+          apply mul_le_mul_of_nonneg_left _ (hp0 x')
+          simp only [hzero x' hx' y hy,sub_zero,abs_of_nonneg (distance_nonneg s l hl _ _)]
+          exact expected_distance_ge s l hl p hp η hbal y
+        _ ≤ _ := Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ C)
+          (fun x' _ _ => mul_nonneg (hp0 x') (hf0 y x'))
+    calc
+      _ ≤ mass p C*(mass p C*(η*L)) := by
+        have hsq : η*η ≤ mass p C*mass p C := mul_le_mul hm hm hη hc0
+        have hmL := mul_le_mul_of_nonneg_right hsq (mul_nonneg hη hL)
+        dsimp only [L] at hmL
+        nlinarith only [hmL]
+      _ = ∑ y ∈ C, p y*(mass p C*(η*L)) := by simp only [mass,Finset.sum_mul]
+      _ ≤ ∑ y ∈ C, p y*∑ x', p x'*∑ x, p x*|distance s l x y-distance s l x' y| :=
+        Finset.sum_le_sum (fun y hyC => mul_le_mul_of_nonneg_left (hy y hyC) (hp0 y))
+      _ ≤ _ := Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ C)
+        (fun y _ _ => mul_nonneg (hp0 y) (Finset.sum_nonneg (fun x' _ => mul_nonneg (hp0 x') (hf0 y x'))))
+  · have hz : ∑ C ∈ s, l C=0 := by
+      apply Finset.sum_eq_zero
+      intro C hC
+      have hnot : ¬0 < l C := fun hc => hn ⟨C,Finset.mem_filter.mpr ⟨hC,hc⟩⟩
+      exact le_antisymm (le_of_not_gt hnot) (hl C hC)
+    rw [hz,mul_zero]
+    exact Finset.sum_nonneg (fun y _ => mul_nonneg (hp0 y)
+      (Finset.sum_nonneg (fun x' _ => mul_nonneg (hp0 x') (hf0 y x'))))
+
+omit [Fintype P] [DecidableEq P] in
+lemma cost_symm (s : Finset (Finset P)) (l : Finset P → ℝ) (p q : P → ℝ) :
+    cost s l p q=cost s l q p := by simp only [cost,abs_sub_comm]
+
+def fluctuation (s : Finset (Finset P)) (l : Finset P → ℝ) (x x' y : P) : ℝ :=
+  |distance s l x y-distance s l x' y|
+
+def doubleMean (s : Finset (Finset P)) (l : Finset P → ℝ) (p : P → ℝ) (y : P) : ℝ :=
+  ∑ x', p x'*∑ x, p x*fluctuation s l x x' y
+
+omit [Fintype P] in
+lemma fluctuation_symm (s : Finset (Finset P)) (l : Finset P → ℝ) (x x' y : P) :
+    fluctuation s l x x' y=fluctuation s l x' x y := abs_sub_comm _ _
+
+lemma sum_le_add_of_abs {a b c : ℝ} (h : |a-b| ≤ c) : a ≤ b+c :=
+  by have := (abs_le.mp h).2; linarith
+
+lemma doubleMean_transfer (s : Finset (Finset P)) (hs : Laminar s)
+    (l : Finset P → ℝ) (hl : ∀ C ∈ s, 0 ≤ l C) (p q : P → ℝ)
+    (hp0 : ∀ x, 0 ≤ p x) (hq0 : ∀ x, 0 ≤ q x)
+    (hp : ∑ x, p x=1) (hq : ∑ x, q x=1) (y : P) :
+    doubleMean s l p y ≤ doubleMean s l q y+2*cost s l p q := by
+  have h1 (x' : P) : (∑ x, p x*fluctuation s l x x' y) ≤
+      (∑ x, q x*fluctuation s l x x' y)+cost s l p q :=
+    sum_le_add_of_abs (first_fluctuation_transfer s hs l hl p q (hp.trans hq.symm) x' y)
+  have h2 (x : P) : (∑ x', p x'*fluctuation s l x x' y) ≤
+      (∑ x', q x'*fluctuation s l x x' y)+cost s l p q := by
+    simpa only [fluctuation,abs_sub_comm] using
+      sum_le_add_of_abs (first_fluctuation_transfer s hs l hl p q (hp.trans hq.symm) x y)
+  calc
+    _ ≤ ∑ x', p x'*((∑ x, q x*fluctuation s l x x' y)+cost s l p q) :=
+      Finset.sum_le_sum (fun x' _ => mul_le_mul_of_nonneg_left (h1 x') (hp0 x'))
+    _ = (∑ x, q x*∑ x', p x'*fluctuation s l x x' y)+cost s l p q := by
+      simp only [mul_add,Finset.sum_add_distrib,← Finset.sum_mul,hp,one_mul]
+      congr 1
+      simp only [Finset.mul_sum]
+      rw [Finset.sum_comm]
+      exact Finset.sum_congr rfl (fun x _ => Finset.sum_congr rfl (fun x' _ => by ring))
+    _ ≤ (∑ x, q x*((∑ x', q x'*fluctuation s l x x' y)+cost s l p q))+cost s l p q := by
+      exact add_le_add (Finset.sum_le_sum (fun x _ => mul_le_mul_of_nonneg_left (h2 x) (hq0 x))) le_rfl
+    _ = doubleMean s l q y+2*cost s l p q := by
+      simp only [doubleMean,mul_add,Finset.sum_add_distrib,← Finset.sum_mul,hq,one_mul]
+      rw [show (∑ x, q x*∑ x', q x'*fluctuation s l x x' y) =
+          ∑ x, q x*∑ x', q x'*fluctuation s l x' x y from by simp_rw [fluctuation_symm s l _ _ y]]
+      ring
+
+lemma doubleMean_fiber (s : Finset (Finset P)) (l : Finset P → ℝ) (p : P → ℝ)
+    (hp0 : ∀ x, 0 ≤ p x) (hp : ∑ x, p x=1) (y : P) (R : ℝ) :
+    doubleMean s l p y ≤ 2*∑ x, p x*|distance s l x y-R| := by
+  have ht (x x' : P) : fluctuation s l x x' y ≤
+      |distance s l x y-R|+|distance s l x' y-R| := by
+    simpa only [fluctuation,abs_sub_comm (R : ℝ)] using abs_sub_le (distance s l x y) R (distance s l x' y)
+  calc
+    _ ≤ ∑ x', p x'*∑ x, p x*(|distance s l x y-R|+|distance s l x' y-R|) := by
+      apply Finset.sum_le_sum
+      intro x' _
+      apply mul_le_mul_of_nonneg_left _ (hp0 x')
+      exact Finset.sum_le_sum (fun x _ => mul_le_mul_of_nonneg_left (ht x x') (hp0 x))
+    _ = _ := by
+      simp only [mul_add,Finset.sum_add_distrib,← Finset.sum_mul,hp,one_mul]
+      ring
+
+lemma last_doubleMean_transfer (s : Finset (Finset P)) (hs : Laminar s)
+    (l : Finset P → ℝ) (hl : ∀ C ∈ s, 0 ≤ l C) (a p q : P → ℝ)
+    (ha0 : ∀ x, 0 ≤ a x) (ha : ∑ x, a x=1) (hpq : ∑ x, p x=∑ x, q x) :
+    (∑ y, p y*doubleMean s l a y) ≤
+      (∑ y, q y*doubleMean s l a y)+2*cost s l p q := by
+  have ht (x x' : P) : (∑ y, p y*fluctuation s l x x' y) ≤
+      (∑ y, q y*fluctuation s l x x' y)+2*cost s l p q :=
+    sum_le_add_of_abs (last_fluctuation_transfer s hs l hl p q hpq x x')
+  have he (b : P → ℝ) : (∑ y, b y*doubleMean s l a y) =
+      ∑ x', a x'*∑ x, a x*∑ y, b y*fluctuation s l x x' y := by
+    simp only [doubleMean,Finset.mul_sum]
+    rw [Finset.sum_comm]
+    apply Finset.sum_congr rfl
+    intro x' _
+    rw [Finset.sum_comm]
+    apply Finset.sum_congr rfl
+    intro x _
+    exact Finset.sum_congr rfl (fun y _ => by ring)
+  rw [he p,he q]
+  calc
+    _ ≤ ∑ x', a x'*∑ x, a x*((∑ y, q y*fluctuation s l x x' y)+2*cost s l p q) := by
+      apply Finset.sum_le_sum
+      intro x' _
+      apply mul_le_mul_of_nonneg_left _ (ha0 x')
+      exact Finset.sum_le_sum (fun x _ => mul_le_mul_of_nonneg_left (ht x x') (ha0 x))
+    _ = _ := by simp only [mul_add,Finset.sum_add_distrib,← Finset.sum_mul,ha,one_mul]
+
+/-- Finite transport to independent sampling with explicit cut costs. -/
+
+lemma iid_transfer (s : Finset (Finset P)) (hs : Laminar s)
+    (l : Finset P → ℝ) (hl : ∀ C ∈ s, 0 ≤ l C) (a t : P → ℝ) (ρ : P → P → ℝ)
+    (ha0 : ∀ x, 0 ≤ a x) (ht0 : ∀ y, 0 ≤ t y) (hρ0 : ∀ y x, 0 ≤ ρ y x)
+    (ha : ∑ x, a x=1) (ht : ∑ y, t y=1) (hρ : ∀ y, ∑ x, ρ y x=1) (R : P → ℝ) :
+    (∑ y, a y*doubleMean s l a y) ≤
+      2*(∑ y, t y*∑ x, ρ y x*|distance s l x y-R y|)+
+      2*(∑ y, t y*cost s l (ρ y) a)+2*cost s l t a := by
+  have hlast := last_doubleMean_transfer s hs l hl a a t ha0 ha (ha.trans ht.symm)
+  rw [cost_symm s l a t] at hlast
+  have hy (y : P) : doubleMean s l a y ≤
+      2*(∑ x, ρ y x*|distance s l x y-R y|)+2*cost s l (ρ y) a := by
+    have h1 := doubleMean_transfer s hs l hl a (ρ y) ha0 (hρ0 y) ha (hρ y) y
+    rw [cost_symm s l a (ρ y)] at h1
+    exact h1.trans (add_le_add (doubleMean_fiber s l (ρ y) (hρ0 y) (hρ y) y (R y)) le_rfl)
+  have hsum := Finset.sum_le_sum (s := Finset.univ) (fun y _ => mul_le_mul_of_nonneg_left (hy y) (ht0 y))
+  have he : (∑ y, t y*(2*(∑ x, ρ y x*|distance s l x y-R y|)+2*cost s l (ρ y) a)) =
+      2*(∑ y, t y*∑ x, ρ y x*|distance s l x y-R y|)+2*(∑ y, t y*cost s l (ρ y) a) := by
+    simp only [mul_add,Finset.sum_add_distrib]
+    rw [Finset.mul_sum,Finset.mul_sum]
+    congr 1 <;> exact Finset.sum_congr rfl (fun y _ => by ring)
+  rw [he] at hsum
+  linarith
+
+end WeakPinned.Cuts
+
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PinnedDistances/UnboundedContradiction.lean b/lean/OAI/Geometry/PinnedDistances/UnboundedContradiction.lean
new file mode 100644
index 0000000000000000000000000000000000000000..7be45e0b27f77a8f93eecb572fb2a82d932b18d8
--- /dev/null
+++ b/lean/OAI/Geometry/PinnedDistances/UnboundedContradiction.lean
@@ -0,0 +1,601 @@
+import OAI.Geometry.PinnedDistances.TreeContradiction
+
+namespace OAI
+
+universe u1 u2 u3 u4 u5 u6 u7 u8 u9 u10 u11 u12
+universe u13 u14 u15 u16 u17 u18 u19 u20 u21 u22 u23
+
+section
+namespace WeakPinned.Hierarchy.Depth
+open MeasureTheory
+variable {P : Type u1} [Fintype P] [DecidableEq P]
+variable (d : Depth P)
+
+lemma transformed_half (r : ℝ) (C : Finset P) (hC : C ∈ d.transformed r) :
+    2*C.card≤Fintype.card P := by
+  rw [transformed,Multiset.mem_add] at hC
+  rcases hC with hC | hC
+  · exact Nat.le_of_not_gt (Finset.mem_filter.mp hC).2
+  · obtain ⟨B,hB,hBC⟩ := Multiset.mem_map.mp hC
+    have hh := (d.mem_giants r B).mp hB
+    subst C
+    rw [Finset.card_compl]
+    have := Finset.card_le_univ B
+    omega
+
+lemma mass_half (C : Finset P) (hC : Fintype.card P<2*C.card) : d.mass C=0 := by
+  have hz (r : ℝ) : d.count r C=0 := by
+    unfold count
+    have hh : C ∉ d.transformed r := fun hc => hC.not_ge (d.transformed_half r C hc)
+    simp [Multiset.count_eq_zero.mpr hh]
+  simp [mass,hz]
+end WeakPinned.Hierarchy.Depth
+
+namespace WeakPinned.Global
+open Hierarchy
+variable {P : Type u2} [Fintype P] [DecidableEq P]
+variable {K : Type u3} [Field K] [NumberField K]
+variable (Z : P → K) (hZ : Function.Injective Z)
+lemma cellMass_half (C : Finset P) (hC : Fintype.card P<2*C.card) : cellMass Z hZ C=0 := by
+  simp only [cellMass,Depth.mass_half _ C hC,MeasureTheory.integral_zero,mul_zero,Finset.sum_const_zero,add_zero]
+
+lemma totalMass_half (Z₂ : P → K) (h₂ : Function.Injective Z₂)
+    (C : Finset P) (hC : 1/2<Variance.cellProb Finset.univ C) :
+    totalMass Z Z₂ hZ h₂ C=0 := by
+  have hn0 : (0:ℝ)<Fintype.card P := by
+    by_contra! hn
+    have hz : Fintype.card P=0 := by exact_mod_cast (le_antisymm hn (Nat.cast_nonneg _))
+    simp [Variance.cellProb,hz] at hC
+    linarith
+  have hh : Fintype.card P<2*C.card := by
+    simp only [Variance.cellProb,Finset.univ_inter,Finset.card_univ] at hC
+    have hl := (lt_div_iff₀ hn0).mp hC
+    exact_mod_cast (show (Fintype.card P:ℝ)<2*C.card by linarith)
+  simp only [totalMass,cellMass_half _ _ C hh,add_zero]
+end WeakPinned.Global
+
+namespace WeakPinned.Variance
+open scoped BigOperators
+open Cuts SmallFiber
+variable {P : Type u4} {I : Type u5} [Fintype P] [Fintype I] [DecidableEq P]
+
+omit [Fintype P] in
+lemma uniform_slice_nonneg (w : I → ℝ) (B : I → Finset P) (src : I → P) (lam : ℝ)
+    (hw : ∀ i, 0≤w i) (C : Finset P) : 0 ≤ slice w (fun i => prob (B i)) src lam C := by
+  unfold slice
+  exact Finset.sum_nonneg (fun i _ => mul_nonneg (hw i) (by
+    split_ifs
+    · simp only [SmallFiber.mass,sum_prob_cell]; exact cellProb_nonneg _ _
+    · exact le_rfl))
+
+omit [Fintype P] in
+lemma edgeCell_le_one (w : I → ℝ) (B : I → Finset P) (src : I → P)
+    (hw : ∀ i, 0≤w i) (hW : (∑ i, w i)≤1) (C : Finset P) :
+    edgeCell w (fun i => prob (B i)) src C≤1 := by
+  rw [edgeCell_formula]
+  apply le_trans _ hW
+  apply Finset.sum_le_sum
+  intro i _
+  by_cases hi : src i ∈ C
+  · simp only [ite_eq_left hi,SmallFiber.mass,sum_prob_cell]
+    nlinarith [mul_le_mul_of_nonneg_left (cellProb_le_one (B i) C) (hw i)]
+  · simp [hi,hw i]
+
+/-- Positive balanced cut length pays all q-overlap except the small-fiber
+slice and cell variance. This is the exact occurrence extraction inequality. -/
+lemma balanced_mass_extraction (m : Finset P → ℝ) (hm : ∀ C, 0 ≤ m C)
+    (hhalf : ∀ C, 1/2<cellProb Finset.univ C → m C=0)
+    (w : I → ℝ) (B : I → Finset P) (src : I → P) (T δ : ℝ)
+    (hw : ∀ i, 0≤w i) (hW : (∑ i, w i)≤1) (hδ : 0<δ) (_hδ4 : δ≤1/4)
+    (hk : ∀ i, 2/((B i).card:ℝ)≤4*δ) :
+    weightedIntegral m (edgeCell w (fun i => prob (B i)) src)≤
+      weightedIntegral m (slice w (fun i => prob (B i)) src (4*δ))+
+      (1/δ)*weightedIntegral m (varianceCell w B T)+
+      (1/δ)*weightedIntegral m (fun C => tent δ (cellProb Finset.univ C)) := by
+  have hpoint (C : Finset P) :
+      m C*edgeCell w (fun i => prob (B i)) src C ≤
+        m C*(slice w (fun i => prob (B i)) src (4*δ) C+
+          (1/δ)*varianceCell w B T C+(1/δ)*tent δ (cellProb Finset.univ C)) := by
+    by_cases hh : 1/2<cellProb Finset.univ C
+    · rw [hhalf C hh]; simp
+    have ha : cellProb Finset.univ C≤1/2 := le_of_not_gt hh
+    apply mul_le_mul_of_nonneg_left _ (hm C)
+    by_cases hs : cellProb Finset.univ C≤2*δ
+    · have hx := small_mass_qCell (B := B) src T (2*δ) hw (by positivity) (fun i => by nlinarith [hk i]) C hs
+      have he : (∑ i, w i*(if src i ∈ C then cellProb (B i) C else 0)) =
+          edgeCell w (fun i => prob (B i)) src C := by
+        rw [edgeCell_formula]; simp only [SmallFiber.mass,sum_prob_cell]
+      have hf : (∑ i, w i*(if src i ∈ C ∧ cellProb (B i) C<2*(2*δ) then cellProb (B i) C else 0)) =
+          slice w (fun i => prob (B i)) src (4*δ) C := by
+        unfold slice
+        simp only [SmallFiber.mass,sum_prob_cell,show (2:ℝ)*(2*δ)=4*δ by ring]
+      rw [he,hf,show (2:ℝ)/(2*δ)=1/δ by ring] at hx
+      exact hx.trans (le_add_of_nonneg_right (mul_nonneg (by positivity) (tent_nonneg _ _)))
+    · have htent : δ≤tent δ (cellProb Finset.univ C) := tent_lower (le_of_not_ge hs) (by linarith)
+      have hh : 1≤(1/δ)*tent δ (cellProb Finset.univ C) := by
+        have := mul_le_mul_of_nonneg_left htent (show 0≤1/δ by positivity)
+        simpa [ne_of_gt hδ] using this
+      have he := (edgeCell_le_one w B src hw hW C).trans hh
+      have hn := uniform_slice_nonneg w B src (4*δ) hw C
+      have hv := mul_nonneg (show 0≤1/δ by positivity) (varianceCell_nonneg w B T hw C)
+      linarith
+  have hh := Finset.sum_le_sum (s:=Finset.univ) (fun C _ => hpoint C)
+  change weightedIntegral m (edgeCell w (fun i => prob (B i)) src)≤_ at hh
+  convert hh using 1
+  simp only [weightedIntegral,mul_add,Finset.sum_add_distrib,Finset.mul_sum]
+  have hcomm (f : Finset P → ℝ) :
+      (1/δ) * ∑ C, m C * f C = ∑ C, m C * ((1/δ)*f C) := by
+    rw [Finset.mul_sum]
+    apply Finset.sum_congr rfl
+    intro C _
+    ring
+  simp only [Finset.mul_sum] at hcomm
+  rw [hcomm, hcomm]
+end WeakPinned.Variance
+
+namespace WeakPinned
+open Filter Variance Global Cuts
+open scoped Topology BigOperators
+
+lemma change_scale_rate (W A X : ℕ → ℝ) (C : ℝ) (hC : 0<C)
+    (hW : ∀ j, 0≤W j) (hA : ∀ j, 0≤A j) (hX : ∀ j, 0≤X j)
+    (hWtop : Tendsto W atTop atTop)
+    (hcomp : ∀ᶠ j in atTop, A j≤C*(W j+1))
+    (hrate : Tendsto (fun j => X j/(W j+A j+1)) atTop (𝓝 0)) :
+    Tendsto (fun j => X j/W j) atTop (𝓝 0) := by
+  have hb : ∀ᶠ j in atTop, X j/W j≤(2+2*C)*(X j/(W j+A j+1)) := by
+    filter_upwards [hWtop.eventually (eventually_ge_atTop (1:ℝ)),hcomp] with j hj hc
+    have hw : 0<W j := by linarith
+    have hd : 0<W j+A j+1 := by linarith [hA j]
+    apply (div_le_iff₀ hw).mpr
+    have hbound : W j+A j+1≤(2+2*C)*W j := by nlinarith
+    have hh := mul_le_mul_of_nonneg_left hbound (hX j)
+    convert (div_le_div_of_nonneg_right hh hd.le) using 1 <;> field_simp
+  exact squeeze_zero' (Eventually.of_forall (fun j => div_nonneg (hX j) (hW j))) hb
+    (by simpa using hrate.const_mul (2+2*C))
+
+lemma balanced_extraction_rate (W A V L Q S a : ℕ → ℝ) (C δ b : ℝ)
+    (hC : 0<C) (hδ : 0<δ) (hb : 0≤b) (hbC : 2*b*C≤1/8)
+    (hW : ∀ j, 0≤W j) (hA : ∀ j, 0≤A j) (hV : ∀ j, 0≤V j)
+    (hWtop : Tendsto W atTop atTop)
+    (hcomp : ∀ᶠ j in atTop, A j≤C*(W j+1))
+    (ha : Tendsto a atTop (𝓝 0))
+    (hvar : Tendsto (fun j => V j/(W j+A j+1)) atTop (𝓝 0))
+    (hQ : ∀ᶠ j in atTop, W j-a j*A j≤Q j)
+    (hS : ∀ᶠ j in atTop, S j≤W j/8+b*A j+32)
+    (hext : ∀ᶠ j in atTop, Q j≤S j+V j/δ+L j/δ) :
+    ∀ᶠ j in atTop, (δ/4)*W j≤L j := by
+  have hv := change_scale_rate W A V C hC hW hA hV hWtop hcomp hvar
+  have ha0 : Tendsto (fun j => |a j|) atTop (𝓝 0) := by simpa using ha.abs
+  filter_upwards [hWtop.eventually (eventually_ge_atTop (256:ℝ)),hcomp,
+    ha0.eventually (gt_mem_nhds (show 0<1/(8*C) by positivity)),
+    hv.eventually (gt_mem_nhds (show 0<δ/8 by positivity)),hQ,hS,hext] with j hw hc ha hv hq hs he
+  have hw0 : 0<W j := by linarith
+  have hac : A j≤2*C*W j := by nlinarith
+  have ha' : a j*A j≤W j/4 := by
+    have hh := mul_le_mul_of_nonneg_right (le_trans (le_abs_self (a j)) ha.le) (hA j)
+    have hh' := mul_le_mul_of_nonneg_left hac (show 0≤1/(8*C) by positivity)
+    have hz : (1/(8*C))*(2*C*W j)=W j/4 := by field_simp; ring
+    rw [hz] at hh'
+    exact hh.trans hh'
+  have hbA : b*A j≤W j/8 := by
+    have h1 := mul_le_mul_of_nonneg_left hac hb
+    have h2 := mul_le_mul_of_nonneg_right hbC hw0.le
+    nlinarith
+  have hv' : V j/δ≤W j/8 := by
+    have hh := (div_lt_iff₀ hw0).mp hv
+    apply (div_le_iff₀ hδ).mpr
+    nlinarith
+  have hL : W j/4≤L j/δ := by linarith
+  have := (le_div_iff₀ hδ).mp hL
+  nlinarith
+
+end WeakPinned
+
+namespace WeakPinned
+open Filter Variance Global Cuts SmallFiber
+open scoped Topology BigOperators
+variable {V : ℕ → Type u6} {I : ℕ → Type u7} {K : ℕ → Type u8} [∀ j, Fintype (V j)] [∀ j, Fintype (I j)] [∀ j, DecidableEq (V j)]
+  [∀ j, Field (K j)] [∀ j, NumberField (K j)]
+variable (G : ∀ j, Core.Fibers (V j) (I j)) (θ γ c : ℝ) (η : ℕ → ℝ)
+  (Z₁ Z₂ : ∀ j, V j → K j) (h₁ : ∀ j, Function.Injective (Z₁ j)) (h₂ : ∀ j, Function.Injective (Z₂ j))
+
+/-- The positive balanced-length estimate of the unbounded branch, on the
+actual varying-field arithmetic models. -/
+theorem arithmetic_balanced_length (hθ : 0<θ) (hγ : 0<γ) (hγ1 : γ<1) (hcγ : c<γ)
+    (hn : Tendsto (fun j => (Fintype.card (V j):ℝ)) atTop atTop)
+    (hη : Tendsto η atTop (𝓝 0)) (hη0 : ∀ j, 0≤η j) (hηb : ∀ j, η j≤θ/2)
+    (hG : ∀ j, PreparedGraph (V j) (I j) (G j) θ γ c (η j))
+    (t : ∀ j, I j → K j) (ht : ∀ j e y, y ∈ (G j).targets e →
+      (Z₁ j ((G j).source e)-Z₁ j y)*(Z₂ j ((G j).source e)-Z₂ j y)=t j e)
+    (hWtop : Tendsto (fun j => overlapScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)) atTop atTop) :
+    ∃ δ : ℝ, 0<δ ∧ δ≤1/8 ∧ ∀ᶠ j in atTop,
+      (δ/4)*overlapScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j) ≤
+        weightedIntegral (totalMass (Z₁ j) (Z₂ j) (h₁ j) (h₂ j))
+          (fun C => tent δ (cellProb Finset.univ C)) := by
+  let C : ℝ := 64+8/θ
+  have hC : 0<C := by dsimp [C]; positivity
+  let τ : ℝ := min 1 (θ/(64*C))
+  have hτ : 0<τ := lt_min (by norm_num) (by positivity)
+  have hτ1 : τ≤1 := min_le_left _ _
+  have hτC : 64*τ*C≤θ := by
+    have hh := (le_div_iff₀ (show 0<64*C by positivity)).mp (min_le_right (1:ℝ) (θ/(64*C)))
+    dsimp [τ]
+    nlinarith
+  let b : ℝ := 4*τ/θ
+  have hb : 0≤b := by dsimp [b]; positivity
+  have hbC : 2*b*C≤1/8 := by
+    dsimp [b]
+    apply (mul_le_mul_iff_left₀ hθ).mp
+    field_simp
+    nlinarith
+  let R : ℝ := 8/τ^2+16/θ
+  have hR : 0<R := by dsimp [R]; positivity
+  let δ : ℝ := min (1/8) ((1/8)/R)
+  have hδ : 0<δ := lt_min (by norm_num) (by positivity)
+  have hδ8 : δ≤1/8 := min_le_left _ _
+  have hδR : δ*R≤1/8 := (le_div_iff₀ hR).mp (min_le_right _ _)
+  refine ⟨δ,hδ,hδ8,?_⟩
+  let W := fun j => overlapScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)
+  let A := fun j => potentialScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)
+  let m := fun j => totalMass (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)
+  let Q := fun j => weightedIntegral (m j) (G j).qCell
+  let S := fun j => weightedIntegral (m j) (slice (G j).fiberWeight
+    (fun e => prob ((G j).targets e)) (G j).source (4*δ))
+  let a := fun j => 2-2*((θ-η j)/(θ+η j/8))
+  have ha : Tendsto a atTop (𝓝 0) := by
+    have hh := (tendsto_const_nhds (x:=(2:ℝ))).sub ((tendsto_const_nhds (x:=(2:ℝ))).mul
+      (((tendsto_const_nhds (x:=θ)).sub hη).div (tendsto_const_nhds.add (hη.div_const 8))
+      (show θ+(0:ℝ)/8≠0 by simpa using hθ.ne')))
+    convert hh using 1
+    simp [hθ.ne']
+  have hcomp := arithmetic_scale_comparison G θ γ c η Z₁ Z₂ h₁ h₂ hθ hγ hγ1 hcγ hn hη hη0 hηb hG t ht
+  apply balanced_extraction_rate W A (fiberVariance G c Z₁ Z₂ h₁ h₂) _ Q S a C δ b hC hδ hb hbC
+    (fun j => overlapScale_nonneg _ _ _ _) (fun j => potentialScale_nonneg _ _ _ _)
+    (fun j => Finset.sum_nonneg (fun B _ => mul_nonneg (totalMass_nonneg _ _ _ _ B)
+      (varianceCell_nonneg _ _ _ (G j).fiberWeight_nonneg B))) hWtop hcomp ha
+    (sampling_variance_rates G θ γ c η Z₁ Z₂ h₁ h₂ hθ hγ hγ1 hcγ hn hη hη0 hηb hG t ht).1
+  · exact Eventually.of_forall (fun j => (hG j).overlap_lower (by linarith [hηb j]) (hη0 j)
+      (m j) (baseConstant (Z₁ j) (Z₂ j) (h₁ j) (h₂ j))
+      (totalPotential (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)) (totalPotential_nonneg _ _ _ _)
+      (totalMass_overlap _ _ _ _ (hG j).card_ge))
+  · filter_upwards [hn.eventually (eventually_ge_atTop (2/τ))] with j hj
+    have hh := (hG j).arithmetic_slice (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)
+      (by linarith [hηb j]) (t j) (ht j) τ (4*δ) 0 (by positivity) (by linarith) hτ hτ1
+      (by have := (div_le_iff₀ hτ).mp hj; nlinarith) (by norm_num)
+    have hi : 1/(θ-η j)≤2/θ := by
+      apply (div_le_div_iff₀ (by linarith [hηb j] : 0<θ-η j) hθ).mpr
+      linarith [hηb j]
+    have h1 : 2*(4*δ)/τ^2+2*(4*δ)/(θ-η j)≤1/8 := by
+      have hh' := mul_le_mul_of_nonneg_left hi (show 0≤8*δ by positivity)
+      dsimp [R] at hδR
+      simp only [div_eq_mul_inv] at hδR hh' ⊢
+      nlinarith
+    have h2 : 2*τ/(θ-η j)≤b := by
+      have hh' := mul_le_mul_of_nonneg_left hi (show 0≤2*τ by positivity)
+      dsimp [b]
+      simp only [div_eq_mul_inv] at hh' ⊢
+      nlinarith
+    have hW0 := overlapScale_nonneg (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)
+    have hA0 := potentialScale_nonneg (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)
+    have h1' := mul_le_mul_of_nonneg_right h1 hW0
+    have h2' := mul_le_mul_of_nonneg_right h2 hA0
+    simp only [neg_zero,zero_div,Real.exp_zero,mul_zero,add_zero,mul_one] at hh
+    dsimp [S,W,A]
+    linarith
+  · have hnγ := (tendsto_rpow_atTop hγ).comp hn
+    filter_upwards [hnγ.eventually (eventually_ge_atTop (2/(4*δ)))] with j hj
+    have hk (i : I j) : 2/(((G j).targets i).card:ℝ)≤4*δ := by
+      have hsize := (hG j).fiber i
+      have hn0 : 0<(((G j).targets i).card:ℝ) := (show 0<2/(4*δ) by positivity).trans_le (hj.trans hsize.le)
+      apply (div_le_iff₀ hn0).mpr
+      have hh := (div_le_iff₀ (show 0<4*δ by positivity)).mp (hj.trans hsize.le)
+      nlinarith
+    have hh := balanced_mass_extraction (m j) (totalMass_nonneg _ _ _ _)
+      (totalMass_half _ _ _ _) (G j).fiberWeight (G j).targets (G j).source
+      ((Fintype.card (V j):ℝ)^(1-c)) δ (G j).fiberWeight_nonneg
+      ((G j).sum_fiberWeight ((hG j).edgeCount_pos (by linarith [hηb j]))).le hδ (by linarith) hk
+    dsimp [Q,S,fiberVariance]
+    have he : (G j).qCell = edgeCell (G j).fiberWeight (fun e => prob ((G j).targets e)) (G j).source := funext (G j).qCell_eq
+    rw [←he] at hh
+    simpa only [m,one_div,mul_comm,div_eq_mul_inv,one_mul] using hh
+end WeakPinned
+noncomputable section
+namespace WeakPinned.Cuts
+open Hierarchy Variance
+variable {P : Type u9} [Fintype P] [DecidableEq P]
+lemma tent_length_occurrence (d : Depth P) (hn : 2≤Fintype.card P) (δ : ℝ) (hδ : 0≤δ)
+    (has : ∀ x, prob (Finset.univ : Finset P) x≤δ) :
+    (∑ C ∈ d.allCuts, d.cutLength (fun B => tent δ (mass (prob Finset.univ) B)) C)=
+      weightedIntegral d.mass (fun C => tent δ (cellProb Finset.univ C)) := by
+  have ha : ∑ x, prob (Finset.univ : Finset P) x=1 :=
+    sum_prob _ (Finset.univ_nonempty_iff.mpr (Fintype.card_pos_iff.mp (by omega)))
+  symm
+  simp only [← sum_prob_cell]
+  change weightedIntegral d.mass (fun C => tent δ (mass (prob Finset.univ) C))=_
+  exact d.symmetric_integral_cut_sum hn _ (fun C hC => tent_zero_left (mass_small _ δ hδ has C hC))
+    (fun C => by rw [mass_complement _ ha,tent_symm])
+end WeakPinned.Cuts
+
+namespace WeakPinned.Global
+open Hierarchy Variance Cuts MeasureTheory
+open scoped BigOperators
+variable {P : Type u10} [Fintype P] [DecidableEq P]
+variable {K : Type u11} [Field K] [NumberField K]
+variable (Z₁ Z₂ : P → K) (h₁ : Function.Injective Z₁) (h₂ : Function.Injective Z₂)
+
+lemma average_swap (f : Depth P → Depth P → ℝ) :
+    average Z₂ Z₁ h₂ h₁ f=average Z₁ Z₂ h₁ h₂ (fun d d' => f d' d) := by
+  classical
+  unfold average placeUnion
+  rw [Finset.union_comm (badPlaces Z₂ h₂)]
+
+omit [NumberField K] in
+include h₁ in
+lemma integrable_weighted_left (hn : 2≤Fintype.card P) (f : Finset P → ℝ) :
+    AverageIntegrable Z₁ Z₂ (fun d _ => weightedIntegral d.mass f) := by
+  intro v
+  apply integrable_finsetSum
+  intro C _
+  exact (integrable_mass_left Z₁ Z₂ h₁ hn C v).mul_const (f C)
+
+omit [NumberField K] in
+include h₂ in
+lemma integrable_weighted_right (hn : 2≤Fintype.card P) (f : Finset P → ℝ) :
+    AverageIntegrable Z₁ Z₂ (fun _ d' => weightedIntegral d'.mass f) := by
+  intro v
+  apply integrable_finsetSum
+  intro C _
+  exact (integrable_mass_right Z₁ Z₂ h₂ hn C v).mul_const (f C)
+
+lemma average_weighted_left (hn : 2≤Fintype.card P) (f : Finset P → ℝ) :
+    average Z₁ Z₂ h₁ h₂ (fun d _ => weightedIntegral d.mass f)=weightedIntegral (cellMass Z₁ h₁) f := by
+  simp only [weightedIntegral,mul_comm _ (f _)]
+  rw [average_sum Z₁ Z₂ h₁ h₂ _ (fun C v => (integrable_mass_left Z₁ Z₂ h₁ hn C v).const_mul (f C))]
+  simp_rw [average_mul,average_mass_left Z₁ Z₂ h₁ h₂ hn]
+
+lemma average_weighted_right (hn : 2≤Fintype.card P) (f : Finset P → ℝ) :
+    average Z₁ Z₂ h₁ h₂ (fun _ d' => weightedIntegral d'.mass f)=weightedIntegral (cellMass Z₂ h₂) f := by
+  simp only [weightedIntegral,mul_comm _ (f _)]
+  rw [average_sum Z₁ Z₂ h₁ h₂ _ (fun C v => (integrable_mass_right Z₁ Z₂ h₂ hn C v).const_mul (f C))]
+  simp_rw [average_mul,average_mass_right Z₁ Z₂ h₁ h₂ hn]
+
+def localTotalIntegral (f : Finset P → ℝ) (d d' : Depth P) : ℝ :=
+  weightedIntegral d.mass f+weightedIntegral d'.mass f
+
+omit [NumberField K] in
+include h₁ h₂ in
+lemma integrable_total_integral (hn : 2≤Fintype.card P) (f : Finset P → ℝ) :
+    AverageIntegrable Z₁ Z₂ (localTotalIntegral f) := fun v =>
+  (integrable_weighted_left Z₁ Z₂ h₁ hn f v).add (integrable_weighted_right Z₁ Z₂ h₂ hn f v)
+
+lemma average_total_integral (hn : 2≤Fintype.card P) (f : Finset P → ℝ) :
+    average Z₁ Z₂ h₁ h₂ (localTotalIntegral f)=weightedIntegral (totalMass Z₁ Z₂ h₁ h₂) f := by
+  change average Z₁ Z₂ h₁ h₂ (fun d d' => weightedIntegral d.mass f+weightedIntegral d'.mass f)=_
+  rw [average_add Z₁ Z₂ h₁ h₂ (integrable_weighted_left Z₁ Z₂ h₁ hn f)
+    (integrable_weighted_right Z₁ Z₂ h₂ hn f),average_weighted_left Z₁ Z₂ h₁ h₂ hn,
+    average_weighted_right Z₁ Z₂ h₁ h₂ hn]
+  simp only [weightedIntegral,totalMass,add_mul,Finset.sum_add_distrib]
+
+end WeakPinned.Global
+
+namespace WeakPinned.Core.Fibers
+open Hierarchy Variance Cuts Global
+variable {V : Type u12} {I : Type u13} [Fintype V] [Fintype I] [DecidableEq V]
+variable (G : Fibers V I)
+lemma two_length_upper (d d' : Depth V) (δ T : ℝ) (hδ : 0≤δ) (hT : 0<T)
+    (hn : 2≤Fintype.card V) (hG : 0<G.edgeCount)
+    (has : ∀ x, prob (Finset.univ : Finset V) x≤δ)
+    (hps : ∀ e x, prob (G.targets e) x≤δ) (hpn : ∀ e, (G.targets e).Nonempty)
+    (hrs : ∀ y x, prob (G.retainedSet T y) x≤δ) (hrn : ∀ y, (G.retainedSet T y).Nonempty) :
+    δ^3*localTotalIntegral (fun C => tent δ (cellProb Finset.univ C)) d d'≤
+      2*(fiberCutDiscrepancy G.fiberWeight (fun e => prob (G.targets e)) G.source δ d d'+
+        fiberCutDiscrepancy G.fiberWeight (fun e => prob (G.targets e)) G.source δ d' d)+
+      6*localTotalIntegral (familyCutError G.fiberWeight G.targets δ) d d'+
+      2*localTotalIntegral (familyCutError (G.retainedWeight T) (G.retainedSet T) δ) d d'+
+      2*G.omittedMass T*localTotalIntegral (fun C => tent δ (cellProb Finset.univ C)) d d' := by
+  have h1 := G.graph_length_upper d d' δ T hδ hT hn hG has hps hpn hrs hrn
+  have h2 := G.graph_length_upper d' d δ T hδ hT hn hG has hps hpn hrs hrn
+  simp only [tent_length_occurrence _ hn δ hδ has,
+    family_charge_occurrence _ hn G.fiberWeight G.targets hpn δ hδ hps has,
+    family_charge_occurrence _ hn (G.retainedWeight T) (G.retainedSet T) hrn δ hδ hrs has] at h1 h2
+  dsimp only [localTotalIntegral]
+  linarith
+end WeakPinned.Core.Fibers
+
+namespace WeakPinned.Global
+open Hierarchy Variance Cuts MeasureTheory
+variable {V : Type u14} {I : Type u15} [Fintype V] [Fintype I] [DecidableEq V]
+variable {K : Type u16} [Field K] [NumberField K]
+variable (G : Core.Fibers V I) (Z₁ Z₂ : V → K) (h₁ : Function.Injective Z₁) (h₂ : Function.Injective Z₂)
+
+lemma arithmetic_length_upper (t : I → K) (ht : ∀ e y, y∈G.targets e →
+      (Z₁ (G.source e)-Z₁ y)*(Z₂ (G.source e)-Z₂ y)=t e)
+    (δ T : ℝ) (hδ : 0≤δ) (hT : 0<T) (hn : 2≤Fintype.card V) (hG : 0<G.edgeCount)
+    (has : ∀ x, prob (Finset.univ : Finset V) x≤δ)
+    (hps : ∀ e x, prob (G.targets e) x≤δ) (hpn : ∀ e, (G.targets e).Nonempty)
+    (hrs : ∀ y x, prob (G.retainedSet T y) x≤δ) (hrn : ∀ y, (G.retainedSet T y).Nonempty) :
+    δ^3*weightedIntegral (totalMass Z₁ Z₂ h₁ h₂) (fun C => tent δ (cellProb Finset.univ C))≤
+      48+6*weightedIntegral (totalMass Z₁ Z₂ h₁ h₂) (familyCutError G.fiberWeight G.targets δ)+
+      2*weightedIntegral (totalMass Z₁ Z₂ h₁ h₂) (familyCutError (G.retainedWeight T) (G.retainedSet T) δ)+
+      2*G.omittedMass T*weightedIntegral (totalMass Z₁ Z₂ h₁ h₂) (fun C => tent δ (cellProb Finset.univ C)) := by
+  have hL := integrable_total_integral Z₁ Z₂ h₁ h₂ hn (fun C => tent δ (cellProb Finset.univ C))
+  have hF := integrable_total_integral Z₁ Z₂ h₁ h₂ hn (familyCutError G.fiberWeight G.targets δ)
+  have hR := integrable_total_integral Z₁ Z₂ h₁ h₂ hn (familyCutError (G.retainedWeight T) (G.retainedSet T) δ)
+  have hp := fun e => sum_prob (G.targets e) (hpn e)
+  have hD := fiber_cut_discrepancy_integrable Z₁ Z₂ h₁ h₂ G.fiberWeight (fun e => prob (G.targets e)) G.source hp δ hδ hps
+  have hD' := fiber_cut_discrepancy_integrable Z₂ Z₁ h₂ h₁ G.fiberWeight (fun e => prob (G.targets e)) G.source hp δ hδ hps
+  have hh := average_mono Z₁ Z₂ h₁ h₂ (fun v => (hL v).const_mul (δ^3))
+    (fun v => (((((hD v).add (hD' v)).const_mul 2).add ((hF v).const_mul 6)).add
+      ((hR v).const_mul 2)).add ((hL v).const_mul (2*G.omittedMass T)))
+    (fun d d' => G.two_length_upper d d' δ T hδ hT hn hG has hps hpn hrs hrn)
+  rw [average_add Z₁ Z₂ h₁ h₂
+    (fun v => ((((hD v).add (hD' v)).const_mul 2).add ((hF v).const_mul 6)).add ((hR v).const_mul 2))
+    (fun v => (hL v).const_mul (2*G.omittedMass T))] at hh
+  rw [average_add Z₁ Z₂ h₁ h₂
+    (fun v => (((hD v).add (hD' v)).const_mul 2).add ((hF v).const_mul 6))
+    (fun v => (hR v).const_mul 2)] at hh
+  rw [average_add Z₁ Z₂ h₁ h₂
+    (fun v => ((hD v).add (hD' v)).const_mul 2) (fun v => (hF v).const_mul 6)] at hh
+  simp only [average_mul] at hh
+  have hpx (e) : prob (G.targets e) (G.source e)=0 := by simp [prob,G.no_loop e]
+  have hte (e) (z) (hz : prob (G.targets e) z≠0) :
+      (Z₁ (G.source e)-Z₁ z)*(Z₂ (G.source e)-Z₂ z)=t e := by
+    apply ht
+    by_contra h
+    exact hz (by simp [prob,h])
+  have hd := arithmetic_fiber_cut Z₁ Z₂ h₁ h₂ G.fiberWeight (fun e => prob (G.targets e)) G.source t
+    G.fiberWeight_nonneg (G.sum_fiberWeight hG) (fun e => Core.Fibers.prob_nn (G.targets e)) hp hpx hte δ hδ hps
+  have hd' := arithmetic_fiber_cut Z₂ Z₁ h₂ h₁ G.fiberWeight (fun e => prob (G.targets e)) G.source t
+    G.fiberWeight_nonneg (G.sum_fiberWeight hG) (fun e => Core.Fibers.prob_nn (G.targets e)) hp hpx
+    (fun e z hz => by rw [mul_comm]; exact hte e z hz) δ hδ hps
+  rw [average_swap] at hd'
+  rw [average_add Z₁ Z₂ h₁ h₂ hD (fun v => hD' v)] at hh
+  simp only [average_total_integral Z₁ Z₂ h₁ h₂ hn] at hh
+  linarith
+end WeakPinned.Global
+noncomputable section
+namespace WeakPinned
+open Filter Variance Core.Fibers
+open scoped Topology
+variable {V : Type u17} {I : Type u18} [Fintype V] [Fintype I] [DecidableEq V]
+
+omit [Fintype V] in
+lemma prob_bound_of_size (B : Finset V) (δ : ℝ) (hδ : 0<δ) (hB : 1/δ≤(B.card:ℝ)) :
+    B.Nonempty ∧ ∀ x, prob B x≤δ := by
+  have hb : 0<(B.card:ℝ) := (one_div_pos.mpr hδ).trans_le hB
+  refine ⟨Finset.card_pos.mp (by exact_mod_cast hb),?_⟩
+  intro x
+  unfold prob
+  split_ifs
+  · exact (div_le_iff₀ hb).mpr (by have := (div_le_iff₀ hδ).mp hB; nlinarith)
+  · exact hδ.le
+
+lemma PreparedGraph.omitted_bound {G : Core.Fibers V I} {θ γ c η : ℝ}
+    (hG : PreparedGraph V I G θ γ c η) (hθ : 0<θ) (hη : η≤θ/2) :
+    G.omittedMass ((Fintype.card V:ℝ)^γ) ≤ (4/θ)*(Fintype.card V:ℝ)^(γ-1) := by
+  let n : ℝ := Fintype.card V
+  have hn : 2≤n := by dsimp only [n]; exact_mod_cast hG.card_ge
+  have hn0 : 0<n := by linarith
+  have hE := hG.edgeCount_pos (by linarith)
+  have hEl : (θ/4)*n^2≤G.edgeCount := by
+    have hp : 0≤(n-1)*n := mul_nonneg (by linarith) hn0.le
+    have hh := mul_nonneg (show 0≤θ-η-θ/2 by linarith) hp
+    have hh' := mul_nonneg (show 0≤θ/4 by positivity) (show 0≤n*(n-2) by positivity)
+    dsimp [n] at *
+    nlinarith [hG.lower]
+  apply (G.small_incoming_mass _ (Real.rpow_nonneg hn0.le γ)).trans
+  change n*n^γ/G.edgeCount≤(4/θ)*n^(γ-1)
+  rw [Real.rpow_sub hn0,Real.rpow_one]
+  apply (div_le_iff₀ hE).mpr
+  have hh := mul_le_mul_of_nonneg_left hEl (show 0≤(4/θ)*(n^γ/n) by positivity)
+  have he : (4/θ)*(n^γ/n)*((θ/4)*n^2)=n*n^γ := by field_simp
+  rw [he] at hh
+  exact hh
+
+variable {Vn : ℕ → Type u19} {In : ℕ → Type u20} [∀ j, Fintype (Vn j)] [∀ j, Fintype (In j)] [∀ j, DecidableEq (Vn j)]
+lemma omitted_rate (G : ∀ j, Core.Fibers (Vn j) (In j)) (θ γ c : ℝ) (η : ℕ → ℝ)
+    (hθ : 0<θ) (hγ1 : γ<1) (hη : ∀ j, η j≤θ/2)
+    (hn : Tendsto (fun j => (Fintype.card (Vn j):ℝ)) atTop atTop)
+    (hG : ∀ j, PreparedGraph (Vn j) (In j) (G j) θ γ c (η j)) :
+    Tendsto (fun j => (G j).omittedMass ((Fintype.card (Vn j):ℝ)^γ)) atTop (𝓝 0) := by
+  apply squeeze_zero (fun j => (G j).omittedMass_nonneg _) (fun j => (hG j).omitted_bound hθ (hη j))
+  have hh := (tendsto_rpow_neg_atTop (show 0<1-γ by linarith)).comp hn
+  have he : -(1-γ)=γ-1 := by ring
+  rw [he] at hh
+  simpa using hh.const_mul (4/θ)
+
+lemma sampling_atoms_eventually (G : ∀ j, Core.Fibers (Vn j) (In j)) (θ γ c : ℝ) (η : ℕ → ℝ)
+    (γ0 : 0<γ) (γ1 : γ<1) (δ : ℝ) (hδ : 0<δ)
+    (hn : Tendsto (fun j => (Fintype.card (Vn j):ℝ)) atTop atTop)
+    (hG : ∀ j, PreparedGraph (Vn j) (In j) (G j) θ γ c (η j)) :
+    ∀ᶠ j in atTop,
+      (∀ x, prob (Finset.univ : Finset (Vn j)) x≤δ) ∧
+      (∀ e, ((G j).targets e).Nonempty) ∧ (∀ e x, prob ((G j).targets e) x≤δ) ∧
+      (∀ y, ((G j).retainedSet ((Fintype.card (Vn j):ℝ)^γ) y).Nonempty) ∧
+      (∀ y x, prob ((G j).retainedSet ((Fintype.card (Vn j):ℝ)^γ) y) x≤δ) := by
+  have hpow := (tendsto_rpow_atTop γ0).comp hn
+  filter_upwards [hn.eventually (eventually_ge_atTop (1/δ)),hpow.eventually (eventually_ge_atTop (1/δ))] with j hnδ hnγ
+  have hn1 : (1:ℝ)≤Fintype.card (Vn j) := by exact_mod_cast (show 1≤Fintype.card (Vn j) by have := (hG j).card_ge; omega)
+  have hs : (Fintype.card (Vn j):ℝ)^γ≤Fintype.card (Vn j) := by
+    simpa using Real.rpow_le_rpow_of_exponent_le hn1 γ1.le
+  have hf (e) := prob_bound_of_size ((G j).targets e) δ hδ (hnγ.trans ((hG j).fiber e).le)
+  have hr (y) := prob_bound_of_size ((G j).retainedSet ((Fintype.card (Vn j):ℝ)^γ) y) δ hδ
+    (hnγ.trans ((G j).retainedSet_size _ hs y))
+  exact ⟨(prob_bound_of_size Finset.univ δ hδ (by simpa using hnδ)).2,
+    fun e => (hf e).1,fun e => (hf e).2,fun y => (hr y).1,fun y => (hr y).2⟩
+end WeakPinned
+noncomputable section
+namespace WeakPinned
+open Filter Variance Global Cuts
+open scoped Topology BigOperators
+
+lemma length_rate_contradiction (W F R L o : ℕ → ℝ) (δ b : ℝ) (hδ : 0<δ) (hb : 0<b)
+    (hW : Tendsto W atTop atTop)
+    (hF : Tendsto (fun j => F j/W j) atTop (𝓝 0))
+    (hR : Tendsto (fun j => R j/W j) atTop (𝓝 0))
+    (ho : Tendsto o atTop (𝓝 0))
+    (hlow : ∀ᶠ j in atTop, b*W j≤L j)
+    (hupp : ∀ᶠ j in atTop, δ^3*L j≤48+6*F j+2*R j+2*o j*L j) : False := by
+  have h48 : Tendsto (fun j => (48:ℝ)/W j) atTop (𝓝 0) := by
+    simpa only [div_eq_mul_inv,mul_zero, Function.comp_apply] using (tendsto_inv_atTop_zero.comp hW).const_mul 48
+  have he : Tendsto (fun j => (48+6*F j+2*R j)/W j) atTop (𝓝 0) := by
+    convert (h48.add (hF.const_mul 6)).add (hR.const_mul 2) using 1
+    · ext j; ring
+    · norm_num
+  have hbound := ho.eventually (gt_mem_nhds (show 0<δ^3/4 by positivity))
+  have hsmall := he.eventually (gt_mem_nhds (show 0<δ^3*b/2 by positivity))
+  obtain ⟨j,hjW,hjo,hje,hjl,hju⟩ := (hW.eventually (eventually_gt_atTop (0:ℝ)) |>.and
+    (hbound.and (hsmall.and (hlow.and hupp)))).exists
+  have hl0 : 0<L j := (mul_pos hb hjW).trans_le hjl
+  have hopen := mul_le_mul_of_nonneg_right hjo.le hl0.le
+  have he' := (div_lt_iff₀ hjW).mp hje
+  have hl' := mul_le_mul_of_nonneg_left hjl (show 0≤δ^3/2 by positivity)
+  nlinarith
+
+variable {V : ℕ → Type u21} {I : ℕ → Type u22} {K : ℕ → Type u23} [∀ j, Fintype (V j)] [∀ j, Fintype (I j)] [∀ j, DecidableEq (V j)]
+  [∀ j, Field (K j)] [∀ j, NumberField (K j)]
+variable (G : ∀ j, Core.Fibers (V j) (I j)) (θ γ c : ℝ) (η : ℕ → ℝ)
+  (Z₁ Z₂ : ∀ j, V j → K j) (h₁ : ∀ j, Function.Injective (Z₁ j)) (h₂ : ∀ j, Function.Injective (Z₂ j))
+
+/-- The entire unbounded regime is impossible for the actual prepared
+arithmetic models, allowing arbitrary and varying number fields. -/
+theorem arithmetic_no_unbounded (hθ : 0<θ) (hγ : 0<γ) (hγ1 : γ<1) (hc : 0<c) (hcγ : c<γ)
+    (hn : Tendsto (fun j => (Fintype.card (V j):ℝ)) atTop atTop)
+    (hη : Tendsto η atTop (𝓝 0)) (hη0 : ∀ j, 0≤η j) (hηb : ∀ j, η j≤θ/2)
+    (hG : ∀ j, PreparedGraph (V j) (I j) (G j) θ γ c (η j))
+    (t : ∀ j, I j → K j) (ht : ∀ j e y, y∈(G j).targets e →
+      (Z₁ j ((G j).source e)-Z₁ j y)*(Z₂ j ((G j).source e)-Z₂ j y)=t j e)
+    (hWtop : Tendsto (fun j => overlapScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)) atTop atTop) : False := by
+  obtain ⟨δ,hδ,_,hlow⟩ := arithmetic_balanced_length G θ γ c η Z₁ Z₂ h₁ h₂ hθ hγ hγ1 hcγ hn hη hη0 hηb hG t ht hWtop
+  let W := fun j => overlapScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)
+  let A := fun j => potentialScale (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)
+  let m := fun j => totalMass (Z₁ j) (Z₂ j) (h₁ j) (h₂ j)
+  let F := fun j => weightedIntegral (m j) (familyCutError (G j).fiberWeight (G j).targets δ)
+  let R := fun j => weightedIntegral (m j) (familyCutError
+    ((G j).retainedWeight ((Fintype.card (V j):ℝ)^γ)) ((G j).retainedSet ((Fintype.card (V j):ℝ)^γ)) δ)
+  have hF0 (j) : 0≤F j := Finset.sum_nonneg (fun C _ => mul_nonneg (totalMass_nonneg _ _ _ _ C)
+    (Finset.sum_nonneg (fun e _ => mul_nonneg ((G j).fiberWeight_nonneg e) (cutError_nonneg _ _ _))))
+  have hR0 (j) : 0≤R j := Finset.sum_nonneg (fun C _ => mul_nonneg (totalMass_nonneg _ _ _ _ C)
+    (Finset.sum_nonneg (fun y _ => mul_nonneg ((G j).retainedWeight_nonneg _ y) (cutError_nonneg _ _ _))))
+  have hcomp := arithmetic_scale_comparison G θ γ c η Z₁ Z₂ h₁ h₂ hθ hγ hγ1 hcγ hn hη hη0 hηb hG t ht
+  have hcut := sampling_cut_approximation_rates G θ γ c η Z₁ Z₂ h₁ h₂ δ hδ hθ hγ hγ1 hc hcγ hn hη hη0 hηb hG t ht
+  have hf := change_scale_rate W A F (64+8/θ) (by positivity)
+    (fun j => overlapScale_nonneg _ _ _ _) (fun j => potentialScale_nonneg _ _ _ _) hF0 hWtop hcomp hcut.1
+  have hr := change_scale_rate W A R (64+8/θ) (by positivity)
+    (fun j => overlapScale_nonneg _ _ _ _) (fun j => potentialScale_nonneg _ _ _ _) hR0 hWtop hcomp hcut.2
+  apply length_rate_contradiction W F R (fun j => weightedIntegral (m j) (fun C => tent δ (cellProb Finset.univ C)))
+    (fun j => (G j).omittedMass ((Fintype.card (V j):ℝ)^γ)) δ (δ/4) hδ (by positivity)
+    hWtop hf hr (omitted_rate G θ γ c η hθ hγ1 hηb hn hG) hlow
+  filter_upwards [sampling_atoms_eventually G θ γ c η hγ hγ1 δ hδ hn hG] with j hj
+  exact arithmetic_length_upper (G j) (Z₁ j) (Z₂ j) (h₁ j) (h₂ j) (t j) (ht j) δ
+    ((Fintype.card (V j):ℝ)^γ) hδ.le (Real.rpow_pos_of_pos (by exact_mod_cast (show 0<Fintype.card (V j) by have := (hG j).card_ge; omega)) γ)
+    (hG j).card_ge ((hG j).edgeCount_pos (by linarith [hηb j])) hj.1 hj.2.2.1 hj.2.1 hj.2.2.2.2 hj.2.2.2.1
+end WeakPinned
+
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PinnedDistances/UniformCore.lean b/lean/OAI/Geometry/PinnedDistances/UniformCore.lean
new file mode 100644
index 0000000000000000000000000000000000000000..a86366e2ab16a3293c73a0b9443e894efabf4439
--- /dev/null
+++ b/lean/OAI/Geometry/PinnedDistances/UniformCore.lean
@@ -0,0 +1,551 @@
+import OAI.Geometry.PinnedDistances.ExtremalCore
+
+namespace OAI
+
+universe u1 u2
+
+section
+noncomputable section
+open NumberField
+open scoped BigOperators
+noncomputable section
+open MeasureTheory
+open scoped Topology
+noncomputable section
+open MeasureTheory
+open scoped Topology ENNReal
+noncomputable section
+open MeasureTheory NumberField
+open scoped BigOperators
+noncomputable section
+open scoped BigOperators
+open MeasureTheory
+noncomputable section
+open Filter
+open scoped Topology
+noncomputable section
+open scoped BigOperators
+
+namespace WeakPinned.Core.Fibers
+open scoped BigOperators
+variable {V : Type u1} {I : Type u2} [Fintype V] [Fintype I] [DecidableEq V]
+variable (G : Fibers V I)
+
+/-- Retain a specified family of the original fibers, restricted to D.
+    This never alters a source or creates a new distance coincidence. -/
+def retained (D : Finset V) (Q : I → Prop) [DecidablePred Q] : Fibers V I where
+  source := G.source
+  targets e := if Q e then G.targets e ∩ D else ∅
+  no_loop e := by
+    split_ifs
+    · exact fun h => G.no_loop e (Finset.mem_inter.mp h).1
+    · simp
+  disjoint e f hef hs := by
+    apply Finset.disjoint_left.mpr
+    intro y hy hf
+    by_cases he : Q e <;> by_cases hh : Q f
+    · simp only [he,hh,ite_eq_left] at hy hf
+      exact (Finset.disjoint_left.mp (G.disjoint e f hef hs))
+        (Finset.mem_inter.mp hy).1 (Finset.mem_inter.mp hf).1
+    all_goals simp [he,hh] at hy hf
+
+lemma retained_card (D : Finset V) (Q : I → Prop) [DecidablePred Q] :
+    (((G.retained D Q).edgeSet (-1) D).card : ℝ) =
+      ∑ e, if Q e ∧ G.source e ∈ D then (G.size D e : ℝ) else 0 := by
+  rw [edgeSet_card]
+  apply Finset.sum_congr rfl
+  intro e _
+  have hsize : (G.retained D Q).size D e = if Q e then G.size D e else 0 := by
+    by_cases hq : Q e <;> simp [size,retained,hq]
+  have ha : (G.retained D Q).active (-1) D e ↔ G.source e ∈ D := by
+    change (G.source e ∈ D ∧ (-1:ℝ)<((G.retained D Q).size D e : ℝ)) ↔ _
+    have hh : (-1:ℝ)<((G.retained D Q).size D e : ℝ) := by have hh := Nat.cast_nonneg (α:=ℝ) ((G.retained D Q).size D e); linarith
+    simp [hh]
+  simp only [ha,hsize]
+  by_cases hq : Q e <;> by_cases hs : G.source e ∈ D <;> simp [hq,hs]
+
+end WeakPinned.Core.Fibers
+
+namespace WeakPinned
+open scoped BigOperators
+
+def heavyFiber (P : Finset Plane) (s K lam : ℝ) (C D : Finset P)
+    (e : fiberIndices P s) : Prop :=
+  (planarFibers P s).active K C e ∧
+    lam*((planarFibers P s).size C e : ℝ) ≤ (planarFibers P s).size D e
+
+instance (P : Finset Plane) (s K lam : ℝ) (C D : Finset P) :
+    DecidablePred (heavyFiber P s K lam C D) := Classical.decPred _
+
+lemma heavySubset_edge_rich (P : Finset Plane) (s K lam r : ℝ) (C D : Finset P)
+    (hlam : 0 ≤ lam) (hK : (D.card : ℝ)^r ≤ lam*K) (x y : P)
+    (hxy : ((planarFibers P s).retained D (heavyFiber P s K lam C D)).edge (-1) D x y) :
+    (x.val,y.val) ∈ richPairs (planeSubset P D) r := by
+  obtain ⟨e,ha,hsrc,hy⟩ := hxy
+  have hxD : x ∈ D := hsrc ▸ ha.1
+  have hyD := (Finset.mem_inter.mp hy).2
+  have hyT := (Finset.mem_inter.mp hy).1
+  change y ∈ (if heavyFiber P s K lam C D e then (planarFibers P s).targets e ∩ D else ∅) at hyT
+  split_ifs at hyT with he
+  · obtain ⟨hactive,hheavy⟩ := he
+    have hsz : (D.card : ℝ)^r ≤ ((planarFibers P s).size D e : ℝ) :=
+      hK.trans ((mul_le_mul_of_nonneg_left (le_of_lt hactive.2) hlam).trans hheavy)
+    have hyT' := (Finset.mem_inter.mp hyT).1
+    have hsrc' : e.val.1=x.val := congrArg Subtype.val hsrc
+    have hne : x.val ≠ y.val := by
+      intro hh
+      have hd := (planar_target_iff P s e y).mp hyT'
+      rw [hsrc',hh,dist_self] at hd
+      exact (index_radius_pos P s e).ne hd
+    apply Finset.mem_filter.mpr
+    refine ⟨Finset.mem_product.mpr ⟨Finset.mem_image.mpr ⟨x,hxD,rfl⟩,
+      Finset.mem_image.mpr ⟨y,hyD,rfl⟩⟩,hne,?_⟩
+    rw [planeSubset_card]
+    have hc := Finset.card_le_card (restricted_targets_subset P s D e y hyT')
+    rw [Finset.card_image_of_injective _ Subtype.val_injective,hsrc'] at hc
+    exact hsz.trans (by exact_mod_cast hc)
+  · simp at hyT
+
+/-- The exact finite counting step underlying both uniform subset estimates
+    for the rich-pair graph. Each ordered pair is counted only once. -/
+theorem heavySubset_count (P : Finset Plane) (s K lam r : ℝ) (C D : Finset P)
+    (hlam : 0 ≤ lam) (hK : (D.card : ℝ)^r ≤ lam*K) :
+    (∑ e, if heavyFiber P s K lam C D e ∧ (planarFibers P s).source e ∈ D
+      then ((planarFibers P s).size D e : ℝ) else 0) ≤
+      (richPairs (planeSubset P D) r).card := by
+  rw [← (planarFibers P s).retained_card]
+  suffices h : (((planarFibers P s).retained D (heavyFiber P s K lam C D)).edgeSet (-1) D).card ≤
+      (richPairs (planeSubset P D) r).card by exact_mod_cast h
+  rw [← Finset.card_image_of_injective _ (subtype_pair_injective P)]
+  apply Finset.card_le_card
+  intro z hz
+  obtain ⟨xy,he,rfl⟩ := Finset.mem_image.mp hz
+  exact heavySubset_edge_rich P s K lam r C D hlam hK xy.1 xy.2 (Finset.mem_filter.mp he).2
+
+end WeakPinned
+
+namespace WeakPinned
+open Filter
+open scoped Topology
+
+def Theta (s : ℝ) : ℝ := limsup (fun n : ℕ => F n s) atTop
+
+lemma F_bdd (s : ℝ) : IsBoundedUnder (· ≤ ·) atTop (fun n : ℕ => F n s) :=
+  isBoundedUnder_of_eventually_le (Eventually.of_forall (fun n => F_le_one n s))
+lemma F_cobdd (s : ℝ) : IsCoboundedUnder (· ≤ ·) atTop (fun n : ℕ => F n s) :=
+  isCoboundedUnder_le_of_le atTop (fun n => F_nonneg n s)
+
+lemma Theta_nonneg (s : ℝ) : 0 ≤ Theta s := by
+  have hh := limsup_le_limsup (f:=atTop) (u:=fun _ : ℕ => (0:ℝ)) (v:=fun n => F n s)
+    (Eventually.of_forall (fun n => F_nonneg n s)) (isCoboundedUnder_le_of_le atTop (fun _ => le_refl (0:ℝ))) (F_bdd s)
+  change 0 ≤ limsup (fun n : ℕ => F n s) atTop
+  simpa only [limsup_const] using hh
+
+lemma Theta_le_one (s : ℝ) : Theta s ≤ 1 :=
+  limsup_le_of_le (F_cobdd s) (Eventually.of_forall (fun n => F_le_one n s))
+
+lemma Theta_antitone : Antitone Theta := by
+  intro s t hst
+  exact limsup_le_limsup (Eventually.of_forall (fun n => F_antitone n hst)) (F_cobdd t) (F_bdd s)
+
+lemma Theta_eq_zero_of_one_le {s : ℝ} (hs : 1 ≤ s) : Theta s=0 := by
+  simp [Theta,F_eq_zero_of_one_le _ hs]
+
+lemma tendsto_F_of_Theta_zero {s : ℝ} (hs : Theta s=0) : Tendsto (fun n : ℕ => F n s) atTop (𝓝 0) := by
+  apply tendsto_order.mpr
+  constructor
+  · intro a ha
+    exact Eventually.of_forall (fun n => ha.trans_le (F_nonneg n s))
+  · intro a ha
+    exact eventually_lt_of_limsup_lt (by change Theta s < a; simpa [hs] using ha) (F_bdd s)
+
+def criticalExponent : ℝ := sSup {s : ℝ | 0 < s ∧ 0 < Theta s}
+
+lemma positive_exponents_bdd : BddAbove {s : ℝ | 0 < s ∧ 0 < Theta s} := by
+  refine ⟨1,?_⟩
+  intro s hs
+  by_contra hn
+  have hz := Theta_eq_zero_of_one_le (le_of_lt (lt_of_not_ge hn))
+  linarith [hs.2]
+
+lemma failed_main_nonempty (h : ¬ MainStatement) : {s : ℝ | 0 < s ∧ 0 < Theta s}.Nonempty := by
+  by_contra hn
+  apply h
+  intro s hs
+  apply tendsto_F_of_Theta_zero
+  have hnp : ¬0 < Theta s := fun hp => hn ⟨s,hs,hp⟩
+  exact le_antisymm (le_of_not_gt hnp) (Theta_nonneg s)
+
+lemma critical_pos (h : ¬ MainStatement) : 0 < criticalExponent := by
+  obtain ⟨s,hs,hθ⟩ := failed_main_nonempty h
+  exact hs.trans_le (le_csSup positive_exponents_bdd ⟨hs,hθ⟩)
+
+lemma critical_le_one (h : ¬ MainStatement) : criticalExponent ≤ 1 :=
+  csSup_le (failed_main_nonempty h) (fun s hs => by
+    by_contra hn
+    have hz := Theta_eq_zero_of_one_le (le_of_lt (lt_of_not_ge hn))
+    linarith [hs.2])
+
+lemma Theta_pos_below_critical (h : ¬ MainStatement) {s : ℝ} (hs : s < criticalExponent) : 0 < Theta s := by
+  obtain ⟨t,ht,hst⟩ := exists_lt_of_lt_csSup (failed_main_nonempty h) hs
+  exact ht.2.trans_le (Theta_antitone (le_of_lt hst))
+
+lemma Theta_zero_above_critical (h : ¬ MainStatement) {s : ℝ} (hs : criticalExponent < s) : Theta s=0 := by
+  apply le_antisymm _ (Theta_nonneg s)
+  by_contra hn
+  have hp : 0 < Theta s := lt_of_not_ge hn
+  have hs0 := (critical_pos h).trans hs
+  have hh : s ≤ criticalExponent := le_csSup positive_exponents_bdd ⟨hs0,hp⟩
+  linarith
+
+/-- A fixed exponent regime obtained from failure of the uniform multiplicity limit. -/
+
+theorem extremal_parameters (h : ¬ MainStatement) :
+    ∃ c s γ : ℝ, 0 < c ∧ c < criticalExponent/3 ∧
+      max c ((1-c)*criticalExponent) < s ∧ s < criticalExponent ∧
+      c < γ ∧ γ < s ∧ ContinuousAt Theta s ∧ 0 < Theta s := by
+  let c := criticalExponent/4
+  have hp := critical_pos h
+  have h1 := critical_le_one h
+  have hc0 : 0 < c := by dsimp [c]; linarith
+  have hc3 : c < criticalExponent/3 := by dsimp [c]; linarith
+  have hcm : c < criticalExponent := by dsimp [c]; linarith
+  have hm : (1-c)*criticalExponent < criticalExponent := by nlinarith
+  have hmax : max c ((1-c)*criticalExponent) < criticalExponent := max_lt hcm hm
+  have hd := (Theta_antitone.countable_not_continuousAt).dense_compl ℝ
+  obtain ⟨s,hs,hr⟩ := hd.exists_mem_open isOpen_Ioo (Set.nonempty_Ioo.mpr hmax)
+  have hcs : c < s := (le_max_left _ _).trans_lt hr.1
+  refine ⟨c,s,(c+s)/2,hc0,hc3,hr.1,hr.2,?_,?_,?_,Theta_pos_below_critical h hr.2⟩
+  · linarith
+  · linarith
+  · simpa only [Set.mem_compl_iff,Set.mem_ofPred_eq,not_not] using hs
+
+lemma extremal_subsequence (s : ℝ) :
+    ∃ N : ℕ → ℕ, Tendsto N atTop atTop ∧ Tendsto (fun i => F (N i) s) atTop (𝓝 (Theta s)) := by
+  obtain ⟨N,hN,hT⟩ := exists_seq_tendsto_limsup (F_cobdd s) (F_bdd s)
+  exact ⟨N,hT,hN⟩
+
+/-- The epsilon-uniform extremal tail bound. The lower cutoff may vary arbitrarily
+    provided it tends to infinity. -/
+lemma extremal_tail {s : ℝ} (hs : ContinuousAt Theta s) {r : ℕ → ℝ} {M : ℕ → ℕ}
+    (hr : Tendsto r atTop (𝓝 s)) (hM : Tendsto M atTop atTop) :
+    ∀ ε > 0, ∀ᶠ i in atTop, ∀ m : ℕ, M i ≤ m → F m (r i) ≤ Theta s+ε := by
+  intro ε hε
+  obtain ⟨δ,hδ,hcont⟩ := Metric.continuousAt_iff.mp hs (ε/2) (by linarith)
+  let t := s-δ/2
+  have ht : t < s := by dsimp [t]; linarith
+  have hdist : dist t s < δ := by
+    rw [Real.dist_eq,abs_of_neg (by dsimp [t]; linarith)]
+    dsimp [t]
+    linarith
+  have hΘ : Theta t < Theta s+ε/2 := by
+    have hh := (abs_lt.mp (show |Theta t-Theta s| < ε/2 from by simpa only [Real.dist_eq] using hcont hdist)).2
+    linarith
+  have hev : ∀ᶠ m : ℕ in atTop, F m t < Theta s+ε :=
+    eventually_lt_of_limsup_lt (lt_trans hΘ (by linarith)) (F_bdd t)
+  obtain ⟨m₀,hm₀⟩ := eventually_atTop.mp hev
+  filter_upwards [hr.eventually_const_lt ht,(tendsto_atTop.mp hM) m₀] with i hi hmi
+  intro m hm
+  exact (F_antitone m (le_of_lt hi)).trans (le_of_lt (hm₀ m (hmi.trans hm)))
+
+end WeakPinned
+
+namespace WeakPinned
+open Filter
+open scoped Topology
+
+def cutoff (s : ℝ) (N : ℕ) : ℝ := (N : ℝ)^s/Real.log N
+
+lemma eventual_power_below_cutoff {s r : ℝ} (hrs : r < s) :
+    ∀ᶠ N : ℕ in atTop, (N : ℝ)^r ≤ cutoff s N := by
+  have ho := (isLittleO_log_rpow_atTop (sub_pos.mpr hrs)).bound (show (0:ℝ)<1 by norm_num)
+  have hn : Tendsto (fun N : ℕ => (N : ℝ)) atTop atTop := tendsto_natCast_atTop_atTop
+  filter_upwards [hn.eventually ho,hn.eventually (eventually_gt_atTop 1)] with N hN hp
+  have hlog : 0 < Real.log N := Real.log_pos hp
+  have hpow : 0 ≤ (N : ℝ)^(s-r) := Real.rpow_nonneg (by linarith) _
+  simp only [Real.norm_eq_abs,abs_of_nonneg hlog.le,abs_of_nonneg hpow,one_mul] at hN
+  unfold cutoff
+  apply (le_div_iff₀ hlog).mpr
+  calc
+    (N : ℝ)^r*Real.log N ≤ (N : ℝ)^r*(N : ℝ)^(s-r) :=
+      mul_le_mul_of_nonneg_left hN (Real.rpow_nonneg (by linarith) _)
+    _ = (N : ℝ)^s := by rw [← Real.rpow_add (by linarith : (0:ℝ)<N)]; congr 1; ring
+
+lemma eventual_cutoff_loss (s η : ℝ) (hη : 0 < η) :
+    ∀ᶠ N : ℕ in atTop, 0 ≤ cutoff s N ∧ cutoff s N < (N : ℝ)^s ∧
+      cutoff s N ≤ (η/8)*(N : ℝ)^s := by
+  have hn : Tendsto (fun N : ℕ => (N : ℝ)) atTop atTop := tendsto_natCast_atTop_atTop
+  have hl := Real.tendsto_log_atTop.comp hn
+  filter_upwards [hl.eventually (eventually_gt_atTop (max 1 (8/η))),
+    hn.eventually (eventually_gt_atTop 1)] with N hlog hN
+  have hlog1 : 1 < Real.log N := (le_max_left _ _).trans_lt hlog
+  have hlogη : 8/η < Real.log N := (le_max_right _ _).trans_lt hlog
+  have hp : 0 < (N : ℝ)^s := Real.rpow_pos_of_pos (by linarith) _
+  have hi : (1:ℝ)/Real.log N ≤ η/8 := by
+    apply (div_le_iff₀ (by linarith : 0 < Real.log N)).mpr
+    have hh : 8 < Real.log N*η := (div_lt_iff₀ hη).mp hlogη
+    nlinarith
+  refine ⟨div_nonneg hp.le (by linarith),?_,?_⟩
+  · exact (div_lt_iff₀ (by linarith : 0 < Real.log N)).mpr (by nlinarith)
+  · unfold cutoff
+    calc
+      (N : ℝ)^s/Real.log N = (1/Real.log N)*(N : ℝ)^s := by ring
+      _ ≤ (η/8)*(N : ℝ)^s := mul_le_mul_of_nonneg_right hi hp.le
+
+lemma continuous_left_tail {s γ ε : ℝ} (hγ : γ < s) (hε : 0 < ε)
+    (hs : ContinuousAt Theta s) :
+    ∃ r : ℝ, γ < r ∧ r < s ∧ ∃ m₁ : ℕ, ∀ m : ℕ, m₁ ≤ m → F m r ≤ Theta s+ε := by
+  obtain ⟨δ,hδ,hcont⟩ := Metric.continuousAt_iff.mp hs (ε/2) (by linarith)
+  obtain ⟨r,hrlo,hrhi⟩ := exists_between (show max γ (s-δ) < s by exact max_lt hγ (by linarith))
+  have hγr : γ < r := (le_max_left _ _).trans_lt hrlo
+  have hdist : dist r s < δ := by
+    rw [Real.dist_eq,abs_of_neg (sub_neg.mpr hrhi)]
+    have hh := (le_max_right γ (s-δ)).trans_lt hrlo
+    linarith
+  have ht : Theta r < Theta s+ε := by
+    have hh := (abs_lt.mp (show |Theta r-Theta s| < ε/2 from by simpa only [Real.dist_eq] using hcont hdist)).2
+    linarith
+  obtain ⟨m₁,hm⟩ := eventually_atTop.mp (eventually_lt_of_limsup_lt ht (F_bdd r))
+  exact ⟨r,hγr,hrhi,m₁,fun m hm₁ => (hm m hm₁).le⟩
+
+lemma half_boundary (N : ℕ) (hN : 4 ≤ N) :
+    let m₀ := (N+1)/2
+    2 ≤ m₀ ∧ m₀ < N ∧
+      (m₀ : ℝ)*((m₀ : ℝ)-1) ≤ (N : ℝ)*((N : ℝ)-1)/2 := by
+  dsimp
+  have hm : 2 ≤ (N+1)/2 := by omega
+  have hlt : (N+1)/2 < N := by omega
+  refine ⟨hm,hlt,?_⟩
+  have h2 : 2*((N+1)/2) ≤ N+1 := by omega
+  have h2R : (2:ℝ)*((N+1)/2 : ℕ) ≤ (N : ℝ)+1 := by exact_mod_cast h2
+  have hNR : (4:ℝ) ≤ N := by exact_mod_cast hN
+  have hmR : (2:ℝ) ≤ ((N+1)/2 : ℕ) := by exact_mod_cast hm
+  nlinarith
+
+/-- Actual core configurations at arbitrarily small density error and arbitrarily
+    large cardinality. No field or position assumptions are introduced. -/
+theorem arbitrarily_large_core {s γ : ℝ} (hγ : 0 < γ) (hγs : γ < s)
+    (hs : ContinuousAt Theta s) (η : ℝ) (hη : 0 < η) (M : ℕ) :
+    ∃ (P : Finset Plane) (C : Finset P), M < C.card ∧
+      (P.card+1)/2 < C.card ∧ (C.card : ℝ)^γ ≤ cutoff s P.card ∧
+      (Theta s-η)*(C.card : ℝ)*((C.card : ℝ)-1) ≤ ((planarFibers P s).edgeSet (cutoff s P.card) C).card ∧
+      (((planarFibers P s).edgeSet (cutoff s P.card) C).card : ℝ) ≤
+        (Theta s+η/8)*(C.card : ℝ)*((C.card : ℝ)-1) ∧
+      (∀ x ∈ C, 2*(Theta s-η)*((C.card : ℝ)-1) ≤
+        (planarFibers P s).degree (cutoff s P.card) C x) := by
+  obtain ⟨r,hγr,hrs,m₁,hm₁⟩ := continuous_left_tail hγs (show 0 < η/8 by positivity) hs
+  obtain ⟨N,hN,hF⟩ := extremal_subsequence s
+  have hlarge : ∀ᶠ i in atTop, 4+2*M+2*m₁ ≤ N i := hN.eventually (eventually_ge_atTop _)
+  have happrox : ∀ᶠ i in atTop, Theta s-η/8 ≤ F (N i) s :=
+    (hF.eventually_const_lt (by linarith : Theta s-η/8 < Theta s)).mono (fun _ h => h.le)
+  obtain ⟨i,hi,hcut,hpow,hinit⟩ := (hlarge.and ((hN.eventually (eventual_cutoff_loss s η hη)).and
+    ((hN.eventually (eventual_power_below_cutoff hrs)).and happrox))).exists
+  obtain ⟨P,hP,hPF⟩ := F_attained (N i) (by omega) s
+  have hn : 4 ≤ P.card := by omega
+  obtain ⟨hm0,hm,hbd⟩ := half_boundary P.card hn
+  have hm1 : m₁ ≤ (P.card+1)/2 := by omega
+  have hη8 := hcut.2.2
+  have hpowP : (P.card : ℝ)^r ≤ cutoff s P.card := by simpa [hP] using hpow
+  have hthreshold : ∀ m : ℕ, (P.card+1)/2 ≤ m → m ≤ P.card → (m : ℝ)^r ≤ cutoff s P.card := by
+    intro m _ hmP
+    exact (Real.rpow_le_rpow (Nat.cast_nonneg m) (by exact_mod_cast hmP) (by linarith)).trans hpowP
+  obtain ⟨C,hC,hlo,hhi,hdeg⟩ := finite_extremal_core P s r (cutoff s P.card) (Theta s) η ((P.card+1)/2)
+    (by omega) hm0 hm hη (by simpa [hP] using hcut.1) (by simpa [hP] using hcut.2.1)
+    (by simpa [hP] using hη8) (by simpa [hPF] using hinit) hbd hthreshold
+    (fun m hm _ => hm₁ m (hm1.trans hm))
+  have hCn : C.card ≤ P.card := by simpa using Finset.card_le_univ C
+  have hCpos : (1:ℝ) ≤ C.card := by exact_mod_cast (show 1 ≤ C.card by omega)
+  refine ⟨P,C,by omega,hC,?_,hlo,hhi,hdeg⟩
+  exact (Real.rpow_le_rpow_of_exponent_le hCpos hγr.le).trans
+    (hthreshold C.card hC.le hCn)
+
+end WeakPinned
+
+namespace WeakPinned
+open scoped BigOperators
+
+lemma restricted_size_lt_card (P : Finset Plane) (s : ℝ) (D : Finset P)
+    (e : fiberIndices P s) (hs : (planarFibers P s).source e ∈ D) :
+    (planarFibers P s).size D e < D.card := by
+  have hh : (planarFibers P s).targets e ∩ D ⊆ D.erase ((planarFibers P s).source e) := by
+    intro y hy
+    obtain ⟨hyT,hyD⟩ := Finset.mem_inter.mp hy
+    refine Finset.mem_erase.mpr ⟨?_,hyD⟩
+    intro he
+    subst y
+    exact (planarFibers P s).no_loop e hyT
+  have hc := Finset.card_le_card hh
+  rw [Finset.card_erase_of_mem hs] at hc
+  have hp := Finset.card_pos.mpr ⟨_,hs⟩
+  change (planarFibers P s).size D e ≤ D.card-1 at hc
+  omega
+
+lemma heavySubset_zero (P : Finset Plane) (s K lam : ℝ) (C D : Finset P)
+    (hlam : 0 ≤ lam) (hsmall : (D.card : ℝ) ≤ lam*K) :
+    (∑ e, if heavyFiber P s K lam C D e ∧ (planarFibers P s).source e ∈ D
+      then ((planarFibers P s).size D e : ℝ) else 0) = 0 := by
+  apply Finset.sum_eq_zero
+  intro e _
+  rw [ite_eq_right]
+  rintro ⟨⟨ha,hh⟩,hs⟩
+  have hk := mul_le_mul_of_nonneg_left (le_of_lt ha.2) hlam
+  have hm : ((planarFibers P s).size D e : ℝ) < D.card := by
+    exact_mod_cast restricted_size_lt_card P s D e hs
+  linarith
+
+/-- A uniform extremal bound valid even for tiny subsets: their contribution
+    is zero, rather than an uncontrolled diagonal error. -/
+theorem heavySubset_uniform (P : Finset Plane) (s K lam r B : ℝ) (C D : Finset P)
+    (hlam : 0 ≤ lam) (hB : 0 ≤ B) (m₁ : ℕ) (hm1 : 2 ≤ m₁)
+    (hT : (m₁ : ℝ) ≤ lam*K)
+    (hthreshold : (D.card : ℝ)^r ≤ lam*K)
+    (hext : ∀ m : ℕ, m₁ ≤ m → m ≤ P.card → F m r ≤ B) :
+    (∑ e, if heavyFiber P s K lam C D e ∧ (planarFibers P s).source e ∈ D
+      then ((planarFibers P s).size D e : ℝ) else 0) ≤
+      B*(D.card : ℝ)*((D.card : ℝ)-1) := by
+  by_cases hm : D.card < m₁
+  · have hDm : (D.card : ℝ) ≤ m₁ := by exact_mod_cast hm.le
+    rw [heavySubset_zero P s K lam C D hlam (hDm.trans hT)]
+    by_cases hzero : D.card=0
+    · simp [hzero]
+    · have hh : (1:ℝ) ≤ D.card := by exact_mod_cast (show 1 ≤ D.card by omega)
+      positivity
+  · have hmge : m₁ ≤ D.card := by omega
+    have hDn : D.card ≤ P.card := by simpa using Finset.card_le_univ D
+    have hh : (2:ℝ) ≤ D.card := by exact_mod_cast hm1.trans hmge
+    have hp : 0 < (D.card : ℝ)*((D.card : ℝ)-1) := by nlinarith
+    have hf := fraction_le_F (planeSubset P D) r (by simpa [planeSubset_card] using hm1.trans hmge)
+    rw [pairFraction,planeSubset_card,div_le_iff₀ hp] at hf
+    have hc := heavySubset_count P s K lam r C D hlam hthreshold
+    have he := mul_le_mul_of_nonneg_right (hext D.card hmge hDn) hp.le
+    nlinarith
+
+lemma eventual_power_below_logsquare {s r : ℝ} (hrs : r < s) :
+    ∀ᶠ N : ℕ in Filter.atTop,
+      (N : ℝ)^r ≤ cutoff s N / Real.log N := by
+  have ho := (isLittleO_log_rpow_rpow_atTop (2:ℝ) (sub_pos.mpr hrs)).bound (show (0:ℝ)<1 by norm_num)
+  have hn : Filter.Tendsto (fun N : ℕ => (N : ℝ)) Filter.atTop Filter.atTop := tendsto_natCast_atTop_atTop
+  filter_upwards [hn.eventually ho,hn.eventually (Filter.eventually_gt_atTop 1)] with N hN hp
+  have hlog : 0 < Real.log N := Real.log_pos hp
+  have hpow : 0 ≤ (N : ℝ)^(s-r) := Real.rpow_nonneg (by linarith) _
+  simp only [Real.norm_eq_abs,Real.rpow_two,one_mul,abs_of_nonneg (sq_nonneg (Real.log (N : ℝ))),abs_of_nonneg hpow] at hN
+  unfold cutoff
+  rw [div_div]
+  apply (le_div_iff₀ (mul_pos hlog hlog)).mpr
+  calc
+    (N : ℝ)^r*(Real.log N*Real.log N) ≤ (N : ℝ)^r*(N : ℝ)^(s-r) := by
+      apply mul_le_mul_of_nonneg_left _ (Real.rpow_nonneg (by linarith) _)
+      nlinarith
+    _ = (N : ℝ)^s := by rw [← Real.rpow_add (by linarith : (0:ℝ)<N)]; congr 1; ring
+
+end WeakPinned
+
+namespace WeakPinned
+open Filter
+open scoped Topology BigOperators
+
+def heavyCount (P : Finset Plane) (s K lam : ℝ) (C D : Finset P) : ℝ :=
+  ∑ e, if heavyFiber P s K lam C D e ∧ (planarFibers P s).source e ∈ D
+    then ((planarFibers P s).size D e : ℝ) else 0
+
+/-- Uniform subset bounds for the finite planar core. -/
+structure PreparedCore (s γ c η : ℝ) (P : Finset Plane) (C : Finset P) : Prop where
+  card_two : 2 ≤ C.card
+  half : (P.card+1)/2 < C.card
+  large_fiber : (C.card : ℝ)^γ ≤ cutoff s P.card
+  lower : (Theta s-η)*(C.card : ℝ)*((C.card : ℝ)-1) ≤
+    ((planarFibers P s).edgeSet (cutoff s P.card) C).card
+  upper : (((planarFibers P s).edgeSet (cutoff s P.card) C).card : ℝ) ≤
+    (Theta s+η/8)*(C.card : ℝ)*((C.card : ℝ)-1)
+  degree : ∀ x ∈ C, 2*(Theta s-η)*((C.card : ℝ)-1) ≤
+    (planarFibers P s).degree (cutoff s P.card) C x
+  subset : ∀ D : Finset P,
+    heavyCount P s (cutoff s P.card) (1/Real.log C.card) C D ≤
+      (Theta s+η)*(D.card : ℝ)*((D.card : ℝ)-1)
+  small : ∀ D : Finset P, (D.card : ℝ) ≤ (C.card : ℝ)^(1-c) →
+    heavyCount P s (cutoff s P.card) (1/Real.log C.card) C D ≤
+      η*(D.card : ℝ)*((D.card : ℝ)-1)
+
+lemma choose_above_critical (hfail : ¬ MainStatement) {s c : ℝ}
+    (hc : c < 1) (hsc : (1-c)*criticalExponent < s) :
+    ∃ t : ℝ, criticalExponent < t ∧ (1-c)*t < s ∧ 0 < t := by
+  have hp : 0 < 1-c := by linarith
+  have hb : criticalExponent < s/(1-c) := (lt_div_iff₀ hp).mpr (by nlinarith)
+  obtain ⟨t,ht,hst⟩ := exists_between hb
+  refine ⟨t,ht,?_,(critical_pos hfail).trans ht⟩
+  have hh := (lt_div_iff₀ hp).mp hst
+  nlinarith
+
+lemma log_cutoff_comparison {N n : ℕ} {s : ℝ} (hn : 2 ≤ n) (hnN : n ≤ N) :
+    cutoff s N / Real.log N ≤ (1/Real.log n)*cutoff s N := by
+  have hnr : (1:ℝ)<n := by exact_mod_cast (show 1<n by omega)
+  have hlog := Real.log_pos hnr
+  have hlogs : Real.log n ≤ Real.log N := Real.log_le_log (by linarith) (by exact_mod_cast hnN)
+  calc
+    cutoff s N / Real.log N ≤ cutoff s N / Real.log n := by
+      apply div_le_div_of_nonneg_left _ hlog hlogs
+      exact div_nonneg (Real.rpow_nonneg (Nat.cast_nonneg N) _) (hlog.le.trans hlogs)
+    _ = (1/Real.log n)*cutoff s N := by ring
+
+/-- Uniformity of both subset estimates is obtained before passing to any
+    arithmetic model. Tiny subsets are treated exactly, with no diagonal error. -/
+theorem arbitrarily_large_prepared (hfail : ¬ MainStatement) {s γ c : ℝ}
+    (hγ : 0 < γ) (hγs : γ < s) (_hc : 0 < c) (hc1 : c < 1)
+    (hsc : (1-c)*criticalExponent < s) (hs : ContinuousAt Theta s)
+    (η : ℝ) (hη : 0 < η) (M : ℕ) :
+    ∃ (P : Finset Plane) (C : Finset P), M < C.card ∧ PreparedCore s γ c η P C := by
+  obtain ⟨r,hγr,hrs,m₁,hm₁⟩ := continuous_left_tail hγs hη hs
+  obtain ⟨t,ht,hst,ht0⟩ := choose_above_critical hfail hc1 hsc
+  obtain ⟨m₂,hm₂⟩ := eventually_atTop.mp
+    ((tendsto_F_of_Theta_zero (Theta_zero_above_critical hfail ht)).eventually_lt_const hη)
+  let m := max 2 (max m₁ m₂)
+  have hm2 : 2 ≤ m := le_max_left _ _
+  have hmr : m₁ ≤ m := (le_max_left _ _).trans (le_max_right _ _)
+  have hmt : m₂ ≤ m := (le_max_right _ _).trans (le_max_right _ _)
+  have hr0 : 0 < r := hγ.trans hγr
+  have hbig : ∀ᶠ N : ℕ in atTop, (m : ℝ) ≤ (N : ℝ)^r :=
+    ((tendsto_rpow_atTop hr0).comp tendsto_natCast_atTop_atTop).eventually (eventually_ge_atTop _)
+  obtain ⟨N₀,hN₀⟩ := eventually_atTop.mp
+    (hbig.and ((eventual_power_below_logsquare hrs).and (eventual_power_below_logsquare hst)))
+  obtain ⟨P,C,hC,hhalf,hpow,hlo,hhi,hdeg⟩ :=
+    arbitrarily_large_core hγ hγs hs η hη (max (max M N₀) 2)
+  have hn : 2 ≤ C.card := by omega
+  have hCn : C.card ≤ P.card := by simpa using Finset.card_le_univ C
+  have hg : N₀ ≤ P.card := by omega
+  obtain ⟨hbigP,hpr,hpt⟩ := hN₀ P.card hg
+  have hcomp := log_cutoff_comparison (s:=s) hn hCn
+  have hlam : 0 ≤ 1/Real.log (C.card : ℝ) := by
+    have hh := Real.log_pos (show (1:ℝ)<C.card by exact_mod_cast (show 1<C.card by omega))
+    positivity
+  have hT : (m : ℝ) ≤ (1/Real.log (C.card : ℝ))*cutoff s P.card := hbigP.trans (hpr.trans hcomp)
+  refine ⟨P,C,by omega,hn,hhalf,hpow,hlo,hhi,hdeg,?_,?_⟩
+  · intro D
+    have hDn : D.card ≤ P.card := by simpa using Finset.card_le_univ D
+    have hDP : (D.card : ℝ)^r ≤ (P.card : ℝ)^r :=
+      Real.rpow_le_rpow (Nat.cast_nonneg _) (by exact_mod_cast hDn) hr0.le
+    exact heavySubset_uniform P s (cutoff s P.card) (1/Real.log C.card) r (Theta s+η) C D hlam
+      (by linarith [Theta_nonneg s]) m hm2 hT (hDP.trans (hpr.trans hcomp))
+      (fun j hj _ => hm₁ j (hmr.trans hj))
+  · intro D hD
+    have hDP : (D.card : ℝ)^t ≤ (P.card : ℝ)^((1-c)*t) := by
+      calc
+        (D.card : ℝ)^t ≤ ((C.card : ℝ)^(1-c))^t :=
+          Real.rpow_le_rpow (Nat.cast_nonneg _) hD ht0.le
+        _ = (C.card : ℝ)^((1-c)*t) := (Real.rpow_mul (Nat.cast_nonneg _) _ _).symm
+        _ ≤ (P.card : ℝ)^((1-c)*t) :=
+          Real.rpow_le_rpow (Nat.cast_nonneg _) (by exact_mod_cast hCn) (by positivity)
+    exact heavySubset_uniform P s (cutoff s P.card) (1/Real.log C.card) t η C D hlam hη.le
+      m hm2 hT (hDP.trans (hpt.trans hcomp)) (fun j hj _ => (hm₂ j (hmt.trans hj)).le)
+
+end WeakPinned
+
+end
+end
+end
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PinnedDistances/Variance.lean b/lean/OAI/Geometry/PinnedDistances/Variance.lean
new file mode 100644
index 0000000000000000000000000000000000000000..078e2a436ca7820dfcf1b23cdc19cd15c828f9ee
--- /dev/null
+++ b/lean/OAI/Geometry/PinnedDistances/Variance.lean
@@ -0,0 +1,741 @@
+import OAI.Geometry.PinnedDistances.ArithmeticHierarchy
+
+namespace OAI
+
+universe u1 u2 u3 u4 u5
+
+section
+noncomputable section
+open NumberField
+open scoped BigOperators
+noncomputable section
+open MeasureTheory
+open scoped Topology
+noncomputable section
+open MeasureTheory
+open scoped Topology ENNReal
+noncomputable section
+open MeasureTheory NumberField
+open scoped BigOperators
+
+/- Finite off-diagonal variance. All diagonal terms are identically zero,
+   not O(1/n): singleton cells may persist forever. -/
+noncomputable section
+open scoped BigOperators
+open MeasureTheory
+namespace WeakPinned.Variance
+
+variable {P : Type u1} [Fintype P] [DecidableEq P]
+
+def prob (B : Finset P) (x : P) : ℝ := if x ∈ B then 1 / B.card else 0
+
+def pairProb (B : Finset P) (x y : P) : ℝ :=
+  if x ∈ B ∧ y ∈ B ∧ x ≠ y then 1 / ((B.card : ℝ) * (B.card - 1)) else 0
+
+/-- Mixed cell probability, including B=P without an undefined outside branch.
+    The outside column formula is never used when B=P. -/
+def mixedProb (B : Finset P) (x y : P) : ℝ :=
+  if x ∈ B ∧ x ≠ y then
+    if y ∈ B then 1 / ((Fintype.card P : ℝ) * (B.card - 1))
+    else 1 / ((Fintype.card P : ℝ) * B.card)
+  else 0
+
+lemma sum_prob (B : Finset P) (hB : B.Nonempty) : ∑ x, prob B x = 1 := by
+  classical
+  rw [show (∑ x, prob B x) = ∑ x ∈ B, 1 / (B.card : ℝ) by simp [prob]]
+  simp [hB.card_ne_zero]
+
+lemma sum_pairProb (B : Finset P) (hB : 2 ≤ B.card) (x : P) :
+    ∑ y, pairProb B x y = prob B x := by
+  classical
+  by_cases hx : x ∈ B
+  · have hk : (B.card : ℝ) ≠ 0 := by exact_mod_cast (by omega : B.card ≠ 0)
+    have hk1 : (B.card : ℝ) - 1 ≠ 0 := by
+      have : (2 : ℝ) ≤ B.card := by exact_mod_cast hB
+      linarith
+    have he : ∀ y, pairProb B x y = if y ∈ B.erase x then
+        1 / ((B.card : ℝ) * (B.card - 1)) else 0 := by
+      intro y
+      simp only [pairProb, hx, true_and, Finset.mem_erase, ne_comm (a := y)]
+      simp only [and_comm]
+    simp_rw [he]
+    simp only [Finset.sum_ite_mem, Finset.univ_inter, Finset.sum_const,
+      nsmul_eq_mul, Finset.card_erase_of_mem hx]
+    rw [Nat.cast_sub (by omega)]
+    simp only [Nat.cast_one, prob, hx, reduceIte]
+    field_simp
+  · simp [pairProb, prob, hx]
+
+omit [Fintype P] in
+lemma pairProb_symm (B : Finset P) (x y : P) : pairProb B x y = pairProb B y x := by
+  simp only [pairProb, and_left_comm, ne_comm]
+
+omit [Fintype P] in
+lemma pairProb_diag (B : Finset P) (x : P) : pairProb B x x = 0 := by
+  simp [pairProb]
+
+omit [DecidableEq P] in
+lemma ambient_card_ge_two (B : Finset P) (hB : 2 ≤ B.card) :
+    2 ≤ Fintype.card P := hB.trans (Finset.card_le_univ B)
+
+lemma mixedProb_decompose (B : Finset P) (hB : 2 ≤ B.card) (x y : P) :
+    mixedProb B x y = (B.card : ℝ) / Fintype.card P * pairProb B x y +
+      prob B x / Fintype.card P * (if y ∈ B then 0 else 1) := by
+  have hk : (B.card : ℝ) ≠ 0 := by exact_mod_cast (by omega : B.card ≠ 0)
+  have hn : (Fintype.card P : ℝ) ≠ 0 := by
+    exact_mod_cast (by have := ambient_card_ge_two B hB; omega : Fintype.card P ≠ 0)
+  have hk1 : (B.card : ℝ) - 1 ≠ 0 := by
+    have : (2 : ℝ) ≤ B.card := by exact_mod_cast hB
+    linarith
+  by_cases hx : x ∈ B <;> by_cases hy : y ∈ B <;> by_cases hxy : x = y
+  all_goals simp_all [mixedProb, pairProb, prob]
+  all_goals field_simp
+
+lemma sum_mixedProb (B : Finset P) (hB : 2 ≤ B.card) (x : P) :
+    ∑ y, mixedProb B x y = prob B x := by
+  have hn : (Fintype.card P : ℝ) ≠ 0 := by
+    exact_mod_cast (by have := ambient_card_ge_two B hB; omega : Fintype.card P ≠ 0)
+  have hout : (∑ y : P, (if y ∈ B then (0 : ℝ) else 1)) =
+      Fintype.card P - (B.card : ℝ) := by
+    have he : ∀ y : P, (if y ∈ B then (0 : ℝ) else 1) =
+        if y ∈ Bᶜ then 1 else 0 := by intro y; simp
+    simp_rw [he]
+    simp only [Finset.sum_ite_mem, Finset.univ_inter, Finset.sum_const,
+      nsmul_eq_mul, mul_one, Finset.card_compl,
+      Nat.cast_sub (Finset.card_le_univ B)]
+  simp_rw [mixedProb_decompose B hB, Finset.sum_add_distrib,
+    ← Finset.mul_sum, sum_pairProb B hB, hout]
+  field_simp
+  ring
+
+lemma sum_mixedProb_left (B : Finset P) (hB : 2 ≤ B.card) (y : P) :
+    ∑ x, mixedProb B x y = 1 / Fintype.card P := by
+  have hk : (B.card : ℝ) ≠ 0 := by exact_mod_cast (by omega : B.card ≠ 0)
+  have hne : B.Nonempty := Finset.card_pos.mp (by omega)
+  simp_rw [mixedProb_decompose B hB, Finset.sum_add_distrib,
+    ← Finset.mul_sum, pairProb_symm B _ y, sum_pairProb B hB,
+    ← Finset.sum_mul, ← Finset.sum_div, sum_prob B hne]
+  by_cases hy : y ∈ B <;> simp [prob, hy]
+  field_simp
+
+lemma mixedProb_diag (B : Finset P) (x : P) : mixedProb B x x = 0 := by
+  simp [mixedProb]
+
+def signedPair (B : Finset P) (x y : P) : ℝ :=
+  pairProb B x y - 2 * mixedProb B x y + pairProb Finset.univ x y
+
+lemma signedPair_diag (B : Finset P) (x : P) : signedPair B x x = 0 := by
+  simp [signedPair, pairProb_diag, mixedProb_diag]
+
+lemma signedPair_row (B : Finset P) (hB : 2 ≤ B.card) (x : P) :
+    (∑ y, signedPair B x y) = 1 / Fintype.card P - prob B x := by
+  have hP : 2 ≤ (Finset.univ : Finset P).card := ambient_card_ge_two B hB
+  simp_rw [signedPair, Finset.sum_add_distrib, Finset.sum_sub_distrib,
+    ← Finset.mul_sum, sum_pairProb B hB, sum_mixedProb B hB,
+    sum_pairProb Finset.univ hP]
+  simp only [prob, Finset.mem_univ, reduceIte, Finset.card_univ]
+  ring
+
+lemma signedPair_col (B : Finset P) (hB : 2 ≤ B.card) (y : P) :
+    (∑ x, signedPair B x y) = prob B y - 1 / Fintype.card P := by
+  have hP : 2 ≤ (Finset.univ : Finset P).card := ambient_card_ge_two B hB
+  simp_rw [signedPair, Finset.sum_add_distrib, Finset.sum_sub_distrib,
+    ← Finset.mul_sum, sum_mixedProb_left B hB, pairProb_symm B _ y,
+    pairProb_symm Finset.univ _ y, sum_pairProb B hB,
+    sum_pairProb Finset.univ hP]
+  simp only [prob, Finset.mem_univ, reduceIte, Finset.card_univ]
+  ring
+
+/-- Exact cancellation against every additive off-diagonal overlap kernel.
+    Only the sum of the two marginals is constrained. -/
+lemma additive_cancel (H L : P → P → ℝ) (b : ℝ) (S : P → ℝ)
+    (hdiag : ∀ x, H x x = 0)
+    (hmass : (∑ x, ∑ y, H x y) = 0)
+    (hmarg : ∀ x, (∑ y, H x y) + ∑ y, H y x = 0)
+    (hL : ∀ x y, x ≠ y → L x y = b + S x + S y) :
+    (∑ x, ∑ y, H x y * L x y) = 0 := by
+  have hterm (x y : P) : H x y * L x y = H x y * (b + S x + S y) := by
+    by_cases h : x = y
+    · subst y; simp [hdiag]
+    · rw [hL x y h]
+  simp_rw [hterm, mul_add, Finset.sum_add_distrib]
+  have hb : (∑ x, ∑ y, H x y * b) = (∑ x, ∑ y, H x y) * b := by
+    simp_rw [Finset.sum_mul]
+  rw [hb, hmass, zero_mul, zero_add]
+  have h₁ : (∑ x, ∑ y, H x y * S x) = ∑ x, (∑ y, H x y) * S x := by
+    simp_rw [Finset.sum_mul]
+  have h₂ : (∑ x, ∑ y, H x y * S y) = ∑ x, (∑ y, H y x) * S x := by
+    rw [Finset.sum_comm]
+    simp_rw [Finset.sum_mul]
+  rw [h₁, h₂, ← Finset.sum_add_distrib]
+  apply Finset.sum_eq_zero
+  intro x _
+  rw [← add_mul, hmarg, zero_mul]
+
+lemma variance_cancel (B : Finset P) (hB : 2 ≤ B.card)
+    (L : P → P → ℝ) (b : ℝ) (S : P → ℝ)
+    (hL : ∀ x y, x ≠ y → L x y = b + S x + S y) :
+    (∑ x, ∑ y, signedPair B x y * L x y) = 0 := by
+  apply additive_cancel _ _ b S (signedPair_diag B) _ _ hL
+  · simp_rw [signedPair_row B hB, Finset.sum_sub_distrib]
+    have hn : (Fintype.card P : ℝ) ≠ 0 := by
+      exact_mod_cast (by have := ambient_card_ge_two B hB; omega : Fintype.card P ≠ 0)
+    rw [sum_prob B (Finset.card_pos.mp (by omega))]
+    simp [hn]
+  · intro x
+    rw [signedPair_row B hB, signedPair_col B hB]
+    ring
+
+/-- Exact cell probabilities, with the diagonal excluded. -/
+def pairCell (B C : Finset P) : ℝ := ∑ x ∈ C, ∑ y ∈ C, pairProb B x y
+
+def mixedCell (B C : Finset P) : ℝ := ∑ x ∈ C, ∑ y ∈ C, mixedProb B x y
+
+def cellProb (B C : Finset P) : ℝ := ((B ∩ C).card : ℝ) / B.card
+
+omit [Fintype P] in
+lemma sum_prob_cell (B C : Finset P) : ∑ x ∈ C, prob B x = cellProb B C := by
+  simp only [prob, Finset.sum_ite_mem, Finset.sum_const, nsmul_eq_mul, mul_one_div,
+    cellProb, Finset.inter_comm]
+
+omit [Fintype P] in
+lemma pairCell_formula (B C : Finset P) :
+    pairCell B C = ((B ∩ C).card : ℝ) * ((B ∩ C).card - 1) /
+      ((B.card : ℝ) * (B.card - 1)) := by
+  classical
+  unfold pairCell pairProb
+  rw [← Finset.sum_product C C (fun xy : P × P =>
+    if xy.1 ∈ B ∧ xy.2 ∈ B ∧ xy.1 ≠ xy.2 then
+      1 / ((B.card : ℝ) * (B.card - 1)) else 0), ← Finset.sum_filter]
+  have hf : (C ×ˢ C).filter (fun xy => xy.1 ∈ B ∧ xy.2 ∈ B ∧ xy.1 ≠ xy.2) =
+      (B ∩ C).offDiag := by
+    ext ⟨x,y⟩
+    simp only [Finset.mem_filter, Finset.mem_product, Finset.mem_offDiag, Finset.mem_inter]
+    tauto
+  rw [hf, Finset.sum_const, nsmul_eq_mul, Finset.offDiag_card, ← Nat.mul_sub_one]
+  by_cases hb : (B ∩ C).card = 0
+  · simp [hb]
+  · rw [Nat.cast_mul, Nat.cast_sub (by omega), Nat.cast_one]
+    ring
+
+omit [Fintype P] in
+lemma sum_outside_cell (B C : Finset P) :
+    (∑ y ∈ C, if y ∈ B then (0 : ℝ) else 1) =
+      (C.card : ℝ) - (B ∩ C).card := by
+  have he : ∀ y, (if y ∈ B then (0 : ℝ) else 1) =
+      if y ∉ B then 1 else 0 := by intro y; split_ifs <;> rfl
+  simp_rw [he]
+  rw [← Finset.sum_filter]
+  rw [show C.filter (fun y => y ∉ B) = C \ B by ext y; simp]
+  simp only [Finset.sum_const, nsmul_eq_mul, mul_one, Finset.card_sdiff,
+    Nat.cast_sub (Finset.card_le_card Finset.inter_subset_right)]
+
+lemma mixedCell_formula (B C : Finset P) (hB : 2 ≤ B.card) :
+    mixedCell B C = (B.card : ℝ) / Fintype.card P * pairCell B C +
+      cellProb B C / Fintype.card P * ((C.card : ℝ) - (B ∩ C).card) := by
+  unfold mixedCell
+  simp_rw [mixedProb_decompose B hB, Finset.sum_add_distrib, ← Finset.mul_sum,
+    sum_outside_cell B C, ← Finset.sum_mul, ← Finset.sum_div, sum_prob_cell B C]
+  rfl
+
+omit [Fintype P] in
+lemma cellProb_nonneg (B C : Finset P) : 0 ≤ cellProb B C := by
+  unfold cellProb
+  positivity
+
+omit [Fintype P] in
+lemma cellProb_le_one (B C : Finset P) : cellProb B C ≤ 1 := by
+  unfold cellProb
+  apply div_le_one_of_le₀
+  · exact_mod_cast Finset.card_le_card (Finset.inter_subset_left (s₂ := C))
+  · positivity
+
+omit [Fintype P] in
+lemma pairCell_nonneg (B C : Finset P) : 0 ≤ pairCell B C := by
+  unfold pairCell pairProb
+  apply Finset.sum_nonneg
+  intro x _
+  apply Finset.sum_nonneg
+  intro y _
+  split_ifs with h
+  · have : 2 ≤ B.card := Finset.one_lt_card.mpr ⟨x,h.1,y,h.2.1,h.2.2⟩
+    have : (2 : ℝ) ≤ B.card := by exact_mod_cast this
+    have : 0 ≤ (B.card : ℝ) - 1 := by linarith
+    positivity
+  · rfl
+
+private lemma pair_fraction (k b : ℝ) (hk : k ≠ 0) (hk1 : k - 1 ≠ 0) :
+    b * (b-1) / (k*(k-1)) = (b/k)^2 - (b/k)*(1-b/k)/(k-1) := by
+  field_simp
+  ring
+
+omit [Fintype P] in
+lemma pairCell_centered (B C : Finset P) (hB : 2 ≤ B.card) :
+    pairCell B C = (cellProb B C)^2 - cellProb B C * (1-cellProb B C) /
+      ((B.card : ℝ) - 1) := by
+  have hk : (B.card : ℝ) ≠ 0 := by exact_mod_cast (by omega : B.card ≠ 0)
+  have hk1 : (B.card : ℝ) - 1 ≠ 0 := by
+    have : (2 : ℝ) ≤ B.card := by exact_mod_cast hB
+    linarith
+  rw [pairCell_formula, cellProb, pair_fraction _ _ hk hk1]
+
+lemma mixedCell_centered (B C : Finset P) (hB : 2 ≤ B.card) :
+    mixedCell B C = cellProb Finset.univ C * cellProb B C -
+      (B.card : ℝ) / Fintype.card P *
+        (cellProb B C * (1-cellProb B C) / ((B.card : ℝ)-1)) := by
+  have hk : (B.card : ℝ) ≠ 0 := by exact_mod_cast (by omega : B.card ≠ 0)
+  rw [mixedCell_formula B C hB, pairCell_centered B C hB]
+  simp only [cellProb, Finset.univ_inter, Finset.card_univ]
+  field_simp
+  ring
+
+lemma mixedCell_le_product (B C : Finset P) (hB : 2 ≤ B.card) :
+    mixedCell B C ≤ cellProb Finset.univ C * cellProb B C := by
+  rw [mixedCell_centered B C hB]
+  have hp := cellProb_nonneg B C
+  have hp1 := cellProb_le_one B C
+  have hk : (2 : ℝ) ≤ B.card := by exact_mod_cast hB
+  have hk0 : 0 ≤ (B.card : ℝ) - 1 := by linarith
+  have hp10 : 0 ≤ 1-cellProb B C := sub_nonneg.mpr hp1
+  have hh : 0 ≤ (B.card : ℝ) / Fintype.card P *
+      (cellProb B C * (1-cellProb B C) / ((B.card : ℝ)-1)) := by positivity
+  linarith
+
+lemma signedCell_formula (B C : Finset P) (hB : 2 ≤ B.card) :
+    (∑ x ∈ C, ∑ y ∈ C, signedPair B x y) =
+      (cellProb B C - cellProb Finset.univ C)^2 +
+      (2 * (B.card : ℝ) / Fintype.card P - 1) *
+        (cellProb B C * (1-cellProb B C) / ((B.card : ℝ)-1)) -
+      cellProb Finset.univ C * (1-cellProb Finset.univ C) /
+        ((Fintype.card P : ℝ)-1) := by
+  have hP : 2 ≤ (Finset.univ : Finset P).card := ambient_card_ge_two B hB
+  simp_rw [signedPair, Finset.sum_add_distrib, Finset.sum_sub_distrib, ← Finset.mul_sum]
+  change pairCell B C - 2 * mixedCell B C + pairCell Finset.univ C = _
+  rw [pairCell_centered B C hB, mixedCell_centered B C hB,
+    pairCell_centered Finset.univ C hP]
+  simp only [Finset.card_univ]
+  ring
+
+private lemma pair_fraction_bounds (n m : ℝ) (hm : 2 ≤ m) (hmn : m ≤ n) :
+    (1/2) * (m/n)^2 ≤ m*(m-1)/(n*(n-1)) ∧
+      m*(m-1)/(n*(n-1)) ≤ (m/n)^2 := by
+  have hn : 0 < n := by linarith
+  have hn1 : 0 < n-1 := by linarith
+  have hmn0 : 0 ≤ m/n := div_nonneg (by linarith) hn.le
+  have hlow : m/(2*n) ≤ (m-1)/(n-1) := by
+    apply (div_le_div_iff₀ (by positivity) hn1).mpr
+    nlinarith [mul_nonneg hn.le (show 0 ≤ m-2 by linarith)]
+  have hupp : (m-1)/(n-1) ≤ m/n := by
+    apply (div_le_div_iff₀ hn1 hn).mpr
+    nlinarith
+  have he : m*(m-1)/(n*(n-1)) = (m/n)*((m-1)/(n-1)) := by
+    field_simp
+  rw [he]
+  constructor
+  · calc
+      (1/2) * (m/n)^2 = (m/n)*(m/(2*n)) := by ring
+      _ ≤ _ := mul_le_mul_of_nonneg_left hlow hmn0
+  · simpa only [sq] using mul_le_mul_of_nonneg_left hupp hmn0
+
+lemma uniform_pair_comparison (C : Finset P) (hC : 2 ≤ C.card) :
+    (1/2) * (cellProb Finset.univ C)^2 ≤ pairCell Finset.univ C ∧
+      pairCell Finset.univ C ≤ (cellProb Finset.univ C)^2 := by
+  rw [pairCell_formula]
+  simp only [cellProb, Finset.univ_inter, Finset.card_univ]
+  apply pair_fraction_bounds
+  · exact_mod_cast hC
+  · exact_mod_cast Finset.card_le_univ C
+
+omit [Fintype P] in
+lemma fiber_pair_lower (B C : Finset P) (hB : 2 ≤ B.card) :
+    (1/2) * (if 2 / (B.card : ℝ) ≤ cellProb B C then (cellProb B C)^2 else 0) ≤
+      pairCell B C := by
+  split_ifs with hp
+  · have hk : 0 < (B.card : ℝ) := by exact_mod_cast (by omega : 0 < B.card)
+    have hb : (2 : ℝ) ≤ (B ∩ C).card :=
+      (div_le_div_iff_of_pos_right hk).mp hp
+    rw [pairCell_formula, cellProb]
+    exact (pair_fraction_bounds _ _ hb
+      (by exact_mod_cast Finset.card_le_card (Finset.inter_subset_left (s₂ := C)))).1
+  · simpa only [mul_zero] using pairCell_nonneg B C
+
+omit [Fintype P] in
+lemma singleton_pairCell (B C : Finset P) (hC : C.card ≤ 1) : pairCell B C = 0 := by
+  unfold pairCell
+  apply Finset.sum_eq_zero
+  intro x hx
+  apply Finset.sum_eq_zero
+  intro y hy
+  have : x = y := Finset.card_le_one.mp hC x hx y hy
+  subst y
+  exact pairProb_diag B x
+
+lemma singleton_mixedCell (B C : Finset P) (hC : C.card ≤ 1) : mixedCell B C = 0 := by
+  unfold mixedCell
+  apply Finset.sum_eq_zero
+  intro x hx
+  apply Finset.sum_eq_zero
+  intro y hy
+  have : x = y := Finset.card_le_one.mp hC x hx y hy
+  subst y
+  exact mixedProb_diag B x
+
+omit [Fintype P] in
+lemma singleton_fiber_term (B C : Finset P) (hB : 2 ≤ B.card) (hC : C.card ≤ 1) :
+    (if 2 / (B.card : ℝ) ≤ cellProb B C then (cellProb B C)^2 else 0) = 0 := by
+  have hk : 0 < (B.card : ℝ) := by exact_mod_cast (by omega : 0 < B.card)
+  have hi : (B ∩ C).card ≤ 1 := (Finset.card_le_card Finset.inter_subset_right).trans hC
+  have hiR : ((B ∩ C).card : ℝ) < 2 := by exact_mod_cast (by omega : (B ∩ C).card < 2)
+  have hp : cellProb B C < 2 / (B.card : ℝ) :=
+    (div_lt_div_iff_of_pos_right hk).mpr hiR
+  simp only [not_le.mpr hp, reduceIte]
+
+def signedCell (B C : Finset P) : ℝ := pairCell B C - 2 * mixedCell B C +
+    pairCell Finset.univ C
+
+lemma signedCell_eq (B C : Finset P) :
+    signedCell B C = ∑ x ∈ C, ∑ y ∈ C, signedPair B x y := by
+  simp only [signedCell, pairCell, mixedCell, signedPair,
+    Finset.sum_add_distrib, Finset.sum_sub_distrib, Finset.mul_sum]
+
+lemma singleton_signedCell (B C : Finset P) (hC : C.card ≤ 1) : signedCell B C = 0 := by
+  simp [signedCell, singleton_pairCell _ _ hC, singleton_mixedCell _ _ hC]
+
+private lemma correction_bound (p k ε : ℝ) (hp : 0 ≤ p) (hp1 : p ≤ 1)
+    (hk : 0 < k-1) (he : 1/(k-1) ≤ ε) :
+    0 ≤ p*(1-p)/(k-1) ∧ p*(1-p)/(k-1) ≤ ε*p := by
+  have hε : 0 ≤ ε := (by positivity : 0 ≤ 1/(k-1)).trans he
+  constructor
+  · exact div_nonneg (mul_nonneg hp (sub_nonneg.mpr hp1)) hk.le
+  · calc
+      p*(1-p)/(k-1) ≤ p/(k-1) := by
+        apply div_le_div_of_nonneg_right _ hk.le
+        nlinarith [sq_nonneg p]
+      _ = p * (1/(k-1)) := by ring
+      _ ≤ p*ε := mul_le_mul_of_nonneg_left he hp
+      _ = _ := mul_comm _ _
+
+/-- The finite large-cell estimate before converting a to pair mass. -/
+lemma signedCell_lower (B C : Finset P) (hB : 2 ≤ B.card) (ε : ℝ)
+    (hk : 1 / ((B.card : ℝ)-1) ≤ ε)
+    (hn : 1 / ((Fintype.card P : ℝ)-1) ≤ ε) :
+    (cellProb B C - cellProb Finset.univ C)^2 -
+      ε * (cellProb B C + cellProb Finset.univ C) ≤ signedCell B C := by
+  have hkr : (2 : ℝ) ≤ B.card := by exact_mod_cast hB
+  have hnr : (2 : ℝ) ≤ Fintype.card P := by exact_mod_cast ambient_card_ge_two B hB
+  have hp := correction_bound (cellProb B C) (B.card : ℝ) ε
+    (cellProb_nonneg B C) (cellProb_le_one B C) (by linarith) hk
+  have ha := correction_bound (cellProb Finset.univ C) (Fintype.card P : ℝ) ε
+    (cellProb_nonneg _ C) (cellProb_le_one _ C) (by linarith) hn
+  rw [signedCell_eq, signedCell_formula B C hB]
+  have hc : 0 ≤ 2 * (B.card : ℝ) / Fintype.card P := by positivity
+  nlinarith [mul_nonneg hc hp.1]
+
+section Family
+variable {I : Type u2} [Fintype I]
+variable (w : I → ℝ) (B : I → Finset P)
+
+lemma mean_cell_bound (C : Finset P) (M : ℝ)
+    (hdom : ∀ x, (∑ i, w i * prob (B i) x) ≤ M * prob Finset.univ x) :
+    (∑ i, w i * cellProb (B i) C) ≤ M * cellProb Finset.univ C := by
+  simp_rw [← sum_prob_cell, Finset.mul_sum]
+  rw [Finset.sum_comm]
+  exact Finset.sum_le_sum fun x _ => hdom x
+
+lemma family_large_raw (C : Finset P) (M ε : ℝ)
+    (hw : ∀ i, 0 ≤ w i) (hW : (∑ i, w i) ≤ 1) (hε : 0 ≤ ε)
+    (hB : ∀ i, 2 ≤ (B i).card)
+    (hk : ∀ i, 1 / (((B i).card : ℝ)-1) ≤ ε)
+    (hn : 1 / ((Fintype.card P : ℝ)-1) ≤ ε)
+    (hmean : (∑ i, w i * cellProb (B i) C) ≤ M * cellProb Finset.univ C) :
+    (∑ i, w i * (cellProb (B i) C - cellProb Finset.univ C)^2) -
+      ε*(M+1)*cellProb Finset.univ C ≤ (∑ i, w i * signedCell (B i) C) := by
+  let a := cellProb Finset.univ C
+  have ha : 0 ≤ a := cellProb_nonneg _ _
+  have hc : (∑ i, w i * cellProb (B i) C) + (∑ i, w i)*a ≤ (M+1)*a := by
+    have := mul_le_mul_of_nonneg_right hW ha
+    dsimp [a] at *
+    linarith
+  calc
+    _ ≤ (∑ i, w i * (cellProb (B i) C - a)^2) -
+        ε*((∑ i, w i * cellProb (B i) C)+(∑ i, w i)*a) := by
+      have := mul_le_mul_of_nonneg_left hc hε
+      dsimp [a] at *
+      nlinarith
+    _ = ∑ i, w i * ((cellProb (B i) C - a)^2 - ε*(cellProb (B i) C+a)) := by
+      simp only [mul_add, Finset.mul_sum, Finset.sum_mul]
+      rw [sub_add_eq_sub_sub, ← Finset.sum_sub_distrib, ← Finset.sum_sub_distrib]
+      apply Finset.sum_congr rfl
+      intro i _
+      ring
+    _ ≤ _ := Finset.sum_le_sum fun i _ => mul_le_mul_of_nonneg_left
+      (signedCell_lower (B i) C (hB i) ε (hk i) hn) (hw i)
+
+lemma family_large (C : Finset P) (M ε τ : ℝ)
+    (hw : ∀ i, 0 ≤ w i) (hW : (∑ i, w i) ≤ 1) (hM : 0 ≤ M) (hε : 0 ≤ ε)
+    (hτ : 0 < τ) (hC : 2 ≤ C.card) (ha : τ ≤ cellProb Finset.univ C)
+    (hB : ∀ i, 2 ≤ (B i).card)
+    (hk : ∀ i, 1 / (((B i).card : ℝ)-1) ≤ ε)
+    (hn : 1 / ((Fintype.card P : ℝ)-1) ≤ ε)
+    (hmean : (∑ i, w i * cellProb (B i) C) ≤ M * cellProb Finset.univ C) :
+    (∑ i, w i * (cellProb (B i) C - cellProb Finset.univ C)^2) -
+      (2*ε*(M+1)/τ)*pairCell Finset.univ C ≤ (∑ i, w i * signedCell (B i) C) := by
+  have hcomp := (uniform_pair_comparison C hC).1
+  have ha0 := cellProb_nonneg (Finset.univ : Finset P) C
+  have hmass : cellProb Finset.univ C ≤ 2*pairCell Finset.univ C/τ := by
+    apply (le_div_iff₀ hτ).mpr
+    nlinarith [mul_le_mul_of_nonneg_right ha ha0]
+  have herr := mul_le_mul_of_nonneg_left hmass (show 0 ≤ ε*(M+1) by positivity)
+  have h := family_large_raw w B C M ε hw hW hε hB hk hn hmean
+  have he : ε*(M+1)*(2*pairCell Finset.univ C/τ) =
+      (2*ε*(M+1)/τ)*pairCell Finset.univ C := by ring
+  rw [he] at herr
+  linarith
+
+lemma family_small (C : Finset P) (M : ℝ)
+    (hw : ∀ i, 0 ≤ w i) (hM : 0 ≤ M) (hC : 2 ≤ C.card)
+    (hB : ∀ i, 2 ≤ (B i).card)
+    (hmean : (∑ i, w i * cellProb (B i) C) ≤ M * cellProb Finset.univ C) :
+    (1/2) * (∑ i, w i * (if 2 / ((B i).card : ℝ) ≤ cellProb (B i) C then
+      (cellProb (B i) C)^2 else 0)) - 4*M*pairCell Finset.univ C ≤
+      (∑ i, w i * signedCell (B i) C) := by
+  let a := cellProb Finset.univ C
+  have ha0 : 0 ≤ a := cellProb_nonneg _ _
+  have hp0 := pairCell_nonneg (Finset.univ : Finset P) C
+  have hc := (uniform_pair_comparison C hC).1
+  have hterm (i : I) :
+      (1/2)*(if 2 / ((B i).card : ℝ) ≤ cellProb (B i) C then
+        (cellProb (B i) C)^2 else 0) - 2*a*cellProb (B i) C ≤ signedCell (B i) C := by
+    have h₁ := fiber_pair_lower (B i) C (hB i)
+    have h₂ := mixedCell_le_product (B i) C (hB i)
+    dsimp [signedCell, a] at *
+    linarith
+  have hs := Finset.sum_le_sum (s := Finset.univ) (fun i _ =>
+    mul_le_mul_of_nonneg_left (hterm i) (hw i))
+  have he : (∑ i, w i * ((1/2)*(if 2 / ((B i).card : ℝ) ≤ cellProb (B i) C then
+        (cellProb (B i) C)^2 else 0) - 2*a*cellProb (B i) C)) =
+      (1/2)*(∑ i, w i * (if 2 / ((B i).card : ℝ) ≤ cellProb (B i) C then
+        (cellProb (B i) C)^2 else 0)) - 2*a*(∑ i, w i * cellProb (B i) C) := by
+    rw [Finset.mul_sum, Finset.mul_sum, ← Finset.sum_sub_distrib]
+    apply Finset.sum_congr rfl
+    intro i _
+    ring
+  rw [he] at hs
+  have hmean' := mul_le_mul_of_nonneg_left hmean (show 0 ≤ 2*a by positivity)
+  have hcomp := mul_le_mul_of_nonneg_left hc (show 0 ≤ 4*M by positivity)
+  dsimp [a] at *
+  nlinarith
+
+end Family
+
+/-- Zero pair overlap is precisely an empty or singleton occurrence. -/
+lemma zero_uniform_pair_iff (C : Finset P) :
+    pairCell Finset.univ C = 0 ↔ C.card ≤ 1 := by
+  refine ⟨?_, singleton_pairCell _ _⟩
+  intro h
+  by_contra! hc
+  have hc2 : 2 ≤ C.card := hc
+  have hn : 0 < (Fintype.card P : ℝ) := by
+    exact_mod_cast (show 0 < Fintype.card P by have := Finset.card_le_univ C; omega)
+  have hm : 0 < (C.card : ℝ) := by exact_mod_cast (by omega : 0 < C.card)
+  have ha : 0 < cellProb Finset.univ C := by
+    simpa only [cellProb, Finset.univ_inter, Finset.card_univ] using div_pos hm hn
+  have hh := (uniform_pair_comparison C hc2).1
+  rw [h] at hh
+  have : 0 < (1/2 : ℝ)*(cellProb Finset.univ C)^2 := by positivity
+  linarith
+
+section CellIntegral
+variable {Ω : Type u3} [MeasurableSpace Ω]
+-- A finite set of vertex labels is discrete. No condition is imposed on the
+-- underlying cell-occurrence measure, which may have infinite singleton tails.
+local instance : MeasurableSpace (Finset P) := ⊤
+local instance : DiscreteMeasurableSpace (Finset P) := ⟨fun _ => trivial⟩
+
+private lemma finite_weighted_bound {T : Type u4} [Fintype T]
+    (f q : T → ℝ) (hq : ∀ t, 0 ≤ q t) (hf : ∀ t, q t = 0 → f t = 0) :
+    ∃ K : ℝ, ∀ t, |f t| ≤ K*q t := by
+  classical
+  refine ⟨∑ t, |f t / q t|, ?_⟩
+  intro t
+  by_cases ht : q t = 0
+  · simp [ht, hf t ht]
+  · calc
+      |f t| = |f t / q t| * q t := by
+        rw [abs_div, abs_of_nonneg (hq t), div_mul_cancel₀ _ ht]
+      _ ≤ _ := mul_le_mul_of_nonneg_right
+        (Finset.single_le_sum (s := Finset.univ) (f := fun t => |f t / q t|)
+          (fun _ _ => abs_nonneg _) (Finset.mem_univ t)) (hq t)
+
+/-- The only integrability input needed for any fixed finite cell function
+    vanishing on singletons is integrability of the off-diagonal pair mass. -/
+lemma integrable_cell_function (C : Ω → Finset P) (μ : Measure Ω)
+    (hC : Measurable C) (hq : Integrable (fun ω => pairCell Finset.univ (C ω)) μ)
+    (f : Finset P → ℝ) (hf : ∀ D, D.card ≤ 1 → f D = 0) :
+    Integrable (fun ω => f (C ω)) μ := by
+  obtain ⟨K,hK⟩ := finite_weighted_bound f (pairCell Finset.univ)
+    (pairCell_nonneg _) (fun D hd => hf D ((zero_uniform_pair_iff D).mp hd))
+  apply (hq.const_mul K).mono' ((measurable_of_finite f).comp hC).aestronglyMeasurable
+  exact Filter.Eventually.of_forall fun ω => by simpa only [Real.norm_eq_abs, Function.comp_apply] using hK (C ω)
+
+def offDiagonalCell (C : Finset P) (x y : P) : ℝ :=
+  if x ≠ y ∧ x ∈ C ∧ y ∈ C then 1 else 0
+
+omit [Fintype P] in
+lemma offDiagonalCell_singleton (C : Finset P) (x y : P) (hC : C.card ≤ 1) :
+    offDiagonalCell C x y = 0 := by
+  unfold offDiagonalCell
+  split_ifs with h
+  · exact False.elim (h.1 (Finset.card_le_one.mp hC x h.2.1 y h.2.2))
+  · rfl
+
+def overlap (C : Ω → Finset P) (μ : Measure Ω) (x y : P) : ℝ :=
+  ∫ ω, offDiagonalCell (C ω) x y ∂μ
+
+lemma cell_sum_eq (H : P → P → ℝ) (hdiag : ∀ x, H x x = 0) (C : Finset P) :
+    (∑ x ∈ C, ∑ y ∈ C, H x y) = ∑ x, ∑ y, H x y * offDiagonalCell C x y := by
+  have he (x y : P) : H x y * offDiagonalCell C x y =
+      if x ∈ C then if y ∈ C then H x y else 0 else 0 := by
+    by_cases hx : x ∈ C <;> by_cases hy : y ∈ C <;> by_cases hxy : x = y
+    all_goals simp_all [offDiagonalCell]
+  simp_rw [he]
+  simp
+
+lemma integral_signedCell (C : Ω → Finset P) (μ : Measure Ω)
+    (hC : Measurable C) (hq : Integrable (fun ω => pairCell Finset.univ (C ω)) μ)
+    (B : Finset P) :
+    (∫ ω, signedCell B (C ω) ∂μ) = ∑ x, ∑ y, signedPair B x y * overlap C μ x y := by
+  have hi (x y : P) : Integrable (fun ω => signedPair B x y * offDiagonalCell (C ω) x y) μ :=
+    (integrable_cell_function C μ hC hq (fun D => offDiagonalCell D x y)
+      (fun D hD => offDiagonalCell_singleton D x y hD)).const_mul _
+  calc
+    _ = ∫ ω, ∑ x, ∑ y, signedPair B x y * offDiagonalCell (C ω) x y ∂μ := by
+      apply integral_congr_ae
+      exact Filter.Eventually.of_forall fun ω => by
+        change signedCell B (C ω) = _
+        rw [signedCell_eq, cell_sum_eq _ (signedPair_diag B)]
+    _ = ∑ x, ∑ y, ∫ ω, signedPair B x y * offDiagonalCell (C ω) x y ∂μ := by
+      rw [integral_finsetSum _ (fun x _ => integrable_finsetSum _ (fun y _ => hi x y))]
+      apply Finset.sum_congr rfl
+      intro x _
+      exact integral_finsetSum _ (fun y _ => hi x y)
+    _ = _ := by simp only [overlap, integral_const_mul]
+
+lemma integrated_cancel (C : Ω → Finset P) (μ : Measure Ω)
+    (hC : Measurable C) (hq : Integrable (fun ω => pairCell Finset.univ (C ω)) μ)
+    (b : ℝ) (S : P → ℝ)
+    (hL : ∀ x y, x ≠ y → overlap C μ x y = b + S x + S y)
+    (B : Finset P) (hB : 2 ≤ B.card) :
+    (∫ ω, signedCell B (C ω) ∂μ) = 0 := by
+  rw [integral_signedCell C μ hC hq]
+  exact variance_cancel B hB _ b S hL
+
+section Quantitative
+variable {I : Type u5} [Fintype I]
+
+def varianceCell (w : I → ℝ) (B : I → Finset P) (T : ℝ) (D : Finset P) : ℝ :=
+  if T < D.card then
+    ∑ i, w i * (cellProb (B i) D - cellProb Finset.univ D)^2
+  else (1/2) * ∑ i, w i *
+    (if 2 / ((B i).card : ℝ) ≤ cellProb (B i) D then (cellProb (B i) D)^2 else 0)
+
+lemma varianceCell_singleton (w : I → ℝ) (B : I → Finset P) (T : ℝ)
+    (hT : 1 ≤ T) (hB : ∀ i, 2 ≤ (B i).card) (D : Finset P) (hD : D.card ≤ 1) :
+    varianceCell w B T D = 0 := by
+  have hd : (D.card : ℝ) ≤ T := (by exact_mod_cast hD : (D.card : ℝ) ≤ 1).trans hT
+  simp [varianceCell, not_lt.mpr hd, singleton_fiber_term _ _ (hB _) hD]
+
+lemma varianceCell_bound (w : I → ℝ) (B : I → Finset P) (T M ε τ : ℝ)
+    (hw : ∀ i, 0 ≤ w i) (hW : (∑ i, w i) ≤ 1)
+    (hT : 1 ≤ T) (hM : 0 ≤ M) (hε : 0 ≤ ε) (hτ : 0 < τ)
+    (hscale : τ * Fintype.card P ≤ T)
+    (hB : ∀ i, 2 ≤ (B i).card)
+    (hk : ∀ i, 1 / (((B i).card : ℝ)-1) ≤ ε)
+    (hn : 1 / ((Fintype.card P : ℝ)-1) ≤ ε)
+    (hdom : ∀ x, (∑ i, w i * prob (B i) x) ≤ M * prob Finset.univ x)
+    (D : Finset P) :
+    varianceCell w B T D ≤ (∑ i, w i * signedCell (B i) D) +
+      (2*ε*(M+1)/τ)*pairCell Finset.univ D +
+      4*M*(if (D.card : ℝ) ≤ T then pairCell Finset.univ D else 0) := by
+  have hmean := mean_cell_bound w B D M hdom
+  have herr : 0 ≤ (2*ε*(M+1)/τ)*pairCell Finset.univ D := by
+    exact mul_nonneg (div_nonneg (mul_nonneg (mul_nonneg (by norm_num) hε)
+      (by linarith)) hτ.le) (pairCell_nonneg _ _)
+  by_cases hD : T < (D.card : ℝ)
+  · have hd2 : 2 ≤ D.card := by
+      have : (1 : ℝ) < D.card := hT.trans_lt hD
+      exact_mod_cast (show 1 < D.card by exact_mod_cast this)
+    have hnp : 0 < (Fintype.card P : ℝ) := by
+      exact_mod_cast (show 0 < Fintype.card P by have := Finset.card_le_univ D; omega)
+    have ha : τ ≤ cellProb Finset.univ D := by
+      simp only [cellProb, Finset.univ_inter, Finset.card_univ]
+      exact (le_div_iff₀ hnp).mpr (hscale.trans hD.le)
+    have hh := family_large w B D M ε τ hw hW hM hε hτ hd2 ha hB hk hn hmean
+    simp only [varianceCell, hD, reduceIte, not_le.mpr hD, mul_zero, add_zero]
+    linarith
+  · simp only [varianceCell, hD, reduceIte, not_lt.mp hD]
+    by_cases hd2 : 2 ≤ D.card
+    · have hh := family_small w B D M hw hM hd2 hB hmean
+      linarith
+    · have hd1 : D.card ≤ 1 := by omega
+      simp [singleton_fiber_term _ _ (hB _) hd1, singleton_signedCell _ _ hd1,
+        singleton_pairCell _ _ hd1]
+
+/-- The integrated off-diagonal variance estimate before specializing the
+    threshold and denominator errors to powers of n. The pair overlap may
+    have infinite singleton tails; no diagonal integral is used. -/
+theorem integrated_variance (C : Ω → Finset P) (μ : Measure Ω)
+    (hC : Measurable C) (hq : Integrable (fun ω => pairCell Finset.univ (C ω)) μ)
+    (b : ℝ) (S : P → ℝ)
+    (hL : ∀ x y, x ≠ y → overlap C μ x y = b + S x + S y)
+    (w : I → ℝ) (B : I → Finset P) (T M ε τ : ℝ)
+    (hw : ∀ i, 0 ≤ w i) (hW : (∑ i, w i) ≤ 1)
+    (hT : 1 ≤ T) (hM : 0 ≤ M) (hε : 0 ≤ ε) (hτ : 0 < τ)
+    (hscale : τ * Fintype.card P ≤ T)
+    (hB : ∀ i, 2 ≤ (B i).card)
+    (hk : ∀ i, 1 / (((B i).card : ℝ)-1) ≤ ε)
+    (hn : 1 / ((Fintype.card P : ℝ)-1) ≤ ε)
+    (hdom : ∀ x, (∑ i, w i * prob (B i) x) ≤ M * prob Finset.univ x) :
+    (∫ ω, varianceCell w B T (C ω) ∂μ) ≤
+      (2*ε*(M+1)/τ)*(∫ ω, pairCell Finset.univ (C ω) ∂μ) +
+      4*M*(∫ ω, if ((C ω).card : ℝ) ≤ T then pairCell Finset.univ (C ω) else 0 ∂μ) := by
+  have hv : Integrable (fun ω => varianceCell w B T (C ω)) μ :=
+    integrable_cell_function C μ hC hq _ (varianceCell_singleton w B T hT hB)
+  have hs (i : I) : Integrable (fun ω => signedCell (B i) (C ω)) μ :=
+    integrable_cell_function C μ hC hq _ (singleton_signedCell _)
+  have hsmall : Integrable (fun ω =>
+      if ((C ω).card : ℝ) ≤ T then pairCell Finset.univ (C ω) else 0) μ := by
+    apply integrable_cell_function C μ hC hq
+      (fun D => if (D.card : ℝ) ≤ T then pairCell Finset.univ D else 0)
+    intro D hD
+    simp [singleton_pairCell _ _ hD]
+  have hsum : Integrable (fun ω => ∑ i, w i * signedCell (B i) (C ω)) μ :=
+    integrable_finsetSum _ fun i _ => (hs i).const_mul _
+  have hzero : (∫ ω, ∑ i, w i * signedCell (B i) (C ω) ∂μ) = 0 := by
+    rw [integral_finsetSum _ fun i _ => (hs i).const_mul _]
+    simp only [integral_const_mul,
+      integrated_cancel C μ hC hq b S hL _ (hB _), mul_zero, Finset.sum_const_zero]
+  have hh := integral_mono hv
+    ((hsum.add (hq.const_mul _)).add (hsmall.const_mul _))
+    (fun ω => varianceCell_bound w B T M ε τ hw hW hT hM hε hτ hscale hB hk hn hdom (C ω))
+  simp only [Pi.add_apply] at hh
+  have h₁ := integral_add (hsum.add (hq.const_mul (2*ε*(M+1)/τ))) (hsmall.const_mul (4*M))
+  have h₂ := integral_add hsum (hq.const_mul (2*ε*(M+1)/τ))
+  simp only [Pi.add_apply] at h₁ h₂
+  rw [h₁, h₂, hzero, zero_add, integral_const_mul, integral_const_mul] at hh
+  exact hh
+
+end Quantitative
+
+end CellIntegral
+
+end WeakPinned.Variance
+
+end
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PinnedDistances/VarianceRates.lean b/lean/OAI/Geometry/PinnedDistances/VarianceRates.lean
new file mode 100644
index 0000000000000000000000000000000000000000..cae7d3331b6758ccfe6097c1e909c33a9b84896a
--- /dev/null
+++ b/lean/OAI/Geometry/PinnedDistances/VarianceRates.lean
@@ -0,0 +1,354 @@
+import OAI.Geometry.PinnedDistances.PartitionIntegrals
+
+namespace OAI
+
+universe u1 u2 u3 u4 u5 u6 u7 u8 u9 u10 u11
+
+section
+noncomputable section
+open scoped BigOperators Topology ZeroAtInfty
+open Filter MeasureTheory
+namespace WeakPinned.Bounded
+
+lemma test_bound_rate (D V A : ℕ → ℝ) (H M : ℝ)
+    (hD : ∀ n,0≤ D n) (hH : 0≤ H)
+    (hV : Tendsto V atTop (𝓝 0)) (hA : Tendsto A atTop (𝓝 0))
+    (h : ∀ η,0<η → ∃ N : ℝ,0≤ N ∧ ∀ δ,0<δ →
+      ∀ n,D n≤ 2*η+H*(V n/δ+N*(δ+(M+1)*A n))) :
+    Tendsto D atTop (𝓝 0) := by
+  apply tendsto_order.mpr
+  constructor
+  · intro a ha
+    exact Eventually.of_forall (fun n => ha.trans_le (hD n))
+  · intro b hb
+    obtain ⟨N,hN,h⟩ := h (b/8) (by positivity)
+    let δ := b/(4*(H*N+1))
+    have hδ : 0<δ := by dsimp [δ]; positivity
+    have he : (H*N+1)*δ=b/4 := by dsimp [δ]; field_simp
+    have hNd : H*N*δ≤ b/4 := by nlinarith
+    have hlt : 2*(b/8)+H*(0/δ+N*(δ+(M+1)*0))<b := by simp only [zero_div,mul_zero,add_zero]; nlinarith
+    have ht := (tendsto_const_nhds.add ((hV.div_const δ).add
+      ((tendsto_const_nhds.add (hA.const_mul (M+1))).const_mul N) |>.const_mul H)).eventually
+      (eventually_lt_nhds hlt)
+    exact ht.mono (fun n hn => (h δ hδ n).trans_lt hn)
+
+lemma zeroAtInfty_tail (f : C₀(ℂ,ℝ)) (η : ℝ) (hη : 0<η) :
+    ∃ R : ℝ,0≤ R ∧ ∀ z,R<‖z‖ → |f z|≤ η := by
+  have hf := (Metric.tendsto_nhds.mp (zero_at_infty f) η hη)
+  obtain ⟨K,hK,hsub⟩ := mem_cocompact.mp hf
+  obtain ⟨R,hR⟩ := hK.isBounded.subset_closedBall (0:ℂ)
+  refine ⟨max R 0,le_max_right _ _,?_⟩
+  intro z hz
+  have hzK : z∉K := by
+    intro hzK
+    have hh := Metric.mem_closedBall.mp (hR hzK)
+    simp only [dist_zero_right] at hh
+    exact (not_le_of_gt (lt_of_le_of_lt (le_max_left _ _) hz)) hh
+  simpa only [Real.dist_eq,sub_zero] using (show dist (f z) 0<η from hsub hzK).le
+
+lemma zeroAtInfty_modulus (f : C₀(ℂ,ℝ)) (η : ℝ) (hη : 0<η) :
+    ∃ ε : ℝ,0<ε ∧ ∀ z w,‖z-w‖<2*ε → |f z-f w|≤ η := by
+  obtain ⟨δ,hδ,hd⟩ := Metric.uniformContinuous_iff.mp (ZeroAtInftyContinuousMap.uniformContinuous f) η hη
+  refine ⟨δ/2,by positivity,?_⟩
+  intro z w hzw
+  have hh := hd (show dist z w<δ by rw [dist_eq_norm]; linarith)
+  exact (by simpa only [Real.dist_eq] using hh.le)
+
+local instance : MeasurableSpace Sphere := borel Sphere
+local instance : BorelSpace Sphere := ⟨rfl⟩
+
+def sphereTest (f : C(Sphere,ℝ)) : C₀(ℂ,ℝ) where
+  toFun := fun z => f (z:Sphere)-f OnePoint.infty
+  continuous_toFun := (f.continuous.comp OnePoint.continuous_coe).sub continuous_const
+  zero_at_infty' := by
+    have hco : Tendsto ((↑) : ℂ → Sphere) (cocompact ℂ) (𝓝 OnePoint.infty) :=
+      OnePoint.tendsto_coe_infty.mono_left cocompact_le_coclosedCompact
+    simpa using ((f.continuous.tendsto OnePoint.infty).comp hco).sub_const (f OnePoint.infty)
+
+lemma sphereTest_bound (f : C(Sphere,ℝ)) (z : ℂ) : |sphereTest f z|≤ 2*‖f‖ := by
+  have h₁ := f.norm_coe_le_norm (z:Sphere)
+  have h₂ := f.norm_coe_le_norm OnePoint.infty
+  change |f (z:Sphere)-f OnePoint.infty|≤ 2*‖f‖
+  simpa only [Real.norm_eq_abs,two_mul] using
+    (norm_sub_le (f (z:Sphere)) (f OnePoint.infty)).trans (add_le_add h₁ h₂)
+
+end WeakPinned.Bounded
+
+end
+end
+section
+noncomputable section
+open scoped BigOperators Topology ZeroAtInfty
+open Filter MeasureTheory
+namespace WeakPinned.Bounded
+open Hierarchy Variance Global NumberField
+variable {P : Type u1} {I : Type u2} [Fintype P] [DecidableEq P] [Fintype I]
+
+lemma testDiscrep_sub_const (w : I → ℝ) (B : I → Finset P) (f : P → ℝ) (c : ℝ)
+    (hn : 0<Fintype.card P) (hB : ∀ i,(B i).Nonempty) :
+    testDiscrep w B (fun x => f x-c)=testDiscrep w B f := by
+  have hh (S : Finset P) (hS : S.Nonempty) :
+      (∑ x,prob S x*(f x-c))=(∑ x,prob S x*f x)-c := by
+    simp_rw [mul_sub]
+    rw [Finset.sum_sub_distrib,← Finset.sum_mul,sum_prob S hS,one_mul]
+  unfold testDiscrep
+  simp_rw [hh _ (hB _),hh Finset.univ (Finset.univ_nonempty_iff.mpr (Fintype.card_pos_iff.mp hn))]
+  simp only [sub_sub_sub_cancel_right]
+
+variable {V : ℕ → Type u3} {J : ℕ → Type u4} {K : ℕ → Type u5}
+  [∀ n,Fintype (V n)] [∀ n,DecidableEq (V n)] [∀ n,Fintype (J n)]
+  [∀ n,Field (K n)] [∀ n,NumberField (K n)]
+
+omit [∀ (n : ℕ), NumberField (K n)] in
+lemma C0_arch_test_rate (Z₁ Z₂ : ∀ n,V n → K n)
+    (h₁ : ∀ n,Function.Injective (Z₁ n)) (h₂ : ∀ n,Function.Injective (Z₂ n))
+    (v : ∀ n,InfinitePlace (K n)) (m : ℕ → ℝ) (b : ℕ → ℂ)
+    (w : ∀ n,J n → ℝ) (B : ∀ n,J n → Finset (V n)) (T : ℕ → ℝ) (M H : ℝ)
+    (hn : ∀ n,2≤ Fintype.card (V n)) (hT : ∀ n,1≤ T n) (hM : 0≤ M) (hH : 0≤ H)
+    (hw : ∀ n i,0≤ w n i) (hW : ∀ n,∑ i,w n i≤ 1) (hB : ∀ n i,2≤ (B n i).card)
+    (hdom : ∀ n x,(∑ i,w n i*prob (B n i) x)≤ M*prob Finset.univ x)
+    (hvar : Tendsto (fun n => archMean (Z₁ n) (Z₂ n) (massEval (varianceCell (w n) (B n) (T n))) (v n)) atTop (𝓝 0))
+    (hsmall : Tendsto (fun n => T n/Fintype.card (V n)) atTop (𝓝 0))
+    (f : C₀(ℂ,ℝ)) (hbound : ∀ z,|f z|≤ H) :
+    Tendsto (fun n => testDiscrep (w n) (B n)
+      (fun x => f ((Real.exp (m n):ℂ)*(v n).embedding (Z₁ n x)+b n))) atTop (𝓝 0) := by
+  apply test_bound_rate _ _ _ H M (fun n => testDiscrep_nonneg _ _ _ (hw n)) hH hvar hsmall
+  intro η hη
+  obtain ⟨R,hR,hout⟩ := zeroAtInfty_tail f η hη
+  obtain ⟨ε,hε,hcont⟩ := zeroAtInfty_modulus f η hη
+  obtain ⟨N,hN⟩ := exists_nat_ge (2*R/(ε/(2*Real.exp 1))+1)
+  refine ⟨((2*N+1)^2:ℕ),by positivity,?_⟩
+  intro δ hδ n
+  have hh := arch_test_bound (Z₁ n) (Z₂ n) (h₁ n) (h₂ n) (v n) (m n) (b n) (hn n)
+    R ε η H M (T n) δ N hR hε hN hη.le hH hM (hT n) hδ
+    (w n) (B n) (hw n) (hW n) (hB n) (hdom n) f hbound hcont hout
+  convert hh using 1
+  ring
+
+local instance : MeasurableSpace Sphere := borel Sphere
+local instance : BorelSpace Sphere := ⟨rfl⟩
+
+omit [∀ (n : ℕ), NumberField (K n)] in
+lemma sphere_arch_test_rate (Z₁ Z₂ : ∀ n,V n → K n)
+    (h₁ : ∀ n,Function.Injective (Z₁ n)) (h₂ : ∀ n,Function.Injective (Z₂ n))
+    (v : ∀ n,InfinitePlace (K n)) (m : ℕ → ℝ) (b : ℕ → ℂ)
+    (w : ∀ n,J n → ℝ) (B : ∀ n,J n → Finset (V n)) (T : ℕ → ℝ) (M : ℝ)
+    (hn : ∀ n,2≤ Fintype.card (V n)) (hT : ∀ n,1≤ T n) (hM : 0≤ M)
+    (hw : ∀ n i,0≤ w n i) (hW : ∀ n,∑ i,w n i≤ 1) (hB : ∀ n i,2≤ (B n i).card)
+    (hdom : ∀ n x,(∑ i,w n i*prob (B n i) x)≤ M*prob Finset.univ x)
+    (hvar : Tendsto (fun n => archMean (Z₁ n) (Z₂ n) (massEval (varianceCell (w n) (B n) (T n))) (v n)) atTop (𝓝 0))
+    (hsmall : Tendsto (fun n => T n/Fintype.card (V n)) atTop (𝓝 0))
+    (f : C(Sphere,ℝ)) :
+    Tendsto (fun n => testDiscrep (w n) (B n)
+      (fun x => f (((Real.exp (m n):ℂ)*(v n).embedding (Z₁ n x)+b n):Sphere))) atTop (𝓝 0) := by
+  have ht := C0_arch_test_rate Z₁ Z₂ h₁ h₂ v m b w B T M (2*‖f‖) hn hT hM (by positivity)
+    hw hW hB hdom hvar hsmall (sphereTest f) (sphereTest_bound f)
+  convert ht using 1
+  funext n
+  symm
+  apply testDiscrep_sub_const _ _ _ _ (by have := hn n; omega)
+  intro i
+  exact Finset.card_pos.mp (by have := hB n i; omega)
+
+end WeakPinned.Bounded
+
+end
+end
+section
+noncomputable section
+open MeasureTheory Filter
+open scoped BigOperators Topology
+namespace WeakPinned.Bounded
+open Variance
+variable {V : Type u6} {I : Type u7} [Fintype V] [DecidableEq V] [Fintype I]
+variable (G : Core.Fibers V I)
+
+omit [Fintype I] in
+lemma incoming_discrep_bound (T H : ℝ) (hT : 0<T) (hn : 2≤Fintype.card V)
+    (f : V → ℝ) (hf : ∀ x,|f x|≤H) :
+    (∑ y,G.incomingWeight y*|(∑ x,G.conditionalSource y x*f x)-∑ x,prob Finset.univ x*f x|)≤
+      testDiscrep (G.retainedWeight T) (G.retainedSet T) f+2*H*G.omittedMass T := by
+  have hA : |∑ x,prob (Finset.univ : Finset V) x*f x|≤H :=
+    avg_abs_bound _ _ _ (Core.Fibers.prob_nn _) (sum_prob _
+      (Finset.univ_nonempty_iff.mpr (Fintype.card_pos_iff.mp (by omega)))) hf
+  unfold testDiscrep Core.Fibers.omittedMass
+  rw [Finset.mul_sum,← Finset.sum_add_distrib]
+  apply Finset.sum_le_sum
+  intro y _
+  by_cases hy : T≤((G.incoming y).card:ℝ)
+  · have hne : (G.incoming y).Nonempty := Finset.card_pos.mp (by exact_mod_cast hT.trans_le hy)
+    simp only [Core.Fibers.retainedWeight,Core.Fibers.retainedSet,Core.Fibers.conditionalSource,
+      ite_eq_left hy,ite_eq_left hne,not_lt_of_ge hy,ite_false,mul_zero,add_zero]
+    exact le_rfl
+  · have hbound : |(∑ x,G.conditionalSource y x*f x)-∑ x,prob Finset.univ x*f x|≤2*H := by
+      have hh := avg_abs_bound _ _ _ (G.conditionalSource_nonneg y) (G.conditionalSource_sum hn y) hf
+      exact (abs_sub _ _).trans (by linarith)
+    simp only [Core.Fibers.retainedWeight,ite_eq_right hy,lt_of_not_ge hy,ite_true,zero_mul,zero_add]
+    nlinarith [mul_le_mul_of_nonneg_left hbound (G.incomingWeight_nonneg y)]
+
+lemma incoming_test_bound (T H C : ℝ) (hT : 0<T) (hn : 2≤Fintype.card V)
+    (f g : V → ℝ) (hf : ∀ x,|f x|≤H) (hg : ∀ y,|g y|≤C) :
+    |(∑ e,G.fiberWeight e*∑ y,prob (G.targets e) y*(f (G.source e)*g y))-
+      (∑ x,prob Finset.univ x*f x)*(∑ y,G.incomingWeight y*g y)|≤
+      C*(testDiscrep (G.retainedWeight T) (G.retainedSet T) f+2*H*G.omittedMass T) := by
+  have hC : 0≤C := (abs_nonneg (g (Classical.choice (Fintype.card_pos_iff.mp (by omega))))).trans (hg _)
+  rw [← G.incoming_eval (fun x y => f x*g y)]
+  have he : (∑ y,G.incomingWeight y*∑ x,G.conditionalSource y x*(f x*g y))-
+      (∑ x,prob Finset.univ x*f x)*(∑ y,G.incomingWeight y*g y)=
+      ∑ y,G.incomingWeight y*((∑ x,G.conditionalSource y x*f x)-∑ x,prob Finset.univ x*f x)*g y := by
+    simp only [Finset.mul_sum,Finset.sum_mul,← Finset.sum_sub_distrib]
+    apply Finset.sum_congr rfl
+    intro y _
+    apply Finset.sum_congr rfl
+    intro x _
+    ring
+  rw [he]
+  calc
+    _ ≤ ∑ y,|G.incomingWeight y*((∑ x,G.conditionalSource y x*f x)-∑ x,prob Finset.univ x*f x)*g y| :=
+      Finset.abs_sum_le_sum_abs _ _
+    _ ≤ C*(∑ y,G.incomingWeight y*|(∑ x,G.conditionalSource y x*f x)-∑ x,prob Finset.univ x*f x|) := by
+      rw [Finset.mul_sum]
+      apply Finset.sum_le_sum
+      intro y _
+      rw [abs_mul,abs_mul,abs_of_nonneg (G.incomingWeight_nonneg y)]
+      nlinarith [mul_le_mul_of_nonneg_left (hg y) (mul_nonneg (G.incomingWeight_nonneg y) (abs_nonneg ((∑ x,G.conditionalSource y x*f x)-∑ x,prob Finset.univ x*f x)))]
+    _ ≤ _ := mul_le_mul_of_nonneg_left (incoming_discrep_bound G T H hT hn f hf) hC
+
+end WeakPinned.Bounded
+
+end
+end
+section
+noncomputable section
+open MeasureTheory Filter
+open scoped BigOperators Topology BoundedContinuousFunction
+namespace WeakPinned.Bounded
+open Variance
+local instance : MeasurableSpace Sphere := borel Sphere
+local instance : BorelSpace Sphere := ⟨rfl⟩
+local instance : MeasurableSingletonClass (Quad Sphere) := inferInstance
+local instance : MeasurableSingletonClass ((Sphere × Sphere) × (Sphere × Sphere)) := inferInstance
+local instance : MeasurableSingletonClass (Expanded Sphere) := Prod.instMeasurableSingletonClass
+variable {V : Type u8} {I : Type u9} [Fintype V] [DecidableEq V] [Fintype I]
+variable (G : Core.Fibers V I)
+
+omit G [Fintype I] in
+lemma integral_uniformLaw (U : V → Sphere) (f : Sphere → ℝ) :
+    (∫ z,f z ∂  uniformLaw U)=∑ x,prob Finset.univ x*f (U x) := by
+  rw [uniformLaw,integral_finiteLaw _ _ (fun _ => inv_nonneg.mpr (Nat.cast_nonneg _))]
+  simp only [prob,Finset.mem_univ,ite_true,Finset.card_univ,one_div]
+
+lemma integral_edgeProbability (U₁ U₂ : V → Sphere) (hG : 0<G.edgeCount)
+    (hB : ∀ e,(G.targets e).Nonempty) (f : Quad Sphere → ℝ) :
+    (∫ q,f q ∂ (edgeProbability G U₁ U₂ hG hB:Measure (Quad Sphere)))=
+      ∑ e,G.fiberWeight e*∑ y,prob (G.targets e) y*f (edgeValue G.source U₁ U₂ (e,y)) := by
+  change (∫ q,f q ∂ finiteLaw _ _)=_
+  rw [integral_finiteLaw _ _ (graph_edgeWeight_nonneg G)]
+  simp only [edgeWeight,Fintype.sum_prod_type,mul_assoc,← Finset.mul_sum]
+
+lemma integral_expandedProbability (U₁ U₂ : V → Sphere) (hG : 0<G.edgeCount)
+    (hB : ∀ e,(G.targets e).Nonempty) (f : Expanded Sphere → ℝ) :
+    (∫ r,f r ∂ (expandedProbability G U₁ U₂ hG hB:Measure (Expanded Sphere)))=
+      ∑ r,tripleWeight G.fiberWeight (fun e => prob (G.targets e)) r*f (tripleValue G.source U₁ U₂ r) := by
+  change (∫ r,f r ∂ finiteLaw (tripleWeight G.fiberWeight (fun e => prob (G.targets e))) (tripleValue G.source U₁ U₂))=_
+  exact integral_finiteLaw _ _ (graph_tripleWeight_nonneg G) f
+
+lemma expanded_test_bound (U₁ U₂ : V → Sphere) (hG : 0<G.edgeCount)
+    (hB : ∀ e,(G.targets e).Nonempty) (hn : 2≤Fintype.card V)
+    (π : Sphere × Sphere → Sphere) (f : Quad Sphere → ℝ) (g h : Sphere → ℝ) (C H₁ H₂ : ℝ)
+    (hC : 0≤C) (hf : ∀ q,|f q|≤C) (hg : ∀ z,|g z|≤H₁) (hh : ∀ z,|h z|≤H₂) :
+    |(∫ r,f r.1*g (π r.2.1)*h (π r.2.2) ∂ (expandedProbability G U₁ U₂ hG hB:Measure (Expanded Sphere)))-
+      (∫ q,f q ∂ (edgeProbability G U₁ U₂ hG hB:Measure (Quad Sphere)))*
+        (∫ z,g z ∂ uniformLaw (fun x => π (U₁ x,U₂ x)))*(∫ z,h z ∂ uniformLaw (fun x => π (U₁ x,U₂ x)))|≤
+      C*(H₂*testDiscrep G.fiberWeight G.targets (fun x => g (π (U₁ x,U₂ x)))+
+        H₁*testDiscrep G.fiberWeight G.targets (fun x => h (π (U₁ x,U₂ x)))) := by
+  rw [integral_expandedProbability,integral_edgeProbability,integral_uniformLaw,integral_uniformLaw]
+  have hA : |∑ x,prob (Finset.univ : Finset V) x*g (π (U₁ x,U₂ x))|≤H₁ := by
+    exact avg_abs_bound _ _ _ (Core.Fibers.prob_nn _) (sum_prob _
+      (Finset.univ_nonempty_iff.mpr (Fintype.card_pos_iff.mp (by omega)))) (fun x => hg _)
+  have hb := mixture_pair_test_bound G.fiberWeight (fun e => prob (G.targets e))
+    (fun e y => f (edgeValue G.source U₁ U₂ (e,y))) (fun x => g (π (U₁ x,U₂ x))) (fun x => h (π (U₁ x,U₂ x)))
+    (∑ x,prob Finset.univ x*g (π (U₁ x,U₂ x))) (∑ x,prob Finset.univ x*h (π (U₁ x,U₂ x)))
+    C H₁ H₂ G.fiberWeight_nonneg (fun e => Core.Fibers.prob_nn _) (fun e => sum_prob _ (hB e))
+    hC (fun e y => hf _) (fun x => hh _) hA
+  simpa only [tripleValue,testDiscrep,edgeWeight,Fintype.sum_prod_type,mul_assoc,← Finset.mul_sum] using hb
+
+lemma integral_target (U₁ U₂ : V → Sphere) (hG : 0<G.edgeCount)
+    (hB : ∀ e,(G.targets e).Nonempty) (g : Sphere × Sphere → ℝ) :
+    (∫ q,g q.2.2 ∂ (edgeProbability G U₁ U₂ hG hB:Measure (Quad Sphere)))=
+      ∑ y,G.incomingWeight y*g (U₁ y,U₂ y) := by
+  rw [integral_edgeProbability]
+  simp only [edgeValue,Finset.mul_sum,← mul_assoc]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro y _
+  rw [← Finset.sum_mul,G.fiber_marginal,G.incoming_marginal]
+
+lemma incoming_law_test_bound (U₁ U₂ : V → Sphere) (hG : 0<G.edgeCount)
+    (hB : ∀ e,(G.targets e).Nonempty) (hn : 2≤Fintype.card V) (T H C : ℝ) (hT : 0<T)
+    (f : Sphere → ℝ) (g : Sphere × Sphere → ℝ) (hf : ∀ z,|f z|≤H) (hg : ∀ z,|g z|≤C) :
+    |(∫ q,f q.1*g q.2.2 ∂ (edgeProbability G U₁ U₂ hG hB:Measure (Quad Sphere)))-
+      (∫ z,f z ∂ uniformLaw U₁)*(∫ q,g q.2.2 ∂ (edgeProbability G U₁ U₂ hG hB:Measure (Quad Sphere)))|≤
+      C*(testDiscrep (G.retainedWeight T) (G.retainedSet T) (fun x => f (U₁ x))+2*H*G.omittedMass T) := by
+  rw [integral_target,integral_edgeProbability,integral_uniformLaw]
+  exact incoming_test_bound G T H C hT hn (fun x => f (U₁ x)) (fun x => g (U₁ x,U₂ x))
+    (fun x => hf _) (fun x => hg _)
+
+end WeakPinned.Bounded
+
+end
+end
+section
+noncomputable section
+open MeasureTheory Filter
+open scoped BigOperators Topology BoundedContinuousFunction
+namespace WeakPinned.Bounded
+local instance : MeasurableSpace Sphere := borel Sphere
+local instance : BorelSpace Sphere := ⟨rfl⟩
+variable {V : ℕ → Type u10} {I : ℕ → Type u11} [∀ n,Fintype (V n)] [∀ n,DecidableEq (V n)] [∀ n,Fintype (I n)]
+variable (G : ∀ n,Core.Fibers (V n) (I n)) (U₁ U₂ : ∀ n,V n → Sphere)
+  (hG : ∀ n,0<(G n).edgeCount) (hB : ∀ n e,((G n).targets e).Nonempty)
+  (hn : ∀ n,2≤Fintype.card (V n))
+
+include hn
+lemma expanded_test_rate (π : Sphere × Sphere → Sphere)
+    (hd : ∀ f : C(Sphere,ℝ),Tendsto (fun n => testDiscrep (G n).fiberWeight (G n).targets
+      (fun x => f (π (U₁ n x,U₂ n x)))) atTop (𝓝 0))
+    (f : Quad Sphere →ᵇ ℝ) (g h : Sphere →ᵇ ℝ) :
+    Tendsto (fun n => (∫ r,f r.1*g (π r.2.1)*h (π r.2.2)
+      ∂ (expandedProbability (G n) (U₁ n) (U₂ n) (hG n) (hB n):Measure (Expanded Sphere)))-
+      (∫ q,f q ∂ (edgeProbability (G n) (U₁ n) (U₂ n) (hG n) (hB n):Measure (Quad Sphere)))*
+        (∫ z,g z ∂ uniformLaw (fun x => π (U₁ n x,U₂ n x)))*
+        (∫ z,h z ∂ uniformLaw (fun x => π (U₁ n x,U₂ n x)))) atTop (𝓝 0) := by
+  apply (tendsto_zero_iff_abs_tendsto_zero _).mpr
+  have ht := ((hd g.toContinuousMap).const_mul ‖h‖ |>.add
+    ((hd h.toContinuousMap).const_mul ‖g‖)).const_mul ‖f‖
+  simp only [mul_zero,zero_add] at ht
+  apply squeeze_zero (fun n => abs_nonneg _) (fun n => ?_) ht
+  exact expanded_test_bound (G n) (U₁ n) (U₂ n) (hG n) (hB n) (hn n) π f g h
+    ‖f‖ ‖g‖ ‖h‖ (norm_nonneg _) (fun q => by simpa only [Real.norm_eq_abs] using f.norm_coe_le_norm q)
+    (fun z => by simpa only [Real.norm_eq_abs] using g.norm_coe_le_norm z)
+    (fun z => by simpa only [Real.norm_eq_abs] using h.norm_coe_le_norm z)
+
+lemma incoming_law_test_rate (T : ℕ → ℝ) (hT : ∀ n,0<T n)
+    (hd : ∀ f : C(Sphere,ℝ),Tendsto (fun n => testDiscrep ((G n).retainedWeight (T n))
+      ((G n).retainedSet (T n)) (fun x => f (U₁ n x))) atTop (𝓝 0))
+    (ho : Tendsto (fun n => (G n).omittedMass (T n)) atTop (𝓝 0))
+    (f : Sphere →ᵇ ℝ) (g : Sphere × Sphere →ᵇ ℝ) :
+    Tendsto (fun n => (∫ q,f q.1*g q.2.2 ∂ (edgeProbability (G n) (U₁ n) (U₂ n) (hG n) (hB n):Measure (Quad Sphere)))-
+      (∫ z,f z ∂ uniformLaw (U₁ n))*(∫ q,g q.2.2 ∂ (edgeProbability (G n) (U₁ n) (U₂ n) (hG n) (hB n):Measure (Quad Sphere))))
+      atTop (𝓝 0) := by
+  apply (tendsto_zero_iff_abs_tendsto_zero _).mpr
+  have ht := ((hd f.toContinuousMap).add (ho.const_mul (2*‖f‖))).const_mul ‖g‖
+  simp only [mul_zero,zero_add] at ht
+  apply squeeze_zero (fun n => abs_nonneg _) (fun n => ?_) ht
+  exact incoming_law_test_bound (G n) (U₁ n) (U₂ n) (hG n) (hB n) (hn n) (T n) ‖f‖ ‖g‖ (hT n) f g
+    (fun z => by simpa only [Real.norm_eq_abs] using f.norm_coe_le_norm z)
+    (fun z => by simpa only [Real.norm_eq_abs] using g.norm_coe_le_norm z)
+
+end WeakPinned.Bounded
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PinnedDistances/WeightedSelection.lean b/lean/OAI/Geometry/PinnedDistances/WeightedSelection.lean
new file mode 100644
index 0000000000000000000000000000000000000000..c5bdc64ad6d686f00f878990d9c9d4ee6489250b
--- /dev/null
+++ b/lean/OAI/Geometry/PinnedDistances/WeightedSelection.lean
@@ -0,0 +1,127 @@
+import OAI.Geometry.PinnedDistances.UnboundedContradiction
+
+namespace OAI
+
+universe u1 u2 u3
+
+section
+noncomputable section
+open scoped BigOperators Topology
+open Filter MeasureTheory
+namespace WeakPinned.Bounded
+
+/-- Simultaneous selection from a normalized weighted family. In particular,
+there is no required lower bound on a nonzero archimedean-place weight. -/
+lemma weighted_select {J : Type u1} [Fintype J] (w T V : J → ℝ) (C ε : ℝ)
+    (hw : ∀ j, 0 < w j) (hW : ∑ j, w j = 1)
+    (hT : ∀ j, 0 ≤ T j) (hV : ∀ j, 0 ≤ V j)
+    (hC : 0 ≤ C) (hε : 0 < ε)
+    (hTC : ∑ j, w j * T j ≤ C) (hVE : ∑ j, w j * V j ≤ ε) :
+    ∃ j, T j ≤ 2*C+1 ∧ V j ≤ 2*ε := by
+  have hne : (Finset.univ : Finset J).Nonempty := by
+    by_contra h
+    rw [Finset.not_nonempty_iff_eq_empty.mp h, Finset.sum_empty] at hW
+    norm_num at hW
+  have hs : ∑ j, w j*(T j+(C+1)/ε*V j) ≤ ∑ j, w j*(2*C+1) := by
+    simp only [mul_add, Finset.sum_add_distrib, ← Finset.sum_mul]
+    rw [hW, one_mul]
+    have hv := mul_le_mul_of_nonneg_left hVE (show 0≤(C+1)/ε by positivity)
+    have he : (∑ j, w j*((C+1)/ε*V j)) = (C+1)/ε*(∑ j,w j*V j) := by
+      simp only [Finset.mul_sum]; apply Finset.sum_congr rfl; intros; ring
+    rw [he]
+    have he' : (C+1)/ε*ε=C+1 := by field_simp
+    rw [he'] at hv
+    linarith
+  obtain ⟨j, _, hj⟩ := Finset.exists_le_of_sum_le hne hs
+  have hj' : T j+(C+1)/ε*V j ≤ 2*C+1 := (mul_le_mul_iff_right₀ (hw j)).mp hj
+  refine ⟨j, (le_add_of_nonneg_right (mul_nonneg (by positivity) (hV j))).trans hj', ?_⟩
+  have hv' : (C+1)/ε*V j ≤ 2*C+1 := by linarith [hT j]
+  have he : (C+1)/ε*(2*ε) = 2*(C+1) := by field_simp
+  by_contra h
+  have hh := mul_lt_mul_of_pos_left (lt_of_not_ge h) (show 0 < (C+1)/ε by positivity)
+  rw [he] at hh
+  linarith
+
+end WeakPinned.Bounded
+
+namespace WeakPinned.Bounded
+open Hierarchy Variance Global NumberField
+variable {P : Type u2} [Fintype P] [DecidableEq P]
+variable {K : Type u3} [Field K] [NumberField K]
+variable (Z₁ Z₂ : P → K) (h₁ : Function.Injective Z₁) (h₂ : Function.Injective Z₂)
+
+omit Z₁ Z₂ h₁ h₂ in
+lemma massEval_nonneg (q : Finset P → ℝ) (hq : ∀ C, 0≤q C) (d d' : Depth P) :
+    0 ≤ massEval q d d' := by
+  exact add_nonneg (Finset.sum_nonneg (fun C _ => mul_nonneg (d.mass_nonneg C) (hq C)))
+    (Finset.sum_nonneg (fun C _ => mul_nonneg (d'.mass_nonneg C) (hq C)))
+
+omit Z₁ Z₂ h₁ h₂ in
+def scaleEval (d d' : Depth P) : ℝ :=
+  massEval (pairCell Finset.univ) d d' + potentialEval (prob Finset.univ) d d'
+
+omit Z₁ Z₂ h₁ h₂ in
+lemma scaleEval_nonneg (d d' : Depth P) : 0 ≤ scaleEval d d' := by
+  apply add_nonneg (massEval_nonneg _ (fun C => pairCell_nonneg _ _) _ _)
+  apply Finset.sum_nonneg
+  intro x _
+  apply mul_nonneg _ (add_nonneg (d.potential_nonneg x) (d'.potential_nonneg x))
+  simp only [prob,Finset.mem_univ,ite_true]; positivity
+
+omit [NumberField K] in
+include h₁ h₂ in
+lemma integrable_scaleEval (hn : 2≤Fintype.card P) :
+    AverageIntegrable Z₁ Z₂ (scaleEval (P:=P)) := by
+  intro v
+  exact (integrable_massEval Z₁ Z₂ h₁ h₂ hn _ v).add
+    (integrable_potentialEval Z₁ Z₂ h₁ h₂ hn _ v)
+
+lemma average_scaleEval (hn : 2≤Fintype.card P) :
+    average Z₁ Z₂ h₁ h₂ scaleEval = overlapScale Z₁ Z₂ h₁ h₂ + potentialScale Z₁ Z₂ h₁ h₂ := by
+  unfold scaleEval
+  rw [average_add Z₁ Z₂ h₁ h₂
+    (integrable_massEval Z₁ Z₂ h₁ h₂ hn _) (integrable_potentialEval Z₁ Z₂ h₁ h₂ hn _),
+    average_massEval Z₁ Z₂ h₁ h₂ hn,average_potentialEval Z₁ Z₂ h₁ h₂ hn]
+  rfl
+
+def archMean (f : Depth P → Depth P → ℝ) (v : InfinitePlace K) : ℝ :=
+  ∫ ω, f (arch Z₁ v ω) (arch Z₂ v ω) ∂ Grid.fullLaw
+
+omit [Fintype P] [NumberField K] in
+lemma archMean_nonneg (f : Depth P → Depth P → ℝ) (hf : ∀ d d', 0≤f d d')
+    (v : InfinitePlace K) : 0≤archMean Z₁ Z₂ f v := integral_nonneg (fun _ => hf _ _)
+
+lemma archMean_le_average (f : Depth P → Depth P → ℝ) (hf : ∀ d d', 0≤f d d') :
+    (∑ v, Arithmetic.archWeight v * archMean Z₁ Z₂ f v) ≤ average Z₁ Z₂ h₁ h₂ f := by
+  apply le_add_of_nonneg_right
+  exact mul_nonneg (by positivity) (Finset.sum_nonneg (fun v _ => hf _ _))
+
+omit Z₁ Z₂ h₁ h₂ in
+lemma archWeight_pos (v : InfinitePlace K) : 0<Arithmetic.archWeight v := by
+  unfold Arithmetic.archWeight
+  apply div_pos
+  · exact_mod_cast (InfinitePlace.mult_pos (w := v))
+  · exact_mod_cast Module.finrank_pos (R:=ℚ) (M:=K)
+
+/-- Both the local scale and any nonnegative sum of variance cells are
+controlled at one genuine complex embedding. -/
+lemma select_arch (hn : 2≤Fintype.card P) (q : Finset P → ℝ) (hq : ∀ C, 0≤q C)
+    (C ε : ℝ) (hC : 0≤C) (hε : 0<ε)
+    (hscale : overlapScale Z₁ Z₂ h₁ h₂ + potentialScale Z₁ Z₂ h₁ h₂ ≤ C)
+    (hvar : weightedIntegral (totalMass Z₁ Z₂ h₁ h₂) q ≤ ε) :
+    ∃ v : InfinitePlace K,
+      archMean Z₁ Z₂ scaleEval v ≤ 2*C+1 ∧ archMean Z₁ Z₂ (massEval q) v ≤ 2*ε := by
+  apply weighted_select Arithmetic.archWeight _ _ C ε archWeight_pos Arithmetic.sum_archWeight
+    (archMean_nonneg Z₁ Z₂ _ scaleEval_nonneg)
+    (archMean_nonneg Z₁ Z₂ _ (massEval_nonneg q hq)) hC hε
+  · exact (archMean_le_average Z₁ Z₂ h₁ h₂ _ scaleEval_nonneg).trans
+      (by rw [average_scaleEval Z₁ Z₂ h₁ h₂ hn]; exact hscale)
+  · exact (archMean_le_average Z₁ Z₂ h₁ h₂ _ (massEval_nonneg q hq)).trans
+      (by rw [average_massEval Z₁ Z₂ h₁ h₂ hn]; exact hvar)
+
+end WeakPinned.Bounded
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PlaneColoring/Accumulation.lean b/lean/OAI/Geometry/PlaneColoring/Accumulation.lean
new file mode 100644
index 0000000000000000000000000000000000000000..fa6cce9d02f5564eab5368d385ab0f5e2aaffe2e
--- /dev/null
+++ b/lean/OAI/Geometry/PlaneColoring/Accumulation.lean
@@ -0,0 +1,401 @@
+import OAI.Geometry.PlaneColoring.Exclusion
+import OAI.Geometry.PlaneColoring.FourTransport
+
+namespace OAI
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+lemma normalized_increment_endpoint_gap {l r a : ℝ} (hlr : l<r)
+    {u δ : ℕ → ℝ} {v : ℝ} (hδ : ∀n,0<δ n)
+    (hv : (a=l ∧ 0<v) ∨ (a=r ∧ v<0))
+    (hu : Tendsto u atTop (𝓝 a))
+    (hs : Tendsto (fun n => (u n-a)/δ n) atTop (𝓝 v)) :
+    ∀ᶠn in atTop,u n∈Ioo l r := by
+  rcases hv with ⟨rfl,hv⟩ | ⟨rfl,hv⟩
+  · exact positive_normalized_increment_enters_gap hlr hδ hv hu hs
+  · filter_upwards [hu.eventually (lt_mem_nhds hlr),hs.eventually (gt_mem_nhds hv)] with n hn hp
+    exact ⟨hn,sub_neg.mp ((div_neg_iff.mp hp).resolve_left (fun h => (not_lt_of_ge (hδ n).le) h.2)).1⟩
+variable (c : ℂ → Fin 5) (hc : Measurable c)
+lemma four_support_positive (hw : WeakColoring 5 c) {x : ℂ} (hx : FourCenter c x)
+    {s : Fin 3} (hs : s≠0) : 0<volume (fourSupport c hc x s) := by
+  apply (four_realState_positive c hc hw hx hs).trans_le
+  apply measure_mono_ae
+  exact (ae_restrict_iff' (four_realState_measurable c hc x s)).mp
+    (volume.restrict (fourRealState c hc x s)).support_mem_ae
+lemma four_other_union (x : ℂ) {s : Fin 3} (hs : s≠0) :
+    fourNonSingleton c hc x=fourSupport c hc x s∪fourSupport c hc x (if s=1 then 2 else 1) := by
+  fin_cases s
+  · exact (hs rfl).elim
+  · rfl
+  · exact Set.union_comm _ _
+lemma four_endpoint_sequence_false (hw : WeakColoring 5 c)
+    {yn : ℕ → ℂ} {x : ℂ} (hyn : ∀n,FourCenter c (yn n)) (hx : FourCenter c x)
+    (hy : Tendsto yn atTop (𝓝 x)) {δ : ℕ → ℝ} {H : RealPlane} (hδ : ∀n,0<δ n)
+    (hdir : Tendsto (fun n => (δ n)⁻¹•Complex.equivRealProdCLM (yn n-x)) atTop (𝓝 H))
+    {l r a : ℝ} (hlr : l<r) (haEnd : a=l ∨ a=r)
+    (hgap : Ioo l r⊆(fourSupport c hc x 1∪fourSupport c hc x 2)ᶜ)
+    {s : Fin 3} (hs : s≠0) (ha : a∈fourSupport c hc x s)
+    (hHa : H.1*Real.cos a+H.2*Real.sin a≠0) : False := by
+  classical
+  let t : Fin 3 := if s=1 then 2 else 1
+  have ht : t≠0 := by dsimp [t]; split <;> decide
+  have hRaN := four_gap_shift_subset c hc hw hx hlr hgap
+    (show a+sixty∈Icc (l+sixty) (r+sixty) from by rcases haEnd with rfl|rfl <;> constructor <;> linarith)
+  have hRa : a+sixty∈fourSupport c hc x t := by
+    change a+sixty∈fourNonSingleton c hc x at hRaN
+    rw [four_other_union c hc x hs] at hRaN
+    exact hRaN.resolve_left (four_support_no_sixty c hc hw hx hs ha)
+  obtain ⟨b0,hb0,b1,hb1,hb0a,hb1a,hv01⟩ := exists_transportVelocity_lt
+    (four_support_positive c hc hw hx ht) hHa
+  have hp : ∃b b',b∈fourSupport c hc x t ∧ b'∈fourSupport c hc x t ∧
+      Real.sin (b-a)≠0 ∧ Real.sin (b'-a)≠0 ∧
+      ((a=l ∧ 0<transportVelocity H a b-transportVelocity H a b') ∨
+       (a=r ∧ transportVelocity H a b-transportVelocity H a b'<0)) := by
+    rcases haEnd with hal|har
+    · exact ⟨b1,b0,hb1,hb0,hb1a,hb0a,Or.inl ⟨hal,sub_pos.mpr hv01⟩⟩
+    · exact ⟨b0,b1,hb0,hb1,hb0a,hb1a,Or.inr ⟨har,sub_neg.mpr hv01⟩⟩
+  obtain ⟨b,b',hb,hb',hab,hab',hvel⟩ := hp
+  obtain ⟨bn,hbn,hbt⟩ := four_support_continuity c hc hw hyn hx hy ht hb'
+  obtain ⟨dn,hdn,hdt⟩ := four_support_continuity c hc hw hyn hx hy ht hRa
+  let Hn := fun n => Complex.equivRealProdCLM (yn n-x)
+  have hH : Tendsto Hn atTop (𝓝 0) := by
+    simpa only [sub_self,map_zero,Function.comp_def] using
+      Complex.equivRealProdCLM.continuous.continuousAt.tendsto.comp (hy.sub_const x)
+  let cn := fun n => (sumTransport hab (Hn n) (a,b)).1
+  have hcn : Tendsto cn atTop (𝓝 a) := (sumTransport_tendsto hab hH tendsto_const_nhds).fst_nhds
+  have hneg : Tendsto (fun n => -Hn n) atTop (𝓝 0) := by simpa only [neg_zero] using hH.neg
+  let un := fun n => (sumTransport hab' (-Hn n) (cn n,bn n)).1
+  have hun : Tendsto un atTop (𝓝 a) :=
+    (sumTransport_tendsto hab' hneg (hcn.prodMk_nhds hbt)).fst_nhds
+  have huGap : ∀ᶠn in atTop,un n∈Ioo l r := normalized_increment_endpoint_gap hlr hδ hvel hun
+    (sumTransport_double_increment hab hab' hH hbt hδ hdir)
+  obtain ⟨P,V,hP,hP0,hV,haV,K,hbranch⟩ := sumTransport_neighborhood hab
+  have hcN : ∀ᶠn in atTop,cn n∈fourNonSingleton c hc (yn n) := by
+    filter_upwards [hH.eventually (hP.mem_nhds hP0)] with n hn
+    exact (four_support_transport c hc hw hx (hyn n) hs ha hb hab hV haV hbranch hn).1
+  obtain ⟨P',V',hP',hP0',hV',haV',K',hbranch'⟩ := sumTransport_neighborhood hab'
+  have hcNot : ∀ᶠn in atTop,cn n∉fourSupport c hc (yn n) s := by
+    filter_upwards [hneg.eventually (hP'.mem_nhds hP0'),
+      (hcn.prodMk_nhds hbt).eventually (hV'.mem_nhds haV'),huGap] with n hn hqn hgn hcu
+    have hcenter : Complex.equivRealProdCLM (x-yn n)= -Hn n := by
+      dsimp [Hn]
+      rw [← map_neg,neg_sub]
+    have htr := four_support_transport_general c hc hw (hyn n) hx hs hab' hcu (hbn n)
+      hV' hqn hbranch' (hcenter.symm ▸ hn)
+    rw [hcenter] at htr
+    exact hgap hgn htr.1
+  have hcT : ∀ᶠn in atTop,cn n∈fourSupport c hc (yn n) t := by
+    filter_upwards [hcN,hcNot] with n hn hnot
+    rw [four_other_union c hc (yn n) hs] at hn
+    exact hn.resolve_left hnot
+  let vn := fun n => if h : cn n∈fourSupport c hc (yn n) t then cn n
+    else (four_support_nonempty c hc hw (hyn n) ht).choose
+  have hvn : ∀n,vn n∈fourSupport c hc (yn n) t := by
+    intro n
+    dsimp [vn]
+    split
+    · assumption
+    · exact (four_support_nonempty c hc hw (hyn n) ht).choose_spec
+  have hvlim : Tendsto vn atTop (𝓝 a) := hcn.congr' (by
+    filter_upwards [hcT] with n hn
+    simp only [vn,dite_eq_left hn])
+  exact four_support_sequence_no_unit c hc hw hyn hx hy ht hvlim hdt hvn hdn (direction_sixty_unit_real a)
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology
+namespace EuclideanFiveColor
+lemma gap_endpoint_nonorthogonal {H : ℝ×ℝ} (hH : H≠0) {a b : ℝ}
+    (hab : 0<b-a) (hab' : b-a<Real.pi) :
+    H.1*Real.cos a+H.2*Real.sin a≠0 ∨ H.1*Real.cos b+H.2*Real.sin b≠0 := by
+  by_contra! h
+  have hs := (Real.sin_pos_of_pos_of_lt_pi hab hab').ne'
+  have h1 : H.1*Real.sin (b-a)=0 := by
+    rw [Real.sin_sub]
+    linear_combination Real.sin b*h.1-Real.sin a*h.2
+  have h2 : H.2*Real.sin (b-a)=0 := by
+    rw [Real.sin_sub]
+    linear_combination -Real.cos b*h.1+Real.cos a*h.2
+  apply hH
+  exact Prod.ext ((mul_eq_zero.mp h1).resolve_right hs) ((mul_eq_zero.mp h2).resolve_right hs)
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+lemma triple_subset_pair {α : Type*} [DecidableEq α] {S : Finset α} {a b d : α}
+    (hS : S.card ≤ 2) (hsub : S ⊆ {a,b,d}) :
+    S ⊆ {a,b} ∨ S ⊆ {a,d} ∨ S ⊆ {b,d} := by
+  by_cases hab : a=b
+  · subst b
+    exact Or.inr (Or.inl (by simpa using hsub))
+  by_cases ha : a∈S
+  · by_cases hb : b∈S
+    · left
+      intro x hx
+      by_contra hn
+      have hxne : x≠a ∧ x≠b := by simpa only [Finset.mem_insert,Finset.mem_singleton,not_or] using hn
+      have : 3 ≤ S.card := by
+        have hh : ({a,b,x} : Finset α) ⊆ S := by simp only [Finset.insert_subset_iff,Finset.singleton_subset_iff]; exact ⟨ha,hb,hx⟩
+        have hc := Finset.card_le_card hh
+        simpa [hab,hxne.1.symm,hxne.2.symm] using hc
+      omega
+    · right; left
+      intro x hx
+      have := hsub hx
+      simp only [Finset.mem_insert,Finset.mem_singleton] at this ⊢
+      rcases this with h|h|h
+      · exact Or.inl h
+      · exact False.elim (hb (h ▸ hx))
+      · exact Or.inr h
+  · right; right
+    intro x hx
+    exact (Finset.mem_insert.mp (hsub hx)).resolve_left (fun h => ha (h ▸ hx))
+lemma triangle_binary (c : ℂ → Fin 5) (hc : Measurable c) (hw : WeakColoring 5 c)
+    {x : ℂ} {a b d i j : Fin 5}
+    (hab : TransitionEdge c a b x) (had : TransitionEdge c a d x) (hbd : TransitionEdge c b d x)
+    (hcomp : Finset.univ\{a,b,d}={i,j}) : BinaryCenter c hc x i j := by
+  filter_upwards [palette_cardinality c hc hw x,edge_palette_not_pair c hc hw hab,
+    edge_palette_not_pair c hc hw had,edge_palette_not_pair c hc hw hbd] with t ht hab had hbd
+  by_contra hn
+  have hnot : i∉palette c hc x t ∧ j∉palette c hc x t := by
+    simpa only [mem_palette,not_or] using hn
+  have hsub : palette c hc x t ⊆ {a,b,d} := by
+    intro s hs
+    by_contra hns
+    have : s∈({i,j} : Finset (Fin 5)) := by rw [← hcomp]; simp [hns]
+    simp only [Finset.mem_insert,Finset.mem_singleton] at this
+    rcases this with rfl|rfl
+    · exact hnot.1 hs
+    · exact hnot.2 hs
+  rcases triple_subset_pair ht.2 hsub with h|h|h
+  · exact hab h
+  · exact had h
+  · exact hbd h
+lemma triangle_centers_separated (c : ℂ → Fin 5) (hc : Measurable c) (hw : WeakColoring 5 c)
+    (a b d i j : Fin 5) (hcomp : Finset.univ\{a,b,d}={i,j}) :
+    ∃ δ>0, ∀ x z : ℂ, TransitionEdge c a b x → TransitionEdge c a d x → TransitionEdge c b d x →
+      TransitionEdge c a b z → TransitionEdge c a d z → TransitionEdge c b d z → x≠z → δ≤dist x z := by
+  obtain ⟨δ,hδ,hsep⟩ := binary_centers_separated c hc hw
+  exact ⟨δ,hδ,fun x z h1 h2 h3 h4 h5 h6 hne => hsep x z i j
+    (triangle_binary c hc hw h1 h2 h3 hcomp) (triangle_binary c hc hw h4 h5 h6 hcomp) hne⟩
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+variable {k : ℕ} (c : ℂ → Fin k)
+lemma transitionRatio_mono {i j : Fin k} {x y : ℂ} {R S : ℝ}
+    (hB : Metric.ball x R ⊆ Metric.ball y S) (h : ℂ) :
+    transitionRatio c i j x R h ≤ transitionRatio c i j y S h := by
+  apply ENNReal.div_le_div_right
+  exact measure_mono (fun _ hq => ⟨hB hq.1,hq.2⟩)
+lemma transitionEdge_closed (i j : Fin k) : IsClosed {x | TransitionEdge c i j x} := by
+  apply isClosed_of_closure_subset
+  intro x hx
+  have hne : i≠j := by
+    obtain ⟨y,_,hy⟩ := mem_closure_iff.mp hx Set.univ isOpen_univ (mem_univ _)
+    exact hy.1
+  refine ⟨hne,fun R hR => ?_⟩
+  obtain ⟨y,hyB,hy⟩ := mem_closure_iff.mp hx (Metric.ball x (R/2)) Metric.isOpen_ball
+    (Metric.mem_ball_self (by linarith))
+  have hB : Metric.ball y (R/2) ⊆ Metric.ball x R := by
+    intro z hz
+    have ht := dist_triangle z y x
+    have hz' : dist z y < R/2 := hz
+    have hy' : dist y x < R/2 := hyB
+    exact lt_of_le_of_lt ht (by linarith)
+  exact (hy.2 (R/2) (by linarith)).trans_le (limsup_le_limsup
+    (Eventually.of_forall (transitionRatio_mono c hB)))
+lemma missing_edge_lim (i j : Fin k) (hne : i≠j) {x : ℂ}
+    (he : ¬TransitionEdge c i j x) : ∃ R>0,
+    Tendsto (transitionRatio c i j x R) (𝓝[≠] (0:ℂ)) (𝓝 0) := by
+  obtain ⟨R,hR,hr⟩ : ∃ R, 0<R ∧ ¬0<limsup (transitionRatio c i j x R) (𝓝[≠] (0:ℂ)) := by
+    have hh : ¬∀ R, 0<R → 0<limsup (transitionRatio c i j x R) (𝓝[≠] (0:ℂ)) :=
+      fun hh => he ⟨hne,hh⟩
+    push Not at hh
+    exact ⟨hh.choose,hh.choose_spec.1,not_lt.mpr hh.choose_spec.2⟩
+  refine ⟨R,hR,?_⟩
+  have hs : limsup (transitionRatio c i j x R) (𝓝[≠] (0:ℂ))=0 := le_antisymm (not_lt.mp hr) (bot_le)
+  exact tendsto_of_le_liminf_of_limsup_le (bot_le) hs.le
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+variable (c : ℂ → Fin 5) (hc : Measurable c)
+include hc
+lemma four_normalized_sequence_false (hw : WeakColoring 5 c)
+    {yn : ℕ → ℂ} {x : ℂ} (hyn : ∀n,FourCenter c (yn n)) (hx : FourCenter c x)
+    (hy : Tendsto yn atTop (𝓝 x)) {δ : ℕ → ℝ} {H : RealPlane} (hδ : ∀n,0<δ n) (hH : H≠0)
+    (hdir : Tendsto (fun n => (δ n)⁻¹•Complex.equivRealProdCLM (yn n-x)) atTop (𝓝 H)) : False := by
+  obtain ⟨a,b,ha,hb,hab,hab',hgap⟩ := four_gap_exists c hc hw hx
+  have hp : b-a<Real.pi := hab'.trans_lt (by dsimp [sixty]; linarith [Real.pi_pos])
+  rcases gap_endpoint_nonorthogonal hH (sub_pos.mpr hab) hp with hda|hdb
+  · rcases ha with ha|ha
+    · exact four_endpoint_sequence_false c hc hw hyn hx hy hδ hdir hab (Or.inl rfl) hgap (by decide) ha hda
+    · exact four_endpoint_sequence_false c hc hw hyn hx hy hδ hdir hab (Or.inl rfl) hgap (by decide) ha hda
+  · rcases hb with hb|hb
+    · exact four_endpoint_sequence_false c hc hw hyn hx hy hδ hdir hab (Or.inr rfl) hgap (by decide) hb hdb
+    · exact four_endpoint_sequence_false c hc hw hyn hx hy hδ hdir hab (Or.inr rfl) hgap (by decide) hb hdb
+lemma four_sequence_no_accumulation (hw : WeakColoring 5 c)
+    {yn : ℕ → ℂ} {x : ℂ} (hyn : ∀n,FourCenter c (yn n)) (hx : FourCenter c x)
+    (hy : Tendsto yn atTop (𝓝 x)) (hne : ∀n,yn n≠x) : False := by
+  let Hn := fun n => Complex.equivRealProdCLM (yn n-x)
+  have hn : ∀n,Hn n≠0 := by
+    intro n h
+    apply hne n
+    apply sub_eq_zero.mp
+    exact Complex.equivRealProdCLM.injective (h.trans (map_zero _).symm)
+  let δ := fun n => ‖Hn n‖
+  have hδ : ∀n,0<δ n := fun n => norm_pos_iff.mpr (hn n)
+  have hd : ∀n,(δ n)⁻¹•Hn n∈Metric.sphere 0 1 := by
+    intro n
+    rw [mem_sphere_zero_iff_norm]
+    exact norm_smul_inv_norm (𝕜 := ℝ) (hn n)
+  obtain ⟨H,hH,φ,hφ,hlim⟩ := (isCompact_sphere (0 : RealPlane) 1).tendsto_subseq hd
+  have hH0 : H≠0 := by
+    rw [mem_sphere_zero_iff_norm] at hH
+    intro hz
+    simp only [hz,norm_zero] at hH
+    norm_num at hH
+  exact four_normalized_sequence_false c hc hw (fun n => hyn (φ n)) hx
+    (hy.comp hφ.tendsto_atTop) (fun n => hδ (φ n)) hH0 hlim
+omit hc in
+lemma fourCenter_closed : IsClosed {x | FourCenter c x} :=
+  (transitionEdge_closed c 0 1).inter ((transitionEdge_closed c 1 2).inter
+    ((transitionEdge_closed c 2 3).inter (transitionEdge_closed c 3 0)))
+lemma fourCenter_discrete (hw : WeakColoring 5 c) : IsDiscrete {x | FourCenter c x} := by
+  apply isDiscrete_iff_discreteTopology.mpr
+  apply discreteTopology_of_noAccPts
+  intro x hx ha
+  obtain ⟨yn,hyn,hy⟩ := mem_closure_iff_seq_limit.mp
+    (accPt_principal_iff_clusterPt.mp ha).mem_closure
+  exact four_sequence_no_accumulation c hc hw (fun n => (hyn n).1) hx hy
+    (fun n => by simpa only [mem_singleton_iff] using (hyn n).2)
+lemma fourCenter_compact_finite (hw : WeakColoring 5 c) {K : Set ℂ} (hK : IsCompact K) :
+    (K∩{x | FourCenter c x}).Finite :=
+  (hK.inter_right (fourCenter_closed c)).finite ((fourCenter_discrete c hc hw).mono inter_subset_right)
+
+end EuclideanFiveColor
+
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+variable {k : ℕ} (c : ℂ → Fin k) (e : Equiv.Perm (Fin k))
+lemma weak_relabel (hw : WeakColoring k c) : WeakColoring k (e ∘ c) := by
+  refine ⟨(measurable_of_finite e).comp_aemeasurable hw.1,?_⟩
+  exact hw.2.mono fun p hp h => hp (e.injective h)
+lemma transitionSet_relabel (i j : Fin k) (x : ℂ) (R : ℝ) (h : ℂ) :
+    transitionSet (e ∘ c) (e i) (e j) x R h=transitionSet c i j x R h := by
+  ext q
+  simp only [transitionSet,Set.mem_ofPred_eq,Function.comp_apply,Equiv.apply_eq_iff_eq]
+lemma transitionEdge_relabel (i j : Fin k) (x : ℂ) :
+    TransitionEdge (e ∘ c) (e i) (e j) x ↔ TransitionEdge c i j x := by
+  have he : transitionRatio (e ∘ c) (e i) (e j) x=transitionRatio c i j x := by
+    funext R h
+    unfold transitionRatio
+    rw [transitionSet_relabel]
+  simp only [TransitionEdge,he,ne_eq,Equiv.apply_eq_iff_eq]
+lemma transitionEdge_symm {i j : Fin k} {x : ℂ} (h : TransitionEdge c i j x) :
+    TransitionEdge c j i x := by
+  have he : transitionRatio c j i x=transitionRatio c i j x := by
+    funext R z
+    have hs : transitionSet c j i x R z=transitionSet c i j x R z := by
+      ext q
+      simp only [transitionSet,Set.mem_ofPred_eq,or_comm]
+    change volume _ / _ = volume _ / _
+    rw [hs]
+  exact ⟨h.1.symm,by simpa only [he] using h.2⟩
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+variable (c : ℂ → Fin 5) (hc : Measurable c)
+def ThreeCenter (x : ℂ) : Prop :=
+  TransitionEdge c 0 1 x ∧ TransitionEdge c 0 2 x ∧ TransitionEdge c 1 2 x
+lemma threeCenter_closed : IsClosed {x | ThreeCenter c x} :=
+  (transitionEdge_closed c 0 1).inter ((transitionEdge_closed c 0 2).inter (transitionEdge_closed c 1 2))
+include hc in
+lemma three_sequence_no_accumulation (hw : WeakColoring 5 c)
+    {yn : ℕ → ℂ} {x : ℂ} (hyn : ∀n,ThreeCenter c (yn n)) (hx : ThreeCenter c x)
+    (hy : Tendsto yn atTop (𝓝 x)) (hne : ∀n,yn n≠x) : False := by
+  obtain ⟨δ,hδ,hsep⟩ := triangle_centers_separated c hc hw 0 1 2 3 4 (by decide)
+  have hd : Tendsto (fun n => dist (yn n) x) atTop (𝓝 0) := by
+    simpa only [dist_self] using hy.dist (tendsto_const_nhds : Tendsto (fun _ : ℕ => x) atTop (𝓝 x))
+  have hev : ∀ᶠn in atTop,dist (yn n) x<δ := hd.eventually (gt_mem_nhds hδ)
+  obtain ⟨n,hn⟩ := hev.exists
+  exact (not_lt_of_ge (hsep (yn n) x (hyn n).1 (hyn n).2.1 (hyn n).2.2 hx.1 hx.2.1 hx.2.2 (hne n))) hn
+include hc in
+lemma threeCenter_discrete (hw : WeakColoring 5 c) : IsDiscrete {x | ThreeCenter c x} := by
+  apply isDiscrete_iff_discreteTopology.mpr
+  apply discreteTopology_of_noAccPts
+  intro x hx ha
+  obtain ⟨yn,hyn,hy⟩ := mem_closure_iff_seq_limit.mp
+    (accPt_principal_iff_clusterPt.mp ha).mem_closure
+  exact three_sequence_no_accumulation c hc hw (fun n => (hyn n).1) hx hy
+    (fun n => by simpa only [mem_singleton_iff] using (hyn n).2)
+include hc in
+lemma threeCenter_compact_finite (hw : WeakColoring 5 c) {K : Set ℂ} (hK : IsCompact K) :
+    (K∩{x | ThreeCenter c x}).Finite :=
+  (hK.inter_right (threeCenter_closed c)).finite ((threeCenter_discrete c hc hw).mono inter_subset_right)
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+variable (c : ℂ → Fin 5) (hc : Measurable c)
+def FiveCenter (x : ℂ) : Prop := ∀i : Fin 5,TransitionEdge c i (i+1) x
+lemma five_palette_admissible (hw : WeakColoring 5 c) {x : ℂ} (hx : FiveCenter c x) :
+    ∀ᵐt∂angularMeasure,FiveAdmissible (palette c hc x t) := by
+  filter_upwards [palette_nonempty c hc x,palette_cardinality c hc hw x,
+    ae_all_iff.mpr (fun i => edge_palette_not_pair c hc hw (hx i))] with t hn hs he
+  exact ⟨hn,hs.2,he⟩
+def fiveState (x : ℂ) (t : Angles) : Fin 5 := fiveCanonical (palette c hc x t)
+lemma palette_eq_measurable (x : ℂ) (p : Finset (Fin 5)) :
+    MeasurableSet {t | palette c hc x t=p} := by
+  simp only [Finset.ext_iff,Set.ofPred_forall]
+  apply MeasurableSet.iInter
+  intro i
+  by_cases hi : i∈p
+  · simpa only [hi,iff_true,mem_palette,Set.ofPred_mem_eq] using paletteSet_measurable c hc x i
+  · simpa only [hi,iff_false,mem_palette,← Set.mem_compl_iff,Set.ofPred_mem_eq] using (paletteSet_measurable c hc x i).compl
+lemma fiveState_measurable (x : ℂ) : Measurable (fiveState c hc x) := by
+  exact Measurable.ite (palette_eq_measurable c hc x _) measurable_const
+    (Measurable.ite (palette_eq_measurable c hc x _) measurable_const
+      (Measurable.ite (palette_eq_measurable c hc x _) measurable_const
+        (Measurable.ite (palette_eq_measurable c hc x _) measurable_const measurable_const)))
+lemma fiveState_diag (hw : WeakColoring 5 c) {x : ℂ} (hx : FiveCenter c x) :
+    ∀ᵐt∂angularMeasure,palette c hc x t=fiveDiag (fiveState c hc x t) :=
+  (five_palette_admissible c hc hw hx).mono fun _ ht => five_admissible_diag _ ht
+lemma fiveState_represents (hw : WeakColoring 5 c) {x : ℂ} (hx : FiveCenter c x) (i : Fin 5) :
+    RealState hc x {t : ℝ | fiveState c hc x (t : Angles)=i} (fiveDiag i) := by
+  refine ⟨measurableSet_eq_fun ((fiveState_measurable c hc x).comp angleMk_quasiMeasurePreserving.measurable) measurable_const,?_⟩
+  intro j hj
+  filter_upwards [angleMk_quasiMeasurePreserving.ae (fiveState_diag c hc hw hx)] with t ht hti
+  change fiveState c hc x (t : Angles)=i at hti
+  apply (mem_palette c hc x (t : Angles) j).mp
+  rw [ht,hti]
+  exact hj
+lemma fiveState_adj (hw : WeakColoring 5 c) {x : ℂ} (hx : FiveCenter c x) :
+    ∀ᵐt∂angularMeasure,fiveAdj (fiveState c hc x t) (fiveState c hc x (t+(sixty : Angles))) := by
+  have hs := fiveState_diag c hc hw hx
+  filter_upwards [hs,(measurePreserving_add_right angularMeasure (sixty : Angles)).quasiMeasurePreserving.ae hs,
+    palette_sixty_disjoint c hc hw x] with t ht hu hd
+  apply (five_diag_disjoint _ _).mp
+  change Disjoint (palette c hc x t) (palette c hc x (t+(sixty : Angles))) at hd
+  rwa [ht,hu] at hd
+end EuclideanFiveColor
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PlaneColoring/AngularBoundary.lean b/lean/OAI/Geometry/PlaneColoring/AngularBoundary.lean
new file mode 100644
index 0000000000000000000000000000000000000000..ae03a677ac5e1be2ef9dc97452cd68367a48d22c
--- /dev/null
+++ b/lean/OAI/Geometry/PlaneColoring/AngularBoundary.lean
@@ -0,0 +1,503 @@
+import OAI.Geometry.PlaneColoring.BoundaryTrace
+
+namespace OAI
+
+noncomputable section
+namespace EuclideanFiveColor
+/-- `0` is the outsider and `1,2` the two diagonal states. -/
+def FourWordValid (w : Fin 6 → Fin 3) : Prop :=
+  (∀j,w j≠w (j+1)) ∧ (∀j,w j≠0 → w (j+3)≠0 → w (j+3)=w j)
+instance (w : Fin 6 → Fin 3) : Decidable (FourWordValid w) := inferInstanceAs (Decidable (_∧_))
+def FourWordMixed (w : Fin 6 → Fin 3) : Prop :=
+  ∃i j,w i=0 ∧ w j=0 ∧ i.val%2≠j.val%2
+instance (w : Fin 6 → Fin 3) : Decidable (FourWordMixed w) := inferInstanceAs (Decidable (∃_i _j,_∧_∧_))
+def FourWordCross (w u : Fin 6 → Fin 3) : Prop :=
+  ∀j,w j≠0 → u (j+1)≠w j ∧ u (j-1)≠w j
+instance (w u : Fin 6 → Fin 3) : Decidable (FourWordCross w u) := inferInstanceAs (Decidable (∀_j,_→_∧_))
+lemma forall_six_word (P : (Fin 6 → Fin 3) → Prop) :
+    (∀w,P w) ↔ ∀a b c d e f : Fin 3,P ![a,b,c,d,e,f] := by
+  constructor
+  · intro h a b c d e f
+    exact h _
+  · intro h w
+    have he : w=![w 0,w 1,w 2,w 3,w 4,w 5] := by
+      ext j
+      fin_cases j <;> rfl
+    rw [he]
+    exact h _ _ _ _ _ _
+lemma four_word_has_outsider : ∀w : Fin 6 → Fin 3,FourWordValid w → ∃j,w j=0 := by
+  simp only [forall_six_word]
+  decide
+lemma four_word_mixed_periodic : ∀w : Fin 6 → Fin 3,FourWordValid w → FourWordMixed w →
+    (∀j,w (j+3)=w j) ∧ w 0≠w 1 ∧ w 1≠w 2 ∧ w 0≠w 2 := by
+  simp only [forall_six_word]
+  decide
+lemma four_word_mixed_isolated : ∀w : Fin 6 → Fin 3,FourWordValid w → FourWordMixed w →
+    ∀u : Fin 6 → Fin 3,FourWordValid u → FourWordCross w u → u=w := by
+  simp only [forall_six_word]
+  decide
+lemma four_word_parity : ∀w : Fin 6 → Fin 3,FourWordValid w → ¬FourWordMixed w →
+    ∃p : Fin 2,∀j,w j=0 → j.val%2=p.val := by
+  simp only [forall_six_word]
+  decide
+lemma four_word_shift_parity : ∀w : Fin 6 → Fin 3,∀p : Fin 2,
+    (∀j,w j=0 → j.val%2=p.val) → ∀j,w (j+1)=0 → j.val%2=(p+1).val := by
+  simp only [forall_six_word]
+  decide
+lemma four_word_opposite_parity : ∀w : Fin 6 → Fin 3,FourWordValid w →
+    (∀j,w j=0 → j.val%2=0) → ∀u : Fin 6 → Fin 3,FourWordValid u →
+    (∀j,u j=0 → j.val%2=1) → FourWordCross w u →
+    ∃a b : Fin 3,a≠0 ∧ b≠0 ∧ a≠b ∧
+      (∀j,j.val%2=1 → w j=a) ∧ (∀j,j.val%2=0 → u j=b) := by
+  simp only [forall_six_word]
+  decide
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Topology
+namespace EuclideanFiveColor
+/-- Uniform local isolation of a value on a dense subset of a connected metric
+space propagates globally. This is the manuscript dense-set chain argument. -/
+lemma dense_isolated_value_constant {E F : Type*} [MetricSpace E] [PreconnectedSpace E]
+    {G : Set E} (hG : Dense G) (f : E → F) {b : F} {ε : ℝ} (hε : 0<ε)
+    (hne : ∃x∈G,f x=b)
+    (hl : ∀x∈G,f x=b → ∀y∈G,dist x y<ε → f y=b) : ∀y∈G,f y=b := by
+  let A : Set E := {x | x∈G ∧ f x=b}
+  have hAne : A.Nonempty := hne
+  have hAopen : IsOpen (closure A) := by
+    apply isOpen_iff_forall_mem_open.mpr
+    intro x hx
+    obtain ⟨a,ha,had⟩ := Metric.mem_closure_iff.mp hx ε hε
+    have hsub : Metric.ball a ε∩G⊆A := by
+      intro y hy
+      exact ⟨hy.2,hl a ha.1 ha.2 y hy.2 (by simpa only [Metric.mem_ball,dist_comm] using hy.1)⟩
+    have hball : Metric.ball a ε⊆closure A := by
+      intro y hy
+      apply closure_mono hsub
+      exact Metric.isOpen_ball.inter_closure ⟨hy,hG y⟩
+    exact ⟨Metric.ball a ε,hball,Metric.isOpen_ball,had⟩
+  have hfull : closure A=univ := (show IsClopen (closure A) from ⟨isClosed_closure,hAopen⟩).eq_univ
+    (hAne.mono subset_closure)
+  intro y hy
+  have hym : y∈closure A := by rw [hfull]; trivial
+  obtain ⟨a,ha,had⟩ := Metric.mem_closure_iff.mp hym ε hε
+  exact hl a ha.1 ha.2 y hy (by simpa only [dist_comm] using had)
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+lemma sixty_six : (6 : ℕ) • (sixty : Angles)=0 := by
+  rw [← AddCircle.coe_nsmul]
+  convert AddCircle.coe_period (p := 2*Real.pi) using 1
+  congr 1
+  simp only [nsmul_eq_mul,sixty]
+  ring
+lemma sixty_three : (3 : ℕ) • (sixty : Angles)=(Real.pi : Angles) := by
+  rw [← AddCircle.coe_nsmul]
+  congr 1
+  simp only [nsmul_eq_mul,sixty]
+  ring
+def sixAngle (j : Fin 6) : Angles := j.val • (sixty : Angles)
+lemma sixAngle_step (j : Fin 6) : sixAngle (j+1)=sixAngle j+(sixty : Angles) := by
+  fin_cases j <;> norm_num [sixAngle,Fin.val_add,succ_nsmul]
+  simpa only [succ_nsmul,zero_nsmul,zero_add] using sixty_six.symm
+lemma sixAngle_half (j : Fin 6) : sixAngle (j+3)=sixAngle j+(Real.pi : Angles) := by
+  rw [show j+3=(j+1+1)+1 from (by decide : ∀j : Fin 6,j+3=(j+1+1)+1) j,sixAngle_step,sixAngle_step,sixAngle_step,← sixty_three]
+  simp only [succ_nsmul,zero_nsmul,zero_add]
+  abel
+lemma direction_add_pi (t : Angles) : direction (t+(Real.pi : Angles))= -direction t := by
+  obtain ⟨a,rfl⟩ := QuotientAddGroup.mk_surjective t
+  rw [← AddCircle.coe_add,direction_real,direction_real,Circle.exp_add]
+  simp
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Topology
+namespace EuclideanFiveColor
+/-- The mixed-parity exceptional words are excluded by actual local isolation
+and sixty-degree covariance on a dense set. -/
+lemma four_words_no_mixed (G : Set Angles) (hG : Dense G) (w : Angles → Fin 6 → Fin 3)
+    (hv : ∀t∈G,FourWordValid (w t))
+    (hshiftG : ∀t∈G,t+(sixty : Angles)∈G)
+    (hshift : ∀t∈G,∀j,w (t+(sixty : Angles)) j=w t (j+1))
+    {ε : ℝ} (hε : 0<ε)
+    (hcross : ∀t∈G,∀u∈G,dist t u<ε → FourWordCross (w t) (w u)) :
+    ∀t∈G,¬FourWordMixed (w t) := by
+  intro t ht hm
+  have he : ∀u∈G,w u=w t := by
+    apply dense_isolated_value_constant hG w hε ⟨t,ht,rfl⟩
+    intro u hu hue v hv' hdist
+    exact (four_word_mixed_isolated (w u) (hv u hu) (by rw [hue]; exact hm)
+      (w v) (hv v hv') (hcross u hu v hv' hdist)).trans hue
+  have hconst := congrFun (he (t+(sixty : Angles)) (hshiftG t ht)) 0
+  rw [hshift t ht 0] at hconst
+  exact (hv t ht).1 0 hconst.symm
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+open scoped RealInnerProductSpace
+namespace EuclideanFiveColor
+/-- The source's strict min/max straddling region, in equivalent witness form. -/
+def Straddle (K : Set ℂ) : Set ℂ := {z | ∃p∈K,∃q∈K,dist z p<1 ∧ 1<dist z q}
+lemma straddle_isOpen (K : Set ℂ) : IsOpen (Straddle K) := by
+  have he : Straddle K=⋃p∈K,⋃q∈K,{z | dist z p<1 ∧ 1<dist z q} := by
+    ext z
+    simp only [Straddle,mem_iUnion,mem_ofPred_eq,exists_prop]
+  rw [he]
+  exact isOpen_iUnion fun _ => isOpen_iUnion fun _ => isOpen_iUnion fun _ => isOpen_iUnion fun _ =>
+    (isOpen_lt (continuous_id.dist continuous_const) continuous_const).inter
+      (isOpen_lt continuous_const (continuous_id.dist continuous_const))
+lemma straddle_connected_unit {K : Set ℂ} (hK : IsPreconnected K) {z : ℂ} (hz : z∈Straddle K) :
+    ∃p∈K,dist z p=1 := by
+  obtain ⟨p,hp,q,hq,hp1,hq1⟩ := hz
+  exact hK.intermediate_value hp hq (continuous_const.dist continuous_id).continuousOn ⟨hp1.le,hq1.le⟩
+lemma approaching_direction_inside {r : ℕ → ℝ} {w : ℕ → ℂ} {v e : ℂ}
+    (hr : ∀n,0<r n) (hr0 : Tendsto r atTop (𝓝 0)) (hw : ∀n,‖w n‖=1)
+    (hwv : Tendsto w atTop (𝓝 v)) (he : ‖e‖=1) (hv : 0<⟪e,v⟫) :
+    ∀ᶠn in atTop,‖e-(r n : ℂ)*w n‖<1 := by
+  have ht : Tendsto (fun n => 2*⟪e,w n⟫-r n) atTop (𝓝 (2*⟪e,v⟫)) := by
+    simpa only [sub_zero] using ((tendsto_const_nhds.inner hwv).const_mul 2).sub hr0
+  filter_upwards [ht.eventually (lt_mem_nhds (by positivity : (0:ℝ)<2*⟪e,v⟫))] with n hn
+  have heq : ‖e-(r n : ℂ)*w n‖^2=1-2*r n*⟪e,w n⟫+(r n)^2 := by
+    change ‖e-r n • w n‖^2= _
+    rw [norm_sub_sq_real,inner_smul_right,norm_smul,Real.norm_eq_abs,abs_of_pos (hr n),he,hw]
+    ring
+  have hpos := mul_pos (hr n) hn
+  nlinarith [norm_nonneg (e-(r n : ℂ)*w n)]
+lemma approaching_direction_outside {r : ℕ → ℝ} {w : ℕ → ℂ} {v e : ℂ}
+    (hr : ∀n,0<r n) (hr0 : Tendsto r atTop (𝓝 0)) (hw : ∀n,‖w n‖=1)
+    (hwv : Tendsto w atTop (𝓝 v)) (he : ‖e‖=1) (hv : ⟪e,v⟫<0) :
+    ∀ᶠn in atTop,1<‖e-(r n : ℂ)*w n‖ := by
+  have ht : Tendsto (fun n => 2*⟪e,w n⟫-r n) atTop (𝓝 (2*⟪e,v⟫)) := by
+    simpa only [sub_zero] using ((tendsto_const_nhds.inner hwv).const_mul 2).sub hr0
+  filter_upwards [ht.eventually (gt_mem_nhds (by linarith : 2*⟪e,v⟫<(0:ℝ)))] with n hn
+  have heq : ‖e-(r n : ℂ)*w n‖^2=1-2*r n*⟪e,w n⟫+(r n)^2 := by
+    change ‖e-r n • w n‖^2= _
+    rw [norm_sub_sq_real,inner_smul_right,norm_smul,Real.norm_eq_abs,abs_of_pos (hr n),he,hw]
+    ring
+  have hneg := mul_neg_of_pos_of_neg (hr n) hn
+  nlinarith [norm_nonneg (e-(r n : ℂ)*w n)]
+lemma outer_straddle_strip {K : Set ℂ} {x e q : ℂ} (hx : x∈K) (hq : q∈K)
+    (_he : ‖e‖=1) (hed : dist (x+e) q<1) :
+    ∃δ>0,∀e' : ℂ,‖e'‖=1 → dist e' e<δ → ∀r : ℝ,1<r → r<1+δ →
+      x+(r : ℂ)*e'∈Straddle K := by
+  have hn : ∀ᶠp : ℝ × ℂ in 𝓝 (1,e),dist (x+(p.1 : ℂ)*p.2) q<1 :=
+    (show Continuous (fun p : ℝ × ℂ => dist (x+(p.1 : ℂ)*p.2) q) by fun_prop).continuousAt.eventually
+      (gt_mem_nhds (by simpa only [Complex.ofReal_one,one_mul] using hed))
+  obtain ⟨δ,hδ,hsub⟩ := Metric.mem_nhds_iff.mp hn
+  refine ⟨δ,hδ,fun e' he' heδ r hr hrδ => ⟨q,hq,x,hx,?_,?_⟩⟩
+  · apply hsub (a := (r,e'))
+    rw [Metric.mem_ball,Prod.dist_eq,max_lt_iff]
+    exact ⟨by rw [Real.dist_eq,abs_of_pos (by linarith : 0<r-1)]; linarith,heδ⟩
+  · rw [dist_eq_norm,add_sub_cancel_left,norm_mul,Complex.norm_real,Real.norm_eq_abs,abs_of_pos (by linarith : 0<r),he',mul_one]
+    exact hr
+lemma inner_straddle_strip {K : Set ℂ} {x e q : ℂ} (hx : x∈K) (hq : q∈K)
+    (_he : ‖e‖=1) (hed : 1<dist (x+e) q) :
+    ∃δ>0,∀e' : ℂ,‖e'‖=1 → dist e' e<δ → ∀r : ℝ,0<r → 1-δ<r → r<1 →
+      x+(r : ℂ)*e'∈Straddle K := by
+  have hn : ∀ᶠp : ℝ × ℂ in 𝓝 (1,e),1<dist (x+(p.1 : ℂ)*p.2) q :=
+    (show Continuous (fun p : ℝ × ℂ => dist (x+(p.1 : ℂ)*p.2) q) by fun_prop).continuousAt.eventually
+      (lt_mem_nhds (by simpa only [Complex.ofReal_one,one_mul] using hed))
+  obtain ⟨δ,hδ,hsub⟩ := Metric.mem_nhds_iff.mp hn
+  refine ⟨δ,hδ,fun e' he' heδ r hr hrδ hr1 => ⟨x,hx,q,hq,?_,?_⟩⟩
+  · rw [dist_eq_norm,add_sub_cancel_left,norm_mul,Complex.norm_real,Real.norm_eq_abs,abs_of_pos hr,he',mul_one]
+    exact hr1
+  · apply hsub (a := (r,e'))
+    rw [Metric.mem_ball,Prod.dist_eq,max_lt_iff]
+    exact ⟨by rw [Real.dist_eq,abs_of_neg (by linarith : r-1<0)]; linarith,heδ⟩
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+open scoped RealInnerProductSpace
+namespace EuclideanFiveColor
+lemma finite_positive_lower_bound {ι : Type*} (E : Finset ι) (d : ι → ℝ)
+    (hd : ∀i∈E,0<d i) : ∃δ>0,δ≤1 ∧ ∀i∈E,δ≤d i := by
+  classical
+  induction E using Finset.induction_on with
+  | empty => exact ⟨1,by norm_num,le_rfl,by simp⟩
+  | @insert a E ha ih =>
+    obtain ⟨δ,hδ,hδ1,hδd⟩ := ih (fun i hi => hd i (Finset.mem_insert_of_mem hi))
+    refine ⟨min δ (d a),lt_min hδ (hd a (Finset.mem_insert_self _ _)),(min_le_left _ _).trans hδ1,?_⟩
+    intro i hi
+    rcases Finset.mem_insert.mp hi with rfl|hi
+    · exact min_le_right _ _
+    · exact (min_le_left _ _).trans (hδd i hi)
+/-- The approach data furnished by the manuscript interface proposition, with no
+assumption of angular strip conclusions. -/
+structure ApproachingContinuum (x : ℂ) (K : Set ℂ) (v : ℂ) where
+  center_mem : x∈K
+  r : ℕ → ℝ
+  w : ℕ → ℂ
+  r_pos : ∀n,0<r n
+  r_tendsto : Tendsto r atTop (𝓝 0)
+  w_unit : ∀n,‖w n‖=1
+  w_tendsto : Tendsto w atTop (𝓝 v)
+  point_mem : ∀n,x+(r n : ℂ)*w n∈K
+lemma ApproachingContinuum.outer_strip {x : ℂ} {K : Set ℂ} {v e : ℂ}
+    (h : ApproachingContinuum x K v) (he : ‖e‖=1) (hv : 0<⟪e,v⟫) :
+    ∃δ>0,∀e' : ℂ,‖e'‖=1 → dist e' e<δ → ∀r : ℝ,1<r → r<1+δ →
+      x+(r : ℂ)*e'∈Straddle K := by
+  obtain ⟨n,hn⟩ := (approaching_direction_inside h.r_pos h.r_tendsto h.w_unit h.w_tendsto he hv).exists
+  apply outer_straddle_strip h.center_mem (h.point_mem n) he
+  simpa only [dist_eq_norm,add_sub_add_left_eq_sub] using hn
+lemma ApproachingContinuum.inner_strip {x : ℂ} {K : Set ℂ} {v e : ℂ}
+    (h : ApproachingContinuum x K v) (he : ‖e‖=1) (hv : ⟪e,v⟫<0) :
+    ∃δ>0,∀e' : ℂ,‖e'‖=1 → dist e' e<δ → ∀r : ℝ,0<r → 1-δ<r → r<1 →
+      x+(r : ℂ)*e'∈Straddle K := by
+  obtain ⟨n,hn⟩ := (approaching_direction_outside h.r_pos h.r_tendsto h.w_unit h.w_tendsto he hv).exists
+  apply inner_straddle_strip h.center_mem (h.point_mem n) he
+  simpa only [dist_eq_norm,add_sub_add_left_eq_sub] using hn
+variable {k : ℕ} (c : ℂ → Fin k) (hc : Measurable c)
+lemma continuum_outer_restriction {ι : Type*} [Fintype ι] (E : Finset ι)
+    {x : ℂ} {K : ι → Set ℂ} {v : ι → ℂ} (hK : ∀i,ApproachingContinuum x (K i) (v i))
+    {J : Finset (Fin k)}
+    (hJ : ∀ᵐz : ℂ∂volume,(∀i∈E,z∈Straddle (K i)) → c z∈J) :
+    ∀ᵐt∂angularMeasure,(∀i∈E,0<⟪direction t,v i⟫) → (palette c hc x t∩J).Nonempty := by
+  classical
+  apply ae_of_locally_ae angularMeasure
+  intro t ht
+  have hex (i : E) := (hK i).outer_strip (direction_norm t) (ht i i.2)
+  choose d hd hs using hex
+  obtain ⟨δ,hδ,_,hδd⟩ := finite_positive_lower_bound Finset.univ d (fun i _ => hd i)
+  let W : Set ℂ := ⋂i : E,Straddle (K i)
+  have ho : IsOpen W := isOpen_iInter_of_finite fun i => straddle_isOpen (K i)
+  have hW : ∀ᵐz : ℂ∂volume,z∈W → c z∈J := hJ.mono fun z hz hzW => hz
+    (fun i hi => mem_iInter.mp hzW ⟨i,hi⟩)
+  exact outer_strip_trace c hc hδ ho hW (fun u hu r hr hrδ => by
+    apply mem_iInter.mpr
+    intro i
+    apply hs i (direction u) (direction_norm u) (hu.trans_le (hδd i (Finset.mem_univ _))) r hr
+    linarith [hδd i (Finset.mem_univ _)])
+lemma continuum_inner_restriction {ι : Type*} [Fintype ι] (E : Finset ι)
+    {x : ℂ} {K : ι → Set ℂ} {v : ι → ℂ} (hK : ∀i,ApproachingContinuum x (K i) (v i))
+    {J : Finset (Fin k)}
+    (hJ : ∀ᵐz : ℂ∂volume,(∀i∈E,z∈Straddle (K i)) → c z∈J) :
+    ∀ᵐt∂angularMeasure,(∀i∈E,⟪direction t,v i⟫<0) → (palette c hc x t∩J).Nonempty := by
+  classical
+  apply ae_of_locally_ae angularMeasure
+  intro t ht
+  have hex (i : E) := (hK i).inner_strip (direction_norm t) (ht i i.2)
+  choose d hd hs using hex
+  obtain ⟨δ,hδ,_,hδd⟩ := finite_positive_lower_bound Finset.univ d (fun i _ => hd i)
+  let W : Set ℂ := ⋂i : E,Straddle (K i)
+  have ho : IsOpen W := isOpen_iInter_of_finite fun i => straddle_isOpen (K i)
+  have hW : ∀ᵐz : ℂ∂volume,z∈W → c z∈J := hJ.mono fun z hz hzW => hz
+    (fun i hi => mem_iInter.mp hzW ⟨i,hi⟩)
+  exact inner_strip_trace c hc hδ ho hW (fun u hu r hr hrδ hr1 => by
+    apply mem_iInter.mpr
+    intro i
+    apply hs i (direction u) (direction_norm u) (hu.trans_le (hδd i (Finset.mem_univ _))) r hr
+    · linarith [hδd i (Finset.mem_univ _)]
+    · exact hr1)
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+variable {k : ℕ} (c : ℂ → Fin k)
+/-- The manuscript's essential spatial color support. -/
+def spatialSupport (i : Fin k) : Set ℂ := (volume.restrict {z | c z=i}).support
+lemma mem_spatialSupport_ae (hc : Measurable c) (i : Fin k) :
+    ∀ᵐz : ℂ∂volume,c z=i → z∈spatialSupport c i :=
+  (ae_restrict_iff' (measurableSet_eq_fun hc measurable_const)).mp
+    (volume.restrict {z | c z=i}).support_mem_ae
+lemma support_exclusion_ae (hc : Measurable c) {i : Fin k} {W : Set ℂ}
+    (h : Disjoint (spatialSupport c i) W) : ∀ᵐz : ℂ∂volume,z∈W → c z≠i := by
+  filter_upwards [mem_spatialSupport_ae c hc i] with z hz hzW hzcol
+  exact Set.disjoint_left.mp h (hz hzcol) hzW
+lemma finite_support_exclusion_ae (hc : Measurable c) {ι : Type*} [Fintype ι]
+    (K : ι → Set ℂ) (F : ι → Finset (Fin k))
+    (h : ∀i,∀j∈F i,Disjoint (spatialSupport c j) (Straddle (K i))) :
+    ∀ᵐz : ℂ∂volume,∀i,z∈Straddle (K i) → c z∉F i := by
+  have hh : ∀i,∀j,∀ᵐz : ℂ∂volume,j∈F i → z∈Straddle (K i) → c z≠j := by
+    intro i j
+    by_cases hj : j∈F i
+    · exact (support_exclusion_ae c hc (h i j hj)).mono fun z hz _ => hz
+    · exact Eventually.of_forall fun _ h => (hj h).elim
+  filter_upwards [ae_all_iff.mpr (fun i => ae_all_iff.mpr (hh i))] with z hz i hi hj
+  exact hz i (c z) hj hi rfl
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+def angleEssential (s : Angles → Bool) (b : Bool) : Set Angles :=
+  {t | ∃U : Set Angles,IsOpen U ∧ t∈U ∧ ∀ᵐu∂angularMeasure,u∈U → s u=b}
+lemma angleEssential_open (s : Angles → Bool) (b : Bool) : IsOpen (angleEssential s b) := by
+  apply isOpen_iff_forall_mem_open.mpr
+  rintro t ⟨U,hU,ht,ha⟩
+  exact ⟨U,fun u hu => ⟨U,hU,hu,ha⟩,hU,ht⟩
+lemma angleEssential_ae (s : Angles → Bool) (b : Bool) :
+    ∀ᵐt∂angularMeasure,t∈angleEssential s b → s t=b := ae_of_locally_ae angularMeasure (fun _ h => h)
+lemma angleEssential_disjoint (s : Angles → Bool) {a b : Bool} (hab : a≠b) :
+    Disjoint (angleEssential s a) (angleEssential s b) := by
+  apply disjoint_left.mpr
+  rintro t ⟨U,hU,ht,ha⟩ ⟨V,hV,ht',hb⟩
+  have hpos := (hU.inter hV).measure_pos angularMeasure ⟨t,ht,ht'⟩
+  obtain ⟨u,hu,huab⟩ := Measure.exists_mem_of_measure_ne_zero_of_ae hpos.ne' (ae_restrict_of_ae (ha.and hb))
+  exact hab ((huab.1 hu.1).symm.trans (huab.2 hu.2))
+def angleBoundary (s : Angles → Bool) : Set Angles := (⋃b,angleEssential s b)ᶜ
+lemma angleBoundary_closed (s : Angles → Bool) : IsClosed (angleBoundary s) :=
+  (isOpen_iUnion (angleEssential_open s)).isClosed_compl
+lemma angleEssential_shift {s : Angles → Bool}
+    (hf : ∀ᵐt∂angularMeasure,s (t+(sixty : Angles))= !(s t)) (t : Angles) (b : Bool) :
+    t+(sixty : Angles)∈angleEssential s (!b) ↔ t∈angleEssential s b := by
+  constructor
+  · rintro ⟨U,hU,ht,ha⟩
+    refine ⟨(fun q : Angles => q+(sixty : Angles)) ⁻¹' U,hU.preimage (continuous_id.add continuous_const),ht,?_⟩
+    filter_upwards [(measurePreserving_add_right angularMeasure (sixty : Angles)).quasiMeasurePreserving.ae ha,hf] with q hq hqf hqU
+    have hh := hq hqU
+    rw [hqf] at hh
+    exact Bool.not_injective hh
+  · rintro ⟨U,hU,ht,ha⟩
+    refine ⟨(fun q : Angles => q-(sixty : Angles)) ⁻¹' U,hU.preimage (continuous_id.sub continuous_const),by simpa,?_⟩
+    have hh := (measurePreserving_add_right angularMeasure (-(sixty : Angles))).quasiMeasurePreserving.ae (ha.and hf)
+    filter_upwards [hh] with q hq hqU
+    have hcol := hq.1 hqU
+    have hflip := hq.2
+    change s (q-(sixty : Angles))=b at hcol
+    simpa only [←sub_eq_add_neg,sub_add_cancel,hcol] using hflip
+lemma angleBoundary_shift {s : Angles → Bool}
+    (hf : ∀ᵐt∂angularMeasure,s (t+(sixty : Angles))= !(s t)) (t : Angles) :
+    t+(sixty : Angles)∈angleBoundary s ↔ t∈angleBoundary s := by
+  simp only [angleBoundary,mem_compl_iff,mem_iUnion,not_exists]
+  constructor
+  · intro h b hb
+    exact h (!b) ((angleEssential_shift hf t b).mpr hb)
+  · intro h b hb
+    apply h (!b)
+    apply (angleEssential_shift hf t (!b)).mp
+    simpa only [Bool.not_not] using hb
+lemma angleBoundary_nonempty {s : Angles → Bool}
+    (hf : ∀ᵐt∂angularMeasure,s (t+(sixty : Angles))= !(s t)) : (angleBoundary s).Nonempty := by
+  by_contra hn
+  have hcov : (univ : Set Angles)⊆angleEssential s false∪angleEssential s true := by
+    intro t _
+    have ht : t∉angleBoundary s := fun h => hn ⟨t,h⟩
+    simp only [angleBoundary,mem_compl_iff,not_not,mem_iUnion] at ht
+    obtain ⟨b,hb⟩ := ht
+    cases b
+    · exact Or.inl hb
+    · exact Or.inr hb
+  have hzero := hcov (mem_univ (0:Angles))
+  have hconst : ∃b,∀ᵐt∂angularMeasure,s t=b := by
+    rcases hzero with h0|h0
+    · have hs := isPreconnected_univ.subset_left_of_subset_union (angleEssential_open s false)
+        (angleEssential_open s true) (angleEssential_disjoint s (by decide)) hcov ⟨0,mem_univ _,h0⟩
+      exact ⟨false,(angleEssential_ae s false).mono fun t ht => ht (hs (mem_univ _))⟩
+    · have hs := isPreconnected_univ.subset_left_of_subset_union (angleEssential_open s true)
+        (angleEssential_open s false) (angleEssential_disjoint s (by decide))
+        (by simpa only [union_comm] using hcov) ⟨0,mem_univ _,h0⟩
+      exact ⟨true,(angleEssential_ae s true).mono fun t ht => ht (hs (mem_univ _))⟩
+  obtain ⟨b,hb⟩ := hconst
+  have hh : ∀ᵐt∂angularMeasure,False := by
+    filter_upwards [hb,(measurePreserving_add_right angularMeasure (sixty : Angles)).quasiMeasurePreserving.ae hb,hf] with t ht ht' he
+    rw [ht,ht'] at he
+    cases b <;> cases he
+  exact hh.exists.choose_spec
+lemma sixAngle_add (i j : Fin 6) : sixAngle (i+j)=sixAngle i+sixAngle j := by
+  induction j using Fin.induction with
+  | zero => simp [sixAngle]
+  | succ j hj =>
+    have he : j.succ=(j.castSucc : Fin 6)+1 := by
+      apply Fin.ext
+      simp only [Fin.val_succ,Fin.val_add,Fin.val_castSucc,Fin.val_one]
+      exact (Nat.mod_eq_of_lt (by omega)).symm
+    rw [he,←add_assoc,sixAngle_step,sixAngle_step,hj]
+    abel
+lemma sixty_order : addOrderOf (sixty : Angles)=6 := by
+  convert (AddCircle.addOrderOf_period_div (p:=2*Real.pi) (by norm_num : 0<(6:ℕ))) using 1
+  congr 2
+  dsimp [sixty]
+  ring
+lemma sixAngle_injective : Function.Injective sixAngle := by
+  intro i j hij
+  apply Fin.ext
+  apply nsmul_injOn_Iio_addOrderOf (x := (sixty : Angles))
+  · simpa only [sixty_order,mem_Iio] using i.isLt
+  · simpa only [sixty_order,mem_Iio] using j.isLt
+  · exact hij
+def sixAngleHom : Fin 6 →+ Angles where
+  toFun := sixAngle
+  map_zero' := rfl
+  map_add' := sixAngle_add
+lemma sixAngle_sub (i j : Fin 6) : sixAngle (i-j)=sixAngle i-sixAngle j := sixAngleHom.map_sub i j
+lemma angleBoundary_six {s : Angles → Bool}
+    (hf : ∀ᵐt∂angularMeasure,s (t+(sixty : Angles))= !(s t)) {t : Angles} (ht : t∈angleBoundary s)
+    (j : Fin 6) : t+sixAngle j∈angleBoundary s := by
+  have hh (n : ℕ) : t+n•(sixty : Angles)∈angleBoundary s := by
+    induction n with
+    | zero => simpa using ht
+    | succ n ih =>
+      simpa only [add_nsmul,one_nsmul,←add_assoc] using (angleBoundary_shift hf (t+n•(sixty : Angles))).mpr ih
+  exact hh j.val
+lemma six_antipodal_vertex_cover : ∀b : Fin 6 → Bool,
+    (∀j,b (j+3)=b j) → (∀j,b j=true ∨ b (j+1)=true) →
+      4≤(Finset.univ.filter fun j => b j=true).card := by decide
+/-- Core finite obstruction behind the one-orbit boundary theorem. -/
+lemma angleBoundary_one_orbit {s : Angles → Bool}
+    (hf : ∀ᵐt∂angularMeasure,s (t+(sixty : Angles))= !(s t)) (B : Finset Angles)
+    (hB : B.card≤6) (hanti : ∀t,t+(Real.pi : Angles)∈B ↔ t∈B)
+    (hl : ∀t,t∉B → t+(sixty : Angles)∉B → ∃b,t∈angleEssential s b) :
+    ∃t,angleBoundary s={u | ∃j : Fin 6,u=t+sixAngle j} := by
+  classical
+  obtain ⟨t,ht⟩ := angleBoundary_nonempty hf
+  let O (q : Angles) : Finset Angles := Finset.univ.image (fun j : Fin 6 => q+sixAngle j)
+  have hcard (q : Angles) (hq : q∈angleBoundary s) : 4≤(B∩O q).card := by
+    let b : Fin 6 → Bool := fun j => decide (q+sixAngle j∈B)
+    have ha (j) : b (j+3)=b j := by
+      simp only [b,sixAngle_half,←add_assoc,hanti]
+    have hc (j) : b j=true ∨ b (j+1)=true := by
+      by_contra hn
+      have hh : q+sixAngle j∉B ∧ q+sixAngle (j+1)∉B := by simpa only [b,decide_eq_true_eq,not_or] using hn
+      obtain ⟨v,hv⟩ := hl (q+sixAngle j) hh.1 (by simpa only [sixAngle_step,←add_assoc] using hh.2)
+      exact (angleBoundary_six hf hq j) (mem_iUnion.mpr ⟨v,hv⟩)
+    have hcount := six_antipodal_vertex_cover b ha hc
+    have hi : Function.Injective (fun j : Fin 6 => q+sixAngle j) := (add_right_injective q).comp sixAngle_injective
+    have him : (Finset.univ.filter fun j => b j=true).image (fun j : Fin 6 => q+sixAngle j)=B∩O q := by
+      ext u
+      simp only [Finset.mem_image,Finset.mem_filter,Finset.mem_univ,true_and,b,decide_eq_true_eq,
+        Finset.mem_inter,O]
+      constructor
+      · rintro ⟨j,hj,rfl⟩; exact ⟨hj,j,rfl⟩
+      · rintro ⟨hu,j,rfl⟩; exact ⟨j,hu,rfl⟩
+    rw [←him,Finset.card_image_of_injective _ hi]
+    exact hcount
+  refine ⟨t,Set.ext fun u => ?_⟩
+  constructor
+  · intro hu
+    by_contra hn
+    have hd : Disjoint (O t) (O u) := by
+      apply Finset.disjoint_left.mpr
+      intro q hqt hqu
+      obtain ⟨i,_,hi⟩ := Finset.mem_image.mp hqt
+      obtain ⟨j,_,hj⟩ := Finset.mem_image.mp hqu
+      apply hn
+      refine ⟨i-j,?_⟩
+      rw [sixAngle_sub]
+      have he : t+sixAngle i=u+sixAngle j := hi.trans hj.symm
+      calc u = (u+sixAngle j)-sixAngle j := by abel
+           _ = (t+sixAngle i)-sixAngle j := by rw [he]
+           _ = t+(sixAngle i-sixAngle j) := by abel
+    have hd' : Disjoint (B∩O t) (B∩O u) := hd.mono Finset.inter_subset_right Finset.inter_subset_right
+    have hh := Finset.card_le_card (show (B∩O t)∪(B∩O u)⊆B from Finset.union_subset Finset.inter_subset_left Finset.inter_subset_left)
+    rw [Finset.card_union_of_disjoint hd'] at hh
+    have hct := hcard t ht
+    have hcu := hcard u hu
+    omega
+  · rintro ⟨j,rfl⟩
+    exact angleBoundary_six hf ht j
+end EuclideanFiveColor
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PlaneColoring/Assembly.lean b/lean/OAI/Geometry/PlaneColoring/Assembly.lean
new file mode 100644
index 0000000000000000000000000000000000000000..bcbfa3ede0ca03b8cda0c2db82fdaec6a2e2e56d
--- /dev/null
+++ b/lean/OAI/Geometry/PlaneColoring/Assembly.lean
@@ -0,0 +1,43 @@
+import Mathlib
+
+namespace OAI
+
+noncomputable section
+
+namespace Problem160
+
+/-- A covering by small cells, whose equally colored distinct centers are far
+apart, gives a proper coloring of the unit-distance graph. The covering need
+not form a partition: classical choice assigns boundary points to one cell. -/
+theorem exists_unit_distance_coloring_of_center_cover
+    {X I : Type*} [PseudoMetricSpace X] {k : ℕ}
+    (center : I → X) (color : I → Fin k) (r : ℝ)
+    (hcover : ∀ x : X, ∃ i : I, dist x (center i) ≤ r)
+    (hdiam : 2 * r < 1)
+    (hsep : ∀ i j : I, i ≠ j → color i = color j →
+      1 + 2 * r < dist (center i) (center j)) :
+    ∃ c : X → Fin k, ∀ x y : X, dist x y = 1 → c x ≠ c y := by
+  classical
+  choose cell hcell using hcover
+  refine ⟨fun x => color (cell x), ?_⟩
+  intro x y hdist hcolor
+  by_cases hsame : cell x = cell y
+  · have htriangle := dist_triangle x (center (cell x)) y
+    have hx := hcell x
+    have hy := hcell y
+    rw [← hsame] at hy
+    rw [dist_comm (center (cell x)) y] at htriangle
+    linarith
+  · have hfar := hsep (cell x) (cell y) hsame hcolor
+    have htriangle₁ := dist_triangle (center (cell x)) x (center (cell y))
+    have htriangle₂ := dist_triangle x y (center (cell y))
+    have hx := hcell x
+    have hy := hcell y
+    rw [dist_comm (center (cell x)) x] at htriangle₁
+    linarith
+
+end Problem160
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PlaneColoring/Basic.lean b/lean/OAI/Geometry/PlaneColoring/Basic.lean
new file mode 100644
index 0000000000000000000000000000000000000000..1268574a95fe674cc78d9db55d0762b8632eefd1
--- /dev/null
+++ b/lean/OAI/Geometry/PlaneColoring/Basic.lean
@@ -0,0 +1,47 @@
+import Mathlib
+
+namespace OAI
+
+noncomputable section
+
+open MeasureTheory
+
+namespace Problem160
+
+abbrev Plane := EuclideanSpace ℝ (Fin 2)
+
+def circleDir (t : ℝ) : Plane :=
+  WithLp.toLp 2 ![Real.cos t, Real.sin t]
+
+def normalizedAngleMeasure : Measure ℝ :=
+  (ENNReal.ofReal (2 * Real.pi))⁻¹ •
+    Measure.restrict volume (Set.Ico 0 (2 * Real.pi))
+
+def circleMeasure : Measure Plane :=
+  Measure.map circleDir normalizedAngleMeasure
+
+def weakPairMeasure (R : ℝ) : Measure (Plane × Plane) :=
+  (Measure.restrict volume (Metric.ball (0 : Plane) R)).prod circleMeasure
+
+def ProperColoring (k : ℕ) : Prop :=
+  ∃ c : Plane → Fin k,
+    ∀ x y : Plane, dist x y = 1 → c x ≠ c y
+
+def WeakMeasurableColoring (k : ℕ) : Prop :=
+  ∃ c : Plane → Fin k,
+    AEMeasurable c volume ∧
+      ∀ R : ℝ, 0 < R →
+        (∑ i : Fin k,
+          (weakPairMeasure R)
+            {p : Plane × Plane |
+              c (Prod.fst p) = i ∧
+              c (Prod.fst p + Prod.snd p) = i}) = 0
+
+def planeChromaticNumber : ℕ :=
+  sInf {k : ℕ | ProperColoring k}
+
+end Problem160
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PlaneColoring/BinaryDynamics.lean b/lean/OAI/Geometry/PlaneColoring/BinaryDynamics.lean
new file mode 100644
index 0000000000000000000000000000000000000000..377efd5ee35520384d2d92cd4a02b2c44c3ab5a7
--- /dev/null
+++ b/lean/OAI/Geometry/PlaneColoring/BinaryDynamics.lean
@@ -0,0 +1,515 @@
+import OAI.Geometry.PlaneColoring.Expansion
+
+namespace OAI
+
+noncomputable section
+open Set Filter Topology
+namespace EuclideanFiveColor
+def binaryQ (d t : ℝ) : ℝ := 1-2*d*Real.cos t+d^2
+def binaryR (d t : ℝ) : ℝ := 2*Real.sqrt (binaryQ d t)*Real.sqrt (1-binaryQ d t/4)
+def binaryB (d t : ℝ) : ℝ := (Real.cos t-d)/binaryQ d t+Real.sin t/binaryR d t
+def binaryM (d t : ℝ) : ℝ := Real.sin t/binaryQ d t+(d-Real.cos t)/binaryR d t
+def binaryCircleMap (d t : ℝ) : ℝ :=
+  t+Real.pi/3+Real.arctan (d*Real.sin t/(1-d*Real.cos t))-Real.arccos (Real.sqrt (binaryQ d t)/2)
+lemma binaryQ_range {d : ℝ} (hd : |d| < 1/2) (t : ℝ) :
+    0 < binaryQ d t ∧ binaryQ d t < 4 ∧ 0 < 1-d*Real.cos t := by
+  have hdc : |d*Real.cos t| ≤ |d| := by
+    simpa only [abs_mul,mul_one] using mul_le_mul_of_nonneg_left (Real.abs_cos_le_one t) (abs_nonneg d)
+  have hdc' := abs_le.mp hdc
+  have hd' := abs_lt.mp hd
+  dsimp [binaryQ]
+  constructor
+  · nlinarith [sq_nonneg d]
+  constructor
+  · nlinarith [sq_abs d,abs_nonneg d]
+  · linarith
+lemma binaryR_pos {d : ℝ} (hd : |d| < 1/2) (t : ℝ) : 0 < binaryR d t := by
+  have h := binaryQ_range hd t
+  unfold binaryR
+  exact mul_pos (mul_pos (by norm_num) (Real.sqrt_pos.mpr h.1)) (Real.sqrt_pos.mpr (by linarith [h.2.1]))
+lemma binary_arctan_identity (d t : ℝ) :
+    (1-d*Real.cos t)^2+(d*Real.sin t)^2=binaryQ d t := by
+  have h := Real.sin_sq_add_cos_sq t
+  dsimp [binaryQ]
+  nlinarith [sq_nonneg d]
+lemma binary_arctan_recip {d : ℝ} (hd : |d| < 1/2) (t : ℝ) :
+    1/(1+(d*Real.sin t/(1-d*Real.cos t))^2)/(1-d*Real.cos t)^2=1/binaryQ d t := by
+  have hq := binaryQ_range hd t
+  have hi := binary_arctan_identity d t
+  field_simp [hq.1.ne',hq.2.2.ne']
+  simp only [mul_pow] at hi
+  rw [hi,div_self hq.1.ne']
+lemma binaryCircleMap_hasDeriv_t {d : ℝ} (hd : |d| < 1/2) (t : ℝ) :
+    HasDerivAt (binaryCircleMap d) (1+d*binaryB d t) t := by
+  have hq := binaryQ_range hd t
+  have hsq := Real.sq_sqrt hq.1.le
+  have hs : 0 < Real.sqrt (binaryQ d t) := Real.sqrt_pos.mpr hq.1
+  have hc : Real.sqrt (binaryQ d t)/2 ≠ -1 := by linarith
+  have hc' : Real.sqrt (binaryQ d t)/2 ≠ 1 := by intro h; nlinarith [hq.2.1]
+  have hqd : HasDerivAt (binaryQ d) (2*d*Real.sin t) t := by
+    convert (((Real.hasDerivAt_cos t).const_mul (2*d)).const_sub 1).add_const (d^2) using 1 <;>
+      first | rfl | ring
+  have hat := (((Real.hasDerivAt_sin t).const_mul d).div
+    (((Real.hasDerivAt_cos t).const_mul d).const_sub 1) hq.2.2.ne').arctan
+  have hac := (Real.hasDerivAt_arccos hc hc').comp t ((hqd.sqrt hq.1.ne').div_const 2)
+  have hh := (((hasDerivAt_id t).add_const (Real.pi/3)).add hat).sub hac
+  change HasDerivAt (binaryCircleMap d) _ t at hh
+  convert hh using 1
+  dsimp [binaryB,binaryR]
+  have hss : 1-(Real.sqrt (binaryQ d t)/2)^2=1-binaryQ d t/4 := by nlinarith [hsq]
+  rw [hss]
+  have hrec := binary_arctan_recip hd t
+  have hnum : d*Real.cos t*(1-d*Real.cos t)-d*Real.sin t*-(d*-Real.sin t)=
+      d*(Real.cos t-d) := by linear_combination -d^2*(Real.sin_sq_add_cos_sq t)
+  rw [hnum]
+  have hrearr : 1/(1+(d*Real.sin t/(1-d*Real.cos t))^2)*
+      (d*(Real.cos t-d)/(1-d*Real.cos t)^2) = d*(Real.cos t-d)/binaryQ d t := by
+    calc
+      _ = (1/(1+(d*Real.sin t/(1-d*Real.cos t))^2)/(1-d*Real.cos t)^2)*(d*(Real.cos t-d)) := by ring
+      _ = d*(Real.cos t-d)/binaryQ d t := by rw [hrec]; ring
+  rw [hrearr]
+  ring
+lemma binaryCircleMap_hasDeriv_d {d : ℝ} (hd : |d| < 1/2) (t : ℝ) :
+    HasDerivAt (fun s => binaryCircleMap s t) (binaryM d t) d := by
+  have hq := binaryQ_range hd t
+  have hsq := Real.sq_sqrt hq.1.le
+  have hs : 0 < Real.sqrt (binaryQ d t) := Real.sqrt_pos.mpr hq.1
+  have hc : Real.sqrt (binaryQ d t)/2 ≠ -1 := by linarith
+  have hc' : Real.sqrt (binaryQ d t)/2 ≠ 1 := by intro h; nlinarith [hq.2.1]
+  have hqd : HasDerivAt (fun s => binaryQ s t) (2*(d-Real.cos t)) d := by
+    convert (((hasDerivAt_id d).mul_const (2*Real.cos t)).const_sub 1).add ((hasDerivAt_id d).pow 2) using 1 <;>
+      first | rfl | (funext s; dsimp only [binaryQ,Pi.add_apply,Pi.pow_apply,id_eq]; ring) |
+        (try dsimp only [id_eq]; ring)
+  have hat := (((hasDerivAt_id d).mul_const (Real.sin t)).div
+    (((hasDerivAt_id d).mul_const (Real.cos t)).const_sub 1) hq.2.2.ne').arctan
+  have hac := (Real.hasDerivAt_arccos hc hc').comp d ((hqd.sqrt hq.1.ne').div_const 2)
+  have hh := (hat.const_add (t+Real.pi/3)).sub hac
+  change HasDerivAt (fun s => binaryCircleMap s t) _ d at hh
+  convert hh using 1
+  dsimp [binaryM,binaryR]
+  have hss : 1-(Real.sqrt (binaryQ d t)/2)^2=1-binaryQ d t/4 := by nlinarith [hsq]
+  rw [hss]
+  have hrec := binary_arctan_recip hd t
+  simp only [one_mul]
+  have hnum : Real.sin t*(1-d*Real.cos t)-d*Real.sin t*-(Real.cos t)=Real.sin t := by ring
+  rw [hnum]
+  have hrearr : 1/(1+(d*Real.sin t/(1-d*Real.cos t))^2)*
+      (Real.sin t/(1-d*Real.cos t)^2) = Real.sin t/binaryQ d t := by
+    calc
+      _ = (1/(1+(d*Real.sin t/(1-d*Real.cos t))^2)/(1-d*Real.cos t)^2)*Real.sin t := by ring
+      _ = Real.sin t/binaryQ d t := by rw [hrec]; ring
+  rw [hrearr]
+  ring
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology
+namespace EuclideanFiveColor
+lemma periodic_reduce_to_period {P : ℝ} (hP : 0 < P) (t : ℝ) :
+    ∃ u ∈ Icc 0 P, ∃ n : ℤ, t=u+(n : ℝ)*P := by
+  refine ⟨Int.fract (t/P)*P,?_,⌊t/P⌋,?_⟩
+  · exact ⟨(Int.fract_div_mul_self_mem_Ico P t hP).1,(Int.fract_div_mul_self_mem_Ico P t hP).2.le⟩
+  · simpa only [zsmul_eq_mul] using (Int.fract_div_mul_self_add_zsmul_eq P t hP.ne').symm
+lemma compact_periodic_uniform {g : ℝ → ℝ → ℝ} {a P : ℝ} (ha : 0 < a) (hP : 0 < P)
+    (hg : ContinuousOn (fun p : ℝ × ℝ => g p.1 p.2) (Icc (-a) a ×ˢ Icc 0 P))
+    (hper : ∀ d, Function.Periodic (g d) P) {ε : ℝ} (hε : 0 < ε) :
+    ∃ δ > 0, δ ≤ a ∧ ∀ d, |d| < δ → ∀ t, |g d t-g 0 t| < ε := by
+  have hu := (isCompact_Icc.prod isCompact_Icc).uniformContinuousOn_of_continuous hg
+  obtain ⟨δ,hδ,hδg⟩ := Metric.uniformContinuousOn_iff.mp hu ε hε
+  refine ⟨min a δ,lt_min ha hδ,min_le_left _ _,?_⟩
+  intro d hd t
+  obtain ⟨u,hu,n,rfl⟩ := periodic_reduce_to_period hP t
+  have hper' : ∀ d, g d (u+(n : ℝ)*P)=g d u := fun d => by
+    simpa only [zsmul_eq_mul] using (hper d).int_mul n u
+  rw [hper' d,hper' 0]
+  have hda := abs_lt.mp (lt_of_lt_of_le hd (min_le_left _ _))
+  have hh := hδg (d,u) ⟨⟨hda.1.le,hda.2.le⟩,hu⟩ (0,u) ⟨⟨by linarith,ha.le⟩,hu⟩
+  rw [Prod.dist_eq,dist_self,dist_zero_right,max_eq_left (norm_nonneg d)] at hh
+  exact hh (lt_of_lt_of_le hd (min_le_right _ _))
+lemma compact_periodic_bounded {g : ℝ → ℝ → ℝ} {a P : ℝ} (_ha : 0 ≤ a) (hP : 0 < P)
+    (hg : ContinuousOn (fun p : ℝ × ℝ => g p.1 p.2) (Icc (-a) a ×ˢ Icc 0 P))
+    (hper : ∀ d, |d| ≤ a → Function.Periodic (g d) P) :
+    ∃ C > 0, ∀ d, |d| ≤ a → ∀ t, |g d t| ≤ C := by
+  obtain ⟨C,hC⟩ := ((isCompact_Icc.prod isCompact_Icc).image_of_continuousOn hg).isBounded |>.exists_pos_norm_le
+  refine ⟨C,hC.1,?_⟩
+  intro d hd t
+  obtain ⟨u,hu,n,rfl⟩ := periodic_reduce_to_period hP t
+  have he : g d (u+(n : ℝ)*P)=g d u := by
+    simpa only [zsmul_eq_mul] using (hper d hd).int_mul n u
+  rw [he]
+  exact hC.2 _ ⟨(d,u),⟨abs_le.mp hd,hu⟩,rfl⟩
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology
+namespace EuclideanFiveColor
+lemma binaryQ_contDiff : ContDiff ℝ 2 (fun p : ℝ × ℝ => binaryQ p.1 p.2) := by
+  unfold binaryQ
+  fun_prop
+lemma binaryB_contDiffAt {p : ℝ × ℝ} (hp : |p.1| < 1/2) :
+    ContDiffAt ℝ 2 (fun q : ℝ × ℝ => binaryB q.1 q.2) p := by
+  have hq := binaryQ_range hp p.2
+  have hr := binaryR_pos hp p.2
+  have hs := Real.sqrt_pos.mpr hq.1
+  have hs' : 0 < 1-binaryQ p.1 p.2/4 := by linarith [hq.2.1]
+  have hQ := binaryQ_contDiff.contDiffAt (x := p)
+  have hR : ContDiffAt ℝ 2 (fun q : ℝ × ℝ => binaryR q.1 q.2) p := by
+    unfold binaryR
+    fun_prop (disch := first | exact hq.1.ne' | exact hs'.ne')
+  unfold binaryB
+  exact ((contDiffAt_snd.cos.sub contDiffAt_fst).div hQ hq.1.ne').add
+    (contDiffAt_snd.sin.div hR hr.ne')
+lemma binaryM_contDiffAt {p : ℝ × ℝ} (hp : |p.1| < 1/2) :
+    ContDiffAt ℝ 2 (fun q : ℝ × ℝ => binaryM q.1 q.2) p := by
+  have hq := binaryQ_range hp p.2
+  have hr := binaryR_pos hp p.2
+  have hs := Real.sqrt_pos.mpr hq.1
+  have hs' : 0 < 1-binaryQ p.1 p.2/4 := by linarith [hq.2.1]
+  have hQ := binaryQ_contDiff.contDiffAt (x := p)
+  have hR : ContDiffAt ℝ 2 (fun q : ℝ × ℝ => binaryR q.1 q.2) p := by
+    unfold binaryR
+    fun_prop (disch := first | exact hq.1.ne' | exact hs'.ne')
+  unfold binaryM
+  exact (contDiffAt_snd.sin.div hQ hq.1.ne').add
+    ((contDiffAt_fst.sub contDiffAt_snd.cos).div hR hr.ne')
+lemma binaryB_periodic (d : ℝ) : Function.Periodic (binaryB d) (2*Real.pi) := by
+  intro t
+  simp only [binaryB,binaryQ,binaryR,Real.cos_add_two_pi,Real.sin_add_two_pi]
+lemma binaryM_periodic (d : ℝ) : Function.Periodic (binaryM d) (2*Real.pi) := by
+  intro t
+  simp only [binaryM,binaryQ,binaryR,Real.cos_add_two_pi,Real.sin_add_two_pi]
+def binaryE (d t : ℝ) : ℝ := fderiv ℝ (fun q : ℝ × ℝ => binaryB q.1 q.2) (d,t) (0,1)
+lemma binaryE_continuousAt {p : ℝ × ℝ} (hp : |p.1| < 1/2) :
+    ContinuousAt (fun q : ℝ × ℝ => binaryE q.1 q.2) p := by
+  exact ((binaryB_contDiffAt hp).continuousAt_fderiv (by norm_num)).clm_apply continuousAt_const
+lemma binaryB_hasDeriv_t {d : ℝ} (hd : |d| < 1/2) (t : ℝ) :
+    HasDerivAt (binaryB d) (binaryE d t) t := by
+  have h := (binaryB_contDiffAt (p := (d,t)) hd).differentiableAt (by norm_num)
+  convert h.hasFDerivAt.comp t ((hasFDerivAt_const d t).prodMk (hasFDerivAt_id t)) |>.hasDerivAt using 1 <;> rfl
+lemma binaryE_periodic (d : ℝ) (hd : |d| < 1/2) : Function.Periodic (binaryE d) (2*Real.pi) := by
+  intro t
+  have h := (binaryB_hasDeriv_t hd (t+2*Real.pi)).comp t ((hasDerivAt_id t).add_const (2*Real.pi))
+  simp only [mul_one] at h
+  have he : (fun s => binaryB d (s+2*Real.pi))=binaryB d := funext (binaryB_periodic d)
+  change HasDerivAt (fun s => binaryB d (s+2*Real.pi)) _ t at h
+  rw [he] at h
+  exact h.unique (binaryB_hasDeriv_t hd t)
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology
+namespace EuclideanFiveColor
+lemma binary_uniform_controls {ε : ℝ} (hε : 0 < ε) :
+    ∃ δ C : ℝ, 0 < δ ∧ δ ≤ 1/4 ∧ 0 < C ∧
+    (∀ d, |d| < δ → ∀ t, |binaryB d t-binaryB 0 t| < ε ∧
+       |binaryM d t-binaryM 0 t| < ε) ∧
+    (∀ d, |d| ≤ δ → ∀ t, |binaryB d t| ≤ C ∧ |binaryM d t| ≤ C ∧ |binaryE d t| ≤ C) := by
+  have hB : ContinuousOn (fun p : ℝ × ℝ => binaryB p.1 p.2)
+      (Icc (-(1/4)) (1/4) ×ˢ Icc 0 (2*Real.pi)) := by
+    intro p hp
+    exact (binaryB_contDiffAt (by rw [abs_lt]; constructor <;> linarith [hp.1.1,hp.1.2])).continuousAt.continuousWithinAt
+  have hM : ContinuousOn (fun p : ℝ × ℝ => binaryM p.1 p.2)
+      (Icc (-(1/4)) (1/4) ×ˢ Icc 0 (2*Real.pi)) := by
+    intro p hp
+    exact (binaryM_contDiffAt (by rw [abs_lt]; constructor <;> linarith [hp.1.1,hp.1.2])).continuousAt.continuousWithinAt
+  have hE : ContinuousOn (fun p : ℝ × ℝ => binaryE p.1 p.2)
+      (Icc (-(1/4)) (1/4) ×ˢ Icc 0 (2*Real.pi)) := by
+    intro p hp
+    exact (binaryE_continuousAt (by rw [abs_lt]; constructor <;> linarith [hp.1.1,hp.1.2])).continuousWithinAt
+  have hP : 0 < 2*Real.pi := by positivity
+  obtain ⟨δB,hδB,hδBa,hδBg⟩ := compact_periodic_uniform (by norm_num : (0:ℝ)<1/4) hP hB binaryB_periodic hε
+  obtain ⟨δM,hδM,hδMa,hδMg⟩ := compact_periodic_uniform (by norm_num : (0:ℝ)<1/4) hP hM binaryM_periodic hε
+  obtain ⟨CB,hCB,hCBg⟩ := compact_periodic_bounded (by norm_num : (0:ℝ)≤1/4) hP hB (fun d _ => binaryB_periodic d)
+  obtain ⟨CM,hCM,hCMg⟩ := compact_periodic_bounded (by norm_num : (0:ℝ)≤1/4) hP hM (fun d _ => binaryM_periodic d)
+  obtain ⟨CE,hCE,hCEg⟩ := compact_periodic_bounded (by norm_num : (0:ℝ)≤1/4) hP hE
+    (fun d hd => binaryE_periodic d (by linarith))
+  refine ⟨min δB δM,CB+CM+CE,lt_min hδB hδM,(min_le_left _ _).trans hδBa,by positivity,?_,?_⟩
+  · intro d hd t
+    exact ⟨hδBg d (hd.trans_le (min_le_left _ _)) t,hδMg d (hd.trans_le (min_le_right _ _)) t⟩
+  · intro d hd t
+    have hda := hd.trans ((min_le_left δB δM).trans hδBa)
+    exact ⟨(hCBg d hda t).trans (by linarith),(hCMg d hda t).trans (by linarith),(hCEg d hda t).trans (by linarith)⟩
+lemma binaryCircleMap_zero (t : ℝ) : binaryCircleMap 0 t=t := by
+  have h : Real.arccos (1/2) = Real.pi/3 := by
+    rw [← Real.cos_pi_div_three,Real.arccos_cos (by positivity) (by linarith [Real.pi_pos])]
+  simp only [binaryCircleMap,binaryQ,mul_zero,zero_mul,sub_zero,zero_pow (by norm_num : (2:ℕ)≠0),add_zero,Real.sqrt_one,one_div,zero_div,Real.arctan_zero]
+  rw [← one_div,h]
+  ring
+lemma binaryCircleMap_displacement {d δ C : ℝ} (hd : 0 ≤ d) (hδ : δ < 1/2) (hdδ : d ≤ δ)
+    (hM : ∀ s, |s| ≤ δ → ∀ t, |binaryM s t| ≤ C) (t : ℝ) :
+    |binaryCircleMap d t-t| ≤ C*d := by
+  have hh : ∀ s ∈ Icc 0 d, HasDerivWithinAt (fun s => binaryCircleMap s t) (binaryM s t) (Icc 0 d) s := by
+    intro s hs
+    exact (binaryCircleMap_hasDeriv_d (lt_of_le_of_lt (by rw [abs_of_nonneg hs.1]; exact hs.2.trans hdδ) hδ) t).hasDerivWithinAt
+  have h := (convex_Icc (0:ℝ) d).norm_image_sub_le_of_norm_hasDerivWithin_le hh
+    (fun s hs => hM s (by rw [abs_of_nonneg hs.1]; exact hs.2.trans hdδ) t) (left_mem_Icc.mpr hd) (right_mem_Icc.mpr hd)
+  simpa only [binaryCircleMap_zero,Real.norm_eq_abs,sub_zero,abs_of_nonneg hd] using h
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology
+namespace EuclideanFiveColor
+lemma binaryR_zero (t : ℝ) : binaryR 0 t=Real.sqrt 3 := by
+  simp only [binaryR,binaryQ,mul_zero,zero_mul,sub_zero,zero_pow (by norm_num : (2:ℕ)≠0),add_zero,Real.sqrt_one,mul_one]
+  have h : (1:ℝ)-1/4=3/(2^2) := by norm_num
+  rw [h,Real.sqrt_div (by norm_num),Real.sqrt_sq (by norm_num)]
+  ring
+lemma binaryB_zero (t : ℝ) : binaryB 0 t=Real.cos t+Real.sin t/Real.sqrt 3 := by
+  simp [binaryB,binaryQ,binaryR_zero]
+lemma binaryM_zero (t : ℝ) : binaryM 0 t=Real.sin t-Real.cos t/Real.sqrt 3 := by
+  simp [binaryM,binaryQ,binaryR_zero,sub_eq_add_neg,neg_div]
+lemma binaryB_zero_shift (t : ℝ) : binaryB 0 (t+Real.pi/3)=Real.cos t-Real.sin t/Real.sqrt 3 := by
+  rw [binaryB_zero,Real.cos_add,Real.sin_add,Real.cos_pi_div_three,Real.sin_pi_div_three]
+  have hr : Real.sqrt 3 ≠ 0 := Real.sqrt_ne_zero'.mpr (by norm_num)
+  have hs := Real.sq_sqrt (by norm_num : (0:ℝ)≤3)
+  field_simp
+  linear_combination -Real.sin t*hs
+lemma binaryM_zero_shift (t : ℝ) : binaryM 0 (t+Real.pi/3)=Real.sin t+Real.cos t/Real.sqrt 3 := by
+  rw [binaryM_zero,Real.cos_add,Real.sin_add,Real.cos_pi_div_three,Real.sin_pi_div_three]
+  have hr : Real.sqrt 3 ≠ 0 := Real.sqrt_ne_zero'.mpr (by norm_num)
+  have hs := Real.sq_sqrt (by norm_num : (0:ℝ)≤3)
+  field_simp
+  linear_combination Real.cos t*hs
+lemma trig_two_large {c s : ℝ} (hcs : c^2+s^2=1) :
+    1/4 ≤ |c+s/Real.sqrt 3| ∨ 1/4 ≤ |c-s/Real.sqrt 3| := by
+  by_contra h
+  push Not at h
+  have h1 := abs_lt.mp h.1
+  have h2 := abs_lt.mp h.2
+  have hc : |c| < 1/4 := by rw [abs_lt]; constructor <;> linarith
+  have hq : |s/Real.sqrt 3| < 1/4 := by rw [abs_lt]; constructor <;> linarith
+  have hr : 0 < Real.sqrt 3 := Real.sqrt_pos.mpr (by norm_num)
+  have hr2 : Real.sqrt 3 < 2 := by nlinarith [Real.sq_sqrt (by norm_num : (0:ℝ)≤3)]
+  rw [abs_div,abs_of_pos hr,div_lt_iff₀ hr] at hq
+  have hs : |s| < 1/2 := by linarith
+  have hc2 : c^2 < (1/4 : ℝ)^2 := by nlinarith [sq_abs c,abs_nonneg c]
+  have hs2 : s^2 < (1/2 : ℝ)^2 := by nlinarith [sq_abs s,abs_nonneg s]
+  nlinarith
+lemma sign_shift_large {g : ℝ → ℝ} (hg : ∀ t, g (t+Real.pi) = -g t)
+    {t r : ℝ} (hr : r ≤ |g t|) : r ≤ g t ∨ r ≤ g (t+Real.pi) := by
+  rw [hg]
+  rcases le_total 0 (g t) with h|h
+  · exact Or.inl (by simpa only [abs_of_nonneg h] using hr)
+  · exact Or.inr (by simpa only [abs_of_nonpos h] using hr)
+lemma binaryB_zero_large (t : ℝ) : ∃ n : ℕ, 1/4 ≤ binaryB 0 (t+(n : ℝ)*(Real.pi/3)) := by
+  have hg : ∀ u, binaryB 0 (u+Real.pi) = -binaryB 0 u := by
+    intro u
+    simp only [binaryB_zero,Real.cos_add_pi,Real.sin_add_pi]
+    ring
+  have hh := trig_two_large (Real.cos_sq_add_sin_sq t)
+  rw [← binaryB_zero,← binaryB_zero_shift] at hh
+  rcases hh with h|h
+  · rcases sign_shift_large hg h with h|h
+    · exact ⟨0,by simpa using h⟩
+    · refine ⟨3,?_⟩
+      convert h using 2; norm_num; ring
+  · rcases sign_shift_large hg h with h|h
+    · exact ⟨1,by simpa using h⟩
+    · refine ⟨4,?_⟩
+      convert h using 2; norm_num; ring
+lemma binaryM_zero_large (t : ℝ) : ∃ n : ℕ, 1/4 ≤ binaryM 0 (t+(n : ℝ)*(Real.pi/3)) := by
+  have hg : ∀ u, binaryM 0 (u+Real.pi) = -binaryM 0 u := by
+    intro u
+    simp only [binaryM_zero,Real.cos_add_pi,Real.sin_add_pi]
+    ring
+  have hh := trig_two_large (Real.sin_sq_add_cos_sq t)
+  rw [← binaryM_zero_shift,← binaryM_zero] at hh
+  rcases hh with h|h
+  · rcases sign_shift_large hg h with h|h
+    · exact ⟨1,by simpa using h⟩
+    · refine ⟨4,?_⟩
+      convert h using 2; norm_num; ring
+  · rcases sign_shift_large hg h with h|h
+    · exact ⟨0,by simpa using h⟩
+    · refine ⟨3,?_⟩
+      convert h using 2; norm_num; ring
+end EuclideanFiveColor
+end
+
+noncomputable section
+open MeasureTheory Filter Topology Set
+open scoped NNReal ENNReal
+namespace EuclideanFiveColor
+lemma expansion_alternation_false {ι : Type*} {F D : ι → ℝ → ℝ}
+    {K : ℝ≥0} {e U L P B κ : ℝ} {A : Set ℝ}
+    (hd : ∀ j x, HasDerivAt (F j) (D j x) x)
+    (hp : ∀ j x, (1/2 : ℝ) ≤ D j x) (hup : ∀ j x, D j x ≤ U)
+    (hK : ∀ j, LipschitzWith K (D j)) (hlarge : ∀ x, ∃ j, 1+e ≤ D j x)
+    (hadvance : ∀ x, ∃ j, x+e ≤ F j x) (hdispl : ∀ j x, F j x ≤ x+B)
+    (he : 0 < e) (hKL : (K : ℝ)*L ≤ e) (hU : 0 < U) (hL : 0 < L) (hP : 0 < P)
+    (hB : B ≤ L/2) (hκ : 0 < κ)
+    (hinv : ∀ j, ∀ᵐ x : ℝ ∂volume, x ∈ A ↔ F j x ∈ A)
+    (hA : MeasurableSet A)
+    (hper : ∀ (n : ℤ) x, x+(n : ℝ)*P ∈ A ↔ x ∈ A)
+    (halt : ∀ᵐ x : ℝ ∂volume, x+κ ∈ A ↔ x ∉ A) : False := by
+  have hpos : volume A ≠ 0 := by
+    intro hz
+    have ha : ∀ᵐ x : ℝ ∂volume, x ∉ A := by
+      rw [ae_iff]
+      convert hz using 2
+      ext x
+      simp only [mem_ofPred_eq,not_not]
+    have hf : ∀ᵐ x : ℝ ∂volume, False := by
+      filter_upwards [ha,(measurePreserving_add_right volume κ).quasiMeasurePreserving.ae ha,halt] with x hx hx' hh
+      exact hx' (hh.mpr hx)
+    exact hf.exists.choose_spec
+  obtain ⟨a,b,hlen,hnull⟩ := expansion_conull_interval hd hp hup hK hlarge he hKL hU hL hP hinv hA hpos hper
+  have hLip : ∀ j, LipschitzWith (Real.toNNReal U) (F j) := by
+    intro j
+    apply lipschitzWith_of_nnnorm_deriv_le (fun x => (hd j x).differentiableAt)
+    intro x
+    rw [(hd j x).deriv]
+    apply NNReal.coe_le_coe.mp
+    rw [Real.coe_toNNReal _ hU.le,coe_nnnorm,Real.norm_eq_abs,
+      abs_of_nonneg (le_trans (by norm_num) (hp j x))]
+    exact hup j x
+  have hmono : ∀ j, Monotone (F j) := by
+    intro j
+    apply StrictMono.monotone
+    apply strictMono_of_deriv_pos
+    intro x
+    rw [(hd j x).deriv]
+    exact lt_of_lt_of_le (by norm_num) (hp j x)
+  obtain ⟨n,hn⟩ := exists_nat_gt ((κ-(b-a))/e)
+  have hnn : κ < b+(n : ℝ)*e-a := by
+    have := (div_lt_iff₀ he).mp hn
+    linarith
+  obtain ⟨b',hb',hnull'⟩ := conull_extend_intervals hLip hmono hadvance hdispl he
+    (by linarith) (hB.trans_eq hlen.symm) hinv hnull n
+  exact conull_interval_alternation_false hκ (by linarith) hnull' halt
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+open scoped NNReal ENNReal
+namespace EuclideanFiveColor
+def binaryConjugate (d : ℝ) (n : ℕ) (t : ℝ) : ℝ :=
+  binaryCircleMap d (t+(n : ℝ)*(Real.pi/3))-(n : ℝ)*(Real.pi/3)
+lemma binaryConjugate_hasDeriv {d : ℝ} (hd : |d|<1/2) (n : ℕ) (t : ℝ) :
+    HasDerivAt (binaryConjugate d n) (1+d*binaryB d (t+(n : ℝ)*(Real.pi/3))) t := by
+  convert ((binaryCircleMap_hasDeriv_t hd (t+(n : ℝ)*(Real.pi/3))).comp t
+    ((hasDerivAt_id t).add_const ((n : ℝ)*(Real.pi/3)))).sub_const ((n : ℝ)*(Real.pi/3)) using 1 <;> first | rfl | simp
+lemma binaryCircleMap_advance {d δ : ℝ} (hd : 0 < d) (hδ : δ ≤ 1/4) (hdδ : d < δ)
+    (hM : ∀ s, |s| < δ → ∀ t, |binaryM s t-binaryM 0 t| < 1/8)
+    {t : ℝ} (ht : 1/4 ≤ binaryM 0 t) : t+d/8 ≤ binaryCircleMap d t := by
+  have hh : ∀ s ∈ Icc 0 d, HasDerivAt (fun s => binaryCircleMap s t) (binaryM s t) s := by
+    intro s hs
+    exact binaryCircleMap_hasDeriv_d (by rw [abs_of_nonneg hs.1]; linarith [hs.2]) t
+  obtain ⟨s,hs,he⟩ := exists_hasDerivAt_eq_slope (fun s => binaryCircleMap s t) (fun s => binaryM s t) hd
+    (fun s hs => (hh s hs).continuousAt.continuousWithinAt) (fun s hs => hh s ⟨hs.1.le,hs.2.le⟩)
+  have hms := (abs_lt.mp (hM s (by rw [abs_of_pos hs.1]; exact hs.2.trans hdδ) t)).1
+  rw [binaryCircleMap_zero,sub_zero] at he
+  have hl : 1/8 ≤ (binaryCircleMap d t-t)/d := by rw [← he]; linarith
+  have := (le_div_iff₀ hd).mp hl
+  linarith
+lemma binary_dynamics_controls : ∃ δ > 0, ∀ d, 0 < d → d < δ →
+    ∃ C L : ℝ, 0 < C ∧ 0 < L ∧ C*d ≤ L/2 ∧
+    (∀ (n : ℕ) t, (1/2 : ℝ) ≤ 1+d*binaryB d (t+(n : ℝ)*(Real.pi/3))) ∧
+    (∀ (n : ℕ) t, 1+d*binaryB d (t+(n : ℝ)*(Real.pi/3)) ≤ 2) ∧
+    (∀ n : ℕ, LipschitzWith (Real.toNNReal (d*C)) (fun t => 1+d*binaryB d (t+(n : ℝ)*(Real.pi/3)))) ∧
+    (∀ t, ∃ n : ℕ, 1+d/8 ≤ 1+d*binaryB d (t+(n : ℝ)*(Real.pi/3))) ∧
+    (∀ t, ∃ n : ℕ, t+d/8 ≤ binaryConjugate d n t) ∧
+    (∀ (n : ℕ) t, binaryConjugate d n t ≤ t+C*d) ∧ d*C*L ≤ d/8 ∧ d < 1/2 := by
+  obtain ⟨δ,C,hδ,hδa,hC,hclose,hbound⟩ := binary_uniform_controls (by norm_num : (0:ℝ)<1/8)
+  let L := 1/(8*C)
+  have hL : 0 < L := by dsimp [L]; positivity
+  refine ⟨min δ (min (1/(2*C)) (L/(2*C))),lt_min hδ (lt_min (by positivity) (by positivity)),?_⟩
+  intro d hd hds
+  have hdδ : d < δ := (lt_min_iff.mp hds).1
+  have hda : d < 1/2 := by linarith
+  have hdC : d*C < 1/2 := by
+    have := (lt_div_iff₀ (by positivity : 0 < 2*C)).mp ((lt_min_iff.mp (lt_min_iff.mp hds).2).1)
+    nlinarith
+  have hdL : C*d ≤ L/2 := by
+    have := (lt_div_iff₀ (by positivity : 0 < 2*C)).mp ((lt_min_iff.mp (lt_min_iff.mp hds).2).2)
+    nlinarith
+  have hbd := hbound d (by rw [abs_of_pos hd]; exact hdδ.le)
+  have hdd : |d|<1/2 := by rwa [abs_of_pos hd]
+  refine ⟨C,L,hC,hL,hdL,?_,?_,?_,?_,?_,?_,?_,hda⟩
+  · intro n t
+    have h := (abs_le.mp (hbd (t+(n : ℝ)*(Real.pi/3))).1).1
+    nlinarith
+  · intro n t
+    have h := (abs_le.mp (hbd (t+(n : ℝ)*(Real.pi/3))).1).2
+    nlinarith
+  · intro n
+    have hder : ∀ t, HasDerivAt (fun t => 1+d*binaryB d (t+(n : ℝ)*(Real.pi/3)))
+        (d*binaryE d (t+(n : ℝ)*(Real.pi/3))) t := by
+      intro t
+      convert (((binaryB_hasDeriv_t hdd (t+(n : ℝ)*(Real.pi/3))).comp t
+        ((hasDerivAt_id t).add_const ((n : ℝ)*(Real.pi/3)))).const_mul d).const_add 1 using 1 <;> first | rfl | simp
+    apply lipschitzWith_of_nnnorm_deriv_le (fun t => (hder t).differentiableAt)
+    intro t
+    rw [(hder t).deriv]
+    apply NNReal.coe_le_coe.mp
+    rw [Real.coe_toNNReal _ (by positivity),coe_nnnorm,norm_mul,Real.norm_eq_abs,abs_of_pos hd,Real.norm_eq_abs]
+    exact mul_le_mul_of_nonneg_left (hbd _).2.2 hd.le
+  · intro t
+    obtain ⟨n,hn⟩ := binaryB_zero_large t
+    refine ⟨n,?_⟩
+    have hh := (abs_lt.mp (hclose d (by rwa [abs_of_pos hd]) (t+(n : ℝ)*(Real.pi/3))).1).1
+    nlinarith
+  · intro t
+    obtain ⟨n,hn⟩ := binaryM_zero_large t
+    refine ⟨n,?_⟩
+    have hh := binaryCircleMap_advance hd hδa hdδ (fun s hs u => (hclose s hs u).2) hn
+    dsimp only [binaryConjugate]
+    linarith
+  · intro n t
+    have hh := binaryCircleMap_displacement hd.le (lt_of_le_of_lt hδa (by norm_num)) hdδ.le
+      (fun s hs u => (hbound s hs u).2.1) (t+(n : ℝ)*(Real.pi/3))
+    dsimp only [binaryConjugate]
+    linarith [(abs_le.mp hh).2]
+  · dsimp [L]
+    field_simp
+    norm_num
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+lemma binaryConjugate_step (d : ℝ) (n : ℕ) (x : ℝ) :
+    binaryConjugate d n (x+Real.pi/3)=binaryConjugate d (n+1) x+Real.pi/3 := by
+  unfold binaryConjugate
+  push_cast
+  have h : x+Real.pi/3+(n : ℝ)*(Real.pi/3)=x+((n : ℝ)+1)*(Real.pi/3) := by ring
+  rw [h]
+  ring
+lemma binary_conjugates_invariant {d : ℝ} {A : Set ℝ}
+    (hq : ∀ n : ℕ, Measure.QuasiMeasurePreserving (binaryConjugate d n) volume volume)
+    (hF : ∀ᵐ x : ℝ ∂volume, x ∈ A ↔ binaryCircleMap d x ∈ A)
+    (halt : ∀ᵐ x : ℝ ∂volume, x+Real.pi/3 ∈ A ↔ x ∉ A) :
+    ∀ n : ℕ, ∀ᵐ x : ℝ ∂volume, x ∈ A ↔ binaryConjugate d n x ∈ A := by
+  intro n
+  induction n with
+  | zero => simpa only [binaryConjugate,Nat.cast_zero,zero_mul,add_zero,sub_zero] using hF
+  | succ n ih =>
+    have hshift := (measurePreserving_add_right volume (Real.pi/3)).quasiMeasurePreserving.ae ih
+    filter_upwards [hshift,halt,(hq (n+1)).ae halt] with x hx ha hb
+    rw [binaryConjugate_step] at hx
+    exact not_iff_not.mp (ha.symm.trans (hx.trans hb))
+lemma binary_alternating_invariant_false : ∃ δ > 0, ∀ d, 0 < d → d < δ →
+    ∀ A : Set ℝ, MeasurableSet A →
+    (∀ (n : ℤ) x, x+(n : ℝ)*(2*Real.pi) ∈ A ↔ x ∈ A) →
+    (∀ᵐ x : ℝ ∂volume, x+Real.pi/3 ∈ A ↔ x ∉ A) →
+    (∀ᵐ x : ℝ ∂volume, x ∈ A ↔ binaryCircleMap d x ∈ A) → False := by
+  obtain ⟨δ,hδ,hctrl⟩ := binary_dynamics_controls
+  refine ⟨δ,hδ,?_⟩
+  intro d hd hdδ A hA hper halt hF
+  obtain ⟨C,L,hC,hL,hB,hlo,hhi,hLip,hlarge,hadvance,hdispl,hKL,hda⟩ := hctrl d hd hdδ
+  have hdd : |d|<1/2 := by rwa [abs_of_pos hd]
+  have hder := fun n t => binaryConjugate_hasDeriv hdd n t
+  have hinv := binary_conjugates_invariant (fun n => deriv_lower_qmp (hder n) (hlo n)) hF halt
+  exact expansion_alternation_false hder hlo hhi hLip hlarge hadvance hdispl (by positivity)
+    (by simpa only [Real.coe_toNNReal _ (mul_pos hd hC).le] using hKL) (by norm_num) hL
+    (by positivity : 0<2*Real.pi) hB (by positivity : 0<Real.pi/3) hinv hA hper halt
+end EuclideanFiveColor
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PlaneColoring/BinaryPalettes.lean b/lean/OAI/Geometry/PlaneColoring/BinaryPalettes.lean
new file mode 100644
index 0000000000000000000000000000000000000000..96c30eedba90fcfa277b5686c36f6d51752084db
--- /dev/null
+++ b/lean/OAI/Geometry/PlaneColoring/BinaryPalettes.lean
@@ -0,0 +1,375 @@
+import OAI.Geometry.PlaneColoring.PaletteTransport
+
+namespace OAI
+
+noncomputable section
+open MeasureTheory Filter Topology Set
+namespace EuclideanFiveColor
+variable {k : ℕ} (c : ℂ → Fin k) (hc : Measurable c)
+def BinaryCenter (x : ℂ) (i j : Fin k) : Prop :=
+  ∀ᵐ t ∂angularMeasure, t ∈ paletteSet c hc x i ∨ t ∈ paletteSet c hc x j
+lemma binary_alternation (hw : WeakColoring k c) {x : ℂ} {i j : Fin k}
+    (hb : BinaryCenter c hc x i j) :
+    ∀ᵐ t ∂angularMeasure, (t+(sixty : Angles) ∈ paletteSet c hc x i ↔ t ∉ paletteSet c hc x i) ∧
+      (t ∈ paletteSet c hc x j ↔ t ∉ paletteSet c hc x i) := by
+  have hb' := (measurePreserving_add_right angularMeasure (sixty : Angles)).quasiMeasurePreserving.ae hb
+  filter_upwards [hb,hb',palette_sixty c hc hw x i,palette_sixty c hc hw x j] with t h h' hi hj
+  constructor
+  · constructor
+    · exact fun ht ht' => hi ht' ht
+    · intro hnot
+      exact h'.resolve_right (hj (h.resolve_left hnot))
+  · constructor
+    · intro ht ht'
+      rcases h' with hpi | hpj
+      · exact hi ht' hpi
+      · exact hj ht hpj
+    · exact h.resolve_left
+lemma binary_pair_distinct (hw : WeakColoring k c) {x : ℂ} {i j : Fin k}
+    (hb : BinaryCenter c hc x i j) : i ≠ j := by
+  intro hij
+  subst j
+  have h := binary_alternation c hc hw hb
+  have hf : ∀ᵐ t ∂angularMeasure, False := by
+    filter_upwards [h,hb] with t ht hb
+    have hp := hb.elim id id
+    exact (ht.2.mp hp) hp
+  exact hf.exists.choose_spec
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+lemma binary_cover_real {k : ℕ} (c : ℂ → Fin k) (hc : Measurable c) {x : ℂ} {i j : Fin k}
+    (hb : BinaryCenter c hc x i j) :
+    ∀ᵐ t : ℝ ∂volume, t ∈ liftPalette c hc x i ∨ t ∈ liftPalette c hc x j :=
+  angleMk_quasiMeasurePreserving.ae hb
+lemma binary_alternation_real {k : ℕ} (c : ℂ → Fin k) (hc : Measurable c) (hw : WeakColoring k c)
+    {x : ℂ} {i j : Fin k} (hb : BinaryCenter c hc x i j) (φ : ℝ) :
+    ∀ᵐ t : ℝ ∂volume, t+Real.pi/3+φ ∈ liftPalette c hc x i ↔ t+φ ∉ liftPalette c hc x i := by
+  have h := angleMk_quasiMeasurePreserving.ae (binary_alternation c hc hw hb)
+  have h' := (measurePreserving_add_right volume φ).quasiMeasurePreserving.ae h
+  filter_upwards [h'] with t ht
+  simpa only [liftPalette,mem_preimage,sixty,← AddCircle.coe_add,show t+φ+Real.pi/3=t+Real.pi/3+φ by ring] using ht.1
+lemma alternating_twostep {A : Set ℝ} {κ : ℝ}
+    (h : ∀ᵐ t : ℝ ∂volume, t+κ ∈ A ↔ t ∉ A) :
+    ∀ᵐ t : ℝ ∂volume, t+2*κ ∈ A ↔ t ∈ A := by
+  have h' := (measurePreserving_add_right volume κ).quasiMeasurePreserving.ae h
+  filter_upwards [h,h'] with t ht ht'
+  rw [show t+κ+κ=t+2*κ by ring] at ht'
+  exact ht'.trans (not_congr ht) |>.trans not_not
+lemma liftPalette_shift_periodic {k : ℕ} (c : ℂ → Fin k) (hc : Measurable c) (x : ℂ) (i : Fin k)
+    (φ : ℝ) (n : ℤ) (t : ℝ) :
+    t+(n : ℝ)*(2*Real.pi)+φ ∈ liftPalette c hc x i ↔ t+φ ∈ liftPalette c hc x i := by
+  have he : ((t+(n : ℝ)*(2*Real.pi)+φ : ℝ) : Angles)=((t+φ : ℝ) : Angles) := by
+    rw [show t+(n : ℝ)*(2*Real.pi)+φ=t+φ+n • (2*Real.pi) by rw [zsmul_eq_mul]; ring,
+      AddCircle.coe_add,AddCircle.coe_zsmul,AddCircle.coe_period,smul_zero,add_zero]
+  change ((t+(n : ℝ)*(2*Real.pi)+φ : ℝ) : Angles) ∈ paletteSet c hc x i ↔ ((t+φ : ℝ) : Angles) ∈ paletteSet c hc x i
+  rw [he]
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+lemma binary_transport_invariant {k : ℕ} (c : ℂ → Fin k) (hc : Measurable c) (hw : WeakColoring k c)
+    {x z : ℂ} {i j : Fin k} (hb : BinaryCenter c hc x i j) (hz : BinaryCenter c hc z i j)
+    {d φ : ℝ} (hd : |d|<1/2) (hD : ∀ t, (1/2 : ℝ) ≤ 1+d*binaryB d t)
+    (hxr : (z-x).re=d*Real.cos φ) (hxi : (z-x).im=d*Real.sin φ) :
+    ∀ᵐ t : ℝ ∂volume, t+φ ∈ liftPalette c hc x i ↔ binaryCircleMap d t+φ ∈ liftPalette c hc x i := by
+  have ha : ∀ r, HasDerivAt (fun t => binaryTarget d t+φ) (1+d*binaryB d (-r)) r :=
+    fun r => (binaryTarget_hasDeriv hd r).add_const φ
+  have hs : ∀ r, HasDerivAt (fun t : ℝ => t+φ) 1 r := fun r => (hasDerivAt_id r).add_const φ
+  have ht : ∀ r, HasDerivAt (fun t => binaryOther d t+φ) (1+d*binaryB d r) r :=
+    fun r => (binaryOther_hasDeriv hd r).add_const φ
+  have hn : ∀ r, Real.sin ((binaryOther d r+φ)-(r+φ))≠0 := by
+    intro r
+    simpa only [add_sub_add_right_eq_sub] using binary_nonparallel hd r
+  have hn' : ∀ r, Real.sin ((r+φ)-(binaryOther d r+φ))≠0 := by
+    intro r
+    rw [show (r+φ)-(binaryOther d r+φ)=-((binaryOther d r+φ)-(r+φ)) by ring,Real.sin_neg,neg_ne_zero]
+    exact hn r
+  have he := binary_geometry_rotated hd hxr hxi
+  have he' : ∀ r, circleTarget (samplingParameter x z 1,binaryTarget d r+φ)=
+      circleSum 1 (binaryOther d r+φ,r+φ) := by
+    intro r
+    rw [he r]
+    simp only [circleSum,Prod.mk.injEq]
+    constructor <;> ring
+  have hp (l : Fin k) := palette_transport_param c hc hw ha (fun r => hD (-r)) hs ht
+    (fun _ => one_ne_zero) hn he l
+  have hp' (l : Fin k) := palette_transport_param c hc hw ha (fun r => hD (-r)) ht hs
+    (fun r => ne_of_gt (lt_of_lt_of_le (by norm_num) (hD r))) hn' he' l
+  have hpat := (deriv_lower_qmp ha (fun r => hD (-r))).ae (binary_cover_real c hc hz)
+  have hpx := (measurePreserving_add_right volume φ).quasiMeasurePreserving.ae (binary_cover_real c hc hb)
+  have hpx' := (deriv_lower_qmp ht hD).ae (binary_cover_real c hc hb)
+  have hsame : ∀ᵐ t : ℝ ∂volume, t+φ ∈ liftPalette c hc x i ↔ binaryOther d t+φ ∈ liftPalette c hc x i := by
+    filter_upwards [hpat,hpx,hpx',hp i,hp j,hp' i,hp' j] with t hy hx hx' hii hjj hi' hj'
+    rcases hy with hy|hy
+    · exact ⟨fun hx => False.elim (hii hy hx),fun hx => False.elim (hi' hy hx)⟩
+    · exact ⟨fun _ => hx'.resolve_right (hj' hy),fun _ => hx.resolve_right (hjj hy)⟩
+  let A : Set ℝ := {t | t+φ ∈ liftPalette c hc x i}
+  have hA : ∀ᵐ t : ℝ ∂volume, t+Real.pi/3 ∈ A ↔ t ∉ A := binary_alternation_real c hc hw hb φ
+  have hp2 := (deriv_lower_qmp (binaryCircleMap_hasDeriv_t hd) hD).ae (alternating_twostep hA)
+  filter_upwards [hsame,hp2] with t ht hp2
+  exact ht.trans hp2
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+lemma binary_centers_separated {k : ℕ} (c : ℂ → Fin k) (hc : Measurable c) (hw : WeakColoring k c) :
+    ∃ δ > 0, ∀ (x z : ℂ) (i j : Fin k), BinaryCenter c hc x i j → BinaryCenter c hc z i j →
+      x ≠ z → δ ≤ dist x z := by
+  obtain ⟨δ₁,hδ₁,hctrl⟩ := binary_dynamics_controls
+  obtain ⟨δ₂,hδ₂,hfalse⟩ := binary_alternating_invariant_false
+  refine ⟨min δ₁ δ₂,lt_min hδ₁ hδ₂,?_⟩
+  intro x z i j hx hz hne
+  by_contra! hdδ
+  let d := ‖z-x‖
+  have hd : 0 < d := norm_pos_iff.mpr (sub_ne_zero.mpr (Ne.symm hne))
+  have hdδ' : d < min δ₁ δ₂ := by simpa only [d,dist_eq_norm,norm_sub_rev] using hdδ
+  obtain ⟨C,L,hC,hL,hB,hlo,hhi,hLip,hlarge,hadvance,hdispl,hKL,hda⟩ :=
+    hctrl d hd (lt_min_iff.mp hdδ').1
+  let φ := Complex.arg (z-x)
+  let A : Set ℝ := {t | t+φ ∈ liftPalette c hc x i}
+  apply hfalse d hd (lt_min_iff.mp hdδ').2 A
+  · exact (liftPalette_measurable c hc x i).preimage (measurable_id.add_const φ)
+  · exact liftPalette_shift_periodic c hc x i φ
+  · exact binary_alternation_real c hc hw hx φ
+  · apply binary_transport_invariant c hc hw hx hz (by rwa [abs_of_pos hd])
+    · intro t
+      simpa only [Nat.cast_zero,zero_mul,add_zero] using hlo 0 t
+    · exact (Complex.norm_mul_cos_arg (z-x)).symm
+    · exact (Complex.norm_mul_sin_arg (z-x)).symm
+end EuclideanFiveColor
+end
+
+noncomputable section
+open MeasureTheory Filter Topology
+namespace EuclideanFiveColor
+instance anglesNontrivial : Nontrivial Angles := by
+  have : Nontrivial Circle := ⟨⟨1,-1,by
+    intro h
+    have hh := congrArg (fun z : Circle => (z : ℂ)) h
+    norm_num at hh⟩⟩
+  exact AddCircle.homeomorphCircle'.toEquiv.nontrivial
+instance angularMeasure_nullSingleton : NullSingletonClass angularMeasure := by
+  exact Measure.IsAddHaarMeasure.nullSingletonClass angularMeasure
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+lemma binary_centers_countable {k : ℕ} (c : ℂ → Fin k) (hc : Measurable c) (hw : WeakColoring k c)
+    (i j : Fin k) : {x | BinaryCenter c hc x i j}.Countable := by
+  obtain ⟨δ,hδ,hsep⟩ := binary_centers_separated c hc hw
+  have hp : {x | BinaryCenter c hc x i j}.PairwiseDisjoint (fun x => Metric.ball x (δ/2)) := by
+    intro x hx z hz hne
+    change Disjoint (Metric.ball x (δ/2)) (Metric.ball z (δ/2))
+    rw [Set.disjoint_left]
+    intro w hwx hwz
+    have := dist_triangle x w z
+    have hxw : dist x w < δ/2 := by simpa only [Metric.mem_ball,dist_comm] using hwx
+    have hwz' : dist w z < δ/2 := hwz
+    have := hsep x z i j hx hz hne
+    linarith
+  exact hp.countable_of_isOpen (fun _ _ => Metric.isOpen_ball)
+    (fun x _ => ⟨x,Metric.mem_ball_self (by linarith)⟩)
+lemma ae_not_binary_circle {k : ℕ} (c : ℂ → Fin k) (hc : Measurable c) (hw : WeakColoring k c)
+    (x : ℂ) (i j : Fin k) : ∀ᵐ t ∂angularMeasure, ¬ BinaryCenter c hc (x+direction t) i j := by
+  have hi : Function.Injective (fun t => x+direction t) := by
+    intro a b hab
+    exact AddCircle.injective_toCircle (by positivity : (2*Real.pi:ℝ)≠0)
+      (Subtype.ext (add_left_cancel hab))
+  have hn := ((binary_centers_countable c hc hw i j).preimage hi).measure_zero angularMeasure
+  exact (measure_eq_zero_iff_ae_notMem).mp hn
+lemma exact_sample_isTrace {k : ℕ} (c : ℂ → Fin k) (hc : Measurable c) {y : ℂ}
+    (hy : TypicalSampling c y 1) : IsTrace c hc y (sampleCode c hc y 1) :=
+  subset_closure ⟨hy,rfl⟩
+lemma trace_positive_le_palette {k : ℕ} (c : ℂ → Fin k) (hc : Measurable c) {y : ℂ}
+    {b : TraceVector k} (hb : IsTrace c hc y b) (i : Fin k) :
+    ∀ᵐ t ∂angularMeasure, 0 < traceValue b i t → t ∈ paletteSet c hc y i :=
+  tracePositive_le_palette c hc y i ⟨b,hb⟩
+lemma trace_sample_ae {k : ℕ} (c : ℂ → Fin k) (hc : Measurable c) (y : ℂ) (i : Fin k) :
+    traceValue (sampleCode c hc y 1) i =ᵐ[angularMeasure] sampleFn c y 1 i := by
+  rw [traceValue_sampleCode]
+  exact sample_ae c hc y 1 i
+lemma exact_sample_palette_fixed {k : ℕ} (c : ℂ → Fin k) (hc : Measurable c) {y : ℂ}
+    (hy : TypicalSampling c y 1) (i : Fin k) :
+    ∀ᵐ t ∂angularMeasure, c (y+direction t)=i → t ∈ paletteSet c hc y i := by
+  have hp := trace_positive_le_palette c hc (exact_sample_isTrace c hc hy) i
+  have hs := trace_sample_ae c hc y i
+  apply (hp.and hs).mono
+  intro t ht hcolor
+  apply ht.1
+  rw [ht.2]
+  change 0 < if c (y+(1:ℂ)*direction t)=i then (1:ℝ) else 0
+  rw [one_mul,ite_eq_left hcolor]
+  exact zero_lt_one
+lemma typical_sample_palette {k : ℕ} (c : ℂ → Fin k) (hc : Measurable c) {y : ℂ}
+    (hy : TypicalSampling c y 1) :
+    ∀ᵐ t ∂angularMeasure, t ∈ paletteSet c hc y (c (y+direction t)) := by
+  exact (ae_all_iff.mpr (exact_sample_palette_fixed c hc hy)).mono
+    (fun t ht => ht (c (y+direction t)) rfl)
+lemma ae_circle_palette_exclusion {k : ℕ} (c : ℂ → Fin k) (hc : Measurable c)
+    (hw : WeakColoring k c) (x : ℂ) :
+    ∀ᵐ t ∂angularMeasure, ∀ᵐ v ∂angularMeasure,
+      c (x+direction t+direction v) ∉ palette c hc x t := by
+  have hh := Measure.ae_ae_of_ae_prod (ae_all_iff.mpr (fun i => palette_first_exclusion c hc hw x i))
+  apply hh.mono
+  intro t ht
+  apply ht.mono
+  intro v hv hmem
+  apply hv (c (x+direction t+direction v)) ((mem_palette c hc x t _).mp hmem)
+  simp only [angularSum,Complex.ofReal_one,one_mul]
+lemma ae_circle_typicalSampling {k : ℕ} (c : ℂ → Fin k) (hc : Measurable c) (x : ℂ) :
+    ∀ᵐ t ∂angularMeasure, TypicalSampling c (x+direction t) 1 := by
+  have htyp := Measure.ae_ae_of_ae_prod ((angularSum_quasiMeasurePreserving x (by norm_num : (1:ℝ)≠0)).ae (ae_density_actual c hc))
+  apply htyp.mono
+  intro t ht
+  refine ⟨by norm_num,?_⟩
+  simpa only [Typical,angularSum,Complex.ofReal_one,one_mul] using ht
+lemma two_complement_pred {α : Type*} [Fintype α] [DecidableEq α] {S : Finset α} {i j a : α}
+    (hcomp : Finset.univ\S={i,j}) (hnot : a∉S) (P : α → Prop) (hp : P a) : P i ∨ P j := by
+  have ha : a∈({i,j} : Finset α) := by rw [← hcomp]; simp only [Finset.mem_sdiff,Finset.mem_univ,true_and]; exact hnot
+  simp only [Finset.mem_insert,Finset.mem_singleton] at ha
+  rcases ha with h|h
+  · exact Or.inl (h ▸ hp)
+  · exact Or.inr (h ▸ hp)
+lemma circle_complement_binary {k : ℕ} (c : ℂ → Fin k) (hc : Measurable c)
+    {x : ℂ} {t : Angles} {S : Finset (Fin k)} {i j : Fin k}
+    (hcomp : Finset.univ\S={i,j}) (hsub : S ⊆ palette c hc x t)
+    (ht : ∀ᵐ v ∂angularMeasure, c (x+direction t+direction v) ∉ palette c hc x t)
+    (hy : TypicalSampling c (x+direction t) 1) : BinaryCenter c hc (x+direction t) i j := by
+  apply (ht.and (typical_sample_palette c hc hy)).mono
+  intro v hv
+  exact two_complement_pred hcomp (fun hq => hv.1 (hsub hq))
+    (fun a => v ∈ paletteSet c hc (x+direction t) a) hv.2
+lemma palette_triple_null (c : ℂ → Fin 5) (hc : Measurable c) (hw : WeakColoring 5 c)
+    (x : ℂ) (S : Finset (Fin 5)) (hS : S.card=3) :
+    ∀ᵐ t ∂angularMeasure, ¬ S ⊆ palette c hc x t := by
+  have hSc : (Finset.univ\S).card=2 := by rw [Finset.card_sdiff_of_subset (Finset.subset_univ S)]; simp [hS]
+  obtain ⟨i,j,_,hcomp⟩ := Finset.card_eq_two.mp hSc
+  filter_upwards [ae_circle_palette_exclusion c hc hw x,ae_circle_typicalSampling c hc x,
+    ae_not_binary_circle c hc hw x i j] with t ht hy hn
+  exact fun hsub => hn (circle_complement_binary c hc hcomp hsub ht hy)
+lemma palette_cardinality (c : ℂ → Fin 5) (hc : Measurable c) (hw : WeakColoring 5 c) (x : ℂ) :
+    ∀ᵐ t ∂angularMeasure, 1 ≤ (palette c hc x t).card ∧ (palette c hc x t).card ≤ 2 := by
+  filter_upwards [palette_nonempty c hc x,ae_all_iff.mpr (fun S : Finset (Fin 5) =>
+    ae_all_iff.mpr (fun hS : S.card=3 => palette_triple_null c hc hw x S hS))] with t ht htriple
+  refine ⟨Nat.one_le_iff_ne_zero.mpr (Finset.card_ne_zero.mpr ht),?_⟩
+  by_contra! hcard
+  obtain ⟨S,hsub,hS⟩ := Finset.exists_subset_card_eq (by omega : 3 ≤ (palette c hc x t).card)
+  exact htriple S hS hsub
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+def triangleAngle : ℝ := Real.arccos (5/6)
+lemma triangleAngle_cos : Real.cos triangleAngle=5/6 := by
+  exact Real.cos_arccos (by norm_num) (by norm_num)
+lemma triangleAngle_pos : 0<triangleAngle := Real.arccos_pos.mpr (by norm_num)
+lemma triangleAngle_lt_pi : triangleAngle<Real.pi := Real.arccos_lt_pi.mpr (by norm_num)
+lemma triangleAngle_ratio_irrational : Irrational (triangleAngle/Real.pi) := by
+  rintro ⟨q,hq⟩
+  have hh := niven (θ := triangleAngle) ⟨q,(div_eq_iff Real.pi_ne_zero).mp hq.symm⟩
+    ⟨5/6,by norm_num only [Rat.cast_div,Rat.cast_ofNat]; exact triangleAngle_cos⟩
+  rw [triangleAngle_cos] at hh
+  norm_num at hh
+lemma triangle_rotation_ergodic :
+    Ergodic (fun t : Angles => ((2*triangleAngle : ℝ) : Angles)+t) angularMeasure := by
+  apply ergodic_add_left_of_denseRange_zsmul _ angularMeasure
+  apply AddCircle.denseRange_zsmul_coe_iff.mpr
+  convert triangleAngle_ratio_irrational using 1
+  field_simp
+lemma equal_triangleAngle_positive {s : Angles → Bool} (hs : Measurable s)
+    (hflip : ∀ᵐ t∂angularMeasure, s (t+(Real.pi/3 : ℝ))= !(s t)) :
+    0<angularMeasure {t | s t=s (t+(triangleAngle : Angles))} := by
+  by_contra hz
+  have hz' : angularMeasure {t | s t=s (t+(triangleAngle : Angles))}=0 := le_antisymm (le_of_not_gt hz) bot_le
+  have hf : ∀ᵐ t∂angularMeasure, s (t+(triangleAngle : Angles))= !(s t) := by
+    filter_upwards [(measure_eq_zero_iff_ae_notMem.mp hz')] with t ht
+    change ¬s t=s (t+(triangleAngle : Angles)) at ht
+    cases h₁ : s t <;> cases h₂ : s (t+(triangleAngle : Angles)) <;> simp_all
+  have hf₂ := (measurePreserving_add_right angularMeasure (triangleAngle : Angles)).quasiMeasurePreserving.ae hf
+  have hinv : (fun t : Angles => s (((2*triangleAngle : ℝ) : Angles)+t))=ᵐ[angularMeasure] s := by
+    filter_upwards [hf,hf₂] with t ht ht'
+    have he : ((2*triangleAngle : ℝ) : Angles)+t=(t+(triangleAngle : Angles))+(triangleAngle : Angles) := by
+      have hh : ((2*triangleAngle : ℝ) : Angles)=(triangleAngle : Angles)+(triangleAngle : Angles) := by
+        calc ((2*triangleAngle : ℝ) : Angles) = ((triangleAngle+triangleAngle : ℝ) : Angles) := congrArg (fun x : ℝ => (x : Angles)) (two_mul triangleAngle)
+          _ = _ := AddCircle.coe_add _ _ _
+      rw [hh]
+      abel
+    rw [he,ht',ht,Bool.not_not]
+  obtain ⟨b,hb⟩ := triangle_rotation_ergodic.ae_eq_const_of_ae_eq_comp₀ hs.nullMeasurable hinv
+  have hb' := (measurePreserving_add_right angularMeasure ((Real.pi/3:ℝ) : Angles)).quasiMeasurePreserving.ae hb
+  have hfalse : ∀ᵐ t∂angularMeasure, False := by
+    filter_upwards [hflip,hb,hb'] with t ht ht' ht''
+    change s t=b at ht'
+    change s (t+((Real.pi/3:ℝ):Angles))=b at ht''
+    rw [ht',ht''] at ht
+    cases b <;> cases ht
+  have hh : (1:ENNReal)=0 := by simpa using (ae_iff.mp hfalse)
+  exact one_ne_zero hh
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+lemma circle_exp_sub_norm_sq (a b : ℝ) :
+    ‖(Circle.exp a : ℂ)-(Circle.exp b : ℂ)‖^2=2-2*Real.cos (a-b) := by
+  rw [← Complex.normSq_eq_norm_sq]
+  simp [Complex.normSq_apply,Circle.coe_exp,Complex.exp_mul_I,← Complex.ofReal_cos,← Complex.ofReal_sin]
+  rw [Real.cos_sub]
+  nlinarith [Real.cos_sq_add_sin_sq a,Real.cos_sq_add_sin_sq b]
+def sourceTriangle (x : ℂ) (t : ℝ) (j : Fin 3) : ℂ :=
+  angularSum x 1 (angleChart (t+2*(j:ℝ)*(Real.pi/3),t+2*(j:ℝ)*(Real.pi/3)+Real.pi+triangleAngle))
+lemma sourceTriangle_eq (x : ℂ) (t : ℝ) (j : Fin 3) :
+    sourceTriangle x t j=x+(Circle.exp t : ℂ)*(Circle.exp (2*(j:ℝ)*(Real.pi/3)) : ℂ)*
+      (1-(Circle.exp triangleAngle : ℂ)) := by
+  simp only [sourceTriangle,angularSum,angleChart,Complex.ofReal_one,one_mul,direction_real,
+    Circle.exp_add,Circle.coe_mul]
+  have hpi : (Circle.exp Real.pi : ℂ)= -1 := by
+    simp [Circle.coe_exp,Complex.exp_mul_I]
+  rw [hpi]
+  ring
+lemma sourceTriangle_pair (x : ℂ) (t : ℝ) (i j : Fin 3) (hne : i≠j) :
+    ‖sourceTriangle x t i-sourceTriangle x t j‖=1 := by
+  have hγ : ‖(1:ℂ)-(Circle.exp triangleAngle : ℂ)‖^2=1/3 := by
+    have hh := circle_exp_sub_norm_sq 0 triangleAngle
+    norm_num only [Circle.exp_zero,Circle.coe_one,zero_sub,Real.cos_neg,triangleAngle_cos] at hh ⊢
+    exact hh
+  have hc₂ : Real.cos (2*(Real.pi/3))= -1/2 := by
+    rw [Real.cos_two_mul,Real.cos_pi_div_three]; norm_num
+  have hc₄ : Real.cos (4*(Real.pi/3))= -1/2 := by
+    rw [show 4*(Real.pi/3)=2*(2*(Real.pi/3)) by ring,Real.cos_two_mul,hc₂]; norm_num
+  have hd : ‖(Circle.exp (2*(i:ℝ)*(Real.pi/3)) : ℂ)-(Circle.exp (2*(j:ℝ)*(Real.pi/3)) : ℂ)‖^2=3 := by
+    rw [circle_exp_sub_norm_sq]
+    fin_cases i <;> fin_cases j <;> try contradiction
+    all_goals norm_num only [Fin.val_zero,Fin.val_one,Fin.val_two,Nat.cast_zero,Nat.cast_one,Nat.cast_ofNat,mul_zero,mul_one,zero_mul,sub_zero,zero_sub]
+    all_goals
+      simp only [show 2*(Real.pi/3)-4*(Real.pi/3)= -(2*(Real.pi/3)) by ring,
+        show 4*(Real.pi/3)-2*(Real.pi/3)=2*(Real.pi/3) by ring,Real.cos_neg,hc₂,hc₄]
+      norm_num
+  have he : sourceTriangle x t i-sourceTriangle x t j=
+      (Circle.exp t : ℂ)*((Circle.exp (2*(i:ℝ)*(Real.pi/3)) : ℂ)-
+        (Circle.exp (2*(j:ℝ)*(Real.pi/3)) : ℂ))*(1-(Circle.exp triangleAngle : ℂ)) := by
+    rw [sourceTriangle_eq,sourceTriangle_eq]; ring
+  rw [he,norm_mul,norm_mul,Circle.norm_coe,one_mul]
+  have hn := norm_nonneg ((Circle.exp (2*(i:ℝ)*(Real.pi/3)) : ℂ)-(Circle.exp (2*(j:ℝ)*(Real.pi/3)) : ℂ))
+  have hm := norm_nonneg ((1:ℂ)-(Circle.exp triangleAngle : ℂ))
+  have heq : (‖(Circle.exp (2*(i:ℝ)*(Real.pi/3)) : ℂ)-(Circle.exp (2*(j:ℝ)*(Real.pi/3)) : ℂ)‖ *
+      ‖(1:ℂ)-(Circle.exp triangleAngle : ℂ)‖)^2=1 := by
+    rw [mul_pow,hd,hγ]; norm_num
+  nlinarith [mul_nonneg hn hm]
+end EuclideanFiveColor
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PlaneColoring/BorelTransfer.lean b/lean/OAI/Geometry/PlaneColoring/BorelTransfer.lean
new file mode 100644
index 0000000000000000000000000000000000000000..34897568b9e5bc28d41006e59435e0ec1c4cf938
--- /dev/null
+++ b/lean/OAI/Geometry/PlaneColoring/BorelTransfer.lean
@@ -0,0 +1,451 @@
+import OAI.Analysis.PlaneSpectrum.Ergodicity
+import OAI.Geometry.PlaneColoring.Coloring
+
+namespace OAI
+
+noncomputable section
+open MeasureTheory Filter Topology
+open scoped ENNReal NNReal InnerProductSpace
+namespace PlaneFiveColor.LabelModel
+open Spectral EuclideanFiveColor EuclideanFiveColor.ContinuousVectors
+variable (k : ℕ) (ω : Space k)
+def colorValue (i : Fin k) (η : Space k) : ℂ := if η.val 0 = i then 1 else 0
+lemma color_memLp (i : Fin k) : MemLp (colorValue k i) 2 (measure k ω) := by
+  refine MemLp.of_bound ?_ 1 ?_
+  · apply Measurable.aestronglyMeasurable
+    apply Measurable.ite _ measurable_const measurable_const
+    exact (measurableSet_singleton i).preimage (show Measurable (fun η : Space k => η.val 0) from
+      ((continuous_apply (0 : E)).comp continuous_subtype_val).measurable)
+  · exact .of_forall fun η => by dsimp [colorValue]; split_ifs <;> simp
+abbrev colorVector (i : Fin k) : Hilbert k ω := (color_memLp k ω i).toLp (colorValue k i)
+lemma color_ae (i : Fin k) : colorVector k ω i =ᵐ[measure k ω] colorValue k i := (color_memLp k ω i).coeFn_toLp
+lemma color_rotation_fixed (i : Fin k) (b : K) :
+    (rotation k ω b : Hilbert k ω →L[ℂ] Hilbert k ω) (colorVector k ω i) = colorVector k ω i := by
+  rw [rotation_apply]
+  apply Lp.ext
+  filter_upwards [Lp.coeFn_compMeasurePreserving (colorVector k ω i) (rotate_preserving k ω b),
+    (rotate_preserving k ω b).quasiMeasurePreserving.ae (color_ae k ω i),color_ae k ω i] with η h1 h2 h3
+  simp only [Function.comp_apply,h1,h2,h3,colorValue,rotate_apply,mul_zero]
+lemma color_unit_correlation (i : Fin k) (b : K) :
+    ⟪colorVector k ω i,op (representation k ω) (kCoeff b) (colorVector k ω i)⟫_ℂ = 0 := by
+  rw [op_apply,L2.inner_def]
+  apply integral_eq_zero_of_ae
+  filter_upwards [color_ae k ω i,Lp.coeFn_compMeasurePreserving (colorVector k ω i) (trans_preserving k ω (kCoeff b)),
+    (trans_preserving k ω (kCoeff b)).quasiMeasurePreserving.ae (color_ae k ω i)] with η h1 h2 h3
+  simp only [h1,h2,Function.comp_apply,h3,colorValue,trans_apply,zero_add]
+  have hp : η.val 0 ≠ η.val (kCoeff b) := η.property 0 (kCoeff b) (by
+    change ‖(0 : ℂ)-(b.val:ℂ)‖ = 1
+    rw [zero_sub,norm_neg]
+    exact Circle.norm_coe b.val)
+  by_cases hi : η.val 0 = i
+  · have hj : η.val (kCoeff b) ≠ i := fun h => hp (hi.trans h.symm)
+    simp [hi,hj]
+  · simp [hi]
+instance continuous_complete : CompleteSpace (space (A := E) (representation k ω)) :=
+  (isClosed_space (A := E) (representation k ω)).completeSpace_coe
+def projected (i : Fin k) : Hilbert k ω := (space (A := E) (representation k ω)).starProjection (colorVector k ω i)
+lemma projected_range (i : Fin k) : ∀ᵐ η ∂measure k ω,
+    ((projected k ω i) η).im = 0 ∧ 0 ≤ ((projected k ω i) η).re ∧ ((projected k ω i) η).re ≤ 1 := by
+  apply projection_range (measure k ω) (representation k ω) (trans k) (trans_preserving k ω) (op_apply k ω)
+  filter_upwards [color_ae k ω i] with η h
+  rw [h]
+  dsimp [colorValue]
+  split_ifs <;> norm_num
+lemma projected_rotation_fixed (i : Fin k) (b : K) :
+    (rotation k ω b : Hilbert k ω →L[ℂ] Hilbert k ω) (projected k ω i) = projected k ω i := by
+  exact projection_covariance_fixed (representation k ω) (rotation k ω b)
+    (Homeomorph.mulLeft₀ (kCoeff b) (kCoeff_ne_zero b)) (mul_zero _)
+    (rotation_covariance k ω b) _ (color_rotation_fixed k ω i b)
+lemma projected_unit_correlation (i : Fin k) (b : K) :
+    ⟪projected k ω i,op (representation k ω) (kCoeff b) (projected k ω i)⟫_ℂ = 0 := by
+  have hb := residual_fourier (representation k ω) (rotation k ω) (rotation_covariance k ω)
+    (colorVector k ω i-projected k ω i)
+    ((space (A := E) (representation k ω)).sub_starProjection_mem_orthogonal (colorVector k ω i))
+    (fun b => by rw [map_sub,color_rotation_fixed,projected_rotation_fixed]) (kCoeff b) (kCoeff_ne_zero b)
+  exact (projection_coefficient (representation k ω) (colorVector k ω i) (kCoeff b) hb).trans (color_unit_correlation k ω i b)
+end PlaneFiveColor.LabelModel
+end
+
+noncomputable section
+open MeasureTheory Filter Topology
+namespace PlaneFiveColor.LabelModel
+open Spectral EuclideanFiveColor EuclideanFiveColor.ContinuousVectors
+variable (k : ℕ) (ω : Space k)
+def oneVector : Hilbert k ω := (memLp_const (1 : ℂ)).toLp (fun _ : Space k => (1 : ℂ))
+lemma one_ae : oneVector k ω =ᵐ[measure k ω] fun _ => (1 : ℂ) := MemLp.coeFn_toLp _
+lemma one_fixed (a : E) : op (representation k ω) a (oneVector k ω) = oneVector k ω := by
+  rw [op_apply]
+  apply Lp.ext
+  filter_upwards [Lp.coeFn_compMeasurePreserving (oneVector k ω) (trans_preserving k ω a),
+    (trans_preserving k ω a).quasiMeasurePreserving.ae (one_ae k ω),one_ae k ω] with η h1 h2 h3
+  exact h1.trans (h2.trans h3.symm)
+lemma one_continuous : oneVector k ω ∈ space (A := E) (representation k ω) := by
+  change Tendsto (fun a : E => op (representation k ω) a (oneVector k ω)) (𝓝 0) (𝓝 (oneVector k ω))
+  simpa only [one_fixed] using (tendsto_const_nhds : Tendsto (fun _ : E => oneVector k ω) (𝓝 0) (𝓝 (oneVector k ω)))
+lemma color_sum : ∑ i : Fin k, colorVector k ω i = oneVector k ω := by
+  apply Lp.ext
+  filter_upwards [Lp.coeFn_fun_finsetSum (Finset.univ) (colorVector k ω),ae_all_iff.mpr (color_ae k ω),one_ae k ω] with η h1 h2 h3
+  rw [h1,h3]
+  simp_rw [h2,colorValue]
+  simp
+lemma projected_sum : ∑ i : Fin k, projected k ω i = oneVector k ω := by
+  change ∑ i : Fin k, (space (A := E) (representation k ω)).starProjection (colorVector k ω i) = _
+  rw [← map_sum,color_sum]
+  exact (space (A := E) (representation k ω)).starProjection_eq_self_iff.mpr (one_continuous k ω)
+end PlaneFiveColor.LabelModel
+end
+
+noncomputable section
+open Filter Topology
+open scoped ComplexConjugate
+namespace PlaneFiveColor.Spectral
+lemma dense_E : Dense (E : Set ℂ) := by
+  intro z
+  have hh : ∀ x y : ℝ, (x : ℂ)+(y : ℂ)*Complex.I ∈ closure (E : Set ℂ) := by
+    intro x y
+    refine (Rat.denseRange_cast : DenseRange ((↑) : ℚ → ℝ)).induction_on₂
+      (p := fun x y : ℝ => (x : ℂ)+(y : ℂ)*Complex.I ∈ closure (E : Set ℂ)) ?_ ?_ x y
+    · exact isClosed_closure.preimage ((Complex.continuous_ofReal.comp continuous_fst).add
+        ((Complex.continuous_ofReal.comp continuous_snd).mul_const Complex.I))
+    · intro q r
+      apply subset_closure
+      apply mem_algebraicClosure_iff.mpr
+      have hq : IsAlgebraic ℚ ((q : ℝ) : ℂ) := by simpa using isAlgebraic_ratCast ℚ (A := ℂ) q
+      have hr : IsAlgebraic ℚ ((r : ℝ) : ℂ) := by simpa using isAlgebraic_ratCast ℚ (A := ℂ) r
+      exact hq.add (hr.mul (mem_algebraicClosure_iff.mp imagUnit.property))
+  simpa only [Complex.re_add_im] using hh z.re z.im
+lemma algebraic_norm (z : E) : IsAlgebraic ℚ (‖(z:ℂ)‖ : ℂ) := by
+  have hz : IsAlgebraic ℚ (z : ℂ) := mem_algebraicClosure_iff.mp z.property
+  have hc : IsAlgebraic ℚ (conj (z : ℂ)) := hz.algHom (Complex.conjAe.restrictScalars ℚ).toAlgHom
+  apply IsAlgebraic.of_pow (n := 2) (by norm_num)
+  simpa only [Complex.mul_conj,Complex.normSq_eq_norm_sq,Complex.ofReal_pow] using hz.mul hc
+end PlaneFiveColor.Spectral
+end
+
+noncomputable section
+open Filter Topology
+open scoped InnerProductSpace
+namespace PlaneFiveColor.Spectral
+open EuclideanFiveColor.ContinuousVectors
+variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
+  (U : DiscreteE →* unitary (H →L[ℂ] H))
+lemma orbit_uniformContinuous {v : H} (hv : v ∈ space (A := E) U) : UniformContinuous (fun a : E => op U a v) := by
+  apply Metric.uniformContinuous_iff.mpr
+  intro ε hε
+  obtain ⟨δ,hδ,h⟩ := Metric.continuousAt_iff.mp (orbit_continuous U hv).continuousAt ε hε
+  refine ⟨δ,hδ,fun a b hab => ?_⟩
+  have ht : dist (op U (a-b) v) v < ε := by
+    have hh := h (show dist (a-b) 0 < δ by simpa only [dist_zero_right,dist_eq_norm,sub_zero] using hab)
+    simpa only [op_zero (A := E) U] using hh
+  calc
+    _ = dist (op U b (op U (a-b) v)) (op U b v) := by
+      rw [← op_add (A := E) U]
+      congr 2
+      abel_nf
+    _ = dist (op U (a-b) v) v := dist_op U b _ _
+    _ < ε := ht
+def orbitExtension (v : H) : ℂ → H := dense_E.extend (fun a : E => op U a v)
+lemma orbitExtension_continuous {v : H} (hv : v ∈ space (A := E) U) : Continuous (orbitExtension U v) :=
+  (dense_E.uniformContinuous_extend (orbit_uniformContinuous U hv)).continuous
+lemma orbitExtension_eq {v : H} (hv : v ∈ space (A := E) U) (a : E) : orbitExtension U v (a : ℂ) = op U a v :=
+  dense_E.extend_eq (orbit_continuous U hv) a
+lemma orbitExtension_inner {v w : H} (hv : v ∈ space (A := E) U) (hw : w ∈ space (A := E) U) (x y : ℂ) :
+    ⟪orbitExtension U v x,orbitExtension U w (x+y)⟫_ℂ = ⟪v,orbitExtension U w y⟫_ℂ := by
+  have hc := orbitExtension_continuous U hv
+  have hd := orbitExtension_continuous U hw
+  refine dense_E.denseRange_val.induction_on₂
+    (p := fun x y : ℂ => ⟪orbitExtension U v x,orbitExtension U w (x+y)⟫_ℂ = ⟪v,orbitExtension U w y⟫_ℂ) ?_ ?_ x y
+  · exact isClosed_eq ((hc.comp continuous_fst).inner (hd.comp (continuous_fst.add continuous_snd)))
+      (continuous_const.inner (hd.comp continuous_snd))
+  · intro a b
+    change ⟪orbitExtension U v (a : ℂ),orbitExtension U w ((a+b : E) : ℂ)⟫_ℂ = _
+    rw [orbitExtension_eq U hv,orbitExtension_eq U hw,orbitExtension_eq U hw,op_add (A := E) U]
+    exact Unitary.inner_map_map (U (Multiplicative.ofAdd a)) v (op U b w)
+end PlaneFiveColor.Spectral
+end
+
+noncomputable section
+open Filter Topology
+namespace PlaneFiveColor.Spectral
+lemma normalized_algebraic (z : E) (hz : z ≠ 0) :
+    ∃ b : K, (b.val : ℂ) = (z : ℂ) / (‖(z:ℂ)‖ : ℂ) := by
+  have hn : ‖(z:ℂ)‖ ≠ 0 := norm_ne_zero_iff.mpr (Subtype.coe_ne_coe.mpr hz)
+  have hb : ‖(z:ℂ) / (‖(z:ℂ)‖ : ℂ)‖ = 1 := by
+    rw [norm_div,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg (norm_nonneg _),div_self hn]
+  have hzalg : IsAlgebraic ℚ (z : ℂ) := mem_algebraicClosure_iff.mp z.property
+  exact ⟨⟨⟨_,mem_sphere_zero_iff_norm.mpr hb⟩,
+    by
+      change IsAlgebraic ℚ ((z : ℂ)/(‖(z:ℂ)‖ : ℂ))
+      simpa only [div_eq_mul_inv] using hzalg.mul (algebraic_norm z).inv⟩,rfl⟩
+lemma circle_closed_induction {p : ℂ → Prop} (hp : IsClosed {z | p z})
+    (halg : ∀ b : K, p (b.val : ℂ)) (u : Circle) : p (u : ℂ) := by
+  obtain ⟨s,hs,ht⟩ := mem_closure_iff_seq_limit.mp (dense_E (u : ℂ))
+  have hne : ∀ᶠ n in atTop, s n ≠ 0 := ht.eventually (isOpen_compl_singleton.mem_nhds (Circle.coe_ne_zero u))
+  have hnorm : Tendsto (fun n => s n / (‖s n‖ : ℂ)) atTop (𝓝 (u : ℂ)) := by
+    have h := ht.div (Complex.continuous_ofReal.continuousAt.tendsto.comp ht.norm)
+      (show (‖(u:ℂ)‖ : ℂ) ≠ 0 by simp [Circle.norm_coe])
+    have he : (u : ℂ) / (‖(u:ℂ)‖ : ℂ) = (u : ℂ) := by simp only [Circle.norm_coe,Complex.ofReal_one,div_one]
+    rw [he] at h
+    convert h using 1
+    rfl
+  apply hp.mem_of_tendsto hnorm
+  filter_upwards [hne] with n hn
+  obtain ⟨b,hb⟩ := normalized_algebraic ⟨s n,hs n⟩ (fun h => hn (congrArg Subtype.val h))
+  rw [← hb]
+  exact halg b
+end PlaneFiveColor.Spectral
+end
+
+noncomputable section
+open MeasureTheory Filter Topology
+open scoped InnerProductSpace
+namespace PlaneFiveColor.LabelModel
+open Spectral EuclideanFiveColor EuclideanFiveColor.ContinuousVectors
+variable (k : ℕ) (ω : Space k)
+lemma projected_continuous (i : Fin k) : projected k ω i ∈ space (A := E) (representation k ω) :=
+  (space (A := E) (representation k ω)).starProjection_apply_mem _
+def fieldVector (i : Fin k) (x : ℂ) : Hilbert k ω := orbitExtension (representation k ω) (projected k ω i) x
+lemma fieldVector_continuous (i : Fin k) : Continuous (fieldVector k ω i) :=
+  orbitExtension_continuous _ (projected_continuous k ω i)
+lemma fieldVector_algebraic (i : Fin k) (a : E) : fieldVector k ω i (a : ℂ) = op (representation k ω) a (projected k ω i) :=
+  orbitExtension_eq _ (projected_continuous k ω i) a
+lemma fieldVector_sum (x : ℂ) : ∑ i : Fin k, fieldVector k ω i x = oneVector k ω := by
+  refine dense_E.denseRange_val.induction_on x (isClosed_eq (continuous_finsetSum _ (fun i _ => fieldVector_continuous k ω i)) continuous_const) ?_
+  intro a
+  simp only [fieldVector_algebraic,← map_sum,projected_sum,one_fixed]
+lemma fieldVector_range (i : Fin k) (x : ℂ) : ∀ᵐ η ∂measure k ω,
+    (fieldVector k ω i x η).im = 0 ∧ 0 ≤ (fieldVector k ω i x η).re ∧ (fieldVector k ω i x η).re ≤ 1 := by
+  have he : clamp_lipschitz.compLp clamp_zero (fieldVector k ω i x) = fieldVector k ω i x := by
+    refine dense_E.denseRange_val.induction_on x (isClosed_eq
+      ((clamp_lipschitz.continuous_compLp clamp_zero).comp (fieldVector_continuous k ω i)) (fieldVector_continuous k ω i)) ?_
+    intro a
+    rw [fieldVector_algebraic,op_apply]
+    apply Lp.ext
+    filter_upwards [clamp_lipschitz.coeFn_compLp clamp_zero (Lp.compMeasurePreserving (trans k a) (trans_preserving k ω a) (projected k ω i)),
+      Lp.coeFn_compMeasurePreserving (projected k ω i) (trans_preserving k ω a),
+      (trans_preserving k ω a).quasiMeasurePreserving.ae (projected_range k ω i)] with η h1 h2 h3
+    simp only [Function.comp_apply] at h1 h2
+    rw [h1,h2]
+    exact clamp_eq h3
+  have ha := clamp_lipschitz.coeFn_compLp clamp_zero (fieldVector k ω i x)
+  rw [he] at ha
+  filter_upwards [ha] with η hη
+  exact hη.symm ▸ clamp_range _
+lemma fieldVector_unit_correlation (i : Fin k) (x : ℂ) (u : Circle) :
+    ⟪fieldVector k ω i x,fieldVector k ω i (x+u)⟫_ℂ = 0 := by
+  rw [show ⟪fieldVector k ω i x,fieldVector k ω i (x+u)⟫_ℂ =
+    ⟪projected k ω i,fieldVector k ω i u⟫_ℂ from
+    orbitExtension_inner _ (projected_continuous k ω i) (projected_continuous k ω i) x u]
+  apply circle_closed_induction (isClosed_eq (continuous_const.inner (fieldVector_continuous k ω i)) continuous_const) _ u
+  intro b
+  change ⟪projected k ω i,fieldVector k ω i ((kCoeff b : E) : ℂ)⟫_ℂ = 0
+  rw [fieldVector_algebraic]
+  exact projected_unit_correlation k ω i b
+end PlaneFiveColor.LabelModel
+end
+
+noncomputable section
+open MeasureTheory Filter Topology TopologicalSpace
+open scoped ENNReal
+namespace EuclideanFiveColor
+variable {T X : Type*} [TopologicalSpace T] [SeparableSpace T] [Nonempty T]
+  [MeasurableSpace T] [BorelSpace T] [MeasurableSpace X] (μ : Measure X)
+  (f : T → Lp ℂ 2 μ) (hf : Continuous f)
+include hf
+omit [MeasurableSpace T] [BorelSpace T] in
+lemma dense_approx (t : T) (ε : ℝ) (hε : 0 < ε) :
+    ∃ n, dist (f (TopologicalSpace.denseSeq T n)) (f t) < ε := by
+  have h := mem_closure_image hf.continuousAt (TopologicalSpace.denseRange_denseSeq T t)
+  rw [← Set.range_comp] at h
+  obtain ⟨n,hn⟩ := Metric.mem_closure_range_iff.mp h ε hε
+  exact ⟨n,by simpa only [Function.comp_apply,dist_comm] using hn⟩
+def approxIndex (n : ℕ) (t : T) : ℕ := Nat.find (dense_approx μ f hf t ((1/2:ℝ)^n) (by positivity))
+lemma index_measurable (n : ℕ) : Measurable (approxIndex μ f hf n) := by
+  apply measurable_find
+  intro j
+  exact isOpen_lt (continuous_const.dist hf) continuous_const |>.measurableSet
+omit [MeasurableSpace T] [BorelSpace T] in
+lemma index_bound (n : ℕ) (t : T) :
+    dist (f (TopologicalSpace.denseSeq T (approxIndex μ f hf n t))) (f t) < (1/2:ℝ)^n := by
+  unfold approxIndex
+  exact Nat.find_spec (dense_approx μ f hf t ((1/2:ℝ)^n) (by positivity))
+def approxField (n : ℕ) (p : T × X) : ℂ :=
+  f (TopologicalSpace.denseSeq T (approxIndex μ f hf n p.1)) p.2
+lemma approxField_measurable (n : ℕ) : Measurable (approxField μ f hf n) := by
+  have h : Measurable (fun p : ℕ × X => f (TopologicalSpace.denseSeq T p.1) p.2) :=
+    measurable_from_prod_countable_right fun j => (Lp.stronglyMeasurable (f (TopologicalSpace.denseSeq T j))).measurable
+  exact h.comp (((index_measurable μ f hf n).comp measurable_fst).prodMk measurable_snd)
+omit [MeasurableSpace T] [BorelSpace T] in
+lemma approxField_tendsto (t : T) : ∀ᵐ x ∂μ,
+    Tendsto (fun n => approxField μ f hf n (t,x)) atTop (𝓝 (f t x)) := by
+  let g : ℕ → Lp ℂ 2 μ := fun n => f (TopologicalSpace.denseSeq T (approxIndex μ f hf n t))-f t
+  have hs : Summable (fun n => ‖g n‖) := summable_geometric_two.of_nonneg_of_le (fun n => norm_nonneg _) (fun n => by simpa only [g,dist_eq_norm] using (index_bound μ f hf n t).le)
+  have ha : ∀ᵐ x ∂μ, Summable (fun n => ‖g n x‖) := by
+    apply summable_norm_of_tsum_eLpNorm_ne_top (by norm_num : (1:ℝ≥0∞) ≤ 2)
+    simpa only [← Lp.enorm_def] using (tsum_enorm_ne_top_iff_summable_norm.mpr hs)
+  have he : ∀ᵐ x ∂μ, ∀ n, g n x = approxField μ f hf n (t,x)-f t x :=
+    ae_all_iff.mpr fun n => Lp.coeFn_sub _ _
+  filter_upwards [ha,he] with x hx he
+  apply tendsto_iff_norm_sub_tendsto_zero.mpr
+  simpa only [he] using hx.tendsto_atTop_zero
+def measurableField (p : T × X) : ℂ :=
+  ⟨limsup (fun n => (approxField μ f hf n p).re) atTop,
+   limsup (fun n => (approxField μ f hf n p).im) atTop⟩
+lemma measurableField_measurable : Measurable (measurableField μ f hf) := by
+  have hr := Measurable.limsup fun n => Complex.continuous_re.measurable.comp (approxField_measurable μ f hf n)
+  have hi := Measurable.limsup fun n => Complex.continuous_im.measurable.comp (approxField_measurable μ f hf n)
+  have hh := (Complex.continuous_ofReal.measurable.comp hr).add ((Complex.continuous_ofReal.measurable.comp hi).mul_const Complex.I)
+  convert hh using 1
+  funext p
+  apply Complex.ext <;> simp [measurableField]
+lemma exists_measurable_field : ∃ F : T × X → ℂ, Measurable F ∧ ∀ t, (fun x => F (t,x)) =ᵐ[μ] f t := by
+  refine ⟨measurableField μ f hf,?_,?_⟩
+  · exact measurableField_measurable μ f hf
+  · intro t
+    filter_upwards [approxField_tendsto μ f hf t] with x hx
+    apply Complex.ext
+    · exact (Complex.continuous_re.continuousAt.tendsto.comp hx).limsup_eq
+    · exact (Complex.continuous_im.continuousAt.tendsto.comp hx).limsup_eq
+end EuclideanFiveColor
+end
+
+noncomputable section
+open MeasureTheory
+open scoped InnerProductSpace ComplexConjugate
+namespace EuclideanFiveColor
+variable {X : Type*} [MeasurableSpace X] (μ : Measure X)
+lemma positive_orthogonal (v w : Lp ℂ 2 μ)
+    (hv : ∀ᵐ x ∂μ, (v x).im = 0 ∧ 0 ≤ (v x).re)
+    (hw : ∀ᵐ x ∂μ, (w x).im = 0 ∧ 0 ≤ (w x).re)
+    (ho : ⟪v,w⟫_ℂ = 0) : ∀ᵐ x ∂μ, (v x).re * (w x).re = 0 := by
+  have hi := L2.integrable_inner (𝕜 := ℂ) v w
+  have he : (fun x => (⟪v x,w x⟫_ℂ).re) =ᵐ[μ] (fun x => (v x).re * (w x).re) := by
+    filter_upwards [hv,hw] with x hx hy
+    simp [RCLike.inner_apply,Complex.mul_re,hx.1,hy.1,mul_comm]
+  have hh : Integrable (fun x => (v x).re * (w x).re) μ := hi.re.congr he
+  apply (integral_eq_zero_iff_of_nonneg_ae _ hh).mp
+  · rw [← integral_congr_ae he]
+    change (∫ x, RCLike.re (⟪v x,w x⟫_ℂ) ∂μ) = 0
+    rw [integral_re hi,← L2.inner_def,ho]
+    rfl
+  · filter_upwards [hv,hw] with x hx hy
+    exact mul_nonneg hx.2 hy.2
+end EuclideanFiveColor
+end
+
+noncomputable section
+open MeasureTheory Filter Topology
+namespace PlaneFiveColor.LabelModel
+open Spectral EuclideanFiveColor EuclideanFiveColor.ContinuousVectors
+variable (k : ℕ) (ω : Space k)
+lemma exists_label_field : ∃ H : Fin k → ℂ × Space k → ℝ,
+    (∀ i, Measurable (H i)) ∧
+    (∀ x : ℂ, ∀ᵐ η ∂measure k ω, ∑ i, H i (x,η) = 1) ∧
+    (∀ x : ℂ, ∀ u : Circle, ∀ᵐ η ∂measure k ω, ∀ i, H i (x,η)*H i (x+u,η) = 0) := by
+  have hex (i : Fin k) := exists_measurable_field (measure k ω) (fieldVector k ω i) (fieldVector_continuous k ω i)
+  choose F hm hae using hex
+  refine ⟨fun i p => (F i p).re,fun i => Complex.continuous_re.measurable.comp (hm i),?_,?_⟩
+  · intro x
+    have hs := Lp.coeFn_fun_finsetSum Finset.univ (fun i : Fin k => fieldVector k ω i x)
+    rw [fieldVector_sum] at hs
+    filter_upwards [ae_all_iff.mpr (fun i => hae i x),hs,one_ae k ω] with η he hs h1
+    have hc : ∑ i : Fin k, F i (x,η) = 1 := by
+      simp_rw [he]
+      exact hs.symm.trans h1
+    have hr := congrArg Complex.re hc
+    simpa only [map_sum,Complex.re_sum,Complex.one_re] using hr
+  · intro x u
+    apply ae_all_iff.mpr
+    intro i
+    have hz := positive_orthogonal (measure k ω) (fieldVector k ω i x) (fieldVector k ω i (x+u))
+      ((fieldVector_range k ω i x).mono fun _ h => ⟨h.1,h.2.1⟩)
+      ((fieldVector_range k ω i (x+u)).mono fun _ h => ⟨h.1,h.2.1⟩)
+      (fieldVector_unit_correlation k ω i x u)
+    filter_upwards [hae i x,hae i (x+u),hz] with η h1 h2 h3
+    simpa only [h1,h2] using h3
+end PlaneFiveColor.LabelModel
+end
+
+noncomputable section
+open MeasureTheory
+namespace EuclideanFiveColor
+variable {X : Type*} [MeasurableSpace X] {k : ℕ} [NeZero k]
+def positiveSupport (f : Fin k → X → ℝ) (x : X) : Finset (Fin k) := Finset.univ.filter fun i => 0 < f i x
+omit [NeZero k] in
+lemma positiveSupport_measurable (f : Fin k → X → ℝ) (hf : ∀ i, Measurable (f i)) : Measurable (positiveSupport f) := by
+  apply measurable_finset_iff.mpr
+  intro i
+  simp only [positiveSupport,Finset.mem_filter,Finset.mem_univ,true_and]
+  exact measurableSet_setOfPred.mp (measurableSet_lt measurable_const (hf i))
+private def pickSupport (s : Finset (Fin k)) : Fin k := if h : s.Nonempty then h.choose else 0
+def positiveSelect (f : Fin k → X → ℝ) (x : X) : Fin k := pickSupport (positiveSupport f x)
+lemma positiveSelect_measurable (f : Fin k → X → ℝ) (hf : ∀ i, Measurable (f i)) : Measurable (positiveSelect f) :=
+  (measurable_of_countable pickSupport).comp (positiveSupport_measurable f hf)
+omit [MeasurableSpace X] in
+lemma positiveSelect_pos (f : Fin k → X → ℝ) (x : X) (hs : ∑ i, f i x = 1) : 0 < f (positiveSelect f x) x := by
+  have hex : ∃ i, 0 < f i x := by
+    by_contra! h
+    have hh := Finset.sum_nonpos (s := Finset.univ) (fun i _ => h i)
+    linarith
+  have hn : (positiveSupport f x).Nonempty := by
+    obtain ⟨i,hi⟩ := hex
+    exact ⟨i,Finset.mem_filter.mpr ⟨Finset.mem_univ _,hi⟩⟩
+  have hmem : positiveSelect f x ∈ positiveSupport f x := by
+    simp only [positiveSelect,pickSupport,dite_eq_left hn]
+    exact hn.choose_spec
+  exact (Finset.mem_filter.mp hmem).2
+end EuclideanFiveColor
+end
+
+noncomputable section
+open MeasureTheory Filter
+namespace EuclideanFiveColor
+variable {Ω S : Type*} [MeasurableSpace Ω] [MeasurableSpace S]
+  (μ : Measure Ω) [IsProbabilityMeasure μ] (ν : Measure S) [SFinite ν]
+  (d : S → ℂ) (hd : Measurable d) {k : ℕ} [NeZero k]
+include hd
+lemma sample_field (H : Fin k → ℂ × Ω → ℝ) (hm : ∀ i, Measurable (H i))
+    (hp : ∀ x : ℂ, ∀ᵐ ω ∂μ, ∑ i, H i (x,ω) = 1)
+    (ho : ∀ x : ℂ, ∀ s : S, ∀ᵐ ω ∂μ, ∀ i, H i (x,ω)*H i (x+d s,ω) = 0) :
+    ∃ c : ℂ → Fin k, Measurable c ∧ ∀ᵐ p ∂(volume : Measure ℂ).prod ν, c p.1 ≠ c (p.1+d p.2) := by
+  let c : ℂ × Ω → Fin k := positiveSelect H
+  have hc : Measurable c := positiveSelect_measurable H hm
+  have ha (p : ℂ × S) : ∀ᵐ ω ∂μ, c (p.1,ω) ≠ c (p.1+d p.2,ω) := by
+    filter_upwards [hp p.1,hp (p.1+d p.2),ho p.1 p.2] with ω h1 h2 h3
+    intro he
+    have hpos1 := positiveSelect_pos H (p.1,ω) h1
+    have hpos2 := positiveSelect_pos H (p.1+d p.2,ω) h2
+    change 0 < H (c (p.1,ω)) (p.1,ω) at hpos1
+    change 0 < H (c (p.1+d p.2,ω)) (p.1+d p.2,ω) at hpos2
+    rw [← he] at hpos2
+    exact (mul_pos hpos1 hpos2).ne' (h3 _)
+  have hm1 : Measurable (fun q : (ℂ × S) × Ω => c (q.1.1,q.2)) :=
+    hc.comp ((measurable_fst.fst).prodMk measurable_snd)
+  have hm2 : Measurable (fun q : (ℂ × S) × Ω => c (q.1.1+d q.1.2,q.2)) :=
+    hc.comp (((measurable_fst.fst).add (hd.comp measurable_fst.snd)).prodMk measurable_snd)
+  have hh : ∀ᵐ ω ∂μ, ∀ᵐ p ∂(volume : Measure ℂ).prod ν, c (p.1,ω) ≠ c (p.1+d p.2,ω) :=
+    (Measure.ae_ae_comm (measurableSet_eq_fun hm1 hm2).compl).mp (.of_forall ha)
+  obtain ⟨ω,hω⟩ := hh.exists
+  exact ⟨fun x => c (x,ω),hc.comp (measurable_id.prodMk measurable_const),hω⟩
+end EuclideanFiveColor
+end
+
+noncomputable section
+open MeasureTheory
+namespace EuclideanFiveColor
+lemma proper_to_borel_weak {k : ℕ} [NeZero k] {c : ℂ → Fin k} (hc : ProperColoring k c) :
+    ∃ d : ℂ → Fin k, Measurable d ∧ WeakColoring k d := by
+  let ω : PlaneFiveColor.LabelModel.Space k := ⟨fun x => c x,fun x y h => hc x y h⟩
+  obtain ⟨H,hm,hp,ho⟩ := PlaneFiveColor.LabelModel.exists_label_field k ω
+  obtain ⟨d,hd,hunit⟩ := sample_field (PlaneFiveColor.LabelModel.measure k ω) angularMeasure
+    direction direction_continuous.measurable
+    H hm hp (fun x t => ho x (AddCircle.toCircle t))
+  exact ⟨d,hd,hd.aemeasurable,hunit⟩
+lemma proper_to_weak {k : ℕ} [NeZero k] {c : ℂ → Fin k} (hc : ProperColoring k c) :
+    ∃ d : ℂ → Fin k, WeakColoring k d := by
+  obtain ⟨d,_,h⟩ := proper_to_borel_weak hc
+  exact ⟨d,h⟩
+end EuclideanFiveColor
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PlaneColoring/BoundaryTrace.lean b/lean/OAI/Geometry/PlaneColoring/BoundaryTrace.lean
new file mode 100644
index 0000000000000000000000000000000000000000..773fc6f48b8dd328bb7b3db733f5a27ce0debea0
--- /dev/null
+++ b/lean/OAI/Geometry/PlaneColoring/BoundaryTrace.lean
@@ -0,0 +1,367 @@
+import OAI.Geometry.PlaneColoring.Rigidity
+import OAI.Geometry.PlaneColoring.States
+
+namespace OAI
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+open scoped RealInnerProductSpace
+namespace EuclideanFiveColor
+variable {k : ℕ} (c : ℂ → Fin k) (hc : Measurable c)
+lemma support_good_sequence {A : Set ℝ} {P : ℝ → Prop} {a : ℝ}
+    (ha : a∈(volume.restrict A).support) (hP : ∀ᵐt∂volume,t∈A → P t) :
+    ∃t : ℕ → ℝ, Tendsto t atTop (𝓝 a) ∧ ∀n,P (t n) := by
+  have hex (n : ℕ) : ∃t,dist t a<1/((n:ℝ)+1) ∧ P t := by
+    have hp : 0<volume (Metric.ball a (1/((n:ℝ)+1))∩A) := by
+      simpa only [Measure.restrict_apply measurableSet_ball] using
+        (Measure.mem_support_iff_forall a).mp ha _ (Metric.ball_mem_nhds a (by positivity))
+    obtain ⟨t,ht,hpt⟩ := ((frequently_ae_mem_iff.mpr hp.ne').and_eventually hP).exists
+    exact ⟨t,ht.1,hpt ht.2⟩
+  choose t hd hpt using hex
+  refine ⟨t,?_,hpt⟩
+  rw [tendsto_iff_dist_tendsto_zero]
+  exact squeeze_zero (fun _ => dist_nonneg) (fun n => (hd n).le) tendsto_one_div_add_atTop_nhds_zero_nat
+lemma state_support_avoiding_trace (hw : WeakColoring k c) {x : ℂ} {A : Set ℝ} {J : Finset (Fin k)}
+    (hA : RealState hc x A J) {a : ℝ} (ha : a∈(volume.restrict A).support) :
+    ∃b : TraceVector k,IsTrace c hc (x+direction (a : Angles)) b ∧ ∀i∈J,traceValue b i=0 := by
+  have he := angleMk_quasiMeasurePreserving.ae (ae_circle_palette_exclusion c hc hw x)
+  have ht := angleMk_quasiMeasurePreserving.ae (ae_circle_typicalSampling c hc x)
+  have hp : ∀ᵐt : ℝ∂volume,t∈A → TypicalSampling c (x+direction (t : Angles)) 1 ∧
+      ∀i∈J,sample c hc (x+direction (t : Angles)) 1 i=0 := by
+    filter_upwards [he,ht,ae_all_iff.mpr (fun i => ae_all_iff.mpr (fun hi : i∈J => hA.2 i hi))] with t he ht hJ hta
+    refine ⟨ht,?_⟩
+    intro i hi
+    apply Lp.ext
+    filter_upwards [sample_ae c hc (x+direction (t : Angles)) 1 i,he,Lp.coeFn_zero ℝ 2 angularMeasure] with v hv he hz
+    rw [hz,hv]
+    have hne : c (x+direction (t : Angles)+direction v)≠i := by
+      intro hh
+      apply he
+      rw [hh]
+      exact (mem_palette c hc x (t : Angles) i).mpr (hJ i hi hta)
+    simp only [sampleFn,Complex.ofReal_one,one_mul,ite_eq_right hne,Pi.zero_apply]
+  obtain ⟨t,ht,hp⟩ := support_good_sequence ha hp
+  obtain ⟨b,_,φ,hφ,hb⟩ := (isCompact_univ : IsCompact (Set.univ : Set (TraceVector k))).isSeqCompact.subseq_of_frequently_in
+    (Eventually.frequently (Eventually.of_forall fun n => Set.mem_univ (sampleCode c hc (x+direction (t n : Angles)) 1)))
+  have hlim : Tendsto (fun n => x+direction (t n : Angles)) atTop (𝓝 (x+direction (a : Angles))) :=
+    tendsto_const_nhds.add (direction_continuous.continuousAt.tendsto.comp ((AddCircle.continuous_mk' (2*Real.pi)).continuousAt.tendsto.comp ht))
+  refine ⟨b,trace_of_realization c hc (fun n => (hp (φ n)).1) (hlim.comp hφ.tendsto_atTop) tendsto_const_nhds hb,?_⟩
+  intro i hi
+  have hh := traceCode_tendsto_iff.mp hb i (traceValue b i)
+  simp only [Function.comp_apply,traceValue_sampleCode,(hp _).2 i hi,inner_zero_left] at hh
+  exact inner_self_eq_zero.mp (tendsto_nhds_unique tendsto_const_nhds hh).symm
+lemma avoiding_trace_palette {x : ℂ} {b : TraceVector k} (hb : IsTrace c hc x b)
+    {J : Finset (Fin k)} (hJ : ∀i∈J,traceValue b i=0) :
+    ∀ᵐt∂angularMeasure,¬palette c hc x t⊆J := by
+  have hs : (fun t => ∑i,traceValue b i t)=ᵐ[angularMeasure](fun _ => (1:ℝ)) := by
+    have hh := Lp.coeFn_fun_finsetSum Finset.univ (traceValue b)
+    rw [(trace_bounds c hc hb).2.2] at hh
+    exact hh.symm.trans (oneL2_ae angularMeasure)
+  have hz (i : Fin k) : ∀ᵐt∂angularMeasure,i∈J → traceValue b i t=0 := by
+    by_cases hi : i∈J
+    · rw [hJ i hi]
+      exact (Lp.coeFn_zero ℝ 2 angularMeasure).mono fun _ hh _ => hh
+    · exact Eventually.of_forall fun _ hh => (hi hh).elim
+  filter_upwards [hs,ae_all_iff.mpr hz,ae_all_iff.mpr (fun i => tracePositive_le_palette c hc x i ⟨b,hb⟩)] with t hs hz hp hsub
+  have hn : ∀i : Fin k,traceValue b i t≤0 := by
+    intro i
+    by_contra! hpos
+    have hm := hsub ((mem_palette c hc x t i).mpr (hp i hpos))
+    linarith [hz i hm]
+  have hh := Finset.sum_nonpos (s := Finset.univ) (fun i _ => hn i)
+  linarith
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory MeasureTheory.Measure
+open scoped RealInnerProductSpace
+namespace EuclideanFiveColor
+variable {k : ℕ} (c : ℂ → Fin k) (hc : Measurable c)
+lemma sample_region_ae (r : ℝ) {W : Set ℂ} {J : Finset (Fin k)}
+    (hW : ∀ᵐz : ℂ∂volume,z∈W → c z∈J) :
+    ∀ᵐy : ℂ∂volume,∀ᵐt∂angularMeasure,y+(r : ℂ)*direction t∈W →
+      c (y+(r : ℂ)*direction t)∈J := by
+  have hq : QuasiMeasurePreserving (fun p : ℂ × Angles => p.1+(r : ℂ)*direction p.2)
+      ((volume : Measure ℂ).prod angularMeasure) volume :=
+    QuasiMeasurePreserving.prod_of_left
+      (measurable_fst.add (measurable_const.mul (direction_continuous.measurable.comp measurable_snd)))
+      (Eventually.of_forall fun t => (measurePreserving_add_right (volume : Measure ℂ)
+        ((r : ℂ)*direction t)).quasiMeasurePreserving)
+  exact Measure.ae_ae_of_ae_prod (hq.ae hW)
+include hc in
+lemma sampling_near_region {x : ℂ} {r : ℝ} (hr : 0<r) {K : Set Angles} (hK : IsCompact K)
+    {W : Set ℂ} (ho : IsOpen W) {J : Finset (Fin k)}
+    (hW : ∀ᵐz : ℂ∂volume,z∈W → c z∈J)
+    (hmem : ∀t∈K,x+(r : ℂ)*direction t∈W) {δ : ℝ} (hδ : 0<δ) :
+    ∃y : ℂ,dist y x<δ ∧ TypicalSampling c y r ∧
+      ∀ᵐt∂angularMeasure,t∈K → c (y+(r : ℂ)*direction t)∈J := by
+  let U : Set (ℂ × Angles) := {p | p.1+(r : ℂ)*direction p.2∈W}
+  have hU : IsOpen U := ho.preimage (continuous_fst.add (continuous_const.mul (direction_continuous.comp continuous_snd)))
+  obtain ⟨V,T,hV,_,hxV,hKT,hVT⟩ := generalized_tube_lemma (isCompact_singleton (x := x)) hK hU
+    (by rintro ⟨y,t⟩ ⟨hy,ht⟩; change y=x at hy; subst y; exact hmem t ht)
+  have hv : x∈V := hxV (mem_singleton x)
+  have hpos : 0<volume (V∩Metric.ball x δ) := (hV.inter Metric.isOpen_ball).measure_pos volume
+    ⟨x,hv,Metric.mem_ball_self hδ⟩
+  obtain ⟨y,hy,hyp,hyW⟩ := ((frequently_ae_mem_iff.mpr hpos.ne').and_eventually
+    ((ae_typicalSampling c hc hr).and (sample_region_ae c r hW))).exists
+  refine ⟨y,hy.2,hyp,hyW.mono fun t ht htK => ht ?_⟩
+  exact hVT (show (y,t)∈V×ˢT from ⟨hy.1,hKT htK⟩)
+lemma weak_limit_zero_on {f : ℕ → RealL2 angularMeasure} {b : RealL2 angularMeasure}
+    {K : Set Angles} (hK : MeasurableSet K)
+    (ht : ∀v,Tendsto (fun n => ⟪f n,v⟫) atTop (𝓝 ⟪b,v⟫))
+    (hb : 0≤ᵐ[angularMeasure] b)
+    (hf : ∀n,∀ᵐt∂angularMeasure,t∈K → f n t=0) :
+    ∀ᵐt∂angularMeasure,t∈K → b t=0 := by
+  have ht' := ht (indicatorConstLp 2 hK (measure_ne_top angularMeasure K) 1)
+  have he (v : RealL2 angularMeasure) :
+      ⟪v,indicatorConstLp 2 hK (measure_ne_top angularMeasure K) 1⟫=∫t in K,v t∂angularMeasure := by
+    rw [real_inner_comm,L2.inner_indicatorConstLp_one]
+  simp only [he] at ht'
+  have hf0 (n) : ∫t in K,f n t∂angularMeasure=0 := by
+    apply integral_eq_zero_of_ae
+    exact (ae_restrict_iff' hK).mpr (hf n)
+  simp only [hf0] at ht'
+  have hzero : ∫t in K,b t∂angularMeasure=0 := (tendsto_nhds_unique tendsto_const_nhds ht').symm
+  have hz := (integral_eq_zero_iff_of_nonneg_ae (ae_restrict_of_ae hb)
+    ((Lp.memLp b).integrable (by norm_num)).integrableOn).mp hzero
+  exact (ae_restrict_iff' hK).mp hz
+lemma trace_region_palette {x : ℂ} {K : Set Angles} (hK : IsCompact K)
+    {W : Set ℂ} (ho : IsOpen W) {J : Finset (Fin k)}
+    (hW : ∀ᵐz : ℂ∂volume,z∈W → c z∈J) {r : ℕ → ℝ}
+    (hr : ∀n,0<r n) (hrt : Tendsto r atTop (𝓝 1))
+    (hmem : ∀n,∀t∈K,x+(r n : ℂ)*direction t∈W) :
+    ∀ᵐt∂angularMeasure,t∈K → (palette c hc x t∩J).Nonempty := by
+  have hex (n : ℕ) := sampling_near_region c hc (hr n) hK ho hW (hmem n)
+    (δ := 1/((n:ℝ)+1)) (by positivity)
+  choose y hy hyt hyJ using hex
+  have hyl : Tendsto y atTop (𝓝 x) := by
+    rw [tendsto_iff_dist_tendsto_zero]
+    exact squeeze_zero (fun _ => dist_nonneg) (fun n => (hy n).le) tendsto_one_div_add_atTop_nhds_zero_nat
+  obtain ⟨b,_,φ,hφ,hb⟩ := (isCompact_univ : IsCompact (Set.univ : Set (TraceVector k))).isSeqCompact.subseq_of_frequently_in
+    (Eventually.frequently (Eventually.of_forall fun n => Set.mem_univ (sampleCode c hc (y n) (r n))))
+  have htrace := trace_of_realization c hc (fun n => hyt (φ n)) (hyl.comp hφ.tendsto_atTop)
+    (hrt.comp hφ.tendsto_atTop) hb
+  have hz (i : Fin k) (hi : i∉J) : ∀ᵐt∂angularMeasure,t∈K → traceValue b i t=0 := by
+    apply weak_limit_zero_on hK.measurableSet
+      (by simpa only [traceValue_sampleCode] using traceCode_tendsto_iff.mp hb i)
+      ((trace_bounds c hc htrace).1 i)
+    intro n
+    filter_upwards [hyJ (φ n),sample_ae c hc (y (φ n)) (r (φ n)) i] with t ht he htK
+    simp only [Function.comp_apply,traceValue_sampleCode]
+    rw [he]
+    unfold sampleFn
+    split_ifs with htcol
+    · exact (hi (htcol ▸ ht htK)).elim
+    · rfl
+  have hs : (fun t => ∑i,traceValue b i t)=ᵐ[angularMeasure](fun _ => (1:ℝ)) := by
+    have hh := Lp.coeFn_fun_finsetSum Finset.univ (traceValue b)
+    rw [(trace_bounds c hc htrace).2.2] at hh
+    exact hh.symm.trans (oneL2_ae angularMeasure)
+  filter_upwards [hs,ae_all_iff.mpr (fun i => ae_all_iff.mpr (hz i)),
+    ae_all_iff.mpr (fun i => tracePositive_le_palette c hc x i ⟨b,htrace⟩)] with t hs hz hp htK
+  by_contra! hn
+  have hn' : ∀i,traceValue b i t≤0 := by
+    intro i
+    by_contra! hpos
+    have hi : i∈palette c hc x t := (mem_palette c hc x t i).mpr (hp i hpos)
+    by_cases hiJ : i∈J
+    · have hh := Finset.mem_inter.mpr ⟨hi,hiJ⟩
+      rw [hn] at hh
+      exact Finset.notMem_empty i hh
+    · linarith [hz i hiJ htK]
+  have hh := Finset.sum_nonpos (s := Finset.univ) (fun i _ => hn' i)
+  linarith
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+variable {k : ℕ} (c : ℂ → Fin k) (hc : Measurable c)
+lemma outer_strip_trace {x : ℂ} {t : Angles} {δ : ℝ} (hδ : 0<δ)
+    {W : Set ℂ} (ho : IsOpen W) {J : Finset (Fin k)}
+    (hW : ∀ᵐz : ℂ∂volume,z∈W → c z∈J)
+    (hstrip : ∀u : Angles,dist (direction u) (direction t)<δ → ∀r : ℝ,
+      1<r → r<1+δ → x+(r : ℂ)*direction u∈W) :
+    ∃U : Set Angles,IsOpen U ∧ t∈U ∧
+      ∀ᵐu∂angularMeasure,u∈U → (palette c hc x u∩J).Nonempty := by
+  let K : Set Angles := {u | dist (direction u) (direction t)≤δ/2}
+  let U : Set Angles := {u | dist (direction u) (direction t)<δ/2}
+  have hK : IsCompact K := (isClosed_le (direction_continuous.dist continuous_const) continuous_const).isCompact
+  let r : ℕ → ℝ := fun n => 1+(δ/2)/(n+1)
+  have hr (n) : 0<r n := by dsimp [r]; positivity
+  have hrt : Tendsto r atTop (𝓝 1) := by
+    simpa only [r,div_eq_mul_inv,one_mul,mul_zero,add_zero] using (tendsto_const_nhds (x := (1:ℝ))).add
+      (tendsto_one_div_add_atTop_nhds_zero_nat.const_mul (δ/2))
+  have hh := trace_region_palette c hc hK ho hW hr hrt (fun n u hu => by
+    apply hstrip u (by dsimp [K] at hu; linarith)
+    · dsimp [r]
+      have : 0<(δ/2)/((n:ℝ)+1) := by positivity
+      linarith
+    · have hn : (1:ℝ)≤(n:ℝ)+1 := by linarith [Nat.cast_nonneg (α := ℝ) n]
+      have hb := div_le_self (by positivity : (0:ℝ)≤δ/2) hn
+      dsimp [r]
+      linarith)
+  refine ⟨U,isOpen_lt (direction_continuous.dist continuous_const) continuous_const,?_,?_⟩
+  · change dist (direction t) (direction t)<δ/2
+    simp only [dist_self]; positivity
+  · exact hh.mono fun u hu hU => hu (show u∈K from (show dist (direction u) (direction t)<δ/2 from hU).le)
+lemma inner_strip_trace {x : ℂ} {t : Angles} {δ : ℝ} (hδ : 0<δ)
+    {W : Set ℂ} (ho : IsOpen W) {J : Finset (Fin k)}
+    (hW : ∀ᵐz : ℂ∂volume,z∈W → c z∈J)
+    (hstrip : ∀u : Angles,dist (direction u) (direction t)<δ → ∀r : ℝ,
+      0<r → 1-δ<r → r<1 → x+(r : ℂ)*direction u∈W) :
+    ∃U : Set Angles,IsOpen U ∧ t∈U ∧
+      ∀ᵐu∂angularMeasure,u∈U → (palette c hc x u∩J).Nonempty := by
+  let K : Set Angles := {u | dist (direction u) (direction t)≤δ/2}
+  let U : Set Angles := {u | dist (direction u) (direction t)<δ/2}
+  have hK : IsCompact K := (isClosed_le (direction_continuous.dist continuous_const) continuous_const).isCompact
+  let ε := min δ (1/2)
+  have he : 0<ε := lt_min hδ (by norm_num)
+  have heδ : ε≤δ := min_le_left _ _
+  have he1 : ε≤1/2 := min_le_right _ _
+  let r : ℕ → ℝ := fun n => 1-(ε/2)/(n+1)
+  have hb (n : ℕ) : (ε/2)/(n+1)≤ε/2 := div_le_self (by positivity) (by linarith [Nat.cast_nonneg (α := ℝ) n])
+  have hr (n) : 0<r n := by dsimp [r]; linarith [hb n]
+  have hrt : Tendsto r atTop (𝓝 1) := by
+    simpa only [r,div_eq_mul_inv,one_mul,mul_zero,sub_zero] using (tendsto_const_nhds (x := (1:ℝ))).sub
+      (tendsto_one_div_add_atTop_nhds_zero_nat.const_mul (ε/2))
+  have hh := trace_region_palette c hc hK ho hW hr hrt (fun n u hu => by
+    apply hstrip u (by dsimp [K] at hu; linarith) (r n) (hr n)
+    · dsimp [r]; linarith [hb n]
+    · dsimp [r]
+      have : 0<(ε/2)/((n:ℝ)+1) := by positivity
+      linarith)
+  refine ⟨U,isOpen_lt (direction_continuous.dist continuous_const) continuous_const,?_,?_⟩
+  · change dist (direction t) (direction t)<δ/2
+    simp only [dist_self]; positivity
+  · exact hh.mono fun u hu hU => hu (show u∈K from (show dist (direction u) (direction t)<δ/2 from hU).le)
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+/-- The source local binary rigidity on arbitrary one-sided strips, allowing
+independent choices of radial side at the two equilateral neighbors. -/
+lemma binary_strip_rigidity {k : ℕ} {c : ℂ → Fin k} (hc : Measurable c) (hw : WeakColoring k c)
+    {a b : Fin k} {U V : Set ℂ} (hU : IsOpen U) (hV : IsOpen V)
+    (hUc : ∀ᵐq : ℂ∂volume,q∈U → c q=a ∨ c q=b)
+    (hVc : ∀ᵐq : ℂ∂volume,q∈V → c q=a ∨ c q=b)
+    {x e : ℂ} (he : ‖e‖=1) {δ : ℝ} (hδ : 0<δ) (bu bv : Bool)
+    (hstripU : ∀e' : ℂ,‖e'‖=1 → dist e' e<δ → ∀r : ℝ,
+      0<r → |r-1|<δ → (if bu then 1<r else r<1) → x+(r:ℂ)*e'∈U)
+    (hstripV : ∀e' : ℂ,‖e'‖=1 → dist e' (e*(Circle.exp (Real.pi/3) : ℂ))<δ → ∀r : ℝ,
+      0<r → |r-1|<δ → (if bv then 1<r else r<1) → x+(r:ℂ)*e'∈V) :
+    ∃i : Fin k,(i=a ∨ i=b) ∧ ∃η>0,∀e' : ℂ,‖e'‖=1 → dist e' e<η → ∀r : ℝ,
+      0<r → |r-1|<η → (if bu then 1<r else r<1) → x+(r:ℂ)*e'∈essentialLocalSet c i := by
+  obtain ⟨v,hv,hpv⟩ := radial_unit_contact he hδ bv hstripV
+  obtain ⟨w,hpw,hw',hd⟩ := open_two_unit_directions hV hv hpv
+  let W : Set ℂ := {p | p+v∈V ∧ p+w∈V}
+  have hW : IsOpen W := (hV.preimage (continuous_id.add continuous_const)).inter
+    (hV.preimage (continuous_id.add continuous_const))
+  obtain ⟨ε,hε,hbox⟩ := polarBox_small hδ hW ⟨hpv,hpw⟩ hstripU he
+  obtain ⟨i,hi,_⟩ := connected_local_ae_constant (polarBox_connected x e bu hε) (fun p hp => by
+    have hp' := hbox hp
+    obtain ⟨r,hr,j,hj⟩ := binary_open_local_rigidity hc hw hU hV hUc hVc hp'.1 hp'.2.1 hp'.2.2 hv hw' hd
+    exact ⟨j,Metric.ball p r,Metric.isOpen_ball,Metric.mem_ball_self hr,hj⟩)
+  have hab : i=a ∨ i=b := by
+    obtain ⟨p,hp⟩ := (polarBox_connected x e bu hε).nonempty
+    have hpos := (hU.inter (essentialLocalSet_open c i)).measure_pos (volume : Measure ℂ)
+      ⟨p,(hbox hp).1,hi hp⟩
+    obtain ⟨q,hq,hqa⟩ := Measure.exists_mem_of_measure_ne_zero_of_ae hpos.ne'
+      (ae_restrict_of_ae (hUc.and (essentialLocalSet_ae c i)))
+    simpa only [hqa.2 hq.2] using hqa.1 hq.1
+  obtain ⟨η,hη,hh⟩ := polarBox_contains_strip he bu hε
+  exact ⟨i,hab,η,hη,fun e' he' hh' r _ hr hside => hi (hh e' he' hh' r hr hside)⟩
+/-- The resulting fixed outsider is actually in the maximal palette on an
+angular neighborhood, by one-sided radial weak traces. -/
+lemma binary_strip_palette_rigidity {k : ℕ} (c : ℂ → Fin k) (hc : Measurable c) (hw : WeakColoring k c)
+    {a b : Fin k} {U V : Set ℂ} (hU : IsOpen U) (hV : IsOpen V)
+    (hUc : ∀ᵐq : ℂ∂volume,q∈U → c q=a ∨ c q=b)
+    (hVc : ∀ᵐq : ℂ∂volume,q∈V → c q=a ∨ c q=b)
+    {x : ℂ} {t : Angles} {δ : ℝ} (hδ : 0<δ) (bu bv : Bool)
+    (hstripU : ∀e' : ℂ,‖e'‖=1 → dist e' (direction t)<δ → ∀r : ℝ,
+      0<r → |r-1|<δ → (if bu then 1<r else r<1) → x+(r:ℂ)*e'∈U)
+    (hstripV : ∀e' : ℂ,‖e'‖=1 → dist e' (direction t*(Circle.exp (Real.pi/3) : ℂ))<δ → ∀r : ℝ,
+      0<r → |r-1|<δ → (if bv then 1<r else r<1) → x+(r:ℂ)*e'∈V) :
+    ∃i : Fin k,(i=a ∨ i=b) ∧ ∃A : Set Angles,IsOpen A ∧ t∈A ∧
+      ∀ᵐu∂angularMeasure,u∈A → i∈palette c hc x u := by
+  obtain ⟨i,hi,η,hη,hs⟩ := binary_strip_rigidity hc hw hU hV hUc hVc (direction_norm t) hδ bu bv hstripU hstripV
+  have hmono : ∀ᵐq : ℂ∂volume,q∈essentialLocalSet c i → c q∈({i} : Finset (Fin k)) :=
+    (essentialLocalSet_ae c i).mono fun q hq hqi => by simpa only [Finset.mem_singleton] using hq hqi
+  have hex : ∃A : Set Angles,IsOpen A ∧ t∈A ∧
+      ∀ᵐu∂angularMeasure,u∈A → (palette c hc x u∩{i}).Nonempty := by
+    cases bu
+    · apply inner_strip_trace c hc hη (essentialLocalSet_open c i) hmono
+      intro u hu r hr hrη hr1
+      apply hs (direction u) (direction_norm u) hu r hr _ hr1
+      rw [abs_lt]; constructor <;> linarith
+    · apply outer_strip_trace c hc hη (essentialLocalSet_open c i) hmono
+      intro u hu r hr1 hrη
+      apply hs (direction u) (direction_norm u) hu r (by linarith) _ hr1
+      rw [abs_lt]; constructor <;> linarith
+  obtain ⟨A,hA,ht,hae⟩ := hex
+  refine ⟨i,hi,A,hA,ht,hae.mono fun u hu hAu => ?_⟩
+  obtain ⟨j,hj⟩ := hu hAu
+  rcases Finset.mem_inter.mp hj with ⟨hjp,hji⟩
+  simpa only [Finset.mem_singleton.mp hji] using hjp
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+open scoped RealInnerProductSpace
+namespace EuclideanFiveColor
+/-- If two unit rays can never be simultaneously positive at a generic angle,
+they are antipodal. This avoids any assumption about multiplicities of ray lines. -/
+lemma unit_antipodes_of_no_positive {v w : ℂ} (hv : ‖v‖=1) (hw : ‖w‖=1)
+    (h : ∀ᵐt∂angularMeasure,¬(0<⟪direction t,v⟫ ∧ 0<⟪direction t,w⟫)) : w= -v := by
+  by_contra hne
+  have hsum : v+w≠0 := by
+    intro hz
+    apply hne
+    apply eq_neg_iff_add_eq_zero.mpr
+    simpa only [add_comm] using hz
+  have hvv : ⟪v,v⟫=(1:ℝ) := by rw [real_inner_self_eq_norm_sq,hv]; norm_num
+  have hww : ⟪w,w⟫=(1:ℝ) := by rw [real_inner_self_eq_norm_sq,hw]; norm_num
+  have hs : 0<⟪v+w,v+w⟫ := by
+    rw [real_inner_self_eq_norm_sq]
+    exact sq_pos_of_pos (norm_pos_iff.mpr hsum)
+  have hc := real_inner_comm v w
+  simp only [inner_add_left,inner_add_right,hvv,hww] at hs
+  have hvpos : 0<⟪v+w,v⟫ := by rw [inner_add_left,hvv]; linarith
+  have hwpos : 0<⟪v+w,w⟫ := by rw [inner_add_left,hww]; linarith
+  let e : ℂ := ‖v+w‖⁻¹ • (v+w)
+  have he : ‖e‖=1 := by
+    simp only [e,norm_smul,Real.norm_eq_abs,abs_inv,abs_of_nonneg (norm_nonneg (v+w))]
+    exact inv_mul_cancel₀ (norm_ne_zero_iff.mpr hsum)
+  have hev : 0<⟪e,v⟫ := by
+    change 0<⟪‖v+w‖⁻¹ • (v+w),_⟫
+    rw [real_inner_smul_left]
+    exact mul_pos (inv_pos.mpr (norm_pos_iff.mpr hsum)) hvpos
+  have hew : 0<⟪e,w⟫ := by
+    change 0<⟪‖v+w‖⁻¹ • (v+w),_⟫
+    rw [real_inner_smul_left]
+    exact mul_pos (inv_pos.mpr (norm_pos_iff.mpr hsum)) hwpos
+  obtain ⟨t,ht⟩ := direction_exists he
+  let U : Set Angles := {t | 0<⟪direction t,v⟫ ∧ 0<⟪direction t,w⟫}
+  have hUv : IsOpen {t : Angles | 0<⟪direction t,v⟫} := isOpen_lt continuous_const (direction_continuous.inner continuous_const)
+  have hUw : IsOpen {t : Angles | 0<⟪direction t,w⟫} := isOpen_lt continuous_const (direction_continuous.inner continuous_const)
+  have hU : IsOpen U := hUv.inter hUw
+  have htU : t∈U := by change _∧_; rw [ht]; exact ⟨hev,hew⟩
+  have hpos := hU.measure_pos angularMeasure ⟨t,htU⟩
+  obtain ⟨u,hu,hu'⟩ := Measure.exists_mem_of_measure_ne_zero_of_ae hpos.ne' (ae_restrict_of_ae h)
+  exact hu' hu
+/-- The source's all-generic split condition already forces opposite edge
+positions to be actual antipodes; the collinear case is retained. -/
+lemma four_split_antipodes (v : Fin 4 → ℂ) (hv : ∀i,‖v i‖=1)
+    (hsplit : ∀ᵐt∂angularMeasure,∀i,¬(0<⟪direction t,v i⟫ ∧ 0<⟪direction t,v (i+2)⟫)) :
+    ∀i,v (i+2)= -v i := by
+  intro i
+  exact unit_antipodes_of_no_positive (hv i) (hv (i+2)) (hsplit.mono fun _ ht => ht i)
+end EuclideanFiveColor
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PlaneColoring/CircleMaps.lean b/lean/OAI/Geometry/PlaneColoring/CircleMaps.lean
new file mode 100644
index 0000000000000000000000000000000000000000..9814c4526bacf3033c69a3b07e0a46025db0788a
--- /dev/null
+++ b/lean/OAI/Geometry/PlaneColoring/CircleMaps.lean
@@ -0,0 +1,517 @@
+import OAI.Geometry.PlaneColoring.Coloring
+
+namespace OAI
+
+noncomputable section
+open Filter Topology Set
+open scoped NNReal
+namespace EuclideanFiveColor
+abbrev RealPlane := ℝ × ℝ
+def circleSum (r : ℝ) (p : RealPlane) : RealPlane :=
+  (r*Real.cos p.1+Real.cos p.2,r*Real.sin p.1+Real.sin p.2)
+def circleSumDeriv (r : ℝ) (p : RealPlane) : RealPlane →L[ℝ] RealPlane :=
+  (r • ((-Real.sin p.1) • ContinuousLinearMap.fst ℝ ℝ ℝ) +
+    (-Real.sin p.2) • ContinuousLinearMap.snd ℝ ℝ ℝ).prod
+  (r • (Real.cos p.1 • ContinuousLinearMap.fst ℝ ℝ ℝ) +
+    Real.cos p.2 • ContinuousLinearMap.snd ℝ ℝ ℝ)
+lemma circleSum_strict (r : ℝ) (p : RealPlane) : HasStrictFDerivAt (circleSum r) (circleSumDeriv r p) p := by
+  have hc1 := (Real.hasStrictDerivAt_cos p.1).comp_hasStrictFDerivAt p
+    (ContinuousLinearMap.fst ℝ ℝ ℝ).hasStrictFDerivAt
+  have hc2 := (Real.hasStrictDerivAt_cos p.2).comp_hasStrictFDerivAt p
+    (ContinuousLinearMap.snd ℝ ℝ ℝ).hasStrictFDerivAt
+  have hs1 := (Real.hasStrictDerivAt_sin p.1).comp_hasStrictFDerivAt p
+    (ContinuousLinearMap.fst ℝ ℝ ℝ).hasStrictFDerivAt
+  have hs2 := (Real.hasStrictDerivAt_sin p.2).comp_hasStrictFDerivAt p
+    (ContinuousLinearMap.snd ℝ ℝ ℝ).hasStrictFDerivAt
+  exact ((hc1.const_smul r).add hc2).prodMk ((hs1.const_smul r).add hs2)
+lemma circleSum_continuous (r : ℝ) : Continuous (circleSum r) :=
+  continuous_iff_continuousAt.mpr fun p => (circleSum_strict r p).continuousAt
+lemma circleSumDeriv_injective {r : ℝ} {p : RealPlane} (h : r*Real.sin (p.2-p.1) ≠ 0) :
+    Function.Injective (circleSumDeriv r p) := by
+  apply (circleSumDeriv r p).toLinearMap.ker_eq_bot.mp
+  apply LinearMap.ker_eq_bot'.mpr
+  intro q hq
+  have h1 := congrArg Prod.fst hq
+  have h2 := congrArg Prod.snd hq
+  change r * (-Real.sin p.1 * q.1) + -Real.sin p.2 * q.2 = 0 at h1
+  change r * (Real.cos p.1 * q.1) + Real.cos p.2 * q.2 = 0 at h2
+  have ha : r*Real.sin (p.2-p.1)*q.1=0 := by
+    rw [Real.sin_sub]
+    linear_combination Real.cos p.2*h1+Real.sin p.2*h2
+  have hb : r*Real.sin (p.2-p.1)*q.2=0 := by
+    rw [Real.sin_sub]
+    linear_combination -(r*Real.cos p.1)*h1-(r*Real.sin p.1)*h2
+  exact Prod.ext ((mul_eq_zero.mp ha).resolve_left h) ((mul_eq_zero.mp hb).resolve_left h)
+def circleSumDerivEquiv {r : ℝ} {p : RealPlane} (h : r*Real.sin (p.2-p.1) ≠ 0) :
+    RealPlane ≃L[ℝ] RealPlane :=
+  ContinuousLinearEquiv.ofBijective (circleSumDeriv r p)
+    (LinearMap.ker_eq_bot.mpr (circleSumDeriv_injective h))
+    (LinearMap.range_eq_top.mpr ((LinearMap.injective_iff_surjective_of_finrank_eq_finrank rfl).mp
+      (circleSumDeriv_injective h)))
+lemma circleSum_local_antilipschitz {r : ℝ} {p : RealPlane} (h : r*Real.sin (p.2-p.1) ≠ 0) :
+    ∃ s : Set RealPlane, IsOpen s ∧ p ∈ s ∧ ∃ K : ℝ≥0, AntilipschitzWith K (s.domRestrict (circleSum r)) := by
+  let L := circleSumDerivEquiv h
+  have hd : HasStrictFDerivAt (circleSum r) (L : RealPlane →L[ℝ] RealPlane) p := circleSum_strict r p
+  obtain ⟨s,hp,hs,ha⟩ := hd.approximates_deriv_on_open_nhds
+  refine ⟨s,hs,hp,_,ha.antilipschitz ?_⟩
+  right
+  have hpos := L.subsingleton_or_nnnorm_symm_pos.resolve_left (not_subsingleton RealPlane)
+  exact half_lt_self (inv_pos.mpr hpos)
+lemma realDirection_lipschitz : LipschitzWith 1 (fun t : ℝ => (Real.cos t,Real.sin t)) := by
+  simpa using Real.lipschitzWith_cos.prodMk Real.lipschitzWith_sin
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+variable {t u : ℝ} (h : Real.sin (u-t) ≠ 0)
+def circleInverseHomeo : OpenPartialHomeomorph RealPlane RealPlane :=
+  (show HasStrictFDerivAt (circleSum 1)
+    (circleSumDerivEquiv (r := 1) (p := (t,u)) (by simpa only [one_mul] using h) : RealPlane →L[ℝ] RealPlane)
+    (t,u) from circleSum_strict 1 (t,u)).toOpenPartialHomeomorph (circleSum 1)
+def circleInverse : RealPlane → RealPlane := by
+  classical
+  exact (circleInverseHomeo h).target.piecewise (circleInverseHomeo h).symm (fun _ => 0)
+lemma circleInverse_measurable : Measurable (circleInverse h) := by
+  classical
+  exact (circleInverseHomeo h).continuousOn_invFun.measurable_piecewise continuous_const.continuousOn
+    (circleInverseHomeo h).open_target.measurableSet
+lemma circleInverse_eq {q : RealPlane} (hq : q ∈ (circleInverseHomeo h).target) :
+    circleInverse h q=(circleInverseHomeo h).symm q := by simp [circleInverse,hq]
+lemma circleInverse_at : circleInverse h (circleSum 1 (t,u))=(t,u) := by
+  have hd : HasStrictFDerivAt (circleSum 1)
+      (circleSumDerivEquiv (r := 1) (p := (t,u)) (by simpa only [one_mul] using h) : RealPlane →L[ℝ] RealPlane)
+      (t,u) := circleSum_strict 1 (t,u)
+  rw [circleInverse_eq h hd.image_mem_toOpenPartialHomeomorph_target]
+  exact hd.localInverse_apply_image
+lemma circleInverse_continuousAt {q : RealPlane} (hq : q ∈ (circleInverseHomeo h).target) :
+    ContinuousAt (circleInverse h) q := by
+  apply ((circleInverseHomeo h).continuousAt_symm hq).congr_of_eventuallyEq
+  filter_upwards [(circleInverseHomeo h).open_target.mem_nhds hq] with y hy
+  exact circleInverse_eq h hy
+lemma circleInverse_strict : HasStrictFDerivAt (circleInverse h)
+    ((circleSumDerivEquiv (r := 1) (p := (t,u)) (by simpa only [one_mul] using h)).symm : RealPlane →L[ℝ] RealPlane)
+      (circleSum 1 (t,u)) := by
+  have hd : HasStrictFDerivAt (circleSum 1)
+      (circleSumDerivEquiv (r := 1) (p := (t,u)) (by simpa only [one_mul] using h) : RealPlane →L[ℝ] RealPlane)
+      (t,u) := circleSum_strict 1 (t,u)
+  apply hd.to_localInverse.congr_of_eventuallyEq
+  filter_upwards [(circleInverseHomeo h).open_target.mem_nhds hd.image_mem_toOpenPartialHomeomorph_target] with q hq
+  exact (circleInverse_eq h hq).symm
+lemma circleInverse_right {q : RealPlane} (hq : q ∈ (circleInverseHomeo h).target) :
+    circleSum 1 (circleInverse h q)=q := by
+  rw [circleInverse_eq h hq]
+  exact (circleInverseHomeo h).right_inv hq
+lemma circleInverse_regular :
+    ∀ᶠ q in 𝓝 (circleSum 1 (t,u)), Real.sin ((circleInverse h q).2-(circleInverse h q).1) ≠ 0 := by
+  have hc := (Real.continuous_sin.continuousAt.comp
+    ((circleInverse_strict h).continuousAt.snd.sub (circleInverse_strict h).continuousAt.fst))
+  have hh : Real.sin ((circleInverse h (circleSum 1 (t,u))).2-(circleInverse h (circleSum 1 (t,u))).1) ≠ 0 := by
+    rwa [circleInverse_at h]
+  exact hc.eventually_ne hh
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology
+open scoped NNReal
+namespace EuclideanFiveColor
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
+omit [CompleteSpace E] in
+lemma triangular_injective (L : E × ℝ →L[ℝ] ℝ) (hc : L (0,1) ≠ 0) :
+    Function.Injective ((ContinuousLinearMap.fst ℝ E ℝ).prod L) := by
+  intro a b hab
+  change (a.1,L a)=(b.1,L b) at hab
+  have hab₁ := congrArg (fun q : E × ℝ => q.1) hab
+  have hab₂ := congrArg (fun q : E × ℝ => q.2) hab
+  dsimp only at hab₁ hab₂
+  have he : a-b=(0,a.2-b.2) := Prod.ext (by simp [hab₁]) rfl
+  have hz : L (0,a.2-b.2)=0 := by rw [← he,map_sub,hab₂,sub_self]
+  have hm : L (0,a.2-b.2)=(a.2-b.2)*L (0,1) := by
+    have h : (0,a.2-b.2)=(a.2-b.2) • ((0,1) : E × ℝ) := by ext <;> simp
+    rw [h,map_smul,smul_eq_mul]
+  rw [hm] at hz
+  exact Prod.ext hab₁ (sub_eq_zero.mp ((mul_eq_zero.mp hz).resolve_right hc))
+omit [CompleteSpace E] in
+lemma triangular_surjective (L : E × ℝ →L[ℝ] ℝ) (hc : L (0,1) ≠ 0) :
+    Function.Surjective ((ContinuousLinearMap.fst ℝ E ℝ).prod L) := by
+  intro b
+  refine ⟨(b.1,(b.2-L (b.1,0))/L (0,1)),Prod.ext rfl ?_⟩
+  change L (b.1,(b.2-L (b.1,0))/L (0,1))=b.2
+  have h (r : ℝ) : L (b.1,r)=L (b.1,0)+r*L (0,1) := by
+    have he : (b.1,r)=(b.1,0)+r • ((0,1) : E × ℝ) := by ext <;> simp
+    rw [he,map_add,map_smul,smul_eq_mul]
+  rw [h,div_mul_cancel₀ _ hc]
+  ring
+/-- Joint strict differentiability and a nonzero fiber derivative give one inverse
+Lipschitz bound simultaneously for every nearby parameter. -/
+lemma strict_fiber_uniform_antilipschitz {S : E × ℝ → ℝ} {L : E × ℝ →L[ℝ] ℝ}
+    {p : E} {t : ℝ} (hS : HasStrictFDerivAt S L (p,t)) (hc : L (0,1) ≠ 0) :
+    ∃ P : Set E, ∃ I : Set ℝ, IsOpen P ∧ p ∈ P ∧ IsOpen I ∧ t ∈ I ∧
+      ∃ K : ℝ≥0, ∀ q ∈ P, AntilipschitzWith K (I.domRestrict (fun u => S (q,u))) := by
+  let T := (ContinuousLinearMap.fst ℝ E ℝ).prod L
+  let H : E × ℝ → E × ℝ := fun q => (q.1,S q)
+  let Q : (E × ℝ) ≃L[ℝ] (E × ℝ) := ContinuousLinearEquiv.ofBijective T
+    (LinearMap.ker_eq_bot.mpr (triangular_injective L hc))
+    (LinearMap.range_eq_top.mpr (triangular_surjective L hc))
+  have hd : HasStrictFDerivAt H (Q : E × ℝ →L[ℝ] E × ℝ) (p,t) :=
+    hasStrictFDerivAt_fst.prodMk hS
+  obtain ⟨V,hpt,hVo,hA⟩ := hd.approximates_deriv_on_open_nhds
+  have hK := hA.antilipschitz (Or.inr (half_lt_self (inv_pos.mpr
+    (Q.subsingleton_or_nnnorm_symm_pos.resolve_left (not_subsingleton (E × ℝ))))))
+  obtain ⟨P,hP,I,hI,hsub⟩ := mem_nhds_prod_iff.mp (hVo.mem_nhds hpt)
+  obtain ⟨P',hP'P,hP'o,hp⟩ := mem_nhds_iff.mp hP
+  obtain ⟨I',hI'I,hI'o,ht⟩ := mem_nhds_iff.mp hI
+  refine ⟨P',I',hP'o,hp,hI'o,ht,(‖(Q.symm : (E × ℝ) →L[ℝ] (E × ℝ))‖₊⁻¹ - ‖(Q.symm : (E × ℝ) →L[ℝ] (E × ℝ))‖₊⁻¹ / 2)⁻¹,?_⟩
+  intro q hq
+  apply AntilipschitzWith.of_le_mul_dist
+  intro a b
+  have ha : (q,a.1) ∈ V := hsub ⟨hP'P hq,hI'I a.2⟩
+  have hb : (q,b.1) ∈ V := hsub ⟨hP'P hq,hI'I b.2⟩
+  have h := hK.le_mul_dist ⟨(q,a.1),ha⟩ ⟨(q,b.1),hb⟩
+  simpa only [Subtype.dist_eq,domRestrict,H,Prod.dist_eq,dist_self,max_eq_right (dist_nonneg)] using h
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Filter Topology Set MeasureTheory
+open scoped NNReal
+namespace EuclideanFiveColor
+abbrev CircleParameter := RealPlane × ℝ
+abbrev CircleJet := CircleParameter × ℝ
+def jetCenter : CircleJet →L[ℝ] RealPlane :=
+  (ContinuousLinearMap.fst ℝ RealPlane ℝ).comp (ContinuousLinearMap.fst ℝ CircleParameter ℝ)
+def jetRadius : CircleJet →L[ℝ] ℝ :=
+  (ContinuousLinearMap.snd ℝ RealPlane ℝ).comp (ContinuousLinearMap.fst ℝ CircleParameter ℝ)
+def jetAngle : CircleJet →L[ℝ] ℝ := ContinuousLinearMap.snd ℝ CircleParameter ℝ
+def circleTarget (q : CircleJet) : RealPlane := q.1.1+q.1.2 • (Real.cos q.2,Real.sin q.2)
+def circleTargetDeriv (q : CircleJet) : CircleJet →L[ℝ] RealPlane :=
+  jetCenter + (q.1.2 • (((-Real.sin q.2) • jetAngle).prod (Real.cos q.2 • jetAngle)) +
+    jetRadius.smulRight (Real.cos q.2,Real.sin q.2))
+lemma circleTarget_strict (q : CircleJet) : HasStrictFDerivAt circleTarget (circleTargetDeriv q) q := by
+  have hc := (Real.hasStrictDerivAt_cos q.2).comp_hasStrictFDerivAt q jetAngle.hasStrictFDerivAt
+  have hs := (Real.hasStrictDerivAt_sin q.2).comp_hasStrictFDerivAt q jetAngle.hasStrictFDerivAt
+  exact jetCenter.hasStrictFDerivAt.add (jetRadius.hasStrictFDerivAt.smul (hc.prodMk hs))
+lemma circleTarget_continuous : Continuous circleTarget :=
+  continuous_iff_continuousAt.mpr fun q => (circleTarget_strict q).continuousAt
+variable {t u : ℝ} (h : Real.sin (u-t) ≠ 0)
+def circleEndpoint (q : CircleJet) : ℝ := (circleInverse h (circleTarget q)).1
+def circleEndpointDeriv (q : CircleJet) : CircleJet →L[ℝ] ℝ :=
+  (ContinuousLinearMap.fst ℝ ℝ ℝ).comp
+    (((circleSumDerivEquiv (r := 1) (p := (t,u)) (by simpa only [one_mul] using h)).symm : RealPlane →L[ℝ] RealPlane).comp
+      (circleTargetDeriv q))
+lemma circleEndpoint_measurable : Measurable (circleEndpoint h) :=
+  measurable_fst.comp ((circleInverse_measurable h).comp circleTarget_continuous.measurable)
+lemma circleEndpoint_strict {q : CircleJet} (he : circleTarget q=circleSum 1 (t,u)) :
+    HasStrictFDerivAt (circleEndpoint h) (circleEndpointDeriv h q) q := by
+  have hi := circleInverse_strict h
+  rw [← he] at hi
+  exact hasStrictFDerivAt_fst.comp q (hi.comp q (circleTarget_strict q))
+lemma inverse_tangent_first_ne {a : ℝ} (ha : Real.sin (a-u) ≠ 0) :
+    ((circleSumDerivEquiv (r := 1) (p := (t,u)) (by simpa only [one_mul] using h)).symm
+      (-Real.sin a,Real.cos a)).1 ≠ 0 := by
+  let L := circleSumDerivEquiv (r := 1) (p := (t,u)) (by simpa only [one_mul] using h)
+  let q := L.symm (-Real.sin a,Real.cos a)
+  intro hq
+  have he : circleSumDeriv 1 (t,u) q=(-Real.sin a,Real.cos a) := L.apply_symm_apply _
+  have h₁ := congrArg Prod.fst he
+  have h₂ := congrArg Prod.snd he
+  change q.1=0 at hq
+  change 1*(-Real.sin t*q.1) + -Real.sin u*q.2 = -Real.sin a at h₁
+  change 1*(Real.cos t*q.1) + Real.cos u*q.2 = Real.cos a at h₂
+  simp only [hq,mul_zero,zero_add] at h₁ h₂
+  apply ha
+  rw [Real.sin_sub]
+  linear_combination (Real.cos u)*h₁+(Real.sin u)*h₂
+lemma circleEndpoint_fiber_ne {q : CircleJet} (hr : q.1.2=1) (ha : Real.sin (q.2-u) ≠ 0) :
+    circleEndpointDeriv h q (0,1) ≠ 0 := by
+  have he : circleTargetDeriv q (0,1)=(-Real.sin q.2,Real.cos q.2) := by
+    simp [circleTargetDeriv,jetCenter,jetRadius,jetAngle,hr]
+  change (((circleSumDerivEquiv (r := 1) (p := (t,u)) (by simpa only [one_mul] using h)).symm
+    (circleTargetDeriv q (0,1))).1) ≠ 0
+  rw [he]
+  exact inverse_tangent_first_ne h ha
+lemma circleEndpoint_uniform {p : CircleParameter} {a : ℝ}
+    (he : circleTarget (p,a)=circleSum 1 (t,u)) (hr : p.2=1) (ha : Real.sin (a-u) ≠ 0) :
+    ∃ P : Set CircleParameter, ∃ I : Set ℝ, IsOpen P ∧ p ∈ P ∧ IsOpen I ∧ a ∈ I ∧
+      ∃ K : ℝ≥0, ∀ q ∈ P, AntilipschitzWith K (I.domRestrict (fun v => circleEndpoint h (q,v))) :=
+  strict_fiber_uniform_antilipschitz (circleEndpoint_strict h he) (circleEndpoint_fiber_ne h hr ha)
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Filter Topology Set MeasureTheory
+open scoped NNReal
+namespace EuclideanFiveColor
+variable {t u : ℝ} (h : Real.sin (u-t) ≠ 0)
+lemma circleEndpoint_good_neighborhood {p : CircleParameter} {a : ℝ}
+    (he : circleTarget (p,a)=circleSum 1 (t,u)) (hr : p.2=1) (ha : Real.sin (a-u) ≠ 0) :
+    ∃ P : Set CircleParameter, ∃ I : Set ℝ, IsOpen P ∧ p ∈ P ∧ IsOpen I ∧ a ∈ I ∧
+      I ⊆ Ioc (a-Real.pi) (a-Real.pi+2*Real.pi) ∧
+      ∃ K : ℝ≥0, (∀ q ∈ P, AntilipschitzWith K (I.domRestrict (fun v => circleEndpoint h (q,v)))) ∧
+        ∀ q ∈ P, ∀ v ∈ I,
+          circleTarget (q,v) ∈ (circleInverseHomeo h).target ∧
+          Real.sin ((circleInverse h (circleTarget (q,v))).2-(circleInverse h (circleTarget (q,v))).1) ≠ 0 ∧
+          circleEndpoint h (q,v) ∈ Metric.closedBall t 1 := by
+  obtain ⟨P₀,I₀,hP₀,hp₀,hI₀,ha₀,K,hK⟩ := circleEndpoint_uniform h he hr ha
+  have hc : ContinuousAt circleTarget (p,a) := circleTarget_continuous.continuousAt
+  have hi := (circleEndpoint_strict h he).continuousAt
+  have hs : ∀ᶠ w in 𝓝 (p,a), circleEndpoint h w ∈ Metric.ball t 1 := by
+    have hb : Metric.ball t 1 ∈ 𝓝 (circleEndpoint h (p,a)) := by
+      have ht : circleEndpoint h (p,a)=t := by simp [circleEndpoint,he,circleInverse_at]
+      rw [ht]
+      exact Metric.ball_mem_nhds _ (by norm_num)
+    exact hi hb
+  have ht : ∀ᶠ w in 𝓝 (p,a), circleTarget w ∈ (circleInverseHomeo h).target := by
+    apply hc
+    rw [he]
+    exact (circleInverseHomeo h).open_target.mem_nhds
+      ((show HasStrictFDerivAt (circleSum 1)
+        (circleSumDerivEquiv (r := 1) (p := (t,u)) (by simpa only [one_mul] using h) : RealPlane →L[ℝ] RealPlane)
+        (t,u) from circleSum_strict 1 (t,u)).image_mem_toOpenPartialHomeomorph_target)
+  have hn : ∀ᶠ w in 𝓝 (p,a), Real.sin ((circleInverse h (circleTarget w)).2-(circleInverse h (circleTarget w)).1) ≠ 0 := by
+    exact hc.eventually (by simpa only [he] using circleInverse_regular h)
+  obtain ⟨P₁,hP₁,I₁,hI₁,hsub⟩ := mem_nhds_prod_iff.mp (ht.and (hn.and hs))
+  obtain ⟨P₂,hP₂,hP₂o,hp₂⟩ := mem_nhds_iff.mp hP₁
+  obtain ⟨I₂,hI₂,hI₂o,ha₂⟩ := mem_nhds_iff.mp hI₁
+  let P := P₀ ∩ P₂
+  let I := I₀ ∩ I₂ ∩ Ioo (a-Real.pi) (a+Real.pi)
+  refine ⟨P,I,hP₀.inter hP₂o,⟨hp₀,hp₂⟩,(hI₀.inter hI₂o).inter isOpen_Ioo,
+    ⟨⟨ha₀,ha₂⟩,by constructor <;> linarith [Real.pi_pos]⟩,?_,K,?_,?_⟩
+  · intro v hv
+    exact ⟨hv.2.1,by linarith [hv.2.2]⟩
+  · intro q hq
+    apply AntilipschitzWith.of_le_mul_dist
+    intro v w
+    exact (hK q hq.1).le_mul_dist ⟨v.1,v.2.1.1⟩ ⟨w.1,w.2.1.1⟩
+  · intro q hq v hv
+    have hh := hsub (show (q,v) ∈ P₁ ×ˢ I₁ from ⟨hP₂ hq.2,hI₂ hv.1.2⟩)
+    exact ⟨hh.1,hh.2.1,Metric.ball_subset_closedBall hh.2.2⟩
+end EuclideanFiveColor
+end
+
+noncomputable section
+open MeasureTheory Filter Topology Set
+namespace EuclideanFiveColor
+def angleChart (p : RealPlane) : Angles × Angles := ((p.1 : Angles),(p.2 : Angles))
+def angleChartMeasure : Measure RealPlane :=
+  volume.restrict (Ioc (0 : ℝ) (2*Real.pi) ×ˢ Ioc (0 : ℝ) (2*Real.pi))
+lemma angleChartMeas_le : angleChartMeasure ≤ volume := Measure.restrict_le_self
+instance : IsFiniteMeasure angleChartMeasure := by
+  change IsFiniteMeasure (volume.restrict _)
+  apply isFiniteMeasure_restrict.mpr
+  apply ne_of_lt
+  apply (measure_mono (Set.prod_mono Ioc_subset_Icc_self Ioc_subset_Icc_self)).trans_lt
+  exact (isCompact_Icc.prod isCompact_Icc).measure_lt_top
+lemma angleChart_preserving : MeasurePreserving angleChart angleChartMeasure
+    ((volume : Measure Angles).prod volume) := by
+  have h := (AddCircle.measurePreserving_mk (2*Real.pi) 0).prod
+    (AddCircle.measurePreserving_mk (2*Real.pi) 0)
+  convert h using 1
+  · rfl
+  · simp only [angleChartMeasure,zero_add,Measure.prod_restrict,← Measure.volume_eq_prod]
+lemma angleChart_measure : (angleChartMeasure.map angleChart) =
+    (ENNReal.ofReal (2*Real.pi))^2 • angularMeasure.prod angularMeasure := by
+  rw [angleChart_preserving.map_eq,AddCircle.volume_eq_smul_haarAddCircle,
+    Measure.prod_smul_left,Measure.prod_smul_right,smul_smul,pow_two]
+  rfl
+lemma integral_angular_eq_chart {f : Angles × Angles → ℝ} (hf : Measurable f) :
+    (∫ p, f p ∂angularMeasure.prod angularMeasure) =
+      ((2*Real.pi)^2)⁻¹ * ∫ q, f (angleChart q) ∂angleChartMeasure := by
+  have h := integral_map (μ := angleChartMeasure) angleChart_preserving.measurable.aemeasurable hf.aestronglyMeasurable
+  rw [angleChart_measure,integral_smul_measure,ENNReal.toReal_pow,
+    ENNReal.toReal_ofReal (by positivity : 0 ≤ 2*Real.pi),smul_eq_mul] at h
+  rw [← h]
+  field_simp
+lemma direction_realPlane (t : ℝ) : Complex.equivRealProdCLM (direction (t : Angles)) =
+    (Real.cos t,Real.sin t) := by
+  rw [direction_real,Circle.coe_exp,Complex.exp_mul_I]
+  ext <;> simp [Complex.cos_ofReal_re,Complex.sin_ofReal_re]
+lemma circleSum_chart (x : ℂ) (r : ℝ) (q : RealPlane) :
+    Complex.equivRealProdCLM (x+(r : ℂ)*direction (angleChart q).1+direction (angleChart q).2) =
+      Complex.equivRealProdCLM x+circleSum r q := by
+  change Complex.equivRealProdCLM (x+(r : ℂ)*direction (q.1 : Angles)+direction (q.2 : Angles)) = _
+  rw [map_add,map_add]
+  have h : (r : ℂ)*direction (q.1 : Angles) = r • direction (q.1 : Angles) := by simp
+  rw [h,map_smul,direction_realPlane,direction_realPlane]
+  ext <;> simp [circleSum,add_assoc]
+end EuclideanFiveColor
+end
+
+noncomputable section
+open MeasureTheory Filter Topology Set
+open scoped NNReal ENNReal
+namespace EuclideanFiveColor
+lemma volume_preimage_bound {F : ℝ × ℝ → ℝ × ℝ} {s : Set (ℝ × ℝ)} {K : ℝ≥0}
+    (hF : AntilipschitzWith K (s.domRestrict F)) (A : Set (ℝ × ℝ)) :
+    volume (F ⁻¹' A ∩ s) ≤ (K : ENNReal)^2 * volume A := by
+  have he : Subtype.val '' ((s.domRestrict F) ⁻¹' A) = F ⁻¹' A ∩ s := by
+    ext p
+    constructor
+    · rintro ⟨q,hq,rfl⟩; exact ⟨hq,q.2⟩
+    · rintro ⟨hp,hs⟩; exact ⟨⟨p,hs⟩,hp,rfl⟩
+  have hi := (isometry_subtype_coe : Isometry (Subtype.val : s → ℝ × ℝ)).hausdorffMeasure_image
+    (Or.inl (by norm_num : (0 : ℝ)≤2)) ((s.domRestrict F) ⁻¹' A)
+  have ha := hF.hausdorffMeasure_preimage_le (by norm_num : (0 : ℝ)≤2) A
+  rw [he,hausdorffMeasure_prod_real] at hi
+  rw [← hi,hausdorffMeasure_prod_real,ENNReal.rpow_two] at ha
+  exact ha
+lemma map_restrict_volume_le {F : ℝ × ℝ → ℝ × ℝ} (hm : Measurable F)
+    {s B : Set (ℝ × ℝ)} (_hB : MeasurableSet B) {K : ℝ≥0}
+    (hF : AntilipschitzWith K (s.domRestrict F)) (himage : MapsTo F s B) :
+    (volume.restrict s).map F ≤ ((K^2 : ℝ≥0) : ENNReal) • volume.restrict B := by
+  apply Measure.le_iff.mpr
+  intro A hA
+  rw [Measure.map_apply hm hA,Measure.restrict_apply (hm hA),Measure.smul_apply,
+    Measure.restrict_apply hA,smul_eq_mul,ENNReal.coe_pow]
+  have he : F ⁻¹' A ∩ s = F ⁻¹' (A ∩ B) ∩ s := by
+    ext x
+    exact ⟨fun hx => ⟨⟨hx.1,himage hx.2⟩,hx.2⟩,fun hx => ⟨hx.1.1,hx.2⟩⟩
+  rw [he]
+  exact volume_preimage_bound hF (A ∩ B)
+lemma antilipschitz_small_perturbation {E X : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+    [PseudoMetricSpace X] {F G : X → E} {K L : ℝ≥0} (hf : AntilipschitzWith K F)
+    (hg : LipschitzWith L G) {a : ℝ} (ha : (K : ℝ)*‖a‖*(L : ℝ) ≤ 1/2) :
+    AntilipschitzWith (2*K) (fun x => F x+a • G x) := by
+  apply AntilipschitzWith.of_le_mul_dist
+  intro x y
+  have h1 := hf.le_mul_dist x y
+  have h2 := hg.dist_le_mul x y
+  have h3 : dist (F x) (F y) ≤ dist (F x+a • G x) (F y+a • G y) + ‖a‖*dist (G x) (G y) := by
+    simpa only [add_sub_cancel_right,dist_smul₀] using
+      dist_sub_sub_le (F x+a • G x) (a • G x) (F y+a • G y) (a • G y)
+  have h4 := mul_le_mul_of_nonneg_left h2 (norm_nonneg a)
+  have h5 := mul_le_mul_of_nonneg_left (h3.trans (add_le_add_right h4 _)) K.2
+  have h6 := mul_le_mul_of_nonneg_right ha (dist_nonneg (x := x) (y := y))
+  simp only [NNReal.coe_mul,NNReal.coe_ofNat]
+  nlinarith
+end EuclideanFiveColor
+end
+
+noncomputable section
+open MeasureTheory Filter Topology Set
+open scoped NNReal
+namespace EuclideanFiveColor
+lemma realDirection_norm_le (t : ℝ) : ‖(Real.cos t,Real.sin t)‖ ≤ (1 : ℝ) := by
+  rw [Prod.norm_def,max_le_iff]
+  exact ⟨by simpa only [Real.norm_eq_abs] using Real.abs_cos_le_one t,
+    by simpa only [Real.norm_eq_abs] using Real.abs_sin_le_one t⟩
+lemma circleSum_decompose (r : ℝ) (q : RealPlane) :
+    circleSum r q = r • (Real.cos q.1,Real.sin q.1) + (Real.cos q.2,Real.sin q.2) := rfl
+lemma circleSum_norm_le (r : ℝ) (q : RealPlane) : ‖circleSum r q‖ ≤ |r|+1 := by
+  rw [circleSum_decompose]
+  calc
+    _ ≤ ‖r • (Real.cos q.1,Real.sin q.1)‖+‖(Real.cos q.2,Real.sin q.2)‖ := norm_add_le _ _
+    _ ≤ |r| * 1+1 := by rw [norm_smul]; exact add_le_add (mul_le_mul_of_nonneg_left (realDirection_norm_le _) (norm_nonneg r)) (realDirection_norm_le _)
+    _ = _ := by ring
+lemma circleSum_perturb_identity (r : ℝ) (q : RealPlane) :
+    circleSum r q = circleSum 1 q + (r-1) • (Real.cos q.1,Real.sin q.1) := by
+  apply Prod.ext
+  · change r*Real.cos q.1+Real.cos q.2 = 1*Real.cos q.1+Real.cos q.2+(r-1)*Real.cos q.1
+    ring
+  · change r*Real.sin q.1+Real.sin q.2 = 1*Real.sin q.1+Real.sin q.2+(r-1)*Real.sin q.1
+    ring
+lemma circleSum_parameter_tendsto {r : ℕ → ℝ} (hr : Tendsto r atTop (𝓝 1)) (q : RealPlane) :
+    Tendsto (fun n => circleSum (r n) q) atTop (𝓝 (circleSum 1 q)) := by
+  exact (hr.mul_const (Real.cos q.1) |>.add_const (Real.cos q.2)).prodMk_nhds
+    (hr.mul_const (Real.sin q.1) |>.add_const (Real.sin q.2))
+lemma circleSum_eventual_image {r : ℕ → ℝ} {y : ℕ → RealPlane} {x : RealPlane}
+    (hr : Tendsto r atTop (𝓝 1)) (hy : Tendsto y atTop (𝓝 x)) :
+    ∀ᶠ n in atTop, ∀ q, y n+circleSum (r n) q ∈ Metric.closedBall x 4 := by
+  filter_upwards [hr.eventually (Metric.ball_mem_nhds (1 : ℝ) (by norm_num : (0 : ℝ)<1)),
+    hy.eventually (Metric.ball_mem_nhds x (by norm_num : (0 : ℝ)<1))] with n hn hyn
+  intro q
+  have hh := circleSum_norm_le (r n) q
+  have hb : |r n| ≤ 2 := by
+    have ht := norm_le_norm_sub_add (r n) (1 : ℝ)
+    rw [Real.norm_eq_abs,Real.norm_eq_abs,Real.norm_eq_abs,abs_one] at ht
+    rw [Real.dist_eq] at hn
+    linarith
+  rw [Metric.mem_closedBall]
+  calc
+    _ ≤ dist (y n+circleSum (r n) q) (y n)+dist (y n) x := dist_triangle _ _ _
+    _ = ‖circleSum (r n) q‖+dist (y n) x := by rw [dist_eq_norm,add_sub_cancel_left]
+    _ ≤ 4 := by linarith
+lemma circleSum_limit_image (x q : RealPlane) : x+circleSum 1 q ∈ Metric.closedBall x 4 := by
+  rw [Metric.mem_closedBall,dist_eq_norm,add_sub_cancel_left]
+  have hh := circleSum_norm_le 1 q
+  norm_num at hh
+  linarith
+lemma circleSum_eventual_antilipschitz {V : Set RealPlane} {K : ℝ≥0}
+    (hK : AntilipschitzWith K (V.domRestrict (circleSum 1)))
+    {r : ℕ → ℝ} (hr : Tendsto r atTop (𝓝 1)) (y : ℕ → RealPlane) :
+    ∀ᶠ n in atTop, AntilipschitzWith (2*K) (V.domRestrict (fun q => y n+circleSum (r n) q)) := by
+  have hD : LipschitzWith 1 (fun q : V => (Real.cos q.1.1,Real.sin q.1.1)) := by
+    have hf : LipschitzWith 1 (fun q : V => q.1.1) := LipschitzWith.of_dist_le_mul (fun q p => by
+      simp only [NNReal.coe_one,one_mul]
+      exact le_max_left _ _)
+    simpa only [Function.comp_def,one_mul] using realDirection_lipschitz.comp hf
+  have ht : Tendsto (fun n => (K : ℝ)*‖r n-1‖) atTop (𝓝 0) := by
+    have hh : Tendsto (fun n => r n-1) atTop (𝓝 0) := by simpa using hr.sub_const 1
+    simpa using hh.norm.const_mul (K : ℝ)
+  filter_upwards [ht.eventually (gt_mem_nhds (by norm_num : (0 : ℝ)<1/2))] with n hn
+  have ha := antilipschitz_small_perturbation hK hD (a := r n-1) (by simpa using le_of_lt hn)
+  apply AntilipschitzWith.of_le_mul_dist
+  intro q p
+  have hh := ha.le_mul_dist q p
+  simpa only [Set.domRestrict_apply,dist_add_left,← circleSum_perturb_identity] using hh
+lemma circleSum_limit_antilipschitz {V : Set RealPlane} {K : ℝ≥0}
+    (hK : AntilipschitzWith K (V.domRestrict (circleSum 1))) (x : RealPlane) :
+    AntilipschitzWith (2*K) (V.domRestrict (fun q => x+circleSum 1 q)) := by
+  apply AntilipschitzWith.of_le_mul_dist
+  intro q p
+  have hh := hK.le_mul_dist q p
+  simpa only [Set.domRestrict_apply,dist_add_left] using hh.trans
+    (mul_le_mul_of_nonneg_right (by simp only [NNReal.coe_mul,NNReal.coe_ofNat]; linarith [K.2] : (K : ℝ)≤ ((2*K : ℝ≥0) : ℝ)) dist_nonneg)
+end EuclideanFiveColor
+end
+
+noncomputable section
+open MeasureTheory Filter Topology Set
+namespace EuclideanFiveColor
+lemma measure_null_of_locally_null_on_conull {X : Type*} [TopologicalSpace X]
+    [HereditarilyLindelofSpace X] [MeasurableSpace X] {μ : Measure X} {R A : Set X}
+    (hR : μ Rᶜ = 0) (h : ∀ x ∈ R, ∃ V : Set X, IsOpen V ∧ x ∈ V ∧ μ (A ∩ V) = 0) :
+    μ A = 0 := by
+  classical
+  choose V hVo hxV hV0 using (fun x : R => h x.1 x.2)
+  obtain ⟨t,ht,hcov⟩ := (HereditarilyLindelofSpace.isLindelof R).elim_countable_subcover V hVo
+    (fun x hx => mem_iUnion.mpr ⟨⟨x,hx⟩,hxV ⟨x,hx⟩⟩)
+  have hu : μ (⋃ i ∈ t, A ∩ V i) = 0 := (measure_biUnion_null_iff ht).mpr (fun i _ => hV0 i)
+  apply measure_mono_null (t := Rᶜ ∪ ⋃ i ∈ t, A ∩ V i) _ (measure_union_null hR hu)
+  intro x hx
+  by_cases hr : x ∈ R
+  · obtain ⟨i,hi,hxi⟩ := mem_iUnion₂.mp (hcov hr)
+    exact Or.inr (mem_iUnion₂.mpr ⟨i,hi,hx,hxi⟩)
+  · exact Or.inl hr
+lemma circleSum_regular_ae : ∀ᵐ p : RealPlane ∂volume, Real.sin (p.2-p.1) ≠ 0 := by
+  rw [Measure.volume_eq_prod]
+  apply (Measure.ae_prod_iff_ae_ae ?_).mpr
+  · exact Eventually.of_forall fun t => by
+      have hs : {s : ℝ | Real.sin (s-t)=0} ⊆ Set.range (fun n : ℤ => t+n*Real.pi) := by
+        intro s h
+        obtain ⟨n,hn⟩ := Real.sin_eq_zero_iff.mp h
+        exact ⟨n,by linarith⟩
+      change volume {s : ℝ | ¬ Real.sin (s-t) ≠ 0} = 0
+      simpa only [not_not] using measure_mono_null hs ((countable_range _).measure_zero volume)
+  · exact (measurableSet_eq_fun (Real.measurable_sin.comp (measurable_snd.sub measurable_fst)) measurable_const).compl
+lemma circleSum_preimage_null {r : ℝ} (hr : r ≠ 0) {A : Set RealPlane} (hA : volume A=0) :
+    volume (circleSum r ⁻¹' A)=0 := by
+  apply measure_null_of_locally_null_on_conull (μ := (volume : Measure RealPlane))
+    (R := {p : RealPlane | Real.sin (p.2-p.1) ≠ 0}) (A := circleSum r ⁻¹' A) circleSum_regular_ae
+  intro p hp
+  obtain ⟨s,hso,hps,K,hK⟩ := circleSum_local_antilipschitz (mul_ne_zero hr hp)
+  refine ⟨s,hso,hps,?_⟩
+  have hh := volume_preimage_bound hK A
+  rw [hA,mul_zero] at hh
+  exact nonpos_iff_eq_zero.mp hh
+lemma circleSum_quasiMeasurePreserving {r : ℝ} (hr : r ≠ 0) :
+    Measure.QuasiMeasurePreserving (circleSum r) volume volume := by
+  refine ⟨(circleSum_continuous r).measurable,Measure.AbsolutelyContinuous.mk (fun A hA hz => ?_)⟩
+  rw [Measure.map_apply (circleSum_continuous r).measurable hA]
+  exact circleSum_preimage_null hr hz
+end EuclideanFiveColor
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PlaneColoring/CircleTransport.lean b/lean/OAI/Geometry/PlaneColoring/CircleTransport.lean
new file mode 100644
index 0000000000000000000000000000000000000000..8e8e76e1a0854604522b98d61626b4c38e5ae022
--- /dev/null
+++ b/lean/OAI/Geometry/PlaneColoring/CircleTransport.lean
@@ -0,0 +1,336 @@
+import OAI.Geometry.PlaneColoring.BinaryDynamics
+import OAI.Geometry.PlaneColoring.CircleMaps
+
+namespace OAI
+
+noncomputable section
+open MeasureTheory Set Filter Topology
+namespace EuclideanFiveColor
+/-- A uniformly bounded nonnegative sequence converging locally in L¹ on a conull open
+cover converges globally in L¹ for a finite inner-regular measure. -/
+lemma tendsto_integral_of_local_cover {X I : Type*} [TopologicalSpace X] [T2Space X]
+    [MeasurableSpace X] [OpensMeasurableSpace X] {μ : Measure X} [IsFiniteMeasure μ]
+    [Measure.InnerRegularCompactLTTop μ] {F : ℕ → X → ℝ}
+    (hF : ∀ n, Integrable (F n) μ) (hb : ∀ n x, 0 ≤ F n x ∧ F n x ≤ 1)
+    (V : I → Set X) (hVo : ∀ i, IsOpen (V i)) (hfull : ∀ᵐ x ∂μ, x ∈ ⋃ i, V i)
+    (hlocal : ∀ i, Tendsto (fun n => ∫ x in V i, F n x ∂μ) atTop (𝓝 0)) :
+    Tendsto (fun n => ∫ x, F n x ∂μ) atTop (𝓝 0) := by
+  classical
+  apply Metric.tendsto_atTop.mpr
+  intro ε hε
+  obtain ⟨K,hKU,hK,hμK⟩ := (isOpen_iUnion hVo).measurableSet.exists_isCompact_sdiff_lt
+    (measure_ne_top μ _) (by exact ne_of_gt (ENNReal.ofReal_pos.mpr (half_pos hε)))
+  have hδ : μ Kᶜ < ENNReal.ofReal (ε/2) := by
+    have he : (Kᶜ : Set X) =ᵐ[μ] ((⋃ i, V i) \ K : Set X) := by
+      filter_upwards [hfull] with x hx
+      exact propext ⟨fun h => ⟨hx,h⟩,fun h => h.2⟩
+    exact (measure_congr he).trans_lt hμK
+  have hδr : μ.real Kᶜ < ε/2 := by
+    have hh := (ENNReal.toReal_lt_toReal (measure_ne_top μ _) (ENNReal.ofReal_ne_top)).mpr hδ
+    simpa only [Measure.real,ENNReal.toReal_ofReal (le_of_lt (half_pos hε))] using hh
+  obtain ⟨J,hJ⟩ := hK.elim_finite_subcover V hVo hKU
+  have hlim : Tendsto (fun n => ∑ i ∈ J, ∫ x in V i, F n x ∂μ) atTop (𝓝 0) := by
+    simpa using tendsto_finsetSum J (fun i _ => hlocal i)
+  obtain ⟨N,hN⟩ := Metric.tendsto_atTop.mp hlim (ε/2) (half_pos hε)
+  refine ⟨N,fun n hn => ?_⟩
+  have hmaj : (∫ x, F n x ∂μ) ≤ (∑ i ∈ J, ∫ x in V i, F n x ∂μ) + μ.real Kᶜ := by
+    have hJJ : Integrable (fun x => ∑ i ∈ J, (V i).indicator (F n) x) μ :=
+      integrable_finsetSum J (fun i _ => (hF n).indicator (hVo i).measurableSet)
+    have hCC : Integrable (Kᶜ.indicator (fun _ => (1 : ℝ))) μ :=
+      (integrable_const _).indicator hK.measurableSet.compl
+    have hi := integral_mono (hF n) (hJJ.add hCC) (fun x => by
+      have hs (i : I) : 0 ≤ (V i).indicator (F n) x := indicator_nonneg (fun y _ => (hb n y).1) x
+      by_cases hx : x ∈ K
+      · obtain ⟨i,hi,hxi⟩ := mem_iUnion₂.mp (hJ hx)
+        have hsum := Finset.single_le_sum (fun j _ => hs j) hi
+        simp only [Pi.add_apply,indicator_of_notMem (show x ∉ Kᶜ from not_not.mpr hx),add_zero]
+        exact (by simpa only [indicator_of_mem hxi] using hsum)
+      · change F n x ≤ (∑ i ∈ J, (V i).indicator (F n) x) + Kᶜ.indicator (fun _ => (1 : ℝ)) x
+        rw [indicator_of_mem hx]
+        linarith [Finset.sum_nonneg (fun i (_ : i ∈ J) => hs i), (hb n x).2])
+    apply hi.trans_eq
+    refine (integral_add hJJ hCC).trans ?_
+    apply congrArg₂ (·+·)
+    · exact (integral_finsetSum J (fun i _ => (hF n).indicator (hVo i).measurableSet)).trans
+        (Finset.sum_congr rfl (fun i _ => integral_indicator (hVo i).measurableSet))
+    · rw [integral_indicator hK.measurableSet.compl]
+      simp only [integral_const,smul_eq_mul,mul_one,Measure.real,Measure.restrict_apply_univ]
+  rw [Real.dist_eq,sub_zero,abs_of_nonneg (integral_nonneg fun x => (hb n x).1)]
+  have hh := hN n hn
+  have hs0 : 0 ≤ ∑ i ∈ J, ∫ x in V i, F n x ∂μ :=
+    Finset.sum_nonneg fun i _ => integral_nonneg (fun x => (hb n x).1)
+  rw [Real.dist_eq,sub_zero,abs_of_nonneg hs0] at hh
+  linarith
+end EuclideanFiveColor
+end
+
+noncomputable section
+open MeasureTheory Filter Topology Set
+open scoped NNReal
+namespace EuclideanFiveColor
+lemma bounded_difference_integrable {X : Type*} [MeasurableSpace X] {μ : Measure X} [IsFiniteMeasure μ]
+    {f g : X → ℝ} (hf : Measurable f) (hg : Measurable g)
+    (hfb : ∀ x, 0 ≤ f x ∧ f x ≤ 1) (hgb : ∀ x, 0 ≤ g x ∧ g x ≤ 1) :
+    Integrable (fun x => ‖f x-g x‖) μ := by
+  apply Integrable.of_bound ((hf.sub hg).norm.aestronglyMeasurable) 1
+  exact Eventually.of_forall fun x => by
+    rw [norm_norm,Real.norm_eq_abs,abs_le]
+    change -1 ≤ f x-g x ∧ f x-g x ≤ 1
+    constructor <;> linarith [(hfb x).1,(hfb x).2,(hgb x).1,(hgb x).2]
+lemma circleSum_tendsto_L1 {μ : Measure RealPlane} [IsFiniteMeasure μ]
+    [μ.InnerRegularCompactLTTop] (hμ : μ ≤ volume)
+    {r : ℕ → ℝ} {y : ℕ → RealPlane} {x : RealPlane}
+    (hr : Tendsto r atTop (𝓝 1)) (hy : Tendsto y atTop (𝓝 x))
+    {f : RealPlane → ℝ} (hf : Measurable f) (hfb : ∀ z, 0 ≤ f z ∧ f z ≤ 1) :
+    Tendsto (fun n => ∫ q, ‖f (y n+circleSum (r n) q)-f (x+circleSum 1 q)‖ ∂μ) atTop (𝓝 0) := by
+  classical
+  let T : ℕ → RealPlane → RealPlane := fun n q => y n+circleSum (r n) q
+  let S : RealPlane → RealPlane := fun q => x+circleSum 1 q
+  have hT (n : ℕ) : Measurable (T n) := (continuous_const.add (circleSum_continuous _)).measurable
+  have hS : Measurable S := (continuous_const.add (circleSum_continuous _)).measurable
+  let R : Set RealPlane := {p | Real.sin (p.2-p.1) ≠ 0}
+  have hex : ∀ p : R, ∃ V : Set RealPlane, IsOpen V ∧ p.1 ∈ V ∧
+      ∃ K : ℝ≥0, AntilipschitzWith K (V.domRestrict (circleSum 1)) := by
+    intro p
+    exact circleSum_local_antilipschitz (by simpa only [one_mul] using (show Real.sin (p.1.2-p.1.1) ≠ 0 from p.2))
+  choose V hVo hpV K hK using hex
+  have hfull : ∀ᵐ q ∂μ, q ∈ ⋃ p : R, V p := by
+    filter_upwards [circleSum_regular_ae.filter_mono (ae_mono hμ)] with q hq
+    exact mem_iUnion.mpr ⟨⟨q,hq⟩,hpV ⟨q,hq⟩⟩
+  apply tendsto_integral_of_local_cover (μ := μ)
+    (fun n => bounded_difference_integrable (hf.comp (hT n)) (hf.comp hS)
+      (fun q => hfb (T n q)) (fun q => hfb (S q)))
+    (fun n q => ⟨norm_nonneg _,by
+      rw [Real.norm_eq_abs,abs_le]
+      change -1 ≤ f (T n q)-f (S q) ∧ f (T n q)-f (S q) ≤ 1
+      constructor <;> linarith [(hfb (T n q)).1,(hfb (T n q)).2,(hfb (S q)).1,(hfb (S q)).2]⟩)
+    V hVo hfull
+  intro p
+  let B := Metric.closedBall x (4 : ℝ)
+  let ν : Measure RealPlane := volume.restrict B
+  have hB : MeasurableSet B := measurableSet_closedBall
+  let : IsFiniteMeasure ν := ⟨by simpa [ν] using (isCompact_closedBall x 4).measure_lt_top⟩
+  have hfi : Integrable f ν := Integrable.of_bound hf.aestronglyMeasurable 1
+    (Eventually.of_forall fun z => by rw [Real.norm_eq_abs,abs_of_nonneg (hfb z).1]; exact (hfb z).2)
+  have hmapT : ∀ᶠ n in atTop, ((μ.restrict (V p)).map (T n)) ≤
+      (((2*K p)^2 : ℝ≥0) : ENNReal) • ν := by
+    filter_upwards [circleSum_eventual_image hr hy,circleSum_eventual_antilipschitz (hK p) hr y] with n hn ha
+    apply (Measure.map_mono (Measure.restrict_mono_measure hμ (V p)) (hT n)).trans
+    exact map_restrict_volume_le (hT n) hB ha (fun q _ => hn q)
+  have hmapS : ((μ.restrict (V p)).map S) ≤ (((2*K p)^2 : ℝ≥0) : ENNReal) • ν := by
+    apply (Measure.map_mono (Measure.restrict_mono_measure hμ (V p)) hS).trans
+    exact map_restrict_volume_le hS hB (circleSum_limit_antilipschitz (hK p) x)
+      (fun q _ => circleSum_limit_image x q)
+  apply tendsto_integral_norm_comp_sub_eventual hT hS hmapT hmapS _ hfi
+  exact Eventually.of_forall fun q => hy.add (circleSum_parameter_tendsto hr q)
+end EuclideanFiveColor
+end
+
+noncomputable section
+open MeasureTheory Filter Topology Set
+namespace EuclideanFiveColor
+def angularSum (x : ℂ) (r : ℝ) (p : Angles × Angles) :=
+  x+(r : ℂ)*direction p.1+direction p.2
+lemma angularSum_measurable (x : ℂ) (r : ℝ) : Measurable (angularSum x r) :=
+  (measurable_const.add (measurable_const.mul (direction_continuous.measurable.comp measurable_fst))).add
+    (direction_continuous.measurable.comp measurable_snd)
+lemma angularSum_chart (x : ℂ) (r : ℝ) (q : RealPlane) :
+    angularSum x r (angleChart q) = Complex.equivRealProdCLM.symm
+      (Complex.equivRealProdCLM x+circleSum r q) := by
+  apply Complex.equivRealProdCLM.injective
+  simpa only [angularSum,ContinuousLinearEquiv.apply_symm_apply] using circleSum_chart x r q
+lemma angularSum_quasiMeasurePreserving (x : ℂ) {r : ℝ} (hr : r ≠ 0) :
+    Measure.QuasiMeasurePreserving (angularSum x r) (angularMeasure.prod angularMeasure) volume := by
+  have hreal : Measure.QuasiMeasurePreserving
+      (fun q => Complex.equivRealProdCLM.symm (Complex.equivRealProdCLM x+circleSum r q)) volume volume :=
+    (Complex.volume_preserving_equiv_real_prod.symm).quasiMeasurePreserving.comp
+      ((measurePreserving_add_left volume (Complex.equivRealProdCLM x)).quasiMeasurePreserving.comp
+        (circleSum_quasiMeasurePreserving hr))
+  refine ⟨angularSum_measurable x r,Measure.AbsolutelyContinuous.mk (fun A hA hz => ?_)⟩
+  rw [Measure.map_apply (angularSum_measurable x r) hA]
+  have hpre : angleChartMeasure (angleChart ⁻¹' (angularSum x r ⁻¹' A))=0 := by
+    apply le_antisymm _ bot_le
+    apply (angleChartMeas_le _).trans
+    have hz' := hreal.preimage_null hz
+    simpa [preimage_preimage,Function.comp_def,angularSum_chart] using le_of_eq hz'
+  have hz' : (ENNReal.ofReal (2*Real.pi))^2 *
+      (angularMeasure.prod angularMeasure) (angularSum x r ⁻¹' A)=0 := by
+    rw [← smul_eq_mul,← Measure.smul_apply,← angleChart_measure,
+      Measure.map_apply angleChart_preserving.measurable ((angularSum_measurable x r) hA)]
+    exact hpre
+  exact (mul_eq_zero.mp hz').resolve_left (by positivity)
+lemma angularSum_tendsto_L1 {r : ℕ → ℝ} {y : ℕ → ℂ} {x : ℂ}
+    (hr : Tendsto r atTop (𝓝 1)) (hy : Tendsto y atTop (𝓝 x))
+    {f : ℂ → ℝ} (hf : Measurable f) (hfb : ∀ z, 0 ≤ f z ∧ f z ≤ 1) :
+    Tendsto (fun n => ∫ p, ‖f (angularSum (y n) (r n) p)-f (angularSum x 1 p)‖
+      ∂angularMeasure.prod angularMeasure) atTop (𝓝 0) := by
+  have h := circleSum_tendsto_L1 (μ := angleChartMeasure) angleChartMeas_le hr
+    (Complex.equivRealProdCLM.continuous.tendsto x |>.comp hy)
+    (hf.comp Complex.equivRealProdCLM.symm.continuous.measurable)
+    (fun z => hfb (Complex.equivRealProdCLM.symm z))
+  have he := h.const_mul (((2*Real.pi)^2)⁻¹)
+  simp only [mul_zero] at he
+  convert he using 1
+  ext n
+  simpa only [Function.comp_def,Pi.sub_apply,angularSum_chart] using
+    (integral_angular_eq_chart (f := fun p => ‖f (angularSum (y n) (r n) p)-f (angularSum x 1 p)‖)
+      (((hf.comp (angularSum_measurable _ _)).sub (hf.comp (angularSum_measurable _ _))).norm))
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Filter Topology Set MeasureTheory
+namespace EuclideanFiveColor
+def samplingParameter (x y : ℂ) (r : ℝ) : CircleParameter := (Complex.equivRealProdCLM (y-x),r)
+lemma samplingParameter_tendsto {x z : ℂ} {y : ℕ → ℂ} {r : ℕ → ℝ}
+    (hy : Tendsto y atTop (𝓝 z)) (hr : Tendsto r atTop (𝓝 1)) :
+    Tendsto (fun n => samplingParameter x (y n) (r n)) atTop (𝓝 (samplingParameter x z 1)) :=
+  (Complex.equivRealProdCLM.continuous.continuousAt.tendsto.comp (hy.sub_const x)).prodMk_nhds hr
+lemma circleTarget_sampling (x y : ℂ) (r f : ℝ) :
+    Complex.equivRealProdCLM x + circleTarget (samplingParameter x y r,f) =
+      Complex.equivRealProdCLM (y+(r : ℂ)*direction (f : Angles)) := by
+  have he : (r : ℂ)*direction (f : Angles)=r • direction (f : Angles) := by simp
+  rw [he,map_add,map_smul,direction_realPlane]
+  simp only [circleTarget,samplingParameter,map_sub]
+  abel
+lemma circleInverse_sampling {t u : ℝ} (h : Real.sin (u-t) ≠ 0) {x y : ℂ} {r f : ℝ}
+    (ht : circleTarget (samplingParameter x y r,f) ∈ (circleInverseHomeo h).target) :
+    y+(r : ℂ)*direction (f : Angles) =
+      angularSum x 1 (angleChart (circleInverse h (circleTarget (samplingParameter x y r,f)))) := by
+  rw [angularSum_chart,circleInverse_right h ht]
+  apply Complex.equivRealProdCLM.injective
+  exact (circleTarget_sampling x y r f).symm
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology
+namespace EuclideanFiveColor
+def binaryPhi (d t : ℝ) : ℝ := t+Real.arctan (d*Real.sin t/(1-d*Real.cos t))
+def binaryBeta (d t : ℝ) : ℝ := Real.arccos (Real.sqrt (binaryQ d t)/2)
+def binaryTarget (d t : ℝ) : ℝ := -binaryCircleMap d (-t)+Real.pi/3
+def binaryOther (d t : ℝ) : ℝ := binaryCircleMap d t+2*(Real.pi/3)
+lemma binaryTarget_eq (d t : ℝ) : binaryTarget d t=binaryPhi d t+binaryBeta d t := by
+  simp only [binaryTarget,binaryCircleMap,binaryPhi,binaryBeta,binaryQ,Real.sin_neg,Real.cos_neg,
+    mul_neg,neg_div,Real.arctan_neg]
+  ring
+lemma binaryOther_eq (d t : ℝ) : binaryOther d t=binaryPhi d t+Real.pi-binaryBeta d t := by
+  dsimp [binaryOther,binaryCircleMap,binaryPhi,binaryBeta]
+  ring
+lemma binaryPhi_trig {d : ℝ} (hd : |d|<1/2) (t : ℝ) :
+    Real.cos (binaryPhi d t)=(Real.cos t-d)/Real.sqrt (binaryQ d t) ∧
+    Real.sin (binaryPhi d t)=Real.sin t/Real.sqrt (binaryQ d t) := by
+  have hq := binaryQ_range hd t
+  have hsq : Real.sqrt (binaryQ d t) ≠ 0 := Real.sqrt_ne_zero'.mpr hq.1
+  have hi := binary_arctan_identity d t
+  have he : 1+(d*Real.sin t/(1-d*Real.cos t))^2=binaryQ d t/(1-d*Real.cos t)^2 := by
+    field_simp [hq.2.2.ne']
+    nlinarith [hi]
+  have hs : Real.sqrt (1+(d*Real.sin t/(1-d*Real.cos t))^2)=Real.sqrt (binaryQ d t)/(1-d*Real.cos t) := by
+    rw [he,Real.sqrt_div hq.1.le,Real.sqrt_sq hq.2.2.le]
+  have hcθ : Real.cos (Real.arctan (d*Real.sin t/(1-d*Real.cos t)))=(1-d*Real.cos t)/Real.sqrt (binaryQ d t) := by
+    rw [Real.cos_arctan,hs]
+    field_simp
+  have hsθ : Real.sin (Real.arctan (d*Real.sin t/(1-d*Real.cos t)))=d*Real.sin t/Real.sqrt (binaryQ d t) := by
+    rw [Real.sin_arctan,hs]
+    field_simp [hq.2.2.ne']
+  dsimp [binaryPhi]
+  simp only [Real.cos_add,Real.sin_add,hcθ,hsθ]
+  constructor
+  · field_simp [hq.2.2.ne',hsq]
+    linear_combination -d*(Real.sin_sq_add_cos_sq t)
+  · field_simp [hq.2.2.ne',hsq]
+    ring
+lemma binaryBeta_trig {d : ℝ} (hd : |d|<1/2) (t : ℝ) :
+    Real.cos (binaryBeta d t)=Real.sqrt (binaryQ d t)/2 ∧
+    Real.sin (binaryBeta d t)=Real.sqrt (1-binaryQ d t/4) := by
+  have hq := binaryQ_range hd t
+  have hs := Real.sq_sqrt hq.1.le
+  have hs' : Real.sqrt (binaryQ d t)/2 ≤ 1 := by nlinarith [Real.sqrt_nonneg (binaryQ d t)]
+  constructor
+  · exact Real.cos_arccos (by linarith [Real.sqrt_nonneg (binaryQ d t)]) hs'
+  · dsimp [binaryBeta]
+    rw [Real.sin_arccos]
+    congr 1
+    nlinarith
+lemma binary_geometry {d : ℝ} (hd : |d|<1/2) (t : ℝ) :
+    d+Real.cos (binaryTarget d t)=Real.cos t+Real.cos (binaryOther d t) ∧
+    Real.sin (binaryTarget d t)=Real.sin t+Real.sin (binaryOther d t) := by
+  have hp := binaryPhi_trig hd t
+  have hb := binaryBeta_trig hd t
+  have hq := binaryQ_range hd t
+  have hs : Real.sqrt (binaryQ d t) ≠ 0 := Real.sqrt_ne_zero'.mpr hq.1
+  rw [binaryTarget_eq,binaryOther_eq,
+    show binaryPhi d t+Real.pi-binaryBeta d t=(binaryPhi d t-binaryBeta d t)+Real.pi by ring,
+    Real.cos_add_pi,Real.sin_add_pi,Real.cos_add,Real.sin_add,Real.cos_sub,Real.sin_sub,hp.1,hp.2,hb.1,hb.2]
+  constructor <;> field_simp <;> ring
+lemma binaryTarget_hasDeriv {d : ℝ} (hd : |d|<1/2) (t : ℝ) :
+    HasDerivAt (binaryTarget d) (1+d*binaryB d (-t)) t := by
+  convert ((binaryCircleMap_hasDeriv_t hd (-t)).comp t (hasDerivAt_neg t)).neg.add_const (Real.pi/3) using 1 <;>
+    first | rfl | ring
+lemma binaryOther_hasDeriv {d : ℝ} (hd : |d|<1/2) (t : ℝ) :
+    HasDerivAt (binaryOther d) (1+d*binaryB d t) t := by
+  exact (binaryCircleMap_hasDeriv_t hd t).add_const (2*(Real.pi/3))
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology
+namespace EuclideanFiveColor
+lemma binary_nonparallel {d : ℝ} (hd : |d|<1/2) (t : ℝ) :
+    Real.sin (binaryOther d t-t) ≠ 0 := by
+  intro hz
+  have hq := binaryQ_range (show |-d|<1/2 by simpa only [abs_neg] using hd) (binaryTarget d t)
+  have hg := binary_geometry hd t
+  have hs := Real.sin_sq_add_cos_sq (binaryOther d t-t)
+  rw [hz] at hs
+  have hi : binaryQ (-d) (binaryTarget d t)=2+2*Real.cos (binaryOther d t-t) := by
+    have hc := Real.sin_sq_add_cos_sq t
+    have hh := Real.sin_sq_add_cos_sq (binaryOther d t)
+    have ht := Real.sin_sq_add_cos_sq (binaryTarget d t)
+    dsimp only [binaryQ]
+    rw [Real.cos_sub]
+    nlinarith [sq_eq_sq_iff_eq_or_eq_neg.mpr (Or.inl hg.1),sq_eq_sq_iff_eq_or_eq_neg.mpr (Or.inl hg.2)]
+  rw [hi] at hq
+  nlinarith [sq_nonneg (Real.cos (binaryOther d t-t))]
+lemma binary_geometry_rotated {x z : ℂ} {d φ : ℝ} (hd : |d|<1/2)
+    (hxr : (z-x).re=d*Real.cos φ) (hxi : (z-x).im=d*Real.sin φ) (t : ℝ) :
+    circleTarget (samplingParameter x z 1,binaryTarget d t+φ)=
+      circleSum 1 (t+φ,binaryOther d t+φ) := by
+  have hg := binary_geometry hd t
+  simp only [circleTarget,samplingParameter,one_smul,circleSum,one_mul]
+  apply Prod.ext
+  · change (z-x).re+Real.cos (binaryTarget d t+φ)=Real.cos (t+φ)+Real.cos (binaryOther d t+φ)
+    rw [hxr,Real.cos_add,Real.cos_add,Real.cos_add]
+    linear_combination Real.cos φ*hg.1-Real.sin φ*hg.2
+  · change (z-x).im+Real.sin (binaryTarget d t+φ)=Real.sin (t+φ)+Real.sin (binaryOther d t+φ)
+    rw [hxi,Real.sin_add,Real.sin_add,Real.sin_add]
+    linear_combination Real.sin φ*hg.1+Real.cos φ*hg.2
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+lemma deriv_lower_surjective {f D : ℝ → ℝ} (hd : ∀ x, HasDerivAt f (D x) x)
+    (hp : ∀ x, (1/2 : ℝ) ≤ D x) : Function.Surjective f := by
+  intro y
+  let R := 2*(|y-f 0|+1)
+  have hR : 0 < R := by dsimp [R]; positivity
+  have h₁ := mul_sub_le_image_sub_of_le_deriv (fun x => (hd x).differentiableAt)
+    (fun x => by rw [(hd x).deriv]; exact hp x) (show -R ≤ 0 by linarith)
+  have h₂ := mul_sub_le_image_sub_of_le_deriv (fun x => (hd x).differentiableAt)
+    (fun x => by rw [(hd x).deriv]; exact hp x) hR.le
+  have hy : y ∈ Icc (f (-R)) (f R) := by
+    constructor <;> dsimp [R] at * <;> linarith [le_abs_self (y-f 0),neg_abs_le (y-f 0)]
+  obtain ⟨x,_,hx⟩ := intermediate_value_Icc (by linarith : -R ≤ R)
+    (continuous_iff_continuousAt.mpr (fun x => (hd x).continuousAt)).continuousOn hy
+  exact ⟨x,hx⟩
+def derivLowerOrderIso {f D : ℝ → ℝ} (hd : ∀ x, HasDerivAt f (D x) x)
+    (hp : ∀ x, (1/2 : ℝ) ≤ D x) : ℝ ≃o ℝ :=
+  (strictMono_of_deriv_pos (fun x => by rw [(hd x).deriv]; linarith [hp x])).orderIsoOfRightInverse f
+    (Function.surjInv (deriv_lower_surjective hd hp)) (Function.rightInverse_surjInv (deriv_lower_surjective hd hp))
+end EuclideanFiveColor
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PlaneColoring/Coloring.lean b/lean/OAI/Geometry/PlaneColoring/Coloring.lean
new file mode 100644
index 0000000000000000000000000000000000000000..6e31f006f54ec11f6248949f8957dfd41aff2adc
--- /dev/null
+++ b/lean/OAI/Geometry/PlaneColoring/Coloring.lean
@@ -0,0 +1,34 @@
+import Mathlib
+
+namespace OAI
+
+namespace EuclideanFiveColor
+
+def ProperColoring (k : ℕ) (c : ℂ → Fin k) : Prop :=
+  ∀ x y : ℂ, ‖x - y‖ = 1 → c x ≠ c y
+
+end EuclideanFiveColor
+
+noncomputable section
+open MeasureTheory
+namespace EuclideanFiveColor
+instance anglePeriodPositive : Fact (0 < 2*Real.pi) := ⟨by positivity⟩
+abbrev Angles := AddCircle (2*Real.pi)
+def angularMeasure : Measure Angles := AddCircle.haarAddCircle
+def direction (t : Angles) : ℂ := AddCircle.toCircle t
+instance : IsProbabilityMeasure angularMeasure := inferInstanceAs (IsProbabilityMeasure (AddCircle.haarAddCircle (T := 2*Real.pi)))
+instance : angularMeasure.IsAddHaarMeasure := inferInstanceAs (AddCircle.haarAddCircle (T := 2*Real.pi)).IsAddHaarMeasure
+lemma direction_continuous : Continuous direction := continuous_subtype_val.comp AddCircle.continuous_toCircle
+@[simp] lemma direction_norm (t : Angles) : ‖direction t‖ = 1 := Circle.norm_coe _
+@[simp] lemma direction_real (t : ℝ) : direction (t : Angles) = (Circle.exp t : ℂ) := by
+  simp [direction,AddCircle.toCircle_apply_mk,Real.pi_ne_zero]
+/-- Exactly the manuscript weak measurable condition, using normalized Haar angular measure.
+A function is understood up to a plane-null set; no regularity is imposed on a proper coloring. -/
+def WeakColoring (k : ℕ) (c : ℂ → Fin k) : Prop :=
+  AEMeasurable c (volume : Measure ℂ) ∧ ∀ᵐ p ∂(volume : Measure ℂ).prod angularMeasure,
+    c p.1 ≠ c (p.1+direction p.2)
+
+end EuclideanFiveColor
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PlaneColoring/Compactness.lean b/lean/OAI/Geometry/PlaneColoring/Compactness.lean
new file mode 100644
index 0000000000000000000000000000000000000000..10e2cd473c4e4cfbd6ac9046db41638435b3914c
--- /dev/null
+++ b/lean/OAI/Geometry/PlaneColoring/Compactness.lean
@@ -0,0 +1,379 @@
+import OAI.Geometry.PlaneColoring.WeakColoring
+
+namespace OAI
+
+noncomputable section
+open Filter Topology
+open scoped RealInnerProductSpace
+namespace EuclideanFiveColor
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E]
+  [TopologicalSpace.SeparableSpace E]
+def weakHilbert (v : E) : WeakDual ℝ E := StrongDual.toWeakDual (InnerProductSpace.toDual ℝ E v)
+omit [TopologicalSpace.SeparableSpace E] in
+@[simp] lemma weakHilbert_apply (v w : E) : weakHilbert v w = ⟪v,w⟫ := rfl
+lemma exists_finite_weak_subseq {ι : Type*} [Finite ι] (f : ℕ → ι → E) (C : ℝ)
+    (hf : ∀ n i, ‖f n i‖ ≤ C) :
+    ∃ b : ι → E, ∃ φ : ℕ → ℕ, StrictMono φ ∧
+      (∀ i, ‖b i‖ ≤ C) ∧ ∀ i v, Tendsto (fun n => ⟪f (φ n) i,v⟫) atTop (𝓝 ⟪b i,v⟫) := by
+  let K : Set (WeakDual ℝ E) := WeakDual.toStrongDual ⁻¹' Metric.closedBall 0 C
+  have hK : IsCompact K := WeakDual.isCompact_closedBall 0 C
+  let : CompactSpace K := isCompact_iff_compactSpace.mp hK
+  let : TopologicalSpace.MetrizableSpace K := WeakDual.metrizable_of_isCompact ℝ E K hK
+  let u : ℕ → ι → K := fun n i => ⟨weakHilbert (f n i), by
+    simpa only [K,Set.mem_preimage,Metric.mem_closedBall,dist_zero_right,weakHilbert,
+      StrongDual.toStrongDual_toWeakDual,LinearIsometryEquiv.norm_map] using hf n i⟩
+  obtain ⟨b,_,φ,hφ,hb⟩ := (isCompact_univ : IsCompact (Set.univ : Set (ι → K))).isSeqCompact.subseq_of_frequently_in
+    (Filter.Eventually.frequently (Filter.Eventually.of_forall (fun n => Set.mem_univ (u n))))
+  refine ⟨fun i => (InnerProductSpace.toDual ℝ E).symm (WeakDual.toStrongDual (b i).1),φ,hφ,?_,?_⟩
+  · intro i
+    simpa only [LinearIsometryEquiv.norm_map,K,Set.mem_preimage,Metric.mem_closedBall,dist_zero_right] using (b i).2
+  · intro i v
+    have ht := ((WeakDual.eval_continuous v).comp continuous_subtype_val).tendsto (b i)
+      |>.comp ((continuous_apply i).tendsto b |>.comp hb)
+    change Tendsto (fun n => ⟪f (φ n) i,v⟫) atTop (𝓝 ((b i).1 v)) at ht
+    simpa only [InnerProductSpace.toDual_symm_apply,WeakDual.toStrongDual_apply] using ht
+end EuclideanFiveColor
+end
+
+noncomputable section
+open MeasureTheory Filter Topology
+open scoped RealInnerProductSpace
+namespace EuclideanFiveColor
+variable {X : Type*} [MeasurableSpace X] (μ : Measure X) [IsFiniteMeasure μ]
+abbrev RealL2 := Lp ℝ 2 μ
+def oneL2 : RealL2 μ := indicatorConstLp 2 MeasurableSet.univ (measure_ne_top μ Set.univ) 1
+lemma oneL2_ae : oneL2 μ =ᵐ[μ] (fun _ => (1 : ℝ)) := by
+  simpa only [oneL2,Set.indicator_univ] using (indicatorConstLp_coeFn (p := 2) (hs := MeasurableSet.univ)
+    (hμs := measure_ne_top μ Set.univ) (c := (1 : ℝ)))
+lemma weak_limit_nonneg {f : ℕ → RealL2 μ} {b : RealL2 μ}
+    (ht : ∀ v, Tendsto (fun n => ⟪f n,v⟫) atTop (𝓝 ⟪b,v⟫))
+    (hf : ∀ n, 0 ≤ᵐ[μ] f n) : 0 ≤ᵐ[μ] b := by
+  apply ae_nonneg_of_forall_setIntegral_nonneg (MemLp.integrable (by norm_num) (Lp.memLp b))
+  intro s hs hμs
+  have h := ht (indicatorConstLp 2 hs hμs.ne 1)
+  have he (v : RealL2 μ) : ⟪v,indicatorConstLp 2 hs hμs.ne 1⟫ = ∫ x in s, v x ∂μ := by
+    rw [real_inner_comm,L2.inner_indicatorConstLp_one]
+  simp_rw [he] at h
+  exact ge_of_tendsto h (Eventually.of_forall fun n => integral_nonneg_of_ae (ae_restrict_of_ae (hf n)))
+lemma weak_limit_le_one {f : ℕ → RealL2 μ} {b : RealL2 μ}
+    (ht : ∀ v, Tendsto (fun n => ⟪f n,v⟫) atTop (𝓝 ⟪b,v⟫))
+    (hf : ∀ n, f n ≤ᵐ[μ] (fun _ => (1 : ℝ))) : b ≤ᵐ[μ] (fun _ => (1 : ℝ)) := by
+  have hh : 0 ≤ᵐ[μ] (oneL2 μ-b) := weak_limit_nonneg μ
+    (f := fun n => oneL2 μ-f n) (fun v => by simpa only [inner_sub_left] using tendsto_const_nhds.sub (ht v)) (fun n => by
+      filter_upwards [Lp.coeFn_sub (oneL2 μ) (f n),oneL2_ae μ,hf n] with x hx h1 hn
+      simpa only [hx,Pi.sub_apply,Pi.zero_apply,h1,sub_nonneg] using hn)
+  filter_upwards [hh,Lp.coeFn_sub (oneL2 μ) b,oneL2_ae μ] with x hx he h1
+  simpa only [he,Pi.sub_apply,Pi.zero_apply,h1,sub_nonneg] using hx
+lemma weak_limit_sum {ι : Type*} [Fintype ι] {f : ℕ → ι → RealL2 μ} {b : ι → RealL2 μ}
+    (ht : ∀ i v, Tendsto (fun n => ⟪f n i,v⟫) atTop (𝓝 ⟪b i,v⟫))
+    (hf : ∀ n, ∑ i, f n i = oneL2 μ) : ∑ i, b i = oneL2 μ := by
+  apply ext_inner_right ℝ
+  intro v
+  have hh := tendsto_finsetSum Finset.univ (fun i _ => ht i v)
+  simp_rw [← sum_inner,hf] at hh
+  exact tendsto_nhds_unique hh tendsto_const_nhds
+end EuclideanFiveColor
+end
+
+noncomputable section
+open MeasureTheory MeasureTheory.Measure Filter Topology
+open scoped RealInnerProductSpace
+namespace EuclideanFiveColor
+variable {k : ℕ} (c : ℂ → Fin k)
+def Typical (z : ℂ) : Prop := DensityOne {w | c w=c z} z
+def TypicalSampling (y : ℂ) (r : ℝ) : Prop :=
+  0 < r ∧ ∀ᵐ t ∂angularMeasure, Typical c (y+(r : ℂ)*direction t)
+lemma ae_typicalSampling (hc : Measurable c) {r : ℝ} (hr : 0<r) :
+    ∀ᵐ y ∂(volume : Measure ℂ), TypicalSampling c y r := by
+  have hq : QuasiMeasurePreserving (fun p : ℂ × Angles => p.1+(r : ℂ)*direction p.2)
+      ((volume : Measure ℂ).prod angularMeasure) volume :=
+    QuasiMeasurePreserving.prod_of_left
+      (measurable_fst.add (measurable_const.mul (direction_continuous.measurable.comp measurable_snd)))
+      (Eventually.of_forall fun t => (measurePreserving_add_right (volume : Measure ℂ)
+        ((r : ℂ)*direction t)).quasiMeasurePreserving)
+  have hh : ∀ᵐ p ∂((volume : Measure ℂ).prod angularMeasure), Typical c (p.1+(r : ℂ)*direction p.2) :=
+    hq.ae (ae_density_actual c hc)
+  exact (Measure.ae_ae_of_ae_prod hh).mono fun y hy => ⟨hr,hy⟩
+def sampleFn (y : ℂ) (r : ℝ) (i : Fin k) (t : Angles) : ℝ :=
+  if c (y+(r : ℂ)*direction t)=i then 1 else 0
+lemma sampleFn_measurable (hc : Measurable c) (y : ℂ) (r : ℝ) (i : Fin k) :
+    Measurable (sampleFn c y r i) := by
+  exact measurable_one.piecewise (measurableSet_eq_fun
+    (hc.comp (measurable_const.add (measurable_const.mul direction_continuous.measurable))) measurable_const) measurable_zero
+lemma sampleFn_nonneg (y : ℂ) (r : ℝ) (i : Fin k) (t : Angles) : 0 ≤ sampleFn c y r i t := by
+  simp only [sampleFn]; split_ifs <;> norm_num
+lemma sampleFn_le_one (y : ℂ) (r : ℝ) (i : Fin k) (t : Angles) : sampleFn c y r i t ≤ 1 := by
+  simp only [sampleFn]; split_ifs <;> norm_num
+lemma sampleFn_memLp (hc : Measurable c) (y : ℂ) (r : ℝ) (i : Fin k) :
+    MemLp (sampleFn c y r i) 2 angularMeasure := by
+  apply MemLp.of_bound (sampleFn_measurable c hc y r i).aestronglyMeasurable 1
+  exact Eventually.of_forall fun t => by rw [Real.norm_eq_abs,abs_of_nonneg (sampleFn_nonneg c y r i t)]; exact sampleFn_le_one c y r i t
+def sample (hc : Measurable c) (y : ℂ) (r : ℝ) (i : Fin k) : RealL2 angularMeasure :=
+  (sampleFn_memLp c hc y r i).toLp _
+lemma sample_ae (hc : Measurable c) (y : ℂ) (r : ℝ) (i : Fin k) :
+    sample c hc y r i =ᵐ[angularMeasure] sampleFn c y r i := MemLp.coeFn_toLp _
+lemma sample_norm_le (hc : Measurable c) (y : ℂ) (r : ℝ) (i : Fin k) :
+    ‖sample c hc y r i‖ ≤ 1 := by
+  have hh : ∀ᵐ t ∂angularMeasure, ‖sample c hc y r i t‖ ≤ 1 := by
+    filter_upwards [sample_ae c hc y r i] with t ht
+    rw [ht,Real.norm_eq_abs,abs_of_nonneg (sampleFn_nonneg c y r i t)]
+    exact sampleFn_le_one c y r i t
+  simpa [measureUnivNNReal] using Lp.norm_le_of_ae_bound (by norm_num : (0 : ℝ) ≤ 1) hh
+lemma sample_sum (hc : Measurable c) (y : ℂ) (r : ℝ) :
+    ∑ i, sample c hc y r i = oneL2 angularMeasure := by
+  apply Lp.ext
+  filter_upwards [Lp.coeFn_fun_finsetSum Finset.univ (fun i => sample c hc y r i),
+    ae_all_iff.mpr (sample_ae c hc y r),oneL2_ae angularMeasure] with t ht ha h1
+  rw [ht,h1]
+  simp only [ha,sampleFn]
+  simp
+lemma sampling_weak_compact (hc : Measurable c) (y : ℕ → ℂ) (r : ℕ → ℝ) :
+    ∃ b : Fin k → RealL2 angularMeasure, ∃ φ : ℕ → ℕ, StrictMono φ ∧
+      (∀ i, 0 ≤ᵐ[angularMeasure] b i) ∧ (∀ i, b i ≤ᵐ[angularMeasure] (fun _ => (1 : ℝ))) ∧
+      (∑ i, b i = oneL2 angularMeasure) ∧
+      ∀ i v, Tendsto (fun n => ⟪sample c hc (y (φ n)) (r (φ n)) i,v⟫) atTop (𝓝 ⟪b i,v⟫) := by
+  let : Fact ((2 : ENNReal) ≠ ⊤) := ⟨by norm_num⟩
+  obtain ⟨b,φ,hφ,_,hb⟩ := exists_finite_weak_subseq
+    (fun n i => sample c hc (y n) (r n) i) 1 (fun n i => sample_norm_le c hc (y n) (r n) i)
+  refine ⟨b,φ,hφ,?_,?_,?_,hb⟩
+  · intro i
+    apply weak_limit_nonneg angularMeasure (hb i)
+    intro n
+    filter_upwards [sample_ae c hc (y (φ n)) (r (φ n)) i] with t ht
+    simpa only [ht,Pi.zero_apply] using sampleFn_nonneg c _ _ i t
+  · intro i
+    apply weak_limit_le_one angularMeasure (hb i)
+    intro n
+    filter_upwards [sample_ae c hc (y (φ n)) (r (φ n)) i] with t ht
+    simpa only [ht] using sampleFn_le_one c _ _ i t
+  · exact weak_limit_sum angularMeasure hb (fun n => sample_sum c hc (y (φ n)) (r (φ n)))
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Filter Topology
+open scoped RealInnerProductSpace
+namespace EuclideanFiveColor.WeakBall
+variable (E : Type*) [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E]
+  [TopologicalSpace.SeparableSpace E]
+abbrev Space := {v : WeakDual ℝ E // WeakDual.toStrongDual v ∈ Metric.closedBall 0 1}
+instance : CompactSpace (Space E) := isCompact_iff_compactSpace.mp (WeakDual.isCompact_closedBall 0 1)
+instance : TopologicalSpace.MetrizableSpace (Space E) :=
+  WeakDual.metrizable_of_isCompact ℝ E _ (WeakDual.isCompact_closedBall 0 1)
+def value (a : Space E) : E := (InnerProductSpace.toDual ℝ E).symm (WeakDual.toStrongDual a.1)
+omit [TopologicalSpace.SeparableSpace E] in
+lemma value_norm (a : Space E) : ‖value E a‖ ≤ 1 := by
+  simpa only [value,LinearIsometryEquiv.norm_map,Metric.mem_closedBall,dist_zero_right] using a.2
+omit [TopologicalSpace.SeparableSpace E] in
+lemma value_eval (a : Space E) (v : E) : ⟪value E a,v⟫ = a.1 v := by
+  simp only [value,InnerProductSpace.toDual_symm_apply,WeakDual.toStrongDual_apply]
+def ofNorm (v : E) (hv : ‖v‖ ≤ 1) : Space E := ⟨weakHilbert v,by
+  simpa only [weakHilbert,StrongDual.toStrongDual_toWeakDual,Metric.mem_closedBall,dist_zero_right,
+    LinearIsometryEquiv.norm_map] using hv⟩
+omit [TopologicalSpace.SeparableSpace E] in
+@[simp] lemma value_ofNorm (v : E) (hv : ‖v‖ ≤ 1) : value E (ofNorm E v hv) = v := by
+  simp only [value,ofNorm,weakHilbert,StrongDual.toStrongDual_toWeakDual,LinearIsometryEquiv.symm_apply_apply]
+omit [TopologicalSpace.SeparableSpace E] in
+lemma continuous_eval (v : E) : Continuous (fun a : Space E => ⟪value E a,v⟫) := by
+  simp_rw [value_eval]
+  exact (WeakDual.eval_continuous v).comp continuous_subtype_val
+omit [TopologicalSpace.SeparableSpace E] in
+lemma tendsto_iff {I : Type*} {l : Filter I} {f : I → Space E} {b : Space E} :
+    Tendsto f l (𝓝 b) ↔ ∀ v : E, Tendsto (fun n => ⟪value E (f n),v⟫) l (𝓝 ⟪value E b,v⟫) := by
+  simp only [value_eval]
+  rw [tendsto_subtype_rng, tendsto_iff_forall_eval_tendsto_topDualPairing]
+  rfl
+end EuclideanFiveColor.WeakBall
+end
+
+noncomputable section
+open MeasureTheory Filter Topology
+open scoped RealInnerProductSpace
+namespace EuclideanFiveColor
+variable {X : Type*} [MeasurableSpace X] {μ : Measure X} [IsFiniteMeasure μ]
+lemma bounded_test_error {f v g : X → ℝ} (hf : AEStronglyMeasurable f μ)
+    (hfb : ∀ᵐ x ∂μ, ‖f x‖ ≤ 1) (hv : Integrable v μ) (hg : Integrable g μ) :
+    dist (∫ x, f x*v x ∂μ) (∫ x, f x*g x ∂μ) ≤ ∫ x, ‖v x-g x‖ ∂μ := by
+  have hft : MemLp f ⊤ μ := MemLp.of_bound hf 1 hfb
+  have hiv : Integrable (fun x => f x*v x) μ := hv.mul_of_top_right hft
+  have hig : Integrable (fun x => f x*g x) μ := hg.mul_of_top_right hft
+  rw [dist_eq_norm,← integral_sub hiv hig]
+  calc
+    _ ≤ ∫ x, ‖f x*v x-f x*g x‖ ∂μ := norm_integral_le_integral_norm _
+    _ ≤ ∫ x, ‖v x-g x‖ ∂μ := by
+      apply integral_mono_ae ((hv.mul_of_top_right hft).sub (hg.mul_of_top_right hft)).norm (hv.sub hg).norm
+      filter_upwards [hfb] with x hx
+      change ‖f x * v x - f x * g x‖ ≤ ‖v x - g x‖
+      rw [← mul_sub,norm_mul]
+      exact mul_le_of_le_one_left (norm_nonneg _) hx
+lemma weak_L2_to_weakStar {f : ℕ → Lp ℝ 2 μ} {b : Lp ℝ 2 μ}
+    (hf : ∀ n, ∀ᵐ x ∂μ, ‖f n x‖ ≤ 1) (hb : ∀ᵐ x ∂μ, ‖b x‖ ≤ 1)
+    (ht : ∀ v : Lp ℝ 2 μ, Tendsto (fun n => ⟪f n,v⟫) atTop (𝓝 ⟪b,v⟫))
+    {v : X → ℝ} (hv : Integrable v μ) :
+    Tendsto (fun n => ∫ x, f n x*v x ∂μ) atTop (𝓝 (∫ x, b x*v x ∂μ)) := by
+  apply Metric.tendsto_atTop.mpr
+  intro ε hε
+  obtain ⟨g,hg,hgm⟩ := (memLp_one_iff_integrable.mpr hv).exists_simpleFunc_eLpNorm_sub_lt (by simp : (1 : ENNReal) ≠ ⊤)
+    (ENNReal.ofReal_ne_zero_iff.mpr (by positivity : 0 < ε/3))
+  obtain ⟨C,hC⟩ := g.exists_forall_norm_le
+  have hg2 : MemLp g 2 μ := MemLp.of_bound g.measurable.aestronglyMeasurable C (Eventually.of_forall hC)
+  have hgi : Integrable (fun x => g x) μ := memLp_one_iff_integrable.mp hgm
+  have herr : (∫ x, ‖v x-g x‖ ∂μ) < ε/3 := by
+    change (∫ x, ‖(v - (fun x => g x)) x‖ ∂μ) < ε/3
+    rw [integral_norm_eq_lintegral_enorm (hv.sub hgi).aestronglyMeasurable,
+      ← eLpNorm_one_eq_lintegral_enorm (hv.sub hgi).aestronglyMeasurable]
+    exact (ENNReal.toReal_lt_toReal (ne_top_of_lt hg) ENNReal.ofReal_ne_top).mpr hg |>.trans_eq (ENNReal.toReal_ofReal (by positivity))
+  have he (w : Lp ℝ 2 μ) : ⟪w,hg2.toLp g⟫ = ∫ x, w x*g x ∂μ := by
+    rw [L2.inner_def]
+    apply integral_congr_ae
+    filter_upwards [hg2.coeFn_toLp] with x hx
+    simp [hx, mul_comm]
+  have htt := ht (hg2.toLp g)
+  simp_rw [he] at htt
+  obtain ⟨N,hN⟩ := Metric.tendsto_atTop.mp htt (ε/3) (by positivity)
+  refine ⟨N,fun n hn => ?_⟩
+  have h1 := bounded_test_error (Lp.aestronglyMeasurable (f n)) (hf n) hv hgi
+  have h2 := bounded_test_error (Lp.aestronglyMeasurable b) hb hv hgi
+  calc
+    _ ≤ dist (∫ x, f n x*v x ∂μ) (∫ x, f n x*g x ∂μ) +
+        dist (∫ x, f n x*g x ∂μ) (∫ x, b x*g x ∂μ) +
+        dist (∫ x, b x*g x ∂μ) (∫ x, b x*v x ∂μ) := by
+      exact dist_triangle4 _ _ _ _
+    _ < ε := by rw [dist_comm (∫ x, b x*g x ∂μ)]; linarith [hN n hn]
+end EuclideanFiveColor
+end
+
+noncomputable section
+open MeasureTheory Filter Topology
+open scoped RealInnerProductSpace
+namespace EuclideanFiveColor
+private instance : Fact ((2 : ENNReal) ≠ ⊤) := ⟨by norm_num⟩
+abbrev TraceVector (k : ℕ) := Fin k → WeakBall.Space (RealL2 angularMeasure)
+def traceValue {k : ℕ} (b : TraceVector k) (i : Fin k) : RealL2 angularMeasure :=
+  WeakBall.value _ (b i)
+variable {k : ℕ} (c : ℂ → Fin k) (hc : Measurable c)
+def sampleCode (y : ℂ) (r : ℝ) : TraceVector k :=
+  fun i => WeakBall.ofNorm _ (sample c hc y r i) (sample_norm_le c hc y r i)
+@[simp] lemma traceValue_sampleCode (y : ℂ) (r : ℝ) (i : Fin k) :
+    traceValue (sampleCode c hc y r) i = sample c hc y r i := WeakBall.value_ofNorm _ _ _
+def sampleSet : Set (ℂ × ℝ × TraceVector k) :=
+  {p | TypicalSampling c p.1 p.2.1 ∧ p.2.2=sampleCode c hc p.1 p.2.1}
+def IsTrace (x : ℂ) (b : TraceVector k) : Prop := (x,1,b) ∈ closure (sampleSet c hc)
+lemma trace_realization {x : ℂ} {b : TraceVector k} (hb : IsTrace c hc x b) :
+    ∃ y : ℕ → ℂ, ∃ r : ℕ → ℝ, (∀ n, TypicalSampling c (y n) (r n)) ∧
+      Tendsto y atTop (𝓝 x) ∧ Tendsto r atTop (𝓝 1) ∧
+      Tendsto (fun n => sampleCode c hc (y n) (r n)) atTop (𝓝 b) := by
+  obtain ⟨p,hp,ht⟩ := mem_closure_iff_seq_limit.mp hb
+  simp only [nhds_prod_eq] at ht
+  refine ⟨fun n => (p n).1,fun n => (p n).2.1,fun n => (hp n).1,ht.fst,ht.snd.fst,?_⟩
+  simpa only [show (fun n => (p n).2.2) = (fun n => sampleCode c hc (p n).1 (p n).2.1) from
+    funext fun n => (hp n).2] using ht.snd.snd
+lemma trace_of_realization {x : ℂ} {b : TraceVector k} {y : ℕ → ℂ} {r : ℕ → ℝ}
+    (htr : ∀ n, TypicalSampling c (y n) (r n)) (hy : Tendsto y atTop (𝓝 x))
+    (hr : Tendsto r atTop (𝓝 1))
+    (hb : Tendsto (fun n => sampleCode c hc (y n) (r n)) atTop (𝓝 b)) : IsTrace c hc x b := by
+  exact mem_closure_of_tendsto (hy.prodMk_nhds (hr.prodMk_nhds hb))
+    (Eventually.of_forall fun n => show (y n,r n,sampleCode c hc (y n) (r n)) ∈ sampleSet c hc from ⟨htr n,rfl⟩)
+lemma exists_trace (x : ℂ) : ∃ b : TraceVector k, IsTrace c hc x b := by
+  have hd : Dense {y : ℂ | TypicalSampling c y 1} := Measure.dense_of_ae
+    (ae_typicalSampling c hc (by norm_num : (0 : ℝ)<1))
+  obtain ⟨y,hy,hyt⟩ := mem_closure_iff_seq_limit.mp (hd x)
+  obtain ⟨b,_,φ,hφ,hb⟩ := (isCompact_univ : IsCompact (Set.univ : Set (TraceVector k))).isSeqCompact.subseq_of_frequently_in
+    (Eventually.frequently (Eventually.of_forall fun n => Set.mem_univ (sampleCode c hc (y n) 1)))
+  exact ⟨b,trace_of_realization c hc (fun n => hy (φ n)) (hyt.comp hφ.tendsto_atTop) tendsto_const_nhds hb⟩
+lemma isClosed_traceGraph : IsClosed {p : ℂ × TraceVector k | IsTrace c hc p.1 p.2} := by
+  exact isClosed_closure.preimage (continuous_fst.prodMk (continuous_const.prodMk continuous_snd))
+lemma isCompact_traces (x : ℂ) : IsCompact {b : TraceVector k | IsTrace c hc x b} := by
+  apply IsClosed.isCompact
+  exact isClosed_closure.preimage (continuous_const.prodMk (continuous_const.prodMk continuous_id))
+lemma trace_limit {x : ℕ → ℂ} {b : ℕ → TraceVector k} {y : ℂ} {a : TraceVector k}
+    (h : ∀ n, IsTrace c hc (x n) (b n)) (hx : Tendsto x atTop (𝓝 y))
+    (hb : Tendsto b atTop (𝓝 a)) : IsTrace c hc y a :=
+  (isClosed_traceGraph c hc).mem_of_tendsto (hx.prodMk_nhds hb) (Eventually.of_forall h)
+lemma traceCode_tendsto_iff {I : Type*} {l : Filter I} {f : I → TraceVector k} {b : TraceVector k} :
+    Tendsto f l (𝓝 b) ↔ ∀ i v, Tendsto (fun n => ⟪traceValue (f n) i,v⟫) l (𝓝 ⟪traceValue b i,v⟫) := by
+  rw [tendsto_pi_nhds]
+  exact forall_congr' fun i => WeakBall.tendsto_iff _
+lemma trace_bounds {x : ℂ} {b : TraceVector k} (hb : IsTrace c hc x b) :
+    (∀ i, 0 ≤ᵐ[angularMeasure] traceValue b i) ∧
+    (∀ i, traceValue b i ≤ᵐ[angularMeasure] (fun _ => (1 : ℝ))) ∧
+    ∑ i, traceValue b i = oneL2 angularMeasure := by
+  obtain ⟨y,r,_,_,_,ht⟩ := trace_realization c hc hb
+  have hw := traceCode_tendsto_iff.mp ht
+  simp only [traceValue_sampleCode] at hw
+  refine ⟨?_,?_,?_⟩
+  · intro i
+    apply weak_limit_nonneg angularMeasure (hw i)
+    intro n
+    filter_upwards [sample_ae c hc (y n) (r n) i] with t he
+    simpa only [he,Pi.zero_apply] using sampleFn_nonneg c (y n) (r n) i t
+  · intro i
+    apply weak_limit_le_one angularMeasure (hw i)
+    intro n
+    filter_upwards [sample_ae c hc (y n) (r n) i] with t he
+    simpa only [he] using sampleFn_le_one c (y n) (r n) i t
+  · exact weak_limit_sum angularMeasure hw (fun n => sample_sum c hc (y n) (r n))
+lemma trace_realization_weakStar {x : ℂ} {b : TraceVector k} (hb : IsTrace c hc x b) :
+    ∃ y : ℕ → ℂ, ∃ r : ℕ → ℝ, (∀ n, TypicalSampling c (y n) (r n)) ∧
+      Tendsto y atTop (𝓝 x) ∧ Tendsto r atTop (𝓝 1) ∧
+      ∀ i (v : Angles → ℝ), Integrable v angularMeasure →
+        Tendsto (fun n => ∫ t, sampleFn c (y n) (r n) i t*v t ∂angularMeasure) atTop
+          (𝓝 (∫ t, traceValue b i t*v t ∂angularMeasure)) := by
+  obtain ⟨y,r,hy,hyt,hrt,hbt⟩ := trace_realization c hc hb
+  refine ⟨y,r,hy,hyt,hrt,fun i v hv => ?_⟩
+  have hbounds := trace_bounds c hc hb
+  have hbn : ∀ᵐ t ∂angularMeasure, ‖traceValue b i t‖ ≤ 1 := by
+    filter_upwards [hbounds.1 i,hbounds.2.1 i] with t ht1 ht2
+    rwa [Real.norm_eq_abs,abs_of_nonneg ht1]
+  have hsn (n : ℕ) : ∀ᵐ t ∂angularMeasure, ‖sample c hc (y n) (r n) i t‖ ≤ 1 := by
+    filter_upwards [sample_ae c hc (y n) (r n) i] with t ht
+    rw [ht,Real.norm_eq_abs,abs_of_nonneg (sampleFn_nonneg c _ _ _ t)]
+    exact sampleFn_le_one c _ _ _ t
+  have hw := weak_L2_to_weakStar hsn hbn
+    (by simpa only [traceValue_sampleCode] using (traceCode_tendsto_iff.mp hbt) i) hv
+  convert hw using 1
+  ext n
+  exact (integral_congr_ae ((sample_ae c hc (y n) (r n) i).mono fun t ht => congrArg (fun a => a*v t) ht)).symm
+end EuclideanFiveColor
+end
+
+noncomputable section
+open MeasureTheory Topology Filter Set
+namespace EuclideanFiveColor
+variable {X I : Type*} [MeasurableSpace X] (μ : Measure X) [IsFiniteMeasure μ]
+/-- A finite-measure essential union is generated by countably many members of the original family. -/
+lemma exists_countable_essential_union (A : I → Set X) (hm : ∀ i, MeasurableSet (A i)) :
+    ∃ J : Set I, J.Countable ∧ ∀ i, A i ≤ᵐ[μ] (⋃ j ∈ J, A j) := by
+  let S : Set ENNReal := {m | ∃ J : Set I, J.Countable ∧ μ (⋃ j ∈ J, A j)=m}
+  have hs : S.Nonempty := ⟨0,∅,Set.countable_empty,by simp⟩
+  obtain ⟨f,_,ht,hf⟩ := exists_seq_tendsto_sSup hs (OrderTop.bddAbove S)
+  choose J hJ he using hf
+  let K : Set I := ⋃ n, J n
+  have hK : K.Countable := Set.countable_iUnion hJ
+  let D : Set X := ⋃ j ∈ K, A j
+  have hD : MeasurableSet D := MeasurableSet.biUnion hK (fun j _ => hm j)
+  have hS : μ D ∈ S := ⟨K,hK,rfl⟩
+  have hmax : μ D = sSup S := le_antisymm (le_sSup hS) (le_of_tendsto ht (Eventually.of_forall (fun n => by
+    rw [← he n]
+    apply measure_mono
+    intro x hx
+    rcases Set.mem_iUnion₂.mp hx with ⟨j,hj,hx⟩
+    exact Set.mem_iUnion₂.mpr ⟨j,Set.mem_iUnion.mpr ⟨n,hj⟩,hx⟩)))
+  refine ⟨K,hK,fun i => ?_⟩
+  have hun : μ (D ∪ A i) ≤ μ D := by
+    rw [hmax]
+    apply le_sSup
+    refine ⟨insert i K,hK.insert i,?_⟩
+    congr 1
+    ext x
+    simp only [Set.mem_iUnion,Set.mem_insert_iff,Set.mem_union,D]
+    aesop
+  have heq : μ (D ∪ A i) = μ D := le_antisymm hun (measure_mono Set.subset_union_left)
+  have hzero : μ ((D ∪ A i) \ D)=0 := by
+    rw [measure_sdiff Set.subset_union_left hD.nullMeasurableSet (measure_ne_top μ D),heq,tsub_self]
+  have hz : μ (A i \ D)=0 := measure_mono_null (by intro x hx; exact ⟨Or.inr hx.1,hx.2⟩) hzero
+  simpa only [ae_le_set] using hz
+end EuclideanFiveColor
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PlaneColoring/ContinuumLimits.lean b/lean/OAI/Geometry/PlaneColoring/ContinuumLimits.lean
new file mode 100644
index 0000000000000000000000000000000000000000..287c13a75787aab22a0c168ad01077cb9895beca
--- /dev/null
+++ b/lean/OAI/Geometry/PlaneColoring/ContinuumLimits.lean
@@ -0,0 +1,353 @@
+import OAI.Geometry.PlaneColoring.GraphRealization
+import OAI.Geometry.PlaneColoring.Interfaces
+import OAI.Geometry.PlaneColoring.Sectors
+
+namespace OAI
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+/-- Vanishing exceptional measure is enough: almost every point is outside the
+exceptional sets infinitely often. No summability or nesting is required. -/
+lemma ae_frequently_notMem_of_measure_tendsto_zero {X : Type*} [MeasurableSpace X]
+    {μ : Measure X} {E : ℕ → Set X}
+    (hE : Tendsto (fun n => μ (E n)) atTop (𝓝 0)) :
+    ∀ᵐ z ∂μ, ∃ᶠ n in atTop, z ∉ E n := by
+  let T : ℕ → Set X := fun N => ⋂ n ≥ N, E n
+  have hT : ∀ N, μ (T N) = 0 := by
+    intro N
+    apply le_antisymm _ (by positivity)
+    apply ge_of_tendsto hE
+    filter_upwards [eventually_ge_atTop N] with n hn
+    exact measure_mono (fun z hz => (mem_iInter.mp (mem_iInter.mp hz n)) hn)
+  have hn : μ (⋃ N, T N) = 0 := measure_iUnion_null hT
+  filter_upwards [measure_eq_zero_iff_ae_notMem.mp hn] with z hz
+  rw [← not_eventually]
+  intro he
+  obtain ⟨N,hN⟩ := eventually_atTop.mp he
+  exact hz (mem_iUnion.mpr ⟨N,mem_iInter.mpr fun n => mem_iInter.mpr (hN n)⟩)
+
+/-- Pointwise differentiation plus vanishing exceptional sets gives the measure
+exclusion used for both the mesh bound and the common continuum exclusions. -/
+lemma ae_ne_one_of_eventual_bound {X : Type*} [MeasurableSpace X]
+    {μ : Measure X} {p : ℕ → X → ℝ} {f : X → ℝ} {E : ℕ → Set X} {U : Set X}
+    {α : ℝ} (hα : 0 < α)
+    (hp : ∀ᵐ z ∂μ, Tendsto (fun n => p n z) atTop (𝓝 (f z)))
+    (hE : Tendsto (fun n => μ (E n)) atTop (𝓝 0))
+    (hb : ∀ z ∈ U, ∀ᶠ n in atTop, z ∉ E n → p n z ≤ 1-α) :
+    ∀ᵐ z ∂μ, z ∈ U → f z ≠ 1 := by
+  filter_upwards [hp,ae_frequently_notMem_of_measure_tendsto_zero hE] with z hz hgood hzU hf
+  have hgt : ∀ᶠ n in atTop, 1-α < p n z :=
+    hz.eventually (Ioi_mem_nhds (by rw [hf]; linarith))
+  obtain ⟨n,hn,hbnd,hlarge⟩ := (hgood.and_eventually ((hb z hzU).and hgt)).exists
+  exact (not_lt_of_ge (hbnd hn)) hlarge
+end EuclideanFiveColor
+end
+
+noncomputable section
+open scoped Classical
+open Set Filter Topology MeasureTheory MeasureTheory.Measure
+namespace EuclideanFiveColor
+lemma colorFn_locallyIntegrable {k : ℕ} {c : ℂ → Fin k} (hc : Measurable c) (i : Fin k) :
+    LocallyIntegrable (colorFn c i) (volume : Measure ℂ) :=
+  (continuous_const.locallyIntegrable (f := fun _ : ℂ => (1:ℝ))).mono
+    (colorFn_measurable hc i).aestronglyMeasurable (Eventually.of_forall fun z => by
+      simpa only [Real.norm_eq_abs,abs_of_nonneg (colorFn_nonneg c i z),abs_one] using colorFn_le_one c i z)
+lemma diskAverage_eq_average_ball {k : ℕ} (c : ℂ → Fin k) (ε : ℝ) (i : Fin k) (q : ℂ) :
+    diskAverage c ε i q = ⨍ z in Metric.ball q ε, colorFn c i z ∂volume := by
+  have he : (fun v : ℂ => q+v) ⁻¹' Metric.ball q ε = Metric.ball 0 ε := by
+    ext v
+    simp [Metric.mem_ball,dist_eq_norm]
+  have hi := (measurePreserving_add_left (volume : Measure ℂ) q).setIntegral_preimage_emb
+    (MeasurableEquiv.addLeft q).measurableEmbedding (colorFn c i) (Metric.ball q ε)
+  rw [he] at hi
+  unfold diskAverage diskLaw
+  rw [integral_smul_measure,ENNReal.toReal_inv,hi,setAverage_eq]
+  simp only [Measure.real,Complex.volume_ball]
+lemma complex_ball_ae_closedBall (q : ℂ) (ε : ℝ) :
+    Metric.ball q ε =ᵐ[volume] Metric.closedBall q ε :=
+  ae_eq_of_subset_of_measure_ge Metric.ball_subset_closedBall
+    (by rw [Complex.volume_closedBall,Complex.volume_ball])
+    measurableSet_ball.nullMeasurableSet ((isCompact_closedBall q ε).measure_ne_top)
+lemma diskAverage_ae_tendsto {k : ℕ} {c : ℂ → Fin k} (hc : Measurable c)
+    {ε : ℕ → ℝ} (hε : Tendsto ε atTop (𝓝 0)) (hpos : ∀ n,0<ε n) (i : Fin k) :
+    ∀ᵐ z : ℂ ∂volume, Tendsto (fun n => diskAverage c (ε n) i z) atTop (𝓝 (colorFn c i z)) := by
+  have hε' : Tendsto ε atTop (𝓝[>] 0) := tendsto_nhdsWithin_iff.mpr
+    ⟨hε,Eventually.of_forall hpos⟩
+  filter_upwards [IsUnifLocDoublingMeasure.ae_tendsto_average volume
+    (colorFn_locallyIntegrable hc i) 1] with z hz
+  have h := hz (fun _ : ℕ => z) ε hε' (Eventually.of_forall fun n => by
+    simp only [one_mul,Metric.mem_closedBall,dist_self]; exact (hpos n).le)
+  convert h using 1
+  ext n
+  rw [diskAverage_eq_average_ball]
+  exact setAverage_congr (complex_ball_ae_closedBall z (ε n))
+
+lemma support_disjoint_open_of_ae {k : ℕ} {c : ℂ → Fin k} (hc : Measurable c)
+    {U : Set ℂ} (hU : IsOpen U) (i : Fin k)
+    (h : ∀ᵐ z ∂(volume : Measure ℂ),z∈U → c z≠i) :
+    Disjoint (spatialSupport c i) U := by
+  have he : Uᶜ ∈ ae (volume.restrict {z | c z=i}) := by
+    change ∀ᵐ z ∂volume.restrict {z | c z=i},z∈Uᶜ
+    rw [ae_restrict_iff' (measurableSet_eq_fun hc measurable_const)]
+    filter_upwards [h] with z hz hczi hzU
+    exact hz hzU hczi
+  have hs := Measure.support_subset_of_isClosed hU.isClosed_compl he
+  exact Set.disjoint_left.mpr fun z hz hzU => hs hz hzU
+
+lemma diskAverage_support_exclusion {k : ℕ} {c : ℂ → Fin k} (hc : Measurable c)
+    {ε : ℕ → ℝ} (hε : Tendsto ε atTop (𝓝 0)) (hpos : ∀ n,0<ε n)
+    {E : ℕ → Set ℂ} (hE : Tendsto (fun n => volume (E n)) atTop (𝓝 0))
+    {α : ℝ} (hα : 0<α) {U : Set ℂ} (hU : IsOpen U) (i : Fin k)
+    (hb : ∀ z∈U,∀ᶠ n in atTop,z∉E n → diskAverage c (ε n) i z≤1-α) :
+    Disjoint (spatialSupport c i) U := by
+  apply support_disjoint_open_of_ae hc hU i
+  filter_upwards [ae_ne_one_of_eventual_bound hα (diskAverage_ae_tendsto hc hε hpos i) hE hb]
+    with z hz hzU hczi
+  exact hz hzU (by simp [colorFn,hczi])
+end EuclideanFiveColor
+end
+
+noncomputable section
+open scoped Classical
+open Set Filter Topology TopologicalSpace
+universe u v
+namespace EuclideanFiveColor
+/-- Connectedness passes to a Vietoris (equivalently compact Hausdorff metric) limit. -/
+theorem nonemptyCompacts_preconnected_limit {X : Type u} [TopologicalSpace X]
+    [T2Space X] [NormalSpace X] {ι : Type v} {l : Filter ι} [NeBot l]
+    {K : ι → NonemptyCompacts X} {L : NonemptyCompacts X}
+    (hlim : Tendsto K l (𝓝 L)) (hc : ∀ᶠn in l,IsPreconnected (K n : Set X)) :
+    IsPreconnected (L : Set X) := by
+  apply (isPreconnected_iff_subset_of_fully_disjoint_closed L.isCompact.isClosed).mpr
+  intro A B hA hB hcov hd
+  by_contra hh
+  push Not at hh
+  obtain ⟨x,hx,hxA⟩ := Set.not_subset.mp hh.1
+  obtain ⟨y,hy,hyB⟩ := Set.not_subset.mp hh.2
+  have hxB := (hcov hx).resolve_left hxA
+  have hyA := (hcov hy).resolve_right hyB
+  obtain ⟨U,V,hU,hV,hAU,hBV,hUV⟩ := normal_separation hA hB hd
+  have hL : (L : Set X)⊆U∪V := hcov.trans (union_subset_union hAU hBV)
+  have hs : ∀ᶠn in l,(K n : Set X)⊆U∪V := hlim.eventually
+    ((NonemptyCompacts.isOpen_subsets_of_isOpen (hU.union hV)).mem_nhds hL)
+  have hu : ∀ᶠn in l,((K n : Set X)∩U).Nonempty := hlim.eventually
+    ((NonemptyCompacts.isOpen_inter_nonempty_of_isOpen hU).mem_nhds ⟨y,hy,hAU hyA⟩)
+  have hv : ∀ᶠn in l,((K n : Set X)∩V).Nonempty := hlim.eventually
+    ((NonemptyCompacts.isOpen_inter_nonempty_of_isOpen hV).mem_nhds ⟨x,hx,hBV hxB⟩)
+  obtain ⟨n,hnc,hns,hnu,hnv⟩ := (hc.and (hs.and (hu.and hv))).exists
+  obtain ⟨z,_,hzU,hzV⟩ := hnc U V hU hV hns hnu hnv
+  exact Set.disjoint_left.mp hUV hzU hzV
+
+lemma nonemptyCompacts_mem_limit {X : Type u} [MetricSpace X]
+    {ι : Type v} {l : Filter ι} [NeBot l]
+    {K : ι → NonemptyCompacts X} {L : NonemptyCompacts X} {x : ι → X} {y : X}
+    (hK : Tendsto K l (𝓝 L)) (hx : Tendsto x l (𝓝 y))
+    (hm : ∀ᶠn in l,x n∈K n) : y∈L := by
+  have hh := (NonemptyCompacts.lipschitz_infDist.continuous.tendsto (y,L)).comp (hx.prodMk_nhds hK)
+  have h0 : (fun n => Metric.infDist (x n) (K n : Set X)) =ᶠ[l] fun _ => 0 :=
+    hm.mono fun n hn => Metric.infDist_zero_of_mem hn
+  have he : Metric.infDist y (L : Set X)=0 := tendsto_nhds_unique hh
+    (tendsto_const_nhds.congr' h0.symm)
+  exact (L.isCompact.isClosed.mem_iff_infDist_zero L.nonempty).mpr he
+
+lemma nonemptyCompacts_inter_closed_limit {X : Type u} [TopologicalSpace X]
+    {ι : Type v} {l : Filter ι} [NeBot l]
+    {K : ι → NonemptyCompacts X} {L : NonemptyCompacts X} {F : Set X}
+    (hF : IsClosed F) (hK : Tendsto K l (𝓝 L))
+    (hm : ∀ᶠn in l,((K n : Set X)∩F).Nonempty) : ((L : Set X)∩F).Nonempty :=
+  (NonemptyCompacts.isClosed_inter_nonempty_of_isClosed hF).mem_of_tendsto hK hm
+
+/-- A finite family of nonempty compact continua in a compact metric space has
+one common subsequence of connected compact limits. -/
+theorem finite_continuum_subsequence {X : Type u} [MetricSpace X] [CompactSpace X]
+    {J : Type v} [Finite J] (K : ℕ → J → NonemptyCompacts X)
+    (hc : ∀n j,IsPreconnected (K n j : Set X)) :
+    ∃L : J → NonemptyCompacts X, ∃φ : ℕ → ℕ, StrictMono φ ∧
+      (∀j,Tendsto (fun n => K (φ n) j) atTop (𝓝 (L j))) ∧
+      (∀j,IsPreconnected (L j : Set X)) := by
+  obtain ⟨L,_,φ,hφ,hlim⟩ := (isCompact_univ : IsCompact (Set.univ : Set (J → NonemptyCompacts X))).tendsto_subseq (fun n => Set.mem_univ (K n))
+  refine ⟨L,φ,hφ,?_,?_⟩
+  · intro j
+    exact (continuous_apply j).continuousAt.tendsto.comp hlim
+  · intro j
+    exact nonemptyCompacts_preconnected_limit
+      ((continuous_apply j).continuousAt.tendsto.comp hlim)
+      (Eventually.of_forall fun n => hc (φ n) j)
+end EuclideanFiveColor
+end
+
+noncomputable section
+open scoped Classical
+open Set Filter Topology MeasureTheory TopologicalSpace
+namespace EuclideanFiveColor
+lemma straddle_eventually_of_compact_limit {K : ℕ → NonemptyCompacts ℂ}
+    {L : NonemptyCompacts ℂ} (hlim : Tendsto K atTop (𝓝 L)) {z : ℂ}
+    (hz : z∈Straddle (L : Set ℂ)) : ∀ᶠ n in atTop,z∈Straddle (K n : Set ℂ) := by
+  obtain ⟨p,hp,q,hq,hp1,hq1⟩ := hz
+  have ha : ∀ᶠn in atTop,((K n : Set ℂ)∩{p : ℂ | dist z p<1}).Nonempty :=
+    hlim.eventually ((NonemptyCompacts.isOpen_inter_nonempty_of_isOpen
+      (isOpen_lt (continuous_const.dist continuous_id) continuous_const)).mem_nhds ⟨p,hp,hp1⟩)
+  have hb : ∀ᶠn in atTop,((K n : Set ℂ)∩{q : ℂ | 1<dist z q}).Nonempty :=
+    hlim.eventually ((NonemptyCompacts.isOpen_inter_nonempty_of_isOpen
+      (isOpen_lt continuous_const (continuous_const.dist continuous_id))).mem_nhds ⟨q,hq,hq1⟩)
+  filter_upwards [ha,hb] with n hn hm
+  obtain ⟨p,hp,hp1⟩ := hn
+  obtain ⟨q,hq,hq1⟩ := hm
+  exact ⟨p,hp,q,hq,hp1,hq1⟩
+/-- The common-exclusion continuum limit argument. The sole exceptional sets are
+actual unit neighborhoods of the holes, and their measures tend to zero. -/
+theorem continuum_limit_support_exclusion {k : ℕ} {c : ℂ → Fin k}
+    (hc : Measurable c) (hw : WeakColoring k c)
+    {K : ℕ → NonemptyCompacts ℂ} {L : NonemptyCompacts ℂ}
+    (hlim : Tendsto K atTop (𝓝 L)) (hconn : ∀n,IsPreconnected (K n : Set ℂ))
+    {ε : ℕ → ℝ} (hε : Tendsto ε atTop (𝓝 0)) (hpos : ∀n,0<ε n)
+    {H E : ℕ → Set ℂ} (hE : Tendsto (fun n => volume (E n)) atTop (𝓝 0))
+    (hnear : ∀n z q,q∈H n → dist z q=1 → z∈E n)
+    {α : ℝ} (hα : 0<α) (i : Fin k)
+    (hprob : ∀n q,q∈K n → q∉H n → α<diskAverage c (ε n) i q) :
+    Disjoint (spatialSupport c i) (Straddle (L : Set ℂ)) := by
+  apply diskAverage_support_exclusion hc hε hpos hE hα (straddle_isOpen _) i
+  intro z hz
+  filter_upwards [straddle_eventually_of_compact_limit hlim hz] with n hn hzE
+  obtain ⟨q,hq,hunit⟩ := straddle_connected_unit (hconn n) hn
+  have hqH : q∉H n := fun hh => hzE (hnear n z q hh hunit)
+  have hb := hprob n q hq hqH
+  have hu : ‖q-z‖=1 := by simpa only [dist_eq_norm,norm_sub_rev] using hunit
+  have hh := diskAverage_unit_pair hc hw (hpos n) hu i
+  linarith
+end EuclideanFiveColor
+end
+
+noncomputable section
+open scoped Classical
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+/-- One interface excludes both labels from the same straddling set and approaches
+its center in an actual unit direction. -/
+def CommonInterface (c : ℂ → Fin 5) (x : ℂ) (i j : Fin 5) : Prop :=
+  ∃K : Set ℂ, ∃v : ℂ, ‖v‖=1 ∧ Nonempty (ApproachingContinuum x K v) ∧
+    Disjoint (spatialSupport c i) (Straddle K) ∧ Disjoint (spatialSupport c j) (Straddle K)
+lemma CommonInterface.symm {c : ℂ → Fin 5} {x : ℂ} {i j : Fin 5}
+    (h : CommonInterface c x i j) : CommonInterface c x j i := by
+  obtain ⟨K,v,hv,hK,hi,hj⟩ := h
+  exact ⟨K,v,hv,hK,hj,hi⟩
+lemma spatialSupport_perm (c : ℂ → Fin 5) (e : Equiv.Perm (Fin 5)) (i : Fin 5) :
+    spatialSupport (e.symm ∘ c) i=spatialSupport c (e i) := by
+  have he : {z : ℂ | (e.symm ∘ c) z=i}={z : ℂ | c z=e i} := by
+    ext z
+    exact e.symm_apply_eq
+  simp only [spatialSupport,he]
+lemma CommonInterface.perm {c : ℂ → Fin 5} {x : ℂ} (e : Equiv.Perm (Fin 5))
+    {i j : Fin 5} (h : CommonInterface c x (e i) (e j)) :
+    CommonInterface (e.symm ∘ c) x i j := by
+  obtain ⟨K,v,hv,hK,hi,hj⟩ := h
+  exact ⟨K,v,hv,hK,by rwa [spatialSupport_perm],by rwa [spatialSupport_perm]⟩
+lemma commonInterface_of_continuum {c : ℂ → Fin 5} {x y : ℂ} {i j : Fin 5}
+    {K : Set ℂ} (hK : IsPreconnected K) (hx : x∈K) (hy : y∈K) (hne : y≠x)
+    (hi : Disjoint (spatialSupport c i) (Straddle K))
+    (hj : Disjoint (spatialSupport c j) (Straddle K)) : CommonInterface c x i j := by
+  obtain ⟨v,hv,ha⟩ := continuum_has_approach hK hx hy hne
+  exact ⟨K,v,hv,ha,hi,hj⟩
+lemma no_triangle_commonInterface {c : ℂ → Fin 5} (hc : Measurable c)
+    (hw : WeakColoring 5 c) {x : ℂ} (h01 : CommonInterface c x 0 1)
+    (h12 : CommonInterface c x 1 2) (h20 : CommonInterface c x 2 0) : False := by
+  let a : Fin 3 → Fin 5 := ![0,1,2]
+  let b : Fin 3 → Fin 5 := ![1,2,0]
+  have h (i : Fin 3) : CommonInterface c x (a i) (b i) := by
+    fin_cases i
+    · exact h01
+    · exact h12
+    · exact h20
+  choose K v hv hK hA hB using h
+  apply no_triangle_interfaces hc hw hv (fun i => Classical.choice (hK i))
+  intro i j hj
+  have he : triangleEdge i={a i,b i} := by fin_cases i <;> decide
+  rw [he] at hj
+  rcases Finset.mem_insert.mp hj with rfl|hj
+  · exact hA i
+  · rw [Finset.mem_singleton] at hj; subst j; exact hB i
+lemma no_four_commonInterface {c : ℂ → Fin 5} (hc : Measurable c)
+    (hw : WeakColoring 5 c) {x : ℂ} (h01 : CommonInterface c x 0 1)
+    (h12 : CommonInterface c x 1 2) (h23 : CommonInterface c x 2 3)
+    (h30 : CommonInterface c x 3 0) : False := by
+  let a : Fin 4 → Fin 5 := ![0,1,2,3]
+  let b : Fin 4 → Fin 5 := ![1,2,3,0]
+  have h (i : Fin 4) : CommonInterface c x (a i) (b i) := by
+    fin_cases i
+    · exact h01
+    · exact h12
+    · exact h23
+    · exact h30
+  choose K v hv hK hA hB using h
+  apply no_four_interfaces hc hw hv (fun i => Classical.choice (hK i))
+  intro i j hj
+  have he : fourEdge i={a i,b i} := by fin_cases i <;> decide
+  rw [he] at hj
+  rcases Finset.mem_insert.mp hj with rfl|hj
+  · exact hA i
+  · rw [Finset.mem_singleton] at hj; subst j; exact hB i
+lemma no_five_commonInterface {c : ℂ → Fin 5} (hc : Measurable c)
+    (hw : WeakColoring 5 c) {x : ℂ} (h : ∀i,CommonInterface c x i (i+1)) : False := by
+  choose K v hv hK hi hj using h
+  exact no_five_exclusion_continua c hc hw hv (fun i => Classical.choice (hK i)) (fun i => ⟨hi i,hj i⟩)
+/-- All actual common-exclusion interfaces at a point form a forest. This
+assembles triangle, four and five cases for arbitrary labels, not just canonical cycles. -/
+theorem commonInterface_graph_acyclic {c : ℂ → Fin 5} (hc : Measurable c)
+    (hw : WeakColoring 5 c) (x : ℂ) (G : SimpleGraph (Fin 5))
+    (hG : ∀i j,G.Adj i j → CommonInterface c x i j) : G.IsAcyclic := by
+  intro v p hp
+  obtain ⟨e,he⟩ := cycle_standard_permutation hp
+  have hm : Measurable (e.symm ∘ c) := (measurable_of_finite e.symm).comp hc
+  have hw' := weak_relabel c e.symm hw
+  have h (i j : Fin 5) (hij : G.Adj (e i) (e j)) : CommonInterface (e.symm ∘ c) x i j :=
+    (hG _ _ hij).perm e
+  rcases he with ⟨h01,h02,h12⟩|⟨h01,h12,h23,h30⟩|hfive
+  · exact no_triangle_commonInterface hm hw' (h _ _ h01) (h _ _ h12) (h _ _ h02).symm
+  · exact no_four_commonInterface hm hw' (h _ _ h01) (h _ _ h12) (h _ _ h23) (h _ _ h30)
+  · exact no_five_commonInterface hm hw' (fun i => h _ _ (hfive i))
+end EuclideanFiveColor
+end
+
+noncomputable section
+open scoped Classical
+open Set Filter Topology TopologicalSpace MeasureTheory
+namespace EuclideanFiveColor
+/-- End-to-end limit of actual connected midpoint fibers, retaining both endpoint
+exclusions on the same limiting continuum. -/
+theorem commonInterface_of_compact_sequence {c : ℂ → Fin 5}
+    (hc : Measurable c) (hw : WeakColoring 5 c) {x : ℂ} {ρ : ℝ} (hρ : 0<ρ)
+    (K : ℕ → NonemptyCompacts ℂ) (hsub : ∀n,(K n : Set ℂ)⊆Metric.closedBall x ρ)
+    (hconn : ∀n,IsPreconnected (K n : Set ℂ))
+    (a : ℕ → ℂ) (ha : ∀n,a n∈K n) (halim : Tendsto a atTop (𝓝 x))
+    (hout : ∀n,((K n : Set ℂ)∩Metric.sphere x ρ).Nonempty)
+    {ε : ℕ → ℝ} (hε : Tendsto ε atTop (𝓝 0)) (hpos : ∀n,0<ε n)
+    {H E : ℕ → Set ℂ} (hE : Tendsto (fun n => volume (E n)) atTop (𝓝 0))
+    (hnear : ∀n z q,q∈H n → dist z q=1 → z∈E n)
+    {α : ℝ} (hα : 0<α) (i j : Fin 5)
+    (hi : ∀n q,q∈K n → q∉H n → α<diskAverage c (ε n) i q)
+    (hj : ∀n q,q∈K n → q∉H n → α<diskAverage c (ε n) j q) :
+    CommonInterface c x i j := by
+  obtain ⟨L,hL,φ,hφ,hlim⟩ := (NonemptyCompacts.isCompact_subsets_of_isCompact
+    (isCompact_closedBall x ρ)).tendsto_subseq hsub
+  have hx : x∈L := nonemptyCompacts_mem_limit hlim (halim.comp hφ.tendsto_atTop)
+    (Eventually.of_forall fun n => ha (φ n))
+  obtain ⟨y,hy,hyρ⟩ := nonemptyCompacts_inter_closed_limit Metric.isClosed_sphere hlim
+    (Eventually.of_forall fun n => hout (φ n))
+  have hne : y≠x := by
+    intro he
+    subst y
+    have he : (0:ℝ)=ρ := by simpa only [Metric.mem_sphere,dist_self] using hyρ
+    linarith
+  apply commonInterface_of_continuum
+    (nonemptyCompacts_preconnected_limit hlim (Eventually.of_forall fun n => hconn (φ n))) hx hy hne
+  · exact continuum_limit_support_exclusion hc hw hlim (fun n => hconn (φ n))
+      (hε.comp hφ.tendsto_atTop) (fun n => hpos (φ n)) (hE.comp hφ.tendsto_atTop)
+      (fun n => hnear (φ n)) hα i (fun n => hi (φ n))
+  · exact continuum_limit_support_exclusion hc hw hlim (fun n => hconn (φ n))
+      (hε.comp hφ.tendsto_atTop) (fun n => hpos (φ n)) (hE.comp hφ.tendsto_atTop)
+      (fun n => hnear (φ n)) hα j (fun n => hj (φ n))
+end EuclideanFiveColor
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PlaneColoring/Covering.lean b/lean/OAI/Geometry/PlaneColoring/Covering.lean
new file mode 100644
index 0000000000000000000000000000000000000000..4cd16782b61ab68a3f572e04839dd32a8511323c
--- /dev/null
+++ b/lean/OAI/Geometry/PlaneColoring/Covering.lean
@@ -0,0 +1,97 @@
+import Mathlib
+
+namespace OAI
+
+noncomputable section
+
+namespace Problem160
+
+private theorem triangular_wedge_bound (s x y : ℝ)
+    (hs : 0 < s) (hsq : s ^ 2 = 3)
+    (hx0 : 0 ≤ x) (hx1 : x ≤ 1 / 2)
+    (hy0 : 0 ≤ y) (hxy : s * y ≤ 1 - x) :
+    x ^ 2 + y ^ 2 ≤ 1 / 3 := by
+  have hnonneg : 0 ≤ (1 - x) - s * y := by linarith
+  have hplus : 0 ≤ (1 - x) + s * y := by
+    nlinarith [mul_nonneg (le_of_lt hs) hy0]
+  have hprod := mul_nonneg hnonneg hplus
+  have hxprod := mul_nonneg hx0 (show 0 ≤ 1 / 2 - x by linarith)
+  nlinarith [sq_nonneg (s * y), sq_nonneg y]
+
+private theorem triangular_rectangle_cover (s x y : ℝ)
+    (hs : 0 < s) (hsq : s ^ 2 = 3)
+    (hx0 : 0 ≤ x) (hx1 : x ≤ 1)
+    (hy0 : 0 ≤ y) (hy1 : y ≤ s / 2) :
+    x ^ 2 + y ^ 2 ≤ 1 / 3 ∨
+    (x - 1) ^ 2 + y ^ 2 ≤ 1 / 3 ∨
+    (x - 1 / 2) ^ 2 + (y - s / 2) ^ 2 ≤ 1 / 3 := by
+  by_cases hx : x ≤ 1 / 2
+  · by_cases hxy : s * y ≤ 1 - x
+    · exact Or.inl (triangular_wedge_bound s x y hs hsq hx0 hx hy0 hxy)
+    · right; right
+      have h := triangular_wedge_bound s (1 / 2 - x) (s / 2 - y) hs hsq
+        (by linarith) (by linarith) (by linarith) (by nlinarith)
+      nlinarith
+  · by_cases hxy : s * y ≤ x
+    · right; left
+      have h := triangular_wedge_bound s (1 - x) y hs hsq
+        (by linarith) (by linarith) hy0 (by linarith)
+      nlinarith
+    · right; right
+      have h := triangular_wedge_bound s (x - 1 / 2) (s / 2 - y) hs hsq
+        (by linarith) (by linarith) (by linarith) (by nlinarith)
+      nlinarith
+
+/-- Every point in the plane is within squared distance `1/3` of the
+unit triangular lattice. The proof gives one of three explicit nearby centers. -/
+theorem triangular_lattice_cover (x y : ℝ) :
+    ∃ m n : ℤ,
+      (x - ((m : ℝ) + (n : ℝ) / 2)) ^ 2 +
+        (y - Real.sqrt 3 * (n : ℝ) / 2) ^ 2 ≤ 1 / 3 := by
+  let s : ℝ := Real.sqrt 3
+  have hs : 0 < s := Real.sqrt_pos.2 (by norm_num)
+  have hsq : s ^ 2 = 3 := Real.sq_sqrt (by norm_num)
+  let n : ℤ := ⌊2 * y / s⌋
+  let m : ℤ := ⌊x - (n : ℝ) / 2⌋
+  let u : ℝ := x - ((m : ℝ) + (n : ℝ) / 2)
+  let v : ℝ := y - s * (n : ℝ) / 2
+  have hn0 : (n : ℝ) ≤ 2 * y / s := Int.floor_le _
+  have hn1 : 2 * y / s < (n : ℝ) + 1 := Int.lt_floor_add_one _
+  have hm0 : (m : ℝ) ≤ x - (n : ℝ) / 2 := Int.floor_le _
+  have hm1 : x - (n : ℝ) / 2 < (m : ℝ) + 1 := Int.lt_floor_add_one _
+  have hu0 : 0 ≤ u := by dsimp [u]; linarith
+  have hu1 : u ≤ 1 := by dsimp [u]; linarith
+  have hv0 : 0 ≤ v := by
+    have h := (le_div_iff₀ hs).mp hn0
+    dsimp [v]
+    nlinarith
+  have hv1 : v ≤ s / 2 := by
+    have h := (div_lt_iff₀ hs).mp hn1
+    dsimp [v]
+    nlinarith
+  rcases triangular_rectangle_cover s u v hs hsq hu0 hu1 hv0 hv1 with h | h | h
+  · exact ⟨m, n, h⟩
+  · refine ⟨m + 1, n, ?_⟩
+    have hu : x - (((m + 1 : ℤ) : ℝ) + (n : ℝ) / 2) = u - 1 := by
+      dsimp [u]
+      push_cast
+      ring
+    rw [hu]
+    exact h
+  · refine ⟨m, n + 1, ?_⟩
+    have hu : x - ((m : ℝ) + ((n + 1 : ℤ) : ℝ) / 2) = u - 1 / 2 := by
+      dsimp [u]
+      push_cast
+      ring
+    have hv : y - Real.sqrt 3 * ((n + 1 : ℤ) : ℝ) / 2 = v - s / 2 := by
+      dsimp [v, s]
+      push_cast
+      ring
+    rw [hu, hv]
+    exact h
+
+end Problem160
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PlaneColoring/CyclicCore.lean b/lean/OAI/Geometry/PlaneColoring/CyclicCore.lean
new file mode 100644
index 0000000000000000000000000000000000000000..487f6e084e421b6abd53b57addd4e6dc25c0550e
--- /dev/null
+++ b/lean/OAI/Geometry/PlaneColoring/CyclicCore.lean
@@ -0,0 +1,280 @@
+import OAI.Geometry.PlaneColoring.FiveSupport
+
+namespace OAI
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+variable (c : ℂ → Fin 5) (hc : Measurable c)
+lemma five_ae_mem_support (x : ℂ) :
+    ∀ᵐt : ℝ∂volume,t∈fiveSupport c hc x (fiveState c hc x (t : Angles)) :=
+  ae_state_support ((fiveState_measurable c hc x).comp angleMk_quasiMeasurePreserving.measurable)
+lemma five_real_diag (hw : WeakColoring 5 c) {x : ℂ} (hx : FiveCenter c x) :
+    ∀ᵐt : ℝ∂volume,palette c hc x (t : Angles)=fiveDiag (fiveState c hc x (t : Angles)) :=
+  angleMk_quasiMeasurePreserving.ae (fiveState_diag c hc hw hx)
+lemma five_local_palette (hw : WeakColoring 5 c) {x : ℂ} (hx : FiveCenter c x) {a : ℝ} {k : Fin 5}
+    (hk : ∀i,a∈fiveSupport c hc x i → k∈fiveDiag i) :
+    ∃V, IsOpen V ∧ a∈V ∧ ∀ᵐt : ℝ∂volume,t∈V → (t : Angles)∈paletteSet c hc x k := by
+  obtain ⟨V,hV,ha,hkV⟩ := stateSupport_local_pred
+    ((fiveState_measurable c hc x).comp angleMk_quasiMeasurePreserving.measurable) hk
+  refine ⟨V,hV,ha,?_⟩
+  filter_upwards [hkV,five_real_diag c hc hw hx] with t ht hd htv
+  apply (mem_palette c hc x (t : Angles) k).mp
+  rw [hd]
+  exact ht htv
+lemma five_endpoint_no_common (hw : WeakColoring 5 c) {x : ℂ} (hx : FiveCenter c x) {a b : ℝ}
+    (hab : Real.sin (b-a)≠0) (hy : FiveCenter c (x+direction (a : Angles)+direction (b : Angles))) :
+    ∀k : Fin 5,∃i,a∈fiveSupport c hc x i ∧ k∉fiveDiag i := by
+  intro k
+  by_contra! h
+  obtain ⟨V,hV,ha,hpal⟩ := five_local_palette c hc hw hx h
+  obtain ⟨U,hU,hyU,hnull⟩ := palette_open_exclusion c hc hw hab hV ha hpal
+  exact transitionEdge_no_open_exclusion c (hy k) hU hyU hnull
+lemma five_limit_contact {x : ℂ} {an : ℕ → ℝ} {a : ℝ} (ha : Tendsto an atTop (𝓝 a))
+    (hn : ∀n,∀k : Fin 5,∃i,an n∈fiveSupport c hc x i ∧ k∉fiveDiag i) :
+    ∃i,a∈fiveSupport c hc x i ∧ a∈fiveSupport c hc x (i+1) := by
+  have hall : ∀k : Fin 5,∃i,a∈fiveSupport c hc x i ∧ k∉fiveDiag i := fun k =>
+    finite_closed_family_no_common_limit (fun _ => Measure.isClosed_support) ha (fun n => hn n k)
+  have hnList : (fiveSupportList c hc x a).Nonempty := by
+    obtain ⟨i,hi⟩ := five_support_cover c hc x a
+    exact ⟨i,(mem_fiveSupportList _ _ _ _ _).mpr hi⟩
+  have hp : ∀k,∃i∈fiveSupportList c hc x a,k∉fiveDiag i := by
+    intro k; obtain ⟨i,hi,hk⟩ := hall k
+    exact ⟨i,(mem_fiveSupportList _ _ _ _ _).mpr hi,hk⟩
+  obtain ⟨i,hi,hi'⟩ := FiveComb.adjacent_oriented _ (five_no_common_adj _ hnList hp)
+  exact ⟨i,(mem_fiveSupportList _ _ _ _ _).mp hi,(mem_fiveSupportList _ _ _ _ _).mp hi'⟩
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology
+namespace EuclideanFiveColor
+lemma chord_angles_sum {z : ℂ} (hz : ‖z‖≤2) :
+    Complex.exp (((z.arg-Real.arccos (‖z‖/2) : ℝ) : ℂ)*Complex.I)+
+      Complex.exp (((z.arg+Real.arccos (‖z‖/2) : ℝ) : ℂ)*Complex.I)=z := by
+  have hc := Real.cos_arccos (x := ‖z‖/2) (by have := norm_nonneg z; linarith) (by linarith)
+  have hre := Complex.norm_mul_cos_arg z
+  have him := Complex.norm_mul_sin_arg z
+  rw [Complex.exp_mul_I,Complex.exp_mul_I]
+  apply Complex.ext
+  · simp only [Complex.add_re,Complex.mul_re,Complex.I_re,Complex.I_im,
+      Complex.cos_ofReal_re,Complex.sin_ofReal_re,Complex.sin_ofReal_im,mul_zero,
+      sub_zero,mul_one]
+    rw [Real.cos_sub,Real.cos_add,hc]
+    nlinarith
+  · simp only [Complex.add_im,Complex.mul_im,Complex.I_re,Complex.I_im,
+      Complex.cos_ofReal_im,Complex.sin_ofReal_re,Complex.sin_ofReal_im,mul_zero,
+      zero_add,add_zero,mul_one]
+    rw [Real.sin_sub,Real.sin_add,hc]
+    nlinarith
+lemma chord_angles_nonparallel {r t : ℝ} (hr : 0<r) (hr2 : r<2) :
+    Real.sin ((t+Real.arccos (r/2))-(t-Real.arccos (r/2)))≠0 := by
+  have ha := Real.arccos_pos.mpr (show r/2<1 by linarith)
+  have hb := Real.arccos_lt_pi_div_two.mpr (show 0<r/2 by linarith)
+  exact (Real.sin_pos_of_pos_of_lt_pi (by linarith) (by linarith)).ne'
+lemma chord_sequence_lifts {z : ℕ → ℂ} (hz : Tendsto z atTop (𝓝 0))
+    (hne : ∀n,z n≠0) (hsmall : ∀n,‖z n‖<2) :
+    ∃ φ : ℕ → ℕ, StrictMono φ ∧ ∃ t : ℝ,∃ a b : ℕ → ℝ,
+      Tendsto a atTop (𝓝 (t-Real.pi/2)) ∧ Tendsto b atTop (𝓝 (t+Real.pi/2)) ∧
+      (∀n,Complex.exp ((a n : ℂ)*Complex.I)+Complex.exp ((b n : ℂ)*Complex.I)=z (φ n)) ∧
+      ∀n,Real.sin (b n-a n)≠0 := by
+  obtain ⟨t,_ht,φ,hφ,harg⟩ := (isCompact_Icc : IsCompact (Icc (-Real.pi) Real.pi)).tendsto_subseq
+    (fun n => ⟨(Complex.neg_pi_lt_arg (z n)).le,Complex.arg_le_pi (z n)⟩)
+  have hn : Tendsto (fun n => ‖z (φ n)‖/2) atTop (𝓝 0) := by
+    simpa only [Function.comp_def,norm_zero,zero_div] using ((hz.comp hφ.tendsto_atTop).norm.div_const 2)
+  have hac : Tendsto (fun n => Real.arccos (‖z (φ n)‖/2)) atTop (𝓝 (Real.pi/2)) := by
+    simpa only [Function.comp_def,Real.arccos_zero] using Real.continuous_arccos.continuousAt.tendsto.comp hn
+  refine ⟨φ,hφ,t,(fun n => (z (φ n)).arg-Real.arccos (‖z (φ n)‖/2)),
+    (fun n => (z (φ n)).arg+Real.arccos (‖z (φ n)‖/2)),harg.sub hac,harg.add hac,?_,?_⟩
+  · exact fun n => chord_angles_sum (hsmall (φ n)).le
+  · exact fun n => chord_angles_nonparallel (norm_pos_iff.mpr (hne (φ n))) (hsmall (φ n))
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+variable (c : ℂ → Fin 5) (hc : Measurable c)
+include hc in
+lemma five_sequence_small_no_accumulation (hw : WeakColoring 5 c)
+    {yn : ℕ → ℂ} {x : ℂ} (hyn : ∀n,FiveCenter c (yn n)) (hx : FiveCenter c x)
+    (hy : Tendsto yn atTop (𝓝 x)) (hne : ∀n,yn n≠x) (hsmall : ∀n,‖yn n-x‖<2) : False := by
+  obtain ⟨φ,hφ,t,a,b,ha,hb,hs,hab⟩ := chord_sequence_lifts
+    (by simpa only [sub_self] using hy.sub_const x) (fun n => sub_ne_zero.mpr (hne n)) hsmall
+  have hcSum (n : ℕ) : x+direction (a n : Angles)+direction (b n : Angles)=yn (φ n) := by
+    rw [direction_real,direction_real,Circle.coe_exp,Circle.coe_exp,add_assoc,hs]
+    abel
+  have haC : ∀n,∀k : Fin 5,∃i,a n∈fiveSupport c hc x i ∧ k∉fiveDiag i := by
+    intro n
+    apply five_endpoint_no_common c hc hw hx (hab n)
+    rw [hcSum]; exact hyn _
+  have hbC : ∀n,∀k : Fin 5,∃i,b n∈fiveSupport c hc x i ∧ k∉fiveDiag i := by
+    intro n
+    apply five_endpoint_no_common c hc hw hx
+    · rw [show a n-b n=-(b n-a n) by ring,Real.sin_neg]
+      exact neg_ne_zero.mpr (hab n)
+    · rw [add_right_comm,hcSum]; exact hyn _
+  obtain ⟨i,hi,hi'⟩ := five_limit_contact c hc ha haC
+  obtain ⟨j,hj,hj'⟩ := five_limit_contact c hc hb hbC
+  exact five_opposite_contacts_false c hc hw hx (by dsimp [sixty]; ring) hi hi' hj hj'
+include hc in
+lemma five_sequence_no_accumulation (hw : WeakColoring 5 c)
+    {yn : ℕ → ℂ} {x : ℂ} (hyn : ∀n,FiveCenter c (yn n)) (hx : FiveCenter c x)
+    (hy : Tendsto yn atTop (𝓝 x)) (hne : ∀n,yn n≠x) : False := by
+  have hsmall : ∀ᶠn in atTop,‖yn n-x‖<2 := by
+    have hn : Tendsto (fun n => ‖yn n-x‖) atTop (𝓝 0) := by
+      simpa only [sub_self,norm_zero] using (hy.sub_const x).norm
+    exact hn.eventually (gt_mem_nhds (by norm_num : (0 : ℝ)<2))
+  obtain ⟨N,hN⟩ := eventually_atTop.mp hsmall
+  exact five_sequence_small_no_accumulation c hc hw (fun n => hyn (n+N)) hx
+    (hy.comp (tendsto_add_atTop_nat N)) (fun n => hne (n+N)) (fun n => hN (n+N) (Nat.le_add_left _ _))
+lemma fiveCenter_closed : IsClosed {x | FiveCenter c x} := by
+  change IsClosed {x | ∀i : Fin 5,TransitionEdge c i (i+1) x}
+  simp only [Set.ofPred_forall]
+  exact isClosed_iInter fun i => transitionEdge_closed c i (i+1)
+include hc in
+lemma fiveCenter_discrete (hw : WeakColoring 5 c) : IsDiscrete {x | FiveCenter c x} := by
+  apply isDiscrete_iff_discreteTopology.mpr
+  apply discreteTopology_of_noAccPts
+  intro x hx ha
+  obtain ⟨yn,hyn,hy⟩ := mem_closure_iff_seq_limit.mp
+    (accPt_principal_iff_clusterPt.mp ha).mem_closure
+  exact five_sequence_no_accumulation c hc hw (fun n => (hyn n).1) hx hy
+    (fun n => by simpa only [mem_singleton_iff] using (hyn n).2)
+include hc in
+lemma fiveCenter_compact_finite (hw : WeakColoring 5 c) {K : Set ℂ} (hK : IsCompact K) :
+    (K∩{x | FiveCenter c x}).Finite :=
+  (hK.inter_right (fiveCenter_closed c)).finite ((fiveCenter_discrete c hc hw).mono inter_subset_right)
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set
+namespace EuclideanFiveColor
+lemma cycle_standard_permutation {G : SimpleGraph (Fin 5)} {v : Fin 5}
+    {p : G.Walk v v} (hp : p.IsCycle) : ∃e : Equiv.Perm (Fin 5),
+    (G.Adj (e 0) (e 1) ∧ G.Adj (e 0) (e 2) ∧ G.Adj (e 1) (e 2)) ∨
+    (G.Adj (e 0) (e 1) ∧ G.Adj (e 1) (e 2) ∧ G.Adj (e 2) (e 3) ∧ G.Adj (e 3) (e 0)) ∨
+    (∀i : Fin 5,G.Adj (e i) (e (i+1))) := by
+  have hl := hp.three_le_length
+  have hinj : Function.Injective (fun i : Fin p.length => p.getVert i) := by
+    intro i j h
+    apply Fin.ext
+    exact hp.getVert_injOn' (by change i.val≤p.length-1; omega)
+      (by change j.val≤p.length-1; omega) h
+  have hle : p.length≤5 := by
+    simpa only [Fintype.card_fin] using Fintype.card_le_of_injective _ hinj
+  obtain ⟨e,he⟩ := Equiv.Perm.exists_extending_pair (Fin.castLE hle)
+    (fun i : Fin p.length => p.getVert i) (Fin.castLE_injective hle) hinj
+  have hem (n : ℕ) (hn : n<p.length) : e ⟨n,lt_of_lt_of_le hn hle⟩=p.getVert n := he ⟨n,hn⟩
+  have h0 : e 0=p.getVert 0 := hem 0 (by omega)
+  have h1 : e 1=p.getVert 1 := hem 1 (by omega)
+  have h2 : e 2=p.getVert 2 := hem 2 (by omega)
+  have h01 : G.Adj (e 0) (e 1) := by rw [h0,h1]; exact p.adj_getVert_succ (by omega)
+  have h12 : G.Adj (e 1) (e 2) := by rw [h1,h2]; exact p.adj_getVert_succ (by omega)
+  refine ⟨e,?_⟩
+  rcases (by omega : p.length=3 ∨ p.length=4 ∨ p.length=5) with hL|hL|hL
+  · left
+    refine ⟨h01,?_,h12⟩
+    have h20 := p.adj_getVert_succ (i := 2) (by omega)
+    have hlast : p.getVert 3=p.getVert 0 := by simpa only [hL,SimpleGraph.Walk.getVert_zero] using p.getVert_length
+    rw [hlast,← h2,← h0] at h20
+    exact h20.symm
+  · right; left
+    have h3 : e 3=p.getVert 3 := hem 3 (by omega)
+    refine ⟨h01,h12,?_,?_⟩
+    · rw [h2,h3]; exact p.adj_getVert_succ (by omega)
+    · have h30 := p.adj_getVert_succ (i := 3) (by omega)
+      have hlast : p.getVert 4=p.getVert 0 := by simpa only [hL,SimpleGraph.Walk.getVert_zero] using p.getVert_length
+      rwa [hlast,← h3,← h0] at h30
+  · right; right
+    have h3 : e 3=p.getVert 3 := hem 3 (by omega)
+    have h4 : e 4=p.getVert 4 := hem 4 (by omega)
+    have h23 : G.Adj (e 2) (e 3) := by rw [h2,h3]; exact p.adj_getVert_succ (by omega)
+    have h34 : G.Adj (e 3) (e 4) := by rw [h3,h4]; exact p.adj_getVert_succ (by omega)
+    have h40 : G.Adj (e 4) (e 0) := by
+      have h40 := p.adj_getVert_succ (i := 4) (by omega)
+      have hlast : p.getVert 5=p.getVert 0 := by simpa only [hL,SimpleGraph.Walk.getVert_zero] using p.getVert_length
+      rwa [hlast,← h4,← h0] at h40
+    intro i
+    fin_cases i
+    · exact h01
+    · exact h12
+    · exact h23
+    · exact h34
+    · exact h40
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+variable (c : ℂ → Fin 5) (hc : Measurable c)
+def transitionGraph (x : ℂ) : SimpleGraph (Fin 5) where
+  Adj i j := TransitionEdge c i j x
+  symm := ⟨fun _ _ h => transitionEdge_symm c h⟩
+  loopless := ⟨fun _ h => h.1 rfl⟩
+
+def CyclicCore : Set ℂ := {x | ¬(transitionGraph c x).IsAcyclic}
+def StandardCore (e : Equiv.Perm (Fin 5)) : Set ℂ :=
+  {x | ThreeCenter (e.symm ∘ c) x ∨ FourCenter (e.symm ∘ c) x ∨ FiveCenter (e.symm ∘ c) x}
+lemma transitionEdge_unlabel (e : Equiv.Perm (Fin 5)) (i j : Fin 5) (x : ℂ) :
+    TransitionEdge (e.symm ∘ c) i j x ↔ TransitionEdge c (e i) (e j) x := by
+  simpa only [Equiv.symm_apply_apply] using transitionEdge_relabel c e.symm (e i) (e j) x
+lemma cyclicCore_subset_standard : CyclicCore c ⊆ ⋃e,StandardCore c e := by
+  intro x hx
+  change ¬∀v (p : (transitionGraph c x).Walk v v),¬p.IsCycle at hx
+  push Not at hx
+  obtain ⟨v,p,hp⟩ := hx
+  obtain ⟨e,he⟩ := cycle_standard_permutation hp
+  refine mem_iUnion.mpr ⟨e,?_⟩
+  change ThreeCenter (e.symm ∘ c) x ∨ FourCenter (e.symm ∘ c) x ∨ FiveCenter (e.symm ∘ c) x
+  simp only [ThreeCenter,FourCenter,FiveCenter,transitionEdge_unlabel]
+  exact he
+lemma cyclicCore_closed : IsClosed (CyclicCore c) := by
+  rw [← isOpen_compl_iff]
+  apply isOpen_iff_mem_nhds.mpr
+  intro x hx
+  have hx' : (transitionGraph c x).IsAcyclic := not_not.mp hx
+  have he : ∀i j : Fin 5,∀ᶠ y in 𝓝 x,
+      TransitionEdge c i j y → TransitionEdge c i j x := by
+    intro i j
+    by_cases hij : TransitionEdge c i j x
+    · exact Filter.Eventually.of_forall fun _ _ => hij
+    · have ho := (transitionEdge_closed c i j).isOpen_compl.mem_nhds hij
+      exact Filter.Eventually.mono ho fun _ hy h => False.elim (hy h)
+  have hall : ∀ᶠ y in 𝓝 x,∀i j : Fin 5,
+      TransitionEdge c i j y → TransitionEdge c i j x :=
+    (Filter.eventually_all.mpr fun i => Filter.eventually_all.mpr (he i))
+  filter_upwards [hall] with y hy
+  change ¬¬(transitionGraph c y).IsAcyclic
+  exact not_not.mpr (hx'.anti (fun {i j} h => hy i j h))
+lemma standardCore_closed (e : Equiv.Perm (Fin 5)) : IsClosed (StandardCore c e) :=
+  (threeCenter_closed (e.symm ∘ c)).union
+    ((fourCenter_closed (e.symm ∘ c)).union (fiveCenter_closed (e.symm ∘ c)))
+include hc in
+lemma standardCore_compact_finite (hw : WeakColoring 5 c) {K : Set ℂ} (hK : IsCompact K)
+    (e : Equiv.Perm (Fin 5)) : (K∩StandardCore c e).Finite := by
+  have hc' : Measurable (e.symm ∘ c) := (measurable_of_finite e.symm).comp hc
+  have hw' := weak_relabel c e.symm hw
+  exact ((threeCenter_compact_finite _ hc' hw' hK).union
+    ((fourCenter_compact_finite _ hc' hw' hK).union
+      (fiveCenter_compact_finite _ hc' hw' hK))).subset (by
+        intro x hx
+        rcases hx.2 with h|h|h
+        · exact Or.inl ⟨hx.1,h⟩
+        · exact Or.inr (Or.inl ⟨hx.1,h⟩)
+        · exact Or.inr (Or.inr ⟨hx.1,h⟩))
+include hc in
+lemma cyclicCore_compact_finite (hw : WeakColoring 5 c) {K : Set ℂ} (hK : IsCompact K) :
+    (K∩CyclicCore c).Finite := by
+  have hf : (⋃e,K∩StandardCore c e).Finite := finite_iUnion fun e =>
+    standardCore_compact_finite c hc hw hK e
+  apply hf.subset
+  intro x hx
+  obtain ⟨e,he⟩ := mem_iUnion.mp (cyclicCore_subset_standard c hx.2)
+  exact mem_iUnion.mpr ⟨e,hx.1,he⟩
+end EuclideanFiveColor
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PlaneColoring/DiskAverages.lean b/lean/OAI/Geometry/PlaneColoring/DiskAverages.lean
new file mode 100644
index 0000000000000000000000000000000000000000..a4f6c191505e54ece748b266724e370338c51ffd
--- /dev/null
+++ b/lean/OAI/Geometry/PlaneColoring/DiskAverages.lean
@@ -0,0 +1,333 @@
+import OAI.Geometry.PlaneColoring.Accumulation
+
+namespace OAI
+
+noncomputable section
+open scoped Classical
+open Set Filter Topology MeasureTheory MeasureTheory.Measure
+namespace EuclideanFiveColor
+def diskLaw (ε : ℝ) : Measure ℂ :=
+  (volume (Metric.ball (0:ℂ) ε))⁻¹ • volume.restrict (Metric.ball 0 ε)
+lemma diskLaw_absolutelyContinuous (ε : ℝ) : diskLaw ε ≪ volume :=
+  (Measure.restrict_le_self (s := Metric.ball (0:ℂ) ε)).absolutelyContinuous.smul_left _
+lemma diskLaw_probability {ε : ℝ} (hε : 0<ε) : IsProbabilityMeasure (diskLaw ε) := by
+  constructor
+  simp only [diskLaw,Measure.smul_apply,Measure.restrict_apply_univ,smul_eq_mul]
+  exact ENNReal.inv_mul_cancel (Metric.measure_ball_pos volume 0 hε).ne'
+    ((measure_mono Metric.ball_subset_closedBall).trans_lt (isCompact_closedBall (0:ℂ) ε).measure_lt_top).ne
+def colorFn {k : ℕ} (c : ℂ → Fin k) (i : Fin k) (z : ℂ) : ℝ := if c z=i then 1 else 0
+lemma colorFn_measurable {k : ℕ} {c : ℂ → Fin k} (hc : Measurable c) (i : Fin k) :
+    Measurable (colorFn c i) :=
+  measurable_one.piecewise (measurableSet_eq_fun hc measurable_const) measurable_zero
+lemma colorFn_nonneg {k : ℕ} (c : ℂ → Fin k) (i : Fin k) (z : ℂ) : 0≤colorFn c i z := by
+  simp only [colorFn]; split_ifs <;> norm_num
+lemma colorFn_le_one {k : ℕ} (c : ℂ → Fin k) (i : Fin k) (z : ℂ) : colorFn c i z≤1 := by
+  simp only [colorFn]; split_ifs <;> norm_num
+lemma colorFn_integrable {k : ℕ} {c : ℂ → Fin k} (hc : Measurable c) (i : Fin k)
+    {μ : Measure ℂ} [IsFiniteMeasure μ] (q : ℂ) : Integrable (fun v => colorFn c i (q+v)) μ := by
+  apply Integrable.of_bound ((colorFn_measurable hc i).comp (measurable_const.add measurable_id)).aestronglyMeasurable 1
+  exact Eventually.of_forall fun v => by
+    change ‖colorFn c i (q+v)‖ ≤ 1
+    rw [Real.norm_eq_abs,abs_of_nonneg (colorFn_nonneg c i (q+v))]
+    exact colorFn_le_one c i (q+v)
+def diskAverage {k : ℕ} (c : ℂ → Fin k) (ε : ℝ) (i : Fin k) (q : ℂ) : ℝ :=
+  ∫ v, colorFn c i (q+v) ∂diskLaw ε
+lemma diskAverage_nonneg {k : ℕ} (c : ℂ → Fin k) (ε : ℝ) (i : Fin k) (q : ℂ) :
+    0≤diskAverage c ε i q := integral_nonneg (fun v => colorFn_nonneg c i (q+v))
+lemma diskAverage_le_one {k : ℕ} {c : ℂ → Fin k} (hc : Measurable c) {ε : ℝ} (hε : 0<ε)
+    (i : Fin k) (q : ℂ) : diskAverage c ε i q≤1 := by
+  have := diskLaw_probability hε
+  simpa only [diskAverage,integral_const,probReal_univ,one_smul] using integral_mono (colorFn_integrable (μ := diskLaw ε) hc i q) (integrable_const (1:ℝ))
+    (fun v => colorFn_le_one c i (q+v))
+lemma diskAverage_sum {k : ℕ} {c : ℂ → Fin k} (hc : Measurable c) {ε : ℝ} (hε : 0<ε)
+    (q : ℂ) : ∑ i,diskAverage c ε i q=1 := by
+  have := diskLaw_probability hε
+  simp only [diskAverage]
+  rw [← integral_finsetSum _ (fun i _ => colorFn_integrable (μ := diskLaw ε) hc i q)]
+  simp [colorFn]
+lemma colorFn_unit_pair {k : ℕ} {c : ℂ → Fin k} (hc : Measurable c) (hw : WeakColoring k c)
+    {q z : ℂ} (hunit : ‖q-z‖=1) (i : Fin k) :
+    ∀ᵐ v ∂(volume : Measure ℂ), colorFn c i (q+v)+colorFn c i (z+v)≤1 := by
+  obtain ⟨t,ht⟩ := direction_exists (u := z-q) (by simpa only [norm_sub_rev] using hunit)
+  have ha := (measurePreserving_add_left (volume : Measure ℂ) q).quasiMeasurePreserving.ae
+    (weak_all_directions c hc hw t)
+  filter_upwards [ha] with v hv
+  have hne : c (q+v)≠c (z+v) := by
+    convert hv using 1
+    rw [ht]
+    congr 1
+    ring
+  simp only [colorFn]
+  split_ifs with h1 h2 h2 <;> try norm_num
+  exact False.elim (hne (h1.trans h2.symm))
+lemma diskAverage_unit_pair {k : ℕ} {c : ℂ → Fin k} (hc : Measurable c) (hw : WeakColoring k c)
+    {ε : ℝ} (hε : 0<ε) {q z : ℂ} (hunit : ‖q-z‖=1) (i : Fin k) :
+    diskAverage c ε i q+diskAverage c ε i z≤1 := by
+  have := diskLaw_probability hε
+  rw [diskAverage,diskAverage,← integral_add (colorFn_integrable (μ := diskLaw ε) hc i q) (colorFn_integrable (μ := diskLaw ε) hc i z)]
+  simpa using integral_mono_ae ((colorFn_integrable (μ := diskLaw ε) hc i q).add (colorFn_integrable (μ := diskLaw ε) hc i z))
+    (integrable_const (1:ℝ)) ((diskLaw_absolutelyContinuous ε).ae_le (colorFn_unit_pair hc hw hunit i))
+end EuclideanFiveColor
+end
+
+noncomputable section
+open scoped Classical symmDiff
+open Set Filter Topology MeasureTheory MeasureTheory.Measure
+namespace EuclideanFiveColor
+lemma volumeReal_ball (q : ℂ) {ε : ℝ} (hε : 0≤ε) : volume.real (Metric.ball q ε)=Real.pi*ε^2 := by
+  simp [Measure.real,Complex.volume_ball,ENNReal.toReal_mul,ENNReal.toReal_pow,ENNReal.toReal_ofReal hε,mul_comm]
+lemma diskAverage_area {k : ℕ} {c : ℂ → Fin k} (hc : Measurable c) (ε : ℝ)
+    (i : Fin k) (q : ℂ) : diskAverage c ε i q=
+      volume.real (Metric.ball q ε ∩ {z | c z=i})/volume.real (Metric.ball (0:ℂ) ε) := by
+  have he : (fun v : ℂ => q+v) ⁻¹' Metric.ball q ε = Metric.ball 0 ε := by
+    ext v
+    simp [Metric.mem_ball,dist_eq_norm]
+  have hi := (measurePreserving_add_left (volume : Measure ℂ) q).setIntegral_preimage_emb
+    (MeasurableEquiv.addLeft q).measurableEmbedding (colorFn c i) (Metric.ball q ε)
+  rw [he] at hi
+  unfold diskAverage diskLaw
+  rw [integral_smul_measure,ENNReal.toReal_inv,hi]
+  have hid : colorFn c i=({z | c z=i} : Set ℂ).indicator (fun _ => (1:ℝ)) := by
+    funext v
+    simp [colorFn,Set.indicator]
+  rw [hid]
+  rw [setIntegral_indicator (measurableSet_eq_fun hc measurable_const)]
+  simp only [integral_const,Measure.real,Measure.restrict_apply_univ,smul_eq_mul,mul_one,div_eq_mul_inv]
+  ring
+lemma ball_ne_top (q : ℂ) (r : ℝ) : volume (Metric.ball q r) ≠ ⊤ :=
+  ((measure_mono Metric.ball_subset_closedBall).trans_lt (isCompact_closedBall q r).measure_lt_top).ne
+lemma symmDiff_balls_subset_annulus (q z : ℂ) (ε : ℝ) :
+    Metric.ball q ε ∆ Metric.ball z ε ⊆
+      Metric.ball q (ε+dist q z) \ Metric.ball q (ε-dist q z) := by
+  intro v hv
+  have h1 := dist_triangle v q z
+  have h2 := dist_triangle v z q
+  rw [dist_comm z q] at h2
+  rcases hv with h|h
+  · refine ⟨?_,?_⟩
+    · change dist v q < ε+dist q z
+      exact lt_of_lt_of_le (show dist v q<ε from h.1) (by linarith [dist_nonneg (x := q) (y := z)])
+    · intro hh
+      have hz : ε≤dist v z := not_lt.mp h.2
+      have hh' : dist v q<ε-dist q z := hh
+      linarith
+  · refine ⟨?_,?_⟩
+    · exact lt_of_le_of_lt h2 (by have hh : dist v z<ε := h.1; linarith)
+    · intro hh
+      have hz : ε≤dist v q := not_lt.mp h.2
+      have hh' : dist v q<ε-dist q z := hh
+      linarith [dist_nonneg (x := q) (y := z)]
+lemma measure_inter_ball_difference (A : Set ℂ) (hA : MeasurableSet A) (q z : ℂ)
+    {ε : ℝ} (hd : dist q z≤ε) :
+    |volume.real (Metric.ball q ε ∩ A)-volume.real (Metric.ball z ε ∩ A)|≤
+      4*Real.pi*ε*dist q z := by
+  have hsub : (Metric.ball q ε ∩ A) ∆ (Metric.ball z ε ∩ A) ⊆
+      Metric.ball q ε ∆ Metric.ball z ε := by
+    intro v hv
+    rcases hv with h|h
+    · exact Or.inl ⟨h.1.1,fun hvz => h.2 ⟨hvz,h.1.2⟩⟩
+    · exact Or.inr ⟨h.1.1,fun hvz => h.2 ⟨hvz,h.1.2⟩⟩
+  have hi := abs_measureReal_sub_le_measureReal_symmDiff' (Metric.isOpen_ball.measurableSet.inter hA).nullMeasurableSet
+    (Metric.isOpen_ball.measurableSet.inter hA).nullMeasurableSet
+    (measure_ne_top_of_subset inter_subset_left (ball_ne_top q ε))
+    (measure_ne_top_of_subset inter_subset_left (ball_ne_top z ε))
+  have hm := measureReal_mono (hsub.trans (symmDiff_balls_subset_annulus q z ε))
+    (measure_ne_top_of_subset sdiff_subset (ball_ne_top q (ε+dist q z)))
+  apply hi.trans (hm.trans_eq ?_)
+  rw [measureReal_sdiff (Metric.ball_subset_ball (by linarith [dist_nonneg (x := q) (y := z)]))
+    measurableSet_ball (ball_ne_top q (ε+dist q z)),volumeReal_ball q (by linarith [dist_nonneg (x := q) (y := z)]),
+    volumeReal_ball q (sub_nonneg.mpr hd)]
+  ring
+lemma diskAverage_lipschitz {k : ℕ} {c : ℂ → Fin k} (hc : Measurable c) {ε : ℝ} (hε : 0<ε)
+    (i : Fin k) : LipschitzWith ⟨4/ε,by positivity⟩ (diskAverage c ε i) := by
+  apply LipschitzWith.of_dist_le_mul
+  intro q z
+  change |diskAverage c ε i q-diskAverage c ε i z|≤(4/ε)*dist q z
+  by_cases hd : dist q z≤ε
+  · rw [diskAverage_area hc,diskAverage_area hc,← sub_div,abs_div,
+      volumeReal_ball 0 hε.le,abs_of_pos (by positivity : 0<Real.pi*ε^2)]
+    apply (div_le_iff₀ (by positivity : 0<Real.pi*ε^2)).mpr
+    have hh := measure_inter_ball_difference {v | c v=i} (measurableSet_eq_fun hc measurable_const) q z hd
+    apply hh.trans_eq
+    field_simp
+  · have hb : |diskAverage c ε i q-diskAverage c ε i z|≤1 := by
+      rw [abs_le]
+      constructor <;> linarith [diskAverage_nonneg c ε i q,diskAverage_nonneg c ε i z,
+        diskAverage_le_one hc hε i q,diskAverage_le_one hc hε i z]
+    apply hb.trans
+    rw [div_mul_eq_mul_div,le_div_iff₀ hε]
+    linarith [not_le.mp hd]
+end EuclideanFiveColor
+end
+
+noncomputable section
+open scoped Classical
+open Set Filter Topology MeasureTheory MeasureTheory.Measure
+namespace EuclideanFiveColor
+variable {k : ℕ} {c : ℂ → Fin k}
+lemma colorFn_norm_le_one (i : Fin k) (z : ℂ) : ‖colorFn c i z‖≤1 := by
+  rw [Real.norm_eq_abs,abs_of_nonneg (colorFn_nonneg c i z)]
+  exact colorFn_le_one c i z
+lemma diskAverage_product {ε : ℝ} (hε : 0<ε) (i j : Fin k) (q : ℂ) :
+    diskAverage c ε i q*diskAverage c ε j q =
+      ∫ (h : ℂ × ℂ), colorFn c i (q+h.1)*colorFn c j (q+h.2) ∂(diskLaw ε).prod (diskLaw ε) := by
+  have := diskLaw_probability hε
+  exact (integral_prod_mul _ _).symm
+lemma disk_overlap_swap (hc : Measurable c) {ε : ℝ} (hε : 0<ε) (i j : Fin k) (x : ℂ) (r : ℝ) :
+    (∫ q in Metric.ball x r, diskAverage c ε i q*diskAverage c ε j q) =
+      ∫ (h : ℂ × ℂ), (∫ q in Metric.ball x r, colorFn c i (q+h.1)*colorFn c j (q+h.2))
+        ∂(diskLaw ε).prod (diskLaw ε) := by
+  have := diskLaw_probability hε
+  have : IsFiniteMeasure (volume.restrict (Metric.ball x r)) := ⟨by
+    simpa only [Measure.restrict_apply_univ] using (ball_ne_top x r).lt_top⟩
+  simp_rw [diskAverage_product hε]
+  apply integral_integral_swap
+  apply Integrable.of_bound
+    (((colorFn_measurable hc i).comp (measurable_fst.add measurable_snd.fst)).mul
+      ((colorFn_measurable hc j).comp (measurable_fst.add measurable_snd.snd))).aestronglyMeasurable 1
+  exact Eventually.of_forall fun z => by
+    change ‖colorFn c i _*colorFn c j _‖≤1
+    rw [norm_mul]
+    exact (mul_le_mul (colorFn_norm_le_one i _) (colorFn_norm_le_one j _) (norm_nonneg _) zero_le_one).trans_eq (one_mul _)
+lemma root_overlap_bound (hc : Measurable c) (i j : Fin k) (x : ℂ) (r R : ℝ) (h₁ h₂ : ℂ)
+    (hr : r+‖h₁‖≤R) :
+    (∫ q in Metric.ball x r, colorFn c i (q+h₁)*colorFn c j (q+h₂))≤
+      volume.real (transitionSet c i j x R (h₂-h₁)) := by
+  let T := transitionSet c i j x R (h₂-h₁)
+  have hT : MeasurableSet T := transitionSet_measurable c hc i j x R (h₂-h₁)
+  have hTf : volume T≠⊤ := (transitionSet_finite c i j x R (h₂-h₁)).ne
+  have hI : Integrable (fun q : ℂ => T.indicator (fun _ => (1:ℝ)) (q+h₁)) volume := by
+    exact ((integrable_indicator_iff hT).mpr (integrableOn_const (C := (1:ℝ)) hTf)).comp_add_right h₁
+  have : IsFiniteMeasure (volume.restrict (Metric.ball x r)) := ⟨by
+    simpa only [Measure.restrict_apply_univ] using (ball_ne_top x r).lt_top⟩
+  have hF : Integrable (fun q : ℂ => colorFn c i (q+h₁)*colorFn c j (q+h₂))
+      (volume.restrict (Metric.ball x r)) := by
+    apply Integrable.of_bound (((colorFn_measurable hc i).comp (measurable_id.add_const h₁)).mul
+      ((colorFn_measurable hc j).comp (measurable_id.add_const h₂))).aestronglyMeasurable 1
+    exact Eventually.of_forall fun q => by
+      change ‖colorFn c i (q+h₁)*colorFn c j (q+h₂)‖≤1
+      rw [norm_mul]
+      simpa using mul_le_mul (colorFn_norm_le_one i (q+h₁)) (colorFn_norm_le_one j (q+h₂)) (norm_nonneg _) zero_le_one
+  have hle : (∫ q in Metric.ball x r, colorFn c i (q+h₁)*colorFn c j (q+h₂))≤
+      ∫ q in Metric.ball x r, T.indicator (fun _ => (1:ℝ)) (q+h₁) := by
+    apply integral_mono_ae hF hI.restrict
+    filter_upwards [ae_restrict_mem measurableSet_ball] with q hq
+    by_cases h1 : c (q+h₁)=i
+    · by_cases h2 : c (q+h₂)=j
+      · have hmem : q+h₁∈T := by
+          refine ⟨?_,Or.inl ⟨h1,?_⟩⟩
+          · change dist (q+h₁) x<R
+            have he : q+h₁-x=(q-x)+h₁ := by ring
+            rw [dist_eq_norm,he]
+            have hq' : ‖q-x‖<r := by simpa only [Metric.mem_ball,dist_eq_norm] using hq
+            exact (norm_add_le _ _).trans_lt (by linarith)
+          · convert h2 using 1; congr 1; ring
+        simp [colorFn,h1,h2,Set.indicator_of_mem hmem]
+      · simp only [colorFn,ite_eq_left h1,ite_eq_right h2,mul_zero]; exact Set.indicator_nonneg (fun _ _ => zero_le_one) _
+    · simp only [colorFn,ite_eq_right h1,zero_mul]; exact Set.indicator_nonneg (fun _ _ => zero_le_one) _
+  apply hle.trans
+  calc
+    (∫ q in Metric.ball x r, T.indicator (fun _ => (1:ℝ)) (q+h₁)) ≤
+      ∫ q, T.indicator (fun _ => (1:ℝ)) (q+h₁) :=
+        integral_mono_measure Measure.restrict_le_self (Eventually.of_forall fun q => Set.indicator_nonneg (fun _ _ => zero_le_one) _) hI
+    _ = volume.real T := by
+      rw [(measurePreserving_add_right volume h₁).integral_comp
+        (MeasurableEquiv.addRight h₁).measurableEmbedding]
+      rw [integral_indicator hT]
+      simp
+lemma root_overlap_zero (i j : Fin k) (hne : i≠j) (x : ℂ) (r : ℝ) (h : ℂ) :
+    (∫ q in Metric.ball x r, colorFn c i (q+h)*colorFn c j (q+h))=0 := by
+  apply integral_eq_zero_of_ae
+  exact Eventually.of_forall fun q => by
+    simp only [colorFn]
+    split_ifs with hi hj hj <;> try norm_num
+    exact False.elim (hne (hi.symm.trans hj))
+end EuclideanFiveColor
+end
+
+noncomputable section
+open scoped Classical
+open Set Filter Topology MeasureTheory MeasureTheory.Measure
+namespace EuclideanFiveColor
+variable {k : ℕ} {c : ℂ → Fin k}
+lemma missing_edge_uniform {i j : Fin k} (hne : i≠j) {x : ℂ} {R : ℝ}
+    (ht : Tendsto (transitionRatio c i j x R) (𝓝[≠] (0:ℂ)) (𝓝 0))
+    {η : ℝ} (hη : 0<η) : ∃ δ>0, ∀ h : ℂ, ‖h‖<δ →
+      volume.real (transitionSet c i j x R h)≤η*‖h‖ := by
+  have hev : ∀ᶠ h in 𝓝[≠] (0:ℂ), transitionRatio c i j x R h<ENNReal.ofReal η :=
+    ht.eventually (gt_mem_nhds (ENNReal.ofReal_pos.mpr hη))
+  obtain ⟨δ,hδ,hd⟩ := Metric.mem_nhdsWithin_iff.mp hev
+  refine ⟨δ,hδ,fun h hh => ?_⟩
+  by_cases hz : h=0
+  · subst h
+    have he : transitionSet c i j x R 0=∅ := by
+      ext q
+      simp only [transitionSet,mem_ofPred_eq,add_zero,mem_empty_iff_false,iff_false]
+      rintro ⟨_,hh|hh⟩
+      · exact hne (hh.1.symm.trans hh.2)
+      · exact hne (hh.2.symm.trans hh.1)
+    simp [he]
+  · have hr : transitionRatio c i j x R h ≤ ENNReal.ofReal η :=
+      le_of_lt (hd ⟨by simpa only [Metric.mem_ball,dist_zero_right] using hh,hz⟩)
+    have hm := (ENNReal.div_le_iff (ENNReal.ofReal_pos.mpr (norm_pos_iff.mpr hz)).ne'
+      ENNReal.ofReal_ne_top).mp hr
+    have hreal := ENNReal.toReal_mono (ENNReal.mul_ne_top ENNReal.ofReal_ne_top ENNReal.ofReal_ne_top) hm
+    simpa only [Measure.real,ENNReal.toReal_mul,ENNReal.toReal_ofReal hη.le,ENNReal.toReal_ofReal (norm_nonneg h)] using hreal
+lemma diskLaw_mem_ball {ε : ℝ} (_hε : 0<ε) : ∀ᵐ h ∂diskLaw ε, h∈Metric.ball (0:ℂ) ε := by
+  rw [diskLaw,Measure.ae_ennreal_smul_measure_iff (ENNReal.inv_ne_zero.mpr (ball_ne_top 0 ε))]
+  exact ae_restrict_mem measurableSet_ball
+lemma disk_overlap_bound (hc : Measurable c) {i j : Fin k} (hne : i≠j)
+    (x : ℂ) (r R : ℝ) {ε δ η : ℝ} (hε : 0<ε) (hη : 0≤η)
+    (hr : r+ε≤R) (hδ : 2*ε≤δ)
+    (hd : ∀ h : ℂ, ‖h‖<δ → volume.real (transitionSet c i j x R h)≤η*‖h‖) :
+    (∫ q in Metric.ball x r, diskAverage c ε i q*diskAverage c ε j q)≤2*η*ε := by
+  have := diskLaw_probability hε
+  rw [disk_overlap_swap hc hε]
+  have hab : ∀ᵐ h ∂(diskLaw ε).prod (diskLaw ε), h.1∈Metric.ball (0:ℂ) ε ∧ h.2∈Metric.ball (0:ℂ) ε := by
+    apply (ae_prod_iff_ae_ae (measurableSet_ball.prod measurableSet_ball)).mpr
+    filter_upwards [diskLaw_mem_ball hε] with h₁ hh₁
+    filter_upwards [diskLaw_mem_ball hε] with h₂ hh₂
+    exact ⟨hh₁,hh₂⟩
+  have hh : ∀ᵐ h ∂(diskLaw ε).prod (diskLaw ε),
+      (∫ q in Metric.ball x r, colorFn c i (q+h.1)*colorFn c j (q+h.2))≤2*η*ε := by
+    filter_upwards [hab] with h hh
+    have ha : ‖h.1‖<ε := by simpa only [Metric.mem_ball,dist_zero_right] using hh.1
+    have hb : ‖h.2‖<ε := by simpa only [Metric.mem_ball,dist_zero_right] using hh.2
+    by_cases he : h.1=h.2
+    · rw [he,root_overlap_zero i j hne]
+      positivity
+    · have hn : ‖h.2-h.1‖ < 2*ε := (norm_sub_le _ _).trans_lt (by linarith)
+      apply (root_overlap_bound hc i j x r R h.1 h.2 (by linarith)).trans
+      apply (hd _ (hn.trans_le hδ)).trans
+      nlinarith
+  have hn : 0≤ᵐ[(diskLaw ε).prod (diskLaw ε)]
+      fun h => ∫ q in Metric.ball x r, colorFn c i (q+h.1)*colorFn c j (q+h.2) :=
+    Eventually.of_forall fun h => integral_nonneg fun q => mul_nonneg (colorFn_nonneg c i _) (colorFn_nonneg c j _)
+  simpa using integral_mono_of_nonneg hn (integrable_const (2*η*ε)) hh
+lemma missing_edge_overlap (hc : Measurable c) {i j : Fin k} (hne : i≠j) {x : ℂ}
+    (he : ¬TransitionEdge c i j x) : ∃ R>0, ∀ r<R,
+      Tendsto (fun ε : ℝ => (∫ q in Metric.ball x r, diskAverage c ε i q*diskAverage c ε j q)/ε)
+        (𝓝[>] 0) (𝓝 0) := by
+  obtain ⟨R,hR,ht⟩ := missing_edge_lim c i j hne he
+  refine ⟨R,hR,fun r hr => ?_⟩
+  apply tendsto_order.2
+  constructor
+  · intro a ha
+    filter_upwards [self_mem_nhdsWithin] with ε hε
+    exact ha.trans_le (div_nonneg (integral_nonneg fun q => mul_nonneg (diskAverage_nonneg c ε i q)
+      (diskAverage_nonneg c ε j q)) (le_of_lt hε))
+  · intro b hb
+    obtain ⟨δ,hδ,hd⟩ := missing_edge_uniform hne ht (show 0<b/4 by linarith)
+    have hv : ∀ᶠ ε : ℝ in 𝓝[>] 0, ε < min (R-r) (δ/2) :=
+      (eventually_lt_nhds (lt_min (by linarith) (by linarith))).filter_mono nhdsWithin_le_nhds
+    filter_upwards [self_mem_nhdsWithin,hv] with ε hε hv
+    have hh := disk_overlap_bound hc hne x r R (show 0<ε from hε) (show 0≤b/4 by linarith)
+      (by have := (lt_min_iff.mp hv).1; linarith) (by have := (lt_min_iff.mp hv).2; linarith) hd
+    apply (div_le_div_of_nonneg_right hh (le_of_lt hε)).trans_lt
+    have hz : ε≠0 := ne_of_gt hε
+    field_simp
+    linarith
+end EuclideanFiveColor
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PlaneColoring/DiskExtension.lean b/lean/OAI/Geometry/PlaneColoring/DiskExtension.lean
new file mode 100644
index 0000000000000000000000000000000000000000..c5cb2eaa4906b9a119a1e0a008d14d9e9ec5b575
--- /dev/null
+++ b/lean/OAI/Geometry/PlaneColoring/DiskExtension.lean
@@ -0,0 +1,611 @@
+import OAI.Geometry.PlaneColoring.GraphRealization
+
+namespace OAI
+
+noncomputable section
+open scoped Classical
+open Set Topology
+namespace EuclideanFiveColor
+structure MarkedDisk where
+  center : ℂ
+  radius : ℝ
+  core : Bool
+namespace MarkedDisk
+def ball (d : MarkedDisk) : Set ℂ := Metric.ball d.center d.radius
+def closedBall (d : MarkedDisk) : Set ℂ := Metric.closedBall d.center d.radius
+def noncoreCost (d : MarkedDisk) : ℝ := if d.core then 0 else d.radius
+def Valid (L : Set ℂ) (d : MarkedDisk) : Prop :=
+  0<d.radius ∧ (d.core=true → ∃x∈L,x∈d.ball)
+
+lemma merge {a b : MarkedDisk} {L : Set ℂ} (ha : a.Valid L) (hb : b.Valid L)
+    (hab : ¬Disjoint a.closedBall b.closedBall) {η : ℝ} (hη : 0<η) :
+    ∃d : MarkedDisk, d.Valid L ∧ a.closedBall⊆d.ball ∧ b.closedBall⊆d.ball ∧
+      d.radius≤a.radius+b.radius+η ∧ d.noncoreCost≤a.noncoreCost+b.noncoreCost+η ∧ d.core=(a.core || b.core) := by
+  obtain ⟨z,hza,hzb⟩ := Set.not_disjoint_iff.mp hab
+  have hd : dist a.center b.center≤a.radius+b.radius :=
+    (dist_triangle a.center z b.center).trans (add_le_add (by simpa [closedBall,dist_comm] using hza) hzb)
+  let t : ℝ := b.radius/(a.radius+b.radius)
+  have hrs : 0<a.radius+b.radius := add_pos ha.1 hb.1
+  have ht0 : 0≤t := (div_pos hb.1 hrs).le
+  have ht1 : t≤1 := (div_le_one hrs).mpr (by linarith [ha.1])
+  let c := AffineMap.lineMap a.center b.center t
+  have hca : dist a.center c≤b.radius := by
+    rw [dist_left_lineMap,Real.norm_of_nonneg ht0]
+    calc t*dist a.center b.center ≤ t*(a.radius+b.radius) := mul_le_mul_of_nonneg_left hd ht0
+         _ = b.radius := by dsimp [t]; field_simp
+  have hcb : dist b.center c≤a.radius := by
+    rw [dist_right_lineMap,Real.norm_of_nonneg (sub_nonneg.mpr ht1)]
+    calc (1-t)*dist a.center b.center ≤ (1-t)*(a.radius+b.radius) :=
+            mul_le_mul_of_nonneg_left hd (sub_nonneg.mpr ht1)
+         _ = a.radius := by dsimp [t]; field_simp; ring
+  let d : MarkedDisk := ⟨c,a.radius+b.radius+η,a.core || b.core⟩
+  have hA : a.closedBall⊆d.ball := by
+    intro q hq
+    have hq' : dist q a.center≤a.radius := hq
+    change dist q c<a.radius+b.radius+η
+    exact lt_of_le_of_lt (dist_triangle q a.center c) (by linarith)
+  have hB : b.closedBall⊆d.ball := by
+    intro q hq
+    have hq' : dist q b.center≤b.radius := hq
+    change dist q c<a.radius+b.radius+η
+    exact lt_of_le_of_lt (dist_triangle q b.center c) (by linarith)
+  refine ⟨d,⟨by dsimp [d]; linarith,?_⟩,hA,hB,le_refl _,?_,rfl⟩
+  · intro hc
+    have hc' : a.core=true ∨ b.core=true := by cases hca : a.core <;> cases hcb : b.core <;> simp_all [d]
+    rcases hc' with hh|hh
+    · obtain ⟨x,hx,hxa⟩ := ha.2 hh
+      exact ⟨x,hx,hA (Metric.ball_subset_closedBall hxa)⟩
+    · obtain ⟨x,hx,hxb⟩ := hb.2 hh
+      exact ⟨x,hx,hB (Metric.ball_subset_closedBall hxb)⟩
+  · dsimp [noncoreCost,d]
+    cases a.core <;> cases b.core <;> simp <;> linarith [ha.1,hb.1]
+end MarkedDisk
+end EuclideanFiveColor
+end
+
+noncomputable section
+open scoped Classical
+open Set Topology
+namespace EuclideanFiveColor
+namespace MarkedDisk
+lemma erased_pair_sum {D : Type*} [DecidableEq D] (S : Finset D) {a b : D}
+    (ha : a∈S) (hb : b∈S) (hab : a≠b) (f : D → ℝ) :
+    (∑x∈(S.erase a).erase b,f x)+f a+f b=∑x∈S,f x := by
+  have h1 := Finset.sum_erase_add S f ha
+  have h2 := Finset.sum_erase_add (S.erase a) f (Finset.mem_erase.mpr ⟨hab.symm,hb⟩)
+  linarith
+lemma merge_sum_bound {D : Type*} [DecidableEq D] (S : Finset D) {a b d : D}
+    (ha : a∈S) (hb : b∈S) (hab : a≠b) (f : D → ℝ) {η : ℝ}
+    (hd0 : 0≤f d) (hd : f d≤f a+f b+η) :
+    (∑x∈insert d ((S.erase a).erase b),f x)≤(∑x∈S,f x)+η := by
+  have hs := erased_pair_sum S ha hb hab f
+  by_cases hmem : d∈(S.erase a).erase b
+  · rw [Finset.insert_eq_of_mem hmem]
+    linarith
+  · rw [Finset.sum_insert hmem]
+    linarith
+
+/-- Finite disk merging with separate total and non-core radius budgets. Strict
+containment at merges, and the retained core witness, are both included. -/
+theorem separate (S : Finset MarkedDisk) {L : Set ℂ} (hS : ∀d∈S,d.Valid L)
+    {η : ℝ} (hη : 0<η) : ∃T : Finset MarkedDisk,
+      (∀d∈T,d.Valid L) ∧ (T : Set MarkedDisk).Pairwise (fun a b => Disjoint a.closedBall b.closedBall) ∧
+      (∀d∈S,∃e∈T,d.ball⊆e.ball ∧ (d.core=true → e.core=true)) ∧
+      (∑d∈T,d.radius)<(∑d∈S,d.radius)+η ∧
+      (∑d∈T,d.noncoreCost)<(∑d∈S,d.noncoreCost)+η := by
+  suffices hh : ∀n : ℕ, ∀S : Finset MarkedDisk, S.card=n →
+      (∀d∈S,d.Valid L) → ∀η : ℝ,0<η → ∃T : Finset MarkedDisk,
+      (∀d∈T,d.Valid L) ∧ (T : Set MarkedDisk).Pairwise (fun a b => Disjoint a.closedBall b.closedBall) ∧
+      (∀d∈S,∃e∈T,d.ball⊆e.ball ∧ (d.core=true → e.core=true)) ∧
+      (∑d∈T,d.radius)<(∑d∈S,d.radius)+η ∧
+      (∑d∈T,d.noncoreCost)<(∑d∈S,d.noncoreCost)+η from hh _ S rfl hS η hη
+  intro n
+  induction n using Nat.strong_induction_on with
+  | h n ih =>
+    intro S hn hS η hη
+    by_cases hp : (S : Set MarkedDisk).Pairwise (fun a b => Disjoint a.closedBall b.closedBall)
+    · exact ⟨S,hS,hp,fun d hd => ⟨d,hd,Subset.rfl,fun h => h⟩,
+        lt_add_of_pos_right _ hη,lt_add_of_pos_right _ hη⟩
+    · simp only [Set.Pairwise,Finset.mem_coe] at hp
+      push Not at hp
+      obtain ⟨a,ha,b,hb,hab,hd⟩ := hp
+      obtain ⟨d,hdv,hda,hdb,hdr,hdc,hdcore⟩ := merge (hS a ha) (hS b hb) hd (half_pos hη)
+      let R := (S.erase a).erase b
+      let S' := insert d R
+      have hR : ∀e∈R,e∈S := by intro e he; exact (Finset.mem_erase.mp (Finset.mem_erase.mp he).2).2
+      have hS' : ∀e∈S',e.Valid L := by
+        intro e he
+        rcases Finset.mem_insert.mp he with rfl|he
+        · exact hdv
+        · exact hS e (hR e he)
+      have hcard : S'.card<n := by
+        have hc₁ := Finset.card_erase_add_one ha
+        have hc₂ := Finset.card_erase_add_one (Finset.mem_erase.mpr ⟨hab.symm,hb⟩)
+        have hc₃ := Finset.card_insert_le d R
+        change R.card+1=(S.erase a).card at hc₂
+        dsimp only [S']
+        omega
+      obtain ⟨T,hT,hdis,hcov,hr,hc⟩ := ih S'.card hcard S' rfl hS' (η/2) (half_pos hη)
+      refine ⟨T,hT,hdis,?_,?_,?_⟩
+      · intro e he
+        have he' : ∃f∈S',e.ball⊆f.ball ∧ (e.core=true → f.core=true) := by
+          by_cases hea : e=a
+          · subst e
+            refine ⟨d,Finset.mem_insert_self _ _,fun _ hq => hda (Metric.ball_subset_closedBall hq),?_⟩
+            intro ha
+            rw [hdcore,ha]
+            simp
+          · by_cases heb : e=b
+            · subst e
+              refine ⟨d,Finset.mem_insert_self _ _,fun _ hq => hdb (Metric.ball_subset_closedBall hq),?_⟩
+              intro hb
+              rw [hdcore,hb]
+              simp
+            · exact ⟨e,Finset.mem_insert_of_mem (by simp [R,he,hea,heb]),Subset.rfl,fun h => h⟩
+        obtain ⟨f,hf,hef,hcore⟩ := he'
+        obtain ⟨g,hg,hfg,hcore'⟩ := hcov f hf
+        exact ⟨g,hg,hef.trans hfg,fun hec => hcore' (hcore hec)⟩
+      · have hb' := merge_sum_bound S ha hb hab radius hdv.1.le hdr
+        change (∑x∈S',x.radius)≤(∑x∈S,x.radius)+η/2 at hb'
+        linarith
+      · have hdc0 : 0≤d.noncoreCost := by unfold noncoreCost; split_ifs <;> linarith [hdv.1]
+        have hb' := merge_sum_bound S ha hb hab noncoreCost hdc0 hdc
+        change (∑x∈S',x.noncoreCost)≤(∑x∈S,x.noncoreCost)+η/2 at hb'
+        linarith
+end MarkedDisk
+end EuclideanFiveColor
+end
+
+noncomputable section
+open scoped Classical
+open Set Topology ContinuousMap
+universe u
+namespace EuclideanFiveColor
+lemma cone_dist (c : ℂ) {r : ℝ} (z : Metric.sphere c r) (t : unitInterval) :
+    dist ((1-t.val) • (z.val-c)+c) c=(1-t.val)*r := by
+  rw [dist_eq_norm,add_sub_cancel_right,norm_smul,Real.norm_of_nonneg (sub_nonneg.mpr t.property.2)]
+  rw [←dist_eq_norm,z.property]
+def diskCone (c : ℂ) {r : ℝ} (hr : 0<r) :
+    C(unitInterval × Metric.sphere c r, Metric.closedBall c r) where
+  toFun p := ⟨(1-p.1.val) • (p.2.val-c)+c,by
+    change dist ((1-p.1.val) • (p.2.val-c)+c) c≤r
+    rw [cone_dist]
+    nlinarith [p.1.property.1]⟩
+  continuous_toFun := (((continuous_const.sub (continuous_subtype_val.comp continuous_fst)).smul
+    ((continuous_subtype_val.comp continuous_snd).sub continuous_const)).add continuous_const).subtype_mk _
+lemma diskCone_surjective (c : ℂ) {r : ℝ} (hr : 0<r) : Function.Surjective (diskCone c hr) := by
+  intro q
+  by_cases hq : q.val=c
+  · obtain ⟨z,hz⟩ := (NormedSpace.sphere_nonempty (x:=c)).mpr hr.le
+    refine ⟨(1,⟨z,hz⟩),?_⟩
+    apply Subtype.ext
+    simpa [diskCone] using hq.symm
+  · let d := dist q.val c
+    have hd : 0<d := dist_pos.mpr hq
+    have hdr : d≤r := q.property
+    let t : unitInterval := ⟨1-d/r,⟨by rw [sub_nonneg,div_le_one hr]; exact hdr,
+      by linarith [div_nonneg hd.le hr.le]⟩⟩
+    have hz : (r/d) • (q.val-c)+c∈Metric.sphere c r := by
+      rw [Metric.mem_sphere,dist_eq_norm,add_sub_cancel_right,norm_smul,Real.norm_of_nonneg (div_pos hr hd).le]
+      rw [←dist_eq_norm]
+      change r/d*d=r
+      field_simp
+    refine ⟨(t,⟨(r/d) • (q.val-c)+c,hz⟩),?_⟩
+    apply Subtype.ext
+    change (1-(1-d/r)) • (((r/d) • (q.val-c)+c)-c)+c=q.val
+    rw [add_sub_cancel_right,smul_smul]
+    have he : (1-(1-d/r))*(r/d)=1 := by field_simp; ring
+    rw [he,one_smul,sub_add_cancel]
+lemma diskCone_fiber (c : ℂ) {r : ℝ} (hr : 0<r)
+    {p q : unitInterval × Metric.sphere c r} (he : diskCone c hr p=diskCone c hr q) :
+    p=q ∨ (p.1=1 ∧ q.1=1) := by
+  have hv := congrArg Subtype.val he
+  have hd := congrArg (fun x : Metric.closedBall c r => dist x.val c) he
+  change dist ((1-p.1.val) • (p.2.val-c)+c) c=dist ((1-q.1.val) • (q.2.val-c)+c) c at hd
+  rw [cone_dist,cone_dist] at hd
+  have ht : p.1=q.1 := Subtype.ext (by nlinarith)
+  by_cases h1 : p.1=1
+  · exact Or.inr ⟨h1,ht.symm.trans h1⟩
+  · left
+    apply Prod.ext ht
+    have hneq : 1-p.1.val≠0 := by
+      intro hh
+      apply h1
+      apply Subtype.ext
+      change p.1.val=1
+      linarith
+    apply Subtype.ext
+    change (1-p.1.val) • (p.2.val-c)+c=(1-q.1.val) • (q.2.val-c)+c at hv
+    rw [←ht] at hv
+    have hsub : p.2.val-c=q.2.val-c := (smul_right_injective ℂ hneq (add_right_cancel hv))
+    exact sub_left_injective hsub
+
+/-- A contraction genuinely extends arbitrary continuous sphere data across a disk. -/
+theorem contractible_disk_extension {Y : Type u} [TopologicalSpace Y] [ContractibleSpace Y]
+    (c : ℂ) {r : ℝ} (hr : 0<r) (f : C(Metric.sphere c r,Y)) :
+    ∃g : C(Metric.closedBall c r,Y), ∀z : Metric.sphere c r,
+      g ⟨z.val,Metric.sphere_subset_closedBall z.property⟩=f z := by
+  let : CompactSpace (Metric.sphere c r) := isCompact_iff_compactSpace.mp (isCompact_sphere c r)
+  obtain ⟨a,⟨H⟩⟩ := id_nullhomotopic Y
+  let F : C(unitInterval × Metric.sphere c r,Y) := H.toContinuousMap.comp
+    (ContinuousMap.prodMk ContinuousMap.fst (f.comp ContinuousMap.snd))
+  have hf : Function.FactorsThrough F (diskCone c hr) := by
+    intro p q he
+    rcases diskCone_fiber c hr he with rfl|⟨hp,hq⟩
+    · rfl
+    · change H (p.1,f p.2)=H (q.1,f q.2)
+      rw [hp,hq]
+      exact (H.map_one_left (f p.2)).trans (H.map_one_left (f q.2)).symm
+  have hquot := IsQuotientMap.of_surjective_continuous (diskCone_surjective c hr) (diskCone c hr).continuous
+  refine ⟨hquot.lift F hf,?_⟩
+  intro z
+  have he : diskCone c hr (0,z)=⟨z.val,Metric.sphere_subset_closedBall z.property⟩ := by
+    apply Subtype.ext
+    simp [diskCone]
+  rw [←he]
+  have hh := DFunLike.congr_fun (hquot.lift_comp F hf) (0,z)
+  change (hquot.lift F hf) (diskCone c hr (0,z))=H (0,f z) at hh
+  exact hh.trans (H.map_zero_left _)
+end EuclideanFiveColor
+end
+
+noncomputable section
+open scoped Classical
+open Set Topology
+universe u
+namespace EuclideanFiveColor
+/-- Linear transfer of a leaf coordinate along its unique incident edge. -/
+def graphLeafMove {V : Type u} (v w : V) (t : ℝ) (p : V → ℝ) : V → ℝ :=
+  p+(t*p v) • (graphVertex w-graphVertex v)
+lemma continuous_graphLeafMove {V : Type u} (v w : V) :
+    Continuous (fun tp : ℝ × (V → ℝ) => graphLeafMove v w tp.1 tp.2) := by
+  apply continuous_pi
+  intro i
+  change Continuous (fun tp : ℝ × (V → ℝ) => tp.2 i + (tp.1 * tp.2 v) * (graphVertex w i-graphVertex v i))
+  fun_prop
+@[simp] lemma graphLeafMove_zero {V : Type u} (v w : V) (p : V → ℝ) : graphLeafMove v w 0 p=p := by
+  simp [graphLeafMove]
+lemma graphLeafMove_of_zero {V : Type u} (v w : V) (p : V → ℝ) (hp : p v=0) (t : ℝ) :
+    graphLeafMove v w t p=p := by simp [graphLeafMove,hp]
+lemma graphLeafMove_edge {V : Type u} {v w : V} (hvw : v≠w) (t s : ℝ) :
+    graphLeafMove v w t (graphEdgePoint v w s)=graphEdgePoint v w (s+t*(1-s)) := by
+  ext i
+  by_cases hi : i=v
+  · subst i; simp [graphLeafMove,graphEdgePoint,graphVertex,hvw]; ring
+  · by_cases hj : i=w
+    · subst i; simp [graphLeafMove,graphEdgePoint,graphVertex,hvw,Ne.symm hvw]
+    · simp [graphLeafMove,graphEdgePoint,graphVertex,hi,hj,hvw]
+lemma graphLeafMove_vertex {V : Type u} {v w : V} (hvw : v≠w) (t : ℝ) :
+    graphLeafMove v w t (graphVertex v)=graphEdgePoint v w t := by
+  simpa using graphLeafMove_edge hvw t 0
+lemma graphLeafMove_mem {V : Type u} {G : SimpleGraph V} {v w : V}
+    (hvw : G.Adj v w) (hleaf : ∀i,G.Adj v i → i=w)
+    {p : V → ℝ} (hp : p∈graphRealization G) {t : ℝ} (ht : t∈Icc (0:ℝ) 1) :
+    graphLeafMove v w t p∈graphRealization G := by
+  obtain ⟨a,b,hab,s,hs,rfl⟩ := hp
+  by_cases ha : a=v
+  · subst a
+    by_cases hb : b=v
+    · subst b
+      rw [graphEdgePoint_self,graphLeafMove_vertex hvw.ne]
+      exact graphEdgePoint_mem (.inr hvw) ht
+    · have : b=w := hleaf b (hab.resolve_left (Ne.symm hb))
+      subst b
+      rw [graphLeafMove_edge hvw.ne]
+      apply graphEdgePoint_mem (.inr hvw)
+      constructor
+      · nlinarith [hs.1,hs.2,ht.1,ht.2]
+      · nlinarith [hs.1,hs.2,ht.1,ht.2]
+  · by_cases hb : b=v
+    · subst b
+      have haw : a=w := hleaf a ((hab.resolve_left ha).symm)
+      subst a
+      rw [graphEdgePoint_swap,graphLeafMove_edge hvw.ne]
+      apply graphEdgePoint_mem (.inr hvw)
+      constructor <;> nlinarith [hs.1,hs.2,ht.1,ht.2]
+    · rw [graphLeafMove_of_zero v w _ (by simp [graphEdgePoint,graphVertex,Ne.symm ha,Ne.symm hb])]
+      exact graphEdgePoint_mem hab hs
+lemma graphLeafMove_one_at {V : Type u} {v w : V} (hvw : v≠w) (p : V → ℝ) :
+    graphLeafMove v w 1 p v=0 := by
+  simp [graphLeafMove,graphVertex,hvw]
+
+def graphDrop {V : Type u} (v : V) (p : V → ℝ) : {i : V // i≠v} → ℝ := fun i => p i
+
+def graphLift {V : Type u} (v : V) (p : {i : V // i≠v} → ℝ) : V → ℝ :=
+  fun i => if hi : i≠v then p ⟨i,hi⟩ else 0
+lemma continuous_graphDrop {V : Type u} (v : V) : Continuous (graphDrop v) :=
+  continuous_pi fun i => continuous_apply i.val
+lemma continuous_graphLift {V : Type u} (v : V) : Continuous (graphLift v) := by
+  apply continuous_pi
+  intro i
+  by_cases hi : i≠v
+  · simpa [graphLift,hi] using (continuous_apply (⟨i,hi⟩ : {i : V // i≠v}))
+  · simpa [graphLift,hi] using (continuous_const : Continuous (fun _ : ({i : V // i≠v} → ℝ) => (0:ℝ)))
+@[simp] lemma graphLift_drop {V : Type u} (v : V) (p : V → ℝ) (hp : p v=0) :
+    graphLift v (graphDrop v p)=p := by
+  ext i
+  by_cases hi : i=v
+  · subst i; simp [graphLift,hp]
+  · simp [graphLift,graphDrop,hi]
+@[simp] lemma graphDrop_lift {V : Type u} (v : V) (p : {i : V // i≠v} → ℝ) :
+    graphDrop v (graphLift v p)=p := by
+  ext i; simp [graphDrop,graphLift,i.property]
+lemma graphDrop_vertex {V : Type u} {v a : V} (ha : a≠v) :
+    graphDrop v (graphVertex a)=graphVertex (⟨a,ha⟩ : {i : V // i≠v}) := by
+  ext i; simp [graphDrop,graphVertex,Subtype.ext_iff]
+lemma graphDrop_edge {V : Type u} {v a b : V} (ha : a≠v) (hb : b≠v) (t : ℝ) :
+    graphDrop v (graphEdgePoint a b t)=graphEdgePoint (⟨a,ha⟩ : {i : V // i≠v}) ⟨b,hb⟩ t := by
+  ext i; simp [graphDrop,graphEdgePoint,graphVertex,Subtype.ext_iff]
+lemma graphLift_vertex {V : Type u} (v : V) (a : {i : V // i≠v}) :
+    graphLift v (graphVertex a)=graphVertex a.val := by
+  ext i
+  by_cases hi : i=v
+  · subst i; simp [graphLift,graphVertex,Ne.symm a.property]
+  · simp [graphLift,graphVertex,hi,Subtype.ext_iff]
+lemma graphLift_edge {V : Type u} (v : V) (a b : {i : V // i≠v}) (t : ℝ) :
+    graphLift v (graphEdgePoint a b t)=graphEdgePoint a.val b.val t := by
+  ext i
+  by_cases hi : i=v
+  · subst i; simp [graphLift,graphEdgePoint,graphVertex,Ne.symm a.property,Ne.symm b.property]
+  · simp [graphLift,graphEdgePoint,graphVertex,hi,Subtype.ext_iff]
+lemma graphLift_mem {V : Type u} {G : SimpleGraph V} (v : V) {p : {i : V // i≠v} → ℝ}
+    (hp : p∈graphRealization (G.induce {i | i≠v})) : graphLift v p∈graphRealization G := by
+  obtain ⟨a,b,hab,t,ht,rfl⟩ := hp
+  rw [graphLift_edge]
+  exact graphEdgePoint_mem (hab.imp (congrArg Subtype.val) id) ht
+lemma graphDrop_mem {V : Type u} {G : SimpleGraph V} (v : V) {p : V → ℝ}
+    (hp : p∈graphRealization G) (hz : p v=0) :
+    graphDrop v p∈graphRealization (G.induce {i | i≠v}) := by
+  obtain ⟨a,b,hab,t,ht,rfl⟩ := hp
+  by_cases ha : a=v
+  · subst a
+    by_cases hb : b=v
+    · subst b
+      simp [graphEdgePoint,graphVertex] at hz
+    · have he : t=1 := by
+        have : 1-t=0 := by simpa [graphEdgePoint,graphVertex,Ne.symm hb] using hz
+        linarith
+      subst t
+      rw [graphEdgePoint_one,graphDrop_vertex hb]
+      exact graphVertex_mem _ _
+  · by_cases hb : b=v
+    · subst b
+      have he : t=0 := by simpa [graphEdgePoint,graphVertex,Ne.symm ha] using hz
+      subst t
+      rw [graphEdgePoint_zero,graphDrop_vertex ha]
+      exact graphVertex_mem _ _
+    · rw [graphDrop_edge ha hb]
+      exact graphEdgePoint_mem (hab.imp (fun h => Subtype.ext h) id) ht
+end EuclideanFiveColor
+end
+
+noncomputable section
+open scoped Classical
+open Set Topology ContinuousMap
+universe u
+namespace EuclideanFiveColor
+
+def graphLeafRetraction {V : Type u} {G : SimpleGraph V} {v w : V}
+    (hvw : G.Adj v w) (hleaf : ∀i,G.Adj v i → i=w) :
+    C(GraphSpace G, GraphSpace (G.induce {i | i≠v})) where
+  toFun p := ⟨graphDrop v (graphLeafMove v w 1 p.val),
+    graphDrop_mem v (graphLeafMove_mem hvw hleaf p.property (by norm_num))
+      (graphLeafMove_one_at hvw.ne p.val)⟩
+  continuous_toFun := (continuous_graphDrop v |>.comp
+    ((continuous_graphLeafMove v w).comp (continuous_const.prodMk continuous_subtype_val))).subtype_mk _
+
+def graphLeafInclusion {V : Type u} (G : SimpleGraph V) (v : V) :
+    C(GraphSpace (G.induce {i | i≠v}), GraphSpace G) where
+  toFun p := ⟨graphLift v p.val,graphLift_mem v p.property⟩
+  continuous_toFun := ((continuous_graphLift v).comp continuous_subtype_val).subtype_mk _
+
+def graphLeafHomotopy {V : Type u} {G : SimpleGraph V} {v w : V}
+    (hvw : G.Adj v w) (hleaf : ∀i,G.Adj v i → i=w) :
+    (ContinuousMap.id (GraphSpace G)).Homotopy
+      ((graphLeafInclusion G v).comp (graphLeafRetraction hvw hleaf)) where
+  toFun tp := ⟨graphLeafMove v w tp.1.val tp.2.val,
+    graphLeafMove_mem hvw hleaf tp.2.property tp.1.property⟩
+  continuous_toFun := ((continuous_graphLeafMove v w).comp
+    ((continuous_subtype_val.comp continuous_fst).prodMk
+      (continuous_subtype_val.comp continuous_snd))).subtype_mk _
+  map_zero_left p := by apply Subtype.ext; exact graphLeafMove_zero v w p.val
+  map_one_left p := by
+    apply Subtype.ext
+    exact (graphLift_drop v _ (graphLeafMove_one_at hvw.ne p.val)).symm
+
+private lemma graphTree_contractible_aux (n : ℕ) :
+    ∀ (V : Type u) [Fintype V] (G : SimpleGraph V), Fintype.card V=n →
+      G.IsTree → ContractibleSpace (GraphSpace G) := by
+  induction n using Nat.strong_induction_on with
+  | h n ih =>
+    intro V inst G hcard hG
+    rcases subsingleton_or_nontrivial V with hSS|hNT
+    · let := hSS
+      obtain ⟨a⟩ := hG.nonempty
+      let : Nonempty (GraphSpace G) := ⟨⟨graphVertex a,graphVertex_mem G a⟩⟩
+      let : Subsingleton (GraphSpace G) := ⟨fun p q => Subtype.ext (by
+        have hp := p.property
+        have hq := q.property
+        simp only [graphRealization_singleton G a,mem_singleton_iff] at hp hq
+        exact hp.trans hq.symm)⟩
+      infer_instance
+    · let := hNT
+      obtain ⟨v,hv⟩ := hG.exists_vert_degree_one_of_nontrivial
+      obtain ⟨w,hvw,hleaf⟩ := SimpleGraph.degree_eq_one_iff_existsUnique_adj.mp hv
+      let W := {i : V // i≠v}
+      let G' : SimpleGraph W := G.induce {i | i≠v}
+      have hs : ({v}ᶜ : Set V)={i | i≠v} := by ext i; simp
+      have hconn : G'.Connected := by
+        have hh := hG.connected.induce_compl_singleton_of_degree_eq_one hv
+        rwa [hs] at hh
+      have hac : G'.IsAcyclic := hG.isAcyclic.induce _
+      have hlt : Fintype.card W<n := by
+        rw [←hcard]
+        apply Fintype.card_lt_of_injective_not_surjective (Subtype.val : W → V) Subtype.val_injective
+        intro hh
+        obtain ⟨i,hi⟩ := hh v
+        exact i.property hi
+      let : ContractibleSpace (GraphSpace G') := ih _ hlt W G' rfl ⟨hconn,hac⟩
+      have hn := ((id_nullhomotopic (GraphSpace G')).comp_left
+        (graphLeafRetraction hvw hleaf)).comp_right (graphLeafInclusion G v)
+      obtain ⟨a,ha⟩ := hn
+      apply (contractible_iff_id_nullhomotopic _).mpr
+      refine ⟨a,ContinuousMap.Homotopic.trans ⟨graphLeafHomotopy hvw hleaf⟩ ?_⟩
+      simpa only [ContinuousMap.id_comp] using ha
+
+/-- Genuine contraction of the geometric probability-simplex realization of a finite tree. -/
+theorem graphTree_contractible {V : Type u} [Fintype V] {G : SimpleGraph V} (hG : G.IsTree) :
+    ContractibleSpace (GraphSpace G) := graphTree_contractible_aux _ V G rfl hG
+end EuclideanFiveColor
+end
+
+noncomputable section
+open scoped Classical
+open Set Topology ContinuousMap
+universe u
+namespace EuclideanFiveColor
+/-- Forest-valued boundary data extend through an actual spanning tree inside the
+complete graph. Isolated labels and empty edge sets are retained. -/
+theorem forest_disk_extension {V : Type u} [Fintype V] [Nonempty V]
+    (c : ℂ) {r : ℝ} (hr : 0<r) (f : C(Metric.sphere c r,V → ℝ))
+    {G : SimpleGraph V} (hG : G.IsAcyclic) (hf : ∀z,f z∈graphRealization G) :
+    ∃g : C(Metric.closedBall c r,GraphSpace (⊤ : SimpleGraph V)), ∀z : Metric.sphere c r,
+      (g ⟨z.val,Metric.sphere_subset_closedBall z.property⟩).val=f z := by
+  obtain ⟨T,hGT,_,hT⟩ := SimpleGraph.connected_top.exists_isTree_le_of_le_of_isAcyclic
+    (show G≤⊤ from le_top) hG
+  let : ContractibleSpace (GraphSpace T) := graphTree_contractible hT
+  let F : C(Metric.sphere c r,GraphSpace T) := ⟨fun z => ⟨f z,graphRealization_mono hGT (hf z)⟩,
+    f.continuous.subtype_mk _⟩
+  obtain ⟨g,hg⟩ := contractible_disk_extension c hr F
+  let g' : C(Metric.closedBall c r,GraphSpace (⊤ : SimpleGraph V)) :=
+    ⟨fun z => ⟨(g z).val,graphRealization_mono le_top (g z).property⟩,
+      (continuous_subtype_val.comp g.continuous).subtype_mk _⟩
+  exact ⟨g',fun z => congrArg Subtype.val (hg z)⟩
+end EuclideanFiveColor
+end
+
+noncomputable section
+open scoped Classical
+open Set Topology ContinuousMap
+universe u v w
+namespace EuclideanFiveColor
+/-- Finite closed-cover pasting, with the actual glued continuous map. -/
+def closedCoverMap {X : Type u} {Y : Type v} {ι : Type w}
+    [TopologicalSpace X] [TopologicalSpace Y] [Finite ι]
+    (S : ι → Set X) (hS : ∀i,IsClosed (S i)) (hc : ⋃i,S i=univ)
+    (f : ∀i,C(S i,Y))
+    (hf : ∀i j x (hi : x∈S i) (hj : x∈S j), f i ⟨x,hi⟩=f j ⟨x,hj⟩) : C(X,Y) where
+  toFun := Set.liftCover S (fun i => f i) hf hc
+  continuous_toFun := (locallyFinite_of_finite S).continuous hc hS (fun i => by
+    rw [continuousOn_iff_continuous_domRestrict]
+    simpa +unfoldPartialApp only [Set.domRestrict,Set.liftCover_coe] using (f i).continuous)
+@[simp] lemma closedCoverMap_apply {X : Type u} {Y : Type v} {ι : Type w}
+    [TopologicalSpace X] [TopologicalSpace Y] [Finite ι]
+    (S : ι → Set X) (hS : ∀i,IsClosed (S i)) (hc : ⋃i,S i=univ)
+    (f : ∀i,C(S i,Y))
+    (hf : ∀i j x (hi : x∈S i) (hj : x∈S j), f i ⟨x,hi⟩=f j ⟨x,hj⟩)
+    {i : ι} (x : S i) : closedCoverMap S hS hc f hf x=f i x := by
+  change Set.liftCover S (fun i => f i) hf hc x=f i x
+  exact Set.liftCover_coe x
+end EuclideanFiveColor
+end
+
+noncomputable section
+open scoped Classical
+open Set Topology ContinuousMap
+universe u
+namespace EuclideanFiveColor
+def diskHoles (S : Finset MarkedDisk) : Set ℂ := ⋃d∈S,d.ball
+def coreHoles (S : Finset MarkedDisk) : Set ℂ := ⋃d∈S,⋃(_ : d.core=true),d.ball
+lemma diskHoles_open (S : Finset MarkedDisk) : IsOpen (diskHoles S) :=
+  isOpen_iUnion fun _ => isOpen_iUnion fun _ => Metric.isOpen_ball
+lemma coreHoles_open (S : Finset MarkedDisk) : IsOpen (coreHoles S) :=
+  isOpen_iUnion fun _ => isOpen_iUnion fun _ => isOpen_iUnion fun _ => Metric.isOpen_ball
+lemma mem_diskHoles {S : Finset MarkedDisk} {z : ℂ} : z∈diskHoles S ↔ ∃d∈S,z∈d.ball := by
+  simp [diskHoles]
+lemma mem_coreHoles {S : Finset MarkedDisk} {z : ℂ} : z∈coreHoles S ↔ ∃d∈S,d.core=true ∧ z∈d.ball := by
+  simp only [coreHoles,mem_iUnion]
+  aesop
+
+/-- Finite closed pasting fills precisely the non-core holes, with actual continuous
+tree extensions on their disks and no change outside the original open holes. -/
+theorem fill_noncore_disks {V : Type u} [Fintype V] [Nonempty V]
+    (Q : Set ℂ) (S : Finset MarkedDisk) (hr : ∀d∈S,0<d.radius)
+    (hdis : (S : Set MarkedDisk).Pairwise (fun a b => Disjoint a.closedBall b.closedBall))
+    (f : C(ℂ,V → ℝ))
+    (hg : ∀z∈Q,z∉diskHoles S → f z∈graphRealization (⊤ : SimpleGraph V))
+    (hb : ∀d∈S,d.core=false → (d.closedBall∩Q).Nonempty →
+      ∃G : SimpleGraph V,G.IsAcyclic ∧ ∀z∈Metric.sphere d.center d.radius,f z∈graphRealization G) :
+    ∃g : C(↥(Q \ coreHoles S),GraphSpace (⊤ : SimpleGraph V)),
+      ∀z : ↥(Q \ coreHoles S),z.val∉diskHoles S → (g z).val=f z.val := by
+  let D : Finset MarkedDisk := S.filter fun d => d.core=false ∧ (d.closedBall∩Q).Nonempty
+  have hD (d : D) : d.val∈S ∧ d.val.core=false ∧ (d.val.closedBall∩Q).Nonempty :=
+    Finset.mem_filter.mp d.property
+  have hex (d : D) : ∃g : C(Metric.closedBall d.val.center d.val.radius,GraphSpace (⊤ : SimpleGraph V)),
+      ∀z : Metric.sphere d.val.center d.val.radius,
+        (g ⟨z.val,Metric.sphere_subset_closedBall z.property⟩).val=f z.val := by
+    obtain ⟨G,hG,hf⟩ := hb d.val (hD d).1 (hD d).2.1 (hD d).2.2
+    exact forest_disk_extension d.val.center (hr _ (hD d).1) (f.restrict _) hG (fun z => hf z.val z.property)
+  choose g hg' using hex
+  let X := Q \ coreHoles S
+  let Z : Option D → Set X := fun i => match i with
+    | none => {z | z.val∉diskHoles S}
+    | some d => {z | z.val∈d.val.closedBall}
+  have hZ : ∀i,IsClosed (Z i) := by
+    intro i
+    cases i with
+    | none => exact (diskHoles_open S).isClosed_compl.preimage continuous_subtype_val
+    | some d => exact Metric.isClosed_closedBall.preimage continuous_subtype_val
+  have hcov : ⋃i,Z i=univ := by
+    apply iUnion_eq_univ_iff.mpr
+    intro z
+    by_cases hz : z.val∈diskHoles S
+    · obtain ⟨d,hd,hzd⟩ := mem_diskHoles.mp hz
+      have hdf : d.core=false := by
+        cases he : d.core
+        · rfl
+        · exact False.elim (z.property.2 (mem_coreHoles.mpr ⟨d,hd,he,hzd⟩))
+      let d' : D := ⟨d,Finset.mem_filter.mpr ⟨hd,hdf,⟨z.val,Metric.ball_subset_closedBall hzd,z.property.1⟩⟩⟩
+      exact ⟨some d',Metric.ball_subset_closedBall hzd⟩
+    · exact ⟨none,hz⟩
+  let f₀ : C(Z none,GraphSpace (⊤ : SimpleGraph V)) :=
+    ⟨fun z => ⟨f z.val.val,hg z.val.val z.val.property.1 z.property⟩,
+      (f.continuous.comp (continuous_subtype_val.comp continuous_subtype_val)).subtype_mk _⟩
+  let ψ (d : D) : C(Z (some d),GraphSpace (⊤ : SimpleGraph V)) :=
+    ⟨fun z => g d ⟨z.val.val,z.property⟩,
+      (g d).continuous.comp ((continuous_subtype_val.comp continuous_subtype_val).subtype_mk _)⟩
+  let φ : ∀i,C(Z i,GraphSpace (⊤ : SimpleGraph V)) := fun i => match i with
+    | none => f₀
+    | some d => ψ d
+  have hbd (d : D) (z : X) (h0 : z∈Z none) (h1 : z∈Z (some d)) :
+      f₀ ⟨z,h0⟩=ψ d ⟨z,h1⟩ := by
+    have hn : z.val∉d.val.ball := fun hz => h0 (mem_diskHoles.mpr ⟨d.val,(hD d).1,hz⟩)
+    have hs : z.val∈Metric.sphere d.val.center d.val.radius := by
+      change dist z.val d.val.center=d.val.radius
+      exact le_antisymm h1 (le_of_not_gt hn)
+    apply Subtype.ext
+    exact (hg' d ⟨z.val,hs⟩).symm
+  have hφ : ∀i j z (hi : z∈Z i) (hj : z∈Z j),φ i ⟨z,hi⟩=φ j ⟨z,hj⟩ := by
+    intro i j z hi hj
+    cases i with
+    | none =>
+      cases j with
+      | none => rfl
+      | some d => exact hbd d z hi hj
+    | some d =>
+      cases j with
+      | none => exact (hbd d z hj hi).symm
+      | some e =>
+        by_cases hde : d=e
+        · subst e; rfl
+        · exact False.elim (Set.disjoint_left.mp (hdis (hD d).1 (hD e).1
+            (fun he => hde (Subtype.ext he))) hi hj)
+  refine ⟨closedCoverMap Z hZ hcov φ hφ,?_⟩
+  intro z hz
+  change (closedCoverMap Z hZ hcov φ hφ z).val=f z.val
+  rw [closedCoverMap_apply Z hZ hcov φ hφ (i:=none) ⟨z,hz⟩]
+  rfl
+end EuclideanFiveColor
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PlaneColoring/Exclusion.lean b/lean/OAI/Geometry/PlaneColoring/Exclusion.lean
new file mode 100644
index 0000000000000000000000000000000000000000..0e600dcb23590097cae2ad894053cacf9e19ef56
--- /dev/null
+++ b/lean/OAI/Geometry/PlaneColoring/Exclusion.lean
@@ -0,0 +1,165 @@
+import OAI.Geometry.PlaneColoring.AngularBoundary
+
+namespace OAI
+
+noncomputable section
+open Set
+namespace EuclideanFiveColor
+def fiveDiag (i : Fin 5) : Finset (Fin 5) := {i,i+2}
+def fiveAdj (i j : Fin 5) : Prop := i+1=j ∨ j+1=i
+instance (i j : Fin 5) : Decidable (fiveAdj i j) := inferInstanceAs (Decidable (_ ∨ _))
+def fiveEdge (i : Fin 5) : Finset (Fin 5) := {i,i+1}
+def FiveAdmissible (p : Finset (Fin 5)) : Prop :=
+  p.Nonempty ∧ p.card≤2 ∧ ∀ i : Fin 5, ¬p⊆fiveEdge i
+def fiveCanonical (p : Finset (Fin 5)) : Fin 5 :=
+  if p=fiveDiag 0 then 0 else if p=fiveDiag 1 then 1 else if p=fiveDiag 2 then 2 else
+    if p=fiveDiag 3 then 3 else 4
+lemma five_admissible_diag : ∀p : Finset (Fin 5), FiveAdmissible p → p=fiveDiag (fiveCanonical p) := by
+  unfold FiveAdmissible
+  decide
+lemma five_diag_disjoint : ∀i j : Fin 5, Disjoint (fiveDiag i) (fiveDiag j) ↔ fiveAdj i j := by decide
+lemma five_diag_nonempty (i : Fin 5) : (fiveDiag i).Nonempty := ⟨i,Finset.mem_insert_self _ _⟩
+lemma five_diag_unique : ∀i j : Fin 5, fiveDiag i=fiveDiag j → i=j := by decide
+lemma five_adj_four : ∀i j : Fin 5, fiveAdj i j → ∃ X : Fin 5, ∀k, k∉fiveDiag i∪fiveDiag j ↔ k=X := by decide
+lemma five_no_common_adj : ∀I : Finset (Fin 5), I.Nonempty → (∀k,∃i∈I,k∉fiveDiag i) →
+    ∃i∈I,∃j∈I,fiveAdj i j := by decide
+lemma five_forced_label : ∀i j X : Fin 5, fiveAdj i j →
+    (∀k,k∉fiveDiag i∪fiveDiag j ↔ k=X) → ∀p : Finset (Fin 5),
+    Disjoint p (fiveDiag i) → ¬p⊆fiveDiag j → X∈p := by decide
+end EuclideanFiveColor
+end
+
+noncomputable section
+namespace EuclideanFiveColor
+/-- Labels allowed on the side selected by `b`; a label survives precisely when
+both of its incident edges have the opposite sign. -/
+def sideAllowed (s : Fin 5 → Bool) (b : Bool) : Finset (Fin 5) :=
+  Finset.univ.filter fun i => s (i-1)≠b ∧ s i≠b
+/-- Both actual one-sided strip restrictions on a palette. -/
+def SideCompatible (s : Fin 5 → Bool) (p : Finset (Fin 5)) : Prop :=
+  p.card≤2 ∧ (p∩sideAllowed s true).Nonempty ∧ (p∩sideAllowed s false).Nonempty
+instance (s : Fin 5 → Bool) (p : Finset (Fin 5)) : Decidable (SideCompatible s p) :=
+  inferInstanceAs (Decidable (_∧_∧_))
+lemma side_compatible_diagonal : ∀s : Fin 5 → Bool,∀p : Finset (Fin 5),
+    SideCompatible s p → p=fiveDiag (fiveCanonical p) := by decide
+lemma side_compatible_options : ∀s : Fin 5 → Bool,∀i j : Fin 5,
+    SideCompatible s (fiveDiag i) → SideCompatible s (fiveDiag j) →
+    j=i ∨ j=i+2 ∨ j=i-2 := by decide
+lemma side_allowed_not (s : Fin 5 → Bool) (b : Bool) :
+    sideAllowed (fun i => !(s i)) b=sideAllowed s (!b) := by
+  ext i
+  simp only [sideAllowed,Finset.mem_filter,Finset.mem_univ,true_and]
+  cases hb : b <;> cases h0 : s (i-1) <;> cases h1 : s i <;> decide
+lemma side_compatible_not (s : Fin 5 → Bool) (p : Finset (Fin 5)) :
+    SideCompatible (fun i => !(s i)) p ↔ SideCompatible s p := by
+  simp only [SideCompatible,side_allowed_not,Bool.not_true,Bool.not_false]
+  tauto
+lemma five_triple_forced : ∀a b c d : Fin 5,
+    fiveAdj a b → fiveAdj b c → fiveAdj c d → (d=a ∨ d=a+2 ∨ d=a-2) →
+    b=a+1 ∧ c=b+1 ∧ d=c+1 ∨ a=b+1 ∧ b=c+1 ∧ c=d+1 := by decide
+lemma five_six_constant_steps_impossible : ∀a b c d e f : Fin 5,
+    (b=a+1 ∧ c=b+1 ∧ d=c+1 ∨ a=b+1 ∧ b=c+1 ∧ c=d+1) →
+    (e=d+1 ∧ f=e+1 ∧ a=f+1 ∨ d=e+1 ∧ e=f+1 ∧ f=a+1) →
+    (c=b+1 ∧ d=c+1 ∧ e=d+1 ∨ b=c+1 ∧ c=d+1 ∧ d=e+1) → False := by decide
+lemma five_word_impossible (p : Fin 6 → Fin 5)
+    (ha : ∀j,fiveAdj (p j) (p (j+1)))
+    (ho : ∀j,p (j+3)=p j ∨ p (j+3)=p j+2 ∨ p (j+3)=p j-2) : False := by
+  have h0 := five_triple_forced (p 0) (p 1) (p 2) (p 3) (ha 0) (ha 1) (ha 2) (ho 0)
+  have h3 := five_triple_forced (p 3) (p 4) (p 5) (p 0) (ha 3) (ha 4) (ha 5) (ho 3)
+  have h1 := five_triple_forced (p 1) (p 2) (p 3) (p 4) (ha 1) (ha 2) (ha 3) (ho 1)
+  exact five_six_constant_steps_impossible _ _ _ _ _ _ h0 h3 h1
+lemma five_strip_orbit_impossible (s : Fin 6 → Fin 5 → Bool) (P : Fin 6 → Finset (Fin 5))
+    (hp : ∀j,SideCompatible (s j) (P j))
+    (hopp : ∀j i,s (j+3) i= !(s j i))
+    (hdis : ∀j,Disjoint (P j) (P (j+1))) : False := by
+  let p : Fin 6 → Fin 5 := fun j => fiveCanonical (P j)
+  have he (j) : P j=fiveDiag (p j) := side_compatible_diagonal (s j) (P j) (hp j)
+  apply five_word_impossible p
+  · intro j
+    exact (five_diag_disjoint _ _).mp (by rw [← he,← he]; exact hdis j)
+  · intro j
+    apply side_compatible_options (s j)
+    · rw [← he]; exact hp j
+    · have ho : s (j+3)=(fun i => !(s j i)) := funext (hopp j)
+      have hh := hp (j+3)
+      rw [ho,side_compatible_not,he] at hh
+      exact hh
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+open scoped RealInnerProductSpace
+namespace EuclideanFiveColor
+def angularSign (v : Fin 5 → ℂ) (t : Angles) (i : Fin 5) : Bool := decide (0<⟪direction t,v i⟫)
+lemma angularSign_opposite {v : Fin 5 → ℂ} {t : Angles} (h : ∀i,⟪direction t,v i⟫≠0) :
+    ∀i,angularSign v (t+(Real.pi : Angles)) i= !(angularSign v t i) := by
+  intro i
+  simp only [angularSign,direction_add_pi,inner_neg_left]
+  by_cases hi : 0<⟪direction t,v i⟫
+  · simp only [hi,show ¬0< -⟪direction t,v i⟫ by linarith,decide_true,decide_false,Bool.not_true]
+  · have hn : ⟪direction t,v i⟫<0 := lt_of_le_of_ne (le_of_not_gt hi) (h i)
+    simp only [hi,show 0< -⟪direction t,v i⟫ by linarith,decide_true,decide_false,Bool.not_false]
+variable (c : ℂ → Fin 5) (hc : Measurable c)
+lemma five_sides_compatible (hw : WeakColoring 5 c) {x : ℂ}
+    {K : Fin 5 → Set ℂ} {v : Fin 5 → ℂ} (hv : ∀i,‖v i‖=1)
+    (hK : ∀i,ApproachingContinuum x (K i) (v i))
+    (hH : ∀i,Disjoint (spatialSupport c i) (Straddle (K i)) ∧
+      Disjoint (spatialSupport c (i+1)) (Straddle (K i))) :
+    ∀ᵐt∂angularMeasure,SideCompatible (angularSign v t) (palette c hc x t) := by
+  let E (s : Fin 5 → Bool) (b : Bool) : Finset (Fin 5) := Finset.univ.filter fun i => s i=b
+  have hex := finite_support_exclusion_ae c hc K (fun i => {i,i+1})
+    (fun i j hj => by
+      rcases Finset.mem_insert.mp hj with rfl|hj
+      · exact (hH _).1
+      · rw [Finset.mem_singleton] at hj; subst j; exact (hH _).2)
+  have hJ (s : Fin 5 → Bool) (b : Bool) :
+      ∀ᵐz : ℂ∂volume,(∀i∈E s b,z∈Straddle (K i)) → c z∈sideAllowed s b := by
+    filter_upwards [hex] with z hz hzE
+    simp only [sideAllowed,Finset.mem_filter,Finset.mem_univ,true_and]
+    constructor
+    · intro he
+      have hmem : c z-1∈E s b := Finset.mem_filter.mpr ⟨Finset.mem_univ _,he⟩
+      apply hz (c z-1) (hzE _ hmem)
+      simp
+    · intro he
+      have hmem : c z∈E s b := Finset.mem_filter.mpr ⟨Finset.mem_univ _,he⟩
+      exact hz (c z) (hzE _ hmem) (Finset.mem_insert_self _ _)
+  have hout (s : Fin 5 → Bool) := continuum_outer_restriction c hc (E s true) hK (hJ s true)
+  have hin (s : Fin 5 → Bool) := continuum_inner_restriction c hc (E s false) hK (hJ s false)
+  filter_upwards [ae_all_iff.mpr hout,ae_all_iff.mpr hin,palette_cardinality c hc hw x,
+    ae_all_iff.mpr (fun i => ae_direction_nonorthogonal (hv i))] with t ho hi hp hn
+  refine ⟨hp.2,ho (angularSign v t) ?_,hi (angularSign v t) ?_⟩
+  · intro i hi
+    have hh : angularSign v t i=true := (Finset.mem_filter.mp hi).2
+    exact of_decide_eq_true hh
+  · intro i hi
+    have hh : angularSign v t i=false := (Finset.mem_filter.mp hi).2
+    have hn' : ¬0<⟪direction t,v i⟫ := of_decide_eq_false hh
+    exact lt_of_le_of_ne (le_of_not_gt hn') (hn i)
+include hc in
+/-- Actual five-cycle angular obstruction under exactly the directional and
+common-support-exclusion data extracted in source Proposition interface-cycle.
+This does not assert that such data have been extracted. -/
+lemma no_five_exclusion_continua (hw : WeakColoring 5 c) {x : ℂ}
+    {K : Fin 5 → Set ℂ} {v : Fin 5 → ℂ} (hv : ∀i,‖v i‖=1)
+    (hK : ∀i,ApproachingContinuum x (K i) (v i))
+    (hH : ∀i,Disjoint (spatialSupport c i) (Straddle (K i)) ∧
+      Disjoint (spatialSupport c (i+1)) (Straddle (K i))) : False := by
+  have hh := (five_sides_compatible c hc hw hv hK hH).and
+    ((palette_sixty_disjoint c hc hw x).and (ae_all_iff.mpr (fun i => ae_direction_nonorthogonal (hv i))))
+  have hshift (j : Fin 6) := (measurePreserving_add_right angularMeasure (sixAngle j)).quasiMeasurePreserving.ae hh
+  obtain ⟨t,ht⟩ := (ae_all_iff.mpr hshift).exists
+  apply five_strip_orbit_impossible
+    (fun j => angularSign v (t+sixAngle j)) (fun j => palette c hc x (t+sixAngle j))
+  · intro j; exact (ht j).1
+  · intro j i
+    rw [sixAngle_half,← add_assoc]
+    exact angularSign_opposite (ht j).2.2 i
+  · intro j
+    rw [sixAngle_step,← add_assoc]
+    exact (ht j).2.1
+end EuclideanFiveColor
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PlaneColoring/Expansion.lean b/lean/OAI/Geometry/PlaneColoring/Expansion.lean
new file mode 100644
index 0000000000000000000000000000000000000000..0db7ab8adebaa5fedcc61fe661fa70f3bdaaf09a
--- /dev/null
+++ b/lean/OAI/Geometry/PlaneColoring/Expansion.lean
@@ -0,0 +1,725 @@
+import Mathlib
+
+namespace OAI
+
+noncomputable section
+open Filter Topology Set
+open scoped NNReal
+namespace EuclideanFiveColor
+lemma positive_log_difference_bound {u v m C : ℝ} (hm : 0 < m)
+    (hu : m ≤ u) (hv : m ≤ v) (hC : 0 ≤ C) (hd : |u-v| ≤ C) :
+    |Real.log u-Real.log v| ≤ C/m := by
+  have hu' : 0 < u := hm.trans_le hu
+  have hv' : 0 < v := hm.trans_le hv
+  have haux : ∀ a b : ℝ, m ≤ a → m ≤ b → a-b ≤ C → Real.log a-Real.log b ≤ C/m := by
+    intro a b ha hb hab
+    have ha' := hm.trans_le ha
+    have hb' := hm.trans_le hb
+    have h := Real.log_le_sub_one_of_pos (div_pos ha' hb')
+    rw [Real.log_div ha'.ne' hb'.ne'] at h
+    have he : a/b-1=(a-b)/b := by field_simp
+    rw [he] at h
+    exact h.trans ((div_le_div_of_nonneg_right hab hb'.le).trans
+      (div_le_div_of_nonneg_left hC hm hb))
+  exact abs_le.mpr ⟨by have h := haux v u hv hu (by linarith [neg_le_abs (u-v)]); linarith,
+    haux u v hu hv (le_trans (le_abs_self _) hd)⟩
+lemma controlled_interval_derivative {f' : ℝ → ℝ} {K : ℝ≥0} {a b e : ℝ}
+    (hlip : LipschitzWith K f') (_hab : a ≤ b) (he : (K : ℝ)*(b-a) ≤ e)
+    (hmid : 1+e ≤ f' ((a+b)/2)) {t : ℝ} (ht : t ∈ Icc a b) :
+    1+e/2 ≤ f' t := by
+  have hdist : |t-(a+b)/2| ≤ (b-a)/2 := by rw [abs_le]; constructor <;> linarith [ht.1,ht.2]
+  have h := hlip.dist_le_mul t ((a+b)/2)
+  rw [Real.dist_eq,Real.dist_eq] at h
+  have hm := mul_le_mul_of_nonneg_left hdist K.2
+  have hn := neg_le_abs (f' t-f' ((a+b)/2))
+  nlinarith
+lemma controlled_interval_expands {f f' : ℝ → ℝ} {K : ℝ≥0} {a b e : ℝ}
+    (hd : ∀ x, HasDerivAt f (f' x) x) (hlip : LipschitzWith K f') (hab : a ≤ b)
+    (he : (K : ℝ)*(b-a) ≤ e) (hmid : 1+e ≤ f' ((a+b)/2)) :
+    (1+e/2)*(b-a) ≤ f b-f a := by
+  apply (convex_Icc a b).mul_sub_le_image_sub_of_le_deriv
+    (fun x _ => (hd x).continuousAt.continuousWithinAt)
+    (fun x _ => (hd x).differentiableAt.differentiableWithinAt) _ a ⟨le_rfl,hab⟩ b ⟨hab,le_rfl⟩ hab
+  intro t ht
+  rw [(hd t).deriv]
+  exact controlled_interval_derivative hlip hab he hmid (interior_subset ht)
+lemma geometric_partial_sum {l : ℕ → ℝ} {q : ℝ} (_hq : 1 < q)
+    (h0 : 0 ≤ l 0) (hstep : ∀ n, q*l n ≤ l (n+1)) (N : ℕ) :
+    (q-1)*(∑ n ∈ Finset.range N, l n) ≤ l N := by
+  have h : (q-1)*(∑ n ∈ Finset.range N, l n) ≤ l N-l 0 := by
+    induction N with
+    | zero => simp
+    | succ n ih =>
+      rw [Finset.sum_range_succ,mul_add]
+      nlinarith [hstep n]
+  linarith
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Filter Topology Set
+open scoped NNReal
+namespace EuclideanFiveColor
+def iterComposition (f : ℕ → ℝ → ℝ) : ℕ → ℝ → ℝ
+  | 0 => id
+  | n+1 => fun x => f n (iterComposition f n x)
+def iterDerivative (f D : ℕ → ℝ → ℝ) : ℕ → ℝ → ℝ
+  | 0 => fun _ => 1
+  | n+1 => fun x => D n (iterComposition f n x)*iterDerivative f D n x
+variable {f D : ℕ → ℝ → ℝ}
+lemma iterComposition_hasDeriv (hd : ∀ n x, HasDerivAt (f n) (D n x) x) (n : ℕ) (x : ℝ) :
+    HasDerivAt (iterComposition f n) (iterDerivative f D n x) x := by
+  induction n with
+  | zero => exact hasDerivAt_id x
+  | succ n ih => exact (hd n _).comp x ih
+lemma iterDerivative_pos (hD : ∀ n x, 0 < D n x) (n : ℕ) (x : ℝ) :
+    0 < iterDerivative f D n x := by
+  induction n with
+  | zero => norm_num [iterDerivative]
+  | succ n ih => exact mul_pos (hD n _) ih
+lemma iterComposition_strictMono (hd : ∀ n x, HasDerivAt (f n) (D n x) x)
+    (hD : ∀ n x, 0 < D n x) (n : ℕ) : StrictMono (iterComposition f n) := by
+  apply strictMono_of_deriv_pos
+  intro x
+  rw [(iterComposition_hasDeriv hd n x).deriv]
+  exact iterDerivative_pos hD n x
+lemma iterDerivative_log_oscillation (hd : ∀ n x, HasDerivAt (f n) (D n x) x)
+    (hD : ∀ n x, (1/2 : ℝ) ≤ D n x) {K : ℝ≥0}
+    (hK : ∀ n, LipschitzWith K (D n)) {a b x y : ℝ} (hx : x ∈ Icc a b) (hy : y ∈ Icc a b) (N : ℕ) :
+    |Real.log (iterDerivative f D N x)-Real.log (iterDerivative f D N y)| ≤
+      2*(K : ℝ)*(∑ n ∈ Finset.range N, (iterComposition f n b-iterComposition f n a)) := by
+  rcases K with ⟨K,hKpos⟩
+  change |Real.log (iterDerivative f D N x)-Real.log (iterDerivative f D N y)| ≤
+    2*K*(∑ n ∈ Finset.range N, (iterComposition f n b-iterComposition f n a))
+  have hp : ∀ n t, 0 < D n t := fun n t => lt_of_lt_of_le (by norm_num) (hD n t)
+  induction N with
+  | zero => simp [iterDerivative]
+  | succ n ih =>
+    have hm := (iterComposition_strictMono hd hp n).monotone
+    have hu := hm hx.1
+    have hv := hm hx.2
+    have hu' := hm hy.1
+    have hv' := hm hy.2
+    have hab : 0 ≤ iterComposition f n b-iterComposition f n a := by linarith
+    have hdist : |iterComposition f n x-iterComposition f n y| ≤
+        iterComposition f n b-iterComposition f n a := by
+      rw [abs_le]; constructor <;> linarith
+    have hkd := (hK n).dist_le_mul (iterComposition f n x) (iterComposition f n y)
+    change |D n (iterComposition f n x)-D n (iterComposition f n y)| ≤
+      K*|iterComposition f n x-iterComposition f n y| at hkd
+    have hC := mul_nonneg hKpos hab
+    have hh := positive_log_difference_bound (by norm_num : (0 : ℝ)<1/2)
+      (hD n (iterComposition f n x)) (hD n (iterComposition f n y)) hC
+      (hkd.trans (mul_le_mul_of_nonneg_left hdist hKpos))
+    simp only [iterDerivative,Real.log_mul (hp n _).ne' (iterDerivative_pos hp n _).ne',
+      Finset.sum_range_succ]
+    have ht := abs_add_le (Real.log (D n (iterComposition f n x))-Real.log (D n (iterComposition f n y)))
+      (Real.log (iterDerivative f D n x)-Real.log (iterDerivative f D n y))
+    have he : Real.log (D n (iterComposition f n x)) + Real.log (iterDerivative f D n x) -
+      (Real.log (D n (iterComposition f n y)) + Real.log (iterDerivative f D n y)) =
+      (Real.log (D n (iterComposition f n x))-Real.log (D n (iterComposition f n y)))+
+      (Real.log (iterDerivative f D n x)-Real.log (iterDerivative f D n y)) := by ring
+    rw [he]
+    exact ht.trans ((add_le_add hh ih).trans_eq (by ring))
+lemma iterDerivative_ratio_bound (hd : ∀ n x, HasDerivAt (f n) (D n x) x)
+    (hD : ∀ n x, (1/2 : ℝ) ≤ D n x) {K : ℝ≥0}
+    (hK : ∀ n, LipschitzWith K (D n)) {a b x y M : ℝ} (hx : x ∈ Icc a b) (hy : y ∈ Icc a b)
+    (N : ℕ) (hM : 2*(K : ℝ)*(∑ n ∈ Finset.range N, (iterComposition f n b-iterComposition f n a)) ≤ M) :
+    iterDerivative f D N x ≤ Real.exp M * iterDerivative f D N y := by
+  have hp : ∀ n t, 0 < D n t := fun n t => lt_of_lt_of_le (by norm_num) (hD n t)
+  have h := (le_abs_self _).trans ((iterDerivative_log_oscillation hd hD hK hx hy N).trans hM)
+  have hex := Real.exp_le_exp.mpr h
+  rw [Real.exp_sub,Real.exp_log (iterDerivative_pos hp N x),Real.exp_log (iterDerivative_pos hp N y)] at hex
+  exact (div_le_iff₀ (iterDerivative_pos hp N y)).mp hex
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Filter Topology Set
+namespace EuclideanFiveColor
+lemma eventually_reaches_threshold {l : ℕ → ℝ} {q L : ℝ} (hq : 1 < q) (h0 : 0 < l 0)
+    (hstep : ∀ n, l n < L → q*l n ≤ l (n+1)) : ∃ N, L ≤ l N := by
+  by_contra hex
+  have hlt : ∀ n, l n < L := fun n => lt_of_not_ge (fun h => hex ⟨n,h⟩)
+  have hp : ∀ n, 0 < l n := by
+    intro n
+    induction n with
+    | zero => exact h0
+    | succ n ih => exact (mul_pos (lt_trans zero_lt_one hq) ih).trans_le (hstep n (hlt n))
+  have hm : Monotone l := monotone_nat_of_le_succ fun n => by nlinarith [hstep n (hlt n),hp n]
+  have hg : ∀ n : ℕ, (n : ℝ)*((q-1)*l 0) ≤ l n := by
+    intro n
+    induction n with
+    | zero => simpa using h0.le
+    | succ n ih =>
+      rw [Nat.cast_add,Nat.cast_one]
+      have hl := hm (Nat.zero_le n)
+      have h := hstep n (hlt n)
+      have ht := mul_le_mul_of_nonneg_left hl (sub_nonneg.mpr hq.le)
+      nlinarith
+  have hc : 0 < (q-1)*l 0 := mul_pos (sub_pos.mpr hq) h0
+  obtain ⟨N,hN⟩ := exists_nat_gt (L/((q-1)*l 0))
+  have hn : L < (N : ℝ)*((q-1)*l 0) := (div_lt_iff₀ hc).mp hN
+  exact (hlt N).not_ge (hn.le.trans (hg N))
+lemma first_expansion_threshold {l : ℕ → ℝ} {q U L : ℝ} (hq : 1 < q) (h0 : 0 < l 0)
+    (hsmall : l 0 < L) (hU : 0 ≤ U)
+    (hstep : ∀ n, l n < L → q*l n ≤ l (n+1))
+    (hupper : ∀ n, l (n+1) ≤ U*l n) :
+    ∃ N, L ≤ l N ∧ l N ≤ U*L ∧ (∀ n < N, l n < L) ∧
+      (q-1)*(∑ n ∈ Finset.range N, l n) ≤ l N := by
+  have hex := eventually_reaches_threshold hq h0 hstep
+  let N := Nat.find hex
+  have hN : L ≤ l N := Nat.find_spec hex
+  have hbefore : ∀ n < N, l n < L := fun n hn => lt_of_not_ge (Nat.find_min hex hn)
+  have hNpos : 0 < N := by
+    by_contra h
+    have hzero : N=0 := Nat.eq_zero_of_not_pos h
+    rw [hzero] at hN
+    exact hsmall.not_ge hN
+  have hup : l N ≤ U*L := by
+    obtain ⟨n,hn⟩ := Nat.exists_eq_succ_of_ne_zero (Nat.ne_of_gt hNpos)
+    rw [hn]
+    exact (hupper n).trans (mul_le_mul_of_nonneg_left (hbefore n (by omega)).le hU)
+  have hsum : ∀ n ≤ N, (q-1)*(∑ i ∈ Finset.range n, l i) ≤ l n-l 0 := by
+    intro n
+    induction n with
+    | zero => intro _; simp
+    | succ n ih =>
+      intro hn
+      have hi := ih (by omega)
+      have hs := hstep n (hbefore n (by omega))
+      rw [Finset.sum_range_succ,mul_add]
+      nlinarith
+  exact ⟨N,hN,hup,hbefore,(hsum N le_rfl).trans (by linarith)⟩
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Filter Topology Set
+open scoped NNReal
+namespace EuclideanFiveColor
+lemma exists_expanding_word {ι : Type*} {F D : ι → ℝ → ℝ} {K : ℝ≥0} {e U L a b : ℝ}
+    (hd : ∀ j x, HasDerivAt (F j) (D j x) x)
+    (hp : ∀ j x, (1/2 : ℝ) ≤ D j x) (hup : ∀ j x, D j x ≤ U)
+    (hK : ∀ j, LipschitzWith K (D j)) (hlarge : ∀ x, ∃ j, 1+e ≤ D j x)
+    (he : 0 < e) (hKL : (K : ℝ)*L ≤ e) (hU : 0 ≤ U)
+    (hab : a < b) (hsmall : b-a < L) :
+    ∃ w : ℕ → ι, ∃ N : ℕ,
+      L ≤ iterComposition (fun n => F (w n)) N b-iterComposition (fun n => F (w n)) N a ∧
+      iterComposition (fun n => F (w n)) N b-iterComposition (fun n => F (w n)) N a ≤ U*L ∧
+      ∀ x ∈ Icc a b, ∀ y ∈ Icc a b,
+        iterDerivative (fun n => F (w n)) (fun n => D (w n)) N x ≤
+          Real.exp (4*(K : ℝ)*U*L/e)*iterDerivative (fun n => F (w n)) (fun n => D (w n)) N y := by
+  choose j hj using hlarge
+  let A : ℕ → ℝ × ℝ := fun n => Nat.rec (a,b)
+    (fun _ p => (F (j ((p.1+p.2)/2)) p.1,F (j ((p.1+p.2)/2)) p.2)) n
+  let w : ℕ → ι := fun n => j (((A n).1+(A n).2)/2)
+  let f : ℕ → ℝ → ℝ := fun n => F (w n)
+  let d : ℕ → ℝ → ℝ := fun n => D (w n)
+  have hd' : ∀ n x, HasDerivAt (f n) (d n x) x := fun n x => hd (w n) x
+  have hp' : ∀ n x, 0 < d n x := fun n x => (by norm_num : (0 : ℝ)<1/2).trans_le (hp (w n) x)
+  have hends : ∀ n, (A n).1=iterComposition f n a ∧ (A n).2=iterComposition f n b := by
+    intro n
+    induction n with
+    | zero => exact ⟨rfl,rfl⟩
+    | succ n ih =>
+      change f n (A n).1=iterComposition f (n+1) a ∧ f n (A n).2=iterComposition f (n+1) b
+      rw [ih.1,ih.2]
+      exact ⟨rfl,rfl⟩
+  let l : ℕ → ℝ := fun n => iterComposition f n b-iterComposition f n a
+  have hpos : ∀ n, 0 < l n := fun n => sub_pos.mpr (iterComposition_strictMono hd' hp' n hab)
+  have hstep : ∀ n, l n < L → (1+e/2)*l n ≤ l (n+1) := by
+    intro n hln
+    apply controlled_interval_expands (hd' n) (hK (w n))
+      (iterComposition_strictMono hd' hp' n hab).le
+    · exact (mul_le_mul_of_nonneg_left hln.le K.2).trans hKL
+    · simpa only [d,w,(hends n).1,(hends n).2] using hj (((A n).1+(A n).2)/2)
+  have hupper : ∀ n, l (n+1) ≤ U*l n := by
+    intro n
+    exact image_sub_le_mul_sub_of_deriv_le (fun x => (hd' n x).differentiableAt)
+      (fun x => by rw [(hd' n x).deriv]; exact hup (w n) x)
+      (iterComposition_strictMono hd' hp' n hab).le
+  obtain ⟨N,hN,hNu,_,hsum⟩ := first_expansion_threshold (by linarith : 1 < 1+e/2)
+    (hpos 0) hsmall hU hstep hupper
+  refine ⟨w,N,hN,hNu,fun x hx y hy => ?_⟩
+  apply iterDerivative_ratio_bound hd' (fun n x => hp (w n) x) (fun n => hK (w n)) hx hy N
+  change 2*(K : ℝ)*(∑ n ∈ Finset.range N, l n) ≤ 4*(K : ℝ)*U*L/e
+  apply (le_div_iff₀ he).mpr
+  have hsum' : e/2*(∑ n ∈ Finset.range N, l n) ≤ U*L := by
+    simpa only [add_sub_cancel_left] using hsum.trans hNu
+  have hmul := mul_le_mul_of_nonneg_left hsum' (by positivity : (0 : ℝ)≤4*(K : ℝ))
+  nlinarith
+end EuclideanFiveColor
+end
+
+noncomputable section
+open MeasureTheory Filter Topology Set
+open scoped NNReal ENNReal
+namespace EuclideanFiveColor
+lemma line_lipschitzOn_image_bound {f : ℝ → ℝ} {s : Set ℝ} {K : ℝ≥0}
+    (hf : LipschitzOnWith K f s) : volume (f '' s) ≤ (K : ℝ≥0∞)*volume s := by
+  have h := hf.hausdorffMeasure_image_le (by norm_num : (0 : ℝ)≤1)
+  simpa only [hausdorffMeasure_real,ENNReal.rpow_one] using h
+lemma interval_defect_image_bound {f : ℝ → ℝ} {a b : ℝ} {K : ℝ≥0} {A : Set ℝ}
+    (hab : a ≤ b) (hc : ContinuousOn f (Icc a b)) (hm : MonotoneOn f (Icc a b))
+    (hK : LipschitzOnWith K f (Icc a b))
+    (hA : ∀ᵐ x : ℝ ∂volume, x ∈ A ↔ f x ∈ A) :
+    volume (Icc (f a) (f b) \ A) ≤ (K : ℝ≥0∞)*volume (Icc a b \ A) := by
+  let E : Set ℝ := {x ∈ Icc a b | ¬(x ∈ A ↔ f x ∈ A)}
+  have hE : volume E=0 := measure_mono_null (fun _ h => h.2) (ae_iff.mp hA)
+  have hEi : volume (f '' E)=0 := by
+    have h := line_lipschitzOn_image_bound (hK.mono (show E ⊆ Icc a b from fun _ h => h.1))
+    rw [hE,mul_zero] at h
+    exact le_antisymm h bot_le
+  have hsub : Icc (f a) (f b) \ A ⊆ f '' (Icc a b \ A) ∪ f '' E := by
+    intro y hy
+    obtain ⟨x,hx,hxy⟩ := (hc.image_Icc_of_monotoneOn hab hm ▸ hy.1 : y ∈ f '' Icc a b)
+    by_cases hxA : x ∈ A
+    · right
+      exact ⟨x,⟨hx,fun h => hy.2 (hxy ▸ h.mp hxA)⟩,hxy⟩
+    · exact Or.inl ⟨x,⟨hx,hxA⟩,hxy⟩
+  calc
+    volume (Icc (f a) (f b) \ A) ≤ volume (f '' (Icc a b \ A) ∪ f '' E) := measure_mono hsub
+    _ ≤ volume (f '' (Icc a b \ A))+volume (f '' E) := measure_union_le _ _
+    _ = volume (f '' (Icc a b \ A)) := by rw [hEi,add_zero]
+    _ ≤ (K : ℝ≥0∞)*volume (Icc a b \ A) := line_lipschitzOn_image_bound (hK.mono sdiff_subset)
+lemma derivative_ratio_lipschitzOn {f D : ℝ → ℝ} {a b R : ℝ} (hab : a < b)
+    (hd : ∀ x, HasDerivAt f (D x) x) (hp : ∀ x, 0 < D x) (hR : 0 ≤ R)
+    (hratio : ∀ x ∈ Icc a b, ∀ y ∈ Icc a b, D x ≤ R*D y) :
+    LipschitzOnWith (Real.toNNReal (R*((f b-f a)/(b-a)))) f (Icc a b) := by
+  obtain ⟨z,hz,he⟩ := exists_hasDerivAt_eq_slope f D hab
+    (fun x _ => (hd x).continuousAt.continuousWithinAt) (fun x _ => hd x)
+  apply (convex_Icc a b).lipschitzOnWith_of_nnnorm_deriv_le
+    (fun x _ => (hd x).differentiableAt)
+  intro x hx
+  rw [(hd x).deriv]
+  apply NNReal.coe_le_coe.mp
+  rw [Real.coe_toNNReal _ (by rw [← he]; exact mul_nonneg hR (hp z).le),coe_nnnorm,Real.norm_eq_abs,abs_of_pos (hp x)]
+  rw [← he]
+  exact hratio x hx z ⟨hz.1.le,hz.2.le⟩
+end EuclideanFiveColor
+end
+
+noncomputable section
+open MeasureTheory Filter Topology Set
+open scoped NNReal ENNReal
+namespace EuclideanFiveColor
+lemma line_volume_preimage_bound {F : ℝ → ℝ} {s : Set (ℝ)} {K : ℝ≥0}
+    (hF : AntilipschitzWith K (s.domRestrict F)) (A : Set (ℝ)) :
+    volume (F ⁻¹' A ∩ s) ≤ (K : ENNReal) * volume A := by
+  have he : Subtype.val '' ((s.domRestrict F) ⁻¹' A) = F ⁻¹' A ∩ s := by
+    ext p
+    constructor
+    · rintro ⟨q,hq,rfl⟩; exact ⟨hq,q.2⟩
+    · rintro ⟨hp,hs⟩; exact ⟨⟨p,hs⟩,hp,rfl⟩
+  have hi := (isometry_subtype_coe : Isometry (Subtype.val : s → ℝ)).hausdorffMeasure_image
+    (Or.inl (by norm_num : (0 : ℝ)≤1)) ((s.domRestrict F) ⁻¹' A)
+  have ha := hF.hausdorffMeasure_preimage_le (by norm_num : (0 : ℝ)≤1) A
+  rw [he,hausdorffMeasure_real] at hi
+  rw [← hi,hausdorffMeasure_real,ENNReal.rpow_one] at ha
+  exact ha
+lemma line_map_restrict_volume_le {F : ℝ → ℝ} (hm : Measurable F)
+    {s B : Set (ℝ)} (_hB : MeasurableSet B) {K : ℝ≥0}
+    (hF : AntilipschitzWith K (s.domRestrict F)) (himage : MapsTo F s B) :
+    (volume.restrict s).map F ≤ (K : ENNReal) • volume.restrict B := by
+  apply Measure.le_iff.mpr
+  intro A hA
+  rw [Measure.map_apply hm hA,Measure.restrict_apply (hm hA),Measure.smul_apply,
+    Measure.restrict_apply hA,smul_eq_mul]
+  have he : F ⁻¹' A ∩ s = F ⁻¹' (A ∩ B) ∩ s := by
+    ext x
+    exact ⟨fun hx => ⟨⟨hx.1,himage hx.2⟩,hx.2⟩,fun hx => ⟨hx.1.1,hx.2⟩⟩
+  rw [he]
+  exact line_volume_preimage_bound hF (A ∩ B)
+end EuclideanFiveColor
+end
+
+noncomputable section
+open MeasureTheory Filter Topology
+open scoped NNReal
+namespace EuclideanFiveColor
+variable {X Y : Type*} [MeasurableSpace X] [TopologicalSpace Y] [NormalSpace Y]
+  [MeasurableSpace Y] [BorelSpace Y] {μ : Measure X} {ν : Measure Y}
+omit [TopologicalSpace Y] [NormalSpace Y] [BorelSpace Y] in
+lemma integrable_comp_of_map_le {T : X → Y} (hT : Measurable T) {C : ℝ≥0}
+    (hmap : μ.map T ≤ (C : ENNReal) • ν) {f : Y → ℝ} (hf : Integrable f ν) :
+    Integrable (fun x => f (T x)) μ := by
+  have hh := (hf.smul_measure (by simp : (C : ENNReal) ≠ ⊤)).mono_measure hmap
+  exact (integrable_map_measure hh.aestronglyMeasurable hT.aemeasurable).mp hh
+omit [TopologicalSpace Y] [NormalSpace Y] [BorelSpace Y] in
+lemma integral_norm_comp_le {T : X → Y} (hT : Measurable T) {C : ℝ≥0}
+    (hmap : μ.map T ≤ (C : ENNReal) • ν) {f : Y → ℝ} (hf : Integrable f ν) :
+    (∫ x, ‖f (T x)‖ ∂μ) ≤ (C : ℝ)*(∫ y, ‖f y‖ ∂ν) := by
+  have hh := (hf.smul_measure (by simp : (C : ENNReal) ≠ ⊤)).mono_measure hmap
+  rw [← integral_map hT.aemeasurable hh.norm.aestronglyMeasurable]
+  calc
+    _ ≤ ∫ y, ‖f y‖ ∂((C : ENNReal) • ν) :=
+      integral_mono_measure hmap (Eventually.of_forall fun y => norm_nonneg _) (hf.norm.smul_measure (by simp))
+    _ = _ := by rw [integral_smul_measure]; simp
+/-- L¹ continuity under measurable perturbations with a common bounded pushforward density. -/
+lemma tendsto_integral_norm_comp_sub [IsFiniteMeasure μ] [ν.WeaklyRegular]
+    {T : ℕ → X → Y} {S : X → Y} {C : ℝ≥0}
+    (hT : ∀ n, Measurable (T n)) (hS : Measurable S)
+    (hmT : ∀ n, μ.map (T n) ≤ (C : ENNReal) • ν) (hmS : μ.map S ≤ (C : ENNReal) • ν)
+    (ht : ∀ᵐ x ∂μ, Tendsto (fun n => T n x) atTop (𝓝 (S x)))
+    {f : Y → ℝ} (hf : Integrable f ν) :
+    Tendsto (fun n => ∫ x, ‖f (T n x)-f (S x)‖ ∂μ) atTop (𝓝 0) := by
+  apply Metric.tendsto_atTop.mpr
+  intro ε hε
+  let δ := ε/(3*((C : ℝ)+1))
+  have hδ : 0<δ := by dsimp [δ]; positivity
+  obtain ⟨g,hg,hgi⟩ := hf.exists_boundedContinuous_integral_sub_le hδ
+  have hgT (n : ℕ) : Integrable (fun x => g (T n x)) μ :=
+    integrable_comp_of_map_le (hT n) (hmT n) hgi
+  have hgS : Integrable (fun x => g (S x)) μ := integrable_comp_of_map_le hS hmS hgi
+  have hfT (n : ℕ) := integrable_comp_of_map_le (hT n) (hmT n) hf
+  have hfS := integrable_comp_of_map_le hS hmS hf
+  have hmid : Tendsto (fun n => ∫ x, ‖g (T n x)-g (S x)‖ ∂μ) atTop (𝓝 0) := by
+    have hp := tendsto_integral_of_dominated_convergence (μ := μ) (fun _ => 2*‖g‖)
+      (fun n => ((hgT n).sub hgS).norm.aestronglyMeasurable) (integrable_const _)
+      (fun n => Eventually.of_forall fun x => by
+        rw [norm_norm]
+        exact (norm_sub_le _ _).trans (by linarith [g.norm_coe_le_norm (T n x),g.norm_coe_le_norm (S x)]))
+      (ht.mono fun x hx => by
+        have hh := (g.continuous.tendsto (S x)).comp hx
+        simpa using (hh.sub tendsto_const_nhds).norm)
+    simpa using hp
+  obtain ⟨N,hN⟩ := Metric.tendsto_atTop.mp hmid (ε/3) (by positivity)
+  refine ⟨N,fun n hn => ?_⟩
+  have h1 : (∫ x, ‖f (T n x)-g (T n x)‖ ∂μ) ≤ (C : ℝ)*δ :=
+    (integral_norm_comp_le (hT n) (hmT n) (hf.sub hgi)).trans (mul_le_mul_of_nonneg_left hg C.2)
+  have h2 : (∫ x, ‖f (S x)-g (S x)‖ ∂μ) ≤ (C : ℝ)*δ :=
+    (integral_norm_comp_le hS hmS (hf.sub hgi)).trans (mul_le_mul_of_nonneg_left hg C.2)
+  have htri : (∫ x, ‖f (T n x)-f (S x)‖ ∂μ) ≤
+      (∫ x, ‖f (T n x)-g (T n x)‖ ∂μ) + (∫ x, ‖g (T n x)-g (S x)‖ ∂μ) +
+      (∫ x, ‖f (S x)-g (S x)‖ ∂μ) := by
+    have ha : Integrable (fun x => ‖f (T n x)-g (T n x)‖) μ := ((hfT n).sub (hgT n)).norm
+    have hb : Integrable (fun x => ‖g (T n x)-g (S x)‖) μ := ((hgT n).sub hgS).norm
+    have hc : Integrable (fun x => ‖f (S x)-g (S x)‖) μ := (hfS.sub hgS).norm
+    have hi := integral_mono ((hfT n).sub hfS).norm ((ha.add hb).add hc) (fun x => by
+      simpa only [Pi.add_apply,Pi.sub_apply,← dist_eq_norm,dist_comm (g (S x))] using
+        dist_triangle4 (f (T n x)) (g (T n x)) (g (S x)) (f (S x)))
+    apply hi.trans_eq
+    exact (integral_add (ha.add hb) hc).trans (congrArg (fun t => t + ∫ x, ‖f (S x)-g (S x)‖ ∂μ) (integral_add ha hb))
+  have hδv : (C : ℝ)*δ ≤ ε/3 := by
+    dsimp [δ]
+    rw [← mul_div_assoc]
+    apply (div_le_iff₀ (by positivity : 0<3*((C : ℝ)+1))).mpr
+    nlinarith [C.2]
+  rw [Real.dist_eq,sub_zero,abs_of_nonneg (integral_nonneg fun x => norm_nonneg _)]
+  have hm := hN n hn
+  rw [Real.dist_eq,sub_zero,abs_of_nonneg (integral_nonneg fun x => norm_nonneg _)] at hm
+  linarith
+end EuclideanFiveColor
+end
+
+noncomputable section
+open MeasureTheory Filter Topology
+open scoped NNReal
+namespace EuclideanFiveColor
+lemma tendsto_integral_norm_comp_sub_eventual {X Y : Type*} [MeasurableSpace X]
+    [TopologicalSpace Y] [NormalSpace Y] [MeasurableSpace Y] [BorelSpace Y]
+    {μ : Measure X} {ν : Measure Y} [IsFiniteMeasure μ] [ν.WeaklyRegular]
+    {T : ℕ → X → Y} {S : X → Y} {C : ℝ≥0}
+    (hT : ∀ n, Measurable (T n)) (hS : Measurable S)
+    (hmT : ∀ᶠ n in atTop, μ.map (T n) ≤ (C : ENNReal) • ν) (hmS : μ.map S ≤ (C : ENNReal) • ν)
+    (ht : ∀ᵐ x ∂μ, Tendsto (fun n => T n x) atTop (𝓝 (S x)))
+    {f : Y → ℝ} (hf : Integrable f ν) :
+    Tendsto (fun n => ∫ x, ‖f (T n x)-f (S x)‖ ∂μ) atTop (𝓝 0) := by
+  obtain ⟨N,hN⟩ := eventually_atTop.mp hmT
+  apply (tendsto_add_atTop_iff_nat N).mp
+  apply tendsto_integral_norm_comp_sub (T := fun n => T (n+N)) (C := C)
+    (fun n => hT (n+N)) hS (fun n => hN _ (Nat.le_add_left N n)) hmS
+  · exact ht.mono fun x hx => (tendsto_add_atTop_iff_nat N).mpr hx
+  · exact hf
+end EuclideanFiveColor
+end
+
+noncomputable section
+open MeasureTheory Set Filter Topology
+open scoped NNReal
+namespace EuclideanFiveColor
+lemma line_antilipschitz_qmp {S : ℝ → ℝ} (hS : Measurable S) {I : Set ℝ} {K : ℝ≥0}
+    (hA : AntilipschitzWith K (I.domRestrict S)) :
+    Measure.QuasiMeasurePreserving S (volume.restrict I) volume := by
+  refine ⟨hS,?_⟩
+  have hle := line_map_restrict_volume_le hS MeasurableSet.univ hA (mapsTo_univ S I)
+  rw [Measure.restrict_univ] at hle
+  exact Measure.absolutelyContinuous_of_le_smul hle
+lemma line_indicator_pullback_L1 {T : ℕ → ℝ → ℝ} {S : ℝ → ℝ} {I B D : Set ℝ} {K : ℝ≥0}
+    [IsFiniteMeasure (volume.restrict I)] [IsFiniteMeasure (volume.restrict B)]
+    (hT : ∀ n, Measurable (T n)) (hS : Measurable S)
+    (hB : IsCompact B) (hD : MeasurableSet D)
+    (hAT : ∀ᶠ n in atTop, AntilipschitzWith K (I.domRestrict (T n)))
+    (hAS : AntilipschitzWith K (I.domRestrict S))
+    (hBT : ∀ᶠ n in atTop, MapsTo (T n) I B) (hBS : MapsTo S I B)
+    (ht : ∀ᵐ x ∂volume.restrict I, Tendsto (fun n => T n x) atTop (𝓝 (S x))) :
+    Tendsto (fun n => ∫ x in I, ‖D.indicator (fun _ => (1 : ℝ)) (T n x)-
+      D.indicator (fun _ => (1 : ℝ)) (S x)‖) atTop (𝓝 0) := by
+  have : (volume.restrict B).WeaklyRegular := Measure.WeaklyRegular.restrict_of_measure_ne_top hB.measure_lt_top.ne
+  apply tendsto_integral_norm_comp_sub_eventual (ν := volume.restrict B) (C := K) hT hS
+  · filter_upwards [hAT,hBT] with n ha hb
+    exact line_map_restrict_volume_le (hT n) hB.measurableSet ha hb
+  · exact line_map_restrict_volume_le hS hB.measurableSet hAS hBS
+  · exact ht
+  · exact (integrable_const (1 : ℝ)).indicator hD
+end EuclideanFiveColor
+end
+
+noncomputable section
+open MeasureTheory Filter Topology Set
+open scoped NNReal ENNReal
+namespace EuclideanFiveColor
+lemma deriv_lower_antilipschitz {f D : ℝ → ℝ} (hd : ∀ x, HasDerivAt f (D x) x)
+    (hp : ∀ x, (1/2 : ℝ) ≤ D x) : AntilipschitzWith 2 f := by
+  apply AntilipschitzWith.of_le_mul_dist
+  intro x y
+  wlog hxy : x ≤ y generalizing x y
+  · simpa only [dist_comm] using this y x (le_of_not_ge hxy)
+  have hm := mul_sub_le_image_sub_of_le_deriv (fun x => (hd x).differentiableAt)
+    (fun x => by rw [(hd x).deriv]; exact hp x) hxy
+  have hmono : f x ≤ f y := le_of_sub_nonneg (by nlinarith)
+  rw [Real.dist_eq,Real.dist_eq,abs_of_nonpos (sub_nonpos.mpr hxy),
+    abs_of_nonpos (sub_nonpos.mpr hmono)]
+  norm_num
+  linarith
+lemma deriv_lower_qmp {f D : ℝ → ℝ} (hd : ∀ x, HasDerivAt f (D x) x)
+    (hp : ∀ x, (1/2 : ℝ) ≤ D x) : Measure.QuasiMeasurePreserving f volume volume := by
+  have hA := (deriv_lower_antilipschitz hd hp).comp
+    ((isometry_subtype_coe : Isometry (Subtype.val : {x : ℝ // x ∈ (univ : Set ℝ)} → ℝ)).antilipschitzWith)
+  simpa only [Measure.restrict_univ] using
+    (line_antilipschitz_qmp (I := univ) (continuous_iff_continuousAt.mpr (fun x => (hd x).continuousAt)).measurable hA)
+lemma iterComposition_invariant {f : ℕ → ℝ → ℝ} {A : Set ℝ}
+    (hq : ∀ n, Measure.QuasiMeasurePreserving (f n) volume volume)
+    (hA : ∀ n, ∀ᵐ x : ℝ ∂volume, x ∈ A ↔ f n x ∈ A) (N : ℕ) :
+    ∀ᵐ x : ℝ ∂volume, x ∈ A ↔ iterComposition f N x ∈ A := by
+  have hq' : ∀ n, Measure.QuasiMeasurePreserving (iterComposition f n) volume volume := by
+    intro n
+    induction n with
+    | zero => exact Measure.QuasiMeasurePreserving.id _
+    | succ n ih => exact (hq n).comp ih
+  induction N with
+  | zero => exact Filter.Eventually.of_forall (fun x => Iff.rfl)
+  | succ N ih =>
+    filter_upwards [ih,(hq' N).ae (hA N)] with x hx hxx
+    exact hx.trans hxx
+lemma expanding_interval_defect {ι : Type*} {F D : ι → ℝ → ℝ}
+    {K : ℝ≥0} {e U L a b : ℝ} {A : Set ℝ}
+    (hd : ∀ j x, HasDerivAt (F j) (D j x) x)
+    (hp : ∀ j x, (1/2 : ℝ) ≤ D j x) (hup : ∀ j x, D j x ≤ U)
+    (hK : ∀ j, LipschitzWith K (D j)) (hlarge : ∀ x, ∃ j, 1+e ≤ D j x)
+    (he : 0 < e) (hKL : (K : ℝ)*L ≤ e) (hU : 0 ≤ U)
+    (hinv : ∀ j, ∀ᵐ x : ℝ ∂volume, x ∈ A ↔ F j x ∈ A)
+    (hab : a < b) (hsmall : b-a < L) :
+    ∃ a' b' : ℝ, L ≤ b'-a' ∧ b'-a' ≤ U*L ∧
+      volume (Icc a' b' \ A) ≤ ENNReal.ofReal (Real.exp (4*(K : ℝ)*U*L/e)*U*L/(b-a))*
+        volume (Icc a b \ A) := by
+  obtain ⟨w,N,hN,hNu,hr⟩ := exists_expanding_word hd hp hup hK hlarge he hKL hU hab hsmall
+  let f := fun n => F (w n)
+  let G := iterComposition f N
+  have hd' := iterComposition_hasDeriv (fun n x => hd (w n) x) N
+  have hp' := iterDerivative_pos (f := f) (fun n x => lt_of_lt_of_le (by norm_num) (hp (w n) x)) N
+  have hm := iterComposition_strictMono (fun n x => hd (w n) x)
+    (fun n x => lt_of_lt_of_le (by norm_num) (hp (w n) x)) N
+  have hLip := derivative_ratio_lipschitzOn hab hd' hp' (Real.exp_pos _).le hr
+  have hAi := iterComposition_invariant (fun n => deriv_lower_qmp (hd (w n)) (hp (w n)))
+    (fun n => hinv (w n)) N
+  refine ⟨G a,G b,hN,hNu,?_⟩
+  apply (interval_defect_image_bound hab.le
+    (fun x _ => (hd' x).continuousAt.continuousWithinAt) (hm.monotone.monotoneOn _) hLip hAi).trans
+  gcongr
+  change ENNReal.ofReal _ ≤ ENNReal.ofReal _
+  apply ENNReal.ofReal_le_ofReal
+  have := mul_le_mul_of_nonneg_left hNu (Real.exp_pos (4*(K : ℝ)*U*L/e)).le
+  calc
+    Real.exp (4*(K : ℝ)*U*L/e)*((G b-G a)/(b-a)) =
+      (Real.exp (4*(K : ℝ)*U*L/e)*(G b-G a))/(b-a) := by ring
+    _ ≤ Real.exp (4*(K : ℝ)*U*L/e)*U*L/(b-a) := by
+      apply div_le_div_of_nonneg_right _ (sub_pos.mpr hab).le
+      simpa only [mul_assoc] using this
+end EuclideanFiveColor
+end
+
+noncomputable section
+open MeasureTheory Filter Topology Set
+namespace EuclideanFiveColor
+lemma periodic_interval_defect {A : Set ℝ} {P : ℝ}
+    (hper : ∀ (n : ℤ) x, x+(n : ℝ)*P ∈ A ↔ x ∈ A) (n : ℤ) (a b : ℝ) :
+    volume (Icc (a-(n : ℝ)*P) (b-(n : ℝ)*P) \ A) = volume (Icc a b \ A) := by
+  have he : Icc (a-(n : ℝ)*P) (b-(n : ℝ)*P) \ A =
+      (fun x : ℝ => x+(n : ℝ)*P) ⁻¹' (Icc a b \ A) := by
+    ext x
+    simp only [mem_preimage,Set.mem_sdiff,mem_Icc,hper]
+    constructor <;> rintro ⟨⟨h₁,h₂⟩,h₃⟩ <;> exact ⟨⟨by linarith,by linarith⟩,h₃⟩
+  rw [he,measure_preimage_add_right]
+lemma periodic_interval_normalize {A : Set ℝ} {P : ℝ} (hP : 0 < P)
+    (hper : ∀ (n : ℤ) x, x+(n : ℝ)*P ∈ A ↔ x ∈ A) (a b : ℝ) :
+    ∃ a' b' : ℝ, a' ∈ Icc 0 P ∧ b'-a'=b-a ∧
+      volume (Icc a' b' \ A)=volume (Icc a b \ A) := by
+  let n : ℤ := ⌊a/P⌋
+  refine ⟨a-(n : ℝ)*P,b-(n : ℝ)*P,?_,by ring,periodic_interval_defect hper n a b⟩
+  have h := Int.fract_div_mul_self_mem_Ico P a hP
+  have he : Int.fract (a/P)*P=a-(n : ℝ)*P := by
+    change (a/P-(⌊a/P⌋ : ℝ))*P=a-(⌊a/P⌋ : ℝ)*P
+    field_simp
+  rw [he] at h
+  exact ⟨h.1,h.2.le⟩
+lemma compact_almost_conull_interval {A : Set ℝ} {P L : ℝ}
+    {a b : ℕ → ℝ} (hL : 0 < L) (ha : ∀ n, a n ∈ Icc 0 P)
+    (hl : ∀ n, L ≤ b n-a n)
+    (hdef : Tendsto (fun n => volume (Icc (a n) (b n) \ A)) atTop (𝓝 0)) :
+    ∃ x y : ℝ, y-x=L/2 ∧ volume (Icc x y \ A)=0 := by
+  obtain ⟨z,hz,φ,hφ,hzlim⟩ := isCompact_Icc.tendsto_subseq ha
+  have he : ∀ᶠ n in atTop, Icc (z+L/4) (z+3*L/4) ⊆ Icc (a (φ n)) (b (φ n)) := by
+    have hdist := (Metric.tendsto_nhds.mp hzlim) (L/4) (by positivity)
+    filter_upwards [hdist] with n hn
+    rw [Real.dist_eq,abs_lt] at hn
+    dsimp only [Function.comp_apply] at hn
+    intro x hx
+    have hh := hl (φ n)
+    constructor <;> linarith [hx.1,hx.2]
+  have hle : ∀ᶠ n in atTop, volume (Icc (z+L/4) (z+3*L/4) \ A) ≤
+      volume (Icc (a (φ n)) (b (φ n)) \ A) := he.mono (fun _ h => measure_mono (sdiff_subset_sdiff_left h))
+  refine ⟨z+L/4,z+3*L/4,by ring,?_⟩
+  exact le_antisymm (ge_of_tendsto (hdef.comp hφ.tendsto_atTop) hle) bot_le
+end EuclideanFiveColor
+end
+
+noncomputable section
+open MeasureTheory Filter Topology Set
+open scoped ENNReal
+namespace EuclideanFiveColor
+lemma exists_density_interval {A : Set ℝ} (hA : MeasurableSet A) (hpos : volume A ≠ 0)
+    {L ε : ℝ} (hL : 0 < L) (hε : 0 < ε) :
+    ∃ a b : ℝ, a < b ∧ b-a < L ∧
+      volume (Icc a b \ A) ≤ ENNReal.ofReal ε*ENNReal.ofReal (b-a) := by
+  have h := Besicovitch.ae_tendsto_measure_inter_div_of_measurableSet (volume : Measure ℝ) hA.compl
+  obtain ⟨x,hx,ht⟩ := Measure.exists_mem_of_measure_ne_zero_of_ae hpos (ae_restrict_of_ae h)
+  have hx' : x ∉ Aᶜ := by simpa using hx
+  simp only [indicator_of_notMem hx'] at ht
+  have he : (0 : ℝ≥0∞) < ENNReal.ofReal ε := ENNReal.ofReal_pos.mpr hε
+  have hev := ht.eventually (gt_mem_nhds he)
+  have hsmall : ∀ᶠ r : ℝ in 𝓝[>] 0, r < L/2 :=
+    (eventually_lt_nhds (by positivity : (0 : ℝ)<L/2)).filter_mono nhdsWithin_le_nhds
+  have hev' : ∀ᶠ r : ℝ in 𝓝[>] 0,
+      0 < r ∧ r < L/2 ∧ volume (Aᶜ ∩ Metric.closedBall x r)/volume (Metric.closedBall x r)<ENNReal.ofReal ε := by
+    filter_upwards [self_mem_nhdsWithin,hev,hsmall] with r hr hh hs
+    exact ⟨hr,hs,hh⟩
+  obtain ⟨r,hr,hs,hh⟩ := hev'.exists
+  refine ⟨x-r,x+r,by linarith,by linarith,?_⟩
+  rw [Real.closedBall_eq_Icc] at hh
+  have heq : Aᶜ ∩ Icc (x-r) (x+r)=Icc (x-r) (x+r) \ A := by ext; simp [and_comm]
+  rw [heq,Real.volume_Icc] at hh
+  exact (ENNReal.div_le_iff (ne_of_gt (ENNReal.ofReal_pos.mpr (by linarith))) ENNReal.ofReal_ne_top).mp hh.le
+end EuclideanFiveColor
+end
+
+noncomputable section
+open MeasureTheory Filter Topology Set
+open scoped NNReal ENNReal
+namespace EuclideanFiveColor
+lemma expansion_almost_conull {ι : Type*} {F D : ι → ℝ → ℝ}
+    {K : ℝ≥0} {e U L : ℝ} {A : Set ℝ}
+    (hd : ∀ j x, HasDerivAt (F j) (D j x) x)
+    (hp : ∀ j x, (1/2 : ℝ) ≤ D j x) (hup : ∀ j x, D j x ≤ U)
+    (hK : ∀ j, LipschitzWith K (D j)) (hlarge : ∀ x, ∃ j, 1+e ≤ D j x)
+    (he : 0 < e) (hKL : (K : ℝ)*L ≤ e) (hU : 0 < U) (hL : 0 < L)
+    (hinv : ∀ j, ∀ᵐ x : ℝ ∂volume, x ∈ A ↔ F j x ∈ A)
+    (hA : MeasurableSet A) (hpos : volume A ≠ 0) {ε : ℝ} (hε : 0 < ε) :
+    ∃ a b : ℝ, L ≤ b-a ∧ volume (Icc a b \ A) ≤ ENNReal.ofReal ε := by
+  let C := Real.exp (4*(K : ℝ)*U*L/e)*U*L
+  have hC : 0 < C := mul_pos (mul_pos (Real.exp_pos _) hU) hL
+  obtain ⟨a,b,hab,hsmall,hdef⟩ := exists_density_interval hA hpos hL (div_pos hε hC)
+  obtain ⟨a',b',hl,_,hbound⟩ := expanding_interval_defect hd hp hup hK hlarge he hKL hU.le hinv hab hsmall
+  refine ⟨a',b',hl,hbound.trans ?_⟩
+  change ENNReal.ofReal (C/(b-a))*volume (Icc a b \ A) ≤ ENNReal.ofReal ε
+  calc
+    _ ≤ ENNReal.ofReal (C/(b-a))*(ENNReal.ofReal (ε/C)*ENNReal.ofReal (b-a)) :=
+      mul_le_mul_right hdef _
+    _ = ENNReal.ofReal ((C/(b-a))*((ε/C)*(b-a))) := by
+      rw [← ENNReal.ofReal_mul (div_pos hε hC).le,← ENNReal.ofReal_mul (div_pos hC (sub_pos.mpr hab)).le]
+    _ = ENNReal.ofReal ε := by congr 1; field_simp [hC.ne', (sub_pos.mpr hab).ne']
+lemma expansion_conull_interval {ι : Type*} {F D : ι → ℝ → ℝ}
+    {K : ℝ≥0} {e U L P : ℝ} {A : Set ℝ}
+    (hd : ∀ j x, HasDerivAt (F j) (D j x) x)
+    (hp : ∀ j x, (1/2 : ℝ) ≤ D j x) (hup : ∀ j x, D j x ≤ U)
+    (hK : ∀ j, LipschitzWith K (D j)) (hlarge : ∀ x, ∃ j, 1+e ≤ D j x)
+    (he : 0 < e) (hKL : (K : ℝ)*L ≤ e) (hU : 0 < U) (hL : 0 < L) (hP : 0 < P)
+    (hinv : ∀ j, ∀ᵐ x : ℝ ∂volume, x ∈ A ↔ F j x ∈ A)
+    (hA : MeasurableSet A) (hpos : volume A ≠ 0)
+    (hper : ∀ (n : ℤ) x, x+(n : ℝ)*P ∈ A ↔ x ∈ A) :
+    ∃ a b : ℝ, b-a=L/2 ∧ volume (Icc a b \ A)=0 := by
+  have hex : ∀ n : ℕ, ∃ a b : ℝ, a ∈ Icc 0 P ∧ L ≤ b-a ∧
+      volume (Icc a b \ A) ≤ ENNReal.ofReal (1/((n : ℝ)+1)) := by
+    intro n
+    obtain ⟨a,b,hl,hdef⟩ := expansion_almost_conull hd hp hup hK hlarge he hKL hU hL hinv hA hpos
+      (by positivity : (0 : ℝ)<1/((n : ℝ)+1))
+    obtain ⟨a',b',ha',hlen,hdef'⟩ := periodic_interval_normalize hP hper a b
+    exact ⟨a',b',ha',by rwa [hlen],hdef'.trans_le hdef⟩
+  choose a b ha hl hdef using hex
+  apply compact_almost_conull_interval hL ha hl
+  have ht : Tendsto (fun n : ℕ => (1 : ℝ)/((n : ℝ)+1)) atTop (𝓝 0) :=
+    tendsto_one_div_add_atTop_nhds_zero_nat
+  exact tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds
+    (by simpa only [Function.comp_def,ENNReal.ofReal_zero] using ENNReal.continuous_ofReal.continuousAt.tendsto.comp ht)
+    (fun _ => zero_le) hdef
+end EuclideanFiveColor
+end
+
+noncomputable section
+open MeasureTheory Filter Topology Set
+open scoped NNReal ENNReal
+namespace EuclideanFiveColor
+lemma conull_extend_intervals {ι : Type*} {F : ι → ℝ → ℝ} {U : ℝ≥0}
+    {e B a b : ℝ} {A : Set ℝ}
+    (hLip : ∀ j, LipschitzWith U (F j)) (hmono : ∀ j, Monotone (F j))
+    (hadvance : ∀ x, ∃ j, x+e ≤ F j x) (hdispl : ∀ j x, F j x ≤ x+B)
+    (he : 0 < e) (hab : a ≤ b) (hB : B ≤ b-a)
+    (hinv : ∀ j, ∀ᵐ x : ℝ ∂volume, x ∈ A ↔ F j x ∈ A)
+    (hnull : volume (Icc a b \ A)=0) (n : ℕ) :
+    ∃ b' : ℝ, b+(n : ℝ)*e ≤ b' ∧ volume (Icc a b' \ A)=0 := by
+  induction n with
+  | zero => exact ⟨b,by simp,hnull⟩
+  | succ n ih =>
+    obtain ⟨b',hb',hnull'⟩ := ih
+    obtain ⟨j,hj⟩ := hadvance b'
+    have hb : b ≤ b' := le_trans (le_add_of_nonneg_right (mul_nonneg (Nat.cast_nonneg n) he.le)) hb'
+    have hfa : F j a ≤ b' := (hdispl j a).trans (by linarith)
+    have himage : volume (Icc (F j a) (F j b') \ A)=0 := by
+      apply le_antisymm _ bot_le
+      exact (interval_defect_image_bound (hab.trans hb) (hLip j).continuous.continuousOn
+        ((hmono j).monotoneOn _) (hLip j).lipschitzOnWith (hinv j)).trans (by simp [hnull'])
+    refine ⟨F j b',?_,?_⟩
+    · push_cast
+      nlinarith
+    · apply measure_mono_null (t := (Icc a b' \ A) ∪ (Icc (F j a) (F j b') \ A))
+      · intro x hx
+        by_cases hxb : x ≤ b'
+        · exact Or.inl ⟨⟨hx.1.1,hxb⟩,hx.2⟩
+        · exact Or.inr ⟨⟨le_trans hfa (le_of_not_ge hxb),hx.1.2⟩,hx.2⟩
+      · exact le_antisymm ((measure_union_le _ _).trans (by rw [hnull',himage,add_zero])) bot_le
+lemma conull_interval_alternation_false {A : Set ℝ} {a b κ : ℝ}
+    (hκ : 0 < κ) (hab : κ < b-a) (hnull : volume (Icc a b \ A)=0)
+    (halt : ∀ᵐ x : ℝ ∂volume, x+κ ∈ A ↔ x ∉ A) : False := by
+  have hmem : ∀ᵐ x : ℝ ∂volume, x ∈ Icc a b → x ∈ A := by
+    rw [ae_iff]
+    exact measure_mono_null (fun x hx => by
+      simp only [Classical.not_imp] at hx
+      exact hx) hnull
+  have hmem' := (measurePreserving_add_right volume κ).quasiMeasurePreserving.ae hmem
+  have hbad : ∀ᵐ x : ℝ ∂volume, x ∉ Icc a (b-κ) := by
+    filter_upwards [hmem,hmem',halt] with x hx hx' ha hxin
+    have hxa : x ∈ A := hx ⟨hxin.1,by linarith [hxin.2]⟩
+    have hxka : x+κ ∈ A := hx' ⟨by linarith [hxin.1],by linarith [hxin.2]⟩
+    exact (ha.mp hxka) hxa
+  have hz : volume (Icc a (b-κ))=0 := by
+    have hh := ae_iff.mp hbad
+    convert hh using 2
+    ext x
+    simp only [mem_Icc,mem_ofPred_eq,not_not]
+  rw [Real.volume_Icc,ENNReal.ofReal_eq_zero] at hz
+  linarith
+end EuclideanFiveColor
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PlaneColoring/Five.lean b/lean/OAI/Geometry/PlaneColoring/Five.lean
new file mode 100644
index 0000000000000000000000000000000000000000..377372bb81c65d351df698936319b809ca2682be
--- /dev/null
+++ b/lean/OAI/Geometry/PlaneColoring/Five.lean
@@ -0,0 +1,131 @@
+import OAI.Geometry.PlaneColoring.BorelTransfer
+import OAI.Geometry.PlaneColoring.Mesh
+
+namespace OAI
+
+noncomputable section
+open scoped Classical
+open Set Topology ContinuousMap
+namespace EuclideanFiveColor
+lemma graphRealization_positive_three {V : Type*} {G : SimpleGraph V} {p : V → ℝ}
+    (hp : p∈graphRealization G) {i j k : V} (hi : 0<p i) (hj : 0<p j) (hk : 0<p k) :
+    i=j ∨ i=k ∨ j=k := by
+  obtain ⟨a,b,_,t,_,rfl⟩ := hp
+  have hpos (v : V) (hv : 0<graphEdgePoint a b t v) : v=a ∨ v=b := by
+    by_contra! hh
+    simp [graphEdgePoint_apply,hh.1,hh.2] at hv
+  rcases hpos i hi with rfl|rfl <;> rcases hpos j hj with rfl|rfl <;>
+    rcases hpos k hk with rfl|rfl <;> simp
+
+/-- The actual positive-coordinate component cover contradicts the planar cover
+obstruction whenever every connected positive region has diameter at most two. -/
+theorem graph_map_small_mesh_impossible
+    (F : C(Metric.closedBall (0:ℂ) 5,GraphSpace (⊤ : SimpleGraph (Fin 5))))
+    (hmesh : ∀(i : Fin 5) (K : Set ℂ),IsPreconnected K → K⊆Metric.closedBall (0:ℂ) 5 →
+      (∀z : Metric.closedBall (0:ℂ) 5,z.val∈K → 0<(F z).val i) →
+      ∀x∈K,∀y∈K,dist x y≤2) : False := by
+  let Q := Metric.closedBall (0:ℂ) 5
+  have : LocallyPathConnectedSpace Q := (convex_closedBall (0:ℂ) (5:ℝ)).locallyPathConnectedSpace
+  let U : Fin 5 → Set Q := fun i => {z | 0<(F z).val i}
+  have hU (i : Fin 5) : IsOpen (U i) :=
+    isOpen_lt continuous_const ((continuous_apply i).comp (continuous_subtype_val.comp F.continuous))
+  let I := {p : Fin 5 × Set Q // ∃x : Q,p.2=connectedComponentIn (U p.1) x}
+  let W : I → Set Q := fun p => p.val.2
+  have hW (p : I) : IsOpen (W p) := by
+    obtain ⟨x,hx⟩ := p.property
+    change IsOpen p.val.2
+    rw [hx]
+    exact (hU _).connectedComponentIn
+  have hpos {p : I} {z : Q} (hz : z∈W p) : 0<(F z).val p.val.1 := by
+    obtain ⟨x,hx⟩ := p.property
+    change z∈p.val.2 at hz
+    rw [hx] at hz
+    exact connectedComponentIn_subset (U p.val.1) x hz
+  have heq {p q : I} {z : Q} (hpq : p.val.1=q.val.1) (hp : z∈W p) (hq : z∈W q) : p=q := by
+    apply Subtype.ext
+    apply Prod.ext hpq
+    obtain ⟨x,hx⟩ := p.property
+    obtain ⟨y,hy⟩ := q.property
+    change z∈p.val.2 at hp
+    change z∈q.val.2 at hq
+    rw [hx,hpq] at hp
+    rw [hy] at hq
+    rw [hx,hy,hpq,connectedComponentIn_eq hp,connectedComponentIn_eq hq]
+  apply PlanarCover.compact_open_cover_impossible Metric.ball_subset_closedBall
+    (isCompact_closedBall (0:ℂ) 5) W hW ?_ ?_ ?_
+  · intro z
+    have he : ∃i,0<(F z).val i := by
+      by_contra! hh
+      have hsum := Finset.sum_nonpos (s := (Finset.univ : Finset (Fin 5))) (fun i _ => hh i)
+      rw [graphRealization_sum (F z).property] at hsum
+      norm_num at hsum
+    obtain ⟨i,hi⟩ := he
+    exact ⟨⟨(i,connectedComponentIn (U i) z),⟨z,rfl⟩⟩,mem_connectedComponentIn hi⟩
+  · intro z p q r hp hq hr
+    rcases graphRealization_positive_three (F z).property (hpos hp) (hpos hq) (hpos hr) with h|h|h
+    · exact Or.inl (heq h hp hq)
+    · exact Or.inr (Or.inl (heq h hp hr))
+    · exact Or.inr (Or.inr (heq h hq hr))
+  · intro p x hx y hy
+    obtain ⟨z,hz⟩ := p.property
+    let K : Set ℂ := Subtype.val '' W p
+    have hK : IsPreconnected K := by
+      apply IsPreconnected.image _ _ continuous_subtype_val.continuousOn
+      change IsPreconnected p.val.2
+      rw [hz]
+      exact isPreconnected_connectedComponentIn
+    have hsub : K⊆Q := by rintro a ⟨b,_,rfl⟩; exact b.property
+    have hprob : ∀z : Q,z.val∈K → 0<(F z).val p.val.1 := by
+      rintro z ⟨b,hb,he⟩
+      have hh : b=z := Subtype.ext he
+      subst b
+      exact hpos hb
+    have hh := hmesh p.val.1 K hK hsub hprob x.val ⟨x,hx,rfl⟩ y.val ⟨y,hy,rfl⟩
+    simpa only [dist_eq_norm] using hh
+end EuclideanFiveColor
+end
+
+noncomputable section
+open scoped Classical
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+/-- Every continuous graph-valued map of the disk has a positive-coordinate
+connected region of diameter greater than two. -/
+lemma graph_map_large_positive
+    (F : C(Metric.closedBall (0:ℂ) 5,GraphSpace (⊤ : SimpleGraph (Fin 5)))) :
+    ∃i : Fin 5,∃K : Set ℂ,IsPreconnected K ∧ K⊆Metric.closedBall (0:ℂ) 5 ∧
+      (∀z : Metric.closedBall (0:ℂ) 5,z.val∈K → 0<(F z).val i) ∧
+      ∃x∈K,∃y∈K,2<dist x y := by
+  by_contra! h
+  exact graph_map_small_mesh_impossible F h
+
+/-- No Borel weak five-coloring exists. The global approximants here are
+constructed from the actual coloring, its transition graph, and its common
+interface limits, rather than supplied as an extra coloring hypothesis. -/
+theorem no_measurable_weak_five_coloring {c : ℂ → Fin 5}
+    (hc : Measurable c) (hw : WeakColoring 5 c) : False := by
+  let t : ℕ → ℝ := fun n => min (1/16) (1/((n:ℝ)+1))
+  have hpos (n : ℕ) : 0<t n := lt_min (by norm_num) (by positivity)
+  have hsmall (n : ℕ) : t n≤1/16 := min_le_left _ _
+  have ht : Tendsto t atTop (𝓝 0) := squeeze_zero (fun n => (hpos n).le)
+    (fun n => min_le_right _ _) tendsto_one_div_add_atTop_nhds_zero_nat
+  let F : ∀n,FullApprox c (1/10) (t n) := fun n =>
+    Classical.choice (fullApprox_exists hc hw (by norm_num) (by norm_num) (hpos n))
+  have hbad : ∃ᶠn in atTop,∃i : Fin 5,∃K : Set ℂ,IsPreconnected K ∧ K⊆Metric.closedBall (0:ℂ) 5 ∧
+      (∀z : Metric.closedBall (0:ℂ) 5,z.val∈K → 0<((F n).map z).val i) ∧
+      ∃x∈K,∃y∈K,2<dist x y :=
+    (Eventually.of_forall (fun n => graph_map_large_positive (F n).map)).frequently
+  obtain ⟨i,hi⟩ := Filter.frequently_exists.mp hbad
+  obtain ⟨φ,hφ,hφbad⟩ := extraction_of_frequently_atTop hi
+  exact FullApprox.no_large_positive_sequence hc hw (by norm_num) (by norm_num)
+    (fun n => F (φ n)) (ht.comp hφ.tendsto_atTop) (fun n => hsmall (φ n)) i hφbad
+
+/-- No unrestricted five-coloring separates all unit-distance pairs in the Euclidean plane. -/
+theorem no_proper_five_coloring : ¬ ∃ c : ℂ → Fin 5, ProperColoring 5 c := by
+  rintro ⟨c,hc⟩
+  obtain ⟨d,hd,hw⟩ := proper_to_borel_weak hc
+  exact no_measurable_weak_five_coloring hd hw
+end EuclideanFiveColor
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PlaneColoring/FiveSupport.lean b/lean/OAI/Geometry/PlaneColoring/FiveSupport.lean
new file mode 100644
index 0000000000000000000000000000000000000000..3bfec05724b5a8c4af61d6d6dbbb3b80e1c980f7
--- /dev/null
+++ b/lean/OAI/Geometry/PlaneColoring/FiveSupport.lean
@@ -0,0 +1,376 @@
+import OAI.Geometry.PlaneColoring.Accumulation
+
+namespace OAI
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+variable {k : ℕ} (c : ℂ → Fin k) (hc : Measurable c)
+lemma palette_neighbor_real (hw : WeakColoring k c) (x : ℂ) (a : ℝ) (i : Fin k) :
+    ∀ᵐt : ℝ∂volume,t∈liftPalette c hc (x+direction (a : Angles)) i → t∉liftPalette c hc x i := by
+  let U : Set ℝ := {t | Real.sin (a-t)≠0}
+  have hU : IsOpen U := isOpen_ne_fun (Real.continuous_sin.comp (continuous_const.sub continuous_id)) continuous_const
+  have hu : ∀ᵐt : ℝ∂volume,t∈U := by
+    filter_upwards [ae_sin_sub_ne_zero a] with t ht
+    change Real.sin (a-t)≠0
+    rwa [← neg_sub t a,Real.sin_neg,neg_ne_zero]
+  have hh := palette_smooth_transport c hc hw (x := x) (z := x+direction (a : Angles)) hU (s := id) (t := fun _ => a)
+    (ds := fun _ => 1) (dt := fun _ => 0) (fun t _ => hasDerivAt_id t)
+    (fun t _ => hasDerivAt_const t a) (fun _ _ => one_ne_zero) (fun _ ht => ht) (fun t _ => ?_) i
+  · exact (ae_restrict_iff' hU.measurableSet).mp hh |>.mp (hu.mono fun t ht hi => hi ht)
+  · simp only [circleTarget,samplingParameter,add_sub_cancel_left,direction_realPlane,circleSum,one_smul,one_mul,id_eq]
+    exact Prod.ext (add_comm _ _) (add_comm _ _)
+lemma palette_neighbor_disjoint (hw : WeakColoring k c) (x : ℂ) (a : ℝ) :
+    ∀ᵐt∂angularMeasure,Disjoint (palette c hc (x+direction (a : Angles)) t) (palette c hc x t) := by
+  have hp (i : Fin k) : ∀ᵐt∂angularMeasure,t∈paletteSet c hc (x+direction (a : Angles)) i → t∉paletteSet c hc x i := by
+    apply ae_angular_of_ae_line
+    · simpa only [imp_iff_not_or,Set.compl_def,Set.union_def,Set.mem_ofPred_eq] using
+        (paletteSet_measurable c hc (x+direction (a : Angles)) i).compl.union (paletteSet_measurable c hc x i).compl
+    · exact palette_neighbor_real c hc hw x a i
+  filter_upwards [ae_all_iff.mpr hp] with t ht
+  exact Finset.disjoint_left.mpr fun i hi hj => ht i ((mem_palette c hc _ t i).mp hi) ((mem_palette c hc _ t i).mp hj)
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+variable {ι : Type*} [Fintype ι] [MeasurableSpace ι] [MeasurableSingletonClass ι]
+def stateSupport (f : ℝ → ι) (i : ι) : Set ℝ := (volume.restrict {t | f t=i}).support
+lemma ae_mem_stateSupport {f : ℝ → ι} (hf : Measurable f) (i : ι) :
+    ∀ᵐt : ℝ∂volume,f t=i → t∈stateSupport f i :=
+  (ae_restrict_iff' (measurableSet_eq_fun hf measurable_const)).mp
+    (volume.restrict {t | f t=i}).support_mem_ae
+lemma ae_state_support {f : ℝ → ι} (hf : Measurable f) :
+    ∀ᵐt : ℝ∂volume,t∈stateSupport f (f t) := by
+  have h : ∀ᵐt : ℝ∂volume,∀i : ι,f t=i → t∈stateSupport f i :=
+    ae_all_iff.mpr (fun i => ae_mem_stateSupport hf i)
+  exact h.mono fun t ht => ht (f t) rfl
+lemma stateSupport_cover {f : ℝ → ι} (hf : Measurable f) (a : ℝ) : ∃i,a∈stateSupport f i := by
+  have hclosed : IsClosed (⋃i,stateSupport f i)  := isClosed_iUnion_of_finite fun _ => Measure.isClosed_support
+  have hae : ∀ᵐt : ℝ∂volume,t∈⋃i,stateSupport f i :=
+    (ae_state_support hf).mono fun t ht => mem_iUnion.mpr ⟨f t,ht⟩
+  have hd := Measure.dense_of_ae hae
+  exact mem_iUnion.mp ((hclosed.closure_eq ▸ hd a) : a∈⋃i,stateSupport f i)
+lemma stateSupport_shift_relation {f : ℝ → ι} (hf : Measurable f) {R : ι → ι → Prop}
+    {d : ℝ} (hR : ∀ᵐt : ℝ∂volume,R (f t) (f (t+d))) {i : ι} {a : ℝ}
+    (ha : a∈stateSupport f i) : ∃j,R i j ∧ a+d∈stateSupport f j := by
+  classical
+  let K : Set ℝ := ⋃j∈({j | R i j} : Set ι),(fun t => t+d)⁻¹'stateSupport f j
+  have hK : IsClosed K := (Set.toFinite _).isClosed_biUnion (fun j _ =>
+    Measure.isClosed_support.preimage (continuous_id.add continuous_const))
+  have hm : ∀ᵐt : ℝ∂volume,f t=i → t∈K := by
+    filter_upwards [hR,(measurePreserving_add_right (volume : Measure ℝ) d).quasiMeasurePreserving.ae (ae_state_support hf)] with t hR ht hti
+    exact mem_iUnion.mpr ⟨f (t+d),mem_iUnion.mpr ⟨hti ▸ hR,ht⟩⟩
+  have hk : a∈K := Measure.support_subset_of_isClosed hK
+    ((ae_restrict_iff' (hf (measurableSet_singleton i))).mpr hm) ha
+  rcases mem_iUnion.mp hk with ⟨j,hj⟩
+  obtain ⟨hr,hs⟩ := mem_iUnion.mp hj
+  exact ⟨j,hr,hs⟩
+omit [Fintype ι] [MeasurableSpace ι] [MeasurableSingletonClass ι] in
+lemma stateSupport_nonempty_measure {f : ℝ → ι} {i : ι} {a : ℝ}
+    (ha : a∈stateSupport f i) : 0<volume {t | f t=i} := by
+  have h := (Measure.mem_support_iff_forall a).mp ha univ (univ_mem)
+  simpa only [Measure.restrict_apply_univ] using h
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+variable (c : ℂ → Fin 5) (hc : Measurable c)
+def fiveRealState (x : ℂ) (i : Fin 5) : Set ℝ := {t | fiveState c hc x (t : Angles)=i}
+def fiveSupport (x : ℂ) (i : Fin 5) : Set ℝ := (volume.restrict (fiveRealState c hc x i)).support
+lemma five_realState_measurable (x : ℂ) (i : Fin 5) : MeasurableSet (fiveRealState c hc x i) :=
+  measurableSet_eq_fun ((fiveState_measurable c hc x).comp angleMk_quasiMeasurePreserving.measurable) measurable_const
+lemma five_support_trace (hw : WeakColoring 5 c) {x : ℂ} (hx : FiveCenter c x) {a : ℝ} {i : Fin 5}
+    (ha : a∈fiveSupport c hc x i) :
+    ∃b : TraceVector 5,IsTrace c hc (x+direction (a : Angles)) b ∧ ∀j∈fiveDiag i,traceValue b j=0 :=
+  state_support_avoiding_trace c hc hw (fiveState_represents c hc hw hx i) ha
+lemma five_support_step_ban (hw : WeakColoring 5 c) {x : ℂ} (hx : FiveCenter c x) {a : ℝ} {i j : Fin 5}
+    (hi : a∈fiveSupport c hc x i) (hj : a∈fiveSupport c hc x j) (hij : fiveAdj i j) :
+    ∀ᵐt∂angularMeasure,¬(fiveState c hc x t=i ∧ fiveState c hc x (t+(sixty : Angles))=j) ∧
+      ¬(fiveState c hc x t=j ∧ fiveState c hc x (t+(sixty : Angles))=i) := by
+  obtain ⟨bi,hbi,hzi⟩ := five_support_trace c hc hw hx hi
+  obtain ⟨bj,hbj,hzj⟩ := five_support_trace c hc hw hx hj
+  obtain ⟨X,hX⟩ := five_adj_four i j hij
+  have hji : fiveAdj j i := hij.symm
+  have hv : ∀ᵐt∂angularMeasure,fiveState c hc x t=i ∨ fiveState c hc x t=j →
+      X∈palette c hc (x+direction (a : Angles)) t := by
+    filter_upwards [avoiding_trace_palette c hc hbi hzi,avoiding_trace_palette c hc hbj hzj,
+      palette_neighbor_disjoint c hc hw x a,fiveState_diag c hc hw hx] with t hni hnj hd he hs
+    rw [he] at hd
+    rcases hs with hi|hj
+    · rw [hi] at hd
+      exact five_forced_label i j X hij hX _ hd hnj
+    · rw [hj] at hd
+      exact five_forced_label j i X hji (by simpa only [Finset.union_comm] using hX) _ hd hni
+  filter_upwards [hv,(measurePreserving_add_right angularMeasure (sixty : Angles)).quasiMeasurePreserving.ae hv,
+    palette_sixty_disjoint c hc hw (x+direction (a : Angles))] with t ht hu hd
+  constructor
+  · rintro ⟨hi,hj⟩
+    exact Finset.disjoint_left.mp hd (ht (Or.inl hi)) (hu (Or.inr hj))
+  · rintro ⟨hj,hi⟩
+    exact Finset.disjoint_left.mp hd (ht (Or.inr hj)) (hu (Or.inl hi))
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+variable (c : ℂ → Fin 5) (hc : Measurable c)
+lemma five_support_cover (x : ℂ) (a : ℝ) : ∃i,a∈fiveSupport c hc x i :=
+  stateSupport_cover ((fiveState_measurable c hc x).comp angleMk_quasiMeasurePreserving.measurable) a
+lemma five_support_next (hw : WeakColoring 5 c) {x : ℂ} (hx : FiveCenter c x) {a : ℝ} {i : Fin 5}
+    (hi : a∈fiveSupport c hc x i) : ∃j,fiveAdj i j ∧ a+sixty∈fiveSupport c hc x j := by
+  apply stateSupport_shift_relation ((fiveState_measurable c hc x).comp angleMk_quasiMeasurePreserving.measurable) _ hi
+  simpa only [Function.comp_def,AddCircle.coe_add] using angleMk_quasiMeasurePreserving.ae (fiveState_adj c hc hw hx)
+lemma five_support_prev (hw : WeakColoring 5 c) {x : ℂ} (hx : FiveCenter c x) {a : ℝ} {i : Fin 5}
+    (hi : a∈fiveSupport c hc x i) : ∃j,fiveAdj i j ∧ a-sixty∈fiveSupport c hc x j := by
+  apply stateSupport_shift_relation ((fiveState_measurable c hc x).comp angleMk_quasiMeasurePreserving.measurable) (d := -sixty) _ hi
+  have hh := angleMk_quasiMeasurePreserving.ae (fiveState_adj c hc hw hx)
+  have hh' := (measurePreserving_add_right (volume : Measure ℝ) (-sixty)).quasiMeasurePreserving.ae hh
+  filter_upwards [hh'] with t ht
+  have he : ((t+-sixty : ℝ) : Angles)+(sixty : Angles)=(t : Angles) := by
+    rw [← AddCircle.coe_add,neg_add_cancel_right]
+  rw [he] at ht
+  exact ht.symm
+lemma five_support_no_sixty (hw : WeakColoring 5 c) {x : ℂ} (hx : FiveCenter c x) {a : ℝ} {i : Fin 5}
+    (hi : a∈fiveSupport c hc x i) : a+sixty∉fiveSupport c hc x i := by
+  obtain ⟨j,hj,hsj⟩ := five_support_next c hc hw hx hi
+  obtain ⟨X,hX⟩ := five_adj_four j i hj.symm
+  have hp : 0<volume (fiveRealState c hc x j) := stateSupport_nonempty_measure hsj
+  intro hs
+  exact strict_state_no_unit_support hc hw hX (fiveState_represents c hc hw hx j)
+    (fiveState_represents c hc hw hx i) hp hi hs (direction_sixty_unit_real a)
+end EuclideanFiveColor
+end
+
+noncomputable section
+namespace EuclideanFiveColor.FiveComb
+-- Independent finite arithmetic facts; the source labels are Z/5Z, represented by Fin 5.
+def Adj (i j : Fin 5) : Prop := i+1=j ∨ j+1=i
+instance (i j : Fin 5) : Decidable (Adj i j) := inferInstanceAs (Decidable (_∨_))
+lemma contact_neighbors : ∀i : Fin 5,∀L : Finset (Fin 5), i∉L → i+1∉L →
+    (∃j∈L,Adj i j) → (∃j∈L,Adj (i+1) j) → i-1∈L ∧ i+2∈L := by decide
+lemma opposite_contacts : ∀i j : Fin 5, Disjoint ({i-1,i+2} : Finset (Fin 5)) {j-1,j+2} →
+    j=i+1 ∨ j+1=i := by decide
+lemma adjacent_oriented : ∀I : Finset (Fin 5), (∃i∈I,∃j∈I,Adj i j) → ∃i,i∈I ∧ i+1∈I := by decide
+lemma neighbors : ∀i j : Fin 5, Adj i j → j=i-1 ∨ j=i+1 := by decide
+lemma two_bans : ∀i j : Fin 5, j=i+1 ∨ j+1=i → ∃s,
+    (s=i ∨ s=i+1) ∧ (s=j ∨ s=j+1) ∧
+    ∀t,Adj s t → ((s=i ∧ t=i+1) ∨ (s=i+1 ∧ t=i)) ∨ ((s=j ∧ t=j+1) ∨ (s=j+1 ∧ t=j)) := by decide
+end EuclideanFiveColor.FiveComb
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+variable (c : ℂ → Fin 5) (hc : Measurable c)
+def fiveSupportList (x : ℂ) (a : ℝ) : Finset (Fin 5) := by
+  classical
+  exact Finset.univ.filter (fun i => a∈fiveSupport c hc x i)
+@[simp] lemma mem_fiveSupportList (x : ℂ) (a : ℝ) (i : Fin 5) :
+    i∈fiveSupportList c hc x a ↔ a∈fiveSupport c hc x i := by simp only [fiveSupportList,Finset.mem_filter,Finset.mem_univ,true_and]
+lemma five_contact_next (hw : WeakColoring 5 c) {x : ℂ} (hx : FiveCenter c x) {a : ℝ} {i : Fin 5}
+    (hi : a∈fiveSupport c hc x i) (hi' : a∈fiveSupport c hc x (i+1)) :
+    i-1∈fiveSupportList c hc x (a+sixty) ∧ i+2∈fiveSupportList c hc x (a+sixty) := by
+  apply FiveComb.contact_neighbors
+  · exact fun h => five_support_no_sixty c hc hw hx hi ((mem_fiveSupportList _ _ _ _ _).mp h)
+  · exact fun h => five_support_no_sixty c hc hw hx hi' ((mem_fiveSupportList _ _ _ _ _).mp h)
+  · obtain ⟨j,hj,hs⟩ := five_support_next c hc hw hx hi
+    exact ⟨j,(mem_fiveSupportList _ _ _ _ _).mpr hs,hj⟩
+  · obtain ⟨j,hj,hs⟩ := five_support_next c hc hw hx hi'
+    exact ⟨j,(mem_fiveSupportList _ _ _ _ _).mpr hs,hj⟩
+lemma five_contact_prev (hw : WeakColoring 5 c) {x : ℂ} (hx : FiveCenter c x) {a : ℝ} {i : Fin 5}
+    (hi : a∈fiveSupport c hc x i) (hi' : a∈fiveSupport c hc x (i+1)) :
+    i-1∈fiveSupportList c hc x (a-sixty) ∧ i+2∈fiveSupportList c hc x (a-sixty) := by
+  apply FiveComb.contact_neighbors
+  · intro h
+    have hh := five_support_no_sixty c hc hw hx ((mem_fiveSupportList _ _ _ _ _).mp h)
+    exact hh (by simpa only [sub_add_cancel] using hi)
+  · intro h
+    have hh := five_support_no_sixty c hc hw hx ((mem_fiveSupportList _ _ _ _ _).mp h)
+    exact hh (by simpa only [sub_add_cancel] using hi')
+  · obtain ⟨j,hj,hs⟩ := five_support_prev c hc hw hx hi
+    exact ⟨j,(mem_fiveSupportList _ _ _ _ _).mpr hs,hj⟩
+  · obtain ⟨j,hj,hs⟩ := five_support_prev c hc hw hx hi'
+    exact ⟨j,(mem_fiveSupportList _ _ _ _ _).mpr hs,hj⟩
+lemma five_opposite_contacts_false (hw : WeakColoring 5 c) {x : ℂ} (hx : FiveCenter c x)
+    {a b : ℝ} (hab : b=a+3*sixty) {i j : Fin 5}
+    (hi : a∈fiveSupport c hc x i) (hi' : a∈fiveSupport c hc x (i+1))
+    (hj : b∈fiveSupport c hc x j) (hj' : b∈fiveSupport c hc x (j+1)) : False := by
+  have hn := five_contact_next c hc hw hx hi hi'
+  have hp := five_contact_prev c hc hw hx hj hj'
+  have hdis : Disjoint ({i-1,i+2} : Finset (Fin 5)) {j-1,j+2} := by
+    apply Finset.disjoint_left.mpr
+    intro k hk hl
+    have hiK : a+sixty∈fiveSupport c hc x k := by
+      apply (mem_fiveSupportList _ _ _ _ _).mp
+      simp only [Finset.mem_insert,Finset.mem_singleton] at hk
+      rcases hk with rfl|rfl <;> tauto
+    have hjK : b-sixty∈fiveSupport c hc x k := by
+      apply (mem_fiveSupportList _ _ _ _ _).mp
+      simp only [Finset.mem_insert,Finset.mem_singleton] at hl
+      rcases hl with rfl|rfl <;> tauto
+    exact five_support_no_sixty c hc hw hx hiK (by convert hjK using 1; rw [hab]; ring)
+  have hji := FiveComb.opposite_contacts i j hdis
+  obtain ⟨s,hs,_hst,hban⟩ := FiveComb.two_bans i j hji
+  have hss : a∈fiveSupport c hc x s := by rcases hs with rfl|rfl <;> assumption
+  have hbanI := five_support_step_ban c hc hw hx hi hi' (Or.inl rfl)
+  have hbanJ := five_support_step_ban c hc hw hx hj hj' (Or.inl rfl)
+  have hnot : ∀ᵐt∂angularMeasure,fiveState c hc x t≠s := by
+    filter_upwards [hbanI,hbanJ,fiveState_adj c hc hw hx] with t hbI hbJ ha he
+    rw [he] at ha
+    rcases hban _ ha with h|h
+    · rcases h with ⟨rfl,ht⟩|⟨rfl,ht⟩
+      · exact hbI.1 ⟨he,ht⟩
+      · exact hbI.2 ⟨he,ht⟩
+    · rcases h with ⟨rfl,ht⟩|⟨rfl,ht⟩
+      · exact hbJ.1 ⟨he,ht⟩
+      · exact hbJ.2 ⟨he,ht⟩
+  have hz : volume (fiveRealState c hc x s)=0 := measure_eq_zero_iff_ae_notMem.mpr (angleMk_quasiMeasurePreserving.ae hnot)
+  exact (stateSupport_nonempty_measure hss).ne' hz
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+open scoped NNReal ENNReal
+lemma plane_lipschitz_image_null {F : RealPlane → RealPlane} {K : ℝ≥0}
+    (hf : LipschitzWith K F) {A : Set RealPlane} (ha : volume A=0) : volume (F '' A)=0 := by
+  have h := hf.hausdorffMeasure_image_le (by norm_num : (0:ℝ)≤2) A
+  rw [hausdorffMeasure_prod_real,ha,mul_zero] at h
+  exact bot_unique h
+variable {k : ℕ} (c : ℂ → Fin k) (hc : Measurable c)
+lemma palette_open_exclusion (hw : WeakColoring k c) {x : ℂ} {i : Fin k} {a b : ℝ}
+    (hab : Real.sin (b-a)≠0) {V : Set ℝ} (hV : IsOpen V) (ha : a∈V)
+    (hpal : ∀ᵐt : ℝ∂volume,t∈V → (t : Angles)∈paletteSet c hc x i) :
+    ∃U : Set ℂ, IsOpen U ∧ x+direction (a : Angles)+direction (b : Angles)∈U ∧
+      volume (U∩{z | c z=i})=0 := by
+  let e := circleInverseHomeo hab
+  let W : Set RealPlane := e.source ∩ Prod.fst ⁻¹' V
+  have hW : IsOpen W := e.open_source.inter (hV.preimage continuous_fst)
+  have haW : (a,b)∈W := by
+    refine ⟨?_,ha⟩
+    exact (show HasStrictFDerivAt (circleSum 1)
+      (circleSumDerivEquiv (r := 1) (p := (a,b)) (by simpa using hab) : RealPlane →L[ℝ] RealPlane)
+      (a,b) from circleSum_strict 1 (a,b)).mem_toOpenPartialHomeomorph_source
+  have hO : IsOpen (circleSum 1 '' W) := e.isOpen_image_of_subset_source hW inter_subset_left
+  let g : RealPlane → ℂ := fun q => x+Complex.equivRealProdCLM.symm q
+  have hgs (q : RealPlane) : g (circleSum 1 q)=angularSum x 1 (angleChart q) := by
+    rw [angularSum_chart]
+    simp only [g,map_add,ContinuousLinearEquiv.symm_apply_apply]
+  have hgc : IsOpenMap g := (Homeomorph.addLeft x).isOpenMap.comp Complex.equivRealProdCLM.symm.toHomeomorph.isOpenMap
+  have hex := anglePair_quasiMeasurePreserving.ae (palette_first_exclusion c hc hw x i)
+  have hp : ∀ᵐq : RealPlane∂volume,q.1∈V → (q.1 : Angles)∈paletteSet c hc x i :=
+    Measure.quasiMeasurePreserving_fst.ae hpal
+  have hnull : volume (W∩{q | c (g (circleSum 1 q))=i})=0 := by
+    apply measure_eq_zero_iff_ae_notMem.mpr
+    filter_upwards [hex,hp] with q hq hp hmem
+    apply hq (hp hmem.1.2)
+    change c (angularSum x 1 (angleChart q))=i
+    rw [← hgs]
+    exact hmem.2
+  have himage := plane_lipschitz_image_null circleSum_one_lipschitz_sumTransport hnull
+  have hnullg : volume (g '' (circleSum 1 '' (W∩{q | c (g (circleSum 1 q))=i})))=0 := by
+    rw [show g '' (circleSum 1 '' (W∩{q | c (g (circleSum 1 q))=i}))=
+      (fun z => Complex.equivRealProdCLM (z-x)) ⁻¹' (circleSum 1 '' (W∩{q | c (g (circleSum 1 q))=i})) by
+        ext z
+        constructor
+        · rintro ⟨p,hp,rfl⟩
+          simpa only [Set.mem_preimage,g,add_sub_cancel_left,ContinuousLinearEquiv.apply_symm_apply] using hp
+        · intro hz
+          refine ⟨Complex.equivRealProdCLM (z-x),hz,?_⟩
+          change x+Complex.equivRealProdCLM.symm (Complex.equivRealProdCLM (z-x))=z
+          rw [ContinuousLinearEquiv.symm_apply_apply]
+          abel]
+    exact ((Complex.volume_preserving_equiv_real_prod.comp (measurePreserving_sub_right volume x)).quasiMeasurePreserving.preimage_null himage)
+  refine ⟨g '' (circleSum 1 '' W),hgc _ hO,?_,?_⟩
+  · refine ⟨circleSum 1 (a,b),⟨(a,b),haW,rfl⟩,?_⟩
+    simpa only [angularSum,Complex.ofReal_one,one_mul,angleChart] using hgs (a,b)
+  · exact measure_mono_null (by rintro z ⟨⟨q,⟨p,hp,rfl⟩,rfl⟩,hz⟩; exact ⟨_,⟨p,⟨hp,hz⟩,rfl⟩,rfl⟩) hnullg
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+variable {k : ℕ} (c : ℂ → Fin k)
+lemma transitionEdge_no_open_exclusion {i j : Fin k} {x : ℂ} (he : TransitionEdge c i j x)
+    {U : Set ℂ} (hU : IsOpen U) (hx : x∈U) (hz : volume (U∩{z | c z=i})=0) : False := by
+  obtain ⟨r,hr,hB⟩ := Metric.isOpen_iff.mp hU x hx
+  have hae : ∀ᵐz : ℂ∂volume,z∈U → c z≠i := by
+    filter_upwards [measure_eq_zero_iff_ae_notMem.mp hz] with z hz hu he
+    exact hz ⟨hu,he⟩
+  have hratio : ∀ᶠh : ℂ in 𝓝[≠] 0, transitionRatio c i j x (r/2) h=0 := by
+    have hh : ∀ᶠh : ℂ in 𝓝[≠] 0,‖h‖<r/2 := by
+      have hb : ∀ᶠh : ℂ in 𝓝[≠] 0,h∈Metric.ball 0 (r/2) :=
+        nhdsWithin_le_nhds (Metric.ball_mem_nhds _ (show 0<r/2 by linarith))
+      filter_upwards [hb] with h hh
+      simpa only [Metric.mem_ball,dist_zero_right] using hh
+    filter_upwards [hh] with h hh
+    have hap := (measurePreserving_add_right volume h).quasiMeasurePreserving.ae hae
+    have hzT : volume (transitionSet c i j x (r/2) h)=0 := by
+      apply measure_eq_zero_iff_ae_notMem.mpr
+      filter_upwards [hae,hap] with q hq hqh ht
+      have hqU : q∈U := hB (Metric.ball_subset_ball (show r/2≤r by linarith) ht.1)
+      have hqhU : q+h∈U := by
+        apply hB
+        have hd := dist_triangle (q+h) q x
+        have hdist : dist (q+h) q=‖h‖ := by simp [dist_eq_norm]
+        rw [hdist] at hd
+        exact hd.trans_lt (by have hqx : dist q x<r/2 := ht.1; linarith)
+      rcases ht.2 with ht|ht
+      · exact hq hqU ht.1
+      · exact hqh hqhU ht.2
+    simp only [transitionRatio,hzT,ENNReal.zero_div]
+  have hh : limsup (transitionRatio c i j x (r/2)) (𝓝[≠] (0:ℂ))=0 := by
+    rw [limsup_congr hratio]
+    exact limsup_const 0
+  have hp := he.2 (r/2) (by linarith)
+  rw [hh] at hp
+  exact lt_irrefl _ hp
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology
+namespace EuclideanFiveColor
+variable {ι X : Type*} [Finite ι] [TopologicalSpace X]
+lemma finite_closed_family_locally_subset {S : ι → Set X} (hS : ∀i,IsClosed (S i)) (a : X) :
+    ∃V,IsOpen V ∧ a∈V ∧ ∀v∈V,∀i,v∈S i → a∈S i := by
+  let V := ⋂i∈({i | a∉S i} : Set ι),(S i)ᶜ
+  have hop : IsOpen V := (Set.toFinite _).isOpen_biInter (fun i _ => (hS i).isOpen_compl)
+  refine ⟨V,hop,?_,?_⟩
+  · exact mem_iInter₂.mpr (fun _ hi => hi)
+  · intro v hv i hi
+    by_contra hn
+    exact mem_iInter₂.mp hv i hn hi
+lemma finite_closed_family_no_common_limit {S : ι → Set X} (hS : ∀i,IsClosed (S i))
+    {P : ι → Prop} {a : X} {an : ℕ → X} (ha : Tendsto an atTop (𝓝 a))
+    (hn : ∀n,∃i,an n∈S i ∧ P i) : ∃i,a∈S i ∧ P i := by
+  obtain ⟨V,hV,haV,hsub⟩ := finite_closed_family_locally_subset hS a
+  have hev : ∀ᶠn in atTop,an n∈V := ha (hV.mem_nhds haV)
+  obtain ⟨n,hnV⟩ := hev.exists
+  obtain ⟨i,hi,hP⟩ := hn n
+  exact ⟨i,hsub _ hnV i hi,hP⟩
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+lemma stateSupport_local_pred {ι : Type*} [Fintype ι] [MeasurableSpace ι] [MeasurableSingletonClass ι]
+    {f : ℝ → ι} (hf : Measurable f) {P : ι → Prop} {a : ℝ}
+    (ha : ∀i,a∈stateSupport f i → P i) :
+    ∃V,IsOpen V ∧ a∈V ∧ ∀ᵐt : ℝ∂volume,t∈V → P (f t) := by
+  obtain ⟨V,hV,haV,hsub⟩ := finite_closed_family_locally_subset
+    (S := stateSupport f) (fun _ => Measure.isClosed_support) a
+  refine ⟨V,hV,haV,?_⟩
+  exact (ae_state_support hf).mono fun t ht hv => ha _ (hsub t hv _ ht)
+end EuclideanFiveColor
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PlaneColoring/FourContinuity.lean b/lean/OAI/Geometry/PlaneColoring/FourContinuity.lean
new file mode 100644
index 0000000000000000000000000000000000000000..28767383517ab9526f3ddec09f77317bc236711f
--- /dev/null
+++ b/lean/OAI/Geometry/PlaneColoring/FourContinuity.lean
@@ -0,0 +1,551 @@
+import OAI.Geometry.PlaneColoring.States
+
+namespace OAI
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+variable (c : ℂ → Fin 5) (hc : Measurable c)
+lemma four_avoiding_traces (hw : WeakColoring 5 c) {x : ℂ} (hx : FourCenter c x) :
+    ∃ b : Fin 4 → TraceVector 5, (∀e,IsTrace c hc x (b e)) ∧
+      ∀e,∀i∈fourEdge e,traceValue (b e) i=0 := by
+  have hh (i j : Fin 5) (he : TransitionEdge c i j x) :
+      ∃ b : TraceVector 5, IsTrace c hc x b ∧ ∀a∈({i,j} : Finset (Fin 5)),traceValue b a=0 := by
+    obtain ⟨b,hb,hi,hj⟩ := transition_avoiding_trace c hc hw he
+    refine ⟨b,hb,?_⟩
+    intro a ha
+    simp only [Finset.mem_insert,Finset.mem_singleton] at ha
+    rcases ha with rfl|rfl <;> assumption
+  have he (e : Fin 4) : ∃ b : TraceVector 5, IsTrace c hc x b ∧ ∀i∈fourEdge e,traceValue b i=0 := by
+    fin_cases e
+    · exact hh 0 1 hx.1
+    · exact hh 1 2 hx.2.1
+    · exact hh 2 3 hx.2.2.1
+    · exact hh 3 0 hx.2.2.2
+  choose b hb hz using he
+  exact ⟨b,hb,hz⟩
+lemma four_state_trace_domination (hw : WeakColoring 5 c) {x : ℂ} (hx : FourCenter c x)
+    (b : Fin 4 → TraceVector 5) (hb : ∀e,IsTrace c hc x (b e))
+    (hz : ∀e,∀i∈fourEdge e,traceValue (b e) i=0) (s : Fin 3) {i : Fin 5} (hi : i∈fourGroup s) :
+    {t | fourState c hc x t=s}.indicator (fun _ => (1:ℝ)) ≤ᵐ[angularMeasure]
+      (fun t => ∑e,traceValue (b e) i t) := by
+  have hs (e : Fin 4) : (fun t => ∑a,traceValue (b e) a t)=ᵐ[angularMeasure] (fun _ => (1:ℝ)) := by
+    have hh := Lp.coeFn_fun_finsetSum Finset.univ (traceValue (b e))
+    rw [(trace_bounds c hc (hb e)).2.2] at hh
+    exact hh.symm.trans (oneL2_ae angularMeasure)
+  have hzero (e : Fin 4) (a : Fin 5) : ∀ᵐt∂angularMeasure,a∈fourEdge e → traceValue (b e) a t=0 := by
+    by_cases ha : a∈fourEdge e
+    · rw [hz e a ha]
+      filter_upwards [Lp.coeFn_zero ℝ 2 angularMeasure] with t ht
+      exact fun _ => ht
+    · exact Eventually.of_forall fun _ => fun h => (ha h).elim
+  filter_upwards [four_palette_admissible c hc hw hx,
+    ae_all_iff.mpr (fun e => ae_all_iff.mpr fun a => (trace_bounds c hc (hb e)).1 a),
+    ae_all_iff.mpr hs,ae_all_iff.mpr (fun e => ae_all_iff.mpr (hzero e)),
+    ae_all_iff.mpr (fun e => ae_all_iff.mpr fun a => tracePositive_le_palette c hc x a ⟨b e,hb e⟩)]
+    with t hadm hnon hsum hzero hpal
+  by_cases hts : fourState c hc x t=s
+  · rw [Set.indicator_of_mem (show t∈{t | fourState c hc x t=s} from hts)]
+    obtain ⟨e,_,hep⟩ := four_trace_edge (palette c hc x t) hadm i (by simpa only [← hts, fourState] using hi)
+    have hj (a : Fin 5) (ha : a≠i) : traceValue (b e) a t=0 := by
+      by_cases he : a∈fourEdge e
+      · exact hzero e a he
+      · apply le_antisymm _ (hnon e a)
+        by_contra! hpos
+        have hp := (mem_palette c hc x t a).mpr (hpal e a hpos)
+        exact he (hep (Finset.mem_sdiff.mpr ⟨hp,by simpa using ha⟩))
+    have heq : traceValue (b e) i t=1 := by
+      rw [← hsum e,Finset.sum_eq_single i _ (by simp)]
+      intro a _ hai
+      exact hj a hai
+    rw [← heq]
+    exact Finset.single_le_sum (fun j _ => hnon j i) (Finset.mem_univ e)
+  · rw [Set.indicator_of_notMem (show t∉{t | fourState c hc x t=s} from hts)]
+    exact Finset.sum_nonneg (fun e _ => hnon e i)
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+open scoped RealInnerProductSpace
+namespace EuclideanFiveColor
+variable {Y : Type*} [MeasurableSpace Y] {μ : Measure Y} [IsProbabilityMeasure μ]
+def indL2 (A : Set Y) (hA : MeasurableSet A) : RealL2 μ :=
+  indicatorConstLp 2 hA (measure_ne_top μ A) 1
+lemma indL2_ae (A : Set Y) (hA : MeasurableSet A) :
+    indL2 (μ := μ) A hA =ᵐ[μ] A.indicator (fun _ => (1:ℝ)) := indicatorConstLp_coeFn
+lemma indL2_nonneg (A : Set Y) (hA : MeasurableSet A) : 0≤ᵐ[μ] indL2 (μ := μ) A hA := by
+  filter_upwards [indL2_ae (μ := μ) A hA] with t ht
+  rw [ht]
+  exact indicator_nonneg (fun _ _ => zero_le_one) _
+lemma indL2_le_one (A : Set Y) (hA : MeasurableSet A) : indL2 (μ := μ) A hA≤ᵐ[μ] (fun _ => (1:ℝ)) := by
+  filter_upwards [indL2_ae (μ := μ) A hA] with t ht
+  rw [ht]
+  exact indicator_le_self' (fun _ _ => zero_le_one) _
+lemma indL2_norm (A : Set Y) (hA : MeasurableSet A) : ‖indL2 (μ := μ) A hA‖≤1 := by
+  have hh : ∀ᵐt∂μ,‖indL2 (μ := μ) A hA t‖≤1 := by
+    filter_upwards [indL2_nonneg (μ := μ) A hA,indL2_le_one (μ := μ) A hA] with t ht hu
+    rwa [Real.norm_eq_abs,abs_of_nonneg ht]
+  simpa [measureUnivNNReal] using Lp.norm_le_of_ae_bound (by norm_num : (0:ℝ)≤1) hh
+lemma indL2_sum {ι : Type*} [Fintype ι] [DecidableEq ι] [MeasurableSpace ι] [MeasurableSingletonClass ι]
+    {s : Y → ι} (hs : Measurable s) :
+    ∑ i, indL2 (μ := μ) {t | s t=i} (measurableSet_eq_fun hs measurable_const)=oneL2 μ := by
+  apply Lp.ext
+  filter_upwards [Lp.coeFn_fun_finsetSum Finset.univ (fun i => indL2 (μ := μ) {t | s t=i} (measurableSet_eq_fun hs measurable_const)),
+    ae_all_iff.mpr (fun i => indL2_ae (μ := μ) {t | s t=i} (measurableSet_eq_fun hs measurable_const)),oneL2_ae μ] with t ht ha hu
+  rw [ht,hu]
+  simp only [ha,Set.indicator_apply,mem_ofPred_eq]
+  simp
+lemma weak_limit_le {f g : ℕ → RealL2 μ} {a b : RealL2 μ}
+    (hf : ∀v,Tendsto (fun n => ⟪f n,v⟫) atTop (𝓝 ⟪a,v⟫))
+    (hg : ∀v,Tendsto (fun n => ⟪g n,v⟫) atTop (𝓝 ⟪b,v⟫))
+    (hle : ∀n,f n≤ᵐ[μ] g n) : a≤ᵐ[μ] b := by
+  have hh : 0≤ᵐ[μ] b-a := weak_limit_nonneg μ (f := fun n => g n-f n)
+    (fun v => by simpa only [inner_sub_left] using (hg v).sub (hf v)) (fun n => by
+      filter_upwards [Lp.coeFn_sub (g n) (f n),hle n] with t ht hh
+      simpa only [ht,Pi.sub_apply,Pi.zero_apply,sub_nonneg] using hh)
+  filter_upwards [hh,Lp.coeFn_sub b a] with t ht he
+  simpa only [he,Pi.sub_apply,Pi.zero_apply,sub_nonneg] using ht
+lemma indL2_weak_L1 {A : ℕ → Set Y} (hA : ∀n,MeasurableSet (A n)) {B : Set Y} (hB : MeasurableSet B)
+    (hw : ∀v : RealL2 μ,Tendsto (fun n => ⟪indL2 (A n) (hA n),v⟫) atTop (𝓝 ⟪indL2 B hB,v⟫)) :
+    Tendsto (fun n => ∫t,‖(A n).indicator (fun _ => (1:ℝ)) t-B.indicator (fun _ => (1:ℝ)) t‖ ∂μ) atTop (𝓝 0) := by
+  have hone (w : RealL2 μ) : ⟪w,oneL2 μ⟫=∫t,w t ∂μ := by
+    rw [real_inner_comm]
+    simpa only [oneL2,setIntegral_univ] using
+      L2.inner_indicatorConstLp_one MeasurableSet.univ (measure_ne_top μ univ) w
+  have hBB : ⟪indL2 (μ := μ) B hB,indL2 B hB⟫=⟪indL2 (μ := μ) B hB,oneL2 μ⟫ := by
+    rw [hone,L2.inner_def]
+    apply integral_congr_ae
+    filter_upwards [indL2_ae (μ := μ) B hB] with t ht
+    rw [ht]
+    by_cases hb : t∈B <;> simp [hb]
+  have he (n : ℕ) : (∫t,‖(A n).indicator (fun _ => (1:ℝ)) t-B.indicator (fun _ => (1:ℝ)) t‖ ∂μ) =
+      ⟪indL2 (μ := μ) (A n) (hA n),oneL2 μ⟫+⟪indL2 (μ := μ) B hB,oneL2 μ⟫-
+      2*⟪indL2 (μ := μ) (A n) (hA n),indL2 (μ := μ) B hB⟫ := by
+    rw [hone,hone,L2.inner_def,← integral_const_mul,← integral_add,
+      ← integral_sub]
+    · apply integral_congr_ae
+      filter_upwards [indL2_ae (μ := μ) (A n) (hA n),indL2_ae (μ := μ) B hB] with t ht hu
+      rw [ht,hu]
+      by_cases ha : t∈A n <;> by_cases hb : t∈B <;> norm_num [ha,hb]
+    · exact ((Lp.memLp _).integrable (by norm_num)).add ((Lp.memLp _).integrable (by norm_num))
+    · exact (L2.integrable_inner _ _).const_mul 2
+    · exact (Lp.memLp _).integrable (by norm_num)
+    · exact (Lp.memLp _).integrable (by norm_num)
+  simp_rw [he]
+  have ht := ((hw (oneL2 μ)).add_const ⟪indL2 (μ := μ) B hB,oneL2 μ⟫).sub
+    ((hw (indL2 (μ := μ) B hB)).const_mul 2)
+  rw [hBB,show ⟪indL2 (μ := μ) B hB, oneL2 μ⟫+⟪indL2 (μ := μ) B hB,oneL2 μ⟫-
+    2*⟪indL2 (μ := μ) B hB,oneL2 μ⟫=0 by ring] at ht
+  exact ht
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+open scoped RealInnerProductSpace
+namespace EuclideanFiveColor
+variable (c : ℂ → Fin 5) (hc : Measurable c)
+def fourStateL2 (x : ℂ) (s : Fin 3) : RealL2 angularMeasure :=
+  indL2 {t | fourState c hc x t=s} (measurableSet_eq_fun (fourState_measurable c hc x) measurable_const)
+lemma four_weak_limit_identify (hw : WeakColoring 5 c)
+    {x : ℕ → ℂ} {y : ℂ} (hx : ∀n,FourCenter c (x n)) (hy : FourCenter c y)
+    (hxt : Tendsto x atTop (𝓝 y)) {f : Fin 3 → RealL2 angularMeasure}
+    (hf : ∀s v,Tendsto (fun n => ⟪fourStateL2 c hc (x n) s,v⟫) atTop (𝓝 ⟪f s,v⟫))
+    (b : ℕ → Fin 4 → TraceVector 5) (a : Fin 4 → TraceVector 5)
+    (hb : ∀n e,IsTrace c hc (x n) (b n e))
+    (hz : ∀n e,∀i∈fourEdge e,traceValue (b n e) i=0)
+    (ht : ∀e,Tendsto (fun n => b n e) atTop (𝓝 (a e))) :
+    ∀s,f s=fourStateL2 c hc y s := by
+  have hat (e : Fin 4) : IsTrace c hc y (a e) := trace_limit c hc (fun n => hb n e) hxt (ht e)
+  have hdom (s : Fin 3) (i : Fin 5) (hi : i∈fourGroup s) :
+      f s≤ᵐ[angularMeasure] (∑e,traceValue (a e) i : RealL2 angularMeasure) := by
+    apply weak_limit_le (hf s) (f := fun n => fourStateL2 c hc (x n) s)
+      (g := fun n => ∑e,traceValue (b n e) i)
+      (fun v => ?_) (fun n => ?_)
+    · simpa only [sum_inner] using tendsto_finsetSum Finset.univ
+        (fun e _ => traceCode_tendsto_iff.mp (ht e) i v)
+    · filter_upwards [indL2_ae (μ := angularMeasure) {t | fourState c hc (x n) t=s}
+        (measurableSet_eq_fun (fourState_measurable c hc (x n)) measurable_const),
+        Lp.coeFn_fun_finsetSum Finset.univ (fun e => traceValue (b n e) i),
+        four_state_trace_domination c hc hw (hx n) (b n) (hb n) (hz n) s hi] with t h₁ h₂ h₃
+      simpa only [fourStateL2,h₁,h₂] using h₃
+  have hnon (s : Fin 3) : 0≤ᵐ[angularMeasure] f s :=
+    weak_limit_nonneg angularMeasure (hf s) (fun n => indL2_nonneg _ _)
+  have hsumf : ∑s,f s=oneL2 angularMeasure :=
+    weak_limit_sum angularMeasure hf (fun n => indL2_sum (fourState_measurable c hc (x n)))
+  have hsupport (s : Fin 3) : ∀ᵐt∂angularMeasure,0<f s t → fourGroup s⊆palette c hc y t := by
+    have hdom' (i : Fin 5) : ∀ᵐt∂angularMeasure,i∈fourGroup s → f s t≤∑e,traceValue (a e) i t := by
+      by_cases hi : i∈fourGroup s
+      · filter_upwards [hdom s i hi,Lp.coeFn_fun_finsetSum Finset.univ (fun e => traceValue (a e) i)] with t ht he
+        exact fun _ => by simpa only [he] using ht
+      · exact Eventually.of_forall fun _ h => (hi h).elim
+    filter_upwards [ae_all_iff.mpr hdom',
+      ae_all_iff.mpr (fun e => ae_all_iff.mpr fun i => tracePositive_le_palette c hc y i ⟨a e,hat e⟩)] with t hdom hpal hp i hi
+    have hsum : 0<∑e,traceValue (a e) i t := hp.trans_le (hdom i hi)
+    have hex : ∃e,0<traceValue (a e) i t := by
+      by_contra! hh
+      exact hsum.not_ge (Finset.sum_nonpos fun e _ => hh e)
+    obtain ⟨e,he⟩ := hex
+    exact (mem_palette c hc y t i).mpr (hpal e i he)
+  intro s
+  apply Lp.ext
+  have hsumae : (fun t => ∑s,f s t)=ᵐ[angularMeasure] (fun _ => (1:ℝ)) := by
+    have hh := Lp.coeFn_fun_finsetSum Finset.univ f
+    rw [hsumf] at hh
+    exact hh.symm.trans (oneL2_ae angularMeasure)
+  filter_upwards [ae_all_iff.mpr hnon,ae_all_iff.mpr hsupport,hsumae,
+    four_palette_admissible c hc hw hy,
+    indL2_ae (μ := angularMeasure) {t | fourState c hc y t=s}
+      (measurableSet_eq_fun (fourState_measurable c hc y) measurable_const)] with t hn hp hsum hadm hi
+  change f s t=indL2 (μ := angularMeasure) {t | fourState c hc y t=s} _ t
+  rw [hi]
+  have hz (v : Fin 3) (hv : v≠fourState c hc y t) : f v t=0 := by
+    apply le_antisymm _ (hn v)
+    by_contra! hh
+    have he := (four_admissible_character (palette c hc y t) hadm v).mp (hp v hh)
+    exact hv he.symm
+  by_cases he : fourState c hc y t=s
+  · rw [Set.indicator_of_mem (show t∈{t | fourState c hc y t=s} from he)]
+    rw [← hsum,Finset.sum_eq_single s _ (by simp)]
+    intro v _ hvs
+    exact hz v (by simpa only [he] using hvs)
+  · rw [Set.indicator_of_notMem (show t∉{t | fourState c hc y t=s} from he)]
+    exact hz s (Ne.symm he)
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+open scoped RealInnerProductSpace
+namespace EuclideanFiveColor
+variable (c : ℂ → Fin 5) (hc : Measurable c)
+def fourStateDistance (x y : ℂ) (s : Fin 3) : ℝ :=
+  ∫t,‖{t | fourState c hc x t=s}.indicator (fun _ => (1:ℝ)) t-
+    {t | fourState c hc y t=s}.indicator (fun _ => (1:ℝ)) t‖ ∂angularMeasure
+lemma four_state_subseq_continuity (hw : WeakColoring 5 c)
+    {x : ℕ → ℂ} {y : ℂ} (hx : ∀n,FourCenter c (x n)) (hy : FourCenter c y)
+    (hxt : Tendsto x atTop (𝓝 y)) :
+    ∃φ : ℕ → ℕ,StrictMono φ ∧ ∀s,Tendsto (fun n => fourStateDistance c hc (x (φ n)) y s) atTop (𝓝 0) := by
+  obtain ⟨f,φ,hφ,_,hf⟩ := exists_finite_weak_subseq (fun n s => fourStateL2 c hc (x n) s) 1
+    (fun _ _ => indL2_norm _ _)
+  choose b hb hz using fun n => four_avoiding_traces c hc hw (hx (φ n))
+  obtain ⟨a,_,ψ,hψ,ha⟩ := (isCompact_univ : IsCompact (univ : Set (Fin 4 → TraceVector 5))).isSeqCompact
+    (fun n => mem_univ (b n))
+  have hid := four_weak_limit_identify c hc hw (fun n => hx (φ (ψ n))) hy
+    (hxt.comp (hφ.tendsto_atTop.comp hψ.tendsto_atTop))
+    (fun s v => (hf s v).comp hψ.tendsto_atTop) (fun n => b (ψ n)) a
+    (fun n => hb (ψ n)) (fun n => hz (ψ n))
+    (fun e => (continuous_apply e).tendsto a |>.comp ha)
+  refine ⟨φ∘ψ,hφ.comp hψ,?_⟩
+  intro s
+  apply indL2_weak_L1 (fun n => measurableSet_eq_fun (fourState_measurable c hc (x (φ (ψ n)))) measurable_const)
+    (measurableSet_eq_fun (fourState_measurable c hc y) measurable_const)
+  intro v
+  have hh := (hf s v).comp hψ.tendsto_atTop
+  rw [hid s] at hh
+  exact hh
+lemma four_state_continuity (hw : WeakColoring 5 c)
+    {x : ℕ → ℂ} {y : ℂ} (hx : ∀n,FourCenter c (x n)) (hy : FourCenter c y)
+    (hxt : Tendsto x atTop (𝓝 y)) (s : Fin 3) :
+    Tendsto (fun n => fourStateDistance c hc (x n) y s) atTop (𝓝 0) := by
+  by_contra hn
+  rw [Metric.tendsto_atTop] at hn
+  push Not at hn
+  obtain ⟨ε,hε,he⟩ := hn
+  choose m hm using he
+  have hmt : Tendsto m atTop atTop := tendsto_atTop_mono (fun n => (hm n).1) tendsto_id
+  obtain ⟨φ,hφ,ht⟩ := four_state_subseq_continuity c hc hw (fun n => hx (m n)) hy (hxt.comp hmt)
+  obtain ⟨N,hN⟩ := Metric.tendsto_atTop.mp (ht s) ε hε
+  exact (hm (φ N)).2.not_gt (hN N le_rfl)
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+open scoped NNReal
+namespace EuclideanFiveColor
+lemma angleMk_compact_bound {E : Set ℝ} (hE : IsCompact E) :
+    ∃ C : ℝ≥0, (volume.restrict E).map ((↑) : ℝ → Angles)≤(C:ENNReal) • angularMeasure := by
+  obtain ⟨F,hF⟩ := hE.elim_finite_subcover (fun a : ℝ => Ioo (a-Real.pi) (a+Real.pi))
+    (fun _ => isOpen_Ioo) (fun a _ => mem_iUnion.mpr ⟨a,by constructor <;> linarith [Real.pi_pos]⟩)
+  let C : ℝ≥0 := ⟨(F.card:ℝ)*(2*Real.pi),by positivity⟩
+  refine ⟨C,Measure.le_iff.mpr (fun A hA => ?_)⟩
+  rw [Measure.map_apply angleMk_quasiMeasurePreserving.measurable hA,
+    Measure.restrict_apply (angleMk_quasiMeasurePreserving.measurable hA),Measure.smul_apply,smul_eq_mul]
+  have hsub : ((↑) : ℝ → Angles) ⁻¹' A∩E ⊆
+      ⋃a∈F,((↑) : ℝ → Angles) ⁻¹' A∩Ioc (a-Real.pi) ((a-Real.pi)+2*Real.pi) := by
+    intro t ht
+    obtain ⟨a,ha,h₁,h₂⟩ := mem_iUnion₂.mp (hF ht.2)
+    exact mem_iUnion₂.mpr ⟨a,ha,ht.1,h₁,by linarith⟩
+  calc
+    _ ≤ volume (⋃a∈F,((↑) : ℝ → Angles) ⁻¹' A∩Ioc (a-Real.pi) ((a-Real.pi)+2*Real.pi)) := measure_mono hsub
+    _ ≤ ∑a∈F,volume (((↑) : ℝ → Angles) ⁻¹' A∩Ioc (a-Real.pi) ((a-Real.pi)+2*Real.pi)) := measure_biUnion_finset_le F _
+    _ = ∑_a∈F,(volume : Measure Angles) A := by
+      apply Finset.sum_congr rfl
+      intro a _
+      exact (AddCircle.add_projection_respects_measure (2*Real.pi) (a-Real.pi) hA).symm
+    _ = (C:ENNReal)*angularMeasure A := by
+      rw [AddCircle.volume_eq_smul_haarAddCircle]
+      simp only [Measure.smul_apply,smul_eq_mul,Finset.sum_const,nsmul_eq_mul]
+      change (F.card:ENNReal)*(ENNReal.ofReal (2*Real.pi)*angularMeasure A)=_
+      rw [← mul_assoc]
+      congr 1
+      rw [show (C:ENNReal)=ENNReal.ofReal ((F.card:ℝ)*(2*Real.pi)) from ENNReal.ofReal_coe_nnreal.symm]
+      rw [ENNReal.ofReal_mul (by positivity : 0≤(F.card:ℝ)),ENNReal.ofReal_natCast]
+lemma angular_compact_L1 {f : ℕ → Angles → ℝ} {g : Angles → ℝ}
+    (hf : ∀n,Integrable (f n) angularMeasure) (hg : Integrable g angularMeasure)
+    (ht : Tendsto (fun n => ∫t,‖f n t-g t‖ ∂angularMeasure) atTop (𝓝 0))
+    {E : Set ℝ} (hE : IsCompact E) :
+    Tendsto (fun n => ∫t in E,‖f n (t : Angles)-g (t : Angles)‖) atTop (𝓝 0) := by
+  obtain ⟨C,hC⟩ := angleMk_compact_bound hE
+  apply squeeze_zero (fun _ => integral_nonneg fun _ => norm_nonneg _)
+    (fun n => integral_norm_comp_le angleMk_quasiMeasurePreserving.measurable hC ((hf n).sub hg))
+  simpa only [mul_zero,Pi.sub_apply] using ht.const_mul (C:ℝ)
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+lemma support_lower_semicontinuity {A : Set ℝ} {B : ℕ → Set ℝ}
+    (hA : MeasurableSet A) (hB : ∀n,MeasurableSet (B n))
+    (ht : ∀E : Set ℝ,IsCompact E → Tendsto
+      (fun n => ∫t in E,‖(B n).indicator (fun _ => (1:ℝ)) t-A.indicator (fun _ => (1:ℝ)) t‖) atTop (𝓝 0))
+    {b : ℝ} (hb : b∈(volume.restrict A).support) {ε : ℝ} (hε : 0<ε) :
+    ∀ᶠn in atTop,∃v∈(volume.restrict (B n)).support,dist v b<ε := by
+  let E := Metric.closedBall b (ε/2)
+  have hp : 0<(volume.restrict E) A := by
+    have hh := (Measure.mem_support_iff_forall b).mp hb (Metric.ball b (ε/2))
+      (Metric.ball_mem_nhds b (by positivity))
+    rw [Measure.restrict_apply measurableSet_ball] at hh
+    rw [Measure.restrict_apply hA]
+    apply hh.trans_le (measure_mono _)
+    exact fun t ht => ⟨ht.2,Metric.ball_subset_closedBall ht.1⟩
+  let : IsFiniteMeasure (volume.restrict E) := ⟨by simpa [E] using (isCompact_closedBall b (ε/2)).measure_lt_top (μ := (volume : Measure ℝ))⟩
+  have hpos := indicator_inter_positive (μ := volume.restrict E) hA hB (fun _ => hA) hp
+    (ht E (isCompact_closedBall _ _)) (by simp)
+  filter_upwards [hpos] with n hn
+  have hpB : (volume.restrict (B n)) E≠0 := by
+    apply ne_of_gt
+    rw [Measure.restrict_apply measurableSet_closedBall]
+    rw [Measure.restrict_apply ((hB n).inter hA)] at hn
+    exact hn.trans_le (measure_mono (fun t ht => ⟨ht.2,ht.1.1⟩))
+  obtain ⟨v,hv,hs⟩ := Measure.exists_mem_of_measure_ne_zero_of_ae hpB (ae_restrict_of_ae (volume.restrict (B n)).support_mem_ae)
+  exact ⟨v,hs,(show dist v b≤ε/2 from hv).trans_lt (by linarith)⟩
+lemma support_sequence_continuity {A : Set ℝ} {B : ℕ → Set ℝ}
+    (hA : MeasurableSet A) (hB : ∀n,MeasurableSet (B n)) (hpB : ∀n,0<volume (B n))
+    (ht : ∀E : Set ℝ,IsCompact E → Tendsto
+      (fun n => ∫t in E,‖(B n).indicator (fun _ => (1:ℝ)) t-A.indicator (fun _ => (1:ℝ)) t‖) atTop (𝓝 0))
+    {b : ℝ} (hb : b∈(volume.restrict A).support) :
+    ∃v : ℕ → ℝ,(∀n,v n∈(volume.restrict (B n)).support) ∧ Tendsto v atTop (𝓝 b) := by
+  have hne (n : ℕ) : (volume.restrict (B n)).support.Nonempty := by
+    apply Measure.nonempty_support
+    intro he
+    have hz := congrArg (fun μ : Measure ℝ => μ univ) he
+    exact (hpB n).ne' (by simpa using hz)
+  choose v hv hd using fun n => (volume.restrict (B n)).isClosed_support.exists_infDist_eq_dist (hne n) b
+  refine ⟨v,hv,Metric.tendsto_atTop.mpr ?_⟩
+  intro ε hε
+  obtain ⟨N,hN⟩ := eventually_atTop.mp (support_lower_semicontinuity hA hB ht hb hε)
+  refine ⟨N,fun n hn => ?_⟩
+  obtain ⟨w,hw,hwb⟩ := hN n hn
+  rw [dist_comm,← hd n]
+  exact (Metric.infDist_le_dist_of_mem hw).trans_lt (by simpa only [dist_comm] using hwb)
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+lemma strict_state_uniform_on_compact {k : ℕ} {c : ℂ → Fin k}
+    (hc : Measurable c) (hw : WeakColoring k c)
+    (x : ℕ → ℂ) (J K : Finset (Fin k)) (X : Fin k)
+    (hlabels : ∀i, i∉J∪K ↔ i=X)
+    (A B : ℕ → Set ℝ) (D : Set ℝ) (hD : MeasurableSet D) (hp : 0<volume D)
+    (hA : ∀n,RealState hc (x n) (A n) J) (hB : ∀n,RealState hc (x n) (B n) K)
+    (hL1 : ∀ E : Set ℝ, IsCompact E →
+      Tendsto (fun n => ∫ v in E, ‖(A n).indicator (fun _ => (1:ℝ)) v-D.indicator (fun _ => (1:ℝ)) v‖)
+        atTop (𝓝 0))
+    {C : Set ℝ} (hC : IsCompact C) :
+    ∃ ε>0, ∀ᶠ n in atTop, ∀b∈C, ∀d∈C,
+      b∈(volume.restrict (B n)).support → d∈(volume.restrict (B n)).support →
+      ε≤|‖direction (b : Angles)-direction (d : Angles)‖-1| := by
+  by_contra! hbad
+  have hex (N : ℕ) : ∃ n≥N, ∃ b∈C, ∃ d∈C,
+      b∈(volume.restrict (B n)).support ∧ d∈(volume.restrict (B n)).support ∧
+      |‖direction (b : Angles)-direction (d : Angles)‖-1|<1/(N+1) := by
+    have hh := hbad (1/((N:ℝ)+1)) (by positivity)
+    simpa only [and_assoc] using (frequently_atTop.mp hh) N
+  choose f hf b hb d hd hbs hds hsmall using hex
+  have hft : Tendsto f atTop atTop := tendsto_atTop_mono hf tendsto_id
+  obtain ⟨q,_hq,φ,hφ,hqt⟩ := (hC.prod hC).isSeqCompact
+    (fun n => show (b n,d n)∈C×ˢC from ⟨hb n,hd n⟩)
+  have hqb : Tendsto (b∘φ) atTop (𝓝 q.1) := (continuous_fst.tendsto q).comp hqt
+  have hqd : Tendsto (d∘φ) atTop (𝓝 q.2) := (continuous_snd.tendsto q).comp hqt
+  have hdir : Continuous (fun v : ℝ => direction (v : Angles)) :=
+    direction_continuous.comp (AddCircle.continuous_mk' (2*Real.pi))
+  have he : ‖direction (q.1 : Angles)-direction (q.2 : Angles)‖=1 := by
+    have hn : Tendsto (fun N => |‖direction (b N : Angles)-direction (d N : Angles)‖-1|) atTop (𝓝 0) := by
+      exact squeeze_zero (fun _ => abs_nonneg _) (fun N => (hsmall N).le)
+        tendsto_one_div_add_atTop_nhds_zero_nat
+    have hl := (((hdir.tendsto q.1).comp hqb).sub ((hdir.tendsto q.2).comp hqd)).norm.sub_const 1 |>.abs
+    have he := tendsto_nhds_unique hl (hn.comp hφ.tendsto_atTop)
+    exact sub_eq_zero.mp (abs_eq_zero.mp he)
+  apply strict_state_sequence_false hc hw (fun n => x (f (φ n))) J K X hlabels
+    (fun n => A (f (φ n))) (fun n => B (f (φ n))) D hD hp
+    (fun n => hA (f (φ n))) (fun n => hB (f (φ n)))
+    (b∘φ) (d∘φ) q.1 q.2 hqb hqd (fun n => hbs (φ n)) (fun n => hds (φ n)) he
+  intro E hE
+  exact (hL1 E hE).comp (hft.comp hφ.tendsto_atTop)
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+lemma angle_preimage_positive {A : Set Angles} (hA : MeasurableSet A) (hp : 0<angularMeasure A) :
+    0<volume (((↑) : ℝ → Angles) ⁻¹' A) := by
+  by_contra! hn
+  have hz := measure_eq_zero_iff_ae_notMem.mp (le_antisymm hn zero_le)
+  have ha : ∀ᵐt∂angularMeasure,t∉A := ae_angular_of_ae_line hA.compl hz
+  exact hp.ne' (measure_eq_zero_iff_ae_notMem.mpr ha)
+variable (c : ℂ → Fin 5) (hc : Measurable c)
+def fourRealState (x : ℂ) (s : Fin 3) : Set ℝ := {t | fourState c hc x (t : Angles)=s}
+def fourSupport (x : ℂ) (s : Fin 3) : Set ℝ := (volume.restrict (fourRealState c hc x s)).support
+lemma four_realState_measurable (x : ℂ) (s : Fin 3) : MeasurableSet (fourRealState c hc x s) :=
+  (measurableSet_eq_fun (fourState_measurable c hc x) measurable_const).preimage
+    angleMk_quasiMeasurePreserving.measurable
+lemma four_realState_positive (hw : WeakColoring 5 c) {x : ℂ} (hx : FourCenter c x)
+    {s : Fin 3} (hs : s≠0) : 0<volume (fourRealState c hc x s) :=
+  angle_preimage_positive (measurableSet_eq_fun (fourState_measurable c hc x) measurable_const)
+    (fourState_diagonal_positive c hc hw hx s hs)
+lemma four_realState_continuity (hw : WeakColoring 5 c)
+    {x : ℕ → ℂ} {y : ℂ} (hx : ∀n,FourCenter c (x n)) (hy : FourCenter c y)
+    (hxt : Tendsto x atTop (𝓝 y)) (s : Fin 3) {E : Set ℝ} (hE : IsCompact E) :
+    Tendsto (fun n => ∫t in E,‖(fourRealState c hc (x n) s).indicator (fun _ => (1:ℝ)) t-
+      (fourRealState c hc y s).indicator (fun _ => (1:ℝ)) t‖) atTop (𝓝 0) := by
+  exact angular_compact_L1
+    (fun n => (integrable_const (1:ℝ)).indicator (measurableSet_eq_fun (fourState_measurable c hc (x n)) measurable_const))
+    ((integrable_const (1:ℝ)).indicator (measurableSet_eq_fun (fourState_measurable c hc y) measurable_const))
+    (four_state_continuity c hc hw hx hy hxt s) hE
+lemma four_support_continuity (hw : WeakColoring 5 c)
+    {x : ℕ → ℂ} {y : ℂ} (hx : ∀n,FourCenter c (x n)) (hy : FourCenter c y)
+    (hxt : Tendsto x atTop (𝓝 y)) {s : Fin 3} (hs : s≠0) {b : ℝ}
+    (hb : b∈fourSupport c hc y s) :
+    ∃v : ℕ → ℝ,(∀n,v n∈fourSupport c hc (x n) s) ∧ Tendsto v atTop (𝓝 b) :=
+  support_sequence_continuity (four_realState_measurable c hc y s)
+    (fun n => four_realState_measurable c hc (x n) s)
+    (fun n => four_realState_positive c hc hw (hx n) hs)
+    (fun _ hE => four_realState_continuity c hc hw hx hy hxt s hE) hb
+lemma four_support_sequence_no_unit (hw : WeakColoring 5 c)
+    {x : ℕ → ℂ} {y : ℂ} (hx : ∀n,FourCenter c (x n)) (hy : FourCenter c y)
+    (hxt : Tendsto x atTop (𝓝 y)) {s : Fin 3} (hs : s≠0)
+    {b d : ℕ → ℝ} {b₀ d₀ : ℝ} (hb : Tendsto b atTop (𝓝 b₀)) (hd : Tendsto d atTop (𝓝 d₀))
+    (hbs : ∀n,b n∈fourSupport c hc (x n) s) (hds : ∀n,d n∈fourSupport c hc (x n) s) :
+    ‖direction (b₀ : Angles)-direction (d₀ : Angles)‖≠1 := by
+  let r : Fin 3 := if s=1 then 2 else 1
+  have hr : r≠0 := by dsimp [r]; split <;> decide
+  have hl : ∀i : Fin 5,i∉fourGroup r∪fourGroup s ↔ i=4 := by
+    exact (show ∀s : Fin 3,s≠0 → ∀i : Fin 5,i∉fourGroup (if s=1 then 2 else 1)∪fourGroup s ↔ i=4 from by decide) s hs
+  intro hu
+  exact strict_state_sequence_false hc hw x (fourGroup r) (fourGroup s) 4 hl
+    (fun n => fourRealState c hc (x n) r) (fun n => fourRealState c hc (x n) s)
+    (fourRealState c hc y r) (four_realState_measurable c hc y r)
+    (four_realState_positive c hc hw hy hr)
+    (fun n => fourState_represents c hc hw (hx n) r) (fun n => fourState_represents c hc hw (hx n) s)
+    b d b₀ d₀ hb hd hbs hds hu (fun _ hE => four_realState_continuity c hc hw hx hy hxt r hE)
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+open scoped NNReal
+namespace EuclideanFiveColor
+lemma plane_antilipschitz_qmp {F : RealPlane → RealPlane} (hF : Measurable F)
+    {V : Set RealPlane} {K : ℝ≥0} (hK : AntilipschitzWith K (V.domRestrict F)) :
+    Measure.QuasiMeasurePreserving F (volume.restrict V) volume := by
+  refine ⟨hF,?_⟩
+  have hh := map_restrict_volume_le hF MeasurableSet.univ hK (mapsTo_univ F V)
+  rw [Measure.restrict_univ] at hh
+  exact Measure.absolutelyContinuous_of_le_smul hh
+lemma support_prod_real {μ ν : Measure ℝ} [SFinite μ] [SFinite ν]
+    {a b : ℝ} (ha : a∈μ.support) (hb : b∈ν.support) : (a,b)∈(μ.prod ν).support := by
+  apply (Measure.mem_support_iff_forall _).mpr
+  intro S hS
+  obtain ⟨I,hI,J,hJ,hIJ⟩ := mem_nhds_prod_iff.mp hS
+  apply (ENNReal.mul_pos ((Measure.mem_support_iff_forall _).mp ha I hI).ne'
+    ((Measure.mem_support_iff_forall _).mp hb J hJ).ne').trans_le
+  rw [← Measure.prod_prod]
+  exact measure_mono hIJ
+lemma continuousAt_maps_support_closed {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]
+    [MeasurableSpace X] {μ : Measure X} {F : X → Y} {C : Set Y} {p : X}
+    (hp : p∈μ.support) (hF : ContinuousAt F p) (hC : IsClosed C)
+    (hae : ∀ᵐq∂μ,F q∈C) : F p∈C := by
+  by_contra hnot
+  have hh : {q | F q∉C}∈𝓝 p := hF (hC.isOpen_compl.mem_nhds hnot)
+  exact ((Measure.mem_support_iff_forall _).mp hp _ hh).ne' hae
+lemma state_support_transport {k : ℕ} {c : ℂ → Fin k}
+    (hc : Measurable c) (hw : WeakColoring k c) {x y : ℂ}
+    {A B C : Set ℝ} {J K : Finset (Fin k)} {X : Fin k}
+    (hA : RealState hc x A J) (hB : RealState hc x B K) (hC : RealState hc y C {X})
+    (hlabels : ∀i,i∉J∪K ↔ i=X)
+    {a b : ℝ} (ha : a∈(volume.restrict A).support) (hb : b∈(volume.restrict B).support)
+    {F : RealPlane → RealPlane} (hF : Measurable F) (hFc : ContinuousAt F (a,b))
+    {V : Set RealPlane} (hV : IsOpen V) (hab : (a,b)∈V) {L : ℝ≥0}
+    (hL : AntilipschitzWith L (V.domRestrict F))
+    (hi : ∀q∈V,Real.sin (q.2-q.1)≠0)
+    (ho : ∀q∈V,Real.sin ((F q).2-(F q).1)≠0)
+    (heq : ∀q∈V,angularSum x 1 (angleChart q)=angularSum y 1 (angleChart (F q))) :
+    F (a,b)∈(volume.restrict Cᶜ).support ×ˢ (volume.restrict Cᶜ).support := by
+  have hdA : ∀ᵐq : RealPlane∂volume,q.1∈A → LineDensityZero Aᶜ q.1 := by
+    simpa only [Measure.volume_eq_prod] using
+      (Measure.quasiMeasurePreserving_fst (μ := (volume : Measure ℝ)) (ν := (volume : Measure ℝ))).ae (ae_lineDensityZero_compl hA.1)
+  have hdB : ∀ᵐq : RealPlane∂volume,q.2∈B → LineDensityZero Bᶜ q.2 := by
+    simpa only [Measure.volume_eq_prod] using
+      (Measure.quasiMeasurePreserving_snd (μ := (volume : Measure ℝ)) (ν := (volume : Measure ℝ))).ae (ae_lineDensityZero_compl hB.1)
+  have hdC : ∀ᵐt : ℝ∂volume,t∉(volume.restrict Cᶜ).support → LineDensityZero Cᶜ t := by
+    have hs : ∀ᵐt : ℝ∂volume,t∈Cᶜ → t∈(volume.restrict Cᶜ).support := by
+      exact (ae_restrict_iff' hC.1.compl).mp (volume.restrict Cᶜ).support_mem_ae
+    filter_upwards [hs,ae_lineDensityZero_compl hC.1] with t ht hd hn
+    apply hd
+    by_contra htC
+    exact hn (ht htC)
+  have hdpair : ∀ᵐq : RealPlane∂volume,
+      (q.1∉(volume.restrict Cᶜ).support → LineDensityZero Cᶜ q.1) ∧
+      (q.2∉(volume.restrict Cᶜ).support → LineDensityZero Cᶜ q.2) := by
+    have h₁ := (Measure.quasiMeasurePreserving_fst (μ := (volume : Measure ℝ)) (ν := (volume : Measure ℝ))).ae hdC
+    have h₂ := (Measure.quasiMeasurePreserving_snd (μ := (volume : Measure ℝ)) (ν := (volume : Measure ℝ))).ae hdC
+    simpa only [Measure.volume_eq_prod] using h₁.and h₂
+  have hforced : ∀ᵐq∂volume.restrict V,q∈A×ˢB →
+      F q∈(volume.restrict Cᶜ).support ×ˢ (volume.restrict Cᶜ).support := by
+    filter_upwards [ae_restrict_of_ae hdA,ae_restrict_of_ae hdB,
+      (plane_antilipschitz_qmp hF hL).ae hdpair,ae_restrict_mem hV.measurableSet] with q hqa hqb hqc hq hAB
+    have hOne : DensityOne {z | c z=X} (angularSum x 1 (angleChart q)) := by
+      simpa only [hlabels] using states_densityOne hc hw hA hB (hqa hAB.1) (hqb hAB.2) (hi q hq)
+    have hnot (hZero : DensityZero {z | c z=X} (angularSum y 1 (angleChart (F q)))) : False := by
+      rw [heq q hq] at hOne
+      have hh := tendsto_nhds_unique hOne hZero
+      exact one_ne_zero hh
+    constructor
+    · by_contra hn
+      exact hnot (state_densityZero hc hw hC (hqc.1 hn) (ho q hq) (Finset.mem_singleton_self _))
+    · by_contra hn
+      apply hnot
+      rw [← angularSum_swap y (F q).1 (F q).2]
+      apply state_densityZero hc hw hC (hqc.2 hn) _ (Finset.mem_singleton_self _)
+      simpa only [← neg_sub (F q).2 (F q).1,Real.sin_neg,neg_ne_zero] using ho q hq
+  have hp : (a,b)∈(volume.restrict (A×ˢB)).support := by
+    rw [Measure.volume_eq_prod,← Measure.prod_restrict]
+    exact support_prod_real ha hb
+  have hpV : (a,b)∈((volume.restrict (A×ˢB)).restrict V).support :=
+    Measure.interior_inter_support ⟨hV.interior_eq.symm ▸ hab,hp⟩
+  apply continuousAt_maps_support_closed hpV hFc
+    ((volume.restrict Cᶜ).isClosed_support.prod (volume.restrict Cᶜ).isClosed_support)
+  filter_upwards [hforced.filter_mono (ae_mono (Measure.restrict_mono_measure Measure.restrict_le_self V)),
+    ae_restrict_of_ae (ae_restrict_mem (hA.1.prod hB.1))] with q hq hAB
+  exact hq hAB
+end EuclideanFiveColor
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PlaneColoring/FourTransport.lean b/lean/OAI/Geometry/PlaneColoring/FourTransport.lean
new file mode 100644
index 0000000000000000000000000000000000000000..22a2e5d1e4879e486b3bda4d34b1b0232a957e03
--- /dev/null
+++ b/lean/OAI/Geometry/PlaneColoring/FourTransport.lean
@@ -0,0 +1,454 @@
+import OAI.Geometry.PlaneColoring.FourContinuity
+
+namespace OAI
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+open scoped NNReal
+namespace EuclideanFiveColor
+lemma circleSum_one_lipschitz_sumTransport : LipschitzWith 2 (circleSum 1) := by
+  apply LipschitzWith.of_dist_le_mul
+  intro p q
+  have hh (g : ℝ → ℝ) (hg : LipschitzWith 1 g) :
+      dist (g p.1+g p.2) (g q.1+g q.2)≤2*max (dist p.1 q.1) (dist p.2 q.2) := by
+    have h1 := hg.dist_le_mul p.1 q.1
+    have h2 := hg.dist_le_mul p.2 q.2
+    norm_num only [NNReal.coe_one,one_mul] at h1 h2
+    have h3 := dist_add_add_le (g p.1) (g p.2) (g q.1) (g q.2)
+    have h4 := le_max_left (dist p.1 q.1) (dist p.2 q.2)
+    have h5 := le_max_right (dist p.1 q.1) (dist p.2 q.2)
+    linarith
+  simpa only [circleSum,one_mul,Prod.dist_eq,NNReal.coe_ofNat] using
+    max_le (hh _ Real.lipschitzWith_cos) (hh _ Real.lipschitzWith_sin)
+
+def sumTransport {a b : ℝ} (h : Real.sin (b-a)≠0) (H : RealPlane) (q : RealPlane) :=
+  circleInverse h (circleSum 1 q-H)
+lemma sumTransport_measurable {a b : ℝ} (h : Real.sin (b-a)≠0) (H : RealPlane) :
+    Measurable (sumTransport h H) :=
+  (circleInverse_measurable h).comp ((circleSum_continuous 1).sub continuous_const).measurable
+lemma sumTransport_neighborhood {a b : ℝ} (h : Real.sin (b-a)≠0) :
+    ∃P V : Set RealPlane,IsOpen P ∧ 0∈P ∧ IsOpen V ∧ (a,b)∈V ∧
+      ∃K : ℝ≥0,∀H∈P,
+        (∀q∈V,ContinuousAt (sumTransport h H) q) ∧
+        AntilipschitzWith K (V.domRestrict (sumTransport h H)) ∧
+        (∀q∈V,Real.sin (q.2-q.1)≠0) ∧
+        (∀q∈V,Real.sin ((sumTransport h H q).2-(sumTransport h H q).1)≠0) ∧
+        ∀q∈V,circleSum 1 (sumTransport h H q)=circleSum 1 q-H := by
+  have hc : Continuous (fun z : RealPlane×RealPlane => circleSum 1 z.2-z.1) :=
+    ((circleSum_continuous 1).comp continuous_snd).sub continuous_fst
+  have hbtarget : circleSum 1 (a,b)∈(circleInverseHomeo h).target :=
+    (circleSum_strict 1 (a,b)).image_mem_toOpenPartialHomeomorph_target
+  have ht : ∀ᶠ z : RealPlane×RealPlane in 𝓝 (0,(a,b)),
+      circleSum 1 z.2-z.1∈(circleInverseHomeo h).target ∧
+      Real.sin ((circleInverse h (circleSum 1 z.2-z.1)).2-(circleInverse h (circleSum 1 z.2-z.1)).1)≠0 := by
+    have hlim : Tendsto (fun z : RealPlane×RealPlane => circleSum 1 z.2-z.1)
+        (𝓝 (0,(a,b))) (𝓝 (circleSum 1 (a,b))) := by simpa only [sub_zero] using hc.tendsto (0,(a,b))
+    have htt : ∀ᶠ q in 𝓝 (circleSum 1 (a,b)),q∈(circleInverseHomeo h).target :=
+      (circleInverseHomeo h).open_target.mem_nhds hbtarget
+    exact hlim.eventually (htt.and (circleInverse_regular h))
+  obtain ⟨P,hP,W,hW,hPW⟩ := mem_nhds_prod_iff.mp ht
+  obtain ⟨P₀,hP₀,hPsub,hPmem⟩ := mem_nhds_iff.mp hP
+  obtain ⟨W₀,hW₀,hWsub,hWmem⟩ := mem_nhds_iff.mp hW
+  obtain ⟨V₀,hV₀,hv₀,L,hL⟩ := circleSum_local_antilipschitz (r := 1) (p := (a,b)) (by simpa only [one_mul] using h)
+  let V := W₀∩V₀∩{q : RealPlane | Real.sin (q.2-q.1)≠0}
+  have hV : IsOpen V := (hWsub.inter hV₀).inter ((Real.continuous_sin.comp (continuous_snd.sub continuous_fst)).isOpen_preimage _ isOpen_ne)
+  refine ⟨P₀,V,hPsub,hPmem,hV,⟨⟨hWmem,hv₀⟩,h⟩,L*2,?_⟩
+  intro H hH
+  have htarget (q : RealPlane) (hq : q∈V) := hPW (a := (H,q)) ⟨hP₀ hH,hW₀ hq.1.1⟩
+  have heq (q : RealPlane) (hq : q∈V) : circleSum 1 (sumTransport h H q)=circleSum 1 q-H :=
+    circleInverse_right h (htarget q hq).1
+  refine ⟨?_,?_,fun _ hq => hq.2,fun q hq => (htarget q hq).2,heq⟩
+  · intro q hq
+    change ContinuousAt (fun z : RealPlane => circleInverse h (circleSum 1 z-H)) q
+    exact (circleInverse_continuousAt h (htarget q hq).1).comp (x := q)
+      (((circleSum_continuous 1).sub (continuous_const (y := H))).continuousAt)
+  · apply AntilipschitzWith.of_le_mul_dist
+    intro q r
+    have hh := hL.le_mul_dist ⟨q,q.2.1.2⟩ ⟨r,r.2.1.2⟩
+    have hd : dist (circleSum 1 q) (circleSum 1 r)=dist (circleSum 1 (sumTransport h H q)) (circleSum 1 (sumTransport h H r)) := by
+      rw [heq q q.2,heq r r.2,dist_sub_right]
+    change dist (q : RealPlane) (r : RealPlane) ≤ L*dist (circleSum 1 q) (circleSum 1 r) at hh
+    rw [hd] at hh
+    have hl := circleSum_one_lipschitz_sumTransport.dist_le_mul (sumTransport h H q) (sumTransport h H r)
+    apply hh.trans
+    have hm := mul_le_mul_of_nonneg_left hl L.2
+    simp only [NNReal.coe_mul,NNReal.coe_ofNat,Set.domRestrict_apply] at *
+    nlinarith
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+open scoped NNReal
+namespace EuclideanFiveColor
+variable (c : ℂ → Fin 5) (hc : Measurable c)
+def fourNonSingleton (x : ℂ) : Set ℝ := fourSupport c hc x 1∪fourSupport c hc x 2
+lemma four_complement_support (x : ℂ) :
+    (volume.restrict (fourRealState c hc x 0)ᶜ).support=fourNonSingleton c hc x := by
+  have he : (fourRealState c hc x 0)ᶜ=fourRealState c hc x 1∪fourRealState c hc x 2 := by
+    ext t
+    change (fourState c hc x (t : Angles)≠0) ↔
+      fourState c hc x (t : Angles)=1 ∨ fourState c hc x (t : Angles)=2
+    exact (show ∀s : Fin 3,s≠0 ↔ s=1 ∨ s=2 from by decide) _
+  have hd : Disjoint (fourRealState c hc x 1) (fourRealState c hc x 2) := by
+    apply Set.disjoint_left.mpr
+    intro t h1 h2
+    exact (by decide : (1 : Fin 3)≠2) (h1.symm.trans h2)
+  rw [he,Measure.restrict_union hd (four_realState_measurable c hc x 2),Measure.support_add]
+  rfl
+lemma four_support_transport (hw : WeakColoring 5 c) {x y : ℂ}
+    (hx : FourCenter c x) (hy : FourCenter c y) {s : Fin 3} (hs : s≠0)
+    {a b : ℝ} (ha : a∈fourSupport c hc x s)
+    (hb : b∈fourSupport c hc x (if s=1 then 2 else 1)) (hab : Real.sin (b-a)≠0)
+    {P V : Set RealPlane} {K : ℝ≥0} (hV : IsOpen V) (habV : (a,b)∈V)
+    (hbranch : ∀H∈P,
+        (∀q∈V,ContinuousAt (sumTransport hab H) q) ∧
+        AntilipschitzWith K (V.domRestrict (sumTransport hab H)) ∧
+        (∀q∈V,Real.sin (q.2-q.1)≠0) ∧
+        (∀q∈V,Real.sin ((sumTransport hab H q).2-(sumTransport hab H q).1)≠0) ∧
+        ∀q∈V,circleSum 1 (sumTransport hab H q)=circleSum 1 q-H)
+    (hxy : Complex.equivRealProdCLM (y-x)∈P) :
+    sumTransport hab (Complex.equivRealProdCLM (y-x)) (a,b)∈
+      fourNonSingleton c hc y×ˢfourNonSingleton c hc y := by
+  obtain ⟨hct,hL,hi,ho,heq⟩ := hbranch _ hxy
+  rw [← four_complement_support c hc y]
+  apply state_support_transport hc hw
+    (fourState_represents c hc hw hx s)
+    (fourState_represents c hc hw hx (if s=1 then 2 else 1))
+    (fourState_represents c hc hw hy 0)
+    ((show ∀s : Fin 3,s≠0 → ∀i : Fin 5,i∉fourGroup s∪fourGroup (if s=1 then 2 else 1) ↔ i=4 from by decide) s hs)
+    ha hb (sumTransport_measurable hab _) (hct _ habV) hV habV hL hi ho
+  intro q hq
+  rw [angularSum_chart,angularSum_chart,heq q hq,map_sub]
+  congr 1
+  abel
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+open scoped NNReal
+namespace EuclideanFiveColor
+lemma four_support_transport_general (coloring : ℂ → Fin 5) (hc : Measurable coloring)
+    (hw : WeakColoring 5 coloring) {x y : ℂ}
+    (hx : FourCenter coloring x) (hy : FourCenter coloring y) {s : Fin 3} (hs : s≠0)
+    {a₀ b₀ : ℝ} (h₀ : Real.sin (b₀-a₀)≠0)
+    {a b : ℝ} (ha : a∈fourSupport coloring hc x s)
+    (hb : b∈fourSupport coloring hc x (if s=1 then 2 else 1))
+    {P V : Set RealPlane} {K : ℝ≥0} (hV : IsOpen V) (habV : (a,b)∈V)
+    (hbranch : ∀H∈P,
+        (∀q∈V,ContinuousAt (sumTransport h₀ H) q) ∧
+        AntilipschitzWith K (V.domRestrict (sumTransport h₀ H)) ∧
+        (∀q∈V,Real.sin (q.2-q.1)≠0) ∧
+        (∀q∈V,Real.sin ((sumTransport h₀ H q).2-(sumTransport h₀ H q).1)≠0) ∧
+        ∀q∈V,circleSum 1 (sumTransport h₀ H q)=circleSum 1 q-H)
+    (hxy : Complex.equivRealProdCLM (y-x)∈P) :
+    sumTransport h₀ (Complex.equivRealProdCLM (y-x)) (a,b)∈
+      fourNonSingleton coloring hc y×ˢfourNonSingleton coloring hc y := by
+  obtain ⟨hct,hL,hi,ho,heq⟩ := hbranch _ hxy
+  rw [← four_complement_support coloring hc y]
+  apply state_support_transport hc hw
+    (fourState_represents coloring hc hw hx s)
+    (fourState_represents coloring hc hw hx (if s=1 then 2 else 1))
+    (fourState_represents coloring hc hw hy 0)
+    ((show ∀s : Fin 3,s≠0 → ∀i : Fin 5,i∉fourGroup s∪fourGroup (if s=1 then 2 else 1) ↔ i=4 from by decide) s hs)
+    ha hb (sumTransport_measurable h₀ _) (hct _ habV) hV habV hL hi ho
+  intro q hq
+  rw [angularSum_chart,angularSum_chart,heq q hq,map_sub]
+  congr 1
+  abel
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+lemma two_closed_shift_separated_not_cover {X : Type*} [TopologicalSpace X]
+    [PreconnectedSpace X] {A B : Set X} (hA : IsClosed A) (hB : IsClosed B)
+    (ha : A.Nonempty) (hb : B.Nonempty) (R : X → X)
+    (hRA : ∀x∈A,R x∉A) (hRB : ∀x∈B,R x∉B) : A∪B≠univ := by
+  intro hcov
+  obtain ⟨x,_,hxA,hxB⟩ := isPreconnected_closed_iff.mp isPreconnected_univ A B hA hB
+    (by rw [hcov]) (by simpa using ha) (by simpa using hb)
+  have hm : R x∈A∪B := hcov.symm ▸ mem_univ _
+  exact hm.elim (hRA x hxA) (hRB x hxB)
+lemma closed_gap {N : Set ℝ} (hN : IsClosed N) {z l r : ℝ}
+    (hz : z∉N) (hl : l∈N) (hr : r∈N) (hlz : l<z) (hzr : z<r) :
+    ∃a b : ℝ,a∈N ∧ b∈N ∧ a<z ∧ z<b ∧ Ioo a b⊆Nᶜ := by
+  have hLC : IsCompact (N∩Icc l z) := isCompact_Icc.inter_left hN
+  have hRC : IsCompact (N∩Icc z r) := isCompact_Icc.inter_left hN
+  obtain ⟨a,ha,ham⟩ := hLC.exists_isMaxOn (f := id) ⟨l,hl,le_rfl,hlz.le⟩ continuous_id.continuousOn
+  obtain ⟨b,hb,hbm⟩ := hRC.exists_isMinOn (f := id) ⟨r,hr,hzr.le,le_rfl⟩ continuous_id.continuousOn
+  have haz : a<z := lt_of_le_of_ne ha.2.2 (fun he => hz (he ▸ ha.1))
+  have hzb : z<b := lt_of_le_of_ne hb.2.1 (fun he => hz (he.symm ▸ hb.1))
+  refine ⟨a,b,ha.1,hb.1,haz,hzb,?_⟩
+  intro t ht hn
+  by_cases htz : t≤z
+  · have hta : t≤a := ham ⟨hn,ha.2.1.trans ht.1.le,htz⟩
+    exact (not_lt_of_ge hta) ht.1
+  · have hbt : b≤t := hbm ⟨hn,(not_le.mp htz).le,ht.2.le.trans hb.2.2⟩
+    exact (not_lt_of_ge hbt) ht.2
+lemma positive_normalized_increment_enters_gap {a b : ℝ} (hab : a<b)
+    {u δ : ℕ → ℝ} {v : ℝ} (hδ : ∀n,0<δ n) (hv : 0<v)
+    (hu : Tendsto u atTop (𝓝 a))
+    (hs : Tendsto (fun n => (u n-a)/δ n) atTop (𝓝 v)) :
+    ∀ᶠn in atTop,u n∈Ioo a b := by
+  filter_upwards [hu.eventually (gt_mem_nhds hab),hs.eventually (lt_mem_nhds hv)] with n hn hp
+  exact ⟨sub_pos.mp ((div_pos_iff.mp hp).resolve_right (fun h => (not_lt_of_ge (hδ n).le) h.2)).1,hn⟩
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+lemma direction_sixty_unit_real (a : ℝ) :
+    ‖direction (a : Angles)-direction ((a+sixty : ℝ) : Angles)‖=1 := by
+  have he : direction (a : Angles)=direction ((a+sixty : ℝ) : Angles)+direction ((a-sixty : ℝ) : Angles) := by
+    apply Complex.equivRealProdCLM.injective
+    simpa only [map_add,direction_realPlane,circleTarget,samplingParameter,sub_self,map_zero,one_smul,zero_add,circleSum,one_mul,Prod.mk_add_mk] using sixty_sum (0:ℂ) a
+  rw [he,add_sub_cancel_left,direction_norm]
+variable (c : ℂ → Fin 5) (hc : Measurable c)
+lemma four_support_nonempty (hw : WeakColoring 5 c) {x : ℂ} (hx : FourCenter c x)
+    {s : Fin 3} (hs : s≠0) : (fourSupport c hc x s).Nonempty := by
+  apply Measure.nonempty_support
+  intro hz
+  have hh := congrArg (fun μ : Measure ℝ => μ univ) hz
+  simp only [Measure.restrict_apply_univ,Measure.coe_zero,Pi.zero_apply] at hh
+  exact (four_realState_positive c hc hw hx hs).ne' hh
+lemma four_support_no_sixty (hw : WeakColoring 5 c) {x : ℂ} (hx : FourCenter c x)
+    {s : Fin 3} (hs : s≠0) {a : ℝ} (ha : a∈fourSupport c hc x s) :
+    a+sixty∉fourSupport c hc x s := by
+  intro hb
+  exact four_support_sequence_no_unit c hc hw (fun _ => hx) hx tendsto_const_nhds hs
+    tendsto_const_nhds tendsto_const_nhds (fun _ => ha) (fun _ => hb) (direction_sixty_unit_real a)
+lemma four_support_union_proper (hw : WeakColoring 5 c) {x : ℂ} (hx : FourCenter c x) :
+    fourSupport c hc x 1∪fourSupport c hc x 2≠univ :=
+  two_closed_shift_separated_not_cover Measure.isClosed_support Measure.isClosed_support
+    (four_support_nonempty c hc hw hx (by decide)) (four_support_nonempty c hc hw hx (by decide))
+    (fun a => a+sixty) (fun _ ha => four_support_no_sixty c hc hw hx (by decide) ha)
+    (fun _ ha => four_support_no_sixty c hc hw hx (by decide) ha)
+lemma four_outside_support_state (x : ℂ) :
+    ∀ᵐt : ℝ∂volume,t∉fourSupport c hc x 1∪fourSupport c hc x 2 →
+      fourState c hc x (t : Angles)=0 := by
+  have h1 : ∀ᵐt : ℝ∂volume,t∈fourRealState c hc x 1 → t∈fourSupport c hc x 1 :=
+    (ae_restrict_iff' (four_realState_measurable c hc x 1)).mp (volume.restrict (fourRealState c hc x 1)).support_mem_ae
+  have h2 : ∀ᵐt : ℝ∂volume,t∈fourRealState c hc x 2 → t∈fourSupport c hc x 2 :=
+    (ae_restrict_iff' (four_realState_measurable c hc x 2)).mp (volume.restrict (fourRealState c hc x 2)).support_mem_ae
+  filter_upwards [h1,h2] with t ht1 ht2 hn
+  have hn1 : fourState c hc x (t : Angles)≠1 := fun he => hn (Or.inl (ht1 he))
+  have hn2 : fourState c hc x (t : Angles)≠2 := fun he => hn (Or.inr (ht2 he))
+  exact (show ∀s : Fin 3,s≠1 → s≠2 → s=0 from by decide) _ hn1 hn2
+lemma four_gap_length (hw : WeakColoring 5 c) {x : ℂ} (hx : FourCenter c x)
+    {a b : ℝ} (hgap : Ioo a b⊆(fourSupport c hc x 1∪fourSupport c hc x 2)ᶜ) : b-a ≤ sixty := by
+  by_contra! hab
+  have hp : 0<volume (Ioo a (b-sixty)) := by
+    rw [Real.volume_Ioo]
+    exact ENNReal.ofReal_pos.mpr (by linarith)
+  have h0 := four_outside_support_state c hc x
+  have hshift := (measurePreserving_add_right (volume : Measure ℝ) sixty).quasiMeasurePreserving.ae h0
+  have hneq := angleMk_quasiMeasurePreserving.ae (fourState_changes c hc hw hx)
+  obtain ⟨t,ht,ht0,hts,htn⟩ := Measure.exists_mem_of_measure_ne_zero_of_ae hp.ne'
+    (ae_restrict_of_ae (h0.and (hshift.and hneq)))
+  have hκ : 0<sixty := by dsimp [sixty]; positivity
+  have he0 := ht0 (hgap ⟨ht.1,by linarith [ht.2]⟩)
+  have he1 := hts (hgap ⟨by linarith [ht.1],by linarith [ht.2]⟩)
+  apply htn
+  simpa only [sixty,AddCircle.coe_add,he0] using he1.symm
+lemma four_gap_exists (hw : WeakColoring 5 c) {x : ℂ} (hx : FourCenter c x) :
+    ∃a b : ℝ,a∈fourSupport c hc x 1∪fourSupport c hc x 2 ∧
+      b∈fourSupport c hc x 1∪fourSupport c hc x 2 ∧ a<b ∧ b-a ≤ sixty ∧
+      Ioo a b⊆(fourSupport c hc x 1∪fourSupport c hc x 2)ᶜ := by
+  let N := fourSupport c hc x 1∪fourSupport c hc x 2
+  have hN : IsClosed N := Measure.isClosed_support.union Measure.isClosed_support
+  have hex : ∃z : ℝ,z∉N := by
+    by_contra! h
+    exact four_support_union_proper c hc hw hx (Set.eq_univ_of_forall h)
+  obtain ⟨z,hz⟩ := hex
+  have hκ : 0<sixty := by dsimp [sixty]; positivity
+  have hn (a b : ℝ) (hab : sixty<b-a) : ∃t∈Ioo a b,t∈N := by
+    by_contra! hbad
+    have hle := four_gap_length c hc hw hx hbad
+    exact (not_lt_of_ge hle) hab
+  obtain ⟨l,hl,hlN⟩ := hn (z-sixty-1) z (by linarith)
+  obtain ⟨r,hr,hrN⟩ := hn z (z+sixty+1) (by linarith)
+  obtain ⟨a,b,ha,hb,haz,hzb,hgap⟩ := closed_gap hN hz hlN hrN hl.2 hr.1
+  exact ⟨a,b,ha,hb,haz.trans hzb,four_gap_length c hc hw hx hgap,hgap⟩
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+lemma open_subset_closed_of_ae {U N : Set ℝ} (hU : IsOpen U) (hN : IsClosed N)
+    (h : ∀ᵐt : ℝ∂volume,t∈U → t∈N) : U⊆N := by
+  apply Set.sdiff_eq_empty.mp
+  apply (hU.sdiff hN).eq_empty_of_measure_zero (μ := volume)
+  apply measure_eq_zero_iff_ae_notMem.mpr
+  filter_upwards [h] with t ht htn
+  exact htn.2 (ht htn.1)
+variable (c : ℂ → Fin 5) (hc : Measurable c)
+lemma four_gap_shift_subset (hw : WeakColoring 5 c) {x : ℂ} (hx : FourCenter c x)
+    {a b : ℝ} (hab : a<b)
+    (hgap : Ioo a b⊆(fourSupport c hc x 1∪fourSupport c hc x 2)ᶜ) :
+    Icc (a+sixty) (b+sixty)⊆fourSupport c hc x 1∪fourSupport c hc x 2 := by
+  have hN : IsClosed (fourSupport c hc x 1∪fourSupport c hc x 2) :=
+    Measure.isClosed_support.union Measure.isClosed_support
+  have hs : Ioo (a+sixty) (b+sixty)⊆fourSupport c hc x 1∪fourSupport c hc x 2 := by
+    apply open_subset_closed_of_ae isOpen_Ioo hN
+    have h0 := four_outside_support_state c hc x
+    have hneq : ∀ᵐt : ℝ∂volume,fourState c hc x (t : Angles)≠fourState c hc x ((t+sixty : ℝ) : Angles) := by
+      simpa only [AddCircle.coe_add,sixty] using angleMk_quasiMeasurePreserving.ae (fourState_changes c hc hw hx)
+    have hm := (measurePreserving_add_right (volume : Measure ℝ) (-sixty)).quasiMeasurePreserving
+    filter_upwards [h0,hm.ae h0,hm.ae hneq] with t ht htm htn htI
+    by_contra hn
+    have he0 := ht hn
+    have hem := htm (hgap ⟨by linarith [htI.1],by linarith [htI.2]⟩)
+    apply htn
+    simp only [neg_add_cancel_right,he0,hem]
+  have hz := closure_minimal hs hN
+  rwa [closure_Ioo (by linarith : a+sixty≠b+sixty)] at hz
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Filter Topology Asymptotics
+namespace EuclideanFiveColor
+lemma strict_increment_limit {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+    [NormedAddCommGroup F] [NormedSpace ℝ F] {f : E → F} {L : E →L[ℝ] F} {p : E}
+    (hf : HasStrictFDerivAt f L p) {u v : ℕ → E} {δ : ℕ → ℝ} {w : E}
+    (hu : Tendsto u atTop (𝓝 p)) (hv : Tendsto v atTop (𝓝 p)) (hδ : ∀n,0<δ n)
+    (hw : Tendsto (fun n => (δ n)⁻¹ • (u n-v n)) atTop (𝓝 w)) :
+    Tendsto (fun n => (δ n)⁻¹ • (f (u n)-f (v n))) atTop (𝓝 (L w)) := by
+  have hb : (fun n => u n-v n) =O[atTop] δ := by
+    apply IsBigO.of_bound (‖w‖+1)
+    filter_upwards [hw.norm.eventually (gt_mem_nhds (lt_add_one ‖w‖))] with n hn
+    rw [norm_smul,Real.norm_eq_abs,abs_inv,abs_of_pos (hδ n)] at hn
+    rw [Real.norm_eq_abs,abs_of_pos (hδ n)]
+    exact ((inv_mul_lt_iff₀ (hδ n)).mp hn).le.trans_eq (mul_comm _ _)
+  have ho : (fun n => f (u n)-f (v n)-L (u n-v n)) =o[atTop] δ :=
+    (hf.isLittleO.comp_tendsto (hu.prodMk_nhds hv)).trans_isBigO hb
+  have hz := ho.tendsto_inv_smul_nhds_zero
+  have hl := L.continuous.tendsto w |>.comp hw
+  have hh := hz.add hl
+  simpa only [Function.comp_def,map_smul,smul_sub,map_sub,sub_add_cancel,zero_add] using hh
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology
+namespace EuclideanFiveColor
+def transportVelocity (H : RealPlane) (a b : ℝ) : ℝ :=
+  -(H.1*Real.cos b+H.2*Real.sin b)/Real.sin (b-a)
+lemma circleInverseDeriv_fst {a b : ℝ} (h : Real.sin (b-a)≠0) (H : RealPlane) :
+    ((circleSumDerivEquiv (r := 1) (p := (a,b)) (by simpa only [one_mul] using h)).symm (-H)).1=
+      transportVelocity H a b := by
+  let L := circleSumDerivEquiv (r := 1) (p := (a,b)) (by simpa only [one_mul] using h)
+  let q := L.symm (-H)
+  have he : circleSumDeriv 1 (a,b) q = -H := L.apply_symm_apply (-H)
+  have h1 := congrArg Prod.fst he
+  have h2 := congrArg Prod.snd he
+  change 1*(-Real.sin a*q.1)+(-Real.sin b)*q.2= -H.1 at h1
+  change 1*(Real.cos a*q.1)+Real.cos b*q.2= -H.2 at h2
+  change q.1= -(H.1*Real.cos b+H.2*Real.sin b)/Real.sin (b-a)
+  apply (eq_div_iff h).mpr
+  rw [Real.sin_sub]
+  linear_combination Real.cos b*h1+Real.sin b*h2
+lemma sumTransport_increment_limit {a b : ℝ} (h : Real.sin (b-a)≠0)
+    {Hn q : ℕ → RealPlane} {δ : ℕ → ℝ} {H : RealPlane}
+    (hH : Tendsto Hn atTop (𝓝 0)) (hq : Tendsto q atTop (𝓝 (a,b)))
+    (hδ : ∀n,0<δ n) (hdir : Tendsto (fun n => (δ n)⁻¹•Hn n) atTop (𝓝 H)) :
+    Tendsto (fun n => ((sumTransport h (Hn n) (q n)).1-(q n).1)/δ n)
+      atTop (𝓝 (transportVelocity H a b)) := by
+  have hv : Tendsto (fun n => circleSum 1 (q n)) atTop (𝓝 (circleSum 1 (a,b))) :=
+    ((circleSum_continuous 1).tendsto (a,b)).comp hq
+  have hu : Tendsto (fun n => circleSum 1 (q n)-Hn n) atTop (𝓝 (circleSum 1 (a,b))) := by
+    simpa only [sub_zero] using hv.sub hH
+  have hdiff : Tendsto (fun n => (δ n)⁻¹•((circleSum 1 (q n)-Hn n)-circleSum 1 (q n)))
+      atTop (𝓝 (-H)) := by
+    simpa only [sub_sub_cancel_left,smul_neg] using hdir.neg
+  have hh := strict_increment_limit (circleInverse_strict h) hu hv hδ hdiff
+  have hid : ∀ᶠ n in atTop,circleInverse h (circleSum 1 (q n))=q n := by
+    have hb : (a,b)∈(circleInverseHomeo h).source :=
+      (circleSum_strict 1 (a,b)).mem_toOpenPartialHomeomorph_source
+    filter_upwards [hq.eventually ((circleInverseHomeo h).open_source.mem_nhds hb)] with n hn
+    have ht : circleSum 1 (q n)∈(circleInverseHomeo h).target := (circleInverseHomeo h).map_source hn
+    rw [circleInverse_eq h ht]
+    exact (circleInverseHomeo h).left_inv hn
+  have hf := hh.fst_nhds
+  simp only [ContinuousLinearEquiv.coe_coe] at hf
+  rw [circleInverseDeriv_fst h H] at hf
+  apply hf.congr'
+  filter_upwards [hid] with n hn
+  simp only [hn,div_eq_inv_mul,sumTransport]
+  rfl
+lemma sumTransport_tendsto {a b : ℝ} (h : Real.sin (b-a)≠0)
+    {Hn q : ℕ → RealPlane} (hH : Tendsto Hn atTop (𝓝 0))
+    (hq : Tendsto q atTop (𝓝 (a,b))) :
+    Tendsto (fun n => sumTransport h (Hn n) (q n)) atTop (𝓝 (a,b)) := by
+  have ht : Tendsto (fun n => circleSum 1 (q n)-Hn n) atTop (𝓝 (circleSum 1 (a,b))) := by
+    simpa only [sub_zero,Function.comp_def] using (((circleSum_continuous 1).tendsto (a,b)).comp hq).sub hH
+  have hi := (circleInverse_strict h).continuousAt.tendsto.comp ht
+  simpa only [sub_zero,circleInverse_at,sumTransport,Function.comp_def] using hi
+lemma transportVelocity_neg (H : RealPlane) (a b : ℝ) :
+    transportVelocity (-H) a b= -transportVelocity H a b := by
+  simp only [transportVelocity,Prod.fst_neg,Prod.snd_neg]
+  ring
+lemma sumTransport_double_increment {a b b' : ℝ}
+    (h : Real.sin (b-a)≠0) (h' : Real.sin (b'-a)≠0)
+    {Hn : ℕ → RealPlane} {δ bn : ℕ → ℝ} {H : RealPlane}
+    (hH : Tendsto Hn atTop (𝓝 0)) (hb : Tendsto bn atTop (𝓝 b'))
+    (hδ : ∀n,0<δ n) (hdir : Tendsto (fun n => (δ n)⁻¹•Hn n) atTop (𝓝 H)) :
+    Tendsto (fun n =>
+      ((sumTransport h' (-Hn n) ((sumTransport h (Hn n) (a,b)).1,bn n)).1-a)/δ n)
+      atTop (𝓝 (transportVelocity H a b-transportVelocity H a b')) := by
+  have hct := sumTransport_tendsto h hH tendsto_const_nhds
+  have hf := sumTransport_increment_limit h hH tendsto_const_nhds hδ hdir
+  have hn : Tendsto (fun n => -Hn n) atTop (𝓝 0) := by simpa only [neg_zero] using hH.neg
+  have hnd : Tendsto (fun n => (δ n)⁻¹•(-Hn n)) atTop (𝓝 (-H)) := by
+    simpa only [smul_neg] using hdir.neg
+  have hr := sumTransport_increment_limit h' hn (hct.fst_nhds.prodMk_nhds hb) hδ hnd
+  have hh := hr.add hf
+  rw [transportVelocity_neg] at hh
+  convert hh using 1
+  · funext n
+    dsimp
+    ring
+  · ring_nf
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+lemma ae_angular_nonparallel (a : ℝ) : ∀ᵐb : ℝ∂volume,Real.sin (b-a)≠0 := by
+  have hs : {b : ℝ | Real.sin (b-a)=0}⊆Set.range (fun n : ℤ => a+n*Real.pi) := by
+    intro b hb
+    obtain ⟨n,hn⟩ := Real.sin_eq_zero_iff.mp hb
+    exact ⟨n,by linarith⟩
+  rw [ae_iff]
+  simpa only [not_not] using measure_mono_null hs ((countable_range _).measure_zero volume)
+lemma transportVelocity_ne {H : RealPlane} {a b b' : ℝ}
+    (ha : H.1*Real.cos a+H.2*Real.sin a≠0)
+    (hb : Real.sin (b-a)≠0) (hb' : Real.sin (b'-a)≠0)
+    (hbb : Real.sin (b'-b)≠0) : transportVelocity H a b≠transportVelocity H a b' := by
+  intro he
+  have hq := (div_eq_div_iff hb hb').mp he
+  have hz : (H.1*Real.cos a+H.2*Real.sin a)*Real.sin (b'-b)=0 := by
+    simp only [Real.sin_sub] at hq ⊢
+    linear_combination -hq
+  exact mul_ne_zero ha hbb hz
+lemma exists_transportVelocity_lt {A : Set ℝ} (hA : 0<volume A)
+    {H : RealPlane} {a : ℝ} (ha : H.1*Real.cos a+H.2*Real.sin a≠0) :
+    ∃b∈A,∃b'∈A,Real.sin (b-a)≠0 ∧ Real.sin (b'-a)≠0 ∧
+      transportVelocity H a b<transportVelocity H a b' := by
+  obtain ⟨b,hb,hba⟩ := Measure.exists_mem_of_measure_ne_zero_of_ae hA.ne'
+    (ae_restrict_of_ae (ae_angular_nonparallel a))
+  obtain ⟨b',hb',hba',hbb⟩ := Measure.exists_mem_of_measure_ne_zero_of_ae hA.ne'
+    (ae_restrict_of_ae ((ae_angular_nonparallel a).and (ae_angular_nonparallel b)))
+  rcases (transportVelocity_ne ha hba hba' hbb).lt_or_gt with hlt | hgt
+  · exact ⟨b,hb,b',hb',hba,hba',hlt⟩
+  · exact ⟨b',hb',b,hb,hba',hba,hgt⟩
+end EuclideanFiveColor
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PlaneColoring/FullApproximation.lean b/lean/OAI/Geometry/PlaneColoring/FullApproximation.lean
new file mode 100644
index 0000000000000000000000000000000000000000..7218a7ae82dfb983b8306a104a2b18c6fa33320c
--- /dev/null
+++ b/lean/OAI/Geometry/PlaneColoring/FullApproximation.lean
@@ -0,0 +1,339 @@
+import OAI.Geometry.PlaneColoring.ContinuumLimits
+import OAI.Geometry.PlaneColoring.GraphPatches
+
+namespace OAI
+
+noncomputable section
+open scoped Classical
+open Set Filter Topology TopologicalSpace MeasureTheory
+namespace EuclideanFiveColor
+/-- Compactness and the angular theorems force all sufficiently fine crossing graphs
+to be forests. This is about actual compact bridges, not assumed interface data. -/
+theorem eventually_forest_of_bridges {c : ℂ → Fin 5} (hc : Measurable c)
+    (hw : WeakColoring 5 c) {x : ℂ} {ρ : ℝ} (hρ : 0<ρ)
+    (D : ℕ → SimpleGraph (Fin 5)) {r ε : ℕ → ℝ}
+    (hr : Tendsto r atTop (𝓝 0)) (hε : Tendsto ε atTop (𝓝 0)) (hpos : ∀n,0<ε n)
+    {H E : ℕ → Set ℂ} (hE : Tendsto (fun n => volume (E n)) atTop (𝓝 0))
+    (hnear : ∀n z q,q∈H n → dist z q=1 → z∈E n)
+    {α : ℝ} (hα : 0<α)
+    (hbridge : ∀n i j,(D n).Adj i j → ∃K : NonemptyCompacts ℂ,
+      (K : Set ℂ)⊆Metric.closedBall x ρ ∧ IsPreconnected (K : Set ℂ) ∧
+      (∃a∈K,dist a x≤r n) ∧ ((K : Set ℂ)∩Metric.sphere x ρ).Nonempty ∧
+      (∀q∈K,q∉H n → α<diskAverage c (ε n) i q) ∧
+      (∀q∈K,q∉H n → α<diskAverage c (ε n) j q)) :
+    ∀ᶠ n in atTop,(D n).IsAcyclic := by
+  have he : ∀i j : Fin 5,∀ᶠ n in atTop,(D n).Adj i j → CommonInterface c x i j := by
+    intro i j
+    by_cases hcommon : CommonInterface c x i j
+    · exact Eventually.of_forall fun _ _ => hcommon
+    · have hn : ∀ᶠ n in atTop,¬(D n).Adj i j := by
+        by_contra hnot
+        have hfreq : ∃ᶠ n in atTop,(D n).Adj i j := hnot
+        obtain ⟨φ,hφ,hφD⟩ := subseq_forall_of_frequently (x:=fun n : ℕ => n) (p:=fun n => (D n).Adj i j) tendsto_id hfreq
+        choose K hK hconn ha hout hi hj using fun n => hbridge (φ n) i j (hφD n)
+        choose a haK har using ha
+        have halim : Tendsto a atTop (𝓝 x) := by
+          rw [tendsto_iff_dist_tendsto_zero]
+          exact squeeze_zero (fun _ => dist_nonneg) har (hr.comp hφ)
+        exact hcommon (commonInterface_of_compact_sequence hc hw hρ K hK hconn a haK halim hout
+          (hε.comp hφ) (fun n => hpos (φ n)) (hE.comp hφ) (fun n => hnear (φ n)) hα i j hi hj)
+      exact hn.mono fun _ hn hd => False.elim (hn hd)
+  have hall : ∀ᶠ n in atTop,∀i j : Fin 5,(D n).Adj i j → CommonInterface c x i j :=
+    eventually_all.mpr fun i => eventually_all.mpr (he i)
+  filter_upwards [hall] with n hn
+  exact commonInterface_graph_acyclic hc hw x (D n) hn
+end EuclideanFiveColor
+end
+
+noncomputable section
+open scoped Classical
+open Set Filter Topology TopologicalSpace MeasureTheory ContinuousMap
+namespace EuclideanFiveColor
+lemma midpoint_threshold_labels {c : ℂ → Fin 5} (hc : Measurable c) {ε α : ℝ}
+    (hε : 0<ε) (hα : α<1/5) {q : ℂ} {i j : Fin 5} (hij : i≠j)
+    (he : thresholdMap c ε α q=graphEdgePoint i j (1/2)) :
+    α<diskAverage c ε i q ∧ α<diskAverage c ε j q := by
+  constructor
+  · apply (thresholdMap_pos hc hε hα q i).mp
+    rw [he]
+    norm_num [graphEdgePoint,graphVertex,hij,hij.symm]
+  · apply (thresholdMap_pos hc hε hα q j).mp
+    rw [he]
+    norm_num [graphEdgePoint,graphVertex,hij,hij.symm]
+/-- The actual continuous annulus maps eventually permit disk extensions. -/
+theorem annulus_maps_eventually_forest {c : ℂ → Fin 5} (hc : Measurable c)
+    (hw : WeakColoring 5 c) {x : ℂ} {R : ℝ} (hR : 0<R) {r ε : ℕ → ℝ}
+    (hr : Tendsto r atTop (𝓝 0)) (hε : Tendsto ε atTop (𝓝 0)) (hpos : ∀n,0<ε n)
+    {H E : ℕ → Set ℂ} (hE : Tendsto (fun n => volume (E n)) atTop (𝓝 0))
+    (hnear : ∀n z q,q∈H n → dist z q=1 → z∈E n)
+    {α : ℝ} (hα : 0<α) (hα' : α<1/5)
+    (f : ∀n,C(planeAnnulus x (r n) R,GraphSpace (⊤ : SimpleGraph (Fin 5))))
+    (hf : ∀n z,z.val∉H n → (f n z).val=thresholdMap c (ε n) α z.val) :
+    ∀ᶠ n in atTop,(crossingGraph (fun z => (f n z).val)
+      (annulusInner x (r n) R) (annulusOuter x (r n) R)).IsAcyclic := by
+  apply eventually_forest_of_bridges hc hw hR _ hr hε hpos hE hnear hα
+  intro n i j hij
+  obtain ⟨K,hK,hconn,hval,hin,hout⟩ := hij.2
+  let L : NonemptyCompacts ℂ := ⟨⟨Subtype.val '' K,hK.image continuous_subtype_val⟩,
+    hconn.nonempty.image _⟩
+  have hsub : (L : Set ℂ)⊆Metric.closedBall x R := by
+    rintro q ⟨z,hz,rfl⟩
+    exact z.property.1
+  have hpre : IsPreconnected (L : Set ℂ) :=
+    hconn.isPreconnected.image _ continuous_subtype_val.continuousOn
+  have hanc : ∃a∈L,dist a x≤r n := by
+    obtain ⟨a,ha,har⟩ := hin
+    exact ⟨a.val,⟨a,ha,rfl⟩,har.le⟩
+  have hout' : ((L : Set ℂ)∩Metric.sphere x R).Nonempty := by
+    obtain ⟨a,ha,haR⟩ := hout
+    exact ⟨a.val,⟨a,ha,rfl⟩,haR⟩
+  refine ⟨L,hsub,hpre,hanc,hout',?_,?_⟩
+  · rintro q ⟨z,hz,rfl⟩ hzH
+    have he : thresholdMap c (ε n) α z.val=graphEdgePoint i j (1/2) := (hf n z hzH).symm.trans (hval hz)
+    exact (midpoint_threshold_labels hc (hpos n) hα' hij.1 he).1
+  · rintro q ⟨z,hz,rfl⟩ hzH
+    have he : thresholdMap c (ε n) α z.val=graphEdgePoint i j (1/2) := (hf n z hzH).symm.trans (hval hz)
+    exact (midpoint_threshold_labels hc (hpos n) hα' hij.1 he).2
+end EuclideanFiveColor
+end
+
+noncomputable section
+open scoped Classical
+open Set Topology ContinuousMap
+universe u
+namespace EuclideanFiveColor
+/-- Actual finite disk pasting from a closed outside domain. Disks may cross the
+boundary of the ambient compact region. -/
+theorem finite_disk_paste {Y : Type u} [TopologicalSpace Y] (Q : Set ℂ)
+    (S : Finset MarkedDisk)
+    (hdis : (S : Set MarkedDisk).Pairwise (fun a b => Disjoint a.closedBall b.closedBall))
+    (f : C(↥(Q \ diskHoles S),Y))
+    (g : ∀d : S,C(d.val.closedBall,Y))
+    (hb : ∀d : S,∀z : Metric.sphere d.val.center d.val.radius,∀hzQ : z.val∈Q,
+      ∀hzS : z.val∉diskHoles S,
+      g d ⟨z.val,Metric.sphere_subset_closedBall z.property⟩=f ⟨z.val,hzQ,hzS⟩) :
+    ∃F : C(Q,Y),∀z : Q,∀hz : z.val∉diskHoles S,F z=f ⟨z.val,z.property,hz⟩ := by
+  let Z : Option S → Set Q := fun i => match i with
+    | none => {z | z.val∉diskHoles S}
+    | some d => {z | z.val∈d.val.closedBall}
+  have hZ : ∀i,IsClosed (Z i) := by
+    intro i
+    cases i with
+    | none => exact (diskHoles_open S).isClosed_compl.preimage continuous_subtype_val
+    | some d => exact Metric.isClosed_closedBall.preimage continuous_subtype_val
+  have hcov : ⋃i,Z i=univ := by
+    apply iUnion_eq_univ_iff.mpr
+    intro z
+    by_cases hz : z.val∈diskHoles S
+    · obtain ⟨d,hd,hzd⟩ := mem_diskHoles.mp hz
+      exact ⟨some ⟨d,hd⟩,Metric.ball_subset_closedBall hzd⟩
+    · exact ⟨none,hz⟩
+  let φ₀ : C(Z none,Y) := ⟨fun z => f ⟨z.val.val,z.val.property,z.property⟩,
+    f.continuous.comp ((continuous_subtype_val.comp continuous_subtype_val).subtype_mk _)⟩
+  let φ₁ (d : S) : C(Z (some d),Y) := ⟨fun z => g d ⟨z.val.val,z.property⟩,
+    (g d).continuous.comp ((continuous_subtype_val.comp continuous_subtype_val).subtype_mk _)⟩
+  let φ : ∀i,C(Z i,Y) := fun i => match i with | none => φ₀ | some d => φ₁ d
+  have hbd (d : S) (z : Q) (h0 : z∈Z none) (h1 : z∈Z (some d)) :
+      φ none ⟨z,h0⟩=φ (some d) ⟨z,h1⟩ := by
+    have hn : z.val∉d.val.ball := fun hz => h0 (mem_diskHoles.mpr ⟨d.val,d.property,hz⟩)
+    have hs : z.val∈Metric.sphere d.val.center d.val.radius :=
+      le_antisymm h1 (le_of_not_gt hn)
+    exact (hb d ⟨z.val,hs⟩ z.property h0).symm
+  have hφ : ∀i j z (hi : z∈Z i) (hj : z∈Z j),φ i ⟨z,hi⟩=φ j ⟨z,hj⟩ := by
+    intro i j z hi hj
+    cases i with
+    | none =>
+      cases j with
+      | none => rfl
+      | some d => exact hbd d z hi hj
+    | some d =>
+      cases j with
+      | none => exact (hbd d z hj hi).symm
+      | some e =>
+        by_cases hde : d=e
+        · subst e; rfl
+        · exact False.elim (Set.disjoint_left.mp (hdis d.property e.property
+            (fun he => hde (Subtype.ext he))) hi hj)
+  refine ⟨closedCoverMap Z hZ hcov φ hφ,fun z hz => ?_⟩
+  exact closedCoverMap_apply Z hZ hcov φ hφ (i:=none) ⟨z,hz⟩
+end EuclideanFiveColor
+end
+
+noncomputable section
+open scoped Classical
+open Set Filter Topology MeasureTheory ContinuousMap
+namespace EuclideanFiveColor
+namespace HoleApprox
+def annulusMap {c : ℂ → Fin 5} {α t R : ℝ} {L : Finset ℂ}
+    (a : HoleApprox c α L t) (hR : R<1) (htR : 2*t<R)
+    (hsep : ∀x∈L,∀y∈L,x≠y → 2*R<dist x y) {x : ℂ} (hx : x∈activeCenters L) :
+    C(planeAnnulus x (2*t) R,GraphSpace (⊤ : SimpleGraph (Fin 5))) :=
+  ⟨fun z => a.map ⟨z.val,a.annulus_in_domain hR htR hsep hx z.property⟩,
+    a.map.continuous.comp (continuous_subtype_val.subtype_mk _)⟩
+/-- Fill all the actual core holes simultaneously when their surrounding crossing
+graphs are forests, retaining the threshold map outside the original and outer disks. -/
+theorem fill_cores {c : ℂ → Fin 5} {α t R : ℝ} {L : Finset ℂ}
+    (a : HoleApprox c α L t) (hR : R<1) (htR : 2*t<R)
+    (hsep : ∀x∈L,∀y∈L,x≠y → 2*R<dist x y)
+    (hforest : ∀x (hx : x∈activeCenters L),
+      (crossingGraph (fun z => (a.annulusMap hR htR hsep hx z).val)
+        (annulusInner x (2*t) R) (annulusOuter x (2*t) R)).IsAcyclic) :
+    ∃F : C(Metric.closedBall (0:ℂ) 5,GraphSpace (⊤ : SimpleGraph (Fin 5))),
+      ∀z : Metric.closedBall (0:ℂ) 5,z.val∉diskHoles a.disks →
+        z.val∉diskHoles (centerDisks (activeCenters L) R) →
+        (F z).val=thresholdMap c a.ε α z.val := by
+  let D := centerDisks (activeCenters L) R
+  have hdis : (D : Set MarkedDisk).Pairwise (fun a b => Disjoint a.closedBall b.closedBall) :=
+    centerDisks_disjoint fun x hx y hy hxy => hsep x (Finset.mem_filter.mp hx).1 y (Finset.mem_filter.mp hy).1 hxy
+  let f : C(↥(Metric.closedBall (0:ℂ) 5 \ diskHoles D),GraphSpace (⊤ : SimpleGraph (Fin 5))) :=
+    ⟨fun z => a.map ⟨z.val,a.outside_centers_in_domain htR (htR.trans hR) z.property⟩,
+      a.map.continuous.comp (continuous_subtype_val.subtype_mk _)⟩
+  have hex (d : D) : ∃g : C(d.val.closedBall,GraphSpace (⊤ : SimpleGraph (Fin 5))),
+      ∀z : Metric.sphere d.val.center d.val.radius,∀hzQ : z.val∈Metric.closedBall (0:ℂ) 5,
+        ∀hzD : z.val∉diskHoles D,
+        g ⟨z.val,Metric.sphere_subset_closedBall z.property⟩=f ⟨z.val,hzQ,hzD⟩ := by
+    obtain ⟨d,hd⟩ := d
+    obtain ⟨x,hx,rfl⟩ := mem_centerDisks.mp hd
+    obtain ⟨g,hg⟩ := annulus_forest_extension htR (a.annulusMap hR htR hsep hx) (hforest x hx)
+    exact ⟨g,fun z _ _ => hg z⟩
+  choose g hg using hex
+  obtain ⟨F,hF⟩ := finite_disk_paste (Metric.closedBall (0:ℂ) 5) D hdis f g hg
+  refine ⟨F,fun z hzS hzD => ?_⟩
+  rw [hF z hzD]
+  exact a.agrees _ hzS
+/-- The common-exclusion limit contradiction applies to the constructed maps, so
+all these actual core fillings exist eventually at any fixed separated outer scale. -/
+theorem eventually_fill_cores {c : ℂ → Fin 5} (hc : Measurable c) (hw : WeakColoring 5 c)
+    {α R : ℝ} (hα : 0<α) (hα' : α<1/5) {L : Finset ℂ} {t : ℕ → ℝ}
+    (a : ∀n,HoleApprox c α L (t n)) (ht : Tendsto t atTop (𝓝 0))
+    (hR0 : 0<R) (hR : R<1) (htR : ∀n,2*t n<R)
+    (hsep : ∀x∈L,∀y∈L,x≠y → 2*R<dist x y) :
+    ∀ᶠ n in atTop,∃F : C(Metric.closedBall (0:ℂ) 5,GraphSpace (⊤ : SimpleGraph (Fin 5))),
+      ∀z : Metric.closedBall (0:ℂ) 5,z.val∉diskHoles (a n).disks →
+        z.val∉diskHoles (centerDisks (activeCenters L) R) →
+        (F z).val=thresholdMap c (a n).ε α z.val := by
+  have hrad : Tendsto (fun n => 2*t n) atTop (𝓝 0) := by simpa using ht.const_mul 2
+  have h (x : activeCenters L) : ∀ᶠ n in atTop,
+      (crossingGraph (fun z => ((a n).annulusMap hR (htR n) hsep x.property z).val)
+        (annulusInner x.val (2*t n) R) (annulusOuter x.val (2*t n) R)).IsAcyclic := by
+    apply annulus_maps_eventually_forest hc hw hR0 hrad (ε_tendsto a ht)
+      (fun n => (a n).ε_pos) (annuli_tendsto a ht)
+      (fun _ _ _ => unit_from_hole_mem_annuli) hα hα'
+      (fun n => (a n).annulusMap hR (htR n) hsep x.property)
+    intro n z hz
+    exact (a n).agrees _ hz
+  have hall := eventually_all.mpr h
+  filter_upwards [hall] with n hn
+  exact (a n).fill_cores hR (htR n) hsep (fun x hx => hn ⟨x,hx⟩)
+end HoleApprox
+end EuclideanFiveColor
+end
+
+noncomputable section
+open scoped Classical
+open Set Topology
+namespace EuclideanFiveColor
+lemma diskHoles_union (S T : Finset MarkedDisk) : diskHoles (S∪T)=diskHoles S∪diskHoles T := by
+  ext z
+  simp only [mem_diskHoles,Finset.mem_union,mem_union]
+  aesop
+lemma disk_radius_sum_union_le (S T : Finset MarkedDisk) (h : ∀d∈S∪T,0≤d.radius) :
+    (∑d∈S∪T,d.radius)≤(∑d∈S,d.radius)+(∑d∈T,d.radius) := by
+  have hs : (∑d∈S∪T,d.radius)+(∑d∈S∩T,d.radius)=(∑d∈S,d.radius)+(∑d∈T,d.radius) := Finset.sum_union_inter
+  have h0 : 0≤∑d∈S∩T,d.radius := Finset.sum_nonneg fun d hd =>
+    h d (Finset.mem_union_left T (Finset.mem_inter.mp hd).1)
+  linarith
+end EuclideanFiveColor
+end
+
+noncomputable section
+open scoped Classical
+open Set Filter Topology MeasureTheory ContinuousMap
+namespace EuclideanFiveColor
+/-- A global graph approximation, with actual small exceptional disks.
+All fields will be constructed, not postulated as coloring hypotheses. -/
+structure FullApprox (c : ℂ → Fin 5) (α t : ℝ) where
+  ε : ℝ
+  ε_pos : 0<ε
+  ε_lt : ε<t
+  disks : Finset MarkedDisk
+  positive : ∀d∈disks,0<d.radius
+  disjoint : (disks : Set MarkedDisk).Pairwise (fun a b => Disjoint a.closedBall b.closedBall)
+  sum_lt : (∑d∈disks,d.radius)<t
+  map : C(Metric.closedBall (0:ℂ) 5,GraphSpace (⊤ : SimpleGraph (Fin 5)))
+  agrees : ∀z : Metric.closedBall (0:ℂ) 5,z.val∉diskHoles disks →
+    (map z).val=thresholdMap c ε α z.val
+/-- The whole producer, using actual annular crossing fibers and all angular
+contradictions: global graph approximants exist at arbitrarily small exception scales. -/
+theorem fullApprox_exists_of_finite_core {c : ℂ → Fin 5} (hc : Measurable c)
+    (hw : WeakColoring 5 c) {α : ℝ} (hα : 0<α) (hα' : α<1/5)
+    (L : Finset ℂ) (hL : ∀x∈Metric.closedBall (0:ℂ) 8,¬(transitionGraph c x).IsAcyclic → x∈L)
+    {t : ℝ} (ht : 0<t) : Nonempty (FullApprox c α t) := by
+  obtain ⟨s,hs,hsep⟩ := finite_centers_separation L
+  let A := activeCenters L
+  let R := min (s/4) (min (1/2) (t/(4*((A.card:ℝ)+1))))
+  have hR0 : 0<R := lt_min (by positivity) (lt_min (by norm_num) (div_pos ht (by positivity)))
+  have hRs : R ≤ s/4 := min_le_left _ _
+  have hRhalf : R≤1/2 := (min_le_right _ _).trans (min_le_left _ _)
+  have hRt : R≤t/(4*((A.card:ℝ)+1)) := (min_le_right _ _).trans (min_le_right _ _)
+  have hR1 : R<1 := lt_of_le_of_lt hRhalf (by norm_num)
+  have hRsep : ∀x∈L,∀y∈L,x≠y → 2*R<dist x y := by
+    intro x hx y hy hxy
+    have hh := hsep x hx y hy hxy
+    linarith
+  have hRbudget : (A.card:ℝ)*R<t/4 := by
+    have hh := (le_div_iff₀ (by positivity : 0<4*((A.card:ℝ)+1))).mp hRt
+    nlinarith
+  let u : ℕ → ℝ := fun n => min (R/4) (min (t/4) (1/((n:ℝ)+1)))
+  have hu : ∀n,0<u n := fun n => lt_min (by positivity) (lt_min (by positivity) (by positivity))
+  have huR : ∀n,2*u n<R := by
+    intro n
+    have hh : u n≤R/4 := min_le_left _ _
+    linarith
+  have hut : ∀n,u n<t := by
+    intro n
+    have hh : u n≤t/4 := (min_le_right _ _).trans (min_le_left _ _)
+    linarith
+  have hult : ∀n,u n<1/2 := by
+    intro n
+    have hh := huR n
+    linarith
+  have hulim : Tendsto u atTop (𝓝 0) :=
+    squeeze_zero (fun n => (hu n).le) (fun n => (min_le_right _ _).trans (min_le_right _ _))
+      tendsto_one_div_add_atTop_nhds_zero_nat
+  let a : ∀n,HoleApprox c α L (u n) := fun n => Classical.choice (holeApprox_exists hc hα hα' L hL (hu n) (hult n))
+  obtain ⟨n,F,hF⟩ := ((HoleApprox.eventually_fill_cores hc hw hα hα' a hulim hR0 hR1 huR hRsep).exists)
+  let D := centerDisks A R
+  have hD : ∀d∈D,d.Valid (L : Set ℂ) := by
+    intro d hd
+    obtain ⟨x,hx,rfl⟩ := mem_centerDisks.mp hd
+    exact ⟨hR0,by simp⟩
+  have hU : ∀d∈(a n).disks∪D,d.Valid (L : Set ℂ) := by
+    intro d hd
+    rcases Finset.mem_union.mp hd with hh|hh
+    · exact (a n).valid d hh
+    · exact hD d hh
+  obtain ⟨S,hS,hdis,hcov,hsum,_⟩ := MarkedDisk.separate ((a n).disks∪D) hU (show 0<t/4 by positivity)
+  have htS : (∑d∈S,d.radius)<t := by
+    have hh := disk_radius_sum_union_le (a n).disks D (fun d hd => (hU d hd).1.le)
+    have hDbudget : (∑d∈D,d.radius)<t/4 := by rw [centerDisks_sum]; exact hRbudget
+    have hun : u n≤t/4 := (min_le_right _ _).trans (min_le_left _ _)
+    linarith [(a n).sum_lt]
+  refine ⟨⟨(a n).ε,(a n).ε_pos,(a n).ε_lt.trans (hut n),S,
+    fun d hd => (hS d hd).1,hdis,htS,F,?_⟩⟩
+  intro z hz
+  have hn : z.val∉diskHoles ((a n).disks∪D) := fun hh => hz ((MarkedDisk.holes_subset_of_cover hcov).1 hh)
+  rw [diskHoles_union] at hn
+  exact hF z (fun hh => hn (Or.inl hh)) (fun hh => hn (Or.inr hh))
+/-- The finite core used above is obtained from the proved actual transition core
+local-finiteness theorem; no regularity of the original coloring is added. -/
+theorem fullApprox_exists {c : ℂ → Fin 5} (hc : Measurable c)
+    (hw : WeakColoring 5 c) {α : ℝ} (hα : 0<α) (hα' : α<1/5) {t : ℝ} (ht : 0<t) :
+    Nonempty (FullApprox c α t) := by
+  let hfin := cyclicCore_compact_finite c hc hw (isCompact_closedBall (0:ℂ) 8)
+  exact fullApprox_exists_of_finite_core hc hw hα hα' hfin.toFinset
+    (fun x hx hcyc => hfin.mem_toFinset.mpr ⟨hx,hcyc⟩) ht
+end EuclideanFiveColor
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PlaneColoring/GraphCuts.lean b/lean/OAI/Geometry/PlaneColoring/GraphCuts.lean
new file mode 100644
index 0000000000000000000000000000000000000000..48ab83b5728ed9be81c0b354b297eb7dac43ba31
--- /dev/null
+++ b/lean/OAI/Geometry/PlaneColoring/GraphCuts.lean
@@ -0,0 +1,314 @@
+import OAI.Geometry.PlaneColoring.DiskExtension
+
+namespace OAI
+
+noncomputable section
+open scoped Classical
+open Set Topology ContinuousMap
+universe u
+namespace EuclideanFiveColor
+lemma graphEdgePoint_index {V : Type u} {i j k : V} {t : ℝ}
+    (h : graphEdgePoint i j t k ≠ 0) : k=i ∨ k=j := by
+  by_contra hh
+  simp only [not_or] at hh
+  exact h (by simp [graphEdgePoint_apply,hh.1,hh.2])
+lemma graphRealization_positive_adj {V : Type u} {G : SimpleGraph V} {p : V → ℝ}
+    (hp : p∈graphRealization G) {i j : V} (hne : i≠j) (hi : 0<p i) (hj : 0<p j) : G.Adj i j := by
+  obtain ⟨a,b,hab,t,ht,rfl⟩ := hp
+  have ha := graphEdgePoint_index (ne_of_gt hi)
+  have hb := graphEdgePoint_index (ne_of_gt hj)
+  rcases ha with rfl|rfl <;> rcases hb with rfl|rfl
+  · exact (hne rfl).elim
+  · exact hab.resolve_left hne
+  · exact (hab.resolve_left hne.symm).symm
+  · exact (hne rfl).elim
+lemma graphRealization_pair_le {V : Type u} {G : SimpleGraph V} {p : V → ℝ}
+    (hp : p∈graphRealization G) {i j : V} (hne : i≠j) : p i+p j≤1 := by
+  obtain ⟨a,b,hab,t,ht,rfl⟩ := hp
+  simp only [graphEdgePoint_apply]
+  split_ifs <;> subst_vars <;> first | contradiction | nlinarith [ht.1,ht.2]
+
+def graphCutRealization {V : Type u} (T : SimpleGraph V) : Set (V → ℝ) :=
+  {p | p∈graphRealization (⊤ : SimpleGraph V) ∧ ∀ i j, i≠j → ¬T.Adj i j →
+    p≠graphEdgePoint i j (1/2)}
+abbrev GraphCutSpace {V : Type u} (T : SimpleGraph V) := ↥(graphCutRealization T)
+lemma graphRealization_subset_cut {V : Type u} (T : SimpleGraph V) :
+    graphRealization T⊆graphCutRealization T := by
+  intro p hp
+  refine ⟨graphRealization_mono le_top hp,?_⟩
+  intro i j hne hn he
+  apply hn
+  apply graphRealization_positive_adj hp hne
+  · rw [he]; norm_num [graphEdgePoint_apply,hne,hne.symm]
+  · rw [he]; norm_num [graphEdgePoint_apply,hne,hne.symm]
+lemma graphCut_dominant {V : Type u} {T : SimpleGraph V} {p : V → ℝ}
+    (hp : p∈graphCutRealization T) (hn : p∉graphRealization T) : ∃i,1/2<p i := by
+  obtain ⟨i,j,hij,t,ht,he⟩ := hp.1
+  have hne : i≠j := by
+    intro h; apply hn; exact ⟨i,j,Or.inl h,t,ht,he⟩
+  have hadj : ¬T.Adj i j := by
+    intro h; apply hn; exact ⟨i,j,Or.inr h,t,ht,he⟩
+  have htm : t≠1/2 := by intro h; exact hp.2 i j hne hadj (he.trans (by rw [h]))
+  rcases lt_or_gt_of_ne htm with hl|hr
+  · refine ⟨i,?_⟩; rw [he]; simp [graphEdgePoint_apply,hne]; linarith
+  · refine ⟨j,?_⟩; rw [he]; simpa [graphEdgePoint_apply,hne.symm] using hr
+lemma graphCut_dominant_unique {V : Type u} {T : SimpleGraph V} {p : V → ℝ}
+    (hp : p∈graphCutRealization T) {i j : V} (hi : 1/2<p i) (hj : 1/2<p j) : i=j := by
+  by_contra hne
+  have hh := graphRealization_pair_le hp.1 hne
+  linarith
+
+def graphCutRetract {V : Type u} (T : SimpleGraph V) (p : GraphCutSpace T) : V → ℝ :=
+  if h : p.val∈graphRealization T then p.val else graphVertex (graphCut_dominant p.property h).choose
+lemma graphCutRetract_tree {V : Type u} {T : SimpleGraph V} (p : GraphCutSpace T)
+    (hp : p.val∈graphRealization T) : graphCutRetract T p=p.val := by
+  simp [graphCutRetract,hp]
+lemma graphCutRetract_dominant {V : Type u} {T : SimpleGraph V} (p : GraphCutSpace T)
+    (hp : p.val∉graphRealization T) {i : V} (hi : 1/2<p.val i) : graphCutRetract T p=graphVertex i := by
+  rw [graphCutRetract,dite_eq_right hp]
+  congr 1
+  exact graphCut_dominant_unique p.property (graphCut_dominant p.property hp).choose_spec hi
+lemma graphCutRetract_mem {V : Type u} (T : SimpleGraph V) (p : GraphCutSpace T) :
+    graphCutRetract T p∈graphRealization T := by
+  unfold graphCutRetract
+  split
+  · assumption
+  · exact graphVertex_mem _ _
+/-- The closed half of a deleted edge, viewed inside the cut graph. -/
+def graphCutStub {V : Type u} (T : SimpleGraph V) (i j : V) : Set (GraphCutSpace T) :=
+  {p | i≠j ∧ ¬T.Adj i j ∧ p.val∈graphEdgePoint i j '' Icc (0:ℝ) (1/2)}
+lemma graphCutStub_retract {V : Type u} {T : SimpleGraph V} {i j : V}
+    {p : GraphCutSpace T} (hp : p∈graphCutStub T i j) : graphCutRetract T p=graphVertex i := by
+  obtain ⟨hne,hadj,t,ht,he⟩ := hp
+  have he' := he.symm
+  have htm : t<1/2 := lt_of_le_of_ne ht.2 (by
+    intro h; exact p.property.2 i j hne hadj (he'.trans (by rw [h])))
+  by_cases hT : p.val∈graphRealization T
+  · rw [graphCutRetract_tree p hT]
+    have ht0 : t=0 := by
+      by_contra hh
+      have hp1 : 0<p.val i := by rw [he']; simp [graphEdgePoint_apply,hne]; linarith
+      have hp2 : 0<p.val j := by rw [he']; simp [graphEdgePoint_apply,hne.symm]; exact lt_of_le_of_ne ht.1 (Ne.symm hh)
+      exact hadj (graphRealization_positive_adj hT hne hp1 hp2)
+    rw [he',ht0,graphEdgePoint_zero]
+  · apply graphCutRetract_dominant p hT
+    rw [he']; simp [graphEdgePoint_apply,hne]; linarith
+lemma graphCut_cover {V : Type u} {T : SimpleGraph V} (p : GraphCutSpace T) :
+    p.val∈graphRealization T ∨ ∃i j,p∈graphCutStub T i j := by
+  by_cases h : p.val∈graphRealization T
+  · exact Or.inl h
+  right
+  obtain ⟨i,j,hij,t,ht,he⟩ := p.property.1
+  have hne : i≠j := by intro hh; exact h ⟨i,j,Or.inl hh,t,ht,he⟩
+  have hadj : ¬T.Adj i j := by intro hh; exact h ⟨i,j,Or.inr hh,t,ht,he⟩
+  by_cases ht' : t≤1/2
+  · exact ⟨i,j,hne,hadj,t,⟨ht.1,ht'⟩,he.symm⟩
+  · refine ⟨j,i,hne.symm,fun hh => hadj hh.symm,1-t,⟨by linarith [ht.2],by linarith⟩,?_⟩
+    exact (graphEdgePoint_swap i j t).symm.trans he.symm
+end EuclideanFiveColor
+end
+
+noncomputable section
+open scoped Classical
+open Set Topology ContinuousMap
+universe u
+namespace EuclideanFiveColor
+lemma graphCutStub_closed {V : Type u} {T : SimpleGraph V} (i j : V) :
+    IsClosed (graphCutStub T i j) := by
+  change IsClosed ({p : GraphCutSpace T | i≠j} ∩
+    ({p : GraphCutSpace T | ¬T.Adj i j} ∩ Subtype.val ⁻¹' (graphEdgePoint i j '' Icc (0:ℝ) (1/2))))
+  exact isClosed_const.inter (isClosed_const.inter ((isCompact_Icc.image (continuous_graphEdgePoint i j)).isClosed.preimage
+    continuous_subtype_val))
+lemma continuous_graphCutRetract {V : Type u} [Finite V] (T : SimpleGraph V) :
+    Continuous (graphCutRetract T) := by
+  let S : Option (V × V) → Set (GraphCutSpace T) := fun a => match a with
+    | none => Subtype.val ⁻¹' graphRealization T
+    | some (i,j) => graphCutStub T i j
+  have hcov : ⋃a,S a=univ := by
+    apply iUnion_eq_univ_iff.mpr
+    intro p
+    rcases graphCut_cover p with hp|⟨i,j,hp⟩
+    · exact ⟨none,hp⟩
+    · exact ⟨some (i,j),hp⟩
+  apply (locallyFinite_of_finite S).continuous hcov
+  · intro a
+    cases a with
+    | none => exact (graphRealization_closed T).preimage continuous_subtype_val
+    | some a => exact graphCutStub_closed a.1 a.2
+  · intro a
+    cases a with
+    | none => exact continuous_subtype_val.continuousOn.congr (fun p hp => graphCutRetract_tree p hp)
+    | some a => exact continuous_const.continuousOn.congr (fun p hp => graphCutStub_retract hp)
+
+lemma graphMidpoint_coord_le {V : Type u} {i j : V} (hne : i≠j) (k : V) :
+    graphEdgePoint i j (1/2) k ≤ 1/2 := by
+  simp only [graphEdgePoint_apply]
+  split_ifs <;> subst_vars <;> first | contradiction | norm_num
+lemma graph_dominant_mem_cut {V : Type u} {T : SimpleGraph V} {p : V → ℝ}
+    (hp : p∈graphRealization (⊤ : SimpleGraph V)) {i : V} (hi : 1/2<p i) :
+    p∈graphCutRealization T := by
+  refine ⟨hp,?_⟩
+  intro j k hjk _ he
+  have hh := graphMidpoint_coord_le hjk i
+  rw [←he] at hh
+  linarith
+lemma graphEdgePoint_lerp_vertex {V : Type u} (i j : V) (t s : ℝ) :
+    (1-s) • graphEdgePoint i j t+s • graphVertex i=graphEdgePoint i j ((1-s)*t) := by
+  ext k
+  simp only [graphEdgePoint,Pi.add_apply,Pi.smul_apply,smul_eq_mul]
+  ring
+lemma graphCut_lerp_mem {V : Type u} (T : SimpleGraph V) (p : GraphCutSpace T)
+    {s : ℝ} (hs : s∈Icc (0:ℝ) 1) :
+    (1-s) • p.val+s • graphCutRetract T p∈graphCutRealization T := by
+  rcases graphCut_cover p with hp|⟨i,j,hp⟩
+  · rw [graphCutRetract_tree p hp,←add_smul]
+    simp
+  · have hr := graphCutStub_retract hp
+    obtain ⟨hne,hadj,t,ht,he⟩ := hp
+    have htm : t<1/2 := lt_of_le_of_ne ht.2 (by
+      intro h; exact p.property.2 i j hne hadj (he.symm.trans (by rw [h])))
+    rw [hr,←he,graphEdgePoint_lerp_vertex]
+    have hu0 : 0≤(1-s)*t := mul_nonneg (by linarith [hs.2]) ht.1
+    have hu1 : (1-s)*t≤t := mul_le_of_le_one_left ht.1 (by linarith [hs.1])
+    have hu : (1-s)*t<1/2 := lt_of_le_of_lt hu1 htm
+    apply graph_dominant_mem_cut (graphEdgePoint_mem (Or.inr (show (⊤ : SimpleGraph V).Adj i j from hne)) ⟨hu0,by linarith⟩) (i := i)
+    simp [graphEdgePoint_apply,hne]
+    linarith
+
+def graphCutRetraction {V : Type u} [Finite V] (T : SimpleGraph V) :
+    C(GraphCutSpace T,GraphSpace T) where
+  toFun p := ⟨graphCutRetract T p,graphCutRetract_mem T p⟩
+  continuous_toFun := (continuous_graphCutRetract T).subtype_mk _
+def graphCutInclusion {V : Type u} (T : SimpleGraph V) : C(GraphSpace T,GraphCutSpace T) where
+  toFun p := ⟨p.val,graphRealization_subset_cut T p.property⟩
+  continuous_toFun := continuous_subtype_val.subtype_mk _
+def graphCutHomotopy {V : Type u} [Finite V] (T : SimpleGraph V) :
+    (ContinuousMap.id (GraphCutSpace T)).Homotopy
+      ((graphCutInclusion T).comp (graphCutRetraction T)) where
+  toFun sp := ⟨(1-sp.1.val) • sp.2.val+sp.1.val • graphCutRetract T sp.2,
+    graphCut_lerp_mem T sp.2 sp.1.property⟩
+  continuous_toFun := (((continuous_const.sub (continuous_subtype_val.comp continuous_fst)).smul
+    (continuous_subtype_val.comp continuous_snd)).add
+    ((continuous_subtype_val.comp continuous_fst).smul
+      ((continuous_graphCutRetract T).comp continuous_snd))).subtype_mk _
+  map_zero_left p := by apply Subtype.ext; simp
+  map_one_left p := by apply Subtype.ext; simp [graphCutInclusion,graphCutRetraction]
+/-- Removing the non-tree edge midpoints leaves a contractible graph (tree with half-open stubs). -/
+theorem graphCut_contractible {V : Type u} [Fintype V] {T : SimpleGraph V} (hT : T.IsTree) :
+    ContractibleSpace (GraphCutSpace T) := by
+  let := graphTree_contractible hT
+  have hn := ((id_nullhomotopic (GraphSpace T)).comp_left
+    (graphCutRetraction T)).comp_right (graphCutInclusion T)
+  obtain ⟨a,ha⟩ := hn
+  apply (contractible_iff_id_nullhomotopic _).mpr
+  refine ⟨a,ContinuousMap.Homotopic.trans ⟨graphCutHomotopy T⟩ ?_⟩
+  simpa only [ContinuousMap.id_comp] using ha
+end EuclideanFiveColor
+end
+
+noncomputable section
+open scoped Classical
+open Set Topology ContinuousMap
+universe u v
+namespace EuclideanFiveColor
+/-- A compact connected fiber joins both designated boundary sets. -/
+def FiberCrosses {X : Type u} {Y : Type v} [TopologicalSpace X]
+    (f : X → Y) (A B : Set X) (b : Y) : Prop :=
+  ∃K : Set X, IsCompact K ∧ IsConnected K ∧ K⊆f ⁻¹' {b} ∧
+    (K∩A).Nonempty ∧ (K∩B).Nonempty
+lemma fiberCrosses_symm {X : Type u} {Y : Type v} [TopologicalSpace X]
+    {f : X → Y} {A B : Set X} {b : Y} (h : FiberCrosses f A B b) : FiberCrosses f B A b := by
+  obtain ⟨K,hK,hC,hf,hA,hB⟩ := h
+  exact ⟨K,hK,hC,hf,hB,hA⟩
+lemma compact_fiber_partition {X : Type u} {Y : Type v} [TopologicalSpace X]
+    [TopologicalSpace Y] [CompactSpace X] [T2Space X] [T1Space Y]
+    (f : C(X,Y)) {A B : Set X} (hA : IsClosed A) (hB : IsClosed B) (b : Y)
+    (hn : ¬FiberCrosses f A B b) :
+    ∃K₀ K₁ : Set X, IsCompact K₀ ∧ IsCompact K₁ ∧ Disjoint K₀ K₁ ∧
+      K₀∪K₁=f ⁻¹' {b} ∧ Disjoint K₀ B ∧ Disjoint K₁ A := by
+  let F : Set X := f ⁻¹' {b}
+  have hF : IsClosed F := isClosed_singleton.preimage f.continuous
+  let : CompactSpace F := isCompact_iff_compactSpace.mp hF.isCompact
+  let I : Set F := Subtype.val ⁻¹' A
+  let O : Set F := Subtype.val ⁻¹' B
+  have hI : IsCompact I := (hA.preimage continuous_subtype_val).isCompact
+  have hO : IsCompact O := (hB.preimage continuous_subtype_val).isCompact
+  have hno : ∀x∈I,Disjoint (connectedComponent x) O := by
+    intro x hx
+    apply disjoint_left.mpr
+    intro y hy hOy
+    apply hn
+    refine ⟨Subtype.val '' connectedComponent x,
+      isClosed_connectedComponent.isCompact.image continuous_subtype_val,
+      isConnected_connectedComponent.image _ continuous_subtype_val.continuousOn,?_,?_,?_⟩
+    · rintro z ⟨w,hw,rfl⟩; exact w.property
+    · exact ⟨x.val,⟨⟨x,mem_connectedComponent,rfl⟩,hx⟩⟩
+    · exact ⟨y.val,⟨⟨y,hy,rfl⟩,hOy⟩⟩
+  obtain ⟨Z₀,Z₁,hZ₀,hZ₁,hd,hcov,hIZ,hOZ⟩ := compact_component_partition hI hO hno
+  refine ⟨Subtype.val '' Z₀,Subtype.val '' Z₁,hZ₀.image continuous_subtype_val,
+    hZ₁.image continuous_subtype_val,?_,?_,?_,?_⟩
+  · apply disjoint_left.mpr
+    rintro x ⟨y,hy,rfl⟩ ⟨z,hz,he⟩
+    have hh : z=y := Subtype.ext he
+    subst z
+    exact disjoint_left.mp hd hy hz
+  · rw [←image_union,hcov]
+    ext x
+    constructor
+    · rintro ⟨y,_,rfl⟩; exact y.property
+    · intro hx; exact ⟨⟨x,hx⟩,mem_univ _,rfl⟩
+  · apply disjoint_left.mpr
+    rintro z ⟨x,hx,rfl⟩ hz
+    exact disjoint_left.mp hd hx (hOZ hz)
+  · apply disjoint_left.mpr
+    rintro z ⟨x,hx,rfl⟩ hz
+    exact disjoint_left.mp hd (hIZ hz) hx
+
+/-- All finitely many non-crossing bad fibers can simultaneously be separated,
+ with the prescribed inner/outer closed sets on opposite sides. -/
+theorem finite_fiber_separation {X : Type u} {Y : Type v} [TopologicalSpace X]
+    [TopologicalSpace Y] [CompactSpace X] [T2Space X] [T1Space Y]
+    (f : C(X,Y)) {A B : Set X} (hA : IsClosed A) (hB : IsClosed B) (hd : Disjoint A B)
+    (S : Finset Y) (hn : ∀b∈S,¬FiberCrosses f A B b) :
+    ∃C₀ C₁ : Set X, IsCompact C₀ ∧ IsCompact C₁ ∧ Disjoint C₀ C₁ ∧
+      A⊆C₀ ∧ B⊆C₁ ∧ f ⁻¹' (↑S : Set Y)⊆C₀∪C₁ := by
+  choose K₀ K₁ hK using (fun b : S => compact_fiber_partition f hA hB b.val (hn b.val b.property))
+  let Z₀ : Set X := ⋃b : S,K₀ b
+  let Z₁ : Set X := ⋃b : S,K₁ b
+  have hz₀ : IsCompact Z₀ := isCompact_iUnion fun b => (hK b).1
+  have hz₁ : IsCompact Z₁ := isCompact_iUnion fun b => (hK b).2.1
+  have hsub₀ (b : S) : K₀ b⊆f ⁻¹' {b.val} := by
+    rw [←(hK b).2.2.2.1]; exact subset_union_left
+  have hsub₁ (b : S) : K₁ b⊆f ⁻¹' {b.val} := by
+    rw [←(hK b).2.2.2.1]; exact subset_union_right
+  have h01 : Disjoint Z₀ Z₁ := by
+    apply disjoint_left.mpr
+    intro x hx hy
+    obtain ⟨b,hb⟩ := mem_iUnion.mp hx
+    obtain ⟨c,hc⟩ := mem_iUnion.mp hy
+    have he : b=c := Subtype.ext ((hsub₀ b hb).symm.trans (hsub₁ c hc))
+    subst c
+    exact disjoint_left.mp (hK b).2.2.1 hb hc
+  have h0B : Disjoint Z₀ B := by
+    apply disjoint_left.mpr
+    intro x hx hB
+    obtain ⟨b,hb⟩ := mem_iUnion.mp hx
+    exact disjoint_left.mp (hK b).2.2.2.2.1 hb hB
+  have h1A : Disjoint Z₁ A := by
+    apply disjoint_left.mpr
+    intro x hx hA
+    obtain ⟨b,hb⟩ := mem_iUnion.mp hx
+    exact disjoint_left.mp (hK b).2.2.2.2.2 hb hA
+  refine ⟨A∪Z₀,B∪Z₁,hA.isCompact.union hz₀,hB.isCompact.union hz₁,?_,subset_union_left,subset_union_left,?_⟩
+  · exact disjoint_union_left.mpr ⟨disjoint_union_right.mpr ⟨hd,h1A.symm⟩,
+      disjoint_union_right.mpr ⟨h0B,h01⟩⟩
+  · intro x hx
+    let b : S := ⟨f x,hx⟩
+    have hh : x∈K₀ b∪K₁ b := by rw [(hK b).2.2.2.1]; rfl
+    rcases hh with hh|hh
+    · exact Or.inl (Or.inr (mem_iUnion.mpr ⟨b,hh⟩))
+    · exact Or.inr (Or.inr (mem_iUnion.mpr ⟨b,hh⟩))
+end EuclideanFiveColor
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PlaneColoring/GraphPatches.lean b/lean/OAI/Geometry/PlaneColoring/GraphPatches.lean
new file mode 100644
index 0000000000000000000000000000000000000000..b3c166e36d46395e5c459e30753614d93b5acd9e
--- /dev/null
+++ b/lean/OAI/Geometry/PlaneColoring/GraphPatches.lean
@@ -0,0 +1,314 @@
+import OAI.Geometry.PlaneColoring.GraphCuts
+import OAI.Geometry.PlaneColoring.HoleApproximation
+
+namespace OAI
+
+noncomputable section
+open scoped Classical
+open Set Topology ContinuousMap
+universe u v
+namespace EuclideanFiveColor
+/-- Patch a map across separated bad fibers. Only the intervening values need lie in
+ a contractible subspace; the map is kept on one closed set and made constant on the other. -/
+theorem contractible_patch {X : Type u} {Y : Type v} [TopologicalSpace X]
+    [TopologicalSpace Y] [NormalSpace X] (D : Set Y) [ContractibleSpace D]
+    (f : C(X,Y)) {A B : Set X} (hA : IsClosed A) (hB : IsClosed B) (hd : Disjoint A B)
+    (hg : ∀x,x∉A → x∉B → f x∈D) :
+    ∃a : D, ∃g : C(X,Y), EqOn g (fun _ => a.val) A ∧ EqOn g f B ∧
+      ∀x,g x=f x ∨ g x∈D := by
+  obtain ⟨t,htB,htA,ht⟩ := exists_continuous_zero_one_of_isClosed hB hA hd.symm
+  obtain ⟨a,⟨H⟩⟩ := id_nullhomotopic D
+  let S : Fin 3 → Set X := ![{x | t x≤1/3},{x | 1/3≤t x ∧ t x≤2/3},{x | 2/3≤t x}]
+  have hS : ∀i,IsClosed (S i) := by
+    intro i
+    fin_cases i
+    · exact isClosed_le t.continuous continuous_const
+    · exact (isClosed_le continuous_const t.continuous).inter
+        (isClosed_le t.continuous continuous_const)
+    · exact isClosed_le continuous_const t.continuous
+  have hc : ⋃i,S i=univ := by
+    apply iUnion_eq_univ_iff.mpr
+    intro x
+    by_cases h0 : t x≤1/3
+    · exact ⟨0,h0⟩
+    by_cases h1 : t x≤2/3
+    · exact ⟨1,⟨(not_le.mp h0).le,h1⟩⟩
+    · exact ⟨2,(not_le.mp h1).le⟩
+  have hgood (x : S 1) : f x.val∈D := by
+    apply hg
+    · intro hx
+      have he : t x.val=1 := htA hx
+      have hh : t x.val≤2/3 := x.property.2
+      linarith
+    · intro hx
+      have he : t x.val=0 := htB hx
+      have hh : 1/3≤t x.val := x.property.1
+      linarith
+  have hτmem (x : S 1) : 3*t x.val-1∈unitInterval := by
+    constructor
+    · linarith [x.property.1]
+    · linarith [x.property.2]
+  let τ : C(S 1,unitInterval) := ⟨fun x => ⟨3*t x.val-1,hτmem x⟩,
+    ((continuous_const.mul (t.continuous.comp continuous_subtype_val)).sub continuous_const).subtype_mk hτmem⟩
+  let φ₁ : C(S 1,Y) := ⟨fun x => (H (τ x,⟨f x.val,hgood x⟩)).val,
+    continuous_subtype_val.comp (H.continuous.comp (τ.continuous.prodMk
+      ((f.continuous.comp continuous_subtype_val).subtype_mk hgood)))⟩
+  let φ : ∀i,C(S i,Y) := fun i => Fin.cases (f.restrict (S 0))
+    (fun j => Fin.cases φ₁ (fun j => ContinuousMap.const (S j.succ.succ) a.val) j) i
+  have h01 (x : X) (h0 : x∈S 0) (h1 : x∈S 1) : φ 0 ⟨x,h0⟩=φ 1 ⟨x,h1⟩ := by
+    have he : t x=1/3 := le_antisymm h0 h1.1
+    have hτ : τ ⟨x,h1⟩=0 := by apply Subtype.ext; change 3*t x-1=0; rw [he]; norm_num
+    change f x=(H (τ ⟨x,h1⟩,⟨f x,hgood ⟨x,h1⟩⟩)).val
+    rw [hτ]
+    exact congrArg Subtype.val (H.map_zero_left ⟨f x,hgood ⟨x,h1⟩⟩) |>.symm
+  have h12 (x : X) (h1 : x∈S 1) (h2 : x∈S 2) : φ 1 ⟨x,h1⟩=φ 2 ⟨x,h2⟩ := by
+    have he : t x=2/3 := le_antisymm h1.2 h2
+    have hτ : τ ⟨x,h1⟩=1 := by apply Subtype.ext; change 3*t x-1=1; rw [he]; norm_num
+    change (H (τ ⟨x,h1⟩,⟨f x,hgood ⟨x,h1⟩⟩)).val=a.val
+    rw [hτ]
+    exact congrArg Subtype.val (H.map_one_left ⟨f x,hgood ⟨x,h1⟩⟩)
+  have hφ : ∀i j x (hi : x∈S i) (hj : x∈S j),φ i ⟨x,hi⟩=φ j ⟨x,hj⟩ := by
+    intro i j x hi hj
+    fin_cases i <;> fin_cases j
+    · rfl
+    · exact h01 x hi hj
+    · have h0 : t x≤1/3 := hi; have h2 : 2/3≤t x := hj; linarith
+    · exact (h01 x hj hi).symm
+    · rfl
+    · exact h12 x hi hj
+    · have h0 : t x≤1/3 := hj; have h2 : 2/3≤t x := hi; linarith
+    · exact (h12 x hj hi).symm
+    · rfl
+  refine ⟨a,closedCoverMap S hS hc φ hφ,?_,?_,?_⟩
+  · intro x hx
+    have hx2 : x∈S 2 := by change 2/3≤t x; rw [htA hx]; norm_num
+    exact closedCoverMap_apply S hS hc φ hφ ⟨x,hx2⟩
+  · intro x hx
+    have hx0 : x∈S 0 := by change t x≤1/3; rw [htB hx]; norm_num
+    exact closedCoverMap_apply S hS hc φ hφ ⟨x,hx0⟩
+  · intro x
+    obtain ⟨i,hi⟩ := iUnion_eq_univ_iff.mp hc x
+    rw [closedCoverMap_apply S hS hc φ hφ ⟨x,hi⟩]
+    fin_cases i
+    · exact Or.inl rfl
+    · exact Or.inr (H (τ ⟨x,hi⟩,⟨f x,hgood ⟨x,hi⟩⟩)).property
+    · exact Or.inr a.property
+end EuclideanFiveColor
+end
+
+noncomputable section
+open scoped Classical
+open Set Topology ContinuousMap
+universe u v
+namespace EuclideanFiveColor
+/-- Edges whose midpoint fibers contain a compact continuum joining the two boundaries. -/
+def crossingGraph {X : Type u} {V : Type v} [TopologicalSpace X]
+    (f : X → V → ℝ) (A B : Set X) : SimpleGraph V where
+  Adj i j := i≠j ∧ FiberCrosses f A B (graphEdgePoint i j (1/2))
+  symm := ⟨by
+    intro i j h
+    refine ⟨h.1.symm,?_⟩
+    have he : graphEdgePoint j i (1/2)=graphEdgePoint i j (1/2) := by
+      convert graphEdgePoint_swap j i (1/2) using 1; norm_num
+    rw [he]; exact h.2⟩
+  loopless := ⟨by intro i h; exact h.1 rfl⟩
+/-- If the crossing edges form a forest, the boundary map can be patched to a constant
+ on the inner closed set, unchanged on the outer closed set. This replaces the graph-word
+ and winding detour for the existential interface-cycle producer. -/
+theorem graph_forest_patch {X : Type u} {V : Type v} [TopologicalSpace X]
+    [CompactSpace X] [T2Space X] [Fintype V] [Nonempty V]
+    (f : C(X,GraphSpace (⊤ : SimpleGraph V))) {A B : Set X}
+    (hA : IsClosed A) (hB : IsClosed B) (hd : Disjoint A B)
+    (hforest : (crossingGraph (fun x => (f x).val) A B).IsAcyclic) :
+    ∃a : GraphSpace (⊤ : SimpleGraph V), ∃g : C(X,GraphSpace (⊤ : SimpleGraph V)),
+      EqOn g (fun _ => a) A ∧ EqOn g f B := by
+  let G := crossingGraph (fun x => (f x).val) A B
+  obtain ⟨T,hGT,_,hT⟩ := SimpleGraph.connected_top.exists_isTree_le_of_le_of_isAcyclic
+    (show G≤⊤ from le_top) hforest
+  let f₀ : C(X,V → ℝ) := ⟨fun x => (f x).val,continuous_subtype_val.comp f.continuous⟩
+  let S : Finset (V → ℝ) := ((Finset.univ : Finset (V×V)).filter
+    (fun a => a.1≠a.2 ∧ ¬T.Adj a.1 a.2)).image (fun a => graphEdgePoint a.1 a.2 (1/2))
+  have hn : ∀b∈S,¬FiberCrosses f₀ A B b := by
+    intro b hb
+    obtain ⟨⟨i,j⟩,hij,rfl⟩ := Finset.mem_image.mp hb
+    simp only [Finset.mem_filter,Finset.mem_univ,true_and] at hij
+    intro hh
+    exact hij.2 (hGT ⟨hij.1,hh⟩)
+  obtain ⟨C₀,C₁,hC₀,hC₁,hCd,hAC,hBC,hSC⟩ := finite_fiber_separation f₀ hA hB hd S hn
+  let : ContractibleSpace (GraphCutSpace T) := graphCut_contractible hT
+  have hg : ∀x,x∉C₀ → x∉C₁ → f₀ x∈graphCutRealization T := by
+    intro x hx₀ hx₁
+    refine ⟨(f x).property,?_⟩
+    intro i j hij hTij he
+    have hm : f₀ x∈S := Finset.mem_image.mpr ⟨(i,j),by simp [hij,hTij],he.symm⟩
+    rcases hSC hm with hh|hh
+    · exact hx₀ hh
+    · exact hx₁ hh
+  obtain ⟨a,g,hgA,hgB,hgR⟩ := contractible_patch (graphCutRealization T) f₀ hC₀.isClosed hC₁.isClosed hCd hg
+  have hgr (x : X) : g x∈graphRealization (⊤ : SimpleGraph V) := by
+    rcases hgR x with hh|hh
+    · rw [hh]; exact (f x).property
+    · exact hh.1
+  let g' : C(X,GraphSpace (⊤ : SimpleGraph V)) := ⟨fun x => ⟨g x,hgr x⟩,g.continuous.subtype_mk hgr⟩
+  refine ⟨⟨a.val,a.property.1⟩,g',?_,?_⟩
+  · intro x hx
+    apply Subtype.ext
+    exact hgA (hAC hx)
+  · intro x hx
+    apply Subtype.ext
+    exact hgB (hBC hx)
+end EuclideanFiveColor
+end
+
+noncomputable section
+open scoped Classical
+open Set Topology ContinuousMap
+universe u
+namespace EuclideanFiveColor
+def planeAnnulus (x : ℂ) (r R : ℝ) : Set ℂ := Metric.closedBall x R \ Metric.ball x r
+lemma planeAnnulus_compact (x : ℂ) (r R : ℝ) : IsCompact (planeAnnulus x r R) :=
+  (isCompact_closedBall x R).diff Metric.isOpen_ball
+lemma mem_planeAnnulus {x z : ℂ} {r R : ℝ} : z∈planeAnnulus x r R ↔ dist z x≤R ∧ r≤dist z x := by
+  simp [planeAnnulus,not_lt]
+def annulusInner (x : ℂ) (r R : ℝ) : Set (planeAnnulus x r R) := {z | dist z.val x=r}
+def annulusOuter (x : ℂ) (r R : ℝ) : Set (planeAnnulus x r R) := {z | dist z.val x=R}
+lemma sphere_mem_annulus {x z : ℂ} {r R : ℝ} (hrR : r≤R) (hz : z∈Metric.sphere x R) :
+    z∈planeAnnulus x r R := by
+  apply mem_planeAnnulus.mpr
+  have hz' : dist z x=R := hz
+  rw [hz']
+  exact ⟨le_rfl,hrR⟩
+/-- A forest crossing graph yields an actual extension of the outer boundary over
+its disk. This uses compact fiber separation and tree contraction, not graph words. -/
+theorem annulus_forest_extension {V : Type u} [Fintype V] [Nonempty V]
+    {x : ℂ} {r R : ℝ} (hrR : r<R)
+    (f : C(planeAnnulus x r R,GraphSpace (⊤ : SimpleGraph V)))
+    (hforest : (crossingGraph (fun z => (f z).val) (annulusInner x r R) (annulusOuter x r R)).IsAcyclic) :
+    ∃g : C(Metric.closedBall x R,GraphSpace (⊤ : SimpleGraph V)),
+      ∀z : Metric.sphere x R,g ⟨z.val,Metric.sphere_subset_closedBall z.property⟩=
+        f ⟨z.val,sphere_mem_annulus hrR.le z.property⟩ := by
+  let : CompactSpace (planeAnnulus x r R) := isCompact_iff_compactSpace.mp (planeAnnulus_compact x r R)
+  have hA : IsClosed (annulusInner x r R) := isClosed_eq (continuous_subtype_val.dist continuous_const) continuous_const
+  have hB : IsClosed (annulusOuter x r R) := isClosed_eq (continuous_subtype_val.dist continuous_const) continuous_const
+  have hd : Disjoint (annulusInner x r R) (annulusOuter x r R) := by
+    apply disjoint_left.mpr
+    intro z hz hz'
+    exact hrR.ne (hz.symm.trans hz')
+  obtain ⟨a,g,hgA,hgB⟩ := graph_forest_patch f hA hB hd hforest
+  let X := Metric.closedBall x R
+  let Z : Bool → Set X := fun b => if b then {z | r≤dist z.val x} else {z | dist z.val x≤r}
+  have hZ : ∀b,IsClosed (Z b) := by
+    intro b
+    cases b
+    · exact isClosed_le (continuous_subtype_val.dist continuous_const) continuous_const
+    · exact isClosed_le continuous_const (continuous_subtype_val.dist continuous_const)
+  have hcov : ⋃b,Z b=univ := by
+    apply iUnion_eq_univ_iff.mpr
+    intro z
+    rcases le_total (dist z.val x) r with hh|hh
+    · exact ⟨false,hh⟩
+    · exact ⟨true,hh⟩
+  let g₁ : C(Z true,GraphSpace (⊤ : SimpleGraph V)) :=
+    ⟨fun z => g ⟨z.val.val,mem_planeAnnulus.mpr ⟨z.val.property,z.property⟩⟩,
+      g.continuous.comp ((continuous_subtype_val.comp continuous_subtype_val).subtype_mk _)⟩
+  let φ : ∀b,C(Z b,GraphSpace (⊤ : SimpleGraph V)) := fun b => Bool.casesOn b (ContinuousMap.const _ a) g₁
+  have heq (z : X) (h0 : z∈Z false) (h1 : z∈Z true) : φ false ⟨z,h0⟩=φ true ⟨z,h1⟩ := by
+    exact (hgA (show (⟨z.val,mem_planeAnnulus.mpr ⟨z.property,h1⟩⟩ : planeAnnulus x r R)∈annulusInner x r R from
+      le_antisymm h0 h1)).symm
+  have hφ : ∀i j z (hi : z∈Z i) (hj : z∈Z j),φ i ⟨z,hi⟩=φ j ⟨z,hj⟩ := by
+    intro i j z hi hj
+    cases i <;> cases j
+    · rfl
+    · exact heq z hi hj
+    · exact (heq z hj hi).symm
+    · rfl
+  refine ⟨closedCoverMap Z hZ hcov φ hφ,fun z => ?_⟩
+  have hz : (⟨z.val,Metric.sphere_subset_closedBall z.property⟩ : X)∈Z true := by
+    change r≤dist z.val x
+    rw [show dist z.val x=R from z.property]
+    exact hrR.le
+  rw [closedCoverMap_apply Z hZ hcov φ hφ (i:=true) ⟨⟨z.val,Metric.sphere_subset_closedBall z.property⟩,hz⟩]
+  exact hgB z.property
+end EuclideanFiveColor
+end
+
+noncomputable section
+open scoped Classical
+open Set Topology
+namespace EuclideanFiveColor
+lemma finite_centers_separation (L : Finset ℂ) : ∃s>0,∀x∈L,∀y∈L,x≠y → s<dist x y := by
+  let P := {p : L×L // p.1≠p.2}
+  obtain ⟨s,hs,hsp⟩ := finite_pos_lt (fun p : P => dist p.val.1.val p.val.2.val)
+    (fun p => dist_pos.mpr (fun he => p.property (Subtype.ext he)))
+  refine ⟨s,hs,fun x hx y hy hxy => ?_⟩
+  exact hsp ⟨(⟨x,hx⟩,⟨y,hy⟩),fun h => hxy (congrArg Subtype.val h)⟩
+def centerDisks (L : Finset ℂ) (R : ℝ) : Finset MarkedDisk := L.image fun x => ⟨x,R,false⟩
+lemma mem_centerDisks {L : Finset ℂ} {R : ℝ} {d : MarkedDisk} :
+    d∈centerDisks L R ↔ ∃x∈L,d=⟨x,R,false⟩ := by simp [centerDisks,eq_comm]
+lemma centerDisks_sum (L : Finset ℂ) (R : ℝ) : (∑d∈centerDisks L R,d.radius)=(L.card:ℝ)*R := by
+  rw [centerDisks,Finset.sum_image (fun x _ y _ h => congrArg MarkedDisk.center h)]
+  simp
+lemma centerDisks_disjoint {L : Finset ℂ} {R : ℝ}
+    (hsep : ∀x∈L,∀y∈L,x≠y → 2*R<dist x y) :
+    (centerDisks L R : Set MarkedDisk).Pairwise (fun a b => Disjoint a.closedBall b.closedBall) := by
+  intro a ha b hb hab
+  obtain ⟨x,hx,rfl⟩ := mem_centerDisks.mp ha
+  obtain ⟨y,hy,rfl⟩ := mem_centerDisks.mp hb
+  have hxy : x≠y := by intro he; subst y; exact hab rfl
+  apply disjoint_left.mpr
+  intro z hz hz'
+  have h1 : dist x z≤R := by simpa only [MarkedDisk.closedBall,Metric.mem_closedBall,dist_comm] using hz
+  have h2 : dist z y≤R := hz'
+  have hh := dist_triangle x z y
+  have hh' := hsep x hx y hy hxy
+  linarith
+lemma mem_center_holes {L : Finset ℂ} {R : ℝ} {z : ℂ} :
+    z∈diskHoles (centerDisks L R) ↔ ∃x∈L,dist z x<R := by
+  rw [mem_diskHoles]
+  constructor
+  · rintro ⟨d,hd,hzd⟩
+    obtain ⟨x,hx,rfl⟩ := mem_centerDisks.mp hd
+    exact ⟨x,hx,hzd⟩
+  · rintro ⟨x,hx,hzx⟩
+    exact ⟨⟨x,R,false⟩,mem_centerDisks.mpr ⟨x,hx,rfl⟩,hzx⟩
+def activeCenters (L : Finset ℂ) : Finset ℂ := L.filter (fun x => dist x 0≤6)
+namespace HoleApprox
+lemma annulus_in_domain {c : ℂ → Fin 5} {α t R : ℝ} {L : Finset ℂ}
+    (a : HoleApprox c α L t) (hR : R<1) (htR : 2*t<R)
+    (hsep : ∀x∈L,∀y∈L,x≠y → 2*R<dist x y) {x : ℂ} (hx : x∈activeCenters L) :
+    planeAnnulus x (2*t) R⊆Metric.closedBall (0:ℂ) 7 \ coreHoles a.disks := by
+  have hx' := Finset.mem_filter.mp hx
+  intro z hz
+  have hz' := mem_planeAnnulus.mp hz
+  constructor
+  · change dist z 0≤7
+    exact (dist_triangle z x 0).trans (by linarith [hx'.2])
+  · intro hh
+    obtain ⟨y,hy⟩ := mem_iUnion.mp (a.cores_near hh)
+    obtain ⟨hyL,hzy⟩ := mem_iUnion.mp hy
+    change dist z y<2*t at hzy
+    by_cases he : x=y
+    · subst y; linarith [hz'.2]
+    · have hh' := hsep x hx'.1 y hyL he
+      have htri := dist_triangle x z y
+      rw [dist_comm x z] at htri
+      linarith [hz'.1]
+lemma outside_centers_in_domain {c : ℂ → Fin 5} {α t R : ℝ} {L : Finset ℂ}
+    (a : HoleApprox c α L t) (htR : 2*t<R) (ht1 : 2*t<1) :
+    Metric.closedBall (0:ℂ) 5 \ diskHoles (centerDisks (activeCenters L) R)⊆
+      Metric.closedBall (0:ℂ) 7 \ coreHoles a.disks := by
+  intro z hz
+  refine ⟨Metric.closedBall_subset_closedBall (by norm_num) hz.1,?_⟩
+  intro hh
+  obtain ⟨y,hy⟩ := mem_iUnion.mp (a.cores_near hh)
+  obtain ⟨hyL,hzy⟩ := mem_iUnion.mp hy
+  change dist z y<2*t at hzy
+  have hy0 : dist y 0≤6 := calc
+    dist y 0≤dist y z+dist z 0 := dist_triangle _ _ _
+    _ ≤ 6 := by rw [dist_comm y z]; have hz0 : dist z 0≤5 := hz.1; linarith
+  exact hz.2 (mem_center_holes.mpr ⟨y,Finset.mem_filter.mpr ⟨hyL,hy0⟩,hzy.trans htR⟩)
+end HoleApprox
+end EuclideanFiveColor
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PlaneColoring/GraphRealization.lean b/lean/OAI/Geometry/PlaneColoring/GraphRealization.lean
new file mode 100644
index 0000000000000000000000000000000000000000..e8384fcfddc9fdf5ceec5d1c90a70066043e0f34
--- /dev/null
+++ b/lean/OAI/Geometry/PlaneColoring/GraphRealization.lean
@@ -0,0 +1,415 @@
+import OAI.Geometry.PlaneColoring.CyclicCore
+import OAI.Geometry.PlaneColoring.PlanarCovers
+
+namespace OAI
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+/-- The directional subsequence in the last paragraph of interface-cycle follows
+from connectedness and an outer contact. Compactness of the continuum is not needed. -/
+lemma continuum_has_approach {x y : ℂ} {K : Set ℂ} (hK : IsPreconnected K)
+    (hx : x∈K) (hy : y∈K) (hne : y≠x) :
+    ∃v : ℂ, ‖v‖=1 ∧ Nonempty (ApproachingContinuum x K v) := by
+  let ρ : ℝ := dist y x
+  have hρ : 0<ρ := dist_pos.mpr hne
+  let r : ℕ → ℝ := fun n => ρ/(n+1)
+  have hr (n) : 0<r n := div_pos hρ (by positivity)
+  have hrle (n) : r n≤ρ := (div_le_iff₀ (by positivity : 0<(n:ℝ)+1)).mpr (by
+    have hn : 0≤(n:ℝ) := Nat.cast_nonneg n
+    nlinarith)
+  have hr0 : Tendsto r atTop (𝓝 0) := by
+    simpa only [r,div_eq_mul_inv,mul_zero,one_mul] using
+      (tendsto_const_nhds (x:=ρ)).mul (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜:=ℝ))
+  have hq (n) : ∃q∈K,dist q x=r n := by
+    have hI := hK.intermediate_value (f:=fun z : ℂ => dist z x) hx hy (continuous_id.dist continuous_const).continuousOn
+    apply hI
+    exact ⟨by simpa using (hr n).le,hrle n⟩
+  choose q hqK hqr using hq
+  let w : ℕ → ℂ := fun n => (q n-x)/(r n : ℂ)
+  have hwn (n) : ‖w n‖=1 := by
+    dsimp [w]
+    rw [norm_div,Complex.norm_real,Real.norm_eq_abs,abs_of_pos (hr n),←dist_eq_norm,hqr]
+    exact div_self (ne_of_gt (hr n))
+  have hwm (n) : w n∈Metric.sphere (0 : ℂ) 1 := by
+    simpa only [Metric.mem_sphere,dist_zero_right] using hwn n
+  obtain ⟨v,hv,φ,hφ,hlim⟩ := (isCompact_sphere (0 : ℂ) 1).tendsto_subseq hwm
+  refine ⟨v,by simpa only [Metric.mem_sphere,dist_zero_right] using hv,?_⟩
+  refine ⟨⟨hx,r∘φ,w∘φ,fun n => hr (φ n),hr0.comp hφ.tendsto_atTop,
+    fun n => hwn (φ n),hlim,?_⟩⟩
+  intro n
+  have he : x+(r (φ n) : ℂ)*w (φ n)=q (φ n) := by
+    dsimp [w]
+    have hn : (r (φ n) : ℂ)≠0 := Complex.ofReal_ne_zero.mpr (ne_of_gt (hr _))
+    field_simp
+    ring
+  exact he ▸ hqK (φ n)
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set
+namespace EuclideanFiveColor
+/-- The compact contact partition used in the manuscript's annular-crossing proof.
+This reuses Mathlib's Hausdorff component quotient and its profinite separation theorem. -/
+lemma compact_component_partition {X : Type*} [TopologicalSpace X] [CompactSpace X] [T2Space X]
+    {I O : Set X} (hI : IsCompact I) (hO : IsCompact O)
+    (hno : ∀x∈I, Disjoint (connectedComponent x) O) :
+    ∃Z₁ Z₂ : Set X, IsCompact Z₁ ∧ IsCompact Z₂ ∧ Disjoint Z₁ Z₂ ∧
+      Z₁∪Z₂=univ ∧ I⊆Z₁ ∧ O⊆Z₂ := by
+  let q : X → ConnectedComponents X := ConnectedComponents.mk
+  have hq : Continuous q := ConnectedComponents.continuous_coe
+  have hid : Disjoint (q '' I) (q '' O) := by
+    apply disjoint_left.mpr
+    rintro z ⟨x,hx,rfl⟩ ⟨y,hy,he⟩
+    have hxy : y∈connectedComponent x := ConnectedComponents.coe_eq_coe'.mp he
+    exact Set.disjoint_left.mp (hno x hx) hxy hy
+  have hsub : q '' I⊆(q '' O)ᶜ := Set.disjoint_left.mp hid
+  obtain ⟨C,hC,hIC,hCO⟩ := exists_clopen_of_closed_subset_open
+    (hI.image hq).isClosed (hO.image hq).isClosed.isOpen_compl hsub
+  refine ⟨q ⁻¹' C,q ⁻¹' Cᶜ,(hC.isClosed.preimage hq).isCompact,
+    (hC.compl.isClosed.preimage hq).isCompact,?_,?_,?_,?_⟩
+  · exact disjoint_compl_right.preimage q
+  · simp only [←preimage_union,union_compl_self,preimage_univ]
+  · intro x hx
+    exact hIC ⟨x,hx,rfl⟩
+  · intro x hx hxc
+    exact hCO hxc ⟨x,hx,rfl⟩
+end EuclideanFiveColor
+end
+
+noncomputable section
+open scoped Classical
+open Set Topology
+universe u
+namespace EuclideanFiveColor
+/-- Standard affine vertices. These are the manuscript's probability-simplex coordinates. -/
+def graphVertex {V : Type u} (i : V) : V → ℝ := fun j => if j=i then 1 else 0
+/-- Affine parameterization of an edge, including its two vertices. -/
+def graphEdgePoint {V : Type u} (i j : V) (t : ℝ) : V → ℝ :=
+  (1-t) • graphVertex i + t • graphVertex j
+/-- The geometric realization of a simple graph in its probability simplex.
+The equal-endpoint option includes isolated vertices. -/
+def graphRealization {V : Type u} (G : SimpleGraph V) : Set (V → ℝ) :=
+  {p | ∃ i j, (i=j ∨ G.Adj i j) ∧ ∃ t ∈ Icc (0:ℝ) 1, p=graphEdgePoint i j t}
+abbrev GraphSpace {V : Type u} (G : SimpleGraph V) := ↥(graphRealization G)
+lemma graphEdgePoint_apply {V : Type u} (i j k : V) (t : ℝ) :
+    graphEdgePoint i j t k = (1-t)*(if k=i then 1 else 0)+t*(if k=j then 1 else 0) := rfl
+@[simp] lemma graphEdgePoint_zero {V : Type u} (i j : V) : graphEdgePoint i j 0=graphVertex i := by
+  simp [graphEdgePoint]
+@[simp] lemma graphEdgePoint_one {V : Type u} (i j : V) : graphEdgePoint i j 1=graphVertex j := by
+  simp [graphEdgePoint]
+@[simp] lemma graphEdgePoint_self {V : Type u} (i : V) (t : ℝ) : graphEdgePoint i i t=graphVertex i := by
+  rw [graphEdgePoint,←add_smul]; simp
+lemma graphEdgePoint_swap {V : Type u} (i j : V) (t : ℝ) :
+    graphEdgePoint i j t=graphEdgePoint j i (1-t) := by
+  simp [graphEdgePoint,add_comm]
+lemma continuous_graphEdgePoint {V : Type u} (i j : V) : Continuous (graphEdgePoint i j) := by
+  exact ((continuous_const.sub continuous_id).smul continuous_const).add
+    (continuous_id.smul continuous_const)
+lemma graphVertex_mem {V : Type u} (G : SimpleGraph V) (i : V) : graphVertex i∈graphRealization G :=
+  ⟨i,i,Or.inl rfl,0,by norm_num,by simp⟩
+lemma graphEdgePoint_mem {V : Type u} {G : SimpleGraph V} {i j : V} (hij : i=j ∨ G.Adj i j)
+    {t : ℝ} (ht : t∈Icc (0:ℝ) 1) : graphEdgePoint i j t∈graphRealization G :=
+  ⟨i,j,hij,t,ht,rfl⟩
+lemma graphRealization_mono {V : Type u} {G H : SimpleGraph V} (h : G≤H) :
+    graphRealization G⊆graphRealization H := by
+  rintro p ⟨i,j,hij,t,ht,rfl⟩
+  exact ⟨i,j,hij.imp id (fun hh => h hh),t,ht,rfl⟩
+lemma graphRealization_nonneg {V : Type u} {G : SimpleGraph V} {p : V → ℝ}
+    (hp : p∈graphRealization G) (i : V) : 0≤p i := by
+  obtain ⟨a,b,_,t,ht,rfl⟩ := hp
+  rw [graphEdgePoint_apply]
+  split_ifs <;> nlinarith [ht.1,ht.2]
+lemma graphRealization_sum {V : Type u} [Fintype V] {G : SimpleGraph V} {p : V → ℝ}
+    (hp : p∈graphRealization G) : ∑ i,p i=1 := by
+  obtain ⟨a,b,_,t,_,rfl⟩ := hp
+  simp [graphEdgePoint,graphVertex,Finset.sum_add_distrib]
+lemma graphRealization_compact {V : Type u} [Finite V] (G : SimpleGraph V) :
+    IsCompact (graphRealization G) := by
+  have he : graphRealization G = ⋃ (i : V) (j : V) (_ : i=j ∨ G.Adj i j),
+      graphEdgePoint i j '' Icc (0:ℝ) 1 := by
+    ext p
+    simp only [mem_iUnion,mem_image,graphRealization,mem_ofPred_eq]
+    constructor
+    · rintro ⟨i,j,hij,t,ht,rfl⟩; exact ⟨i,j,hij,t,ht,rfl⟩
+    · rintro ⟨i,j,hij,t,ht,rfl⟩; exact ⟨i,j,hij,t,ht,rfl⟩
+  rw [he]
+  exact isCompact_iUnion fun i => isCompact_iUnion fun j => isCompact_iUnion fun _ =>
+    isCompact_Icc.image (continuous_graphEdgePoint i j)
+lemma graphRealization_closed {V : Type u} [Finite V] (G : SimpleGraph V) :
+    IsClosed (graphRealization G) := (graphRealization_compact G).isClosed
+instance graphSpace_compact {V : Type u} [Finite V] (G : SimpleGraph V) : CompactSpace (GraphSpace G) :=
+  isCompact_iff_compactSpace.mp (graphRealization_compact G)
+lemma graphRealization_singleton {V : Type u} [Subsingleton V] (G : SimpleGraph V) (a : V) :
+    graphRealization G={graphVertex a} := by
+  ext p
+  constructor
+  · rintro ⟨i,j,_,t,_,rfl⟩
+    simp [Subsingleton.elim i a,Subsingleton.elim j a]
+  · rintro rfl
+    exact graphVertex_mem G a
+end EuclideanFiveColor
+end
+
+noncomputable section
+open scoped Classical
+open Set Topology
+universe u
+namespace EuclideanFiveColor
+lemma graphRealization_of_two_support {V : Type u} [Fintype V]
+    {G : SimpleGraph V} {p : V → ℝ} (hn : ∀i,0≤p i) (hs : ∑i,p i=1)
+    {a b : V} (hab : a=b ∨ G.Adj a b) (hsupp : ∀j,j≠a → j≠b → p j=0) :
+    p∈graphRealization G := by
+  by_cases he : a=b
+  · subst b
+    have ha : p a=1 := by
+      have hh : ∑i,p i=p a := Finset.sum_eq_single a (fun i _ hi => hsupp i hi hi) (by simp)
+      linarith
+    have hp : p=graphVertex a := by
+      funext i
+      by_cases hi : i=a
+      · subst i; simp [graphVertex,ha]
+      · simp [graphVertex,hi,hsupp i hi hi]
+    rw [hp]; exact graphVertex_mem _ _
+  · have he' : b≠a := Ne.symm he
+    have hs' : p a+p b=1 := by
+      have hh : ∑i∈({a,b} : Finset V),p i=∑i,p i := by
+        apply Finset.sum_subset (Finset.subset_univ _)
+        intro i _ hi
+        simp only [Finset.mem_insert,Finset.mem_singleton,not_or] at hi
+        exact hsupp i hi.1 hi.2
+      simpa [he] using hh.trans hs
+    refine ⟨a,b,hab,p b,⟨hn _,by linarith [hn a]⟩,?_⟩
+    funext i
+    rw [graphEdgePoint_apply]
+    by_cases hi : i=a
+    · subst i; simp [he]; linarith
+    · by_cases hj : i=b
+      · subst i; simp [he']
+      · simp [hi,hj,hsupp i hi hj]
+
+/-- A simplex vector whose positive labels form a clique in a forest is genuinely
+in that forest's affine realization, rather than just in the complete graph. -/
+theorem simplex_mem_forest {V : Type u} [Fintype V] {G : SimpleGraph V}
+    (hG : G.IsAcyclic) {p : V → ℝ} (hn : ∀i,0≤p i) (hs : ∑i,p i=1)
+    (hcl : ∀i j,i≠j → 0<p i → 0<p j → G.Adj i j) : p∈graphRealization G := by
+  let S : Finset V := Finset.univ.filter fun i => p i≠0
+  have hmem (i : V) : i∈S ↔ 0<p i := by simp [S,lt_iff_le_and_ne,hn i,ne_comm]
+  have hsum : ∑i∈S,p i=1 := by
+    have he : ∑i∈S,p i=∑i,p i := by
+      apply Finset.sum_subset (Finset.filter_subset _ _)
+      intro i _ hi
+      simpa [S] using hi
+    exact he.trans hs
+  have hne : S.Nonempty := by
+    by_contra he
+    rw [Finset.not_nonempty_iff_eq_empty.mp he,Finset.sum_empty] at hsum
+    norm_num at hsum
+  have hcard : S.card≤2 := by
+    by_contra hh
+    have h3 : 3≤S.card := by omega
+    obtain ⟨T,hTS,hT⟩ := Finset.exists_subset_card_eq h3
+    apply hG.cliqueFree (le_refl 3) T
+    refine ⟨?_,hT⟩
+    intro i hi j hj hij
+    exact hcl i j hij ((hmem i).mp (hTS hi)) ((hmem j).mp (hTS hj))
+  by_cases h1 : S.card=1
+  · obtain ⟨a,hSa⟩ := Finset.card_eq_one.mp h1
+    apply graphRealization_of_two_support hn hs (G:=G) (a:=a) (b:=a) (Or.inl rfl)
+    intro j hj _
+    by_contra hp
+    have hm : j∈S := by simp [S,hp]
+    rw [hSa,Finset.mem_singleton] at hm
+    exact hj hm
+  · have h2 : S.card=2 := by have := hne.card_pos; omega
+    obtain ⟨a,b,hab,hS⟩ := Finset.card_eq_two.mp h2
+    have ha : 0<p a := (hmem _).mp (by rw [hS]; simp)
+    have hb : 0<p b := (hmem _).mp (by rw [hS]; simp)
+    apply graphRealization_of_two_support hn hs (Or.inr (hcl a b hab ha hb))
+    intro j hja hjb
+    by_contra hp
+    have hm : j∈S := by simp [S,hp]
+    rw [hS] at hm
+    simp [hja,hjb] at hm
+end EuclideanFiveColor
+end
+
+noncomputable section
+open scoped Classical
+open Set Topology MeasureTheory
+namespace EuclideanFiveColor
+/-- The manuscript's thresholded and renormalized disk average (equation 22). -/
+def thresholdWeight (c : ℂ → Fin 5) (ε α : ℝ) (q : ℂ) (i : Fin 5) : ℝ :=
+  max (diskAverage c ε i q-α) 0
+def thresholdTotal (c : ℂ → Fin 5) (ε α : ℝ) (q : ℂ) : ℝ :=
+  ∑i,thresholdWeight c ε α q i
+def thresholdMap (c : ℂ → Fin 5) (ε α : ℝ) (q : ℂ) (i : Fin 5) : ℝ :=
+  thresholdWeight c ε α q i / thresholdTotal c ε α q
+lemma thresholdTotal_pos {c : ℂ → Fin 5} (hc : Measurable c) {ε α : ℝ}
+    (hε : 0<ε) (hα : α<1/5) (q : ℂ) : 0<thresholdTotal c ε α q := by
+  have hs := diskAverage_sum hc hε q
+  have hh : 1-5*α ≤ thresholdTotal c ε α q := by
+    have hh := Finset.sum_le_sum (s := (Finset.univ : Finset (Fin 5)))
+      (fun i _ => le_max_left (diskAverage c ε i q-α) 0)
+    simpa [thresholdTotal,thresholdWeight,Finset.sum_sub_distrib,hs] using hh
+  linarith
+lemma thresholdMap_nonneg {c : ℂ → Fin 5} (hc : Measurable c) {ε α : ℝ}
+    (hε : 0<ε) (hα : α<1/5) (q : ℂ) (i : Fin 5) : 0≤thresholdMap c ε α q i :=
+  div_nonneg (le_max_right _ _) (thresholdTotal_pos hc hε hα q).le
+lemma thresholdMap_sum {c : ℂ → Fin 5} (hc : Measurable c) {ε α : ℝ}
+    (hε : 0<ε) (hα : α<1/5) (q : ℂ) : ∑i,thresholdMap c ε α q i=1 := by
+  unfold thresholdMap
+  rw [←Finset.sum_div]
+  exact div_self (thresholdTotal_pos hc hε hα q).ne'
+lemma thresholdMap_pos {c : ℂ → Fin 5} (hc : Measurable c) {ε α : ℝ}
+    (hε : 0<ε) (hα : α<1/5) (q : ℂ) (i : Fin 5) :
+    0<thresholdMap c ε α q i ↔ α<diskAverage c ε i q := by
+  rw [thresholdMap,div_pos_iff_of_pos_right (thresholdTotal_pos hc hε hα q)]
+  simp [thresholdWeight]
+lemma continuous_thresholdMap {c : ℂ → Fin 5} (hc : Measurable c) {ε α : ℝ}
+    (hε : 0<ε) (hα : α<1/5) : Continuous (thresholdMap c ε α) := by
+  have hw (i : Fin 5) : Continuous (fun q => thresholdWeight c ε α q i) :=
+    ((diskAverage_lipschitz hc hε i).continuous.sub continuous_const).max continuous_const
+  have ht : Continuous (thresholdTotal c ε α) := continuous_finsetSum _ (fun i _ => hw i)
+  exact continuous_pi fun i => (hw i).div ht (fun q => (thresholdTotal_pos hc hε hα q).ne')
+lemma thresholdMap_mem_forest {c : ℂ → Fin 5} (hc : Measurable c) {ε α : ℝ}
+    (hε : 0<ε) (hα : α<1/5) (q : ℂ) {G : SimpleGraph (Fin 5)} (hG : G.IsAcyclic)
+    (hb : ∀i j,i≠j → α<diskAverage c ε i q → α<diskAverage c ε j q → G.Adj i j) :
+    thresholdMap c ε α q∈graphRealization G := by
+  apply simplex_mem_forest hG (thresholdMap_nonneg hc hε hα q) (thresholdMap_sum hc hε hα q)
+  intro i j hij hi hj
+  exact hb i j hij ((thresholdMap_pos hc hε hα q i).mp hi) ((thresholdMap_pos hc hε hα q j).mp hj)
+end EuclideanFiveColor
+end
+
+noncomputable section
+open scoped Classical
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+lemma finite_pos_lt {ι : Type*} [Finite ι] (R : ι → ℝ) (hR : ∀i,0<R i) :
+    ∃s : ℝ,0<s ∧ ∀i,s<R i := by
+  have hh : ∀ᶠ s : ℝ in 𝓝 0,∀i,s<R i := Filter.eventually_all.mpr fun i => gt_mem_nhds (hR i)
+  obtain ⟨δ,hδ,hd⟩ := Metric.mem_nhds_iff.mp hh
+  refine ⟨δ/2,half_pos hδ,hd ?_⟩
+  change dist (δ/2) (0:ℝ)<δ
+  rw [Real.dist_eq,sub_zero,abs_of_pos (half_pos hδ)]
+  exact half_lt_self hδ
+lemma finite_small_scale {ι : Type*} [Finite ι] (P : ι → ℝ → Prop)
+    (h : ∀i,∃δ>0,∀ε,0<ε → ε<δ → P i ε) : ∃δ>0,∀ε,0<ε → ε<δ → ∀i,P i ε := by
+  choose δ hδ hd using h
+  obtain ⟨s,hs,hsi⟩ := finite_pos_lt δ hδ
+  exact ⟨s,hs,fun ε hε hεs i => hd i ε hε (hεs.trans (hsi i))⟩
+
+/-- A genuine local forest chart; its exception disks have arbitrarily small total
+radius at sufficiently small averaging scale. -/
+theorem forest_chart {c : ℂ → Fin 5} (hc : Measurable c) {α : ℝ}
+    (hα : 0<α) (hα' : α<1/5) (x : ℂ) (hf : (transitionGraph c x).IsAcyclic) :
+    ∃s>0,∀η>0,∃δ>0,∀ε,0<ε → ε<δ → ∃C : Finset ℂ,
+      (C.card:ℝ)*(α*ε/4)<η ∧ ∀q∈Metric.ball x s,
+      q∉⋃z∈C,Metric.ball z (α*ε/4) → thresholdMap c ε α q∈graphRealization (transitionGraph c x) := by
+  have hall : ∀p : Fin 5 × Fin 5,∃R>0, p.1≠p.2 → ¬TransitionEdge c p.1 p.2 x →
+      ∀s<R,∀η>0,∃δ>0,∀ε,0<ε → ε<δ → ∃C : Finset ℂ,
+        (∀q∈Metric.ball x s,α<diskAverage c ε p.1 q → α<diskAverage c ε p.2 q →
+          ∃z∈C,q∈Metric.ball z (α*ε/4)) ∧ (C.card:ℝ)*(α*ε/4)<η := by
+    intro p
+    by_cases hp : p.1≠p.2 ∧ ¬TransitionEdge c p.1 p.2 x
+    · obtain ⟨R,hR,hcov⟩ := missing_edge_disk_cover hc hp.1 hp.2
+      exact ⟨R,hR,fun _ _ s hs η hη => hcov s hs α hα η hη⟩
+    · exact ⟨1,by norm_num,fun h₁ h₂ => False.elim (hp ⟨h₁,h₂⟩)⟩
+  choose R hR hcov using hall
+  obtain ⟨s,hs,hsR⟩ := finite_pos_lt R hR
+  refine ⟨s,hs,fun η hη => ?_⟩
+  let P (p : Fin 5 × Fin 5) (ε : ℝ) : Prop := ∃C : Finset ℂ,
+    (C.card:ℝ)*(α*ε/4)<η/25 ∧ (p.1≠p.2 → ¬TransitionEdge c p.1 p.2 x →
+      ∀q∈Metric.ball x s,α<diskAverage c ε p.1 q → α<diskAverage c ε p.2 q →
+        ∃z∈C,q∈Metric.ball z (α*ε/4))
+  have hP : ∀p,∃δ>0,∀ε,0<ε → ε<δ → P p ε := by
+    intro p
+    by_cases hp : p.1≠p.2 ∧ ¬TransitionEdge c p.1 p.2 x
+    · obtain ⟨δ,hδ,hd⟩ := hcov p hp.1 hp.2 s (hsR p) (η/25) (by positivity)
+      refine ⟨δ,hδ,fun ε hε hεδ => ?_⟩
+      obtain ⟨C,hC,hCr⟩ := hd ε hε hεδ
+      exact ⟨C,hCr,fun _ _ => hC⟩
+    · refine ⟨1,by norm_num,fun ε _ _ => ⟨∅,by simpa using (show 0<η/25 by positivity),?_⟩⟩
+      exact fun h₁ h₂ => False.elim (hp ⟨h₁,h₂⟩)
+  obtain ⟨δ,hδ,hd⟩ := finite_small_scale P hP
+  refine ⟨δ,hδ,fun ε hε hεδ => ?_⟩
+  choose C hCr hC using hd ε hε hεδ
+  let T := Finset.univ.biUnion C
+  have hr : 0≤α*ε/4 := by positivity
+  refine ⟨T,?_,?_⟩
+  · have hcard : (T.card:ℝ)≤∑p : Fin 5 × Fin 5,((C p).card:ℝ) := by
+      exact_mod_cast (Finset.card_biUnion_le (s := Finset.univ) (t := C))
+    have hsum : (∑p : Fin 5 × Fin 5,((C p).card:ℝ)*(α*ε/4))<∑_p : Fin 5 × Fin 5,η/25 :=
+      Finset.sum_lt_sum_of_nonempty Finset.univ_nonempty (fun p _ => hCr p)
+    have hb := mul_le_mul_of_nonneg_right hcard hr
+    rw [Finset.sum_mul] at hb
+    have hsum' : (∑p : Fin 5 × Fin 5,((C p).card:ℝ)*(α*ε/4))<25*(η/25) := by simpa using hsum
+    exact hb.trans_lt (by linarith [hsum'])
+  · intro q hq hqT
+    apply thresholdMap_mem_forest hc hε hα' q hf
+    intro i j hij hi hj
+    by_contra he
+    obtain ⟨z,hz,hqz⟩ := hC (i,j) hij he q hq hi hj
+    apply hqT
+    exact mem_iUnion.mpr ⟨z,mem_iUnion.mpr ⟨Finset.mem_biUnion.mpr ⟨(i,j),Finset.mem_univ _,hz⟩,hqz⟩⟩
+end EuclideanFiveColor
+end
+
+noncomputable section
+open scoped Classical
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+/-- Compact local forest control with one Lebesgue radius, chosen before the hole
+budget and the disk-average scale. The resulting forest controls the WHOLE nearby
+ball off the common exception disks, not merely the compact set. -/
+theorem compact_forest_charts {c : ℂ → Fin 5} (hc : Measurable c) {α : ℝ}
+    (hα : 0<α) (hα' : α<1/5) (M : Set ℂ) (hM : IsCompact M)
+    (hf : ∀x∈M,(transitionGraph c x).IsAcyclic) :
+    ∃l>0,∀η>0,∃δ>0,∀ε,0<ε → ε<δ → ∃C : Finset ℂ,
+      (C.card:ℝ)*(α*ε/4)<η ∧ ∀a∈M,∃G : SimpleGraph (Fin 5),G.IsAcyclic ∧
+      ∀q∈Metric.ball a l,q∉⋃z∈C,Metric.ball z (α*ε/4) → thresholdMap c ε α q∈graphRealization G := by
+  choose s hs hchart using fun x : M => forest_chart hc hα hα' x.val (hf x.val x.property)
+  have hcover : M⊆⋃x : M,Metric.ball x.val (s x) := by
+    intro x hx
+    exact mem_iUnion.mpr ⟨⟨x,hx⟩,Metric.mem_ball_self (hs _)⟩
+  obtain ⟨S,hS⟩ := hM.elim_finite_subcover (fun x : M => Metric.ball x.val (s x))
+    (fun _ => Metric.isOpen_ball) hcover
+  have hS' : M⊆⋃x : S,Metric.ball x.val.val (s x.val) := by
+    intro x hx
+    obtain ⟨a,ha⟩ := mem_iUnion.mp (hS hx)
+    obtain ⟨haS,hxa⟩ := mem_iUnion.mp ha
+    exact mem_iUnion.mpr ⟨⟨a,haS⟩,hxa⟩
+  obtain ⟨l,hl,hleb⟩ := lebesgue_number_lemma_of_metric hM (fun _ => Metric.isOpen_ball) hS'
+  refine ⟨l,hl,fun η hη => ?_⟩
+  let β := η/((S.card:ℝ)+1)
+  have hβ : 0<β := div_pos hη (by positivity)
+  let P (x : S) (ε : ℝ) : Prop := ∃C : Finset ℂ,
+    (C.card:ℝ)*(α*ε/4)<β ∧ ∀q∈Metric.ball x.val.val (s x.val),
+      q∉⋃z∈C,Metric.ball z (α*ε/4) → thresholdMap c ε α q∈graphRealization (transitionGraph c x.val.val)
+  obtain ⟨δ,hδ,hd⟩ := finite_small_scale P (fun x => hchart x.val β hβ)
+  refine ⟨δ,hδ,fun ε hε hεδ => ?_⟩
+  choose C hCr hC using hd ε hε hεδ
+  let T := Finset.univ.biUnion C
+  refine ⟨T,?_,?_⟩
+  · have hcard : (T.card:ℝ)≤∑p : S,((C p).card:ℝ) := by
+      exact_mod_cast (Finset.card_biUnion_le (s := Finset.univ) (t := C))
+    have hb := mul_le_mul_of_nonneg_right hcard (show 0≤α*ε/4 by positivity)
+    rw [Finset.sum_mul] at hb
+    have hsum := Finset.sum_le_sum (s:=(Finset.univ : Finset S)) (fun p _ => (hCr p).le)
+    have hsum' : (∑p:S,((C p).card:ℝ)*(α*ε/4))≤(S.card:ℝ)*β := by simpa using hsum
+    have hβeq : β*((S.card:ℝ)+1)=η := div_mul_cancel₀ η (by positivity)
+    exact hb.trans hsum' |>.trans_lt (by nlinarith)
+  · intro a ha
+    obtain ⟨x,hx⟩ := hleb a ha
+    refine ⟨transitionGraph c x.val.val,hf x.val.val x.val.property,fun q hq hqT => ?_⟩
+    apply hC x q (hx hq)
+    intro hh
+    obtain ⟨z,hz⟩ := mem_iUnion.mp hh
+    obtain ⟨hzC,hqz⟩ := mem_iUnion.mp hz
+    exact hqT (mem_iUnion.mpr ⟨z,mem_iUnion.mpr ⟨Finset.mem_biUnion.mpr ⟨x,Finset.mem_univ _,hzC⟩,hqz⟩⟩)
+end EuclideanFiveColor
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PlaneColoring/HoleApproximation.lean b/lean/OAI/Geometry/PlaneColoring/HoleApproximation.lean
new file mode 100644
index 0000000000000000000000000000000000000000..808947d4b29895f9f151ab9749f13ec2746ce7f4
--- /dev/null
+++ b/lean/OAI/Geometry/PlaneColoring/HoleApproximation.lean
@@ -0,0 +1,328 @@
+import OAI.Geometry.PlaneColoring.DiskExtension
+
+namespace OAI
+
+noncomputable section
+open scoped Classical
+open Set Topology
+namespace EuclideanFiveColor
+namespace MarkedDisk
+lemma radius_le_sum {S : Finset MarkedDisk} (hS : ∀d∈S,0≤d.radius) {d : MarkedDisk} (hd : d∈S) :
+    d.radius≤∑e∈S,e.radius := Finset.single_le_sum hS hd
+lemma noncoreCost_nonneg {d : MarkedDisk} (hd : 0≤d.radius) : 0≤d.noncoreCost := by
+  unfold noncoreCost
+  split <;> positivity
+lemma noncore_radius_le_sum {S : Finset MarkedDisk} (hS : ∀d∈S,0≤d.radius)
+    {d : MarkedDisk} (hd : d∈S) (hc : d.core=false) : d.radius≤∑e∈S,e.noncoreCost := by
+  have hh := Finset.single_le_sum (fun e he => noncoreCost_nonneg (hS e he)) hd
+  simpa [noncoreCost,hc] using hh
+lemma sphere_avoids_holes {S : Finset MarkedDisk}
+    (hdis : (S : Set MarkedDisk).Pairwise (fun a b => Disjoint a.closedBall b.closedBall))
+    {d : MarkedDisk} (hd : d∈S) {z : ℂ} (hz : z∈Metric.sphere d.center d.radius) : z∉diskHoles S := by
+  rintro hh
+  obtain ⟨e,he,hze⟩ := mem_diskHoles.mp hh
+  by_cases hde : d=e
+  · subst e
+    have hh' : dist z d.center<d.radius := hze
+    have hz' : dist z d.center=d.radius := hz
+    linarith
+  · exact Set.disjoint_left.mp (hdis hd he hde) (Metric.sphere_subset_closedBall hz)
+      (Metric.ball_subset_closedBall hze)
+lemma closedBall_near_core {L : Set ℂ} {d : MarkedDisk} (hd : d.Valid L) (hc : d.core=true) :
+    ∃x∈L,d.closedBall⊆Metric.ball x (2*d.radius) := by
+  obtain ⟨x,hx,hxd⟩ := hd.2 hc
+  refine ⟨x,hx,fun z hz => ?_⟩
+  have hx' : dist d.center x<d.radius := by simpa only [ball,Metric.mem_ball,dist_comm] using hxd
+  exact lt_of_le_of_lt (dist_triangle z d.center x) (by dsimp [closedBall] at hz; change dist z d.center≤d.radius at hz; linarith)
+lemma sphere_norm_bound {d : MarkedDisk} {R : ℝ}
+    (hmeet : (d.closedBall∩Metric.closedBall (0:ℂ) R).Nonempty) {z : ℂ}
+    (hz : z∈Metric.sphere d.center d.radius) : dist z 0≤2*d.radius+R := by
+  obtain ⟨a,ha,haR⟩ := hmeet
+  have hza : dist z a≤2*d.radius := calc
+    dist z a≤dist z d.center+dist d.center a := dist_triangle _ _ _
+    _ ≤ 2*d.radius := by
+      have hh : dist z d.center=d.radius := hz
+      have ha' : dist d.center a≤d.radius := by simpa only [closedBall,Metric.mem_closedBall,dist_comm] using ha
+      linarith
+  exact (dist_triangle z a 0).trans (by change dist a 0≤R at haR; linarith)
+end MarkedDisk
+end EuclideanFiveColor
+end
+
+noncomputable section
+open scoped Classical
+open Set Topology
+namespace EuclideanFiveColor
+namespace MarkedDisk
+def initialDisks (L C : Finset ℂ) (δ u : ℝ) : Finset MarkedDisk :=
+  L.image (fun x => ⟨x,δ,true⟩) ∪ C.image (fun x => ⟨x,u,false⟩)
+lemma mem_initialDisks {L C : Finset ℂ} {δ u : ℝ} {d : MarkedDisk} :
+    d∈initialDisks L C δ u ↔ (∃x∈L,d=⟨x,δ,true⟩) ∨ (∃x∈C,d=⟨x,u,false⟩) := by
+  simp [initialDisks,eq_comm]
+lemma initialDisks_valid (L C : Finset ℂ) {δ u : ℝ} (hδ : 0<δ) (hu : 0<u) :
+    ∀d∈initialDisks L C δ u,d.Valid (L : Set ℂ) := by
+  intro d hd
+  rcases mem_initialDisks.mp hd with ⟨x,hx,rfl⟩|⟨x,hx,rfl⟩
+  · exact ⟨hδ,fun _ => ⟨x,hx,Metric.mem_ball_self hδ⟩⟩
+  · exact ⟨hu,by simp⟩
+lemma initialDisks_sum (L C : Finset ℂ) (δ u : ℝ) :
+    (∑d∈initialDisks L C δ u,d.radius)=(L.card:ℝ)*δ+(C.card:ℝ)*u ∧
+    (∑d∈initialDisks L C δ u,d.noncoreCost)=(C.card:ℝ)*u := by
+  have hd : Disjoint (L.image (fun x => (⟨x,δ,true⟩ : MarkedDisk)))
+      (C.image (fun x => (⟨x,u,false⟩ : MarkedDisk))) := by
+    apply Finset.disjoint_left.mpr
+    intro d hd he
+    obtain ⟨x,hx,hxd⟩ := Finset.mem_image.mp hd
+    obtain ⟨y,hy,hyd⟩ := Finset.mem_image.mp he
+    have hh := congrArg MarkedDisk.core (hxd.trans hyd.symm)
+    exact Bool.noConfusion hh
+  have hi (r : ℝ) (b : Bool) : Function.Injective (fun x : ℂ => (⟨x,r,b⟩ : MarkedDisk)) := by
+    intro x y h; exact congrArg MarkedDisk.center h
+  simp only [initialDisks,Finset.sum_union hd]
+  rw [Finset.sum_image (fun x _ y _ h => hi δ true h),Finset.sum_image (fun x _ y _ h => hi u false h),
+      Finset.sum_image (fun x _ y _ h => hi δ true h),Finset.sum_image (fun x _ y _ h => hi u false h)]
+  simp [noncoreCost]
+lemma initial_core_cover (L C : Finset ℂ) (δ u : ℝ) {x z : ℂ} (hx : x∈L)
+    (hz : z∈Metric.ball x δ) : z∈coreHoles (initialDisks L C δ u) := by
+  exact mem_coreHoles.mpr ⟨⟨x,δ,true⟩,mem_initialDisks.mpr (Or.inl ⟨x,hx,rfl⟩),rfl,hz⟩
+lemma initial_marked_cover (L C : Finset ℂ) (δ u : ℝ) {x z : ℂ} (hx : x∈C)
+    (hz : z∈Metric.ball x u) : z∈diskHoles (initialDisks L C δ u) := by
+  exact mem_diskHoles.mpr ⟨⟨x,u,false⟩,mem_initialDisks.mpr (Or.inr ⟨x,hx,rfl⟩),hz⟩
+lemma holes_subset_of_cover {S T : Finset MarkedDisk}
+    (hcov : ∀d∈S,∃e∈T,d.ball⊆e.ball ∧ (d.core=true → e.core=true)) :
+    diskHoles S⊆diskHoles T ∧ coreHoles S⊆coreHoles T := by
+  constructor
+  · intro z hz
+    obtain ⟨d,hd,hzd⟩ := mem_diskHoles.mp hz
+    obtain ⟨e,he,hde,_⟩ := hcov d hd
+    exact mem_diskHoles.mpr ⟨e,he,hde hzd⟩
+  · intro z hz
+    obtain ⟨d,hd,hcd,hzd⟩ := mem_coreHoles.mp hz
+    obtain ⟨e,he,hde,hc⟩ := hcov d hd
+    exact mem_coreHoles.mpr ⟨e,he,hc hcd,hde hzd⟩
+end MarkedDisk
+end EuclideanFiveColor
+end
+
+noncomputable section
+open scoped Classical
+open Set Filter Topology MeasureTheory ContinuousMap
+namespace EuclideanFiveColor
+open MarkedDisk
+/-- Actual small-hole maps on a fixed disk, with arbitrarily small total radii.
+The core witnesses lie in the finite actual cyclic core; all other holes have been
+filled by genuine tree extensions. -/
+theorem small_hole_map {c : ℂ → Fin 5} (hc : Measurable c) {α : ℝ}
+    (hα : 0<α) (hα' : α<1/5) (L : Finset ℂ)
+    (hL : ∀x∈Metric.closedBall (0:ℂ) 8,¬(transitionGraph c x).IsAcyclic → x∈L)
+    {t : ℝ} (ht : 0<t) (ht' : t<1/2) :
+    ∃e>0,∀ε,0<ε → ε<e → ∃S : Finset MarkedDisk,
+      (∀d∈S,d.Valid (L : Set ℂ)) ∧
+      (S : Set MarkedDisk).Pairwise (fun a b => Disjoint a.closedBall b.closedBall) ∧
+      (∑d∈S,d.radius)<t ∧
+      ∃g : C(↥(Metric.closedBall (0:ℂ) 7 \ coreHoles S),GraphSpace (⊤ : SimpleGraph (Fin 5))),
+        ∀z : ↥(Metric.closedBall (0:ℂ) 7 \ coreHoles S),z.val∉diskHoles S →
+          (g z).val=thresholdMap c ε α z.val := by
+  let δ : ℝ := t/(4*((L.card:ℝ)+1))
+  have hδ : 0<δ := div_pos ht (by positivity)
+  have hδeq : δ*(4*((L.card:ℝ)+1))=t := div_mul_cancel₀ _ (by positivity)
+  have hδbudget : (L.card:ℝ)*δ<t/4 := by nlinarith
+  let M := Metric.closedBall (0:ℂ) 8 \ ⋃x∈L,Metric.ball x δ
+  have hM : IsCompact M := (isCompact_closedBall (0:ℂ) 8).diff
+    (isOpen_iUnion fun _ => isOpen_iUnion fun _ => Metric.isOpen_ball)
+  have hf : ∀x∈M,(transitionGraph c x).IsAcyclic := by
+    intro x hx
+    by_contra hn
+    exact hx.2 (mem_iUnion.mpr ⟨x,mem_iUnion.mpr ⟨hL x hx.1 hn,Metric.mem_ball_self hδ⟩⟩)
+  obtain ⟨l,hl,hcharts⟩ := compact_forest_charts hc hα hα' M hM hf
+  let κ := min (t/4) (l/6)
+  have hκ : 0<κ := lt_min (by positivity) (by positivity)
+  have hκt : κ≤t/4 := min_le_left _ _
+  have hκl : κ≤l/6 := min_le_right _ _
+  obtain ⟨e,he,hC⟩ := hcharts κ hκ
+  refine ⟨e,he,fun ε hε hεe => ?_⟩
+  obtain ⟨C,hCr,hCv⟩ := hC ε hε hεe
+  let u := α*ε/4
+  have hu : 0<u := by dsimp [u]; positivity
+  obtain ⟨S,hS,hdis,hcov,hsum,hcost⟩ := MarkedDisk.separate
+    (initialDisks L C δ u) (initialDisks_valid L C hδ hu) hκ
+  have htot : (∑d∈S,d.radius)<t := by
+    rw [(initialDisks_sum L C δ u).1] at hsum
+    change (C.card:ℝ)*u<κ at hCr
+    linarith
+  have hnrad : ∀d∈S,d.core=false → 2*d.radius<l := by
+    rw [(initialDisks_sum L C δ u).2] at hcost
+    intro d hd hdf
+    have hd' := noncore_radius_le_sum (fun a ha => (hS a ha).1.le) hd hdf
+    change (C.card:ℝ)*u<κ at hCr
+    linarith
+  have hcover := holes_subset_of_cover hcov
+  have hMof (z : ℂ) (hz : z∈Metric.closedBall (0:ℂ) 8) (hzS : z∉diskHoles S) : z∈M := by
+    refine ⟨hz,?_⟩
+    intro hh
+    obtain ⟨x,hx⟩ := mem_iUnion.mp hh
+    obtain ⟨hxL,hzx⟩ := mem_iUnion.mp hx
+    exact hzS (hcover.1 (mem_diskHoles.mpr ⟨⟨x,δ,true⟩,
+      mem_initialDisks.mpr (Or.inl ⟨x,hxL,rfl⟩),hzx⟩))
+  have hmarked (z : ℂ) (hz : z∉diskHoles S) : z∉⋃x∈C,Metric.ball x u := by
+    intro hh
+    obtain ⟨x,hx⟩ := mem_iUnion.mp hh
+    obtain ⟨hxC,hzx⟩ := mem_iUnion.mp hx
+    exact hz (hcover.1 (initial_marked_cover L C δ u hxC hzx))
+  let f : C(ℂ,Fin 5 → ℝ) := ⟨thresholdMap c ε α,continuous_thresholdMap hc hε hα'⟩
+  have hg : ∀z∈Metric.closedBall (0:ℂ) 7,z∉diskHoles S →
+      f z∈graphRealization (⊤ : SimpleGraph (Fin 5)) := by
+    intro z hz hzS
+    have hzM := hMof z (Metric.closedBall_subset_closedBall (by norm_num) hz) hzS
+    obtain ⟨G,hG,hq⟩ := hCv z hzM
+    exact graphRealization_mono le_top (hq z (Metric.mem_ball_self hl) (hmarked z hzS))
+  have hb : ∀d∈S,d.core=false → (d.closedBall∩Metric.closedBall (0:ℂ) 7).Nonempty →
+      ∃G : SimpleGraph (Fin 5),G.IsAcyclic ∧ ∀z∈Metric.sphere d.center d.radius,
+        f z∈graphRealization G := by
+    intro d hd hdf hmeet
+    have hr := (hS d hd).1
+    have hsph (z : ℂ) (hz : z∈Metric.sphere d.center d.radius) : z∈Metric.closedBall (0:ℂ) 8 := by
+      have hzr := sphere_norm_bound hmeet hz
+      have hd' := radius_le_sum (fun a ha => (hS a ha).1.le) hd
+      change dist z 0≤8
+      linarith
+    obtain ⟨a,ha⟩ := (NormedSpace.sphere_nonempty (x:=d.center)).mpr hr.le
+    have haM := hMof a (hsph a ha) (sphere_avoids_holes hdis hd ha)
+    obtain ⟨G,hG,hq⟩ := hCv a haM
+    refine ⟨G,hG,fun z hz => hq z ?_ (hmarked z (sphere_avoids_holes hdis hd hz))⟩
+    have hza : dist z a≤2*d.radius := calc
+      dist z a≤dist z d.center+dist d.center a := dist_triangle _ _ _
+      _ = 2*d.radius := by
+        have hz' : dist z d.center=d.radius := hz
+        have ha' : dist d.center a=d.radius := by simpa only [Metric.mem_sphere,dist_comm] using ha
+        rw [hz',ha']; ring
+    exact hza.trans_lt (hnrad d hd hdf)
+  obtain ⟨g,hg'⟩ := fill_noncore_disks (Metric.closedBall (0:ℂ) 7) S
+    (fun d hd => (hS d hd).1) hdis f hg hb
+  exact ⟨S,hS,hdis,htot,g,hg'⟩
+end EuclideanFiveColor
+end
+
+noncomputable section
+open scoped Classical ENNReal
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+/-- Exception annuli for exact unit distances from the small closed disks. -/
+def diskAnnulus (d : MarkedDisk) : Set ℂ :=
+  Metric.closedBall d.center (1+d.radius) \ Metric.ball d.center (1-d.radius)
+def holeAnnuli (S : Finset MarkedDisk) : Set ℂ := ⋃d∈S,diskAnnulus d
+lemma diskAnnulus_measurable (d : MarkedDisk) : MeasurableSet (diskAnnulus d) :=
+  Metric.isClosed_closedBall.measurableSet.diff Metric.isOpen_ball.measurableSet
+lemma unit_from_disk_mem_annulus {d : MarkedDisk} {z q : ℂ}
+    (hq : q∈d.closedBall) (hzq : dist z q=1) : z∈diskAnnulus d := by
+  have hq' : dist q d.center≤d.radius := hq
+  constructor
+  · change dist z d.center≤1+d.radius
+    exact (dist_triangle z q d.center).trans (by linarith)
+  · change ¬ dist z d.center<1-d.radius
+    have hh := dist_triangle z d.center q
+    rw [dist_comm d.center q,hzq] at hh
+    linarith
+lemma volume_diskAnnulus {d : MarkedDisk} (h0 : 0≤d.radius) (h1 : d.radius≤1) :
+    volume (diskAnnulus d)=ENNReal.ofReal (4*Real.pi*d.radius) := by
+  have hsub : Metric.ball d.center (1-d.radius)⊆Metric.closedBall d.center (1+d.radius) :=
+    Metric.ball_subset_closedBall.trans (Metric.closedBall_subset_closedBall (by linarith))
+  have hvol (r : ℝ) (hr : 0≤r) : ENNReal.ofReal r ^ 2 * (NNReal.pi : ℝ≥0∞)=ENNReal.ofReal (r^2*Real.pi) := by
+    rw [ENNReal.ofReal_mul (sq_nonneg r),ENNReal.ofReal_pow hr]
+    congr 1
+    rw [←NNReal.coe_real_pi,ENNReal.ofReal_coe_nnreal]
+  rw [diskAnnulus,measure_sdiff hsub Metric.isOpen_ball.measurableSet.nullMeasurableSet (by finiteness),
+    Complex.volume_closedBall,Complex.volume_ball,hvol _ (by linarith),hvol _ (by linarith),
+    ←ENNReal.ofReal_sub _ (by positivity : 0≤(1-d.radius)^2*Real.pi)]
+  congr 1
+  ring
+
+lemma volume_holeAnnuli_le {S : Finset MarkedDisk}
+    (hS : ∀d∈S,0≤d.radius ∧ d.radius≤1) :
+    volume (holeAnnuli S)≤ENNReal.ofReal (4*Real.pi*(∑d∈S,d.radius)) := by
+  calc
+    volume (holeAnnuli S)≤∑d∈S,volume (diskAnnulus d) := measure_biUnion_finset_le S _
+    _ = ∑d∈S,ENNReal.ofReal (4*Real.pi*d.radius) := by
+      apply Finset.sum_congr rfl
+      intro d hd
+      exact volume_diskAnnulus (hS d hd).1 (hS d hd).2
+    _ = ENNReal.ofReal (∑d∈S,4*Real.pi*d.radius) := by
+      rw [ENNReal.ofReal_sum_of_nonneg (fun d hd => mul_nonneg (by positivity) (hS d hd).1)]
+    _ = ENNReal.ofReal (4*Real.pi*(∑d∈S,d.radius)) := by rw [Finset.mul_sum]
+lemma unit_from_hole_mem_annuli {S : Finset MarkedDisk} {z q : ℂ}
+    (hq : q∈diskHoles S) (hzq : dist z q=1) : z∈holeAnnuli S := by
+  obtain ⟨d,hd,hqd⟩ := mem_diskHoles.mp hq
+  exact mem_iUnion.mpr ⟨d,mem_iUnion.mpr ⟨hd,unit_from_disk_mem_annulus (Metric.ball_subset_closedBall hqd) hzq⟩⟩
+lemma holeAnnuli_tendsto_zero (S : ℕ → Finset MarkedDisk)
+    (hS : ∀n d,d∈S n → 0≤d.radius)
+    (hlim : Tendsto (fun n => ∑d∈S n,d.radius) atTop (𝓝 0)) :
+    Tendsto (fun n => volume (holeAnnuli (S n))) atTop (𝓝 0) := by
+  have hsmall : ∀ᶠ n in atTop,(∑d∈S n,d.radius)≤1 :=
+    hlim.eventually (eventually_le_nhds (by norm_num : (0:ℝ)<1))
+  have hbound : ∀ᶠ n in atTop,volume (holeAnnuli (S n))≤ENNReal.ofReal (4*Real.pi*(∑d∈S n,d.radius)) := by
+    filter_upwards [hsmall] with n hn
+    apply volume_holeAnnuli_le
+    exact fun d hd => ⟨hS n d hd,(MarkedDisk.radius_le_sum (hS n) hd).trans hn⟩
+  have ht : Tendsto (fun n => ENNReal.ofReal (4*Real.pi*(∑d∈S n,d.radius))) atTop (𝓝 0) := by
+    simpa [Function.comp_def] using ENNReal.continuous_ofReal.continuousAt.tendsto.comp (hlim.const_mul (4*Real.pi))
+  apply ENNReal.tendsto_nhds_zero.mpr
+  intro a ha
+  filter_upwards [hbound,ENNReal.tendsto_nhds_zero.mp ht a ha] with n hn hn'
+  exact hn.trans hn'
+end EuclideanFiveColor
+end
+
+noncomputable section
+open scoped Classical
+open Set Filter Topology MeasureTheory ContinuousMap
+namespace EuclideanFiveColor
+/-- Data actually constructed by the small-hole lemma. -/
+structure HoleApprox (c : ℂ → Fin 5) (α : ℝ) (L : Finset ℂ) (t : ℝ) where
+  ε : ℝ
+  ε_pos : 0<ε
+  ε_lt : ε<t
+  disks : Finset MarkedDisk
+  valid : ∀d∈disks,d.Valid (L : Set ℂ)
+  disjoint : (disks : Set MarkedDisk).Pairwise (fun a b => Disjoint a.closedBall b.closedBall)
+  sum_lt : (∑d∈disks,d.radius)<t
+  map : C(↥(Metric.closedBall (0:ℂ) 7 \ coreHoles disks),GraphSpace (⊤ : SimpleGraph (Fin 5)))
+  agrees : ∀z : ↥(Metric.closedBall (0:ℂ) 7 \ coreHoles disks),z.val∉diskHoles disks →
+    (map z).val=thresholdMap c ε α z.val
+lemma holeApprox_exists {c : ℂ → Fin 5} (hc : Measurable c) {α : ℝ}
+    (hα : 0<α) (hα' : α<1/5) (L : Finset ℂ)
+    (hL : ∀x∈Metric.closedBall (0:ℂ) 8,¬(transitionGraph c x).IsAcyclic → x∈L)
+    {t : ℝ} (ht : 0<t) (ht' : t<1/2) : Nonempty (HoleApprox c α L t) := by
+  obtain ⟨e,he,h⟩ := small_hole_map hc hα hα' L hL ht ht'
+  let ε := min (e/2) (t/2)
+  have hε : 0<ε := lt_min (by positivity) (by positivity)
+  have hεe : ε<e := lt_of_le_of_lt (min_le_left _ _) (by linarith)
+  have hεt : ε<t := lt_of_le_of_lt (min_le_right _ _) (by linarith)
+  obtain ⟨S,hS,hdis,hsum,g,hg⟩ := h ε hε hεe
+  exact ⟨⟨ε,hε,hεt,S,hS,hdis,hsum,g,hg⟩⟩
+namespace HoleApprox
+lemma cores_near {c : ℂ → Fin 5} {α t : ℝ} {L : Finset ℂ}
+    (a : HoleApprox c α L t) : coreHoles a.disks⊆⋃x∈L,Metric.ball x (2*t) := by
+  intro z hz
+  obtain ⟨d,hd,hcd,hzd⟩ := mem_coreHoles.mp hz
+  obtain ⟨x,hx,hsub⟩ := MarkedDisk.closedBall_near_core (a.valid d hd) hcd
+  have hrad := MarkedDisk.radius_le_sum (fun e he => (a.valid e he).1.le) hd
+  exact mem_iUnion.mpr ⟨x,mem_iUnion.mpr ⟨hx,
+    Metric.ball_subset_ball (by linarith [a.sum_lt]) (hsub (Metric.ball_subset_closedBall hzd))⟩⟩
+lemma ε_tendsto {c : ℂ → Fin 5} {α : ℝ} {L : Finset ℂ} {t : ℕ → ℝ}
+    (a : ∀n,HoleApprox c α L (t n)) (ht : Tendsto t atTop (𝓝 0)) :
+    Tendsto (fun n => (a n).ε) atTop (𝓝 0) :=
+  squeeze_zero (fun n => (a n).ε_pos.le) (fun n => (a n).ε_lt.le) ht
+lemma sum_tendsto {c : ℂ → Fin 5} {α : ℝ} {L : Finset ℂ} {t : ℕ → ℝ}
+    (a : ∀n,HoleApprox c α L (t n)) (ht : Tendsto t atTop (𝓝 0)) :
+    Tendsto (fun n => ∑d∈(a n).disks,d.radius) atTop (𝓝 0) :=
+  squeeze_zero (fun n => Finset.sum_nonneg fun d hd => ((a n).valid d hd).1.le)
+    (fun n => (a n).sum_lt.le) ht
+lemma annuli_tendsto {c : ℂ → Fin 5} {α : ℝ} {L : Finset ℂ} {t : ℕ → ℝ}
+    (a : ∀n,HoleApprox c α L (t n)) (ht : Tendsto t atTop (𝓝 0)) :
+    Tendsto (fun n => volume (holeAnnuli (a n).disks)) atTop (𝓝 0) :=
+  holeAnnuli_tendsto_zero _ (fun n d hd => ((a n).valid d hd).1.le) (sum_tendsto a ht)
+end HoleApprox
+end EuclideanFiveColor
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PlaneColoring/Interfaces.lean b/lean/OAI/Geometry/PlaneColoring/Interfaces.lean
new file mode 100644
index 0000000000000000000000000000000000000000..f3c7964fd9178a2d1bb39330f0bdb25380b1c078
--- /dev/null
+++ b/lean/OAI/Geometry/PlaneColoring/Interfaces.lean
@@ -0,0 +1,527 @@
+import OAI.Geometry.PlaneColoring.AngularBoundary
+
+namespace OAI
+
+noncomputable section
+open Set
+namespace EuclideanFiveColor
+/-- The four cyclic edges, with outsider label `4`. This definition is local to
+this interface argument and agrees extensionally with `fourEdge`. -/
+def interfaceFourEdge (i : Fin 4) : Finset (Fin 5) :=
+  if i=0 then {0,1} else if i=1 then {1,2} else if i=2 then {2,3} else {3,0}
+def interfaceFourAllowed (s : Fin 4 → Bool) (b : Bool) : Finset (Fin 5) :=
+  Finset.univ.filter fun j => ∀i, s i=b → j∉interfaceFourEdge i
+def InterfaceFourCompatible (s : Fin 4 → Bool) (p : Finset (Fin 5)) : Prop :=
+  p.Nonempty ∧ p.card≤2 ∧
+    (p∩interfaceFourAllowed s true).Nonempty ∧ (p∩interfaceFourAllowed s false).Nonempty
+def InterfaceFourSplit (s : Fin 4 → Bool) : Prop := s 0≠s 2 ∧ s 1≠s 3
+def interfaceFourDiagonal (s : Fin 4 → Bool) : Fin 3 := if s 0=s 1 then 2 else 1
+lemma interface_four_not_edge : ∀s : Fin 4 → Bool, ∀p : Finset (Fin 5),
+    InterfaceFourCompatible s p → ∀i, ¬p⊆interfaceFourEdge i := by
+  unfold InterfaceFourCompatible interfaceFourAllowed
+  decide
+lemma interface_four_split : ∀s : Fin 4 → Bool, ∀p : Finset (Fin 5),
+    InterfaceFourCompatible s p → 4∉p → InterfaceFourSplit s ∧
+      (if 4∈p then (0:Fin 3) else if 0∈p then 1 else 2)=interfaceFourDiagonal s := by
+  unfold InterfaceFourCompatible InterfaceFourSplit interfaceFourAllowed interfaceFourDiagonal
+  decide
+lemma interface_four_split_neg (s : Fin 4 → Bool) :
+    InterfaceFourSplit (fun i => !(s i)) ↔ InterfaceFourSplit s := by
+  simp [InterfaceFourSplit]
+lemma interface_four_diagonal_neg (s : Fin 4 → Bool) :
+    interfaceFourDiagonal (fun i => !(s i))=interfaceFourDiagonal s := by
+  simp [interfaceFourDiagonal]
+lemma interface_four_diagonal_ne_zero (s : Fin 4 → Bool) : interfaceFourDiagonal s≠0 := by
+  unfold interfaceFourDiagonal
+  split <;> decide
+lemma interface_four_split_no_opposite : ∀s : Fin 4 → Bool,
+    InterfaceFourSplit s → ∀i,¬(s i=true ∧ s (i+2)=true) := by
+  unfold InterfaceFourSplit
+  decide
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+open scoped RealInnerProductSpace
+namespace EuclideanFiveColor
+def fourInterfaceSign (v : Fin 4 → ℂ) (t : Angles) (i : Fin 4) : Bool :=
+  decide (0<⟪direction t,v i⟫)
+lemma interfaceFourEdge_eq (i : Fin 4) : interfaceFourEdge i=fourEdge i := rfl
+lemma fourInterfaceSign_opposite {v : Fin 4 → ℂ} {t : Angles}
+    (hn : ∀i,⟪direction t,v i⟫≠0) :
+    fourInterfaceSign v (t+(Real.pi : Angles))=fun i => !(fourInterfaceSign v t i) := by
+  funext i
+  simp only [fourInterfaceSign,direction_add_pi,inner_neg_left]
+  by_cases hi : 0<⟪direction t,v i⟫
+  · simp only [hi,show ¬0< -⟪direction t,v i⟫ by linarith,decide_true,decide_false,Bool.not_true]
+  · have hneg : ⟪direction t,v i⟫<0 := lt_of_le_of_ne (le_of_not_gt hi) (hn i)
+    simp only [hi,show 0< -⟪direction t,v i⟫ by linarith,decide_true,decide_false,Bool.not_false]
+variable {c : ℂ → Fin 5} (hc : Measurable c)
+lemma four_interface_compatible (hw : WeakColoring 5 c) {x : ℂ}
+    {K : Fin 4 → Set ℂ} {v : Fin 4 → ℂ} (hv : ∀i,‖v i‖=1)
+    (hK : ∀i,ApproachingContinuum x (K i) (v i))
+    (hH : ∀i,∀j∈fourEdge i,Disjoint (spatialSupport c j) (Straddle (K i))) :
+    ∀ᵐt∂angularMeasure,InterfaceFourCompatible (fourInterfaceSign v t) (palette c hc x t) := by
+  classical
+  let E (s : Fin 4 → Bool) (b : Bool) : Finset (Fin 4) := Finset.univ.filter fun i => s i=b
+  have hex := finite_support_exclusion_ae c hc K fourEdge hH
+  have hJ (s : Fin 4 → Bool) (b : Bool) :
+      ∀ᵐz : ℂ∂volume,(∀i∈E s b,z∈Straddle (K i)) → c z∈interfaceFourAllowed s b := by
+    filter_upwards [hex] with z hz hzE
+    simp only [interfaceFourAllowed,Finset.mem_filter,Finset.mem_univ,true_and]
+    intro i hi
+    exact hz i (hzE i (Finset.mem_filter.mpr ⟨Finset.mem_univ _,hi⟩))
+  have hout (s : Fin 4 → Bool) := continuum_outer_restriction c hc (E s true) hK (hJ s true)
+  have hin (s : Fin 4 → Bool) := continuum_inner_restriction c hc (E s false) hK (hJ s false)
+  filter_upwards [ae_all_iff.mpr hout,ae_all_iff.mpr hin,palette_nonempty c hc x,
+    palette_cardinality c hc hw x,ae_all_iff.mpr (fun i => ae_direction_nonorthogonal (hv i))]
+    with t ho hi hp hcard hn
+  refine ⟨hp,hcard.2,ho (fourInterfaceSign v t) ?_,hi (fourInterfaceSign v t) ?_⟩
+  · intro i hi
+    exact of_decide_eq_true (Finset.mem_filter.mp hi).2
+  · intro i hi
+    have hn' : ¬0<⟪direction t,v i⟫ := of_decide_eq_false (Finset.mem_filter.mp hi).2
+    exact lt_of_le_of_ne (le_of_not_gt hn') (hn i)
+lemma four_interface_admissible {x : ℂ} {v : Fin 4 → ℂ}
+    (h : ∀ᵐt∂angularMeasure,InterfaceFourCompatible (fourInterfaceSign v t) (palette c hc x t)) :
+    ∀ᵐt∂angularMeasure,FourAdmissible (palette c hc x t) := by
+  filter_upwards [h] with t ht
+  exact ⟨ht.1,ht.2.1,interface_four_not_edge _ _ ht⟩
+lemma four_state_group_of_admissible {x : ℂ}
+    (h : ∀ᵐt∂angularMeasure,FourAdmissible (palette c hc x t)) :
+    ∀ᵐt∂angularMeasure,fourGroup (fourState c hc x t)⊆palette c hc x t :=
+  h.mono fun _ ht => four_admissible_group _ ht
+lemma four_state_represents_of_group {x : ℂ}
+    (hg : ∀ᵐt∂angularMeasure,fourGroup (fourState c hc x t)⊆palette c hc x t)
+    (s : Fin 3) :
+    RealState hc x {t : ℝ | fourState c hc x (t : Angles)=s} (fourGroup s) := by
+  refine ⟨measurableSet_eq_fun ((fourState_measurable c hc x).comp angleMk_quasiMeasurePreserving.measurable) measurable_const,?_⟩
+  intro i hi
+  filter_upwards [angleMk_quasiMeasurePreserving.ae hg] with t ht hts
+  change fourState c hc x (t : Angles)=s at hts
+  exact (mem_palette c hc x (t : Angles) i).mp (ht (by simpa only [hts] using hi))
+lemma four_state_changes_of_group (hw : WeakColoring 5 c) {x : ℂ}
+    (hg : ∀ᵐt∂angularMeasure,fourGroup (fourState c hc x t)⊆palette c hc x t) :
+    ∀ᵐt∂angularMeasure,fourState c hc x t≠fourState c hc x (t+(sixty : Angles)) := by
+  filter_upwards [hg,(measurePreserving_add_right angularMeasure (sixty : Angles)).quasiMeasurePreserving.ae hg,
+    palette_sixty_disjoint c hc hw x] with t ht hu hd he
+  obtain ⟨i,hi⟩ := fourGroup_nonempty (fourState c hc x t)
+  exact Finset.disjoint_left.mp hd (ht hi) (hu (by simpa only [he] using hi))
+lemma four_state_diagonal_positive_of_group (hw : WeakColoring 5 c) {x : ℂ}
+    (hg : ∀ᵐt∂angularMeasure,fourGroup (fourState c hc x t)⊆palette c hc x t)
+    (s : Fin 3) (hs : s≠0) : 0<angularMeasure {t | fourState c hc x t=s} := by
+  by_contra! hn
+  have hnot : ∀ᵐt∂angularMeasure,fourState c hc x t≠s :=
+    measure_eq_zero_iff_ae_notMem.mp (le_antisymm hn zero_le)
+  let r : Fin 3 := if s=1 then 2 else 1
+  have hr : r≠0 ∧ r≠s ∧ ∀t : Fin 3,t≠0 → t≠s → t=r := by
+    exact (show ∀s : Fin 3,s≠0 → (if s=1 then (2:Fin 3) else 1)≠0 ∧ (if s=1 then (2:Fin 3) else 1)≠s ∧
+      ∀t : Fin 3,t≠0 → t≠s → t=(if s=1 then 2 else 1) from by decide) s hs
+  let f : Angles → Bool := fun t => if fourState c hc x t=0 then false else true
+  have hf : Measurable f := Measurable.ite
+    (measurableSet_eq_fun (fourState_measurable c hc x) measurable_const) measurable_const measurable_const
+  have hrep (v : Bool) : RealState hc x {t : ℝ | f (t : Angles)=v} (if v then fourGroup r else fourGroup 0) := by
+    refine ⟨measurableSet_eq_fun (hf.comp angleMk_quasiMeasurePreserving.measurable) measurable_const,?_⟩
+    intro i hi
+    filter_upwards [angleMk_quasiMeasurePreserving.ae hg,angleMk_quasiMeasurePreserving.ae hnot]
+      with t ht htn htv
+    change f (t : Angles)=v at htv
+    apply (mem_palette c hc x (t : Angles) i).mp
+    apply ht
+    cases v
+    · have hz : fourState c hc x (t : Angles)=0 := by simpa [f] using htv
+      simpa only [hz,Bool.false_eq_true,ite_false] using hi
+    · have hz : fourState c hc x (t : Angles)=r := hr.2.2 _ (by simpa [f] using htv) htn
+      simpa only [hz,ite_true] using hi
+  have hcard : (Finset.univ\(fourGroup 0∪fourGroup r)).card≤2 := by
+    exact (show ∀s : Fin 3,s≠0 → (Finset.univ\(fourGroup 0∪fourGroup (if s=1 then 2 else 1))).card≤2 from by decide) s hs
+  apply two_state_impossible hc hw x (fourGroup 0) (fourGroup r) hcard f hf (hrep false) (hrep true)
+  filter_upwards [hnot,(measurePreserving_add_right angularMeasure (sixty : Angles)).quasiMeasurePreserving.ae hnot,
+      four_state_changes_of_group hc hw hg] with t ht hu hne
+  exact three_exclude_flip _ _ s hs ht hu hne
+lemma four_interface_state_rule {x : ℂ} {v : Fin 4 → ℂ}
+    (h : ∀ᵐt∂angularMeasure,InterfaceFourCompatible (fourInterfaceSign v t) (palette c hc x t)) :
+    ∀ᵐt∂angularMeasure,fourState c hc x t≠0 →
+      InterfaceFourSplit (fourInterfaceSign v t) ∧ fourState c hc x t=interfaceFourDiagonal (fourInterfaceSign v t) := by
+  filter_upwards [h] with t ht hn
+  have ho : 4∉palette c hc x t := by
+    intro ho
+    exact hn (by simp only [fourState,fourCanonical,ho,ite_true])
+  exact interface_four_split _ _ ht ho
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+lemma angle_state_preimage_positive {A : Set Angles} (hA : MeasurableSet A) (hp : 0<angularMeasure A) :
+    0<volume (((↑) : ℝ → Angles) ⁻¹' A) := by
+  by_contra! hn
+  have hz := measure_eq_zero_iff_ae_notMem.mp (le_antisymm hn zero_le)
+  have ha : ∀ᵐt∂angularMeasure,t∉A := ae_angular_of_ae_line hA.compl hz
+  exact hp.ne' (measure_eq_zero_iff_ae_notMem.mpr ha)
+/-- Every point of an angular essential support has a real essential-support lift.
+The compact fundamental domain is used only to establish the support inclusion. -/
+lemma angle_support_lift {A : Set Angles} (hA : MeasurableSet A) {t : Angles}
+    (ht : t∈(angularMeasure.restrict A).support) :
+    ∃b : ℝ,(b : Angles)=t ∧ b∈(volume.restrict (((↑) : ℝ → Angles) ⁻¹' A)).support := by
+  let B : Set ℝ := ((↑) : ℝ → Angles) ⁻¹' A
+  let S : Set ℝ := (volume.restrict B).support∩Icc 0 (2*Real.pi)
+  let F : Set Angles := ((↑) : ℝ → Angles) '' S
+  have hm : Continuous ((↑) : ℝ → Angles) := AddCircle.continuous_mk' _
+  have hS : IsCompact S := isCompact_Icc.inter_left Measure.isClosed_support
+  have hF : IsClosed F := (hS.image hm).isClosed
+  have ha : ∀ᵐt∂((volume : Measure Angles).restrict A),t∈F := by
+    rw [← (AddCircle.measurePreserving_mk (2*Real.pi) 0).map_eq,
+      Measure.restrict_map hm.measurable hA]
+    apply (ae_map_iff hm.measurable.aemeasurable hF.measurableSet).mpr
+    have hs : ∀ᵐb∂((volume.restrict (Ioc 0 (0+2*Real.pi))).restrict B),b∈(volume.restrict B).support := by
+      apply ae_mono (Measure.restrict_mono_measure Measure.restrict_le_self B)
+      exact Measure.support_mem_ae
+    have hi : ∀ᵐb∂((volume.restrict (Ioc 0 (0+2*Real.pi))).restrict B),b∈Ioc 0 (0+2*Real.pi) :=
+      ae_restrict_of_ae (ae_restrict_mem measurableSet_Ioc)
+    filter_upwards [hs,hi] with b hb hib
+    exact ⟨b,⟨hb,hib.1.le,by simpa only [zero_add] using hib.2⟩,rfl⟩
+  have ha' : ∀ᵐt∂(angularMeasure.restrict A),t∈F := by
+    rw [AddCircle.volume_eq_smul_haarAddCircle,Measure.restrict_smul] at ha
+    exact (Measure.ae_ennreal_smul_measure_iff (by positivity : ENNReal.ofReal (2*Real.pi)≠0)).mp ha
+  obtain ⟨b,hb,hbt⟩ := Measure.support_subset_of_isClosed hF ha' ht
+  exact ⟨b,hbt,hb.1⟩
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+open scoped RealInnerProductSpace
+namespace EuclideanFiveColor
+lemma direction_inner_real (t : ℝ) (v : ℂ) :
+    ⟪direction (t : Angles),v⟫=Real.cos t*v.re+Real.sin t*v.im := by
+  rw [Complex.inner,direction_real,Circle.coe_exp,Complex.exp_ofReal_mul_I]
+  simp only [Complex.mul_re,Complex.conj_re,Complex.conj_im,Complex.add_re,
+    Complex.ofReal_re,Complex.ofReal_im,Complex.I_re,Complex.I_im,Complex.add_im,Complex.mul_im]
+  ring
+lemma three_direction_moment (t : Angles) (v w : ℂ) :
+    ⟪direction t,v⟫*⟪direction t,w⟫+
+      ⟪direction (t+(sixty : Angles)),v⟫*⟪direction (t+(sixty : Angles)),w⟫+
+      ⟪direction (t+2•(sixty : Angles)),v⟫*⟪direction (t+2•(sixty : Angles)),w⟫=
+      (3/2:ℝ)*⟪v,w⟫ := by
+  obtain ⟨a,rfl⟩ := QuotientAddGroup.mk_surjective t
+  rw [← AddCircle.coe_nsmul,← AddCircle.coe_add,← AddCircle.coe_add]
+  simp only [direction_inner_real,nsmul_eq_mul,sixty,Real.cos_add,Real.sin_add,
+    Real.cos_pi_div_three,Real.sin_pi_div_three]
+  norm_num only [Nat.cast_ofNat]
+  rw [show 2*(Real.pi/3)=Real.pi/3+Real.pi/3 by ring]
+  simp only [Real.cos_add,Real.sin_add,Real.cos_pi_div_three,Real.sin_pi_div_three,
+    Complex.inner,Complex.mul_re,Complex.conj_re,Complex.conj_im]
+  have hs : Real.sqrt 3^2=3 := Real.sq_sqrt (by norm_num)
+  have hc := Real.sin_sq_add_cos_sq a
+  ring_nf
+  have hs3 : Real.sqrt 3^3=3*Real.sqrt 3 := by rw [pow_succ,hs]
+  have hs4 : Real.sqrt 3^4=9 := by rw [show 4=2*2 from rfl,pow_mul,hs]; norm_num
+  simp only [hs,hs3,hs4]
+  linear_combination (3/2:ℝ)*(v.re*w.re+v.im*w.im)*hc
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+open scoped RealInnerProductSpace
+namespace EuclideanFiveColor
+lemma direction_sixty_unit (t : Angles) : ‖direction t-direction (t+(sixty : Angles))‖=1 := by
+  obtain ⟨a,rfl⟩ := QuotientAddGroup.mk_surjective t
+  rw [← AddCircle.coe_add,direction_real,direction_real]
+  have he : (Circle.exp a : ℂ)-(Circle.exp (a+sixty) : ℂ)=(Circle.exp (a-sixty) : ℂ) := by
+    simp only [Circle.coe_exp,Complex.exp_ofReal_mul_I]
+    apply Complex.ext <;> simp only [Complex.sub_re,Complex.sub_im,Complex.add_re,Complex.add_im,
+      Complex.mul_re,Complex.mul_im,Complex.ofReal_re,Complex.ofReal_im,Complex.I_re,Complex.I_im,
+      Real.cos_add,Real.cos_sub,Real.sin_add,Real.sin_sub,sixty,Real.cos_pi_div_three] <;> ring
+  rw [he]
+  exact Circle.norm_coe _
+lemma compact_no_unit_margin {F : Set Angles} (hF : IsCompact F)
+    (hu : ∀t∈F,∀u∈F,‖direction t-direction u‖≠1) :
+    ∃ε : ℝ,0<ε ∧ ∀t∈F,∀u,dist t u<ε → u+(sixty : Angles)∉F ∧ u-(sixty : Angles)∉F := by
+  let T : Set Angles := {u | u+(sixty : Angles)∉F ∧ u-(sixty : Angles)∉F}
+  have ho : IsOpen T := (hF.isClosed.isOpen_compl.preimage (continuous_id.add_const _)).inter
+    (hF.isClosed.isOpen_compl.preimage (continuous_id.sub continuous_const))
+  have hs : F⊆T := by
+    intro t ht
+    refine ⟨fun hh => hu t ht _ hh (direction_sixty_unit t),fun hh => ?_⟩
+    apply hu (t-(sixty : Angles)) hh t ht
+    simpa only [sub_add_cancel] using direction_sixty_unit (t-(sixty : Angles))
+  obtain ⟨ε,hε,hsub⟩ := hF.exists_thickening_subset_open ho hs
+  exact ⟨ε,hε,fun t ht u htu => hsub (Metric.mem_thickening_iff.mpr ⟨t,ht,by simpa only [dist_comm] using htu⟩)⟩
+
+variable {c : ℂ → Fin 5} (hc : Measurable c)
+def interfaceStateSet (x : ℂ) (s : Fin 3) : Set Angles := {t | fourState c hc x t=s}
+def interfaceStateSupport (x : ℂ) (s : Fin 3) : Set Angles :=
+  (angularMeasure.restrict (interfaceStateSet hc x s)).support
+lemma interfaceStateSet_measurable (x : ℂ) (s : Fin 3) : MeasurableSet (interfaceStateSet hc x s) :=
+  measurableSet_eq_fun (fourState_measurable c hc x) measurable_const
+lemma interfaceStateSupport_ae (x : ℂ) :
+    ∀ᵐt∂angularMeasure,∀s : Fin 3,fourState c hc x t=s → t∈interfaceStateSupport hc x s := by
+  apply ae_all_iff.mpr
+  intro s
+  exact (ae_restrict_iff' (interfaceStateSet_measurable hc x s)).mp Measure.support_mem_ae
+lemma interfaceStateSupport_no_unit (hw : WeakColoring 5 c) {x : ℂ}
+    (hg : ∀ᵐt∂angularMeasure,fourGroup (fourState c hc x t)⊆palette c hc x t)
+    {s : Fin 3} (hs : s≠0) {t u : Angles}
+    (ht : t∈interfaceStateSupport hc x s) (hu : u∈interfaceStateSupport hc x s) :
+    ‖direction t-direction u‖≠1 := by
+  let r : Fin 3 := if s=1 then 2 else 1
+  have hr : r≠0 := by dsimp [r]; split <;> decide
+  have hl : ∀i : Fin 5,i∉fourGroup r∪fourGroup s ↔ i=4 := by
+    exact (show ∀s : Fin 3,s≠0 → ∀i : Fin 5,i∉fourGroup (if s=1 then 2 else 1)∪fourGroup s ↔ i=4 from by decide) s hs
+  obtain ⟨b,rfl,hb⟩ := angle_support_lift (interfaceStateSet_measurable hc x s) ht
+  obtain ⟨d,rfl,hd⟩ := angle_support_lift (interfaceStateSet_measurable hc x s) hu
+  exact strict_state_no_unit_support hc hw hl (four_state_represents_of_group hc hg r)
+    (four_state_represents_of_group hc hg s)
+    (angle_state_preimage_positive (interfaceStateSet_measurable hc x r)
+      (four_state_diagonal_positive_of_group hc hw hg r hr)) hb hd
+lemma interfaceStateSupport_margin (hw : WeakColoring 5 c) {x : ℂ}
+    (hg : ∀ᵐt∂angularMeasure,fourGroup (fourState c hc x t)⊆palette c hc x t) :
+    ∃ε : ℝ,0<ε ∧ ∀s : Fin 3,s≠0 → ∀t∈interfaceStateSupport hc x s,∀u,dist t u<ε →
+      u+(sixty : Angles)∉interfaceStateSupport hc x s ∧ u-(sixty : Angles)∉interfaceStateSupport hc x s := by
+  have h (s : Fin 3) (hs : s≠0) := compact_no_unit_margin
+    (show IsCompact (interfaceStateSupport hc x s) from Measure.isClosed_support.isCompact)
+    (fun t ht u hu => interfaceStateSupport_no_unit hc hw hg hs ht hu)
+  obtain ⟨ε₁,hε₁,h₁⟩ := h 1 (by decide)
+  obtain ⟨ε₂,hε₂,h₂⟩ := h 2 (by decide)
+  refine ⟨min ε₁ ε₂,lt_min hε₁ hε₂,?_⟩
+  intro s hs t ht u htu
+  fin_cases s
+  · exact (hs rfl).elim
+  · exact h₁ t ht u (htu.trans_le (min_le_left _ _))
+  · exact h₂ t ht u (htu.trans_le (min_le_right _ _))
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+open scoped RealInnerProductSpace
+namespace EuclideanFiveColor
+lemma four_diagonal_product_neg {v : Fin 4 → ℂ} {t : Angles}
+    (hn : ∀i,⟪direction t,v i⟫≠0) (hd : interfaceFourDiagonal (fourInterfaceSign v t)=1) :
+    ⟪direction t,v 0⟫*⟪direction t,v 1⟫<0 := by
+  by_cases h0 : 0<⟪direction t,v 0⟫ <;> by_cases h1 : 0<⟪direction t,v 1⟫
+  · simp only [interfaceFourDiagonal,fourInterfaceSign,h0,h1] at hd
+    norm_num at hd
+  · exact mul_neg_of_pos_of_neg h0 (lt_of_le_of_ne (le_of_not_gt h1) (hn 1))
+  · exact mul_neg_of_neg_of_pos (lt_of_le_of_ne (le_of_not_gt h0) (hn 0)) h1
+  · simp only [interfaceFourDiagonal,fourInterfaceSign,h0,h1] at hd
+    norm_num at hd
+lemma four_diagonal_product_pos {v : Fin 4 → ℂ} {t : Angles}
+    (hn : ∀i,⟪direction t,v i⟫≠0) (hd : interfaceFourDiagonal (fourInterfaceSign v t)=2) :
+    0<⟪direction t,v 0⟫*⟪direction t,v 1⟫ := by
+  by_cases h0 : 0<⟪direction t,v 0⟫ <;> by_cases h1 : 0<⟪direction t,v 1⟫
+  · exact mul_pos h0 h1
+  · simp only [interfaceFourDiagonal,fourInterfaceSign,h0,h1,decide_true,decide_false] at hd
+    norm_num at hd
+  · simp only [interfaceFourDiagonal,fourInterfaceSign,h0,h1,decide_true,decide_false] at hd
+    norm_num at hd
+  · exact mul_pos_of_neg_of_neg (lt_of_le_of_ne (le_of_not_gt h0) (hn 0))
+      (lt_of_le_of_ne (le_of_not_gt h1) (hn 1))
+lemma four_diagonal_orbit_half {v : Fin 4 → ℂ} {t : Angles} {j : Fin 6}
+    (hn : ∀i,⟪direction (t+sixAngle j),v i⟫≠0) :
+    interfaceFourDiagonal (fourInterfaceSign v (t+sixAngle (j+3)))=
+      interfaceFourDiagonal (fourInterfaceSign v (t+sixAngle j)) := by
+  rw [sixAngle_half,← add_assoc,fourInterfaceSign_opposite hn,interface_four_diagonal_neg]
+lemma four_diagonal_uniform_of_parity {v : Fin 4 → ℂ} {t : Angles} {a : Fin 3} {p : Fin 2}
+    (hn : ∀j i,⟪direction (t+sixAngle j),v i⟫≠0)
+    (hd : ∀j : Fin 6,j.val%2≠p.val → interfaceFourDiagonal (fourInterfaceSign v (t+sixAngle j))=a) :
+    ∀j : Fin 6,interfaceFourDiagonal (fourInterfaceSign v (t+sixAngle j))=a := by
+  intro j
+  by_cases hj : j.val%2≠p.val
+  · exact hd j hj
+  have hh : (j+3).val%2≠p.val := by
+    have h : ∀p : Fin 2,∀j : Fin 6,¬j.val%2≠p.val → (j+3).val%2≠p.val := by decide
+    exact h p j hj
+  rw [← four_diagonal_orbit_half (hn j)]
+  exact hd (j+3) hh
+lemma four_uniform_diagonal_one {v : Fin 4 → ℂ} {t : Angles}
+    (hn : ∀j i,⟪direction (t+sixAngle j),v i⟫≠0)
+    (hd : ∀j : Fin 6,interfaceFourDiagonal (fourInterfaceSign v (t+sixAngle j))=1) :
+    ⟪v 0,v 1⟫<0 := by
+  have h0 := four_diagonal_product_neg (hn 0) (hd 0)
+  have h1 := four_diagonal_product_neg (hn 1) (hd 1)
+  have h2 := four_diagonal_product_neg (hn 2) (hd 2)
+  norm_num only [sixAngle,Fin.val_zero,Fin.val_one,Fin.val_two,zero_nsmul,one_nsmul,add_zero] at h0 h1 h2
+  have hm := three_direction_moment t (v 0) (v 1)
+  linarith
+lemma four_uniform_diagonal_two {v : Fin 4 → ℂ} {t : Angles}
+    (hn : ∀j i,⟪direction (t+sixAngle j),v i⟫≠0)
+    (hd : ∀j : Fin 6,interfaceFourDiagonal (fourInterfaceSign v (t+sixAngle j))=2) :
+    0<⟪v 0,v 1⟫ := by
+  have h0 := four_diagonal_product_pos (hn 0) (hd 0)
+  have h1 := four_diagonal_product_pos (hn 1) (hd 1)
+  have h2 := four_diagonal_product_pos (hn 2) (hd 2)
+  norm_num only [sixAngle,Fin.val_zero,Fin.val_one,Fin.val_two,zero_nsmul,one_nsmul,add_zero] at h0 h1 h2
+  have hm := three_direction_moment t (v 0) (v 1)
+  linarith
+/-- The three-direction quadratic moment replaces the source's projective arc
+length argument. No assumption about antipodal ray pairs or the collinear case
+is required: two opposite uniform diagonal triples are already contradictory. -/
+lemma four_opposite_uniform_diagonals_false {v : Fin 4 → ℂ} {t u : Angles}
+    (hnt : ∀j i,⟪direction (t+sixAngle j),v i⟫≠0)
+    (hnu : ∀j i,⟪direction (u+sixAngle j),v i⟫≠0)
+    {a b : Fin 3} (ha : a≠0) (hb : b≠0) (hab : a≠b)
+    (hdt : ∀j : Fin 6,interfaceFourDiagonal (fourInterfaceSign v (t+sixAngle j))=a)
+    (hdu : ∀j : Fin 6,interfaceFourDiagonal (fourInterfaceSign v (u+sixAngle j))=b) : False := by
+  have hab' : (a=1 ∧ b=2) ∨ (a=2 ∧ b=1) :=
+    (show ∀a b : Fin 3,a≠0 → b≠0 → a≠b → (a=1 ∧ b=2) ∨ (a=2 ∧ b=1) from by decide) a b ha hb hab
+  rcases hab' with ⟨rfl,rfl⟩|⟨rfl,rfl⟩
+  · exact (four_uniform_diagonal_one hnt hdt).not_gt (four_uniform_diagonal_two hnu hdu)
+  · exact (four_uniform_diagonal_one hnu hdu).not_gt (four_uniform_diagonal_two hnt hdt)
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+open scoped RealInnerProductSpace
+namespace EuclideanFiveColor
+lemma sixAngle_prev (j : Fin 6) : sixAngle (j-1)=sixAngle j-(sixty : Angles) := by
+  have h := sixAngle_step (j-1)
+  simpa only [sub_add_cancel] using eq_sub_iff_add_eq.mpr h.symm
+variable {c : ℂ → Fin 5} (hc : Measurable c)
+def fourInterfaceWord (x : ℂ) (t : Angles) (j : Fin 6) : Fin 3 :=
+  fourState c hc x (t+sixAngle j)
+lemma fourInterfaceWord_shift (x : ℂ) (t : Angles) (j : Fin 6) :
+    fourInterfaceWord hc x (t+(sixty : Angles)) j=fourInterfaceWord hc x t (j+1) := by
+  simp only [fourInterfaceWord,sixAngle_step]
+  congr 1
+  abel
+/-- Construction of the dense conull shift-invariant set and its actual
+six-position words, from the previously established genuine state system. -/
+lemma four_interface_word_data (hw : WeakColoring 5 c) {x : ℂ} {v : Fin 4 → ℂ}
+    (hv : ∀i,‖v i‖=1)
+    (hcompat : ∀ᵐt∂angularMeasure,InterfaceFourCompatible (fourInterfaceSign v t) (palette c hc x t)) :
+    ∃G : Set Angles,Dense G ∧ (∀t∈G,t+(sixty : Angles)∈G) ∧
+      (∀t∈G,∀j i,⟪direction (t+sixAngle j),v i⟫≠0) ∧
+      (∀t∈G,∀j,fourInterfaceWord hc x t j≠0 → fourInterfaceWord hc x t j=
+        interfaceFourDiagonal (fourInterfaceSign v (t+sixAngle j))) ∧
+      (∀t∈G,FourWordValid (fourInterfaceWord hc x t)) ∧
+      ∃ε : ℝ,0<ε ∧ ∀t∈G,∀u∈G,dist t u<ε →
+        FourWordCross (fourInterfaceWord hc x t) (fourInterfaceWord hc x u) := by
+  have hg := four_state_group_of_admissible hc (four_interface_admissible hc hcompat)
+  obtain ⟨ε,hε,hmargin⟩ := interfaceStateSupport_margin hc hw hg
+  let Q (t : Angles) : Prop := (∀i,⟪direction t,v i⟫≠0) ∧
+    (fourState c hc x t≠0 → fourState c hc x t=interfaceFourDiagonal (fourInterfaceSign v t)) ∧
+    fourState c hc x t≠fourState c hc x (t+(sixty : Angles)) ∧
+    ∀s : Fin 3,fourState c hc x t=s → t∈interfaceStateSupport hc x s
+  have hQ : ∀ᵐt∂angularMeasure,Q t := by
+    filter_upwards [ae_all_iff.mpr (fun i => ae_direction_nonorthogonal (hv i)),
+      four_interface_state_rule hc hcompat,four_state_changes_of_group hc hw hg,
+      interfaceStateSupport_ae hc x] with t hn hr hs hF
+    exact ⟨hn,fun h => (hr h).2,hs,hF⟩
+  let G : Set Angles := {t | ∀j : Fin 6,Q (t+sixAngle j)}
+  have hGae : ∀ᵐt∂angularMeasure,t∈G := ae_all_iff.mpr fun j =>
+    (measurePreserving_add_right angularMeasure (sixAngle j)).quasiMeasurePreserving.ae hQ
+  have hGshift : ∀t∈G,t+(sixty : Angles)∈G := by
+    intro t ht j
+    have he : t+(sixty : Angles)+sixAngle j=t+sixAngle (j+1) := by
+      rw [sixAngle_step]
+      abel
+    rw [he]
+    exact ht (j+1)
+  refine ⟨G,Measure.dense_of_ae hGae,hGshift,fun t ht j => (ht j).1,fun t ht j => (ht j).2.1,?_,ε,hε,?_⟩
+  · intro t ht
+    refine ⟨?_,?_⟩
+    · intro j
+      simpa only [fourInterfaceWord,sixAngle_step,← add_assoc] using (ht j).2.2.1
+    · intro j hj hj3
+      dsimp only [fourInterfaceWord]
+      rw [(ht (j+3)).2.1 hj3,(ht j).2.1 hj]
+      exact four_diagonal_orbit_half (ht j).1
+  · intro t ht u hu htu j hj
+    have hm := hmargin (fourInterfaceWord hc x t j) hj (t+sixAngle j)
+      ((ht j).2.2.2 _ rfl) (u+sixAngle j) (by simpa only [dist_add_right] using htu)
+    constructor
+    · intro he
+      apply hm.1
+      have hh := (hu (j+1)).2.2.2 _ he
+      simpa only [sixAngle_step,← add_assoc] using hh
+    · intro he
+      apply hm.2
+      have hh := (hu (j-1)).2.2.2 _ he
+      simpa only [sixAngle_prev,add_sub_assoc] using hh
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+open scoped RealInnerProductSpace
+namespace EuclideanFiveColor
+lemma four_word_opposite_geometry_false {v : Fin 4 → ℂ} {t u : Angles}
+    {w z : Fin 6 → Fin 3} (hw : FourWordValid w) (hz : FourWordValid z)
+    (hzero : ∀j,w j=0 → j.val%2=0) (hzero' : ∀j,z j=0 → j.val%2=1)
+    (hcross : FourWordCross w z)
+    (hnt : ∀j i,⟪direction (t+sixAngle j),v i⟫≠0)
+    (hnu : ∀j i,⟪direction (u+sixAngle j),v i⟫≠0)
+    (hdt : ∀j,w j≠0 → w j=interfaceFourDiagonal (fourInterfaceSign v (t+sixAngle j)))
+    (hdu : ∀j,z j≠0 → z j=interfaceFourDiagonal (fourInterfaceSign v (u+sixAngle j))) : False := by
+  obtain ⟨a,b,ha,hb,hab,hwa,hzb⟩ := four_word_opposite_parity w hw hzero z hz hzero' hcross
+  apply four_opposite_uniform_diagonals_false hnt hnu ha hb hab
+  · apply four_diagonal_uniform_of_parity (p:=0) hnt
+    intro j hj
+    have hh := hwa j (by norm_num only [Fin.val_zero] at hj; omega)
+    exact (hdt j (fun h => ha (hh.symm.trans h))).symm.trans hh
+  · apply four_diagonal_uniform_of_parity (p:=1) hnu
+    intro j hj
+    have hh := hzb j (by norm_num only [Fin.val_one] at hj; omega)
+    exact (hdu j (fun h => hb (hh.symm.trans h))).symm.trans hh
+
+lemma no_four_word_geometry (v : Fin 4 → ℂ) (G : Set Angles) (hG : Dense G)
+    (w : Angles → Fin 6 → Fin 3) (hw : ∀t∈G,FourWordValid (w t))
+    (hshiftG : ∀t∈G,t+(sixty : Angles)∈G)
+    (hshift : ∀t∈G,∀j,w (t+(sixty : Angles)) j=w t (j+1))
+    (hn : ∀t∈G,∀j i,⟪direction (t+sixAngle j),v i⟫≠0)
+    (hd : ∀t∈G,∀j,w t j≠0 → w t j=interfaceFourDiagonal (fourInterfaceSign v (t+sixAngle j)))
+    {ε : ℝ} (hε : 0<ε)
+    (hcross : ∀t∈G,∀u∈G,dist t u<ε → FourWordCross (w t) (w u)) : False := by
+  classical
+  have hnomix := four_words_no_mixed G hG w hw hshiftG hshift hε hcross
+  have hpar (t : Angles) (ht : t∈G) := four_word_parity (w t) (hw t ht) (hnomix t ht)
+  let p : Angles → Fin 2 := fun t => if ht : t∈G then (hpar t ht).choose else 0
+  have hp (t : Angles) (ht : t∈G) : ∀j,w t j=0 → j.val%2=(p t).val := by
+    simpa only [p,dite_eq_left ht] using (hpar t ht).choose_spec
+  have hclose (t : Angles) (ht : t∈G) (u : Angles) (hu : u∈G) (htu : dist t u<ε) : p t=p u := by
+    by_contra hne
+    have hcases : (p t=0 ∧ p u=1) ∨ (p t=1 ∧ p u=0) :=
+      (show ∀a b : Fin 2,a≠b → (a=0 ∧ b=1) ∨ (a=1 ∧ b=0) from by decide) _ _ hne
+    rcases hcases with ⟨hpt,hpu⟩|⟨hpt,hpu⟩
+    · exact four_word_opposite_geometry_false (hw t ht) (hw u hu)
+        (by simpa only [hpt,Fin.val_zero] using hp t ht)
+        (by simpa only [hpu,Fin.val_one] using hp u hu)
+        (hcross t ht u hu htu) (hn t ht) (hn u hu) (hd t ht) (hd u hu)
+    · exact four_word_opposite_geometry_false (hw u hu) (hw t ht)
+        (by simpa only [hpu,Fin.val_zero] using hp u hu)
+        (by simpa only [hpt,Fin.val_one] using hp t ht)
+        (hcross u hu t ht (by simpa only [dist_comm] using htu))
+        (hn u hu) (hn t ht) (hd u hu) (hd t ht)
+  obtain ⟨t,ht⟩ := hG.nonempty
+  have hconst : ∀u∈G,p u=p t := by
+    apply dense_isolated_value_constant hG p hε ⟨t,ht,rfl⟩
+    intro u hu hut z hz huz
+    exact (hclose u hu z hz huz).symm.trans hut
+  obtain ⟨j,hj⟩ := four_word_has_outsider (w (t+(sixty : Angles))) (hw _ (hshiftG t ht))
+  have hh := four_word_shift_parity (w t) (p t) (hp t ht) j
+    (by simpa only [hshift t ht j] using hj)
+  have hh' := hp _ (hshiftG t ht) j hj
+  rw [hconst _ (hshiftG t ht)] at hh'
+  have he : (p t).val=(p t+1).val := hh'.symm.trans hh
+  exact (show ∀a : Fin 2,a.val≠(a+1).val from by decide) (p t) he
+
+/-- The actual length-four interface cycle is impossible. All hypotheses are
+exactly approach directions and common straddling exclusion of both endpoints. -/
+lemma no_four_interfaces {c : ℂ → Fin 5} (hc : Measurable c) (hw : WeakColoring 5 c)
+    {x : ℂ} {K : Fin 4 → Set ℂ} {v : Fin 4 → ℂ} (hv : ∀i,‖v i‖=1)
+    (hK : ∀i,ApproachingContinuum x (K i) (v i))
+    (hH : ∀i,∀j∈fourEdge i,Disjoint (spatialSupport c j) (Straddle (K i))) : False := by
+  obtain ⟨G,hG,hshiftG,hn,hd,hw',ε,hε,hcross⟩ := four_interface_word_data hc hw hv
+    (four_interface_compatible hc hw hv hK hH)
+  exact no_four_word_geometry v G hG (fourInterfaceWord hc x) hw' hshiftG
+    (fun t _ j => fourInterfaceWord_shift hc x t j) hn hd hε hcross
+end EuclideanFiveColor
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PlaneColoring/Lattice.lean b/lean/OAI/Geometry/PlaneColoring/Lattice.lean
new file mode 100644
index 0000000000000000000000000000000000000000..bbebe2a89c619158aa93f14d79fec556a9297431
--- /dev/null
+++ b/lean/OAI/Geometry/PlaneColoring/Lattice.lean
@@ -0,0 +1,66 @@
+import Mathlib
+
+namespace OAI
+
+/-! Integer arithmetic for the classical seven-color triangular lattice. -/
+
+namespace Problem160
+namespace SevenLattice
+
+/-- Squared Euclidean norm in a triangular lattice of unit spacing. -/
+def normSq (a b : ℤ) : ℤ := a ^ 2 + a * b + b ^ 2
+
+/-- The index-seven coloring of triangular-lattice sites. -/
+def color (a b : ℤ) : ZMod 7 := (a : ZMod 7) + 3 * (b : ZMod 7)
+
+lemma normSq_nonneg (a b : ℤ) : 0 ≤ normSq a b := by
+  unfold normSq
+  nlinarith [sq_nonneg (2 * a + b), sq_nonneg b]
+
+lemma normSq_pos {a b : ℤ} (h : a ≠ 0 ∨ b ≠ 0) : 0 < normSq a b := by
+  unfold normSq
+  rcases h with ha | hb
+  · have ha' : 0 < a ^ 2 := sq_pos_of_ne_zero ha
+    nlinarith [sq_nonneg (a + 2 * b)]
+  · have hb' : 0 < b ^ 2 := sq_pos_of_ne_zero hb
+    nlinarith [sq_nonneg (2 * a + b)]
+
+lemma seven_dvd_normSq {a b : ℤ} (h : 7 ∣ a + 3 * b) : 7 ∣ normSq a b := by
+  have hi : normSq a b = (a + 3 * b) * (a - 2 * b) + 7 * b ^ 2 := by
+    unfold normSq
+    ring
+  rw [hi]
+  exact dvd_add (dvd_mul_of_dvd_left h _) (dvd_mul_right _ _)
+
+lemma normSq_ge_seven {a b : ℤ} (h : 7 ∣ a + 3 * b)
+    (hn : a ≠ 0 ∨ b ≠ 0) : 7 ≤ normSq a b := by
+  exact Int.le_of_dvd (normSq_pos hn) (seven_dvd_normSq h)
+
+lemma color_eq_iff_dvd {a b c d : ℤ} :
+    color a b = color c d ↔ 7 ∣ (a - c) + 3 * (b - d) := by
+  have hd := ZMod.intCast_zmod_eq_zero_iff_dvd ((a - c) + 3 * (b - d)) 7
+  norm_num only [Nat.cast_ofNat] at hd
+  rw [← hd]
+  change color a b = color c d ↔ (((a - c) + 3 * (b - d) : ℤ) : ZMod 7) = 0
+  simp only [Int.cast_add, Int.cast_sub, Int.cast_mul, Int.cast_ofNat]
+  unfold color
+  constructor <;> intro h <;> linear_combination h
+
+lemma same_color_separation {a b c d : ℤ} (hc : color a b = color c d)
+    (hne : (a, b) ≠ (c, d)) : 7 ≤ normSq (a - c) (b - d) := by
+  apply normSq_ge_seven (color_eq_iff_dvd.mp hc)
+  by_contra h
+  push Not at h
+  apply hne
+  apply Prod.ext <;> omega
+
+/-- The colors can be used directly as the `Fin 7` target of a coloring. -/
+def colorFin (a b : ℤ) : Fin 7 := color a b
+
+lemma colorFin_injective_value {a b c d : ℤ}
+    (h : colorFin a b = colorFin c d) : color a b = color c d := h
+
+end SevenLattice
+end Problem160
+
+end OAI
diff --git a/lean/OAI/Geometry/PlaneColoring/Main.lean b/lean/OAI/Geometry/PlaneColoring/Main.lean
new file mode 100644
index 0000000000000000000000000000000000000000..1df81c07905687feaaa0fbef285338e13c717216
--- /dev/null
+++ b/lean/OAI/Geometry/PlaneColoring/Main.lean
@@ -0,0 +1 @@
+import OAI.Geometry.PlaneColoring.Seven
diff --git a/lean/OAI/Geometry/PlaneColoring/MeasurableAssembly.lean b/lean/OAI/Geometry/PlaneColoring/MeasurableAssembly.lean
new file mode 100644
index 0000000000000000000000000000000000000000..a89d3d71d3c61ab66c5ea561b56cd5484456c231
--- /dev/null
+++ b/lean/OAI/Geometry/PlaneColoring/MeasurableAssembly.lean
@@ -0,0 +1,57 @@
+import Mathlib
+
+namespace OAI
+
+noncomputable section
+
+namespace Problem160
+
+/-- A countable cover admits a measurable least-index choice, so separated
+small cells give a measurable proper coloring, not merely an arbitrary one. -/
+theorem exists_measurable_unit_distance_coloring_of_center_cover
+    {X I : Type*} [PseudoMetricSpace X] [MeasurableSpace X] [BorelSpace X]
+    [Countable I] [Nonempty I] {k : ℕ}
+    (center : I → X) (color : I → Fin k) (r : ℝ)
+    (hcover : ∀ x : X, ∃ i : I, dist x (center i) ≤ r)
+    (hdiam : 2 * r < 1)
+    (hsep : ∀ i j : I, i ≠ j → color i = color j →
+      1 + 2 * r < dist (center i) (center j)) :
+    ∃ c : X → Fin k, Measurable c ∧
+      ∀ x y : X, dist x y = 1 → c x ≠ c y := by
+  classical
+  obtain ⟨e, he⟩ := exists_surjective_nat I
+  have hcoverNat : ∀ x : X, ∃ n : ℕ, dist x (center (e n)) ≤ r := by
+    intro x
+    obtain ⟨i, hi⟩ := hcover x
+    obtain ⟨n, rfl⟩ := he i
+    exact ⟨n, hi⟩
+  let pick : X → ℕ := fun x => Nat.find (hcoverNat x)
+  have hpick : Measurable pick := by
+    apply measurable_find hcoverNat
+    intro n
+    exact (isClosed_le (continuous_id.dist continuous_const) continuous_const).measurableSet
+  let cell : X → I := fun x => e (pick x)
+  have hcell (x : X) : dist x (center (cell x)) ≤ r := Nat.find_spec (hcoverNat x)
+  refine ⟨fun x => color (cell x), ?_, ?_⟩
+  · exact (measurable_of_countable (fun n => color (e n))).comp hpick
+  · intro x y hdist hcolor
+    by_cases hsame : cell x = cell y
+    · have htriangle := dist_triangle x (center (cell x)) y
+      have hx := hcell x
+      have hy := hcell y
+      rw [← hsame] at hy
+      rw [dist_comm (center (cell x)) y] at htriangle
+      linarith
+    · have hfar := hsep (cell x) (cell y) hsame hcolor
+      have htriangle₁ := dist_triangle (center (cell x)) x (center (cell y))
+      have htriangle₂ := dist_triangle x y (center (cell y))
+      have hx := hcell x
+      have hy := hcell y
+      rw [dist_comm (center (cell x)) x] at htriangle₁
+      linarith
+
+end Problem160
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PlaneColoring/MeasurableProper.lean b/lean/OAI/Geometry/PlaneColoring/MeasurableProper.lean
new file mode 100644
index 0000000000000000000000000000000000000000..666106a2d44f4d31b3562f997c6a1ec4eaad5a65
--- /dev/null
+++ b/lean/OAI/Geometry/PlaneColoring/MeasurableProper.lean
@@ -0,0 +1,68 @@
+import OAI.Geometry.PlaneColoring.Basic
+
+namespace OAI
+
+noncomputable section
+
+open MeasureTheory
+
+namespace Problem160
+
+@[continuity, fun_prop]
+theorem continuous_circleDir : Continuous circleDir := by
+  unfold circleDir
+  fun_prop
+
+@[fun_prop]
+theorem measurable_circleDir : Measurable circleDir :=
+  continuous_circleDir.measurable
+
+instance normalizedAngleMeasure_isProbabilityMeasure :
+    IsProbabilityMeasure normalizedAngleMeasure where
+  measure_univ := by
+    have hpos : 0 < (2 : ℝ) * Real.pi := by positivity
+    simp only [normalizedAngleMeasure, Measure.smul_apply, Measure.restrict_apply_univ,
+      Real.volume_Ico, sub_zero, smul_eq_mul]
+    exact ENNReal.inv_mul_cancel (ne_of_gt (ENNReal.ofReal_pos.mpr hpos))
+      ENNReal.ofReal_ne_top
+
+instance circleMeasure_isProbabilityMeasure : IsProbabilityMeasure circleMeasure :=
+  inferInstanceAs (IsProbabilityMeasure (Measure.map circleDir normalizedAngleMeasure))
+
+@[simp]
+theorem norm_circleDir (t : ℝ) : ‖circleDir t‖ = 1 := by
+  have h : ‖circleDir t‖ ^ 2 = 1 := by
+    rw [EuclideanSpace.real_norm_sq_eq]
+    simp [circleDir, Fin.sum_univ_two]
+  nlinarith [norm_nonneg (circleDir t)]
+
+theorem ae_circleMeasure_norm : ∀ᵐ u ∂circleMeasure, ‖u‖ = 1 := by
+  unfold circleMeasure
+  rw [ae_map_iff measurable_circleDir.aemeasurable]
+  · exact Filter.Eventually.of_forall norm_circleDir
+  · exact isClosed_eq continuous_norm continuous_const |>.measurableSet
+
+theorem ae_weakPairMeasure_unit (R : ℝ) :
+    ∀ᵐ p ∂weakPairMeasure R, ‖p.2‖ = 1 := by
+  exact Measure.quasiMeasurePreserving_snd.ae ae_circleMeasure_norm
+
+theorem weakMeasurableColoring_of_aemeasurable_proper {k : ℕ}
+    (c : Plane → Fin k) (hc : AEMeasurable c volume)
+    (hproper : ∀ x y : Plane, dist x y = 1 → c x ≠ c y) :
+    WeakMeasurableColoring k := by
+  refine ⟨c, hc, ?_⟩
+  intro R hR
+  apply Finset.sum_eq_zero
+  intro i hi
+  apply measure_mono_null (t := {p : Plane × Plane | ‖p.2‖ ≠ 1})
+  · intro p hp hnorm
+    have hd : dist p.1 (p.1 + p.2) = 1 := by
+      simpa [dist_eq_norm] using hnorm
+    exact hproper _ _ hd (hp.1.trans hp.2.symm)
+  · exact ae_iff.mp (ae_weakPairMeasure_unit R)
+
+end Problem160
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PlaneColoring/Mesh.lean b/lean/OAI/Geometry/PlaneColoring/Mesh.lean
new file mode 100644
index 0000000000000000000000000000000000000000..019fbacaad551666bb89ed9b31044135a0a07b04
--- /dev/null
+++ b/lean/OAI/Geometry/PlaneColoring/Mesh.lean
@@ -0,0 +1,154 @@
+import OAI.Geometry.PlaneColoring.FullApproximation
+
+namespace OAI
+
+noncomputable section
+open scoped Classical
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+/-- A point of a connected set with a distant partner can be moved to the
+boundary of its (unique) small hole. Pairwise disjoint closed disks make the
+new point lie outside every open hole. No path regularity is required. -/
+lemma connected_escape_holes {S : Finset MarkedDisk} {t : ℝ}
+    (hS : ∀d∈S,0≤d.radius)
+    (hdis : (S : Set MarkedDisk).Pairwise (fun a b => Disjoint a.closedBall b.closedBall))
+    (hsum : (∑d∈S,d.radius)<t) {K : Set ℂ} (hK : IsPreconnected K)
+    {a b : ℂ} (ha : a∈K) (hb : b∈K) (hab : 2*t<dist a b) :
+    ∃a'∈K,a'∉diskHoles S ∧ dist a a'≤2*t := by
+  have ht : 0<t := (Finset.sum_nonneg hS).trans_lt hsum
+  by_cases haS : a∈diskHoles S
+  · obtain ⟨d,hd,had⟩ := mem_diskHoles.mp haS
+    have hrad : d.radius<t := (MarkedDisk.radius_le_sum hS hd).trans_lt hsum
+    have had' : dist a d.center<d.radius := had
+    have hbd : d.radius<dist b d.center := by
+      by_contra! hh
+      have htri := dist_triangle a d.center b
+      rw [dist_comm d.center b] at htri
+      linarith
+    obtain ⟨p,hp,hpd⟩ := hK.intermediate_value ha hb
+      (continuous_id.dist continuous_const).continuousOn ⟨had'.le,hbd.le⟩
+    refine ⟨p,hp,MarkedDisk.sphere_avoids_holes hdis hd hpd,?_⟩
+    change dist p d.center=d.radius at hpd
+    have htri := dist_triangle a d.center p
+    rw [dist_comm d.center p,hpd] at htri
+    linarith
+  · exact ⟨a,ha,haS,by simpa using (mul_pos (by norm_num : (0:ℝ)<2) ht).le⟩
+
+/-- Positive averages on shrinking disks force every limit center into the
+actual measure support of that color. -/
+lemma support_of_positive_diskAverage_limit {k : ℕ} {c : ℂ → Fin k} (hc : Measurable c)
+    {ε : ℕ → ℝ} (hε : Tendsto ε atTop (𝓝 0)) {a : ℕ → ℂ} {x : ℂ}
+    (ha : Tendsto a atTop (𝓝 x)) (i : Fin k)
+    (hp : ∀n,0<diskAverage c (ε n) i (a n)) : x∈spatialSupport c i := by
+  apply (Measure.mem_support_iff_forall x).mpr
+  intro U hU
+  obtain ⟨r,hr,hrU⟩ := Metric.mem_nhds_iff.mp hU
+  have hnear : ∀ᶠ n in atTop,dist (a n) x<r/2 := ha.eventually (Metric.ball_mem_nhds x (by positivity))
+  have hsmall : ∀ᶠ n in atTop,ε n<r/2 := hε.eventually (gt_mem_nhds (by positivity : (0:ℝ)<r/2))
+  obtain ⟨n,han,hεn⟩ := (hnear.and hsmall).exists
+  have hsub : Metric.ball (a n) (ε n)⊆U := by
+    intro z hz
+    apply hrU
+    have hh : dist z (a n)<ε n := hz
+    have htri := dist_triangle z (a n) x
+    change dist z x<r
+    linarith
+  have hpos := hp n
+  rw [diskAverage_area hc] at hpos
+  apply pos_iff_ne_zero.mpr
+  intro hzero
+  have hzero' : volume (U∩{z | c z=i})=0 := by
+    simpa only [spatialSupport,Measure.restrict_apply' (measurableSet_eq_fun hc measurable_const)] using hzero
+  have hball : volume (Metric.ball (a n) (ε n)∩{z | c z=i})=0 :=
+    measure_mono_null (inter_subset_inter_left _ hsub) hzero'
+  simp only [Measure.real,hball,ENNReal.toReal_zero,zero_div,lt_self_iff_false] at hpos
+end EuclideanFiveColor
+end
+
+noncomputable section
+open scoped Classical
+open Set Filter Topology MeasureTheory ContinuousMap
+namespace EuclideanFiveColor
+namespace FullApprox
+lemma ε_tendsto {c : ℂ → Fin 5} {α : ℝ} {t : ℕ → ℝ}
+    (a : ∀n,FullApprox c α (t n)) (ht : Tendsto t atTop (𝓝 0)) :
+    Tendsto (fun n => (a n).ε) atTop (𝓝 0) :=
+  squeeze_zero (fun n => (a n).ε_pos.le) (fun n => (a n).ε_lt.le) ht
+lemma annuli_tendsto {c : ℂ → Fin 5} {α : ℝ} {t : ℕ → ℝ}
+    (a : ∀n,FullApprox c α (t n)) (ht : Tendsto t atTop (𝓝 0)) :
+    Tendsto (fun n => volume (holeAnnuli (a n).disks)) atTop (𝓝 0) := by
+  apply holeAnnuli_tendsto_zero _ (fun n d hd => ((a n).positive d hd).le)
+  exact squeeze_zero (fun n => Finset.sum_nonneg fun d hd => ((a n).positive d hd).le)
+    (fun n => (a n).sum_lt.le) ht
+lemma positive_average {c : ℂ → Fin 5} (hc : Measurable c) {α t : ℝ} (hα : α<1/5)
+    (a : FullApprox c α t) (i : Fin 5) (q : Metric.closedBall (0:ℂ) 5)
+    (hq : q.val∉diskHoles a.disks) (hp : 0<(a.map q).val i) : α<diskAverage c a.ε i q.val := by
+  rw [a.agrees q hq] at hp
+  exact (thresholdMap_pos hc a.ε_pos hα q.val i).mp hp
+
+/-- A continuous graph approximant cannot have positive-coordinate connected
+sets of macroscopic diameter at every shrinking scale. This is the manuscript
+mesh argument, with connectedness replacing the stronger path assumption. -/
+theorem no_large_positive_sequence {c : ℂ → Fin 5} (hc : Measurable c) (hw : WeakColoring 5 c)
+    {α : ℝ} (hα : 0<α) (hα' : α<1/5) {t : ℕ → ℝ}
+    (F : ∀n,FullApprox c α (t n)) (htlim : Tendsto t atTop (𝓝 0))
+    (htsmall : ∀n,t n≤1/16) (i : Fin 5)
+    (hbad : ∀n,∃K : Set ℂ, IsPreconnected K ∧ K⊆Metric.closedBall (0:ℂ) 5 ∧
+      (∀z : Metric.closedBall (0:ℂ) 5,z.val∈K → 0<((F n).map z).val i) ∧
+      ∃x∈K,∃y∈K,2<dist x y) : False := by
+  choose K hK hsub hprob x hx y hy hxy using hbad
+  have hxy' (n : ℕ) : 2*t n<dist (x n) (y n) := by linarith [htsmall n,hxy n]
+  have hyx' (n : ℕ) : 2*t n<dist (y n) (x n) := by rw [dist_comm]; exact hxy' n
+  choose a haK haH hax using fun n => connected_escape_holes
+    (fun d hd => ((F n).positive d hd).le) (F n).disjoint (F n).sum_lt (hK n) (hx n) (hy n) (hxy' n)
+  choose b hbK hbH hby using fun n => connected_escape_holes
+    (fun d hd => ((F n).positive d hd).le) (F n).disjoint (F n).sum_lt (hK n) (hy n) (hx n) (hyx' n)
+  have hab (n : ℕ) : 3/2≤dist (a n) (b n) := by
+    have h1 := dist_triangle (x n) (a n) (y n)
+    have h2 := dist_triangle (a n) (b n) (y n)
+    rw [dist_comm (b n) (y n)] at h2
+    linarith [hxy n,hax n,hby n,htsmall n]
+  obtain ⟨p,hp,φ,hφ,hlim⟩ := ((isCompact_closedBall (0:ℂ) 5).prod
+    (isCompact_closedBall (0:ℂ) 5)).tendsto_subseq (x := fun n => (a n,b n)) (fun n => ⟨hsub n (haK n),hsub n (hbK n)⟩)
+  have halim : Tendsto (fun n => a (φ n)) atTop (𝓝 p.1) := (continuous_fst.tendsto p).comp hlim
+  have hblim : Tendsto (fun n => b (φ n)) atTop (𝓝 p.2) := (continuous_snd.tendsto p).comp hlim
+  have hpdist : 3/2≤dist p.1 p.2 := ge_of_tendsto (halim.dist hblim) (Eventually.of_forall (fun n => hab (φ n)))
+  have hεlim := (ε_tendsto F htlim).comp hφ.tendsto_atTop
+  have hE := (annuli_tendsto F htlim).comp hφ.tendsto_atTop
+  have hasup : p.1∈spatialSupport c i := support_of_positive_diskAverage_limit hc hεlim halim i
+    (fun n => hα.trans (positive_average hc hα' (F (φ n)) i ⟨a (φ n),hsub _ (haK _)⟩
+      (haH _) (hprob _ _ (haK _))))
+  have hexc : Disjoint (spatialSupport c i) (Metric.ball p.1 (1/8)) := by
+    apply diskAverage_support_exclusion hc hεlim (fun n => (F (φ n)).ε_pos) hE hα Metric.isOpen_ball i
+    intro z hz
+    have hane : ∀ᶠn in atTop,dist (a (φ n)) p.1<1/8 :=
+      halim.eventually (Metric.ball_mem_nhds _ (by norm_num))
+    have hbne : ∀ᶠn in atTop,dist (b (φ n)) p.2<1/8 :=
+      hblim.eventually (Metric.ball_mem_nhds _ (by norm_num))
+    filter_upwards [hane,hbne] with n han hbn hzE
+    have hza : dist z (a (φ n))<1 := by
+      have hh := dist_triangle z p.1 (a (φ n))
+      rw [dist_comm p.1 (a (φ n))] at hh
+      change dist z p.1<1/8 at hz
+      linarith
+    have hzb : 1<dist z (b (φ n)) := by
+      have h1 := dist_triangle p.1 z p.2
+      have h2 := dist_triangle z (b (φ n)) p.2
+      rw [dist_comm p.1 z] at h1
+      change dist z p.1<1/8 at hz
+      linarith
+    obtain ⟨q,hq,hunit⟩ := straddle_connected_unit (hK (φ n))
+      ⟨a (φ n),haK _,b (φ n),hbK _,hza,hzb⟩
+    have hqH : q∉diskHoles (F (φ n)).disks := fun hh => hzE (unit_from_hole_mem_annuli hh hunit)
+    have hqprob := positive_average hc hα' (F (φ n)) i ⟨q,hsub _ hq⟩ hqH (hprob _ _ hq)
+    have hu : ‖q-z‖=1 := by simpa only [dist_eq_norm,norm_sub_rev] using hunit
+    have hh := diskAverage_unit_pair hc hw (F (φ n)).ε_pos hu i
+    change diskAverage c (F (φ n)).ε i z≤1-α
+    change α<diskAverage c (F (φ n)).ε i q at hqprob
+    linarith
+  exact Set.disjoint_left.mp hexc hasup (Metric.mem_ball_self (by norm_num))
+end FullApprox
+end EuclideanFiveColor
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PlaneColoring/Metric.lean b/lean/OAI/Geometry/PlaneColoring/Metric.lean
new file mode 100644
index 0000000000000000000000000000000000000000..b74ab445a1fd9164dbf9f14621cb9acbf94ad049
--- /dev/null
+++ b/lean/OAI/Geometry/PlaneColoring/Metric.lean
@@ -0,0 +1,72 @@
+import OAI.Geometry.PlaneColoring.Basic
+
+namespace OAI
+
+noncomputable section
+
+namespace Problem160
+
+/-- Triangular lattice with nearest-neighbor spacing `4/5`. -/
+def triangularLattice (a b : ℤ) : Plane :=
+  WithLp.toLp 2 ![(4 / 5 : ℝ) * ((a : ℝ) + (b : ℝ)/2),
+    (2 * Real.sqrt 3 / 5 : ℝ) * b]
+
+@[simp] theorem triangularLattice_zero (a b : ℤ) :
+    triangularLattice a b 0 = (4 / 5 : ℝ) * ((a : ℝ) + (b : ℝ)/2) := rfl
+
+@[simp] theorem triangularLattice_one (a b : ℤ) :
+    triangularLattice a b 1 = (2 * Real.sqrt 3 / 5 : ℝ) * b := rfl
+
+theorem triangularLattice_norm_sq (a b : ℤ) :
+    ‖triangularLattice a b‖ ^ 2 =
+      (16 / 25 : ℝ) * ((a : ℝ)^2 + (a : ℝ)*b + (b : ℝ)^2) := by
+  rw [EuclideanSpace.real_norm_sq_eq]
+  simp only [Fin.sum_univ_two, triangularLattice_zero, triangularLattice_one]
+  ring_nf
+  norm_num
+  ring
+
+theorem triangularLattice_sub (a b c d : ℤ) :
+    triangularLattice a b - triangularLattice c d = triangularLattice (a-c) (b-d) := by
+  ext i
+  fin_cases i <;> simp [triangularLattice] <;> ring
+
+theorem triangularLattice_dist_sq (a b c d : ℤ) :
+    dist (triangularLattice a b) (triangularLattice c d) ^ 2 =
+      (16 / 25 : ℝ) * (((a-c : ℤ) : ℝ)^2 +
+        ((a-c : ℤ) : ℝ)*(b-d : ℤ) + ((b-d : ℤ) : ℝ)^2) := by
+  rw [dist_eq_norm, triangularLattice_sub, triangularLattice_norm_sq]
+
+theorem triangularLattice_dist_gt_two (a b c d : ℤ)
+    (h : 7 ≤ (a-c)^2 + (a-c)*(b-d) + (b-d)^2) :
+    (2 : ℝ) < dist (triangularLattice a b) (triangularLattice c d) := by
+  have hr : (7 : ℝ) ≤ ((a-c : ℤ) : ℝ)^2 +
+      ((a-c : ℤ) : ℝ)*(b-d : ℤ) + ((b-d : ℤ) : ℝ)^2 := by
+    exact_mod_cast h
+  have heq := triangularLattice_dist_sq a b c d
+  have hd := dist_nonneg (x := triangularLattice a b) (y := triangularLattice c d)
+  nlinarith
+
+theorem dist_triangularLattice_sq (x : Plane) (a b : ℤ) :
+    dist x (triangularLattice a b)^2 =
+      (16 / 25 : ℝ) *
+        (((5 / 4 : ℝ) * x 0 - ((a : ℝ) + (b : ℝ)/2))^2 +
+         ((5 / 4 : ℝ) * x 1 - Real.sqrt 3 * b / 2)^2) := by
+  rw [dist_eq_norm, EuclideanSpace.real_norm_sq_eq]
+  simp only [Fin.sum_univ_two, PiLp.sub_apply, triangularLattice_zero,
+    triangularLattice_one]
+  ring
+
+theorem dist_triangularLattice_le_of_coordinates (x : Plane) (a b : ℤ)
+    (h : ((5 / 4 : ℝ) * x 0 - ((a : ℝ) + (b : ℝ)/2))^2 +
+         ((5 / 4 : ℝ) * x 1 - Real.sqrt 3 * b / 2)^2 ≤ (1 / 3 : ℝ)) :
+    dist x (triangularLattice a b) ≤ (7 / 15 : ℝ) := by
+  have heq := dist_triangularLattice_sq x a b
+  have hd := dist_nonneg (x := x) (y := triangularLattice a b)
+  nlinarith
+
+end Problem160
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PlaneColoring/Moser.lean b/lean/OAI/Geometry/PlaneColoring/Moser.lean
new file mode 100644
index 0000000000000000000000000000000000000000..c215fe51831de770114226fc606f0cc65afbefa7
--- /dev/null
+++ b/lean/OAI/Geometry/PlaneColoring/Moser.lean
@@ -0,0 +1,439 @@
+import Mathlib
+
+namespace OAI
+
+noncomputable section
+namespace EuclideanFiveColor
+def ThreeLabelRegion (x y : ℝ) : Prop :=
+  0<x^2+y^2 ∧ x^2+y^2<4 ∧
+    y^2*(3*x^2-y^2)^2 < (x^2+y^2)^3*(1-(x^2+y^2)/4)*(x^2+y^2-1)^2
+lemma square_coordinate_bounds {x c h : ℝ} (_hh : 0≤h) (hc : h≤|c|) (hx : |x-c|≤h) :
+    (|c|-h)^2≤x^2 ∧ x^2≤(|c|+h)^2 := by
+  have hu : |x|≤|c|+h := by
+    calc |x|=|c+(x-c)| := by congr 1; ring
+         _≤|c|+|x-c| := abs_add_le _ _
+         _≤|c|+h := by linarith
+  have hl : |c|-h≤|x| := by
+    have ht : |c|≤|x|+|x-c| := by
+      calc |c|=|x-(x-c)| := by congr 1; ring
+           _≤|x|+|x-c| := abs_sub _ _
+    linarith
+  constructor
+  · simpa only [sq_abs] using pow_le_pow_left₀ (sub_nonneg.mpr hc) hl 2
+  · simpa only [sq_abs] using pow_le_pow_left₀ (abs_nonneg x) hu 2
+lemma region_of_rational_bounds {x y a b Y H d : ℝ}
+    (ha : 0<a) (hb : b<4) (hY : 0≤Y) (_hH : 0≤H) (hd : 0≤d)
+    (hl : a≤x^2+y^2) (hu : x^2+y^2≤b)
+    (hy : y^2≤Y) (habs : |3*x^2-y^2|≤H) (hg : d≤|x^2+y^2-1|)
+    (hnum : Y*H^2<a^3*(1-b/4)*d^2) : ThreeLabelRegion x y := by
+  refine ⟨lt_of_lt_of_le ha hl,lt_of_le_of_lt hu hb,?_⟩
+  calc y^2*(3*x^2-y^2)^2 ≤ Y*H^2 := by
+         have hpow : (3*x^2-y^2)^2≤H^2 := by
+           simpa only [sq_abs] using pow_le_pow_left₀ (abs_nonneg _) habs 2
+         exact mul_le_mul hy hpow (sq_nonneg _) hY
+       _ < a^3*(1-b/4)*d^2 := hnum
+       _ ≤ (x^2+y^2)^3*(1-(x^2+y^2)/4)*(x^2+y^2-1)^2 := by
+         have hnonneg : 0≤1-b/4 := by linarith
+         have hgap : d^2≤(x^2+y^2-1)^2 := by
+           simpa only [sq_abs] using pow_le_pow_left₀ hd hg 2
+         have hpos : 0≤(x^2+y^2)^3*(1-(x^2+y^2)/4) :=
+           mul_nonneg (pow_nonneg (le_trans ha.le hl) _) (by linarith)
+         gcongr
+lemma abs_le_of_bounds {x l u c h : ℝ} (hl : l≤x) (hu : x≤u)
+    (hc1 : c-h≤l) (hc2 : u≤c+h) : |x-c|≤h := by
+  rw [abs_le]; constructor <;> linarith
+end EuclideanFiveColor
+end
+
+noncomputable section
+namespace EuclideanFiveColor
+lemma sin_sub_add_product (a b : ℝ) :
+    Real.sin (a-b)*Real.sin (a+b)=Real.sin a^2-Real.sin b^2 := by
+  rw [Real.sin_sub,Real.sin_add]
+  have ha : Real.cos a^2=1-Real.sin a^2 := by linarith [Real.sin_sq_add_cos_sq a]
+  have hb : Real.cos b^2=1-Real.sin b^2 := by linarith [Real.sin_sq_add_cos_sq b]
+  calc _ = Real.sin a^2*Real.cos b^2-Real.cos a^2*Real.sin b^2 := by ring
+       _ = _ := by rw [ha,hb]; ring
+lemma sin_three_using_cos (a : ℝ) :
+    Real.sin (3*a)=Real.sin a*(4*Real.cos a^2-1) := by
+  rw [Real.sin_three_mul]
+  nlinarith [congrArg (fun x : ℝ => Real.sin a*x) (Real.sin_sq_add_cos_sq a)]
+/-- Exact strict angular form of the manuscript's polynomial region. -/
+lemma polynomial_region_angles {z : ℂ} (hz : ThreeLabelRegion z.re z.im) :
+    ∃ t u : ℝ, (Circle.exp t : ℂ)+(Circle.exp u : ℂ)=z ∧
+      Real.sin (u-t)≠0 ∧ Real.sin (3*t)*Real.sin (3*u)<0 := by
+  let r : ℝ := ‖z‖
+  let p : ℝ := z.arg
+  let d : ℝ := Real.arccos (r/2)
+  have hr2 : r^2=z.re^2+z.im^2 := by
+    change ‖z‖^2=_
+    rw [Complex.sq_norm,Complex.normSq_apply]
+    ring
+  have hr0 : 0<r := by
+    have hn : 0≤r := norm_nonneg z
+    nlinarith [hz.1]
+  have hrlt : r<2 := by nlinarith [hz.2.1]
+  have hdcos : Real.cos d=r/2 := Real.cos_arccos (by linarith) (by linarith)
+  have hd0 : 0<d := Real.arccos_pos.mpr (by linarith)
+  have hdlt : d<Real.pi/2 := Real.arccos_lt_pi_div_two.mpr (by positivity)
+  have hdsin : 0<Real.sin d := Real.sin_pos_of_pos_of_lt_pi hd0 (by linarith [Real.pi_pos])
+  have hdsq : Real.sin d^2=1-r^2/4 := by
+    have hh := Real.sin_sq_add_cos_sq d
+    rw [hdcos] at hh
+    nlinarith
+  have hdp : Real.sin (3*d)=Real.sin d*(r^2-1) := by
+    rw [sin_three_using_cos,hdcos]; ring
+  have hpysin : r*Real.sin p=z.im := Complex.norm_mul_sin_arg z
+  have hpxcos : r*Real.cos p=z.re := Complex.norm_mul_cos_arg z
+  have hpy : r^3*Real.sin (3*p)=z.im*(3*z.re^2-z.im^2) := by
+    rw [Real.sin_three_mul]
+    calc _ = 3*r^2*(r*Real.sin p)-4*(r*Real.sin p)^3 := by ring
+         _ = _ := by rw [hr2,hpysin]; ring
+  have hP : z.im^2*(3*z.re^2-z.im^2)^2=r^6*Real.sin (3*p)^2 := by
+    calc _ = (z.im*(3*z.re^2-z.im^2))^2 := by ring
+         _ = (r^3*Real.sin (3*p))^2 := by rw [hpy]
+         _ = _ := by ring
+  have hP' : (z.re^2+z.im^2)^3*(1-(z.re^2+z.im^2)/4)*(z.re^2+z.im^2-1)^2=
+      r^6*Real.sin (3*d)^2 := by
+    rw [←hr2,hdp]
+    rw [show 1-r^2/4=Real.sin d^2 from hdsq.symm]
+    ring
+  have hlt : Real.sin (3*p)^2<Real.sin (3*d)^2 := by
+    have hh := hz.2.2
+    rw [hP,hP'] at hh
+    exact (mul_lt_mul_iff_right₀ (pow_pos hr0 6)).mp hh
+  refine ⟨p-d,p+d,?_,?_,?_⟩
+  · apply Complex.ext
+    · simp only [Complex.add_re,Circle.coe_exp,Complex.exp_mul_I,
+        ←Complex.ofReal_cos,←Complex.ofReal_sin,Complex.add_re,Complex.mul_re,
+        Complex.ofReal_re,Complex.I_re,Complex.ofReal_im,Complex.I_im,mul_zero,
+        sub_zero,zero_mul,add_zero]
+      rw [Real.cos_sub,Real.cos_add,hdcos]
+      nlinarith
+    · simp only [Complex.add_im,Circle.coe_exp,Complex.exp_mul_I,
+        ←Complex.ofReal_cos,←Complex.ofReal_sin,Complex.add_im,Complex.mul_im,
+        Complex.ofReal_re,Complex.I_re,Complex.ofReal_im,Complex.I_im,mul_zero,
+        mul_one,add_zero,zero_add]
+      rw [Real.sin_sub,Real.sin_add,hdcos]
+      nlinarith
+  · rw [show p+d-(p-d)=2*d by ring,Real.sin_two_mul,hdcos]
+    exact ne_of_gt (by positivity)
+  · rw [mul_sub,mul_add,sin_sub_add_product]
+    linarith
+end EuclideanFiveColor
+end
+
+noncomputable section
+namespace EuclideanFiveColor
+def moserA : ℂ := ((Real.sqrt 3 : ℂ)+Complex.I)/2
+def moserB : ℂ := ((Real.sqrt 3 : ℂ)-Complex.I)/2
+def moserT : ℂ := Real.sqrt 3
+def moserU : ℂ := (5+Complex.I*(Real.sqrt 11 : ℂ))/6
+def moserPoint : Fin 7 → ℂ := ![0,moserA,moserB,moserT,moserU*moserA,moserU*moserB,moserU*moserT]
+@[simp] lemma moserPoint_0 : moserPoint 0=0 := rfl
+@[simp] lemma moserPoint_1 : moserPoint 1=moserA := rfl
+@[simp] lemma moserPoint_2 : moserPoint 2=moserB := rfl
+@[simp] lemma moserPoint_3 : moserPoint 3=moserT := rfl
+@[simp] lemma moserPoint_4 : moserPoint 4=moserU*moserA := rfl
+@[simp] lemma moserPoint_5 : moserPoint 5=moserU*moserB := rfl
+@[simp] lemma moserPoint_6 : moserPoint 6=moserU*moserT := rfl
+def moserEdges : Finset (Fin 7 × Fin 7) :=
+  {(0,1),(0,2),(1,2),(3,1),(3,2),(0,4),(0,5),(4,5),(6,4),(6,5),(3,6)}
+lemma complex_norm_one_of_sq {z : ℂ} (h : z.re^2+z.im^2=1) : ‖z‖=1 := by
+  have he : ‖z‖^2=1 := by rw [← Complex.normSq_eq_norm_sq,Complex.normSq_apply]; nlinarith
+  nlinarith [norm_nonneg z]
+lemma moser_edge_unit {i j : Fin 7} (h : (i,j)∈moserEdges) : ‖moserPoint i-moserPoint j‖=1 := by
+  have h3 := Real.sq_sqrt (show (0:ℝ)≤3 by norm_num)
+  have h11 := Real.sq_sqrt (show (0:ℝ)≤11 by norm_num)
+  simp only [moserEdges,Finset.mem_insert,Finset.mem_singleton,Prod.mk.injEq] at h
+  rcases h with h|h|h|h|h|h|h|h|h|h|h <;> rcases h with ⟨rfl,rfl⟩
+  all_goals
+    apply complex_norm_one_of_sq
+    norm_num [moserA,moserB,moserT,moserU,Complex.mul_re,Complex.mul_im,
+      Complex.div_ofReal_re,Complex.div_ofReal_im,Complex.sub_re,Complex.sub_im,
+      Complex.add_re,Complex.add_im] <;>
+      (ring_nf; norm_num [h3,h11])
+lemma three_color_diamond {a b p q : Fin 3} (hab : a≠b)
+    (hpa : p≠a) (hpb : p≠b) (hqa : q≠a) (hqb : q≠b) : p=q := by
+  fin_cases a <;> fin_cases b <;> fin_cases p <;> fin_cases q <;> simp_all
+lemma moser_not_three_colorable (f : Fin 7 → Fin 3)
+    (hf : ∀i j,(i,j)∈moserEdges → f i≠f j) : False := by
+  have h03 : f 0=f 3 := three_color_diamond
+    (hf 1 2 (by decide)) (hf 0 1 (by decide)) (hf 0 2 (by decide))
+    (hf 3 1 (by decide)) (hf 3 2 (by decide))
+  have h06 : f 0=f 6 := three_color_diamond
+    (hf 4 5 (by decide)) (hf 0 4 (by decide)) (hf 0 5 (by decide))
+    (hf 6 4 (by decide)) (hf 6 5 (by decide))
+  exact hf 3 6 (by decide) (h03.symm.trans h06)
+def moserRotation : ℂ := (35+12*Complex.I)/37
+def placedMoser (i : Fin 7) : ℂ := moserRotation*(moserPoint i+(-290+149*Complex.I)/250)
+lemma moserRotation_unit : ‖moserRotation‖=1 := by
+  apply complex_norm_one_of_sq
+  norm_num [moserRotation,Complex.div_ofReal_re,Complex.div_ofReal_im]
+lemma placedMoser_edge_unit {i j : Fin 7} (h : (i,j)∈moserEdges) : ‖placedMoser i-placedMoser j‖=1 := by
+  have he : placedMoser i-placedMoser j=moserRotation*(moserPoint i-moserPoint j) := by
+    unfold placedMoser; ring
+  rw [he,norm_mul,moserRotation_unit,moser_edge_unit h,one_mul]
+end EuclideanFiveColor
+end
+
+noncomputable section
+namespace EuclideanFiveColor
+def certificateX : Fin 7 → ℝ := ![-1.2906,-0.6335,-0.3092,0.3478,-1.1598,-0.3666,-0.2358]
+def certificateY : Fin 7 → ℝ := ![0.1876,0.9414,-0.0045,0.7493,1.1790,0.5700,1.5614]
+lemma radical_bounds : 1.73205<Real.sqrt 3 ∧ Real.sqrt 3<1.73206 ∧
+    3.31662<Real.sqrt 11 ∧ Real.sqrt 11<3.31663 := by
+  have h3 := Real.sq_sqrt (show (0:ℝ)≤3 by norm_num)
+  have h11 := Real.sq_sqrt (show (0:ℝ)≤11 by norm_num)
+  have hn3 := Real.sqrt_nonneg (3:ℝ)
+  have hn11 := Real.sqrt_nonneg (11:ℝ)
+  constructor
+  · nlinarith
+  constructor
+  · nlinarith
+  constructor <;> nlinarith
+lemma radical_product_bounds : (1.73205*3.31662 : ℝ)≤Real.sqrt 3*Real.sqrt 11 ∧
+    Real.sqrt 3*Real.sqrt 11≤1.73206*3.31663 := by
+  obtain ⟨ha,hb,hc,hd⟩ := radical_bounds
+  constructor <;> gcongr
+lemma seven_point_box_0 :
+    |(placedMoser 0).re-certificateX 0|≤(0.00011 : ℝ) ∧
+    |(placedMoser 0).im-certificateY 0|≤(0.00011 : ℝ) := by
+  obtain ⟨ha,hb,hc,hd⟩ := radical_bounds
+  obtain ⟨hp,hq⟩ := radical_product_bounds
+  change |(placedMoser 0).re-(-1.2906)|≤0.00011 ∧ |(placedMoser 0).im-(0.1876)|≤0.00011
+  norm_num [placedMoser,moserRotation,moserA,moserB,moserT,moserU,
+    Complex.mul_re,Complex.mul_im,Complex.div_ofReal_re,Complex.div_ofReal_im,
+    Complex.add_re,Complex.add_im,Complex.sub_re,Complex.sub_im]
+lemma seven_point_box_1 :
+    |(placedMoser 1).re-certificateX 1|≤(0.00011 : ℝ) ∧
+    |(placedMoser 1).im-certificateY 1|≤(0.00011 : ℝ) := by
+  obtain ⟨ha,hb,hc,hd⟩ := radical_bounds
+  obtain ⟨hp,hq⟩ := radical_product_bounds
+  change |(placedMoser 1).re-(-0.6335)|≤0.00011 ∧ |(placedMoser 1).im-(0.9414)|≤0.00011
+  norm_num [placedMoser,moserRotation,moserA,moserB,moserT,moserU,
+    Complex.mul_re,Complex.mul_im,Complex.div_ofReal_re,Complex.div_ofReal_im,
+    Complex.add_re,Complex.add_im,Complex.sub_re,Complex.sub_im]
+  constructor <;> rw [abs_le] <;> constructor <;> nlinarith
+lemma seven_point_box_2 :
+    |(placedMoser 2).re-certificateX 2|≤(0.00011 : ℝ) ∧
+    |(placedMoser 2).im-certificateY 2|≤(0.00011 : ℝ) := by
+  obtain ⟨ha,hb,hc,hd⟩ := radical_bounds
+  obtain ⟨hp,hq⟩ := radical_product_bounds
+  change |(placedMoser 2).re-(-0.3092)|≤0.00011 ∧ |(placedMoser 2).im-(-0.0045)|≤0.00011
+  norm_num [placedMoser,moserRotation,moserA,moserB,moserT,moserU,
+    Complex.mul_re,Complex.mul_im,Complex.div_ofReal_re,Complex.div_ofReal_im,
+    Complex.add_re,Complex.add_im,Complex.sub_re,Complex.sub_im]
+  constructor <;> rw [abs_le] <;> constructor <;> nlinarith
+lemma seven_point_box_3 :
+    |(placedMoser 3).re-certificateX 3|≤(0.00011 : ℝ) ∧
+    |(placedMoser 3).im-certificateY 3|≤(0.00011 : ℝ) := by
+  obtain ⟨ha,hb,hc,hd⟩ := radical_bounds
+  obtain ⟨hp,hq⟩ := radical_product_bounds
+  change |(placedMoser 3).re-(0.3478)|≤0.00011 ∧ |(placedMoser 3).im-(0.7493)|≤0.00011
+  norm_num [placedMoser,moserRotation,moserA,moserB,moserT,moserU,
+    Complex.mul_re,Complex.mul_im,Complex.div_ofReal_re,Complex.div_ofReal_im,
+    Complex.add_re,Complex.add_im,Complex.sub_re,Complex.sub_im]
+  constructor <;> rw [abs_le] <;> constructor <;> nlinarith
+lemma seven_point_box_4 :
+    |(placedMoser 4).re-certificateX 4|≤(0.00011 : ℝ) ∧
+    |(placedMoser 4).im-certificateY 4|≤(0.00011 : ℝ) := by
+  obtain ⟨ha,hb,hc,hd⟩ := radical_bounds
+  obtain ⟨hp,hq⟩ := radical_product_bounds
+  change |(placedMoser 4).re-(-1.1598)|≤0.00011 ∧ |(placedMoser 4).im-(1.1790)|≤0.00011
+  norm_num [placedMoser,moserRotation,moserA,moserB,moserT,moserU,
+    Complex.mul_re,Complex.mul_im,Complex.div_ofReal_re,Complex.div_ofReal_im,
+    Complex.add_re,Complex.add_im,Complex.sub_re,Complex.sub_im]
+  constructor <;> rw [abs_le] <;> constructor <;> nlinarith
+lemma seven_point_box_5 :
+    |(placedMoser 5).re-certificateX 5|≤(0.00011 : ℝ) ∧
+    |(placedMoser 5).im-certificateY 5|≤(0.00011 : ℝ) := by
+  obtain ⟨ha,hb,hc,hd⟩ := radical_bounds
+  obtain ⟨hp,hq⟩ := radical_product_bounds
+  change |(placedMoser 5).re-(-0.3666)|≤0.00011 ∧ |(placedMoser 5).im-(0.5700)|≤0.00011
+  norm_num [placedMoser,moserRotation,moserA,moserB,moserT,moserU,
+    Complex.mul_re,Complex.mul_im,Complex.div_ofReal_re,Complex.div_ofReal_im,
+    Complex.add_re,Complex.add_im,Complex.sub_re,Complex.sub_im]
+  constructor <;> rw [abs_le] <;> constructor <;> nlinarith
+lemma seven_point_box_6 :
+    |(placedMoser 6).re-certificateX 6|≤(0.00011 : ℝ) ∧
+    |(placedMoser 6).im-certificateY 6|≤(0.00011 : ℝ) := by
+  obtain ⟨ha,hb,hc,hd⟩ := radical_bounds
+  obtain ⟨hp,hq⟩ := radical_product_bounds
+  change |(placedMoser 6).re-(-0.2358)|≤0.00011 ∧ |(placedMoser 6).im-(1.5614)|≤0.00011
+  norm_num [placedMoser,moserRotation,moserA,moserB,moserT,moserU,
+    Complex.mul_re,Complex.mul_im,Complex.div_ofReal_re,Complex.div_ofReal_im,
+    Complex.add_re,Complex.add_im,Complex.sub_re,Complex.sub_im]
+  constructor <;> rw [abs_le] <;> constructor <;> nlinarith
+lemma seven_point_boxes (i : Fin 7) :
+    |(placedMoser i).re-certificateX i|≤(0.00011 : ℝ) ∧
+    |(placedMoser i).im-certificateY i|≤(0.00011 : ℝ) := by
+  fin_cases i
+  · exact seven_point_box_0
+  · exact seven_point_box_1
+  · exact seven_point_box_2
+  · exact seven_point_box_3
+  · exact seven_point_box_4
+  · exact seven_point_box_5
+  · exact seven_point_box_6
+end EuclideanFiveColor
+end
+
+noncomputable section
+namespace EuclideanFiveColor
+/-- Exact rational interval certificate for the manuscript's seven placed vertices. -/
+lemma seven_point_certificate (i : Fin 7) :
+    ThreeLabelRegion (placedMoser i).re (placedMoser i).im := by
+  obtain ⟨hx,hy⟩ := seven_point_boxes i
+  fin_cases i
+  · change ThreeLabelRegion (placedMoser 0).re (placedMoser 0).im
+    change |(placedMoser 0).re-(-1.2906)|≤0.00011 at hx
+    change |(placedMoser 0).im-(0.1876)|≤0.00011 at hy
+    have hxx := square_coordinate_bounds (by norm_num : (0:ℝ)≤0.00011)
+      (by norm_num : (0.00011:ℝ)≤|(-1.2906:ℝ)|) hx
+    have hyy := square_coordinate_bounds (by norm_num : (0:ℝ)≤0.00011)
+      (by norm_num : (0.00011:ℝ)≤|(0.1876:ℝ)|) hy
+    norm_num at hxx hyy
+    apply region_of_rational_bounds (a := 1.70) (b := 1.71) (Y := 0.0353) (H := 4.963) (d := 0.70)
+    · norm_num
+    · norm_num
+    · norm_num
+    · norm_num
+    · norm_num
+    · nlinarith [hxx.1,hyy.1]
+    · nlinarith [hxx.2,hyy.2]
+    · nlinarith [hyy.2]
+    · rw [abs_le]; constructor <;> nlinarith [hxx.1,hxx.2,hyy.1,hyy.2]
+    · have hh := le_abs_self ((placedMoser 0).re^2+(placedMoser 0).im^2-1)
+      nlinarith [hxx.1,hxx.2,hyy.1,hyy.2]
+    · norm_num
+  · change ThreeLabelRegion (placedMoser 1).re (placedMoser 1).im
+    change |(placedMoser 1).re-(-0.6335)|≤0.00011 at hx
+    change |(placedMoser 1).im-(0.9414)|≤0.00011 at hy
+    have hxx := square_coordinate_bounds (by norm_num : (0:ℝ)≤0.00011)
+      (by norm_num : (0.00011:ℝ)≤|(-0.6335:ℝ)|) hx
+    have hyy := square_coordinate_bounds (by norm_num : (0:ℝ)≤0.00011)
+      (by norm_num : (0.00011:ℝ)≤|(0.9414:ℝ)|) hy
+    norm_num at hxx hyy
+    apply region_of_rational_bounds (a := 1.28) (b := 1.29) (Y := 0.887) (H := 0.319) (d := 0.28)
+    · norm_num
+    · norm_num
+    · norm_num
+    · norm_num
+    · norm_num
+    · nlinarith [hxx.1,hyy.1]
+    · nlinarith [hxx.2,hyy.2]
+    · nlinarith [hyy.2]
+    · rw [abs_le]; constructor <;> nlinarith [hxx.1,hxx.2,hyy.1,hyy.2]
+    · have hh := le_abs_self ((placedMoser 1).re^2+(placedMoser 1).im^2-1)
+      nlinarith [hxx.1,hxx.2,hyy.1,hyy.2]
+    · norm_num
+  · change ThreeLabelRegion (placedMoser 2).re (placedMoser 2).im
+    change |(placedMoser 2).re-(-0.3092)|≤0.00011 at hx
+    change |(placedMoser 2).im-(-0.0045)|≤0.00011 at hy
+    have hxx := square_coordinate_bounds (by norm_num : (0:ℝ)≤0.00011)
+      (by norm_num : (0.00011:ℝ)≤|(-0.3092:ℝ)|) hx
+    have hyy := square_coordinate_bounds (by norm_num : (0:ℝ)≤0.00011)
+      (by norm_num : (0.00011:ℝ)≤|(-0.0045:ℝ)|) hy
+    norm_num at hxx hyy
+    apply region_of_rational_bounds (a := 0.095) (b := 0.096) (Y := 0.000022) (H := 0.287) (d := 0.904)
+    · norm_num
+    · norm_num
+    · norm_num
+    · norm_num
+    · norm_num
+    · nlinarith [hxx.1,hyy.1]
+    · nlinarith [hxx.2,hyy.2]
+    · nlinarith [hyy.2]
+    · rw [abs_le]; constructor <;> nlinarith [hxx.1,hxx.2,hyy.1,hyy.2]
+    · have hh := neg_le_abs ((placedMoser 2).re^2+(placedMoser 2).im^2-1)
+      nlinarith [hxx.1,hxx.2,hyy.1,hyy.2]
+    · norm_num
+  · change ThreeLabelRegion (placedMoser 3).re (placedMoser 3).im
+    change |(placedMoser 3).re-(0.3478)|≤0.00011 at hx
+    change |(placedMoser 3).im-(0.7493)|≤0.00011 at hy
+    have hxx := square_coordinate_bounds (by norm_num : (0:ℝ)≤0.00011)
+      (by norm_num : (0.00011:ℝ)≤|(0.3478:ℝ)|) hx
+    have hyy := square_coordinate_bounds (by norm_num : (0:ℝ)≤0.00011)
+      (by norm_num : (0.00011:ℝ)≤|(0.7493:ℝ)|) hy
+    norm_num at hxx hyy
+    apply region_of_rational_bounds (a := 0.682) (b := 0.683) (Y := 0.562) (H := 0.199) (d := 0.317)
+    · norm_num
+    · norm_num
+    · norm_num
+    · norm_num
+    · norm_num
+    · nlinarith [hxx.1,hyy.1]
+    · nlinarith [hxx.2,hyy.2]
+    · nlinarith [hyy.2]
+    · rw [abs_le]; constructor <;> nlinarith [hxx.1,hxx.2,hyy.1,hyy.2]
+    · have hh := neg_le_abs ((placedMoser 3).re^2+(placedMoser 3).im^2-1)
+      nlinarith [hxx.1,hxx.2,hyy.1,hyy.2]
+    · norm_num
+  · change ThreeLabelRegion (placedMoser 4).re (placedMoser 4).im
+    change |(placedMoser 4).re-(-1.1598)|≤0.00011 at hx
+    change |(placedMoser 4).im-(1.1790)|≤0.00011 at hy
+    have hxx := square_coordinate_bounds (by norm_num : (0:ℝ)≤0.00011)
+      (by norm_num : (0.00011:ℝ)≤|(-1.1598:ℝ)|) hx
+    have hyy := square_coordinate_bounds (by norm_num : (0:ℝ)≤0.00011)
+      (by norm_num : (0.00011:ℝ)≤|(1.1790:ℝ)|) hy
+    norm_num at hxx hyy
+    apply region_of_rational_bounds (a := 2.73) (b := 2.74) (Y := 1.391) (H := 2.647) (d := 1.73)
+    · norm_num
+    · norm_num
+    · norm_num
+    · norm_num
+    · norm_num
+    · nlinarith [hxx.1,hyy.1]
+    · nlinarith [hxx.2,hyy.2]
+    · nlinarith [hyy.2]
+    · rw [abs_le]; constructor <;> nlinarith [hxx.1,hxx.2,hyy.1,hyy.2]
+    · have hh := le_abs_self ((placedMoser 4).re^2+(placedMoser 4).im^2-1)
+      nlinarith [hxx.1,hxx.2,hyy.1,hyy.2]
+    · norm_num
+  · change ThreeLabelRegion (placedMoser 5).re (placedMoser 5).im
+    change |(placedMoser 5).re-(-0.3666)|≤0.00011 at hx
+    change |(placedMoser 5).im-(0.5700)|≤0.00011 at hy
+    have hxx := square_coordinate_bounds (by norm_num : (0:ℝ)≤0.00011)
+      (by norm_num : (0.00011:ℝ)≤|(-0.3666:ℝ)|) hx
+    have hyy := square_coordinate_bounds (by norm_num : (0:ℝ)≤0.00011)
+      (by norm_num : (0.00011:ℝ)≤|(0.5700:ℝ)|) hy
+    norm_num at hxx hyy
+    apply region_of_rational_bounds (a := 0.459) (b := 0.46) (Y := 0.326) (H := 0.079) (d := 0.54)
+    · norm_num
+    · norm_num
+    · norm_num
+    · norm_num
+    · norm_num
+    · nlinarith [hxx.1,hyy.1]
+    · nlinarith [hxx.2,hyy.2]
+    · nlinarith [hyy.2]
+    · rw [abs_le]; constructor <;> nlinarith [hxx.1,hxx.2,hyy.1,hyy.2]
+    · have hh := neg_le_abs ((placedMoser 5).re^2+(placedMoser 5).im^2-1)
+      nlinarith [hxx.1,hxx.2,hyy.1,hyy.2]
+    · norm_num
+  · change ThreeLabelRegion (placedMoser 6).re (placedMoser 6).im
+    change |(placedMoser 6).re-(-0.2358)|≤0.00011 at hx
+    change |(placedMoser 6).im-(1.5614)|≤0.00011 at hy
+    have hxx := square_coordinate_bounds (by norm_num : (0:ℝ)≤0.00011)
+      (by norm_num : (0.00011:ℝ)≤|(-0.2358:ℝ)|) hx
+    have hyy := square_coordinate_bounds (by norm_num : (0:ℝ)≤0.00011)
+      (by norm_num : (0.00011:ℝ)≤|(1.5614:ℝ)|) hy
+    norm_num at hxx hyy
+    apply region_of_rational_bounds (a := 2.493) (b := 2.495) (Y := 2.44) (H := 2.273) (d := 1.493)
+    · norm_num
+    · norm_num
+    · norm_num
+    · norm_num
+    · norm_num
+    · nlinarith [hxx.1,hyy.1]
+    · nlinarith [hxx.2,hyy.2]
+    · nlinarith [hyy.2]
+    · rw [abs_le]; constructor <;> nlinarith [hxx.1,hxx.2,hyy.1,hyy.2]
+    · have hh := le_abs_self ((placedMoser 6).re^2+(placedMoser 6).im^2-1)
+      nlinarith [hxx.1,hxx.2,hyy.1,hyy.2]
+    · norm_num
+end EuclideanFiveColor
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PlaneColoring/PaletteDensity.lean b/lean/OAI/Geometry/PlaneColoring/PaletteDensity.lean
new file mode 100644
index 0000000000000000000000000000000000000000..c06bb17af276fd9a2ce94ca0c4653baeecca6314
--- /dev/null
+++ b/lean/OAI/Geometry/PlaneColoring/PaletteDensity.lean
@@ -0,0 +1,485 @@
+import OAI.Geometry.PlaneColoring.CircleTransport
+import OAI.Geometry.PlaneColoring.Palettes
+
+namespace OAI
+
+noncomputable section
+open MeasureTheory Filter Topology
+namespace EuclideanFiveColor
+lemma weak_strong_disjoint {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y]
+    {μ : Measure X} {ν : Measure Y} [IsProbabilityMeasure μ] [IsProbabilityMeasure ν]
+    {F : ℕ → X → ℝ} {b : X → ℝ} {G : ℕ → X × Y → ℝ} {g : X × Y → ℝ}
+    (hF : ∀ n, Measurable (F n)) (hb : Measurable b)
+    (hG : ∀ n, Measurable (G n)) (hg : Measurable g)
+    (hFb : ∀ n x, 0 ≤ F n x ∧ F n x ≤ 1)
+    (hbb : ∀ᵐ x ∂μ, 0 ≤ b x ∧ b x ≤ 1)
+    (hGb : ∀ n p, 0 ≤ G n p ∧ G n p ≤ 1)
+    (hgb : ∀ p, 0 ≤ g p ∧ g p ≤ 1)
+    (hw : ∀ v : X → ℝ, Integrable v μ →
+      Tendsto (fun n => ∫ x, F n x*v x ∂μ) atTop (𝓝 (∫ x, b x*v x ∂μ)))
+    (hs : Tendsto (fun n => ∫ p, ‖G n p-g p‖ ∂μ.prod ν) atTop (𝓝 0))
+    (hz : ∀ n, ∀ᵐ p ∂μ.prod ν, F n p.1*G n p=0) :
+    ∀ᵐ p ∂μ.prod ν, b p.1*g p=0 := by
+  have hgi : Integrable g (μ.prod ν) := Integrable.of_bound hg.aestronglyMeasurable 1
+    (Eventually.of_forall fun p => by rw [Real.norm_eq_abs,abs_of_nonneg (hgb p).1]; exact (hgb p).2)
+  have hFi (n : ℕ) : Integrable (fun p : X × Y => F n p.1*g p) (μ.prod ν) := by
+    apply Integrable.of_bound (((hF n).comp measurable_fst).mul hg).aestronglyMeasurable 1
+    exact Eventually.of_forall fun p => by
+      change ‖F n p.1*g p‖ ≤ 1
+      rw [Real.norm_eq_abs,abs_of_nonneg (mul_nonneg (hFb n p.1).1 (hgb p).1)]
+      exact (mul_le_of_le_one_left (hgb p).1 (hFb n p.1).2).trans (hgb p).2
+  have hbi : Integrable (fun p : X × Y => b p.1*g p) (μ.prod ν) := by
+    apply Integrable.of_bound ((hb.comp measurable_fst).mul hg).aestronglyMeasurable 1
+    filter_upwards [Measure.quasiMeasurePreserving_fst.ae hbb] with p hp
+    change ‖b p.1*g p‖ ≤ 1
+    rw [Real.norm_eq_abs,abs_of_nonneg (mul_nonneg hp.1 (hgb p).1)]
+    exact (mul_le_of_le_one_left (hgb p).1 hp.2).trans (hgb p).2
+  have herr (n : ℕ) : Integrable (fun p => ‖G n p-g p‖) (μ.prod ν) := by
+    apply Integrable.of_bound ((hG n).sub hg).norm.aestronglyMeasurable 1
+    exact Eventually.of_forall fun p => by
+      rw [norm_norm,Real.norm_eq_abs,abs_le]
+      change -1 ≤ G n p-g p ∧ G n p-g p ≤ 1
+      constructor <;> linarith [(hGb n p).1,(hGb n p).2,(hgb p).1,(hgb p).2]
+  have hi0 : Tendsto (fun n => ∫ p, F n p.1*g p ∂μ.prod ν) atTop (𝓝 0) := by
+    apply squeeze_zero (fun n => integral_nonneg fun p => mul_nonneg (hFb n p.1).1 (hgb p).1) _ hs
+    intro n
+    apply integral_mono_ae (hFi n) (herr n)
+    filter_upwards [hz n] with p hp
+    rw [Real.norm_eq_abs]
+    calc
+      F n p.1*g p = F n p.1*(g p-G n p) := by rw [mul_sub,hp,sub_zero]
+      _ ≤ |g p-G n p| := by
+        calc
+          _ ≤ F n p.1*|g p-G n p| := mul_le_mul_of_nonneg_left (le_abs_self _) (hFb n p.1).1
+          _ ≤ 1*|g p-G n p| := mul_le_mul_of_nonneg_right (hFb n p.1).2 (abs_nonneg _)
+          _ = _ := one_mul _
+      _ = |G n p-g p| := abs_sub_comm _ _
+  let v : X → ℝ := fun x => ∫ y, g (x,y) ∂ν
+  have hvi : Integrable v μ := hgi.integral_prod_left
+  have hlim := hw v hvi
+  have hconvert (f : X → ℝ) (hi : Integrable (fun p : X × Y => f p.1*g p) (μ.prod ν)) :
+      (∫ p, f p.1*g p ∂μ.prod ν) = ∫ x, f x*v x ∂μ := by
+    rw [integral_prod _ hi]
+    apply integral_congr_ae
+    exact Eventually.of_forall fun x => by
+      exact integral_const_mul (f x) (fun y => g (x,y))
+  have he0 : (∫ p, b p.1*g p ∂μ.prod ν)=0 := by
+    have ht : Tendsto (fun n => ∫ p, F n p.1*g p ∂μ.prod ν) atTop
+        (𝓝 (∫ p, b p.1*g p ∂μ.prod ν)) := by
+      simpa only [hconvert b hbi,hconvert (F _) (hFi _)] using hlim
+    exact tendsto_nhds_unique ht hi0
+  exact (integral_eq_zero_iff_of_nonneg_ae
+    ((Measure.quasiMeasurePreserving_fst.ae hbb).mono fun p hp => mul_nonneg hp.1 (hgb p).1) hbi).mp he0
+end EuclideanFiveColor
+end
+
+noncomputable section
+open MeasureTheory Filter Topology
+namespace EuclideanFiveColor
+variable {k : ℕ} (c : ℂ → Fin k) (hc : Measurable c)
+def colorIndicator (i : Fin k) (z : ℂ) : ℝ := if c z=i then 1 else 0
+include hc in
+lemma colorIndicator_measurable (i : Fin k) : Measurable (colorIndicator c i) :=
+  measurable_one.piecewise (measurableSet_eq_fun hc measurable_const) measurable_zero
+lemma colorIndicator_bounds (i : Fin k) (z : ℂ) : 0 ≤ colorIndicator c i z ∧ colorIndicator c i z ≤ 1 := by
+  simp only [colorIndicator]; split_ifs <;> norm_num
+include hc in
+lemma sample_pair_disjoint (hw : WeakColoring k c) {y : ℂ} {r : ℝ}
+    (hy : TypicalSampling c y r) (i : Fin k) :
+    ∀ᵐ p ∂angularMeasure.prod angularMeasure,
+      sampleFn c y r i p.1*colorIndicator c i (angularSum y r p)=0 := by
+  have h1 : ∀ᵐ p ∂angularMeasure.prod angularMeasure, Typical c (y+(r : ℂ)*direction p.1) :=
+    Measure.quasiMeasurePreserving_fst.ae hy.2
+  have h2 := (angularSum_quasiMeasurePreserving y hy.1.ne').ae (ae_density_actual c hc)
+  filter_upwards [h1,h2] with p hp1 hp2
+  by_cases ha : c (y+(r : ℂ)*direction p.1)=i
+  · have hb : c (angularSum y r p) ≠ i := by
+      intro hb
+      have hdist : ‖(y+(r : ℂ)*direction p.1)-angularSum y r p‖=1 := by
+        simp [angularSum]
+      apply weak_density_proper c hc hw i hdist
+      · simpa only [Typical,ha] using hp1
+      · simpa only [hb] using hp2
+    simp [sampleFn,colorIndicator,ha,hb]
+  · simp [sampleFn,ha]
+lemma trace_pair_disjoint (hw : WeakColoring k c) {x : ℂ} {b : TraceVector k}
+    (hb : IsTrace c hc x b) (i : Fin k) :
+    ∀ᵐ p ∂angularMeasure.prod angularMeasure,
+      traceValue b i p.1*colorIndicator c i (angularSum x 1 p)=0 := by
+  obtain ⟨y,r,ht,hy,hr,hweak⟩ := trace_realization_weakStar c hc hb
+  have hbnd := trace_bounds c hc hb
+  apply weak_strong_disjoint (F := fun n => sampleFn c (y n) (r n) i)
+    (G := fun n p => colorIndicator c i (angularSum (y n) (r n) p))
+    (fun n => sampleFn_measurable c hc (y n) (r n) i) (Lp.stronglyMeasurable _).measurable
+    (fun n => (colorIndicator_measurable c hc i).comp (angularSum_measurable _ _))
+    ((colorIndicator_measurable c hc i).comp (angularSum_measurable _ _))
+    (fun n t => ⟨sampleFn_nonneg c _ _ _ t,sampleFn_le_one c _ _ _ t⟩)
+    (Eventually.and (hbnd.1 i) (hbnd.2.1 i))
+    (fun n p => colorIndicator_bounds c i _) (fun p => colorIndicator_bounds c i _)
+    (hweak i) (angularSum_tendsto_L1 hr hy (colorIndicator_measurable c hc i) (colorIndicator_bounds c i))
+    (fun n => sample_pair_disjoint c hc hw (ht n) i)
+lemma palette_first_exclusion (hw : WeakColoring k c) (x : ℂ) (i : Fin k) :
+    ∀ᵐ p ∂angularMeasure.prod angularMeasure,
+      p.1 ∈ paletteSet c hc x i → c (angularSum x 1 p) ≠ i := by
+  have hall : ∀ᵐ p ∂angularMeasure.prod angularMeasure, ∀ b ∈ paletteGenerators c hc x i,
+      traceValue b.1 i p.1*colorIndicator c i (angularSum x 1 p)=0 :=
+    (ae_ball_iff (paletteGenerators_countable c hc x i)).mpr
+      (fun b _ => trace_pair_disjoint c hc hw b.2 i)
+  filter_upwards [hall] with p hp
+  intro hm he
+  obtain ⟨b,hb,hpos⟩ := Set.mem_iUnion₂.mp hm
+  have hzero := hp b hb
+  have hval : colorIndicator c i (angularSum x 1 p)=1 := by simp [colorIndicator,he]
+  rw [hval,mul_one] at hzero
+  exact (show 0 < traceValue b.1 i p.1 from hpos).ne' hzero
+lemma palette_second_exclusion (hw : WeakColoring k c) (x : ℂ) (i : Fin k) :
+    ∀ᵐ p ∂angularMeasure.prod angularMeasure,
+      p.2 ∈ paletteSet c hc x i → c (angularSum x 1 p) ≠ i := by
+  have h := (Measure.measurePreserving_swap (μ := angularMeasure) (ν := angularMeasure)).quasiMeasurePreserving.ae
+    (palette_first_exclusion c hc hw x i)
+  simp only [Prod.swap,angularSum,Complex.ofReal_one,one_mul] at h ⊢
+  filter_upwards [h] with p hp
+  convert hp using 1; rw [add_right_comm]
+lemma palette_transfer_pair (hw : WeakColoring k c) (x : ℂ) :
+    ∀ᵐ p ∂angularMeasure.prod angularMeasure,
+      c (angularSum x 1 p) ∉ palette c hc x p.1 ∪ palette c hc x p.2 := by
+  classical
+  filter_upwards [ae_all_iff.mpr (fun i => (palette_first_exclusion c hc hw x i).and
+    (palette_second_exclusion c hc hw x i))] with p hp
+  simpa only [Finset.mem_union,mem_palette,not_or] using
+    And.intro (fun h => (hp _).1 h rfl) (fun h => (hp _).2 h rfl)
+end EuclideanFiveColor
+
+end
+
+noncomputable section
+open MeasureTheory Set
+namespace EuclideanFiveColor
+lemma angleMk_quasiMeasurePreserving :
+    Measure.QuasiMeasurePreserving ((↑) : ℝ → Angles) volume angularMeasure := by
+  have hm : Measurable ((↑) : ℝ → Angles) := (AddCircle.measurePreserving_mk (2*Real.pi) 0).measurable
+  refine ⟨hm,Measure.AbsolutelyContinuous.mk (fun A hA hz => ?_)⟩
+  rw [Measure.map_apply hm hA]
+  have hzv : (volume : Measure Angles) A=0 := by
+    rw [AddCircle.volume_eq_smul_haarAddCircle,Measure.smul_apply]
+    change ENNReal.ofReal (2*Real.pi) * angularMeasure A=0
+    rw [hz,mul_zero]
+  have hh (n : ℤ) : volume (((↑) : ℝ → Angles) ⁻¹' A ∩
+      Ioc (n • (2*Real.pi)) ((n+1) • (2*Real.pi)))=0 := by
+    have h := (AddCircle.measurePreserving_mk (2*Real.pi) (n • (2*Real.pi))).quasiMeasurePreserving.preimage_null hzv
+    rw [Measure.restrict_apply' measurableSet_Ioc] at h
+    simpa only [add_zsmul,one_zsmul] using h
+  have hcov := iUnion_Ioc_zsmul (by positivity : 0 < 2*Real.pi)
+  have he : ((↑) : ℝ → Angles) ⁻¹' A = ⋃ n : ℤ,
+      ((↑) : ℝ → Angles) ⁻¹' A ∩ Ioc (n • (2*Real.pi)) ((n+1) • (2*Real.pi)) := by
+    rw [← inter_iUnion,hcov,inter_univ]
+  rw [he]
+  exact measure_iUnion_null hh
+lemma anglePair_quasiMeasurePreserving :
+    Measure.QuasiMeasurePreserving (fun p : ℝ × ℝ => ((p.1 : Angles),(p.2 : Angles)))
+      volume (angularMeasure.prod angularMeasure) := by
+  convert QuasiMeasurePreserving.prodMap angleMk_quasiMeasurePreserving
+    angleMk_quasiMeasurePreserving using 1; rfl
+end EuclideanFiveColor
+end
+
+noncomputable section
+open MeasureTheory Filter Topology Set
+open scoped NNReal
+namespace EuclideanFiveColor
+def FlatDensityZero (A : Set (ℝ × ℝ)) (p : ℝ × ℝ) : Prop :=
+  Tendsto (fun r : ℝ => volume.real (A ∩ Metric.closedBall p r)/r^2) (𝓝[>] 0) (𝓝 0)
+def LineDensityZero (D : Set ℝ) (t : ℝ) : Prop :=
+  Tendsto (fun r : ℝ => volume.real (D ∩ Metric.closedBall t r)/r) (𝓝[>] 0) (𝓝 0)
+lemma volume_image_bound {F : ℝ × ℝ → ℝ × ℝ} {s : Set (ℝ × ℝ)} {K : ℝ≥0}
+    (hF : LipschitzWith K (s.domRestrict F)) (A : Set (ℝ × ℝ)) (hA : A ⊆ s) :
+    volume (F '' A) ≤ (K : ENNReal)^2 * volume A := by
+  let B : Set s := Subtype.val ⁻¹' A
+  have he1 : (s.domRestrict F) '' B = F '' A := by
+    ext y
+    constructor
+    · rintro ⟨z,hz,rfl⟩; exact ⟨z.1,hz,rfl⟩
+    · rintro ⟨z,hz,rfl⟩; exact ⟨⟨z,hA hz⟩,hz,rfl⟩
+  have he2 : Subtype.val '' B = A := by
+    ext z
+    exact ⟨fun ⟨q,hq,he⟩ => he ▸ hq,fun hz => ⟨⟨z,hA hz⟩,hz,rfl⟩⟩
+  have hi := (isometry_subtype_coe : Isometry (Subtype.val : s → ℝ × ℝ)).hausdorffMeasure_image
+    (Or.inl (by norm_num : (0 : ℝ)≤2)) B
+  have ha := hF.hausdorffMeasure_image_le (by norm_num : (0 : ℝ)≤2) B
+  rw [he1,hausdorffMeasure_prod_real,ENNReal.rpow_two] at ha
+  rw [he2,hausdorffMeasure_prod_real] at hi
+  rwa [← hi] at ha
+lemma FlatDensityZero.mono_ae {A B : Set (ℝ × ℝ)} {p : ℝ × ℝ}
+    (hb : FlatDensityZero B p) (hAB : A ≤ᵐ[volume] B) : FlatDensityZero A p := by
+  apply squeeze_zero (fun r => div_nonneg (measureReal_nonneg) (sq_nonneg r)) _ hb
+  intro r
+  apply div_le_div_of_nonneg_right _ (sq_nonneg r)
+  apply ENNReal.toReal_mono
+  · exact ne_top_of_le_ne_top (isCompact_closedBall p r).measure_ne_top (measure_mono inter_subset_right)
+  · exact measure_mono_ae (hAB.mono fun z hz h => ⟨hz h.1,h.2⟩)
+lemma flatDensityZero_prod {D : Set ℝ} {t u : ℝ} (hD : LineDensityZero D t) :
+    FlatDensityZero (D ×ˢ univ) (t,u) := by
+  have h := hD.const_mul 2
+  simp only [mul_zero] at h
+  apply h.congr'
+  filter_upwards [self_mem_nhdsWithin] with r hr
+  have hp : 0 < r := hr
+  rw [← closedBall_prod_same,prod_inter_prod,univ_inter,Measure.volume_eq_prod]
+  simp only [Measure.real,Measure.prod_prod,ENNReal.toReal_mul,Real.volume_closedBall,
+    ENNReal.toReal_ofReal (by positivity : 0 ≤ 2*r)]
+  change 2*(volume.real (D ∩ Metric.closedBall t r)/r) =
+    volume.real (D ∩ Metric.closedBall t r)*(2*r)/r^2
+  field_simp [hp.ne']
+lemma tendsto_pos_mul_nhdsWithin {a : ℝ} (ha : 0<a) :
+    Tendsto (fun r : ℝ => a*r) (𝓝[>] 0) (𝓝[>] 0) := by
+  apply tendsto_nhdsWithin_iff.mpr
+  constructor
+  · simpa using (tendsto_const_nhds.mul (tendsto_id.mono_left nhdsWithin_le_nhds : Tendsto (fun r : ℝ => r) (𝓝[>] 0) (𝓝 0)))
+  · filter_upwards [self_mem_nhdsWithin] with r hr
+    exact mul_pos ha hr
+lemma FlatDensityZero.image {D s : Set (ℝ × ℝ)} {p : ℝ × ℝ} {F : ℝ × ℝ → ℝ × ℝ}
+    {K L : ℝ≥0} (hD : FlatDensityZero D p) (hps : p ∈ s)
+    (hF : LipschitzWith K (s.domRestrict F)) (hA : AntilipschitzWith L (s.domRestrict F))
+    (hL : 0 < L) : FlatDensityZero (F '' (D ∩ s)) (F p) := by
+  have h := (hD.comp (tendsto_pos_mul_nhdsWithin (show 0 < (L : ℝ) from hL))).const_mul
+    ((K : ℝ)^2*(L : ℝ)^2)
+  simp only [mul_zero] at h
+  apply squeeze_zero' (Eventually.of_forall fun r => div_nonneg measureReal_nonneg (sq_nonneg r)) _ h
+  filter_upwards [self_mem_nhdsWithin] with r hr
+  have hp : 0 < r := hr
+  let E : Set (ℝ × ℝ) := D ∩ s ∩ Metric.closedBall p ((L : ℝ)*r)
+  have he : F '' (D ∩ s) ∩ Metric.closedBall (F p) r ⊆ F '' E := by
+    rintro z ⟨⟨q,hq,rfl⟩,hz⟩
+    refine ⟨q,⟨hq,?_⟩,rfl⟩
+    exact (hA.le_mul_dist ⟨q,hq.2⟩ ⟨p,hps⟩).trans
+      (mul_le_mul_of_nonneg_left hz L.2)
+  have hm := (measure_mono he).trans (volume_image_bound hF E (fun q hq => hq.1.2))
+  have hsub : E ⊆ D ∩ Metric.closedBall p ((L : ℝ)*r) := fun q hq => ⟨hq.1.1,hq.2⟩
+  have hfin : volume (D ∩ Metric.closedBall p ((L : ℝ)*r)) ≠ ⊤ :=
+    ne_top_of_le_ne_top (isCompact_closedBall _ _).measure_ne_top (measure_mono inter_subset_right)
+  have hm' := hm.trans (mul_le_mul (le_refl _) (measure_mono hsub) bot_le bot_le)
+  have hreal := ENNReal.toReal_mono (ENNReal.mul_ne_top (by finiteness) hfin) hm'
+  simp only [ENNReal.toReal_mul,ENNReal.toReal_pow,ENNReal.coe_toReal] at hreal
+  calc
+    _ ≤ (K : ℝ)^2*volume.real (D ∩ Metric.closedBall p ((L : ℝ)*r))/r^2 :=
+      div_le_div_of_nonneg_right hreal (sq_nonneg r)
+    _ = _ := by
+      change _ = ((K : ℝ)^2*(L : ℝ)^2)*
+        (volume.real (D ∩ Metric.closedBall p ((L : ℝ)*r))/((L : ℝ)*r)^2)
+      have hLr : (L : ℝ) ≠ 0 := ne_of_gt hL
+      field_simp [hp.ne',hLr]
+end EuclideanFiveColor
+end
+
+noncomputable section
+open MeasureTheory Filter Topology Set
+open scoped NNReal
+namespace EuclideanFiveColor
+lemma FlatDensityZero.of_locally_le {A B s : Set (ℝ × ℝ)} {p : ℝ × ℝ}
+    (hb : FlatDensityZero B p) (hs : s ∈ 𝓝 p) (hAB : (A ∩ s : Set (ℝ × ℝ)) ≤ᵐ[volume] B) :
+    FlatDensityZero A p := by
+  obtain ⟨δ,hδ,hsδ⟩ := Metric.mem_nhds_iff.mp hs
+  apply squeeze_zero' (Eventually.of_forall fun r => div_nonneg measureReal_nonneg (sq_nonneg r)) _ hb
+  filter_upwards [(eventually_lt_nhds hδ).filter_mono nhdsWithin_le_nhds] with r hr
+  apply div_le_div_of_nonneg_right _ (sq_nonneg r)
+  apply ENNReal.toReal_mono
+  · exact ne_top_of_le_ne_top (isCompact_closedBall p r).measure_ne_top (measure_mono inter_subset_right)
+  · apply measure_mono_ae
+    filter_upwards [hAB] with q hq
+    intro hqa
+    exact ⟨hq ⟨hqa.1,hsδ (Metric.closedBall_subset_ball hr hqa.2)⟩,hqa.2⟩
+lemma FlatDensityZero.transport {A D s : Set (ℝ × ℝ)} {p : ℝ × ℝ}
+    {F : ℝ × ℝ → ℝ × ℝ} {K L : ℝ≥0} (hD : FlatDensityZero D p)
+    (hps : p ∈ s) (hs : F '' s ∈ 𝓝 (F p))
+    (hF : LipschitzWith K (s.domRestrict F)) (hA : AntilipschitzWith L (s.domRestrict F))
+    (hL : 0 < L) (hAD : (F ⁻¹' A ∩ s : Set (ℝ × ℝ)) ≤ᵐ[volume] D) : FlatDensityZero A (F p) := by
+  let N := (F ⁻¹' A ∩ s) \ D
+  have hN : volume N=0 := measure_eq_zero_iff_ae_notMem.mpr (hAD.mono fun q hq h => h.2 (hq h.1))
+  have himg : volume (F '' N)=0 := by
+    apply le_antisymm _ (zero_le : (0 : ENNReal) ≤ volume (F '' N))
+    simpa only [hN,mul_zero] using volume_image_bound hF N (fun q hq => hq.1.2)
+  apply (hD.image hps hF hA hL).of_locally_le hs
+  filter_upwards [measure_eq_zero_iff_ae_notMem.mp himg] with q hq
+  rintro ⟨hqa,⟨t,ht,rfl⟩⟩
+  by_cases hd : t ∈ D
+  · exact ⟨t,⟨hd,ht⟩,rfl⟩
+  · exact (hq ⟨t,⟨⟨hqa,ht⟩,hd⟩,rfl⟩).elim
+end EuclideanFiveColor
+end
+
+noncomputable section
+open MeasureTheory Filter Topology Set
+open scoped NNReal
+namespace EuclideanFiveColor
+lemma circleSum_one_lipschitz : LipschitzWith 2 (circleSum 1) := by
+  apply LipschitzWith.of_dist_le_mul
+  intro a b
+  have h₁ := realDirection_lipschitz.dist_le_mul a.1 b.1
+  have h₂ := realDirection_lipschitz.dist_le_mul a.2 b.2
+  have h := dist_add_add_le (Real.cos a.1,Real.sin a.1) (Real.cos a.2,Real.sin a.2)
+    (Real.cos b.1,Real.sin b.1) (Real.cos b.2,Real.sin b.2)
+  have he (p : RealPlane) : circleSum 1 p =
+      (Real.cos p.1,Real.sin p.1)+(Real.cos p.2,Real.sin p.2) := by simp [circleSum]
+  rw [he a,he b]
+  norm_num only [NNReal.coe_ofNat,NNReal.coe_one,one_mul] at *
+  have ha : dist a.1 b.1 ≤ dist a b := by rw [Prod.dist_eq]; exact le_max_left _ _
+  have hb : dist a.2 b.2 ≤ dist a b := by rw [Prod.dist_eq]; exact le_max_right _ _
+  linarith
+lemma circleSum_densityZero {D : Set ℝ} {A : Set RealPlane} {t u : ℝ} (x : RealPlane)
+    (hD : LineDensityZero D t) (hu : Real.sin (u-t) ≠ 0)
+    (hAD : ∀ᵐ q : RealPlane ∂volume, x+circleSum 1 q ∈ A → q.1 ∈ D) :
+    FlatDensityZero A (x+circleSum 1 (t,u)) := by
+  let F : RealPlane → RealPlane := fun q => x+circleSum 1 q
+  have hreg : (1 : ℝ)*Real.sin (u-t) ≠ 0 := by simpa only [one_mul] using hu
+  obtain ⟨s,hso,hps,K,hK⟩ := circleSum_local_antilipschitz (r := 1) (p := (t,u)) hreg
+  have ha : AntilipschitzWith (K+1) (s.domRestrict F) := by
+    apply AntilipschitzWith.of_le_mul_dist
+    intro a b
+    have h := hK.le_mul_dist a b
+    simp only [domRestrict] at h
+    simp only [domRestrict,F,dist_add_left,NNReal.coe_add,NNReal.coe_one]
+    nlinarith [dist_nonneg (x := circleSum 1 a.1) (y := circleSum 1 b.1)]
+  let L := circleSumDerivEquiv (r := 1) (p := (t,u)) hreg
+  have hd : HasStrictFDerivAt F (L : RealPlane →L[ℝ] RealPlane) (t,u) :=
+    (circleSum_strict 1 (t,u)).const_add x
+  have him : F '' s ∈ 𝓝 (F (t,u)) := by
+    rw [← hd.map_nhds_eq_of_equiv]
+    exact image_mem_map (hso.mem_nhds hps)
+  have hl : LipschitzWith 2 F := LipschitzWith.of_dist_le_mul fun a b => by
+    simpa only [F,dist_add_left] using circleSum_one_lipschitz.dist_le_mul a b
+  apply (flatDensityZero_prod hD).transport hps him
+    (show LipschitzWith 2 (s.domRestrict F) from fun a b => hl a.1 b.1) ha (by positivity)
+  filter_upwards [hAD] with q hq
+  rintro ⟨hm,_⟩
+  exact ⟨hq hm,mem_univ _⟩
+end EuclideanFiveColor
+end
+
+noncomputable section
+open MeasureTheory Filter Topology Set
+namespace EuclideanFiveColor
+lemma complex_ball_flat_ball {z w : ℂ} {r : ℝ} (h : w ∈ Metric.closedBall z r) :
+    Complex.equivRealProdCLM w ∈ Metric.closedBall (Complex.equivRealProdCLM z) r := by
+  rw [Metric.mem_closedBall,dist_eq_norm] at h ⊢
+  change max ‖w.re-z.re‖ ‖w.im-z.im‖ ≤ r
+  simpa only [Real.norm_eq_abs,Complex.sub_re,Complex.sub_im] using
+    max_le ((Complex.abs_re_le_norm (w-z)).trans h) ((Complex.abs_im_le_norm (w-z)).trans h)
+lemma flat_to_complex_densityZero {A : Set ℂ} {z : ℂ}
+    (h : FlatDensityZero (Complex.equivRealProdCLM.symm ⁻¹' A) (Complex.equivRealProdCLM z)) :
+    Tendsto (fun r : ℝ => volume (A ∩ Metric.closedBall z r)/volume (Metric.closedBall z r))
+      (𝓝[>] 0) (𝓝 (0 : ENNReal)) := by
+  have ht := h.const_mul (Real.pi)⁻¹
+  simp only [mul_zero] at ht
+  have he := ENNReal.continuous_ofReal.continuousAt.tendsto.comp ht
+  simp only [ENNReal.ofReal_zero] at he
+  apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds he
+    (Eventually.of_forall fun _ => zero_le) ?_
+  filter_upwards [self_mem_nhdsWithin] with r hr
+  have hp : 0 < r := hr
+  let B := Complex.equivRealProdCLM.symm ⁻¹' A ∩ Metric.closedBall (Complex.equivRealProdCLM z) r
+  have hm : volume (A ∩ Metric.closedBall z r) ≤ (volume : Measure (ℝ × ℝ)) B := by
+    apply (measure_mono (t := Complex.equivRealProdCLM ⁻¹' B) ?_).trans
+      (Complex.volume_preserving_equiv_real_prod.measure_preimage_le B)
+    intro w hw
+    exact ⟨by simpa only [Set.mem_preimage,ContinuousLinearEquiv.symm_apply_apply] using hw.1,complex_ball_flat_ball hw.2⟩
+  have hfin : (volume : Measure (ℝ × ℝ)) B ≠ ⊤ :=
+    ne_top_of_le_ne_top (isCompact_closedBall _ _).measure_ne_top (measure_mono inter_subset_right)
+  have hfinA : volume (A ∩ Metric.closedBall z r) ≠ ⊤ := ne_top_of_le_ne_top (isCompact_closedBall z r).measure_ne_top
+    (measure_mono (s := A ∩ Metric.closedBall z r) inter_subset_right)
+  apply (ENNReal.toReal_le_toReal (ENNReal.div_ne_top hfinA
+    (Metric.measure_closedBall_pos volume z hp).ne') ENNReal.ofReal_ne_top).mp
+  rw [ENNReal.toReal_div,Complex.volume_closedBall,ENNReal.toReal_mul,ENNReal.toReal_pow,
+    ENNReal.toReal_ofReal hp.le,ENNReal.coe_toReal,NNReal.coe_real_pi,
+    ENNReal.toReal_ofReal (by positivity)]
+  calc
+    _ ≤ volume.real B/(r^2*Real.pi) := div_le_div_of_nonneg_right (ENNReal.toReal_mono hfin hm) (by positivity)
+    _ = _ := by dsimp [B]; ring
+lemma flat_density_not_one {A : Set ℂ} {z : ℂ}
+    (h : FlatDensityZero (Complex.equivRealProdCLM.symm ⁻¹' A) (Complex.equivRealProdCLM z)) :
+    ¬ DensityOne A z := by
+  intro hd
+  have he := tendsto_nhds_unique (flat_to_complex_densityZero h) hd
+  exact zero_ne_one he
+end EuclideanFiveColor
+end
+
+noncomputable section
+open MeasureTheory Filter Topology Set
+namespace EuclideanFiveColor
+lemma ae_lineDensityZero_compl {D : Set ℝ} (hD : MeasurableSet D) :
+    ∀ᵐ t : ℝ ∂volume, t ∈ D → LineDensityZero Dᶜ t := by
+  have hh := Besicovitch.ae_tendsto_measure_inter_div_of_measurableSet volume hD.compl
+  filter_upwards [hh] with t ht htd
+  rw [Set.indicator_of_notMem (show t ∉ Dᶜ from by simpa only [mem_compl_iff,not_not] using htd)] at ht
+  have hreal := (ENNReal.tendsto_toReal (by simp : (0 : ENNReal) ≠ ⊤)).comp ht
+  simp only [ENNReal.toReal_zero,Function.comp_def,ENNReal.toReal_div] at hreal
+  have h := hreal.const_mul 2
+  simp only [mul_zero] at h
+  apply h.congr'
+  filter_upwards [self_mem_nhdsWithin] with r hr
+  have hp : 0 < r := hr
+  rw [Real.volume_closedBall,ENNReal.toReal_ofReal (by positivity : 0 ≤ 2*r)]
+  change 2*(volume.real (Dᶜ ∩ Metric.closedBall t r)/(2*r)) = _
+  field_simp
+end EuclideanFiveColor
+end
+
+noncomputable section
+open MeasureTheory Filter Topology Set
+namespace EuclideanFiveColor
+variable {k : ℕ} (c : ℂ → Fin k) (hc : Measurable c)
+def liftPalette (x : ℂ) (i : Fin k) : Set ℝ :=
+  ((↑) : ℝ → Angles) ⁻¹' paletteSet c hc x i
+lemma liftPalette_measurable (x : ℂ) (i : Fin k) : MeasurableSet (liftPalette c hc x i) :=
+  (paletteSet_measurable c hc x i).preimage angleMk_quasiMeasurePreserving.measurable
+lemma palette_spatial_densityZero (hw : WeakColoring k c) {x : ℂ} {i : Fin k} {t u : ℝ}
+    (ht : LineDensityZero (liftPalette c hc x i)ᶜ t) (hu : Real.sin (u-t) ≠ 0) :
+    Tendsto (fun r : ℝ => volume ({z | c z=i} ∩ Metric.closedBall (angularSum x 1 (angleChart (t,u))) r)/
+      volume (Metric.closedBall (angularSum x 1 (angleChart (t,u))) r))
+        (𝓝[>] 0) (𝓝 (0 : ENNReal)) := by
+  apply flat_to_complex_densityZero
+  have he := anglePair_quasiMeasurePreserving.ae (palette_first_exclusion c hc hw x i)
+  have hAD : ∀ᵐ q : RealPlane ∂volume, Complex.equivRealProdCLM x+circleSum 1 q ∈
+      Complex.equivRealProdCLM.symm ⁻¹' {z | c z=i} → q.1 ∈ (liftPalette c hc x i)ᶜ := by
+    filter_upwards [he] with q hq hm hd
+    apply hq hd
+    change c (Complex.equivRealProdCLM.symm (Complex.equivRealProdCLM x+circleSum 1 q))=i at hm
+    change c (angularSum x 1 (angleChart q))=i
+    rwa [angularSum_chart]
+  have hb := circleSum_densityZero (Complex.equivRealProdCLM x) ht hu hAD
+  rwa [← circleSum_chart x 1 (t,u)] at hb
+lemma palette_spatial_not_densityOne (hw : WeakColoring k c) {x : ℂ} {i : Fin k} {t u : ℝ}
+    (ht : LineDensityZero (liftPalette c hc x i)ᶜ t) (hu : Real.sin (u-t) ≠ 0) :
+    ¬ DensityOne {z | c z=i} (angularSum x 1 (angleChart (t,u))) := by
+  intro hd
+  exact zero_ne_one (tendsto_nhds_unique (palette_spatial_densityZero c hc hw ht hu) hd)
+end EuclideanFiveColor
+end
+
+noncomputable section
+open MeasureTheory Filter Topology Set
+namespace EuclideanFiveColor
+variable {k : ℕ} (c : ℂ → Fin k) (hc : Measurable c)
+lemma palette_sample_branch_exclusion (hw : WeakColoring k c) {x y : ℂ} {r : ℝ}
+    (hy : TypicalSampling c y r) (i : Fin k) {I : Set ℝ} (hI : MeasurableSet I) {s t : ℝ → ℝ}
+    (hq : Measure.QuasiMeasurePreserving s (volume.restrict I) volume)
+    (hn : ∀ f ∈ I, Real.sin (t f-s f) ≠ 0)
+    (he : ∀ f ∈ I, y+(r : ℂ)*direction (f : Angles)=angularSum x 1 (angleChart (s f,t f))) :
+    ∀ᵐ f : ℝ ∂volume.restrict I,
+      sampleFn c y r i (f : Angles)*(liftPalette c hc x i).indicator (fun _ => (1 : ℝ)) (s f)=0 := by
+  have hd := hq.ae (ae_lineDensityZero_compl (liftPalette_measurable c hc x i))
+  have ht := ae_restrict_of_ae (s := I) (angleMk_quasiMeasurePreserving.ae hy.2)
+  filter_upwards [hd,ht,ae_restrict_mem hI] with f hfd hft hf
+  by_cases hs : s f ∈ liftPalette c hc x i
+  · have hzero := palette_spatial_not_densityOne c hc hw (hfd hs) (hn f hf)
+    have hi : c (y+(r : ℂ)*direction (f : Angles)) ≠ i := by
+      intro hi
+      apply hzero
+      rw [← he f hf]
+      simpa only [Typical,hi] using hft
+    simp only [sampleFn,ite_eq_right hi,zero_mul]
+  · rw [indicator_of_notMem hs,mul_zero]
+end EuclideanFiveColor
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PlaneColoring/PaletteTransport.lean b/lean/OAI/Geometry/PlaneColoring/PaletteTransport.lean
new file mode 100644
index 0000000000000000000000000000000000000000..dab3fbec7feaef6bf5a7ac839097062028692943
--- /dev/null
+++ b/lean/OAI/Geometry/PlaneColoring/PaletteTransport.lean
@@ -0,0 +1,458 @@
+import OAI.Geometry.PlaneColoring.PaletteDensity
+
+namespace OAI
+
+noncomputable section
+open MeasureTheory Filter Topology Set
+namespace EuclideanFiveColor
+lemma angleRep_preserving (a : ℝ) :
+    MeasurePreserving (fun t : Angles => (AddCircle.equivIoc (2*Real.pi) a t : ℝ))
+      volume (volume.restrict (Ioc a (a+2*Real.pi))) :=
+  (measurePreserving_subtype_coe measurableSet_Ioc).comp (AddCircle.measurePreserving_equivIoc _)
+lemma weak_lift_Ioc {F : ℕ → Angles → ℝ} {b : Angles → ℝ}
+    (hw : ∀ v : Angles → ℝ, Integrable v angularMeasure →
+      Tendsto (fun n => ∫ t, F n t*v t ∂angularMeasure) atTop (𝓝 (∫ t, b t*v t ∂angularMeasure)))
+    (a : ℝ) (v : ℝ → ℝ) (hv : Integrable v (volume.restrict (Ioc a (a+2*Real.pi)))) :
+    Tendsto (fun n => ∫ t in Ioc a (a+2*Real.pi), F n (t : Angles)*v t) atTop
+      (𝓝 (∫ t in Ioc a (a+2*Real.pi), b (t : Angles)*v t)) := by
+  let V : Angles → ℝ := AddCircle.liftIoc (2*Real.pi) a v
+  have hvi : Integrable V (volume : Measure Angles) := (angleRep_preserving a).integrable_comp_of_integrable hv
+  have hva : Integrable V angularMeasure := by
+    rw [AddCircle.volume_eq_smul_haarAddCircle] at hvi
+    exact (integrable_smul_measure (by positivity : ENNReal.ofReal (2*Real.pi) ≠ 0) (ENNReal.ofReal_ne_top)).mp hvi
+  have h := (hw V hva).const_mul (2*Real.pi)
+  have he (f : Angles → ℝ) : (∫ t in Ioc a (a+2*Real.pi), f (t : Angles)*v t) =
+      (2*Real.pi)*(∫ t, f t*V t ∂angularMeasure) := by
+    have hpre := AddCircle.integral_preimage (2*Real.pi) a (fun t => f t*V t)
+    have hi : (∫ t in Ioc a (a+2*Real.pi), f (t : Angles)*v t) =
+        ∫ t in Ioc a (a+2*Real.pi), f (t : Angles)*V (t : Angles) := by
+      apply setIntegral_congr_fun measurableSet_Ioc
+      intro t ht
+      change f (t : Angles)*v t=f (t : Angles)*V (t : Angles)
+      rw [show V (t : Angles)=v t from AddCircle.liftIoc_coe_apply ht]
+    rw [hi,hpre,AddCircle.volume_eq_smul_haarAddCircle,integral_smul_measure,
+      ENNReal.toReal_ofReal (by positivity : 0 ≤ 2*Real.pi),smul_eq_mul]
+    rfl
+  simpa only [← he] using h
+lemma weak_restrict {X : Type*} [MeasurableSpace X] {μ : Measure X}
+    {F : ℕ → X → ℝ} {b : X → ℝ}
+    (hw : ∀ v : X → ℝ, Integrable v μ →
+      Tendsto (fun n => ∫ t, F n t*v t ∂μ) atTop (𝓝 (∫ t, b t*v t ∂μ)))
+    {I : Set X} (hI : MeasurableSet I) :
+    ∀ v : X → ℝ, Integrable v (μ.restrict I) →
+      Tendsto (fun n => ∫ t, F n t*v t ∂μ.restrict I) atTop
+        (𝓝 (∫ t, b t*v t ∂μ.restrict I)) := by
+  intro v hv
+  have hvi : Integrable (I.indicator v) μ := (integrable_indicator_iff hI).mpr hv
+  have h := hw (I.indicator v) hvi
+  have he (f : X → ℝ) : (fun t => f t*I.indicator v t)=I.indicator (fun t => f t*v t) := by
+    funext t
+    by_cases ht : t ∈ I <;> simp [ht]
+  simpa only [he,integral_indicator hI] using h
+lemma weak_lift_set {F : ℕ → Angles → ℝ} {b : Angles → ℝ}
+    (hw : ∀ v : Angles → ℝ, Integrable v angularMeasure →
+      Tendsto (fun n => ∫ t, F n t*v t ∂angularMeasure) atTop (𝓝 (∫ t, b t*v t ∂angularMeasure)))
+    {I : Set ℝ} (hI : MeasurableSet I) {a : ℝ} (hIa : I ⊆ Ioc a (a+2*Real.pi)) :
+    ∀ v : ℝ → ℝ, Integrable v (volume.restrict I) →
+      Tendsto (fun n => ∫ t in I, F n (t : Angles)*v t) atTop
+        (𝓝 (∫ t in I, b (t : Angles)*v t)) := by
+  have h := weak_restrict (weak_lift_Ioc hw a) hI
+  simpa only [Measure.restrict_restrict hI,inter_eq_left.mpr hIa] using h
+end EuclideanFiveColor
+end
+
+noncomputable section
+open MeasureTheory Filter Topology
+namespace EuclideanFiveColor
+lemma weak_strong_disjoint_single {X : Type*} [MeasurableSpace X]
+    {μ : Measure X} [IsFiniteMeasure μ]
+    {F : ℕ → X → ℝ} {b : X → ℝ} {G : ℕ → X → ℝ} {g : X → ℝ}
+    (hF : ∀ n, Measurable (F n)) (hb : Measurable b)
+    (hG : ∀ n, Measurable (G n)) (hg : Measurable g)
+    (hFb : ∀ n x, 0 ≤ F n x ∧ F n x ≤ 1)
+    (hbb : ∀ᵐ x ∂μ, 0 ≤ b x ∧ b x ≤ 1)
+    (hGb : ∀ n p, 0 ≤ G n p ∧ G n p ≤ 1)
+    (hgb : ∀ p, 0 ≤ g p ∧ g p ≤ 1)
+    (hw : ∀ v : X → ℝ, Integrable v μ →
+      Tendsto (fun n => ∫ x, F n x*v x ∂μ) atTop (𝓝 (∫ x, b x*v x ∂μ)))
+    (hs : Tendsto (fun n => ∫ p, ‖G n p-g p‖ ∂μ) atTop (𝓝 0))
+    (hz : ∀ n, ∀ᵐ p ∂μ, F n p*G n p=0) :
+    ∀ᵐ p ∂μ, b p*g p=0 := by
+  have hgi : Integrable g (μ) := Integrable.of_bound hg.aestronglyMeasurable 1
+    (Eventually.of_forall fun p => by rw [Real.norm_eq_abs,abs_of_nonneg (hgb p).1]; exact (hgb p).2)
+  have hFi (n : ℕ) : Integrable (fun p : X => F n p*g p) (μ) := by
+    apply Integrable.of_bound ((hF n).mul hg).aestronglyMeasurable 1
+    exact Eventually.of_forall fun p => by
+      change ‖F n p*g p‖ ≤ 1
+      rw [Real.norm_eq_abs,abs_of_nonneg (mul_nonneg (hFb n p).1 (hgb p).1)]
+      exact (mul_le_of_le_one_left (hgb p).1 (hFb n p).2).trans (hgb p).2
+  have hbi : Integrable (fun p : X => b p*g p) (μ) := by
+    apply Integrable.of_bound (hb.mul hg).aestronglyMeasurable 1
+    filter_upwards [hbb] with p hp
+    change ‖b p*g p‖ ≤ 1
+    rw [Real.norm_eq_abs,abs_of_nonneg (mul_nonneg hp.1 (hgb p).1)]
+    exact (mul_le_of_le_one_left (hgb p).1 hp.2).trans (hgb p).2
+  have herr (n : ℕ) : Integrable (fun p => ‖G n p-g p‖) (μ) := by
+    apply Integrable.of_bound ((hG n).sub hg).norm.aestronglyMeasurable 1
+    exact Eventually.of_forall fun p => by
+      rw [norm_norm,Real.norm_eq_abs,abs_le]
+      change -1 ≤ G n p-g p ∧ G n p-g p ≤ 1
+      constructor <;> linarith [(hGb n p).1,(hGb n p).2,(hgb p).1,(hgb p).2]
+  have hi0 : Tendsto (fun n => ∫ p, F n p*g p ∂μ) atTop (𝓝 0) := by
+    apply squeeze_zero (fun n => integral_nonneg fun p => mul_nonneg (hFb n p).1 (hgb p).1) _ hs
+    intro n
+    apply integral_mono_ae (hFi n) (herr n)
+    filter_upwards [hz n] with p hp
+    rw [Real.norm_eq_abs]
+    calc
+      F n p*g p = F n p*(g p-G n p) := by rw [mul_sub,hp,sub_zero]
+      _ ≤ |g p-G n p| := by
+        calc
+          _ ≤ F n p*|g p-G n p| := mul_le_mul_of_nonneg_left (le_abs_self _) (hFb n p).1
+          _ ≤ 1*|g p-G n p| := mul_le_mul_of_nonneg_right (hFb n p).2 (abs_nonneg _)
+          _ = _ := one_mul _
+      _ = |G n p-g p| := abs_sub_comm _ _
+  have he0 : (∫ p, b p*g p ∂μ)=0 := tendsto_nhds_unique (hw g hgi) hi0
+  exact (integral_eq_zero_iff_of_nonneg_ae
+    (hbb.mono fun p hp => mul_nonneg hp.1 (hgb p).1) hbi).mp he0
+lemma weak_strong_disjoint_single_eventual {X : Type*} [MeasurableSpace X]
+    {μ : Measure X} [IsFiniteMeasure μ]
+    {F : ℕ → X → ℝ} {b : X → ℝ} {G : ℕ → X → ℝ} {g : X → ℝ}
+    (hF : ∀ n, Measurable (F n)) (hb : Measurable b)
+    (hG : ∀ n, Measurable (G n)) (hg : Measurable g)
+    (hFb : ∀ n x, 0 ≤ F n x ∧ F n x ≤ 1)
+    (hbb : ∀ᵐ x ∂μ, 0 ≤ b x ∧ b x ≤ 1)
+    (hGb : ∀ n p, 0 ≤ G n p ∧ G n p ≤ 1)
+    (hgb : ∀ p, 0 ≤ g p ∧ g p ≤ 1)
+    (hw : ∀ v : X → ℝ, Integrable v μ →
+      Tendsto (fun n => ∫ x, F n x*v x ∂μ) atTop (𝓝 (∫ x, b x*v x ∂μ)))
+    (hs : Tendsto (fun n => ∫ p, ‖G n p-g p‖ ∂μ) atTop (𝓝 0))
+    (hz : ∀ᶠ n in atTop, ∀ᵐ p ∂μ, F n p*G n p=0) :
+    ∀ᵐ p ∂μ, b p*g p=0 := by
+  obtain ⟨N,hN⟩ := eventually_atTop.mp hz
+  exact weak_strong_disjoint_single (fun n => hF (n+N)) hb (fun n => hG (n+N)) hg
+    (fun n => hFb (n+N)) hbb (fun n => hGb (n+N)) hgb
+    (fun v hv => (tendsto_add_atTop_iff_nat N).mpr (hw v hv))
+    ((tendsto_add_atTop_iff_nat N).mpr hs) (fun n => hN _ (Nat.le_add_left N n))
+end EuclideanFiveColor
+end
+
+noncomputable section
+open MeasureTheory Filter Topology Set
+open scoped NNReal
+namespace EuclideanFiveColor
+variable {k : ℕ} (c : ℂ → Fin k) (hc : Measurable c)
+lemma palette_trace_branch (hw : WeakColoring k c) {x z : ℂ} {t u a : ℝ}
+    (h : Real.sin (u-t) ≠ 0) {P : Set CircleParameter} {I : Set ℝ} {K : ℝ≥0}
+    (hP : P ∈ 𝓝 (samplingParameter x z 1))
+    (hI : MeasurableSet I) (hIa : I ⊆ Ioc a (a+2*Real.pi))
+    (hA : ∀ q ∈ P, AntilipschitzWith K (I.domRestrict (fun v => circleEndpoint h (q,v))))
+    (hB : ∀ q ∈ P, ∀ v ∈ I, circleTarget (q,v) ∈ (circleInverseHomeo h).target ∧
+      Real.sin ((circleInverse h (circleTarget (q,v))).2-(circleInverse h (circleTarget (q,v))).1) ≠ 0 ∧
+      circleEndpoint h (q,v) ∈ Metric.closedBall t 1)
+    {b : TraceVector k} (hb : IsTrace c hc z b) (i : Fin k) :
+    ∀ᵐ f : ℝ ∂volume.restrict I, traceValue b i (f : Angles)*
+      (liftPalette c hc x i).indicator (fun _ => (1 : ℝ)) (circleEndpoint h (samplingParameter x z 1,f))=0 := by
+  have : IsFiniteMeasure (volume.restrict I) := isFiniteMeasure_restrict.mpr
+    (ne_of_lt ((measure_mono (hIa.trans Ioc_subset_Icc_self)).trans_lt isCompact_Icc.measure_lt_top))
+  have : IsFiniteMeasure (volume.restrict (Metric.closedBall t 1)) :=
+    isFiniteMeasure_restrict.mpr (isCompact_closedBall t 1).measure_lt_top.ne
+  have hp : samplingParameter x z 1 ∈ P := mem_of_mem_nhds hP
+  obtain ⟨y,r,hy,hyt,hrt,hwt⟩ := trace_realization_weakStar c hc hb
+  let p : ℕ → CircleParameter := fun n => samplingParameter x (y n) (r n)
+  let p₀ := samplingParameter x z 1
+  let T : ℕ → ℝ → ℝ := fun n f => circleEndpoint h (p n,f)
+  let S : ℝ → ℝ := fun f => circleEndpoint h (p₀,f)
+  have hpt : Tendsto p atTop (𝓝 p₀) := samplingParameter_tendsto hyt hrt
+  have hpn : ∀ᶠ n in atTop, p n ∈ P := hpt hP
+  have hTm (n : ℕ) : Measurable (T n) :=
+    (circleEndpoint_measurable h).comp (measurable_const.prodMk measurable_id)
+  have hSm : Measurable S := (circleEndpoint_measurable h).comp (measurable_const.prodMk measurable_id)
+  let D := liftPalette c hc x i
+  have hDm : MeasurableSet D := liftPalette_measurable c hc x i
+  have hDind : Measurable (D.indicator (fun _ => (1 : ℝ))) := measurable_const.indicator hDm
+  have hs : Tendsto (fun n => ∫ f in I, ‖D.indicator (fun _ => (1 : ℝ)) (T n f)-
+      D.indicator (fun _ => (1 : ℝ)) (S f)‖) atTop (𝓝 0) := by
+    apply line_indicator_pullback_L1 hTm hSm (isCompact_closedBall t 1) hDm
+    · exact hpn.mono fun n hn => hA (p n) hn
+    · exact hA p₀ hp
+    · exact hpn.mono fun n hn f hf => (hB (p n) hn f hf).2.2
+    · exact fun f hf => (hB p₀ hp f hf).2.2
+    · filter_upwards [ae_restrict_mem hI] with f hf
+      exact ((circleInverse_continuousAt h (hB p₀ hp f hf).1).fst.comp
+        circleTarget_continuous.continuousAt).tendsto.comp (hpt.prodMk_nhds tendsto_const_nhds)
+  have hz : ∀ᶠ n in atTop, ∀ᵐ f : ℝ ∂volume.restrict I,
+      sampleFn c (y n) (r n) i (f : Angles)*D.indicator (fun _ => (1 : ℝ)) (T n f)=0 := by
+    filter_upwards [hpn] with n hn
+    exact palette_sample_branch_exclusion c hc hw (hy n) i hI
+      (line_antilipschitz_qmp (hTm n) (hA (p n) hn))
+      (fun f hf => (hB (p n) hn f hf).2.1)
+      (fun f hf => circleInverse_sampling h (hB (p n) hn f hf).1)
+  have hbb : ∀ᵐ f : ℝ ∂volume.restrict I, 0 ≤ traceValue b i (f : Angles) ∧ traceValue b i (f : Angles) ≤ 1 :=
+    ae_restrict_of_ae (angleMk_quasiMeasurePreserving.ae
+      (((trace_bounds c hc hb).1 i).and ((trace_bounds c hc hb).2.1 i)))
+  have hgi (f : ℝ) : 0 ≤ D.indicator (fun _ => (1 : ℝ)) f ∧ D.indicator (fun _ => (1 : ℝ)) f ≤ 1 := by
+    by_cases hf : f ∈ D <;> simp [hf]
+  exact weak_strong_disjoint_single_eventual
+    (fun n => (sampleFn_measurable c hc (y n) (r n) i).comp angleMk_quasiMeasurePreserving.measurable)
+    ((Lp.stronglyMeasurable (traceValue b i)).measurable.comp angleMk_quasiMeasurePreserving.measurable)
+    (fun n => hDind.comp (hTm n)) (hDind.comp hSm)
+    (fun n f => ⟨sampleFn_nonneg c _ _ _ _,sampleFn_le_one c _ _ _ _⟩) hbb
+    (fun n f => hgi (T n f)) (fun f => hgi (S f))
+    (weak_lift_set (hwt i) hI hIa) hs hz
+end EuclideanFiveColor
+end
+
+noncomputable section
+open MeasureTheory Filter Topology Set
+namespace EuclideanFiveColor
+variable {k : ℕ} (c : ℂ → Fin k) (hc : Measurable c)
+lemma palette_branch_local (hw : WeakColoring k c) {x z : ℂ} {t u a : ℝ}
+    (h : Real.sin (u-t) ≠ 0)
+    (he : circleTarget (samplingParameter x z 1,a)=circleSum 1 (t,u))
+    (ha : Real.sin (a-u) ≠ 0) :
+    ∃ I : Set ℝ, IsOpen I ∧ a ∈ I ∧ ∀ i : Fin k,
+      ∀ᵐ f : ℝ ∂volume.restrict I,
+        f ∈ liftPalette c hc z i → circleEndpoint h (samplingParameter x z 1,f) ∉ liftPalette c hc x i := by
+  obtain ⟨P,I,hP,hp,hI,haI,hIa,K,hA,hB⟩ := circleEndpoint_good_neighborhood h he rfl ha
+  refine ⟨I,hI,haI,fun i => ?_⟩
+  have ht : ∀ᵐ f : ℝ ∂volume.restrict I, ∀ b ∈ paletteGenerators c hc z i,
+      traceValue b.1 i (f : Angles)*(liftPalette c hc x i).indicator (fun _ => (1 : ℝ))
+        (circleEndpoint h (samplingParameter x z 1,f))=0 := by
+    apply (ae_ball_iff (paletteGenerators_countable c hc z i)).mpr
+    intro b _
+    exact palette_trace_branch c hc hw h (hP.mem_nhds hp) hI.measurableSet hIa hA hB b.2 i
+  filter_upwards [ht] with f hf hfm hxs
+  obtain ⟨b,hb,hbf⟩ := mem_iUnion₂.mp (show (f : Angles) ∈ paletteSet c hc z i from hfm)
+  have hz := hf b hb
+  rw [indicator_of_mem hxs,mul_one] at hz
+  exact (ne_of_gt hbf) hz
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+lemma circleInverse_left {t u : ℝ} (h : Real.sin (u-t) ≠ 0) {q : RealPlane}
+    (hq : q ∈ (circleInverseHomeo h).source) : circleInverse h (circleSum 1 q)=q := by
+  have ht : circleSum 1 q ∈ (circleInverseHomeo h).target := (circleInverseHomeo h).map_source hq
+  rw [circleInverse_eq h ht]
+  exact (circleInverseHomeo h).left_inv hq
+lemma circleEndpoint_eventuallyEq {x z : ℂ} {a : ℝ} {s t : ℝ → ℝ}
+    (hs : ContinuousAt s a) (ht : ContinuousAt t a) (h : Real.sin (t a-s a) ≠ 0)
+    (he : ∀ᶠ f in 𝓝 a, circleTarget (samplingParameter x z 1,f)=circleSum 1 (s f,t f)) :
+    ∀ᶠ f in 𝓝 a, circleEndpoint h (samplingParameter x z 1,f)=s f := by
+  have hp : (s a,t a) ∈ (circleInverseHomeo h).source :=
+    (show HasStrictFDerivAt (circleSum 1)
+      (circleSumDerivEquiv (r := 1) (p := (s a,t a)) (by simpa only [one_mul] using h) : RealPlane →L[ℝ] RealPlane)
+      (s a,t a) from circleSum_strict 1 (s a,t a)).mem_toOpenPartialHomeomorph_source
+  have hf := (hs.prodMk ht) ((circleInverseHomeo h).open_source.mem_nhds hp)
+  filter_upwards [hf,he] with f hf he
+  simp only [circleEndpoint,he,circleInverse_left h hf]
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+lemma ae_of_locally_ae {X : Type*} [TopologicalSpace X] [SecondCountableTopology X]
+    [MeasurableSpace X] (μ : Measure X) {S : Set X} {P : X → Prop}
+    (h : ∀x∈S,∃U : Set X,IsOpen U ∧ x∈U ∧ ∀ᵐy∂μ,y∈U → P y) :
+    ∀ᵐx∂μ,x∈S → P x := by
+  choose U ho hm ha using fun x : S => h x x.2
+  obtain ⟨T,hT,hcov⟩ := (HereditarilyLindelofSpace.isLindelof S).elim_countable_subcover U ho
+    (by intro x hx; exact mem_iUnion.mpr ⟨⟨x,hx⟩,hm ⟨x,hx⟩⟩)
+  have hh : ∀ᵐy∂μ,∀x∈T,y∈U x → P y := (ae_ball_iff hT).mpr (fun x _ => ha x)
+  filter_upwards [hh] with y hy hs
+  obtain ⟨x,hx,hyU⟩ := mem_iUnion₂.mp (hcov hs)
+  exact hy x hx hyU
+lemma ae_restrict_of_locally_ae {X : Type*} [TopologicalSpace X] [SecondCountableTopology X]
+    [MeasurableSpace X] [BorelSpace X] {μ : Measure X} {S : Set X} {P : X → Prop}
+    (hS : MeasurableSet S)
+    (h : ∀x∈S,∃U : Set X,IsOpen U ∧ x∈U ∧ ∀ᵐy∂μ.restrict U,P y) :
+    ∀ᵐx∂μ.restrict S,P x := by
+  apply (ae_restrict_iff' hS).mpr
+  apply ae_of_locally_ae μ
+  intro x hx
+  obtain ⟨U,hU,hxU,ha⟩ := h x hx
+  exact ⟨U,hU,hxU,(ae_restrict_iff' hU.measurableSet).mp ha⟩
+end EuclideanFiveColor
+end
+
+noncomputable section
+open MeasureTheory Filter Topology Set
+namespace EuclideanFiveColor
+variable {k : ℕ} (c : ℂ → Fin k) (hc : Measurable c)
+lemma palette_transport_real (hw : WeakColoring k c) {x z : ℂ} {U : Set ℝ} (hU : IsOpen U)
+    {s t : ℝ → ℝ} (hs : ContinuousOn s U) (ht : ContinuousOn t U)
+    (hnp : ∀ a ∈ U, Real.sin (t a-s a) ≠ 0)
+    (hss : ∀ a ∈ U, Real.sin (a-t a) ≠ 0)
+    (he : ∀ a ∈ U, circleTarget (samplingParameter x z 1,a)=circleSum 1 (s a,t a)) (i : Fin k) :
+    ∀ᵐ f : ℝ ∂volume.restrict U, f ∈ liftPalette c hc z i → s f ∉ liftPalette c hc x i := by
+  apply ae_restrict_of_locally_ae hU.measurableSet
+  intro a ha
+  obtain ⟨I,hI,haI,hb⟩ := palette_branch_local c hc hw (hnp a ha) (he a ha) (hss a ha)
+  have heq := circleEndpoint_eventuallyEq (hs.continuousAt (hU.mem_nhds ha))
+    (ht.continuousAt (hU.mem_nhds ha)) (hnp a ha)
+    ((show ∀ᶠ f in 𝓝 a, f ∈ U from hU.mem_nhds ha).mono fun f hf => he f hf)
+  obtain ⟨V,hV,hVo,haV⟩ := mem_nhds_iff.mp heq
+  refine ⟨I ∩ V,hI.inter hVo,⟨haI,haV⟩,?_⟩
+  have hb' := ae_restrict_of_ae_restrict_of_subset (s := I ∩ V) (t := I) inter_subset_left (hb i)
+  filter_upwards [hb',ae_restrict_mem (hI.inter hVo).measurableSet] with f hf hfI
+  rw [hV hfI.2] at hf
+  exact hf
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Filter Topology Set MeasureTheory
+namespace EuclideanFiveColor
+lemma endpoint_fiber_nonzero {x z : ℂ} {a ds dt : ℝ} {s t : ℝ → ℝ}
+    (hs : HasDerivAt s ds a) (ht : HasDerivAt t dt a) (hds : ds ≠ 0)
+    (hn : Real.sin (t a-s a) ≠ 0)
+    (he : ∀ᶠ f in 𝓝 a, circleTarget (samplingParameter x z 1,f)=circleSum 1 (s f,t f)) :
+    Real.sin (a-t a) ≠ 0 := by
+  have hv : HasDerivAt (fun f => circleTarget (samplingParameter x z 1,f))
+      (-Real.sin a,Real.cos a) a := by
+    simpa only [circleTarget,samplingParameter,one_smul] using
+      ((Real.hasDerivAt_cos a).prodMk (Real.hasDerivAt_sin a)).const_add (Complex.equivRealProdCLM (z-x))
+  have hc₁ := (Real.hasDerivAt_cos (s a)).comp a hs
+  have hc₂ := (Real.hasDerivAt_cos (t a)).comp a ht
+  have hs₁ := (Real.hasDerivAt_sin (s a)).comp a hs
+  have hs₂ := (Real.hasDerivAt_sin (t a)).comp a ht
+  have hw : HasDerivAt (fun f => circleSum 1 (s f,t f))
+      (-Real.sin (s a)*ds+-Real.sin (t a)*dt,Real.cos (s a)*ds+Real.cos (t a)*dt) a := by
+    have h := (hc₁.add hc₂).prodMk (hs₁.add hs₂)
+    convert h using 1; simp [circleSum,Function.comp_def]
+  have hd := (hv.congr_of_eventuallyEq (Filter.EventuallyEq.symm he)).unique hw
+  have h₁ := congrArg Prod.fst hd
+  have h₂ := congrArg Prod.snd hd
+  dsimp only at h₁ h₂
+  have hid : Real.sin (a-t a)=ds*Real.sin (s a-t a) := by
+    rw [Real.sin_sub,Real.sin_sub]
+    linear_combination -(Real.cos (t a))*h₁-(Real.sin (t a))*h₂
+  rw [hid]
+  apply mul_ne_zero hds
+  rw [show s a-t a=-(t a-s a) by ring,Real.sin_neg,neg_ne_zero]
+  exact hn
+end EuclideanFiveColor
+end
+
+noncomputable section
+open MeasureTheory Filter Topology Set
+namespace EuclideanFiveColor
+variable {k : ℕ} (c : ℂ → Fin k) (hc : Measurable c)
+lemma palette_smooth_transport (hw : WeakColoring k c) {x z : ℂ} {U : Set ℝ} (hU : IsOpen U)
+    {s t ds dt : ℝ → ℝ} (hs : ∀ a ∈ U, HasDerivAt s (ds a) a)
+    (ht : ∀ a ∈ U, HasDerivAt t (dt a) a) (hds : ∀ a ∈ U, ds a ≠ 0)
+    (hnp : ∀ a ∈ U, Real.sin (t a-s a) ≠ 0)
+    (he : ∀ a ∈ U, circleTarget (samplingParameter x z 1,a)=circleSum 1 (s a,t a)) (i : Fin k) :
+    ∀ᵐ f : ℝ ∂volume.restrict U, f ∈ liftPalette c hc z i → s f ∉ liftPalette c hc x i := by
+  apply palette_transport_real c hc hw hU (fun a ha => (hs a ha).continuousAt.continuousWithinAt)
+    (fun a ha => (ht a ha).continuousAt.continuousWithinAt) hnp ?_ he i
+  intro a ha
+  exact endpoint_fiber_nonzero (hs a ha) (ht a ha) (hds a ha) (hnp a ha)
+    ((show ∀ᶠ f in 𝓝 a, f ∈ U from hU.mem_nhds ha).mono fun f hf => he f hf)
+end EuclideanFiveColor
+end
+
+noncomputable section
+open MeasureTheory Set Filter Topology
+namespace EuclideanFiveColor
+lemma palette_transport_param {k : ℕ} (c : ℂ → Fin k) (hc : Measurable c) (hw : WeakColoring k c)
+    {x z : ℂ} {a s t Da Ds Dt : ℝ → ℝ}
+    (ha : ∀ r, HasDerivAt a (Da r) r) (hpos : ∀ r, (1/2 : ℝ) ≤ Da r)
+    (hs : ∀ r, HasDerivAt s (Ds r) r) (ht : ∀ r, HasDerivAt t (Dt r) r)
+    (hDs : ∀ r, Ds r ≠ 0) (hnp : ∀ r, Real.sin (t r-s r) ≠ 0)
+    (he : ∀ r, circleTarget (samplingParameter x z 1,a r)=circleSum 1 (s r,t r)) (i : Fin k) :
+    ∀ᵐ r : ℝ ∂volume, a r ∈ liftPalette c hc z i → s r ∉ liftPalette c hc x i := by
+  let e := derivLowerOrderIso ha hpos
+  have heq : ∀ r, e r=a r := fun _ => rfl
+  have hinv : ∀ y, HasDerivAt e.symm (Da (e.symm y))⁻¹ y := by
+    intro y
+    apply HasDerivAt.of_local_left_inverse e.symm.continuous.continuousAt (ha (e.symm y))
+      (ne_of_gt (lt_of_lt_of_le (by norm_num) (hpos _)))
+    exact Filter.Eventually.of_forall fun r => by rw [← heq,e.apply_symm_apply]
+  have h := palette_smooth_transport c hc hw (x := x) (z := z) isOpen_univ
+    (s := s ∘ e.symm) (t := t ∘ e.symm)
+    (fun y _ => (hs _).comp y (hinv y)) (fun y _ => (ht _).comp y (hinv y))
+    (fun y _ => mul_ne_zero (hDs _) (inv_ne_zero (ne_of_gt (lt_of_lt_of_le (by norm_num) (hpos _)))))
+    (fun y _ => hnp _) (fun y _ => by simpa only [Function.comp_apply,← heq,e.apply_symm_apply] using he (e.symm y)) i
+  rw [Measure.restrict_univ] at h
+  have h' := (deriv_lower_qmp ha hpos).ae h
+  simpa only [Function.comp_apply,← heq,e.symm_apply_apply] using h'
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Filter Topology Set MeasureTheory
+namespace EuclideanFiveColor
+def sixty : ℝ := Real.pi/3
+lemma sin_sixty_ne : Real.sin sixty ≠ 0 := by
+  rw [sixty,Real.sin_pi_div_three]
+  positivity
+lemma sin_twice_sixty_ne : Real.sin (2*sixty) ≠ 0 := by
+  rw [Real.sin_two_mul,sixty,Real.sin_pi_div_three,Real.cos_pi_div_three]
+  positivity
+lemma sixty_sum (x : ℂ) (a : ℝ) :
+    circleTarget (samplingParameter x x 1,a)=circleSum 1 (a+sixty,a-sixty) := by
+  simp only [circleTarget,samplingParameter,sub_self,map_zero,one_smul,zero_add]
+  ext <;> simp only [circleSum,one_mul,Real.cos_add,Real.cos_sub,Real.sin_add,Real.sin_sub]
+  · rw [sixty,Real.cos_pi_div_three]
+    ring
+  · rw [sixty,Real.cos_pi_div_three]
+    ring
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Filter Topology Set MeasureTheory
+namespace EuclideanFiveColor
+lemma ae_angular_of_ae_line {p : Angles → Prop} (hp : MeasurableSet {t | p t})
+    (h : ∀ᵐ t : ℝ ∂volume, p (t : Angles)) : ∀ᵐ t ∂angularMeasure, p t := by
+  have hh : ∀ᵐ t ∂(volume : Measure Angles), p t := by
+    rw [← (AddCircle.measurePreserving_mk (2*Real.pi) 0).map_eq]
+    exact (ae_map_iff (AddCircle.measurePreserving_mk (2*Real.pi) 0).measurable.aemeasurable hp).mpr
+      (ae_restrict_of_ae h)
+  rw [AddCircle.volume_eq_smul_haarAddCircle] at hh
+  exact (Measure.ae_ennreal_smul_measure_iff (by positivity : ENNReal.ofReal (2*Real.pi) ≠ 0)).mp hh
+lemma ae_sin_sub_ne_zero (t : ℝ) : ∀ᵐ a : ℝ ∂volume, Real.sin (a-t) ≠ 0 := by
+  have hs : {a : ℝ | Real.sin (a-t)=0} ⊆ range (fun n : ℤ => t+n*Real.pi) := by
+    intro a ha
+    obtain ⟨n,hn⟩ := Real.sin_eq_zero_iff.mp ha
+    exact ⟨n,by linarith⟩
+  change volume {a : ℝ | ¬Real.sin (a-t) ≠ 0}=0
+  simpa only [not_not] using measure_mono_null hs ((countable_range _).measure_zero volume)
+end EuclideanFiveColor
+end
+
+noncomputable section
+open MeasureTheory Filter Topology Set
+namespace EuclideanFiveColor
+variable {k : ℕ} (c : ℂ → Fin k) (hc : Measurable c)
+lemma palette_sixty_real (hw : WeakColoring k c) (x : ℂ) (i : Fin k) :
+    ∀ᵐ f : ℝ ∂volume, f ∈ liftPalette c hc x i → f+sixty ∉ liftPalette c hc x i := by
+  have h := palette_transport_real c hc hw (x := x) (z := x) isOpen_univ
+    (s := fun a => a+sixty) (t := fun a => a-sixty)
+    (continuous_id.add continuous_const).continuousOn (continuous_id.sub continuous_const).continuousOn
+    (fun a _ => by
+      rw [show (a-sixty)-(a+sixty)=-(2*sixty) by ring,Real.sin_neg,neg_ne_zero]
+      exact sin_twice_sixty_ne)
+    (fun a _ => by simpa only [sub_sub_cancel] using sin_sixty_ne)
+    (fun a _ => sixty_sum x a) i
+  simpa only [Measure.restrict_univ] using h
+lemma palette_sixty (hw : WeakColoring k c) (x : ℂ) (i : Fin k) :
+    ∀ᵐ f ∂angularMeasure, f ∈ paletteSet c hc x i → f+(sixty : Angles) ∉ paletteSet c hc x i := by
+  apply ae_angular_of_ae_line
+  · simpa only [imp_iff_not_or,Set.compl_def,Set.union_def,Set.preimage,Function.id_def,Set.mem_ofPred_eq] using (paletteSet_measurable c hc x i).compl.union
+      (((paletteSet_measurable c hc x i).preimage (measurable_id.add_const (sixty : Angles))).compl)
+  · simpa only [liftPalette,mem_preimage,AddCircle.coe_add] using palette_sixty_real c hc hw x i
+lemma palette_sixty_disjoint (hw : WeakColoring k c) (x : ℂ) :
+    ∀ᵐ f ∂angularMeasure, Disjoint (palette c hc x f) (palette c hc x (f+(sixty : Angles))) := by
+  filter_upwards [ae_all_iff.mpr (fun i => palette_sixty c hc hw x i)] with f hf
+  exact Finset.disjoint_left.mpr fun i hi hj => hf i ((mem_palette c hc x f i).mp hi)
+    ((mem_palette c hc x (f+(sixty : Angles)) i).mp hj)
+end EuclideanFiveColor
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PlaneColoring/Palettes.lean b/lean/OAI/Geometry/PlaneColoring/Palettes.lean
new file mode 100644
index 0000000000000000000000000000000000000000..9ea57bc871baed5237d7b3cca892cb628057bbf9
--- /dev/null
+++ b/lean/OAI/Geometry/PlaneColoring/Palettes.lean
@@ -0,0 +1,124 @@
+import OAI.Geometry.PlaneColoring.Transitions
+
+namespace OAI
+
+noncomputable section
+open MeasureTheory Filter Topology
+namespace EuclideanFiveColor
+variable {k : ℕ} (c : ℂ → Fin k) (hc : Measurable c)
+abbrev Traces (x : ℂ) := {b : TraceVector k // IsTrace c hc x b}
+def tracePositive (x : ℂ) (i : Fin k) (b : Traces c hc x) : Set Angles :=
+  {t | 0 < traceValue b.1 i t}
+lemma tracePositive_measurable (x : ℂ) (i : Fin k) (b : Traces c hc x) :
+    MeasurableSet (tracePositive c hc x i b) :=
+  measurableSet_lt measurable_const (Lp.stronglyMeasurable _).measurable
+def paletteGenerators (x : ℂ) (i : Fin k) : Set (Traces c hc x) :=
+  (exists_countable_essential_union angularMeasure (tracePositive c hc x i) (tracePositive_measurable c hc x i)).choose
+lemma paletteGenerators_countable (x : ℂ) (i : Fin k) : (paletteGenerators c hc x i).Countable :=
+  (exists_countable_essential_union angularMeasure (tracePositive c hc x i) (tracePositive_measurable c hc x i)).choose_spec.1
+def paletteSet (x : ℂ) (i : Fin k) : Set Angles :=
+  ⋃ b ∈ paletteGenerators c hc x i, tracePositive c hc x i b
+lemma paletteSet_measurable (x : ℂ) (i : Fin k) : MeasurableSet (paletteSet c hc x i) :=
+  MeasurableSet.biUnion (paletteGenerators_countable c hc x i) (fun b _ => tracePositive_measurable c hc x i b)
+lemma tracePositive_le_palette (x : ℂ) (i : Fin k) (b : Traces c hc x) :
+    tracePositive c hc x i b ≤ᵐ[angularMeasure] paletteSet c hc x i :=
+  (exists_countable_essential_union angularMeasure (tracePositive c hc x i) (tracePositive_measurable c hc x i)).choose_spec.2 b
+lemma paletteSet_minimal (x : ℂ) (i : Fin k) {A : Set Angles}
+    (hA : ∀ b : Traces c hc x, tracePositive c hc x i b ≤ᵐ[angularMeasure] A) :
+    paletteSet c hc x i ≤ᵐ[angularMeasure] A := by
+  have hh : ∀ᵐ t ∂angularMeasure, ∀ b ∈ paletteGenerators c hc x i, t ∈ tracePositive c hc x i b → t ∈ A :=
+    (ae_ball_iff (paletteGenerators_countable c hc x i)).mpr (fun b _ => hA b)
+  filter_upwards [hh] with t ht
+  intro hm
+  rcases Set.mem_iUnion₂.mp hm with ⟨b,hb,hbt⟩
+  exact ht b hb hbt
+def palette (x : ℂ) (t : Angles) : Finset (Fin k) := by
+  classical
+  exact Finset.univ.filter (fun i => t ∈ paletteSet c hc x i)
+@[simp] lemma mem_palette (x : ℂ) (t : Angles) (i : Fin k) :
+    i ∈ palette c hc x t ↔ t ∈ paletteSet c hc x i := by simp [palette]
+lemma palette_nonempty (x : ℂ) : ∀ᵐ t ∂angularMeasure, (palette c hc x t).Nonempty := by
+  obtain ⟨b,hb⟩ := exists_trace c hc x
+  let B : Traces c hc x := ⟨b,hb⟩
+  have hsum := (trace_bounds c hc hb).2.2
+  have hs : (fun t => ∑ i, traceValue b i t) =ᵐ[angularMeasure] (fun _ => (1 : ℝ)) := by
+    have hh := Lp.coeFn_fun_finsetSum Finset.univ (traceValue b)
+    rw [hsum] at hh
+    exact hh.symm.trans (oneL2_ae angularMeasure)
+  filter_upwards [hs,ae_all_iff.mpr (fun i => tracePositive_le_palette c hc x i B)] with t ht hP
+  have he : ∃ i : Fin k, 0 < traceValue b i t := by
+    by_contra! hh
+    have hz : (∑ i, traceValue b i t) ≤ 0 := Finset.sum_nonpos (fun i _ => hh i)
+    linarith
+  obtain ⟨i,hi⟩ := he
+  exact ⟨i,(mem_palette c hc x t i).mpr ((hP i) hi)⟩
+end EuclideanFiveColor
+end
+
+noncomputable section
+open MeasureTheory Filter Topology
+open scoped RealInnerProductSpace
+namespace EuclideanFiveColor
+variable {k : ℕ} (c : ℂ → Fin k) (hc : Measurable c)
+lemma inner_oneL2_integral (f : RealL2 angularMeasure) :
+    ⟪f,oneL2 angularMeasure⟫ = ∫ t, f t ∂angularMeasure := by
+  rw [real_inner_comm]
+  exact (L2.inner_indicatorConstLp_one MeasurableSet.univ (measure_ne_top angularMeasure Set.univ) f).trans
+    (by simp)
+lemma sample_inner_one (i : Fin k) (q : ℂ) :
+    ⟪sample c hc q 1 i,oneL2 angularMeasure⟫ = colorMass c i q := by
+  rw [inner_oneL2_integral]
+  exact integral_congr_ae (sample_ae c hc q 1 i)
+lemma transition_avoiding_trace (hw : WeakColoring k c) {i j : Fin k} {x : ℂ}
+    (he : TransitionEdge c i j x) :
+    ∃ b : TraceVector k, IsTrace c hc x b ∧ traceValue b i=0 ∧ traceValue b j=0 := by
+  obtain ⟨q,hq⟩ := Classical.axiomOfChoice (fun n : ℕ => transition_small_mass c hc hw he
+    (R := 1/((n:ℝ)+1)) (ε := 1/((n:ℝ)+1)) (by positivity) (by positivity))
+  have hqt : Tendsto q atTop (𝓝 x) := by
+    rw [tendsto_iff_dist_tendsto_zero]
+    apply squeeze_zero (fun n => dist_nonneg)
+      (fun n => (hq n).1.le)
+    exact tendsto_one_div_add_atTop_nhds_zero_nat
+  obtain ⟨b,_,φ,hφ,hb⟩ := (isCompact_univ : IsCompact (Set.univ : Set (TraceVector k))).isSeqCompact.subseq_of_frequently_in
+    (Eventually.frequently (Eventually.of_forall fun n => Set.mem_univ (sampleCode c hc (q n) 1)))
+  have htrace : IsTrace c hc x b := trace_of_realization c hc
+    (fun n => (hq (φ n)).2.1) (hqt.comp hφ.tendsto_atTop) tendsto_const_nhds hb
+  have hmass (a : Fin k) (ha : a=i ∨ a=j) : ∫ t, traceValue b a t ∂angularMeasure = 0 := by
+    have ht := (traceCode_tendsto_iff.mp hb) a (oneL2 angularMeasure)
+    simp only [Function.comp_apply,traceValue_sampleCode,sample_inner_one] at ht
+    rw [inner_oneL2_integral] at ht
+    have ht' : Tendsto (fun n => colorMass c a (q n)) atTop (𝓝 0) := by
+      apply squeeze_zero (fun n => colorMass_nonneg c a (q n)) (fun n => ?_)
+        tendsto_one_div_add_atTop_nhds_zero_nat
+      rcases ha with rfl | rfl
+      · exact (le_add_of_nonneg_right (colorMass_nonneg c j (q n))).trans (hq n).2.2.le
+      · exact (le_add_of_nonneg_left (colorMass_nonneg c i (q n))).trans (hq n).2.2.le
+    exact tendsto_nhds_unique ht (ht'.comp hφ.tendsto_atTop)
+  have hzero (a : Fin k) (ha : a=i ∨ a=j) : traceValue b a=0 := by
+    apply Lp.ext
+    have hi : Integrable (traceValue b a) angularMeasure := (Lp.memLp _).integrable (by norm_num)
+    have hz := (integral_eq_zero_iff_of_nonneg_ae ((trace_bounds c hc htrace).1 a) hi).mp (hmass a ha)
+    exact hz.trans (Lp.coeFn_zero ℝ 2 angularMeasure).symm
+  exact ⟨b,htrace,hzero i (Or.inl rfl),hzero j (Or.inr rfl)⟩
+lemma edge_palette_not_pair (hw : WeakColoring k c) {i j : Fin k} {x : ℂ}
+    (he : TransitionEdge c i j x) : ∀ᵐ t ∂angularMeasure, ¬ palette c hc x t ⊆ {i,j} := by
+  obtain ⟨b,hb,hi,hj⟩ := transition_avoiding_trace c hc hw he
+  have hs : (fun t => ∑ a, traceValue b a t) =ᵐ[angularMeasure] (fun _ => (1 : ℝ)) := by
+    have hh := Lp.coeFn_fun_finsetSum Finset.univ (traceValue b)
+    rw [(trace_bounds c hc hb).2.2] at hh
+    exact hh.symm.trans (oneL2_ae angularMeasure)
+  have hzi : traceValue b i =ᵐ[angularMeasure] (fun _ => (0:ℝ)) := by rw [hi]; exact Lp.coeFn_zero _ _ _
+  have hzj : traceValue b j =ᵐ[angularMeasure] (fun _ => (0:ℝ)) := by rw [hj]; exact Lp.coeFn_zero _ _ _
+  filter_upwards [hs,hzi,hzj,ae_all_iff.mpr (fun a => tracePositive_le_palette c hc x a ⟨b,hb⟩)] with t ht hi0 hj0 hp hsub
+  have hnon : ∀ a : Fin k, traceValue b a t ≤ 0 := by
+    intro a
+    by_contra! ha
+    have ham := hsub ((mem_palette c hc x t a).mpr (hp a ha))
+    simp only [Finset.mem_insert,Finset.mem_singleton] at ham
+    rcases ham with rfl | rfl <;> linarith
+  have hh := Finset.sum_nonpos (s := Finset.univ) (fun a _ => hnon a)
+  linarith
+end EuclideanFiveColor
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PlaneColoring/PlanarCovers.lean b/lean/OAI/Geometry/PlaneColoring/PlanarCovers.lean
new file mode 100644
index 0000000000000000000000000000000000000000..63b9f996a82d11c410e191c4fdef1f7aa1eba075
--- /dev/null
+++ b/lean/OAI/Geometry/PlaneColoring/PlanarCovers.lean
@@ -0,0 +1,402 @@
+import OAI.Geometry.PlaneColoring.DiskAverages
+
+namespace OAI
+
+noncomputable section
+open Set
+open scoped NNReal ENNReal
+namespace EuclideanFiveColor
+lemma finite_packed_cover {X : Type*} [MetricSpace X] {A : Set X} (hA : TotallyBounded A)
+    {ρ : ℝ} (hρ : 0<ρ) : ∃ C : Finset X, (↑C : Set X)⊆A ∧
+      (∀ x∈A, ∃ c∈C, dist x c≤ρ) ∧
+      (↑C : Set X).Pairwise (fun x y => ρ<dist x y) := by
+  classical
+  let e : ℝ≥0 := ⟨ρ,hρ.le⟩
+  obtain ⟨D,_,hD,hcover⟩ := Metric.exists_finite_isCover_of_totallyBounded
+    (s := A) (ε := e/2) (by exact div_ne_zero (ne_of_gt (show (0 : ℝ≥0)<e from hρ)) (by norm_num) : e/2≠0) hA
+  have hp : Metric.packingNumber e A≠⊤ := by
+    have hh := (Metric.packingNumber_two_mul_le_externalCoveringNumber (e/2) A).trans
+      hcover.externalCoveringNumber_le_encard
+    have he : 2*(e/2)=e := by ring
+    rw [he] at hh
+    exact ne_top_of_le_ne_top hD.encard_lt_top.ne hh
+  let C := Metric.maximalSeparatedSet e A
+  have hCf : C.Finite := Set.encard_ne_top_iff.mp (by
+    change (Metric.maximalSeparatedSet e A).encard≠⊤
+    rwa [Metric.encard_maximalSeparatedSet hp])
+  refine ⟨hCf.toFinset,?_,?_,?_⟩
+  · simpa only [Set.Finite.coe_toFinset] using (Metric.maximalSeparatedSet_subset (ε := e) (A := A))
+  · intro x hx
+    obtain ⟨c,hc,hd⟩ := Metric.isCover_maximalSeparatedSet hp hx
+    refine ⟨c,hCf.mem_toFinset.mpr hc,?_⟩
+    change edist x c≤(e : ℝ≥0∞) at hd
+    exact (ENNReal.ofReal_le_coe (b := e)).mp (by simpa only [edist_dist] using hd)
+  · intro x hx y hy hne
+    have hd := Metric.isSeparated_maximalSeparatedSet (ε := e) (A := A)
+      (hCf.mem_toFinset.mp hx) (hCf.mem_toFinset.mp hy) hne
+    exact (ENNReal.coe_lt_ofReal (a := e)).mp (by simpa only [edist_dist] using hd)
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory Function
+namespace EuclideanFiveColor
+variable {k : ℕ} {c : ℂ → Fin k}
+lemma diskAverage_product_integrable (hc : Measurable c) {ε : ℝ} (hε : 0<ε)
+    (i j : Fin k) (x : ℂ) (r : ℝ) :
+    IntegrableOn (fun q => diskAverage c ε i q*diskAverage c ε j q) (Metric.ball x r) := by
+  let : IsFiniteMeasure (volume.restrict (Metric.ball x r)) := ⟨by simpa using (lt_top_iff_ne_top.mpr (ball_ne_top x r))⟩
+  apply Integrable.of_bound (((diskAverage_lipschitz hc hε i).continuous.mul
+    (diskAverage_lipschitz hc hε j).continuous).aestronglyMeasurable) 1
+  exact Eventually.of_forall fun q => by
+    change ‖diskAverage c ε i q*diskAverage c ε j q‖≤1
+    rw [Real.norm_eq_abs,abs_of_nonneg (mul_nonneg (diskAverage_nonneg c ε i q) (diskAverage_nonneg c ε j q))]
+    exact (mul_le_of_le_one_left (diskAverage_nonneg c ε j q)
+      (diskAverage_le_one hc hε i q)).trans (diskAverage_le_one hc hε j q)
+lemma finite_disk_overlap_cover (hc : Measurable c) {ε α : ℝ} (hε : 0<ε) (hα : 0<α)
+    (i j : Fin k) (x : ℂ) {s r : ℝ} (hr : s+α*ε/16≤r) :
+    ∃ C : Finset ℂ, (∀ q∈Metric.ball x s,
+        α<diskAverage c ε i q → α<diskAverage c ε j q →
+        ∃ z∈C, q∈Metric.ball z (α*ε/4)) ∧
+      (C.card:ℝ)*(α*ε/4) ≤ (256/(Real.pi*α^3))*
+        ((∫ q in Metric.ball x r, diskAverage c ε i q*diskAverage c ε j q)/ε) := by
+  classical
+  let ρ := α*ε/8
+  have hρ : 0<ρ := by dsimp [ρ]; positivity
+  let A := {q | q∈Metric.ball x s ∧ α<diskAverage c ε i q ∧ α<diskAverage c ε j q}
+  obtain ⟨C,hCA,hcover,hsep⟩ := finite_packed_cover
+    (A := A) ((isCompact_closedBall x s).totallyBounded.subset
+      (fun _ hq => Metric.ball_subset_closedBall hq.1)) hρ
+  have hsub : ∀ z∈C, Metric.ball z (ρ/2)⊆Metric.ball x r := by
+    intro z hz q hq
+    have hz' := (hCA hz).1
+    have hq' : dist q z<ρ/2 := hq
+    have hz'' : dist z x<s := hz'
+    have hd := dist_triangle q z x
+    change dist q x<r
+    dsimp [ρ] at hq'
+    linarith
+  have hlow : ∀ z∈C, ∀ q∈Metric.ball z (ρ/2),
+      α^2/4 ≤ diskAverage c ε i q*diskAverage c ε j q := by
+    intro z hz q hq
+    have hp (a : Fin k) (ha : α<diskAverage c ε a z) : α/2≤diskAverage c ε a q := by
+      have hh := (diskAverage_lipschitz hc hε a).dist_le_mul q z
+      change |diskAverage c ε a q-diskAverage c ε a z|≤(4/ε)*dist q z at hh
+      have hh' := (abs_le.mp hh).1
+      have hd : dist q z<ρ/2 := hq
+      have hmul : (4/ε)*dist q z<α/2 := by
+        calc (4/ε)*dist q z < (4/ε)*(ρ/2) := mul_lt_mul_of_pos_left hd (by positivity)
+          _ = α/4 := by dsimp [ρ]; field_simp; ring
+          _ < α/2 := by linarith
+      linarith
+    have hi := hp i (hCA hz).2.1
+    have hj := hp j (hCA hz).2.2
+    calc α^2/4 = (α/2)*(α/2) := by ring
+      _ ≤ _ := mul_le_mul hi hj (by positivity) (diskAverage_nonneg c ε i q)
+  have hdj : (↑C : Set ℂ).Pairwise (Disjoint on fun z => Metric.ball z (ρ/2)) := by
+    intro z hz w hw hne
+    apply Set.disjoint_left.mpr
+    intro q hq hq'
+    have hh := dist_triangle z q w
+    have h₁ : dist q z<ρ/2 := hq
+    have h₂ : dist q w<ρ/2 := hq'
+    rw [dist_comm z q] at hh
+    linarith [hsep hz hw hne]
+  have hint : (C.card:ℝ)*(Real.pi*(ρ/2)^2*(α^2/4)) ≤
+      ∫ q in Metric.ball x r, diskAverage c ε i q*diskAverage c ε j q := by
+    calc (C.card:ℝ)*(Real.pi*(ρ/2)^2*(α^2/4)) =
+        ∑ z∈C, ∫ _q in Metric.ball z (ρ/2), (α^2/4) := by
+          simp only [setIntegral_const,smul_eq_mul,volumeReal_ball _ (by positivity : 0≤ρ/2)]
+          simp
+      _ ≤ ∑ z∈C, ∫ q in Metric.ball z (ρ/2), diskAverage c ε i q*diskAverage c ε j q := by
+        apply Finset.sum_le_sum
+        intro z hz
+        apply setIntegral_mono_on (integrableOn_const (ball_ne_top z (ρ/2)))
+          (diskAverage_product_integrable hc hε i j z (ρ/2)) measurableSet_ball
+        exact hlow z hz
+      _ = ∫ q in ⋃ z∈C, Metric.ball z (ρ/2), diskAverage c ε i q*diskAverage c ε j q := by
+        exact (integral_biUnion_finset C (fun _ _ => measurableSet_ball) hdj
+          (fun z _ => diskAverage_product_integrable hc hε i j z (ρ/2))).symm
+      _ ≤ _ := setIntegral_mono_set (diskAverage_product_integrable hc hε i j x r)
+        (Eventually.of_forall fun q => mul_nonneg (diskAverage_nonneg c ε i q) (diskAverage_nonneg c ε j q))
+        (Eventually.of_forall (fun q hq => by
+          obtain ⟨z,hz,hq⟩ := mem_iUnion₂.mp hq
+          exact hsub z hz hq))
+  refine ⟨C,?_,?_⟩
+  · intro q hq hi hj
+    obtain ⟨z,hz,hd⟩ := hcover q ⟨hq,hi,hj⟩
+    refine ⟨z,hz,?_⟩
+    change dist q z<α*ε/4
+    dsimp [ρ] at hd
+    nlinarith [mul_pos hα hε]
+  · have hconst : 0<Real.pi*α^3 := by positivity
+    rw [← mul_div_assoc (256/(Real.pi*α^3)) _ ε]
+    apply (le_div_iff₀ hε).mpr
+    rw [div_mul_eq_mul_div]
+    apply (le_div_iff₀ hconst).mpr
+    dsimp [ρ] at hint
+    nlinarith [Real.pi_pos]
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+variable {k : ℕ} {c : ℂ → Fin k}
+lemma missing_edge_disk_cover (hc : Measurable c) {i j : Fin k} (hne : i≠j) {x : ℂ}
+    (he : ¬TransitionEdge c i j x) : ∃ R>0, ∀ s<R, ∀ α>0, ∀ η>0,
+      ∃ δ>0, ∀ ε, 0<ε → ε<δ → ∃ C : Finset ℂ,
+        (∀ q∈Metric.ball x s, α<diskAverage c ε i q → α<diskAverage c ε j q →
+          ∃ z∈C, q∈Metric.ball z (α*ε/4)) ∧ (C.card:ℝ)*(α*ε/4)<η := by
+  obtain ⟨R,hR,ht⟩ := missing_edge_overlap hc hne he
+  refine ⟨R,hR,fun s hs α hα η hη => ?_⟩
+  let r := (s+R)/2
+  have hsr : s<r := by dsimp [r]; linarith
+  have hrR : r<R := by dsimp [r]; linarith
+  have hlim := (ht r hrR).const_mul (256/(Real.pi*α^3))
+  rw [mul_zero] at hlim
+  have he₁ := hlim.eventually (gt_mem_nhds hη)
+  have he₂ : ∀ᶠ ε : ℝ in 𝓝[>] 0, s+α*ε/16<r := by
+    have ht' : Tendsto (fun ε : ℝ => s+α*ε/16) (𝓝[>] 0) (𝓝 s) := by
+      have hid : Tendsto (fun ε : ℝ => ε) (𝓝[>] 0) (𝓝 0) := nhdsWithin_le_nhds
+      simpa using (tendsto_const_nhds (x := s)).add ((hid.const_mul α).div_const 16)
+    exact ht'.eventually (gt_mem_nhds hsr)
+  obtain ⟨δ,hδ,hd⟩ := Metric.mem_nhdsWithin_iff.mp (he₁.and he₂)
+  refine ⟨δ,hδ,fun ε hε hεδ => ?_⟩
+  have hε' := hd (show ε∈Metric.ball 0 δ∩Ioi 0 from ⟨by simpa [Real.dist_eq,abs_of_pos hε] using hεδ,hε⟩)
+  obtain ⟨C,hC,hb⟩ := finite_disk_overlap_cover hc hε hα i j x hε'.2.le
+  exact ⟨C,hC,hb.trans_lt hε'.1⟩
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology
+namespace PlanarCover
+lemma ball_simplyConnected (y : ℂ) {r : ℝ} (hr : 0 < r) :
+    IsSimplyConnected (Metric.ball y r) := by
+  let := (convex_ball y r).contractibleSpace (Metric.nonempty_ball.mpr hr)
+  change SimplyConnectedSpace (Metric.ball y r)
+  infer_instance
+lemma hit_center {F : ℂ → ℂ} {y : ℂ}
+    (hc : ContinuousOn F (Metric.ball y 4))
+    (hb : ∀ z ∈ Metric.ball y 4, ‖F z-z‖ ≤ 2) :
+    y ∈ F '' Metric.ball y 4 := by
+  by_contra hn
+  have hzero : 0 ∉ (fun z => F z-y) '' Metric.ball y 4 := by
+    rintro ⟨z,hz,he⟩
+    apply hn
+    exact ⟨z,hz,sub_eq_zero.mp he⟩
+  obtain ⟨L,hLc,hL⟩ := Complex.exists_continuousOn_eqOn_exp_comp
+    (ball_simplyConnected y (by norm_num : (0:ℝ)<4)) Metric.isOpen_ball
+    (hc.sub continuousOn_const) hzero
+  let e : ℝ → ℂ := fun t => Complex.exp (t*Complex.I)
+  have ec : Continuous e := by fun_prop
+  have en (t : ℝ) : ‖e t‖ = 1 := by simp [e,Complex.norm_exp]
+  let γ : ℝ → ℂ := fun t => y+3*e t
+  have γc : Continuous γ := by fun_prop
+  have γmem (t : ℝ) : γ t ∈ Metric.ball y 4 := by
+    norm_num [γ,Metric.mem_ball,dist_eq_norm,en]
+  let H : ℝ → ℂ := fun t => (F (γ t)-y) / e t
+  have Hc : Continuous H := (hc.comp_continuous γc γmem).sub continuous_const |>.div ec (fun t => Complex.exp_ne_zero _)
+  have Hnear (t : ℝ) : ‖H t-3‖ ≤ 2 := by
+    have he : e t ≠ 0 := Complex.exp_ne_zero _
+    have h : H t-3 = (F (γ t)-γ t)/e t := by dsimp [H,γ]; field_simp; ring
+    rw [h,norm_div,en,div_one]
+    exact hb _ (γmem t)
+  have Hpos (t : ℝ) : 0 < (H t).re := by
+    have : |(H t-3).re| ≤ 2 := (Complex.abs_re_le_norm (H t-3)).trans (Hnear t)
+    have := (abs_le.mp this).1
+    norm_num at this
+    linarith
+  have Hslit (t : ℝ) : H t ∈ Complex.slitPlane := Or.inl (Hpos t)
+  have logHc : Continuous (fun t => Complex.log (H t)) :=
+    Hc.clog Hslit
+  let d : ℝ → ℂ := fun t => L (γ t) - Complex.log (H t) - t*Complex.I
+  have dc : Continuous d := ((hLc.comp_continuous γc γmem).sub logHc).sub (by fun_prop)
+  have de (t : ℝ) : Complex.exp (d t) = 1 := by
+    have hH : H t ≠ 0 := fun hz => by have := Hpos t; simp [hz] at this
+    have hLe : Complex.exp (L (γ t)) = F (γ t)-y := hL (γmem t)
+    simp only [d,Complex.exp_sub,Complex.exp_log hH,hLe]
+    change (F (γ t)-y) / ((F (γ t)-y)/e t) / e t = 1
+    have hF : F (γ t)-y ≠ 0 := fun hz => hzero ⟨_,γmem t,hz⟩
+    have he : e t ≠ 0 := Complex.exp_ne_zero _
+    field_simp
+  have hconst : d 0 = d (2*Real.pi) :=
+    Complex.isCoveringMap_exp.constOn_of_comp isPreconnected_univ dc.continuousOn
+      (fun a _ b _ => Subtype.ext ((de a).trans (de b).symm)) (mem_univ _) (mem_univ _)
+  have e0 : e 0 = 1 := by simp [e]
+  have e2 : e (2*Real.pi) = 1 := by simp [e,Complex.exp_two_pi_mul_I]
+  have γeq : γ (2*Real.pi) = γ 0 := by dsimp [γ]; rw [e0,e2]
+  have Heq : H (2*Real.pi) = H 0 := by dsimp [H]; rw [γeq,e0,e2]
+  dsimp [d] at hconst
+  rw [γeq,Heq] at hconst
+  have hi := congrArg Complex.im hconst
+  norm_num at hi
+  nlinarith [Real.pi_pos]
+end PlanarCover
+end
+
+noncomputable section
+open Set MeasureTheory MeasureTheory.Measure
+namespace PlanarCover
+lemma affinePair_null (a b : ℂ) : volume (affineSpan ℝ ({a,b} : Set ℂ) : Set ℂ)=0 := by
+  apply addHaar_affineSubspace volume
+  intro he
+  have hd : Submodule.span ℝ ({a-b} : Set ℂ)=⊤ := by
+    change Submodule.span ℝ ({a -ᵥ b} : Set ℂ)=⊤
+    rw [← vectorSpan_pair ℝ a b,← direction_affineSpan,he]
+    exact AffineSubspace.direction_top _ _ _
+  have hf := finrank_span_le_card (R := ℝ) ({a-b} : Set ℂ)
+  rw [hd,finrank_top,Complex.finrank_real_complex] at hf
+  norm_num at hf
+lemma hit_center_subtype {Q : Set ℂ} {y : ℂ} (hQ : Metric.ball y 4 ⊆ Q)
+    {F : Q → ℂ} (hc : Continuous F) (hb : ∀ z, ‖F z-z.val‖ ≤ 2) :
+    y ∈ Set.range F := by
+  classical
+  let g : ℂ → ℂ := fun z => if hz:z∈Q then F ⟨z,hz⟩ else 0
+  have ge (z : ℂ) (hz : z∈Q) : g z=F ⟨z,hz⟩ := dite_eq_left hz
+  have hgc : ContinuousOn g (Metric.ball y 4) := by
+    rw [continuousOn_iff_continuous_domRestrict]
+    have hh := hc.comp (continuous_subtype_val.subtype_mk (fun z : Metric.ball y 4 => hQ z.property))
+    convert hh using 1
+    ext z
+    exact ge _ (hQ z.property)
+  obtain ⟨z,hz,he⟩ := hit_center hgc (fun z hz => by rw [ge _ (hQ hz)]; exact hb _)
+  exact ⟨⟨z,hQ hz⟩,by rwa [ge _ (hQ hz)] at he⟩
+lemma small_support_pair {ι : Type*} [DecidableEq ι] {s : Finset ι}
+    (hs : s.Nonempty) (hc : s.card≤2) : ∃ a b, s ⊆ {a,b} := by
+  by_cases h1 : s.card=1
+  · obtain ⟨a,rfl⟩ := Finset.card_eq_one.mp h1
+    exact ⟨a,a,by simp⟩
+  · have h2 : s.card=2 := by have := hs.card_pos; omega
+    obtain ⟨a,b,_,rfl⟩ := Finset.card_eq_two.mp h2
+    exact ⟨a,b,Finset.Subset.refl _⟩
+lemma barycenter_two {ι : Type*} [Fintype ι] (p : ι → ℂ) (w : ι → ℝ)
+    (hn : ∀ i,0≤w i) (hs : ∑ i,w i=1)
+    (hcard : (Finset.univ.filter (fun i => w i≠0)).card≤2) :
+    ∑ i,w i • p i ∈ ⋃ a, ⋃ b, (affineSpan ℝ ({p a,p b} : Set ℂ) : Set ℂ) := by
+  classical
+  let s := Finset.univ.filter (fun i => w i≠0)
+  have hsw : ∑ i ∈ s, w i = ∑ i, w i := by
+    apply Finset.sum_subset (Finset.filter_subset _ _)
+    intro i _ hi
+    simpa [s] using hi
+  have hsv : ∑ i∈s,w i=1 := hsw.trans hs
+  have hne : s.Nonempty := by
+    by_contra he
+    rw [Finset.not_nonempty_iff_eq_empty.mp he,Finset.sum_empty] at hsv
+    exact zero_ne_one hsv
+  obtain ⟨a,b,hab⟩ := small_support_pair hne hcard
+  refine mem_iUnion.mpr ⟨a,mem_iUnion.mpr ⟨b,?_⟩⟩
+  have hm := (affineSpan ℝ ({p a,p b} : Set ℂ)).convex.sum_mem
+    (z := p) (fun i _ => hn i) hsv (fun i hi => mem_affineSpan ℝ (by
+      have hh := hab hi
+      simp only [Finset.mem_insert,Finset.mem_singleton] at hh
+      rcases hh with rfl|rfl <;> simp))
+  have hh : ∑ i ∈ s, w i • p i = ∑ i, w i • p i := by
+    apply Finset.sum_subset (Finset.filter_subset _ _)
+    intro i _ hi
+    have hi0 : w i = 0 := by simpa [s] using hi
+    simp [hi0]
+  rwa [hh] at hm
+lemma finite_partition_impossible {ι : Type*} [Fintype ι] {Q : Set ℂ}
+    (hQ : Metric.ball (0:ℂ) 5 ⊆ Q) (p : ι → ℂ) (w : ι → C(Q,ℝ))
+    (hn : ∀ i z,0≤w i z) (hs : ∀ z,∑ i,w i z=1)
+    (hm : ∀ z,(Finset.univ.filter (fun i => w i z≠0)).card≤2)
+    (hd : ∀ i z, w i z≠0 → ‖p i-z.val‖≤2) : False := by
+  classical
+  let F : Q → ℂ := fun z => ∑ i,w i z • p i
+  have Fc : Continuous F := by fun_prop
+  have Fb (z : Q) : ‖F z-z.val‖≤2 := by
+    have he : F z-z.val=∑ i,w i z • (p i-z.val) := by
+      simp only [smul_sub,Finset.sum_sub_distrib,← Finset.sum_smul,hs,one_smul,F]
+    rw [he]
+    calc
+      ‖∑ i,w i z • (p i-z.val)‖ ≤ ∑ i,‖w i z • (p i-z.val)‖ := norm_sum_le _ _
+      _ ≤ ∑ i,w i z*2 := by
+        apply Finset.sum_le_sum
+        intro i _
+        by_cases hi : w i z=0
+        · simp [hi]
+        · rw [norm_smul,Real.norm_eq_abs,abs_of_nonneg (hn i z)]
+          exact mul_le_mul_of_nonneg_left (hd i z hi) (hn i z)
+      _ = 2 := by rw [← Finset.sum_mul,hs]; ring
+  have hsub : Metric.ball (0:ℂ) 1 ⊆ Set.range F := by
+    intro y hy
+    apply hit_center_subtype (F := F) ?_ Fc Fb
+    intro z hz
+    apply hQ
+    rw [Metric.mem_ball,dist_zero_right] at hy ⊢
+    have hh := norm_add_le (z-y) y
+    rw [sub_add_cancel] at hh
+    have hz' : ‖z-y‖<4 := by simpa [Metric.mem_ball,dist_eq_norm] using hz
+    exact hh.trans_lt (by linarith)
+  have hnull : volume (⋃ a, ⋃ b, (affineSpan ℝ ({p a,p b} : Set ℂ) : Set ℂ))=0 :=
+    measure_iUnion_null fun a => measure_iUnion_null fun b => affinePair_null _ _
+  have hfsub : Set.range F ⊆ ⋃ a, ⋃ b, (affineSpan ℝ ({p a,p b} : Set ℂ) : Set ℂ) := by
+    rintro _ ⟨z,rfl⟩
+    exact barycenter_two p (fun i => w i z) (fun i => hn i z) (hs z) (hm z)
+  have hz := measure_mono_null (hsub.trans hfsub) hnull
+  exact (Metric.isOpen_ball.measure_pos volume (Metric.nonempty_ball.mpr (by norm_num : (0:ℝ)<1))).ne' hz
+end PlanarCover
+end
+
+noncomputable section
+open scoped Classical
+open Set MeasureTheory
+namespace PlanarCover
+lemma finite_open_cover_impossible {ι : Type*} [Fintype ι] {Q : Set ℂ}
+    (hQ : Metric.ball (0:ℂ) 5 ⊆ Q) (U : ι → Set Q)
+    (ho : ∀ i, IsOpen (U i)) (hcov : ∀ z : Q, ∃ i, z∈U i)
+    (hm : ∀ z : Q, (Finset.univ.filter (fun i => z∈U i)).card≤2)
+    (hd : ∀ i, ∀ x∈U i, ∀ y∈U i, ‖x.val-y.val‖≤2) : False := by
+  classical
+  obtain ⟨w,hw⟩ := PartitionOfUnity.exists_isSubordinate isClosed_univ U ho (by
+    intro z _
+    exact mem_iUnion.mpr (hcov z))
+  have hz0 : (0:ℂ)∈Q := hQ (Metric.mem_ball_self (by norm_num))
+  let p : ι → Q := fun i => if h : (U i).Nonempty then h.choose else ⟨0,hz0⟩
+  apply finite_partition_impossible hQ (fun i => (p i).val) w (w.nonneg) ?_ ?_ ?_
+  · intro z
+    simpa only [finsum_eq_sum_of_fintype] using w.sum_eq_one (mem_univ z)
+  · intro z
+    apply le_trans (Finset.card_le_card ?_) (hm z)
+    intro i hi
+    simp only [Finset.mem_filter,Finset.mem_univ,true_and] at hi ⊢
+    exact hw i (subset_closure hi)
+  · intro i z hz
+    have hzu : z∈U i := hw i (subset_closure hz)
+    have hp : p i∈U i := by
+      dsimp [p]
+      rw [dite_eq_left ⟨z,hzu⟩]
+      exact Classical.choose_spec _
+    exact hd i _ hp z hzu
+lemma compact_open_cover_impossible {ι : Type*} {Q : Set ℂ}
+    (hQ : Metric.ball (0:ℂ) 5 ⊆ Q) (hQc : IsCompact Q) (U : ι → Set Q)
+    (ho : ∀ i, IsOpen (U i)) (hcov : ∀ z : Q, ∃ i, z∈U i)
+    (hm : ∀ z : Q, ∀ i j l, z∈U i → z∈U j → z∈U l → i=j ∨ i=l ∨ j=l)
+    (hd : ∀ i, ∀ x∈U i, ∀ y∈U i, ‖x.val-y.val‖≤2) : False := by
+  classical
+  have : CompactSpace Q := isCompact_iff_compactSpace.mp hQc
+  obtain ⟨s,hs⟩ := isCompact_univ.elim_finite_subcover U ho (by
+    intro z _
+    exact mem_iUnion.mpr (hcov z))
+  apply finite_open_cover_impossible hQ (fun i : s => U i.val) (fun i => ho i.val) ?_ ?_ ?_
+  · intro z
+    rcases mem_iUnion.mp (hs (mem_univ z)) with ⟨i,hi⟩
+    rcases mem_iUnion.mp hi with ⟨his,hiz⟩
+    exact ⟨⟨i,his⟩,hiz⟩
+  · intro z
+    by_contra! hcard
+    obtain ⟨i,hi,j,hj,l,hl,hij,hil,hjl⟩ := Finset.two_lt_card.mp hcard
+    simp only [Finset.mem_filter,Finset.mem_univ,true_and] at hi hj hl
+    rcases hm z i.val j.val l.val hi hj hl with h|h|h
+    · exact hij (Subtype.ext h)
+    · exact hil (Subtype.ext h)
+    · exact hjl (Subtype.ext h)
+  · intro i
+    exact hd i.val
+end PlanarCover
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PlaneColoring/Rigidity.lean b/lean/OAI/Geometry/PlaneColoring/Rigidity.lean
new file mode 100644
index 0000000000000000000000000000000000000000..407c49a8368fe189b0e3a383bb7b95e84a16176f
--- /dev/null
+++ b/lean/OAI/Geometry/PlaneColoring/Rigidity.lean
@@ -0,0 +1,443 @@
+import OAI.Geometry.PlaneColoring.PaletteTransport
+
+namespace OAI
+
+noncomputable section
+open Set Filter Topology
+namespace EuclideanFiveColor
+/-- Common unit neighbor for the transverse chord of length `t`. -/
+def evenOffset (v : ℂ) (t : ℝ) : ℂ :=
+  ((Real.sqrt (1-t^2/4) : ℂ)+((t/2 : ℝ) : ℂ)*Complex.I)*v
+lemma norm_even_wing {t : ℝ} (ht : |t|<2) :
+    ‖(Real.sqrt (1-t^2/4) : ℂ)+((t/2 : ℝ) : ℂ)*Complex.I‖=1 := by
+  have ht2 : t^2≤4 := by
+    have hh := mul_self_le_mul_self (abs_nonneg t) ht.le
+    nlinarith [sq_abs t]
+  have hsq := Real.sq_sqrt (show 0≤1-t^2/4 by linarith)
+  apply (sq_eq_sq₀ (norm_nonneg _) (by norm_num : 0≤(1:ℝ))).mp
+  rw [Complex.sq_norm,Complex.normSq_apply]
+  norm_num only [Complex.add_re,Complex.ofReal_re,Complex.mul_re,Complex.ofReal_im,
+    Complex.I_re,Complex.I_im,Complex.add_im,Complex.mul_im,mul_zero,zero_mul,sub_zero,
+    add_zero,zero_add,mul_one]
+  nlinarith
+lemma evenOffset_distances {v : ℂ} (hv : ‖v‖=1) {t : ℝ} (ht : |t|<2) :
+    ‖evenOffset v t‖=1 ∧ ‖evenOffset v t-(t:ℂ)*(Complex.I*v)‖=1 := by
+  constructor
+  · rw [evenOffset,norm_mul,norm_even_wing ht,hv,mul_one]
+  · have he : evenOffset v t-(t:ℂ)*(Complex.I*v)=evenOffset v (-t) := by
+      simp only [evenOffset,neg_sq,neg_div,Complex.ofReal_neg,Complex.ofReal_div,Complex.ofReal_ofNat]
+      ring
+    rw [he,evenOffset,norm_mul,norm_even_wing (by simpa only [abs_neg] using ht),hv,mul_one]
+@[fun_prop] lemma continuous_evenOffset (v : ℂ) : Continuous (evenOffset v) := by
+  unfold evenOffset
+  fun_prop
+lemma evenOffset_zero (v : ℂ) : evenOffset v 0=v := by simp [evenOffset]
+def transverseDet (v w : ℂ) : ℝ :=
+  (Complex.I*v).re*(Complex.I*w).im-(Complex.I*w).re*(Complex.I*v).im
+def transverseFirst (v w h : ℂ) : ℝ :=
+  (h.re*(Complex.I*w).im-(Complex.I*w).re*h.im)/transverseDet v w
+def transverseSecond (v w h : ℂ) : ℝ :=
+  ((Complex.I*v).re*h.im-h.re*(Complex.I*v).im)/transverseDet v w
+lemma transverse_decompose {v w : ℂ} (hd : transverseDet v w≠0) (h : ℂ) :
+    (transverseFirst v w h : ℂ)*(Complex.I*v)+
+      (transverseSecond v w h : ℂ)*(Complex.I*w)=h := by
+  apply Complex.ext <;>
+    simp only [Complex.add_re,Complex.mul_re,Complex.ofReal_re,Complex.ofReal_im,
+      Complex.add_im,Complex.mul_im,zero_mul,sub_zero,add_zero,
+      transverseFirst,transverseSecond] <;>
+    field_simp <;> dsimp [transverseDet] <;>
+    simp only [Complex.mul_re,Complex.mul_im,Complex.I_re,Complex.I_im,zero_mul,one_mul,zero_sub,zero_add] <;> ring
+@[fun_prop] lemma continuous_transverseFirst (v w : ℂ) : Continuous (transverseFirst v w) := by
+  unfold transverseFirst
+  fun_prop
+@[fun_prop] lemma continuous_transverseSecond (v w : ℂ) : Continuous (transverseSecond v w) := by
+  unfold transverseSecond
+  fun_prop
+lemma transverseFirst_zero (v w : ℂ) : transverseFirst v w 0=0 := by simp [transverseFirst]
+lemma transverseSecond_zero (v w : ℂ) : transverseSecond v w 0=0 := by simp [transverseSecond]
+def evenPath (v w h : ℂ) : Fin 5 → ℂ := ![0,
+  evenOffset v (transverseFirst v w h),
+  (transverseFirst v w h : ℂ)*(Complex.I*v),
+  (transverseFirst v w h : ℂ)*(Complex.I*v)+evenOffset w (transverseSecond v w h),h]
+lemma evenPath_edges {v w h : ℂ} (hv : ‖v‖=1) (hw : ‖w‖=1)
+    (hd : transverseDet v w≠0) (h₁ : |transverseFirst v w h|<2)
+    (h₂ : |transverseSecond v w h|<2) :
+    ∀i : Fin 4, ‖evenPath v w h i.castSucc-evenPath v w h i.succ‖=1 := by
+  have he₁ := evenOffset_distances hv h₁
+  have he₂ := evenOffset_distances hw h₂
+  have hdec := transverse_decompose hd h
+  intro i
+  fin_cases i
+  · change ‖0-evenOffset v (transverseFirst v w h)‖=1
+    simpa only [zero_sub,norm_neg] using he₁.1
+  · exact he₁.2
+  · change ‖(transverseFirst v w h : ℂ)*(Complex.I*v)-
+        ((transverseFirst v w h : ℂ)*(Complex.I*v)+evenOffset w (transverseSecond v w h))‖=1
+    simpa only [sub_add_cancel_left,norm_neg] using he₂.1
+  · change ‖(transverseFirst v w h : ℂ)*(Complex.I*v)+evenOffset w (transverseSecond v w h)-h‖=1
+    calc
+      _ = ‖evenOffset w (transverseSecond v w h)-(transverseSecond v w h : ℂ)*(Complex.I*w)‖ :=
+        congrArg norm (by linear_combination hdec)
+      _ = 1 := he₂.2
+/-- Actual four-unit-edge paths stay inside the prescribed two open sets for
+all small starting-point and displacement perturbations. -/
+lemma evenPath_local {U V : Set ℂ} (hU : IsOpen U) (hV : IsOpen V)
+    {p v w : ℂ} (hp : p∈U) (hpv : p+v∈V) (hpw : p+w∈V) :
+    ∃r>0,∃δ>0,∀q h : ℂ,q∈Metric.ball p r → ‖h‖<δ →
+      |transverseFirst v w h|<2 ∧ |transverseSecond v w h|<2 ∧
+      (∀i : Fin 5,(q+evenPath v w h i)∈if i.val%2=0 then U else V) := by
+  let P : ℂ × ℂ → Prop := fun z => |transverseFirst v w z.2|<2 ∧
+    |transverseSecond v w z.2|<2 ∧
+    (∀i : Fin 5,(z.1+evenPath v w z.2 i)∈if i.val%2=0 then U else V)
+  have hm : ∀ᶠz in 𝓝 (p,(0:ℂ)),P z := by
+    have ha : ∀ᶠz : ℂ × ℂ in 𝓝 (p,(0:ℂ)), |transverseFirst v w z.2|<2 := by
+      apply ((continuous_transverseFirst v w).comp continuous_snd).abs.continuousAt.eventually_lt continuousAt_const
+      simp [transverseFirst_zero]
+    have hb : ∀ᶠz : ℂ × ℂ in 𝓝 (p,(0:ℂ)), |transverseSecond v w z.2|<2 := by
+      apply ((continuous_transverseSecond v w).comp continuous_snd).abs.continuousAt.eventually_lt continuousAt_const
+      simp [transverseSecond_zero]
+    have hc (i : Fin 5) : ∀ᶠz : ℂ × ℂ in 𝓝 (p,(0:ℂ)),
+        (z.1+evenPath v w z.2 i)∈if i.val%2=0 then U else V := by
+      have hco : Continuous (fun z : ℂ × ℂ => z.1+evenPath v w z.2 i) := by
+        fin_cases i <;> dsimp [evenPath] <;> fun_prop
+      apply hco.continuousAt.eventually
+      fin_cases i <;> norm_num [evenPath,transverseFirst_zero,transverseSecond_zero,evenOffset_zero]
+      · exact hU.mem_nhds hp
+      · exact hV.mem_nhds hpv
+      · exact hU.mem_nhds hp
+      · exact hV.mem_nhds hpw
+      · exact hU.mem_nhds hp
+    exact ha.and (hb.and (eventually_all.mpr hc))
+  obtain ⟨ε,hε,hb⟩ := Metric.mem_nhds_iff.mp hm
+  refine ⟨ε,hε,ε,hε,?_⟩
+  intro q h hq hh
+  exact hb (show (q,h)∈Metric.ball (p,(0:ℂ)) ε by
+    simpa only [Metric.mem_ball,Prod.dist_eq,dist_zero_right,max_lt_iff] using And.intro hq hh)
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+/-- Local invariance for every sufficiently small fixed translation forces
+local essential constancy. This is the Fubini/shear step of angular-binary-local. -/
+lemma local_ae_translation_constant {ι : Type*} [MeasurableSpace ι] [MeasurableSingletonClass ι]
+    [Countable ι] (c : ℂ → ι) (hc : Measurable c) {p : ℂ} {r δ : ℝ}
+    (hr : 0<r) (hδ : 0<δ)
+    (hshift : ∀h : ℂ,‖h‖<δ → ∀ᵐq : ℂ∂volume,q∈Metric.ball p r → c (q+h)=c q) :
+    ∃s>0,s≤r ∧ ∃i,∀ᵐq : ℂ∂volume,q∈Metric.ball p s → c q=i := by
+  let s : ℝ := min r δ/4
+  have hs : 0<s := by dsimp [s]; positivity
+  have hsr : s≤r := by dsimp [s]; linarith [min_le_left r δ,lt_min hr hδ]
+  have hsd : 2*s<δ := by dsimp [s]; linarith [min_le_right r δ,lt_min hr hδ]
+  have hpair : ∀ᵐz : ℂ × ℂ∂volume.prod volume,
+      ‖z.1‖<δ → z.2∈Metric.ball p r → c (z.2+z.1)=c z.2 := by
+    apply (Measure.ae_prod_iff_ae_ae ?_).mpr
+    · exact Eventually.of_forall fun h => by
+        by_cases hh : ‖h‖<δ
+        · exact (hshift h hh).mono fun q hq _ => hq
+        · exact Eventually.of_forall fun _ hh' => (hh hh').elim
+    · have hx : MeasurableSet {z : ℂ × ℂ | ‖z.1‖<δ} := measurableSet_lt measurable_fst.norm measurable_const
+      have hy : MeasurableSet {z : ℂ × ℂ | z.2∈Metric.ball p r} := Metric.isOpen_ball.measurableSet.preimage measurable_snd
+      have hz : MeasurableSet {z : ℂ × ℂ | c (z.2+z.1)=c z.2} :=
+        measurableSet_eq_fun (hc.comp (measurable_snd.add measurable_fst)) (hc.comp measurable_snd)
+      exact hx.imp (hy.imp hz)
+  have hd := (measurePreserving_sub_prod (volume : Measure ℂ) volume).quasiMeasurePreserving.ae hpair
+  have he : ∀ᵐz : ℂ × ℂ∂volume.prod volume,
+      z.1∈Metric.ball p s → z.2∈Metric.ball p s → c z.1=c z.2 := by
+    filter_upwards [hd] with z hz h1 h2
+    have hdist : ‖z.1-z.2‖<δ := by
+      rw [←dist_eq_norm]
+      have h := dist_triangle z.1 p z.2
+      have h2' : dist p z.2<s := by simpa only [Metric.mem_ball,dist_comm] using h2
+      have h1' : dist z.1 p<s := h1
+      linarith
+    have hh := hz hdist (Metric.ball_subset_ball hsr h2)
+    simpa only [add_sub_cancel] using hh
+  have hae := Measure.ae_ae_of_ae_prod he
+  obtain ⟨q,hq,hqae⟩ := Measure.exists_mem_of_measure_ne_zero_of_ae
+    (Metric.measure_ball_pos volume p hs).ne' (ae_restrict_of_ae hae)
+  refine ⟨s,hs,hsr,c q,?_⟩
+  exact hqae.mono fun z hz hzs => (hz hq hzs).symm
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+lemma binary_two_edges {ι : Type*} {a b u v w : ι} (hu : u=a ∨ u=b)
+    (hv : v=a ∨ v=b) (hw : w=a ∨ w=b) (huv : u≠v) (hvw : v≠w) : u=w := by
+  rcases hu with rfl|rfl <;> rcases hv with rfl|rfl <;> rcases hw with rfl|rfl <;> tauto
+lemma ae_translated_unit_pair {k : ℕ} {c : ℂ → Fin k} (hc : Measurable c) (hw : WeakColoring k c)
+    {a b : ℂ} (hab : ‖a-b‖=1) : ∀ᵐq : ℂ∂volume,c (q+a)≠c (q+b) := by
+  obtain ⟨t,ht⟩ := direction_exists (u:=b-a) (by simpa only [norm_sub_rev] using hab)
+  have hh := (measurePreserving_add_right volume a).quasiMeasurePreserving.ae (weak_all_directions c hc hw t)
+  filter_upwards [hh] with q hq
+  simpa only [ht,show q+a+(b-a)=q+b by ring] using hq
+/-- Local essential constancy inside two-color open regions, from the source's
+actual four-edge paths. The directions are genuine unit directions landing in V. -/
+lemma binary_open_local_rigidity {k : ℕ} {c : ℂ → Fin k} (hc : Measurable c) (hw : WeakColoring k c)
+    {a b : Fin k} {U V : Set ℂ} (hU : IsOpen U) (hV : IsOpen V)
+    (hUc : ∀ᵐq : ℂ∂volume,q∈U → c q=a ∨ c q=b)
+    (hVc : ∀ᵐq : ℂ∂volume,q∈V → c q=a ∨ c q=b)
+    {p v w : ℂ} (hp : p∈U) (hpv : p+v∈V) (hpw : p+w∈V)
+    (hv : ‖v‖=1) (hw' : ‖w‖=1) (hd : transverseDet v w≠0) :
+    ∃r>0,∃i : Fin k,∀ᵐq : ℂ∂volume,q∈Metric.ball p r → c q=i := by
+  obtain ⟨r,hr,δ,hδ,hinside⟩ := evenPath_local hU hV hp hpv hpw
+  have hshift (h : ℂ) (hh : ‖h‖<δ) :
+      ∀ᵐq : ℂ∂volume,q∈Metric.ball p r → c (q+h)=c q := by
+    have hsmall := hinside p h (Metric.mem_ball_self hr) hh
+    have hedges := evenPath_edges hv hw' hd hsmall.1 hsmall.2.1
+    have hcm (i : Fin 5) := (measurePreserving_add_right volume (evenPath v w h i)).quasiMeasurePreserving.ae (hUc.and hVc)
+    have hne (i : Fin 4) := ae_translated_unit_pair hc hw (hedges i)
+    filter_upwards [ae_all_iff.mpr hcm,ae_all_iff.mpr hne] with q hqc hqp hq
+    have hi := (hinside q h hq hh).2.2
+    have hcol (i : Fin 5) : c (q+evenPath v w h i)=a ∨ c (q+evenPath v w h i)=b := by
+      have hii := hi i
+      by_cases he : i.val%2=0
+      · exact (hqc i).1 (by simpa only [he,ite_true] using hii)
+      · exact (hqc i).2 (by simpa only [he,ite_false] using hii)
+    have he02 := binary_two_edges (hcol 0) (hcol 1) (hcol 2) (hqp 0) (hqp 1)
+    have he24 := binary_two_edges (hcol 2) (hcol 3) (hcol 4) (hqp 2) (hqp 3)
+    change c (q+0)=c (q+evenPath v w h 2) at he02
+    change c (q+evenPath v w h 2)=c (q+h) at he24
+    simpa only [add_zero] using (he02.trans he24).symm
+  obtain ⟨s,hs,_,i,hi⟩ := local_ae_translation_constant c hc hr hδ hshift
+  exact ⟨s,hs,i,hi⟩
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology
+namespace EuclideanFiveColor
+lemma transverseDet_mul_unit {v : ℂ} (hv : ‖v‖=1) (z : ℂ) :
+    transverseDet v (z*v)=z.im := by
+  have hh : v.re^2+v.im^2=1 := by
+    calc
+      v.re^2+v.im^2 = ‖v‖^2 := by rw [Complex.sq_norm,Complex.normSq_apply]; ring
+      _ = 1 := by rw [hv]; norm_num
+  calc
+    transverseDet v (z*v) = (v.re^2+v.im^2)*z.im := by
+      simp only [transverseDet,Complex.mul_re,Complex.mul_im,Complex.I_re,Complex.I_im]
+      ring
+    _ = z.im := by rw [hh,one_mul]
+/-- An open set meeting a unit circle contains two nonparallel unit directions
+from its center. No regularity of the color classes is involved. -/
+lemma open_two_unit_directions {V : Set ℂ} (hV : IsOpen V) {p v : ℂ}
+    (hv : ‖v‖=1) (hpv : p+v∈V) :
+    ∃w : ℂ,p+w∈V ∧ ‖w‖=1 ∧ transverseDet v w≠0 := by
+  have ht : Continuous (fun t : ℝ => p+(Circle.exp t : ℂ)*v) := by fun_prop
+  have hm : (fun t : ℝ => p+(Circle.exp t : ℂ)*v) ⁻¹' V∈𝓝 0 := by
+    apply ht.continuousAt.preimage_mem_nhds
+    simpa only [Circle.exp_zero,Circle.coe_one,one_mul] using hV.mem_nhds hpv
+  obtain ⟨δ,hδ,hd⟩ := Metric.mem_nhds_iff.mp hm
+  let t : ℝ := min δ Real.pi/2
+  have hp := Real.pi_pos
+  have hmin : 0 < min δ Real.pi := lt_min hδ hp
+  have htp : 0<t := by dsimp [t]; positivity
+  have htd : t<δ := by dsimp [t]; linarith [min_le_left δ Real.pi]
+  have htπ : t<Real.pi := by dsimp [t]; linarith [min_le_right δ Real.pi]
+  refine ⟨(Circle.exp t : ℂ)*v,hd ?_,by rw [norm_mul,Circle.norm_coe,hv,one_mul],?_⟩
+  · simpa only [Metric.mem_ball,dist_zero_right,Real.norm_eq_abs,abs_of_pos htp] using htd
+  · rw [transverseDet_mul_unit hv]
+    simpa [Circle.coe_exp,Complex.exp_mul_I,←Complex.ofReal_cos,←Complex.ofReal_sin] using
+      (ne_of_gt (Real.sin_pos_of_pos_of_lt_pi htp htπ))
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology
+namespace EuclideanFiveColor
+lemma double_cos_circle (t : ℝ) :
+    (2*Real.cos t : ℂ)*(Circle.exp t : ℂ)-1=(Circle.exp (2*t) : ℂ) := by
+  apply Complex.ext <;>
+    simp only [Circle.coe_exp,Complex.exp_mul_I,←Complex.ofReal_cos,←Complex.ofReal_sin,
+      Complex.add_re,Complex.add_im,Complex.mul_re,Complex.mul_im,Complex.sub_re,
+      Complex.sub_im,Complex.ofReal_re,Complex.ofReal_im,Complex.I_re,Complex.I_im,
+      Complex.one_re,Complex.one_im,Complex.re_ofNat,Complex.im_ofNat,
+      Real.cos_two_mul,Real.sin_two_mul] <;> ring
+/-- The unit circle from one endpoint penetrates either prescribed radial side
+near its equilateral neighbor. This is the constructive version of the source IFT step. -/
+lemma radial_unit_contact {x e : ℂ} (he : ‖e‖=1) {δ : ℝ} (hδ : 0<δ)
+    {V : Set ℂ} (b : Bool)
+    (hstrip : ∀e' : ℂ,‖e'‖=1 → dist e' (e*(Circle.exp (Real.pi/3) : ℂ))<δ →
+      ∀r : ℝ,0<r → |r-1|<δ → (if b then 1<r else r<1) → x+(r:ℂ)*e'∈V) :
+    ∃v : ℂ,‖v‖=1 ∧ (x+e)+v∈V := by
+  let a : ℝ := Real.pi/3
+  let r : ℝ → ℝ := fun t => 2*Real.cos t
+  let E : ℝ → ℂ := fun t => e*(Circle.exp t : ℂ)
+  have hr : Continuous r := by fun_prop
+  have hE : Continuous E := by fun_prop
+  have hra : r a=1 := by norm_num [r,a,Real.cos_pi_div_three]
+  have ha : 0<a ∧ a<Real.pi := by dsimp [a]; constructor <;> linarith [Real.pi_pos]
+  have hn : ∀ᶠt in 𝓝 a,0<r t ∧ |r t-1|<δ ∧ dist (E t) (E a)<δ ∧ 0<t ∧ t<Real.pi := by
+    have h₁ : ∀ᶠt in 𝓝 a,0<r t := continuousAt_const.eventually_lt hr.continuousAt (by rw [hra]; norm_num)
+    have h₂ : ∀ᶠt in 𝓝 a,|r t-1|<δ :=
+      (hr.sub continuous_const).abs.continuousAt.eventually_lt_const (by simpa only [Pi.sub_apply,hra,sub_self,abs_zero] using hδ)
+    have h₃ : ∀ᶠt in 𝓝 a,dist (E t) (E a)<δ :=
+      (hE.dist continuous_const).continuousAt.eventually_lt_const (by simpa only [dist_self] using hδ)
+    exact h₁.and (h₂.and (h₃.and ((eventually_gt_nhds ha.1).and (eventually_lt_nhds ha.2))))
+  have hex : ∃t,(0<r t ∧ |r t-1|<δ ∧ dist (E t) (E a)<δ) ∧ (if b then 1<r t else r t<1) := by
+    cases b
+    · obtain ⟨t,ht,hat⟩ := ((hn.filter_mono nhdsWithin_le_nhds).and
+          (self_mem_nhdsWithin : ∀ᶠt in 𝓝[>] a,t∈Ioi a)).exists
+      refine ⟨t,⟨ht.1,ht.2.1,ht.2.2.1⟩,?_⟩
+      have hc := Real.cos_lt_cos_of_nonneg_of_le_pi ha.1.le ht.2.2.2.2.le hat
+      dsimp [r]
+      have hh : 2*Real.cos a=1 := hra
+      linarith
+    · obtain ⟨t,ht,hta⟩ := ((hn.filter_mono nhdsWithin_le_nhds).and
+          (self_mem_nhdsWithin : ∀ᶠt in 𝓝[<] a,t∈Iio a)).exists
+      refine ⟨t,⟨ht.1,ht.2.1,ht.2.2.1⟩,?_⟩
+      have hc := Real.cos_lt_cos_of_nonneg_of_le_pi ht.2.2.2.1.le ha.2.le hta
+      dsimp [r]
+      have hh : 2*Real.cos a=1 := hra
+      linarith
+  obtain ⟨t,ht,hside⟩ := hex
+  have hEt : ‖E t‖=1 := by
+    change ‖e*(Circle.exp t : ℂ)‖=1
+    rw [norm_mul,he,Circle.norm_coe,one_mul]
+  have hmem := hstrip (E t) hEt ht.2.2 (r t) ht.1 ht.2.1 hside
+  let v : ℂ := (r t:ℂ)*E t-e
+  have hve : v=e*(Circle.exp (2*t) : ℂ) := by
+    change ((2*Real.cos t : ℝ):ℂ)*(e*(Circle.exp t : ℂ))-e=e*(Circle.exp (2*t) : ℂ)
+    calc
+      _ = e*((2*Real.cos t : ℂ)*(Circle.exp t : ℂ)-1) := by push_cast; ring
+      _ = _ := by rw [double_cos_circle]
+  refine ⟨v,by rw [hve,norm_mul,he,Circle.norm_coe,one_mul],?_⟩
+  have hvq : (x+e)+v=x+(r t:ℂ)*E t := by dsimp [v]; ring
+  exact hvq ▸ hmem
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology
+namespace EuclideanFiveColor
+def radialSide (b : Bool) (ε : ℝ) : Set ℝ := if b then Ioo 1 (1+ε) else Ioo (1-ε) 1
+def polarBox (x e : ℂ) (b : Bool) (ε : ℝ) : Set ℂ :=
+  (fun rt : ℝ × ℝ => x+(rt.1:ℂ)*(e*(Circle.exp rt.2 : ℂ))) ''
+    (radialSide b ε ×ˢ Ioo (-ε) ε)
+lemma radialSide_connected (b : Bool) {ε : ℝ} (hε : 0<ε) : IsConnected (radialSide b ε) := by
+  cases b <;> dsimp [radialSide] <;> apply isConnected_Ioo <;> linarith
+lemma radialSide_abs {b : Bool} {ε r : ℝ} (hr : r∈radialSide b ε) : |r-1|<ε := by
+  cases b <;> dsimp [radialSide] at hr
+  all_goals rw [abs_lt]; constructor <;> linarith [hr.1,hr.2]
+lemma mem_radialSide {b : Bool} {ε r : ℝ} (hr : |r-1|<ε)
+    (hb : if b then 1<r else r<1) : r∈radialSide b ε := by
+  rw [abs_lt] at hr
+  cases b <;> dsimp [radialSide] at * <;> constructor <;> linarith [hr.1,hr.2]
+lemma polarBox_connected (x e : ℂ) (b : Bool) {ε : ℝ} (hε : 0<ε) :
+    IsConnected (polarBox x e b ε) := by
+  apply ((radialSide_connected b hε).prod (isConnected_Ioo (by linarith : -ε<ε))).image
+  fun_prop
+/-- Arbitrarily small connected polar boxes on either side of a unit circle. -/
+lemma polarBox_small {x e : ℂ} {b : Bool} {δ : ℝ} (hδ : 0<δ)
+    {U W : Set ℂ} (hW : IsOpen W) (hp : x+e∈W)
+    (hstrip : ∀e' : ℂ,‖e'‖=1 → dist e' e<δ → ∀r : ℝ,
+      0<r → |r-1|<δ → (if b then 1<r else r<1) → x+(r:ℂ)*e'∈U)
+    (he : ‖e‖=1) : ∃ε>0,polarBox x e b ε⊆U∩W := by
+  let F : ℝ × ℝ → ℂ := fun rt => x+(rt.1:ℂ)*(e*(Circle.exp rt.2 : ℂ))
+  let E : ℝ × ℝ → ℂ := fun rt => e*(Circle.exp rt.2 : ℂ)
+  have hF : Continuous F := by fun_prop
+  have hE : Continuous E := by fun_prop
+  have hF0 : F (1,0)=x+e := by simp [F]
+  have hE0 : E (1,0)=e := by simp [E]
+  have hn : ∀ᶠrt in 𝓝 ((1,0) : ℝ × ℝ),F rt∈W ∧ dist (E rt) e<δ ∧ 0<rt.1 ∧ |rt.1-1|<δ := by
+    have h₁ : ∀ᶠrt in 𝓝 ((1,0) : ℝ × ℝ),F rt∈W := hF.continuousAt.preimage_mem_nhds (hW.mem_nhds (hF0 ▸ hp))
+    have h₂ : ∀ᶠrt in 𝓝 ((1,0) : ℝ × ℝ),dist (E rt) e<δ :=
+      (hE.dist continuous_const).continuousAt.eventually_lt_const (by simpa [hE0] using hδ)
+    have h₃ : ∀ᶠrt in 𝓝 ((1,0) : ℝ × ℝ),0<rt.1 :=
+      continuousAt_const.eventually_lt continuous_fst.continuousAt (by norm_num)
+    have h₄ : ∀ᶠrt in 𝓝 ((1,0) : ℝ × ℝ),|rt.1-1|<δ :=
+      (continuous_fst.sub continuous_const).abs.continuousAt.eventually_lt_const (by simpa using hδ)
+    exact h₁.and (h₂.and (h₃.and h₄))
+  obtain ⟨ε,hε,hh⟩ := Metric.mem_nhds_iff.mp hn
+  refine ⟨ε,hε,?_⟩
+  rintro q ⟨⟨r,t⟩,⟨hr,ht⟩,rfl⟩
+  have hrt : (r,t)∈Metric.ball ((1,0) : ℝ × ℝ) ε := by
+    rw [Metric.mem_ball,Prod.dist_eq,max_lt_iff]
+    exact ⟨by simpa [Real.dist_eq] using radialSide_abs hr,by simpa [Real.dist_eq,abs_lt] using ht⟩
+  have hz := hh hrt
+  refine ⟨hstrip (E (r,t)) ?_ hz.2.1 r hz.2.2.1 hz.2.2.2 ?_,hz.1⟩
+  · change ‖e*(Circle.exp t : ℂ)‖=1
+    rw [norm_mul,he,Circle.norm_coe,one_mul]
+  · cases b <;> dsimp [radialSide] at hr ⊢
+    · exact hr.2
+    · exact hr.1
+/-- A polar box contains a complete one-sided radial strip, not just selected rays. -/
+lemma polarBox_contains_strip {x e : ℂ} (he : ‖e‖=1) (b : Bool)
+    {ε : ℝ} (hε : 0<ε) : ∃δ>0,∀e' : ℂ,‖e'‖=1 → dist e' e<δ → ∀r : ℝ,
+      |r-1|<δ → (if b then 1<r else r<1) → x+(r:ℂ)*e'∈polarBox x e b ε := by
+  have he0 : e≠0 := by intro h; simp [h] at he
+  have ha : ContinuousAt (fun z : ℂ => Complex.arg (z/e)) e := by
+    have hh : ContinuousAt Complex.arg (e/e) := Complex.continuousAt_arg (by
+      simpa only [div_self he0] using Complex.one_mem_slitPlane)
+    exact hh.comp (f := fun z : ℂ => z/e) (by fun_prop)
+  have hn : ∀ᶠz in 𝓝 e,|Complex.arg (z/e)|<ε :=
+    ha.abs.eventually_lt_const (by simpa [div_self he0] using hε)
+  obtain ⟨δ,hδ,hh⟩ := Metric.mem_nhds_iff.mp hn
+  refine ⟨min δ ε,lt_min hδ hε,?_⟩
+  intro e' he' hdist r hr hb
+  let t : ℝ := Complex.arg (e'/e)
+  have ht : -ε<t ∧ t<ε := abs_lt.mp (hh (hdist.trans_le (min_le_left _ _)))
+  have heq : e*(Circle.exp t : ℂ)=e' := by
+    have hnorm : ‖e'/e‖=1 := by rw [norm_div,he',he,div_one]
+    have h := Complex.norm_mul_exp_arg_mul_I (e'/e)
+    rw [hnorm,Complex.ofReal_one,one_mul] at h
+    change e*Complex.exp ((t:ℂ)*Complex.I)=e'
+    rw [show Complex.exp ((t:ℂ)*Complex.I)=e'/e from h]
+    field_simp
+  refine ⟨(r,t),⟨mem_radialSide (hr.trans_le (min_le_right _ _)) hb,ht⟩,?_⟩
+  change x+(r:ℂ)*(e*(Circle.exp t : ℂ))=x+(r:ℂ)*e'
+  rw [heq]
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+/-- Points having a fixed essential color on some ambient neighborhood. -/
+def essentialLocalSet {ι : Type*} (c : ℂ → ι) (i : ι) : Set ℂ :=
+  {p | ∃U : Set ℂ,IsOpen U ∧ p∈U ∧ ∀ᵐq : ℂ∂volume,q∈U → c q=i}
+lemma essentialLocalSet_open {ι : Type*} (c : ℂ → ι) (i : ι) : IsOpen (essentialLocalSet c i) := by
+  apply isOpen_iff_forall_mem_open.mpr
+  rintro p ⟨U,hU,hp,hae⟩
+  exact ⟨U,fun q hq => ⟨U,hU,hq,hae⟩,hU,hp⟩
+lemma essentialLocalSet_disjoint {ι : Type*} (c : ℂ → ι) {i j : ι} (hne : i≠j) :
+    Disjoint (essentialLocalSet c i) (essentialLocalSet c j) := by
+  apply disjoint_left.mpr
+  rintro p ⟨U,hU,hp,hu⟩ ⟨V,hV,hp',hv⟩
+  have hpos := (hU.inter hV).measure_pos (volume : Measure ℂ) ⟨p,hp,hp'⟩
+  obtain ⟨q,hq,hqa⟩ := Measure.exists_mem_of_measure_ne_zero_of_ae hpos.ne' (ae_restrict_of_ae (hu.and hv))
+  exact hne ((hqa.1 hq.1).symm.trans (hqa.2 hq.2))
+lemma essentialLocalSet_ae {ι : Type*} (c : ℂ → ι) (i : ι) :
+    ∀ᵐq : ℂ∂volume,q∈essentialLocalSet c i → c q=i :=
+  ae_of_locally_ae volume (fun _ h => h)
+/-- Local essential constancy on a connected set has one global essential value.
+The connected set need not be open, which permits polar product images directly. -/
+lemma connected_local_ae_constant {ι : Type*} {c : ℂ → ι} {S : Set ℂ}
+    (hS : IsConnected S) (hl : ∀p∈S,∃i,p∈essentialLocalSet c i) :
+    ∃i,S⊆essentialLocalSet c i ∧ (∀ᵐq : ℂ∂volume,q∈S → c q=i) := by
+  classical
+  obtain ⟨p,hp⟩ := hS.nonempty
+  obtain ⟨i,hi⟩ := hl p hp
+  let V : Set ℂ := ⋃j≠i,essentialLocalSet c j
+  have hv : IsOpen V := isOpen_iUnion fun j => isOpen_iUnion fun _ => essentialLocalSet_open c j
+  have hd : Disjoint (essentialLocalSet c i) V := by
+    apply disjoint_iUnion_right.mpr
+    intro j
+    apply disjoint_iUnion_right.mpr
+    intro hji
+    exact essentialLocalSet_disjoint c (Ne.symm hji)
+  have hsuv : S⊆essentialLocalSet c i∪V := by
+    intro q hq
+    obtain ⟨j,hj⟩ := hl q hq
+    by_cases hji : j=i
+    · exact Or.inl (hji ▸ hj)
+    · exact Or.inr (mem_iUnion₂.mpr ⟨j,hji,hj⟩)
+  have hs := hS.isPreconnected.subset_left_of_subset_union (essentialLocalSet_open c i) hv hd hsuv ⟨p,hp,hi⟩
+  exact ⟨i,hs,(essentialLocalSet_ae c i).mono fun q hq hs' => hq (hs hs')⟩
+end EuclideanFiveColor
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PlaneColoring/Sectors.lean b/lean/OAI/Geometry/PlaneColoring/Sectors.lean
new file mode 100644
index 0000000000000000000000000000000000000000..6965c9e8db1387ab9caa31459c32ce6c51725ea1
--- /dev/null
+++ b/lean/OAI/Geometry/PlaneColoring/Sectors.lean
@@ -0,0 +1,453 @@
+import OAI.Geometry.PlaneColoring.AngularBoundary
+import OAI.Geometry.PlaneColoring.Moser
+
+namespace OAI
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+open scoped RealInnerProductSpace
+namespace EuclideanFiveColor
+def triangleEdge (i : Fin 3) : Finset (Fin 5) := if i=0 then {0,1} else if i=1 then {1,2} else {2,0}
+lemma triangle_majority_cover : ∀s : Fin 3 → Bool,∃b : Bool,
+    ∀j∈({0,1,2} : Finset (Fin 5)),∃i : Fin 3,s i=b ∧ j∈triangleEdge i := by decide
+lemma triangle_outsider : ∀s : Fin 3 → Bool,∀b : Bool,
+    (∀j∈({0,1,2} : Finset (Fin 5)),∃i : Fin 3,s i=b ∧ j∈triangleEdge i) →
+    ∀j : Fin 5,(∀i : Fin 3,s i=b → j∉triangleEdge i) → j=3 ∨ j=4 := by decide
+lemma ApproachingContinuum.side_strip {x : ℂ} {K : Set ℂ} {v e : ℂ}
+    (h : ApproachingContinuum x K v) (he : ‖e‖=1) (hn : ⟪e,v⟫≠0) (b : Bool)
+    (hs : decide (0<⟪e,v⟫)=b) : ∃δ>0,∀e' : ℂ,‖e'‖=1 → dist e' e<δ → ∀r : ℝ,
+      0<r → |r-1|<δ → (if b then 1<r else r<1) → x+(r:ℂ)*e'∈Straddle K := by
+  cases b
+  · have hneg : ⟪e,v⟫<0 := lt_of_le_of_ne (le_of_not_gt (of_decide_eq_false hs)) hn
+    obtain ⟨δ,hδ,hh⟩ := h.inner_strip he hneg
+    refine ⟨δ,hδ,fun e' he' hd r hr hab hr1 => hh e' he' hd r hr ?_ hr1⟩
+    have := (abs_lt.mp hab).1
+    linarith
+  · obtain ⟨δ,hδ,hh⟩ := h.outer_strip he (of_decide_eq_true hs)
+    refine ⟨δ,hδ,fun e' he' hd r _ hab hr1 => hh e' he' hd r hr1 ?_⟩
+    have := (abs_lt.mp hab).2
+    linarith
+/-- Three interface edges supply an actual outsider-only open strip in every
+nonexceptional direction, with the radial side chosen by the majority sign. -/
+lemma triangle_outsider_strip {c : ℂ → Fin 5} (hc : Measurable c)
+    {x : ℂ} {K : Fin 3 → Set ℂ} {v : Fin 3 → ℂ}
+    (hK : ∀i,ApproachingContinuum x (K i) (v i))
+    (hH : ∀i,∀j∈triangleEdge i,Disjoint (spatialSupport c j) (Straddle (K i)))
+    {e : ℂ} (he : ‖e‖=1) (hn : ∀i,⟪e,v i⟫≠0) :
+    ∃b : Bool,∃δ>0,∃W : Set ℂ,IsOpen W ∧
+      (∀ᵐz : ℂ∂volume,z∈W → c z=3 ∨ c z=4) ∧
+      ∀e' : ℂ,‖e'‖=1 → dist e' e<δ → ∀r : ℝ,
+        0<r → |r-1|<δ → (if b then 1<r else r<1) → x+(r:ℂ)*e'∈W := by
+  classical
+  let s : Fin 3 → Bool := fun i => decide (0<⟪e,v i⟫)
+  obtain ⟨b,hb⟩ := triangle_majority_cover s
+  let E : Finset (Fin 3) := Finset.univ.filter fun i => s i=b
+  have hex (i : E) := (hK i).side_strip he (hn i) b ((Finset.mem_filter.mp i.2).2)
+  choose d hd hs using hex
+  obtain ⟨δ,hδ,_,hδd⟩ := finite_positive_lower_bound Finset.univ d (fun i _ => hd i)
+  let W : Set ℂ := ⋂i : E,Straddle (K i)
+  refine ⟨b,δ,hδ,W,isOpen_iInter_of_finite (fun i => straddle_isOpen (K i)),?_,?_⟩
+  · filter_upwards [finite_support_exclusion_ae c hc K triangleEdge hH] with z hz hzW
+    apply triangle_outsider s b hb (c z)
+    intro i hi
+    exact hz i (mem_iInter.mp hzW ⟨i,Finset.mem_filter.mpr ⟨Finset.mem_univ _,hi⟩⟩)
+  · intro e' he' heδ r hr hrδ hside
+    apply mem_iInter.mpr
+    intro i
+    exact hs i e' he' (heδ.trans_le (hδd i (Finset.mem_univ _))) r hr
+      (hrδ.trans_le (hδd i (Finset.mem_univ _))) hside
+/-- Actual local triangle palette rigidity, without terminal angular sectors
+or boundary information as hypotheses. -/
+lemma triangle_palette_local {c : ℂ → Fin 5} (hc : Measurable c) (hw : WeakColoring 5 c)
+    {x : ℂ} {K : Fin 3 → Set ℂ} {v : Fin 3 → ℂ}
+    (hK : ∀i,ApproachingContinuum x (K i) (v i))
+    (hH : ∀i,∀j∈triangleEdge i,Disjoint (spatialSupport c j) (Straddle (K i)))
+    {t : Angles} (hn : ∀i,⟪direction t,v i⟫≠0)
+    (hnR : ∀i,⟪direction t*(Circle.exp (Real.pi/3) : ℂ),v i⟫≠0) :
+    ∃i : Fin 5,(i=3 ∨ i=4) ∧ ∃A : Set Angles,IsOpen A ∧ t∈A ∧
+      ∀ᵐu∂angularMeasure,u∈A → i∈palette c hc x u := by
+  obtain ⟨bu,δu,hδu,U,hU,hUc,hsU⟩ := triangle_outsider_strip hc hK hH (direction_norm t) hn
+  have hunit : ‖direction t*(Circle.exp (Real.pi/3) : ℂ)‖=1 := by
+    rw [norm_mul,direction_norm,Circle.norm_coe,one_mul]
+  obtain ⟨bv,δv,hδv,V,hV,hVc,hsV⟩ := triangle_outsider_strip hc hK hH hunit hnR
+  apply binary_strip_palette_rigidity c hc hw hU hV hUc hVc (lt_min hδu hδv) bu bv
+  · intro e' he' hd r hr hrd hb
+    exact hsU e' he' (hd.trans_le (min_le_left _ _)) r hr (hrd.trans_le (min_le_left _ _)) hb
+  · intro e' he' hd r hr hrd hb
+    exact hsV e' he' (hd.trans_le (min_le_right _ _)) r hr (hrd.trans_le (min_le_right _ _)) hb
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+open scoped RealInnerProductSpace
+namespace EuclideanFiveColor
+lemma triangle_binary_from_continua {c : ℂ → Fin 5} (hc : Measurable c)
+    {x : ℂ} {K : Fin 3 → Set ℂ} {v : Fin 3 → ℂ} (hv : ∀i,‖v i‖=1)
+    (hK : ∀i,ApproachingContinuum x (K i) (v i))
+    (hH : ∀i,∀j∈triangleEdge i,Disjoint (spatialSupport c j) (Straddle (K i))) :
+    BinaryCenter c hc x 3 4 := by
+  have hh : ∀ᵐt∂angularMeasure,(∀i,⟪direction t,v i⟫≠0) →
+      t∈paletteSet c hc x 3 ∨ t∈paletteSet c hc x 4 := by
+    apply ae_of_locally_ae angularMeasure
+    intro t ht
+    obtain ⟨b,δ,hδ,W,hW,hWc,hs⟩ := triangle_outsider_strip hc hK hH (direction_norm t) ht
+    have hcols : ∀ᵐz : ℂ∂volume,z∈W → c z∈({3,4} : Finset (Fin 5)) :=
+      hWc.mono fun z hz hzw => by simpa only [Finset.mem_insert,Finset.mem_singleton] using hz hzw
+    have hex : ∃A : Set Angles,IsOpen A ∧ t∈A ∧
+        ∀ᵐu∂angularMeasure,u∈A → (palette c hc x u∩{3,4}).Nonempty := by
+      cases b
+      · apply inner_strip_trace c hc hδ hW hcols
+        intro u hu r hr hrδ hr1
+        apply hs (direction u) (direction_norm u) hu r hr _ hr1
+        rw [abs_lt]; constructor <;> linarith
+      · apply outer_strip_trace c hc hδ hW hcols
+        intro u hu r hr1 hrδ
+        apply hs (direction u) (direction_norm u) hu r (by linarith) _ hr1
+        rw [abs_lt]; constructor <;> linarith
+    obtain ⟨A,hA,htA,hae⟩ := hex
+    refine ⟨A,hA,htA,hae.mono fun u hu hAu => ?_⟩
+    obtain ⟨i,hi⟩ := hu hAu
+    rcases Finset.mem_inter.mp hi with ⟨hip,hi⟩
+    rcases Finset.mem_insert.mp hi with rfl|hi
+    · exact Or.inl ((mem_palette c hc x u 3).mp hip)
+    · rw [Finset.mem_singleton] at hi
+      subst i
+      exact Or.inr ((mem_palette c hc x u 4).mp hip)
+  filter_upwards [hh,ae_all_iff.mpr (fun i => ae_direction_nonorthogonal (hv i))] with t ht hn
+  exact ht hn
+def triangleState (c : ℂ → Fin 5) (hc : Measurable c) (x : ℂ) (t : Angles) : Bool := by
+  classical
+  exact if t∈paletteSet c hc x 3 then false else true
+lemma triangleState_measurable (c : ℂ → Fin 5) (hc : Measurable c) (x : ℂ) :
+    Measurable (triangleState c hc x) := Measurable.ite (paletteSet_measurable c hc x 3) measurable_const measurable_const
+lemma triangleState_flip {c : ℂ → Fin 5} (hc : Measurable c) (hw : WeakColoring 5 c) {x : ℂ}
+    (hb : BinaryCenter c hc x 3 4) :
+    ∀ᵐt∂angularMeasure,triangleState c hc x (t+(sixty : Angles))= !(triangleState c hc x t) := by
+  filter_upwards [binary_alternation c hc hw hb] with t ht
+  by_cases hp : t∈paletteSet c hc x 3
+  · have hn : t+(sixty : Angles)∉paletteSet c hc x 3 := fun h => (ht.1.mp h) hp
+    simp only [triangleState,hp,hn,ite_true,ite_false,Bool.not_false]
+  · have hn : t+(sixty : Angles)∈paletteSet c hc x 3 := ht.1.mpr hp
+    simp only [triangleState,hp,hn,ite_true,ite_false,Bool.not_true]
+lemma triangleState_local {c : ℂ → Fin 5} (hc : Measurable c) (hw : WeakColoring 5 c)
+    {x : ℂ} {K : Fin 3 → Set ℂ} {v : Fin 3 → ℂ} (hv : ∀i,‖v i‖=1)
+    (hK : ∀i,ApproachingContinuum x (K i) (v i))
+    (hH : ∀i,∀j∈triangleEdge i,Disjoint (spatialSupport c j) (Straddle (K i)))
+    {t : Angles} (hn : ∀i,⟪direction t,v i⟫≠0)
+    (hnR : ∀i,⟪direction t*(Circle.exp (Real.pi/3) : ℂ),v i⟫≠0) :
+    ∃b,t∈angleEssential (triangleState c hc x) b := by
+  obtain ⟨i,hi,A,hA,htA,hae⟩ := triangle_palette_local hc hw hK hH hn hnR
+  have hb := binary_alternation c hc hw (triangle_binary_from_continua hc hv hK hH)
+  rcases hi with rfl|rfl
+  · refine ⟨false,A,hA,htA,hae.mono fun u hu hAu => ?_⟩
+    have hp := (mem_palette c hc x u 3).mp (hu hAu)
+    simp only [triangleState,hp,ite_true]
+  · refine ⟨true,A,hA,htA,?_⟩
+    filter_upwards [hae,hb] with u hu hu' hAu
+    have hp := hu'.2.mp ((mem_palette c hc x u 4).mp (hu hAu))
+    simp only [triangleState,hp,ite_false]
+lemma direction_injective : Function.Injective direction := fun _ _ h =>
+  AddCircle.injective_toCircle (by positivity : (2*Real.pi:ℝ)≠0) (Subtype.ext h)
+lemma direction_plus_sixty (t : Angles) :
+    direction (t+(sixty : Angles))=direction t*(Circle.exp (Real.pi/3) : ℂ) := by
+  obtain ⟨a,rfl⟩ := QuotientAddGroup.mk_surjective t
+  rw [←AddCircle.coe_add,direction_real,direction_real,Circle.exp_add]
+  rfl
+lemma triple_orthogonal_cover (v : Fin 3 → ℂ) (hv : ∀i,‖v i‖=1) :
+    ∃B : Finset Angles,B.card≤6 ∧ (∀t,t+(Real.pi : Angles)∈B ↔ t∈B) ∧
+      ∀t,t∉B → ∀i,⟪direction t,v i⟫≠0 := by
+  classical
+  have hex (i : Fin 3) : ∃t : Angles,direction t=Complex.I*v i := direction_exists
+    (by rw [norm_mul,Complex.norm_I,hv,one_mul])
+  choose a ha using hex
+  let B : Finset Angles := Finset.univ.biUnion fun i : Fin 3 => {a i,a i+(Real.pi : Angles)}
+  have hmem (t : Angles) : t∈B ↔ ∃i : Fin 3,t=a i ∨ t=a i+(Real.pi : Angles) := by
+    simp only [B,Finset.mem_biUnion,Finset.mem_univ,true_and,Finset.mem_insert,Finset.mem_singleton]
+  have hdouble (t : Angles) : t+(Real.pi : Angles)+(Real.pi : Angles)=t := by
+    apply direction_injective
+    simp only [direction_add_pi,neg_neg]
+  refine ⟨B,?_,?_,?_⟩
+  · calc B.card ≤ ∑i : Fin 3,({a i,a i+(Real.pi : Angles)} : Finset Angles).card := Finset.card_biUnion_le
+         _ ≤ ∑_i : Fin 3,2 := Finset.sum_le_sum fun _ _ => by simpa using (Finset.card_insert_le (a _) ({a _+(Real.pi : Angles)} : Finset Angles))
+         _ = 6 := by norm_num
+  · intro t
+    rw [hmem,hmem]
+    constructor
+    · rintro ⟨i,hi|hi⟩
+      · exact ⟨i,Or.inr (by rw [←hi,hdouble])⟩
+      · exact ⟨i,Or.inl (add_right_cancel hi)⟩
+    · rintro ⟨i,rfl|rfl⟩
+      · exact ⟨i,Or.inr rfl⟩
+      · exact ⟨i,Or.inl (hdouble _)⟩
+  · intro t ht i hinner
+    apply ht
+    rw [hmem]
+    rcases complex_unit_orthogonal (direction_norm t) (hv i) hinner with he|he
+    · exact ⟨i,Or.inl (direction_injective (he.trans (ha i).symm))⟩
+    · refine ⟨i,Or.inr (direction_injective ?_)⟩
+      rw [direction_add_pi,ha,he]
+      ring
+/-- The actual three-interface data force the source's unique six-position
+boundary orbit. This theorem does not assume local rigidity or orbit finiteness. -/
+lemma triangle_boundary_one_orbit {c : ℂ → Fin 5} (hc : Measurable c) (hw : WeakColoring 5 c)
+    {x : ℂ} {K : Fin 3 → Set ℂ} {v : Fin 3 → ℂ} (hv : ∀i,‖v i‖=1)
+    (hK : ∀i,ApproachingContinuum x (K i) (v i))
+    (hH : ∀i,∀j∈triangleEdge i,Disjoint (spatialSupport c j) (Straddle (K i))) :
+    ∃t,angleBoundary (triangleState c hc x)={u | ∃j : Fin 6,u=t+sixAngle j} := by
+  obtain ⟨B,hB,ha,hn⟩ := triple_orthogonal_cover v hv
+  apply angleBoundary_one_orbit (triangleState_flip hc hw (triangle_binary_from_continua hc hv hK hH)) B hB ha
+  intro t ht htR
+  apply triangleState_local hc hw hv hK hH (hn t ht)
+  simpa only [direction_plus_sixty] using hn (t+(sixty : Angles)) htR
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+lemma angleEssential_connected {s : Angles → Bool} {U : Set Angles}
+    (hU : IsPreconnected U) (hne : U.Nonempty) (hbound : Disjoint U (angleBoundary s)) :
+    ∃b,U⊆angleEssential s b := by
+  have hcov : U⊆angleEssential s false∪angleEssential s true := by
+    intro t ht
+    have hn := disjoint_left.mp hbound ht
+    simp only [angleBoundary,mem_compl_iff,not_not,mem_iUnion] at hn
+    obtain ⟨b,hb⟩ := hn
+    cases b
+    · exact Or.inl hb
+    · exact Or.inr hb
+  obtain ⟨t,ht⟩ := hne
+  rcases hcov ht with hb|hb
+  · exact ⟨false,hU.subset_left_of_subset_union (angleEssential_open s false) (angleEssential_open s true)
+      (angleEssential_disjoint s (by decide)) hcov ⟨t,ht,hb⟩⟩
+  · exact ⟨true,hU.subset_left_of_subset_union (angleEssential_open s true) (angleEssential_open s false)
+      (angleEssential_disjoint s (by decide)) (by simpa only [union_comm] using hcov) ⟨t,ht,hb⟩⟩
+lemma one_orbit_first_arc {s : Angles → Bool} {φ : ℝ}
+    (hboundary : angleBoundary s={u | ∃j : Fin 6,u=(φ : Angles)+sixAngle j}) :
+    ∃b,∀t∈Ioo φ (φ+Real.pi/3),(t : Angles)∈angleEssential s b := by
+  let U : Set Angles := ((↑) : ℝ → Angles) '' Ioo φ (φ+Real.pi/3)
+  have hU : IsPreconnected U := isPreconnected_Ioo.image _ continuous_quotient_mk'.continuousOn
+  have hne : U.Nonempty := (nonempty_Ioo.mpr (by linarith [Real.pi_pos] : φ<φ+Real.pi/3)).image _
+  have hd : Disjoint U (angleBoundary s) := by
+    apply disjoint_left.mpr
+    rintro q ⟨t,ht,rfl⟩ hq
+    rw [hboundary] at hq
+    obtain ⟨j,hj⟩ := hq
+    have he : (t : Angles)=((φ+(j.val:ℝ)*(Real.pi/3) : ℝ) : Angles) := by
+      have hx : sixAngle j=((j.val:ℝ)*(Real.pi/3) : ℝ) := by
+        rw [sixAngle,←AddCircle.coe_nsmul]
+        simp only [nsmul_eq_mul,sixty]
+      simpa only [hx,AddCircle.coe_add] using hj
+    have htI : t∈Ico φ (φ+2*Real.pi) := ⟨ht.1.le,by linarith [Real.pi_pos,ht.2]⟩
+    have hjI : φ+(j.val:ℝ)*(Real.pi/3)∈Ico φ (φ+2*Real.pi) := by
+      have hj0 : 0≤(j.val:ℝ) := Nat.cast_nonneg _
+      have hj6 : (j.val:ℝ)<6 := by exact_mod_cast j.isLt
+      constructor
+      · nlinarith [Real.pi_pos]
+      · nlinarith [Real.pi_pos]
+    have hv := (AddCircle.coe_eq_coe_iff_of_mem_Ico htI hjI).mp he
+    by_cases hz : j.val=0
+    · simp only [hz,Nat.cast_zero,zero_mul,add_zero] at hv
+      linarith [ht.1]
+    · have hj1 : (1:ℝ)≤j.val := by exact_mod_cast Nat.one_le_iff_ne_zero.mpr hz
+      nlinarith [ht.2,Real.pi_pos]
+  obtain ⟨b,hb⟩ := angleEssential_connected hU hne hd
+  exact ⟨b,fun t ht => hb ⟨t,ht,rfl⟩⟩
+lemma angleEssential_real_periodic {s : Angles → Bool}
+    (hf : ∀ᵐt∂angularMeasure,s (t+(sixty : Angles))= !(s t)) (b : Bool) :
+    Function.Periodic (fun t : ℝ => (t : Angles)∈angleEssential s b) (2*(Real.pi/3)) := by
+  intro t
+  apply propext
+  have hh := angleEssential_shift hf ((t : Angles)+(sixty : Angles)) (!b)
+  rw [Bool.not_not] at hh
+  have hh' := angleEssential_shift hf (t : Angles) b
+  have he : ((t+2*(Real.pi/3) : ℝ) : Angles) = (t : Angles)+(sixty : Angles)+(sixty : Angles) := by
+    calc ((t+2*(Real.pi/3) : ℝ) : Angles) = ((t+sixty+sixty : ℝ) : Angles) := by
+           congr 1
+           dsimp [sixty]
+           ring
+         _ = _ := by rw [AddCircle.coe_add,AddCircle.coe_add]
+  dsimp only
+  rw [he]
+  exact hh.trans hh'
+lemma one_orbit_sectors {s : Angles → Bool} {φ : ℝ}
+    (hf : ∀ᵐt∂angularMeasure,s (t+(sixty : Angles))= !(s t))
+    (hboundary : angleBoundary s={u | ∃j : Fin 6,u=(φ : Angles)+sixAngle j}) :
+    ∃b,(∀t,0<Real.sin (3*(t-φ)) → (t : Angles)∈angleEssential s b) ∧
+      (∀t,Real.sin (3*(t-φ))<0 → (t : Angles)∈angleEssential s (!b)) := by
+  obtain ⟨b,hb⟩ := one_orbit_first_arc hboundary
+  have hb' (t : ℝ) (ht : t∈Ioo (φ+Real.pi/3) (φ+2*(Real.pi/3))) :
+      (t : Angles)∈angleEssential s (!b) := by
+    have hh := (angleEssential_shift hf ((t-Real.pi/3 : ℝ) : Angles) b).mpr
+      (hb (t-Real.pi/3) ⟨by linarith [ht.1],by linarith [ht.2]⟩)
+    simpa only [sixty,←AddCircle.coe_add,sub_add_cancel] using hh
+  have hreduce (t : ℝ) : ∃r : ℝ,∃n : ℤ,0≤r ∧ r<2*(Real.pi/3) ∧
+      t=φ+r+n*(2*(Real.pi/3)) ∧ Real.sin (3*(t-φ))=Real.sin (3*r) := by
+    let n : ℤ := ⌊(t-φ)/(2*(Real.pi/3))⌋
+    let r : ℝ := (t-φ)-n*(2*(Real.pi/3))
+    have he : t=φ+r+n*(2*(Real.pi/3)) := by dsimp [r]; ring
+    refine ⟨r,n,Int.sub_floor_div_mul_nonneg _ (by positivity),Int.sub_floor_div_mul_lt _ (by positivity),he,?_⟩
+    have he' : 3*(t-φ)=3*r+(n:ℝ)*(2*Real.pi) := by rw [he]; ring
+    rw [he',Real.sin_add_int_mul_two_pi]
+  refine ⟨b,?_,?_⟩
+  · intro t ht
+    obtain ⟨r,n,hr0,hrP,he,hsin⟩ := hreduce t
+    rw [hsin] at ht
+    have hrpos : 0<r := by
+      by_contra hn
+      have hr : r=0 := le_antisymm (le_of_not_gt hn) hr0
+      simp only [hr,mul_zero,Real.sin_zero,lt_self_iff_false] at ht
+    have hra : r<Real.pi/3 := by
+      by_contra hn
+      have hh := Real.sin_nonpos_of_nonpos_of_neg_pi_le
+        (by linarith : 3*r-2*Real.pi≤0) (by linarith : -Real.pi≤3*r-2*Real.pi)
+      rw [Real.sin_sub_two_pi] at hh
+      exact (not_lt_of_ge hh) ht
+    have hy := hb (φ+r) ⟨by linarith,by linarith⟩
+    rw [he]
+    exact (angleEssential_real_periodic hf b).int_mul n (φ+r) ▸ hy
+  · intro t ht
+    obtain ⟨r,n,hr0,hrP,he,hsin⟩ := hreduce t
+    rw [hsin] at ht
+    have hra : Real.pi/3<r := by
+      by_contra hn
+      have hh := Real.sin_nonneg_of_nonneg_of_le_pi (by linarith : 0≤3*r) (by linarith : 3*r≤Real.pi)
+      exact (not_lt_of_ge hh) ht
+    have hy := hb' (φ+r) ⟨by linarith,by linarith⟩
+    rw [he]
+    exact (angleEssential_real_periodic hf (!b)).int_mul n (φ+r) ▸ hy
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set MeasureTheory
+namespace EuclideanFiveColor
+lemma moser_density_impossible {k : ℕ} {c : ℂ → Fin k}
+    (hc : Measurable c) (hw : WeakColoring k c) (x v : ℂ) (hv : ‖v‖=1)
+    (J : Finset (Fin k)) (hJ : J.card≤3) :
+    ¬(∀ i, DensityOne {q | c q∈J} (x+v*placedMoser i)) := by
+  intro hD
+  obtain ⟨q,hq,hp⟩ := finite_density_realization hc hw (fun i => x+v*placedMoser i)
+    (fun _ => J) hD
+  have hh : Fintype.card J≤Fintype.card (Fin 3) := by simpa using hJ
+  obtain ⟨f⟩ := Function.Embedding.nonempty_of_card_le hh
+  apply moser_not_three_colorable (fun i => f ⟨c (q+(x+v*placedMoser i)),hq i⟩)
+  intro i j hij he
+  have hd : ‖(x+v*placedMoser i)-(x+v*placedMoser j)‖=1 := by
+    rw [show (x+v*placedMoser i)-(x+v*placedMoser j)=v*(placedMoser i-placedMoser j) by ring,
+      norm_mul,hv,placedMoser_edge_unit hij,one_mul]
+  exact hp i j hd (congrArg Subtype.val (f.injective he))
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+lemma lineDensityZero_compl_of_mem_nhds {A : Set ℝ} {t : ℝ} (hA : A∈𝓝 t) :
+    LineDensityZero Aᶜ t := by
+  obtain ⟨ε,hε,hsub⟩ := Metric.mem_nhds_iff.mp hA
+  apply tendsto_const_nhds.congr'
+  filter_upwards [(eventually_lt_nhds hε).filter_mono nhdsWithin_le_nhds] with r hr
+  have hempty : Aᶜ∩Metric.closedBall t r=∅ := by
+    ext q
+    simp only [mem_empty_iff_false,iff_false]
+    rintro ⟨hq,hqr⟩
+    exact hq (hsub (lt_of_le_of_lt hqr hr))
+  simp [hempty]
+def positiveSector (φ : ℝ) : Set ℝ := {t | 0<Real.sin (3*(t-φ))}
+def negativeSector (φ : ℝ) : Set ℝ := {t | Real.sin (3*(t-φ))<0}
+lemma positiveSector_open (φ : ℝ) : IsOpen (positiveSector φ) := by
+  apply isOpen_lt <;> fun_prop
+lemma negativeSector_open (φ : ℝ) : IsOpen (negativeSector φ) := by
+  apply isOpen_lt <;> fun_prop
+variable {k : ℕ} {c : ℂ → Fin k} (hc : Measurable c)
+lemma polynomial_region_density (hw : WeakColoring k c) {x : ℂ} {a b : Fin k} {φ : ℝ}
+    (hpos : RealState hc x (positiveSector φ) {a})
+    (hneg : RealState hc x (negativeSector φ) {b})
+    {z : ℂ} (hz : ThreeLabelRegion z.re z.im) :
+    DensityOne {q | c q∉({a,b} : Finset (Fin k))} (x+(Circle.exp φ:ℂ)*z) := by
+  obtain ⟨t,u,hs,hn,ht⟩ := polynomial_region_angles hz
+  have hpoint : angularSum x 1 (angleChart (φ+t,φ+u))=x+(Circle.exp φ:ℂ)*z := by
+    simp only [angularSum,angleChart,Complex.ofReal_one,one_mul,direction_real,Circle.exp_add,Circle.coe_mul]
+    rw [add_assoc,←mul_add,hs]
+  have hnon : Real.sin ((φ+u)-(φ+t))≠0 := by
+    simpa only [show φ+u-(φ+t)=u-t by ring] using hn
+  rcases mul_neg_iff.mp ht with ⟨ht,hu⟩|⟨ht,hu⟩
+  · have htm : φ+t∈positiveSector φ := by simpa [positiveSector] using ht
+    have hum : φ+u∈negativeSector φ := by simpa [negativeSector] using hu
+    have hd := states_densityOne hc hw hpos hneg
+      (lineDensityZero_compl_of_mem_nhds ((positiveSector_open φ).mem_nhds htm))
+      (lineDensityZero_compl_of_mem_nhds ((negativeSector_open φ).mem_nhds hum)) hnon
+    rw [hpoint] at hd
+    simpa only [Finset.singleton_union] using hd
+  · have htm : φ+t∈negativeSector φ := by simpa [negativeSector] using ht
+    have hum : φ+u∈positiveSector φ := by simpa [positiveSector] using hu
+    have hd := states_densityOne hc hw hneg hpos
+      (lineDensityZero_compl_of_mem_nhds ((negativeSector_open φ).mem_nhds htm))
+      (lineDensityZero_compl_of_mem_nhds ((positiveSector_open φ).mem_nhds hum)) hnon
+    rw [hpoint] at hd
+    simpa only [Finset.singleton_union,Finset.pair_comm b a] using hd
+/-- The exact six-sector outcome of triangle rigidity is impossible, for every center and phase. -/
+lemma no_alternating_sectors {c : ℂ → Fin 5} (hc : Measurable c) (hw : WeakColoring 5 c)
+    {x : ℂ} {a b : Fin 5} (hab : a≠b) (φ : ℝ)
+    (hpos : RealState hc x (positiveSector φ) {a})
+    (hneg : RealState hc x (negativeSector φ) {b}) : False := by
+  let J : Finset (Fin 5) := {a,b}ᶜ
+  have hJ : J.card≤3 := by
+    change ({a,b}ᶜ : Finset (Fin 5)).card≤3
+    rw [Finset.card_compl,Finset.card_pair hab]
+    decide
+  apply moser_density_impossible hc hw x (Circle.exp φ) (Circle.norm_coe _) J hJ
+  intro i
+  simpa only [J,Finset.mem_compl] using polynomial_region_density hc hw hpos hneg
+    (seven_point_certificate i)
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+lemma triangleState_palette {c : ℂ → Fin 5} (hc : Measurable c) (hw : WeakColoring 5 c)
+    {x : ℂ} (hb : BinaryCenter c hc x 3 4) (b : Bool) :
+    ∀ᵐt∂angularMeasure,triangleState c hc x t=b → t∈paletteSet c hc x (if b then 4 else 3) := by
+  classical
+  filter_upwards [binary_alternation c hc hw hb] with t ht he
+  cases b
+  · simp only [Bool.false_eq_true,ite_false]
+    by_contra hn
+    simp only [triangleState,hn,ite_false] at he
+    exact Bool.noConfusion he
+  · simp only [ite_true]
+    apply ht.2.mpr
+    intro hp
+    simp only [triangleState,hp,ite_true] at he
+    exact Bool.noConfusion he
+/-- Complete actual triangle-interface endgame, including boundary rigidity,
+sector extraction and the exact seven-point unit-distance certificate. -/
+lemma no_triangle_interfaces {c : ℂ → Fin 5} (hc : Measurable c) (hw : WeakColoring 5 c)
+    {x : ℂ} {K : Fin 3 → Set ℂ} {v : Fin 3 → ℂ} (hv : ∀i,‖v i‖=1)
+    (hK : ∀i,ApproachingContinuum x (K i) (v i))
+    (hH : ∀i,∀j∈triangleEdge i,Disjoint (spatialSupport c j) (Straddle (K i))) : False := by
+  have hbinary := triangle_binary_from_continua hc hv hK hH
+  obtain ⟨q,hq⟩ := triangle_boundary_one_orbit hc hw hv hK hH
+  obtain ⟨φ,rfl⟩ := QuotientAddGroup.mk_surjective q
+  obtain ⟨b,hpos,hneg⟩ := one_orbit_sectors (triangleState_flip hc hw hbinary) hq
+  have hstate (d : Bool) (A : Set ℝ) (hA : MeasurableSet A)
+      (hAd : ∀t∈A,(t : Angles)∈angleEssential (triangleState c hc x) d) :
+      RealState hc x A {if d then 4 else 3} := by
+    refine ⟨hA,?_⟩
+    intro i hi
+    rw [Finset.mem_singleton] at hi
+    subst i
+    have hh := angleMk_quasiMeasurePreserving.ae
+      ((angleEssential_ae (triangleState c hc x) d).and (triangleState_palette hc hw hbinary d))
+    filter_upwards [hh] with t ht hAt
+    exact ht.2 (ht.1 (hAd t hAt))
+  have hp := hstate b (positiveSector φ) (positiveSector_open φ).measurableSet hpos
+  have hn := hstate (!b) (negativeSector φ) (negativeSector_open φ).measurableSet hneg
+  exact no_alternating_sectors hc hw (by cases b <;> decide) φ hp hn
+end EuclideanFiveColor
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PlaneColoring/Seven.lean b/lean/OAI/Geometry/PlaneColoring/Seven.lean
new file mode 100644
index 0000000000000000000000000000000000000000..b23cd2d0189d3ec3829b81af92afd9d9a4ce5dc1
--- /dev/null
+++ b/lean/OAI/Geometry/PlaneColoring/Seven.lean
@@ -0,0 +1,68 @@
+import OAI.Geometry.PlaneColoring.Assembly
+import OAI.Geometry.PlaneColoring.Lattice
+import OAI.Geometry.PlaneColoring.Metric
+import OAI.Geometry.PlaneColoring.Covering
+import OAI.Geometry.PlaneColoring.MeasurableAssembly
+import OAI.Geometry.PlaneColoring.MeasurableProper
+
+namespace OAI
+
+noncomputable section
+
+namespace Problem160
+
+/-- The elementary covering-radius statement is the only geometric input
+needed to turn the index-seven lattice coloring into a plane coloring. -/
+theorem properColoring_seven_of_normalized_cover
+    (hcover : ∀ x y : ℝ, ∃ a b : ℤ,
+      (x - ((a : ℝ) + (b : ℝ) / 2)) ^ 2 +
+        (y - Real.sqrt 3 * b / 2) ^ 2 ≤ (1 / 3 : ℝ)) :
+    ProperColoring 7 := by
+  apply exists_unit_distance_coloring_of_center_cover
+    (fun p : ℤ × ℤ => triangularLattice p.1 p.2)
+    (fun p : ℤ × ℤ => SevenLattice.colorFin p.1 p.2) (7 / 15 : ℝ)
+  · intro x
+    obtain ⟨a, b, hab⟩ := hcover ((5 / 4 : ℝ) * x 0) ((5 / 4 : ℝ) * x 1)
+    exact ⟨(a, b), dist_triangularLattice_le_of_coordinates x a b hab⟩
+  · norm_num
+  · intro i j hij hc
+    have hq := SevenLattice.same_color_separation
+      (SevenLattice.colorFin_injective_value hc) hij
+    have hsep := triangularLattice_dist_gt_two i.1 i.2 j.1 j.2 hq
+    linarith
+
+/-- The classical seven-color upper bound, with all boundary points assigned
+using the closed triangular-lattice cover. -/
+theorem properColoring_seven : ProperColoring 7 :=
+  properColoring_seven_of_normalized_cover triangular_lattice_cover
+
+/-- The seven-color construction admits a measurable choice on its boundary
+points: select the first covering center in a countable enumeration. -/
+theorem exists_measurable_proper_seven :
+    ∃ c : Plane → Fin 7, Measurable c ∧
+      ∀ x y : Plane, dist x y = 1 → c x ≠ c y := by
+  apply exists_measurable_unit_distance_coloring_of_center_cover
+    (fun p : ℤ × ℤ => triangularLattice p.1 p.2)
+    (fun p : ℤ × ℤ => SevenLattice.colorFin p.1 p.2) (7 / 15 : ℝ)
+  · intro x
+    obtain ⟨a, b, hab⟩ := triangular_lattice_cover
+      ((5 / 4 : ℝ) * x 0) ((5 / 4 : ℝ) * x 1)
+    exact ⟨(a, b), dist_triangularLattice_le_of_coordinates x a b hab⟩
+  · norm_num
+  · intro i j hij hc
+    have hq := SevenLattice.same_color_separation
+      (SevenLattice.colorFin_injective_value hc) hij
+    have hsep := triangularLattice_dist_gt_two i.1 i.2 j.1 j.2 hq
+    linarith
+
+/-- The classical measurable seven-color construction satisfies the exact
+weak pair-measure condition. -/
+theorem weakMeasurableColoring_seven : WeakMeasurableColoring 7 := by
+  obtain ⟨c, hc, hproper⟩ := exists_measurable_proper_seven
+  exact weakMeasurableColoring_of_aemeasurable_proper c hc.aemeasurable hproper
+
+end Problem160
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PlaneColoring/States.lean b/lean/OAI/Geometry/PlaneColoring/States.lean
new file mode 100644
index 0000000000000000000000000000000000000000..d4b162d37aa7493b00981ef696b89a60715c48eb
--- /dev/null
+++ b/lean/OAI/Geometry/PlaneColoring/States.lean
@@ -0,0 +1,764 @@
+import OAI.Geometry.PlaneColoring.BinaryPalettes
+
+namespace OAI
+
+noncomputable section
+open Set
+namespace EuclideanFiveColor
+def fourGroup (s : Fin 3) : Finset (Fin 5) := if s=0 then {4} else if s=1 then {0,2} else {1,3}
+def fourEdge (e : Fin 4) : Finset (Fin 5) :=
+  if e=0 then {0,1} else if e=1 then {1,2} else if e=2 then {2,3} else {3,0}
+def fourCanonical (p : Finset (Fin 5)) : Fin 3 := if 4∈p then 0 else if 0∈p then 1 else 2
+def FourAdmissible (p : Finset (Fin 5)) : Prop :=
+  p.Nonempty ∧ p.card≤2 ∧ ∀ e : Fin 4, ¬p⊆fourEdge e
+lemma fourGroup_nonempty (s : Fin 3) : (fourGroup s).Nonempty := by
+  fin_cases s <;> simp [fourGroup]
+lemma four_admissible_group : ∀ p : Finset (Fin 5), FourAdmissible p → fourGroup (fourCanonical p)⊆p := by
+  unfold FourAdmissible
+  decide
+lemma four_group_unique : ∀ p : Finset (Fin 5), p.card≤2 → ∀ s t : Fin 3,
+    fourGroup s⊆p → fourGroup t⊆p → s=t := by decide
+lemma four_admissible_character : ∀ p : Finset (Fin 5), FourAdmissible p → ∀ s : Fin 3,
+    fourGroup s⊆p ↔ fourCanonical p=s := by
+  intro p hp s
+  constructor
+  · intro hs
+    exact (four_group_unique p hp.2.1 s (fourCanonical p) hs (four_admissible_group p hp)).symm
+  · rintro rfl
+    exact four_admissible_group p hp
+lemma four_trace_edge : ∀ p : Finset (Fin 5), FourAdmissible p → ∀ i∈fourGroup (fourCanonical p),
+    ∃ e : Fin 4, i∉fourEdge e ∧ p\{i}⊆fourEdge e := by
+  unfold FourAdmissible
+  decide
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+lemma LineDensityZero.mono_ae {A B : Set ℝ} {t : ℝ}
+    (hB : LineDensityZero B t) (hAB : A≤ᵐ[volume] B) : LineDensityZero A t := by
+  apply squeeze_zero' ?_ ?_ hB
+  · filter_upwards [self_mem_nhdsWithin] with r hr
+    exact div_nonneg measureReal_nonneg (le_of_lt hr)
+  · filter_upwards [self_mem_nhdsWithin] with r hr
+    apply div_le_div_of_nonneg_right _ (le_of_lt hr)
+    apply ENNReal.toReal_mono
+    · exact ne_top_of_le_ne_top (isCompact_closedBall t r).measure_ne_top (measure_mono inter_subset_right)
+    · exact measure_mono_ae (hAB.mono fun _ hz h => ⟨hz h.1,h.2⟩)
+def DensityZero (A : Set ℂ) (x : ℂ) : Prop :=
+  Tendsto (fun r : ℝ => volume (A∩Metric.closedBall x r)/volume (Metric.closedBall x r))
+    (𝓝[>] 0) (𝓝 (0:ENNReal))
+lemma DensityZero.mono {A B : Set ℂ} {x : ℂ} (hB : DensityZero B x) (hAB : A⊆B) :
+    DensityZero A x := by
+  apply tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds hB (fun _ => zero_le)
+  intro r
+  exact ENNReal.div_le_div_right (measure_mono (inter_subset_inter_left _ hAB)) _
+lemma DensityZero.union {A B : Set ℂ} {x : ℂ} (hA : DensityZero A x) (hB : DensityZero B x) :
+    DensityZero (A∪B) x := by
+  have hh := hA.add hB
+  rw [add_zero] at hh
+  apply tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds hh (fun _ => zero_le)
+  intro r
+  dsimp only
+  rw [← ENNReal.add_div,union_inter_distrib_right]
+  exact ENNReal.div_le_div_right (measure_union_le _ _) _
+lemma densityZero_empty (x : ℂ) : DensityZero ∅ x := by
+  simp only [DensityZero,empty_inter,measure_empty,ENNReal.zero_div]
+  exact tendsto_const_nhds
+lemma DensityZero.biUnion_finset {ι : Type*} (J : Finset ι) {A : ι → Set ℂ} {x : ℂ}
+    (hA : ∀ i∈J,DensityZero (A i) x) : DensityZero (⋃ i∈J,A i) x := by
+  classical
+  induction J using Finset.induction_on with
+  | empty => simpa using densityZero_empty x
+  | @insert i J hi ih =>
+    rw [Finset.set_biUnion_insert]
+    exact (hA i (Finset.mem_insert_self _ _)).union
+      (ih (fun j hj => hA j (Finset.mem_insert_of_mem hj)))
+lemma DensityZero.compl {A : Set ℂ} {x : ℂ} (hA : MeasurableSet A) (hz : DensityZero A x) :
+    DensityOne Aᶜ x := by
+  have ht := ENNReal.Tendsto.sub (tendsto_const_nhds (x := (1:ENNReal))) hz (Or.inl (by finiteness))
+  rw [tsub_zero] at ht
+  apply ht.congr'
+  filter_upwards [self_mem_nhdsWithin] with r hr
+  have hfin : volume (A∩Metric.closedBall x r)≠⊤ :=
+    ne_top_of_le_ne_top (isCompact_closedBall x r).measure_ne_top (measure_mono inter_subset_right)
+  have hp := (Metric.measure_closedBall_pos volume x hr).ne'
+  have he : Aᶜ∩Metric.closedBall x r=Metric.closedBall x r\(A∩Metric.closedBall x r) := by
+    ext z; simp; tauto
+  rw [he,measure_sdiff inter_subset_right (hA.inter measurableSet_closedBall).nullMeasurableSet hfin,
+    ENNReal.sub_div (fun _ _ => hp),ENNReal.div_self hp (isCompact_closedBall x r).measure_ne_top]
+variable {k : ℕ} {c : ℂ → Fin k} (hc : Measurable c)
+def RealState (x : ℂ) (A : Set ℝ) (J : Finset (Fin k)) : Prop :=
+  MeasurableSet A ∧ ∀ i∈J, A≤ᵐ[volume] liftPalette c hc x i
+lemma state_densityZero (hw : WeakColoring k c) {x : ℂ} {A : Set ℝ} {J : Finset (Fin k)}
+    (hA : RealState hc x A J) {t u : ℝ} (ht : LineDensityZero Aᶜ t)
+    (hu : Real.sin (u-t)≠0) {i : Fin k} (hi : i∈J) :
+    DensityZero {z | c z=i} (angularSum x 1 (angleChart (t,u))) := by
+  apply palette_spatial_densityZero c hc hw _ hu
+  apply ht.mono_ae
+  filter_upwards [hA.2 i hi] with v hv
+  exact mt hv
+lemma angularSum_swap (x : ℂ) (t u : ℝ) :
+    angularSum x 1 (angleChart (u,t))=angularSum x 1 (angleChart (t,u)) := by
+  simp only [angularSum,angleChart,Complex.ofReal_one,one_mul]
+  ring
+lemma states_densityOne (hw : WeakColoring k c) {x : ℂ} {A B : Set ℝ} {J K : Finset (Fin k)}
+    (hA : RealState hc x A J) (hB : RealState hc x B K) {t u : ℝ}
+    (ht : LineDensityZero Aᶜ t) (hu : LineDensityZero Bᶜ u) (hne : Real.sin (u-t)≠0) :
+    DensityOne {z | c z∉J∪K} (angularSum x 1 (angleChart (t,u))) := by
+  have hz : DensityZero (⋃ i∈J∪K, {z | c z=i}) (angularSum x 1 (angleChart (t,u))) := by
+    apply DensityZero.biUnion_finset
+    intro i hi
+    rcases Finset.mem_union.mp hi with hi|hi
+    · exact state_densityZero hc hw hA ht hne hi
+    · rw [← angularSum_swap x t u]
+      apply state_densityZero hc hw hB hu _ hi
+      rwa [← neg_sub u t,Real.sin_neg,neg_ne_zero]
+  have hm : MeasurableSet (⋃ i∈J∪K, {z | c z=i}) :=
+    (J∪K).measurableSet_biUnion fun _ _ => measurableSet_eq_fun hc measurable_const
+  convert hz.compl hm using 1
+  ext z
+  simp only [mem_compl_iff,mem_iUnion,mem_ofPred_eq]
+  aesop
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+lemma DensityOne.compl {A : Set ℂ} {x : ℂ} (hA : MeasurableSet A) (h : DensityOne A x) :
+    DensityZero Aᶜ x := by
+  have ht := ENNReal.Tendsto.sub (tendsto_const_nhds (x := (1:ENNReal))) h (Or.inl (by finiteness))
+  rw [tsub_self] at ht
+  apply ht.congr'
+  filter_upwards [self_mem_nhdsWithin] with r hr
+  have hfin : volume (A∩Metric.closedBall x r)≠⊤ :=
+    ne_top_of_le_ne_top (isCompact_closedBall x r).measure_ne_top (measure_mono inter_subset_right)
+  have hp := (Metric.measure_closedBall_pos volume x hr).ne'
+  have he : Aᶜ∩Metric.closedBall x r=Metric.closedBall x r\(A∩Metric.closedBall x r) := by
+    ext z; simp; tauto
+  rw [he,measure_sdiff inter_subset_right (hA.inter measurableSet_closedBall).nullMeasurableSet hfin,
+    ENNReal.sub_div (fun _ _ => hp),ENNReal.div_self hp (isCompact_closedBall x r).measure_ne_top]
+lemma DensityOne.inter {A B : Set ℂ} {x : ℂ} (hA : MeasurableSet A) (hB : MeasurableSet B)
+    (ha : DensityOne A x) (hb : DensityOne B x) : DensityOne (A∩B) x := by
+  have hh := ((ha.compl hA).union (hb.compl hB)).compl (hA.compl.union hB.compl)
+  simpa using hh
+lemma DensityOne.measure_ne_zero {A : Set ℂ} {x : ℂ} (h : DensityOne A x) : volume A≠0 := by
+  intro hz
+  have hzero : (fun r => volume (A∩Metric.closedBall x r)/volume (Metric.closedBall x r))=fun _ : ℝ => (0:ENNReal) := by
+    funext r
+    rw [measure_mono_null inter_subset_left hz,ENNReal.zero_div]
+  change Tendsto _ _ _ at h
+  rw [hzero] at h
+  exact zero_ne_one (tendsto_nhds_unique tendsto_const_nhds h)
+lemma densityOne_univ (x : ℂ) : DensityOne univ x := by
+  apply tendsto_const_nhds.congr'
+  filter_upwards [self_mem_nhdsWithin] with r hr
+  simp only [univ_inter]
+  exact (ENNReal.div_self (Metric.measure_closedBall_pos volume x hr).ne'
+    (isCompact_closedBall x r).measure_ne_top).symm
+lemma DensityOne.iInter {ι : Type*} [Finite ι] {A : ι → Set ℂ} {x : ℂ}
+    (hm : ∀i,MeasurableSet (A i)) (h : ∀i,DensityOne (A i) x) : DensityOne (⋂i,A i) x := by
+  classical
+  let := Fintype.ofFinite ι
+  have hh (J : Finset ι) : DensityOne (⋂i∈J,A i) x := by
+    induction J using Finset.induction_on with
+    | empty => simpa using densityOne_univ x
+    | @insert i J hi ih =>
+      simp only [Finset.set_biInter_insert]
+      exact (h i).inter (hm i) (J.measurableSet_biInter fun j _ => hm j) ih
+  simpa using hh Finset.univ
+variable {k : ℕ} {c : ℂ → Fin k}
+lemma finite_density_realization (hc : Measurable c) (hw : WeakColoring k c)
+    {ι : Type*} [Finite ι] (z : ι → ℂ) (J : ι → Finset (Fin k))
+    (hz : ∀v,DensityOne {q | c q∈J v} (z v)) :
+    ∃q : ℂ, (∀v,c (q+z v)∈J v) ∧
+      ∀v w, ‖z v-z w‖=1 → c (q+z v)≠c (q+z w) := by
+  have hmem : DensityOne (⋂v,{q | c (q+z v)∈J v}) 0 := by
+    apply DensityOne.iInter
+    · intro v
+      exact (J v).finite_toSet.measurableSet.preimage (hc.comp (measurable_id.add_const _))
+    · intro v
+      simpa only [zero_add,preimage_ofPred_eq] using (densityOne_translate (A := {q | c q∈J v}) (x := 0) (u := z v)).mpr (by simpa using hz v)
+  have hproper : ∀ᵐ q : ℂ ∂volume, ∀v w, ‖z v-z w‖=1 → c (q+z v)≠c (q+z w) := by
+    apply ae_all_iff.mpr
+    intro v
+    apply ae_all_iff.mpr
+    intro w
+    by_cases hh : ‖z v-z w‖=1
+    · obtain ⟨t,ht⟩ := direction_exists (u := z w-z v) (by simpa only [norm_sub_rev] using hh)
+      have hd := (measurePreserving_add_right volume (z v)).quasiMeasurePreserving.ae
+        (weak_all_directions c hc hw t)
+      filter_upwards [hd] with q hq _
+      simpa only [ht,show q+z v+(z w-z v)=q+z w by ring] using hq
+    · exact Eventually.of_forall fun _ => fun h => (hh h).elim
+  obtain ⟨q,hq,hqp⟩ := Measure.exists_mem_of_measure_ne_zero_of_ae hmem.measure_ne_zero (ae_restrict_of_ae hproper)
+  exact ⟨q,fun v => mem_iInter.mp hq v,hqp⟩
+lemma two_label_triangle_impossible (hc : Measurable c) (hw : WeakColoring k c)
+    (z : Fin 3 → ℂ) (J : Finset (Fin k)) (hJ : J.card≤2)
+    (hd : ∀v w,v≠w → ‖z v-z w‖=1) :
+    ¬(∀v,DensityOne {q | c q∈J} (z v)) := by
+  intro hz
+  obtain ⟨q,hq,hp⟩ := finite_density_realization hc hw z (fun _ => J) hz
+  have hinj : Function.Injective (fun v : Fin 3 => (⟨c (q+z v),hq v⟩ : J)) := by
+    intro v w he
+    by_contra hne
+    exact hp v w (hd v w hne) (congrArg Subtype.val he)
+  have hcard := Fintype.card_le_of_injective _ hinj
+  simp only [Fintype.card_fin,Fintype.card_coe] at hcard
+  omega
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+lemma bool_flip_two {s : ℝ → Bool} {κ : ℝ}
+    (h : ∀ᵐt∂volume,s (t+κ)= !(s t)) : ∀ᵐt∂volume,s (t+2*κ)=s t := by
+  have h' := (measurePreserving_add_right volume κ).quasiMeasurePreserving.ae h
+  filter_upwards [h,h'] with t ht ht'
+  rw [show t+κ+κ=t+2*κ by ring,ht,Bool.not_not] at ht'
+  exact ht'
+lemma bool_flip_three {s : ℝ → Bool} {κ : ℝ}
+    (h : ∀ᵐt∂volume,s (t+κ)= !(s t)) : ∀ᵐt∂volume,s (t+3*κ)= !(s t) := by
+  have h' := (measurePreserving_add_right volume (2*κ)).quasiMeasurePreserving.ae h
+  filter_upwards [bool_flip_two h,h'] with t ht ht'
+  rwa [show t+2*κ+κ=t+3*κ by ring,ht] at ht'
+lemma bool_flip_even {s : ℝ → Bool} {κ : ℝ}
+    (h : ∀ᵐt∂volume,s (t+κ)= !(s t)) (n : ℕ) : ∀ᵐt∂volume,s (t+2*n*κ)=s t := by
+  induction n with
+  | zero => simp
+  | succ n ih =>
+    have h' := (measurePreserving_add_right volume (2*n*κ)).quasiMeasurePreserving.ae (bool_flip_two h)
+    filter_upwards [ih,h'] with t ht ht'
+    rw [ht] at ht'
+    convert ht' using 1
+    norm_num
+    congr 1
+    ring
+lemma triangleAngle_nonparallel : Real.sin (Real.pi+triangleAngle)≠0 := by
+  rw [add_comm,Real.sin_add_pi,neg_ne_zero]
+  exact (Real.sin_pos_of_pos_of_lt_pi triangleAngle_pos triangleAngle_lt_pi).ne'
+lemma two_state_impossible {k : ℕ} {c : ℂ → Fin k} (hc : Measurable c) (hw : WeakColoring k c)
+    (x : ℂ) (J K : Finset (Fin k)) (hcard : (Finset.univ\(J∪K)).card≤2)
+    (s : Angles → Bool) (hs : Measurable s)
+    (hJ : RealState hc x {t : ℝ | s (t : Angles)=false} J)
+    (hK : RealState hc x {t : ℝ | s (t : Angles)=true} K)
+    (hflip : ∀ᵐt∂angularMeasure,s (t+(Real.pi/3 : ℝ))= !(s t)) : False := by
+  let f : ℝ → Bool := fun t => s (t : Angles)
+  have hf : Measurable f := hs.comp angleMk_quasiMeasurePreserving.measurable
+  have hflip' : ∀ᵐt : ℝ ∂volume, f (t+Real.pi/3)= !(f t) := by
+    have hh := angleMk_quasiMeasurePreserving.ae hflip
+    simpa only [← AddCircle.coe_add] using hh
+  have heven (n : ℕ) := bool_flip_even hflip' n
+  have hπ : ∀ᵐt : ℝ ∂volume, f (t+Real.pi)= !(f t) := by
+    simpa only [show 3*(Real.pi/3)=Real.pi by ring] using bool_flip_three hflip'
+  have hden : ∀ᵐt : ℝ ∂volume, ∀b : Bool, f t=b → LineDensityZero {a : ℝ | f a≠b} t := by
+    apply ae_all_iff.mpr
+    intro b
+    exact ae_lineDensityZero_compl (measurableSet_eq_fun hf measurable_const)
+  have hgood : ∀ᵐt : ℝ ∂volume, ∀j : Fin 3,
+      f (t+2*(j:ℝ)*(Real.pi/3))=f t ∧
+      f (t+2*(j:ℝ)*(Real.pi/3)+Real.pi+triangleAngle)= !(f (t+triangleAngle)) ∧
+      (∀b : Bool,f (t+2*(j:ℝ)*(Real.pi/3))=b →
+        LineDensityZero {a : ℝ | f a≠b} (t+2*(j:ℝ)*(Real.pi/3))) ∧
+      (∀b : Bool,f (t+2*(j:ℝ)*(Real.pi/3)+Real.pi+triangleAngle)=b →
+        LineDensityZero {a : ℝ | f a≠b} (t+2*(j:ℝ)*(Real.pi/3)+Real.pi+triangleAngle)) := by
+    apply ae_all_iff.mpr
+    intro j
+    have h₁ := heven j.val
+    have h₂ := (measurePreserving_add_right volume triangleAngle).quasiMeasurePreserving.ae (heven j.val)
+    have h₃ := (measurePreserving_add_right volume (triangleAngle+2*(j:ℝ)*(Real.pi/3))).quasiMeasurePreserving.ae hπ
+    have h₄ := (measurePreserving_add_right volume (2*(j:ℝ)*(Real.pi/3))).quasiMeasurePreserving.ae hden
+    have h₅ := (measurePreserving_add_right volume (2*(j:ℝ)*(Real.pi/3)+Real.pi+triangleAngle)).quasiMeasurePreserving.ae hden
+    filter_upwards [h₁,h₂,h₃,h₄,h₅] with t ht₁ ht₂ ht₃ ht₄ ht₅
+    refine ⟨ht₁,?_,ht₄,?_⟩
+    · change f (t+(triangleAngle+2*(j:ℝ)*(Real.pi/3))+Real.pi)= !(f (t+(triangleAngle+2*(j:ℝ)*(Real.pi/3)))) at ht₃
+      rw [show t+(triangleAngle+2*(j:ℝ)*(Real.pi/3))=t+triangleAngle+2*(j:ℝ)*(Real.pi/3) by ring,ht₂] at ht₃
+      simpa only [show t+triangleAngle+2*(j:ℝ)*(Real.pi/3)+Real.pi = t+2*(j:ℝ)*(Real.pi/3)+Real.pi+triangleAngle by ring] using ht₃
+    · simpa only [show t+(2*(j:ℝ)*(Real.pi/3)+Real.pi+triangleAngle)=t+2*(j:ℝ)*(Real.pi/3)+Real.pi+triangleAngle by ring] using ht₅
+  have hfreq : ∃ᶠt : ℝ in ae volume, f t=f (t+triangleAngle) := by
+    by_contra hh
+    have hn : ∀ᵐt : ℝ ∂volume, f t≠f (t+triangleAngle) := not_frequently.mp hh
+    have ha : ∀ᵐt∂angularMeasure,s t≠s (t+(triangleAngle : Angles)) := by
+      apply ae_angular_of_ae_line
+      · exact (measurableSet_eq_fun hs (hs.comp (measurable_id.add_const _))).compl
+      · simpa only [f,← AddCircle.coe_add] using hn
+    exact (ne_of_gt (equal_triangleAngle_positive hs hflip))
+      (measure_eq_zero_iff_ae_notMem.mpr ha)
+  obtain ⟨t,hteq,htgood⟩ := (hfreq.and_eventually hgood).exists
+  apply two_label_triangle_impossible hc hw (sourceTriangle x t) (Finset.univ\(J∪K)) hcard
+    (sourceTriangle_pair x t)
+  intro j
+  obtain ⟨h₁,h₂,h₃,h₄⟩ := htgood j
+  rw [← hteq] at h₂
+  have hne : Real.sin ((t+2*(j:ℝ)*(Real.pi/3)+Real.pi+triangleAngle)-(t+2*(j:ℝ)*(Real.pi/3)))≠0 := by
+    rw [show (t+2*(j:ℝ)*(Real.pi/3)+Real.pi+triangleAngle)-(t+2*(j:ℝ)*(Real.pi/3))=Real.pi+triangleAngle by ring]
+    exact triangleAngle_nonparallel
+  have hd : DensityOne {q | c q∉J∪K} (sourceTriangle x t j) := by
+    cases hb : f t
+    · have ha := h₃ false (h₁.trans hb)
+      have hb' := h₄ true (by simpa [hb] using h₂)
+      exact states_densityOne hc hw hJ hK ha hb' hne
+    · have ha := h₃ true (h₁.trans hb)
+      have hb' := h₄ false (by simpa [hb] using h₂)
+      simpa only [Finset.union_comm,sourceTriangle] using states_densityOne hc hw hK hJ ha hb' hne
+  simpa only [Finset.mem_sdiff,Finset.mem_univ,true_and] using hd
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+variable (c : ℂ → Fin 5) (hc : Measurable c)
+def FourCenter (x : ℂ) : Prop :=
+  TransitionEdge c 0 1 x ∧ TransitionEdge c 1 2 x ∧ TransitionEdge c 2 3 x ∧ TransitionEdge c 3 0 x
+lemma four_palette_admissible (hw : WeakColoring 5 c) {x : ℂ} (hx : FourCenter c x) :
+    ∀ᵐ t∂angularMeasure, FourAdmissible (palette c hc x t) := by
+  filter_upwards [palette_nonempty c hc x,palette_cardinality c hc hw x,
+    edge_palette_not_pair c hc hw hx.1,edge_palette_not_pair c hc hw hx.2.1,
+    edge_palette_not_pair c hc hw hx.2.2.1,edge_palette_not_pair c hc hw hx.2.2.2] with t ht hcard h01 h12 h23 h30
+  refine ⟨ht,hcard.2,?_⟩
+  intro e
+  fin_cases e <;> simp only [fourEdge] <;> assumption
+def fourState (x : ℂ) (t : Angles) : Fin 3 := fourCanonical (palette c hc x t)
+lemma fourState_measurable (x : ℂ) : Measurable (fourState c hc x) := by
+  change Measurable (fun t => if 4∈palette c hc x t then (0:Fin 3) else if 0∈palette c hc x t then 1 else 2)
+  exact Measurable.ite (by simpa only [mem_palette,Set.ofPred_mem_eq] using paletteSet_measurable c hc x 4) measurable_const
+    (Measurable.ite (by simpa only [mem_palette,Set.ofPred_mem_eq] using paletteSet_measurable c hc x 0) measurable_const measurable_const)
+lemma fourState_group (hw : WeakColoring 5 c) {x : ℂ} (hx : FourCenter c x) :
+    ∀ᵐ t∂angularMeasure, fourGroup (fourState c hc x t)⊆palette c hc x t :=
+  (four_palette_admissible c hc hw hx).mono fun _ ht => four_admissible_group _ ht
+lemma fourState_represents (hw : WeakColoring 5 c) {x : ℂ} (hx : FourCenter c x) (s : Fin 3) :
+    RealState hc x {t : ℝ | fourState c hc x (t : Angles)=s} (fourGroup s) := by
+  refine ⟨measurableSet_eq_fun ((fourState_measurable c hc x).comp angleMk_quasiMeasurePreserving.measurable) measurable_const,?_⟩
+  intro i hi
+  have hh := angleMk_quasiMeasurePreserving.ae (fourState_group c hc hw hx)
+  filter_upwards [hh] with t ht hts
+  change fourState c hc x (t : Angles)=s at hts
+  exact (mem_palette c hc x (t : Angles) i).mp (ht (by simpa only [hts] using hi))
+lemma fourState_changes (hw : WeakColoring 5 c) {x : ℂ} (hx : FourCenter c x) :
+    ∀ᵐ t∂angularMeasure, fourState c hc x t≠fourState c hc x (t+(Real.pi/3 : ℝ)) := by
+  have hgp := fourState_group c hc hw hx
+  filter_upwards [hgp,(measurePreserving_add_right angularMeasure (sixty : Angles)).quasiMeasurePreserving.ae hgp,
+    palette_sixty_disjoint c hc hw x] with t ht hu hd he
+  obtain ⟨i,hi⟩ := fourGroup_nonempty (fourState c hc x t)
+  have hi' : i∈palette c hc x (t+(sixty : Angles)) := hu (by simpa only [sixty,he] using hi)
+  exact Finset.disjoint_left.mp hd (ht hi) hi'
+lemma three_exclude_flip : ∀ (s t o : Fin 3), o≠0 → s≠o → t≠o → s≠t →
+    (if t=0 then false else true) = !(if s=0 then false else true) := by decide
+lemma fourState_diagonal_positive (hw : WeakColoring 5 c) {x : ℂ} (hx : FourCenter c x)
+    (s : Fin 3) (hs : s≠0) : 0<angularMeasure {t | fourState c hc x t=s} := by
+  by_contra! hn
+  have hnot : ∀ᵐt∂angularMeasure, fourState c hc x t≠s :=
+    measure_eq_zero_iff_ae_notMem.mp (le_antisymm hn zero_le)
+  let r : Fin 3 := if s=1 then 2 else 1
+  have hr : r≠0 ∧ r≠s ∧ ∀t : Fin 3,t≠0 → t≠s → t=r := by
+    exact (show ∀s : Fin 3,s≠0 → (if s=1 then (2:Fin 3) else 1)≠0 ∧ (if s=1 then (2:Fin 3) else 1)≠s ∧
+      ∀t : Fin 3,t≠0 → t≠s → t=(if s=1 then 2 else 1) from by decide) s hs
+  let f : Angles → Bool := fun t => if fourState c hc x t=0 then false else true
+  have hf : Measurable f := Measurable.ite
+    (measurableSet_eq_fun (fourState_measurable c hc x) measurable_const) measurable_const measurable_const
+  have hrep (v : Bool) : RealState hc x {t : ℝ | f (t : Angles)=v} (if v then fourGroup r else fourGroup 0) := by
+    refine ⟨measurableSet_eq_fun (hf.comp angleMk_quasiMeasurePreserving.measurable) measurable_const,?_⟩
+    intro i hi
+    filter_upwards [angleMk_quasiMeasurePreserving.ae (fourState_group c hc hw hx),
+      angleMk_quasiMeasurePreserving.ae hnot] with t ht htn htv
+    change f (t : Angles)=v at htv
+    apply (mem_palette c hc x (t : Angles) i).mp
+    apply ht
+    cases v
+    · have hz : fourState c hc x (t : Angles)=0 := by simpa [f] using htv
+      simpa only [hz,Bool.false_eq_true,ite_false] using hi
+    · have hz : fourState c hc x (t : Angles)=r := hr.2.2 _ (by simpa [f] using htv) htn
+      simpa only [hz,ite_true] using hi
+  have hcard : (Finset.univ\(fourGroup 0∪fourGroup r)).card≤2 := by
+    exact (show ∀s : Fin 3,s≠0 → (Finset.univ\(fourGroup 0∪fourGroup (if s=1 then 2 else 1))).card≤2 from by decide) s hs
+  apply two_state_impossible hc hw x (fourGroup 0) (fourGroup r) hcard f hf (hrep false) (hrep true)
+  filter_upwards [hnot,(measurePreserving_add_right angularMeasure (Real.pi/3 : ℝ)).quasiMeasurePreserving.ae hnot,
+      fourState_changes c hc hw hx] with t ht hu hne
+  exact three_exclude_flip _ _ s hs ht hu hne
+end EuclideanFiveColor
+end
+
+noncomputable section
+open MeasureTheory Filter Topology Set
+open scoped NNReal
+namespace EuclideanFiveColor
+lemma varying_pullback_L1 {X Y : Type*} [MeasurableSpace X]
+    [TopologicalSpace Y] [NormalSpace Y] [MeasurableSpace Y] [BorelSpace Y]
+    {μ : Measure X} {ν : Measure Y} [IsFiniteMeasure μ] [ν.WeaklyRegular]
+    {T : ℕ → X → Y} {S : X → Y} {C : ℝ≥0}
+    (hT : ∀ n, Measurable (T n)) (hS : Measurable S)
+    (hmT : ∀ᶠ n in atTop, μ.map (T n) ≤ (C : ENNReal) • ν) (hmS : μ.map S ≤ (C : ENNReal) • ν)
+    (ht : ∀ᵐ x ∂μ, Tendsto (fun n => T n x) atTop (𝓝 (S x)))
+    {f : Y → ℝ} {g : ℕ → Y → ℝ} (hf : Integrable f ν) (hg : ∀ n, Integrable (g n) ν)
+    (hgf : Tendsto (fun n => ∫ y, ‖g n y-f y‖ ∂ν) atTop (𝓝 0)) :
+    Tendsto (fun n => ∫ x, ‖g n (T n x)-f (S x)‖ ∂μ) atTop (𝓝 0) := by
+  have hfixed := tendsto_integral_norm_comp_sub_eventual hT hS hmT hmS ht hf
+  have hbound : Tendsto (fun n => (C : ℝ)*(∫ y, ‖g n y-f y‖ ∂ν)+
+      ∫ x, ‖f (T n x)-f (S x)‖ ∂μ) atTop (𝓝 0) := by
+    simpa only [mul_zero,add_zero] using (hgf.const_mul (C : ℝ)).add hfixed
+  apply squeeze_zero' (Eventually.of_forall fun n => integral_nonneg fun _ => norm_nonneg _) ?_ hbound
+  filter_upwards [hmT] with n hn
+  have hgT := integrable_comp_of_map_le (hT n) hn (hg n)
+  have hfT := integrable_comp_of_map_le (hT n) hn hf
+  have hfS := integrable_comp_of_map_le hS hmS hf
+  calc
+    _ ≤ ∫ x, ‖g n (T n x)-f (T n x)‖+‖f (T n x)-f (S x)‖ ∂μ :=
+      integral_mono (hgT.sub hfS).norm ((hgT.sub hfT).norm.add (hfT.sub hfS).norm)
+        (fun x => norm_sub_le_norm_sub_add_norm_sub _ _ _)
+    _ = (∫ x, ‖g n (T n x)-f (T n x)‖ ∂μ)+(∫ x, ‖f (T n x)-f (S x)‖ ∂μ) :=
+      integral_add (hgT.sub hfT).norm (hfT.sub hfS).norm
+    _ ≤ _ := add_le_add (integral_norm_comp_le (hT n) hn ((hg n).sub hf)) le_rfl
+lemma indicator_inter_positive {X : Type*} [MeasurableSpace X] {μ : Measure X} [IsFiniteMeasure μ]
+    {A : Set X} {B D : ℕ → Set X} (hA : MeasurableSet A)
+    (hB : ∀ n,MeasurableSet (B n)) (hD : ∀ n,MeasurableSet (D n)) (hp : 0<μ A)
+    (hBA : Tendsto (fun n => ∫ x, ‖(B n).indicator (fun _ => (1:ℝ)) x-A.indicator (fun _ => (1:ℝ)) x‖ ∂μ) atTop (𝓝 0))
+    (hDA : Tendsto (fun n => ∫ x, ‖(D n).indicator (fun _ => (1:ℝ)) x-A.indicator (fun _ => (1:ℝ)) x‖ ∂μ) atTop (𝓝 0)) :
+    ∀ᶠ n in atTop, 0<μ (B n∩D n) := by
+  have hpos : 0<(μ A).toReal := ENNReal.toReal_pos hp.ne' (measure_ne_top _ _)
+  have ht := hBA.add hDA
+  rw [add_zero] at ht
+  filter_upwards [ht.eventually (gt_mem_nhds hpos)] with n hn
+  by_contra hh
+  have hz : μ (B n∩D n)=0 := nonpos_iff_eq_zero.mp (le_of_not_gt hh)
+  have hiA : Integrable (A.indicator (fun _ => (1:ℝ))) μ := (integrable_const _).indicator hA
+  have hiB : Integrable ((B n).indicator (fun _ => (1:ℝ))) μ := (integrable_const _).indicator (hB n)
+  have hiD : Integrable ((D n).indicator (fun _ => (1:ℝ))) μ := (integrable_const _).indicator (hD n)
+  have hi := integral_mono_ae hiA ((hiB.sub hiA).norm.add (hiD.sub hiA).norm) ?_
+  · simp only [Pi.add_apply] at hi
+    rw [integral_add (hiB.sub hiA).norm (hiD.sub hiA).norm,integral_indicator_const (1:ℝ) hA,smul_eq_mul,mul_one] at hi
+    exact (not_lt_of_ge hi) hn
+  · filter_upwards [measure_eq_zero_iff_ae_notMem.mp hz] with x hx
+    by_cases ha : x∈A
+    · by_cases hb : x∈B n
+      · have hd : x∉D n := fun hd => hx ⟨hb,hd⟩
+        simp [ha,hb,hd]
+      · by_cases hd : x∈D n <;> simp [ha,hb,hd]
+    · simp only [Pi.add_apply,Set.indicator_of_notMem ha]
+      positivity
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+open scoped NNReal
+namespace EuclideanFiveColor
+lemma circleEndpoint_identity_neighborhood {a u : ℝ} (h : Real.sin (u-a)≠0) :
+    ∃ I : Set ℝ, IsOpen I ∧ a∈I ∧
+      ∀ v∈I, circleTarget (samplingParameter 0 (direction (u : Angles)) 1,v)∈(circleInverseHomeo h).target ∧
+        circleEndpoint h (samplingParameter 0 (direction (u : Angles)) 1,v)=v := by
+  let V := (circleInverseHomeo h).source
+  have ha : (a,u)∈V := (circleSum_strict 1 (a,u)).mem_toOpenPartialHomeomorph_source
+  let I := (fun v : ℝ => (v,u)) ⁻¹' V
+  refine ⟨I,(circleInverseHomeo h).open_source.preimage (continuous_id.prodMk continuous_const),ha,?_⟩
+  intro v hv
+  have he : circleTarget (samplingParameter 0 (direction (u : Angles)) 1,v)=circleSum 1 (v,u) := by
+    simp [circleTarget,samplingParameter,circleSum,add_comm]
+  have hv' : (v,u)∈(circleInverseHomeo h).source := hv
+  have ht : circleSum 1 (v,u)∈(circleInverseHomeo h).target := (circleInverseHomeo h).map_source hv'
+  refine ⟨he ▸ ht,?_⟩
+  rw [circleEndpoint,he,circleInverse_eq h ht]
+  exact congrArg Prod.fst ((circleInverseHomeo h).left_inv hv')
+lemma near_identity_circle_in {a u : ℝ} (h : Real.sin (u-a)≠0)
+    {w : ℕ → ℂ} (hw : Tendsto w atTop (𝓝 (direction (u : Angles))))
+    {V : Set ℝ} (hV : IsOpen V) (haV : a∈V) :
+    ∃ δ : ℝ, 0<δ ∧ δ≤1 ∧ ∃ C : ℝ≥0, 1≤C ∧ ∃ T : ℕ → ℝ → ℝ,
+      (∀ n,Measurable (T n)) ∧
+      (∀ v∈Metric.ball a δ,Tendsto (fun n => T n v) atTop (𝓝 v)) ∧
+      (∀ᶠ n in atTop, AntilipschitzWith C ((Metric.ball a δ).domRestrict (T n)) ∧
+        MapsTo (T n) (Metric.ball a δ) (Metric.closedBall a 1) ∧
+        MapsTo (T n) (Metric.ball a δ) V ∧
+        ∀ v∈Metric.ball a δ, ‖w n+direction (v : Angles)-direction (T n v : Angles)‖=1) := by
+  let p := samplingParameter 0 (direction (u : Angles)) 1
+  have he : circleTarget (p,a)=circleSum 1 (a,u) := by
+    simp [p,circleTarget,samplingParameter,circleSum,add_comm]
+  have ha : Real.sin (a-u)≠0 := by rwa [← neg_sub u a,Real.sin_neg,neg_ne_zero]
+  obtain ⟨P,I,hP,hp,hI,hai,hIp,K,hK,hctrl⟩ := circleEndpoint_good_neighborhood h he rfl ha
+  obtain ⟨J,hJ,haj,hid⟩ := circleEndpoint_identity_neighborhood h
+  have hcv : ∀ᶠ q in 𝓝 (p,a),circleEndpoint h q∈V := by
+    apply (circleEndpoint_strict h he).continuousAt
+    simpa only [circleEndpoint,he,circleInverse_at] using hV.mem_nhds haV
+  obtain ⟨P₁,hP₁,I₁,hI₁,hsubV⟩ := mem_nhds_prod_iff.mp hcv
+  obtain ⟨δ,hδ,hsub⟩ := Metric.mem_nhds_iff.mp
+    (inter_mem ((hI.inter hJ).mem_nhds ⟨hai,haj⟩) hI₁)
+  let ε := min δ 1
+  have hε : 0<ε := lt_min hδ (by norm_num)
+  let T : ℕ → ℝ → ℝ := fun n v => circleEndpoint h (samplingParameter 0 (w n) 1,v)
+  have hparam := samplingParameter_tendsto (x := 0) hw tendsto_const_nhds
+  have hT (n : ℕ) : Measurable (T n) :=
+    (circleEndpoint_measurable h).comp (measurable_const.prodMk measurable_id)
+  refine ⟨ε,hε,min_le_right _ _,max K 1,le_max_right _ _,T,hT,?_,?_⟩
+  · intro v hv
+    have hvI := (hsub ((Metric.ball_subset_ball (min_le_left δ 1)) hv)).1
+    have hh := (circleInverse_continuousAt h (hid v hvI.2).1).fst.tendsto.comp
+      (circleTarget_continuous.continuousAt.tendsto.comp (hparam.prodMk_nhds tendsto_const_nhds))
+    have hid' := (hid v hvI.2).2
+    change Tendsto (fun n => T n v) atTop (𝓝 (circleEndpoint h (p,v))) at hh
+    simpa only [p,hid'] using hh
+  · filter_upwards [hparam.eventually (hP.mem_nhds hp),hparam.eventually hP₁] with n hn hnV
+    refine ⟨?_,?_,?_,?_⟩
+    · apply AntilipschitzWith.of_le_mul_dist
+      intro v z
+      have hv := (hsub ((Metric.ball_subset_ball (min_le_left δ 1)) v.2)).1.1
+      have hz := (hsub ((Metric.ball_subset_ball (min_le_left δ 1)) z.2)).1.1
+      exact ((hK _ hn).le_mul_dist ⟨v,hv⟩ ⟨z,hz⟩).trans
+        (mul_le_mul_of_nonneg_right (by exact_mod_cast le_max_left K 1) dist_nonneg)
+    · intro v hv
+      exact (hctrl _ hn v (hsub ((Metric.ball_subset_ball (min_le_left δ 1)) hv)).1.1).2.2
+    · intro v hv
+      exact hsubV ⟨hnV,(hsub ((Metric.ball_subset_ball (min_le_left δ 1)) hv)).2⟩
+    · intro v hv
+      have ht := (hctrl _ hn v (hsub ((Metric.ball_subset_ball (min_le_left δ 1)) hv)).1.1).1
+      have hh := circleInverse_sampling h ht
+      simp only [Complex.ofReal_one,one_mul,angularSum,zero_add,angleChart] at hh
+      change w n+direction (v : Angles)=direction (T n v : Angles)+
+        direction (((circleInverse h (circleTarget (samplingParameter 0 (w n) 1,v))).2) : Angles) at hh
+      rw [hh,add_sub_cancel_left]
+      exact Circle.norm_coe _
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology
+namespace EuclideanFiveColor
+lemma nonparallel_neighborhood {a b : ℝ} (h : Real.sin (b-a)≠0)
+    {s : ℕ → ℝ} (hs : Tendsto s atTop (𝓝 b)) :
+    ∃ V : Set ℝ, IsOpen V ∧ a∈V ∧ ∀ᶠ n in atTop, ∀ v∈V, Real.sin (s n-v)≠0 := by
+  have hf : Continuous (fun q : ℝ×ℝ => Real.sin (q.2-q.1)) :=
+    Real.continuous_sin.comp (continuous_snd.sub continuous_fst)
+  have ho : IsOpen {q : ℝ×ℝ | Real.sin (q.2-q.1)≠0} :=
+    isOpen_compl_singleton.preimage hf
+  have hh : {q : ℝ×ℝ | Real.sin (q.2-q.1)≠0}∈𝓝 (a,b) := ho.mem_nhds h
+  obtain ⟨ε,hε,hsub⟩ := Metric.mem_nhds_iff.mp hh
+  refine ⟨Metric.ball a ε,Metric.isOpen_ball,Metric.mem_ball_self hε,?_⟩
+  filter_upwards [hs.eventually (Metric.ball_mem_nhds b hε)] with n hn
+  intro v hv
+  exact @hsub (v,s n) (show max (dist v a) (dist (s n) b)<ε from max_lt hv hn)
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+open scoped NNReal
+namespace EuclideanFiveColor
+lemma strict_state_sequence_local_false {k : ℕ} {c : ℂ → Fin k}
+    (hc : Measurable c) (hw : WeakColoring k c)
+    (x : ℕ → ℂ) (J K : Finset (Fin k)) (X : Fin k)
+    (hlabels : ∀i, i∉J∪K ↔ i=X)
+    (A B : ℕ → Set ℝ) (D : Set ℝ) (hD : MeasurableSet D)
+    (hA : ∀n,RealState hc (x n) (A n) J) (hB : ∀n,RealState hc (x n) (B n) K)
+    (b d : ℕ → ℝ) (b₀ d₀ a u : ℝ)
+    (hb : Tendsto b atTop (𝓝 b₀)) (hd : Tendsto d atTop (𝓝 d₀))
+    (hbden : ∀n,LineDensityZero (B n)ᶜ (b n)) (hdden : ∀n,LineDensityZero (B n)ᶜ (d n))
+    (hu : direction (u : Angles)=direction (b₀ : Angles)-direction (d₀ : Angles))
+    (hab : Real.sin (b₀-a)≠0) (had : Real.sin (d₀-a)≠0) (hau : Real.sin (u-a)≠0)
+    (hpos : ∀ε>0,0<volume (D∩Metric.ball a ε))
+    (hL1 : ∀ E : Set ℝ, IsCompact E →
+      Tendsto (fun n => ∫ v in E, ‖(A n).indicator (fun _ => (1:ℝ)) v-D.indicator (fun _ => (1:ℝ)) v‖)
+        atTop (𝓝 0)) : False := by
+  let w : ℕ → ℂ := fun n => direction (b n : Angles)-direction (d n : Angles)
+  have hdir : Continuous (fun v : ℝ => direction (v : Angles)) := direction_continuous.comp (AddCircle.continuous_mk' (2*Real.pi))
+  have hwt : Tendsto w atTop (𝓝 (direction (u : Angles))) := by
+    rw [hu]
+    exact (hdir.tendsto b₀ |>.comp hb).sub (hdir.tendsto d₀ |>.comp hd)
+  obtain ⟨V,hV,haV,hparb⟩ := nonparallel_neighborhood hab hb
+  obtain ⟨W,hW,haW,hpard⟩ := nonparallel_neighborhood had hd
+  obtain ⟨δ,hδ,hδ1,C,hC,T,hT,hTlim,hctrl⟩ := near_identity_circle_in hau hwt hW haW
+  obtain ⟨η,hη,hηV⟩ := Metric.mem_nhds_iff.mp (hV.mem_nhds haV)
+  let ε := min δ η
+  have hε : 0<ε := lt_min hδ hη
+  let I := Metric.ball a ε
+  let E := Metric.closedBall a 1
+  have hIE : I⊆E := (Metric.ball_subset_ball ((min_le_left δ η).trans hδ1)).trans Metric.ball_subset_closedBall
+  have hIδ : I⊆Metric.ball a δ := Metric.ball_subset_ball (min_le_left δ η)
+  have hIV : I⊆V := (Metric.ball_subset_ball (min_le_right δ η)).trans hηV
+  let μ := volume.restrict I
+  let ν := volume.restrict E
+  have hEc : IsCompact E := isCompact_closedBall a 1
+  have : IsFiniteMeasure ν := isFiniteMeasure_restrict.mpr hEc.measure_ne_top
+  have : IsFiniteMeasure μ := isFiniteMeasure_restrict.mpr
+    (ne_top_of_le_ne_top hEc.measure_ne_top (measure_mono hIE))
+  have : ν.WeaklyRegular := Measure.WeaklyRegular.restrict_of_measure_ne_top hEc.measure_lt_top.ne
+  have hS : AntilipschitzWith C (I.domRestrict id) := by
+    apply AntilipschitzWith.of_le_mul_dist
+    intro v z
+    change dist v.1 z.1≤(C:ℝ)*dist v.1 z.1
+    simpa only [one_mul] using mul_le_mul_of_nonneg_right (show (1:ℝ)≤C by exact_mod_cast hC) (dist_nonneg (x := v.1) (y := z.1))
+  have hmS : μ.map id≤(C:ENNReal) •ν := line_map_restrict_volume_le measurable_id hEc.measurableSet hS hIE
+  have hmT : ∀ᶠ n in atTop, μ.map (T n)≤(C:ENNReal) •ν := by
+    filter_upwards [hctrl] with n hn
+    apply line_map_restrict_volume_le (hT n) hEc.measurableSet
+      (AntilipschitzWith.of_le_mul_dist fun v z => hn.1.le_mul_dist ⟨v,hIδ v.2⟩ ⟨z,hIδ z.2⟩)
+    exact hn.2.1.mono_left hIδ
+  have hiD : Integrable (D.indicator (fun _ => (1:ℝ))) ν := (integrable_const _).indicator hD
+  have hiA (n : ℕ) : Integrable ((A n).indicator (fun _ => (1:ℝ))) ν := (integrable_const _).indicator (hA n).1
+  have hlimT : ∀ᵐ v ∂μ,Tendsto (fun n => T n v) atTop (𝓝 (id v)) := by
+    filter_upwards [ae_restrict_mem measurableSet_ball] with v hv
+    exact hTlim v (hIδ hv)
+  have hcomp := varying_pullback_L1 hT measurable_id hmT hmS hlimT hiD hiA (hL1 E hEc)
+  have hdirect : Tendsto (fun n => ∫ v, ‖(A n).indicator (fun _ => (1:ℝ)) v-D.indicator (fun _ => (1:ℝ)) v‖ ∂μ)
+      atTop (𝓝 0) := by
+    apply squeeze_zero' (Eventually.of_forall fun n => integral_nonneg fun v => norm_nonneg _) _
+      (by simpa only [mul_zero] using (hL1 E hEc).const_mul (C:ℝ))
+    exact Eventually.of_forall fun n => integral_norm_comp_le measurable_id hmS ((hiA n).sub hiD)
+  have hpre (n : ℕ) : MeasurableSet (T n ⁻¹' A n) := (hA n).1.preimage (hT n)
+  have hcompeq (n : ℕ) (v : ℝ) : (T n ⁻¹' A n).indicator (fun _ => (1:ℝ)) v =
+      (A n).indicator (fun _ => (1:ℝ)) (T n v) := by
+    by_cases hv : T n v∈A n <;> simp [hv]
+  have hpositive : ∀ᶠ n in atTop,0<μ (A n∩T n ⁻¹' A n) := by
+    apply indicator_inter_positive hD (fun n => (hA n).1) hpre ?_ hdirect
+    · simpa only [hcompeq,id_eq] using hcomp
+    · simpa only [μ,Measure.restrict_apply hD,I] using hpos ε hε
+  obtain ⟨n,hn,hbp,hdp,hp⟩ := (hctrl.and (hparb.and (hpard.and hpositive))).exists
+  have hqmp : Measure.QuasiMeasurePreserving (T n) μ volume :=
+    line_antilipschitz_qmp (hT n)
+      (AntilipschitzWith.of_le_mul_dist fun v z => hn.1.le_mul_dist ⟨v,hIδ v.2⟩ ⟨z,hIδ z.2⟩)
+  have hden := ae_lineDensityZero_compl (hA n).1
+  have hgood : ∀ᵐ v ∂μ,v∈I ∧ (v∈A n → LineDensityZero (A n)ᶜ v) ∧
+      (T n v∈A n → LineDensityZero (A n)ᶜ (T n v)) := by
+    filter_upwards [ae_restrict_mem measurableSet_ball,ae_restrict_of_ae hden,hqmp.ae hden] with v hv hd₁ hd₂
+    exact ⟨hv,hd₁,hd₂⟩
+  obtain ⟨v,hvA,hvI,hvden,hTvden⟩ := ((frequently_ae_mem_iff.mpr hp.ne').and_eventually hgood).exists
+  have h₁ := states_densityOne hc hw (hA n) (hB n) (hvden hvA.1) (hbden n) (hbp v (hIV hvI))
+  have h₂ := states_densityOne hc hw (hA n) (hB n) (hTvden hvA.2) (hdden n)
+    (hdp (T n v) (hn.2.2.1 (hIδ hvI)))
+  have he : {z | c z∉J∪K}={z | c z=X} := by ext z; exact hlabels (c z)
+  rw [he] at h₁ h₂
+  apply weak_density_proper c hc hw X _ h₁ h₂
+  have hdist := hn.2.2.2 v (hIδ hvI)
+  convert hdist using 1
+  congr 1
+  simp only [angularSum,angleChart,Complex.ofReal_one,one_mul,w]
+  ring
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+lemma positive_state_support_point {D : Set ℝ} (_hD : MeasurableSet D) (hp : 0<volume D)
+    (b d u : ℝ) : ∃ a : ℝ, Real.sin (b-a)≠0 ∧ Real.sin (d-a)≠0 ∧ Real.sin (u-a)≠0 ∧
+      ∀ ε>0,0<volume (D∩Metric.ball a ε) := by
+  have hs : ∀ᵐ a ∂volume.restrict D, a∈(volume.restrict D).support ∧
+      Real.sin (a-b)≠0 ∧ Real.sin (a-d)≠0 ∧ Real.sin (a-u)≠0 := by
+    filter_upwards [(volume.restrict D).support_mem_ae,
+      ae_restrict_of_ae (ae_sin_sub_ne_zero b),ae_restrict_of_ae (ae_sin_sub_ne_zero d),
+      ae_restrict_of_ae (ae_sin_sub_ne_zero u)] with a ha hb hd hu
+    exact ⟨ha,hb,hd,hu⟩
+  obtain ⟨a,_,ha,hb,hd,hu⟩ := Measure.exists_mem_of_measure_ne_zero_of_ae hp.ne' hs
+  have hneg (v : ℝ) (hv : Real.sin (a-v)≠0) : Real.sin (v-a)≠0 := by
+    rwa [← neg_sub a v,Real.sin_neg,neg_ne_zero]
+  refine ⟨a,hneg b hb,hneg d hd,hneg u hu,?_⟩
+  intro ε hε
+  have hh := (Measure.mem_support_iff_forall a).mp ha (Metric.ball a ε) (Metric.ball_mem_nhds a hε)
+  simpa only [Measure.restrict_apply measurableSet_ball,inter_comm] using hh
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace EuclideanFiveColor
+lemma support_density_direction {B : Set ℝ} (hB : MeasurableSet B) {b : ℝ}
+    (hb : b∈(volume.restrict B).support) {ε : ℝ} (hε : 0<ε) :
+    ∃ t∈B, dist t b<ε ∧ LineDensityZero Bᶜ t := by
+  have hp : 0<volume (B∩Metric.ball b ε) := by
+    have hh := (Measure.mem_support_iff_forall b).mp hb (Metric.ball b ε) (Metric.ball_mem_nhds b hε)
+    simpa only [Measure.restrict_apply measurableSet_ball,inter_comm] using hh
+  obtain ⟨t,ht,hd⟩ := ((frequently_ae_mem_iff.mpr hp.ne').and_eventually (ae_lineDensityZero_compl hB)).exists
+  exact ⟨t,ht.1,ht.2,hd ht.1⟩
+lemma support_density_sequence {B : ℕ → Set ℝ} (hB : ∀n,MeasurableSet (B n))
+    {b : ℕ → ℝ} {b₀ : ℝ} (hb : Tendsto b atTop (𝓝 b₀))
+    (hs : ∀n,b n∈(volume.restrict (B n)).support) :
+    ∃ d : ℕ → ℝ, Tendsto d atTop (𝓝 b₀) ∧ ∀n,LineDensityZero (B n)ᶜ (d n) := by
+  have hp (n : ℕ) : 0<(1:ℝ)/(n+1) := by positivity
+  choose d _hmem hdist hden using fun n => support_density_direction (hB n) (hs n) (hp n)
+  refine ⟨d,?_,hden⟩
+  apply hb.congr_dist
+  simp only [dist_comm (b _) (d _)]
+  apply squeeze_zero (fun _ => dist_nonneg) (fun n => (hdist n).le)
+  exact tendsto_one_div_add_atTop_nhds_zero_nat
+lemma strict_state_sequence_false {k : ℕ} {c : ℂ → Fin k}
+    (hc : Measurable c) (hw : WeakColoring k c)
+    (x : ℕ → ℂ) (J K : Finset (Fin k)) (X : Fin k)
+    (hlabels : ∀i, i∉J∪K ↔ i=X)
+    (A B : ℕ → Set ℝ) (D : Set ℝ) (hD : MeasurableSet D) (hp : 0<volume D)
+    (hA : ∀n,RealState hc (x n) (A n) J) (hB : ∀n,RealState hc (x n) (B n) K)
+    (b d : ℕ → ℝ) (b₀ d₀ : ℝ)
+    (hb : Tendsto b atTop (𝓝 b₀)) (hd : Tendsto d atTop (𝓝 d₀))
+    (hbs : ∀n,b n∈(volume.restrict (B n)).support)
+    (hds : ∀n,d n∈(volume.restrict (B n)).support)
+    (hu : ‖direction (b₀ : Angles)-direction (d₀ : Angles)‖=1)
+    (hL1 : ∀ E : Set ℝ, IsCompact E →
+      Tendsto (fun n => ∫ v in E, ‖(A n).indicator (fun _ => (1:ℝ)) v-D.indicator (fun _ => (1:ℝ)) v‖)
+        atTop (𝓝 0)) : False := by
+  obtain ⟨b',hb',hbden⟩ := support_density_sequence (fun n => (hB n).1) hb hbs
+  obtain ⟨d',hd',hdden⟩ := support_density_sequence (fun n => (hB n).1) hd hds
+  obtain ⟨u,hu⟩ := direction_exists hu
+  obtain ⟨u,rfl⟩ := QuotientAddGroup.mk_surjective u
+  obtain ⟨a,hab,had,hau,hpos⟩ := positive_state_support_point hD hp b₀ d₀ u
+  exact strict_state_sequence_local_false hc hw x J K X hlabels A B D hD hA hB
+    b' d' b₀ d₀ a u hb' hd' hbden hdden hu hab had hau hpos hL1
+lemma strict_state_no_unit_support {k : ℕ} {c : ℂ → Fin k}
+    (hc : Measurable c) (hw : WeakColoring k c)
+    {x : ℂ} {J K : Finset (Fin k)} {X : Fin k}
+    (hlabels : ∀i, i∉J∪K ↔ i=X)
+    {A B : Set ℝ} (hA : RealState hc x A J) (hB : RealState hc x B K)
+    (hp : 0<volume A) {b d : ℝ}
+    (hb : b∈(volume.restrict B).support) (hd : d∈(volume.restrict B).support) :
+    ‖direction (b : Angles)-direction (d : Angles)‖≠1 := by
+  intro hu
+  apply strict_state_sequence_false hc hw (fun _ => x) J K X hlabels (fun _ => A) (fun _ => B)
+    A hA.1 hp (fun _ => hA) (fun _ => hB) (fun _ => b) (fun _ => d) b d tendsto_const_nhds
+    tendsto_const_nhds (fun _ => hb) (fun _ => hd) hu
+  intro E _
+  simp only [sub_self,norm_zero,integral_zero]
+  exact tendsto_const_nhds
+end EuclideanFiveColor
+end
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+open scoped RealInnerProductSpace
+namespace EuclideanFiveColor
+lemma complex_unit_orthogonal {e v : ℂ} (he : ‖e‖=1) (hv : ‖v‖=1) (h : ⟪e,v⟫=0) :
+    e=Complex.I*v ∨ e= -Complex.I*v := by
+  have hr : (e*star v).re=0 := by
+    rw [real_inner_comm,Complex.inner] at h
+    exact h
+  have hn : ‖e*star v‖=1 := by rw [norm_mul,norm_star,he,hv,one_mul]
+  have hs : (e*star v).im^2=1 := by
+    have hh := Complex.normSq_eq_norm_sq (e*star v)
+    rw [hn] at hh
+    simp only [Complex.normSq_apply,hr,zero_mul,zero_add,one_pow] at hh
+    nlinarith
+  have him : (e*star v).im=1 ∨ (e*star v).im= -1 := by
+    exact sq_eq_sq_iff_eq_or_eq_neg.mp (by simpa only [one_pow] using hs)
+  have hmul : star v*v=1 := by
+    change (starRingEnd ℂ) v*v=1
+    rw [Complex.conj_mul']
+    simp only [hv,one_pow,Complex.ofReal_one]
+  rcases him with him|him
+  · left
+    have hz : e*star v=Complex.I := Complex.ext (by simpa using hr) (by simpa using him)
+    calc e = (e*star v)*v := by rw [mul_assoc,hmul,mul_one]
+         _ = Complex.I*v := congrArg (·*v) hz
+  · right
+    have hz : e*star v= -Complex.I := Complex.ext (by simpa using hr) (by simpa using him)
+    calc e = (e*star v)*v := by rw [mul_assoc,hmul,mul_one]
+         _ = -Complex.I*v := congrArg (·*v) hz
+lemma ae_direction_nonorthogonal {v : ℂ} (hv : ‖v‖=1) :
+    ∀ᵐt∂angularMeasure,⟪direction t,v⟫≠0 := by
+  have hd : Function.Injective direction := fun _ _ h =>
+    AddCircle.injective_toCircle (by positivity : (2*Real.pi:ℝ)≠0) (Subtype.ext h)
+  have hf : {t | ⟪direction t,v⟫=0}.Finite := by
+    apply ((Set.toFinite {Complex.I*v,-Complex.I*v}).preimage (f := direction) hd.injOn).subset
+    intro t ht
+    rcases complex_unit_orthogonal (direction_norm t) hv ht with ht|ht <;> simp [ht]
+  exact measure_eq_zero_iff_ae_notMem.mp (hf.measure_zero angularMeasure)
+end EuclideanFiveColor
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PlaneColoring/Transitions.lean b/lean/OAI/Geometry/PlaneColoring/Transitions.lean
new file mode 100644
index 0000000000000000000000000000000000000000..6257a624e429115ef214f8846bfb16716f47caf9
--- /dev/null
+++ b/lean/OAI/Geometry/PlaneColoring/Transitions.lean
@@ -0,0 +1,150 @@
+import OAI.Analysis.PlaneSpectrum.CircleOperator
+
+namespace OAI
+
+noncomputable section
+open MeasureTheory MeasureTheory.Measure Filter Topology Set
+namespace EuclideanFiveColor
+variable {k : ℕ} (c : ℂ → Fin k)
+def transitionSet (i j : Fin k) (x : ℂ) (R : ℝ) (h : ℂ) : Set ℂ :=
+  {q | q ∈ Metric.ball x R ∧ ((c q=i ∧ c (q+h)=j) ∨ (c q=j ∧ c (q+h)=i))}
+lemma transitionSet_measurable (hc : Measurable c) (i j : Fin k) (x : ℂ) (R : ℝ) (h : ℂ) :
+    MeasurableSet (transitionSet c i j x R h) := by
+  exact measurableSet_ball.inter (((measurableSet_eq_fun hc measurable_const).inter
+    (measurableSet_eq_fun (hc.comp (measurable_id.add_const h)) measurable_const)).union
+    ((measurableSet_eq_fun hc measurable_const).inter
+    (measurableSet_eq_fun (hc.comp (measurable_id.add_const h)) measurable_const)))
+lemma transitionSet_finite (i j : Fin k) (x : ℂ) (R : ℝ) (h : ℂ) :
+    volume (transitionSet c i j x R h) < ⊤ :=
+  (measure_mono (fun _ hq => Metric.ball_subset_closedBall hq.1)).trans_lt
+    (isCompact_closedBall x R).measure_lt_top
+def transitionRatio (i j : Fin k) (x : ℂ) (R : ℝ) (h : ℂ) : ENNReal :=
+  volume (transitionSet c i j x R h) / ENNReal.ofReal ‖h‖
+def TransitionEdge (i j : Fin k) (x : ℂ) : Prop :=
+  i ≠ j ∧ ∀ R : ℝ, 0<R → 0 < limsup (transitionRatio c i j x R) (𝓝[≠] (0:ℂ))
+lemma transition_abundance {i j : Fin k} {x : ℂ} (he : TransitionEdge c i j x)
+    {R : ℝ} (hR : 0<R) : ∃ a : ℝ, 0<a ∧
+      ∃ᶠ h in 𝓝[≠] (0:ℂ), a*‖h‖ < volume.real (transitionSet c i j x R h) := by
+  obtain ⟨a,_,ha,hal⟩ := ENNReal.lt_iff_exists_real_btwn.mp (he.2 R hR)
+  have ha0 : 0<a := ENNReal.ofReal_pos.mp ha
+  refine ⟨a,ha0,?_⟩
+  have hf := frequently_lt_of_lt_limsup (h := hal)
+  refine (hf.and_eventually self_mem_nhdsWithin).mono ?_
+  intro h hh
+  have hne : h≠0 := hh.2
+  have hn : ENNReal.ofReal ‖h‖≠0 := (ENNReal.ofReal_pos.mpr (norm_pos_iff.mpr hne)).ne'
+  have hm : ENNReal.ofReal a*ENNReal.ofReal ‖h‖ < volume (transitionSet c i j x R h) :=
+    (ENNReal.lt_div_iff_mul_lt (Or.inl hn) (Or.inl ENNReal.ofReal_ne_top)).mp hh.1
+  rw [← ENNReal.ofReal_mul ha0.le] at hm
+  exact (ENNReal.ofReal_lt_iff_lt_toReal (mul_nonneg ha0.le (norm_nonneg h)) (transitionSet_finite c i j x R h).ne).mp hm
+lemma colorMass_own_zero (hc : Measurable c) (hw : WeakColoring k c) {q : ℂ}
+    (hq : Typical c q) (hs : TypicalSampling c q 1) : colorMass c (c q) q=0 := by
+  apply integral_eq_zero_of_ae
+  filter_upwards [hs.2] with t ht
+  simp only [Complex.ofReal_one,one_mul] at ht
+  have hn : c (q+direction t)≠c q := by
+    intro he
+    have hd : ‖q-(q+direction t)‖=1 := by
+      simp only [sub_add_cancel_left,norm_neg]
+      exact Circle.norm_coe _
+    exact weak_density_proper c hc hw (c q) hd hq (by simpa only [Typical,he] using ht)
+  simp only [sampleFn,Complex.ofReal_one,one_mul,ite_eq_right hn,Pi.zero_apply]
+lemma exists_energy_small {X : Type*} [MeasurableSpace X] (μ : Measure X)
+    {T : Set X} (hT : MeasurableSet T) (hfin : μ T < ⊤) {E : X → ℝ}
+    (hE : Integrable E μ) (hn : ∀ᵐ q ∂μ, 0≤E q) {P : X → Prop} (hP : ∀ᵐ q ∂μ, P q)
+    {d : ℝ} (hi : (∫ q, E q ∂μ) < d*μ.real T) : ∃ q ∈ T, P q ∧ E q<d := by
+  by_contra! h
+  have : IsFiniteMeasure (μ.restrict T) := ⟨by simpa only [Measure.restrict_apply_univ] using hfin⟩
+  have hge : ∀ᵐ q ∂μ.restrict T, d≤E q := by
+    filter_upwards [ae_restrict_mem hT,ae_restrict_of_ae hP] with q hq hp
+    exact h q hq hp
+  have hl := integral_mono_ae (integrable_const d) hE.restrict hge
+  have hr := integral_mono_measure (μ.restrict_le_self (s := T)) hn hE
+  have he : (∫ _q in T, d ∂μ)=d*μ.real T := by simp [mul_comm]
+  rw [he] at hl
+  exact (not_lt_of_ge (hl.trans hr)) hi
+end EuclideanFiveColor
+end
+
+noncomputable section
+open MeasureTheory MeasureTheory.Measure Filter Topology Set
+namespace EuclideanFiveColor
+variable {k : ℕ} (c : ℂ → Fin k) (hc : Measurable c)
+def colorError (i : Fin k) (x : ℂ) (R : ℝ) (h : ℂ) : ComplexPlaneL2 :=
+  let g := (colorCutoff_memLp c hc i x R).toLp (colorCutoff c i x R)
+  circleOperator (g-planeTranslate h g)
+lemma colorError_ae (i : Fin k) (x : ℂ) (R : ℝ) {h : ℂ} (hh : ‖h‖<1) :
+    ∀ᵐ q ∂(volume : Measure ℂ), q ∈ Metric.ball x R →
+      colorError c hc i x R h q = ((colorMass c i q-colorMass c i (q+h) : ℝ) : ℂ) := by
+  filter_upwards [circleOperator_difference_ae (colorCutoff_memLp c hc i x R) h] with q hq hB
+  exact hq.trans (colorCutoff_circleDifference c hc i x q h R hB hh)
+lemma colorError_smoothing (i : Fin k) (x : ℂ) (R : ℝ) :
+    Tendsto (fun h : ℂ => ‖colorError c hc i x R h‖^2 / ‖h‖) (𝓝 0) (𝓝 0) :=
+  circle_smoothing_L2 _
+lemma transition_mass_energy (hw : WeakColoring k c) (i j : Fin k) (x : ℂ) (R : ℝ)
+    {h : ℂ} (hh : ‖h‖<1) : ∀ᵐ q ∂(volume : Measure ℂ), q ∈ transitionSet c i j x R h →
+      (colorMass c i q+colorMass c j q)^2 ≤
+        ‖colorError c hc i x R h q‖^2+‖colorError c hc j x R h q‖^2 := by
+  filter_upwards [colorError_ae c hc i x R hh,colorError_ae c hc j x R hh,
+    ae_density_actual c hc,ae_typicalSampling c hc (by norm_num : (0:ℝ)<1),
+    (measurePreserving_add_right volume h).quasiMeasurePreserving.ae (ae_density_actual c hc),
+    (measurePreserving_add_right volume h).quasiMeasurePreserving.ae
+      (ae_typicalSampling c hc (by norm_num : (0:ℝ)<1))] with q hi hj hq hqs hqh hqhs hT
+  have he_i : ‖colorError c hc i x R h q‖^2=(colorMass c i q-colorMass c i (q+h))^2 := by
+    rw [hi hT.1,Complex.norm_real,Real.norm_eq_abs,sq_abs]
+  have he_j : ‖colorError c hc j x R h q‖^2=(colorMass c j q-colorMass c j (q+h))^2 := by
+    rw [hj hT.1,Complex.norm_real,Real.norm_eq_abs,sq_abs]
+  rw [he_i,he_j]
+  have h0 := colorMass_own_zero c hc hw hq hqs
+  have h1 := colorMass_own_zero c hc hw hqh hqhs
+  rcases hT.2 with ⟨hqi,hqj⟩ | ⟨hqj,hqi⟩
+  · rw [hqi] at h0
+    rw [hqj] at h1
+    rw [h0,h1]
+    nlinarith [sq_nonneg (colorMass c i (q+h))]
+  · rw [hqj] at h0
+    rw [hqi] at h1
+    rw [h0,h1]
+    nlinarith [sq_nonneg (colorMass c j (q+h))]
+include hc in
+lemma transition_small_mass (hw : WeakColoring k c) {i j : Fin k} {x : ℂ}
+    (he : TransitionEdge c i j x) {R ε : ℝ} (hR : 0<R) (hε : 0<ε) :
+    ∃ q ∈ Metric.ball x R, TypicalSampling c q 1 ∧ colorMass c i q+colorMass c j q<ε := by
+  obtain ⟨a,ha,hfreq⟩ := transition_abundance c he hR
+  have hlim : Tendsto (fun h : ℂ =>
+      (‖colorError c hc i x R h‖^2+‖colorError c hc j x R h‖^2)/‖h‖) (𝓝 0) (𝓝 0) := by
+    simp only [add_div]
+    simpa only [add_zero] using (colorError_smoothing c hc i x R).add (colorError_smoothing c hc j x R)
+  have hsmall : ∀ᶠ h in 𝓝[≠] (0:ℂ),
+      (‖colorError c hc i x R h‖^2+‖colorError c hc j x R h‖^2)/‖h‖<ε^2*a :=
+    (hlim.eventually (gt_mem_nhds (by positivity))).filter_mono nhdsWithin_le_nhds
+  have hnorm : ∀ᶠ h in 𝓝[≠] (0:ℂ), ‖h‖<1 :=
+    (continuous_norm.tendsto (0:ℂ)).eventually (by simpa using (gt_mem_nhds (by norm_num : (0:ℝ)<1)))
+      |>.filter_mono nhdsWithin_le_nhds
+  obtain ⟨h,hab,hsm,hhn,hne⟩ := (hfreq.and_eventually (hsmall.and (hnorm.and self_mem_nhdsWithin))).exists
+  have hnpos : 0<‖h‖ := norm_pos_iff.mpr hne
+  let E : ℂ → ℝ := fun q => ‖colorError c hc i x R h q‖^2+‖colorError c hc j x R h q‖^2
+  have hEi : Integrable (fun q => ‖colorError c hc i x R h q‖^2) volume :=
+    (memLp_two_iff_integrable_sq_norm (Lp.aestronglyMeasurable _)).mp (Lp.memLp _)
+  have hEj : Integrable (fun q => ‖colorError c hc j x R h q‖^2) volume :=
+    (memLp_two_iff_integrable_sq_norm (Lp.aestronglyMeasurable _)).mp (Lp.memLp _)
+  have hInt : (∫ q, E q ∂volume) < ε^2*volume.real (transitionSet c i j x R h) := by
+    rw [show (∫ q, E q ∂volume) = ‖colorError c hc i x R h‖^2+‖colorError c hc j x R h‖^2 by
+      dsimp only [E]
+      rw [integral_add hEi hEj,← planeL2_norm_sq,← planeL2_norm_sq]]
+    have hs := (div_lt_iff₀ hnpos).mp hsm
+    have hab' := mul_lt_mul_of_pos_left hab (sq_pos_of_pos hε)
+    nlinarith
+  obtain ⟨q,hq,hqs,henergy⟩ := exists_energy_small volume (transitionSet_measurable c hc i j x R h)
+    (transitionSet_finite c i j x R h) (hEi.add hEj)
+    (Eventually.of_forall fun q => add_nonneg (sq_nonneg _) (sq_nonneg _))
+    ((ae_typicalSampling c hc (by norm_num : (0:ℝ)<1)).and (transition_mass_energy c hc hw i j x R hhn)) hInt
+  refine ⟨q,hq.1,hqs.1,?_⟩
+  have hm := hqs.2 hq
+  change E q<ε^2 at henergy
+  change (colorMass c i q+colorMass c j q)^2≤E q at hm
+  nlinarith
+end EuclideanFiveColor
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PlaneColoring/WeakColoring.lean b/lean/OAI/Geometry/PlaneColoring/WeakColoring.lean
new file mode 100644
index 0000000000000000000000000000000000000000..4b283483cf1e369d2cd0780812faa94538559d03
--- /dev/null
+++ b/lean/OAI/Geometry/PlaneColoring/WeakColoring.lean
@@ -0,0 +1,176 @@
+import OAI.Geometry.PlaneColoring.Coloring
+
+namespace OAI
+
+noncomputable section
+open MeasureTheory Filter Topology
+open scoped RealInnerProductSpace
+namespace EuclideanFiveColor
+abbrev PlaneL2 := Lp ℝ 2 (volume : Measure ℂ)
+def shiftL2 (z : ℂ) (f : PlaneL2) : PlaneL2 :=
+  Lp.compMeasurePreserving (fun x => x+z) (measurePreserving_add_right volume z) f
+lemma shiftL2_ae (z : ℂ) (f : PlaneL2) : shiftL2 z f =ᵐ[volume] fun x => f (x+z) :=
+  Lp.coeFn_compMeasurePreserving f _
+lemma shiftL2_continuous (f : PlaneL2) : Continuous (fun z => shiftL2 z f) := by
+  let g : C(ℂ,C(ℂ,ℂ)) := (ContinuousMap.mk (fun p : ℂ × ℂ => p.2+p.1) (by fun_prop)).curry
+  exact continuous_const.compMeasurePreservingLp g.continuous (fun z => measurePreserving_add_right volume z) (by simp)
+lemma correlation_continuous (f : PlaneL2) : Continuous (fun t : ℝ => ⟪f,shiftL2 (Circle.exp t : ℂ) f⟫) := by
+  exact continuous_const.inner ((shiftL2_continuous f).comp (continuous_subtype_val.comp Circle.exp.continuous))
+lemma product_zero_of_inner_zero (f g : PlaneL2) (hf : ∀ᵐ x ∂volume, 0 ≤ f x)
+    (hg : ∀ᵐ x ∂volume, 0 ≤ g x) (h : ⟪f,g⟫ = 0) :
+    ∀ᵐ x ∂volume, f x*g x=0 := by
+  have hi : Integrable (fun x : ℂ => f x*g x) volume := by
+    simpa [RCLike.inner_apply,mul_comm] using (L2.integrable_inner (𝕜 := ℝ) f g)
+  have hn : ∀ᵐ x ∂volume, 0 ≤ f x*g x := by
+    filter_upwards [hf,hg] with x hx hy
+    exact mul_nonneg hx hy
+  rw [L2.inner_def] at h
+  apply (integral_eq_zero_iff_of_nonneg_ae hn hi).mp
+  simpa [RCLike.inner_apply,mul_comm] using h
+/-- AE exclusions over any full-support parameter measure upgrade to every parameter. -/
+lemma all_parameters_product_zero {T : Type*} [TopologicalSpace T] [MeasurableSpace T]
+    (μ : Measure T) [μ.IsOpenPosMeasure] (d : T → ℂ) (hd : Continuous d)
+    (f : PlaneL2) (hf : ∀ᵐ x ∂volume, 0 ≤ f x)
+    (ho : ∀ᵐ t ∂μ, ∀ᵐ x ∂(volume : Measure ℂ), f x*f (x+d t)=0) :
+    ∀ t : T, ∀ᵐ x ∂(volume : Measure ℂ), f x*f (x+d t)=0 := by
+  have hc : Continuous (fun t : T => ⟪f,shiftL2 (d t) f⟫) :=
+    continuous_const.inner ((shiftL2_continuous f).comp hd)
+  have hz : (fun t : T => ⟪f,shiftL2 (d t) f⟫) =ᵐ[μ] fun _ => 0 := by
+    filter_upwards [ho] with t ht
+    rw [L2.inner_def]
+    apply integral_eq_zero_of_ae
+    filter_upwards [shiftL2_ae (d t) f,ht] with x hx hh
+    simpa [RCLike.inner_apply,mul_comm,hx] using hh
+  have he := (hc.ae_eq_iff_eq μ continuous_const).mp hz
+  intro t
+  have hg : ∀ᵐ x ∂volume, 0 ≤ shiftL2 (d t) f x := by
+    filter_upwards [shiftL2_ae (d t) f,
+      (measurePreserving_add_right volume (d t)).quasiMeasurePreserving.ae hf] with x hx hh
+    simpa only [hx] using hh
+  have h0 := product_zero_of_inner_zero f (shiftL2 (d t) f) hf hg (congrFun he t)
+  filter_upwards [h0,shiftL2_ae (d t) f] with x hx hh
+  simpa only [hh] using hx
+end EuclideanFiveColor
+end
+
+noncomputable section
+open MeasureTheory Filter Topology
+namespace EuclideanFiveColor
+variable {k : ℕ} (c : ℂ → Fin k) (hc : Measurable c)
+def truncatedClass (i : Fin k) (n : ℕ) : Set ℂ := {z | c z=i} ∩ Metric.closedBall 0 n
+include hc
+lemma truncatedClass_measurable (i : Fin k) (n : ℕ) : MeasurableSet (truncatedClass c i n) :=
+  (measurableSet_eq_fun hc measurable_const).inter measurableSet_closedBall
+omit hc in
+lemma truncatedClass_finite (i : Fin k) (n : ℕ) : volume (truncatedClass c i n) ≠ ⊤ := by
+  exact ne_top_of_le_ne_top ((isCompact_closedBall (0 : ℂ) (n : ℝ)).measure_ne_top)
+    (measure_mono Set.inter_subset_right)
+def truncatedVector (i : Fin k) (n : ℕ) : PlaneL2 :=
+  (memLp_indicator_const 2 (truncatedClass_measurable c hc i n) (1 : ℝ) (Or.inr (truncatedClass_finite c i n))).toLp _
+lemma truncatedVector_ae (i : Fin k) (n : ℕ) :
+    truncatedVector c hc i n =ᵐ[volume] (truncatedClass c i n).indicator (fun _ => (1 : ℝ)) :=
+  MemLp.coeFn_toLp _
+lemma weak_all_directions (hweak : WeakColoring k c) :
+    ∀ t : Angles, ∀ᵐ z ∂(volume : Measure ℂ), c z ≠ c (z+direction t) := by
+  have ho : ∀ᵐ t ∂angularMeasure, ∀ᵐ z ∂(volume : Measure ℂ), c z ≠ c (z+direction t) := by
+    apply (Measure.ae_ae_comm ?_).mp (Measure.ae_ae_of_ae_prod hweak.2)
+    exact (measurableSet_eq_fun (hc.comp measurable_fst)
+      (hc.comp (measurable_fst.add (direction_continuous.measurable.comp measurable_snd)))).compl
+  have hz (i : Fin k) (n : ℕ) : ∀ t : Angles, ∀ᵐ z ∂(volume : Measure ℂ),
+      truncatedVector c hc i n z*truncatedVector c hc i n (z+direction t)=0 := by
+    apply all_parameters_product_zero angularMeasure direction direction_continuous
+    · filter_upwards [truncatedVector_ae c hc i n] with z hz
+      rw [hz]
+      exact Set.indicator_nonneg (fun _ _ => zero_le_one) _
+    · filter_upwards [ho] with t ht
+      filter_upwards [ht,truncatedVector_ae c hc i n,
+        (measurePreserving_add_right volume (direction t)).quasiMeasurePreserving.ae (truncatedVector_ae c hc i n)] with z hne h1 h2
+      rw [h1,h2]
+      by_cases h : z ∈ truncatedClass c i n
+      · have hh : z+direction t ∉ truncatedClass c i n := fun hh => hne (h.1.trans hh.1.symm)
+        simp [Set.indicator_of_notMem hh]
+      · simp [Set.indicator_of_notMem h]
+  intro t
+  have hi : ∀ᵐ z ∂(volume : Measure ℂ), ∀ i : Fin k, ∀ n : ℕ,
+      ((truncatedClass c i n).indicator (fun _ => (1 : ℝ)) z)*
+      ((truncatedClass c i n).indicator (fun _ => (1 : ℝ)) (z+direction t))=0 := by
+    apply ae_all_iff.mpr
+    intro i
+    apply ae_all_iff.mpr
+    intro n
+    filter_upwards [hz i n t,truncatedVector_ae c hc i n,
+      (measurePreserving_add_right volume (direction t)).quasiMeasurePreserving.ae (truncatedVector_ae c hc i n)] with z h0 h1 h2
+    simpa only [h1,h2] using h0
+  filter_upwards [hi] with z hz
+  intro he
+  obtain ⟨n,hn⟩ := exists_nat_gt (max ‖z‖ ‖z+direction t‖)
+  have h1 : z ∈ truncatedClass c (c z) n := ⟨rfl,by
+    simpa only [Metric.mem_closedBall,dist_zero_right] using (lt_of_le_of_lt (le_max_left _ _) hn).le⟩
+  have h2 : z+direction t ∈ truncatedClass c (c z) n := ⟨he.symm,by
+    simpa only [Metric.mem_closedBall,dist_zero_right] using (lt_of_le_of_lt (le_max_right _ _) hn).le⟩
+  have h0 := hz (c z) n
+  simp [Set.indicator_of_mem h1,Set.indicator_of_mem h2] at h0
+end EuclideanFiveColor
+end
+
+noncomputable section
+open MeasureTheory Filter Topology Set
+namespace EuclideanFiveColor
+def DensityOne (A : Set ℂ) (x : ℂ) : Prop :=
+  Tendsto (fun r : ℝ => volume (A ∩ Metric.closedBall x r) / volume (Metric.closedBall x r))
+    (𝓝[>] 0) (𝓝 (1 : ENNReal))
+lemma densityOne_disjoint {A B : Set ℂ} (hB : MeasurableSet B)
+    (hd : AEDisjoint volume A B) {x : ℂ} (hA : DensityOne A x) : ¬ DensityOne B x := by
+  intro hb
+  have hh := hA.add hb
+  have hle : (1 : ENNReal)+1 ≤ 1 := le_of_tendsto hh (by
+    filter_upwards [self_mem_nhdsWithin] with r hr
+    rw [← ENNReal.add_div,← measure_union₀ (hB.inter measurableSet_closedBall).nullMeasurableSet
+      (hd.mono inter_subset_left inter_subset_left)]
+    apply (ENNReal.div_le_iff (Metric.measure_closedBall_pos volume x hr).ne'
+      ((isCompact_closedBall x r).measure_ne_top)).mpr
+    simpa only [one_mul] using measure_mono (union_subset inter_subset_right inter_subset_right))
+  norm_num at hle
+lemma densityOne_translate {A : Set ℂ} {x u : ℂ} :
+    DensityOne ((fun z : ℂ => z+u) ⁻¹' A) x ↔ DensityOne A (x+u) := by
+  have he (r : ℝ) : (fun z : ℂ => z+u) ⁻¹' (Metric.closedBall (x+u) r) = Metric.closedBall x r := by
+    ext z
+    simp only [Set.mem_preimage,Metric.mem_closedBall,dist_eq_norm,add_sub_add_right_eq_sub]
+  have h1 (r : ℝ) : volume (((fun z : ℂ => z+u) ⁻¹' A) ∩ Metric.closedBall x r) =
+      volume (A ∩ Metric.closedBall (x+u) r) := by
+    rw [← he,← Set.preimage_inter,measure_preimage_add_right]
+  have h2 (r : ℝ) : volume (Metric.closedBall x r) = volume (Metric.closedBall (x+u) r) := by
+    rw [← he,measure_preimage_add_right]
+  simp only [DensityOne,h1,h2]
+lemma direction_exists {u : ℂ} (hu : ‖u‖=1) : ∃ t, direction t=u := by
+  let v : Circle := ⟨u,mem_sphere_zero_iff_norm.mpr hu⟩
+  obtain ⟨t,ht⟩ := Circle.exp_surjective v
+  exact ⟨(t : Angles),by simpa only [direction_real] using congrArg Subtype.val ht⟩
+lemma weak_density_proper {k : ℕ} (c : ℂ → Fin k) (hc : Measurable c) (hw : WeakColoring k c)
+    (i : Fin k) {x y : ℂ} (hxy : ‖x-y‖=1) (hx : DensityOne {z | c z=i} x) :
+    ¬ DensityOne {z | c z=i} y := by
+  let u := y-x
+  obtain ⟨t,ht⟩ := direction_exists (u := u) (by simpa only [u,norm_sub_rev] using hxy)
+  have hd : AEDisjoint volume {z | c z=i} ((fun z : ℂ => z+u) ⁻¹' {z | c z=i}) := by
+    apply (measure_eq_zero_iff_ae_notMem).mpr
+    filter_upwards [weak_all_directions c hc hw t] with z hz
+    rintro ⟨h1,h2⟩
+    exact hz (by simpa only [ht] using h1.trans h2.symm)
+  have hh := densityOne_disjoint ((measurableSet_eq_fun hc measurable_const).preimage
+    (measurable_id.add_const u)) hd hx
+  simp only [id_eq] at hh
+  rw [densityOne_translate] at hh
+  simpa only [u,add_sub_cancel] using hh
+lemma ae_density_actual {k : ℕ} (c : ℂ → Fin k) (hc : Measurable c) :
+    ∀ᵐ x ∂(volume : Measure ℂ), DensityOne {z | c z=c x} x := by
+  have hh : ∀ᵐ x ∂(volume : Measure ℂ), ∀ i : Fin k,
+      Tendsto (fun r : ℝ => volume ({z | c z=i} ∩ Metric.closedBall x r) / volume (Metric.closedBall x r))
+        (𝓝[>] 0) (𝓝 (({z | c z=i} : Set ℂ).indicator (1 : ℂ → ENNReal) x)) :=
+    ae_all_iff.mpr fun i => Besicovitch.ae_tendsto_measure_inter_div_of_measurableSet volume
+      (measurableSet_eq_fun hc measurable_const)
+  filter_upwards [hh] with x hx
+  simpa only [DensityOne,Set.indicator_of_mem (show x ∈ {z | c z=c x} from rfl),Pi.one_apply] using hx (c x)
+end EuclideanFiveColor
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PolarProducts/CanonicalCoordinates.lean b/lean/OAI/Geometry/PolarProducts/CanonicalCoordinates.lean
new file mode 100644
index 0000000000000000000000000000000000000000..1a4d9504e024e7a2991fd6272d920993738fbb17
--- /dev/null
+++ b/lean/OAI/Geometry/PolarProducts/CanonicalCoordinates.lean
@@ -0,0 +1,168 @@
+import OAI.Geometry.PolarProducts.PlanarSquare
+
+namespace OAI
+
+universe u131 u132 u133
+
+section
+namespace CanonicalLinear
+noncomputable section
+open scoped InnerProductSpace
+variable {κ : Type u131} [Fintype κ] [DecidableEq κ]
+abbrev E (κ : Type u132) := EuclideanSpace ℝ κ
+
+def coordinate (j : κ) : E κ →L[ℝ] ℝ := PiLp.proj 2 (fun _ : κ => ℝ) j
+
+def adaptedMap (u v : E κ) (j : κ) : E κ →L[ℝ] E κ :=
+  ContinuousLinearMap.id ℝ (E κ) - (1 / u j) • (coordinate j).smulRight u +
+    (innerSL ℝ v).smulRight (EuclideanSpace.single j 1)
+
+@[simp] theorem adaptedMap_apply (u v q : E κ) (j : κ) :
+    adaptedMap u v j q = q - (q j / u j) • u + (inner (𝕜 := ℝ) v q) • EuclideanSpace.single j 1 := by
+  simp [adaptedMap, coordinate, div_eq_mul_inv, smul_smul, mul_comm]
+
+theorem adaptedMap_coordinate (u v q : E κ) (j : κ) (huj : u j ≠ 0) :
+    adaptedMap u v j q j = inner (𝕜 := ℝ) v q := by
+  simp [adaptedMap_apply, PiLp.sub_apply, PiLp.add_apply, PiLp.smul_apply, div_mul_cancel₀ _ huj]
+
+theorem adaptedMap_u {u v : E κ} (j : κ) (huj : u j ≠ 0)
+    (hvu : inner (𝕜 := ℝ) v u = 1) : adaptedMap u v j u = EuclideanSpace.single j 1 := by
+  simp [adaptedMap_apply, div_self huj, hvu]
+
+theorem adaptedMap_injective {u v : E κ} (j : κ) (huj : u j ≠ 0)
+    (hvu : inner (𝕜 := ℝ) v u = 1) : Function.Injective (adaptedMap u v j) := by
+  apply LinearMap.ker_eq_bot.mp
+  rw [LinearMap.ker_eq_bot']
+  intro q hq
+  change adaptedMap u v j q = 0 at hq
+  have hi : inner (𝕜 := ℝ) v q = 0 := by
+    have hh := congrArg (fun x : E κ => x j) hq
+    simpa only [adaptedMap_coordinate u v q j huj, PiLp.zero_apply] using hh
+  have he : q = (q j/u j) • u := by
+    simpa only [adaptedMap_apply, hi, zero_smul, add_zero, sub_eq_zero] using hq
+  have ht : q j/u j = 0 := by
+    have hh := congrArg (fun x : E κ => inner (𝕜 := ℝ) v x) he
+    simpa only [real_inner_smul_right, hvu, mul_one, hi] using hh.symm
+  simp only [ht, zero_smul] at he
+  exact he
+
+def adapted {u v : E κ} (j : κ) (huj : u j ≠ 0)
+    (hvu : inner (𝕜 := ℝ) v u = 1) : E κ ≃L[ℝ] E κ :=
+  ContinuousLinearEquiv.ofBijective (adaptedMap u v j)
+    (LinearMap.ker_eq_bot.mpr (adaptedMap_injective j huj hvu))
+    (LinearMap.range_eq_top.mpr (LinearMap.surjective_of_injective (adaptedMap_injective j huj hvu)))
+
+@[simp] theorem adapted_apply {u v : E κ} (j : κ) (huj : u j ≠ 0)
+    (hvu : inner (𝕜 := ℝ) v u = 1) (q : E κ) :
+    adapted j huj hvu q = adaptedMap u v j q := rfl
+
+theorem adapted_first {u v : E κ} (j : κ) (huj : u j ≠ 0)
+    (hvu : inner (𝕜 := ℝ) v u = 1) (q : E κ) :
+    adapted j huj hvu q j = inner (𝕜 := ℝ) v q := adaptedMap_coordinate u v q j huj
+
+theorem adapted_inverse_single {u v : E κ} (j : κ) (huj : u j ≠ 0)
+    (hvu : inner (𝕜 := ℝ) v u = 1) :
+    (adapted j huj hvu).symm (EuclideanSpace.single j 1) = u := by
+  apply (adapted j huj hvu).injective
+  simpa only [ContinuousLinearEquiv.apply_symm_apply, adapted_apply] using (adaptedMap_u j huj hvu).symm
+
+def cotangent (A : E κ ≃L[ℝ] E κ) : (E κ × E κ) →L[ℝ] (E κ × E κ) :=
+  (A.toContinuousLinearMap.comp (ContinuousLinearMap.fst ℝ (E κ) (E κ))).prod
+    (A.symm.toContinuousLinearMap.adjoint.comp (ContinuousLinearMap.snd ℝ (E κ) (E κ)))
+
+omit [DecidableEq κ] in
+@[simp] theorem cotangent_apply (A : E κ ≃L[ℝ] E κ) (x : E κ × E κ) :
+    cotangent A x = (A x.1, A.symm.toContinuousLinearMap.adjoint x.2) := rfl
+
+omit [DecidableEq κ] in
+theorem cotangent_preserves (A : E κ ≃L[ℝ] E κ) (x y : E κ × E κ) :
+    inner (𝕜 := ℝ) (cotangent A x).1 (cotangent A y).2 -
+      inner (𝕜 := ℝ) (cotangent A y).1 (cotangent A x).2 =
+    inner (𝕜 := ℝ) x.1 y.2 - inner (𝕜 := ℝ) y.1 x.2 := by
+  simp only [cotangent_apply, ContinuousLinearMap.adjoint_inner_right,
+    ContinuousLinearEquiv.coe_coe, ContinuousLinearEquiv.symm_apply_apply]
+
+omit [DecidableEq κ] in
+theorem cotangent_injective (A : E κ ≃L[ℝ] E κ) : Function.Injective (cotangent A) := by
+  intro x y hxy
+  apply Prod.ext
+  · exact A.injective (congrArg Prod.fst hxy)
+  · apply ext_inner_left ℝ
+    intro z
+    obtain ⟨w,rfl⟩ := A.symm.surjective z
+    have hh := congrArg (fun t : E κ × E κ => inner (𝕜 := ℝ) w t.2) hxy
+    simpa only [cotangent_apply, ContinuousLinearMap.adjoint_inner_right,
+      ContinuousLinearEquiv.coe_coe] using hh
+
+theorem cotangent_first {u v : E κ} (j : κ) (huj : u j ≠ 0)
+    (hvu : inner (𝕜 := ℝ) v u = 1) (x : E κ × E κ) :
+    (cotangent (adapted j huj hvu) x).1 j = inner (𝕜 := ℝ) v x.1 ∧
+    (cotangent (adapted j huj hvu) x).2 j = inner (𝕜 := ℝ) u x.2 := by
+  constructor
+  · exact adapted_first j huj hvu x.1
+  · calc
+      _ = inner (𝕜 := ℝ) (EuclideanSpace.single j 1) (cotangent (adapted j huj hvu) x).2 := by simp only [EuclideanSpace.inner_single_left, map_one, one_mul]
+      _ = _ := by
+        simp only [cotangent_apply, ContinuousLinearMap.adjoint_inner_right,
+          ContinuousLinearEquiv.coe_coe, adapted_inverse_single]
+
+end
+end CanonicalLinear
+end
+
+section
+namespace UpperCoordinates
+noncomputable section
+open Set
+open scoped ContDiff
+variable {κ : Type u133} [Fintype κ] [DecidableEq κ]
+open ComplexCoordinates CanonicalLinear
+
+def phaseForm (x y : R κ × R κ) : ℝ :=
+  inner (𝕜 := ℝ) x.1 y.2 - inner (𝕜 := ℝ) y.1 x.2
+
+omit [DecidableEq κ] in
+theorem form_parts (v w : C κ) : PlanarCylinder.form v w = phaseForm (parts v) (parts w) := by
+  simp only [PlanarCylinder.form, PlanarSquare.form, phaseForm, parts_apply, PiLp.inner_apply,
+    re_apply, im_apply, RCLike.inner_apply', RCLike.conj_to_real, ← Finset.sum_sub_distrib]
+
+omit [DecidableEq κ] in
+theorem form_parts_symm (x y : R κ × R κ) :
+    PlanarCylinder.form (parts.symm x) (parts.symm y) = phaseForm x y := by
+  rw [form_parts, ContinuousLinearEquiv.apply_symm_apply, ContinuousLinearEquiv.apply_symm_apply]
+
+def targetCoordinates (A : E κ ≃L[ℝ] E κ) : (E κ × E κ) →L[ℝ] C κ :=
+  parts.symm.toContinuousLinearMap.comp (cotangent A)
+
+omit [DecidableEq κ] in
+theorem targetCoordinates_preserves (A : E κ ≃L[ℝ] E κ) (x y : E κ × E κ) :
+    PlanarCylinder.form (targetCoordinates A x) (targetCoordinates A y) = phaseForm x y := by
+  change PlanarCylinder.form (parts.symm (cotangent A x)) (parts.symm (cotangent A y)) = _
+  rw [form_parts_symm]
+  exact cotangent_preserves A x y
+
+omit [DecidableEq κ] in
+theorem targetCoordinates_injective (A : E κ ≃L[ℝ] E κ) :
+    Function.Injective (targetCoordinates A) :=
+  parts.symm.injective.comp (cotangent_injective A)
+
+theorem targetCoordinates_first {u v : E κ} (j : κ) (huj : u j ≠ 0)
+    (hvu : inner (𝕜 := ℝ) v u = 1) (x : E κ × E κ) :
+    (targetCoordinates (adapted j huj hvu) x j).re = inner (𝕜 := ℝ) v x.1 ∧
+    (targetCoordinates (adapted j huj hvu) x j).im = inner (𝕜 := ℝ) u x.2 := by
+  have hh := cotangent_first j huj hvu x
+  have hp := parts.apply_symm_apply (cotangent (adapted j huj hvu) x)
+  have hre : re (parts.symm (cotangent (adapted j huj hvu) x)) = (cotangent (adapted j huj hvu) x).1 :=
+    congrArg Prod.fst hp
+  have him : im (parts.symm (cotangent (adapted j huj hvu) x)) = (cotangent (adapted j huj hvu) x).2 :=
+    congrArg Prod.snd hp
+  change re (parts.symm (cotangent (adapted j huj hvu) x)) j = _ ∧
+    im (parts.symm (cotangent (adapted j huj hvu) x)) j = _
+  rw [hre, him]
+  exact hh
+
+end
+end UpperCoordinates
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PolarProducts/CanonicalCylinder.lean b/lean/OAI/Geometry/PolarProducts/CanonicalCylinder.lean
new file mode 100644
index 0000000000000000000000000000000000000000..df6958f27ca1b843ec6afc30b63bf6b2609d63a8
--- /dev/null
+++ b/lean/OAI/Geometry/PolarProducts/CanonicalCylinder.lean
@@ -0,0 +1,116 @@
+import OAI.Geometry.PolarProducts.Nonsqueezing
+
+namespace OAI
+
+universe u128
+
+section
+namespace ConvexPolar
+open Set Filter
+open scoped Topology
+noncomputable section
+variable {E : Type u128} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+
+theorem inner_lt_one_of_mem_interior {K : Set E} {p x : E}
+    (hp : p ∈ polar K) (hp0 : p ≠ 0) (hx : x ∈ interior K) :
+    inner (𝕜 := ℝ) p x < 1 := by
+  obtain ⟨r, hr, hball⟩ := Metric.mem_nhds_iff.mp (mem_interior_iff_mem_nhds.mp hx)
+  let t := r/(2*‖p‖)
+  have ht : 0 < t := div_pos hr (mul_pos (by norm_num) (norm_pos_iff.mpr hp0))
+  have htn : t*‖p‖ = r/2 := by dsimp [t]; field_simp
+  have hy : x+t • p ∈ K := by
+    apply hball
+    rw [Metric.mem_ball, dist_eq_norm, add_sub_cancel_left, norm_smul,
+      Real.norm_eq_abs, abs_of_pos ht, htn]
+    exact half_lt_self hr
+  have hh := hp _ hy
+  rw [inner_add_left, real_inner_smul_left, real_inner_self_eq_norm_sq, real_inner_comm] at hh
+  have hpos : 0 < t*‖p‖^2 := mul_pos ht (sq_pos_of_pos (norm_pos_iff.mpr hp0))
+  linarith
+
+theorem abs_inner_lt_one_of_mem_interior {K : Set E}
+    (hs : ∀ x, x ∈ K ↔ -x ∈ K) {p x : E}
+    (hp : p ∈ polar K) (hp0 : p ≠ 0) (hx : x ∈ interior K) :
+    |inner (𝕜 := ℝ) p x| < 1 := by
+  have hpneg : -p ∈ polar K := by
+    intro q hq
+    have hh := hp (-q) ((hs q).mp hq)
+    simpa only [inner_neg_left, inner_neg_right] using hh
+  have hpos := inner_lt_one_of_mem_interior hp hp0 hx
+  have hneg := inner_lt_one_of_mem_interior hpneg (neg_ne_zero.mpr hp0) hx
+  rw [inner_neg_left] at hneg
+  exact abs_lt.mpr ⟨by linarith, hpos⟩
+
+theorem polar_symmetric {K : Set E} (hs : ∀ x, x ∈ K ↔ -x ∈ K) :
+    ∀ p, p ∈ polar K ↔ -p ∈ polar K := by
+  have hh : ∀ p ∈ polar K, -p ∈ polar K := by
+    intro p hp x hx
+    simpa only [inner_neg_left, inner_neg_right] using hp (-x) ((hs x).mp hx)
+  intro p
+  exact ⟨hh p, fun hp => by simpa using hh (-p) hp⟩
+
+theorem exists_support_pair {K : Set E}
+    (hc : IsCompact K) (h0 : (0 : E) ∈ interior K) {v : E} (hv : v ≠ 0) :
+    ∃ q ∈ K, ∃ p ∈ polar K, inner (𝕜 := ℝ) p q = 1 := by
+  obtain ⟨q, hq, hmax⟩ := hc.exists_isMaxOn ⟨0, interior_subset h0⟩
+    (show ContinuousOn (fun x : E => inner (𝕜 := ℝ) v x) K from by fun_prop)
+  obtain ⟨r, hr, hball⟩ := Metric.mem_nhds_iff.mp (mem_interior_iff_mem_nhds.mp h0)
+  have hx : (r/(2*‖v‖)) • v ∈ K := by
+    apply hball
+    rw [Metric.mem_ball, dist_zero_right, norm_smul, Real.norm_eq_abs,
+      abs_of_pos (div_pos hr (mul_pos (by norm_num) (norm_pos_iff.mpr hv)))]
+    have he : r/(2*‖v‖)*‖v‖ = r/2 := by field_simp
+    rw [he]
+    exact half_lt_self hr
+  have hspos : 0 < inner (𝕜 := ℝ) v q := by
+    have hh : inner (𝕜 := ℝ) v ((r/(2*‖v‖)) • v) ≤ inner (𝕜 := ℝ) v q := hmax hx
+    have he : inner (𝕜 := ℝ) v ((r/(2*‖v‖)) • v) = r/2*‖v‖ := by
+      rw [real_inner_smul_right, real_inner_self_eq_norm_sq]
+      field_simp
+    rw [he] at hh
+    exact (mul_pos (half_pos hr) (norm_pos_iff.mpr hv)).trans_le hh
+  refine ⟨q, hq, (inner (𝕜 := ℝ) v q)⁻¹ • v, ?_, ?_⟩
+  · intro x hx
+    rw [real_inner_smul_right, show inner (𝕜 := ℝ) x v = inner (𝕜 := ℝ) v x from real_inner_comm _ _, ← div_eq_inv_mul]
+    exact (div_le_one hspos).mpr (hmax hx)
+  · rw [real_inner_smul_left, inv_mul_cancel₀ hspos.ne']
+
+end
+end ConvexPolar
+
+namespace SymmetricPolar
+open Set
+noncomputable section
+variable {n : ℕ}
+
+def canonicalSquareCylinder (q₀ p₀ : Position n) : Set (Phase n) :=
+  {z | |inner (𝕜 := ℝ) p₀ z.1| < 1 ∧ |inner (𝕜 := ℝ) q₀ z.2| < 1}
+
+theorem polarProduct_subset_squareCylinder {K : Set (Position n)}
+    (hK : IsSymmetricConvexBody K) {q₀ p₀ : Position n}
+    (hq : q₀ ∈ K) (hp : p₀ ∈ polar K) (hqp : inner (𝕜 := ℝ) p₀ q₀ = 1) :
+    polarProduct K ⊆ canonicalSquareCylinder q₀ p₀ := by
+  have hp0 : p₀ ≠ 0 := by intro he; simp [he] at hqp
+  have hq0 : q₀ ≠ 0 := by intro he; simp [he] at hqp
+  intro z hz
+  have hqp' : q₀ ∈ ConvexPolar.polar (ConvexPolar.polar K) := by
+    intro p hp
+    rw [real_inner_comm]
+    exact hp _ hq
+  exact ⟨ConvexPolar.abs_inner_lt_one_of_mem_interior hK.2.2.2 hp hp0 hz.1,
+    ConvexPolar.abs_inner_lt_one_of_mem_interior (ConvexPolar.polar_symmetric hK.2.2.2) hqp' hq0 hz.2⟩
+
+theorem exists_squareCylinder {K : Set (Position n)} (hK : IsSymmetricConvexBody K)
+    (hn : 0 < n) : ∃ q₀ p₀ : Position n, inner (𝕜 := ℝ) p₀ q₀ = 1 ∧
+      polarProduct K ⊆ canonicalSquareCylinder q₀ p₀ := by
+  let v : Position n := (EuclideanSpace.basisFun (Fin n) ℝ) ⟨0, hn⟩
+  have hv : v ≠ 0 := (EuclideanSpace.basisFun (Fin n) ℝ).toBasis.ne_zero _
+  obtain ⟨q₀, hq, p₀, hp, hqp⟩ := ConvexPolar.exists_support_pair hK.1
+    (ConvexPolar.zero_mem_interior_of_symmetric hK.2.1 hK.2.2.1 hK.2.2.2) hv
+  exact ⟨q₀, p₀, hqp, polarProduct_subset_squareCylinder hK hq hp hqp⟩
+
+end
+end SymmetricPolar
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PolarProducts/CompleteFlow.lean b/lean/OAI/Geometry/PolarProducts/CompleteFlow.lean
new file mode 100644
index 0000000000000000000000000000000000000000..668639922acce5dcdd0e6631a6a9ca28595f6392
--- /dev/null
+++ b/lean/OAI/Geometry/PolarProducts/CompleteFlow.lean
@@ -0,0 +1,272 @@
+import OAI.Geometry.PolarProducts.LocalODE
+
+namespace OAI
+
+universe u7 u8 u9 u10 u11 u12 u13
+
+section LowerBoundInline
+open Set Filter Function
+open scoped Topology ContDiff NNReal
+open Set Filter Metric
+open scoped Topology ContDiff
+open Set Filter Function MeasureTheory Metric
+open scoped Topology ContDiff NNReal
+
+namespace SmoothODE
+
+open Set Filter Function MeasureTheory Metric
+open scoped Topology ContDiff NNReal
+
+variable {E : Type u7} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  [CompleteSpace E] [ProperSpace E]
+
+noncomputable section
+
+def variation {V : E → E} {K : ℝ≥0} (hV : LipschitzWith K V)
+    (h : ℝ) (hh : ‖h‖₊ * K < 1) (x v : E) (t : ℝ) : E :=
+  v + h • ∫ s in 0..t,
+    fderiv ℝ V (extend (trajectory hV h hh x) s) (extend (tangentPath hV h hh x v) s)
+
+theorem variation_coe {V : E → E} {K : ℝ≥0} (hV : LipschitzWith K V)
+    (hVs : ContDiff ℝ ∞ V) (h : ℝ) (hh : ‖h‖₊ * K < 1) (x v : E) (t : Time) :
+    variation hV h hh x v t = tangentPath hV h hh x v t :=
+  (congrArg (fun u : C(Time, E) => u t) (tangentPath_eq hV hVs h hh x v)).symm
+
+omit [ProperSpace E] in
+@[simp] theorem variation_zero {V : E → E} {K : ℝ≥0} (hV : LipschitzWith K V)
+    (h : ℝ) (hh : ‖h‖₊ * K < 1) (x v : E) : variation hV h hh x v 0 = v := by
+  simp [variation]
+
+omit [ProperSpace E] in
+theorem continuous_variation {V : E → E} {K : ℝ≥0} (hV : LipschitzWith K V)
+    (hVs : ContDiff ℝ ∞ V) (h : ℝ) (hh : ‖h‖₊ * K < 1) (x v : E) :
+    Continuous (variation hV h hh x v) := by
+  have hc := ((hVs.continuous_fderiv (by simp)).comp
+    (continuous_extend (trajectory hV h hh x))).clm_apply
+      (continuous_extend (tangentPath hV h hh x v))
+  exact continuous_const.add
+    ((intervalIntegral.continuous_primitive (fun a b => hc.intervalIntegrable a b) 0).const_smul h)
+
+theorem hasDerivAt_variation {V : E → E} {K : ℝ≥0} (hV : LipschitzWith K V)
+    (hVs : ContDiff ℝ ∞ V) (h : ℝ) (hh : ‖h‖₊ * K < 1) (x v : E) (t : Time) :
+    HasDerivAt (variation hV h hh x v)
+      (h • fderiv ℝ V (curve hV h hh x t) (variation hV h hh x v t)) (t : ℝ) := by
+  have hc := ((hVs.continuous_fderiv (by simp)).comp
+    (continuous_extend (trajectory hV h hh x))).clm_apply
+      (continuous_extend (tangentPath hV h hh x v))
+  have hd := intervalIntegral.integral_hasDerivAt_right
+    (hc.intervalIntegrable 0 t) (hc.stronglyMeasurableAtFilter volume (𝓝 (t : ℝ)))
+      hc.continuousAt
+  rw [curve_coe, variation_coe hV hVs]
+  change HasDerivAt (fun s : ℝ => v + h • ∫ r in 0..s,
+    fderiv ℝ V (extend (trajectory hV h hh x) r) (extend (tangentPath hV h hh x v) r))
+    _ (t : ℝ)
+  simpa only [Pi.smul_apply, Function.comp_apply, extend_coe] using (hd.const_smul h).const_add v
+
+theorem tangentPath_one {V : E → E} {K : ℝ≥0} (hV : LipschitzWith K V)
+    (hVs : ContDiff ℝ ∞ V) (h : ℝ) (hh : ‖h‖₊ * K < 1) (x v : E) :
+    tangentPath hV h hh x v ⟨1, by constructor <;> norm_num⟩ = fderiv ℝ (step hV h hh) x v := by
+  let t : Time := ⟨1, by constructor <;> norm_num⟩
+  have hd := (ContinuousMap.evalCLM ℝ (M := E) t).hasFDerivAt.comp x
+    ((contDiff_trajectory hV hVs h hh).differentiable (by simp) x).hasFDerivAt
+  exact (congrArg (fun L : E →L[ℝ] E => L v) hd.fderiv).symm
+
+theorem variation_one {V : E → E} {K : ℝ≥0} (hV : LipschitzWith K V)
+    (hVs : ContDiff ℝ ∞ V) (h : ℝ) (hh : ‖h‖₊ * K < 1) (x v : E) :
+    variation hV h hh x v 1 = fderiv ℝ (step hV h hh) x v :=
+  (variation_coe hV hVs h hh x v ⟨1, by constructor <;> norm_num⟩).trans
+    (tangentPath_one hV hVs h hh x v)
+
+end
+end SmoothODE
+
+namespace SmoothODE
+
+open Set Filter Function MeasureTheory Metric
+open scoped Topology ContDiff NNReal
+
+variable {E : Type u8} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  [CompleteSpace E] [ProperSpace E]
+
+noncomputable section
+
+theorem step_preserves_tensor {V : E → E} {K : ℝ≥0} (hV : LipschitzWith K V)
+    (hVs : ContDiff ℝ ∞ V) (h : ℝ) (hh : ‖h‖₊ * K < 1)
+    {B : E → E →L[ℝ] E →L[ℝ] ℝ} (hB : Differentiable ℝ B)
+    (hInv : ∀ z a b, fderiv ℝ B z (V z) a b +
+      B z (fderiv ℝ V z a) b + B z a (fderiv ℝ V z b) = 0) (x a b : E) :
+    B (step hV h hh x) (fderiv ℝ (step hV h hh) x a)
+      (fderiv ℝ (step hV h hh) x b) = B x a b := by
+  let F : ℝ → ℝ := fun t => B (curve hV h hh x t)
+    (variation hV h hh x a t) (variation hV h hh x b t)
+  have hc : Continuous F :=
+    ((hB.continuous.comp (continuous_curve hV h hh x)).clm_apply
+      (continuous_variation hV hVs h hh x a)).clm_apply
+      (continuous_variation hV hVs h hh x b)
+  have hd : ∀ t ∈ Ico (0 : ℝ) 1, HasDerivWithinAt F 0 (Ici t) t := by
+    intro t ht
+    let T : Time := ⟨t, ht.1, ht.2.le⟩
+    have hv := hasDerivAt_curve hV h hh x T
+    have ha := hasDerivAt_variation hV hVs h hh x a T
+    have hb := hasDerivAt_variation hV hVs h hh x b T
+    have hdb : HasDerivAt (fun s : ℝ => B (curve hV h hh x s))
+        (fderiv ℝ B (curve hV h hh x t) (h • V (curve hV h hh x t))) t :=
+      (hB (curve hV h hh x t)).hasFDerivAt.comp_hasDerivAt (l := B)
+        (l' := fderiv ℝ B (curve hV h hh x t)) t hv
+    have hdf := (hdb.clm_apply ha).clm_apply hb
+    have hz :
+        ((fderiv ℝ B (curve hV h hh x t) (h • V (curve hV h hh x t)))
+          (variation hV h hh x a t) +
+          B (curve hV h hh x t)
+            (h • fderiv ℝ V (curve hV h hh x t) (variation hV h hh x a t)))
+          (variation hV h hh x b t) +
+          B (curve hV h hh x t) (variation hV h hh x a t)
+            (h • fderiv ℝ V (curve hV h hh x t) (variation hV h hh x b t)) = 0 := by
+      simp only [map_smul, smul_apply,
+        add_apply, smul_eq_mul]
+      calc
+        _ = h * (fderiv ℝ B (curve hV h hh x t) (V (curve hV h hh x t))
+            (variation hV h hh x a t) (variation hV h hh x b t) +
+          B (curve hV h hh x t)
+            (fderiv ℝ V (curve hV h hh x t) (variation hV h hh x a t))
+            (variation hV h hh x b t) +
+          B (curve hV h hh x t) (variation hV h hh x a t)
+            (fderiv ℝ V (curve hV h hh x t) (variation hV h hh x b t))) := by ring
+        _ = 0 := by rw [hInv, mul_zero]
+    exact (hdf.congr_deriv hz).hasDerivWithinAt
+  have he := constant_of_has_deriv_right_zero hc.continuousOn hd 1
+    (show 1 ∈ Icc (0 : ℝ) 1 by constructor <;> norm_num)
+  simpa only [F, curve_one, variation_one hV hVs, curve_zero, variation_zero] using he
+
+end
+end SmoothODE
+
+namespace SmoothODE
+
+open Set Filter Function MeasureTheory Metric
+open scoped Topology ContDiff NNReal
+
+variable {E : Type u9} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  [CompleteSpace E] [ProperSpace E]
+
+noncomputable section
+
+omit [CompleteSpace E] [ProperSpace E] in
+
+theorem contDiff_homeomorph_pow (f : E ≃ₜ E) (hf : ContDiff ℝ ∞ (f : E → E))
+    (n : ℕ) : ContDiff ℝ ∞ ((f ^ n : E ≃ₜ E) : E → E) := by
+  induction n with
+  | zero =>
+    rw [pow_zero]
+    change ContDiff ℝ ∞ (id : E → E)
+    exact contDiff_id
+  | succ n ih =>
+    rw [pow_succ']
+    change ContDiff ℝ ∞ ((f : E → E) ∘ (f ^ n : E ≃ₜ E))
+    exact hf.comp ih
+
+omit [CompleteSpace E] [ProperSpace E] in
+
+theorem homeomorph_pow_preserves_tensor (f : E ≃ₜ E)
+    (hf : ContDiff ℝ ∞ (f : E → E))
+    {B : E → E →L[ℝ] E →L[ℝ] ℝ}
+    (hB : ∀ x a b, B (f x) (fderiv ℝ (f : E → E) x a)
+      (fderiv ℝ (f : E → E) x b) = B x a b) (n : ℕ) (x a b : E) :
+    B ((f ^ n) x) (fderiv ℝ ((f ^ n : E ≃ₜ E) : E → E) x a)
+      (fderiv ℝ ((f ^ n : E ≃ₜ E) : E → E) x b) = B x a b := by
+  induction n with
+  | zero =>
+    rw [pow_zero]
+    change B x (fderiv ℝ (id : E → E) x a) (fderiv ℝ (id : E → E) x b) = _
+    simp
+  | succ n ih =>
+    rw [pow_succ']
+    change B (f ((f ^ n) x)) (fderiv ℝ ((f : E → E) ∘ (f ^ n : E ≃ₜ E)) x a)
+      (fderiv ℝ ((f : E → E) ∘ (f ^ n : E ≃ₜ E)) x b) = _
+    rw [fderiv_comp x (hf.differentiable (by simp) _)
+      ((contDiff_homeomorph_pow f hf n).differentiable (by simp) _)]
+    simp only [ContinuousLinearMap.comp_apply]
+    rw [hB]
+    exact ih
+
+omit [ProperSpace E] in
+
+theorem curve_linear_component {F : Type u10} [NormedAddCommGroup F] [NormedSpace ℝ F]
+    {V : E → E} {K : ℝ≥0} (hV : LipschitzWith K V)
+    (h : ℝ) (hh : ‖h‖₊ * K < 1) (L : E →L[ℝ] F) (v : F)
+    (hL : ∀ z, L (V z) = v) (x : E) (t : Time) :
+    L (curve hV h hh x t) = L x + (t : ℝ) • h • v := by
+  let G : ℝ → F := fun s => L (curve hV h hh x s) - s • h • v
+  have hc : Continuous G :=
+    (L.continuous.comp (continuous_curve hV h hh x)).sub (continuous_id.smul continuous_const)
+  have hd : ∀ s ∈ Ico (0 : ℝ) 1, HasDerivWithinAt G 0 (Ici s) s := by
+    intro s hs
+    have hd := L.hasFDerivAt.comp_hasDerivAt s
+      (hasDerivAt_curve hV h hh x ⟨s, hs.1, hs.2.le⟩)
+    have hlin := (hasDerivAt_id s).smul_const (h • v)
+    have hh' : HasDerivAt G 0 s := by
+      change HasDerivAt ((L ∘ curve hV h hh x) - fun s : ℝ => s • h • v) 0 s
+      simpa only [id_eq, map_smul, hL, one_smul, sub_self] using hd.sub hlin
+    exact hh'.hasDerivWithinAt
+  have he := constant_of_has_deriv_right_zero hc.continuousOn hd (t : ℝ) t.property
+  simpa only [G, curve_zero, zero_smul, sub_zero, sub_eq_iff_eq_add] using he
+
+omit [ProperSpace E] in
+@[simp] theorem step_linear_component {F : Type u11} [NormedAddCommGroup F] [NormedSpace ℝ F]
+    {V : E → E} {K : ℝ≥0} (hV : LipschitzWith K V)
+    (h : ℝ) (hh : ‖h‖₊ * K < 1) (L : E →L[ℝ] F) (v : F)
+    (hL : ∀ z, L (V z) = v) (x : E) :
+    L (step hV h hh x) = L x + h • v := by
+  simpa only [curve_one, one_smul] using
+    curve_linear_component hV h hh L v hL x (⟨1, by constructor <;> norm_num⟩ : Time)
+
+omit [CompleteSpace E] [ProperSpace E] in
+
+theorem homeomorph_pow_linear_component {F : Type u12} [NormedAddCommGroup F] [NormedSpace ℝ F]
+    (f : E ≃ₜ E) (L : E →L[ℝ] F) (v : F) (hL : ∀ x, L (f x) = L x + v)
+    (n : ℕ) (x : E) : L ((f ^ n) x) = L x + n • v := by
+  induction n with
+  | zero => simp
+  | succ n ih =>
+    simp only [pow_succ', Homeomorph.mul_apply, hL, ih, succ_nsmul]
+    abel
+
+theorem exists_transport {V : E → E} {K : ℝ≥0} (hV : LipschitzWith K V)
+    (hVs : ContDiff ℝ ∞ V) {B : E → E →L[ℝ] E →L[ℝ] ℝ}
+    (hB : Differentiable ℝ B)
+    (hInv : ∀ z a b, fderiv ℝ B z (V z) a b + B z (fderiv ℝ V z a) b +
+      B z a (fderiv ℝ V z b) = 0)
+    {F : Type u13} [NormedAddCommGroup F] [NormedSpace ℝ F]
+    (L : E →L[ℝ] F) (v : F) (hL : ∀ x, L (V x) = v) (T : ℝ) :
+    ∃ f : E ≃ₜ E, ContDiff ℝ ∞ (f : E → E) ∧ ContDiff ℝ ∞ (f.symm : E → E) ∧
+      (∀ x a b, B (f x) (fderiv ℝ (f : E → E) x a) (fderiv ℝ (f : E → E) x b) = B x a b) ∧
+      (∀ x, L (f x) = L x + T • v) := by
+  obtain ⟨N, hN⟩ := exists_nat_gt (max (‖T‖ * (K : ℝ)) 0)
+  have hN0 : (0 : ℝ) < N := lt_of_le_of_lt (le_max_right _ _) hN
+  have hNne : (N : ℝ) ≠ 0 := ne_of_gt hN0
+  let h : ℝ := T / N
+  have hh : ‖h‖₊ * K < 1 := by
+    change ‖h‖ * (K : ℝ) < (1 : ℝ)
+    dsimp [h]
+    rw [abs_div, abs_of_pos hN0, div_mul_eq_mul_div]
+    exact (div_lt_one hN0).mpr ((le_max_left _ _).trans_lt hN)
+  let f := stepHomeomorph hV hVs h hh
+  have hf : ContDiff ℝ ∞ (f : E → E) := contDiff_step hV hVs h hh
+  have hfi : ContDiff ℝ ∞ (f.symm : E → E) := contDiff_step hV hVs (-h) (by simpa using hh)
+  refine ⟨f ^ N, contDiff_homeomorph_pow f hf N, ?_, ?_, ?_⟩
+  · change ContDiff ℝ ∞ ((f ^ N)⁻¹ : E ≃ₜ E)
+    rw [← inv_pow]
+    exact contDiff_homeomorph_pow f⁻¹ hfi N
+  · exact homeomorph_pow_preserves_tensor f hf (step_preserves_tensor hV hVs h hh hB hInv) N
+  · intro x
+    have he := homeomorph_pow_linear_component f L (h • v)
+      (step_linear_component hV h hh L v hL) N x
+    have hs : (N : ℝ) * h = T := by dsimp [h]; field_simp
+    simpa only [nsmul_eq_mul, ← Nat.cast_smul_eq_nsmul ℝ, smul_smul, hs] using he
+
+end
+end SmoothODE
+
+end LowerBoundInline
+
+end OAI
diff --git a/lean/OAI/Geometry/PolarProducts/Contraction.lean b/lean/OAI/Geometry/PolarProducts/Contraction.lean
new file mode 100644
index 0000000000000000000000000000000000000000..e09a0c708c76ff62d70d2f1ab13b2eb88156576d
--- /dev/null
+++ b/lean/OAI/Geometry/PolarProducts/Contraction.lean
@@ -0,0 +1,85 @@
+import OAI.Geometry.PolarProducts.Model
+
+namespace OAI
+
+universe u1 u2
+
+section LowerBoundInline
+
+open Set Filter Function
+open scoped Topology ContDiff NNReal
+
+namespace SmoothContraction
+
+variable {E : Type u1} {F : Type u2} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
+  [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F]
+
+noncomputable section
+
+def fixedPoint {f : E × F → F} {L : ℝ≥0}
+    (hL : L < 1) (hLip : ∀ x, LipschitzWith L (fun y => f (x, y))) (x : E) : F :=
+  ContractingWith.fixedPoint (fun y => f (x, y)) ⟨hL, hLip x⟩
+
+omit [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
+  [NormedSpace ℝ F] in
+theorem fixedPoint_eq {f : E × F → F} {L : ℝ≥0}
+    (hL : L < 1) (hLip : ∀ x, LipschitzWith L (fun y => f (x, y))) (x : E) :
+    f (x, fixedPoint hL hLip x) = fixedPoint hL hLip x :=
+  (show ContractingWith L (fun y => f (x, y)) from ⟨hL, hLip x⟩).fixedPoint_isFixedPt
+
+omit [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
+  [NormedSpace ℝ F] in
+theorem eq_fixedPoint {f : E × F → F} {L : ℝ≥0}
+    (hL : L < 1) (hLip : ∀ x, LipschitzWith L (fun y => f (x, y))) (x : E) {y : F}
+    (hy : f (x, y) = y) : y = fixedPoint hL hLip x :=
+  (show ContractingWith L (fun y => f (x, y)) from ⟨hL, hLip x⟩).fixedPoint_unique hy
+
+omit [CompleteSpace E] in
+
+theorem partial_isInvertible {f : E × F → F} {L : ℝ≥0}
+    (hf : Differentiable ℝ f) (hL : L < 1)
+    (hLip : ∀ x, LipschitzWith L (fun y => f (x, y))) (u : E × F) :
+    ((fderiv ℝ (fun p : E × F => p.2 - f p) u).comp
+      (ContinuousLinearMap.inr ℝ E F)).IsInvertible := by
+  let A := (fderiv ℝ f u).comp (ContinuousLinearMap.inr ℝ E F)
+  have hA : HasFDerivAt (fun y => f (u.1, y)) A u.2 :=
+    (hf u).hasFDerivAt.comp u.2 (hasFDerivAt_prodMk_right u.1 u.2)
+  have hnorm : ‖A‖ < 1 := by
+    rw [← hA.fderiv]
+    exact (norm_fderiv_le_of_lipschitz ℝ (hLip u.1)).trans_lt (by exact_mod_cast hL)
+  have heq : (fderiv ℝ (fun p : E × F => p.2 - f p) u).comp
+      (ContinuousLinearMap.inr ℝ E F) = 1 - A := by
+    change (fderiv ℝ (Prod.snd - f) u).comp _ = _
+    rw [(hasFDerivAt_snd.sub (hf u).hasFDerivAt).fderiv]
+    ext y
+    simp [A]
+  rw [heq]
+  obtain ⟨v, hv⟩ := isUnit_one_sub_of_norm_lt_one hnorm
+  exact ⟨ContinuousLinearEquiv.ofUnit v, hv⟩
+
+theorem contDiff_fixedPoint {f : E × F → F} {L : ℝ≥0}
+    (hf : ContDiff ℝ ∞ f) (hL : L < 1)
+    (hLip : ∀ x, LipschitzWith L (fun y => f (x, y))) :
+    ContDiff ℝ ∞ (fixedPoint hL hLip) := by
+  rw [contDiff_iff_contDiffAt]
+  intro x
+  let y := fixedPoint hL hLip x
+  let g : E × F → F := fun p => p.2 - f p
+  have hg : ContDiffAt ℝ ∞ g (x, y) := (contDiff_snd.sub hf).contDiffAt
+  have hi := partial_isInvertible (hf.differentiable (by simp)) hL hLip (x, y)
+  let ψ := hg.implicitFunction (by simp) hi
+  have hψ : ContDiffAt ℝ ∞ ψ x := hg.contDiffAt_implicitFunction (by simp) hi
+  have hzero : g (x, y) = 0 := sub_eq_zero.mpr (fixedPoint_eq hL hLip x).symm
+  have hnear : ψ =ᶠ[𝓝 x] fixedPoint hL hLip := by
+    filter_upwards [hg.eventually_apply_implicitFunction (by simp) hi] with x' hx'
+    apply eq_fixedPoint hL hLip x'
+    have : g (x', ψ x') = 0 := hx'.trans hzero
+    exact (sub_eq_zero.mp this).symm
+  exact hψ.congr_of_eventuallyEq hnear.symm
+
+end
+end SmoothContraction
+
+end LowerBoundInline
+
+end OAI
diff --git a/lean/OAI/Geometry/PolarProducts/EmbeddingInverse.lean b/lean/OAI/Geometry/PolarProducts/EmbeddingInverse.lean
new file mode 100644
index 0000000000000000000000000000000000000000..df44989c6e16ab3daef19da42a5db89969d34552
--- /dev/null
+++ b/lean/OAI/Geometry/PolarProducts/EmbeddingInverse.lean
@@ -0,0 +1,187 @@
+import OAI.Geometry.PolarProducts.HamiltonianGrowth
+
+namespace OAI
+
+universe u121 u122
+
+section NonsqueezingInline
+
+namespace EmbeddingInverse
+noncomputable section
+open Set Filter
+open scoped ContDiff Topology
+variable {E : Type u121} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
+
+ def localEquiv {U : Set E} (hU : IsOpen U) (f : E → E)
+    (hf : ContDiffOn ℝ ∞ f U) (hi : InjOn f U)
+    (hD : ∀ x ∈ U, (fderiv ℝ f x).IsInvertible) : OpenPartialHomeomorph E E :=
+  OpenPartialHomeomorph.ofContinuousOpenRestrict (hi.toPartialEquiv f U) hf.continuousOn
+    (LocalEmbedding.isOpenEmbedding hU hf hi hD).isOpenMap hU
+
+@[simp] theorem localEquiv_apply {U : Set E} (hU : IsOpen U) (f : E → E)
+    (hf : ContDiffOn ℝ ∞ f U) (hi : InjOn f U)
+    (hD : ∀ x ∈ U, (fderiv ℝ f x).IsInvertible) (x : E) :
+    localEquiv hU f hf hi hD x = f x := rfl
+
+@[simp] theorem localEquiv_source {U : Set E} (hU : IsOpen U) (f : E → E)
+    (hf : ContDiffOn ℝ ∞ f U) (hi : InjOn f U)
+    (hD : ∀ x ∈ U, (fderiv ℝ f x).IsInvertible) :
+    (localEquiv hU f hf hi hD).source = U := rfl
+
+@[simp] theorem localEquiv_target {U : Set E} (hU : IsOpen U) (f : E → E)
+    (hf : ContDiffOn ℝ ∞ f U) (hi : InjOn f U)
+    (hD : ∀ x ∈ U, (fderiv ℝ f x).IsInvertible) :
+    (localEquiv hU f hf hi hD).target = f '' U := rfl
+
+ theorem contDiffAt_inverse (e : OpenPartialHomeomorph E E)
+    (he : ContDiffOn ℝ ∞ e e.source)
+    (hD : ∀ x ∈ e.source, (fderiv ℝ e x).IsInvertible)
+    {y : E} (hy : y ∈ e.target) : ContDiffAt ℝ ∞ e.symm y := by
+  have hs := he.contDiffAt (e.open_source.mem_nhds (e.map_target hy))
+  obtain ⟨D,hD⟩ := hD _ (e.map_target hy)
+  apply e.contDiffAt_symm hy (f₀' := D) _ hs
+  rw [hD]
+  exact (hs.differentiableAt (by simp)).hasFDerivAt
+
+ def extendFunction (e : OpenPartialHomeomorph E E) (f : E → ℝ) (y : E) : ℝ := by
+  classical
+  exact if y ∈ e.target then f (e.symm y) else 0
+
+omit [NormedSpace ℝ E] [CompleteSpace E] in
+ theorem extendFunction_apply (e : OpenPartialHomeomorph E E) (f : E → ℝ)
+    {x : E} (hx : x ∈ e.source) : extendFunction e f (e x) = f x := by
+  simp only [extendFunction, ite_eq_left (e.map_source hx), e.left_inv hx]
+
+omit [NormedSpace ℝ E] [CompleteSpace E] in
+ theorem extendFunction_zero (e : OpenPartialHomeomorph E E) (f : E → ℝ)
+    (S : Set E) (hz : ∀ x ∉ S, f x = 0) {y : E} (hy : y ∉ e '' S) :
+    extendFunction e f y = 0 := by
+  unfold extendFunction
+  split_ifs with hyt
+  · apply hz
+    intro hin
+    exact hy ⟨e.symm y,hin,e.right_inv hyt⟩
+  · rfl
+
+ theorem contDiff_extendFunction (e : OpenPartialHomeomorph E E)
+    (he : ContDiffOn ℝ ∞ e e.source)
+    (hD : ∀ x ∈ e.source, (fderiv ℝ e x).IsInvertible)
+    {f : E → ℝ} (hf : ContDiff ℝ ∞ f) {S : Set E} (hS : IsCompact S)
+    (hSU : S ⊆ e.source) (hz : ∀ x ∉ S, f x = 0) :
+    ContDiff ℝ ∞ (extendFunction e f) := by
+  rw [contDiff_iff_contDiffAt]
+  intro y
+  by_cases hy : y ∈ e.target
+  · apply (hf.contDiffAt.comp y (contDiffAt_inverse e he hD hy)).congr_of_eventuallyEq
+    filter_upwards [e.open_target.mem_nhds hy] with z hz
+    simp only [extendFunction, hz, ite_true, Function.comp_apply]
+  · have hT : IsClosed (e '' S) := (hS.image_of_continuousOn (e.continuousOn.mono hSU)).isClosed
+    have hyt : y ∈ (e '' S)ᶜ := by
+      rintro ⟨x,hx,rfl⟩
+      exact hy (e.map_source (hSU hx))
+    apply (contDiffAt_const (c := (0 : ℝ))).congr_of_eventuallyEq
+    filter_upwards [hT.isOpen_compl.mem_nhds hyt] with z hz'
+    exact extendFunction_zero e f S hz hz'
+
+omit [NormedSpace ℝ E] [CompleteSpace E] in
+ theorem extendFunction_mem_range (e : OpenPartialHomeomorph E E) (f : E → ℝ) (y : E) :
+    extendFunction e f y = 0 ∨ ∃ x ∈ e.source, extendFunction e f y = f x := by
+  by_cases hy : y ∈ e.target
+  · exact Or.inr ⟨e.symm y,e.map_target hy, by simp only [extendFunction,hy,ite_true]⟩
+  · exact Or.inl (by simp only [extendFunction,hy,ite_false])
+
+omit [NormedSpace ℝ E] [CompleteSpace E] in
+ theorem eventually_of_comp (e : OpenPartialHomeomorph E E) {x : E} (hx : x ∈ e.source)
+    {P : E → Prop} (hP : ∀ᶠ z in 𝓝 x, P (e z)) : ∀ᶠ y in 𝓝 (e x), P y := by
+   rw [← e.map_nhds_eq hx]
+   exact hP
+
+end
+end EmbeddingInverse
+
+namespace ComplexSymplectic
+noncomputable section
+open Set Filter
+open scoped ContDiff Topology
+open FourierPolynomial HamiltonianODE
+variable {κ : Type u122} [Fintype κ]
+
+ def form (v w : Vector κ) : ℝ := inner (𝕜 := ℝ) (Complex.I • v) w
+ def Preserves (A : Vector κ →L[ℝ] Vector κ) : Prop := ∀ v w, form (A v) (A w) = form v w
+
+ theorem form_ext {v w : Vector κ} (h : ∀ z, form v z = form w z) : v = w := by
+   exact smul_right_injective _ Complex.I_ne_zero (ext_inner_right ℝ h)
+
+ theorem form_I (v w : Vector κ) : form (Complex.I • v) w = -inner (𝕜 := ℝ) v w := by
+   simp [form, smul_smul]
+
+ theorem injective {A : Vector κ →L[ℝ] Vector κ} (hA : Preserves A) : Function.Injective A := by
+   intro v w hvw
+   apply form_ext
+   intro z
+   calc
+     form v z = form (A v) (A z) := (hA v z).symm
+     _ = form (A w) (A z) := by rw [hvw]
+     _ = form w z := hA w z
+
+ theorem invertible {A : Vector κ →L[ℝ] Vector κ} (hA : Preserves A) : A.IsInvertible := by
+   have hi := injective hA
+   exact ⟨ContinuousLinearEquiv.ofBijective A (LinearMap.ker_eq_bot.mpr hi)
+     (LinearMap.range_eq_top.mpr (LinearMap.surjective_of_injective hi)), rfl⟩
+
+ theorem push_gradient {A : Vector κ →L[ℝ] Vector κ} (hA : Preserves A) (a b : Vector κ)
+    (hh : ∀ v, inner (𝕜 := ℝ) b (A v) = inner (𝕜 := ℝ) a v) :
+    A (Complex.I • a) = Complex.I • b := by
+   apply form_ext
+   intro w
+   obtain ⟨v,rfl⟩ := (LinearMap.surjective_of_injective (injective hA)) w
+   change form (A (Complex.I • a)) (A v) = form (Complex.I • b) (A v)
+   rw [hA, form_I, form_I, hh]
+
+ theorem push_hamiltonian (e : OpenPartialHomeomorph (Vector κ) (Vector κ))
+    (he : ContDiffOn ℝ ∞ e e.source)
+    (hω : ∀ x ∈ e.source, Preserves (fderiv ℝ e x))
+    {H G : Vector κ → ℝ} (hH : ContDiff ℝ ∞ H) (hG : ContDiff ℝ ∞ G)
+    (hcomp : ∀ x ∈ e.source, H (e x) = G x) {x : Vector κ} (hx : x ∈ e.source) :
+    fderiv ℝ e x (Complex.I • gradient G x) = Complex.I • gradient H (e x) := by
+   apply push_gradient (hω x hx)
+   intro v
+   have hed := ((he.contDiffAt (e.open_source.mem_nhds hx)).differentiableAt (by simp)).hasFDerivAt
+   have hhd := ((hH.differentiable (by simp) (e x)).hasGradientAt.hasFDerivAt).comp x hed
+   have hgd := ((hG.differentiable (by simp) x).hasGradientAt.hasFDerivAt)
+   have heq : H ∘ e =ᶠ[𝓝 x] G := by
+     filter_upwards [e.open_source.mem_nhds hx] with z hz
+     exact hcomp z hz
+   have hh := hhd.unique (hgd.congr_of_eventuallyEq heq)
+   exact congrArg (fun D : Vector κ →L[ℝ] ℝ => D v) hh
+
+ theorem inverse_curve (e : OpenPartialHomeomorph (Vector κ) (Vector κ))
+    (he : ContDiffOn ℝ ∞ e e.source)
+    (hω : ∀ x ∈ e.source, Preserves (fderiv ℝ e x))
+    {H G : Vector κ → ℝ} (hH : ContDiff ℝ ∞ H) (hG : ContDiff ℝ ∞ G)
+    (hcomp : ∀ x ∈ e.source, H (e x) = G x)
+    {x : ℝ → Vector κ} (hxt : ∀ t, x t ∈ e.target)
+    (hx : ∀ t, HasDerivAt x (Complex.I • gradient H (x t)) t) (t : ℝ) :
+    HasDerivAt (fun s => e.symm (x s)) (Complex.I • gradient G (e.symm (x t))) t := by
+   let y := e.symm (x t)
+   have hy : y ∈ e.source := e.map_target (hxt t)
+   obtain ⟨D,hD⟩ := invertible (hω y hy)
+   have hed : HasFDerivAt e D.toContinuousLinearMap y := by
+     rw [hD]
+     exact ((he.contDiffAt (e.open_source.mem_nhds hy)).differentiableAt (by simp)).hasFDerivAt
+   have hcd := (e.hasFDerivAt_symm (hxt t) hed).comp_hasDerivAt t (hx t)
+   apply hcd.congr_deriv
+   apply D.injective
+   simp only [ContinuousLinearEquiv.coe_coe, ContinuousLinearEquiv.apply_symm_apply]
+   have hh := push_hamiltonian e he hω hH hG hcomp hy
+   rw [e.right_inv (hxt t)] at hh
+   change Complex.I • gradient H (x t) = D.toContinuousLinearMap (Complex.I • gradient G y)
+   rw [hD]
+   exact hh.symm
+
+end
+end ComplexSymplectic
+
+end NonsqueezingInline
+
+end OAI
diff --git a/lean/OAI/Geometry/PolarProducts/ExteriorHamiltonian.lean b/lean/OAI/Geometry/PolarProducts/ExteriorHamiltonian.lean
new file mode 100644
index 0000000000000000000000000000000000000000..274b94c160652a9be4b34bd2851e691adbcdf6d0
--- /dev/null
+++ b/lean/OAI/Geometry/PolarProducts/ExteriorHamiltonian.lean
@@ -0,0 +1,377 @@
+import OAI.Geometry.PolarProducts.SmoothProfiles
+
+namespace OAI
+
+universe u123 u124 u125 u126
+
+section NonsqueezingInline
+
+namespace PushedBall
+noncomputable section
+open Set Filter
+open scoped ContDiff Topology
+open FourierPolynomial HamiltonianODE SmoothProfiles EmbeddingInverse
+variable {κ : Type u123} [Fintype κ]
+
+ def domain (L : ℝ) : Set (Vector κ) := {z | ‖z‖^2 < L}
+ def core (L ε : ℝ) : Set (Vector κ) := Metric.closedBall 0 (Real.sqrt (L-ε))
+ def radial (r L ε : ℝ) (z : Vector κ) : ℝ := ballProfile r L ε (‖z‖^2)
+
+ theorem isOpen_domain (L : ℝ) : IsOpen (domain (κ := κ) L) :=
+   isOpen_lt (continuous_norm.pow 2) continuous_const
+
+ theorem core_compact (L ε : ℝ) : IsCompact (core (κ := κ) L ε) := isCompact_closedBall _ _
+
+ theorem mem_core_iff {L ε : ℝ} (hL : 0 ≤ L-ε) (z : Vector κ) :
+    z ∈ core L ε ↔ ‖z‖^2 ≤ L-ε := by
+   simp only [core, Metric.mem_closedBall, dist_zero_right]
+   exact (Real.le_sqrt (norm_nonneg z) hL)
+
+ theorem core_subset_domain {L ε : ℝ} (hε : 0 < ε) (hL : ε ≤ L) :
+    core (κ := κ) L ε ⊆ domain L := by
+   intro z hz
+   have hh := (mem_core_iff (sub_nonneg.mpr hL) z).mp hz
+   change ‖z‖^2 < L
+   linarith
+
+ theorem contDiff_radial (r L ε : ℝ) : ContDiff ℝ ∞ (radial (κ := κ) r L ε) :=
+   (contDiff_ballProfile r L ε).comp ((contDiff_id : ContDiff ℝ ∞ (id : Vector κ → Vector κ)).norm_sq ℝ)
+
+ theorem radial_eq_height {r L ε : ℝ} (hε : 0 < ε) (hL : ε ≤ L)
+    {z : Vector κ} (hz : z ∉ core L ε) : radial r L ε z = ballHeight r L ε := by
+   apply ballProfile_plateau r L hε
+   exact le_of_lt (lt_of_not_ge ((mem_core_iff (sub_nonneg.mpr hL) z).not.mp hz))
+
+ def pushed (e : OpenPartialHomeomorph (Vector κ) (Vector κ)) (r L ε : ℝ) (z : Vector κ) : ℝ :=
+   extendFunction e (fun w => radial r L ε w-ballHeight r L ε) z+ballHeight r L ε
+
+ theorem contDiff_pushed (e : OpenPartialHomeomorph (Vector κ) (Vector κ))
+    (he : ContDiffOn ℝ ∞ e e.source) {L : ℝ} (hs : e.source = domain L)
+    (hD : ∀ x ∈ e.source, (fderiv ℝ e x).IsInvertible)
+    {r ε : ℝ} (hε : 0 < ε) (hL : ε ≤ L) : ContDiff ℝ ∞ (pushed e r L ε) := by
+   apply (contDiff_extendFunction e he hD ((contDiff_radial r L ε).sub contDiff_const)
+     (core_compact L ε) _ _).add contDiff_const
+   · rw [hs]; exact core_subset_domain hε hL
+   · intro z hz
+     rw [radial_eq_height hε hL hz, sub_self]
+
+ theorem pushed_apply (e : OpenPartialHomeomorph (Vector κ) (Vector κ)) (r L ε : ℝ)
+    {x : Vector κ} (hx : x ∈ e.source) : pushed e r L ε (e x) = radial r L ε x := by
+   simp only [pushed, extendFunction_apply e _ hx, sub_add_cancel]
+
+ theorem pushed_plateau (e : OpenPartialHomeomorph (Vector κ) (Vector κ))
+    {r L ε : ℝ} (hε : 0 < ε) (hL : ε ≤ L) {x : Vector κ} (hx : x ∉ e '' core L ε) :
+    pushed e r L ε x = ballHeight r L ε := by
+   rw [pushed, extendFunction_zero e _ (core L ε) (fun z hz => by rw [radial_eq_height hε hL hz,sub_self]) hx,
+     zero_add]
+
+ theorem pushed_bounds (e : OpenPartialHomeomorph (Vector κ) (Vector κ))
+    {r L ε : ℝ} (hr : 0 ≤ r) (hε : 0 < ε) (x : Vector κ) :
+    0 ≤ pushed e r L ε x ∧ pushed e r L ε x ≤ ballHeight r L ε := by
+   rcases extendFunction_mem_range e (fun w => radial r L ε w-ballHeight r L ε) x with hz | ⟨w,_,hw⟩
+   · simp only [pushed,hz,zero_add,le_refl,and_true]
+     exact ballProfile_nonneg hr L hε (L-ε)
+   · simp only [pushed,hw,sub_add_cancel]
+     exact ⟨ballProfile_nonneg hr L hε _,ballProfile_le_height hr L hε _⟩
+
+ theorem zero_mem_domain {L : ℝ} (hL : 0 < L) : (0 : Vector κ) ∈ domain L := by
+   simpa only [domain, mem_ofPred_eq, norm_zero, zero_pow (by norm_num : (2 : ℕ) ≠ 0)] using hL
+
+ theorem radial_flat {r L ε : ℝ} (hε : 0 < ε) :
+    ∀ᶠ z in 𝓝 (0 : Vector κ), radial r L ε z = 0 := by
+   have hsmall : ∀ᶠ z in 𝓝 (0 : Vector κ), ‖z‖^2 < ε :=
+     (isOpen_domain ε).mem_nhds (zero_mem_domain hε)
+   filter_upwards [hsmall] with z hz
+   exact ballProfile_zero r L hε hz.le
+
+ theorem pushed_flat (e : OpenPartialHomeomorph (Vector κ) (Vector κ))
+    {r L ε : ℝ} (hs : e.source = domain L) (hε : 0 < ε) (hL : ε ≤ L) :
+    ∀ᶠ z in 𝓝 (e 0), pushed e r L ε z = 0 := by
+   have h0 : (0 : Vector κ) ∈ e.source := by
+     rw [hs]; exact zero_mem_domain (hε.trans_le hL)
+   apply eventually_of_comp e h0
+   filter_upwards [e.open_source.mem_nhds h0,radial_flat (r := r) (L := L) hε] with z hz hflat
+   rw [pushed_apply e r L ε hz]
+   exact hflat
+
+ theorem radial_slow {r L ε : ℝ} (hr : 0 ≤ r) (hrπ : r < Real.pi)
+    {x : ℝ → Vector κ}
+    (hx : ∀ t, HasDerivAt x (Complex.I • gradient (radial r L ε) (x t)) t)
+    (hp : x 1 = x 0) (t : ℝ) : x t = x 0 :=
+   slow_radial_constant (hasDerivAt_ballProfile r L ε) (ballSlope_nonneg hr L ε)
+     (fun s => (ballSlope_le hr L ε s).trans_lt hrπ) hx hp t
+
+end
+end PushedBall
+
+namespace ExteriorHamiltonian
+noncomputable section
+open Set Filter
+open scoped ContDiff Topology
+open FourierPolynomial DiagonalQuadratic HamiltonianODE HamiltonianGrowth SmoothProfiles
+variable {κ : Type u124} [Fintype κ]
+
+ def potential (d : κ → ℝ) (T δ : ℝ) (z : Vector κ) : ℝ := exterior δ (energy d z-T)
+ def full (H₀ : Vector κ → ℝ) (d : κ → ℝ) (T δ : ℝ) (z : Vector κ) : ℝ :=
+   H₀ z+potential d T δ z
+
+ theorem contDiff_potential (d : κ → ℝ) (T δ : ℝ) : ContDiff ℝ ∞ (potential d T δ) :=
+   (contDiff_exterior δ).comp ((contDiff_energy d).sub contDiff_const)
+
+ theorem contDiff_full {H₀ : Vector κ → ℝ} (hH₀ : ContDiff ℝ ∞ H₀) (d : κ → ℝ) (T δ : ℝ) :
+    ContDiff ℝ ∞ (full H₀ d T δ) := hH₀.add (contDiff_potential d T δ)
+
+ theorem gradient_potential (d : κ → ℝ) (T δ : ℝ) (z : Vector κ) :
+    gradient (potential d T δ) z = (2*exteriorSlope δ (energy d z-T)) • diag d z := by
+   change gradient (fun w : Vector κ => exterior δ (energy d w-T)) z = _
+   convert! gradient_comp_energy d z ((hasDerivAt_exterior δ (energy d z-T)).comp (energy d z)
+     ((hasDerivAt_id (energy d z)).sub_const T)) using 1
+   simp only [mul_one]
+
+ theorem gradient_sum {H G : Vector κ → ℝ} (hH : ContDiff ℝ ∞ H) (hG : ContDiff ℝ ∞ G)
+    (z : Vector κ) : gradient (fun w => H w+G w) z = gradient H z+gradient G z := by
+   apply HasGradientAt.gradient
+   rw [hasGradientAt_iff_hasFDerivAt]
+   have hh := ((hH.differentiable (by simp) z).hasGradientAt.hasFDerivAt).add
+     ((hG.differentiable (by simp) z).hasGradientAt.hasFDerivAt)
+   convert! hh using 1
+   ext w
+   change inner (𝕜 := ℝ) (gradient H z+gradient G z) w =
+     inner (𝕜 := ℝ) (gradient H z) w+inner (𝕜 := ℝ) (gradient G z) w
+   exact inner_add_left _ _ _
+
+ theorem gradient_full {H₀ : Vector κ → ℝ} (hH₀ : ContDiff ℝ ∞ H₀)
+    (d : κ → ℝ) (T δ : ℝ) (z : Vector κ) :
+    gradient (full H₀ d T δ) z = gradient H₀ z+(2*exteriorSlope δ (energy d z-T)) • diag d z := by
+   change gradient (fun w => H₀ w+potential d T δ w) z = _
+   rw [gradient_sum hH₀ (contDiff_potential d T δ), gradient_potential]
+
+ theorem gradient_at_max {H : Vector κ → ℝ} {m : ℝ} (hm : ∀ z, H z ≤ m)
+    {z : Vector κ} (hz : H z = m) : gradient H z = 0 := by
+   have he : IsLocalMax H z := Filter.Eventually.of_forall (fun w => by rw [hz]; exact hm w)
+   simp only [gradient,he.fderiv_eq_zero,map_zero]
+
+ theorem gradient_at_min {H : Vector κ → ℝ} {m : ℝ} (hm : ∀ z, m ≤ H z)
+    {z : Vector κ} (hz : H z = m) : gradient H z = 0 := by
+   have he : IsLocalMin H z := Filter.Eventually.of_forall (fun w => by rw [hz]; exact hm w)
+   simp only [gradient,he.fderiv_eq_zero,map_zero]
+
+ theorem potential_zero (d : κ → ℝ) (T : ℝ) {δ : ℝ} (hδ : 0 < δ) {z : Vector κ}
+    (hz : energy d z ≤ T) : potential d T δ z = 0 := exterior_zero hδ (sub_nonpos.mpr hz)
+
+ theorem potential_nonneg (d : κ → ℝ) (T : ℝ) {δ : ℝ} (hδ : 0 < δ) (z : Vector κ) :
+    0 ≤ potential d T δ z := exterior_nonneg hδ _
+
+ theorem compact_sublevel (d : κ → ℝ) {b C : ℝ} (hb : 0 < b) (hd : ∀ j, b ≤ d j) (hC : 0 ≤ C) :
+    IsCompact {z : Vector κ | energy d z ≤ C} := by
+   apply (isCompact_closedBall (0 : Vector κ) (Real.sqrt (C/b))).of_isClosed_subset
+     (isClosed_le (contDiff_energy d).continuous continuous_const)
+   intro z hz
+   simp only [Metric.mem_closedBall,dist_zero_right]
+   apply (Real.le_sqrt (norm_nonneg z) (div_nonneg hC hb.le)).mpr
+   apply (le_div_iff₀ hb).mpr
+   have he := energy_lower d b hd z
+   change energy d z ≤ C at hz
+   nlinarith
+
+ theorem bounded_remainder {H₀ : Vector κ → ℝ} (hH₀ : ContDiff ℝ ∞ H₀)
+    {m : ℝ} (hm : ∀ z, H₀ z ≤ m) {S : Set (Vector κ)} (hS : IsCompact S)
+    (he : ∀ z ∉ S, H₀ z = m) (d : κ → ℝ) {b T δ : ℝ}
+    (hb : 0 < b) (hd : ∀ j, b ≤ d j) (hδ : 0 < δ) (hT : 0 ≤ T+δ) :
+    ∃ B : ℝ, 0 ≤ B ∧ ∀ z,
+      ‖gradient (full H₀ d T δ) z-(2 : ℝ) • diag d z‖ ≤ B := by
+   apply bound_remainder_of_compact (contDiff_full hH₀ d T δ) d
+     (hS.union (compact_sublevel d hb hd hT))
+   intro z hz
+   have hzS : z ∉ S := fun h => hz (Or.inl h)
+   have hzQ : T+δ < energy d z := lt_of_not_ge (fun h => hz (Or.inr h))
+   rw [gradient_full hH₀, gradient_at_max hm (he z hzS),
+     exteriorSlope_one hδ (by linarith), mul_one, zero_add]
+
+ theorem full_flat {H₀ : Vector κ → ℝ} (d : κ → ℝ) {T δ : ℝ} (hδ : 0 < δ)
+    {p : Vector κ} (hf : ∀ᶠ z in 𝓝 p, H₀ z = 0) (hp : energy d p < T) :
+    ∀ᶠ z in 𝓝 p, full H₀ d T δ z = 0 := by
+   have hq : ∀ᶠ z in 𝓝 p, energy d z < T :=
+     (isOpen_lt (contDiff_energy d).continuous continuous_const).mem_nhds hp
+   filter_upwards [hf,hq] with z hz hze
+   simp only [full,hz,potential_zero d T hδ hze.le,add_zero]
+
+end
+end ExteriorHamiltonian
+
+namespace ExteriorHamiltonian
+noncomputable section
+open Set Filter MeasureTheory
+open scoped ContDiff Topology
+open FourierPolynomial DiagonalQuadratic HamiltonianODE SmoothProfiles HamiltonianPeriodic
+variable {κ : Type u125} [Fintype κ]
+
+ theorem action_nonpos {H : Vector κ → ℝ} {x : ℝ → Vector κ}
+    (h : ∀ t, inner (𝕜 := ℝ) (gradient H (x t)) (x t)/2-H (x t) ≤ 0) :
+    actionIntegral H x ≤ 0 := by
+   rw [actionIntegral,intervalIntegral.integral_of_le (by norm_num : (0:ℝ) ≤ 1)]
+   exact integral_nonpos h
+
+ theorem gradient_of_constant_curve {H : Vector κ → ℝ} {x : ℝ → Vector κ}
+    (hx : ∀ t, HasDerivAt x (Complex.I • gradient H (x t)) t)
+    (hc : ∀ t, x t = x 0) (t : ℝ) : gradient H (x t) = 0 := by
+   have he : x = fun _ => x 0 := funext hc
+   have hd0 : HasDerivAt x 0 t := by
+     convert! hasDerivAt_const t (x 0) using 1
+   have hd := (hx t).unique hd0
+   exact (smul_eq_zero.mp hd).resolve_left Complex.I_ne_zero
+
+ theorem action_nonpos_of_constant {H : Vector κ → ℝ} (hH0 : ∀ z, 0 ≤ H z)
+    {x : ℝ → Vector κ} (hx : ∀ t, HasDerivAt x (Complex.I • gradient H (x t)) t)
+    (hc : ∀ t, x t = x 0) : actionIntegral H x ≤ 0 := by
+   apply action_nonpos
+   intro t
+   rw [gradient_of_constant_curve hx hc t,inner_zero_left,zero_div,zero_sub]
+   exact neg_nonpos.mpr (hH0 (x t))
+
+ theorem high_orbit_nonpos (j₀ : κ) {H₀ : Vector κ → ℝ} (hH₀ : ContDiff ℝ ∞ H₀)
+    {m : ℝ} (hm : ∀ z, H₀ z ≤ m) (hH₀0 : ∀ z, 0 ≤ H₀ z)
+    (d : κ → ℝ) (hd : ∀ j, 0 ≤ d j) (hda : Real.pi < d j₀)
+    (hdb : ∀ j, j ≠ j₀ → d j < Real.pi) (hda2 : d j₀ < 2*Real.pi)
+    {T δ : ℝ} (hδ : 0 < δ) (hmag : Real.pi*(T+δ) < m*d j₀)
+    {x : ℝ → Vector κ}
+    (hx : ∀ t, HasDerivAt x (Complex.I • gradient (full H₀ d T δ) (x t)) t)
+    (hp : x 1 = x 0) (hplateau : ∀ t, H₀ (x t) = m) :
+    actionIntegral (full H₀ d T δ) x ≤ 0 := by
+   have he (t : ℝ) : gradient (full H₀ d T δ) (x t) =
+       (2*exteriorSlope δ (energy d (x t)-T)) • diag d (x t) := by
+     rw [gradient_full hH₀,gradient_at_max hm (hplateau t),zero_add]
+   have hode (t : ℝ) := (hx t).congr_deriv (congrArg (fun z : Vector κ => Complex.I • z) (he t))
+   have hQ (t : ℝ) := quadratic_constant d hode t 0
+   let v := exteriorSlope δ (energy d (x 0)-T)
+   have hv : 0 ≤ v := exteriorSlope_nonneg _ _
+   have hv1 : v ≤ 1 := exteriorSlope_le_one _ _
+   have hode' (t : ℝ) : HasDerivAt x (Complex.I • ((2*v) • diag d (x t))) t := by
+     simpa only [hQ t] using hode t
+   rcases diagonal_either j₀ d hv hv1 hd hdb hda2 hode' hp with heq | hc
+   · have ha0 : 0 < d j₀ := Real.pi_pos.trans hda
+     have hvlt : v < 1 := by nlinarith
+     have hQt : energy d (x 0) < T+δ := by
+       by_contra hn
+       have hz : v = 1 := exteriorSlope_one hδ (by linarith)
+       linarith
+     have hvT : v*(T+δ) < m := by
+       apply (mul_lt_mul_iff_left₀ ha0).mp
+       calc
+         (v*(T+δ))*d j₀ = Real.pi*(T+δ) := by rw [← heq]; ring
+         _ < m*d j₀ := hmag
+     apply action_nonpos
+     intro t
+     rw [he t,hQ t,real_inner_smul_left,← energy_eq_inner, hQ t]
+     have hg := potential_nonneg d T hδ (x t)
+     change (2*v)*energy d (x 0)/2-(H₀ (x t)+potential d T δ (x t)) ≤ 0
+     rw [hplateau t]
+     nlinarith [mul_le_mul_of_nonneg_left hQt.le hv]
+   · apply action_nonpos_of_constant _ hx hc
+     intro z
+     exact add_nonneg (hH₀0 z) (potential_nonneg d T hδ z)
+
+ theorem zero_vector_at_plateau {H₀ : Vector κ → ℝ} (hH₀ : ContDiff ℝ ∞ H₀)
+    {m : ℝ} (hm : ∀ z, H₀ z ≤ m) (d : κ → ℝ) {T δ : ℝ} (hδ : 0 < δ)
+    {U : Set (Vector κ)} (hU : ∀ z ∈ U, energy d z ≤ T)
+    (he : ∀ z ∉ U, H₀ z = m) {z : Vector κ} (hz : full H₀ d T δ z = m) :
+    gradient (full H₀ d T δ) z = 0 := by
+   have hp : potential d T δ z = 0 := by
+     by_cases hu : z ∈ U
+     · exact potential_zero d T hδ (hU z hu)
+     · have hh := he z hu
+       change H₀ z+potential d T δ z=m at hz
+       linarith
+   have h0 : H₀ z = m := by simpa only [full,hp,add_zero] using hz
+   change gradient (fun w => H₀ w+potential d T δ w) z = 0
+   rw [gradient_sum hH₀ (contDiff_potential d T δ),gradient_at_max hm h0,
+     gradient_at_min (potential_nonneg d T hδ) hp,add_zero]
+
+ theorem all_orbits_nonpos (j₀ : κ) {H₀ : Vector κ → ℝ} (hH₀ : ContDiff ℝ ∞ H₀)
+    {m : ℝ} (hm : ∀ z, H₀ z ≤ m) (hH₀0 : ∀ z, 0 ≤ H₀ z)
+    (d : κ → ℝ) (hd : ∀ j, 0 ≤ d j) (hda : Real.pi < d j₀)
+    (hdb : ∀ j, j ≠ j₀ → d j < Real.pi) (hda2 : d j₀ < 2*Real.pi)
+    {T δ : ℝ} (hδ : 0 < δ) (hmag : Real.pi*(T+δ) < m*d j₀)
+    {U : Set (Vector κ)} (hU : ∀ z ∈ U, energy d z ≤ T)
+    (he : ∀ z ∉ U, H₀ z = m)
+    (hslow : ∀ x : ℝ → Vector κ, (∀ t, x t ∈ U) → x 1 = x 0 →
+      (∀ t, HasDerivAt x (Complex.I • gradient (full H₀ d T δ) (x t)) t) → ∀ t, x t = x 0)
+    {x : ℝ → Vector κ}
+    (hx : ∀ t, HasDerivAt x (Complex.I • gradient (full H₀ d T δ) (x t)) t)
+    (hp : x 1 = x 0) : actionIntegral (full H₀ d T δ) x ≤ 0 := by
+   have hH0 (z : Vector κ) : 0 ≤ full H₀ d T δ z :=
+     add_nonneg (hH₀0 z) (potential_nonneg d T hδ z)
+   have hE (t : ℝ) := energy_constant (contDiff_full hH₀ d T δ) hx t 0
+   rcases lt_trichotomy (full H₀ d T δ (x 0)) m with hl | heq | hh
+   · have hxt (t : ℝ) : x t ∈ U := by
+       by_contra hn
+       have hz := he (x t) hn
+       have hg := potential_nonneg d T hδ (x t)
+       have ht : full H₀ d T δ (x t) < m := by rw [hE t]; exact hl
+       change H₀ (x t)+potential d T δ (x t) < m at ht
+       linarith
+     exact action_nonpos_of_constant hH0 hx (hslow x hxt hp hx)
+   · apply action_nonpos_of_constant hH0 hx
+     intro s
+     apply zero_derivative_constant _ s 0
+     intro t
+     have hz := zero_vector_at_plateau hH₀ hm d hδ hU he ((hE t).trans heq)
+     simpa only [hz,smul_zero] using hx t
+   · apply high_orbit_nonpos j₀ hH₀ hm hH₀0 d hd hda hdb hda2 hδ hmag hx hp
+     intro t
+     apply he
+     intro hu
+     have hpz := potential_zero d T hδ (hU (x t) hu)
+     have ht : m < full H₀ d T δ (x t) := by rw [hE t]; exact hh
+     simp only [full,hpz,add_zero] at ht
+     exact (not_lt_of_ge (hm (x t))) ht
+
+end
+end ExteriorHamiltonian
+
+namespace PushedBall
+noncomputable section
+open Set Filter
+open scoped ContDiff Topology
+open FourierPolynomial DiagonalQuadratic HamiltonianODE SmoothProfiles ExteriorHamiltonian
+variable {κ : Type u126} [Fintype κ]
+
+ theorem pushed_full_comp (e : OpenPartialHomeomorph (Vector κ) (Vector κ))
+    (r L ε : ℝ) (d : κ → ℝ) {T δ : ℝ} (hδ : 0 < δ)
+    (hQ : ∀ z ∈ e.target, energy d z ≤ T) {z : Vector κ} (hz : z ∈ e.source) :
+    full (pushed e r L ε) d T δ (e z) = radial r L ε z := by
+   rw [full,pushed_apply e r L ε hz,potential_zero d T hδ (hQ _ (e.map_source hz)),add_zero]
+
+ theorem pushed_full_slow (e : OpenPartialHomeomorph (Vector κ) (Vector κ))
+    (he : ContDiffOn ℝ ∞ e e.source) (hω : ∀ z ∈ e.source, ComplexSymplectic.Preserves (fderiv ℝ e z))
+    {r L ε : ℝ} (hs : e.source = domain L) (hr : 0 ≤ r) (hrπ : r < Real.pi)
+    (hε : 0 < ε) (hL : ε ≤ L) (d : κ → ℝ) {T δ : ℝ} (hδ : 0 < δ)
+    (hQ : ∀ z ∈ e.target, energy d z ≤ T)
+    {x : ℝ → Vector κ} (hxt : ∀ t, x t ∈ e.target) (hp : x 1 = x 0)
+    (hx : ∀ t, HasDerivAt x (Complex.I • gradient (full (pushed e r L ε) d T δ) (x t)) t)
+    (t : ℝ) : x t = x 0 := by
+   have hD (z) (hz : z ∈ e.source) := ComplexSymplectic.invertible (hω z hz)
+   have hH := contDiff_full (contDiff_pushed e he hs hD (r := r) hε hL) d T δ
+   have hy (s : ℝ) := ComplexSymplectic.inverse_curve e he hω hH (contDiff_radial r L ε)
+     (fun z hz => pushed_full_comp e r L ε d hδ hQ hz) hxt hx s
+   have hp' : e.symm (x 1) = e.symm (x 0) := congrArg e.symm hp
+   have hc := radial_slow hr hrπ hy hp' t
+   have hh := congrArg e hc
+   simpa only [e.right_inv (hxt t),e.right_inv (hxt 0)] using hh
+
+ theorem pushed_plateau_outside (e : OpenPartialHomeomorph (Vector κ) (Vector κ))
+    {r L ε : ℝ} (hs : e.source = domain L) (hε : 0 < ε) (hL : ε ≤ L)
+    {z : Vector κ} (hz : z ∉ e.target) : pushed e r L ε z = ballHeight r L ε := by
+   apply pushed_plateau e hε hL
+   rintro ⟨x,hx,rfl⟩
+   apply hz
+   apply e.map_source
+   rw [hs]
+   exact core_subset_domain hε hL hx
+
+end
+end PushedBall
+
+end NonsqueezingInline
+
+end OAI
diff --git a/lean/OAI/Geometry/PolarProducts/FiniteDescent.lean b/lean/OAI/Geometry/PolarProducts/FiniteDescent.lean
new file mode 100644
index 0000000000000000000000000000000000000000..84ccbf67296d13cfc4debcbbaef84c0ddb1df96d
--- /dev/null
+++ b/lean/OAI/Geometry/PolarProducts/FiniteDescent.lean
@@ -0,0 +1,329 @@
+import OAI.Geometry.PolarProducts.SmoothDegree
+
+namespace OAI
+
+universe u80 u81 u82 u83
+
+section NonsqueezingInline
+section
+open Set Filter Function MeasureTheory
+open scoped Topology ContDiff
+
+namespace SmoothLinking
+noncomputable section
+open Set Filter Function
+open scoped Topology ContDiff
+variable {E : Type u80} {F : Type u81} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  [NormedAddCommGroup F] [InnerProductSpace ℝ F]
+
+def conePoint (e : F) (z : E × ℝ) : E × F := (z.1, z.2 • e)
+
+def degreeMap (η : E × F → E × F) (e : F) (ρ : ℝ) (z : E × ℝ) : E × ℝ :=
+  if 0 < z.2+ρ then
+    ((η (z.1, (z.2+ρ) • e)).1, (‖(η (z.1, (z.2+ρ) • e)).2‖^2-ρ^2)/(z.2+2*ρ))
+  else z
+
+omit [NormedAddCommGroup E] [NormedSpace ℝ E] in
+theorem degreeMap_eq_self {η : E × F → E × F} {e : F} (he : ‖e‖ = 1)
+    {ρ : ℝ} (hρ : 0 < ρ) {z : E × ℝ} (hz : 0 < z.2+ρ)
+    (hη : η (z.1, (z.2+ρ) • e) = (z.1, (z.2+ρ) • e)) :
+    degreeMap η e ρ z = z := by
+  simp only [degreeMap, ite_eq_left hz, hη, norm_smul, he, mul_one,
+    Real.norm_eq_abs, sq_abs]
+  apply Prod.ext
+  · rfl
+  change ((z.2+ρ)^2-ρ^2)/(z.2+2*ρ) = z.2
+  apply (div_eq_iff (by linarith : z.2+2*ρ ≠ 0)).mpr
+  ring
+
+theorem contDiffAt_degreeMap_pos {η : E × F → E × F} (hη : ContDiff ℝ ∞ η)
+    (e : F) {ρ : ℝ} (hρ : 0 < ρ) {z : E × ℝ} (hz : 0 < z.2+ρ) :
+    ContDiffAt ℝ ∞ (degreeMap η e ρ) z := by
+  have hinput : ContDiff ℝ ∞ (fun w : E × ℝ => (w.1, (w.2+ρ) • e)) :=
+    contDiff_fst.prodMk ((contDiff_snd.add contDiff_const).smul contDiff_const)
+  have hout := hη.comp hinput
+  have hn : z.2+2*ρ ≠ 0 := by linarith
+  have hd : ContDiffAt ℝ ∞ (fun w : E × ℝ =>
+      ((η (w.1, (w.2+ρ) • e)).1, (‖(η (w.1, (w.2+ρ) • e)).2‖^2-ρ^2)/(w.2+2*ρ))) z := by
+    exact (contDiff_fst.comp hout).contDiffAt.prodMk
+      (((contDiff_norm_sq ℝ).comp (contDiff_snd.comp hout)).contDiffAt.sub contDiffAt_const |>.div
+        (contDiffAt_snd.add contDiffAt_const) hn)
+  apply hd.congr_of_eventuallyEq
+  filter_upwards [(isOpen_lt continuous_const (continuous_snd.add continuous_const)).mem_nhds hz] with w hw
+  change 0 < w.2+ρ at hw
+  simp only [degreeMap, ite_eq_left hw]
+
+theorem contDiff_degreeMap {η : E × F → E × F} (hη : ContDiff ℝ ∞ η)
+    {e : F} (he : ‖e‖ = 1) {ρ ε : ℝ} (hρ : 0 < ρ) (hε : 0 < ε)
+    (hfixed : ∀ (y : E) (s : ℝ), 0 < s → s < ε → η (y, s • e) = (y, s • e)) :
+    ContDiff ℝ ∞ (degreeMap η e ρ) := by
+  apply contDiff_iff_contDiffAt.mpr
+  intro z
+  by_cases hz : 0 < z.2+ρ
+  · exact contDiffAt_degreeMap_pos hη e hρ hz
+  have hz' : z.2+ρ < ε := lt_of_le_of_lt (le_of_not_gt hz) hε
+  apply (contDiffAt_id : ContDiffAt ℝ ∞ (id : E × ℝ → E × ℝ) z).congr_of_eventuallyEq
+  filter_upwards [(isOpen_lt (continuous_snd.add continuous_const) continuous_const).mem_nhds hz'] with w hw
+  by_cases hp : 0 < w.2+ρ
+  · exact degreeMap_eq_self he hρ hp (hfixed w.1 (w.2+ρ) hp hw)
+  · simp only [degreeMap, ite_eq_right hp, id_eq]
+
+omit [NormedSpace ℝ E] in
+theorem norm_translate_le (z : E × ℝ) {ρ : ℝ} (hρ : 0 ≤ ρ) :
+    ‖z‖ ≤ ‖(z.1, z.2+ρ)‖ + ρ := by
+  have he : z = (z.1, z.2+ρ) - (0,ρ) := by ext <;> simp
+  calc
+    ‖z‖ = ‖(z.1, z.2+ρ) - (0,ρ)‖ := congrArg norm he
+    _ ≤ ‖(z.1, z.2+ρ)‖+‖((0,ρ) : E × ℝ)‖ := norm_sub_le _ _
+    _ = _ := by simp [Prod.norm_def, abs_of_nonneg hρ, hρ]
+
+omit [NormedSpace ℝ E] in
+theorem degreeMap_identity_outside {η : E × F → E × F} {e : F} (he : ‖e‖ = 1)
+    {ρ R : ℝ} (hρ : 0 < ρ)
+    (hfixed : ∀ (y : E) (s : ℝ), 0 < s → R ≤ ‖(y,s)‖ → η (y, s • e) = (y, s • e))
+    (z : E × ℝ) (hz : R+ρ ≤ ‖z‖) : degreeMap η e ρ z = z := by
+  by_cases hp : 0 < z.2+ρ
+  · apply degreeMap_eq_self he hρ hp
+    apply hfixed _ _ hp
+    have hn := norm_translate_le z hρ.le
+    linarith
+  · simp only [degreeMap, ite_eq_right hp]
+
+theorem sphere_linking [FiniteDimensional ℝ E]
+    {η : E × F → E × F} (hη : ContDiff ℝ ∞ η) {e : F} (he : ‖e‖ = 1)
+    {ρ ε R : ℝ} (hρ : 0 < ρ) (hε : 0 < ε) (hR : ρ < R)
+    (hnear : ∀ (y : E) (s : ℝ), 0 < s → s < ε → η (y, s • e) = (y, s • e))
+    (hfar : ∀ (y : E) (s : ℝ), 0 < s → R ≤ ‖(y,s)‖ → η (y, s • e) = (y, s • e)) :
+    ∃ (y : E) (s : ℝ), 0 < s ∧ ‖(y,s)‖ < R ∧ (η (y, s • e)).1 = 0 ∧ ‖(η (y, s • e)).2‖ = ρ := by
+  obtain ⟨z, hz⟩ := SmoothDegree.exists_zero (contDiff_degreeMap hη he hρ hε hnear)
+    (show 0 < R+ρ by linarith) (degreeMap_identity_outside he hρ hfar)
+  have hp : 0 < z.2+ρ := by
+    by_contra h
+    have heq : z = 0 := by simpa only [degreeMap, ite_eq_right h] using hz
+    simp only [heq, Prod.snd_zero, zero_add] at h
+    exact h hρ
+  have hfst : (η (z.1, (z.2+ρ) • e)).1 = 0 := by
+    simpa only [degreeMap, ite_eq_left hp, Prod.fst_zero, Prod.fst] using congrArg Prod.fst hz
+  have hsnd : ‖(η (z.1, (z.2+ρ) • e)).2‖ = ρ := by
+    have h := congrArg Prod.snd hz
+    simp only [degreeMap, ite_eq_left hp, Prod.snd_zero] at h
+    have hd : z.2+2*ρ ≠ 0 := by linarith
+    have hsq : ‖(η (z.1, (z.2+ρ) • e)).2‖^2 = ρ^2 :=
+      sub_eq_zero.mp ((div_eq_zero_iff).mp h |>.resolve_right hd)
+    nlinarith [norm_nonneg (η (z.1, (z.2+ρ) • e)).2]
+  refine ⟨z.1, z.2+ρ, hp, ?_, hfst, hsnd⟩
+  by_contra! hn
+  have hid := hfar z.1 (z.2+ρ) hp hn
+  rw [hid] at hfst hsnd
+  simp only [norm_smul, he, mul_one, Real.norm_eq_abs,
+    abs_of_pos hp] at hfst hsnd
+  rw [hfst, hsnd] at hn
+  have : ‖((0 : E),ρ)‖ = ρ := by simp [Prod.norm_def, abs_of_pos hρ, hρ.le]
+  rw [this] at hn
+  exact hR.not_ge hn
+
+end
+end SmoothLinking
+
+open Set Filter Function MeasureTheory Metric
+open scoped Topology ContDiff NNReal
+
+namespace FlowDescent
+noncomputable section
+open Set Filter Function MeasureTheory Metric
+open scoped Topology ContDiff NNReal
+variable {E : Type u82} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
+
+ theorem curve_displacement {V : E → E} {K : ℝ≥0} (hV : LipschitzWith K V)
+    {h : ℝ} (hh : ‖h‖₊ * K < 1) (h0 : 0 ≤ h) {L : ℝ}
+    (hL : ∀ x, ‖V x‖ ≤ L) (x : E) {t : ℝ} (ht : t ∈ Icc (0 : ℝ) 1) :
+    ‖SmoothODE.curve hV h hh x t - x‖ ≤ h*L*t := by
+  have hb (s : ℝ) (_hs : s ∈ Icc (0 : ℝ) 1) :
+      ‖h • V (SmoothODE.curve hV h hh x s)‖ ≤ h*L := by
+    rw [norm_smul, Real.norm_eq_abs, abs_of_nonneg h0]
+    exact mul_le_mul_of_nonneg_left (hL _) h0
+  have hm := Convex.norm_image_sub_le_of_norm_hasDerivWithin_le
+    (fun s hs => (SmoothODE.hasDerivAt_curve hV h hh x ⟨s,hs⟩).hasDerivWithinAt)
+    hb (convex_Icc (0 : ℝ) 1) (show (0 : ℝ) ∈ Icc (0 : ℝ) 1 by simp) ht
+  simpa only [SmoothODE.curve_zero, sub_zero, Real.norm_eq_abs, abs_of_nonneg ht.1] using hm
+
+ theorem curve_energy_deriv {V : E → E} {K : ℝ≥0} (hV : LipschitzWith K V)
+    {h : ℝ} (hh : ‖h‖₊ * K < 1) {Φ : E → ℝ} (hΦ : Differentiable ℝ Φ)
+    (x : E) {t : ℝ} (ht : t ∈ Icc (0 : ℝ) 1) :
+    HasDerivAt (fun s => Φ (SmoothODE.curve hV h hh x s))
+      (h * fderiv ℝ Φ (SmoothODE.curve hV h hh x t) (V (SmoothODE.curve hV h hh x t))) t := by
+  have hd := (hΦ (SmoothODE.curve hV h hh x t)).hasFDerivAt.comp_hasDerivAt t
+    (SmoothODE.hasDerivAt_curve hV h hh x ⟨t,ht⟩)
+  simpa only [Function.comp_def, map_smul, smul_eq_mul] using hd
+
+ theorem curve_energy_antitone {V : E → E} {K : ℝ≥0} (hV : LipschitzWith K V)
+    {h : ℝ} (hh : ‖h‖₊ * K < 1) (h0 : 0 ≤ h) {Φ : E → ℝ} (hΦ : Differentiable ℝ Φ)
+    (hdec : ∀ x, fderiv ℝ Φ x (V x) ≤ 0) (x : E) :
+    AntitoneOn (fun t => Φ (SmoothODE.curve hV h hh x t)) (Icc (0 : ℝ) 1) := by
+  apply antitoneOn_of_hasDerivWithinAt_nonpos (convex_Icc (0 : ℝ) 1)
+    (hΦ.continuous.comp (SmoothODE.continuous_curve hV h hh x)).continuousOn
+    (fun t ht => (curve_energy_deriv hV hh hΦ x (interior_subset ht)).hasDerivWithinAt)
+  intro t _
+  exact mul_nonpos_of_nonneg_of_nonpos h0 (hdec _)
+
+ theorem step_energy_le {V : E → E} {K : ℝ≥0} (hV : LipschitzWith K V)
+    {h : ℝ} (hh : ‖h‖₊ * K < 1) (h0 : 0 ≤ h) {Φ : E → ℝ} (hΦ : Differentiable ℝ Φ)
+    (hdec : ∀ x, fderiv ℝ Φ x (V x) ≤ 0) (x : E) :
+    Φ (SmoothODE.step hV h hh x) ≤ Φ x := by
+  simpa only [SmoothODE.curve_one, SmoothODE.curve_zero] using
+    (curve_energy_antitone hV hh h0 hΦ hdec x (by simp) (by simp) zero_le_one)
+
+ theorem step_energy_drop {V : E → E} {K : ℝ≥0} (hV : LipschitzWith K V)
+    {h : ℝ} (hh : ‖h‖₊ * K < 1) (h0 : 0 ≤ h) {Φ : E → ℝ} (hΦ : Differentiable ℝ Φ)
+    (hdec : ∀ x, fderiv ℝ Φ x (V x) ≤ 0) {L M a b : ℝ}
+    (hL : ∀ x, ‖V x‖ ≤ L)
+    (hrate : ∀ x, ‖x‖ ≤ M → a ≤ Φ x → Φ x ≤ b → fderiv ℝ Φ x (V x) ≤ -1)
+    (x : E) (hx : ‖x‖+h*L ≤ M) (hxb : Φ x ≤ b) :
+    Φ (SmoothODE.step hV h hh x) ≤ max a (Φ x-h) := by
+  by_cases hxa : Φ (SmoothODE.step hV h hh x) ≤ a
+  · exact hxa.trans (le_max_left _ _)
+  have ha := le_of_lt (lt_of_not_ge hxa)
+  have ham := curve_energy_antitone hV hh h0 hΦ hdec x
+  have hb (t : ℝ) (ht : t ∈ Icc (0 : ℝ) 1) :
+      fderiv ℝ Φ (SmoothODE.curve hV h hh x t) (V (SmoothODE.curve hV h hh x t)) ≤ -1 := by
+    apply hrate
+    · have hdn := curve_displacement hV hh h0 hL x ht
+      have hLn : 0 ≤ L := le_trans (norm_nonneg (V x)) (hL x)
+      have hn := norm_le_norm_sub_add (SmoothODE.curve hV h hh x t) x
+      have htime := mul_le_mul_of_nonneg_left ht.2 (mul_nonneg h0 hLn)
+      linarith
+    · exact ha.trans (by simpa only [SmoothODE.curve_one] using ham ht (by simp) ht.2)
+    · exact (show Φ (SmoothODE.curve hV h hh x t) ≤ Φ x from by
+        simpa only [SmoothODE.curve_zero] using ham (by simp) ht ht.1).trans hxb
+  have hm : AntitoneOn (fun t => Φ (SmoothODE.curve hV h hh x t)+h*t) (Icc (0 : ℝ) 1) := by
+    apply antitoneOn_of_hasDerivWithinAt_nonpos (convex_Icc (0 : ℝ) 1)
+      (((hΦ.continuous.comp (SmoothODE.continuous_curve hV h hh x)).add
+        (continuous_const.mul continuous_id)).continuousOn)
+      (fun t ht => ((curve_energy_deriv hV hh hΦ x (interior_subset ht)).add
+        ((hasDerivAt_id t).const_mul h)).hasDerivWithinAt)
+    intro t ht
+    have := mul_le_mul_of_nonneg_left (hb t (interior_subset ht)) h0
+    linarith
+  have hend := hm (by simp) (by simp) zero_le_one
+  simp only [SmoothODE.curve_one, SmoothODE.curve_zero, mul_one, mul_zero, add_zero] at hend
+  exact (by linarith : Φ (SmoothODE.step hV h hh x) ≤ Φ x-h).trans (le_max_right _ _)
+
+end
+end FlowDescent
+
+namespace FlowDescent
+noncomputable section
+open Set Filter Function MeasureTheory Metric
+open scoped Topology ContDiff NNReal
+variable {E : Type u83} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
+
+ omit [NormedSpace ℝ E] [CompleteSpace E] in
+ theorem iterate_norm_le {f : E → E} {D : ℝ} (hD : ∀ x, ‖f x-x‖ ≤ D) (j : ℕ) (x : E) :
+    ‖f^[j] x‖ ≤ ‖x‖+j*D := by
+  induction j with
+  | zero => simp
+  | succ j ih =>
+    rw [Function.iterate_succ_apply']
+    have hn := norm_le_norm_sub_add (f (f^[j] x)) (f^[j] x)
+    have hd := hD (f^[j] x)
+    push_cast
+    linarith
+
+ omit [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] in
+ theorem iterate_energy_le {f : E → E} {Φ : E → ℝ} (hf : ∀ x, Φ (f x) ≤ Φ x)
+    (j : ℕ) (x : E) : Φ (f^[j] x) ≤ Φ x := by
+  induction j with
+  | zero => rfl
+  | succ j ih =>
+    rw [Function.iterate_succ_apply']
+    exact (hf (f^[j] x)).trans ih
+
+ omit [NormedSpace ℝ E] [CompleteSpace E] in
+ theorem iterate_energy_drop {f : E → E} {Φ : E → ℝ} {h D M a b : ℝ} (h0 : 0 ≤ h)
+    (hD0 : 0 ≤ D) (hD : ∀ x, ‖f x-x‖ ≤ D) (hdec : ∀ x, Φ (f x) ≤ Φ x)
+    (hdrop : ∀ x, ‖x‖+D ≤ M → Φ x ≤ b → Φ (f x) ≤ max a (Φ x-h))
+    (N : ℕ) (x : E) (hx : ‖x‖+N*D ≤ M) (hb : Φ x ≤ b) :
+    Φ (f^[N] x) ≤ max a (Φ x-N*h) := by
+  suffices hs : ∀ j, j ≤ N → Φ (f^[j] x) ≤ max a (Φ x-j*h) by exact hs N le_rfl
+  intro j
+  induction j with
+  | zero => intro _; simp
+  | succ j ih =>
+    intro hj
+    have hij := ih (Nat.le_trans (Nat.le_succ j) hj)
+    have hn := iterate_norm_le hD j x
+    have hb' := (iterate_energy_le hdec j x).trans hb
+    have hjR : (j : ℝ)+1 ≤ N := by exact_mod_cast hj
+    have hx' : ‖f^[j] x‖+D ≤ M := by
+      have := mul_le_mul_of_nonneg_right hjR hD0
+      nlinarith
+    have hd := hdrop (f^[j] x) hx' hb'
+    rw [Function.iterate_succ_apply']
+    apply hd.trans
+    apply max_le (le_max_left _ _)
+    apply (sub_le_sub_right hij h).trans
+    rw [← max_sub_sub_right]
+    apply max_le
+    · exact (sub_le_self a h0).trans (le_max_left _ _)
+    · apply le_trans _ (le_max_right _ _)
+      push_cast
+      linarith
+
+ omit [CompleteSpace E] in
+ theorem contDiff_iterate {f : E → E} (hf : ContDiff ℝ ∞ f) (j : ℕ) :
+    ContDiff ℝ ∞ (f^[j]) := by
+  induction j with
+  | zero => exact contDiff_id
+  | succ j ih =>
+    convert! hf.comp ih using 1
+    ext x
+    exact Function.iterate_succ_apply' f j x
+
+ theorem exists_descent_map [ProperSpace E] {V : E → E} {K : ℝ≥0}
+    (hV : LipschitzWith K V) (hVs : ContDiff ℝ ∞ V) {Φ : E → ℝ}
+    (hΦ : Differentiable ℝ Φ) (hdec : ∀ x, fderiv ℝ Φ x (V x) ≤ 0)
+    {L M a b B T : ℝ} (hT : 0 < T) (hba : b-T ≤ a)
+    (hL : ∀ x, ‖V x‖ ≤ L) (hM : B+T*L ≤ M)
+    (hrate : ∀ x, ‖x‖ ≤ M → a ≤ Φ x → Φ x ≤ b → fderiv ℝ Φ x (V x) ≤ -1) :
+    ∃ η : E → E, ContDiff ℝ ∞ η ∧ (∀ x, Φ (η x) ≤ Φ x) ∧
+      (∀ x, V x = 0 → η x = x) ∧
+      ∀ x, ‖x‖ ≤ B → Φ x ≤ b → Φ (η x) ≤ a := by
+  obtain ⟨N, hN⟩ := exists_nat_gt (max (T*(K : ℝ)) 0)
+  have hN0 : (0 : ℝ) < N := (le_max_right _ _).trans_lt hN
+  have hNne : (N : ℝ) ≠ 0 := hN0.ne'
+  let h := T/N
+  have h0 : 0 ≤ h := (div_pos hT hN0).le
+  have hh : ‖h‖₊ * K < 1 := by
+    change ‖h‖*(K : ℝ) < (1 : ℝ)
+    rw [Real.norm_eq_abs, abs_of_nonneg h0]
+    dsimp [h]
+    rw [div_mul_eq_mul_div]
+    exact (div_lt_one hN0).mpr ((le_max_left _ _).trans_lt hN)
+  have hNh : (N : ℝ)*h = T := by dsimp [h]; field_simp
+  let f := SmoothODE.step hV h hh
+  have hf : ContDiff ℝ ∞ f := SmoothODE.contDiff_step hV hVs h hh
+  have hfd (x : E) : Φ (f x) ≤ Φ x := step_energy_le hV hh h0 hΦ hdec x
+  have hfD (x : E) : ‖f x-x‖ ≤ h*L := by
+    simpa only [SmoothODE.curve_one, mul_one] using
+      (curve_displacement hV hh h0 hL x (t := 1) (by simp))
+  have hL0 : 0 ≤ L := (norm_nonneg (V 0)).trans (hL 0)
+  refine ⟨f^[N], contDiff_iterate hf N, iterate_energy_le hfd N, ?_, ?_⟩
+  · intro x hx
+    have hfix : f x = x := by
+      simpa only [SmoothODE.curve_one] using
+        (SmoothODE.curve_of_eq_zero hV h hh x hx (by simp : (1 : ℝ) ∈ Icc (0 : ℝ) 1))
+    exact Function.iterate_fixed hfix N
+  · intro x hx hb
+    have hbound : ‖x‖+(N : ℝ)*(h*L) ≤ M := by nlinarith [mul_assoc (N : ℝ) h L]
+    have hd := iterate_energy_drop h0 (mul_nonneg h0 hL0) hfD hfd
+      (fun y hy hby => step_energy_drop hV hh h0 hΦ hdec hL hrate y hy hby) N x hbound hb
+    rw [hNh] at hd
+    exact hd.trans (max_le le_rfl (by linarith))
+
+end
+end FlowDescent
+
+end
+end NonsqueezingInline
+
+end OAI
diff --git a/lean/OAI/Geometry/PolarProducts/FourierIntegrals.lean b/lean/OAI/Geometry/PolarProducts/FourierIntegrals.lean
new file mode 100644
index 0000000000000000000000000000000000000000..1da217f97c636736243a00d4e98acf291164b9d4
--- /dev/null
+++ b/lean/OAI/Geometry/PolarProducts/FourierIntegrals.lean
@@ -0,0 +1,269 @@
+import OAI.Geometry.PolarProducts.GradientLinking
+
+namespace OAI
+
+universe u89 u90 u91 u92 u93 u94 u95 u96 u97 u98 u99
+
+section NonsqueezingInline
+
+namespace FourierPolynomial
+noncomputable section
+open MeasureTheory AddCircle Finset
+open scoped ComplexConjugate
+
+local instance : Fact (0 < (1 : ℝ)) := ⟨zero_lt_one⟩
+abbrev Time := AddCircle (1 : ℝ)
+abbrev μ : Measure Time := AddCircle.haarAddCircle
+
+def eval {ι : Type u89} [Fintype ι] (k : ι → ℤ) (a : ι → ℂ) (t : Time) : ℂ :=
+  ∑ i, a i * fourier (k i) t
+
+ theorem continuous_eval {ι : Type u90} [Fintype ι] (k : ι → ℤ) (a : ι → ℂ) :
+    Continuous (eval k a) := by
+  exact continuous_finsetSum _ (fun i _ => continuous_const.mul (fourier (k i)).continuous)
+
+ theorem integral_fourier (k : ℤ) :
+    (∫ t : Time, fourier k t ∂μ) = if k = 0 then 1 else 0 := by
+  have h := congrFun (fourierCoeff_fourier (T := (1 : ℝ)) k) 0
+  simpa only [fourierCoeff, neg_zero, fourier_zero, one_smul, Pi.single_apply,
+    eq_comm (a := (0 : ℤ))] using h
+
+ theorem integrable_continuous {G : Type u91} [NormedAddCommGroup G]
+    {f : Time → G} (hf : Continuous f) : Integrable f μ := by
+  simpa only [integrableOn_univ] using hf.continuousOn.integrableOn_compact isCompact_univ
+
+ theorem integral_conj_fourier_mul (k l : ℤ) :
+    (∫ t : Time, conj (fourier k t) * fourier l t ∂μ) = if k = l then 1 else 0 := by
+  simp_rw [← fourier_neg, ← fourier_add]
+  rw [integral_fourier]
+  simp only [neg_add_eq_zero]
+
+ theorem integral_conj_eval_mul {ι : Type u92} {κ : Type u93} [Fintype ι] [Fintype κ]
+    (k : ι → ℤ) (l : κ → ℤ) (a : ι → ℂ) (b : κ → ℂ) :
+    (∫ t : Time, conj (eval k a t) * eval l b t ∂μ) =
+      ∑ i, ∑ j, if k i = l j then conj (a i) * b j else 0 := by
+  classical
+  have he (t : Time) : conj (eval k a t) * eval l b t =
+      ∑ i, ∑ j, conj (a i)*conj (fourier (k i) t)*(b j*fourier (l j) t) := by
+    change conj (∑ i, a i*fourier (k i) t) * eval l b t = _
+    rw [map_sum, Finset.sum_mul]
+    simp only [map_mul, eval, Finset.mul_sum]
+  simp_rw [he]
+  rw [integral_finsetSum]
+  · apply Finset.sum_congr rfl
+    intro i _
+    rw [integral_finsetSum]
+    · apply Finset.sum_congr rfl
+      intro j _
+      simp_rw [show ∀ t : Time, conj (a i) * conj (fourier (k i) t) *
+          (b j * fourier (l j) t) =
+          (conj (a i)*b j) * (conj (fourier (k i) t)*fourier (l j) t) by intro t; ring]
+      rw [integral_const_mul, integral_conj_fourier_mul]
+      split_ifs <;> simp
+    · intro j _
+      exact integrable_continuous ((continuous_const.mul ((fourier (k i)).continuous.star)).mul
+        (continuous_const.mul (fourier (l j)).continuous))
+  · intro i _
+    apply integrable_finsetSum
+    intro j _
+    exact integrable_continuous ((continuous_const.mul ((fourier (k i)).continuous.star)).mul
+      (continuous_const.mul (fourier (l j)).continuous))
+
+ theorem norm_sq_re (z : ℂ) : ‖z‖^2 = (conj z*z).re := by
+  simp [Complex.sq_norm, Complex.normSq_apply, Complex.mul_re]
+
+ theorem integral_norm_sq {ι : Type u94} [Fintype ι] (k : ι → ℤ) (hk : Function.Injective k)
+    (a : ι → ℂ) : (∫ t : Time, ‖eval k a t‖^2 ∂μ) = ∑ i, ‖a i‖^2 := by
+  classical
+  simp_rw [norm_sq_re]
+  have hI : Integrable (fun t : Time => conj (eval k a t) * eval k a t) μ :=
+    integrable_continuous ((continuous_eval k a).star.mul (continuous_eval k a))
+  change (∫ t : Time, RCLike.re (conj (eval k a t)*eval k a t) ∂μ) = _
+  rw [integral_re hI, integral_conj_eval_mul]
+  simp only [hk.eq_iff]
+  simp
+
+ theorem coe_integral_norm_fourth {ι : Type u95} [Fintype ι] (k : ι → ℤ) (a : ι → ℂ) :
+    (((∫ t : Time, ‖eval k a t‖^4 ∂μ) : ℝ) : ℂ) =
+      ∑ p : ι × ι, ∑ q : ι × ι,
+        if k p.1+k p.2 = k q.1+k q.2 then conj (a p.1*a p.2)*(a q.1*a q.2) else 0 := by
+  let k' : ι × ι → ℤ := fun p => k p.1+k p.2
+  let a' : ι × ι → ℂ := fun p => a p.1*a p.2
+  have heval (t : Time) : eval k' a' t = eval k a t * eval k a t := by
+    simp only [eval, k', a', Fintype.sum_prod_type, fourier_add, Finset.sum_mul, Finset.mul_sum]
+    apply Finset.sum_congr rfl
+    intro i _
+    apply Finset.sum_congr rfl
+    intro j _
+    ring
+  have hintegrand (t : Time) : (↑(‖eval k a t‖^4) : ℂ) =
+      conj (eval k' a' t)*eval k' a' t := by
+    rw [heval, map_mul, mul_mul_mul_comm, ← Complex.normSq_eq_conj_mul_self,
+      ← Complex.sq_norm]
+    push_cast
+    ring
+  rw [← integral_complex_ofReal]
+  simp_rw [hintegrand]
+  exact integral_conj_eval_mul k' k' a' a'
+
+end
+end FourierPolynomial
+
+namespace WeightedSchur
+noncomputable section
+open Finset
+
+ theorem two_mul_le_weighted {w v : ℝ} (hw : 0 < w) (hv : 0 < v) (x y : ℝ) :
+    2*x*y ≤ w*x^2/v + v*y^2/w := by
+  rw [div_add_div _ _ hv.ne' hw.ne', le_div_iff₀ (mul_pos hv hw)]
+  nlinarith [sq_nonneg (w*x-v*y)]
+
+ theorem sum_le {ι : Type u96} [Fintype ι] (r : ι → ι → Prop) [DecidableRel r]
+    (hr : ∀ {i j}, r i j → r j i) (w : ι → ℝ) (hw : ∀ i, 0 < w i) (x : ι → ℝ)
+    {C : ℝ} (hrow : ∀ i, (∑ j, if r i j then 1/w j else 0) ≤ C) :
+    (∑ i, ∑ j, if r i j then x i*x j else 0) ≤ C * ∑ i, w i*x i^2 := by
+  have hp (i j : ι) : 2*(if r i j then x i*x j else 0) ≤
+      (if r i j then w i*x i^2/w j else 0) +
+      (if r j i then w j*x j^2/w i else 0) := by
+    by_cases hij : r i j
+    · simp only [ite_eq_left hij, ite_eq_left (hr hij)]
+      simpa only [mul_assoc] using two_mul_le_weighted (hw i) (hw j) (x i) (x j)
+    · have hji : ¬r j i := fun h => hij (hr h)
+      simp only [ite_eq_right hij, ite_eq_right hji, mul_zero, add_zero, le_refl]
+  have hs := Finset.sum_le_sum (fun i (_ : i ∈ (univ : Finset ι)) =>
+    Finset.sum_le_sum (fun j (_ : j ∈ (univ : Finset ι)) => hp i j))
+  simp only [Finset.sum_add_distrib, ← Finset.mul_sum] at hs
+  have hcomm : (∑ i : ι, ∑ j : ι, if r j i then w j*x j^2/w i else 0) =
+      ∑ i : ι, ∑ j : ι, if r i j then w i*x i^2/w j else 0 := Finset.sum_comm
+  rw [hcomm] at hs
+  have hh : (∑ i : ι, ∑ j : ι, if r i j then w i*x i^2/w j else 0) ≤
+      C * ∑ i, w i*x i^2 := by
+    calc
+      _ = ∑ i, w i*x i^2 * (∑ j, if r i j then 1/w j else 0) := by
+        apply Finset.sum_congr rfl
+        intro i _
+        rw [Finset.mul_sum]
+        apply Finset.sum_congr rfl
+        intro j _
+        split_ifs <;> ring
+      _ ≤ ∑ i, w i*x i^2*C := Finset.sum_le_sum (fun i _ =>
+        mul_le_mul_of_nonneg_left (hrow i) (mul_nonneg (hw i).le (sq_nonneg _)))
+      _ = _ := by rw [← Finset.sum_mul]; ring
+  linarith
+
+ theorem reciprocal_pair_rows {ι : Type u97} [Fintype ι] (k : ι → ℕ)
+    (hk : Function.Injective k) (hk0 : ∀ i, 0 < k i) (p : ι × ι) :
+    (∑ q : ι × ι, if k p.1+k p.2 = k q.1+k q.2 then
+      1 / ((k q.1 : ℝ)*(k q.2 : ℝ)) else 0) ≤ 1 := by
+  classical
+  let m := k p.1+k p.2
+  have hm : 2 ≤ m := by dsimp [m]; have := hk0 p.1; have := hk0 p.2; omega
+  let Q := univ.filter (fun q : ι × ι => k q.1+k q.2 = m)
+  have hQ (q : ι × ι) : q ∈ Q ↔ k q.1+k q.2 = m := by simp [Q]
+  have hmaps : Set.MapsTo (fun q : ι × ι => k q.1) (↑Q : Set (ι × ι))
+      (↑(Icc 1 (m-1)) : Set ℕ) := by
+    intro q hq
+    have he := (hQ q).mp hq
+    have h0 := hk0 q.1
+    have h1 := hk0 q.2
+    simp only [mem_coe, mem_Icc]
+    omega
+  have hinj : Set.InjOn (fun q : ι × ι => k q.1) (↑Q : Set (ι × ι)) := by
+    intro q hq z hz he
+    change k q.1 = k z.1 at he
+    have hq' := (hQ q).mp hq
+    have hz' := (hQ z).mp hz
+    apply Prod.ext (hk he)
+    apply hk
+    omega
+  have hc : Q.card ≤ m-1 := by
+    have hc' := Finset.card_le_card_of_injOn (fun q : ι × ι => k q.1) hmaps hinj
+    simpa only [Nat.card_Icc, Nat.add_sub_cancel] using hc'
+  have hd : 0 < (m : ℝ)-1 := by
+    have hmR : (2 : ℝ) ≤ m := by exact_mod_cast hm
+    linarith
+  have hterm (q : ι × ι) (hq : q ∈ Q) :
+      1/((k q.1 : ℝ)*(k q.2 : ℝ)) ≤ 1/((m : ℝ)-1) := by
+    apply one_div_le_one_div_of_le hd
+    have h0 : 1 ≤ (k q.1 : ℝ) := by exact_mod_cast (show 1 ≤ k q.1 from hk0 q.1)
+    have h1 : 1 ≤ (k q.2 : ℝ) := by exact_mod_cast (show 1 ≤ k q.2 from hk0 q.2)
+    have he : (k q.1 : ℝ)+(k q.2 : ℝ) = m := by exact_mod_cast (hQ q).mp hq
+    nlinarith [mul_nonneg (sub_nonneg.mpr h0) (sub_nonneg.mpr h1)]
+  calc
+    _ = ∑ q ∈ Q, 1/((k q.1 : ℝ)*(k q.2 : ℝ)) := by
+      rw [Finset.sum_filter]
+      apply Finset.sum_congr rfl
+      intro q _
+      simp only [m, eq_comm (a := k q.1+k q.2)]
+    _ ≤ ∑ _q ∈ Q, 1/((m : ℝ)-1) := Finset.sum_le_sum hterm
+    _ = (Q.card : ℝ)/((m : ℝ)-1) := by simp [div_eq_mul_inv]
+    _ ≤ 1 := by
+      rw [div_le_one hd]
+      have hcR : (Q.card : ℝ) ≤ (m-1 : ℕ) := by exact_mod_cast hc
+      rw [Nat.cast_sub (by omega : 1 ≤ m), Nat.cast_one] at hcR
+      exact hcR
+
+end
+end WeightedSchur
+
+namespace FourierPolynomial
+noncomputable section
+open MeasureTheory AddCircle Finset
+open scoped ComplexConjugate
+local instance : Fact (0 < (1 : ℝ)) := ⟨zero_lt_one⟩
+
+ theorem integral_norm_fourth_le_pairs {ι : Type u98} [Fintype ι] (k : ι → ℤ) (a : ι → ℂ) :
+    (∫ t : Time, ‖eval k a t‖^4 ∂μ) ≤
+      ∑ p : ι × ι, ∑ q : ι × ι,
+        if k p.1+k p.2 = k q.1+k q.2 then
+          (‖a p.1‖*‖a p.2‖)*(‖a q.1‖*‖a q.2‖) else 0 := by
+  classical
+  have hI : 0 ≤ ∫ t : Time, ‖eval k a t‖^4 ∂μ := integral_nonneg (fun _ => by positivity)
+  calc
+    _ = ‖(((∫ t : Time, ‖eval k a t‖^4 ∂μ) : ℝ) : ℂ)‖ := by
+      rw [Complex.norm_real, Real.norm_eq_abs, abs_of_nonneg hI]
+    _ = ‖∑ p : ι × ι, ∑ q : ι × ι,
+        if k p.1+k p.2 = k q.1+k q.2 then conj (a p.1*a p.2)*(a q.1*a q.2) else 0‖ :=
+      congrArg norm (coe_integral_norm_fourth k a)
+    _ ≤ ∑ p : ι × ι, ∑ q : ι × ι,
+        ‖if k p.1+k p.2 = k q.1+k q.2 then conj (a p.1*a p.2)*(a q.1*a q.2) else 0‖ :=
+      (norm_sum_le _ _).trans (Finset.sum_le_sum (fun _ _ => norm_sum_le _ _))
+    _ = _ := by
+      apply Finset.sum_congr rfl
+      intro p _
+      apply Finset.sum_congr rfl
+      intro q _
+      split_ifs <;> simp only [norm_mul, Complex.norm_conj, norm_zero]
+
+ theorem positive_L4 {ι : Type u99} [Fintype ι] (k : ι → ℕ)
+    (hk : Function.Injective k) (hk0 : ∀ i, 0 < k i) (a : ι → ℂ) :
+    (∫ t : Time, ‖eval (fun i => (k i : ℤ)) a t‖^4 ∂μ) ≤
+      (∑ i, (k i : ℝ)*‖a i‖^2)^2 := by
+  classical
+  have hrows := WeightedSchur.reciprocal_pair_rows k hk hk0
+  let w : ι × ι → ℝ := fun p => (k p.1 : ℝ)*(k p.2 : ℝ)
+  have hw (p : ι × ι) : 0 < w p := by
+    exact mul_pos (by exact_mod_cast hk0 p.1) (by exact_mod_cast hk0 p.2)
+  have hS := WeightedSchur.sum_le
+    (fun p q : ι × ι => k p.1+k p.2 = k q.1+k q.2)
+    (fun h => h.symm) w hw (fun p => ‖a p.1‖*‖a p.2‖) hrows
+  have hpair := integral_norm_fourth_le_pairs (fun i => (k i : ℤ)) a
+  simp only [← Nat.cast_add, Int.natCast_inj] at hpair
+  apply hpair.trans
+  apply (hS.trans_eq _)
+  simp only [one_mul, Fintype.sum_prod_type, w]
+  calc
+    _ = ∑ i, ∑ j, ((k i : ℝ)*‖a i‖^2)*((k j : ℝ)*‖a j‖^2) := by
+      apply Finset.sum_congr rfl
+      intro i _
+      apply Finset.sum_congr rfl
+      intro j _
+      ring
+    _ = _ := by rw [← Finset.sum_mul_sum]; ring
+
+end
+end FourierPolynomial
+
+end NonsqueezingInline
+
+end OAI
diff --git a/lean/OAI/Geometry/PolarProducts/FourierPaths.lean b/lean/OAI/Geometry/PolarProducts/FourierPaths.lean
new file mode 100644
index 0000000000000000000000000000000000000000..b093fd77de79985855e256ac6a3e4a437b31ac93
--- /dev/null
+++ b/lean/OAI/Geometry/PolarProducts/FourierPaths.lean
@@ -0,0 +1,328 @@
+import OAI.Geometry.PolarProducts.FourierIntegrals
+
+namespace OAI
+
+universe u100 u101 u102 u103 u104 u105 u106 u107 u108 u109 u110 u111
+
+section NonsqueezingInline
+
+namespace FourierPolynomial
+noncomputable section
+open MeasureTheory AddCircle Finset
+open scoped ComplexConjugate ContDiff
+local instance : Fact (0 < (1 : ℝ)) := ⟨zero_lt_one⟩
+ theorem real_inner_eq_complex_re {α : Type u100} [Fintype α] (a b : EuclideanSpace ℂ α) :
+    inner (𝕜 := ℝ) a b = RCLike.re (inner (𝕜 := ℂ) a b) := by
+  simp only [PiLp.inner_apply, real_inner_eq_re_inner, map_sum]
+
+variable {ι : Type u101} {κ : Type u102} [Fintype ι] [Fintype κ]
+
+abbrev Vector (κ : Type u103) := EuclideanSpace ℂ κ
+abbrev Coeff (ι : Type u104) (κ : Type u105) := EuclideanSpace ℂ (ι × κ)
+
+def mode (a : Coeff ι κ) (i : ι) : Vector κ :=
+  (EuclideanSpace.equiv κ ℂ).symm (fun j => a (i,j))
+
+omit [Fintype ι] [Fintype κ] in
+@[simp] theorem mode_apply (a : Coeff ι κ) (i : ι) (j : κ) : mode a i j = a (i,j) := rfl
+
+def loop (k : ι → ℤ) (a : Coeff ι κ) : C(Time, Vector κ) where
+  toFun t := ∑ i, fourier (k i) t • mode a i
+  continuous_toFun := continuous_finsetSum _ (fun i _ =>
+    (fourier (k i)).continuous.smul continuous_const)
+
+@[simp] theorem loop_apply (k : ι → ℤ) (a : Coeff ι κ) (t : Time) :
+    loop k a t = ∑ i, fourier (k i) t • mode a i := rfl
+
+ theorem loop_coord (k : ι → ℤ) (a : Coeff ι κ) (t : Time) (j : κ) :
+    loop k a t j = eval k (fun i => a (i,j)) t := by
+  simp [loop_apply, eval, mul_comm]
+
+ def loopCLM (k : ι → ℤ) : Coeff ι κ →L[ℝ] C(Time, Vector κ) :=
+  LinearMap.toContinuousLinearMap {
+    toFun := loop k
+    map_add' := by
+      intro a b
+      ext t j
+      simp only [ContinuousMap.add_apply, PiLp.add_apply, loop_coord, eval,
+        add_mul, Finset.sum_add_distrib]
+    map_smul' := by
+      intro r a
+      ext t j
+      simp only [ContinuousMap.smul_apply, PiLp.smul_apply, loop_coord, eval,
+        RingHom.id_apply, RCLike.real_smul_eq_coe_mul, mul_assoc, ← Finset.mul_sum] }
+
+@[simp] theorem loopCLM_apply (k : ι → ℤ) (a : Coeff ι κ) : loopCLM k a = loop k a := rfl
+
+ theorem integral_loop_norm_sq (k : ι → ℤ) (hk : Function.Injective k) (a : Coeff ι κ) :
+    (∫ t : Time, ‖loop k a t‖^2 ∂μ) = ‖a‖^2 := by
+  classical
+  simp_rw [EuclideanSpace.norm_sq_eq, loop_coord]
+  rw [integral_finsetSum]
+  · simp_rw [integral_norm_sq k hk]
+    rw [Finset.sum_comm, Fintype.sum_prod_type]
+  · intro j _
+    exact integrable_continuous ((continuous_eval k _).norm.pow 2)
+
+ theorem integral_loop_coord_sq (k : ι → ℤ) (hk : Function.Injective k)
+    (a : Coeff ι κ) (j : κ) :
+    (∫ t : Time, ‖loop k a t j‖^2 ∂μ) = ∑ i, ‖a (i,j)‖^2 := by
+  simp_rw [loop_coord]
+  exact integral_norm_sq k hk _
+
+end
+end FourierPolynomial
+
+namespace FourierPolynomial
+noncomputable section
+open MeasureTheory AddCircle Finset
+open scoped ComplexConjugate ContDiff
+local instance : Fact (0 < (1 : ℝ)) := ⟨zero_lt_one⟩
+variable {ι : Type u106} {κ : Type u107} [Fintype ι] [Fintype κ]
+
+ theorem vector_norm_fourth_le (z : Vector κ) :
+    ‖z‖^4 ≤ (Fintype.card κ : ℝ) * ∑ j, ‖z j‖^4 := by
+  have h := Finset.sum_mul_sq_le_sq_mul_sq (univ : Finset κ)
+    (fun j => ‖z j‖^2) (fun _ => (1 : ℝ))
+  simp only [mul_one, one_pow, sum_const, card_univ, nsmul_eq_mul, mul_one, ← pow_mul] at h
+  calc
+    _ = (∑ j, ‖z j‖^2)^2 := by rw [← EuclideanSpace.norm_sq_eq]; ring
+    _ ≤ _ := by simpa only [mul_comm] using h
+
+ theorem integral_loop_norm_fourth_le (k : ι → ℕ)
+    (hk : Function.Injective k) (hk0 : ∀ i, 0 < k i) (a : Coeff ι κ) :
+    (∫ t : Time, ‖loop (fun i => (k i : ℤ)) a t‖^4 ∂μ) ≤
+      (Fintype.card κ : ℝ) * (∑ p : ι × κ, (k p.1 : ℝ)*‖a p‖^2)^2 := by
+  classical
+  let L := loop (fun i => (k i : ℤ)) a
+  have hcont : Continuous (fun t : Time => (Fintype.card κ : ℝ)*∑ j, ‖L t j‖^4) :=
+    continuous_const.mul (continuous_finsetSum _ (fun j _ =>
+      (((PiLp.continuous_apply 2 _ j).comp L.continuous).norm.pow 4)))
+  calc
+    _ ≤ ∫ t : Time, (Fintype.card κ : ℝ)*∑ j, ‖L t j‖^4 ∂μ :=
+      integral_mono (integrable_continuous (L.continuous.norm.pow 4))
+        (integrable_continuous hcont) (fun t => vector_norm_fourth_le (L t))
+    _ = (Fintype.card κ : ℝ)*∑ j, ∫ t : Time, ‖L t j‖^4 ∂μ := by
+      rw [integral_const_mul, integral_finsetSum]
+      intro j _
+      exact integrable_continuous (((PiLp.continuous_apply 2 _ j).comp L.continuous).norm.pow 4)
+    _ ≤ (Fintype.card κ : ℝ)*∑ j, (∑ i, (k i : ℝ)*‖a (i,j)‖^2)^2 := by
+      apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg _)
+      apply Finset.sum_le_sum
+      intro j _
+      simp only [L, loop_coord]
+      exact positive_L4 k hk hk0 _
+    _ ≤ (Fintype.card κ : ℝ)*(∑ j, ∑ i, (k i : ℝ)*‖a (i,j)‖^2)^2 := by
+      apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg _)
+      exact Finset.sum_sq_le_sq_sum_of_nonneg (fun j _ =>
+        Finset.sum_nonneg (fun i _ => mul_nonneg (Nat.cast_nonneg _) (sq_nonneg _)))
+    _ = _ := by rw [Finset.sum_comm, Fintype.sum_prod_type]
+
+end
+end FourierPolynomial
+
+namespace FourierPolynomial
+noncomputable section
+open MeasureTheory AddCircle Finset
+open scoped ComplexConjugate ContDiff
+local instance : Fact (0 < (1 : ℝ)) := ⟨zero_lt_one⟩
+variable {ι : Type u108} {κ : Type u109} [Fintype ι] [Fintype κ]
+
+ theorem fourierCoeff_eval (k : ι → ℤ) (a : ι → ℂ) (j : ℤ) :
+    fourierCoeff (eval k a) j = ∑ i, if j = k i then a i else 0 := by
+  classical
+  simp only [fourierCoeff, eval, smul_eq_mul, Finset.mul_sum]
+  rw [integral_finsetSum]
+  · apply Finset.sum_congr rfl
+    intro i _
+    have he (t : Time) : fourier (-j) t * (a i * fourier (k i) t) =
+        a i*(conj (fourier j t)*fourier (k i) t) := by rw [fourier_neg]; ring
+    simp_rw [he]
+    rw [integral_const_mul, integral_conj_fourier_mul]
+    split_ifs <;> simp only [mul_one, mul_zero]
+  · intro i _
+    exact integrable_continuous ((fourier (-j)).continuous.mul
+      (continuous_const.mul (fourier (k i)).continuous))
+
+ def project (k : ι → ℤ) (f : C(Time, Vector κ)) : Coeff ι κ :=
+  (EuclideanSpace.equiv (ι × κ) ℂ).symm (fun p => fourierCoeff (fun t => f t p.2) (k p.1))
+
+omit [Fintype ι] [Fintype κ] in
+@[simp] theorem project_apply (k : ι → ℤ) (f : C(Time, Vector κ)) (p : ι × κ) :
+    project k f p = fourierCoeff (fun t => f t p.2) (k p.1) := rfl
+
+ theorem project_loop (k : ι → ℤ) (hk : Function.Injective k) (a : Coeff ι κ) :
+    project k (loop k a) = a := by
+  classical
+  ext p
+  simp only [project_apply, loop_coord, fourierCoeff_eval, hk.eq_iff]
+  simp
+
+ theorem integral_fourier_mul_conj (k : ℤ) (f : C(Time, ℂ)) :
+    (∫ t : Time, conj (f t) * fourier k t ∂μ) = conj (fourierCoeff f k) := by
+  rw [fourierCoeff, ← integral_conj]
+  apply integral_congr_ae
+  filter_upwards [] with t
+  simp only [smul_eq_mul, map_mul, fourier_neg, starRingEnd_self_apply]
+  ring
+
+ theorem integral_inner_loop (k : ι → ℤ) (f : C(Time, Vector κ)) (a : Coeff ι κ) :
+    (∫ t : Time, inner (𝕜 := ℝ) (f t) (loop k a t) ∂μ) =
+      inner (𝕜 := ℝ) (project k f) a := by
+  classical
+  have hstage (j : κ) :
+      (∫ t : Time, conj (f t j)*eval k (fun i => a (i,j)) t ∂μ) =
+        ∑ i, conj (project k f (i,j))*a (i,j) := by
+    simp only [eval, Finset.mul_sum]
+    rw [integral_finsetSum]
+    · apply Finset.sum_congr rfl
+      intro i _
+      have he (t : Time) : conj (f t j) * (a (i,j) * fourier (k i) t) =
+          (conj (f t j) * fourier (k i) t) * a (i,j) := by ring
+      simp_rw [he]
+      rw [integral_mul_const]
+      exact congrArg (fun z => z*a (i,j))
+        (integral_fourier_mul_conj (k i)
+          ⟨fun t => f t j, (PiLp.continuous_apply 2 _ j).comp f.continuous⟩)
+    · intro i _
+      exact integrable_continuous
+        ((((PiLp.continuous_apply 2 _ j).comp f.continuous).star).mul
+          (continuous_const.mul (fourier (k i)).continuous))
+  have hcomplex : (∫ t : Time, inner (𝕜 := ℂ) (f t) (loop k a t) ∂μ) =
+      inner (𝕜 := ℂ) (project k f) a := by
+    change (∫ t : Time, ∑ j, loop k a t j * conj (f t j) ∂μ) =
+      ∑ p : ι × κ, a p * conj (project k f p)
+    simp_rw [loop_coord, mul_comm (eval k _ _)]
+    rw [integral_finsetSum]
+    · simp_rw [hstage]
+      rw [Fintype.sum_prod_type, Finset.sum_comm]
+      congr 1
+      ext j
+      apply Finset.sum_congr rfl
+      intro i _
+      ring
+    · intro j _
+      exact integrable_continuous
+        ((((PiLp.continuous_apply 2 _ j).comp f.continuous).star).mul
+          (continuous_eval k _))
+  simp only [real_inner_eq_complex_re]
+  rw [integral_re (integrable_continuous
+    (show Continuous (fun t => inner (𝕜 := ℂ) (f t) (loop k a t)) from
+      f.continuous.inner (loop k a).continuous)), hcomplex]
+
+end
+end FourierPolynomial
+
+namespace FourierPolynomial
+noncomputable section
+open MeasureTheory AddCircle Finset
+open scoped ComplexConjugate ContDiff
+local instance : Fact (0 < (1 : ℝ)) := ⟨zero_lt_one⟩
+variable {ι : Type u110} {κ : Type u111} [Fintype ι] [Fintype κ]
+
+ theorem project_bessel (k : ι → ℤ) (hk : Function.Injective k) (f : C(Time, Vector κ)) :
+    ‖project k f‖^2 ≤ ∫ t : Time, ‖f t‖^2 ∂μ := by
+  let a := project k f
+  have hnon : 0 ≤ ∫ t : Time, ‖f t-loop k a t‖^2 ∂μ :=
+    integral_nonneg (fun _ => sq_nonneg _)
+  have h1 : Integrable (fun t : Time => ‖f t‖^2) μ :=
+    integrable_continuous (f.continuous.norm.pow 2)
+  have h2 : Integrable (fun t : Time => inner (𝕜 := ℝ) (f t) (loop k a t)) μ :=
+    integrable_continuous (f.continuous.inner (loop k a).continuous)
+  have h3 : Integrable (fun t : Time => ‖loop k a t‖^2) μ :=
+    integrable_continuous ((loop k a).continuous.norm.pow 2)
+  simp_rw [norm_sub_sq_real] at hnon
+  rw [integral_add, integral_sub, integral_const_mul,
+    integral_inner_loop, integral_loop_norm_sq k hk, real_inner_self_eq_norm_sq] at hnon
+  rotate_left
+  · exact h1
+  · exact h2.const_mul 2
+  · exact h1.sub (h2.const_mul 2)
+  · exact h3
+  dsimp [a] at *
+  linarith
+
+ theorem project_bounded (k : ι → ℤ) (hk : Function.Injective k)
+    (f : C(Time, Vector κ)) {B : ℝ} (hB : 0 ≤ B) (hf : ∀ t, ‖f t‖ ≤ B) :
+    ‖project k f‖ ≤ B := by
+  have hi : (∫ t : Time, ‖f t‖^2 ∂μ) ≤ B^2 := by
+    calc
+      _ ≤ ∫ _ : Time, B^2 ∂μ := integral_mono
+        (integrable_continuous (f.continuous.norm.pow 2)) (integrable_const _) (fun t =>
+          pow_le_pow_left₀ (norm_nonneg _) (hf t) 2)
+      _ = _ := by simp
+  have h := (project_bessel k hk f).trans hi
+  nlinarith [norm_nonneg (project k f)]
+
+end
+end FourierPolynomial
+
+namespace FourierPolynomial
+noncomputable section
+open MeasureTheory AddCircle Finset
+open scoped ComplexConjugate ContDiff
+local instance : Fact (0 < (1 : ℝ)) := ⟨zero_lt_one⟩
+variable {ι κ : Type} [Fintype ι] [Fintype κ]
+
+ def average : C(Time, ℝ) →L[ℝ] ℝ := by
+  let L : C(Time, ℝ) →ₗ[ℝ] ℝ := {
+    toFun := fun f => ∫ t : Time, f t ∂μ
+    map_add' := fun f g => integral_add (integrable_continuous f.continuous)
+      (integrable_continuous g.continuous)
+    map_smul' := fun r f => by simp only [ContinuousMap.smul_apply,
+      integral_smul, RingHom.id_apply] }
+  apply L.mkContinuous 1
+  intro f
+  change ‖∫ t : Time, f t ∂μ‖ ≤ 1*‖f‖
+  simpa using norm_integral_le_of_norm_le_const (μ := μ)
+    (Filter.Eventually.of_forall (fun t => f.norm_coe_le_norm t))
+
+@[simp] theorem average_apply (f : C(Time, ℝ)) : average f = ∫ t : Time, f t ∂μ := rfl
+
+ def potential (k : ι → ℤ) (H : Vector κ → ℝ) (a : Coeff ι κ) : ℝ :=
+  ∫ t : Time, H (loop k a t) ∂μ
+
+ theorem contDiff_potential (k : ι → ℤ) {H : Vector κ → ℝ} (hH : ContDiff ℝ ∞ H) :
+    ContDiff ℝ ∞ (potential k H) := by
+  exact average.contDiff.comp ((SmoothPaths.contDiff_superpose hH).comp (loopCLM k).contDiff)
+
+ theorem contDiff_gradient_function {H : Vector κ → ℝ} (hH : ContDiff ℝ ∞ H) :
+    ContDiff ℝ ∞ (gradient H) :=
+  (InnerProductSpace.toDual ℝ _).symm.toContinuousLinearEquiv.contDiff.comp
+    (hH.fderiv_right (by simp))
+
+ def gradLoop (k : ι → ℤ) {H : Vector κ → ℝ} (hH : ContDiff ℝ ∞ H) (a : Coeff ι κ) :
+    C(Time, Vector κ) :=
+  ⟨fun t => gradient H (loop k a t), (contDiff_gradient_function hH).continuous.comp
+    (loop k a).continuous⟩
+
+ theorem hasGradientAt_potential (k : ι → ℤ) {H : Vector κ → ℝ}
+    (hH : ContDiff ℝ ∞ H) (a : Coeff ι κ) :
+    HasGradientAt (potential k H) (project k (gradLoop k hH a)) a := by
+  have hl : HasFDerivAt (loop k) (loopCLM k) a := by
+    change HasFDerivAt (loopCLM k) (loopCLM k) a
+    exact (loopCLM (κ := κ) k).hasFDerivAt (x := a)
+  have hS := (SmoothPaths.hasFDerivAt_superpose
+    (hH.differentiable (by simp)) (hH.fderiv_right (m := ∞) (by simp)).continuous (loop k a)).comp
+      a hl
+  have hi := average.hasFDerivAt.comp a hS
+  rw [hasGradientAt_iff_hasFDerivAt]
+  change HasFDerivAt (potential k H) _ a at hi
+  apply hi.congr_fderiv
+  ext b
+  change (∫ t : Time, fderiv ℝ H (loop k a t) (loop k b t) ∂μ) =
+    inner (𝕜 := ℝ) (project k (gradLoop k hH a)) b
+  simp_rw [← inner_gradient_left]
+  exact integral_inner_loop k (gradLoop k hH a) b
+
+ theorem gradient_potential (k : ι → ℤ) {H : Vector κ → ℝ}
+    (hH : ContDiff ℝ ∞ H) (a : Coeff ι κ) :
+    gradient (potential k H) a = project k (gradLoop k hH a) :=
+  (hasGradientAt_potential k hH a).gradient
+
+end
+end FourierPolynomial
+
+end NonsqueezingInline
+
+end OAI
diff --git a/lean/OAI/Geometry/PolarProducts/GalerkinBounds.lean b/lean/OAI/Geometry/PolarProducts/GalerkinBounds.lean
new file mode 100644
index 0000000000000000000000000000000000000000..dbaa5092242a6c0a684d473a07aa3e4d1e499fd0
--- /dev/null
+++ b/lean/OAI/Geometry/PolarProducts/GalerkinBounds.lean
@@ -0,0 +1,415 @@
+import OAI.Geometry.PolarProducts.QuadraticAction
+
+namespace OAI
+
+universe u113
+
+section NonsqueezingInline
+
+namespace FourierPolynomial
+noncomputable section
+open MeasureTheory AddCircle Finset
+open scoped ComplexConjugate ContDiff
+local instance : Fact (0 < (1 : ℝ)) := ⟨zero_lt_one⟩
+variable {ι ν κ : Type} [Fintype ι] [Fintype ν] [Fintype κ]
+open DiagonalQuadratic
+
+ def leftCoeff (a : Coeff (ι ⊕ ν) κ) : Coeff ι κ :=
+  (EuclideanSpace.equiv (ι × κ) ℂ).symm (fun p => a (Sum.inl p.1,p.2))
+
+ def rightCoeff (a : Coeff (ι ⊕ ν) κ) : Coeff ν κ :=
+  (EuclideanSpace.equiv (ν × κ) ℂ).symm (fun p => a (Sum.inr p.1,p.2))
+
+ def joinCoeff (a : Coeff ι κ) (b : Coeff ν κ) : Coeff (ι ⊕ ν) κ :=
+  (EuclideanSpace.equiv ((ι ⊕ ν) × κ) ℂ).symm (fun p => Sum.elim
+    (fun i => a (i,p.2)) (fun j => b (j,p.2)) p.1)
+
+omit [Fintype ι] [Fintype ν] [Fintype κ] in
+@[simp] theorem leftCoeff_apply (a : Coeff (ι ⊕ ν) κ) (p : ι × κ) :
+    leftCoeff a p = a (Sum.inl p.1,p.2) := rfl
+omit [Fintype ι] [Fintype ν] [Fintype κ] in
+@[simp] theorem rightCoeff_apply (a : Coeff (ι ⊕ ν) κ) (p : ν × κ) :
+    rightCoeff a p = a (Sum.inr p.1,p.2) := rfl
+omit [Fintype ι] [Fintype ν] [Fintype κ] in
+@[simp] theorem joinCoeff_inl (a : Coeff ι κ) (b : Coeff ν κ) (i : ι) (j : κ) :
+    joinCoeff a b (Sum.inl i,j) = a (i,j) := rfl
+omit [Fintype ι] [Fintype ν] [Fintype κ] in
+@[simp] theorem joinCoeff_inr (a : Coeff ι κ) (b : Coeff ν κ) (i : ν) (j : κ) :
+    joinCoeff a b (Sum.inr i,j) = b (i,j) := rfl
+
+ theorem norm_joinCoeff_sq (a : Coeff ι κ) (b : Coeff ν κ) :
+    ‖joinCoeff a b‖^2 = ‖a‖^2+‖b‖^2 := by
+  simp only [EuclideanSpace.norm_sq_eq, Fintype.sum_prod_type, Fintype.sum_sum_type,
+    joinCoeff_inl, joinCoeff_inr]
+
+ theorem energy_joinCoeff (w : (ι ⊕ ν) × κ → ℝ) (a : Coeff ι κ) (b : Coeff ν κ) :
+    energy w (joinCoeff a b) = energy (fun p => w (Sum.inl p.1,p.2)) a +
+      energy (fun p => w (Sum.inr p.1,p.2)) b := by
+  simp only [energy, Fintype.sum_prod_type, Fintype.sum_sum_type,
+    joinCoeff_inl, joinCoeff_inr]
+
+ theorem loop_joinCoeff (k : ι → ℤ) (l : ν → ℤ) (a : Coeff ι κ) (b : Coeff ν κ) :
+    loop (Sum.elim k l) (joinCoeff a b) = loop k a + loop l b := by
+  ext t j
+  simp only [ContinuousMap.add_apply, PiLp.add_apply, loop_coord, eval,
+    Fintype.sum_sum_type, Sum.elim_inl, Sum.elim_inr, joinCoeff_inl, joinCoeff_inr]
+
+ def splitEquiv (d : ν → ℝ) (hd : ∀ i, d i ≠ 0) :
+    Coeff (ι ⊕ ν) κ ≃L[ℝ] Coeff ι κ × Coeff ν κ :=
+  LinearEquiv.toContinuousLinearEquiv {
+    toFun := fun a => (leftCoeff a, diag (fun p => d p.1) (rightCoeff a))
+    invFun := fun p => joinCoeff p.1 (diag (fun q => (d q.1)⁻¹) p.2)
+    left_inv := by
+      intro a
+      ext ⟨i,j⟩
+      cases i with
+      | inl i => rfl
+      | inr i => simp [hd i]
+    right_inv := by
+      intro ⟨a,b⟩
+      ext p
+      · rfl
+      · simp [hd p.1]
+    map_add' := by
+      intro a b
+      ext p
+      · rfl
+      · simp [smul_add]
+    map_smul' := by
+      intro r a
+      ext p
+      · rfl
+      · simp
+        ring }
+
+ theorem splitEquiv_symm (d : ν → ℝ) (hd : ∀ i, d i ≠ 0)
+    (a : Coeff ι κ) (b : Coeff ν κ) :
+    (splitEquiv d hd).symm (a,b) = joinCoeff a (diag (fun q => (d q.1)⁻¹) b) := rfl
+
+ def positiveScale (k : ν → ℕ) (hk : ∀ i, 0 < k i) :
+    Coeff (ι ⊕ ν) κ ≃L[ℝ] Coeff ι κ × Coeff ν κ :=
+  splitEquiv (fun i => Real.sqrt (k i : ℝ)) (fun i =>
+    (Real.sqrt_pos.2 (by exact_mod_cast hk i)).ne')
+
+ theorem energy_scaled (k : ν → ℕ) (hk : ∀ i, 0 < k i) (b : Coeff ν κ) :
+    energy (fun p => (k p.1 : ℝ))
+      (diag (fun p => (Real.sqrt (k p.1 : ℝ))⁻¹) b) = ‖b‖^2 := by
+  classical
+  simp only [energy, EuclideanSpace.norm_sq_eq]
+  apply sum_congr rfl
+  intro p _
+  have hp : 0 < (k p.1 : ℝ) := by exact_mod_cast hk p.1
+  rw [diag_apply, norm_smul, Real.norm_eq_abs, abs_inv, abs_of_nonneg (Real.sqrt_nonneg _),
+    mul_pow, inv_pow, Real.sq_sqrt hp.le]
+  field_simp
+
+end
+end FourierPolynomial
+
+namespace FourierPolynomial
+noncomputable section
+open MeasureTheory AddCircle Finset
+open scoped ComplexConjugate ContDiff
+local instance : Fact (0 < (1 : ℝ)) := ⟨zero_lt_one⟩
+variable {ι ν κ : Type} [Fintype ι] [Fintype ν] [Fintype κ]
+open DiagonalQuadratic
+
+ theorem action_positive_lower (k : ι → ℤ) (l : ν → ℕ)
+    (hl : Function.Injective l) (hl0 : ∀ i, 0 < l i)
+    {H : Vector κ → ℝ} (hH : Continuous H) {C : ℝ} (hC : 0 ≤ C)
+    (hup : ∀ z, H z ≤ C*‖z‖^4) (b : Coeff ν κ) :
+    Real.pi*‖b‖^2-C*(Fintype.card κ : ℝ)*‖b‖^4 ≤
+      action (Sum.elim k (fun i => (l i : ℤ))) H ((positiveScale l hl0).symm (0,b)) := by
+  let v := diag (fun p : ν × κ => (Real.sqrt (l p.1 : ℝ))⁻¹) b
+  have hkin : energy (fun p : (ι ⊕ ν) × κ =>
+      Real.pi*((Sum.elim k (fun i => (l i : ℤ)) p.1 : ℤ) : ℝ)) (joinCoeff 0 v) = Real.pi*‖b‖^2 := by
+    rw [energy_joinCoeff]
+    simp only [Sum.elim_inl, Sum.elim_inr, Int.cast_natCast]
+    have hz : energy (fun p : ι × κ => Real.pi*(k p.1 : ℝ)) 0 = 0 := by simp [energy]
+    rw [hz, zero_add, energy_mul, energy_scaled l hl0]
+  have hp := potential_upper_fourth l hl hl0 hH hC hup v
+  change _ ≤ C*(Fintype.card κ : ℝ)*(energy (fun p : ν × κ => (l p.1 : ℝ)) v)^2 at hp
+  rw [energy_scaled l hl0] at hp
+  have hlo : potential (Sum.elim k (fun i => (l i : ℤ))) H (joinCoeff 0 v) =
+      potential (fun i => (l i : ℤ)) H v := by
+    unfold potential
+    rw [loop_joinCoeff, loop_zero, zero_add]
+  change _ ≤ energy _ (joinCoeff 0 v)-potential _ H (joinCoeff 0 v)
+  rw [hkin, hlo]
+  nlinarith
+
+ theorem cone_scaled_single [DecidableEq ν] [DecidableEq κ]
+    (l : ν → ℕ) (hl0 : ∀ i, 0 < l i) (i₀ : ν) (hl1 : l i₀ = 1) (j₀ : κ)
+    (a : Coeff ι κ) (s : ℝ) :
+    (positiveScale l hl0).symm (a,s • PiLp.single 2 (i₀,j₀) (1 : ℂ)) =
+      joinCoeff a (s • PiLp.single 2 (i₀,j₀) (1 : ℂ)) := by
+  apply congrArg (joinCoeff a)
+  ext p
+  simp only [diag_apply, PiLp.smul_apply, PiLp.single_apply]
+  split_ifs with hp
+  · subst p
+    simp [hl1]
+  · simp
+
+ theorem action_cone_upper [DecidableEq ν] [DecidableEq κ]
+    (k : ι → ℤ) (hk0 : ∀ i, k i ≤ 0) (l : ν → ℕ) (hl0 : ∀ i, 0 < l i)
+    (i₀ : ν) (hl1 : l i₀ = 1) (j₀ : κ)
+    {H : Vector κ → ℝ} (hH : ∀ z, 0 ≤ H z) (a : Coeff ι κ) (s : ℝ) :
+    action (Sum.elim k (fun i => (l i : ℤ))) H
+      ((positiveScale l hl0).symm (a,s • PiLp.single 2 (i₀,j₀) (1 : ℂ))) ≤ Real.pi*s^2 := by
+  rw [cone_scaled_single l hl0 i₀ hl1 j₀]
+  have hn := energy_nonpos (fun p : ι × κ => Real.pi*(k p.1 : ℝ))
+    (fun p => mul_nonpos_of_nonneg_of_nonpos Real.pi_pos.le (by exact_mod_cast hk0 p.1)) a
+  have hp := potential_nonneg (Sum.elim k (fun i => (l i : ℤ))) hH
+    (joinCoeff a (s • PiLp.single 2 (i₀,j₀) (1 : ℂ)))
+  unfold action
+  rw [energy_joinCoeff]
+  simp only [Sum.elim_inl, Sum.elim_inr, Int.cast_natCast, energy_smul,
+    energy_single, hl1, Nat.cast_one, norm_one, one_pow, mul_one]
+  nlinarith
+
+ theorem action_cone_far [DecidableEq ν] [DecidableEq κ]
+    (k : ι → ℤ) (hk0 : ∀ i, k i ≤ 0) (l : ν → ℕ) (hl0 : ∀ i, 0 < l i)
+    (hkl : Function.Injective (Sum.elim k (fun i => (l i : ℤ))))
+    (i₀ : ν) (hl1 : l i₀ = 1) (j₀ : κ)
+    {H : Vector κ → ℝ} (hH : Continuous H) (d : κ → ℝ) (C η : ℝ)
+    (hd : ∀ j, η ≤ d j) (hd0 : Real.pi+η ≤ d j₀)
+    (hlo : ∀ z, energy d z-C ≤ H z) (a : Coeff ι κ) (s : ℝ) :
+    action (Sum.elim k (fun i => (l i : ℤ))) H
+      ((positiveScale l hl0).symm (a,s • PiLp.single 2 (i₀,j₀) (1 : ℂ))) ≤
+      C-η*(‖a‖^2+s^2) := by
+  rw [cone_scaled_single l hl0 i₀ hl1 j₀]
+  have hn := energy_nonpos (fun p : ι × κ => Real.pi*(k p.1 : ℝ))
+    (fun p => mul_nonpos_of_nonneg_of_nonpos Real.pi_pos.le (by exact_mod_cast hk0 p.1)) a
+  have hdlo := energy_lower (fun p : ι × κ => d p.2) η (fun p => hd p.2) a
+  have hp := potential_lower (Sum.elim k (fun i => (l i : ℤ))) hkl hH d C hlo
+    (joinCoeff a (s • PiLp.single 2 (i₀,j₀) (1 : ℂ)))
+  unfold action
+  rw [energy_joinCoeff]
+  rw [energy_joinCoeff] at hp
+  simp only [Sum.elim_inl, Sum.elim_inr, Int.cast_natCast, energy_smul,
+    energy_single, hl1, Nat.cast_one, norm_one, one_pow, mul_one] at hp ⊢
+  nlinarith [mul_nonneg (sub_nonneg.mpr hd0) (sq_nonneg s)]
+
+end
+end FourierPolynomial
+
+namespace FourierPolynomial
+noncomputable section
+open MeasureTheory AddCircle Finset
+open scoped ComplexConjugate ContDiff
+local instance : Fact (0 < (1 : ℝ)) := ⟨zero_lt_one⟩
+variable {ι κ : Type} [Fintype ι] [Fintype κ]
+open DiagonalQuadratic
+
+ def velocityCoeff (k : ι → ℤ) (a : Coeff ι κ) : Coeff ι κ :=
+   Complex.I • ((2 : ℝ) • diag (fun p => Real.pi*(k p.1 : ℝ)) a)
+
+omit [Fintype ι] [Fintype κ] in
+ theorem mode_velocityCoeff (k : ι → ℤ) (a : Coeff ι κ) (i : ι) :
+    mode (velocityCoeff k a) i = (2*(Real.pi : ℂ)*Complex.I*(k i : ℂ)) • mode a i := by
+  ext j
+  simp only [mode_apply, velocityCoeff, PiLp.smul_apply, diag_apply,
+    RCLike.real_smul_eq_coe_mul, smul_eq_mul]
+  simp only [map_mul, map_intCast, map_ofNat]
+  change Complex.I * ((2 : ℂ) * (((Real.pi : ℂ)*(k i : ℂ))*a (i,j))) =
+    (2*(Real.pi : ℂ)*Complex.I*(k i : ℂ))*a (i,j)
+  ring
+
+ theorem hasDerivAt_loop (k : ι → ℤ) (a : Coeff ι κ) (t : ℝ) :
+    HasDerivAt (fun s : ℝ => loop k a (s : Time)) (loop k (velocityCoeff k a) (t : Time)) t := by
+  have hs := HasDerivAt.fun_sum (u := univ) (fun i _ =>
+    (hasDerivAt_fourier (1 : ℝ) (k i) t).smul_const (mode a i))
+  change HasDerivAt (fun s : ℝ => loop k a (s : Time)) _ t at hs
+  apply hs.congr_deriv
+  simp only [loop_apply, mode_velocityCoeff, smul_smul, div_one, Complex.ofReal_one]
+  apply sum_congr rfl
+  intro i _
+  congr 1
+  ring
+
+ theorem norm_velocityCoeff_critical (k : ι → ℤ) {H : Vector κ → ℝ}
+    (hH : ContDiff ℝ ∞ H) (a : Coeff ι κ) (ha : gradient (action k H) a = 0) :
+    ‖velocityCoeff k a‖ = ‖project k (gradLoop k hH a)‖ := by
+  rw [gradient_action k hH, sub_eq_zero] at ha
+  simp only [velocityCoeff, ha, norm_smul, Complex.norm_I, one_mul]
+
+ theorem norm_project_grad (k : ι → ℤ) (hk : Function.Injective k)
+    {H : Vector κ → ℝ} (hH : ContDiff ℝ ∞ H) (d : κ → ℝ) {B D : ℝ}
+    (hB : 0 ≤ B) (hD : 0 ≤ D) (hd : ∀ j, |d j| ≤ D)
+    (hrem : ∀ z, ‖gradient H z-(2 : ℝ) • diag d z‖ ≤ B) (a : Coeff ι κ) :
+    ‖project k (gradLoop k hH a)‖ ≤ B+2*D*‖a‖ := by
+  have hb : ‖project k (remainderLoop k hH d a)‖ ≤ B :=
+    project_bounded k hk _ hB (fun t => by rw [remainderLoop_apply]; exact hrem _)
+  have he : project k (gradLoop k hH a) = project k (remainderLoop k hH d a) +
+      (2 : ℝ) • diag (fun p => d p.2) a := by
+    rw [remainderLoop, project_sub, project_loop k hk]
+    abel
+  have hn := norm_add_le (project k (remainderLoop k hH d a))
+    ((2 : ℝ) • diag (fun p => d p.2) a)
+  rw [← he, norm_smul, Real.norm_eq_abs, abs_of_pos (by norm_num : (0:ℝ) < 2)] at hn
+  have hdiag := norm_diag_upper (fun p : ι × κ => d p.2) hD (fun p => hd p.2) a
+  nlinarith
+
+ theorem intervalIntegral_eq_average {E : Type u113} [NormedAddCommGroup E] [NormedSpace ℝ E]
+    (f : Time → E) : (∫ t in (0:ℝ)..1, f (t : Time)) = ∫ t : Time, f t ∂μ := by
+  rw [AddCircle.integral_haarAddCircle]
+  simpa only [zero_add, inv_one, one_smul] using AddCircle.intervalIntegral_preimage (1 : ℝ) 0 f
+
+ def meanAction {H : Vector κ → ℝ} (_hH : ContDiff ℝ ∞ H) (x : C(Time,Vector κ)) : ℝ :=
+    ∫ t : Time, inner (𝕜 := ℝ) (gradient H (x t)) (x t)/2-H (x t) ∂μ
+
+ theorem contDiff_meanAction {H : Vector κ → ℝ} (hH : ContDiff ℝ ∞ H) :
+    ContDiff ℝ ∞ (meanAction hH) := by
+  have hg : ContDiff ℝ ∞ (fun z : Vector κ => inner (𝕜 := ℝ) (gradient H z) z/2-H z) :=
+    (((contDiff_gradient_function hH).inner ℝ contDiff_id).div_const 2).sub hH
+  exact average.contDiff.comp (SmoothPaths.contDiff_superpose hg)
+
+ theorem critical_action_eq (k : ι → ℤ) {H : Vector κ → ℝ}
+    (hH : ContDiff ℝ ∞ H) (a : Coeff ι κ) (ha : gradient (action k H) a = 0) :
+    action k H a = meanAction hH (loop k a) := by
+  have he : (2 : ℝ) • diag (fun p : ι × κ => Real.pi*(k p.1 : ℝ)) a = project k (gradLoop k hH a) := by
+    simpa only [gradient_action k hH, sub_eq_zero] using ha
+  have hi := integral_inner_loop k (gradLoop k hH a) a
+  rw [← he, real_inner_smul_left, ← energy_eq_inner] at hi
+  have hgi : Integrable (fun t : Time => inner (𝕜 := ℝ) (gradient H (loop k a t)) (loop k a t)) μ :=
+    integrable_continuous ((gradLoop k hH a).continuous.inner (loop k a).continuous)
+  have hpi : Integrable (fun t : Time => H (loop k a t)) μ :=
+    integrable_continuous (hH.continuous.comp (loop k a).continuous)
+  unfold meanAction
+  rw [integral_sub (hgi.div_const 2) hpi, integral_div]
+  change action k H a = (∫ t : Time, inner (𝕜 := ℝ) (gradLoop k hH a t) (loop k a t) ∂μ)/2-potential k H a
+  rw [hi]
+  unfold action
+  ring
+
+end
+end FourierPolynomial
+
+namespace FourierPolynomial
+noncomputable section
+open MeasureTheory AddCircle Finset
+open scoped ComplexConjugate ContDiff
+local instance : Fact (0 < (1 : ℝ)) := ⟨zero_lt_one⟩
+variable {ι ν κ : Type} [Fintype ι] [Fintype ν] [Fintype κ]
+open DiagonalQuadratic
+
+ def linkRadius (κ : Type) [Fintype κ] (M : ℝ) : ℝ :=
+  Real.sqrt (Real.pi/(2*(M*(Fintype.card κ : ℝ)+1)))
+
+ def linkLevel (κ : Type) [Fintype κ] (M : ℝ) : ℝ :=
+  Real.pi/4*(linkRadius κ M)^2
+
+ theorem linkRadius_pos {M : ℝ} (hM : 0 ≤ M) : 0 < linkRadius κ M := by
+  unfold linkRadius
+  apply Real.sqrt_pos.2
+  exact div_pos Real.pi_pos (by positivity)
+
+ theorem linkRadius_sq {M : ℝ} (hM : 0 ≤ M) :
+    (linkRadius κ M)^2 = Real.pi/(2*(M*(Fintype.card κ : ℝ)+1)) :=
+  Real.sq_sqrt (by positivity)
+
+ theorem linkLevel_pos {M : ℝ} (hM : 0 ≤ M) : 0 < linkLevel κ M := by
+  unfold linkLevel
+  exact mul_pos (div_pos Real.pi_pos (by norm_num)) (sq_pos_of_pos (linkRadius_pos hM))
+
+ theorem linkLevel_bound {M : ℝ} (hM : 0 ≤ M) :
+    2*linkLevel κ M ≤ Real.pi*(linkRadius κ M)^2-
+      M*(Fintype.card κ : ℝ)*(linkRadius κ M)^4 := by
+  have hden : 0 < 2*(M*(Fintype.card κ : ℝ)+1) := by positivity
+  have he : 2*(M*(Fintype.card κ : ℝ)+1)*(linkRadius κ M)^2 = Real.pi := by
+    rw [linkRadius_sq hM]
+    field_simp
+  have hr : 0 ≤ (linkRadius κ M)^2 := sq_nonneg _
+  have hm : M*(Fintype.card κ : ℝ)*(linkRadius κ M)^2 ≤ Real.pi/2 := by nlinarith
+  have hb := mul_le_mul_of_nonneg_right hm hr
+  unfold linkLevel
+  nlinarith
+
+ theorem exists_link_epsilon {a : ℝ} (ha : 0 < a) :
+    ∃ ε > 0, ∀ s : ℝ, 0 < s → s < ε → Real.pi*s^2 ≤ a/2 := by
+  let ε := Real.sqrt (a/(2*Real.pi))
+  have hε : 0 < ε := Real.sqrt_pos.2 (div_pos ha (by positivity))
+  have hεsq : ε^2 = a/(2*Real.pi) := Real.sq_sqrt (by positivity)
+  refine ⟨ε,hε,fun s hs hse => ?_⟩
+  have hs2 : s^2 ≤ ε^2 := pow_le_pow_left₀ hs.le hse.le 2
+  have hm := mul_le_mul_of_nonneg_left hs2 Real.pi_pos.le
+  rw [hεsq] at hm
+  have hid : Real.pi*(a/(2*Real.pi)) = a/2 := by field_simp
+  rwa [hid] at hm
+
+ theorem exists_link_far {ρ η : ℝ} (hρ : 0 < ρ) (hη : 0 < η) (C : ℝ) :
+    ∃ R : ℝ, ρ < R ∧ ∀ u s : ℝ, 0 ≤ u → R ≤ max u |s| → C-η*(u^2+s^2) ≤ 0 := by
+  let R := max (ρ+1) (max 1 ((|C|+1)/η))
+  have hR : ρ < R := lt_of_lt_of_le (by linarith) (le_max_left _ _)
+  have hR1 : 1 ≤ R := (le_max_left _ _).trans (le_max_right _ _)
+  have hRc : (|C|+1)/η ≤ R := (le_max_right _ _).trans (le_max_right _ _)
+  have hReq : |C|+1 ≤ η*R := by
+    have := (div_le_iff₀ hη).mp hRc
+    linarith
+  refine ⟨R,hR,fun u s _ hfar => ?_⟩
+  have hsq : R^2 ≤ u^2+s^2 := by
+    rcases le_max_iff.mp hfar with hy | hss
+    · have := pow_le_pow_left₀ (by linarith : 0 ≤ R) hy 2
+      nlinarith [sq_nonneg s]
+    · have := pow_le_pow_left₀ (by linarith : 0 ≤ R) hss 2
+      rw [sq_abs] at this
+      nlinarith [sq_nonneg u]
+  have hh : R ≤ R^2 := by nlinarith
+  have hm := mul_le_mul_of_nonneg_left (hh.trans hsq) hη.le
+  linarith [le_abs_self C]
+ theorem finite_critical [DecidableEq ν] [DecidableEq κ]
+    (k : ι → ℤ) (hk0 : ∀ i, k i ≤ 0) (l : ν → ℕ)
+    (hl : Function.Injective l) (hl0 : ∀ i, 0 < l i)
+    (hkl : Function.Injective (Sum.elim k (fun i => (l i : ℤ))))
+    (i₀ : ν) (hl1 : l i₀ = 1) (j₀ : κ)
+    {H : Vector κ → ℝ} (hH : ContDiff ℝ ∞ H) (hH0 : ∀ z, 0 ≤ H z)
+    (d₀ d : κ → ℝ) {C M η γ B : ℝ}
+    (hM : 0 ≤ M) (hη : 0 < η) (hγ : 0 < γ) (hB : 0 ≤ B)
+    (hup : ∀ z, H z ≤ M*‖z‖^4)
+    (hd : ∀ j, η ≤ d₀ j) (hd0 : Real.pi+η ≤ d₀ j₀)
+    (hlo : ∀ z, energy d₀ z-C ≤ H z)
+    (hgap : ∀ i j, γ ≤ |Real.pi*((Sum.elim k (fun i => (l i : ℤ)) i : ℤ) : ℝ)-d j|)
+    (hrem : ∀ z, ‖gradient H z-(2 : ℝ) • diag d z‖ ≤ B) :
+    ∃ x : Coeff (ι ⊕ ν) κ,
+      linkLevel κ M ≤ action (Sum.elim k (fun i => (l i : ℤ))) H x ∧
+      action (Sum.elim k (fun i => (l i : ℤ))) H x ≤ max C (linkLevel κ M) ∧
+      gradient (action (Sum.elim k (fun i => (l i : ℤ))) H) x = 0 := by
+  have ha : 0 < linkLevel κ M := linkLevel_pos hM
+  have hρ : 0 < linkRadius κ M := linkRadius_pos hM
+  obtain ⟨ε,hε,hnear⟩ := exists_link_epsilon ha
+  obtain ⟨R,hR,hfar⟩ := exists_link_far hρ hη C
+  let e : Coeff ν κ := PiLp.single 2 (i₀,j₀) (1 : ℂ)
+  have he : ‖e‖ = 1 := by simp [e]
+  apply FiniteLinking.exists_critical
+    (H := Coeff (ι ⊕ ν) κ) (E := Coeff ι κ) (F := Coeff ν κ)
+    (Φ := action (Sum.elim k (fun i => (l i : ℤ))) H)
+    (positiveScale l hl0) (contDiff_action (Sum.elim k (fun i => (l i : ℤ))) hH) he ha
+    (le_max_right _ _) hρ hε hR
+  ·
+    intro z hz
+    have hb := action_positive_lower k l hl hl0 hH.continuous hM hup z
+    rw [hz] at hb
+    exact (linkLevel_bound hM).trans hb
+  ·
+    intro y s hs hse
+    exact (action_cone_upper k hk0 l hl0 i₀ hl1 j₀ hH0 y s).trans (hnear s hs hse)
+  ·
+    intro y s _ hys
+    have hs := hfar ‖y‖ s (norm_nonneg y) (by simpa only [Prod.norm_def, Real.norm_eq_abs] using hys)
+    exact (action_cone_far k hk0 l hl0 hkl i₀ hl1 j₀ hH.continuous d₀ C η hd hd0 hlo y s).trans
+      (hs.trans (by positivity))
+  ·
+    intro y s _ _
+    have hnon : 0 ≤ η*(‖y‖^2+s^2) := mul_nonneg hη.le (add_nonneg (sq_nonneg _) (sq_nonneg _))
+    exact (action_cone_far k hk0 l hl0 hkl i₀ hl1 j₀ hH.continuous d₀ C η hd hd0 hlo y s).trans
+      ((sub_le_self C hnon).trans (le_max_left _ _))
+  ·
+    exact gradient_coercive _ hkl hH d hγ hB hgap hrem
+
+end
+end FourierPolynomial
+
+end NonsqueezingInline
+
+end OAI
diff --git a/lean/OAI/Geometry/PolarProducts/GradientLinking.lean b/lean/OAI/Geometry/PolarProducts/GradientLinking.lean
new file mode 100644
index 0000000000000000000000000000000000000000..19284b8b757e839e0941dece5a91316c2973694b
--- /dev/null
+++ b/lean/OAI/Geometry/PolarProducts/GradientLinking.lean
@@ -0,0 +1,233 @@
+import OAI.Geometry.PolarProducts.FiniteDescent
+
+namespace OAI
+
+universe u84 u85 u86 u87 u88
+
+section NonsqueezingInline
+section
+open Set Filter Function MeasureTheory
+open scoped Topology ContDiff
+open Set Filter Function MeasureTheory Metric
+open scoped Topology ContDiff NNReal
+
+namespace GradientDescent
+noncomputable section
+open Set Filter Function Metric
+open scoped Topology ContDiff
+variable {E : Type u84} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E]
+
+ theorem contDiff_gradient {Φ : E → ℝ} (hΦ : ContDiff ℝ ∞ Φ) :
+    ContDiff ℝ ∞ (gradient Φ) := by
+  exact (InnerProductSpace.toDual ℝ E).symm.toContinuousLinearEquiv.contDiff.comp
+    (hΦ.fderiv_right (by simp))
+
+ def field (Φ σ : E → ℝ) (ε : ℝ) (x : E) : E :=
+   (-(2*σ x/(‖gradient Φ x‖^2+ε^2))) • gradient Φ x
+
+ theorem denom_pos (Φ : E → ℝ) {ε : ℝ} (hε : 0 < ε) (x : E) :
+    0 < ‖gradient Φ x‖^2+ε^2 := add_pos_of_nonneg_of_pos (sq_nonneg _) (sq_pos_of_pos hε)
+
+ theorem contDiff_field {Φ σ : E → ℝ} (hΦ : ContDiff ℝ ∞ Φ) (hσ : ContDiff ℝ ∞ σ)
+    {ε : ℝ} (hε : 0 < ε) : ContDiff ℝ ∞ (field Φ σ ε) := by
+  have hg := contDiff_gradient hΦ
+  exact ((contDiff_const.mul hσ).div (((contDiff_norm_sq ℝ).comp hg).add contDiff_const)
+    (fun x => (denom_pos Φ hε x).ne')).neg.smul hg
+
+ theorem field_eq_zero {Φ σ : E → ℝ} {ε : ℝ} {x : E} (hσ : σ x = 0) :
+    field Φ σ ε x = 0 := by simp [field, hσ]
+
+ theorem field_rate (Φ σ : E → ℝ) (ε : ℝ) (x : E) :
+    fderiv ℝ Φ x (field Φ σ ε x) = -(2*σ x/(‖gradient Φ x‖^2+ε^2))*‖gradient Φ x‖^2 := by
+  rw [field, map_smul, smul_eq_mul, ← inner_gradient_left]
+  rw [real_inner_self_eq_norm_sq]
+
+ theorem field_nonincrease (Φ : E → ℝ) {σ : E → ℝ} (hσ : ∀ x, 0 ≤ σ x)
+    {ε : ℝ} (hε : 0 < ε) (x : E) : fderiv ℝ Φ x (field Φ σ ε x) ≤ 0 := by
+  rw [field_rate]
+  apply mul_nonpos_of_nonpos_of_nonneg
+  · exact neg_nonpos.mpr (div_nonneg (mul_nonneg (by norm_num) (hσ x)) (denom_pos Φ hε x).le)
+  · exact sq_nonneg _
+
+ theorem field_decrease (Φ : E → ℝ) {σ : E → ℝ} {ε : ℝ} (hε : 0 < ε)
+    {x : E} (hσ : σ x = 1) (hx : ε ≤ ‖gradient Φ x‖) :
+    fderiv ℝ Φ x (field Φ σ ε x) ≤ -1 := by
+  rw [field_rate, hσ, mul_one]
+  have hd := denom_pos Φ hε x
+  have he : -(2/(‖gradient Φ x‖^2+ε^2))*‖gradient Φ x‖^2 =
+      -(2*‖gradient Φ x‖^2/(‖gradient Φ x‖^2+ε^2)) := by ring
+  rw [he, neg_le_neg_iff, le_div_iff₀ hd]
+  nlinarith [norm_nonneg (gradient Φ x)]
+
+ theorem field_norm_le (Φ : E → ℝ) {σ : E → ℝ} (hσ : ∀ x, σ x ∈ Icc (0 : ℝ) 1)
+    {ε : ℝ} (hε : 0 < ε) (x : E) : ‖field Φ σ ε x‖ ≤ 1/ε := by
+  have hden := denom_pos Φ hε x
+  rw [field, norm_smul, Real.norm_eq_abs, abs_neg,
+    abs_of_nonneg (div_nonneg (mul_nonneg (by norm_num) (hσ x).1) hden.le)]
+  rw [div_mul_eq_mul_div, div_le_div_iff₀ hden hε]
+  have hs := mul_le_mul_of_nonneg_right (hσ x).2 (norm_nonneg (gradient Φ x))
+  nlinarith [sq_nonneg (‖gradient Φ x‖-ε)]
+
+ theorem field_hasCompactSupport {Φ σ : E → ℝ} (hσ : HasCompactSupport σ) (ε : ℝ) :
+    HasCompactSupport (field Φ σ ε) := by
+  apply HasCompactSupport.of_support_subset_isCompact hσ.isCompact
+  intro x hx
+  by_contra hn
+  apply hx
+  exact field_eq_zero (image_eq_zero_of_notMem_tsupport hn)
+
+end
+end GradientDescent
+
+namespace GradientDescent
+noncomputable section
+open Set Filter Function Metric
+open scoped Topology ContDiff NNReal
+variable {E : Type u85} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E]
+
+ theorem exists_uniform_gradient_lower {Φ : E → ℝ} (hΦ : ContDiff ℝ ∞ Φ)
+    {a b : ℝ} (hnocrit : ∀ x, a ≤ Φ x → Φ x ≤ b → gradient Φ x ≠ 0)
+    (hcoercive : ∃ R : ℝ, ∀ x, R ≤ ‖x‖ → 1 ≤ ‖gradient Φ x‖) :
+    ∃ ε : ℝ, 0 < ε ∧ ∀ x, a ≤ Φ x → Φ x ≤ b → ε ≤ ‖gradient Φ x‖ := by
+  obtain ⟨R,hR⟩ := hcoercive
+  let S := closedBall (0 : E) R ∩ Φ ⁻¹' Icc a b
+  have hS : IsCompact S := (isCompact_closedBall (0 : E) R).inter_right
+    (isClosed_Icc.preimage hΦ.continuous)
+  by_cases hne : S.Nonempty
+  · obtain ⟨x,hx,hm⟩ := hS.exists_isMinOn hne (contDiff_gradient hΦ).continuous.norm.continuousOn
+    have hxg : 0 < ‖gradient Φ x‖ := norm_pos_iff.mpr (hnocrit x hx.2.1 hx.2.2)
+    refine ⟨min 1 ‖gradient Φ x‖, lt_min zero_lt_one hxg, ?_⟩
+    intro y ha hb
+    by_cases hy : ‖y‖ ≤ R
+    · exact (min_le_right _ _).trans (hm ⟨by simpa using hy, ha, hb⟩)
+    · exact (min_le_left _ _).trans (hR y (le_of_lt (lt_of_not_ge hy)))
+  · refine ⟨1, zero_lt_one, ?_⟩
+    intro y ha hb
+    apply hR
+    by_contra h
+    exact hne ⟨y, by simpa using (le_of_lt (lt_of_not_ge h)), ha, hb⟩
+
+ def energyCutoff (Φ : E → ℝ) (a : ℝ) (x : E) : ℝ :=
+   Real.smoothTransition ((2*Φ x-a)/a)
+
+ omit [FiniteDimensional ℝ E] in
+ theorem energyCutoff_contDiff {Φ : E → ℝ} (hΦ : ContDiff ℝ ∞ Φ) (a : ℝ) :
+    ContDiff ℝ ∞ (energyCutoff Φ a) := by
+  exact Real.smoothTransition.contDiff.comp
+    (((contDiff_const.mul hΦ).sub contDiff_const).div_const a)
+
+ omit [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] in
+ theorem energyCutoff_mem (Φ : E → ℝ) (a : ℝ) (x : E) :
+    energyCutoff Φ a x ∈ Icc (0 : ℝ) 1 :=
+  ⟨Real.smoothTransition.nonneg _, Real.smoothTransition.le_one _⟩
+
+ omit [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] in
+ theorem energyCutoff_one (Φ : E → ℝ) {a : ℝ} (ha : 0 < a) {x : E} (hx : a ≤ Φ x) :
+    energyCutoff Φ a x = 1 := by
+  apply Real.smoothTransition.one_of_one_le
+  rw [le_div_iff₀ ha]
+  linarith
+
+ omit [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] in
+ theorem energyCutoff_zero (Φ : E → ℝ) {a : ℝ} (ha : 0 < a) {x : E} (hx : Φ x ≤ a/2) :
+    energyCutoff Φ a x = 0 := by
+  apply Real.smoothTransition.zero_of_nonpos
+  exact div_nonpos_of_nonpos_of_nonneg (by linarith) ha.le
+
+ theorem exists_cutoff_field {Φ : E → ℝ} (hΦ : ContDiff ℝ ∞ Φ)
+    {a b ε M : ℝ} (ha : 0 < a) (hε : 0 < ε) (hM : 0 < M)
+    (hgrad : ∀ x, a ≤ Φ x → Φ x ≤ b → ε ≤ ‖gradient Φ x‖) :
+    ∃ (V : E → E) (K : ℝ≥0), ContDiff ℝ ∞ V ∧ LipschitzWith K V ∧
+      (∀ x, ‖V x‖ ≤ 1/ε) ∧ (∀ x, fderiv ℝ Φ x (V x) ≤ 0) ∧
+      (∀ x, Φ x ≤ a/2 → V x = 0) ∧
+      (∀ x, ‖x‖ ≤ M → a ≤ Φ x → Φ x ≤ b → fderiv ℝ Φ x (V x) ≤ -1) := by
+  let χ : ContDiffBump (0 : E) := ⟨M, M+1, hM, by linarith⟩
+  let σ : E → ℝ := fun x => χ x * energyCutoff Φ a x
+  have hσ : ContDiff ℝ ∞ σ := χ.contDiff.mul (energyCutoff_contDiff hΦ a)
+  have hσc : HasCompactSupport σ := χ.hasCompactSupport.mul_right
+  have hσm (x : E) : σ x ∈ Icc (0 : ℝ) 1 := by
+    refine ⟨mul_nonneg χ.nonneg (energyCutoff_mem Φ a x).1, ?_⟩
+    exact (mul_le_of_le_one_left (energyCutoff_mem Φ a x).1 χ.le_one).trans
+      (energyCutoff_mem Φ a x).2
+  let V := field Φ σ ε
+  have hV : ContDiff ℝ ∞ V := contDiff_field hΦ hσ hε
+  obtain ⟨K,hK⟩ := ContDiff.lipschitzWith_of_hasCompactSupport
+    (field_hasCompactSupport hσc ε) hV (by simp)
+  refine ⟨V,K,hV,hK,field_norm_le Φ hσm hε,field_nonincrease Φ (fun x => (hσm x).1) hε,?_,?_⟩
+  · intro x hx
+    apply field_eq_zero
+    simp only [σ, energyCutoff_zero Φ ha hx, mul_zero]
+  · intro x hx hxa hxb
+    apply field_decrease Φ hε _ (hgrad x hxa hxb)
+    have hχ : χ x = 1 := χ.one_of_mem_closedBall (by simpa using hx)
+    simp only [σ, hχ, energyCutoff_one Φ ha hxa, mul_one]
+
+end
+end GradientDescent
+
+namespace FiniteLinking
+noncomputable section
+open Set Filter Function Metric
+open scoped Topology ContDiff NNReal
+variable {H : Type u86} {E : Type u87} {F : Type u88} [NormedAddCommGroup H] [InnerProductSpace ℝ H]
+  [FiniteDimensional ℝ H] [NormedAddCommGroup E] [NormedSpace ℝ E]
+  [FiniteDimensional ℝ E] [NormedAddCommGroup F] [InnerProductSpace ℝ F]
+
+ theorem exists_critical (l : H ≃L[ℝ] E × F) {Φ : H → ℝ} (hΦ : ContDiff ℝ ∞ Φ)
+    {e : F} (he : ‖e‖ = 1) {a b ρ ε R : ℝ}
+    (ha : 0 < a) (hba : a ≤ b) (hρ : 0 < ρ) (hε : 0 < ε) (hR : ρ < R)
+    (hpos : ∀ z : F, ‖z‖ = ρ → 2*a ≤ Φ (l.symm (0,z)))
+    (hnear : ∀ (y : E) (s : ℝ), 0 < s → s < ε → Φ (l.symm (y,s • e)) ≤ a/2)
+    (hfar : ∀ (y : E) (s : ℝ), 0 < s → R ≤ ‖(y,s)‖ → Φ (l.symm (y,s • e)) ≤ a/2)
+    (hupper : ∀ (y : E) (s : ℝ), 0 < s → ‖(y,s)‖ < R → Φ (l.symm (y,s • e)) ≤ b)
+    (hcoercive : ∃ C : ℝ, ∀ x, C ≤ ‖x‖ → 1 ≤ ‖gradient Φ x‖) :
+    ∃ x : H, a ≤ Φ x ∧ Φ x ≤ b ∧ gradient Φ x = 0 := by
+  by_contra! hn
+  obtain ⟨δ,hδ,hgrad⟩ := GradientDescent.exists_uniform_gradient_lower hΦ hn hcoercive
+  let B := ‖l.symm.toContinuousLinearMap‖ * R
+  let T := b-a+1
+  let M := B+T*(1/δ)+1
+  have hR0 : 0 < R := hρ.trans hR
+  have hB : 0 ≤ B := by dsimp [B]; positivity
+  have hT : 0 < T := by dsimp [T]; linarith
+  have hM : 0 < M := by
+    have : 0 < T*(1/δ) := mul_pos hT (one_div_pos.mpr hδ)
+    dsimp [M]
+    linarith
+  obtain ⟨V,K,hVs,hV,hL,hdec,hfix,hrate⟩ :=
+    GradientDescent.exists_cutoff_field hΦ ha hδ hM hgrad
+  obtain ⟨η,hη,_,hηfix,hηlow⟩ := FlowDescent.exists_descent_map hV hVs
+    (hΦ.differentiable (by simp)) hdec hT (show b-T ≤ a by dsimp [T]; linarith)
+    hL (show B+T*(1/δ) ≤ M by dsimp [M]; linarith) hrate
+  let ψ : E × F → E × F := l ∘ η ∘ l.symm
+  have hψ : ContDiff ℝ ∞ ψ := l.contDiff.comp (hη.comp l.symm.contDiff)
+  have hψeq (y : E) (s : ℝ) (hx : Φ (l.symm (y,s • e)) ≤ a/2) :
+      ψ (y,s • e) = (y,s • e) := by
+    simp only [ψ, Function.comp_apply, hηfix _ (hfix _ hx), l.apply_symm_apply]
+  obtain ⟨y,s,hs,hys,hfst,hsnd⟩ := SmoothLinking.sphere_linking hψ he hρ hε hR
+    (fun y s hs heps => hψeq y s (hnear y s hs heps))
+    (fun y s hs hnorm => hψeq y s (hfar y s hs hnorm))
+  have hncone : ‖(y,s • e)‖ = ‖(y,s)‖ := by
+    simp only [Prod.norm_def, norm_smul, he, mul_one, Real.norm_eq_abs]
+  have hxB : ‖l.symm (y,s • e)‖ ≤ B :=
+    (l.symm.toContinuousLinearMap.le_opNorm _).trans (by
+      rw [hncone]
+      exact mul_le_mul_of_nonneg_left hys.le (norm_nonneg l.symm.toContinuousLinearMap))
+  have hlow := hηlow (l.symm (y,s • e)) hxB (hupper y s hs hys)
+  have hhigh := hpos (ψ (y,s • e)).2 hsnd
+  have heq : l.symm (0,(ψ (y,s • e)).2) = η (l.symm (y,s • e)) := by
+    rw [← hfst]
+    change l.symm (ψ (y,s • e)) = _
+    simp only [ψ, Function.comp_apply, l.symm_apply_apply]
+  rw [heq] at hhigh
+  linarith
+
+end
+end FiniteLinking
+
+end
+
+end NonsqueezingInline
+
+end OAI
diff --git a/lean/OAI/Geometry/PolarProducts/HamiltonianGrowth.lean b/lean/OAI/Geometry/PolarProducts/HamiltonianGrowth.lean
new file mode 100644
index 0000000000000000000000000000000000000000..9f2134602373f79733bc1bc9c393569499813452
--- /dev/null
+++ b/lean/OAI/Geometry/PolarProducts/HamiltonianGrowth.lean
@@ -0,0 +1,236 @@
+import OAI.Geometry.PolarProducts.HamiltonianODE
+
+namespace OAI
+
+universe u119 u120
+
+section NonsqueezingInline
+
+namespace HamiltonianGrowth
+noncomputable section
+open Set Filter
+open scoped ContDiff Topology
+open FourierPolynomial DiagonalQuadratic HamiltonianODE
+variable {κ : Type u119} [Fintype κ]
+
+ theorem contDiff_gradient {H : Vector κ → ℝ} (hH : ContDiff ℝ ∞ H) :
+    ContDiff ℝ ∞ (gradient H) :=
+   (InnerProductSpace.toDual ℝ _).symm.toContinuousLinearEquiv.contDiff.comp
+     (hH.fderiv_right (by simp))
+
+ theorem gradient_translate {H : Vector κ → ℝ} (hH : ContDiff ℝ ∞ H) (p z : Vector κ) :
+    gradient (fun w => H (w+p)) z = gradient H (z+p) := by
+   apply HasGradientAt.gradient
+   rw [hasGradientAt_iff_hasFDerivAt]
+   have hh := ((hH.differentiable (by simp) (z+p)).hasGradientAt.hasFDerivAt).comp z
+       ((hasFDerivAt_id z).add_const p)
+   convert! hh using 1
+
+ theorem energy_upper (d : κ → ℝ) (D : ℝ) (hd : ∀ j, d j ≤ D) (z : Vector κ) :
+    energy d z ≤ D*‖z‖^2 := by
+   simp only [energy, EuclideanSpace.norm_sq_eq, Finset.mul_sum]
+   exact Finset.sum_le_sum (fun j _ => mul_le_mul_of_nonneg_right (hd j) (sq_nonneg _))
+
+ theorem energy_sub_uniform (d : κ → ℝ) (η : ℝ) (z : Vector κ) :
+    energy (fun j => d j-η) z = energy d z-η*‖z‖^2 := by
+   simp only [energy, sub_mul, Finset.sum_sub_distrib, EuclideanSpace.norm_sq_eq, Finset.mul_sum]
+
+ theorem quadratic_error {H : Vector κ → ℝ} (hH : ContDiff ℝ ∞ H) (h0 : H 0 = 0)
+    (d : κ → ℝ) {B : ℝ} (hb : ∀ z, ‖gradient H z-(2 : ℝ) • diag d z‖ ≤ B) (z : Vector κ) :
+    |H z-energy d z| ≤ B*‖z‖ := by
+   have hr (w : Vector κ) : HasGradientAt (fun v => H v-energy d v)
+       (gradient H w-(2 : ℝ) • diag d w) w := by
+     rw [hasGradientAt_iff_hasFDerivAt]
+     have hh := ((hH.differentiable (by simp) w).hasGradientAt.hasFDerivAt).sub
+       (hasGradientAt_energy d w).hasFDerivAt
+     convert! hh using 1
+     ext v
+     change inner (𝕜 := ℝ) (gradient H w-(2 : ℝ) • diag d w) v =
+       inner (𝕜 := ℝ) (gradient H w) v-inner (𝕜 := ℝ) ((2 : ℝ) • diag d w) v
+     exact inner_sub_left _ _ _
+   have bound (w : Vector κ) : ‖fderiv ℝ (fun v => H v-energy d v) w‖ ≤ B := by
+     rw [(hr w).hasFDerivAt.fderiv]
+     simpa only [LinearIsometryEquiv.norm_map] using hb w
+   have hm := (convex_univ : Convex ℝ (univ : Set (Vector κ))).norm_image_sub_le_of_norm_fderiv_le
+     (fun w _ => (hr w).hasFDerivAt.differentiableAt) (fun w _ => bound w)
+     (mem_univ (0 : Vector κ)) (mem_univ z)
+   simpa [h0, energy] using hm
+
+ theorem quartic_bound {H : Vector κ → ℝ} {A B r : ℝ}
+    (hA : 0 ≤ A) (hB : 0 ≤ B) (hr : 0 < r)
+    (hz : ∀ z, ‖z‖ < r → H z = 0) (hu : ∀ z, H z ≤ A*‖z‖^2+B*‖z‖) :
+    ∃ M : ℝ, 0 ≤ M ∧ ∀ z, H z ≤ M*‖z‖^4 := by
+   refine ⟨A/r^2+B/r^3, by positivity, ?_⟩
+   intro z
+   by_cases hzr : ‖z‖ < r
+   · rw [hz z hzr]; positivity
+   have hnr : r ≤ ‖z‖ := le_of_not_gt hzr
+   have h2 : r^2*‖z‖^2 ≤ ‖z‖^4 := by
+     have := mul_le_mul_of_nonneg_right (pow_le_pow_left₀ hr.le hnr 2) (sq_nonneg ‖z‖)
+     nlinarith only [this]
+   have h3 : r^3*‖z‖ ≤ ‖z‖^4 := by
+     have := mul_le_mul_of_nonneg_right (pow_le_pow_left₀ hr.le hnr 3) (norm_nonneg z)
+     nlinarith only [this]
+   have ha : A*‖z‖^2 ≤ (A/r^2)*‖z‖^4 := by
+     rw [div_mul_eq_mul_div, le_div_iff₀ (pow_pos hr 2)]
+     nlinarith only [mul_le_mul_of_nonneg_left h2 hA]
+   have hb : B*‖z‖ ≤ (B/r^3)*‖z‖^4 := by
+     rw [div_mul_eq_mul_div, le_div_iff₀ (pow_pos hr 3)]
+     nlinarith only [mul_le_mul_of_nonneg_left h3 hB]
+   calc
+     H z ≤ A*‖z‖^2+B*‖z‖ := hu z
+     _ ≤ (A/r^2+B/r^3)*‖z‖^4 := by nlinarith only [ha,hb]
+
+ theorem lower_bound {H : Vector κ → ℝ} (hH : ContDiff ℝ ∞ H) (h0 : H 0 = 0)
+    (d : κ → ℝ) {B η : ℝ} (hη : 0 < η)
+    (hb : ∀ z, ‖gradient H z-(2 : ℝ) • diag d z‖ ≤ B) (z : Vector κ) :
+    energy (fun j => d j-η) z-B^2/(4*η) ≤ H z := by
+   have he := (abs_le.mp (quadratic_error hH h0 d hb z)).1
+   have hc : (4*η)*(B^2/(4*η)) = B^2 := by field_simp
+   rw [energy_sub_uniform]
+   nlinarith [sq_nonneg (2*η*‖z‖-B)]
+
+ theorem bounded_remainder_translate {H : Vector κ → ℝ} (hH : ContDiff ℝ ∞ H)
+    (d : κ → ℝ) {B : ℝ} (hb : ∀ z, ‖gradient H z-(2 : ℝ) • diag d z‖ ≤ B)
+    (p z : Vector κ) :
+    ‖gradient (fun w => H (w+p)) z-(2 : ℝ) • diag d z‖ ≤ B+‖(2 : ℝ) • diag d p‖ := by
+   rw [gradient_translate hH]
+   have hd : diag d (z+p) = diag d z+diag d p := (diagCLM d).map_add z p
+   have he : gradient H (z+p)-(2 : ℝ) • diag d z =
+       (gradient H (z+p)-(2 : ℝ) • diag d (z+p))+(2 : ℝ) • diag d p := by
+     rw [hd, smul_add]; abel
+   rw [he]
+   have hh := norm_add_le (gradient H (z+p)-(2 : ℝ) • diag d (z+p)) ((2 : ℝ) • diag d p)
+   linarith [hb (z+p)]
+
+ theorem bound_remainder_of_compact {H : Vector κ → ℝ} (hH : ContDiff ℝ ∞ H)
+    (d : κ → ℝ) {S : Set (Vector κ)} (hS : IsCompact S)
+    (he : ∀ z ∉ S, gradient H z = (2 : ℝ) • diag d z) :
+    ∃ B : ℝ, 0 ≤ B ∧ ∀ z, ‖gradient H z-(2 : ℝ) • diag d z‖ ≤ B := by
+   have hc : Continuous (fun z => gradient H z-(2 : ℝ) • diag d z) :=
+     (contDiff_gradient hH).continuous.sub ((diagCLM d).continuous.const_smul (2 : ℝ))
+   obtain ⟨B,hB⟩ := hS.exists_bound_of_continuousOn hc.continuousOn
+   refine ⟨max B 0,le_max_right _ _,?_⟩
+   intro z
+   by_cases hz : z ∈ S
+   · exact (hB z hz).trans (le_max_left _ _)
+   · rw [he z hz, sub_self, norm_zero]
+     exact le_max_right _ _
+
+end
+end HamiltonianGrowth
+
+namespace HamiltonianPeriodic
+noncomputable section
+open MeasureTheory Set Filter
+open scoped ContDiff Topology
+open FourierPolynomial DiagonalQuadratic HamiltonianODE HamiltonianGrowth
+variable {κ : Type u120} [Fintype κ]
+
+ def actionIntegral (H : Vector κ → ℝ) (x : ℝ → Vector κ) : ℝ :=
+   ∫ t in (0:ℝ)..1, inner (𝕜 := ℝ) (gradient H (x t)) (x t)/2-H (x t)
+
+ theorem integral_gradient_pairing {H : Vector κ → ℝ} (hH : ContDiff ℝ ∞ H)
+    {x : ℝ → Vector κ} (hx : ∀ t, HasDerivAt x (Complex.I • gradient H (x t)) t)
+    (hp : x 1 = x 0) (p : Vector κ) :
+    (∫ t in (0:ℝ)..1, inner (𝕜 := ℝ) (gradient H (x t)) p) = 0 := by
+   have hc : Continuous x := continuous_iff_continuousAt.mpr (fun t => (hx t).continuousAt)
+   have hi : IntervalIntegrable (fun t => inner (𝕜 := ℝ) (gradient H (x t)) p) volume 0 1 :=
+     (((contDiff_gradient hH).continuous.comp hc).inner continuous_const).intervalIntegrable 0 1
+   have hd (t : ℝ) : HasDerivAt (fun s => inner (𝕜 := ℝ) (x s) (Complex.I • p))
+       (inner (𝕜 := ℝ) (gradient H (x t)) p) t := by
+     simpa only [inner_zero_right, add_zero, zero_add, inner_I_I] using
+       (hx t).inner ℝ (hasDerivAt_const t (Complex.I • p))
+   have he := intervalIntegral.integral_eq_sub_of_hasDerivAt (fun t _ => hd t) hi
+   simpa only [hp, sub_self] using he
+
+ theorem action_translate {H : Vector κ → ℝ} (hH : ContDiff ℝ ∞ H) (p : Vector κ)
+    {y : ℝ → Vector κ}
+    (hy : ∀ t, HasDerivAt y (Complex.I • gradient (fun z => H (z+p)) (y t)) t)
+    (hp : y 1 = y 0) :
+    actionIntegral H (fun t => y t+p) = actionIntegral (fun z => H (z+p)) y := by
+   have hc : Continuous y := continuous_iff_continuousAt.mpr (fun t => (hy t).continuousAt)
+   have hxt (t : ℝ) : HasDerivAt (fun s => y s+p) (Complex.I • gradient H (y t+p)) t := by
+     simpa only [gradient_translate hH] using (hy t).add_const p
+   have hz := integral_gradient_pairing hH hxt (by simp only [hp]) p
+   have hg : Continuous (fun t => gradient H (y t+p)) :=
+     (contDiff_gradient hH).continuous.comp (hc.add continuous_const)
+   have hi1 : IntervalIntegrable (fun t => inner (𝕜 := ℝ) (gradient H (y t+p)) (y t)/2-H (y t+p)) volume 0 1 :=
+     (((hg.inner hc).div_const 2).sub (hH.continuous.comp (hc.add continuous_const))).intervalIntegrable _ _
+   have hi2 : IntervalIntegrable (fun t => inner (𝕜 := ℝ) (gradient H (y t+p)) p/2) volume 0 1 :=
+     ((hg.inner continuous_const).div_const 2).intervalIntegrable _ _
+   unfold actionIntegral
+   simp_rw [gradient_translate hH]
+   have he (t : ℝ) : inner (𝕜 := ℝ) (gradient H (y t+p)) (y t+p)/2-H (y t+p) =
+       (inner (𝕜 := ℝ) (gradient H (y t+p)) (y t)/2-H (y t+p))+
+       inner (𝕜 := ℝ) (gradient H (y t+p)) p/2 := by rw [inner_add_right]; ring
+   simp_rw [he]
+   rw [intervalIntegral.integral_add hi1 hi2, intervalIntegral.integral_div, hz, zero_div, add_zero]
+
+end
+end HamiltonianPeriodic
+
+namespace HamiltonianPeriodic
+noncomputable section
+open MeasureTheory Set Filter
+open scoped ContDiff Topology
+open FourierPolynomial DiagonalQuadratic HamiltonianODE HamiltonianGrowth
+variable {κ : Type} [Fintype κ] [DecidableEq κ]
+
+ theorem exists_positive_flat (j₀ : κ) {H : Vector κ → ℝ}
+    (hH : ContDiff ℝ ∞ H) (hH0 : ∀ z, 0 ≤ H z)
+    (hflat : ∀ᶠ z in 𝓝 (0 : Vector κ), H z = 0)
+    (d : κ → ℝ) {η γ B D : ℝ}
+    (hη : 0 < η) (hγ : 0 < γ) (hB : 0 ≤ B) (hD : 0 ≤ D)
+    (hd : ∀ j, 2*η ≤ d j) (hd0 : Real.pi+2*η ≤ d j₀) (hdD : ∀ j, |d j| ≤ D)
+    (hgap : ∀ m : ℤ, ∀ j, γ ≤ |Real.pi*(m : ℝ)-d j|)
+    (hrem : ∀ z, ‖gradient H z-(2 : ℝ) • diag d z‖ ≤ B) :
+    ∃ x : ℝ → Vector κ, x 1 = x 0 ∧ 0 < actionIntegral H x ∧
+      ∀ t : ℝ, HasDerivAt x (Complex.I • gradient H (x t)) t := by
+   have h0 := hflat.self_of_nhds
+   obtain ⟨r,hr,hzr⟩ := Metric.eventually_nhds_iff.mp hflat
+   have hu (z : Vector κ) : H z ≤ D*‖z‖^2+B*‖z‖ := by
+     have he := (abs_le.mp (quadratic_error hH h0 d hrem z)).2
+     have hd' := energy_upper d D (fun j => (le_abs_self _).trans (hdD j)) z
+     linarith
+   obtain ⟨M,hM,hup⟩ := quartic_bound hD hB hr (fun z hz => hzr (by simpa only [dist_zero_right] using hz)) hu
+   obtain ⟨x,ha,hx⟩ := exists_positive_periodic j₀ hH hH0 (fun j => d j-η) d hM hη hγ hB hD hup
+     (fun j => by linarith [hd j]) (by linarith)
+     (lower_bound hH h0 d hη hrem) hdD hgap hrem
+   refine ⟨fun t => x (t : Time),?_,?_,hx⟩
+   · simp only [AddCircle.coe_period, AddCircle.coe_zero]
+   · change 0 < ∫ t in (0:ℝ)..1, (fun s : Time => inner (𝕜 := ℝ) (gradient H (x s)) (x s)/2-H (x s)) (t : Time)
+     have he := intervalIntegral_eq_average (fun s : Time => inner (𝕜 := ℝ) (gradient H (x s)) (x s)/2-H (x s))
+     exact he.symm ▸ ha
+
+ theorem exists_positive_at_flat_point (j₀ : κ) {H : Vector κ → ℝ}
+    (hH : ContDiff ℝ ∞ H) (hH0 : ∀ z, 0 ≤ H z) (p : Vector κ)
+    (hflat : ∀ᶠ z in 𝓝 p, H z = 0)
+    (d : κ → ℝ) {η γ B D : ℝ}
+    (hη : 0 < η) (hγ : 0 < γ) (hB : 0 ≤ B) (hD : 0 ≤ D)
+    (hd : ∀ j, 2*η ≤ d j) (hd0 : Real.pi+2*η ≤ d j₀) (hdD : ∀ j, |d j| ≤ D)
+    (hgap : ∀ m : ℤ, ∀ j, γ ≤ |Real.pi*(m : ℝ)-d j|)
+    (hrem : ∀ z, ‖gradient H z-(2 : ℝ) • diag d z‖ ≤ B) :
+    ∃ x : ℝ → Vector κ, x 1 = x 0 ∧ 0 < actionIntegral H x ∧
+      ∀ t : ℝ, HasDerivAt x (Complex.I • gradient H (x t)) t := by
+   have ht : ContDiff ℝ ∞ (fun z : Vector κ => H (z+p)) := hH.comp (contDiff_id.add contDiff_const)
+   have hz : ∀ᶠ z in 𝓝 (0 : Vector κ), H (z+p) = 0 := by
+     have hc : Tendsto (fun z : Vector κ => z+p) (𝓝 0) (𝓝 p) := by
+       have hc : Continuous (fun z : Vector κ => z+p) := continuous_id.add continuous_const
+       have hc0 : Tendsto (fun z : Vector κ => z+p) (𝓝 0) (𝓝 ((0 : Vector κ)+p)) := hc.continuousAt
+       simpa only [zero_add] using hc0
+     exact hc.eventually hflat
+   obtain ⟨y,hp,ha,hy⟩ := exists_positive_flat j₀ ht (fun z => hH0 (z+p)) hz d hη hγ
+     (add_nonneg hB (norm_nonneg _)) hD hd hd0 hdD hgap (bounded_remainder_translate hH d hrem p)
+   refine ⟨fun t => y t+p,by simp only [hp],?_,?_⟩
+   · rw [action_translate hH p hy hp]
+     exact ha
+   · intro t
+     simpa only [gradient_translate hH] using (hy t).add_const p
+
+end
+end HamiltonianPeriodic
+
+end NonsqueezingInline
+
+end OAI
diff --git a/lean/OAI/Geometry/PolarProducts/HamiltonianODE.lean b/lean/OAI/Geometry/PolarProducts/HamiltonianODE.lean
new file mode 100644
index 0000000000000000000000000000000000000000..7fca9180ad13780fe7ac26918134dec02cc80c98
--- /dev/null
+++ b/lean/OAI/Geometry/PolarProducts/HamiltonianODE.lean
@@ -0,0 +1,206 @@
+import OAI.Geometry.PolarProducts.PeriodicSolutions
+
+namespace OAI
+
+universe u116 u117 u118
+
+section NonsqueezingInline
+
+namespace HamiltonianODE
+noncomputable section
+open MeasureTheory AddCircle Finset Set Filter
+open scoped ComplexConjugate ContDiff Topology
+local instance : Fact (0 < (1 : ℝ)) := ⟨zero_lt_one⟩
+open FourierPolynomial DiagonalQuadratic
+variable {κ : Type u116} [Fintype κ]
+
+ theorem inner_I_self (z : Vector κ) : inner (𝕜 := ℝ) z (Complex.I • z) = 0 := by
+  rw [real_inner_eq_complex_re, inner_smul_right, inner_self_eq_norm_sq_to_K]
+  simp [pow_two]
+
+ theorem inner_I_left (z w : Vector κ) :
+    inner (𝕜 := ℝ) (Complex.I • z) w = -inner (𝕜 := ℝ) z (Complex.I • w) := by
+  simp only [real_inner_eq_complex_re, inner_smul_left, inner_smul_right]
+  simp [Complex.mul_re]
+
+ theorem inner_I_I (z w : Vector κ) :
+    inner (𝕜 := ℝ) (Complex.I • z) (Complex.I • w) = inner (𝕜 := ℝ) z w := by
+  rw [inner_I_left, smul_smul]
+  simp
+
+ theorem zero_derivative_constant {E : Type u117} [NormedAddCommGroup E] [NormedSpace ℝ E]
+    {x : ℝ → E} (hx : ∀ t, HasDerivAt x 0 t) (s t : ℝ) : x s = x t := by
+  exact is_const_of_deriv_eq_zero (fun t => (hx t).differentiableAt) (fun t => (hx t).deriv) s t
+
+ theorem energy_constant {H : Vector κ → ℝ} (hH : ContDiff ℝ ∞ H) {x : ℝ → Vector κ}
+    (hx : ∀ t, HasDerivAt x (Complex.I • gradient H (x t)) t) (s t : ℝ) : H (x s) = H (x t) := by
+  apply zero_derivative_constant (x := H ∘ x) _ s t
+  intro u
+  have hh := ((hH.differentiable (by simp) (x u)).hasGradientAt.hasFDerivAt).comp_hasDerivAt u (hx u)
+  simpa only [InnerProductSpace.toDual_apply_apply, inner_I_self] using hh
+
+ theorem quadratic_constant (d : κ → ℝ) {a : ℝ → ℝ} {x : ℝ → Vector κ}
+    (hx : ∀ t, HasDerivAt x (Complex.I • ((2*a t) • diag d (x t))) t) (s t : ℝ) :
+    energy d (x s) = energy d (x t) := by
+  apply zero_derivative_constant (x := energy d ∘ x) _ s t
+  intro u
+  have hh := (hasGradientAt_energy d (x u)).hasFDerivAt.comp_hasDerivAt u (hx u)
+  simpa only [InnerProductSpace.toDual_apply_apply, real_inner_smul_left, smul_comm Complex.I (2*a u),
+    real_inner_smul_right, inner_I_self, mul_zero] using hh
+
+ theorem linear_solution {x : ℝ → ℂ} (v : ℂ)
+    (hx : ∀ t, HasDerivAt x (v*x t) t) (t : ℝ) :
+    x t = Complex.exp (v*(t : ℂ))*x 0 := by
+  have hd (s : ℝ) : HasDerivAt (fun u : ℝ => Complex.exp (-v*(u : ℂ))*x u) 0 s := by
+    have he := ((Complex.ofRealCLM.hasDerivAt (x := s)).const_mul (-v)).cexp
+    have hh := he.mul (hx s)
+    apply hh.congr_deriv
+    simp only [Complex.ofRealCLM_apply, Complex.ofReal_one, mul_one]
+    ring
+  have hc := zero_derivative_constant hd t 0
+  simp only [Complex.ofReal_zero, mul_zero, Complex.exp_zero, one_mul] at hc
+  have he := congrArg (fun z : ℂ => Complex.exp (v*(t:ℂ))*z) hc
+  rw [← mul_assoc, ← Complex.exp_add] at he
+  have hz : v*(t:ℂ)+ -v*(t:ℂ) = 0 := by ring
+  simpa only [hz, Complex.exp_zero, one_mul] using he
+
+ theorem periodic_frequency {x : ℝ → ℂ} (a : ℝ)
+    (hx : ∀ t, HasDerivAt x ((2*(a:ℂ)*Complex.I)*x t) t)
+    (hp : x 1 = x 0) (hzero : x 0 ≠ 0) : ∃ m : ℤ, a = Real.pi*m := by
+  have he := linear_solution (2*(a:ℂ)*Complex.I) hx 1
+  rw [Complex.ofReal_one, mul_one, hp] at he
+  have hh : Complex.exp (2*(a:ℂ)*Complex.I) = 1 := by
+    apply mul_right_cancel₀ hzero
+    simpa only [one_mul] using he.symm
+  obtain ⟨m,hm⟩ := Complex.exp_eq_one_iff.mp hh
+  refine ⟨m,?_⟩
+  have hi := congrArg Complex.im hm
+  simp [Complex.mul_im, Complex.mul_re] at hi
+  linarith
+
+ theorem slow_linear_constant {x : ℝ → ℂ} {a : ℝ} (ha : 0 ≤ a) (haπ : a < Real.pi)
+    (hx : ∀ t, HasDerivAt x ((2*(a:ℂ)*Complex.I)*x t) t)
+    (hp : x 1 = x 0) (t : ℝ) : x t = x 0 := by
+  by_cases hz : x 0 = 0
+  · simp only [linear_solution _ hx t, hz, mul_zero]
+  obtain ⟨m,hm⟩ := periodic_frequency a hx hp hz
+  have hm0 : 0 ≤ m := by
+    have : 0 ≤ (m:ℝ) := (mul_nonneg_iff_of_pos_left Real.pi_pos).mp (hm ▸ ha)
+    exact_mod_cast this
+  have hm1 : m < 1 := by
+    have : (m:ℝ) < 1 := (mul_lt_mul_iff_right₀ Real.pi_pos).mp (by simpa [mul_comm] using hm ▸ haπ)
+    exact_mod_cast this
+  have hmz : m = 0 := by omega
+  have haz : a = 0 := by simp [hm,hmz]
+  simp only [linear_solution _ hx t, haz, Complex.ofReal_zero, mul_zero, zero_mul,
+    Complex.exp_zero, one_mul]
+
+end
+end HamiltonianODE
+
+namespace HamiltonianODE
+noncomputable section
+open Set Filter
+open scoped ContDiff Topology
+open FourierPolynomial DiagonalQuadratic
+variable {κ : Type u118} [Fintype κ]
+
+omit [Fintype κ] in
+ theorem diag_one (z : Vector κ) : diag (fun _ => 1) z = z := by
+   ext j
+   simp
+
+ theorem energy_one (z : Vector κ) : energy (fun _ => 1) z = ‖z‖^2 := by
+   simp only [energy, one_mul, EuclideanSpace.norm_sq_eq]
+
+ theorem gradient_comp_energy (d : κ → ℝ) {f : ℝ → ℝ} {v : ℝ} (z : Vector κ)
+    (hf : HasDerivAt f v (energy d z)) :
+    gradient (fun w => f (energy d w)) z = (2*v) • diag d z := by
+   apply HasGradientAt.gradient
+   rw [hasGradientAt_iff_hasFDerivAt]
+   have h := hf.comp_hasFDerivAt z (hasGradientAt_energy d z).hasFDerivAt
+   apply h.congr_fderiv
+   ext w
+   change v * inner (𝕜 := ℝ) ((2 : ℝ) • diag d z) w =
+     inner (𝕜 := ℝ) ((2*v) • diag d z) w
+   simp only [real_inner_smul_left]
+   ring
+
+ theorem coordinate_derivative (d : κ → ℝ) (v : ℝ) {x : ℝ → Vector κ}
+    (hx : ∀ t, HasDerivAt x (Complex.I • ((2*v) • diag d (x t))) t) (j : κ) (t : ℝ) :
+    HasDerivAt (fun s => x s j) ((2*((v*d j : ℝ):ℂ)*Complex.I)*x t j) t := by
+   have h := (((PiLp.proj (𝕜 := ℂ) 2 (fun _ : κ => ℂ) j).restrictScalars ℝ).hasFDerivAt).comp_hasDerivAt t (hx t)
+   apply h.congr_deriv
+   change Complex.I * ((2*v) • (d j • x t j)) = (2*((v*d j : ℝ):ℂ)*Complex.I)*x t j
+   simp only [Complex.real_smul, Complex.ofReal_mul, Complex.ofReal_ofNat]
+   ring
+
+ theorem slow_diagonal_constant (d : κ → ℝ) (v : ℝ)
+    (hv : ∀ j, 0 ≤ v*d j) (hvπ : ∀ j, v*d j < Real.pi)
+    {x : ℝ → Vector κ}
+    (hx : ∀ t, HasDerivAt x (Complex.I • ((2*v) • diag d (x t))) t)
+    (hp : x 1 = x 0) (t : ℝ) : x t = x 0 := by
+   ext j
+   exact slow_linear_constant (hv j) (hvπ j) (coordinate_derivative d v hx j)
+     (congrArg (fun z : Vector κ => z j) hp) t
+
+ theorem slow_radial_constant {f v : ℝ → ℝ}
+    (hf : ∀ s, HasDerivAt f (v s) s)
+    (hv : ∀ s, 0 ≤ v s) (hvπ : ∀ s, v s < Real.pi)
+    {x : ℝ → Vector κ}
+    (hx : ∀ t, HasDerivAt x (Complex.I • gradient (fun z : Vector κ => f (‖z‖^2)) (x t)) t)
+    (hp : x 1 = x 0) (t : ℝ) : x t = x 0 := by
+   have he (s : ℝ) : gradient (fun z : Vector κ => f (‖z‖^2)) (x s) =
+       (2*v (‖x s‖^2)) • diag (fun _ => 1) (x s) := by
+     simpa only [energy_one] using gradient_comp_energy (fun _ => 1) (x s) (hf (energy (fun _ => 1) (x s)))
+   have hode (s : ℝ) := (hx s).congr_deriv (congrArg (fun z : Vector κ => Complex.I • z) (he s))
+   have hc (s : ℝ) : ‖x s‖^2 = ‖x 0‖^2 := by
+     simpa only [energy_one] using quadratic_constant (fun _ => 1) hode s 0
+   apply slow_diagonal_constant (fun _ => 1) (v (‖x 0‖^2)) (by simpa only [mul_one] using fun _ : κ => hv (‖x 0‖^2))
+     (by simpa only [mul_one] using fun _ : κ => hvπ (‖x 0‖^2)) _ hp t
+   intro s
+   simpa only [hc s] using hode s
+
+ theorem moderate_linear_either {x : ℝ → ℂ} {a : ℝ} (ha : 0 ≤ a) (ha2π : a < 2*Real.pi)
+    (hx : ∀ t, HasDerivAt x ((2*(a:ℂ)*Complex.I)*x t) t)
+    (hp : x 1 = x 0) : a = Real.pi ∨ ∀ t, x t = x 0 := by
+   by_cases hz : x 0 = 0
+   · exact Or.inr (fun t => by simp only [linear_solution _ hx t,hz,mul_zero])
+   obtain ⟨m,hm⟩ := periodic_frequency a hx hp hz
+   have hm0 : 0 ≤ m := by
+     have : 0 ≤ (m:ℝ) := (mul_nonneg_iff_of_pos_left Real.pi_pos).mp (hm ▸ ha)
+     exact_mod_cast this
+   have hm2 : m < 2 := by
+     have : (m:ℝ) < 2 := by nlinarith [Real.pi_pos]
+     exact_mod_cast this
+   have he : m = 0 ∨ m = 1 := by omega
+   rcases he with he | he
+   · have haz : a = 0 := by simp [hm,he]
+     exact Or.inr (fun t => by simp only [linear_solution _ hx t,haz,Complex.ofReal_zero,
+       mul_zero,zero_mul,Complex.exp_zero,one_mul])
+   · exact Or.inl (by simpa [he] using hm)
+
+ theorem diagonal_either (j₀ : κ) (d : κ → ℝ) {v : ℝ}
+    (hv : 0 ≤ v) (hv1 : v ≤ 1) (hd : ∀ j, 0 ≤ d j)
+    (hdπ : ∀ j, j ≠ j₀ → d j < Real.pi) (hd2π : d j₀ < 2*Real.pi)
+    {x : ℝ → Vector κ}
+    (hx : ∀ t, HasDerivAt x (Complex.I • ((2*v) • diag d (x t))) t)
+    (hp : x 1 = x 0) : v*d j₀ = Real.pi ∨ ∀ t, x t = x 0 := by
+   have hvd (j : κ) : v*d j ≤ d j := by nlinarith [hd j]
+   rcases moderate_linear_either (mul_nonneg hv (hd j₀)) (lt_of_le_of_lt (hvd j₀) hd2π)
+     (coordinate_derivative d v hx j₀) (congrArg (fun z : Vector κ => z j₀) hp) with he | he
+   · exact Or.inl he
+   · right
+     intro t
+     ext j
+     by_cases hj : j = j₀
+     · subst j; exact he t
+     · exact slow_linear_constant (mul_nonneg hv (hd j)) (lt_of_le_of_lt (hvd j) (hdπ j hj))
+         (coordinate_derivative d v hx j) (congrArg (fun z : Vector κ => z j) hp) t
+
+end
+end HamiltonianODE
+
+end NonsqueezingInline
+
+end OAI
diff --git a/lean/OAI/Geometry/PolarProducts/HolomorphicBalls.lean b/lean/OAI/Geometry/PolarProducts/HolomorphicBalls.lean
new file mode 100644
index 0000000000000000000000000000000000000000..8c7c010369c39314a7dfe5faf594e848f3046c32
--- /dev/null
+++ b/lean/OAI/Geometry/PolarProducts/HolomorphicBalls.lean
@@ -0,0 +1,350 @@
+import OAI.Geometry.PolarProducts.RadialMap
+
+namespace OAI
+
+universe u41 u42 u43 u44 u45 u46 u47 u48 u49
+
+section LowerBoundInline
+open Set Filter Function
+open scoped Topology ContDiff NNReal
+open Set Filter Metric
+open scoped Topology ContDiff
+open Set Filter Function MeasureTheory Metric
+open scoped Topology ContDiff NNReal
+open Set Filter Function
+open scoped Topology ContDiff
+open Set Filter Function
+open scoped Topology ContDiff NNReal
+open Set Filter
+open scoped Topology ContDiff
+open Set Filter Function
+open scoped Topology ContDiff
+open Set Filter Function
+open scoped ContDiff Topology
+open Set MeasureTheory
+open scoped ContDiff Interval Topology
+open Set
+open scoped Topology ContDiff
+
+open Set
+namespace SingularComparison
+noncomputable section
+
+theorem exists_inner_radius {μ α R : ℝ} (hμ : 0 ≤ μ) (hμα : μ < α)
+    (hR : 0 < R) (M C : ℝ) :
+    ∃ r : ℝ, 0 < r ∧ r < R ∧ ∀ (t δ H : ℝ), 0 < t → t ≤ r^2 → 0 < δ →
+      H ≤ α * Real.log t + C → H + 2 ≤ μ * Real.log (t + δ) - M := by
+  let L := (-M - C - 2) / (α - μ)
+  obtain ⟨r, hr, hrsmall⟩ := exists_between (show (0 : ℝ) < min R (Real.sqrt (Real.exp L)) by
+    exact lt_min hR (Real.sqrt_pos.mpr (Real.exp_pos L)))
+  refine ⟨r, hr, hrsmall.trans_le (min_le_left _ _), ?_⟩
+  intro t δ H ht htr hδ hH
+  have hre : r < Real.sqrt (Real.exp L) := hrsmall.trans_le (min_le_right _ _)
+  have hr2 : r^2 ≤ Real.exp L := by
+    have hs := Real.sq_sqrt (Real.exp_pos L).le
+    nlinarith [Real.sqrt_nonneg (Real.exp L)]
+  have htL : Real.log t ≤ L := by
+    calc
+      Real.log t ≤ Real.log (Real.exp L) := Real.log_le_log ht (htr.trans hr2)
+      _ = L := Real.log_exp L
+  have hlog : Real.log t ≤ Real.log (t + δ) := Real.log_le_log ht (by linarith)
+  have hcoef : (α - μ) * Real.log t ≤ -M - C - 2 := by
+    calc
+      _ ≤ (α - μ) * L := mul_le_mul_of_nonneg_left htL (sub_nonneg.mpr hμα.le)
+      _ = -M - C - 2 := by dsimp [L]; field_simp [sub_ne_zero.mpr hμα.ne']
+  have hmono := mul_le_mul_of_nonneg_left hlog hμ
+  linarith
+
+theorem outer_comparison {μ R δ H L M : ℝ} (hμ : 0 ≤ μ) (hδ : 0 < δ)
+    (hδ1 : δ ≤ 1) (hH : L ≤ H)
+    (hM : μ * Real.log (R^2 + 1) - L + 2 < M) :
+    μ * Real.log (R^2 + δ) - M + 2 < H := by
+  have hp : 0 < R^2 + δ := add_pos_of_nonneg_of_pos (sq_nonneg R) hδ
+  have hlog := Real.log_le_log hp (show R^2 + δ ≤ R^2 + 1 by linarith)
+  have hh := mul_le_mul_of_nonneg_left hlog hμ
+  linarith
+
+theorem radius_bound {a μ r δ : ℝ} (ha : 0 < a) (_ham : a < μ) (hr : 0 < r)
+    (hδ : 0 < δ) (hδb : δ < r^2 * (μ / a - 1)) :
+    a < r^2 * (1 + μ / (r^2 + δ)) := by
+  have hd : 0 < r^2 + δ := add_pos_of_nonneg_of_pos (sq_nonneg r) hδ
+  have hr2 : 0 < r^2 := sq_pos_of_pos hr
+  have hδa : δ * a < r^2 * (μ - a) := by
+    calc
+      δ * a < (r^2 * (μ / a - 1)) * a := mul_lt_mul_of_pos_right hδb ha
+      _ = r^2 * (μ - a) := by field_simp
+  have hfrac : a < r^2 * μ / (r^2 + δ) := by
+    apply (lt_div_iff₀ hd).mpr
+    nlinarith
+  calc
+    a < r^2 * μ / (r^2 + δ) := hfrac
+    _ < r^2 * (1 + μ / (r^2 + δ)) := by
+      rw [mul_add, mul_one, mul_div_assoc]
+      linarith
+
+end
+end SingularComparison
+
+namespace ComplexPotential
+
+open Set Filter RadialSymplectic
+open scoped ContDiff Topology
+
+noncomputable section
+
+variable {E : Type u41} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E]
+
+theorem exists_ball_from_logarithmic_pole (J : E →L[ℝ] E)
+    (hsk : ∀ a b, inner (𝕜 := ℝ) a (J b) = -inner (𝕜 := ℝ) (J a) b)
+    (hJ : ∀ a, ‖J a‖ = ‖a‖) (hJJ : ∀ a, J (J a) = -a)
+    {U : Set E} (hU : IsOpen U) (hU0 : (0 : E) ∈ U)
+    {τ H : E → ℝ} (hτ : ∀ x ∈ U, IsPSHAt J τ x) (hτ0 : τ 0 = 0)
+    (hH : ∀ x ∈ U, x ≠ 0 → IsPSHAt J H x)
+    (hcompact : ∀ s : ℝ, 0 < s → s < 1 → IsCompact {x | x ∈ U ∧ τ x ≤ s})
+    {s₀ α C ρ : ℝ} (hs₀ : s₀ < 1) (hρ : 0 < ρ)
+    (htail : ∀ x ∈ U, s₀ ≤ τ x → H x = τ x)
+    (hpole : ∀ x ∈ U, x ≠ 0 → ‖x‖ ≤ ρ → H x ≤ α * Real.log (‖x‖^2) + C)
+    {a : ℝ} (ha : 0 < a) (haα : a < α) :
+    ∃ F : E ≃ₜ E, ContDiff ℝ ∞ (F : E → E) ∧ ContDiff ℝ ∞ (F.symm : E → E) ∧
+      (∀ x : E, ‖x‖^2 < a → F x ∈ U) ∧
+      ∀ x : E, ‖x‖^2 < a → ∀ v w : E,
+        ddc J (fun y => ‖y‖^2 + τ y) (F x) (fderiv ℝ (F : E → E) x v)
+          (fderiv ℝ (F : E → E) x w) = standardTensor J v w := by
+  obtain ⟨μ, haμ, hμα⟩ := exists_between haα
+  have hμ : 0 < μ := ha.trans haμ
+  have hn : {x : E | x ∈ U ∧ τ x < 1/2} ∈ 𝓝 (0 : E) := by
+    exact inter_mem (hU.mem_nhds hU0)
+      ((hτ 0 hU0).1.continuousAt.eventually_lt continuousAt_const (by norm_num [hτ0]))
+  obtain ⟨ε, hε, hεball⟩ := Metric.mem_nhds_iff.mp hn
+  let R := ε / 2
+  have hR : 0 < R := half_pos hε
+  have hRball (x : E) (hx : ‖x‖ ≤ R) : x ∈ U ∧ τ x < 1/2 := by
+    apply hεball
+    rw [Metric.mem_ball, dist_zero_right]
+    exact hx.trans_lt (half_lt_self hε)
+  have hsp : IsCompact {x : E | ‖x‖ = R} := by
+    simpa only [Metric.sphere, dist_zero_right] using isCompact_sphere (0 : E) R
+  have hcont : ContinuousOn H {x : E | ‖x‖ = R} := by
+    intro x hx
+    apply ((hH x (hRball x hx.le).1 ?_).1.continuousAt).continuousWithinAt
+    intro hz
+    have : R = 0 := by simpa [hz] using hx.symm
+    exact hR.ne' this
+  obtain ⟨L, hL⟩ := hsp.bddBelow_image hcont
+  let M := μ * Real.log (R^2 + 1) - L + 3
+  obtain ⟨r, hr, hrlt, hrdom⟩ := SingularComparison.exists_inner_radius hμ.le hμα
+    (lt_min hR hρ) M C
+  have hrR : r < R := hrlt.trans_le (min_le_left _ _)
+  have hrρ : r < ρ := hrlt.trans_le (min_le_right _ _)
+  have hδmax : 0 < min (1 : ℝ) (r^2 * (μ / a - 1)) := by
+    apply lt_min zero_lt_one
+    exact mul_pos (sq_pos_of_pos hr) (sub_pos.mpr ((one_lt_div ha).mpr haμ))
+  obtain ⟨δ, hδ, hδlt⟩ := exists_between hδmax
+  have hδ1 : δ ≤ 1 := (hδlt.trans_le (min_le_left _ _)).le
+  have hδb : δ < r^2 * (μ / a - 1) := hδlt.trans_le (min_le_right _ _)
+  let P : E → ℝ := fun x => μ * logNormSq δ x - M
+  have hP (x : E) : IsPSHAt J P x :=
+    ((isPSHAt_logNormSq J hsk hJ hJJ hδ.le
+      (add_pos_of_nonneg_of_pos (sq_nonneg _) hδ)).const_smul hμ.le).sub_const M
+  have hinner (x : E) (hx0 : x ≠ 0) (hx : ‖x‖ ≤ r) : H x + 1 ≤ P x := by
+    have hxu := (hRball x (hx.trans hrR.le)).1
+    have hh := hrdom (‖x‖^2) δ (H x) (sq_pos_of_pos (norm_pos_iff.mpr hx0))
+      ((sq_le_sq₀ (norm_nonneg _) hr.le).mpr hx) hδ
+      (hpole x hxu hx0 (hx.trans hrρ.le))
+    change H x + 1 ≤ μ * Real.log (‖x‖^2 + δ) - M
+    linarith
+  have houter (x : E) (_hx : x ∈ U) (hnorm : ‖x‖ = R) : P x + 1 < H x := by
+    have hh := SingularComparison.outer_comparison hμ.le hδ hδ1
+      (hL ⟨x, hnorm, rfl⟩) (show μ * Real.log (R^2 + 1) - L + 2 < M by dsimp [M]; linarith)
+    change μ * Real.log (‖x‖^2 + δ) - M + 1 < H x
+    rw [hnorm]
+    linarith
+  let T := gluedPotential R H P
+  have hT (x : E) (hx : x ∈ U) : IsPSHAt J T x :=
+    isPSHAt_gluedPotential J hJJ hr hrR H P hH (fun x _ => hP x) hinner houter hx
+  obtain ⟨σ, hσmin, hσ1⟩ := exists_between (show max s₀ (1/2 : ℝ) < 1 from
+    max_lt hs₀ (by norm_num))
+  have hσ0 : 0 < σ := lt_trans (by norm_num : (0 : ℝ) < 1/2)
+    ((le_max_right _ _).trans_lt hσmin)
+  let K : Set E := {x | x ∈ U ∧ τ x ≤ σ}
+  have hK : IsCompact K := hcompact σ hσ0 hσ1
+  have heq (x : E) (hx : x ∈ U) (hxK : x ∉ K) : T x = τ x := by
+    have hστ : σ < τ x := lt_of_not_ge (fun h => hxK ⟨hx, h⟩)
+    have hnorm : R ≤ ‖x‖ := by
+      by_contra hh
+      have hhalf := (hRball x (not_le.mp hh).le).2
+      linarith [(le_max_right s₀ (1/2 : ℝ)).trans_lt hσmin]
+    exact (gluedPotential_outer R H P hnorm).trans (htail x hx
+      (((le_max_left _ _).trans_lt hσmin).le.trans hστ.le))
+  obtain ⟨Q, hQ, hQi, _hout, hQmap, hQpull⟩ := exists_potential_transport J hsk hJ
+    hU hK (fun _ hx => hx.1) hτ hT heq
+  let F := (radialHomeomorph (E := E) hμ hδ).symm.trans Q
+  have hF : ContDiff ℝ ∞ (F : E → E) := hQ.comp (contDiff_radialContraction hμ hδ)
+  have hFi : ContDiff ℝ ∞ (F.symm : E → E) := (contDiff_radialExpansion hμ hδ).comp hQi
+  have hmap (x : E) (hx : ‖x‖^2 < a) : ‖radialContraction μ δ x‖ < r := by
+    apply (sq_lt_sq₀ (norm_nonneg _) hr.le).mp
+    apply radialContraction_mapsTo hμ hδ _ hx
+    calc
+      a ≤ r^2 * (1 + μ / (r^2 + δ)) := (SingularComparison.radius_bound ha haμ hr hδ hδb).le
+      _ = radiusSq μ δ (r^2) := by unfold radiusSq; ring
+  have hUmap (x : E) (hx : ‖x‖^2 < a) : radialContraction μ δ x ∈ U :=
+    (hRball _ ((hmap x hx).trans hrR).le).1
+  refine ⟨F, hF, hFi, fun x hx => (hQmap _).mpr (hUmap x hx), ?_⟩
+  intro x hx v w
+  change ddc J (fun y => ‖y‖^2 + τ y) (Q (radialContraction μ δ x))
+    (fderiv ℝ ((Q : E → E) ∘ radialContraction μ δ) x v)
+    (fderiv ℝ ((Q : E → E) ∘ radialContraction μ δ) x w) = _
+  rw [fderiv_comp x (hQ.differentiable (by simp) _) ((contDiff_radialContraction hμ hδ).differentiable
+    (by simp) x)]
+  simp only [ContinuousLinearMap.comp_apply]
+  rw [hQpull _ (hUmap x hx)]
+  have heT : (fun z : E => ‖z‖^2 + T z) =ᶠ[𝓝 (radialContraction μ δ x)]
+      fun z => modelPotential μ δ z - M := by
+    filter_upwards [continuous_norm.continuousAt.eventually_lt continuousAt_const (hmap x hx)] with z hz
+    dsimp only [T]
+    rw [gluedPotential_inner hrR H P hinner hz.le]
+    dsimp [P, logNormSq, modelPotential]
+    ring
+  rw [ddc_congr J heT, ddc_sub_const J (contDiff_modelPotential μ hδ).contDiffAt,
+    radialContraction_pullback J hsk hμ hδ]
+
+end
+end ComplexPotential
+
+namespace TensorCalculus
+
+open Set Filter
+open scoped Topology ContDiff
+
+variable {E : Type u42} {F : Type u43} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  [NormedAddCommGroup F] [NormedSpace ℝ F]
+
+noncomputable section
+
+local instance pullbackOneTensorGroup : NormedAddCommGroup (F →L[ℝ] ℝ) := inferInstance
+local instance pullbackOneTensorSpace : NormedSpace ℝ (F →L[ℝ] ℝ) := inferInstance
+local instance pullbackTwoTensorGroup : NormedAddCommGroup (F →L[ℝ] F →L[ℝ] ℝ) := inferInstance
+local instance pullbackTwoTensorSpace : NormedSpace ℝ (F →L[ℝ] F →L[ℝ] ℝ) := inferInstance
+
+def pullbackOneForm (f : E → F) (γ : F → F →L[ℝ] ℝ) (x : E) : E →L[ℝ] ℝ :=
+  (γ (f x)).comp (fderiv ℝ f x)
+
+@[simp] theorem pullbackOneForm_apply (f : E → F) (γ : F → F →L[ℝ] ℝ) (x a : E) :
+    pullbackOneForm f γ x a = γ (f x) (fderiv ℝ f x a) := rfl
+
+theorem fderiv_pullbackOneForm {f : E → F} {γ : F → F →L[ℝ] ℝ} {x : E}
+    (hf : DifferentiableAt ℝ f x) (hd : DifferentiableAt ℝ (fderiv ℝ f) x)
+    (hγ : DifferentiableAt ℝ γ (f x)) (a b : E) :
+    fderiv ℝ (pullbackOneForm f γ) x a b =
+      fderiv ℝ γ (f x) (fderiv ℝ f x a) (fderiv ℝ f x b) +
+        γ (f x) (fderiv ℝ (fderiv ℝ f) x a b) := by
+  unfold pullbackOneForm
+  rw [fderiv_clm_comp (c := fun y => γ (f y)) (hγ.comp x hf) hd]
+  rw [show fderiv ℝ (fun y => γ (f y)) x = (fderiv ℝ γ (f x)).comp (fderiv ℝ f x) by
+    exact fderiv_comp x hγ hf]
+  simp only [add_apply, ContinuousLinearMap.comp_apply, ContinuousLinearMap.flip_apply,
+    ContinuousLinearMap.compL_apply]
+  ring
+
+theorem exteriorD_pullbackOneForm {f : E → F} {γ : F → F →L[ℝ] ℝ} {x : E}
+    (hf : ContDiffAt ℝ ∞ f x) (hγ : DifferentiableAt ℝ γ (f x)) (a b : E) :
+    exteriorD (pullbackOneForm f γ) x a b =
+      exteriorD γ (f x) (fderiv ℝ f x a) (fderiv ℝ f x b) := by
+  have hd := (hf.fderiv_right (m := ∞) (by simp)).differentiableAt (by simp)
+  have hf' := hf.differentiableAt (by simp)
+  have hs := hf.isSymmSndFDerivAt (by
+    rw [minSmoothness_of_isRCLikeNormedField]
+    exact WithTop.coe_le_coe.mpr le_top)
+  simp only [exteriorD_apply, fderiv_pullbackOneForm hf' hd hγ]
+  rw [hs.eq a b]
+  ring
+
+end
+end TensorCalculus
+
+namespace ComplexPotential
+
+open TensorCalculus Set Filter
+open scoped ContDiff Topology
+
+variable {E : Type u44} {F : Type u45} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  [NormedAddCommGroup F] [NormedSpace ℝ F]
+
+noncomputable section
+
+theorem dc_comp (J : E →L[ℝ] E) (K : F →L[ℝ] F) {f : E → F} {φ : F → ℝ} {x : E}
+    (hf : DifferentiableAt ℝ f x) (hφ : DifferentiableAt ℝ φ (f x))
+    (hCR : ∀ a, fderiv ℝ f x (J a) = K (fderiv ℝ f x a)) :
+    dc J (φ ∘ f) x = pullbackOneForm f (dc K φ) x := by
+  ext a
+  rw [dc_apply, fderiv_comp x hφ hf, ContinuousLinearMap.comp_apply, hCR,
+    pullbackOneForm_apply, dc_apply]
+
+theorem ddc_comp (J : E →L[ℝ] E) (K : F →L[ℝ] F) {f : E → F} {φ : F → ℝ} {x : E}
+    (hf : ContDiffAt ℝ ∞ f x) (hφ : ContDiffAt ℝ ∞ φ (f x))
+    (hCR : ∀ᶠ y in 𝓝 x, ∀ a, fderiv ℝ f y (J a) = K (fderiv ℝ f y a)) (a b : E) :
+    ddc J (φ ∘ f) x a b = ddc K φ (f x) (fderiv ℝ f x a) (fderiv ℝ f x b) := by
+  have h1 : (1 : ℕ∞ω) ≤ ∞ := WithTop.coe_le_coe.mpr le_top
+  have he : dc J (φ ∘ f) =ᶠ[𝓝 x] pullbackOneForm f (dc K φ) := by
+    filter_upwards [(hf.of_le h1).eventually (by simp),
+      hf.continuousAt ((hφ.of_le h1).eventually (by simp)), hCR] with y hy hz hc
+    exact dc_comp J K (hy.differentiableAt one_ne_zero)
+      ((show ContDiffAt ℝ (1 : ℕ∞ω) φ (f y) from hz).differentiableAt one_ne_zero) hc
+  unfold ddc exteriorD
+  rw [he.fderiv_eq]
+  exact exteriorD_pullbackOneForm hf ((contDiffAt_dc K hφ).differentiableAt (by simp)) a b
+
+end
+end ComplexPotential
+
+namespace ComplexPotential
+
+open Set Filter TensorCalculus
+open scoped ContDiff Topology
+
+noncomputable section
+
+variable {E : Type u46} {F : Type u47} [NormedAddCommGroup E] [NormedSpace ℂ E]
+  [NormedAddCommGroup F] [NormedSpace ℂ F]
+
+def complexStructure (E : Type u48) [NormedAddCommGroup E] [NormedSpace ℂ E] : E →L[ℝ] E :=
+  (Complex.I • ContinuousLinearMap.id ℂ E).restrictScalars ℝ
+
+@[simp] theorem complexStructure_apply (a : E) : complexStructure E a = Complex.I • a := rfl
+
+theorem complexStructure_sq (a : E) : complexStructure E (complexStructure E a) = -a := by
+  simp [complexStructure_apply, smul_smul]
+
+theorem complexStructure_norm (a : E) : ‖complexStructure E a‖ = ‖a‖ := by
+  simp [complexStructure_apply, norm_smul]
+
+theorem fderiv_commutes_complexStructure {f : E → F} {x : E}
+    (hf : DifferentiableAt ℂ f x) (a : E) :
+    fderiv ℝ f x (complexStructure E a) = complexStructure F (fderiv ℝ f x a) := by
+  rw [hf.fderiv_restrictScalars ℝ]
+  simp [complexStructure_apply]
+
+theorem complexStructure_skew {G : Type u49} [NormedAddCommGroup G]
+    [InnerProductSpace ℂ G] (a b : G) :
+    let : InnerProductSpace ℝ G := InnerProductSpace.complexToReal
+    inner (𝕜 := ℝ) a (complexStructure G b) = -inner (𝕜 := ℝ) (complexStructure G a) b := by
+  let : InnerProductSpace ℝ G := InnerProductSpace.complexToReal
+  simp [real_inner_eq_re_inner ℂ, inner_smul_right, inner_smul_left]
+
+theorem IsPSHAt.holomorphic_comp {f : E → F} {φ : F → ℝ} {x : E}
+    (hφ : IsPSHAt (complexStructure F) φ (f x))
+    (hf : ContDiffAt ℝ ∞ f x) (hhol : ∀ᶠ y in 𝓝 x, DifferentiableAt ℂ f y) :
+    IsPSHAt (complexStructure E) (φ ∘ f) x := by
+  refine ⟨hφ.1.comp x hf, fun a => ?_⟩
+  rw [ddc_comp (complexStructure E) (complexStructure F) hf hφ.1
+    (hhol.mono (fun y hy => fderiv_commutes_complexStructure hy)),
+    fderiv_commutes_complexStructure hhol.self_of_nhds]
+  exact hφ.2 _
+
+end
+end ComplexPotential
+
+end LowerBoundInline
+
+end OAI
diff --git a/lean/OAI/Geometry/PolarProducts/HorizontalIntegrals.lean b/lean/OAI/Geometry/PolarProducts/HorizontalIntegrals.lean
new file mode 100644
index 0000000000000000000000000000000000000000..79335d059df52078d67bf6b6ab5ff35d0063a5d8
--- /dev/null
+++ b/lean/OAI/Geometry/PolarProducts/HorizontalIntegrals.lean
@@ -0,0 +1,315 @@
+import OAI.Geometry.PolarProducts.LensEstimates
+
+namespace OAI
+
+universe u52 u53 u54
+
+section LowerBoundInline
+open Set Filter Function
+open scoped Topology ContDiff NNReal
+open Set Filter Metric
+open scoped Topology ContDiff
+open Set Filter Function MeasureTheory Metric
+open scoped Topology ContDiff NNReal
+open Set Filter Function
+open scoped Topology ContDiff
+open Set Filter Function
+open scoped Topology ContDiff NNReal
+open Set Filter
+open scoped Topology ContDiff
+open Set Filter Function
+open scoped Topology ContDiff
+open Set Filter Function
+open scoped ContDiff Topology
+open Set MeasureTheory
+open scoped ContDiff Interval Topology
+open Set
+open scoped Topology ContDiff
+open Set
+open Set MeasureTheory
+open scoped ContDiff Interval Topology
+open Set Filter Complex
+open scoped Topology ContDiff
+
+namespace SmoothPaths
+open Set Filter MeasureTheory
+open scoped Topology ContDiff
+noncomputable section
+
+universe u
+variable {E F : Type u} {H : Type u52} [NormedAddCommGroup E] [NormedSpace ℝ E] [ProperSpace E]
+  [NormedAddCommGroup F] [NormedSpace ℝ F]
+  [NormedAddCommGroup H] [NormedSpace ℝ H]
+
+def integralPath : C(SmoothODE.Time, F) →L[ℝ] F :=
+  (ContinuousMap.evalCLM ℝ ⟨1, by norm_num⟩).comp SmoothODE.primitiveCLM
+
+@[simp] theorem integralPath_apply (u : C(SmoothODE.Time, F)) :
+    integralPath u = ∫ t in (0 : ℝ)..1, SmoothODE.extend u t := rfl
+
+theorem contDiff_integralPath {f : E → F} (hf : ContDiff ℝ ∞ f)
+    (L : H →L[ℝ] C(SmoothODE.Time, E)) :
+    ContDiff ℝ ∞ (fun z => integralPath (superpose hf.continuous (L z))) :=
+  (integralPath (F := F)).contDiff.comp ((contDiff_superpose hf).comp L.contDiff)
+
+end
+end SmoothPaths
+
+namespace SmoothExtension
+open Set Filter
+open scoped Topology ContDiff Manifold
+noncomputable section
+
+variable {E : Type u53} {F : Type u54} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  [FiniteDimensional ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F]
+
+theorem exists_global_near_closed {U K : Set E} (hU : IsOpen U) (hK : IsClosed K)
+    (hKU : K ⊆ U) {f : E → F} (hf : ContDiffOn ℝ ∞ f U) :
+    ∃ f₁ : E → F, ContDiff ℝ ∞ f₁ ∧ f₁ =ᶠ[𝓝ˢ K] f := by
+  obtain ⟨b, hb0, hb1, _⟩ := exists_contMDiffMap_zero_one_nhds_of_isClosed
+    (𝓘(ℝ, E)) hU.isClosed_compl hK
+    (disjoint_left.mpr (by intro x hx hxK; exact hx (hKU hxK))) (n := ⊤)
+  have hb : ContDiff ℝ ∞ b := contMDiff_iff_contDiff.mp b.contMDiff
+  refine ⟨fun x => b x • f x, ?_, ?_⟩
+  · apply contDiff_iff_contDiffAt.mpr
+    intro x
+    by_cases hx : x ∈ U
+    · exact hb.contDiffAt.smul (hf.contDiffAt (hU.mem_nhds hx))
+    · have hh : ∀ᶠ y in 𝓝 x, b y = 0 := hb0.filter_mono (nhds_le_nhdsSet hx)
+      apply (contDiffAt_const (c := (0 : F))).congr_of_eventuallyEq
+      filter_upwards [hh] with y hy
+      simp [hy]
+  · filter_upwards [hb1] with x hx
+    simp [hx]
+
+end
+end SmoothExtension
+
+namespace PlanarLens
+open Set Filter Complex MeasureTheory
+open scoped Topology ContDiff
+noncomputable section
+
+def density (k : ℕ) (z : ℂ) : ℝ := ‖g z‖^(2*k-2) * ‖deriv g z‖^2
+
+def J (k : ℕ) (v t : ℝ) : ℝ := (k : ℝ)^2 * ∫ h in 0..v, density k ((h : ℂ)+(t : ℂ)*I)
+
+theorem density_nonneg (k : ℕ) (z : ℂ) : 0 ≤ density k z := by
+  unfold density; positivity
+
+theorem contDiffOn_density (k : ℕ) : ContDiffOn ℝ ∞ (density k) D := by
+  have hg : ContDiffOn ℝ ∞ g D := (analyticOnNhd_g.contDiffOn isOpen_D.uniqueDiffOn).restrict_scalars ℝ
+  have hd : ContDiffOn ℝ ∞ (deriv g) D :=
+    (analyticOnNhd_g.deriv.contDiffOn isOpen_D.uniqueDiffOn).restrict_scalars ℝ
+  have he : density k = fun z => (‖g z‖^2)^(k-1) * ‖deriv g z‖^2 := by
+    funext z
+    simp only [density, ← pow_mul]
+    congr 2
+    omega
+  rw [he]
+  exact ((hg.norm_sq ℂ).pow (k-1)).mul (hd.norm_sq ℂ)
+
+theorem continuousOn_density (k : ℕ) : ContinuousOn (density k) D :=
+  (contDiffOn_density k).continuousOn
+
+theorem integrable_density {v t : ℝ} (hz : (v : ℂ)+(t : ℂ)*I ∈ D) (k : ℕ) :
+    IntervalIntegrable (fun h : ℝ => density k ((h : ℂ)+(t : ℂ)*I)) volume 0 v := by
+  apply ContinuousOn.intervalIntegrable
+  apply (continuousOn_density k).comp (by fun_prop)
+  intro h hh
+  exact horizontal_segment_mem_signed hz hh
+
+theorem hasDerivAt_J {v t : ℝ} (hz : (v : ℂ)+(t : ℂ)*I ∈ D) (k : ℕ) :
+    HasDerivAt (fun v => J k v t) ((k : ℝ)^2*density k ((v : ℂ)+(t : ℂ)*I)) v := by
+  have hc : ContinuousAt (fun h : ℝ => density k ((h : ℂ)+(t : ℂ)*I)) v :=
+    ((continuousOn_density k).continuousAt (isOpen_D.mem_nhds hz)).comp (f := fun h : ℝ => (h : ℂ)+(t : ℂ)*I)
+      (show ContinuousAt (fun h : ℝ => (h : ℂ)+(t : ℂ)*I) v from by fun_prop)
+  have hm : StronglyMeasurableAtFilter
+      (fun h : ℝ => density k ((h : ℂ)+(t : ℂ)*I)) (𝓝 v) volume := by
+    apply ContinuousAt.stronglyMeasurableAtFilter
+      (isOpen_D.preimage (show Continuous (fun h : ℝ => (h : ℂ)+(t : ℂ)*I) from by fun_prop))
+      (fun h hh => ((continuousOn_density k).continuousAt (isOpen_D.mem_nhds hh)).comp (f := fun a : ℝ => (a : ℂ)+(t : ℂ)*I)
+        (show ContinuousAt (fun a : ℝ => (a : ℂ)+(t : ℂ)*I) h from by fun_prop))
+      v hz
+  exact (intervalIntegral.integral_hasDerivAt_right (integrable_density hz k) hm hc).const_mul _
+
+@[simp] theorem J_zero (k : ℕ) (t : ℝ) : J k 0 t = 0 := by simp [J]
+
+theorem J_nonneg {v t : ℝ} (hv : 0 ≤ v) (k : ℕ) : 0 ≤ J k v t := by
+  apply mul_nonneg (sq_nonneg _)
+  exact intervalIntegral.integral_nonneg hv (fun _ _ => density_nonneg _ _)
+
+def horizontalPath : ℂ →L[ℝ] C(SmoothODE.Time, ℂ) := by
+  let L : ℂ →ₗ[ℝ] C(SmoothODE.Time, ℂ) :=
+    { toFun := fun z => ⟨fun a => ((a : ℝ)*z.re : ℝ) + (z.im : ℂ)*I, by fun_prop⟩
+      map_add' := by intro z w; ext a; simp; ring
+      map_smul' := by intro c z; ext a; simp; ring }
+  apply L.mkContinuous 2
+  intro z
+  apply (ContinuousMap.norm_le _ (by positivity)).mpr
+  intro a
+  change ‖(((a : ℝ)*z.re : ℝ) : ℂ)+(z.im : ℂ)*I‖ ≤ _
+  calc
+    _ ≤ ‖(((a : ℝ)*z.re : ℝ) : ℂ)‖ + ‖(z.im : ℂ)*I‖ := norm_add_le _ _
+    _ = |(a : ℝ)| * |z.re| + |z.im| := by simp
+    _ ≤ |z.re| + |z.im| := by
+      rw [abs_of_nonneg a.2.1]
+      nlinarith [a.2.2, abs_nonneg z.re]
+    _ ≤ 2*‖z‖ := by linarith [abs_re_le_norm z, abs_im_le_norm z]
+
+@[simp] theorem horizontalPath_apply (z : ℂ) (a : SmoothODE.Time) :
+    horizontalPath z a = (((a : ℝ)*z.re : ℝ) : ℂ)+(z.im : ℂ)*I := rfl
+
+theorem J_affine (k : ℕ) (v t : ℝ) :
+    J k v t = (k : ℝ)^2*v*∫ a in (0 : ℝ)..1, density k (((a*v : ℝ) : ℂ)+(t : ℂ)*I) := by
+  rw [J, mul_assoc]
+  congr 1
+  have hh := intervalIntegral.smul_integral_comp_mul_right
+    (fun h : ℝ => density k ((h : ℂ)+(t : ℂ)*I)) v (a := 0) (b := 1)
+  simpa only [smul_eq_mul, zero_mul, one_mul] using hh.symm
+
+theorem contDiffOn_J (k : ℕ) : ContDiffOn ℝ ∞ (fun z : ℂ => J k z.re z.im) D := by
+  intro z hz
+  let C := (fun a : ℝ => (((a*z.re : ℝ) : ℂ)+(z.im : ℂ)*I)) '' Icc (0 : ℝ) 1
+  have hCc : IsCompact C := isCompact_Icc.image (by fun_prop)
+  have hCD : C ⊆ D := by
+    rintro _ ⟨a, ha, rfl⟩
+    apply horizontal_segment_mem_signed (v := z.re) (t := z.im)
+      (by simpa only [re_add_im] using hz)
+    by_cases hv : 0 ≤ z.re
+    · rw [uIcc_of_le hv]
+      constructor <;> nlinarith [ha.1, ha.2]
+    · rw [uIcc_of_ge (le_of_not_ge hv)]
+      constructor <;> nlinarith [ha.1, ha.2]
+  obtain ⟨f, hf, he⟩ := SmoothExtension.exists_global_near_closed isOpen_D hCc.isClosed hCD
+    (contDiffOn_density k)
+  have hglob : ContDiff ℝ ∞ (fun w : ℂ =>
+      (k : ℝ)^2*w.re*SmoothPaths.integralPath (SmoothPaths.superpose hf.continuous (horizontalPath w))) :=
+    (contDiff_const.mul Complex.reCLM.contDiff).mul
+      (SmoothPaths.contDiff_integralPath hf horizontalPath)
+  have heq : ∀ᶠ w in 𝓝 z, ∀ a ∈ Icc (0 : ℝ) 1,
+      f ((((a*w.re : ℝ) : ℂ)+(w.im : ℂ)*I)) = density k ((((a*w.re : ℝ) : ℂ)+(w.im : ℂ)*I)) := by
+    apply isCompact_Icc.eventually_forall_of_forall_eventually
+    intro a ha
+    have hs : ∀ᶠ y in 𝓝 ((((a*z.re : ℝ) : ℂ)+(z.im : ℂ)*I)), f y = density k y :=
+      he.filter_mono (nhds_le_nhdsSet (mem_image_of_mem _ ha))
+    exact (show ContinuousAt (fun p : ℂ×ℝ => (((p.2*p.1.re : ℝ) : ℂ)+(p.1.im : ℂ)*I)) (z,a)
+      from by fun_prop) hs
+  apply ContDiffAt.contDiffWithinAt
+  apply hglob.contDiffAt.congr_of_eventuallyEq
+  filter_upwards [heq] with w hw
+  rw [J_affine, SmoothPaths.integralPath_apply]
+  congr 1
+  apply intervalIntegral.integral_congr
+  intro a ha
+  rw [uIcc_of_le (by norm_num : (0 : ℝ) ≤ 1)] at ha
+  have hp : projIcc (0 : ℝ) 1 zero_le_one a = ⟨a, ha⟩ := by exact projIcc_of_mem zero_le_one ha
+  simp only [SmoothODE.extend, hp, SmoothPaths.superpose_apply, horizontalPath_apply]
+  exact (hw a ha).symm
+
+end
+end PlanarLens
+
+namespace PlanarLens
+open Set Filter Complex MeasureTheory
+open scoped Topology ContDiff
+noncomputable section
+
+def rho (t h : ℝ) : ℝ := ‖g ((h : ℂ)+(t : ℂ)*I)‖
+def rhoD (t h : ℝ) : ℝ := A (g ((h : ℂ)+(t : ℂ)*I)) /
+  (rho t h * ‖deriv F (g ((h : ℂ)+(t : ℂ)*I))‖^2)
+
+theorem rho_pos {t h : ℝ} (hz : (h : ℂ)+(t : ℂ)*I ∈ D) (hh : 0 < h) : 0 < rho t h := by
+  apply norm_pos_iff.mpr
+  intro he
+  have hre := re_g_pos hz (by simpa using hh)
+  rw [he] at hre
+  simp at hre
+
+theorem hasDerivAt_rho {t h : ℝ} (hz : (h : ℂ)+(t : ℂ)*I ∈ D) (hh : 0 < h) :
+    HasDerivAt (rho t) (rhoD t h) h := by
+  have hn := rho_pos hz hh
+  have hd := (horizontal_normSq_deriv hz).sqrt (ne_of_gt (sq_pos_of_pos hn))
+  have he : (fun x : ℝ => Real.sqrt (‖g ((x : ℂ)+(t : ℂ)*I)‖^2)) = rho t := by
+    funext x; exact Real.sqrt_sq (norm_nonneg _)
+  rw [he] at hd
+  convert hd using 1
+  rw [Real.sqrt_sq (norm_nonneg _)]
+  dsimp [rhoD, rho]
+  ring
+
+theorem rhoD_pos {t h : ℝ} (hz : (h : ℂ)+(t : ℂ)*I ∈ D) (hh : 0 < h) : 0 < rhoD t h := by
+  apply div_pos (A_pos (norm_g_lt hz) (re_g_pos hz (by simpa using hh)))
+  exact mul_pos (rho_pos hz hh) (sq_pos_of_pos (norm_pos_iff.mpr (deriv_F_ne_zero (norm_g_lt hz))))
+
+theorem continuousOn_rho {t v : ℝ} (hz : (v : ℂ)+(t : ℂ)*I ∈ D) (hv : 0 ≤ v) :
+    ContinuousOn (rho t) (Icc 0 v) := by
+  apply (continuousOn_g.norm).comp (by fun_prop)
+  intro h hh
+  exact horizontal_segment_mem hz hv hh
+
+theorem density_radial {k : ℕ} (hk : 1 ≤ k) {t h : ℝ}
+    (hz : (h : ℂ)+(t : ℂ)*I ∈ D) (hh : 0 < h) :
+    density k ((h : ℂ)+(t : ℂ)*I) =
+      (rho t h)^(2*k-1) * (1/A (g ((h : ℂ)+(t : ℂ)*I))) * rhoD t h := by
+  have hA := ne_of_gt (A_pos (norm_g_lt hz) (re_g_pos hz (by simpa using hh)))
+  have hr := ne_of_gt (rho_pos hz hh)
+  have hF := norm_ne_zero_iff.mpr (deriv_F_ne_zero (norm_g_lt hz))
+  have he : deriv g ((h : ℂ)+(t : ℂ)*I) = (deriv F (g ((h : ℂ)+(t : ℂ)*I)))⁻¹ := by
+    rw [← deriv_g_F (norm_g_lt hz), F_g hz]
+  rw [density, he, norm_inv]
+  rw [show 2*k-1 = (2*k-2)+1 by omega, pow_succ]
+  dsimp [rhoD, rho] at *
+  field_simp
+  rw [pow_succ]
+
+theorem deriv_F_neg {w : ℂ} (hw : ‖w‖ < 1) : deriv F (-w) = deriv F w := by
+  rw [(hasDerivAt_F (by simpa using hw)).deriv, (hasDerivAt_F hw).deriv]
+  congr 1
+  apply tsum_congr
+  intro n
+  simp only [termDeriv]
+  rw [(show Even (2*n) from ⟨n, by omega⟩).neg_pow]
+
+theorem deriv_F_conj {w : ℂ} (hw : ‖w‖ < 1) :
+    deriv F ((starRingEnd ℂ) w) = (starRingEnd ℂ) (deriv F w) := by
+  have he : (fun n : ℕ => termDeriv n ((starRingEnd ℂ) w)) =
+      fun n : ℕ => conjCLE (termDeriv n w) := by
+    funext n
+    simp [termDeriv, map_ofNat]
+  rw [(hasDerivAt_F (by simpa using hw)).deriv, (hasDerivAt_F hw).deriv]
+  simp only [he, ← conjCLE.map_tsum]
+  simp [map_ofNat]
+
+theorem density_neg_conj {z : ℂ} (hz : z ∈ D) (k : ℕ) :
+    density k (-(starRingEnd ℂ) z) = density k z := by
+  have hd (w : ℂ) (hw : w ∈ D) : deriv g w = (deriv F (g w))⁻¹ := by
+    rw [← deriv_g_F (norm_g_lt hw), F_g hw]
+  rw [density, density, hd _ (neg_conj_mem_D hz), hd _ hz, g_neg_conj hz,
+    deriv_F_neg (by simpa using norm_g_lt hz), deriv_F_conj (norm_g_lt hz)]
+  simp
+
+theorem J_neg {v t : ℝ} (hz : (v : ℂ)+(t : ℂ)*I ∈ D) (k : ℕ) :
+    J k (-v) t = -J k v t := by
+  have he (h : ℝ) : -(starRingEnd ℂ) ((h : ℂ)+(t : ℂ)*I) = ((-h : ℝ) : ℂ)+(t : ℂ)*I := by
+    apply Complex.ext <;> simp
+  have hsub := intervalIntegral.integral_comp_neg
+    (fun h : ℝ => density k ((h : ℂ)+(t : ℂ)*I)) (a := 0) (b := v)
+  simp only [neg_zero] at hsub
+  have hi : (∫ h in (0 : ℝ)..v, density k (((-h : ℝ) : ℂ)+(t : ℂ)*I)) =
+      ∫ h in (0 : ℝ)..v, density k ((h : ℂ)+(t : ℂ)*I) := by
+    apply intervalIntegral.integral_congr
+    intro h hh
+    dsimp only
+    rw [← he, density_neg_conj (horizontal_segment_mem_signed hz hh)]
+  rw [hi, intervalIntegral.integral_symm (a := 0) (b := -v)] at hsub
+  rw [J, J, ← mul_neg]
+  congr 1
+  linarith
+
+end
+end PlanarLens
+
+end LowerBoundInline
+
+end OAI
diff --git a/lean/OAI/Geometry/PolarProducts/LensArea.lean b/lean/OAI/Geometry/PolarProducts/LensArea.lean
new file mode 100644
index 0000000000000000000000000000000000000000..9fe41839868014881996e2447e8feb1027b509a9
--- /dev/null
+++ b/lean/OAI/Geometry/PolarProducts/LensArea.lean
@@ -0,0 +1,455 @@
+import OAI.Geometry.PolarProducts.LensMap
+
+namespace OAI
+
+section LowerBoundInline
+open Set Filter Function
+open scoped Topology ContDiff NNReal
+open Set Filter Metric
+open scoped Topology ContDiff
+open Set Filter Function MeasureTheory Metric
+open scoped Topology ContDiff NNReal
+open Set Filter Function
+open scoped Topology ContDiff
+open Set Filter Function
+open scoped Topology ContDiff NNReal
+open Set Filter
+open scoped Topology ContDiff
+open Set Filter Function
+open scoped Topology ContDiff
+open Set Filter Function
+open scoped ContDiff Topology
+open Set MeasureTheory
+open scoped ContDiff Interval Topology
+open Set
+open scoped Topology ContDiff
+open Set
+open Set MeasureTheory
+open scoped ContDiff Interval Topology
+open Set Filter Complex
+open scoped Topology ContDiff
+
+namespace PlanarLens
+open Set Filter Complex
+open scoped Topology ContDiff
+noncomputable section
+
+def D : Set ℂ := F '' Metric.ball 0 1
+
+def g : ℂ → ℂ := Function.invFunOn F (Metric.ball 0 1)
+
+theorem g_F {w : ℂ} (hw : ‖w‖ < 1) : g (F w) = w :=
+  F_injOn.leftInvOn_invFunOn (by simpa using hw)
+
+theorem F_g {z : ℂ} (hz : z ∈ D) : F (g z) = z :=
+  Function.invFunOn_eq hz
+
+theorem norm_g_lt {z : ℂ} (hz : z ∈ D) : ‖g z‖ < 1 := by
+  have h := Function.invFunOn_mem hz
+  simpa [g] using h
+
+theorem zero_mem_D : (0 : ℂ) ∈ D := ⟨0, by simp, F_zero⟩
+
+@[simp] theorem g_zero : g 0 = 0 := by
+  simpa using g_F (by simp : ‖(0 : ℂ)‖ < 1)
+
+theorem g_eq_zero_iff {z : ℂ} (hz : z ∈ D) : g z = 0 ↔ z = 0 := by
+  constructor
+  · intro h
+    rw [← F_g hz, h, F_zero]
+  · rintro rfl; exact g_zero
+
+theorem deriv_F_ne_zero {w : ℂ} (hw : ‖w‖ < 1) : deriv F w ≠ 0 := by
+  intro h
+  have hh := deriv_F_re_pos hw
+  rw [h, zero_re] at hh
+  exact lt_irrefl _ hh
+
+theorem analyticOnNhd_g : AnalyticOnNhd ℂ g D := by
+  rintro z ⟨w, hw, rfl⟩
+  have hw' : ‖w‖ < 1 := by simpa using hw
+  apply (analyticAt_comp_iff_of_deriv_ne_zero (analyticOnNhd_F w hw)
+    (deriv_F_ne_zero hw')).mp
+  apply analyticAt_id.congr
+  filter_upwards [Metric.isOpen_ball.mem_nhds hw] with v hv
+  exact (g_F (by simpa using hv)).symm
+
+theorem isOpen_D : IsOpen D := by
+  rcases analyticOnNhd_F.is_constant_or_isOpen (convex_ball (0 : ℂ) 1).isPreconnected with h | h
+  · obtain ⟨c, hc⟩ := h
+    have hz : (0 : ℂ) ∈ Metric.ball 0 1 := by simp
+    have hh : (1/2 : ℂ) ∈ Metric.ball 0 1 := by norm_num [Metric.mem_ball, norm_div]
+    have he := F_injOn hz hh ((hc 0 hz).trans (hc _ hh).symm)
+    norm_num at he
+  · exact h _ Subset.rfl Metric.isOpen_ball
+
+theorem isBounded_D : Bornology.IsBounded D := by
+  apply ((isCompact_closedBall (0 : ℂ) 1).image_of_continuousOn continuousOn_F_closedBall).isBounded.subset
+  exact image_mono Metric.ball_subset_closedBall
+
+theorem continuousOn_g : ContinuousOn g D := analyticOnNhd_g.continuousOn
+
+theorem g_injOn : InjOn g D := by
+  intro z hz w hw he
+  rw [← F_g hz, ← F_g hw, he]
+
+theorem deriv_g_F {w : ℂ} (hw : ‖w‖ < 1) : deriv g (F w) = (deriv F w)⁻¹ := by
+  have hFw : F w ∈ D := ⟨w, by simpa using hw, rfl⟩
+  have hg := (analyticOnNhd_g _ hFw).differentiableAt.hasDerivAt
+  have hf := (hasDerivAt_F hw).differentiableAt.hasDerivAt
+  have hc := hg.comp w hf
+  have he : g ∘ F =ᶠ[𝓝 w] id := by
+    filter_upwards [Metric.isOpen_ball.mem_nhds (by simpa using hw : w ∈ Metric.ball 0 1)] with z hz
+    exact g_F (by simpa using hz)
+  have hid := hc.congr_of_eventuallyEq he.symm
+  have hm : deriv g (F w) * deriv F w = 1 := hid.unique (hasDerivAt_id w)
+  apply mul_right_cancel₀ (deriv_F_ne_zero hw)
+  rw [hm, inv_mul_cancel₀ (deriv_F_ne_zero hw)]
+
+theorem deriv_g_ne_zero {z : ℂ} (hz : z ∈ D) : deriv g z ≠ 0 := by
+  rw [← F_g hz, deriv_g_F (norm_g_lt hz)]
+  exact inv_ne_zero (deriv_F_ne_zero (norm_g_lt hz))
+
+theorem re_g_pos {z : ℂ} (hz : z ∈ D) (hr : 0 < z.re) : 0 < (g z).re := by
+  by_contra h
+  rcases (le_of_not_gt h).eq_or_lt with he | he
+  · have hh := re_F_eq_zero he
+    rw [F_g hz] at hh
+    linarith
+  · have hh := re_F_neg (norm_g_lt hz) he
+    rw [F_g hz] at hh
+    linarith
+
+end
+end PlanarLens
+
+namespace PlanarLens
+open Set Filter Complex
+open scoped Topology ContDiff
+noncomputable section
+
+def T (r : ℝ) : ℝ := (F ((r : ℂ)*I)).im
+
+theorem F_imaginary (r : ℝ) : F ((r : ℂ)*I) = (T r : ℂ)*I := by
+  apply Complex.ext <;> simp [T, re_F_imaginary]
+
+@[simp] theorem T_zero : T 0 = 0 := by simp [T]
+
+@[simp] theorem T_neg (r : ℝ) : T (-r) = -T r := by simp [T]
+
+theorem summable_odd_sq : Summable (fun n : ℕ => 1/(2*(n : ℝ)+1)^2) := by
+  have h := hasSum_zeta_two.summable.comp_injective (i := fun n : ℕ => 2*n+1)
+    (by intro n m h; dsimp at h; omega)
+  simpa only [Function.comp_def, Nat.cast_add, Nat.cast_mul, Nat.cast_ofNat, Nat.cast_one] using h
+
+theorem tsum_odd_sq : (∑' n : ℕ, 1/(2*(n : ℝ)+1)^2) = Real.pi^2/8 := by
+  have he : HasSum (fun n : ℕ => 1/((2*n : ℕ) : ℝ)^2) (Real.pi^2/24) := by
+    have hh : HasSum (fun n : ℕ => (1/4 : ℝ) * (1/(n : ℝ)^2))
+        ((1/4 : ℝ)*(Real.pi^2/6)) := hasSum_zeta_two.mul_left _
+    have eqf : (fun n : ℕ => 1/((2*n : ℕ) : ℝ)^2) =
+        (fun n : ℕ => (1/4 : ℝ)*(1/(n : ℝ)^2)) := by
+      funext n
+      push_cast
+      simp only [mul_pow, div_mul_eq_div_div]
+      norm_num
+      ring
+    rw [eqf, show Real.pi^2/24 = (1/4 : ℝ)*(Real.pi^2/6) by ring]
+    exact hh
+  have ho : HasSum (fun n : ℕ => 1/((2*n+1 : ℕ) : ℝ)^2)
+      (∑' n : ℕ, 1/(2*(n : ℝ)+1)^2) := by
+    simpa only [Nat.cast_add, Nat.cast_mul, Nat.cast_ofNat, Nat.cast_one] using summable_odd_sq.hasSum
+  have h := (HasSum.even_add_odd (f := fun n : ℕ => 1/(n : ℝ)^2) he ho).unique hasSum_zeta_two
+  linarith
+
+@[simp] theorem F_I : F (I : ℂ) = (I : ℂ) := by
+  have he : (fun n : ℕ => term n I) = (fun n : ℕ => (I : ℂ) * ((1/(2*(n : ℝ)+1)^2 : ℝ) : ℂ)) := by
+    funext n
+    have hp : (-1 : ℂ)^n * (I : ℂ)^(2*n+1) = (I : ℂ) := by
+      rw [pow_succ, pow_mul, I_sq]
+      have h1 : (-1 : ℂ)^n * (-1)^n = 1 := by rw [← mul_pow]; simp
+      linear_combination I * h1
+    simp only [term, hp, ofReal_div, ofReal_one, ofReal_pow, ofReal_add, ofReal_mul,
+      ofReal_ofNat, ofReal_natCast]
+    ring
+  rw [F, he, tsum_mul_left, ← Complex.ofReal_tsum, tsum_odd_sq]
+  push_cast
+  field_simp [Real.pi_ne_zero]
+
+@[simp] theorem T_one : T 1 = 1 := by simp [T]
+
+theorem continuousOn_T : ContinuousOn T (Icc (-1) 1) := by
+  have hh : ContinuousOn (fun r : ℝ => F ((r : ℂ)*I)) (Icc (-1) 1) := by
+    apply continuousOn_F_closedBall.comp (by fun_prop)
+    intro r hr
+    simpa [Metric.mem_closedBall, norm_mul, norm_real, Real.norm_eq_abs, abs_le] using hr
+  exact Complex.continuous_im.comp_continuousOn hh
+
+theorem hasDerivAt_T {r : ℝ} (hr : |r| < 1) :
+    HasDerivAt T (deriv F ((r : ℂ)*I)).re r := by
+  have hf := (hasDerivAt_F (w := (r : ℂ)*I) (by simpa [norm_mul] using hr)).differentiableAt.hasDerivAt
+  have hc := hf.comp (r : ℂ) ((hasDerivAt_id _).mul_const I)
+  have hm := Complex.imCLM.hasFDerivAt.comp_hasDerivAt r hc.comp_ofReal
+  convert hm using 1 <;> first | rfl | simp
+
+theorem strictMonoOn_T : StrictMonoOn T (Icc (-1) 1) := by
+  apply strictMonoOn_of_hasDerivWithinAt_pos (convex_Icc _ _) continuousOn_T
+  · intro r hr
+    have h : |r| < 1 := by simpa only [interior_Icc, mem_Ioo, abs_lt] using hr
+    exact (hasDerivAt_T h).hasDerivWithinAt
+  · intro r hr
+    have h : |r| < 1 := by simpa only [interior_Icc, mem_Ioo, abs_lt] using hr
+    exact deriv_F_re_pos (by simpa [norm_mul] using h)
+
+theorem T_pos {r : ℝ} (hr : 0 < r) (hr1 : r ≤ 1) : 0 < T r := by
+  have h := strictMonoOn_T (by norm_num : (0 : ℝ) ∈ Icc (-1) 1) ⟨by linarith, hr1⟩ hr
+  simpa using h
+
+theorem T_lt_one {r : ℝ} (hr : -1 ≤ r) (hr1 : r < 1) : T r < 1 := by
+  have h := strictMonoOn_T ⟨hr, hr1.le⟩ (by norm_num : (1 : ℝ) ∈ Icc (-1) 1) hr1
+  simpa using h
+
+theorem exists_T_eq {t : ℝ} (ht : |t| < 1) : ∃ r ∈ Ioo (-1) 1, T r = t := by
+  have hh : t ∈ Icc (T (-1)) (T 1) := by simpa using
+    ⟨(abs_lt.mp ht).1.le, (abs_lt.mp ht).2.le⟩
+  obtain ⟨r, hr, he⟩ := intermediate_value_Icc (by norm_num : (-1 : ℝ) ≤ 1) continuousOn_T hh
+  refine ⟨r, ⟨hr.1.lt_of_ne ?_, hr.2.lt_of_ne ?_⟩, he⟩
+  · intro h
+    rw [← h] at he
+    have h' : T (-1) = -1 := by simp
+    rw [h'] at he
+    linarith [abs_lt.mp ht |>.1]
+  · intro h
+    rw [h, T_one] at he
+    linarith [abs_lt.mp ht |>.2]
+
+end
+end PlanarLens
+
+namespace PlanarLens
+open Set Filter Complex
+open scoped Topology ContDiff
+noncomputable section
+
+def A (w : ℂ) : ℝ := (w * deriv F w).re
+
+theorem A_pos {w : ℂ} (hw : ‖w‖ < 1) (hr : 0 < w.re) : 0 < A w := by
+  rw [A, re_mul_deriv_F hw]
+  apply mul_pos (by positivity)
+  apply Real.arctan_pos.mpr
+  apply div_pos (by linarith)
+  nlinarith [norm_nonneg w]
+
+theorem hasDerivAt_horizontal_g {v t : ℝ} (hz : (v : ℂ)+(t : ℂ)*I ∈ D) :
+    HasDerivAt (fun h : ℝ => g ((h : ℂ)+(t : ℂ)*I))
+      (deriv g ((v : ℂ)+(t : ℂ)*I)) v := by
+  have hg := (analyticOnNhd_g _ hz).differentiableAt.hasDerivAt
+  have hc := hg.comp (v : ℂ) ((hasDerivAt_id _).add_const ((t : ℂ)*I))
+  convert hc.comp_ofReal using 1 <;> first | rfl | simp only [mul_one]
+
+theorem horizontal_normSq_deriv {v t : ℝ} (hz : (v : ℂ)+(t : ℂ)*I ∈ D) :
+    HasDerivAt (fun h : ℝ => ‖g ((h : ℂ)+(t : ℂ)*I)‖^2)
+      (2*A (g ((v : ℂ)+(t : ℂ)*I)) / ‖deriv F (g ((v : ℂ)+(t : ℂ)*I))‖^2) v := by
+  have hh := (hasDerivAt_horizontal_g hz).norm_sq
+  have he : deriv g ((v : ℂ)+(t : ℂ)*I) = (deriv F (g ((v : ℂ)+(t : ℂ)*I)))⁻¹ := by
+    rw [← deriv_g_F (norm_g_lt hz), F_g hz]
+  rw [he] at hh
+  convert hh using 1
+  rw [real_inner_eq_re_inner ℂ, RCLike.inner_apply]
+  simp only [A, Complex.sq_norm, Complex.mul_re, Complex.inv_re, Complex.inv_im,
+    Complex.conj_re, Complex.conj_im, RCLike.re_to_complex]
+  ring
+
+theorem horizontal_normSq_deriv_pos {v t : ℝ} (hz : (v : ℂ)+(t : ℂ)*I ∈ D)
+    (hv : 0 < v) :
+    0 < 2*A (g ((v : ℂ)+(t : ℂ)*I)) / ‖deriv F (g ((v : ℂ)+(t : ℂ)*I))‖^2 := by
+  apply div_pos
+  · exact mul_pos (by norm_num) (A_pos (norm_g_lt hz) (re_g_pos hz (by simpa using hv)))
+  · exact sq_pos_of_pos (norm_pos_iff.mpr (deriv_F_ne_zero (norm_g_lt hz)))
+
+theorem horizontal_segment_mem {v t : ℝ} (hz : (v : ℂ)+(t : ℂ)*I ∈ D)
+    (_hv : 0 ≤ v) {h : ℝ} (hh : h ∈ Icc 0 v) : (h : ℂ)+(t : ℂ)*I ∈ D := by
+  let R := ‖g ((v : ℂ)+(t : ℂ)*I)‖
+  have hR : R < 1 := norm_g_lt hz
+  let C := F '' Metric.closedBall (0 : ℂ) R
+  have hCD : C ⊆ D := by
+    apply image_mono
+    intro w hw
+    have hw' : ‖w‖ ≤ R := by simpa using hw
+    simpa using hw'.trans_lt hR
+  have hCc : IsCompact C := by
+    apply (isCompact_closedBall (0 : ℂ) R).image_of_continuousOn
+    exact continuousOn_F_closedBall.mono (Metric.closedBall_subset_closedBall hR.le)
+  have hCm {a : ℝ} : (a : ℂ)+(t : ℂ)*I ∈ C ↔
+      (a : ℂ)+(t : ℂ)*I ∈ D ∧ ‖g ((a : ℂ)+(t : ℂ)*I)‖ ≤ R := by
+    constructor
+    · rintro ⟨w, hw, he⟩
+      have hw' : ‖w‖ ≤ R := by simpa using hw
+      refine ⟨hCD ⟨w, hw, he⟩, ?_⟩
+      rw [← he, g_F (hw'.trans_lt hR)]
+      exact hw'
+    · rintro ⟨ha, hr⟩
+      exact ⟨g _, by simpa using hr, F_g ha⟩
+  let S := Icc h v ∩ (fun a : ℝ => (a : ℂ)+(t : ℂ)*I) ⁻¹' C
+  have hSc : IsCompact S := isCompact_Icc.inter_right (hCc.isClosed.preimage (by fun_prop))
+  have hvS : v ∈ S := ⟨⟨hh.2, le_rfl⟩, hCm.mpr ⟨hz, le_rfl⟩⟩
+  obtain ⟨a, ha, hmin⟩ := hSc.exists_isMinOn ⟨v, hvS⟩ (continuous_id.continuousOn)
+  have haD := (hCm.mp ha.2).1
+  have haR := (hCm.mp ha.2).2
+  have hha : h ≤ a := ha.1.1
+  by_cases he : a = h
+  · simpa only [he] using haD
+  have hha' : h < a := lt_of_le_of_ne hha (Ne.symm he)
+  have ha0 : 0 < a := hh.1.trans_lt hha'
+  have hopen : IsOpen ((fun b : ℝ => (b : ℂ)+(t : ℂ)*I) ⁻¹' D) :=
+    isOpen_D.preimage (by fun_prop)
+  obtain ⟨ε, hε, hball⟩ := Metric.isOpen_iff.mp hopen a haD
+  let d := min (a-h) ε / 2
+  have hd : 0 < d := by dsimp [d]; positivity
+  have hdh : d < a-h := by dsimp [d]; linarith [min_le_left (a-h) ε]
+  have hdε : d < ε := by dsimp [d]; linarith [min_le_right (a-h) ε]
+  let b := a-d
+  have hhb : h < b := by dsimp [b]; linarith
+  have hba : b < a := by dsimp [b]; linarith
+  have hseg : ∀ x ∈ Icc b a, (x : ℂ)+(t : ℂ)*I ∈ D := by
+    intro x hx
+    apply hball
+    rw [Metric.mem_ball, Real.dist_eq, abs_of_nonpos (by linarith [hx.2] : x-a ≤ 0)]
+    have hxb : a-d ≤ x := hx.1
+    linarith
+  have hmono : StrictMonoOn (fun x : ℝ => ‖g ((x : ℂ)+(t : ℂ)*I)‖^2) (Icc b a) := by
+    apply strictMonoOn_of_hasDerivWithinAt_pos (convex_Icc b a)
+      (fun x hx => (horizontal_normSq_deriv (hseg x hx)).continuousAt.continuousWithinAt)
+      (fun x hx => (horizontal_normSq_deriv (hseg x (interior_subset hx))).hasDerivWithinAt)
+    intro x hx
+    have hx' := interior_subset hx
+    apply horizontal_normSq_deriv_pos (hseg x hx')
+    linarith [hx'.1, hh.1]
+  have hnorm : ‖g ((b : ℂ)+(t : ℂ)*I)‖ ≤ R := by
+    apply le_of_sq_le_sq _ (norm_nonneg _)
+    have hm := hmono ⟨le_rfl, hba.le⟩ ⟨hba.le, le_rfl⟩ hba
+    have hR0 : 0 ≤ R := norm_nonneg _
+    nlinarith [norm_nonneg (g ((a : ℂ)+(t : ℂ)*I))]
+  have hbS : b ∈ S := ⟨⟨hhb.le, hba.le.trans ha.1.2⟩,
+    hCm.mpr ⟨hseg b ⟨le_rfl, hba.le⟩, hnorm⟩⟩
+  have ham : a ≤ b := hmin hbS
+  exact False.elim (not_le_of_gt hba ham)
+
+theorem neg_conj_mem_D {z : ℂ} (hz : z ∈ D) : -(starRingEnd ℂ) z ∈ D := by
+  rcases hz with ⟨w, hw, rfl⟩
+  refine ⟨-(starRingEnd ℂ) w, by simpa using hw, ?_⟩
+  simp
+
+theorem g_neg_conj {z : ℂ} (hz : z ∈ D) : g (-(starRingEnd ℂ) z) = -(starRingEnd ℂ) (g z) := by
+  apply F_injOn (by simpa using norm_g_lt (neg_conj_mem_D hz))
+    (by simpa using norm_g_lt hz)
+  rw [F_g (neg_conj_mem_D hz)]
+  simp [F_g hz]
+
+theorem conj_mem_D {z : ℂ} (hz : z ∈ D) : (starRingEnd ℂ) z ∈ D := by
+  rcases hz with ⟨w, hw, rfl⟩
+  exact ⟨(starRingEnd ℂ) w, by simpa using hw, by simp⟩
+
+theorem g_conj {z : ℂ} (hz : z ∈ D) : g ((starRingEnd ℂ) z) = (starRingEnd ℂ) (g z) := by
+  apply F_injOn (by simpa using norm_g_lt (conj_mem_D hz))
+    (by simpa using norm_g_lt hz)
+  simp [F_g (conj_mem_D hz), F_g hz]
+
+theorem horizontal_segment_mem_signed {v t : ℝ} (hz : (v : ℂ)+(t : ℂ)*I ∈ D)
+    {h : ℝ} (hh : h ∈ uIcc 0 v) : (h : ℂ)+(t : ℂ)*I ∈ D := by
+  rcases le_total 0 v with hv | hv
+  · rw [uIcc_of_le hv] at hh
+    exact horizontal_segment_mem hz hv hh
+  · rw [uIcc_of_ge hv] at hh
+    have he (a : ℝ) : -((starRingEnd ℂ) ((a : ℂ)+(t : ℂ)*I)) = ((-a : ℝ) : ℂ)+(t : ℂ)*I := by
+      apply Complex.ext <;> simp
+    have hz' := neg_conj_mem_D hz
+    rw [he] at hz'
+    have hh' : -h ∈ Icc 0 (-v) := ⟨by linarith [hh.2], by linarith [hh.1]⟩
+    have hhD := horizontal_segment_mem hz' (by linarith) hh'
+    have hneg := neg_conj_mem_D hhD
+    simpa only [he, neg_neg] using hneg
+
+theorem axis_mem_D {v t : ℝ} (hz : (v : ℂ)+(t : ℂ)*I ∈ D) : (t : ℂ)*I ∈ D := by
+  simpa using horizontal_segment_mem_signed hz (left_mem_uIcc (a := 0) (b := v))
+
+theorem g_axis_re_zero {t : ℝ} (ht : (t : ℂ)*I ∈ D) : (g ((t : ℂ)*I)).re = 0 := by
+  have he : -(starRingEnd ℂ) ((t : ℂ)*I) = (t : ℂ)*I := by simp
+  have hh := g_neg_conj ht
+  rw [he] at hh
+  have hr := congr_arg Complex.re hh
+  simp only [neg_re, conj_re] at hr
+  linarith
+
+theorem axis_image {t : ℝ} (ht : (t : ℂ)*I ∈ D) :
+    ∃ r ∈ Ioo (-1 : ℝ) 1, g ((t : ℂ)*I) = (r : ℂ)*I ∧ T r = t := by
+  let r := (g ((t : ℂ)*I)).im
+  have he : g ((t : ℂ)*I) = (r : ℂ)*I := by
+    apply Complex.ext <;> simp [r, g_axis_re_zero ht]
+  have hr : |r| < 1 := by
+    have hh := norm_g_lt ht
+    rw [he] at hh
+    simpa [norm_mul] using hh
+  refine ⟨r, abs_lt.mp hr, he, ?_⟩
+  have hh := congr_arg Complex.im (F_g ht)
+  rw [he] at hh
+  simpa [T] using hh
+
+theorem im_mem_Ioo {z : ℂ} (hz : z ∈ D) : z.im ∈ Ioo (-1 : ℝ) 1 := by
+  have hz' : (z.re : ℂ)+(z.im : ℂ)*I ∈ D := by simpa using hz
+  obtain ⟨r, hr, _, ht⟩ := axis_image (axis_mem_D hz')
+  rw [← ht]
+  refine ⟨?_, T_lt_one hr.1.le hr.2⟩
+  have h := strictMonoOn_T (by norm_num : (-1 : ℝ) ∈ Icc (-1) 1) ⟨hr.1.le, hr.2.le⟩ hr.1
+  simpa using h
+
+theorem imaginary_mem_D {t : ℝ} (ht : |t| < 1) : (t : ℂ)*I ∈ D := by
+  obtain ⟨r, hr, he⟩ := exists_T_eq ht
+  exact ⟨(r : ℂ)*I, by simpa [Metric.mem_ball, norm_mul, abs_lt] using hr,
+    by rw [F_imaginary r, he]⟩
+
+theorem horizontal_radius_strictMono {v t : ℝ} (hz : (v : ℂ)+(t : ℂ)*I ∈ D)
+    (hv : 0 ≤ v) : StrictMonoOn (fun h : ℝ => ‖g ((h : ℂ)+(t : ℂ)*I)‖) (Icc 0 v) := by
+  have hm : StrictMonoOn (fun h : ℝ => ‖g ((h : ℂ)+(t : ℂ)*I)‖^2) (Icc 0 v) := by
+    apply strictMonoOn_of_hasDerivWithinAt_pos (convex_Icc 0 v)
+      (fun h hh => (horizontal_normSq_deriv (horizontal_segment_mem hz hv hh)).continuousAt.continuousWithinAt)
+      (fun h hh => (horizontal_normSq_deriv (horizontal_segment_mem hz hv (interior_subset hh))).hasDerivWithinAt)
+    intro h hh
+    apply horizontal_normSq_deriv_pos (horizontal_segment_mem hz hv (interior_subset hh))
+    have hh' : h ∈ Ioo 0 v := by simpa only [interior_Icc] using hh
+    exact hh'.1
+  intro a ha b hb hab
+  have hh := hm ha hb hab
+  nlinarith [norm_nonneg (g ((a : ℂ)+(t : ℂ)*I)), norm_nonneg (g ((b : ℂ)+(t : ℂ)*I))]
+
+theorem T_norm_g_axis {t : ℝ} (ht : (t : ℂ)*I ∈ D) : T ‖g ((t : ℂ)*I)‖ = |t| := by
+  obtain ⟨r, hr, he, hT⟩ := axis_image ht
+  rw [he]
+  simp only [norm_mul, norm_real, norm_I, mul_one, Real.norm_eq_abs]
+  rcases le_total 0 r with hr0 | hr0
+  · rw [abs_of_nonneg hr0, ← hT, abs_of_nonneg]
+    have h := strictMonoOn_T.monotoneOn (by norm_num : (0 : ℝ) ∈ Icc (-1) 1)
+      ⟨hr.1.le, hr.2.le⟩ hr0
+    simpa using h
+  · rw [abs_of_nonpos hr0, T_neg, ← hT, abs_of_nonpos]
+    have h := strictMonoOn_T.monotoneOn ⟨hr.1.le, hr.2.le⟩
+      (by norm_num : (0 : ℝ) ∈ Icc (-1) 1) hr0
+    simpa using h
+
+theorem vertical_gap_pos {z : ℂ} (hz : z ∈ D) (hr : 0 < z.re) :
+    0 < T ‖g z‖ - |z.im| := by
+  have hz' : (z.re : ℂ)+(z.im : ℂ)*I ∈ D := by simpa using hz
+  have hnorm := horizontal_radius_strictMono hz' hr.le
+    (left_mem_Icc.mpr hr.le) (right_mem_Icc.mpr hr.le) hr
+  simp only [ofReal_zero, zero_add, re_add_im] at hnorm
+  rw [← T_norm_g_axis (axis_mem_D hz')]
+  have hh := strictMonoOn_T
+    ⟨by linarith [norm_nonneg (g ((z.im : ℂ)*I))], (norm_g_lt (axis_mem_D hz')).le⟩
+    ⟨by linarith [norm_nonneg (g z)], (norm_g_lt hz).le⟩ hnorm
+  linarith
+
+end
+end PlanarLens
+
+end LowerBoundInline
+
+end OAI
diff --git a/lean/OAI/Geometry/PolarProducts/LensBounds.lean b/lean/OAI/Geometry/PolarProducts/LensBounds.lean
new file mode 100644
index 0000000000000000000000000000000000000000..41dae90641f3946068a25be0378234b738113343
--- /dev/null
+++ b/lean/OAI/Geometry/PolarProducts/LensBounds.lean
@@ -0,0 +1,246 @@
+import OAI.Geometry.PolarProducts.TipEstimates
+
+namespace OAI
+
+section LowerBoundInline
+open Set Filter Function
+open scoped Topology ContDiff NNReal
+open Set Filter Metric
+open scoped Topology ContDiff
+open Set Filter Function MeasureTheory Metric
+open scoped Topology ContDiff NNReal
+open Set Filter Function
+open scoped Topology ContDiff
+open Set Filter Function
+open scoped Topology ContDiff NNReal
+open Set Filter
+open scoped Topology ContDiff
+open Set Filter Function
+open scoped Topology ContDiff
+open Set Filter Function
+open scoped ContDiff Topology
+open Set MeasureTheory
+open scoped ContDiff Interval Topology
+open Set
+open scoped Topology ContDiff
+open Set
+open Set MeasureTheory
+open scoped ContDiff Interval Topology
+open Set Filter Complex
+open scoped Topology ContDiff
+open MeasureTheory intervalIntegral Set
+open scoped Real
+
+namespace PlanarLens
+open Set Filter Complex MeasureTheory
+open scoped Topology ContDiff
+noncomputable section
+
+theorem exists_inner_bound {a : ℝ} (ha0 : 0 ≤ a) (ha1 : a < 1) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ (k : ℕ) (t v : ℝ),
+      (v : ℂ)+(t : ℂ)*I ∈ D → 0 ≤ v → rho t v ≤ a →
+      (k : ℝ)*∫ h in 0..v, density k ((h : ℂ)+(t : ℂ)*I) ≤
+        C*(k : ℝ)*a^(2*k-2) := by
+  let Cset := F '' Metric.closedBall (0 : ℂ) a
+  have hCD : Cset ⊆ D := by
+    apply image_mono
+    intro w hw
+    have hw' : ‖w‖ ≤ a := by simpa using hw
+    simpa using hw'.trans_lt ha1
+  have hCc : IsCompact Cset := (isCompact_closedBall (0 : ℂ) a).image_of_continuousOn
+    (continuousOn_F_closedBall.mono (Metric.closedBall_subset_closedBall ha1.le))
+  obtain ⟨N, hN⟩ := hCc.exists_bound_of_continuousOn
+    (analyticOnNhd_g.deriv.continuousOn.mono hCD)
+  obtain ⟨B, hB0, hB⟩ := isBounded_D.exists_pos_norm_le
+  let M := max N 0
+  have hM : 0 ≤ M := le_max_right _ _
+  refine ⟨B*M^2, mul_nonneg hB0.le (sq_nonneg _), ?_⟩
+  intro k t v hz hv hra
+  have hvB : v ≤ B := by
+    have hh := (abs_re_le_norm ((v : ℂ)+(t : ℂ)*I)).trans (hB _ hz)
+    simpa only [add_re, ofReal_re, mul_re, I_re, I_im, ofReal_im, mul_zero, zero_mul, sub_self, add_zero, abs_of_nonneg hv] using hh
+  have hpi : IntervalIntegrable (fun h : ℝ => a^(2*k-2)*M^2) volume 0 v := intervalIntegrable_const
+  have hii := integrable_density hz k
+  have hcmp : ∫ h in 0..v, density k ((h : ℂ)+(t : ℂ)*I) ≤ v*(a^(2*k-2)*M^2) := by
+    have hh : ∫ h in 0..v, density k ((h : ℂ)+(t : ℂ)*I) ≤
+        ∫ h in 0..v, a^(2*k-2)*M^2 := by
+      apply intervalIntegral.integral_mono_on hv hii hpi
+      intro h hh
+      have hhz := horizontal_segment_mem hz hv hh
+      have hr : rho t h ≤ a := ((horizontal_radius_strictMono hz hv).monotoneOn hh
+        (right_mem_Icc.mpr hv) hh.2).trans hra
+      have hc : (h : ℂ)+(t : ℂ)*I ∈ Cset :=
+        ⟨g _, by simpa [rho] using hr, F_g hhz⟩
+      have hn : ‖deriv g ((h : ℂ)+(t : ℂ)*I)‖ ≤ M := (hN _ hc).trans (le_max_left _ _)
+      exact mul_le_mul (pow_le_pow_left₀ (norm_nonneg _) hr _) (pow_le_pow_left₀ (norm_nonneg _) hn _)
+        (sq_nonneg _) (pow_nonneg ha0 _)
+    simpa only [intervalIntegral.integral_const, sub_zero, smul_eq_mul] using hh
+  calc
+    _ ≤ (k : ℝ)*(v*(a^(2*k-2)*M^2)) := mul_le_mul_of_nonneg_left hcmp (Nat.cast_nonneg k)
+    _ ≤ (k : ℝ)*(B*(a^(2*k-2)*M^2)) := mul_le_mul_of_nonneg_left
+      (mul_le_mul_of_nonneg_right hvB (mul_nonneg (pow_nonneg ha0 _) (sq_nonneg _))) (Nat.cast_nonneg k)
+    _ = _ := by ring
+
+theorem exists_uniform_error_bound {a : ℝ} (ha : 1/2 ≤ a) (ha1 : a < 1) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ (k : ℕ), 1 ≤ k → ∀ (t v : ℝ),
+      (v : ℂ)+(t : ℂ)*I ∈ D → 0 ≤ v →
+      (k : ℝ)*∫ h in 0..v, density k ((h : ℂ)+(t : ℂ)*I) ≤
+      Real.pi/4*(rho t v)^(2*k) + C*(k : ℝ)*a^(2*k-2) +
+        Real.pi^3/Real.sqrt (tailSlope a)*(1+Real.pi/Real.exp 1) := by
+  have ha0 : 0 ≤ a := by linarith
+  obtain ⟨C, hC, hinner⟩ := exists_inner_bound ha0 ha1
+  refine ⟨C, hC, ?_⟩
+  intro k hk t v hz hv
+  have hE : 0 ≤ Real.pi^3/Real.sqrt (tailSlope a)*(1+Real.pi/Real.exp 1) := by positivity
+  have hCterm : 0 ≤ C*(k : ℝ)*a^(2*k-2) := by positivity
+  have hR : 0 ≤ Real.pi/4*(rho t v)^(2*k) := by
+    exact mul_nonneg (by positivity) (pow_nonneg (norm_nonneg _) _)
+  by_cases hr : rho t v ≤ a
+  · exact (hinner k t v hz hv hr).trans (by linarith)
+  by_cases hl : a ≤ rho t 0
+  · have hh := outer_integral_bound hk hz (le_refl 0) hv ha hl
+    have hn : 0 ≤ Real.pi/4*(rho t 0)^(2*k) :=
+      mul_nonneg (by positivity) (pow_nonneg (norm_nonneg _) _)
+    nlinarith
+  have haI : a ∈ Icc (rho t 0) (rho t v) := ⟨(lt_of_not_ge hl).le, (lt_of_not_ge hr).le⟩
+  obtain ⟨u, hu, heu⟩ := intermediate_value_Icc hv (continuousOn_rho hz hv) haI
+  have huD := horizontal_segment_mem hz hv hu
+  have hleft := hinner k t u huD hu.1 heu.le
+  have hright := outer_integral_bound hk hz hu.1 hu.2 ha heu.ge
+  have hIuv : IntervalIntegrable (fun h : ℝ => density k ((h : ℂ)+(t : ℂ)*I)) volume u v := by
+    exact (integrable_density hz k).mono_set (uIcc_subset_uIcc (by rwa [uIcc_of_le hv]) right_mem_uIcc)
+  have hadd := intervalIntegral.integral_add_adjacent_intervals (integrable_density huD k) hIuv
+  have hn : 0 ≤ Real.pi/4*(rho t u)^(2*k) :=
+    mul_nonneg (by positivity) (pow_nonneg (norm_nonneg _) _)
+  rw [← hadd]
+  nlinarith
+
+end
+end PlanarLens
+
+namespace PlanarLens
+open Set Filter Complex MeasureTheory
+open scoped Topology ContDiff
+noncomputable section
+
+theorem J_div_order {k : ℕ} (hk : 1 ≤ k) {v t : ℝ} (hv : 0 ≤ v) :
+    |J k v t|/(k : ℝ) = (k : ℝ)*∫ h in 0..v, density k ((h : ℂ)+(t : ℂ)*I) := by
+  have hk0 : (k : ℝ) ≠ 0 := by exact_mod_cast (by omega : k ≠ 0)
+  rw [abs_of_nonneg (J_nonneg hv k), J]
+  field_simp
+
+theorem rho_neg {v t : ℝ} (hz : (v : ℂ)+(t : ℂ)*I ∈ D) : rho t (-v) = rho t v := by
+  have he : -(starRingEnd ℂ) ((v : ℂ)+(t : ℂ)*I) = ((-v : ℝ) : ℂ)+(t : ℂ)*I := by
+    apply Complex.ext <;> simp
+  rw [rho, rho, ← he, g_neg_conj hz]
+  simp
+
+def excess (k : ℕ) (z : ℂ) : ℝ := |J k z.re z.im|/(k : ℝ) - Real.pi/4*‖g z‖^(2*k)
+
+def epsilon (k : ℕ) : ℝ := max 0 (sSup (excess k '' D))
+
+theorem epsilon_nonneg (k : ℕ) : 0 ≤ epsilon k := le_max_left _ _
+
+theorem exists_excess_bound {a : ℝ} (ha : 1/2 ≤ a) (ha1 : a < 1) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ (k : ℕ), 1 ≤ k → ∀ z ∈ D,
+      excess k z ≤ C*(k : ℝ)*a^(2*k-2) +
+        Real.pi^3/Real.sqrt (tailSlope a)*(1+Real.pi/Real.exp 1) := by
+  obtain ⟨C, hC, hb⟩ := exists_uniform_error_bound ha ha1
+  refine ⟨C, hC, ?_⟩
+  intro k hk z hz
+  have hz' : (z.re : ℂ)+(z.im : ℂ)*I ∈ D := by simpa only [re_add_im] using hz
+  have hpos (v t : ℝ) (hv : 0 ≤ v) (hh : (v : ℂ)+(t : ℂ)*I ∈ D) :
+      |J k v t|/(k : ℝ) - Real.pi/4*(rho t v)^(2*k) ≤
+        C*(k : ℝ)*a^(2*k-2) + Real.pi^3/Real.sqrt (tailSlope a)*(1+Real.pi/Real.exp 1) := by
+    rw [J_div_order hk hv]
+    linarith [hb k hk t v hh hv]
+  by_cases hv : 0 ≤ z.re
+  · simpa only [excess, rho, re_add_im] using hpos z.re z.im hv hz'
+  · have hnz : ((-z.re : ℝ) : ℂ)+(z.im : ℂ)*I ∈ D := by
+      convert neg_conj_mem_D hz using 1
+      apply Complex.ext <;> simp
+    have hh := hpos (-z.re) z.im (by linarith) hnz
+    rw [J_neg hz', abs_neg, rho_neg hz'] at hh
+    simpa only [excess, rho, re_add_im] using hh
+
+theorem bddAbove_excess {k : ℕ} (hk : 1 ≤ k) : BddAbove (excess k '' D) := by
+  obtain ⟨C, _, hC⟩ := exists_excess_bound (a := 3/4) (by norm_num) (by norm_num)
+  refine ⟨C*(k : ℝ)*(3/4)^(2*k-2) + Real.pi^3/Real.sqrt (tailSlope (3/4))*(1+Real.pi/Real.exp 1), ?_⟩
+  rintro _ ⟨z, hz, rfl⟩
+  exact hC k hk z hz
+
+theorem excess_le_epsilon {k : ℕ} (hk : 1 ≤ k) {z : ℂ} (hz : z ∈ D) : excess k z ≤ epsilon k :=
+  (le_csSup (bddAbove_excess hk) (mem_image_of_mem _ hz)).trans (le_max_right _ _)
+
+theorem epsilon_le_of_excess_le {k : ℕ} {B : ℝ} (hB : 0 ≤ B) (h : ∀ z ∈ D, excess k z ≤ B) :
+    epsilon k ≤ B := by
+  apply max_le hB
+  apply csSup_le ⟨excess k 0, mem_image_of_mem _ zero_mem_D⟩
+  rintro _ ⟨z, hz, rfl⟩
+  exact h z hz
+
+theorem tendsto_inner_error {a : ℝ} (ha : 0 < a) (ha1 : a < 1) (C : ℝ) :
+    Tendsto (fun k : ℕ => C*(k : ℝ)*a^(2*k-2)) atTop (𝓝 0) := by
+  have hq : |a^2| < 1 := by rw [abs_of_nonneg (sq_nonneg _)]; nlinarith
+  have ht := (tendsto_pow_const_mul_const_pow_of_abs_lt_one 1 hq).const_mul (C/a^2)
+  simp only [pow_one, mul_zero] at ht
+  apply ht.congr'
+  filter_upwards [eventually_ge_atTop (1 : ℕ)] with k hk
+  have hp : a^(2*k-2)*a^2 = (a^2)^k := by
+    rw [← pow_add, Nat.sub_add_cancel (by omega : 2 ≤ 2*k), pow_mul]
+  have ha2 : a^2 ≠ 0 := pow_ne_zero _ ha.ne'
+  apply mul_right_cancel₀ ha2
+  calc
+    ((C/a^2)*((k : ℝ)*(a^2)^k))*a^2 = C*(k : ℝ)*(a^2)^k := by field_simp
+    _ = C*(k : ℝ)*(a^(2*k-2)*a^2) := by rw [hp]
+    _ = C*(k : ℝ)*a^(2*k-2)*a^2 := by ring
+
+theorem exists_small_tail (e : ℝ) (he : 0 < e) :
+    ∃ a ∈ Ioo (1/2 : ℝ) 1,
+      Real.pi^3/Real.sqrt (tailSlope a)*(1+Real.pi/Real.exp 1) < e := by
+  let H := Real.pi^3*(1+Real.pi/Real.exp 1)
+  have hH : 0 < H := by dsimp [H]; positivity
+  obtain ⟨a, ha, hL⟩ := exists_tail_cutoff ((H/e)^2)
+  have ha0 : 0 < a := by linarith [ha.1]
+  have hs : 0 < Real.sqrt (tailSlope a) := Real.sqrt_pos.2 (tailSlope_pos ha0 ha.2)
+  have hq : H/e < Real.sqrt (tailSlope a) := by
+    rw [← Real.sqrt_sq (div_nonneg hH.le he.le)]
+    exact Real.sqrt_lt_sqrt (sq_nonneg _) hL
+  refine ⟨a, ha, ?_⟩
+  have hh : H < e * Real.sqrt (tailSlope a) := by
+    have := (div_lt_iff₀ he).mp hq
+    nlinarith
+  have hh' := (div_lt_iff₀ hs).mpr hh
+  convert hh' using 1
+  first | rfl | (dsimp only [H]; ring)
+
+theorem tendsto_epsilon : Tendsto epsilon atTop (𝓝 0) := by
+  apply tendsto_order.mpr
+  constructor
+  · intro c hc
+    exact Filter.Eventually.of_forall (fun k => hc.trans_le (epsilon_nonneg k))
+  · intro e he
+    obtain ⟨a, ha, htail⟩ := exists_small_tail (e/2) (by linarith)
+    obtain ⟨C, hC, hbound⟩ := exists_excess_bound ha.1.le ha.2
+    have ha0 : 0 < a := by linarith [ha.1]
+    have ht := (tendsto_inner_error ha0 ha.2 C).eventually (eventually_lt_nhds (by linarith : (0 : ℝ) < e/2))
+    filter_upwards [ht, eventually_ge_atTop (1 : ℕ)] with k hinner hk
+    have hE := epsilon_le_of_excess_le (k := k)
+      (B := C*(k : ℝ)*a^(2*k-2)+Real.pi^3/Real.sqrt (tailSlope a)*(1+Real.pi/Real.exp 1))
+      (by positivity) (hbound k hk)
+    linarith
+
+theorem uniform_slice {k : ℕ} (hk : 1 ≤ k) {v t : ℝ} (hz : (v : ℂ)+(t : ℂ)*I ∈ D) :
+    |J k v t|/(k : ℝ) ≤ Real.pi/4*‖g ((v : ℂ)+(t : ℂ)*I)‖^(2*k)+epsilon k := by
+  have hh := excess_le_epsilon hk hz
+  have hre : ((v : ℂ)+(t : ℂ)*I).re = v := by simp
+  have him : ((v : ℂ)+(t : ℂ)*I).im = t := by simp
+  simp only [excess, hre, him] at hh
+  linarith
+
+end
+end PlanarLens
+
+end LowerBoundInline
+
+end OAI
diff --git a/lean/OAI/Geometry/PolarProducts/LensEstimates.lean b/lean/OAI/Geometry/PolarProducts/LensEstimates.lean
new file mode 100644
index 0000000000000000000000000000000000000000..49d26b3274a8f72c4ad8dfdc84ebdede02cf1cb7
--- /dev/null
+++ b/lean/OAI/Geometry/PolarProducts/LensEstimates.lean
@@ -0,0 +1,399 @@
+import OAI.Geometry.PolarProducts.LensArea
+
+namespace OAI
+
+section LowerBoundInline
+open Set Filter Function
+open scoped Topology ContDiff NNReal
+open Set Filter Metric
+open scoped Topology ContDiff
+open Set Filter Function MeasureTheory Metric
+open scoped Topology ContDiff NNReal
+open Set Filter Function
+open scoped Topology ContDiff
+open Set Filter Function
+open scoped Topology ContDiff NNReal
+open Set Filter
+open scoped Topology ContDiff
+open Set Filter Function
+open scoped Topology ContDiff
+open Set Filter Function
+open scoped ContDiff Topology
+open Set MeasureTheory
+open scoped ContDiff Interval Topology
+open Set
+open scoped Topology ContDiff
+open Set
+open Set MeasureTheory
+open scoped ContDiff Interval Topology
+open Set Filter Complex
+open scoped Topology ContDiff
+
+namespace PlanarLens
+open Set Filter Complex MeasureTheory intervalIntegral
+open scoped Topology ContDiff
+noncomputable section
+
+theorem arctan_le_self {x : ℝ} (hx : 0 ≤ x) : Real.arctan x ≤ x := by
+  have hc : Continuous (fun t : ℝ => 1/(1+t^2)) := by
+    apply continuous_const.div (by fun_prop)
+    intro t; positivity
+  have hi := integral_mono_on (μ := volume) hx (hc.intervalIntegrable 0 x)
+    (intervalIntegrable_const (c := (1 : ℝ))) (fun t _ => by
+      apply (div_le_one (by positivity : 0 < 1+t^2)).mpr
+      nlinarith [sq_nonneg t])
+  simpa [integral_one_div_one_add_sq] using hi
+
+theorem half_le_arctan {x : ℝ} (hx : 0 ≤ x) (hx1 : x ≤ 1) : x/2 ≤ Real.arctan x := by
+  have hc : Continuous (fun t : ℝ => 1/(1+t^2)) := by
+    apply continuous_const.div (by fun_prop)
+    intro t; positivity
+  have hi := integral_mono_on (μ := volume) hx (intervalIntegrable_const (c := (1/2 : ℝ)))
+    (hc.intervalIntegrable 0 x) (fun t ht => by
+      apply (le_div_iff₀ (by positivity : 0 < 1+t^2)).mpr
+      have h0 := ht.1
+      have h1 := ht.2.trans hx1
+      nlinarith)
+  simpa [integral_one_div_one_add_sq, div_eq_mul_inv] using hi
+
+theorem reciprocal_arctan {x : ℝ} (hx : 0 < x) :
+    1 / Real.arctan x ≤ 2 / Real.pi + 2/x := by
+  have ha : 0 < Real.arctan x := Real.arctan_pos.mpr hx
+  rcases le_total x 1 with hx1 | hx1
+  · have hh := half_le_arctan hx.le hx1
+    have hdiv : 1 / Real.arctan x ≤ 2/x := by
+      apply (div_le_div_iff₀ ha hx).mpr
+      linarith
+    apply hdiv.trans
+    have hp : 0 < 2 / Real.pi := by positivity
+    linarith
+  · have hh : Real.pi / 2 - Real.arctan x ≤ 1/x := by
+      rw [← Real.arctan_inv_of_pos hx]
+      simpa only [one_div] using arctan_le_self (inv_nonneg.mpr hx.le)
+    have hA : Real.pi / 4 ≤ Real.arctan x := by
+      simpa using Real.arctan_strictMono.monotone hx1
+    have hπA : 1 ≤ Real.pi * Real.arctan x := by
+      have hp := Real.pi_gt_three
+      nlinarith [mul_le_mul_of_nonneg_left hA Real.pi_pos.le]
+    have hfirst : (Real.pi/2-Real.arctan x) / Real.arctan x ≤ (1/x)/Real.arctan x :=
+      div_le_div_of_nonneg_right hh ha.le
+    have hsecond : (2/Real.pi)*((1/x)/Real.arctan x) ≤ 2/x := by
+      apply (le_div_iff₀ hx).mpr
+      have he : (2/Real.pi)*((1/x)/Real.arctan x)*x = 2/(Real.pi*Real.arctan x) := by
+        field_simp
+      rw [he]
+      apply (div_le_iff₀ (mul_pos Real.pi_pos ha)).mpr
+      linarith
+    have hthird := mul_le_mul_of_nonneg_left hfirst
+      (by positivity : 0 ≤ 2/Real.pi)
+    have he : (2/Real.pi)*((Real.pi/2-Real.arctan x)/Real.arctan x) =
+        1/Real.arctan x - 2/Real.pi := by field_simp
+    rw [he] at hthird
+    linarith
+
+def circleTop (r s : ℝ) : ℂ := (r : ℂ) * ((Real.sin s : ℂ)+(Real.cos s : ℂ)*I)
+
+@[simp] theorem circleTop_re (r s : ℝ) : (circleTop r s).re = r * Real.sin s := by
+  simp only [circleTop, mul_re, add_re, ofReal_re, ofReal_im, I_im, I_re,
+    mul_zero, mul_one, sub_zero, zero_mul, add_zero]
+
+@[simp] theorem circleTop_im (r s : ℝ) : (circleTop r s).im = r * Real.cos s := by
+  simp only [circleTop, mul_im, add_re, add_im, ofReal_re, ofReal_im, mul_re, I_im, I_re,
+    mul_zero, mul_one, zero_add, zero_mul, add_zero]
+
+theorem norm_circleTop (r s : ℝ) : ‖circleTop r s‖ = |r| := by
+  have hh : ‖circleTop r s‖^2 = r^2 := by
+    rw [Complex.sq_norm, normSq_apply, circleTop_re, circleTop_im]
+    nlinarith [Real.sin_sq_add_cos_sq s]
+  nlinarith [sq_abs r, norm_nonneg (circleTop r s), abs_nonneg r]
+
+@[simp] theorem circleTop_zero (r : ℝ) : circleTop r 0 = (r : ℂ)*I := by
+  simp [circleTop]
+
+@[simp] theorem circleTop_pi_half (r : ℝ) : circleTop r (Real.pi/2) = (r : ℂ) := by
+  simp [circleTop]
+
+theorem continuous_circleTop (r : ℝ) : Continuous (circleTop r) := by
+  unfold circleTop; fun_prop
+
+theorem hasDerivAt_F_im_circleTop {r : ℝ} (hr : |r| < 1) (s : ℝ) :
+    HasDerivAt (fun s => (F (circleTop r s)).im) (-A (circleTop r s)) s := by
+  have hnorm : ‖(r : ℂ)*(Complex.sin s+Complex.cos s*I)‖ < 1 := by
+    rw [← Complex.ofReal_sin, ← Complex.ofReal_cos]
+    change ‖circleTop r s‖ < 1
+    rw [norm_circleTop]
+    exact hr
+  have hd := ((Complex.hasDerivAt_sin (s : ℂ)).add
+    ((Complex.hasDerivAt_cos (s : ℂ)).mul_const I)).const_mul (r : ℂ)
+  have hf := (hasDerivAt_F hnorm).differentiableAt.hasDerivAt
+  have hh := hf.comp (s : ℂ) hd
+  have hm := Complex.imCLM.hasFDerivAt.comp_hasDerivAt s hh.comp_ofReal
+  convert hm using 1 <;> try rfl
+  · funext x
+    simp only [circleTop, ofReal_sin, ofReal_cos, Function.comp_apply, Pi.add_apply,
+      Complex.imCLM_apply]
+  · simp only [Complex.imCLM_apply, A, circleTop, ← ofReal_sin, ← ofReal_cos,
+      mul_re, mul_im, add_re, add_im, ofReal_re, ofReal_im, I_re, I_im,
+      mul_zero, mul_one, add_zero, zero_add, zero_sub, neg_re, neg_im]
+    ring
+
+theorem continuous_A_circleTop {r : ℝ} (hr : |r| < 1) :
+    Continuous (fun s => A (circleTop r s)) := by
+  rw [continuous_iff_continuousAt]
+  intro s
+  have hs : circleTop r s ∈ Metric.ball (0 : ℂ) 1 := by
+    simpa [Metric.mem_ball, norm_circleTop] using hr
+  exact Complex.continuous_re.continuousAt.comp
+    (((continuous_circleTop r).continuousAt).mul
+      (((analyticOnNhd_F _ hs).deriv.continuousAt).comp (continuous_circleTop r).continuousAt))
+
+theorem gap_integral {r : ℝ} (hr : |r| < 1) (s : ℝ) :
+    T r - (F (circleTop r s)).im = ∫ x in 0..s, A (circleTop r x) := by
+  have hh := integral_eq_sub_of_hasDerivAt (fun x _ => hasDerivAt_F_im_circleTop hr x)
+    ((continuous_A_circleTop hr).neg.intervalIntegrable 0 s)
+  rw [intervalIntegral.integral_neg, circleTop_zero] at hh
+  change -(∫ x in 0..s, A (circleTop r x)) = (F (circleTop r s)).im - T r at hh
+  linarith
+
+theorem A_circleTop_le {r s : ℝ} (hr : 0 ≤ r) (hr1 : r < 1)
+    (hs : 0 ≤ s) (hs1 : s ≤ Real.pi/2) :
+    A (circleTop r s) ≤ (8*r/(Real.pi^2*(1-r^2)))*s := by
+  have hrabs : |r| < 1 := by simpa only [abs_of_nonneg hr] using hr1
+  have hnorm : ‖circleTop r s‖ < 1 := by simpa only [norm_circleTop] using hrabs
+  rw [A, re_mul_deriv_F hnorm, circleTop_re, norm_circleTop, sq_abs]
+  have hd : 0 < 1-r^2 := by nlinarith
+  have hsin : 0 ≤ Real.sin s := Real.sin_nonneg_of_mem_Icc ⟨hs, by linarith [Real.pi_pos]⟩
+  have hx : 0 ≤ 2*(r*Real.sin s)/(1-r^2) := by positivity
+  calc
+    _ ≤ (4/Real.pi^2) * (2*(r*Real.sin s)/(1-r^2)) := by
+      exact mul_le_mul_of_nonneg_left (arctan_le_self hx) (by positivity)
+    _ ≤ (4/Real.pi^2) * (2*(r*s)/(1-r^2)) := by
+      gcongr
+      exact Real.sin_le hs
+    _ = _ := by field_simp; ring
+
+theorem gap_le_square {r s : ℝ} (hr : 0 ≤ r) (hr1 : r < 1)
+    (hs : 0 ≤ s) (hs1 : s ≤ Real.pi/2) :
+    T r - (F (circleTop r s)).im ≤ s^2/(1-r) := by
+  have hrabs : |r| < 1 := by simpa only [abs_of_nonneg hr] using hr1
+  rw [gap_integral hrabs]
+  have hh := integral_mono_on (μ := volume) hs ((continuous_A_circleTop hrabs).intervalIntegrable 0 s)
+    ((show Continuous (fun x : ℝ => (8*r/(Real.pi^2*(1-r^2)))*x) from by fun_prop).intervalIntegrable 0 s)
+    (fun x hx => A_circleTop_le hr hr1 hx.1 (hx.2.trans hs1))
+  have he : (∫ x in 0..s, (8*r/(Real.pi^2*(1-r^2)))*x) =
+      (4*r/(Real.pi^2*(1-r^2)))*s^2 := by
+    rw [intervalIntegral.integral_const_mul, integral_id]
+    ring
+  rw [he] at hh
+  apply hh.trans
+  have hd : 0 < 1-r^2 := by nlinarith
+  have h1 : 0 < 1-r := by linarith
+  have hco : 4*r/(Real.pi^2*(1-r^2)) ≤ 1/(1-r) := by
+    apply (div_le_div_iff₀ (mul_pos (sq_pos_of_pos Real.pi_pos) hd) h1).mpr
+    have hp := Real.pi_gt_three
+    have heq : 1-r^2 = (1-r)*(1+r) := by ring
+    rw [heq]
+    have haux : 4*r ≤ Real.pi^2*(1+r) := by
+      have hp4 : 0 ≤ Real.pi^2-4 := by nlinarith
+      nlinarith [mul_nonneg hp4 hr]
+    nlinarith [mul_nonneg (sub_nonneg.mpr haux) h1.le]
+  simpa only [one_div, div_eq_mul_inv, mul_comm, one_mul, mul_one] using mul_le_mul_of_nonneg_right hco (sq_nonneg s)
+
+theorem reciprocal_A_circleTop {r s : ℝ} (hr : 1/2 ≤ r) (hr1 : r < 1)
+    (hs : 0 < s) (hs1 : s ≤ Real.pi/2) :
+    1 / A (circleTop r s) ≤ Real.pi/2 + Real.pi^3*(1-r)/s := by
+  have hr0 : 0 < r := by linarith
+  have hd : 0 < 1-r^2 := by nlinarith
+  have hsin : 0 < Real.sin s := Real.sin_pos_of_pos_of_lt_pi hs (by linarith [Real.pi_pos])
+  have hnorm : ‖circleTop r s‖ < 1 := by simpa [norm_circleTop, abs_of_pos hr0] using hr1
+  let X := 2*(r*Real.sin s)/(1-r^2)
+  have hX : 0 < X := by dsimp [X]; positivity
+  have hA : 0 < Real.arctan X := Real.arctan_pos.mpr hX
+  have he : 1 / A (circleTop r s) = (Real.pi^2/4) * (1/Real.arctan X) := by
+    rw [A, re_mul_deriv_F hnorm, circleTop_re, norm_circleTop, sq_abs]
+    dsimp [X]
+    field_simp
+  rw [he]
+  have hh := mul_le_mul_of_nonneg_left (reciprocal_arctan hX)
+    (by positivity : 0 ≤ Real.pi^2/4)
+  apply hh.trans
+  have hratio : (1+r)/r ≤ 4 := by
+    apply (div_le_iff₀ hr0).mpr
+    linarith
+  have hsinbound : (1-r)/Real.sin s ≤ (Real.pi/2)*((1-r)/s) := by
+    apply (div_le_iff₀ hsin).mpr
+    have hx := Real.mul_le_sin hs.le hs1
+    have hy : s ≤ (Real.pi/2)*Real.sin s := by
+      have hp := mul_le_mul_of_nonneg_left hx (Real.pi_pos.le)
+      field_simp at hp
+      linarith
+    have hz := mul_le_mul_of_nonneg_left hy (show 0 ≤ (1-r)/s by positivity)
+    have heq : (1-r)/s*s = 1-r := by field_simp
+    rw [heq] at hz
+    nlinarith
+  calc
+    _ = Real.pi/2 + (Real.pi^2/4)*((1+r)/r)*((1-r)/Real.sin s) := by
+      dsimp [X]
+      field_simp
+      ring
+    _ ≤ Real.pi/2 + Real.pi^2*((1-r)/Real.sin s) := by
+      gcongr
+      nlinarith [mul_le_mul_of_nonneg_left hratio (show 0 ≤ Real.pi^2/4 by positivity)]
+    _ ≤ Real.pi/2 + Real.pi^2*((Real.pi/2)*((1-r)/s)) := by gcongr
+    _ ≤ Real.pi/2 + Real.pi^3*(1-r)/s := by
+      have hp : 0 ≤ Real.pi^3*(1-r)/s := by positivity
+      have heq : Real.pi^2*((Real.pi/2)*((1-r)/s)) =
+          (Real.pi^3*(1-r)/s)/2 := by ring
+      rw [heq]
+      linarith
+
+theorem circleTop_arg {w : ℂ} (hr : 0 < w.re) (hi : 0 ≤ w.im) :
+    ∃ s ∈ Ioc (0 : ℝ) (Real.pi/2), circleTop ‖w‖ s = w := by
+  let s := Real.pi/2-w.arg
+  have ha0 : 0 ≤ w.arg := Complex.arg_nonneg_iff.mpr hi
+  have ha1 : w.arg < Real.pi/2 := Complex.arg_lt_pi_div_two_iff.mpr (Or.inl hr)
+  refine ⟨s, ⟨by dsimp [s]; linarith, by dsimp [s]; linarith⟩, ?_⟩
+  apply Complex.ext
+  · rw [circleTop_re]
+    dsimp [s]
+    rw [Real.sin_pi_div_two_sub, Complex.norm_mul_cos_arg]
+  · rw [circleTop_im]
+    dsimp [s]
+    rw [Real.cos_pi_div_two_sub, Complex.norm_mul_sin_arg]
+
+lemma div_le_sqrt_of_gap {d δ s : ℝ} (hd : 0 < d) (hδ : 0 < δ) (hs : 0 < s)
+    (hg : δ ≤ s^2/d) : d/s ≤ Real.sqrt (d/δ) := by
+  have hg' : δ*d ≤ s^2 := (le_div_iff₀ hd).mp hg
+  have he : (d/s)^2 ≤ d/δ := by
+    rw [div_pow]
+    apply (div_le_div_iff₀ (sq_pos_of_pos hs) hδ).mpr
+    nlinarith [mul_le_mul_of_nonneg_left hg' hd.le]
+  exact (Real.le_sqrt (by positivity) (by positivity)).mpr he
+
+theorem reciprocal_A_slice {w : ℂ} (hw : ‖w‖ < 1) (hr : 0 < w.re)
+    (hradius : 1/2 ≤ ‖w‖) :
+    1 / A w ≤ Real.pi/2 + Real.pi^3 *
+      Real.sqrt ((1-‖w‖)/(T ‖w‖ - |(F w).im|)) := by
+  have hFw : F w ∈ D := ⟨w, by simpa using hw, rfl⟩
+  have hdelta : 0 < T ‖w‖ - |(F w).im| := by
+    have hh := vertical_gap_pos hFw (re_F_pos hw hr)
+    simpa only [g_F hw] using hh
+  have upper (z : ℂ) (hz : ‖z‖ < 1) (hzr : 0 < z.re) (hzi : 0 ≤ z.im)
+      (hzlow : 1/2 ≤ ‖z‖) (hgap : 0 < T ‖z‖ - |(F z).im|) :
+      1 / A z ≤ Real.pi/2 + Real.pi^3 *
+        Real.sqrt ((1-‖z‖)/(T ‖z‖ - |(F z).im|)) := by
+    obtain ⟨s, hs, he⟩ := circleTop_arg hzr hzi
+    have hh := reciprocal_A_circleTop hzlow hz hs.1 hs.2
+    rw [he] at hh
+    apply hh.trans
+    have hgs := gap_le_square (norm_nonneg z) hz hs.1.le hs.2
+    rw [he] at hgs
+    have hg : T ‖z‖ - |(F z).im| ≤ s^2/(1-‖z‖) := by
+      linarith [le_abs_self (F z).im]
+    have hb := div_le_sqrt_of_gap (sub_pos.mpr hz) hgap hs.1 hg
+    have hm := mul_le_mul_of_nonneg_left hb (show 0 ≤ Real.pi^3 by positivity)
+    have heq : Real.pi^3*((1-‖z‖)/s) = Real.pi^3*(1-‖z‖)/s := by ring
+    rw [heq] at hm
+    linarith
+  rcases le_total 0 w.im with hi | hi
+  · exact upper w hw hr hi hradius hdelta
+  · have hn : ‖(starRingEnd ℂ) w‖ = ‖w‖ := by simp
+    have ha : A ((starRingEnd ℂ) w) = A w := by
+      rw [A, re_mul_deriv_F (by simpa using hw), A, re_mul_deriv_F hw]
+      simp
+    have hh := upper ((starRingEnd ℂ) w) (by simpa using hw) (by simpa using hr)
+      (by simpa using neg_nonneg.mpr hi) (by simpa using hradius) (by simpa using hdelta)
+    simpa [ha] using hh
+
+end
+end PlanarLens
+
+namespace PlanarLens
+open Set Filter Complex
+open scoped Topology ContDiff
+noncomputable section
+
+theorem deriv_T_eq {r : ℝ} (hr : 0 < r) (hr1 : r < 1) :
+    deriv T r = (8 / Real.pi^2) * (Real.artanh r / r) := by
+  have habs : |r| < 1 := by simpa only [abs_of_pos hr] using hr1
+  rw [(hasDerivAt_T habs).deriv]
+  have hh := congr_arg Complex.im (mul_deriv_F (w := (r : ℂ)*I) (by simpa [norm_mul] using habs))
+  have hc : cayley ((r : ℂ)*I) = (((1-r)/(1+r) : ℝ) : ℂ) := by
+    unfold cayley
+    push_cast
+    rw [mul_assoc, I_mul_I]
+    ring
+  have hai : (Complex.arctan ((r : ℂ)*I)).im = Real.artanh r := by
+    rw [Complex.arctan]
+    change ((-I/2) * Complex.log (cayley ((r : ℂ)*I))).im = _
+    rw [hc, mul_im]
+    have hp : 0 < (1-r)/(1+r) := by positivity
+    have hre : (-I/2 : ℂ).re = 0 := by norm_num [div_re]
+    have him : (-I/2 : ℂ).im = -1/2 := by norm_num [div_im]
+    rw [hre, him, zero_mul, zero_add]
+    rw [Complex.log_re, norm_real, Real.norm_eq_abs, abs_of_pos hp,
+      Real.artanh_eq_half_log ⟨by linarith, hr1.le⟩]
+    have hlog : Real.log ((1-r)/(1+r)) = -Real.log ((1+r)/(1-r)) := by
+      rw [show (1-r)/(1+r) = ((1+r)/(1-r))⁻¹ by rw [inv_div], Real.log_inv]
+    rw [hlog]
+    norm_num
+  simp only [mul_im, mul_re, ofReal_re, ofReal_im, I_re, I_im, mul_zero, mul_one,
+    zero_mul, zero_sub, add_zero, neg_zero, zero_add] at hh
+  rw [hai] at hh
+  rw [mul_comm] at hh
+  have hh := (eq_div_iff hr.ne').mpr hh
+  calc
+    _ = ((8 / Real.pi^2) * Real.artanh r)/r := hh
+    _ = _ := by ring
+
+theorem deriv_T_lower {a r : ℝ} (ha : 0 < a) (har : a ≤ r) (hr : r < 1) :
+    (8/Real.pi^2) * Real.artanh a ≤ deriv T r := by
+  have hr0 : 0 < r := ha.trans_le har
+  rw [deriv_T_eq hr0 hr]
+  apply mul_le_mul_of_nonneg_left _ (by positivity)
+  apply (Real.artanh_le_artanh (by linarith) hr har).trans
+  exact le_div_self (Real.artanh_nonneg hr0.le) hr0 hr.le
+
+theorem T_tail_gap {a r s : ℝ} (ha : 0 < a) (har : a ≤ r) (hrs : r ≤ s)
+    (hs : s < 1) :
+    (8/Real.pi^2)*Real.artanh a * (s-r) ≤ T s-T r := by
+  apply Convex.mul_sub_le_image_sub_of_le_deriv (convex_Icc r s)
+    (continuousOn_T.mono (by intro x hx; exact ⟨by linarith [hx.1], hx.2.trans hs.le⟩))
+    (by intro x hx
+        have hxabs : |x| < 1 := by
+          simp only [interior_Icc, mem_Ioo] at hx
+          exact abs_lt.mpr ⟨by linarith, by linarith⟩
+        exact (hasDerivAt_T hxabs).differentiableAt.differentiableWithinAt)
+    (by intro x hx
+        simp only [interior_Icc, mem_Ioo] at hx
+        exact deriv_T_lower ha (har.trans hx.1.le) (hx.2.trans hs))
+    r ⟨le_rfl, hrs⟩ s ⟨hrs, le_rfl⟩ hrs
+
+theorem exists_tail_cutoff (M : ℝ) : ∃ a ∈ Ioo (1/2 : ℝ) 1,
+    M < (8/Real.pi^2)*Real.artanh a := by
+  let y := max (Real.artanh (1/2)) ((Real.pi^2/8)*M) + 1
+  let a := Real.tanh y
+  have ha : a ∈ Ioo (-1 : ℝ) 1 := ⟨Real.neg_one_lt_tanh y, Real.tanh_lt_one y⟩
+  have hy : Real.artanh a = y := Real.artanh_tanh y
+  have hhalf : (1/2 : ℝ) < a := by
+    apply (Real.artanh_lt_artanh_iff (by norm_num) ha).mp
+    rw [hy]
+    dsimp [y]
+    have := le_max_left (Real.artanh (1/2)) ((Real.pi^2/8)*M)
+    linarith
+  refine ⟨a, ⟨hhalf, ha.2⟩, ?_⟩
+  rw [hy]
+  have hyy : (Real.pi^2/8)*M < y := by
+    dsimp [y]
+    have := le_max_right (Real.artanh (1/2)) ((Real.pi^2/8)*M)
+    linarith
+  have hm := mul_lt_mul_of_pos_left hyy (by positivity : 0 < 8/Real.pi^2)
+  have he : (8/Real.pi^2)*((Real.pi^2/8)*M) = M := by field_simp
+  rwa [he] at hm
+
+end
+end PlanarLens
+
+end LowerBoundInline
+
+end OAI
diff --git a/lean/OAI/Geometry/PolarProducts/LensMap.lean b/lean/OAI/Geometry/PolarProducts/LensMap.lean
new file mode 100644
index 0000000000000000000000000000000000000000..66770dae6622d7ec08c082464cf49ca57443ff36
--- /dev/null
+++ b/lean/OAI/Geometry/PolarProducts/LensMap.lean
@@ -0,0 +1,406 @@
+import OAI.Geometry.PolarProducts.LogProfile
+
+namespace OAI
+
+section LowerBoundInline
+open Set Filter Function
+open scoped Topology ContDiff NNReal
+open Set Filter Metric
+open scoped Topology ContDiff
+open Set Filter Function MeasureTheory Metric
+open scoped Topology ContDiff NNReal
+open Set Filter Function
+open scoped Topology ContDiff
+open Set Filter Function
+open scoped Topology ContDiff NNReal
+open Set Filter
+open scoped Topology ContDiff
+open Set Filter Function
+open scoped Topology ContDiff
+open Set Filter Function
+open scoped ContDiff Topology
+open Set MeasureTheory
+open scoped ContDiff Interval Topology
+open Set
+open scoped Topology ContDiff
+open Set
+open Set MeasureTheory
+open scoped ContDiff Interval Topology
+
+open Set Filter Complex
+open scoped Topology ContDiff
+
+namespace PlanarLens
+noncomputable section
+
+def term (n : ℕ) (w : ℂ) : ℂ := (-1)^n * w^(2*n+1) / (2*n+1)^2
+
+def termDeriv (n : ℕ) (w : ℂ) : ℂ := (-1)^n * w^(2*n) / (2*n+1)
+
+def F (w : ℂ) : ℂ := (8 / (Real.pi^2) : ℝ) * ∑' n : ℕ, term n w
+
+theorem hasDerivAt_term (n : ℕ) (w : ℂ) : HasDerivAt (term n) (termDeriv n w) w := by
+  have hn : (2 * (n : ℂ) + 1) ≠ 0 := by
+    exact_mod_cast (show 2 * n + 1 ≠ 0 by omega)
+  convert (((hasDerivAt_id w).pow (2*n+1)).const_mul ((-1 : ℂ)^n)).div_const
+    ((2*(n : ℂ)+1)^2) using 1 <;> try rfl
+  simp only [termDeriv, Nat.cast_add, Nat.cast_mul, Nat.cast_ofNat, Nat.cast_one, Nat.add_sub_cancel]
+  field_simp [hn]
+  rfl
+
+@[simp] theorem term_zero (n : ℕ) : term n 0 = 0 := by simp [term]
+
+@[simp] theorem F_zero : F 0 = 0 := by simp [F]
+
+theorem norm_termDeriv_le {r : ℝ} (hr : 0 ≤ r) (hr1 : r ≤ 1)
+    {w : ℂ} (hw : ‖w‖ ≤ r) (n : ℕ) : ‖termDeriv n w‖ ≤ r^n := by
+  have hn : (1 : ℝ) ≤ 2 * n + 1 := by have := Nat.cast_nonneg (α := ℝ) n; linarith
+  calc
+    ‖termDeriv n w‖ = ‖w‖^(2*n) / (2*n+1) := by
+      simp only [termDeriv, norm_div, norm_mul, norm_pow, norm_neg, norm_one, one_pow, one_mul]
+      congr 1
+      norm_cast
+    _ ≤ r^(2*n) := (div_le_self (pow_nonneg (norm_nonneg _) _) hn).trans
+      (pow_le_pow_left₀ (norm_nonneg _) hw _)
+    _ ≤ r^n := pow_le_pow_of_le_one hr hr1 (by omega)
+
+theorem summable_termDeriv {w : ℂ} (hw : ‖w‖ < 1) : Summable (fun n => termDeriv n w) := by
+  exact Summable.of_norm_bounded (summable_geometric_of_lt_one (norm_nonneg w) hw)
+    (norm_termDeriv_le (norm_nonneg w) hw.le le_rfl)
+
+theorem hasDerivAt_F {w : ℂ} (hw : ‖w‖ < 1) :
+    HasDerivAt F ((8 / (Real.pi^2) : ℝ) * ∑' n : ℕ, termDeriv n w) w := by
+  obtain ⟨r, hwr, hr1⟩ := exists_between hw
+  have hr : 0 < r := (norm_nonneg w).trans_lt hwr
+  have hh := hasDerivAt_tsum_of_isPreconnected
+    (summable_geometric_of_lt_one hr.le hr1) Metric.isOpen_ball
+    (convex_ball (0 : ℂ) r).isPreconnected
+    (fun n z _ => hasDerivAt_term n z)
+    (fun n z hz => norm_termDeriv_le hr.le hr1.le ((by simpa [Metric.mem_ball] using hz : ‖z‖ < r).le) n)
+    (show (0 : ℂ) ∈ Metric.ball 0 r by simpa using hr)
+    (show Summable (fun n => term n (0 : ℂ)) by simp)
+    (show w ∈ Metric.ball 0 r by simpa using hwr)
+  exact hh.const_mul _
+
+theorem analyticOnNhd_F : AnalyticOnNhd ℂ F (Metric.ball (0 : ℂ) 1) := by
+  apply DifferentiableOn.analyticOnNhd _ Metric.isOpen_ball
+  intro w hw
+  exact (hasDerivAt_F (by simpa using hw)).differentiableAt.differentiableWithinAt
+
+theorem mul_deriv_F {w : ℂ} (hw : ‖w‖ < 1) :
+    w * deriv F w = (8 / (Real.pi^2) : ℝ) * Complex.arctan w := by
+  rw [(hasDerivAt_F hw).deriv, mul_left_comm, ← tsum_mul_left]
+  congr 1
+  have he : (fun n : ℕ => w * termDeriv n w) =
+      (fun n : ℕ => (-1)^n * w^(2*n+1) / (2*n+1)) := by
+    funext n
+    dsimp [termDeriv]
+    rw [pow_succ]
+    ring
+  rw [he]
+  simpa only [Nat.cast_add, Nat.cast_mul, Nat.cast_ofNat, Nat.cast_one] using (Complex.hasSum_arctan hw).tsum_eq
+
+theorem norm_term_le {w : ℂ} (hw : ‖w‖ ≤ 1) (n : ℕ) :
+    ‖term n w‖ ≤ 1 / ((n : ℝ)+1)^2 := by
+  have hn : (0 : ℝ) < n + 1 := by positivity
+  calc
+    ‖term n w‖ = ‖w‖^(2*n+1) / (2*(n : ℝ)+1)^2 := by
+      simp only [term, norm_div, norm_mul, norm_pow, norm_neg, norm_one, one_pow, one_mul]
+      congr 2
+      norm_cast
+    _ ≤ 1 / (2*(n : ℝ)+1)^2 := div_le_div_of_nonneg_right
+      (pow_le_one₀ (norm_nonneg _) hw) (sq_nonneg _)
+    _ ≤ 1 / ((n : ℝ)+1)^2 := one_div_le_one_div_of_le (sq_pos_of_pos hn)
+      (pow_le_pow_left₀ hn.le (by have := Nat.cast_nonneg (α := ℝ) n; linarith) 2)
+
+theorem summable_bound : Summable (fun n : ℕ => 1 / ((n : ℝ)+1)^2) := by
+  simpa only [Nat.cast_add, Nat.cast_one] using
+    (summable_nat_add_iff 1).2 (Real.summable_one_div_nat_pow.2 (by norm_num : 1 < 2))
+
+theorem summable_term {w : ℂ} (hw : ‖w‖ ≤ 1) : Summable (fun n => term n w) :=
+  Summable.of_norm_bounded summable_bound (norm_term_le hw)
+
+theorem continuousOn_F_closedBall : ContinuousOn F (Metric.closedBall 0 1) := by
+  apply ContinuousOn.const_mul _ _
+  apply continuousOn_tsum (fun n => (by unfold term; fun_prop : Continuous (term n)).continuousOn)
+    summable_bound
+  intro n w hw
+  exact norm_term_le (by simpa using hw) n
+
+theorem F_norm_le {w : ℂ} (hw : ‖w‖ ≤ 1) :
+    ‖F w‖ ≤ (8 / Real.pi^2) * ∑' n : ℕ, 1 / ((n : ℝ)+1)^2 := by
+  have hc : 0 ≤ 8 / Real.pi^2 := by positivity
+  rw [F, norm_mul, Complex.norm_real, Real.norm_eq_abs, abs_of_nonneg hc]
+  apply mul_le_mul_of_nonneg_left _ hc
+  exact (norm_tsum_le_tsum_norm (summable_term hw).norm).trans
+    (Summable.tsum_le_tsum (norm_term_le hw) (summable_term hw).norm summable_bound)
+
+@[simp] theorem F_neg (w : ℂ) : F (-w) = -F w := by
+  have he : (fun n : ℕ => term n (-w)) = fun n : ℕ => -term n w := by
+    funext n
+    simp only [term]
+    rw [(show Odd (2*n+1) by exact ⟨n, by omega⟩).neg_pow]
+    ring
+  simp only [F, he, tsum_neg, mul_neg]
+
+@[simp] theorem F_conj (w : ℂ) : F ((starRingEnd ℂ) w) = (starRingEnd ℂ) (F w) := by
+  have he : (fun n : ℕ => term n ((starRingEnd ℂ) w)) = fun n : ℕ => conjCLE (term n w) := by
+    funext n
+    simp [term, map_ofNat]
+  simp only [F, he, ← conjCLE.map_tsum]
+  simp [map_ofNat]
+
+def cayley (w : ℂ) : ℂ := (1 + w*I)/(1-w*I)
+
+theorem cayley_re (w : ℂ) : (cayley w).re = (1 - ‖w‖^2) / normSq (1-w*I) := by
+  rw [cayley, div_re, Complex.sq_norm]
+  simp only [add_re, one_re, mul_re, I_re, mul_zero, mul_im, I_im, mul_one,
+    zero_sub, sub_re, add_im, one_im, sub_im, add_zero, zero_add, normSq_apply]
+  ring
+
+theorem cayley_im (w : ℂ) : (cayley w).im = 2*w.re / normSq (1-w*I) := by
+  rw [cayley, div_im]
+  simp only [add_re, one_re, mul_re, I_re, mul_zero, mul_im, I_im, mul_one,
+    zero_sub, sub_re, add_im, one_im, sub_im, add_zero, zero_add]
+  ring
+
+theorem cayley_re_pos {w : ℂ} (hw : ‖w‖ < 1) : 0 < (cayley w).re := by
+  have hn : 1-w*I ≠ 0 := by
+    intro h
+    have he := congr_arg norm (sub_eq_zero.mp h)
+    simp only [norm_one, norm_mul, norm_I, mul_one] at he
+    linarith
+  rw [cayley_re]
+  exact div_pos (by nlinarith [norm_nonneg w]) (normSq_pos.mpr hn)
+
+theorem arctan_re_eq {w : ℂ} (hw : ‖w‖ < 1) :
+    (Complex.arctan w).re = (1/2 : ℝ) * Real.arctan (2*w.re/(1-‖w‖^2)) := by
+  have harg := abs_lt.mp (Complex.abs_arg_lt_pi_div_two_iff.mpr (Or.inl (cayley_re_pos hw)))
+  have ht := Real.arctan_tan harg.1 harg.2
+  rw [Complex.tan_arg, cayley_re, cayley_im] at ht
+  have hn : normSq (1-w*I) ≠ 0 := by
+    apply ne_of_gt
+    have h := cayley_re_pos hw
+    rw [cayley_re] at h
+    exact (div_pos_iff.mp h).resolve_right (by rintro ⟨_, hneg⟩; exact (normSq_nonneg _).not_gt hneg) |>.2
+  have he : (2*w.re/normSq (1-w*I)) / ((1-‖w‖^2)/normSq (1-w*I)) =
+      2*w.re/(1-‖w‖^2) := by field_simp
+  rw [he] at ht
+  rw [Complex.arctan, mul_re]
+  have hre : (-I / 2 : ℂ).re = 0 := by norm_num [div_re]
+  have him : (-I / 2 : ℂ).im = -1/2 := by norm_num [div_im]
+  rw [hre, him, zero_mul, log_im]
+  change 0 - (-1/2) * (cayley w).arg = _
+  rw [← ht]
+  ring
+
+theorem re_mul_deriv_F {w : ℂ} (hw : ‖w‖ < 1) :
+    (w * deriv F w).re = (4 / Real.pi^2) * Real.arctan (2*w.re/(1-‖w‖^2)) := by
+  rw [mul_deriv_F hw, re_ofReal_mul, arctan_re_eq hw]
+  ring
+
+end
+end PlanarLens
+
+namespace PlanarLens
+open Set Filter Complex
+open scoped Topology ContDiff
+noncomputable section
+
+theorem norm_one_add_mul_I_sq (w : ℂ) : ‖1+w*I‖^2 = 1-2*w.im+‖w‖^2 := by
+  simp only [Complex.sq_norm, normSq_apply, add_re, one_re, mul_re, I_re, I_im,
+    mul_zero, mul_one, zero_sub, add_im, one_im, mul_im, zero_add, add_zero]
+  ring
+
+theorem norm_one_sub_mul_I_sq (w : ℂ) : ‖1-w*I‖^2 = 1+2*w.im+‖w‖^2 := by
+  simp only [Complex.sq_norm, normSq_apply, sub_re, one_re, mul_re, I_re, I_im,
+    mul_zero, mul_one, zero_sub, sub_im, one_im, mul_im, add_zero]
+  ring
+
+theorem one_add_mul_I_ne_zero {w : ℂ} (hw : ‖w‖ < 1) : 1+w*I ≠ 0 := by
+  intro h
+  have he : w*I = -1 := eq_neg_of_add_eq_zero_right h
+  have hh := congr_arg norm he
+  simp only [norm_mul, norm_I, mul_one, norm_neg, norm_one] at hh
+  linarith
+
+theorem one_sub_mul_I_ne_zero {w : ℂ} (hw : ‖w‖ < 1) : 1-w*I ≠ 0 := by
+  have h := one_add_mul_I_ne_zero (w := -w) (by simpa using hw)
+  simpa only [neg_mul, sub_eq_add_neg] using h
+
+theorem arctan_im_eq (w : ℂ) :
+    (Complex.arctan w).im = -(1/2 : ℝ) * Real.log (‖1+w*I‖/‖1-w*I‖) := by
+  rw [Complex.arctan, mul_im]
+  have hre : (-I / 2 : ℂ).re = 0 := by norm_num [div_re]
+  have him : (-I / 2 : ℂ).im = -1/2 := by norm_num [div_im]
+  rw [hre, him, zero_mul, zero_add, log_re, norm_div]
+  ring
+
+theorem arctan_im_pos {w : ℂ} (hw : ‖w‖ < 1) (hi : 0 < w.im) :
+    0 < (Complex.arctan w).im := by
+  have ha : 0 < ‖1+w*I‖ := norm_pos_iff.mpr (one_add_mul_I_ne_zero hw)
+  have hb : 0 < ‖1-w*I‖ := norm_pos_iff.mpr (one_sub_mul_I_ne_zero hw)
+  have hab : ‖1+w*I‖ < ‖1-w*I‖ := by
+    have h1 := norm_one_add_mul_I_sq w
+    have h2 := norm_one_sub_mul_I_sq w
+    nlinarith [norm_nonneg (1+w*I), norm_nonneg (1-w*I)]
+  rw [arctan_im_eq]
+  have := Real.log_neg (div_pos ha hb) ((div_lt_one hb).mpr hab)
+  linarith
+
+theorem arctan_im_neg {w : ℂ} (hw : ‖w‖ < 1) (hi : w.im < 0) :
+    (Complex.arctan w).im < 0 := by
+  have hb : 0 < ‖1-w*I‖ := norm_pos_iff.mpr (one_sub_mul_I_ne_zero hw)
+  have hab : ‖1-w*I‖ < ‖1+w*I‖ := by
+    have h1 := norm_one_add_mul_I_sq w
+    have h2 := norm_one_sub_mul_I_sq w
+    nlinarith [norm_nonneg (1+w*I), norm_nonneg (1-w*I)]
+  rw [arctan_im_eq]
+  have := Real.log_pos ((one_lt_div hb).mpr hab)
+  linarith
+
+theorem arctan_im_mul_nonneg {w : ℂ} (hw : ‖w‖ < 1) :
+    0 ≤ (Complex.arctan w).im * w.im := by
+  rcases lt_trichotomy w.im 0 with hi | hi | hi
+  · exact mul_nonneg_of_nonpos_of_nonpos (arctan_im_neg hw hi).le hi.le
+  · simp [hi]
+  · exact (mul_pos (arctan_im_pos hw hi) hi).le
+
+theorem arctan_re_mul_nonneg {w : ℂ} (hw : ‖w‖ < 1) :
+    0 ≤ (Complex.arctan w).re * w.re := by
+  rw [arctan_re_eq hw]
+  have hn : 0 < 1-‖w‖^2 := by nlinarith [norm_nonneg w]
+  rcases lt_trichotomy w.re 0 with hi | hi | hi
+  · have hr : Real.arctan (2*w.re/(1-‖w‖^2)) < 0 :=
+      Real.arctan_lt_zero.mpr (div_neg_of_neg_of_pos (by linarith) hn)
+    exact mul_nonneg_of_nonpos_of_nonpos (by linarith) hi.le
+  · simp [hi]
+  · have hr : 0 < Real.arctan (2*w.re/(1-‖w‖^2)) :=
+      Real.arctan_pos.mpr (div_pos (by linarith) hn)
+    positivity
+
+theorem arctan_div_re_pos {w : ℂ} (hw : ‖w‖ < 1) (hw0 : w ≠ 0) :
+    0 < (Complex.arctan w / w).re := by
+  rw [div_re, ← add_div]
+  apply div_pos _ (Complex.normSq_pos.mpr hw0)
+  have hr := arctan_re_mul_nonneg hw
+  have hi := arctan_im_mul_nonneg hw
+  by_cases hre : w.re = 0
+  · have him : w.im ≠ 0 := by
+      intro h; apply hw0; exact Complex.ext (by simpa using hre) (by simpa using h)
+    rcases lt_or_gt_of_ne him with him | him
+    · have := mul_pos_of_neg_of_neg (arctan_im_neg hw him) him
+      linarith
+    · have := mul_pos (arctan_im_pos hw him) him
+      linarith
+  · have hn : 0 < 1-‖w‖^2 := by nlinarith [norm_nonneg w]
+    have hp : 0 < (Complex.arctan w).re * w.re := by
+      rw [arctan_re_eq hw]
+      rcases lt_or_gt_of_ne hre with hre | hre
+      · have h := Real.arctan_lt_zero.mpr (div_neg_of_neg_of_pos
+          (by linarith : 2*w.re < 0) hn)
+        exact mul_pos_of_neg_of_neg (by linarith) hre
+      · have h := Real.arctan_pos.mpr (div_pos (by linarith : 0 < 2*w.re) hn)
+        positivity
+    linarith
+
+@[simp] theorem deriv_F_zero : deriv F 0 = (8 / Real.pi^2 : ℝ) := by
+  rw [(hasDerivAt_F (by simp : ‖(0 : ℂ)‖ < 1)).deriv]
+  have he : (∑' n : ℕ, termDeriv n 0) = 1 := by
+    rw [tsum_eq_single 0]
+    · simp [termDeriv]
+    · intro n hn
+      have h : 2*n ≠ 0 := by omega
+      simp [termDeriv, h]
+  rw [he, mul_one]
+
+theorem deriv_F_re_pos {w : ℂ} (hw : ‖w‖ < 1) : 0 < (deriv F w).re := by
+  by_cases hw0 : w = 0
+  · rw [hw0, deriv_F_zero, ofReal_re]
+    positivity
+  · have he : deriv F w = (8 / Real.pi^2 : ℝ) * (Complex.arctan w / w) := by
+      apply (mul_left_cancel₀ hw0)
+      rw [mul_deriv_F hw]
+      field_simp
+    rw [he, re_ofReal_mul]
+    exact mul_pos (by positivity) (arctan_div_re_pos hw hw0)
+
+theorem F_injOn : InjOn F (Metric.ball 0 1) := by
+  intro x hx y hy hxy
+  by_contra hn
+  have hd : y-x ≠ 0 := sub_ne_zero.mpr (Ne.symm hn)
+  have hseg : ∀ t ∈ Icc (0 : ℝ) 1, ‖x+(t : ℂ)*(y-x)‖ < 1 := by
+    intro t ht
+    have hh := (convex_ball (0 : ℂ) 1).add_smul_sub_mem hx hy ht
+    simpa only [Metric.mem_ball, dist_zero_right, Complex.real_smul] using hh
+  let H : ℝ → ℝ := fun t => (F (x+(t : ℂ)*(y-x))/(y-x)).re
+  have hh : ∀ t ∈ Icc (0 : ℝ) 1,
+      HasDerivAt H (deriv F (x+(t : ℂ)*(y-x))).re t := by
+    intro t ht
+    have hf := (hasDerivAt_F (hseg t ht)).differentiableAt.hasDerivAt
+    have hc := ((hasDerivAt_id (t : ℂ)).mul_const (y-x)).const_add x
+    have hm := (hf.comp (t : ℂ) hc).div_const (y-x)
+    have he : deriv F (x+(t : ℂ)*(y-x)) * (1*(y-x)) / (y-x) =
+        deriv F (x+(t : ℂ)*(y-x)) := by field_simp
+    rw [he] at hm
+    exact hm.real_of_complex
+  have hmono : StrictMonoOn H (Icc (0 : ℝ) 1) := by
+    apply strictMonoOn_of_hasDerivWithinAt_pos (convex_Icc 0 1)
+      (fun t ht => (hh t ht).continuousAt.continuousWithinAt)
+      (fun t ht => (hh t (interior_subset ht)).hasDerivWithinAt)
+    intro t ht
+    exact deriv_F_re_pos (hseg t (interior_subset ht))
+  have hlt := hmono (by norm_num : (0 : ℝ) ∈ Icc 0 1)
+    (by norm_num : (1 : ℝ) ∈ Icc 0 1) (by norm_num : (0 : ℝ) < 1)
+  simp only [H, ofReal_zero, zero_mul, add_zero, ofReal_one, one_mul, add_sub_cancel] at hlt
+  rw [hxy] at hlt
+  exact lt_irrefl _ hlt
+
+theorem re_F_imaginary (t : ℝ) : (F ((t : ℂ)*I)).re = 0 := by
+  have h : (starRingEnd ℂ) ((t : ℂ)*I) = -((t : ℂ)*I) := by simp
+  have he := F_conj ((t : ℂ)*I)
+  rw [h, F_neg] at he
+  have hre := congr_arg Complex.re he
+  simp only [neg_re, conj_re] at hre
+  linarith
+
+theorem re_F_pos {w : ℂ} (hw : ‖w‖ < 1) (hr : 0 < w.re) : 0 < (F w).re := by
+  let H : ℝ → ℝ := fun t => (F ((t : ℂ)+(w.im : ℂ)*I)).re
+  have hseg : ∀ t ∈ Icc 0 w.re, ‖(t : ℂ)+(w.im : ℂ)*I‖ < 1 := by
+    intro t ht
+    have he : ‖(t : ℂ)+(w.im : ℂ)*I‖^2 = t^2+w.im^2 := by
+      rw [Complex.sq_norm]
+      simp [normSq_apply, pow_two]
+    have hh : ‖w‖^2 = w.re^2+w.im^2 := by
+      rw [Complex.sq_norm]
+      simp [normSq_apply, pow_two]
+    have hn : ‖w‖^2 < 1 := by nlinarith [norm_nonneg w]
+    nlinarith [norm_nonneg ((t : ℂ)+(w.im : ℂ)*I), ht.1, ht.2]
+  have hh : ∀ t ∈ Icc 0 w.re, HasDerivAt H (deriv F ((t : ℂ)+(w.im : ℂ)*I)).re t := by
+    intro t ht
+    have hf := (hasDerivAt_F (hseg t ht)).differentiableAt.hasDerivAt
+    have hm := hf.comp (t : ℂ) ((hasDerivAt_id _).add_const ((w.im : ℂ)*I))
+    convert hm.real_of_complex using 1 <;> first | rfl | simp
+  have hmono : StrictMonoOn H (Icc 0 w.re) := by
+    apply strictMonoOn_of_hasDerivWithinAt_pos (convex_Icc 0 w.re)
+      (fun t ht => (hh t ht).continuousAt.continuousWithinAt)
+      (fun t ht => (hh t (interior_subset ht)).hasDerivWithinAt)
+    exact fun t ht => deriv_F_re_pos (hseg t (interior_subset ht))
+  have hlt := hmono ⟨le_rfl, hr.le⟩ ⟨hr.le, le_rfl⟩ hr
+  have he : (w.re : ℂ)+(w.im : ℂ)*I = w := by simp
+  simpa only [H, ofReal_zero, zero_add, re_F_imaginary, he] using hlt
+
+theorem re_F_neg {w : ℂ} (hw : ‖w‖ < 1) (hr : w.re < 0) : (F w).re < 0 := by
+  have h := re_F_pos (w := -w) (by simpa using hw) (by simpa using neg_pos.mpr hr)
+  simpa only [F_neg, neg_re, neg_pos] using h
+
+theorem re_F_eq_zero {w : ℂ} (hr : w.re = 0) : (F w).re = 0 := by
+  have he : w = (w.im : ℂ)*I := by apply Complex.ext <;> simp [hr]
+  rw [he, re_F_imaginary]
+
+end
+end PlanarLens
+
+end LowerBoundInline
+
+end OAI
diff --git a/lean/OAI/Geometry/PolarProducts/LocalODE.lean b/lean/OAI/Geometry/PolarProducts/LocalODE.lean
new file mode 100644
index 0000000000000000000000000000000000000000..f087035940b94c8c5c031f0b438c56b93f97bed8
--- /dev/null
+++ b/lean/OAI/Geometry/PolarProducts/LocalODE.lean
@@ -0,0 +1,320 @@
+import OAI.Geometry.PolarProducts.SmoothPaths
+
+namespace OAI
+
+universe u3 u4 u5 u6
+
+section LowerBoundInline
+open Set Filter Function
+open scoped Topology ContDiff NNReal
+open Set Filter Metric
+open scoped Topology ContDiff
+
+open Set Filter Function MeasureTheory Metric
+open scoped Topology ContDiff NNReal
+
+namespace SmoothODE
+
+abbrev Time := Icc (0 : ℝ) 1
+
+variable {E : Type u3} [NormedAddCommGroup E] [NormedSpace ℝ E]
+
+noncomputable section
+
+def extend (u : C(Time, E)) : ℝ → E := fun t => u (projIcc 0 1 zero_le_one t)
+
+omit [NormedSpace ℝ E] in
+theorem continuous_extend (u : C(Time, E)) : Continuous (extend u) :=
+  u.continuous.comp continuous_projIcc
+
+omit [NormedSpace ℝ E] in
+@[simp] theorem extend_coe (u : C(Time, E)) (t : Time) : extend u t = u t := by
+  simp [extend]
+
+def primitive (u : C(Time, E)) : C(Time, E) :=
+  ⟨fun t => ∫ s in 0..(t : ℝ), extend u s,
+    (intervalIntegral.continuous_primitive
+      (fun a b => (continuous_extend u).intervalIntegrable a b) 0).comp continuous_subtype_val⟩
+
+@[simp] theorem primitive_apply (u : C(Time, E)) (t : Time) :
+    primitive u t = ∫ s in 0..(t : ℝ), extend u s := rfl
+
+theorem norm_primitive_le (u : C(Time, E)) : ‖primitive u‖ ≤ ‖u‖ := by
+  apply (ContinuousMap.norm_le _ (norm_nonneg _)).mpr
+  intro t
+  have h := intervalIntegral.norm_integral_le_of_norm_le_const
+    (a := (0 : ℝ)) (b := (t : ℝ)) (C := ‖u‖)
+    (fun s _ => u.norm_coe_le_norm (projIcc 0 1 zero_le_one s))
+  change ‖∫ s in 0..(t : ℝ), extend u s‖ ≤ ‖u‖
+  refine h.trans ?_
+  rw [sub_zero, abs_of_nonneg t.2.1]
+  exact mul_le_of_le_one_right (norm_nonneg _) t.2.2
+
+def primitiveCLM : C(Time, E) →L[ℝ] C(Time, E) := by
+  let L : C(Time, E) →ₗ[ℝ] C(Time, E) :=
+    { toFun := primitive
+      map_add' := by
+        intro u v
+        ext t
+        change (∫ s in 0..(t : ℝ), extend u s + extend v s) = _
+        exact intervalIntegral.integral_add
+          ((continuous_extend u).intervalIntegrable _ _)
+          ((continuous_extend v).intervalIntegrable _ _)
+      map_smul' := by
+        intro c u
+        ext t
+        exact intervalIntegral.integral_smul c (extend u) }
+  exact L.mkContinuous 1 (by
+    intro u
+    change ‖primitive u‖ ≤ 1 * ‖u‖
+    simpa using norm_primitive_le u)
+
+@[simp] theorem primitiveCLM_apply (u : C(Time, E)) : primitiveCLM u = primitive u := rfl
+
+theorem primitiveCLM_norm_le : ‖primitiveCLM (E := E)‖ ≤ 1 :=
+  ContinuousLinearMap.opNorm_le_bound _ zero_le_one fun u => by
+    simpa using norm_primitive_le u
+
+end
+end SmoothODE
+
+namespace SmoothODE
+
+open Set Filter Function MeasureTheory Metric
+open scoped Topology ContDiff NNReal
+
+variable {E : Type u4} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  [CompleteSpace E]
+
+noncomputable section
+
+def picard {V : E → E} (hV : Continuous V) (h : ℝ) (x : E)
+    (u : C(Time, E)) : C(Time, E) :=
+  ContinuousMap.const Time x + h • primitive (SmoothPaths.superpose hV u)
+
+omit [NormedSpace ℝ E] [CompleteSpace E] in
+theorem lipschitz_superpose {V : E → E} {K : ℝ≥0}
+    (hV : LipschitzWith K V) :
+    LipschitzWith K (SmoothPaths.superpose (X := Time) hV.continuous) := by
+  apply LipschitzWith.of_dist_le_mul
+  intro u v
+  apply (ContinuousMap.dist_le (mul_nonneg K.coe_nonneg dist_nonneg)).mpr
+  intro t
+  exact (hV.dist_le_mul (u t) (v t)).trans
+    (mul_le_mul_of_nonneg_left (ContinuousMap.dist_apply_le_dist (f := u) (g := v) t) K.coe_nonneg)
+
+omit [CompleteSpace E] in
+theorem lipschitz_picard {V : E → E} {K : ℝ≥0} (hV : LipschitzWith K V)
+    (h : ℝ) (x : E) : LipschitzWith (‖h‖₊ * K) (picard hV.continuous h x) := by
+  have hp : LipschitzWith 1 (primitive (E := E)) := by
+    change LipschitzWith 1 (primitiveCLM (E := E))
+    exact ContinuousLinearMap.opNorm_le_iff_lipschitz.mp primitiveCLM_norm_le
+  have hi := (lipschitzWith_smul h).comp (hp.comp (lipschitz_superpose hV))
+  apply LipschitzWith.of_dist_le_mul
+  intro u v
+  simpa [picard] using hi.dist_le_mul u v
+
+omit [CompleteSpace E] in
+
+theorem contDiff_picard [ProperSpace E] {V : E → E} (hV : ContDiff ℝ ∞ V) (h : ℝ) :
+    ContDiff ℝ ∞ (fun p : E × C(Time, E) => picard hV.continuous h p.1 p.2) := by
+  have hc : ContDiff ℝ ∞ (fun p : E × C(Time, E) => ContinuousMap.const Time p.1) :=
+    (ContinuousLinearMap.const ℝ Time : E →L[ℝ] C(Time, E)).contDiff.comp contDiff_fst
+  have hp : ContDiff ℝ ∞ (fun p : E × C(Time, E) =>
+      primitive (SmoothPaths.superpose hV.continuous p.2)) :=
+    primitiveCLM.contDiff.comp ((SmoothPaths.contDiff_superpose hV).comp contDiff_snd)
+  exact hc.add (hp.const_smul h)
+
+def trajectory {V : E → E} {K : ℝ≥0} (hV : LipschitzWith K V)
+    (h : ℝ) (hh : ‖h‖₊ * K < 1) (x : E) : C(Time, E) :=
+  SmoothContraction.fixedPoint (f := fun p : E × C(Time, E) => picard hV.continuous h p.1 p.2) hh (lipschitz_picard hV h) x
+
+theorem trajectory_eq {V : E → E} {K : ℝ≥0} (hV : LipschitzWith K V)
+    (h : ℝ) (hh : ‖h‖₊ * K < 1) (x : E) :
+    picard hV.continuous h x (trajectory hV h hh x) = trajectory hV h hh x :=
+  SmoothContraction.fixedPoint_eq (f := fun p : E × C(Time, E) => picard hV.continuous h p.1 p.2) hh (lipschitz_picard hV h) x
+
+theorem contDiff_trajectory [ProperSpace E] {V : E → E} {K : ℝ≥0} (hV : LipschitzWith K V)
+    (hVs : ContDiff ℝ ∞ V) (h : ℝ) (hh : ‖h‖₊ * K < 1) :
+    ContDiff ℝ ∞ (trajectory hV h hh) :=
+  SmoothContraction.contDiff_fixedPoint (f := fun p : E × C(Time, E) => picard hV.continuous h p.1 p.2) (contDiff_picard hVs h) hh (lipschitz_picard hV h)
+
+def curve {V : E → E} {K : ℝ≥0} (hV : LipschitzWith K V)
+    (h : ℝ) (hh : ‖h‖₊ * K < 1) (x : E) (t : ℝ) : E :=
+  x + h • ∫ s in 0..t, V (extend (trajectory hV h hh x) s)
+
+@[simp] theorem curve_coe {V : E → E} {K : ℝ≥0} (hV : LipschitzWith K V)
+    (h : ℝ) (hh : ‖h‖₊ * K < 1) (x : E) (t : Time) :
+    curve hV h hh x t = trajectory hV h hh x t :=
+  congrArg (fun u : C(Time, E) => u t) (trajectory_eq hV h hh x)
+
+@[simp] theorem curve_zero {V : E → E} {K : ℝ≥0} (hV : LipschitzWith K V)
+    (h : ℝ) (hh : ‖h‖₊ * K < 1) (x : E) : curve hV h hh x 0 = x := by
+  simp [curve]
+
+theorem continuous_curve {V : E → E} {K : ℝ≥0} (hV : LipschitzWith K V)
+    (h : ℝ) (hh : ‖h‖₊ * K < 1) (x : E) : Continuous (curve hV h hh x) := by
+  have hc := hV.continuous.comp (continuous_extend (trajectory hV h hh x))
+  exact continuous_const.add
+    ((intervalIntegral.continuous_primitive (fun a b => hc.intervalIntegrable a b) 0).const_smul h)
+
+theorem hasDerivAt_curve {V : E → E} {K : ℝ≥0} (hV : LipschitzWith K V)
+    (h : ℝ) (hh : ‖h‖₊ * K < 1) (x : E) (t : Time) :
+    HasDerivAt (curve hV h hh x) (h • V (curve hV h hh x t)) (t : ℝ) := by
+  have hc := hV.continuous.comp (continuous_extend (trajectory hV h hh x))
+  have hd := intervalIntegral.integral_hasDerivAt_right
+    (hc.intervalIntegrable 0 t) (hc.stronglyMeasurableAtFilter volume (𝓝 (t : ℝ)))
+      hc.continuousAt
+  rw [curve_coe]
+  change HasDerivAt (fun s : ℝ =>
+    x + h • ∫ r in 0..s, V (extend (trajectory hV h hh x) r)) _ (t : ℝ)
+  simpa only [Pi.smul_apply, Function.comp_apply, extend_coe] using
+    (hd.const_smul h).const_add x
+
+def step {V : E → E} {K : ℝ≥0} (hV : LipschitzWith K V)
+    (h : ℝ) (hh : ‖h‖₊ * K < 1) (x : E) : E :=
+  trajectory hV h hh x ⟨1, by constructor <;> norm_num⟩
+
+theorem contDiff_step [ProperSpace E] {V : E → E} {K : ℝ≥0}
+    (hV : LipschitzWith K V) (hVs : ContDiff ℝ ∞ V)
+    (h : ℝ) (hh : ‖h‖₊ * K < 1) : ContDiff ℝ ∞ (step hV h hh) :=
+  (ContinuousMap.evalCLM ℝ (M := E) (⟨1, by constructor <;> norm_num⟩ : Time)).contDiff.comp
+    (contDiff_trajectory hV hVs h hh)
+
+@[simp] theorem curve_one {V : E → E} {K : ℝ≥0} (hV : LipschitzWith K V)
+    (h : ℝ) (hh : ‖h‖₊ * K < 1) (x : E) : curve hV h hh x 1 = step hV h hh x :=
+  curve_coe hV h hh x ⟨1, by constructor <;> norm_num⟩
+
+end
+end SmoothODE
+
+namespace SmoothODE
+
+open Set Filter Function MeasureTheory Metric
+open scoped Topology ContDiff NNReal
+
+variable {E : Type u5} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  [CompleteSpace E]
+
+noncomputable section
+
+theorem curve_unique {V : E → E} {K : ℝ≥0} (hV : LipschitzWith K V)
+    (h : ℝ) (hh : ‖h‖₊ * K < 1) (x : E) {f : ℝ → E}
+    (hf : ContinuousOn f (Icc 0 1))
+    (hf' : ∀ t ∈ Ico (0 : ℝ) 1, HasDerivWithinAt f (h • V (f t)) (Ici t) t)
+    (h0 : f 0 = x) : EqOn f (curve hV h hh x) (Icc 0 1) := by
+  apply ODE_solution_unique_of_mem_Icc_right
+    (v := fun _ y => h • V y) (s := fun _ => univ)
+    (K := ‖h‖₊ * K)
+  · intro _ _
+    exact ((lipschitzWith_smul h).comp hV).lipschitzOnWith
+  · exact hf
+  · exact hf'
+  · intro _ _; trivial
+  · exact (continuous_curve hV h hh x).continuousOn
+  · intro t ht
+    exact (hasDerivAt_curve hV h hh x ⟨t, ht.1, ht.2.le⟩).hasDerivWithinAt
+  · intro _ _; trivial
+  · simpa using h0
+
+theorem curve_reverse {V : E → E} {K : ℝ≥0} (hV : LipschitzWith K V)
+    (h : ℝ) (hh : ‖h‖₊ * K < 1) (x : E) :
+    EqOn (fun t => curve hV h hh x (1 - t))
+      (curve hV (-h) (by simpa using hh) (step hV h hh x)) (Icc 0 1) := by
+  apply curve_unique hV (-h) (by simpa using hh) (step hV h hh x)
+  · exact ((continuous_curve hV h hh x).comp (continuous_const.sub continuous_id)).continuousOn
+  · intro t ht
+    have hr : 1 - t ∈ Icc (0 : ℝ) 1 := ⟨by linarith [ht.2], by linarith [ht.1]⟩
+    have hd := (hasDerivAt_curve hV h hh x ⟨1 - t, hr⟩).scomp t
+      ((hasDerivAt_id t).const_sub 1)
+    have he : HasDerivAt (fun t => curve hV h hh x (1 - t))
+        ((-h) • V (curve hV h hh x (1 - t))) t := by
+      simpa only [Function.comp_def, id_eq, neg_smul, one_smul] using hd
+    exact he.hasDerivWithinAt
+  · simp
+
+theorem step_neg_step {V : E → E} {K : ℝ≥0} (hV : LipschitzWith K V)
+    (h : ℝ) (hh : ‖h‖₊ * K < 1) (x : E) :
+    step hV (-h) (by simpa using hh) (step hV h hh x) = x := by
+  have he := curve_reverse hV h hh x (show 1 ∈ Icc (0 : ℝ) 1 by constructor <;> norm_num)
+  simpa using he.symm
+
+def stepHomeomorph [ProperSpace E] {V : E → E} {K : ℝ≥0}
+    (hV : LipschitzWith K V) (hVs : ContDiff ℝ ∞ V)
+    (h : ℝ) (hh : ‖h‖₊ * K < 1) : E ≃ₜ E where
+  toFun := step hV h hh
+  invFun := step hV (-h) (by simpa using hh)
+  left_inv := step_neg_step hV h hh
+  right_inv x := by
+    simpa only [neg_neg] using step_neg_step hV (-h) (by simpa using hh) x
+  continuous_toFun := (contDiff_step hV hVs h hh).continuous
+  continuous_invFun := (contDiff_step hV hVs (-h) (by simpa using hh)).continuous
+
+@[simp] theorem stepHomeomorph_apply [ProperSpace E] {V : E → E} {K : ℝ≥0}
+    (hV : LipschitzWith K V) (hVs : ContDiff ℝ ∞ V)
+    (h : ℝ) (hh : ‖h‖₊ * K < 1) (x : E) :
+    stepHomeomorph hV hVs h hh x = step hV h hh x := rfl
+
+theorem curve_of_eq_zero {V : E → E} {K : ℝ≥0} (hV : LipschitzWith K V)
+    (h : ℝ) (hh : ‖h‖₊ * K < 1) (x : E) (hx : V x = 0) :
+    EqOn (curve hV h hh x) (fun _ => x) (Icc 0 1) := by
+  apply EqOn.symm
+  apply curve_unique hV h hh x continuousOn_const
+  · intro t _
+    simpa only [hx, smul_zero] using (hasDerivWithinAt_const t (Ici t) x)
+  · rfl
+
+@[simp] theorem step_of_eq_zero {V : E → E} {K : ℝ≥0} (hV : LipschitzWith K V)
+    (h : ℝ) (hh : ‖h‖₊ * K < 1) (x : E) (hx : V x = 0) :
+    step hV h hh x = x := by
+  have he := curve_of_eq_zero hV h hh x hx (show 1 ∈ Icc (0 : ℝ) 1 by constructor <;> norm_num)
+  simpa using he
+
+end
+end SmoothODE
+
+namespace SmoothODE
+
+open Set Filter Function MeasureTheory Metric
+open scoped Topology ContDiff NNReal
+
+variable {E : Type u6} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  [CompleteSpace E] [ProperSpace E]
+
+noncomputable section
+
+def tangentPath {V : E → E} {K : ℝ≥0} (hV : LipschitzWith K V)
+    (h : ℝ) (hh : ‖h‖₊ * K < 1) (x v : E) : C(Time, E) :=
+  fderiv ℝ (trajectory hV h hh) x v
+
+theorem tangentPath_eq {V : E → E} {K : ℝ≥0} (hV : LipschitzWith K V)
+    (hVs : ContDiff ℝ ∞ V) (h : ℝ) (hh : ‖h‖₊ * K < 1) (x v : E) :
+    tangentPath hV h hh x v =
+      ContinuousMap.const Time v + h • primitive
+        (SmoothPaths.applyField
+          (SmoothPaths.superpose (hVs.continuous_fderiv (by simp)) (trajectory hV h hh x))
+          (tangentPath hV h hh x v)) := by
+  let Q := trajectory hV h hh
+  let DQ := fderiv ℝ Q x
+  let A := SmoothPaths.applyField
+    (SmoothPaths.superpose (hVs.continuous_fderiv (by simp)) (Q x))
+  have hQ : HasFDerivAt Q DQ x :=
+    (contDiff_trajectory hV hVs h hh).differentiable (by simp) x |>.hasFDerivAt
+  have hS := (SmoothPaths.hasFDerivAt_superpose
+    (hVs.differentiable (by simp)) (hVs.continuous_fderiv (by simp)) (Q x)).comp x hQ
+  have hP := (primitiveCLM (E := E)).hasFDerivAt.comp x hS
+  have hC := (ContinuousLinearMap.const ℝ Time : E →L[ℝ] C(Time, E)).hasFDerivAt (x := x)
+  have hD := hC.add (hP.const_smul h)
+  have hD' : HasFDerivAt Q
+      ((ContinuousLinearMap.const ℝ Time : E →L[ℝ] C(Time, E)) +
+        h • primitiveCLM.comp (A.comp DQ)) x := by
+    apply hD.congr_of_eventuallyEq
+    exact Filter.Eventually.of_forall (fun y => (trajectory_eq hV h hh y).symm)
+  exact congrArg (fun L : E →L[ℝ] C(Time, E) => L v) (hQ.unique hD')
+
+end
+end SmoothODE
+
+end LowerBoundInline
+
+end OAI
diff --git a/lean/OAI/Geometry/PolarProducts/LogProfile.lean b/lean/OAI/Geometry/PolarProducts/LogProfile.lean
new file mode 100644
index 0000000000000000000000000000000000000000..e70eabbe8c8fe94c00a508b8828e79547555de22
--- /dev/null
+++ b/lean/OAI/Geometry/PolarProducts/LogProfile.lean
@@ -0,0 +1,282 @@
+import OAI.Geometry.PolarProducts.HolomorphicBalls
+
+namespace OAI
+
+universe u50 u51
+
+section LowerBoundInline
+open Set Filter Function
+open scoped Topology ContDiff NNReal
+open Set Filter Metric
+open scoped Topology ContDiff
+open Set Filter Function MeasureTheory Metric
+open scoped Topology ContDiff NNReal
+open Set Filter Function
+open scoped Topology ContDiff
+open Set Filter Function
+open scoped Topology ContDiff NNReal
+open Set Filter
+open scoped Topology ContDiff
+open Set Filter Function
+open scoped Topology ContDiff
+open Set Filter Function
+open scoped ContDiff Topology
+open Set MeasureTheory
+open scoped ContDiff Interval Topology
+open Set
+open scoped Topology ContDiff
+open Set
+
+open Set MeasureTheory
+open scoped ContDiff Interval Topology
+
+namespace SmoothLogProfile
+noncomputable section
+
+variable (b l u : ℝ)
+
+def cutoff (t : ℝ) : ℝ := Real.smoothTransition ((t - l) / (u - l))
+
+def slope (t : ℝ) : ℝ := b + cutoff l u t * (Real.exp t - b)
+
+def profile (t : ℝ) : ℝ := Real.exp u + ∫ s in u..t, slope b l u s
+
+theorem cutoff_smooth : ContDiff ℝ ∞ (cutoff l u) :=
+  Real.smoothTransition.contDiff.comp ((contDiff_id.sub contDiff_const).div_const _)
+
+theorem slope_smooth : ContDiff ℝ ∞ (slope b l u) :=
+  contDiff_const.add ((cutoff_smooth l u).mul (Real.contDiff_exp.sub contDiff_const))
+
+theorem cutoff_mem (t : ℝ) : cutoff l u t ∈ Icc (0 : ℝ) 1 :=
+  ⟨Real.smoothTransition.nonneg _, Real.smoothTransition.le_one _⟩
+
+theorem cutoff_zero (hlu : l < u) {t : ℝ} (ht : t ≤ l) : cutoff l u t = 0 := by
+  apply Real.smoothTransition.zero_of_nonpos
+  exact div_nonpos_of_nonpos_of_nonneg (sub_nonpos.mpr ht) (sub_nonneg.mpr hlu.le)
+
+theorem cutoff_one (hlu : l < u) {t : ℝ} (ht : u ≤ t) : cutoff l u t = 1 := by
+  apply Real.smoothTransition.one_of_one_le
+  exact (le_div_iff₀ (sub_pos.mpr hlu)).mpr (by linarith)
+
+theorem cutoff_monotone (hlu : l < u) : Monotone (cutoff l u) := by
+  intro x y hxy
+  exact Real.smoothTransition.monotone
+    (div_le_div_of_nonneg_right (sub_le_sub_right hxy l) (sub_nonneg.mpr hlu.le))
+
+theorem slope_left (hlu : l < u) {t : ℝ} (ht : t ≤ l) : slope b l u t = b := by
+  simp [slope, cutoff_zero l u hlu ht]
+
+theorem slope_right (hlu : l < u) {t : ℝ} (ht : u ≤ t) : slope b l u t = Real.exp t := by
+  simp [slope, cutoff_one l u hlu ht]
+
+theorem slope_pos (hb : 0 < b) (t : ℝ) : 0 < slope b l u t := by
+  have hc := cutoff_mem l u t
+  have he := Real.exp_pos t
+  by_cases h : cutoff l u t = 1
+  · simpa [slope, h] using he
+  have hp : 0 < (1 - cutoff l u t) * b := mul_pos (sub_pos.mpr (lt_of_le_of_ne hc.2 h)) hb
+  have hq : 0 ≤ cutoff l u t * Real.exp t := mul_nonneg hc.1 he.le
+  unfold slope
+  nlinarith
+
+theorem slope_monotone (hlu : l < u) (hbl : b ≤ Real.exp l) : Monotone (slope b l u) := by
+  intro x y hxy
+  by_cases hyl : y ≤ l
+  · rw [slope_left b l u hlu (hxy.trans hyl), slope_left b l u hlu hyl]
+  have hly : l ≤ y := (not_le.mp hyl).le
+  have hey : b ≤ Real.exp y := hbl.trans (Real.exp_le_exp.mpr hly)
+  by_cases hxl : x ≤ l
+  · rw [slope_left b l u hlu hxl]
+    exact le_add_of_nonneg_right (mul_nonneg (cutoff_mem l u y).1 (sub_nonneg.mpr hey))
+  have hlx : l ≤ x := (not_le.mp hxl).le
+  have hex : b ≤ Real.exp x := hbl.trans (Real.exp_le_exp.mpr hlx)
+  unfold slope
+  apply add_le_add le_rfl
+  exact mul_le_mul (cutoff_monotone l u hlu hxy)
+    (sub_le_sub_right (Real.exp_le_exp.mpr hxy) b) (sub_nonneg.mpr hex) (cutoff_mem l u y).1
+
+theorem hasDerivAt_profile (t : ℝ) : HasDerivAt (profile b l u) (slope b l u t) t := by
+  have hd : HasDerivAt (fun t => ∫ s in u..t, slope b l u s) (slope b l u t) t := by
+    apply intervalIntegral.integral_hasDerivAt_right
+    · exact (slope_smooth b l u).continuous.intervalIntegrable _ _
+    · exact (slope_smooth b l u).continuous.stronglyMeasurableAtFilter _ _
+    · exact (slope_smooth b l u).continuous.continuousAt
+  exact hd.const_add _
+
+@[simp] theorem deriv_profile : deriv (profile b l u) = slope b l u :=
+  funext (fun t => (hasDerivAt_profile b l u t).deriv)
+
+theorem profile_smooth : ContDiff ℝ ∞ (profile b l u) := by
+  rw [contDiff_infty_iff_deriv]
+  exact ⟨fun t => (hasDerivAt_profile b l u t).differentiableAt,
+    deriv_profile b l u ▸ slope_smooth b l u⟩
+
+theorem deriv_profile_pos (hb : 0 < b) (t : ℝ) : 0 < deriv (profile b l u) t := by
+  rw [deriv_profile]
+  exact slope_pos b l u hb t
+
+theorem second_deriv_profile_nonneg (hlu : l < u) (hbl : b ≤ Real.exp l) (t : ℝ) :
+    0 ≤ deriv (deriv (profile b l u)) t := by
+  rw [deriv_profile]
+  exact (slope_monotone b l u hlu hbl).deriv_nonneg
+
+theorem profile_sub (s t : ℝ) :
+    profile b l u t - profile b l u s = ∫ x in s..t, slope b l u x := by
+  have hadd := intervalIntegral.integral_add_adjacent_intervals
+    ((slope_smooth b l u).continuous.intervalIntegrable (μ := volume) u s)
+    ((slope_smooth b l u).continuous.intervalIntegrable s t)
+  unfold profile
+  linarith
+
+theorem profile_right (hlu : l < u) {t : ℝ} (ht : u ≤ t) :
+    profile b l u t = Real.exp t := by
+  have he : (∫ s in u..t, slope b l u s) = Real.exp t - Real.exp u := by
+    calc
+      _ = ∫ s in u..t, Real.exp s := by
+        apply intervalIntegral.integral_congr
+        intro s hs
+        exact slope_right b l u hlu (le_trans (by simp [ht]) hs.1)
+      _ = Real.exp t - Real.exp u := by simp
+  simp [profile, he]
+
+theorem profile_left (hlu : l < u) {t : ℝ} (ht : t ≤ l) :
+    profile b l u t = b * t + (profile b l u l - b * l) := by
+  have he : (∫ s in t..l, slope b l u s) = (l - t) * b := by
+    calc
+      _ = ∫ s in t..l, b := by
+        apply intervalIntegral.integral_congr
+        intro s hs
+        exact slope_left b l u hlu (le_trans hs.2 (by simp [ht]))
+      _ = (l - t) * b := by simp
+  have hdiff := profile_sub b l u t l
+  rw [he] at hdiff
+  nlinarith
+
+end
+end SmoothLogProfile
+
+namespace ComplexPotential
+
+open Set Filter Asymptotics
+open scoped Topology ContDiff
+noncomputable section
+
+variable {E : Type u50} {F : Type u51} [NormedAddCommGroup E] [InnerProductSpace ℂ E]
+  [NormedAddCommGroup F] [InnerProductSpace ℂ F]
+  [FiniteDimensional ℂ E] [FiniteDimensional ℂ F]
+
+theorem exists_ball_from_holomorphic_zero {Ω : Set E} (hΩ : IsOpen Ω) (hΩ0 : (0 : E) ∈ Ω)
+    {f : E → F} (hf : AnalyticOnNhd ℂ f Ω)
+    (hfzero : ∀ x ∈ Ω, f x = 0 ↔ x = 0)
+    {k : ℕ} (hk : 1 ≤ k) (ho : f =O[𝓝 (0 : E)] (fun x : E => ‖x‖^k))
+    (hcompact : ∀ s : ℝ, 0 < s → s < 1 → IsCompact {x | x ∈ Ω ∧ ‖f x‖^2 ≤ s})
+    {a : ℝ} (ha : 0 < a) (hak : a < (k : ℝ)) :
+    let : InnerProductSpace ℝ E := InnerProductSpace.complexToReal
+    ∃ Q : E ≃ₜ E, ContDiff ℝ ∞ (Q : E → E) ∧ ContDiff ℝ ∞ (Q.symm : E → E) ∧
+      (∀ x : E, ‖x‖^2 < a → Q x ∈ Ω ∧ ‖f (Q x)‖^2 < 1) ∧
+      ∀ x : E, ‖x‖^2 < a → ∀ v w : E,
+        ddc (complexStructure E) (fun y => ‖y‖^2 + ‖f y‖^2) (Q x)
+          (fderiv ℝ (Q : E → E) x v) (fderiv ℝ (Q : E → E) x w) =
+            standardTensor (complexStructure E) v w := by
+  let : InnerProductSpace ℝ E := InnerProductSpace.complexToReal
+  let : InnerProductSpace ℝ F := InnerProductSpace.complexToReal
+  let τ : E → ℝ := fun x => ‖f x‖^2
+  let U : Set E := {x | x ∈ Ω ∧ τ x < 1}
+  have hτsmooth (x : E) (hx : x ∈ Ω) : ContDiffAt ℝ ∞ τ x :=
+    (contDiff_norm_sq ℝ).contDiffAt.comp x ((hf x hx).contDiffAt.restrict_scalars ℝ)
+  have hτcont : ContinuousOn τ Ω := fun x hx => (hτsmooth x hx).continuousAt.continuousWithinAt
+  have hU : IsOpen U := by
+    rw [isOpen_iff_mem_nhds]
+    intro x hx
+    exact inter_mem (hΩ.mem_nhds hx.1)
+      ((hτsmooth x hx.1).continuousAt.eventually_lt continuousAt_const hx.2)
+  have hf0 : f 0 = 0 := (hfzero 0 hΩ0).mpr rfl
+  have hτ0 : τ 0 = 0 := by simp [τ, hf0]
+  have hU0 : (0 : E) ∈ U := ⟨hΩ0, by rw [hτ0]; norm_num⟩
+  have hτpsh (x : E) (hx : x ∈ U) : IsPSHAt (complexStructure E) τ x := by
+    have hn : IsPSHAt (complexStructure F) (fun z : F => ‖z‖^2) (f x) := by
+      refine ⟨(contDiff_norm_sq ℝ).contDiffAt, fun v => ?_⟩
+      rw [ddc_norm_sq _ (complexStructure_skew), standardTensor_complex_line _ complexStructure_norm]
+      exact sq_nonneg _
+    apply hn.holomorphic_comp ((hf x hx.1).contDiffAt.restrict_scalars ℝ)
+    filter_upwards [hΩ.mem_nhds hx.1] with y hy
+    exact (hf y hy).differentiableAt
+  have hpos (x : E) (hx : x ∈ U) (hx0 : x ≠ 0) : 0 < τ x := by
+    exact sq_pos_of_pos (norm_pos_iff.mpr (fun hh => hx0 ((hfzero x hx.1).mp hh)))
+  have hlogpsh (x : E) (hx : x ∈ U) (hx0 : x ≠ 0) :
+      IsPSHAt (complexStructure E) (Real.log ∘ τ) x := by
+    have hn := isPSHAt_logNormSq (complexStructure F) complexStructure_skew
+      complexStructure_norm complexStructure_sq (δ := 0) le_rfl
+        (show 0 < ‖f x‖^2 + 0 by simpa [τ] using hpos x hx hx0)
+    have hh := hn.holomorphic_comp ((hf x hx.1).contDiffAt.restrict_scalars ℝ) (by
+      filter_upwards [hΩ.mem_nhds hx.1] with y hy
+      exact (hf y hy).differentiableAt)
+    simpa only [logNormSq, add_zero, Function.comp_def, τ] using hh
+  have hkpos : (0 : ℝ) < k := by exact_mod_cast (Nat.zero_lt_of_lt hk)
+  obtain ⟨b, hab, hb1⟩ := exists_between ((div_lt_one hkpos).mpr hak)
+  have hb : 0 < b := (div_pos ha hkpos).trans hab
+  obtain ⟨l, hbl, hl0⟩ := exists_between (Real.log_neg hb hb1)
+  obtain ⟨u, hlu, hu0⟩ := exists_between hl0
+  have hble : b ≤ Real.exp l := by
+    exact ((Real.log_le_iff_le_exp hb).mp hbl.le)
+  let H : E → ℝ := SmoothLogProfile.profile b l u ∘ Real.log ∘ τ
+  have hH (x : E) (hx : x ∈ U) (hx0 : x ≠ 0) : IsPSHAt (complexStructure E) H x := by
+    exact (hlogpsh x hx hx0).scalar_comp _ complexStructure_sq
+      (SmoothLogProfile.profile_smooth b l u).contDiffAt
+      (SmoothLogProfile.deriv_profile_pos b l u hb _).le
+      (SmoothLogProfile.second_deriv_profile_nonneg b l u hlu hble _)
+  have htail (x : E) (hx : x ∈ U) (ht : Real.exp u ≤ τ x) : H x = τ x := by
+    have htpos : 0 < τ x := (Real.exp_pos u).trans_le ht
+    have hlog : u ≤ Real.log (τ x) := by
+      simpa using Real.log_le_log (Real.exp_pos u) ht
+    change SmoothLogProfile.profile b l u (Real.log (τ x)) = τ x
+    rw [SmoothLogProfile.profile_right b l u hlu hlog, Real.exp_log htpos]
+  obtain ⟨C, hC, hCb⟩ := ho.exists_pos
+  have hev : ∀ᶠ x in 𝓝 (0 : E), ‖f x‖ ≤ C * ‖x‖^k ∧ τ x < Real.exp l := by
+    filter_upwards [hCb.bound,
+      (hτsmooth 0 hΩ0).continuousAt.eventually_lt continuousAt_const
+        (show τ 0 < Real.exp l by rw [hτ0]; exact Real.exp_pos l)] with x hx ht
+    exact ⟨by simpa only [Real.norm_eq_abs, abs_of_nonneg (pow_nonneg (norm_nonneg _) _)] using hx, ht⟩
+  obtain ⟨ε, hε, hεprop⟩ := Metric.eventually_nhds_iff.mp hev
+  let ρ := ε / 2
+  have hρ : 0 < ρ := half_pos hε
+  let C₁ := b * (2 * Real.log C) + (SmoothLogProfile.profile b l u l - b * l)
+  have hpole (x : E) (hx : x ∈ U) (hx0 : x ≠ 0) (hnorm : ‖x‖ ≤ ρ) :
+      H x ≤ (b * k) * Real.log (‖x‖^2) + C₁ := by
+    have hdist : dist x 0 < ε := by rw [dist_zero_right]; exact hnorm.trans_lt (half_lt_self hε)
+    obtain ⟨hbound, hsmall⟩ := hεprop hdist
+    have hxp : 0 < ‖x‖ := norm_pos_iff.mpr hx0
+    have hfxp : 0 < ‖f x‖ := norm_pos_iff.mpr (fun hh => hx0 ((hfzero x hx.1).mp hh))
+    have hlogl : Real.log (τ x) ≤ l := (Real.log_le_iff_le_exp (hpos x hx hx0)).mpr hsmall.le
+    change SmoothLogProfile.profile b l u (Real.log (τ x)) ≤ _
+    rw [SmoothLogProfile.profile_left b l u hlu hlogl]
+    have hlg := Real.log_le_log hfxp hbound
+    rw [Real.log_mul hC.ne' (pow_pos hxp k).ne', Real.log_pow] at hlg
+    have hmul := mul_le_mul_of_nonneg_left hlg hb.le
+    dsimp [τ, C₁]
+    rw [Real.log_pow, Real.log_pow]
+    norm_num only [Nat.cast_ofNat]
+    nlinarith
+  apply exists_ball_from_logarithmic_pole (complexStructure E) complexStructure_skew
+    complexStructure_norm complexStructure_sq hU hU0 hτpsh hτ0 hH
+  · intro s hs hs1
+    have he : {x | x ∈ U ∧ τ x ≤ s} = {x | x ∈ Ω ∧ ‖f x‖^2 ≤ s} := by
+      ext x
+      constructor
+      · exact fun hx => ⟨hx.1.1, hx.2⟩
+      · exact fun hx => ⟨⟨hx.1, hx.2.trans_lt hs1⟩, hx.2⟩
+    rw [he]
+    exact hcompact s hs hs1
+  · exact Real.exp_lt_one_iff.mpr hu0
+  · exact hρ
+  · exact htail
+  · exact hpole
+  · exact ha
+  · exact (div_lt_iff₀ hkpos).mp hab
+
+end
+end ComplexPotential
+
+end LowerBoundInline
+
+end OAI
diff --git a/lean/OAI/Geometry/PolarProducts/LowerBound.lean b/lean/OAI/Geometry/PolarProducts/LowerBound.lean
new file mode 100644
index 0000000000000000000000000000000000000000..824ecfb7fc08500ebbb5886d8cbab506051a3f28
--- /dev/null
+++ b/lean/OAI/Geometry/PolarProducts/LowerBound.lean
@@ -0,0 +1,64 @@
+import OAI.Geometry.PolarProducts.PolarApproximation
+
+namespace OAI
+
+section LowerBoundInline
+open Set Filter Function
+open scoped Topology ContDiff NNReal
+open Set Filter Metric
+open scoped Topology ContDiff
+open Set Filter Function MeasureTheory Metric
+open scoped Topology ContDiff NNReal
+open Set Filter Function
+open scoped Topology ContDiff
+open Set Filter Function
+open scoped Topology ContDiff NNReal
+open Set Filter
+open scoped Topology ContDiff
+open Set Filter Function
+open scoped Topology ContDiff
+open Set Filter Function
+open scoped ContDiff Topology
+open Set MeasureTheory
+open scoped ContDiff Interval Topology
+open Set
+open scoped Topology ContDiff
+open Set
+open Set MeasureTheory
+open scoped ContDiff Interval Topology
+open Set Filter Complex
+open scoped Topology ContDiff
+open MeasureTheory intervalIntegral Set
+open scoped Real
+open Set
+open scoped ContDiff ENNReal
+
+namespace SymmetricPolar
+open Set
+open scoped ENNReal
+noncomputable section
+variable {n : ℕ}
+
+theorem gromovWidth_mono {U V : Set (Phase n)} (h : U ⊆ V) :
+    gromovWidth U ≤ gromovWidth V := by
+  apply sSup_le_sSup
+  rintro _ ⟨c, ⟨hc, he⟩, rfl⟩
+  exact ⟨c, ⟨hc, he.mono_target h⟩, rfl⟩
+
+theorem four_le_gromovWidth_polarProduct {K : Set (Position n)} (hK : IsSymmetricConvexBody K) :
+    4 ≤ gromovWidth (polarProduct K) := by
+  apply ENNReal.le_of_forall_pos_nnreal_lt
+  intro r hr hr4
+  have hrreal : (0 : ℝ) < r := by exact_mod_cast hr
+  have hr4real : (r : ℝ) < 4 := by
+    apply (ENNReal.ofReal_lt_ofReal_iff (by norm_num : (0 : ℝ) < 4)).mp
+    simpa using hr4
+  apply le_sSup
+  refine ⟨(r : ℝ), ⟨hrreal, symmetric_polar_embedding hK hrreal hr4real⟩, ?_⟩
+  simp
+
+end
+end SymmetricPolar
+end LowerBoundInline
+
+end OAI
diff --git a/lean/OAI/Geometry/PolarProducts/Main.lean b/lean/OAI/Geometry/PolarProducts/Main.lean
new file mode 100644
index 0000000000000000000000000000000000000000..1b51f466a7102e0851387484f9ed44a0c8e8ff4f
--- /dev/null
+++ b/lean/OAI/Geometry/PolarProducts/Main.lean
@@ -0,0 +1,19 @@
+import OAI.Geometry.PolarProducts.UpperBound
+
+namespace OAI
+
+namespace SymmetricPolar
+
+theorem symmetric_polar_main {n : ℕ} (hn : 2 ≤ n)
+    (K : Set (Position n)) (hK : IsSymmetricConvexBody K) :
+    gromovWidth (polarProduct K) = 4 ∧
+      ∀ c : ℝ, 0 < c → c < 4 →
+        HasSymplecticEmbedding (capacityBall n c) (polarProduct K) := by
+  refine ⟨le_antisymm (gromovWidth_polarProduct_le_four hK (by omega))
+    (four_le_gromovWidth_polarProduct hK), ?_⟩
+  intro c hc hc4
+  exact symmetric_polar_embedding hK hc hc4
+
+end SymmetricPolar
+
+end OAI
diff --git a/lean/OAI/Geometry/PolarProducts/Model.lean b/lean/OAI/Geometry/PolarProducts/Model.lean
new file mode 100644
index 0000000000000000000000000000000000000000..9cce72ffd69d1fc749b295f211f3298ba40a8230
--- /dev/null
+++ b/lean/OAI/Geometry/PolarProducts/Model.lean
@@ -0,0 +1,44 @@
+import Mathlib
+
+namespace OAI
+
+section
+open Set
+open scoped ContDiff ENNReal
+
+namespace SymmetricPolar
+
+abbrev Position (n : ℕ) := EuclideanSpace ℝ (Fin n)
+abbrev Phase (n : ℕ) := Position n × Position n
+
+def IsSymmetricConvexBody {n : ℕ} (K : Set (Position n)) : Prop :=
+  IsCompact K ∧ Convex ℝ K ∧ (interior K).Nonempty ∧
+    ∀ x, x ∈ K ↔ -x ∈ K
+
+def polar {n : ℕ} (K : Set (Position n)) : Set (Position n) :=
+  {p | ∀ q ∈ K, inner (𝕜 := ℝ) q p ≤ 1}
+
+def polarProduct {n : ℕ} (K : Set (Position n)) : Set (Phase n) :=
+  interior K ×ˢ interior (polar K)
+
+def capacityBall (n : ℕ) (c : ℝ) : Set (Phase n) :=
+  {z | Real.pi * (‖z.1‖ ^ 2 + ‖z.2‖ ^ 2) < c}
+
+noncomputable def omega0 {n : ℕ} (v w : Phase n) : ℝ :=
+  inner (𝕜 := ℝ) v.1 w.2 - inner (𝕜 := ℝ) w.1 v.2
+
+def HasSymplecticEmbedding {n : ℕ} (U V : Set (Phase n)) : Prop :=
+  ∃ e : Phase n → Phase n,
+    ContDiffOn ℝ ∞ e U ∧
+    Topology.IsEmbedding (fun z : U => e z) ∧
+    MapsTo e U V ∧
+    ∀ z ∈ U, ∀ v w : Phase n,
+      omega0 (fderiv ℝ e z v) (fderiv ℝ e z w) = omega0 v w
+
+noncomputable def gromovWidth {n : ℕ} (U : Set (Phase n)) : ℝ≥0∞ :=
+  sSup (ENNReal.ofReal '' {c : ℝ | 0 < c ∧ HasSymplecticEmbedding (capacityBall n c) U})
+
+end SymmetricPolar
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PolarProducts/MomentumBounds.lean b/lean/OAI/Geometry/PolarProducts/MomentumBounds.lean
new file mode 100644
index 0000000000000000000000000000000000000000..2646cee0ed84add3af7935c92e1ad62ddac9f6ea
--- /dev/null
+++ b/lean/OAI/Geometry/PolarProducts/MomentumBounds.lean
@@ -0,0 +1,304 @@
+import OAI.Geometry.PolarProducts.StripEmbedding
+
+namespace OAI
+
+universe u70 u71 u72 u73
+
+section LowerBoundInline
+open Set Filter Function
+open scoped Topology ContDiff NNReal
+open Set Filter Metric
+open scoped Topology ContDiff
+open Set Filter Function MeasureTheory Metric
+open scoped Topology ContDiff NNReal
+open Set Filter Function
+open scoped Topology ContDiff
+open Set Filter Function
+open scoped Topology ContDiff NNReal
+open Set Filter
+open scoped Topology ContDiff
+open Set Filter Function
+open scoped Topology ContDiff
+open Set Filter Function
+open scoped ContDiff Topology
+open Set MeasureTheory
+open scoped ContDiff Interval Topology
+open Set
+open scoped Topology ContDiff
+open Set
+open Set MeasureTheory
+open scoped ContDiff Interval Topology
+open Set Filter Complex
+open scoped Topology ContDiff
+open MeasureTheory intervalIntegral Set
+open scoped Real
+
+namespace PolarStrips
+open Set Filter ComplexCoordinates
+open scoped Topology ContDiff
+noncomputable section
+variable {ι : Type u70} {κ : Type u71} [Fintype ι] [Fintype κ]
+
+theorem position_mem_interior (b : κ → R ι) (k : ℕ) {z : C ι}
+    (hz : z ∈ domain (fun j => functional (b j))) :
+    (phase b k z).1 ∈ interior (ConvexPolar.stripBody b) := by
+  apply ConvexPolar.mem_interior_stripBody
+  intro j
+  change |inner (𝕜 := ℝ) (b j) (im z)| < 1
+  rw [← im_functional]
+  exact abs_lt.mpr (PlanarLens.im_mem_Ioo (hz j))
+
+theorem norm_re_le (z : C ι) : ‖re z‖ ≤ ‖z‖ := by
+  have hh := norm_sq_parts z
+  nlinarith [sq_nonneg ‖im z‖, norm_nonneg (re z), norm_nonneg z]
+
+theorem momentum_bound (b : κ → R ι) {k : ℕ} (hk : 1 ≤ k) {z : C ι}
+    (hz : z ∈ domain (fun j => functional (b j))) {y : R ι}
+    (hy : y ∈ ConvexPolar.stripBody b) {A B : ℝ}
+    (hzA : ‖z‖ ≤ A) (hyB : ‖y‖ ≤ B) :
+    inner (𝕜 := ℝ) y (phase b k z).2 / (k : ℝ) ≤
+      A*B/(k : ℝ) + Real.pi/4*‖tuple (fun j => functional (b j)) k z‖^2 +
+        (Fintype.card κ : ℝ)*PlanarLens.epsilon k := by
+  have hkpos : (0 : ℝ) < k := by exact_mod_cast (Nat.zero_lt_of_lt hk)
+  have hu : inner (𝕜 := ℝ) y (-re z) ≤ A*B := by
+    have hh := real_inner_le_norm y (-re z)
+    rw [norm_neg] at hh
+    have hm := mul_le_mul hyB ((norm_re_le z).trans hzA) (norm_nonneg _) (le_trans (norm_nonneg _) hyB)
+    nlinarith
+  have hj (j : κ) : -(PlanarLens.primitive k (functional (b j) z)*inner (𝕜 := ℝ) y (b j)) ≤
+      |PlanarLens.primitive k (functional (b j) z)| := by
+    have hh : |inner (𝕜 := ℝ) y (b j)| ≤ 1 := by rw [real_inner_comm]; exact hy j
+    calc
+      _ ≤ |PlanarLens.primitive k (functional (b j) z)*inner (𝕜 := ℝ) y (b j)| := neg_le_abs _
+      _ = |PlanarLens.primitive k (functional (b j) z)| * |inner (𝕜 := ℝ) y (b j)| := abs_mul _ _
+      _ ≤ _ := by simpa using mul_le_mul_of_nonneg_left hh (abs_nonneg (PlanarLens.primitive k (functional (b j) z)))
+  have hsum : inner (𝕜 := ℝ) y (phase b k z).2 ≤ A*B+∑ j, |PlanarLens.primitive k (functional (b j) z)| := by
+    have hh := Finset.sum_le_sum (fun j (_ : j ∈ Finset.univ) => hj j)
+    simp only [phase, inner_sub_right, inner_sum, real_inner_smul_right, Finset.sum_neg_distrib] at ⊢ hh
+    linarith
+  have hslice (j : κ) : |PlanarLens.primitive k (functional (b j) z)|/(k : ℝ) ≤
+      Real.pi/4*‖PlanarLens.g (functional (b j) z)‖^(2*k)+PlanarLens.epsilon k := by
+    simpa only [PlanarLens.primitive, Complex.re_add_im] using
+      PlanarLens.uniform_slice hk (v := (functional (b j) z).re) (t := (functional (b j) z).im)
+        (by simpa only [Complex.re_add_im] using hz j)
+  calc
+    _ ≤ (A*B+∑ j, |PlanarLens.primitive k (functional (b j) z)|)/(k : ℝ) :=
+      div_le_div_of_nonneg_right hsum hkpos.le
+    _ = A*B/(k : ℝ) + ∑ j, |PlanarLens.primitive k (functional (b j) z)|/(k : ℝ) := by
+      rw [add_div, Finset.sum_div]
+    _ ≤ A*B/(k : ℝ) + ∑ j, (Real.pi/4*‖PlanarLens.g (functional (b j) z)‖^(2*k)+PlanarLens.epsilon k) :=
+      add_le_add le_rfl (Finset.sum_le_sum (fun j _ => hslice j))
+    _ = _ := by rw [Finset.sum_add_distrib, ← Finset.mul_sum, ← norm_tuple_sq]; simp [add_assoc]
+
+theorem exists_uniform_momentum_bound (b : κ → R ι)
+    (hb : Function.Injective (fun x : R ι => fun j => inner (𝕜 := ℝ) (b j) x)) :
+    ∃ C B : ℝ, 0 ≤ C ∧ 0 ≤ B ∧ (∀ y ∈ ConvexPolar.stripBody b, ‖y‖ ≤ B) ∧
+      ∀ k : ℕ, 1 ≤ k → ∀ z ∈ domain (fun j => functional (b j)),
+        ∀ y ∈ ConvexPolar.stripBody b,
+          inner (𝕜 := ℝ) y (phase b k z).2 / (k : ℝ) ≤
+            C/(k : ℝ)+Real.pi/4*‖tuple (fun j => functional (b j)) k z‖^2 +
+              (Fintype.card κ : ℝ)*PlanarLens.epsilon k := by
+  obtain ⟨A, hA⟩ := (isBounded_domain _ (functional_injective b hb)).exists_norm_le
+  obtain ⟨B, hB⟩ := (ConvexPolar.isCompact_stripBody b hb).isBounded.exists_norm_le
+  have hA0 : 0 ≤ A := (norm_nonneg (0 : C ι)).trans (hA 0 (zero_mem_domain _))
+  have hB0 : 0 ≤ B := (norm_nonneg (0 : R ι)).trans (hB 0 (interior_subset (ConvexPolar.zero_mem_interior_stripBody b)))
+  refine ⟨A*B, B, mul_nonneg hA0 hB0, hB0, hB, ?_⟩
+  intro k hk z hz y hy
+  exact momentum_bound b hk hz hy (hA z hz) (hB y hy)
+
+theorem exists_scaled_image (b : κ → R ι)
+    (hb : Function.Injective (fun x : R ι => fun j => inner (𝕜 := ℝ) (b j) x))
+    {η : ℝ} (hη : 0 < η) :
+    ∃ k : ℕ, 1 ≤ k ∧ ∀ z : C ι,
+      z ∈ domain (fun j => functional (b j)) → ‖tuple (fun j => functional (b j)) k z‖^2 < 1 →
+      (phase b k z).1 ∈ interior (ConvexPolar.stripBody b) ∧
+        (((1+η)*Real.pi*(k : ℝ)/4)⁻¹ • (phase b k z).2) ∈ interior (ConvexPolar.polar (ConvexPolar.stripBody b)) := by
+  obtain ⟨C, B, _hC, hB, hbound, hmoment⟩ := exists_uniform_momentum_bound b hb
+  have ht : Tendsto (fun k : ℕ => C/(k : ℝ)+(Fintype.card κ : ℝ)*PlanarLens.epsilon k) atTop (𝓝 0) := by
+    simpa using (tendsto_const_div_atTop_nhds_zero_nat C).add (PlanarLens.tendsto_epsilon.const_mul (Fintype.card κ : ℝ))
+  have hsmall := ht.eventually (eventually_lt_nhds (show (0 : ℝ) < η*Real.pi/4 by positivity))
+  obtain ⟨k, hk, hksmall⟩ := (eventually_ge_atTop (1 : ℕ)).and hsmall |>.exists
+  have hkpos : (0 : ℝ) < k := by exact_mod_cast (Nat.zero_lt_of_lt hk)
+  let T := (1+η)*Real.pi/4
+  have hT : 0 < T := by dsimp [T]; positivity
+  have hTsmall : C/(k : ℝ)+Real.pi/4+(Fintype.card κ : ℝ)*PlanarLens.epsilon k < T := by dsimp [T]; nlinarith
+  refine ⟨k, hk, fun z hz hτ => ⟨position_mem_interior b k hz, ?_⟩⟩
+  apply ConvexPolar.mem_interior_polar_of_bound hB hbound
+    (r := (C/(k : ℝ)+Real.pi/4+(Fintype.card κ : ℝ)*PlanarLens.epsilon k)/T)
+  · exact (div_lt_one hT).mpr hTsmall
+  · intro y hy
+    have hh := hmoment k hk z hz y hy
+    have hp : inner (𝕜 := ℝ) y (phase b k z).2/(k : ℝ) ≤
+        C/(k : ℝ)+Real.pi/4+(Fintype.card κ : ℝ)*PlanarLens.epsilon k := by
+      have hm := mul_le_mul_of_nonneg_left hτ.le (by positivity : (0 : ℝ) ≤ Real.pi/4)
+      linarith
+    have hd := div_le_div_of_nonneg_right hp hT.le
+    calc
+      _ = inner (𝕜 := ℝ) y (phase b k z).2/(k : ℝ)/T := by
+        rw [real_inner_smul_right]
+        dsimp only [T]
+        field_simp
+      _ ≤ _ := hd
+
+end
+end PolarStrips
+
+open Set
+open scoped ContDiff ENNReal
+
+namespace SymmetricPolar
+
+end SymmetricPolar
+
+namespace LocalEmbedding
+open Set Filter
+open scoped Topology ContDiff
+noncomputable section
+variable {E : Type u72} {F : Type u73} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
+  [NormedAddCommGroup F] [NormedSpace ℝ F]
+
+theorem open_image {U : Set E} (hU : IsOpen U) {f : E → F}
+    (hf : ContDiffOn ℝ ∞ f U)
+    (hD : ∀ x ∈ U, (fderiv ℝ f x).IsInvertible) : IsOpen (f '' U) := by
+  rw [isOpen_iff_mem_nhds]
+  rintro _ ⟨x, hx, rfl⟩
+  have hs := hf.contDiffAt (hU.mem_nhds hx)
+  obtain ⟨e, heq⟩ := hD x hx
+  have he : HasFDerivAt f e.toContinuousLinearMap x := by
+    rw [heq]
+    exact (hs.differentiableAt (by simp)).hasFDerivAt
+  rw [← (hs.hasStrictFDerivAt' he (by simp)).map_nhds_eq_of_equiv]
+  change f ⁻¹' (f '' U) ∈ 𝓝 x
+  exact mem_of_superset (hU.mem_nhds hx) (fun y hy => mem_image_of_mem f hy)
+
+theorem isOpenEmbedding {U : Set E} (hU : IsOpen U) {f : E → F}
+    (hf : ContDiffOn ℝ ∞ f U) (hi : InjOn f U)
+    (hD : ∀ x ∈ U, (fderiv ℝ f x).IsInvertible) :
+    Topology.IsOpenEmbedding (fun x : U => f x) := by
+  apply Topology.IsOpenEmbedding.of_continuous_injective_isOpenMap hf.continuousOn.domRestrict
+  · intro x y h; exact Subtype.ext (hi x.2 y.2 h)
+  · intro V hV
+    have hUV : Subtype.val '' V ⊆ U := by rintro _ ⟨x, _, rfl⟩; exact x.2
+    have huV : IsOpen (Subtype.val '' V) := hU.isOpenMap_subtype_val _ hV
+    have ho := open_image huV (hf.mono hUV) (fun x hx => hD x (hUV hx))
+    change IsOpen ((fun x : U => f x) '' V)
+    rw [image_image] at ho
+    exact ho
+
+end
+end LocalEmbedding
+
+namespace SymmetricPolar
+open Set Filter
+open scoped ContDiff Topology
+noncomputable section
+variable {n : ℕ}
+
+theorem isOpen_capacityBall (n : ℕ) (c : ℝ) : IsOpen (capacityBall n c) :=
+  isOpen_lt (by fun_prop) continuous_const
+
+def diagScale (a b : ℝ) : Phase n →L[ℝ] Phase n :=
+  (a • ContinuousLinearMap.fst ℝ (Position n) (Position n)).prod
+    (b • ContinuousLinearMap.snd ℝ (Position n) (Position n))
+
+@[simp] theorem diagScale_apply (a b : ℝ) (z : Phase n) : diagScale a b z = (a • z.1, b • z.2) := rfl
+
+theorem diagScale_injective {a b : ℝ} (ha : a ≠ 0) (hb : b ≠ 0) :
+    Function.Injective (diagScale a b : Phase n → Phase n) := by
+  intro x y h
+  exact Prod.ext ((smul_right_injective _ ha) (congrArg Prod.fst h))
+    ((smul_right_injective _ hb) (congrArg Prod.snd h))
+
+theorem omega0_diagScale (a b : ℝ) (v w : Phase n) :
+    omega0 (diagScale a b v) (diagScale a b w) = (a*b)*omega0 v w := by
+  simp only [omega0, diagScale_apply, real_inner_smul_left, real_inner_smul_right]
+  ring
+
+theorem omega0_nondegenerate (v : Phase n) (h : ∀ w : Phase n, omega0 v w = 0) : v = 0 := by
+  have hh := h (-v.2, v.1)
+  simp only [omega0, inner_neg_left, real_inner_self_eq_norm_sq, sub_neg_eq_add] at hh
+  have h1 : v.1 = 0 := norm_eq_zero.mp (by nlinarith [norm_nonneg v.1, sq_nonneg ‖v.2‖])
+  have h2 : v.2 = 0 := norm_eq_zero.mp (by nlinarith [norm_nonneg v.2, sq_nonneg ‖v.1‖])
+  exact Prod.ext h1 h2
+
+theorem isInvertible_of_preserves_omega0 (L : Phase n →L[ℝ] Phase n)
+    (hL : ∀ v w, omega0 (L v) (L w) = omega0 v w) : L.IsInvertible := by
+  have hi : Function.Injective L := by
+    intro v w hvw
+    have hz : L (v-w) = 0 := by rw [map_sub, hvw, sub_self]
+    apply sub_eq_zero.mp
+    apply omega0_nondegenerate
+    intro a
+    rw [← hL, hz]
+    simp [omega0]
+  have hs : Function.Surjective L := LinearMap.surjective_of_injective hi
+  exact ⟨ContinuousLinearEquiv.ofBijective L (LinearMap.ker_eq_bot.mpr hi)
+    (LinearMap.range_eq_top.mpr hs), rfl⟩
+
+theorem hasSymplecticEmbedding_of_injOn {U V : Set (Phase n)} (hU : IsOpen U)
+    {e : Phase n → Phase n} (he : ContDiffOn ℝ ∞ e U) (hi : InjOn e U)
+    (hm : MapsTo e U V) (hω : ∀ z ∈ U, ∀ v w,
+      omega0 (fderiv ℝ e z v) (fderiv ℝ e z w) = omega0 v w) :
+    HasSymplecticEmbedding U V := by
+  refine ⟨e, he, ?_, hm, hω⟩
+  exact (LocalEmbedding.isOpenEmbedding hU he hi (fun z hz => isInvertible_of_preserves_omega0 _ (hω z hz))).isEmbedding
+
+theorem HasSymplecticEmbedding.mono_target {U V W : Set (Phase n)}
+    (h : HasSymplecticEmbedding U V) (hVW : V ⊆ W) : HasSymplecticEmbedding U W := by
+  rcases h with ⟨e, he, hi, hm, hω⟩
+  exact ⟨e, he, hi, hm.mono_right hVW, hω⟩
+
+theorem HasSymplecticEmbedding.injOn {U : Set (Phase n)} {e : Phase n → Phase n}
+    (he : Topology.IsEmbedding (fun z : U => e z)) : InjOn e U := by
+  intro x hx y hy h
+  exact congrArg Subtype.val (he.injective (show e (⟨x,hx⟩ : U) = e (⟨y,hy⟩ : U) from h))
+
+theorem mapsTo_scaled_ball {lam c : ℝ} (hlam : 0 < lam) :
+    MapsTo (diagScale (Real.sqrt lam) (Real.sqrt lam) : Phase n → Phase n)
+      (capacityBall n c) (capacityBall n (lam*c)) := by
+  intro z hz
+  change Real.pi*(‖Real.sqrt lam • z.1‖^2+‖Real.sqrt lam • z.2‖^2) < lam*c
+  have hsq : |Real.sqrt lam|^2 = lam := by rw [sq_abs, Real.sq_sqrt hlam.le]
+  simp only [norm_smul, Real.norm_eq_abs, mul_pow, hsq]
+  have hh := mul_lt_mul_of_pos_left hz hlam
+  nlinarith
+
+theorem embedding_of_scaled_ball_map {lam c a b : ℝ} (hlam : 0 < lam) (hab : a*b = lam⁻¹)
+    {V : Set (Phase n)} {e : Phase n → Phase n}
+    (he : ContDiffOn ℝ ∞ e (capacityBall n (lam*c)))
+    (hi : InjOn e (capacityBall n (lam*c)))
+    (hm : MapsTo (fun z => diagScale a b (e z)) (capacityBall n (lam*c)) V)
+    (hω : ∀ z ∈ capacityBall n (lam*c), ∀ v w,
+      omega0 (fderiv ℝ e z v) (fderiv ℝ e z w) = omega0 v w) :
+    HasSymplecticEmbedding (capacityBall n c) V := by
+  let D : Phase n →L[ℝ] Phase n := diagScale (Real.sqrt lam) (Real.sqrt lam)
+  let L : Phase n →L[ℝ] Phase n := diagScale a b
+  let F : Phase n → Phase n := fun z => L (e (D z))
+  have hD : MapsTo D (capacityBall n c) (capacityBall n (lam*c)) := mapsTo_scaled_ball hlam
+  have hLinj : Function.Injective L := diagScale_injective (left_ne_zero_of_mul (by rw [hab]; exact inv_ne_zero hlam.ne'))
+    (right_ne_zero_of_mul (by rw [hab]; exact inv_ne_zero hlam.ne'))
+  have hDinj : Function.Injective D := diagScale_injective (Real.sqrt_pos.mpr hlam).ne' (Real.sqrt_pos.mpr hlam).ne'
+  have hs (z : Phase n) (hz : z ∈ capacityBall n c) : ContDiffAt ℝ ∞ e (D z) :=
+    he.contDiffAt ((isOpen_capacityBall _ _).mem_nhds (hD hz))
+  apply hasSymplecticEmbedding_of_injOn (isOpen_capacityBall n c) (e := F)
+  · intro z hz
+    exact (L.contDiff.contDiffAt.comp z ((hs z hz).comp z D.contDiff.contDiffAt)).contDiffWithinAt
+  · intro z hz w hw h
+    exact hDinj (hi (hD hz) (hD hw) (hLinj h))
+  · intro z hz
+    exact hm (hD hz)
+  · intro z hz v w
+    have hd : HasFDerivAt F (L.comp ((fderiv ℝ e (D z)).comp D)) z :=
+      L.hasFDerivAt.comp z (((hs z hz).differentiableAt (by simp)).hasFDerivAt.comp z D.hasFDerivAt)
+    rw [hd.fderiv]
+    change omega0 (diagScale a b (fderiv ℝ e (D z) (D v)))
+      (diagScale a b (fderiv ℝ e (D z) (D w))) = _
+    rw [omega0_diagScale, hω _ (hD hz), hab, omega0_diagScale]
+    rw [← pow_two, Real.sq_sqrt hlam.le, ← mul_assoc, inv_mul_cancel₀ hlam.ne', one_mul]
+
+end
+end SymmetricPolar
+
+end LowerBoundInline
+
+end OAI
diff --git a/lean/OAI/Geometry/PolarProducts/MoserField.lean b/lean/OAI/Geometry/PolarProducts/MoserField.lean
new file mode 100644
index 0000000000000000000000000000000000000000..01f2324cdce2fe6af91c90a82e306a91b02c65e0
--- /dev/null
+++ b/lean/OAI/Geometry/PolarProducts/MoserField.lean
@@ -0,0 +1,299 @@
+import OAI.Geometry.PolarProducts.TensorCalculus
+
+namespace OAI
+
+universe u23 u24
+
+section LowerBoundInline
+open Set Filter Function
+open scoped Topology ContDiff NNReal
+open Set Filter Metric
+open scoped Topology ContDiff
+open Set Filter Function MeasureTheory Metric
+open scoped Topology ContDiff NNReal
+open Set Filter Function
+open scoped Topology ContDiff
+
+open Set Filter Function
+open scoped Topology ContDiff NNReal
+
+namespace Moser
+
+variable {E : Type u23} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
+
+noncomputable section
+
+def spatialField (A : (ℝ × E) → E →L[ℝ] E →L[ℝ] ℝ)
+    (γ : E → E →L[ℝ] ℝ) (ρ : ℝ → ℝ) (z : ℝ × E) : E :=
+  -(ρ z.1) • (A z).inverse (γ z.2)
+
+omit [CompleteSpace E] in
+theorem spatialField_eq_zero_of_oneForm_eq_zero
+    (A : (ℝ × E) → E →L[ℝ] E →L[ℝ] ℝ)
+    (γ : E → E →L[ℝ] ℝ) (ρ : ℝ → ℝ) (z : ℝ × E) (hz : γ z.2 = 0) :
+    spatialField A γ ρ z = 0 := by simp [spatialField, hz]
+
+omit [CompleteSpace E] in
+theorem contract_spatialField {A : (ℝ × E) → E →L[ℝ] E →L[ℝ] ℝ}
+    (γ : E → E →L[ℝ] ℝ) (ρ : ℝ → ℝ) (z : ℝ × E) (hA : (A z).IsInvertible) :
+    A z (spatialField A γ ρ z) = -(ρ z.1) • γ z.2 := by
+  simp only [spatialField, map_smul, hA.self_apply_inverse]
+
+omit [CompleteSpace E] in
+
+theorem oneForm_spatialField {A : (ℝ × E) → E →L[ℝ] E →L[ℝ] ℝ}
+    (γ : E → E →L[ℝ] ℝ) (ρ : ℝ → ℝ) (z : ℝ × E)
+    (hA : (A z).IsInvertible) (hsk : ∀ a b, A z a b = -A z b a) :
+    γ z.2 (spatialField A γ ρ z) = 0 := by
+  let v := (A z).inverse (γ z.2)
+  have he : A z v = γ z.2 := hA.self_apply_inverse _
+  have hh := hsk v v
+  have hz : A z v v = 0 := by linarith
+  have hγv : γ z.2 v = 0 := by rw [← he]; exact hz
+  simpa [spatialField, v, map_smul, smul_eq_mul] using
+    congrArg (fun t : ℝ => -(ρ z.1) * t) hγv
+
+theorem contDiff_spatialField {A : (ℝ × E) → E →L[ℝ] E →L[ℝ] ℝ}
+    {γ : E → E →L[ℝ] ℝ} {ρ : ℝ → ℝ} {U : Set E}
+    (hA : ∀ z, z.2 ∈ U → ContDiffAt ℝ ∞ A z)
+    (hi : ∀ z, z.2 ∈ U → (A z).IsInvertible)
+    (hγ : ContDiff ℝ ∞ γ) (hρ : ContDiff ℝ ∞ ρ) (hs : tsupport γ ⊆ U) :
+    ContDiff ℝ ∞ (spatialField A γ ρ) := by
+  rw [contDiff_iff_contDiffAt]
+  intro z
+  by_cases hz : z.2 ∈ U
+  · have hv : ContDiffAt ℝ ∞ (fun x => (A x).inverse) z :=
+      (hi z hz).contDiffAt_map_inverse.comp z (hA z hz)
+    exact ((hρ.contDiffAt.comp z contDiffAt_fst).neg).smul
+      (hv.clm_apply (hγ.contDiffAt.comp z contDiffAt_snd))
+  · have ht : z.2 ∉ tsupport γ := fun hc => hz (hs hc)
+    have he : ∀ᶠ y in 𝓝 z, y.2 ∉ tsupport γ :=
+      continuous_snd.continuousAt ((isClosed_tsupport γ).isOpen_compl.mem_nhds ht)
+    have heq : spatialField A γ ρ =ᶠ[𝓝 z] fun _ => 0 := he.mono (fun y hy =>
+      spatialField_eq_zero_of_oneForm_eq_zero A γ ρ y (image_eq_zero_of_notMem_tsupport hy))
+    exact contDiffAt_const.congr_of_eventuallyEq heq
+
+omit [CompleteSpace E] in
+
+theorem hasCompactSupport_spatialField
+    (A : (ℝ × E) → E →L[ℝ] E →L[ℝ] ℝ)
+    {γ : E → E →L[ℝ] ℝ} {ρ : ℝ → ℝ}
+    (hγ : HasCompactSupport γ) (hρ : HasCompactSupport ρ) :
+    HasCompactSupport (spatialField A γ ρ) := by
+  apply HasCompactSupport.of_support_subset_isCompact (hρ.prod hγ)
+  intro z hz
+  constructor
+  · by_contra ht
+    apply hz
+    simp [spatialField, image_eq_zero_of_notMem_tsupport ht]
+  · by_contra hx
+    apply hz
+    exact spatialField_eq_zero_of_oneForm_eq_zero A γ ρ z
+      (image_eq_zero_of_notMem_tsupport hx)
+
+theorem exists_lipschitz_spaceTimeField
+    {A : (ℝ × E) → E →L[ℝ] E →L[ℝ] ℝ}
+    {γ : E → E →L[ℝ] ℝ} {ρ : ℝ → ℝ} {U : Set E}
+    (hA : ∀ z, z.2 ∈ U → ContDiffAt ℝ ∞ A z)
+    (hi : ∀ z, z.2 ∈ U → (A z).IsInvertible)
+    (hγ : ContDiff ℝ ∞ γ) (hρ : ContDiff ℝ ∞ ρ) (hs : tsupport γ ⊆ U)
+    (hcγ : HasCompactSupport γ) (hcρ : HasCompactSupport ρ) :
+    ∃ K : ℝ≥0, LipschitzWith K (fun z => ((1 : ℝ), spatialField A γ ρ z)) := by
+  obtain ⟨K, hK⟩ := ContDiff.lipschitzWith_of_hasCompactSupport
+    (hasCompactSupport_spatialField A hcγ hcρ)
+    (contDiff_spatialField hA hi hγ hρ hs) (by simp)
+  exact ⟨max 0 K, (LipschitzWith.const (1 : ℝ)).prodMk hK⟩
+
+end
+end Moser
+
+namespace Moser
+
+open Set Filter Function TensorCalculus
+open scoped Topology ContDiff
+
+variable {E : Type u24} [NormedAddCommGroup E] [NormedSpace ℝ E]
+
+noncomputable section
+
+local instance oneTensorGroup : NormedAddCommGroup (E →L[ℝ] ℝ) := inferInstance
+local instance oneTensorSpace : NormedSpace ℝ (E →L[ℝ] ℝ) := inferInstance
+local instance twoTensorGroup : NormedAddCommGroup (E →L[ℝ] E →L[ℝ] ℝ) := inferInstance
+local instance twoTensorSpace : NormedSpace ℝ (E →L[ℝ] E →L[ℝ] ℝ) := inferInstance
+
+def extendedOneForm (γ : E → E →L[ℝ] ℝ) (β : ℝ → ℝ) (z : ℝ × E) :
+    (ℝ × E) →L[ℝ] ℝ :=
+  β z.1 • (γ z.2).comp (ContinuousLinearMap.snd ℝ ℝ E)
+
+@[simp] theorem extendedOneForm_apply (γ : E → E →L[ℝ] ℝ) (β : ℝ → ℝ)
+    (z a : ℝ × E) : extendedOneForm γ β z a = β z.1 * γ z.2 a.2 := rfl
+
+theorem contDiff_extendedOneForm {γ : E → E →L[ℝ] ℝ} {β : ℝ → ℝ}
+    (hγ : ContDiff ℝ ∞ γ) (hβ : ContDiff ℝ ∞ β) :
+    ContDiff ℝ ∞ (extendedOneForm γ β) := by
+  exact (hβ.comp contDiff_fst).smul ((hγ.comp contDiff_snd).clm_comp contDiff_const)
+
+theorem fderiv_extendedOneForm {γ : E → E →L[ℝ] ℝ} {β : ℝ → ℝ} {z : ℝ × E}
+    (hγ : DifferentiableAt ℝ γ z.2) (hβ : DifferentiableAt ℝ β z.1)
+    (a b : ℝ × E) :
+    fderiv ℝ (extendedOneForm γ β) z a b =
+      deriv β z.1 * a.1 * γ z.2 b.2 + β z.1 * fderiv ℝ γ z.2 a.2 b.2 := by
+  have hα : DifferentiableAt ℝ (extendedOneForm γ β) z :=
+    (hβ.comp z differentiableAt_fst).smul
+      ((hγ.comp z differentiableAt_snd).clm_comp (differentiableAt_const _))
+  have he : fderiv ℝ (fun y => extendedOneForm γ β y b) z a =
+      fderiv ℝ (extendedOneForm γ β) z a b := by
+    rw [fderiv_clm_apply hα (differentiableAt_const b)]; simp
+  rw [← he]
+  simp only [extendedOneForm_apply]
+  have hb : DifferentiableAt ℝ (fun y : ℝ × E => β y.1) z :=
+    hβ.comp z differentiableAt_fst
+  have hg : DifferentiableAt ℝ (fun y : ℝ × E => γ y.2 b.2) z :=
+    (hγ.comp z differentiableAt_snd).clm_apply (differentiableAt_const b.2)
+  rw [fderiv_fun_mul hb hg]
+  rw [fderiv_fun_comp z hβ differentiableAt_fst]
+  have hg' : DifferentiableAt ℝ (fun y : ℝ × E => γ y.2) z :=
+    hγ.comp z differentiableAt_snd
+  rw [fderiv_clm_apply hg' (differentiableAt_const b.2)]
+  rw [fderiv_fun_comp z hγ differentiableAt_snd]
+  simp [fderiv_eq_smul_deriv, fderiv_fst, fderiv_snd, mul_comm, add_comm]
+
+def spatialTensor (W : E → TwoTensor E) (γ : E → E →L[ℝ] ℝ)
+    (β : ℝ → ℝ) (z : ℝ × E) : TwoTensor E :=
+  W z.2 + β z.1 • exteriorD γ z.2
+
+def spaceTimeTensor (W : E → TwoTensor E) (γ : E → E →L[ℝ] ℝ)
+    (β : ℝ → ℝ) (z : ℝ × E) : TwoTensor (ℝ × E) :=
+  pullLinear (ContinuousLinearMap.snd ℝ ℝ E) (W z.2) +
+    exteriorD (extendedOneForm γ β) z
+
+theorem contDiffAt_spatialTensor {W : E → TwoTensor E} {γ : E → E →L[ℝ] ℝ}
+    {β : ℝ → ℝ} {z : ℝ × E} (hW : ContDiffAt ℝ ∞ W z.2)
+    (hγ : ContDiff ℝ ∞ γ) (hβ : ContDiff ℝ ∞ β) :
+    ContDiffAt ℝ ∞ (spatialTensor W γ β) z := by
+  have hd : ContDiff ℝ ∞ (exteriorD γ) :=
+    contDiff_exteriorD hγ
+  exact (hW.comp z contDiffAt_snd).add
+    ((hβ.contDiffAt.comp z contDiffAt_fst).smul (hd.contDiffAt.comp z contDiffAt_snd))
+
+theorem contDiffAt_spaceTimeTensor {W : E → TwoTensor E} {γ : E → E →L[ℝ] ℝ}
+    {β : ℝ → ℝ} {z : ℝ × E} (hW : ContDiffAt ℝ ∞ W z.2)
+    (hγ : ContDiff ℝ ∞ γ) (hβ : ContDiff ℝ ∞ β) :
+    ContDiffAt ℝ ∞ (spaceTimeTensor W γ β) z := by
+  have hα := contDiff_extendedOneForm hγ hβ
+  have hd : ContDiff ℝ ∞ (exteriorD (extendedOneForm γ β)) :=
+    contDiff_exteriorD hα
+  exact (contDiffAt_pullLinear (ContinuousLinearMap.snd ℝ ℝ E)
+    (hW.comp z contDiffAt_snd)).add hd.contDiffAt
+
+theorem spaceTimeTensor_apply {W : E → TwoTensor E} {γ : E → E →L[ℝ] ℝ}
+    {β : ℝ → ℝ} {z : ℝ × E} (hγ : DifferentiableAt ℝ γ z.2)
+    (hβ : DifferentiableAt ℝ β z.1) (a b : ℝ × E) :
+    spaceTimeTensor W γ β z a b = spatialTensor W γ β z a.2 b.2 +
+      deriv β z.1 * (a.1 * γ z.2 b.2 - b.1 * γ z.2 a.2) := by
+  simp only [spaceTimeTensor, add_apply, pullLinear_apply,
+    ContinuousLinearMap.coe_snd', exteriorD_apply, fderiv_extendedOneForm hγ hβ,
+    spatialTensor, smul_apply, smul_eq_mul]
+  ring
+
+theorem spaceTimeTensor_skew (W : E → TwoTensor E) (γ : E → E →L[ℝ] ℝ)
+    (β : ℝ → ℝ) {z : ℝ × E} (hW : ∀ a b, W z.2 a b = -W z.2 b a) (a b : ℝ × E) :
+    spaceTimeTensor W γ β z a b = -spaceTimeTensor W γ β z b a := by
+  simp only [spaceTimeTensor, add_apply, pullLinear_apply,
+    ContinuousLinearMap.coe_snd']
+  rw [hW a.2 b.2, exteriorD_skew (extendedOneForm γ β) z a b]
+  ring
+
+theorem spaceTimeTensor_closed {W : E → TwoTensor E} {γ : E → E →L[ℝ] ℝ}
+    {β : ℝ → ℝ} {z : ℝ × E} (hW : ContDiffAt ℝ ∞ W z.2)
+    (hγ : ContDiff ℝ ∞ γ) (hβ : ContDiff ℝ ∞ β)
+    (hclosed : ∀ a b c, fderiv ℝ W z.2 a b c +
+      fderiv ℝ W z.2 b c a + fderiv ℝ W z.2 c a b = 0) (a b c : ℝ × E) :
+    fderiv ℝ (spaceTimeTensor W γ β) z a b c +
+      fderiv ℝ (spaceTimeTensor W γ β) z b c a +
+      fderiv ℝ (spaceTimeTensor W γ β) z c a b = 0 := by
+  have hα := contDiff_extendedOneForm hγ hβ
+  have hd : ContDiff ℝ ∞ (exteriorD (extendedOneForm γ β)) :=
+    contDiff_exteriorD hα
+  have hp : ContDiffAt ℝ ∞ (fun y : ℝ × E =>
+      pullLinear (ContinuousLinearMap.snd ℝ ℝ E) (W y.2)) z :=
+    contDiffAt_pullLinear (ContinuousLinearMap.snd ℝ ℝ E) (hW.comp z contDiffAt_snd)
+  have he := pullLinear_closed (ContinuousLinearMap.snd ℝ ℝ E) hW hclosed a b c
+  have hf := exteriorD_closed (x := z) hα.contDiffAt a b c
+  unfold spaceTimeTensor
+  simp only [fderiv_fun_add (hp.differentiableAt (by simp))
+    (hd.differentiable (by simp) z), add_apply]
+  dsimp only [ContinuousLinearMap.coe_snd'] at he
+  linarith
+
+theorem spaceTimeTensor_kernel {W : E → TwoTensor E} {γ : E → E →L[ℝ] ℝ}
+    {β : ℝ → ℝ} {z : ℝ × E} (hγ : DifferentiableAt ℝ γ z.2)
+    (hβ : DifferentiableAt ℝ β z.1)
+    (hi : (spatialTensor W γ β z).IsInvertible)
+    (hsk : ∀ a b, W z.2 a b = -W z.2 b a) :
+    spaceTimeTensor W γ β z (1, spatialField (spatialTensor W γ β) γ (deriv β) z) = 0 := by
+  have hs : ∀ a b, spatialTensor W γ β z a b = -spatialTensor W γ β z b a := by
+    intro a b
+    simp only [spatialTensor, add_apply, smul_apply,
+      smul_eq_mul]
+    rw [hsk a b, exteriorD_skew γ z.2 a b]; ring
+  have hc := contract_spatialField γ (deriv β) z hi
+  have hz := oneForm_spatialField γ (deriv β) z hi hs
+  apply ContinuousLinearMap.ext
+  intro a
+  rw [spaceTimeTensor_apply hγ hβ]
+  simp only [one_mul, hz, mul_zero, sub_zero]
+  have hc' := congrArg (fun L : E →L[ℝ] ℝ => L a.2) hc
+  simp only [smul_apply, smul_eq_mul] at hc'
+  simp only [zero_apply]
+  linarith
+
+end
+end Moser
+
+open Set Filter
+open scoped Topology ContDiff
+
+namespace Moser
+
+noncomputable section
+
+def timeCutoff : ℝ → ℝ := Real.smoothTransition
+
+theorem timeCutoff_smooth : ContDiff ℝ ∞ timeCutoff := Real.smoothTransition.contDiff
+
+theorem timeCutoff_mem (t : ℝ) : timeCutoff t ∈ Icc (0 : ℝ) 1 :=
+  ⟨Real.smoothTransition.nonneg t, Real.smoothTransition.le_one t⟩
+
+@[simp] theorem timeCutoff_zero : timeCutoff 0 = 0 := Real.smoothTransition.zero
+
+@[simp] theorem timeCutoff_one : timeCutoff 1 = 1 := Real.smoothTransition.one
+
+theorem timeCutoff_deriv_smooth : ContDiff ℝ ∞ (deriv timeCutoff) :=
+  (show ContDiff ℝ (∞ + 1) timeCutoff by simpa using timeCutoff_smooth).deriv'
+
+theorem timeCutoff_deriv_hasCompactSupport : HasCompactSupport (deriv timeCutoff) := by
+  apply HasCompactSupport.of_support_subset_isCompact (isCompact_Icc (a := (0 : ℝ)) (b := 1))
+  intro t ht
+  by_contra h
+  have hd : deriv timeCutoff t = 0 := by
+    rw [mem_Icc, not_and_or] at h
+    rcases h with h | h
+    · have hn : ∀ᶠ y in 𝓝 t, y < 0 := Iio_mem_nhds (lt_of_not_ge h)
+      have he : timeCutoff =ᶠ[𝓝 t] fun _ => 0 :=
+        hn.mono (fun y hy =>
+          Real.smoothTransition.zero_of_nonpos hy.le)
+      rw [he.deriv_eq, deriv_const]
+    · have hn : ∀ᶠ y in 𝓝 t, 1 < y := Ioi_mem_nhds (lt_of_not_ge h)
+      have he : timeCutoff =ᶠ[𝓝 t] fun _ => 1 :=
+        hn.mono (fun y hy =>
+          Real.smoothTransition.one_of_one_le hy.le)
+      rw [he.deriv_eq, deriv_const]
+  exact ht hd
+
+end
+end Moser
+
+end LowerBoundInline
+
+end OAI
diff --git a/lean/OAI/Geometry/PolarProducts/MoserFlow.lean b/lean/OAI/Geometry/PolarProducts/MoserFlow.lean
new file mode 100644
index 0000000000000000000000000000000000000000..fc7401628dc006d4f8ae8ae60d2b032461d71633
--- /dev/null
+++ b/lean/OAI/Geometry/PolarProducts/MoserFlow.lean
@@ -0,0 +1,211 @@
+import OAI.Geometry.PolarProducts.MoserField
+
+namespace OAI
+
+universe u25 u26
+
+section LowerBoundInline
+open Set Filter Function
+open scoped Topology ContDiff NNReal
+open Set Filter Metric
+open scoped Topology ContDiff
+open Set Filter Function MeasureTheory Metric
+open scoped Topology ContDiff NNReal
+open Set Filter Function
+open scoped Topology ContDiff
+open Set Filter Function
+open scoped Topology ContDiff NNReal
+open Set Filter
+open scoped Topology ContDiff
+
+open Set Filter Function
+open scoped Topology ContDiff
+
+namespace SmoothODE
+
+variable {E : Type u25} [NormedAddCommGroup E] [NormedSpace ℝ E]
+
+noncomputable section
+
+def fiberHomeomorph (f : (ℝ × E) ≃ₜ (ℝ × E)) (T : ℝ)
+    (hTime : ∀ z, (f z).1 = z.1 + T) (t : ℝ) : E ≃ₜ E where
+  toFun x := (f (t, x)).2
+  invFun y := (f.symm (t + T, y)).2
+  left_inv x := by
+    have he : (t + T, (f (t, x)).2) = f (t, x) := by
+      ext
+      · exact (hTime (t, x)).symm
+      · rfl
+    change (f.symm (t + T, (f (t, x)).2)).2 = x
+    rw [he, f.symm_apply_apply]
+  right_inv y := by
+    have he : (f.symm (t + T, y)).1 = t := by
+      have h := hTime (f.symm (t + T, y))
+      rw [f.apply_symm_apply] at h
+      dsimp at h
+      linarith
+    have he' : (t, (f.symm (t + T, y)).2) = f.symm (t + T, y) := by
+      ext
+      · exact he.symm
+      · rfl
+    change (f (t, (f.symm (t + T, y)).2)).2 = y
+    rw [he', f.apply_symm_apply]
+  continuous_toFun := continuous_snd.comp (f.continuous.comp (continuous_const.prodMk continuous_id))
+  continuous_invFun := continuous_snd.comp
+    (f.symm.continuous.comp (continuous_const.prodMk continuous_id))
+
+omit [NormedSpace ℝ E] in
+@[simp] theorem fiberHomeomorph_apply (f : (ℝ × E) ≃ₜ (ℝ × E)) (T : ℝ)
+    (hTime : ∀ z, (f z).1 = z.1 + T) (t : ℝ) (x : E) :
+    fiberHomeomorph f T hTime t x = (f (t, x)).2 := rfl
+
+omit [NormedSpace ℝ E] in
+@[simp] theorem fiberHomeomorph_symm_apply (f : (ℝ × E) ≃ₜ (ℝ × E)) (T : ℝ)
+    (hTime : ∀ z, (f z).1 = z.1 + T) (t : ℝ) (x : E) :
+    (fiberHomeomorph f T hTime t).symm x = (f.symm (t + T, x)).2 := rfl
+
+theorem contDiff_fiberHomeomorph (f : (ℝ × E) ≃ₜ (ℝ × E)) (T : ℝ)
+    (hTime : ∀ z, (f z).1 = z.1 + T) (t : ℝ)
+    (hf : ContDiff ℝ ∞ (f : (ℝ × E) → (ℝ × E))) :
+    ContDiff ℝ ∞ (fiberHomeomorph f T hTime t : E → E) :=
+  (hf.comp (contDiff_const.prodMk contDiff_id)).snd
+
+theorem contDiff_fiberHomeomorph_symm (f : (ℝ × E) ≃ₜ (ℝ × E)) (T : ℝ)
+    (hTime : ∀ z, (f z).1 = z.1 + T) (t : ℝ)
+    (hf : ContDiff ℝ ∞ (f.symm : (ℝ × E) → (ℝ × E))) :
+    ContDiff ℝ ∞ ((fiberHomeomorph f T hTime t).symm : E → E) :=
+  (hf.comp (contDiff_const.prodMk contDiff_id)).snd
+
+theorem fderiv_fiberHomeomorph (f : (ℝ × E) ≃ₜ (ℝ × E)) (T : ℝ)
+    (hTime : ∀ z, (f z).1 = z.1 + T) (t : ℝ)
+    (hf : ContDiff ℝ ∞ (f : (ℝ × E) → (ℝ × E))) (x a : E) :
+    fderiv ℝ (f : (ℝ × E) → (ℝ × E)) (t, x) (0, a) =
+      (0, fderiv ℝ (fiberHomeomorph f T hTime t : E → E) x a) := by
+  have hi : HasFDerivAt (fun y : E => (t, y)) (ContinuousLinearMap.inr ℝ ℝ E) x := by
+    have hh := (hasFDerivAt_const (𝕜 := ℝ) t x).prodMk (hasFDerivAt_id (𝕜 := ℝ) x)
+    have he : (0 : E →L[ℝ] ℝ).prod (ContinuousLinearMap.id ℝ E) =
+        ContinuousLinearMap.inr ℝ ℝ E := by
+      ext y <;> rfl
+    rw [he] at hh
+    exact hh
+  have hd := (hf.differentiable (by simp) (t, x)).hasFDerivAt.comp x hi
+  change HasFDerivAt (fun y : E => f (t, y)) _ x at hd
+  have he : (fun y : E => f (t, y)) =
+      fun y => (t + T, fiberHomeomorph f T hTime t y) := by
+    funext y
+    exact Prod.ext (hTime (t, y)) rfl
+  have hd' := (hasFDerivAt_const (𝕜 := ℝ) (t + T) x).prodMk
+    ((contDiff_fiberHomeomorph f T hTime t hf).differentiable (by simp) x).hasFDerivAt
+  have hu := congrArg (fun L : E →L[ℝ] (ℝ × E) => L a)
+    ((he ▸ hd).unique hd')
+  simpa using hu
+
+omit [NormedSpace ℝ E] in
+
+theorem fiberHomeomorph_mapsTo (f : (ℝ × E) ≃ₜ (ℝ × E)) (T : ℝ)
+    (hTime : ∀ z, (f z).1 = z.1 + T) (t : ℝ) {U : Set E}
+    (hOut : ∀ z, z.2 ∉ U → (f z).2 = z.2) :
+    ∀ x, fiberHomeomorph f T hTime t x ∈ U ↔ x ∈ U := by
+  intro x
+  constructor
+  · intro hx
+    by_contra hc
+    rw [fiberHomeomorph_apply, hOut (t, x) hc] at hx
+    exact hc hx
+  · intro hx
+    by_contra hc
+    have he : fiberHomeomorph f T hTime t (fiberHomeomorph f T hTime t x) =
+        fiberHomeomorph f T hTime t x := hOut _ hc
+    have hu := (fiberHomeomorph f T hTime t).injective he
+    exact hc (hu.symm ▸ hx)
+
+end
+end SmoothODE
+
+namespace Moser
+
+open Set Filter Function TensorCalculus SmoothODE
+open scoped Topology ContDiff NNReal
+
+variable {E : Type u26} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  [CompleteSpace E] [ProperSpace E]
+
+noncomputable section
+
+theorem compactlySupported_deformation {U : Set E} (hU : IsOpen U)
+    {W : E → TwoTensor E} {γ : E → E →L[ℝ] ℝ}
+    (hW : ∀ x ∈ U, ContDiffAt ℝ ∞ W x)
+    (hsk : ∀ x ∈ U, ∀ a b, W x a b = -W x b a)
+    (hclosed : ∀ x ∈ U, ∀ a b c,
+      fderiv ℝ W x a b c + fderiv ℝ W x b c a + fderiv ℝ W x c a b = 0)
+    (hγ : ContDiff ℝ ∞ γ) (hcγ : HasCompactSupport γ) (hs : tsupport γ ⊆ U)
+    (hi : ∀ x ∈ U, ∀ s ∈ Icc (0 : ℝ) 1, (W x + s • exteriorD γ x).IsInvertible) :
+    ∃ F : E ≃ₜ E, ContDiff ℝ ∞ (F : E → E) ∧ ContDiff ℝ ∞ (F.symm : E → E) ∧
+      (∀ x, x ∉ U → F x = x) ∧ (∀ x, F x ∈ U ↔ x ∈ U) ∧
+      ∀ x ∈ U, ∀ a b,
+        (W (F x) + exteriorD γ (F x)) (fderiv ℝ (F : E → E) x a)
+          (fderiv ℝ (F : E → E) x b) = W x a b := by
+  let A := spatialTensor W γ timeCutoff
+  let Y := spatialField A γ (deriv timeCutoff)
+  let V : (ℝ × E) → (ℝ × E) := fun z => (1, Y z)
+  let B := spaceTimeTensor W γ timeCutoff
+  have hA : ∀ z, z.2 ∈ U → ContDiffAt ℝ ∞ A z := by
+    intro z hz
+    exact contDiffAt_spatialTensor (hW z.2 hz) hγ timeCutoff_smooth
+  have hAi : ∀ z, z.2 ∈ U → (A z).IsInvertible := by
+    intro z hz
+    exact hi z.2 hz _ (timeCutoff_mem z.1)
+  have hY : ContDiff ℝ ∞ Y :=
+    contDiff_spatialField hA hAi hγ timeCutoff_deriv_smooth hs
+  have hV : ContDiff ℝ ∞ V := contDiff_const.prodMk hY
+  obtain ⟨K, hK⟩ := exists_lipschitz_spaceTimeField hA hAi hγ
+    timeCutoff_deriv_smooth hs hcγ timeCutoff_deriv_hasCompactSupport
+  have hOut : ∀ z, z.2 ∉ U → (V z).2 = 0 := by
+    intro z hz
+    exact spatialField_eq_zero_of_oneForm_eq_zero A γ (deriv timeCutoff) z
+      (image_eq_zero_of_notMem_tsupport (fun hc => hz (hs hc)))
+  have hB : ∀ z, z.2 ∈ U → DifferentiableAt ℝ B z := by
+    intro z hz
+    exact (contDiffAt_spaceTimeTensor (hW z.2 hz) hγ timeCutoff_smooth).differentiableAt
+      (by simp)
+  have hIv : ∀ z, z.2 ∈ U → ∀ a b, fderiv ℝ B z (V z) a b +
+      B z (fderiv ℝ V z a) b + B z a (fderiv ℝ V z b) = 0 := by
+    intro z hz a b
+    have hn : ∀ᶠ y in 𝓝 z, y.2 ∈ U := continuous_snd.continuousAt (hU.mem_nhds hz)
+    apply invariant_of_closed_kernel_at (hB z hz) (hV.differentiable (by simp) z)
+      (hn.mono (fun y hy => spaceTimeTensor_skew W γ timeCutoff (hsk y.2 hy)))
+      (spaceTimeTensor_closed (hW z.2 hz) hγ timeCutoff_smooth (hclosed z.2 hz))
+      (hn.mono (fun y hy => spaceTimeTensor_kernel (hγ.differentiable (by simp) y.2)
+        (timeCutoff_smooth.differentiable (by simp) y.1) (hAi y hy) (hsk y.2 hy)))
+  obtain ⟨f, hf, hfi, htime, hout, htrans⟩ :=
+    exists_spaceTime_transport hK hV (fun _ => rfl) hOut hB hIv 1
+  let F := fiberHomeomorph f 1 htime 0
+  have hF : ContDiff ℝ ∞ (F : E → E) := contDiff_fiberHomeomorph f 1 htime 0 hf
+  have hFi : ContDiff ℝ ∞ (F.symm : E → E) := contDiff_fiberHomeomorph_symm f 1 htime 0 hfi
+  refine ⟨F, hF, hFi, ?_, fiberHomeomorph_mapsTo f 1 htime 0 hout, ?_⟩
+  · intro x hx
+    exact hout (0, x) hx
+  · intro x hx a b
+    have hh := (htrans (0, x) hx).2 (0, a) (0, b)
+    rw [fderiv_fiberHomeomorph f 1 htime 0 hf x a,
+      fderiv_fiberHomeomorph f 1 htime 0 hf x b] at hh
+    have he : f (0, x) = (1, F x) := by
+      apply Prod.ext
+      · simpa using htime (0, x)
+      · rfl
+    rw [he] at hh
+    have hb (z : ℝ × E) (a b : E) : B z (0, a) (0, b) = A z a b := by
+      dsimp only [B, A]
+      rw [spaceTimeTensor_apply (hγ.differentiable (by simp) z.2)
+        (timeCutoff_smooth.differentiable (by simp) z.1)]
+      simp
+    rw [hb, hb] at hh
+    simpa only [A, spatialTensor, timeCutoff_zero, timeCutoff_one,
+      one_smul, zero_smul, add_zero, add_apply, smul_apply, smul_eq_mul, zero_mul] using hh
+
+end
+end Moser
+
+end LowerBoundInline
+
+end OAI
diff --git a/lean/OAI/Geometry/PolarProducts/Nonsqueezing.lean b/lean/OAI/Geometry/PolarProducts/Nonsqueezing.lean
new file mode 100644
index 0000000000000000000000000000000000000000..8c1f86ccb64c7128f617b8f67449d0e3fd9f61d6
--- /dev/null
+++ b/lean/OAI/Geometry/PolarProducts/Nonsqueezing.lean
@@ -0,0 +1,161 @@
+import OAI.Geometry.PolarProducts.ExteriorHamiltonian
+
+namespace OAI
+
+universe u127
+
+section NonsqueezingInline
+
+namespace CylinderParameters
+noncomputable section
+open Set Filter Finset
+open scoped ContDiff Topology
+open FourierPolynomial DiagonalQuadratic HamiltonianODE
+variable {κ : Type u127} [Fintype κ] [DecidableEq κ]
+
+ def weights (j₀ : κ) (a b : ℝ) (j : κ) : ℝ := if j=j₀ then a else b
+
+ theorem energy_weights (j₀ : κ) (a b : ℝ) (z : Vector κ) :
+    energy (weights j₀ a b) z = (a-b)*‖z j₀‖^2+b*‖z‖^2 := by
+   have hh (j : κ) : weights j₀ a b j*‖z j‖^2 =
+       b*‖z j‖^2+(if j=j₀ then (a-b)*‖z j‖^2 else 0) := by
+     by_cases hj : j=j₀ <;> simp only [weights,hj,ite_true,ite_false] <;> ring
+   simp only [energy,hh,Finset.sum_add_distrib,Finset.sum_ite_eq',Finset.mem_univ,ite_true,
+     ← Finset.mul_sum,← EuclideanSpace.norm_sq_eq]
+   ring
+
+ theorem energy_weights_upper (j₀ : κ) {a b R M : ℝ} (hb : 0 ≤ b) (hab : b ≤ a)
+    (hR : 0 ≤ R) {z : Vector κ} (hz : ‖z‖^2 ≤ M) (hz₀ : ‖z j₀‖^2 ≤ R) :
+    energy (weights j₀ a b) z ≤ a*R+b*M := by
+   rw [energy_weights]
+   have h1 := mul_le_mul_of_nonneg_left hz (show 0 ≤ b from hb)
+   have h2 := mul_le_mul_of_nonneg_left hz₀ (sub_nonneg.mpr hab)
+   nlinarith
+
+ theorem gap_first {m : ℤ} : Real.pi/2 ≤ |Real.pi*(m:ℝ)-3*Real.pi/2| := by
+   rcases le_or_gt m 1 with hm | hm
+   · have hm' : (m:ℝ) ≤ 1 := by exact_mod_cast hm
+     apply (le_abs.mpr (Or.inr ?_))
+     nlinarith [Real.pi_pos]
+   · have hm' : (2:ℝ) ≤ m := by exact_mod_cast (show 2 ≤ m by omega)
+     apply (le_abs.mpr (Or.inl ?_))
+     nlinarith [Real.pi_pos]
+
+ theorem gap_small {b : ℝ} (hb : 0 < b) (hbπ : b ≤ Real.pi/2) (m : ℤ) :
+    b ≤ |Real.pi*(m:ℝ)-b| := by
+   rcases le_or_gt m 0 with hm | hm
+   · have hm' : (m:ℝ) ≤ 0 := by exact_mod_cast hm
+     apply (le_abs.mpr (Or.inr ?_))
+     nlinarith [Real.pi_pos]
+   · have hm' : (1:ℝ) ≤ m := by exact_mod_cast (show 1 ≤ m by omega)
+     apply (le_abs.mpr (Or.inl ?_))
+     nlinarith [Real.pi_pos]
+
+omit [Fintype κ] in
+ theorem weights_bounds (j₀ : κ) {b : ℝ} (hb : 0 < b) (hbπ : b ≤ Real.pi/4) :
+    (∀ j, b ≤ weights j₀ (3*Real.pi/2) b j) ∧
+    Real.pi+b ≤ weights j₀ (3*Real.pi/2) b j₀ ∧
+    (∀ j, |weights j₀ (3*Real.pi/2) b j| ≤ 3*Real.pi/2) ∧
+    (∀ j, j ≠ j₀ → weights j₀ (3*Real.pi/2) b j < Real.pi) ∧
+    (∀ m : ℤ, ∀ j, b ≤ |Real.pi*(m:ℝ)-weights j₀ (3*Real.pi/2) b j|) := by
+   have hab : b ≤ 3*Real.pi/2 := by linarith [Real.pi_pos]
+   refine ⟨?_,?_,?_,?_,?_⟩
+   · intro j; unfold weights; split_ifs
+     · exact hab
+     · exact le_rfl
+   · simp only [weights,ite_true]; linarith [Real.pi_pos]
+   · intro j; unfold weights; split_ifs
+     · rw [abs_of_pos (by positivity)]
+     · rw [abs_of_pos hb]; exact hab
+   · intro j hj; simp only [weights,hj,ite_false]; linarith [Real.pi_pos]
+   · intro m j; unfold weights; split_ifs
+     · exact (show b ≤ Real.pi/2 by linarith [Real.pi_pos]).trans gap_first
+     · exact gap_small hb (by linarith [Real.pi_pos]) m
+
+ theorem exists_parameters (j₀ : κ) {R M m : ℝ} (hR : 0 ≤ R) (hM : 0 ≤ M)
+    (hRm : Real.pi*R < m) :
+    ∃ b T δ : ℝ, 0 < b ∧ b ≤ Real.pi/4 ∧ 0 < δ ∧ 0 < T ∧
+      Real.pi*(T+δ) < m*(3*Real.pi/2) ∧
+      ∀ z : Vector κ, ‖z‖^2 ≤ M → ‖z j₀‖^2 ≤ R →
+        energy (weights j₀ (3*Real.pi/2) b) z < T := by
+   let a := 3*Real.pi/2
+   have ha : 0 < a := by dsimp [a]; positivity
+   let τ := (m-Real.pi*R)/(4*Real.pi)
+   have hτ : 0 < τ := div_pos (sub_pos.mpr hRm) (by positivity)
+   have hτeq : 4*Real.pi*τ=m-Real.pi*R := by dsimp [τ]; field_simp
+   let b := min (Real.pi/4) (a*τ/(M+1))
+   have hb : 0 < b := lt_min (by positivity) (div_pos (mul_pos ha hτ) (by linarith))
+   have hbπ : b ≤ Real.pi/4 := min_le_left _ _
+   have hbb : b*(M+1) ≤ a*τ := (le_div_iff₀ (by linarith : 0 < M+1)).mp (min_le_right _ _)
+   have hbM : b*M < a*τ := by nlinarith
+   refine ⟨b,a*(R+τ),a*τ,hb,hbπ,mul_pos ha hτ,mul_pos ha (by linarith),?_,?_⟩
+   · change Real.pi*(a*(R+τ)+a*τ) < m*a
+     have hh : Real.pi*(R+2*τ)< m := by nlinarith
+     nlinarith [mul_lt_mul_of_pos_left hh ha]
+   · intro z hz hz₀
+     have hab : b ≤ a := by dsimp [a]; linarith [Real.pi_pos]
+     have he := energy_weights_upper j₀ hb.le hab hR hz hz₀
+     change energy (weights j₀ a b) z < a*(R+τ)
+     nlinarith
+
+end
+end CylinderParameters
+
+namespace CylinderNonsqueezing
+noncomputable section
+open Set Filter
+open scoped ContDiff Topology
+open FourierPolynomial DiagonalQuadratic HamiltonianODE SmoothProfiles HamiltonianPeriodic
+open ExteriorHamiltonian PushedBall CylinderParameters
+variable {κ : Type} [Fintype κ] [DecidableEq κ]
+
+ theorem radius_le_of_embedding (j₀ : κ) {L R M : ℝ} (hL : 0 < L) (hR : 0 ≤ R) (hM : 0 ≤ M)
+    (e : OpenPartialHomeomorph (Vector κ) (Vector κ))
+    (he : ContDiffOn ℝ ∞ e e.source) (hs : e.source = domain L)
+    (hω : ∀ z ∈ e.source, ComplexSymplectic.Preserves (fderiv ℝ e z))
+    (hbounded : ∀ z ∈ e.target, ‖z‖^2 ≤ M)
+    (hcylinder : ∀ z ∈ e.target, ‖z j₀‖^2 ≤ R) : L ≤ R := by
+   by_contra hn
+   have hRL : R < L := lt_of_not_ge hn
+   obtain ⟨r,ε,hr,hrπ,hε,hεL,hm⟩ := exists_ball_parameters (mul_nonneg Real.pi_pos.le hR) hL
+     (mul_lt_mul_of_pos_left hRL Real.pi_pos)
+   let m := ballHeight r L ε
+   have hεL' : ε ≤ L := by linarith
+   have hRm : Real.pi*R < m := hm.trans_le (ballHeight_lower hr.le L hε hεL.le)
+   obtain ⟨b,T,δ,hb,hbπ,hδ,hT,hmag,hcontain⟩ := exists_parameters j₀ hR hM hRm
+   let d := weights j₀ (3*Real.pi/2) b
+   obtain ⟨hdb,hdb0,hdD,hdπ,hgap⟩ := weights_bounds j₀ hb hbπ
+   have hQ (z : Vector κ) (hz : z ∈ e.target) : energy d z < T :=
+     hcontain z (hbounded z hz) (hcylinder z hz)
+   have hD (z) (hz : z ∈ e.source) := ComplexSymplectic.invertible (hω z hz)
+   let H₀ := pushed e r L ε
+   have hH₀ : ContDiff ℝ ∞ H₀ := contDiff_pushed e he hs hD hε hεL'
+   have hH₀0 (z : Vector κ) : 0 ≤ H₀ z := (pushed_bounds e hr.le hε z).1
+   have hH₀m (z : Vector κ) : H₀ z ≤ m := (pushed_bounds e hr.le hε z).2
+   have hS : IsCompact (e '' core L ε) := (core_compact L ε).image_of_continuousOn
+     (e.continuousOn.mono (by rw [hs]; exact core_subset_domain hε hεL'))
+   obtain ⟨B,hB,hrem⟩ := bounded_remainder hH₀ hH₀m hS
+     (fun z hz => pushed_plateau e hε hεL' hz) d hb hdb hδ (by linarith : 0 ≤ T+δ)
+   have h0 : (0 : Vector κ) ∈ e.source := by rw [hs]; exact zero_mem_domain hL
+   have hflat := full_flat d hδ (pushed_flat e (r := r) hs hε hεL') (hQ _ (e.map_source h0))
+   have hnonneg (z : Vector κ) : 0 ≤ full H₀ d T δ z := add_nonneg (hH₀0 z) (potential_nonneg d T hδ z)
+   obtain ⟨x,hp,ha,hx⟩ := exists_positive_at_flat_point j₀ (contDiff_full hH₀ d T δ) hnonneg (e 0)
+     hflat d (div_pos hb (by norm_num : (0:ℝ) < 2)) hb hB
+     (by positivity : 0 ≤ 3*Real.pi/2)
+     (fun j => by dsimp [d]; linarith [hdb j])
+     (by dsimp [d]; linarith) hdD hgap hrem
+   have hn := all_orbits_nonpos j₀ hH₀ hH₀m hH₀0 d (fun j => hb.le.trans (hdb j))
+     (by dsimp [d,weights]; simp only [ite_true]; linarith [Real.pi_pos]) hdπ
+     (by dsimp [d,weights]; simp only [ite_true]; linarith [Real.pi_pos]) hδ
+     (by simpa only [d,weights,ite_true] using hmag)
+     (fun z hz => (hQ z hz).le)
+     (fun z hz => pushed_plateau_outside e hs hε hεL' hz)
+     (fun x hxt hp hx t => pushed_full_slow e he hω hs hr.le hrπ hε hεL' d hδ
+       (fun z hz => (hQ z hz).le) hxt hp hx t) hx hp
+   exact (not_lt_of_ge hn) ha
+
+end
+end CylinderNonsqueezing
+end NonsqueezingInline
+
+end OAI
diff --git a/lean/OAI/Geometry/PolarProducts/PeriodicFourier.lean b/lean/OAI/Geometry/PolarProducts/PeriodicFourier.lean
new file mode 100644
index 0000000000000000000000000000000000000000..ba3873584cc89fe0aa9e0b44237794433eb3ea79
--- /dev/null
+++ b/lean/OAI/Geometry/PolarProducts/PeriodicFourier.lean
@@ -0,0 +1,301 @@
+import OAI.Geometry.PolarProducts.GalerkinBounds
+
+namespace OAI
+
+universe u114 u115
+
+section NonsqueezingInline
+
+namespace L2Paths
+noncomputable section
+open Set intervalIntegral
+open MeasureTheory (volume)
+open scoped Topology BoundedContinuousFunction
+variable {E : Type u114} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
+
+ theorem integral_sq_le (f : ℝ → ℝ) (hf : Continuous f) {a b : ℝ} (hab : a ≤ b) :
+    (∫ t in a..b, f t)^2 ≤ (b-a)*(∫ t in a..b, (f t)^2) := by
+  rcases hab.eq_or_lt with he | hab
+  · subst b
+    simp
+  let avg := ∫ t in a..b, f t
+  let L := b-a
+  have hL : 0 < L := sub_pos.mpr hab
+  have hnon : 0 ≤ ∫ t in a..b, (L*f t-avg)^2 :=
+    integral_nonneg_of_forall hab.le (fun _ => sq_nonneg _)
+  have he (t : ℝ) : (L*f t-avg)^2 = L^2*(f t)^2-(2*L*avg)*f t+avg^2 := by ring
+  simp_rw [he] at hnon
+  rw [integral_add, integral_sub, integral_const_mul, integral_const_mul, integral_const] at hnon
+  · simp only [smul_eq_mul] at hnon
+    change 0 ≤ L^2*(∫ t in a..b, (f t)^2)-(2*L*avg)*avg+L*avg^2 at hnon
+    change avg^2 ≤ L*(∫ t in a..b, (f t)^2)
+    nlinarith
+  · exact (hf.pow 2).intervalIntegrable _ _ |>.const_mul _
+  · exact hf.intervalIntegrable _ _ |>.const_mul _
+  · exact (((hf.pow 2).const_mul _).sub (hf.const_mul _)).intervalIntegrable _ _
+  · exact intervalIntegrable_const
+
+ theorem norm_sub_sq_le (x v : ℝ → E) (hv : Continuous v)
+    (hd : ∀ t, HasDerivAt x (v t) t) {a b : ℝ} (hab : a ≤ b) :
+    ‖x b-x a‖^2 ≤ (b-a)*(∫ t in a..b, ‖v t‖^2) := by
+  have he := integral_eq_sub_of_hasDerivAt (fun t _ => hd t) (hv.intervalIntegrable a b)
+  have hn : ‖x b-x a‖ ≤ ∫ t in a..b, ‖v t‖ := by
+    rw [← he]
+    exact norm_integral_le_integral_norm hab
+  have hi := integral_sq_le (fun t => ‖v t‖) hv.norm hab
+  have hs : ‖x b-x a‖^2 ≤ (∫ t in a..b, ‖v t‖)^2 :=
+    pow_le_pow_left₀ (norm_nonneg _) hn 2
+  exact hs.trans hi
+
+ theorem norm_sub_sq_le_global (x v : ℝ → E) (hv : Continuous v)
+    (hd : ∀ t, HasDerivAt x (v t) t) {D : ℝ}
+    (hD : (∫ t in (0:ℝ)..1, ‖v t‖^2) ≤ D^2)
+    {a b : ℝ} (ha : a ∈ Icc (0:ℝ) 1) (hb : b ∈ Icc (0:ℝ) 1) :
+    ‖x b-x a‖^2 ≤ D^2*|b-a| := by
+  wlog hab : a ≤ b generalizing a b
+  · have h := this hb ha (le_of_not_ge hab)
+    rwa [norm_sub_rev, abs_sub_comm] at h
+  have hi := integral_mono_interval (μ := volume) ha.1 hab hb.2
+    (Filter.Eventually.of_forall (fun t => sq_nonneg ‖v t‖)) ((hv.norm.pow 2).intervalIntegrable 0 1)
+  have hc := norm_sub_sq_le x v hv hd hab
+  have hm := mul_le_mul_of_nonneg_left (hi.trans hD) (sub_nonneg.mpr hab)
+  rw [abs_of_nonneg (sub_nonneg.mpr hab)]
+  exact hc.trans (by nlinarith)
+
+ theorem norm_bound (x v : ℝ → E) (hv : Continuous v)
+    (hd : ∀ t, HasDerivAt x (v t) t) {A D : ℝ} (hA : 0 ≤ A) (hD : 0 ≤ D)
+    (hxm : (∫ t in (0:ℝ)..1, ‖x t‖^2) ≤ A^2)
+    (hvm : (∫ t in (0:ℝ)..1, ‖v t‖^2) ≤ D^2)
+    {t : ℝ} (ht : t ∈ Icc (0:ℝ) 1) : ‖x t‖ ≤ A+D := by
+  have hx : Continuous x := continuous_iff_continuousAt.mpr (fun s => (hd s).continuousAt)
+  have hpoint (s : ℝ) (hs : s ∈ Icc (0:ℝ) 1) : ‖x t‖ ≤ ‖x s‖+D := by
+    have hdiff := norm_sub_sq_le_global x v hv hd hvm hs ht
+    have hab : |t-s| ≤ 1 := (abs_le).mpr ⟨by linarith [ht.1,hs.2], by linarith [ht.2,hs.1]⟩
+    have hmul := mul_le_mul_of_nonneg_left hab (sq_nonneg D)
+    have hn : ‖x t-x s‖ ≤ D := by nlinarith [norm_nonneg (x t-x s)]
+    have htri := norm_sub_le (x t-x s) (-x s)
+    rw [sub_neg_eq_add, sub_add_cancel, norm_neg] at htri
+    linarith
+  have hi : ∫ s in (0:ℝ)..1, ‖x t‖ ≤ ∫ s in (0:ℝ)..1, ‖x s‖+D :=
+    integral_mono_on (by norm_num) intervalIntegrable_const
+      ((hx.norm.add continuous_const).intervalIntegrable _ _) hpoint
+  rw [integral_const, integral_add (hx.norm.intervalIntegrable _ _) intervalIntegrable_const,
+    integral_const] at hi
+  simp only [sub_zero, one_smul] at hi
+  have hcs := integral_sq_le (fun s => ‖x s‖) hx.norm (show (0:ℝ) ≤ 1 by norm_num)
+  simp only [sub_zero, one_mul] at hcs
+  have hmean : ∫ t_1 in (0:ℝ)..1, ‖x t_1‖ ≤ A := by
+    nlinarith [integral_nonneg_of_forall (μ := volume) (show (0:ℝ) ≤ 1 by norm_num) (fun s => norm_nonneg (x s))]
+  linarith
+
+abbrev Segment := Icc (0 : ℝ) 1
+
+omit [NormedSpace ℝ E] [CompleteSpace E] in
+ theorem exists_subsequence [ProperSpace E] (u : ℕ → C(Segment,E)) {D R : ℝ}
+    (hmod : ∀ n (t s : Segment), dist (u n t) (u n s)^2 ≤ D^2*dist t s)
+    (hbound : ∀ n (t : Segment), ‖u n t‖ ≤ R) :
+    ∃ x : C(Segment,E), ∃ φ : ℕ → ℕ, StrictMono φ ∧
+      Filter.Tendsto (u ∘ φ) Filter.atTop (𝓝 x) := by
+  let eqv := ContinuousMap.isometryEquivBoundedOfCompact Segment E
+  let v : ℕ → (Segment →ᵇ E) := fun n => eqv (u n)
+  have heq : Equicontinuous ((↑) : Set.range v → Segment → E) := by
+    apply UniformEquicontinuous.equicontinuous
+    rw [Metric.uniformEquicontinuous_iff]
+    intro ε hε
+    refine ⟨ε^2/(D^2+1), by positivity, fun t s hts f => ?_⟩
+    rcases f.property with ⟨n,hn⟩
+    have hm := hmod n t s
+    have hd : (D^2+1)*dist t s < ε^2 := by
+      have := (lt_div_iff₀ (by positivity : 0 < D^2+1)).mp hts
+      linarith
+    rw [← hn]
+    change dist (u n t) (u n s) < ε
+    nlinarith [(dist_nonneg : 0 ≤ dist (u n t) (u n s)), (dist_nonneg : 0 ≤ dist t s)]
+  have hc : IsCompact (closure (Set.range v)) :=
+    BoundedContinuousFunction.arzela_ascoli (Metric.closedBall (0:E) R)
+      (isCompact_closedBall _ _) (Set.range v)
+      (fun f t hf => by
+        rcases hf with ⟨n,rfl⟩
+        change dist (u n t) (0:E) ≤ R
+        simpa only [dist_zero_right] using hbound n t) heq
+  obtain ⟨x,hx,φ,hφ,ht⟩ := hc.tendsto_subseq (fun n => subset_closure (Set.mem_range_self n))
+  refine ⟨eqv.symm x,φ,hφ,?_⟩
+  exact eqv.symm.continuous.continuousAt.tendsto.comp ht
+
+local instance : Fact (0 < (1 : ℝ)) := ⟨zero_lt_one⟩
+abbrev Time := AddCircle (1 : ℝ)
+
+def restriction (x : C(Time,E)) : C(Segment,E) :=
+  ⟨fun t => x (t.val : Time), x.continuous.comp (continuous_quot_mk.comp continuous_subtype_val)⟩
+
+omit [NormedSpace ℝ E] [CompleteSpace E] in
+ theorem restriction_isometry : Isometry (restriction (E := E)) := by
+  rw [isometry_iff_dist_eq]
+  intro x y
+  apply le_antisymm
+  · apply (ContinuousMap.dist_le (dist_nonneg : 0 ≤ dist x y)).2
+    intro t
+    exact ContinuousMap.dist_apply_le_dist (t.val : Time)
+  · apply (ContinuousMap.dist_le (dist_nonneg : 0 ≤ dist (restriction x) (restriction y))).2
+    intro t
+    let r := AddCircle.equivIoc (1 : ℝ) 0 t
+    have hr : r.val ∈ Set.Icc (0:ℝ) 1 := ⟨r.property.1.le, by simpa using r.property.2⟩
+    have h := ContinuousMap.dist_apply_le_dist (f := restriction x) (g := restriction y) ⟨r.val,hr⟩
+    simpa only [restriction, ContinuousMap.coe_mk, r, AddCircle.coe_equivIoc] using h
+
+omit [NormedSpace ℝ E] in
+ theorem exists_periodic_subsequence [ProperSpace E] (u : ℕ → C(Time,E)) {D R : ℝ}
+    (hmod : ∀ n (t s : Segment), dist (u n (t.val : Time)) (u n (s.val : Time))^2 ≤ D^2*dist t s)
+    (hbound : ∀ n (t : Segment), ‖u n (t.val : Time)‖ ≤ R) :
+    ∃ x : C(Time,E), ∃ φ : ℕ → ℕ, StrictMono φ ∧
+      Filter.Tendsto (u ∘ φ) Filter.atTop (𝓝 x) := by
+  obtain ⟨x,φ,hφ,ht⟩ := exists_subsequence (fun n => restriction (u n)) hmod hbound
+  have hc' : Cauchy (Filter.map restriction (Filter.map (u ∘ φ) Filter.atTop)) := by
+    rw [Filter.map_map]
+    exact ht.cauchySeq
+  have hc : CauchySeq (u ∘ φ) :=
+    restriction_isometry.isUniformInducing.cauchy_map_iff.mp hc'
+  obtain ⟨y,hy⟩ := cauchySeq_tendsto_of_complete hc
+  exact ⟨y,φ,hφ,hy⟩
+
+end
+end L2Paths
+
+namespace PeriodicFourier
+noncomputable section
+open MeasureTheory AddCircle Finset Set
+open scoped ComplexConjugate Topology
+local instance : Fact (0 < (1 : ℝ)) := ⟨zero_lt_one⟩
+abbrev Time := AddCircle (1 : ℝ)
+abbrev μ : Measure Time := AddCircle.haarAddCircle
+open FourierPolynomial (integrable_continuous)
+
+ def coefficient (n : ℤ) : C(Time,ℂ) →L[ℂ] ℂ := by
+  let L : C(Time,ℂ) →ₗ[ℂ] ℂ := {
+    toFun := fun f => fourierCoeff f n
+    map_add' := fun f g => congrFun (fourierCoeff.add
+      (integrable_continuous f.continuous) (integrable_continuous g.continuous)) n
+    map_smul' := fun c f => fourierCoeff.const_smul f c n }
+  apply L.mkContinuous 1
+  intro f
+  change ‖∫ t : Time, fourier (-n) t • f t ∂μ‖ ≤ 1*‖f‖
+  have h := norm_integral_le_of_norm_le_const (μ := μ) (C := ‖f‖)
+    (Filter.Eventually.of_forall (fun t : Time => show ‖fourier (-n) t • f t‖ ≤ ‖f‖ by
+      simpa only [norm_smul, fourier_apply, Circle.norm_coe, one_mul] using f.norm_coe_le_norm t))
+  simpa using h
+
+@[simp] theorem coefficient_apply (n : ℤ) (f : C(Time,ℂ)) : coefficient n f = fourierCoeff f n := rfl
+
+ theorem coefficient_ext {f g : C(Time,ℂ)} (h : ∀ n, coefficient n f = coefficient n g) : f = g := by
+  have hz (n : ℤ) : fourierCoeff (f-g) n = 0 := by
+    change coefficient n (f-g) = 0
+    rw [map_sub, h, sub_self]
+  have hs : Summable (fourierCoeff (f-g)) := by
+    simpa only [funext hz] using (summable_zero : Summable (fun _ : ℤ => (0:ℂ)))
+  have hh := hasSum_fourier_series_of_summable hs
+  simp only [hz, zero_smul] at hh
+  have he : f-g = 0 := hh.unique hasSum_zero
+  exact sub_eq_zero.mp he
+
+ theorem coefficient_const (n : ℤ) (c : ℂ) :
+    coefficient n (ContinuousMap.const Time c) = if n = 0 then c else 0 := by
+  rw [coefficient_apply, fourierCoeff]
+  simp only [ContinuousMap.coe_const, Function.const_apply, smul_eq_mul]
+  rw [integral_mul_const, FourierPolynomial.integral_fourier]
+  simp only [neg_eq_zero]
+  split_ifs <;> simp
+
+ theorem constant_of_nonzero_coeff {f : C(Time,ℂ)}
+    (h : ∀ n : ℤ, n ≠ 0 → coefficient n f = 0) :
+    f = ContinuousMap.const Time (coefficient 0 f) := by
+  apply coefficient_ext
+  intro n
+  rw [coefficient_const]
+  by_cases hn : n = 0
+  · subst n; simp
+  · simp [hn,h n hn]
+
+ theorem continuous_lift {E : Type u115} [TopologicalSpace E] {f : ℝ → E}
+    (hf : Continuous f) (hp : Function.Periodic f 1) : Continuous hp.lift :=
+  hf.quotient_liftOn' _
+
+ def primitive (v : C(Time,ℂ)) (t : ℝ) : ℂ := ∫ s in (0:ℝ)..t, v (s : Time)
+
+ theorem hasDerivAt_primitive (v : C(Time,ℂ)) (t : ℝ) :
+    HasDerivAt (primitive v) (v (t : Time)) t := by
+  have hv : Continuous (fun s : ℝ => v (s : Time)) := v.continuous.comp continuous_quot_mk
+  exact intervalIntegral.integral_hasDerivAt_right (hv.intervalIntegrable _ _) hv.stronglyMeasurable.stronglyMeasurableAtFilter hv.continuousAt
+
+ theorem periodic_primitive (v : C(Time,ℂ)) (hzero : coefficient 0 v = 0) :
+    Function.Periodic (primitive v) 1 := by
+  have hv : Continuous (fun s : ℝ => v (s : Time)) := v.continuous.comp continuous_quot_mk
+  have hp : Function.Periodic (fun s : ℝ => v (s : Time)) 1 := by
+    intro t
+    simp
+  have hi : ∫ s in (0:ℝ)..1, v (s : Time) = 0 := by
+    have h := hzero
+    rw [coefficient_apply, fourierCoeff_eq_intervalIntegral _ _ 0] at h
+    simpa only [one_div_one, one_smul, zero_add, neg_zero, fourier_zero] using h
+  intro t
+  unfold primitive
+  rw [hp.intervalIntegral_add_eq_add 0 t (fun a b => hv.intervalIntegrable a b)]
+  simp only [zero_add, hi, add_zero]
+
+ def periodicPrimitive (v : C(Time,ℂ)) (hzero : coefficient 0 v = 0) : C(Time,ℂ) :=
+  ⟨(periodic_primitive v hzero).lift,
+    continuous_lift (continuous_iff_continuousAt.mpr (fun t => (hasDerivAt_primitive v t).continuousAt)) _⟩
+
+@[simp] theorem periodicPrimitive_coe (v : C(Time,ℂ)) (hzero : coefficient 0 v = 0) (t : ℝ) :
+    periodicPrimitive v hzero (t : Time) = primitive v t := rfl
+
+ theorem coefficient_eq_on (f : C(Time,ℂ)) (n : ℤ) :
+    fourierCoeffOn (show (0:ℝ) < 1 by norm_num) (fun t : ℝ => f (t : Time)) n = coefficient n f := by
+  rw [fourierCoeffOn_eq_integral, coefficient_apply, fourierCoeff_eq_intervalIntegral _ _ 0]
+  simp
+
+ theorem derivative_coefficient (v : C(Time,ℂ)) (hzero : coefficient 0 v = 0)
+    {n : ℤ} (hn : n ≠ 0) :
+    (2*(Real.pi : ℂ)*Complex.I*(n : ℂ)) * coefficient n (periodicPrimitive v hzero) = coefficient n v := by
+  have hp := periodic_primitive v hzero
+  have hi := fourierCoeffOn_of_hasDerivAt (show (0:ℝ) < 1 by norm_num) hn
+    (fun t _ => hasDerivAt_primitive v t)
+    ((v.continuous.comp continuous_quot_mk).intervalIntegrable 0 1)
+  have ht : primitive v 1 = primitive v 0 := by simpa using hp 0
+  simp only [ht, sub_self, mul_zero, Complex.ofReal_one, Complex.ofReal_zero, sub_zero, one_mul, zero_sub] at hi
+  change fourierCoeffOn _ (fun t : ℝ => periodicPrimitive v hzero (t : Time)) n =
+    1 / (-2 * (Real.pi : ℂ) * Complex.I * (n : ℂ)) * -fourierCoeffOn _ (fun t : ℝ => v (t : Time)) n at hi
+  rw [coefficient_eq_on, coefficient_eq_on] at hi
+  rw [hi]
+  have hπ : (Real.pi : ℂ) ≠ 0 := by exact_mod_cast Real.pi_ne_zero
+  have hn' : (n : ℂ) ≠ 0 := by exact_mod_cast hn
+  field_simp
+
+ theorem hasDerivAt_of_coefficients (x v : C(Time,ℂ))
+    (h : ∀ n : ℤ, coefficient n v = (2*(Real.pi : ℂ)*Complex.I*(n : ℂ))*coefficient n x)
+    (t : ℝ) : HasDerivAt (fun s : ℝ => x (s : Time)) (v (t : Time)) t := by
+  have hz : coefficient 0 v = 0 := by simpa using h 0
+  let y := periodicPrimitive v hz
+  have hc : x-y = ContinuousMap.const Time (coefficient 0 (x-y)) := by
+    apply constant_of_nonzero_coeff
+    intro n hn
+    rw [map_sub]
+    have he := derivative_coefficient v hz hn
+    rw [h n] at he
+    have hn' : (2*(Real.pi : ℂ)*Complex.I*(n : ℂ)) ≠ 0 := by
+      apply mul_ne_zero (mul_ne_zero (mul_ne_zero (by norm_num) (by exact_mod_cast Real.pi_ne_zero)) Complex.I_ne_zero)
+      exact_mod_cast hn
+    have heq := (mul_left_cancel₀ hn') he
+    change coefficient n x-coefficient n (periodicPrimitive v hz) = 0
+    exact sub_eq_zero.mpr heq.symm
+  have hx (s : ℝ) : x (s : Time) = primitive v s+coefficient 0 (x-y) := by
+    have he := congrArg (fun f : C(Time,ℂ) => f (s : Time)) hc
+    change x (s : Time)-primitive v s = coefficient 0 (x-y) at he
+    exact sub_eq_iff_eq_add'.mp he
+  simpa only [← hx] using (hasDerivAt_primitive v t).add_const (coefficient 0 (x-y))
+
+end
+end PeriodicFourier
+
+end NonsqueezingInline
+
+end OAI
diff --git a/lean/OAI/Geometry/PolarProducts/PeriodicSolutions.lean b/lean/OAI/Geometry/PolarProducts/PeriodicSolutions.lean
new file mode 100644
index 0000000000000000000000000000000000000000..1bdd05984cd926b6a5217f3178b1ee2bcbc7c8f8
--- /dev/null
+++ b/lean/OAI/Geometry/PolarProducts/PeriodicSolutions.lean
@@ -0,0 +1,230 @@
+import OAI.Geometry.PolarProducts.PeriodicFourier
+
+namespace OAI
+
+section NonsqueezingInline
+
+namespace FourierPolynomial
+noncomputable section
+open MeasureTheory AddCircle Finset Set Filter
+open scoped ComplexConjugate ContDiff Topology
+local instance : Fact (0 < (1 : ℝ)) := ⟨zero_lt_one⟩
+variable {ι κ : Type} [Fintype ι] [Fintype κ]
+open DiagonalQuadratic
+
+ theorem critical_H1_bounds (k : ι → ℤ) (hk : Function.Injective k)
+    {H : Vector κ → ℝ} (hH : ContDiff ℝ ∞ H) (d : κ → ℝ) {γ B D : ℝ}
+    (hγ : 0 < γ) (hB : 0 ≤ B) (hD : 0 ≤ D) (hd : ∀ j, |d j| ≤ D)
+    (hgap : ∀ i j, γ ≤ |Real.pi*(k i : ℝ)-d j|)
+    (hrem : ∀ z, ‖gradient H z-(2 : ℝ) • diag d z‖ ≤ B) (a : Coeff ι κ)
+    (ha : gradient (action k H) a = 0) :
+    (∫ t in (0:ℝ)..1, ‖loop k a (t : Time)‖^2) ≤ (B/(2*γ))^2 ∧
+    (∫ t in (0:ℝ)..1, ‖loop k (velocityCoeff k a) (t : Time)‖^2) ≤
+      (B+2*D*(B/(2*γ)))^2 := by
+  have hn := critical_norm_bound k hk hH d hγ hB hgap hrem a ha
+  have hv := norm_project_grad k hk hH d hB hD hd hrem a
+  rw [← norm_velocityCoeff_critical k hH a ha] at hv
+  have hd' : 0 ≤ 2*D := mul_nonneg (by norm_num) hD
+  have hvel : ‖velocityCoeff k a‖ ≤ B+2*D*(B/(2*γ)) :=
+    hv.trans (by linarith [mul_le_mul_of_nonneg_left hn hd'])
+  rw [intervalIntegral_eq_average (fun t : Time => ‖loop k a t‖^2),
+    intervalIntegral_eq_average (fun t : Time => ‖loop k (velocityCoeff k a) t‖^2),
+    integral_loop_norm_sq k hk, integral_loop_norm_sq k hk]
+  exact ⟨pow_le_pow_left₀ (norm_nonneg _) hn 2, pow_le_pow_left₀ (norm_nonneg _) hvel 2⟩
+
+ theorem critical_modulus (k : ι → ℤ) (hk : Function.Injective k)
+    {H : Vector κ → ℝ} (hH : ContDiff ℝ ∞ H) (d : κ → ℝ) {γ B D : ℝ}
+    (hγ : 0 < γ) (hB : 0 ≤ B) (hD : 0 ≤ D) (hd : ∀ j, |d j| ≤ D)
+    (hgap : ∀ i j, γ ≤ |Real.pi*(k i : ℝ)-d j|)
+    (hrem : ∀ z, ‖gradient H z-(2 : ℝ) • diag d z‖ ≤ B) (a : Coeff ι κ)
+    (ha : gradient (action k H) a = 0) (t s : L2Paths.Segment) :
+    dist (loop k a (t.val : Time)) (loop k a (s.val : Time))^2 ≤
+      (B+2*D*(B/(2*γ)))^2*dist t s := by
+  have hb := (critical_H1_bounds k hk hH d hγ hB hD hd hgap hrem a ha).2
+  have h := L2Paths.norm_sub_sq_le_global
+    (fun t : ℝ => loop k a (t : Time)) (fun t : ℝ => loop k (velocityCoeff k a) (t : Time))
+    ((loop k (velocityCoeff k a)).continuous.comp continuous_quot_mk)
+    (hasDerivAt_loop k a) hb s.property t.property
+  simpa only [dist_eq_norm, Subtype.dist_eq, Real.dist_eq, Real.norm_eq_abs] using h
+
+ theorem critical_loop_bound (k : ι → ℤ) (hk : Function.Injective k)
+    {H : Vector κ → ℝ} (hH : ContDiff ℝ ∞ H) (d : κ → ℝ) {γ B D : ℝ}
+    (hγ : 0 < γ) (hB : 0 ≤ B) (hD : 0 ≤ D) (hd : ∀ j, |d j| ≤ D)
+    (hgap : ∀ i j, γ ≤ |Real.pi*(k i : ℝ)-d j|)
+    (hrem : ∀ z, ‖gradient H z-(2 : ℝ) • diag d z‖ ≤ B) (a : Coeff ι κ)
+    (ha : gradient (action k H) a = 0) (t : L2Paths.Segment) :
+    ‖loop k a (t.val : Time)‖ ≤ B/(2*γ)+(B+2*D*(B/(2*γ))) := by
+  have hb := critical_H1_bounds k hk hH d hγ hB hD hd hgap hrem a ha
+  apply L2Paths.norm_bound
+    (fun t : ℝ => loop k a (t : Time)) (fun t : ℝ => loop k (velocityCoeff k a) (t : Time))
+    ((loop k (velocityCoeff k a)).continuous.comp continuous_quot_mk)
+    (hasDerivAt_loop k a) (by positivity) (by positivity) hb.1 hb.2 t.property
+
+ theorem critical_subsequence {α : ℕ → Type} [∀ n, Fintype (α n)]
+    (k : (n : ℕ) → α n → ℤ) (hk : ∀ n, Function.Injective (k n))
+    {H : Vector κ → ℝ} (hH : ContDiff ℝ ∞ H) (d : κ → ℝ) {γ B D : ℝ}
+    (hγ : 0 < γ) (hB : 0 ≤ B) (hD : 0 ≤ D) (hd : ∀ j, |d j| ≤ D)
+    (hgap : ∀ n i j, γ ≤ |Real.pi*(k n i : ℝ)-d j|)
+    (hrem : ∀ z, ‖gradient H z-(2 : ℝ) • diag d z‖ ≤ B)
+    (a : (n : ℕ) → Coeff (α n) κ) (ha : ∀ n, gradient (action (k n) H) (a n) = 0) :
+    ∃ x : C(Time,Vector κ), ∃ φ : ℕ → ℕ, StrictMono φ ∧
+      Tendsto (fun n => loop (k (φ n)) (a (φ n))) atTop (𝓝 x) := by
+  exact L2Paths.exists_periodic_subsequence (fun n => loop (k n) (a n))
+    (fun n t s => critical_modulus (k n) (hk n) hH d hγ hB hD hd (hgap n) hrem (a n) (ha n) t s)
+    (fun n t => critical_loop_bound (k n) (hk n) hH d hγ hB hD hd (hgap n) hrem (a n) (ha n) t)
+
+ def coordPath (j : κ) : C(Time,Vector κ) →L[ℂ] C(Time,ℂ) :=
+   ContinuousLinearMap.compLeftContinuous ℂ Time (PiLp.proj 2 (fun _ : κ => ℂ) j)
+
+@[simp] theorem coordPath_apply (j : κ) (x : C(Time,Vector κ)) (t : Time) : coordPath j x t = x t j := rfl
+
+ def gradPath {H : Vector κ → ℝ} (hH : ContDiff ℝ ∞ H) : C(Time,Vector κ) → C(Time,Vector κ) :=
+   SmoothPaths.superpose (contDiff_gradient_function hH).continuous
+
+ theorem contDiff_gradPath {H : Vector κ → ℝ} (hH : ContDiff ℝ ∞ H) : ContDiff ℝ ∞ (gradPath hH) :=
+   SmoothPaths.contDiff_superpose (contDiff_gradient_function hH)
+
+ theorem critical_coefficient (k : ι → ℤ) (hk : Function.Injective k)
+    {H : Vector κ → ℝ} (hH : ContDiff ℝ ∞ H) (a : Coeff ι κ)
+    (ha : gradient (action k H) a = 0) (i : ι) (j : κ) :
+    PeriodicFourier.coefficient (k i) (coordPath j (gradPath hH (loop k a))) =
+      (2*(Real.pi : ℂ)*(k i : ℂ))*PeriodicFourier.coefficient (k i) (coordPath j (loop k a)) := by
+  have he : (2 : ℝ) • diag (fun p : ι × κ => Real.pi*(k p.1 : ℝ)) a = project k (gradLoop k hH a) := by
+    simpa only [gradient_action k hH, sub_eq_zero] using ha
+  change project k (gradLoop k hH a) (i,j) = (2*(Real.pi : ℂ)*(k i : ℂ))*project k (loop k a) (i,j)
+  rw [project_loop k hk, ← he]
+  simp only [PiLp.smul_apply, diag_apply, RCLike.real_smul_eq_coe_mul, map_mul, map_intCast, map_ofNat]
+  change (2:ℂ)*((Real.pi : ℂ)*(k i : ℂ)*a (i,j)) = (2*(Real.pi : ℂ)*(k i : ℂ))*a (i,j)
+  ring
+
+ theorem limit_coefficient {α : ℕ → Type} [∀ n, Fintype (α n)]
+    (k : (n : ℕ) → α n → ℤ) (hk : ∀ n, Function.Injective (k n))
+    (hex : ∀ m : ℤ, ∀ᶠ n in atTop, m ∈ Set.range (k n))
+    {H : Vector κ → ℝ} (hH : ContDiff ℝ ∞ H)
+    (a : (n : ℕ) → Coeff (α n) κ) (ha : ∀ n, gradient (action (k n) H) (a n) = 0)
+    (φ : ℕ → ℕ) (hφ : StrictMono φ) (x : C(Time,Vector κ))
+    (ht : Tendsto (fun n => loop (k (φ n)) (a (φ n))) atTop (𝓝 x)) (m : ℤ) (j : κ) :
+    PeriodicFourier.coefficient m (coordPath j (gradPath hH x)) =
+      (2*(Real.pi : ℂ)*(m : ℂ))*PeriodicFourier.coefficient m (coordPath j x) := by
+  have hg := (contDiff_gradPath hH).continuous.continuousAt.tendsto.comp ht
+  have hc := (PeriodicFourier.coefficient m).continuous.comp (coordPath j).continuous
+  have hleft := hc.continuousAt.tendsto.comp hg
+  have hright := (tendsto_const_nhds (x := (2*(Real.pi : ℂ)*(m : ℂ)))).mul
+    (hc.continuousAt.tendsto.comp ht)
+  apply tendsto_nhds_unique_of_eventuallyEq hleft hright
+  filter_upwards [hφ.tendsto_atTop.eventually (hex m)] with n hn
+  rcases hn with ⟨i,rfl⟩
+  exact critical_coefficient (k (φ n)) (hk (φ n)) hH (a (φ n)) (ha (φ n)) i j
+
+ theorem hasDerivAt_of_critical_coefficients {H : Vector κ → ℝ} (hH : ContDiff ℝ ∞ H)
+    (x : C(Time,Vector κ))
+    (h : ∀ m j, PeriodicFourier.coefficient m (coordPath j (gradPath hH x)) =
+      (2*(Real.pi : ℂ)*(m : ℂ))*PeriodicFourier.coefficient m (coordPath j x)) (t : ℝ) :
+    HasDerivAt (fun s : ℝ => x (s : Time)) (Complex.I • gradient H (x (t : Time))) t := by
+  have hj (j : κ) : HasDerivAt (fun s : ℝ => x (s : Time) j)
+      (Complex.I • gradient H (x (t : Time)) j) t := by
+    have hc (m : ℤ) : PeriodicFourier.coefficient m (Complex.I • coordPath j (gradPath hH x)) =
+        (2*(Real.pi : ℂ)*Complex.I*(m : ℂ))*PeriodicFourier.coefficient m (coordPath j x) := by
+      rw [map_smul, h]
+      simp only [smul_eq_mul]
+      ring
+    exact PeriodicFourier.hasDerivAt_of_coefficients (coordPath j x)
+      (Complex.I • coordPath j (gradPath hH x)) hc t
+  have hp := hasDerivAt_pi.mpr hj
+  exact (PiLp.continuousLinearEquiv 2 ℝ (fun _ : κ => ℂ)).symm.hasFDerivAt.comp_hasDerivAt t hp
+
+ theorem positive_action_limit {α : ℕ → Type} [∀ n, Fintype (α n)]
+    (k : (n : ℕ) → α n → ℤ) {H : Vector κ → ℝ} (hH : ContDiff ℝ ∞ H)
+    (a : (n : ℕ) → Coeff (α n) κ) (ha : ∀ n, gradient (action (k n) H) (a n) = 0)
+    {L : ℝ} (hL : ∀ n, L ≤ action (k n) H (a n))
+    (φ : ℕ → ℕ) (x : C(Time,Vector κ))
+    (ht : Tendsto (fun n => loop (k (φ n)) (a (φ n))) atTop (𝓝 x)) :
+    L ≤ meanAction hH x := by
+  apply ge_of_tendsto' ((contDiff_meanAction hH).continuous.continuousAt.tendsto.comp ht)
+  intro n
+  change L ≤ meanAction hH (loop (k (φ n)) (a (φ n)))
+  rw [← critical_action_eq _ hH _ (ha (φ n))]
+  exact hL (φ n)
+
+end
+end FourierPolynomial
+
+namespace HamiltonianPeriodic
+noncomputable section
+open MeasureTheory AddCircle Finset Set Filter
+open scoped ComplexConjugate ContDiff Topology
+local instance : Fact (0 < (1 : ℝ)) := ⟨zero_lt_one⟩
+open DiagonalQuadratic FourierPolynomial
+
+ def negativeMode (N : ℕ) (i : Fin (N+1)) : ℤ := -(i.val : ℤ)
+ def positiveMode (N : ℕ) (i : Fin (N+1)) : ℕ := i.val+1
+ def frequency (N : ℕ) : Fin (N+1) ⊕ Fin (N+1) → ℤ :=
+   Sum.elim (negativeMode N) (fun i => (positiveMode N i : ℤ))
+
+ theorem negativeMode_nonpos (N : ℕ) (i : Fin (N+1)) : negativeMode N i ≤ 0 := by
+  simp [negativeMode]
+
+ theorem positiveMode_pos (N : ℕ) (i : Fin (N+1)) : 0 < positiveMode N i := Nat.succ_pos _
+
+ theorem positiveMode_injective (N : ℕ) : Function.Injective (positiveMode N) := by
+  intro i j hij
+  apply Fin.ext
+  simpa [positiveMode] using hij
+
+ theorem frequency_injective (N : ℕ) : Function.Injective (frequency N) := by
+  intro i j hij
+  cases i <;> cases j
+  · simp only [frequency, Sum.elim_inl, negativeMode, neg_inj, Nat.cast_inj] at hij
+    exact congrArg Sum.inl (Fin.ext hij)
+  · simp only [frequency, Sum.elim_inl, Sum.elim_inr, negativeMode, positiveMode, Nat.cast_add, Nat.cast_one] at hij
+    omega
+  · simp only [frequency, Sum.elim_inl, Sum.elim_inr, negativeMode, positiveMode, Nat.cast_add, Nat.cast_one] at hij
+    omega
+  · simp only [frequency, Sum.elim_inr, Nat.cast_inj] at hij
+    exact congrArg Sum.inr (positiveMode_injective N hij)
+
+ theorem frequency_exhausts (m : ℤ) : ∀ᶠ N in atTop, m ∈ Set.range (frequency N) := by
+  cases m with
+  | ofNat m =>
+    cases m with
+    | zero =>
+      filter_upwards [] with N
+      exact ⟨Sum.inl ⟨0,Nat.succ_pos N⟩, by simp [frequency,negativeMode]⟩
+    | succ m =>
+      filter_upwards [eventually_ge_atTop m] with N hN
+      exact ⟨Sum.inr ⟨m,by omega⟩, by simp [frequency,positiveMode]⟩
+  | negSucc m =>
+    filter_upwards [eventually_ge_atTop (m+1)] with N hN
+    exact ⟨Sum.inl ⟨m+1,by omega⟩, by simp [frequency,negativeMode,Int.negSucc_eq]⟩
+
+ variable {κ : Type} [Fintype κ] [DecidableEq κ]
+
+ theorem exists_positive_periodic
+    (j₀ : κ) {H : Vector κ → ℝ} (hH : ContDiff ℝ ∞ H) (hH0 : ∀ z, 0 ≤ H z)
+    (d₀ d : κ → ℝ) {C M η γ B D : ℝ}
+    (hM : 0 ≤ M) (hη : 0 < η) (hγ : 0 < γ) (hB : 0 ≤ B) (hD : 0 ≤ D)
+    (hup : ∀ z, H z ≤ M*‖z‖^4)
+    (hd : ∀ j, η ≤ d₀ j) (hd0 : Real.pi+η ≤ d₀ j₀)
+    (hlo : ∀ z, energy d₀ z-C ≤ H z) (hdD : ∀ j, |d j| ≤ D)
+    (hgap : ∀ m : ℤ, ∀ j, γ ≤ |Real.pi*(m : ℝ)-d j|)
+    (hrem : ∀ z, ‖gradient H z-(2 : ℝ) • diag d z‖ ≤ B) :
+    ∃ x : C(Time,Vector κ), 0 < meanAction hH x ∧
+      ∀ t : ℝ, HasDerivAt (fun s : ℝ => x (s : Time))
+        (Complex.I • gradient H (x (t : Time))) t := by
+  have he (N : ℕ) := finite_critical (negativeMode N) (negativeMode_nonpos N)
+    (positiveMode N) (positiveMode_injective N) (positiveMode_pos N) (frequency_injective N)
+    (⟨0,Nat.succ_pos N⟩ : Fin (N+1)) rfl j₀ hH hH0 d₀ d hM hη hγ hB hup hd hd0 hlo
+    (fun i j => hgap (frequency N i) j) hrem
+  choose a ha hupper hz using he
+  obtain ⟨x,φ,hφ,ht⟩ := critical_subsequence frequency frequency_injective hH d hγ hB hD hdD
+    (fun n i j => hgap (frequency n i) j) hrem a hz
+  refine ⟨x,lt_of_lt_of_le (linkLevel_pos hM) (positive_action_limit frequency hH a hz ha φ x ht),?_⟩
+  exact hasDerivAt_of_critical_coefficients hH x
+    (limit_coefficient frequency frequency_injective frequency_exhausts hH a hz φ hφ x ht)
+
+end
+end HamiltonianPeriodic
+
+end NonsqueezingInline
+
+end OAI
diff --git a/lean/OAI/Geometry/PolarProducts/PlanarSquare.lean b/lean/OAI/Geometry/PolarProducts/PlanarSquare.lean
new file mode 100644
index 0000000000000000000000000000000000000000..180278fc3f71194d170495a7dfd927bb3a994653
--- /dev/null
+++ b/lean/OAI/Geometry/PolarProducts/PlanarSquare.lean
@@ -0,0 +1,234 @@
+import OAI.Geometry.PolarProducts.CanonicalCylinder
+
+namespace OAI
+
+universe u129 u130
+
+section
+namespace PlanarSquare
+noncomputable section
+open Set Filter
+open scoped ContDiff Topology
+
+def square : Set ℂ := {z | |z.re| < 1 ∧ |z.im| < 1}
+def radius (z : ℂ) : ℝ := Real.sqrt (2*(z.im+1)/Real.pi)
+def angle : ℂ →L[ℝ] ℂ := Complex.reCLM.smulRight (-(Real.pi : ℂ)*Complex.I)
+def rotation (z : ℂ) : ℂ := Complex.exp (angle z)
+def map (z : ℂ) : ℂ := (radius z : ℂ)*rotation z
+
+theorem radius_pos {z : ℂ} (hz : -1 < z.im) : 0 < radius z := by
+  apply Real.sqrt_pos.2
+  exact div_pos (by linarith) Real.pi_pos
+
+theorem radius_sq {z : ℂ} (hz : -1 < z.im) : radius z ^ 2 = 2*(z.im+1)/Real.pi :=
+  Real.sq_sqrt (div_nonneg (by linarith) Real.pi_pos.le)
+
+@[simp] theorem angle_re (z : ℂ) : (angle z).re = 0 := by simp [angle, Complex.mul_re, Complex.mul_im]
+@[simp] theorem angle_im (z : ℂ) : (angle z).im = -Real.pi*z.re := by
+  simp [angle, Complex.mul_re, Complex.mul_im, mul_comm]
+
+@[simp] theorem norm_rotation (z : ℂ) : ‖rotation z‖ = 1 := by simp [rotation, Complex.norm_exp]
+
+theorem norm_map_sq {z : ℂ} (hz : -1 < z.im) : ‖map z‖^2 = 2*(z.im+1)/Real.pi := by
+  rw [map, norm_mul, norm_rotation, mul_one, Complex.norm_real, Real.norm_eq_abs, sq_abs, radius_sq hz]
+
+theorem map_bound {z : ℂ} (hz : z ∈ square) : ‖map z‖^2 < 4/Real.pi := by
+  rw [norm_map_sq (abs_lt.mp hz.2).1]
+  exact (div_lt_div_iff_of_pos_right Real.pi_pos).2 (by linarith [(abs_lt.mp hz.2).2])
+
+theorem map_injOn : InjOn map square := by
+  intro z hz w hw hzw
+  have him : z.im = w.im := by
+    have hh := congrArg (fun t : ℂ => ‖t‖^2) hzw
+    rw [norm_map_sq (abs_lt.mp hz.2).1, norm_map_sq (abs_lt.mp hw.2).1] at hh
+    have := (div_left_inj' Real.pi_ne_zero).mp hh
+    linarith
+  have hrad : radius z = radius w := by simp only [radius, him]
+  have hrot : rotation z = rotation w := by
+    change (radius z : ℂ)*rotation z = (radius w : ℂ)*rotation w at hzw
+    rw [← hrad] at hzw
+    exact mul_left_cancel₀ (Complex.ofReal_ne_zero.mpr (radius_pos (abs_lt.mp hz.2).1).ne') hzw
+  have hab (x : ℂ) (hx : x ∈ square) : -Real.pi < (angle x).im ∧ (angle x).im ≤ Real.pi := by
+    rw [angle_im]
+    have ht := mul_lt_mul_of_pos_left (abs_lt.mp hx.1).2 Real.pi_pos
+    have hb := mul_lt_mul_of_pos_left (abs_lt.mp hx.1).1 Real.pi_pos
+    constructor <;> linarith
+  have hangle := Complex.exp_inj_of_neg_pi_lt_of_le_pi (hab z hz).1 (hab z hz).2
+    (hab w hw).1 (hab w hw).2 hrot
+  have hre : z.re = w.re := by
+    have hh := congrArg Complex.im hangle
+    simp only [angle_im] at hh
+    nlinarith [Real.pi_pos]
+  exact Complex.ext hre him
+
+theorem contDiffAt_map {z : ℂ} (hz : -1 < z.im) : ContDiffAt ℝ ∞ map z := by
+  have hrad : ContDiffAt ℝ ∞ radius z := by
+    apply ContDiffAt.sqrt
+    · exact (contDiffAt_const.mul (Complex.imCLM.contDiff.contDiffAt.add contDiffAt_const)).div_const Real.pi
+    · exact ne_of_gt (div_pos (by linarith) Real.pi_pos)
+  exact (Complex.ofRealCLM.contDiff.contDiffAt.comp z hrad).mul (angle.contDiff.contDiffAt.cexp)
+
+theorem hasFDerivAt_radius {z : ℂ} (hz : -1 < z.im) :
+    HasFDerivAt radius ((1/(Real.pi*radius z)) • Complex.imCLM) z := by
+  have hbase : HasFDerivAt (fun w : ℂ => 2*(w.im+1)/Real.pi)
+      ((2/Real.pi) • Complex.imCLM) z := by
+    have hf : (fun w : ℂ => 2*(w.im+1)/Real.pi) =
+        (fun w : ℂ => (2/Real.pi)*(Complex.imCLM w+1)) := by funext w; simp only [Complex.imCLM_apply]; ring
+    rw [hf]
+    exact (Complex.imCLM.hasFDerivAt.add_const 1).const_mul (2/Real.pi)
+  have hs := hbase.sqrt (ne_of_gt (div_pos (by linarith) Real.pi_pos))
+  rw [smul_smul] at hs
+  change HasFDerivAt radius (((1/(2*radius z))*(2/Real.pi)) • Complex.imCLM) z at hs
+  have hcoef : (1/(2*radius z))*(2/Real.pi) = 1/(Real.pi*radius z) := by
+    field_simp
+  rwa [hcoef] at hs
+
+def derivative (z : ℂ) : ℂ →L[ℝ] ℂ :=
+  (radius z : ℂ) • (rotation z • angle) +
+    rotation z • (Complex.ofRealCLM.comp ((1/(Real.pi*radius z)) • Complex.imCLM))
+
+theorem hasFDerivAt_map {z : ℂ} (hz : -1 < z.im) : HasFDerivAt map (derivative z) z := by
+  exact ((Complex.ofRealCLM.hasFDerivAt.comp z (hasFDerivAt_radius hz)).mul
+    angle.hasFDerivAt.cexp)
+
+theorem derivative_apply (z v : ℂ) :
+    derivative z v = rotation z * (((v.im/(Real.pi*radius z) : ℝ) : ℂ) -
+      Complex.I * ((Real.pi*radius z*v.re : ℝ) : ℂ)) := by
+  simp [derivative, angle, Complex.real_smul, smul_eq_mul, div_eq_mul_inv]
+  ring
+
+def form (v w : ℂ) : ℝ := v.re*w.im - w.re*v.im
+
+theorem form_mul (a v w : ℂ) : form (a*v) (a*w) = ‖a‖^2*form v w := by
+  simp only [form, Complex.mul_re, Complex.mul_im, Complex.sq_norm, Complex.normSq_apply]
+  ring
+
+theorem derivative_preserves {z : ℂ} (hz : -1 < z.im) (v w : ℂ) :
+    form (derivative z v) (derivative z w) = form v w := by
+  simp only [derivative_apply, form_mul, norm_rotation, one_pow, one_mul]
+  simp only [form, Complex.sub_re, Complex.sub_im, Complex.ofReal_re, Complex.ofReal_im,
+    Complex.mul_re, Complex.mul_im, Complex.I_re, Complex.I_im]
+  have hr := (radius_pos hz).ne'
+  field_simp
+  ring
+
+end
+end PlanarSquare
+end
+
+section
+namespace PlanarCylinder
+noncomputable section
+open Set Filter
+open scoped ContDiff Topology
+variable {κ : Type u129} [Fintype κ] [DecidableEq κ]
+abbrev V (κ : Type u130) := EuclideanSpace ℂ κ
+
+def domain (j₀ : κ) : Set (V κ) := {z | z j₀ ∈ PlanarSquare.square}
+def map (j₀ : κ) (z : V κ) : V κ :=
+  (EuclideanSpace.equiv κ ℂ).symm (fun j => if j = j₀ then PlanarSquare.map (z j) else z j)
+
+omit [Fintype κ] in
+@[simp] theorem map_apply (j₀ j : κ) (z : V κ) :
+    map j₀ z j = if j = j₀ then PlanarSquare.map (z j) else z j := rfl
+
+omit [Fintype κ] in
+@[simp] theorem map_first (j₀ : κ) (z : V κ) : map j₀ z j₀ = PlanarSquare.map (z j₀) := by simp
+
+omit [Fintype κ] in
+theorem map_injOn (j₀ : κ) : InjOn (map j₀) (domain j₀) := by
+  intro z hz w hw hzw
+  ext j
+  have hh := congrArg (fun v : V κ => v j) hzw
+  by_cases hj : j = j₀
+  · subst j
+    apply PlanarSquare.map_injOn hz hw
+    simpa only [map_first] using hh
+  · simpa only [map_apply, ite_eq_right hj] using hh
+
+def derivative (j₀ : κ) (z : V κ) : V κ →L[ℝ] V κ :=
+  (PiLp.continuousLinearEquiv 2 ℝ (fun _ : κ => ℂ)).symm.toContinuousLinearMap.comp
+    (ContinuousLinearMap.pi (fun j => if j = j₀ then
+      (PlanarSquare.derivative (z j)).comp (PiLp.proj 2 (fun _ : κ => ℂ) j)
+      else (PiLp.proj 2 (fun _ : κ => ℂ) j)))
+
+omit [Fintype κ] in
+@[simp] theorem derivative_apply (j₀ j : κ) (z v : V κ) :
+    derivative j₀ z v j = if j = j₀ then PlanarSquare.derivative (z j) (v j) else v j := by
+  change (if j = j₀ then (PlanarSquare.derivative (z j)).comp (PiLp.proj 2 (fun _ : κ => ℂ) j)
+    else (PiLp.proj 2 (fun _ : κ => ℂ) j)) v = _
+  split_ifs <;> rfl
+
+theorem hasFDerivAt_map (j₀ : κ) {z : V κ} (hz : -1 < (z j₀).im) :
+    HasFDerivAt (map j₀) (derivative j₀ z) z := by
+  apply (PiLp.hasFDerivAt_toLp (𝕜 := ℝ) 2 _).comp
+  apply hasFDerivAt_pi.mpr
+  intro j
+  by_cases hj : j = j₀
+  · subst j
+    simp only [ite_true]
+    exact (PlanarSquare.hasFDerivAt_map hz).comp z (PiLp.hasFDerivAt_apply (𝕜 := ℝ) 2 z j₀)
+  · simp only [ite_eq_right hj]
+    exact PiLp.hasFDerivAt_apply (𝕜 := ℝ) 2 z j
+
+theorem contDiffAt_map (j₀ : κ) {z : V κ} (hz : -1 < (z j₀).im) :
+    ContDiffAt ℝ ∞ (map j₀) z := by
+  apply (contDiffAt_piLp 2).mpr
+  intro j
+  by_cases hj : j = j₀
+  · subst j
+    simp only [map_first]
+    exact (PlanarSquare.contDiffAt_map hz).comp z
+      (show ContDiffAt ℝ ∞ (fun w : V κ => w j₀) z from contDiffAt_piLp_apply 2)
+  · simp only [map_apply, ite_eq_right hj]
+    exact contDiffAt_piLp_apply 2
+
+def form (v w : V κ) : ℝ := ∑ j, PlanarSquare.form (v j) (w j)
+
+omit [DecidableEq κ] in
+theorem form_eq_inner (v w : V κ) : form v w = inner (𝕜 := ℝ) (Complex.I • v) w := by
+  simp only [form, PiLp.inner_apply, PiLp.smul_apply, smul_eq_mul]
+  apply Finset.sum_congr rfl
+  intro j _
+  rw [real_inner_eq_re_inner, RCLike.inner_apply']
+  change PlanarSquare.form (v j) (w j) = ((starRingEnd ℂ) (Complex.I * v j) * w j).re
+  simp only [PlanarSquare.form, Complex.mul_re, Complex.mul_im, Complex.I_re, Complex.I_im,
+    Complex.conj_re, Complex.conj_im]
+  ring
+
+theorem derivative_preserves (j₀ : κ) {z : V κ} (hz : -1 < (z j₀).im) (v w : V κ) :
+    form (derivative j₀ z v) (derivative j₀ z w) = form v w := by
+  unfold form
+  apply Finset.sum_congr rfl
+  intro j _
+  by_cases hj : j = j₀
+  · subst j
+    simp only [derivative_apply, ite_true]
+    exact PlanarSquare.derivative_preserves hz _ _
+  · simp only [derivative_apply, ite_eq_right hj]
+
+omit [Fintype κ] in
+theorem first_bound (j₀ : κ) {z : V κ} (hz : z ∈ domain j₀) :
+    ‖map j₀ z j₀‖^2 < 4/Real.pi := by
+  rw [map_first]
+  exact PlanarSquare.map_bound hz
+
+theorem norm_bound (j₀ : κ) {z : V κ} (hz : z ∈ domain j₀) :
+    ‖map j₀ z‖^2 ≤ ‖z‖^2 + 4/Real.pi := by
+  rw [EuclideanSpace.norm_sq_eq, EuclideanSpace.norm_sq_eq]
+  calc
+    _ ≤ ∑ j, (‖z j‖^2 + if j = j₀ then 4/Real.pi else 0) := by
+      apply Finset.sum_le_sum
+      intro j _
+      by_cases hj : j = j₀
+      · subst j
+        simp only [ite_true]
+        exact (first_bound j₀ hz).le.trans (le_add_of_nonneg_left (sq_nonneg _))
+      · simp [map_apply, ite_eq_right hj]
+    _ = _ := by simp [Finset.sum_add_distrib]
+
+end
+end PlanarCylinder
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PolarProducts/PolarApproximation.lean b/lean/OAI/Geometry/PolarProducts/PolarApproximation.lean
new file mode 100644
index 0000000000000000000000000000000000000000..3fa05f7cfd645265820ee3ee61c3032f887292ed
--- /dev/null
+++ b/lean/OAI/Geometry/PolarProducts/PolarApproximation.lean
@@ -0,0 +1,341 @@
+import OAI.Geometry.PolarProducts.MomentumBounds
+
+namespace OAI
+
+universe u74 u75 u76 u77
+
+section LowerBoundInline
+open Set Filter Function
+open scoped Topology ContDiff NNReal
+open Set Filter Metric
+open scoped Topology ContDiff
+open Set Filter Function MeasureTheory Metric
+open scoped Topology ContDiff NNReal
+open Set Filter Function
+open scoped Topology ContDiff
+open Set Filter Function
+open scoped Topology ContDiff NNReal
+open Set Filter
+open scoped Topology ContDiff
+open Set Filter Function
+open scoped Topology ContDiff
+open Set Filter Function
+open scoped ContDiff Topology
+open Set MeasureTheory
+open scoped ContDiff Interval Topology
+open Set
+open scoped Topology ContDiff
+open Set
+open Set MeasureTheory
+open scoped ContDiff Interval Topology
+open Set Filter Complex
+open scoped Topology ContDiff
+open MeasureTheory intervalIntegral Set
+open scoped Real
+open Set
+open scoped ContDiff ENNReal
+
+namespace SymmetricPolar
+open Set Filter ComplexCoordinates ComplexPotential PolarStrips
+open scoped Topology ContDiff
+noncomputable section
+variable {n : ℕ} {κ : Type u74} [Fintype κ]
+
+theorem exists_positive_slack {c d : ℝ} (hc : 0 < c) (hcd : c < d) :
+    ∃ η : ℝ, 0 < η ∧ (1+η)*c < d := by
+  let η := (d-c)/(2*c)
+  have hη : 0 < η := div_pos (sub_pos.mpr hcd) (mul_pos (by norm_num) hc)
+  have he : η*c = (d-c)/2 := by dsimp [η]; field_simp
+  exact ⟨η, hη, by nlinarith⟩
+
+theorem finite_strips_embedding (b : κ → Position n)
+    (hb : Function.Injective (fun x : Position n => fun j => inner (𝕜 := ℝ) (b j) x))
+    {c : ℝ} (hc : 0 < c) (hc4 : c < 4) :
+    HasSymplecticEmbedding (capacityBall n c)
+      (interior (ConvexPolar.stripBody b) ×ˢ interior (ConvexPolar.polar (ConvexPolar.stripBody b))) := by
+  let : InnerProductSpace ℝ (C (Fin n)) := InnerProductSpace.complexToReal
+  obtain ⟨η, hη, hηc⟩ := exists_positive_slack hc hc4
+  obtain ⟨k, hk, hscaled⟩ := exists_scaled_image b hb hη
+  let ℓ := fun j => functional (b j)
+  have hℓ : Function.Injective (fun z : C (Fin n) => fun j => ℓ j z) := functional_injective b hb
+  have hkpos : (0 : ℝ) < k := by exact_mod_cast (Nat.zero_lt_of_lt hk)
+  let S := (1+η)*Real.pi*(k : ℝ)/4
+  have hS : 0 < S := by dsimp [S]; positivity
+  let a := S*c/Real.pi
+  have ha : 0 < a := div_pos (mul_pos hS hc) Real.pi_pos
+  have hak : a < (k : ℝ) := by
+    apply (div_lt_iff₀ Real.pi_pos).mpr
+    have hh := mul_lt_mul_of_pos_right hηc (mul_pos Real.pi_pos hkpos)
+    dsimp [S]
+    nlinarith
+  obtain ⟨Q, hQ, _hQi, hQm, hQω⟩ :=
+    exists_ball_from_holomorphic_zero (isOpen_domain ℓ) (zero_mem_domain ℓ)
+      (analyticOnNhd_tuple ℓ k) (fun x hx => tuple_zero_iff ℓ hℓ hk hx) hk (tuple_order ℓ k)
+      (fun s hs hs1 => compact_sublevel ℓ hℓ hk hs hs1) ha hak
+  let i : Phase n →L[ℝ] C (Fin n) := parts.symm.toContinuousLinearMap
+  have hi (z : Phase n) (hz : z ∈ capacityBall n (S*c)) : ‖i z‖^2 < a := by
+    apply (lt_div_iff₀ Real.pi_pos).mpr
+    have he : ‖i z‖^2 = ‖z.1‖^2+‖z.2‖^2 := by
+      have hh := norm_sq_parts (parts.symm z)
+      have hre : re (parts.symm z) = z.1 := congrArg Prod.fst (parts.apply_symm_apply z)
+      have him : im (parts.symm z) = z.2 := congrArg Prod.snd (parts.apply_symm_apply z)
+      simpa only [i, ContinuousLinearEquiv.coe_coe, hre, him] using hh
+    rw [he, mul_comm]
+    exact hz
+  let e : Phase n → Phase n := fun z => phase b k (Q (i z))
+  have hs (z : Phase n) (hz : z ∈ capacityBall n (S*c)) : ContDiffAt ℝ ∞ e z := by
+    have hphase : ContDiffAt ℝ ∞ (phase b k) (Q (i z)) :=
+      contDiffAt_phase b k (hQm _ (hi z hz)).1
+    have hh := hphase.comp z (hQ.contDiffAt.comp z i.contDiff.contDiffAt)
+    simpa only [Function.comp_def, e] using hh
+  apply embedding_of_scaled_ball_map hS (a := 1) (b := S⁻¹) (by simp) (e := e)
+  · exact fun z hz => (hs z hz).contDiffWithinAt
+  · intro z hz w hw he
+    have hq : Q (i z) = Q (i w) := injOn_phase b k (hQm _ (hi z hz)).1 (hQm _ (hi w hw)).1 he
+    exact parts.symm.injective (Q.injective hq)
+  · intro z hz
+    have hm := hscaled (Q (i z)) (hQm _ (hi z hz)).1 (hQm _ (hi z hz)).2
+    simpa only [diagScale_apply, one_smul, e, S, Set.mem_prod] using hm
+  · intro z hz v w
+    have hphase : HasFDerivAt (phase b k) (fderiv ℝ (phase b k) (Q (i z))) (Q (i z)) :=
+      ((contDiffAt_phase b k (hQm _ (hi z hz)).1).differentiableAt (by simp)).hasFDerivAt
+    have hd : HasFDerivAt e ((fderiv ℝ (phase b k) (Q (i z))).comp
+        ((fderiv ℝ (Q : C (Fin n) → C (Fin n)) (i z)).comp i)) z := by
+      have hh := hphase.comp z ((hQ.differentiable (by simp) (i z)).hasFDerivAt.comp z i.hasFDerivAt)
+      simpa only [Function.comp_def, e] using hh
+    rw [hd.fderiv]
+    change phaseForm (fderiv ℝ (phase b k) (Q (i z)) (fderiv ℝ (Q : C (Fin n) → C (Fin n)) (i z) (i v)))
+      (fderiv ℝ (phase b k) (Q (i z)) (fderiv ℝ (Q : C (Fin n) → C (Fin n)) (i z) (i w))) = _
+    rw [phase_pullback b k (hQm _ (hi z hz)).1, hQω _ (hi z hz), standardTensor_parts]
+    change phaseForm (parts (parts.symm v)) (parts (parts.symm w)) = _
+    rw [parts.apply_symm_apply, parts.apply_symm_apply]
+    rfl
+
+end
+end SymmetricPolar
+
+namespace ConvexPolar
+open Set Filter
+open scoped Topology
+noncomputable section
+variable {E : Type u75} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+
+theorem zero_mem_interior_of_symmetric {K : Set E} (hK : Convex ℝ K)
+    (hi : (interior K).Nonempty) (hs : ∀ x, x ∈ K ↔ -x ∈ K) : (0 : E) ∈ interior K := by
+  obtain ⟨x, hx⟩ := hi
+  have hn : -x ∈ interior K := by
+    apply mem_interior_iff_mem_nhds.mpr
+    have hh : K ∈ 𝓝 (-(-x)) := by simpa using mem_interior_iff_mem_nhds.mp hx
+    have hh' := continuous_neg.continuousAt.preimage_mem_nhds hh
+    have he : (fun y : E => -y) ⁻¹' K = K := by ext y; exact (hs y).symm
+    rwa [he] at hh'
+  have hm := hK.interior hx hn (by norm_num : (0 : ℝ) ≤ 1/2) (by norm_num : (0 : ℝ) ≤ 1/2)
+    (by norm_num : (1/2 : ℝ)+1/2 = 1)
+  simpa using hm
+
+theorem isClosed_polar (K : Set E) : IsClosed (polar K) := by
+  have he : polar K = ⋂ q ∈ K, {p : E | inner (𝕜 := ℝ) q p ≤ 1} := by ext p; simp [polar]
+  rw [he]
+  exact isClosed_biInter (fun q _ => isClosed_le (by fun_prop) continuous_const)
+
+theorem polar_abs_pairing {K : Set E} (hs : ∀ x, x ∈ K ↔ -x ∈ K)
+    {p : E} (hp : p ∈ polar K) {x : E} (hx : x ∈ K) : |inner (𝕜 := ℝ) p x| ≤ 1 := by
+  apply abs_le.mpr
+  have hn := hp (-x) ((hs x).mp hx)
+  rw [inner_neg_left, real_inner_comm] at hn
+  exact ⟨by linarith, by rw [real_inner_comm]; exact hp x hx⟩
+
+theorem norm_smul_normalized {x : E} (hx : x ≠ 0) {r : ℝ} (hr : 0 ≤ r) :
+    ‖(r/‖x‖) • x‖ = r := by
+  rw [norm_smul, Real.norm_eq_abs, abs_of_nonneg (div_nonneg hr (norm_nonneg _)), div_mul_cancel₀ _ (norm_ne_zero_iff.mpr hx)]
+
+theorem normalized_inner {x : E} (hx : x ≠ 0) (r : ℝ) :
+    inner (𝕜 := ℝ) ((r/‖x‖) • x) x = r*‖x‖ := by
+  rw [real_inner_smul_left, real_inner_self_eq_norm_sq]
+  field_simp
+
+theorem norm_le_of_mem_polar {K : Set E} {r : ℝ} (hr : 0 < r)
+    (hball : Metric.closedBall (0 : E) r ⊆ K) {p : E} (hp : p ∈ polar K) : ‖p‖ ≤ r⁻¹ := by
+  by_cases hp0 : p = 0
+  · simp [hp0, hr.le]
+  have hq : (r/‖p‖) • p ∈ K := hball (by simpa using (norm_smul_normalized hp0 hr.le).le)
+  have hh := hp _ hq
+  rw [normalized_inner hp0] at hh
+  have hh' : ‖p‖ ≤ r⁻¹ * (1 : ℝ) := (le_inv_mul_iff₀ hr).mpr hh
+  simpa only [mul_one] using hh'
+
+theorem isCompact_polar [FiniteDimensional ℝ E] {K : Set E} (h0 : (0 : E) ∈ interior K) : IsCompact (polar K) := by
+  obtain ⟨r, hr, hball⟩ := Metric.mem_nhds_iff.mp (mem_interior_iff_mem_nhds.mp h0)
+  have hr2 : 0 < r/2 := half_pos hr
+  have hclosed : Metric.closedBall (0 : E) (r/2) ⊆ K := by
+    intro x hx
+    apply hball
+    exact (Metric.mem_closedBall.mp hx).trans_lt (half_lt_self hr)
+  apply (isCompact_closedBall (0 : E) ((r/2)⁻¹)).of_isClosed_subset (isClosed_polar K)
+  intro p hp
+  simpa using norm_le_of_mem_polar hr2 hclosed hp
+
+theorem mem_polar_polar_iff [FiniteDimensional ℝ E] {K : Set E} (hK : Convex ℝ K) (hc : IsClosed K)
+    (h0 : (0 : E) ∈ K) (x : E) : x ∈ polar (polar K) ↔ x ∈ K := by
+  constructor
+  · intro hx
+    by_contra hn
+    obtain ⟨f, u, hfu, hux⟩ := geometric_hahn_banach_closed_point hK hc hn
+    have hu : 0 < u := by simpa using hfu 0 h0
+    let v : E := (InnerProductSpace.toDual ℝ E).symm f
+    have hv (y : E) : inner (𝕜 := ℝ) y v = f y := by
+      rw [real_inner_comm]
+      exact InnerProductSpace.toDual_symm_apply
+    have hp : u⁻¹ • v ∈ polar K := by
+      intro y hy
+      rw [real_inner_smul_right, hv, ← div_eq_inv_mul]
+      exact (div_le_one hu).mpr (hfu y hy).le
+    have hh := hx _ hp
+    rw [real_inner_smul_left, real_inner_comm, hv, ← div_eq_inv_mul] at hh
+    exact (not_le_of_gt hux) ((div_le_one hu).mp hh)
+  · intro hx p hp
+    rw [real_inner_comm]
+    exact hp x hx
+
+theorem closedBall_subset_polar_of_norm_bound {K : Set E} {B : ℝ} (hB : 0 ≤ B)
+    (hbound : ∀ x ∈ K, ‖x‖ ≤ B) : Metric.closedBall (0 : E) ((B+1)⁻¹) ⊆ polar K := by
+  intro p hp x hx
+  have hp' : ‖p‖ ≤ (B+1)⁻¹ := by simpa using hp
+  have hi := real_inner_le_norm x p
+  have hm := mul_le_mul (hbound x hx) hp' (norm_nonneg _) hB
+  have hr : 0 < B+1 := by linarith
+  have hb : B*(B+1)⁻¹ ≤ 1 := (div_le_one hr).mpr (by linarith)
+  exact hi.trans (hm.trans hb)
+
+end
+end ConvexPolar
+
+namespace ConvexPolar
+open Set Filter
+open scoped Topology
+noncomputable section
+variable {E : Type u76} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+variable {κ : Type u77}
+
+theorem pairing_le_of_net {P : Set E} {b : κ → E} {δ M : ℝ}
+    (hnet : ∀ p ∈ P, ∃ j, ‖p-b j‖ ≤ δ) {x : E}
+    (hx : ∀ j, inner (𝕜 := ℝ) (b j) x ≤ M) {p : E} (hp : p ∈ P) :
+    inner (𝕜 := ℝ) p x ≤ M+δ*‖x‖ := by
+  obtain ⟨j, hj⟩ := hnet p hp
+  have hh := real_inner_le_norm (p-b j) x
+  rw [inner_sub_left] at hh
+  have hm := mul_le_mul_of_nonneg_right hj (norm_nonneg x)
+  linarith [hx j]
+
+theorem net_norm_bound {P : Set E} {b : κ → E} {δ M r : ℝ}
+    (hr : 0 ≤ r) (hM : 0 ≤ M) (hball : Metric.closedBall (0 : E) r ⊆ P)
+    (hnet : ∀ p ∈ P, ∃ j, ‖p-b j‖ ≤ δ) {x : E}
+    (hx : ∀ j, inner (𝕜 := ℝ) (b j) x ≤ M) : (r-δ)*‖x‖ ≤ M := by
+  by_cases hx0 : x = 0
+  · simpa [hx0] using hM
+  have hp : (r/‖x‖) • x ∈ P := hball (by simpa using (norm_smul_normalized hx0 hr).le)
+  have hh := pairing_le_of_net hnet hx hp
+  rw [normalized_inner hx0] at hh
+  nlinarith
+
+theorem finite_polar_approximation [FiniteDimensional ℝ E] {K : Set E}
+    (hc : IsCompact K) (hK : Convex ℝ K) (hi : (interior K).Nonempty)
+    (hs : ∀ x, x ∈ K ↔ -x ∈ K) {α : ℝ} (hα : 0 < α) :
+    ∃ t : Finset E,
+      Function.Injective (fun x : E => fun j : t => inner (𝕜 := ℝ) (j : E) x) ∧
+      K ⊆ stripBody (fun j : t => (j : E)) ∧
+      (∀ x ∈ stripBody (fun j : t => (j : E)), (1+α)⁻¹ • x ∈ K) ∧
+      polar (stripBody (fun j : t => (j : E))) ⊆ polar K := by
+  classical
+  have h0 := zero_mem_interior_of_symmetric hK hi hs
+  obtain ⟨B, hB, hbound⟩ := hc.isBounded.exists_pos_norm_le
+  let r := (B+1)⁻¹
+  have hr : 0 < r := inv_pos.mpr (by linarith)
+  let δ := α*r/(1+α)
+  have hlam : 0 < 1+α := by linarith
+  have hδ : 0 < δ := div_pos (mul_pos hα hr) hlam
+  have hrδ : 0 < r-δ := by
+    apply sub_pos.mpr
+    apply (div_lt_iff₀ hlam).mpr
+    nlinarith
+  have hδeq : δ = α*(r-δ) := by dsimp [δ]; field_simp; ring
+  have hball : Metric.closedBall (0 : E) r ⊆ polar K :=
+    closedBall_subset_polar_of_norm_bound hB.le hbound
+  obtain ⟨t, htK, ht, hcover⟩ := (isCompact_polar h0).finite_cover_balls hδ
+  let b : ht.toFinset → E := fun j => j
+  have hb (j : ht.toFinset) : b j ∈ polar K := htK (by simpa only [b, Set.Finite.mem_toFinset] using j.2)
+  have hnet (p : E) (hp : p ∈ polar K) : ∃ j : ht.toFinset, ‖p-b j‖ ≤ δ := by
+    obtain ⟨q, hq⟩ := mem_iUnion.mp (hcover hp)
+    obtain ⟨hqmem, hdist⟩ := mem_iUnion.mp hq
+    refine ⟨⟨q, by simpa using hqmem⟩, ?_⟩
+    simpa only [dist_eq_norm, b] using (Metric.mem_ball.mp hdist).le
+  have hKK' : K ⊆ stripBody b := by
+    intro x hx j
+    exact polar_abs_pairing hs (hb j) hx
+  have hspans : Function.Injective (fun x : E => fun j => inner (𝕜 := ℝ) (b j) x) := by
+    intro x y hxy
+    have hz (j) : inner (𝕜 := ℝ) (b j) (x-y) ≤ 0 := by
+      rw [inner_sub_right]
+      have hh : inner (𝕜 := ℝ) (b j) x = inner (𝕜 := ℝ) (b j) y := congrFun hxy j
+      rw [hh, sub_self]
+    have hh := net_norm_bound hr.le le_rfl hball hnet hz
+    have he : ‖x-y‖ = 0 := by nlinarith [norm_nonneg (x-y)]
+    exact sub_eq_zero.mp (norm_eq_zero.mp he)
+  refine ⟨ht.toFinset, hspans, hKK', ?_, polar_antitone hKK'⟩
+  intro x hx
+  apply (mem_polar_polar_iff hK hc.isClosed (interior_subset h0) _).mp
+  have hx' (j) : inner (𝕜 := ℝ) (b j) x ≤ 1 := (le_abs_self _).trans (hx j)
+  have hh := net_norm_bound hr.le (by norm_num : (0 : ℝ) ≤ 1) hball hnet hx'
+  have hd : δ*‖x‖ ≤ α := by nlinarith
+  intro p hp
+  rw [real_inner_smul_right, ← div_eq_inv_mul]
+  apply (div_le_one hlam).mpr
+  have hh' := pairing_le_of_net hnet hx' hp
+  linarith
+
+end
+end ConvexPolar
+
+namespace SymmetricPolar
+open Set Filter
+open scoped Topology ContDiff Pointwise
+noncomputable section
+variable {n : ℕ}
+
+theorem symmetric_polar_embedding {K : Set (Position n)} (hK : IsSymmetricConvexBody K)
+    {c : ℝ} (hc : 0 < c) (hc4 : c < 4) :
+    HasSymplecticEmbedding (capacityBall n c) (polarProduct K) := by
+  obtain ⟨α, hα, hαc⟩ := exists_positive_slack hc hc4
+  let lam := 1+α
+  have hlam : 0 < lam := by dsimp [lam]; linarith
+  rcases hK with ⟨hcompact, hconvex, hinterior, hsymm⟩
+  obtain ⟨t, htinj, hKK', hK'K, hpolar⟩ :=
+    ConvexPolar.finite_polar_approximation hcompact hconvex hinterior hsymm hα
+  let b : t → Position n := fun j => j
+  have hsubset : ConvexPolar.stripBody b ⊆ lam • K := by
+    intro x hx
+    refine ⟨lam⁻¹ • x, hK'K x hx, ?_⟩
+    change lam • (lam⁻¹ • x) = x
+    rw [smul_smul, mul_inv_cancel₀ hlam.ne', one_smul]
+  have hposition {q : Position n} (hq : q ∈ interior (ConvexPolar.stripBody b)) :
+      lam⁻¹ • q ∈ interior K := by
+    have hh := interior_mono hsubset hq
+    rw [interior_smul₀ hlam.ne'] at hh
+    obtain ⟨w, hw, rfl⟩ := hh
+    simpa only [smul_smul, inv_mul_cancel₀ hlam.ne', one_smul] using hw
+  obtain ⟨e, he, hi, hm, hω⟩ := finite_strips_embedding b htinj (mul_pos hlam hc) hαc
+  apply embedding_of_scaled_ball_map hlam (a := lam⁻¹) (b := 1) (by simp) he
+    (HasSymplecticEmbedding.injOn hi) ?_ hω
+  intro z hz
+  obtain ⟨hq, hp⟩ := hm hz
+  change lam⁻¹ • (e z).1 ∈ interior K ∧ (1 : ℝ) • (e z).2 ∈ interior (polar K)
+  refine ⟨hposition hq, ?_⟩
+  change (1 : ℝ) • (e z).2 ∈ interior (ConvexPolar.polar K)
+  simpa only [one_smul] using interior_mono hpolar hp
+
+end
+end SymmetricPolar
+
+end LowerBoundInline
+
+end OAI
diff --git a/lean/OAI/Geometry/PolarProducts/PotentialDeformation.lean b/lean/OAI/Geometry/PolarProducts/PotentialDeformation.lean
new file mode 100644
index 0000000000000000000000000000000000000000..a2599cf68b9ff948b4baa4c0053ae44aa20d0a2b
--- /dev/null
+++ b/lean/OAI/Geometry/PolarProducts/PotentialDeformation.lean
@@ -0,0 +1,349 @@
+import OAI.Geometry.PolarProducts.PotentialForms
+
+namespace OAI
+
+universe u30 u31 u32 u33 u34 u35
+
+section LowerBoundInline
+open Set Filter Function
+open scoped Topology ContDiff NNReal
+open Set Filter Metric
+open scoped Topology ContDiff
+open Set Filter Function MeasureTheory Metric
+open scoped Topology ContDiff NNReal
+open Set Filter Function
+open scoped Topology ContDiff
+open Set Filter Function
+open scoped Topology ContDiff NNReal
+open Set Filter
+open scoped Topology ContDiff
+open Set Filter Function
+open scoped Topology ContDiff
+
+open Set Filter Function
+open scoped ContDiff Topology
+
+namespace CompactExtension
+noncomputable section
+
+variable {E : Type u30} {F : Type u31} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  [NormedAddCommGroup F] [NormedSpace ℝ F]
+
+def extend (U : Set E) (f : E → F) : E → F := by
+  classical
+  exact U.indicator f
+
+omit [NormedSpace ℝ E] [NormedSpace ℝ F] in
+theorem eventuallyEq_extend {U : Set E} (hU : IsOpen U) (f : E → F) {x : E} (hx : x ∈ U) :
+    extend U f =ᶠ[𝓝 x] f := by
+  classical
+  filter_upwards [hU.mem_nhds hx] with y hy
+  exact indicator_of_mem hy f
+
+omit [NormedSpace ℝ E] [NormedSpace ℝ F] in
+theorem tsupport_extend_subset {U C : Set E} (hC : IsClosed C) {f : E → F}
+    (hf : ∀ x ∈ U, x ∉ C → f x = 0) : tsupport (extend U f) ⊆ C := by
+  classical
+  apply closure_minimal _ hC
+  intro x hx
+  by_contra hn
+  have hh : extend U f x = 0 := by
+    by_cases hU : x ∈ U
+    · exact (indicator_of_mem hU f).trans (hf x hU hn)
+    · exact indicator_of_notMem hU f
+  exact hx hh
+
+theorem contDiff_extend {U C : Set E} (hU : IsOpen U) (hC : IsClosed C) (hCU : C ⊆ U)
+    {f : E → F} (hf : ∀ x ∈ U, ContDiffAt ℝ ∞ f x)
+    (hzero : ∀ x ∈ U, x ∉ C → f x = 0) : ContDiff ℝ ∞ (extend U f) := by
+  rw [contDiff_iff_contDiffAt]
+  intro x
+  by_cases hx : x ∈ U
+  · exact (hf x hx).congr_of_eventuallyEq (eventuallyEq_extend hU f hx)
+  · have hxc : x ∉ C := fun hc => hx (hCU hc)
+    apply contDiffAt_const.congr_of_eventuallyEq
+    filter_upwards [hC.isOpen_compl.mem_nhds hxc] with y hy
+    exact image_eq_zero_of_notMem_tsupport (fun ht => hy (tsupport_extend_subset hC hzero ht))
+
+omit [NormedSpace ℝ E] [NormedSpace ℝ F] in
+theorem hasCompactSupport_extend {U C : Set E} (hC : IsCompact C) {f : E → F}
+    (hzero : ∀ x ∈ U, x ∉ C → f x = 0) : HasCompactSupport (extend U f) :=
+  hC.of_isClosed_subset (isClosed_tsupport _) (tsupport_extend_subset hC.isClosed hzero)
+
+end
+end CompactExtension
+
+namespace ComplexPotential
+
+open TensorCalculus Set Filter Function
+open scoped ContDiff Topology
+
+noncomputable section
+
+variable {E : Type u32} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E]
+
+local instance moserOneGroup : NormedAddCommGroup (E →L[ℝ] ℝ) := inferInstance
+local instance moserOneSpace : NormedSpace ℝ (E →L[ℝ] ℝ) := inferInstance
+local instance moserTwoGroup : NormedAddCommGroup (TwoTensor E) := inferInstance
+local instance moserTwoSpace : NormedSpace ℝ (TwoTensor E) := inferInstance
+
+omit [FiniteDimensional ℝ E] in
+theorem tsupport_dc_subset (J : E →L[ℝ] E) (φ : E → ℝ) :
+    tsupport (dc J φ) ⊆ tsupport φ :=
+  (tsupport_comp_subset (map_zero (dcOp J)) (fderiv ℝ φ)).trans (tsupport_fderiv_subset ℝ)
+
+omit [FiniteDimensional ℝ E] in
+theorem fderiv_homeomorph_apply_symm (F : E ≃ₜ E)
+    (hF : ContDiff ℝ ∞ (F : E → E)) (hFi : ContDiff ℝ ∞ (F.symm : E → E)) (x a : E) :
+    fderiv ℝ (F : E → E) (F.symm x) (fderiv ℝ (F.symm : E → E) x a) = a := by
+  have he : (F : E → E) ∘ (F.symm : E → E) = id := funext F.apply_symm_apply
+  have hd := fderiv_comp x (hF.differentiable (by simp) (F.symm x))
+    (hFi.differentiable (by simp) x)
+  rw [he, fderiv_id] at hd
+  exact (congrArg (fun L : E →L[ℝ] E => L a) hd).symm
+
+theorem exists_potential_transport (J : E →L[ℝ] E)
+    (hsk : ∀ a b, inner (𝕜 := ℝ) a (J b) = -inner (𝕜 := ℝ) (J a) b)
+    (hJ : ∀ a, ‖J a‖ = ‖a‖)
+    {U C : Set E} (hU : IsOpen U) (hC : IsCompact C) (hCU : C ⊆ U)
+    {τ T : E → ℝ} (hτ : ∀ x ∈ U, IsPSHAt J τ x) (hT : ∀ x ∈ U, IsPSHAt J T x)
+    (heq : ∀ x ∈ U, x ∉ C → T x = τ x) :
+    ∃ F : E ≃ₜ E, ContDiff ℝ ∞ (F : E → E) ∧ ContDiff ℝ ∞ (F.symm : E → E) ∧
+      (∀ x, x ∉ U → F x = x) ∧ (∀ x, F x ∈ U ↔ x ∈ U) ∧
+      ∀ x ∈ U, ∀ a b,
+        ddc J (fun y => ‖y‖^2 + τ y) (F x) (fderiv ℝ (F : E → E) x a)
+          (fderiv ℝ (F : E → E) x b) = ddc J (fun y => ‖y‖^2 + T y) x a b := by
+  let g := CompactExtension.extend U (fun y => T y - τ y)
+  have hz : ∀ x ∈ U, x ∉ C → T x - τ x = 0 := by
+    intro x hx hxc
+    rw [heq x hx hxc, sub_self]
+  have hg : ContDiff ℝ ∞ g := CompactExtension.contDiff_extend hU hC.isClosed hCU
+    (fun x hx => (hT x hx).1.sub (hτ x hx).1) hz
+  have hgs : tsupport g ⊆ C := CompactExtension.tsupport_extend_subset hC.isClosed hz
+  let γ := dc J g
+  let W := ddc J (fun y => ‖y‖^2 + τ y)
+  have hγ : ContDiff ℝ ∞ γ := contDiff_dc J hg
+  have hsγ : tsupport γ ⊆ C := (tsupport_dc_subset J g).trans hgs
+  have hcγ : HasCompactSupport γ := hC.of_isClosed_subset (isClosed_tsupport _) hsγ
+  have hdiff (x : E) (hx : x ∈ U) :
+      exteriorD γ x = ddc J T x - ddc J τ x := by
+    change ddc J g x = _
+    rw [ddc_congr J (CompactExtension.eventuallyEq_extend hU _ hx)]
+    exact ddc_sub J (hT x hx).1 (hτ x hx).1
+  have hnorm (x : E) (hx : x ∈ U) : W x = standardTensor J + ddc J τ x := by
+    dsimp [W]
+    rw [ddc_add J (contDiff_norm_sq ℝ).contDiffAt (hτ x hx).1, ddc_norm_sq J hsk]
+  have hW : ∀ x ∈ U, ContDiffAt ℝ ∞ W x := fun x hx =>
+    contDiffAt_ddc J ((contDiff_norm_sq ℝ).contDiffAt.add (hτ x hx).1)
+  have hWi : ∀ x ∈ U, ∀ s ∈ Icc (0 : ℝ) 1, (W x + s • exteriorD γ x).IsInvertible := by
+    intro x hx s hs
+    apply isInvertible_of_positive_complex_line _ J
+    intro a ha
+    rw [hnorm x hx, hdiff x hx]
+    simp only [add_apply, smul_apply, sub_apply, smul_eq_mul,
+      standardTensor_complex_line J hJ]
+    have hn : 0 < ‖a‖^2 := sq_pos_of_pos (norm_pos_iff.mpr ha)
+    have h₀ := mul_nonneg (sub_nonneg.mpr hs.2) ((hτ x hx).2 a)
+    have h₁ := mul_nonneg hs.1 ((hT x hx).2 a)
+    nlinarith
+  obtain ⟨F, hF, hFi, hout, hmap, hpull⟩ := Moser.compactlySupported_deformation hU hW
+    (fun x _ a b => ddc_skew J _ x a b)
+    (fun x hx a b c => ddc_closed J ((contDiff_norm_sq ℝ).contDiffAt.add (hτ x hx).1) a b c)
+    hγ hcγ (hsγ.trans hCU) hWi
+  refine ⟨F.symm, hFi, hF, ?_, ?_, ?_⟩
+  · intro x hx
+    exact (F.symm_apply_eq.mpr (hout x hx).symm)
+  · intro x
+    simpa using (hmap (F.symm x)).symm
+  · intro x hx a b
+    have hix : F.symm x ∈ U := (hmap (F.symm x)).mp (by simpa using hx)
+    have hh := hpull (F.symm x) hix (fderiv ℝ (F.symm : E → E) x a)
+      (fderiv ℝ (F.symm : E → E) x b)
+    rw [F.apply_symm_apply, fderiv_homeomorph_apply_symm F hF hFi,
+      fderiv_homeomorph_apply_symm F hF hFi, hnorm x hx, hdiff x hx] at hh
+    have hend : standardTensor J + ddc J τ x + (ddc J T x - ddc J τ x) =
+        ddc J (fun y => ‖y‖^2 + T y) x := by
+      rw [ddc_add J (contDiff_norm_sq ℝ).contDiffAt (hT x hx).1, ddc_norm_sq J hsk]
+      abel
+    rw [hend] at hh
+    exact hh.symm
+
+end
+end ComplexPotential
+
+namespace ComplexPotential
+
+open TensorCalculus Set Filter
+open scoped ContDiff Topology
+
+variable {E : Type u33} [NormedAddCommGroup E] [NormedSpace ℝ E]
+
+noncomputable section
+
+theorem dc_scalar_comp (J : E →L[ℝ] E) {φ : E → ℝ} {h : ℝ → ℝ} {x : E}
+    (hφ : DifferentiableAt ℝ φ x) (hh : DifferentiableAt ℝ h (φ x)) :
+    dc J (h ∘ φ) x = deriv h (φ x) • dc J φ x := by
+  have hd := (hh.hasDerivAt.comp_hasFDerivAt x hφ.hasFDerivAt).fderiv
+  ext a
+  rw [dc_apply, hd]
+  simp only [smul_apply, smul_eq_mul, dc_apply]
+  ring
+
+theorem exteriorD_smul_one {γ : E → E →L[ℝ] ℝ} {s : E → ℝ} {x : E}
+    (hs : DifferentiableAt ℝ s x) (hγ : DifferentiableAt ℝ γ x) (a b : E) :
+    exteriorD (fun y => s y • γ y) x a b = s x * exteriorD γ x a b +
+      fderiv ℝ s x a * γ x b - fderiv ℝ s x b * γ x a := by
+  simp only [exteriorD_apply, fderiv_fun_smul hs hγ, add_apply, smul_apply,
+    ContinuousLinearMap.smulRight_apply, smul_eq_mul]
+  ring
+
+theorem ddc_scalar_comp (J : E →L[ℝ] E) {φ : E → ℝ} {h : ℝ → ℝ} {x : E}
+    (hφ : ContDiffAt ℝ ∞ φ x) (hh : ContDiffAt ℝ ∞ h (φ x)) (a b : E) :
+    ddc J (h ∘ φ) x a b = deriv h (φ x) * ddc J φ x a b +
+      deriv (deriv h) (φ x) *
+        (fderiv ℝ φ x b * fderiv ℝ φ x (J a) -
+          fderiv ℝ φ x a * fderiv ℝ φ x (J b)) / 4 := by
+  have h1 : (1 : ℕ∞ω) ≤ ∞ := WithTop.coe_le_coe.mpr le_top
+  have he : dc J (h ∘ φ) =ᶠ[𝓝 x] fun y => deriv h (φ y) • dc J φ y := by
+    filter_upwards [(hφ.of_le h1).eventually (by simp),
+      hφ.continuousAt ((hh.of_le h1).eventually (by simp))] with y hy hz
+    exact dc_scalar_comp J (hy.differentiableAt one_ne_zero)
+      ((show ContDiffAt ℝ (1 : ℕ∞ω) h (φ y) from hz).differentiableAt one_ne_zero)
+  have hdh := (hh.derivWithin (m := ∞) (by simp)).differentiableAt (by simp)
+  have hs : DifferentiableAt ℝ (fun y => deriv h (φ y)) x :=
+    hdh.comp x (hφ.differentiableAt (by simp))
+  have hd : fderiv ℝ (fun y => deriv h (φ y)) x =
+      deriv (deriv h) (φ x) • fderiv ℝ φ x :=
+    (hdh.hasDerivAt.comp_hasFDerivAt x (hφ.differentiableAt (by simp)).hasFDerivAt).fderiv
+  have heD : exteriorD (dc J (h ∘ φ)) x =
+      exteriorD (fun y => deriv h (φ y) • dc J φ y) x := by
+    unfold exteriorD
+    rw [he.fderiv_eq]
+  rw [ddc, heD, exteriorD_smul_one hs ((contDiffAt_dc J hφ).differentiableAt (by simp))]
+  rw [hd]
+  simp only [smul_apply, smul_eq_mul, dc_apply]
+  change _ = deriv h (φ x) * exteriorD (dc J φ) x a b + _
+  ring
+
+theorem ddc_scalar_comp_complex_line (J : E →L[ℝ] E) (hJ : ∀ a, J (J a) = -a)
+    {φ : E → ℝ} {h : ℝ → ℝ} {x : E}
+    (hφ : ContDiffAt ℝ ∞ φ x) (hh : ContDiffAt ℝ ∞ h (φ x)) (a : E) :
+    ddc J (h ∘ φ) x a (J a) = deriv h (φ x) * ddc J φ x a (J a) +
+      deriv (deriv h) (φ x) * ((fderiv ℝ φ x a)^2 + (fderiv ℝ φ x (J a))^2) / 4 := by
+  rw [ddc_scalar_comp J hφ hh, hJ, map_neg]
+  ring
+
+theorem IsPSHAt.scalar_comp (J : E →L[ℝ] E) (hJ : ∀ a, J (J a) = -a)
+    {φ : E → ℝ} {h : ℝ → ℝ} {x : E} (hφ : IsPSHAt J φ x)
+    (hh : ContDiffAt ℝ ∞ h (φ x)) (hmono : 0 ≤ deriv h (φ x))
+    (hconv : 0 ≤ deriv (deriv h) (φ x)) : IsPSHAt J (h ∘ φ) x := by
+  refine ⟨hh.comp x hφ.1, fun a => ?_⟩
+  rw [ddc_scalar_comp_complex_line J hJ hφ.1 hh]
+  exact add_nonneg (mul_nonneg hmono (hφ.2 a))
+    (div_nonneg (mul_nonneg hconv (add_nonneg (sq_nonneg _) (sq_nonneg _))) (by norm_num))
+
+end
+end ComplexPotential
+
+namespace ComplexPotential
+
+variable {E : Type u34} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+
+theorem complex_line_inner_sq_le (J : E →L[ℝ] E)
+    (hsk : ∀ a b, inner (𝕜 := ℝ) a (J b) = -inner (𝕜 := ℝ) (J a) b)
+    (hJ : ∀ a, ‖J a‖ = ‖a‖) (x a : E) :
+    (inner (𝕜 := ℝ) x a)^2 + (inner (𝕜 := ℝ) x (J a))^2 ≤ ‖x‖^2 * ‖a‖^2 := by
+  let p := inner (𝕜 := ℝ) x a
+  let q := inner (𝕜 := ℝ) x (J a)
+  have hxa : inner (𝕜 := ℝ) (J x) a = -q := by
+    dsimp [q]
+    linarith [hsk x a]
+  have hxj : inner (𝕜 := ℝ) x (J x) = 0 := by
+    linarith [hsk x x, real_inner_comm x (J x)]
+  have hjx : inner (𝕜 := ℝ) (J x) x = 0 := by rw [real_inner_comm, hxj]
+  let w := p • x - q • J x
+  have hw : inner (𝕜 := ℝ) w a = p^2 + q^2 := by
+    dsimp [w]
+    simp only [inner_sub_left, real_inner_smul_left, hxa]
+    change p * p - q * (-q) = _
+    ring
+  have hww : inner (𝕜 := ℝ) w w = (p^2 + q^2) * ‖x‖^2 := by
+    dsimp [w]
+    simp only [inner_sub_left, inner_sub_right, real_inner_smul_left,
+      real_inner_smul_right, hxj, hjx]
+    rw [real_inner_self_eq_norm_sq, real_inner_self_eq_norm_sq, hJ]
+    ring
+  have hcs := real_inner_mul_inner_self_le w a
+  rw [hw, hww, real_inner_self_eq_norm_sq] at hcs
+  change p^2 + q^2 ≤ ‖x‖^2 * ‖a‖^2
+  rcases eq_or_lt_of_le (add_nonneg (sq_nonneg p) (sq_nonneg q)) with hs | hs
+  · rw [← hs]
+    positivity
+  · exact (mul_le_mul_iff_right₀ hs).mp (by nlinarith [hcs])
+
+end ComplexPotential
+
+namespace ComplexPotential
+
+open Set Filter
+open scoped ContDiff Topology
+
+noncomputable section
+
+@[simp] theorem deriv_log_add (δ t : ℝ) :
+    deriv (fun s => Real.log (s + δ)) t = (t + δ)⁻¹ := by
+  rw [deriv_comp_add_const, Real.deriv_log]
+
+theorem second_deriv_log_add (δ t : ℝ) (ht : t + δ ≠ 0) :
+    deriv (deriv (fun s => Real.log (s + δ))) t = -1 / (t + δ)^2 := by
+  have he : deriv (fun s => Real.log (s + δ)) = fun t => (t + δ)⁻¹ :=
+    funext (deriv_log_add δ)
+  rw [he]
+  exact ((hasDerivAt_id t).add_const δ |>.inv ht).deriv
+
+variable {E : Type u35} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+
+def logNormSq (δ : ℝ) (x : E) : ℝ := Real.log (‖x‖^2 + δ)
+
+theorem contDiffAt_logNormSq (δ : ℝ) {x : E} (hx : ‖x‖^2 + δ ≠ 0) :
+    ContDiffAt ℝ ∞ (logNormSq δ) x :=
+  ((contDiff_norm_sq ℝ).contDiffAt.add contDiffAt_const).log hx
+
+theorem ddc_logNormSq_complex_line (J : E →L[ℝ] E)
+    (hsk : ∀ a b, inner (𝕜 := ℝ) a (J b) = -inner (𝕜 := ℝ) (J a) b)
+    (hJ : ∀ a, ‖J a‖ = ‖a‖) (hJJ : ∀ a, J (J a) = -a)
+    (δ : ℝ) {x : E} (hx : ‖x‖^2 + δ ≠ 0) (a : E) :
+    ddc J (logNormSq δ) x a (J a) = ‖a‖^2 / (‖x‖^2 + δ) -
+      ((inner (𝕜 := ℝ) x a)^2 + (inner (𝕜 := ℝ) x (J a))^2) / (‖x‖^2 + δ)^2 := by
+  have hl : ContDiffAt ℝ ∞ (fun t : ℝ => Real.log (t + δ)) (‖x‖^2) :=
+    (contDiffAt_id.add contDiffAt_const).log hx
+  have he := ddc_scalar_comp_complex_line J hJJ (contDiff_norm_sq ℝ).contDiffAt hl a
+  rw [ddc_norm_sq J hsk, standardTensor_complex_line J hJ,
+    deriv_log_add, second_deriv_log_add δ _ hx, fderiv_norm_sq_apply] at he
+  simp only [smul_apply, innerSL_apply_apply] at he
+  change ddc J (logNormSq δ) x a (J a) = _ at he
+  rw [he]
+  ring
+
+theorem isPSHAt_logNormSq (J : E →L[ℝ] E)
+    (hsk : ∀ a b, inner (𝕜 := ℝ) a (J b) = -inner (𝕜 := ℝ) (J a) b)
+    (hJ : ∀ a, ‖J a‖ = ‖a‖) (hJJ : ∀ a, J (J a) = -a)
+    {δ : ℝ} (hδ : 0 ≤ δ) {x : E} (hx : 0 < ‖x‖^2 + δ) :
+    IsPSHAt J (logNormSq δ) x := by
+  refine ⟨contDiffAt_logNormSq δ hx.ne', fun a => ?_⟩
+  rw [ddc_logNormSq_complex_line J hsk hJ hJJ δ hx.ne']
+  have hineq := complex_line_inner_sq_le J hsk hJ x a
+  have hnum : (inner (𝕜 := ℝ) x a)^2 + (inner (𝕜 := ℝ) x (J a))^2 ≤
+      ‖a‖^2 * (‖x‖^2 + δ) := by
+    nlinarith [mul_nonneg (sq_nonneg ‖a‖) hδ]
+  apply sub_nonneg.mpr
+  apply (div_le_div_iff₀ (sq_pos_of_pos hx) hx).mpr
+  nlinarith [mul_le_mul_of_nonneg_right hnum hx.le]
+
+end
+end ComplexPotential
+
+end LowerBoundInline
+
+end OAI
diff --git a/lean/OAI/Geometry/PolarProducts/PotentialForms.lean b/lean/OAI/Geometry/PolarProducts/PotentialForms.lean
new file mode 100644
index 0000000000000000000000000000000000000000..c3970ca558f2f42f9ed85443094af65946c7b484
--- /dev/null
+++ b/lean/OAI/Geometry/PolarProducts/PotentialForms.lean
@@ -0,0 +1,308 @@
+import OAI.Geometry.PolarProducts.MoserFlow
+
+namespace OAI
+
+universe u27 u28 u29
+
+section LowerBoundInline
+open Set Filter Function
+open scoped Topology ContDiff NNReal
+open Set Filter Metric
+open scoped Topology ContDiff
+open Set Filter Function MeasureTheory Metric
+open scoped Topology ContDiff NNReal
+open Set Filter Function
+open scoped Topology ContDiff
+open Set Filter Function
+open scoped Topology ContDiff NNReal
+open Set Filter
+open scoped Topology ContDiff
+open Set Filter Function
+open scoped Topology ContDiff
+
+namespace ComplexPotential
+
+open TensorCalculus Set Filter
+open scoped Topology ContDiff
+
+variable {E : Type u27} [NormedAddCommGroup E] [NormedSpace ℝ E]
+
+noncomputable section
+
+local instance oneTensorGroup : NormedAddCommGroup (E →L[ℝ] ℝ) := inferInstance
+local instance oneTensorSpace : NormedSpace ℝ (E →L[ℝ] ℝ) := inferInstance
+local instance twoTensorGroup : NormedAddCommGroup (E →L[ℝ] E →L[ℝ] ℝ) := inferInstance
+local instance twoTensorSpace : NormedSpace ℝ (E →L[ℝ] E →L[ℝ] ℝ) := inferInstance
+
+def dcOp (J : E →L[ℝ] E) : (E →L[ℝ] ℝ) →L[ℝ] E →L[ℝ] ℝ :=
+  (-1 / 4 : ℝ) • (ContinuousLinearMap.compL ℝ E E ℝ).flip J
+
+def dc (J : E →L[ℝ] E) (φ : E → ℝ) (x : E) : E →L[ℝ] ℝ :=
+  dcOp J (fderiv ℝ φ x)
+
+def ddc (J : E →L[ℝ] E) (φ : E → ℝ) : E → TwoTensor E := exteriorD (dc J φ)
+
+@[simp] theorem dcOp_apply (J : E →L[ℝ] E) (L : E →L[ℝ] ℝ) (a : E) :
+    dcOp J L a = -(L (J a)) / 4 := by
+  simp only [dcOp, smul_apply, ContinuousLinearMap.flip_apply,
+    ContinuousLinearMap.compL_apply, ContinuousLinearMap.comp_apply, smul_eq_mul]
+  ring
+
+@[simp] theorem dc_apply (J : E →L[ℝ] E) (φ : E → ℝ) (x a : E) :
+    dc J φ x a = -(fderiv ℝ φ x (J a)) / 4 := dcOp_apply J _ a
+
+theorem contDiffAt_dc (J : E →L[ℝ] E) {φ : E → ℝ} {x : E}
+    (hφ : ContDiffAt ℝ ∞ φ x) : ContDiffAt ℝ ∞ (dc J φ) x := by
+  change ContDiffAt ℝ ∞ (dcOp J ∘ fderiv ℝ φ) x
+  exact (dcOp J).contDiff.contDiffAt.comp x (hφ.fderiv_right (m := ∞) (by simp))
+
+theorem contDiff_dc (J : E →L[ℝ] E) {φ : E → ℝ}
+    (hφ : ContDiff ℝ ∞ φ) : ContDiff ℝ ∞ (dc J φ) := by
+  rw [contDiff_iff_contDiffAt]
+  exact fun x => contDiffAt_dc J (hφ.contDiffAt (x := x))
+
+theorem fderiv_dc (J : E →L[ℝ] E) {φ : E → ℝ} {x : E}
+    (hφ : DifferentiableAt ℝ (fderiv ℝ φ) x) (a b : E) :
+    fderiv ℝ (dc J φ) x a b = -(fderiv ℝ (fderiv ℝ φ) x a (J b)) / 4 := by
+  change fderiv ℝ (dcOp J ∘ fderiv ℝ φ) x a b = _
+  rw [fderiv_comp x (dcOp J).differentiableAt hφ]
+  simp
+
+theorem ddc_apply (J : E →L[ℝ] E) {φ : E → ℝ} {x : E}
+    (hφ : DifferentiableAt ℝ (fderiv ℝ φ) x) (a b : E) :
+    ddc J φ x a b =
+      (fderiv ℝ (fderiv ℝ φ) x b (J a) - fderiv ℝ (fderiv ℝ φ) x a (J b)) / 4 := by
+  simp only [ddc, exteriorD_apply, fderiv_dc J hφ]
+  ring
+
+theorem ddc_skew (J : E →L[ℝ] E) (φ : E → ℝ) (x a b : E) :
+    ddc J φ x a b = -ddc J φ x b a := exteriorD_skew _ _ _ _
+
+theorem contDiffAt_ddc (J : E →L[ℝ] E) {φ : E → ℝ} {x : E}
+    (hφ : ContDiffAt ℝ ∞ φ x) : ContDiffAt ℝ ∞ (ddc J φ) x := by
+  change ContDiffAt ℝ ∞ (alt ∘ fderiv ℝ (dc J φ)) x
+  exact (alt (E := E)).contDiff.contDiffAt.comp x
+    ((contDiffAt_dc J hφ).fderiv_right (m := ∞) (by simp))
+
+theorem contDiff_ddc (J : E →L[ℝ] E) {φ : E → ℝ}
+    (hφ : ContDiff ℝ ∞ φ) : ContDiff ℝ ∞ (ddc J φ) :=
+  contDiff_exteriorD (contDiff_dc J hφ)
+
+theorem ddc_closed (J : E →L[ℝ] E) {φ : E → ℝ} {x : E}
+    (hφ : ContDiffAt ℝ ∞ φ x) (a b c : E) :
+    fderiv ℝ (ddc J φ) x a b c + fderiv ℝ (ddc J φ) x b c a +
+      fderiv ℝ (ddc J φ) x c a b = 0 := exteriorD_closed (contDiffAt_dc J hφ) a b c
+
+theorem ddc_complex_line (J : E →L[ℝ] E) (hJ : ∀ a, J (J a) = -a)
+    {φ : E → ℝ} {x : E} (hφ : DifferentiableAt ℝ (fderiv ℝ φ) x) (a : E) :
+    ddc J φ x a (J a) =
+      (fderiv ℝ (fderiv ℝ φ) x a a + fderiv ℝ (fderiv ℝ φ) x (J a) (J a)) / 4 := by
+  rw [ddc_apply J hφ, hJ]
+  simp only [map_neg]
+  ring
+
+def IsPSHAt (J : E →L[ℝ] E) (φ : E → ℝ) (x : E) : Prop :=
+  ContDiffAt ℝ ∞ φ x ∧ ∀ a, 0 ≤ ddc J φ x a (J a)
+
+theorem dc_congr (J : E →L[ℝ] E) {φ ψ : E → ℝ} {x : E}
+    (h : φ =ᶠ[𝓝 x] ψ) : dc J φ x = dc J ψ x := by
+  rw [dc, dc, h.fderiv_eq]
+
+theorem ddc_congr (J : E →L[ℝ] E) {φ ψ : E → ℝ} {x : E}
+    (h : φ =ᶠ[𝓝 x] ψ) : ddc J φ x = ddc J ψ x := by
+  have he : dc J φ =ᶠ[𝓝 x] dc J ψ := h.eventuallyEq_nhds.mono (fun y hy => dc_congr J hy)
+  simpa only [ddc, exteriorD] using congrArg alt he.fderiv_eq
+
+theorem dc_add (J : E →L[ℝ] E) {φ ψ : E → ℝ} {x : E}
+    (hφ : DifferentiableAt ℝ φ x) (hψ : DifferentiableAt ℝ ψ x) :
+    dc J (fun y => φ y + ψ y) x = dc J φ x + dc J ψ x := by
+  simp only [dc, fderiv_fun_add hφ hψ, map_add]
+
+theorem ddc_add (J : E →L[ℝ] E) {φ ψ : E → ℝ} {x : E}
+    (hφ : ContDiffAt ℝ ∞ φ x) (hψ : ContDiffAt ℝ ∞ ψ x) :
+    ddc J (fun y => φ y + ψ y) x = ddc J φ x + ddc J ψ x := by
+  have h1 : (1 : ℕ∞ω) ≤ ∞ := WithTop.coe_le_coe.mpr le_top
+  have he : dc J (fun y => φ y + ψ y) =ᶠ[𝓝 x] fun y => dc J φ y + dc J ψ y := by
+    filter_upwards [(hφ.of_le h1).eventually (by simp),
+      (hψ.of_le h1).eventually (by simp)] with y hy hz
+    exact dc_add J (hy.differentiableAt (by simp)) (hz.differentiableAt (by simp))
+  unfold ddc exteriorD
+  rw [he.fderiv_eq, fderiv_fun_add
+    ((contDiffAt_dc J hφ).differentiableAt (by simp))
+    ((contDiffAt_dc J hψ).differentiableAt (by simp)), map_add]
+
+end
+end ComplexPotential
+
+namespace ComplexPotential
+
+open TensorCalculus
+open scoped ContDiff Topology
+
+variable {E : Type u28} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+
+noncomputable section
+
+def standardTensor (J : E →L[ℝ] E) : TwoTensor E := (innerSL ℝ).comp J
+
+@[simp] theorem standardTensor_apply (J : E →L[ℝ] E) (a b : E) :
+    standardTensor J a b = inner (𝕜 := ℝ) (J a) b := rfl
+
+theorem dc_norm_sq (J : E →L[ℝ] E)
+    (hsk : ∀ a b, inner (𝕜 := ℝ) a (J b) = -inner (𝕜 := ℝ) (J a) b) (x : E) :
+    dc J (fun x : E => ‖x‖^2) x = (1 / 2 : ℝ) • standardTensor J x := by
+  ext a
+  rw [dc_apply, fderiv_norm_sq_apply]
+  simp only [smul_apply, smul_eq_mul, innerSL_apply_apply, standardTensor_apply]
+  rw [hsk]
+  ring
+
+theorem ddc_norm_sq (J : E →L[ℝ] E)
+    (hsk : ∀ a b, inner (𝕜 := ℝ) a (J b) = -inner (𝕜 := ℝ) (J a) b) (x : E) :
+    ddc J (fun x : E => ‖x‖^2) x = standardTensor J := by
+  have he : dc J (fun x : E => ‖x‖^2) = fun x => (1 / 2 : ℝ) • standardTensor J x :=
+    funext (dc_norm_sq J hsk)
+  apply ContinuousLinearMap.ext
+  intro a
+  apply ContinuousLinearMap.ext
+  intro b
+  rw [ddc, he, exteriorD_apply]
+  have hd : fderiv ℝ (fun x => (1 / 2 : ℝ) • standardTensor J x) x =
+      (1 / 2 : ℝ) • standardTensor J := by
+    exact ((standardTensor J).hasFDerivAt.const_smul (1 / 2 : ℝ)).fderiv
+  rw [hd]
+  simp only [smul_apply, smul_eq_mul, standardTensor_apply]
+  have hh : inner (𝕜 := ℝ) (J b) a = -inner (𝕜 := ℝ) (J a) b := by
+    rw [real_inner_comm, hsk]
+  rw [hh]
+  ring
+
+theorem standardTensor_complex_line (J : E →L[ℝ] E) (hJ : ∀ a, ‖J a‖ = ‖a‖) (a : E) :
+    standardTensor J a (J a) = ‖a‖^2 := by
+  rw [standardTensor_apply, real_inner_self_eq_norm_sq, hJ]
+
+theorem isInvertible_of_positive_complex_line [FiniteDimensional ℝ E]
+    (B : TwoTensor E) (J : E →L[ℝ] E) (hp : ∀ a : E, a ≠ 0 → 0 < B a (J a)) :
+    B.IsInvertible := by
+  have hi : Function.Injective B := by
+    intro a b hab
+    have hz : B (a - b) = 0 := by rw [map_sub, hab, sub_self]
+    by_contra hn
+    have hpos := hp (a - b) (sub_ne_zero.mpr hn)
+    rw [hz, zero_apply] at hpos
+    exact (lt_irrefl 0) hpos
+  let e := (InnerProductSpace.toDual ℝ E).toContinuousLinearEquiv
+  let L : E →L[ℝ] E := e.symm.toContinuousLinearMap.comp B
+  have hL : Function.Injective L := e.symm.injective.comp hi
+  have hsL : Function.Surjective L := LinearMap.surjective_of_injective hL
+  have hs : Function.Surjective B := by
+    intro v
+    obtain ⟨x, hx⟩ := hsL (e.symm v)
+    refine ⟨x, ?_⟩
+    exact e.symm.injective hx
+  exact ⟨ContinuousLinearEquiv.ofBijective B (LinearMap.ker_eq_bot.mpr hi)
+    (LinearMap.range_eq_top.mpr hs), rfl⟩
+
+theorem isInvertible_ddc_norm_sq_add [FiniteDimensional ℝ E]
+    (J : E →L[ℝ] E) (hsk : ∀ a b, inner (𝕜 := ℝ) a (J b) = -inner (𝕜 := ℝ) (J a) b)
+    (hJ : ∀ a, ‖J a‖ = ‖a‖) {φ : E → ℝ} {x : E} (hφ : IsPSHAt J φ x) :
+    (ddc J (fun y => ‖y‖^2 + φ y) x).IsInvertible := by
+  apply isInvertible_of_positive_complex_line _ J
+  intro a ha
+  rw [ddc_add J (contDiff_norm_sq ℝ).contDiffAt hφ.1, add_apply, add_apply,
+    ddc_norm_sq J hsk, standardTensor_complex_line J hJ]
+  exact add_pos_of_pos_of_nonneg (sq_pos_of_pos (norm_pos_iff.mpr ha)) (hφ.2 a)
+
+end
+end ComplexPotential
+
+namespace ComplexPotential
+
+open TensorCalculus Set Filter
+open scoped ContDiff Topology
+
+variable {E : Type u29} [NormedAddCommGroup E] [NormedSpace ℝ E]
+
+noncomputable section
+
+local instance linearOneGroup : NormedAddCommGroup (E →L[ℝ] ℝ) := inferInstance
+local instance linearOneSpace : NormedSpace ℝ (E →L[ℝ] ℝ) := inferInstance
+local instance linearTwoGroup : NormedAddCommGroup (E →L[ℝ] E →L[ℝ] ℝ) := inferInstance
+local instance linearTwoSpace : NormedSpace ℝ (E →L[ℝ] E →L[ℝ] ℝ) := inferInstance
+
+@[simp] theorem dc_const (J : E →L[ℝ] E) (c : ℝ) (x : E) :
+    dc J (fun _ => c) x = 0 := by
+  simp [dc]
+
+@[simp] theorem ddc_const (J : E →L[ℝ] E) (c : ℝ) (x : E) :
+    ddc J (fun _ => c) x = 0 := by
+  simp [ddc, exteriorD, show dc J (fun _ : E => c) = fun _ => 0 by funext y; simp]
+
+theorem dc_const_smul (J : E →L[ℝ] E) (c : ℝ) {φ : E → ℝ} {x : E}
+    (hφ : DifferentiableAt ℝ φ x) :
+    dc J (fun y => c * φ y) x = c • dc J φ x := by
+  ext a
+  rw [dc_apply, fderiv_const_mul hφ]
+  simp only [smul_apply, smul_eq_mul, dc_apply]
+  ring
+
+theorem ddc_const_smul (J : E →L[ℝ] E) (c : ℝ) {φ : E → ℝ} {x : E}
+    (hφ : ContDiffAt ℝ ∞ φ x) :
+    ddc J (fun y => c * φ y) x = c • ddc J φ x := by
+  have h1 : (1 : ℕ∞ω) ≤ ∞ := WithTop.coe_le_coe.mpr le_top
+  have he : dc J (fun y => c * φ y) =ᶠ[𝓝 x] fun y => c • dc J φ y := by
+    filter_upwards [(hφ.of_le h1).eventually (by simp)] with y hy
+    exact dc_const_smul J c (hy.differentiableAt one_ne_zero)
+  unfold ddc exteriorD
+  rw [he.fderiv_eq]
+  have hd : fderiv ℝ (fun y => c • dc J φ y) x = c • fderiv ℝ (dc J φ) x :=
+    (((contDiffAt_dc J hφ).differentiableAt (by simp)).hasFDerivAt.const_smul c).fderiv
+  rw [hd, map_smul]
+
+theorem ddc_neg (J : E →L[ℝ] E) {φ : E → ℝ} {x : E}
+    (hφ : ContDiffAt ℝ ∞ φ x) :
+    ddc J (fun y => -φ y) x = -ddc J φ x := by
+  simpa only [neg_one_mul, neg_one_smul] using ddc_const_smul J (-1) hφ
+
+theorem ddc_sub (J : E →L[ℝ] E) {φ ψ : E → ℝ} {x : E}
+    (hφ : ContDiffAt ℝ ∞ φ x) (hψ : ContDiffAt ℝ ∞ ψ x) :
+    ddc J (fun y => φ y - ψ y) x = ddc J φ x - ddc J ψ x := by
+  simp_rw [sub_eq_add_neg]
+  rw [ddc_add J hφ hψ.neg, ddc_neg J hψ]
+
+theorem ddc_add_const (J : E →L[ℝ] E) {φ : E → ℝ} {x : E}
+    (hφ : ContDiffAt ℝ ∞ φ x) (c : ℝ) :
+    ddc J (fun y => φ y + c) x = ddc J φ x := by
+  rw [ddc_add J hφ contDiffAt_const, ddc_const, add_zero]
+
+theorem ddc_sub_const (J : E →L[ℝ] E) {φ : E → ℝ} {x : E}
+    (hφ : ContDiffAt ℝ ∞ φ x) (c : ℝ) :
+    ddc J (fun y => φ y - c) x = ddc J φ x := by
+  rw [ddc_sub J hφ contDiffAt_const, ddc_const, sub_zero]
+
+theorem IsPSHAt.add {J : E →L[ℝ] E} {φ ψ : E → ℝ} {x : E}
+    (hφ : IsPSHAt J φ x) (hψ : IsPSHAt J ψ x) :
+    IsPSHAt J (fun y => φ y + ψ y) x := by
+  refine ⟨hφ.1.add hψ.1, fun a => ?_⟩
+  rw [ddc_add J hφ.1 hψ.1, add_apply, add_apply]
+  exact add_nonneg (hφ.2 a) (hψ.2 a)
+
+theorem IsPSHAt.const_smul {J : E →L[ℝ] E} {φ : E → ℝ} {x : E}
+    (hφ : IsPSHAt J φ x) {c : ℝ} (hc : 0 ≤ c) :
+    IsPSHAt J (fun y => c * φ y) x := by
+  refine ⟨contDiffAt_const.mul hφ.1, fun a => ?_⟩
+  rw [ddc_const_smul J c hφ.1]
+  simpa only [smul_apply, smul_eq_mul] using mul_nonneg hc (hφ.2 a)
+
+theorem IsPSHAt.sub_const {J : E →L[ℝ] E} {φ : E → ℝ} {x : E}
+    (hφ : IsPSHAt J φ x) (c : ℝ) : IsPSHAt J (fun y => φ y - c) x := by
+  refine ⟨hφ.1.sub contDiffAt_const, fun a => ?_⟩
+  rw [ddc_sub_const J hφ.1 c]
+  exact hφ.2 a
+
+end
+end ComplexPotential
+
+end LowerBoundInline
+
+end OAI
diff --git a/lean/OAI/Geometry/PolarProducts/QuadraticAction.lean b/lean/OAI/Geometry/PolarProducts/QuadraticAction.lean
new file mode 100644
index 0000000000000000000000000000000000000000..8d73bb6c834f166bf721870f1dd6952beeb5b354
--- /dev/null
+++ b/lean/OAI/Geometry/PolarProducts/QuadraticAction.lean
@@ -0,0 +1,330 @@
+import OAI.Geometry.PolarProducts.FourierPaths
+
+namespace OAI
+
+universe u112
+
+section NonsqueezingInline
+
+namespace DiagonalQuadratic
+noncomputable section
+open Finset
+open scoped ComplexConjugate ContDiff
+variable {α : Type u112} [Fintype α]
+
+ def diag (w : α → ℝ) (a : EuclideanSpace ℂ α) : EuclideanSpace ℂ α :=
+  (EuclideanSpace.equiv α ℂ).symm (fun p => w p • a p)
+
+omit [Fintype α] in
+@[simp] theorem diag_apply (w : α → ℝ) (a : EuclideanSpace ℂ α) (p : α) :
+    diag w a p = w p • a p := rfl
+
+ def diagCLM (w : α → ℝ) : EuclideanSpace ℂ α →L[ℝ] EuclideanSpace ℂ α :=
+  LinearMap.toContinuousLinearMap {
+    toFun := diag w
+    map_add' := by intros; ext; simp [smul_add]
+    map_smul' := by intros; ext; simp; ring }
+
+@[simp] theorem diagCLM_apply (w : α → ℝ) (a : EuclideanSpace ℂ α) : diagCLM w a = diag w a := rfl
+
+ theorem real_inner_eq_sum (a b : EuclideanSpace ℂ α) :
+    inner (𝕜 := ℝ) a b = ∑ p, inner (𝕜 := ℝ) (a p) (b p) := by
+  exact PiLp.inner_apply a b
+
+ theorem inner_diag (w : α → ℝ) (a b : EuclideanSpace ℂ α) :
+    inner (𝕜 := ℝ) (diag w a) b = ∑ p, w p * inner (𝕜 := ℝ) (a p) (b p) := by
+  rw [real_inner_eq_sum]
+  simp only [diag_apply, real_inner_smul_left]
+
+ theorem self_adjoint (w : α → ℝ) (a b : EuclideanSpace ℂ α) :
+    inner (𝕜 := ℝ) (diag w a) b = inner (𝕜 := ℝ) a (diag w b) := by
+  rw [inner_diag, real_inner_eq_sum]
+  simp only [diag_apply, real_inner_smul_right]
+
+ def energy (w : α → ℝ) (a : EuclideanSpace ℂ α) : ℝ := ∑ p, w p * ‖a p‖^2
+
+ theorem energy_eq_inner (w : α → ℝ) (a : EuclideanSpace ℂ α) :
+    energy w a = inner (𝕜 := ℝ) (diag w a) a := by
+  rw [inner_diag]
+  simp only [energy, real_inner_self_eq_norm_sq]
+
+ theorem contDiff_energy (w : α → ℝ) : ContDiff ℝ ∞ (energy w) := by
+  simp_rw [funext (energy_eq_inner w)]
+  exact (diagCLM w).contDiff.inner ℝ contDiff_id
+
+ theorem hasGradientAt_energy (w : α → ℝ) (a : EuclideanSpace ℂ α) :
+    HasGradientAt (energy w) ((2 : ℝ) • diag w a) a := by
+  rw [hasGradientAt_iff_hasFDerivAt]
+  have h := (diagCLM w).hasFDerivAt.inner ℝ (hasFDerivAt_id a)
+  simp only [diagCLM_apply] at h
+  change HasFDerivAt (fun x => inner (𝕜 := ℝ) (diag w x) x) _ a at h
+  simp_rw [← energy_eq_inner] at h
+  apply h.congr_fderiv
+  ext b
+  change inner (𝕜 := ℝ) (diag w a) b + inner (𝕜 := ℝ) (diag w b) a =
+    inner (𝕜 := ℝ) ((2 : ℝ) • diag w a) b
+  rw [real_inner_smul_left, self_adjoint w b a]
+  rw [real_inner_comm b (diag w a)]
+  ring
+
+ theorem gradient_energy (w : α → ℝ) (a : EuclideanSpace ℂ α) :
+    gradient (energy w) a = (2 : ℝ) • diag w a := (hasGradientAt_energy w a).gradient
+
+ theorem norm_diag_sq (w : α → ℝ) (a : EuclideanSpace ℂ α) :
+    ‖diag w a‖^2 = ∑ p, (w p)^2*‖a p‖^2 := by
+  simp only [EuclideanSpace.norm_sq_eq, diag_apply, norm_smul, mul_pow, Real.norm_eq_abs,
+    sq_abs]
+
+ theorem norm_diag_lower (w : α → ℝ) {d : ℝ} (hd : 0 ≤ d) (hw : ∀ p, d ≤ |w p|)
+    (a : EuclideanSpace ℂ α) : d*‖a‖ ≤ ‖diag w a‖ := by
+  have h : (d*‖a‖)^2 ≤ ‖diag w a‖^2 := by
+    rw [mul_pow, norm_diag_sq, EuclideanSpace.norm_sq_eq, mul_sum]
+    apply sum_le_sum
+    intro p _
+    apply mul_le_mul_of_nonneg_right _ (sq_nonneg _)
+    calc
+      d^2 ≤ |w p|^2 := pow_le_pow_left₀ hd (hw p) 2
+      _ = _ := sq_abs _
+  nlinarith [norm_nonneg (diag w a), mul_nonneg hd (norm_nonneg a)]
+
+ theorem norm_diag_upper (w : α → ℝ) {d : ℝ} (hd : 0 ≤ d) (hw : ∀ p, |w p| ≤ d)
+    (a : EuclideanSpace ℂ α) : ‖diag w a‖ ≤ d*‖a‖ := by
+  have h : ‖diag w a‖^2 ≤ (d*‖a‖)^2 := by
+    rw [mul_pow, norm_diag_sq, EuclideanSpace.norm_sq_eq, mul_sum]
+    apply sum_le_sum
+    intro p _
+    apply mul_le_mul_of_nonneg_right _ (sq_nonneg _)
+    calc
+      (w p)^2 = |w p|^2 := (sq_abs _).symm
+      _ ≤ _ := pow_le_pow_left₀ (abs_nonneg _) (hw p) 2
+  nlinarith [norm_nonneg (diag w a), mul_nonneg hd (norm_nonneg a)]
+
+ theorem energy_smul (w : α → ℝ) (r : ℝ) (a : EuclideanSpace ℂ α) :
+    energy w (r • a) = r^2*energy w a := by
+  simp only [energy, PiLp.smul_apply, norm_smul, Real.norm_eq_abs, mul_pow, sq_abs,
+    mul_sum]
+  apply sum_congr rfl
+  intro p _
+  ring
+
+ theorem energy_mul (w : α → ℝ) (r : ℝ) (a : EuclideanSpace ℂ α) :
+    energy (fun p => r*w p) a = r*energy w a := by
+  simp only [energy, mul_sum, mul_assoc]
+
+ theorem energy_lower (w : α → ℝ) (d : ℝ) (hw : ∀ p, d ≤ w p)
+    (a : EuclideanSpace ℂ α) : d*‖a‖^2 ≤ energy w a := by
+  simp only [EuclideanSpace.norm_sq_eq, mul_sum, energy]
+  exact sum_le_sum (fun p _ => mul_le_mul_of_nonneg_right (hw p) (sq_nonneg _))
+
+ theorem energy_nonpos (w : α → ℝ) (hw : ∀ p, w p ≤ 0)
+    (a : EuclideanSpace ℂ α) : energy w a ≤ 0 :=
+  sum_nonpos (fun p _ => mul_nonpos_of_nonpos_of_nonneg (hw p) (sq_nonneg _))
+
+ theorem energy_single [DecidableEq α] (w : α → ℝ) (p : α) (z : ℂ) :
+    energy w (PiLp.single 2 p z) = w p * ‖z‖^2 := by
+  classical
+  unfold energy
+  rw [sum_eq_single p]
+  · simp
+  · intro q _ hq
+    simp [hq]
+  · simp
+
+end
+end DiagonalQuadratic
+
+namespace FourierPolynomial
+noncomputable section
+open MeasureTheory AddCircle Finset
+open scoped ComplexConjugate ContDiff
+local instance : Fact (0 < (1 : ℝ)) := ⟨zero_lt_one⟩
+variable {ι κ : Type} [Fintype ι] [Fintype κ]
+open DiagonalQuadratic
+
+ def action (k : ι → ℤ) (H : Vector κ → ℝ) (a : Coeff ι κ) : ℝ :=
+   energy (fun p => Real.pi*(k p.1 : ℝ)) a - potential k H a
+
+ theorem contDiff_action (k : ι → ℤ) {H : Vector κ → ℝ} (hH : ContDiff ℝ ∞ H) :
+    ContDiff ℝ ∞ (action k H) :=
+  (contDiff_energy _).sub (contDiff_potential k hH)
+
+ theorem hasGradientAt_action (k : ι → ℤ) {H : Vector κ → ℝ} (hH : ContDiff ℝ ∞ H)
+    (a : Coeff ι κ) :
+    HasGradientAt (action k H)
+      ((2 : ℝ) • diag (fun p => Real.pi*(k p.1 : ℝ)) a - project k (gradLoop k hH a)) a := by
+  have h := (hasGradientAt_energy (fun p : ι × κ => Real.pi*(k p.1 : ℝ)) a).hasFDerivAt.sub
+    (hasGradientAt_potential k hH a).hasFDerivAt
+  rw [hasGradientAt_iff_hasFDerivAt]
+  change HasFDerivAt (action k H) _ a at h
+  apply h.congr_fderiv
+  ext b
+  simp only [map_sub]
+
+ theorem gradient_action (k : ι → ℤ) {H : Vector κ → ℝ} (hH : ContDiff ℝ ∞ H)
+    (a : Coeff ι κ) : gradient (action k H) a =
+      (2 : ℝ) • diag (fun p => Real.pi*(k p.1 : ℝ)) a - project k (gradLoop k hH a) :=
+  (hasGradientAt_action k hH a).gradient
+
+ theorem potential_nonneg (k : ι → ℤ) {H : Vector κ → ℝ} (hH : ∀ z, 0 ≤ H z)
+    (a : Coeff ι κ) : 0 ≤ potential k H a := integral_nonneg (fun _ => hH _)
+
+ theorem integral_energy (k : ι → ℤ) (hk : Function.Injective k) (d : κ → ℝ)
+    (a : Coeff ι κ) :
+    (∫ t : Time, energy d (loop k a t) ∂μ) = energy (fun p => d p.2) a := by
+  classical
+  simp only [energy]
+  rw [integral_finsetSum]
+  · simp_rw [integral_const_mul, integral_loop_coord_sq k hk, mul_sum]
+    rw [Fintype.sum_prod_type, sum_comm]
+  · intro j _
+    exact integrable_continuous (continuous_const.mul
+      ((((PiLp.continuous_apply 2 _ j).comp (loop k a).continuous).norm.pow 2)))
+
+ theorem potential_lower (k : ι → ℤ) (hk : Function.Injective k) {H : Vector κ → ℝ}
+    (hH : Continuous H) (d : κ → ℝ) (C : ℝ) (hlo : ∀ z, energy d z-C ≤ H z)
+    (a : Coeff ι κ) : energy (fun p => d p.2) a-C ≤ potential k H a := by
+  have hi : Integrable (fun t : Time => energy d (loop k a t)) μ :=
+    integrable_continuous ((contDiff_energy d).continuous.comp (loop k a).continuous)
+  calc
+    _ = ∫ t : Time, energy d (loop k a t)-C ∂μ := by
+      rw [integral_sub hi (integrable_const _), integral_energy k hk]
+      simp
+    _ ≤ _ := integral_mono (hi.sub (integrable_const _))
+      (integrable_continuous (hH.comp (loop k a).continuous)) (fun t => hlo _)
+
+ theorem potential_upper_fourth (k : ι → ℕ) (hk : Function.Injective k)
+    (hk0 : ∀ i, 0 < k i) {H : Vector κ → ℝ} (hH : Continuous H)
+    {C : ℝ} (hC : 0 ≤ C) (hup : ∀ z, H z ≤ C*‖z‖^4) (a : Coeff ι κ) :
+    potential (fun i => (k i : ℤ)) H a ≤
+      C*(Fintype.card κ : ℝ)*(∑ p : ι × κ, (k p.1 : ℝ)*‖a p‖^2)^2 := by
+  calc
+    _ ≤ ∫ t : Time, C*‖loop (fun i => (k i : ℤ)) a t‖^4 ∂μ :=
+      integral_mono (integrable_continuous (hH.comp (loop _ a).continuous))
+        (integrable_continuous (continuous_const.mul ((loop _ a).continuous.norm.pow 4)))
+        (fun t => hup _)
+    _ = C*(∫ t : Time, ‖loop (fun i => (k i : ℤ)) a t‖^4 ∂μ) := integral_const_mul _ _
+    _ ≤ _ := by simpa only [mul_assoc] using
+      mul_le_mul_of_nonneg_left (integral_loop_norm_fourth_le k hk hk0 a) hC
+
+end
+end FourierPolynomial
+
+namespace FourierPolynomial
+noncomputable section
+open MeasureTheory AddCircle Finset
+open scoped ComplexConjugate ContDiff
+local instance : Fact (0 < (1 : ℝ)) := ⟨zero_lt_one⟩
+variable {ι κ : Type} [Fintype ι] [Fintype κ]
+open DiagonalQuadratic
+
+omit [Fintype ι] in
+ theorem project_add (k : ι → ℤ) (f g : C(Time, Vector κ)) :
+    project k (f+g) = project k f + project k g := by
+  ext p
+  change fourierCoeff ((fun t => f t p.2)+(fun t => g t p.2)) (k p.1) = _
+  have hf : Integrable (fun t : Time => f t p.2) μ :=
+    integrable_continuous ((PiLp.continuous_apply 2 _ p.2).comp f.continuous)
+  have hg : Integrable (fun t : Time => g t p.2) μ :=
+    integrable_continuous ((PiLp.continuous_apply 2 _ p.2).comp g.continuous)
+  rw [fourierCoeff.add hf hg]
+  rfl
+
+omit [Fintype ι] in
+ theorem project_sub (k : ι → ℤ) (f g : C(Time, Vector κ)) :
+    project k (f-g) = project k f - project k g := by
+  ext p
+  simp only [project_apply, ContinuousMap.sub_apply, PiLp.sub_apply, fourierCoeff,
+    smul_sub]
+  rw [integral_sub]
+  · exact integrable_continuous ((fourier (-k p.1)).continuous.smul
+      ((PiLp.continuous_apply 2 _ p.2).comp f.continuous))
+  · exact integrable_continuous ((fourier (-k p.1)).continuous.smul
+      ((PiLp.continuous_apply 2 _ p.2).comp g.continuous))
+
+ theorem loop_diag (k : ι → ℤ) (d : κ → ℝ) (a : Coeff ι κ) (t : Time) :
+    loop k (diag (fun p => d p.2) a) t = diag d (loop k a t) := by
+  ext j
+  simp only [loop_coord, eval, diag_apply, RCLike.real_smul_eq_coe_mul, mul_assoc,
+    ← mul_sum]
+
+ theorem loop_smul (k : ι → ℤ) (r : ℝ) (a : Coeff ι κ) :
+    loop k (r • a) = r • loop k a := (loopCLM k).map_smul r a
+
+ theorem loop_zero (k : ι → ℤ) : loop k (0 : Coeff ι κ) = 0 := (loopCLM k).map_zero
+
+end
+end FourierPolynomial
+
+namespace FourierPolynomial
+noncomputable section
+open MeasureTheory AddCircle Finset
+open scoped ComplexConjugate ContDiff
+local instance : Fact (0 < (1 : ℝ)) := ⟨zero_lt_one⟩
+variable {ι κ : Type} [Fintype ι] [Fintype κ]
+open DiagonalQuadratic
+
+ def remainderLoop (k : ι → ℤ) {H : Vector κ → ℝ} (hH : ContDiff ℝ ∞ H)
+    (d : κ → ℝ) (a : Coeff ι κ) : C(Time, Vector κ) :=
+  gradLoop k hH a - loop k ((2 : ℝ) • diag (fun p => d p.2) a)
+
+ theorem remainderLoop_apply (k : ι → ℤ) {H : Vector κ → ℝ} (hH : ContDiff ℝ ∞ H)
+    (d : κ → ℝ) (a : Coeff ι κ) (t : Time) :
+    remainderLoop k hH d a t = gradient H (loop k a t)-(2 : ℝ) • diag d (loop k a t) := by
+  simp only [remainderLoop, ContinuousMap.sub_apply, gradLoop, ContinuousMap.coe_mk,
+    loop_smul, ContinuousMap.smul_apply, loop_diag]
+
+ theorem gradient_action_remainder (k : ι → ℤ) (hk : Function.Injective k)
+    {H : Vector κ → ℝ} (hH : ContDiff ℝ ∞ H) (d : κ → ℝ) (a : Coeff ι κ) :
+    gradient (action k H) a =
+      (2 : ℝ) • diag (fun p => Real.pi*(k p.1 : ℝ)-d p.2) a-project k (remainderLoop k hH d a) := by
+  rw [gradient_action k hH, remainderLoop, project_sub, project_loop k hk]
+  have he : diag (fun p : ι × κ => Real.pi*(k p.1 : ℝ)-d p.2) a =
+      diag (fun p : ι × κ => Real.pi*(k p.1 : ℝ)) a-diag (fun p : ι × κ => d p.2) a := by
+    ext p
+    simp only [diag_apply, PiLp.sub_apply, sub_smul]
+  rw [he, smul_sub]
+  abel
+
+ theorem norm_gradient_lower (k : ι → ℤ) (hk : Function.Injective k)
+    {H : Vector κ → ℝ} (hH : ContDiff ℝ ∞ H) (d : κ → ℝ) {γ B : ℝ}
+    (hγ : 0 ≤ γ) (hB : 0 ≤ B) (hgap : ∀ i j, γ ≤ |Real.pi*(k i : ℝ)-d j|)
+    (hrem : ∀ z, ‖gradient H z-(2 : ℝ) • diag d z‖ ≤ B) (a : Coeff ι κ) :
+    2*γ*‖a‖ ≤ ‖gradient (action k H) a‖+B := by
+  have hb : ‖project k (remainderLoop k hH d a)‖ ≤ B :=
+    project_bounded k hk _ hB (fun t => by rw [remainderLoop_apply]; exact hrem _)
+  have hd := norm_diag_lower (fun p : ι × κ => Real.pi*(k p.1 : ℝ)-d p.2) hγ
+    (fun p => hgap p.1 p.2) a
+  have ht := norm_sub_le (gradient (action k H) a) (-project k (remainderLoop k hH d a))
+  rw [sub_neg_eq_add, norm_neg] at ht
+  have he : gradient (action k H) a + project k (remainderLoop k hH d a) =
+      (2 : ℝ) • diag (fun p => Real.pi*(k p.1 : ℝ)-d p.2) a := by
+    rw [gradient_action_remainder k hk hH]; abel
+  rw [he, norm_smul, Real.norm_eq_abs, abs_of_pos (by norm_num : (0 : ℝ) < 2)] at ht
+  nlinarith
+
+ theorem gradient_coercive (k : ι → ℤ) (hk : Function.Injective k)
+    {H : Vector κ → ℝ} (hH : ContDiff ℝ ∞ H) (d : κ → ℝ) {γ B : ℝ}
+    (hγ : 0 < γ) (hB : 0 ≤ B) (hgap : ∀ i j, γ ≤ |Real.pi*(k i : ℝ)-d j|)
+    (hrem : ∀ z, ‖gradient H z-(2 : ℝ) • diag d z‖ ≤ B) :
+    ∃ C : ℝ, ∀ a : Coeff ι κ, C ≤ ‖a‖ → 1 ≤ ‖gradient (action k H) a‖ := by
+  refine ⟨(B+1)/(2*γ), fun a ha => ?_⟩
+  have h := norm_gradient_lower k hk hH d hγ.le hB hgap hrem a
+  have hm := (div_le_iff₀ (mul_pos (by norm_num) hγ)).mp ha
+  nlinarith
+
+ theorem critical_norm_bound (k : ι → ℤ) (hk : Function.Injective k)
+    {H : Vector κ → ℝ} (hH : ContDiff ℝ ∞ H) (d : κ → ℝ) {γ B : ℝ}
+    (hγ : 0 < γ) (hB : 0 ≤ B) (hgap : ∀ i j, γ ≤ |Real.pi*(k i : ℝ)-d j|)
+    (hrem : ∀ z, ‖gradient H z-(2 : ℝ) • diag d z‖ ≤ B) (a : Coeff ι κ)
+    (ha : gradient (action k H) a = 0) : ‖a‖ ≤ B/(2*γ) := by
+  have h := norm_gradient_lower k hk hH d hγ.le hB hgap hrem a
+  rw [ha, norm_zero, zero_add] at h
+  rw [le_div_iff₀ (mul_pos (by norm_num) hγ)]
+  nlinarith
+
+end
+end FourierPolynomial
+
+end NonsqueezingInline
+
+end OAI
diff --git a/lean/OAI/Geometry/PolarProducts/RadialMap.lean b/lean/OAI/Geometry/PolarProducts/RadialMap.lean
new file mode 100644
index 0000000000000000000000000000000000000000..62d5264a24928f7c243d43bc59e5b79de056e683
--- /dev/null
+++ b/lean/OAI/Geometry/PolarProducts/RadialMap.lean
@@ -0,0 +1,363 @@
+import OAI.Geometry.PolarProducts.RegularizedMaximum
+
+namespace OAI
+
+universe u38 u39 u40
+
+section LowerBoundInline
+open Set Filter Function
+open scoped Topology ContDiff NNReal
+open Set Filter Metric
+open scoped Topology ContDiff
+open Set Filter Function MeasureTheory Metric
+open scoped Topology ContDiff NNReal
+open Set Filter Function
+open scoped Topology ContDiff
+open Set Filter Function
+open scoped Topology ContDiff NNReal
+open Set Filter
+open scoped Topology ContDiff
+open Set Filter Function
+open scoped Topology ContDiff
+open Set Filter Function
+open scoped ContDiff Topology
+open Set MeasureTheory
+open scoped ContDiff Interval Topology
+
+namespace RadialSymplectic
+
+open Set Filter Function TensorCalculus
+open scoped Topology ContDiff
+
+variable {E : Type u38} [NormedAddCommGroup E] [NormedSpace ℝ E]
+
+noncomputable section
+
+local instance oneTensorGroup : NormedAddCommGroup (E →L[ℝ] ℝ) := inferInstance
+local instance oneTensorSpace : NormedSpace ℝ (E →L[ℝ] ℝ) := inferInstance
+local instance twoTensorGroup : NormedAddCommGroup (E →L[ℝ] E →L[ℝ] ℝ) := inferInstance
+local instance twoTensorSpace : NormedSpace ℝ (E →L[ℝ] E →L[ℝ] ℝ) := inferInstance
+
+def radialMap (α : E → ℝ) (x : E) : E := α x • x
+
+def radialPrimitive (B : TwoTensor E) (α : E → ℝ) (x : E) : E →L[ℝ] ℝ :=
+  (α x ^ 2 / 2) • B x
+
+theorem fderiv_radialMap {α : E → ℝ} {x : E} (hα : DifferentiableAt ℝ α x) (a : E) :
+    fderiv ℝ (radialMap α) x a = α x • a + (fderiv ℝ α x a) • x := by
+  have hh := hα.hasFDerivAt.smul (hasFDerivAt_id x)
+  have he := congrArg (fun L : E →L[ℝ] E => L a) hh.fderiv
+  change fderiv ℝ (radialMap α) x a = _ at he
+  simpa only [add_apply, smul_apply, ContinuousLinearMap.id_apply,
+    ContinuousLinearMap.smulRight_apply, id_eq] using he
+
+theorem fderiv_radialPrimitive (B : TwoTensor E) {α : E → ℝ} {x : E}
+    (hα : DifferentiableAt ℝ α x) (a b : E) :
+    fderiv ℝ (radialPrimitive B α) x a b =
+      (α x ^ 2 / 2) * B a b + α x * fderiv ℝ α x a * B x b := by
+  have hs : HasFDerivAt (fun y => α y ^ 2 / 2) (α x • fderiv ℝ α x) x := by
+    convert! (hα.hasFDerivAt.pow 2).mul_const (1 / 2 : ℝ) using 1 <;>
+      simp [div_eq_mul_inv, smul_smul]
+  have hh := hs.smul B.hasFDerivAt
+  have he := congrArg (fun L : E →L[ℝ] E →L[ℝ] ℝ => L a b) hh.fderiv
+  change fderiv ℝ (radialPrimitive B α) x a b = _ at he
+  simpa only [add_apply, smul_apply, ContinuousLinearMap.smulRight_apply,
+    smul_eq_mul, mul_assoc] using he
+
+theorem radial_pullback (B : TwoTensor E) (hsk : ∀ a b, B a b = -B b a)
+    {α : E → ℝ} {x : E} (hα : DifferentiableAt ℝ α x) (a b : E) :
+    B (fderiv ℝ (radialMap α) x a) (fderiv ℝ (radialMap α) x b) =
+      exteriorD (radialPrimitive B α) x a b := by
+  have hz : B x x = 0 := by have hh := hsk x x; linarith
+  simp only [fderiv_radialMap hα, exteriorD_apply, fderiv_radialPrimitive B hα,
+    map_add, map_smul, add_apply, smul_apply, smul_eq_mul, hz, mul_zero, add_zero]
+  rw [hsk a x, hsk b a]
+  ring
+
+end
+end RadialSymplectic
+
+open Set
+open scoped Topology ContDiff
+
+namespace RadialSymplectic
+
+noncomputable section
+
+def radiusSq (μ δ s : ℝ) : ℝ := s + μ * (s / (s + δ))
+
+def inverseScaleSq (μ δ t : ℝ) : ℝ :=
+  2 * δ / (μ + δ - t + Real.sqrt ((μ + δ - t) ^ 2 + 4 * δ * t))
+
+theorem radiusSq_strictMonoOn {μ δ : ℝ} (hμ : 0 ≤ μ) (hδ : 0 < δ) :
+    StrictMonoOn (radiusSq μ δ) (Ici 0) := by
+  intro s hs t ht hst
+  have hsδ : 0 < s + δ := add_pos_of_nonneg_of_pos hs hδ
+  have htδ : 0 < t + δ := add_pos_of_nonneg_of_pos ht hδ
+  have hf : s / (s + δ) ≤ t / (t + δ) := by
+    apply (div_le_div_iff₀ hsδ htδ).mpr
+    nlinarith
+  have hh := mul_le_mul_of_nonneg_left hf hμ
+  dsimp only [radiusSq]
+  linarith
+
+theorem radial_discriminant_pos {μ δ : ℝ} (hμ : 0 < μ) (hδ : 0 < δ) (t : ℝ) :
+    0 < (μ + δ - t) ^ 2 + 4 * δ * t := by
+  nlinarith [sq_nonneg (t + δ - μ), mul_pos hμ hδ]
+
+theorem inverseScaleSq_den_pos {μ δ t : ℝ} (hμ : 0 < μ) (hδ : 0 < δ)
+    (_ht : 0 ≤ t) : 0 < μ + δ - t + Real.sqrt ((μ + δ - t) ^ 2 + 4 * δ * t) := by
+  have hsq := Real.sq_sqrt (radial_discriminant_pos hμ hδ t).le
+  have hr := Real.sqrt_nonneg ((μ + δ - t) ^ 2 + 4 * δ * t)
+  by_contra hc
+  have hc' : Real.sqrt ((μ + δ - t) ^ 2 + 4 * δ * t) ≤ -(μ + δ - t) := by
+    linarith
+  have htt : 0 < t := by linarith
+  have hmul := mul_self_le_mul_self hr hc'
+  nlinarith [mul_pos hδ htt]
+
+theorem inverseScaleSq_pos {μ δ t : ℝ} (hμ : 0 < μ) (hδ : 0 < δ) (ht : 0 ≤ t) :
+    0 < inverseScaleSq μ δ t :=
+  div_pos (mul_pos (by norm_num) hδ) (inverseScaleSq_den_pos hμ hδ ht)
+
+theorem inverseScaleSq_quadratic {μ δ t : ℝ} (hμ : 0 < μ) (hδ : 0 < δ)
+    (ht : 0 ≤ t) :
+    t * (inverseScaleSq μ δ t) ^ 2 + (μ + δ - t) * inverseScaleSq μ δ t - δ = 0 := by
+  have hD := (inverseScaleSq_den_pos hμ hδ ht).ne'
+  have hsq := Real.sq_sqrt (radial_discriminant_pos hμ hδ t).le
+  unfold inverseScaleSq
+  generalize Real.sqrt ((μ + δ - t) ^ 2 + 4 * δ * t) = R at hD hsq ⊢
+  field_simp [hD]
+  nlinarith
+
+theorem radiusSq_inverse {μ δ t : ℝ} (hμ : 0 < μ) (hδ : 0 < δ) (ht : 0 ≤ t) :
+    radiusSq μ δ (inverseScaleSq μ δ t * t) = t := by
+  have hq := inverseScaleSq_pos hμ hδ ht
+  have hquad := inverseScaleSq_quadratic hμ hδ ht
+  have hd : 0 < inverseScaleSq μ δ t * t + δ :=
+    add_pos_of_nonneg_of_pos (mul_nonneg hq.le ht) hδ
+  unfold radiusSq
+  field_simp [hd.ne']
+  nlinarith [mul_eq_zero_of_left hquad t]
+
+theorem inverseScaleSq_factor {μ δ t : ℝ} (hμ : 0 < μ) (hδ : 0 < δ) (ht : 0 ≤ t) :
+    (1 + μ / (inverseScaleSq μ δ t * t + δ)) * inverseScaleSq μ δ t = 1 := by
+  have hq := inverseScaleSq_pos hμ hδ ht
+  have hquad := inverseScaleSq_quadratic hμ hδ ht
+  have hd : 0 < inverseScaleSq μ δ t * t + δ :=
+    add_pos_of_nonneg_of_pos (mul_nonneg hq.le ht) hδ
+  field_simp [hd.ne']
+  nlinarith
+
+theorem contDiffAt_inverseScaleSq {μ δ t : ℝ} (hμ : 0 < μ) (hδ : 0 < δ) (ht : 0 ≤ t) :
+    ContDiffAt ℝ ∞ (inverseScaleSq μ δ) t := by
+  have hs : ContDiffAt ℝ ∞ (fun y => Real.sqrt ((μ + δ - y) ^ 2 + 4 * δ * y)) t :=
+    (Real.contDiffAt_sqrt (radial_discriminant_pos hμ hδ t).ne').comp t
+      (((contDiffAt_const.sub contDiffAt_id).pow 2).add (contDiffAt_const.mul contDiffAt_id))
+  exact contDiffAt_const.div ((contDiffAt_const.sub contDiffAt_id).add hs)
+    (inverseScaleSq_den_pos hμ hδ ht).ne'
+
+end
+end RadialSymplectic
+
+namespace RadialSymplectic
+
+open Set
+open scoped Topology ContDiff
+
+variable {E : Type u39} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+
+noncomputable section
+
+def expansionScale (μ δ : ℝ) (x : E) : ℝ := Real.sqrt (1 + μ / (‖x‖ ^ 2 + δ))
+def radialExpansion (μ δ : ℝ) : E → E := radialMap (expansionScale μ δ)
+def contractionScale (μ δ : ℝ) (x : E) : ℝ :=
+  Real.sqrt (inverseScaleSq μ δ (‖x‖ ^ 2))
+def radialContraction (μ δ : ℝ) : E → E := radialMap (contractionScale μ δ)
+
+omit [InnerProductSpace ℝ E] in
+theorem expansionScale_pos {μ δ : ℝ} (hμ : 0 < μ) (hδ : 0 < δ) (x : E) :
+    0 < expansionScale μ δ x := by
+  unfold expansionScale
+  apply Real.sqrt_pos.mpr
+  have hd : 0 < ‖x‖ ^ 2 + δ := add_pos_of_nonneg_of_pos (sq_nonneg _) hδ
+  positivity
+
+omit [InnerProductSpace ℝ E] in
+theorem expansionScale_sq {μ δ : ℝ} (hμ : 0 < μ) (hδ : 0 < δ) (x : E) :
+    expansionScale μ δ x ^ 2 = 1 + μ / (‖x‖ ^ 2 + δ) := by
+  unfold expansionScale
+  apply Real.sq_sqrt
+  have hd : 0 < ‖x‖ ^ 2 + δ := add_pos_of_nonneg_of_pos (sq_nonneg _) hδ
+  positivity
+
+theorem norm_sq_radialExpansion {μ δ : ℝ} (hμ : 0 < μ) (hδ : 0 < δ) (x : E) :
+    ‖radialExpansion μ δ x‖ ^ 2 = radiusSq μ δ (‖x‖ ^ 2) := by
+  simp only [radialExpansion, radialMap, norm_smul, Real.norm_eq_abs, mul_pow, sq_abs]
+  rw [expansionScale_sq hμ hδ]
+  unfold radiusSq
+  ring
+
+theorem radialExpansion_injective {μ δ : ℝ} (hμ : 0 < μ) (hδ : 0 < δ) :
+    Function.Injective (radialExpansion μ δ : E → E) := by
+  intro x y h
+  have hh := congrArg (fun z : E => ‖z‖ ^ 2) h
+  rw [norm_sq_radialExpansion hμ hδ, norm_sq_radialExpansion hμ hδ] at hh
+  have hnorm := (radiusSq_strictMonoOn hμ.le hδ).injOn (sq_nonneg ‖x‖) (sq_nonneg ‖y‖) hh
+  have hscale : expansionScale μ δ x = expansionScale μ δ y := by
+    unfold expansionScale
+    rw [hnorm]
+  dsimp only [radialExpansion, radialMap] at h
+  rw [hscale] at h
+  exact (smul_right_injective E (expansionScale_pos hμ hδ y).ne') h
+
+omit [InnerProductSpace ℝ E] in
+theorem contractionScale_pos {μ δ : ℝ} (hμ : 0 < μ) (hδ : 0 < δ) (x : E) :
+    0 < contractionScale μ δ x :=
+  Real.sqrt_pos.mpr (inverseScaleSq_pos hμ hδ (sq_nonneg _))
+
+theorem norm_sq_radialContraction {μ δ : ℝ} (hμ : 0 < μ) (hδ : 0 < δ) (x : E) :
+    ‖radialContraction μ δ x‖ ^ 2 = inverseScaleSq μ δ (‖x‖ ^ 2) * ‖x‖ ^ 2 := by
+  simp only [radialContraction, radialMap, norm_smul, Real.norm_eq_abs, mul_pow, sq_abs]
+  rw [contractionScale, Real.sq_sqrt (inverseScaleSq_pos hμ hδ (sq_nonneg _)).le]
+
+theorem expansionScale_mul_contractionScale {μ δ : ℝ} (hμ : 0 < μ) (hδ : 0 < δ)
+    (x : E) : expansionScale μ δ (radialContraction μ δ x) * contractionScale μ δ x = 1 := by
+  have hp := expansionScale_pos hμ hδ (radialContraction μ δ x)
+  have hq := contractionScale_pos hμ hδ x
+  have hsq : (expansionScale μ δ (radialContraction μ δ x) * contractionScale μ δ x)^2 = 1 := by
+    rw [mul_pow, expansionScale_sq hμ hδ, contractionScale,
+      Real.sq_sqrt (inverseScaleSq_pos hμ hδ (sq_nonneg _)).le,
+      norm_sq_radialContraction hμ hδ]
+    exact inverseScaleSq_factor hμ hδ (sq_nonneg _)
+  nlinarith [mul_pos hp hq]
+
+theorem radialExpansion_radialContraction {μ δ : ℝ} (hμ : 0 < μ) (hδ : 0 < δ)
+    (x : E) : radialExpansion μ δ (radialContraction μ δ x) = x := by
+  change expansionScale μ δ (radialContraction μ δ x) • (contractionScale μ δ x • x) = x
+  rw [smul_smul, expansionScale_mul_contractionScale hμ hδ, one_smul]
+
+theorem radialContraction_radialExpansion {μ δ : ℝ} (hμ : 0 < μ) (hδ : 0 < δ)
+    (x : E) : radialContraction μ δ (radialExpansion μ δ x) = x :=
+  radialExpansion_injective hμ hδ (radialExpansion_radialContraction hμ hδ _)
+
+theorem contDiff_expansionScale {μ δ : ℝ} (hμ : 0 < μ) (hδ : 0 < δ) :
+    ContDiff ℝ ∞ (expansionScale μ δ : E → ℝ) := by
+  have hn : ContDiff ℝ ∞ (fun x : E => ‖x‖^2 + δ) := (contDiff_norm_sq ℝ).add contDiff_const
+  have hd (x : E) : ‖x‖ ^ 2 + δ ≠ 0 := (add_pos_of_nonneg_of_pos (sq_nonneg _) hδ).ne'
+  have hp (x : E) : 1 + μ / (‖x‖ ^ 2 + δ) ≠ 0 := by
+    have hpp : 0 < ‖x‖ ^ 2 + δ := add_pos_of_nonneg_of_pos (sq_nonneg _) hδ
+    positivity
+  exact (contDiff_const.add (contDiff_const.div hn hd)).sqrt hp
+
+theorem contDiff_contractionScale {μ δ : ℝ} (hμ : 0 < μ) (hδ : 0 < δ) :
+    ContDiff ℝ ∞ (contractionScale μ δ : E → ℝ) := by
+  rw [contDiff_iff_contDiffAt]
+  intro x
+  have hi : ContDiffAt ℝ ∞ (fun z : E => inverseScaleSq μ δ (‖z‖ ^ 2)) x :=
+    (contDiffAt_inverseScaleSq hμ hδ (sq_nonneg ‖x‖)).comp (f := fun z : E => ‖z‖^2) x
+      (contDiff_norm_sq ℝ).contDiffAt
+  exact (Real.contDiffAt_sqrt (inverseScaleSq_pos hμ hδ (sq_nonneg ‖x‖)).ne').comp
+    (f := fun z : E => inverseScaleSq μ δ (‖z‖ ^ 2)) x hi
+
+theorem contDiff_radialExpansion {μ δ : ℝ} (hμ : 0 < μ) (hδ : 0 < δ) :
+    ContDiff ℝ ∞ (radialExpansion μ δ : E → E) :=
+  (contDiff_expansionScale hμ hδ).smul contDiff_id
+
+theorem contDiff_radialContraction {μ δ : ℝ} (hμ : 0 < μ) (hδ : 0 < δ) :
+    ContDiff ℝ ∞ (radialContraction μ δ : E → E) :=
+  (contDiff_contractionScale hμ hδ).smul contDiff_id
+
+def radialHomeomorph {μ δ : ℝ} (hμ : 0 < μ) (hδ : 0 < δ) : E ≃ₜ E where
+  toFun := radialExpansion μ δ
+  invFun := radialContraction μ δ
+  left_inv := radialContraction_radialExpansion hμ hδ
+  right_inv := radialExpansion_radialContraction hμ hδ
+  continuous_toFun := (contDiff_radialExpansion hμ hδ).continuous
+  continuous_invFun := (contDiff_radialContraction hμ hδ).continuous
+
+theorem radialContraction_mapsTo {μ δ a r : ℝ} (hμ : 0 < μ) (hδ : 0 < δ)
+    (hbound : a ≤ radiusSq μ δ (r^2)) :
+    MapsTo (radialContraction μ δ : E → E) {x | ‖x‖^2 < a} {x | ‖x‖^2 < r^2} := by
+  intro x hx
+  have hi := radiusSq_inverse hμ hδ (sq_nonneg ‖x‖)
+  have hh : radiusSq μ δ (‖radialContraction μ δ x‖^2) < radiusSq μ δ (r^2) := by
+    rw [norm_sq_radialContraction hμ hδ, hi]
+    exact lt_of_lt_of_le hx hbound
+  exact ((radiusSq_strictMonoOn hμ.le hδ).lt_iff_lt
+    (show ‖radialContraction μ δ x‖^2 ∈ Ici (0 : ℝ) from sq_nonneg ‖radialContraction μ δ x‖)
+    (show r^2 ∈ Ici (0 : ℝ) from sq_nonneg r)).mp hh
+
+end
+end RadialSymplectic
+
+namespace RadialSymplectic
+
+open ComplexPotential TensorCalculus
+open scoped ContDiff
+
+noncomputable section
+
+variable {E : Type u40} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+
+def modelPotential (μ δ : ℝ) (x : E) : ℝ := ‖x‖^2 + μ * Real.log (‖x‖^2 + δ)
+
+theorem contDiff_modelPotential (μ : ℝ) {δ : ℝ} (hδ : 0 < δ) :
+    ContDiff ℝ ∞ (modelPotential μ δ : E → ℝ) :=
+  (contDiff_norm_sq ℝ).add (contDiff_const.mul
+    (((contDiff_norm_sq ℝ).add contDiff_const).log
+      (fun x => (add_pos_of_nonneg_of_pos (sq_nonneg ‖x‖) hδ).ne')))
+
+theorem dc_modelPotential (J : E →L[ℝ] E)
+    (hsk : ∀ a b, inner (𝕜 := ℝ) a (J b) = -inner (𝕜 := ℝ) (J a) b)
+    {μ δ : ℝ} (hμ : 0 < μ) (hδ : 0 < δ) (x : E) :
+    dc J (modelPotential μ δ) x = radialPrimitive (standardTensor J) (expansionScale μ δ) x := by
+  let h : ℝ → ℝ := fun t => t + μ * Real.log (t + δ)
+  have hp : ‖x‖^2 + δ ≠ 0 := (add_pos_of_nonneg_of_pos (sq_nonneg _) hδ).ne'
+  have hd : HasDerivAt h (1 + μ * (‖x‖^2 + δ)⁻¹) (‖x‖^2) := by
+    convert (hasDerivAt_id (‖x‖^2)).add
+      ((((hasDerivAt_id (‖x‖^2)).add_const δ).log hp).const_mul μ) using 1 <;> first | rfl | simp [one_div]
+  change dc J (h ∘ (fun y : E => ‖y‖^2)) x = _
+  rw [dc_scalar_comp J ((contDiff_norm_sq ℝ).differentiable (show (∞ : ℕ∞ω) ≠ 0 by simp) x)
+    hd.differentiableAt, hd.deriv, dc_norm_sq J hsk, smul_smul]
+  unfold radialPrimitive
+  rw [expansionScale_sq hμ hδ]
+  congr 1
+  ring
+
+theorem radialExpansion_pullback (J : E →L[ℝ] E)
+    (hsk : ∀ a b, inner (𝕜 := ℝ) a (J b) = -inner (𝕜 := ℝ) (J a) b)
+    {μ δ : ℝ} (hμ : 0 < μ) (hδ : 0 < δ) (x a b : E) :
+    standardTensor J (fderiv ℝ (radialExpansion μ δ : E → E) x a)
+      (fderiv ℝ (radialExpansion μ δ : E → E) x b) = ddc J (modelPotential μ δ) x a b := by
+  have hsk' (a b : E) : standardTensor J a b = -standardTensor J b a := by
+    change inner (𝕜 := ℝ) (J a) b = -inner (𝕜 := ℝ) (J b) a
+    rw [← real_inner_comm (J a) b, hsk]
+  have he : dc J (modelPotential μ δ) = radialPrimitive (standardTensor J) (expansionScale μ δ) :=
+    funext (dc_modelPotential J hsk hμ hδ)
+  rw [ddc, he]
+  exact radial_pullback (standardTensor J) hsk' ((contDiff_expansionScale hμ hδ).differentiable
+    (by simp) x) a b
+
+theorem radialContraction_pullback (J : E →L[ℝ] E)
+    (hsk : ∀ a b, inner (𝕜 := ℝ) a (J b) = -inner (𝕜 := ℝ) (J a) b)
+    {μ δ : ℝ} (hμ : 0 < μ) (hδ : 0 < δ) (x a b : E) :
+    ddc J (modelPotential μ δ) (radialContraction μ δ x)
+      (fderiv ℝ (radialContraction μ δ : E → E) x a)
+      (fderiv ℝ (radialContraction μ δ : E → E) x b) = standardTensor J a b := by
+  have hd (v : E) : fderiv ℝ (radialExpansion μ δ : E → E) (radialContraction μ δ x)
+      (fderiv ℝ (radialContraction μ δ : E → E) x v) = v := by
+    have he : (radialExpansion μ δ : E → E) ∘ radialContraction μ δ = id :=
+      funext (radialExpansion_radialContraction hμ hδ)
+    have hh := fderiv_comp x ((contDiff_radialExpansion hμ hδ).differentiable (by simp)
+      (radialContraction μ δ x)) ((contDiff_radialContraction hμ hδ).differentiable (by simp) x)
+    rw [he, fderiv_id] at hh
+    exact (congrArg (fun L : E →L[ℝ] E => L v) hh).symm
+  rw [← radialExpansion_pullback J hsk hμ hδ, hd, hd]
+
+end
+end RadialSymplectic
+
+end LowerBoundInline
+
+end OAI
diff --git a/lean/OAI/Geometry/PolarProducts/RegularizedMaximum.lean b/lean/OAI/Geometry/PolarProducts/RegularizedMaximum.lean
new file mode 100644
index 0000000000000000000000000000000000000000..5bd7d08731f28a967cf357bfe25d565ee7ac1978
--- /dev/null
+++ b/lean/OAI/Geometry/PolarProducts/RegularizedMaximum.lean
@@ -0,0 +1,248 @@
+import OAI.Geometry.PolarProducts.PotentialDeformation
+
+namespace OAI
+
+universe u36 u37
+
+section LowerBoundInline
+open Set Filter Function
+open scoped Topology ContDiff NNReal
+open Set Filter Metric
+open scoped Topology ContDiff
+open Set Filter Function MeasureTheory Metric
+open scoped Topology ContDiff NNReal
+open Set Filter Function
+open scoped Topology ContDiff
+open Set Filter Function
+open scoped Topology ContDiff NNReal
+open Set Filter
+open scoped Topology ContDiff
+open Set Filter Function
+open scoped Topology ContDiff
+open Set Filter Function
+open scoped ContDiff Topology
+
+open Set MeasureTheory
+open scoped ContDiff Interval Topology
+
+namespace SmoothPositivePart
+
+noncomputable section
+
+def primitive (t : ℝ) : ℝ := ∫ s in (0 : ℝ)..t, Real.smoothTransition s
+
+theorem hasDerivAt_primitive (t : ℝ) : HasDerivAt primitive (Real.smoothTransition t) t := by
+  apply intervalIntegral.integral_hasDerivAt_right
+  · exact Real.smoothTransition.continuous.intervalIntegrable _ _
+  · exact ContinuousAt.stronglyMeasurableAtFilter isOpen_univ
+      (fun _ _ => Real.smoothTransition.continuousAt) t (mem_univ t)
+  · exact Real.smoothTransition.continuousAt
+
+@[simp] theorem deriv_primitive : deriv primitive = Real.smoothTransition :=
+  funext (fun t => (hasDerivAt_primitive t).deriv)
+
+theorem primitive_smooth : ContDiff ℝ ∞ primitive := by
+  rw [contDiff_infty_iff_deriv]
+  exact ⟨fun t => (hasDerivAt_primitive t).differentiableAt,
+    deriv_primitive ▸ Real.smoothTransition.contDiff⟩
+
+theorem primitive_of_nonpos {t : ℝ} (ht : t ≤ 0) : primitive t = 0 := by
+  calc
+    primitive t = ∫ s in (0 : ℝ)..t, (0 : ℝ) := by
+      apply intervalIntegral.integral_congr
+      intro s hs
+      apply Real.smoothTransition.zero_of_nonpos
+      exact le_trans hs.2 (max_eq_left ht).le
+    _ = 0 := intervalIntegral.integral_zero
+
+theorem primitive_one_mem : primitive 1 ∈ Icc (0 : ℝ) 1 := by
+  constructor
+  · apply intervalIntegral.integral_nonneg (by norm_num)
+    intro s _
+    exact Real.smoothTransition.nonneg s
+  · calc
+      primitive 1 ≤ ∫ s in (0 : ℝ)..1, (1 : ℝ) := by
+        apply intervalIntegral.integral_mono_on (by norm_num)
+        · exact Real.smoothTransition.continuous.intervalIntegrable _ _
+        · exact continuous_const.intervalIntegrable _ _
+        · intro s _
+          exact Real.smoothTransition.le_one s
+      _ = 1 := by simp
+
+theorem primitive_of_one_le {t : ℝ} (ht : 1 ≤ t) : primitive t = t + (primitive 1 - 1) := by
+  have he : (∫ s in (1 : ℝ)..t, Real.smoothTransition s) = t - 1 := by
+    calc
+      _ = ∫ s in (1 : ℝ)..t, (1 : ℝ) := by
+        apply intervalIntegral.integral_congr
+        intro s hs
+        exact Real.smoothTransition.one_of_one_le (le_trans (by simp [ht]) hs.1)
+      _ = t - 1 := by simp
+  have hadd := intervalIntegral.integral_add_adjacent_intervals
+    (Real.smoothTransition.continuous.intervalIntegrable (μ := volume) (0 : ℝ) 1)
+    (Real.smoothTransition.continuous.intervalIntegrable (1 : ℝ) t)
+  change primitive 1 + _ = primitive t at hadd
+  rw [he] at hadd
+  linarith
+
+def shift : ℝ := primitive 1 - 1
+
+theorem shift_mem : shift ∈ Icc (-1 : ℝ) 0 := by
+  have h := primitive_one_mem
+  change -1 ≤ primitive 1 - 1 ∧ primitive 1 - 1 ≤ 0
+  constructor <;> linarith [h.1, h.2]
+
+def positivePart (t : ℝ) : ℝ := primitive (t - shift)
+
+theorem positivePart_smooth : ContDiff ℝ ∞ positivePart :=
+  primitive_smooth.comp (contDiff_id.sub contDiff_const)
+
+theorem positivePart_zero {t : ℝ} (ht : t ≤ -1) : positivePart t = 0 := by
+  apply primitive_of_nonpos
+  linarith [shift_mem.1]
+
+theorem positivePart_eq_self {t : ℝ} (ht : 1 ≤ t) : positivePart t = t := by
+  rw [positivePart, primitive_of_one_le (by linarith [shift_mem.2])]
+  unfold shift
+  ring
+
+theorem deriv_positivePart (t : ℝ) : deriv positivePart t = Real.smoothTransition (t - shift) := by
+  change deriv (fun t => primitive (t - shift)) t = _
+  simpa only [mul_one, Function.comp_def, id_eq] using
+    ((hasDerivAt_primitive (t - shift)).comp t ((hasDerivAt_id t).sub_const shift)).deriv
+
+theorem deriv_positivePart_mem (t : ℝ) : deriv positivePart t ∈ Icc (0 : ℝ) 1 := by
+  rw [deriv_positivePart]
+  exact ⟨Real.smoothTransition.nonneg _, Real.smoothTransition.le_one _⟩
+
+theorem second_deriv_positivePart_nonneg (t : ℝ) : 0 ≤ deriv (deriv positivePart) t := by
+  apply Monotone.deriv_nonneg
+  intro x y hxy
+  rw [deriv_positivePart, deriv_positivePart]
+  exact Real.smoothTransition.monotone (sub_le_sub_right hxy _)
+
+def smoothMax (h p : ℝ) : ℝ := p + positivePart (h - p)
+
+theorem smoothMax_left {h p : ℝ} (hp : p + 1 ≤ h) : smoothMax h p = h := by
+  rw [smoothMax, positivePart_eq_self (by linarith)]
+  ring
+
+theorem smoothMax_right {h p : ℝ} (hp : h + 1 ≤ p) : smoothMax h p = p := by
+  rw [smoothMax, positivePart_zero (by linarith), add_zero]
+
+end
+end SmoothPositivePart
+
+namespace ComplexPotential
+
+open TensorCalculus SmoothPositivePart Set Filter
+open scoped ContDiff Topology
+
+variable {E : Type u36} [NormedAddCommGroup E] [NormedSpace ℝ E]
+
+noncomputable section
+
+local instance maxOneGroup : NormedAddCommGroup (E →L[ℝ] ℝ) := inferInstance
+local instance maxOneSpace : NormedSpace ℝ (E →L[ℝ] ℝ) := inferInstance
+local instance maxTwoGroup : NormedAddCommGroup (E →L[ℝ] E →L[ℝ] ℝ) := inferInstance
+local instance maxTwoSpace : NormedSpace ℝ (E →L[ℝ] E →L[ℝ] ℝ) := inferInstance
+
+theorem IsPSHAt.smoothMax {J : E →L[ℝ] E} (hJ : ∀ a, J (J a) = -a)
+    {φ ψ : E → ℝ} {x : E} (hφ : IsPSHAt J φ x) (hψ : IsPSHAt J ψ x) :
+    IsPSHAt J (fun y => smoothMax (φ y) (ψ y)) x := by
+  have hdiff := hφ.1.sub hψ.1
+  have hs : ContDiffAt ℝ ∞ (positivePart ∘ fun y => φ y - ψ y) x :=
+    positivePart_smooth.contDiffAt.comp x hdiff
+  refine ⟨hψ.1.add hs, fun a => ?_⟩
+  change 0 ≤ ddc J (fun y => ψ y + (positivePart ∘ fun z => φ z - ψ z) y) x a (J a)
+  rw [ddc_add J hψ.1 hs, add_apply, add_apply,
+    ddc_scalar_comp_complex_line J hJ hdiff positivePart_smooth.contDiffAt,
+    ddc_sub J hφ.1 hψ.1, sub_apply, sub_apply]
+  have hβ := deriv_positivePart_mem (φ x - ψ x)
+  have h2 := second_deriv_positivePart_nonneg (φ x - ψ x)
+  have hpos : 0 ≤ (1 - deriv positivePart (φ x - ψ x)) * ddc J ψ x a (J a) +
+      deriv positivePart (φ x - ψ x) * ddc J φ x a (J a) :=
+    add_nonneg (mul_nonneg (sub_nonneg.mpr hβ.2) (hψ.2 a))
+      (mul_nonneg hβ.1 (hφ.2 a))
+  have hquad : 0 ≤ deriv (deriv positivePart) (φ x - ψ x) *
+      ((fderiv ℝ (fun y => φ y - ψ y) x a)^2 +
+        (fderiv ℝ (fun y => φ y - ψ y) x (J a))^2) / 4 := by positivity
+  nlinarith
+
+theorem IsPSHAt.congr {J : E →L[ℝ] E} {φ ψ : E → ℝ} {x : E}
+    (hφ : IsPSHAt J φ x) (he : ψ =ᶠ[𝓝 x] φ) : IsPSHAt J ψ x := by
+  refine ⟨hφ.1.congr_of_eventuallyEq he, fun a => ?_⟩
+  rw [ddc_congr J he]
+  exact hφ.2 a
+
+end
+end ComplexPotential
+
+namespace ComplexPotential
+
+open SmoothPositivePart Set Filter
+open scoped ContDiff Topology
+
+variable {E : Type u37} [NormedAddCommGroup E] [NormedSpace ℝ E]
+
+noncomputable section
+
+omit [NormedSpace ℝ E] in
+
+def gluedPotential (R : ℝ) (H P : E → ℝ) (x : E) : ℝ := by
+  classical
+  exact if ‖x‖ < R then if x = 0 then P x else smoothMax (H x) (P x) else H x
+
+omit [NormedSpace ℝ E] in
+theorem gluedPotential_outer (R : ℝ) (H P : E → ℝ) {x : E} (hx : R ≤ ‖x‖) :
+    gluedPotential R H P x = H x := by
+  classical
+  simp [gluedPotential, not_lt.mpr hx]
+
+omit [NormedSpace ℝ E] in
+theorem gluedPotential_inner {r R : ℝ} (hrR : r < R) (H P : E → ℝ)
+    (hdom : ∀ x : E, x ≠ 0 → ‖x‖ ≤ r → H x + 1 ≤ P x) {x : E} (hx : ‖x‖ ≤ r) :
+    gluedPotential R H P x = P x := by
+  classical
+  have hR : ‖x‖ < R := hx.trans_lt hrR
+  by_cases hzero : x = 0
+  · rw [gluedPotential, ite_eq_left hR, ite_eq_left hzero]
+  · rw [gluedPotential, ite_eq_left hR, ite_eq_right hzero, smoothMax_right (hdom x hzero hx)]
+
+theorem isPSHAt_gluedPotential (J : E →L[ℝ] E) (hJ : ∀ a, J (J a) = -a)
+    {U : Set E} {r R : ℝ} (hr : 0 < r) (hrR : r < R) (H P : E → ℝ)
+    (hH : ∀ x ∈ U, x ≠ 0 → IsPSHAt J H x)
+    (hP : ∀ x ∈ U, IsPSHAt J P x)
+    (hinner : ∀ x : E, x ≠ 0 → ‖x‖ ≤ r → H x + 1 ≤ P x)
+    (houter : ∀ x ∈ U, ‖x‖ = R → P x + 1 < H x)
+    {x : E} (hx : x ∈ U) : IsPSHAt J (gluedPotential R H P) x := by
+  classical
+  have hRpos : 0 < R := hr.trans hrR
+  by_cases hzero : x = 0
+  · have hn : ‖x‖ < r := by simpa [hzero] using hr
+    apply (hP x hx).congr
+    filter_upwards [continuous_norm.continuousAt.eventually_lt continuousAt_const hn] with y hy
+    exact gluedPotential_inner hrR H P hinner hy.le
+  have hHx := hH x hx hzero
+  have hPx := hP x hx
+  rcases lt_trichotomy ‖x‖ R with hn | hn | hn
+  · apply (hHx.smoothMax hJ hPx).congr
+    filter_upwards [continuous_norm.continuousAt.eventually_lt continuousAt_const hn,
+      eventually_ne_nhds hzero] with y hy hy0
+    simp [gluedPotential, hy, hy0]
+  · apply hHx.congr
+    have hlt := houter x hx hn
+    filter_upwards [(hPx.1.continuousAt.add continuousAt_const).eventually_lt
+      hHx.1.continuousAt hlt, eventually_ne_nhds hzero] with y hy hy0
+    by_cases hny : ‖y‖ < R
+    · simp [gluedPotential, hny, hy0, smoothMax_left hy.le]
+    · simp [gluedPotential, hny]
+  · apply hHx.congr
+    filter_upwards [continuousAt_const.eventually_lt continuous_norm.continuousAt hn] with y hy
+    exact gluedPotential_outer R H P hy.le
+
+end
+end ComplexPotential
+
+end LowerBoundInline
+
+end OAI
diff --git a/lean/OAI/Geometry/PolarProducts/SmoothDegree.lean b/lean/OAI/Geometry/PolarProducts/SmoothDegree.lean
new file mode 100644
index 0000000000000000000000000000000000000000..30ec2683e138cb70c06209134d718790527231e1
--- /dev/null
+++ b/lean/OAI/Geometry/PolarProducts/SmoothDegree.lean
@@ -0,0 +1,362 @@
+import OAI.Geometry.PolarProducts.LowerBound
+
+namespace OAI
+
+universe u78 u79
+
+section NonsqueezingInline
+section
+
+open Set Filter Function MeasureTheory
+open scoped Topology ContDiff
+
+namespace SmoothDegree
+noncomputable section
+
+abbrev Space (d : ℕ) := Fin d → ℝ
+
+def basis (d : ℕ) : Fin d → Space d := fun i => Pi.single i 1
+
+def vol (d : ℕ) : (Space d) [⋀^Fin d]→L[ℝ] ℝ :=
+  { Matrix.detRowAlternating with cont := continuous_id.matrix_det }
+
+@[simp] theorem vol_apply (d : ℕ) (v : Fin d → Space d) : vol d v = Matrix.det v := rfl
+
+@[simp] theorem vol_basis (d : ℕ) : vol d (basis d) = 1 := by
+  have he : basis d = (1 : Matrix (Fin d) (Fin d) ℝ) := by
+    ext i j
+    simp [basis, Matrix.one_apply, Pi.single_apply, eq_comm]
+  change Matrix.det (basis d) = 1
+  rw [he, Matrix.det_one]
+
+ theorem vol_update_basis {d : ℕ} (i : Fin d) (v : Space d) :
+    vol d (Function.update (basis d) i v) = v i := by
+  change Matrix.det (Matrix.updateRow (basis d) i v) = v i
+  have hv : v = ∑ j, v j • (basis d) j := by
+    ext k
+    simp [basis, Pi.single_apply]
+  calc
+    _ = Matrix.det (Matrix.updateRow (basis d) i (∑ j, v j • (basis d) j)) :=
+      congrArg (fun w => Matrix.det (Matrix.updateRow (basis d) i w)) hv
+    _ = v i • Matrix.det (basis d) := Matrix.det_updateRow_sum (basis d) i v
+    _ = v i := by rw [show Matrix.det (basis d) = 1 from vol_basis d]; simp
+
+ theorem contract_basis {n : ℕ} (i : Fin (n+1)) (v : Space (n+1)) :
+    (vol (n+1)).curryLeft v (i.removeNth (basis (n+1))) = (-1 : ℝ)^i.val * v i := by
+  rw [ContinuousAlternatingMap.curryLeft_apply_apply,
+    ← ContinuousAlternatingMap.neg_one_pow_smul_map_insertNth,
+    Fin.insertNth_removeNth, vol_update_basis]
+  simp only [zsmul_eq_mul, Int.cast_pow, Int.cast_neg, Int.cast_one]
+
+ theorem extDeriv_contract_basis {n : ℕ} {V : Space (n+1) → Space (n+1)}
+    {x : Space (n+1)} {V' : Space (n+1) →L[ℝ] Space (n+1)} (hV : HasFDerivAt V V' x) :
+    extDeriv (fun y => (vol (n+1)).curryLeft (V y)) x (basis (n+1)) =
+      ∑ i, V' (basis (n+1) i) i := by
+  have hc : DifferentiableAt ℝ (fun y => (vol (n+1)).curryLeft (V y)) x :=
+    (vol (n+1)).curryLeft.differentiableAt.comp x hV.differentiableAt
+  rw [extDeriv_apply hc]
+  apply Finset.sum_congr rfl
+  intro i hi
+  have he : (fun y => (vol (n+1)).curryLeft (V y) (i.removeNth (basis (n+1)))) =
+      fun y => (-1 : ℝ)^i.val * (V y) i := funext fun y => contract_basis i (V y)
+  have hd : HasFDerivAt (fun y => (-1 : ℝ)^i.val * (V y) i)
+      (((-1 : ℝ)^i.val) • ((ContinuousLinearMap.proj i).comp V')) x :=
+    (((ContinuousLinearMap.proj i : Space (n+1) →L[ℝ] ℝ).hasFDerivAt).comp x hV).const_mul _
+  rw [he, hd.fderiv]
+  simp only [smul_apply, ContinuousLinearMap.comp_apply,
+    ContinuousLinearMap.proj_apply, smul_eq_mul, zsmul_eq_mul, Int.cast_pow, Int.cast_neg, Int.cast_one]
+  rw [← mul_assoc, ← pow_add, show i.val+i.val = 2*i.val by omega,
+    pow_mul, neg_one_sq, one_pow, one_mul]
+
+ theorem extDeriv_contract_eq_zero {n : ℕ} {V : Space (n+1) → Space (n+1)}
+    {x : Space (n+1)} (hV : DifferentiableAt ℝ V x)
+    (hdiv : ∑ i, fderiv ℝ V x (basis (n+1) i) i = 0) :
+    extDeriv (fun y => (vol (n+1)).curryLeft (V y)) x = 0 := by
+  apply ContinuousAlternatingMap.toAlternatingMap_injective
+  apply ((extDeriv (fun y => (vol (n+1)).curryLeft (V y)) x).toAlternatingMap.map_basis_eq_zero_iff
+    (Pi.basisFun ℝ (Fin (n+1)))).mp
+  have he : (Pi.basisFun ℝ (Fin (n+1)) : Fin (n+1) → Space (n+1)) = basis (n+1) := by
+    ext i j; simp [basis, Pi.basisFun_apply]
+  change extDeriv (fun y => (vol (n+1)).curryLeft (V y)) x _ = 0
+  rw [he, extDeriv_contract_basis hV.hasFDerivAt, hdiv]
+
+ def sqsum {d : ℕ} (x : Space d) : ℝ := ∑ i, x i ^ 2
+
+ theorem sqsum_pos {d : ℕ} {x : Space d} (hx : x ≠ 0) : 0 < sqsum x := by
+  obtain ⟨i, hi⟩ : ∃ i, x i ≠ 0 := by contrapose! hx; ext i; exact hx i
+  exact Finset.sum_pos' (fun i _ => sq_nonneg (x i)) ⟨i, Finset.mem_univ _, sq_pos_of_ne_zero hi⟩
+
+ def dot {d : ℕ} (x : Space d) : Space d →L[ℝ] ℝ :=
+   ∑ i, x i • ContinuousLinearMap.proj i
+
+ @[simp] theorem dot_apply {d : ℕ} (x v : Space d) : dot x v = ∑ i, x i * v i := by
+  simp [dot]
+
+ theorem hasFDerivAt_sqsum {d : ℕ} (x : Space d) :
+    HasFDerivAt sqsum ((2 : ℝ) • dot x) x := by
+  have h := HasFDerivAt.sum (u := Finset.univ)
+    (fun i _ => ((ContinuousLinearMap.proj i : Space d →L[ℝ] ℝ).hasFDerivAt (x := x)).pow 2)
+  convert! h using 1
+  · ext y
+    simp [sqsum]
+  · ext v
+    simp [dot, Finset.mul_sum, mul_assoc]
+
+ def radial {d : ℕ} (x : Space d) : Space d :=
+   (sqsum x ^ (-(d : ℝ)/2)) • x
+
+ theorem hasFDerivAt_radial {d : ℕ} {x : Space d} (hx : x ≠ 0) :
+    HasFDerivAt radial
+      ((((-(d : ℝ)/2) * sqsum x ^ (-(d : ℝ)/2 - 1)) • ((2 : ℝ) • dot x)).smulRight x +
+      (sqsum x ^ (-(d : ℝ)/2)) • ContinuousLinearMap.id ℝ (Space d)) x := by
+  convert! ((hasFDerivAt_sqsum x).rpow_const (p := -(d : ℝ)/2)
+      (Or.inl (sqsum_pos hx).ne')).smul (hasFDerivAt_id x) using 1
+  exact add_comm _ _
+
+ theorem radial_divergence {d : ℕ} {x : Space d} (hx : x ≠ 0) :
+    ∑ i, fderiv ℝ (@radial d) x (basis d i) i = 0 := by
+  rw [(hasFDerivAt_radial hx).fderiv]
+  have hb (i : Fin d) : dot x (basis d i) = x i := by
+    simp [dot_apply, basis, Pi.single_apply, eq_comm]
+  simp only [add_apply, Pi.add_apply, smul_apply,
+    ContinuousLinearMap.smulRight_apply, ContinuousLinearMap.id_apply, Pi.smul_apply,
+    smul_eq_mul, hb]
+  simp only [basis, Pi.single_eq_same, mul_one, Finset.sum_add_distrib,
+    Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul]
+  have he : (∑ i : Fin d, (-(d : ℝ)/2 * sqsum x ^ (-(d : ℝ)/2-1) * (2*x i)) * x i) =
+      -(d : ℝ) * (sqsum x ^ (-(d : ℝ)/2-1) * sqsum x) := by
+    simp only [sqsum, Finset.mul_sum]
+    apply Finset.sum_congr rfl
+    intro i hi
+    ring
+  rw [he]
+  have hp : sqsum x ^ (-(d : ℝ)/2-1) * sqsum x = sqsum x ^ (-(d : ℝ)/2) := by
+    calc
+      _ = sqsum x ^ (-(d : ℝ)/2-1) * sqsum x ^ (1 : ℝ) := by rw [Real.rpow_one]
+      _ = sqsum x ^ (-(d : ℝ)/2-1+1) := (Real.rpow_add (sqsum_pos hx) _ _).symm
+      _ = _ := by congr 1; ring
+  rw [hp]
+  ring
+
+ def radialForm (n : ℕ) (x : Space (n+1)) : (Space (n+1)) [⋀^Fin n]→L[ℝ] ℝ :=
+   (vol (n+1)).curryLeft (radial x)
+
+ theorem differentiableAt_radialForm {n : ℕ} {x : Space (n+1)} (hx : x ≠ 0) :
+    DifferentiableAt ℝ (radialForm n) x :=
+   (vol (n+1)).curryLeft.differentiableAt.comp x (hasFDerivAt_radial hx).differentiableAt
+
+ theorem closed_radialForm {n : ℕ} {x : Space (n+1)} (hx : x ≠ 0) :
+    extDeriv (radialForm n) x = 0 :=
+   extDeriv_contract_eq_zero (hasFDerivAt_radial hx).differentiableAt (radial_divergence hx)
+
+ def pullForm {n : ℕ} (f : Space (n+1) → Space (n+1)) (x : Space (n+1)) :
+    (Space (n+1)) [⋀^Fin n]→L[ℝ] ℝ :=
+   (radialForm n (f x)).compContinuousLinearMap (fderiv ℝ f x)
+
+ theorem differentiableAt_pullForm {n : ℕ} {f : Space (n+1) → Space (n+1)}
+    (hf : ContDiff ℝ ∞ f) {x : Space (n+1)} (hx : f x ≠ 0) :
+    DifferentiableAt ℝ (pullForm f) x := by
+  apply DifferentiableAt.continuousAlternatingMapCompContinuousLinearMap
+  · exact (differentiableAt_radialForm hx).comp x (hf.differentiable (by simp) x)
+  · exact (hf.contDiffAt.fderiv_right (m := ∞) (by simp)).differentiableAt (by simp)
+
+ theorem closed_pullForm {n : ℕ} {f : Space (n+1) → Space (n+1)}
+    (hf : ContDiff ℝ ∞ f) {x : Space (n+1)} (hx : f x ≠ 0) :
+    extDeriv (pullForm f) x = 0 := by
+  change extDeriv (fun y => (radialForm n (f y)).compContinuousLinearMap (fderiv ℝ f y)) x = 0
+  rw [extDeriv_pullback (differentiableAt_radialForm hx) hf.contDiffAt (by
+    simp only [minSmoothness_of_isRCLikeNormedField]
+    change ((2 : ℕ∞) : WithTop ℕ∞) ≤ ((⊤ : ℕ∞) : WithTop ℕ∞)
+    exact WithTop.coe_le_coe.mpr le_top),
+    closed_radialForm hx]
+  ext v
+  rfl
+
+ def formVector {n : ℕ} (η : Space (n+1) → (Space (n+1)) [⋀^Fin n]→L[ℝ] ℝ)
+    (x : Space (n+1)) : Space (n+1) :=
+   fun i => (-1 : ℝ)^i.val * η x (i.removeNth (basis (n+1)))
+
+ theorem differentiableAt_formVector {n : ℕ}
+    {η : Space (n+1) → (Space (n+1)) [⋀^Fin n]→L[ℝ] ℝ} {x : Space (n+1)}
+    (hη : DifferentiableAt ℝ η x) : DifferentiableAt ℝ (formVector η) x := by
+  apply differentiableAt_pi.mpr
+  intro i
+  exact ((ContinuousAlternatingMap.apply ℝ _ _ (i.removeNth (basis (n+1)))).differentiableAt.comp x hη).const_mul _
+
+ theorem divergence_formVector {n : ℕ}
+    {η : Space (n+1) → (Space (n+1)) [⋀^Fin n]→L[ℝ] ℝ} {x : Space (n+1)}
+    (hη : DifferentiableAt ℝ η x) :
+    ∑ i, fderiv ℝ (formVector η) x (basis (n+1) i) i = extDeriv η x (basis (n+1)) := by
+  rw [extDeriv_apply hη]
+  apply Finset.sum_congr rfl
+  intro i hi
+  have hd := (differentiableAt_formVector hη).hasFDerivAt
+  have hei : fderiv ℝ (fun y => formVector η y i) x =
+      (ContinuousLinearMap.proj i).comp (fderiv ℝ (formVector η) x) :=
+    ((ContinuousLinearMap.proj i : Space (n+1) →L[ℝ] ℝ).hasFDerivAt.comp x hd).fderiv
+  have hw : DifferentiableAt ℝ (fun y => η y (i.removeNth (basis (n+1)))) x :=
+    (hη.hasFDerivAt.continuousAlternatingMap_apply_const _).differentiableAt
+  have he : fderiv ℝ (fun y => formVector η y i) x =
+      (-1 : ℝ)^i.val • fderiv ℝ (fun y => η y (i.removeNth (basis (n+1)))) x :=
+    (hw.hasFDerivAt.const_mul _).fderiv
+  have := congrArg (fun l : Space (n+1) →L[ℝ] ℝ => l (basis (n+1) i)) (hei.symm.trans he)
+  simpa only [ContinuousLinearMap.comp_apply, ContinuousLinearMap.proj_apply, smul_apply,
+    smul_eq_mul, zsmul_eq_mul, Int.cast_pow, Int.cast_neg, Int.cast_one] using this
+
+ theorem formVector_radialForm {n : ℕ} (x : Space (n+1)) :
+    formVector (radialForm n) x = radial x := by
+  ext i
+  change (-1 : ℝ)^i.val * (vol (n+1)).curryLeft (radial x) (i.removeNth (basis (n+1))) = _
+  rw [contract_basis, ← mul_assoc, ← pow_add,
+    show i.val+i.val = 2*i.val by omega, pow_mul, neg_one_sq, one_pow, one_mul]
+
+ theorem pullForm_eventually_id {n : ℕ} {f : Space (n+1) → Space (n+1)} {x : Space (n+1)}
+    (h : f =ᶠ[𝓝 x] id) : pullForm f x = radialForm n x := by
+  unfold pullForm
+  rw [h.eq_of_nhds, h.fderiv_eq, fderiv_id]
+  ext v
+  rfl
+
+ theorem positive_integral_cube {n : ℕ} (g : Space n → ℝ) (hg : Continuous g)
+    (hp : ∀ x, 0 < g x) {R : ℝ} (hR : 0 < R) :
+    0 < ∫ x in Icc (fun _ : Fin n => -R) (fun _ => R), g x := by
+  apply (setIntegral_pos_iff_support_of_nonneg_ae
+    (Eventually.of_forall (fun x => (hp x).le)) hg.integrableOn_Icc).mpr
+  have hs : Function.support g = univ := by
+    ext x
+    simp only [Function.mem_support, Set.mem_univ, iff_true]
+    exact (hp x).ne'
+  rw [hs, univ_inter, Real.volume_Icc_pi]
+  apply pos_iff_ne_zero.mpr
+  exact Finset.prod_ne_zero_iff.mpr fun _ _ => ENNReal.ofReal_ne_zero_iff.mpr (by linarith)
+
+ theorem continuous_face {n : ℕ} {G : Space (n+1) → Space (n+1)} (hG : Continuous G)
+    (i : Fin (n+1)) (a : ℝ) : Continuous (fun y : Space n => G (i.insertNth a y) i) := by
+  exact (continuous_apply i).comp (hG.comp (continuous_const.finInsertNth i continuous_id))
+
+ theorem radial_front_pos {n : ℕ} (i : Fin (n+1)) (y : Space n) {S : ℝ} (hS : 0 < S) :
+    0 < radial (i.insertNth S y : Space (n+1)) i := by
+  simp only [radial, Pi.smul_apply, smul_eq_mul, Fin.insertNth_apply_same]
+  refine mul_pos (Real.rpow_pos_of_pos (sqsum_pos ?_) _) hS
+  intro he
+  have he' := congrFun he i
+  exact hS.ne' (by simpa only [Fin.insertNth_apply_same, Pi.zero_apply] using he')
+
+ theorem radial_back_neg {n : ℕ} (i : Fin (n+1)) (y : Space n) {S : ℝ} (hS : 0 < S) :
+    radial (i.insertNth (-S) y : Space (n+1)) i < 0 := by
+  simp only [radial, Pi.smul_apply, smul_eq_mul, Fin.insertNth_apply_same]
+  refine mul_neg_of_pos_of_neg (Real.rpow_pos_of_pos (sqsum_pos ?_) _) (neg_neg_of_pos hS)
+  intro he
+  have he' := congrFun he i
+  simp only [Fin.insertNth_apply_same, Pi.zero_apply, neg_eq_zero] at he'
+  exact hS.ne' he'
+
+ theorem face_norm_bound {n : ℕ} (i : Fin (n+1)) (y : Space n) (a : ℝ) :
+    |a| ≤ ‖(i.insertNth a y : Space (n+1))‖ := by
+  simpa only [Fin.insertNth_apply_same, Real.norm_eq_abs] using
+    (norm_le_pi_norm (i.insertNth a y : Space (n+1)) i)
+
+ theorem positive_flux {n : ℕ} {G : Space (n+1) → Space (n+1)} (hG : Continuous G)
+    {R S : ℝ} (hS : 0 < S) (hRS : R < S)
+    (hGrad : ∀ x, R < ‖x‖ → G x = radial x) (i : Fin (n+1)) :
+    0 < (∫ y in Icc (fun _ : Fin n => -S) (fun _ => S), G (i.insertNth S y) i) -
+      ∫ y in Icc (fun _ : Fin n => -S) (fun _ => S), G (i.insertNth (-S) y) i := by
+  have hfront (y : Space n) : 0 < G (i.insertNth S y) i := by
+    rw [hGrad _ (lt_of_lt_of_le hRS (by simpa only [abs_of_pos hS] using face_norm_bound i y S))]
+    exact radial_front_pos i y hS
+  have hback (y : Space n) : G (i.insertNth (-S) y) i < 0 := by
+    rw [hGrad _ (lt_of_lt_of_le hRS (by simpa only [abs_neg, abs_of_pos hS] using face_norm_bound i y (-S)))]
+    exact radial_back_neg i y hS
+  have hp := positive_integral_cube _ (continuous_face hG i S) hfront hS
+  have hn := positive_integral_cube (fun y => -G (i.insertNth (-S) y) i)
+    (continuous_face hG i (-S)).neg (fun y => neg_pos.mpr (hback y)) hS
+  rw [integral_neg] at hn
+  linarith
+
+ theorem divergence_obstruction {n : ℕ} {G : Space (n+1) → Space (n+1)}
+    (hG : Differentiable ℝ G) (hdiv : ∀ x, ∑ i, fderiv ℝ G x (basis (n+1) i) i = 0)
+    {R : ℝ} (hR : 0 < R) (hGrad : ∀ x, R < ‖x‖ → G x = radial x) : False := by
+  let S := R+1
+  have hS : 0 < S := by dsimp [S]; linarith
+  have hstokes := integral_divergence_of_hasFDerivAt_off_countable
+    (fun _ : Fin (n+1) => -S) (fun _ => S) (fun _ => by linarith)
+    G (fderiv ℝ G) ∅ countable_empty hG.continuous.continuousOn
+    (fun x _ => (hG x).hasFDerivAt)
+    (show IntegrableOn (fun x => ∑ i, fderiv ℝ G x (basis (n+1) i) i)
+      (Icc (fun _ : Fin (n+1) => -S) (fun _ => S)) volume from by
+      simpa only [hdiv] using (integrableOn_zero : IntegrableOn (fun _ : Space (n+1) => (0 : ℝ))
+        (Icc (fun _ => -S) (fun _ => S)) volume))
+  change (∫ x in Icc (fun _ : Fin (n+1) => -S) (fun _ => S),
+      ∑ i, fderiv ℝ G x (basis (n+1) i) i) = _ at hstokes
+  simp only [hdiv, integral_zero] at hstokes
+  have hp := Finset.sum_pos (s := Finset.univ)
+    (fun i _ => positive_flux hG.continuous hS (show R < S by dsimp [S]; linarith) hGrad i)
+    Finset.univ_nonempty
+  exact (ne_of_gt hp) hstokes.symm
+
+ theorem exists_zero_of_identity_outside {n : ℕ} {f : Space (n+1) → Space (n+1)}
+    (hf : ContDiff ℝ ∞ f) {R : ℝ} (hR : 0 < R)
+    (hId : ∀ x, R ≤ ‖x‖ → f x = x) : ∃ x, f x = 0 := by
+  by_contra hnone
+  push Not at hnone
+  let G := formVector (pullForm f)
+  have hGdiff : Differentiable ℝ G := fun x =>
+    differentiableAt_formVector (differentiableAt_pullForm hf (hnone x))
+  have hdiv (x : Space (n+1)) : ∑ i, fderiv ℝ G x (basis (n+1) i) i = 0 := by
+    rw [divergence_formVector (differentiableAt_pullForm hf (hnone x)),
+      closed_pullForm hf (hnone x)]
+    rfl
+  have hevent {x : Space (n+1)} (hx : R < ‖x‖) : f =ᶠ[𝓝 x] id := by
+    have hopen : IsOpen {y : Space (n+1) | R < ‖y‖} := isOpen_lt continuous_const continuous_norm
+    filter_upwards [hopen.mem_nhds hx] with y hy
+    exact hId y hy.le
+  have hGrad {x : Space (n+1)} (hx : R < ‖x‖) : G x = radial x := by
+    change formVector (pullForm f) x = radial x
+    have he := pullForm_eventually_id (hevent hx)
+    have hrewrite : formVector (pullForm f) x = formVector (radialForm n) x := by
+      ext i
+      simp only [formVector, he]
+    rw [hrewrite, formVector_radialForm]
+  exact divergence_obstruction hGdiff hdiv hR (fun x hx => hGrad hx)
+
+ theorem exists_zero_space {d : ℕ} {f : Space d → Space d}
+    (hf : ContDiff ℝ ∞ f) {R : ℝ} (hR : 0 < R)
+    (hId : ∀ x, R ≤ ‖x‖ → f x = x) : ∃ x, f x = 0 := by
+  cases d with
+  | zero => exact ⟨0, Subsingleton.elim _ _⟩
+  | succ n => exact exists_zero_of_identity_outside hf hR hId
+
+ theorem exists_zero {E : Type u78} [NormedAddCommGroup E] [NormedSpace ℝ E]
+    [FiniteDimensional ℝ E] {f : E → E} (hf : ContDiff ℝ ∞ f)
+    {R : ℝ} (hR : 0 < R) (hId : ∀ x, R ≤ ‖x‖ → f x = x) : ∃ x, f x = 0 := by
+  let b := (Module.finBasis ℝ E).equivFunL
+  let C := ‖b.toContinuousLinearMap‖
+  have hC : 0 ≤ C := norm_nonneg _
+  let S := C*R+1
+  have hS : 0 < S := by dsimp [S]; positivity
+  let g : Space (Module.finrank ℝ E) → Space (Module.finrank ℝ E) := fun y => b (f (b.symm y))
+  have hg : ContDiff ℝ ∞ g := b.contDiff.comp (hf.comp b.symm.contDiff)
+  have hgId (y : Space (Module.finrank ℝ E)) (hy : S ≤ ‖y‖) : g y = y := by
+    have hbnd : ‖y‖ ≤ C*‖b.symm y‖ := by
+      simpa only [ContinuousLinearEquiv.coe_coe, b.apply_symm_apply] using
+        (b.toContinuousLinearMap.le_opNorm (b.symm y))
+    have hn : R ≤ ‖b.symm y‖ := by dsimp [S] at hy; nlinarith
+    dsimp [g]
+    rw [hId _ hn, b.apply_symm_apply]
+  obtain ⟨y, hy⟩ := exists_zero_space hg hS hgId
+  refine ⟨b.symm y, b.injective ?_⟩
+  simpa only [map_zero] using hy
+
+ theorem exists_zero_of_compactSupport {E : Type u79} [NormedAddCommGroup E] [NormedSpace ℝ E]
+    [FiniteDimensional ℝ E] {f : E → E} (hf : ContDiff ℝ ∞ f)
+    (hc : HasCompactSupport (fun x => f x-x)) : ∃ x, f x = 0 := by
+  obtain ⟨R, hR, h⟩ := hc.exists_pos_le_norm
+  exact exists_zero hf hR (fun x hx => sub_eq_zero.mp (h x hx))
+
+end
+end SmoothDegree
+
+end
+end NonsqueezingInline
+
+end OAI
diff --git a/lean/OAI/Geometry/PolarProducts/SmoothFlow.lean b/lean/OAI/Geometry/PolarProducts/SmoothFlow.lean
new file mode 100644
index 0000000000000000000000000000000000000000..88700988e045ce0a35e44a7caaa923fff4f8b323
--- /dev/null
+++ b/lean/OAI/Geometry/PolarProducts/SmoothFlow.lean
@@ -0,0 +1,259 @@
+import OAI.Geometry.PolarProducts.CompleteFlow
+
+namespace OAI
+
+universe u14 u15 u16
+
+section LowerBoundInline
+open Set Filter Function
+open scoped Topology ContDiff NNReal
+open Set Filter Metric
+open scoped Topology ContDiff
+open Set Filter Function MeasureTheory Metric
+open scoped Topology ContDiff NNReal
+
+namespace SmoothODE
+
+open Set Filter Function MeasureTheory Metric
+open scoped Topology ContDiff NNReal
+
+variable {E : Type u14} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
+
+noncomputable section
+
+theorem curve_injective {V : E → E} {K : ℝ≥0} (hV : LipschitzWith K V)
+    (h : ℝ) (hh : ‖h‖₊ * K < 1) (t : Time) :
+    Function.Injective (fun x => curve hV h hh x t) := by
+  intro x y he
+  have heq : EqOn (curve hV h hh x) (curve hV h hh y) (Icc 0 t) := by
+    apply ODE_solution_unique_of_mem_Icc_left
+      (v := fun _ y => h • V y) (s := fun _ => univ) (K := ‖h‖₊ * K)
+    · intro _ _
+      exact ((lipschitzWith_smul h).comp hV).lipschitzOnWith
+    · exact (continuous_curve hV h hh x).continuousOn
+    · intro s hs
+      exact (hasDerivAt_curve hV h hh x ⟨s, hs.1.le, hs.2.trans t.property.2⟩).hasDerivWithinAt
+    · intro _ _; trivial
+    · exact (continuous_curve hV h hh y).continuousOn
+    · intro s hs
+      exact (hasDerivAt_curve hV h hh y ⟨s, hs.1.le, hs.2.trans t.property.2⟩).hasDerivWithinAt
+    · intro _ _; trivial
+    · exact he
+  simpa only [curve_zero] using heq (show 0 ∈ Icc (0 : ℝ) t from ⟨le_rfl, t.property.1⟩)
+
+theorem curve_of_const_along {V : E → E} {K : ℝ≥0} (hV : LipschitzWith K V)
+    (h : ℝ) (hh : ‖h‖₊ * K < 1) (x v : E)
+    (hc : ∀ t : Time, V (x + (t : ℝ) • h • v) = v) :
+    EqOn (curve hV h hh x) (fun t => x + t • h • v) (Icc 0 1) := by
+  apply EqOn.symm
+  apply curve_unique hV h hh x
+  · exact (continuous_const.add (continuous_id.smul continuous_const)).continuousOn
+  · intro t ht
+    have he := hc ⟨t, ht.1, ht.2.le⟩
+    have hd := ((hasDerivAt_id t).smul_const (h • v)).const_add x
+    simpa only [he, one_smul, id_eq] using hd.hasDerivWithinAt (s := Ici t)
+  · simp
+
+theorem curve_spaceTime_outside {V : ℝ × E → ℝ × E} {K : ℝ≥0}
+    (hV : LipschitzWith K V) (h : ℝ) (hh : ‖h‖₊ * K < 1)
+    {U : Set E} (hTime : ∀ z, (V z).1 = 1)
+    (hOut : ∀ z, z.2 ∉ U → (V z).2 = 0) (x : ℝ × E) (hx : x.2 ∉ U) (t : Time) :
+    curve hV h hh x t = (x.1 + (t : ℝ) * h, x.2) := by
+  have hc : ∀ s : Time, V (x + (s : ℝ) • h • (1, (0 : E))) = (1, 0) := by
+    intro s
+    apply Prod.ext
+    · exact hTime _
+    · exact hOut _ (by simpa using hx)
+  have he := curve_of_const_along hV h hh x (1, (0 : E)) hc t.property
+  calc
+    curve hV h hh x t = x + (t : ℝ) • h • (1, (0 : E)) := he
+    _ = (x.1 + (t : ℝ) * h, x.2) := by ext <;> simp [smul_eq_mul]
+
+theorem curve_spaceTime_mapsTo {V : ℝ × E → ℝ × E} {K : ℝ≥0}
+    (hV : LipschitzWith K V) (h : ℝ) (hh : ‖h‖₊ * K < 1)
+    {U : Set E} (hTime : ∀ z, (V z).1 = 1)
+    (hOut : ∀ z, z.2 ∉ U → (V z).2 = 0) (x : ℝ × E) (hx : x.2 ∈ U) (t : Time) :
+    (curve hV h hh x t).2 ∈ U := by
+  by_contra hout
+  let y : ℝ × E := (x.1, (curve hV h hh x t).2)
+  have he : curve hV h hh y t = curve hV h hh x t := by
+    rw [curve_spaceTime_outside hV h hh hTime hOut y hout]
+    apply Prod.ext
+    · have he := curve_linear_component hV h hh (ContinuousLinearMap.fst ℝ ℝ E) 1 hTime x t
+      simpa only [ContinuousLinearMap.coe_fst', smul_eq_mul, mul_one, y] using he.symm
+    · rfl
+  have hy := curve_injective hV h hh t he
+  have hxy := congrArg Prod.snd hy
+  exact hout (hxy ▸ hx)
+
+end
+end SmoothODE
+
+namespace SmoothODE
+
+open Set Filter Function MeasureTheory Metric
+open scoped Topology ContDiff NNReal
+
+variable {E : Type u15} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  [CompleteSpace E] [ProperSpace E]
+
+noncomputable section
+
+theorem step_preserves_tensor_on {V : E → E} {K : ℝ≥0} (hV : LipschitzWith K V)
+    (hVs : ContDiff ℝ ∞ V) (h : ℝ) (hh : ‖h‖₊ * K < 1)
+    {B : E → E →L[ℝ] E →L[ℝ] ℝ} {U : Set E}
+    (hB : ∀ z ∈ U, DifferentiableAt ℝ B z)
+    (hInv : ∀ z ∈ U, ∀ a b, fderiv ℝ B z (V z) a b +
+      B z (fderiv ℝ V z a) b + B z a (fderiv ℝ V z b) = 0) (x : E) (hpath : ∀ t : Time, curve hV h hh x t ∈ U) (a b : E) :
+    B (step hV h hh x) (fderiv ℝ (step hV h hh) x a)
+      (fderiv ℝ (step hV h hh) x b) = B x a b := by
+  let F : ℝ → ℝ := fun t => B (curve hV h hh x t)
+    (variation hV h hh x a t) (variation hV h hh x b t)
+  have hBc : ContinuousOn B U := fun z hz => (hB z hz).continuousAt.continuousWithinAt
+  have hc : ContinuousOn F (Icc 0 1) :=
+    ((hBc.comp (continuous_curve hV h hh x).continuousOn (fun t ht => hpath ⟨t, ht⟩)).clm_apply
+      (continuous_variation hV hVs h hh x a).continuousOn).clm_apply
+      (continuous_variation hV hVs h hh x b).continuousOn
+  have hd : ∀ t ∈ Ico (0 : ℝ) 1, HasDerivWithinAt F 0 (Ici t) t := by
+    intro t ht
+    let T : Time := ⟨t, ht.1, ht.2.le⟩
+    have hv := hasDerivAt_curve hV h hh x T
+    have ha := hasDerivAt_variation hV hVs h hh x a T
+    have hb := hasDerivAt_variation hV hVs h hh x b T
+    have hdb : HasDerivAt (fun s : ℝ => B (curve hV h hh x s))
+        (fderiv ℝ B (curve hV h hh x t) (h • V (curve hV h hh x t))) t :=
+      (hB (curve hV h hh x t) (hpath T)).hasFDerivAt.comp_hasDerivAt (l := B)
+        (l' := fderiv ℝ B (curve hV h hh x t)) t hv
+    have hdf := (hdb.clm_apply ha).clm_apply hb
+    have hz :
+        ((fderiv ℝ B (curve hV h hh x t) (h • V (curve hV h hh x t)))
+          (variation hV h hh x a t) +
+          B (curve hV h hh x t)
+            (h • fderiv ℝ V (curve hV h hh x t) (variation hV h hh x a t)))
+          (variation hV h hh x b t) +
+          B (curve hV h hh x t) (variation hV h hh x a t)
+            (h • fderiv ℝ V (curve hV h hh x t) (variation hV h hh x b t)) = 0 := by
+      simp only [map_smul, smul_apply,
+        add_apply, smul_eq_mul]
+      calc
+        _ = h * (fderiv ℝ B (curve hV h hh x t) (V (curve hV h hh x t))
+            (variation hV h hh x a t) (variation hV h hh x b t) +
+          B (curve hV h hh x t)
+            (fderiv ℝ V (curve hV h hh x t) (variation hV h hh x a t))
+            (variation hV h hh x b t) +
+          B (curve hV h hh x t) (variation hV h hh x a t)
+            (fderiv ℝ V (curve hV h hh x t) (variation hV h hh x b t))) := by ring
+        _ = 0 := by rw [hInv _ (hpath T), mul_zero]
+    exact (hdf.congr_deriv hz).hasDerivWithinAt
+  have he := constant_of_has_deriv_right_zero hc hd 1
+    (show 1 ∈ Icc (0 : ℝ) 1 by constructor <;> norm_num)
+  simpa only [F, curve_one, variation_one hV hVs, curve_zero, variation_zero] using he
+
+end
+end SmoothODE
+
+namespace SmoothODE
+
+open Set Filter Function MeasureTheory Metric
+open scoped Topology ContDiff NNReal
+
+variable {E : Type u16} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  [CompleteSpace E] [ProperSpace E]
+
+noncomputable section
+
+omit [CompleteSpace E] [ProperSpace E] in
+
+theorem homeomorph_pow_preserves_tensor_on (f : E ≃ₜ E)
+    (hf : ContDiff ℝ ∞ (f : E → E)) {U : Set E} (hU : MapsTo f U U)
+    {B : E → E →L[ℝ] E →L[ℝ] ℝ}
+    (hB : ∀ x ∈ U, ∀ a b, B (f x) (fderiv ℝ (f : E → E) x a)
+      (fderiv ℝ (f : E → E) x b) = B x a b) (n : ℕ) (x : E) (hx : x ∈ U) :
+    (f ^ n) x ∈ U ∧ ∀ a b,
+      B ((f ^ n) x) (fderiv ℝ ((f ^ n : E ≃ₜ E) : E → E) x a)
+        (fderiv ℝ ((f ^ n : E ≃ₜ E) : E → E) x b) = B x a b := by
+  induction n with
+  | zero =>
+    refine ⟨hx, ?_⟩
+    intro a b
+    change B x (fderiv ℝ (id : E → E) x a) (fderiv ℝ (id : E → E) x b) = _
+    simp
+  | succ n ih =>
+    rw [pow_succ']
+    refine ⟨hU ih.1, ?_⟩
+    intro a b
+    change B (f ((f ^ n) x)) (fderiv ℝ ((f : E → E) ∘ (f ^ n : E ≃ₜ E)) x a)
+      (fderiv ℝ ((f : E → E) ∘ (f ^ n : E ≃ₜ E)) x b) = _
+    rw [fderiv_comp x (hf.differentiable (by simp) _)
+      ((contDiff_homeomorph_pow f hf n).differentiable (by simp) _)]
+    simp only [ContinuousLinearMap.comp_apply]
+    rw [hB _ ih.1]
+    exact ih.2 a b
+
+theorem exists_spaceTime_transport {V : ℝ × E → ℝ × E} {K : ℝ≥0}
+    (hV : LipschitzWith K V) (hVs : ContDiff ℝ ∞ V) {U : Set E}
+    (hTime : ∀ z, (V z).1 = 1) (hOut : ∀ z, z.2 ∉ U → (V z).2 = 0)
+    {B : (ℝ × E) → (ℝ × E) →L[ℝ] (ℝ × E) →L[ℝ] ℝ}
+    (hB : ∀ z, z.2 ∈ U → DifferentiableAt ℝ B z)
+    (hInv : ∀ z, z.2 ∈ U → ∀ a b, fderiv ℝ B z (V z) a b +
+      B z (fderiv ℝ V z a) b + B z a (fderiv ℝ V z b) = 0) (T : ℝ) :
+    ∃ f : (ℝ × E) ≃ₜ (ℝ × E),
+      ContDiff ℝ ∞ (f : (ℝ × E) → (ℝ × E)) ∧
+      ContDiff ℝ ∞ (f.symm : (ℝ × E) → (ℝ × E)) ∧
+      (∀ x, (f x).1 = x.1 + T) ∧
+      (∀ x, x.2 ∉ U → (f x).2 = x.2) ∧
+      (∀ x, x.2 ∈ U → (f x).2 ∈ U ∧ ∀ a b,
+        B (f x) (fderiv ℝ (f : (ℝ × E) → (ℝ × E)) x a)
+          (fderiv ℝ (f : (ℝ × E) → (ℝ × E)) x b) = B x a b) := by
+  obtain ⟨N, hN⟩ := exists_nat_gt (max (‖T‖ * (K : ℝ)) 0)
+  have hN0 : (0 : ℝ) < N := lt_of_le_of_lt (le_max_right _ _) hN
+  have hNne : (N : ℝ) ≠ 0 := ne_of_gt hN0
+  let h : ℝ := T / N
+  have hh : ‖h‖₊ * K < 1 := by
+    change ‖h‖ * (K : ℝ) < (1 : ℝ)
+    dsimp [h]
+    rw [abs_div, abs_of_pos hN0, div_mul_eq_mul_div]
+    exact (div_lt_one hN0).mpr ((le_max_left _ _).trans_lt hN)
+  let f := stepHomeomorph hV hVs h hh
+  have hf : ContDiff ℝ ∞ (f : (ℝ × E) → (ℝ × E)) := contDiff_step hV hVs h hh
+  have hfi : ContDiff ℝ ∞ (f.symm : (ℝ × E) → (ℝ × E)) :=
+    contDiff_step hV hVs (-h) (by simpa using hh)
+  have hm : ∀ x, x.2 ∈ U → (f x).2 ∈ U := by
+    intro x hx
+    simpa only [Prod.mk.eta, curve_one, f, stepHomeomorph_apply] using
+      curve_spaceTime_mapsTo hV h hh hTime hOut x hx ⟨1, by constructor <;> norm_num⟩
+  have ho : ∀ x, x.2 ∉ U → (f x).2 = x.2 := by
+    intro x hx
+    have he := curve_spaceTime_outside hV h hh hTime hOut x hx ⟨1, by constructor <;> norm_num⟩
+    simpa only [curve_one, f, stepHomeomorph_apply] using congrArg Prod.snd he
+  refine ⟨f ^ N, contDiff_homeomorph_pow f hf N, ?_, ?_, ?_, ?_⟩
+  · change ContDiff ℝ ∞ ((f ^ N)⁻¹ : (ℝ × E) ≃ₜ (ℝ × E))
+    rw [← inv_pow]
+    exact contDiff_homeomorph_pow f⁻¹ hfi N
+  · intro x
+    have hs : (N : ℝ) * h = T := by dsimp [h]; field_simp
+    have he := homeomorph_pow_linear_component f (ContinuousLinearMap.fst ℝ ℝ E) h
+      (by intro z; simpa only [smul_eq_mul, mul_one, f, stepHomeomorph_apply] using
+        step_linear_component hV h hh (ContinuousLinearMap.fst ℝ ℝ E) 1 hTime z) N x
+    simpa only [ContinuousLinearMap.coe_fst', nsmul_eq_mul, hs] using he
+  · intro x hx
+    have hi : ∀ n : ℕ, ((f ^ n) x).2 = x.2 := by
+      intro n
+      induction n with
+      | zero => rfl
+      | succ n ih =>
+        rw [pow_succ', Homeomorph.mul_apply, ho _ (by simpa only [ih] using hx), ih]
+    exact hi N
+  · intro x hx
+    apply homeomorph_pow_preserves_tensor_on f hf (U := Prod.snd ⁻¹' U)
+      (fun _ hz => hm _ hz) (fun z hz a b => ?_) N x hx
+    exact step_preserves_tensor_on hV hVs h hh (U := Prod.snd ⁻¹' U)
+      (fun y hy => hB y hy) (fun y hy => hInv y hy) z
+      (fun t => curve_spaceTime_mapsTo hV h hh hTime hOut z hz t) a b
+
+end
+end SmoothODE
+
+end LowerBoundInline
+
+end OAI
diff --git a/lean/OAI/Geometry/PolarProducts/SmoothPaths.lean b/lean/OAI/Geometry/PolarProducts/SmoothPaths.lean
new file mode 100644
index 0000000000000000000000000000000000000000..f8a1243cf8854de4a363dce8f7f5f96eaa0267b3
--- /dev/null
+++ b/lean/OAI/Geometry/PolarProducts/SmoothPaths.lean
@@ -0,0 +1,136 @@
+import OAI.Geometry.PolarProducts.Contraction
+
+namespace OAI
+
+section LowerBoundInline
+open Set Filter Function
+open scoped Topology ContDiff NNReal
+
+open Set Filter Metric
+open scoped Topology ContDiff
+
+namespace SmoothPaths
+
+universe u v
+
+variable {X : Type v} {E F : Type u} [TopologicalSpace X] [CompactSpace X]
+  [NormedAddCommGroup E] [NormedSpace ℝ E]
+  [NormedAddCommGroup F] [NormedSpace ℝ F]
+
+noncomputable section
+
+def applyField (A : C(X, E →L[ℝ] F)) : C(X, E) →L[ℝ] C(X, F) := by
+  let L : C(X, E) →ₗ[ℝ] C(X, F) :=
+    { toFun := fun u => ⟨fun t => A t (u t), A.continuous.clm_apply u.continuous⟩
+      map_add' := by intros; ext; simp
+      map_smul' := by intros; ext; simp }
+  apply L.mkContinuous ‖A‖
+  intro u
+  apply (ContinuousMap.norm_le (L u) (mul_nonneg (norm_nonneg A) (norm_nonneg u))).mpr
+  intro t
+  exact ((A t).le_opNorm (u t)).trans
+    (mul_le_mul (A.norm_coe_le_norm t) (u.norm_coe_le_norm t)
+      (norm_nonneg _) (norm_nonneg _))
+
+@[simp] theorem applyField_apply (A : C(X, E →L[ℝ] F)) (u : C(X, E)) (t : X) :
+    applyField A u t = A t (u t) := rfl
+
+def applyFieldCLM : C(X, E →L[ℝ] F) →L[ℝ] C(X, E) →L[ℝ] C(X, F) := by
+  let L : C(X, E →L[ℝ] F) →ₗ[ℝ] (C(X, E) →L[ℝ] C(X, F)) :=
+    { toFun := fun A => applyField A
+      map_add' := by intros; ext; simp
+      map_smul' := by intros; ext; simp }
+  have hL : ∀ A : C(X, E →L[ℝ] F), ‖L A‖ ≤ 1 * ‖A‖ := by
+    intro A
+    rw [one_mul]
+    apply (applyField A).opNorm_le_bound (norm_nonneg A)
+    intro u
+    apply (ContinuousMap.norm_le (applyField A u)
+      (mul_nonneg (norm_nonneg A) (norm_nonneg u))).mpr
+    intro t
+    exact ((A t).le_opNorm (u t)).trans
+      (mul_le_mul (A.norm_coe_le_norm t) (u.norm_coe_le_norm t)
+        (norm_nonneg _) (norm_nonneg _))
+  exact LinearMap.mkContinuous (σ := RingHom.id ℝ)
+    (E := C(X, E →L[ℝ] F)) (F := C(X, E) →L[ℝ] C(X, F)) L 1 hL
+
+@[simp] theorem applyFieldCLM_apply (A : C(X, E →L[ℝ] F)) :
+    applyFieldCLM A = applyField A := rfl
+
+def superpose {f : E → F} (hf : Continuous f) (u : C(X, E)) : C(X, F) :=
+  ⟨fun t => f (u t), hf.comp u.continuous⟩
+
+omit [CompactSpace X] [NormedSpace ℝ E] [NormedSpace ℝ F] in
+@[simp] theorem superpose_apply {f : E → F} (hf : Continuous f)
+    (u : C(X, E)) (t : X) : superpose hf u t = f (u t) := rfl
+
+omit [CompactSpace X] [NormedSpace ℝ E] [NormedSpace ℝ F] in
+theorem continuous_superpose {f : E → F} (hf : Continuous f) :
+    Continuous (superpose (X := X) hf) :=
+  ContinuousMap.continuous_postcomp ⟨f, hf⟩
+
+theorem hasFDerivAt_superpose [ProperSpace E] {f : E → F}
+    (hf : Differentiable ℝ f) (hf' : Continuous (fderiv ℝ f)) (u : C(X, E)) :
+    HasFDerivAt (superpose hf.continuous)
+      (applyField (superpose hf' u)) u := by
+  rw [hasFDerivAt_iff_isLittleO_nhds_zero, Asymptotics.isLittleO_iff]
+  intro ε hε
+  have huc : UniformContinuousOn (fderiv ℝ f) (closedBall (0 : E) (‖u‖ + 2)) :=
+    (isCompact_closedBall (0 : E) (‖u‖ + 2)).uniformContinuousOn_of_continuous hf'.continuousOn
+  obtain ⟨δ, hδ, hbound⟩ := Metric.uniformContinuousOn_iff.mp huc ε hε
+  filter_upwards [Metric.ball_mem_nhds (0 : C(X, E)) (lt_min hδ zero_lt_one)] with h hh
+  have hhδ : ‖h‖ < δ := (mem_ball_zero_iff.mp hh).trans_le (min_le_left _ _)
+  have hh1 : ‖h‖ < 1 := (mem_ball_zero_iff.mp hh).trans_le (min_le_right _ _)
+  apply (ContinuousMap.norm_le _ (mul_nonneg hε.le (norm_nonneg _))).mpr
+  intro t
+  have hu : ‖u t‖ ≤ ‖u‖ := u.norm_coe_le_norm t
+  have hsmall : ‖h t‖ < δ := (h.norm_coe_le_norm t).trans_lt hhδ
+  have hsmall1 : ‖h t‖ < 1 := (h.norm_coe_le_norm t).trans_lt hh1
+  have hut : u t ∈ closedBall (0 : E) (‖u‖ + 2) := by
+    rw [mem_closedBall_zero_iff]; linarith
+  have hall : ∀ y ∈ ball (u t) (min δ 1),
+      ‖fderiv ℝ f y - fderiv ℝ f (u t)‖ ≤ ε := by
+    intro y hy
+    have hdist : dist y (u t) < δ := hy.trans_le (min_le_left _ _)
+    have hdist1 : dist y (u t) < 1 := hy.trans_le (min_le_right _ _)
+    have hyball : y ∈ closedBall (0 : E) (‖u‖ + 2) := by
+      rw [mem_closedBall_zero_iff]
+      have hn := norm_le_norm_sub_add y (u t)
+      rw [← dist_eq_norm] at hn
+      linarith
+    exact le_of_lt (by simpa only [dist_eq_norm] using hbound y hyball (u t) hut hdist)
+  have hu0 : u t ∈ ball (u t) (min δ 1) := mem_ball_self (lt_min hδ zero_lt_one)
+  have huh : u t + h t ∈ ball (u t) (min δ 1) := by
+    simp only [mem_ball, dist_eq_norm, add_sub_cancel_left]
+    exact lt_min hsmall hsmall1
+  have he := (convex_ball (u t) (min δ 1)).norm_image_sub_le_of_norm_fderiv_le'
+    (fun _ _ => hf _) hall hu0 huh
+  simp only [add_sub_cancel_left] at he
+  exact he.trans (mul_le_mul_of_nonneg_left (h.norm_coe_le_norm t) hε.le)
+
+theorem contDiff_superpose_nat [ProperSpace E] (n : ℕ) {f : E → F}
+    (hf : ContDiff ℝ n f) : ContDiff ℝ n (superpose (X := X) hf.continuous) := by
+  induction n generalizing F with
+  | zero => exact contDiff_zero.mpr (continuous_superpose hf.continuous)
+  | succ n ih =>
+    have hn : (↑(n + 1) : WithTop ℕ∞) ≠ 0 := by simp
+    have hd : Differentiable ℝ f := hf.differentiable hn
+    have hcd : ContDiff ℝ n (fderiv ℝ f) := hf.fderiv_right (by simp)
+    have hc : ContDiff ℝ n (fun u : C(X, E) => applyField (superpose hcd.continuous u)) :=
+      by simpa only [Function.comp_def, applyFieldCLM_apply] using
+        (applyFieldCLM (X := X) (E := E) (F := F)).contDiff.comp (ih hcd)
+    apply contDiff_succ_iff_hasFDerivAt.mpr
+    exact ⟨_, hc, hasFDerivAt_superpose hd hcd.continuous⟩
+
+theorem contDiff_superpose [ProperSpace E] {f : E → F}
+    (hf : ContDiff ℝ ∞ f) : ContDiff ℝ ∞ (superpose (X := X) hf.continuous) := by
+  apply contDiff_infty.mpr
+  intro n
+  exact contDiff_superpose_nat n (contDiff_infty.mp hf n)
+
+end
+end SmoothPaths
+
+end LowerBoundInline
+
+end OAI
diff --git a/lean/OAI/Geometry/PolarProducts/SmoothProfiles.lean b/lean/OAI/Geometry/PolarProducts/SmoothProfiles.lean
new file mode 100644
index 0000000000000000000000000000000000000000..d3cc4cf217a22bd572c17418fdc2c050f25a6e59
--- /dev/null
+++ b/lean/OAI/Geometry/PolarProducts/SmoothProfiles.lean
@@ -0,0 +1,221 @@
+import OAI.Geometry.PolarProducts.EmbeddingInverse
+
+namespace OAI
+
+section NonsqueezingInline
+
+namespace SmoothProfiles
+noncomputable section
+open MeasureTheory Set Filter
+open scoped ContDiff Topology
+
+ def primitive (v : ℝ → ℝ) (s : ℝ) : ℝ := ∫ t in (0:ℝ)..s, v t
+
+ theorem hasDerivAt_primitive {v : ℝ → ℝ} (hv : Continuous v) (s : ℝ) :
+    HasDerivAt (primitive v) (v s) s :=
+  intervalIntegral.integral_hasDerivAt_right (hv.intervalIntegrable _ _)
+    hv.stronglyMeasurable.stronglyMeasurableAtFilter hv.continuousAt
+
+ theorem contDiff_primitive {v : ℝ → ℝ} (hv : ContDiff ℝ ∞ v) : ContDiff ℝ ∞ (primitive v) := by
+  rw [contDiff_infty_iff_deriv]
+  exact ⟨fun x => (hasDerivAt_primitive hv.continuous x).differentiableAt,
+    by simpa only [(funext (fun x => (hasDerivAt_primitive hv.continuous x).deriv))] using hv⟩
+
+ theorem primitive_zero {v : ℝ → ℝ} (hz : ∀ t ≤ 0, v t = 0) {s : ℝ} (hs : s ≤ 0) :
+    primitive v s = 0 := by
+  rw [primitive, intervalIntegral.integral_symm]
+  have hi : ∫ t in s..0, v t = 0 := by
+    calc
+      _ = ∫ _t in s..0, (0 : ℝ) := intervalIntegral.integral_congr (fun t ht =>
+        hz t (by rw [Set.uIcc_of_le hs] at ht; exact ht.2))
+      _ = _ := by simp
+  simp only [hi, neg_zero]
+
+ theorem primitive_nonneg {v : ℝ → ℝ} (_hv : Continuous v)
+    (hn : ∀ t, 0 ≤ v t) (hz : ∀ t ≤ 0, v t = 0) (s : ℝ) : 0 ≤ primitive v s := by
+  by_cases hs : s ≤ 0
+  · simp only [primitive_zero hz hs, le_refl]
+  · exact intervalIntegral.integral_nonneg (by linarith) (fun t _ => hn t)
+
+ theorem primitive_mono {v : ℝ → ℝ} (hv : Continuous v) (hn : ∀ t, 0 ≤ v t) :
+    Monotone (primitive v) := by
+  exact monotone_of_deriv_nonneg (fun t => (hasDerivAt_primitive hv t).differentiableAt)
+    (fun t => by simpa only [(hasDerivAt_primitive hv t).deriv] using hn t)
+
+ def exteriorSlope (δ : ℝ) (s : ℝ) : ℝ := Real.smoothTransition (s/δ)
+ def exterior (δ : ℝ) : ℝ → ℝ := primitive (exteriorSlope δ)
+
+ theorem contDiff_exteriorSlope (δ : ℝ) : ContDiff ℝ ∞ (exteriorSlope δ) :=
+   Real.smoothTransition.contDiff.comp (contDiff_id.div_const δ)
+
+ theorem contDiff_exterior (δ : ℝ) : ContDiff ℝ ∞ (exterior δ) :=
+   contDiff_primitive (contDiff_exteriorSlope δ)
+
+ theorem hasDerivAt_exterior (δ s : ℝ) : HasDerivAt (exterior δ) (exteriorSlope δ s) s :=
+   hasDerivAt_primitive (contDiff_exteriorSlope δ).continuous s
+
+ theorem exteriorSlope_nonneg (δ s : ℝ) : 0 ≤ exteriorSlope δ s := Real.smoothTransition.nonneg _
+ theorem exteriorSlope_le_one (δ s : ℝ) : exteriorSlope δ s ≤ 1 := Real.smoothTransition.le_one _
+ theorem exteriorSlope_zero {δ : ℝ} (hδ : 0 < δ) {s : ℝ} (hs : s ≤ 0) : exteriorSlope δ s = 0 :=
+   Real.smoothTransition.zero_of_nonpos (div_nonpos_of_nonpos_of_nonneg hs hδ.le)
+ theorem exteriorSlope_one {δ : ℝ} (hδ : 0 < δ) {s : ℝ} (hs : δ ≤ s) : exteriorSlope δ s = 1 :=
+   Real.smoothTransition.one_of_one_le ((one_le_div hδ).mpr hs)
+
+ theorem exterior_zero {δ : ℝ} (hδ : 0 < δ) {s : ℝ} (hs : s ≤ 0) : exterior δ s = 0 :=
+   primitive_zero (fun _ ht => exteriorSlope_zero hδ ht) hs
+
+ theorem exterior_nonneg {δ : ℝ} (hδ : 0 < δ) (s : ℝ) : 0 ≤ exterior δ s :=
+   primitive_nonneg (contDiff_exteriorSlope δ).continuous (exteriorSlope_nonneg δ)
+     (fun _ ht => exteriorSlope_zero hδ ht) s
+
+ theorem exterior_affine {δ : ℝ} (hδ : 0 < δ) {s : ℝ} (hs : δ ≤ s) :
+     exterior δ s = s-δ+exterior δ δ := by
+   have hc := (contDiff_exteriorSlope δ).continuous
+   have hi := intervalIntegral.integral_add_adjacent_intervals (hc.intervalIntegrable (μ := volume) 0 δ)
+     (hc.intervalIntegrable (μ := volume) δ s)
+   change exterior δ δ+(∫ t in δ..s, exteriorSlope δ t) = exterior δ s at hi
+   have he : (∫ t in δ..s, exteriorSlope δ t) = s-δ := by
+     calc
+       _ = ∫ _t in δ..s, (1 : ℝ) := intervalIntegral.integral_congr (fun t ht =>
+         exteriorSlope_one hδ (by rw [Set.uIcc_of_le hs] at ht; exact ht.1))
+       _ = _ := by simp
+   rw [he] at hi
+   linarith
+
+ theorem exterior_lower {δ : ℝ} (hδ : 0 < δ) (s : ℝ) : s-δ ≤ exterior δ s := by
+   by_cases hs : δ ≤ s
+   · rw [exterior_affine hδ hs]; linarith [exterior_nonneg hδ δ]
+   · linarith [exterior_nonneg hδ s]
+
+ theorem exterior_upper {δ : ℝ} (hδ : 0 < δ) (s : ℝ) : exterior δ s ≤ max s 0 := by
+   by_cases hs : s ≤ 0
+   · rw [exterior_zero hδ hs, max_eq_right hs]
+   · have hs0 := le_of_not_ge hs
+     calc
+       exterior δ s ≤ ∫ _t in (0:ℝ)..s, (1 : ℝ) :=
+         intervalIntegral.integral_mono_on hs0 ((contDiff_exteriorSlope δ).continuous.intervalIntegrable _ _)
+           (continuous_const.intervalIntegrable _ _) (fun t _ => exteriorSlope_le_one δ t)
+       _ = max s 0 := by simp [max_eq_left hs0]
+
+ theorem primitive_zero_below {v : ℝ → ℝ} {b : ℝ} (hb : 0 ≤ b)
+    (hz : ∀ t ≤ b, v t = 0) {s : ℝ} (hs : s ≤ b) : primitive v s = 0 := by
+  unfold primitive
+  calc
+    _ = ∫ _t in (0:ℝ)..s, (0 : ℝ) := intervalIntegral.integral_congr (fun t ht =>
+      hz t (ht.2.trans (max_le hb hs)))
+    _ = _ := by simp
+
+ theorem primitive_const_above {v : ℝ → ℝ} (hv : Continuous v) {b : ℝ}
+    (hz : ∀ t, b ≤ t → v t = 0) {s : ℝ} (hs : b ≤ s) : primitive v s = primitive v b := by
+  have hi := intervalIntegral.integral_add_adjacent_intervals
+    (hv.intervalIntegrable (μ := volume) 0 b) (hv.intervalIntegrable (μ := volume) b s)
+  have he : (∫ t in b..s, v t) = 0 := by
+    calc
+      _ = ∫ _t in b..s, (0 : ℝ) := intervalIntegral.integral_congr (fun t ht =>
+        hz t (by rw [Set.uIcc_of_le hs] at ht; exact ht.1))
+      _ = _ := by simp
+  change primitive v b+_ = primitive v s at hi
+  simpa only [he, add_zero] using hi.symm
+
+ def ballSlope (r L ε s : ℝ) : ℝ :=
+    r*Real.smoothTransition ((s-ε)/ε)*Real.smoothTransition ((L-ε-s)/ε)
+ def ballProfile (r L ε : ℝ) : ℝ → ℝ := primitive (ballSlope r L ε)
+ def ballHeight (r L ε : ℝ) : ℝ := ballProfile r L ε (L-ε)
+
+ theorem contDiff_ballSlope (r L ε : ℝ) : ContDiff ℝ ∞ (ballSlope r L ε) :=
+    (contDiff_const.mul (Real.smoothTransition.contDiff.comp ((contDiff_id.sub contDiff_const).div_const ε))).mul
+      (Real.smoothTransition.contDiff.comp ((contDiff_const.sub contDiff_id).div_const ε))
+
+ theorem contDiff_ballProfile (r L ε : ℝ) : ContDiff ℝ ∞ (ballProfile r L ε) :=
+    contDiff_primitive (contDiff_ballSlope r L ε)
+
+ theorem hasDerivAt_ballProfile (r L ε s : ℝ) :
+    HasDerivAt (ballProfile r L ε) (ballSlope r L ε s) s :=
+    hasDerivAt_primitive (contDiff_ballSlope r L ε).continuous s
+
+ theorem ballSlope_nonneg {r : ℝ} (hr : 0 ≤ r) (L ε s : ℝ) : 0 ≤ ballSlope r L ε s :=
+    mul_nonneg (mul_nonneg hr (Real.smoothTransition.nonneg _)) (Real.smoothTransition.nonneg _)
+
+ theorem ballSlope_le {r : ℝ} (hr : 0 ≤ r) (L ε s : ℝ) : ballSlope r L ε s ≤ r := by
+    have h1 := Real.smoothTransition.le_one ((s-ε)/ε)
+    have h2 := Real.smoothTransition.le_one ((L-ε-s)/ε)
+    have h0 := Real.smoothTransition.nonneg ((s-ε)/ε)
+    have hp := mul_le_mul_of_nonneg_left h2 (mul_nonneg hr h0)
+    have hq := mul_le_mul_of_nonneg_left h1 hr
+    dsimp [ballSlope]
+    nlinarith
+
+ theorem ballSlope_zero_left (r L : ℝ) {ε : ℝ} (hε : 0 < ε) {s : ℝ} (hs : s ≤ ε) :
+    ballSlope r L ε s = 0 := by
+    unfold ballSlope
+    rw [Real.smoothTransition.zero_of_nonpos (div_nonpos_of_nonpos_of_nonneg (by linarith) hε.le)]
+    ring
+
+ theorem ballSlope_zero_right (r L : ℝ) {ε : ℝ} (hε : 0 < ε) {s : ℝ} (hs : L-ε ≤ s) :
+    ballSlope r L ε s = 0 := by
+    unfold ballSlope
+    rw [show Real.smoothTransition ((L-ε-s)/ε) = 0 from
+      Real.smoothTransition.zero_of_nonpos (div_nonpos_of_nonpos_of_nonneg (by linarith) hε.le)]
+    ring
+
+ theorem ballSlope_middle (r L : ℝ) {ε : ℝ} (hε : 0 < ε) {s : ℝ}
+    (hs : 2*ε ≤ s) (ht : s ≤ L-2*ε) : ballSlope r L ε s = r := by
+    unfold ballSlope
+    rw [Real.smoothTransition.one_of_one_le ((one_le_div hε).mpr (by linarith)),
+      Real.smoothTransition.one_of_one_le ((one_le_div hε).mpr (by linarith))]
+    ring
+
+ theorem ballProfile_zero (r L : ℝ) {ε : ℝ} (hε : 0 < ε) {s : ℝ} (hs : s ≤ ε) :
+    ballProfile r L ε s = 0 :=
+    primitive_zero_below hε.le (fun _ ht => ballSlope_zero_left r L hε ht) hs
+
+ theorem ballProfile_plateau (r L : ℝ) {ε : ℝ} (hε : 0 < ε) {s : ℝ} (hs : L-ε ≤ s) :
+    ballProfile r L ε s = ballHeight r L ε :=
+    primitive_const_above (contDiff_ballSlope r L ε).continuous
+      (fun _ ht => ballSlope_zero_right r L hε ht) hs
+
+ theorem ballProfile_nonneg {r : ℝ} (hr : 0 ≤ r) (L : ℝ) {ε : ℝ} (hε : 0 < ε) (s : ℝ) :
+    0 ≤ ballProfile r L ε s :=
+    primitive_nonneg (contDiff_ballSlope r L ε).continuous (ballSlope_nonneg hr L ε)
+      (fun _ ht => ballSlope_zero_left r L hε (ht.trans hε.le)) s
+
+ theorem ballProfile_le_height {r : ℝ} (hr : 0 ≤ r) (L : ℝ) {ε : ℝ} (hε : 0 < ε) (s : ℝ) :
+    ballProfile r L ε s ≤ ballHeight r L ε := by
+    by_cases hs : L-ε ≤ s
+    · rw [ballProfile_plateau r L hε hs]
+    · exact primitive_mono (contDiff_ballSlope r L ε).continuous (ballSlope_nonneg hr L ε) (le_of_not_ge hs)
+
+ theorem ballHeight_lower {r : ℝ} (hr : 0 ≤ r) (L : ℝ) {ε : ℝ} (hε : 0 < ε) (hL : 4*ε ≤ L) :
+    r*(L-4*ε) ≤ ballHeight r L ε := by
+    have hm := primitive_mono (contDiff_ballSlope r L ε).continuous (ballSlope_nonneg hr L ε)
+    have hn := ballProfile_nonneg hr L hε (2*ε)
+    have hc := (contDiff_ballSlope r L ε).continuous
+    have hi := intervalIntegral.integral_add_adjacent_intervals
+      (hc.intervalIntegrable (μ := volume) 0 (2*ε)) (hc.intervalIntegrable (μ := volume) (2*ε) (L-2*ε))
+    have he : (∫ t in (2*ε)..(L-2*ε), ballSlope r L ε t) = r*(L-4*ε) := by
+      calc
+        _ = ∫ _t in (2*ε)..(L-2*ε), r := intervalIntegral.integral_congr (fun t ht => by
+          rw [Set.uIcc_of_le (by linarith)] at ht
+          exact ballSlope_middle r L hε ht.1 ht.2)
+        _ = _ := by simp; ring
+    change ballProfile r L ε (2*ε)+_ = ballProfile r L ε (L-2*ε) at hi
+    rw [he] at hi
+    have hmono : ballProfile r L ε (L-2*ε) ≤ ballHeight r L ε := hm (by linarith)
+    linarith
+
+ theorem exists_ball_parameters {A L : ℝ} (hA : 0 ≤ A) (hL : 0 < L) (hAL : A < Real.pi*L) :
+    ∃ r ε : ℝ, 0 < r ∧ r < Real.pi ∧ 0 < ε ∧ 4*ε < L ∧ A < r*(L-4*ε) := by
+    obtain ⟨r,hrA,hrπ⟩ := exists_between ((div_lt_iff₀ hL).mpr (by simpa [mul_comm] using hAL))
+    have hr : 0 < r := lt_of_le_of_lt (div_nonneg hA hL.le) hrA
+    have hrl : A < r*L := (div_lt_iff₀ hL).mp hrA
+    let ε := (r*L-A)/(8*r)
+    have hε : 0 < ε := div_pos (sub_pos.mpr hrl) (by positivity)
+    have he : 8*r*ε = r*L-A := by dsimp [ε]; field_simp
+    refine ⟨r,ε,hr,hrπ,hε,?_,?_⟩ <;> nlinarith
+
+end
+end SmoothProfiles
+
+end NonsqueezingInline
+
+end OAI
diff --git a/lean/OAI/Geometry/PolarProducts/StripEmbedding.lean b/lean/OAI/Geometry/PolarProducts/StripEmbedding.lean
new file mode 100644
index 0000000000000000000000000000000000000000..2b059d70f5928eff182ec01d9bb2a10c35beca4a
--- /dev/null
+++ b/lean/OAI/Geometry/PolarProducts/StripEmbedding.lean
@@ -0,0 +1,326 @@
+import OAI.Geometry.PolarProducts.StripMaps
+
+namespace OAI
+
+universe u61 u62 u63 u64 u65 u66 u67 u68 u69
+
+section LowerBoundInline
+open Set Filter Function
+open scoped Topology ContDiff NNReal
+open Set Filter Metric
+open scoped Topology ContDiff
+open Set Filter Function MeasureTheory Metric
+open scoped Topology ContDiff NNReal
+open Set Filter Function
+open scoped Topology ContDiff
+open Set Filter Function
+open scoped Topology ContDiff NNReal
+open Set Filter
+open scoped Topology ContDiff
+open Set Filter Function
+open scoped Topology ContDiff
+open Set Filter Function
+open scoped ContDiff Topology
+open Set MeasureTheory
+open scoped ContDiff Interval Topology
+open Set
+open scoped Topology ContDiff
+open Set
+open Set MeasureTheory
+open scoped ContDiff Interval Topology
+open Set Filter Complex
+open scoped Topology ContDiff
+open MeasureTheory intervalIntegral Set
+open scoped Real
+
+namespace PolarStrips
+open Set ComplexCoordinates
+open scoped ContDiff
+noncomputable section
+variable {ι : Type u61} {κ : Type u62} [Fintype ι] [Fintype κ]
+
+def phase (b : κ → R ι) (k : ℕ) (z : C ι) : R ι × R ι :=
+  (im z, -re z - ∑ j, PlanarLens.primitive k (functional (b j) z) • b j)
+
+theorem contDiffAt_phase (b : κ → R ι) (k : ℕ) {z : C ι}
+    (hz : z ∈ domain (fun j => functional (b j))) : ContDiffAt ℝ ∞ (phase b k) z := by
+  change ContDiffAt ℝ ∞ (fun z : C ι => (ComplexCoordinates.im z, -ComplexCoordinates.re z - ∑ j, PlanarLens.primitive k (functional (b j) z) • b j)) z
+  refine (ComplexCoordinates.im : C ι →L[ℝ] R ι).contDiff.contDiffAt.prodMk
+    ((ComplexCoordinates.re : C ι →L[ℝ] R ι).contDiff.contDiffAt.neg.sub ?_)
+  apply ContDiffAt.sum
+  intro j _
+  exact ((PlanarLens.contDiffAt_primitive (hz j) k).comp z
+    ((functional (b j)).restrictScalars ℝ).contDiff.contDiffAt).smul contDiffAt_const
+
+theorem momentum_monotonic (b : κ → R ι) (k : ℕ) {z w : C ι}
+    (hz : z ∈ domain (fun j => functional (b j)))
+    (hw : w ∈ domain (fun j => functional (b j))) (hx : im z = im w) :
+    inner (𝕜 := ℝ) ((phase b k w).2-(phase b k z).2) (re w-re z) ≤ -‖re w-re z‖^2 := by
+  have hj (j : κ) : 0 ≤
+      (PlanarLens.primitive k (functional (b j) w)-PlanarLens.primitive k (functional (b j) z)) *
+      inner (𝕜 := ℝ) (b j) (re w-re z) := by
+    have ht : (functional (b j) z).im = (functional (b j) w).im := by rw [im_functional, im_functional, hx]
+    have hv : (((functional (b j) z).re : ℂ)+((functional (b j) w).im : ℂ)*Complex.I) ∈ PlanarLens.D := by
+      rw [← ht, Complex.re_add_im]; exact hz j
+    have hh := PlanarLens.J_product_nonneg (w := (functional (b j) w).re) hv (by simpa only [Complex.re_add_im] using hw j) k
+    simpa only [PlanarLens.primitive, ht, inner_sub_right, ← re_functional] using hh
+  have he : (phase b k w).2-(phase b k z).2 = -(re w-re z) -
+      ∑ j, (PlanarLens.primitive k (functional (b j) w)-PlanarLens.primitive k (functional (b j) z)) • b j := by
+    simp only [phase, sub_smul, Finset.sum_sub_distrib]
+    abel
+  rw [he, inner_sub_left, inner_neg_left, real_inner_self_eq_norm_sq]
+  have hsum : 0 ≤ inner (𝕜 := ℝ)
+      (∑ j, (PlanarLens.primitive k (functional (b j) w)-PlanarLens.primitive k (functional (b j) z)) • b j)
+      (re w-re z) := by
+    simp only [sum_inner, real_inner_smul_left]
+    exact Finset.sum_nonneg (fun j _ => hj j)
+  linarith
+
+theorem injOn_phase (b : κ → R ι) (k : ℕ) :
+    InjOn (phase b k) (domain (fun j => functional (b j))) := by
+  intro z hz w hw he
+  have hx : im z = im w := congrArg Prod.fst he
+  have hp : (phase b k w).2-(phase b k z).2 = 0 := sub_eq_zero.mpr (congrArg Prod.snd he).symm
+  have hh := momentum_monotonic b k hz hw hx
+  rw [hp, inner_zero_left] at hh
+  have hr : re w = re z := sub_eq_zero.mp (norm_eq_zero.mp (by nlinarith [norm_nonneg (re w-re z)]))
+  apply parts.injective
+  exact Prod.ext hr.symm hx
+
+end
+end PolarStrips
+
+namespace ComplexPotential
+open ComplexCoordinates Set Filter
+open scoped Topology ContDiff
+noncomputable section
+
+variable {E : Type u63} {F : Type u64} [NormedAddCommGroup E] [InnerProductSpace ℂ E]
+  [NormedAddCommGroup F] [InnerProductSpace ℂ F] [CompleteSpace F]
+
+theorem ddc_holomorphic_norm_sq {U : Set E} (hU : IsOpen U) {f : E → F}
+    (hf : AnalyticOnNhd ℂ f U) {z : E} (hz : z ∈ U) (v w : E) :
+    let : InnerProductSpace ℝ E := InnerProductSpace.complexToReal
+    let : InnerProductSpace ℝ F := InnerProductSpace.complexToReal
+    ddc (complexStructure E) (fun x => ‖f x‖^2) z v w =
+      standardTensor (complexStructure F) (fderiv ℝ f z v) (fderiv ℝ f z w) := by
+  let : InnerProductSpace ℝ E := InnerProductSpace.complexToReal
+  let : InnerProductSpace ℝ F := InnerProductSpace.complexToReal
+  have he := ddc_comp (complexStructure E) (complexStructure F)
+    ((hf z hz).contDiffAt.restrict_scalars ℝ) (contDiff_norm_sq ℝ).contDiffAt
+    (show ∀ᶠ x in 𝓝 z, ∀ a, fderiv ℝ f x (complexStructure E a) = complexStructure F (fderiv ℝ f x a) by
+      filter_upwards [hU.mem_nhds hz] with x hx
+      exact fderiv_commutes_complexStructure (hf x hx).differentiableAt) v w
+  rw [ddc_norm_sq (complexStructure F) complexStructure_skew] at he
+  exact he
+
+variable {ι : Type u65} [Fintype ι]
+
+def phaseForm (v w : R ι × R ι) : ℝ :=
+  inner (𝕜 := ℝ) v.1 w.2-inner (𝕜 := ℝ) w.1 v.2
+
+theorem standardTensor_parts (v w : C ι) :
+    let : InnerProductSpace ℝ (C ι) := InnerProductSpace.complexToReal
+    standardTensor (complexStructure (C ι)) v w = phaseForm (parts v) (parts w) := by
+  let : InnerProductSpace ℝ (C ι) := InnerProductSpace.complexToReal
+  dsimp only
+  rw [standardTensor_apply, complexStructure_apply, real_inner_parts]
+  simp only [re_I_smul, im_I_smul, inner_neg_left, phaseForm, parts_apply]
+  rw [real_inner_comm (im v) (re w)]
+  ring
+
+theorem standardTensor_coordinates (v w : C ι) :
+    let : InnerProductSpace ℝ (C ι) := InnerProductSpace.complexToReal
+    standardTensor (complexStructure (C ι)) v w =
+      ∑ i, ((v i).re*(w i).im-(w i).re*(v i).im) := by
+  dsimp only
+  rw [standardTensor_parts]
+  simp [phaseForm, EuclideanSpace.inner_eq_star_dotProduct, dotProduct, Finset.sum_sub_distrib, mul_comm]
+
+theorem scalar_wedge_mul (c v w : ℂ) :
+    (c*v).re*(c*w).im-(c*w).re*(c*v).im =
+      ‖c‖^2*(v.re*w.im-w.re*v.im) := by
+  rw [Complex.sq_norm]
+  simp only [Complex.mul_re, Complex.mul_im, Complex.normSq_apply]
+  ring
+
+end
+end ComplexPotential
+
+namespace PolarStrips
+open Set Filter ComplexCoordinates ComplexPotential
+open scoped Topology ContDiff
+noncomputable section
+variable {ι : Type u66} {κ : Type u67} [Fintype ι] [Fintype κ]
+
+theorem fderiv_tuple_apply (ℓ : κ → C ι →L[ℂ] ℂ) (k : ℕ) {z : C ι}
+    (hz : z ∈ domain ℓ) (v : C ι) (j : κ) :
+    fderiv ℝ (tuple ℓ k) z v j =
+      (k : ℂ)*PlanarLens.g (ℓ j z)^(k-1)*deriv PlanarLens.g (ℓ j z)*ℓ j v := by
+  have ht := ((analyticOnNhd_tuple ℓ k z hz).differentiableAt.restrictScalars ℝ).hasFDerivAt
+  have hp := (((PiLp.proj (𝕜 := ℂ) 2 (fun _ : κ => ℂ) j).restrictScalars ℝ).hasFDerivAt.comp z ht)
+  have hg := ((PlanarLens.analyticOnNhd_g _ (hz j)).differentiableAt.hasDerivAt.pow k).hasFDerivAt
+  have hc := (hg.comp z (ℓ j).hasFDerivAt).restrictScalars ℝ
+  have he := hc.unique hp
+  have hh := congrArg (fun L : C ι →L[ℝ] ℂ => L v) he
+  change (ℓ j v)*((k : ℂ)*PlanarLens.g (ℓ j z)^(k-1)*deriv PlanarLens.g (ℓ j z)) =
+    fderiv ℝ (tuple ℓ k) z v j at hh
+  simpa only [mul_comm, mul_left_comm, mul_assoc] using hh.symm
+
+theorem norm_coefficient (k : ℕ) (w : ℂ) :
+    ‖(k : ℂ)*PlanarLens.g w^(k-1)*deriv PlanarLens.g w‖^2 =
+      (k : ℝ)^2*PlanarLens.density k w := by
+  simp only [norm_mul, norm_pow, Complex.norm_natCast, mul_pow, ← pow_mul, PlanarLens.density]
+  have he : (k-1)*2 = 2*k-2 := by omega
+  rw [he]
+  ring
+
+theorem fderiv_phase_apply (b : κ → R ι) (k : ℕ) {z : C ι}
+    (hz : z ∈ domain (fun j => functional (b j))) (v : C ι) :
+    fderiv ℝ (phase b k) z v = (im v, -re v - ∑ j,
+      fderiv ℝ (PlanarLens.primitive k) (functional (b j) z) (functional (b j) v) • b j) := by
+  let L : κ → C ι →L[ℝ] R ι := fun j =>
+    ((fderiv ℝ (PlanarLens.primitive k) (functional (b j) z)).comp
+      ((functional (b j)).restrictScalars ℝ)).smulRight (b j)
+  have hj (j : κ) : HasFDerivAt (fun x : C ι => PlanarLens.primitive k (functional (b j) x) • b j) (L j) z :=
+    (((PlanarLens.contDiffAt_primitive (hz j) k).differentiableAt (by simp)).hasFDerivAt.comp z
+      ((functional (b j)).restrictScalars ℝ).hasFDerivAt).smul_const (b j)
+  have hs : HasFDerivAt (fun x : C ι => ∑ j, PlanarLens.primitive k (functional (b j) x) • b j)
+      (∑ j, L j) z := by
+    simpa only [Finset.sum_fn] using (HasFDerivAt.sum (u := Finset.univ) (fun j _ => hj j))
+  have hd : HasFDerivAt (phase b k)
+      ((im : C ι →L[ℝ] R ι).prod (-(re : C ι →L[ℝ] R ι)-∑ j, L j)) z :=
+    im.hasFDerivAt.prodMk (re.hasFDerivAt.neg.sub hs)
+  rw [hd.fderiv]
+  simp only [ContinuousLinearMap.prod_apply, sub_apply,
+    neg_apply, sum_apply, L,
+    ContinuousLinearMap.smulRight_apply, ContinuousLinearMap.comp_apply]
+  rfl
+
+theorem phase_form_formula (b : κ → R ι) (k : ℕ) {z : C ι}
+    (hz : z ∈ domain (fun j => functional (b j))) (v w : C ι) :
+    phaseForm (fderiv ℝ (phase b k) z v) (fderiv ℝ (phase b k) z w) =
+      phaseForm (parts v) (parts w) + ∑ j,
+        (k : ℝ)^2*PlanarLens.density k (functional (b j) z) *
+          ((functional (b j) v).re*(functional (b j) w).im -
+            (functional (b j) w).re*(functional (b j) v).im) := by
+  rw [fderiv_phase_apply b k hz, fderiv_phase_apply b k hz]
+  have he (a : C ι) (j : κ) : inner (𝕜 := ℝ) (im a) (b j) = (functional (b j) a).im := by
+    rw [real_inner_comm, im_functional]
+  have hf : phaseForm (im v, -re v - ∑ j,
+        fderiv ℝ (PlanarLens.primitive k) (functional (b j) z) (functional (b j) v) • b j)
+      (im w, -re w - ∑ j,
+        fderiv ℝ (PlanarLens.primitive k) (functional (b j) z) (functional (b j) w) • b j) =
+      phaseForm (parts v) (parts w) + ∑ j,
+        (fderiv ℝ (PlanarLens.primitive k) (functional (b j) z) (functional (b j) v) * (functional (b j) w).im -
+         fderiv ℝ (PlanarLens.primitive k) (functional (b j) z) (functional (b j) w) * (functional (b j) v).im) := by
+    simp only [phaseForm, parts_apply, inner_sub_right, inner_neg_right, inner_sum,
+      real_inner_smul_right, he, Finset.sum_sub_distrib]
+    rw [real_inner_comm (im v) (re w), real_inner_comm (im w) (re v)]
+    ring
+  rw [hf]
+  congr 1
+  apply Finset.sum_congr rfl
+  intro j _
+  rw [PlanarLens.fderiv_primitive_apply (hz j) k (functional (b j) v),
+    PlanarLens.fderiv_primitive_apply (hz j) k (functional (b j) w)]
+  ring
+
+theorem phase_pullback (b : κ → R ι) (k : ℕ) {z : C ι}
+    (hz : z ∈ domain (fun j => functional (b j))) (v w : C ι) :
+    let : InnerProductSpace ℝ (C ι) := InnerProductSpace.complexToReal
+    phaseForm (fderiv ℝ (phase b k) z v) (fderiv ℝ (phase b k) z w) =
+      ddc (complexStructure (C ι)) (fun x => ‖x‖^2 + ‖tuple (fun j => functional (b j)) k x‖^2) z v w := by
+  let : InnerProductSpace ℝ (C ι) := InnerProductSpace.complexToReal
+  let : InnerProductSpace ℝ (C κ) := InnerProductSpace.complexToReal
+  let ℓ := fun j => functional (b j)
+  have hτ : ContDiffAt ℝ ∞ (fun x => ‖tuple ℓ k x‖^2) z :=
+    ((analyticOnNhd_tuple ℓ k z hz).contDiffAt.restrict_scalars ℝ).norm_sq ℂ
+  dsimp only
+  rw [ddc_add _ (contDiff_norm_sq ℝ).contDiffAt hτ,
+    add_apply, add_apply,
+    ddc_norm_sq _ complexStructure_skew,
+    ddc_holomorphic_norm_sq (isOpen_domain ℓ) (analyticOnNhd_tuple ℓ k) hz,
+    standardTensor_parts v w,
+    standardTensor_coordinates (fderiv ℝ (tuple ℓ k) z v) (fderiv ℝ (tuple ℓ k) z w),
+    phase_form_formula b k hz]
+  congr 1
+  apply Finset.sum_congr rfl
+  intro j _
+  rw [fderiv_tuple_apply ℓ k hz v j, fderiv_tuple_apply ℓ k hz w j, scalar_wedge_mul, norm_coefficient]
+
+end
+end PolarStrips
+
+namespace ConvexPolar
+open Set Filter
+open scoped Topology
+noncomputable section
+variable {E : Type u68} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+
+def polar (K : Set E) : Set E := {p | ∀ q ∈ K, inner (𝕜 := ℝ) q p ≤ 1}
+
+theorem polar_antitone {K L : Set E} (h : K ⊆ L) : polar L ⊆ polar K :=
+  fun _ hp q hq => hp q (h hq)
+
+theorem mem_interior_polar_of_bound {K : Set E} {p : E} {B r : ℝ}
+    (hB : 0 ≤ B) (hbound : ∀ q ∈ K, ‖q‖ ≤ B) (hr : r < 1)
+    (hp : ∀ q ∈ K, inner (𝕜 := ℝ) q p ≤ r) : p ∈ interior (polar K) := by
+  let δ := (1-r)/(B+1)
+  have hδ : 0 < δ := div_pos (by linarith) (by linarith)
+  apply mem_interior_iff_mem_nhds.mpr
+  apply mem_of_superset (Metric.ball_mem_nhds p hδ)
+  intro w hw q hq
+  have hw' : ‖w-p‖ < δ := by simpa only [Metric.mem_ball, dist_eq_norm] using hw
+  have hinner := real_inner_le_norm q (w-p)
+  have hbq := hbound q hq
+  have hmul : ‖q‖*‖w-p‖ ≤ (B+1)*δ := by
+    calc
+      ‖q‖*‖w-p‖ ≤ B*δ := mul_le_mul hbq hw'.le (norm_nonneg _) hB
+      _ ≤ (B+1)*δ := mul_le_mul_of_nonneg_right (by linarith) hδ.le
+  have hδeq : (B+1)*δ = 1-r := by dsimp [δ]; field_simp
+  rw [inner_sub_right] at hinner
+  linarith [hp q hq]
+
+variable {κ : Type u69}
+
+def stripBody (b : κ → E) : Set E := {x | ∀ j, |inner (𝕜 := ℝ) (b j) x| ≤ 1}
+
+variable [Fintype κ]
+
+theorem mem_interior_stripBody (b : κ → E) {x : E}
+    (hx : ∀ j, |inner (𝕜 := ℝ) (b j) x| < 1) : x ∈ interior (stripBody b) := by
+  let U : Set E := {y | ∀ j, |inner (𝕜 := ℝ) (b j) y| < 1}
+  have hopen : IsOpen U := by
+    have he : U = ⋂ j, {y | |inner (𝕜 := ℝ) (b j) y| < 1} := by ext y; simp [U]
+    rw [he]
+    apply isOpen_iInter_of_finite
+    intro j
+    exact isOpen_lt (by fun_prop) continuous_const
+  exact mem_interior_iff_mem_nhds.mpr
+    (mem_of_superset (hopen.mem_nhds hx) (fun y hy j => (hy j).le))
+
+theorem zero_mem_interior_stripBody (b : κ → E) : (0 : E) ∈ interior (stripBody b) :=
+  mem_interior_stripBody b (by intro j; simp)
+
+variable [FiniteDimensional ℝ E]
+
+omit [Fintype κ] in
+theorem isCompact_stripBody (b : κ → E)
+    (hb : Function.Injective (fun x : E => fun j => inner (𝕜 := ℝ) (b j) x)) :
+    IsCompact (stripBody b) := by
+  let L : E →L[ℝ] (κ → ℝ) := ContinuousLinearMap.pi (fun j => innerSL ℝ (b j))
+  have hL : Topology.IsClosedEmbedding L :=
+    LinearMap.isClosedEmbedding_of_injective (LinearMap.ker_eq_bot.mpr hb)
+  have hc := hL.isCompact_preimage (isCompact_pi_infinite (fun _ : κ => isCompact_Icc (a := (-1 : ℝ)) (b := 1)))
+  convert hc using 1
+  ext x
+  simp only [stripBody, mem_ofPred_eq, mem_preimage, ContinuousLinearMap.pi_apply, innerSL_apply_apply,
+    mem_Icc, ← abs_le, L]
+
+end
+end ConvexPolar
+
+end LowerBoundInline
+
+end OAI
diff --git a/lean/OAI/Geometry/PolarProducts/StripMaps.lean b/lean/OAI/Geometry/PolarProducts/StripMaps.lean
new file mode 100644
index 0000000000000000000000000000000000000000..38fed6a5084ef8b0272515b1d60a85a71fcdbfba
--- /dev/null
+++ b/lean/OAI/Geometry/PolarProducts/StripMaps.lean
@@ -0,0 +1,325 @@
+import OAI.Geometry.PolarProducts.LensBounds
+
+namespace OAI
+
+universe u55 u56 u57 u58 u59 u60
+
+section LowerBoundInline
+open Set Filter Function
+open scoped Topology ContDiff NNReal
+open Set Filter Metric
+open scoped Topology ContDiff
+open Set Filter Function MeasureTheory Metric
+open scoped Topology ContDiff NNReal
+open Set Filter Function
+open scoped Topology ContDiff
+open Set Filter Function
+open scoped Topology ContDiff NNReal
+open Set Filter
+open scoped Topology ContDiff
+open Set Filter Function
+open scoped Topology ContDiff
+open Set Filter Function
+open scoped ContDiff Topology
+open Set MeasureTheory
+open scoped ContDiff Interval Topology
+open Set
+open scoped Topology ContDiff
+open Set
+open Set MeasureTheory
+open scoped ContDiff Interval Topology
+open Set Filter Complex
+open scoped Topology ContDiff
+open MeasureTheory intervalIntegral Set
+open scoped Real
+
+namespace PolarStrips
+open Set Filter Asymptotics
+open scoped Topology ContDiff
+noncomputable section
+
+variable {E : Type u55} [NormedAddCommGroup E] [NormedSpace ℂ E]
+variable {ι : Type u56} [Fintype ι]
+
+def domain (ℓ : ι → E →L[ℂ] ℂ) : Set E := {z | ∀ j, ℓ j z ∈ PlanarLens.D}
+
+def tuple (ℓ : ι → E →L[ℂ] ℂ) (k : ℕ) (z : E) : EuclideanSpace ℂ ι :=
+  (EuclideanSpace.equiv ι ℂ).symm (fun j => PlanarLens.g (ℓ j z)^k)
+
+omit [Fintype ι] in
+@[simp] theorem tuple_apply (ℓ : ι → E →L[ℂ] ℂ) (k : ℕ) (z : E) (j : ι) :
+    tuple ℓ k z j = PlanarLens.g (ℓ j z)^k := rfl
+
+theorem norm_tuple_sq (ℓ : ι → E →L[ℂ] ℂ) (k : ℕ) (z : E) :
+    ‖tuple ℓ k z‖^2 = ∑ j, ‖PlanarLens.g (ℓ j z)‖^(2*k) := by
+  rw [EuclideanSpace.norm_sq_eq]
+  apply Finset.sum_congr rfl
+  intro j _
+  change ‖PlanarLens.g (ℓ j z)^k‖^2 = _
+  rw [norm_pow, ← pow_mul, Nat.mul_comm k 2]
+
+theorem isOpen_domain (ℓ : ι → E →L[ℂ] ℂ) : IsOpen (domain ℓ) := by
+  have he : domain ℓ = ⋂ j, (ℓ j) ⁻¹' PlanarLens.D := by ext z; simp [domain]
+  rw [he]
+  exact isOpen_iInter_of_finite (fun j => PlanarLens.isOpen_D.preimage (ℓ j).continuous)
+
+omit [Fintype ι] in
+theorem zero_mem_domain (ℓ : ι → E →L[ℂ] ℂ) : (0 : E) ∈ domain ℓ := by
+  intro j
+  simpa using PlanarLens.zero_mem_D
+
+theorem analyticOnNhd_tuple (ℓ : ι → E →L[ℂ] ℂ) (k : ℕ) :
+    AnalyticOnNhd ℂ (tuple ℓ k) (domain ℓ) := by
+  intro z hz
+  apply ((EuclideanSpace.equiv ι ℂ).symm.toContinuousLinearMap.analyticAt _).comp
+  exact AnalyticAt.pi (fun j => ((PlanarLens.analyticOnNhd_g _ (hz j)).comp ((ℓ j).analyticAt z)).pow k)
+
+omit [Fintype ι] in
+theorem tuple_zero_iff (ℓ : ι → E →L[ℂ] ℂ)
+    (hℓ : Function.Injective (fun z => fun j => ℓ j z)) {k : ℕ} (hk : 1 ≤ k)
+    {z : E} (hz : z ∈ domain ℓ) : tuple ℓ k z = 0 ↔ z = 0 := by
+  constructor
+  · intro hf
+    apply hℓ
+    funext j
+    have hj := congrArg (fun w : EuclideanSpace ℂ ι => w j) hf
+    have hg : PlanarLens.g (ℓ j z) = 0 := (pow_eq_zero_iff (by omega : k ≠ 0)).mp (by simpa using hj)
+    simpa using (PlanarLens.g_eq_zero_iff (hz j)).mp hg
+  · rintro rfl
+    ext j
+    simp only [tuple_apply, map_zero, PlanarLens.g_zero, zero_pow (by omega : k ≠ 0)]
+    rfl
+
+theorem tuple_order (ℓ : ι → E →L[ℂ] ℂ) (k : ℕ) :
+    tuple ℓ k =O[𝓝 (0 : E)] (fun z : E => ‖z‖^k) := by
+  have hc (j : ι) : (fun z => PlanarLens.g (ℓ j z)) =O[𝓝 (0 : E)] (fun z : E => ‖z‖) := by
+    have ha : AnalyticAt ℂ (fun z => PlanarLens.g (ℓ j z)) (0 : E) :=
+      (PlanarLens.analyticOnNhd_g _ (by simpa using PlanarLens.zero_mem_D)).comp ((ℓ j).analyticAt 0)
+    have hh := ha.differentiableAt.isBigO_sub.norm_right
+    simpa only [map_zero, PlanarLens.g_zero, sub_zero] using hh
+  have hp : (fun z j => PlanarLens.g (ℓ j z)^k) =O[𝓝 (0 : E)] (fun z : E => ‖z‖^k) :=
+    isBigO_pi.mpr (fun j => (hc j).pow k)
+  exact ((EuclideanSpace.equiv ι ℂ).symm.toContinuousLinearMap.isBigO_comp
+    (fun z j => PlanarLens.g (ℓ j z)^k) _).trans hp
+
+variable [FiniteDimensional ℂ E]
+
+omit [Fintype ι] in
+
+theorem isBounded_domain (ℓ : ι → E →L[ℂ] ℂ)
+    (hℓ : Function.Injective (fun z => fun j => ℓ j z)) : Bornology.IsBounded (domain ℓ) := by
+  let L : E →L[ℂ] (ι → ℂ) := ContinuousLinearMap.pi ℓ
+  have hL : Topology.IsClosedEmbedding L :=
+    LinearMap.isClosedEmbedding_of_injective (LinearMap.ker_eq_bot.mpr hℓ)
+  let C := PlanarLens.F '' Metric.closedBall (0 : ℂ) 1
+  have hC : IsCompact C := (isCompact_closedBall (0 : ℂ) 1).image_of_continuousOn
+    PlanarLens.continuousOn_F_closedBall
+  have hc : IsCompact (L ⁻¹' {f : ι → ℂ | ∀ j, f j ∈ C}) :=
+    hL.isCompact_preimage (isCompact_pi_infinite (fun _ => hC))
+  apply hc.isBounded.subset
+  intro z hz j
+  exact image_mono Metric.ball_subset_closedBall (hz j)
+
+omit [Fintype ι] in
+
+theorem compact_inner_preimage (ℓ : ι → E →L[ℂ] ℂ)
+    (hℓ : Function.Injective (fun z => fun j => ℓ j z)) {r : ℝ} (hr : r < 1) :
+    IsCompact {z : E | ∀ j, ℓ j z ∈ PlanarLens.F '' Metric.closedBall (0 : ℂ) r} := by
+  let L : E →L[ℂ] (ι → ℂ) := ContinuousLinearMap.pi ℓ
+  have hL : Topology.IsClosedEmbedding L :=
+    LinearMap.isClosedEmbedding_of_injective (LinearMap.ker_eq_bot.mpr hℓ)
+  have hC : IsCompact (PlanarLens.F '' Metric.closedBall (0 : ℂ) r) :=
+    (isCompact_closedBall (0 : ℂ) r).image_of_continuousOn
+      (PlanarLens.continuousOn_F_closedBall.mono (Metric.closedBall_subset_closedBall hr.le))
+  have hc := hL.isCompact_preimage (isCompact_pi_infinite (fun _ : ι => hC))
+  exact hc
+
+theorem compact_sublevel (ℓ : ι → E →L[ℂ] ℂ)
+    (hℓ : Function.Injective (fun z => fun j => ℓ j z)) {k : ℕ} (hk : 1 ≤ k)
+    {s : ℝ} (hs : 0 < s) (hs1 : s < 1) :
+    IsCompact {z : E | z ∈ domain ℓ ∧ ‖tuple ℓ k z‖^2 ≤ s} := by
+  have hk0 : 2*k ≠ 0 := by omega
+  have hval : (s+1)/2 ∈ Icc ((0 : ℝ)^(2*k)) ((1 : ℝ)^(2*k)) := by
+    simp only [zero_pow hk0, one_pow]
+    constructor <;> linarith
+  obtain ⟨r, hr, hre⟩ := intermediate_value_Icc (by norm_num : (0 : ℝ) ≤ 1)
+    (f := fun r : ℝ => r^(2*k)) (by fun_prop) hval
+  dsimp only at hre
+  have hr0 : 0 ≤ r := hr.1
+  have hr1 : r < 1 := by
+    by_contra hn
+    have her : r = 1 := le_antisymm hr.2 (le_of_not_gt hn)
+    rw [her, one_pow] at hre
+    linarith
+  let C := {z : E | ∀ j, ℓ j z ∈ PlanarLens.F '' Metric.closedBall (0 : ℂ) r}
+  have hC : IsCompact C := compact_inner_preimage ℓ hℓ hr1
+  have hCD : C ⊆ domain ℓ := by
+    intro z hz j
+    rcases hz j with ⟨w, hw, he⟩
+    refine ⟨w, ?_, he⟩
+    have hw' : ‖w‖ ≤ r := by simpa using hw
+    simpa using hw'.trans_lt hr1
+  have hsub : ∀ z ∈ domain ℓ, ‖tuple ℓ k z‖^2 ≤ s → z ∈ C := by
+    intro z hz hzs j
+    have hcoord : ‖PlanarLens.g (ℓ j z)‖^(2*k) ≤ ‖tuple ℓ k z‖^2 := by
+      have hh := pow_le_pow_left₀ (norm_nonneg _) (PiLp.norm_apply_le (tuple ℓ k z) j) 2
+      simpa only [tuple_apply, norm_pow, ← pow_mul, Nat.mul_comm k 2] using hh
+    have hp : ‖PlanarLens.g (ℓ j z)‖^(2*k) ≤ r^(2*k) := by linarith
+    have hn : ‖PlanarLens.g (ℓ j z)‖ ≤ r := (pow_le_pow_iff_left₀ (norm_nonneg _) hr0 hk0).mp hp
+    exact ⟨PlanarLens.g (ℓ j z), by simpa using hn, PlanarLens.F_g (hz j)⟩
+  let : CompactSpace C := isCompact_iff_compactSpace.mp hC
+  have hcont : Continuous (fun z : C => ‖tuple ℓ k z‖^2) :=
+    ((analyticOnNhd_tuple ℓ k).continuousOn.mono hCD).domRestrict.norm.pow 2
+  have hclosed : IsClosed {z : C | ‖tuple ℓ k z‖^2 ≤ s} := isClosed_le hcont continuous_const
+  have hc := hclosed.isCompact.image continuous_subtype_val
+  convert hc using 1
+  ext z
+  constructor
+  · intro hz
+    exact ⟨⟨z, hsub z hz.1 hz.2⟩, hz.2, rfl⟩
+  · rintro ⟨z, hz, rfl⟩
+    exact ⟨hCD z.2, hz⟩
+
+end
+end PolarStrips
+
+namespace ComplexCoordinates
+open scoped ComplexConjugate ContDiff
+noncomputable section
+variable {ι : Type u57} [Fintype ι]
+abbrev R (ι : Type u58) := EuclideanSpace ℝ ι
+abbrev C (ι : Type u59) := EuclideanSpace ℂ ι
+
+def re : C ι →L[ℝ] R ι :=
+  LinearMap.toContinuousLinearMap {
+    toFun := fun z => (EuclideanSpace.equiv ι ℝ).symm (fun i => (z i).re)
+    map_add' := by intro x y; ext i; simp
+    map_smul' := by intro a x; ext i; simp }
+
+def im : C ι →L[ℝ] R ι :=
+  LinearMap.toContinuousLinearMap {
+    toFun := fun z => (EuclideanSpace.equiv ι ℝ).symm (fun i => (z i).im)
+    map_add' := by intro x y; ext i; simp
+    map_smul' := by intro a x; ext i; simp }
+
+@[simp] theorem re_apply (z : C ι) (i : ι) : re z i = (z i).re := rfl
+@[simp] theorem im_apply (z : C ι) (i : ι) : im z i = (z i).im := rfl
+
+def parts : C ι ≃L[ℝ] R ι × R ι :=
+  LinearEquiv.toContinuousLinearEquiv {
+    toFun := fun z => (re z, im z)
+    invFun := fun p => (EuclideanSpace.equiv ι ℂ).symm (fun i => (p.1 i : ℂ)+(p.2 i : ℂ)*Complex.I)
+    left_inv := by intro z; ext i; simp [Complex.re_add_im]
+    right_inv := by intro p; ext i <;> simp
+    map_add' := by intros; simp
+    map_smul' := by intros; simp }
+
+@[simp] theorem parts_apply (z : C ι) : parts z = (re z, im z) := rfl
+@[simp] theorem parts_symm_apply (x y : R ι) (i : ι) :
+    parts.symm (x,y) i = (x i : ℂ)+(y i : ℂ)*Complex.I := rfl
+
+def functional (b : R ι) : C ι →L[ℂ] ℂ :=
+  LinearMap.toContinuousLinearMap {
+    toFun := fun z => ∑ i, (b i : ℂ)*z i
+    map_add' := by intro x y; simp [mul_add, Finset.sum_add_distrib]
+    map_smul' := by intro a x; simp [mul_left_comm, Finset.mul_sum] }
+
+@[simp] theorem functional_apply (b : R ι) (z : C ι) : functional b z = ∑ i, (b i : ℂ)*z i := rfl
+
+theorem re_functional (b : R ι) (z : C ι) :
+    (functional b z).re = inner (𝕜 := ℝ) b (re z) := by
+  simp [functional_apply, EuclideanSpace.inner_eq_star_dotProduct, dotProduct, mul_comm]
+
+theorem im_functional (b : R ι) (z : C ι) :
+    (functional b z).im = inner (𝕜 := ℝ) b (im z) := by
+  simp [functional_apply, EuclideanSpace.inner_eq_star_dotProduct, dotProduct, mul_comm]
+
+theorem norm_sq_parts (z : C ι) : ‖z‖^2 = ‖re z‖^2+‖im z‖^2 := by
+  simp only [EuclideanSpace.norm_sq_eq, re_apply, im_apply, Real.norm_eq_abs, sq_abs,
+    Complex.sq_norm, Complex.normSq_apply, ← pow_two, Finset.sum_add_distrib]
+
+theorem real_inner_parts (z w : C ι) :
+    let : InnerProductSpace ℝ (C ι) := InnerProductSpace.complexToReal
+    inner (𝕜 := ℝ) z w = inner (𝕜 := ℝ) (re z) (re w)+inner (𝕜 := ℝ) (im z) (im w) := by
+  let : InnerProductSpace ℝ (C ι) := InnerProductSpace.complexToReal
+  simp [real_inner_eq_re_inner, EuclideanSpace.inner_eq_star_dotProduct, dotProduct,
+    Finset.sum_add_distrib]
+
+@[simp] theorem re_I_smul (z : C ι) : re (Complex.I • z) = -im z := by
+  ext i; simp [Complex.mul_re]
+@[simp] theorem im_I_smul (z : C ι) : im (Complex.I • z) = re z := by
+  ext i; simp [Complex.mul_im]
+
+theorem functional_injective {κ : Type u60} (b : κ → R ι)
+    (hb : Function.Injective (fun x : R ι => fun j => inner (𝕜 := ℝ) (b j) x)) :
+    Function.Injective (fun z : C ι => fun j => functional (b j) z) := by
+  intro z w h
+  apply parts.injective
+  apply Prod.ext
+  · apply hb
+    funext j
+    have hh := congrArg Complex.re (congrFun h j)
+    simpa only [re_functional, parts_apply] using hh
+  · apply hb
+    funext j
+    have hh := congrArg Complex.im (congrFun h j)
+    simpa only [im_functional, parts_apply] using hh
+
+end
+end ComplexCoordinates
+
+namespace PlanarLens
+open Set Filter Complex MeasureTheory
+open scoped Topology ContDiff
+noncomputable section
+
+def primitive (k : ℕ) (z : ℂ) : ℝ := J k z.re z.im
+
+theorem contDiffAt_primitive {z : ℂ} (hz : z ∈ D) (k : ℕ) :
+    ContDiffAt ℝ ∞ (primitive k) z :=
+  (contDiffOn_J k).contDiffAt (isOpen_D.mem_nhds hz)
+
+theorem J_mono {v w t : ℝ} (hv : (v : ℂ)+(t : ℂ)*I ∈ D)
+    (hw : (w : ℂ)+(t : ℂ)*I ∈ D) (hvw : v ≤ w) (k : ℕ) : J k v t ≤ J k w t := by
+  have hi := (integrable_density hv k).symm.trans (integrable_density hw k)
+  have he := intervalIntegral.integral_add_adjacent_intervals (integrable_density hv k) hi
+  have hp : 0 ≤ ∫ h in v..w, density k ((h : ℂ)+(t : ℂ)*I) :=
+    intervalIntegral.integral_nonneg hvw (fun _ _ => density_nonneg _ _)
+  unfold J
+  apply mul_le_mul_of_nonneg_left _ (sq_nonneg _)
+  linarith
+
+theorem J_product_nonneg {v w t : ℝ} (hv : (v : ℂ)+(t : ℂ)*I ∈ D)
+    (hw : (w : ℂ)+(t : ℂ)*I ∈ D) (k : ℕ) :
+    0 ≤ (J k w t-J k v t)*(w-v) := by
+  rcases le_total v w with h | h
+  · exact mul_nonneg (sub_nonneg.mpr (J_mono hv hw h k)) (sub_nonneg.mpr h)
+  · exact mul_nonneg_of_nonpos_of_nonpos (sub_nonpos.mpr (J_mono hw hv h k)) (sub_nonpos.mpr h)
+
+theorem fderiv_primitive_one {z : ℂ} (hz : z ∈ D) (k : ℕ) :
+    fderiv ℝ (primitive k) z 1 = (k : ℝ)^2*density k z := by
+  have hp := (contDiffAt_primitive hz k).differentiableAt (by simp)
+  have hd : HasDerivAt (fun v : ℝ => (v : ℂ)+(z.im : ℂ)*I) 1 z.re :=
+    by simpa using ((hasDerivAt_id (z.re : ℂ)).comp_ofReal.add_const ((z.im : ℂ)*I))
+  have hp' : HasFDerivAt (primitive k) (fderiv ℝ (primitive k) z) ((z.re : ℂ)+(z.im : ℂ)*I) := by
+    simpa only [Complex.re_add_im] using hp.hasFDerivAt
+  have hh := hp'.comp_hasDerivAt z.re hd
+  have hj := hasDerivAt_J (v := z.re) (t := z.im) (by simpa only [Complex.re_add_im] using hz) k
+  have hh' : HasDerivAt (fun v => J k v z.im) (fderiv ℝ (primitive k) z 1) z.re := by
+    simpa [primitive, Function.comp_def] using hh
+  simpa only [Complex.re_add_im] using hh'.unique hj
+
+theorem fderiv_primitive_apply {z : ℂ} (hz : z ∈ D) (k : ℕ) (h : ℂ) :
+    fderiv ℝ (primitive k) z h =
+      (k : ℝ)^2*density k z * h.re + fderiv ℝ (primitive k) z I * h.im := by
+  have he : h = h.re • (1 : ℂ)+h.im • I := by apply Complex.ext <;> simp
+  calc
+    fderiv ℝ (primitive k) z h = fderiv ℝ (primitive k) z (h.re • (1 : ℂ)+h.im • I) := congrArg _ he
+    _ = _ := by simp only [map_add, map_smul, smul_eq_mul, fderiv_primitive_one hz k]; ring
+
+end
+end PlanarLens
+
+end LowerBoundInline
+
+end OAI
diff --git a/lean/OAI/Geometry/PolarProducts/TensorCalculus.lean b/lean/OAI/Geometry/PolarProducts/TensorCalculus.lean
new file mode 100644
index 0000000000000000000000000000000000000000..4d4064314d3df02875a0f2e761fa35425a4897f3
--- /dev/null
+++ b/lean/OAI/Geometry/PolarProducts/TensorCalculus.lean
@@ -0,0 +1,225 @@
+import OAI.Geometry.PolarProducts.SmoothFlow
+
+namespace OAI
+
+universe u17 u18 u19 u20 u21 u22
+
+section LowerBoundInline
+open Set Filter Function
+open scoped Topology ContDiff NNReal
+open Set Filter Metric
+open scoped Topology ContDiff
+open Set Filter Function MeasureTheory Metric
+open scoped Topology ContDiff NNReal
+
+open Set Filter Function
+open scoped Topology ContDiff
+
+namespace TensorCalculus
+
+variable {E : Type u17} [NormedAddCommGroup E] [NormedSpace ℝ E]
+
+noncomputable section
+
+local instance oneTensorGroup : NormedAddCommGroup (E →L[ℝ] ℝ) := inferInstance
+local instance oneTensorSpace : NormedSpace ℝ (E →L[ℝ] ℝ) := inferInstance
+local instance twoTensorGroup : NormedAddCommGroup (E →L[ℝ] E →L[ℝ] ℝ) := inferInstance
+local instance twoTensorSpace : NormedSpace ℝ (E →L[ℝ] E →L[ℝ] ℝ) := inferInstance
+
+abbrev TwoTensor (E : Type u18) [NormedAddCommGroup E] [NormedSpace ℝ E] :=
+  E →L[ℝ] E →L[ℝ] ℝ
+
+def evalTensor (a b : E) : TwoTensor E →L[ℝ] ℝ :=
+  (ContinuousLinearMap.apply ℝ ℝ b).comp
+    (ContinuousLinearMap.apply ℝ (E →L[ℝ] ℝ) a)
+
+@[simp] theorem evalTensor_apply (a b : E) (B : TwoTensor E) :
+    evalTensor a b B = B a b := rfl
+
+theorem fderiv_evalTensor {D : Type u19} [NormedAddCommGroup D] [NormedSpace ℝ D]
+    {B : D → TwoTensor E} {x : D} (hB : DifferentiableAt ℝ B x)
+    (d : D) (a b : E) : fderiv ℝ (fun y => B y a b) x d = fderiv ℝ B x d a b := by
+  rw [fderiv_clm_apply (hB.clm_apply (differentiableAt_const a))
+    (differentiableAt_const b)]
+  rw [fderiv_clm_apply hB (differentiableAt_const a)]
+  simp
+
+theorem fderiv_skew {B : E → TwoTensor E} (hB : Differentiable ℝ B)
+    (hsk : ∀ x a b, B x a b = -B x b a) (x d a b : E) :
+    fderiv ℝ B x d a b = -fderiv ℝ B x d b a := by
+  have he : (fun y => B y a b) = fun y => -B y b a := funext (fun y => hsk y a b)
+  have he' := congrArg (fun f : E → ℝ => fderiv ℝ f x d) he
+  rw [fderiv_fun_neg] at he'
+  simpa only [fderiv_evalTensor (hB x), neg_apply] using he'
+
+theorem derivative_kernel {B : E → TwoTensor E} {V : E → E}
+    (hB : Differentiable ℝ B) (hV : Differentiable ℝ V)
+    (hker : ∀ x, B x (V x) = 0) (x a b : E) :
+    B x (fderiv ℝ V x a) b + fderiv ℝ B x a (V x) b = 0 := by
+  have hd : HasFDerivAt (fun y => B y (V y))
+      ((B x).comp (fderiv ℝ V x) + (fderiv ℝ B x).flip (V x)) x :=
+    (hB x).hasFDerivAt.clm_apply (hV x).hasFDerivAt
+  have hz : HasFDerivAt (fun y => B y (V y)) 0 x :=
+    (hasFDerivAt_const (𝕜 := ℝ) (0 : E →L[ℝ] ℝ) x).congr_of_eventuallyEq
+      (Filter.Eventually.of_forall hker)
+  have he := congrArg (fun L : E →L[ℝ] E →L[ℝ] ℝ => L a b) (hd.unique hz)
+  simpa using he
+
+theorem invariant_of_closed_kernel {B : E → TwoTensor E} {V : E → E}
+    (hB : Differentiable ℝ B) (hV : Differentiable ℝ V)
+    (hsk : ∀ x a b, B x a b = -B x b a)
+    (hclosed : ∀ x a b c, fderiv ℝ B x a b c +
+      fderiv ℝ B x b c a + fderiv ℝ B x c a b = 0)
+    (hker : ∀ x, B x (V x) = 0) (x a b : E) :
+    fderiv ℝ B x (V x) a b + B x (fderiv ℝ V x a) b +
+      B x a (fderiv ℝ V x b) = 0 := by
+  have h1 := derivative_kernel hB hV hker x a b
+  have h2 := derivative_kernel hB hV hker x b a
+  have h3 := hclosed x (V x) a b
+  have h4 := fderiv_skew hB hsk x a b (V x)
+  have h5 := hsk x a (fderiv ℝ V x b)
+  linarith
+
+theorem fderiv_skew_at {B : E → TwoTensor E} {x : E} (hB : DifferentiableAt ℝ B x)
+    (hsk : ∀ᶠ y in 𝓝 x, ∀ a b, B y a b = -B y b a) (d a b : E) :
+    fderiv ℝ B x d a b = -fderiv ℝ B x d b a := by
+  have he : (fun y => B y a b) =ᶠ[𝓝 x] (fun y => -B y b a) :=
+    hsk.mono (fun y hy => hy a b)
+  have he' := congrArg (fun L : E →L[ℝ] ℝ => L d) he.fderiv_eq
+  rw [fderiv_fun_neg] at he'
+  simpa only [fderiv_evalTensor hB, neg_apply] using he'
+
+theorem derivative_kernel_at {B : E → TwoTensor E} {V : E → E} {x : E}
+    (hB : DifferentiableAt ℝ B x) (hV : DifferentiableAt ℝ V x)
+    (hker : ∀ᶠ y in 𝓝 x, B y (V y) = 0) (a b : E) :
+    B x (fderiv ℝ V x a) b + fderiv ℝ B x a (V x) b = 0 := by
+  have hd : HasFDerivAt (fun y => B y (V y))
+      ((B x).comp (fderiv ℝ V x) + (fderiv ℝ B x).flip (V x)) x :=
+    hB.hasFDerivAt.clm_apply hV.hasFDerivAt
+  have hz : HasFDerivAt (fun y => B y (V y)) 0 x :=
+    (hasFDerivAt_const (𝕜 := ℝ) (0 : E →L[ℝ] ℝ) x).congr_of_eventuallyEq hker
+  have he := congrArg (fun L : E →L[ℝ] E →L[ℝ] ℝ => L a b) (hd.unique hz)
+  simpa using he
+
+theorem invariant_of_closed_kernel_at {B : E → TwoTensor E} {V : E → E} {x : E}
+    (hB : DifferentiableAt ℝ B x) (hV : DifferentiableAt ℝ V x)
+    (hsk : ∀ᶠ y in 𝓝 x, ∀ a b, B y a b = -B y b a)
+    (hclosed : ∀ a b c, fderiv ℝ B x a b c +
+      fderiv ℝ B x b c a + fderiv ℝ B x c a b = 0)
+    (hker : ∀ᶠ y in 𝓝 x, B y (V y) = 0) (a b : E) :
+    fderiv ℝ B x (V x) a b + B x (fderiv ℝ V x a) b +
+      B x a (fderiv ℝ V x b) = 0 := by
+  have h1 := derivative_kernel_at hB hV hker a b
+  have h2 := derivative_kernel_at hB hV hker b a
+  have h3 := hclosed (V x) a b
+  have h4 := fderiv_skew_at hB hsk a b (V x)
+  have h5 := (hsk.self_of_nhds) a (fderiv ℝ V x b)
+  linarith
+
+def alt : (E →L[ℝ] E →L[ℝ] ℝ) →L[ℝ] (E →L[ℝ] E →L[ℝ] ℝ) :=
+  ContinuousLinearMap.id ℝ (E →L[ℝ] E →L[ℝ] ℝ) -
+    (ContinuousLinearMap.flipₗᵢ ℝ E E ℝ).toLinearIsometry.toContinuousLinearMap
+
+@[simp] theorem alt_apply (B : TwoTensor E) (a b : E) :
+    alt B a b = B a b - B b a := rfl
+
+def exteriorD (γ : E → E →L[ℝ] ℝ) (x : E) : TwoTensor E :=
+  alt (fderiv ℝ γ x)
+
+@[simp] theorem exteriorD_apply (γ : E → E →L[ℝ] ℝ) (x a b : E) :
+    exteriorD γ x a b = fderiv ℝ γ x a b - fderiv ℝ γ x b a := rfl
+
+theorem exteriorD_skew (γ : E → E →L[ℝ] ℝ) (x a b : E) :
+    exteriorD γ x a b = -exteriorD γ x b a := by
+  simp only [exteriorD_apply]; ring
+
+theorem fderiv_exteriorD {γ : E → E →L[ℝ] ℝ} {x : E}
+    (hγ : DifferentiableAt ℝ (fderiv ℝ γ) x) (d a b : E) :
+    fderiv ℝ (exteriorD γ) x d a b =
+      fderiv ℝ (fderiv ℝ γ) x d a b - fderiv ℝ (fderiv ℝ γ) x d b a := by
+  change fderiv ℝ (alt ∘ fderiv ℝ γ) x d a b = _
+  rw [fderiv_comp x (ContinuousLinearMap.differentiableAt (alt (E := E))) hγ]
+  simp
+
+theorem contDiff_exteriorD {γ : E → E →L[ℝ] ℝ}
+    (hγ : ContDiff ℝ ∞ γ) : ContDiff ℝ ∞ (exteriorD γ) := by
+  change ContDiff ℝ ∞ ((alt (E := E)) ∘ fderiv ℝ γ)
+  exact (alt (E := E)).contDiff.comp (hγ.fderiv_right (m := ∞) (by simp))
+
+theorem exteriorD_closed {γ : E → E →L[ℝ] ℝ} {x : E}
+    (hγ : ContDiffAt ℝ ∞ γ x) (a b c : E) :
+    fderiv ℝ (exteriorD γ) x a b c + fderiv ℝ (exteriorD γ) x b c a +
+      fderiv ℝ (exteriorD γ) x c a b = 0 := by
+  have hd : DifferentiableAt ℝ (fderiv ℝ γ) x :=
+    (hγ.fderiv_right (by simp : (∞ : ℕ∞ω) + 1 ≤ ∞)).differentiableAt (by simp)
+  have hs := hγ.isSymmSndFDerivAt (by
+    rw [minSmoothness_of_isRCLikeNormedField]
+    change ((2 : ℕ∞) : WithTop ℕ∞) ≤ (⊤ : ℕ∞)
+    exact WithTop.coe_le_coe.mpr le_top)
+  simp only [fderiv_exteriorD hd]
+  have h1 := congrArg (fun L : E →L[ℝ] ℝ => L c) (hs.eq a b)
+  have h2 := congrArg (fun L : E →L[ℝ] ℝ => L a) (hs.eq b c)
+  have h3 := congrArg (fun L : E →L[ℝ] ℝ => L b) (hs.eq c a)
+  linarith
+
+end
+end TensorCalculus
+
+namespace TensorCalculus
+
+open Set Filter Function
+open scoped Topology ContDiff
+
+variable {E : Type u20} {F : Type u21} {D : Type u22} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  [NormedAddCommGroup F] [NormedSpace ℝ F]
+  [NormedAddCommGroup D] [NormedSpace ℝ D]
+
+noncomputable section
+
+def pullLinear (L : F →L[ℝ] E) (B : TwoTensor E) : TwoTensor F :=
+  B.bilinearComp L L
+
+@[simp] theorem pullLinear_apply (L : F →L[ℝ] E) (B : TwoTensor E) (a b : F) :
+    pullLinear L B a b = B (L a) (L b) := rfl
+
+theorem contDiffAt_pullLinear (L : F →L[ℝ] E) {B : D → TwoTensor E} {z : D}
+    (hB : ContDiffAt ℝ ∞ B z) :
+    ContDiffAt ℝ ∞ (fun x => pullLinear L (B x)) z := by
+  let P : (E →L[ℝ] ℝ) →L[ℝ] (F →L[ℝ] ℝ) :=
+    (ContinuousLinearMap.compL ℝ F E ℝ).flip L
+  change ContDiffAt ℝ ∞ (fun x => P.comp ((B x).comp L)) z
+  exact contDiffAt_const.clm_comp (hB.clm_comp contDiffAt_const)
+
+theorem fderiv_pullLinear_eval (L : F →L[ℝ] E) {B : D → TwoTensor E} {z : D}
+    (hB : DifferentiableAt ℝ B z)
+    (hP : DifferentiableAt ℝ (fun x => pullLinear L (B x)) z) (d : D) (a b : F) :
+    fderiv ℝ (fun x => pullLinear L (B x)) z d a b =
+      fderiv ℝ B z d (L a) (L b) := by
+  rw [← fderiv_evalTensor hP, ← fderiv_evalTensor hB]
+  rfl
+
+theorem pullLinear_closed (L : F →L[ℝ] E) {B : E → TwoTensor E} {z : F}
+    (hB : ContDiffAt ℝ ∞ B (L z))
+    (hclosed : ∀ a b c, fderiv ℝ B (L z) a b c +
+      fderiv ℝ B (L z) b c a + fderiv ℝ B (L z) c a b = 0) (a b c : F) :
+    fderiv ℝ (fun x => pullLinear L (B (L x))) z a b c +
+      fderiv ℝ (fun x => pullLinear L (B (L x))) z b c a +
+      fderiv ℝ (fun x => pullLinear L (B (L x))) z c a b = 0 := by
+  have hs : ContDiffAt ℝ ∞ (fun x => B (L x)) z := hB.comp z L.contDiff.contDiffAt
+  have ht := contDiffAt_pullLinear L hs
+  have he (d u v : F) :
+      fderiv ℝ (fun x => pullLinear L (B (L x))) z d u v =
+        fderiv ℝ B (L z) (L d) (L u) (L v) := by
+    rw [fderiv_pullLinear_eval L (hs.differentiableAt (by simp))
+      (ht.differentiableAt (by simp))]
+    rw [fderiv_fun_comp z (hB.differentiableAt (by simp)) L.differentiableAt]
+    simp
+  simp only [he]
+  exact hclosed (L a) (L b) (L c)
+
+end
+end TensorCalculus
+
+end LowerBoundInline
+
+end OAI
diff --git a/lean/OAI/Geometry/PolarProducts/TipEstimates.lean b/lean/OAI/Geometry/PolarProducts/TipEstimates.lean
new file mode 100644
index 0000000000000000000000000000000000000000..acb5bb2ae47e546a0d36702361582b0800cbc813
--- /dev/null
+++ b/lean/OAI/Geometry/PolarProducts/TipEstimates.lean
@@ -0,0 +1,400 @@
+import OAI.Geometry.PolarProducts.HorizontalIntegrals
+
+namespace OAI
+
+section LowerBoundInline
+open Set Filter Function
+open scoped Topology ContDiff NNReal
+open Set Filter Metric
+open scoped Topology ContDiff
+open Set Filter Function MeasureTheory Metric
+open scoped Topology ContDiff NNReal
+open Set Filter Function
+open scoped Topology ContDiff
+open Set Filter Function
+open scoped Topology ContDiff NNReal
+open Set Filter
+open scoped Topology ContDiff
+open Set Filter Function
+open scoped Topology ContDiff
+open Set Filter Function
+open scoped ContDiff Topology
+open Set MeasureTheory
+open scoped ContDiff Interval Topology
+open Set
+open scoped Topology ContDiff
+open Set
+open Set MeasureTheory
+open scoped ContDiff Interval Topology
+open Set Filter Complex
+open scoped Topology ContDiff
+
+open MeasureTheory intervalIntegral Set
+open scoped Real
+
+namespace SymmetricPolar
+noncomputable section
+
+def endpointKernel (H y : ℝ) : ℝ := Real.sqrt (y / (H - y))
+
+lemma endpointKernel_eq_rpow {H y : ℝ} (hy : 0 ≤ y) (hyH : y ≤ H) :
+    endpointKernel H y = y ^ (1 / 2 : ℝ) * (H - y) ^ (-(1 / 2 : ℝ)) := by
+  rw [endpointKernel, Real.sqrt_div hy, Real.sqrt_eq_rpow, Real.sqrt_eq_rpow,
+    Real.rpow_neg (sub_nonneg.mpr hyH), div_eq_mul_inv]
+
+lemma intervalIntegrable_endpointKernel {H : ℝ} (hH : 0 ≤ H) :
+    IntervalIntegrable (endpointKernel H) volume 0 H := by
+  have hi : IntervalIntegrable (fun y : ℝ => (H - y) ^ (-(1 / 2 : ℝ))) volume 0 H := by
+    simpa using ((intervalIntegral.intervalIntegrable_rpow' (a := 0) (b := H)
+      (by norm_num : (-1 : ℝ) < -(1 / 2 : ℝ))).comp_sub_left H).symm
+  have hm := hi.continuousOn_mul Real.continuous_sqrt.continuousOn
+  apply IntervalIntegrable.congr (f := fun y => Real.sqrt y * (H - y) ^ (-(1 / 2 : ℝ))) _ hm
+  intro y hy
+  rw [uIoc_of_le hH] at hy
+  dsimp only
+  rw [endpointKernel_eq_rpow hy.1.le hy.2, Real.sqrt_eq_rpow]
+
+private lemma beta_three_halves_one_half :
+    Complex.betaIntegral (3 / 2) (1 / 2) = (Real.pi : ℂ) / 2 := by
+  rw [Complex.betaIntegral_eq_Gamma_mul_div _ _ (by norm_num) (by norm_num)]
+  have hG : Complex.Gamma (2 : ℂ) = 1 := by
+    norm_num [show (2 : ℂ) = 1 + 1 by norm_num, Complex.Gamma_add_one]
+  have hreflect := Complex.Gamma_mul_Gamma_one_sub (1 / 2)
+  norm_num at hreflect
+  rw [show (Real.pi : ℂ) * (1 / 2) = (Real.pi : ℂ) / 2 by ring,
+    Complex.sin_pi_div_two, div_one] at hreflect
+  rw [show (3 / 2 : ℂ) + 1 / 2 = 2 by norm_num, hG, div_one,
+    show (3 / 2 : ℂ) = 1 / 2 + 1 by norm_num,
+    Complex.Gamma_add_one _ (by norm_num)]
+  rw [mul_assoc, hreflect]
+  ring
+
+lemma integral_endpointKernel {H : ℝ} (hH : 0 < H) :
+    (∫ y in 0..H, endpointKernel H y) = Real.pi * H / 2 := by
+  apply Complex.ofReal_injective
+  rw [← intervalIntegral.integral_ofReal]
+  have h := Complex.betaIntegral_scaled (3 / 2) (1 / 2) hH
+  norm_num only at h
+  rw [beta_three_halves_one_half] at h
+  have hc : (∫ y in 0..H, (endpointKernel H y : ℂ)) =
+      ∫ y in 0..H, (y : ℂ) ^ ((3 / 2 : ℂ) - 1) *
+        ((H : ℂ) - y) ^ ((1 / 2 : ℂ) - 1) := by
+    apply intervalIntegral.integral_congr
+    intro y hy
+    rw [uIcc_of_le hH.le] at hy
+    dsimp only
+    rw [endpointKernel_eq_rpow hy.1 hy.2, Complex.ofReal_mul,
+      Complex.ofReal_cpow hy.1, Complex.ofReal_cpow (sub_nonneg.mpr hy.2),
+      Complex.ofReal_sub]
+    norm_num
+  rw [hc]
+  norm_num only
+  rw [h]
+  norm_num
+  ring
+
+lemma endpointKernel_le_one {H y : ℝ} (hH : 0 < H) (_hy : 0 ≤ y)
+    (hyH : y ≤ H / 2) : endpointKernel H y ≤ 1 := by
+  apply Real.sqrt_le_one.mpr
+  exact (div_le_one (by linarith : 0 < H - y)).mpr (by linarith)
+
+lemma intervalIntegrable_dampedEndpointKernel {H : ℝ} (hH : 0 ≤ H) :
+    IntervalIntegrable (fun y => Real.exp (-y) * endpointKernel H y) volume 0 H :=
+  (intervalIntegrable_endpointKernel hH).continuousOn_mul (by fun_prop)
+
+lemma integral_dampedEndpointKernel_le {H : ℝ} (hH : 0 < H) :
+    (∫ y in 0..H, Real.exp (-y) * endpointKernel H y) ≤
+      1 + Real.pi / Real.exp 1 := by
+  have hhalf : 0 ≤ H / 2 := by positivity
+  have hhalfH : H / 2 ≤ H := by linarith
+  have hsfirst : uIcc 0 (H / 2) ⊆ uIcc 0 H := by
+    simpa only [uIcc_of_le hhalf, uIcc_of_le hH.le] using
+      Icc_subset_Icc le_rfl hhalfH
+  have hslast : uIcc (H / 2) H ⊆ uIcc 0 H := by
+    simpa only [uIcc_of_le hhalfH, uIcc_of_le hH.le] using
+      Icc_subset_Icc hhalf le_rfl
+  have hk := intervalIntegrable_endpointKernel hH.le
+  have hf := intervalIntegrable_dampedEndpointKernel hH.le
+  have hfirst : (∫ y in 0..H / 2, Real.exp (-y) * endpointKernel H y) ≤ 1 := by
+    calc
+      _ ≤ ∫ y in 0..H / 2, Real.exp (-y) := by
+        apply intervalIntegral.integral_mono_on hhalf (hf.mono_set hsfirst)
+          ((by fun_prop : Continuous (fun y : ℝ => Real.exp (-y))).intervalIntegrable _ _)
+        intro y hy
+        exact mul_le_of_le_one_right (Real.exp_pos _).le
+          (endpointKernel_le_one hH hy.1 hy.2)
+      _ ≤ 1 := by
+        rw [intervalIntegral.integral_comp_neg, integral_exp]
+        simp only [neg_zero, Real.exp_zero]
+        linarith [Real.exp_pos (-(H / 2))]
+  have hlast : (∫ y in H / 2..H, Real.exp (-y) * endpointKernel H y) ≤
+      Real.exp (-(H / 2)) * (Real.pi * H / 2) := by
+    calc
+      _ ≤ ∫ y in H / 2..H, Real.exp (-(H / 2)) * endpointKernel H y := by
+        apply intervalIntegral.integral_mono_on hhalfH (hf.mono_set hslast)
+          ((hk.mono_set hslast).const_mul _)
+        intro y hy
+        exact mul_le_mul_of_nonneg_right (Real.exp_le_exp.mpr (neg_le_neg hy.1))
+          (Real.sqrt_nonneg _)
+      _ = Real.exp (-(H / 2)) * ∫ y in H / 2..H, endpointKernel H y :=
+        intervalIntegral.integral_const_mul _ _
+      _ ≤ Real.exp (-(H / 2)) * ∫ y in 0..H, endpointKernel H y := by
+        apply mul_le_mul_of_nonneg_left _ (Real.exp_pos _).le
+        apply intervalIntegral.integral_mono_interval hhalf hhalfH le_rfl _ hk
+        exact Filter.Eventually.of_forall (fun y => Real.sqrt_nonneg _)
+      _ = _ := by rw [integral_endpointKernel hH]
+  have hexp : (H / 2) * Real.exp (-(H / 2)) ≤ 1 / Real.exp 1 := by
+    calc
+      _ ≤ Real.exp (H / 2 - 1) * Real.exp (-(H / 2)) := by
+        apply mul_le_mul_of_nonneg_right _ (Real.exp_pos _).le
+        linarith [Real.add_one_le_exp (H / 2 - 1)]
+      _ = Real.exp (-1) := by rw [← Real.exp_add]; congr 1; ring
+      _ = _ := by rw [Real.exp_neg, one_div]
+  rw [← intervalIntegral.integral_add_adjacent_intervals
+    (hf.mono_set hsfirst) (hf.mono_set hslast)]
+  have hp := mul_le_mul_of_nonneg_left hexp Real.pi_pos.le
+  have hp' : Real.exp (-(H / 2)) * (Real.pi * H / 2) ≤ Real.pi / Real.exp 1 := by
+    calc
+      _ = Real.pi * (H / 2 * Real.exp (-(H / 2))) := by ring
+      _ ≤ Real.pi * (1 / Real.exp 1) := hp
+      _ = _ := by ring
+  exact add_le_add hfirst (hlast.trans hp')
+
+def movingEndpointIntegral (k : ℕ) (a : ℝ) : ℝ :=
+  (k : ℝ) * ∫ r in a..1,
+    r ^ (2 * k - 1) * Real.sqrt ((1 - r) / (r - a))
+
+lemma movingEndpointIntegral_eq {k : ℕ} (hk : 0 < k) (a : ℝ) :
+    movingEndpointIntegral k a = ∫ y in 0..(k : ℝ) * (1 - a),
+      (1 - y / k) ^ (2 * k - 1) * endpointKernel ((k : ℝ) * (1 - a)) y := by
+  have hk0 : (k : ℝ) ≠ 0 := by exact_mod_cast hk.ne'
+  let f : ℝ → ℝ := fun y =>
+    (1 - y / k) ^ (2 * k - 1) * endpointKernel ((k : ℝ) * (1 - a)) y
+  have hsub (r : ℝ) : f ((k : ℝ) - k * r) =
+      r ^ (2 * k - 1) * Real.sqrt ((1 - r) / (r - a)) := by
+    dsimp [f, endpointKernel]
+    have hbase : 1 - ((k : ℝ) - k * r) / k = r := by field_simp; ring
+    rw [hbase]
+    congr 2
+    rw [show (k : ℝ) - k * r = k * (1 - r) by ring,
+      show (k : ℝ) * (1 - a) - k * (1 - r) = k * (r - a) by ring,
+      mul_div_mul_left _ _ hk0]
+  calc
+    _ = (k : ℝ) • ∫ r in a..1, f ((k : ℝ) - k * r) := by
+      simp only [movingEndpointIntegral, smul_eq_mul, hsub]
+    _ = ∫ y in (k : ℝ) - k * 1..(k : ℝ) - k * a, f y :=
+      intervalIntegral.smul_integral_comp_sub_mul f _ _
+    _ = _ := by
+      rw [show (k : ℝ) - k * 1 = 0 by ring,
+        show (k : ℝ) - k * a = k * (1 - a) by ring]
+
+lemma movingEndpointIntegral_le {k : ℕ} (hk : 1 ≤ k) {a : ℝ}
+    (ha : 0 ≤ a) (ha1 : a < 1) :
+    movingEndpointIntegral k a ≤ 1 + Real.pi / Real.exp 1 := by
+  have hkR : (0 : ℝ) < k := by exact_mod_cast (by omega : 0 < k)
+  have hH : 0 < (k : ℝ) * (1 - a) := mul_pos hkR (sub_pos.mpr ha1)
+  rw [movingEndpointIntegral_eq (by omega : 0 < k) a]
+  apply le_trans _ (integral_dampedEndpointKernel_le hH)
+  apply intervalIntegral.integral_mono_on hH.le
+    ((intervalIntegrable_endpointKernel hH.le).continuousOn_mul (by fun_prop))
+    (intervalIntegrable_dampedEndpointKernel hH.le)
+  intro y hy
+  apply mul_le_mul_of_nonneg_right _ (Real.sqrt_nonneg _)
+  have hyk : y ≤ (k : ℝ) := by nlinarith [mul_nonneg hkR.le ha, hy.2]
+  have hb0 : 0 ≤ 1 - y / (k : ℝ) := sub_nonneg.mpr ((div_le_one hkR).mpr hyk)
+  have hb1 : 1 - y / (k : ℝ) ≤ 1 := sub_le_self _ (div_nonneg hy.1 hkR.le)
+  exact (pow_le_pow_of_le_one hb0 hb1 (by omega : k ≤ 2 * k - 1)).trans
+    (Real.one_sub_div_pow_le_exp_neg hyk)
+
+lemma movingEndpointIntegral_le_linear {k : ℕ} (hk : 1 ≤ k) {a : ℝ}
+    (ha : 0 ≤ a) (ha1 : a < 1) :
+    movingEndpointIntegral k a ≤ Real.pi * ((k : ℝ) * (1 - a)) / 2 := by
+  have hkR : (0 : ℝ) < k := by exact_mod_cast (by omega : 0 < k)
+  have hH : 0 < (k : ℝ) * (1 - a) := mul_pos hkR (sub_pos.mpr ha1)
+  rw [movingEndpointIntegral_eq (by omega : 0 < k) a,
+    ← integral_endpointKernel hH]
+  apply intervalIntegral.integral_mono_on hH.le
+    ((intervalIntegrable_endpointKernel hH.le).continuousOn_mul (by fun_prop))
+    (intervalIntegrable_endpointKernel hH.le)
+  intro y hy
+  apply mul_le_of_le_one_left (Real.sqrt_nonneg _)
+  have hyk : y ≤ (k : ℝ) := by nlinarith [mul_nonneg hkR.le ha, hy.2]
+  have hb0 : 0 ≤ 1 - y / (k : ℝ) := sub_nonneg.mpr ((div_le_one hkR).mpr hyk)
+  have hb1 : 1 - y / (k : ℝ) ≤ 1 := sub_le_self _ (div_nonneg hy.1 hkR.le)
+  exact pow_le_one₀ hb0 hb1
+
+end
+end SymmetricPolar
+
+namespace PlanarLens
+open Set Filter Complex MeasureTheory
+open scoped Topology ContDiff
+noncomputable section
+
+def tailSlope (a : ℝ) : ℝ := 8/Real.pi^2 * Real.artanh a
+
+theorem tailSlope_pos {a : ℝ} (ha : 0 < a) (ha1 : a < 1) : 0 < tailSlope a := by
+  apply mul_pos (by positivity)
+  exact Real.artanh_pos ⟨ha, ha1⟩
+
+theorem T_rho_ge_abs {t v u : ℝ} (hz : (v : ℂ)+(t : ℂ)*I ∈ D)
+    (hv : 0 ≤ v) (hu : u ∈ Icc 0 v) : |t| ≤ T (rho t u) := by
+  have hmono := (horizontal_radius_strictMono hz hv).monotoneOn
+  have h0 := horizontal_segment_mem hz hv (left_mem_Icc.mpr hv)
+  have hU := horizontal_segment_mem hz hv hu
+  have hle := hmono (left_mem_Icc.mpr hv) hu hu.1
+  have hT := strictMonoOn_T.monotoneOn
+    (show rho t 0 ∈ Icc (-1) 1 from ⟨(le_trans (by norm_num) (norm_nonneg _)), (norm_g_lt h0).le⟩)
+    (show rho t u ∈ Icc (-1) 1 from ⟨le_trans (by norm_num) (norm_nonneg _), (norm_g_lt hU).le⟩) hle
+  have he : T (rho t 0) = |t| := by simpa [rho] using T_norm_g_axis (axis_mem_D hz)
+  rwa [he] at hT
+
+theorem reciprocal_A_tail {w : ℂ} (hw : ‖w‖ < 1) (hwr : 0 < w.re)
+    {a b : ℝ} (ha : 1/2 ≤ a) (hab : a ≤ b) (hbw : b < ‖w‖)
+    (ht : |(F w).im| ≤ T b) :
+    1/A w ≤ Real.pi/2 + Real.pi^3 / Real.sqrt (tailSlope a) *
+      Real.sqrt ((1-‖w‖)/(‖w‖-b)) := by
+  have ha0 : 0 < a := by linarith
+  have ha1 : a < 1 := hab.trans_lt (hbw.trans hw)
+  have hL := tailSlope_pos ha0 ha1
+  have hgap := T_tail_gap ha0 hab hbw.le hw
+  have hnum : 0 ≤ 1-‖w‖ := by linarith
+  have hden : 0 < tailSlope a * (‖w‖-b) := mul_pos hL (sub_pos.mpr hbw)
+  have hgap' : tailSlope a * (‖w‖-b) ≤ T ‖w‖ - |(F w).im| := by
+    dsimp [tailSlope] at *
+    linarith
+  apply (reciprocal_A_slice hw hwr (ha.trans (hab.trans hbw.le))).trans
+  apply add_le_add_right
+  calc
+    Real.pi^3 * Real.sqrt ((1-‖w‖)/(T ‖w‖ - |(F w).im|))
+        ≤ Real.pi^3 * Real.sqrt ((1-‖w‖)/(tailSlope a*(‖w‖-b))) := by
+      apply mul_le_mul_of_nonneg_left _ (by positivity)
+      apply Real.sqrt_le_sqrt
+      exact div_le_div_of_nonneg_left hnum hden hgap'
+    _ = Real.pi^3 / Real.sqrt (tailSlope a) * Real.sqrt ((1-‖w‖)/(‖w‖-b)) := by
+      rw [show (1-‖w‖)/(tailSlope a*(‖w‖-b)) = ((1-‖w‖)/(‖w‖-b))/(tailSlope a) by rw [div_div, mul_comm (‖w‖-b)],
+        Real.sqrt_div (div_nonneg hnum (sub_nonneg.mpr hbw.le))]
+      ring
+
+theorem integrable_radialError {b : ℝ} (hb : b ≤ 1) (k : ℕ) :
+    IntervalIntegrable (fun r : ℝ => r^(2*k-1)*Real.sqrt ((1-r)/(r-b))) volume b 1 := by
+  have h := (SymmetricPolar.intervalIntegrable_endpointKernel (sub_nonneg.mpr hb)).comp_sub_left 1
+  have he : (fun r : ℝ => SymmetricPolar.endpointKernel (1-b) (1-r)) =
+      fun r : ℝ => Real.sqrt ((1-r)/(r-b)) := by
+    funext r
+    dsimp [SymmetricPolar.endpointKernel]
+    congr 2
+    ring
+  rw [he] at h
+  have hh : IntervalIntegrable (fun r : ℝ => Real.sqrt ((1-r)/(r-b))) volume b 1 := by
+    convert h.symm using 1 <;> ring
+  exact hh.continuousOn_mul (by fun_prop)
+
+theorem integral_radialLeading {k : ℕ} (hk : 1 ≤ k) (b R : ℝ) :
+    (k : ℝ) * ∫ r in b..R, r^(2*k-1)*(Real.pi/2) =
+      Real.pi/4 * (R^(2*k)-b^(2*k)) := by
+  have hk0 : (k : ℝ) ≠ 0 := by exact_mod_cast (by omega : k ≠ 0)
+  have hd (r : ℝ) : HasDerivAt (fun r : ℝ => r^(2*k)/(2*k)) (r^(2*k-1)) r := by
+    convert ((hasDerivAt_id r).pow (2*k)).div_const ((2:ℝ)*k) using 1 <;>
+      first | rfl | (simp only [Nat.cast_mul, Nat.cast_ofNat, id_eq, mul_one]; field_simp [hk0])
+  have hi := intervalIntegral.integral_eq_sub_of_hasDerivAt
+    (fun r (_ : r ∈ uIcc b R) => hd r)
+    (show IntervalIntegrable (fun r : ℝ => r^(2*k-1)) volume b R from (by fun_prop : Continuous _).intervalIntegrable _ _)
+  rw [intervalIntegral.integral_mul_const, hi]
+  field_simp
+  ring
+
+theorem outer_integral_bound {k : ℕ} (hk : 1 ≤ k) {t v u a : ℝ}
+    (hz : (v : ℂ)+(t : ℂ)*I ∈ D) (hu : 0 ≤ u) (huv : u ≤ v)
+    (ha : 1/2 ≤ a) (hau : a ≤ rho t u) :
+    (k : ℝ)*∫ h in u..v, density k ((h : ℂ)+(t : ℂ)*I) ≤
+      Real.pi/4*((rho t v)^(2*k)-(rho t u)^(2*k)) +
+        Real.pi^3/Real.sqrt (tailSlope a)*(1+Real.pi/Real.exp 1) := by
+  have hv : 0 ≤ v := hu.trans huv
+  have huI : u ∈ Icc 0 v := ⟨hu, huv⟩
+  have huD := horizontal_segment_mem hz hv huI
+  have hb0 : 0 ≤ rho t u := norm_nonneg _
+  have hb1 : rho t u < 1 := norm_g_lt huD
+  have hR1 : rho t v < 1 := norm_g_lt hz
+  have hrho := horizontal_radius_strictMono hz hv
+  have hbR : rho t u ≤ rho t v := hrho.monotoneOn huI (right_mem_Icc.mpr hv) huv
+  have ha1 : a < 1 := hau.trans_lt hb1
+  have hL : 0 < tailSlope a := tailSlope_pos (by linarith) ha1
+  have hc : 0 ≤ Real.pi^3/Real.sqrt (tailSlope a) := by positivity
+  let P : ℝ → ℝ := fun r => r^(2*k-1)*(Real.pi/2) +
+    (Real.pi^3/Real.sqrt (tailSlope a))*(r^(2*k-1)*Real.sqrt ((1-r)/(r-rho t u)))
+  have hradE := integrable_radialError hb1.le k
+  have hradE' : IntervalIntegrable
+      (fun r : ℝ => r^(2*k-1)*Real.sqrt ((1-r)/(r-rho t u))) volume (rho t u) (rho t v) :=
+    hradE.mono_set (uIcc_subset_uIcc (left_mem_uIcc) (by
+      rw [uIcc_of_le hb1.le]; exact ⟨hbR, hR1.le⟩))
+  have hP : IntervalIntegrable P volume (rho t u) (rho t v) :=
+    ((by fun_prop : Continuous (fun r : ℝ => r^(2*k-1)*(Real.pi/2))).intervalIntegrable _ _).add
+      (hradE'.const_mul _)
+  have hcont : ContinuousOn (rho t) (uIcc u v) := by
+    rw [uIcc_of_le huv]
+    exact (continuousOn_rho hz hv).mono (fun _ hx => ⟨hu.trans hx.1, hx.2⟩)
+  have hder : ∀ h ∈ Ioo (min u v) (max u v), HasDerivAt (rho t) (rhoD t h) h := by
+    rw [min_eq_left huv, max_eq_right huv]
+    intro h hh
+    exact hasDerivAt_rho (horizontal_segment_mem hz hv ⟨hu.trans hh.1.le, hh.2.le⟩)
+      (hu.trans_lt hh.1)
+  have hpos : ∀ h ∈ Ioo (min u v) (max u v), 0 ≤ rhoD t h := by
+    rw [min_eq_left huv, max_eq_right huv]
+    intro h hh
+    exact (rhoD_pos (horizontal_segment_mem hz hv ⟨hu.trans hh.1.le, hh.2.le⟩)
+      (hu.trans_lt hh.1)).le
+  have hPc : IntervalIntegrable (fun h => P (rho t h)*rhoD t h) volume u v :=
+    (intervalIntegral.integrable_comp_mul_deriv_iff_of_deriv_nonneg hcont hder hpos).mpr hP
+  have hdens : IntervalIntegrable (fun h : ℝ => density k ((h : ℂ)+(t : ℂ)*I)) volume u v := by
+    apply ContinuousOn.intervalIntegrable
+    apply (continuousOn_density k).comp (by fun_prop)
+    intro h hh
+    rw [uIcc_of_le huv] at hh
+    exact horizontal_segment_mem hz hv ⟨hu.trans hh.1, hh.2⟩
+  have hcmp : ∫ h in u..v, density k ((h : ℂ)+(t : ℂ)*I) ≤ ∫ r in rho t u..rho t v, P r := by
+    rw [← intervalIntegral.integral_comp_mul_deriv_of_deriv_nonneg hcont hder hpos (g := P)]
+    apply intervalIntegral.integral_mono_ae_restrict huv hdens hPc
+    change ∀ᵐ (h : ℝ) ∂volume.restrict (Icc u v), density k ((h : ℂ)+(t : ℂ)*I) ≤ P (rho t h)*rhoD t h
+    rw [ae_restrict_iff' measurableSet_Icc]
+    filter_upwards [volume.ae_ne u] with h hne hh
+    have hhu : u < h := lt_of_le_of_ne hh.1 (Ne.symm hne)
+    have hh0 : 0 < h := hu.trans_lt hhu
+    have hhz := horizontal_segment_mem hz hv ⟨hh0.le, hh.2⟩
+    have hbr : rho t u < rho t h := hrho huI ⟨hh0.le, hh.2⟩ hhu
+    have hA := reciprocal_A_tail (norm_g_lt hhz) (re_g_pos hhz (by simpa using hh0))
+      ha hau hbr (by simpa only [F_g hhz, add_im, ofReal_im, mul_im, ofReal_re, I_im,
+        I_re, mul_one, mul_zero, add_zero, zero_add] using T_rho_ge_abs hz hv huI)
+    rw [density_radial hk hhz hh0]
+    have hmul := mul_le_mul_of_nonneg_right
+      (mul_le_mul_of_nonneg_left hA (pow_nonneg (norm_nonneg (g ((h : ℂ)+(t : ℂ)*I))) (2*k-1))) (rhoD_pos hhz hh0).le
+    convert hmul using 1 <;>
+      first | rfl | (dsimp only [P, rho]; ring)
+  have hlead := integral_radialLeading hk (rho t u) (rho t v)
+  have herr : (k : ℝ)*∫ r in rho t u..rho t v,
+      r^(2*k-1)*Real.sqrt ((1-r)/(r-rho t u)) ≤ 1+Real.pi/Real.exp 1 := by
+    apply le_trans _ (SymmetricPolar.movingEndpointIntegral_le hk hb0 hb1)
+    apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg k)
+    apply intervalIntegral.integral_mono_interval le_rfl hbR hR1.le _ hradE
+    change ∀ᵐ (r : ℝ) ∂volume.restrict (Ioc (rho t u) 1), 0 ≤ r^(2*k-1)*Real.sqrt ((1-r)/(r-rho t u))
+    rw [ae_restrict_iff' measurableSet_Ioc]
+    exact Filter.Eventually.of_forall (fun r hr => mul_nonneg
+      (pow_nonneg (hb0.trans hr.1.le) _) (Real.sqrt_nonneg _))
+  have hh := mul_le_mul_of_nonneg_left hcmp (Nat.cast_nonneg k)
+  dsimp [P] at hh
+  rw [intervalIntegral.integral_add
+    ((by fun_prop : Continuous (fun r : ℝ => r^(2*k-1)*(Real.pi/2))).intervalIntegrable _ _)
+    (hradE'.const_mul _), intervalIntegral.integral_const_mul, mul_add, hlead] at hh
+  apply hh.trans
+  apply add_le_add_right
+  calc
+    _ = (Real.pi^3/Real.sqrt (tailSlope a))*((k : ℝ)*∫ r in rho t u..rho t v,
+        r^(2*k-1)*Real.sqrt ((1-r)/(r-rho t u))) := by ring
+    _ ≤ _ := mul_le_mul_of_nonneg_left herr hc
+
+end
+end PlanarLens
+
+end LowerBoundInline
+
+end OAI
diff --git a/lean/OAI/Geometry/PolarProducts/UpperBound.lean b/lean/OAI/Geometry/PolarProducts/UpperBound.lean
new file mode 100644
index 0000000000000000000000000000000000000000..280f3145286b0f4449c88a9ce6fd4420bfe33075
--- /dev/null
+++ b/lean/OAI/Geometry/PolarProducts/UpperBound.lean
@@ -0,0 +1,121 @@
+import OAI.Geometry.PolarProducts.CanonicalCoordinates
+
+namespace OAI
+
+section
+namespace CylinderNonsqueezing
+noncomputable section
+open Set
+open scoped ContDiff
+open FourierPolynomial
+variable {κ : Type} [Fintype κ] [DecidableEq κ]
+
+theorem radius_le_of_map (j₀ : κ) {L R M : ℝ} (hL : 0 < L) (hR : 0 ≤ R) (hM : 0 ≤ M)
+    (f : Vector κ → Vector κ) (hf : ContDiffOn ℝ ∞ f (PushedBall.domain L))
+    (hi : InjOn f (PushedBall.domain L))
+    (hω : ∀ z ∈ PushedBall.domain L, ComplexSymplectic.Preserves (fderiv ℝ f z))
+    (hbounded : ∀ z ∈ PushedBall.domain L, ‖f z‖^2 ≤ M)
+    (hcylinder : ∀ z ∈ PushedBall.domain L, ‖f z j₀‖^2 ≤ R) : L ≤ R := by
+  let hU := PushedBall.isOpen_domain (κ := κ) L
+  have hD (z) (hz : z ∈ PushedBall.domain L) := ComplexSymplectic.invertible (hω z hz)
+  let e := EmbeddingInverse.localEquiv hU f hf hi hD
+  exact radius_le_of_embedding j₀ hL hR hM e hf rfl hω
+    (by rintro _ ⟨z,hz,rfl⟩; exact hbounded z hz)
+    (by rintro _ ⟨z,hz,rfl⟩; exact hcylinder z hz)
+
+end
+end CylinderNonsqueezing
+end
+
+section
+namespace SymmetricPolar
+noncomputable section
+open Set Filter
+open scoped ContDiff Topology ENNReal
+open ComplexCoordinates UpperCoordinates
+variable {n : ℕ}
+
+theorem capacity_le_four_of_embedding {K : Set (Position n)} (hK : IsSymmetricConvexBody K)
+    (hn : 0 < n) {c : ℝ} (hc : 0 < c)
+    (hemb : HasSymplecticEmbedding (capacityBall n c) (polarProduct K)) : c ≤ 4 := by
+  obtain ⟨u,v,hvu,hcyl⟩ := exists_squareCylinder hK hn
+  have hu : u ≠ 0 := by intro h; simp [h] at hvu
+  obtain ⟨j,huj⟩ : ∃ j : Fin n, u j ≠ 0 := by
+    by_contra h
+    apply hu
+    ext j
+    exact not_not.mp ((not_exists.mp h) j)
+  let A := CanonicalLinear.adapted j huj hvu
+  let T := targetCoordinates A
+  obtain ⟨e,he,hei,heK,heω⟩ := hemb
+  let f : C (Fin n) → C (Fin n) := fun z => PlanarCylinder.map j (T (e (parts z)))
+  have hball {z : C (Fin n)} (hz : z ∈ PushedBall.domain (c/Real.pi)) :
+      parts z ∈ capacityBall n c := by
+    change Real.pi*(‖re z‖^2+‖im z‖^2) < c
+    rw [← norm_sq_parts]
+    exact (lt_div_iff₀ Real.pi_pos).mp hz |> fun h => by simpa [mul_comm] using h
+  have hfirst (z : C (Fin n)) (hz : z ∈ PushedBall.domain (c/Real.pi)) :
+      T (e (parts z)) ∈ PlanarCylinder.domain j := by
+    have hh := hcyl (heK (hball hz))
+    have hi := targetCoordinates_first j huj hvu (e (parts z))
+    change |(T (e (parts z)) j).re| < 1 ∧ |(T (e (parts z)) j).im| < 1
+    rw [hi.1, hi.2]
+    exact hh
+  have hsm (z : C (Fin n)) (hz : z ∈ PushedBall.domain (c/Real.pi)) : ContDiffAt ℝ ∞ f z := by
+    have hez := he.contDiffAt ((isOpen_lt (by fun_prop) continuous_const).mem_nhds (hball hz))
+    have hP : ContDiffAt ℝ ∞ (parts : C (Fin n) → Phase n) z :=
+      (parts (ι := Fin n)).contDiff.contDiffAt
+    have hT : ContDiffAt ℝ ∞ (fun w => T (e (parts w))) z :=
+      T.contDiff.contDiffAt.comp z (hez.comp z hP)
+    have hout := (PlanarCylinder.contDiffAt_map j (abs_lt.mp (hfirst z hz).2).1).comp z hT
+    exact hout
+  have hfi : InjOn f (PushedBall.domain (c/Real.pi)) := by
+    intro z hz w hw hzw
+    apply parts.injective
+    apply HasSymplecticEmbedding.injOn hei (hball hz) (hball hw)
+    apply targetCoordinates_injective A
+    exact PlanarCylinder.map_injOn j (hfirst z hz) (hfirst w hw) hzw
+  have hfω (z : C (Fin n)) (hz : z ∈ PushedBall.domain (c/Real.pi)) :
+      ComplexSymplectic.Preserves (fderiv ℝ f z) := by
+    have hez := he.contDiffAt ((isOpen_lt (by fun_prop) continuous_const).mem_nhds (hball hz))
+    have hed := (hez.differentiableAt (by simp)).hasFDerivAt
+    have hd := (PlanarCylinder.hasFDerivAt_map j (abs_lt.mp (hfirst z hz).2).1).comp z
+      (T.hasFDerivAt.comp z (hed.comp z (parts (ι := Fin n)).hasFDerivAt))
+    have hfd : fderiv ℝ f z = (PlanarCylinder.derivative j (T (e (parts z)))).comp
+        (T.comp ((fderiv ℝ e (parts z)).comp parts.toContinuousLinearMap)) := hd.fderiv
+    intro a b
+    change inner (𝕜 := ℝ) (Complex.I • fderiv ℝ f z a) (fderiv ℝ f z b) =
+      inner (𝕜 := ℝ) (Complex.I • a) b
+    rw [← PlanarCylinder.form_eq_inner, ← PlanarCylinder.form_eq_inner, hfd]
+    simp only [ContinuousLinearMap.comp_apply, ContinuousLinearEquiv.coe_coe]
+    rw [PlanarCylinder.derivative_preserves j (abs_lt.mp (hfirst z hz).2).1,
+      targetCoordinates_preserves, form_parts]
+    exact heω _ (hball hz) (parts a) (parts b)
+  have hpK : IsCompact (polar K) := ConvexPolar.isCompact_polar
+    (ConvexPolar.zero_mem_interior_of_symmetric hK.2.1 hK.2.2.1 hK.2.2.2)
+  obtain ⟨B,hB,hTB⟩ := ((hK.1.prod hpK).image T.continuous).isBounded.exists_pos_norm_le
+  have hbnd (z : C (Fin n)) (hz : z ∈ PushedBall.domain (c/Real.pi)) :
+      ‖f z‖^2 ≤ B^2 + 4/Real.pi := by
+    have hh := heK (hball hz)
+    have hT := hTB _ ⟨e (parts z), ⟨interior_subset hh.1,interior_subset hh.2⟩, rfl⟩
+    have hsq := (sq_le_sq₀ (norm_nonneg _) hB.le).mpr hT
+    have hnorm := PlanarCylinder.norm_bound j (hfirst z hz)
+    change ‖PlanarCylinder.map j (T (e (parts z)))‖^2 ≤ _
+    linarith
+  have hle := CylinderNonsqueezing.radius_le_of_map j (div_pos hc Real.pi_pos)
+    (by positivity : (0:ℝ) ≤ 4/Real.pi) (by positivity : (0:ℝ) ≤ B^2+4/Real.pi)
+    f (fun z hz => (hsm z hz).contDiffWithinAt) hfi hfω hbnd
+    (fun z hz => (PlanarCylinder.first_bound j (hfirst z hz)).le)
+  exact (div_le_div_iff_of_pos_right Real.pi_pos).mp hle
+
+theorem gromovWidth_polarProduct_le_four {K : Set (Position n)}
+    (hK : IsSymmetricConvexBody K) (hn : 0 < n) : gromovWidth (polarProduct K) ≤ 4 := by
+  apply sSup_le
+  rintro _ ⟨c,⟨hc,he⟩,rfl⟩
+  exact (ENNReal.ofReal_le_ofReal (capacity_le_four_of_embedding hK hn hc he)).trans_eq (by norm_num)
+
+end
+end SymmetricPolar
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/ABPIntegralBound.lean b/lean/OAI/Geometry/PrescribedPotential/ABPIntegralBound.lean
new file mode 100644
index 0000000000000000000000000000000000000000..4a4279ccde3b9e3960fda9cb7800ad7fb470cc6e
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/ABPIntegralBound.lean
@@ -0,0 +1,88 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.GlobalSobolev
+import OAI.Geometry.PrescribedPotential.LocalPotentialABP
+
+namespace OAI
+
+/-! A B P Integral Bound. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Metric Filter Topology MeasureTheory Matrix
+open scoped ContDiff InnerProductSpace
+namespace PotentialABP
+variable {d : ℕ}
+local notation "E" => EuclideanSpace ℂ (Fin d)
+local instance abpIntegralRealIP (d : ℕ) : InnerProductSpace ℝ (EuclideanSpace ℂ (Fin d)) :=
+  InnerProductSpace.rclikeToReal ℂ (EuclideanSpace ℂ (Fin d))
+local notation "e" => EuclideanSpace.equiv (Fin d) ℂ
+
+theorem abp_integral_minimum_bound
+    (μ : Measure E) [μ.IsAddHaarMeasure]
+    {f : (Fin d → ℂ) → ℝ} (hf : ContDiff ℝ ∞ f)
+    (G : E → Matrix (Fin d) (Fin d) ℂ)
+    {c : E} {r δ K B : ℝ} (hr : 0 < r) (hδ : 0 < δ) (hK : 0 < K)
+    (hmin : ∀ x ∈ closedBall c r, f (e c) ≤ f (e x))
+    (hG : ∀ x ∈ closedBall c r, (G x).IsHermitian)
+    (hGδ : ∀ x ∈ closedBall c r, ∀ v : E,
+      δ*‖v‖^2 ≤ inner ℝ ((G x).toEuclideanLin v) v)
+    (hdet : ∀ x ∈ closedBall c r,
+      |(G x + PotentialKaehler.potentialMatrix f (e x)).det.re| ≤ K)
+    {U : Set E} (hU : MeasurableSet U) (hsub : closedBall c r ⊆ U)
+    (hn : ∀ x ∈ U, f (e x) ≤ 0)
+    (hi : IntegrableOn (fun x => -f (e x)) U μ)
+    (hB : (∫ x in U, -f (e x) ∂μ) ≤ B) :
+    -f (e c) ≤ δ*r^2/2 + B*((4:ℝ)^(2*d)*K^2) /
+      μ.real (closedBall (0:E) (δ*r/2)) := by
+  let S : Set E := {x | x ∈ closedBall c r ∧ f (e x) ≤ f (e c)+δ*r^2/2}
+  let C : ℝ := (4:ℝ)^(2*d)*K^2
+  let V : ℝ := μ.real (closedBall (0:E) (δ*r/2))
+  have hC : 0 < C := by positivity
+  have hV : 0 < V := ENNReal.toReal_pos
+    (measure_closedBall_pos μ _ (by positivity)).ne'
+    (isCompact_closedBall _ _).measure_lt_top.ne
+  have hSc : IsClosed S := isClosed_closedBall.inter
+    (isClosed_le (hf.continuous.comp (EuclideanSpace.equiv (Fin d) ℂ).continuous) continuous_const)
+  have hSsub : S ⊆ U := fun x hx => hsub hx.1
+  have hSi : IntegrableOn (fun x => -f (e x)) S μ := hi.mono_set hSsub
+  have hSfin : μ S ≠ ⊤ :=
+    ((measure_mono (show S ⊆ closedBall c r from fun _ hx => hx.1)).trans_lt
+      (isCompact_closedBall _ _).measure_lt_top).ne
+  have hABP := local_potential_sublevel_volume μ hf G hr hδ hK.le hmin hG hGδ hdet
+  have hABPr : V ≤ C * μ.real S := by
+    have hne : ENNReal.ofReal C * μ S ≠ ⊤ := ENNReal.mul_ne_top ENNReal.ofReal_ne_top hSfin
+    have hh := ENNReal.toReal_mono hne hABP
+    simpa only [V,C,S,measureReal_def,ENNReal.toReal_mul,ENNReal.toReal_ofReal hC.le] using hh
+  have hBn : 0 ≤ B := (setIntegral_nonneg hU (fun x hx => neg_nonneg.mpr (hn x hx))).trans hB
+  have hIS : (-f (e c)-δ*r^2/2)*μ.real S ≤ B := by
+    have hc : IntegrableOn (fun _ : E => -f (e c)-δ*r^2/2) S μ :=
+      integrableOn_const hSfin
+    have hm := setIntegral_mono_on hc hSi hSc.measurableSet (fun x (hx : x ∈ S) => by
+      have hh := hx.2
+      linarith)
+    have hm' := setIntegral_mono_set hi
+      (by
+        filter_upwards [ae_restrict_mem hU] with x hx
+        exact neg_nonneg.mpr (hn x hx))
+      (Filter.Eventually.of_forall hSsub)
+    have hh := hm.trans (hm'.trans hB)
+    simpa only [setIntegral_const, smul_eq_mul, mul_comm] using hh
+  by_cases hm : 0 ≤ -f (e c)-δ*r^2/2
+  · have hmul := mul_le_mul_of_nonneg_left hABPr hm
+    have hmul' := mul_le_mul_of_nonneg_left hIS hC.le
+    have hh : (-f (e c)-δ*r^2/2)*V ≤ B*C := by nlinarith
+    have hd := (le_div_iff₀ hV).mpr hh
+    change -f (e c) ≤ δ*r^2/2 + B*C/V
+    linarith
+  · have hdiv : 0 ≤ B*C/V := div_nonneg (mul_nonneg hBn hC.le) hV.le
+    change -f (e c) ≤ δ*r^2/2 + B*C/V
+    linarith
+end PotentialABP
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/AffineFamilyJets.lean b/lean/OAI/Geometry/PrescribedPotential/AffineFamilyJets.lean
new file mode 100644
index 0000000000000000000000000000000000000000..10f670d04e81f1f631794de2c1a9d684266247c3
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/AffineFamilyJets.lean
@@ -0,0 +1,45 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.QuasilinearJetBasics
+
+namespace OAI
+
+/-! Affine Family Jets. -/
+
+section
+
+noncomputable section
+open Set Filter Topology Finset
+open scoped ContDiff
+namespace HigherJet
+variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  [NormedAddCommGroup F] [NormedSpace ℝ F]
+lemma affine_family_jets {U : Set E} (hU : IsOpen U) {f g : E → F}
+    (hf : ContDiffOn ℝ ∞ f U) (hg : ContDiffOn ℝ ∞ g U)
+    (m : ℕ) {K : Set E} (hK : IsCompact K) (hKU : K ⊆ U) :
+    ∃ C : ℝ, ∀ j, j ≤ m → ∀ t ∈ Icc (0:ℝ) 1, ∀ x ∈ K,
+      ‖iteratedFDeriv ℝ j (fun y => (1-t) • f y-t • g y) x‖ ≤ C := by
+  obtain ⟨A,hA,hfa⟩ := compact_coefficient_jets hK (fun x hx => hf.contDiffAt (hU.mem_nhds (hKU hx))) m
+  obtain ⟨B,hB,hgb⟩ := compact_coefficient_jets hK (fun x hx => hg.contDiffAt (hU.mem_nhds (hKU hx))) m
+  refine ⟨A+B,fun j hj t ht x hx => ?_⟩
+  have hf' := hf.contDiffAt (hU.mem_nhds (hKU hx))
+  have hg' := hg.contDiffAt (hU.mem_nhds (hKU hx))
+  have hn : (j:ℕ∞ω) ≤ ∞ := WithTop.coe_le_coe.mpr le_top
+  have he : iteratedFDeriv ℝ j ((1-t) • f-t • g) x =
+      (1-t) • iteratedFDeriv ℝ j f x-t • iteratedFDeriv ℝ j g x := by
+    calc
+      _ = iteratedFDeriv ℝ j ((1-t) • f) x-iteratedFDeriv ℝ j (t • g) x :=
+        iteratedFDeriv_sub_apply ((hf'.const_smul (1-t)).of_le hn) ((hg'.const_smul t).of_le hn)
+      _ = _ := by rw [iteratedFDeriv_const_smul_apply (hf'.of_le hn),iteratedFDeriv_const_smul_apply (hg'.of_le hn)]
+  change ‖iteratedFDeriv ℝ j ((1-t) • f-t • g) x‖ ≤ _
+  rw [he]
+  apply (norm_sub_le _ _).trans
+  simp only [norm_smul,Real.norm_eq_abs,abs_of_nonneg ht.1,abs_of_nonneg (sub_nonneg.mpr ht.2)]
+  have ha := mul_le_mul_of_nonneg_left (hfa j hj x hx) (sub_nonneg.mpr ht.2)
+  have hb := mul_le_mul_of_nonneg_left (hgb j hj x hx) ht.1
+  nlinarith [ht.1,ht.2]
+end HigherJet
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/AnalyticSupport.lean b/lean/OAI/Geometry/PrescribedPotential/AnalyticSupport.lean
new file mode 100644
index 0000000000000000000000000000000000000000..03d609943d0516e7070bea438ecf28de40950416
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/AnalyticSupport.lean
@@ -0,0 +1,2523 @@
+import Mathlib
+
+namespace OAI
+
+/-! Analytic Support. -/
+
+section
+
+ 
+
+noncomputable section
+open Set MeasureTheory Filter Complex
+open scoped Topology
+
+namespace ProjectiveKaehler
+
+ 
+
+def compactIntegral {K : Type*} [TopologicalSpace K] [CompactSpace K]
+    [MeasurableSpace K] [OpensMeasurableSpace K] (μ : Measure K) [IsFiniteMeasure μ] :
+    C(K, ℂ) →L[ℂ] ℂ :=
+  LinearMap.mkContinuous
+    { toFun := fun f => ∫ x, f x ∂μ
+      map_add' := by
+        intro f g
+        exact integral_add
+          (f.continuous.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace f))
+          (g.continuous.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace g))
+      map_smul' := by
+        intro c f
+        exact integral_smul c f }
+    (μ.real univ) (by
+      intro f
+      change ‖∫ x, f x ∂μ‖ ≤ μ.real univ * ‖f‖
+      rw [mul_comm]
+      exact norm_integral_le_of_norm_le_const
+        (μ := μ) (ae_of_all _ fun x => f.norm_coe_le_norm x))
+
+abbrev Angle := Icc (0 : ℝ) (2 * Real.pi)
+
+def angleMeasure : Measure Angle := Measure.comap Subtype.val volume
+
+instance : IsFiniteMeasure angleMeasure := by
+  refine ⟨?_⟩
+  rw [angleMeasure, Measure.comap_apply Subtype.val Subtype.val_injective
+    (fun s hs => (MeasurableEmbedding.subtype_coe measurableSet_Icc).measurableSet_image.mpr hs)
+    volume MeasurableSet.univ]
+  simpa only [Set.image_univ, Subtype.range_coe_subtype, ofPred_mem_eq] using
+    (measure_Icc_lt_top (μ := volume) (a := (0 : ℝ)) (b := 2 * Real.pi))
+
+abbrev Boundary (n : ℕ) := Fin n → Angle
+
+def boundaryCons (n : ℕ) : C(Angle × Boundary n, Boundary (n + 1)) where
+  toFun p := Fin.cons p.1 p.2
+  continuous_toFun := by
+    apply continuous_pi
+    intro i
+    refine Fin.cases ?_ (fun j => ?_) i
+    · simpa using continuous_fst
+    · exact (continuous_apply j).comp continuous_snd
+
+ 
+
+def integrateFirst {K : Type*} [TopologicalSpace K] [CompactSpace K]
+    (L : C(K, ℂ) →L[ℂ] ℂ) : C(Angle × K, ℂ) →L[ℂ] ℂ :=
+  LinearMap.mkContinuous
+    { toFun := fun f => ∫ t, L (f.curry t) ∂angleMeasure
+      map_add' := by
+        intro f g
+        change (∫ t, L ((f + g).curry t) ∂angleMeasure) = _
+        have h : ∀ t, (f + g).curry t = f.curry t + g.curry t := fun _ => by
+          ext; rfl
+        simp_rw [h, map_add]
+        apply integral_add
+        · exact (L.continuous.comp f.curry.continuous).integrable_of_hasCompactSupport
+            (HasCompactSupport.of_compactSpace _)
+        · exact (L.continuous.comp g.curry.continuous).integrable_of_hasCompactSupport
+            (HasCompactSupport.of_compactSpace _)
+      map_smul' := by
+        intro c f
+        have h : ∀ t, (c • f).curry t = c • f.curry t := fun _ => by
+          ext; rfl
+        simp_rw [h, map_smul]
+        exact integral_smul c _ }
+    (‖L‖ * angleMeasure.real univ) (by
+      intro f
+      calc
+        ‖∫ t, L (f.curry t) ∂angleMeasure‖ ≤
+            (‖L‖ * ‖f‖) * angleMeasure.real univ :=
+          norm_integral_le_of_norm_le_const (ae_of_all _ fun t =>
+            (L.le_opNorm _).trans (mul_le_mul_of_nonneg_left
+              (((f.curry t).norm_le (norm_nonneg f)).mpr
+                (fun y => f.norm_coe_le_norm (t, y))) (norm_nonneg L)))
+        _ = (‖L‖ * angleMeasure.real univ) * ‖f‖ := by ring)
+
+ 
+def boundaryIntegral : (n : ℕ) → C(Boundary n, ℂ) →L[ℂ] ℂ
+  | 0 => ContinuousMap.evalCLM ℂ (fun i => Fin.elim0 i)
+  | n + 1 => (integrateFirst (boundaryIntegral n)).comp
+      (ContinuousMap.compCLM ℂ ℂ (boundaryCons n))
+
+@[simp] theorem boundaryIntegral_zero (f : C(Boundary 0, ℂ)) :
+    boundaryIntegral 0 f = f (fun i => Fin.elim0 i) := rfl
+
+@[simp] theorem boundaryIntegral_succ (n : ℕ) (f : C(Boundary (n + 1), ℂ)) :
+    boundaryIntegral (n + 1) f =
+      ∫ t, boundaryIntegral n ((f.comp (boundaryCons n)).curry t) ∂angleMeasure := rfl
+
+ 
+def boundaryPoint {n : ℕ} (c : Fin n → ℂ) (R : ℝ) : C(Boundary n, Fin n → ℂ) where
+  toFun t i := circleMap (c i) R (t i).val
+  continuous_toFun := by
+    apply continuous_pi
+    intro i
+    exact (continuous_circleMap _ _).comp (continuous_subtype_val.comp (continuous_apply i))
+
+ 
+
+def denominator {n : ℕ} (c : Fin n → ℂ) (R : ℝ) (i : Fin n)
+    (z : Fin n → ℂ) : C(Boundary n, ℂ) :=
+  ⟨fun t => circleMap (c i) R (t i).val - z i, by
+    exact ((continuous_circleMap _ _).comp
+      (continuous_subtype_val.comp (continuous_apply i))).sub continuous_const⟩
+
+lemma analyticAt_denominator {n : ℕ} (c : Fin n → ℂ) (R : ℝ) (i : Fin n)
+    (z : Fin n → ℂ) : AnalyticAt ℂ (denominator c R i) z := by
+  have h : denominator c R i = fun w => denominator c R i 0 -
+      (ContinuousLinearMap.const ℂ (Boundary n)) (w i) := by
+    funext w
+    ext t
+    simp [denominator]
+  rw [h]
+  have hi : AnalyticAt ℂ (fun w : Fin n → ℂ => w i) z :=
+    (ContinuousLinearMap.proj i : (Fin n → ℂ) →L[ℂ] ℂ).analyticAt z
+  exact analyticAt_const.sub
+    (AnalyticAt.comp (f := fun w : Fin n → ℂ => w i)
+      ((ContinuousLinearMap.const ℂ (Boundary n)).analyticAt (z i)) hi)
+
+lemma isUnit_denominator {n : ℕ} {c z : Fin n → ℂ} {R : ℝ} (hR : 0 < R)
+    (hz : z ∈ Metric.ball c R) (i : Fin n) : IsUnit (denominator c R i z) := by
+  apply (ContinuousMap.isUnit_iff_forall_ne_zero _).mpr
+  intro t he
+  rw [ball_pi c hR] at hz
+  have hi : z i ∈ Metric.ball (c i) R := hz i (Set.mem_univ i)
+  have hs := circleMap_mem_sphere (c i) hR.le (t i).val
+  have he' : circleMap (c i) R (t i).val = z i := sub_eq_zero.mp he
+  rw [he'] at hs
+  exact (ne_of_lt hi) hs
+
+lemma analyticAt_inverseDenominator {n : ℕ} {c z : Fin n → ℂ} {R : ℝ}
+    (hR : 0 < R) (hz : z ∈ Metric.ball c R) (i : Fin n) :
+    AnalyticAt ℂ (fun w => Ring.inverse (denominator c R i w)) z := by
+  have h : AnalyticAt ℂ Ring.inverse (denominator c R i z) := by
+    simpa only [IsUnit.unit_spec] using
+      (analyticAt_inverse (𝕜 := ℂ) (isUnit_denominator hR hz i).unit)
+  exact h.comp (analyticAt_denominator c R i z)
+
+lemma inverseDenominator_apply {n : ℕ} {c z : Fin n → ℂ} {R : ℝ}
+    (hR : 0 < R) (hz : z ∈ Metric.ball c R) (i : Fin n) (t : Boundary n) :
+    Ring.inverse (denominator c R i z) t = (circleMap (c i) R (t i).val - z i)⁻¹ := by
+  have h := congrArg (fun F : C(Boundary n, ℂ) => F t)
+    (Ring.mul_inverse_cancel (denominator c R i z) (isUnit_denominator hR hz i))
+  change (circleMap (c i) R (t i).val - z i) * _ = 1 at h
+  exact eq_inv_of_mul_eq_one_right h
+
+lemma boundaryPoint_mem_closedBall {n : ℕ} (c : Fin n → ℂ) {R : ℝ} (hR : 0 ≤ R)
+    (t : Boundary n) : boundaryPoint c R t ∈ Metric.closedBall c R := by
+  rw [closedBall_pi c hR]
+  intro i _
+  exact circleMap_mem_closedBall (c i) hR (t i).val
+
+def boundaryValue {n : ℕ} {c : Fin n → ℂ} {R : ℝ} (hR : 0 ≤ R)
+    {f : (Fin n → ℂ) → ℂ} (hf : ContinuousOn f (Metric.closedBall c R)) :
+    C(Boundary n, ℂ) :=
+  ⟨fun t => f (boundaryPoint c R t),
+    hf.comp_continuous (boundaryPoint c R).continuous (boundaryPoint_mem_closedBall c hR)⟩
+
+def boundarySpeed {n : ℕ} (R : ℝ) : C(Boundary n, ℂ) :=
+  ∏ i : Fin n, ⟨fun t => circleMap 0 R (t i).val * I, by
+    exact ((continuous_circleMap 0 R).comp
+      (continuous_subtype_val.comp (continuous_apply i))).mul continuous_const⟩
+
+def cauchyTransform {n : ℕ} (c : Fin n → ℂ) (R : ℝ) (F : C(Boundary n, ℂ))
+    (z : Fin n → ℂ) : ℂ :=
+  boundaryIntegral n (boundarySpeed R * (∏ i, Ring.inverse (denominator c R i z)) * F)
+
+lemma analyticAt_cauchyTransform {n : ℕ} {c z : Fin n → ℂ} {R : ℝ}
+    (hR : 0 < R) (hz : z ∈ Metric.ball c R) (F : C(Boundary n, ℂ)) :
+    AnalyticAt ℂ (cauchyTransform c R F) z := by
+  have hp : AnalyticAt ℂ (fun w => ∏ i, Ring.inverse (denominator c R i w)) z :=
+    Finset.analyticAt_fun_prod Finset.univ (fun i _ => analyticAt_inverseDenominator hR hz i)
+  have h : AnalyticAt ℂ (fun w => boundarySpeed R *
+      (∏ i, Ring.inverse (denominator c R i w)) * F) z :=
+    (analyticAt_const.mul hp).mul analyticAt_const
+  exact AnalyticAt.comp (f := fun w => boundarySpeed R *
+      (∏ i, Ring.inverse (denominator c R i w)) * F)
+    ((boundaryIntegral n).analyticAt _) h
+
+abbrev circleKernel (c : ℂ) (R : ℝ) (z : ℂ) (t : ℝ) : ℂ :=
+  (circleMap 0 R t * I) * (circleMap c R t - z)⁻¹
+
+lemma cauchyIntegrand_apply {n : ℕ} {c z : Fin n → ℂ} {R : ℝ}
+    (hR : 0 < R) (hz : z ∈ Metric.ball c R) (F : C(Boundary n, ℂ)) (t : Boundary n) :
+    (boundarySpeed (n := n) R * (∏ i, Ring.inverse (denominator c R i z)) * F) t =
+      (∏ i, circleKernel (c i) R (z i) (t i).val) * F t := by
+  simp only [ContinuousMap.mul_apply, boundarySpeed, ContinuousMap.prod_apply,
+    ContinuousMap.coe_mk, inverseDenominator_apply hR hz]
+  rw [← Finset.prod_mul_distrib]
+
+lemma circleCauchy {c z : ℂ} {R : ℝ} {f : ℂ → ℂ}
+    (hf : DifferentiableOn ℂ f (Metric.closedBall c R)) (hz : z ∈ Metric.ball c R) :
+    (∫ t : Angle, circleKernel c R z t.val * f (circleMap c R t.val) ∂angleMeasure) =
+      (2 * Real.pi * I) * f z := by
+  have h := hf.circleIntegral_sub_inv_smul hz
+  rw [circleIntegral_def_Icc] at h
+  rw [angleMeasure, integral_subtype_comap (s := Angle) measurableSet_Icc
+    (fun t => circleKernel c R z t * f (circleMap c R t))]
+  simpa only [deriv_circleMap, smul_eq_mul, mul_assoc] using h
+
+lemma differentiable_cons_left {n : ℕ} (y : Fin n → ℂ) :
+    Differentiable ℂ (fun a : ℂ => (Fin.cons a y : Fin (n + 1) → ℂ)) := by
+  apply differentiable_pi.mpr
+  intro i
+  refine Fin.cases ?_ (fun j => ?_) i
+  · simp
+  · simp
+
+lemma differentiable_cons_right {n : ℕ} (a : ℂ) :
+    Differentiable ℂ (fun y : Fin n → ℂ => (Fin.cons a y : Fin (n + 1) → ℂ)) := by
+  apply differentiable_pi.mpr
+  intro i
+  refine Fin.cases ?_ (fun j => ?_) i
+  · simp
+  · exact (ContinuousLinearMap.proj j : (Fin n → ℂ) →L[ℂ] ℂ).differentiable
+
+lemma cons_mem_closedBall {n : ℕ} {c : Fin (n + 1) → ℂ} {R : ℝ}
+    (hR : 0 ≤ R) {a : ℂ} {y : Fin n → ℂ}
+    (ha : a ∈ Metric.closedBall (c 0) R)
+    (hy : y ∈ Metric.closedBall (Fin.tail c) R) :
+    Fin.cons a y ∈ Metric.closedBall c R := by
+  rw [closedBall_pi c hR]
+  rw [closedBall_pi (Fin.tail c) hR] at hy
+  intro i _
+  refine Fin.cases ?_ (fun j => ?_) i
+  · exact ha
+  · exact hy j (Set.mem_univ j)
+
+lemma tail_mem_ball {n : ℕ} {c z : Fin (n + 1) → ℂ} {R : ℝ}
+    (hR : 0 < R) (hz : z ∈ Metric.ball c R) :
+    Fin.tail z ∈ Metric.ball (Fin.tail c) R := by
+  rw [ball_pi c hR] at hz
+  rw [ball_pi (Fin.tail c) hR]
+  exact fun i _ => hz i.succ (Set.mem_univ _)
+
+lemma boundaryPoint_cons {n : ℕ} (c : Fin (n + 1) → ℂ) (R : ℝ)
+    (a : Angle) (t : Boundary n) :
+    boundaryPoint c R (Fin.cons a t) =
+      Fin.cons (circleMap (c 0) R a.val) (boundaryPoint (Fin.tail c) R t) := by
+  ext i
+  refine Fin.cases ?_ (fun j => ?_) i <;> rfl
+
+ 
+
+lemma boundaryIntegral_cauchy (n : ℕ) {c z : Fin n → ℂ} {R : ℝ}
+    {f : (Fin n → ℂ) → ℂ} (hR : 0 < R)
+    (hf : DifferentiableOn ℂ f (Metric.closedBall c R)) (hz : z ∈ Metric.ball c R)
+    (F : C(Boundary n, ℂ))
+    (hF : ∀ t, F t = (∏ i, circleKernel (c i) R (z i) (t i).val) *
+      f (boundaryPoint c R t)) :
+    boundaryIntegral n F = (2 * Real.pi * I) ^ n * f z := by
+  induction n with
+  | zero =>
+    rw [boundaryIntegral_zero, hF]
+    simp only [Finset.univ_eq_empty, Finset.prod_empty, one_mul, pow_zero]
+    congr 1
+    exact Subsingleton.elim _ _
+  | succ n ih =>
+    have hz0 : z 0 ∈ Metric.ball (c 0) R := by
+      rw [ball_pi c hR] at hz
+      exact hz 0 (Set.mem_univ _)
+    have hzt := tail_mem_ball hR hz
+    rw [boundaryIntegral_succ]
+    have inner : ∀ a : Angle,
+        boundaryIntegral n ((F.comp (boundaryCons n)).curry a) =
+          (2 * Real.pi * I) ^ n *
+            (circleKernel (c 0) R (z 0) a.val *
+              f (Fin.cons (circleMap (c 0) R a.val) (Fin.tail z))) := by
+      intro a
+      apply ih (c := Fin.tail c) (z := Fin.tail z)
+        (f := fun y => circleKernel (c 0) R (z 0) a.val *
+          f (Fin.cons (circleMap (c 0) R a.val) y))
+      · exact (hf.comp (differentiable_cons_right _).differentiableOn
+          (fun y hy => cons_mem_closedBall hR.le
+            (circleMap_mem_closedBall (c 0) hR.le a.val) hy)).const_mul _
+      · exact hzt
+      · intro t
+        change F (Fin.cons a t) = _
+        rw [hF, Fin.prod_univ_succ, boundaryPoint_cons]
+        simp only [Fin.cons_zero, Fin.cons_succ, Fin.tail]
+        ring
+    simp_rw [inner]
+    rw [integral_const_mul]
+    have hslice : DifferentiableOn ℂ (fun a => f (Fin.cons a (Fin.tail z)))
+        (Metric.closedBall (c 0) R) :=
+      hf.comp (differentiable_cons_left _).differentiableOn
+        (fun a ha => cons_mem_closedBall hR.le ha (Metric.ball_subset_closedBall hzt))
+    rw [circleCauchy hslice hz0, Fin.cons_self_tail, pow_succ]
+    ring
+
+lemma cauchyTransform_eq {n : ℕ} {c z : Fin n → ℂ} {R : ℝ}
+    {f : (Fin n → ℂ) → ℂ} (hR : 0 < R)
+    (hf : DifferentiableOn ℂ f (Metric.closedBall c R)) (hz : z ∈ Metric.ball c R) :
+    cauchyTransform c R (boundaryValue hR.le hf.continuousOn) z =
+      (2 * Real.pi * I) ^ n * f z := by
+  apply boundaryIntegral_cauchy n hR hf hz
+  intro t
+  exact cauchyIntegrand_apply hR hz _ t
+
+lemma analyticAt_of_differentiableOn_closedBall {n : ℕ} {c z : Fin n → ℂ} {R : ℝ}
+    {f : (Fin n → ℂ) → ℂ} (hR : 0 < R)
+    (hf : DifferentiableOn ℂ f (Metric.closedBall c R)) (hz : z ∈ Metric.ball c R) :
+    AnalyticAt ℂ f z := by
+  have h : AnalyticAt ℂ (fun w => ((2 * Real.pi * I) ^ n)⁻¹ *
+      cauchyTransform c R (boundaryValue hR.le hf.continuousOn) w) z :=
+    analyticAt_const.mul (analyticAt_cauchyTransform hR hz _)
+  apply h.congr
+  filter_upwards [Metric.isOpen_ball.mem_nhds hz] with w hw
+  rw [cauchyTransform_eq hR hf hw]
+  rw [← mul_assoc, inv_mul_cancel₀ (pow_ne_zero n Complex.two_pi_I_ne_zero), one_mul]
+
+ 
+
+theorem analyticOnNhd_of_differentiableOn {n : ℕ} {s : Set (Fin n → ℂ)}
+    {f : (Fin n → ℂ) → ℂ} (hs : IsOpen s) (hf : DifferentiableOn ℂ f s) :
+    AnalyticOnNhd ℂ f s := by
+  intro z hz
+  obtain ⟨R, hR, hsub⟩ := Metric.nhds_basis_closedBall.mem_iff.mp (hs.mem_nhds hz)
+  exact analyticAt_of_differentiableOn_closedBall hR (hf.mono hsub)
+    (Metric.mem_ball_self hR)
+
+ 
+theorem analyticOnNhd_pi_of_differentiableOn {n m : ℕ} {s : Set (Fin n → ℂ)}
+    {f : (Fin n → ℂ) → (Fin m → ℂ)} (hs : IsOpen s) (hf : DifferentiableOn ℂ f s) :
+    AnalyticOnNhd ℂ f s := by
+  intro z hz
+  apply analyticAt_pi_iff.mpr
+  intro i
+  have hi : DifferentiableOn ℂ (fun w => f w i) s :=
+    (ContinuousLinearMap.proj i : (Fin m → ℂ) →L[ℂ] ℂ).differentiable.comp_differentiableOn hf
+  exact analyticOnNhd_of_differentiableOn hs hi z hz
+
+theorem contDiffOn_real_pi_of_differentiableOn {n m : ℕ} {s : Set (Fin n → ℂ)}
+    {f : (Fin n → ℂ) → (Fin m → ℂ)} (hs : IsOpen s) (hf : DifferentiableOn ℂ f s) :
+    ContDiffOn ℝ ⊤ f s := by
+  intro z hz
+  exact ((analyticOnNhd_pi_of_differentiableOn hs hf z hz).contDiffAt.restrict_scalars ℝ).contDiffWithinAt
+
+end ProjectiveKaehler
+
+ 
+
+noncomputable section
+open Matrix
+open scoped ComplexOrder Matrix.Norms.Elementwise
+
+namespace ProjectiveKaehler
+
+variable {m n : ℕ}
+
+ 
+def mass (x : Fin m → ℂ) : ℂ := star x ⬝ᵥ x
+
+lemma mass_ne_zero {x : Fin m → ℂ} (hx : x ≠ 0) : mass x ≠ 0 := by
+  simpa only [mass, ne_eq, dotProduct_star_self_eq_zero] using hx
+
+lemma mass_pos {x : Fin m → ℂ} (hx : x ≠ 0) : 0 < mass x :=
+  lt_of_le_of_ne (dotProduct_star_self_nonneg x) (Ne.symm (mass_ne_zero hx))
+
+@[simp] lemma star_mass (x : Fin m → ℂ) : star (mass x) = mass x := by
+  exact (Matrix.star_dotProduct x x).symm
+
+ 
+def horizontal (x : Fin m → ℂ) : Matrix (Fin m) (Fin m) ℂ :=
+  1 - (mass x)⁻¹ • vecMulVec x (star x)
+
+lemma horizontal_mulVec (x u : Fin m → ℂ) :
+    horizontal x *ᵥ u = u - ((mass x)⁻¹ * (star x ⬝ᵥ u)) • x := by
+  simp [horizontal, sub_mulVec, smul_mulVec, vecMulVec_mulVec, smul_smul]
+
+lemma horizontal_mulVec_eq_zero_iff (x u : Fin m → ℂ) :
+    horizontal x *ᵥ u = 0 ↔ u = ((mass x)⁻¹ * (star x ⬝ᵥ u)) • x := by
+  rw [horizontal_mulVec, sub_eq_zero]
+
+lemma horizontal_self {x : Fin m → ℂ} (hx : x ≠ 0) : horizontal x *ᵥ x = 0 := by
+  rw [horizontal_mulVec]
+  change x - ((mass x)⁻¹ * mass x) • x = 0
+  rw [inv_mul_cancel₀ (mass_ne_zero hx), one_smul, sub_self]
+
+lemma horizontal_eq_conjTranspose (x : Fin m → ℂ) : (horizontal x)ᴴ = horizontal x := by
+  simp [horizontal, conjTranspose_sub, conjTranspose_smul, conjTranspose_vecMulVec]
+
+lemma horizontal_idempotent {x : Fin m → ℂ} (hx : x ≠ 0) :
+    horizontal x * horizontal x = horizontal x := by
+  have hu : ∀ u, horizontal x *ᵥ (horizontal x *ᵥ u) = horizontal x *ᵥ u := by
+    intro u
+    have h := congrArg (horizontal x).mulVec (horizontal_mulVec x u)
+    simpa only [mulVec_sub, mulVec_smul, horizontal_self hx, smul_zero, sub_zero] using h
+  apply Matrix.ext_of_mulVec_single
+  intro j
+  simpa only [mulVec_mulVec] using hu (Pi.single j 1)
+
+ 
+
+def fsMatrix (x : Fin m → ℂ) (D : Matrix (Fin m) (Fin n) ℂ) :
+    Matrix (Fin n) (Fin n) ℂ :=
+  (mass x)⁻¹ • ((horizontal x * D)ᴴ * (horizontal x * D))
+
+lemma fsMatrix_posDef {x : Fin m → ℂ} {D : Matrix (Fin m) (Fin n) ℂ}
+    (hx : x ≠ 0) (hD : Function.Injective (horizontal x * D).mulVec) :
+    (fsMatrix x D).PosDef := by
+  exact (Matrix.PosDef.conjTranspose_mul_self _ hD).smul (inv_pos.mpr (mass_pos hx))
+
+lemma fsMatrix_eq {x : Fin m → ℂ} (hx : x ≠ 0) (D : Matrix (Fin m) (Fin n) ℂ) :
+    fsMatrix x D = (mass x)⁻¹ • (Dᴴ * horizontal x * D) := by
+  unfold fsMatrix
+  congr 1
+  rw [conjTranspose_mul, horizontal_eq_conjTranspose, Matrix.mul_assoc,
+    ← Matrix.mul_assoc (horizontal x) (horizontal x) D, horizontal_idempotent hx,
+    ← Matrix.mul_assoc]
+
+@[simp] lemma mass_smul (a : ℂ) (x : Fin m → ℂ) :
+    mass (a • x) = (star a * a) * mass x := by
+  simp only [mass, star_smul, smul_dotProduct, dotProduct_smul, smul_eq_mul]
+  ring
+
+lemma horizontal_smul {a : ℂ} (ha : a ≠ 0) {x : Fin m → ℂ} (_hx : x ≠ 0) :
+    horizontal (a • x) = horizontal x := by
+  unfold horizontal
+  rw [mass_smul, star_smul, smul_vecMulVec, vecMulVec_smul, smul_smul, smul_smul]
+  congr 2
+  have hs : star a ≠ 0 := star_ne_zero.mpr ha
+  field_simp
+
+lemma horizontal_gauge {x : Fin m → ℂ} (hx : x ≠ 0) (a : ℂ)
+    (D : Matrix (Fin m) (Fin n) ℂ) (r : Fin n → ℂ) :
+    horizontal x * (a • D + vecMulVec x r) = a • (horizontal x * D) := by
+  rw [Matrix.mul_add, Matrix.mul_smul, mul_vecMulVec, horizontal_self hx]
+  simp
+
+ 
+
+lemma fsMatrix_gauge {a : ℂ} (ha : a ≠ 0) {x : Fin m → ℂ} (hx : x ≠ 0)
+    (D : Matrix (Fin m) (Fin n) ℂ) (r : Fin n → ℂ) :
+    fsMatrix (a • x) (a • D + vecMulVec x r) = fsMatrix x D := by
+  simp only [fsMatrix, mass_smul, horizontal_smul ha hx, horizontal_gauge hx,
+    conjTranspose_smul, Matrix.smul_mul, Matrix.mul_smul, smul_smul]
+  congr 1
+  have hs : star a ≠ 0 := star_ne_zero.mpr ha
+  field_simp
+
+lemma fsMatrix_comp {k : ℕ} (x : Fin m → ℂ) (D : Matrix (Fin m) (Fin n) ℂ)
+    (L : Matrix (Fin n) (Fin k) ℂ) :
+    fsMatrix x (D * L) = Lᴴ * fsMatrix x D * L := by
+  simp only [fsMatrix, ← Matrix.mul_assoc (horizontal x) D L, conjTranspose_mul,
+    Matrix.mul_smul, Matrix.smul_mul]
+  congr 1
+  simp only [Matrix.mul_assoc]
+
+lemma fsMatrix_pair {x : Fin m → ℂ} (hx : x ≠ 0) (D : Matrix (Fin m) (Fin n) ℂ)
+    (u v : Fin n → ℂ) :
+    star u ⬝ᵥ (fsMatrix x D *ᵥ v) =
+      (mass x)⁻¹ * (star (D *ᵥ u) ⬝ᵥ (D *ᵥ v)) - (mass x)⁻¹ ^ 2 *
+        ((star (D *ᵥ u) ⬝ᵥ x) * (star x ⬝ᵥ (D *ᵥ v))) := by
+  rw [fsMatrix_eq hx D, smul_mulVec, dotProduct_smul, smul_eq_mul]
+  rw [← mulVec_mulVec, ← mulVec_mulVec, dotProduct_mulVec, ← star_mulVec]
+  rw [horizontal_mulVec, dotProduct_sub, dotProduct_smul, smul_eq_mul]
+  ring
+
+ 
+def jacobianMatrix (f : (Fin n → ℂ) → (Fin m → ℂ)) (z : Fin n → ℂ) :
+    Matrix (Fin m) (Fin n) ℂ := LinearMap.toMatrix' (fderiv ℂ f z).toLinearMap
+
+@[simp] lemma jacobianMatrix_mulVec (f : (Fin n → ℂ) → (Fin m → ℂ))
+    (z v : Fin n → ℂ) : jacobianMatrix f z *ᵥ v = fderiv ℂ f z v :=
+  LinearMap.toMatrix'_mulVec _ _
+
+lemma normalized_derivative {f : (Fin n → ℂ) → (Fin m → ℂ)} {z : Fin n → ℂ}
+    (hf : DifferentiableAt ℂ f z) {a : Fin m} (ha : f z a ≠ 0)
+    (v : Fin n → ℂ) (b : Fin m) :
+    fderiv ℂ (fun w j => f w j / f w a) z v b =
+      (fderiv ℂ f z v b * f z a - f z b * fderiv ℂ f z v a) / (f z a)^2 := by
+  have hd (i : Fin m) : HasFDerivAt (fun w => f w i)
+      ((ContinuousLinearMap.proj i).comp (fderiv ℂ f z)) z :=
+    (ContinuousLinearMap.proj i : (Fin m → ℂ) →L[ℂ] ℂ).hasFDerivAt.comp z hf.hasFDerivAt
+  have hdiv (i : Fin m) := (hd i).mul ((hasDerivAt_inv ha).comp_hasFDerivAt z (hd a))
+  simp only [Pi.mul_def, Function.comp_def] at hdiv
+  simp only [div_eq_mul_inv]
+  rw [fderiv_pi (fun i => (hdiv i).differentiableAt)]
+  change fderiv ℂ (fun w => f w b * (f w a)⁻¹) z v = _
+  rw [(hdiv b).fderiv]
+  simp only [_root_.add_apply, _root_.smul_apply,
+    ContinuousLinearMap.comp_apply, ContinuousLinearMap.proj_apply,
+    smul_eq_mul]
+  field_simp
+  ring
+
+ 
+
+lemma fsMatrix_posDef_of_immersion {f : (Fin n → ℂ) → (Fin m → ℂ)} {z : Fin n → ℂ}
+    (hf : DifferentiableAt ℂ f z) (hx : f z ≠ 0)
+    (hImm : ∃ a, f z a ≠ 0 ∧
+      Function.Injective (fderiv ℂ (fun w b => f w b / f w a) z)) :
+    (fsMatrix (f z) (jacobianMatrix f z)).PosDef := by
+  apply fsMatrix_posDef hx
+  obtain ⟨a, ha, hImm⟩ := hImm
+  have hk : ∀ v, (horizontal (f z) * jacobianMatrix f z) *ᵥ v = 0 → v = 0 := by
+    intro v hv
+    rw [← mulVec_mulVec, jacobianMatrix_mulVec, horizontal_mulVec_eq_zero_iff] at hv
+    have hn : fderiv ℂ (fun w b => f w b / f w a) z v = 0 := by
+      ext b
+      rw [normalized_derivative hf ha, hv]
+      simp only [Pi.smul_apply, smul_eq_mul, Pi.zero_apply]
+      ring
+    apply hImm
+    simpa only [map_zero] using hn
+  intro u v huv
+  apply sub_eq_zero.mp
+  apply hk
+  rw [mulVec_sub, huv, sub_self]
+
+end ProjectiveKaehler
+
+namespace Anticanonical
+
+abbrev Coordinates (d : ℕ) := Fin d → ℂ
+
+ 
+def coordinateOverlap {X : Type*} [TopologicalSpace X] {d : ℕ}
+    (e e' : OpenPartialHomeomorph X (Coordinates d)) : Set (Coordinates d) :=
+  e.target ∩ e.symm ⁻¹' e'.source
+
+ 
+
+structure ComplexAtlas (d : ℕ) (X : Type*) [TopologicalSpace X] where
+  count : ℕ
+  chart : Fin count → OpenPartialHomeomorph X (Coordinates d)
+  covers : ∀ x : X, ∃ i, x ∈ (chart i).source
+  holomorphic : ∀ i j, DifferentiableOn ℂ ((chart j) ∘ (chart i).symm)
+    (coordinateOverlap (chart i) (chart j))
+
+namespace ComplexAtlas
+
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] (A : ComplexAtlas d X)
+
+ 
+def transition (i j : Fin A.count) : Coordinates d → Coordinates d :=
+  A.chart j ∘ (A.chart i).symm
+
+ 
+
+def jacobian (i j : Fin A.count) (x : X) : ℂ :=
+  Matrix.det (LinearMap.toMatrix'
+    (fderiv ℂ (A.transition i j) (A.chart i x)).toLinearMap)
+
+ 
+
+structure ProjectiveEmbedding where
+  ambientDimension : ℕ
+  map : X → Projectivization ℂ (Fin (ambientDimension + 1) → ℂ)
+  injective : Function.Injective map
+  lift : Fin A.count → Coordinates d → (Fin (ambientDimension + 1) → ℂ)
+  holomorphic_lift : ∀ i, DifferentiableOn ℂ (lift i) (A.chart i).target
+  lift_ne_zero : ∀ i z, z ∈ (A.chart i).target → lift i z ≠ 0
+  represents : ∀ i z (hz : z ∈ (A.chart i).target),
+    map ((A.chart i).symm z) = Projectivization.mk ℂ (lift i z) (lift_ne_zero i z hz)
+  immersion : ∀ i z, z ∈ (A.chart i).target →
+    ∃ a : Fin (ambientDimension + 1), lift i z a ≠ 0 ∧
+      Function.Injective (fderiv ℂ (fun w b ↦ lift i w b / lift i w a) z)
+
+end ComplexAtlas
+end Anticanonical
+namespace Anticanonical.ComplexAtlas
+
+open Filter Topology
+
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] (A : ComplexAtlas d X)
+
+lemma isOpen_coordinateOverlap (i j : Fin A.count) :
+    IsOpen (coordinateOverlap (A.chart i) (A.chart j)) :=
+  ((A.chart i).symm.trans (A.chart j)).open_source
+
+lemma coordinate_mem_overlap (i j : Fin A.count) {x : X}
+    (hi : x ∈ (A.chart i).source) (hj : x ∈ (A.chart j).source) :
+    A.chart i x ∈ coordinateOverlap (A.chart i) (A.chart j) := by
+  refine ⟨(A.chart i).mapsTo hi, ?_⟩
+  change (A.chart i).symm (A.chart i x) ∈ (A.chart j).source
+  simpa only [(A.chart i).left_inv hi] using hj
+
+lemma transition_at (i j : Fin A.count) {x : X} (hi : x ∈ (A.chart i).source) :
+    A.transition i j (A.chart i x) = A.chart j x := by
+  simp only [transition, Function.comp_apply, (A.chart i).left_inv hi]
+
+lemma differentiableAt_transition (i j : Fin A.count) {x : X}
+    (hi : x ∈ (A.chart i).source) (hj : x ∈ (A.chart j).source) :
+    DifferentiableAt ℂ (A.transition i j) (A.chart i x) :=
+  (A.holomorphic i j).differentiableAt
+    ((A.isOpen_coordinateOverlap i j).mem_nhds (A.coordinate_mem_overlap i j hi hj))
+
+end Anticanonical.ComplexAtlas
+namespace Anticanonical.ComplexAtlas
+
+open Matrix
+open scoped ComplexOrder Matrix.Norms.Elementwise
+
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] (A : ComplexAtlas d X)
+
+def derivativeMatrix (i j : Fin A.count) (x : X) : Matrix (Fin d) (Fin d) ℂ :=
+  LinearMap.toMatrix' (fderiv ℂ (A.transition i j) (A.chart i x)).toLinearMap
+
+ 
+
+def fundamentalForm (g : Matrix (Fin d) (Fin d) ℂ)
+    (u v : Coordinates d) : ℝ :=
+  (star u ⬝ᵥ (g *ᵥ v)).im
+
+ 
+
+structure KaehlerMetric where
+  matrix : Fin A.count → Coordinates d → Matrix (Fin d) (Fin d) ℂ
+  smooth : ∀ i, ContDiffOn ℝ ⊤ (matrix i) (A.chart i).target
+  positive : ∀ i z, z ∈ (A.chart i).target → (matrix i z).PosDef
+  compatibility : ∀ i j x, x ∈ (A.chart i).source → x ∈ (A.chart j).source →
+    matrix i (A.chart i x) =
+      (A.derivativeMatrix i j x)ᴴ * matrix j (A.chart j x) * A.derivativeMatrix i j x
+  closed : ∀ i z, z ∈ (A.chart i).target → ∀ u v w : Coordinates d,
+    fderiv ℝ (fun y ↦ fundamentalForm (matrix i y) v w) z u +
+      fderiv ℝ (fun y ↦ fundamentalForm (matrix i y) w u) z v +
+      fderiv ℝ (fun y ↦ fundamentalForm (matrix i y) u v) z w = 0
+
+end Anticanonical.ComplexAtlas
+
+namespace ProjectiveKaehler
+open Set Matrix
+open scoped ComplexOrder Matrix.Norms.Elementwise
+
+variable {d m n k : ℕ} {s : Set (Fin d → ℂ)}
+
+lemma smooth_conj {f : (Fin d → ℂ) → ℂ} (hf : ContDiffOn ℝ ⊤ f s) :
+    ContDiffOn ℝ ⊤ (fun z => star (f z)) s :=
+  Complex.conjCLE.contDiff.comp_contDiffOn hf
+
+lemma smooth_mass {f : (Fin d → ℂ) → (Fin m → ℂ)} (hf : ContDiffOn ℝ ⊤ f s) :
+    ContDiffOn ℝ ⊤ (fun z => mass (f z)) s := by
+  change ContDiffOn ℝ ⊤ (fun z => ∑ i, star (f z i) * f z i) s
+  apply ContDiffOn.sum
+  intro i _
+  exact (smooth_conj (contDiffOn_pi.mp hf i)).mul (contDiffOn_pi.mp hf i)
+
+lemma smooth_horizontal {f : (Fin d → ℂ) → (Fin m → ℂ)}
+    (hf : ContDiffOn ℝ ⊤ f s) (hne : ∀ z ∈ s, f z ≠ 0) :
+    ContDiffOn ℝ ⊤ (fun z => horizontal (f z)) s := by
+  have hi := (smooth_mass hf).inv fun z hz => mass_ne_zero (hne z hz)
+  apply contDiffOn_pi.mpr
+  intro i
+  apply contDiffOn_pi.mpr
+  intro j
+  exact contDiffOn_const.sub (hi.mul ((contDiffOn_pi.mp hf i).mul
+    (smooth_conj (contDiffOn_pi.mp hf j))))
+
+lemma smooth_matrix_mul {F : (Fin d → ℂ) → Matrix (Fin m) (Fin n) ℂ}
+    {G : (Fin d → ℂ) → Matrix (Fin n) (Fin k) ℂ}
+    (hF : ContDiffOn ℝ ⊤ F s) (hG : ContDiffOn ℝ ⊤ G s) :
+    ContDiffOn ℝ ⊤ (fun z => F z * G z) s := by
+  apply contDiffOn_pi.mpr
+  intro i
+  apply contDiffOn_pi.mpr
+  intro j
+  change ContDiffOn ℝ ⊤ (fun z => ∑ l, F z i l * G z l j) s
+  apply ContDiffOn.sum
+  intro l _
+  exact (contDiffOn_pi.mp (contDiffOn_pi.mp hF i) l).mul
+    (contDiffOn_pi.mp (contDiffOn_pi.mp hG l) j)
+
+lemma smooth_conjTranspose {F : (Fin d → ℂ) → Matrix (Fin m) (Fin n) ℂ}
+    (hF : ContDiffOn ℝ ⊤ F s) : ContDiffOn ℝ ⊤ (fun z => (F z)ᴴ) s := by
+  apply contDiffOn_pi.mpr
+  intro i
+  apply contDiffOn_pi.mpr
+  intro j
+  exact smooth_conj (contDiffOn_pi.mp (contDiffOn_pi.mp hF j) i)
+
+lemma smooth_fsMatrix {f : (Fin d → ℂ) → (Fin m → ℂ)}
+    {D : (Fin d → ℂ) → Matrix (Fin m) (Fin n) ℂ}
+    (hf : ContDiffOn ℝ ⊤ f s) (hD : ContDiffOn ℝ ⊤ D s) (hne : ∀ z ∈ s, f z ≠ 0) :
+    ContDiffOn ℝ ⊤ (fun z => fsMatrix (f z) (D z)) s := by
+  have hQ := smooth_matrix_mul (smooth_horizontal hf hne) hD
+  exact ((smooth_mass hf).inv (fun z hz => mass_ne_zero (hne z hz))).smul
+    (smooth_matrix_mul (smooth_conjTranspose hQ) hQ)
+
+lemma smooth_jacobianMatrix {f : (Fin d → ℂ) → (Fin m → ℂ)}
+    (hs : IsOpen s) (hf : DifferentiableOn ℂ f s) :
+    ContDiffOn ℝ ⊤ (jacobianMatrix f) s := by
+  have hd : ContDiffOn ℂ ⊤ (fderiv ℂ f) s :=
+    (analyticOnNhd_pi_of_differentiableOn hs hf).fderiv.contDiffOn hs.uniqueDiffOn
+  apply contDiffOn_pi.mpr
+  intro i
+  apply contDiffOn_pi.mpr
+  intro j
+  have hc : ContDiffOn ℂ ⊤ (fun z => fderiv ℂ f z (Pi.single j 1)) s :=
+    hd.clm_apply contDiffOn_const
+  exact (contDiffOn_pi.mp hc i).restrict_scalars ℝ
+
+end ProjectiveKaehler
+
+namespace Anticanonical.ComplexAtlas
+open ProjectiveKaehler
+open scoped ComplexOrder Matrix.Norms.Elementwise
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+
+ 
+def ProjectiveEmbedding.backgroundMatrix (P : A.ProjectiveEmbedding) (i : Fin A.count)
+    (z : Coordinates d) : Matrix (Fin d) (Fin d) ℂ :=
+  fsMatrix (P.lift i z) (jacobianMatrix (P.lift i) z)
+
+lemma ProjectiveEmbedding.backgroundMatrix_smooth (P : A.ProjectiveEmbedding) (i : Fin A.count) :
+    ContDiffOn ℝ ⊤ (P.backgroundMatrix i) (A.chart i).target := by
+  exact smooth_fsMatrix
+    (contDiffOn_real_pi_of_differentiableOn (A.chart i).open_target (P.holomorphic_lift i))
+    (smooth_jacobianMatrix (A.chart i).open_target (P.holomorphic_lift i))
+    (P.lift_ne_zero i)
+
+lemma ProjectiveEmbedding.backgroundMatrix_positive (P : A.ProjectiveEmbedding)
+    (i : Fin A.count) {z : Coordinates d} (hz : z ∈ (A.chart i).target) :
+    (P.backgroundMatrix i z).PosDef := by
+  exact fsMatrix_posDef_of_immersion
+    ((P.holomorphic_lift i).differentiableAt ((A.chart i).open_target.mem_nhds hz))
+    (P.lift_ne_zero i z hz) (P.immersion i z hz)
+
+end Anticanonical.ComplexAtlas
+
+namespace ProjectiveKaehler
+open Filter Topology Matrix
+variable {m n k : ℕ}
+
+lemma jacobianMatrix_comp {f : (Fin n → ℂ) → (Fin m → ℂ)}
+    {g : (Fin k → ℂ) → (Fin n → ℂ)} {z : Fin k → ℂ}
+    (hf : DifferentiableAt ℂ f (g z)) (hg : DifferentiableAt ℂ g z) :
+    jacobianMatrix (f ∘ g) z = jacobianMatrix f (g z) * jacobianMatrix g z := by
+  unfold jacobianMatrix
+  rw [fderiv_comp z hf hg]
+  exact LinearMap.toMatrix'_comp _ _
+
+lemma jacobianMatrix_smul {f : (Fin n → ℂ) → (Fin m → ℂ)}
+    {c : (Fin n → ℂ) → ℂ} {z : Fin n → ℂ}
+    (hc : DifferentiableAt ℂ c z) (hf : DifferentiableAt ℂ f z) :
+    jacobianMatrix (fun w => c w • f w) z = c z • jacobianMatrix f z +
+      vecMulVec (f z) (fun j => fderiv ℂ c z (Pi.single j 1)) := by
+  change LinearMap.toMatrix' (fderiv ℂ (c • f) z).toLinearMap = _
+  rw [fderiv_smul hc hf]
+  ext i j
+  simp [LinearMap.toMatrix'_apply, jacobianMatrix, vecMulVec, mul_comm]
+
+ 
+
+lemma fsMatrix_eq_of_eventually_proportional
+    {f g : (Fin n → ℂ) → (Fin m → ℂ)} {z : Fin n → ℂ}
+    (hf : DifferentiableAt ℂ f z) (hg : DifferentiableAt ℂ g z)
+    (hfn : f z ≠ 0) (hgn : g z ≠ 0)
+    (hp : ∀ᶠ w in 𝓝 z, ∃ c : ℂ, c • g w = f w) :
+    fsMatrix (f z) (jacobianMatrix f z) = fsMatrix (g z) (jacobianMatrix g z) := by
+  classical
+  obtain ⟨a, ha⟩ : ∃ a, g z a ≠ 0 := by
+    by_contra h
+    push Not at h
+    exact hgn (funext h)
+  let c : (Fin n → ℂ) → ℂ := fun w => f w a / g w a
+  have hd : DifferentiableAt ℂ c z := by
+    simpa only [c, div_eq_mul_inv, Pi.mul_def, Pi.inv_def] using
+      (differentiableAt_pi.mp hf a).mul ((differentiableAt_pi.mp hg a).inv ha)
+  have he : f =ᶠ[𝓝 z] (fun w => c w • g w) := by
+    filter_upwards [hp, (differentiableAt_pi.mp hg a).continuousAt.eventually_ne ha] with w hw hwa
+    obtain ⟨b, hb⟩ := hw
+    have hwa' : f w a = b * g w a := by
+      simpa only [Pi.smul_apply, smul_eq_mul] using (congrFun hb a).symm
+    have hc : c w = b := by simp [c, hwa', hwa]
+    rw [hc, hb]
+  have hc : c z ≠ 0 := by
+    intro h
+    apply hfn
+    simpa only [h, zero_smul] using he.self_of_nhds
+  have hJ : jacobianMatrix f z = jacobianMatrix (fun w => c w • g w) z := by
+    unfold jacobianMatrix
+    rw [he.fderiv_eq]
+  rw [he.self_of_nhds, hJ, jacobianMatrix_smul hd hg]
+  exact fsMatrix_gauge hc hgn _ _
+
+end ProjectiveKaehler
+
+namespace Anticanonical.ComplexAtlas
+open ProjectiveKaehler Filter Topology Matrix
+open scoped ComplexOrder Matrix.Norms.Elementwise
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+
+lemma ProjectiveEmbedding.backgroundMatrix_compatibility (P : A.ProjectiveEmbedding)
+    (i j : Fin A.count) (x : X) (hi : x ∈ (A.chart i).source)
+    (hj : x ∈ (A.chart j).source) :
+    P.backgroundMatrix i (A.chart i x) =
+      (A.derivativeMatrix i j x)ᴴ * P.backgroundMatrix j (A.chart j x) *
+        A.derivativeMatrix i j x := by
+  let z := A.chart i x
+  have hzi : z ∈ (A.chart i).target := (A.chart i).mapsTo hi
+  have hzj : A.transition i j z ∈ (A.chart j).target := by
+    rw [A.transition_at i j hi]
+    exact (A.chart j).mapsTo hj
+  have hdi := (P.holomorphic_lift i).differentiableAt ((A.chart i).open_target.mem_nhds hzi)
+  have hdj := (P.holomorphic_lift j).differentiableAt ((A.chart j).open_target.mem_nhds hzj)
+  have hdt := A.differentiableAt_transition i j hi hj
+  have he := fsMatrix_eq_of_eventually_proportional hdi (hdj.comp z hdt)
+    (P.lift_ne_zero i z hzi) (P.lift_ne_zero j _ hzj) ?_
+  · change fsMatrix (P.lift i z) (jacobianMatrix (P.lift i) z) = _
+    rw [he, jacobianMatrix_comp hdj hdt, fsMatrix_comp]
+    simp only [Function.comp_apply, A.transition_at i j hi, derivativeMatrix,
+      jacobianMatrix, backgroundMatrix, z]
+  · filter_upwards [(A.isOpen_coordinateOverlap i j).mem_nhds
+        (A.coordinate_mem_overlap i j hi hj)] with w hw
+    have hwj : A.transition i j w ∈ (A.chart j).target := (A.chart j).mapsTo hw.2
+    have heq : (A.chart j).symm (A.transition i j w) = (A.chart i).symm w :=
+      (A.chart j).left_inv hw.2
+    have hproj : Projectivization.mk ℂ (P.lift i w) (P.lift_ne_zero i w hw.1) =
+        Projectivization.mk ℂ (P.lift j (A.transition i j w)) (P.lift_ne_zero j _ hwj) := by
+      rw [← P.represents i w hw.1, ← P.represents j _ hwj, heq]
+    obtain ⟨c, hc⟩ := (Projectivization.mk_eq_mk_iff ℂ _ _ _ _).mp hproj
+    exact ⟨(c : ℂ), hc⟩
+
+end Anticanonical.ComplexAtlas
+
+ 
+
+noncomputable section
+namespace ProjectiveKaehler.Closedness
+open Matrix Filter Topology
+open scoped ComplexOrder Matrix.Norms.Elementwise
+
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+variable {m : ℕ}
+
+def pair (x y : Fin m → ℂ) : ℂ := star x ⬝ᵥ y
+
+@[simp] lemma star_pair (x y : Fin m → ℂ) : star (pair x y) = pair y x := by
+  simp [pair, dotProduct, star_sum, star_mul, mul_comm]
+
+lemma pair_self_ne_zero {x : Fin m → ℂ} (hx : x ≠ 0) : pair x x ≠ 0 := by
+  simpa only [pair, ne_eq, dotProduct_star_self_eq_zero] using hx
+
+lemma smooth_pair {f g : E → (Fin m → ℂ)} {z : E}
+    (hf : ContDiffAt ℝ ⊤ f z) (hg : ContDiffAt ℝ ⊤ g z) :
+    ContDiffAt ℝ ⊤ (fun y => pair (f y) (g y)) z := by
+  unfold pair dotProduct
+  apply ContDiffAt.sum
+  intro i _
+  exact (Complex.conjCLE.contDiff.contDiffAt.comp z (contDiffAt_pi.mp hf i)).mul
+    (contDiffAt_pi.mp hg i)
+
+lemma fderiv_pair_apply {f g : E → (Fin m → ℂ)} {z : E}
+    (hf : DifferentiableAt ℝ f z) (hg : DifferentiableAt ℝ g z) (u : E) :
+    fderiv ℝ (fun y => pair (f y) (g y)) z u =
+      pair (fderiv ℝ f z u) (g z) + pair (f z) (fderiv ℝ g z u) := by
+  have h (i : Fin m) := (differentiableAt_pi.mp hf i).hasFDerivAt.star.mul
+    (differentiableAt_pi.mp hg i).hasFDerivAt
+  have hs := HasFDerivAt.fun_sum (u := Finset.univ) (fun i _ => h i)
+  simp only [Pi.mul_def] at hs
+  change fderiv ℝ (fun y => ∑ i, star (f y i) * g y i) z u = _
+  rw [hs.fderiv]
+  simp only [_root_.sum_apply, Finset.sum_add_distrib, _root_.add_apply,
+    _root_.smul_apply, ContinuousLinearMap.comp_apply, smul_eq_mul]
+  simp only [fderiv_apply hf, fderiv_apply hg]
+  simp only [ContinuousLinearMap.comp_apply, ContinuousLinearMap.proj_apply]
+  change (∑ i, star (f z i) * fderiv ℝ g z u i) +
+    (∑ i, g z i * star (fderiv ℝ f z u i)) = _
+  simp only [pair, dotProduct, Pi.star_apply]
+  rw [add_comm]
+  congr 1
+  apply Finset.sum_congr rfl
+  intro i _
+  exact mul_comm _ _
+
+lemma fderiv_eval_derivative {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F]
+    {f : E → F} {z : E} (hf : ContDiffAt ℝ ⊤ f z) (u v : E) :
+    fderiv ℝ (fun y => fderiv ℝ f y v) z u = fderiv ℝ (fderiv ℝ f) z u v := by
+  rw [fderiv_clm_apply ((hf.fderiv_right (m := ⊤) (by simp)).differentiableAt (by simp))
+    (differentiableAt_const v)]
+  simp
+
+lemma second_derivative_symmetric {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F]
+    {f : E → F} {z : E} (hf : ContDiffAt ℝ ⊤ f z) (u v : E) :
+    fderiv ℝ (fun y => fderiv ℝ f y v) z u =
+      fderiv ℝ (fun y => fderiv ℝ f y u) z v := by
+  rw [fderiv_eval_derivative hf, fderiv_eval_derivative hf]
+  exact hf.isSymmSndFDerivAt (by simp) u v
+
+ 
+def beta (f : E → (Fin m → ℂ)) (w y : E) : ℝ :=
+  ((pair (f y) (f y))⁻¹ * pair (f y) (fderiv ℝ f y w)).im
+
+lemma smooth_beta {f : E → (Fin m → ℂ)} {z : E} (hf : ContDiffAt ℝ ⊤ f z)
+    (hne : f z ≠ 0) (w : E) : ContDiffAt ℝ ⊤ (beta f w) z := by
+  have hd := (hf.fderiv_right (m := ⊤) (by simp)).clm_apply (contDiffAt_const (c := w))
+  exact Complex.imCLM.contDiff.contDiffAt.comp z
+    (((smooth_pair hf hf).inv (pair_self_ne_zero hne)).mul (smooth_pair hf hd))
+
+lemma fderiv_inv_complex_apply {q : E → ℂ} {z : E}
+    (hq : DifferentiableAt ℝ q z) (hn : q z ≠ 0) (u : E) :
+    fderiv ℝ (fun y => (q y)⁻¹) z u = -(q z)⁻¹ ^ 2 * fderiv ℝ q z u := by
+  have h := ((hasDerivAt_inv hn).hasFDerivAt.restrictScalars ℝ).comp z hq.hasFDerivAt
+  simp only [Function.comp_def] at h
+  rw [h.fderiv]
+  simp [ContinuousLinearMap.comp_apply, inv_pow, mul_comm]
+
+lemma fderiv_im_apply {q : E → ℂ} {z : E}
+    (hq : DifferentiableAt ℝ q z) (u : E) :
+    fderiv ℝ (fun y => (q y).im) z u = (fderiv ℝ q z u).im := by
+  have h := Complex.imCLM.hasFDerivAt.comp z hq.hasFDerivAt
+  exact congrArg (fun l : E →L[ℝ] ℝ => l u) h.fderiv
+
+lemma fderiv_beta {f : E → (Fin m → ℂ)} {z : E} (hf : ContDiffAt ℝ ⊤ f z)
+    (hne : f z ≠ 0) (u w : E) :
+    fderiv ℝ (beta f w) z u =
+      (-(pair (f z) (f z))⁻¹ ^ 2 *
+        (pair (fderiv ℝ f z u) (f z) + pair (f z) (fderiv ℝ f z u)) *
+          pair (f z) (fderiv ℝ f z w) +
+        (pair (f z) (f z))⁻¹ *
+          (pair (fderiv ℝ f z u) (fderiv ℝ f z w) +
+            pair (f z) (fderiv ℝ (fderiv ℝ f) z u w))).im := by
+  have hD : ContDiffAt ℝ ⊤ (fun y => fderiv ℝ f y w) z :=
+    (hf.fderiv_right (by simp)).clm_apply contDiffAt_const
+  have hQ := (smooth_pair hf hf).differentiableAt (by simp)
+  have hR := (smooth_pair hf hD).differentiableAt (by simp)
+  have hI := hQ.inv (pair_self_ne_zero hne)
+  unfold beta
+  have hP : DifferentiableAt ℝ (fun y => (pair (f y) (f y))⁻¹ *
+      pair (f y) (fderiv ℝ f y w)) z := by
+    simpa only [Pi.mul_def, Pi.inv_def] using hI.mul hR
+  rw [fderiv_im_apply hP]
+  have hM := fderiv_mul hI hR
+  simp only [Pi.mul_def, Pi.inv_def] at hM
+  rw [hM]
+  simp only [_root_.add_apply, _root_.smul_apply, smul_eq_mul]
+  rw [fderiv_inv_complex_apply hQ (pair_self_ne_zero hne)]
+  rw [fderiv_pair_apply (hf.differentiableAt (by simp)) (hf.differentiableAt (by simp)),
+    fderiv_pair_apply (hf.differentiableAt (by simp)) (hD.differentiableAt (by simp)),
+    fderiv_eval_derivative hf]
+  congr 1
+  ring
+
+lemma anti_beta_algebra (r a b c t : ℂ) (hr : star r = r) :
+    (-r ^ 2 * (star a + a) * b + r * (c + t)).im -
+      (-r ^ 2 * (star b + b) * a + r * (star c + t)).im =
+        2 * (r * c - r ^ 2 * (star a * b)).im := by
+  rw [← Complex.sub_im]
+  have h : (-r ^ 2 * (star a + a) * b + r * (c + t)) -
+      (-r ^ 2 * (star b + b) * a + r * (star c + t)) =
+      (r * c - r ^ 2 * (star a * b)) - star (r * c - r ^ 2 * (star a * b)) := by
+    simp only [star_sub, star_mul, star_pow, star_star, hr]
+    ring
+  rw [h]
+  simp only [Complex.sub_im, Complex.star_def, Complex.conj_im]
+  ring
+
+ 
+
+def omega (f : E → (Fin m → ℂ)) (v w y : E) : ℝ :=
+  ((pair (f y) (f y))⁻¹ * pair (fderiv ℝ f y v) (fderiv ℝ f y w) -
+    (pair (f y) (f y))⁻¹ ^ 2 *
+      (pair (fderiv ℝ f y v) (f y) * pair (f y) (fderiv ℝ f y w))).im
+
+lemma beta_antisym {f : E → (Fin m → ℂ)} {z : E} (hf : ContDiffAt ℝ ⊤ f z)
+    (hne : f z ≠ 0) (v w : E) :
+    fderiv ℝ (beta f w) z v - fderiv ℝ (beta f v) z w = 2 * omega f v w z := by
+  rw [fderiv_beta hf hne, fderiv_beta hf hne,
+    (hf.isSymmSndFDerivAt (by simp) w v)]
+  unfold omega
+  rw [← star_pair (f z) (fderiv ℝ f z v),
+    ← star_pair (f z) (fderiv ℝ f z w),
+    ← star_pair (fderiv ℝ f z v) (fderiv ℝ f z w)]
+  apply anti_beta_algebra
+  simp
+
+lemma cyclic_antisym_derivative {b : E → E → ℝ} {z : E}
+    (hb : ∀ v, ContDiffAt ℝ ⊤ (b v) z) (u v w : E) :
+    fderiv ℝ (fun y => fderiv ℝ (b w) y v - fderiv ℝ (b v) y w) z u +
+      fderiv ℝ (fun y => fderiv ℝ (b u) y w - fderiv ℝ (b w) y u) z v +
+      fderiv ℝ (fun y => fderiv ℝ (b v) y u - fderiv ℝ (b u) y v) z w = 0 := by
+  have hd (a c : E) : DifferentiableAt ℝ (fun y => fderiv ℝ (b a) y c) z :=
+    (((hb a).fderiv_right (m := ⊤) (by simp)).clm_apply contDiffAt_const).differentiableAt
+      (by simp)
+  simp only [fderiv_fun_sub (hd _ _) (hd _ _), _root_.sub_apply]
+  rw [second_derivative_symmetric (hb w) u v, second_derivative_symmetric (hb v) u w,
+    second_derivative_symmetric (hb u) v w]
+  ring
+
+lemma smooth_omega {f : E → (Fin m → ℂ)} {z : E} (hf : ContDiffAt ℝ ⊤ f z)
+    (hne : f z ≠ 0) (v w : E) : ContDiffAt ℝ ⊤ (omega f v w) z := by
+  have hD (a : E) : ContDiffAt ℝ ⊤ (fun y => fderiv ℝ f y a) z :=
+    (hf.fderiv_right (by simp)).clm_apply contDiffAt_const
+  have hI := (smooth_pair hf hf).inv (pair_self_ne_zero hne)
+  exact Complex.imCLM.contDiff.contDiffAt.comp z
+    ((hI.mul (smooth_pair (hD v) (hD w))).sub
+      ((hI.pow 2).mul ((smooth_pair (hD v) hf).mul (smooth_pair hf (hD w)))))
+
+ 
+
+lemma omega_closed {f : E → (Fin m → ℂ)} {U : Set E} (hU : IsOpen U)
+    (hf : ContDiffOn ℝ ⊤ f U) (hne : ∀ y ∈ U, f y ≠ 0)
+    {z : E} (hz : z ∈ U) (u v w : E) :
+    fderiv ℝ (omega f v w) z u + fderiv ℝ (omega f w u) z v +
+      fderiv ℝ (omega f u v) z w = 0 := by
+  have hfc := hf.contDiffAt (hU.mem_nhds hz)
+  have hbc (a : E) := smooth_beta hfc (hne z hz) a
+  have he (a b c : E) :
+      fderiv ℝ (fun y => fderiv ℝ (beta f b) y a - fderiv ℝ (beta f a) y b) z c =
+        2 * fderiv ℝ (omega f a b) z c := by
+    have heq : (fun y => fderiv ℝ (beta f b) y a - fderiv ℝ (beta f a) y b)
+        =ᶠ[𝓝 z] (fun y => 2 * omega f a b y) := by
+      filter_upwards [hU.mem_nhds hz] with y hy
+      exact beta_antisym (hf.contDiffAt (hU.mem_nhds hy)) (hne y hy) a b
+    rw [heq.fderiv_eq, fderiv_const_mul
+      ((smooth_omega hfc (hne z hz) a b).differentiableAt (by simp))]
+    rfl
+  have hc := cyclic_antisym_derivative hbc u v w
+  rw [he, he, he] at hc
+  linarith
+
+end ProjectiveKaehler.Closedness
+
+namespace ProjectiveKaehler
+open Anticanonical.ComplexAtlas Matrix
+variable {m n : ℕ}
+
+lemma fundamentalForm_fs_eq_omega {f : (Fin n → ℂ) → (Fin m → ℂ)}
+    {z : Fin n → ℂ} (hf : DifferentiableAt ℂ f z) (hne : f z ≠ 0) (v w : Fin n → ℂ) :
+    fundamentalForm (fsMatrix (f z) (jacobianMatrix f z)) v w =
+      Closedness.omega f v w z := by
+  have hd (a : Fin n → ℂ) : fderiv ℝ f z a = fderiv ℂ f z a := by
+    rw [(hf.hasFDerivAt.restrictScalars ℝ).fderiv]
+    rfl
+  unfold fundamentalForm
+  rw [fsMatrix_pair hne]
+  simp only [jacobianMatrix_mulVec, Closedness.omega, Closedness.pair, mass, hd]
+
+end ProjectiveKaehler
+
+namespace Anticanonical.ComplexAtlas
+open ProjectiveKaehler Filter Topology Matrix
+open scoped ComplexOrder Matrix.Norms.Elementwise
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+
+lemma ProjectiveEmbedding.backgroundMatrix_closed (P : A.ProjectiveEmbedding)
+    (i : Fin A.count) {z : Coordinates d} (hz : z ∈ (A.chart i).target)
+    (u v w : Coordinates d) :
+    fderiv ℝ (fun y => fundamentalForm (P.backgroundMatrix i y) v w) z u +
+      fderiv ℝ (fun y => fundamentalForm (P.backgroundMatrix i y) w u) z v +
+      fderiv ℝ (fun y => fundamentalForm (P.backgroundMatrix i y) u v) z w = 0 := by
+  have hs := contDiffOn_real_pi_of_differentiableOn (A.chart i).open_target
+    (P.holomorphic_lift i)
+  have he (a b : Coordinates d) :
+      (fun y => fundamentalForm (P.backgroundMatrix i y) a b)
+        =ᶠ[𝓝 z] Closedness.omega (P.lift i) a b := by
+    filter_upwards [(A.chart i).open_target.mem_nhds hz] with y hy
+    exact fundamentalForm_fs_eq_omega
+      ((P.holomorphic_lift i).differentiableAt ((A.chart i).open_target.mem_nhds hy))
+      (P.lift_ne_zero i y hy) a b
+  rw [(he v w).fderiv_eq, (he w u).fderiv_eq, (he u v).fderiv_eq]
+  exact Closedness.omega_closed (A.chart i).open_target hs (P.lift_ne_zero i) hz u v w
+
+ 
+
+def ProjectiveEmbedding.backgroundKaehlerMetric (P : A.ProjectiveEmbedding) : A.KaehlerMetric where
+  matrix := P.backgroundMatrix
+  smooth := P.backgroundMatrix_smooth
+  positive := fun i _ hz => P.backgroundMatrix_positive i hz
+  compatibility := P.backgroundMatrix_compatibility
+  closed := fun i _ hz => P.backgroundMatrix_closed i hz
+
+theorem ProjectiveEmbedding.nonemptyKaehlerMetric (P : A.ProjectiveEmbedding) :
+    Nonempty A.KaehlerMetric := ⟨P.backgroundKaehlerMetric⟩
+
+end Anticanonical.ComplexAtlas
+
+ 
+
+namespace Anticanonical.SourceSmooth
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+open Matrix
+variable {d : ℕ} {X : Type*} [TopologicalSpace X]
+
+ 
+
+def levi (φ : Coordinates d → ℝ) (z v : Coordinates d) : ℝ :=
+  (fderiv ℝ (fderiv ℝ φ) z v) v +
+    (fderiv ℝ (fderiv ℝ φ) z (Complex.I • v)) (Complex.I • v)
+
+ 
+structure SemipositiveAnticanonicalMetric (A : ComplexAtlas d X) where
+  weight : Fin A.count → Coordinates d → ℝ
+  smooth : ∀ i, ContDiffOn ℝ ∞ (weight i) (A.chart i).target
+  compatibility : ∀ i j x, x ∈ (A.chart i).source → x ∈ (A.chart j).source →
+    weight j (A.chart j x) =
+      weight i (A.chart i x) + Real.log (‖A.jacobian i j x‖ ^ 2)
+  semipositive : ∀ i z, z ∈ (A.chart i).target →
+    ∀ v : Coordinates d, 0 ≤ levi (weight i) z v
+
+ 
+structure KaehlerMetric (A : ComplexAtlas d X) where
+  matrix : Fin A.count → Coordinates d → Matrix (Fin d) (Fin d) ℂ
+  smooth : ∀ i, ContDiffOn ℝ ∞ (matrix i) (A.chart i).target
+  positive : ∀ i z, z ∈ (A.chart i).target → (matrix i z).PosDef
+  compatibility : ∀ i j x, x ∈ (A.chart i).source → x ∈ (A.chart j).source →
+    matrix i (A.chart i x) =
+      (A.derivativeMatrix i j x)ᴴ * matrix j (A.chart j x) * A.derivativeMatrix i j x
+  closed : ∀ i z, z ∈ (A.chart i).target → ∀ u v w : Coordinates d,
+    fderiv ℝ (fun y ↦ ComplexAtlas.fundamentalForm (matrix i y) v w) z u +
+      fderiv ℝ (fun y ↦ ComplexAtlas.fundamentalForm (matrix i y) w u) z v +
+      fderiv ℝ (fun y ↦ ComplexAtlas.fundamentalForm (matrix i y) u v) z w = 0
+
+namespace KaehlerMetric
+variable {A : ComplexAtlas d X}
+def volumeCoefficient (g : KaehlerMetric A) (i : Fin A.count) (z : Coordinates d) : ℝ :=
+  (g.matrix i z).det.re
+
+def ricciPotential (g : KaehlerMetric A) (i : Fin A.count) (z : Coordinates d) : ℝ :=
+  -Real.log (g.volumeCoefficient i z)
+
+def ricciQuadratic (g : KaehlerMetric A) (i : Fin A.count) (z v : Coordinates d) : ℝ :=
+  levi (g.ricciPotential i) z v
+end KaehlerMetric
+
+ 
+
+def PrescribedRicci : Prop :=
+  ∀ (d : ℕ) (X : Type) [TopologicalSpace X] [T2Space X]
+    [CompactSpace X] [ConnectedSpace X] (A : ComplexAtlas d X),
+    Nonempty A.ProjectiveEmbedding →
+    ∀ h : SemipositiveAnticanonicalMetric A,
+      ∃ g : KaehlerMetric A, ∀ i z, z ∈ (A.chart i).target →
+        ∀ v : Coordinates d, g.ricciQuadratic i z v = levi (h.weight i) z v
+
+end Anticanonical.SourceSmooth
+
+namespace Anticanonical.SourceSmooth
+open scoped ContDiff
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+
+ 
+
+def projectiveBackground (P : A.ProjectiveEmbedding) : KaehlerMetric A where
+  matrix := P.backgroundMatrix
+  smooth := fun i => (P.backgroundMatrix_smooth i).of_le (by simp)
+  positive := fun i _ hz => P.backgroundMatrix_positive i hz
+  compatibility := P.backgroundMatrix_compatibility
+  closed := fun i _ hz => P.backgroundMatrix_closed i hz
+
+end Anticanonical.SourceSmooth
+
+ 
+
+noncomputable section
+namespace PotentialKaehler
+open Matrix
+open scoped Matrix.Norms.Elementwise ComplexOrder ContDiff
+
+variable {n : ℕ}
+abbrev V (n : ℕ) := Fin n → ℂ
+abbrev RealBilinear (n : ℕ) := V n →L[ℝ] V n →L[ℝ] ℝ
+
+ 
+
+def hermitianPart (B : RealBilinear n) (u v : V n) : ℂ :=
+  ⟨(B u v + B (Complex.I • u) (Complex.I • v)) / 4,
+    (B (Complex.I • u) v - B u (Complex.I • v)) / 4⟩
+
+lemma complex_smul_decompose (a : ℂ) (v : V n) :
+    a • v = a.re • v + a.im • (Complex.I • v) := by
+  ext i
+  change a * v i = a.re • v i + a.im • (Complex.I * v i)
+  rw [Complex.real_smul, Complex.real_smul, ← mul_assoc, ← add_mul, Complex.re_add_im]
+
+lemma hermitianPart_add_left (B : RealBilinear n) (u u' v : V n) :
+    hermitianPart B (u + u') v = hermitianPart B u v + hermitianPart B u' v := by
+  apply Complex.ext <;> simp [hermitianPart, smul_add, map_add] <;> ring
+
+lemma hermitianPart_add_right (B : RealBilinear n) (u v v' : V n) :
+    hermitianPart B u (v + v') = hermitianPart B u v + hermitianPart B u v' := by
+  apply Complex.ext <;> simp [hermitianPart, smul_add, map_add] <;> ring
+
+lemma hermitianPart_real_smul_left (B : RealBilinear n) (a : ℝ) (u v : V n) :
+    hermitianPart B (a • u) v = a • hermitianPart B u v := by
+  apply Complex.ext <;> simp [hermitianPart, smul_comm Complex.I a, map_smul, smul_eq_mul] <;> ring
+
+lemma hermitianPart_real_smul_right (B : RealBilinear n) (a : ℝ) (u v : V n) :
+    hermitianPart B u (a • v) = a • hermitianPart B u v := by
+  apply Complex.ext <;> simp [hermitianPart, smul_comm Complex.I a, map_smul, smul_eq_mul] <;> ring
+
+lemma hermitianPart_I_left (B : RealBilinear n) (u v : V n) :
+    hermitianPart B (Complex.I • u) v = -Complex.I * hermitianPart B u v := by
+  apply Complex.ext <;> simp [hermitianPart, smul_smul, Complex.I_mul_I, map_neg] <;> ring
+
+lemma hermitianPart_I_right (B : RealBilinear n) (u v : V n) :
+    hermitianPart B u (Complex.I • v) = Complex.I * hermitianPart B u v := by
+  apply Complex.ext <;> simp [hermitianPart, smul_smul, Complex.I_mul_I, map_neg] <;> ring
+
+lemma hermitianPart_smul_left (B : RealBilinear n) (a : ℂ) (u v : V n) :
+    hermitianPart B (a • u) v = star a * hermitianPart B u v := by
+  rw [complex_smul_decompose, hermitianPart_add_left,
+    hermitianPart_real_smul_left, hermitianPart_real_smul_left, hermitianPart_I_left]
+  apply Complex.ext <;> simp [Complex.mul_re, Complex.mul_im]
+
+lemma hermitianPart_smul_right (B : RealBilinear n) (a : ℂ) (u v : V n) :
+    hermitianPart B u (a • v) = a * hermitianPart B u v := by
+  rw [complex_smul_decompose, hermitianPart_add_right,
+    hermitianPart_real_smul_right, hermitianPart_real_smul_right, hermitianPart_I_right]
+  apply Complex.ext <;> simp [Complex.mul_re, Complex.mul_im]
+  ring
+
+ 
+def hermitianPartForm (B : RealBilinear n) : V n →ₗ⋆[ℂ] V n →ₗ[ℂ] ℂ where
+  toFun u :=
+    { toFun := hermitianPart B u
+      map_add' := hermitianPart_add_right B u
+      map_smul' := fun a v => hermitianPart_smul_right B a u v }
+  map_add' u u' := by
+    apply LinearMap.ext
+    intro v
+    exact hermitianPart_add_left B u u' v
+  map_smul' a u := by
+    apply LinearMap.ext
+    intro v
+    exact hermitianPart_smul_left B a u v
+
+lemma hermitianPart_star (B : RealBilinear n) (hB : ∀ u v, B u v = B v u) (u v : V n) :
+    star (hermitianPart B u v) = hermitianPart B v u := by
+  apply Complex.ext <;> simp [hermitianPart, hB u v, hB (Complex.I • u) (Complex.I • v),
+    hB (Complex.I • u) v, hB u (Complex.I • v)]
+  ring
+
+def hermitianPartMatrix (B : RealBilinear n) : Matrix (Fin n) (Fin n) ℂ :=
+  LinearMap.toMatrix₂ (Pi.basisFun ℂ (Fin n)) (Pi.basisFun ℂ (Fin n)) (hermitianPartForm B)
+
+lemma hermitianPartMatrix_apply (B : RealBilinear n) (i j : Fin n) :
+    hermitianPartMatrix B i j = hermitianPart B (Pi.single i 1) (Pi.single j 1) := by
+  simp [hermitianPartMatrix, LinearMap.toMatrix₂_apply, hermitianPartForm]
+
+lemma hermitianPartMatrix_pair (B : RealBilinear n) (u v : V n) :
+    star u ⬝ᵥ (hermitianPartMatrix B *ᵥ v) = hermitianPart B u v := by
+  simpa [hermitianPartMatrix, hermitianPartForm] using
+    (star_dotProduct_toMatrix₂_mulVec (B := hermitianPartForm B) (Pi.basisFun ℂ (Fin n)) u v)
+
+lemma hermitianPartMatrix_hermitian (B : RealBilinear n) (hB : ∀ u v, B u v = B v u) :
+    (hermitianPartMatrix B).IsHermitian := by
+  apply (LinearMap.isSymm_iff_isHermitian_toMatrix (Pi.basisFun ℂ (Fin n))).mp
+  exact ⟨hermitianPart_star B hB⟩
+
+ 
+def potentialMatrix (φ : V n → ℝ) (z : V n) : Matrix (Fin n) (Fin n) ℂ :=
+  hermitianPartMatrix (fderiv ℝ (fderiv ℝ φ) z)
+
+lemma potentialMatrix_hermitian {φ : V n → ℝ} {z : V n}
+    (hφ : ContDiffAt ℝ ∞ φ z) : (potentialMatrix φ z).IsHermitian := by
+  apply hermitianPartMatrix_hermitian
+  exact hφ.isSymmSndFDerivAt (by simp only [minSmoothness_of_isRCLikeNormedField]; change ((2 : ℕ∞) : WithTop ℕ∞) ≤ ((⊤ : ℕ∞) : WithTop ℕ∞); exact WithTop.coe_le_coe.mpr le_top)
+
+lemma smooth_hermitianPart {B : V n → RealBilinear n} {z : V n}
+    (hB : ContDiffAt ℝ ∞ B z) (u v : V n) :
+    ContDiffAt ℝ ∞ (fun y => hermitianPart (B y) u v) z := by
+  have hs (a b : V n) : ContDiffAt ℝ ∞ (fun y => B y a b) z :=
+    (hB.clm_apply contDiffAt_const).clm_apply contDiffAt_const
+  have hr := ((hs u v).add (hs (Complex.I • u) (Complex.I • v))).div_const 4
+  have hi := ((hs (Complex.I • u) v).sub (hs u (Complex.I • v))).div_const 4
+  simpa only [hermitianPart, Complex.mk_eq_add_mul_I, Function.comp_def, Complex.ofRealCLM_apply] using
+    (Complex.ofRealCLM.contDiff.contDiffAt.comp z hr).add
+      ((Complex.ofRealCLM.contDiff.contDiffAt.comp z hi).mul (contDiffAt_const (c := Complex.I)))
+
+lemma potentialMatrix_smooth {φ : V n → ℝ} {z : V n}
+    (hφ : ContDiffAt ℝ ∞ φ z) : ContDiffAt ℝ ∞ (potentialMatrix φ) z := by
+  apply contDiffAt_pi.mpr
+  intro i
+  apply contDiffAt_pi.mpr
+  intro j
+  simpa only [potentialMatrix, hermitianPartMatrix_apply] using
+    smooth_hermitianPart ((hφ.fderiv_right (m := ∞) (by simp)).fderiv_right (m := ∞) (by simp))
+      (Pi.single i 1) (Pi.single j 1)
+
+section Derivatives
+variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  [NormedAddCommGroup F] [NormedSpace ℝ F]
+
+lemma fderiv_eval_derivative {f : E → F} {z : E} (hf : ContDiffAt ℝ ∞ f z) (u v : E) :
+    fderiv ℝ (fun y => fderiv ℝ f y v) z u = fderiv ℝ (fderiv ℝ f) z u v := by
+  rw [fderiv_clm_apply ((hf.fderiv_right (m := ∞) (by simp)).differentiableAt (by simp))
+    (differentiableAt_const v)]
+  simp
+
+lemma second_derivative_symmetric {f : E → F} {z : E}
+    (hf : ContDiffAt ℝ ∞ f z) (u v : E) :
+    fderiv ℝ (fun y => fderiv ℝ f y v) z u =
+      fderiv ℝ (fun y => fderiv ℝ f y u) z v := by
+  rw [fderiv_eval_derivative hf, fderiv_eval_derivative hf]
+  exact hf.isSymmSndFDerivAt (by simp only [minSmoothness_of_isRCLikeNormedField]; change ((2 : ℕ∞) : WithTop ℕ∞) ≤ ((⊤ : ℕ∞) : WithTop ℕ∞); exact WithTop.coe_le_coe.mpr le_top) u v
+
+lemma cyclic_antisym_derivative {b : E → E → ℝ} {z : E}
+    (hb : ∀ v, ContDiffAt ℝ ∞ (b v) z) (u v w : E) :
+    fderiv ℝ (fun y => fderiv ℝ (b w) y v - fderiv ℝ (b v) y w) z u +
+      fderiv ℝ (fun y => fderiv ℝ (b u) y w - fderiv ℝ (b w) y u) z v +
+      fderiv ℝ (fun y => fderiv ℝ (b v) y u - fderiv ℝ (b u) y v) z w = 0 := by
+  have hd (a c : E) : DifferentiableAt ℝ (fun y => fderiv ℝ (b a) y c) z :=
+    (((hb a).fderiv_right (m := ∞) (by simp)).clm_apply contDiffAt_const).differentiableAt
+      (by simp)
+  simp only [fderiv_fun_sub (hd _ _) (hd _ _), _root_.sub_apply]
+  rw [second_derivative_symmetric (hb w) u v, second_derivative_symmetric (hb v) u w,
+    second_derivative_symmetric (hb u) v w]
+  ring
+end Derivatives
+
+ 
+def potentialPrimitive (φ : V n → ℝ) (v z : V n) : ℝ :=
+  -(fderiv ℝ φ z (Complex.I • v)) / 4
+
+def potentialForm (φ : V n → ℝ) (u v z : V n) : ℝ :=
+  (star u ⬝ᵥ (potentialMatrix φ z *ᵥ v)).im
+
+lemma potentialPrimitive_smooth {φ : V n → ℝ} {z : V n}
+    (hφ : ContDiffAt ℝ ∞ φ z) (v : V n) :
+    ContDiffAt ℝ ∞ (potentialPrimitive φ v) z :=
+  (((hφ.fderiv_right (by simp)).clm_apply contDiffAt_const).neg).div_const 4
+
+lemma potentialPrimitive_derivative {φ : V n → ℝ} {z : V n}
+    (hφ : ContDiffAt ℝ ∞ φ z) (u v : V n) :
+    fderiv ℝ (potentialPrimitive φ v) z u =
+      -(fderiv ℝ (fderiv ℝ φ) z u (Complex.I • v)) / 4 := by
+  have hd := ((hφ.fderiv_right (m := ∞) (by simp)).clm_apply
+    (contDiffAt_const (c := Complex.I • v))).differentiableAt (by simp)
+  unfold potentialPrimitive
+  simp only [div_eq_mul_inv]
+  have hm := fderiv_mul_const (hd.neg) (4 : ℝ)⁻¹
+  simp only [Pi.neg_apply] at hm
+  rw [hm]
+  rw [fderiv_neg]
+  simp only [_root_.smul_apply, _root_.neg_apply, smul_eq_mul,
+    fderiv_eval_derivative hφ]
+  ring
+
+lemma potentialPrimitive_antisym {φ : V n → ℝ} {z : V n}
+    (hφ : ContDiffAt ℝ ∞ φ z) (u v : V n) :
+    fderiv ℝ (potentialPrimitive φ v) z u -
+      fderiv ℝ (potentialPrimitive φ u) z v = potentialForm φ u v z := by
+  rw [potentialPrimitive_derivative hφ, potentialPrimitive_derivative hφ]
+  simp only [potentialForm, potentialMatrix, hermitianPartMatrix_pair, hermitianPart]
+  rw [hφ.isSymmSndFDerivAt (by simp only [minSmoothness_of_isRCLikeNormedField]; change ((2 : ℕ∞) : WithTop ℕ∞) ≤ ((⊤ : ℕ∞) : WithTop ℕ∞); exact WithTop.coe_le_coe.mpr le_top) v (Complex.I • u)]
+  ring
+
+lemma potentialMatrix_closed {φ : V n → ℝ} {U : Set (V n)}
+    (hU : IsOpen U) (hφ : ContDiffOn ℝ ∞ φ U) {z : V n} (hz : z ∈ U)
+    (u v w : V n) :
+    fderiv ℝ (potentialForm φ v w) z u + fderiv ℝ (potentialForm φ w u) z v +
+      fderiv ℝ (potentialForm φ u v) z w = 0 := by
+  have he (a b : V n) :
+      (fun y => fderiv ℝ (potentialPrimitive φ b) y a -
+        fderiv ℝ (potentialPrimitive φ a) y b) =ᶠ[nhds z] potentialForm φ a b := by
+    filter_upwards [hU.mem_nhds hz] with y hy
+    exact potentialPrimitive_antisym (hφ.contDiffAt (hU.mem_nhds hy)) a b
+  have hc := cyclic_antisym_derivative
+    (fun a => potentialPrimitive_smooth (hφ.contDiffAt (hU.mem_nhds hz)) a) u v w
+  rwa [(he _ _).fderiv_eq, (he _ _).fderiv_eq, (he _ _).fderiv_eq] at hc
+
+section ChainRule
+variable {E F G : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  [NormedAddCommGroup F] [NormedSpace ℝ F] [NormedAddCommGroup G] [NormedSpace ℝ G]
+
+omit [NormedSpace ℝ E] [NormedSpace ℝ F] in
+lemma eventually_differentiableAt {𝕜 : Type*} [NontriviallyNormedField 𝕜]
+    [NormedSpace 𝕜 E] [NormedSpace 𝕜 F] {f : E → F} {z : E}
+    (hf : ContDiffAt 𝕜 ∞ f z) :
+    ∀ᶠ y in nhds z, DifferentiableAt 𝕜 f y := by
+  have h1 : ContDiffAt 𝕜 1 f z := (contDiffAt_infty.mp hf) 1
+  exact (h1.eventually (by simp)).mono fun y hy => hy.differentiableAt (by simp)
+
+lemma second_derivative_comp {f : E → F} {g : F → G} {z : E}
+    (hf : ContDiffAt ℝ ∞ f z) (hg : ContDiffAt ℝ ∞ g (f z)) (u v : E) :
+    fderiv ℝ (fderiv ℝ (g ∘ f)) z u v =
+      fderiv ℝ (fderiv ℝ g) (f z) (fderiv ℝ f z u) (fderiv ℝ f z v) +
+        fderiv ℝ g (f z) (fderiv ℝ (fderiv ℝ f) z u v) := by
+  have hfd := hf.differentiableAt (by simp)
+  have hgd := hg.differentiableAt (by simp)
+  have hDf := (hf.fderiv_right (m := ∞) (by simp)).differentiableAt (by simp)
+  have hDg := (hg.fderiv_right (m := ∞) (by simp)).differentiableAt (by simp)
+  have he : fderiv ℝ (g ∘ f) =ᶠ[nhds z] fun y =>
+      (fderiv ℝ g (f y)).comp (fderiv ℝ f y) := by
+    filter_upwards [eventually_differentiableAt hf,
+      hf.continuousAt.tendsto.eventually (eventually_differentiableAt hg)] with y hy hy'
+    exact fderiv_comp y hy' hy
+  rw [he.fderiv_eq]
+  have hm := fderiv_clm_comp (hDg.comp z hfd) hDf
+  simp only [Function.comp_def] at hm
+  rw [hm]
+  have hc := fderiv_comp z hDg hfd
+  simp only [Function.comp_def] at hc
+  rw [hc]
+  simp only [_root_.add_apply, ContinuousLinearMap.comp_apply,
+    ContinuousLinearMap.compL_apply, ContinuousLinearMap.flip_apply]
+  exact add_comm _ _
+end ChainRule
+
+lemma fderiv_complex_smul {m : ℕ} {f : V n → V m} {z : V n}
+    (hf : DifferentiableAt ℂ f z) (a : ℂ) (v : V n) :
+    fderiv ℝ f z (a • v) = a • fderiv ℝ f z v := by
+  rw [hf.fderiv_restrictScalars (𝕜 := ℝ)]
+  exact map_smul (fderiv ℂ f z) a v
+
+lemma second_derivative_complex_smul_right {m : ℕ} {f : V n → V m} {z : V n}
+    (hf : ContDiffAt ℂ ∞ f z) (a : ℂ) (u v : V n) :
+    fderiv ℝ (fderiv ℝ f) z u (a • v) =
+      a • fderiv ℝ (fderiv ℝ f) z u v := by
+  have hr := hf.restrict_scalars ℝ
+  have hd := ((hr.fderiv_right (m := ∞) (by simp)).clm_apply
+    (contDiffAt_const (c := v))).differentiableAt (by simp)
+  have he : (fun y => fderiv ℝ f y (a • v)) =ᶠ[nhds z]
+      (fun y => a • fderiv ℝ f y v) := by
+    filter_upwards [eventually_differentiableAt hf] with y hy
+    exact fderiv_complex_smul hy a v
+  have he' := congrArg (fun l : V n →L[ℝ] V m => l u) he.fderiv_eq
+  rw [fderiv_eval_derivative hr] at he'
+  have hs : fderiv ℝ (fun y => a • fderiv ℝ f y v) z =
+      a • fderiv ℝ (fun y => fderiv ℝ f y v) z := fderiv_const_smul hd a
+  rw [hs] at he'
+  simpa only [_root_.smul_apply, fderiv_eval_derivative hr] using he'
+
+lemma second_derivative_complex_smul_left {m : ℕ} {f : V n → V m} {z : V n}
+    (hf : ContDiffAt ℂ ∞ f z) (a : ℂ) (u v : V n) :
+    fderiv ℝ (fderiv ℝ f) z (a • u) v =
+      a • fderiv ℝ (fderiv ℝ f) z u v := by
+  have hs := (hf.restrict_scalars ℝ).isSymmSndFDerivAt
+    (by simp only [minSmoothness_of_isRCLikeNormedField]
+        change ((2 : ℕ∞) : WithTop ℕ∞) ≤ ((⊤ : ℕ∞) : WithTop ℕ∞)
+        exact WithTop.coe_le_coe.mpr le_top)
+  rw [hs (a • u) v, second_derivative_complex_smul_right hf, hs v u]
+
+ 
+
+lemma hermitianPart_comp {m : ℕ} {f : V n → V m} {φ : V m → ℝ} {z : V n}
+    (hf : ContDiffAt ℂ ∞ f z) (hφ : ContDiffAt ℝ ∞ φ (f z)) (u v : V n) :
+    hermitianPart (fderiv ℝ (fderiv ℝ (φ ∘ f)) z) u v =
+      hermitianPart (fderiv ℝ (fderiv ℝ φ) (f z))
+        (fderiv ℝ f z u) (fderiv ℝ f z v) := by
+  have hr := hf.restrict_scalars ℝ
+  have hd := hf.differentiableAt (by simp)
+  apply Complex.ext <;>
+    simp only [hermitianPart, second_derivative_comp hr hφ,
+      fderiv_complex_smul hd, second_derivative_complex_smul_left hf,
+      second_derivative_complex_smul_right hf, smul_smul, Complex.I_mul_I,
+      neg_smul, one_smul, map_neg] <;> ring
+
+lemma matrix_pair_pullback {m : ℕ} (H : Matrix (Fin m) (Fin m) ℂ)
+    (D : Matrix (Fin m) (Fin n) ℂ) (u v : V n) :
+    star u ⬝ᵥ ((Dᴴ * H * D) *ᵥ v) = star (D *ᵥ u) ⬝ᵥ (H *ᵥ (D *ᵥ v)) := by
+  rw [← mulVec_mulVec, ← mulVec_mulVec, dotProduct_mulVec, ← star_mulVec]
+
+lemma matrix_eq_of_pair (H K : Matrix (Fin n) (Fin n) ℂ)
+    (h : ∀ u v : V n, star u ⬝ᵥ (H *ᵥ v) = star u ⬝ᵥ (K *ᵥ v)) : H = K := by
+  ext i j
+  simpa using h (Pi.single i 1) (Pi.single j 1)
+
+lemma potentialMatrix_congr {φ ψ : V n → ℝ} {z : V n}
+    (he : φ =ᶠ[nhds z] ψ) : potentialMatrix φ z = potentialMatrix ψ z := by
+  unfold potentialMatrix
+  rw [(he.fderiv (𝕜 := ℝ)).fderiv_eq]
+
+lemma potentialMatrix_comp {m : ℕ} {f : V n → V m} {φ : V m → ℝ} {z : V n}
+    (hf : ContDiffAt ℂ ∞ f z) (hφ : ContDiffAt ℝ ∞ φ (f z)) :
+    potentialMatrix (φ ∘ f) z =
+      (LinearMap.toMatrix' (fderiv ℂ f z).toLinearMap)ᴴ *
+        potentialMatrix φ (f z) * LinearMap.toMatrix' (fderiv ℂ f z).toLinearMap := by
+  apply matrix_eq_of_pair
+  intro u v
+  rw [matrix_pair_pullback]
+  simp only [potentialMatrix, hermitianPartMatrix_pair, LinearMap.toMatrix'_mulVec]
+  have he := hermitianPart_comp hf hφ u v
+  rw [(hf.differentiableAt (by simp)).fderiv_restrictScalars (𝕜 := ℝ)] at he
+  exact he
+
+end PotentialKaehler
+
+namespace Anticanonical.SourceSmooth
+open Matrix Filter
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+
+ 
+structure SmoothRealFunction (A : ComplexAtlas d X) where
+  value : X → ℝ
+  smooth : ∀ i, ContDiffOn ℝ ∞ (value ∘ (A.chart i).symm) (A.chart i).target
+
+namespace SmoothRealFunction
+
+def localExpression (φ : SmoothRealFunction A) (i : Fin A.count) : Coordinates d → ℝ :=
+  φ.value ∘ (A.chart i).symm
+
+ 
+def hessian (φ : SmoothRealFunction A) (i : Fin A.count) (z : Coordinates d) :
+    Matrix (Fin d) (Fin d) ℂ := PotentialKaehler.potentialMatrix (φ.localExpression i) z
+
+lemma hessian_smooth (φ : SmoothRealFunction A) (i : Fin A.count) :
+    ContDiffOn ℝ ∞ (φ.hessian i) (A.chart i).target := by
+  intro z hz
+  exact (PotentialKaehler.potentialMatrix_smooth
+    ((φ.smooth i).contDiffAt ((A.chart i).open_target.mem_nhds hz))).contDiffWithinAt
+
+lemma hessian_hermitian (φ : SmoothRealFunction A) (i : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart i).target) : (φ.hessian i z).IsHermitian :=
+  PotentialKaehler.potentialMatrix_hermitian
+    ((φ.smooth i).contDiffAt ((A.chart i).open_target.mem_nhds hz))
+
+lemma hessian_compatibility (φ : SmoothRealFunction A) (i j : Fin A.count) {x : X}
+    (hi : x ∈ (A.chart i).source) (hj : x ∈ (A.chart j).source) :
+    φ.hessian i (A.chart i x) =
+      (A.derivativeMatrix i j x)ᴴ * φ.hessian j (A.chart j x) * A.derivativeMatrix i j x := by
+  have hz := A.coordinate_mem_overlap i j hi hj
+  have ho := A.isOpen_coordinateOverlap i j
+  have hf : ContDiffAt ℂ ∞ (A.transition i j) (A.chart i x) :=
+    (ProjectiveKaehler.analyticOnNhd_pi_of_differentiableOn ho (A.holomorphic i j) _ hz).contDiffAt
+  have hφ : ContDiffAt ℝ ∞ (φ.localExpression j) (A.transition i j (A.chart i x)) := by
+    rw [A.transition_at i j hi]
+    exact (φ.smooth j).contDiffAt ((A.chart j).open_target.mem_nhds ((A.chart j).mapsTo hj))
+  have he : φ.localExpression i =ᶠ[nhds (A.chart i x)]
+      φ.localExpression j ∘ A.transition i j := by
+    filter_upwards [ho.mem_nhds hz] with y hy
+    dsimp only [localExpression, ComplexAtlas.transition, Function.comp_apply]
+    rw [(A.chart j).left_inv hy.2]
+  unfold hessian
+  rw [PotentialKaehler.potentialMatrix_congr he,
+    PotentialKaehler.potentialMatrix_comp hf hφ, A.transition_at i j hi]
+  rfl
+
+lemma hessian_closed (φ : SmoothRealFunction A) (i : Fin A.count) {z : Coordinates d}
+    (hz : z ∈ (A.chart i).target) (u v w : Coordinates d) :
+    fderiv ℝ (fun y => ComplexAtlas.fundamentalForm (φ.hessian i y) v w) z u +
+      fderiv ℝ (fun y => ComplexAtlas.fundamentalForm (φ.hessian i y) w u) z v +
+      fderiv ℝ (fun y => ComplexAtlas.fundamentalForm (φ.hessian i y) u v) z w = 0 :=
+  PotentialKaehler.potentialMatrix_closed (A.chart i).open_target (φ.smooth i) hz u v w
+
+end SmoothRealFunction
+
+lemma fundamentalForm_add (H K : Matrix (Fin d) (Fin d) ℂ) (u v : Coordinates d) :
+    ComplexAtlas.fundamentalForm (H + K) u v =
+      ComplexAtlas.fundamentalForm H u v + ComplexAtlas.fundamentalForm K u v := by
+  simp [ComplexAtlas.fundamentalForm, add_mulVec, dotProduct_add]
+
+lemma fundamentalForm_smooth {g : Coordinates d → Matrix (Fin d) (Fin d) ℂ}
+    {z : Coordinates d} (hg : ContDiffAt ℝ ∞ g z) (u v : Coordinates d) :
+    ContDiffAt ℝ ∞ (fun y => ComplexAtlas.fundamentalForm (g y) u v) z := by
+  apply Complex.imCLM.contDiff.contDiffAt.comp z
+  apply ContDiffAt.sum
+  intro i _
+  apply contDiffAt_const.mul
+  change ContDiffAt ℝ ∞ (fun y => ∑ j, g y i j * v j) z
+  apply ContDiffAt.sum
+  intro j _
+  exact (contDiffAt_pi.mp (contDiffAt_pi.mp hg i) j).mul contDiffAt_const
+
+namespace KaehlerMetric
+
+ 
+
+def PositivePotential (g : KaehlerMetric A) (φ : SmoothRealFunction A) : Prop :=
+  ∀ i z, z ∈ (A.chart i).target → (g.matrix i z + φ.hessian i z).PosDef
+
+ 
+
+def deform (g : KaehlerMetric A) (φ : SmoothRealFunction A) (hp : g.PositivePotential φ) :
+    KaehlerMetric A where
+  matrix i z := g.matrix i z + φ.hessian i z
+  smooth i := (g.smooth i).add (φ.hessian_smooth i)
+  positive := hp
+  compatibility i j x hi hj := by
+    rw [g.compatibility i j x hi hj, φ.hessian_compatibility i j hi hj]
+    simp only [mul_add, add_mul]
+  closed i z hz u v w := by
+    have hg := (g.smooth i).contDiffAt ((A.chart i).open_target.mem_nhds hz)
+    have hφ := (φ.hessian_smooth i).contDiffAt ((A.chart i).open_target.mem_nhds hz)
+    simp only [fundamentalForm_add]
+    simp only [fderiv_fun_add
+      ((fundamentalForm_smooth hg _ _).differentiableAt (by simp))
+      ((fundamentalForm_smooth hφ _ _).differentiableAt (by simp)), _root_.add_apply]
+    have h1 := g.closed i z hz u v w
+    have h2 := φ.hessian_closed i hz u v w
+    linarith
+
+end KaehlerMetric
+end Anticanonical.SourceSmooth
+
+ 
+
+open Matrix Polynomial
+open scoped Matrix.Norms.Elementwise
+noncomputable section
+namespace MongeAmpere
+variable {n : Type*} [Fintype n] [DecidableEq n]
+
+lemma hasDerivAt_det_one_add_smul (M : Matrix n n ℂ) :
+    HasDerivAt (fun t : ℂ => (1 + t • M).det) M.trace 0 := by
+  have hp := (Matrix.det (1 + (Polynomial.X : ℂ[X]) • M.map Polynomial.C)).hasDerivAt (0 : ℂ)
+  rw [Matrix.derivative_det_one_add_X_smul] at hp
+  simpa [eval_det, ← Matrix.smul_eq_mul_diagonal] using hp
+
+lemma hasDerivAt_det_add_smul (H K : Matrix n n ℂ) (hH : IsUnit H.det) :
+    HasDerivAt (fun t : ℂ => (H + t • K).det)
+      (H.det * (H⁻¹ * K).trace) 0 := by
+  have hd := (hasDerivAt_det_one_add_smul (H⁻¹ * K)).const_mul H.det
+  apply hd.congr_of_eventuallyEq
+  filter_upwards [] with t
+  rw [← Matrix.det_mul]
+  congr 1
+  simp [Matrix.mul_add, ← Matrix.mul_assoc,
+    Matrix.mul_nonsing_inv _ hH]
+
+open scoped ComplexOrder
+
+ 
+
+lemma hasDerivAt_logdet_add_smul (H K : Matrix n n ℂ) (hH : H.PosDef) :
+    HasDerivAt (fun t : ℝ => Real.log ((H + (t : ℂ) • K).det.re))
+      (H⁻¹ * K).trace.re 0 := by
+  have hd := (hasDerivAt_det_add_smul H K (isUnit_iff_ne_zero.mpr (ne_of_gt hH.det_pos))).real_of_complex
+  have hp := Complex.pos_iff.mp hH.det_pos
+  have hne : (H + ((0 : ℝ) : ℂ) • K).det.re ≠ 0 := by
+    simpa using ne_of_gt hp.1
+  have hl := hd.log hne
+  simpa [Complex.mul_re, hp.2.symm, hp.1.ne', mul_div_cancel_left₀] using hl
+
+ 
+
+lemma linearized_symbol_pos (H : Matrix n n ℂ) (hH : H.PosDef)
+    (v : n → ℂ) (hv : v ≠ 0) :
+    0 < (H⁻¹ * Matrix.vecMulVec v (star v)).trace.re := by
+  rw [Matrix.mul_vecMulVec, Matrix.trace_vecMulVec, dotProduct_comm]
+  exact hH.inv.re_dotProduct_pos hv
+
+end MongeAmpere
+
+namespace Anticanonical.SourceSmooth
+open Matrix
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+
+namespace SmoothRealFunction
+
+def constant (c : ℝ) : SmoothRealFunction A where
+  value _ := c
+  smooth _ := contDiffOn_const
+
+def realSMul (c : ℝ) (φ : SmoothRealFunction A) : SmoothRealFunction A where
+  value x := c * φ.value x
+  smooth i := contDiffOn_const.mul (φ.smooth i)
+
+lemma hessian_constant (c : ℝ) (i : Fin A.count) (z : Coordinates d) :
+    (constant (A := A) c).hessian i z = 0 := by
+  ext a b
+  apply Complex.ext <;> simp [hessian, localExpression, constant, Function.comp_def, PotentialKaehler.potentialMatrix,
+    PotentialKaehler.hermitianPartMatrix_apply, PotentialKaehler.hermitianPart]
+
+lemma hessian_realSMul (c : ℝ) (φ : SmoothRealFunction A)
+    (i : Fin A.count) (z : Coordinates d) :
+    (φ.realSMul c).hessian i z = (c : ℂ) • φ.hessian i z := by
+  unfold hessian PotentialKaehler.potentialMatrix
+  have he : (φ.realSMul c).localExpression i = c • φ.localExpression i := rfl
+  rw [he, fderiv_const_smul_field (𝕜 := ℝ) (f := φ.localExpression i) c]
+  rw [fderiv_const_smul_field (𝕜 := ℝ) (f := fderiv ℝ (φ.localExpression i)) c]
+  ext a b
+  simp only [PotentialKaehler.hermitianPartMatrix_apply, Matrix.smul_apply]
+  apply Complex.ext <;>
+    simp [PotentialKaehler.hermitianPart, Pi.smul_apply, smul_eq_mul,
+      Complex.mul_re, Complex.mul_im] <;> ring
+
+end SmoothRealFunction
+
+namespace KaehlerMetric
+
+ 
+
+def linearizedMongeAmpere (g : KaehlerMetric A) (ψ : SmoothRealFunction A)
+    (i : Fin A.count) (z : Coordinates d) : ℝ :=
+  ((g.matrix i z)⁻¹ * ψ.hessian i z).trace.re
+
+ 
+theorem mongeAmpere_directional_derivative (g : KaehlerMetric A)
+    (ψ : SmoothRealFunction A) (i : Fin A.count) {z : Coordinates d}
+    (hz : z ∈ (A.chart i).target) :
+    HasDerivAt (fun t : ℝ =>
+      Real.log ((g.matrix i z + (ψ.realSMul t).hessian i z).det.re) -
+        Real.log (g.volumeCoefficient i z)) (g.linearizedMongeAmpere ψ i z) 0 := by
+  simp only [SmoothRealFunction.hessian_realSMul]
+  exact (MongeAmpere.hasDerivAt_logdet_add_smul _ _ (g.positive i z hz)).sub_const _
+
+lemma linearizedMongeAmpere_constant (g : KaehlerMetric A) (c : ℝ) :
+    g.linearizedMongeAmpere (SmoothRealFunction.constant c) = 0 := by
+  funext i z
+  simp [linearizedMongeAmpere, SmoothRealFunction.hessian_constant]
+
+ 
+
+theorem linearizedMongeAmpere_not_injective [Nonempty X] (g : KaehlerMetric A) :
+    ¬ Function.Injective g.linearizedMongeAmpere := by
+  intro h
+  have hc : SmoothRealFunction.constant (A := A) 0 = SmoothRealFunction.constant 1 :=
+    h (by rw [linearizedMongeAmpere_constant, linearizedMongeAmpere_constant])
+  let x : X := Classical.choice inferInstance
+  have he := congrArg (fun φ : SmoothRealFunction A => φ.value x) hc
+  simp [SmoothRealFunction.constant] at he
+
+end KaehlerMetric
+end Anticanonical.SourceSmooth
+
+namespace EllipticKernel
+open Filter Set
+open scoped ContDiff
+
+lemma second_deriv_nonpos_at_localMax {f : ℝ → ℝ} {x : ℝ}
+    (hmax : IsLocalMax f x) (hc : ContinuousAt f x) : deriv (deriv f) x ≤ 0 := by
+  by_contra h
+  have hpos : 0 < deriv (deriv f) x := lt_of_not_ge h
+  have hmin := isLocalMin_of_deriv_deriv_pos hpos hmax.deriv_eq_zero hc
+  have he : f =ᶠ[nhds x] fun _ => f x := by
+    filter_upwards [hmax, hmin] with y hy hy'
+    exact le_antisymm hy hy'
+  have hd : deriv f =ᶠ[nhds x] fun _ => 0 := by
+    simpa using he.deriv
+  have hdd : deriv (deriv f) x = 0 := by
+    simpa using hd.deriv_eq
+  exact (ne_of_gt hpos) hdd
+
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+
+lemma second_fderiv_nonpos_at_localMax {f : E → ℝ} {x : E}
+    (hf : ContDiffAt ℝ ∞ f x) (hmax : IsLocalMax f x) (v : E) :
+    fderiv ℝ (fderiv ℝ f) x v v ≤ 0 := by
+  let r : ℝ → E := fun t => x + t • v
+  have hr : ContDiff ℝ ∞ r := contDiff_const.add (contDiff_id.smul contDiff_const)
+  have hr0 : r 0 = x := by simp [r]
+  have hfr : ContDiffAt ℝ ∞ (f ∘ r) 0 := (show ContDiffAt ℝ ∞ f (r 0) by simpa only [hr0] using hf).comp 0 hr.contDiffAt
+  have hmr : IsLocalMax (f ∘ r) 0 :=
+    (hr0.symm ▸ hmax).comp_continuous hr.continuous.continuousAt
+  have hle := second_deriv_nonpos_at_localMax hmr hfr.continuousAt
+  have hd2 : deriv (deriv (f ∘ r)) 0 = fderiv ℝ (fderiv ℝ (f ∘ r)) 0 1 1 :=
+    PotentialKaehler.fderiv_eval_derivative hfr 1 1
+  rw [hd2, PotentialKaehler.second_derivative_comp hr.contDiffAt
+    (by simpa only [hr0] using hf) 1 1] at hle
+  have hDr : fderiv ℝ r = fun _ => ContinuousLinearMap.smulRight (ContinuousLinearMap.id ℝ ℝ) v := by
+    funext t
+    simpa [r] using
+      ((hasFDerivAt_id t).smul_const v).const_add x |>.fderiv
+  rw [hDr] at hle
+  simpa [hr0] using hle
+
+end EllipticKernel
+
+namespace EllipticKernel
+open Matrix
+open scoped MatrixOrder ComplexOrder Matrix.Norms.Elementwise ContDiff
+
+lemma trace_mul_nonneg {n : Type*} [Fintype n] [DecidableEq n]
+    {A B : Matrix n n ℂ} (hA : A.PosSemidef) (hB : B.PosSemidef) :
+    0 ≤ (A * B).trace.re := by
+  let S := CFC.sqrt A
+  have hS : S.PosSemidef := (CFC.sqrt_nonneg A).posSemidef
+  have hsquare : S * S = A := by simpa [S, pow_two] using CFC.sq_sqrt A hA.nonneg
+  have htr := Complex.nonneg_iff.mp (hB.mul_mul_conjTranspose_same S).trace_nonneg
+  have heq : (S * B * Sᴴ).trace = (A * B).trace := by
+    rw [hS.isHermitian.eq, Matrix.trace_mul_cycle, hsquare]
+  rw [heq] at htr
+  exact htr.1
+
+lemma negative_potentialMatrix_posSemidef_at_localMax {n : ℕ}
+    {f : PotentialKaehler.V n → ℝ} {x : PotentialKaehler.V n}
+    (hf : ContDiffAt ℝ ∞ f x) (hmax : IsLocalMax f x) :
+    (-PotentialKaehler.potentialMatrix f x).PosSemidef := by
+  refine Matrix.posSemidef_iff_dotProduct_mulVec.mpr
+    ⟨(PotentialKaehler.potentialMatrix_hermitian hf).neg, ?_⟩
+  intro v
+  rw [neg_mulVec, dotProduct_neg]
+  rw [PotentialKaehler.potentialMatrix,
+    PotentialKaehler.hermitianPartMatrix_pair]
+  apply Complex.nonneg_iff.mpr
+  constructor
+  · simp only [Complex.neg_re, PotentialKaehler.hermitianPart]
+    have h1 := second_fderiv_nonpos_at_localMax hf hmax v
+    have h2 := second_fderiv_nonpos_at_localMax hf hmax (Complex.I • v)
+    linarith
+  · simp only [Complex.neg_im, PotentialKaehler.hermitianPart]
+    have hs := hf.isSymmSndFDerivAt (by
+      simp only [minSmoothness_of_isRCLikeNormedField]
+      change ((2 : ℕ∞) : WithTop ℕ∞) ≤ ((⊤ : ℕ∞) : WithTop ℕ∞)
+      exact WithTop.coe_le_coe.mpr le_top) (Complex.I • v) v
+    rw [hs]
+    ring
+
+lemma trace_potentialMatrix_nonpos_at_localMax {n : ℕ}
+    {f : PotentialKaehler.V n → ℝ} {x : PotentialKaehler.V n}
+    (hf : ContDiffAt ℝ ∞ f x) (hmax : IsLocalMax f x)
+    (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef) :
+    (H⁻¹ * PotentialKaehler.potentialMatrix f x).trace.re ≤ 0 := by
+  have ht := trace_mul_nonneg hH.inv.posSemidef
+    (negative_potentialMatrix_posSemidef_at_localMax hf hmax)
+  simpa using ht
+
+end EllipticKernel
+
+ 
+
+noncomputable section
+open Filter Set
+open scoped ContDiff
+namespace EllipticKernel
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+abbrev Bilin (E : Type*) [NormedAddCommGroup E] [NormedSpace ℝ E] :=
+  E →L[ℝ] E →L[ℝ] ℝ
+abbrev SecondOrder (E : Type*) [NormedAddCommGroup E] [NormedSpace ℝ E] :=
+  Bilin E →L[ℝ] ℝ
+
+local instance barrierBilinNormedAddCommGroup : NormedAddCommGroup (Bilin E) := inferInstance
+local instance barrierBilinNormedSpace : NormedSpace ℝ (Bilin E) := inferInstance
+
+def realInner : Bilin E := innerSL ℝ (E := E)
+
+lemma realInner_apply (u v : E) : realInner u v = inner ℝ u v := by
+  exact innerSL_apply_apply ℝ u v
+
+def rankOne (v : E) : Bilin E := (realInner v).smulRight (realInner v)
+def barrier (α : ℝ) (y : E) (z : E) : ℝ := Real.exp (-α * ‖z-y‖^2)
+
+lemma barrier_smooth (α : ℝ) (y : E) : ContDiff ℝ ∞ (barrier α y) :=
+  (contDiff_const.mul ((contDiff_id.sub contDiff_const).norm_sq ℝ)).exp
+
+lemma barrier_hasFDerivAt (α : ℝ) (y z : E) :
+    HasFDerivAt (barrier α y)
+      ((Real.exp (-α * ‖z-y‖^2) * (-2*α)) • realInner (z-y)) z := by
+  have h := ((((hasFDerivAt_id z).sub_const y).norm_sq).const_mul (-α)).exp
+  change HasFDerivAt (fun x => Real.exp (-α * ‖x-y‖^2)) _ z
+  convert! h using 1
+  ext v
+  simp only [_root_.smul_apply, realInner_apply, ContinuousLinearMap.comp_apply,
+    ContinuousLinearMap.id_apply, id_eq, innerSL_apply_apply, smul_eq_mul]
+  ring
+
+lemma barrier_fderiv (α : ℝ) (y : E) :
+    fderiv ℝ (barrier α y) = fun z =>
+      (Real.exp (-α * ‖z-y‖^2) * (-2*α)) • realInner (z-y) := by
+  funext z
+  exact (barrier_hasFDerivAt α y z).fderiv
+
+lemma barrier_hessian (α : ℝ) (y z : E) :
+    fderiv ℝ (fderiv ℝ (barrier α y)) z =
+      Real.exp (-α * ‖z-y‖^2) •
+        ((4*α^2) • rankOne (z-y) - (2*α) • (realInner (E := E))) := by
+  rw [barrier_fderiv]
+  have h₁ := (barrier_hasFDerivAt α y z).mul_const (-2*α)
+  have h₂ := (realInner (E := E)).hasFDerivAt.comp z ((hasFDerivAt_id z).sub_const y)
+  have hh := h₁.smul h₂
+  have he := hh.fderiv
+  simp only [Function.comp_def, id_eq] at he
+  change fderiv ℝ (fun z => (barrier α y z * (-2*α)) • realInner (z-y)) z = _
+  convert! he using 1
+  ext u v
+  simp [rankOne, barrier, realInner, ContinuousLinearMap.smulRight_apply]
+  ring
+end EllipticKernel
+
+namespace EllipticKernel
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+
+local instance rankOneBilinNormedAddCommGroup : NormedAddCommGroup (Bilin E) := inferInstance
+local instance rankOneBilinNormedSpace : NormedSpace ℝ (Bilin E) := inferInstance
+
+lemma continuous_rankOne : Continuous (rankOne (E := E)) := by
+  unfold rankOne
+  exact ((ContinuousLinearMap.smulRightL ℝ E (E →L[ℝ] ℝ)).continuous.comp
+    (realInner (E := E)).continuous).clm_apply (realInner (E := E)).continuous
+
+ 
+
+lemma exists_positive_barrier {s : Set E} (hs : IsCompact s)
+    (ℒ : E → SecondOrder E) (hℒ : ContinuousOn ℒ s) (y : E)
+    (hpos : ∀ z ∈ s, 0 < ℒ z (rankOne (z-y))) :
+    ∃ α : ℝ, 0 < α ∧ ∀ z ∈ s,
+      0 < ℒ z (fderiv ℝ (fderiv ℝ (barrier α y)) z) := by
+  by_cases hne : s.Nonempty
+  · let a : E → ℝ := fun z => ℒ z (rankOne (z-y))
+    let b : E → ℝ := fun z => ℒ z (realInner (E := E))
+    have ha : ContinuousOn a s :=
+      hℒ.clm_apply (continuous_rankOne.comp (continuous_id.sub continuous_const)).continuousOn
+    have hb : ContinuousOn b s := hℒ.clm_apply continuousOn_const
+    have hq : ContinuousOn (fun z => b z / (2*a z)) s :=
+      hb.div (continuousOn_const.mul ha) (fun z hz => mul_ne_zero (by norm_num) (ne_of_gt (hpos z hz)))
+    obtain ⟨z₀, hz₀, hmax⟩ := hs.exists_isMaxOn hne hq
+    let α := max (b z₀ / (2*a z₀)) 0 + 1
+    have hα : 0 < α := by dsimp [α]; positivity
+    refine ⟨α, hα, ?_⟩
+    intro z hz
+    have haq : b z / (2*a z) < α := lt_of_le_of_lt (hmax hz)
+      (by dsimp [α]; linarith [le_max_left (b z₀ / (2*a z₀)) 0])
+    have haz : 0 < a z := hpos z hz
+    have hba : b z < 2*α*a z := by
+      have := (div_lt_iff₀ (by positivity : 0 < 2*a z)).mp haq
+      nlinarith
+    rw [barrier_hessian]
+    simp only [map_smul, map_sub, smul_eq_mul]
+    change 0 < Real.exp (-α*‖z-y‖^2) * (4*α^2 * a z - 2*α * b z)
+    apply mul_pos (Real.exp_pos _)
+    nlinarith [mul_pos hα (sub_pos.mpr hba)]
+  · exact ⟨1, by norm_num, fun z hz => False.elim (hne ⟨z, hz⟩)⟩
+
+end EllipticKernel
+
+namespace EllipticKernel
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+
+lemma hessian_add {f g : E → ℝ} {z : E}
+    (hf : ContDiffAt ℝ ∞ f z) (hg : ContDiffAt ℝ ∞ g z) :
+    fderiv ℝ (fderiv ℝ (f+g)) z =
+      fderiv ℝ (fderiv ℝ f) z + fderiv ℝ (fderiv ℝ g) z := by
+  have ef : ∀ᶠ x in nhds z, DifferentiableAt ℝ f x :=
+    (((contDiffAt_infty.mp hf) 1).eventually (by simp)).mono
+      (fun _ h => h.differentiableAt (by simp))
+  have eg : ∀ᶠ x in nhds z, DifferentiableAt ℝ g x :=
+    (((contDiffAt_infty.mp hg) 1).eventually (by simp)).mono
+      (fun _ h => h.differentiableAt (by simp))
+  have he : fderiv ℝ (f+g) =ᶠ[nhds z] (fderiv ℝ f + fderiv ℝ g) := by
+    filter_upwards [ef, eg] with x hx hx'
+    exact fderiv_add hx hx'
+  rw [he.fderiv_eq]
+  exact fderiv_add ((hf.fderiv_right (m := ∞) (by simp)).differentiableAt (by simp))
+    ((hg.fderiv_right (m := ∞) (by simp)).differentiableAt (by simp))
+
+lemma hessian_add_smul_sub_const {f g : E → ℝ} {z : E}
+    (hf : ContDiffAt ℝ ∞ f z) (hg : ContDiffAt ℝ ∞ g z) (ε c : ℝ) :
+    fderiv ℝ (fderiv ℝ (fun z => f z + ε * (g z-c))) z =
+      fderiv ℝ (fderiv ℝ f) z + ε • fderiv ℝ (fderiv ℝ g) z := by
+  change fderiv ℝ (fderiv ℝ (f + ε • (fun z => g z-c))) z = _
+  rw [hessian_add hf (show ContDiffAt ℝ ∞ (ε • (fun z => g z-c)) z from
+    (hg.sub contDiffAt_const).const_smul ε)]
+  rw [fderiv_const_smul_field (𝕜 := ℝ) (f := fun z => g z-c) ε]
+  rw [fderiv_const_smul_field (𝕜 := ℝ) (f := fderiv ℝ (fun z => g z-c)) ε]
+  have he : fderiv ℝ (fun z => g z-c) = fderiv ℝ g := by
+    funext x; exact fderiv_sub_const c
+  rw [he]
+  rfl
+
+end EllipticKernel
+
+ 
+noncomputable section
+open Set Filter Metric
+namespace EllipticKernel
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [ProperSpace E]
+
+omit [NormedSpace ℝ E] in
+lemma annulus_le_of_no_localMax {u : E → ℝ} {y : E} {r M : ℝ}
+    (hu : ContinuousOn u (closedBall y r \ ball y (r/2)))
+    (hinner : ∀ z, dist z y = r/2 → u z ≤ M)
+    (houter : ∀ z, dist z y = r → u z ≤ M)
+    (hmax : ∀ z, r/2 < dist z y → dist z y < r → ¬IsLocalMax u z) :
+    ∀ z ∈ closedBall y r \ ball y (r/2), u z ≤ M := by
+  intro z hz
+  have hc : IsCompact (closedBall y r \ ball y (r/2)) :=
+    (isCompact_closedBall y r).diff isOpen_ball
+  obtain ⟨p, hp, hpm⟩ := hc.exists_isMaxOn ⟨z, hz⟩ hu
+  have hp₁ : dist p y ≤ r := hp.1
+  have hp₂ : r/2 ≤ dist p y := le_of_not_gt hp.2
+  have hbound : u p ≤ M := by
+    by_cases hlow : dist p y = r/2
+    · exact hinner p hlow
+    by_cases hhigh : dist p y = r
+    · exact houter p hhigh
+    have hlow' : r/2 < dist p y := lt_of_le_of_ne hp₂ (Ne.symm hlow)
+    have hhigh' : dist p y < r := lt_of_le_of_ne hp₁ hhigh
+    have hn : closedBall y r \ ball y (r/2) ∈ nhds p := by
+      have h₁ : {q : E | dist q y < r} ∈ nhds p :=
+        (isOpen_lt (continuous_id.dist continuous_const) continuous_const).mem_nhds hhigh'
+      have h₂ : {q : E | r/2 < dist q y} ∈ nhds p :=
+        (isOpen_lt continuous_const (continuous_id.dist continuous_const)).mem_nhds hlow'
+      filter_upwards [h₁, h₂] with q hq₁ hq₂
+      exact ⟨le_of_lt hq₁, not_lt_of_gt hq₂⟩
+    exact False.elim (hmax p hlow' hhigh' (hpm.isLocalMax hn))
+  exact le_trans (hpm hz) hbound
+
+end EllipticKernel
+
+namespace EllipticKernel
+open Set Metric Filter
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [ProperSpace E]
+
+ 
+
+omit [NormedSpace ℝ E] in
+lemma tangent_ball {f : E → ℝ} {U : Set E} {x y : E} {δ M : ℝ}
+    (hδ : 0 < δ) (hU : closedBall x δ ⊆ U) (hf : ContinuousOn f U)
+    (hle : ∀ z ∈ U, f z ≤ M) (hx : f x = M)
+    (hy : y ∈ ball x (δ/4)) (hyf : f y < M) :
+    ∃ r : ℝ, 0 < r ∧ closedBall y r ⊆ U ∧
+      (∀ z ∈ ball y r, f z < M) ∧
+      ∃ p : E, dist p y = r ∧ f p = M := by
+  let S := closedBall x (δ/2) ∩ f ⁻¹' {M}
+  have hhalf : closedBall x (δ/2) ⊆ U :=
+    subset_trans (closedBall_subset_closedBall (by linarith)) hU
+  have hSclosed : IsClosed S :=
+    (hf.mono hhalf).preimage_isClosed_of_isClosed isClosed_closedBall isClosed_singleton
+  have hScompact : IsCompact S := (isCompact_closedBall x (δ/2)).of_isClosed_subset
+    hSclosed inter_subset_left
+  have hxS : x ∈ S := ⟨by simp only [mem_closedBall, dist_self]; linarith, hx⟩
+  obtain ⟨p, hp, hmin⟩ := hScompact.exists_isMinOn ⟨x, hxS⟩
+    (continuous_id.dist continuous_const).continuousOn
+  let r := dist p y
+  have hrle : r ≤ dist x y := hmin hxS
+  have hye : dist y x < δ/4 := hy
+  have hrsmall : r < δ/4 := lt_of_le_of_lt hrle (by simpa [dist_comm] using hye)
+  have hpval : f p = M := hp.2
+  have hpy : p ≠ y := by intro h; rw [h] at hpval; linarith
+  have hr : 0 < r := dist_pos.mpr hpy
+  have hsub : closedBall y r ⊆ closedBall x (δ/2) := by
+    intro z hz
+    have hz' : dist z y ≤ r := hz
+    have htri := dist_triangle z y x
+    show dist z x ≤ δ/2
+    linarith
+  refine ⟨r, hr, subset_trans hsub hhalf, ?_, p, rfl, hpval⟩
+  intro z hz
+  have hzhalf := hsub (ball_subset_closedBall hz)
+  have hzU := hhalf hzhalf
+  refine lt_of_le_of_ne (hle z hzU) ?_
+  intro heq
+  have hzS : z ∈ S := ⟨hzhalf, heq⟩
+  have hzd : r ≤ dist z y := hmin hzS
+  exact (not_lt_of_ge hzd) hz
+
+end EllipticKernel
+
+namespace EllipticKernel
+open Set Filter Metric
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [ProperSpace E]
+local instance tangentMaxBilinNormedAddCommGroup : NormedAddCommGroup (Bilin E) := inferInstance
+local instance tangentMaxBilinNormedSpace : NormedSpace ℝ (Bilin E) := inferInstance
+
+ 
+
+lemma no_tangent_max {f : E → ℝ} {U : Set E} (hU : IsOpen U)
+    (hf : ContDiffOn ℝ ∞ f U) (ℒ : E → SecondOrder E) (hℒ : ContinuousOn ℒ U)
+    (hell : ∀ z ∈ U, ∀ v : E, v ≠ 0 → 0 < ℒ z (rankOne v))
+    (htest : ∀ z ∈ U, ∀ u : E → ℝ, ContDiffAt ℝ ∞ u z → IsLocalMax u z →
+      ℒ z (fderiv ℝ (fderiv ℝ u) z) ≤ 0)
+    (hharm : ∀ z ∈ U, ℒ z (fderiv ℝ (fderiv ℝ f) z) = 0)
+    {M r : ℝ} {y p : E} (hle : ∀ z ∈ U, f z ≤ M)
+    (hr : 0 < r) (hball : closedBall y r ⊆ U)
+    (hstrict : ∀ z ∈ ball y r, f z < M)
+    (hp : dist p y = r) (hfp : f p = M) : False := by
+  let K := closedBall y r \ ball y (r/2)
+  have hK : IsCompact K := (isCompact_closedBall y r).diff isOpen_ball
+  have hKU : K ⊆ U := fun _ hz => hball hz.1
+  obtain ⟨α, hα, hbar⟩ := exists_positive_barrier hK ℒ (hℒ.mono hKU) y (by
+    intro z hz
+    apply hell z (hKU hz)
+    intro he
+    have hzy : z = y := sub_eq_zero.mp he
+    have hh : ¬dist z y < r/2 := hz.2
+    rw [hzy, dist_self] at hh
+    linarith)
+  have hhalf : closedBall y (r/2) ⊆ ball y r := closedBall_subset_ball (by linarith)
+  obtain ⟨q, hq, hqm⟩ := (isCompact_closedBall y (r/2)).exists_isMaxOn
+    ⟨y, by simp only [mem_closedBall, dist_self]; linarith⟩
+    (hf.continuousOn.mono (subset_trans hhalf (subset_trans ball_subset_closedBall hball)))
+  have hqf : f q < M := hstrict q (hhalf hq)
+  let ε := M - f q
+  have hε : 0 < ε := sub_pos.mpr hqf
+  let c := Real.exp (-α*r^2)
+  let u : E → ℝ := fun z => f z + ε*(barrier α y z-c)
+  have hus : ContDiffOn ℝ ∞ u U :=
+    hf.add (contDiffOn_const.mul ((barrier_smooth α y).contDiffOn.sub contDiffOn_const))
+  have hble : ∀ z : E, barrier α y z-c ≤ 1 := by
+    intro z
+    have he : barrier α y z ≤ 1 := Real.exp_le_one_iff.mpr
+      (mul_nonpos_of_nonpos_of_nonneg (by linarith) (sq_nonneg _))
+    have hc : 0 < c := Real.exp_pos _
+    linarith
+  have houter : ∀ z : E, dist z y = r → u z ≤ M := by
+    intro z hz
+    have hzU := hball (show z ∈ closedBall y r from le_of_eq hz)
+    have hb : barrier α y z = c := by simp [barrier, ← dist_eq_norm, hz, c]
+    simpa [u, hb] using hle z hzU
+  have hinner : ∀ z : E, dist z y = r/2 → u z ≤ M := by
+    intro z hz
+    have hfz : f z ≤ f q := hqm (show z ∈ closedBall y (r/2) from le_of_eq hz)
+    have hh := mul_le_mul_of_nonneg_left (hble z) hε.le
+    dsimp [u, ε] at *
+    linarith
+  have hno : ∀ z, r/2 < dist z y → dist z y < r → ¬IsLocalMax u z := by
+    intro z hz₁ hz₂ hm
+    have hzK : z ∈ K := ⟨le_of_lt hz₂, not_lt_of_gt hz₁⟩
+    have hzU := hKU hzK
+    have hleu := htest z hzU u (hus.contDiffAt (hU.mem_nhds hzU)) hm
+    have he : fderiv ℝ (fderiv ℝ u) z =
+        fderiv ℝ (fderiv ℝ f) z + ε • fderiv ℝ (fderiv ℝ (barrier α y)) z :=
+      hessian_add_smul_sub_const (hf.contDiffAt (hU.mem_nhds hzU))
+        (barrier_smooth α y).contDiffAt ε c
+    rw [he, map_add, map_smul, hharm z hzU, zero_add, smul_eq_mul] at hleu
+    exact (not_lt_of_ge hleu) (mul_pos hε (hbar z hzK))
+  have huK : ∀ z ∈ K, u z ≤ M := annulus_le_of_no_localMax
+    (hus.continuousOn.mono hKU) hinner houter hno
+  have hpU := hball (show p ∈ closedBall y r from le_of_eq hp)
+  have hbp : barrier α y p = c := by simp [barrier, ← dist_eq_norm, hp, c]
+  have hup : u p = M := by simp [u, hbp, hfp]
+  have humax : IsLocalMax u p := by
+    have hn : {z : E | r/2 < dist z y} ∈ nhds p :=
+      (isOpen_lt continuous_const (continuous_id.dist continuous_const)).mem_nhds
+        (by change r/2 < dist p y; rw [hp]; linarith)
+    filter_upwards [hU.mem_nhds hpU, hn] with z hzU hz
+    rw [hup]
+    by_cases hd : dist z y ≤ r
+    · exact huK z ⟨hd, not_lt_of_gt hz⟩
+    have hd' : r ≤ dist z y := le_of_lt (lt_of_not_ge hd)
+    have hsq : r^2 ≤ ‖z-y‖^2 := by rw [← dist_eq_norm]; nlinarith [dist_nonneg (x := z) (y := y)]
+    have hb : barrier α y z ≤ c := Real.exp_le_exp.mpr (by nlinarith)
+    have hfz := hle z hzU
+    have hh : ε * (barrier α y z-c) ≤ 0 := mul_nonpos_of_nonneg_of_nonpos hε.le (sub_nonpos.mpr hb)
+    dsimp [u]
+    linarith
+  have hfmax : IsLocalMax f p := by
+    filter_upwards [hU.mem_nhds hpU] with z hz
+    simpa [hfp] using hle z hz
+  have hdu := (hf.contDiffAt (hU.mem_nhds hpU)).differentiableAt (by simp) |>.hasFDerivAt
+  have hdb := ((barrier_hasFDerivAt α y p).sub_const c).const_mul ε
+  have he := (hdu.add hdb).fderiv
+  change fderiv ℝ u p = _ at he
+  rw [humax.fderiv_eq_zero, hfmax.fderiv_eq_zero] at he
+  have hv := congrArg (fun L : E →L[ℝ] ℝ => L (p-y)) he
+  have hn : ‖p-y‖^2 > 0 := by rw [← dist_eq_norm, hp]; positivity
+  have hneg : ε * (Real.exp (-α*‖p-y‖^2) * (-2*α)) * ‖p-y‖^2 < 0 := by
+    exact mul_neg_of_neg_of_pos (mul_neg_of_pos_of_neg hε
+      (mul_neg_of_pos_of_neg (Real.exp_pos _) (by linarith))) hn
+  have hv' : 0 = ε * (Real.exp (-α*‖p-y‖^2) * (-2*α)) * ‖p-y‖^2 := by
+    simpa only [_root_.smul_apply, _root_.zero_apply, zero_add, _root_.add_apply, realInner_apply,
+      real_inner_self_eq_norm_sq, smul_eq_mul, mul_assoc] using hv
+  linarith
+
+end EllipticKernel
+
+namespace EllipticKernel
+open Set Filter Metric
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [ProperSpace E]
+local instance harmonicMaxBilinNormedAddCommGroup : NormedAddCommGroup (Bilin E) := inferInstance
+local instance harmonicMaxBilinNormedSpace : NormedSpace ℝ (Bilin E) := inferInstance
+
+theorem harmonic_eventually_eq_of_max {f : E → ℝ} {U : Set E} (hU : IsOpen U)
+    (hf : ContDiffOn ℝ ∞ f U) (ℒ : E → SecondOrder E) (hℒ : ContinuousOn ℒ U)
+    (hell : ∀ z ∈ U, ∀ v : E, v ≠ 0 → 0 < ℒ z (rankOne v))
+    (htest : ∀ z ∈ U, ∀ u : E → ℝ, ContDiffAt ℝ ∞ u z → IsLocalMax u z →
+      ℒ z (fderiv ℝ (fderiv ℝ u) z) ≤ 0)
+    (hharm : ∀ z ∈ U, ℒ z (fderiv ℝ (fderiv ℝ f) z) = 0)
+    {x : E} {M : ℝ} (hx : x ∈ U) (hfx : f x = M) (hle : ∀ z ∈ U, f z ≤ M) :
+    f =ᶠ[nhds x] fun _ => M := by
+  obtain ⟨a, ha, hball⟩ := Metric.mem_nhds_iff.mp (hU.mem_nhds hx)
+  let δ := a/2
+  have hδ : 0 < δ := half_pos ha
+  have hδU : closedBall x δ ⊆ U :=
+    subset_trans (closedBall_subset_ball (by dsimp [δ]; linarith)) hball
+  filter_upwards [ball_mem_nhds x (show 0 < δ/4 by positivity)] with y hy
+  by_contra hyf
+  have hyU : y ∈ U := hδU (mem_closedBall.mpr (by
+    have h := mem_ball.mp hy
+    linarith))
+  have hlt : f y < M := lt_of_le_of_ne (hle y hyU) hyf
+  obtain ⟨r, hr, hrU, hstrict, p, hp, hfp⟩ :=
+    tangent_ball hδ hδU hf.continuousOn hle hfx hy hlt
+  exact no_tangent_max hU hf ℒ hℒ hell htest hharm hle hr hrU hstrict hp hfp
+
+end EllipticKernel
+
+namespace EllipticKernel
+open Matrix
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+abbrev EC (n : ℕ) := EuclideanSpace ℂ (Fin n)
+variable {n : ℕ}
+local instance coordinateBilinNormedAddCommGroup : NormedAddCommGroup (Bilin (EC n)) := inferInstance
+local instance coordinateBilinNormedSpace : NormedSpace ℝ (Bilin (EC n)) := inferInstance
+
+ 
+
+def coordinateEquiv (n : ℕ) : EC n ≃L[ℝ] PotentialKaehler.V n :=
+  PiLp.continuousLinearEquiv 2 ℝ (fun _ : Fin n => ℂ)
+
+lemma coordinateEquiv_apply (z : EC n) : coordinateEquiv n z = WithLp.ofLp z := rfl
+lemma coordinateEquiv_symm_apply (z : PotentialKaehler.V n) :
+    (coordinateEquiv n).symm z = WithLp.toLp 2 z := rfl
+
+def pullBilin (B : Bilin (EC n)) : PotentialKaehler.RealBilinear n :=
+  ((B.comp (coordinateEquiv n).symm.toContinuousLinearMap).flip.comp
+    (coordinateEquiv n).symm.toContinuousLinearMap).flip
+
+lemma pullBilin_apply (B : Bilin (EC n)) (u v : PotentialKaehler.V n) :
+    pullBilin B u v = B ((coordinateEquiv n).symm u) ((coordinateEquiv n).symm v) := rfl
+
+lemma pullBilin_add (B C : Bilin (EC n)) : pullBilin (B+C) = pullBilin B + pullBilin C := by
+  ext u v
+  simp only [pullBilin_apply, _root_.add_apply]
+
+lemma pullBilin_smul (c : ℝ) (B : Bilin (EC n)) : pullBilin (c • B) = c • pullBilin B := by
+  ext u v
+  simp only [pullBilin_apply, _root_.smul_apply]
+
+lemma hermitianPartMatrix_add (B C : PotentialKaehler.RealBilinear n) :
+    PotentialKaehler.hermitianPartMatrix (B+C) =
+      PotentialKaehler.hermitianPartMatrix B + PotentialKaehler.hermitianPartMatrix C := by
+  ext i j
+  apply Complex.ext <;>
+    simp [PotentialKaehler.hermitianPartMatrix_apply, PotentialKaehler.hermitianPart]
+    <;> ring
+
+lemma hermitianPartMatrix_smul (c : ℝ) (B : PotentialKaehler.RealBilinear n) :
+    PotentialKaehler.hermitianPartMatrix (c • B) = c • PotentialKaehler.hermitianPartMatrix B := by
+  ext i j
+  apply Complex.ext <;>
+    simp [PotentialKaehler.hermitianPartMatrix_apply, PotentialKaehler.hermitianPart]
+    <;> ring
+
+def traceBilinLinear (H : Matrix (Fin n) (Fin n) ℂ) : Bilin (EC n) →ₗ[ℝ] ℝ where
+  toFun B := (H⁻¹ * PotentialKaehler.hermitianPartMatrix (pullBilin B)).trace.re
+  map_add' B C := by rw [pullBilin_add, hermitianPartMatrix_add, mul_add, trace_add, Complex.add_re]
+  map_smul' c B := by rw [pullBilin_smul, hermitianPartMatrix_smul, Matrix.mul_smul, trace_smul]; simp
+
+def traceBilin (H : Matrix (Fin n) (Fin n) ℂ) : SecondOrder (EC n) :=
+  (traceBilinLinear H).toContinuousLinearMap
+
+lemma traceBilin_apply (H : Matrix (Fin n) (Fin n) ℂ) (B : Bilin (EC n)) :
+    traceBilin H B = (H⁻¹ * PotentialKaehler.hermitianPartMatrix (pullBilin B)).trace.re := rfl
+
+lemma traceBilin_continuousOn {U : Set (EC n)}
+    (H : EC n → Matrix (Fin n) (Fin n) ℂ) (hH : ContinuousOn H U)
+    (hpos : ∀ z ∈ U, (H z).PosDef) : ContinuousOn (fun z => traceBilin (H z)) U := by
+  apply continuousOn_clm_apply.mpr
+  intro B
+  change ContinuousOn (fun z => ((H z)⁻¹ * PotentialKaehler.hermitianPartMatrix (pullBilin B)).trace.re) U
+  have hi : ContinuousOn (fun z => (H z)⁻¹) U := by
+    intro z hz
+    apply (continuousAt_matrix_inv (H z) ?_).comp_continuousWithinAt (hH z hz)
+    obtain ⟨u, hu⟩ := isUnit_iff_ne_zero.mpr (ne_of_gt (hpos z hz).det_pos)
+    rw [← hu]
+    exact NormedRing.inverse_continuousAt u
+  apply Complex.continuous_re.comp_continuousOn
+  unfold Matrix.trace
+  apply continuousOn_finsetSum
+  intro i hi'
+  exact continuousOn_pi.mp (continuousOn_pi.mp (hi.mul continuousOn_const) i) i
+
+lemma pullBilin_hessian {u : EC n → ℝ} {z : EC n} (hu : ContDiffAt ℝ ∞ u z) :
+    pullBilin (fderiv ℝ (fderiv ℝ u) z) =
+      fderiv ℝ (fderiv ℝ (u ∘ (coordinateEquiv n).symm)) (coordinateEquiv n z) := by
+  let s := (coordinateEquiv n).symm
+  have hu' : ContDiffAt ℝ ∞ u (s (coordinateEquiv n z)) := by simpa [s] using hu
+  have hd : fderiv ℝ s = fun _ => s.toContinuousLinearMap := by
+    funext w; exact s.hasFDerivAt.fderiv
+  ext a b
+  rw [PotentialKaehler.second_derivative_comp s.contDiff.contDiffAt hu' a b, hd]
+  simp [pullBilin_apply, s]
+
+lemma traceBilin_localMax (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    {u : EC n → ℝ} {z : EC n} (hu : ContDiffAt ℝ ∞ u z) (hmax : IsLocalMax u z) :
+    traceBilin H (fderiv ℝ (fderiv ℝ u) z) ≤ 0 := by
+  rw [traceBilin_apply, pullBilin_hessian hu]
+  let s := (coordinateEquiv n).symm
+  have hu' : ContDiffAt ℝ ∞ u (s (coordinateEquiv n z)) := by simpa [s] using hu
+  have hm' : IsLocalMax u (s (coordinateEquiv n z)) := by simpa [s] using hmax
+  exact trace_potentialMatrix_nonpos_at_localMax
+    (hu'.comp _ s.contDiff.contDiffAt) (hm'.comp_continuous s.continuous.continuousAt) H hH
+
+end EllipticKernel
+
+namespace EllipticKernel
+open Matrix
+open scoped ComplexOrder Matrix.Norms.Elementwise ContDiff
+variable {n : ℕ}
+
+lemma realInner_coordinate_single (v : EC n) (i : Fin n) (c : ℂ) :
+    realInner v ((coordinateEquiv n).symm (Pi.single i c)) = (c * star (v i)).re := by
+  rw [realInner_apply, coordinateEquiv_symm_apply, PiLp.inner_apply]
+  simp [Pi.single_apply, apply_ite Complex.re, Complex.inner]
+
+lemma pull_rankOne (v : EC n) :
+    PotentialKaehler.hermitianPartMatrix (pullBilin (rankOne v)) =
+      (1/4 : ℝ) • Matrix.vecMulVec (coordinateEquiv n v) (star (coordinateEquiv n v)) := by
+  ext i j
+  apply Complex.ext <;>
+    simp [PotentialKaehler.hermitianPartMatrix_apply, PotentialKaehler.hermitianPart,
+      pullBilin_apply, rankOne, ContinuousLinearMap.smulRight_apply,
+      realInner_coordinate_single, ← Pi.single_smul, coordinateEquiv_apply,
+      Complex.mul_re, Complex.mul_im, Matrix.vecMulVec_apply] <;> ring
+
+lemma traceBilin_symbol (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (v : EC n) (hv : v ≠ 0) : 0 < traceBilin H (rankOne v) := by
+  have hve : coordinateEquiv n v ≠ 0 := by
+    intro h
+    apply hv
+    exact (coordinateEquiv n).injective (by simpa using h)
+  have hp := MongeAmpere.linearized_symbol_pos H hH (coordinateEquiv n v) hve
+  rw [traceBilin_apply, pull_rankOne, Matrix.mul_smul, trace_smul]
+  simpa only [Complex.smul_re, smul_eq_mul] using mul_pos (by norm_num : (0 : ℝ) < 1/4) hp
+
+end EllipticKernel
+
+namespace Anticanonical.SourceSmooth
+open EllipticKernel Set Filter
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+
+lemma SmoothRealFunction.continuous (φ : SmoothRealFunction A) : Continuous φ.value := by
+  apply continuous_iff_continuousAt.mpr
+  intro x
+  obtain ⟨i, hi⟩ := A.covers x
+  have hci : ContinuousAt (φ.localExpression i) (A.chart i x) :=
+    ((φ.smooth i).contDiffAt ((A.chart i).open_target.mem_nhds ((A.chart i).map_source hi))).continuousAt
+  apply (hci.comp ((A.chart i).continuousAt hi)).congr_of_eventuallyEq
+  filter_upwards [(A.chart i).eventually_left_inverse hi] with y hy
+  simp [SmoothRealFunction.localExpression, hy]
+
+lemma SmoothRealFunction.euclidean_smooth (φ : SmoothRealFunction A) (i : Fin A.count) :
+    ContDiffOn ℝ ∞ (φ.localExpression i ∘ coordinateEquiv d)
+      ((coordinateEquiv d) ⁻¹' (A.chart i).target) :=
+  (φ.smooth i).comp (coordinateEquiv d).contDiff.contDiffOn (fun _ h => h)
+
+lemma KaehlerMetric.traceBilin_localExpression (g : KaehlerMetric A) (φ : SmoothRealFunction A)
+    (i : Fin A.count) {z : EC d} (hz : coordinateEquiv d z ∈ (A.chart i).target) :
+    traceBilin (g.matrix i (coordinateEquiv d z))
+      (fderiv ℝ (fderiv ℝ (φ.localExpression i ∘ coordinateEquiv d)) z) =
+        g.linearizedMongeAmpere φ i (coordinateEquiv d z) := by
+  have hu := (φ.euclidean_smooth i).contDiffAt
+    (((A.chart i).open_target.preimage (coordinateEquiv d).continuous).mem_nhds hz)
+  have he : (φ.localExpression i ∘ coordinateEquiv d) ∘ (coordinateEquiv d).symm =
+      φ.localExpression i := by
+    funext w
+    simp
+  rw [traceBilin_apply, pullBilin_hessian hu, he]
+  rfl
+
+ 
+
+lemma KaehlerMetric.harmonic_eventually_constant (g : KaehlerMetric A)
+    (φ : SmoothRealFunction A)
+    (hφ : ∀ (i : Fin A.count) (z : Coordinates d), z ∈ (A.chart i).target →
+      g.linearizedMongeAmpere φ i z = 0)
+    {x : X} {M : ℝ} (hx : φ.value x = M) (hmax : ∀ y : X, φ.value y ≤ M) :
+    φ.value =ᶠ[nhds x] fun _ => M := by
+  obtain ⟨i, hi⟩ := A.covers x
+  let e := coordinateEquiv d
+  let U := e ⁻¹' (A.chart i).target
+  let u := φ.localExpression i ∘ e
+  let H := fun z => g.matrix i (e z)
+  have hU : IsOpen U := (A.chart i).open_target.preimage e.continuous
+  have hu : ContDiffOn ℝ ∞ u U := φ.euclidean_smooth i
+  have hH : ContinuousOn H U :=
+    (g.smooth i).continuousOn.comp e.continuous.continuousOn (fun _ h => h)
+  have hp : ∀ z ∈ U, (H z).PosDef := fun z hz => g.positive i (e z) hz
+  have hh : ∀ z ∈ U, traceBilin (H z) (fderiv ℝ (fderiv ℝ u) z) = 0 := by
+    intro z hz
+    exact (g.traceBilin_localExpression φ i hz).trans (hφ i (e z) hz)
+  have hz : e.symm (A.chart i x) ∈ U := by
+    simpa [U] using (A.chart i).map_source hi
+  have hux : u (e.symm (A.chart i x)) = M := by
+    simpa [u, SmoothRealFunction.localExpression, (A.chart i).left_inv hi] using hx
+  have heq := harmonic_eventually_eq_of_max hU hu (fun z => traceBilin (H z))
+    (traceBilin_continuousOn H hH hp)
+    (fun z hz v hv => traceBilin_symbol (H z) (hp z hz) v hv)
+    (fun z hz u hu hm => traceBilin_localMax (H z) (hp z hz) hu hm)
+    hh hz hux (fun z _ => hmax ((A.chart i).symm (e z)))
+  have ht : Tendsto (fun y => e.symm (A.chart i y)) (nhds x)
+      (nhds (e.symm (A.chart i x))) :=
+    e.symm.continuous.continuousAt.comp ((A.chart i).continuousAt hi)
+  filter_upwards [ht.eventually heq, (A.chart i).eventually_left_inverse hi] with y hy hiy
+  simpa [u, SmoothRealFunction.localExpression, hiy] using hy
+
+ 
+
+theorem linearizedMongeAmpere_kernel (d : ℕ) (X : Type*) [TopologicalSpace X]
+    [T2Space X] [CompactSpace X] [ConnectedSpace X] (A : ComplexAtlas d X)
+    (g : KaehlerMetric A) (φ : SmoothRealFunction A)
+    (hφ : ∀ (i : Fin A.count) (z : Coordinates d), z ∈ (A.chart i).target →
+      g.linearizedMongeAmpere φ i z = 0) :
+    ∃ c : ℝ, ∀ x : X, φ.value x = c := by
+  obtain ⟨x, _, hx⟩ := isCompact_univ.exists_isMaxOn Set.univ_nonempty φ.continuous.continuousOn
+  have hmax : ∀ y, φ.value y ≤ φ.value x := fun y => hx (mem_univ y)
+  let S := {y : X | φ.value y = φ.value x}
+  have hs : IsClosed S := isClosed_eq φ.continuous continuous_const
+  have ho : IsOpen S := by
+    rw [isOpen_iff_mem_nhds]
+    intro y hy
+    exact g.harmonic_eventually_constant φ hφ hy hmax
+  have hall : S = univ := (show IsClopen S from ⟨hs, ho⟩).eq_univ ⟨x, rfl⟩
+  exact ⟨φ.value x, fun y => show y ∈ S from hall ▸ mem_univ y⟩
+
+end Anticanonical.SourceSmooth
+
+namespace Anticanonical.SourceSmooth
+open Set Filter
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+
+namespace SmoothRealFunction
+ 
+def normalizeAt (φ : SmoothRealFunction A) (x₀ : X) : SmoothRealFunction A where
+  value x := φ.value x - φ.value x₀
+  smooth i := (φ.smooth i).sub contDiffOn_const
+
+@[simp] lemma normalizeAt_value (φ : SmoothRealFunction A) (x₀ : X) :
+    (φ.normalizeAt x₀).value x₀ = 0 := sub_self _
+
+lemma hessian_normalizeAt (φ : SmoothRealFunction A) (x₀ : X) :
+    (φ.normalizeAt x₀).hessian = φ.hessian := by
+  funext i z
+  have he : (φ.normalizeAt x₀).localExpression i =
+      fun w => φ.localExpression i w - φ.value x₀ := rfl
+  have hd : fderiv ℝ (fun w => φ.localExpression i w - φ.value x₀) =
+      fderiv ℝ (φ.localExpression i) := by
+    funext w; exact fderiv_sub_const _
+  simp only [hessian, he, PotentialKaehler.potentialMatrix, hd]
+end SmoothRealFunction
+
+namespace KaehlerMetric
+lemma linearizedMongeAmpere_realSMul (g : KaehlerMetric A)
+    (φ : SmoothRealFunction A) (c : ℝ) (i : Fin A.count) (z : Coordinates d) :
+    g.linearizedMongeAmpere (φ.realSMul c) i z = c * g.linearizedMongeAmpere φ i z := by
+  simp only [linearizedMongeAmpere, SmoothRealFunction.hessian_realSMul,
+    Matrix.mul_smul, Matrix.trace_smul, smul_eq_mul, Complex.re_ofReal_mul]
+
+lemma linearizedMongeAmpere_normalizeAt (g : KaehlerMetric A)
+    (φ : SmoothRealFunction A) (x₀ : X) :
+    g.linearizedMongeAmpere (φ.normalizeAt x₀) = g.linearizedMongeAmpere φ := by
+  funext i z
+  unfold linearizedMongeAmpere
+  rw [SmoothRealFunction.hessian_normalizeAt]
+
+lemma linearized_nonpos_at_globalMax (g : KaehlerMetric A)
+    (φ : SmoothRealFunction A) {x : X} (hx : ∀ y : X, φ.value y ≤ φ.value x)
+    (i : Fin A.count) (hi : x ∈ (A.chart i).source) :
+    g.linearizedMongeAmpere φ i (A.chart i x) ≤ 0 := by
+  have hs := (φ.smooth i).contDiffAt
+    ((A.chart i).open_target.mem_nhds ((A.chart i).map_source hi))
+  have hm : IsLocalMax (φ.localExpression i) (A.chart i x) := by
+    apply Filter.Eventually.of_forall
+    intro z
+    simpa [SmoothRealFunction.localExpression, (A.chart i).left_inv hi] using
+      hx ((A.chart i).symm z)
+  exact EllipticKernel.trace_potentialMatrix_nonpos_at_localMax hs hm _
+    (g.positive i _ ((A.chart i).map_source hi))
+
+ 
+
+theorem no_positive_constant_rhs [CompactSpace X] [Nonempty X]
+    (g : KaehlerMetric A) (φ : SmoothRealFunction A) {c : ℝ} (hc : 0 < c) :
+    ¬ (∀ i z, z ∈ (A.chart i).target → g.linearizedMongeAmpere φ i z = c) := by
+  intro h
+  obtain ⟨x, _, hx⟩ := isCompact_univ.exists_isMaxOn univ_nonempty φ.continuous.continuousOn
+  obtain ⟨i, hi⟩ := A.covers x
+  have hn := g.linearized_nonpos_at_globalMax φ (fun y => hx (mem_univ y)) i hi
+  rw [h i _ ((A.chart i).map_source hi)] at hn
+  exact not_le_of_gt hc hn
+
+ 
+
+lemma constant_rhs_eq_zero [CompactSpace X] [Nonempty X]
+    (g : KaehlerMetric A) (φ : SmoothRealFunction A) {c : ℝ}
+    (hc : ∀ i z, z ∈ (A.chart i).target → g.linearizedMongeAmpere φ i z = c) : c = 0 := by
+  have hn : c ≤ 0 := le_of_not_gt (fun h => g.no_positive_constant_rhs φ h hc)
+  have hp : 0 ≤ c := by
+    by_contra h
+    have hc' : 0 < -c := by linarith
+    apply g.no_positive_constant_rhs (φ.realSMul (-1)) hc'
+    intro i z hz
+    rw [linearizedMongeAmpere_realSMul, hc i z hz]
+    ring
+  exact le_antisymm hn hp
+
+ 
+
+theorem normalized_augmented_kernel [T2Space X] [CompactSpace X] [ConnectedSpace X]
+    (g : KaehlerMetric A) (φ : SmoothRealFunction A) (x₀ : X) (c : ℝ)
+    (hx : φ.value x₀ = 0)
+    (he : ∀ i z, z ∈ (A.chart i).target → g.linearizedMongeAmpere φ i z + c = 0) :
+    c = 0 ∧ ∀ x, φ.value x = 0 := by
+  have hc : -c = 0 := g.constant_rhs_eq_zero φ (by
+    intro i z hz
+    linarith [he i z hz])
+  have hc₀ : c = 0 := neg_eq_zero.mp hc
+  obtain ⟨k, hk⟩ := linearizedMongeAmpere_kernel d X A g φ (by
+    intro i z hz
+    simpa [hc₀] using he i z hz)
+  refine ⟨hc₀, fun x => ?_⟩
+  rw [hk x, ← hk x₀, hx]
+end KaehlerMetric
+
+ 
+
+def HasNormalizedPoissonSolution {d : ℕ} {X : Type*} [TopologicalSpace X]
+    {A : ComplexAtlas d X} (g : KaehlerMetric A) (x₀ : X)
+    (f : SmoothRealFunction A) : Prop :=
+  ∃ (φ : SmoothRealFunction A) (c : ℝ), φ.value x₀ = 0 ∧
+    ∀ i z, z ∈ (A.chart i).target →
+      g.linearizedMongeAmpere φ i z + c = f.localExpression i z
+
+end Anticanonical.SourceSmooth
+
+end
+end
+end
+end
+end
+end
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/AtlasLocalization.lean b/lean/OAI/Geometry/PrescribedPotential/AtlasLocalization.lean
new file mode 100644
index 0000000000000000000000000000000000000000..b258131f11abf817e868a504e4f18e41e5b66a90
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/AtlasLocalization.lean
@@ -0,0 +1,89 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.AnalyticSupport
+
+namespace OAI
+
+/-! Atlas Localization. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Manifold IsManifold
+open scoped ContDiff
+
+namespace Anticanonical.ComplexAtlas
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] (A : ComplexAtlas d X)
+
+ 
+@[instance_reducible] def chartedSpace : ChartedSpace (Coordinates d) X where
+  atlas := Set.range A.chart
+  chartAt x := A.chart (A.covers x).choose
+  mem_chart_source x := (A.covers x).choose_spec
+  chart_mem_atlas point := ⟨(A.covers point).choose, rfl⟩
+
+lemma real_smooth_transition (i j : Fin A.count) :
+    ContDiffOn ℝ ∞ (A.transition i j) (coordinateOverlap (A.chart i) (A.chart j)) := by
+  intro z hz
+  exact ((ProjectiveKaehler.analyticOnNhd_pi_of_differentiableOn
+    (A.isOpen_coordinateOverlap i j) (A.holomorphic i j) z hz).contDiffAt.restrict_scalars ℝ).contDiffWithinAt
+
+ 
+theorem isManifold : letI := A.chartedSpace; IsManifold 𝓘(ℝ, Coordinates d) ∞ X := by
+  let := A.chartedSpace
+  apply isManifold_of_contDiffOn
+  rintro e e' ⟨i, rfl⟩ ⟨j, rfl⟩
+  simpa only [modelWithCornersSelf_coe, modelWithCornersSelf_coe_symm,
+    Function.id_comp, Function.comp_id, Set.preimage_id, Set.range_id, Set.inter_univ, OpenPartialHomeomorph.coe_trans,
+    OpenPartialHomeomorph.trans_source, OpenPartialHomeomorph.symm_source,
+    transition, coordinateOverlap] using
+      A.real_smooth_transition i j
+
+ 
+
+theorem contMDiff_iff (f : X → ℝ) : letI := A.chartedSpace;
+    ContMDiff 𝓘(ℝ, Coordinates d) 𝓘(ℝ) ∞ f ↔
+      ∀ i, ContDiffOn ℝ ∞ (f ∘ (A.chart i).symm) (A.chart i).target := by
+  let := A.chartedSpace
+  let := A.isManifold
+  constructor
+  · intro hf i
+    have he : A.chart i ∈ maximalAtlas 𝓘(ℝ, Coordinates d) ∞ X :=
+      (contDiffGroupoid ∞ 𝓘(ℝ, Coordinates d)).subset_maximalAtlas ⟨i, rfl⟩
+    exact (hf.comp_contMDiffOn (contMDiffOn_symm_of_mem_maximalAtlas he)).contDiffOn
+  · intro hf x
+    rw [contMDiffAt_iff_source]
+    have hchart : chartAt (Coordinates d) x = A.chart (A.covers x).choose := rfl
+    have hx := (A.chart (A.covers x).choose).mapsTo (A.covers x).choose_spec
+    have hh := ((hf (A.covers x).choose).contDiffAt
+      ((A.chart (A.covers x).choose).open_target.mem_nhds hx)).contMDiffAt
+    simpa only [extChartAt_coe_symm, extChartAt_coe, hchart, modelWithCornersSelf_coe,
+      modelWithCornersSelf_coe_symm, Function.comp_id, Function.id_comp,
+      Set.range_id, contMDiffWithinAt_univ, chartAt, chartedSpace] using hh
+
+ 
+
+theorem exists_smooth_partition [T2Space X] [CompactSpace X] :
+    ∃ ρ : Fin A.count → SourceSmooth.SmoothRealFunction A,
+      (∀ i x, 0 ≤ (ρ i).value x) ∧
+      (∀ x, ∑ i, (ρ i).value x = 1) ∧
+      (∀ i, tsupport (ρ i).value ⊆ (A.chart i).source) := by
+  let := A.chartedSpace
+  let := A.isManifold
+  obtain ⟨ρ, hρ⟩ := SmoothPartitionOfUnity.exists_isSubordinate
+    𝓘(ℝ, Coordinates d) (s := (Set.univ : Set X)) isClosed_univ
+    (fun i => (A.chart i).source) (fun i => (A.chart i).open_source)
+    (by intro x _; obtain ⟨i, hi⟩ := A.covers x; exact Set.mem_iUnion.mpr ⟨i, hi⟩)
+  refine ⟨fun i => ⟨ρ i, (A.contMDiff_iff (ρ i)).mp (ρ i).contMDiff⟩, ?_, ?_, ?_⟩
+  · exact fun i x => ρ.nonneg i x
+  · intro x
+    simpa only [finsum_eq_sum_of_fintype] using ρ.sum_eq_one (Set.mem_univ x)
+  · exact hρ
+
+end Anticanonical.ComplexAtlas
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/AugmentedVolume.lean b/lean/OAI/Geometry/PrescribedPotential/AugmentedVolume.lean
new file mode 100644
index 0000000000000000000000000000000000000000..e771ab9224baac8904c2d36ed824a6dd6fbafea2
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/AugmentedVolume.lean
@@ -0,0 +1,91 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.KaehlerClosedDerivatives
+import OAI.Geometry.PrescribedPotential.PatchCutoffs
+import OAI.Geometry.PrescribedPotential.RealVolume
+import OAI.Geometry.PrescribedPotential.RealPoisson
+import OAI.Geometry.PrescribedPotential.RealPoissonEquivalence
+
+namespace OAI
+
+/-! Augmented Volume. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology
+open scoped ContDiff Classical
+namespace ExponentialAugment
+variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  [NormedAddCommGroup F] [NormedSpace ℝ F]
+lemma derivative (V : E → F) (L : E →L[ℝ] F) (C : ℝ →L[ℝ] F) (e : E →L[ℝ] ℝ)
+    (hv : HasStrictFDerivAt V L 0) (h0 : V 0 = C 1) :
+    HasStrictFDerivAt (fun z : E × ℝ => (Real.exp (-z.2) • V z.1, e z.1))
+      (((L ∘L ContinuousLinearMap.fst ℝ E ℝ) - (C ∘L ContinuousLinearMap.snd ℝ E ℝ)).prod
+        (e ∘L ContinuousLinearMap.fst ℝ E ℝ)) 0 := by
+  let π₁ := ContinuousLinearMap.fst ℝ E ℝ
+  let π₂ := ContinuousLinearMap.snd ℝ E ℝ
+  have hv' : HasStrictFDerivAt (fun z : E × ℝ => V z.1) (L ∘L π₁) 0 :=
+    hv.comp 0 π₁.hasStrictFDerivAt
+  have he : HasStrictFDerivAt (fun z : E × ℝ => Real.exp (-z.2)) (-π₂) 0 := by
+    simpa [π₂] using ((-π₂).hasStrictFDerivAt (x := (0 : E × ℝ))).exp
+  have hh := he.smul hv'
+  have hn := (e ∘L π₁).hasStrictFDerivAt (x := (0 : E × ℝ))
+  have hj := hh.prodMk hn
+  have hlin : (Real.exp (-(0 : E × ℝ).2) • (L ∘L π₁) +
+      (-π₂).smulRight (V (0 : E × ℝ).1)).prod (e ∘L π₁) =
+      ((L ∘L π₁) - (C ∘L π₂)).prod (e ∘L π₁) := by
+    apply ContinuousLinearMap.ext
+    intro z
+    apply Prod.ext
+    · change Real.exp (-0) • L z.1 + (-z.2) • V 0 = L z.1 - C z.2
+      rw [neg_zero, Real.exp_zero, one_smul, h0, ← map_smul]
+      simp [sub_eq_add_neg]
+    · rfl
+  rw [hlin] at hj
+  exact hj
+end ExponentialAugment
+
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+local instance augmentedVolumeNormedAddCommGroup (s : ℝ) :
+    NormedAddCommGroup (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedAddCommGroup
+local instance augmentedVolumeNormedSpace (s : ℝ) : NormedSpace ℝ (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedSpace
+local instance augmentedVolumeIsTopologicalAddGroup (s : ℝ) :
+    IsTopologicalAddGroup (D.localizers.RealSobolev s) :=
+  Submodule.isTopologicalAddGroup _
+local instance augmentedVolumeContinuousSMul (s : ℝ) :
+    ContinuousSMul ℝ (D.localizers.RealSobolev s) :=
+  SMulMemClass.continuousSMul _
+
+def augmentedVolume (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) (x₀ : X)
+    (z : D.localizers.RealSobolev ((k : ℝ)+2) × ℝ) :
+    D.localizers.RealSobolev (k : ℝ) × ℝ :=
+  (Real.exp (-z.2) • D.realVolume k hk z.1, D.realEvaluation ((k : ℝ)+2) x₀ z.1)
+
+lemma augmentedVolume_zero (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) (x₀ : X) :
+    D.augmentedVolume k hk x₀ 0 = (D.realConstants (k : ℝ) 1, 0) := by
+  simp [augmentedVolume, D.realVolume_zero]
+
+lemma augmentedVolume_hasStrictFDerivAt_zero (k : ℕ)
+    (hk : Module.finrank ℝ (EC d) < k) (x₀ : X) :
+    HasStrictFDerivAt (D.augmentedVolume k hk x₀) (D.realAugmentedL k x₀) 0 := by
+  exact ExponentialAugment.derivative (D.realVolume k hk) (D.realLOrder k)
+    (D.realConstants (k : ℝ)) (D.realEvaluation ((k : ℝ)+2) x₀)
+    (D.realVolume_hasStrictFDerivAt_zero k hk) (D.realVolume_zero k hk)
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/BallSubmean.lean b/lean/OAI/Geometry/PrescribedPotential/BallSubmean.lean
new file mode 100644
index 0000000000000000000000000000000000000000..389c444cd103d465e802066a92467c5955978e36
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/BallSubmean.lean
@@ -0,0 +1,135 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CircleRadialCalculus
+import OAI.Geometry.PrescribedPotential.KaehlerClosedDerivatives
+
+namespace OAI
+
+/-! Ball Submean. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Metric MeasureTheory
+open scoped ContDiff InnerProductSpace
+namespace PotentialABP
+
+section Affine
+variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  [NormedAddCommGroup F] [NormedSpace ℝ F]
+
+lemma hessian_affine_comp {f : F → ℝ} (hf : ContDiff ℝ ∞ f)
+    (L : E →L[ℝ] F) (c : F) (z u v : E) :
+    fderiv ℝ (fderiv ℝ (fun y => f (c + L y))) z u v =
+      fderiv ℝ (fderiv ℝ f) (c + L z) (L u) (L v) := by
+  have hd : fderiv ℝ (fun y => c + L y) = fun _ => L := by
+    funext y
+    exact (L.hasFDerivAt.const_add c).fderiv
+  have hc : ContDiff ℝ ∞ (fun y => c + L y) := contDiff_const.add L.contDiff
+  change fderiv ℝ (fderiv ℝ (f ∘ (fun y => c + L y))) z u v = _
+  rw [PotentialKaehler.second_derivative_comp hc.contDiffAt hf.contDiffAt, hd]
+  simp
+
+end Affine
+
+section ComplexHilbert
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℂ E]
+  [FiniteDimensional ℂ E] [MeasurableSpace E] [BorelSpace E]
+local instance ballSubmeanRealInnerProductSpace : InnerProductSpace ℝ E :=
+  InnerProductSpace.rclikeToReal ℂ E
+
+private def unitScalarIso (a : ℂ) (ha : ‖a‖ = 1) : E ≃ₗᵢ[ℝ] E where
+  toLinearEquiv := (LinearEquiv.smulOfNeZero ℂ E a (by
+    intro h; simp [h] at ha)).restrictScalars ℝ
+  norm_map' x := by simp [LinearEquiv.smulOfNeZero_apply, norm_smul, ha]
+
+lemma ball_integral_unit_smul (f : E → ℝ) (r : ℝ) (a : ℂ) (ha : ‖a‖ = 1) :
+    ∫ x in ball (0:E) r, f (a • x) = ∫ x in ball (0:E) r, f x := by
+  let e : E ≃ₗᵢ[ℝ] E := unitScalarIso a ha
+  have hb : e ⁻¹' ball (0:E) r = ball (0:E) r := by
+    ext x
+    simp only [mem_preimage, mem_ball, dist_zero_right, e.norm_map]
+  have hm := e.measurePreserving.restrict_preimage (s := ball (0:E) r) isOpen_ball.measurableSet
+  rw [hb] at hm
+  exact hm.integral_comp e.toHomeomorph.measurableEmbedding f
+
+lemma ball_integral_translate (f : E → ℝ) (c : E) (r : ℝ) :
+    ∫ x in ball (0:E) r, f (c+x) = ∫ x in ball c r, f x := by
+  have hb : (fun x : E => c+x) ⁻¹' ball c r = ball (0:E) r := by
+    ext x
+    simp [mem_ball, dist_eq_norm]
+  have hm := (measurePreserving_add_left (volume : Measure E) c).restrict_preimage
+    (s := ball c r) isOpen_ball.measurableSet
+  rw [hb] at hm
+  exact hm.integral_comp (Homeomorph.addLeft c).measurableEmbedding f
+
+omit [FiniteDimensional ℂ E] [MeasurableSpace E] [BorelSpace E] in
+lemma complexLine_hessian {f : E → ℝ} (hf : ContDiff ℝ ∞ f)
+    (c v : E) (z a b : ℂ) :
+    fderiv ℝ (fderiv ℝ (fun w : ℂ => f (c + w • v))) z a b =
+      fderiv ℝ (fderiv ℝ f) (c + z • v) (a • v) (b • v) := by
+  let L : ℂ →L[ℝ] E := ((ContinuousLinearMap.id ℂ ℂ).smulRight v).restrictScalars ℝ
+  exact hessian_affine_comp hf L c z a b
+
+ 
+
+theorem smooth_ball_submean {f : E → ℝ} (hf : ContDiff ℝ ∞ f)
+    (c : E) {R : ℝ} (hR : 0 ≤ R)
+    (hlevi : ∀ x ∈ closedBall c R, ∀ v : E,
+      0 ≤ fderiv ℝ (fderiv ℝ f) x v v +
+        fderiv ℝ (fderiv ℝ f) x (Complex.I • v) (Complex.I • v)) :
+    (volume.real (ball (0:E) R)) * f c ≤ ∫ x in ball c R, f x := by
+  let F : ℝ → E → ℝ := fun θ v => f (c + circleMap 0 1 θ • v)
+  have hF : Continuous (Function.uncurry F) := hf.continuous.comp
+    (continuous_const.add (((continuous_circleMap 0 1).comp continuous_fst).smul continuous_snd))
+  have hi : Integrable (Function.uncurry F)
+      ((volume.restrict (uIoc 0 (2*Real.pi))).prod
+        (volume.restrict (ball (0:E) R))) := by
+    rw [Measure.prod_restrict]
+    apply (hF.continuousOn.integrableOn_compact
+      (isCompact_uIcc.prod (isCompact_closedBall (0:E) R))).mono_set
+    exact prod_mono uIoc_subset_uIcc ball_subset_closedBall
+  have hv (v : E) (hv : v ∈ ball (0:E) R) :
+      2*Real.pi*f c ≤ ∫ θ in 0..2*Real.pi, F θ v := by
+    have hc : ContDiff ℝ ∞ (fun z : ℂ => f (c + z • v)) :=
+      hf.comp (contDiff_const.add (contDiff_id.smul contDiff_const))
+    have h := smooth_circle_submean hc 0 (by norm_num : (0:ℝ) ≤ 1) ?_
+    · simp only [zero_smul, add_zero] at h
+      unfold Real.circleAverage at h
+      simp only [smul_eq_mul] at h
+      exact (le_inv_mul_iff₀ (by positivity : 0 < 2*Real.pi)).mp h
+    · intro z hz
+      rw [complexLine_hessian hf, complexLine_hessian hf, one_smul]
+      apply hlevi
+      have hz' : ‖z‖ ≤ 1 := by simpa [mem_closedBall] using hz
+      have hv' : ‖v‖ ≤ R := (by simpa [mem_ball] using hv : ‖v‖ < R).le
+      simp only [mem_closedBall, dist_eq_norm, add_sub_cancel_left, norm_smul]
+      exact (mul_le_mul_of_nonneg_left hv' (norm_nonneg z)).trans
+        (mul_le_of_le_one_left hR hz')
+  have hmean : (2*Real.pi) * (volume.real (ball (0:E) R) * f c) ≤
+      ∫ v in ball (0:E) R, ∫ θ in 0..2*Real.pi, F θ v := by
+    have hi' : IntegrableOn (fun v => ∫ θ in 0..2*Real.pi, F θ v) (ball (0:E) R) := by
+      simpa only [intervalIntegral.integral_of_le (by positivity : (0:ℝ) ≤ 2*Real.pi),
+        uIoc_of_le (by positivity : (0:ℝ) ≤ 2*Real.pi), Function.uncurry_def,
+        IntegrableOn] using hi.integral_prod_right
+    have hiC : IntegrableOn (fun _ : E => 2*Real.pi*f c) (ball (0:E) R) :=
+      (continuous_const.continuousOn.integrableOn_compact (isCompact_closedBall (0:E) R)).mono_set
+        ball_subset_closedBall
+    have hm := setIntegral_mono_on hiC hi' isOpen_ball.measurableSet hv
+    simpa only [setIntegral_const, smul_eq_mul, mul_left_comm] using hm
+  rw [← intervalIntegral_integral_swap hi] at hmean
+  have he (θ : ℝ) : (∫ v in ball (0:E) R, F θ v) = ∫ x in ball c R, f x := by
+    exact (ball_integral_unit_smul (fun x => f (c+x)) R (circleMap 0 1 θ)
+      (by simp)).trans (ball_integral_translate f c R)
+  simp_rw [he] at hmean
+  simp only [intervalIntegral.integral_const, sub_zero, smul_eq_mul] at hmean
+  nlinarith [Real.pi_pos]
+
+end ComplexHilbert
+end PotentialABP
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/BernsteinEnergyInterior.lean b/lean/OAI/Geometry/PrescribedPotential/BernsteinEnergyInterior.lean
new file mode 100644
index 0000000000000000000000000000000000000000..d49c44696506d2ad744b7df0dab260b1b4fbb552
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/BernsteinEnergyInterior.lean
@@ -0,0 +1,123 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.BernsteinHigherRescue
+import OAI.Geometry.PrescribedPotential.BernsteinLocalC2
+import OAI.Geometry.PrescribedPotential.RealCutoffBounds
+
+namespace OAI
+
+/-! Bernstein Energy Interior. -/
+
+section
+
+ 
+noncomputable section
+open Set Filter Topology
+open scoped ContDiff
+namespace HigherJet
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  {ι : Type*} [Fintype ι]
+
+lemma quartic_energy_interior {K U : Set E} (hK : IsCompact K) (hU : IsOpen U) (hKU : K ⊆ U)
+    {χ : E → ℝ} (hχ : ContDiff ℝ ∞ χ) (hχval : ∀ x ∈ K, 0 ≤ χ x ∧ χ x ≤ 1)
+    (hχint : ∀ x ∈ K, χ x ≠ 0 → x ∈ interior K)
+    {A R B : ℝ} (hA : 1 ≤ A) (hR : 1 ≤ R) (hB : 0 ≤ B) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ S T : E → ℝ, ContDiffOn ℝ ∞ S U → ContDiffOn ℝ ∞ T U →
+      ∀ v : E → ι → E, (∀ x ∈ K, ∀ i, ‖v x i‖ ≤ B) →
+      (∀ x ∈ K, 0 ≤ S x ∧ S x ≤ R ∧ 0 ≤ T x) →
+      (∀ x ∈ K, ∃ D : ℝ, 0 ≤ D ∧ T x ≤ A*D ∧ D ≤ A*T x ∧
+        D-R ≤ frameLaplace (v x) S x ∧
+        -R*(1+T x*Real.sqrt (T x)) ≤ frameLaplace (v x) T x ∧
+        ‖frameGradient (v x) S x‖^2 ≤ 4*S x*D) →
+      ∀ z ∈ K, χ z=1 → T z ≤ C := by
+  classical
+  obtain ⟨C0,hC0,hcut⟩ := cutoff_uniform_frames (ι := ι) hK χ hχ hB
+  let C1 := 1+C0+R+A*R+4*A^3*R
+  have hA0 : 0 ≤ A := by linarith
+  have hR0 : 0 ≤ R := by linarith
+  have hC1 : 1 ≤ C1 := by dsimp [C1]; nlinarith [mul_nonneg hA0 hR0, show 0 ≤ 4*A^3*R by positivity]
+  have hc0 : C0 ≤ C1 := by dsimp [C1]; nlinarith [mul_nonneg hA0 hR0, show 0 ≤ 4*A^3*R by positivity]
+  have hr1 : R ≤ C1 := by dsimp [C1]; nlinarith [mul_nonneg hA0 hR0, show 0 ≤ 4*A^3*R by positivity]
+  have har : A*R ≤ C1 := by dsimp [C1]; nlinarith [mul_nonneg hA0 hR0, show 0 ≤ 4*A^3*R by positivity]
+  have hgradC : 4*A^3*R ≤ C1 := by dsimp [C1]; nlinarith [mul_nonneg hA0 hR0, show 0 ≤ 4*A^3*R by positivity]
+  let C := (8*C1+1+A*R)*(2*quarticConstant C1 (8*C1+1+A*R)+2)^2
+  refine ⟨C,by dsimp [C]; positivity,?_⟩
+  intro S T hS hT v hv hvalues hineq z hz hz1
+  apply quartic_cutoff_uniform hK hU hKU (contDiffOn_const.mul hS) hT (hχ.pow 2).contDiffOn
+    hC1 (mul_nonneg hA0 hR0) (S := fun y => A*S y) (η := fun y => (χ y)^2) ?_ ?_ v ?_ z hz (by rw [hz1]; norm_num)
+  · intro x hx
+    obtain ⟨hs,hsR,ht⟩ := hvalues x hx
+    obtain ⟨hχ0,hχ1⟩ := hχval x hx
+    exact ⟨mul_nonneg hA0 hs,mul_le_mul_of_nonneg_left hsR hA0,ht,sq_nonneg _,by nlinarith⟩
+  · intro x hx hn
+    exact hχint x hx (fun he => hn (by rw [he]; ring))
+  · intro x hx _
+    obtain ⟨D,hD,htD,hDt,hLS,hLT,hGS⟩ := hineq x hx
+    obtain ⟨hs,hsR,ht⟩ := hvalues x hx
+    obtain ⟨hLχ,hGχ⟩ := hcut x hx (v x) (hv x hx)
+    refine ⟨?_,?_,(neg_le_neg hc0).trans hLχ,hGχ.trans (mul_le_mul_of_nonneg_right hc0 (sq_nonneg _)),?_⟩
+    · rw [frameLaplace_const_mul hU hS (hKU hx)]
+      have hh := mul_le_mul_of_nonneg_left hLS hA0
+      nlinarith only [hh,htD,har]
+    · have hh := mul_le_mul_of_nonneg_right hr1 (show 0 ≤ 1+T x*Real.sqrt (T x) by positivity)
+      nlinarith only [hh,hLT]
+    · rw [frameGradient_const_mul ((hS.contDiffAt (hU.mem_nhds (hKU hx))).differentiableAt (by simp)),norm_smul,Real.norm_eq_abs,mul_pow,sq_abs]
+      calc
+        _ ≤ A^2*(4*S x*D) := mul_le_mul_of_nonneg_left hGS (sq_nonneg A)
+        _ ≤ A^2*(4*R*(A*T x)) := mul_le_mul_of_nonneg_left
+          (mul_le_mul (mul_le_mul_of_nonneg_left hsR (by norm_num)) hDt hD (by positivity)) (sq_nonneg A)
+        _ = (4*A^3*R)*T x := by ring
+        _ ≤ C1*T x := mul_le_mul_of_nonneg_right hgradC ht
+
+lemma linear_energy_interior {K U : Set E} (hK : IsCompact K) (hU : IsOpen U) (hKU : K ⊆ U)
+    {χ : E → ℝ} (hχ : ContDiff ℝ ∞ χ) (hχval : ∀ x ∈ K, 0 ≤ χ x ∧ χ x ≤ 1)
+    (hχint : ∀ x ∈ K, χ x ≠ 0 → x ∈ interior K)
+    {A R B : ℝ} (hA : 1 ≤ A) (hR : 1 ≤ R) (hB : 0 ≤ B) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ S T : E → ℝ, ContDiffOn ℝ ∞ S U → ContDiffOn ℝ ∞ T U →
+      ∀ v : E → ι → E, (∀ x ∈ K, ∀ i, ‖v x i‖ ≤ B) →
+      (∀ x ∈ K, 0 ≤ S x ∧ S x ≤ R ∧ 0 ≤ T x) →
+      (∀ x ∈ K, ∃ D Q : ℝ, 0 ≤ D ∧ 0 ≤ Q ∧ T x ≤ A*D ∧
+        D-R ≤ frameLaplace (v x) S x ∧
+        Q-R*(1+T x) ≤ frameLaplace (v x) T x ∧
+        ‖frameGradient (v x) T x‖^2 ≤ 4*T x*Q) →
+      ∀ z ∈ K, χ z=1 → T z ≤ C := by
+  classical
+  obtain ⟨C0,hC0,hcut⟩ := cutoff_uniform_frames (ι := ι) hK χ hχ hB
+  let C1 := 1+C0+R+A*R
+  have hA0 : 0 ≤ A := by linarith
+  have hR0 : 0 ≤ R := by linarith
+  have hC1 : 0 ≤ C1 := by dsimp [C1]; nlinarith [mul_nonneg hA0 hR0, show 0 ≤ 4*A^3*R by positivity]
+  have hc0 : C0 ≤ C1 := by dsimp [C1]; nlinarith [mul_nonneg hA0 hR0, show 0 ≤ 4*A^3*R by positivity]
+  have hr1 : R ≤ C1 := by dsimp [C1]; nlinarith [mul_nonneg hA0 hR0, show 0 ≤ 4*A^3*R by positivity]
+  have har : A*R ≤ C1 := by dsimp [C1]; nlinarith [mul_nonneg hA0 hR0, show 0 ≤ 4*A^3*R by positivity]
+  let C := 2*C1*(1+(12*C1+1)/1)+((12*C1+1)/1)*(A*R)
+  refine ⟨C,by dsimp [C]; positivity,?_⟩
+  intro S T hS hT v hv hvalues hineq
+  choose D Q hD hQ htD hLS hLT hGT using hineq
+  let Q' : E → ℝ := fun x => if hx : x ∈ K then Q x hx else 0
+  intro z hz hz1
+  apply linear_cutoff_uniform hK hU hKU (contDiffOn_const.mul hS) hT (hχ.pow 2).contDiffOn
+    hC1 (mul_nonneg hA0 hR0) (show (0:ℝ)<1 by norm_num)
+    (S := fun y => A*S y) (η := fun y => (χ y)^2) ?_ ?_ v Q' ?_ z hz (by rw [hz1]; norm_num)
+  · intro x hx
+    obtain ⟨hs,hsR,ht⟩ := hvalues x hx
+    obtain ⟨hχ0,hχ1⟩ := hχval x hx
+    exact ⟨mul_nonneg hA0 hs,mul_le_mul_of_nonneg_left hsR hA0,ht,sq_nonneg _,by nlinarith⟩
+  · intro x hx hn
+    exact hχint x hx (fun he => hn (by rw [he]; ring))
+  · intro x hx _
+    have hqx : Q' x=Q x hx := dite_eq_left hx
+    rw [hqx]
+    obtain ⟨hLχ,hGχ⟩ := hcut x hx (v x) (hv x hx)
+    refine ⟨hQ x hx,?_,?_,(neg_le_neg hc0).trans hLχ,hGχ.trans (mul_le_mul_of_nonneg_right hc0 (sq_nonneg _)),hGT x hx⟩
+    · rw [frameLaplace_const_mul hU hS (hKU hx)]
+      have hh := mul_le_mul_of_nonneg_left (hLS x hx) hA0
+      have ht := htD x hx
+      nlinarith only [hh,ht,har]
+    · have hh := mul_le_mul_of_nonneg_right hr1 (show 0 ≤ 1+T x by linarith [(hvalues x hx).2.2])
+      linarith only [hLT x hx,hh]
+end HigherJet
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/BernsteinHigherRescue.lean b/lean/OAI/Geometry/PrescribedPotential/BernsteinHigherRescue.lean
new file mode 100644
index 0000000000000000000000000000000000000000..8c0ea06245e48c1d327476fc3c67de6a2d4672a5
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/BernsteinHigherRescue.lean
@@ -0,0 +1,91 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CalabiCutoffAlgebra
+import OAI.Geometry.PrescribedPotential.RealFrameScalar
+
+namespace OAI
+
+/-! Bernstein Higher Rescue. -/
+
+section
+
+ 
+noncomputable section
+open Set Filter Topology
+open scoped ContDiff
+namespace HigherJet
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  {ι : Type*} [Fintype ι]
+
+lemma frameLaplace_const_mul {U : Set E} (hU : IsOpen U) {f : E → ℝ}
+    (hf : ContDiffOn ℝ ∞ f U) {x : E} (hx : x ∈ U) (v : ι → E) (c : ℝ) :
+    frameLaplace v (fun y => c*f y) x = c*frameLaplace v f x := by
+  rw [frameLaplace_mul hU contDiffOn_const hf hx,frameLaplace_const,frameGradient_const]
+  simp
+
+lemma linear_cutoff_uniform {K U : Set E} (hK : IsCompact K) (hU : IsOpen U) (hKU : K ⊆ U)
+    {S T η : E → ℝ} (hS : ContDiffOn ℝ ∞ S U) (hT : ContDiffOn ℝ ∞ T U)
+    (hη : ContDiffOn ℝ ∞ η U) {C R δ : ℝ} (hC : 0 ≤ C) (_hR : 0 ≤ R) (hδ : 0 < δ)
+    (hvalues : ∀ x ∈ K, 0 ≤ S x ∧ S x ≤ R ∧ 0 ≤ T x ∧ 0 ≤ η x ∧ η x ≤ 1)
+    (hinterior : ∀ x ∈ K, η x ≠ 0 → x ∈ interior K)
+    (v : E → ι → E) (Q : E → ℝ)
+    (hineq : ∀ x ∈ K, η x ≠ 0 →
+      0 ≤ Q x ∧ δ*T x-C ≤ frameLaplace (v x) S x ∧
+      Q x-C*(1+T x) ≤ frameLaplace (v x) T x ∧
+      -C ≤ frameLaplace (v x) η x ∧
+      ‖frameGradient (v x) η x‖^2 ≤ C*η x ∧
+      ‖frameGradient (v x) T x‖^2 ≤ 4*T x*Q x) :
+    ∀ z ∈ K, η z = 1 → T z ≤ 2*C*(1+(12*C+1)/δ)+((12*C+1)/δ)*R := by
+  let A := (12*C+1)/δ
+  have hA : 0 ≤ A := div_nonneg (by positivity) hδ.le
+  let W : E → ℝ := fun x => η x*T x+A*S x
+  have hW : ContDiffOn ℝ ∞ W U := (hη.mul hT).add (contDiffOn_const.mul hS)
+  intro z hz hηz
+  obtain ⟨x,hx,hmax⟩ := hK.exists_isMaxOn ⟨z,hz⟩ (hW.continuousOn.mono hKU)
+  obtain ⟨hSx,hSRx,hTx,hηx,hη1x⟩ := hvalues x hx
+  have hL : T z ≤ W x := by
+    have h := hmax hz
+    change η z*T z+A*S z ≤ W x at h
+    rw [hηz,one_mul] at h
+    exact (le_add_of_nonneg_right (mul_nonneg hA (hvalues z hz).1)).trans h
+  have hB : W x ≤ 2*C*(1+A)+A*R := by
+    by_cases hn : η x=0
+    · dsimp only [W]
+      rw [hn,zero_mul,zero_add]
+      exact (mul_le_mul_of_nonneg_left hSRx hA).trans (le_add_of_nonneg_left (by positivity))
+    have hm : IsLocalMax W x := by
+      filter_upwards [mem_interior_iff_mem_nhds.mp (hinterior x hx hn)] with y hy
+      exact hmax hy
+    have hmaxL := frameLaplace_nonpos_of_localMax (hW.contDiffAt (hU.mem_nhds (hKU hx))) hm (v x)
+    change frameLaplace (v x) (fun y => η y*T y+A*S y) x ≤ 0 at hmaxL
+    rw [frameLaplace_add hU (hη.mul hT) (contDiffOn_const.mul hS) (hKU hx),
+      frameLaplace_mul hU hη hT (hKU hx),frameLaplace_const_mul hU hS (hKU hx)] at hmaxL
+    obtain ⟨hQ,hLS,hLT,hLη,hGη,hGT⟩ := hineq x hx hn
+    have hcauchy : (inner ℝ (frameGradient (v x) η x) (frameGradient (v x) T x))^2 ≤
+        ‖frameGradient (v x) η x‖^2*‖frameGradient (v x) T x‖^2 := by
+      simpa only [sq_abs,mul_pow] using
+        (sq_le_sq₀ (abs_nonneg (inner ℝ (frameGradient (v x) η x) (frameGradient (v x) T x)))
+          (mul_nonneg (norm_nonneg _) (norm_nonneg _))).mpr
+          (abs_real_inner_le_norm (frameGradient (v x) η x) (frameGradient (v x) T x))
+    have hcross : (2*inner ℝ (frameGradient (v x) η x) (frameGradient (v x) T x))^2 ≤
+        8*(2*C)*η x*T x*Q x := by
+      have h := mul_le_mul hGη hGT (sq_nonneg _) (mul_nonneg hC hηx)
+      nlinarith only [hcauchy,h]
+    have hh : T x ≤ 2*C*(1+A) := by
+      have hls : Q x-(2*C)*(1+T x) ≤ frameLaplace (v x) T x := by
+        have : 0 ≤ C*(1+T x) := by positivity
+        linarith only [hLT,this]
+      have hlt : δ*T x-2*C ≤ frameLaplace (v x) S x := by linarith only [hLS,hC]
+      have hle : -(2*C) ≤ frameLaplace (v x) η x := by linarith only [hLη,hC]
+      have ha : (6*(2*C)+1)/δ = A := by dsimp [A]; congr 1; ring
+      simpa only [ha] using CalabiEstimate.cutoff_max_bound hTx hQ hηx hη1x
+        (show 0 ≤ 2*C by positivity) hδ hls hlt hle hcross (by simpa only [ha] using hmaxL)
+    dsimp only [W]
+    exact add_le_add ((mul_le_mul_of_nonneg_right hη1x hTx).trans (by simpa using hh))
+      (mul_le_mul_of_nonneg_left hSRx hA)
+  exact hL.trans hB
+end HigherJet
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/BernsteinLocalC2.lean b/lean/OAI/Geometry/PrescribedPotential/BernsteinLocalC2.lean
new file mode 100644
index 0000000000000000000000000000000000000000..1378d4e98623e0473cfcc8c6a5f9d6ce84fa192a
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/BernsteinLocalC2.lean
@@ -0,0 +1,91 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.BernsteinProductMaximum
+import OAI.Geometry.PrescribedPotential.BernsteinQuarticAlgebra
+
+namespace OAI
+
+/-! Bernstein Local C2. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology
+open scoped ContDiff
+namespace HigherJet
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  {ι : Type*} [Fintype ι]
+
+def quarticConstant (C B : ℝ) : ℝ := B*C+C+3*B*C+4*C^2+1
+
+lemma quartic_cutoff_uniform {K U : Set E} (hK : IsCompact K) (hU : IsOpen U) (hKU : K ⊆ U)
+    {S T η : E → ℝ} (hS : ContDiffOn ℝ ∞ S U) (hT : ContDiffOn ℝ ∞ T U)
+    (hη : ContDiffOn ℝ ∞ η U) {C R : ℝ} (hC : 1 ≤ C) (hR : 0 ≤ R)
+    (hvalues : ∀ x ∈ K, 0 ≤ S x ∧ S x ≤ R ∧ 0 ≤ T x ∧ 0 ≤ η x ∧ η x ≤ 1)
+    (hinterior : ∀ x ∈ K, η x ≠ 0 → x ∈ interior K)
+    (v : E → ι → E)
+    (hineq : ∀ x ∈ K, η x ≠ 0 →
+      T x-C ≤ frameLaplace (v x) S x ∧
+      -C*(1+T x*Real.sqrt (T x)) ≤ frameLaplace (v x) T x ∧
+      -C ≤ frameLaplace (v x) η x ∧
+      ‖frameGradient (v x) η x‖^2 ≤ C*η x ∧
+      ‖frameGradient (v x) S x‖^2 ≤ C*T x) :
+    ∀ z ∈ K, η z = 1 →
+      T z ≤ (8*C+1+R)*(2*quarticConstant C (8*C+1+R)+2)^2 := by
+  let A := 8*C+1
+  let B := A+R
+  let D := quarticConstant C B
+  have hA : 1 ≤ A := by dsimp [A]; linarith
+  have hB : 0 ≤ B := by dsimp [B]; linarith
+  have hD : 1 ≤ D := by
+    have h : 0 ≤ B*C+C+3*B*C+4*C^2 := by positivity
+    dsimp [D,quarticConstant]; linarith
+  let W : E → ℝ := fun x => η x*(A+S x)*T x
+  have hW : ContDiffOn ℝ ∞ W U := (hη.mul (contDiffOn_const.add hS)).mul hT
+  intro z hz hηz
+  obtain ⟨x,hx,hmax⟩ := hK.exists_isMaxOn ⟨z,hz⟩ (hW.continuousOn.mono hKU)
+  obtain ⟨hSx,hSRx,hTx,hηx,hη1x⟩ := hvalues x hx
+  obtain ⟨hSz,_,hTz,_,_⟩ := hvalues z hz
+  have hL : A*T z ≤ W x := by
+    have hh := hmax hz
+    change η z*(A+S z)*T z ≤ W x at hh
+    rw [hηz,one_mul] at hh
+    nlinarith only [hh,mul_nonneg hSz hTz]
+  have hAB : A+S x ≤ B := by dsimp [B]; linarith
+  have hax : 0 < A+S x := by linarith
+  by_cases hn : η x=0
+  · have he : W x=0 := by dsimp [W]; rw [hn]; ring
+    rw [he] at hL
+    have : T z ≤ 0 := by nlinarith only [hL,hA,hTz]
+    exact this.trans (mul_nonneg (by dsimp [A,B] at hB; exact hB) (sq_nonneg _))
+  have hηpos : 0 < η x := lt_of_le_of_ne hηx (Ne.symm hn)
+  have hm : IsLocalMax W x := by
+    filter_upwards [mem_interior_iff_mem_nhds.mp (hinterior x hx hn)] with y hy
+    exact hmax hy
+  have hnum := quartic_test_at_max hU hη hS hT (hKU hx) (v x) A hn (ne_of_gt hax) hm
+  obtain ⟨hLS,hLT,hLη,hGη,hGS⟩ := hineq x hx hn
+  have hcross : (inner ℝ (frameGradient (v x) η x) (frameGradient (v x) S x))^2 ≤
+      ‖frameGradient (v x) η x‖^2*‖frameGradient (v x) S x‖^2 := by
+    simpa only [sq_abs,mul_pow] using
+      (sq_le_sq₀ (abs_nonneg (inner ℝ (frameGradient (v x) η x) (frameGradient (v x) S x)))
+        (mul_nonneg (norm_nonneg _) (norm_nonneg _))).mpr
+        (abs_real_inner_le_norm (frameGradient (v x) η x) (frameGradient (v x) S x))
+  have hb := BernsteinEstimate.quartic_max_reduction hηpos hη1x hTx
+    (Real.sqrt_nonneg _) (Real.sq_sqrt hTx) hC (by dsimp only [A]; linarith only [hSx]) hAB
+    hLS hLT hLη (sq_nonneg _) (sq_nonneg _) hGη hGS hcross hnum
+  have hbound : η x*T x ≤ (2*D+2)^2 :=
+    BernsteinEstimate.weighted_quartic_bound hηx hη1x hTx hD hb
+  have hWB : W x ≤ B*(2*D+2)^2 := by
+    calc
+      _ = (A+S x)*(η x*T x) := by dsimp [W]; ring
+      _ ≤ B*(2*D+2)^2 := mul_le_mul hAB hbound (mul_nonneg hηx hTx) hB
+  change T z ≤ B*(2*D+2)^2
+  have hAT : T z ≤ A*T z := le_mul_of_one_le_left hTz hA
+  exact hAT.trans (hL.trans hWB)
+end HigherJet
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/BernsteinProductMaximum.lean b/lean/OAI/Geometry/PrescribedPotential/BernsteinProductMaximum.lean
new file mode 100644
index 0000000000000000000000000000000000000000..98f379d8be3de8275e74cb5870f08d72466d1a8f
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/BernsteinProductMaximum.lean
@@ -0,0 +1,78 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.RealFrameScalar
+import OAI.Geometry.PrescribedPotential.WirtingerCalculus
+
+namespace OAI
+
+/-! Bernstein Product Maximum. -/
+
+section
+
+ 
+noncomputable section
+open Set Filter Topology Finset
+open scoped ContDiff
+namespace BernsteinEstimate
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V]
+
+lemma product_gradient_cross_identity {η a T : ℝ} (hη : η ≠ 0) (ha : a ≠ 0)
+    (X Y Z : V) (h : (a*T) • X+(η*T) • Y+(η*a) • Z=0) :
+    2*η*inner ℝ Y Z+2*a*inner ℝ X Z+2*T*inner ℝ X Y =
+      -2*η*T/a*‖Y‖^2-2*a*T/η*‖X‖^2-2*T*inner ℝ X Y := by
+  have hx := congrArg (fun W => inner ℝ X W) h
+  have hy := congrArg (fun W => inner ℝ Y W) h
+  simp only [inner_add_right,inner_smul_right,inner_zero_right,real_inner_self_eq_norm_sq] at hx hy
+  rw [real_inner_comm X Y] at hy
+  have hx' := congrArg (fun r : ℝ => 2*a*r) hx
+  have hy' := congrArg (fun r : ℝ => 2*η*r) hy
+  field_simp
+  nlinarith only [hx',hy']
+end BernsteinEstimate
+
+namespace HigherJet
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  {ι : Type*} [Fintype ι]
+
+lemma quartic_test_at_max {U : Set E} (hU : IsOpen U) {η S T : E → ℝ}
+    (hηs : ContDiffOn ℝ ∞ η U) (hSs : ContDiffOn ℝ ∞ S U) (hTs : ContDiffOn ℝ ∞ T U)
+    {x : E} (hx : x ∈ U) (v : ι → E) (A : ℝ)
+    (hηx : η x ≠ 0) (hax : A+S x ≠ 0)
+    (hmax : IsLocalMax (fun y => η y*(A+S y)*T y) x) :
+    η x*(A+S x)*frameLaplace v T x+η x*T x*frameLaplace v S x+
+      (A+S x)*T x*frameLaplace v η x -
+      2*η x*T x/(A+S x)*‖frameGradient v S x‖^2 -
+      2*(A+S x)*T x/η x*‖frameGradient v η x‖^2 -
+      2*T x*inner ℝ (frameGradient v η x) (frameGradient v S x) ≤ 0 := by
+  let a : E → ℝ := fun y => A+S y
+  have has : ContDiffOn ℝ ∞ a U := contDiffOn_const.add hSs
+  have hηa := (hηs.contDiffAt (hU.mem_nhds hx)).differentiableAt (by simp)
+  have hSa := (hSs.contDiffAt (hU.mem_nhds hx)).differentiableAt (by simp)
+  have hTa := (hTs.contDiffAt (hU.mem_nhds hx)).differentiableAt (by simp)
+  have haa := (has.contDiffAt (hU.mem_nhds hx)).differentiableAt (by simp)
+  have hg : frameGradient v a x = frameGradient v S x := by
+    rw [show a = fun y => A+S y from rfl,frameGradient_add (differentiableAt_const A) hSa,frameGradient_const,zero_add]
+  have hl : frameLaplace v a x = frameLaplace v S x := by
+    rw [show a = fun y => A+S y from rfl,frameLaplace_add hU contDiffOn_const hSs hx,frameLaplace_const,zero_add]
+  have hgrad := frameGradient_eq_zero_of_localMax hmax v
+  have hm := frameLaplace_nonpos_of_localMax
+    (((hηs.mul has).mul hTs).contDiffAt (hU.mem_nhds hx)) hmax v
+  change frameGradient v (fun y => (η y*a y)*T y) x = 0 at hgrad
+  rw [frameGradient_mul (f := fun y => η y*a y) (hηa.mul haa) hTa,
+    frameGradient_mul hηa haa,hg,smul_add,smul_smul,smul_smul] at hgrad
+  have hcross := BernsteinEstimate.product_gradient_cross_identity (T := T x) hηx hax
+    (frameGradient v η x) (frameGradient v S x) (frameGradient v T x)
+    (by simpa only [mul_comm,mul_left_comm,mul_assoc,a] using hgrad)
+  change frameLaplace v (fun y => (η y*a y)*T y) x ≤ 0 at hm
+  rw [frameLaplace_mul hU (f := fun y => η y*a y) (hηs.mul has) hTs hx,
+    frameLaplace_mul hU hηs has hx,frameGradient_mul hηa haa,hg,hl,
+    inner_add_left,inner_smul_left,inner_smul_left] at hm
+  dsimp only [a] at hm
+  simp only [starRingEnd_apply,star_trivial] at hm
+  simp only [neg_mul,neg_div] at hcross
+  nlinarith only [hm,hcross]
+end HigherJet
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/BernsteinQuarticAlgebra.lean b/lean/OAI/Geometry/PrescribedPotential/BernsteinQuarticAlgebra.lean
new file mode 100644
index 0000000000000000000000000000000000000000..e9d9fc213e26a610077f1134760f52d5e1339538
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/BernsteinQuarticAlgebra.lean
@@ -0,0 +1,94 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CalabiCutoffAlgebra
+
+namespace OAI
+
+/-! Bernstein Quartic Algebra. -/
+
+section
+
+ 
+namespace BernsteinEstimate
+
+lemma quadratic_root_bound {u C : ℝ} (hu : 0 ≤ u) (hC : 1 ≤ C)
+    (h : u^2/2 ≤ C*u+2*C) : u^2 ≤ (2*C+2)^2 := by
+  have hle : u ≤ 2*C+2 := by
+    by_contra hn
+    have hh : 2*C+2 < u := lt_of_not_ge hn
+    nlinarith [mul_pos (by linarith : 0 < u-(2*C+2)) (by linarith : 0 < u+2)]
+  exact pow_le_pow_left₀ hu hle 2
+
+lemma weighted_quartic_bound {η T C : ℝ} (hη : 0 ≤ η) (hη1 : η ≤ 1)
+    (hT : 0 ≤ T) (hC : 1 ≤ C)
+    (h : η*T^2/2 ≤ C*η*T*Real.sqrt T+C*T+C) :
+    η*T ≤ (2*C+2)^2 := by
+  by_cases ht : T ≤ 1
+  · have he := mul_le_mul hη1 ht hT (by norm_num : (0:ℝ) ≤ 1)
+    nlinarith
+  have ht' : 0 < T := by linarith
+  have h1 : η*T/2 ≤ C*η*Real.sqrt T+2*C := by
+    have hc : C ≤ C*T := le_mul_of_one_le_right (by linarith) (le_of_not_ge ht)
+    have hh : T*(η*T/2) ≤ T*(C*η*Real.sqrt T+2*C) := by nlinarith only [h,hc]
+    exact (mul_le_mul_iff_right₀ ht').mp hh
+  let u := Real.sqrt (η*T)
+  have hu : 0 ≤ u := Real.sqrt_nonneg _
+  have hu2 : u^2 = η*T := Real.sq_sqrt (mul_nonneg hη hT)
+  have hs : η*Real.sqrt T ≤ u := by
+    apply (sq_le_sq₀ (mul_nonneg hη (Real.sqrt_nonneg _)) hu).mp
+    rw [mul_pow,Real.sq_sqrt hT,hu2]
+    exact mul_le_mul_of_nonneg_right (by nlinarith only [hη,hη1] : η^2 ≤ η) hT
+  have h2 : u^2/2 ≤ C*u+2*C := by
+    have hmul := mul_le_mul_of_nonneg_left hs (by linarith : 0 ≤ C)
+    nlinarith only [h1,hmul,hu2]
+  rw [← hu2]
+  exact quadratic_root_bound hu hC h2
+
+ 
+
+lemma quartic_max_reduction {η T r a B C LS LT Lη xs ys z : ℝ}
+    (hη : 0 < η) (hη1 : η ≤ 1) (hT : 0 ≤ T) (hr : 0 ≤ r) (_hr2 : r^2=T)
+    (hC : 1 ≤ C) (ha : 8*C+1 ≤ a) (haB : a ≤ B)
+    (hLS : T-C ≤ LS) (hLT : -C*(1+T*r) ≤ LT) (hLη : -C ≤ Lη)
+    (_hxs : 0 ≤ xs) (hys : 0 ≤ ys) (hxsB : xs ≤ C*η) (hysB : ys ≤ C*T)
+    (hz : z^2 ≤ xs*ys)
+    (hmax : η*a*LT+η*T*LS+a*T*Lη-2*η*T/a*ys-2*a*T/η*xs-2*T*z ≤ 0) :
+    η*T^2/2 ≤ (B*C+C+3*B*C+4*C^2+1)*η*T*r +
+      (B*C+C+3*B*C+4*C^2+1)*T + (B*C+C+3*B*C+4*C^2+1) := by
+  have hC0 : 0 ≤ C := by linarith
+  have ha0 : 0 < a := by linarith
+  have hB0 : 0 ≤ B := by linarith
+  have hyterm : 2*η*T/a*ys ≤ η*T^2/4 := by
+    rw [div_mul_eq_mul_div]
+    apply (div_le_iff₀ ha0).mpr
+    have hh := mul_le_mul_of_nonneg_left hysB (by positivity : 0 ≤ 2*η*T)
+    have haa : 8*C ≤ a := by linarith
+    have hhh := mul_le_mul_of_nonneg_right haa (by positivity : 0 ≤ η*T^2)
+    nlinarith only [hh,hhh]
+  have hxterm : 2*a*T/η*xs ≤ 2*B*C*T := by
+    calc
+      _ ≤ 2*a*T/η*(C*η) := mul_le_mul_of_nonneg_left hxsB (by positivity)
+      _ = 2*a*C*T := by field_simp
+      _ ≤ 2*B*C*T := by nlinarith only [mul_le_mul_of_nonneg_right haB (mul_nonneg hC0 hT)]
+  have hzz : (-2*T*z)^2 ≤ 8*(C^2*T)*(η*T^2/2) := by
+    have hh := hz.trans (mul_le_mul hxsB hysB hys (mul_nonneg hC0 hη.le))
+    have hh' := mul_le_mul_of_nonneg_left hh (by positivity : 0 ≤ 4*T^2)
+    nlinarith only [hh']
+  have hcross := CalabiEstimate.cross_absorb (by positivity : 0 ≤ C^2*T)
+    (by positivity : 0 ≤ η*T^2/2) hzz
+  have h1 := mul_le_mul_of_nonneg_left hLT (by positivity : 0 ≤ η*a)
+  have h2 := mul_le_mul_of_nonneg_left hLS (by positivity : 0 ≤ η*T)
+  have h3 := mul_le_mul_of_nonneg_left hLη (by positivity : 0 ≤ a*T)
+  have hab0 := mul_le_mul_of_nonneg_right haB (by positivity : 0 ≤ η*C)
+  have hab1 := mul_le_mul_of_nonneg_right haB (by positivity : 0 ≤ η*C*T*r)
+  have hab2 := mul_le_mul_of_nonneg_right haB (by positivity : 0 ≤ C*T)
+  have heet := mul_le_mul_of_nonneg_right hη1 (by positivity : 0 ≤ C*T)
+  have heec := mul_le_mul_of_nonneg_right hη1 (by positivity : 0 ≤ B*C)
+  have herr0 : 0 ≤ (C+3*B*C+4*C^2+1)*η*T*r := by positivity
+  have herr1 : 0 ≤ (B*C+1)*T := by positivity
+  have herr2 : 0 ≤ C+3*B*C+4*C^2+1 := by positivity
+  nlinarith only [hmax,h1,h2,h3,hyterm,hxterm,hcross,hab0,hab1,hab2,heet,heec,herr0,herr1,herr2]
+end BernsteinEstimate
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/BootstrapCalculus.lean b/lean/OAI/Geometry/PrescribedPotential/BootstrapCalculus.lean
new file mode 100644
index 0000000000000000000000000000000000000000..708a543093f11843f97c4c9d28de847172cf817e
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/BootstrapCalculus.lean
@@ -0,0 +1,62 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CircleRadialCalculus
+import OAI.Geometry.PrescribedPotential.KaehlerClosedDerivatives
+
+namespace OAI
+
+/-! Bootstrap Calculus. -/
+
+section
+
+ 
+noncomputable section
+namespace FixedBootstrapCalculus
+variable {E F G H : Type*}
+  [NormedAddCommGroup E] [NormedSpace ℝ E]
+  [NormedAddCommGroup F] [NormedSpace ℝ F]
+  [NormedAddCommGroup G] [NormedSpace ℝ G]
+  [NormedAddCommGroup H] [NormedSpace ℝ H]
+
+lemma derivative_compatibility (B : E → E →L[ℝ] F) (C : G → G →L[ℝ] H)
+    (L : E →L[ℝ] G) (M : F →L[ℝ] H)
+    (hB : Differentiable ℝ B) (hC : Differentiable ℝ C)
+    (he : ∀ y v, C (L y) (L v) = M (B y v)) (u w v : E) :
+    fderiv ℝ C (L u) (L w) (L v) = M (fderiv ℝ B u w v) := by
+  have hL := ((hC (L u)).hasFDerivAt.comp u L.hasFDerivAt).clm_apply
+    (hasFDerivAt_const (L v) u)
+  have hR := M.hasFDerivAt.comp u
+    ((hB u).hasFDerivAt.clm_apply (hasFDerivAt_const v u))
+  have he' : (fun y => C (L y) (L v)) = (fun y => M (B y v)) := funext (fun y => he y v)
+  simp only [Function.comp_def] at hL hR
+  rw [he'] at hL
+  have hh := congrArg (fun T : E →L[ℝ] H => T w) (hL.unique hR)
+  simpa only [add_apply, ContinuousLinearMap.comp_apply,
+    zero_apply, map_zero, zero_add, ContinuousLinearMap.flip_apply] using hh
+
+lemma differentiated_commutator (B : E → E →L[ℝ] F)
+    (L P : G →L[ℝ] E) (Q M : H →L[ℝ] F) (V N : G → H)
+    (hB : Differentiable ℝ B) (hV : Differentiable ℝ V) (hN : Differentiable ℝ N)
+    (he : ∀ y, B (L y) (P y) = Q (V y) + M (N y)) (u w : G) :
+    B (L u) (P w) + fderiv ℝ B (L u) (L w) (P u) =
+      Q (fderiv ℝ V u w) + M (fderiv ℝ N u w) := by
+  have hL := ((hB (L u)).hasFDerivAt.comp u L.hasFDerivAt).clm_apply P.hasFDerivAt
+  have hR := (Q.hasFDerivAt.comp u (hV u).hasFDerivAt).add
+    (M.hasFDerivAt.comp u (hN u).hasFDerivAt)
+  have he' : (fun y => B (L y) (P y)) = (fun y => Q (V y) + M (N y)) := funext he
+  simp only [Function.comp_def] at hL hR
+  rw [he'] at hL
+  have hh := congrArg (fun T : G →L[ℝ] F => T w) (hL.unique hR)
+  simpa only [add_apply, ContinuousLinearMap.comp_apply,
+    ContinuousLinearMap.flip_apply] using hh
+lemma lowered_remainder (M : H →L[ℝ] F) (a b c : F) (r e : H)
+    (h : a + b = c + M r) (hb : b = M e) : a = c + M (r-e) := by
+  rw [hb] at h
+  rw [map_sub]
+  exact (eq_sub_iff_add_eq.mpr h).trans (by abel)
+
+end FixedBootstrapCalculus
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/BoundedBilinearCompletion.lean b/lean/OAI/Geometry/PrescribedPotential/BoundedBilinearCompletion.lean
new file mode 100644
index 0000000000000000000000000000000000000000..a8beafc44358647e08840506bbf1da98e57996ef
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/BoundedBilinearCompletion.lean
@@ -0,0 +1,62 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CircleRadialCalculus
+import OAI.Geometry.PrescribedPotential.GlobalExtension
+import OAI.Geometry.PrescribedPotential.KaehlerClosedDerivatives
+
+namespace OAI
+
+/-! Bounded Bilinear Completion. -/
+
+section
+
+ 
+
+noncomputable section
+namespace BoundedBilinearCompletion
+variable {V H : Type*} [AddCommGroup V] [Module ℝ V]
+  [NormedAddCommGroup H] [NormedSpace ℝ H] [CompleteSpace H]
+  (e : V →ₗ[ℝ] H) (he : DenseRange e) (B : V →ₗ[ℝ] V →ₗ[ℝ] V)
+  {C : ℝ} (hC : 0 ≤ C) (hB : ∀ u v, ‖e (B u v)‖ ≤ C*(‖e u‖*‖e v‖))
+
+def right (u : V) : H →L[ℝ] H := (e.comp (B u)).extendOfNorm e
+
+include he hB in
+lemma right_eq (u v : V) : right e B u (e v) = e (B u v) :=
+  LinearMap.extendOfNorm_eq he ⟨C*‖e u‖, fun v => by simpa only [mul_assoc, LinearMap.comp_apply] using hB u v⟩ v
+
+include he hC hB in
+lemma right_bound (u : V) : ‖right e B u‖ ≤ C*‖e u‖ :=
+  LinearMap.opNorm_extendOfNorm_le he (mul_nonneg hC (norm_nonneg _))
+    (fun v => by simpa only [mul_assoc, LinearMap.comp_apply] using hB u v)
+
+def rightLinear : V →ₗ[ℝ] (H →L[ℝ] H) where
+  toFun := right e B
+  map_add' u v := by
+    apply DFunLike.coe_injective
+    apply he.equalizer (by fun_prop) (by fun_prop)
+    funext w
+    simp only [Function.comp_apply, add_apply, right_eq e he B hB,
+      map_add, LinearMap.add_apply]
+  map_smul' c u := by
+    apply DFunLike.coe_injective
+    apply he.equalizer (by fun_prop) (by fun_prop)
+    funext w
+    simp only [Function.comp_apply, smul_apply, right_eq e he B hB,
+      map_smul, LinearMap.smul_apply, RingHom.id_apply]
+
+def extend : H →L[ℝ] H →L[ℝ] H := (rightLinear e he B hB).extendOfNorm e
+
+include hC in
+lemma extend_eq (u v : V) : extend e he B hB (e u) (e v) = e (B u v) := by
+  rw [extend, LinearMap.extendOfNorm_eq he ⟨C, fun u => right_bound e he B hC hB u⟩]
+  exact right_eq e he B hB u v
+
+include hC in
+lemma extend_bound : ‖extend e he B hB‖ ≤ C :=
+  LinearMap.opNorm_extendOfNorm_le he hC (fun u => right_bound e he B hC hB u)
+end BoundedBilinearCompletion
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/BoundedNormalFrames.lean b/lean/OAI/Geometry/PrescribedPotential/BoundedNormalFrames.lean
new file mode 100644
index 0000000000000000000000000000000000000000..300f3673f6cd40750f80812ac98d6c12f077ad56
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/BoundedNormalFrames.lean
@@ -0,0 +1,108 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.FrameCompactBounds
+
+namespace OAI
+
+/-! Bounded Normal Frames. -/
+
+section
+
+noncomputable section
+open Set Metric Filter Topology Matrix
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace KaehlerCalculus
+variable {n : ℕ}
+
+lemma frameGram_compact_lower {X : Type*} [TopologicalSpace X]
+    {L : Set X} (hL : IsCompact L) (G : X → Matrix (Fin n) (Fin n) ℂ)
+    (hG : ContinuousOn G L) (hp : ∀ x ∈ L, (G x).PosDef) :
+    ∃ δ : ℝ, 0 < δ ∧ ∀ x ∈ L, ∀ B, δ*‖B‖^2 ≤ frameGram (G x) B := by
+  apply compact_homogeneous_positive hL (fun x B => frameGram (G x) B)
+  · apply continuousOn_trace_re
+    exact ((continuousOn_matrix_star continuous_snd.continuousOn).mul
+      (hG.comp continuous_fst.continuousOn (fun _ h => h.1))).mul continuous_snd.continuousOn
+  · exact fun x hx B hB => frameGram_pos (hp x hx) hB
+  · exact fun x _ r B => frameGram_smul (G x) B r
+
+lemma unnormalize_matrix {C D M : Matrix (Fin n) (Fin n) ℂ}
+    (hCD : C*D = 1) (hM : Cᴴ*M*C = 1) : M = Dᴴ*D := by
+  calc
+    M = (C*D)ᴴ*M*(C*D) := by rw [hCD]; simp
+    _ = Dᴴ*(Cᴴ*M*C)*D := by simp only [Matrix.conjTranspose_mul,mul_assoc]
+    _ = Dᴴ*D := by rw [hM,mul_one]
+
+lemma normalizing_frame_gram_le {C G M : Matrix (Fin n) (Fin n) ℂ}
+    {R : ℝ} (h : ((R:ℂ) • M-G).PosSemidef) (hNorm : Cᴴ*M*C = 1) :
+    frameGram G C ≤ R*n := by
+  have hh := (Complex.nonneg_iff.mp (h.conjTranspose_mul_mul_same C).trace_nonneg).1
+  rw [mul_sub,sub_mul,Matrix.mul_smul,Matrix.smul_mul,hNorm,Matrix.trace_sub,Matrix.trace_smul] at hh
+  simpa only [Complex.sub_re,smul_eq_mul,Complex.mul_re,Complex.ofReal_re,Complex.ofReal_im,
+    zero_mul,sub_zero,Matrix.trace_one,Fintype.card_fin,Complex.natCast_re,sub_nonneg,frameGram] using hh
+
+lemma metric_trace_le {G M : Matrix (Fin n) (Fin n) ℂ} {K : ℝ}
+    (h : ((K:ℂ) • G-M).PosSemidef) : M.trace.re ≤ K*G.trace.re := by
+  have hh := (Complex.nonneg_iff.mp h.trace_nonneg).1
+  rw [Matrix.trace_sub,Matrix.trace_smul] at hh
+  simpa only [Complex.sub_re,smul_eq_mul,Complex.mul_re,Complex.ofReal_re,Complex.ofReal_im,
+    zero_mul,sub_zero,sub_nonneg] using hh
+
+lemma compact_normalizing_frames {X : Type*} [TopologicalSpace X]
+    {L : Set X} (hL : IsCompact L) (G : X → Matrix (Fin n) (Fin n) ℂ)
+    (hG : ContinuousOn G L) (hp : ∀ x ∈ L, (G x).PosDef)
+    {R K : ℝ} (hR : 0 ≤ R) (hK : 0 ≤ K) :
+    ∃ B : ℝ, 0 ≤ B ∧ ∀ x ∈ L, ∀ M C D : Matrix (Fin n) (Fin n) ℂ,
+      ((R:ℂ) • M-G x).PosSemidef → ((K:ℂ) • G x-M).PosSemidef →
+      C*D = 1 → Cᴴ*M*C = 1 → ‖C‖ ≤ B ∧ ‖D‖ ≤ B := by
+  obtain ⟨a,ha,hGa⟩ := frameGram_compact_lower hL G hG hp
+  obtain ⟨b,hb,hIb⟩ := frameGram_compact_lower (n := n)
+    (X := Unit) (L := univ) isCompact_univ (fun _ => 1) continuousOn_const (fun _ _ => Matrix.PosDef.one)
+  obtain ⟨T,hT⟩ := hL.exists_bound_of_continuousOn (continuousOn_trace_re hG)
+  let Tc := max T 0
+  let B := max (R*n/a+1) (K*Tc/b+1)
+  have htc : 0 ≤ Tc := le_max_right _ _
+  refine ⟨B,?_,?_⟩
+  · exact (by dsimp [B]; exact (by positivity : 0 ≤ R*n/a+1).trans (le_max_left _ _))
+  intro x hx M C D hlow hupp hCD hNorm
+  have htr : (G x).trace.re ≤ Tc := by
+    exact (le_abs_self _).trans ((hT x hx).trans (le_max_left _ _))
+  have hc := (hGa x hx C).trans (normalizing_frame_gram_le hlow hNorm)
+  have hc2 : ‖C‖^2 ≤ R*n/a := (le_div_iff₀ ha).mpr (by simpa only [mul_comm] using hc)
+  have hc1 : ‖C‖ ≤ R*n/a+1 := by nlinarith [sq_nonneg (‖C‖-1)]
+  have hd := hIb () (mem_univ _) D
+  have hDgram : frameGram 1 D = M.trace.re := by
+    rw [unnormalize_matrix hCD hNorm]
+    simp only [frameGram,mul_one]
+  rw [hDgram] at hd
+  have hd2 : ‖D‖^2 ≤ K*Tc/b := (le_div_iff₀ hb).mpr (by
+    have hm := (metric_trace_le hupp).trans (mul_le_mul_of_nonneg_left htr hK)
+    exact (by simpa only [mul_comm] using hd.trans hm))
+  have hd1 : ‖D‖ ≤ K*Tc/b+1 := by nlinarith [sq_nonneg (‖D‖-1)]
+  exact ⟨hc1.trans (le_max_left _ _),hd1.trans (le_max_right _ _)⟩
+
+def boundedFramePairs (B : ℝ) : Set (Matrix (Fin n) (Fin n) ℂ × Matrix (Fin n) (Fin n) ℂ) :=
+  {p | ‖p.1‖ ≤ B ∧ ‖p.2‖ ≤ B ∧ p.1*p.2 = 1 ∧ p.2*p.1 = 1}
+
+lemma boundedFramePairs_compact (B : ℝ) : IsCompact (boundedFramePairs (n := n) B) := by
+  have he : boundedFramePairs (n := n) B =
+      (closedBall (0:Matrix (Fin n) (Fin n) ℂ) B ×ˢ closedBall 0 B) ∩
+        {p | p.1*p.2 = 1 ∧ p.2*p.1 = 1} := by
+    ext p
+    simp only [boundedFramePairs,mem_ofPred_eq,mem_inter_iff,mem_prod,mem_closedBall,dist_zero_right]
+    tauto
+  rw [he]
+  apply (isCompact_closedBall _ _ |>.prod (isCompact_closedBall _ _)).inter_right
+  have hc : Continuous (fun p : Matrix (Fin n) (Fin n) ℂ × Matrix (Fin n) (Fin n) ℂ => p.1*p.2) :=
+    continuous_fst.matrix_mul continuous_snd
+  have hd : Continuous (fun p : Matrix (Fin n) (Fin n) ℂ × Matrix (Fin n) (Fin n) ℂ => p.2*p.1) :=
+    continuous_snd.matrix_mul continuous_fst
+  exact (isClosed_eq hc continuous_const).inter (isClosed_eq hd continuous_const)
+
+lemma boundedFramePairs_units {B : ℝ} {p : Matrix (Fin n) (Fin n) ℂ × Matrix (Fin n) (Fin n) ℂ}
+    (hp : p ∈ boundedFramePairs B) : IsUnit p.1 ∧ IsUnit p.2 := by
+  exact ⟨⟨⟨p.1,p.2,hp.2.2.1,hp.2.2.2⟩,rfl⟩,⟨⟨p.2,p.1,hp.2.2.2,hp.2.2.1⟩,rfl⟩⟩
+end KaehlerCalculus
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/BoundedPathContinuation.lean b/lean/OAI/Geometry/PrescribedPotential/BoundedPathContinuation.lean
new file mode 100644
index 0000000000000000000000000000000000000000..3a53d7a6a45c83d2b29418925f68c223cda98cc7
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/BoundedPathContinuation.lean
@@ -0,0 +1,87 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.NormalizedPathCompactness
+import OAI.Geometry.PrescribedPotential.SmoothPathOpenness
+
+namespace OAI
+
+/-! Bounded Path Continuation. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology
+open scoped ContDiff Classical
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  [ConnectedSpace X] {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+ 
+
+def PathSobolevBounds (h : SemipositiveAnticanonicalMetric A) (x₀ : X) : Prop :=
+  ∀ k : ℕ, ∃ C : ℝ, ∀ (t b : ℝ) (φ : SmoothRealFunction A),
+    t ∈ Icc 0 1 → φ.value x₀ = 0 → SolvesVolumePath g h t φ b →
+      ‖D.localizers.embed (((2*k : ℕ) : ℝ)+2) (Smooth.ofReal φ)‖ ≤ C
+
+omit [ConnectedSpace X] in
+lemma smoothVolumePathTimes_relatively_closed (h : SemipositiveAnticanonicalMetric A)
+    (x₀ : X) (hbound : D.PathSobolevBounds h x₀) :
+    IsClosed ((Subtype.val : Icc (0:ℝ) 1 → ℝ) ⁻¹' SmoothVolumePathTimes g h x₀) := by
+  apply IsSeqClosed.isClosed
+  intro t tLim ht htLim
+  choose φ b hn hs using ht
+  have hB : ∀ k : ℕ, ∃ C : ℝ, ∀ n : ℕ,
+      ‖D.localizers.embed (((2*k : ℕ) : ℝ)+2) (RealSmooth.ofReal (φ n)).val‖ ≤ C := by
+    intro k
+    obtain ⟨C,hC⟩ := hbound k
+    exact ⟨C,fun n => hC (t n).val (b n) (φ n) (t n).property (hn n) (hs n)⟩
+  have ht' : Tendsto (fun n => (t n).val) atTop (𝓝 tLim.val) :=
+    (continuous_subtype_val.tendsto tLim).comp htLim
+  obtain ⟨fLim,bLim,r,hr,hb,hf,hnLim,hsLim⟩ := D.normalized_path_sequence_compact h x₀
+    (fun n => (t n).val) b tLim.val ht' (fun n => (t n).property)
+    (fun n => RealSmooth.ofReal (φ n)) (by simpa only [RealSmooth.source_ofReal] using hn)
+    (by simpa only [RealSmooth.source_ofReal] using hs) hB
+  exact ⟨fLim.source,bLim,hnLim,hsLim⟩
+
+ 
+
+theorem smoothVolumePathTimes_full_of_bounds (h : SemipositiveAnticanonicalMetric A)
+    (x₀ : X) (hbound : D.PathSobolevBounds h x₀) :
+    Icc 0 1 ⊆ SmoothVolumePathTimes g h x₀ := by
+  let S : Set (Icc (0:ℝ) 1) := Subtype.val ⁻¹' SmoothVolumePathTimes g h x₀
+  have hc : IsClopen S := ⟨D.smoothVolumePathTimes_relatively_closed h x₀ hbound,
+    (smoothVolumePathTimes_isOpen g h x₀).preimage continuous_subtype_val⟩
+  have hn : S.Nonempty := ⟨⟨0,by constructor <;> norm_num⟩,
+    smoothVolumePathTimes_zero g h x₀⟩
+  have he := hc.eq_univ hn
+  intro t ht
+  have hh : (⟨t,ht⟩ : Icc (0:ℝ) 1) ∈ S := by rw [he]; trivial
+  exact hh
+
+end GluingData
+end GlobalElliptic
+
+namespace Anticanonical.SourceSmooth
+open GlobalElliptic
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  [ConnectedSpace X] {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+
+ 
+
+theorem prescribedPotential_of_path_bounds (P : A.ProjectiveEmbedding)
+    (h : SemipositiveAnticanonicalMetric A) (D : GluingData (projectiveBackground P) ι)
+    (x₀ : X) (hbound : D.PathSobolevBounds h x₀) : HasPrescribedVolumePotential P h := by
+  obtain ⟨φ,b,_,hs⟩ := D.smoothVolumePathTimes_full_of_bounds h x₀ hbound
+    (by constructor <;> norm_num : (1:ℝ) ∈ Icc 0 1)
+  exact volumePath_one P h φ b hs
+
+end Anticanonical.SourceSmooth
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/BoundedPoisson.lean b/lean/OAI/Geometry/PrescribedPotential/BoundedPoisson.lean
new file mode 100644
index 0000000000000000000000000000000000000000..49f9695d183790f140e1850edf4474f65539f26b
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/BoundedPoisson.lean
@@ -0,0 +1,148 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.ConstantProjection
+import OAI.Geometry.PrescribedPotential.PatchCutoffs
+import OAI.Geometry.PrescribedPotential.NormalizedPoisson
+
+namespace OAI
+
+/-! Bounded Poisson. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology
+open scoped ContDiff Classical
+namespace GlobalElliptic
+namespace LinearNormalization
+variable {H : Type*} [NormedAddCommGroup H] [NormedSpace ℝ H]
+
+def coefficient (C : ℂ →L[ℝ] H) (hC : Function.Injective C) (P : H →L[ℝ] H)
+    (hP : ∀ u, P u ∈ (C : ℂ →ₗ[ℝ] H).range) : H →L[ℝ] ℂ := by
+  let S := (C : ℂ →ₗ[ℝ] H).range
+  letI : FiniteDimensional ℝ S := LinearMap.finiteDimensional_range _
+  let e : ℂ ≃ₗ[ℝ] S := LinearEquiv.ofInjective (C : ℂ →ₗ[ℝ] H) hC
+  let l : S →L[ℝ] ℂ := LinearMap.toContinuousLinearMap e.symm.toLinearMap
+  exact l ∘L P.codRestrict S hP
+
+lemma coefficient_spec (C : ℂ →L[ℝ] H) (hC : Function.Injective C) (P : H →L[ℝ] H)
+    (hP : ∀ u, P u ∈ (C : ℂ →ₗ[ℝ] H).range) (u : H) :
+    C (coefficient C hC P hP u) = P u :=
+  LinearEquiv.ofInjective_symm_apply (C : ℂ →ₗ[ℝ] H) (h := hC) ⟨P u, hP u⟩
+
+end LinearNormalization
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  [ConnectedSpace X] {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace Localizers
+variable (D : Localizers A ι)
+
+def constantsOrder (s : ℝ) : ℂ →L[ℝ] D.Sobolev s :=
+  LinearMap.toContinuousLinearMap ((D.embed s).comp Smooth.constLinear)
+
+omit [ConnectedSpace X] in
+@[simp] lemma constantsOrder_apply (s : ℝ) (c : ℂ) :
+    D.constantsOrder s c = D.embed s (Smooth.const c) := rfl
+
+lemma constants_injective : Function.Injective D.constants := by
+  intro c e h
+  have hh := D.embed_injective 0 h
+  let x : X := Classical.arbitrary X
+  exact congrArg (fun u : Smooth A => u x) hh
+
+variable {D}
+
+ 
+def ConstantProjection.coefficient (P : D.ConstantProjection) : D.Sobolev 0 →L[ℝ] ℂ :=
+  LinearNormalization.coefficient D.constants D.constants_injective P.proj (by
+    intro u
+    obtain ⟨c, hc⟩ := P.value u
+    exact ⟨c, hc.symm⟩)
+
+lemma ConstantProjection.coefficient_spec (P : D.ConstantProjection) (u : D.Sobolev 0) :
+    D.constants (P.coefficient u) = P.proj u :=
+  LinearNormalization.coefficient_spec _ _ _ _ u
+
+lemma ConstantProjection.coefficient_constants (P : D.ConstantProjection) (c : ℂ) :
+    P.coefficient (D.constants c) = c := by
+  apply D.constants_injective
+  rw [P.coefficient_spec]
+  exact P.fixed c
+
+end Localizers
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+lemma fredholm_injective (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
+    (P : D.localizers.ConstantProjection) :
+    Function.Injective (1 - D.fredholmOperator m hm he P :
+      D.localizers.Sobolev 0 →L[ℝ] D.localizers.Sobolev 0) := by
+  intro u v huv
+  have hz : (1 - D.fredholmOperator m hm he P) (u-v) = 0 := by
+    rw [map_sub, huv, sub_self]
+  have hh : D.fredholmOperator m hm he P (u-v) = u-v := by
+    exact (sub_eq_zero.mp hz).symm
+  exact sub_eq_zero.mp (D.fredholm_kernel m hm he P (u-v) hh)
+
+def fredholmEquiv (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
+    (P : D.localizers.ConstantProjection) :
+    D.localizers.Sobolev 0 ≃L[ℝ] D.localizers.Sobolev 0 :=
+  ContinuousLinearEquiv.ofBijective (1 - D.fredholmOperator m hm he P)
+    (LinearMap.ker_eq_bot.mpr (D.fredholm_injective m hm he P))
+    (LinearMap.range_eq_top.mpr (D.fredholm_surjective m hm he P))
+
+@[simp] lemma fredholmEquiv_apply (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
+    (P : D.localizers.ConstantProjection) (u : D.localizers.Sobolev 0) :
+    D.fredholmEquiv m hm he P u =
+      u - ((m^2 : ℝ) • D.resolventZero m hm he u - P.proj u) := rfl
+
+def poissonInput (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
+    (P : D.localizers.ConstantProjection) : D.localizers.Sobolev 0 →L[ℝ] D.localizers.Sobolev 0 :=
+  - (D.fredholmEquiv m hm he P).symm.toContinuousLinearMap
+
+lemma poissonInput_equation (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
+    (P : D.localizers.ConstantProjection) (f : D.localizers.Sobolev 0) :
+    D.poissonInput m hm he P f = -f - P.proj (D.poissonInput m hm he P f) +
+      (m^2 : ℝ) • D.resolventZero m hm he (D.poissonInput m hm he P f) := by
+  have h : D.fredholmEquiv m hm he P (D.poissonInput m hm he P f) = -f := by
+    change D.fredholmEquiv m hm he P (- (D.fredholmEquiv m hm he P).symm f) = -f
+    rw [map_neg, ContinuousLinearEquiv.apply_symm_apply]
+  rw [D.fredholmEquiv_apply] at h
+  calc
+    _ = -f + ((m^2 : ℝ) • D.resolventZero m hm he (D.poissonInput m hm he P f) -
+      P.proj (D.poissonInput m hm he P f)) := sub_eq_iff_eq_add.mp h
+    _ = _ := by abel
+
+def poissonRaw (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
+    (P : D.localizers.ConstantProjection) : D.localizers.Sobolev 0 →L[ℝ] D.localizers.Sobolev 2 :=
+  D.completedResolvent m hm he ∘L D.poissonInput m hm he P
+
+def poissonResidue (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
+    (P : D.localizers.ConstantProjection) : D.localizers.Sobolev 0 →L[ℝ] ℂ :=
+  P.coefficient ∘L D.poissonInput m hm he P
+
+lemma poissonRaw_equation (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
+    (P : D.localizers.ConstantProjection) (f : D.localizers.Sobolev 0) :
+    D.completedL (D.poissonRaw m hm he P f) = f + D.localizers.constants (D.poissonResidue m hm he P f) := by
+  have h := D.completedResolvent_equation m hm he (D.poissonInput m hm he P f)
+  have hw := D.poissonInput_equation m hm he P f
+  change (m^2 : ℝ) • D.resolventZero m hm he (D.poissonInput m hm he P f) -
+    D.completedL (D.poissonRaw m hm he P f) = D.poissonInput m hm he P f at h
+  change _ = f + D.localizers.constants (P.coefficient (D.poissonInput m hm he P f))
+  rw [P.coefficient_spec]
+  have hh := h.trans hw
+  apply sub_right_inj.mp
+  calc
+    (m^2 : ℝ) • D.resolventZero m hm he (D.poissonInput m hm he P f) -
+        D.completedL (D.poissonRaw m hm he P f) = _ := hh
+    _ = (m^2 : ℝ) • D.resolventZero m hm he (D.poissonInput m hm he P f) -
+        (f + P.proj (D.poissonInput m hm he P f)) := by abel
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/CalabiBochner.lean b/lean/OAI/Geometry/PrescribedPotential/CalabiBochner.lean
new file mode 100644
index 0000000000000000000000000000000000000000..ac0e96132d6e7ade24637dc8737b94c441836b61
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/CalabiBochner.lean
@@ -0,0 +1,163 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CalabiCovariant
+import OAI.Geometry.PrescribedPotential.CalabiPairCalculus
+
+namespace OAI
+
+/-! Calabi Bochner. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace KaehlerCalculus
+variable {n : ℕ}
+
+lemma tensorPairAt_sum_left {α : Type*} (s : Finset α)
+    (P M : Matrix (Fin n) (Fin n) ℂ) (T : α → ConnectionTensor n) (U : ConnectionTensor n) :
+    tensorPairAt P M (fun i => ∑ a ∈ s, T a i) U = ∑ a ∈ s, tensorPairAt P M (T a) U := by
+  simp only [tensorPairAt,Matrix.sum_mul,Matrix.trace_sum,Finset.mul_sum]
+  calc
+    _ = ∑ i, ∑ a ∈ s, ∑ j, P i j*(T a i*P*(U j)ᴴ*M).trace := by
+      apply Finset.sum_congr rfl
+      intro i _
+      rw [Finset.sum_comm]
+    _ = _ := Finset.sum_comm
+
+lemma tensorPairAt_sum_right {α : Type*} (s : Finset α)
+    (P M : Matrix (Fin n) (Fin n) ℂ) (T : ConnectionTensor n) (U : α → ConnectionTensor n) :
+    tensorPairAt P M T (fun i => ∑ a ∈ s, U a i) = ∑ a ∈ s, tensorPairAt P M T (U a) := by
+  simp only [tensorPairAt,Matrix.conjTranspose_sum,Matrix.sum_mul,
+    Matrix.trace_sum,Finset.mul_sum]
+  calc
+    _ = ∑ i, ∑ a ∈ s, ∑ j, P i j*(T i*P*(U a j)ᴴ*M).trace := by
+      apply Finset.sum_congr rfl
+      intro i _
+      rw [Finset.sum_comm]
+    _ = _ := Finset.sum_comm
+
+namespace LocalKaehlerField
+variable (K : LocalKaehlerField n)
+
+lemma covHol_bar_commute {T : Fin n → V n → Matrix (Fin n) (Fin n) ℂ}
+    (hT : ∀ i, ContDiffOn ℝ ∞ (T i) K.domain) {z : V n} (hz : z ∈ K.domain) (k i : Fin n) :
+    K.covHol (fun j => mderiv Complex.I (e k) (T j)) k i z =
+      mderiv Complex.I (e k) (K.covHol T k i) z - K.curvature k k z*T i z +
+        T i z*K.curvature k k z + ∑ q, K.curvature k k z q i • T q z := by
+  have hn := K.isOpen.mem_nhds hz
+  have ht (j) := (hT j).contDiffAt hn
+  have hg := K.connection_smooth k |>.contDiffAt hn
+  have hs := ContDiffAt.sum (fun q (_ : q ∈ (Finset.univ : Finset (Fin n))) =>
+    matrix_smooth_smul (entry_smooth hg q i) (ht q))
+  unfold covHol
+  rw [mderiv_sub (((mderiv_smooth (ht i) _ _).add (matrix_smooth_mul hg (ht i))).sub
+      (matrix_smooth_mul (ht i) hg)) hs,
+    mderiv_sub ((mderiv_smooth (ht i) _ _).add (matrix_smooth_mul hg (ht i)))
+      (matrix_smooth_mul (ht i) hg),
+    mderiv_add (mderiv_smooth (ht i) _ _) (matrix_smooth_mul hg (ht i)),
+    mderiv_mul hg (ht i),mderiv_mul (ht i) hg,
+    mderiv_sum Finset.univ (fun q _ => matrix_smooth_smul (entry_smooth hg q i) (ht q))]
+  simp only [mderiv_smul (entry_smooth hg _ i) (ht _),Finset.sum_add_distrib,
+    mderiv_comm (ht i) Complex.I (-Complex.I) (e k) (e k)]
+  change _ = _ - (K.curvature k k z*T i z) + _ + _
+  unfold curvature
+  have he : (∑ q, wderiv Complex.I (e k) (fun y => K.connection k y q i) z • T q z) =
+      ∑ q, mderiv Complex.I (e k) (K.connection k) z q i • T q z := rfl
+  rw [he]
+  abel
+
+lemma calabi_curvature_divergence_at_one {z : V n} (hz : z ∈ K.domain)
+    (hM : K.matrix z = 1) (i : Fin n) :
+    (∑ k, K.covHol (K.curvature k) k i z) =
+      mderiv (-Complex.I) (e i) K.ricciHessian z - K.ricciHessian z*K.connection i z := by
+  have he (k : Fin n) : K.covHol (K.curvature k) k i z =
+      mderiv Complex.I (e k) (K.covHol K.connection k i) z - K.curvature k k z*K.connection i z +
+        K.connection i z*K.curvature k k z + ∑ q, K.curvature k k z q i • K.connection q z :=
+    K.covHol_bar_commute (T := K.connection) K.connection_smooth hz k i
+  simp_rw [he]
+  simp only [Finset.sum_add_distrib,Finset.sum_sub_distrib,← Matrix.sum_mul,← Matrix.mul_sum]
+  rw [K.calabi_connection_laplacian_at_one hz hM i,K.curvature_contraction_at_one hz hM]
+  have hs : (∑ k, ∑ q, K.curvature k k z q i • K.connection q z) =
+      ∑ q, K.ricciHessian z q i • K.connection q z := by
+    rw [Finset.sum_comm]
+    apply Finset.sum_congr rfl
+    intro q _
+    rw [← Finset.sum_smul]
+    congr 1
+    simpa only [Matrix.sum_apply] using congrArg
+      (fun A : Matrix (Fin n) (Fin n) ℂ => A q i) (K.curvature_contraction_at_one hz hM)
+  rw [hs]
+  abel
+
+def calabiNorm (z : V n) : ℝ := (K.tensorPair K.connection K.connection z).re
+
+lemma calabiNorm_smooth : ContDiffOn ℝ ∞ K.calabiNorm K.domain :=
+  Complex.reCLM.contDiff.comp_contDiffOn (K.tensorPair_smoothOn K.connection_smooth K.connection_smooth)
+
+lemma calabiNorm_complex {z : V n} (hz : z ∈ K.domain) :
+    (K.calabiNorm z : ℂ) = K.tensorPair K.connection K.connection z := by
+  apply Complex.ext
+  · rfl
+  · exact (tensorPairAt_self_real (K.positive z hz).isHermitian.inv
+      (K.positive z hz).isHermitian _).symm
+
+lemma calabiNorm_second {z : V n} (hz : z ∈ K.domain) (k : Fin n) :
+    dzbar (e k) (dz (e k) (fun y => (K.calabiNorm y : ℂ))) z =
+      K.tensorPair (fun i => mderiv Complex.I (e k) (K.covHol K.connection k i)) K.connection z +
+      K.tensorPair (K.covHol K.connection k) (K.covHol K.connection k) z +
+      K.tensorPair (K.curvature k) (K.curvature k) z +
+      K.tensorPair K.connection (K.covHol (K.curvature k) k) z := by
+  have hn := K.isOpen.mem_nhds hz
+  have hc : (fun y => (K.calabiNorm y : ℂ)) =ᶠ[𝓝 z] K.tensorPair K.connection K.connection := by
+    filter_upwards [hn] with y hy using K.calabiNorm_complex hy
+  have hd : dz (e k) (fun y => (K.calabiNorm y : ℂ)) =ᶠ[𝓝 z]
+      (fun y => K.tensorPair (K.covHol K.connection k) K.connection y +
+        K.tensorPair K.connection (K.curvature k) y) := by
+    filter_upwards [hn] with y hy
+    have he : (fun y => (K.calabiNorm y : ℂ)) =ᶠ[𝓝 y] K.tensorPair K.connection K.connection := by
+      filter_upwards [K.isOpen.mem_nhds hy] with x hx using K.calabiNorm_complex hx
+    change wderiv (-Complex.I) (e k) _ y = _
+    rw [wderiv_congr he]
+    exact K.tensorPair_hol K.connection_smooth K.connection_smooth hy k
+  change wderiv Complex.I (e k) _ z = _
+  rw [wderiv_congr hd,wderiv_add
+    ((K.tensorPair_smoothOn (T := K.covHol K.connection k) (U := K.connection)
+      (K.covHol_smooth (T := K.connection) K.connection_smooth k) K.connection_smooth).contDiffAt hn
+      |>.differentiableAt (by simp))
+    ((K.tensorPair_smoothOn (T := K.connection) (U := K.curvature k) K.connection_smooth (K.curvature_smooth k)).contDiffAt hn
+      |>.differentiableAt (by simp))]
+  change dzbar (e k) _ z + dzbar (e k) _ z = _
+  rw [K.tensorPair_bar (T := K.covHol K.connection k) (U := K.connection)
+    (K.covHol_smooth (T := K.connection) K.connection_smooth k) K.connection_smooth hz,
+    K.tensorPair_bar (T := K.connection) (U := K.curvature k) K.connection_smooth (K.curvature_smooth k) hz]
+  exact (add_assoc _ _ _).symm
+
+lemma calabi_bochner_at_one {z : V n} (hz : z ∈ K.domain) (hM : K.matrix z = 1) :
+    (PotentialKaehler.potentialMatrix K.calabiNorm z).trace =
+      (∑ k, tensorPairAt 1 1 (fun i => K.covHol K.connection k i z)
+        (fun i => K.covHol K.connection k i z)) +
+      (∑ k, tensorPairAt 1 1 (fun i => K.curvature k i z) (fun i => K.curvature k i z)) +
+      tensorPairAt 1 1 (fun i => mderiv (-Complex.I) (e i) K.ricciHessian z -
+        K.connection i z*K.ricciHessian z - ∑ q, K.ricciHessian z q i • K.connection q z)
+        (fun i => K.connection i z) +
+      tensorPairAt 1 1 (fun i => K.connection i z)
+        (fun i => mderiv (-Complex.I) (e i) K.ricciHessian z - K.ricciHessian z*K.connection i z) := by
+  change (∑ k, PotentialKaehler.potentialMatrix K.calabiNorm z k k) = _
+  simp_rw [potentialMatrix_eq_dzbar_dz (K.calabiNorm_smooth.contDiffAt (K.isOpen.mem_nhds hz))]
+  change (∑ k, dzbar (e k) (dz (e k) (fun y => (K.calabiNorm y : ℂ))) z) = _
+  simp_rw [K.calabiNorm_second hz]
+  simp only [tensorPair,hM,inv_one,Finset.sum_add_distrib]
+  rw [← tensorPairAt_sum_left,← tensorPairAt_sum_right]
+  simp_rw [K.calabi_connection_laplacian_at_one hz hM,K.calabi_curvature_divergence_at_one hz hM]
+  abel
+
+end LocalKaehlerField
+end KaehlerCalculus
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/CalabiConnection.lean b/lean/OAI/Geometry/PrescribedPotential/CalabiConnection.lean
new file mode 100644
index 0000000000000000000000000000000000000000..69fcc4333f4510018a66e22f8a182c0e48650cfb
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/CalabiConnection.lean
@@ -0,0 +1,107 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.KaehlerSecondDerivatives
+
+namespace OAI
+
+/-! Calabi Connection. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace KaehlerCalculus
+
+structure LocalKaehlerField (n : ℕ) where
+  domain : Set (V n)
+  isOpen : IsOpen domain
+  matrix : V n → Matrix (Fin n) (Fin n) ℂ
+  smooth : ContDiffOn ℝ ∞ matrix domain
+  positive : ∀ z ∈ domain, (matrix z).PosDef
+  closed : ∀ y ∈ domain, ∀ u v w : V n,
+    fderiv ℝ (fun q => Anticanonical.ComplexAtlas.fundamentalForm (matrix q) v w) y u +
+    fderiv ℝ (fun q => Anticanonical.ComplexAtlas.fundamentalForm (matrix q) w u) y v +
+    fderiv ℝ (fun q => Anticanonical.ComplexAtlas.fundamentalForm (matrix q) u v) y w = 0
+
+namespace LocalKaehlerField
+variable {n : ℕ} (K : LocalKaehlerField n)
+
+def connection (j : Fin n) (z : V n) : Matrix (Fin n) (Fin n) ℂ :=
+  (K.matrix z)⁻¹ * mderiv (-Complex.I) (e j) K.matrix z
+
+def curvature (i j : Fin n) : V n → Matrix (Fin n) (Fin n) ℂ :=
+  mderiv Complex.I (e i) (K.connection j)
+
+def logDet : V n → ℝ := fun y => Real.log (K.matrix y).det.re
+
+def ricciHessian (z : V n) : Matrix (Fin n) (Fin n) ℂ :=
+  PotentialKaehler.potentialMatrix K.logDet z
+
+lemma inverse_smooth : ContDiffOn ℝ ∞ (fun y => (K.matrix y)⁻¹) K.domain :=
+  MatrixSmoothGeneral.inverse K.smooth (fun y hy => ne_of_gt (K.positive y hy).det_pos)
+
+lemma connection_smooth (j : Fin n) : ContDiffOn ℝ ∞ (K.connection j) K.domain := by
+  apply K.isOpen.contDiffOn_iff.mpr
+  intro z hz
+  exact matrix_smooth_mul (K.inverse_smooth.contDiffAt (K.isOpen.mem_nhds hz))
+    (mderiv_smooth (K.smooth.contDiffAt (K.isOpen.mem_nhds hz)) _ _)
+
+lemma logDet_smooth : ContDiffOn ℝ ∞ K.logDet K.domain := by
+  apply K.isOpen.contDiffOn_iff.mpr
+  intro z hz
+  exact (Complex.reCLM.contDiff.contDiffAt.comp z
+    (determinant_smooth.contDiffAt.comp z (K.smooth.contDiffAt (K.isOpen.mem_nhds hz)))).log
+      (ne_of_gt (Complex.pos_iff.mp (K.positive z hz).det_pos).1)
+
+lemma connection_symmetric {z : V n} (hz : z ∈ K.domain) (i j k : Fin n) :
+    K.connection i z k j = K.connection j z k i := by
+  unfold connection
+  simp only [Matrix.mul_apply]
+  apply Finset.sum_congr rfl
+  intro a _
+  congr 1
+  exact closed_dz_symmetry K.isOpen K.smooth (fun y hy => (K.positive y hy).isHermitian)
+    K.closed hz a j i
+
+lemma curvature_formula {z : V n} (hz : z ∈ K.domain) (i j : Fin n) :
+    K.curvature i j z = (K.matrix z)⁻¹ *
+      (mderiv Complex.I (e i) (mderiv (-Complex.I) (e j) K.matrix) z -
+        mderiv Complex.I (e i) K.matrix z * (K.matrix z)⁻¹ *
+          mderiv (-Complex.I) (e j) K.matrix z) := by
+  have hs := K.smooth.contDiffAt (K.isOpen.mem_nhds hz)
+  unfold curvature connection
+  rw [mderiv_mul (K.inverse_smooth.contDiffAt (K.isOpen.mem_nhds hz))
+    (mderiv_smooth hs _ _), mderiv_inverse K.isOpen K.smooth
+      (fun y hy => ne_of_gt (K.positive y hy).det_pos) hz]
+  noncomm_ring
+
+lemma ricciHessian_trace {z : V n} (hz : z ∈ K.domain) (i j : Fin n) :
+    K.ricciHessian z i j = (K.curvature i j z).trace := by
+  rw [ricciHessian,potentialMatrix_eq_dzbar_dz (K.logDet_smooth.contDiffAt (K.isOpen.mem_nhds hz))]
+  change dzbar (e i) (dz (e j) (fun y => (K.logDet y:ℂ))) z = _
+  have he : dz (e j) (fun y => (K.logDet y:ℂ)) =ᶠ[𝓝 z]
+      (fun y => (K.connection j y).trace) := by
+    filter_upwards [K.isOpen.mem_nhds hz] with y hy
+    exact wderiv_logdet K.isOpen K.smooth K.positive hy _ _
+  rw [show dzbar (e i) (dz (e j) (fun y => (K.logDet y:ℂ))) z =
+    dzbar (e i) (fun y => (K.connection j y).trace) z from wderiv_congr he _ _]
+  exact mderiv_trace (K.connection_smooth j |>.contDiffAt (K.isOpen.mem_nhds hz)) _ _
+
+lemma curvature_contraction_at_one {z : V n} (hz : z ∈ K.domain)
+    (hM : K.matrix z = 1) : ∑ k, K.curvature k k z = K.ricciHessian z := by
+  have hc := ricci_cancellation_at_one K.isOpen K.smooth K.positive K.closed hz hM
+  have he : (∑ k, K.curvature k k z) =
+      (∑ k, mderiv Complex.I (e k) (mderiv (-Complex.I) (e k) K.matrix) z) -
+      ∑ k, mderiv Complex.I (e k) K.matrix z*mderiv (-Complex.I) (e k) K.matrix z := by
+    simp only [K.curvature_formula hz, hM,inv_one,one_mul,mul_one,Finset.sum_sub_distrib]
+  rw [he,hc,add_sub_cancel_right]
+  rfl
+end LocalKaehlerField
+end KaehlerCalculus
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/CalabiCoordinateBound.lean b/lean/OAI/Geometry/PrescribedPotential/CalabiCoordinateBound.lean
new file mode 100644
index 0000000000000000000000000000000000000000..05c97bba67acb7c7e650ccf5ebcf2ac60030d554
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/CalabiCoordinateBound.lean
@@ -0,0 +1,160 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CalabiCoordinateCoercivity
+import OAI.Geometry.PrescribedPotential.CircleRadialCalculus
+import OAI.Geometry.PrescribedPotential.HolomorphicDirection
+import OAI.Geometry.PrescribedPotential.TracePositivity
+
+namespace OAI
+
+/-! Calabi Coordinate Bound. -/
+
+section
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace KaehlerCalculus
+variable {n : ℕ}
+
+lemma matrix_mul_elementwise_bound (M N : Matrix (Fin n) (Fin n) ℂ) :
+    ‖M*N‖ ≤ n*‖M‖*‖N‖ := by
+  apply (Matrix.norm_le_iff (by positivity)).mpr
+  intro i j
+  rw [Matrix.mul_apply]
+  calc
+    _ ≤ ∑ k, ‖M i k*N k j‖ := norm_sum_le _ _
+    _ ≤ ∑ _k : Fin n, ‖M‖*‖N‖ := by
+      apply Finset.sum_le_sum
+      intro k _
+      rw [norm_mul]
+      exact mul_le_mul (Matrix.norm_entry_le_entrywise_sup_norm M)
+        (Matrix.norm_entry_le_entrywise_sup_norm N) (norm_nonneg _) (norm_nonneg _)
+    _ = _ := by simp [mul_assoc]
+
+lemma hol_matrix_stencil_bound (v : V n) (T : ConnectionTensor n) :
+    ‖∑ i, v i • T i‖ ≤ n*‖v‖*‖T‖ := by
+  calc
+    _ ≤ ∑ i, ‖v i • T i‖ := norm_sum_le _ _
+    _ ≤ ∑ _i : Fin n, ‖v‖*‖T‖ := by
+      apply Finset.sum_le_sum
+      intro i _
+      rw [norm_smul]
+      exact mul_le_mul (norm_le_pi_norm v i) (norm_le_pi_norm T i) (norm_nonneg _) (norm_nonneg _)
+    _ = _ := by simp [mul_assoc]
+
+namespace LocalKaehlerField
+variable (K : LocalKaehlerField n)
+lemma holDerivative_connection {z : V n} (hz : z ∈ K.domain) (i : Fin n) :
+    mderiv (-Complex.I) (e i) K.matrix z = K.matrix z*K.connection i z := by
+  rw [connection,← mul_assoc,Matrix.mul_nonsing_inv _
+    (isUnit_iff_ne_zero.mpr (ne_of_gt (K.positive z hz).det_pos)),one_mul]
+
+lemma fderiv_eq_hol_add_star {z : V n} (hz : z ∈ K.domain) (v : V n) :
+    fderiv ℝ K.matrix z v = mderiv (-Complex.I) v K.matrix z +
+      (mderiv (-Complex.I) v K.matrix z)ᴴ := by
+  rw [← mderiv_bar_eq_conjTranspose K.isOpen K.smooth
+    (fun y hy => (K.positive y hy).isHermitian) hz v]
+  have hd := (K.smooth.contDiffAt (K.isOpen.mem_nhds hz)).differentiableAt (by simp)
+  ext i j
+  have he : fderiv ℝ K.matrix z v i j = fderiv ℝ (fun y => K.matrix y i j) z v := by
+    change fderiv ℝ (fun y a b => K.matrix y a b) z v i j = _
+    rw [fderiv_pi (fun a => differentiableAt_pi.mp hd a)]
+    simp only [ContinuousLinearMap.pi_apply]
+    rw [fderiv_pi (fun b => differentiableAt_pi.mp (differentiableAt_pi.mp hd i) b)]
+    rfl
+  rw [he]
+  change _ = wderiv (-Complex.I) v (fun y => K.matrix y i j) z +
+    wderiv Complex.I v (fun y => K.matrix y i j) z
+  unfold wderiv
+  ring
+
+lemma matrix_fderiv_bound {z : V n} (hz : z ∈ K.domain) :
+    ‖fderiv ℝ (fun y i j => K.matrix y i j) z‖ ≤ 2*n^2*‖K.matrix z‖*‖fun i => K.connection i z‖ := by
+  apply ContinuousLinearMap.opNorm_le_bound _ (by positivity)
+  intro v
+  have he : fderiv ℝ (fun y i j => K.matrix y i j) z v =
+      (fun i j => (mderiv (-Complex.I) v K.matrix z +
+        (mderiv (-Complex.I) v K.matrix z)ᴴ) i j) := K.fderiv_eq_hol_add_star hz v
+  rw [he]
+  change ‖mderiv (-Complex.I) v K.matrix z + (mderiv (-Complex.I) v K.matrix z)ᴴ‖ ≤ _
+  have ht : mderiv (-Complex.I) v K.matrix z =
+      K.matrix z*(∑ i, v i • K.connection i z) := by
+    rw [mderiv_hol_coordinates,Matrix.mul_sum]
+    apply Finset.sum_congr rfl
+    intro i _
+    rw [K.holDerivative_connection hz,Matrix.mul_smul]
+  have hb := (matrix_mul_elementwise_bound (K.matrix z) (∑ i, v i • K.connection i z)).trans
+    (mul_le_mul_of_nonneg_left (hol_matrix_stencil_bound v (fun i => K.connection i z)) (by positivity))
+  rw [← ht] at hb
+  calc
+    _ ≤ ‖mderiv (-Complex.I) v K.matrix z‖ + ‖(mderiv (-Complex.I) v K.matrix z)ᴴ‖ := norm_add_le _ _
+    _ ≤ 2*(n*‖K.matrix z‖*(n*‖v‖*‖fun i => K.connection i z‖)) := by
+      rw [Matrix.norm_conjTranspose]
+      linarith
+    _ = _ := by ring
+end LocalKaehlerField
+
+ 
+
+theorem calabi_coordinate_connection_uniform {X : Type*} [TopologicalSpace X]
+    {L : Set X} (hL : IsCompact L) (G : X → Matrix (Fin n) (Fin n) ℂ)
+    (hG : ContinuousOn G L) (hp : ∀ x ∈ L, (G x).PosDef)
+    {R H C : ℝ} (hR : 0 ≤ R) (hH : 0 ≤ H) (hC : 0 ≤ C) :
+    ∃ B : ℝ, 0 ≤ B ∧ ∀ x ∈ L, ∀ (K : LocalKaehlerField n) z, z ∈ K.domain →
+      ((R:ℂ) • K.matrix z-G x).PosSemidef → ((H:ℂ) • G x-K.matrix z).PosSemidef →
+      K.calabiNorm z ≤ C → ‖K.matrix z‖ ≤ B ∧ ‖fun i => K.connection i z‖ ≤ B := by
+  obtain ⟨b,hb,hframe⟩ := compact_normalizing_frames hL G hG hp hR hH
+  obtain ⟨δ,hδ,hcoercive⟩ := boundedFrame_tensor_coercivity (n := n) b
+  let M := (n:ℝ)*b^2
+  let T := C/δ+1
+  let B := max M T
+  have hM : 0 ≤ M := by dsimp [M]; positivity
+  have hT : 0 ≤ T := by dsimp [T]; positivity
+  refine ⟨B,hM.trans (le_max_left _ _),?_⟩
+  intro x hx K z hz hlow hupp hcal
+  obtain ⟨F,hF,hN⟩ := whitening (K.matrix z) (K.positive z hz)
+  have hFI : F*F⁻¹=1 := Matrix.mul_nonsing_inv _ ((Matrix.isUnit_iff_isUnit_det _).mp hF)
+  have hIF : F⁻¹*F=1 := Matrix.nonsing_inv_mul _ ((Matrix.isUnit_iff_isUnit_det _).mp hF)
+  obtain ⟨hf,hfi⟩ := hframe x hx (K.matrix z) F F⁻¹ hlow hupp hFI hN
+  have hpair : (F,F⁻¹) ∈ boundedFramePairs b := ⟨hf,hfi,hFI,hIF⟩
+  have hcoer : ∀ U : ConnectionTensor n, δ*‖U‖^2 ≤ tensorSquare (tensorTransport F F⁻¹ U) :=
+    hcoercive (F,F⁻¹) hpair
+  have hc : δ*‖fun i => K.connection i z‖^2 ≤ C :=
+    (connection_norm_bound_of_frame (K := K) (z := z) (C := F) (D := F⁻¹)
+      (δ := δ) hz hFI hIF hN hδ hcoer).trans hcal
+  have hconn : ‖fun i => K.connection i z‖ ≤ T := by
+    have hsq : ‖fun i => K.connection i z‖^2 ≤ C/δ :=
+      (le_div_iff₀ hδ).mpr (by simpa only [mul_comm] using hc)
+    dsimp [T]
+    nlinarith only [hsq, sq_nonneg (‖fun i => K.connection i z‖-1)]
+  have hm : ‖K.matrix z‖ ≤ M := by
+    rw [unnormalize_matrix hFI hN]
+    have hh := matrix_mul_elementwise_bound (F⁻¹)ᴴ F⁻¹
+    rw [Matrix.norm_conjTranspose] at hh
+    have hs := (sq_le_sq₀ (norm_nonneg _) hb).mpr hfi
+    dsimp [M]
+    nlinarith only [hh,hs,Nat.cast_nonneg (α := ℝ) n]
+  exact ⟨hm.trans (le_max_left _ _),hconn.trans (le_max_right _ _)⟩
+
+theorem calabi_coordinate_C1_uniform {X : Type*} [TopologicalSpace X]
+    {L : Set X} (hL : IsCompact L) (G : X → Matrix (Fin n) (Fin n) ℂ)
+    (hG : ContinuousOn G L) (hp : ∀ x ∈ L, (G x).PosDef)
+    {R H C : ℝ} (hR : 0 ≤ R) (hH : 0 ≤ H) (hC : 0 ≤ C) :
+    ∃ B : ℝ, 0 ≤ B ∧ ∀ x ∈ L, ∀ (K : LocalKaehlerField n) z, z ∈ K.domain →
+      ((R:ℂ) • K.matrix z-G x).PosSemidef → ((H:ℂ) • G x-K.matrix z).PosSemidef →
+      K.calabiNorm z ≤ C → ‖K.matrix z‖ ≤ B ∧
+        ‖fderiv ℝ (fun y i j => K.matrix y i j) z‖ ≤ B := by
+  obtain ⟨B,hB,hbound⟩ := calabi_coordinate_connection_uniform hL G hG hp hR hH hC
+  refine ⟨max B (2*n^2*B*B),hB.trans (le_max_left _ _),?_⟩
+  intro x hx K z hz hl hu hs
+  obtain ⟨hm,hc⟩ := hbound x hx K z hz hl hu hs
+  refine ⟨hm.trans (le_max_left _ _),?_⟩
+  exact (K.matrix_fderiv_bound hz).trans ((mul_le_mul
+    (mul_le_mul_of_nonneg_left hm (by positivity)) hc (norm_nonneg _) (by positivity)).trans
+      (le_max_right _ _))
+end KaehlerCalculus
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/CalabiCoordinateCoercivity.lean b/lean/OAI/Geometry/PrescribedPotential/CalabiCoordinateCoercivity.lean
new file mode 100644
index 0000000000000000000000000000000000000000..0b925404e6c0597f9b4ccf81703310017af31915
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/CalabiCoordinateCoercivity.lean
@@ -0,0 +1,118 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.BoundedNormalFrames
+import OAI.Geometry.PrescribedPotential.CalabiBochner
+import OAI.Geometry.PrescribedPotential.CalabiConnection
+import OAI.Geometry.PrescribedPotential.CalabiFrameAlgebra
+import OAI.Geometry.PrescribedPotential.CalabiNormAlgebra
+
+namespace OAI
+
+/-! Calabi Coordinate Coercivity. -/
+
+section
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace KaehlerCalculus
+variable {n : ℕ}
+
+def tensorTransport (C D : Matrix (Fin n) (Fin n) ℂ) (T : ConnectionTensor n) : ConnectionTensor n :=
+  fun i => D*(∑ p, C p i • T p)*C
+
+lemma tensorTransport_smul (C D : Matrix (Fin n) (Fin n) ℂ) (r : ℝ) (T : ConnectionTensor n) :
+    tensorTransport C D (r • T) = r • tensorTransport C D T := by
+  funext i
+  simp only [tensorTransport,Pi.smul_apply,smul_comm (_:ℂ) r,← Finset.smul_sum,
+    Matrix.mul_smul,Matrix.smul_mul]
+
+lemma tensorTransport_zero (C D : Matrix (Fin n) (Fin n) ℂ) :
+    tensorTransport C D (0 : ConnectionTensor n) = 0 := by
+  funext i
+  simp [tensorTransport]
+
+lemma tensorTransport_undo {C D : Matrix (Fin n) (Fin n) ℂ}
+    (hCD : C*D=1) (_hDC : D*C=1) (T : ConnectionTensor n) :
+    tensorTransport D C (tensorTransport C D T) = T := by
+  funext i
+  unfold tensorTransport
+  have he : (∑ p, D p i • (D*(∑ q, C q p • T q)*C)) =
+      D*(∑ q, (C*D) q i • T q)*C := by
+    calc
+      _ = ∑ p, D*(∑ q, (D p i*C q p) • T q)*C := by
+        apply Finset.sum_congr rfl
+        intro p _
+        simp only [← smul_smul,← Finset.smul_sum,Matrix.mul_smul,Matrix.smul_mul]
+      _ = D*(∑ p, ∑ q, (D p i*C q p) • T q)*C := by
+        rw [Matrix.mul_sum,Matrix.sum_mul]
+      _ = _ := by
+        congr 2
+        rw [Finset.sum_comm]
+        apply Finset.sum_congr rfl
+        intro q _
+        simp only [Matrix.mul_apply,Finset.sum_smul]
+        apply Finset.sum_congr rfl
+        intro p _
+        rw [mul_comm]
+
+  rw [he,hCD]
+  simp only [Matrix.one_apply,ite_smul,one_smul,zero_smul,Finset.sum_ite_eq',Finset.mem_univ,ite_true]
+  simp only [mul_assoc,hCD,mul_one]
+  rw [← mul_assoc,hCD,one_mul]
+
+lemma continuous_tensorTransport : Continuous
+    (fun q : (Matrix (Fin n) (Fin n) ℂ × Matrix (Fin n) (Fin n) ℂ) × ConnectionTensor n =>
+      tensorTransport q.1.1 q.1.2 q.2) := by
+  apply continuous_pi
+  intro i
+  let P := (Matrix (Fin n) (Fin n) ℂ × Matrix (Fin n) (Fin n) ℂ) × ConnectionTensor n
+  have hC : Continuous (fun q : P => q.1.1) := continuous_fst.comp continuous_fst
+  have hD : Continuous (fun q : P => q.1.2) := continuous_snd.comp continuous_fst
+  have hT (p : Fin n) : Continuous (fun q : P => q.2 p) := (continuous_apply p).comp continuous_snd
+  have hc (p : Fin n) : Continuous (fun q : P => q.1.1 p i) :=
+    (continuous_apply i).comp ((continuous_apply p).comp hC)
+  exact (hD.matrix_mul (continuous_finsetSum _ (fun p _ => (hc p).smul (hT p)))).matrix_mul hC
+
+lemma boundedFrame_tensor_coercivity (B : ℝ) :
+    ∃ δ : ℝ, 0 < δ ∧ ∀ p ∈ boundedFramePairs (n := n) B, ∀ T : ConnectionTensor n,
+      δ*‖T‖^2 ≤ tensorSquare (tensorTransport p.1 p.2 T) := by
+  apply compact_homogeneous_positive (boundedFramePairs_compact B)
+    (fun p T => tensorSquare (tensorTransport p.1 p.2 T))
+  · exact (continuous_tensorSquare.comp continuous_tensorTransport).continuousOn
+  · intro p hp T hT
+    apply tensorSquare_pos
+    intro he
+    apply hT
+    rw [← tensorTransport_undo hp.2.2.1 hp.2.2.2 T,he,tensorTransport_zero]
+  · intro p _ r T
+    rw [tensorTransport_smul,tensorSquare_smul]
+
+lemma connection_norm_bound_of_frame {K : LocalKaehlerField n} {z : V n}
+    (_hz : z ∈ K.domain) {C D : Matrix (Fin n) (Fin n) ℂ}
+    (hCD : C*D=1) (hDC : D*C=1) (hN : Cᴴ*K.matrix z*C=1)
+    {δ : ℝ} (_hδ : 0 < δ)
+    (hc : ∀ T : ConnectionTensor n, δ*‖T‖^2 ≤ tensorSquare (tensorTransport C D T)) :
+    δ*‖fun i => K.connection i z‖^2 ≤ K.calabiNorm z := by
+  have hC : IsUnit C := ⟨⟨C,D,hCD,hDC⟩,rfl⟩
+  have hi : C⁻¹=D := Matrix.inv_eq_right_inv hCD
+  have hI : C⁻¹*(K.matrix z)⁻¹*(Cᴴ)⁻¹=1 := by
+    rw [← Matrix.mul_inv_rev,← Matrix.mul_inv_rev,← mul_assoc,hN,inv_one]
+  have he := tensorPairAt_changeBasis C (K.matrix z)⁻¹ (K.matrix z) hC
+    (fun i => K.connection i z) (fun i => K.connection i z)
+  rw [hI,hN] at he
+  have ht : pullConnectionTensor C (fun i => K.connection i z) =
+      tensorTransport C D (fun i => K.connection i z) := by
+    unfold pullConnectionTensor tensorTransport
+    rw [hi]
+  rw [ht] at he
+  have hs : tensorSquare (tensorTransport C D (fun i => K.connection i z)) = K.calabiNorm z := by
+    change (tensorPairAt 1 1 _ _).re = _
+    exact congrArg Complex.re he
+  rw [← hs]
+  exact hc _
+end KaehlerCalculus
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/CalabiCovariant.lean b/lean/OAI/Geometry/PrescribedPotential/CalabiCovariant.lean
new file mode 100644
index 0000000000000000000000000000000000000000..3189aeebb2465cc9c0de857a236bd979b45db80c
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/CalabiCovariant.lean
@@ -0,0 +1,156 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CalabiConnection
+import OAI.Geometry.PrescribedPotential.CalabiRicciContraction
+import OAI.Geometry.PrescribedPotential.MatrixTensorCalculus
+
+namespace OAI
+
+/-! Calabi Covariant. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace KaehlerCalculus.LocalKaehlerField
+variable {n : ℕ} (K : LocalKaehlerField n)
+
+lemma curvature_smooth (i j : Fin n) : ContDiffOn ℝ ∞ (K.curvature i j) K.domain := by
+  apply K.isOpen.contDiffOn_iff.mpr
+  intro z hz
+  exact mderiv_smooth (K.connection_smooth j |>.contDiffAt (K.isOpen.mem_nhds hz)) _ _
+
+lemma ricciHessian_smooth : ContDiffOn ℝ ∞ K.ricciHessian K.domain := by
+  apply K.isOpen.contDiffOn_iff.mpr
+  intro z hz
+  exact PotentialKaehler.potentialMatrix_smooth (K.logDet_smooth.contDiffAt (K.isOpen.mem_nhds hz))
+
+def covHol (T : Fin n → V n → Matrix (Fin n) (Fin n) ℂ) (k i : Fin n) (z : V n) :
+    Matrix (Fin n) (Fin n) ℂ :=
+  mderiv (-Complex.I) (e k) (T i) z + K.connection k z*T i z - T i z*K.connection k z -
+    ∑ q, (K.connection k z q i) • T q z
+
+lemma covHol_smooth {T : Fin n → V n → Matrix (Fin n) (Fin n) ℂ}
+    (hT : ∀ i, ContDiffOn ℝ ∞ (T i) K.domain) (k i : Fin n) :
+    ContDiffOn ℝ ∞ (K.covHol T k i) K.domain := by
+  apply K.isOpen.contDiffOn_iff.mpr
+  intro z hz
+  have hΓ := K.connection_smooth k |>.contDiffAt (K.isOpen.mem_nhds hz)
+  have ht (j) := (hT j).contDiffAt (K.isOpen.mem_nhds hz)
+  exact (((mderiv_smooth (ht i) _ _).add (matrix_smooth_mul hΓ (ht i))).sub
+    (matrix_smooth_mul (ht i) hΓ)).sub
+      (ContDiffAt.sum (fun q _ => matrix_smooth_smul (entry_smooth hΓ q i) (ht q)))
+
+lemma covHol_connection_swap {z : V n} (hz : z ∈ K.domain) (k i : Fin n) :
+    K.covHol K.connection k i z = mderiv (-Complex.I) (e i) (K.connection k) z -
+      ∑ q, (K.connection k z q i) • K.connection q z := by
+  have he := K.connection_hol_flat hz k i
+  unfold covHol
+  rw [sub_eq_iff_eq_add] at he
+  rw [he]
+  abel
+
+lemma derivative_curvature_contraction_at_one {z : V n} (hz : z ∈ K.domain)
+    (hM : K.matrix z = 1) (i : Fin n) :
+    (∑ k, mderiv (-Complex.I) (e i) (K.curvature k k) z) =
+      mderiv (-Complex.I) (e i) (fun y => (K.matrix y)⁻¹*K.ricciHessian y) z +
+        ∑ p, ∑ q, (K.connection i z p q) • K.curvature q p z := by
+  have hnh := K.isOpen.mem_nhds hz
+  have hI := K.inverse_smooth.contDiffAt hnh
+  have hV (p q) := K.curvature_smooth p q |>.contDiffAt hnh
+  have hC := K.connection_smooth i |>.contDiffAt hnh
+  have he : (fun y => ∑ p, ∑ q, ((K.matrix y)⁻¹ p q) • K.curvature q p y) =ᶠ[𝓝 z]
+      (fun y => (K.matrix y)⁻¹*K.ricciHessian y) := by
+    filter_upwards [hnh] with y hy
+    exact K.curvature_contraction hy
+  have hh := mderiv_congr he (-Complex.I) (e i)
+  have hd : mderiv (-Complex.I) (e i) (fun y => (K.matrix y)⁻¹) z = -K.connection i z := by
+    rw [mderiv_inverse K.isOpen K.smooth (fun y hy => ne_of_gt (K.positive y hy).det_pos) hz,
+      connection,hM]
+    simp only [inv_one,neg_mul,one_mul,mul_one]
+  have hf (p q) : mderiv (-Complex.I) (e i)
+      (fun y => ((K.matrix y)⁻¹ p q) • K.curvature q p y) z =
+      -(K.connection i z p q) • K.curvature q p z +
+        ((1 : Matrix (Fin n) (Fin n) ℂ) p q) • mderiv (-Complex.I) (e i) (K.curvature q p) z := by
+    rw [mderiv_smul (entry_smooth hI p q) (hV q p)]
+    change mderiv (-Complex.I) (e i) (fun y => (K.matrix y)⁻¹) z p q • _ + _ = _
+    rw [hd,hM,inv_one]
+    rfl
+  rw [mderiv_sum Finset.univ (fun p _ => ContDiffAt.sum (fun q _ =>
+    matrix_smooth_smul (entry_smooth hI p q) (hV q p)))] at hh
+  simp only [mderiv_sum Finset.univ (fun q _ => matrix_smooth_smul (entry_smooth hI _ q) (hV q _)),hf,
+    Finset.sum_add_distrib,neg_smul,Finset.sum_neg_distrib] at hh
+  have hone : (∑ p, ∑ q, ((1 : Matrix (Fin n) (Fin n) ℂ) p q) •
+      mderiv (-Complex.I) (e i) (K.curvature q p) z) =
+      ∑ k, mderiv (-Complex.I) (e i) (K.curvature k k) z := by
+    simp [Matrix.one_apply]
+  rw [hone] at hh
+  have hh' := congrArg (fun A => A + ∑ p, ∑ q, (K.connection i z p q) • K.curvature q p z) hh
+  convert hh' using 1
+  abel
+
+lemma bar_covHol_connection {z : V n} (hz : z ∈ K.domain) (k i : Fin n) :
+    mderiv Complex.I (e k) (K.covHol K.connection k i) z =
+      mderiv (-Complex.I) (e i) (K.curvature k k) z -
+        ∑ q, ((K.curvature k k z q i) • K.connection q z +
+          (K.connection k z q i) • K.curvature k q z) := by
+  have hnh := K.isOpen.mem_nhds hz
+  have hG (j) := K.connection_smooth j |>.contDiffAt hnh
+  have he : (K.covHol K.connection k i) =ᶠ[𝓝 z]
+      (fun y => mderiv (-Complex.I) (e i) (K.connection k) y -
+        ∑ q, (K.connection k y q i) • K.connection q y) := by
+    filter_upwards [hnh] with y hy
+    exact K.covHol_connection_swap hy k i
+  rw [mderiv_congr he, mderiv_sub (mderiv_smooth (hG k) _ _)
+    (ContDiffAt.sum (fun q _ => matrix_smooth_smul (entry_smooth (hG k) q i) (hG q))),
+    mderiv_comm (hG k) Complex.I (-Complex.I) (e k) (e i),
+    mderiv_sum Finset.univ (fun q _ => matrix_smooth_smul (entry_smooth (hG k) q i) (hG q))]
+  congr 1
+  apply Finset.sum_congr rfl
+  intro q _
+  exact mderiv_smul (entry_smooth (hG k) q i) (hG q) _ _
+
+ 
+
+lemma calabi_connection_laplacian_at_one {z : V n} (hz : z ∈ K.domain)
+    (hM : K.matrix z = 1) (i : Fin n) :
+    (∑ k, mderiv Complex.I (e k) (K.covHol K.connection k i) z) =
+      mderiv (-Complex.I) (e i) K.ricciHessian z -
+        K.connection i z * K.ricciHessian z -
+        ∑ q, (K.ricciHessian z q i) • K.connection q z := by
+  have hc := K.curvature_contraction_at_one hz hM
+  have h1 : (∑ k, ∑ q, (K.curvature k k z q i) • K.connection q z) =
+      ∑ q, (K.ricciHessian z q i) • K.connection q z := by
+    rw [Finset.sum_comm]
+    apply Finset.sum_congr rfl
+    intro q _
+    rw [← Finset.sum_smul]
+    congr 1
+    simpa only [Matrix.sum_apply] using congrArg (fun A : Matrix (Fin n) (Fin n) ℂ => A q i) hc
+  have h2 : (∑ k, ∑ q, (K.connection k z q i) • K.curvature k q z) =
+      ∑ p, ∑ q, (K.connection i z p q) • K.curvature q p z := by
+    rw [Finset.sum_comm]
+    apply Finset.sum_congr rfl
+    intro q _
+    apply Finset.sum_congr rfl
+    intro k _
+    rw [K.connection_symmetric hz]
+  have h3 : mderiv (-Complex.I) (e i) (fun y => (K.matrix y)⁻¹*K.ricciHessian y) z =
+      mderiv (-Complex.I) (e i) K.ricciHessian z - K.connection i z*K.ricciHessian z := by
+    rw [mderiv_mul (K.inverse_smooth.contDiffAt (K.isOpen.mem_nhds hz))
+      (K.ricciHessian_smooth.contDiffAt (K.isOpen.mem_nhds hz)),
+      mderiv_inverse K.isOpen K.smooth (fun y hy => ne_of_gt (K.positive y hy).det_pos) hz,
+      connection,hM]
+    simp only [inv_one,neg_mul,one_mul,mul_one]
+    abel
+  simp only [K.bar_covHol_connection hz, Finset.sum_sub_distrib, Finset.sum_add_distrib]
+  rw [h1,h2,K.derivative_curvature_contraction_at_one hz hM,h3]
+  abel
+end KaehlerCalculus.LocalKaehlerField
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/CalabiCutoffAlgebra.lean b/lean/OAI/Geometry/PrescribedPotential/CalabiCutoffAlgebra.lean
new file mode 100644
index 0000000000000000000000000000000000000000..24c948d902dece76d443d36437c5260d7e91f99f
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/CalabiCutoffAlgebra.lean
@@ -0,0 +1,47 @@
+import Mathlib
+
+namespace OAI
+
+/-! Calabi Cutoff Algebra. -/
+
+section
+
+ 
+
+namespace CalabiEstimate
+lemma cross_absorb {x y r : ℝ} (hx : 0 ≤ x) (hy : 0 ≤ y)
+    (hr : r^2 ≤ 8*x*y) : -(y/2+4*x) ≤ r := by
+  have hab : |r| ≤ y/2+4*x := by
+    apply (sq_le_sq₀ (abs_nonneg _) (by positivity)).mp
+    rw [sq_abs]
+    nlinarith [sq_nonneg (y/2-4*x)]
+  exact (neg_le_neg hab).trans (neg_abs_le r)
+
+lemma cutoff_max_bound {S Q η C δ LS LT Lη cross : ℝ}
+    (hS : 0 ≤ S) (hQ : 0 ≤ Q) (hη : 0 ≤ η) (hη1 : η ≤ 1)
+    (hC : 0 ≤ C) (hδ : 0 < δ)
+    (hLS : Q-C*(1+S) ≤ LS) (hLT : δ*S-C ≤ LT)
+    (hLη : -C ≤ Lη) (hcross : cross^2 ≤ 8*C*η*S*Q)
+    (hmax : η*LS+S*Lη+cross+((6*C+1)/δ)*LT ≤ 0) :
+    S ≤ C*(1+(6*C+1)/δ) := by
+  let A := (6*C+1)/δ
+  have hA : 0 ≤ A := div_nonneg (by positivity) hδ.le
+  have hAd : A*δ = 6*C+1 := by dsimp [A]; field_simp
+  have hx : -(η*Q/2+4*(C*S)) ≤ cross := by
+    apply cross_absorb (mul_nonneg hC hS) (mul_nonneg hη hQ)
+    nlinarith [hcross]
+  have h1 := mul_le_mul_of_nonneg_left hLS hη
+  have h2 := mul_le_mul_of_nonneg_left hLη hS
+  have h3 := mul_le_mul_of_nonneg_left hLT hA
+  have h4 := mul_le_mul_of_nonneg_right hη1 (mul_nonneg hC (by linarith : 0 ≤ 1+S))
+  have h5 := mul_nonneg hη hQ
+  have he : A*(δ*S-C) = (6*C+1)*S-A*C := by rw [mul_sub,← mul_assoc,hAd]
+  rw [he] at h3
+  change η*LS+S*Lη+cross+A*LT ≤ 0 at hmax
+  change S ≤ C*(1+A)
+  nlinarith
+end CalabiEstimate
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/CalabiCutoffFrames.lean b/lean/OAI/Geometry/PrescribedPotential/CalabiCutoffFrames.lean
new file mode 100644
index 0000000000000000000000000000000000000000..bf06d697bdde1c66725cc38364875826fe39730d
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/CalabiCutoffFrames.lean
@@ -0,0 +1,128 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.FrameCompactBounds
+import OAI.Geometry.PrescribedPotential.GlobalSobolev
+import OAI.Geometry.PrescribedPotential.HolomorphicDirection
+import OAI.Geometry.PrescribedPotential.ScalarFrameCalculus
+import OAI.Geometry.PrescribedPotential.ScalarProductLaplacian
+
+namespace OAI
+
+/-! Calabi Cutoff Frames. -/
+
+section
+
+noncomputable section
+open Set Metric Filter Topology Matrix
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace KaehlerCalculus
+variable {n : ℕ}
+
+lemma holGradientSquare_sq {f : V n → ℝ} {z : V n} (hf : ContDiffAt ℝ ∞ f z) :
+    holGradientSquare (fun y => (f y)^2) z = 4*(f z)^2*holGradientSquare f z := by
+  have hfc : ContDiffAt ℝ ∞ (fun y => (f y:ℂ)) z := Complex.ofRealCLM.contDiff.contDiffAt.comp z hf
+  have hd (k : Fin n) : dz (e k) (fun y => ((f y)^2:ℂ)) z = (2*(f z:ℂ))*dz (e k) (fun y => (f y:ℂ)) z := by
+    simp only [pow_two]
+    change wderiv (-Complex.I) (e k) (fun y => (f y:ℂ)*(f y:ℂ)) z = _
+    rw [wderiv_mul (hfc.differentiableAt (by simp)) (hfc.differentiableAt (by simp))]
+    change dz (e k) _ z*(f z:ℂ)+(f z:ℂ)*dz (e k) _ z = _
+    ring
+  rw [holGradientSquare_eq,holGradientSquare_eq,Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro k _
+  have hco : (fun y => (((f y)^2:ℝ):ℂ)) = (fun y => (f y:ℂ)^2) := by funext y; simp
+  rw [hco,hd,norm_mul,norm_mul,Complex.norm_ofNat,Complex.norm_real,Real.norm_eq_abs,mul_pow,mul_pow,sq_abs]
+  ring
+
+def frameGradientSquare (f : V n → ℝ) (z : V n) (C : Matrix (Fin n) (Fin n) ℂ) : ℝ :=
+  ∑ k, ‖∑ p, C p k*dz (e p) (fun y => (f y:ℂ)) z‖^2
+
+lemma frameGradientSquare_continuous {f : V n → ℝ} (hf : ContDiff ℝ ∞ f) :
+    Continuous (fun q : V n × Matrix (Fin n) (Fin n) ℂ => frameGradientSquare f q.1 q.2) := by
+  have hfc : ContDiff ℝ ∞ (fun y => (f y:ℂ)) := Complex.ofRealCLM.contDiff.comp hf
+  unfold frameGradientSquare
+  apply continuous_finsetSum
+  intro k _
+  apply Continuous.pow
+  apply Continuous.norm
+  apply continuous_finsetSum
+  intro p _
+  exact ((continuous_apply_apply p k).comp continuous_snd).mul
+    ((contDiff_iff_contDiffAt.mpr (fun z => wderiv_smooth hfc.contDiffAt _ _)).continuous.comp continuous_fst)
+
+lemma holGradientSquare_linear (B : V n →L[ℂ] V n) {f : V n → ℝ} {w : V n}
+    (hf : ContDiffAt ℝ ∞ f (B w)) :
+    holGradientSquare (fun y => f (B y)) w = frameGradientSquare f (B w) (frameMatrix B) := by
+  have hfc : ContDiffAt ℝ ∞ (fun y => (f y:ℂ)) (B w) := Complex.ofRealCLM.contDiff.contDiffAt.comp (B w) hf
+  rw [holGradientSquare_eq]
+  unfold frameGradientSquare
+  apply Finset.sum_congr rfl
+  intro k _
+  congr 2
+  change wderiv (-Complex.I) (e k) (fun y => (f (B y):ℂ)) w = _
+  rw [wderiv_comp_linear B (hfc.differentiableAt (by simp))]
+  change dz (B (e k)) (fun y => (f y:ℂ)) (B w) = _
+  rw [dz_direction_coordinates]
+  rfl
+
+lemma frameGradientSquare_bounded {L : Set (V n)} (hL : IsCompact L) {f : V n → ℝ}
+    (hf : ContDiff ℝ ∞ f) (B : ℝ) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ z ∈ L, ∀ b : Matrix (Fin n) (Fin n) ℂ, ‖b‖ ≤ B → frameGradientSquare f z b ≤ C := by
+  obtain ⟨C,hC⟩ := (hL.prod (isCompact_closedBall (0:Matrix (Fin n) (Fin n) ℂ) B)).exists_bound_of_continuousOn
+    (frameGradientSquare_continuous hf).continuousOn
+  refine ⟨max C 0,le_max_right _ _,?_⟩
+  intro z hz b hb
+  have hp : (z,b) ∈ L ×ˢ closedBall (0:Matrix (Fin n) (Fin n) ℂ) B :=
+    ⟨hz,by simpa only [mem_closedBall,dist_zero_right] using hb⟩
+  have hh := hC (z,b) hp
+  change ‖frameGradientSquare f z b‖ ≤ C at hh
+  rw [Real.norm_eq_abs] at hh
+  exact (le_abs_self _).trans (hh.trans (le_max_left _ _))
+
+lemma frameHessianTrace_continuous {f : V n → ℝ} (hf : ContDiff ℝ ∞ f) :
+    Continuous (fun q : V n × Matrix (Fin n) (Fin n) ℂ =>
+      (q.2ᴴ*PotentialKaehler.potentialMatrix f q.1*q.2).trace.re) := by
+  have hH : Continuous (PotentialKaehler.potentialMatrix f) :=
+    (contDiff_iff_contDiffAt.mpr (fun z => PotentialKaehler.potentialMatrix_smooth hf.contDiffAt)).continuous
+  exact continuousOn_univ.mp (continuousOn_trace_re
+    (((continuousOn_matrix_star continuous_snd.continuousOn).mul (hH.comp continuous_fst).continuousOn).mul continuous_snd.continuousOn))
+
+lemma frameHessianTrace_bounded {L : Set (V n)} (hL : IsCompact L) {f : V n → ℝ}
+    (hf : ContDiff ℝ ∞ f) (B : ℝ) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ z ∈ L, ∀ b : Matrix (Fin n) (Fin n) ℂ, ‖b‖ ≤ B →
+      -C ≤ (bᴴ*PotentialKaehler.potentialMatrix f z*b).trace.re := by
+  obtain ⟨C,hC⟩ := (hL.prod (isCompact_closedBall (0:Matrix (Fin n) (Fin n) ℂ) B)).exists_bound_of_continuousOn
+    (frameHessianTrace_continuous hf).continuousOn
+  refine ⟨max C 0,le_max_right _ _,?_⟩
+  intro z hz b hb
+  have hp : (z,b) ∈ L ×ˢ closedBall (0:Matrix (Fin n) (Fin n) ℂ) B :=
+    ⟨hz,by simpa only [mem_closedBall,dist_zero_right] using hb⟩
+  have hh := hC (z,b) hp
+  change ‖(bᴴ*PotentialKaehler.potentialMatrix f z*b).trace.re‖ ≤ C at hh
+  rw [Real.norm_eq_abs] at hh
+  exact (neg_le_neg (hh.trans (le_max_left _ _))).trans (neg_abs_le _)
+
+lemma cutoff_frame_bounds {L : Set (V n)} (hL : IsCompact L) (χ : V n → ℝ)
+    (hχ : ContDiff ℝ ∞ χ) (B : ℝ) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ (b : V n →L[ℂ] V n) (w : V n), b w ∈ L → ‖frameMatrix b‖ ≤ B →
+      -C ≤ (PotentialKaehler.potentialMatrix (fun y => (χ (b y))^2) w).trace.re ∧
+      holGradientSquare (fun y => (χ (b y))^2) w ≤ C*(χ (b w))^2 := by
+  obtain ⟨C1,hC1,h1⟩ := frameHessianTrace_bounded hL (hχ.pow 2) B
+  obtain ⟨C2,hC2,h2⟩ := frameGradientSquare_bounded hL hχ B
+  let C := max C1 (4*C2)
+  refine ⟨C,hC1.trans (le_max_left _ _),?_⟩
+  intro b w hw hb
+  constructor
+  · rw [potentialMatrix_linear b (hχ.contDiffAt.pow 2)]
+    exact (neg_le_neg (le_max_left C1 (4*C2))).trans (h1 _ hw _ hb)
+  · have hs : ContDiffAt ℝ ∞ (fun y => χ (b y)) w := hχ.contDiffAt.comp w (b.restrictScalars ℝ).contDiff.contDiffAt
+    rw [holGradientSquare_sq hs,holGradientSquare_linear b hχ.contDiffAt]
+    have hh := h2 (b w) hw (frameMatrix b) hb
+    have hC : 4*C2 ≤ C := le_max_right _ _
+    have hmul := mul_le_mul_of_nonneg_right (by linarith : 4*frameGradientSquare χ (b w) (frameMatrix b) ≤ C) (sq_nonneg (χ (b w)))
+    nlinarith
+end KaehlerCalculus
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/CalabiCutoffPoint.lean b/lean/OAI/Geometry/PrescribedPotential/CalabiCutoffPoint.lean
new file mode 100644
index 0000000000000000000000000000000000000000..852b492e37aa8f9f635cb8fcb3af121a55c963a2
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/CalabiCutoffPoint.lean
@@ -0,0 +1,66 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CalabiCutoffAlgebra
+import OAI.Geometry.PrescribedPotential.CalabiPointInequalities
+import OAI.Geometry.PrescribedPotential.CalabiRescue
+import OAI.Geometry.PrescribedPotential.CalabiResidualBound
+import OAI.Geometry.PrescribedPotential.PotentialLinearCalculus
+import OAI.Geometry.PrescribedPotential.ScalarProductLaplacian
+
+namespace OAI
+
+/-! Calabi Cutoff Point. -/
+
+section
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace KaehlerCalculus.LocalKaehlerField
+variable {n : ℕ} (K : LocalKaehlerField n)
+
+lemma calabi_cutoff_max_at_one {z : V n} (hz : z ∈ K.domain) (hM : K.matrix z = 1)
+    (D : Matrix (Fin n) (Fin n) ℂ) {η : V n → ℝ} (hηs : ContDiffAt ℝ ∞ η z)
+    {C δ : ℝ} (hC : 0 ≤ C) (hδ : 0 < δ) (hη : 0 ≤ η z) (hη1 : η z ≤ 1)
+    (hηlap : -C ≤ (PotentialKaehler.potentialMatrix η z).trace.re)
+    (hηgrad : holGradientSquare η z ≤ C*η z)
+    (hRes : -C*(1+K.calabiNorm z) ≤ calabiResidual (K.ricciHessian z)
+      (fun i => mderiv (-Complex.I) (e i) K.ricciHessian z) (fun i => K.connection i z))
+    (hRic : -C ≤ (Dᴴ*K.ricciHessian z*D).trace.re)
+    (hRescue : δ*K.calabiNorm z ≤ rescueSquare D (fun i => K.connection i z))
+    (hmax : IsLocalMax (fun y => η y*K.calabiNorm y+((6*C+1)/δ)*K.weightedTrace D y) z) :
+    K.calabiNorm z ≤ C*(1+(6*C+1)/δ) := by
+  have hn := K.isOpen.mem_nhds hz
+  have hSs := K.calabiNorm_smooth.contDiffAt hn
+  have hTs := (K.weightedTrace_smooth D).contDiffAt hn
+  have hS : 0 ≤ K.calabiNorm z := by rw [K.calabiNorm_at_one hM]; exact tensorSquare_nonneg _
+  have hQ := K.calabiGood_nonneg z
+  have hgrad : holGradientSquare K.calabiNorm z ≤ 2*K.calabiNorm z*K.calabiGood z := by
+    rw [holGradientSquare_eq]
+    exact K.calabi_gradient_sum_at_one hz hM
+  have hcross : (gradientCross η K.calabiNorm z)^2 ≤ 8*C*η z*K.calabiNorm z*K.calabiGood z := by
+    have hh := gradientCross_sq η K.calabiNorm z
+    have ha := mul_le_mul hηgrad hgrad (sq_nonneg ‖holGradient K.calabiNorm z‖)
+      (mul_nonneg hC hη)
+    dsimp only [holGradientSquare] at ha hh
+    nlinarith
+  have hLS : K.calabiGood z-C*(1+K.calabiNorm z) ≤
+      (PotentialKaehler.potentialMatrix K.calabiNorm z).trace.re := by
+    rw [K.calabi_bochner_real_at_one hz hM]
+    linarith
+  have hLT : δ*K.calabiNorm z-C ≤
+      (PotentialKaehler.potentialMatrix (K.weightedTrace D) z).trace.re := by
+    rw [K.weightedTrace_laplacian_at_one D hz hM]
+    linarith
+  have ht := EllipticKernel.trace_potentialMatrix_nonpos_at_localMax
+    ((hηs.mul hSs).add (contDiffAt_const.mul hTs)) hmax (1 : Matrix (Fin n) (Fin n) ℂ) Matrix.PosDef.one
+  rw [inv_one,one_mul,potentialMatrix_affine (hηs.mul hSs) hTs,Matrix.trace_add,Matrix.trace_smul,
+    Complex.add_re] at ht
+  simp only [smul_eq_mul,Complex.mul_re,Complex.ofReal_re,Complex.ofReal_im,zero_mul,sub_zero] at ht
+  rw [potentialMatrix_trace_mul hηs hSs] at ht
+  exact CalabiEstimate.cutoff_max_bound hS hQ hη hη1 hC hδ hLS hLT hηlap hcross ht
+end KaehlerCalculus.LocalKaehlerField
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/CalabiFrameAlgebra.lean b/lean/OAI/Geometry/PrescribedPotential/CalabiFrameAlgebra.lean
new file mode 100644
index 0000000000000000000000000000000000000000..4542ba6e596a528865f237b5624f8208d79e28f3
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/CalabiFrameAlgebra.lean
@@ -0,0 +1,89 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CalabiTensorAlgebra
+
+namespace OAI
+
+/-! Calabi Frame Algebra. -/
+
+section
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace KaehlerCalculus
+variable {n : ℕ}
+
+def pullConnectionTensor (C : Matrix (Fin n) (Fin n) ℂ) (T : ConnectionTensor n) : ConnectionTensor n :=
+  fun i => C⁻¹*(∑ p, C p i • T p)*C
+
+lemma tensor_trace_changeBasis (C P M T U : Matrix (Fin n) (Fin n) ℂ) (hC : IsUnit C) :
+    ((C⁻¹*T*C)*(C⁻¹*P*(Cᴴ)⁻¹)*(C⁻¹*U*C)ᴴ*(Cᴴ*M*C)).trace = (T*P*Uᴴ*M).trace := by
+  have hc : C*C⁻¹=1 := Matrix.mul_nonsing_inv _ (Matrix.isUnit_iff_isUnit_det _ |>.mp hC)
+  have hd : (Cᴴ)⁻¹*Cᴴ=1 := by
+    rw [← Matrix.conjTranspose_nonsing_inv,← Matrix.conjTranspose_mul,hc,Matrix.conjTranspose_one]
+  simp only [Matrix.conjTranspose_mul,Matrix.conjTranspose_nonsing_inv,Matrix.mul_assoc]
+  rw [← Matrix.mul_assoc C C⁻¹,hc,Matrix.one_mul,
+    ← Matrix.mul_assoc (Cᴴ)⁻¹ Cᴴ,hd,Matrix.one_mul,
+    ← Matrix.mul_assoc (Cᴴ)⁻¹ Cᴴ,hd,Matrix.one_mul]
+  calc
+    _ = ((C⁻¹*(T*P*Uᴴ*M))*C).trace := by simp only [mul_assoc]
+    _ = ((C*C⁻¹)*(T*P*Uᴴ*M)).trace := Matrix.trace_mul_cycle _ _ _
+    _ = _ := by simp only [hc,one_mul,mul_assoc]
+
+lemma tensor_pair_mixed (P M Q C : Matrix (Fin n) (Fin n) ℂ) (T U : ConnectionTensor n) :
+    (∑ i, ∑ j, Q i j * ((∑ p, C p i • T p)*P*(∑ q, C q j • U q)ᴴ*M).trace) =
+      ∑ p, ∑ q, (C*Q*Cᴴ) p q*(T p*P*(U q)ᴴ*M).trace := by
+  have he (i j) : ((∑ p, C p i • T p)*P*(∑ q, C q j • U q)ᴴ*M).trace =
+      ∑ p, ∑ q, C p i * star (C q j) * (T p*P*(U q)ᴴ*M).trace := by
+    simp only [Matrix.sum_mul,Matrix.conjTranspose_sum,Matrix.trace_sum,
+      Matrix.smul_mul,Matrix.mul_smul,Matrix.conjTranspose_smul,Matrix.trace_smul,
+      smul_eq_mul,Finset.mul_sum]
+    rw [Finset.sum_comm]
+    apply Finset.sum_congr rfl
+    intro p _
+    apply Finset.sum_congr rfl
+    intro q _
+    ring
+  simp_rw [he]
+  simp only [Matrix.mul_apply,Matrix.conjTranspose_apply,Finset.mul_sum,Finset.sum_mul]
+  calc
+    _ = ∑ p, ∑ q, ∑ i, ∑ j,
+        Q i j * (C p i * star (C q j) * (T p*P*(U q)ᴴ*M).trace) := by
+      conv_lhs => arg 2; ext i; rw [Finset.sum_comm]; arg 2; ext p; rw [Finset.sum_comm]
+      rw [Finset.sum_comm]
+      apply Finset.sum_congr rfl
+      intro p _
+      rw [Finset.sum_comm]
+    _ = _ := by
+      apply Finset.sum_congr rfl
+      intro p _
+      apply Finset.sum_congr rfl
+      intro q _
+      rw [Finset.sum_comm]
+      apply Finset.sum_congr rfl
+      intro j _
+      apply Finset.sum_congr rfl
+      intro i _
+      ring
+
+lemma tensorPairAt_changeBasis (C P M : Matrix (Fin n) (Fin n) ℂ) (hC : IsUnit C)
+    (T U : ConnectionTensor n) :
+    tensorPairAt (C⁻¹*P*(Cᴴ)⁻¹) (Cᴴ*M*C) (pullConnectionTensor C T) (pullConnectionTensor C U) =
+      tensorPairAt P M T U := by
+  unfold tensorPairAt pullConnectionTensor
+  simp_rw [tensor_trace_changeBasis C P M _ _ hC]
+  rw [tensor_pair_mixed]
+  have hc : C*C⁻¹=1 := Matrix.mul_nonsing_inv _ (Matrix.isUnit_iff_isUnit_det _ |>.mp hC)
+  have hd : (Cᴴ)⁻¹*Cᴴ=1 := by
+    rw [← Matrix.conjTranspose_nonsing_inv,← Matrix.conjTranspose_mul,hc,Matrix.conjTranspose_one]
+  have he : C*(C⁻¹*P*(Cᴴ)⁻¹)*Cᴴ = P := by
+    simp only [mul_assoc]
+    rw [← mul_assoc C C⁻¹,hc,one_mul]
+    simp only [hd,mul_one]
+  simp only [he]
+end KaehlerCalculus
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/CalabiFrameCalculus.lean b/lean/OAI/Geometry/PrescribedPotential/CalabiFrameCalculus.lean
new file mode 100644
index 0000000000000000000000000000000000000000..be7a926f457e4ce3d97f8cb03a1f9f1fc42b498f
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/CalabiFrameCalculus.lean
@@ -0,0 +1,75 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CalabiBochner
+import OAI.Geometry.PrescribedPotential.CalabiFrameAlgebra
+import OAI.Geometry.PrescribedPotential.HolomorphicDirection
+import OAI.Geometry.PrescribedPotential.MatrixFrameNormalization
+import OAI.Geometry.PrescribedPotential.ScalarFrameCalculus
+
+namespace OAI
+
+/-! Calabi Frame Calculus. -/
+
+section
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace KaehlerCalculus.LocalKaehlerField
+variable {n : ℕ} (K : LocalKaehlerField n)
+
+def pull (B : V n →L[ℂ] V n) (hB : IsUnit (frameMatrix B)) : LocalKaehlerField n where
+  domain := B ⁻¹' K.domain
+  isOpen := K.isOpen.preimage B.continuous
+  matrix := pullMetric B K.matrix
+  smooth := pullMetric_smooth K.isOpen B K.smooth
+  positive := pullMetric_positive B hB K.positive
+  closed := pullMetric_closed K.isOpen B K.smooth K.closed
+
+lemma pull_connection (B : V n →L[ℂ] V n) (hB : IsUnit (frameMatrix B))
+    {w : V n} (hw : B w ∈ K.domain) (i : Fin n) :
+    (K.pull B hB).connection i w = pullConnectionTensor (frameMatrix B) (fun p => K.connection p (B w)) i := by
+  let C := frameMatrix B
+  have hc : C*C⁻¹ = 1 := Matrix.mul_nonsing_inv _ (Matrix.isUnit_iff_isUnit_det _ |>.mp hB)
+  have hd : (Cᴴ)⁻¹*Cᴴ = 1 := by
+    rw [← Matrix.conjTranspose_nonsing_inv,← Matrix.conjTranspose_mul,hc,Matrix.conjTranspose_one]
+  change (pullMetric B K.matrix w)⁻¹*mderiv (-Complex.I) (e i) (pullMetric B K.matrix) w = _
+  rw [mderiv_pullMetric B (K.smooth.contDiffAt (K.isOpen.mem_nhds hw))]
+  unfold pullMetric
+  change (Cᴴ*K.matrix (B w)*C)⁻¹*(Cᴴ*mderiv (-Complex.I) (B (e i)) K.matrix (B w)*C) = _
+  simp only [Matrix.mul_inv_rev,mul_assoc]
+  rw [← mul_assoc (Cᴴ)⁻¹ Cᴴ,hd,one_mul,mderiv_hol_coordinates]
+  have he (p) : B (e i) p = C p i := rfl
+  simp only [he,pullConnectionTensor,connection,Matrix.mul_sum,Matrix.mul_smul,Matrix.sum_mul,Matrix.smul_mul,mul_assoc,C]
+
+lemma pull_calabiNorm (B : V n →L[ℂ] V n) (hB : IsUnit (frameMatrix B))
+    {w : V n} (hw : B w ∈ K.domain) :
+    (K.pull B hB).calabiNorm w = K.calabiNorm (B w) := by
+  unfold calabiNorm tensorPair
+  simp_rw [K.pull_connection B hB hw]
+  change (tensorPairAt (pullMetric B K.matrix w)⁻¹ (pullMetric B K.matrix w) _ _).re = _
+  unfold pullMetric
+  simp only [Matrix.mul_inv_rev]
+  exact congrArg Complex.re (by
+    simpa only [mul_assoc] using tensorPairAt_changeBasis (frameMatrix B) (K.matrix (B w))⁻¹
+      (K.matrix (B w)) hB (fun p => K.connection p (B w)) (fun p => K.connection p (B w)))
+
+lemma pull_ricciHessian (B : V n →L[ℂ] V n) (hB : IsUnit (frameMatrix B))
+    {w : V n} (hw : B w ∈ K.domain) :
+    (K.pull B hB).ricciHessian w = pullMetric B K.ricciHessian w :=
+  potentialMatrix_logdet_pullMetric K.isOpen B hB K.smooth K.positive hw
+
+lemma pull_ricciHessian_hol (B : V n →L[ℂ] V n) (hB : IsUnit (frameMatrix B))
+    {w : V n} (hw : B w ∈ K.domain) (i : Fin n) :
+    mderiv (-Complex.I) (e i) (K.pull B hB).ricciHessian w =
+      (frameMatrix B)ᴴ * mderiv (-Complex.I) (B (e i)) K.ricciHessian (B w) * frameMatrix B := by
+  have he : (K.pull B hB).ricciHessian =ᶠ[𝓝 w] pullMetric B K.ricciHessian := by
+    filter_upwards [B.continuous.continuousAt.preimage_mem_nhds (K.isOpen.mem_nhds hw)] with y hy
+    exact K.pull_ricciHessian B hB hy
+  rw [mderiv_congr he]
+  exact mderiv_pullMetric B (K.ricciHessian_smooth.contDiffAt (K.isOpen.mem_nhds hw)) _ _
+end KaehlerCalculus.LocalKaehlerField
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/CalabiFrameScalars.lean b/lean/OAI/Geometry/PrescribedPotential/CalabiFrameScalars.lean
new file mode 100644
index 0000000000000000000000000000000000000000..14967aff36de9477581ae9a1aea08ad2072abf18
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/CalabiFrameScalars.lean
@@ -0,0 +1,65 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CalabiFrameCalculus
+import OAI.Geometry.PrescribedPotential.CalabiPointInequalities
+import OAI.Geometry.PrescribedPotential.CalabiRicciFrames
+import OAI.Geometry.PrescribedPotential.HolomorphicDirection
+import OAI.Geometry.PrescribedPotential.MatrixFrameNormalization
+
+namespace OAI
+
+/-! Calabi Frame Scalars. -/
+
+section
+
+noncomputable section
+open Set Metric Filter Topology Matrix
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace KaehlerCalculus.LocalKaehlerField
+variable {n : ℕ} (K : LocalKaehlerField n)
+
+lemma calabiNorm_nonneg {z : V n} (hz : z ∈ K.domain) : 0 ≤ K.calabiNorm z := by
+  obtain ⟨B,w,hB,hw,hM⟩ := normalized_frame K.matrix z (K.positive z hz)
+  have hwU : B w ∈ K.domain := hw ▸ hz
+  have h := K.pull_calabiNorm B hB hwU
+  rw [hw] at h
+  rw [← h,(K.pull B hB).calabiNorm_at_one hM]
+  exact tensorSquare_nonneg _
+
+lemma pull_weightedTrace_undo (B : V n →L[ℂ] V n) (hB : IsUnit (frameMatrix B))
+    (D : Matrix (Fin n) (Fin n) ℂ) (hCD : frameMatrix B*D = 1) (w : V n) :
+    (K.pull B hB).weightedTrace D w = (K.matrix (B w)).trace.re := by
+  change (Dᴴ*((frameMatrix B)ᴴ*K.matrix (B w)*frameMatrix B)*D).trace.re = _
+  have he : Dᴴ*((frameMatrix B)ᴴ*K.matrix (B w)*frameMatrix B)*D =
+      (frameMatrix B*D)ᴴ*K.matrix (B w)*(frameMatrix B*D) := by
+    simp only [Matrix.conjTranspose_mul,mul_assoc]
+  rw [he,hCD,Matrix.conjTranspose_one,one_mul,mul_one]
+
+lemma pull_ricciHessian_jet (B : V n →L[ℂ] V n) (hB : IsUnit (frameMatrix B))
+    {w : V n} (hw : B w ∈ K.domain) :
+    (fun i => mderiv (-Complex.I) (e i) (K.pull B hB).ricciHessian w) =
+      frameRicciJet (fun i => mderiv (-Complex.I) (e i) K.ricciHessian (B w)) (frameMatrix B) := by
+  funext i
+  rw [K.pull_ricciHessian_hol B hB hw,mderiv_hol_coordinates]
+  rfl
+
+lemma pull_cutoff_max (B : V n →L[ℂ] V n) (hB : IsUnit (frameMatrix B))
+    {z w : V n} (hz : z ∈ K.domain) (hw : B w = z)
+    (D : Matrix (Fin n) (Fin n) ℂ) (hCD : frameMatrix B*D = 1)
+    (χ : V n → ℝ) (a : ℝ)
+    (hmax : IsLocalMax (fun y => (χ y)^2*K.calabiNorm y+a*(K.matrix y).trace.re) z) :
+    IsLocalMax (fun y => (χ (B y))^2*(K.pull B hB).calabiNorm y+a*(K.pull B hB).weightedTrace D y) w := by
+  have ht : Tendsto B (𝓝 w) (𝓝 z) := hw ▸ B.continuous.continuousAt
+  have hnormw := K.pull_calabiNorm B hB (show B w ∈ K.domain by rw [hw]; exact hz)
+  have htracew := K.pull_weightedTrace_undo B hB D hCD w
+  change ∀ᶠ y in 𝓝 w, _ ≤ _
+  dsimp only
+  rw [hnormw,htracew,hw]
+  filter_upwards [ht.eventually hmax,ht.eventually (K.isOpen.mem_nhds hz)] with y hy hym
+  rw [K.pull_calabiNorm B hB hym,K.pull_weightedTrace_undo B hB D hCD]
+  exact hy
+end KaehlerCalculus.LocalKaehlerField
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/CalabiLocalUniform.lean b/lean/OAI/Geometry/PrescribedPotential/CalabiLocalUniform.lean
new file mode 100644
index 0000000000000000000000000000000000000000..56fe6f45e1a1173d2ee1d86c5e52d9f9ce288e61
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/CalabiLocalUniform.lean
@@ -0,0 +1,153 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CalabiCutoffFrames
+import OAI.Geometry.PrescribedPotential.CalabiCutoffPoint
+import OAI.Geometry.PrescribedPotential.CalabiFrameScalars
+import OAI.Geometry.PrescribedPotential.CalabiPointInequalities
+import OAI.Geometry.PrescribedPotential.CalabiRicciFrames
+import OAI.Geometry.PrescribedPotential.MatrixFrameNormalization
+
+namespace OAI
+
+/-! Calabi Local Uniform. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Metric Filter Topology Matrix
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace KaehlerCalculus
+variable {n : ℕ}
+
+lemma calabi_cutoff_max_uniform {Y : Type*} [TopologicalSpace Y]
+    {P : Set Y} {L : Set (V n)} (G : V n → Matrix (Fin n) (Fin n) ℂ)
+    (H : Y × V n → Matrix (Fin n) (Fin n) ℂ) (J : Y × V n → ConnectionTensor n)
+    (χ : V n → ℝ) (hχ : ContDiff ℝ ∞ χ)
+    {R K B C δ : ℝ} (hC : 0 ≤ C) (hδ : 0 < δ)
+    (hframe : ∀ z ∈ L, ∀ M C D : Matrix (Fin n) (Fin n) ℂ,
+      ((R:ℂ) • M-G z).PosSemidef → ((K:ℂ) • G z-M).PosSemidef →
+      C*D = 1 → Cᴴ*M*C = 1 → ‖C‖ ≤ B ∧ ‖D‖ ≤ B)
+    (hcut : ∀ (b : V n →L[ℂ] V n) w, b w ∈ L → ‖frameMatrix b‖ ≤ B →
+      -C ≤ (PotentialKaehler.potentialMatrix (fun y => (χ (b y))^2) w).trace.re ∧
+      holGradientSquare (fun y => (χ (b y))^2) w ≤ C*(χ (b w))^2)
+    (hric : ∀ x ∈ P ×ˢ L, ∀ p ∈ boundedFramePairs (n := n) B, ∀ T : ConnectionTensor n,
+      -C*(1+tensorSquare T) ≤ calabiResidual (p.1ᴴ*H x*p.1) (frameRicciJet (J x) p.1) T ∧
+      -C ≤ (p.2ᴴ*(p.1ᴴ*H x*p.1)*p.2).trace.re ∧
+      δ*tensorSquare T ≤ rescueSquare p.2 T)
+    (M : LocalKaehlerField n) {t : Y} (ht : t ∈ P) {z : V n} (hz : z ∈ L)
+    (hzM : z ∈ M.domain)
+    (hell : ((R:ℂ) • M.matrix z-G z).PosSemidef ∧ ((K:ℂ) • G z-M.matrix z).PosSemidef)
+    (hH : M.ricciHessian z = H (t,z))
+    (hJ : (fun i => mderiv (-Complex.I) (e i) M.ricciHessian z) = J (t,z))
+    (hχ0 : 0 ≤ χ z) (hχ1 : χ z ≤ 1)
+    (hmax : IsLocalMax (fun y => (χ y)^2*M.calabiNorm y+((6*C+1)/δ)*(M.matrix y).trace.re) z) :
+    M.calabiNorm z ≤ C*(1+(6*C+1)/δ) := by
+  obtain ⟨b,w,hb,hw,hnorm⟩ := normalized_frame M.matrix z (M.positive z hzM)
+  let D := (frameMatrix b)⁻¹
+  have hCD : frameMatrix b*D = 1 := Matrix.mul_nonsing_inv _ (Matrix.isUnit_iff_isUnit_det _ |>.mp hb)
+  have hDC : D*frameMatrix b = 1 := Matrix.nonsing_inv_mul _ (Matrix.isUnit_iff_isUnit_det _ |>.mp hb)
+  have hnorm' : (frameMatrix b)ᴴ*M.matrix z*frameMatrix b = 1 := by
+    simpa only [pullMetric,hw] using hnorm
+  obtain ⟨hbc,hdc⟩ := hframe z hz (M.matrix z) (frameMatrix b) D hell.1 hell.2 hCD hnorm'
+  have hp : (frameMatrix b,D) ∈ boundedFramePairs B := ⟨hbc,hdc,hCD,hDC⟩
+  have hwM : b w ∈ M.domain := hw ▸ hzM
+  have hwL : b w ∈ L := hw ▸ hz
+  have hS := M.pull_calabiNorm b hb hwM
+  have hS' : tensorSquare (fun i => (M.pull b hb).connection i w) = M.calabiNorm z := by
+    rw [← (M.pull b hb).calabiNorm_at_one hnorm,hS,hw]
+  obtain ⟨hr1,hr2,hr3⟩ := hric (t,z) ⟨ht,hz⟩ (frameMatrix b,D) hp (fun i => (M.pull b hb).connection i w)
+  have hh : (M.pull b hb).ricciHessian w = (frameMatrix b)ᴴ*H (t,z)*frameMatrix b := by
+    rw [M.pull_ricciHessian b hb hwM]
+    simp only [pullMetric,hw,hH]
+  have hj : (fun i => mderiv (-Complex.I) (e i) (M.pull b hb).ricciHessian w) = frameRicciJet (J (t,z)) (frameMatrix b) := by
+    rw [M.pull_ricciHessian_jet b hb hwM,hw,hJ]
+  obtain ⟨hc1,hc2⟩ := hcut b w hwL hbc
+  have hχs : ContDiffAt ℝ ∞ (fun y => (χ (b y))^2) w :=
+    (hχ.contDiffAt.comp w (b.restrictScalars ℝ).contDiff.contDiffAt).pow 2
+  have hχb1 : (χ (b w))^2 ≤ 1 := by rw [hw]; nlinarith
+  have hnormM : (M.pull b hb).matrix w = 1 := hnorm
+  have hhmax := M.pull_cutoff_max b hb hzM hw D hCD χ ((6*C+1)/δ) hmax
+  have hbound := (M.pull b hb).calabi_cutoff_max_at_one hwM hnormM D hχs hC hδ
+    (sq_nonneg _) hχb1 hc1 hc2
+    (by rw [hh,hj,(M.pull b hb).calabiNorm_at_one hnormM]; exact hr1)
+    (by rw [hh]; exact hr2)
+    (by rw [(M.pull b hb).calabiNorm_at_one hnormM]; exact hr3) hhmax
+  rwa [hS,hw] at hbound
+
+ 
+
+theorem calabi_interior_uniform {Y : Type*} [TopologicalSpace Y]
+    {P : Set Y} (hP : IsCompact P) {L : Set (V n)} (hL : IsCompact L)
+    (G : V n → Matrix (Fin n) (Fin n) ℂ)
+    (hG : ContinuousOn G L) (hGp : ∀ z ∈ L, (G z).PosDef)
+    (H : Y × V n → Matrix (Fin n) (Fin n) ℂ) (J : Y × V n → ConnectionTensor n)
+    (hH : ContinuousOn H (P ×ˢ L)) (hJ : ContinuousOn J (P ×ˢ L))
+    (χ : V n → ℝ) (hχ : ContDiff ℝ ∞ χ)
+    (hχ0 : ∀ z ∈ L, 0 ≤ χ z) (hχ1 : ∀ z ∈ L, χ z ≤ 1)
+    (hχint : ∀ z ∈ L, χ z ≠ 0 → z ∈ interior L)
+    {R K : ℝ} (hR : 0 ≤ R) (hK : 0 ≤ K) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ (M : LocalKaehlerField n) (t : Y),
+      t ∈ P → L ⊆ M.domain →
+      (∀ z ∈ L, ((R:ℂ) • M.matrix z-G z).PosSemidef ∧ ((K:ℂ) • G z-M.matrix z).PosSemidef) →
+      (∀ z ∈ L, M.ricciHessian z = H (t,z)) →
+      (∀ z ∈ L, (fun i => mderiv (-Complex.I) (e i) M.ricciHessian z) = J (t,z)) →
+      ∀ z ∈ L, χ z = 1 → M.calabiNorm z ≤ C := by
+  obtain ⟨B,hB,hframe⟩ := compact_normalizing_frames hL G hG hGp hR hK
+  obtain ⟨C1,δ,hC1,hδ,hric⟩ := calabi_frame_uniform (hP.prod hL) H J hH hJ B
+  obtain ⟨C2,hC2,hcut⟩ := cutoff_frame_bounds hL χ hχ B
+  obtain ⟨Tb,hTb⟩ := hL.exists_bound_of_continuousOn (continuousOn_trace_re hG)
+  let C := max C1 C2
+  let T := K*max Tb 0
+  let a := (6*C+1)/δ
+  have hC : 0 ≤ C := hC1.trans (le_max_left _ _)
+  have hT : 0 ≤ T := mul_nonneg hK (le_max_right _ _)
+  have ha : 0 ≤ a := div_nonneg (by positivity) hδ.le
+  refine ⟨C*(1+a)+a*T,by positivity,?_⟩
+  intro M t ht hdom hell hRic hJet z hz hχz
+  have htr (y) (hy : y ∈ L) : (M.matrix y).trace.re ≤ T := by
+    exact (metric_trace_le (hell y hy).2).trans (mul_le_mul_of_nonneg_left
+      ((le_abs_self _).trans ((hTb y hy).trans (le_max_left _ _))) hK)
+  let F : V n → ℝ := fun y => (χ y)^2*M.calabiNorm y+a*(M.matrix y).trace.re
+  have hFc : ContinuousOn F L :=
+    ((hχ.continuous.continuousOn.pow 2).mul (M.calabiNorm_smooth.continuousOn.mono hdom)).add
+      (continuousOn_const.mul (continuousOn_trace_re (M.smooth.continuousOn.mono hdom)))
+  obtain ⟨y,hy,hmax⟩ := hL.exists_isMaxOn ⟨z,hz⟩ hFc
+  have hFz : M.calabiNorm z ≤ F z := by
+    dsimp only [F]
+    rw [hχz,one_pow,one_mul]
+    have ht0 := (Complex.nonneg_iff.mp (M.positive z (hdom hz)).posSemidef.trace_nonneg).1
+    exact le_add_of_nonneg_right (mul_nonneg ha ht0)
+  have hFzy := hmax hz
+  have hFbound : F y ≤ C*(1+a)+a*T := by
+    by_cases hzero : χ y = 0
+    · dsimp only [F]
+      rw [hzero,zero_pow (by decide),zero_mul,zero_add]
+      exact (mul_le_mul_of_nonneg_left (htr y hy) ha).trans (le_add_of_nonneg_left (mul_nonneg hC (by positivity)))
+    have hlocal : IsLocalMax F y := by
+      filter_upwards [mem_interior_iff_mem_nhds.mp (hχint y hy hzero)] with v hv
+      exact hmax hv
+    have hcut' (b : V n →L[ℂ] V n) w (hw : b w ∈ L) (hb : ‖frameMatrix b‖ ≤ B) := hcut b w hw hb
+    have hs : M.calabiNorm y ≤ C*(1+a) := by
+      apply calabi_cutoff_max_uniform G H J χ hχ hC hδ hframe
+        (fun b w hw hb => ⟨(neg_le_neg (le_max_right C1 C2)).trans (hcut' b w hw hb).1,
+          (hcut' b w hw hb).2.trans (mul_le_mul_of_nonneg_right (le_max_right C1 C2) (sq_nonneg _))⟩)
+        (fun x hx p hp U => ?_) M ht hy (hdom hy) (hell y hy) (hRic y hy) (hJet y hy)
+        (hχ0 y hy) (hχ1 y hy) hlocal
+      obtain ⟨h1,h2,h3⟩ := hric x hx p hp U
+      refine ⟨?_,(neg_le_neg (le_max_left C1 C2)).trans h2,h3⟩
+      have he : -C*(1+tensorSquare U) ≤ -C1*(1+tensorSquare U) := by
+        simpa only [neg_mul] using (neg_le_neg (mul_le_mul_of_nonneg_right
+          (le_max_left C1 C2) (add_nonneg zero_le_one (tensorSquare_nonneg U))))
+      exact he.trans h1
+    have hs0 := M.calabiNorm_nonneg (hdom hy)
+    have hc : (χ y)^2 ≤ 1 := by nlinarith [hχ0 y hy,hχ1 y hy]
+    have hh := (mul_le_mul_of_nonneg_right hc hs0).trans (by simpa using hs)
+    exact add_le_add hh (mul_le_mul_of_nonneg_left (htr y hy) ha)
+  exact hFz.trans (hFzy.trans hFbound)
+end KaehlerCalculus
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/CalabiNormAlgebra.lean b/lean/OAI/Geometry/PrescribedPotential/CalabiNormAlgebra.lean
new file mode 100644
index 0000000000000000000000000000000000000000..79223259b70af0159338342aeedfd4bbe7c8eb29
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/CalabiNormAlgebra.lean
@@ -0,0 +1,91 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CalabiTensorAlgebra
+
+namespace OAI
+
+/-! Calabi Norm Algebra. -/
+
+section
+
+ 
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace KaehlerCalculus
+variable {n : ℕ}
+
+def tensorVector (T : ConnectionTensor n) : EuclideanSpace ℂ (Fin n × Fin n × Fin n) :=
+  WithLp.toLp 2 (fun p => T p.1 p.2.1 p.2.2)
+
+def tensorSquare (T : ConnectionTensor n) : ℝ := (tensorPairAt 1 1 T T).re
+
+lemma tensorPairAt_one_inner (T U : ConnectionTensor n) :
+    tensorPairAt 1 1 T U = inner ℂ (tensorVector U) (tensorVector T) := by
+  rw [tensorPairAt_one]
+  simp only [tensorVector,PiLp.inner_apply,Fintype.sum_prod_type,
+    Matrix.trace,Matrix.diag_apply,Matrix.mul_apply,Matrix.conjTranspose_apply,
+    RCLike.inner_apply]
+  rfl
+
+lemma tensorSquare_norm (T : ConnectionTensor n) : tensorSquare T = ‖tensorVector T‖^2 := by
+  rw [tensorSquare,tensorPairAt_one_inner]
+  exact (norm_sq_eq_re_inner (𝕜 := ℂ) (tensorVector T)).symm
+
+lemma tensorSquare_nonneg (T : ConnectionTensor n) : 0 ≤ tensorSquare T := by
+  rw [tensorSquare_norm]
+  positivity
+
+lemma tensorSquare_pos {T : ConnectionTensor n} (hT : T ≠ 0) : 0 < tensorSquare T := by
+  rw [tensorSquare_norm]
+  apply sq_pos_of_pos
+  apply norm_pos_iff.mpr
+  intro he
+  apply hT
+  funext i
+  ext a b
+  exact congrArg (fun v : EuclideanSpace ℂ (Fin n × Fin n × Fin n) => v (i,a,b)) he
+
+lemma tensorPairAt_cauchy (T U : ConnectionTensor n) :
+    ‖tensorPairAt 1 1 T U‖^2 ≤ tensorSquare T*tensorSquare U := by
+  rw [tensorPairAt_one_inner,tensorSquare_norm,tensorSquare_norm]
+  have hh := norm_inner_le_norm (𝕜 := ℂ) (tensorVector U) (tensorVector T)
+  have hs := sq_le_sq₀ (norm_nonneg _) (mul_nonneg (norm_nonneg _) (norm_nonneg _)) |>.mpr hh
+  simpa only [mul_pow,mul_comm] using hs
+
+lemma tensorSquare_smul (r : ℝ) (T : ConnectionTensor n) :
+    tensorSquare (r • T) = r^2*tensorSquare T := by
+  have he : tensorVector (r • T) = r • tensorVector T := rfl
+  rw [tensorSquare_norm,he,norm_smul,mul_pow,Real.norm_eq_abs,sq_abs,tensorSquare_norm]
+
+lemma continuous_tensorSquare : Continuous (tensorSquare (n := n)) := by
+  rw [show tensorSquare (n := n) = fun T => ‖tensorVector T‖^2 from funext tensorSquare_norm]
+  apply Continuous.pow
+  apply Continuous.norm
+  exact (PiLp.continuous_toLp 2 _).comp (continuous_pi (fun p =>
+    (continuous_apply p.2.2).comp ((continuous_apply p.2.1).comp (continuous_apply p.1))))
+
+lemma continuous_tensorVector : Continuous (tensorVector (n := n)) := by
+  exact (PiLp.continuous_toLp 2 _).comp (continuous_pi (fun p =>
+    (continuous_apply p.2.2).comp ((continuous_apply p.2.1).comp (continuous_apply p.1))))
+
+lemma tensorPairAt_one_continuous : Continuous
+    (fun q : ConnectionTensor n × ConnectionTensor n => tensorPairAt 1 1 q.1 q.2) := by
+  simp_rw [tensorPairAt_one_inner]
+  exact Continuous.inner (𝕜 := ℂ) (continuous_tensorVector.comp continuous_snd)
+    (continuous_tensorVector.comp continuous_fst)
+
+lemma tensorPairAt_one_smul (r s : ℝ) (T U : ConnectionTensor n) :
+    tensorPairAt 1 1 (r • T) (s • U) = ((r*s : ℝ) : ℂ)*tensorPairAt 1 1 T U := by
+  simp only [tensorPairAt_one,Pi.smul_apply]
+  change (∑ i, (((r:ℂ) • T i)*((s:ℂ) • U i)ᴴ).trace) = _
+  simp only [Matrix.conjTranspose_smul,Complex.star_def,Complex.conj_ofReal,
+    smul_mul_assoc,mul_smul_comm,smul_smul,Matrix.trace_smul,smul_eq_mul,
+    ← Complex.ofReal_mul,← Finset.mul_sum]
+  rw [mul_comm s r]
+
+end KaehlerCalculus
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/CalabiPairCalculus.lean b/lean/OAI/Geometry/PrescribedPotential/CalabiPairCalculus.lean
new file mode 100644
index 0000000000000000000000000000000000000000..2afe6f50d3e5872ca75811b9450d34389fcd559d
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/CalabiPairCalculus.lean
@@ -0,0 +1,128 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CalabiConnection
+import OAI.Geometry.PrescribedPotential.CalabiCovariant
+import OAI.Geometry.PrescribedPotential.CalabiTensorAlgebra
+import OAI.Geometry.PrescribedPotential.MatrixTensorCalculus
+
+namespace OAI
+
+/-! Calabi Pair Calculus. -/
+
+section
+
+ 
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace KaehlerCalculus
+variable {n : ℕ}
+
+lemma calabi_pair_matrix_trace_smooth {M : V n → Matrix (Fin n) (Fin n) ℂ} {z : V n}
+    (hM : ContDiffAt ℝ ∞ M z) : ContDiffAt ℝ ∞ (fun y => (M y).trace) z :=
+  ContDiffAt.sum (fun i _ => entry_smooth hM i i)
+
+lemma tensorPair_smooth {P M : V n → Matrix (Fin n) (Fin n) ℂ}
+    {T U : Fin n → V n → Matrix (Fin n) (Fin n) ℂ} {z : V n}
+    (hP : ContDiffAt ℝ ∞ P z) (hM : ContDiffAt ℝ ∞ M z)
+    (hT : ∀ i, ContDiffAt ℝ ∞ (T i) z) (hU : ∀ i, ContDiffAt ℝ ∞ (U i) z) :
+    ContDiffAt ℝ ∞ (fun y => tensorPairAt (P y) (M y) (fun i => T i y) (fun i => U i y)) z := by
+  exact ContDiffAt.sum (fun i _ => ContDiffAt.sum (fun j _ =>
+    (entry_smooth hP i j).mul (calabi_pair_matrix_trace_smooth (matrix_smooth_mul
+      (matrix_smooth_mul (matrix_smooth_mul (hT i) hP) (matrix_smooth_conjTranspose (hU j))) hM))))
+
+namespace LocalKaehlerField
+variable (K : LocalKaehlerField n)
+
+def tensorPair (T U : Fin n → V n → Matrix (Fin n) (Fin n) ℂ) (z : V n) : ℂ :=
+  tensorPairAt (K.matrix z)⁻¹ (K.matrix z) (fun i => T i z) (fun i => U i z)
+
+lemma tensorPair_smoothOn {T U : Fin n → V n → Matrix (Fin n) (Fin n) ℂ}
+    (hT : ∀ i, ContDiffOn ℝ ∞ (T i) K.domain) (hU : ∀ i, ContDiffOn ℝ ∞ (U i) K.domain) :
+    ContDiffOn ℝ ∞ (K.tensorPair T U) K.domain := by
+  apply K.isOpen.contDiffOn_iff.mpr
+  intro z hz
+  exact tensorPair_smooth (K.inverse_smooth.contDiffAt (K.isOpen.mem_nhds hz))
+    (K.smooth.contDiffAt (K.isOpen.mem_nhds hz))
+    (fun i => (hT i).contDiffAt (K.isOpen.mem_nhds hz))
+    (fun i => (hU i).contDiffAt (K.isOpen.mem_nhds hz))
+
+lemma matrix_hol {z : V n} (hz : z ∈ K.domain) (k : Fin n) :
+    mderiv (-Complex.I) (e k) K.matrix z = K.matrix z*K.connection k z := by
+  rw [connection,← mul_assoc,Matrix.mul_nonsing_inv _
+    (isUnit_iff_ne_zero.mpr (ne_of_gt (K.positive z hz).det_pos)),one_mul]
+
+lemma inverse_hol {z : V n} (hz : z ∈ K.domain) (k : Fin n) :
+    mderiv (-Complex.I) (e k) (fun y => (K.matrix y)⁻¹) z =
+      -K.connection k z*(K.matrix z)⁻¹ := by
+  rw [mderiv_inverse K.isOpen K.smooth (fun y hy => ne_of_gt (K.positive y hy).det_pos) hz]
+  simp only [connection,neg_mul]
+
+lemma tensorPair_hol {T U : Fin n → V n → Matrix (Fin n) (Fin n) ℂ}
+    (hT : ∀ i, ContDiffOn ℝ ∞ (T i) K.domain) (hU : ∀ i, ContDiffOn ℝ ∞ (U i) K.domain)
+    {z : V n} (hz : z ∈ K.domain) (k : Fin n) :
+    dz (e k) (K.tensorPair T U) z =
+      K.tensorPair (K.covHol T k) U z +
+        K.tensorPair T (fun j => mderiv Complex.I (e k) (U j)) z := by
+  have hn := K.isOpen.mem_nhds hz
+  have hP := K.inverse_smooth.contDiffAt hn
+  have hM := K.smooth.contDiffAt hn
+  have ht (i) := (hT i).contDiffAt hn
+  have hu (i) := (hU i).contDiffAt hn
+  have hQ (i j) := matrix_smooth_mul (matrix_smooth_mul
+    (matrix_smooth_mul (ht i) hP) (matrix_smooth_conjTranspose (hu j))) hM
+  have he (i j) : wderiv (-Complex.I) (e k)
+      (fun y => ((K.matrix y)⁻¹ i j) * (T i y*(K.matrix y)⁻¹*(U j y)ᴴ*K.matrix y).trace) z =
+      (-(K.connection k z*(K.matrix z)⁻¹) i j *
+        (T i z*(K.matrix z)⁻¹*(U j z)ᴴ*K.matrix z).trace +
+        (K.matrix z)⁻¹ i j *
+          (mderiv (-Complex.I) (e k) (T i) z*(K.matrix z)⁻¹*(U j z)ᴴ*K.matrix z -
+           T i z*K.connection k z*(K.matrix z)⁻¹*(U j z)ᴴ*K.matrix z +
+           T i z*(K.matrix z)⁻¹*(mderiv Complex.I (e k) (U j) z)ᴴ*K.matrix z +
+           T i z*(K.matrix z)⁻¹*(U j z)ᴴ*K.matrix z*K.connection k z).trace) := by
+    rw [wderiv_mul ((entry_smooth hP i j).differentiableAt (by simp))
+      ((calabi_pair_matrix_trace_smooth (hQ i j)).differentiableAt (by simp)),mderiv_trace (hQ i j)]
+    change mderiv (-Complex.I) (e k) (fun y => (K.matrix y)⁻¹) z i j * _ + _ = _
+    rw [K.inverse_hol hz]
+    simp only [neg_mul,Matrix.neg_apply]
+    congr 1
+    congr 1
+    congr 1
+    rw [mderiv_mul (matrix_smooth_mul (matrix_smooth_mul (ht i) hP)
+      (matrix_smooth_conjTranspose (hu j))) hM,
+      mderiv_mul (matrix_smooth_mul (ht i) hP) (matrix_smooth_conjTranspose (hu j)),
+      mderiv_mul (ht i) hP,mderiv_conjTranspose (hu j),K.inverse_hol hz,K.matrix_hol hz]
+    noncomm_ring
+  change wderiv (-Complex.I) (e k) (fun y => ∑ i, ∑ j,
+    ((K.matrix y)⁻¹ i j)*(T i y*(K.matrix y)⁻¹*(U j y)ᴴ*K.matrix y).trace) z = _
+  rw [wderiv_sum Finset.univ (fun i _ => (ContDiffAt.sum (fun j _ =>
+    (entry_smooth hP i j).mul (calabi_pair_matrix_trace_smooth (hQ i j)))).differentiableAt (by simp))]
+  simp only [wderiv_sum Finset.univ (fun j _ => ((entry_smooth hP _ j).mul
+    (calabi_pair_matrix_trace_smooth (hQ _ j))).differentiableAt (by simp)),he]
+  exact covariant_pair_algebra (K.matrix z)⁻¹ (K.matrix z) (K.connection k z)
+    (fun i => T i z) (fun i => U i z) (fun i => mderiv (-Complex.I) (e k) (T i) z)
+    (fun i => mderiv Complex.I (e k) (U i) z)
+
+lemma tensorPair_bar {T U : Fin n → V n → Matrix (Fin n) (Fin n) ℂ}
+    (hT : ∀ i, ContDiffOn ℝ ∞ (T i) K.domain) (hU : ∀ i, ContDiffOn ℝ ∞ (U i) K.domain)
+    {z : V n} (hz : z ∈ K.domain) (k : Fin n) :
+    dzbar (e k) (K.tensorPair T U) z =
+      K.tensorPair (fun i => mderiv Complex.I (e k) (T i)) U z +
+        K.tensorPair T (K.covHol U k) z := by
+  have hc {T U : Fin n → V n → Matrix (Fin n) (Fin n) ℂ} {y : V n} (hy : y ∈ K.domain) :
+      star (K.tensorPair T U y) = K.tensorPair U T y :=
+    tensorPairAt_conj (K.positive y hy).isHermitian.inv (K.positive y hy).isHermitian _ _
+  have he : K.tensorPair T U =ᶠ[𝓝 z] (fun y => star (K.tensorPair U T y)) := by
+    filter_upwards [K.isOpen.mem_nhds hz] with y hy using (hc hy).symm
+  change wderiv Complex.I (e k) (K.tensorPair T U) z = _
+  rw [wderiv_congr he]
+  change dzbar (e k) (fun y => star (K.tensorPair U T y)) z = _
+  rw [dzbar_conj (((K.tensorPair_smoothOn hU hT).contDiffAt
+    (K.isOpen.mem_nhds hz)).differentiableAt (by simp)),K.tensorPair_hol hU hT hz]
+  rw [star_add,hc hz,hc hz,add_comm]
+end LocalKaehlerField
+end KaehlerCalculus
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/CalabiPointInequalities.lean b/lean/OAI/Geometry/PrescribedPotential/CalabiPointInequalities.lean
new file mode 100644
index 0000000000000000000000000000000000000000..8f136c4cf49f0d8bf485754827963f91a3c2585d
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/CalabiPointInequalities.lean
@@ -0,0 +1,79 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CalabiBochner
+import OAI.Geometry.PrescribedPotential.CalabiResidualBound
+
+namespace OAI
+
+/-! Calabi Point Inequalities. -/
+
+section
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace KaehlerCalculus.LocalKaehlerField
+variable {n : ℕ} (K : LocalKaehlerField n)
+
+def calabiGood (z : V n) : ℝ :=
+  (∑ k, tensorSquare (fun i => K.covHol K.connection k i z)) +
+    ∑ k, tensorSquare (fun i => K.curvature k i z)
+
+lemma calabiGood_nonneg (z : V n) : 0 ≤ K.calabiGood z :=
+  add_nonneg (Finset.sum_nonneg (fun _ _ => tensorSquare_nonneg _))
+    (Finset.sum_nonneg (fun _ _ => tensorSquare_nonneg _))
+
+lemma calabiNorm_at_one {z : V n} (hM : K.matrix z = 1) :
+    K.calabiNorm z = tensorSquare (fun i => K.connection i z) := by
+  simp only [calabiNorm,tensorPair,hM,inv_one,tensorSquare]
+
+lemma calabi_bochner_real_at_one {z : V n} (hz : z ∈ K.domain) (hM : K.matrix z = 1) :
+    (PotentialKaehler.potentialMatrix K.calabiNorm z).trace.re = K.calabiGood z +
+      calabiResidual (K.ricciHessian z) (fun i => mderiv (-Complex.I) (e i) K.ricciHessian z)
+        (fun i => K.connection i z) := by
+  rw [K.calabi_bochner_at_one hz hM]
+  have he : (fun i => mderiv (-Complex.I) (e i) K.ricciHessian z -
+      K.connection i z*K.ricciHessian z - ∑ q, K.ricciHessian z q i • K.connection q z) =
+      (fun i => mderiv (-Complex.I) (e i) K.ricciHessian z) -
+        ricciRight (K.ricciHessian z) (fun i => K.connection i z) := by
+    funext i
+    exact (sub_add_eq_sub_sub _ _ _).symm
+  rw [he]
+  simp only [Complex.add_re,Complex.re_sum,calabiGood,tensorSquare,calabiResidual,Complex.add_re]
+  exact add_assoc _ _ _
+
+lemma calabi_gradient_at_one {z : V n} (hz : z ∈ K.domain) (hM : K.matrix z = 1) (k : Fin n) :
+    ‖dz (e k) (fun y => (K.calabiNorm y : ℂ)) z‖^2 ≤ 2*K.calabiNorm z *
+      (tensorSquare (fun i => K.covHol K.connection k i z)+tensorSquare (fun i => K.curvature k i z)) := by
+  have he : (fun y => (K.calabiNorm y : ℂ)) =ᶠ[𝓝 z] K.tensorPair K.connection K.connection := by
+    filter_upwards [K.isOpen.mem_nhds hz] with y hy using K.calabiNorm_complex hy
+  have hd : dz (e k) (fun y => (K.calabiNorm y : ℂ)) z =
+      tensorPairAt 1 1 (fun i => K.covHol K.connection k i z) (fun i => K.connection i z) +
+      tensorPairAt 1 1 (fun i => K.connection i z) (fun i => K.curvature k i z) := by
+    change wderiv (-Complex.I) (e k) _ z = _
+    rw [wderiv_congr he]
+    change dz (e k) (K.tensorPair K.connection K.connection) z = _
+    rw [K.tensorPair_hol (T := K.connection) (U := K.connection) K.connection_smooth K.connection_smooth hz]
+    simp only [tensorPair,hM,inv_one]
+    rfl
+  rw [hd,K.calabiNorm_at_one hM]
+  have h1 := tensorPairAt_cauchy (fun i => K.covHol K.connection k i z) (fun i => K.connection i z)
+  have h2 := tensorPairAt_cauchy (fun i => K.connection i z) (fun i => K.curvature k i z)
+  have hh := norm_add_le
+    (tensorPairAt 1 1 (fun i => K.covHol K.connection k i z) (fun i => K.connection i z))
+    (tensorPairAt 1 1 (fun i => K.connection i z) (fun i => K.curvature k i z))
+  have hs := sq_le_sq₀ (norm_nonneg _) (add_nonneg (norm_nonneg _) (norm_nonneg _)) |>.mpr hh
+  nlinarith [sq_nonneg (‖tensorPairAt 1 1 (fun i => K.covHol K.connection k i z) (fun i => K.connection i z)‖-
+    ‖tensorPairAt 1 1 (fun i => K.connection i z) (fun i => K.curvature k i z)‖)]
+
+lemma calabi_gradient_sum_at_one {z : V n} (hz : z ∈ K.domain) (hM : K.matrix z = 1) :
+    (∑ k, ‖dz (e k) (fun y => (K.calabiNorm y : ℂ)) z‖^2) ≤ 2*K.calabiNorm z*K.calabiGood z := by
+  calc
+    _ ≤ ∑ k, 2*K.calabiNorm z*(tensorSquare (fun i => K.covHol K.connection k i z)+
+      tensorSquare (fun i => K.curvature k i z)) := Finset.sum_le_sum (fun k _ => K.calabi_gradient_at_one hz hM k)
+    _ = _ := by rw [← Finset.mul_sum,Finset.sum_add_distrib]; rfl
+end KaehlerCalculus.LocalKaehlerField
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/CalabiRescue.lean b/lean/OAI/Geometry/PrescribedPotential/CalabiRescue.lean
new file mode 100644
index 0000000000000000000000000000000000000000..f04047418b401019308cfd5e2ad3eee9896d6ef4
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/CalabiRescue.lean
@@ -0,0 +1,138 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CalabiConnection
+import OAI.Geometry.PrescribedPotential.CalabiNormAlgebra
+import OAI.Geometry.PrescribedPotential.FrameCompactBounds
+import OAI.Geometry.PrescribedPotential.LinearFrameCalculus
+
+namespace OAI
+
+/-! Calabi Rescue. -/
+
+section
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace KaehlerCalculus
+variable {n : ℕ}
+
+def rescueSquare (D : Matrix (Fin n) (Fin n) ℂ) (T : ConnectionTensor n) : ℝ :=
+  ∑ k, frameGram 1 (T k*D)
+
+lemma rescueSquare_pos {D : Matrix (Fin n) (Fin n) ℂ} (hD : IsUnit D)
+    {T : ConnectionTensor n} (hT : T ≠ 0) : 0 < rescueSquare D T := by
+  obtain ⟨k,hk⟩ : ∃ k, T k ≠ 0 := by
+    by_contra h
+    apply hT
+    funext k
+    exact not_not.mp (fun hk => h ⟨k,hk⟩)
+  have hkd : T k*D ≠ 0 := by
+    intro he
+    apply hk
+    exact hD.mul_right_cancel (by simpa using he)
+  unfold rescueSquare
+  apply Finset.sum_pos'
+  · intro i _
+    simpa only [frameGram,mul_one] using (Complex.nonneg_iff.mp (Matrix.posSemidef_conjTranspose_mul_self (T i*D)).trace_nonneg).1
+  · exact ⟨k,Finset.mem_univ _,frameGram_pos Matrix.PosDef.one hkd⟩
+
+lemma rescueSquare_smul (D : Matrix (Fin n) (Fin n) ℂ) (T : ConnectionTensor n) (r : ℝ) :
+    rescueSquare D (r • T) = r^2*rescueSquare D T := by
+  simp only [rescueSquare,Pi.smul_apply,Matrix.smul_mul]
+  change (∑ k, frameGram 1 (r • (T k*D))) = _
+  simp_rw [frameGram_smul]
+  rw [Finset.mul_sum]
+
+lemma rescueSquare_continuous : Continuous (fun p : Matrix (Fin n) (Fin n) ℂ × ConnectionTensor n =>
+    rescueSquare p.1 p.2) := by
+  unfold rescueSquare
+  apply continuous_finsetSum
+  intro k _
+  have hc : Continuous (fun p : Matrix (Fin n) (Fin n) ℂ × ConnectionTensor n => p.2 k*p.1) :=
+    ((continuous_apply k).comp continuous_snd).mul continuous_fst
+  apply continuousOn_univ.mp
+  exact continuousOn_trace_re (((continuousOn_matrix_star hc.continuousOn).mul
+    (continuous_const (y := (1 : Matrix (Fin n) (Fin n) ℂ))).continuousOn).mul hc.continuousOn)
+
+lemma rescueSquare_compact_lower {X : Type*} [TopologicalSpace X] {L : Set X} (hL : IsCompact L)
+    (D : X → Matrix (Fin n) (Fin n) ℂ) (hD : ContinuousOn D L) (hu : ∀ x ∈ L, IsUnit (D x)) :
+    ∃ δ : ℝ, 0 < δ ∧ ∀ x ∈ L, ∀ T, δ*tensorSquare T ≤ rescueSquare (D x) T := by
+  obtain ⟨a,ha,hbound⟩ := compact_homogeneous_positive hL (fun x T => rescueSquare (D x) T)
+    ((rescueSquare_continuous.comp_continuousOn (f := fun p : X × ConnectionTensor n => (D p.1,p.2))
+      ((hD.comp continuous_fst.continuousOn (fun _ h => h.1)).prodMk continuous_snd.continuousOn)))
+    (fun x hx T hn => rescueSquare_pos (hu x hx) hn) (fun x _ r T => rescueSquare_smul (D x) T r)
+  obtain ⟨B,hB,hSq⟩ := compact_homogeneous_bound (K := (univ : Set Unit)) isCompact_univ 2 (by decide)
+    (fun (_ : Unit) (T : ConnectionTensor n) => tensorSquare T)
+    (continuous_tensorSquare.comp continuous_snd).continuousOn (fun _ _ r T => tensorSquare_smul r T)
+  have hBp : 0 < B+1 := by linarith
+  refine ⟨a/(B+1),div_pos ha hBp,?_⟩
+  intro x hx T
+  have hS : tensorSquare T ≤ (B+1)*‖T‖^2 := by
+    have hh := (le_abs_self (tensorSquare T)).trans (hSq () (mem_univ _) T)
+    nlinarith [sq_nonneg ‖T‖]
+  have hh := mul_le_mul_of_nonneg_left hS (div_nonneg ha.le hBp.le)
+  have he : a/(B+1)*((B+1)*‖T‖^2) = a*‖T‖^2 := by field_simp
+  exact (he ▸ hh).trans (hbound x hx T)
+
+namespace LocalKaehlerField
+variable (K : LocalKaehlerField n)
+
+def weightedTrace (D : Matrix (Fin n) (Fin n) ℂ) (z : V n) : ℝ := (Dᴴ*K.matrix z*D).trace.re
+
+lemma weightedTrace_smooth (D : Matrix (Fin n) (Fin n) ℂ) :
+    ContDiffOn ℝ ∞ (K.weightedTrace D) K.domain := by
+  apply K.isOpen.contDiffOn_iff.mpr
+  intro z hz
+  have hs := matrix_smooth_mul (matrix_smooth_mul (contDiffAt_const (c := Dᴴ))
+    (K.smooth.contDiffAt (K.isOpen.mem_nhds hz))) (contDiffAt_const (c := D))
+  exact Complex.reCLM.contDiff.contDiffAt.comp z (ContDiffAt.sum
+    (fun i (_ : i ∈ (Finset.univ : Finset (Fin n))) => entry_smooth hs i i))
+
+lemma weightedTrace_complex (D : Matrix (Fin n) (Fin n) ℂ) {z : V n} (hz : z ∈ K.domain) :
+    (K.weightedTrace D z : ℂ) = (Dᴴ*K.matrix z*D).trace := by
+  apply Complex.ext
+  · rfl
+  · exact (Complex.nonneg_iff.mp ((K.positive z hz).posSemidef.conjTranspose_mul_mul_same D).trace_nonneg).2
+
+lemma weightedTrace_laplacian_at_one (D : Matrix (Fin n) (Fin n) ℂ)
+    {z : V n} (hz : z ∈ K.domain) (hM : K.matrix z = 1) :
+    (PotentialKaehler.potentialMatrix (K.weightedTrace D) z).trace.re =
+      (Dᴴ*K.ricciHessian z*D).trace.re + rescueSquare D (fun k => K.connection k z) := by
+  have hn := K.isOpen.mem_nhds hz
+  have hs := K.smooth.contDiffAt hn
+  have ht := K.weightedTrace_smooth D |>.contDiffAt hn
+  have he : (fun y => (K.weightedTrace D y:ℂ)) =ᶠ[𝓝 z] (fun y => (Dᴴ*K.matrix y*D).trace) := by
+    filter_upwards [hn] with y hy using K.weightedTrace_complex D hy
+  have hd (k : Fin n) : dz (e k) (fun y => (K.weightedTrace D y:ℂ)) =ᶠ[𝓝 z]
+      (fun y => (Dᴴ*mderiv (-Complex.I) (e k) K.matrix y*D).trace) := by
+    filter_upwards [hn] with y hy
+    have hey : (fun y => (K.weightedTrace D y:ℂ)) =ᶠ[𝓝 y] (fun y => (Dᴴ*K.matrix y*D).trace) := by
+      filter_upwards [K.isOpen.mem_nhds hy] with x hx using K.weightedTrace_complex D hx
+    change wderiv (-Complex.I) _ _ _ = _
+    rw [wderiv_congr hey,mderiv_trace (matrix_smooth_mul (matrix_smooth_mul contDiffAt_const
+      (K.smooth.contDiffAt (K.isOpen.mem_nhds hy))) contDiffAt_const),
+      mderiv_congruence (K.smooth.contDiffAt (K.isOpen.mem_nhds hy))]
+  have hdd (k : Fin n) : (PotentialKaehler.potentialMatrix (K.weightedTrace D) z) k k =
+      (Dᴴ*mderiv Complex.I (e k) (mderiv (-Complex.I) (e k) K.matrix) z*D).trace := by
+    rw [potentialMatrix_eq_dzbar_dz ht]
+    change wderiv Complex.I (e k) (dz (e k) (fun y => (K.weightedTrace D y:ℂ))) z = _
+    rw [wderiv_congr (hd k),mderiv_trace (matrix_smooth_mul (matrix_smooth_mul contDiffAt_const
+      (mderiv_smooth hs _ _)) contDiffAt_const),mderiv_congruence (mderiv_smooth hs _ _)]
+  have hc := ricci_cancellation_at_one K.isOpen K.smooth K.positive K.closed hz hM
+  change (∑ k, (PotentialKaehler.potentialMatrix (K.weightedTrace D) z) k k).re = _
+  simp_rw [hdd]
+  rw [← Matrix.trace_sum,← Matrix.sum_mul,← Matrix.mul_sum,hc,mul_add,add_mul,
+    Matrix.trace_add,Complex.add_re]
+  congr 1
+  simp only [Matrix.mul_sum,Matrix.sum_mul,Matrix.trace_sum,Complex.re_sum,rescueSquare,frameGram,
+    connection,hM,inv_one,one_mul,Matrix.conjTranspose_mul,mul_assoc]
+  apply Finset.sum_congr rfl
+  intro k _
+  rw [mderiv_bar_eq_conjTranspose K.isOpen K.smooth (fun y hy => (K.positive y hy).isHermitian) hz]
+end LocalKaehlerField
+end KaehlerCalculus
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/CalabiResidualBound.lean b/lean/OAI/Geometry/PrescribedPotential/CalabiResidualBound.lean
new file mode 100644
index 0000000000000000000000000000000000000000..ee8e64cbae76c3d166e1ac11c6c112a02b2efe0d
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/CalabiResidualBound.lean
@@ -0,0 +1,144 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CalabiNormAlgebra
+import OAI.Geometry.PrescribedPotential.CompactHomogeneousBounds
+
+namespace OAI
+
+/-! Calabi Residual Bound. -/
+
+section
+
+ 
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace KaehlerCalculus
+variable {n : ℕ}
+
+def ricciRight (H : Matrix (Fin n) (Fin n) ℂ) (T : ConnectionTensor n) : ConnectionTensor n :=
+  fun i => T i*H + ∑ q, H q i • T q
+
+def ricciLeft (H : Matrix (Fin n) (Fin n) ℂ) (T : ConnectionTensor n) : ConnectionTensor n :=
+  fun i => H*T i
+
+def calabiResidual (H : Matrix (Fin n) (Fin n) ℂ) (D T : ConnectionTensor n) : ℝ :=
+  (tensorPairAt 1 1 (D-ricciRight H T) T + tensorPairAt 1 1 T (D-ricciLeft H T)).re
+
+lemma continuous_ricciRight : Continuous
+    (fun q : Matrix (Fin n) (Fin n) ℂ × ConnectionTensor n => ricciRight q.1 q.2) := by
+  apply continuous_pi
+  intro i
+  apply Continuous.add
+  · exact ((continuous_apply i).comp continuous_snd).mul continuous_fst
+  · apply continuous_finsetSum
+    intro q _
+    exact ((continuous_apply_apply q i).comp continuous_fst).smul
+      ((continuous_apply q).comp continuous_snd)
+
+lemma continuous_ricciLeft : Continuous
+    (fun q : Matrix (Fin n) (Fin n) ℂ × ConnectionTensor n => ricciLeft q.1 q.2) := by
+  exact continuous_pi (fun i => continuous_fst.mul ((continuous_apply i).comp continuous_snd))
+
+lemma ricciRight_smul (H : Matrix (Fin n) (Fin n) ℂ) (r : ℝ) (T : ConnectionTensor n) :
+    ricciRight H (r • T) = r • ricciRight H T := by
+  funext i
+  simp only [ricciRight,Pi.smul_apply,smul_mul_assoc,smul_add,Finset.smul_sum]
+  congr 1
+  apply Finset.sum_congr rfl
+  intro q _
+  exact smul_comm _ _ _
+
+lemma ricciLeft_smul (H : Matrix (Fin n) (Fin n) ℂ) (r : ℝ) (T : ConnectionTensor n) :
+    ricciLeft H (r • T) = r • ricciLeft H T := by
+  funext i
+  change H*(r • T i) = r • (H*T i)
+  exact mul_smul_comm _ _ _
+
+lemma calabiResidual_homogeneous (H : Matrix (Fin n) (Fin n) ℂ)
+    (D T : ConnectionTensor n) (r : ℝ) :
+    calabiResidual H (r • D) (r • T) = r^2*calabiResidual H D T := by
+  unfold calabiResidual
+  rw [ricciRight_smul,ricciLeft_smul,← smul_sub,← smul_sub,tensorPairAt_one_smul,tensorPairAt_one_smul]
+  simp only [← mul_add,Complex.mul_re,Complex.ofReal_re,Complex.ofReal_im,zero_mul,sub_zero]
+  ring
+
+lemma continuous_calabiResidual : Continuous
+    (fun p : (Matrix (Fin n) (Fin n) ℂ × ConnectionTensor n) × ConnectionTensor n =>
+      calabiResidual p.1.1 p.1.2 p.2) := by
+  have hH := continuous_fst.comp (continuous_fst (X := Matrix (Fin n) (Fin n) ℂ × ConnectionTensor n)
+    (Y := ConnectionTensor n))
+  have hD := continuous_snd.comp (continuous_fst (X := Matrix (Fin n) (Fin n) ℂ × ConnectionTensor n)
+    (Y := ConnectionTensor n))
+  have hT := continuous_snd (X := Matrix (Fin n) (Fin n) ℂ × ConnectionTensor n) (Y := ConnectionTensor n)
+  have hR : Continuous (fun p : (Matrix (Fin n) (Fin n) ℂ × ConnectionTensor n) × ConnectionTensor n =>
+      ricciRight p.1.1 p.2) := continuous_ricciRight.comp (f := fun p : (Matrix (Fin n) (Fin n) ℂ × ConnectionTensor n) × ConnectionTensor n => (p.1.1,p.2)) (hH.prodMk hT)
+  have hL : Continuous (fun p : (Matrix (Fin n) (Fin n) ℂ × ConnectionTensor n) × ConnectionTensor n =>
+      ricciLeft p.1.1 p.2) := continuous_ricciLeft.comp (f := fun p : (Matrix (Fin n) (Fin n) ℂ × ConnectionTensor n) × ConnectionTensor n => (p.1.1,p.2)) (hH.prodMk hT)
+  have hA : Continuous (fun p : (Matrix (Fin n) (Fin n) ℂ × ConnectionTensor n) × ConnectionTensor n =>
+      tensorPairAt 1 1 (p.1.2-ricciRight p.1.1 p.2) p.2) :=
+    tensorPairAt_one_continuous.comp (f := fun p : (Matrix (Fin n) (Fin n) ℂ × ConnectionTensor n) × ConnectionTensor n => (p.1.2-ricciRight p.1.1 p.2,p.2))
+      ((hD.sub hR).prodMk hT)
+  have hB : Continuous (fun p : (Matrix (Fin n) (Fin n) ℂ × ConnectionTensor n) × ConnectionTensor n =>
+      tensorPairAt 1 1 p.2 (p.1.2-ricciLeft p.1.1 p.2)) :=
+    tensorPairAt_one_continuous.comp (f := fun p : (Matrix (Fin n) (Fin n) ℂ × ConnectionTensor n) × ConnectionTensor n => (p.2,p.1.2-ricciLeft p.1.1 p.2))
+      (hT.prodMk (hD.sub hL))
+  exact Complex.continuous_re.comp (f := fun p : (Matrix (Fin n) (Fin n) ℂ × ConnectionTensor n) × ConnectionTensor n =>
+    tensorPairAt 1 1 (p.1.2-ricciRight p.1.1 p.2) p.2 +
+      tensorPairAt 1 1 p.2 (p.1.2-ricciLeft p.1.1 p.2)) (hA.add hB)
+
+lemma tensorSquare_controls_norm : ∃ δ : ℝ, 0 < δ ∧ ∀ T : ConnectionTensor n, δ*‖T‖^2 ≤ tensorSquare T := by
+  obtain ⟨δ,hδ,h⟩ := compact_homogeneous_positive (K := (univ : Set Unit)) isCompact_univ
+    (fun (_ : Unit) (T : ConnectionTensor n) => tensorSquare T)
+    (continuous_tensorSquare.comp continuous_snd).continuousOn
+    (fun _ _ _ hT => tensorSquare_pos hT) (fun _ _ r T => tensorSquare_smul r T)
+  exact ⟨δ,hδ,fun T => h () (mem_univ _) T⟩
+
+lemma calabiResidual_compact_bound {X : Type*} [TopologicalSpace X] {L : Set X} (hL : IsCompact L)
+    (H : X → Matrix (Fin n) (Fin n) ℂ) (D : X → ConnectionTensor n)
+    (hH : ContinuousOn H L) (hD : ContinuousOn D L) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ x ∈ L, ∀ T : ConnectionTensor n,
+      |calabiResidual (H x) (D x) T| ≤ C*(1+tensorSquare T) := by
+  let E := ℝ × ConnectionTensor n
+  let F : X → E → ℝ := fun x v => calabiResidual (H x) (v.1 • D x) v.2
+  have hc : ContinuousOn (fun q : X × E => F q.1 q.2) (L ×ˢ Metric.closedBall 0 1) := by
+    have hx := continuous_fst.continuousOn (s := L ×ˢ Metric.closedBall (0:E) 1)
+    have h1 := (continuous_fst.comp continuous_snd).continuousOn
+      (s := L ×ˢ Metric.closedBall (0:E) 1)
+    have h2 := (continuous_snd.comp continuous_snd).continuousOn
+      (s := L ×ˢ Metric.closedBall (0:E) 1)
+    exact continuous_calabiResidual.comp_continuousOn
+      (f := fun q : X × E => ((H q.1,q.2.1 • D q.1),q.2.2))
+      (((hH.comp hx (fun _ h => h.1)).prodMk (h1.smul (hD.comp hx (fun _ h => h.1)))).prodMk h2)
+  have hh (x : X) (_ : x ∈ L) (r : ℝ) (v : E) : F x (r • v) = r^2*F x v := by
+    change calabiResidual (H x) ((r*v.1) • D x) (r • v.2) = _
+    rw [mul_smul,calabiResidual_homogeneous]
+  obtain ⟨B,hB,hbound⟩ := compact_homogeneous_bound hL 2 (by decide) F hc hh
+  obtain ⟨δ,hδ,hnorm⟩ := tensorSquare_controls_norm (n := n)
+  refine ⟨B*(1+δ⁻¹),mul_nonneg hB (by positivity),?_⟩
+  intro x hx T
+  have hb := hbound x hx (1,T)
+  change |calabiResidual (H x) ((1:ℝ) • D x) T| ≤ B*‖((1:ℝ),T)‖^2 at hb
+  rw [one_smul] at hb
+  have hp : ‖((1:ℝ),T)‖^2 ≤ 1+‖T‖^2 := by
+    rw [Prod.norm_def,Real.norm_eq_abs,abs_one]
+    rcases le_total 1 ‖T‖ with ht | ht
+    · rw [max_eq_right ht]
+      linarith
+    · rw [max_eq_left ht]
+      nlinarith [sq_nonneg ‖T‖]
+  have ht : ‖T‖^2 ≤ δ⁻¹*tensorSquare T := by
+    have hh := mul_le_mul_of_nonneg_left (hnorm T) (inv_nonneg.mpr hδ.le)
+    simpa only [← mul_assoc,inv_mul_cancel₀ hδ.ne',one_mul] using hh
+  have hs := tensorSquare_nonneg T
+  have hh : B*(1+‖T‖^2) ≤ B*(1+δ⁻¹)*(1+tensorSquare T) := by
+    have hh : 1+‖T‖^2 ≤ (1+δ⁻¹)*(1+tensorSquare T) := by
+      have hd : 0 ≤ δ⁻¹ := inv_nonneg.mpr hδ.le
+      nlinarith
+    simpa only [mul_assoc] using mul_le_mul_of_nonneg_left hh hB
+  exact hb.trans ((mul_le_mul_of_nonneg_left hp hB).trans hh)
+end KaehlerCalculus
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/CalabiRicciContraction.lean b/lean/OAI/Geometry/PrescribedPotential/CalabiRicciContraction.lean
new file mode 100644
index 0000000000000000000000000000000000000000..3f1faa64b424e5942abb6f58505d2f76caa59ccc
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/CalabiRicciContraction.lean
@@ -0,0 +1,94 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CalabiConnection
+
+namespace OAI
+
+/-! Calabi Ricci Contraction. -/
+
+section
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace KaehlerCalculus.LocalKaehlerField
+variable {n : ℕ} (K : LocalKaehlerField n)
+
+lemma lower_curvature_formula {z : V n} (hz : z ∈ K.domain) (i j : Fin n) :
+    K.matrix z * K.curvature i j z =
+      mderiv Complex.I (e i) (mderiv (-Complex.I) (e j) K.matrix) z -
+        mderiv Complex.I (e i) K.matrix z * (K.matrix z)⁻¹ *
+          mderiv (-Complex.I) (e j) K.matrix z := by
+  rw [K.curvature_formula hz,← mul_assoc,Matrix.mul_nonsing_inv _
+    (isUnit_iff_ne_zero.mpr (ne_of_gt (K.positive z hz).det_pos)),one_mul]
+
+lemma lower_curvature_symmetry {z : V n} (hz : z ∈ K.domain) (i j a b : Fin n) :
+    (K.matrix z * K.curvature i j z) a b = (K.matrix z * K.curvature a b z) i j := by
+  rw [K.lower_curvature_formula hz,K.lower_curvature_formula hz]
+  simp only [Matrix.sub_apply]
+  congr 1
+  · exact closed_mixed_symmetry K.isOpen K.smooth
+      (fun y hy => (K.positive y hy).isHermitian) K.closed hz a b i j
+  · simp only [Matrix.mul_apply,Finset.sum_mul]
+    apply Finset.sum_congr rfl
+    intro k _
+    apply Finset.sum_congr rfl
+    intro l _
+    change (dzbar (e i) (fun y => K.matrix y a l) z * (K.matrix z)⁻¹ l k) *
+      dz (e j) (fun y => K.matrix y k b) z =
+        (dzbar (e a) (fun y => K.matrix y i l) z * (K.matrix z)⁻¹ l k) *
+          dz (e b) (fun y => K.matrix y k j) z
+    rw [closed_dzbar_symmetry K.isOpen K.smooth
+      (fun y hy => (K.positive y hy).isHermitian) K.closed hz a l i,
+      closed_dz_symmetry K.isOpen K.smooth
+      (fun y hy => (K.positive y hy).isHermitian) K.closed hz k b j]
+
+lemma ricciHessian_contraction {z : V n} (hz : z ∈ K.domain) :
+    K.ricciHessian z = ∑ p, ∑ q, ((K.matrix z)⁻¹ p q) •
+      (K.matrix z * K.curvature q p z) := by
+  have hi : (K.matrix z)⁻¹*K.matrix z = 1 := Matrix.nonsing_inv_mul _
+    (isUnit_iff_ne_zero.mpr (ne_of_gt (K.positive z hz).det_pos))
+  ext a b
+  rw [K.ricciHessian_trace hz]
+  have he : K.curvature a b z = (K.matrix z)⁻¹*(K.matrix z*K.curvature a b z) := by
+    rw [← mul_assoc,hi,one_mul]
+  conv_lhs => rw [he]
+  simp only [Matrix.trace,Matrix.diag_apply,Matrix.mul_apply,Matrix.sum_apply,
+    Matrix.smul_apply,smul_eq_mul]
+  apply Finset.sum_congr rfl
+  intro p _
+  apply Finset.sum_congr rfl
+  intro q _
+  congr 1
+  exact K.lower_curvature_symmetry hz a b q p
+
+lemma curvature_contraction {z : V n} (hz : z ∈ K.domain) :
+    (∑ p, ∑ q, ((K.matrix z)⁻¹ p q) • K.curvature q p z) =
+      (K.matrix z)⁻¹ * K.ricciHessian z := by
+  rw [K.ricciHessian_contraction hz]
+  simp only [Matrix.mul_sum,Matrix.mul_smul,← Matrix.mul_assoc,
+    Matrix.nonsing_inv_mul _ (isUnit_iff_ne_zero.mpr (ne_of_gt (K.positive z hz).det_pos)),one_mul]
+
+lemma connection_hol_derivative {z : V n} (hz : z ∈ K.domain) (i j : Fin n) :
+    mderiv (-Complex.I) (e i) (K.connection j) z =
+      (K.matrix z)⁻¹ * mderiv (-Complex.I) (e i) (mderiv (-Complex.I) (e j) K.matrix) z -
+        K.connection i z * K.connection j z := by
+  have hs := K.smooth.contDiffAt (K.isOpen.mem_nhds hz)
+  unfold connection
+  rw [mderiv_mul (K.inverse_smooth.contDiffAt (K.isOpen.mem_nhds hz))
+    (mderiv_smooth hs _ _), mderiv_inverse K.isOpen K.smooth
+      (fun y hy => ne_of_gt (K.positive y hy).det_pos) hz]
+  noncomm_ring
+
+lemma connection_hol_flat {z : V n} (hz : z ∈ K.domain) (i j : Fin n) :
+    mderiv (-Complex.I) (e i) (K.connection j) z -
+      mderiv (-Complex.I) (e j) (K.connection i) z =
+      K.connection j z*K.connection i z-K.connection i z*K.connection j z := by
+  rw [K.connection_hol_derivative hz,K.connection_hol_derivative hz,
+    mderiv_comm (K.smooth.contDiffAt (K.isOpen.mem_nhds hz)) (-Complex.I) (-Complex.I) (e i) (e j)]
+  abel
+end KaehlerCalculus.LocalKaehlerField
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/CalabiRicciFrames.lean b/lean/OAI/Geometry/PrescribedPotential/CalabiRicciFrames.lean
new file mode 100644
index 0000000000000000000000000000000000000000..7ce7b1c3756327a3c998cf7d4c4ef3a6459ada24
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/CalabiRicciFrames.lean
@@ -0,0 +1,94 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.BoundedNormalFrames
+import OAI.Geometry.PrescribedPotential.CalabiRescue
+import OAI.Geometry.PrescribedPotential.CalabiResidualBound
+
+namespace OAI
+
+/-! Calabi Ricci Frames. -/
+
+section
+
+noncomputable section
+open Set Metric Filter Topology Matrix
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace KaehlerCalculus
+variable {n : ℕ}
+
+def frameRicciJet (J : ConnectionTensor n) (C : Matrix (Fin n) (Fin n) ℂ) : ConnectionTensor n :=
+  fun i => Cᴴ*(∑ p, C p i • J p)*C
+
+lemma frameRicciJet_continuous : Continuous
+    (fun q : ConnectionTensor n × Matrix (Fin n) (Fin n) ℂ => frameRicciJet q.1 q.2) := by
+  apply continuous_pi
+  intro i
+  apply Continuous.matrix_mul _ continuous_snd
+  apply Continuous.matrix_mul (continuousOn_univ.mp (continuousOn_matrix_star continuous_snd.continuousOn))
+  apply continuous_finsetSum
+  intro p _
+  exact ((continuous_apply_apply p i).comp continuous_snd).smul ((continuous_apply p).comp continuous_fst)
+
+lemma frameRicci_continuousOn {X : Type*} [TopologicalSpace X] {L : Set X}
+    (H : X → Matrix (Fin n) (Fin n) ℂ) (hH : ContinuousOn H L) (B : ℝ) :
+    ContinuousOn (fun q : X × (Matrix (Fin n) (Fin n) ℂ × Matrix (Fin n) (Fin n) ℂ) =>
+      q.2.1ᴴ*H q.1*q.2.1) (L ×ˢ boundedFramePairs B) := by
+  have hc : Continuous (fun q : X × (Matrix (Fin n) (Fin n) ℂ × Matrix (Fin n) (Fin n) ℂ) => q.2.1) :=
+    continuous_fst.comp continuous_snd
+  exact ((continuousOn_matrix_star hc.continuousOn).mul
+    (hH.comp continuous_fst.continuousOn (fun _ h => h.1))).mul hc.continuousOn
+
+lemma frameRicciJet_continuousOn {X : Type*} [TopologicalSpace X] {L : Set X}
+    (J : X → ConnectionTensor n) (hJ : ContinuousOn J L) (B : ℝ) :
+    ContinuousOn (fun q : X × (Matrix (Fin n) (Fin n) ℂ × Matrix (Fin n) (Fin n) ℂ) =>
+      frameRicciJet (J q.1) q.2.1) (L ×ˢ boundedFramePairs B) := by
+  apply frameRicciJet_continuous.comp_continuousOn
+    (f := fun q : X × (Matrix (Fin n) (Fin n) ℂ × Matrix (Fin n) (Fin n) ℂ) => (J q.1,q.2.1))
+  exact (hJ.comp continuous_fst.continuousOn (fun _ h => h.1)).prodMk
+    (continuous_fst.comp continuous_snd).continuousOn
+
+lemma frameRicciTrace_continuousOn {X : Type*} [TopologicalSpace X] {L : Set X}
+    (H : X → Matrix (Fin n) (Fin n) ℂ) (hH : ContinuousOn H L) (B : ℝ) :
+    ContinuousOn (fun q : X × (Matrix (Fin n) (Fin n) ℂ × Matrix (Fin n) (Fin n) ℂ) =>
+      (q.2.2ᴴ*(q.2.1ᴴ*H q.1*q.2.1)*q.2.2).trace.re) (L ×ˢ boundedFramePairs B) := by
+  have hc : Continuous (fun q : X × (Matrix (Fin n) (Fin n) ℂ × Matrix (Fin n) (Fin n) ℂ) => q.2.2) :=
+    continuous_snd.comp continuous_snd
+  exact continuousOn_trace_re (((continuousOn_matrix_star hc.continuousOn).mul
+    (frameRicci_continuousOn H hH B)).mul hc.continuousOn)
+
+lemma calabi_frame_uniform {X : Type*} [TopologicalSpace X] {L : Set X} (hL : IsCompact L)
+    (H : X → Matrix (Fin n) (Fin n) ℂ) (J : X → ConnectionTensor n)
+    (hH : ContinuousOn H L) (hJ : ContinuousOn J L) (B : ℝ) :
+    ∃ C δ : ℝ, 0 ≤ C ∧ 0 < δ ∧ ∀ x ∈ L,
+      ∀ p ∈ boundedFramePairs (n := n) B, ∀ T : ConnectionTensor n,
+      -C*(1+tensorSquare T) ≤ calabiResidual (p.1ᴴ*H x*p.1) (frameRicciJet (J x) p.1) T ∧
+      -C ≤ (p.2ᴴ*(p.1ᴴ*H x*p.1)*p.2).trace.re ∧
+      δ*tensorSquare T ≤ rescueSquare p.2 T := by
+  let E := Matrix (Fin n) (Fin n) ℂ × Matrix (Fin n) (Fin n) ℂ
+  let Q := L ×ˢ boundedFramePairs (n := n) B
+  have hQ : IsCompact Q := hL.prod (boundedFramePairs_compact B)
+  let H' : X × E → Matrix (Fin n) (Fin n) ℂ := fun q => q.2.1ᴴ*H q.1*q.2.1
+  let J' : X × E → ConnectionTensor n := fun q => frameRicciJet (J q.1) q.2.1
+  have hh : ContinuousOn H' Q := frameRicci_continuousOn H hH B
+  have hj : ContinuousOn J' Q := frameRicciJet_continuousOn J hJ B
+  obtain ⟨C1,hC1p,hres⟩ := calabiResidual_compact_bound hQ H' J' hh hj
+  have ht : ContinuousOn (fun q : X × E => (q.2.2ᴴ*H' q*q.2.2).trace.re) Q :=
+    frameRicciTrace_continuousOn H hH B
+  obtain ⟨C2,htrace⟩ := hQ.exists_bound_of_continuousOn ht
+  obtain ⟨δ,hδ,hresc⟩ := rescueSquare_compact_lower (boundedFramePairs_compact (n := n) B)
+    Prod.snd continuous_snd.continuousOn (fun p hp => (boundedFramePairs_units hp).2)
+  let C := max C1 C2
+  refine ⟨C,δ,hC1p.trans (le_max_left _ _),hδ,?_⟩
+  intro x hx p hp T
+  have hq : (x,p) ∈ Q := ⟨hx,hp⟩
+  refine ⟨?_,?_,hresc p hp T⟩
+  · have hh := hres (x,p) hq T
+    have hr := (neg_le_neg hh).trans (neg_abs_le (calabiResidual (H' (x,p)) (J' (x,p)) T))
+    have he := (neg_le_neg (mul_le_mul_of_nonneg_right (le_max_left C1 C2) (add_nonneg zero_le_one (tensorSquare_nonneg T)))).trans hr
+    simpa only [neg_mul,C,H',J'] using he
+  · exact (neg_le_neg ((htrace (x,p) hq).trans (le_max_right C1 C2))).trans (neg_abs_le _)
+end KaehlerCalculus
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/CalabiTensorAlgebra.lean b/lean/OAI/Geometry/PrescribedPotential/CalabiTensorAlgebra.lean
new file mode 100644
index 0000000000000000000000000000000000000000..fa262023057192cf7546d4be37b957e97da81b10
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/CalabiTensorAlgebra.lean
@@ -0,0 +1,100 @@
+import Mathlib
+
+namespace OAI
+
+/-! Calabi Tensor Algebra. -/
+
+section
+
+ 
+
+noncomputable section
+open Matrix
+open scoped ComplexOrder MatrixOrder
+namespace KaehlerCalculus
+variable {n : ℕ}
+abbrev ConnectionTensor (n : ℕ) := Fin n → Matrix (Fin n) (Fin n) ℂ
+
+def tensorPairAt (P M : Matrix (Fin n) (Fin n) ℂ) (T U : ConnectionTensor n) : ℂ :=
+  ∑ i, ∑ j, P i j * (T i*P*(U j)ᴴ*M).trace
+
+lemma tensorPairAt_one (T U : ConnectionTensor n) : tensorPairAt 1 1 T U =
+    ∑ i, (T i*(U i)ᴴ).trace := by
+  simp [tensorPairAt,Matrix.one_apply]
+
+lemma tensorPairAt_add_left (P M : Matrix (Fin n) (Fin n) ℂ) (T U W : ConnectionTensor n) :
+    tensorPairAt P M (fun i => T i+U i) W = tensorPairAt P M T W+tensorPairAt P M U W := by
+  simp only [tensorPairAt,add_mul,Matrix.trace_add,mul_add,Finset.sum_add_distrib]
+
+lemma tensorPairAt_sub_left (P M : Matrix (Fin n) (Fin n) ℂ) (T U W : ConnectionTensor n) :
+    tensorPairAt P M (fun i => T i-U i) W = tensorPairAt P M T W-tensorPairAt P M U W := by
+  simp only [tensorPairAt,sub_mul,Matrix.trace_sub,mul_sub,Finset.sum_sub_distrib]
+
+lemma tensorPairAt_conj {P M : Matrix (Fin n) (Fin n) ℂ} (hP : P.IsHermitian)
+    (hM : M.IsHermitian) (T U : ConnectionTensor n) :
+    star (tensorPairAt P M T U) = tensorPairAt P M U T := by
+  simp only [tensorPairAt,star_sum,star_mul,← Matrix.trace_conjTranspose,
+    Matrix.conjTranspose_mul,Matrix.conjTranspose_conjTranspose,hM.eq,hP.eq]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro i _
+  apply Finset.sum_congr rfl
+  intro j _
+  rw [← hP.apply i j]
+  have hc : (M*(U i*(P*(T j)ᴴ))).trace = (U i*P*(T j)ᴴ*M).trace := by
+    simpa only [mul_assoc] using Matrix.trace_mul_comm M (U i*P*(T j)ᴴ)
+  rw [hc]
+  ring
+
+lemma tensorPairAt_self_real {P M : Matrix (Fin n) (Fin n) ℂ} (hP : P.IsHermitian)
+    (hM : M.IsHermitian) (T : ConnectionTensor n) :
+    (tensorPairAt P M T T).im = 0 := by
+  have he := congrArg Complex.im (tensorPairAt_conj hP hM T T)
+  simp only [Complex.star_def,Complex.conj_im] at he
+  linarith
+
+lemma matrix_contraction_reindex (P C : Matrix (Fin n) (Fin n) ℂ)
+    (Q : Fin n → Fin n → ℂ) :
+    (∑ i, ∑ j, (C*P) i j*Q i j) = ∑ i, ∑ j, P i j * ∑ q, C q i*Q q j := by
+  simp only [Matrix.mul_apply,Finset.sum_mul,Finset.mul_sum]
+  calc
+    (∑ i, ∑ j, ∑ q, C i q*P q j*Q i j) =
+        ∑ q, ∑ j, ∑ i, C i q*P q j*Q i j := by
+      trans ∑ i, ∑ q, ∑ j, C i q*P q j*Q i j
+      · apply Finset.sum_congr rfl
+        intro i _
+        rw [Finset.sum_comm]
+      rw [Finset.sum_comm]
+      apply Finset.sum_congr rfl
+      intro q _
+      rw [Finset.sum_comm]
+    _ = _ := by
+      apply Finset.sum_congr rfl
+      intro q _
+      apply Finset.sum_congr rfl
+      intro j _
+      apply Finset.sum_congr rfl
+      intro i _
+      ring
+
+lemma covariant_pair_algebra (P M C : Matrix (Fin n) (Fin n) ℂ)
+    (T U A B : ConnectionTensor n) :
+    (∑ i, ∑ j, (-(C*P) i j*(T i*P*(U j)ᴴ*M).trace+
+      P i j*(A i*P*(U j)ᴴ*M-T i*C*P*(U j)ᴴ*M+
+        T i*P*(B j)ᴴ*M+T i*P*(U j)ᴴ*M*C).trace)) =
+      tensorPairAt P M (fun i => A i+C*T i-T i*C-∑ q, C q i • T q) U +
+        tensorPairAt P M T B := by
+  have hc (i j) : (C*T i*P*(U j)ᴴ*M).trace = (T i*P*(U j)ᴴ*M*C).trace := by
+    simpa only [mul_assoc] using Matrix.trace_mul_comm C (T i*P*(U j)ᴴ*M)
+  simp only [tensorPairAt,sub_mul,add_mul,Matrix.sum_mul,Matrix.smul_mul,
+    Matrix.trace_add,Matrix.trace_sub,Matrix.trace_sum,Matrix.trace_smul,smul_eq_mul,
+    hc,mul_add,mul_sub,Finset.sum_add_distrib,Finset.sum_sub_distrib,
+    neg_mul,Finset.sum_neg_distrib]
+  rw [matrix_contraction_reindex P C (fun i j => (T i*P*(U j)ᴴ*M).trace)]
+  ring
+end KaehlerCalculus
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/ChartParametrix.lean b/lean/OAI/Geometry/PrescribedPotential/ChartParametrix.lean
new file mode 100644
index 0000000000000000000000000000000000000000..555d9717fdd3e380e479e2cad1e7bfe13ba5d43f
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/ChartParametrix.lean
@@ -0,0 +1,135 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.MetricLocalization
+
+namespace OAI
+
+/-! Chart Parametrix. -/
+
+section
+
+ 
+
+noncomputable section
+open Filter Set Topology MeasureTheory TemperedDistribution LineDeriv
+open scoped SchwartzMap BoundedContinuousFunction ComplexOrder MatrixOrder Matrix.Norms.Elementwise ContDiff
+
+namespace MetricLocalization
+open EllipticKernel FrozenPoisson SobolevChart
+variable {n : ℕ}
+
+abbrev BasisIndex (n : ℕ) := Fin (Module.finrank ℝ (EC n))
+
+def chartDifferential (c : BasisIndex n → BasisIndex n → EC n →ᵇ ℂ) :
+    𝓢'(EC n, ℂ) →L[ℂ] 𝓢'(EC n, ℂ) :=
+  ∑ k, ∑ l, smulLeftCLM ℂ (c k l) ∘L
+    lineDerivOpCLM ℂ 𝓢'(EC n, ℂ) (stdOrthonormalBasis ℝ (EC n) k) ∘L
+    lineDerivOpCLM ℂ 𝓢'(EC n, ℂ) (stdOrthonormalBasis ℝ (EC n) l)
+
+def extendedCoefficient (H : Matrix (Fin n) (Fin n) ℂ)
+    (a : BasisIndex n → BasisIndex n → EC n →ᵇ ℂ)
+    (k l : BasisIndex n) : EC n →ᵇ ℂ :=
+  BoundedContinuousFunction.const _
+    (traceBilin H (rankTwo (stdOrthonormalBasis ℝ (EC n) k)
+      (stdOrthonormalBasis ℝ (EC n) l)) : ℂ) + a k l
+
+lemma extendedCoefficient_temperate (H : Matrix (Fin n) (Fin n) ℂ)
+    (a : BasisIndex n → BasisIndex n → EC n →ᵇ ℂ)
+    (ha : ∀ k l, (a k l : EC n → ℂ).HasTemperateGrowth) (k l : BasisIndex n) :
+    (extendedCoefficient H a k l : EC n → ℂ).HasTemperateGrowth := by
+  change ((fun _ : EC n => (traceBilin H (rankTwo
+    (stdOrthonormalBasis ℝ (EC n) k) (stdOrthonormalBasis ℝ (EC n) l)) : ℂ)) +
+    (a k l : EC n → ℂ)).HasTemperateGrowth
+  apply Function.HasTemperateGrowth.add
+  · fun_prop
+  · exact ha k l
+
+lemma chartDifferential_extension (H : Matrix (Fin n) (Fin n) ℂ)
+    (a : BasisIndex n → BasisIndex n → EC n →ᵇ ℂ)
+    (ha : ∀ k l, (a k l : EC n → ℂ).HasTemperateGrowth) (u : L2 (EC n)) :
+    chartDifferential (extendedCoefficient H a) (realize 2 u) =
+      frozenDifferential H (realize 2 u) +
+        (perturbation (stdOrthonormalBasis ℝ (EC n)) a u : 𝓢'(EC n, ℂ)) := by
+  rw [perturbation_distribution _ a ha]
+  simp only [chartDifferential, frozenDifferential, differentialWithBasis,
+    _root_.sum_apply, ContinuousLinearMap.comp_apply,
+    _root_.smul_apply, ← Finset.sum_add_distrib]
+  apply Finset.sum_congr rfl
+  intro k _
+  apply Finset.sum_congr rfl
+  intro l _
+  change smulLeftCLM ℂ ((fun _ : EC n => (traceBilin H (rankTwo
+    (stdOrthonormalBasis ℝ (EC n) k) (stdOrthonormalBasis ℝ (EC n) l)) : ℂ)) +
+    (a k l : EC n → ℂ)) _ = _
+  rw [smulLeftCLM_add (by fun_prop) (ha k l)]
+  simp
+
+ 
+
+theorem chart_parametrix {U : Set (EC n)} (hU : IsOpen U)
+    (H : EC n → Matrix (Fin n) (Fin n) ℂ) (hH : ContDiffOn ℝ ∞ H U)
+    (hh : ∀ y ∈ U, (H y).PosDef) {x : EC n} (hx : x ∈ U) :
+    ∃ (V : Set (EC n)) (c : BasisIndex n → BasisIndex n → EC n →ᵇ ℂ)
+      (R : L2 (EC n) →L[ℂ] L2 (EC n)) (C : ℝ),
+      IsOpen V ∧ x ∈ V ∧ V ⊆ U ∧ 0 < C ∧
+      (∀ k l, (c k l : EC n → ℂ).HasTemperateGrowth) ∧
+      (∀ y ∈ V, ∀ k l,
+        c k l y = (traceBilin (H y) (rankTwo
+          (stdOrthonormalBasis ℝ (EC n) k) (stdOrthonormalBasis ℝ (EC n) l)) : ℂ)) ∧
+      (∀ f, realize 2 (R f) - chartDifferential c (realize 2 (R f)) = (f : 𝓢'(EC n, ℂ))) ∧
+      (∀ f, ‖R f‖ ≤ C * ‖f‖) ∧
+      (∀ (u f : L2 (EC n)), realize 2 u - chartDifferential c (realize 2 u) = (f : 𝓢'(EC n, ℂ)) →
+        u = R f) := by
+  let b := stdOrthonormalBasis ℝ (EC n)
+  obtain ⟨a, hat, _, has, hag⟩ := local_coefficient_extension hU H hH hh hx b
+  obtain ⟨V, hV, hVo, hxV⟩ := mem_nhds_iff.mp (inter_mem (hU.mem_nhds hx) hag)
+  let R := localResolvent (H x) (hh x hx) b a has
+  let C := ellipticBound (H x) (hh x hx) /
+    (1 - perturbationBound b a * ellipticBound (H x) (hh x hx))
+  have hC : 0 < C := div_pos (ellipticBound_pos _ _) (sub_pos.mpr has)
+  refine ⟨V, extendedCoefficient (H x) a, R, C, hVo, hxV,
+    fun y hy => (hV hy).1, hC, extendedCoefficient_temperate _ _ hat, ?_, ?_, ?_, ?_⟩
+  · intro y hy k l
+    change (_ : ℂ) + a k l y = _
+    rw [(hV hy).2 k l]
+    simp [b]
+  · intro f
+    rw [chartDifferential_extension _ _ hat, ← sub_sub]
+    exact localResolvent_equation _ _ _ _ has f
+  · intro f
+    exact localResolvent_bound _ _ _ _ has f
+  · intro u f hu
+    rw [chartDifferential_extension _ _ hat, ← sub_sub] at hu
+    exact perturbed_unique _ _ _ _ has u _ f hu (localResolvent_equation _ _ _ _ has f)
+
+end MetricLocalization
+
+namespace Anticanonical.SourceSmooth
+open EllipticKernel MetricLocalization SobolevChart FrozenPoisson
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+
+ 
+
+theorem KaehlerMetric.chart_parametrix (g : KaehlerMetric A)
+    (i : Fin A.count) {x : EC d} (hx : coordinateEquiv d x ∈ (A.chart i).target) :
+    ∃ (V : Set (EC d)) (c : BasisIndex d → BasisIndex d → EC d →ᵇ ℂ)
+      (R : L2 (EC d) →L[ℂ] L2 (EC d)) (C : ℝ),
+      IsOpen V ∧ x ∈ V ∧ V ⊆ (coordinateEquiv d) ⁻¹' (A.chart i).target ∧ 0 < C ∧
+      (∀ k l, (c k l : EC d → ℂ).HasTemperateGrowth) ∧
+      (∀ y ∈ V, ∀ k l,
+        c k l y = (traceBilin (g.matrix i (coordinateEquiv d y)) (rankTwo
+          (stdOrthonormalBasis ℝ (EC d) k) (stdOrthonormalBasis ℝ (EC d) l)) : ℂ)) ∧
+      (∀ f, realize 2 (R f) - chartDifferential c (realize 2 (R f)) = (f : 𝓢'(EC d, ℂ))) ∧
+      (∀ f, ‖R f‖ ≤ C * ‖f‖) ∧
+      (∀ (u f : L2 (EC d)), realize 2 u - chartDifferential c (realize 2 u) = (f : 𝓢'(EC d, ℂ)) →
+        u = R f) := by
+  exact MetricLocalization.chart_parametrix
+    ((A.chart i).open_target.preimage (coordinateEquiv d).continuous)
+    _ ((g.smooth i).comp (coordinateEquiv d).contDiff.contDiffOn (fun _ h => h))
+    (fun y hy => g.positive i _ hy) hx
+
+end Anticanonical.SourceSmooth
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/ChartTransport.lean b/lean/OAI/Geometry/PrescribedPotential/ChartTransport.lean
new file mode 100644
index 0000000000000000000000000000000000000000..3246f8ad7845d0430070e4014f5b68ee877d6c31
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/ChartTransport.lean
@@ -0,0 +1,158 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.LocalRegularity
+import OAI.Geometry.PrescribedPotential.LocalizedPullback
+
+namespace OAI
+
+/-! Chart Transport. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory LineDeriv
+open scoped ContDiff SchwartzMap Classical BoundedContinuousFunction
+
+namespace GlobalElliptic
+open SobolevChart
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+
+ 
+structure ChartCutoff (U : Set E) where
+  val : 𝓢(E, ℂ)
+  compact : HasCompactSupport (val : E → ℂ)
+  support_sub : tsupport (val : E → ℂ) ⊆ U
+
+instance {U : Set E} : CoeFun (ChartCutoff U) (fun _ => E → ℂ) := ⟨fun κ => κ.val⟩
+
+namespace ChartCutoff
+variable {U : Set E} (κ : ChartCutoff U)
+
+lemma product_smooth {h : E → ℂ} (hU : IsOpen U) (hh : ContDiffOn ℝ ∞ h U) :
+    ContDiff ℝ ∞ (fun x => κ x * h x) := by
+  rw [contDiff_iff_contDiffAt]
+  intro x
+  by_cases hx : x ∈ U
+  · exact (κ.val.smooth ⊤).contDiffAt.mul (hh.contDiffAt (hU.mem_nhds hx))
+  · have ht : x ∉ tsupport (κ : E → ℂ) := fun ht => hx (κ.support_sub ht)
+    apply (contDiffAt_const (c := (0 : ℂ))).congr_of_eventuallyEq
+    filter_upwards [notMem_tsupport_iff_eventuallyEq.mp ht] with y hy
+    simp [hy]
+
+lemma product_compact (h : E → ℂ) : HasCompactSupport (fun x => κ x * h x) := by
+  apply HasCompactSupport.of_support_subset_isCompact κ.compact
+  intro x hx
+  exact subset_tsupport _ (left_ne_zero_of_mul hx)
+
+def product {h : E → ℂ} (hU : IsOpen U) (hh : ContDiffOn ℝ ∞ h U) : 𝓢(E, ℂ) :=
+  (κ.product_compact h).toSchwartzMap (κ.product_smooth hU hh)
+
+@[simp] lemma product_apply {h : E → ℂ} (hU : IsOpen U) (hh : ContDiffOn ℝ ∞ h U) (x : E) :
+    κ.product hU hh x = κ x * h x := rfl
+
+lemma product_tsupport {h : E → ℂ} (hU : IsOpen U) (hh : ContDiffOn ℝ ∞ h U) :
+    tsupport (κ.product hU hh : E → ℂ) ⊆ tsupport (κ : E → ℂ) := by
+  apply closure_mono
+  exact fun x hx => left_ne_zero_of_mul hx
+
+ 
+def mulOn {h : E → ℂ} (hU : IsOpen U) (hh : ContDiffOn ℝ ∞ h U) : ChartCutoff U where
+  val := κ.product hU hh
+  compact := κ.product_compact h
+  support_sub := (κ.product_tsupport hU hh).trans κ.support_sub
+
+lemma deriv_tsupport (v : E) :
+    tsupport (⇑(∂_{v} κ.val : 𝓢(E, ℂ))) ⊆ tsupport (κ : E → ℂ) := by
+  intro x hx
+  by_contra hn
+  have hh := notMem_tsupport_iff_eventuallyEq.mp hn
+  have hd : (fun y => (∂_{v} κ.val : 𝓢(E, ℂ)) y) =ᶠ[𝓝 x] 0 := by
+    filter_upwards [hh.eventually_nhds] with y hy
+    simp only [SchwartzMap.lineDerivOp_apply_eq_fderiv]
+    have hy' : (κ : E → ℂ) =ᶠ[𝓝 y] 0 := hy
+    rw [hy'.fderiv_eq]
+    simp
+  exact (notMem_tsupport_iff_eventuallyEq.mpr hd) hx
+
+def deriv (v : E) : ChartCutoff U where
+  val := ∂_{v} κ.val
+  compact := (κ.compact.of_isClosed_subset (isClosed_tsupport _) (κ.deriv_tsupport v))
+  support_sub := (κ.deriv_tsupport v).trans κ.support_sub
+
+end ChartCutoff
+
+variable (e : OpenPartialHomeomorph E E) (he : ContDiffOn ℝ ∞ e e.source)
+
+ 
+def chartTransport (κ : ChartCutoff e.source) : 𝓢(E, ℂ) →ₗ[ℝ] 𝓢(E, ℂ) where
+  toFun f := κ.product e.open_source ((f.smooth ⊤).comp_contDiffOn he)
+  map_add' f h := by
+    ext x
+    simp [mul_add]
+  map_smul' c f := by
+    ext x
+    simp [mul_left_comm]
+
+@[simp] lemma chartTransport_apply (κ : ChartCutoff e.source) (f : 𝓢(E, ℂ)) (x : E) :
+    chartTransport e he κ f x = κ x * f (e x) := rfl
+
+ 
+variable [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+
+theorem chartTransport_l2_bound (hs : ContDiffOn ℝ ∞ e.symm e.target)
+    (κ : ChartCutoff e.source) : ∃ C : ℝ, 0 ≤ C ∧ ∀ f : 𝓢(E, ℂ),
+      ‖schwartzCoord 0 (chartTransport e he κ f)‖ ≤ C * ‖schwartzCoord 0 f‖ := by
+  let K := tsupport (κ : E → ℂ)
+  obtain ⟨C, hC, hmeas, hmap⟩ := chart_map_restrict_bound e he hs κ.compact κ.support_sub
+  let B : L2 E →L[ℝ] L2 E :=
+    (Lp.extendZero (μ := (volume : Measure E)) (p := 2) (F := ℂ) (isClosed_tsupport _).measurableSet).toContinuousLinearMap ∘L
+      Lp.pullbackBounded (K.indicator e) hmeas ENNReal.ofReal_ne_top hmap
+  let T : L2 E →L[ℝ] L2 E := (multiply κ.val.toBoundedContinuousFunction).restrictScalars ℝ ∘L B
+  refine ⟨‖T‖, norm_nonneg _, fun f => ?_⟩
+  have hc (u : 𝓢(E, ℂ)) : schwartzCoord 0 u = u.toLp 2 volume := by
+    apply realize_injective 0
+    simpa [realize] using realize_schwartzCoord 0 u
+  rw [hc, hc]
+  have heq : T (f.toLp 2 volume) = (chartTransport e he κ f).toLp 2 volume := by
+    apply Lp.ext
+    have hb := Lp.coeFn_pullbackBounded (K.indicator e) hmeas ENNReal.ofReal_ne_top hmap
+      (f.toLp 2 volume)
+    have hz := Lp.coeFn_extendZero (μ := (volume : Measure E))
+      (isClosed_tsupport (κ : E → ℂ)).measurableSet
+      (Lp.pullbackBounded (K.indicator e) hmeas ENNReal.ofReal_ne_top hmap (f.toLp 2 volume))
+    have hf := (f.memLp 2 volume).coeFn_toLp
+    have hfc := (ae_of_ae_map hmeas.aemeasurable
+      (ae_mono hmap (Measure.ae_smul_measure hf _)))
+    have ht := (chartTransport e he κ f).coeFn_toLp (p := 2) (μ := volume)
+    have hm : (fun x => multiply κ.val.toBoundedContinuousFunction (B (f.toLp 2 volume)) x) =ᵐ[volume]
+        fun x => κ x * B (f.toLp 2 volume) x := by
+      filter_upwards [(ContinuousLinearMap.mul ℂ ℂ).coeFn_holder (r := 2)
+        (κ.val.toBoundedContinuousFunction.memLp_top.toLp κ.val.toBoundedContinuousFunction : Lp ℂ ⊤ volume)
+        (B (f.toLp 2 volume)),
+        (κ.val.toBoundedContinuousFunction.memLp_top (μ := volume)).coeFn_toLp] with x h₁ h₂
+      exact h₁.trans (by simp only [ContinuousLinearMap.mul_apply', h₂]; rfl)
+    filter_upwards [hm, hz, (ae_restrict_iff' (isClosed_tsupport _).measurableSet).mp hb,
+      (ae_restrict_iff' (isClosed_tsupport _).measurableSet).mp hfc, ht]
+      with x h₁ h₂ h₃ h₄ h₅
+    change multiply κ.val.toBoundedContinuousFunction (B (f.toLp 2 volume)) x = _
+    rw [h₁, h₅]
+    change B (f.toLp 2 volume) x = _ at h₂
+    rw [h₂]
+    by_cases hx : x ∈ K
+    · rw [indicator_of_mem hx, h₃ hx]
+      have hh := h₄ hx
+      change (f.toLp 2 volume) (K.indicator e x) = f (K.indicator e x) at hh
+      rw [hh, indicator_of_mem hx]
+      rfl
+    · have hk : κ x = 0 := image_eq_zero_of_notMem_tsupport hx
+      simp [chartTransport_apply, hk]
+  rw [← heq]
+  exact T.le_opNorm _
+
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/ChartTransportHigher.lean b/lean/OAI/Geometry/PrescribedPotential/ChartTransportHigher.lean
new file mode 100644
index 0000000000000000000000000000000000000000..2758a807d188a4ca4677739e2e2153b4e6693a15
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/ChartTransportHigher.lean
@@ -0,0 +1,127 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.ChartTransport
+import OAI.Geometry.PrescribedPotential.CoreBounds
+
+namespace OAI
+
+/-! Chart Transport Higher. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory LineDeriv TemperedDistribution
+open scoped ContDiff SchwartzMap Classical Laplacian
+
+namespace GlobalElliptic
+open SobolevChart
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E]
+variable (e : OpenPartialHomeomorph E E) (he : ContDiffOn ℝ ∞ e e.source)
+
+include he in
+lemma transportCoefficient_smooth (v : E) (j : Fin (Module.finrank ℝ E)) :
+    ContDiffOn ℝ ∞ (fun x => ((stdOrthonormalBasis ℝ E).repr (fderiv ℝ e x v) j : ℂ)) e.source := by
+  apply Complex.ofRealCLM.contDiff.comp_contDiffOn
+  apply ((EuclideanSpace.proj j).contDiff.comp (stdOrthonormalBasis ℝ E).repr.contDiff).comp_contDiffOn
+  exact (he.fderiv_of_isOpen e.open_source (by simp)).clm_apply contDiffOn_const
+
+def transportCoefficient (κ : ChartCutoff e.source) (v : E) (j : Fin (Module.finrank ℝ E)) :
+    ChartCutoff e.source := κ.mulOn e.open_source (transportCoefficient_smooth e he v j)
+
+@[simp] lemma transportCoefficient_apply (κ : ChartCutoff e.source) (v : E)
+    (j : Fin (Module.finrank ℝ E)) (x : E) :
+    transportCoefficient e he κ v j x =
+      κ x * ((stdOrthonormalBasis ℝ E).repr (fderiv ℝ e x v) j : ℂ) := rfl
+
+omit [FiniteDimensional ℝ E] in
+lemma chartTransport_tsupport (κ : ChartCutoff e.source) (f : 𝓢(E, ℂ)) :
+    tsupport (chartTransport e he κ f : E → ℂ) ⊆ tsupport (κ : E → ℂ) :=
+  κ.product_tsupport e.open_source ((f.smooth ⊤).comp_contDiffOn he)
+
+ 
+
+lemma chartTransport_deriv (κ : ChartCutoff e.source) (v : E) (f : 𝓢(E, ℂ)) :
+    ∂_{v} (chartTransport e he κ f) = chartTransport e he (κ.deriv v) f +
+      ∑ j, chartTransport e he (transportCoefficient e he κ v j)
+        (∂_{stdOrthonormalBasis ℝ E j} f) := by
+  ext x
+  by_cases hx : x ∈ e.source
+  · have hed := (he.contDiffAt (e.open_source.mem_nhds hx)).differentiableAt (by simp)
+    have hfd := f.differentiableAt.comp x hed
+    simp only [SchwartzMap.lineDerivOp_apply_eq_fderiv, chartTransport_apply,
+      _root_.add_apply, _root_.sum_apply, transportCoefficient_apply]
+    change fderiv ℝ (fun y => κ y * f (e y)) x v =
+      fderiv ℝ κ.val x v * f (e x) + _
+    have hm := fderiv_fun_mul κ.val.differentiableAt hfd
+    have hc := fderiv_comp x f.differentiableAt hed
+    dsimp only [Function.comp_def] at hm hc
+    rw [hm, hc]
+    simp only [_root_.add_apply, _root_.smul_apply,
+      ContinuousLinearMap.comp_apply, smul_eq_mul]
+    have hb := congrArg (fun z => fderiv ℝ f (e x) z)
+      ((stdOrthonormalBasis ℝ E).sum_repr (fderiv ℝ e x v))
+    simp only [map_sum, map_smul, Complex.real_smul] at hb
+    rw [← hb, Finset.mul_sum]
+    ring_nf
+  · have hk : x ∉ tsupport (κ : E → ℂ) := fun ht => hx (κ.support_sub ht)
+    have ht : x ∉ tsupport (chartTransport e he κ f : E → ℂ) :=
+      fun ht => hk (chartTransport_tsupport e he κ f ht)
+    have hder : (∂_{v} (chartTransport e he κ f) : 𝓢(E, ℂ)) x = 0 := by
+      rw [SchwartzMap.lineDerivOp_apply_eq_fderiv,
+        (notMem_tsupport_iff_eventuallyEq.mp ht).fderiv_eq]
+      simp
+    have hk' : κ.deriv v x = 0 := image_eq_zero_of_notMem_tsupport
+      (fun h => hk (κ.deriv_tsupport v h))
+    simp only [_root_.add_apply, _root_.sum_apply, chartTransport_apply,
+      transportCoefficient_apply, hder, image_eq_zero_of_notMem_tsupport hk, hk', zero_mul,
+      Finset.sum_const_zero, add_zero]
+
+variable [MeasurableSpace E] [BorelSpace E]
+
+lemma word_deriv_comm (ws : List E) (v : E) (f : 𝓢(E, ℂ)) :
+    schwartzWord ws (∂_{v} f) = ∂_{v} (schwartzWord ws f) := by
+  induction ws with
+  | nil => rfl
+  | cons w ws ih =>
+    change ∂_{w} (schwartzWord ws (∂_{v} f)) = ∂_{v} (∂_{w} (schwartzWord ws f))
+    rw [ih, schwartz_deriv_comm]
+
+include he in
+lemma chartTransport_word_bound (hs : ContDiffOn ℝ ∞ e.symm e.target) (ws : List E)
+    (κ : ChartCutoff e.source) {s : ℝ} (hws : (ws.length : ℝ) ≤ s) :
+    CoreBound s 0 (fun f => schwartzWord ws (chartTransport e he κ f)) := by
+  induction ws generalizing κ s with
+  | nil =>
+    have hzero : CoreBound 0 0 (chartTransport e he κ) := chartTransport_l2_bound e he hs κ
+    exact hzero.comp (CoreBound.id (by simpa using hws))
+  | cons w ws ih =>
+    have hw : (ws.length : ℝ) + 1 ≤ s := by simpa only [List.length_cons, Nat.cast_add, Nat.cast_one] using hws
+    have hfirst := ih (κ.deriv w) (s := s) (by linarith)
+    have hrest (j : Fin (Module.finrank ℝ E)) :=
+      (ih (transportCoefficient e he κ w j) (s := s - 1) (by linarith)).comp
+        (coreBound_deriv (stdOrthonormalBasis ℝ E j) (s := s) (t := s - 1) le_rfl)
+    have hh := hfirst.add (CoreBound.sum Finset.univ _ (fun j _ => hrest j))
+    convert hh using 1
+    funext f
+    change ∂_{w} (schwartzWord ws (chartTransport e he κ f)) = _
+    rw [← word_deriv_comm, chartTransport_deriv, map_add, map_sum]
+    rfl
+
+ 
+
+theorem chartTransport_even_bound (hs : ContDiffOn ℝ ∞ e.symm e.target)
+    (κ : ChartCutoff e.source) (k : ℕ) :
+    CoreBound (2 * (k : ℝ)) (2 * (k : ℝ)) (chartTransport e he κ) := by
+  apply coreBound_of_words k
+  intro ws hw
+  apply chartTransport_word_bound e he hs ws κ
+  exact_mod_cast hw
+
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/ChernLuFrameInequality.lean b/lean/OAI/Geometry/PrescribedPotential/ChernLuFrameInequality.lean
new file mode 100644
index 0000000000000000000000000000000000000000..acc2af1c6898cff2f31242e1a496203af6c169f2
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/ChernLuFrameInequality.lean
@@ -0,0 +1,56 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.FrameCompactBounds
+import OAI.Geometry.PrescribedPotential.FrameSecondCalculus
+import OAI.Geometry.PrescribedPotential.MatrixFrameNormalization
+import OAI.Geometry.PrescribedPotential.ScalarFrameCalculus
+
+namespace OAI
+
+/-! Chern Lu Frame Inequality. -/
+
+section
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace KaehlerCalculus
+variable {n : ℕ}
+
+lemma inverseTrace_log_inequality_frame [Nonempty (Fin n)]
+    {U : Set (V n)} (hU : IsOpen U)
+    (B : V n →L[ℂ] V n) (hB : IsUnit (frameMatrix B))
+    {M G : V n → Matrix (Fin n) (Fin n) ℂ} (hM : ContDiffOn ℝ ∞ M U)
+    (hp : ∀ y ∈ U, (M y).PosDef)
+    (hclosed : ∀ y ∈ U, ∀ u v w : V n,
+      fderiv ℝ (fun q => Anticanonical.ComplexAtlas.fundamentalForm (M q) v w) y u +
+      fderiv ℝ (fun q => Anticanonical.ComplexAtlas.fundamentalForm (M q) w u) y v +
+      fderiv ℝ (fun q => Anticanonical.ComplexAtlas.fundamentalForm (M q) u v) y w = 0)
+    (hG : ContDiffOn ℝ ∞ G U) (hGp : ∀ y ∈ U, (G y).PosDef)
+    {w : V n} (hw : B w ∈ U) (hM1 : pullMetric B M w = 1) :
+    frameResidual (PotentialKaehler.potentialMatrix (fun y => Real.log (M y).det.re) (B w))
+      (G (B w)) (curvatureDiagonal G (B w)) (frameMatrix B) ≤
+      inverseTrace M G (B w) *
+        (PotentialKaehler.potentialMatrix (fun y => Real.log (inverseTrace M G (B y))) w).trace.re := by
+  have hi := inverseTrace_log_inequality_at_one (hU.preimage B.continuous)
+    (pullMetric_smooth hU B hM) (pullMetric_positive B hB hp)
+    (pullMetric_closed hU B hM hclosed) (pullMetric_smooth hU B hG) (pullMetric_positive B hB hGp) hw hM1
+  rw [potentialMatrix_logdet_pullMetric hU B hB hM hp hw] at hi
+  simp_rw [curvatureDiagonal_pullMetric hU B hB hG hw] at hi
+  have hs : inverseTrace (pullMetric B M) (pullMetric B G) = fun y => inverseTrace M G (B y) :=
+    funext (fun y => inverseTrace_pullMetric B hB M G y)
+  rw [hs] at hi
+  simpa only [frameResidual,pullMetric,frameMatrix_mulVec] using hi
+
+lemma inverseTrace_eq_frameGram (B : V n →L[ℂ] V n) (hB : IsUnit (frameMatrix B))
+    (M G : V n → Matrix (Fin n) (Fin n) ℂ) (w : V n) (hM1 : pullMetric B M w = 1) :
+    inverseTrace M G (B w) = frameGram (G (B w)) (frameMatrix B) := by
+  rw [← inverseTrace_pullMetric B hB M G w]
+  unfold inverseTrace
+  rw [hM1,inv_one,one_mul]
+  rfl
+end KaehlerCalculus
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/ChernLuIdentity.lean b/lean/OAI/Geometry/PrescribedPotential/ChernLuIdentity.lean
new file mode 100644
index 0000000000000000000000000000000000000000..3d2d015bf84f3a1d513d9be7721642e07d627b40
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/ChernLuIdentity.lean
@@ -0,0 +1,77 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.InverseTraceCalculus
+import OAI.Geometry.PrescribedPotential.KaehlerSecondDerivatives
+
+namespace OAI
+
+/-! Chern Lu Identity. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace KaehlerCalculus
+variable {n : ℕ}
+
+lemma sum_ricci_cancellation (A B C Y Z W : Fin n → Matrix (Fin n) (Fin n) ℂ)
+    (G H : Matrix (Fin n) (Fin n) ℂ)
+    (hc : ∑ k, C k = H + ∑ k, B k*A k) :
+    ∑ k, ((B k*A k+A k*B k-C k)*G-A k*Z k-B k*Y k+W k) =
+      -(H*G) + ∑ k, (A k*B k*G-A k*Z k-B k*Y k+W k) := by
+  simp only [add_mul,sub_mul,Finset.sum_add_distrib,Finset.sum_sub_distrib,
+    ← Matrix.sum_mul]
+  rw [hc]
+  noncomm_ring
+
+lemma inverse_trace_ricci_identity_at_one {U : Set (V n)} (hU : IsOpen U)
+    {M G : V n → Matrix (Fin n) (Fin n) ℂ}
+    (hM : ContDiffOn ℝ ∞ M U) (hp : ∀ y ∈ U, (M y).PosDef)
+    (hclosed : ∀ y ∈ U, ∀ u v w : V n,
+      fderiv ℝ (fun q => Anticanonical.ComplexAtlas.fundamentalForm (M q) v w) y u +
+      fderiv ℝ (fun q => Anticanonical.ComplexAtlas.fundamentalForm (M q) w u) y v +
+      fderiv ℝ (fun q => Anticanonical.ComplexAtlas.fundamentalForm (M q) u v) y w = 0)
+    (hG : ContDiffOn ℝ ∞ G U) (hGh : ∀ y ∈ U, (G y).IsHermitian)
+    {z : V n} (hz : z ∈ U) (hMz : M z = 1) :
+    ∑ k, dzbar (e k) (dz (e k) (fun y => ((M y)⁻¹*G y).trace)) z =
+      -(PotentialKaehler.potentialMatrix (fun y => Real.log (M y).det.re) z*G z).trace +
+      ∑ k, (mderiv (-Complex.I) (e k) M z*(mderiv (-Complex.I) (e k) M z)ᴴ*G z -
+        mderiv (-Complex.I) (e k) M z*(mderiv (-Complex.I) (e k) G z)ᴴ -
+        (mderiv (-Complex.I) (e k) M z)ᴴ*mderiv (-Complex.I) (e k) G z +
+        mderiv Complex.I (e k) (mderiv (-Complex.I) (e k) G) z).trace := by
+  let A := fun k => mderiv (-Complex.I) (e k) M z
+  let B := fun k => mderiv Complex.I (e k) M z
+  let C := fun k => mderiv Complex.I (e k) (mderiv (-Complex.I) (e k) M) z
+  let Y := fun k => mderiv (-Complex.I) (e k) G z
+  let Z := fun k => mderiv Complex.I (e k) G z
+  let W := fun k => mderiv Complex.I (e k) (mderiv (-Complex.I) (e k) G) z
+  let H := PotentialKaehler.potentialMatrix (fun y => Real.log (M y).det.re) z
+  have hc : ∑ k, C k = H + ∑ k, B k*A k :=
+    ricci_cancellation_at_one hU hM hp hclosed hz hMz
+  have hn : ∀ y ∈ U, (M y).det ≠ 0 := fun y hy => ne_of_gt (hp y hy).det_pos
+  have hsG := hG.contDiffAt (hU.mem_nhds hz)
+  have hfirst : (∑ k, dzbar (e k) (dz (e k) (fun y => ((M y)⁻¹*G y).trace)) z) =
+      ∑ k, ((B k*A k+A k*B k-C k)*G z-A k*Z k-B k*Y k+W k).trace := by
+    apply Finset.sum_congr rfl
+    intro k _
+    exact second_wderiv_inverse_trace_at_one hU hM hn hz hMz hsG
+      (-Complex.I) Complex.I (e k) (e k)
+  have hsecond := congrArg Matrix.trace (sum_ricci_cancellation A B C Y Z W (G z) H hc)
+  rw [Matrix.trace_add,Matrix.trace_neg,Matrix.trace_sum,Matrix.trace_sum] at hsecond
+  have hlast : (∑ k, (A k*B k*G z-A k*Z k-B k*Y k+W k).trace) =
+      ∑ k, (A k*(A k)ᴴ*G z-A k*(Y k)ᴴ-(A k)ᴴ*Y k+W k).trace := by
+    apply Finset.sum_congr rfl
+    intro k _
+    have hb : B k = (A k)ᴴ := mderiv_bar_eq_conjTranspose hU hM
+      (fun y hy => (hp y hy).isHermitian) hz (e k)
+    have hz' : Z k = (Y k)ᴴ := mderiv_bar_eq_conjTranspose hU hG hGh hz (e k)
+    rw [hb,hz']
+  exact hfirst.trans (hsecond.trans (congrArg (fun q => -(H*G z).trace+q) hlast))
+end KaehlerCalculus
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/ChernLuLocalInequality.lean b/lean/OAI/Geometry/PrescribedPotential/ChernLuLocalInequality.lean
new file mode 100644
index 0000000000000000000000000000000000000000..69e733d1f0d0e43862d7a0b77e92fd83c8bacf25
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/ChernLuLocalInequality.lean
@@ -0,0 +1,103 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.InverseTraceFunction
+import OAI.Geometry.PrescribedPotential.LogHessianCalculus
+import OAI.Geometry.PrescribedPotential.MatrixTraceInequalities
+
+namespace OAI
+
+/-! Chern Lu Local Inequality. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace KaehlerCalculus
+variable {n : ℕ}
+
+def curvatureDiagonal (G : V n → Matrix (Fin n) (Fin n) ℂ) (z v : V n) :=
+  mderiv Complex.I v (mderiv (-Complex.I) v G) z -
+    (mderiv (-Complex.I) v G z)ᴴ*(G z)⁻¹*mderiv (-Complex.I) v G z
+
+lemma trace_gradient_sum_bound (G : Matrix (Fin n) (Fin n) ℂ) (hG : G.PosDef)
+    (X Y W : Fin n → Matrix (Fin n) (Fin n) ℂ) :
+    ∑ k, ‖(Y k-X k*G).trace‖^2 ≤ G.trace.re *
+      ((∑ k, (X k*(X k)ᴴ*G-X k*(Y k)ᴴ-(X k)ᴴ*Y k+W k).trace.re) -
+        ∑ k, (W k-(Y k)ᴴ*G⁻¹*Y k).trace.re) := by
+  have h := Finset.sum_le_sum (s := Finset.univ)
+    (fun k _ => trace_gradient_square_bound hG (X k) (Y k))
+  rw [← Finset.mul_sum] at h
+  have he : ((∑ k, (X k*(X k)ᴴ*G-X k*(Y k)ᴴ-(X k)ᴴ*Y k+W k).trace.re) -
+      ∑ k, (W k-(Y k)ᴴ*G⁻¹*Y k).trace.re) =
+      ∑ k, ((X k*(X k)ᴴ*G).trace.re-(X k*(Y k)ᴴ).trace.re-
+        ((X k)ᴴ*Y k).trace.re+((Y k)ᴴ*G⁻¹*Y k).trace.re) := by
+    rw [← Finset.sum_sub_distrib]
+    apply Finset.sum_congr rfl
+    intro k _
+    simp only [Matrix.trace_add,Matrix.trace_sub,Complex.add_re,Complex.sub_re]
+    ring
+  rw [he]
+  exact h
+
+lemma log_hessian_trace {U : Set (V n)} (hU : IsOpen U)
+    {f : V n → ℝ} (hf : ContDiffOn ℝ ∞ f U) (hp : ∀ y ∈ U, f y ≠ 0)
+    {z : V n} (hz : z ∈ U) :
+    (PotentialKaehler.potentialMatrix (fun y => Real.log (f y)) z).trace.re =
+      (f z)⁻¹*(PotentialKaehler.potentialMatrix f z).trace.re -
+      (f z)^(-2 : ℤ)*(∑ k, ‖dz (e k) (fun y => (f y : ℂ)) z‖^2) := by
+  simp only [Matrix.trace,Matrix.diag_apply,Complex.re_sum]
+  simp_rw [log_hessian_diagonal hU hf hp hz]
+  rw [Finset.sum_sub_distrib,← Finset.mul_sum,← Finset.mul_sum]
+
+lemma scalar_log_cancellation {s d l r q c a : ℝ} (hs : 0 < s)
+    (hl : l = s⁻¹*d - s^(-2 : ℤ)*a) (hd : d = -r+q)
+    (ha : a ≤ s*(q-c)) : -r+c ≤ s*l := by
+  have he : s^2*l = s*d-a := by
+    rw [hl]
+    simp only [_root_.zpow_neg,zpow_ofNat]
+    field_simp
+  have hh : s*(-r+c) ≤ s^2*l := by rw [he,hd]; nlinarith
+  have hf : s^2*l = s*(s*l) := by ring
+  rw [hf] at hh
+  exact (mul_le_mul_iff_right₀ hs).mp hh
+
+lemma inverseTrace_log_inequality_at_one [Nonempty (Fin n)]
+    {U : Set (V n)} (hU : IsOpen U)
+    {M G : V n → Matrix (Fin n) (Fin n) ℂ} (hM : ContDiffOn ℝ ∞ M U)
+    (hp : ∀ y ∈ U, (M y).PosDef)
+    (hclosed : ∀ y ∈ U, ∀ u v w : V n,
+      fderiv ℝ (fun q => Anticanonical.ComplexAtlas.fundamentalForm (M q) v w) y u +
+      fderiv ℝ (fun q => Anticanonical.ComplexAtlas.fundamentalForm (M q) w u) y v +
+      fderiv ℝ (fun q => Anticanonical.ComplexAtlas.fundamentalForm (M q) u v) y w = 0)
+    (hG : ContDiffOn ℝ ∞ G U) (hGp : ∀ y ∈ U, (G y).PosDef)
+    {z : V n} (hz : z ∈ U) (hMz : M z = 1) :
+    -(PotentialKaehler.potentialMatrix (fun y => Real.log (M y).det.re) z*G z).trace.re +
+      ∑ k, (curvatureDiagonal G z (e k)).trace.re ≤
+      inverseTrace M G z *
+        (PotentialKaehler.potentialMatrix (fun y => Real.log (inverseTrace M G y)) z).trace.re := by
+  let X := fun k => mderiv (-Complex.I) (e k) M z
+  let Y := fun k => mderiv (-Complex.I) (e k) G z
+  let W := fun k => mderiv Complex.I (e k) (mderiv (-Complex.I) (e k) G) z
+  have hGh : ∀ y ∈ U, (G y).IsHermitian := fun y hy => (hGp y hy).isHermitian
+  have hpos : ∀ y ∈ U, 0 < inverseTrace M G y := fun y hy => inverseTrace_pos (hp y hy) (hGp y hy)
+  have hl := log_hessian_trace hU (inverseTrace_smooth hU hM hp hG)
+    (fun y hy => ne_of_gt (hpos y hy)) hz
+  have hd := inverseTrace_second_at_one hU hM hp hclosed hG hGh hz hMz
+  have hg := trace_gradient_sum_bound (G z) (hGp z hz) X Y W
+  have hs : inverseTrace M G z = (G z).trace.re := by simp [inverseTrace,hMz]
+  have he : (∑ k, ‖dz (e k) (fun y => (inverseTrace M G y : ℂ)) z‖^2) =
+      ∑ k, ‖(Y k-X k*G z).trace‖^2 := by
+    apply Finset.sum_congr rfl
+    intro k _
+    rw [inverseTrace_dz_at_one hU hM hp hG hGh hz hMz]
+  rw [he] at hl
+  rw [← hs] at hg
+  exact scalar_log_cancellation (hpos z hz) hl hd hg
+end KaehlerCalculus
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/ChernLuMaximum.lean b/lean/OAI/Geometry/PrescribedPotential/ChernLuMaximum.lean
new file mode 100644
index 0000000000000000000000000000000000000000..41c3aeaad39b059cfd4d9633715bff6f0b576f13
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/ChernLuMaximum.lean
@@ -0,0 +1,102 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.ChernLuFrameInequality
+import OAI.Geometry.PrescribedPotential.PotentialLinearCalculus
+
+namespace OAI
+
+/-! Chern Lu Maximum. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace KaehlerCalculus
+variable {n : ℕ}
+
+lemma pullMetric_potential_relation (B : V n →L[ℂ] V n)
+    (M G : V n → Matrix (Fin n) (Fin n) ℂ) (f : V n → ℝ) (w : V n)
+    (hf : ContDiffAt ℝ ∞ f (B w))
+    (he : M (B w) = G (B w)+PotentialKaehler.potentialMatrix f (B w)) :
+    pullMetric B M w = pullMetric B G w+PotentialKaehler.potentialMatrix (fun y => f (B y)) w := by
+  rw [potentialMatrix_linear B hf]
+  unfold pullMetric
+  rw [he,mul_add,add_mul]
+
+lemma trace_potential_in_frame (B : V n →L[ℂ] V n) (hB : IsUnit (frameMatrix B))
+    (M G : V n → Matrix (Fin n) (Fin n) ℂ) (f : V n → ℝ) (w : V n)
+    (hf : ContDiffAt ℝ ∞ f (B w))
+    (he : M (B w) = G (B w)+PotentialKaehler.potentialMatrix f (B w))
+    (hM1 : pullMetric B M w = 1) :
+    (PotentialKaehler.potentialMatrix (fun y => f (B y)) w).trace.re =
+      (n:ℝ)-inverseTrace M G (B w) := by
+  have hh := congrArg (fun H : Matrix (Fin n) (Fin n) ℂ => H.trace.re)
+    (pullMetric_potential_relation B M G f w hf he)
+  rw [hM1,Matrix.trace_one,Matrix.trace_add,Complex.add_re] at hh
+  have hs := inverseTrace_eq_frameGram B hB M G w hM1
+  change inverseTrace M G (B w) = (pullMetric B G w).trace.re at hs
+  simp only [Fintype.card_fin,Complex.natCast_re] at hh
+  linarith
+
+lemma scalar_trace_maximum {C S L D : ℝ} (hp : 0 < S)
+    (hr : -C*S^2 ≤ S*L) (hm : L-(C+1)*(D-S) ≤ 0) : S ≤ (C+1)*D := by
+  have he : S*(S-(C+1)*D) ≤ 0 := by nlinarith
+  by_contra h
+  have hmul := mul_pos hp (sub_pos.mpr (lt_of_not_ge h))
+  linarith
+
+lemma inverseTrace_at_localMax_bound [Nonempty (Fin n)]
+    {U : Set (V n)} (hU : IsOpen U)
+    {M G : V n → Matrix (Fin n) (Fin n) ℂ} (hM : ContDiffOn ℝ ∞ M U)
+    (hp : ∀ y ∈ U, (M y).PosDef)
+    (hclosed : ∀ y ∈ U, ∀ u v w : V n,
+      fderiv ℝ (fun q => Anticanonical.ComplexAtlas.fundamentalForm (M q) v w) y u +
+      fderiv ℝ (fun q => Anticanonical.ComplexAtlas.fundamentalForm (M q) w u) y v +
+      fderiv ℝ (fun q => Anticanonical.ComplexAtlas.fundamentalForm (M q) u v) y w = 0)
+    (hG : ContDiffOn ℝ ∞ G U) (hGp : ∀ y ∈ U, (G y).PosDef)
+    {f : V n → ℝ} (hf : ContDiffOn ℝ ∞ f U)
+    {z : V n} (hz : z ∈ U)
+    (he : M z = G z+PotentialKaehler.potentialMatrix f z)
+    (C : ℝ) (hbound : ∀ B : Matrix (Fin n) (Fin n) ℂ,
+      -C*(frameGram (G z) B)^2 ≤ frameResidual
+        (PotentialKaehler.potentialMatrix (fun y => Real.log (M y).det.re) z)
+        (G z) (curvatureDiagonal G z) B)
+    (hmax : IsLocalMax (fun y => Real.log (inverseTrace M G y)-(C+1)*f y) z) :
+    inverseTrace M G z ≤ (C+1)*n := by
+  obtain ⟨B,w,hB,hw,hM1⟩ := normalized_frame M z (hp z hz)
+  have hwU : B w ∈ U := hw ▸ hz
+  have hfs := hf.contDiffAt (hU.mem_nhds hz)
+  have hSpos : 0 < inverseTrace M G z := inverseTrace_pos (hp z hz) (hGp z hz)
+  have hSs := (inverseTrace_smooth hU hM hp hG).contDiffAt (hU.mem_nhds hz)
+  have hlog := hSs.log (ne_of_gt hSpos)
+  have hbℝ := (B.restrictScalars ℝ).contDiff.contDiffAt (x := w) (n := ∞)
+  have hfc : ContDiffAt ℝ ∞ (fun y => f (B y)) w := by
+    exact (hw ▸ hfs).comp w hbℝ
+  have hlc : ContDiffAt ℝ ∞ (fun y => Real.log (inverseTrace M G (B y))) w := by
+    exact (hw ▸ hlog).comp w hbℝ
+  have hmaxc : IsLocalMax (fun y => Real.log (inverseTrace M G (B y))-(C+1)*f (B y)) w := by
+    have ht : Tendsto B (𝓝 w) (𝓝 z) := hw ▸ B.continuous.continuousAt
+    change ∀ᶠ y in 𝓝 w, Real.log (inverseTrace M G (B y))-(C+1)*f (B y) ≤
+      Real.log (inverseTrace M G (B w))-(C+1)*f (B w)
+    rw [hw]
+    exact ht.eventually hmax
+  have ht := EllipticKernel.trace_potentialMatrix_nonpos_at_localMax
+    (hlc.sub (contDiffAt_const.mul hfc)) hmaxc (1 : Matrix (Fin n) (Fin n) ℂ) Matrix.PosDef.one
+  rw [inv_one,one_mul,potentialMatrix_trace_sub_smul hlc hfc] at ht
+  have hphi := trace_potential_in_frame B hB M G f w (hw ▸ hfs) (hw ▸ he) hM1
+  rw [hphi,hw] at ht
+  have hi := inverseTrace_log_inequality_frame hU B hB hM hp hclosed hG hGp hwU hM1
+  rw [hw] at hi
+  have hb := hbound (frameMatrix B)
+  have hgram := inverseTrace_eq_frameGram B hB M G w hM1
+  rw [hw] at hgram
+  rw [← hgram] at hb
+  exact scalar_trace_maximum hSpos (hb.trans hi) ht
+end KaehlerCalculus
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/CircleRadialCalculus.lean b/lean/OAI/Geometry/PrescribedPotential/CircleRadialCalculus.lean
new file mode 100644
index 0000000000000000000000000000000000000000..2979e2f25711e929cfeccd32b831188b361093d9
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/CircleRadialCalculus.lean
@@ -0,0 +1,199 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CircleSubmeanCalculus
+
+namespace OAI
+
+/-! Circle Radial Calculus. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Metric MeasureTheory
+open scoped ContDiff
+namespace PotentialABP
+
+private def v (θ : ℝ) : ℂ := circleMap 0 1 θ
+private def w (θ : ℝ) : ℂ := v θ * Complex.I
+
+private lemma dv (θ : ℝ) : HasDerivAt v (w θ) θ := hasDerivAt_circleMap 0 1 θ
+private lemma dw (θ : ℝ) : HasDerivAt w (-v θ) θ := by
+  convert (dv θ).mul_const Complex.I using 1 <;> first | rfl | simp [w, mul_assoc]
+
+private lemma cv : Continuous v := continuous_circleMap 0 1
+private lemma cw : Continuous w := cv.mul continuous_const
+
+ 
+def circleIntegralReal (f : ℂ → ℝ) (c : ℂ) (r : ℝ) : ℝ :=
+  ∫ θ in 0..2*Real.pi, f (c + r • v θ)
+def radialIntegral (f : ℂ → ℝ) (c : ℂ) (r : ℝ) : ℝ :=
+  ∫ θ in 0..2*Real.pi, fderiv ℝ f (c + r • v θ) (v θ)
+def secondRadialIntegral (f : ℂ → ℝ) (c : ℂ) (r : ℝ) : ℝ :=
+  ∫ θ in 0..2*Real.pi, fderiv ℝ (fderiv ℝ f) (c + r • v θ) (v θ) (v θ)
+def angularHessianIntegral (f : ℂ → ℝ) (c : ℂ) (r : ℝ) : ℝ :=
+  ∫ θ in 0..2*Real.pi, fderiv ℝ (fderiv ℝ f) (c + r • v θ) (w θ) (w θ)
+
+lemma circleIntegralReal_eq_circleAverage (f : ℂ → ℝ) (c : ℂ) (r : ℝ) :
+    circleIntegralReal f c r = 2*Real.pi * Real.circleAverage f c r := by
+  unfold circleIntegralReal Real.circleAverage
+  simp only [smul_eq_mul, ← mul_assoc, mul_inv_cancel₀ (by positivity : (2:ℝ)*Real.pi ≠ 0), one_mul]
+  congr 1
+  funext θ
+  congr 1
+  simp [v, circleMap, Complex.real_smul]
+
+lemma circleIntegralReal_hasDerivAt {f : ℂ → ℝ} (hf : ContDiff ℝ ∞ f) (c : ℂ) (r : ℝ) :
+    HasDerivAt (circleIntegralReal f c) (radialIntegral f c r) r := by
+  refine hasDerivAt_intervalIntegral_continuous
+    (F := fun s θ => f (c + s • v θ))
+    (F' := fun s θ => fderiv ℝ f (c + s • v θ) (v θ)) ?_ ?_ ?_ 0 (2*Real.pi) r
+  · exact hf.continuous.comp (continuous_const.add (continuous_fst.smul (cv.comp continuous_snd)))
+  · exact ((hf.continuous_fderiv (by simp)).comp
+      (continuous_const.add (continuous_fst.smul (cv.comp continuous_snd)))).clm_apply (cv.comp continuous_snd)
+  · intro s θ
+    simpa only [one_smul, Function.comp_def, id_eq] using
+      (hf.differentiable (by simp) _).hasFDerivAt.comp_hasDerivAt s
+        (((hasDerivAt_id s).smul_const (v θ)).const_add c)
+
+lemma radialIntegral_hasDerivAt {f : ℂ → ℝ} (hf : ContDiff ℝ ∞ f) (c : ℂ) (r : ℝ) :
+    HasDerivAt (radialIntegral f c) (secondRadialIntegral f c r) r := by
+  have hdf := hf.fderiv_right (m := ∞) (by simp)
+  refine hasDerivAt_intervalIntegral_continuous
+    (F := fun s θ => fderiv ℝ f (c + s • v θ) (v θ))
+    (F' := fun s θ => fderiv ℝ (fderiv ℝ f) (c + s • v θ) (v θ) (v θ))
+    ?_ ?_ ?_ 0 (2*Real.pi) r
+  · exact (hdf.continuous.comp
+      (continuous_const.add (continuous_fst.smul (cv.comp continuous_snd)))).clm_apply (cv.comp continuous_snd)
+  · exact (((hdf.continuous_fderiv (by simp)).comp
+      (continuous_const.add (continuous_fst.smul (cv.comp continuous_snd)))).clm_apply
+      (cv.comp continuous_snd)).clm_apply (cv.comp continuous_snd)
+  · intro s θ
+    have h := ((hdf.differentiable (by simp) _).hasFDerivAt.comp_hasDerivAt s
+      (((hasDerivAt_id s).smul_const (v θ)).const_add c)).clm_apply (hasDerivAt_const s (v θ))
+    simpa only [one_smul, Function.comp_def, id_eq, map_zero, add_zero] using h
+
+lemma radialIntegral_eq_radius_angular {f : ℂ → ℝ} (hf : ContDiff ℝ ∞ f) (c : ℂ) (r : ℝ) :
+    radialIntegral f c r = r * angularHessianIntegral f c r := by
+  have hdf := hf.fderiv_right (m := ∞) (by simp)
+  let T : ℝ → ℝ := fun θ => fderiv ℝ f (c + r • v θ) (w θ)
+  let T' : ℝ → ℝ := fun θ => r * fderiv ℝ (fderiv ℝ f) (c + r • v θ) (w θ) (w θ)
+    - fderiv ℝ f (c + r • v θ) (v θ)
+  have ht (θ : ℝ) : HasDerivAt T (T' θ) θ := by
+    have h := ((hdf.differentiable (by simp) _).hasFDerivAt.comp_hasDerivAt θ
+      (((dv θ).const_smul r).const_add c)).clm_apply (dw θ)
+    simpa only [T, T', Function.comp_def, map_smul, map_neg, smul_eq_mul, sub_eq_add_neg, Pi.smul_apply, smul_apply] using h
+  have hct : Continuous T' := by
+    apply Continuous.sub
+    · exact continuous_const.mul (((hdf.continuous_fderiv (by simp)).comp
+        (continuous_const.add (cv.const_smul r))).clm_apply cw |>.clm_apply cw)
+    · exact (hdf.continuous.comp (continuous_const.add (cv.const_smul r))).clm_apply cv
+  have he := intervalIntegral.integral_eq_sub_of_hasDerivAt
+    (fun θ _ => ht θ) (hct.intervalIntegrable 0 (2*Real.pi))
+  have hp : T (2*Real.pi) = T 0 := by
+    have hv : v (2*Real.pi) = v 0 := by
+      simpa only [v, zero_add] using periodic_circleMap 0 1 0
+    simp only [T, w, hv]
+  rw [hp, sub_self] at he
+  have hc1 : Continuous (fun θ => fderiv ℝ (fderiv ℝ f) (c + r • v θ) (w θ) (w θ)) :=
+    (((hdf.continuous_fderiv (by simp)).comp
+      (continuous_const.add (cv.const_smul r))).clm_apply cw).clm_apply cw
+  have hc2 : Continuous (fun θ => fderiv ℝ f (c + r • v θ) (v θ)) :=
+    (hdf.continuous.comp (continuous_const.add (cv.const_smul r))).clm_apply cv
+  dsimp only [T'] at he
+  have hi1 : IntervalIntegrable (fun θ => r * fderiv ℝ (fderiv ℝ f)
+      (c + r • v θ) (w θ) (w θ)) volume 0 (2*Real.pi) :=
+    (continuous_const.mul hc1).intervalIntegrable _ _
+  rw [intervalIntegral.integral_sub hi1 (hc2.intervalIntegrable _ _),
+    intervalIntegral.integral_const_mul] at he
+  exact (sub_eq_zero.mp he).symm
+
+lemma radialIntegral_continuous {f : ℂ → ℝ} (hf : ContDiff ℝ ∞ f) (c : ℂ) :
+    Continuous (radialIntegral f c) :=
+  continuous_iff_continuousAt.mpr (fun r => (radialIntegral_hasDerivAt hf c r).continuousAt)
+lemma circleIntegralReal_continuous {f : ℂ → ℝ} (hf : ContDiff ℝ ∞ f) (c : ℂ) :
+    Continuous (circleIntegralReal f c) :=
+  continuous_iff_continuousAt.mpr (fun r => (circleIntegralReal_hasDerivAt hf c r).continuousAt)
+
+private lemma hessian_circle_pair (B : ℂ →L[ℝ] ℂ →L[ℝ] ℝ) (θ : ℝ) :
+    B (v θ) (v θ) + B (w θ) (w θ) = B 1 1 + B Complex.I Complex.I := by
+  have hv : v θ = Real.cos θ • (1:ℂ) + Real.sin θ • Complex.I := by
+    apply Complex.ext <;> simp [v, circleMap_zero_re, circleMap_zero_im, -Complex.ofReal_cos, -Complex.ofReal_sin]
+  have hw : w θ = (-Real.sin θ) • (1:ℂ) + Real.cos θ • Complex.I := by
+    apply Complex.ext <;> simp [w, v, circleMap_zero_re, circleMap_zero_im, Complex.mul_re, Complex.mul_im, -Complex.ofReal_cos, -Complex.ofReal_sin]
+  rw [hv, hw]
+  calc
+    _ = (Real.sin θ ^ 2 + Real.cos θ ^ 2) * (B 1 1 + B Complex.I Complex.I) := by
+      simp only [map_add, map_smul, add_apply,
+        smul_apply, smul_eq_mul]
+      ring
+    _ = _ := by rw [Real.sin_sq_add_cos_sq, one_mul]
+
+ 
+
+theorem smooth_circle_submean {f : ℂ → ℝ} (hf : ContDiff ℝ ∞ f)
+    (c : ℂ) {R : ℝ} (hR : 0 ≤ R)
+    (htrace : ∀ z ∈ closedBall c R,
+      0 ≤ fderiv ℝ (fderiv ℝ f) z 1 1 +
+        fderiv ℝ (fderiv ℝ f) z Complex.I Complex.I) :
+    f c ≤ Real.circleAverage f c R := by
+  have hdf := hf.fderiv_right (m := ∞) (by simp)
+  have hpos (r : ℝ) (hr : r ∈ Icc 0 R) :
+      0 ≤ secondRadialIntegral f c r + angularHessianIntegral f c r := by
+    have hc1 : Continuous (fun θ => fderiv ℝ (fderiv ℝ f) (c + r • v θ) (v θ) (v θ)) :=
+      (((hdf.continuous_fderiv (by simp)).comp
+        (continuous_const.add (cv.const_smul r))).clm_apply cv).clm_apply cv
+    have hc2 : Continuous (fun θ => fderiv ℝ (fderiv ℝ f) (c + r • v θ) (w θ) (w θ)) :=
+      (((hdf.continuous_fderiv (by simp)).comp
+        (continuous_const.add (cv.const_smul r))).clm_apply cw).clm_apply cw
+    dsimp only [secondRadialIntegral, angularHessianIntegral]
+    rw [← intervalIntegral.integral_add (hc1.intervalIntegrable _ _) (hc2.intervalIntegrable _ _)]
+    apply intervalIntegral.integral_nonneg_of_forall (by positivity)
+    intro θ
+    rw [hessian_circle_pair]
+    apply htrace
+    simp only [mem_closedBall, dist_eq_norm, add_sub_cancel_left, norm_smul,
+      v, norm_circleMap_zero, abs_one, mul_one, Real.norm_eq_abs, abs_of_nonneg hr.1]
+    exact hr.2
+  let J : ℝ → ℝ := fun r => r * radialIntegral f c r
+  have hdJ (r : ℝ) : HasDerivAt J
+      (radialIntegral f c r + r * secondRadialIntegral f c r) r := by
+    have hh := (hasDerivAt_id r).mul (radialIntegral_hasDerivAt hf c r)
+    change HasDerivAt (fun s => s * radialIntegral f c s) _ r at hh
+    simpa only [J, id_eq, one_mul] using hh
+  have hmJ : MonotoneOn J (Icc 0 R) := by
+    apply monotoneOn_of_hasDerivWithinAt_nonneg (convex_Icc 0 R)
+      ((continuous_id.mul (radialIntegral_continuous hf c)).continuousOn)
+      (fun r _ => (hdJ r).hasDerivWithinAt)
+    intro r hr
+    have hr' := interior_subset hr
+    rw [radialIntegral_eq_radius_angular hf, ← mul_add]
+    exact mul_nonneg hr'.1 (by simpa only [add_comm] using hpos r hr')
+  have hmF : MonotoneOn (circleIntegralReal f c) (Icc 0 R) := by
+    apply monotoneOn_of_hasDerivWithinAt_nonneg (convex_Icc 0 R)
+      (circleIntegralReal_continuous hf c).continuousOn
+      (fun r _ => (circleIntegralReal_hasDerivAt hf c r).hasDerivWithinAt)
+    intro r hr
+    have hr' : r ∈ Ioo 0 R := by simpa only [interior_Icc] using hr
+    have hj := hmJ ⟨le_rfl,hR⟩ ⟨hr'.1.le,hr'.2.le⟩ hr'.1.le
+    simp only [J, zero_mul] at hj
+    nlinarith [hr'.1]
+  have he := hmF ⟨le_rfl,hR⟩ ⟨hR,le_rfl⟩ hR
+  have hF0 : circleIntegralReal f c 0 = 2*Real.pi*f c := by
+    simp [circleIntegralReal]
+  have hFR : Real.circleAverage f c R = (2*Real.pi)⁻¹ * circleIntegralReal f c R := by
+    unfold Real.circleAverage circleIntegralReal
+    congr 2
+    funext θ
+    congr 1
+    simp [v, circleMap, Complex.real_smul]
+  rw [hF0] at he
+  rw [hFR]
+  exact (le_inv_mul_iff₀ (by positivity : 0 < 2*Real.pi)).mpr (by simpa only [mul_comm] using he)
+
+end PotentialABP
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/CircleSubmeanCalculus.lean b/lean/OAI/Geometry/PrescribedPotential/CircleSubmeanCalculus.lean
new file mode 100644
index 0000000000000000000000000000000000000000..954022613a8df086b46b91edb7f0e29ca5fba652
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/CircleSubmeanCalculus.lean
@@ -0,0 +1,41 @@
+import Mathlib
+
+namespace OAI
+
+/-! Circle Submean Calculus. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Metric MeasureTheory
+namespace PotentialABP
+
+lemma hasDerivAt_intervalIntegral_continuous
+    {F F' : ℝ → ℝ → ℝ}
+    (hF : Continuous (Function.uncurry F))
+    (hF' : Continuous (Function.uncurry F'))
+    (hderiv : ∀ r θ, HasDerivAt (fun s => F s θ) (F' r θ) r)
+    (a b r : ℝ) :
+    HasDerivAt (fun s => ∫ θ in a..b, F s θ) (∫ θ in a..b, F' r θ) r := by
+  obtain ⟨C, hC⟩ := ((isCompact_closedBall r 1).prod isCompact_uIcc).bddAbove_image
+    hF'.norm.continuousOn
+  apply (intervalIntegral.hasDerivAt_integral_of_dominated_loc_of_deriv_le
+    (bound := fun _ => C) (s := ball r 1) (ball_mem_nhds r (by norm_num)) ?_ ?_ ?_ ?_ ?_ ?_).2
+  · filter_upwards [] with s
+    exact (hF.comp (continuous_const.prodMk continuous_id)).aestronglyMeasurable
+  · exact (hF.comp (continuous_const.prodMk continuous_id)).intervalIntegrable a b
+  · exact (hF'.comp (continuous_const.prodMk continuous_id)).aestronglyMeasurable
+  · filter_upwards [] with θ hθ s hs
+    exact hC ⟨(s,θ), ⟨ball_subset_closedBall hs, uIoc_subset_uIcc hθ⟩, rfl⟩
+  · exact intervalIntegrable_const
+  · filter_upwards [] with θ _ s _
+    exact hderiv s θ
+
+end PotentialABP
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/ClosedRangeSmooth.lean b/lean/OAI/Geometry/PrescribedPotential/ClosedRangeSmooth.lean
new file mode 100644
index 0000000000000000000000000000000000000000..aca5911d4598638cdb38077ce58282abca73558f
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/ClosedRangeSmooth.lean
@@ -0,0 +1,103 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.AugmentedVolume
+import OAI.Geometry.PrescribedPotential.RealNonlinearDerivative
+
+namespace OAI
+
+/-! Closed Range Smooth. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology
+open scoped ContDiff Classical
+namespace ClosedRangeSmooth
+universe u
+variable {E F G : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  [NormedAddCommGroup F] [NormedSpace ℝ F]
+  [NormedAddCommGroup G] [NormedSpace ℝ G]
+
+lemma isometry_derivative_iff (e : F →ₗᵢ[ℝ] G) (f : E → F) (L : E →L[ℝ] F) (x : E) :
+    HasFDerivAt f L x ↔ HasFDerivAt (fun y => e (f y)) (e.toContinuousLinearMap ∘L L) x := by
+  simp only [hasFDerivAt_iff_isLittleO]
+  conv_lhs => rw [← Asymptotics.isLittleO_norm_left]
+  conv_rhs => rw [← Asymptotics.isLittleO_norm_left]
+  simp only [ContinuousLinearMap.comp_apply, LinearIsometry.coe_toContinuousLinearMap,
+    ← map_sub, e.norm_map]
+
+lemma lift_derivative [CompleteSpace F] (e : F →ₗᵢ[ℝ] G) (f : E → F)
+    {x : E} {J : E →L[ℝ] G} (hJ : HasFDerivAt (fun y => e (f y)) J x) :
+    ∃ L : E →L[ℝ] F, HasFDerivAt f L x ∧ e.toContinuousLinearMap ∘L L = J := by
+  let S := e.range
+  have hS : IsClosed (S : Set G) := e.isometry.isClosedEmbedding.isClosed_range
+  let r : E → S := fun y => e.equivRange (f y)
+  have hmem : ∀ v, J v ∈ S :=
+    ClosedDerivative.derivative_mem S hS r hJ
+  let L := e.equivRange.symm.toContinuousLinearEquiv.toContinuousLinearMap ∘L J.codRestrict S hmem
+  have hcomp : e.toContinuousLinearMap ∘L L = J := by
+    ext v
+    change e (e.equivRange.symm ⟨J v,hmem v⟩) = J v
+    exact congrArg Subtype.val (e.equivRange.apply_symm_apply ⟨J v,hmem v⟩)
+  refine ⟨L,(isometry_derivative_iff e f L x).mpr ?_,hcomp⟩
+  rwa [hcomp]
+
+ 
+
+lemma contDiff_of_isometry (n : ℕ) [CompleteSpace F] (e : F →ₗᵢ[ℝ] G) (f : E → F)
+    (hf : ContDiff ℝ n (fun y => e (f y))) : ContDiff ℝ n f := by
+  induction n generalizing F G with
+  | zero =>
+    rw [Nat.cast_zero, contDiff_zero] at hf ⊢
+    exact e.isEmbedding.continuous_iff.mpr hf
+  | succ n ih =>
+    rw [Nat.cast_succ, contDiff_succ_iff_hasFDerivAt] at hf ⊢
+    obtain ⟨J,hJ,hd⟩ := hf
+    choose L hL he using fun x => lift_derivative e f (hd x)
+    refine ⟨L,?_,hL⟩
+    let ep : (E →L[ℝ] F) →ₗᵢ[ℝ] (E →L[ℝ] G) := e.postcomp
+    apply ih ep L
+    convert hJ using 1
+    funext x
+    exact he x
+
+lemma contDiff_infty_of_isometry [CompleteSpace F] (e : F →ₗᵢ[ℝ] G) (f : E → F)
+    (hf : ContDiff ℝ ∞ (fun y => e (f y))) : ContDiff ℝ ∞ f := by
+  apply contDiff_infty.mpr
+  intro n
+  exact contDiff_of_isometry n e f (contDiff_infty.mp hf n)
+end ClosedRangeSmooth
+
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+local instance closedRangeNormedGroup (s : ℝ) : NormedAddCommGroup (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedAddCommGroup
+local instance closedRangeNormedSpace (s : ℝ) : NormedSpace ℝ (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedSpace
+local instance closedRangeTopologicalGroup (s : ℝ) : IsTopologicalAddGroup (D.localizers.RealSobolev s) :=
+  Submodule.isTopologicalAddGroup _
+local instance closedRangeContinuousSMul (s : ℝ) : ContinuousSMul ℝ (D.localizers.RealSobolev s) :=
+  SMulMemClass.continuousSMul _
+
+lemma realVolume_contDiff (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) :
+    ContDiff ℝ ∞ (D.realVolume k hk) :=
+  ClosedRangeSmooth.contDiff_infty_of_isometry
+    (D.localizers.realCompletion (k : ℝ)).subtypeₗᵢ _ (D.realVolume_ambient_contDiff k hk)
+
+lemma augmentedVolume_contDiff (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) (x₀ : X) :
+    ContDiff ℝ ∞ (D.augmentedVolume k hk x₀) := by
+  apply ContDiff.prodMk
+  · exact (contDiff_snd.neg.exp).smul ((D.realVolume_contDiff k hk).comp contDiff_fst)
+  · exact (D.realEvaluation ((k : ℝ)+2) x₀).contDiff.comp contDiff_fst
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/CommutatorSupport.lean b/lean/OAI/Geometry/PrescribedPotential/CommutatorSupport.lean
new file mode 100644
index 0000000000000000000000000000000000000000..4e465fbe53a72f65e017c19a804dc388069195df
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/CommutatorSupport.lean
@@ -0,0 +1,142 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.FrozenEllipticJets
+import OAI.Geometry.PrescribedPotential.NonlinearHessianCommutator
+import OAI.Geometry.PrescribedPotential.NonlinearRemainderStrong
+import OAI.Geometry.PrescribedPotential.RealSobolev
+import OAI.Geometry.PrescribedPotential.RegularityCoordinates
+
+namespace OAI
+
+/-! Commutator Support. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Matrix LineDeriv
+open scoped ContDiff SchwartzMap Classical Matrix.Norms.Elementwise
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+
+lemma hessianLocalizedError_zero {κ : 𝓢(EC d, ℂ)} (v : EC d) (f : 𝓢(EC d, ℂ))
+    {y : EC d} (hy : y ∉ tsupport (κ : EC d → ℂ)) (i j : Fin d) :
+    hessianLocalizedError κ v i j f y = 0 := by
+  have hz := notMem_tsupport_iff_eventuallyEq.mp hy
+  have he : (SchwartzMap.smulLeftCLM ℂ κ (∂_{v} f) : EC d → ℂ) =ᶠ[𝓝 y]
+      (0 : 𝓢(EC d, ℂ)) := by
+    filter_upwards [hz] with z hzz
+    rw [SchwartzMap.smulLeftCLM_apply_apply κ.hasTemperateGrowth]
+    change κ z * _ = 0
+    simp only [hzz,Pi.zero_apply,zero_mul]
+  have hh := congrArg (fun u : 𝓢(EC d, ℂ) => u y) (hessianLocalizedError_identity κ f v i j)
+  rw [hessianEntrySchwartz_congr he,map_zero,_root_.zero_apply] at hh
+  simp only [_root_.add_apply] at hh
+  rw [SchwartzMap.smulLeftCLM_apply_apply κ.hasTemperateGrowth] at hh
+  change 0 = κ y * _ + hessianLocalizedError κ v i j f y at hh
+  simp only [hz.eq_of_nhds,Pi.zero_apply,zero_mul,zero_add] at hh
+  exact hh.symm
+
+omit [T2Space X] [CompactSpace X] in
+lemma source_hessian_zero_of_local {φ : SmoothRealFunction A} {x : X}
+    (he : φ.value =ᶠ[𝓝 x] (fun _ => 0)) (q : Fin A.count)
+    (hx : x ∈ (A.chart q).source) : φ.hessian q (A.chart q x) = 0 := by
+  have ht := (A.chart q).mapsTo hx
+  have hc : Tendsto (A.chart q).symm (𝓝 (A.chart q x)) (𝓝 x) := by
+    simpa only [(A.chart q).left_inv hx] using
+      ((A.chart q).symm.continuousAt ht).tendsto
+  have hh : φ.localExpression q =ᶠ[𝓝 (A.chart q x)] (fun _ => 0) := he.comp_tendsto hc
+  have h2 := (hh.fderiv (𝕜 := ℝ)).fderiv_eq (𝕜 := ℝ)
+  change PotentialKaehler.hermitianPartMatrix (fderiv ℝ (fderiv ℝ (φ.localExpression q))
+    (A.chart q x)) = 0
+  rw [h2]
+  ext i j
+  simp [PotentialKaehler.hermitianPartMatrix_apply,PotentialKaehler.hermitianPart]
+  rfl
+
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+def derivativeSupport (p : ι) : Set X :=
+  (A.euclideanChart (D.patch p).index).symm '' tsupport (D.cutoff p : EC d → ℂ)
+
+lemma localizedDerivative_tsupport (p : ι) (v : EC d) (f : Smooth A) :
+    tsupport (D.localizedDerivative p v f : X → ℂ) ⊆ D.derivativeSupport p := by
+  rw [D.localizedDerivative_regularity]
+  exact (globalize_tsupport _ _ _).trans (image_mono (D.cutoff p).realPart_support)
+
+omit [T2Space X] [CompactSpace X] in
+lemma derivativeSupport_source {p : ι} {x : X} (hx : x ∈ D.derivativeSupport p) :
+    x ∈ (A.euclideanChart (D.patch p).index).source ∧
+      A.euclideanChart (D.patch p).index x ∈ tsupport (D.cutoff p : EC d → ℂ) := by
+  obtain ⟨y,hy,rfl⟩ := hx
+  have ht := (D.cutoff p).support_sub hy
+  exact ⟨(A.euclideanChart (D.patch p).index).symm.mapsTo ht,
+    by rwa [(A.euclideanChart (D.patch p).index).right_inv ht]⟩
+
+omit [T2Space X] [CompactSpace X] in
+lemma outsideDerivativeSupport_coord {p : ι} {x : X} (hx : x ∉ D.derivativeSupport p)
+    (hs : x ∈ (A.euclideanChart (D.patch p).index).source) :
+    A.euclideanChart (D.patch p).index x ∉ tsupport (D.cutoff p : EC d → ℂ) := by
+  intro hh
+  exact hx ⟨_,hh,(A.euclideanChart (D.patch p).index).left_inv hs⟩
+
+lemma localizedDerivative_zero {p : ι} {x : X} (hx : x ∉ D.derivativeSupport p)
+    (v : EC d) (f : Smooth A) : D.localizedDerivative p v f x = 0 :=
+  image_eq_zero_of_notMem_tsupport (fun hh => hx (D.localizedDerivative_tsupport p v f hh))
+
+lemma regularityError_zero {p : ι} {x : X} (hx : x ∉ D.derivativeSupport p)
+    (v : EC d) (f : Smooth A) (i j : Fin d) : D.regularityError p v i j f x = 0 := by
+  change globalize (D.patch p).index (D.regularityCutoff p) _ x = 0
+  rw [globalize_apply]
+  split_ifs with hs
+  · have hh := D.outsideDerivativeSupport_coord hx hs
+    have hz := hessianLocalizedError_zero v
+      (localize A (D.patch p).index (cutoffGlobal (D.patch p).index (D.regularityOuter p))
+        (cutoffGlobal_support _ _) f)
+      (fun hh' => hh ((D.cutoff p).realPart_support hh')) i j
+    exact mul_eq_zero_of_right _ hz
+  · rfl
+
+lemma nonlinearRemainder_zero {p : ι} {x : X} (hx : x ∉ D.derivativeSupport p)
+    (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) (v : EC d) (f : Smooth A) :
+    D.localizers.strong (k : ℝ)
+      (D.nonlinearRemainder k hk p v (D.localizers.embed ((k : ℝ)+2) f)) x = 0 := by
+  rw [D.nonlinearRemainder_strong_embed]
+  have hm : (fun (i j : Fin d) => D.regularityError p v i j f x -
+      D.localizedDerivative p v (D.regularityMetric p i j) x) = 0 := by
+    ext i j
+    rw [D.regularityError_zero hx,D.localizedDerivative_zero hx,sub_self]
+    rfl
+  let M : Matrix (Fin d) (Fin d) ℂ := fun i j =>
+    D.regularityMetric p i j x + D.regularityHessian p i j f x
+  change D.regularityInverseDet p x * NonlinearHessian.detDifferential M _ -
+    D.localizedDerivative p v (D.regularityInverseDet p) x * M.det = 0
+  rw [hm]
+  change D.regularityInverseDet p x * NonlinearHessian.detDifferential M (0 : Matrix (Fin d) (Fin d) ℂ) -
+    D.localizedDerivative p v (D.regularityInverseDet p) x * M.det = 0
+  rw [(NonlinearHessian.detDifferential M).map_zero,D.localizedDerivative_zero hx,
+    mul_zero,zero_mul,sub_self]
+
+lemma localizedDerivative_source_hessian_zero {p : ι} {x : X}
+    (hx : x ∉ D.derivativeSupport p) (v : EC d) (φ : SmoothRealFunction A)
+    (q : Fin A.count) (hq : x ∈ (A.chart q).source) :
+    ((D.realLocalizedDerivative p v (RealSmooth.ofReal φ)).source).hessian q (A.chart q x) = 0 := by
+  apply source_hessian_zero_of_local _ q hq
+  have hn : x ∉ tsupport (D.localizedDerivative p v (Smooth.ofReal φ) : X → ℂ) :=
+    fun hh => hx (D.localizedDerivative_tsupport p v (Smooth.ofReal φ) hh)
+  have he := notMem_tsupport_iff_eventuallyEq.mp hn
+  filter_upwards [he] with y hy
+  change (D.localizedDerivative p v (Smooth.ofReal φ) y).re = 0
+  rw [hy]
+  rfl
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/CompactAlternative.lean b/lean/OAI/Geometry/PrescribedPotential/CompactAlternative.lean
new file mode 100644
index 0000000000000000000000000000000000000000..38ca07817450bca69bc11637b291d97cb1d317f2
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/CompactAlternative.lean
@@ -0,0 +1,34 @@
+import Mathlib
+
+namespace OAI
+
+/-! Compact Alternative. -/
+
+section
+
+ 
+noncomputable section
+namespace GlobalElliptic
+variable {H : Type*} [NormedAddCommGroup H] [NormedSpace ℝ H] [CompleteSpace H]
+
+lemma compact_one_sub_surjective (T : H →L[ℝ] H) (hT : IsCompactOperator T)
+    (hker : ∀ u : H, T u = u → u = 0) :
+    Function.Surjective (1 - T : H →L[ℝ] H) := by
+  have hn : ¬ Module.End.HasEigenvalue (T : Module.End ℝ H) 1 := by
+    intro hev
+    obtain ⟨u, hu⟩ := hev.exists_hasEigenvector
+    apply hu.2
+    apply hker u
+    have heq : T u = (1 : ℝ) • u := Module.End.HasEigenvector.apply_eq_smul hu
+    exact heq.trans (one_smul ℝ u)
+  have hr := (hT.hasEigenvalue_or_mem_resolventSet (by norm_num : (1 : ℝ) ≠ 0)).resolve_left hn
+  rw [spectrum.mem_resolventSet_iff] at hr
+  have hu : IsUnit (1 - T : H →L[ℝ] H) := by simpa only [map_one] using hr
+  exact (ContinuousLinearMap.isUnit_iff_bijective.mp hu).2
+
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/CompactCoefficientBootstrap.lean b/lean/OAI/Geometry/PrescribedPotential/CompactCoefficientBootstrap.lean
new file mode 100644
index 0000000000000000000000000000000000000000..6fd2187e4bd697e5099cab65f158595adc870f95
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/CompactCoefficientBootstrap.lean
@@ -0,0 +1,70 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.QuasilinearFamilyBootstrap
+
+namespace OAI
+
+/-! Compact Coefficient Bootstrap. -/
+
+section
+
+noncomputable section
+open Set Filter Topology Finset Module
+open scoped ContDiff
+namespace HigherJet
+variable {E F G : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [NormedAddCommGroup F] [InnerProductSpace ℝ F]
+  [NormedAddCommGroup G] [NormedSpace ℝ G]
+  [FiniteDimensional ℝ E] [FiniteDimensional ℝ F]
+  {ι α : Type*} [Fintype ι]
+local instance compactHessianNorm : NormedAddCommGroup (E →L[ℝ] E →L[ℝ] F) := inferInstance
+local instance compactHessianSpace : NormedSpace ℝ (E →L[ℝ] E →L[ℝ] F) := inferInstance
+lemma quasilinear_compact_coefficient_bootstrap (B : G →L[ℝ] (E →L[ℝ] E →L[ℝ] F) →L[ℝ] F)
+    {U : Set E} (hU : IsOpen U) (e : OrthonormalBasis ι ℝ E)
+    (u r : α → E → F) (a : F → G) (b : F × (E →L[ℝ] F) → F)
+    (τ : α → E → E ≃L[ℝ] E)
+    (hu : ∀ s, ContDiffOn ℝ ∞ (u s) U) (hr : ∀ s, ContDiffOn ℝ ∞ (r s) U)
+    (ha : ∀ s y, y ∈ U → ContDiffAt ℝ ∞ a (u s y))
+    (hb : ∀ s y, y ∈ U → ContDiffAt ℝ ∞ b (firstJet (u s) y))
+    (heq : ∀ s y, y ∈ U → B (a (u s y)) (hessian (u s) y) = r s y+b (firstJet (u s) y))
+    (hframe : ∀ s x, x ∈ U → ∀ H : E →L[ℝ] E →L[ℝ] F,
+      B (a (u s x)) H = ∑ i, H (τ s x (e i)) (τ s x (e i)))
+    (hframes : ∀ K, IsCompact K → K ⊆ U → ∃ C : ℝ, 0 ≤ C ∧ ∀ s x, x ∈ K →
+      ‖(τ s x).toContinuousLinearMap‖ ≤ C ∧ ‖(τ s x).symm.toContinuousLinearMap‖ ≤ C)
+    (hcompact : ∀ K, IsCompact K → K ⊆ U → ∃ P : Set (F × (E →L[ℝ] F)), IsCompact P ∧
+      (∀ s x, x ∈ K → firstJet (u s) x ∈ P) ∧
+      (∀ p ∈ P, ContDiffAt ℝ ∞ a p.1 ∧ ContDiffAt ℝ ∞ b p))
+    (hrbounds : ∀ m K, IsCompact K → K ⊆ U → ∃ C : ℝ, ∀ j, j ≤ m → ∀ s x, x ∈ K →
+      ‖iteratedFDeriv ℝ j (r s) x‖ ≤ C) :
+    ∀ m, 1 ≤ m → ∀ K, IsCompact K → K ⊆ U →
+      ∃ C : ℝ, 0 ≤ C ∧ ∀ s x, x ∈ K → ‖iteratedFDeriv ℝ m (u s) x‖ ≤ C := by
+  apply quasilinear_family_bootstrap B hU e u r a b τ hu hr ha hb heq hframe hframes
+  intro m K hK hKU
+  obtain ⟨P,hP,hmem,hsm⟩ := hcompact K hK hKU
+  obtain ⟨A,hA,ha'⟩ := compact_coefficient_jets (hP.image continuous_fst)
+    (fun y hy => by obtain ⟨p,hp,rfl⟩ := hy; exact (hsm p hp).1) m
+  obtain ⟨Q,hQ,hb'⟩ := compact_coefficient_jets hP (fun y hy => (hsm y hy).2) m
+  obtain ⟨J,hJ⟩ := hP.exists_bound_of_continuousOn (continuous_snd.continuousOn : ContinuousOn (fun p : F × (E →L[ℝ] F) => p.2) P)
+  obtain ⟨R,hR⟩ := hrbounds m K hK hKU
+  let T : ℝ := ∑ j ∈ range (m+1), (Fintype.card (OrderedFinpartition j) : ℝ)
+  have hT : 0 ≤ T := Finset.sum_nonneg (fun _ _ => Nat.cast_nonneg _)
+  let C := 1+‖B‖+A+Q+max J 0+max R 0+T
+  have hC : 1 ≤ C := by dsimp [C]; linarith [norm_nonneg B,le_max_right J 0,le_max_right R 0]
+  have hcB : ‖B‖ ≤ C := by dsimp [C]; linarith [le_max_right J 0,le_max_right R 0]
+  have hcA : A ≤ C := by dsimp [C]; linarith [norm_nonneg B,le_max_right J 0,le_max_right R 0]
+  have hcQ : Q ≤ C := by dsimp [C]; linarith [norm_nonneg B,le_max_right J 0,le_max_right R 0]
+  have hcJ : J ≤ C := by dsimp [C]; linarith [norm_nonneg B,le_max_left J 0,le_max_right R 0]
+  have hcR : R ≤ C := by dsimp [C]; linarith [norm_nonneg B,le_max_right J 0,le_max_left R 0]
+  have hcT : T ≤ C := by dsimp [C]; linarith [norm_nonneg B,le_max_right J 0,le_max_right R 0]
+  refine ⟨C,hC,hcB,fun j hj => ?_,fun s x hx => ?_⟩
+  · exact (Finset.single_le_sum (fun i _ => Nat.cast_nonneg (α := ℝ) (Fintype.card (OrderedFinpartition i)))
+      (mem_range.mpr (by omega))).trans hcT
+  · have hp := hmem s x hx
+    refine ⟨?_,fun j hj => (ha' j hj _ ⟨firstJet (u s) x,hp,rfl⟩).trans hcA,
+      fun j hj => (hb' j hj _ hp).trans hcQ,fun j hj => (hR j hj s x hx).trans hcR⟩
+    simpa only [norm_iteratedFDeriv_one,firstJet] using (hJ _ hp).trans hcJ
+end HigherJet
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/CompactHomogeneousBounds.lean b/lean/OAI/Geometry/PrescribedPotential/CompactHomogeneousBounds.lean
new file mode 100644
index 0000000000000000000000000000000000000000..f4101f6192cdcfb2a5eddd74e78b1830391696dc
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/CompactHomogeneousBounds.lean
@@ -0,0 +1,83 @@
+import Mathlib
+
+namespace OAI
+
+/-! Compact Homogeneous Bounds. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Metric
+namespace KaehlerCalculus
+variable {X E : Type*} [TopologicalSpace X]
+  [NormedAddCommGroup E] [NormedSpace ℝ E] [ProperSpace E]
+
+lemma compact_homogeneous_bound {K : Set X} (hK : IsCompact K)
+    (p : ℕ) (hp : 0 < p) (F : X → E → ℝ)
+    (hF : ContinuousOn (fun q : X × E => F q.1 q.2) (K ×ˢ closedBall 0 1))
+    (hhom : ∀ x ∈ K, ∀ r : ℝ, ∀ v, F x (r • v) = r^p*F x v) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ x ∈ K, ∀ v, |F x v| ≤ C*‖v‖^p := by
+  obtain ⟨C,hC⟩ := (hK.prod (isCompact_closedBall (0:E) 1)).exists_bound_of_continuousOn hF
+  refine ⟨max C 0,le_max_right _ _,?_⟩
+  intro x hx v
+  by_cases hv : v = 0
+  · subst v
+    have hh := hhom x hx 0 (0:E)
+    simp only [zero_smul,zero_pow (Nat.ne_of_gt hp),zero_mul] at hh
+    simp [hh,zero_pow (Nat.ne_of_gt hp)]
+  have hn : 0 < ‖v‖ := norm_pos_iff.mpr hv
+  have hw : ‖(‖v‖⁻¹ : ℝ) • v‖ = 1 := by
+    rw [norm_smul,Real.norm_eq_abs,abs_of_pos (inv_pos.mpr hn),inv_mul_cancel₀ hn.ne']
+  have hb := hC (x,(‖v‖⁻¹ : ℝ) • v) ⟨hx,by simpa [mem_closedBall,dist_zero_right] using hw.le⟩
+  change ‖F x ((‖v‖⁻¹:ℝ) • v)‖ ≤ C at hb
+  rw [hhom x hx,Real.norm_eq_abs,abs_mul,abs_of_nonneg (pow_nonneg (le_of_lt (inv_pos.mpr hn)) _)] at hb
+  have hh := mul_le_mul_of_nonneg_left (hb.trans (le_max_left C 0)) (pow_nonneg (norm_nonneg v) p)
+  have he : ‖v‖^p*((‖v‖⁻¹)^p*|F x v|) = |F x v| := by
+    rw [← mul_assoc,← mul_pow,mul_inv_cancel₀ hn.ne',one_pow,one_mul]
+  rw [he] at hh
+  simpa only [mul_comm] using hh
+
+lemma compact_homogeneous_positive {K : Set X} (hK : IsCompact K)
+    (F : X → E → ℝ)
+    (hF : ContinuousOn (fun q : X × E => F q.1 q.2) (K ×ˢ sphere 0 1))
+    (hpos : ∀ x ∈ K, ∀ v ≠ 0, 0 < F x v)
+    (hhom : ∀ x ∈ K, ∀ r : ℝ, ∀ v, F x (r • v) = r^2*F x v) :
+    ∃ δ : ℝ, 0 < δ ∧ ∀ x ∈ K, ∀ v, δ*‖v‖^2 ≤ F x v := by
+  let L := K ×ˢ sphere (0:E) 1
+  have hLc : IsCompact L := hK.prod (isCompact_sphere _ _)
+  have hp : ∀ q ∈ L, 0 < F q.1 q.2 := by
+    rintro ⟨x,v⟩ ⟨hx,hv⟩
+    apply hpos x hx v
+    intro hv0
+    simp [hv0] at hv
+  have hδ : ∃ δ : ℝ, 0 < δ ∧ ∀ q ∈ L, δ ≤ F q.1 q.2 := by
+    by_cases hn : L.Nonempty
+    · obtain ⟨q,hq,hmin⟩ := hLc.exists_isMinOn hn hF
+      exact ⟨_,hp q hq,fun y hy => hmin hy⟩
+    · exact ⟨1,by norm_num,fun q hq => (hn ⟨q,hq⟩).elim⟩
+  obtain ⟨δ,hδ,hb⟩ := hδ
+  refine ⟨δ,hδ,?_⟩
+  intro x hx v
+  by_cases hv : v = 0
+  · subst v
+    have hh := hhom x hx 0 (0:E)
+    simp only [zero_smul,zero_pow (by decide : 2 ≠ 0),zero_mul] at hh
+    simp [hh]
+  have hn : 0 < ‖v‖ := norm_pos_iff.mpr hv
+  have hw : ‖(‖v‖⁻¹ : ℝ) • v‖ = 1 := by
+    rw [norm_smul,Real.norm_eq_abs,abs_of_pos (inv_pos.mpr hn),inv_mul_cancel₀ hn.ne']
+  have h := hb (x,(‖v‖⁻¹:ℝ) • v) ⟨hx,by simpa [mem_sphere,dist_zero_right] using hw⟩
+  rw [hhom x hx] at h
+  have hh := mul_le_mul_of_nonneg_left h (sq_nonneg ‖v‖)
+  have he : ‖v‖^2*((‖v‖⁻¹)^2*F x v) = F x v := by
+    rw [← mul_assoc,← mul_pow,mul_inv_cancel₀ hn.ne',one_pow,one_mul]
+  rw [he] at hh
+  simpa only [mul_comm] using hh
+end KaehlerCalculus
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/CompactIntegralPropagation.lean b/lean/OAI/Geometry/PrescribedPotential/CompactIntegralPropagation.lean
new file mode 100644
index 0000000000000000000000000000000000000000..8e6023add0bd3ccb462b90965ebd69a87ce5fab9
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/CompactIntegralPropagation.lean
@@ -0,0 +1,104 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.WirtingerCalculus
+
+namespace OAI
+
+/-! Compact Integral Propagation. -/
+
+section
+
+ 
+
+noncomputable section
+open Set MeasureTheory
+namespace PotentialABP
+variable {X I : Type*} [TopologicalSpace X] [ConnectedSpace X]
+  [MeasurableSpace X] [Fintype I]
+
+ 
+
+omit [TopologicalSpace X] [ConnectedSpace X] in
+lemma exists_le_total_div (μ : Measure X) [IsFiniteMeasure μ]
+    {u : X → ℝ} (hu : Integrable u μ) (hn : ∀ x, 0 ≤ u x)
+    {s : Set X} (hs : 0 < μ.real s) {B : ℝ} (hB : ∫ x, u x ∂μ ≤ B) :
+    ∃ x ∈ s, u x ≤ B / μ.real s := by
+  have hs0 : μ s ≠ 0 := by
+    intro h; simp [measureReal_def, h] at hs
+  obtain ⟨x, hx, hv⟩ := exists_le_setAverage hs0 (measure_ne_top μ s) hu.integrableOn
+  refine ⟨x, hx, hv.trans ?_⟩
+  rw [setAverage_eq, smul_eq_mul, ← div_eq_inv_mul]
+  apply div_le_div_of_nonneg_right _ hs.le
+  exact (setIntegral_le_integral hu (Filter.Eventually.of_forall hn)).trans hB
+
+ 
+
+theorem finite_cover_integral_bound
+    (U : I → Set X) (hopen : ∀ i, IsOpen (U i))
+    (hne : ∀ i, (U i).Nonempty) (hcover : ∀ x, ∃ i, x ∈ U i)
+    (μ : I → Measure X) [∀ i, IsFiniteMeasure (μ i)]
+    (hover : ∀ i j, (U i ∩ U j).Nonempty → 0 < (μ i).real (U j))
+    (F : (X → ℝ) → Prop)
+    (hn : ∀ u, F u → ∀ x, 0 ≤ u x)
+    (hint : ∀ u, F u → ∀ i, Integrable u (μ i))
+    (C : I → ℝ) (hC : ∀ i, 0 ≤ C i)
+    (hlocal : ∀ u, F u → ∀ i y, y ∈ U i →
+      (∫ x, u x ∂μ i) ≤ C i * (u y + 1)) :
+    ∃ B : ℝ, ∀ u, F u → (∃ x, u x = 0) → ∀ i,
+      (∫ x, u x ∂μ i) ≤ B := by
+  classical
+  have hstart (s : I) : ∀ j, ∃ B : ℝ, ∀ u, F u →
+      (∃ x ∈ U s, u x = 0) → (∫ x, u x ∂μ j) ≤ B := by
+    let Good : I → Prop := fun j => ∃ B : ℝ, ∀ u, F u →
+      (∃ x ∈ U s, u x = 0) → (∫ x, u x ∂μ j) ≤ B
+    have hGs : Good s := by
+      refine ⟨C s, ?_⟩
+      rintro u hu ⟨x, hx, hz⟩
+      simpa [hz] using hlocal u hu s x hx
+    have hstep {i j : I} (hi : Good i) (hij : (U i ∩ U j).Nonempty) : Good j := by
+      obtain ⟨B, hB⟩ := hi
+      refine ⟨C j * (B / (μ i).real (U j) + 1), ?_⟩
+      intro u hu hz
+      obtain ⟨y, hy, hyB⟩ := exists_le_total_div (μ i) (hint u hu i)
+        (hn u hu) (hover i j hij) (hB u hu hz)
+      exact (hlocal u hu j y hy).trans
+        (mul_le_mul_of_nonneg_left (by linarith : u y + 1 ≤ B / (μ i).real (U j) + 1) (hC j))
+    let V : Set X := ⋃ i, ⋃ (_ : Good i), U i
+    have hVo : IsOpen V := isOpen_iUnion fun i => isOpen_iUnion fun _ => hopen i
+    have hVc : IsClosed V := by
+      have he : Vᶜ = ⋃ i, ⋃ (_ : ¬ Good i), U i := by
+        ext x
+        simp only [V, mem_compl_iff, mem_iUnion]
+        constructor
+        · intro hx
+          obtain ⟨i, hi⟩ := hcover x
+          exact ⟨i, fun hg => hx ⟨i, hg, hi⟩, hi⟩
+        · rintro ⟨i, hgi, hxi⟩ ⟨j, hgj, hxj⟩
+          exact hgi (hstep hgj ⟨x, hxj, hxi⟩)
+      apply isOpen_compl_iff.mp
+      rw [he]
+      exact isOpen_iUnion fun i => isOpen_iUnion fun _ => hopen i
+    have hV : V = univ := (show IsClopen V from ⟨hVc, hVo⟩).eq_univ (by
+      obtain ⟨x, hx⟩ := hne s
+      exact ⟨x, mem_iUnion.mpr ⟨s, mem_iUnion.mpr ⟨hGs, hx⟩⟩⟩)
+    intro j
+    obtain ⟨x, hx⟩ := hne j
+    have hxV : x ∈ V := by rw [hV]; trivial
+    obtain ⟨i, hgi, hxi⟩ := (by simpa only [V, mem_iUnion] using hxV :
+      ∃ i, ∃ _ : Good i, x ∈ U i)
+    exact hstep hgi ⟨x, hxi, hx⟩
+  choose B hB using hstart
+  refine ⟨∑ s, ∑ j, max (B s j) 0, ?_⟩
+  intro u hu hz i
+  obtain ⟨x, hx⟩ := hz
+  obtain ⟨s, hs⟩ := hcover x
+  apply (hB s i u hu ⟨x, hs, hx⟩).trans
+  apply (le_max_left _ _).trans
+  apply (Finset.single_le_sum (fun j _ => le_max_right _ _) (Finset.mem_univ i)).trans
+  exact Finset.single_le_sum (fun s _ => Finset.sum_nonneg fun j _ => le_max_right _ _)
+    (Finset.mem_univ s)
+end PotentialABP
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/CompactSmoothCutoff.lean b/lean/OAI/Geometry/PrescribedPotential/CompactSmoothCutoff.lean
new file mode 100644
index 0000000000000000000000000000000000000000..3b773a8a0d13c72fa7f9f2e16ffe1cfff03bef47
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/CompactSmoothCutoff.lean
@@ -0,0 +1,52 @@
+import Mathlib
+
+namespace OAI
+
+/-! Compact Smooth Cutoff. -/
+
+section
+
+noncomputable section
+open Set Filter Topology Function
+open scoped ContDiff
+namespace HigherJet
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E]
+
+lemma compact_smooth_cutoff {K U : Set E} (hK : IsCompact K) (hU : IsOpen U) (hKU : K ⊆ U) :
+    ∃ L : Set E, ∃ χ : E → ℝ, IsCompact L ∧ L ⊆ U ∧ K ⊆ L ∧ ContDiff ℝ ∞ χ ∧
+      (∀ x, 0 ≤ χ x ∧ χ x ≤ 1) ∧ (∀ x ∈ K, χ x=1) ∧
+      (∀ x ∈ L, χ x≠0 → x ∈ interior L) := by
+  classical
+  obtain ⟨L,hL,hKL,hLU⟩ := exists_compact_between hK hU hKU
+  obtain ⟨f,hfs,hfd,hfv⟩ := isOpen_interior.exists_contDiff_support_eq (n := ⊤) (s := interior L)
+  obtain ⟨g,hgs,hgd,hgv⟩ := hK.isClosed.isOpen_compl.exists_contDiff_support_eq (n := ⊤)
+  have hfp (x : E) : 0 ≤ f x := (hfv ⟨x,rfl⟩).1
+  have hgp (x : E) : 0 ≤ g x := (hgv ⟨x,rfl⟩).1
+  have hd (x : E) : 0 < f x+g x := by
+    by_cases hx : x ∈ K
+    · have hf : f x ≠ 0 := by rw [← mem_support,hfs]; exact hKL hx
+      have hp := lt_of_le_of_ne (hfp x) (Ne.symm hf)
+      linarith [hgp x]
+    · have hg : g x ≠ 0 := by rw [← mem_support,hgs]; exact hx
+      have hp := lt_of_le_of_ne (hgp x) (Ne.symm hg)
+      linarith [hfp x]
+  refine ⟨L,fun x => f x/(f x+g x),hL,hLU,hKL.trans interior_subset,
+    hfd.div (hfd.add hgd) (fun x => ne_of_gt (hd x)),?_,?_,?_⟩
+  · intro x
+    exact ⟨div_nonneg (hfp x) (hd x).le,(div_le_one (hd x)).mpr (by linarith [hgp x])⟩
+  · intro x hx
+    have hg : g x=0 := by
+      by_contra hn
+      have hh : x ∈ Kᶜ := by rw [← hgs]; exact hn
+      exact hh hx
+    change f x/(f x+g x)=1
+    rw [hg,add_zero,div_self (ne_of_gt (by linarith [hd x]))]
+  · intro x _ hn
+    rw [← hfs]
+    exact fun he => hn (by change f x/(f x+g x)=0; rw [he,zero_div])
+end HigherJet
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/CompletedProduct.lean b/lean/OAI/Geometry/PrescribedPotential/CompletedProduct.lean
new file mode 100644
index 0000000000000000000000000000000000000000..6efe7e32ac2a0aa4f3fbdd5592e739d993715f92
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/CompletedProduct.lean
@@ -0,0 +1,84 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.BoundedBilinearCompletion
+import OAI.Geometry.PrescribedPotential.GlobalProduct
+import OAI.Geometry.PrescribedPotential.GlobalStrongEmbedding
+import OAI.Geometry.PrescribedPotential.PatchCutoffs
+
+namespace OAI
+
+/-! Completed Product. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology
+open scoped ContDiff Classical BoundedContinuousFunction
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+
+def Smooth.mulBilinear : Smooth A →ₗ[ℝ] Smooth A →ₗ[ℝ] Smooth A where
+  toFun f :=
+    { toFun := f.mul
+      map_add' h l := by ext x; change f x*(h x+l x) = _; exact mul_add _ _ _
+      map_smul' c h := by ext x; change f x*(c • h x) = c • (f x*h x); simp [mul_left_comm] }
+  map_add' f h := by ext l x; change (f x+h x)*l x = _; exact add_mul _ _ _
+  map_smul' c f := by ext h x; change (c • f x)*h x = c • (f x*h x); simp [mul_assoc]
+
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+ 
+def product (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) :
+    D.localizers.Sobolev (k : ℝ) →L[ℝ]
+      D.localizers.Sobolev (k : ℝ) →L[ℝ] D.localizers.Sobolev (k : ℝ) :=
+  BoundedBilinearCompletion.extend (D.localizers.embed (k : ℝ))
+    (D.localizers.embed_dense (k : ℝ)) Smooth.mulBilinear
+    (D.smooth_product_bound k hk).choose_spec.2
+
+lemma product_embed (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) (f h : Smooth A) :
+    D.product k hk (D.localizers.embed (k : ℝ) f) (D.localizers.embed (k : ℝ) h) =
+      D.localizers.embed (k : ℝ) (f.mul h) :=
+  BoundedBilinearCompletion.extend_eq _ _ _ (D.smooth_product_bound k hk).choose_spec.1
+    (D.smooth_product_bound k hk).choose_spec.2 f h
+
+lemma product_strong (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (u v : D.localizers.Sobolev (k : ℝ)) :
+    D.localizers.strong (k : ℝ) (D.product k hk u v) =
+      D.localizers.strong (k : ℝ) u * D.localizers.strong (k : ℝ) v := by
+  have hs : (Module.finrank ℝ (EC d) : ℝ) < 2*(k : ℝ) := by
+    have hh : (Module.finrank ℝ (EC d) : ℝ) < (k : ℝ) := by exact_mod_cast hk
+    linarith [Nat.cast_nonneg (α := ℝ) k]
+  have hf (f : Smooth A) : ∀ v,
+      D.localizers.strong (k : ℝ) (D.product k hk (D.localizers.embed (k : ℝ) f) v) =
+        D.localizers.strong (k : ℝ) (D.localizers.embed (k : ℝ) f)*D.localizers.strong (k : ℝ) v := by
+    apply congr_fun
+    apply (D.localizers.embed_dense (k : ℝ)).equalizer (by fun_prop) (by fun_prop)
+    funext h
+    dsimp only [Function.comp_apply]
+    rw [D.product_embed, D.localizers.strong_embed _ hs, D.localizers.strong_embed _ hs,
+      D.localizers.strong_embed _ hs]
+    rfl
+  have he : (fun u => D.localizers.strong (k : ℝ) (D.product k hk u v)) =
+      (fun u => D.localizers.strong (k : ℝ) u*D.localizers.strong (k : ℝ) v) := by
+    apply (D.localizers.embed_dense (k : ℝ)).equalizer (by fun_prop) (by fun_prop)
+    funext f
+    exact hf f v
+  exact congr_fun he u
+
+ 
+
+lemma product_contDiff (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) :
+    ContDiff ℝ ∞ (fun z : D.localizers.Sobolev (k : ℝ) × D.localizers.Sobolev (k : ℝ) =>
+      D.product k hk z.1 z.2) :=
+  ((D.product k hk).contDiff.comp contDiff_fst).clm_apply contDiff_snd
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/CompletedRealification.lean b/lean/OAI/Geometry/PrescribedPotential/CompletedRealification.lean
new file mode 100644
index 0000000000000000000000000000000000000000..7cb7e1954866c4f3bf25b7479632703cc8701a45
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/CompletedRealification.lean
@@ -0,0 +1,100 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.GlobalConjugation
+
+namespace OAI
+
+/-! Completed Realification. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology
+open scoped ContDiff Classical
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+local instance realifyNG (s : ℝ) : NormedAddCommGroup (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedAddCommGroup
+local instance realifyNS (s : ℝ) : NormedSpace ℝ (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedSpace
+local instance realifyTG (s : ℝ) : IsTopologicalAddGroup (D.localizers.RealSobolev s) :=
+  Submodule.isTopologicalAddGroup _
+local instance realifyCS (s : ℝ) : ContinuousSMul ℝ (D.localizers.RealSobolev s) :=
+  SMulMemClass.continuousSMul _
+
+def completedRealify (k : ℕ) : D.localizers.Sobolev (k : ℝ) →L[ℝ] D.localizers.Sobolev (k : ℝ) :=
+  D.localizers.extendCore (k : ℝ) (k : ℝ) Smooth.realify
+
+lemma completedRealify_embed (k : ℕ) (f : Smooth A) :
+    D.completedRealify k (D.localizers.embed (k : ℝ) f) = D.localizers.embed (k : ℝ) (Smooth.realify f) :=
+  D.localizers.extendCore_embed (D.realify_integer_bound k) f
+
+lemma completedRealify_mem (k : ℕ) (u : D.localizers.Sobolev (k : ℝ)) :
+    D.completedRealify k u ∈ D.localizers.realCompletion (k : ℝ) := by
+  have hf (f : Smooth A) : D.completedRealify k (D.localizers.embed (k : ℝ) f) ∈
+      D.localizers.realCompletion (k : ℝ) := by
+    rw [D.completedRealify_embed]
+    exact (D.localizers.realEmbed (k : ℝ) ⟨Smooth.realify f,Smooth.realify_real f⟩).property
+  exact (D.localizers.embed_dense (k : ℝ)).induction_on u
+    ((Submodule.isClosed_topologicalClosure _).preimage (D.completedRealify k).continuous) hf
+
+def realProjection (k : ℕ) : D.localizers.Sobolev (k : ℝ) →L[ℝ] D.localizers.RealSobolev (k : ℝ) :=
+  (D.completedRealify k).codRestrict _ (D.completedRealify_mem k)
+
+lemma completedRealify_of_real (k : ℕ) (u : D.localizers.RealSobolev (k : ℝ)) :
+    D.completedRealify k u.val = u.val := by
+  have he : (fun u : D.localizers.RealSobolev (k : ℝ) => D.completedRealify k u.val) =
+      (fun u => u.val) := by
+    apply (D.localizers.realEmbed_dense (k : ℝ)).equalizer (by fun_prop) continuous_subtype_val
+    funext f
+    dsimp only [Function.comp_apply]
+    rw [Localizers.realEmbed_val,D.completedRealify_embed,Smooth.realify_eq_of_real]
+  exact congr_fun he u
+
+lemma realProjection_of_real (k : ℕ) (u : D.localizers.RealSobolev (k : ℝ)) :
+    D.realProjection k u.val = u := Subtype.ext (D.completedRealify_of_real k u)
+
+lemma completedRealify_lower (k l : ℕ) (hlk : l ≤ k) (u : D.localizers.Sobolev (k : ℝ)) :
+    D.localizers.lower (k : ℝ) (l : ℝ) (D.completedRealify k u) =
+      D.completedRealify l (D.localizers.lower (k : ℝ) (l : ℝ) u) := by
+  have hkl : (l : ℝ) ≤ (k : ℝ) := by exact_mod_cast hlk
+  have he : D.localizers.lower (k : ℝ) (l : ℝ) ∘ D.completedRealify k =
+      D.completedRealify l ∘ D.localizers.lower (k : ℝ) (l : ℝ) := by
+    apply (D.localizers.embed_dense (k : ℝ)).equalizer (by fun_prop) (by fun_prop)
+    funext f
+    dsimp only [Function.comp_apply]
+    rw [D.completedRealify_embed,D.localizers.lower_embed hkl,
+      D.localizers.lower_embed hkl,D.completedRealify_embed]
+  exact congr_fun he u
+
+ 
+
+lemma mem_real_of_lower (k l : ℕ) (hlk : l ≤ k) (u : D.localizers.Sobolev (k : ℝ))
+    (hu : D.localizers.lower (k : ℝ) (l : ℝ) u ∈ D.localizers.realCompletion (l : ℝ)) :
+    u ∈ D.localizers.realCompletion (k : ℝ) := by
+  have he : D.completedRealify k u = u := by
+    apply D.localizers.lower_injective (by exact_mod_cast hlk : (l : ℝ) ≤ (k : ℝ))
+    rw [D.completedRealify_lower k l hlk]
+    exact D.completedRealify_of_real l ⟨_,hu⟩
+  rw [← he]
+  exact D.completedRealify_mem k u
+
+lemma mem_real_of_lower_eq (k l : ℕ) (s : ℝ) (hs : s = (l : ℝ))
+    (hlk : l ≤ k) (u : D.localizers.Sobolev (k : ℝ))
+    (hu : D.localizers.lower (k : ℝ) s u ∈ D.localizers.realCompletion s) :
+    u ∈ D.localizers.realCompletion (k : ℝ) := by
+  subst s
+  exact D.mem_real_of_lower k l hlk u hu
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/CompletedResolvent.lean b/lean/OAI/Geometry/PrescribedPotential/CompletedResolvent.lean
new file mode 100644
index 0000000000000000000000000000000000000000..d712b4c7d81f0aa5f6719a43043c9df5e79ff095
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/CompletedResolvent.lean
@@ -0,0 +1,164 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.ComplexOperator
+import OAI.Geometry.PrescribedPotential.GlobalBounds
+import OAI.Geometry.PrescribedPotential.GlobalError
+import OAI.Geometry.PrescribedPotential.PatchCutoffs
+
+namespace OAI
+
+/-! Completed Resolvent. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory
+open scoped SchwartzMap ContDiff Classical
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace Localizers
+variable (D : Localizers A ι)
+
+ 
+
+lemma extendCore_compat {s t a b : ℝ} (has : a ≤ s) (hbt : b ≤ t)
+    (T : Smooth A →ₗ[ℝ] Smooth A) (hT : D.BoundedCore s t T)
+    (hB : D.BoundedCore a b T) (u : D.Sobolev s) :
+    D.lower t b (D.extendCore s t T u) =
+      D.extendCore a b T (D.lower s a u) := by
+  have hS : D.BoundedCore s b T := hT.target_mono hbt
+  rw [D.lower_extendCore hbt T hT hS, D.extendCore_lower has T hB hS]
+
+ 
+def AllRegular (u : D.Sobolev 0) : Prop :=
+  ∀ k : ℕ, ∃ v : D.Sobolev (k : ℝ), D.lower (k : ℝ) 0 v = u
+
+lemma allRegular_embed (f : Smooth A) : D.AllRegular (D.embed 0 f) := by
+  intro k
+  exact ⟨D.embed (k : ℝ) f, D.lower_embed (Nat.cast_nonneg _) f⟩
+
+lemma allRegular_of_gain (T : Smooth A →ₗ[ℝ] Smooth A)
+    (h0 : D.BoundedCore 0 0 T)
+    (hgain : ∀ k : ℕ, D.BoundedCore (k : ℝ) ((k : ℝ) + 1) T)
+    (f : Smooth A) (u : D.Sobolev 0)
+    (hu : u = D.embed 0 f + D.extendCore 0 0 T u) : D.AllRegular u := by
+  intro k
+  induction k with
+  | zero =>
+    rw [Nat.cast_zero]
+    exact ⟨u, by rw [D.lower_self]; rfl⟩
+  | succ k ih =>
+    rw [Nat.cast_add, Nat.cast_one]
+    obtain ⟨v, hv⟩ := ih
+    have h := D.extendCore_compat (Nat.cast_nonneg k)
+      (by positivity : (0 : ℝ) ≤ (k : ℝ) + 1) T (hgain k) h0 v
+    refine ⟨D.embed (k + 1 : ℝ) f + D.extendCore (k : ℝ) (k + 1 : ℝ) T v, ?_⟩
+    rw [map_add, D.lower_embed (by positivity), h, hv, ← hu]
+
+end Localizers
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+ 
+def completedL : D.localizers.Sobolev 2 →L[ℝ] D.localizers.Sobolev 0 :=
+  D.localizers.extendCore 2 0 (complexL g)
+
+def shiftedOperator (m : ℝ) : D.localizers.Sobolev 2 →L[ℝ] D.localizers.Sobolev 0 :=
+  (m^2 : ℝ) • D.localizers.lower 2 0 - D.completedL
+
+def completedParametrix (m : ℝ) (hm : 1 ≤ m) :
+    D.localizers.Sobolev 0 →L[ℝ] D.localizers.Sobolev 2 :=
+  D.localizers.extendCore 0 2 (D.parametrix m hm)
+
+def completedError (m : ℝ) (hm : 1 ≤ m) :
+    D.localizers.Sobolev 0 →L[ℝ] D.localizers.Sobolev 0 :=
+  D.localizers.extendCore 0 0 (D.error m hm)
+
+lemma completedParametrix_bound (m : ℝ) (hm : 1 ≤ m) :
+    D.localizers.BoundedCore 0 2 (D.parametrix m hm) := by
+  have hb := D.parametrix_bound m hm 0
+  rw [Nat.cast_zero, zero_add] at hb
+  exact hb
+
+lemma completedParametrix_equation (m : ℝ) (hm : 1 ≤ m) (u : D.localizers.Sobolev 0) :
+    D.shiftedOperator m (D.completedParametrix m hm u) = u - D.completedError m hm u := by
+  have he : D.shiftedOperator m ∘ D.completedParametrix m hm =
+      fun u => u - D.completedError m hm u := by
+    apply (D.localizers.embed_dense 0).equalizer (by fun_prop) (by fun_prop)
+    funext f
+    simp only [Function.comp_apply, shiftedOperator, completedParametrix, completedL,
+      completedError, _root_.sub_apply, _root_.smul_apply,
+      D.localizers.extendCore_embed (D.completedParametrix_bound m hm),
+      D.localizers.extendCore_embed D.complexL_bound,
+      D.localizers.extendCore_embed (D.error_bound_zero m hm),
+      D.localizers.lower_embed (by norm_num : (0 : ℝ) ≤ 2)]
+    rw [← map_smul, ← map_sub, D.parametrix_equation, map_sub]
+  exact congr_fun he u
+
+lemma exists_completedError_small :
+    ∃ m : ℝ, ∃ hm : 1 ≤ m, ‖D.completedError m hm‖ < 1 := by
+  obtain ⟨m, hm, q, hq, hq1, hb⟩ := D.exists_contractive_shift
+  exact ⟨m, hm, (D.localizers.extendCore_norm hq hb).trans_lt hq1⟩
+
+ 
+
+def errorInverse (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1) :
+    D.localizers.Sobolev 0 →L[ℝ] D.localizers.Sobolev 0 :=
+  ↑((Units.oneSub (D.completedError m hm) he)⁻¹)
+
+lemma errorInverse_equation (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
+    (f : D.localizers.Sobolev 0) :
+    D.errorInverse m hm he f - D.completedError m hm (D.errorInverse m hm he f) = f := by
+  let U := Units.oneSub (D.completedError m hm) he
+  change ((↑U : D.localizers.Sobolev 0 →L[ℝ] D.localizers.Sobolev 0) * ↑U⁻¹) f = f
+  rw [Units.mul_inv, _root_.one_apply_eq_self]
+
+def completedResolvent (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1) :
+    D.localizers.Sobolev 0 →L[ℝ] D.localizers.Sobolev 2 :=
+  D.completedParametrix m hm ∘L D.errorInverse m hm he
+
+theorem completedResolvent_equation (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
+    (f : D.localizers.Sobolev 0) :
+    D.shiftedOperator m (D.completedResolvent m hm he f) = f := by
+  rw [completedResolvent, ContinuousLinearMap.comp_apply, D.completedParametrix_equation,
+    D.errorInverse_equation]
+
+lemma errorInverse_regular (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
+    (f : Smooth A) : D.localizers.AllRegular (D.errorInverse m hm he (D.localizers.embed 0 f)) := by
+  apply D.localizers.allRegular_of_gain (D.error m hm) (D.error_bound_zero m hm)
+    (D.error_bound_integer m hm) f
+  exact sub_eq_iff_eq_add.mp (D.errorInverse_equation m hm he _)
+
+lemma parametrix_regular (m : ℝ) (hm : 1 ≤ m) (u : D.localizers.Sobolev 0)
+    (hu : D.localizers.AllRegular u) :
+    D.localizers.AllRegular (D.localizers.lower 2 0 (D.completedParametrix m hm u)) := by
+  intro k
+  obtain ⟨v, hv⟩ := hu k
+  have hb := D.parametrix_bound m hm k
+  refine ⟨D.localizers.lower ((k : ℝ) + 2) (k : ℝ)
+    (D.localizers.extendCore (k : ℝ) ((k : ℝ) + 2) (D.parametrix m hm) v), ?_⟩
+  rw [← ContinuousLinearMap.comp_apply,
+    D.localizers.lower_comp (by linarith : (k : ℝ) ≤ (k : ℝ) + 2) (Nat.cast_nonneg k)]
+  rw [D.localizers.extendCore_compat (Nat.cast_nonneg k)
+    (by positivity : (0 : ℝ) ≤ (k : ℝ) + 2) _ hb
+      ((D.completedParametrix_bound m hm).target_mono (by norm_num : (0 : ℝ) ≤ 2)), hv]
+  exact (D.localizers.lower_extendCore (by norm_num : (0 : ℝ) ≤ 2) _
+    (D.completedParametrix_bound m hm)
+    ((D.completedParametrix_bound m hm).target_mono (by norm_num : (0 : ℝ) ≤ 2)) u).symm
+
+theorem completedResolvent_regular (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
+    (f : Smooth A) :
+    D.localizers.AllRegular (D.localizers.lower 2 0
+      (D.completedResolvent m hm he (D.localizers.embed 0 f))) :=
+  D.parametrix_regular m hm _ (D.errorInverse_regular m hm he f)
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/CompletedVolume.lean b/lean/OAI/Geometry/PrescribedPotential/CompletedVolume.lean
new file mode 100644
index 0000000000000000000000000000000000000000..c9e0b0d1227861beeea78db192c3cb650aa30e52
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/CompletedVolume.lean
@@ -0,0 +1,99 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.GlobalHessian
+import OAI.Geometry.PrescribedPotential.GlobalMetricEntries
+import OAI.Geometry.PrescribedPotential.PatchCutoffs
+import OAI.Geometry.PrescribedPotential.PotentialDensity
+import OAI.Geometry.PrescribedPotential.RealSobolev
+import OAI.Geometry.PrescribedPotential.SobolevDeterminant
+
+namespace OAI
+
+/-! Completed Volume. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff Classical BoundedContinuousFunction
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+def completedVolume (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (u : D.localizers.Sobolev ((k : ℝ)+2)) : D.localizers.Sobolev (k : ℝ) :=
+  ∑ p, D.product k hk (D.localizers.embed (k : ℝ) (D.determinantWeight p))
+    (D.determinant k hk d (fun i j => D.localizers.embed (k : ℝ) (D.metricEntry p i j) +
+      D.completedHessian k p i j u))
+
+lemma completedVolume_contDiff (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) :
+    ContDiff ℝ ∞ (D.completedVolume k hk) := by
+  apply ContDiff.sum
+  intro p _
+  exact (D.product k hk (D.localizers.embed (k : ℝ) (D.determinantWeight p))).contDiff.comp
+    (D.determinant_contDiff k hk d _ (fun i j => contDiff_const.add (D.completedHessian k p i j).contDiff))
+
+lemma completedVolume_embed (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (φ : SmoothRealFunction A) :
+    D.completedVolume k hk (D.localizers.embed ((k : ℝ)+2) (Smooth.ofReal φ)) =
+      D.localizers.embed (k : ℝ) (Smooth.ofReal (g.potentialDensity φ)) := by
+  have hs : (Module.finrank ℝ (EC d) : ℝ) < 2*(k : ℝ) := by
+    have hh : (Module.finrank ℝ (EC d) : ℝ) < (k : ℝ) := by exact_mod_cast hk
+    linarith [Nat.cast_nonneg (α := ℝ) k]
+  apply D.localizers.strong_injective (k : ℝ) hs
+  ext x
+  simp only [completedVolume, map_sum, BoundedContinuousFunction.sum_apply]
+  rw [D.localizers.strong_embed _ hs]
+  have hterm (p : ι) : D.localizers.strong (k : ℝ)
+      (D.product k hk (D.localizers.embed (k : ℝ) (D.determinantWeight p))
+        (D.determinant k hk d (fun i j => D.localizers.embed (k : ℝ) (D.metricEntry p i j) +
+          D.completedHessian k p i j (D.localizers.embed ((k : ℝ)+2) (Smooth.ofReal φ))))) x =
+      D.localizers.weight p x * ((g.potentialDensity φ).value x : ℂ) := by
+    rw [D.product_strong, BoundedContinuousFunction.mul_apply, D.localizers.strong_embed _ hs]
+    change D.determinantWeight p x * _ = _
+    by_cases hz : D.localizers.weight p x = 0
+    · rw [D.determinantWeight_zero p hz, hz, zero_mul, zero_mul]
+    · have hx : x ∈ tsupport (D.localizers.weight p : X → ℂ) := subset_tsupport _ hz
+      rw [D.determinantWeight_apply p hx]
+      erw [D.determinant_strong]
+      have hm : (fun i j => D.localizers.strong (k : ℝ)
+          (D.localizers.embed (k : ℝ) (D.metricEntry p i j) +
+            D.completedHessian k p i j (D.localizers.embed ((k : ℝ)+2) (Smooth.ofReal φ))) x) =
+          g.matrix (D.patch p).index (A.chart (D.patch p).index x) +
+            φ.hessian (D.patch p).index (A.chart (D.patch p).index x) := by
+        ext i j
+        rw [D.completedHessian_embed, map_add, BoundedContinuousFunction.add_apply,
+          D.localizers.strong_embed _ hs, D.localizers.strong_embed _ hs]
+        change D.metricEntry p i j x + D.localizedHessian p i j (Smooth.ofReal φ) x = _
+        rw [D.metricEntry_apply p i j hx, D.localizedHessian_source p i j φ hx]
+        rfl
+      rw [hm, g.potentialDensity_complex φ (D.patch p).index (by simpa using D.patch_source p hx)]
+      unfold KaehlerMetric.volumePolynomial
+      ring
+  simp_rw [hterm]
+  rw [← Finset.sum_mul, D.localizers.sum_one, one_mul]
+  rfl
+
+lemma completedVolume_real (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) :
+    ∀ u ∈ D.localizers.realCompletion ((k : ℝ)+2),
+      D.completedVolume k hk u ∈ D.localizers.realCompletion (k : ℝ) := by
+  apply ClosedCore.maps_closure_range (D.localizers.realCoreMap ((k : ℝ)+2))
+    (D.localizers.realCoreMap (k : ℝ)) (D.completedVolume k hk)
+    (D.completedVolume_contDiff k hk).continuous
+  intro f
+  refine ⟨RealSmooth.ofReal (g.potentialDensity f.source), ?_⟩
+  change D.completedVolume k hk (D.localizers.embed ((k : ℝ)+2) f.val) = _
+  rw [← f.ofReal_source, D.completedVolume_embed]
+  rfl
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/ComplexContactDeterminant.lean b/lean/OAI/Geometry/PrescribedPotential/ComplexContactDeterminant.lean
new file mode 100644
index 0000000000000000000000000000000000000000..c354a6e5457af75161c3a99aee7e6010148da2f3
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/ComplexContactDeterminant.lean
@@ -0,0 +1,66 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.GradientJacobian
+
+namespace OAI
+
+/-! Complex Contact Determinant. -/
+
+section
+
+ 
+
+noncomputable section
+open Matrix Set Filter Topology
+open scoped InnerProductSpace ContDiff ComplexOrder
+namespace PotentialABP
+variable {d : ℕ}
+local notation "E" => EuclideanSpace ℂ (Fin d)
+local instance complexContactDeterminantRealInnerProductSpace (d : ℕ) :
+    InnerProductSpace ℝ (EuclideanSpace ℂ (Fin d)) :=
+  InnerProductSpace.rclikeToReal ℂ (EuclideanSpace ℂ (Fin d))
+
+lemma real_det_hermitian (H : Matrix (Fin d) (Fin d) ℂ) (hH : H.IsHermitian) :
+    (H.toEuclideanLin.restrictScalars ℝ).det = H.det.re ^ 2 := by
+  rw [LinearMap.det_restrictScalars]
+  have hm : H.toEuclideanLin.det = H.det := LinearMap.det_toLpLin 2 H
+  rw [hm, Algebra.norm_complex_eq]
+  have him : H.det.im = 0 := by
+    have he : star H.det = H.det := by
+      rw [← Matrix.det_conjTranspose, hH]
+    exact (Complex.conj_eq_iff_im).mp he
+  simp [Complex.normSq_apply, him, pow_two]
+
+ 
+
+lemma gradient_det_le_hermitian {u : E → ℝ} (hu : ContDiff ℝ ∞ u) {x : E}
+    (hB : ∀ v, 0 ≤ fderiv ℝ (fderiv ℝ u) x v v)
+    (H : Matrix (Fin d) (Fin d) ℂ) (hH : H.IsHermitian)
+    (hdom : ∀ v : E, fderiv ℝ (fderiv ℝ u) x v v ≤
+      4 * inner ℝ (H.toEuclideanLin v) v) :
+    |(fderiv ℝ (gradient u) x).det| ≤ (4:ℝ)^(2*d) * H.det.re^2 := by
+  let L := (fderiv ℝ (gradient u) x).toLinearMap
+  let M : E →ₗ[ℝ] E := (4:ℝ) • H.toEuclideanLin.restrictScalars ℝ
+  have hL : L.IsPositive := gradient_hessian_positive hu hB
+  have hM : (M-L).IsPositive := by
+    refine ⟨?_,?_⟩
+    · exact (LinearMap.IsSymmetric.smul (by simp)
+        (Matrix.isSymmetric_toEuclideanLin_iff.mpr hH).restrictScalars).sub hL.1
+    · intro v
+      change 0 ≤ inner ℝ ((4:ℝ) • H.toEuclideanLin v - fderiv ℝ (gradient u) x v) v
+      rw [inner_sub_left, real_inner_smul_left, inner_fderiv_gradient hu]
+      exact sub_nonneg.mpr (hdom v)
+  have hn := positive_det_nonneg hL
+  have hm := positive_det_mono hL hM
+  change |L.det| ≤ _
+  rw [abs_of_nonneg hn]
+  convert hm using 1
+  dsimp only [M]
+  rw [LinearMap.det_smul, real_det_hermitian H hH,
+    finrank_real_of_complex, finrank_euclideanSpace, Fintype.card_fin]
+
+end PotentialABP
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/ComplexHessian.lean b/lean/OAI/Geometry/PrescribedPotential/ComplexHessian.lean
new file mode 100644
index 0000000000000000000000000000000000000000..e79c6c8e931ae711a30c46a5a9f881c87bdb1ab9
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/ComplexHessian.lean
@@ -0,0 +1,108 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CoreBounds
+
+namespace OAI
+
+/-! Complex Hessian. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology MeasureTheory LineDeriv
+open scoped ContDiff SchwartzMap Classical
+namespace SobolevChart
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+
+lemma schwartzCoord_smul (s : ℝ) (c : ℂ) (f : 𝓢(E, ℂ)) :
+    schwartzCoord s (c • f) = c • schwartzCoord s f := by
+  apply realize_injective s
+  rw [map_smul, realize_schwartzCoord, realize_schwartzCoord]
+  exact (SchwartzMap.toTemperedDistributionCLM E ℂ volume).map_smul c f
+
+lemma CoreBound.smul {s t : ℝ} {T : 𝓢(E, ℂ) → 𝓢(E, ℂ)}
+    (hT : CoreBound s t T) (c : ℂ) : CoreBound s t (fun f => c • T f) := by
+  obtain ⟨C,hC,h⟩ := hT
+  refine ⟨‖c‖*C,mul_nonneg (norm_nonneg _) hC,fun f => ?_⟩
+  rw [schwartzCoord_smul, norm_smul, mul_assoc]
+  exact mul_le_mul_of_nonneg_left (h f) (norm_nonneg c)
+
+omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] in
+lemma complex_smooth_second_deriv (f : 𝓢(E, ℂ)) (v w x : E) :
+    (∂_{v} (∂_{w} f)) x = fderiv ℝ (fderiv ℝ f) x v w := by
+  simp only [SchwartzMap.lineDerivOp_apply_eq_fderiv]
+  change fderiv ℝ (fun y => fderiv ℝ f y w) x v = _
+  exact PotentialKaehler.fderiv_eval_derivative f.smooth'.contDiffAt v w
+
+end SobolevChart
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+
+lemma second_postcomp_general {E F G : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+    [NormedAddCommGroup F] [NormedSpace ℝ F] [NormedAddCommGroup G] [NormedSpace ℝ G]
+    {f : E → F} {x : E} (hf : ContDiffAt ℝ ∞ f x) (p : F →L[ℝ] G) (v w : E) :
+    fderiv ℝ (fderiv ℝ (p ∘ f)) x v w = p (fderiv ℝ (fderiv ℝ f) x v w) := by
+  have hN : ∀ᶠ y in 𝓝 x, DifferentiableAt ℝ f y := by
+    filter_upwards [(hf.of_le (by simp : (1 : ℕ∞ω) ≤ ∞)).eventually (by simp)] with y hy
+    exact hy.differentiableAt (by simp)
+  have he : fderiv ℝ (p ∘ f) =ᶠ[𝓝 x] (fun y => p.comp (fderiv ℝ f y)) := by
+    filter_upwards [hN] with y hy
+    exact (p.hasFDerivAt.comp y hy.hasFDerivAt).fderiv
+  rw [he.fderiv_eq, fderiv_clm_comp (differentiableAt_const p)
+    ((hf.fderiv_right (m := ∞) (by simp)).differentiableAt (by simp))]
+  simp
+
+variable {d : ℕ}
+def hessianDirection (i : Fin d) : EC d := (coordinateEquiv d).symm (Pi.single i 1)
+def hessianIDirection (i : Fin d) : EC d := (coordinateEquiv d).symm (Complex.I • Pi.single i 1)
+
+def hessianEntrySchwartz (i j : Fin d) : 𝓢(EC d, ℂ) →L[ℂ] 𝓢(EC d, ℂ) :=
+  (1/4 : ℂ) •
+    ((lineDerivOpCLM ℂ 𝓢(EC d, ℂ) (hessianDirection i) ∘L
+      lineDerivOpCLM ℂ 𝓢(EC d, ℂ) (hessianDirection j)) +
+    (lineDerivOpCLM ℂ 𝓢(EC d, ℂ) (hessianIDirection i) ∘L
+      lineDerivOpCLM ℂ 𝓢(EC d, ℂ) (hessianIDirection j)) +
+    Complex.I • ((lineDerivOpCLM ℂ 𝓢(EC d, ℂ) (hessianIDirection i) ∘L
+      lineDerivOpCLM ℂ 𝓢(EC d, ℂ) (hessianDirection j)) +
+    (-1 : ℂ) • (lineDerivOpCLM ℂ 𝓢(EC d, ℂ) (hessianDirection i) ∘L
+      lineDerivOpCLM ℂ 𝓢(EC d, ℂ) (hessianIDirection j))))
+
+lemma hessianEntrySchwartz_apply (i j : Fin d) (f : 𝓢(EC d, ℂ)) (x : EC d) :
+    hessianEntrySchwartz i j f x = (1/4 : ℂ) *
+      (fderiv ℝ (fderiv ℝ f) x (hessianDirection i) (hessianDirection j) +
+       fderiv ℝ (fderiv ℝ f) x (hessianIDirection i) (hessianIDirection j) +
+       Complex.I * (fderiv ℝ (fderiv ℝ f) x (hessianIDirection i) (hessianDirection j) -
+         fderiv ℝ (fderiv ℝ f) x (hessianDirection i) (hessianIDirection j))) := by
+  simp only [hessianEntrySchwartz, smul_apply, add_apply,
+    ContinuousLinearMap.comp_apply, lineDerivOpCLM_apply,
+    smul_eq_mul, complex_smooth_second_deriv]
+  ring
+
+lemma hessianEntrySchwartz_bound (s : ℝ) (i j : Fin d) :
+    CoreBound (s+2) s (hessianEntrySchwartz i j) := by
+  have hb (v w : EC d) : CoreBound (s+2) s (fun f : 𝓢(EC d, ℂ) => ∂_{v} (∂_{w} f)) :=
+    (coreBound_deriv v (s := s+1) (t := s) (by linarith)).comp
+      (coreBound_deriv w (s := s+2) (t := s+1) (by linarith))
+  exact ((hb _ _).add (hb _ _) |>.add (((hb _ _).add ((hb _ _).smul (-1))).smul Complex.I)).smul (1/4)
+
+lemma hessianEntrySchwartz_real {u : EC d → ℝ} {x : EC d} (hu : ContDiffAt ℝ ∞ u x)
+    (f : 𝓢(EC d, ℂ)) (he : (f : EC d → ℂ) =ᶠ[𝓝 x] Complex.ofRealCLM ∘ u) (i j : Fin d) :
+    hessianEntrySchwartz i j f x =
+      PotentialKaehler.hermitianPartMatrix (pullBilin (fderiv ℝ (fderiv ℝ u) x)) i j := by
+  rw [hessianEntrySchwartz_apply]
+  have he2 : fderiv ℝ (fderiv ℝ f) x = fderiv ℝ (fderiv ℝ (Complex.ofRealCLM ∘ u)) x :=
+    (he.fderiv (𝕜 := ℝ)).fderiv_eq
+  rw [he2]
+  simp only [second_postcomp_general hu, Complex.ofRealCLM_apply,
+    PotentialKaehler.hermitianPartMatrix_apply, PotentialKaehler.hermitianPart,
+    pullBilin_apply, hessianDirection, hessianIDirection]
+  apply Complex.ext <;> simp [Complex.mul_re, Complex.mul_im] <;> ring
+
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/ComplexKernel.lean b/lean/OAI/Geometry/PrescribedPotential/ComplexKernel.lean
new file mode 100644
index 0000000000000000000000000000000000000000..ecf5fe786fa2f3e57a73a2bfc1b518356aa568af
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/ComplexKernel.lean
@@ -0,0 +1,67 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.ComplexOperator
+
+namespace OAI
+
+/-! Complex Kernel. -/
+
+section
+
+ 
+
+noncomputable section
+namespace GlobalElliptic
+open Anticanonical SourceSmooth
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+
+lemma complexL_re (g : KaehlerMetric A) (f : Smooth A) (x : X) :
+    (complexL g f x).re = (g.laplacian (f.part Complex.reCLM)).value x := by
+  change (((g.laplacian (f.part Complex.reCLM)).value x : ℂ) +
+    Complex.I * ((g.laplacian (f.part Complex.imCLM)).value x : ℂ)).re = _
+  simp
+
+lemma complexL_im (g : KaehlerMetric A) (f : Smooth A) (x : X) :
+    (complexL g f x).im = (g.laplacian (f.part Complex.imCLM)).value x := by
+  change (((g.laplacian (f.part Complex.reCLM)).value x : ℂ) +
+    Complex.I * ((g.laplacian (f.part Complex.imCLM)).value x : ℂ)).im = _
+  simp
+
+lemma complexL_constant_rhs_zero [CompactSpace X] [Nonempty X]
+    (g : KaehlerMetric A) (f : Smooth A) {c : ℂ}
+    (hf : complexL g f = Smooth.const c) : c = 0 := by
+  have hr : c.re = 0 := g.constant_rhs_eq_zero (f.part Complex.reCLM) (by
+    intro i z hz
+    rw [← g.laplacian_localExpression _ i hz]
+    change (g.laplacian (f.part Complex.reCLM)).value ((A.chart i).symm z) = _
+    rw [← complexL_re, hf]
+    rfl)
+  have hi : c.im = 0 := g.constant_rhs_eq_zero (f.part Complex.imCLM) (by
+    intro i z hz
+    rw [← g.laplacian_localExpression _ i hz]
+    change (g.laplacian (f.part Complex.imCLM)).value ((A.chart i).symm z) = _
+    rw [← complexL_im, hf]
+    rfl)
+  exact Complex.ext hr hi
+
+lemma complexL_kernel [T2Space X] [CompactSpace X] [ConnectedSpace X]
+    (g : KaehlerMetric A) (f : Smooth A) (hf : complexL g f = 0) :
+    ∃ c : ℂ, f = Smooth.const c := by
+  obtain ⟨r, hr⟩ := linearizedMongeAmpere_kernel d X A g (f.part Complex.reCLM) (by
+    intro i z hz
+    rw [← g.laplacian_localExpression _ i hz]
+    change (g.laplacian (f.part Complex.reCLM)).value ((A.chart i).symm z) = _
+    rw [← complexL_re, hf, Smooth.zero_apply, Complex.zero_re])
+  obtain ⟨s, hs⟩ := linearizedMongeAmpere_kernel d X A g (f.part Complex.imCLM) (by
+    intro i z hz
+    rw [← g.laplacian_localExpression _ i hz]
+    change (g.laplacian (f.part Complex.imCLM)).value ((A.chart i).symm z) = _
+    rw [← complexL_im, hf, Smooth.zero_apply, Complex.zero_im])
+  refine ⟨⟨r, s⟩, Smooth.ext (fun x => ?_)⟩
+  exact Complex.ext (hr x) (hs x)
+
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/ComplexOperator.lean b/lean/OAI/Geometry/PrescribedPotential/ComplexOperator.lean
new file mode 100644
index 0000000000000000000000000000000000000000..d9581e4d2b5ecdd585929b421c48f3b4f130f0d9
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/ComplexOperator.lean
@@ -0,0 +1,203 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.GlobalPartition
+import OAI.Geometry.PrescribedPotential.GlobalOperator
+
+namespace OAI
+
+/-! Complex Operator. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology
+open scoped ContDiff Classical
+
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel FrozenPoisson
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+
+lemma second_postcomp {f : E → ℂ} {x : E} (hf : ContDiffAt ℝ ∞ f x)
+    (p : ℂ →L[ℝ] ℝ) (v w : E) :
+    fderiv ℝ (fderiv ℝ (p ∘ f)) x v w = p (fderiv ℝ (fderiv ℝ f) x v w) := by
+  have hN : ∀ᶠ y in 𝓝 x, DifferentiableAt ℝ f y := by
+    filter_upwards [(hf.of_le (by simp : (1 : ℕ∞ω) ≤ ∞)).eventually (by simp)] with y hy
+    exact hy.differentiableAt (by simp)
+  have he : fderiv ℝ (p ∘ f) =ᶠ[𝓝 x] (fun y => p.comp (fderiv ℝ f y)) := by
+    filter_upwards [hN] with y hy
+    exact (p.hasFDerivAt.comp y hy.hasFDerivAt).fderiv
+  rw [he.fderiv_eq, fderiv_clm_comp (differentiableAt_const p)
+    ((hf.fderiv_right (m := ∞) (by simp)).differentiableAt (by simp))]
+  simp
+
+lemma second_add {f h : E → ℂ} {x : E} (hf : ContDiffAt ℝ ∞ f x)
+    (hh : ContDiffAt ℝ ∞ h x) :
+    fderiv ℝ (fderiv ℝ (fun y => f y + h y)) x =
+      fderiv ℝ (fderiv ℝ f) x + fderiv ℝ (fderiv ℝ h) x := by
+  have hN (u : E → ℂ) (hu : ContDiffAt ℝ ∞ u x) :
+      ∀ᶠ y in 𝓝 x, DifferentiableAt ℝ u y := by
+    filter_upwards [(hu.of_le (by simp : (1 : ℕ∞ω) ≤ ∞)).eventually (by simp)] with y hy
+    exact hy.differentiableAt (by simp)
+  have he : fderiv ℝ (fun y => f y + h y) =ᶠ[𝓝 x]
+      (fun y => fderiv ℝ f y + fderiv ℝ h y) := by
+    filter_upwards [hN f hf, hN h hh] with y hy hz
+    exact fderiv_add hy hz
+  rw [he.fderiv_eq]
+  exact fderiv_add ((hf.fderiv_right (m := ∞) (by simp)).differentiableAt (by simp))
+    ((hh.fderiv_right (m := ∞) (by simp)).differentiableAt (by simp))
+
+lemma second_real_smul {f : E → ℂ} {x : E} (hf : ContDiffAt ℝ ∞ f x) (c : ℝ) :
+    fderiv ℝ (fderiv ℝ (fun y => c • f y)) x =
+      c • fderiv ℝ (fderiv ℝ f) x := by
+  have hN : ∀ᶠ y in 𝓝 x, DifferentiableAt ℝ f y := by
+    filter_upwards [(hf.of_le (by simp : (1 : ℕ∞ω) ≤ ∞)).eventually (by simp)] with y hy
+    exact hy.differentiableAt (by simp)
+  have he : fderiv ℝ (fun y => c • f y) =ᶠ[𝓝 x]
+      (fun y => c • fderiv ℝ f y) := by
+    filter_upwards [hN] with y hy
+    exact fderiv_const_smul hy c
+  rw [he.fderiv_eq]
+  exact fderiv_const_smul ((hf.fderiv_right (m := ∞) (by simp)).differentiableAt (by simp)) c
+
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+
+ 
+def Smooth.part (f : Smooth A) (p : ℂ →L[ℝ] ℝ) : SmoothRealFunction A where
+  value x := p (f x)
+  smooth i := by
+    have hh := p.contDiff.comp_contDiffOn (f.smooth i)
+    have hc := hh.comp (coordinateEquiv d).symm.contDiff.contDiffOn (by
+      intro z hz
+      simpa only [ComplexAtlas.euclideanChart_target, mem_preimage,
+        ContinuousLinearEquiv.apply_symm_apply] using hz)
+    convert hc using 1
+    ext z
+    simp only [Function.comp_apply, ComplexAtlas.euclideanChart_symm_apply,
+      ContinuousLinearEquiv.apply_symm_apply]
+
+@[simp] lemma Smooth.part_value (f : Smooth A) (p : ℂ →L[ℝ] ℝ) (x : X) :
+    (f.part p).value x = p (f x) := rfl
+
+ 
+def complexLValue (g : KaehlerMetric A) (f : Smooth A) : Smooth A :=
+  Smooth.ofReal (g.laplacian (f.part Complex.reCLM)) +
+    Smooth.mul (Smooth.const Complex.I) (Smooth.ofReal (g.laplacian (f.part Complex.imCLM)))
+
+ 
+def localL (g : KaehlerMetric A) (i : Fin A.count) (f : EC d → ℂ) (y : EC d) : ℂ :=
+  ∑ k, ∑ l, (traceBilin (g.matrix i (coordinateEquiv d y))
+    (rankTwo (stdOrthonormalBasis ℝ (EC d) k) (stdOrthonormalBasis ℝ (EC d) l)) : ℂ) *
+      fderiv ℝ (fderiv ℝ f) y (stdOrthonormalBasis ℝ (EC d) k) (stdOrthonormalBasis ℝ (EC d) l)
+
+lemma part_laplacian (g : KaehlerMetric A) (f : Smooth A)
+    (p : ℂ →L[ℝ] ℝ) (i : Fin A.count) {x : X} (hx : x ∈ (A.euclideanChart i).source) :
+    (g.laplacian (f.part p)).value x =
+      ∑ k, ∑ l, traceBilin (g.matrix i (coordinateEquiv d (A.euclideanChart i x)))
+        (rankTwo (stdOrthonormalBasis ℝ (EC d) k) (stdOrthonormalBasis ℝ (EC d) l)) *
+        p (fderiv ℝ (fderiv ℝ (f ∘ (A.euclideanChart i).symm))
+          (A.euclideanChart i x) (stdOrthonormalBasis ℝ (EC d) k) (stdOrthonormalBasis ℝ (EC d) l)) := by
+  have hxc : x ∈ (A.chart i).source := by simpa using hx
+  change g.laplacianValue (f.part p) x = _
+  rw [g.laplacianValue_local (f.part p) i hxc]
+  have hy : coordinateEquiv d (A.euclideanChart i x) ∈ (A.chart i).target := by
+    simpa only [ComplexAtlas.euclideanChart_apply, ContinuousLinearEquiv.apply_symm_apply]
+      using (A.chart i).mapsTo hxc
+  have ht := g.traceBilin_localExpression (f.part p) i hy
+  rw [ComplexAtlas.euclideanChart_apply, ContinuousLinearEquiv.apply_symm_apply] at ht
+  rw [← ht]
+  rw [bilin_expansion (stdOrthonormalBasis ℝ (EC d))
+    (fderiv ℝ (fderiv ℝ ((f.part p).localExpression i ∘ coordinateEquiv d))
+      ((coordinateEquiv d).symm (A.chart i x))) ]
+  simp only [map_sum, map_smul, smul_eq_mul]
+  apply Finset.sum_congr rfl
+  intro k _
+  apply Finset.sum_congr rfl
+  intro l _
+  rw [mul_comm]
+  congr 1
+  exact second_postcomp ((f.smooth i).contDiffAt
+    ((A.euclideanChart i).open_target.mem_nhds ((A.euclideanChart i).mapsTo hx))) p _ _
+
+lemma complexLValue_local (g : KaehlerMetric A) (f : Smooth A)
+    (i : Fin A.count) {x : X} (hx : x ∈ (A.euclideanChart i).source) :
+    complexLValue g f x = localL g i (f ∘ (A.euclideanChart i).symm) (A.euclideanChart i x) := by
+  change ((g.laplacian (f.part Complex.reCLM)).value x : ℂ) +
+    Complex.I * ((g.laplacian (f.part Complex.imCLM)).value x : ℂ) = _
+  rw [part_laplacian g f Complex.reCLM i hx, part_laplacian g f Complex.imCLM i hx]
+  simp only [localL, Complex.ofReal_sum, Complex.ofReal_mul, Finset.mul_sum, ← Finset.sum_add_distrib]
+  apply Finset.sum_congr rfl
+  intro k _
+  apply Finset.sum_congr rfl
+  intro l _
+  simp only [Complex.reCLM_apply, Complex.imCLM_apply]
+  rw [mul_left_comm Complex.I, ← mul_add, mul_comm Complex.I, Complex.re_add_im]
+
+lemma localL_congr (g : KaehlerMetric A) (i : Fin A.count)
+    {f h : EC d → ℂ} {x : EC d} (he : f =ᶠ[𝓝 x] h) :
+    localL g i f x = localL g i h x := by
+  simp only [localL, he.fderiv.fderiv_eq]
+
+lemma complexLValue_add (g : KaehlerMetric A) (f h : Smooth A) :
+    complexLValue g (f + h) = complexLValue g f + complexLValue g h := by
+  apply Smooth.ext
+  intro x
+  obtain ⟨i, hi⟩ := A.covers x
+  have hx : x ∈ (A.euclideanChart i).source := by simpa using hi
+  simp only [Smooth.add_apply, complexLValue_local g _ i hx, localL]
+  have hf := (f.smooth i).contDiffAt ((A.euclideanChart i).open_target.mem_nhds
+    ((A.euclideanChart i).mapsTo hx))
+  have hh := (h.smooth i).contDiffAt ((A.euclideanChart i).open_target.mem_nhds
+    ((A.euclideanChart i).mapsTo hx))
+  rw [show ((f + h : Smooth A) ∘ (A.euclideanChart i).symm) =
+    (fun y => (f ∘ (A.euclideanChart i).symm) y + (h ∘ (A.euclideanChart i).symm) y) from rfl,
+    second_add hf hh]
+  simp only [add_apply, mul_add, Finset.sum_add_distrib]
+
+lemma complexLValue_smul (g : KaehlerMetric A) (c : ℝ) (f : Smooth A) :
+    complexLValue g (c • f) = c • complexLValue g f := by
+  apply Smooth.ext
+  intro x
+  obtain ⟨i, hi⟩ := A.covers x
+  have hx : x ∈ (A.euclideanChart i).source := by simpa using hi
+  simp only [Smooth.smul_apply, complexLValue_local g _ i hx, localL]
+  have hf := (f.smooth i).contDiffAt ((A.euclideanChart i).open_target.mem_nhds
+    ((A.euclideanChart i).mapsTo hx))
+  rw [show ((c • f : Smooth A) ∘ (A.euclideanChart i).symm) =
+    (fun y => c • (f ∘ (A.euclideanChart i).symm) y) from rfl,
+    second_real_smul hf c]
+  simp only [smul_apply, mul_smul_comm, Finset.smul_sum]
+
+ 
+def complexL (g : KaehlerMetric A) : Smooth A →ₗ[ℝ] Smooth A where
+  toFun := complexLValue g
+  map_add' := complexLValue_add g
+  map_smul' := complexLValue_smul g
+
+lemma complexL_tsupport (g : KaehlerMetric A) (f : Smooth A) :
+    tsupport (complexL g f : X → ℂ) ⊆ tsupport (f : X → ℂ) := by
+  apply closure_minimal _ (isClosed_tsupport _)
+  intro x hx
+  by_contra hn
+  obtain ⟨i, hi⟩ := A.covers x
+  let e := A.euclideanChart i
+  have he : x ∈ e.source := by simpa [e] using hi
+  have hzero : f =ᶠ[𝓝 x] (fun _ => (0 : ℂ)) :=
+    Filter.eventuallyEq_iff_exists_mem.mpr
+      ⟨(tsupport (f : X → ℂ))ᶜ, (isClosed_tsupport _).isOpen_compl.mem_nhds hn,
+        fun y hy => image_eq_zero_of_notMem_tsupport hy⟩
+  have hz : (f ∘ e.symm) =ᶠ[𝓝 (e x)] (fun _ => 0) := by
+    have hc := (e.symm.continuousAt (e.mapsTo he)).tendsto
+    rw [e.left_inv he] at hc
+    exact hc.eventually hzero
+  have hv := complexLValue_local g f i he
+  change complexLValue g f x ≠ 0 at hx
+  rw [hv, localL_congr g i hz] at hx
+  exact hx (by simp [localL])
+
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/ConstantProjection.lean b/lean/OAI/Geometry/PrescribedPotential/ConstantProjection.lean
new file mode 100644
index 0000000000000000000000000000000000000000..3e501ea68b71e52944fce58b748c9328b1c13f84
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/ConstantProjection.lean
@@ -0,0 +1,62 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.GlobalPartition
+
+namespace OAI
+
+/-! Constant Projection. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+open scoped SchwartzMap ContDiff Classical
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X]
+  {A : ComplexAtlas d X}
+
+def Smooth.constLinear : ℂ →ₗ[ℝ] Smooth A where
+  toFun := Smooth.const
+  map_add' := by intro c e; rfl
+  map_smul' := by intro c e; rfl
+
+namespace Localizers
+variable [T2Space X] [CompactSpace X] {ι : Type*} [Fintype ι] (D : Localizers A ι)
+
+def constants : ℂ →L[ℝ] D.Sobolev 0 :=
+  LinearMap.toContinuousLinearMap ((D.embed 0).comp Smooth.constLinear)
+
+@[simp] lemma constants_apply (c : ℂ) :
+    D.constants c = D.embed 0 (Smooth.const c) := rfl
+
+ 
+structure ConstantProjection where
+  proj : D.Sobolev 0 →L[ℝ] D.Sobolev 0
+  compact : IsCompactOperator proj
+  value : ∀ u, ∃ c : ℂ, proj u = D.embed 0 (Smooth.const c)
+  fixed : ∀ c : ℂ, proj (D.embed 0 (Smooth.const c)) = D.embed 0 (Smooth.const c)
+
+theorem constantProjection_exists : Nonempty D.ConstantProjection := by
+  let S := (D.constants : ℂ →ₗ[ℝ] D.Sobolev 0).range
+  let : FiniteDimensional ℝ S := LinearMap.finiteDimensional_range _
+  obtain ⟨r, hr⟩ := Submodule.ClosedComplemented.of_finiteDimensional S
+  let P := S.subtypeL ∘L r
+  refine ⟨⟨P, ?_, ?_, ?_⟩⟩
+  · exact ((isCompactOperator_id (E := S)).comp_clm r).clm_comp S.subtypeL
+  · intro u
+    obtain ⟨c, hc⟩ := (r u).property
+    exact ⟨c, hc.symm⟩
+  · intro c
+    have hc : D.embed 0 (Smooth.const c) ∈ S := ⟨c, rfl⟩
+    have h := congrArg Subtype.val (hr ⟨D.embed 0 (Smooth.const c), hc⟩)
+    exact h
+
+end Localizers
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/ContactSetABP.lean b/lean/OAI/Geometry/PrescribedPotential/ContactSetABP.lean
new file mode 100644
index 0000000000000000000000000000000000000000..a3062905a58d97053300d76f77e10ef5c8f57ba8
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/ContactSetABP.lean
@@ -0,0 +1,186 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.AnalyticSupport
+import OAI.Geometry.PrescribedPotential.CircleRadialCalculus
+
+namespace OAI
+
+/-! Contact Set A B P. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Metric MeasureTheory
+open scoped ContDiff InnerProductSpace
+namespace PotentialABP
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E]
+
+ 
+def contacts (u : E → ℝ) (c : E) (r a : ℝ) : Set E :=
+  {x | x ∈ closedBall c r ∧ ‖gradient u x‖ ≤ a ∧
+    ∀ y ∈ closedBall c r, u x + inner ℝ (gradient u x) (y-x) ≤ u y}
+
+lemma gradient_continuous {u : E → ℝ} (hu : ContDiff ℝ ∞ u) :
+    Continuous (gradient u) := by
+  exact (InnerProductSpace.toDual ℝ E).symm.continuous.comp
+    (hu.continuous_fderiv (by simp))
+
+lemma contacts_isClosed {u : E → ℝ} (hu : ContDiff ℝ ∞ u) (c : E) (r a : ℝ) :
+    IsClosed (contacts u c r a) := by
+  have hg := gradient_continuous hu
+  change IsClosed (closedBall c r ∩ ({x | ‖gradient u x‖ ≤ a} ∩
+    {x | ∀ y ∈ closedBall c r, u x + inner ℝ (gradient u x) (y-x) ≤ u y}))
+  apply isClosed_closedBall.inter
+  apply IsClosed.inter (isClosed_le hg.norm continuous_const)
+  simp only [ofPred_forall]
+  apply isClosed_iInter
+  intro y
+  apply isClosed_iInter
+  intro _
+  exact isClosed_le (hu.continuous.add (hg.inner (continuous_const.sub continuous_id)))
+    continuous_const
+
+lemma contacts_sublevel {u : E → ℝ} {c : E} {r a : ℝ} (hr : 0 ≤ r) (ha : 0 ≤ a)
+    {x : E} (hx : x ∈ contacts u c r a) : u x ≤ u c + a*r := by
+  have h := hx.2.2 c (mem_closedBall_self hr)
+  have hinner : - (a*r) ≤ inner ℝ (gradient u x) (c-x) := by
+    have hnorm : ‖c-x‖ ≤ r := by simpa [dist_eq_norm, norm_sub_rev] using hx.1
+    have hb := abs_real_inner_le_norm (gradient u x) (c-x)
+    have hc := mul_le_mul hx.2.1 hnorm (norm_nonneg _) ha
+    exact le_trans (neg_le_neg hc) ((abs_le.mp hb).1)
+  linarith
+
+lemma contacts_interior {u : E → ℝ} {c : E} {r a b : ℝ}
+    (hr : 0 < r) (ha : 0 ≤ a) (hgap : a*r < b)
+    (hboundary : ∀ x, dist x c = r → u c + b ≤ u x) :
+    contacts u c r a ⊆ ball c r := by
+  intro x hx
+  have hs := contacts_sublevel hr.le ha hx
+  have hd : dist x c ≤ r := hx.1
+  apply lt_of_le_of_ne hd
+  intro he
+  linarith [hboundary x he]
+
+lemma tilted_minimum_gradient {u : E → ℝ} (hu : ContDiff ℝ ∞ u)
+    {c : E} {r : ℝ} {x p : E} (hx : x ∈ ball c r)
+    (hm : IsMinOn (fun y => u y - inner ℝ p (y-c)) (closedBall c r) x) :
+    gradient u x = p := by
+  have hloc : IsLocalMin (fun y => u y - inner ℝ p (y-c)) x := by
+    filter_upwards [isOpen_ball.mem_nhds hx] with y hy
+    exact hm (ball_subset_closedBall hy)
+  have hp : HasFDerivAt (fun y : E => inner ℝ p (y-c))
+      (InnerProductSpace.toDual ℝ E p) x := by
+    simpa only [ContinuousLinearMap.comp_id, Function.comp_def, id_eq, InnerProductSpace.toDual_apply_apply] using
+      (InnerProductSpace.toDual ℝ E p).hasFDerivAt.comp x
+        ((hasFDerivAt_id x).sub_const c)
+  have hd := ((hu.differentiable (by simp) x).hasFDerivAt.sub hp).fderiv
+  change fderiv ℝ (fun y => u y - inner ℝ p (y-c)) x = _ at hd
+  rw [hloc.fderiv_eq_zero] at hd
+  apply (InnerProductSpace.toDual ℝ E).injective
+  rw [toDual_gradient]
+  exact sub_eq_zero.mp hd.symm
+
+ 
+
+lemma gradient_image_contacts {u : E → ℝ} (hu : ContDiff ℝ ∞ u)
+    {c : E} {r a b : ℝ} (hr : 0 < r) (ha : 0 ≤ a) (hgap : a*r < b)
+    (hboundary : ∀ x, dist x c = r → u c + b ≤ u x) :
+    closedBall (0:E) a ⊆ gradient u '' contacts u c r a := by
+  intro p hp
+  have hp' : ‖p‖ ≤ a := by simpa using hp
+  have hcont : Continuous (fun y => u y - inner ℝ p (y-c)) :=
+    hu.continuous.sub (continuous_const.inner (continuous_id.sub continuous_const))
+  obtain ⟨x,hx,hm⟩ := (isCompact_closedBall c r).exists_isMinOn
+    ⟨c,mem_closedBall_self hr.le⟩
+    hcont.continuousOn
+  have hmin : u x - inner ℝ p (x-c) ≤ u c - inner ℝ p (c-c) :=
+    hm (mem_closedBall_self hr.le)
+  have hupper : u x ≤ u c + a*r := by
+    have hh := (real_inner_le_norm p (x-c)).trans
+      (mul_le_mul hp' (by simpa [dist_eq_norm] using hx) (norm_nonneg _) ha)
+    simp only [sub_self, inner_zero_right, sub_zero] at hmin
+    linarith
+  have hxi : x ∈ ball c r := by
+    change dist x c < r
+    apply lt_of_le_of_ne (show dist x c ≤ r from hx)
+    intro he
+    linarith [hboundary x he]
+  have hg := tilted_minimum_gradient hu hxi hm
+  refine ⟨x,⟨hx,by simpa [hg] using hp',?_⟩,hg⟩
+  intro y hy
+  have he : u x - inner ℝ p (x-c) ≤ u y - inner ℝ p (y-c) := hm hy
+  rw [hg]
+  have hi : inner ℝ p (y-x) = inner ℝ p (y-c) - inner ℝ p (x-c) := by
+    rw [← inner_sub_right]
+    congr 1
+    abel
+  rw [hi]
+  linarith
+
+lemma gradient_contDiff {u : E → ℝ} (hu : ContDiff ℝ ∞ u) :
+    ContDiff ℝ ∞ (gradient u) := by
+  exact (InnerProductSpace.toDual ℝ E).symm.contDiff.comp
+    (hu.fderiv_right (by simp))
+
+ 
+
+lemma contact_measure_estimate [MeasurableSpace E] [BorelSpace E]
+    (μ : Measure E) [μ.IsAddHaarMeasure]
+    {u : E → ℝ} (hu : ContDiff ℝ ∞ u) {c : E} {r a b C : ℝ}
+    (hr : 0 < r) (ha : 0 ≤ a) (hgap : a*r < b)
+    (hboundary : ∀ x, dist x c = r → u c + b ≤ u x)
+    (hdet : ∀ x ∈ contacts u c r a,
+      |(fderiv ℝ (gradient u) x).det| ≤ C) :
+    μ (closedBall (0:E) a) ≤ ENNReal.ofReal C * μ (contacts u c r a) := by
+  have hs := (contacts_isClosed hu c r a).measurableSet
+  calc
+    μ (closedBall (0:E) a) ≤ μ (gradient u '' contacts u c r a) :=
+      measure_mono (gradient_image_contacts hu hr ha hgap hboundary)
+    _ ≤ ∫⁻ x in contacts u c r a, ENNReal.ofReal |(fderiv ℝ (gradient u) x).det| ∂μ :=
+      addHaar_image_le_lintegral_abs_det_fderiv μ hs
+        (fun x _ => ((gradient_contDiff hu).differentiable (by simp) x).hasFDerivAt.hasFDerivWithinAt)
+    _ ≤ ∫⁻ _ in contacts u c r a, ENNReal.ofReal C ∂μ :=
+      setLIntegral_mono' hs (fun x hx => ENNReal.ofReal_le_ofReal (hdet x hx))
+    _ = ENNReal.ofReal C * μ (contacts u c r a) := by simp
+
+ 
+lemma contacts_hessian_nonneg {u : E → ℝ} (hu : ContDiff ℝ ∞ u)
+    {c : E} {r a : ℝ} {x : E} (hx : x ∈ contacts u c r a)
+    (hxi : x ∈ ball c r) (v : E) :
+    0 ≤ fderiv ℝ (fderiv ℝ u) x v v := by
+  let p := gradient u x
+  let l : E → ℝ := fun y => u x + inner ℝ p (y-x)
+  have hl : HasFDerivAt l (InnerProductSpace.toDual ℝ E p) x := by
+    simpa [l, ContinuousLinearMap.comp_id, Function.comp_def, id_eq,
+      InnerProductSpace.toDual_apply_apply] using
+      ((InnerProductSpace.toDual ℝ E p).hasFDerivAt.comp x
+        ((hasFDerivAt_id x).sub_const x)).const_add (u x)
+  have hld (y : E) : fderiv ℝ l y = InnerProductSpace.toDual ℝ E p := by
+    simpa [l, ContinuousLinearMap.comp_id, Function.comp_def, id_eq,
+      InnerProductSpace.toDual_apply_apply] using
+      ((InnerProductSpace.toDual ℝ E p).hasFDerivAt.comp y
+        ((hasFDerivAt_id y).sub_const x)).const_add (u x) |>.fderiv
+  have hlc : ContDiff ℝ ∞ l := contDiff_const.add
+    (ContDiff.inner ℝ contDiff_const (contDiff_id.sub contDiff_const))
+  have hmax : IsLocalMax (fun y => l y - u y) x := by
+    filter_upwards [isOpen_ball.mem_nhds hxi] with y hy
+    have hh := hx.2.2 y (ball_subset_closedBall hy)
+    simpa [l, p] using (sub_nonpos.mpr hh)
+  have hs := EllipticKernel.second_fderiv_nonpos_at_localMax
+    (hlc.sub hu).contDiffAt hmax v
+  have he : fderiv ℝ (fun y => l y - u y) =
+      fun y => InnerProductSpace.toDual ℝ E p - fderiv ℝ u y := by
+    funext y
+    change fderiv ℝ (l - u) y = _
+    rw [fderiv_sub (hlc.differentiable (by simp) y) (hu.differentiable (by simp) y), hld]
+  rw [he, fderiv_const_sub] at hs
+  simpa using hs
+
+end PotentialABP
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/CoordinateBalls.lean b/lean/OAI/Geometry/PrescribedPotential/CoordinateBalls.lean
new file mode 100644
index 0000000000000000000000000000000000000000..d4b98dab6de55eb91142e26c12d8f29af741b6de
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/CoordinateBalls.lean
@@ -0,0 +1,124 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.KaehlerClosedDerivatives
+import OAI.Geometry.PrescribedPotential.ParametrixAtlas
+
+namespace OAI
+
+/-! Coordinate Balls. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Metric Filter Topology MeasureTheory
+open scoped ContDiff
+namespace Anticanonical.SourceSmooth
+open EllipticKernel
+local instance coordinateBallRealIP (d : ℕ) : InnerProductSpace ℝ (EC d) :=
+  InnerProductSpace.rclikeToReal ℂ (EC d)
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+
+structure CoordinateBall (A : ComplexAtlas d X) where
+  index : Fin A.count
+  center : EC d
+  radius : ℝ
+  radius_pos : 0 < radius
+  closure_sub : closedBall center (4*radius) ⊆ (A.euclideanChart index).target
+
+namespace CoordinateBall
+variable (p : CoordinateBall A)
+def source : Set X := (A.euclideanChart p.index).source ∩
+  (A.euclideanChart p.index) ⁻¹' ball p.center p.radius
+
+lemma ball_sub : ball p.center p.radius ⊆ (A.euclideanChart p.index).target := by
+  apply subset_trans _ p.closure_sub
+  exact (ball_subset_closedBall.trans (closedBall_subset_closedBall (by linarith [p.radius_pos])))
+
+lemma isOpen_source : IsOpen p.source :=
+  (A.euclideanChart p.index).isOpen_inter_preimage isOpen_ball
+
+lemma source_nonempty : p.source.Nonempty := by
+  have hc : p.center ∈ (A.euclideanChart p.index).target :=
+    p.ball_sub (mem_ball_self p.radius_pos)
+  refine ⟨(A.euclideanChart p.index).symm p.center,
+    (A.euclideanChart p.index).symm.mapsTo hc, ?_⟩
+  change A.euclideanChart p.index ((A.euclideanChart p.index).symm p.center) ∈ ball p.center p.radius
+  rw [(A.euclideanChart p.index).right_inv hc]
+  exact mem_ball_self p.radius_pos
+
+lemma exists_at (x : X) : ∃ p : CoordinateBall A, x ∈ p.source := by
+  obtain ⟨i, hi⟩ := A.covers x
+  let e := A.euclideanChart i
+  have hxe : x ∈ e.source := by simpa [e] using hi
+  have hxt : e x ∈ e.target := e.mapsTo hxe
+  obtain ⟨r, hr, hsub⟩ := Metric.nhds_basis_closedBall.mem_iff.mp (e.open_target.mem_nhds hxt)
+  refine ⟨⟨i, e x, r/4, by positivity, ?_⟩, hxe, ?_⟩
+  · convert hsub using 1; ring_nf
+  · exact mem_ball_self (by positivity)
+
+theorem finite_cover [CompactSpace X] :
+    ∃ S : Finset (CoordinateBall A), ∀ x, ∃ p ∈ S, x ∈ p.source := by
+  classical
+  obtain ⟨S, hS⟩ := isCompact_univ.elim_finite_subcover
+    (fun p : CoordinateBall A => p.source) (fun p => p.isOpen_source) (by
+      intro x _
+      obtain ⟨p, hp⟩ := exists_at (A := A) x
+      exact mem_iUnion.mpr ⟨p, hp⟩)
+  refine ⟨S, fun x => ?_⟩
+  obtain ⟨q, hq, hxq⟩ := mem_iUnion₂.mp (hS (mem_univ x))
+  exact ⟨q, hq, hxq⟩
+
+section Measure
+variable [MeasurableSpace X] [BorelSpace X]
+
+def measure : Measure X :=
+  Measure.map (A.euclideanChart p.index).symm (volume.restrict (ball p.center p.radius))
+
+instance : IsFiniteMeasure p.measure := by
+  have : IsFiniteMeasure (volume.restrict (ball p.center p.radius)) :=
+    isFiniteMeasure_restrict.mpr isBounded_ball.measure_lt_top.ne
+  unfold measure
+  infer_instance
+
+lemma symm_aemeasurable : AEMeasurable (A.euclideanChart p.index).symm
+    (volume.restrict (ball p.center p.radius)) :=
+  ((A.euclideanChart p.index).continuousOn_symm.mono p.ball_sub).aemeasurable
+    isOpen_ball.measurableSet
+
+lemma integral_measure (f : X → ℝ) (hf : Continuous f) :
+    (∫ x, f x ∂p.measure) =
+      ∫ z in ball p.center p.radius, f ((A.euclideanChart p.index).symm z) := by
+  exact integral_map p.symm_aemeasurable hf.aestronglyMeasurable
+
+lemma measure_open_pos {U : Set X} (hU : IsOpen U) (hnear : (p.source ∩ U).Nonempty) :
+    0 < p.measure.real U := by
+  have he : ((A.euclideanChart p.index).symm ⁻¹' U) ∩ ball p.center p.radius =
+      ball p.center p.radius ∩
+        ((A.euclideanChart p.index).target ∩ (A.euclideanChart p.index).symm ⁻¹' U) := by
+    ext z
+    simp only [mem_inter_iff, mem_preimage]
+    constructor
+    · rintro ⟨hu, hz⟩; exact ⟨hz, p.ball_sub hz, hu⟩
+    · rintro ⟨hz, _, hu⟩; exact ⟨hu, hz⟩
+  have ho : IsOpen (((A.euclideanChart p.index).symm ⁻¹' U) ∩ ball p.center p.radius) := by
+    rw [he]
+    exact isOpen_ball.inter ((A.euclideanChart p.index).symm.isOpen_inter_preimage hU)
+  have hn : (((A.euclideanChart p.index).symm ⁻¹' U) ∩ ball p.center p.radius).Nonempty := by
+    obtain ⟨x, hx, hxu⟩ := hnear
+    refine ⟨A.euclideanChart p.index x, ?_, hx.2⟩
+    simpa only [mem_preimage, (A.euclideanChart p.index).left_inv hx.1] using hxu
+  have hm : 0 < p.measure U := by
+    change 0 < Measure.map _ _ U
+    rw [Measure.map_apply_of_aemeasurable p.symm_aemeasurable hU.measurableSet,
+      Measure.restrict_apply' isOpen_ball.measurableSet]
+    exact ho.measure_pos volume hn
+  exact ENNReal.toReal_pos hm.ne' (measure_ne_top p.measure U)
+end Measure
+end CoordinateBall
+end Anticanonical.SourceSmooth
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/CoordinateEllipticity.lean b/lean/OAI/Geometry/PrescribedPotential/CoordinateEllipticity.lean
new file mode 100644
index 0000000000000000000000000000000000000000..1a73ebf79d95a7e939898da305eedabc1709cf5d
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/CoordinateEllipticity.lean
@@ -0,0 +1,125 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CircleRadialCalculus
+import OAI.Geometry.PrescribedPotential.CoordinateBalls
+
+namespace OAI
+
+/-! Coordinate Ellipticity. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Metric Filter Topology MeasureTheory Matrix
+open scoped ContDiff InnerProductSpace ComplexOrder Matrix.Norms.Elementwise
+namespace PotentialABP
+variable {d : ℕ}
+local notation "E" => EuclideanSpace ℂ (Fin d)
+local instance ellipticityRealIP (d : ℕ) : InnerProductSpace ℝ (EuclideanSpace ℂ (Fin d)) :=
+  InnerProductSpace.rclikeToReal ℂ (EuclideanSpace ℂ (Fin d))
+
+lemma matrix_quadratic_pos {H : Matrix (Fin d) (Fin d) ℂ} (hH : H.PosDef)
+    {v : E} (hv : v ≠ 0) : 0 < inner ℝ (H.toEuclideanLin v) v := by
+  rw [real_inner_comm, real_inner_eq_re_inner (𝕜 := ℂ), EuclideanSpace.inner_eq_star_dotProduct]
+  rw [dotProduct_comm]
+  apply hH.re_dotProduct_pos
+  intro h
+  apply hv
+  exact (EuclideanSpace.equiv (Fin d) ℂ).injective (by simpa using h)
+
+ 
+
+lemma compact_matrix_lower_bound {K : Set E} (hK : IsCompact K)
+    {G : E → Matrix (Fin d) (Fin d) ℂ}
+    (hcont : ContinuousOn (fun x => (Matrix.toEuclideanCLM (𝕜 := ℂ) (n := Fin d)) (G x)) K)
+    (hpos : ∀ x ∈ K, (G x).PosDef) :
+    ∃ δ : ℝ, 0 < δ ∧ ∀ x ∈ K, ∀ v : E,
+      δ*‖v‖^2 ≤ inner ℝ ((G x).toEuclideanLin v) v := by
+  let Q : E × E → ℝ := fun z => inner ℝ ((G z.1).toEuclideanLin z.2) z.2
+  let L := K ×ˢ sphere (0:E) 1
+  have hLc : IsCompact L := hK.prod (isCompact_sphere _ _)
+  have hQc : ContinuousOn Q L := by
+    have hc := hcont.comp continuous_fst.continuousOn (show MapsTo Prod.fst L K from fun z hz => hz.1)
+    exact (hc.clm_apply continuous_snd.continuousOn).inner continuous_snd.continuousOn
+  have hp : ∀ z ∈ L, 0 < Q z := by
+    rintro ⟨x,v⟩ ⟨hx,hv⟩
+    apply matrix_quadratic_pos (hpos x hx)
+    intro hv0
+    change v = 0 at hv0
+    have hw : ‖v‖ = 1 := by simpa [mem_sphere, dist_zero_right] using hv
+    rw [hv0, norm_zero] at hw
+    norm_num at hw
+  have hδ : ∃ δ : ℝ, 0 < δ ∧ ∀ z ∈ L, δ ≤ Q z := by
+    by_cases hn : L.Nonempty
+    · obtain ⟨z, hz, hzmin⟩ := hLc.exists_isMinOn hn hQc
+      exact ⟨Q z, hp z hz, fun y hy => hzmin hy⟩
+    · exact ⟨1, by norm_num, fun z hz => (hn ⟨z,hz⟩).elim⟩
+  obtain ⟨δ,hδ,hbound⟩ := hδ
+  refine ⟨δ,hδ,?_⟩
+  intro x hx v
+  by_cases hv : v = 0
+  · simp [hv]
+  have hn : 0 < ‖v‖ := norm_pos_iff.mpr hv
+  have hw : ‖(‖v‖⁻¹ : ℝ) • v‖ = 1 := by
+    rw [norm_smul, Real.norm_eq_abs, abs_of_pos (inv_pos.mpr hn), inv_mul_cancel₀ hn.ne']
+  have hb := hbound (x,(‖v‖⁻¹ : ℝ) • v) ⟨hx, by simpa [mem_sphere, dist_zero_right] using hw⟩
+  have he : Q (x,(‖v‖⁻¹ : ℝ) • v) = ‖v‖⁻¹*‖v‖⁻¹ * inner ℝ ((G x).toEuclideanLin v) v := by
+    dsimp only [Q]
+    have hmap : (G x).toEuclideanLin ((‖v‖⁻¹ : ℝ) • v) =
+        (‖v‖⁻¹ : ℝ) • (G x).toEuclideanLin v :=
+      ((G x).toEuclideanLin.restrictScalars ℝ).map_smul _ _
+    rw [hmap, real_inner_smul_left, real_inner_smul_right]
+    ring
+  rw [he] at hb
+  have hh := mul_le_mul_of_nonneg_left hb (sq_nonneg ‖v‖)
+  have hc : ‖v‖^2 * (‖v‖⁻¹ * ‖v‖⁻¹ * inner ℝ ((G x).toEuclideanLin v) v) =
+      inner ℝ ((G x).toEuclideanLin v) v := by field_simp
+  rw [hc] at hh
+  nlinarith
+end PotentialABP
+
+namespace Anticanonical.SourceSmooth
+open EllipticKernel PotentialABP
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+local instance coordinateEllipticityRealIP (d : ℕ) : InnerProductSpace ℝ (EC d) :=
+  InnerProductSpace.rclikeToReal ℂ (EC d)
+namespace CoordinateBall
+variable (p : CoordinateBall A)
+
+lemma metric_continuousOn (g : KaehlerMetric A) :
+    ContinuousOn (fun z : EC d =>
+      (Matrix.toEuclideanCLM (𝕜 := ℂ) (n := Fin d)) (g.matrix p.index (coordinateEquiv d z)))
+      (closedBall p.center (3*p.radius)) := by
+  apply (Matrix.toEuclideanCLM (𝕜 := ℂ) (n := Fin d)).toAlgEquiv.toLinearMap.continuous_of_finiteDimensional.comp_continuousOn
+  have hs : ContinuousOn (fun z : EC d => g.matrix p.index (coordinateEquiv d z))
+      (A.euclideanChart p.index).target := by
+    apply continuousOn_pi.mpr
+    intro i
+    apply continuousOn_pi.mpr
+    intro j
+    have hij := (contDiffOn_pi.mp (contDiffOn_pi.mp (g.smooth p.index) i) j).continuousOn
+    rw [ComplexAtlas.euclideanChart_target]
+    convert hij.comp (coordinateEquiv d).continuous.continuousOn
+      (show Set.MapsTo (coordinateEquiv d)
+        ((coordinateEquiv d) ⁻¹' (A.chart p.index).target) (A.chart p.index).target from
+          fun _ hz => hz) using 1
+    rfl
+  exact hs.mono (fun z hz => p.closure_sub
+    (closedBall_subset_closedBall (by linarith [p.radius_pos]) hz))
+
+lemma metric_lower_bound (g : KaehlerMetric A) : ∃ δ : ℝ, 0 < δ ∧
+    ∀ z ∈ closedBall p.center (3*p.radius), ∀ v : EC d,
+      δ*‖v‖^2 ≤ inner ℝ ((g.matrix p.index (coordinateEquiv d z)).toEuclideanLin v) v := by
+  apply compact_matrix_lower_bound (isCompact_closedBall _ _) (p.metric_continuousOn g)
+  intro z hz
+  apply g.positive
+  have ht := p.closure_sub (closedBall_subset_closedBall (by linarith [p.radius_pos]) hz)
+  simpa only [ComplexAtlas.euclideanChart_target, mem_preimage] using ht
+end CoordinateBall
+end Anticanonical.SourceSmooth
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/CoordinateExtension.lean b/lean/OAI/Geometry/PrescribedPotential/CoordinateExtension.lean
new file mode 100644
index 0000000000000000000000000000000000000000..b96871911eea8f8b9c2204ddf4730debf7d6ef56
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/CoordinateExtension.lean
@@ -0,0 +1,54 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CoordinateBalls
+import OAI.Geometry.PrescribedPotential.CoordinatePotential
+
+namespace OAI
+
+/-! Coordinate Extension. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Metric Filter Topology
+open scoped ContDiff
+namespace Anticanonical.SourceSmooth
+open EllipticKernel
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+namespace CoordinateBall
+variable (p : CoordinateBall A)
+
+def coordExtension (φ : SmoothRealFunction A) : Coordinates d → ℝ :=
+  p.extension φ ∘ (coordinateEquiv d).symm
+
+lemma coordExtension_smooth (φ : SmoothRealFunction A) :
+    ContDiff ℝ ∞ (p.coordExtension φ) := by
+  exact (p.extension_smooth φ).comp
+    ((coordinateEquiv d).symm.toContinuousLinearMap.restrictScalars ℝ).contDiff
+
+lemma coordExtension_apply (φ : SmoothRealFunction A) (z : EC d) :
+    p.coordExtension φ (coordinateEquiv d z) = p.extension φ z := by
+  simp only [coordExtension, Function.comp_apply, ContinuousLinearEquiv.symm_apply_apply]
+
+lemma coordExtension_eventuallyEq (φ : SmoothRealFunction A) {z : EC d}
+    (hz : z ∈ closedBall p.center (3*p.radius)) :
+    p.coordExtension φ =ᶠ[𝓝 (coordinateEquiv d z)] φ.localExpression p.index := by
+  have he := (p.extension_eventuallyEq φ hz).comp_tendsto
+    ((coordinateEquiv d).symm.continuousAt.tendsto)
+  simpa only [coordExtension, SmoothRealFunction.localExpression, Function.comp_def,
+    ComplexAtlas.euclideanChart_symm_apply, ContinuousLinearEquiv.apply_symm_apply,
+    ContinuousLinearEquiv.symm_apply_apply, coordinateEquiv, PiLp.coe_continuousLinearEquiv, PiLp.coe_symm_continuousLinearEquiv, WithLp.ofLp_toLp] using he
+
+lemma coordExtension_hessian (φ : SmoothRealFunction A) {z : EC d}
+    (hz : z ∈ closedBall p.center (3*p.radius)) :
+    PotentialKaehler.potentialMatrix (p.coordExtension φ) (coordinateEquiv d z) =
+      φ.hessian p.index (coordinateEquiv d z) :=
+  PotentialKaehler.potentialMatrix_congr (p.coordExtension_eventuallyEq φ hz)
+end CoordinateBall
+end Anticanonical.SourceSmooth
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/CoordinateInnerCover.lean b/lean/OAI/Geometry/PrescribedPotential/CoordinateInnerCover.lean
new file mode 100644
index 0000000000000000000000000000000000000000..c68523cfd515a2ec2162af74a0d7a44e5c0833a9
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/CoordinateInnerCover.lean
@@ -0,0 +1,66 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CoordinateBalls
+
+namespace OAI
+
+/-! Coordinate Inner Cover. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Metric Filter Topology
+namespace Anticanonical.SourceSmooth
+open EllipticKernel
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+namespace CoordinateBall
+variable (p : CoordinateBall A)
+
+def half : CoordinateBall A where
+  index := p.index
+  center := p.center
+  radius := p.radius/2
+  radius_pos := by have hp := p.radius_pos; positivity
+  closure_sub := (closedBall_subset_closedBall (by linarith [p.radius_pos])).trans p.closure_sub
+
+lemma half_source_sub : p.half.source ⊆ p.source := by
+  rintro x ⟨hx,hb⟩
+  exact ⟨hx, ball_subset_ball (by change p.radius/2 ≤ p.radius; linarith [p.radius_pos]) hb⟩
+
+lemma centered_small_closedBall_subset {c : EC d} (hc : c ∈ ball p.center (p.radius/2)) :
+    closedBall c (p.radius/2) ⊆ ball p.center p.radius := by
+  intro z hz
+  have hc' : dist c p.center < p.radius/2 := hc
+  have hz' : dist z c ≤ p.radius/2 := hz
+  change dist z p.center < p.radius
+  linarith [dist_triangle z c p.center]
+
+lemma half_exists_at (x : X) : ∃ p : CoordinateBall A, x ∈ p.half.source := by
+  obtain ⟨i,hi⟩ := A.covers x
+  let e := A.euclideanChart i
+  have hs : x ∈ e.source := by simpa [e] using hi
+  obtain ⟨r,hr,hsub⟩ := Metric.nhds_basis_closedBall.mem_iff.mp
+    (e.open_target.mem_nhds (e.mapsTo hs))
+  refine ⟨⟨i,e x,r/4,by positivity,?_⟩,hs,?_⟩
+  · convert hsub using 1; ring_nf
+  · exact mem_ball_self (by change 0 < r/4/2; positivity)
+
+theorem finite_inner_cover [CompactSpace X] :
+    ∃ S : Finset (CoordinateBall A), ∀ x, ∃ p ∈ S, x ∈ p.half.source := by
+  classical
+  obtain ⟨S,hS⟩ := isCompact_univ.elim_finite_subcover
+    (fun p : CoordinateBall A => p.half.source) (fun p => p.half.isOpen_source) (by
+      intro x _
+      obtain ⟨p,hp⟩ := half_exists_at (A := A) x
+      exact mem_iUnion.mpr ⟨p,hp⟩)
+  refine ⟨S, fun x => ?_⟩
+  obtain ⟨p,hp,hx⟩ := mem_iUnion₂.mp (hS (mem_univ x))
+  exact ⟨p,hp,hx⟩
+end CoordinateBall
+end Anticanonical.SourceSmooth
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/CoordinateMinimumBound.lean b/lean/OAI/Geometry/PrescribedPotential/CoordinateMinimumBound.lean
new file mode 100644
index 0000000000000000000000000000000000000000..f89aff0808475ad43be1addd3389c46a98a8f69f
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/CoordinateMinimumBound.lean
@@ -0,0 +1,88 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.ABPIntegralBound
+import OAI.Geometry.PrescribedPotential.CoordinateBalls
+import OAI.Geometry.PrescribedPotential.CoordinateEllipticity
+import OAI.Geometry.PrescribedPotential.CoordinateExtension
+import OAI.Geometry.PrescribedPotential.CoordinateInnerCover
+
+namespace OAI
+
+/-! Coordinate Minimum Bound. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Metric Filter Topology MeasureTheory Matrix
+open scoped ContDiff InnerProductSpace ComplexOrder
+namespace Anticanonical.SourceSmooth
+open EllipticKernel PotentialABP
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+local instance coordinateMinimumRealIP (d : ℕ) : InnerProductSpace ℝ (EC d) :=
+  InnerProductSpace.rclikeToReal ℂ (EC d)
+namespace CoordinateBall
+variable [MeasurableSpace X] [BorelSpace X] (p : CoordinateBall A)
+
+lemma minimum_bound (g : KaehlerMetric A) (B : ℝ) {K : ℝ} (hK : 0 < K) :
+    ∃ C : ℝ, ∀ φ : SmoothRealFunction A,
+      (∀ x, φ.value x ≤ 0) → (∫ x, -φ.value x ∂p.measure) ≤ B →
+      (∀ z ∈ closedBall p.center (3*p.radius),
+        |(g.matrix p.index (coordinateEquiv d z) + φ.hessian p.index (coordinateEquiv d z)).det.re| ≤ K) →
+      ∀ x ∈ p.half.source, (∀ y, φ.value x ≤ φ.value y) → -φ.value x ≤ C := by
+  obtain ⟨δ,hδ,hδb⟩ := p.metric_lower_bound g
+  let r := p.radius/2
+  have hr : 0 < r := by dsimp only [r]; exact half_pos p.radius_pos
+  let C := δ*r^2/2+B*((4:ℝ)^(2*d)*K^2)/volume.real (closedBall (0:EC d) (δ*r/2))
+  refine ⟨C, ?_⟩
+  intro φ hn hB hdet x hx hmin
+  let c := A.euclideanChart p.index x
+  have hcb : c ∈ ball p.center (p.radius/2) := hx.2
+  have hsub := p.centered_small_closedBall_subset hcb
+  have hball : ball p.center p.radius ⊆ closedBall p.center (3*p.radius) :=
+    ball_subset_closedBall.trans (closedBall_subset_closedBall (by linarith [p.radius_pos]))
+  have hsub3 : closedBall c r ⊆ closedBall p.center (3*p.radius) := hsub.trans hball
+  have he (z) (hz : z ∈ closedBall p.center (3*p.radius)) :
+      p.coordExtension φ ((EuclideanSpace.equiv (Fin d) ℂ) z) =
+        φ.value ((A.euclideanChart p.index).symm z) := by
+    change p.coordExtension φ (coordinateEquiv d z) = _
+    rw [p.coordExtension_apply, p.extension_eq φ hz]
+  have hec : p.coordExtension φ ((EuclideanSpace.equiv (Fin d) ℂ) c) = φ.value x := by
+    rw [he c (hsub3 (mem_closedBall_self hr.le)), (A.euclideanChart p.index).left_inv hx.1]
+  have hi : IntegrableOn
+      (fun z : EC d => -p.coordExtension φ ((EuclideanSpace.equiv (Fin d) ℂ) z))
+      (ball p.center p.radius) volume := by
+    exact (((p.coordExtension_smooth φ).continuous.comp
+      (EuclideanSpace.equiv (Fin d) ℂ).continuous).neg.continuousOn.integrableOn_compact
+        (isCompact_closedBall p.center p.radius)).mono_set ball_subset_closedBall
+  have hBi : (∫ z in ball p.center p.radius,
+      -p.coordExtension φ ((EuclideanSpace.equiv (Fin d) ℂ) z)) ≤ B := by
+    rw [p.integral_measure (fun x => -φ.value x) φ.continuous.neg] at hB
+    apply (setIntegral_congr_fun isOpen_ball.measurableSet (fun z hz => ?_)).trans_le hB
+    rw [he z (hball hz)]
+  have hh := abp_integral_minimum_bound volume (p.coordExtension_smooth φ)
+    (fun z => g.matrix p.index (coordinateEquiv d z)) hr hδ hK
+    (by intro z hz; rw [hec, he z (hsub3 hz)]; exact hmin _)
+    (by
+      intro z hz
+      apply (g.positive _ _ ?_).isHermitian
+      have ht := p.closure_sub (closedBall_subset_closedBall (by linarith [p.radius_pos]) (hsub3 hz))
+      simpa only [ComplexAtlas.euclideanChart_target, mem_preimage] using ht)
+    (fun z hz v => hδb z (hsub3 hz) v)
+    (by
+      intro z hz
+      change |(g.matrix p.index (coordinateEquiv d z) +
+        PotentialKaehler.potentialMatrix (p.coordExtension φ) (coordinateEquiv d z)).det.re| ≤ K
+      rw [p.coordExtension_hessian φ (hsub3 hz)]
+      exact hdet z (hsub3 hz))
+    isOpen_ball.measurableSet hsub
+    (by intro z hz; rw [he z (hball hz)]; exact hn _) hi hBi
+  rw [hec] at hh
+  exact hh
+end CoordinateBall
+end Anticanonical.SourceSmooth
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/CoordinatePotential.lean b/lean/OAI/Geometry/PrescribedPotential/CoordinatePotential.lean
new file mode 100644
index 0000000000000000000000000000000000000000..a7e1696ebdfca764a8415d024d3c7e88540214e1
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/CoordinatePotential.lean
@@ -0,0 +1,158 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CircleRadialCalculus
+import OAI.Geometry.PrescribedPotential.CoordinateBalls
+
+namespace OAI
+
+/-! Coordinate Potential. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Metric Filter Topology MeasureTheory Matrix
+open scoped ContDiff InnerProductSpace ComplexOrder Matrix.Norms.Elementwise
+namespace PotentialABP
+variable {d : ℕ}
+local notation "E" => EuclideanSpace ℂ (Fin d)
+local instance coordinatePotentialRealIP (d : ℕ) : InnerProductSpace ℝ (EuclideanSpace ℂ (Fin d)) :=
+  InnerProductSpace.rclikeToReal ℂ (EuclideanSpace ℂ (Fin d))
+local notation "e" => EuclideanSpace.equiv (Fin d) ℂ
+
+lemma hessian_coord_comp_at {f : (Fin d → ℂ) → ℝ} {x : E}
+    (hf : ContDiffAt ℝ ∞ f (e x)) (v w : E) :
+    fderiv ℝ (fderiv ℝ (f ∘ e)) x v w =
+      fderiv ℝ (fderiv ℝ f) (e x) (e v) (e w) := by
+  let er := (EuclideanSpace.equiv (Fin d) ℂ).toContinuousLinearMap.restrictScalars ℝ
+  have her : ContDiff ℝ ∞ e := er.contDiff
+  have hd : fderiv ℝ e = fun _ => (er : E →L[ℝ] (Fin d → ℂ)) := by
+    funext y
+    exact er.hasFDerivAt.fderiv
+  rw [PotentialKaehler.second_derivative_comp her.contDiffAt hf v w, hd]
+  simp
+  rfl
+
+lemma levi_coord_pair_at {f : (Fin d → ℂ) → ℝ} {x : E}
+    (hf : ContDiffAt ℝ ∞ f (e x)) (v : E) :
+    fderiv ℝ (fderiv ℝ (f ∘ e)) x v v +
+      fderiv ℝ (fderiv ℝ (f ∘ e)) x (Complex.I • v) (Complex.I • v) =
+        4 * inner ℝ ((PotentialKaehler.potentialMatrix f (e x)).toEuclideanLin v) v := by
+  rw [hessian_coord_comp_at hf, hessian_coord_comp_at hf]
+  rw [real_inner_comm, real_inner_eq_re_inner (𝕜 := ℂ), EuclideanSpace.inner_eq_star_dotProduct]
+  rw [dotProduct_comm]
+  change _ = 4 * (star (e v) ⬝ᵥ ((PotentialKaehler.potentialMatrix f (e x)) *ᵥ (e v))).re
+  rw [PotentialKaehler.potentialMatrix, PotentialKaehler.hermitianPartMatrix_pair]
+  simp only [map_smul, PotentialKaehler.hermitianPart]
+  ring
+
+lemma matrix_quadratic_le (H : Matrix (Fin d) (Fin d) ℂ) (v : E) :
+    inner ℝ (H.toEuclideanLin v) v ≤ ‖(Matrix.toEuclideanCLM (𝕜 := ℂ) (n := Fin d)) H‖ * ‖v‖^2 := by
+  apply (real_inner_le_norm _ _).trans
+  have h := ((Matrix.toEuclideanCLM (𝕜 := ℂ) (n := Fin d)) H).le_opNorm v
+  change ‖H.toEuclideanLin v‖ ≤ ‖(Matrix.toEuclideanCLM (𝕜 := ℂ) (n := Fin d)) H‖ * ‖v‖ at h
+  nlinarith [mul_le_mul_of_nonneg_right h (norm_nonneg v)]
+
+lemma matrix_quadratic_nonneg {H : Matrix (Fin d) (Fin d) ℂ} (hH : H.PosSemidef) (v : E) :
+    0 ≤ inner ℝ (H.toEuclideanLin v) v := by
+  rw [real_inner_eq_re_inner (𝕜 := ℂ)]
+  exact (Matrix.isPositive_toEuclideanLin_iff.mpr hH).re_inner_nonneg_left v
+end PotentialABP
+
+namespace Anticanonical.SourceSmooth
+open EllipticKernel PotentialABP
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+local instance sourceCoordinateRealIP (d : ℕ) : InnerProductSpace ℝ (EC d) :=
+  InnerProductSpace.rclikeToReal ℂ (EC d)
+namespace CoordinateBall
+variable (p : CoordinateBall A)
+
+def bump : ContDiffBump p.center where
+  rIn := 7*p.radius/2
+  rOut := 4*p.radius
+  rIn_pos := by linarith [p.radius_pos]
+  rIn_lt_rOut := by linarith [p.radius_pos]
+
+def extension (φ : SmoothRealFunction A) : EC d → ℝ :=
+  fun z => p.bump z * φ.value ((A.euclideanChart p.index).symm z)
+
+lemma extension_smooth (φ : SmoothRealFunction A) : ContDiff ℝ ∞ (p.extension φ) := by
+  apply MetricLocalization.smooth_bump_smul p.bump
+  intro z hz
+  have hz' : z ∈ (A.euclideanChart p.index).target := p.closure_sub
+    (by simpa only [bump, ContDiffBump.tsupport_eq] using hz)
+  exact (φ.euclidean_smooth p.index).contDiffAt
+    (by simpa only [ComplexAtlas.euclideanChart_target] using
+      (A.euclideanChart p.index).open_target.mem_nhds hz')
+
+lemma extension_eventuallyEq (φ : SmoothRealFunction A) {z : EC d}
+    (hz : z ∈ closedBall p.center (3*p.radius)) :
+    p.extension φ =ᶠ[𝓝 z] (φ.value ∘ (A.euclideanChart p.index).symm) := by
+  have hz' : z ∈ ball p.center (7*p.radius/2) := by
+    have hd : dist z p.center ≤ 3*p.radius := hz
+    change dist z p.center < 7*p.radius/2
+    linarith [p.radius_pos]
+  filter_upwards [isOpen_ball.mem_nhds hz'] with y hy
+  have hone : p.bump y = 1 := p.bump.one_of_mem_closedBall
+    (show dist y p.center ≤ 7*p.radius/2 from (show dist y p.center < 7*p.radius/2 from hy).le)
+  simp [extension, hone]
+
+lemma extension_eq (φ : SmoothRealFunction A) {z : EC d}
+    (hz : z ∈ closedBall p.center (3*p.radius)) :
+    p.extension φ z = φ.value ((A.euclideanChart p.index).symm z) :=
+  (p.extension_eventuallyEq φ hz).eq_of_nhds
+
+lemma metric_bound (g : KaehlerMetric A) : ∃ C : ℝ, 0 ≤ C ∧
+    ∀ z ∈ closedBall p.center (3*p.radius), ∀ v : EC d,
+      inner ℝ ((g.matrix p.index (coordinateEquiv d z)).toEuclideanLin v) v ≤ C*‖v‖^2 := by
+  have hc : ContinuousOn (fun z : EC d =>
+      (Matrix.toEuclideanCLM (𝕜 := ℂ) (n := Fin d)) (g.matrix p.index (coordinateEquiv d z)))
+      (closedBall p.center (3*p.radius)) := by
+    apply (Matrix.toEuclideanCLM (𝕜 := ℂ) (n := Fin d)).toAlgEquiv.toLinearMap.continuous_of_finiteDimensional.comp_continuousOn
+    have hs : ContinuousOn (fun z : EC d => g.matrix p.index (coordinateEquiv d z))
+        (A.euclideanChart p.index).target := by
+      apply continuousOn_pi.mpr
+      intro i
+      apply continuousOn_pi.mpr
+      intro j
+      have hij := (contDiffOn_pi.mp (contDiffOn_pi.mp (g.smooth p.index) i) j).continuousOn
+      rw [ComplexAtlas.euclideanChart_target]
+      convert hij.comp (coordinateEquiv d).continuous.continuousOn
+        (show Set.MapsTo (coordinateEquiv d)
+          ((coordinateEquiv d) ⁻¹' (A.chart p.index).target) (A.chart p.index).target from
+            fun _ hz => hz) using 1; rfl
+    exact hs.mono (fun z hz => p.closure_sub
+      (closedBall_subset_closedBall (by linarith [p.radius_pos]) hz))
+  obtain ⟨C, hC⟩ := (isCompact_closedBall p.center (3*p.radius)).exists_bound_of_continuousOn hc
+  refine ⟨max C 0, le_max_right _ _, fun z hz v => ?_⟩
+  exact (matrix_quadratic_le _ v).trans
+    (mul_le_mul_of_nonneg_right ((hC z hz).trans (le_max_left _ _)) (sq_nonneg _))
+
+lemma extension_levi_lower (g : KaehlerMetric A) (φ : SmoothRealFunction A)
+    (hφ : g.PositivePotential φ) {C : ℝ}
+    (hC : ∀ z ∈ closedBall p.center (3*p.radius), ∀ v : EC d,
+      inner ℝ ((g.matrix p.index (coordinateEquiv d z)).toEuclideanLin v) v ≤ C*‖v‖^2)
+    {z : EC d} (hz : z ∈ closedBall p.center (3*p.radius)) (v : EC d) :
+    -(4*C*‖v‖^2) ≤ fderiv ℝ (fderiv ℝ (p.extension φ)) z v v +
+      fderiv ℝ (fderiv ℝ (p.extension φ)) z (Complex.I • v) (Complex.I • v) := by
+  have hzt : z ∈ (A.euclideanChart p.index).target :=
+    p.closure_sub (closedBall_subset_closedBall (by linarith [p.radius_pos]) hz)
+  have hzt' : coordinateEquiv d z ∈ (A.chart p.index).target := by simpa only [ComplexAtlas.euclideanChart_target, mem_preimage] using hzt
+  have he := ((p.extension_eventuallyEq φ hz).fderiv (𝕜 := ℝ)).fderiv_eq (𝕜 := ℝ)
+  rw [he]
+  have hl := levi_coord_pair_at ((φ.smooth p.index).contDiffAt
+    ((A.chart p.index).open_target.mem_nhds hzt')) v
+  have hn := matrix_quadratic_nonneg (hφ p.index _ hzt').posSemidef v
+  rw [map_add, LinearMap.add_apply, inner_add_left] at hn
+  change fderiv ℝ (fderiv ℝ (φ.value ∘ (A.euclideanChart p.index).symm)) z v v +
+      fderiv ℝ (fderiv ℝ (φ.value ∘ (A.euclideanChart p.index).symm)) z (Complex.I • v) (Complex.I • v) =
+      4*inner ℝ ((φ.hessian p.index (coordinateEquiv d z)).toEuclideanLin v) v at hl
+  rw [hl]
+  linarith [hC z hz v]
+end CoordinateBall
+end Anticanonical.SourceSmooth
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/CoreBounds.lean b/lean/OAI/Geometry/PrescribedPotential/CoreBounds.lean
new file mode 100644
index 0000000000000000000000000000000000000000..55016a0c7e34d16524081d6bc81ba59490d76deb
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/CoreBounds.lean
@@ -0,0 +1,161 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.LocalRegularity
+
+namespace OAI
+
+/-! Core Bounds. -/
+
+section
+
+ 
+
+noncomputable section
+open Filter Topology MeasureTheory LineDeriv TemperedDistribution
+open scoped ContDiff SchwartzMap Classical Laplacian
+
+namespace SobolevChart
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+
+ 
+def CoreBound (s t : ℝ) (T : 𝓢(E, ℂ) → 𝓢(E, ℂ)) : Prop :=
+  ∃ C : ℝ, 0 ≤ C ∧ ∀ f, ‖schwartzCoord t (T f)‖ ≤ C * ‖schwartzCoord s f‖
+
+lemma CoreBound.comp {s t r : ℝ} {S T : 𝓢(E, ℂ) → 𝓢(E, ℂ)}
+    (hS : CoreBound t r S) (hT : CoreBound s t T) : CoreBound s r (S ∘ T) := by
+  obtain ⟨C, hC, hc⟩ := hS
+  obtain ⟨D, hD, hd⟩ := hT
+  refine ⟨C * D, mul_nonneg hC hD, fun f => ?_⟩
+  exact (hc (T f)).trans (by simpa [mul_assoc] using mul_le_mul_of_nonneg_left (hd f) hC)
+
+lemma CoreBound.add {s t : ℝ} {S T : 𝓢(E, ℂ) → 𝓢(E, ℂ)}
+    (hS : CoreBound s t S) (hT : CoreBound s t T) :
+    CoreBound s t (fun f => S f + T f) := by
+  obtain ⟨C, hC, hc⟩ := hS
+  obtain ⟨D, hD, hd⟩ := hT
+  refine ⟨C + D, add_nonneg hC hD, fun f => ?_⟩
+  rw [schwartzCoord_add, add_mul]
+  exact (norm_add_le _ _).trans (add_le_add (hc f) (hd f))
+
+lemma CoreBound.sum {ι : Type*} {s t : ℝ} (I : Finset ι)
+    (T : ι → 𝓢(E, ℂ) → 𝓢(E, ℂ)) (hh : ∀ i ∈ I, CoreBound s t (T i)) :
+    CoreBound s t (fun f => ∑ i ∈ I, T i f) := by
+  choose C hC h using fun i : I => hh i i.property
+  refine ⟨∑ i : I, C i, Finset.sum_nonneg (fun i _ => hC i), fun f => ?_⟩
+  rw [schwartzCoord_sum]
+  calc
+    _ ≤ ∑ i ∈ I, ‖schwartzCoord t (T i f)‖ := norm_sum_le _ _
+    _ = ∑ i : I, ‖schwartzCoord t (T i f)‖ := (Finset.sum_coe_sort I _).symm
+    _ ≤ ∑ i : I, C i * ‖schwartzCoord s f‖ := Finset.sum_le_sum (fun i _ => h i f)
+    _ = _ := (Finset.sum_mul ..).symm
+
+lemma CoreBound.id {s t : ℝ} (hst : t ≤ s) : CoreBound (E := E) s t id := by
+  let B := liftOperator (E := E) s t (ContinuousLinearMap.id ℂ _) (fun _ hu => hu.mono hst)
+  refine ⟨‖B‖, norm_nonneg _, fun f => ?_⟩
+  have hb : B (schwartzCoord s f) = schwartzCoord t f := by
+    apply realize_injective t
+    exact (realize_liftOperator s t _ _ _).trans (by simp only [realize_schwartzCoord, ContinuousLinearMap.id_apply])
+  rw [Function.id_def, ← hb]
+  exact B.le_opNorm _
+
+lemma coreBound_deriv {s t : ℝ} (v : E) (hst : t ≤ s - 1) :
+    CoreBound s t (fun f : 𝓢(E, ℂ) => ∂_{v} f) := by
+  let B := liftOperator s t (lineDerivOpCLM ℂ 𝓢'(E, ℂ) v)
+    (fun _ hu => hu.lineDerivOp.mono hst)
+  refine ⟨‖B‖, norm_nonneg _, fun f => ?_⟩
+  have hb : B (schwartzCoord s f) = schwartzCoord t (∂_{v} f) := by
+    apply realize_injective t
+    exact (realize_liftOperator s t _ _ _).trans (by
+      rw [realize_schwartzCoord, realize_schwartzCoord]
+      exact lineDerivOp_toTemperedDistributionCLM_eq f v)
+  rw [← hb]
+  exact B.le_opNorm _
+
+lemma coreBound_word (ws : List E) {s t : ℝ} (hst : t ≤ s - ws.length) :
+    CoreBound s t (schwartzWord ws) := by
+  induction ws generalizing t with
+  | nil => exact CoreBound.id (by simpa using hst)
+  | cons v ws ih =>
+    have hw : CoreBound s (t + 1) (schwartzWord ws) := ih (by simp only [List.length_cons, Nat.cast_add, Nat.cast_one] at hst; linarith)
+    exact (coreBound_deriv v (s := t + 1) (t := t) (by linarith)).comp hw
+
+lemma schwartzCoord_distribution (s : ℝ) (f : 𝓢(E, ℂ)) :
+    (schwartzCoord s f : 𝓢'(E, ℂ)) = besselPotential E ℂ s (f : 𝓢'(E, ℂ)) := by
+  rw [← realize_schwartzCoord s f, bessel_realize]
+
+lemma schwartzCoord_raise_two (s : ℝ) (f : 𝓢(E, ℂ)) :
+    schwartzCoord (s + 2) f = schwartzCoord s f - (((2 * Real.pi)^2 : ℝ) : ℂ)⁻¹ •
+      ∑ j, schwartzCoord s (∂_{stdOrthonormalBasis ℝ E j} (∂_{stdOrthonormalBasis ℝ E j} f)) := by
+  apply l2_injective
+  simp only [Lp.toTemperedDistributionCLM_apply, map_sub, map_smul, map_sum,
+    schwartzCoord_distribution]
+  rw [add_comm s 2, ← besselPotential_besselPotential_apply 2 s,
+    bessel_two_laplacian, map_sub, map_smul]
+  congr 2
+  rw [TemperedDistribution.laplacian_eq_sum (stdOrthonormalBasis ℝ E), map_sum]
+  apply Finset.sum_congr rfl
+  intro j _
+  rw [← lineDerivOp_toTemperedDistributionCLM_eq,
+    ← lineDerivOp_toTemperedDistributionCLM_eq]
+
+lemma schwartzCoord_raise_two_norm (s : ℝ) (f : 𝓢(E, ℂ)) :
+    ‖schwartzCoord (s + 2) f‖ ≤ ‖schwartzCoord s f‖ +
+      ‖(((2 * Real.pi)^2 : ℝ) : ℂ)⁻¹‖ * ∑ j, ‖schwartzCoord s
+        (∂_{stdOrthonormalBasis ℝ E j} (∂_{stdOrthonormalBasis ℝ E j} f))‖ := by
+  let u := schwartzCoord s f
+  let D (j : Fin (Module.finrank ℝ E)) := schwartzCoord s
+    (∂_{stdOrthonormalBasis ℝ E j} (∂_{stdOrthonormalBasis ℝ E j} f))
+  let c : ℂ := (((2 * Real.pi)^2 : ℝ) : ℂ)⁻¹
+  rw [schwartzCoord_raise_two]
+  change ‖u - c • ∑ j, D j‖ ≤ ‖u‖ + ‖c‖ * ∑ j, ‖D j‖
+  have hs := norm_smul c (∑ j, D j)
+  calc
+    _ ≤ ‖u‖ + ‖c • ∑ j, D j‖ := norm_sub_le _ _
+    _ = ‖u‖ + ‖c‖ * ‖∑ j, D j‖ := congrArg (fun z : ℝ => ‖u‖ + z) hs
+    _ ≤ _ := add_le_add_right (mul_le_mul_of_nonneg_left (norm_sum_le _ _) (norm_nonneg c)) _
+
+lemma coreBound_raise_two {s t : ℝ} (T : 𝓢(E, ℂ) → 𝓢(E, ℂ))
+    (hT : CoreBound s t T)
+    (hD : ∀ j, CoreBound s t (fun f => ∂_{stdOrthonormalBasis ℝ E j}
+      (∂_{stdOrthonormalBasis ℝ E j} (T f)))) : CoreBound s (t + 2) T := by
+  obtain ⟨C, hC, hc⟩ := hT
+  choose D hD0 hdb using hD
+  let c : ℝ := ‖(((2 * Real.pi)^2 : ℝ) : ℂ)⁻¹‖
+  have hc0 : 0 ≤ c := norm_nonneg _
+  refine ⟨C + c * ∑ j, D j, add_nonneg hC (mul_nonneg hc0 (Finset.sum_nonneg (fun j _ => hD0 j))), fun f => ?_⟩
+  apply (schwartzCoord_raise_two_norm t (T f)).trans
+  have hd := Finset.sum_le_sum (s := Finset.univ) (fun j _ => hdb j f)
+  have hsum := add_le_add (hc f) (mul_le_mul_of_nonneg_left hd hc0)
+  convert hsum using 1; rw [← Finset.sum_mul]; ring
+
+omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] in
+lemma schwartzWord_append (vs ws : List E) (f : 𝓢(E, ℂ)) :
+    schwartzWord (vs ++ ws) f = schwartzWord vs (schwartzWord ws f) := by
+  induction vs with
+  | nil => rfl
+  | cons v vs ih => simpa only [List.cons_append, schwartzWord, ContinuousLinearMap.comp_apply] using congrArg (lineDerivOpCLM ℂ 𝓢(E, ℂ) v) ih
+
+ 
+
+lemma coreBound_of_words (k : ℕ) {s : ℝ} (T : 𝓢(E, ℂ) → 𝓢(E, ℂ))
+    (h : ∀ ws : List E, ws.length ≤ 2 * k → CoreBound s 0 (fun f => schwartzWord ws (T f))) :
+    CoreBound s (2 * (k : ℝ)) T := by
+  induction k generalizing T with
+  | zero => simpa [schwartzWord] using h [] (by simp)
+  | succ k ih =>
+    have ht := ih T (fun ws hw => h ws (by omega))
+    have hd (j : Fin (Module.finrank ℝ E)) := ih
+      (fun f => ∂_{stdOrthonormalBasis ℝ E j} (∂_{stdOrthonormalBasis ℝ E j} (T f)))
+      (fun ws hw => by
+        have hh := h (ws ++ [stdOrthonormalBasis ℝ E j, stdOrthonormalBasis ℝ E j])
+          (by simp only [List.length_append, List.length_cons, List.length_nil] ; omega)
+        simpa only [schwartzWord_append, schwartzWord, ContinuousLinearMap.comp_apply,
+          ContinuousLinearMap.id_apply, lineDerivOpCLM_apply] using hh)
+    convert coreBound_raise_two T ht hd using 1; push_cast; ring
+
+end SobolevChart
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/CutoffHigher.lean b/lean/OAI/Geometry/PrescribedPotential/CutoffHigher.lean
new file mode 100644
index 0000000000000000000000000000000000000000..e876ce5b5bb569e4409167b0f97deeabd14d5c7e
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/CutoffHigher.lean
@@ -0,0 +1,174 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.IntegerBounds
+import OAI.Geometry.PrescribedPotential.PatchDifferential
+
+namespace OAI
+
+/-! Cutoff Higher. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory TemperedDistribution LineDeriv
+open scoped SchwartzMap ContDiff Classical BoundedContinuousFunction
+
+namespace SobolevChart
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+
+lemma schwartzCoord_product_zero (κ f : 𝓢(E, ℂ)) :
+    schwartzCoord 0 (SchwartzMap.smulLeftCLM ℂ κ f) =
+      multiply κ.toBoundedContinuousFunction (schwartzCoord 0 f) := by
+  apply l2_injective
+  simp only [Lp.toTemperedDistributionCLM_apply]
+  rw [schwartzCoord_distribution, schwartz_temperate_product_distribution κ.hasTemperateGrowth,
+    besselPotential_zero, ContinuousLinearMap.id_apply,
+    multiply_distribution _ κ.hasTemperateGrowth, schwartzCoord_distribution,
+    besselPotential_zero, ContinuousLinearMap.id_apply]
+  rfl
+
+lemma coreBound_coefficient_zero (c : E →ᵇ ℂ)
+    (hc : (c : E → ℂ).HasTemperateGrowth) :
+    CoreBound 0 0 (SchwartzMap.smulLeftCLM ℂ c) := by
+  refine ⟨‖c‖, norm_nonneg _, fun f => ?_⟩
+  have he : schwartzCoord 0 (SchwartzMap.smulLeftCLM ℂ c f) =
+      multiply c (schwartzCoord 0 f) := by
+    apply l2_injective
+    simp only [Lp.toTemperedDistributionCLM_apply]
+    rw [schwartzCoord_distribution, schwartz_temperate_product_distribution hc,
+      besselPotential_zero, ContinuousLinearMap.id_apply,
+      multiply_distribution c hc, schwartzCoord_distribution,
+      besselPotential_zero, ContinuousLinearMap.id_apply]
+  rw [he]
+  exact multiply_norm_le _ _
+
+lemma coreBound_product_zero (κ : 𝓢(E, ℂ)) :
+    CoreBound 0 0 (SchwartzMap.smulLeftCLM ℂ κ) := by
+  refine ⟨‖κ.toBoundedContinuousFunction‖, norm_nonneg _, fun f => ?_⟩
+  rw [schwartzCoord_product_zero]
+  exact multiply_norm_le _ _
+
+lemma coreBound_product_word (ws : List E) (κ : 𝓢(E, ℂ)) {s : ℝ}
+    (hws : (ws.length : ℝ) ≤ s) :
+    CoreBound s 0 (fun f => schwartzWord ws (SchwartzMap.smulLeftCLM ℂ κ f)) := by
+  induction ws generalizing κ s with
+  | nil => exact (coreBound_product_zero κ).comp (CoreBound.id (by simpa using hws))
+  | cons w ws ih =>
+    have hw : (ws.length : ℝ) + 1 ≤ s := by
+      simpa only [List.length_cons, Nat.cast_add, Nat.cast_one] using hws
+    have hfirst := ih (∂_{w} κ) (s := s) (by linarith)
+    have hrest := (ih κ (s := s - 1) (by linarith)).comp
+      (coreBound_deriv w (s := s) (t := s - 1) le_rfl)
+    convert hfirst.add hrest using 1
+    funext f
+    change ∂_{w} (schwartzWord ws (SchwartzMap.smulLeftCLM ℂ κ f)) = _
+    rw [← GlobalElliptic.word_deriv_comm, schwartz_deriv_product, map_add]
+    rfl
+
+lemma coreBound_product_integer (κ : 𝓢(E, ℂ)) (k : ℕ) :
+    CoreBound (k : ℝ) (k : ℝ) (SchwartzMap.smulLeftCLM ℂ κ) := by
+  have hh := coreBound_of_words_integer k (s := (k : ℝ)) (t := 0)
+    (SchwartzMap.smulLeftCLM ℂ κ)
+    (fun ws hw => coreBound_product_word ws κ (by exact_mod_cast hw))
+  simpa only [zero_add] using hh
+
+omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] in
+lemma schwartz_second_product (v w : E) (κ f : 𝓢(E, ℂ)) :
+    ∂_{v} (∂_{w} (SchwartzMap.smulLeftCLM ℂ κ f)) =
+      SchwartzMap.smulLeftCLM ℂ κ (∂_{v} (∂_{w} f)) +
+        (SchwartzMap.smulLeftCLM ℂ (∂_{v} κ : 𝓢(E, ℂ)) (∂_{w} f) +
+          SchwartzMap.smulLeftCLM ℂ (∂_{w} κ : 𝓢(E, ℂ)) (∂_{v} f) +
+            SchwartzMap.smulLeftCLM ℂ (∂_{v} (∂_{w} κ) : 𝓢(E, ℂ)) f) := by
+  rw [schwartz_deriv_product, lineDerivOp_add, schwartz_deriv_product, schwartz_deriv_product]
+  abel
+
+omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] in
+lemma schwartz_product_assoc {c : E → ℂ} (hc : c.HasTemperateGrowth) (κ f : 𝓢(E, ℂ)) :
+    SchwartzMap.smulLeftCLM ℂ c (SchwartzMap.smulLeftCLM ℂ κ f) =
+      SchwartzMap.smulLeftCLM ℂ (SchwartzMap.smulLeftCLM ℂ c κ : 𝓢(E, ℂ)) f := by
+  ext x
+  simp only [SchwartzMap.smulLeftCLM_apply_apply hc,
+    SchwartzMap.smulLeftCLM_apply_apply κ.hasTemperateGrowth,
+    SchwartzMap.smulLeftCLM_apply_apply (SchwartzMap.smulLeftCLM ℂ c κ).hasTemperateGrowth,
+    smul_eq_mul, mul_assoc]
+
+omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] in
+lemma schwartz_product_comm {c : E → ℂ} (hc : c.HasTemperateGrowth) (κ f : 𝓢(E, ℂ)) :
+    SchwartzMap.smulLeftCLM ℂ c (SchwartzMap.smulLeftCLM ℂ κ f) =
+      SchwartzMap.smulLeftCLM ℂ κ (SchwartzMap.smulLeftCLM ℂ c f) := by
+  ext x
+  simp only [SchwartzMap.smulLeftCLM_apply_apply hc,
+    SchwartzMap.smulLeftCLM_apply_apply κ.hasTemperateGrowth, smul_eq_mul]
+  ring
+end SobolevChart
+
+namespace MetricLocalization
+open EllipticKernel SobolevChart
+variable {n : ℕ}
+
+lemma schwartzDifferential_bound
+    (c : BasisIndex n → BasisIndex n → EC n →ᵇ ℂ)
+    (hc : ∀ i j, (c i j : EC n → ℂ).HasTemperateGrowth) :
+    CoreBound 2 0 (schwartzDifferential c) := by
+  change CoreBound 2 0 (fun f => schwartzDifferential c f)
+  simp only [schwartzDifferential, _root_.sum_apply,
+    ContinuousLinearMap.comp_apply, lineDerivOpCLM_apply]
+  apply CoreBound.sum Finset.univ
+  intro i _
+  apply CoreBound.sum Finset.univ
+  intro j _
+  exact (coreBound_coefficient_zero (c i j) (hc i j)).comp
+    ((coreBound_deriv _ (s := 1) (t := 0) (by norm_num)).comp
+      (coreBound_deriv _ (s := 2) (t := 1) (by norm_num)))
+
+lemma schwartzCutoffError_firstOrder
+    (c : BasisIndex n → BasisIndex n → EC n →ᵇ ℂ)
+    (hc : ∀ i j, (c i j : EC n → ℂ).HasTemperateGrowth) (κ f : 𝓢(EC n, ℂ)) :
+    schwartzCutoffError c κ f = ∑ i, ∑ j,
+      (SchwartzMap.smulLeftCLM ℂ
+        (SchwartzMap.smulLeftCLM ℂ (c i j) (∂_{stdOrthonormalBasis ℝ (EC n) i} κ) : 𝓢(EC n, ℂ))
+        (∂_{stdOrthonormalBasis ℝ (EC n) j} f) +
+      SchwartzMap.smulLeftCLM ℂ
+        (SchwartzMap.smulLeftCLM ℂ (c i j) (∂_{stdOrthonormalBasis ℝ (EC n) j} κ) : 𝓢(EC n, ℂ))
+        (∂_{stdOrthonormalBasis ℝ (EC n) i} f) +
+      SchwartzMap.smulLeftCLM ℂ
+        (SchwartzMap.smulLeftCLM ℂ (c i j)
+          (∂_{stdOrthonormalBasis ℝ (EC n) i} (∂_{stdOrthonormalBasis ℝ (EC n) j} κ)) : 𝓢(EC n, ℂ)) f) := by
+  simp only [schwartzCutoffError, schwartzDifferential, _root_.sub_apply,
+    _root_.sum_apply, ContinuousLinearMap.comp_apply, lineDerivOpCLM_apply,
+    map_sum, ← Finset.sum_sub_distrib]
+  apply Finset.sum_congr rfl
+  intro i _
+  apply Finset.sum_congr rfl
+  intro j _
+  rw [schwartz_second_product]
+  simp only [map_add, schwartz_product_comm (hc i j) κ,
+    schwartz_product_assoc (hc i j)]
+  abel
+
+ 
+theorem schwartzCutoffError_bound_integer
+    (c : BasisIndex n → BasisIndex n → EC n →ᵇ ℂ)
+    (hc : ∀ i j, (c i j : EC n → ℂ).HasTemperateGrowth) (κ : 𝓢(EC n, ℂ)) (k : ℕ) :
+    CoreBound ((k : ℝ) + 1) (k : ℝ) (schwartzCutoffError c κ) := by
+  change CoreBound ((k : ℝ) + 1) (k : ℝ) (fun f => schwartzCutoffError c κ f)
+  simp_rw [schwartzCutoffError_firstOrder c hc κ]
+  apply CoreBound.sum Finset.univ
+  intro i _
+  apply CoreBound.sum Finset.univ
+  intro j _
+  apply CoreBound.add
+  · apply CoreBound.add
+    · exact (coreBound_product_integer _ k).comp
+        (coreBound_deriv _ (s := (k : ℝ) + 1) (t := (k : ℝ)) (by linarith))
+    · exact (coreBound_product_integer _ k).comp
+        (coreBound_deriv _ (s := (k : ℝ) + 1) (t := (k : ℝ)) (by linarith))
+  · exact (coreBound_product_integer _ k).comp (CoreBound.id (by linarith))
+end MetricLocalization
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/DerivativeEllipticGain.lean b/lean/OAI/Geometry/PrescribedPotential/DerivativeEllipticGain.lean
new file mode 100644
index 0000000000000000000000000000000000000000..2415d45ff9ec3a8c7d54461150aed31c1c2efb44
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/DerivativeEllipticGain.lean
@@ -0,0 +1,101 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.NonlinearScale
+import OAI.Geometry.PrescribedPotential.WeakNonlinearCommutator
+import OAI.Geometry.PrescribedPotential.TwoScaleRegularity
+
+namespace OAI
+
+/-! Derivative Elliptic Gain. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology
+open scoped ContDiff Classical
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  [ConnectedSpace X] {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+local instance (s : ℝ) : NormedAddCommGroup (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedAddCommGroup
+local instance (s : ℝ) : NormedSpace ℝ (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedSpace
+local instance (s : ℝ) : IsTopologicalAddGroup (D.localizers.RealSobolev s) :=
+  Submodule.isTopologicalAddGroup _
+local instance (s : ℝ) : ContinuousSMul ℝ (D.localizers.RealSobolev s) :=
+  SMulMemClass.continuousSMul _
+
+omit [ConnectedSpace X] in
+lemma nonlinear_commutator_solution (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (p : ι) (v : EC d) (u : D.localizers.RealSobolev (((k+1 : ℕ) : ℝ)+2))
+    (F : RealSmooth A)
+    (hF : D.realVolume (k+1) (by omega) u = D.localizers.realEmbed ((k+1 : ℕ) : ℝ) F) :
+    D.realVolumeDerivative k hk
+      (D.localizers.realLower (((k+1 : ℕ) : ℝ)+2) ((k : ℝ)+2) (by push_cast; linarith) u)
+      (D.potentialDerivative k p v u) =
+      D.localizers.realLower ((k+1 : ℕ) : ℝ) (k : ℝ) (by push_cast; linarith)
+        (D.localizers.realEmbed ((k+1 : ℕ) : ℝ) (D.realLocalizedDerivative p v F) +
+          D.realNonlinearRemainder (k+1) (by omega) p v u) := by
+  have hh := D.nonlinear_commutator_weak k hk p v u
+  rw [hF,D.densityDerivative_embed] at hh
+  rw [hh]
+  rw [map_add,Localizers.realLower_embed]
+
+omit [ConnectedSpace X] in
+lemma linearizedVolume_apply_zero_scalar (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (x₀ : X) (u w : D.localizers.RealSobolev ((k : ℝ)+2)) :
+    D.linearizedVolume k hk x₀ u (w,0) =
+      (D.realVolumeDerivative k hk u w, D.realEvaluation ((k : ℝ)+2) x₀ w) := by
+  rw [D.linearizedVolume_apply]
+  simp only [map_zero,sub_zero]
+
+ 
+
+theorem nonlinear_first_derivative_gain (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (x₀ : X) :
+    ∀ᶠ u in 𝓝 (0 : D.localizers.RealSobolev (((k+1 : ℕ) : ℝ)+2)),
+      ∀ (F : RealSmooth A),
+        D.realVolume (k+1) (by omega) u = D.localizers.realEmbed ((k+1 : ℕ) : ℝ) F →
+      ∀ (p : ι) (v : EC d),
+        ∃ w : D.localizers.RealSobolev (((k+1 : ℕ) : ℝ)+2),
+          D.localizers.realLower (((k+1 : ℕ) : ℝ)+2) ((k : ℝ)+2)
+            (by push_cast; linarith) w = D.potentialDerivative k p v u := by
+  let : NormedAddCommGroup (D.localizers.RealSobolev (((k+1 : ℕ) : ℝ)+2)) := inferInstance
+  let : NormedSpace ℝ (D.localizers.RealSobolev (((k+1 : ℕ) : ℝ)+2)) := inferInstance
+  let : NormedAddCommGroup (D.localizers.RealSobolev ((k : ℝ)+2)) := inferInstance
+  let : NormedSpace ℝ (D.localizers.RealSobolev ((k : ℝ)+2)) := inferInstance
+  let : NormedAddCommGroup (D.localizers.RealSobolev ((k+1 : ℕ) : ℝ)) := inferInstance
+  let : NormedSpace ℝ (D.localizers.RealSobolev ((k+1 : ℕ) : ℝ)) := inferInstance
+  let : NormedAddCommGroup (D.localizers.RealSobolev (k : ℝ)) := inferInstance
+  let : NormedSpace ℝ (D.localizers.RealSobolev (k : ℝ)) := inferInstance
+  filter_upwards [D.linearizedVolume_local_gain (k+1) k (by omega) hk (by omega) x₀]
+    with u hu
+  intro F hF p v
+  let w : D.localizers.RealSobolev ((k : ℝ)+2) × ℝ := (D.potentialDerivative k p v u,0)
+  let f : D.localizers.RealSobolev ((k+1 : ℕ) : ℝ) × ℝ :=
+    (D.localizers.realEmbed ((k+1 : ℕ) : ℝ) (D.realLocalizedDerivative p v F) +
+      D.realNonlinearRemainder (k+1) (by omega) p v u,
+      D.realEvaluation ((k : ℝ)+2) x₀ (D.potentialDerivative k p v u))
+  have he : D.linearizedVolume k hk x₀
+      (D.localizers.realLower (((k+1 : ℕ) : ℝ)+2) ((k : ℝ)+2) (by push_cast; linarith) u) w =
+      D.pairLower ((k+1 : ℕ) : ℝ) (k : ℝ) (by push_cast; linarith) f := by
+    change D.linearizedVolume k hk x₀ _ (D.potentialDerivative k p v u,0) = _
+    rw [D.linearizedVolume_apply_zero_scalar,D.pairLower_apply]
+    apply Prod.ext
+    · exact D.nonlinear_commutator_solution k hk p v u F hF
+    · rfl
+  obtain ⟨z,hz,_⟩ := hu w f he
+  refine ⟨z.1,?_⟩
+  exact congrArg Prod.fst hz
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/DerivativeLocalization.lean b/lean/OAI/Geometry/PrescribedPotential/DerivativeLocalization.lean
new file mode 100644
index 0000000000000000000000000000000000000000..36a197a60d81bc7e6952dfe4272214f11a762ac7
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/DerivativeLocalization.lean
@@ -0,0 +1,138 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.LocalizedDerivations
+
+namespace OAI
+
+/-! Derivative Localization. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory TemperedDistribution LineDeriv
+open scoped ContDiff SchwartzMap Classical
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+
+lemma localize_tsupport_image (q : Fin A.count) (ρ : Smooth A)
+    (hρ : tsupport (ρ : X → ℂ) ⊆ (A.euclideanChart q).source) (f : Smooth A) :
+    tsupport (localize A q ρ hρ f : EC d → ℂ) ⊆
+      (A.euclideanChart q) '' tsupport (ρ : X → ℂ) := by
+  apply closure_minimal (localizeFun_support A q ρ f)
+  exact ((isClosed_tsupport _).isCompact.image_of_continuousOn
+    ((A.euclideanChart q).continuousOn.mono hρ)).isClosed
+
+lemma schwartz_derivative_zero_of_not_tsupport (f : 𝓢(EC d,ℂ)) (v : EC d) {y : EC d}
+    (hy : y ∉ tsupport (f : EC d → ℂ)) : (∂_{v} f : 𝓢(EC d,ℂ)) y = 0 := by
+  rw [SchwartzMap.lineDerivOp_apply_eq_fderiv,
+    (notMem_tsupport_iff_eventuallyEq.mp hy).fderiv_eq]
+  simp
+
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+lemma localizedDerivative_zero_off_function (p : ι) (v : EC d) (f : Smooth A) {x : X}
+    (hx : x ∉ tsupport (f : X → ℂ)) : D.localizedDerivative p v f x = 0 := by
+  rw [D.localizedDerivative_apply]
+  split_ifs with hs
+  · have ht := (A.euclideanChart (D.patch p).index).mapsTo hs
+    have hc : Tendsto (A.euclideanChart (D.patch p).index).symm
+        (𝓝 (A.euclideanChart (D.patch p).index x)) (𝓝 x) := by
+      simpa only [(A.euclideanChart (D.patch p).index).left_inv hs] using
+        ((A.euclideanChart (D.patch p).index).symm.continuousAt ht).tendsto
+    have hh := (notMem_tsupport_iff_eventuallyEq.mp hx).comp_tendsto hc
+    rw [hh.fderiv_eq]
+    simp
+  · rfl
+
+lemma localizedDerivative_function_tsupport (p : ι) (v : EC d) (f : Smooth A) :
+    tsupport (D.localizedDerivative p v f : X → ℂ) ⊆ tsupport (f : X → ℂ) := by
+  apply closure_minimal _ (isClosed_tsupport _)
+  intro x hx
+  by_contra hh
+  exact hx (D.localizedDerivative_zero_off_function p v f hh)
+
+lemma derivativeWeight_support (p : ι) (v : EC d) :
+    tsupport (D.localizedDerivative p v (D.localizers.weight p) : X → ℂ) ⊆
+      (A.euclideanChart (D.patch p).index).source := by
+  exact (D.localizedDerivative_function_tsupport p v _).trans
+    (by rw [← D.index_eq]; exact D.localizers.support_sub p)
+
+lemma forcing_derivative (p : ι) (v : EC d) (f : Smooth A) :
+    (∂_{v} (D.forcing p f) : 𝓢(EC d,ℂ)) =
+      localize A (D.patch p).index (D.localizedDerivative p v (D.localizers.weight p))
+        (D.derivativeWeight_support p v) f + D.forcing p (D.localizedDerivative p v f) := by
+  let q := (D.patch p).index
+  let e := A.euclideanChart q
+  let ρ := D.localizers.weight p
+  have hρ : tsupport (ρ : X → ℂ) ⊆ e.source := by
+    change tsupport (D.localizers.weight p : X → ℂ) ⊆
+      (A.euclideanChart (D.patch p).index).source
+    rw [← D.index_eq]
+    exact D.localizers.support_sub p
+  have heq (h : Smooth A) : D.forcing p h = localize A q ρ hρ h := by
+    unfold forcing
+    simp only [D.index_eq]
+    rfl
+  ext y
+  by_cases hy : y ∈ e '' tsupport (ρ : X → ℂ)
+  · obtain ⟨x,hx,rfl⟩ := hy
+    have hs := hρ hx
+    have ht := e.mapsTo hs
+    have hf : (D.forcing p f : EC d → ℂ) =ᶠ[𝓝 (e x)]
+        (fun y => ρ (e.symm y) * f (e.symm y)) := by
+      filter_upwards [e.open_target.mem_nhds ht] with z hz
+      rw [heq,localize_apply,localizeFun_apply A ρ f hz]
+    have hdρ := ((ρ.smooth q).contDiffAt (e.open_target.mem_nhds ht)).differentiableAt (by simp)
+    have hdf := ((f.smooth q).contDiffAt (e.open_target.mem_nhds ht)).differentiableAt (by simp)
+    change DifferentiableAt ℝ (fun y => ρ (e.symm y)) (e x) at hdρ
+    change DifferentiableAt ℝ (fun y => f (e.symm y)) (e x) at hdf
+    have hloc (σ : Smooth A)
+        (hσ : tsupport (σ : X → ℂ) ⊆ e.source) (fh : Smooth A) :
+        localize A q σ hσ fh (e x) = σ x * fh x := by
+      rw [localize_apply,localizeFun_apply A σ fh ht,e.left_inv hs]
+    rw [SchwartzMap.lineDerivOp_apply_eq_fderiv,hf.fderiv_eq,
+      fderiv_fun_mul hdρ hdf]
+    change ρ (e.symm (e x)) * fderiv ℝ (fun y => f (e.symm y)) (e x) v +
+      f (e.symm (e x)) * fderiv ℝ (fun y => ρ (e.symm y)) (e x) v =
+      localize A q (D.localizedDerivative p v ρ) (D.derivativeWeight_support p v) f (e x) +
+      D.forcing p (D.localizedDerivative p v f) (e x)
+    rw [hloc,heq,hloc,e.left_inv hs,D.localizedDerivative_apply,
+      D.localizedDerivative_apply]
+    have hs' : x ∈ (A.euclideanChart (D.patch p).index).source := hs
+    simp only [ite_eq_left hs']
+    change _ = (D.cutoff p).realPart (e x) *
+      fderiv ℝ (fun y => ρ (e.symm y)) (e x) v * f x +
+      ρ x * ((D.cutoff p).realPart (e x) * fderiv ℝ (fun y => f (e.symm y)) (e x) v)
+    rw [ChartCutoff.realPart_apply,D.cutoff_one p x hx]
+    simp only [Complex.one_re,Complex.ofReal_one,one_mul]
+    ring
+  · have hz : y ∉ tsupport (D.forcing p f : EC d → ℂ) := by
+      rw [heq]
+      exact fun hh => hy (localize_tsupport_image q ρ hρ f hh)
+    rw [schwartz_derivative_zero_of_not_tsupport _ v hz]
+    have hz₁ : localize A q (D.localizedDerivative p v ρ) (D.derivativeWeight_support p v) f y = 0 := by
+      apply image_eq_zero_of_notMem_tsupport
+      intro hh
+      apply hy
+      obtain ⟨x,hx,hxy⟩ := localize_tsupport_image q (D.localizedDerivative p v ρ)
+        (D.derivativeWeight_support p v) f hh
+      exact ⟨x,D.localizedDerivative_function_tsupport p v ρ hx,hxy⟩
+    have hz₂ : D.forcing p (D.localizedDerivative p v f) y = 0 := by
+      rw [heq]
+      apply image_eq_zero_of_notMem_tsupport
+      exact fun hh => hy (localize_tsupport_image q ρ hρ _ hh)
+    change 0 = localize A q (D.localizedDerivative p v ρ) (D.derivativeWeight_support p v) f y +
+      D.forcing p (D.localizedDerivative p v f) y
+    rw [hz₁,hz₂,add_zero]
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/DeterminantOrder.lean b/lean/OAI/Geometry/PrescribedPotential/DeterminantOrder.lean
new file mode 100644
index 0000000000000000000000000000000000000000..ef6b6c0b90cd6dd280bdd00152051d01335a9c3f
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/DeterminantOrder.lean
@@ -0,0 +1,62 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.AnalyticSupport
+
+namespace OAI
+
+/-! Determinant Order. -/
+
+section
+
+ 
+
+noncomputable section
+open Matrix Filter Set
+open scoped ComplexOrder MatrixOrder Classical
+namespace MongeAmpere
+variable {n : Type*} [Fintype n] [DecidableEq n]
+
+lemma hasDerivAt_det_real_shift (A B : Matrix n n ℂ) (t : ℝ)
+    (ht : (A + (t : ℂ) • B).PosDef) :
+    HasDerivAt (fun s : ℝ => (A + (s : ℂ) • B).det.re)
+      ((A + (t : ℂ) • B).det * ((A + (t : ℂ) • B)⁻¹ * B).trace).re t := by
+  have hd := (hasDerivAt_det_add_smul (A + (t : ℂ) • B) B
+    (isUnit_iff_ne_zero.mpr ht.det_pos.ne')).real_of_complex
+  have hc := hd.comp_of_eq t ((hasDerivAt_id t).sub_const t) (by simp)
+  simp only [mul_one] at hc
+  apply hc.congr_of_eventuallyEq
+  filter_upwards [] with s
+  dsimp only [Function.comp_apply, id_eq]
+  simp only [Complex.ofReal_sub, sub_smul]
+  congr 2
+  abel
+
+lemma det_re_add_posSemidef (A B : Matrix n n ℂ) (hA : A.PosDef) (hB : B.PosSemidef) :
+    A.det.re ≤ (A+B).det.re := by
+  let f : ℝ → ℝ := fun t => (A + (t : ℂ) • B).det.re
+  have hp (t : ℝ) (ht : 0 ≤ t) : (A + (t : ℂ) • B).PosDef :=
+    hA.add_posSemidef (hB.smul (Complex.nonneg_iff.mpr ⟨ht,rfl⟩))
+  have hd (t : ℝ) (ht : 0 ≤ t) := hasDerivAt_det_real_shift A B t (hp t ht)
+  have hm : MonotoneOn f (Icc 0 1) := by
+    apply monotoneOn_of_deriv_nonneg (convex_Icc 0 1)
+    · intro t ht
+      exact (hd t ht.1).continuousAt.continuousWithinAt
+    · intro t ht
+      exact (hd t (interior_subset ht).1).differentiableAt.differentiableWithinAt
+    · intro t ht
+      rw [(hd t (interior_subset ht).1).deriv]
+      have hdet := Complex.pos_iff.mp (hp t (interior_subset ht).1).det_pos
+      rw [Complex.mul_re, hdet.2.symm, zero_mul, sub_zero]
+      exact mul_nonneg hdet.1.le (EllipticKernel.trace_mul_nonneg
+        (hp t (interior_subset ht).1).inv.posSemidef hB)
+  simpa [f] using hm (show (0 : ℝ) ∈ Icc 0 1 by constructor <;> norm_num)
+    (show (1 : ℝ) ∈ Icc 0 1 by constructor <;> norm_num) (by norm_num)
+
+lemma det_re_mono {A B : Matrix n n ℂ} (hA : A.PosDef) (hBA : (B-A).PosSemidef) :
+    A.det.re ≤ B.det.re := by
+  simpa using det_re_add_posSemidef A (B-A) hA hBA
+end MongeAmpere
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/DirectionalJetEnergy.lean b/lean/OAI/Geometry/PrescribedPotential/DirectionalJetEnergy.lean
new file mode 100644
index 0000000000000000000000000000000000000000..8eac68b88ab0d8b2fdf1a41fed6d1dcf5387ca17
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/DirectionalJetEnergy.lean
@@ -0,0 +1,73 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.JetNormEquivalence
+import OAI.Geometry.PrescribedPotential.RealJetEnergy
+
+namespace OAI
+
+/-! Directional Jet Energy. -/
+
+section
+
+ 
+noncomputable section
+open Set Filter Topology Finset Module
+open scoped ContDiff
+namespace HigherJet
+variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  [NormedAddCommGroup F] [InnerProductSpace ℝ F]
+  {ι : Type*} [Fintype ι]
+
+def jetFamily (e : ι → E) (m : ℕ) (u : E → F) (σ : Fin m → ι) : E → F :=
+  word (List.ofFn (fun i => e (σ i))) u
+
+def jetEnergy (e : ι → E) (m : ℕ) (u : E → F) : E → ℝ :=
+  familyEnergy (jetFamily e m u)
+
+omit [Fintype ι] in
+lemma jetFamily_smoothOn {U : Set E} (hU : IsOpen U) {u : E → F}
+    (hu : ContDiffOn ℝ ∞ u U) (e : ι → E) (m : ℕ) (σ : Fin m → ι) :
+    ContDiffOn ℝ ∞ (jetFamily e m u σ) U := word_smoothOn hU hu _
+
+lemma jetEnergy_smoothOn {U : Set E} (hU : IsOpen U) {u : E → F}
+    (hu : ContDiffOn ℝ ∞ u U) (e : ι → E) (m : ℕ) :
+    ContDiffOn ℝ ∞ (jetEnergy e m u) U := familyEnergy_smoothOn (jetFamily_smoothOn hU hu e m)
+
+lemma jetEnergy_nonneg (e : ι → E) (m : ℕ) (u : E → F) (x : E) :
+    0 ≤ jetEnergy e m u x := Finset.sum_nonneg (fun _ _ => sq_nonneg _)
+
+lemma jetArray_energy {U : Set E} (hU : IsOpen U) {u : E → F}
+    (hu : ContDiffOn ℝ ∞ u U) (e : ι → E) (m : ℕ) {x : E} (hx : x ∈ U) :
+    ‖jetArray e m (iteratedFDeriv ℝ m u x)‖^2 = jetEnergy e m u x := by
+  rw [PiLp.norm_sq_eq_of_L2]
+  apply Finset.sum_congr rfl
+  intro σ _
+  congr 1
+  exact congrArg norm (word_ofFn_eq hU hu m (fun i => e (σ i)) hx).symm
+
+lemma norm_iteratedFDeriv_le_jetEnergy [FiniteDimensional ℝ E] [FiniteDimensional ℝ F]
+    (e : Basis ι ℝ E) (m : ℕ) :
+    ∃ C : ℝ, 0 < C ∧ ∀ (U : Set E), IsOpen U → ∀ u : E → F, ContDiffOn ℝ ∞ u U →
+      ∀ x ∈ U, ‖iteratedFDeriv ℝ m u x‖ ≤ C*Real.sqrt (jetEnergy e m u x) := by
+  obtain ⟨C,hC,hb⟩ := jetArray_norm_control (F := F) e m
+  refine ⟨C,hC,fun U hU u hu x hx => ?_⟩
+  have he : Real.sqrt (jetEnergy e m u x) = ‖jetArray e m (iteratedFDeriv ℝ m u x)‖ := by
+    rw [← jetArray_energy hU hu e m hx,Real.sqrt_sq (norm_nonneg _)]
+  rw [he]
+  exact hb _
+
+lemma family_laplace_norm_bound {κ : Type*} [Fintype κ]
+    {v : ι → E} {f : κ → E → F} {x : E} {C : ℝ} (hC : 0 ≤ C)
+    (hb : ∀ j, ‖frameLaplace v (f j) x‖ ≤ C) :
+    Real.sqrt (∑ j, ‖frameLaplace v (f j) x‖^2) ≤ (Fintype.card κ+1)*C := by
+  apply (Real.sqrt_le_iff).mpr
+  refine ⟨by positivity,?_⟩
+  calc
+    _ ≤ ∑ _j : κ, C^2 := Finset.sum_le_sum (fun j _ => pow_le_pow_left₀ (norm_nonneg _) (hb j) 2)
+    _ = (Fintype.card κ : ℝ)*C^2 := by simp
+    _ ≤ _ := by nlinarith [mul_nonneg (sq_nonneg C) (sq_nonneg (Fintype.card κ : ℝ))]
+end HigherJet
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/EncodedMetricEquation.lean b/lean/OAI/Geometry/PrescribedPotential/EncodedMetricEquation.lean
new file mode 100644
index 0000000000000000000000000000000000000000..b128ed3026802235d6e272d6469d031f6869d7a5
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/EncodedMetricEquation.lean
@@ -0,0 +1,77 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CalabiConnection
+import OAI.Geometry.PrescribedPotential.MetricQuasilinearEquation
+import OAI.Geometry.PrescribedPotential.PrincipalComponents
+import OAI.Geometry.PrescribedPotential.QuasilinearJetBasics
+
+namespace OAI
+
+/-! Encoded Metric Equation. -/
+
+section
+
+noncomputable section
+open Set Filter Topology Matrix Finset
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace MetricSystem
+open EllipticKernel FrozenPoisson HigherJet KaehlerCalculus
+variable {n : ℕ}
+local instance equationRealHMIP : InnerProductSpace ℝ (HM n) := InnerProductSpace.rclikeToReal ℂ (HM n)
+local instance equationRealECIP : InnerProductSpace ℝ (EC n) := InnerProductSpace.rclikeToReal ℂ (EC n)
+
+lemma encoded_W {M : V n → Mat n} {z : V n} (hM : ContDiffAt ℝ ∞ M z)
+    (a : ℂ) (v : V n) :
+    W a v (fderiv ℝ (fun x => encode n (M (coordinateEquiv n x))) ((coordinateEquiv n).symm z)) =
+      mderiv a v M z := by
+  have hd := ((encode n).hasFDerivAt.comp ((coordinateEquiv n).symm z)
+    (((by simpa only [ContinuousLinearEquiv.apply_symm_apply] using hM :
+      ContDiffAt ℝ ∞ M (coordinateEquiv n ((coordinateEquiv n).symm z))).differentiableAt (by simp)).hasFDerivAt.comp
+      ((coordinateEquiv n).symm z) (coordinateEquiv n).hasFDerivAt))
+  have hd' := hd.fderiv
+  simp only [Function.comp_def] at hd'
+  have hw := congrArg (W a v) hd'
+  calc
+    _ = _ := hw
+    _ = _ := by
+      rw [mderiv_eq hM]
+      simp only [W,ContinuousLinearMap.comp_apply,ContinuousLinearEquiv.coe_coe,
+        ContinuousLinearEquiv.apply_symm_apply]
+      change (1/2:ℂ) • ((encode n).symm (encode n ((fderiv ℝ M z) v))+a • (encode n).symm (encode n ((fderiv ℝ M z) (Complex.I • v)))) = _
+      rw [ContinuousLinearEquiv.symm_apply_apply,ContinuousLinearEquiv.symm_apply_apply]
+
+lemma encoded_quadratic {M : V n → Mat n} {z : V n} (hM : ContDiffAt ℝ ∞ M z) :
+    quadraticCoefficient (encode n (M z), fderiv ℝ (fun y => encode n (M (coordinateEquiv n y))) ((coordinateEquiv n).symm z)) =
+      encode n (∑ i, ∑ j, (M z)⁻¹ i j • (mderiv Complex.I (e j) M z*(M z)⁻¹*mderiv (-Complex.I) (e i) M z)) := by
+  unfold quadraticCoefficient
+  simp only [inverseCoefficient,ContinuousLinearEquiv.symm_apply_apply]
+  apply congrArg (encode n)
+  apply sum_congr rfl
+  intro i _
+  apply sum_congr rfl
+  intro j _
+  rw [encoded_W hM,encoded_W hM]
+
+local instance equationHessianNorm : NormedAddCommGroup (EC n →L[ℝ] EC n →L[ℝ] HM n) := inferInstance
+local instance equationHessianSpace : NormedSpace ℝ (EC n →L[ℝ] EC n →L[ℝ] HM n) := inferInstance
+
+lemma encoded_metric_equation {ι : Type*} [Fintype ι] (basis : OrthonormalBasis ι ℝ (EC n))
+    (K : LocalKaehlerField n) {x : EC n} (hx : coordinateEquiv n x ∈ K.domain) :
+    principalBilinear basis (inverseCoefficient (encode n (K.matrix (coordinateEquiv n x))))
+      (hessian (fun y => encode n (K.matrix (coordinateEquiv n y))) x) =
+      encode n (K.ricciHessian (coordinateEquiv n x))+
+      quadraticCoefficient (firstJet (fun y => encode n (K.matrix (coordinateEquiv n y))) x) := by
+  have hK := K.smooth.contDiffAt (K.isOpen.mem_nhds hx)
+  have he := encoded_principal basis hK (K.positive _ hx)
+  simp only [ContinuousLinearEquiv.symm_apply_apply] at he
+  have hQ := encoded_quadratic hK
+  simp only [ContinuousLinearEquiv.symm_apply_apply] at hQ
+  change principalBilinear basis (K.matrix (coordinateEquiv n x))⁻¹
+    (fderiv ℝ (fderiv ℝ (fun y => encode n (K.matrix (coordinateEquiv n y)))) x) = _
+  rw [he,K.metric_quasilinear_equation hx,map_add]
+  exact congrArg (fun y => encode n (K.ricciHessian (coordinateEquiv n x))+y) hQ.symm
+end MetricSystem
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/EvenRellich.lean b/lean/OAI/Geometry/PrescribedPotential/EvenRellich.lean
new file mode 100644
index 0000000000000000000000000000000000000000..76d17fa7fc4353e98ef28cf2775cad08735781f8
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/EvenRellich.lean
@@ -0,0 +1,104 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CompletedResolvent
+import OAI.Geometry.PrescribedPotential.GlobalRellich
+import OAI.Geometry.PrescribedPotential.HigherMetricOperator
+import OAI.Geometry.PrescribedPotential.HigherResolvent
+import OAI.Geometry.PrescribedPotential.QuantitativeResolvent
+
+namespace OAI
+
+/-! Even Rellich. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology
+open scoped ContDiff Classical
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+def shiftedOrder (m : ℝ) (k : ℕ) :
+    D.localizers.Sobolev ((k : ℝ)+2) →L[ℝ] D.localizers.Sobolev (k : ℝ) :=
+  (m^2 : ℝ) • D.localizers.lower ((k : ℝ)+2) (k : ℝ) - D.completedLOrder k
+
+lemma shiftedOrder_embed (m : ℝ) (k : ℕ) (f : Smooth A) :
+    D.shiftedOrder m k (D.localizers.embed ((k : ℝ)+2) f) =
+      D.localizers.embed (k : ℝ) ((m^2 : ℝ) • f - complexL g f) := by
+  simp only [shiftedOrder,sub_apply,smul_apply,
+    D.localizers.lower_embed (by linarith : (k : ℝ) ≤ (k : ℝ)+2),
+    D.completedLOrder_embed, map_sub, map_smul]
+
+lemma shiftedOrder_lower (m : ℝ) (k : ℕ) (u : D.localizers.Sobolev ((k : ℝ)+2)) :
+    D.localizers.lower (k : ℝ) 0 (D.shiftedOrder m k u) =
+      D.shiftedOperator m (D.localizers.lower ((k : ℝ)+2) 2 u) := by
+  simp only [shiftedOrder,shiftedOperator,sub_apply,
+    smul_apply,map_sub,map_smul,D.completedLOrder_lower]
+  rw [D.localizers.lower_lower (by linarith : (k : ℝ) ≤ (k : ℝ)+2) (Nat.cast_nonneg k),
+    D.localizers.lower_lower (by linarith [Nat.cast_nonneg (α := ℝ) k] :
+      (2 : ℝ) ≤ (k : ℝ)+2) (by norm_num : (0 : ℝ) ≤ 2)]
+
+lemma resolventOrder_left (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
+    (k : ℕ) (R : D.localizers.Sobolev (k : ℝ) →L[ℝ] D.localizers.Sobolev ((k : ℝ)+2))
+    (hR : ∀ f, D.localizers.lower ((k : ℝ)+2) 2 (R f) =
+      D.completedResolvent m hm he (D.localizers.lower (k : ℝ) 0 f))
+    (u : D.localizers.Sobolev ((k : ℝ)+2)) : R (D.shiftedOrder m k u) = u := by
+  apply D.localizers.lower_injective (by linarith [Nat.cast_nonneg (α := ℝ) k] :
+    (2 : ℝ) ≤ (k : ℝ)+2)
+  rw [hR,D.shiftedOrder_lower,D.completedResolvent_left]
+
+lemma lower_resolvent_factor (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
+    (k j : ℕ) (hjk : (j : ℝ) = (k : ℝ)+2)
+    (R : D.localizers.Sobolev (k : ℝ) →L[ℝ] D.localizers.Sobolev ((k : ℝ)+2))
+    (hR : ∀ f, D.localizers.lower ((k : ℝ)+2) 2 (R f) =
+      D.completedResolvent m hm he (D.localizers.lower (k : ℝ) 0 f)) :
+    D.localizers.lower ((j : ℝ)+2) (j : ℝ) =
+      (D.localizers.lower ((k : ℝ)+2) (j : ℝ) ∘L R) ∘L
+        D.localizers.lower ((k : ℝ)+2) (k : ℝ) ∘L
+        (D.localizers.lower (j : ℝ) ((k : ℝ)+2) ∘L D.shiftedOrder m j) := by
+  apply DFunLike.coe_injective
+  apply (D.localizers.embed_dense ((j : ℝ)+2)).equalizer (by fun_prop) (by fun_prop)
+  funext f
+  simp only [ContinuousLinearMap.coe_comp,Function.comp_apply,D.shiftedOrder_embed,
+    D.localizers.lower_embed (by linarith only [hjk] : (j : ℝ) ≤ (j : ℝ)+2),
+    D.localizers.lower_embed (by linarith only [hjk] : (k : ℝ)+2 ≤ (j : ℝ)),
+    D.localizers.lower_embed (by linarith : (k : ℝ) ≤ (k : ℝ)+2)]
+  rw [← D.shiftedOrder_embed m k,D.resolventOrder_left m hm he k R hR]
+  exact (D.localizers.lower_embed (by linarith only [hjk] : (j : ℝ) ≤ (k : ℝ)+2) f).symm
+
+lemma lower_compact_step (k j : ℕ) (hjk : (j : ℝ) = (k : ℝ)+2)
+    (hK : IsCompactOperator (D.localizers.lower ((k : ℝ)+2) (k : ℝ))) :
+    IsCompactOperator (D.localizers.lower ((j : ℝ)+2) (j : ℝ)) := by
+  obtain ⟨m,hm,he⟩ := D.exists_completedError_small
+  obtain ⟨R,hR⟩ := D.completedResolvent_order m hm he k
+  rw [D.lower_resolvent_factor m hm he k j hjk R hR]
+  exact (hK.comp_clm (D.localizers.lower (j : ℝ) ((k : ℝ)+2) ∘L D.shiftedOrder m j)).clm_comp
+    (D.localizers.lower ((k : ℝ)+2) (j : ℝ) ∘L R)
+
+ 
+
+def CompactLower (s t : ℝ) : Prop := IsCompactOperator (D.localizers.lower s t)
+
+ 
+theorem lower_even_compact (n : ℕ) :
+    IsCompactOperator (D.localizers.lower (((2*n : ℕ) : ℝ)+2) ((2*n : ℕ) : ℝ)) := by
+  change D.CompactLower (((2*n : ℕ) : ℝ)+2) ((2*n : ℕ) : ℝ)
+  induction n with
+  | zero =>
+    simp only [mul_zero,Nat.cast_zero,zero_add]
+    exact D.lower_compact
+  | succ n ih =>
+    exact D.lower_compact_step (2*n) (2*(n+1)) (by push_cast; ring) ih
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/FamilyBernsteinBootstrap.lean b/lean/OAI/Geometry/PrescribedPotential/FamilyBernsteinBootstrap.lean
new file mode 100644
index 0000000000000000000000000000000000000000..7880aa8f3635c19e4412257cefab9c01145cb4b0
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/FamilyBernsteinBootstrap.lean
@@ -0,0 +1,109 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.BernsteinEnergyInterior
+import OAI.Geometry.PrescribedPotential.CompactSmoothCutoff
+
+namespace OAI
+
+/-! Family Bernstein Bootstrap. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology
+open scoped ContDiff
+namespace HigherJet
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E]
+  {ι α : Type*} [Fintype ι]
+
+def EnergyBound (S : ℕ → α → E → ℝ) (m : ℕ) (K : Set E) : Prop :=
+  ∃ C : ℝ, 0 ≤ C ∧ ∀ a x, x ∈ K → S m a x ≤ C
+
+lemma family_bernstein_bootstrap {U : Set E} (hU : IsOpen U)
+    (S D : ℕ → α → E → ℝ) (v : α → E → ι → E)
+    (hsmooth : ∀ m a, ContDiffOn ℝ ∞ (S m a) U)
+    (hpos : ∀ m a x, x ∈ U → 0 ≤ S m a x ∧ 0 ≤ D m a x)
+    (hgrad : ∀ m a x, x ∈ U → ‖frameGradient (v a x) (S m a) x‖^2 ≤ 4*S m a x*D m a x)
+    (hframes : ∀ K, IsCompact K → K ⊆ U → ∃ B : ℝ, 0 ≤ B ∧ ∀ a x, x ∈ K → ∀ i, ‖v a x i‖ ≤ B)
+    (hcoerce : ∀ m, 1 ≤ m → ∀ K, IsCompact K → K ⊆ U → ∃ A : ℝ, 1 ≤ A ∧
+      ∀ a x, x ∈ K → S (m+1) a x ≤ A*D m a x ∧ D m a x ≤ A*S (m+1) a x)
+    (hfirst : ∀ K, IsCompact K → K ⊆ U → EnergyBound S 1 K)
+    (hbounded : ∀ m, 1 ≤ m → ∀ K, IsCompact K → K ⊆ U →
+      (∀ j, 1 ≤ j → j ≤ m → EnergyBound S j K) →
+      ∃ R : ℝ, 0 ≤ R ∧ ∀ a x, x ∈ K → D m a x-R ≤ frameLaplace (v a x) (S m a) x)
+    (hcubic : ∀ K, IsCompact K → K ⊆ U → ∃ R : ℝ, 0 ≤ R ∧ ∀ a x, x ∈ K →
+      -R*(1+S 2 a x*Real.sqrt (S 2 a x)) ≤ frameLaplace (v a x) (S 2 a) x)
+    (hlinear : ∀ m, 3 ≤ m → ∀ K, IsCompact K → K ⊆ U →
+      (∀ j, 1 ≤ j → j < m → EnergyBound S j K) →
+      ∃ R : ℝ, 0 ≤ R ∧ ∀ a x, x ∈ K →
+        D m a x-R*(1+S m a x) ≤ frameLaplace (v a x) (S m a) x) :
+    ∀ m, 1 ≤ m → ∀ K, IsCompact K → K ⊆ U → EnergyBound S m K := by
+  intro m
+  induction m using Nat.strong_induction_on with
+  | h m ih =>
+    intro hm K hK hKU
+    by_cases hm1 : m=1
+    · subst m
+      exact hfirst K hK hKU
+    obtain ⟨L,χ,hL,hLU,hKL,hχ,hχval,hχone,hχint⟩ := compact_smooth_cutoff hK hU hKU
+    obtain ⟨B,hB,hv⟩ := hframes L hL hLU
+    have hχvals : ∀ x ∈ L, 0 ≤ χ x ∧ χ x ≤ 1 := fun x _ => hχval x
+    by_cases hm2 : m=2
+    · subst m
+      obtain ⟨A,hA,hctrl⟩ := hcoerce 1 le_rfl L hL hLU
+      obtain ⟨R0,hR0,hSb⟩ := hfirst L hL hLU
+      obtain ⟨R1,hR1,hLS⟩ := hbounded 1 le_rfl L hL hLU (fun j hj hj1 => by
+        have he : j=1 := by omega
+        subst j
+        exact hfirst L hL hLU)
+      obtain ⟨R2,hR2,hLT⟩ := hcubic L hL hLU
+      let R := 1+R0+R1+R2
+      have hR : 1 ≤ R := by dsimp [R]; linarith
+      have hr0 : R0 ≤ R := by dsimp [R]; linarith
+      have hr1 : R1 ≤ R := by dsimp [R]; linarith
+      have hr2 : R2 ≤ R := by dsimp [R]; linarith
+      obtain ⟨C,hC,hbound⟩ := quartic_energy_interior (ι := ι) hL hU hLU hχ hχvals hχint hA hR hB
+      refine ⟨C,hC,fun a z hz => hbound (S 1 a) (S 2 a) (hsmooth 1 a) (hsmooth 2 a) (v a)
+        (fun x hx => hv a x hx) ?_ ?_ z (hKL hz) (hχone z hz)⟩
+      · intro x hx
+        exact ⟨(hpos 1 a x (hLU hx)).1,(hSb a x hx).trans hr0,(hpos 2 a x (hLU hx)).1⟩
+      · intro x hx
+        have hc := hctrl a x hx
+        refine ⟨D 1 a x,(hpos 1 a x (hLU hx)).2,hc.1,hc.2,?_,?_,hgrad 1 a x (hLU hx)⟩
+        · linarith only [hLS a x hx,hr1]
+        · have hq := mul_le_mul_of_nonneg_right hr2
+            (show 0 ≤ 1+S 2 a x*Real.sqrt (S 2 a x) by have := (hpos 2 a x (hLU hx)).1; positivity)
+          nlinarith only [hq,hLT a x hx]
+    · have hm3 : 3 ≤ m := by omega
+      have hiL : ∀ j, 1 ≤ j → j < m → EnergyBound S j L := fun j hj hjm => ih j hjm hj L hL hLU
+      have hpred : 1 ≤ m-1 := by omega
+      have he : m-1+1=m := by omega
+      obtain ⟨A,hA,hctrl⟩ := hcoerce (m-1) hpred L hL hLU
+      obtain ⟨R0,hR0,hSb⟩ := hiL (m-1) hpred (by omega)
+      obtain ⟨R1,hR1,hLS⟩ := hbounded (m-1) hpred L hL hLU
+        (fun j hj hjm => hiL j hj (by omega))
+      obtain ⟨R2,hR2,hLT⟩ := hlinear m hm3 L hL hLU hiL
+      let R := 1+R0+R1+R2
+      have hR : 1 ≤ R := by dsimp [R]; linarith
+      have hr0 : R0 ≤ R := by dsimp [R]; linarith
+      have hr1 : R1 ≤ R := by dsimp [R]; linarith
+      have hr2 : R2 ≤ R := by dsimp [R]; linarith
+      obtain ⟨C,hC,hbound⟩ := linear_energy_interior (ι := ι) hL hU hLU hχ hχvals hχint hA hR hB
+      refine ⟨C,hC,fun a z hz => hbound (S (m-1) a) (S m a) (hsmooth (m-1) a) (hsmooth m a) (v a)
+        (fun x hx => hv a x hx) ?_ ?_ z (hKL hz) (hχone z hz)⟩
+      · intro x hx
+        exact ⟨(hpos (m-1) a x (hLU hx)).1,(hSb a x hx).trans hr0,(hpos m a x (hLU hx)).1⟩
+      · intro x hx
+        have hc := (hctrl a x hx).1
+        rw [he] at hc
+        refine ⟨D (m-1) a x,D m a x,(hpos (m-1) a x (hLU hx)).2,(hpos m a x (hLU hx)).2,hc,?_,?_,hgrad m a x (hLU hx)⟩
+        · linarith only [hLS a x hx,hr1]
+        · have hq := mul_le_mul_of_nonneg_right hr2 (show 0 ≤ 1+S m a x by linarith [(hpos m a x (hLU hx)).1])
+          linarith only [hq,hLT a x hx]
+end HigherJet
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/FiniteSobolevEmbedding.lean b/lean/OAI/Geometry/PrescribedPotential/FiniteSobolevEmbedding.lean
new file mode 100644
index 0000000000000000000000000000000000000000..976b1fcd19b5a3355370638af48dcf2b4cbb9f78
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/FiniteSobolevEmbedding.lean
@@ -0,0 +1,90 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CircleRadialCalculus
+import OAI.Geometry.PrescribedPotential.StrongSobolevEmbedding
+
+namespace OAI
+
+/-! Finite Sobolev Embedding. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology MeasureTheory FourierTransform TemperedDistribution
+open scoped SchwartzMap BoundedContinuousFunction ContDiff
+namespace SobolevChart
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+
+lemma fourier_moment_of_bessel {u : 𝓢'(E, ℂ)}
+    {v : Lp ℂ 1 (volume : Measure E)} (hv : 𝓕 u = (v : 𝓢'(E,ℂ))) (n : ℕ)
+    (w : Lp ℂ 1 (volume : Measure E))
+    (hw' : smulLeftCLM ℂ (fun x : E => (((1 + ‖x‖ ^ 2) ^ n : ℝ) : ℂ))
+      (𝓕 u) = (w : 𝓢'(E, ℂ))) : Integrable (fun x => ‖x‖^n * ‖v x‖) := by
+  let b := inverseWeight (E := E) n
+  have hb := b.memLp_top (μ := volume)
+  have he : (hb.toLp b : Lp ℂ ⊤ volume) • w = v := by
+    apply (LinearMap.ker_eq_bot.mp (Lp.ker_toTemperedDistributionCLM_eq_bot
+      (F := ℂ) (E := E) (μ := volume) (p := 1)))
+    change (((hb.toLp b : Lp ℂ ⊤ volume) • w : Lp ℂ 1 volume) : 𝓢'(E, ℂ)) = (v : 𝓢'(E, ℂ))
+    rw [Lp.toTemperedDistribution_smul_eq (inverseWeight_temperate n) hb,
+      ← hw', smulLeftCLM_smulLeftCLM_apply (by fun_prop) (inverseWeight_temperate n)]
+    have hc : (fun x : E => (((1 + ‖x‖ ^ 2) ^ n : ℝ) : ℂ)) * (b : E → ℂ) = fun _ => 1 := by
+      ext x
+      exact weight_cancel n x
+    rw [hc, smulLeftCLM_const, one_smul, hv]
+  have he' : ∀ᵐ x, ((1 + ‖x‖ ^ 2) ^ n : ℝ) * ‖v x‖ = ‖w x‖ := by
+    have hh := Lp.coeFn_lpSMul (r := 1) (hb.toLp b : Lp ℂ ⊤ volume) w
+    rw [he] at hh
+    filter_upwards [hh, hb.coeFn_toLp] with x hx hb'
+    have hx' : (((1 + ‖x‖ ^ 2) ^ n : ℝ) : ℂ) * v x = w x := by
+      rw [hx, Pi.smul_apply', hb', smul_eq_mul, ← mul_assoc, weight_cancel, one_mul]
+    simpa only [norm_mul, Complex.norm_real, Real.norm_eq_abs,
+      abs_of_nonneg (show 0 ≤ (1 + ‖x‖ ^ 2) ^ n by positivity)] using congrArg norm hx'
+  apply (L1.integrable_coeFn w).norm.mono'
+    (((continuous_norm.pow n).aestronglyMeasurable).mul (Lp.aestronglyMeasurable v).norm)
+  filter_upwards [he'] with x hx
+  change ‖‖x‖ ^ n * ‖v x‖‖ ≤ ‖w x‖
+  rw [Real.norm_of_nonneg (mul_nonneg (pow_nonneg (norm_nonneg x) n) (norm_nonneg (v x))), ← hx]
+  apply mul_le_mul_of_nonneg_right _ (norm_nonneg _)
+  apply pow_le_pow_left₀ (norm_nonneg _)
+  nlinarith [sq_nonneg (‖x‖ - 1 / 2)]
+
+lemma finite_fourier_moments {s : ℝ} {u : 𝓢'(E,ℂ)} (hu : MemSobolev s 2 u)
+    (N : ℕ) (hs : (Module.finrank ℝ E : ℝ) < 2*(s - 2*N)) :
+    ∃ v : Lp ℂ 1 (volume : Measure E), 𝓕 u = (v : 𝓢'(E,ℂ)) ∧
+      ∀ n : ℕ, n ≤ N → Integrable (fun x => ‖x‖^n * ‖v x‖) := by
+  obtain ⟨v,hv⟩ := hu.fourier_memL1 (by linarith [Nat.cast_nonneg (α := ℝ) N])
+  refine ⟨v,hv,fun n hn => ?_⟩
+  have hm : MemSobolev (s - 2*n) 2 (besselPotential E ℂ (2*n) u) := by
+    rw [memSobolev_besselPotential_iff]
+    simpa only [add_sub_cancel] using hu
+  obtain ⟨w,hw⟩ := hm.fourier_memL1 (by
+    have hh : (n : ℝ) ≤ N := by exact_mod_cast hn
+    linarith)
+  apply fourier_moment_of_bessel hv n w
+  rw [fourier_besselPotential_eq_smulLeftCLM_fourier_apply] at hw
+  simpa only [mul_div_cancel_left₀ (n : ℝ) (by norm_num : (2 : ℝ) ≠ 0),
+    Real.rpow_natCast] using hw
+
+lemma strongEmbedding_contDiff (s : ℝ) (hs : (Module.finrank ℝ E : ℝ) < 2*s)
+    (N : ℕ) (hN : (Module.finrank ℝ E : ℝ) < 2*(s - 2*N)) (u : L2 E) :
+    ContDiff ℝ N (strongEmbedding s hs u : E → ℂ) := by
+  obtain ⟨v,hv,hm⟩ := finite_fourier_moments (realize_memSobolev s u) N hN
+  have he : strongEmbedding s hs u = Real.Lp.fourierTransformInv v := by
+    apply boundedDistribution_injective
+    rw [strongEmbedding_distribution, boundedDistributionCLM_apply,
+      fourierInv_L1_distribution, ← hv, fourierInv_fourier_eq]
+  rw [he]
+  change ContDiff ℝ N (fun x : E => 𝓕 (v : E → ℂ) (-x))
+  have hf : ContDiff ℝ N (𝓕 (v : E → ℂ)) := Real.contDiff_fourier (N := N) (by
+    intro n hn
+    exact hm n (by exact_mod_cast hn))
+  exact hf.comp contDiff_neg
+end SobolevChart
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/FixedScaleBootstrap.lean b/lean/OAI/Geometry/PrescribedPotential/FixedScaleBootstrap.lean
new file mode 100644
index 0000000000000000000000000000000000000000..0165e409f2614be06941bbb11f87defd1c8aa780
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/FixedScaleBootstrap.lean
@@ -0,0 +1,255 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.NonlinearSobolevGain
+import OAI.Geometry.PrescribedPotential.LinearizedCommutator
+
+namespace OAI
+
+/-! Fixed Scale Bootstrap. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology
+open scoped ContDiff Classical
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  [ConnectedSpace X] {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+local instance fixedBootNG (s : ℝ) : NormedAddCommGroup (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedAddCommGroup
+local instance fixedBootNS (s : ℝ) : NormedSpace ℝ (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedSpace
+local instance fixedBootTG (s : ℝ) : IsTopologicalAddGroup (D.localizers.RealSobolev s) :=
+  Submodule.isTopologicalAddGroup _
+local instance fixedBootCS (s : ℝ) : ContinuousSMul ℝ (D.localizers.RealSobolev s) :=
+  SMulMemClass.continuousSMul _
+
+ 
+
+def LinearizedGain (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (u : D.localizers.RealSobolev (((k+1 : ℕ) : ℝ)+2)) : Prop :=
+  ∀ (w : D.localizers.RealSobolev ((k : ℝ)+2))
+      (f : D.localizers.RealSobolev ((k+1 : ℕ) : ℝ)),
+    D.realVolumeDerivative k hk
+      (D.localizers.realLower (((k+1 : ℕ) : ℝ)+2) ((k : ℝ)+2) (by push_cast; linarith) u) w =
+      D.localizers.realLower ((k+1 : ℕ) : ℝ) (k : ℝ) (by push_cast; linarith) f →
+    ∃ z : D.localizers.RealSobolev (((k+1 : ℕ) : ℝ)+2),
+      D.localizers.realLower (((k+1 : ℕ) : ℝ)+2) ((k : ℝ)+2) (by push_cast; linarith) z = w
+
+ 
+lemma linearizedGain_nhds (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) (x₀ : X) :
+    ∀ᶠ u in 𝓝 (0 : D.localizers.RealSobolev (((k+1 : ℕ) : ℝ)+2)),
+      D.LinearizedGain k hk u := by
+  filter_upwards [D.linearizedVolume_local_gain (k+1) k (by omega) hk (by omega) x₀]
+    with u hu
+  intro w f hf
+  have he : D.linearizedVolume k hk x₀
+      (D.localizers.realLower (((k+1 : ℕ) : ℝ)+2) ((k : ℝ)+2) (by push_cast; linarith) u)
+      (w,0) = D.pairLower ((k+1 : ℕ) : ℝ) (k : ℝ) (by push_cast; linarith)
+        (f,D.realEvaluation ((k : ℝ)+2) x₀ w) := by
+    rw [D.linearizedVolume_apply_zero_scalar,D.pairLower_apply]
+    exact Prod.ext hf rfl
+  obtain ⟨z,hz,_⟩ := hu (w,0) (f,D.realEvaluation ((k : ℝ)+2) x₀ w) he
+  exact ⟨z.1,congrArg Prod.fst hz⟩
+
+ 
+omit [ConnectedSpace X] in
+lemma linearized_derivative_equation (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (p : ι) (v : EC d) (u : D.localizers.RealSobolev (((k+2 : ℕ) : ℝ)+2))
+    (w : D.localizers.RealSobolev (((k+1 : ℕ) : ℝ)+2))
+    (f : D.localizers.RealSobolev ((k+2 : ℕ) : ℝ))
+    (hf : D.realVolumeDerivative (k+1) (by omega)
+      (D.localizers.realLower (((k+2 : ℕ) : ℝ)+2) (((k+1 : ℕ) : ℝ)+2)
+        (by push_cast; linarith) u) w =
+      D.localizers.realLower ((k+2 : ℕ) : ℝ) ((k+1 : ℕ) : ℝ) (by push_cast; linarith) f) :
+    let u₁ := D.localizers.realLower (((k+2 : ℕ) : ℝ)+2) (((k+1 : ℕ) : ℝ)+2)
+      (by push_cast; linarith) u
+    D.realVolumeDerivative k hk
+      (D.localizers.realLower (((k+1 : ℕ) : ℝ)+2) ((k : ℝ)+2) (by push_cast; linarith) u₁)
+      (D.potentialDerivative k p v w) =
+    D.localizers.realLower ((k+1 : ℕ) : ℝ) (k : ℝ) (by push_cast; linarith)
+      (D.densityDerivative (k+1) p v f +
+        (D.realRemainderDerivative (k+1) (by omega) p v u₁ w -
+          D.realVolumeSecond (k+1) (by omega) u₁ w
+            (D.potentialDerivative (k+1) p v u))) := by
+  let : NormedAddCommGroup (D.localizers.RealSobolev ((k+1 : ℕ) : ℝ)) := inferInstance
+  let : NormedSpace ℝ (D.localizers.RealSobolev ((k+1 : ℕ) : ℝ)) := inferInstance
+  let : NormedAddCommGroup (D.localizers.RealSobolev (k : ℝ)) := inferInstance
+  let : NormedSpace ℝ (D.localizers.RealSobolev (k : ℝ)) := inferInstance
+  have hh := D.linearized_commutator_weak k hk p v u w
+  dsimp only at hh ⊢
+  rw [hf, D.densityDerivative_lower (k+1) k (by omega)] at hh
+  simpa only [map_add] using hh
+
+ 
+
+omit [ConnectedSpace X] in
+lemma linearizedGain_succ (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (u : D.localizers.RealSobolev (((k+2 : ℕ) : ℝ)+2))
+    (hG : D.LinearizedGain k hk
+      (D.localizers.realLower (((k+2 : ℕ) : ℝ)+2) (((k+1 : ℕ) : ℝ)+2)
+        (by push_cast; linarith) u)) :
+    D.LinearizedGain (k+1) (by omega) u := by
+  intro w f hf
+  let u₁ := D.localizers.realLower (((k+2 : ℕ) : ℝ)+2) (((k+1 : ℕ) : ℝ)+2)
+    (by push_cast; linarith) u
+  have hDer (p : ι) (v : EC d) : ∃ z : D.localizers.RealSobolev (((k+1 : ℕ) : ℝ)+2),
+      D.localizers.realLower (((k+1 : ℕ) : ℝ)+2) ((k : ℝ)+2)
+        (by push_cast; linarith) z = D.potentialDerivative k p v w := by
+    exact hG _ _ (D.linearized_derivative_equation k hk p v u w f hf)
+  obtain ⟨z,hz⟩ := D.derivative_lifts_gain k w hDer
+  refine ⟨D.localizers.realLower ((k+4 : ℕ) : ℝ) (((k+1+1 : ℕ) : ℝ)+2)
+    (by push_cast; linarith) z, ?_⟩
+  rw [Localizers.realLower_lower]
+  exact hz
+
+ 
+omit [ConnectedSpace X] in
+lemma nonlinear_gain_of_linearizedGain (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (u : D.localizers.RealSobolev (((k+1 : ℕ) : ℝ)+2))
+    (hG : D.LinearizedGain k hk u) (F : RealSmooth A)
+    (hF : D.realVolume (k+1) (by omega) u = D.localizers.realEmbed ((k+1 : ℕ) : ℝ) F) :
+    ∃ z : D.localizers.RealSobolev (((k+2 : ℕ) : ℝ)+2),
+      D.localizers.realLower (((k+2 : ℕ) : ℝ)+2) (((k+1 : ℕ) : ℝ)+2)
+        (by push_cast; linarith) z = u := by
+  have hDer (p : ι) (v : EC d) : ∃ z : D.localizers.RealSobolev (((k+1 : ℕ) : ℝ)+2),
+      D.localizers.realLower (((k+1 : ℕ) : ℝ)+2) ((k : ℝ)+2)
+        (by push_cast; linarith) z = D.potentialDerivative k p v u := by
+    apply hG (D.potentialDerivative k p v u)
+      (D.localizers.realEmbed ((k+1 : ℕ) : ℝ) (D.realLocalizedDerivative p v F) +
+        D.realNonlinearRemainder (k+1) (by omega) p v u)
+    exact D.nonlinear_commutator_solution k hk p v u F hF
+  obtain ⟨z,hz⟩ := D.derivative_lifts_gain k u hDer
+  refine ⟨D.localizers.realLower ((k+4 : ℕ) : ℝ) (((k+2 : ℕ) : ℝ)+2)
+    (by push_cast; linarith) z, ?_⟩
+  rw [Localizers.realLower_lower]
+  exact hz
+
+ 
+omit [ConnectedSpace X] in
+lemma realVolume_lift_solution (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (u : D.localizers.RealSobolev ((k : ℝ)+2))
+    (v : D.localizers.RealSobolev (((k+1 : ℕ) : ℝ)+2))
+    (hv : D.localizers.realLower (((k+1 : ℕ) : ℝ)+2) ((k : ℝ)+2) (by push_cast; linarith) v = u)
+    (F : RealSmooth A) (hF : D.realVolume k hk u = D.localizers.realEmbed (k : ℝ) F) :
+    D.realVolume (k+1) (by omega) v = D.localizers.realEmbed ((k+1 : ℕ) : ℝ) F := by
+  apply D.localizers.realLower_injective
+    (by push_cast; linarith : (k : ℝ) ≤ ((k+1 : ℕ) : ℝ))
+  exact (D.realVolume_lower (k+1) k (by omega) hk (by omega) v).symm.trans
+    ((congrArg (D.realVolume k hk) hv).trans
+      (hF.trans (D.localizers.realLower_embed (by push_cast; linarith only) F).symm))
+
+ 
+
+omit [ConnectedSpace X] in
+lemma nonlinear_all_order_lift (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (u : D.localizers.RealSobolev (((k+1 : ℕ) : ℝ)+2))
+    (hG : D.LinearizedGain k hk u) (F : RealSmooth A)
+    (hF : D.realVolume (k+1) (by omega) u = D.localizers.realEmbed ((k+1 : ℕ) : ℝ) F)
+    (l : ℕ) (hl : k ≤ l) :
+    ∃ z : D.localizers.RealSobolev (((l+1 : ℕ) : ℝ)+2),
+      D.localizers.realLower (((l+1 : ℕ) : ℝ)+2) (((k+1 : ℕ) : ℝ)+2)
+        (by exact_mod_cast Nat.add_le_add_right (Nat.add_le_add_right hl 1) 2) z = u ∧
+      D.LinearizedGain l (by omega) z ∧
+      D.realVolume (l+1) (by omega) z = D.localizers.realEmbed ((l+1 : ℕ) : ℝ) F := by
+  induction l, hl using Nat.le_induction with
+  | base =>
+    refine ⟨u,?_,hG,hF⟩
+    apply Subtype.ext
+    exact congrArg (fun T : D.localizers.Sobolev (((k+1 : ℕ) : ℝ)+2) →L[ℝ]
+      D.localizers.Sobolev (((k+1 : ℕ) : ℝ)+2) => T u.val) (D.localizers.lower_self _)
+  | succ l hl ih =>
+    obtain ⟨z,hz,hGz,hFz⟩ := ih
+    obtain ⟨v,hv⟩ := D.nonlinear_gain_of_linearizedGain l (by omega) z hGz F hFz
+    refine ⟨v,?_,?_,?_⟩
+    · calc
+        D.localizers.realLower (((l+1+1 : ℕ) : ℝ)+2) (((k+1 : ℕ) : ℝ)+2) _ v =
+            D.localizers.realLower (((l+1 : ℕ) : ℝ)+2) (((k+1 : ℕ) : ℝ)+2) _
+              (D.localizers.realLower (((l+2 : ℕ) : ℝ)+2) (((l+1 : ℕ) : ℝ)+2) _ v) :=
+          (D.localizers.realLower_lower _ _ v).symm
+        _ = u := by rw [hv,hz]
+    · apply D.linearizedGain_succ l (by omega) v
+      rwa [hv]
+    · exact D.realVolume_lift_solution (l+1) (by omega) z v hv F hFz
+
+omit [ConnectedSpace X] in
+lemma allRegular_of_real_lifts (k : ℕ)
+    (u : D.localizers.RealSobolev (((k+1 : ℕ) : ℝ)+2))
+    (hL : ∀ m : ℕ, ∃ z : D.localizers.RealSobolev (((k+m+1 : ℕ) : ℝ)+2),
+      D.localizers.realLower (((k+m+1 : ℕ) : ℝ)+2) (((k+1 : ℕ) : ℝ)+2)
+        (by push_cast; linarith only [Nat.cast_nonneg (α := ℝ) m]) z = u) :
+    D.localizers.AllRegular
+      (D.localizers.realLower (((k+1 : ℕ) : ℝ)+2) 0 (by positivity) u).val := by
+  intro m
+  obtain ⟨z,hz⟩ := hL m
+  refine ⟨D.localizers.lower (((k+m+1 : ℕ) : ℝ)+2) (m : ℝ) z.val,?_⟩
+  have hm : (m : ℝ) ≤ ((k+m+1 : ℕ) : ℝ)+2 := by
+    push_cast
+    linarith only [Nat.cast_nonneg (α := ℝ) k]
+  have hh := congrArg
+    (D.localizers.realLower (((k+1 : ℕ) : ℝ)+2) 0 (by positivity)) hz
+  have hh' := (D.localizers.realLower_lower _ _ z).symm.trans hh
+  exact (D.localizers.lower_lower hm (Nat.cast_nonneg m) z.val).trans
+    (congrArg Subtype.val hh')
+
+omit [ConnectedSpace X] in
+lemma real_smooth_of_allRegular (s : ℝ) (hs : 0 ≤ s)
+    (hStrong : (Module.finrank ℝ (EC d) : ℝ) < 2*s)
+    (u : D.localizers.RealSobolev s)
+    (hreg : D.localizers.AllRegular (D.localizers.realLower s 0 hs u).val) :
+    ∃ φ : RealSmooth A, D.localizers.realEmbed s φ = u := by
+  obtain ⟨f,hf⟩ := D.allRegular_smooth hreg
+  have he : D.localizers.embed s f = u.val := by
+    apply D.localizers.lower_injective hs
+    exact (D.localizers.lower_embed hs f).trans hf
+  have hr (x : X) : (f x).im = 0 := by
+    have hh := D.localizers.real_strong hStrong u x
+    rw [← he,D.localizers.strong_embed s hStrong f] at hh
+    exact hh
+  exact ⟨⟨f,hr⟩,Subtype.ext he⟩
+
+ 
+
+omit [ConnectedSpace X] in
+lemma nonlinear_smooth_of_linearizedGain (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (u : D.localizers.RealSobolev (((k+1 : ℕ) : ℝ)+2))
+    (hG : D.LinearizedGain k hk u) (F : RealSmooth A)
+    (hF : D.realVolume (k+1) (by omega) u = D.localizers.realEmbed ((k+1 : ℕ) : ℝ) F) :
+    ∃ φ : RealSmooth A, D.localizers.realEmbed (((k+1 : ℕ) : ℝ)+2) φ = u := by
+  have hStrong : (Module.finrank ℝ (EC d) : ℝ) < 2 * (((k+1 : ℕ) : ℝ)+2) := by
+    have hk' : (Module.finrank ℝ (EC d) : ℝ) < (k : ℝ) := by exact_mod_cast hk
+    push_cast
+    linarith only [hk',Nat.cast_nonneg (α := ℝ) k]
+  apply D.real_smooth_of_allRegular _ (by positivity) hStrong u
+  apply D.allRegular_of_real_lifts k u
+  intro m
+  obtain ⟨z,hz,_,_⟩ := D.nonlinear_all_order_lift k hk u hG F hF (k+m) (by omega)
+  exact ⟨z,hz⟩
+
+ 
+
+def FixedSpatialSmoothness (s : ℕ) (hs : Module.finrank ℝ (EC d) < s) : Prop :=
+  ∀ᶠ u in 𝓝 (0 : D.localizers.RealSobolev (((s+1 : ℕ) : ℝ)+2)),
+    ∀ F : RealSmooth A,
+      D.realVolume (s+1) (by omega) u = D.localizers.realEmbed ((s+1 : ℕ) : ℝ) F →
+      ∃ φ : RealSmooth A, D.localizers.realEmbed (((s+1 : ℕ) : ℝ)+2) φ = u
+
+ 
+
+theorem fixed_spatial_smoothness (s : ℕ) (hs : Module.finrank ℝ (EC d) < s) (x₀ : X) :
+    D.FixedSpatialSmoothness s hs := by
+  filter_upwards [D.linearizedGain_nhds s hs x₀] with u hu
+  intro F hF
+  exact D.nonlinear_smooth_of_linearizedGain s hs u hu F hF
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/FourierKernel.lean b/lean/OAI/Geometry/PrescribedPotential/FourierKernel.lean
new file mode 100644
index 0000000000000000000000000000000000000000..0c5b6350a9a9f46aeaa4a55dab630524c0782875
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/FourierKernel.lean
@@ -0,0 +1,94 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.SobolevParametrix
+
+namespace OAI
+
+/-! Fourier Kernel. -/
+
+section
+
+ 
+noncomputable section
+open MeasureTheory Set Filter Topology FourierTransform
+open scoped BoundedContinuousFunction SchwartzMap Classical Real ComplexInnerProductSpace
+namespace EllipticCompact
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+open SobolevChart
+
+lemma phase_sub_bound (a b : ℝ) :
+    ‖((Real.fourierChar a : Circle) : ℂ) - (Real.fourierChar b : Circle)‖ ≤ 2 * Real.pi * ‖a - b‖ := by
+  have he : ((Real.fourierChar a : Circle) : ℂ) - (Real.fourierChar b : Circle) =
+      (((Real.fourierChar (a-b) : Circle) : ℂ) - 1) * (Real.fourierChar b : Circle) := by
+    rw [sub_mul, one_mul, ← Circle.coe_mul, ← AddChar.map_add_eq_mul]
+    rw [sub_add_cancel]
+  rw [he, norm_mul, Circle.norm_coe, mul_one, Real.fourierChar_apply]
+  have hb := Real.norm_exp_I_mul_ofReal_sub_one_le (x := 2 * Real.pi * (a-b))
+  rw [mul_comm (Complex.I)] at hb
+  exact hb.trans (by simp only [norm_mul, Real.norm_eq_abs, abs_of_pos Real.pi_pos]; norm_num)
+
+omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] in
+lemma phase_inner_sub_bound (x z y : E) :
+    ‖((Real.fourierChar (inner ℝ y x) : Circle) : ℂ) -
+      (Real.fourierChar (inner ℝ y z) : Circle)‖ ≤ 2 * Real.pi * ‖y‖ * ‖x-z‖ := by
+  apply (phase_sub_bound _ _).trans
+  rw [← inner_sub_right]
+  calc
+    _ ≤ (2 * Real.pi) * (‖y‖ * ‖x-z‖) :=
+      mul_le_mul_of_nonneg_left (norm_inner_le_norm y (x-z))
+        (mul_nonneg (by norm_num) Real.pi_pos.le)
+    _ = _ := by ring
+
+def kernelSectionFun (χ : 𝓢(E, ℂ)) (x y : E) : ℂ :=
+  star (((Real.fourierChar (inner ℝ y x) : Circle) : ℂ) * χ y)
+
+lemma kernelSection_memLp (χ : 𝓢(E, ℂ)) (x : E) :
+    MemLp (kernelSectionFun χ x) 2 (volume : Measure E) := by
+  apply (χ.memLp (p := 2) (μ := volume)).mono
+    (show Continuous (kernelSectionFun χ x) by
+      unfold kernelSectionFun
+      exact ((continuous_subtype_val.comp (Real.continuous_fourierChar.comp
+        (continuous_id.inner continuous_const))).mul χ.continuous).star
+      ).aestronglyMeasurable
+  filter_upwards [] with y
+  simp only [kernelSectionFun, norm_star, norm_mul, Circle.norm_coe, one_mul, le_refl]
+
+def kernelSection (χ : 𝓢(E, ℂ)) (x : E) : L2 E :=
+  (kernelSection_memLp χ x).toLp (kernelSectionFun χ x)
+
+lemma kernelSection_ae (χ : 𝓢(E, ℂ)) (x : E) :
+    kernelSection χ x =ᵐ[volume] kernelSectionFun χ x :=
+  (kernelSection_memLp χ x).coeFn_toLp
+
+lemma kernelSection_continuous (χ : 𝓢(E, ℂ)) (hχ : HasCompactSupport χ) :
+    Continuous (kernelSection χ) := by
+  obtain ⟨R, hR, hbound⟩ := hχ.isBounded.exists_pos_norm_le
+  have hb (x z : E) : ‖kernelSection χ x - kernelSection χ z‖ ≤
+      (2 * Real.pi * R * ‖χ.toLp 2‖) * ‖x-z‖ := by
+    calc
+      ‖kernelSection χ x - kernelSection χ z‖ ≤
+          (2 * Real.pi * R * ‖x-z‖) * ‖χ.toLp 2‖ := by
+        apply Lp.norm_le_mul_norm_of_ae_le_mul
+        filter_upwards [Lp.coeFn_sub (kernelSection χ x) (kernelSection χ z),
+          kernelSection_ae χ x, kernelSection_ae χ z, χ.coeFn_toLp 2 volume] with y hs hx hz hχy
+        rw [hs, Pi.sub_apply, hx, hz, hχy]
+        simp only [kernelSectionFun, ← star_sub, ← sub_mul, norm_star, norm_mul]
+        by_cases hy : χ y = 0
+        · simp [hy]
+        · apply mul_le_mul_of_nonneg_right _ (norm_nonneg _)
+          exact (phase_inner_sub_bound x z y).trans
+            (mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left
+              (hbound y (subset_tsupport _ hy)) (by positivity)) (norm_nonneg _))
+      _ = _ := by ring
+  apply LipschitzWith.continuous (K := ⟨2 * Real.pi * R * ‖χ.toLp 2‖, by positivity⟩)
+  apply LipschitzWith.of_dist_le_mul
+  intro x z
+  rw [dist_eq_norm, dist_eq_norm]
+  exact hb x z
+
+end EllipticCompact
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/FrameCompactBounds.lean b/lean/OAI/Geometry/PrescribedPotential/FrameCompactBounds.lean
new file mode 100644
index 0000000000000000000000000000000000000000..767b2ede601536a37a4717f12e34b48072caddd6
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/FrameCompactBounds.lean
@@ -0,0 +1,132 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CompactHomogeneousBounds
+import OAI.Geometry.PrescribedPotential.WirtingerCalculus
+import OAI.Geometry.PrescribedPotential.KaehlerClosedDerivatives
+
+namespace OAI
+
+/-! Frame Compact Bounds. -/
+
+section
+
+ 
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff ComplexOrder MatrixOrder Matrix.Norms.Elementwise
+namespace KaehlerCalculus
+variable {n : ℕ}
+
+def frameGram (G B : Matrix (Fin n) (Fin n) ℂ) : ℝ := (Bᴴ*G*B).trace.re
+
+def frameResidual (A G : Matrix (Fin n) (Fin n) ℂ)
+    (R : V n → Matrix (Fin n) (Fin n) ℂ) (B : Matrix (Fin n) (Fin n) ℂ) : ℝ :=
+  -(Bᴴ*A*B*(Bᴴ*G*B)).trace.re + ∑ k, (Bᴴ*R (B*ᵥe k)*B).trace.re
+
+lemma frameGram_pos {G B : Matrix (Fin n) (Fin n) ℂ} (hG : G.PosDef) (hB : B ≠ 0) :
+    0 < frameGram G B := by
+  obtain ⟨Q,hQ,rfl⟩ := CStarAlgebra.isStrictlyPositive_iff_eq_star_mul_self.mp hG.isStrictlyPositive
+  have hn : Q*B ≠ 0 := by
+    intro he
+    apply hB
+    exact hQ.mul_left_cancel (by simpa using he)
+  have he : Bᴴ*(Qᴴ*Q)*B = (Q*B)ᴴ*(Q*B) := by
+    simp only [Matrix.conjTranspose_mul,Matrix.mul_assoc]
+  change 0 < (Bᴴ*(Qᴴ*Q)*B).trace.re
+  rw [he]
+  have hp := (Matrix.posSemidef_conjTranspose_mul_self (Q*B)).trace_nonneg
+  have hne : ((Q*B)ᴴ*(Q*B)).trace ≠ 0 := by
+    intro he0
+    exact hn ((Matrix.trace_conjTranspose_mul_self_eq_zero_iff (A := Q*B)).mp he0)
+  exact (Complex.pos_iff.mp (lt_of_le_of_ne hp (Ne.symm hne))).1
+
+lemma frameGram_smul (G B : Matrix (Fin n) (Fin n) ℂ) (r : ℝ) :
+    frameGram G (r • B) = r^2*frameGram G B := by
+  change frameGram G ((r:ℂ) • B) = _
+  unfold frameGram
+  simp only [Matrix.conjTranspose_smul,Complex.star_def,Complex.conj_ofReal,
+    smul_mul_assoc,mul_smul_comm,smul_smul,Matrix.trace_smul,smul_eq_mul,← Complex.ofReal_mul,
+    Complex.mul_re,Complex.ofReal_re,Complex.ofReal_im,zero_mul,sub_zero]
+  ring
+
+lemma frameResidual_smul (A G : Matrix (Fin n) (Fin n) ℂ)
+    (R : V n → Matrix (Fin n) (Fin n) ℂ)
+    (hR : ∀ r : ℝ, ∀ v, R (r • v) = (r:ℂ)^2 • R v)
+    (B : Matrix (Fin n) (Fin n) ℂ) (r : ℝ) :
+    frameResidual A G R (r • B) = r^4*frameResidual A G R B := by
+  unfold frameResidual
+  simp only [Matrix.smul_mulVec,hR]
+  change -(((r:ℂ) • B)ᴴ*A*((r:ℂ) • B)*(((r:ℂ) • B)ᴴ*G*((r:ℂ) • B))).trace.re +
+    (∑ k, (((r:ℂ) • B)ᴴ*((r:ℂ)^2 • R (B*ᵥe k))*((r:ℂ) • B)).trace.re) = _
+  simp only [Matrix.conjTranspose_smul,Complex.star_def,Complex.conj_ofReal,
+    smul_mul_assoc,mul_smul_comm,smul_smul,Matrix.trace_smul,smul_eq_mul]
+  simp only [← Complex.ofReal_pow,← Complex.ofReal_mul,Complex.mul_re,
+    Complex.ofReal_re,Complex.ofReal_im,zero_mul,sub_zero]
+  rw [← Finset.mul_sum]
+  ring
+
+lemma continuousOn_trace_re {X : Type*} [TopologicalSpace X] {K : Set X}
+    {F : X → Matrix (Fin n) (Fin n) ℂ} (hF : ContinuousOn F K) :
+    ContinuousOn (fun x => (F x).trace.re) K := by
+  simp only [Matrix.trace,Matrix.diag_apply,Complex.re_sum]
+  exact continuousOn_finsetSum _ (fun k _ =>
+    Complex.continuous_re.comp_continuousOn (continuousOn_pi.mp (continuousOn_pi.mp hF k) k))
+
+lemma continuousOn_matrix_star {X : Type*} [TopologicalSpace X] {K : Set X}
+    {F : X → Matrix (Fin n) (Fin n) ℂ} (hF : ContinuousOn F K) :
+    ContinuousOn (fun x => (F x)ᴴ) K := by
+  exact continuousOn_pi.mpr (fun i => continuousOn_pi.mpr (fun j =>
+    (continuousOn_pi.mp (continuousOn_pi.mp hF j) i).star))
+
+lemma compact_frame_bound {X : Type*} [TopologicalSpace X] {K : Set X} (hK : IsCompact K)
+    (A G : X → Matrix (Fin n) (Fin n) ℂ) (R : X → V n → Matrix (Fin n) (Fin n) ℂ)
+    (hA : ContinuousOn A K) (hG : ContinuousOn G K) (hp : ∀ x ∈ K, (G x).PosDef)
+    (hR : ContinuousOn (fun q : X × V n => R q.1 q.2) (K ×ˢ univ))
+    (hRh : ∀ x ∈ K, ∀ r : ℝ, ∀ v, R x (r • v) = (r:ℂ)^2 • R x v) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ x ∈ K, ∀ B,
+      -C*(frameGram (G x) B)^2 ≤ frameResidual (A x) (G x) (R x) B := by
+  let E := Matrix (Fin n) (Fin n) ℂ
+  have hcgram : ContinuousOn (fun q : X × E => frameGram (G q.1) q.2) (K ×ˢ univ) := by
+    apply continuousOn_trace_re
+    exact ((continuousOn_matrix_star continuous_snd.continuousOn).mul
+      (hG.comp continuous_fst.continuousOn (fun _ h => h.1))).mul continuous_snd.continuousOn
+  have hcres : ContinuousOn (fun q : X × E => frameResidual (A q.1) (G q.1) (R q.1) q.2) (K ×ˢ univ) := by
+    have hb := continuous_snd.continuousOn (s := K ×ˢ (univ : Set E))
+    have hbs := continuousOn_matrix_star hb
+    have ha := hA.comp continuous_fst.continuousOn (s := K ×ˢ (univ : Set E)) (fun _ h => h.1)
+    have hg := hG.comp continuous_fst.continuousOn (s := K ×ˢ (univ : Set E)) (fun _ h => h.1)
+    apply ContinuousOn.add
+    · exact (continuousOn_trace_re (((hbs.mul ha).mul hb).mul ((hbs.mul hg).mul hb))).neg
+    · apply continuousOn_finsetSum
+      intro k _
+      apply continuousOn_trace_re
+      apply ContinuousOn.mul (ContinuousOn.mul hbs _) hb
+      apply hR.comp (f := fun q : X × E => (q.1,q.2*ᵥe k))
+      · exact continuous_fst.continuousOn.prodMk
+          (continuous_snd.matrix_mulVec continuous_const).continuousOn
+      · intro q hq
+        exact ⟨hq.1,mem_univ _⟩
+  obtain ⟨δ,hδ,hgram⟩ := compact_homogeneous_positive hK (fun x B => frameGram (G x) B)
+    (hcgram.mono (fun _ h => ⟨h.1,mem_univ _⟩))
+    (fun x hx B hB => frameGram_pos (hp x hx) hB) (fun x _ r B => frameGram_smul (G x) B r)
+  obtain ⟨C,hC,hres⟩ := compact_homogeneous_bound hK 4 (by norm_num)
+    (fun x B => frameResidual (A x) (G x) (R x) B)
+    (hcres.mono (fun _ h => ⟨h.1,mem_univ _⟩))
+    (fun x hx r B => frameResidual_smul _ _ _ (hRh x hx) B r)
+  refine ⟨C/δ^2,div_nonneg hC (sq_nonneg δ),?_⟩
+  intro x hx B
+  have hg := hgram x hx B
+  have hr := neg_abs_le (frameResidual (A x) (G x) (R x) B) |>.trans' (neg_le_neg (hres x hx B))
+  have hq : 0 ≤ frameGram (G x) B := (mul_nonneg hδ.le (sq_nonneg _)).trans hg
+  have hh : δ^2*‖B‖^4 ≤ (frameGram (G x) B)^2 := by
+    have hh := sq_le_sq₀ (mul_nonneg hδ.le (sq_nonneg ‖B‖)) hq |>.mpr hg
+    simpa only [mul_pow,← pow_mul] using hh
+  have hmul := mul_le_mul_of_nonneg_left hh (div_nonneg hC (sq_nonneg δ))
+  have he : C/δ^2*(δ^2*‖B‖^4) = C*‖B‖^4 := by field_simp
+  rw [he] at hmul
+  linarith
+end KaehlerCalculus
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/FrameSecondCalculus.lean b/lean/OAI/Geometry/PrescribedPotential/FrameSecondCalculus.lean
new file mode 100644
index 0000000000000000000000000000000000000000..02ed8393520b3561b1b611724718b7908de7697c
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/FrameSecondCalculus.lean
@@ -0,0 +1,58 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.ChernLuLocalInequality
+import OAI.Geometry.PrescribedPotential.LinearFrameCalculus
+
+namespace OAI
+
+/-! Frame Second Calculus. -/
+
+section
+
+ 
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace KaehlerCalculus
+variable {n : ℕ}
+
+lemma mderiv_pullMetric_second {U : Set (V n)} (hU : IsOpen U)
+    (B : V n →L[ℂ] V n) {M : V n → Matrix (Fin n) (Fin n) ℂ}
+    (hM : ContDiffOn ℝ ∞ M U) {z : V n} (hz : B z ∈ U)
+    (a b : ℂ) (u v : V n) :
+    mderiv a u (mderiv b v (pullMetric B M)) z =
+      (frameMatrix B)ᴴ*mderiv a (B u) (mderiv b (B v) M) (B z)*frameMatrix B := by
+  have he : mderiv b v (pullMetric B M) =ᶠ[𝓝 z] pullMetric B (mderiv b (B v) M) := by
+    filter_upwards [B.continuous.continuousAt.preimage_mem_nhds (hU.mem_nhds hz)] with y hy
+    exact mderiv_pullMetric B (hM.contDiffAt (hU.mem_nhds hy)) b v
+  rw [mderiv_congr he,mderiv_pullMetric B (mderiv_smooth (hM.contDiffAt (hU.mem_nhds hz)) b (B v))]
+
+lemma congruence_inverse_cancel (B G Y : Matrix (Fin n) (Fin n) ℂ)
+    (hB : IsUnit B) :
+    (Bᴴ*Y*B)ᴴ*(Bᴴ*G*B)⁻¹*(Bᴴ*Y*B) = Bᴴ*(Yᴴ*G⁻¹*Y)*B := by
+  have hb := (Matrix.isUnit_iff_isUnit_det B).mp hB
+  simp only [Matrix.conjTranspose_mul,Matrix.conjTranspose_conjTranspose,
+    Matrix.mul_inv_rev,← Matrix.conjTranspose_nonsing_inv]
+  have hc : B*B⁻¹ = 1 := Matrix.mul_nonsing_inv _ hb
+  have hd : (B⁻¹)ᴴ*Bᴴ = 1 := by rw [← Matrix.conjTranspose_mul,hc,Matrix.conjTranspose_one]
+  simp only [Matrix.mul_assoc]
+  rw [← Matrix.mul_assoc B B⁻¹,hc,Matrix.one_mul]
+  rw [← Matrix.mul_assoc (B⁻¹)ᴴ Bᴴ,hd,Matrix.one_mul]
+
+lemma curvatureDiagonal_pullMetric {U : Set (V n)} (hU : IsOpen U)
+    (B : V n →L[ℂ] V n) (hB : IsUnit (frameMatrix B))
+    {M : V n → Matrix (Fin n) (Fin n) ℂ}
+    (hM : ContDiffOn ℝ ∞ M U) {z : V n} (hz : B z ∈ U) (v : V n) :
+    curvatureDiagonal (pullMetric B M) z v =
+      (frameMatrix B)ᴴ*curvatureDiagonal M (B z) (B v)*frameMatrix B := by
+  unfold curvatureDiagonal
+  rw [mderiv_pullMetric_second hU B hM hz,
+    mderiv_pullMetric B (hM.contDiffAt (hU.mem_nhds hz))]
+  unfold pullMetric
+  rw [congruence_inverse_cancel _ _ _ hB]
+  simp only [mul_sub,sub_mul]
+end KaehlerCalculus
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/FredholmKernel.lean b/lean/OAI/Geometry/PrescribedPotential/FredholmKernel.lean
new file mode 100644
index 0000000000000000000000000000000000000000..989bdb3907728401b3f5d489c0859df8d8cf3ab3
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/FredholmKernel.lean
@@ -0,0 +1,70 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.ComplexKernel
+import OAI.Geometry.PrescribedPotential.PatchCutoffs
+import OAI.Geometry.PrescribedPotential.ConstantProjection
+import OAI.Geometry.PrescribedPotential.FredholmRegularity
+import OAI.Geometry.PrescribedPotential.GlobalSmooth
+
+namespace OAI
+
+/-! Fredholm Kernel. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory
+open scoped SchwartzMap ContDiff Classical
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  [ConnectedSpace X] {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+def fredholmOperator (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
+    (P : D.localizers.ConstantProjection) : D.localizers.Sobolev 0 →L[ℝ] D.localizers.Sobolev 0 :=
+  (m^2 : ℝ) • D.resolventZero m hm he - P.proj
+
+lemma fredholm_kernel (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
+    (P : D.localizers.ConstantProjection) (w : D.localizers.Sobolev 0)
+    (hw : D.fredholmOperator m hm he P w = w) : w = 0 := by
+  obtain ⟨c, hc⟩ := P.value w
+  have hw' : w = D.localizers.embed 0 (-Smooth.const c) +
+      (m^2 : ℝ) • D.resolventZero m hm he w := by
+    change (m^2 : ℝ) • D.resolventZero m hm he w - P.proj w = w at hw
+    rw [hc] at hw
+    rw [map_neg]
+    calc
+      w = (m^2 : ℝ) • D.resolventZero m hm he w -
+          D.localizers.embed 0 (Smooth.const c) := hw.symm
+      _ = _ := by abel
+  obtain ⟨u, hu⟩ := D.allRegular_smooth (D.resolvent_feedback_regular m hm he w _ hw')
+  have hs := D.smooth_shifted_equation m hm he w u hu
+  have hL : complexL g u = Smooth.const c := by
+    apply D.localizers.embed_injective 0
+    have h := hs.trans hw'
+    rw [map_sub, map_smul, map_neg, ← hu] at h
+    apply sub_right_inj.mp
+    calc
+      (m^2 : ℝ) • D.localizers.embed 0 u - D.localizers.embed 0 (complexL g u) =
+          -D.localizers.embed 0 (Smooth.const c) + (m^2 : ℝ) • D.localizers.embed 0 u := h
+      _ = (m^2 : ℝ) • D.localizers.embed 0 u - D.localizers.embed 0 (Smooth.const c) := by abel
+  have hc0 := complexL_constant_rhs_zero g u hL
+  have hL0 : complexL g u = 0 := by rw [hL, hc0]; rfl
+  obtain ⟨a, ha⟩ := complexL_kernel g u hL0
+  have hwc : w = D.localizers.embed 0 (Smooth.const ((m^2 : ℝ) • a)) := by
+    rw [hL0, sub_zero, ha] at hs
+    exact hs.symm
+  have hpw : P.proj w = w := by rw [hwc, P.fixed]
+  rw [← hpw, hc, hc0]
+  exact map_zero (D.localizers.embed 0)
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/FredholmRegularity.lean b/lean/OAI/Geometry/PrescribedPotential/FredholmRegularity.lean
new file mode 100644
index 0000000000000000000000000000000000000000..b3c4496c8f5d420e8b153fdfa8868784ff2a8802
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/FredholmRegularity.lean
@@ -0,0 +1,135 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CompletedResolvent
+import OAI.Geometry.PrescribedPotential.GlobalCompletion
+
+namespace OAI
+
+/-! Fredholm Regularity. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory
+open scoped SchwartzMap ContDiff Classical
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace Localizers
+variable {D : Localizers A ι}
+
+lemma BoundedCore.add {s t : ℝ} {S T : Smooth A →ₗ[ℝ] Smooth A}
+    (hS : D.BoundedCore s t S) (hT : D.BoundedCore s t T) :
+    D.BoundedCore s t (S+T) := by
+  obtain ⟨B, hB, hb⟩ := hS
+  obtain ⟨C, hC, hc⟩ := hT
+  refine ⟨B+C, add_nonneg hB hC, fun f => ?_⟩
+  simp only [LinearMap.add_apply, map_add]
+  exact (norm_add_le _ _).trans (by simpa only [add_mul] using add_le_add (hb f) (hc f))
+
+lemma BoundedCore.smul {s t : ℝ} {T : Smooth A →ₗ[ℝ] Smooth A}
+    (hT : D.BoundedCore s t T) (c : ℝ) : D.BoundedCore s t (c • T) := by
+  obtain ⟨C, hC, hc⟩ := hT
+  refine ⟨‖c‖*C, mul_nonneg (norm_nonneg _) hC, fun f => ?_⟩
+  simp only [LinearMap.smul_apply, map_smul, norm_smul, mul_assoc]
+  exact mul_le_mul_of_nonneg_left (hc f) (norm_nonneg c)
+
+lemma extendCore_add {s t : ℝ} {S T : Smooth A →ₗ[ℝ] Smooth A}
+    (hS : D.BoundedCore s t S) (hT : D.BoundedCore s t T) :
+    D.extendCore s t (S+T) = D.extendCore s t S + D.extendCore s t T := by
+  apply DFunLike.coe_injective
+  apply (D.embed_dense s).equalizer (by fun_prop) (by fun_prop)
+  funext f
+  simp only [Function.comp_apply, _root_.add_apply, D.extendCore_embed hS,
+    D.extendCore_embed hT, D.extendCore_embed (hS.add hT), LinearMap.add_apply, map_add]
+
+lemma extendCore_smul {s t : ℝ} {T : Smooth A →ₗ[ℝ] Smooth A}
+    (hT : D.BoundedCore s t T) (c : ℝ) :
+    D.extendCore s t (c • T) = c • D.extendCore s t T := by
+  apply DFunLike.coe_injective
+  apply (D.embed_dense s).equalizer (by fun_prop) (by fun_prop)
+  funext f
+  simp only [Function.comp_apply, _root_.smul_apply, D.extendCore_embed hT,
+    D.extendCore_embed (hT.smul c), LinearMap.smul_apply, map_smul]
+
+end Localizers
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+def resolventZero (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1) :
+    D.localizers.Sobolev 0 →L[ℝ] D.localizers.Sobolev 0 :=
+  D.localizers.lower 2 0 ∘L D.completedResolvent m hm he
+
+def feedbackCore (m : ℝ) (hm : 1 ≤ m) : Smooth A →ₗ[ℝ] Smooth A :=
+  D.error m hm + (m^2 : ℝ) • D.parametrix m hm
+
+lemma feedbackCore_bound (m : ℝ) (hm : 1 ≤ m) (k : ℕ) :
+    D.localizers.BoundedCore (k : ℝ) ((k : ℝ)+1) (D.feedbackCore m hm) := by
+  have hQ : D.localizers.BoundedCore (k : ℝ) ((k : ℝ)+2) (D.parametrix m hm) :=
+    D.parametrix_bound m hm k
+  exact (D.error_bound_integer m hm k).add ((hQ.target_mono (by linarith)).smul (m^2))
+
+lemma feedbackCore_bound_zero (m : ℝ) (hm : 1 ≤ m) :
+    D.localizers.BoundedCore 0 0 (D.feedbackCore m hm) := by
+  have hb := D.feedbackCore_bound m hm 0
+  rw [Nat.cast_zero, zero_add] at hb
+  exact hb.target_mono (by norm_num)
+
+lemma completed_feedback (m : ℝ) (hm : 1 ≤ m) (v : D.localizers.Sobolev 0) :
+    D.localizers.extendCore 0 0 (D.feedbackCore m hm) v =
+      D.completedError m hm v + (m^2 : ℝ) •
+        D.localizers.lower 2 0 (D.completedParametrix m hm v) := by
+  have hQ := (D.completedParametrix_bound m hm).target_mono (by norm_num : (0:ℝ) ≤ 2)
+  rw [feedbackCore, Localizers.extendCore_add (D.error_bound_zero m hm) (hQ.smul (m^2)),
+    Localizers.extendCore_smul hQ]
+  simp only [_root_.add_apply, _root_.smul_apply]
+  change _ = (D.localizers.extendCore 0 0 (D.error m hm)) v + (m^2 : ℝ) •
+    D.localizers.lower 2 0 (D.localizers.extendCore 0 2 (D.parametrix m hm) v)
+  rw [D.localizers.lower_extendCore (by norm_num : (0:ℝ) ≤ 2) _
+    (D.completedParametrix_bound m hm) hQ v]
+
+ 
+
+theorem resolvent_feedback_regular (m : ℝ) (hm : 1 ≤ m)
+    (he : ‖D.completedError m hm‖ < 1) (w : D.localizers.Sobolev 0)
+    (f : Smooth A) (hw : w = D.localizers.embed 0 f + (m^2 : ℝ) • D.resolventZero m hm he w) :
+    D.localizers.AllRegular (D.resolventZero m hm he w) := by
+  let v := D.errorInverse m hm he w
+  have hv : v - D.completedError m hm v =
+      D.localizers.embed 0 f + (m^2 : ℝ) •
+        D.localizers.lower 2 0 (D.completedParametrix m hm v) :=
+    (D.errorInverse_equation m hm he w).trans hw
+  have hvreg : D.localizers.AllRegular v := by
+    apply D.localizers.allRegular_of_gain (D.feedbackCore m hm)
+      (D.feedbackCore_bound_zero m hm) (D.feedbackCore_bound m hm) f
+    rw [D.completed_feedback]
+    calc
+      v = D.localizers.embed 0 f + (m^2 : ℝ) •
+          D.localizers.lower 2 0 (D.completedParametrix m hm v) +
+          D.completedError m hm v := sub_eq_iff_eq_add.mp hv
+      _ = _ := by abel
+  exact D.parametrix_regular m hm v hvreg
+
+lemma smooth_shifted_equation (m : ℝ) (hm : 1 ≤ m)
+    (he : ‖D.completedError m hm‖ < 1) (w : D.localizers.Sobolev 0)
+    (u : Smooth A) (hu : D.localizers.embed 0 u = D.resolventZero m hm he w) :
+    D.localizers.embed 0 ((m^2 : ℝ) • u - complexL g u) = w := by
+  have h2 : D.localizers.embed 2 u = D.completedResolvent m hm he w := by
+    apply D.localizers.lower_injective (by norm_num : (0 : ℝ) ≤ 2)
+    rw [D.localizers.lower_embed (by norm_num : (0 : ℝ) ≤ 2)]
+    exact hu
+  have h := D.completedResolvent_equation m hm he w
+  rw [← h2] at h
+  simpa only [shiftedOperator, completedL, _root_.sub_apply,
+    _root_.smul_apply, D.localizers.lower_embed (by norm_num : (0 : ℝ) ≤ 2),
+    D.localizers.extendCore_embed D.complexL_bound, map_sub, map_smul] using h
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/FrequencyApprox.lean b/lean/OAI/Geometry/PrescribedPotential/FrequencyApprox.lean
new file mode 100644
index 0000000000000000000000000000000000000000..f3f12d04e3a99caeb026f5aef650d1e7810c8e14
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/FrequencyApprox.lean
@@ -0,0 +1,163 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.SobolevEmbedding
+import OAI.Geometry.PrescribedPotential.TruncatedRellich
+
+namespace OAI
+
+/-! Frequency Approx. -/
+
+section
+
+ 
+
+noncomputable section
+open MeasureTheory Set Filter Topology FourierTransform
+open scoped BoundedContinuousFunction SchwartzMap Classical ContDiff
+namespace EllipticCompact
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E]
+open SobolevChart
+
+omit [FiniteDimensional ℝ E] in
+lemma inverseWeight_one_apply (x : E) :
+    inverseWeight 1 x = ((((1 + ‖x‖ ^ 2 : ℝ)⁻¹) : ℝ) : ℂ) := by
+  change (((1 + ‖x‖ ^ 2) ^ (-((1 : ℕ) : ℝ)) : ℝ) : ℂ) = _
+  rw [Nat.cast_one, Real.rpow_neg_one]
+
+omit [FiniteDimensional ℝ E] in
+lemma inverseWeight_one_norm (x : E) :
+    ‖inverseWeight 1 x‖ = 1 / (1 + ‖x‖ ^ 2) := by
+  rw [inverseWeight_one_apply, Complex.norm_real, Real.norm_eq_abs,
+    abs_of_pos (inv_pos.mpr (by positivity)), one_div]
+
+def frequencyBump (n : ℕ) : ContDiffBump (0 : E) where
+  rIn := (n : ℝ) + 1
+  rOut := (n : ℝ) + 2
+  rIn_pos := by positivity
+  rIn_lt_rOut := by linarith
+
+lemma frequencyCutoff_support (n : ℕ) :
+    HasCompactSupport (fun x : E => (frequencyBump (E := E) n x : ℂ) * inverseWeight 1 x) := by
+  have hb : HasCompactSupport (fun x : E => (frequencyBump (E := E) n x : ℂ)) :=
+    (frequencyBump (E := E) n).hasCompactSupport.comp_left (g := Complex.ofReal) Complex.ofReal_zero
+  exact hb.mul_right
+
+def frequencyCutoff (n : ℕ) : 𝓢(E, ℂ) := (frequencyCutoff_support (E := E) n).toSchwartzMap
+  ((Complex.ofRealCLM.contDiff.comp (frequencyBump (E := E) n).contDiff).mul (inverseWeight_temperate 1).1)
+
+lemma frequencyCutoff_apply (n : ℕ) (x : E) :
+    frequencyCutoff (E := E) n x = (frequencyBump (E := E) n x : ℂ) * inverseWeight 1 x := rfl
+
+lemma frequencyCutoff_compact (n : ℕ) : HasCompactSupport (frequencyCutoff (E := E) n) :=
+  frequencyCutoff_support n
+
+lemma frequencyCutoff_bound (n : ℕ) :
+    ‖(frequencyCutoff (E := E) n).toBoundedContinuousFunction - inverseWeight 1‖ ≤
+      1 / ((n : ℝ) + 1) := by
+  apply (BoundedContinuousFunction.norm_le (by positivity : 0 ≤ 1 / ((n : ℝ) + 1))).mpr
+  intro x
+  change ‖frequencyCutoff (E := E) n x - inverseWeight 1 x‖ ≤ _
+  rw [frequencyCutoff_apply]
+  by_cases hx : ‖x‖ ≤ (n : ℝ) + 1
+  · have hx' : x ∈ Metric.closedBall (0 : E) (frequencyBump (E := E) n).rIn := by
+      simpa only [Metric.mem_closedBall, dist_zero_right, frequencyBump] using hx
+    rw [(frequencyBump (E := E) n).one_of_mem_closedBall hx', Complex.ofReal_one, one_mul,
+      sub_self, norm_zero]
+    positivity
+  · have hR : 0 < (n : ℝ) + 1 := by positivity
+    have hx' : (n : ℝ) + 1 ≤ ‖x‖ := le_of_lt (lt_of_not_ge hx)
+    have hden : (n : ℝ) + 1 ≤ 1 + ‖x‖ ^ 2 := by
+      have hn0 : (0 : ℝ) ≤ n := Nat.cast_nonneg n
+      nlinarith [norm_nonneg x]
+    have hb' : ‖(frequencyBump (E := E) n x : ℂ) - 1‖ ≤ 1 := by
+      rw [← Complex.ofReal_one, ← Complex.ofReal_sub, Complex.norm_real, Real.norm_eq_abs]
+      exact abs_le.mpr ⟨by linarith [(frequencyBump (E := E) n).nonneg' x],
+        by linarith [(frequencyBump (E := E) n).le_one (x := x)]⟩
+    calc
+      _ = ‖((frequencyBump (E := E) n x : ℂ) - 1) * inverseWeight 1 x‖ := by congr 1; ring
+      _ = ‖(frequencyBump (E := E) n x : ℂ) - 1‖ * ‖inverseWeight 1 x‖ := norm_mul _ _
+      _ ≤ 1 * ‖inverseWeight 1 x‖ := mul_le_mul_of_nonneg_right hb' (norm_nonneg _)
+      _ ≤ 1 / ((n : ℝ) + 1) := by
+        rw [one_mul, inverseWeight_one_norm]
+        exact one_div_le_one_div_of_le hR hden
+
+lemma inverseWeight_schwartz_approx {ε : ℝ} (hε : 0 < ε) :
+    ∃ χ : 𝓢(E, ℂ), HasCompactSupport χ ∧
+      ‖χ.toBoundedContinuousFunction - inverseWeight 1‖ < ε := by
+  obtain ⟨n, hn⟩ := exists_nat_one_div_lt hε
+  exact ⟨frequencyCutoff n, frequencyCutoff_compact n, (frequencyCutoff_bound n).trans_lt hn⟩
+
+variable [MeasurableSpace E] [BorelSpace E]
+
+lemma multiply_sub (g h : E →ᵇ ℂ) : multiply (g-h) = multiply g - multiply h := by
+  ext1 u
+  apply Lp.ext
+  filter_upwards [multiply_ae (g-h) u, multiply_ae g u, multiply_ae h u,
+    Lp.coeFn_sub (multiply g u) (multiply h u)] with x h₁ h₂ h₃ h₄
+  change (multiply (g-h) u : E → ℂ) x = ((multiply g u - multiply h u : L2 E) : E → ℂ) x
+  rw [h₁, h₄, Pi.sub_apply, h₂, h₃]
+  simp only [BoundedContinuousFunction.sub_apply, sub_mul]
+
+lemma multiplier_sub (g h : E →ᵇ ℂ) : multiplier (g-h) = multiplier g - multiplier h := by
+  ext1 u
+  simp only [multiplier, ContinuousLinearMap.comp_apply, multiply_sub,
+    sub_apply, map_sub]
+
+lemma localized_multiplier_diff_bound (κ g h : E →ᵇ ℂ) :
+    ‖multiply κ ∘L multiplier g - multiply κ ∘L multiplier h‖ ≤ ‖κ‖ * ‖g-h‖ := by
+  apply ContinuousLinearMap.opNorm_le_bound _ (by positivity)
+  intro u
+  simp only [sub_apply, ContinuousLinearMap.comp_apply]
+  rw [← map_sub]
+  have he : multiplier g u - multiplier h u = multiplier (g-h) u := by
+    rw [multiplier_sub, sub_apply]
+  rw [he]
+  calc
+    ‖multiply κ (multiplier (g-h) u)‖ ≤ ‖κ‖ * ‖multiplier (g-h) u‖ := multiply_norm_le _ _
+    _ ≤ ‖κ‖ * (‖g-h‖ * ‖u‖) := mul_le_mul_of_nonneg_left (multiplier_norm_le _ _) (norm_nonneg _)
+    _ = _ := by ring
+
+lemma compact_localized_frequency (κ χ : 𝓢(E, ℂ))
+    (hκ : HasCompactSupport κ) (hχ : HasCompactSupport χ) :
+    IsCompactOperator (multiply κ.toBoundedContinuousFunction ∘L
+      multiplier χ.toBoundedContinuousFunction) := by
+  have h := (compact_truncated κ χ hκ hχ).comp_clm (fourierCLM ℂ (L2 E))
+  simpa only [multiplier, ContinuousLinearMap.coe_comp, Function.comp_assoc] using h
+
+lemma localized_frequency_distance (κ χ : 𝓢(E, ℂ)) {ε : ℝ}
+    (happ : ‖χ.toBoundedContinuousFunction - inverseWeight 1‖ <
+      ε / (‖κ.toBoundedContinuousFunction‖ + 1)) :
+    dist (multiply κ.toBoundedContinuousFunction ∘L multiplier (inverseWeight 1))
+      (multiply κ.toBoundedContinuousFunction ∘L multiplier χ.toBoundedContinuousFunction) < ε := by
+  have hB : 0 < ‖κ.toBoundedContinuousFunction‖ + 1 := by positivity
+  rw [dist_comm, dist_eq_norm]
+  calc
+    _ ≤ ‖κ.toBoundedContinuousFunction‖ * ‖χ.toBoundedContinuousFunction - inverseWeight 1‖ :=
+      localized_multiplier_diff_bound _ _ _
+    _ ≤ (‖κ.toBoundedContinuousFunction‖ + 1) * ‖χ.toBoundedContinuousFunction - inverseWeight 1‖ :=
+      mul_le_mul_of_nonneg_right (by linarith) (norm_nonneg _)
+    _ < (‖κ.toBoundedContinuousFunction‖ + 1) * (ε / (‖κ.toBoundedContinuousFunction‖ + 1)) :=
+      mul_lt_mul_of_pos_left happ hB
+    _ = ε := mul_div_cancel₀ _ (ne_of_gt hB)
+
+ 
+theorem localized_bessel_compact (κ : 𝓢(E, ℂ)) (hκ : HasCompactSupport κ) :
+    IsCompactOperator (multiply κ.toBoundedContinuousFunction ∘L multiplier (inverseWeight 1)) := by
+  have hc : IsClosed {T : L2 E →L[ℂ] L2 E | IsCompactOperator T} :=
+    isClosed_setOfPred_isCompactOperator
+  change (multiply κ.toBoundedContinuousFunction ∘L multiplier (inverseWeight 1)) ∈
+    {T : L2 E →L[ℂ] L2 E | IsCompactOperator T}
+  rw [← hc.closure_eq]
+  apply Metric.mem_closure_iff.mpr
+  intro ε hε
+  have hB : 0 < ‖κ.toBoundedContinuousFunction‖ + 1 := by positivity
+  obtain ⟨χ, hχ, happ⟩ := inverseWeight_schwartz_approx (E := E) (div_pos hε hB)
+  exact ⟨multiply κ.toBoundedContinuousFunction ∘L multiplier χ.toBoundedContinuousFunction,
+    compact_localized_frequency κ χ hκ hχ, localized_frequency_distance κ χ happ⟩
+
+end EllipticCompact
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/FrozenEllipticJets.lean b/lean/OAI/Geometry/PrescribedPotential/FrozenEllipticJets.lean
new file mode 100644
index 0000000000000000000000000000000000000000..96cfdb3bace21cb0225b07971c3b138126400d2b
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/FrozenEllipticJets.lean
@@ -0,0 +1,85 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.RealJetEnergy
+
+namespace OAI
+
+/-! Frozen Elliptic Jets. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Finset
+open scoped ContDiff
+namespace HigherJet
+variable {E F G : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  [NormedAddCommGroup F] [InnerProductSpace ℝ F]
+  [NormedAddCommGroup G] [NormedSpace ℝ G]
+  {ι : Type*} [Fintype ι]
+
+def hessian (u : E → F) : E → E →L[ℝ] E →L[ℝ] F := fderiv ℝ (fderiv ℝ u)
+
+lemma hessian_smoothOn {U : Set E} (hU : IsOpen U) {u : E → F}
+    (hu : ContDiffOn ℝ ∞ u U) : ContDiffOn ℝ ∞ (hessian u) U :=
+  (hu.fderiv_of_isOpen hU (m := ∞) (by simp)).fderiv_of_isOpen hU (m := ∞) (by simp)
+
+lemma word_sum {U : Set E} (hU : IsOpen U) (u : ι → E → F)
+    (hu : ∀ i, ContDiffOn ℝ ∞ (u i) U) (ws : List E) {x : E} (hx : x ∈ U) :
+    word ws (fun y => ∑ i, u i y) x = ∑ i, word ws (u i) x := by
+  induction ws generalizing x with
+  | nil => rfl
+  | cons w ws ih =>
+    change dir w (word ws (fun y => ∑ i, u i y)) x = _
+    rw [dir_congr (by filter_upwards [hU.mem_nhds hx] with y hy using ih hy)]
+    exact dir_sum _ (fun i => ((word_smoothOn hU (hu i) ws).contDiffAt
+      (hU.mem_nhds hx)).differentiableAt (by simp)) w
+
+lemma word_sub {U : Set E} (hU : IsOpen U) {f g : E → F}
+    (hf : ContDiffOn ℝ ∞ f U) (hg : ContDiffOn ℝ ∞ g U)
+    (ws : List E) {x : E} (hx : x ∈ U) :
+    word ws (fun y => f y-g y) x = word ws f x-word ws g x := by
+  induction ws generalizing x with
+  | nil => rfl
+  | cons w ws ih =>
+    change dir w (word ws (fun y => f y-g y)) x = _
+    rw [dir_congr (by filter_upwards [hU.mem_nhds hx] with y hy using ih hy)]
+    exact congrArg (fun T : E →L[ℝ] F => T w)
+      ((((word_smoothOn hU hf ws).contDiffAt (hU.mem_nhds hx)).differentiableAt (by simp)).hasFDerivAt.sub
+       (((word_smoothOn hU hg ws).contDiffAt (hU.mem_nhds hx)).differentiableAt (by simp)).hasFDerivAt).fderiv
+
+lemma frameLaplace_word {U : Set E} (hU : IsOpen U) {u : E → F}
+    (hu : ContDiffOn ℝ ∞ u U) (v : ι → E) (ws : List E) {x : E} (hx : x ∈ U) :
+    frameLaplace v (word ws u) x = word ws (frameLaplace v u) x := by
+  rw [show frameLaplace v u = fun y => ∑ i, dir (v i) (dir (v i) u) y from rfl,
+    word_sum hU _ (fun i => dir_smoothOn hU (dir_smoothOn hU hu _) _) ws hx]
+  apply Finset.sum_congr rfl
+  intro i _
+  exact (word_dir_dir_comm hU hu _ _ ws hx).symm
+
+lemma frozen_equation_word (B : G →L[ℝ] (E →L[ℝ] E →L[ℝ] F) →L[ℝ] F)
+    {U : Set E} (hU : IsOpen U) {u r : E → F} {a : E → G}
+    (hu : ContDiffOn ℝ ∞ u U) (hr : ContDiffOn ℝ ∞ r U)
+    (ha : ContDiffOn ℝ ∞ a U) (heq : ∀ y ∈ U, B (a y) (hessian u y) = r y)
+    {x : E} (hx : x ∈ U) (v : ι → E)
+    (hframe : ∀ H : E →L[ℝ] E →L[ℝ] F, B (a x) H = ∑ i, H (v i) (v i))
+    (ws : List E) :
+    frameLaplace v (word ws u) x = word ws r x-
+      word ws (fun y => B (a y-a x) (hessian u y)) x := by
+  have hg : frameLaplace v u =ᶠ[𝓝 x]
+      (fun y => r y-B (a y-a x) (hessian u y)) := by
+    filter_upwards [hU.mem_nhds hx] with y hy
+    rw [← heq y hy,map_sub,sub_apply,sub_sub_cancel]
+    rw [hframe]
+    apply Finset.sum_congr rfl
+    intro i _
+    exact dir_dir (hu.contDiffAt (hU.mem_nhds hy)) _ _
+  rw [frameLaplace_word hU hu v ws hx,word_congr hg]
+  exact word_sub hU hr ((B.contDiff.comp_contDiffOn (ha.sub contDiffOn_const)).clm_apply (hessian_smoothOn hU hu)) ws hx
+
+end HigherJet
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/FrozenPoisson.lean b/lean/OAI/Geometry/PrescribedPotential/FrozenPoisson.lean
new file mode 100644
index 0000000000000000000000000000000000000000..fc7862940e3be03d9a10ef8cead499999e6bbad6
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/FrozenPoisson.lean
@@ -0,0 +1,428 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.AnalyticSupport
+
+namespace OAI
+
+/-! Frozen Poisson. -/
+
+section
+
+ 
+
+noncomputable section
+open Matrix
+open scoped ComplexOrder MatrixOrder SchwartzMap Real
+
+namespace FrozenPoisson
+open EllipticKernel
+variable {n : ℕ}
+
+ 
+lemma posDef_factor (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef) :
+    ∃ B : Matrix (Fin n) (Fin n) ℂ, IsUnit B ∧ H = star B * B := by
+  obtain ⟨B, hB⟩ := CStarAlgebra.nonneg_iff_eq_star_mul_self.mp hH.posSemidef.nonneg
+  refine ⟨B, ?_, hB⟩
+  rw [Matrix.isUnit_iff_isUnit_det]
+  apply isUnit_iff_ne_zero.mpr
+  intro hb
+  have he := congrArg Matrix.det hB
+  rw [Matrix.det_mul, hb, mul_zero] at he
+  exact ne_of_gt hH.det_pos he
+
+ 
+
+def symbol (H : Matrix (Fin n) (Fin n) ℂ) (ξ : EC n) : ℝ :=
+  (2 * Real.pi)^2 * traceBilin H (rankOne ξ)
+
+lemma trace_rankOne_factor (H B : Matrix (Fin n) (Fin n) ℂ)
+    (hB : H⁻¹ = star B * B) (ξ : EC n) :
+    traceBilin H (rankOne ξ) = (1/4 : ℝ) * ‖Matrix.toEuclideanCLM (𝕜 := ℂ) (n := Fin n) B ξ‖^2 := by
+  rw [traceBilin_apply, pull_rankOne, Matrix.mul_smul, Matrix.trace_smul]
+  simp only [Complex.smul_re, smul_eq_mul]
+  congr 1
+  rw [Matrix.mul_vecMulVec, Matrix.trace_vecMulVec, dotProduct_comm, hB,
+    ← Matrix.mulVec_mulVec, Matrix.dotProduct_mulVec]
+  change (star (coordinateEquiv n ξ) ᵥ* Bᴴ ⬝ᵥ B *ᵥ coordinateEquiv n ξ).re = _
+  rw [← Matrix.star_mulVec, dotProduct_comm]
+  have he := EuclideanSpace.inner_eq_star_dotProduct
+    (Matrix.toEuclideanCLM (𝕜 := ℂ) (n := Fin n) B ξ) (Matrix.toEuclideanCLM (𝕜 := ℂ) (n := Fin n) B ξ)
+  rw [Matrix.ofLp_toEuclideanCLM] at he
+  rw [coordinateEquiv_apply, ← he]
+  rw [inner_self_eq_norm_sq_to_K]
+  norm_cast
+
+lemma symbol_eq_norm_sq (H B : Matrix (Fin n) (Fin n) ℂ)
+    (hB : H⁻¹ = star B * B) (ξ : EC n) :
+    symbol H ξ = ‖Real.pi • Matrix.toEuclideanCLM (𝕜 := ℂ) (n := Fin n) B ξ‖^2 := by
+  rw [symbol, trace_rankOne_factor H B hB, norm_smul, Real.norm_eq_abs,
+    abs_of_pos Real.pi_pos]
+  ring
+
+ 
+def factorMap (B : Matrix (Fin n) (Fin n) ℂ) : EC n →L[ℝ] EC n :=
+  Real.pi • (Matrix.toEuclideanCLM (𝕜 := ℂ) (n := Fin n) B).restrictScalars ℝ
+
+lemma factorMap_injective (B : Matrix (Fin n) (Fin n) ℂ) (hB : IsUnit B) :
+    Function.Injective (factorMap B) := by
+  intro x y h
+  have he : Matrix.toEuclideanCLM (𝕜 := ℂ) (n := Fin n) B x =
+      Matrix.toEuclideanCLM (𝕜 := ℂ) (n := Fin n) B y := by
+    have hh := congrArg (fun w : EC n => (Real.pi⁻¹ : ℝ) • w) h
+    simpa [factorMap, smul_smul, Real.pi_ne_zero] using hh
+  apply WithLp.ofLp_injective
+  apply Matrix.mulVec_injective_iff_isUnit.mpr hB
+  exact congrArg WithLp.ofLp he
+
+lemma symbol_eq_factorMap (H B : Matrix (Fin n) (Fin n) ℂ)
+    (hB : H⁻¹ = star B * B) : symbol H = fun ξ => ‖factorMap B ξ‖^2 :=
+  funext (symbol_eq_norm_sq H B hB)
+
+lemma symbol_nonneg (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef) (ξ : EC n) :
+    0 ≤ symbol H ξ := by
+  obtain ⟨B, _, hB⟩ := posDef_factor H⁻¹ hH.inv
+  rw [symbol_eq_factorMap H B hB]
+  positivity
+
+lemma symbol_temperate (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef) :
+    (symbol H).HasTemperateGrowth := by
+  obtain ⟨B, _, hB⟩ := posDef_factor H⁻¹ hH.inv
+  rw [symbol_eq_factorMap H B hB]
+  exact (Function.hasTemperateGrowth_norm_sq _).comp (factorMap B).hasTemperateGrowth
+
+ 
+
+def inverseSymbol (H : Matrix (Fin n) (Fin n) ℂ) (ξ : EC n) : ℂ :=
+  ((1 + symbol H ξ)⁻¹ : ℝ)
+
+lemma inverseSymbol_temperate (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef) :
+    (inverseSymbol H).HasTemperateGrowth := by
+  obtain ⟨B, _, hB⟩ := posDef_factor H⁻¹ hH.inv
+  have he : inverseSymbol H = Complex.ofReal ∘
+      ((fun ξ : EC n => (1 + ‖ξ‖^2)^(-1 : ℝ)) ∘ factorMap B) := by
+    funext ξ
+    simp only [Function.comp_apply, Real.rpow_neg_one, inverseSymbol,
+      symbol_eq_factorMap H B hB]
+  rw [he]
+  exact Complex.hasTemperateGrowth_ofReal.comp
+    ((Function.hasTemperateGrowth_one_add_norm_sq_rpow _ (-1)).comp
+      (factorMap B).hasTemperateGrowth)
+
+ 
+
+lemma inverseSymbol_weighted_bounded (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef) :
+    ∃ C, ∀ ξ : EC n, ‖(1 + ‖ξ‖^2 : ℝ) * inverseSymbol H ξ‖ ≤ C := by
+  obtain ⟨B, hBu, hB⟩ := posDef_factor H⁻¹ hH.inv
+  obtain ⟨K, _, hK⟩ := (factorMap B).toLinearMap.injective_iff_antilipschitz.mp
+    (factorMap_injective B hBu)
+  refine ⟨1 + (K : ℝ)^2, fun ξ => ?_⟩
+  have hle := hK.le_mul_norm (map_zero (factorMap B)) ξ
+  have hsq := pow_le_pow_left₀ (norm_nonneg ξ) hle 2
+  change ‖ξ‖^2 ≤ ((K : ℝ) * ‖factorMap B ξ‖)^2 at hsq
+  rw [mul_pow] at hsq
+  simp only [inverseSymbol, symbol_eq_factorMap H B hB, ← Complex.ofReal_mul,
+    Complex.norm_real, Real.norm_eq_abs]
+  rw [abs_of_nonneg (by positivity), mul_inv_le_iff₀ (by positivity)]
+  nlinarith [sq_nonneg (K : ℝ), sq_nonneg ‖factorMap B ξ‖]
+
+open TemperedDistribution
+
+ 
+def shifted (H : Matrix (Fin n) (Fin n) ℂ) :
+    𝓢'(EC n, ℂ) →L[ℂ] 𝓢'(EC n, ℂ) :=
+  fourierMultiplierCLM ℂ (fun ξ => ((1 + symbol H ξ : ℝ) : ℂ))
+
+ 
+def resolvent (H : Matrix (Fin n) (Fin n) ℂ) :
+    𝓢'(EC n, ℂ) →L[ℂ] 𝓢'(EC n, ℂ) :=
+  fourierMultiplierCLM ℂ (inverseSymbol H)
+
+lemma forward_temperate (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef) :
+    (fun ξ => ((1 + symbol H ξ : ℝ) : ℂ)).HasTemperateGrowth :=
+  Complex.hasTemperateGrowth_ofReal.comp
+    ((Function.HasTemperateGrowth.const 1).add (symbol_temperate H hH))
+
+lemma shifted_resolvent (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (u : 𝓢'(EC n, ℂ)) : shifted H (resolvent H u) = u := by
+  rw [shifted, resolvent,
+    fourierMultiplierCLM_fourierMultiplierCLM_apply (inverseSymbol_temperate H hH)
+      (forward_temperate H hH)]
+  have he : inverseSymbol H * (fun ξ => ((1 + symbol H ξ : ℝ) : ℂ)) =
+      fun _ => (1 : ℂ) := by
+    funext ξ
+    simp only [Pi.mul_apply, inverseSymbol, ← Complex.ofReal_mul]
+    rw [inv_mul_cancel₀ (by linarith [symbol_nonneg H hH ξ]), Complex.ofReal_one]
+  rw [he, fourierMultiplierCLM_const]
+  simp
+
+lemma resolvent_shifted (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (u : 𝓢'(EC n, ℂ)) : resolvent H (shifted H u) = u := by
+  rw [shifted, resolvent,
+    fourierMultiplierCLM_fourierMultiplierCLM_apply (forward_temperate H hH)
+      (inverseSymbol_temperate H hH), mul_comm]
+  have he : inverseSymbol H * (fun ξ => ((1 + symbol H ξ : ℝ) : ℂ)) =
+      fun _ => (1 : ℂ) := by
+    funext ξ
+    simp only [Pi.mul_apply, inverseSymbol, ← Complex.ofReal_mul]
+    rw [inv_mul_cancel₀ (by linarith [symbol_nonneg H hH ξ]), Complex.ofReal_one]
+  rw [he, fourierMultiplierCLM_const]
+  simp
+
+ 
+theorem resolvent_memSobolev (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    {s : ℝ} {u : 𝓢'(EC n, ℂ)} (hu : MemSobolev s 2 u) :
+    MemSobolev (s + 2) 2 (resolvent H u) := by
+  rw [add_comm s 2, ← memSobolev_besselPotential_iff]
+  have he : besselPotential (EC n) ℂ 2 (resolvent H u) =
+      fourierMultiplierCLM ℂ (fun ξ => (1 + ‖ξ‖^2 : ℝ) * inverseSymbol H ξ) u := by
+    rw [besselPotential, resolvent,
+      fourierMultiplierCLM_fourierMultiplierCLM_apply (inverseSymbol_temperate H hH)
+        (by fun_prop)]
+    congr 2
+    ext ξ
+    simp [mul_comm]
+  rw [he]
+  apply hu.fourierMultiplierCLM_of_bounded _ (inverseSymbol_weighted_bounded H hH)
+  have hi := inverseSymbol_temperate H hH
+  fun_prop
+
+ 
+
+theorem frozen_sobolev_resolvent (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    {s : ℝ} (f : 𝓢'(EC n, ℂ)) (hf : MemSobolev s 2 f) :
+    ∃! u : 𝓢'(EC n, ℂ), shifted H u = f ∧ MemSobolev (s + 2) 2 u := by
+  refine ⟨resolvent H f, ⟨shifted_resolvent H hH f, resolvent_memSobolev H hH hf⟩, ?_⟩
+  intro u hu
+  rw [← hu.1, resolvent_shifted H hH]
+
+ 
+open LineDeriv
+
+section BilinearExpansion
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+variable {ι : Type*} [Fintype ι]
+
+def rankTwo (v w : E) : Bilin E := (realInner v).smulRight (realInner w)
+
+lemma bilin_expansion (b : OrthonormalBasis ι ℝ E) (B : Bilin E) :
+    B = ∑ i, ∑ j, B (b i) (b j) • rankTwo (b i) (b j) := by
+  ext x y
+  conv_lhs => rw [← b.sum_repr' x, ← b.sum_repr' y]
+  simp only [map_sum, map_smul, _root_.sum_apply,
+    _root_.smul_apply, smul_eq_mul, rankTwo,
+    ContinuousLinearMap.smulRight_apply, realInner_apply, Finset.mul_sum]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro i _
+  apply Finset.sum_congr rfl
+  intro j _
+  ring
+
+lemma second_symbol_expansion (b : OrthonormalBasis ι ℝ E) (L : SecondOrder E) (ξ : E) :
+    L (rankOne ξ) = ∑ i, ∑ j,
+      L (rankTwo (b i) (b j)) * inner ℝ ξ (b i) * inner ℝ ξ (b j) := by
+  rw [bilin_expansion b (rankOne ξ)]
+  simp only [map_sum, map_smul, rankOne, ContinuousLinearMap.smulRight_apply,
+    _root_.smul_apply, realInner_apply, smul_eq_mul]
+  apply Finset.sum_congr rfl
+  intro i _
+  apply Finset.sum_congr rfl
+  intro j _
+  ring
+end BilinearExpansion
+
+section Differential
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+
+lemma second_deriv_multiplier (v w : E) (u : 𝓢'(E, ℂ)) :
+    ∂_{v} (∂_{w} u) = fourierMultiplierCLM ℂ
+      (fun ξ => (-(2 * Real.pi)^2 : ℂ) * (inner ℝ ξ v : ℂ) * (inner ℝ ξ w : ℂ)) u := by
+  rw [lineDeriv_eq_fourierMultiplierCLM, lineDeriv_eq_fourierMultiplierCLM,
+    ContinuousLinearMap.map_smul,
+    fourierMultiplierCLM_fourierMultiplierCLM_apply (by fun_prop) (by fun_prop),
+    smul_smul]
+  have hg : ((fun ξ => (inner ℝ ξ w : ℂ)) * (fun ξ => (inner ℝ ξ v : ℂ))).HasTemperateGrowth := by
+    fun_prop
+  rw [← _root_.smul_apply, ← fourierMultiplierCLM_smul hg]
+  congr 2
+  ext ξ
+  simp only [Pi.mul_apply, Pi.smul_apply, smul_eq_mul]
+  have hi := Complex.I_sq
+  calc
+    _ = ((2 * Real.pi : ℂ)^2 * Complex.I^2) *
+          (inner ℝ ξ w : ℂ) * (inner ℝ ξ v : ℂ) := by ring
+    _ = _ := by rw [hi]; ring
+
+lemma multiplier_sub {g₁ g₂ : E → ℂ}
+    (h₁ : g₁.HasTemperateGrowth) (h₂ : g₂.HasTemperateGrowth) (u : 𝓢'(E, ℂ)) :
+    fourierMultiplierCLM ℂ (g₁ - g₂) u =
+      fourierMultiplierCLM ℂ g₁ u - fourierMultiplierCLM ℂ g₂ u := by
+  simp only [fourierMultiplierCLM_apply]
+  rw [smulLeftCLM_sub h₁ h₂, _root_.sub_apply]
+  simp only [sub_eq_add_neg, FourierTransform.fourierInv_add, FourierTransform.fourierInv_neg]
+
+end Differential
+
+section DifferentialExpansion
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+variable {ι : Type*} [Fintype ι]
+
+ 
+
+def differentialWithBasis (b : OrthonormalBasis ι ℝ E) (L : SecondOrder E) :
+    𝓢'(E, ℂ) →L[ℂ] 𝓢'(E, ℂ) :=
+  ∑ i, ∑ j, (L (rankTwo (b i) (b j)) : ℂ) •
+    (lineDerivOpCLM ℂ 𝓢'(E, ℂ) (b i) ∘L lineDerivOpCLM ℂ 𝓢'(E, ℂ) (b j))
+
+lemma differentialWithBasis_fourier (b : OrthonormalBasis ι ℝ E)
+    (L : SecondOrder E) (u : 𝓢'(E, ℂ)) :
+    differentialWithBasis b L u = fourierMultiplierCLM ℂ
+      (fun ξ => (-(2 * Real.pi)^2 : ℂ) * (L (rankOne ξ) : ℂ)) u := by
+  let a : ι → ι → E → ℂ := fun i j ξ => (L (rankTwo (b i) (b j)) : ℂ) *
+    ((-(2 * Real.pi)^2 : ℂ) * (inner ℝ ξ (b i) : ℂ) * (inner ℝ ξ (b j) : ℂ))
+  have ha i j : (a i j).HasTemperateGrowth := by dsimp [a]; fun_prop
+  have ht i j : (L (rankTwo (b i) (b j)) : ℂ) • ∂_{b i} (∂_{b j} u) =
+      fourierMultiplierCLM ℂ (a i j) u := by
+    rw [second_deriv_multiplier, ← _root_.smul_apply,
+      ← fourierMultiplierCLM_smul (by fun_prop)]
+    rfl
+  have hm : fourierMultiplierCLM ℂ (fun ξ => ∑ i, ∑ j, a i j ξ) =
+      ∑ i, ∑ j, fourierMultiplierCLM ℂ (a i j) := by
+    rw [fourierMultiplierCLM_sum ℂ (by intro i _; fun_prop)]
+    apply Finset.sum_congr rfl
+    intro i _
+    exact fourierMultiplierCLM_sum ℂ (fun j _ => ha i j)
+  simp only [differentialWithBasis, _root_.sum_apply, _root_.smul_apply,
+    ContinuousLinearMap.comp_apply, lineDerivOpCLM_apply, ht]
+  have hsum : (∑ i, ∑ j, fourierMultiplierCLM ℂ (a i j)) u =
+      ∑ i, ∑ j, fourierMultiplierCLM ℂ (a i j) u := by
+    simp only [_root_.sum_apply]
+  rw [← hsum, ← hm]
+  congr 2
+  ext ξ
+  rw [second_symbol_expansion b L ξ]
+  simp only [Complex.ofReal_sum, Complex.ofReal_mul, Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro i _
+  apply Finset.sum_congr rfl
+  intro j _
+  dsimp [a]
+  ring
+
+end DifferentialExpansion
+
+ 
+def frozenDifferential (H : Matrix (Fin n) (Fin n) ℂ) :
+    𝓢'(EC n, ℂ) →L[ℂ] 𝓢'(EC n, ℂ) :=
+  differentialWithBasis (stdOrthonormalBasis ℝ (EC n)) (traceBilin H)
+
+lemma frozenDifferential_fourier (H : Matrix (Fin n) (Fin n) ℂ) (u : 𝓢'(EC n, ℂ)) :
+    frozenDifferential H u = fourierMultiplierCLM ℂ (fun ξ => (-symbol H ξ : ℝ)) u := by
+  rw [frozenDifferential, differentialWithBasis_fourier]
+  congr 2
+  ext ξ
+  simp only [symbol, Complex.ofReal_neg, Complex.ofReal_mul, Complex.ofReal_pow,
+    Complex.ofReal_ofNat, neg_mul]
+
+lemma shifted_eq_identity_sub (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (u : 𝓢'(EC n, ℂ)) : shifted H u = u - frozenDifferential H u := by
+  rw [frozenDifferential_fourier]
+  have hn : (fun ξ => ((-symbol H ξ : ℝ) : ℂ)).HasTemperateGrowth :=
+    Complex.hasTemperateGrowth_ofReal.comp (symbol_temperate H hH).neg
+  have he := multiplier_sub (Function.HasTemperateGrowth.const (1 : ℂ)) hn u
+  simpa only [Pi.sub_def, Complex.ofReal_neg, sub_neg_eq_add, shifted,
+    Complex.ofReal_add, Complex.ofReal_one, fourierMultiplierCLM_const,
+    _root_.smul_apply, ContinuousLinearMap.id_apply, one_smul] using he
+
+ 
+
+theorem frozen_differential_sobolev_resolvent
+    (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    {s : ℝ} (f : 𝓢'(EC n, ℂ)) (hf : MemSobolev s 2 f) :
+    ∃! u : 𝓢'(EC n, ℂ), u - frozenDifferential H u = f ∧ MemSobolev (s + 2) 2 u := by
+  simpa only [← shifted_eq_identity_sub H hH] using frozen_sobolev_resolvent H hH f hf
+
+section SmoothComparison
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+variable {ι : Type*} [Fintype ι]
+
+ 
+def smoothDifferentialWithBasis (b : OrthonormalBasis ι ℝ E) (L : SecondOrder E) :
+    𝓢(E, ℝ) →L[ℝ] 𝓢(E, ℝ) :=
+  ∑ i, ∑ j, L (rankTwo (b i) (b j)) •
+    (lineDerivOpCLM ℝ 𝓢(E, ℝ) (b i) ∘L lineDerivOpCLM ℝ 𝓢(E, ℝ) (b j))
+
+lemma smooth_second_deriv (f : 𝓢(E, ℝ)) (v w x : E) :
+    (∂_{v} (∂_{w} f)) x = fderiv ℝ (fderiv ℝ f) x v w := by
+  simp only [SchwartzMap.lineDerivOp_apply_eq_fderiv]
+  change fderiv ℝ (fun y => fderiv ℝ f y w) x v = _
+  exact PotentialKaehler.fderiv_eval_derivative f.smooth'.contDiffAt v w
+
+lemma smoothDifferentialWithBasis_apply (b : OrthonormalBasis ι ℝ E)
+    (L : SecondOrder E) (f : 𝓢(E, ℝ)) (x : E) :
+    smoothDifferentialWithBasis b L f x = L (fderiv ℝ (fderiv ℝ f) x) := by
+  rw [bilin_expansion b (fderiv ℝ (fderiv ℝ f) x)]
+  simp only [smoothDifferentialWithBasis, _root_.sum_apply, _root_.smul_apply,
+    ContinuousLinearMap.comp_apply, lineDerivOpCLM_apply, smooth_second_deriv, map_sum, map_smul, smul_eq_mul]
+  apply Finset.sum_congr rfl
+  intro i _
+  apply Finset.sum_congr rfl
+  intro j _
+  ring
+
+def complexify : 𝓢(E, ℝ) →L[ℝ] 𝓢(E, ℂ) :=
+  SchwartzMap.postcompCLM Complex.ofRealCLM
+
+lemma complexify_deriv (f : 𝓢(E, ℝ)) (v : E) :
+    complexify (∂_{v} f) = ∂_{v} (complexify f) := by
+  ext x
+  change Complex.ofReal (fderiv ℝ f x v) =
+    fderiv ℝ (Complex.ofRealCLM ∘ f) x v
+  rw [(Complex.ofRealCLM.hasFDerivAt.comp x (f.hasFDerivAt x)).fderiv]
+  rfl
+
+variable [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+
+lemma weak_complexify_deriv (f : 𝓢(E, ℝ)) (v : E) :
+    (complexify (∂_{v} f) : 𝓢'(E, ℂ)) = ∂_{v} (complexify f : 𝓢'(E, ℂ)) := by
+  rw [complexify_deriv]
+  exact (TemperedDistribution.lineDerivOp_toTemperedDistributionCLM_eq _ v).symm
+
+lemma differentialWithBasis_real (b : OrthonormalBasis ι ℝ E)
+    (L : SecondOrder E) (f : 𝓢(E, ℝ)) :
+    differentialWithBasis b L (complexify f : 𝓢'(E, ℂ)) =
+      (complexify (smoothDifferentialWithBasis b L f) : 𝓢'(E, ℂ)) := by
+  simp only [differentialWithBasis, smoothDifferentialWithBasis,
+    _root_.sum_apply, _root_.smul_apply, ContinuousLinearMap.comp_apply,
+    lineDerivOpCLM_apply, map_sum, map_smul]
+  simp only [ContinuousLinearMap.map_smul_of_tower]
+  apply Finset.sum_congr rfl
+  intro i _
+  apply Finset.sum_congr rfl
+  intro j _
+  rw [weak_complexify_deriv, weak_complexify_deriv]
+  apply ContinuousLinearMap.ext
+  intro h
+  exact Complex.coe_smul _ _
+
+end SmoothComparison
+
+ 
+
+lemma frozenDifferential_real_trace (H : Matrix (Fin n) (Fin n) ℂ) (f : 𝓢(EC n, ℝ)) :
+    ∃ l : 𝓢(EC n, ℝ),
+      (∀ x, l x = traceBilin H (fderiv ℝ (fderiv ℝ f) x)) ∧
+      frozenDifferential H (SchwartzMap.toTemperedDistributionCLM (EC n) ℂ
+        MeasureTheory.volume (complexify f)) =
+      SchwartzMap.toTemperedDistributionCLM (EC n) ℂ MeasureTheory.volume (complexify l) := by
+  refine ⟨smoothDifferentialWithBasis (stdOrthonormalBasis ℝ (EC n)) (traceBilin H) f, ?_, ?_⟩
+  · exact fun x => smoothDifferentialWithBasis_apply _ _ _ x
+  · exact differentialWithBasis_real _ _ _
+
+end FrozenPoisson
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/GlobalBounds.lean b/lean/OAI/Geometry/PrescribedPotential/GlobalBounds.lean
new file mode 100644
index 0000000000000000000000000000000000000000..cdeacd46aa99afb1f4e20cfe956fe46a0dd43dc5
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/GlobalBounds.lean
@@ -0,0 +1,119 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CutoffHigher
+import OAI.Geometry.PrescribedPotential.GlobalCompletion
+import OAI.Geometry.PrescribedPotential.GlobalError
+import OAI.Geometry.PrescribedPotential.GlobalParametrixEquation
+
+namespace OAI
+
+/-! Global Bounds. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory TemperedDistribution LineDeriv
+open scoped SchwartzMap ContDiff Classical BoundedContinuousFunction ComplexOrder MatrixOrder
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart MetricLocalization
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace Localizers
+variable {D : Localizers A ι}
+
+lemma BoundedCore.comp {s t r : ℝ} {S T : Smooth A →ₗ[ℝ] Smooth A}
+    (hS : D.BoundedCore t r S) (hT : D.BoundedCore s t T) :
+    D.BoundedCore s r (S.comp T) := by
+  obtain ⟨C, hC, hc⟩ := hS
+  obtain ⟨B, hB, hb⟩ := hT
+  refine ⟨C * B, mul_nonneg hC hB, fun f => ?_⟩
+  exact (hc (T f)).trans (by simpa only [mul_assoc] using mul_le_mul_of_nonneg_left (hb f) hC)
+
+lemma BoundedCore.source_mono {s t r : ℝ} {T : Smooth A →ₗ[ℝ] Smooth A}
+    (hT : D.BoundedCore s t T) (hrs : s ≤ r) : D.BoundedCore r t T := by
+  simpa only [LinearMap.comp_id] using hT.comp (D.boundedCore_lower hrs)
+
+lemma BoundedCore.target_mono {s t r : ℝ} {T : Smooth A →ₗ[ℝ] Smooth A}
+    (hT : D.BoundedCore s t T) (hrt : r ≤ t) : D.BoundedCore s r T := by
+  simpa only [LinearMap.id_comp] using (D.boundedCore_lower hrt).comp hT
+end Localizers
+
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+lemma error_bound_zero (m : ℝ) (hm : 1 ≤ m) :
+    D.localizers.BoundedCore 0 0 (D.error m hm) := by
+  obtain ⟨C, hC, hc⟩ := D.error_bound
+  exact ⟨C / m, div_nonneg hC (le_trans (by norm_num) hm), hc m hm⟩
+
+lemma error_piece_bound_integer (m : ℝ) (hm : 1 ≤ m) (k : ℕ) (p : ι) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ f : Smooth A,
+      ‖D.localizers.embed ((k : ℝ) + 1) (globalize (D.patch p).index (D.outerCutoff p)
+        ((D.patch p).cutoffError (D.cutoff p).val
+          ((D.patch p).localInverse m hm (D.forcing p f))))‖ ≤
+        C * ‖D.localizers.embed (k : ℝ) f‖ := by
+  obtain ⟨B, hB, hb⟩ := globalize_integer_bound D.localizers (D.patch p).index (D.outerCutoff p) (k + 1)
+  have he : CoreBound ((k : ℝ) + 2) ((k : ℝ) + 1)
+      ((D.patch p).cutoffError (D.cutoff p).val) := by
+    convert schwartzCutoffError_bound_integer (D.patch p).coefficients
+      (D.patch p).coefficients_temperate (D.cutoff p).val (k + 1) using 1 <;> push_cast <;> first | rfl | ring
+  obtain ⟨C, hC, hc⟩ := he.comp ((D.patch p).localInverse_bound m hm k)
+  refine ⟨B * C, mul_nonneg hB hC, fun f => ?_⟩
+  have hh := hb ((D.patch p).cutoffError (D.cutoff p).val
+    ((D.patch p).localInverse m hm (D.forcing p f)))
+  rw [Nat.cast_add, Nat.cast_one] at hh
+  apply hh.trans
+  have hr := (hc (D.forcing p f)).trans
+    (mul_le_mul_of_nonneg_left (D.forcing_bound p (k : ℝ) f) hC)
+  simpa only [mul_assoc, Function.comp_apply, ParametrixPatch.differential] using mul_le_mul_of_nonneg_left hr hB
+
+ 
+theorem error_bound_integer (m : ℝ) (hm : 1 ≤ m) (k : ℕ) :
+    D.localizers.BoundedCore (k : ℝ) ((k : ℝ) + 1) (D.error m hm) := by
+  choose C hC hc using D.error_piece_bound_integer m hm k
+  refine ⟨∑ p, C p, Finset.sum_nonneg (fun p _ => hC p), fun f => ?_⟩
+  rw [error_apply, map_sum, Finset.sum_mul]
+  exact (norm_sum_le _ _).trans (Finset.sum_le_sum (fun p _ => hc p f))
+
+lemma complexL_reconstruction (f : Smooth A) :
+    complexL g f = ∑ p, globalize (D.patch p).index (D.outerCutoff p)
+      ((D.patch p).differential (SchwartzMap.smulLeftCLM ℂ (D.cutoff p).val (D.forcing p f))) := by
+  conv_lhs => rw [← D.forcing_reconstruction f, map_sum]
+  apply Finset.sum_congr rfl
+  intro p _
+  exact complexL_globalize (D.patch p) (D.cutoff p) (D.outerCutoff p)
+    (D.cutoff_support p) (D.outerCutoff_one p) _
+
+lemma complexL_piece_bound (p : ι) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ f : Smooth A,
+      ‖D.localizers.embed 0 (globalize (D.patch p).index (D.outerCutoff p)
+        ((D.patch p).differential (SchwartzMap.smulLeftCLM ℂ (D.cutoff p).val (D.forcing p f))))‖ ≤
+        C * ‖D.localizers.embed 2 f‖ := by
+  obtain ⟨B, hB, hb⟩ := globalize_integer_bound D.localizers (D.patch p).index (D.outerCutoff p) 0
+  have hk : CoreBound 2 2 (SchwartzMap.smulLeftCLM ℂ (D.cutoff p).val) :=
+    coreBound_product_integer (D.cutoff p).val 2
+  obtain ⟨C, hC, hc⟩ := (schwartzDifferential_bound (D.patch p).coefficients
+    (D.patch p).coefficients_temperate).comp hk
+  refine ⟨B * C, mul_nonneg hB hC, fun f => ?_⟩
+  have hh := hb ((D.patch p).differential (SchwartzMap.smulLeftCLM ℂ (D.cutoff p).val (D.forcing p f)))
+  rw [Nat.cast_zero] at hh
+  apply hh.trans
+  have hr := (hc (D.forcing p f)).trans
+    (mul_le_mul_of_nonneg_left (D.forcing_bound p 2 f) hC)
+  simpa only [mul_assoc, Function.comp_apply, ParametrixPatch.differential] using mul_le_mul_of_nonneg_left hr hB
+
+ 
+theorem complexL_bound : D.localizers.BoundedCore 2 0 (complexL g) := by
+  choose C hC hc using D.complexL_piece_bound
+  refine ⟨∑ p, C p, Finset.sum_nonneg (fun p _ => hC p), fun f => ?_⟩
+  rw [D.complexL_reconstruction, map_sum, Finset.sum_mul]
+  exact (norm_sum_le _ _).trans (Finset.sum_le_sum (fun p _ => hc p f))
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/GlobalCompletion.lean b/lean/OAI/Geometry/PrescribedPotential/GlobalCompletion.lean
new file mode 100644
index 0000000000000000000000000000000000000000..b2f86fd6fca71e125226e948fa0e1bee3ffff6e8
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/GlobalCompletion.lean
@@ -0,0 +1,126 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CoreBounds
+import OAI.Geometry.PrescribedPotential.GlobalPartition
+
+namespace OAI
+
+/-! Global Completion. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+open scoped SchwartzMap ContDiff Classical
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace Localizers
+variable (D : Localizers A ι)
+
+def BoundedCore (s t : ℝ) (T : Smooth A →ₗ[ℝ] Smooth A) : Prop :=
+  ∃ C : ℝ, 0 ≤ C ∧ ∀ f, ‖D.embed t (T f)‖ ≤ C * ‖D.embed s f‖
+
+def extendCore (s t : ℝ) (T : Smooth A →ₗ[ℝ] Smooth A) :
+    D.Sobolev s →L[ℝ] D.Sobolev t :=
+  ((D.embed t).comp T).extendOfNorm (D.embed s)
+
+lemma extendCore_embed {s t : ℝ} {T : Smooth A →ₗ[ℝ] Smooth A}
+    (hT : D.BoundedCore s t T) (f : Smooth A) :
+    D.extendCore s t T (D.embed s f) = D.embed t (T f) :=
+  LinearMap.extendOfNorm_eq (D.embed_dense s) (by
+    obtain ⟨C, _, h⟩ := hT
+    exact ⟨C, h⟩) f
+
+lemma extendCore_norm {s t : ℝ} {T : Smooth A →ₗ[ℝ] Smooth A}
+    {C : ℝ} (hC : 0 ≤ C) (hT : ∀ f, ‖D.embed t (T f)‖ ≤ C * ‖D.embed s f‖) :
+    ‖D.extendCore s t T‖ ≤ C := by
+  apply ContinuousLinearMap.opNorm_le_bound _ hC
+  exact LinearMap.norm_extendOfNorm_apply_le (D.embed_dense s) C hT
+
+lemma boundedCore_lower {s t : ℝ} (hst : t ≤ s) :
+    D.BoundedCore s t (LinearMap.id : Smooth A →ₗ[ℝ] Smooth A) := by
+  obtain ⟨C, hC, hc⟩ := CoreBound.id (E := EC d) hst
+  refine ⟨C, hC, fun f => ?_⟩
+  change ‖D.coordinates t f‖ ≤ C * ‖D.coordinates s f‖
+  apply (pi_norm_le_iff_of_nonneg (mul_nonneg hC (norm_nonneg _))).mpr
+  intro p
+  exact (hc (localize A (D.index p) (D.weight p) (D.support_sub p) f)).trans
+    (mul_le_mul_of_nonneg_left (norm_le_pi_norm (D.coordinates s f) p) hC)
+
+def lower (s t : ℝ) : D.Sobolev s →L[ℝ] D.Sobolev t :=
+  D.extendCore s t LinearMap.id
+
+lemma lower_embed {s t : ℝ} (hst : t ≤ s) (f : Smooth A) :
+    D.lower s t (D.embed s f) = D.embed t f :=
+  D.extendCore_embed (D.boundedCore_lower hst) f
+
+lemma lower_self (s : ℝ) : D.lower s s = ContinuousLinearMap.id ℝ _ := by
+  apply DFunLike.coe_injective
+  apply (D.embed_dense s).equalizer (D.lower s s).continuous continuous_id
+  funext f
+  exact D.lower_embed le_rfl f
+
+lemma lower_comp {s t r : ℝ} (hst : t ≤ s) (htr : r ≤ t) :
+    (D.lower t r).comp (D.lower s t) = D.lower s r := by
+  apply DFunLike.coe_injective
+  apply (D.embed_dense s).equalizer (by fun_prop) (by fun_prop)
+  funext f
+  simp only [Function.comp_apply, ContinuousLinearMap.comp_apply,
+    D.lower_embed hst, D.lower_embed htr, D.lower_embed (htr.trans hst)]
+
+ 
+def distribution (s : ℝ) (p : ι) : D.Sobolev s →L[ℝ] 𝓢'(EC d, ℂ) :=
+  (realize s).restrictScalars ℝ ∘L (ContinuousLinearMap.proj p) ∘L (D.closedSpace s).subtypeL
+
+lemma distribution_apply (s : ℝ) (p : ι) (u : D.Sobolev s) :
+    D.distribution s p u = realize s (u.val p) := rfl
+
+lemma distribution_embed (s : ℝ) (p : ι) (f : Smooth A) :
+    D.distribution s p (D.embed s f) =
+      SchwartzMap.toTemperedDistributionCLM (EC d) ℂ volume
+        (localize A (D.index p) (D.weight p) (D.support_sub p) f) :=
+  realize_schwartzCoord s _
+
+lemma distribution_lower {s t : ℝ} (hst : t ≤ s) (p : ι) (u : D.Sobolev s) :
+    D.distribution t p (D.lower s t u) = D.distribution s p u := by
+  have he : (D.distribution t p) ∘ D.lower s t = D.distribution s p := by
+    apply (D.embed_dense s).equalizer (by fun_prop) (by fun_prop)
+    funext f
+    simp only [Function.comp_apply, D.lower_embed hst, distribution_embed]
+  exact congr_fun he u
+
+lemma lower_injective {s t : ℝ} (hst : t ≤ s) : Function.Injective (D.lower s t) := by
+  intro u v huv
+  apply Subtype.ext
+  funext p
+  apply realize_injective s
+  change D.distribution s p u = D.distribution s p v
+  rw [← D.distribution_lower hst p u, ← D.distribution_lower hst p v, huv]
+
+lemma lower_extendCore {s t r : ℝ} (htr : r ≤ t) (T : Smooth A →ₗ[ℝ] Smooth A)
+    (hT : D.BoundedCore s t T) (hR : D.BoundedCore s r T) (u : D.Sobolev s) :
+    D.lower t r (D.extendCore s t T u) = D.extendCore s r T u := by
+  have he : (D.lower t r) ∘ D.extendCore s t T = D.extendCore s r T := by
+    apply (D.embed_dense s).equalizer (by fun_prop) (by fun_prop)
+    funext f
+    simp only [Function.comp_apply, D.extendCore_embed hT, D.extendCore_embed hR, D.lower_embed htr]
+  exact congr_fun he u
+
+lemma extendCore_lower {s t r : ℝ} (hst : t ≤ s) (T : Smooth A →ₗ[ℝ] Smooth A)
+    (hT : D.BoundedCore t r T) (hS : D.BoundedCore s r T) (u : D.Sobolev s) :
+    D.extendCore t r T (D.lower s t u) = D.extendCore s r T u := by
+  have he : (D.extendCore t r T) ∘ D.lower s t = D.extendCore s r T := by
+    apply (D.embed_dense s).equalizer (by fun_prop) (by fun_prop)
+    funext f
+    simp only [Function.comp_apply, D.lower_embed hst, D.extendCore_embed hT, D.extendCore_embed hS]
+  exact congr_fun he u
+end Localizers
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/GlobalConjugation.lean b/lean/OAI/Geometry/PrescribedPotential/GlobalConjugation.lean
new file mode 100644
index 0000000000000000000000000000000000000000..69f071afd8cccb349783a7050cd3d913f5549b50
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/GlobalConjugation.lean
@@ -0,0 +1,115 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.GlobalCompletion
+import OAI.Geometry.PrescribedPotential.GlobalParametrix
+import OAI.Geometry.PrescribedPotential.RealSobolev
+import OAI.Geometry.PrescribedPotential.SchwartzConjugation
+
+namespace OAI
+
+/-! Global Conjugation. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory LineDeriv
+open scoped ContDiff SchwartzMap Classical
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+namespace ChartCutoff
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] {U : Set E}
+def conjugate (κ : ChartCutoff U) : ChartCutoff U where
+  val := schwartzConj κ.val
+  compact := by
+    apply HasCompactSupport.of_support_subset_isCompact κ.compact
+    intro x hx
+    exact subset_tsupport _ (fun h => hx (by simp only [schwartzConj_apply,h,star_zero]))
+  support_sub := by
+    apply Subset.trans _ κ.support_sub
+    apply closure_mono
+    intro x hx h
+    exact hx (by simp only [schwartzConj_apply,h,star_zero])
+@[simp] lemma conjugate_apply (κ : ChartCutoff U) (x : E) : κ.conjugate x = star (κ x) := rfl
+end ChartCutoff
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+namespace Smooth
+
+def conjugate : Smooth A →ₗ[ℝ] Smooth A where
+  toFun f := ⟨fun x => star (f x),fun i => Complex.conjCLE.toContinuousLinearMap.contDiff.comp_contDiffOn (f.smooth i)⟩
+  map_add' f h := by ext x; exact star_add _ _
+  map_smul' c f := by ext x; exact map_smul Complex.conjCLE.toContinuousLinearMap c (f x)
+@[simp] lemma conjugate_apply (f : Smooth A) (x : X) : conjugate f x = star (f x) := rfl
+
+def realify : Smooth A →ₗ[ℝ] Smooth A := (1/2 : ℝ) • (LinearMap.id + conjugate)
+lemma realify_apply (f : Smooth A) (x : X) : realify f x = ((f x).re : ℂ) := by
+  change (1/2 : ℝ) • (f x + star (f x)) = _
+  apply Complex.ext <;> simp
+  ring
+lemma realify_real (f : Smooth A) : realify f ∈ realSmoothSpace A := by
+  intro x
+  rw [realify_apply]
+  rfl
+lemma realify_eq_of_real (f : RealSmooth A) : realify f.val = f.val := by
+  ext x
+  rw [realify_apply]
+  exact Complex.ext rfl (f.property x).symm
+end Smooth
+variable [T2Space X] {ι : Type*} [Fintype ι]
+lemma conjugate_globalize (i : Fin A.count) (κ : ChartCutoff (A.euclideanChart i).target)
+    (f : 𝓢(EC d,ℂ)) :
+    Smooth.conjugate (globalize i κ f) = globalize i κ.conjugate (schwartzConj f) := by
+  ext x
+  rw [Smooth.conjugate_apply,globalize_apply,globalize_apply]
+  split_ifs <;> simp only [ChartCutoff.conjugate_apply,schwartzConj_apply,star_mul',star_zero]
+namespace GluingData
+variable [CompactSpace X] {g : KaehlerMetric A} (D : GluingData g ι)
+
+lemma conjugate_integer_bound (k : ℕ) : D.localizers.BoundedCore (k : ℝ) (k : ℝ) Smooth.conjugate := by
+  have hb (p : ι) := globalize_integer_bound D.localizers (D.patch p).index (D.cutoff p).conjugate k
+  choose B hB hb using hb
+  obtain ⟨C,hC,hc⟩ := schwartzConj_integer_bound (E := EC d) k
+  refine ⟨∑ p, B p*C,Finset.sum_nonneg (fun p _ => mul_nonneg (hB p) hC),fun f => ?_⟩
+  have he : Smooth.conjugate f = ∑ p, globalize (D.patch p).index (D.cutoff p).conjugate
+      (schwartzConj (D.forcing p f)) := by
+    conv_lhs => rw [← D.forcing_reconstruction f]
+    rw [map_sum]
+    apply Finset.sum_congr rfl
+    intro p _
+    exact conjugate_globalize _ _ _
+  rw [he,map_sum,Finset.sum_mul]
+  apply (norm_sum_le _ _).trans
+  apply Finset.sum_le_sum
+  intro p _
+  exact (hb p _).trans ((mul_le_mul_of_nonneg_left (hc _) (hB p)).trans (by
+    simpa only [mul_assoc] using mul_le_mul_of_nonneg_left (D.forcing_bound p (k : ℝ) f)
+      (mul_nonneg (hB p) hC)))
+
+lemma embed_realify (k : ℕ) (f : Smooth A) :
+    D.localizers.embed (k : ℝ) (Smooth.realify f) =
+      (1/2 : ℝ) • (D.localizers.embed (k : ℝ) f + D.localizers.embed (k : ℝ) (Smooth.conjugate f)) := by
+  change D.localizers.embed (k : ℝ) ((1/2 : ℝ) • (f + Smooth.conjugate f)) = _
+  rw [map_smul,map_add]
+
+lemma half_add_norm_le {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+    (u v : E) (C : ℝ) (hC : ‖v‖ ≤ C*‖u‖) :
+    ‖(1/2 : ℝ) • (u+v)‖ ≤ ((1/2 : ℝ)*(1+C))*‖u‖ := by
+  rw [norm_smul]
+  norm_num only [norm_div,norm_one,Real.norm_ofNat]
+  calc
+    _ ≤ (1/2 : ℝ)*(‖u‖+‖v‖) := mul_le_mul_of_nonneg_left (norm_add_le _ _) (by norm_num)
+    _ ≤ (1/2 : ℝ)*(‖u‖+C*‖u‖) := mul_le_mul_of_nonneg_left (add_le_add le_rfl hC) (by norm_num)
+    _ = _ := by ring
+
+lemma realify_integer_bound (k : ℕ) : D.localizers.BoundedCore (k : ℝ) (k : ℝ) Smooth.realify := by
+  obtain ⟨C,hC,hb⟩ := D.conjugate_integer_bound k
+  refine ⟨(1/2 : ℝ)*(1+C),mul_nonneg (by norm_num) (by linarith),fun f => ?_⟩
+  rw [D.embed_realify]
+  exact half_add_norm_le _ _ C (hb f)
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/GlobalError.lean b/lean/OAI/Geometry/PrescribedPotential/GlobalError.lean
new file mode 100644
index 0000000000000000000000000000000000000000..680f4da0733a9df40896e4762c79f971a652e94b
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/GlobalError.lean
@@ -0,0 +1,70 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.GlobalParametrixEquation
+
+namespace OAI
+
+/-! Global Error. -/
+
+section
+
+ 
+noncomputable section
+open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory
+open scoped SchwartzMap ContDiff Classical ComplexOrder MatrixOrder
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart MetricLocalization
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+lemma error_piece_bound (p : ι) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ m (hm : 1 ≤ m) (f : Smooth A),
+      ‖D.localizers.embed 0 (globalize (D.patch p).index (D.outerCutoff p)
+        ((D.patch p).cutoffError (D.cutoff p).val
+          ((D.patch p).localInverse m hm (D.forcing p f))))‖ ≤
+        C / m * ‖D.localizers.embed 0 f‖ := by
+  obtain ⟨B, hB, hb⟩ := globalize_integer_bound D.localizers (D.patch p).index (D.outerCutoff p) 0
+  let c := cutoffErrorConstant (D.patch p).matrix (D.patch p).positive
+    (coefficientBCF (D.patch p).coefficient) (D.cutoff p).val
+  refine ⟨B * |c|, mul_nonneg hB (abs_nonneg _), fun m hm f => ?_⟩
+  have hm0 : 0 ≤ m := le_trans (by norm_num) hm
+  have hb' := hb ((D.patch p).cutoffError (D.cutoff p).val
+    ((D.patch p).localInverse m hm (D.forcing p f)))
+  rw [Nat.cast_zero] at hb'
+  apply hb'.trans
+  calc
+    _ ≤ B * (c / m * ‖schwartzCoord 0 (D.forcing p f)‖) :=
+      mul_le_mul_of_nonneg_left ((D.patch p).cutoffError_bound m hm (D.cutoff p).val (D.forcing p f)) hB
+    _ ≤ B * (|c| / m * ‖D.localizers.embed 0 f‖) := by
+      apply mul_le_mul_of_nonneg_left _ hB
+      exact mul_le_mul (div_le_div_of_nonneg_right (le_abs_self _) hm0)
+        (D.forcing_bound p 0 f) (norm_nonneg _) (div_nonneg (abs_nonneg _) hm0)
+    _ = _ := by ring
+
+ 
+theorem error_bound :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ m (hm : 1 ≤ m) (f : Smooth A),
+      ‖D.localizers.embed 0 (D.error m hm f)‖ ≤ C / m * ‖D.localizers.embed 0 f‖ := by
+  choose C hC hc using D.error_piece_bound
+  refine ⟨∑ p, C p, Finset.sum_nonneg (fun p _ => hC p), fun m hm f => ?_⟩
+  rw [error_apply, map_sum, Finset.sum_div, Finset.sum_mul]
+  exact (norm_sum_le _ _).trans (Finset.sum_le_sum (fun p _ => hc p m hm f))
+
+ 
+
+theorem exists_contractive_shift :
+    ∃ m : ℝ, ∃ hm : 1 ≤ m, ∃ q : ℝ, 0 ≤ q ∧ q < 1 ∧
+      ∀ f : Smooth A, ‖D.localizers.embed 0 (D.error m hm f)‖ ≤ q * ‖D.localizers.embed 0 f‖ := by
+  obtain ⟨C, hC, hc⟩ := D.error_bound
+  have hm : 1 ≤ C + 1 := by linarith
+  refine ⟨C + 1, hm, C / (C + 1), div_nonneg hC (by linarith), ?_, hc _ hm⟩
+  exact (div_lt_one (by linarith : 0 < C + 1)).mpr (by linarith)
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/GlobalExtension.lean b/lean/OAI/Geometry/PrescribedPotential/GlobalExtension.lean
new file mode 100644
index 0000000000000000000000000000000000000000..12ef970ce3e636c72ff5f34c75350b1d03c551a0
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/GlobalExtension.lean
@@ -0,0 +1,124 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.ChartTransport
+import OAI.Geometry.PrescribedPotential.GlobalSobolev
+
+namespace OAI
+
+/-! Global Extension. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory
+open scoped ContDiff SchwartzMap Classical
+
+namespace GlobalElliptic
+open Anticanonical EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X]
+  {A : ComplexAtlas d X}
+
+ 
+def extendFun (i : Fin A.count) (f : EC d → ℂ) (x : X) : ℂ :=
+  if x ∈ (A.euclideanChart i).source then f (A.euclideanChart i x) else 0
+
+lemma extendFun_apply {i : Fin A.count} (f : EC d → ℂ) {x : X}
+    (hx : x ∈ (A.euclideanChart i).source) :
+    extendFun (A := A) i f x = f (A.euclideanChart i x) := ite_eq_left hx
+
+lemma extendFun_support (i : Fin A.count) (f : EC d → ℂ) :
+    Function.support (extendFun (A := A) i f) ⊆
+      (A.euclideanChart i).symm '' tsupport f := by
+  intro x hx
+  have hn : extendFun (A := A) i f x ≠ 0 := hx
+  by_cases hi : x ∈ (A.euclideanChart i).source
+  · rw [extendFun_apply f hi] at hn
+    exact ⟨A.euclideanChart i x, subset_tsupport f hn, (A.euclideanChart i).left_inv hi⟩
+  · exact (hn (ite_eq_right hi)).elim
+
+variable [T2Space X]
+
+lemma extendFun_smooth (i : Fin A.count) (f : EC d → ℂ)
+    (hf : ContDiff ℝ ∞ f) (hc : HasCompactSupport f)
+    (hs : tsupport f ⊆ (A.euclideanChart i).target) :
+    ∀ j, ContDiffOn ℝ ∞ (extendFun (A := A) i f ∘ (A.euclideanChart j).symm)
+      (A.euclideanChart j).target := by
+  intro j y hy
+  let e := A.euclideanChart i
+  let q := A.euclideanChart j
+  let K := e.symm '' tsupport f
+  have hK : IsCompact K := hc.image_of_continuousOn (e.symm.continuousOn.mono hs)
+  have hKi : K ⊆ e.source := by rintro x ⟨z, hz, rfl⟩; exact e.symm.mapsTo (hs hz)
+  by_cases hi : q.symm y ∈ e.source
+  · let t := q.symm.trans e
+    have ht : y ∈ t.source := ⟨hy, hi⟩
+    have hsm := (hf.contDiffAt (x := t y)).comp y
+      ((A.euclidean_transition_smooth j i).contDiffAt (t.open_source.mem_nhds ht))
+    apply (hsm.congr_of_eventuallyEq ?_).contDiffWithinAt
+    filter_upwards [t.open_source.mem_nhds ht] with z hz
+    exact ite_eq_left hz.2
+  · have hnot : q.symm y ∉ K := fun hk => hi (hKi hk)
+    have hn := (q.symm.continuousAt hy) (hK.isClosed.isOpen_compl.mem_nhds hnot)
+    apply ((contDiffAt_const (c := (0 : ℂ))).congr_of_eventuallyEq ?_).contDiffWithinAt
+    filter_upwards [hn] with z hz
+    exact Function.notMem_support.mp (fun hp => hz (extendFun_support i f hp))
+
+ 
+def extend (i : Fin A.count) (f : EC d → ℂ) (hf : ContDiff ℝ ∞ f)
+    (hc : HasCompactSupport f) (hs : tsupport f ⊆ (A.euclideanChart i).target) : Smooth A :=
+  ⟨extendFun i f, extendFun_smooth i f hf hc hs⟩
+
+lemma extend_tsupport (i : Fin A.count) (f : EC d → ℂ) (hf : ContDiff ℝ ∞ f)
+    (hc : HasCompactSupport f) (hs : tsupport f ⊆ (A.euclideanChart i).target) :
+    tsupport (extend i f hf hc hs : X → ℂ) ⊆ (A.euclideanChart i).symm '' tsupport f := by
+  apply closure_minimal (extendFun_support i f)
+  exact (hc.image_of_continuousOn ((A.euclideanChart i).symm.continuousOn.mono hs)).isClosed
+
+ 
+def globalize (i : Fin A.count) (κ : ChartCutoff (A.euclideanChart i).target) :
+    𝓢(EC d, ℂ) →ₗ[ℝ] Smooth A where
+  toFun f := extend i (fun y => κ y * f y)
+    ((κ.val.smooth ⊤).mul (f.smooth ⊤)) (κ.product_compact f)
+    ((κ.product_tsupport (A.euclideanChart i).open_target (f.smooth ⊤).contDiffOn).trans κ.support_sub)
+  map_add' f h := by
+    apply Smooth.ext
+    intro x
+    change extendFun i (fun y => κ y * (f + h) y) x =
+      extendFun i (fun y => κ y * f y) x + extendFun i (fun y => κ y * h y) x
+    by_cases hx : x ∈ (A.euclideanChart i).source
+    · simp only [extendFun, ite_eq_left hx, add_apply]
+      exact mul_add _ _ _
+    · simp only [extendFun, ite_eq_right hx, add_zero]
+  map_smul' c f := by
+    apply Smooth.ext
+    intro x
+    change extendFun i (fun y => κ y * (c • f) y) x =
+      c • extendFun i (fun y => κ y * f y) x
+    by_cases hx : x ∈ (A.euclideanChart i).source
+    · simp only [extendFun, ite_eq_left hx, smul_apply,
+        Complex.real_smul]
+      ring
+    · simp only [extendFun, ite_eq_right hx, smul_zero]
+
+@[simp] lemma globalize_apply (i : Fin A.count) (κ : ChartCutoff (A.euclideanChart i).target)
+    (f : 𝓢(EC d, ℂ)) (x : X) :
+    globalize i κ f x = if x ∈ (A.euclideanChart i).source then
+      κ (A.euclideanChart i x) * f (A.euclideanChart i x) else 0 := rfl
+
+lemma globalize_tsupport (i : Fin A.count) (κ : ChartCutoff (A.euclideanChart i).target)
+    (f : 𝓢(EC d, ℂ)) :
+    tsupport (globalize i κ f : X → ℂ) ⊆
+      (A.euclideanChart i).symm '' tsupport (κ : EC d → ℂ) := by
+  exact (extend_tsupport i (fun y => κ y * f y)
+    ((κ.val.smooth ⊤).mul (f.smooth ⊤)) (κ.product_compact f)
+    ((κ.product_tsupport (A.euclideanChart i).open_target (f.smooth ⊤).contDiffOn).trans
+      κ.support_sub)).trans (image_mono
+    (κ.product_tsupport (A.euclideanChart i).open_target (f.smooth ⊤).contDiffOn))
+
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/GlobalFiniteEmbedding.lean b/lean/OAI/Geometry/PrescribedPotential/GlobalFiniteEmbedding.lean
new file mode 100644
index 0000000000000000000000000000000000000000..42e3297d2ffe3c26d1cc598fbfe6d70d0dabab96
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/GlobalFiniteEmbedding.lean
@@ -0,0 +1,69 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.FiniteSobolevEmbedding
+import OAI.Geometry.PrescribedPotential.GlobalPartition
+import OAI.Geometry.PrescribedPotential.GlobalStrongEmbedding
+import OAI.Geometry.PrescribedPotential.KaehlerClosedDerivatives
+
+namespace OAI
+
+/-! Global Finite Embedding. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology
+open scoped ContDiff Classical
+namespace GlobalElliptic
+open Anticanonical EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace Localizers
+variable (D : Localizers A ι)
+
+omit [T2Space X] [CompactSpace X] in
+lemma exists_weight_ne_zero (x : X) : ∃ p, D.weight p x ≠ 0 := by
+  by_contra! hn
+  have h := D.sum_one x
+  simp only [hn, Finset.sum_const_zero] at h
+  exact zero_ne_one h
+
+lemma strong_contDiffOn (s : ℝ) (hs : (Module.finrank ℝ (EC d) : ℝ) < 2*s)
+    (N : ℕ) (hN : (Module.finrank ℝ (EC d) : ℝ) < 2*(s - 2*N))
+    (u : D.Sobolev s) (i : Fin A.count) :
+    ContDiffOn ℝ N ((D.strong s u) ∘ (A.euclideanChart i).symm)
+      (A.euclideanChart i).target := by
+  intro z hz
+  let q := A.euclideanChart i
+  let x := q.symm z
+  obtain ⟨p,hp⟩ := D.exists_weight_ne_zero x
+  let e := A.euclideanChart (D.index p)
+  have hx : x ∈ e.source := D.support_sub p (subset_tsupport _ hp)
+  let t := q.symm.trans e
+  have ht : z ∈ t.source := ⟨hz,hx⟩
+  have hf := (strongEmbedding_contDiff s hs N hN (u.val p)).contDiffAt (x := t z)
+  have htrans := ((A.euclidean_transition_smooth i (D.index p)).contDiffAt
+    (t.open_source.mem_nhds ht)).of_le (show (N : ℕ∞ω) ≤ ∞ from ENat.natCast_le_of_coe_top_le_withTop le_rfl N)
+  have hw := ((D.weight p).smooth i).contDiffAt (q.open_target.mem_nhds hz)
+  have hn : (D.weight p ∘ q.symm) z ≠ 0 := hp
+  have hsm := (hf.comp z htrans).mul ((hw.of_le (show (N : ℕ∞ω) ≤ ∞ from ENat.natCast_le_of_coe_top_le_withTop le_rfl N)).inv hn)
+  apply (hsm.congr_of_eventuallyEq ?_).contDiffWithinAt
+  have hne : ∀ᶠ y in 𝓝 z, (D.weight p ∘ q.symm) y ≠ 0 :=
+    hw.continuousAt.eventually_ne hn
+  filter_upwards [t.open_source.mem_nhds ht, hne] with y hy hny
+  change D.strong s u (q.symm y) = strongEmbedding s hs (u.val p) (e (q.symm y)) /
+    D.weight p (q.symm y)
+  rw [D.strong_localized s hs]
+  have hyv : q.symm y ∈ e.source := hy.2
+  change D.strong s u (q.symm y) = (if e (q.symm y) ∈ e.target then
+    D.weight p (e.symm (e (q.symm y))) * D.strong s u (e.symm (e (q.symm y))) else 0) / _
+  rw [ite_eq_left (e.mapsTo hyv), e.left_inv hyv]
+  exact (mul_div_cancel_left₀ _ hny).symm
+end Localizers
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/GlobalHessian.lean b/lean/OAI/Geometry/PrescribedPotential/GlobalHessian.lean
new file mode 100644
index 0000000000000000000000000000000000000000..3070bd00bcbfa14ddf3aa57e31a33256f78d748d
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/GlobalHessian.lean
@@ -0,0 +1,124 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CircleRadialCalculus
+import OAI.Geometry.PrescribedPotential.ComplexHessian
+import OAI.Geometry.PrescribedPotential.GlobalCompletion
+import OAI.Geometry.PrescribedPotential.GlobalParametrix
+import OAI.Geometry.PrescribedPotential.GlobalProduct
+import OAI.Geometry.PrescribedPotential.KaehlerClosedDerivatives
+
+namespace OAI
+
+/-! Global Hessian. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory
+open scoped ContDiff SchwartzMap Classical
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+
+omit [CompactSpace X] in
+lemma cutoffGlobal_support (i : Fin A.count) (κ : ChartCutoff (A.euclideanChart i).target) :
+    tsupport (cutoffGlobal i κ : X → ℂ) ⊆ (A.euclideanChart i).source := by
+  intro x hx
+  obtain ⟨y,hy,rfl⟩ := cutoffGlobal_tsupport i κ hx
+  exact (A.euclideanChart i).symm.mapsTo (κ.support_sub hy)
+
+lemma localize_cutoff_eventually (i : Fin A.count)
+    (κ η : ChartCutoff (A.euclideanChart i).target)
+    (hη : ∀ y ∈ tsupport (κ : EC d → ℂ), η y = 1)
+    (f : Smooth A) {y : EC d} (hy : y ∈ (A.euclideanChart i).target) (hκ : κ y ≠ 0) :
+    (localize A i (cutoffGlobal i η) (cutoffGlobal_support i η) f : EC d → ℂ)
+      =ᶠ[𝓝 y] f ∘ (A.euclideanChart i).symm := by
+  have hn : ∀ᶠ z in 𝓝 y, κ z ≠ 0 :=
+    (isOpen_ne_fun κ.val.continuous continuous_const).mem_nhds hκ
+  filter_upwards [(A.euclideanChart i).open_target.mem_nhds hy, hn] with z hz hnz
+  rw [localize_apply, localizeFun_apply A _ _ hz]
+  have hs : (A.euclideanChart i).symm z ∈ (A.euclideanChart i).source :=
+    (A.euclideanChart i).symm.mapsTo hz
+  change (if (A.euclideanChart i).symm z ∈ (A.euclideanChart i).source then
+      η (A.euclideanChart i ((A.euclideanChart i).symm z)) else 0) * f ((A.euclideanChart i).symm z) = _
+  rw [ite_eq_left hs, (A.euclideanChart i).right_inv hz, hη z (subset_tsupport _ hnz), one_mul]
+  rfl
+
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+def localizedHessian (p : ι) (i j : Fin d) : Smooth A →ₗ[ℝ] Smooth A :=
+  (globalize (D.patch p).index (D.cutoff p)).comp
+    (((hessianEntrySchwartz i j).restrictScalars ℝ).toLinearMap.comp
+      (localizeLinear A (D.patch p).index (cutoffGlobal (D.patch p).index (D.outerCutoff p))
+        (cutoffGlobal_support _ _)))
+
+lemma localizedHessian_bound (k : ℕ) (p : ι) (i j : Fin d) :
+    D.localizers.BoundedCore ((k : ℝ)+2) (k : ℝ) (D.localizedHessian p i j) := by
+  let q := (D.patch p).index
+  let ρ := cutoffGlobal q (D.outerCutoff p)
+  let hp := cutoffGlobal_support q (D.outerCutoff p)
+  obtain ⟨B,hB,hb⟩ := globalize_integer_bound D.localizers q (D.cutoff p) k
+  obtain ⟨C,hC,hc⟩ := hessianEntrySchwartz_bound (k : ℝ) i j
+  obtain ⟨E,hE,he⟩ := D.localization_integer_bound q ρ hp (k+2)
+  rw [Nat.cast_add, Nat.cast_ofNat] at he
+  refine ⟨B*(C*E),mul_nonneg hB (mul_nonneg hC hE),fun f => ?_⟩
+  let v := localize A q ρ hp f
+  change ‖D.localizers.embed (k : ℝ) (globalize q (D.cutoff p) (hessianEntrySchwartz i j v))‖ ≤ _
+  calc
+    _ ≤ B*‖schwartzCoord (k : ℝ) (hessianEntrySchwartz i j v)‖ := hb _
+    _ ≤ B*(C*‖schwartzCoord ((k : ℝ)+2) v‖) := mul_le_mul_of_nonneg_left (hc v) hB
+    _ ≤ B*(C*(E*‖D.localizers.embed ((k : ℝ)+2) f‖)) :=
+      mul_le_mul_of_nonneg_left (mul_le_mul_of_nonneg_left (he f) hC) hB
+    _ = _ := by ring
+
+lemma localizedHessian_source (p : ι) (i j : Fin d) (φ : SmoothRealFunction A)
+    {x : X} (hx : x ∈ tsupport (D.localizers.weight p : X → ℂ)) :
+    D.localizedHessian p i j (Smooth.ofReal φ) x =
+      φ.hessian (D.patch p).index (A.chart (D.patch p).index x) i j := by
+  let q := (D.patch p).index
+  let e := A.euclideanChart q
+  have hs : x ∈ e.source := by
+    have hh := D.localizers.support_sub p hx
+    rwa [D.index_eq] at hh
+  have ht : e x ∈ e.target := e.mapsTo hs
+  have hc : D.cutoff p (e x) = 1 := D.cutoff_one p x hx
+  have he := localize_cutoff_eventually q (D.cutoff p) (D.outerCutoff p)
+    (D.outerCutoff_one p) (Smooth.ofReal φ) ht (by rw [hc]; exact one_ne_zero)
+  let u : EC d → ℝ := φ.localExpression q ∘ coordinateEquiv d
+  have hu : ContDiffAt ℝ ∞ u (e x) := (φ.euclidean_smooth q).contDiffAt (by simpa only [e, ComplexAtlas.euclideanChart_target] using e.open_target.mem_nhds ht)
+  have hre : (fun y => Smooth.ofReal φ (e.symm y)) = Complex.ofRealCLM ∘ u := rfl
+  change globalize q (D.cutoff p) (hessianEntrySchwartz i j
+    (localize A q (cutoffGlobal q (D.outerCutoff p)) (cutoffGlobal_support _ _) (Smooth.ofReal φ))) x = _
+  rw [globalize_apply, ite_eq_left hs, hc, one_mul,
+    hessianEntrySchwartz_real hu _ (he.trans (Filter.Eventually.of_forall (congr_fun hre))) i j,
+    pullBilin_hessian hu]
+  have hcomp : u ∘ (coordinateEquiv d).symm = φ.localExpression q := by
+    funext z; simp [u]
+  rw [hcomp]
+  change PotentialKaehler.hermitianPartMatrix
+    (fderiv ℝ (fderiv ℝ (φ.localExpression q)) (coordinateEquiv d (e x))) i j = _
+  have heq : coordinateEquiv d (e x) = A.chart q x := by
+    change coordinateEquiv d ((coordinateEquiv d).symm (A.chart q x)) = _
+    exact (coordinateEquiv d).apply_symm_apply _
+  rw [heq]
+  rfl
+
+def completedHessian (k : ℕ) (p : ι) (i j : Fin d) :
+    D.localizers.Sobolev ((k : ℝ)+2) →L[ℝ] D.localizers.Sobolev (k : ℝ) :=
+  D.localizers.extendCore ((k : ℝ)+2) (k : ℝ) (D.localizedHessian p i j)
+
+lemma completedHessian_embed (k : ℕ) (p : ι) (i j : Fin d) (f : Smooth A) :
+    D.completedHessian k p i j (D.localizers.embed ((k : ℝ)+2) f) =
+      D.localizers.embed (k : ℝ) (D.localizedHessian p i j f) :=
+  D.localizers.extendCore_embed (D.localizedHessian_bound k p i j) f
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/GlobalMetricEntries.lean b/lean/OAI/Geometry/PrescribedPotential/GlobalMetricEntries.lean
new file mode 100644
index 0000000000000000000000000000000000000000..2f866aff2ce29ad6b4a916192db2c47cf3eb0f49
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/GlobalMetricEntries.lean
@@ -0,0 +1,107 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.GlobalTransport
+import OAI.Geometry.PrescribedPotential.PatchCutoffs
+
+namespace OAI
+
+/-! Global Metric Entries. -/
+
+section
+
+ 
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff SchwartzMap Classical ComplexOrder Matrix.Norms.Elementwise
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+
+def chartFunction (q : Fin A.count) (κ : ChartCutoff (A.euclideanChart q).target)
+    (f : EC d → ℂ) (hf : ContDiffOn ℝ ∞ f (A.euclideanChart q).target) : Smooth A :=
+  cutoffGlobal q (κ.mulOn (A.euclideanChart q).open_target hf)
+
+lemma chartFunction_apply (q : Fin A.count) (κ : ChartCutoff (A.euclideanChart q).target)
+    (f : EC d → ℂ) (hf : ContDiffOn ℝ ∞ f (A.euclideanChart q).target)
+    {x : X} (hx : x ∈ (A.euclideanChart q).source) :
+    chartFunction q κ f hf x = κ (A.euclideanChart q x) * f (A.euclideanChart q x) := by
+  change (if x ∈ (A.euclideanChart q).source then _ else 0) = _
+  rw [ite_eq_left hx]
+  rfl
+
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+omit [T2Space X] in
+lemma metric_euclidean_smooth (g : KaehlerMetric A) (q : Fin A.count) :
+    ContDiffOn ℝ ∞ (fun y : EC d => g.matrix q (coordinateEquiv d y))
+      (A.euclideanChart q).target := by
+  rw [ComplexAtlas.euclideanChart_target]
+  exact (g.smooth q).comp (coordinateEquiv d).contDiff.contDiffOn (fun _ h => h)
+
+omit [T2Space X] in
+lemma inverseDet_euclidean_smooth (g : KaehlerMetric A) (q : Fin A.count) :
+    ContDiffOn ℝ ∞ (fun y : EC d => (g.matrix q (coordinateEquiv d y)).det⁻¹)
+      (A.euclideanChart q).target := by
+  have hd : ContDiffOn ℝ ∞ (fun y : EC d => (g.matrix q (coordinateEquiv d y)).det)
+      (A.euclideanChart q).target := by
+    simp only [Matrix.det_apply']
+    apply ContDiffOn.sum
+    intro σ _
+    apply contDiffOn_const.mul
+    apply contDiffOn_prod
+    intro j _
+    exact contDiffOn_pi.mp (contDiffOn_pi.mp (metric_euclidean_smooth g q) (σ j)) j
+  apply hd.inv
+  intro y hy
+  exact ne_of_gt (g.positive q (coordinateEquiv d y) (by simpa using hy)).det_pos
+
+def metricEntry (p : ι) (i j : Fin d) : Smooth A :=
+  chartFunction (D.patch p).index (D.cutoff p)
+    (fun y => g.matrix (D.patch p).index (coordinateEquiv d y) i j)
+    (contDiffOn_pi.mp (contDiffOn_pi.mp (metric_euclidean_smooth g (D.patch p).index) i) j)
+
+def inverseDet (p : ι) : Smooth A :=
+  chartFunction (D.patch p).index (D.cutoff p)
+    (fun y => (g.matrix (D.patch p).index (coordinateEquiv d y)).det⁻¹)
+    (inverseDet_euclidean_smooth g (D.patch p).index)
+
+def determinantWeight (p : ι) : Smooth A := (D.localizers.weight p).mul (D.inverseDet p)
+
+omit [T2Space X] in
+lemma patch_source (p : ι) {x : X} (hx : x ∈ tsupport (D.localizers.weight p : X → ℂ)) :
+    x ∈ (A.euclideanChart (D.patch p).index).source := by
+  have hh := D.localizers.support_sub p hx
+  rwa [D.index_eq] at hh
+
+lemma metricEntry_apply (p : ι) (i j : Fin d) {x : X}
+    (hx : x ∈ tsupport (D.localizers.weight p : X → ℂ)) :
+    D.metricEntry p i j x = g.matrix (D.patch p).index (A.chart (D.patch p).index x) i j := by
+  rw [metricEntry, chartFunction_apply _ _ _ _ (D.patch_source p hx), D.cutoff_one p x hx, one_mul,
+    ComplexAtlas.euclideanChart_apply, ContinuousLinearEquiv.apply_symm_apply]
+
+lemma inverseDet_apply (p : ι) {x : X}
+    (hx : x ∈ tsupport (D.localizers.weight p : X → ℂ)) :
+    D.inverseDet p x = (g.matrix (D.patch p).index (A.chart (D.patch p).index x)).det⁻¹ := by
+  rw [inverseDet, chartFunction_apply _ _ _ _ (D.patch_source p hx), D.cutoff_one p x hx, one_mul,
+    ComplexAtlas.euclideanChart_apply, ContinuousLinearEquiv.apply_symm_apply]
+
+lemma determinantWeight_apply (p : ι) {x : X}
+    (hx : x ∈ tsupport (D.localizers.weight p : X → ℂ)) :
+    D.determinantWeight p x = D.localizers.weight p x *
+      (g.matrix (D.patch p).index (A.chart (D.patch p).index x)).det⁻¹ := by
+  change D.localizers.weight p x * D.inverseDet p x = _
+  rw [D.inverseDet_apply p hx]
+
+lemma determinantWeight_zero (p : ι) {x : X} (hx : D.localizers.weight p x = 0) :
+    D.determinantWeight p x = 0 := by
+  change D.localizers.weight p x * _ = 0
+  rw [hx, zero_mul]
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/GlobalNonlinearCommutator.lean b/lean/OAI/Geometry/PrescribedPotential/GlobalNonlinearCommutator.lean
new file mode 100644
index 0000000000000000000000000000000000000000..b3ebefc85cdd763e3a1a65ef49af0bb2f7234268
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/GlobalNonlinearCommutator.lean
@@ -0,0 +1,63 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CommutatorSupport
+import OAI.Geometry.PrescribedPotential.NonlinearCoreCommutator
+import OAI.Geometry.PrescribedPotential.PatchCutoffs
+import OAI.Geometry.PrescribedPotential.RealSobolev
+
+namespace OAI
+
+/-! Global Nonlinear Commutator. -/
+
+section
+
+ 
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff Classical Matrix.Norms.Elementwise
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+lemma nonlinear_commutator_core_strong (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (p : ι) (v : EC d) (φ : SmoothRealFunction A) (x : X) :
+    D.localizers.strong (k : ℝ)
+      (fderiv ℝ (D.completedVolume k hk) (D.localizers.embed ((k : ℝ)+2) (Smooth.ofReal φ))
+        (D.localizers.embed ((k : ℝ)+2) (D.localizedDerivative p v (Smooth.ofReal φ)))) x =
+      D.localizedDerivative p v (Smooth.ofReal (g.potentialDensity φ)) x +
+      D.localizers.strong (k : ℝ)
+        (D.nonlinearRemainder k hk p v (D.localizers.embed ((k : ℝ)+2) (Smooth.ofReal φ))) x := by
+  by_cases hx : x ∈ D.derivativeSupport p
+  · obtain ⟨hs,hκ⟩ := D.derivativeSupport_source hx
+    exact D.nonlinear_commutator_on_support k hk p v φ hs hκ
+  · obtain ⟨q,hq⟩ := A.covers x
+    let ψ := (D.realLocalizedDerivative p v (RealSmooth.ofReal φ)).source
+    have hψ : Smooth.ofReal ψ = D.localizedDerivative p v (Smooth.ofReal φ) :=
+      RealSmooth.ofReal_source _
+    rw [← hψ,D.completedVolumeDerivative_source k hk φ ψ q hq,
+      D.localizedDerivative_source_hessian_zero hx v φ q hq,
+      map_zero,mul_zero,D.localizedDerivative_zero hx,D.nonlinearRemainder_zero hx,add_zero]
+
+lemma nonlinear_commutator_core (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (p : ι) (v : EC d) (φ : SmoothRealFunction A) :
+    fderiv ℝ (D.completedVolume k hk) (D.localizers.embed ((k : ℝ)+2) (Smooth.ofReal φ))
+        (D.localizers.embed ((k : ℝ)+2) (D.localizedDerivative p v (Smooth.ofReal φ))) =
+      D.localizers.embed (k : ℝ) (D.localizedDerivative p v (Smooth.ofReal (g.potentialDensity φ))) +
+      D.nonlinearRemainder k hk p v (D.localizers.embed ((k : ℝ)+2) (Smooth.ofReal φ)) := by
+  have hs : (Module.finrank ℝ (EC d) : ℝ) < 2*(k : ℝ) := by
+    have hh : (Module.finrank ℝ (EC d) : ℝ) < (k : ℝ) := by exact_mod_cast hk
+    linarith [Nat.cast_nonneg (α := ℝ) k]
+  apply D.localizers.strong_injective (k : ℝ) hs
+  ext x
+  rw [map_add,D.localizers.strong_embed _ hs]
+  exact D.nonlinear_commutator_core_strong k hk p v φ x
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/GlobalOperator.lean b/lean/OAI/Geometry/PrescribedPotential/GlobalOperator.lean
new file mode 100644
index 0000000000000000000000000000000000000000..88749435afdda51721f4131decaaeae2c3afd053
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/GlobalOperator.lean
@@ -0,0 +1,134 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.AnalyticSupport
+
+namespace OAI
+
+/-! Global Operator. -/
+
+section
+
+ 
+
+noncomputable section
+open Matrix Filter Topology
+open scoped ContDiff ComplexOrder MatrixOrder Matrix.Norms.Elementwise
+
+namespace MatrixSmoothGeneral
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {n : ℕ}
+
+lemma determinant {U : Set E} {H : E → Matrix (Fin n) (Fin n) ℂ}
+    (hH : ContDiffOn ℝ ∞ H U) : ContDiffOn ℝ ∞ (fun z => (H z).det) U := by
+  classical
+  simp_rw [Matrix.det_apply']
+  apply ContDiffOn.sum
+  intro σ _
+  apply contDiffOn_const.mul
+  apply contDiffOn_prod
+  intro i _
+  exact contDiffOn_pi.mp (contDiffOn_pi.mp hH (σ i)) i
+
+lemma adjugate {U : Set E} {H : E → Matrix (Fin n) (Fin n) ℂ}
+    (hH : ContDiffOn ℝ ∞ H U) : ContDiffOn ℝ ∞ (fun z => (H z).adjugate) U := by
+  classical
+  apply contDiffOn_pi.mpr
+  intro i
+  apply contDiffOn_pi.mpr
+  intro j
+  simp_rw [Matrix.adjugate_apply]
+  apply determinant
+  apply contDiffOn_pi.mpr
+  intro k
+  apply contDiffOn_pi.mpr
+  intro l
+  by_cases hk : k = j
+  · subst k
+    simp only [Matrix.updateRow_self]
+    exact contDiffOn_const
+  · simp only [Matrix.updateRow_ne hk]
+    exact contDiffOn_pi.mp (contDiffOn_pi.mp hH k) l
+
+lemma inverse {U : Set E} {H : E → Matrix (Fin n) (Fin n) ℂ}
+    (hH : ContDiffOn ℝ ∞ H U) (hh : ∀ x ∈ U, (H x).det ≠ 0) :
+    ContDiffOn ℝ ∞ (fun z => (H z)⁻¹) U := by
+  simp_rw [Matrix.inv_def, Ring.inverse_eq_inv']
+  exact ((determinant hH).inv hh).smul (adjugate hH)
+end MatrixSmoothGeneral
+
+namespace Anticanonical.SourceSmooth.KaehlerMetric
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+
+lemma transition_det_isUnit (g : KaehlerMetric A) (i j : Fin A.count) {x : X}
+    (hi : x ∈ (A.chart i).source) (hj : x ∈ (A.chart j).source) :
+    IsUnit (A.derivativeMatrix i j x).det := by
+  apply isUnit_iff_ne_zero.mpr
+  intro h
+  have he := congrArg Matrix.det (g.compatibility i j x hi hj)
+  rw [Matrix.det_mul, h, mul_zero] at he
+  exact ne_of_gt (g.positive i (A.chart i x) ((A.chart i).mapsTo hi)).det_pos he
+
+lemma linearizedMongeAmpere_compatibility (g : KaehlerMetric A) (φ : SmoothRealFunction A)
+    (i j : Fin A.count) {x : X}
+    (hi : x ∈ (A.chart i).source) (hj : x ∈ (A.chart j).source) :
+    g.linearizedMongeAmpere φ i (A.chart i x) =
+      g.linearizedMongeAmpere φ j (A.chart j x) := by
+  let D := A.derivativeMatrix i j x
+  have hD : IsUnit D.det := g.transition_det_isUnit i j hi hj
+  have hs : IsUnit Dᴴ.det := by rw [det_conjTranspose]; exact hD.star
+  unfold linearizedMongeAmpere
+  rw [g.compatibility i j x hi hj, φ.hessian_compatibility i j hi hj]
+  change (((Dᴴ * g.matrix j (A.chart j x) * D)⁻¹ *
+    (Dᴴ * φ.hessian j (A.chart j x) * D)).trace).re = _
+  have he : (Dᴴ * g.matrix j (A.chart j x) * D)⁻¹ *
+      (Dᴴ * φ.hessian j (A.chart j x) * D) =
+      D⁻¹ * (g.matrix j (A.chart j x))⁻¹ * φ.hessian j (A.chart j x) * D := by
+    rw [Matrix.mul_inv_rev, Matrix.mul_inv_rev]
+    simp only [mul_assoc, nonsing_inv_mul_cancel_left Dᴴ _ hs]
+  rw [he, Matrix.mul_assoc D⁻¹ _ _, trace_mul_cycle, mul_nonsing_inv D hD, one_mul]
+
+ 
+
+def laplacianValue (g : KaehlerMetric A) (φ : SmoothRealFunction A) (x : X) : ℝ :=
+  g.linearizedMongeAmpere φ (A.covers x).choose (A.chart (A.covers x).choose x)
+
+lemma laplacianValue_local (g : KaehlerMetric A) (φ : SmoothRealFunction A)
+    (i : Fin A.count) {x : X} (hx : x ∈ (A.chart i).source) :
+    g.laplacianValue φ x = g.linearizedMongeAmpere φ i (A.chart i x) :=
+  g.linearizedMongeAmpere_compatibility φ _ i (A.covers x).choose_spec hx
+
+lemma linearizedMongeAmpere_smooth (g : KaehlerMetric A) (φ : SmoothRealFunction A)
+    (i : Fin A.count) : ContDiffOn ℝ ∞ (g.linearizedMongeAmpere φ i) (A.chart i).target := by
+  apply Complex.reCLM.contDiff.comp_contDiffOn
+  simp only [Matrix.trace, Matrix.diag_apply]
+  apply ContDiffOn.sum
+  intro j _
+  simp only [Matrix.mul_apply]
+  apply ContDiffOn.sum
+  intro k _
+  apply ContDiffOn.mul
+  · exact contDiffOn_pi.mp (contDiffOn_pi.mp
+      (MatrixSmoothGeneral.inverse (g.smooth i)
+        (fun z hz => ne_of_gt (g.positive i z hz).det_pos)) j) k
+  · exact contDiffOn_pi.mp (contDiffOn_pi.mp (φ.hessian_smooth i) k) j
+
+ 
+
+def laplacian (g : KaehlerMetric A) (φ : SmoothRealFunction A) : SmoothRealFunction A where
+  value := g.laplacianValue φ
+  smooth i := by
+    apply (g.linearizedMongeAmpere_smooth φ i).congr
+    intro z hz
+    dsimp only [Function.comp_apply]
+    rw [g.laplacianValue_local φ i ((A.chart i).mapsTo_symm hz), (A.chart i).right_inv hz]
+
+lemma laplacian_localExpression (g : KaehlerMetric A) (φ : SmoothRealFunction A)
+    (i : Fin A.count) {z : Coordinates d} (hz : z ∈ (A.chart i).target) :
+    (g.laplacian φ).localExpression i z = g.linearizedMongeAmpere φ i z := by
+  change g.laplacianValue φ ((A.chart i).symm z) = _
+  rw [g.laplacianValue_local φ i ((A.chart i).mapsTo_symm hz), (A.chart i).right_inv hz]
+
+end Anticanonical.SourceSmooth.KaehlerMetric
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/GlobalParametrix.lean b/lean/OAI/Geometry/PrescribedPotential/GlobalParametrix.lean
new file mode 100644
index 0000000000000000000000000000000000000000..0881e55903332739a5899e0f39c070d351d97d80
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/GlobalParametrix.lean
@@ -0,0 +1,145 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.GlobalTransport
+import OAI.Geometry.PrescribedPotential.IntegerBounds
+import OAI.Geometry.PrescribedPotential.PatchCutoffs
+
+namespace OAI
+
+/-! Global Parametrix. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory
+open scoped SchwartzMap ContDiff Classical ComplexOrder MatrixOrder
+
+namespace Anticanonical.SourceSmooth.ParametrixPatch
+open EllipticKernel SobolevChart FrozenPoisson ParameterRegularity
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+  {g : KaehlerMetric A}
+
+abbrev matrix (p : ParametrixPatch g) := g.matrix p.index (coordinateEquiv d p.center)
+lemma positive (p : ParametrixPatch g) : p.matrix.PosDef :=
+  g.positive p.index _ (by simpa using p.mem_target)
+
+ 
+
+def localInverse (p : ParametrixPatch g) (m : ℝ) (hm : 1 ≤ m) :
+    𝓢(EC d, ℂ) →ₗ[ℝ] 𝓢(EC d, ℂ) :=
+  (schwartzLocal p.matrix p.positive m hm (stdOrthonormalBasis ℝ (EC d))
+    p.coefficient p.small).restrictScalars ℝ
+
+lemma localInverse_bound (p : ParametrixPatch g) (m : ℝ) (hm : 1 ≤ m) (k : ℕ) :
+    CoreBound (k : ℝ) ((k : ℝ) + 2) (p.localInverse m hm) :=
+  schwartzLocal_integer_bound p.matrix p.positive m hm (stdOrthonormalBasis ℝ (EC d))
+    p.coefficient p.small k
+end Anticanonical.SourceSmooth.ParametrixPatch
+
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+
+theorem globalize_integer_bound (D : Localizers A ι)
+    (i : Fin A.count) (κ : ChartCutoff (A.euclideanChart i).target) (k : ℕ) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ f : 𝓢(EC d, ℂ),
+      ‖D.embed (k : ℝ) (globalize i κ f)‖ ≤ C * ‖schwartzCoord (k : ℝ) f‖ := by
+  have hB (j : ι) : CoreBound (k : ℝ) (k : ℝ)
+      (fun f => localize A (D.index j) (D.weight j) (D.support_sub j) (globalize i κ f)) := by
+    simp_rw [localize_globalize]
+    exact chartTransport_integer_bound _ (A.euclidean_transition_smooth (D.index j) i)
+      (A.euclidean_transition_smooth i (D.index j)) _ k
+  choose C hC h using hB
+  refine ⟨∑ j, C j, Finset.sum_nonneg (fun j _ => hC j), fun f => ?_⟩
+  change ‖D.coordinates (k : ℝ) (globalize i κ f)‖ ≤ _
+  apply (pi_norm_le_iff_of_nonneg (mul_nonneg (Finset.sum_nonneg (fun j _ => hC j)) (norm_nonneg _))).mpr
+  intro j
+  exact (h j f).trans (mul_le_mul_of_nonneg_right
+    (Finset.single_le_sum (fun j _ => hC j) (Finset.mem_univ j)) (norm_nonneg _))
+
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+def forcing (p : ι) : Smooth A →ₗ[ℝ] 𝓢(EC d, ℂ) :=
+  localizeLinear A (D.localizers.index p) (D.localizers.weight p) (D.localizers.support_sub p)
+
+lemma forcing_bound (p : ι) (s : ℝ) (f : Smooth A) :
+    ‖schwartzCoord s (D.forcing p f)‖ ≤ ‖D.localizers.embed s f‖ :=
+  norm_le_pi_norm (D.localizers.coordinates s f) p
+
+ 
+def parametrix (m : ℝ) (hm : 1 ≤ m) : Smooth A →ₗ[ℝ] Smooth A :=
+  ∑ p, (globalize (D.patch p).index (D.cutoff p)).comp
+    (((D.patch p).localInverse m hm).comp (D.forcing p))
+
+lemma parametrix_apply (m : ℝ) (hm : 1 ≤ m) (f : Smooth A) :
+    D.parametrix m hm f = ∑ p, globalize (D.patch p).index (D.cutoff p)
+      ((D.patch p).localInverse m hm (D.forcing p f)) := by
+  simp only [parametrix, LinearMap.sum_apply, LinearMap.comp_apply]
+
+lemma parametrix_piece_bound (m : ℝ) (hm : 1 ≤ m) (k : ℕ) (p : ι) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ f : Smooth A,
+      ‖D.localizers.embed ((k : ℝ) + 2) (globalize (D.patch p).index (D.cutoff p)
+        ((D.patch p).localInverse m hm (D.forcing p f)))‖ ≤
+        C * ‖D.localizers.embed (k : ℝ) f‖ := by
+  obtain ⟨B, hB, hb⟩ := globalize_integer_bound D.localizers (D.patch p).index (D.cutoff p) (k + 2)
+  obtain ⟨C, hC, hc⟩ := (D.patch p).localInverse_bound m hm k
+  refine ⟨B * C, mul_nonneg hB hC, fun f => ?_⟩
+  have hh := hb ((D.patch p).localInverse m hm (D.forcing p f))
+  rw [Nat.cast_add, Nat.cast_ofNat] at hh
+  apply hh.trans
+  have hr := (hc (D.forcing p f)).trans
+    (mul_le_mul_of_nonneg_left (D.forcing_bound p (k : ℝ) f) hC)
+  simpa only [mul_assoc] using mul_le_mul_of_nonneg_left hr hB
+
+ 
+lemma parametrix_bound (m : ℝ) (hm : 1 ≤ m) (k : ℕ) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ f : Smooth A,
+      ‖D.localizers.embed ((k : ℝ) + 2) (D.parametrix m hm f)‖ ≤
+        C * ‖D.localizers.embed (k : ℝ) f‖ := by
+  choose C hC hc using D.parametrix_piece_bound m hm k
+  refine ⟨∑ p, C p, Finset.sum_nonneg (fun p _ => hC p), fun f => ?_⟩
+  rw [parametrix_apply, map_sum, Finset.sum_mul]
+  apply (norm_sum_le _ _).trans
+  exact Finset.sum_le_sum (fun p _ => hc p f)
+
+ 
+
+lemma forcing_reconstruction (f : Smooth A) :
+    (∑ p, globalize (D.patch p).index (D.cutoff p) (D.forcing p f)) = f := by
+  apply Smooth.ext
+  intro x
+  have he (p : ι) : globalize (D.patch p).index (D.cutoff p) (D.forcing p f) x =
+      D.localizers.weight p x * f x := by
+    by_cases hx : x ∈ tsupport (D.localizers.weight p : X → ℂ)
+    · have hp := (D.support_patch p hx).1
+      rw [globalize_apply, ite_eq_left hp, D.cutoff_one p x hx, one_mul]
+      change localizeFun A (D.localizers.index p) (D.localizers.weight p) f
+        (A.euclideanChart (D.patch p).index x) = _
+      rw [D.index_eq, localizeFun_apply A _ _ ((A.euclideanChart (D.patch p).index).mapsTo hp),
+        (A.euclideanChart (D.patch p).index).left_inv hp]
+    · have hz := image_eq_zero_of_notMem_tsupport hx
+      rw [hz, zero_mul, globalize_apply]
+      split_ifs with hp
+      · change D.cutoff p _ * (localizeFun A (D.localizers.index p) (D.localizers.weight p) f _) = 0
+        rw [D.index_eq, localizeFun_apply A _ _ ((A.euclideanChart (D.patch p).index).mapsTo hp),
+          (A.euclideanChart (D.patch p).index).left_inv hp, hz, zero_mul, mul_zero]
+      · rfl
+  change (∑ p, (globalize (D.patch p).index (D.cutoff p) (D.forcing p f) : Smooth A)) x = _
+  have hev (S : Finset ι) (F : ι → Smooth A) : (∑ p ∈ S, F p) x = ∑ p ∈ S, F p x := by
+    induction S using Finset.induction_on with
+    | empty => simp
+    | @insert p S hp ih => simp only [Finset.sum_insert hp, Smooth.add_apply, ih]
+  rw [hev]
+  simp_rw [he]
+  rw [← Finset.sum_mul, D.localizers.sum_one, one_mul]
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/GlobalParametrixEquation.lean b/lean/OAI/Geometry/PrescribedPotential/GlobalParametrixEquation.lean
new file mode 100644
index 0000000000000000000000000000000000000000..b6c699fbbe8e63d022467e524fc92ff4c550df4d
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/GlobalParametrixEquation.lean
@@ -0,0 +1,140 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.GlobalParametrix
+import OAI.Geometry.PrescribedPotential.PatchDifferential
+
+namespace OAI
+
+/-! Global Parametrix Equation. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory TemperedDistribution LineDeriv
+open scoped SchwartzMap ContDiff Classical ComplexOrder MatrixOrder
+
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart MetricLocalization
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X]
+  {A : ComplexAtlas d X}
+
+lemma globalize_local (i : Fin A.count)
+    (κ : ChartCutoff (A.euclideanChart i).target) (u : 𝓢(EC d, ℂ))
+    {y : EC d} (hy : y ∈ (A.euclideanChart i).target) :
+    (globalize i κ u ∘ (A.euclideanChart i).symm) =ᶠ[𝓝 y]
+      (SchwartzMap.smulLeftCLM ℂ κ.val u : EC d → ℂ) := by
+  filter_upwards [(A.euclideanChart i).open_target.mem_nhds hy] with z hz
+  have hs : (A.euclideanChart i).symm z ∈ (A.euclideanChart i).source :=
+    (A.euclideanChart i).map_target hz
+  simp only [Function.comp_apply, globalize_apply, ite_eq_left hs,
+    (A.euclideanChart i).right_inv hz,
+    SchwartzMap.smulLeftCLM_apply_apply κ.val.hasTemperateGrowth, smul_eq_mul]
+
+lemma globalize_absorb (i : Fin A.count)
+    (κ η : ChartCutoff (A.euclideanChart i).target)
+    (hη : ∀ y ∈ tsupport (κ : EC d → ℂ), η y = 1) (u : 𝓢(EC d, ℂ)) :
+    globalize i η (SchwartzMap.smulLeftCLM ℂ κ.val u) = globalize i κ u := by
+  apply Smooth.ext
+  intro x
+  simp only [globalize_apply, SchwartzMap.smulLeftCLM_apply_apply κ.val.hasTemperateGrowth,
+    smul_eq_mul]
+  split_ifs with hx
+  · by_cases hy : A.euclideanChart i x ∈ tsupport (κ : EC d → ℂ)
+    · rw [hη _ hy, one_mul]
+    · rw [image_eq_zero_of_notMem_tsupport hy, zero_mul, mul_zero]
+  · rfl
+
+lemma complexL_globalize {g : KaehlerMetric A} (p : ParametrixPatch g)
+    (κ η : ChartCutoff (A.euclideanChart p.index).target)
+    (hκ : tsupport (κ : EC d → ℂ) ⊆ p.domain)
+    (hη : ∀ y ∈ tsupport (κ : EC d → ℂ), η y = 1) (u : 𝓢(EC d, ℂ)) :
+    complexL g (globalize p.index κ u) =
+      globalize p.index η (p.differential (SchwartzMap.smulLeftCLM ℂ κ.val u)) := by
+  apply Smooth.ext
+  intro x
+  let e := A.euclideanChart p.index
+  have hDs : tsupport (p.differential (SchwartzMap.smulLeftCLM ℂ κ.val u) : EC d → ℂ) ⊆
+      tsupport (κ : EC d → ℂ) :=
+    (schwartzDifferential_tsupport _ _).trans
+      ((SchwartzMap.tsupport_smulLeftCLM_subset _ _).trans inter_subset_right)
+  by_cases hx : x ∈ e.symm '' tsupport (κ : EC d → ℂ)
+  · obtain ⟨y, hy, rfl⟩ := hx
+    have hys := κ.support_sub hy
+    have hsrc : e.symm y ∈ (A.euclideanChart p.index).source := e.map_target hys
+    change complexLValue g _ _ = _
+    rw [complexLValue_local g _ p.index hsrc]
+    change localL g p.index _ (e (e.symm y)) = _
+    rw [e.right_inv hys,
+      localL_congr g p.index (globalize_local p.index κ u hys),
+      ← p.differential_local _ (hκ hy), globalize_apply, ite_eq_left hsrc, e.right_inv hys,
+      hη y hy, one_mul]
+  · have hv : complexL g (globalize p.index κ u) x = 0 :=
+      image_eq_zero_of_notMem_tsupport (fun ht => hx
+        (globalize_tsupport p.index κ u (complexL_tsupport g _ ht)))
+    rw [hv, globalize_apply]
+    split_ifs with hi
+    · have hn : e x ∉ tsupport (κ : EC d → ℂ) := fun hy => hx ⟨e x, hy, e.left_inv hi⟩
+      rw [image_eq_zero_of_notMem_tsupport (fun ht => hn (hDs ht)), mul_zero]
+    · rfl
+
+omit [T2Space X] in
+lemma local_shift_cutoff_equation {g : KaehlerMetric A} (p : ParametrixPatch g)
+    (m : ℝ) (hm : 1 ≤ m) (κ f : 𝓢(EC d, ℂ)) :
+    (m^2 : ℝ) • SchwartzMap.smulLeftCLM ℂ κ (p.localInverse m hm f) -
+      p.differential (SchwartzMap.smulLeftCLM ℂ κ (p.localInverse m hm f)) =
+      SchwartzMap.smulLeftCLM ℂ κ f - p.cutoffError κ (p.localInverse m hm f) := by
+  have he := congrArg (SchwartzMap.smulLeftCLM ℂ κ) (p.localInverse_equation m hm f)
+  simp only [map_sub, map_smul] at he
+  rw [← he]
+  simp only [ParametrixPatch.cutoffError, schwartzCutoffError,
+    _root_.sub_apply, ContinuousLinearMap.comp_apply]
+  rw [← Complex.ofReal_pow, Complex.coe_smul]
+  simp only [ParametrixPatch.differential]
+  abel
+
+lemma global_shift_piece_equation {g : KaehlerMetric A} (p : ParametrixPatch g)
+    (κ η : ChartCutoff (A.euclideanChart p.index).target)
+    (hκ : tsupport (κ : EC d → ℂ) ⊆ p.domain)
+    (hη : ∀ y ∈ tsupport (κ : EC d → ℂ), η y = 1)
+    (m : ℝ) (hm : 1 ≤ m) (f : 𝓢(EC d, ℂ)) :
+    (m^2 : ℝ) • globalize p.index κ (p.localInverse m hm f) -
+      complexL g (globalize p.index κ (p.localInverse m hm f)) =
+      globalize p.index κ f - globalize p.index η (p.cutoffError κ.val (p.localInverse m hm f)) := by
+  have hh := congrArg (globalize p.index η) (local_shift_cutoff_equation p m hm κ.val f)
+  simp only [map_sub, map_smul, globalize_absorb p.index κ η hη] at hh
+  rw [complexL_globalize p κ η hκ hη]
+  exact hh
+
+variable [CompactSpace X] {ι : Type*} [Fintype ι]
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+def error (m : ℝ) (hm : 1 ≤ m) : Smooth A →ₗ[ℝ] Smooth A :=
+  ∑ p, (globalize (D.patch p).index (D.outerCutoff p)).comp
+    ((((D.patch p).cutoffError (D.cutoff p).val).restrictScalars ℝ).toLinearMap.comp
+      (((D.patch p).localInverse m hm).comp (D.forcing p)))
+
+lemma error_apply (m : ℝ) (hm : 1 ≤ m) (f : Smooth A) :
+    D.error m hm f = ∑ p, globalize (D.patch p).index (D.outerCutoff p)
+      ((D.patch p).cutoffError (D.cutoff p).val ((D.patch p).localInverse m hm (D.forcing p f))) := by
+  simp only [error, LinearMap.sum_apply, LinearMap.comp_apply,
+    ContinuousLinearMap.coe_coe]
+  rfl
+
+ 
+theorem parametrix_equation (m : ℝ) (hm : 1 ≤ m) (f : Smooth A) :
+    (m^2 : ℝ) • D.parametrix m hm f - complexL g (D.parametrix m hm f) =
+      f - D.error m hm f := by
+  rw [parametrix_apply, map_sum, Finset.smul_sum, ← Finset.sum_sub_distrib]
+  simp_rw [global_shift_piece_equation (D.patch _) (D.cutoff _) (D.outerCutoff _)
+    (D.cutoff_support _) (D.outerCutoff_one _) m hm]
+  rw [Finset.sum_sub_distrib, forcing_reconstruction, error_apply]
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/GlobalPartition.lean b/lean/OAI/Geometry/PrescribedPotential/GlobalPartition.lean
new file mode 100644
index 0000000000000000000000000000000000000000..41ae08308201329e76a140e7526751fe6a0714cb
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/GlobalPartition.lean
@@ -0,0 +1,154 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.GlobalSobolev
+import OAI.Geometry.PrescribedPotential.ParametrixAtlas
+
+namespace OAI
+
+/-! Global Partition. -/
+
+section
+
+noncomputable section
+open Set Filter Topology MeasureTheory Manifold IsManifold
+open scoped ContDiff SchwartzMap Classical
+
+namespace GlobalElliptic
+open Anticanonical EllipticKernel SobolevChart SourceSmooth
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+
+ 
+def Smooth.ofReal (f : SmoothRealFunction A) : Smooth A :=
+  ⟨fun x => (f.value x : ℂ), fun i => by
+    apply Complex.ofRealCLM.contDiff.comp_contDiffOn
+    exact (f.smooth i).comp (coordinateEquiv d).contDiff.contDiffOn
+      (fun y hy => by simpa using hy)⟩
+
+@[simp] lemma Smooth.ofReal_apply (f : SmoothRealFunction A) (x : X) :
+    Smooth.ofReal f x = (f.value x : ℂ) := rfl
+
+lemma Smooth.ofReal_tsupport (f : SmoothRealFunction A) :
+    tsupport (Smooth.ofReal f : X → ℂ) = tsupport f.value := by
+  apply congrArg closure
+  ext x
+  simp [Function.mem_support]
+
+ 
+
+structure Localizers (A : ComplexAtlas d X) (ι : Type*) [Fintype ι] where
+  index : ι → Fin A.count
+  weight : ι → Smooth A
+  support_sub : ∀ i, tsupport (weight i : X → ℂ) ⊆ (A.euclideanChart (index i)).source
+  sum_one : ∀ x, ∑ i, weight i x = 1
+
+namespace Localizers
+variable {ι : Type*} [Fintype ι] (D : Localizers A ι)
+
+ 
+def coordinates [T2Space X] [CompactSpace X] (s : ℝ) :
+    Smooth A →ₗ[ℝ] (ι → L2 (EC d)) where
+  toFun f i := schwartzCoord s (localize A (D.index i) (D.weight i) (D.support_sub i) f)
+  map_add' f h := by
+    funext i
+    change schwartzCoord s (localizeLinear A (D.index i) (D.weight i) (D.support_sub i) (f + h)) = _
+    rw [map_add, schwartzCoord_add]
+    rfl
+  map_smul' c f := by
+    funext i
+    apply realize_injective s
+    change realize s (schwartzCoord s (localizeLinear A (D.index i) (D.weight i) (D.support_sub i) (c • f))) =
+      realize s (c • schwartzCoord s (localizeLinear A (D.index i) (D.weight i) (D.support_sub i) f))
+    rw [map_smul, realize_schwartzCoord]
+    change SchwartzMap.toTemperedDistributionCLM _ _ volume (c • localizeLinear A (D.index i) (D.weight i) (D.support_sub i) f) = _
+    rw [ContinuousLinearMap.map_smul_of_tower, ContinuousLinearMap.map_smul_of_tower, realize_schwartzCoord]
+
+lemma coordinates_injective [T2Space X] [CompactSpace X] (s : ℝ) :
+    Function.Injective (D.coordinates s) := by
+  intro f h heq
+  apply Smooth.ext
+  intro x
+  have hi (i : ι) : D.weight i x * f x = D.weight i x * h x := by
+    have ht := congrArg (fun v : ι → L2 (EC d) => realize s (v i)) heq
+    dsimp only [coordinates, LinearMap.coe_mk, AddHom.coe_mk] at ht
+    rw [realize_schwartzCoord, realize_schwartzCoord] at ht
+    have he := schwartz_distribution_injective ht
+    by_cases hx : x ∈ (A.euclideanChart (D.index i)).source
+    · have htx := (A.euclideanChart (D.index i)).mapsTo hx
+      have hh := congrArg (fun u : 𝓢(EC d, ℂ) => u (A.euclideanChart (D.index i) x)) he
+      simpa only [localize_apply, localizeFun_apply A _ _ htx,
+        (A.euclideanChart (D.index i)).left_inv hx] using hh
+    · have hz : D.weight i x = 0 := image_eq_zero_of_notMem_tsupport
+        (fun ht => hx (D.support_sub i ht))
+      simp [hz]
+  have hsum := Finset.sum_congr (s₁ := Finset.univ) rfl (fun i _ => hi i)
+  simpa only [← Finset.sum_mul, D.sum_one, one_mul] using hsum
+
+ 
+
+def closedSpace [T2Space X] [CompactSpace X] (s : ℝ) :
+    Submodule ℝ (ι → L2 (EC d)) := (LinearMap.range (D.coordinates s)).topologicalClosure
+
+abbrev Sobolev [T2Space X] [CompactSpace X] (s : ℝ) := ↥(D.closedSpace s)
+
+instance [T2Space X] [CompactSpace X] (s : ℝ) : CompleteSpace (D.Sobolev s) :=
+  (LinearMap.range (D.coordinates s)).isClosed_topologicalClosure.completeSpace_coe
+
+ 
+def embed [T2Space X] [CompactSpace X] (s : ℝ) : Smooth A →ₗ[ℝ] D.Sobolev s :=
+  (D.coordinates s).codRestrict (D.closedSpace s)
+    (fun f => Submodule.le_topologicalClosure _ (LinearMap.mem_range_self _ f))
+
+lemma embed_injective [T2Space X] [CompactSpace X] (s : ℝ) :
+    Function.Injective (D.embed s) := by
+  intro f h hh
+  exact D.coordinates_injective s (congrArg Subtype.val hh)
+
+lemma embed_dense [T2Space X] [CompactSpace X] (s : ℝ) : DenseRange (D.embed s) := by
+  change Dense (Set.range (D.embed s))
+  rw [Subtype.dense_iff, ← Set.range_comp]
+  change closure (Set.range (D.coordinates s)) ⊆ closure (Set.range (D.coordinates s))
+  exact subset_rfl
+
+end Localizers
+
+ 
+
+theorem exists_patch_localizers [T2Space X] [CompactSpace X] (g : KaehlerMetric A) :
+    ∃ (S : Finset (ParametrixPatch g)) (D : Localizers A S),
+      (∀ p : S, D.index p = p.val.index) ∧
+      (∀ p : S, tsupport (D.weight p : X → ℂ) ⊆ p.val.source) := by
+  obtain ⟨S, hS⟩ := ParametrixPatch.finite_cover g
+  let := A.chartedSpace
+  let := A.isManifold
+  obtain ⟨ρ, hρ⟩ := SmoothPartitionOfUnity.exists_isSubordinate
+    𝓘(ℝ, Coordinates d) (s := (Set.univ : Set X)) isClosed_univ
+    (fun p : S => p.val.source) (fun p => p.val.isOpen_source) (by
+      intro x _
+      obtain ⟨p, hp, hx⟩ := hS x
+      exact mem_iUnion.mpr ⟨⟨p, hp⟩, hx⟩)
+  let f (p : S) : SmoothRealFunction A :=
+    ⟨ρ p, (A.contMDiff_iff (ρ p)).mp (ρ p).contMDiff⟩
+  let D : Localizers A S :=
+    { index := fun p => p.val.index
+      weight := fun p => Smooth.ofReal (f p)
+      support_sub := by
+        intro p
+        rw [Smooth.ofReal_tsupport]
+        exact (hρ p).trans (fun x hx => hx.1)
+      sum_one := by
+        intro x
+        change (∑ p : S, ((ρ p x : ℝ) : ℂ)) = 1
+        rw [← Complex.ofReal_sum]
+        exact_mod_cast (show ∑ p : S, ρ p x = 1 by
+          simpa only [finsum_eq_sum_of_fintype] using ρ.sum_eq_one (mem_univ x)) }
+  refine ⟨S, D, fun _ => rfl, ?_⟩
+  intro p
+  change tsupport (Smooth.ofReal (f p) : X → ℂ) ⊆ _
+  rw [Smooth.ofReal_tsupport]
+  exact hρ p
+
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/GlobalProduct.lean b/lean/OAI/Geometry/PrescribedPotential/GlobalProduct.lean
new file mode 100644
index 0000000000000000000000000000000000000000..d4014bda2d41e845d7bdce71ce3446337de311f6
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/GlobalProduct.lean
@@ -0,0 +1,115 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.GlobalTransport
+import OAI.Geometry.PrescribedPotential.PatchCutoffs
+import OAI.Geometry.PrescribedPotential.SobolevProduct
+
+namespace OAI
+
+/-! Global Product. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory
+open scoped SchwartzMap ContDiff Classical
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+lemma localization_integer_bound (i : Fin A.count) (ρ : Smooth A)
+    (hρ : tsupport (ρ : X → ℂ) ⊆ (A.euclideanChart i).source) (k : ℕ) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ f : Smooth A,
+      ‖schwartzCoord (k : ℝ) (localize A i ρ hρ f)‖ ≤ C*‖D.localizers.embed (k : ℝ) f‖ := by
+  have hb (p : ι) : CoreBound (k : ℝ) (k : ℝ) (fun v =>
+      localize A i ρ hρ (globalize (D.patch p).index (D.cutoff p) v)) := by
+    simp_rw [localize_globalize]
+    exact chartTransport_integer_bound _ (A.euclidean_transition_smooth i (D.patch p).index)
+      (A.euclidean_transition_smooth (D.patch p).index i) _ k
+  choose C hC hc using hb
+  refine ⟨∑ p, C p, Finset.sum_nonneg (fun p _ => hC p), fun f => ?_⟩
+  have he : localize A i ρ hρ f = ∑ p, localize A i ρ hρ
+      (globalize (D.patch p).index (D.cutoff p) (D.forcing p f)) := by
+    change localizeLinear A i ρ hρ f = _
+    conv_lhs => rw [← D.forcing_reconstruction f]
+    rw [map_sum]
+    rfl
+  rw [he, schwartzCoord_sum, Finset.sum_mul]
+  apply (norm_sum_le _ _).trans
+  apply Finset.sum_le_sum
+  intro p _
+  exact (hc p (D.forcing p f)).trans
+    (mul_le_mul_of_nonneg_left (D.forcing_bound p (k : ℝ) f) (hC p))
+
+def outerWeight (p : ι) : Smooth A := cutoffGlobal (D.patch p).index (D.cutoff p)
+
+omit [CompactSpace X] in
+lemma outerWeight_support (p : ι) :
+    tsupport (D.outerWeight p : X → ℂ) ⊆ (A.euclideanChart (D.patch p).index).source := by
+  intro x hx
+  obtain ⟨y,hy,rfl⟩ := cutoffGlobal_tsupport (D.patch p).index (D.cutoff p) hx
+  exact (A.euclideanChart (D.patch p).index).symm.mapsTo ((D.cutoff p).support_sub hy)
+
+lemma product_localization (p : ι) (f h : Smooth A) :
+    D.forcing p (f.mul h) = SchwartzMap.smulLeftCLM ℂ (D.forcing p f)
+      (localize A (D.patch p).index (D.outerWeight p) (D.outerWeight_support p) h) := by
+  ext y
+  rw [SchwartzMap.smulLeftCLM_apply_apply (D.forcing p f).hasTemperateGrowth]
+  change localizeFun A (D.localizers.index p) (D.localizers.weight p) (f.mul h) y =
+    localizeFun A (D.localizers.index p) (D.localizers.weight p) f y *
+      localizeFun A (D.patch p).index (D.outerWeight p) h y
+  rw [D.index_eq]
+  by_cases hy : y ∈ (A.euclideanChart (D.patch p).index).target
+  · rw [localizeFun_apply A _ _ hy, localizeFun_apply A _ _ hy, localizeFun_apply A _ _ hy]
+    by_cases hx : (A.euclideanChart (D.patch p).index).symm y ∈
+        tsupport (D.localizers.weight p : X → ℂ)
+    · have hw : D.outerWeight p ((A.euclideanChart (D.patch p).index).symm y) = 1 := by
+        change (if (A.euclideanChart (D.patch p).index).symm y ∈
+          (A.euclideanChart (D.patch p).index).source then _ else 0) = 1
+        have hys : (A.euclideanChart (D.patch p).index).symm y ∈
+            (A.euclideanChart (D.patch p).index).source :=
+          (A.euclideanChart (D.patch p).index).symm.mapsTo hy
+        rw [ite_eq_left hys]
+        exact D.cutoff_one p _ hx
+      rw [hw, one_mul]
+      exact (mul_assoc _ _ _).symm
+    · rw [image_eq_zero_of_notMem_tsupport hx]
+      simp
+  · simp only [localizeFun, ite_eq_right hy, zero_mul]
+
+theorem smooth_product_bound (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ f h : Smooth A,
+      ‖D.localizers.embed (k : ℝ) (f.mul h)‖ ≤
+        C*(‖D.localizers.embed (k : ℝ) f‖*‖D.localizers.embed (k : ℝ) h‖) := by
+  obtain ⟨B,hB,hb⟩ := schwartz_sobolev_product_bound (E := EC d) k hk
+  have hh (p : ι) := D.localization_integer_bound (D.patch p).index (D.outerWeight p)
+    (D.outerWeight_support p) k
+  choose C hC hc using hh
+  refine ⟨B*∑ p, C p, mul_nonneg hB (Finset.sum_nonneg (fun p _ => hC p)), fun f h => ?_⟩
+  change ‖D.localizers.coordinates (k : ℝ) (f.mul h)‖ ≤ _
+  apply (pi_norm_le_iff_of_nonneg (mul_nonneg (mul_nonneg hB
+    (Finset.sum_nonneg (fun p _ => hC p))) (mul_nonneg (norm_nonneg _) (norm_nonneg _)))).mpr
+  intro p
+  change ‖schwartzCoord (k : ℝ) (D.forcing p (f.mul h))‖ ≤ _
+  rw [D.product_localization]
+  apply (hb (D.forcing p f) (localize A (D.patch p).index (D.outerWeight p)
+    (D.outerWeight_support p) h)).trans
+  calc
+    _ ≤ B*(‖D.localizers.embed (k : ℝ) f‖*(C p*‖D.localizers.embed (k : ℝ) h‖)) :=
+      mul_le_mul_of_nonneg_left (mul_le_mul (D.forcing_bound p (k : ℝ) f) (hc p h)
+        (norm_nonneg _) (norm_nonneg _)) hB
+    _ ≤ B*(‖D.localizers.embed (k : ℝ) f‖*((∑ q, C q)*‖D.localizers.embed (k : ℝ) h‖)) := by
+      gcongr
+      exact Finset.single_le_sum (fun q _ => hC q) (Finset.mem_univ p)
+    _ = _ := by ring
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/GlobalRellich.lean b/lean/OAI/Geometry/PrescribedPotential/GlobalRellich.lean
new file mode 100644
index 0000000000000000000000000000000000000000..2166eeb3e613737a94eb964ec533d0a76642cdf0
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/GlobalRellich.lean
@@ -0,0 +1,102 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.FrequencyApprox
+import OAI.Geometry.PrescribedPotential.GlobalSmooth
+import OAI.Geometry.PrescribedPotential.PatchCutoffs
+
+namespace OAI
+
+/-! Global Rellich. -/
+
+section
+
+ 
+noncomputable section
+open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory FourierTransform TemperedDistribution
+open scoped SchwartzMap ContDiff Classical BoundedContinuousFunction
+namespace SobolevChart
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+
+lemma realize_zero (u : L2 E) : realize 0 u = (u : 𝓢'(E, ℂ)) := by
+  simp only [realize, neg_zero, ContinuousLinearMap.comp_apply,
+    besselPotential_zero, ContinuousLinearMap.id_apply, Lp.toTemperedDistributionCLM_apply]
+
+lemma bessel_lower_two (u : L2 E) :
+    (multiplier (inverseWeight 1) u : 𝓢'(E, ℂ)) = realize 2 u := by
+  rw [multiplier_distribution _ (inverseWeight_temperate 1)]
+  simp only [realize, ContinuousLinearMap.comp_apply, Lp.toTemperedDistributionCLM_apply]
+  rw [besselPotential]
+  have heq : (inverseWeight (E := E) 1 : E → ℂ) =
+      (fun x : E => (((1+‖x‖^2)^(-(2:ℝ)/2) : ℝ) : ℂ)) := by
+    funext x
+    change (((1+‖x‖^2)^(-((1:ℕ):ℝ)) : ℝ) : ℂ) = _
+    norm_num
+  rw [heq]
+
+end SobolevChart
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+lemma cutoff_coordinate (p : ι) (u : D.localizers.Sobolev 0) :
+    multiply (D.cutoff p).val.toBoundedContinuousFunction (u.val p) = u.val p := by
+  apply l2_injective
+  simp only [Lp.toTemperedDistributionCLM_apply]
+  rw [multiply_distribution _ (D.cutoff p).val.hasTemperateGrowth]
+  have hd : D.localizers.distribution 0 p u = (u.val p : 𝓢'(EC d, ℂ)) := realize_zero _
+  rw [← hd]
+  exact D.cutoff_distribution p u
+
+lemma lower_coordinate (p : ι) (u : D.localizers.Sobolev 2) :
+    (D.localizers.lower 2 0 u).val p = multiplier (inverseWeight 1) (u.val p) := by
+  apply l2_injective
+  simp only [Lp.toTemperedDistributionCLM_apply]
+  rw [bessel_lower_two]
+  rw [← realize_zero]
+  exact D.localizers.distribution_lower (by norm_num : (0 : ℝ) ≤ 2) p u
+
+ 
+
+theorem lower_compact : IsCompactOperator (D.localizers.lower 2 0) := by
+  let T (p : ι) : D.localizers.Sobolev 2 →L[ℝ] D.localizers.Sobolev 0 :=
+    D.localizers.globalizeL2 (D.patch p).index (D.cutoff p) ∘L
+      (multiply (D.cutoff p).val.toBoundedContinuousFunction ∘L
+        multiplier (inverseWeight 1)).restrictScalars ℝ ∘L
+      (ContinuousLinearMap.proj p) ∘L (D.localizers.closedSpace 2).subtypeL
+  have hT (p : ι) : IsCompactOperator (T p) := by
+    have ht := ((EllipticCompact.localized_bessel_compact (D.cutoff p).val (D.cutoff p).compact).comp_clm
+      ((ContinuousLinearMap.proj p) ∘L (D.localizers.closedSpace 2).subtypeL)).clm_comp
+      (D.localizers.globalizeL2 (D.patch p).index (D.cutoff p))
+    simpa only [T, ContinuousLinearMap.coe_comp, ContinuousLinearMap.coe_restrictScalars',
+      Function.comp_assoc] using ht
+  have he : D.localizers.lower 2 0 = ∑ p, T p := by
+    ext1 u
+    rw [← D.reconstruct_eq (D.localizers.lower 2 0 u), D.reconstruct_apply]
+    simp only [sum_apply]
+    apply Finset.sum_congr rfl
+    intro p _
+    change D.localizers.globalizeL2 (D.patch p).index (D.cutoff p)
+      ((D.localizers.lower 2 0 u).val p) =
+      D.localizers.globalizeL2 (D.patch p).index (D.cutoff p)
+        (multiply (D.cutoff p).val.toBoundedContinuousFunction
+          (multiplier (inverseWeight 1) (u.val p)))
+    rw [← D.lower_coordinate p u, D.cutoff_coordinate]
+  rw [he]
+  classical
+  have hh (S : Finset ι) : IsCompactOperator (∑ p ∈ S, T p : D.localizers.Sobolev 2 →L[ℝ] D.localizers.Sobolev 0) := by
+    induction S using Finset.induction_on with
+    | empty => simpa using (isCompactOperator_zero : IsCompactOperator (0 : D.localizers.Sobolev 2 →L[ℝ] D.localizers.Sobolev 0))
+    | @insert p S hp ih =>
+      simpa only [Finset.sum_insert hp, FunLike.coe_add] using (hT p).add ih
+  exact hh Finset.univ
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/GlobalSmooth.lean b/lean/OAI/Geometry/PrescribedPotential/GlobalSmooth.lean
new file mode 100644
index 0000000000000000000000000000000000000000..1db71c4988f8796fbbbad3353ab87fac56eef2a5
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/GlobalSmooth.lean
@@ -0,0 +1,164 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CompletedResolvent
+import OAI.Geometry.PrescribedPotential.GlobalParametrix
+import OAI.Geometry.PrescribedPotential.PatchDifferential
+
+namespace OAI
+
+/-! Global Smooth. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory TemperedDistribution
+open scoped SchwartzMap ContDiff Classical BoundedContinuousFunction
+namespace SobolevChart
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+
+lemma schwartzCoord_order_zero (f : 𝓢(E, ℂ)) :
+    schwartzCoord 0 f = SchwartzMap.toLpCLM ℝ ℂ 2 volume f := by
+  apply l2_injective
+  simp only [Lp.toTemperedDistributionCLM_apply]
+  rw [schwartzCoord_distribution, besselPotential_zero, ContinuousLinearMap.id_apply]
+  exact (Lp.toTemperedDistribution_toLp_eq f).symm
+
+end SobolevChart
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace Localizers
+variable (D : Localizers A ι)
+
+lemma allRegular_distribution {u : D.Sobolev 0} (hu : D.AllRegular u) (p : ι) :
+    ∀ k : ℕ, MemSobolev (2 * (k : ℝ)) 2 (D.distribution 0 p u) := by
+  intro k
+  obtain ⟨v, hv⟩ := hu (2 * k)
+  have h : MemSobolev ((2 * k : ℕ) : ℝ) 2 (D.distribution 0 p u) := by
+    rw [← hv, D.distribution_lower (Nat.cast_nonneg _)]
+    exact realize_memSobolev _ _
+  simpa only [Nat.cast_mul, Nat.cast_ofNat] using h
+
+ 
+
+def globalizeL2 (i : Fin A.count) (κ : ChartCutoff (A.euclideanChart i).target) :
+    L2 (EC d) →L[ℝ] D.Sobolev 0 :=
+  ((D.embed 0).comp (globalize i κ)).extendOfNorm
+    (SchwartzMap.toLpCLM ℝ ℂ 2 volume).toLinearMap
+
+lemma globalizeL2_schwartz (i : Fin A.count) (κ : ChartCutoff (A.euclideanChart i).target)
+    (f : 𝓢(EC d, ℂ)) :
+    D.globalizeL2 i κ (schwartzCoord 0 f) = D.embed 0 (globalize i κ f) := by
+  rw [schwartzCoord_order_zero]
+  have hb := globalize_integer_bound D i κ 0
+  rw [Nat.cast_zero] at hb
+  obtain ⟨C, hC, h⟩ := hb
+  have hn : ∀ f : 𝓢(EC d, ℂ),
+      ‖D.embed 0 (globalize i κ f)‖ ≤ C * ‖SchwartzMap.toLpCLM ℝ ℂ 2 volume f‖ := by
+    intro f
+    rw [← schwartzCoord_order_zero]
+    exact h f
+  exact LinearMap.extendOfNorm_eq (SchwartzMap.denseRange_toLpCLM (by norm_num)) ⟨C, hn⟩ f
+
+end Localizers
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+lemma cutoff_forcing (p : ι) (f : Smooth A) :
+    SchwartzMap.smulLeftCLM ℂ (D.cutoff p).val (D.forcing p f) = D.forcing p f := by
+  ext y
+  rw [SchwartzMap.smulLeftCLM_apply_apply (D.cutoff p).val.hasTemperateGrowth, smul_eq_mul]
+  by_cases hy : D.forcing p f y = 0
+  · rw [hy, mul_zero]
+  · have hs := localizeFun_support A (D.localizers.index p) (D.localizers.weight p) f hy
+    obtain ⟨x, hx, he⟩ := hs
+    rw [D.index_eq] at he
+    rw [← he, D.cutoff_one p x hx, one_mul]
+
+lemma cutoff_distribution (p : ι) (u : D.localizers.Sobolev 0) :
+    smulLeftCLM ℂ (D.cutoff p).val (D.localizers.distribution 0 p u) =
+      D.localizers.distribution 0 p u := by
+  have he : smulLeftCLM ℂ (D.cutoff p).val ∘ D.localizers.distribution 0 p =
+      D.localizers.distribution 0 p := by
+    apply (D.localizers.embed_dense 0).equalizer (by fun_prop) (by fun_prop)
+    funext f
+    simp only [Function.comp_apply, D.localizers.distribution_embed]
+    change smulLeftCLM ℂ (D.cutoff p).val (SchwartzMap.toTemperedDistributionCLM (EC d) ℂ volume (D.forcing p f)) = _
+    rw [← schwartz_temperate_product_distribution (D.cutoff p).val.hasTemperateGrowth,
+      D.cutoff_forcing]
+    rfl
+  exact congr_fun he u
+
+ 
+
+lemma allRegular_coordinate {u : D.localizers.Sobolev 0} (hu : D.localizers.AllRegular u) (p : ι) :
+    ∃ f : 𝓢(EC d, ℂ), schwartzCoord 0 f = u.val p := by
+  obtain ⟨F, hF, he⟩ := all_even_smooth_representative (D.localizers.allRegular_distribution hu p)
+  let f := SchwartzMap.smulLeftCLM ℂ F (D.cutoff p).val
+  have hf : SchwartzMap.toTemperedDistributionCLM (EC d) ℂ volume f =
+      D.localizers.distribution 0 p u := by
+    rw [schwartz_mul_temperate_distribution _ hF, temperateDistribution_bounded, he,
+      D.cutoff_distribution]
+  refine ⟨f, ?_⟩
+  apply realize_injective 0
+  rw [realize_schwartzCoord, hf]
+  rfl
+
+def reconstruct : D.localizers.Sobolev 0 →L[ℝ] D.localizers.Sobolev 0 :=
+  ∑ p, (D.localizers.globalizeL2 (D.patch p).index (D.cutoff p)) ∘L
+    (ContinuousLinearMap.proj p) ∘L (D.localizers.closedSpace 0).subtypeL
+
+lemma reconstruct_apply (u : D.localizers.Sobolev 0) :
+    D.reconstruct u = ∑ p, D.localizers.globalizeL2 (D.patch p).index (D.cutoff p) (u.val p) := by
+  simp only [reconstruct, _root_.sum_apply, ContinuousLinearMap.comp_apply,
+    ContinuousLinearMap.proj_apply, Submodule.subtypeL_apply]
+
+lemma reconstruct_eq (u : D.localizers.Sobolev 0) : D.reconstruct u = u := by
+  have he : D.reconstruct = ContinuousLinearMap.id ℝ _ := by
+    apply DFunLike.coe_injective
+    apply (D.localizers.embed_dense 0).equalizer (by fun_prop) continuous_id
+    funext f
+    change D.reconstruct (D.localizers.embed 0 f) = D.localizers.embed 0 f
+    rw [D.reconstruct_apply]
+    change (∑ p, D.localizers.globalizeL2 (D.patch p).index (D.cutoff p)
+      (schwartzCoord 0 (D.forcing p f))) = D.localizers.embed 0 f
+    simp_rw [D.localizers.globalizeL2_schwartz]
+    rw [← map_sum, D.forcing_reconstruction]
+  rw [he, ContinuousLinearMap.id_apply]
+
+ 
+
+theorem allRegular_smooth {u : D.localizers.Sobolev 0} (hu : D.localizers.AllRegular u) :
+    ∃ f : Smooth A, D.localizers.embed 0 f = u := by
+  choose f hf using D.allRegular_coordinate hu
+  refine ⟨∑ p, globalize (D.patch p).index (D.cutoff p) (f p), ?_⟩
+  rw [map_sum, ← D.reconstruct_eq u, D.reconstruct_apply]
+  apply Finset.sum_congr rfl
+  intro p _
+  rw [← hf p, D.localizers.globalizeL2_schwartz]
+
+ 
+
+theorem exists_smooth_shifted (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
+    (f : Smooth A) : ∃ u : Smooth A, (m^2 : ℝ) • u - complexL g u = f := by
+  obtain ⟨u, hu⟩ := D.allRegular_smooth (D.completedResolvent_regular m hm he f)
+  have h2 : D.localizers.embed 2 u = D.completedResolvent m hm he (D.localizers.embed 0 f) := by
+    apply D.localizers.lower_injective (by norm_num : (0 : ℝ) ≤ 2)
+    rw [D.localizers.lower_embed (by norm_num : (0 : ℝ) ≤ 2), hu]
+  refine ⟨u, D.localizers.embed_injective 0 ?_⟩
+  have h := D.completedResolvent_equation m hm he (D.localizers.embed 0 f)
+  rw [← h2] at h
+  simpa only [shiftedOperator, completedL, _root_.sub_apply,
+    _root_.smul_apply, D.localizers.lower_embed (by norm_num : (0 : ℝ) ≤ 2),
+    D.localizers.extendCore_embed D.complexL_bound, map_sub, map_smul] using h
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/GlobalSobolev.lean b/lean/OAI/Geometry/PrescribedPotential/GlobalSobolev.lean
new file mode 100644
index 0000000000000000000000000000000000000000..c97e51ac8a85b8be785111a7bed3570a2e3dcf37
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/GlobalSobolev.lean
@@ -0,0 +1,139 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.LocalizedCompact
+import OAI.Geometry.PrescribedPotential.ParametrixAtlas
+
+namespace OAI
+
+/-! Global Sobolev. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology MeasureTheory Manifold IsManifold
+open scoped ContDiff SchwartzMap BoundedContinuousFunction Classical
+
+namespace GlobalElliptic
+open Anticanonical EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] (A : ComplexAtlas d X)
+
+ 
+def smoothSpace : Submodule ℝ (X → ℂ) where
+  carrier := {f | ∀ i, ContDiffOn ℝ ∞ (f ∘ (A.euclideanChart i).symm)
+    (A.euclideanChart i).target}
+  zero_mem' := by intro i; exact contDiffOn_const
+  add_mem' := by intro f h hf hh i; exact (hf i).add (hh i)
+  smul_mem' := by intro c f hf i; exact contDiffOn_const.smul (hf i)
+
+abbrev Smooth := ↥(smoothSpace A)
+
+instance : FunLike (Smooth A) X ℂ where
+  coe := Subtype.val
+  coe_injective := Subtype.val_injective
+
+namespace Smooth
+variable {A}
+
+@[simp] lemma add_apply (f h : Smooth A) (x : X) : (f + h) x = f x + h x := rfl
+@[simp] lemma smul_apply (c : ℝ) (f : Smooth A) (x : X) : (c • f) x = c • f x := rfl
+@[simp] lemma zero_apply (x : X) : (0 : Smooth A) x = 0 := rfl
+
+lemma smooth (f : Smooth A) (i : Fin A.count) :
+    ContDiffOn ℝ ∞ (f ∘ (A.euclideanChart i).symm) (A.euclideanChart i).target := f.property i
+
+lemma continuous (f : Smooth A) : Continuous (f : X → ℂ) := by
+  rw [continuous_iff_continuousAt]
+  intro x
+  obtain ⟨i, hi⟩ := A.covers x
+  let e := A.euclideanChart i
+  have hx : x ∈ e.source := by simpa [e] using hi
+  have he := e.continuousAt hx
+  have hf := ((f.smooth i).continuousOn.continuousAt (e.open_target.mem_nhds (e.mapsTo hx))).comp he
+  apply hf.congr
+  filter_upwards [e.open_source.mem_nhds hx] with y hy
+  exact congrArg f (e.left_inv hy)
+
+@[ext] lemma ext {f h : Smooth A} (H : ∀ x, f x = h x) : f = h := Subtype.ext (funext H)
+
+def const (c : ℂ) : Smooth A := ⟨fun _ => c, fun _ => contDiffOn_const⟩
+
+ 
+def mul (f h : Smooth A) : Smooth A := ⟨fun x => f x * h x, fun i => (f.smooth i).mul (h.smooth i)⟩
+
+end Smooth
+
+ 
+def localizeFun (i : Fin A.count) (ρ f : Smooth A) (y : EC d) : ℂ :=
+  if y ∈ (A.euclideanChart i).target then
+    ρ ((A.euclideanChart i).symm y) * f ((A.euclideanChart i).symm y) else 0
+
+lemma localizeFun_apply {i : Fin A.count} (ρ f : Smooth A) {y : EC d}
+    (hy : y ∈ (A.euclideanChart i).target) :
+    localizeFun A i ρ f y = ρ ((A.euclideanChart i).symm y) *
+      f ((A.euclideanChart i).symm y) := ite_eq_left hy
+
+lemma localizeFun_support (i : Fin A.count) (ρ f : Smooth A) :
+    Function.support (localizeFun A i ρ f) ⊆ (A.euclideanChart i) '' tsupport (ρ : X → ℂ) := by
+  intro y hy
+  have hn : localizeFun A i ρ f y ≠ 0 := hy
+  by_cases ht : y ∈ (A.euclideanChart i).target
+  · rw [localizeFun_apply A ρ f ht] at hn
+    exact ⟨(A.euclideanChart i).symm y,
+      subset_tsupport _ (left_ne_zero_of_mul hn), (A.euclideanChart i).right_inv ht⟩
+  · exact (hn (ite_eq_right ht)).elim
+
+lemma localizeFun_smooth_compact [T2Space X] [CompactSpace X]
+    (i : Fin A.count) (ρ f : Smooth A)
+    (hρ : tsupport (ρ : X → ℂ) ⊆ (A.euclideanChart i).source) :
+    ContDiff ℝ ∞ (localizeFun A i ρ f) ∧ HasCompactSupport (localizeFun A i ρ f) := by
+  let e := A.euclideanChart i
+  let K := e '' tsupport (ρ : X → ℂ)
+  have hK : IsCompact K := (isClosed_tsupport _).isCompact.image_of_continuousOn
+    (e.continuousOn.mono hρ)
+  have hs : Function.support (localizeFun A i ρ f) ⊆ K := localizeFun_support A i ρ f
+  constructor
+  · rw [contDiff_iff_contDiffAt]
+    intro y
+    by_cases hy : y ∈ e.target
+    · have hc := ((ρ.smooth i).mul (f.smooth i)).contDiffAt (e.open_target.mem_nhds hy)
+      apply hc.congr_of_eventuallyEq
+      filter_upwards [e.open_target.mem_nhds hy] with z hz
+      exact localizeFun_apply A ρ f hz
+    · have hyK : y ∉ K := by
+        rintro ⟨x, hx, rfl⟩
+        exact hy (e.mapsTo (hρ hx))
+      apply (contDiffAt_const (c := (0 : ℂ))).congr_of_eventuallyEq
+      filter_upwards [hK.isClosed.isOpen_compl.mem_nhds hyK] with z hz
+      exact Function.notMem_support.mp (fun h => hz (hs h))
+  · exact HasCompactSupport.of_support_subset_isCompact hK hs
+
+ 
+def localize [T2Space X] [CompactSpace X] (i : Fin A.count) (ρ : Smooth A)
+    (hρ : tsupport (ρ : X → ℂ) ⊆ (A.euclideanChart i).source) (f : Smooth A) : 𝓢(EC d, ℂ) :=
+  (localizeFun_smooth_compact A i ρ f hρ).2.toSchwartzMap
+    (localizeFun_smooth_compact A i ρ f hρ).1
+
+@[simp] lemma localize_apply [T2Space X] [CompactSpace X] (i : Fin A.count) (ρ : Smooth A)
+    (hρ : tsupport (ρ : X → ℂ) ⊆ (A.euclideanChart i).source) (f : Smooth A) (y : EC d) :
+    localize A i ρ hρ f y = localizeFun A i ρ f y := rfl
+
+ 
+def localizeLinear [T2Space X] [CompactSpace X] (i : Fin A.count) (ρ : Smooth A)
+    (hρ : tsupport (ρ : X → ℂ) ⊆ (A.euclideanChart i).source) : Smooth A →ₗ[ℝ] 𝓢(EC d, ℂ) where
+  toFun := localize A i ρ hρ
+  map_add' f h := by
+    ext y
+    simp only [localize_apply, localizeFun, add_apply]
+    split_ifs <;> simp [mul_add]
+  map_smul' c f := by
+    ext y
+    simp only [localize_apply, localizeFun, smul_apply]
+    split_ifs <;> simp [mul_left_comm]
+
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/GlobalStrongEmbedding.lean b/lean/OAI/Geometry/PrescribedPotential/GlobalStrongEmbedding.lean
new file mode 100644
index 0000000000000000000000000000000000000000..f71815d7ea5c8efbd495cf6c0bd1d97fcad269e1
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/GlobalStrongEmbedding.lean
@@ -0,0 +1,108 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CircleRadialCalculus
+import OAI.Geometry.PrescribedPotential.SobolevProduct
+
+namespace OAI
+
+/-! Global Strong Embedding. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory
+open scoped SchwartzMap ContDiff Classical BoundedContinuousFunction
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+
+def Smooth.bounded (f : Smooth A) : X →ᵇ ℂ :=
+  BoundedContinuousFunction.mkOfCompact ⟨f, f.continuous⟩
+
+def Smooth.boundedLinear : Smooth A →ₗ[ℝ] (X →ᵇ ℂ) where
+  toFun := Smooth.bounded
+  map_add' _ _ := rfl
+  map_smul' _ _ := rfl
+
+namespace Localizers
+variable (D : Localizers A ι)
+
+lemma smooth_bounded_estimate (s : ℝ) (hs : (Module.finrank ℝ (EC d) : ℝ) < 2*s) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ f : Smooth A, ‖f.bounded‖ ≤ C*‖D.embed s f‖ := by
+  let B := ‖strongEmbedding (E := EC d) s hs‖
+  refine ⟨(Fintype.card ι : ℝ)*B, mul_nonneg (Nat.cast_nonneg _) (norm_nonneg _), fun f => ?_⟩
+  apply (BoundedContinuousFunction.norm_le (mul_nonneg
+    (mul_nonneg (Nat.cast_nonneg _) (norm_nonneg _)) (norm_nonneg _))).mpr
+  intro x
+  change ‖f x‖ ≤ _
+  have he : f x = ∑ p : ι, D.weight p x*f x := by rw [← Finset.sum_mul, D.sum_one, one_mul]
+  rw [he]
+  apply (norm_sum_le _ _).trans
+  calc
+    _ ≤ ∑ p : ι, B*‖D.embed s f‖ := by
+      apply Finset.sum_le_sum
+      intro p _
+      by_cases hx : x ∈ tsupport (D.weight p : X → ℂ)
+      · have hxs := D.support_sub p hx
+        have hp := strongEmbedding_norm s hs
+          (schwartzCoord s (localize A (D.index p) (D.weight p) (D.support_sub p) f))
+          (A.euclideanChart (D.index p) x)
+        rw [strongEmbedding_schwartz] at hp
+        change ‖localizeFun A (D.index p) (D.weight p) f (A.euclideanChart (D.index p) x)‖ ≤ _ at hp
+        rw [localizeFun_apply A _ _ ((A.euclideanChart (D.index p)).mapsTo hxs),
+          (A.euclideanChart (D.index p)).left_inv hxs] at hp
+        exact hp.trans (mul_le_mul_of_nonneg_left
+          (norm_le_pi_norm (D.coordinates s f) p) (norm_nonneg _))
+      · rw [image_eq_zero_of_notMem_tsupport hx, zero_mul, norm_zero]
+        positivity
+    _ = _ := by simp only [Finset.sum_const, Finset.card_univ, nsmul_eq_mul]; ring
+
+def strong (s : ℝ) : D.Sobolev s →L[ℝ] (X →ᵇ ℂ) :=
+  Smooth.boundedLinear.extendOfNorm (D.embed s)
+
+lemma strong_embed (s : ℝ) (hs : (Module.finrank ℝ (EC d) : ℝ) < 2*s) (f : Smooth A) :
+    D.strong s (D.embed s f) = f.bounded := by
+  obtain ⟨C,_,hc⟩ := D.smooth_bounded_estimate s hs
+  exact LinearMap.extendOfNorm_eq (D.embed_dense s) ⟨C,hc⟩ f
+
+lemma strong_localized (s : ℝ) (hs : (Module.finrank ℝ (EC d) : ℝ) < 2*s)
+    (u : D.Sobolev s) (p : ι) (y : EC d) :
+    strongEmbedding s hs (u.val p) y = if y ∈ (A.euclideanChart (D.index p)).target then
+      D.weight p ((A.euclideanChart (D.index p)).symm y) *
+        D.strong s u ((A.euclideanChart (D.index p)).symm y) else 0 := by
+  have he : (fun u : D.Sobolev s => strongEmbedding s hs (u.val p) y) =
+      (fun u : D.Sobolev s => if y ∈ (A.euclideanChart (D.index p)).target then
+        D.weight p ((A.euclideanChart (D.index p)).symm y)*
+          D.strong s u ((A.euclideanChart (D.index p)).symm y) else 0) := by
+    apply (D.embed_dense s).equalizer
+    · exact ((BoundedContinuousFunction.lipschitz_eval_const y).continuous.comp
+        ((strongEmbedding s hs).continuous.comp ((continuous_apply p).comp continuous_subtype_val)))
+    · split_ifs <;> fun_prop
+    funext f
+    dsimp only [Function.comp_apply]
+    rw [D.strong_embed s hs]
+    change strongEmbedding s hs (schwartzCoord s (localize A (D.index p)
+      (D.weight p) (D.support_sub p) f)) y = _
+    rw [strongEmbedding_schwartz]
+    rfl
+  exact congr_fun he u
+
+lemma strong_injective (s : ℝ) (hs : (Module.finrank ℝ (EC d) : ℝ) < 2*s) :
+    Function.Injective (D.strong s) := by
+  intro u v huv
+  apply Subtype.ext
+  funext p
+  apply realize_injective s
+  rw [← strongEmbedding_distribution s hs, ← strongEmbedding_distribution s hs]
+  congr 1
+  ext y
+  rw [D.strong_localized s hs, D.strong_localized s hs, huv]
+end Localizers
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/GlobalTransport.lean b/lean/OAI/Geometry/PrescribedPotential/GlobalTransport.lean
new file mode 100644
index 0000000000000000000000000000000000000000..9b72c9c68a54e143822e87eb5d10e642f55673d1
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/GlobalTransport.lean
@@ -0,0 +1,109 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.ChartTransportHigher
+import OAI.Geometry.PrescribedPotential.GlobalExtension
+import OAI.Geometry.PrescribedPotential.GlobalPartition
+
+namespace OAI
+
+/-! Global Transport. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory
+open scoped ContDiff SchwartzMap Classical
+
+namespace GlobalElliptic
+open Anticanonical EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X}
+
+lemma localize_tsupport (i : Fin A.count) (ρ f : Smooth A)
+    (hρ : tsupport (ρ : X → ℂ) ⊆ (A.euclideanChart i).source) :
+    tsupport (localize A i ρ hρ f : EC d → ℂ) ⊆
+      (A.euclideanChart i) '' (tsupport (ρ : X → ℂ) ∩ tsupport (f : X → ℂ)) := by
+  let e := A.euclideanChart i
+  have hK : IsCompact (tsupport (ρ : X → ℂ) ∩ tsupport (f : X → ℂ)) :=
+    (isClosed_tsupport _).isCompact.inter_right (isClosed_tsupport _)
+  apply closure_minimal _ (hK.image_of_continuousOn (e.continuousOn.mono
+    (fun x hx => hρ hx.1))).isClosed
+  intro y hy
+  have hn : localize A i ρ hρ f y ≠ 0 := hy
+  by_cases ht : y ∈ e.target
+  · rw [localize_apply, localizeFun_apply A ρ f ht] at hn
+    exact ⟨e.symm y, ⟨subset_tsupport _ (left_ne_zero_of_mul hn),
+      subset_tsupport _ (right_ne_zero_of_mul hn)⟩, e.right_inv ht⟩
+  · exact (hn (ite_eq_right ht)).elim
+
+def cutoffGlobal (i : Fin A.count) (κ : ChartCutoff (A.euclideanChart i).target) : Smooth A :=
+  extend i κ.val (κ.val.smooth ⊤) κ.compact κ.support_sub
+
+omit [CompactSpace X] in
+lemma cutoffGlobal_tsupport (i : Fin A.count) (κ : ChartCutoff (A.euclideanChart i).target) :
+    tsupport (cutoffGlobal i κ : X → ℂ) ⊆
+      (A.euclideanChart i).symm '' tsupport (κ : EC d → ℂ) :=
+  extend_tsupport i κ.val (κ.val.smooth ⊤) κ.compact κ.support_sub
+
+ 
+def overlapCutoff (i j : Fin A.count) (κ : ChartCutoff (A.euclideanChart i).target)
+    (ρ : Smooth A) (hρ : tsupport (ρ : X → ℂ) ⊆ (A.euclideanChart j).source) :
+    ChartCutoff ((A.euclideanChart j).symm.trans (A.euclideanChart i)).source where
+  val := localize A j ρ hρ (cutoffGlobal i κ)
+  compact := (localizeFun_smooth_compact A j ρ _ hρ).2
+  support_sub := by
+    intro y hy
+    obtain ⟨x, ⟨hxρ, hxf⟩, rfl⟩ := localize_tsupport j ρ _ hρ hy
+    have hxj := hρ hxρ
+    have hxi : x ∈ (A.euclideanChart i).source := by
+      obtain ⟨z, hz, rfl⟩ := cutoffGlobal_tsupport i κ hxf
+      exact (A.euclideanChart i).symm.mapsTo (κ.support_sub hz)
+    refine ⟨(A.euclideanChart j).mapsTo hxj, ?_⟩
+    simpa only [OpenPartialHomeomorph.symm_symm, mem_preimage,
+      (A.euclideanChart j).left_inv hxj] using hxi
+
+@[simp] lemma overlapCutoff_apply (i j : Fin A.count)
+    (κ : ChartCutoff (A.euclideanChart i).target) (ρ : Smooth A)
+    (hρ : tsupport (ρ : X → ℂ) ⊆ (A.euclideanChart j).source) (y : EC d) :
+    overlapCutoff i j κ ρ hρ y = if y ∈ (A.euclideanChart j).target then
+      ρ ((A.euclideanChart j).symm y) *
+        (if (A.euclideanChart j).symm y ∈ (A.euclideanChart i).source then
+          κ (A.euclideanChart i ((A.euclideanChart j).symm y)) else 0) else 0 := rfl
+
+lemma localize_globalize (i j : Fin A.count) (κ : ChartCutoff (A.euclideanChart i).target)
+    (ρ : Smooth A) (hρ : tsupport (ρ : X → ℂ) ⊆ (A.euclideanChart j).source)
+    (f : 𝓢(EC d, ℂ)) :
+    localize A j ρ hρ (globalize i κ f) =
+      chartTransport ((A.euclideanChart j).symm.trans (A.euclideanChart i))
+        (A.euclidean_transition_smooth j i) (overlapCutoff i j κ ρ hρ) f := by
+  ext y
+  simp only [localize_apply, localizeFun, globalize_apply, chartTransport_apply,
+    overlapCutoff_apply, OpenPartialHomeomorph.trans_apply]
+  split_ifs <;> simp_all only [mul_zero, zero_mul, mul_assoc]
+
+ 
+
+theorem globalize_even_bound {ι : Type*} [Fintype ι] (D : Localizers A ι)
+    (i : Fin A.count) (κ : ChartCutoff (A.euclideanChart i).target) (k : ℕ) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ f : 𝓢(EC d, ℂ),
+      ‖D.embed (2 * (k : ℝ)) (globalize i κ f)‖ ≤ C * ‖schwartzCoord (2 * (k : ℝ)) f‖ := by
+  have hB (j : ι) : CoreBound (2 * (k : ℝ)) (2 * (k : ℝ))
+      (fun f => localize A (D.index j) (D.weight j) (D.support_sub j) (globalize i κ f)) := by
+    simp_rw [localize_globalize]
+    exact chartTransport_even_bound _ (A.euclidean_transition_smooth (D.index j) i)
+      (A.euclidean_transition_smooth i (D.index j)) _ k
+  choose C hC h using hB
+  refine ⟨∑ j, C j, Finset.sum_nonneg (fun j _ => hC j), fun f => ?_⟩
+  change ‖D.coordinates (2 * (k : ℝ)) (globalize i κ f)‖ ≤ _
+  apply (pi_norm_le_iff_of_nonneg (mul_nonneg (Finset.sum_nonneg (fun j _ => hC j)) (norm_nonneg _))).mpr
+  intro j
+  exact (h j f).trans (mul_le_mul_of_nonneg_right
+    (Finset.single_le_sum (fun j _ => hC j) (Finset.mem_univ j)) (norm_nonneg _))
+
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/GradientJacobian.lean b/lean/OAI/Geometry/PrescribedPotential/GradientJacobian.lean
new file mode 100644
index 0000000000000000000000000000000000000000..d27b629baef3bf6bf5a4674f6aafe217423fc089
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/GradientJacobian.lean
@@ -0,0 +1,68 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CircleRadialCalculus
+import OAI.Geometry.PrescribedPotential.RealDeterminantOrder
+
+namespace OAI
+
+/-! Gradient Jacobian. -/
+
+section
+
+ 
+
+noncomputable section
+open Matrix Set Filter Topology
+open scoped InnerProductSpace ContDiff
+namespace PotentialABP
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E]
+
+lemma positive_det_nonneg {L : E →ₗ[ℝ] E} (hL : L.IsPositive) : 0 ≤ L.det := by
+  let b := stdOrthonormalBasis ℝ E
+  rw [← LinearMap.det_toMatrix b.toBasis]
+  exact ((LinearMap.posSemidef_toMatrix_iff b).mpr hL).det_nonneg
+
+lemma positive_det_mono {L M : E →ₗ[ℝ] E}
+    (hL : L.IsPositive) (hML : (M-L).IsPositive) : L.det ≤ M.det := by
+  let b := stdOrthonormalBasis ℝ E
+  rw [← LinearMap.det_toMatrix b.toBasis L, ← LinearMap.det_toMatrix b.toBasis M]
+  apply real_det_mono ((LinearMap.posSemidef_toMatrix_iff b).mpr hL)
+  rw [← map_sub]
+  exact (LinearMap.posSemidef_toMatrix_iff b).mpr hML
+
+lemma inner_fderiv_gradient {u : E → ℝ} (hu : ContDiff ℝ ∞ u) (x v w : E) :
+    inner ℝ (fderiv ℝ (gradient u) x v) w = fderiv ℝ (fderiv ℝ u) x v w := by
+  have hd := (InnerProductSpace.toDual ℝ E).symm.hasFDerivAt.comp x
+    ((hu.fderiv_right (m := ∞) (by simp)).differentiable (by simp) x).hasFDerivAt
+  have he := hd.fderiv
+  change fderiv ℝ (gradient u) x = _ at he
+  rw [he]
+  change inner ℝ ((InnerProductSpace.toDual ℝ E).symm (fderiv ℝ (fderiv ℝ u) x v)) w = _
+  rw [← InnerProductSpace.toDual_apply_apply]
+  simp
+
+lemma gradient_hessian_symmetric {u : E → ℝ} (hu : ContDiff ℝ ∞ u) (x : E) :
+    (fderiv ℝ (gradient u) x).toLinearMap.IsSymmetric := by
+  intro v w
+  change inner ℝ (fderiv ℝ (gradient u) x v) w = inner ℝ v (fderiv ℝ (gradient u) x w)
+  rw [inner_fderiv_gradient hu, real_inner_comm, inner_fderiv_gradient hu]
+  exact hu.contDiffAt.isSymmSndFDerivAt (by
+    simp only [minSmoothness_of_isRCLikeNormedField]
+    change ((2 : ℕ∞) : WithTop ℕ∞) ≤ ((⊤ : ℕ∞) : WithTop ℕ∞)
+    exact WithTop.coe_le_coe.mpr le_top) v w
+
+lemma gradient_hessian_positive {u : E → ℝ} (hu : ContDiff ℝ ∞ u) {x : E}
+    (hB : ∀ v, 0 ≤ fderiv ℝ (fderiv ℝ u) x v v) :
+    (fderiv ℝ (gradient u) x).toLinearMap.IsPositive := by
+  refine ⟨gradient_hessian_symmetric hu x, ?_⟩
+  intro v
+  change 0 ≤ inner ℝ (fderiv ℝ (gradient u) x v) v
+  rw [inner_fderiv_gradient hu]
+  exact hB v
+
+end PotentialABP
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/HessianCommutator.lean b/lean/OAI/Geometry/PrescribedPotential/HessianCommutator.lean
new file mode 100644
index 0000000000000000000000000000000000000000..ff895326134aeef1c924bbddf5582019e8056316
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/HessianCommutator.lean
@@ -0,0 +1,83 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CircleRadialCalculus
+import OAI.Geometry.PrescribedPotential.ComplexHessian
+import OAI.Geometry.PrescribedPotential.CutoffHigher
+
+namespace OAI
+
+/-! Hessian Commutator. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory LineDeriv
+open scoped ContDiff SchwartzMap Classical
+namespace SobolevChart
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+
+def secondLocalizedError (κ : 𝓢(E, ℂ)) (v a b : E) :
+    𝓢(E, ℂ) →L[ℂ] 𝓢(E, ℂ) :=
+  SchwartzMap.smulLeftCLM ℂ (∂_{a} κ : 𝓢(E, ℂ)) ∘L
+    (lineDerivOpCLM ℂ 𝓢(E, ℂ) b ∘L lineDerivOpCLM ℂ 𝓢(E, ℂ) v) +
+  SchwartzMap.smulLeftCLM ℂ (∂_{b} κ : 𝓢(E, ℂ)) ∘L
+    (lineDerivOpCLM ℂ 𝓢(E, ℂ) a ∘L lineDerivOpCLM ℂ 𝓢(E, ℂ) v) +
+  SchwartzMap.smulLeftCLM ℂ (∂_{a} (∂_{b} κ) : 𝓢(E, ℂ)) ∘L
+    lineDerivOpCLM ℂ 𝓢(E, ℂ) v
+
+lemma secondLocalizedError_identity (κ f : 𝓢(E, ℂ)) (v a b : E) :
+    ∂_{a} (∂_{b} (SchwartzMap.smulLeftCLM ℂ κ (∂_{v} f))) =
+      SchwartzMap.smulLeftCLM ℂ κ (∂_{v} (∂_{a} (∂_{b} f))) +
+        secondLocalizedError κ v a b f := by
+  rw [schwartz_second_product]
+  have hd : ∂_{a} (∂_{b} (∂_{v} f)) = ∂_{v} (∂_{a} (∂_{b} f)) := by
+    rw [schwartz_deriv_comm b v, schwartz_deriv_comm a v]
+  rw [hd]
+  rfl
+
+lemma secondLocalizedError_bound (κ : 𝓢(E, ℂ)) (v a b : E) (k : ℕ) :
+    CoreBound ((k : ℝ)+2) (k : ℝ) (secondLocalizedError κ v a b) := by
+  have hd (w : E) : CoreBound ((k : ℝ)+2) (k : ℝ)
+      (fun f : 𝓢(E, ℂ) => ∂_{w} (∂_{v} f)) :=
+    (coreBound_deriv w (s := (k : ℝ)+1) (t := (k : ℝ)) (by linarith)).comp
+      (coreBound_deriv v (s := (k : ℝ)+2) (t := (k : ℝ)+1) (by linarith))
+  exact ((coreBound_product_integer (∂_{a} κ) k).comp (hd b)).add
+    ((coreBound_product_integer (∂_{b} κ) k).comp (hd a)) |>.add
+      ((coreBound_product_integer (∂_{a} (∂_{b} κ)) k).comp
+        (coreBound_deriv v (s := (k : ℝ)+2) (t := (k : ℝ)) (by linarith)))
+end SobolevChart
+
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ}
+
+def hessianLocalizedError (κ : 𝓢(EC d, ℂ)) (v : EC d) (i j : Fin d) :
+    𝓢(EC d, ℂ) →L[ℂ] 𝓢(EC d, ℂ) :=
+  (1/4 : ℂ) • (secondLocalizedError κ v (hessianDirection i) (hessianDirection j) +
+    secondLocalizedError κ v (hessianIDirection i) (hessianIDirection j) +
+      Complex.I • (secondLocalizedError κ v (hessianIDirection i) (hessianDirection j) +
+        (-1 : ℂ) • secondLocalizedError κ v (hessianDirection i) (hessianIDirection j)))
+
+lemma hessianLocalizedError_identity (κ f : 𝓢(EC d, ℂ)) (v : EC d) (i j : Fin d) :
+    hessianEntrySchwartz i j (SchwartzMap.smulLeftCLM ℂ κ (∂_{v} f)) =
+      SchwartzMap.smulLeftCLM ℂ κ (∂_{v} (hessianEntrySchwartz i j f)) +
+        hessianLocalizedError κ v i j f := by
+  simp only [hessianEntrySchwartz, hessianLocalizedError, smul_apply,
+    add_apply, ContinuousLinearMap.comp_apply, lineDerivOpCLM_apply,
+    secondLocalizedError_identity, lineDerivOp_add, lineDerivOp_smul, map_smul, map_add]
+  module
+
+lemma hessianLocalizedError_bound (κ : 𝓢(EC d, ℂ)) (v : EC d) (i j : Fin d) (k : ℕ) :
+    CoreBound ((k : ℝ)+2) (k : ℝ) (hessianLocalizedError κ v i j) := by
+  exact ((secondLocalizedError_bound κ v _ _ k).add
+    (secondLocalizedError_bound κ v _ _ k) |>.add
+      (((secondLocalizedError_bound κ v _ _ k).add
+        ((secondLocalizedError_bound κ v _ _ k).smul (-1))).smul Complex.I)).smul (1/4)
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/HigherDirectionalJets.lean b/lean/OAI/Geometry/PrescribedPotential/HigherDirectionalJets.lean
new file mode 100644
index 0000000000000000000000000000000000000000..18a5b29bacd502665c5d390bf17a1355c8607d8a
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/HigherDirectionalJets.lean
@@ -0,0 +1,105 @@
+import Mathlib
+
+namespace OAI
+
+/-! Higher Directional Jets. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology
+open scoped ContDiff
+namespace HigherJet
+variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  [NormedAddCommGroup F] [NormedSpace ℝ F]
+
+def dir (v : E) (f : E → F) (x : E) : F := fderiv ℝ f x v
+
+def word : List E → (E → F) → E → F
+  | [], f => f
+  | v::ws, f => dir v (word ws f)
+
+lemma dir_smoothAt {f : E → F} {x : E} (hf : ContDiffAt ℝ ∞ f x) (v : E) :
+    ContDiffAt ℝ ∞ (dir v f) x :=
+  (hf.fderiv_right (m := ∞) (by simp)).clm_apply contDiffAt_const
+
+lemma dir_smoothOn {U : Set E} (hU : IsOpen U) {f : E → F}
+    (hf : ContDiffOn ℝ ∞ f U) (v : E) : ContDiffOn ℝ ∞ (dir v f) U :=
+  (hf.fderiv_of_isOpen hU (m := ∞) (by simp)).clm_apply contDiffOn_const
+
+lemma word_smoothOn {U : Set E} (hU : IsOpen U) {f : E → F}
+    (hf : ContDiffOn ℝ ∞ f U) (ws : List E) : ContDiffOn ℝ ∞ (word ws f) U := by
+  induction ws with
+  | nil => exact hf
+  | cons v ws ih => exact dir_smoothOn hU ih v
+
+lemma dir_congr {f g : E → F} {x : E} (h : f =ᶠ[𝓝 x] g) (v : E) :
+    dir v f x = dir v g x := congrArg (fun T : E →L[ℝ] F => T v) h.fderiv_eq
+
+lemma word_congr {f g : E → F} {x : E} (h : f =ᶠ[𝓝 x] g) (ws : List E) :
+    word ws f x = word ws g x := by
+  induction ws generalizing x with
+  | nil => exact h.self_of_nhds
+  | cons v ws ih =>
+    exact dir_congr (h.eventually_nhds.mono (fun y hy => ih hy)) v
+
+lemma dir_dir {f : E → F} {x : E} (hf : ContDiffAt ℝ ∞ f x) (u v : E) :
+    dir u (dir v f) x = fderiv ℝ (fderiv ℝ f) x u v := by
+  have hd := (hf.fderiv_right (m := ∞) (by simp)).differentiableAt (by simp)
+  exact congrArg (fun T : E →L[ℝ] F => T u)
+    ((ContinuousLinearMap.apply ℝ F v).hasFDerivAt.comp x hd.hasFDerivAt).fderiv
+
+lemma dir_comm {f : E → F} {x : E} (hf : ContDiffAt ℝ ∞ f x) (u v : E) :
+    dir u (dir v f) x = dir v (dir u f) x := by
+  rw [dir_dir hf,dir_dir hf]
+  exact hf.isSymmSndFDerivAt (by
+    simp only [minSmoothness_of_isRCLikeNormedField]
+    exact WithTop.coe_le_coe.mpr le_top) u v
+
+lemma dir_word_comm {U : Set E} (hU : IsOpen U) {f : E → F}
+    (hf : ContDiffOn ℝ ∞ f U) (v : E) (ws : List E) {x : E} (hx : x ∈ U) :
+    dir v (word ws f) x = word ws (dir v f) x := by
+  induction ws generalizing x with
+  | nil => rfl
+  | cons w ws ih =>
+    change dir v (dir w (word ws f)) x = dir w (word ws (dir v f)) x
+    rw [dir_comm ((word_smoothOn hU hf ws).contDiffAt (hU.mem_nhds hx))]
+    exact dir_congr (by filter_upwards [hU.mem_nhds hx] with y hy using ih hy) w
+
+lemma word_dir_dir_comm {U : Set E} (hU : IsOpen U) {f : E → F}
+    (hf : ContDiffOn ℝ ∞ f U) (v w : E) (ws : List E) {x : E} (hx : x ∈ U) :
+    word ws (dir v (dir w f)) x = dir v (dir w (word ws f)) x := by
+  rw [← dir_word_comm hU (dir_smoothOn hU hf w) v ws hx]
+  exact dir_congr (by
+    filter_upwards [hU.mem_nhds hx] with y hy
+    exact (dir_word_comm hU hf w ws hy).symm) v
+
+lemma word_ofFn_eq {U : Set E} (hU : IsOpen U) {f : E → F}
+    (hf : ContDiffOn ℝ ∞ f U) (m : ℕ) (v : Fin m → E) {x : E} (hx : x ∈ U) :
+    word (List.ofFn v) f x = iteratedFDeriv ℝ m f x v := by
+  induction m generalizing x with
+  | zero => rfl
+  | succ m ih =>
+    rw [List.ofFn_succ]
+    change dir (v 0) (word (List.ofFn (fun i => v i.succ)) f) x = _
+    have he : word (List.ofFn (fun i => v i.succ)) f =ᶠ[𝓝 x]
+        (fun y => iteratedFDeriv ℝ m f y (fun i => v i.succ)) := by
+      filter_upwards [hU.mem_nhds hx] with y hy using ih _ hy
+    rw [dir_congr he]
+    exact (DifferentiableAt.iteratedFDeriv_succ_apply_left'
+      (((hf.contDiffAt (hU.mem_nhds hx)).iteratedFDeriv_right
+        (m := ∞) (i := m) (by exact WithTop.coe_le_coe.mpr le_top)).differentiableAt (by simp))).symm
+
+lemma norm_word_ofFn_le {U : Set E} (hU : IsOpen U) {f : E → F}
+    (hf : ContDiffOn ℝ ∞ f U) (m : ℕ) (v : Fin m → E) {x : E} (hx : x ∈ U) :
+    ‖word (List.ofFn v) f x‖ ≤ ‖iteratedFDeriv ℝ m f x‖*∏ i, ‖v i‖ := by
+  rw [word_ofFn_eq hU hf m v hx]
+  exact (iteratedFDeriv ℝ m f x).le_opNorm v
+end HigherJet
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/HigherJetCommutator.lean b/lean/OAI/Geometry/PrescribedPotential/HigherJetCommutator.lean
new file mode 100644
index 0000000000000000000000000000000000000000..f3c59f54460972522676af0904eaea0c43bbf3e6
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/HigherJetCommutator.lean
@@ -0,0 +1,87 @@
+import Mathlib
+
+namespace OAI
+
+/-! Higher Jet Commutator. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Finset
+open scoped ContDiff
+namespace HigherJet
+variable {E F G H : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  [NormedAddCommGroup F] [NormedSpace ℝ F]
+  [NormedAddCommGroup G] [NormedSpace ℝ G]
+  [NormedAddCommGroup H] [NormedSpace ℝ H]
+
+lemma bilinear_derivative_bound_on (B : F →L[ℝ] G →L[ℝ] H)
+    {U : Set E} (hU : IsOpen U) {f : E → F} {g : E → G}
+    (hf : ContDiffOn ℝ ∞ f U) (hg : ContDiffOn ℝ ∞ g U)
+    {x : E} (hx : x ∈ U) (m : ℕ) :
+    ‖iteratedFDeriv ℝ m (fun y => B (f y) (g y)) x‖ ≤
+      ‖B‖*∑ i ∈ range (m+1), (m.choose i : ℝ)*‖iteratedFDeriv ℝ i f x‖*
+        ‖iteratedFDeriv ℝ (m-i) g x‖ := by
+  have hb := B.norm_iteratedFDerivWithin_le_of_bilinear hf hg hU.uniqueDiffOn hx
+    (show (m:ℕ∞ω) ≤ ∞ from WithTop.coe_le_coe.mpr le_top)
+  simpa only [iteratedFDerivWithin_of_isOpen _ hU hx] using hb
+
+lemma frozen_derivative_norm {f : E → F} {x : E} (hf : ContDiffAt ℝ ∞ f x) (m : ℕ) :
+    ‖iteratedFDeriv ℝ m (fun y => f y-f x) x‖ =
+      if m=0 then 0 else ‖iteratedFDeriv ℝ m f x‖ := by
+  by_cases hm : m=0
+  · subst m
+    simp
+  · rw [ite_eq_right hm]
+    change ‖iteratedFDeriv ℝ m (f - fun _ => f x) x‖ = _
+    rw [iteratedFDeriv_sub_apply (g := fun _ => f x)
+      (hf.of_le (show (m:ℕ∞ω) ≤ ∞ from WithTop.coe_le_coe.mpr le_top)) contDiffAt_const,
+      iteratedFDeriv_const_of_ne hm,Pi.zero_apply,sub_zero]
+
+lemma frozen_product_jet_bound (B : F →L[ℝ] G →L[ℝ] H)
+    {U : Set E} (hU : IsOpen U) {a : E → F} {v : E → G}
+    (ha : ContDiffOn ℝ ∞ a U) (hv : ContDiffOn ℝ ∞ v U)
+    {x : E} (hx : x ∈ U) (m : ℕ) :
+    ‖iteratedFDeriv ℝ m (fun y => B (a y-a x) (v y)) x‖ ≤
+      ‖B‖*∑ i ∈ range (m+1), (m.choose i : ℝ)*
+        (if i=0 then 0 else ‖iteratedFDeriv ℝ i a x‖)*
+        ‖iteratedFDeriv ℝ (m-i) v x‖ := by
+  have hb := bilinear_derivative_bound_on B hU (f := fun y => a y-a x) (ha.sub contDiffOn_const) hv hx m
+  simpa only [frozen_derivative_norm (ha.contDiffAt (hU.mem_nhds hx))] using hb
+
+ 
+
+lemma frozen_product_tame_bound (B : F →L[ℝ] G →L[ℝ] H)
+    {U : Set E} (hU : IsOpen U) {a : E → F} {v : E → G}
+    (ha : ContDiffOn ℝ ∞ a U) (hv : ContDiffOn ℝ ∞ v U)
+    {x : E} (hx : x ∈ U) (m : ℕ) {C T : ℝ} (hC : 0 ≤ C) (hT : 0 ≤ T)
+    (hproducts : ∀ i, 0 < i → i ≤ m →
+      ‖iteratedFDeriv ℝ i a x‖*‖iteratedFDeriv ℝ (m-i) v x‖ ≤ C*T) :
+    ‖iteratedFDeriv ℝ m (fun y => B (a y-a x) (v y)) x‖ ≤
+      ‖B‖*(2:ℝ)^m*C*T := by
+  apply (frozen_product_jet_bound B hU ha hv hx m).trans
+  have hb : (∑ i ∈ range (m+1), (m.choose i : ℝ)*
+      (if i=0 then 0 else ‖iteratedFDeriv ℝ i a x‖)*
+        ‖iteratedFDeriv ℝ (m-i) v x‖) ≤
+      ∑ i ∈ range (m+1), (m.choose i : ℝ)*(C*T) := by
+    apply Finset.sum_le_sum
+    intro i hi
+    by_cases hi0 : i=0
+    · simp only [hi0,ite_true,mul_zero,zero_mul]
+      positivity
+    · rw [ite_eq_right hi0,mul_assoc]
+      exact mul_le_mul_of_nonneg_left (hproducts i (Nat.pos_of_ne_zero hi0)
+        (Nat.le_of_lt_succ (mem_range.mp hi))) (Nat.cast_nonneg _)
+  have he : (∑ i ∈ range (m+1), (m.choose i : ℝ)*(C*T)) = 2^m*(C*T) := by
+    rw [← Finset.sum_mul]
+    congr 1
+    exact_mod_cast Nat.sum_range_choose m
+  exact (mul_le_mul_of_nonneg_left (hb.trans_eq he) (norm_nonneg B)).trans_eq (by ring)
+end HigherJet
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/HigherJetComposition.lean b/lean/OAI/Geometry/PrescribedPotential/HigherJetComposition.lean
new file mode 100644
index 0000000000000000000000000000000000000000..96368b6f86ff8dc20cc4e755699edbc6ddeebd3a
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/HigherJetComposition.lean
@@ -0,0 +1,100 @@
+import Mathlib
+
+namespace OAI
+
+/-! Higher Jet Composition. -/
+
+section
+
+ 
+
+noncomputable section
+open Finset
+open scoped ContDiff
+namespace HigherJet
+
+lemma prod_one_exception {ι : Type*} [Fintype ι] [DecidableEq ι]
+    (f : ι → ℝ) {B T : ℝ} (hB : 1 ≤ B) (hT : 1 ≤ T)
+    (hf : ∀ i, 0 ≤ f i) (hb : ∀ i, f i ≤ T ∨ f i ≤ B)
+    (hunique : ∀ i j, B < f i → B < f j → i = j) :
+    ∏ i, f i ≤ B^(Fintype.card ι)*T := by
+  by_cases he : ∃ i, B < f i
+  · obtain ⟨j,hj⟩ := he
+    have h_bound (i : ι) : f i ≤ B*(if i=j then T else 1) := by
+      by_cases hij : i=j
+      · subst i
+        simp only [ite_true]
+        obtain hi | hi := hb j
+        · exact hi.trans (le_mul_of_one_le_left (by linarith) hB)
+        · exact hi.trans (le_mul_of_one_le_right (by linarith) hT)
+      · rw [ite_eq_right hij,mul_one]
+        exact le_of_not_gt (fun hi => hij (hunique i j hi hj))
+    exact (Finset.prod_le_prod₀ (fun index _ => hf index) (fun index _ => h_bound index)).trans_eq
+      (by rw [Finset.prod_mul_distrib]; simp)
+  · have hh (i) : f i ≤ B := le_of_not_gt (fun hi => he ⟨i,hi⟩)
+    exact (Finset.prod_le_prod₀ (fun index _ => hf index) (fun index _ => hh index)).trans
+      (by simpa using le_mul_of_one_le_right (pow_nonneg (by linarith : 0 ≤ B) _) hT)
+
+lemma OrderedFinpartition.large_part_unique {m r : ℕ}
+    (c : OrderedFinpartition m) (hr : m < 2*r) (i j : Fin c.length)
+    (hi : r ≤ c.partSize i) (hj : r ≤ c.partSize j) : i=j := by
+  by_contra hij
+  have hsum : ∑ k, c.partSize k = m := by
+    simpa using c.sum_sigma_eq_sum (fun _ => (1 : ℕ))
+  have htwo : c.partSize i+c.partSize j ≤ ∑ k, c.partSize k := by
+    simpa [hij] using (Finset.sum_le_sum_of_subset_of_nonneg
+      (show ({i,j} : Finset (Fin c.length)) ⊆ univ from subset_univ _)
+      (fun _ _ _ => Nat.zero_le _) : ∑ k ∈ ({i,j} : Finset (Fin c.length)), c.partSize k ≤
+        ∑ k, c.partSize k)
+  omega
+
+lemma OrderedFinpartition.prod_jet_bound {m r : ℕ}
+    (c : OrderedFinpartition m) (hr : m < 2*r)
+    (a : ℕ → ℝ) {B T : ℝ} (hB : 1 ≤ B) (hT : 1 ≤ T)
+    (ha : ∀ j, 0 ≤ a j) (hlow : ∀ j, 0 < j → j < r → a j ≤ B)
+    (hhigh : ∀ j, r ≤ j → j ≤ m → a j ≤ T) :
+    ∏ i, a (c.partSize i) ≤ B^m*T := by
+  have hb (i : Fin c.length) : a (c.partSize i) ≤ T ∨ a (c.partSize i) ≤ B := by
+    by_cases hi : c.partSize i < r
+    · exact Or.inr (hlow _ (c.partSize_pos i) hi)
+    · exact Or.inl (hhigh _ (le_of_not_gt hi) (c.partSize_le i))
+  have hu (i j : Fin c.length) (hi : B < a (c.partSize i)) (hj : B < a (c.partSize j)) : i=j := by
+    have hi' : r ≤ c.partSize i := le_of_not_gt (fun hh => (not_lt_of_ge (hlow _ (c.partSize_pos i) hh)) hi)
+    have hj' : r ≤ c.partSize j := le_of_not_gt (fun hh => (not_lt_of_ge (hlow _ (c.partSize_pos j) hh)) hj)
+    exact OrderedFinpartition.large_part_unique c hr i j hi' hj'
+  exact (prod_one_exception (fun i => a (c.partSize i)) hB hT
+    (fun _ => ha _) hb hu).trans (mul_le_mul_of_nonneg_right
+      (pow_le_pow_right₀ hB (by simpa using c.length_le)) (by linarith))
+
+variable {E F G : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  [NormedAddCommGroup F] [NormedSpace ℝ F]
+  [NormedAddCommGroup G] [NormedSpace ℝ G]
+
+ 
+
+lemma composition_top_jet_bound {f : E → F} {g : F → G} {x : E}
+    (hf : ContDiffAt ℝ ∞ f x) (hg : ContDiffAt ℝ ∞ g (f x))
+    {m r : ℕ} (hr : m < 2*r) {B T C : ℝ} (hB : 1 ≤ B) (hT : 1 ≤ T) (hC : 0 ≤ C)
+    (hlow : ∀ j, 0 < j → j < r → ‖iteratedFDeriv ℝ j f x‖ ≤ B)
+    (hhigh : ∀ j, r ≤ j → j ≤ m → ‖iteratedFDeriv ℝ j f x‖ ≤ T)
+    (hgjet : ∀ j, j ≤ m → ‖iteratedFDeriv ℝ j g (f x)‖ ≤ C) :
+    ‖iteratedFDeriv ℝ m (g ∘ f) x‖ ≤
+      (Fintype.card (OrderedFinpartition m) : ℝ)*C*B^m*T := by
+  rw [iteratedFDeriv_comp (i := m) hg hf (by exact WithTop.coe_le_coe.mpr le_top),FormalMultilinearSeries.taylorComp]
+  calc
+    _ ≤ ∑ c : OrderedFinpartition m, ‖c.compAlongOrderedFinpartition
+      (iteratedFDeriv ℝ c.length g (f x)) (fun i => iteratedFDeriv ℝ (c.partSize i) f x)‖ := norm_sum_le _ _
+    _ ≤ ∑ _c : OrderedFinpartition m, C*(B^m*T) := by
+      apply Finset.sum_le_sum
+      intro c _
+      exact c.norm_compAlongOrderedFinpartition_le _ _ |>.trans
+        (mul_le_mul (hgjet _ c.length_le)
+          (OrderedFinpartition.prod_jet_bound c hr _ hB hT (fun _ => norm_nonneg _) hlow hhigh)
+          (Finset.prod_nonneg (fun _ _ => norm_nonneg _)) hC)
+    _ = _ := by simp [mul_assoc]
+end HigherJet
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/HigherMetricOperator.lean b/lean/OAI/Geometry/PrescribedPotential/HigherMetricOperator.lean
new file mode 100644
index 0000000000000000000000000000000000000000..2dd2081143b7c6fe14882a27474c1c97cd88fe34
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/HigherMetricOperator.lean
@@ -0,0 +1,97 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CompletedResolvent
+import OAI.Geometry.PrescribedPotential.CutoffHigher
+import OAI.Geometry.PrescribedPotential.GlobalParametrix
+
+namespace OAI
+
+/-! Higher Metric Operator. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory TemperedDistribution LineDeriv
+open scoped SchwartzMap ContDiff Classical BoundedContinuousFunction
+namespace MetricLocalization
+open EllipticKernel SobolevChart
+variable {n : ℕ}
+
+lemma differential_cutoff_bound_integer
+    (c : BasisIndex n → BasisIndex n → EC n →ᵇ ℂ)
+    (hc : ∀ i j, (c i j : EC n → ℂ).HasTemperateGrowth) (κ : 𝓢(EC n, ℂ)) (k : ℕ) :
+    CoreBound ((k : ℝ)+2) (k : ℝ) (fun f =>
+      schwartzDifferential c (SchwartzMap.smulLeftCLM ℂ κ f)) := by
+  have hd (i j : BasisIndex n) : CoreBound ((k : ℝ)+2) (k : ℝ) (fun f =>
+      SchwartzMap.smulLeftCLM ℂ (c i j) (∂_{stdOrthonormalBasis ℝ (EC n) i}
+        (∂_{stdOrthonormalBasis ℝ (EC n) j} (SchwartzMap.smulLeftCLM ℂ κ f)))) := by
+    simp_rw [schwartz_second_product, map_add, schwartz_product_assoc (hc i j)]
+    apply CoreBound.add
+    · exact (coreBound_product_integer _ k).comp
+        ((coreBound_deriv _ (s := (k : ℝ)+1) (t := (k : ℝ)) (by linarith)).comp
+          (coreBound_deriv _ (s := (k : ℝ)+2) (t := (k : ℝ)+1) (by linarith)))
+    · apply CoreBound.add
+      · apply CoreBound.add
+        · exact (coreBound_product_integer _ k).comp
+            (coreBound_deriv _ (s := (k : ℝ)+2) (t := (k : ℝ)) (by linarith))
+        · exact (coreBound_product_integer _ k).comp
+            (coreBound_deriv _ (s := (k : ℝ)+2) (t := (k : ℝ)) (by linarith))
+      · exact (coreBound_product_integer _ k).comp (CoreBound.id (by linarith))
+  change CoreBound ((k : ℝ)+2) (k : ℝ) (fun f => schwartzDifferential c (SchwartzMap.smulLeftCLM ℂ κ f))
+  simp only [schwartzDifferential, sum_apply,
+    ContinuousLinearMap.comp_apply, lineDerivOpCLM_apply]
+  exact CoreBound.sum Finset.univ _ (fun i _ => CoreBound.sum Finset.univ _ (fun j _ => hd i j))
+end MetricLocalization
+
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart MetricLocalization
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+lemma complexL_piece_bound_integer (k : ℕ) (p : ι) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ f : Smooth A,
+      ‖D.localizers.embed (k : ℝ) (globalize (D.patch p).index (D.outerCutoff p)
+        ((D.patch p).differential (SchwartzMap.smulLeftCLM ℂ (D.cutoff p).val (D.forcing p f))))‖ ≤
+        C*‖D.localizers.embed ((k : ℝ)+2) f‖ := by
+  obtain ⟨B,hB,hb⟩ := globalize_integer_bound D.localizers (D.patch p).index (D.outerCutoff p) k
+  obtain ⟨C,hC,hc⟩ := differential_cutoff_bound_integer (D.patch p).coefficients
+    (D.patch p).coefficients_temperate (D.cutoff p).val k
+  refine ⟨B*C,mul_nonneg hB hC,fun f => ?_⟩
+  apply (hb ((D.patch p).differential (SchwartzMap.smulLeftCLM ℂ (D.cutoff p).val (D.forcing p f)))).trans
+  have hr := (hc (D.forcing p f)).trans
+    (mul_le_mul_of_nonneg_left (D.forcing_bound p ((k : ℝ)+2) f) hC)
+  simpa only [mul_assoc, ParametrixPatch.differential] using mul_le_mul_of_nonneg_left hr hB
+
+lemma complexL_bound_integer (k : ℕ) :
+    D.localizers.BoundedCore ((k : ℝ)+2) (k : ℝ) (complexL g) := by
+  choose C hC hc using D.complexL_piece_bound_integer k
+  refine ⟨∑ p,C p,Finset.sum_nonneg (fun p _ => hC p),fun f => ?_⟩
+  rw [D.complexL_reconstruction, map_sum, Finset.sum_mul]
+  exact (norm_sum_le _ _).trans (Finset.sum_le_sum (fun p _ => hc p f))
+
+def completedLOrder (k : ℕ) : D.localizers.Sobolev ((k : ℝ)+2) →L[ℝ] D.localizers.Sobolev (k : ℝ) :=
+  D.localizers.extendCore ((k : ℝ)+2) (k : ℝ) (complexL g)
+
+lemma completedLOrder_embed (k : ℕ) (f : Smooth A) :
+    D.completedLOrder k (D.localizers.embed ((k : ℝ)+2) f) = D.localizers.embed (k : ℝ) (complexL g f) :=
+  D.localizers.extendCore_embed (D.complexL_bound_integer k) f
+
+lemma completedLOrder_lower (k : ℕ) (u : D.localizers.Sobolev ((k : ℝ)+2)) :
+    D.localizers.lower (k : ℝ) 0 (D.completedLOrder k u) =
+      D.completedL (D.localizers.lower ((k : ℝ)+2) 2 u) := by
+  have hk : (0 : ℝ) ≤ k := Nat.cast_nonneg k
+  change D.localizers.lower (k : ℝ) 0 (D.localizers.extendCore _ _ _ u) = _
+  rw [D.localizers.lower_extendCore hk _ (D.complexL_bound_integer k)
+    ((D.complexL_bound_integer k).target_mono hk)]
+  exact (D.localizers.extendCore_lower (by linarith) (complexL g) D.complexL_bound
+    ((D.complexL_bound_integer k).target_mono hk) u).symm
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/HigherPoisson.lean b/lean/OAI/Geometry/PrescribedPotential/HigherPoisson.lean
new file mode 100644
index 0000000000000000000000000000000000000000..68a85d4351f2378f3bee7c7a6337aed1b912a96d
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/HigherPoisson.lean
@@ -0,0 +1,131 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.BoundedPoisson
+import OAI.Geometry.PrescribedPotential.HigherResolvent
+import OAI.Geometry.PrescribedPotential.PatchCutoffs
+
+namespace OAI
+
+/-! Higher Poisson. -/
+
+section
+
+ 
+noncomputable section
+open Set Filter Topology
+open scoped ContDiff Classical
+namespace LinearNormalization
+variable {U V W : Type*}
+  [NormedAddCommGroup U] [NormedSpace ℝ U]
+  [NormedAddCommGroup V] [NormedSpace ℝ V]
+  [NormedAddCommGroup W] [NormedSpace ℝ W]
+lemma feedback_lift (q : V →L[ℝ] U) (p : W →L[ℝ] V) (r : W →L[ℝ] U)
+    (T : U →L[ℝ] U) (S : V →L[ℝ] V) (C : U →L[ℝ] U) (K : W →L[ℝ] W)
+    (R : U →L[ℝ] U) (R' : V →L[ℝ] W) (a : ℝ)
+    (hp : ∀ x, q (p x) = r x) (hS : ∀ x, q (S x) = T (q x))
+    (hK : ∀ x, r (K x) = C (r x)) (hR : ∀ x, r (R' x) = R (q x))
+    (hT : ∀ x, T x = -x - C x + a • R (T x)) (x : W) :
+    r (-x - K x + a • R' (S (p x))) = T (r x) := by
+  rw [map_add, map_sub, map_neg, map_smul, hK, hR, hS, hp]
+  exact (hT _).symm
+end LinearNormalization
+
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  [ConnectedSpace X] {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+omit [ConnectedSpace X] in
+lemma resolventGain_order (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1) (k : ℕ) :
+    ∃ R : D.localizers.Sobolev (k : ℝ) →L[ℝ] D.localizers.Sobolev ((k : ℝ)+1),
+      ∀ f, D.localizers.lower ((k : ℝ)+1) 0 (R f) =
+        D.resolventZero m hm he (D.localizers.lower (k : ℝ) 0 f) := by
+  obtain ⟨R,hR⟩ := D.completedResolvent_order m hm he k
+  refine ⟨D.localizers.lower ((k : ℝ)+2) ((k : ℝ)+1) ∘L R, fun f => ?_⟩
+  simp only [ContinuousLinearMap.comp_apply]
+  rw [D.localizers.lower_lower (by linarith : (k : ℝ)+1 ≤ (k : ℝ)+2) (by positivity : (0:ℝ) ≤ (k : ℝ)+1)]
+  rw [← D.localizers.lower_lower (by linarith [Nat.cast_nonneg (α := ℝ) k] : (2:ℝ) ≤ (k : ℝ)+2)
+    (by norm_num : (0:ℝ) ≤ 2), hR]
+  rfl
+
+def poissonCorrectionOrder (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
+    (P : D.localizers.ConstantProjection) (s : ℝ) :
+    D.localizers.Sobolev s →L[ℝ] D.localizers.Sobolev s :=
+  D.localizers.constantsOrder s ∘L D.poissonResidue m hm he P ∘L D.localizers.lower s 0
+
+lemma poissonCorrectionOrder_lower (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
+    (P : D.localizers.ConstantProjection) {s : ℝ} (hs : 0 ≤ s) (f : D.localizers.Sobolev s) :
+    D.localizers.lower s 0 (D.poissonCorrectionOrder m hm he P s f) =
+      P.proj (D.poissonInput m hm he P (D.localizers.lower s 0 f)) := by
+  change D.localizers.lower s 0 (D.localizers.embed s (Smooth.const
+    (P.coefficient (D.poissonInput m hm he P (D.localizers.lower s 0 f))))) = _
+  rw [D.localizers.lower_embed hs]
+  exact P.coefficient_spec _
+
+lemma poissonInput_step (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
+    (P : D.localizers.ConstantProjection) (k : ℕ)
+    (W : D.localizers.Sobolev (k : ℝ) →L[ℝ] D.localizers.Sobolev (k : ℝ))
+    (hW : ∀ f, D.localizers.lower (k : ℝ) 0 (W f) =
+      D.poissonInput m hm he P (D.localizers.lower (k : ℝ) 0 f)) :
+    ∃ V : D.localizers.Sobolev ((k : ℝ)+1) →L[ℝ] D.localizers.Sobolev ((k : ℝ)+1),
+      ∀ f, D.localizers.lower ((k : ℝ)+1) 0 (V f) =
+        D.poissonInput m hm he P (D.localizers.lower ((k : ℝ)+1) 0 f) := by
+  obtain ⟨R,hR⟩ := D.resolventGain_order m hm he k
+  let W' : D.localizers.Sobolev ((k : ℝ)+1) →L[ℝ] D.localizers.Sobolev ((k : ℝ)+1) :=
+    -ContinuousLinearMap.id ℝ _ -
+      D.poissonCorrectionOrder m hm he P ((k : ℝ)+1) +
+      (m^2 : ℝ) • (R ∘L W ∘L D.localizers.lower ((k : ℝ)+1) (k : ℝ))
+  refine ⟨W', fun f => ?_⟩
+  change D.localizers.lower ((k : ℝ)+1) 0
+    (-f - D.poissonCorrectionOrder m hm he P ((k : ℝ)+1) f +
+     (m^2 : ℝ) • R (W (D.localizers.lower ((k : ℝ)+1) (k : ℝ) f))) = _
+  apply LinearNormalization.feedback_lift
+    (D.localizers.lower (k : ℝ) 0) (D.localizers.lower ((k : ℝ)+1) (k : ℝ))
+    (D.localizers.lower ((k : ℝ)+1) 0) (D.poissonInput m hm he P) W
+    (P.proj ∘L D.poissonInput m hm he P)
+    (D.poissonCorrectionOrder m hm he P ((k : ℝ)+1))
+    (D.resolventZero m hm he) R (m^2)
+  · exact fun f => D.localizers.lower_lower (by linarith : (k : ℝ) ≤ (k : ℝ)+1)
+      (Nat.cast_nonneg k) f
+  · exact hW
+  · exact D.poissonCorrectionOrder_lower m hm he P (by positivity : (0:ℝ) ≤ (k : ℝ)+1)
+  · exact hR
+  · exact D.poissonInput_equation m hm he P
+
+ 
+
+theorem poissonInput_order (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
+    (P : D.localizers.ConstantProjection) (k : ℕ) :
+    ∃ W : D.localizers.Sobolev (k : ℝ) →L[ℝ] D.localizers.Sobolev (k : ℝ),
+      ∀ f, D.localizers.lower (k : ℝ) 0 (W f) =
+        D.poissonInput m hm he P (D.localizers.lower (k : ℝ) 0 f) := by
+  induction k with
+  | zero =>
+    rw [Nat.cast_zero]
+    refine ⟨D.poissonInput m hm he P, fun f => ?_⟩
+    simp only [D.localizers.lower_self, ContinuousLinearMap.id_apply]
+  | succ k ih =>
+    rw [Nat.cast_succ]
+    obtain ⟨W,hW⟩ := ih
+    exact D.poissonInput_step m hm he P k W hW
+
+theorem poissonRaw_order (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
+    (P : D.localizers.ConstantProjection) (k : ℕ) :
+    ∃ R : D.localizers.Sobolev (k : ℝ) →L[ℝ] D.localizers.Sobolev ((k : ℝ)+2),
+      ∀ f, D.localizers.lower ((k : ℝ)+2) 2 (R f) =
+        D.poissonRaw m hm he P (D.localizers.lower (k : ℝ) 0 f) := by
+  obtain ⟨W,hW⟩ := D.poissonInput_order m hm he P k
+  obtain ⟨R,hR⟩ := D.completedResolvent_order m hm he k
+  refine ⟨R ∘L W, fun f => ?_⟩
+  simp only [ContinuousLinearMap.comp_apply]
+  rw [hR, hW]
+  rfl
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/HigherResolvent.lean b/lean/OAI/Geometry/PrescribedPotential/HigherResolvent.lean
new file mode 100644
index 0000000000000000000000000000000000000000..717e39fd133cacae2616a1a5b87109ff32a9c2a8
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/HigherResolvent.lean
@@ -0,0 +1,77 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CompletedResolvent
+import OAI.Geometry.PrescribedPotential.PatchCutoffs
+
+namespace OAI
+
+/-! Higher Resolvent. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology
+open scoped ContDiff Classical
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace Localizers
+variable (D : Localizers A ι)
+
+lemma lower_lower {s t r : ℝ} (hst : t ≤ s) (htr : r ≤ t) (u : D.Sobolev s) :
+    D.lower t r (D.lower s t u) = D.lower s r u :=
+  congrArg (fun L : D.Sobolev s →L[ℝ] D.Sobolev r => L u) (D.lower_comp hst htr)
+
+end Localizers
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+ 
+
+theorem errorInverse_order (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
+    (k : ℕ) : ∃ R : D.localizers.Sobolev (k : ℝ) →L[ℝ] D.localizers.Sobolev (k : ℝ),
+      ∀ f, D.localizers.lower (k : ℝ) 0 (R f) =
+        D.errorInverse m hm he (D.localizers.lower (k : ℝ) 0 f) := by
+  induction k with
+  | zero =>
+    rw [Nat.cast_zero]
+    refine ⟨D.errorInverse m hm he, fun f => ?_⟩
+    simp only [D.localizers.lower_self, ContinuousLinearMap.id_apply]
+  | succ k ih =>
+    rw [Nat.cast_succ]
+    obtain ⟨R, hR⟩ := ih
+    let U := D.localizers.extendCore (k : ℝ) ((k : ℝ)+1) (D.error m hm)
+    let R' : D.localizers.Sobolev ((k : ℝ)+1) →L[ℝ] D.localizers.Sobolev ((k : ℝ)+1) :=
+      ContinuousLinearMap.id ℝ _ + U ∘L R ∘L D.localizers.lower ((k : ℝ)+1) (k : ℝ)
+    refine ⟨R', fun f => ?_⟩
+    change D.localizers.lower ((k : ℝ)+1) 0
+      (f + U (R (D.localizers.lower ((k : ℝ)+1) (k : ℝ) f))) = _
+    rw [map_add, D.localizers.extendCore_compat (Nat.cast_nonneg k) (by positivity : (0:ℝ) ≤ (k : ℝ)+1)
+      (D.error m hm) (D.error_bound_integer m hm k) (D.error_bound_zero m hm), hR,
+      D.localizers.lower_lower (by linarith : (k : ℝ) ≤ (k : ℝ)+1) (Nat.cast_nonneg k)]
+    have h := D.errorInverse_equation m hm he (D.localizers.lower ((k : ℝ)+1) 0 f)
+    change _ = D.errorInverse m hm he (D.localizers.lower ((k : ℝ)+1) 0 f)
+    exact (sub_eq_iff_eq_add.mp h).symm
+
+ 
+theorem completedResolvent_order (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
+    (k : ℕ) : ∃ R : D.localizers.Sobolev (k : ℝ) →L[ℝ] D.localizers.Sobolev ((k : ℝ)+2),
+      ∀ f, D.localizers.lower ((k : ℝ)+2) 2 (R f) =
+        D.completedResolvent m hm he (D.localizers.lower (k : ℝ) 0 f) := by
+  obtain ⟨R, hR⟩ := D.errorInverse_order m hm he k
+  refine ⟨D.localizers.extendCore (k : ℝ) ((k : ℝ)+2) (D.parametrix m hm) ∘L R, fun f => ?_⟩
+  simp only [ContinuousLinearMap.comp_apply]
+  rw [D.localizers.extendCore_compat (Nat.cast_nonneg k)
+    (by linarith [Nat.cast_nonneg (α := ℝ) k] : (2:ℝ) ≤ (k : ℝ)+2)
+    (D.parametrix m hm) (D.parametrix_bound m hm k) (D.completedParametrix_bound m hm), hR]
+  rfl
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/HolomorphicDirection.lean b/lean/OAI/Geometry/PrescribedPotential/HolomorphicDirection.lean
new file mode 100644
index 0000000000000000000000000000000000000000..1f89c7f8e3a3866b9f8755c5aa7c33c90b660688
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/HolomorphicDirection.lean
@@ -0,0 +1,67 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.KaehlerClosedDerivatives
+import OAI.Geometry.PrescribedPotential.MatrixWirtinger
+
+namespace OAI
+
+/-! Holomorphic Direction. -/
+
+section
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace KaehlerCalculus
+variable {n : ℕ}
+
+lemma wderiv_direction_sum {α : Type*} (s : Finset α) (a : ℂ) (v : α → V n)
+    (f : V n → ℂ) (z : V n) :
+    wderiv a (∑ i ∈ s, v i) f z = ∑ i ∈ s, wderiv a (v i) f z := by
+  simp only [wderiv,Finset.smul_sum,map_sum,Finset.mul_sum]
+  rw [← Finset.sum_div,Finset.sum_add_distrib]
+
+lemma dz_direction_smul (c : ℂ) (v : V n) (f : V n → ℂ) (z : V n) :
+    dz (c • v) f z = c*dz v f z := by
+  have he : c • v = c.re • v + c.im • (Complex.I • v) := by
+    ext j
+    change c*v j = (c.re:ℂ)*v j+(c.im:ℂ)*(Complex.I*v j)
+    conv_lhs => rw [← Complex.re_add_im c]
+    ring
+  have hi : Complex.I • (c • v) = -c.im • v + c.re • (Complex.I • v) := by
+    ext j
+    change Complex.I*(c*v j) = ((-c.im:ℝ):ℂ)*v j+(c.re:ℂ)*(Complex.I*v j)
+    conv_lhs => rw [← Complex.re_add_im c]
+    simp only [Complex.ofReal_neg]
+    ring_nf
+    simp only [Complex.I_sq]
+    ring
+  unfold dz wderiv
+  rw [hi,he]
+  simp only [map_add,map_smul,Complex.real_smul,Complex.ofReal_neg]
+  conv_rhs => arg 1; rw [← Complex.re_add_im c]
+  ring_nf
+  simp only [Complex.I_sq]
+  ring
+
+lemma dz_direction_coordinates (v : V n) (f : V n → ℂ) (z : V n) :
+    dz v f z = ∑ i, v i*dz (e i) f z := by
+  have he : v = ∑ i, v i • e i := by ext j; simp [e,Pi.single_apply]
+  calc
+    _ = dz (∑ i, v i • e i) f z := congrArg (fun u => dz u f z) he
+    _ = _ := by
+      unfold dz
+      rw [wderiv_direction_sum]
+      apply Finset.sum_congr rfl
+      intro i _
+      exact dz_direction_smul (v i) (e i) f z
+
+lemma mderiv_hol_coordinates (v : V n) (M : V n → Matrix (Fin n) (Fin n) ℂ) (z : V n) :
+    mderiv (-Complex.I) v M z = ∑ i, v i • mderiv (-Complex.I) (e i) M z := by
+  ext a b
+  simpa only [Matrix.sum_apply,Matrix.smul_apply,smul_eq_mul,mderiv,dz] using dz_direction_coordinates v (fun y => M y a b) z
+end KaehlerCalculus
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/IntegerBounds.lean b/lean/OAI/Geometry/PrescribedPotential/IntegerBounds.lean
new file mode 100644
index 0000000000000000000000000000000000000000..2f0a633958acd11c8f46ef067b7775aaaced6e6c
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/IntegerBounds.lean
@@ -0,0 +1,146 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.ChartTransportHigher
+import OAI.Geometry.PrescribedPotential.ParameterCoreBounds
+
+namespace OAI
+
+/-! Integer Bounds. -/
+
+section
+
+ 
+
+noncomputable section
+open Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory TemperedDistribution LineDeriv
+open scoped SchwartzMap ContDiff Classical Laplacian
+
+namespace SobolevChart
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+
+def raiseZero (s : ℝ) : L2 E →L[ℂ] L2 E :=
+  liftOperator s (s + 1) (besselPotential E ℂ (-2)) (fun u hu => by
+    rw [memSobolev_besselPotential_iff]
+    exact hu.mono (by linarith))
+
+def raiseDeriv (s : ℝ) (v : E) : L2 E →L[ℂ] L2 E :=
+  liftOperator s (s + 1) ((besselPotential E ℂ (-2)).comp (lineDerivOpCLM ℂ 𝓢'(E, ℂ) v))
+    (fun u hu => by
+      change MemSobolev (s + 1) 2 (besselPotential E ℂ (-2) (∂_{v} u))
+      rw [memSobolev_besselPotential_iff]
+      exact hu.lineDerivOp.mono (by linarith))
+
+lemma realize_raiseZero (s : ℝ) (u : L2 E) :
+    realize (s + 1) (raiseZero s u) = besselPotential E ℂ (-2) (realize s u) :=
+  realize_liftOperator _ _ _ _ _
+
+lemma realize_raiseDeriv (s : ℝ) (v : E) (u : L2 E) :
+    realize (s + 1) (raiseDeriv s v u) = besselPotential E ℂ (-2) (∂_{v} (realize s u)) :=
+  realize_liftOperator _ _ _ _ _
+
+lemma schwartzCoord_raise_one (s : ℝ) (f : 𝓢(E, ℂ)) :
+    schwartzCoord (s + 1) f = raiseZero s (schwartzCoord s f) -
+      (((2 * Real.pi)^2 : ℝ) : ℂ)⁻¹ • ∑ j,
+        raiseDeriv s (stdOrthonormalBasis ℝ E j)
+          (schwartzCoord s (∂_{stdOrthonormalBasis ℝ E j} f)) := by
+  apply realize_injective (s + 1)
+  simp only [map_sub, map_smul, map_sum, realize_raiseZero, realize_raiseDeriv,
+    realize_schwartzCoord]
+  have he := congrArg (besselPotential E ℂ (-2)) (bessel_two_laplacian (f : 𝓢'(E, ℂ)))
+  rw [besselPotential_besselPotential_apply] at he
+  norm_num only [neg_add_cancel, besselPotential_zero, ContinuousLinearMap.id_apply] at he
+  rw [map_sub, map_smul, TemperedDistribution.laplacian_eq_sum (stdOrthonormalBasis ℝ E), map_sum] at he
+  rw [he]
+  congr 2
+  apply Finset.sum_congr rfl
+  intro j _
+  rw [← lineDerivOp_toTemperedDistributionCLM_eq]
+
+lemma coreBound_raise_one {s t : ℝ} (T : 𝓢(E, ℂ) → 𝓢(E, ℂ))
+    (hT : CoreBound s t T)
+    (hD : ∀ j, CoreBound s t (fun f => ∂_{stdOrthonormalBasis ℝ E j} (T f))) :
+    CoreBound s (t + 1) T := by
+  obtain ⟨C, hC, hc⟩ := hT
+  choose D hD0 hdb using hD
+  let c : ℝ := ‖(((2 * Real.pi)^2 : ℝ) : ℂ)⁻¹‖
+  refine ⟨‖raiseZero (E := E) t‖ * C + c * ∑ j, ‖raiseDeriv t (stdOrthonormalBasis ℝ E j)‖ * D j,
+    add_nonneg (mul_nonneg (norm_nonneg _) hC)
+      (mul_nonneg (norm_nonneg _) (Finset.sum_nonneg (fun j _ => mul_nonneg (norm_nonneg _) (hD0 j)))), fun f => ?_⟩
+  rw [schwartzCoord_raise_one]
+  calc
+    _ ≤ ‖raiseZero t (schwartzCoord t (T f))‖ +
+        c * ‖∑ j, raiseDeriv t (stdOrthonormalBasis ℝ E j)
+          (schwartzCoord t (∂_{stdOrthonormalBasis ℝ E j} (T f)))‖ := by
+      simpa only [norm_smul] using norm_sub_le
+        (raiseZero t (schwartzCoord t (T f)))
+        ((((2 * Real.pi)^2 : ℝ) : ℂ)⁻¹ • ∑ j, raiseDeriv t (stdOrthonormalBasis ℝ E j)
+          (schwartzCoord t (∂_{stdOrthonormalBasis ℝ E j} (T f))))
+    _ ≤ (‖raiseZero (E := E) t‖ * C) * ‖schwartzCoord s f‖ +
+        c * ∑ j, (‖raiseDeriv t (stdOrthonormalBasis ℝ E j)‖ * D j) * ‖schwartzCoord s f‖ := by
+      apply add_le_add
+      · exact ((raiseZero t).le_opNorm _).trans (by
+          simpa only [mul_assoc] using
+            (mul_le_mul_of_nonneg_left (hc f) (norm_nonneg (raiseZero (E := E) t))))
+      · apply mul_le_mul_of_nonneg_left _ (norm_nonneg _)
+        apply (norm_sum_le _ _).trans
+        apply Finset.sum_le_sum
+        intro j _
+        exact ((raiseDeriv t (stdOrthonormalBasis ℝ E j)).le_opNorm _).trans (by
+          simpa only [mul_assoc] using mul_le_mul_of_nonneg_left (hdb j f)
+            (norm_nonneg (raiseDeriv t (stdOrthonormalBasis ℝ E j))))
+    _ = _ := by rw [← Finset.sum_mul]; ring
+
+lemma coreBound_of_words_integer (k : ℕ) {s t : ℝ} (T : 𝓢(E, ℂ) → 𝓢(E, ℂ))
+    (h : ∀ ws : List E, ws.length ≤ k → CoreBound s t (fun f => schwartzWord ws (T f))) :
+    CoreBound s (t + k) T := by
+  induction k generalizing T with
+  | zero => simpa [schwartzWord] using h [] (by simp)
+  | succ k ih =>
+    have ht := ih T (fun ws hw => h ws (by omega))
+    have hd (j : Fin (Module.finrank ℝ E)) := ih
+      (fun f => ∂_{stdOrthonormalBasis ℝ E j} (T f))
+      (fun ws hw => by
+        have hh := h (ws ++ [stdOrthonormalBasis ℝ E j])
+          (by simp only [List.length_append, List.length_cons, List.length_nil]; omega)
+        simpa only [schwartzWord_append, schwartzWord, ContinuousLinearMap.comp_apply,
+          ContinuousLinearMap.id_apply, lineDerivOpCLM_apply] using hh)
+    convert coreBound_raise_one T ht hd using 1; push_cast; ring
+
+end SobolevChart
+
+namespace GlobalElliptic
+open SobolevChart
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+
+ 
+
+theorem chartTransport_integer_bound (e : OpenPartialHomeomorph E E)
+    (he : ContDiffOn ℝ ∞ e e.source) (hs : ContDiffOn ℝ ∞ e.symm e.target)
+    (κ : ChartCutoff e.source) (k : ℕ) :
+    CoreBound (k : ℝ) (k : ℝ) (chartTransport e he κ) := by
+  have hh := coreBound_of_words_integer k (s := (k : ℝ)) (t := 0) (chartTransport e he κ)
+    (fun ws hw => chartTransport_word_bound e he hs ws κ (by exact_mod_cast hw))
+  simpa only [zero_add] using hh
+end GlobalElliptic
+
+namespace FrozenPoisson.ParameterRegularity
+open SobolevChart EllipticKernel
+open scoped ComplexOrder MatrixOrder
+variable {n : ℕ} {ι : Type*} [Fintype ι]
+
+theorem schwartzLocal_integer_bound (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (t : ℝ) (ht : 1 ≤ t) (v : ι → EC n) (a : SmoothCoefficients ι (EC n))
+    (hsmall : perturbationBound v (coefficientBCF a) * ellipticBound H hH < 1) (k : ℕ) :
+    CoreBound (k : ℝ) ((k : ℝ) + 2) (schwartzLocal H hH t ht v a hsmall) := by
+  rw [add_comm]
+  apply coreBound_of_words_integer k
+  intro ws hw
+  apply schwartzLocal_word_bound H hH t ht v a hsmall ws
+  exact_mod_cast hw
+end FrozenPoisson.ParameterRegularity
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/InverseTraceCalculus.lean b/lean/OAI/Geometry/PrescribedPotential/InverseTraceCalculus.lean
new file mode 100644
index 0000000000000000000000000000000000000000..3d0d899df24ea99ccaec13c48ad1c70395558063
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/InverseTraceCalculus.lean
@@ -0,0 +1,83 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.MatrixWirtinger
+
+namespace OAI
+
+/-! Inverse Trace Calculus. -/
+
+section
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace KaehlerCalculus
+variable {n : ℕ}
+
+lemma second_mderiv_inverse_at_one {U : Set (V n)} (hU : IsOpen U)
+    {M : V n → Matrix (Fin n) (Fin n) ℂ} (hM : ContDiffOn ℝ ∞ M U)
+    (hn : ∀ y ∈ U, (M y).det ≠ 0) {z : V n} (hz : z ∈ U) (hMz : M z = 1)
+    (a b : ℂ) (u v : V n) :
+    mderiv b u (mderiv a v (fun y => (M y)⁻¹)) z =
+      mderiv b u M z*mderiv a v M z + mderiv a v M z*mderiv b u M z -
+        mderiv b u (mderiv a v M) z := by
+  have hs := hM.contDiffAt (hU.mem_nhds hz)
+  have hi := (MatrixSmoothGeneral.inverse hM hn).contDiffAt (hU.mem_nhds hz)
+  have hd := mderiv_smooth hs a v
+  have he : mderiv a v (fun y => (M y)⁻¹) =ᶠ[𝓝 z]
+      (fun y => -(M y)⁻¹*mderiv a v M y*(M y)⁻¹) := by
+    filter_upwards [hU.mem_nhds hz] with y hy
+    exact mderiv_inverse hU hM hn hy a v
+  rw [mderiv_congr he,mderiv_mul (matrix_smooth_mul hi.neg hd) hi,
+    mderiv_mul hi.neg hd,mderiv_neg hi,mderiv_inverse hU hM hn hz,hMz]
+  simp only [inv_one,neg_mul,one_mul,mul_one,neg_neg]
+  noncomm_ring
+
+lemma second_mderiv_mul {M N : V n → Matrix (Fin n) (Fin n) ℂ} {z : V n}
+    (hM : ContDiffAt ℝ ∞ M z) (hN : ContDiffAt ℝ ∞ N z)
+    (a b : ℂ) (u v : V n) :
+    mderiv b u (mderiv a v (fun y => M y*N y)) z =
+    mderiv b u (mderiv a v M) z*N z + mderiv a v M z*mderiv b u N z +
+      mderiv b u M z*mderiv a v N z + M z*mderiv b u (mderiv a v N) z := by
+  have he : mderiv a v (fun y => M y*N y) =ᶠ[𝓝 z]
+      (fun y => mderiv a v M y*N y+M y*mderiv a v N y) := by
+    filter_upwards [(hM.of_le (show (1 : WithTop ℕ∞) ≤ ∞ by simp)).eventually (by decide),
+      (hN.of_le (show (1 : WithTop ℕ∞) ≤ ∞ by simp)).eventually (by decide)] with y hyM hyN
+    exact mderiv_mul_differentiable (hyM.differentiableAt (by simp))
+      (hyN.differentiableAt (by simp)) a v
+  rw [mderiv_congr he,mderiv_add (matrix_smooth_mul (mderiv_smooth hM a v) hN)
+    (matrix_smooth_mul hM (mderiv_smooth hN a v)),
+    mderiv_mul (mderiv_smooth hM a v) hN,mderiv_mul hM (mderiv_smooth hN a v)]
+  abel
+
+lemma second_wderiv_trace {M : V n → Matrix (Fin n) (Fin n) ℂ} {z : V n}
+    (hM : ContDiffAt ℝ ∞ M z) (a b : ℂ) (u v : V n) :
+    wderiv b u (wderiv a v (fun y => (M y).trace)) z =
+      (mderiv b u (mderiv a v M) z).trace := by
+  have he : wderiv a v (fun y => (M y).trace) =ᶠ[𝓝 z]
+      (fun y => (mderiv a v M y).trace) := by
+    filter_upwards [(hM.of_le (show (1 : WithTop ℕ∞) ≤ ∞ by simp)).eventually (by decide)] with y hy
+    exact mderiv_trace_differentiable (hy.differentiableAt (by simp)) a v
+  rw [wderiv_congr he,mderiv_trace (mderiv_smooth hM a v)]
+
+lemma second_wderiv_inverse_trace_at_one {U : Set (V n)} (hU : IsOpen U)
+    {M G : V n → Matrix (Fin n) (Fin n) ℂ} (hM : ContDiffOn ℝ ∞ M U)
+    (hn : ∀ y ∈ U, (M y).det ≠ 0) {z : V n} (hz : z ∈ U) (hMz : M z = 1)
+    (hG : ContDiffAt ℝ ∞ G z) (a b : ℂ) (u v : V n) :
+    wderiv b u (wderiv a v (fun y => ((M y)⁻¹*G y).trace)) z =
+      ((mderiv b u M z*mderiv a v M z + mderiv a v M z*mderiv b u M z -
+        mderiv b u (mderiv a v M) z)*G z -
+        mderiv a v M z*mderiv b u G z - mderiv b u M z*mderiv a v G z +
+        mderiv b u (mderiv a v G) z).trace := by
+  have hi := (MatrixSmoothGeneral.inverse hM hn).contDiffAt (hU.mem_nhds hz)
+  rw [second_wderiv_trace (matrix_smooth_mul hi hG),second_mderiv_mul hi hG,
+    second_mderiv_inverse_at_one hU hM hn hz hMz,
+    mderiv_inverse hU hM hn hz,mderiv_inverse hU hM hn hz,hMz]
+  congr 1
+  simp only [inv_one,neg_mul,one_mul,mul_one]
+  noncomm_ring
+end KaehlerCalculus
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/InverseTraceFunction.lean b/lean/OAI/Geometry/PrescribedPotential/InverseTraceFunction.lean
new file mode 100644
index 0000000000000000000000000000000000000000..91cf49c685ee6331fee5e545a8182fe49400d49a
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/InverseTraceFunction.lean
@@ -0,0 +1,102 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.ChernLuIdentity
+import OAI.Geometry.PrescribedPotential.TracePositivity
+
+namespace OAI
+
+/-! Inverse Trace Function. -/
+
+section
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace KaehlerCalculus
+variable {n : ℕ}
+
+def inverseTrace (M G : V n → Matrix (Fin n) (Fin n) ℂ) (z : V n) : ℝ :=
+  ((M z)⁻¹*G z).trace.re
+
+lemma matrix_trace_smooth {M : V n → Matrix (Fin n) (Fin n) ℂ} {z : V n}
+    (hM : ContDiffAt ℝ ∞ M z) : ContDiffAt ℝ ∞ (fun y => (M y).trace) z :=
+  ContDiffAt.sum (fun k _ => entry_smooth hM k k)
+
+lemma inverseTrace_smooth {U : Set (V n)} (hU : IsOpen U)
+    {M G : V n → Matrix (Fin n) (Fin n) ℂ} (hM : ContDiffOn ℝ ∞ M U)
+    (hp : ∀ y ∈ U, (M y).PosDef) (hG : ContDiffOn ℝ ∞ G U) :
+    ContDiffOn ℝ ∞ (inverseTrace M G) U := by
+  intro z hz
+  apply ContDiffAt.contDiffWithinAt
+  apply Complex.reCLM.contDiff.contDiffAt.comp z
+  exact matrix_trace_smooth (matrix_smooth_mul
+    ((MatrixSmoothGeneral.inverse hM (fun y hy => ne_of_gt (hp y hy).det_pos)).contDiffAt (hU.mem_nhds hz))
+    (hG.contDiffAt (hU.mem_nhds hz)))
+
+lemma inverseTrace_real {M G : V n → Matrix (Fin n) (Fin n) ℂ} {z : V n}
+    (hM : (M z).PosDef) (hG : (G z).IsHermitian) :
+    (inverseTrace M G z : ℂ) = ((M z)⁻¹*G z).trace :=
+  trace_mul_real hM.inv.isHermitian hG
+
+lemma inverseTrace_pos [Nonempty (Fin n)] {M G : V n → Matrix (Fin n) (Fin n) ℂ}
+    {z : V n} (hM : (M z).PosDef) (hG : (G z).PosDef) :
+    0 < inverseTrace M G z := trace_mul_pos hM.inv hG
+
+lemma second_wderiv_congr {f g : V n → ℂ} {z : V n} (he : f =ᶠ[𝓝 z] g)
+    (a b : ℂ) (u v : V n) :
+    wderiv b u (wderiv a v f) z = wderiv b u (wderiv a v g) z := by
+  exact wderiv_congr (he.eventually_nhds.mono (fun _ h => wderiv_congr h a v)) b u
+
+lemma inverseTrace_dz_at_one {U : Set (V n)} (hU : IsOpen U)
+    {M G : V n → Matrix (Fin n) (Fin n) ℂ} (hM : ContDiffOn ℝ ∞ M U)
+    (hp : ∀ y ∈ U, (M y).PosDef) (hG : ContDiffOn ℝ ∞ G U)
+    (hGh : ∀ y ∈ U, (G y).IsHermitian) {z : V n} (hz : z ∈ U) (hMz : M z = 1)
+    (v : V n) : dz v (fun y => (inverseTrace M G y : ℂ)) z =
+      (mderiv (-Complex.I) v G z-mderiv (-Complex.I) v M z*G z).trace := by
+  have he : (fun y => (inverseTrace M G y : ℂ)) =ᶠ[𝓝 z]
+      (fun y => ((M y)⁻¹*G y).trace) := by
+    filter_upwards [hU.mem_nhds hz] with y hy
+    exact inverseTrace_real (hp y hy) (hGh y hy)
+  have hi := (MatrixSmoothGeneral.inverse hM (fun y hy => ne_of_gt (hp y hy).det_pos)).contDiffAt (hU.mem_nhds hz)
+  have hg := hG.contDiffAt (hU.mem_nhds hz)
+  change wderiv (-Complex.I) v _ z = _
+  rw [wderiv_congr he,mderiv_trace (matrix_smooth_mul hi hg),mderiv_mul hi hg,
+    mderiv_inverse hU hM (fun y hy => ne_of_gt (hp y hy).det_pos) hz,hMz]
+  simp only [inv_one,neg_mul,one_mul,mul_one]
+  congr 1
+  abel
+
+lemma inverseTrace_second_at_one {U : Set (V n)} (hU : IsOpen U)
+    {M G : V n → Matrix (Fin n) (Fin n) ℂ} (hM : ContDiffOn ℝ ∞ M U)
+    (hp : ∀ y ∈ U, (M y).PosDef)
+    (hclosed : ∀ y ∈ U, ∀ u v w : V n,
+      fderiv ℝ (fun q => Anticanonical.ComplexAtlas.fundamentalForm (M q) v w) y u +
+      fderiv ℝ (fun q => Anticanonical.ComplexAtlas.fundamentalForm (M q) w u) y v +
+      fderiv ℝ (fun q => Anticanonical.ComplexAtlas.fundamentalForm (M q) u v) y w = 0)
+    (hG : ContDiffOn ℝ ∞ G U) (hGh : ∀ y ∈ U, (G y).IsHermitian)
+    {z : V n} (hz : z ∈ U) (hMz : M z = 1) :
+    (PotentialKaehler.potentialMatrix (inverseTrace M G) z).trace.re =
+      -(PotentialKaehler.potentialMatrix (fun y => Real.log (M y).det.re) z*G z).trace.re +
+      ∑ k, (mderiv (-Complex.I) (e k) M z*(mderiv (-Complex.I) (e k) M z)ᴴ*G z -
+        mderiv (-Complex.I) (e k) M z*(mderiv (-Complex.I) (e k) G z)ᴴ -
+        (mderiv (-Complex.I) (e k) M z)ᴴ*mderiv (-Complex.I) (e k) G z +
+        mderiv Complex.I (e k) (mderiv (-Complex.I) (e k) G) z).trace.re := by
+  have he : (fun y => (inverseTrace M G y : ℂ)) =ᶠ[𝓝 z]
+      (fun y => ((M y)⁻¹*G y).trace) := by
+    filter_upwards [hU.mem_nhds hz] with y hy
+    exact inverseTrace_real (hp y hy) (hGh y hy)
+  have hs := (inverseTrace_smooth hU hM hp hG).contDiffAt (hU.mem_nhds hz)
+  have htr : (PotentialKaehler.potentialMatrix (inverseTrace M G) z).trace =
+      ∑ k, dzbar (e k) (dz (e k) (fun y => ((M y)⁻¹*G y).trace)) z := by
+    apply Finset.sum_congr rfl
+    intro k _
+    change PotentialKaehler.potentialMatrix (inverseTrace M G) z k k = _
+    rw [potentialMatrix_eq_dzbar_dz hs]
+    exact second_wderiv_congr he _ _ _ _
+  rw [htr,inverse_trace_ricci_identity_at_one hU hM hp hclosed hG hGh hz hMz]
+  simp only [Complex.add_re,Complex.neg_re,Complex.re_sum]
+end KaehlerCalculus
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/JetDissipation.lean b/lean/OAI/Geometry/PrescribedPotential/JetDissipation.lean
new file mode 100644
index 0000000000000000000000000000000000000000..a1561355ca42d8c3a463cd1dbd9e7aa6b5a7d925
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/JetDissipation.lean
@@ -0,0 +1,98 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.DirectionalJetEnergy
+
+namespace OAI
+
+/-! Jet Dissipation. -/
+
+section
+
+ 
+noncomputable section
+open Set Filter Topology Finset Module
+open scoped ContDiff
+namespace HigherJet
+variable {E F : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [NormedAddCommGroup F] [InnerProductSpace ℝ F]
+  {ι : Type*} [Fintype ι]
+
+omit [Fintype ι] in
+lemma jetFamily_cons (e : ι → E) (m : ℕ) (u : E → F) (i : ι) (σ : Fin m → ι) :
+    jetFamily e (m+1) u (Fin.cons i σ) = dir (e i) (jetFamily e m u σ) := by
+  simp only [jetFamily,List.ofFn_succ,Fin.cons_zero,Fin.cons_succ,word]
+
+lemma jetEnergy_succ (e : ι → E) (m : ℕ) (u : E → F) (x : E) :
+    jetEnergy e (m+1) u x = familyDissipation e (jetFamily e m u) x := by
+  unfold jetEnergy familyEnergy familyDissipation frameGradientSq
+  rw [← Equiv.sum_comp (Fin.consEquiv (fun _ : Fin (m+1) => ι))
+    (fun σ => ‖jetFamily e (m+1) u σ x‖^2)]
+  simp only [Fintype.sum_prod_type]
+  change (∑ i : ι, ∑ σ : Fin m → ι, ‖jetFamily e (m+1) u (Fin.cons i σ) x‖^2) = _
+  simp only [jetFamily_cons]
+  exact Finset.sum_comm
+
+variable [FiniteDimensional ℝ E] [FiniteDimensional ℝ F]
+def linearArray (e : ι → E) : (E →L[ℝ] F) →ₗ[ℝ] PiLp 2 (fun _ : ι => F) where
+  toFun T := WithLp.toLp 2 (fun i => T (e i))
+  map_add' T R := by ext i; rfl
+  map_smul' c T := by ext i; rfl
+
+omit [Fintype ι] [FiniteDimensional ℝ E] [FiniteDimensional ℝ F] in
+lemma linearArray_injective (e : Basis ι ℝ E) :
+    Function.Injective (linearArray (F := F) e) := by
+  intro T R h
+  apply ContinuousLinearMap.coe_injective
+  exact e.ext (fun i => congrArg (fun a : PiLp 2 (fun _ : ι => F) => a i) h)
+
+lemma linearArray_norm_control (e : Basis ι ℝ E) :
+    ∃ C : ℝ, 0 < C ∧ ∀ T : E →L[ℝ] F, ‖T‖ ≤ C*‖linearArray e T‖ := by
+  obtain ⟨C,hC,hb⟩ := (linearArray (F := F) e).injective_iff_antilipschitz.mp (linearArray_injective e)
+  refine ⟨C,by exact_mod_cast hC,fun T => ?_⟩
+  simpa only [map_zero,dist_zero_right] using hb.le_mul_dist T 0
+
+lemma linear_frame_coercivity (e : OrthonormalBasis ι ℝ E) :
+    ∃ C : ℝ, 0 < C ∧ ∀ (T : E →L[ℝ] F) (τ : E ≃L[ℝ] E) (B : ℝ),
+      0 ≤ B → ‖τ.symm.toContinuousLinearMap‖ ≤ B →
+      ∑ i, ‖T (e i)‖^2 ≤ C*B^2*(∑ i, ‖T (τ (e i))‖^2) := by
+  obtain ⟨A,hA,hb⟩ := linearArray_norm_control (F := F) e.toBasis
+  refine ⟨(Fintype.card ι+1)*A^2,by positivity,fun T τ B hB hτ => ?_⟩
+  have he : (T.comp τ.toContinuousLinearMap).comp τ.symm.toContinuousLinearMap = T := by ext x; simp
+  have ht : ‖T‖ ≤ A*‖linearArray e (T.comp τ.toContinuousLinearMap)‖*B := by
+    calc
+      _ = ‖(T.comp τ.toContinuousLinearMap).comp τ.symm.toContinuousLinearMap‖ := congrArg norm he.symm
+      _ ≤ ‖T.comp τ.toContinuousLinearMap‖*‖τ.symm.toContinuousLinearMap‖ := ContinuousLinearMap.opNorm_comp_le _ _
+      _ ≤ (A*‖linearArray e (T.comp τ.toContinuousLinearMap)‖)*B :=
+        mul_le_mul (hb _) hτ (norm_nonneg _) (by positivity)
+  have hs : ‖T‖^2 ≤ A^2*B^2*(∑ i, ‖T (τ (e i))‖^2) := by
+    have h := pow_le_pow_left₀ (norm_nonneg T) ht 2
+    rw [mul_pow,mul_pow,PiLp.norm_sq_eq_of_L2] at h
+    simpa only [linearArray,LinearMap.coe_mk,AddHom.coe_mk,PiLp.toLp_apply,
+      ContinuousLinearMap.comp_apply,ContinuousLinearEquiv.coe_coe,mul_assoc,mul_left_comm,mul_comm] using h
+  calc
+    _ ≤ ∑ _i : ι, ‖T‖^2 := by
+      apply Finset.sum_le_sum
+      intro i _
+      exact pow_le_pow_left₀ (norm_nonneg _) (by simpa only [e.norm_eq_one,mul_one] using T.le_opNorm (e i)) 2
+    _ = (Fintype.card ι : ℝ)*‖T‖^2 := by simp
+    _ ≤ (Fintype.card ι+1)*(A^2*B^2*(∑ i, ‖T (τ (e i))‖^2)) := by
+      have h1 := mul_le_mul_of_nonneg_left hs (by positivity : 0 ≤ (Fintype.card ι : ℝ))
+      have h2 : 0 ≤ A^2*B^2*(∑ i, ‖T (τ (e i))‖^2) := by positivity
+      nlinarith only [h1,h2]
+    _ = _ := by ring
+
+lemma jetEnergy_frame_coercivity (e : OrthonormalBasis ι ℝ E) :
+    ∃ C : ℝ, 0 < C ∧ ∀ (m : ℕ) (u : E → F) (x : E) (τ : E ≃L[ℝ] E) (B : ℝ),
+      0 ≤ B → ‖τ.symm.toContinuousLinearMap‖ ≤ B →
+      jetEnergy e (m+1) u x ≤ C*B^2*familyDissipation (fun i => τ (e i)) (jetFamily e m u) x := by
+  obtain ⟨C,hC,hb⟩ := linear_frame_coercivity (F := F) e
+  refine ⟨C,hC,fun m u x τ B hB hτ => ?_⟩
+  rw [jetEnergy_succ]
+  unfold familyDissipation frameGradientSq dir
+  rw [Finset.mul_sum]
+  exact Finset.sum_le_sum (fun σ _ => hb (fderiv ℝ (jetFamily e m u σ) x) τ B hB hτ)
+end HigherJet
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/JetEnergyCoercivity.lean b/lean/OAI/Geometry/PrescribedPotential/JetEnergyCoercivity.lean
new file mode 100644
index 0000000000000000000000000000000000000000..6ff80e0e7a00ca0706c8035946dd333cbd0ad7fd
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/JetEnergyCoercivity.lean
@@ -0,0 +1,85 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.RealJetEnergy
+
+namespace OAI
+
+/-! Jet Energy Coercivity. -/
+
+section
+
+ 
+noncomputable section
+open Set Filter Topology Finset
+open scoped ContDiff
+namespace HigherJet
+variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  [NormedAddCommGroup F] [InnerProductSpace ℝ F]
+  {ι κ : Type*} [Fintype ι] [Fintype κ]
+
+lemma family_inner_lower (f g : κ → F) :
+    -Real.sqrt (∑ j, ‖f j‖^2)*Real.sqrt (∑ j, ‖g j‖^2) ≤ ∑ j, inner ℝ (f j) (g j) := by
+  let a : PiLp 2 (fun _ : κ => F) := WithLp.toLp 2 f
+  let b : PiLp 2 (fun _ : κ => F) := WithLp.toLp 2 g
+  have ha : Real.sqrt (∑ j, ‖f j‖^2) = ‖a‖ := by
+    have he : ‖a‖^2 = ∑ j, ‖f j‖^2 := by rw [PiLp.norm_sq_eq_of_L2]
+    rw [← he,Real.sqrt_sq (norm_nonneg _)]
+  have hb : Real.sqrt (∑ j, ‖g j‖^2) = ‖b‖ := by
+    have he : ‖b‖^2 = ∑ j, ‖g j‖^2 := by rw [PiLp.norm_sq_eq_of_L2]
+    rw [← he,Real.sqrt_sq (norm_nonneg _)]
+  rw [ha,hb]
+  simpa only [neg_mul,PiLp.inner_apply,a,b,PiLp.toLp_apply] using
+    (neg_le_neg (abs_real_inner_le_norm a b)).trans (neg_abs_le (inner ℝ a b))
+
+lemma familyEnergy_laplace_lower {U : Set E} (hU : IsOpen U) {f : κ → E → F}
+    (hf : ∀ j, ContDiffOn ℝ ∞ (f j) U) {x : E} (hx : x ∈ U) (v : ι → E) :
+    2*familyDissipation v f x - 2*Real.sqrt (familyEnergy f x)*
+      Real.sqrt (∑ j, ‖frameLaplace v (f j) x‖^2) ≤ frameLaplace v (familyEnergy f) x := by
+  rw [familyEnergy_laplace hU hf hx]
+  have hb := family_inner_lower (fun j => f j x) (fun j => frameLaplace v (f j) x)
+  change -Real.sqrt (familyEnergy f x)*_ ≤ _ at hb
+  linarith only [hb]
+end HigherJet
+
+namespace BernsteinEstimate
+lemma energy_first_numeric {r d B C L : ℝ} (hr : 0 ≤ r) (hd : 0 ≤ d)
+    (hC : 0 ≤ C) (hrC : r ≤ C) (hB : B ≤ C*(1+d))
+    (hL : 2*d^2-2*r*B ≤ L) :
+    d^2-(2*C^2+C^4) ≤ L := by
+  have h1 := mul_le_mul_of_nonneg_left hB (by positivity : 0 ≤ 2*r)
+  have h2 := mul_le_mul_of_nonneg_right hrC (by positivity : 0 ≤ 2*C*(1+d))
+  nlinarith only [hL,h1,h2,sq_nonneg (d-C^2)]
+
+lemma energy_cubic_numeric {r d B C L : ℝ} (hr : 0 ≤ r) (_hd : 0 ≤ d)
+    (hC : 0 ≤ C) (hB : B ≤ C*(1+r^2+d))
+    (hL : 2*d^2-2*r*B ≤ L) :
+    d^2-(4*C+2*C^2)*(1+r^3) ≤ L := by
+  have hrb : r ≤ 1+r^3 := by
+    by_cases h : r ≤ 1
+    · nlinarith [pow_nonneg hr 3]
+    · have h' : 1 ≤ r := le_of_not_ge h
+      have hh := pow_le_pow_left₀ (by norm_num : (0:ℝ) ≤ 1) h' 2
+      nlinarith [mul_nonneg hr (by linarith : 0 ≤ r^2-1)]
+  have hr2b : r^2 ≤ 1+r^3 := by
+    by_cases h : r ≤ 1
+    · nlinarith [pow_nonneg hr 3]
+    · nlinarith [mul_nonneg (sq_nonneg r) (by linarith : 0 ≤ r-1)]
+  have h1 := mul_le_mul_of_nonneg_left hB (by positivity : 0 ≤ 2*r)
+  have h2 := mul_le_mul_of_nonneg_left hrb (by positivity : 0 ≤ 2*C)
+  have h3 := mul_le_mul_of_nonneg_left hr2b (sq_nonneg C)
+  have h4 : 0 ≤ 2*C+C^2*(1+r^3) := by positivity
+  nlinarith only [hL,h1,h2,h3,h4,sq_nonneg (d-C*r)]
+
+lemma energy_linear_numeric {r d B C L : ℝ} (hr : 0 ≤ r) (_hd : 0 ≤ d)
+    (hC : 0 ≤ C) (hB : B ≤ C*(1+r+d))
+    (hL : 2*d^2-2*r*B ≤ L) :
+    d^2-(4*C+C^2)*(1+r^2) ≤ L := by
+  have h1 := mul_le_mul_of_nonneg_left hB (by positivity : 0 ≤ 2*r)
+  have h2 := mul_le_mul_of_nonneg_left (sq_nonneg (r-1)) hC
+  have h3 : 0 ≤ 3*C+C^2+C*r^2 := by positivity
+  nlinarith only [hL,h1,h2,h3,sq_nonneg (d-C*r)]
+end BernsteinEstimate
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/JetEnergyEstimates.lean b/lean/OAI/Geometry/PrescribedPotential/JetEnergyEstimates.lean
new file mode 100644
index 0000000000000000000000000000000000000000..c636a93bbfd152c6bb847f07a3d3f2f94cfd3ae0
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/JetEnergyEstimates.lean
@@ -0,0 +1,77 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.JetEnergyUniform
+
+namespace OAI
+
+/-! Jet Energy Estimates. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Finset Module
+open scoped ContDiff
+namespace HigherJet
+variable {E F : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [NormedAddCommGroup F] [InnerProductSpace ℝ F]
+  {ι : Type*} [Fintype ι]
+
+lemma energy_first_of_residual {U : Set E} (hU : IsOpen U) {u : E → F}
+    (hu : ContDiffOn ℝ ∞ u U) (e : ι → E) {x : E} (hx : x ∈ U) (v : ι → E)
+    {A R : ℝ} (hA : 1 ≤ A) (hR : 0 ≤ R)
+    (hS : Real.sqrt (jetEnergy e 1 u x) ≤ A)
+    (hN : ‖iteratedFDeriv ℝ 2 u x‖ ≤ A*Real.sqrt (familyDissipation v (jetFamily e 1 u) x))
+    (hr : ∀ σ, ‖frameLaplace v (jetFamily e 1 u σ) x‖ ≤ R*(1+‖iteratedFDeriv ℝ 2 u x‖)) :
+    familyDissipation v (jetFamily e 1 u) x -
+      (2*(A+(Fintype.card (Fin 1 → ι)+1)*R*A)^2+(A+(Fintype.card (Fin 1 → ι)+1)*R*A)^4) ≤
+      frameLaplace v (jetEnergy e 1 u) x := by
+  apply BernsteinEstimate.first_energy_scaled (W := Real.sqrt (∑ σ, ‖frameLaplace v (jetFamily e 1 u σ) x‖^2)) (jetEnergy_nonneg _ _ _ _)
+    (familyDissipation_nonneg _ _ _) hA (show 0 ≤ (Fintype.card (Fin 1 → ι)+1)*R by positivity) hS
+  · exact (family_laplace_norm_bound (show 0 ≤ R*(1+‖iteratedFDeriv ℝ 2 u x‖) by positivity) hr).trans
+      (by simpa only [mul_assoc] using mul_le_mul_of_nonneg_left (mul_le_mul_of_nonneg_left
+        (add_le_add (le_refl 1) hN) hR) (show 0 ≤ (Fintype.card (Fin 1 → ι)+1 : ℝ) by positivity))
+  · exact familyEnergy_laplace_lower hU (jetFamily_smoothOn hU hu e 1) hx v
+
+lemma energy_second_of_residual {U : Set E} (hU : IsOpen U) {u : E → F}
+    (hu : ContDiffOn ℝ ∞ u U) (e : ι → E) {x : E} (hx : x ∈ U) (v : ι → E)
+    {A R : ℝ} (hA : 1 ≤ A) (hR : 0 ≤ R)
+    (hN : ‖iteratedFDeriv ℝ 2 u x‖ ≤ A*Real.sqrt (jetEnergy e 2 u x))
+    (hD : ‖iteratedFDeriv ℝ 3 u x‖ ≤ A*Real.sqrt (familyDissipation v (jetFamily e 2 u) x))
+    (hr : ∀ σ, ‖frameLaplace v (jetFamily e 2 u σ) x‖ ≤
+      R*(1+‖iteratedFDeriv ℝ 2 u x‖^2+‖iteratedFDeriv ℝ 3 u x‖)) :
+    familyDissipation v (jetFamily e 2 u) x -
+      (4*((Fintype.card (Fin 2 → ι)+1)*R*A^2)+2*((Fintype.card (Fin 2 → ι)+1)*R*A^2)^2)*
+        (1+jetEnergy e 2 u x*Real.sqrt (jetEnergy e 2 u x)) ≤
+      frameLaplace v (jetEnergy e 2 u) x := by
+  apply BernsteinEstimate.cubic_energy_scaled (W := Real.sqrt (∑ σ, ‖frameLaplace v (jetFamily e 2 u σ) x‖^2)) (jetEnergy_nonneg _ _ _ _)
+    (familyDissipation_nonneg _ _ _) hA (show 0 ≤ (Fintype.card (Fin 2 → ι)+1)*R by positivity)
+  · have hn2 := pow_le_pow_left₀ (norm_nonneg _) hN 2
+    rw [mul_pow,Real.sq_sqrt (jetEnergy_nonneg _ _ _ _)] at hn2
+    exact (family_laplace_norm_bound (show 0 ≤ R*(1+‖iteratedFDeriv ℝ 2 u x‖^2+‖iteratedFDeriv ℝ 3 u x‖) by positivity) hr).trans
+      (by simpa only [mul_assoc] using mul_le_mul_of_nonneg_left (mul_le_mul_of_nonneg_left
+        (add_le_add (add_le_add (le_refl 1) hn2) hD) hR) (show 0 ≤ (Fintype.card (Fin 2 → ι)+1 : ℝ) by positivity))
+  · exact familyEnergy_laplace_lower hU (jetFamily_smoothOn hU hu e 2) hx v
+
+lemma energy_high_of_residual {U : Set E} (hU : IsOpen U) {u : E → F}
+    (hu : ContDiffOn ℝ ∞ u U) (e : ι → E) (m : ℕ) {x : E} (hx : x ∈ U) (v : ι → E)
+    {A R : ℝ} (hA : 1 ≤ A) (hR : 0 ≤ R)
+    (hN : ‖iteratedFDeriv ℝ m u x‖ ≤ A*Real.sqrt (jetEnergy e m u x))
+    (hD : ‖iteratedFDeriv ℝ (m+1) u x‖ ≤ A*Real.sqrt (familyDissipation v (jetFamily e m u) x))
+    (hr : ∀ σ, ‖frameLaplace v (jetFamily e m u σ) x‖ ≤
+      R*(1+‖iteratedFDeriv ℝ m u x‖+‖iteratedFDeriv ℝ (m+1) u x‖)) :
+    familyDissipation v (jetFamily e m u) x -
+      (4*((Fintype.card (Fin m → ι)+1)*R*A)+((Fintype.card (Fin m → ι)+1)*R*A)^2)*
+        (1+jetEnergy e m u x) ≤ frameLaplace v (jetEnergy e m u) x := by
+  apply BernsteinEstimate.linear_energy_scaled (W := Real.sqrt (∑ σ, ‖frameLaplace v (jetFamily e m u σ) x‖^2)) (jetEnergy_nonneg _ _ _ _)
+    (familyDissipation_nonneg _ _ _) hA (show 0 ≤ (Fintype.card (Fin m → ι)+1)*R by positivity)
+  · exact (family_laplace_norm_bound (show 0 ≤ R*(1+‖iteratedFDeriv ℝ m u x‖+‖iteratedFDeriv ℝ (m+1) u x‖) by positivity) hr).trans
+      (by simpa only [mul_assoc] using mul_le_mul_of_nonneg_left (mul_le_mul_of_nonneg_left
+        (add_le_add (add_le_add (le_refl 1) hN) hD) hR) (show 0 ≤ (Fintype.card (Fin m → ι)+1 : ℝ) by positivity))
+  · exact familyEnergy_laplace_lower hU (jetFamily_smoothOn hU hu e m) hx v
+end HigherJet
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/JetEnergyUniform.lean b/lean/OAI/Geometry/PrescribedPotential/JetEnergyUniform.lean
new file mode 100644
index 0000000000000000000000000000000000000000..a170b9fc3d2f3d3f12054c1480295eac0cc8e4fe
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/JetEnergyUniform.lean
@@ -0,0 +1,118 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.JetDissipation
+import OAI.Geometry.PrescribedPotential.JetEnergyCoercivity
+import OAI.Geometry.PrescribedPotential.RealFrameScalar
+
+namespace OAI
+
+/-! Jet Energy Uniform. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Finset Module
+open scoped ContDiff
+namespace HigherJet
+variable {E F : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [NormedAddCommGroup F] [InnerProductSpace ℝ F]
+  {ι : Type*} [Fintype ι]
+
+lemma jetEnergy_le_norm {U : Set E} (hU : IsOpen U) {u : E → F}
+    (hu : ContDiffOn ℝ ∞ u U) (e : OrthonormalBasis ι ℝ E) (m : ℕ) {x : E} (hx : x ∈ U) :
+    jetEnergy e m u x ≤ (Fintype.card (Fin m → ι) : ℝ)*‖iteratedFDeriv ℝ m u x‖^2 := by
+  calc
+    _ ≤ ∑ _σ : Fin m → ι, ‖iteratedFDeriv ℝ m u x‖^2 := by
+      apply Finset.sum_le_sum
+      intro σ _
+      apply pow_le_pow_left₀ (norm_nonneg _)
+      simpa only [jetFamily,e.norm_eq_one,Finset.prod_const_one,mul_one] using norm_word_ofFn_le hU hu m (fun i => e (σ i)) hx
+    _ = _ := by simp
+
+lemma jetEnergy_gradient_bound {U : Set E} (hU : IsOpen U) {u : E → F}
+    (hu : ContDiffOn ℝ ∞ u U) (e : ι → E) (m : ℕ) {x : E} (hx : x ∈ U) (v : ι → E) :
+    ‖frameGradient v (jetEnergy e m u) x‖^2 ≤
+      4*jetEnergy e m u x*familyDissipation v (jetFamily e m u) x := by
+  rw [norm_frameGradient_sq]
+  exact familyEnergy_gradient_bound hU (jetFamily_smoothOn hU hu e m) hx v
+
+lemma familyDissipation_nonneg (v : ι → E) {κ : Type*} [Fintype κ] (f : κ → E → F) (x : E) :
+    0 ≤ familyDissipation v f x := sum_nonneg (fun _ _ => sum_nonneg (fun _ _ => sq_nonneg _))
+
+variable [FiniteDimensional ℝ E] [FiniteDimensional ℝ F]
+lemma jetEnergy_frame_upper (e : OrthonormalBasis ι ℝ E) :
+    ∃ C : ℝ, 0 < C ∧ ∀ (m : ℕ) (u : E → F) (x : E) (τ : E ≃L[ℝ] E) (B : ℝ),
+      0 ≤ B → ‖τ.toContinuousLinearMap‖ ≤ B →
+      familyDissipation (fun i => τ (e i)) (jetFamily e m u) x ≤ C*B^2*jetEnergy e (m+1) u x := by
+  obtain ⟨A,hA,hb⟩ := linearArray_norm_control (F := F) e.toBasis
+  refine ⟨(Fintype.card ι+1)*A^2,by positivity,fun m u x τ B hB hτ => ?_⟩
+  rw [jetEnergy_succ]
+  unfold familyDissipation frameGradientSq dir
+  rw [Finset.mul_sum]
+  apply sum_le_sum
+  intro σ _
+  let T := fderiv ℝ (jetFamily e m u σ) x
+  have ht : ‖T‖^2 ≤ A^2*(∑ i, ‖T (e i)‖^2) := by
+    have hh := pow_le_pow_left₀ (norm_nonneg T) (hb T) 2
+    rw [mul_pow,PiLp.norm_sq_eq_of_L2] at hh
+    exact hh
+  calc
+    _ ≤ ∑ _i : ι, (‖T‖*B)^2 := by
+      apply sum_le_sum
+      intro i _
+      apply pow_le_pow_left₀ (norm_nonneg _)
+      exact (T.le_opNorm _).trans (mul_le_mul_of_nonneg_left
+        ((τ.toContinuousLinearMap.le_opNorm (e i)).trans (by simpa only [e.norm_eq_one,mul_one] using hτ)) (norm_nonneg _))
+    _ = (Fintype.card ι : ℝ)*(‖T‖*B)^2 := by simp
+    _ ≤ ((Fintype.card ι : ℝ)+1)*A^2*B^2*(∑ i, ‖T (e i)‖^2) := by
+      have h1 := mul_le_mul_of_nonneg_left (mul_le_mul_of_nonneg_right ht (sq_nonneg B)) (show 0 ≤ (Fintype.card ι : ℝ) by positivity)
+      have h2 : 0 ≤ A^2*B^2*(∑ i, ‖T (e i)‖^2) := by positivity
+      nlinarith only [h1,h2]
+
+end HigherJet
+namespace BernsteinEstimate
+lemma first_energy_scaled {S D A R W L : ℝ} (_hS : 0 ≤ S) (hD : 0 ≤ D)
+    (hA : 1 ≤ A) (hR : 0 ≤ R) (hSbound : Real.sqrt S ≤ A)
+    (hW : W ≤ R*(1+A*Real.sqrt D)) (hL : 2*D-2*Real.sqrt S*W ≤ L) :
+    D-(2*(A+R*A)^2+(A+R*A)^4) ≤ L := by
+  have hc : 0 ≤ A+R*A := by positivity
+  have hr : R*(1+A*Real.sqrt D) ≤ (A+R*A)*(1+Real.sqrt D) := by
+    nlinarith only [hA,mul_nonneg hR (by linarith : 0 ≤ A-1),mul_nonneg (show 0 ≤ A by linarith) (Real.sqrt_nonneg D)]
+  have hh := energy_first_numeric (Real.sqrt_nonneg S) (Real.sqrt_nonneg D) hc
+    (hSbound.trans (le_add_of_nonneg_right (mul_nonneg hR (by linarith)))) (hW.trans hr)
+    (by simpa only [Real.sq_sqrt hD] using hL)
+  simpa only [Real.sq_sqrt hD] using hh
+
+lemma cubic_energy_scaled {S D A R W L : ℝ} (hS : 0 ≤ S) (hD : 0 ≤ D)
+    (hA : 1 ≤ A) (hR : 0 ≤ R)
+    (hW : W ≤ R*(1+A^2*S+A*Real.sqrt D)) (hL : 2*D-2*Real.sqrt S*W ≤ L) :
+    D-(4*(R*A^2)+2*(R*A^2)^2)*(1+S*Real.sqrt S) ≤ L := by
+  have hA0 : 0 ≤ A := by linarith
+  have hAA : A ≤ A^2 := by nlinarith
+  have hW' : W ≤ R*A^2*(1+(Real.sqrt S)^2+Real.sqrt D) := by
+    rw [Real.sq_sqrt hS]
+    have hh : 1+A^2*S+A*Real.sqrt D ≤ A^2*(1+S+Real.sqrt D) := by
+      nlinarith only [hAA,hA,mul_nonneg (by linarith : 0 ≤ A^2-A) (Real.sqrt_nonneg D)]
+    exact hW.trans ((mul_le_mul_of_nonneg_left hh hR).trans_eq (by ring))
+  have hh := energy_cubic_numeric (Real.sqrt_nonneg S) (Real.sqrt_nonneg D)
+    (show 0 ≤ R*A^2 by positivity) hW' (by simpa only [Real.sq_sqrt hD] using hL)
+  have he : (Real.sqrt S)^3 = S*Real.sqrt S := by rw [pow_succ,Real.sq_sqrt hS]
+  simpa only [Real.sq_sqrt hD,he] using hh
+
+lemma linear_energy_scaled {S D A R W L : ℝ} (hS : 0 ≤ S) (hD : 0 ≤ D)
+    (hA : 1 ≤ A) (hR : 0 ≤ R)
+    (hW : W ≤ R*(1+A*Real.sqrt S+A*Real.sqrt D)) (hL : 2*D-2*Real.sqrt S*W ≤ L) :
+    D-(4*(R*A)+(R*A)^2)*(1+S) ≤ L := by
+  have hW' : W ≤ R*A*(1+Real.sqrt S+Real.sqrt D) := by
+    apply hW.trans
+    nlinarith only [mul_nonneg hR (by linarith : 0 ≤ A-1)]
+  have hh := energy_linear_numeric (Real.sqrt_nonneg S) (Real.sqrt_nonneg D)
+    (show 0 ≤ R*A by positivity) hW' (by simpa only [Real.sq_sqrt hD] using hL)
+  simpa only [Real.sq_sqrt hD,Real.sq_sqrt hS] using hh
+end BernsteinEstimate
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/JetNormEquivalence.lean b/lean/OAI/Geometry/PrescribedPotential/JetNormEquivalence.lean
new file mode 100644
index 0000000000000000000000000000000000000000..dc1d89e0c44ea38c4ba1316b9e696cc224e317c1
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/JetNormEquivalence.lean
@@ -0,0 +1,52 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.KaehlerClosedDerivatives
+
+namespace OAI
+
+/-! Jet Norm Equivalence. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Finset Module
+open scoped ContDiff
+namespace HigherJet
+variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  [NormedAddCommGroup F] [InnerProductSpace ℝ F]
+  [FiniteDimensional ℝ E] [FiniteDimensional ℝ F]
+  {ι : Type*} [Fintype ι]
+
+def jetArray (e : ι → E) (m : ℕ) :
+    ContinuousMultilinearMap ℝ (fun _ : Fin m => E) F →ₗ[ℝ]
+      PiLp 2 (fun _ : Fin m → ι => F) where
+  toFun T := WithLp.toLp 2 (fun σ => T (fun i => e (σ i)))
+  map_add' T R := by ext σ; rfl
+  map_smul' c T := by ext σ; rfl
+
+omit [FiniteDimensional ℝ E] [FiniteDimensional ℝ F] [Fintype ι] in
+lemma jetArray_injective (e : Basis ι ℝ E) (m : ℕ) :
+    Function.Injective (jetArray (F := F) e m) := by
+  intro T R h
+  have he : T.toMultilinearMap = R.toMultilinearMap := Module.Basis.ext_multilinear (fun _ => e)
+    (fun σ => congrArg (fun V : PiLp 2 (fun _ : Fin m → ι => F) => V σ) h)
+  ext v
+  exact congrArg (fun L => L v) he
+
+omit [FiniteDimensional ℝ E] in
+lemma jetArray_norm_control (e : Basis ι ℝ E) (m : ℕ) :
+    ∃ C : ℝ, 0 < C ∧ ∀ T : ContinuousMultilinearMap ℝ (fun _ : Fin m => E) F,
+      ‖T‖ ≤ C*‖jetArray e m T‖ := by
+  let : FiniteDimensional ℝ (ContinuousMultilinearMap ℝ (fun _ : Fin m => E) F) :=
+    FiniteDimensional.of_injective (jetArray (F := F) e m) (jetArray_injective e m)
+  obtain ⟨C,hC,hb⟩ := (jetArray (F := F) e m).injective_iff_antilipschitz.mp (jetArray_injective e m)
+  refine ⟨C,by exact_mod_cast hC,fun T => ?_⟩
+  simpa only [map_zero,dist_zero_right] using hb.le_mul_dist T 0
+
+end HigherJet
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/KaehlerClosedDerivatives.lean b/lean/OAI/Geometry/PrescribedPotential/KaehlerClosedDerivatives.lean
new file mode 100644
index 0000000000000000000000000000000000000000..5d7835ac94e0fbd623ba83b4f36d8ca02000aa1f
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/KaehlerClosedDerivatives.lean
@@ -0,0 +1,94 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.WirtingerCalculus
+
+namespace OAI
+
+/-! Kaehler Closed Derivatives. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace KaehlerCalculus
+open Anticanonical
+variable {n : ℕ}
+def e (i : Fin n) : V n := Pi.single i 1
+
+lemma fundamental_e (M : Matrix (Fin n) (Fin n) ℂ) (i j : Fin n) :
+    ComplexAtlas.fundamentalForm M (e i) (e j) = (M i j).im := by
+  simp [ComplexAtlas.fundamentalForm,e]
+lemma fundamental_Ie (M : Matrix (Fin n) (Fin n) ℂ) (i j : Fin n) :
+    ComplexAtlas.fundamentalForm M (Complex.I • e i) (e j) = -(M i j).re := by
+  simp [ComplexAtlas.fundamentalForm,e,star_smul,smul_dotProduct]
+lemma fundamental_eI (M : Matrix (Fin n) (Fin n) ℂ) (i j : Fin n) :
+    ComplexAtlas.fundamentalForm M (e i) (Complex.I • e j) = (M i j).re := by
+  simp [ComplexAtlas.fundamentalForm,e,Matrix.mulVec_smul,dotProduct_smul]
+lemma fundamental_IeI (M : Matrix (Fin n) (Fin n) ℂ) (i j : Fin n) :
+    ComplexAtlas.fundamentalForm M (Complex.I • e i) (Complex.I • e j) = (M i j).im := by
+  simp [ComplexAtlas.fundamentalForm,e,Matrix.mulVec_smul,star_smul,
+    smul_dotProduct,dotProduct_smul]
+
+lemma fderiv_re {f : V n → ℂ} {z : V n} (hf : DifferentiableAt ℝ f z) (v : V n) :
+    fderiv ℝ (fun y => (f y).re) z v = (fderiv ℝ f z v).re := by
+  exact congrArg (fun T : V n →L[ℝ] ℝ => T v)
+    ((Complex.reCLM.hasFDerivAt.comp z hf.hasFDerivAt).fderiv)
+lemma fderiv_im {f : V n → ℂ} {z : V n} (hf : DifferentiableAt ℝ f z) (v : V n) :
+    fderiv ℝ (fun y => (f y).im) z v = (fderiv ℝ f z v).im := by
+  exact congrArg (fun T : V n →L[ℝ] ℝ => T v)
+    ((Complex.imCLM.hasFDerivAt.comp z hf.hasFDerivAt).fderiv)
+
+lemma closed_dz_symmetry {U : Set (V n)} (hU : IsOpen U)
+    {M : V n → Matrix (Fin n) (Fin n) ℂ}
+    (hM : ContDiffOn ℝ ∞ M U) (hh : ∀ y ∈ U, (M y).IsHermitian)
+    (hclosed : ∀ y ∈ U, ∀ u v w : V n,
+      fderiv ℝ (fun q => ComplexAtlas.fundamentalForm (M q) v w) y u +
+      fderiv ℝ (fun q => ComplexAtlas.fundamentalForm (M q) w u) y v +
+      fderiv ℝ (fun q => ComplexAtlas.fundamentalForm (M q) u v) y w = 0)
+    {z : V n} (hz : z ∈ U) (i j k : Fin n) :
+    dz (e k) (fun y => M y i j) z = dz (e j) (fun y => M y i k) z := by
+  have hd (a b : Fin n) : DifferentiableAt ℝ (fun y => M y a b) z :=
+    ((contDiffOn_pi.mp (contDiffOn_pi.mp hM a) b).contDiffAt
+      (hU.mem_nhds hz)).differentiableAt (by simp)
+  have hre (a b : Fin n) : (fun y => (M y a b).re) =ᶠ[𝓝 z] (fun y => (M y b a).re) := by
+    filter_upwards [hU.mem_nhds hz] with y hy
+    simpa only [Complex.star_def,Complex.conj_re] using congrArg Complex.re ((hh y hy).apply b a)
+  have him (a b : Fin n) : (fun y => (M y a b).im) =ᶠ[𝓝 z] (fun y => -(M y b a).im) := by
+    filter_upwards [hU.mem_nhds hz] with y hy
+    have he := congrArg Complex.im ((hh y hy).apply a b)
+    simpa only [Complex.star_def,Complex.conj_im] using he.symm
+  have hRe (a b : Fin n) (v : V n) :
+      (fderiv ℝ (fun y => M y a b) z v).re = (fderiv ℝ (fun y => M y b a) z v).re := by
+    rw [← fderiv_re (hd a b), (hre a b).fderiv_eq, fderiv_re (hd b a)]
+  have hIm (a b : Fin n) (v : V n) :
+      (fderiv ℝ (fun y => M y a b) z v).im = -(fderiv ℝ (fun y => M y b a) z v).im := by
+    rw [← fderiv_im (hd a b), (him a b).fderiv_eq, fderiv_fun_neg,
+      _root_.neg_apply, fderiv_im (hd b a)]
+  have h1 := hclosed z hz (Complex.I • e k) (e i) (e j)
+  have h2 := hclosed z hz (Complex.I • e j) (e i) (e k)
+  have h3 := hclosed z hz (e k) (e i) (e j)
+  have h4 := hclosed z hz (Complex.I • e k) (e i) (Complex.I • e j)
+  simp only [fundamental_e,fundamental_Ie,fundamental_eI,fundamental_IeI] at h1 h2 h3 h4
+  have hneg (a b : Fin n) : DifferentiableAt ℝ (fun y => (M y a b).re) z :=
+    Complex.reCLM.differentiableAt.comp z (hd a b)
+  simp only [fderiv_fun_neg,_root_.neg_apply,fderiv_re (hd _ _),fderiv_im (hd _ _)] at h1 h2 h3 h4
+  have h5 := hRe j k (e i)
+  have h6 := hRe k i (e j)
+  have h7 := hRe j i (e k)
+  have h8 := hRe k i (Complex.I • e j)
+  have h9 := hIm k i (e j)
+  apply Complex.ext
+  · simp only [dz,wderiv,Complex.div_ofNat_re,Complex.add_re,neg_mul,
+      Complex.neg_re,Complex.I_mul_re]
+    linarith
+  · simp only [dz,wderiv,Complex.div_ofNat_im,Complex.add_im,neg_mul,
+      Complex.neg_im,Complex.I_mul_im]
+    linarith
+end KaehlerCalculus
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/KaehlerSecondDerivatives.lean b/lean/OAI/Geometry/PrescribedPotential/KaehlerSecondDerivatives.lean
new file mode 100644
index 0000000000000000000000000000000000000000..214a3be7b6f59f21b1b1169c0ada45b526548477
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/KaehlerSecondDerivatives.lean
@@ -0,0 +1,99 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.MatrixLogDet
+import OAI.Geometry.PrescribedPotential.WirtingerConjugate
+
+namespace OAI
+
+/-! Kaehler Second Derivatives. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace KaehlerCalculus
+variable {n : ℕ}
+
+lemma closed_mixed_symmetry {U : Set (V n)} (hU : IsOpen U)
+    {M : V n → Matrix (Fin n) (Fin n) ℂ}
+    (hM : ContDiffOn ℝ ∞ M U) (hh : ∀ y ∈ U, (M y).IsHermitian)
+    (hclosed : ∀ y ∈ U, ∀ u v w : V n,
+      fderiv ℝ (fun q => Anticanonical.ComplexAtlas.fundamentalForm (M q) v w) y u +
+      fderiv ℝ (fun q => Anticanonical.ComplexAtlas.fundamentalForm (M q) w u) y v +
+      fderiv ℝ (fun q => Anticanonical.ComplexAtlas.fundamentalForm (M q) u v) y w = 0)
+    {z : V n} (hz : z ∈ U) (i j k l : Fin n) :
+    dzbar (e k) (dz (e l) (fun y => M y i j)) z =
+      dzbar (e i) (dz (e j) (fun y => M y k l)) z := by
+  have hs := hM.contDiffAt (hU.mem_nhds hz)
+  have he1 : dz (e l) (fun y => M y i j) =ᶠ[𝓝 z] dz (e j) (fun y => M y i l) := by
+    filter_upwards [hU.mem_nhds hz] with y hy
+    exact closed_dz_symmetry hU hM hh hclosed hy i j l
+  rw [show dzbar (e k) (dz (e l) (fun y => M y i j)) z =
+    dzbar (e k) (dz (e j) (fun y => M y i l)) z from wderiv_congr he1 _ _]
+  rw [show dzbar (e k) (dz (e j) (fun y => M y i l)) z =
+    dz (e j) (dzbar (e k) (fun y => M y i l)) z from wderiv_comm (entry_smooth hs i l) _ _ _ _]
+  have he2 : dzbar (e k) (fun y => M y i l) =ᶠ[𝓝 z] dzbar (e i) (fun y => M y k l) := by
+    filter_upwards [hU.mem_nhds hz] with y hy
+    exact closed_dzbar_symmetry hU hM hh hclosed hy i l k
+  rw [show dz (e j) (dzbar (e k) (fun y => M y i l)) z =
+    dz (e j) (dzbar (e i) (fun y => M y k l)) z from wderiv_congr he2 _ _]
+  exact wderiv_comm (entry_smooth hs k l) _ _ _ _
+
+lemma closed_derivative_quadratic {U : Set (V n)} (hU : IsOpen U)
+    {M : V n → Matrix (Fin n) (Fin n) ℂ}
+    (hM : ContDiffOn ℝ ∞ M U) (hh : ∀ y ∈ U, (M y).IsHermitian)
+    (hclosed : ∀ y ∈ U, ∀ u v w : V n,
+      fderiv ℝ (fun q => Anticanonical.ComplexAtlas.fundamentalForm (M q) v w) y u +
+      fderiv ℝ (fun q => Anticanonical.ComplexAtlas.fundamentalForm (M q) w u) y v +
+      fderiv ℝ (fun q => Anticanonical.ComplexAtlas.fundamentalForm (M q) u v) y w = 0)
+    {z : V n} (hz : z ∈ U) (i j : Fin n) :
+    ∑ k, (mderiv Complex.I (e k) M z*mderiv (-Complex.I) (e k) M z) i j =
+      (mderiv Complex.I (e i) M z*mderiv (-Complex.I) (e j) M z).trace := by
+  classical
+  simp only [Matrix.mul_apply,Matrix.trace,Matrix.diag_apply,mderiv]
+  apply Finset.sum_congr rfl
+  intro k _
+  apply Finset.sum_congr rfl
+  intro l _
+  change dzbar (e k) (fun y => M y i l) z * dz (e k) (fun y => M y l j) z =
+    dzbar (e i) (fun y => M y k l) z * dz (e j) (fun y => M y l k) z
+  rw [closed_dzbar_symmetry hU hM hh hclosed hz i l k,
+    closed_dz_symmetry hU hM hh hclosed hz l j k]
+
+lemma ricci_cancellation_at_one {U : Set (V n)} (hU : IsOpen U)
+    {M : V n → Matrix (Fin n) (Fin n) ℂ}
+    (hM : ContDiffOn ℝ ∞ M U) (hp : ∀ y ∈ U, (M y).PosDef)
+    (hclosed : ∀ y ∈ U, ∀ u v w : V n,
+      fderiv ℝ (fun q => Anticanonical.ComplexAtlas.fundamentalForm (M q) v w) y u +
+      fderiv ℝ (fun q => Anticanonical.ComplexAtlas.fundamentalForm (M q) w u) y v +
+      fderiv ℝ (fun q => Anticanonical.ComplexAtlas.fundamentalForm (M q) u v) y w = 0)
+    {z : V n} (hz : z ∈ U) (hMz : M z = 1) :
+    (∑ k, mderiv Complex.I (e k) (mderiv (-Complex.I) (e k) M) z) =
+      PotentialKaehler.potentialMatrix (fun y => Real.log (M y).det.re) z +
+        ∑ k, mderiv Complex.I (e k) M z*mderiv (-Complex.I) (e k) M z := by
+  have hh : ∀ y ∈ U, (M y).IsHermitian := fun y hy => (hp y hy).isHermitian
+  ext i j
+  simp only [Matrix.sum_apply,Matrix.add_apply]
+  rw [closed_derivative_quadratic hU hM hh hclosed hz i j]
+  have hs := hM.contDiffAt (hU.mem_nhds hz)
+  have hl : ContDiffAt ℝ ∞ (fun y => Real.log (M y).det.re) z :=
+    (Complex.reCLM.contDiff.contDiffAt.comp z (determinant_smooth.contDiffAt.comp z hs)).log
+      (ne_of_gt (Complex.pos_iff.mp (hp z hz).det_pos).1)
+  rw [potentialMatrix_eq_dzbar_dz hl]
+  have he := second_wderiv_logdet_at_one hU hM hp hz hMz (-Complex.I) Complex.I (e i) (e j)
+  change dzbar (e i) (dz (e j) (fun y => (Real.log (M y).det.re : ℂ))) z = _ at he
+  change ∑ k, dzbar (e k) (dz (e k) (fun y => M y i j)) z =
+    dzbar (e i) (dz (e j) (fun y => (Real.log (M y).det.re : ℂ))) z + _
+  rw [he,sub_add_cancel]
+  unfold Matrix.trace
+  apply Finset.sum_congr rfl
+  intro k _
+  exact closed_mixed_symmetry hU hM hh hclosed hz i j k k
+end KaehlerCalculus
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/KernelCompact.lean b/lean/OAI/Geometry/PrescribedPotential/KernelCompact.lean
new file mode 100644
index 0000000000000000000000000000000000000000..746377e23a63844ccd7fad37088cfeec14431f58
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/KernelCompact.lean
@@ -0,0 +1,68 @@
+import Mathlib
+
+namespace OAI
+
+/-! Kernel Compact. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology
+open scoped BoundedContinuousFunction
+namespace EllipticCompact
+variable {K H : Type*} [TopologicalSpace K] [CompactSpace K]
+  [NormedAddCommGroup H] [NormedSpace ℂ H]
+
+def kernelOperator (g : K →ᵇ (H →L[ℂ] ℂ)) : H →L[ℂ] (K →ᵇ ℂ) :=
+  LinearMap.mkContinuous
+    { toFun := fun u => BoundedContinuousFunction.mkOfCompact
+        ⟨fun x => g x u, by fun_prop⟩
+      map_add' := by intro u v; ext x; exact map_add (g x) u v
+      map_smul' := by intro c u; ext x; exact map_smul (g x) c u }
+    ‖g‖ (fun u => by
+      change ‖BoundedContinuousFunction.mkOfCompact ⟨fun x => g x u, _⟩‖ ≤ ‖g‖ * ‖u‖
+      apply (BoundedContinuousFunction.norm_le (mul_nonneg (norm_nonneg g) (norm_nonneg u))).mpr
+      intro x
+      exact ((g x).le_opNorm u).trans (mul_le_mul_of_nonneg_right (g.norm_coe_le_norm x) (norm_nonneg _)))
+
+lemma kernelOperator_apply (g : K →ᵇ (H →L[ℂ] ℂ)) (u : H) (x : K) :
+    kernelOperator g u x = g x u := rfl
+
+ 
+
+theorem kernelOperator_compact (g : K →ᵇ (H →L[ℂ] ℂ)) :
+    IsCompactOperator (kernelOperator g) := by
+  let S := kernelOperator g '' Metric.closedBall (0 : H) 1
+  have heq : Equicontinuous ((↑) : S → K → ℂ) := by
+    intro x
+    apply Metric.equicontinuousAt_of_continuity_modulus
+      (b := fun y => ‖g x - g y‖)
+    · have h : ContinuousAt (fun y => ‖g x - g y‖) x := by fun_prop
+      simpa using h.tendsto
+    · filter_upwards [] with y F
+      obtain ⟨u, hu, hF⟩ := F.property
+      have hn : ‖u‖ ≤ 1 := by simpa only [Metric.mem_closedBall, dist_zero_right] using hu
+      change dist (F.val x) (F.val y) ≤ _
+      rw [← hF, kernelOperator_apply, kernelOperator_apply, dist_eq_norm,
+        ← sub_apply]
+      exact ((g x - g y).le_opNorm u).trans (by simpa only [mul_one] using
+        (mul_le_mul_of_nonneg_left hn (norm_nonneg (g x - g y))))
+  have hc : IsCompact (closure S) := BoundedContinuousFunction.arzela_ascoli
+    (Metric.closedBall (0 : ℂ) ‖g‖) (isCompact_closedBall _ _) S (by
+      intro F x hF
+      obtain ⟨u, hu, rfl⟩ := hF
+      have hn : ‖u‖ ≤ 1 := by simpa only [Metric.mem_closedBall, dist_zero_right] using hu
+      rw [Metric.mem_closedBall, dist_zero_right, kernelOperator_apply]
+      exact ((g x).le_opNorm u).trans ((mul_le_mul_of_nonneg_right (g.norm_coe_le_norm x) (norm_nonneg _)).trans
+        (by simpa only [mul_one] using mul_le_mul_of_nonneg_left hn (norm_nonneg g)))) heq
+  apply (isCompactOperator_iff_exists_mem_nhds_image_subset_compact _).mpr
+  exact ⟨Metric.closedBall 0 1, Metric.closedBall_mem_nhds _ (by norm_num), closure S, hc,
+    subset_closure⟩
+end EllipticCompact
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/LinearFrameCalculus.lean b/lean/OAI/Geometry/PrescribedPotential/LinearFrameCalculus.lean
new file mode 100644
index 0000000000000000000000000000000000000000..afe766a83714ab221b6dbfadc6cc59ccd93f5061
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/LinearFrameCalculus.lean
@@ -0,0 +1,98 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.MatrixWirtinger
+
+namespace OAI
+
+/-! Linear Frame Calculus. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace KaehlerCalculus
+variable {n : ℕ}
+
+def frameMatrix (B : V n →L[ℂ] V n) : Matrix (Fin n) (Fin n) ℂ :=
+  LinearMap.toMatrix' B.toLinearMap
+
+def pullMetric (B : V n →L[ℂ] V n) (M : V n → Matrix (Fin n) (Fin n) ℂ)
+    (y : V n) : Matrix (Fin n) (Fin n) ℂ := (frameMatrix B)ᴴ*M (B y)*frameMatrix B
+
+lemma frameMatrix_mulVec (B : V n →L[ℂ] V n) (v : V n) : frameMatrix B*ᵥv = B v :=
+  LinearMap.toMatrix'_mulVec B.toLinearMap v
+
+lemma pullMetric_smooth {U : Set (V n)} (hU : IsOpen U)
+    (B : V n →L[ℂ] V n) {M : V n → Matrix (Fin n) (Fin n) ℂ}
+    (hM : ContDiffOn ℝ ∞ M U) : ContDiffOn ℝ ∞ (pullMetric B M) (B ⁻¹' U) := by
+  intro y hy
+  have hm := (hM.contDiffAt (hU.mem_nhds hy)).comp y (B.restrictScalars ℝ).contDiff.contDiffAt
+  exact (matrix_smooth_mul (matrix_smooth_mul contDiffAt_const hm) contDiffAt_const).contDiffWithinAt
+
+lemma wderiv_comp_linear (B : V n →L[ℂ] V n) {f : V n → ℂ} {z : V n}
+    (hf : DifferentiableAt ℝ f (B z)) (a : ℂ) (v : V n) :
+    wderiv a v (fun y => f (B y)) z = wderiv a (B v) f (B z) := by
+  have he := fderiv_fun_comp z hf (B.restrictScalars ℝ).differentiableAt
+  have hb : fderiv ℝ (fun y => B y) z = B.restrictScalars ℝ := (B.restrictScalars ℝ).fderiv
+  rw [hb] at he
+  simp only [wderiv,he,ContinuousLinearMap.comp_apply]
+  change (fderiv ℝ f (B z) (B v) + a*fderiv ℝ f (B z) (B (Complex.I • v)))/2 = _
+  rw [map_smul]
+
+lemma mderiv_comp_linear (B : V n →L[ℂ] V n) {M : V n → Matrix (Fin n) (Fin n) ℂ}
+    {z : V n} (hM : ContDiffAt ℝ ∞ M (B z)) (a : ℂ) (v : V n) :
+    mderiv a v (fun y => M (B y)) z = mderiv a (B v) M (B z) := by
+  ext i j
+  exact wderiv_comp_linear B ((entry_smooth hM i j).differentiableAt (by simp)) a v
+
+lemma mderiv_congruence {M : V n → Matrix (Fin n) (Fin n) ℂ} {z : V n}
+    (hM : ContDiffAt ℝ ∞ M z) (C D : Matrix (Fin n) (Fin n) ℂ) (a : ℂ) (v : V n) :
+    mderiv a v (fun y => C*M y*D) z = C*mderiv a v M z*D := by
+  rw [mderiv_mul (matrix_smooth_mul contDiffAt_const hM) contDiffAt_const,
+    mderiv_mul contDiffAt_const hM,mderiv_const,mderiv_const]
+  simp only [zero_mul,mul_zero,zero_add,add_zero]
+
+lemma mderiv_pullMetric (B : V n →L[ℂ] V n) {M : V n → Matrix (Fin n) (Fin n) ℂ}
+    {z : V n} (hM : ContDiffAt ℝ ∞ M (B z)) (a : ℂ) (v : V n) :
+    mderiv a v (pullMetric B M) z =
+      (frameMatrix B)ᴴ*mderiv a (B v) M (B z)*frameMatrix B := by
+  unfold pullMetric
+  rw [mderiv_congruence (show ContDiffAt ℝ ∞ (fun y => M (B y)) z from
+    hM.comp z (B.restrictScalars ℝ).contDiff.contDiffAt),
+    mderiv_comp_linear B hM]
+
+lemma fundamental_pullMetric (B : V n →L[ℂ] V n)
+    (M : V n → Matrix (Fin n) (Fin n) ℂ) (z u v : V n) :
+    Anticanonical.ComplexAtlas.fundamentalForm (pullMetric B M z) u v =
+      Anticanonical.ComplexAtlas.fundamentalForm (M (B z)) (B u) (B v) := by
+  unfold Anticanonical.ComplexAtlas.fundamentalForm pullMetric
+  rw [PotentialKaehler.matrix_pair_pullback,frameMatrix_mulVec,frameMatrix_mulVec]
+
+lemma pullMetric_closed {U : Set (V n)} (hU : IsOpen U)
+    (B : V n →L[ℂ] V n) {M : V n → Matrix (Fin n) (Fin n) ℂ}
+    (hM : ContDiffOn ℝ ∞ M U)
+    (hclosed : ∀ y ∈ U, ∀ u v w : V n,
+      fderiv ℝ (fun q => Anticanonical.ComplexAtlas.fundamentalForm (M q) v w) y u +
+      fderiv ℝ (fun q => Anticanonical.ComplexAtlas.fundamentalForm (M q) w u) y v +
+      fderiv ℝ (fun q => Anticanonical.ComplexAtlas.fundamentalForm (M q) u v) y w = 0) :
+    ∀ y ∈ B ⁻¹' U, ∀ u v w : V n,
+      fderiv ℝ (fun q => Anticanonical.ComplexAtlas.fundamentalForm (pullMetric B M q) v w) y u +
+      fderiv ℝ (fun q => Anticanonical.ComplexAtlas.fundamentalForm (pullMetric B M q) w u) y v +
+      fderiv ℝ (fun q => Anticanonical.ComplexAtlas.fundamentalForm (pullMetric B M q) u v) y w = 0 := by
+  intro y hy u v w
+  simp only [fundamental_pullMetric]
+  have hd (a b : V n) := (Anticanonical.SourceSmooth.fundamentalForm_smooth
+    (hM.contDiffAt (hU.mem_nhds hy)) a b).differentiableAt (by simp)
+  have hb : fderiv ℝ (fun y => B y) y = B.restrictScalars ℝ := (B.restrictScalars ℝ).fderiv
+  simp only [fderiv_fun_comp y (hd _ _) (B.restrictScalars ℝ).differentiableAt,
+    hb,ContinuousLinearMap.comp_apply]
+  exact hclosed (B y) hy (B u) (B v) (B w)
+
+end KaehlerCalculus
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/LinearizedCommutator.lean b/lean/OAI/Geometry/PrescribedPotential/LinearizedCommutator.lean
new file mode 100644
index 0000000000000000000000000000000000000000..4203359bba3728c03a5a724c3e6f1e481e7b4116
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/LinearizedCommutator.lean
@@ -0,0 +1,208 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.BootstrapCalculus
+import OAI.Geometry.PrescribedPotential.ClosedRangeSmooth
+import OAI.Geometry.PrescribedPotential.RealMapSmooth
+import OAI.Geometry.PrescribedPotential.WeakNonlinearCommutator
+
+namespace OAI
+
+/-! Linearized Commutator. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology
+open scoped ContDiff Classical
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+local instance linComNG (s : ℝ) : NormedAddCommGroup (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedAddCommGroup
+local instance linComNS (s : ℝ) : NormedSpace ℝ (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedSpace
+local instance linComTG (s : ℝ) : IsTopologicalAddGroup (D.localizers.RealSobolev s) :=
+  Submodule.isTopologicalAddGroup _
+local instance linComCS (s : ℝ) : ContinuousSMul ℝ (D.localizers.RealSobolev s) :=
+  SMulMemClass.continuousSMul _
+
+lemma realVolumeDerivative_eq_fderiv (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) :
+    D.realVolumeDerivative k hk = fderiv ℝ (D.realVolume k hk) := by
+  funext u
+  exact (D.realVolume_hasStrictFDerivAt k hk u).hasFDerivAt.fderiv.symm
+
+lemma realVolumeDerivative_contDiff (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) :
+    ContDiff ℝ ∞ (D.realVolumeDerivative k hk) := by
+  rw [D.realVolumeDerivative_eq_fderiv k hk]
+  exact (D.realVolume_contDiff k hk).fderiv_right (m := ∞) (by simp)
+
+def realVolumeSecond (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (u w v : D.localizers.RealSobolev ((k : ℝ)+2)) : D.localizers.RealSobolev (k : ℝ) := by
+  letI : NormedAddCommGroup (D.localizers.RealSobolev ((k : ℝ)+2)) := inferInstance
+  letI : NormedSpace ℝ (D.localizers.RealSobolev ((k : ℝ)+2)) := inferInstance
+  letI : NormedAddCommGroup (D.localizers.RealSobolev (k : ℝ)) := inferInstance
+  letI : NormedSpace ℝ (D.localizers.RealSobolev (k : ℝ)) := inferInstance
+  exact fderiv ℝ (D.realVolumeDerivative k hk) u w v
+
+def realRemainderDerivative (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (p : ι) (v : EC d) (u : D.localizers.RealSobolev ((k : ℝ)+2)) :
+    D.localizers.RealSobolev ((k : ℝ)+2) →L[ℝ] D.localizers.RealSobolev (k : ℝ) :=
+  fderiv ℝ (D.realNonlinearRemainder k hk p v) u
+
+lemma realVolumeSecond_lower (k l : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (hl : Module.finrank ℝ (EC d) < l) (hlk : l ≤ k)
+    (u w v : D.localizers.RealSobolev ((k : ℝ)+2)) :
+    D.realVolumeSecond l hl
+        (D.localizers.realLower ((k : ℝ)+2) ((l : ℝ)+2)
+          (by exact_mod_cast Nat.add_le_add_right hlk 2) u)
+        (D.localizers.realLower ((k : ℝ)+2) ((l : ℝ)+2)
+          (by exact_mod_cast Nat.add_le_add_right hlk 2) w)
+        (D.localizers.realLower ((k : ℝ)+2) ((l : ℝ)+2)
+          (by exact_mod_cast Nat.add_le_add_right hlk 2) v) =
+      D.localizers.realLower (k : ℝ) (l : ℝ) (by exact_mod_cast hlk)
+        (D.realVolumeSecond k hk u w v) := by
+  let : NormedAddCommGroup (D.localizers.RealSobolev ((k : ℝ)+2)) := inferInstance
+  let : NormedSpace ℝ (D.localizers.RealSobolev ((k : ℝ)+2)) := inferInstance
+  let : NormedAddCommGroup (D.localizers.RealSobolev ((l : ℝ)+2)) := inferInstance
+  let : NormedSpace ℝ (D.localizers.RealSobolev ((l : ℝ)+2)) := inferInstance
+  let : NormedAddCommGroup (D.localizers.RealSobolev (k : ℝ)) := inferInstance
+  let : NormedSpace ℝ (D.localizers.RealSobolev (k : ℝ)) := inferInstance
+  let : NormedAddCommGroup (D.localizers.RealSobolev (l : ℝ)) := inferInstance
+  let : NormedSpace ℝ (D.localizers.RealSobolev (l : ℝ)) := inferInstance
+  let L := D.localizers.realLower ((k : ℝ)+2) ((l : ℝ)+2)
+    (by exact_mod_cast Nat.add_le_add_right hlk 2)
+  let M := D.localizers.realLower (k : ℝ) (l : ℝ) (by exact_mod_cast hlk)
+  have he : ∀ y z, D.realVolumeDerivative l hl (L y) (L z) =
+      M (D.realVolumeDerivative k hk y z) := by
+    intro y z
+    exact DFunLike.congr_fun (D.realVolumeDerivative_lower k l hk hl hlk y) z
+  have hB := (D.realVolumeDerivative_contDiff k hk).differentiable (by simp)
+  have hC := (D.realVolumeDerivative_contDiff l hl).differentiable (by simp)
+  exact FixedBootstrapCalculus.derivative_compatibility
+    (D.realVolumeDerivative k hk) (D.realVolumeDerivative l hl) L M hB hC he u w v
+
+lemma potentialDerivative_lower (k l : ℕ) (hlk : l ≤ k) (p : ι) (v : EC d)
+    (u : D.localizers.RealSobolev (((k+1 : ℕ) : ℝ)+2)) :
+    D.potentialDerivative l p v
+      (D.localizers.realLower (((k+1 : ℕ) : ℝ)+2) (((l+1 : ℕ) : ℝ)+2)
+        (by exact_mod_cast Nat.add_le_add_right (Nat.add_le_add_right hlk 1) 2) u) =
+    D.localizers.realLower ((k : ℝ)+2) ((l : ℝ)+2)
+      (by exact_mod_cast Nat.add_le_add_right hlk 2) (D.potentialDerivative k p v u) := by
+  have he : (fun u => D.potentialDerivative l p v
+      (D.localizers.realLower (((k+1 : ℕ) : ℝ)+2) (((l+1 : ℕ) : ℝ)+2)
+        (by exact_mod_cast Nat.add_le_add_right (Nat.add_le_add_right hlk 1) 2) u)) =
+      (fun u => D.localizers.realLower ((k : ℝ)+2) ((l : ℝ)+2)
+        (by exact_mod_cast Nat.add_le_add_right hlk 2) (D.potentialDerivative k p v u)) := by
+    apply (D.localizers.realEmbed_dense (((k+1 : ℕ) : ℝ)+2)).equalizer
+      (by fun_prop) (by fun_prop)
+    funext f
+    simp only [Function.comp_apply, Localizers.realLower_embed, potentialDerivative_embed]
+  exact congr_fun he u
+
+lemma densityDerivative_lower (k l : ℕ) (hlk : l ≤ k) (p : ι) (v : EC d)
+    (u : D.localizers.RealSobolev ((k+1 : ℕ) : ℝ)) :
+    D.densityDerivative l p v
+      (D.localizers.realLower ((k+1 : ℕ) : ℝ) ((l+1 : ℕ) : ℝ)
+        (by exact_mod_cast Nat.add_le_add_right hlk 1) u) =
+    D.localizers.realLower (k : ℝ) (l : ℝ) (by exact_mod_cast hlk)
+      (D.densityDerivative k p v u) := by
+  have he : (fun u => D.densityDerivative l p v
+      (D.localizers.realLower ((k+1 : ℕ) : ℝ) ((l+1 : ℕ) : ℝ)
+        (by exact_mod_cast Nat.add_le_add_right hlk 1) u)) =
+      (fun u => D.localizers.realLower (k : ℝ) (l : ℝ) (by exact_mod_cast hlk)
+        (D.densityDerivative k p v u)) := by
+    apply (D.localizers.realEmbed_dense ((k+1 : ℕ) : ℝ)).equalizer
+      (by fun_prop) (by fun_prop)
+    funext f
+    simp only [Function.comp_apply, Localizers.realLower_embed, densityDerivative_embed]
+  exact congr_fun he u
+
+ 
+lemma nonlinear_commutator_linearization (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (p : ι) (v : EC d) (u w : D.localizers.RealSobolev (((k+1 : ℕ) : ℝ)+2)) :
+    D.realVolumeDerivative k hk
+        (D.localizers.realLower (((k+1 : ℕ) : ℝ)+2) ((k : ℝ)+2) (by push_cast; linarith) u)
+        (D.potentialDerivative k p v w) +
+      D.realVolumeSecond k hk
+        (D.localizers.realLower (((k+1 : ℕ) : ℝ)+2) ((k : ℝ)+2) (by push_cast; linarith) u)
+        (D.localizers.realLower (((k+1 : ℕ) : ℝ)+2) ((k : ℝ)+2) (by push_cast; linarith) w)
+        (D.potentialDerivative k p v u) =
+    D.densityDerivative k p v (D.realVolumeDerivative (k+1) (by omega) u w) +
+      D.localizers.realLower ((k+1 : ℕ) : ℝ) (k : ℝ) (by push_cast; linarith)
+        (D.realRemainderDerivative (k+1) (by omega) p v u w) := by
+  let : NormedAddCommGroup (D.localizers.RealSobolev (((k+1 : ℕ) : ℝ)+2)) := inferInstance
+  let : NormedSpace ℝ (D.localizers.RealSobolev (((k+1 : ℕ) : ℝ)+2)) := inferInstance
+  let : NormedAddCommGroup (D.localizers.RealSobolev ((k : ℝ)+2)) := inferInstance
+  let : NormedSpace ℝ (D.localizers.RealSobolev ((k : ℝ)+2)) := inferInstance
+  let : NormedAddCommGroup (D.localizers.RealSobolev ((k+1 : ℕ) : ℝ)) := inferInstance
+  let : NormedSpace ℝ (D.localizers.RealSobolev ((k+1 : ℕ) : ℝ)) := inferInstance
+  let : NormedAddCommGroup (D.localizers.RealSobolev (k : ℝ)) := inferInstance
+  let : NormedSpace ℝ (D.localizers.RealSobolev (k : ℝ)) := inferInstance
+  let : NormedAddCommGroup (D.localizers.RealSobolev ((k : ℝ)+2) →L[ℝ]
+    D.localizers.RealSobolev (k : ℝ)) := inferInstance
+  let : NormedSpace ℝ (D.localizers.RealSobolev ((k : ℝ)+2) →L[ℝ]
+    D.localizers.RealSobolev (k : ℝ)) := inferInstance
+  have hF : fderiv ℝ (D.realVolume (k+1) (by omega)) u =
+      D.realVolumeDerivative (k+1) (by omega) u :=
+    (D.realVolume_hasStrictFDerivAt (k+1) (by omega) u).hasFDerivAt.fderiv
+  have hh := FixedBootstrapCalculus.differentiated_commutator
+    (D.realVolumeDerivative k hk)
+    (D.localizers.realLower (((k+1 : ℕ) : ℝ)+2) ((k : ℝ)+2) (by push_cast; linarith))
+    (D.potentialDerivative k p v) (D.densityDerivative k p v)
+    (D.localizers.realLower ((k+1 : ℕ) : ℝ) (k : ℝ) (by push_cast; linarith))
+    (D.realVolume (k+1) (by omega)) (D.realNonlinearRemainder (k+1) (by omega) p v)
+    ((D.realVolumeDerivative_contDiff k hk).differentiable (by simp))
+    ((D.realVolume_contDiff (k+1) (by omega)).differentiable (by simp))
+    ((D.realNonlinearRemainder_contDiff (k+1) (by omega) p v).differentiable (by simp))
+    (D.nonlinear_commutator_weak k hk p v) u w
+  rw [hF] at hh
+  exact hh
+
+lemma secondPotential_lower (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (p : ι) (v : EC d) (u : D.localizers.RealSobolev (((k+2 : ℕ) : ℝ)+2))
+    (w : D.localizers.RealSobolev (((k+1 : ℕ) : ℝ)+2)) :
+    let u₁ := D.localizers.realLower (((k+2 : ℕ) : ℝ)+2) (((k+1 : ℕ) : ℝ)+2) (by push_cast; linarith) u
+    let L := D.localizers.realLower (((k+1 : ℕ) : ℝ)+2) ((k : ℝ)+2) (by push_cast; linarith)
+    D.realVolumeSecond k hk (L u₁) (L w) (D.potentialDerivative k p v u₁) =
+      D.localizers.realLower ((k+1 : ℕ) : ℝ) (k : ℝ) (by push_cast; linarith)
+        (D.realVolumeSecond (k+1) (by omega) u₁ w (D.potentialDerivative (k+1) p v u)) := by
+  dsimp only
+  let u₁ := D.localizers.realLower (((k+2 : ℕ) : ℝ)+2) (((k+1 : ℕ) : ℝ)+2) (by push_cast; linarith) u
+  let L := D.localizers.realLower (((k+1 : ℕ) : ℝ)+2) ((k : ℝ)+2) (by push_cast; linarith)
+  have hd := D.potentialDerivative_lower (k+1) k (by omega) p v u
+  have ht := congrArg (D.realVolumeSecond k hk (L u₁) (L w)) hd
+  exact ht.trans (D.realVolumeSecond_lower (k+1) k (by omega) hk (by omega)
+    u₁ w (D.potentialDerivative (k+1) p v u))
+
+ 
+
+lemma linearized_commutator_weak (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (p : ι) (v : EC d) (u : D.localizers.RealSobolev (((k+2 : ℕ) : ℝ)+2))
+    (w : D.localizers.RealSobolev (((k+1 : ℕ) : ℝ)+2)) :
+    let u₁ := D.localizers.realLower (((k+2 : ℕ) : ℝ)+2) (((k+1 : ℕ) : ℝ)+2) (by push_cast; linarith) u
+    D.realVolumeDerivative k hk
+        (D.localizers.realLower (((k+1 : ℕ) : ℝ)+2) ((k : ℝ)+2) (by push_cast; linarith) u₁)
+        (D.potentialDerivative k p v w) =
+    D.densityDerivative k p v (D.realVolumeDerivative (k+1) (by omega) u₁ w) +
+      D.localizers.realLower ((k+1 : ℕ) : ℝ) (k : ℝ) (by push_cast; linarith)
+        (D.realRemainderDerivative (k+1) (by omega) p v u₁ w -
+          D.realVolumeSecond (k+1) (by omega) u₁ w
+            (D.potentialDerivative (k+1) p v u)) := by
+  dsimp only
+  let u₁ := D.localizers.realLower (((k+2 : ℕ) : ℝ)+2) (((k+1 : ℕ) : ℝ)+2) (by push_cast; linarith) u
+  exact FixedBootstrapCalculus.lowered_remainder
+    (D.localizers.realLower ((k+1 : ℕ) : ℝ) (k : ℝ) (by push_cast; linarith)) _ _ _ _ _
+    (D.nonlinear_commutator_linearization k hk p v u₁ w)
+    (D.secondPotential_lower k hk p v u w)
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/LiteralWeakHessian.lean b/lean/OAI/Geometry/PrescribedPotential/LiteralWeakHessian.lean
new file mode 100644
index 0000000000000000000000000000000000000000..1c05a2dc090c5781fd20fed82e0de81003345b59
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/LiteralWeakHessian.lean
@@ -0,0 +1,203 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.GlobalHessian
+import OAI.Geometry.PrescribedPotential.GlobalStrongEmbedding
+import OAI.Geometry.PrescribedPotential.StrongDerivatives
+
+namespace OAI
+
+/-! Literal Weak Hessian. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory LineDeriv
+open scoped ContDiff SchwartzMap Classical BoundedContinuousFunction
+namespace SobolevChart
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+
+def schwartzCoordLinear (s : ℝ) : 𝓢(E,ℂ) →ₗ[ℝ] L2 E where
+  toFun := schwartzCoord s
+  map_add' := schwartzCoord_add s
+  map_smul' c f := by
+    change schwartzCoord s (c • f) = c • schwartzCoord s f
+    rw [RCLike.real_smul_eq_coe_smul (K := ℂ), schwartzCoord_smul,
+      RCLike.real_smul_eq_coe_smul (K := ℂ)]
+
+end SobolevChart
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+
+ 
+def literalComplexHessian (f : EC d → ℂ) (x : EC d) (i j : Fin d) : ℂ :=
+  (1/4 : ℂ) * (fderiv ℝ (fderiv ℝ f) x (hessianDirection i) (hessianDirection j) +
+    fderiv ℝ (fderiv ℝ f) x (hessianIDirection i) (hessianIDirection j) +
+    Complex.I * (fderiv ℝ (fderiv ℝ f) x (hessianIDirection i) (hessianDirection j) -
+      fderiv ℝ (fderiv ℝ f) x (hessianDirection i) (hessianIDirection j)))
+
+lemma literalComplexHessian_schwartz (f : 𝓢(EC d,ℂ)) (x : EC d) (i j : Fin d) :
+    literalComplexHessian f x i j = hessianEntrySchwartz i j f x :=
+  (hessianEntrySchwartz_apply i j f x).symm
+
+lemma literalComplexHessian_congr {f h : EC d → ℂ} {x : EC d}
+    (he : f =ᶠ[𝓝 x] h) (i j : Fin d) :
+    literalComplexHessian f x i j = literalComplexHessian h x i j := by
+  unfold literalComplexHessian
+  rw [(he.fderiv (𝕜 := ℝ)).fderiv_eq]
+
+lemma literalComplexHessian_strong_continuous (s : ℝ)
+    (hs : (Module.finrank ℝ (EC d) : ℝ) < 2*s)
+    (hN : (Module.finrank ℝ (EC d) : ℝ) < 2*(s-4)) (x : EC d) (i j : Fin d) :
+    Continuous (fun u : L2 (EC d) => literalComplexHessian (strongEmbedding s hs u) x i j) := by
+  have h1 : (Module.finrank ℝ (EC d) : ℝ) < 2*(s-1) := by linarith
+  have h2 : (Module.finrank ℝ (EC d) : ℝ) < 2*((s-1)-1) := by linarith
+  simp only [literalComplexHessian, strong_second_fderiv s hs h1 h2 hN]
+  fun_prop
+
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+def completedLocalize (s : ℝ) (i : Fin A.count) (ρ : Smooth A)
+    (hρ : tsupport (ρ : X → ℂ) ⊆ (A.euclideanChart i).source) :
+    D.localizers.Sobolev s →L[ℝ] L2 (EC d) :=
+  ((schwartzCoordLinear s).comp (localizeLinear A i ρ hρ)).extendOfNorm (D.localizers.embed s)
+
+lemma completedLocalize_embed (k : ℕ) (i : Fin A.count) (ρ : Smooth A)
+    (hρ : tsupport (ρ : X → ℂ) ⊆ (A.euclideanChart i).source) (f : Smooth A) :
+    D.completedLocalize (k : ℝ) i ρ hρ (D.localizers.embed (k : ℝ) f) =
+      schwartzCoord (k : ℝ) (localize A i ρ hρ f) := by
+  obtain ⟨C,_,hC⟩ := D.localization_integer_bound i ρ hρ k
+  exact LinearMap.extendOfNorm_eq (D.localizers.embed_dense (k : ℝ)) ⟨C,hC⟩ f
+
+lemma completedLocalize_plus_two_embed (k : ℕ) (i : Fin A.count) (ρ : Smooth A)
+    (hρ : tsupport (ρ : X → ℂ) ⊆ (A.euclideanChart i).source) (f : Smooth A) :
+    D.completedLocalize ((k : ℝ)+2) i ρ hρ (D.localizers.embed ((k : ℝ)+2) f) =
+      schwartzCoord ((k : ℝ)+2) (localize A i ρ hρ f) := by
+  obtain ⟨C,_,hC⟩ := D.localization_integer_bound i ρ hρ (k+2)
+  rw [Nat.cast_add, Nat.cast_ofNat] at hC
+  exact LinearMap.extendOfNorm_eq (D.localizers.embed_dense ((k : ℝ)+2)) ⟨C,hC⟩ f
+
+lemma completedLocalize_strong (k : ℕ)
+    (hs : (Module.finrank ℝ (EC d) : ℝ) < 2*(k : ℝ))
+    (i : Fin A.count) (ρ : Smooth A)
+    (hρ : tsupport (ρ : X → ℂ) ⊆ (A.euclideanChart i).source)
+    (u : D.localizers.Sobolev (k : ℝ)) (y : EC d) :
+    strongEmbedding (k : ℝ) hs (D.completedLocalize (k : ℝ) i ρ hρ u) y =
+      if y ∈ (A.euclideanChart i).target then
+        ρ ((A.euclideanChart i).symm y) * D.localizers.strong (k : ℝ) u ((A.euclideanChart i).symm y)
+      else 0 := by
+  have he : (fun u : D.localizers.Sobolev (k : ℝ) =>
+      strongEmbedding (k : ℝ) hs (D.completedLocalize (k : ℝ) i ρ hρ u) y) =
+      (fun u => if y ∈ (A.euclideanChart i).target then
+        ρ ((A.euclideanChart i).symm y) * D.localizers.strong (k : ℝ) u ((A.euclideanChart i).symm y)
+      else 0) := by
+    apply (D.localizers.embed_dense (k : ℝ)).equalizer
+    · fun_prop
+    · split_ifs <;> fun_prop
+    funext f
+    dsimp only [Function.comp_apply]
+    rw [D.completedLocalize_embed, strongEmbedding_schwartz, D.localizers.strong_embed _ hs]
+    rfl
+  exact congr_fun he u
+
+lemma completedLocalize_plus_two_strong (k : ℕ)
+    (hs : (Module.finrank ℝ (EC d) : ℝ) < 2*((k : ℝ)+2))
+    (i : Fin A.count) (ρ : Smooth A)
+    (hρ : tsupport (ρ : X → ℂ) ⊆ (A.euclideanChart i).source)
+    (u : D.localizers.Sobolev ((k : ℝ)+2)) (y : EC d) :
+    strongEmbedding ((k : ℝ)+2) hs (D.completedLocalize ((k : ℝ)+2) i ρ hρ u) y =
+      if y ∈ (A.euclideanChart i).target then
+        ρ ((A.euclideanChart i).symm y) * D.localizers.strong ((k : ℝ)+2) u ((A.euclideanChart i).symm y)
+      else 0 := by
+  have he : (fun u : D.localizers.Sobolev ((k : ℝ)+2) =>
+      strongEmbedding ((k : ℝ)+2) hs (D.completedLocalize ((k : ℝ)+2) i ρ hρ u) y) =
+      (fun u => if y ∈ (A.euclideanChart i).target then
+        ρ ((A.euclideanChart i).symm y) * D.localizers.strong ((k : ℝ)+2) u ((A.euclideanChart i).symm y)
+      else 0) := by
+    apply (D.localizers.embed_dense ((k : ℝ)+2)).equalizer
+    · fun_prop
+    · split_ifs <;> fun_prop
+    funext f
+    dsimp only [Function.comp_apply]
+    rw [D.completedLocalize_plus_two_embed, strongEmbedding_schwartz,
+      D.localizers.strong_embed _ hs]
+    rfl
+  exact congr_fun he u
+
+lemma completedHessian_strong_localize (k : ℕ)
+    (hk : (Module.finrank ℝ (EC d) : ℝ) < 2*(k : ℝ))
+    (hs : (Module.finrank ℝ (EC d) : ℝ) < 2*((k : ℝ)+2))
+    (hN : (Module.finrank ℝ (EC d) : ℝ) < 2*((k : ℝ)+2-4))
+    (u : D.localizers.Sobolev ((k : ℝ)+2)) (p : ι) (i j : Fin d)
+    {x : X} (hx : x ∈ (A.euclideanChart (D.patch p).index).source) :
+    D.localizers.strong (k : ℝ) (D.completedHessian k p i j u) x =
+      D.cutoff p (A.euclideanChart (D.patch p).index x) *
+      literalComplexHessian (strongEmbedding ((k : ℝ)+2) hs
+        (D.completedLocalize ((k : ℝ)+2) (D.patch p).index
+          (cutoffGlobal (D.patch p).index (D.outerCutoff p)) (cutoffGlobal_support _ _) u))
+          (A.euclideanChart (D.patch p).index x) i j := by
+  have he : (fun u : D.localizers.Sobolev ((k : ℝ)+2) =>
+      D.localizers.strong (k : ℝ) (D.completedHessian k p i j u) x) =
+      (fun u => D.cutoff p (A.euclideanChart (D.patch p).index x) *
+      literalComplexHessian (strongEmbedding ((k : ℝ)+2) hs
+        (D.completedLocalize ((k : ℝ)+2) (D.patch p).index
+          (cutoffGlobal (D.patch p).index (D.outerCutoff p)) (cutoffGlobal_support _ _) u))
+          (A.euclideanChart (D.patch p).index x) i j) := by
+    apply (D.localizers.embed_dense ((k : ℝ)+2)).equalizer
+    · fun_prop
+    · exact continuous_const.mul ((literalComplexHessian_strong_continuous _ hs hN _ i j).comp
+        (D.completedLocalize ((k : ℝ)+2) _ _ _).continuous)
+    funext f
+    dsimp only [Function.comp_apply]
+    rw [D.completedHessian_embed, D.localizers.strong_embed _ hk,
+      D.completedLocalize_plus_two_embed, strongEmbedding_schwartz]
+    change globalize (D.patch p).index (D.cutoff p) _ x =
+      D.cutoff p (A.euclideanChart (D.patch p).index x) *
+        literalComplexHessian (localize A (D.patch p).index
+          (cutoffGlobal (D.patch p).index (D.outerCutoff p)) (cutoffGlobal_support _ _) f)
+          (A.euclideanChart (D.patch p).index x) i j
+    rw [literalComplexHessian_schwartz, globalize_apply, ite_eq_left hx]
+    rfl
+  exact congr_fun he u
+
+lemma completedHessian_literal (k : ℕ)
+    (hk : (Module.finrank ℝ (EC d) : ℝ) < 2*(k : ℝ))
+    (hs : (Module.finrank ℝ (EC d) : ℝ) < 2*((k : ℝ)+2))
+    (hN : (Module.finrank ℝ (EC d) : ℝ) < 2*((k : ℝ)+2-4))
+    (u : D.localizers.Sobolev ((k : ℝ)+2)) (p : ι) (i j : Fin d)
+    {x : X} (hx : x ∈ tsupport (D.localizers.weight p : X → ℂ)) :
+    D.localizers.strong (k : ℝ) (D.completedHessian k p i j u) x =
+      literalComplexHessian ((D.localizers.strong ((k : ℝ)+2) u) ∘
+        (A.euclideanChart (D.patch p).index).symm)
+        (A.euclideanChart (D.patch p).index x) i j := by
+  let q := (D.patch p).index
+  let e := A.euclideanChart q
+  have hx' : x ∈ e.source := by
+    have hh := D.localizers.support_sub p hx
+    rwa [D.index_eq] at hh
+  have hcut := D.cutoff_one p x hx
+  rw [D.completedHessian_strong_localize k hk hs hN u p i j hx', hcut, one_mul]
+  apply literalComplexHessian_congr
+  have ht : e x ∈ e.target := e.mapsTo hx'
+  have hn : ∀ᶠ y in 𝓝 (e x), D.cutoff p y ≠ 0 :=
+    (isOpen_ne_fun (D.cutoff p).val.continuous continuous_const).mem_nhds
+      (by change D.cutoff p (e x) ≠ 0; rw [hcut]; exact one_ne_zero)
+  filter_upwards [e.open_target.mem_nhds ht, hn] with y hy hny
+  rw [D.completedLocalize_plus_two_strong, ite_eq_left hy]
+  change (if e.symm y ∈ e.source then D.outerCutoff p (e (e.symm y)) else 0) *
+    D.localizers.strong ((k : ℝ)+2) u (e.symm y) = _
+  rw [ite_eq_left (e.mapsTo_symm hy), e.right_inv hy,
+    D.outerCutoff_one p y (subset_tsupport _ hny), one_mul]
+  rfl
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/LocalPotentialABP.lean b/lean/OAI/Geometry/PrescribedPotential/LocalPotentialABP.lean
new file mode 100644
index 0000000000000000000000000000000000000000..1ad91248352905fa973a6c0876354fb89c8b9bbe
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/LocalPotentialABP.lean
@@ -0,0 +1,117 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.ComplexContactDeterminant
+import OAI.Geometry.PrescribedPotential.ContactSetABP
+import OAI.Geometry.PrescribedPotential.QuadraticContact
+
+namespace OAI
+
+/-! Local Potential A B P. -/
+
+section
+
+ 
+
+noncomputable section
+open Matrix Set Filter Topology Metric MeasureTheory
+open scoped InnerProductSpace ContDiff
+namespace PotentialABP
+variable {d : ℕ}
+local notation "E" => EuclideanSpace ℂ (Fin d)
+local instance localPotentialABPRealInnerProductSpace (d : ℕ) :
+    InnerProductSpace ℝ (EuclideanSpace ℂ (Fin d)) :=
+  InnerProductSpace.rclikeToReal ℂ (EuclideanSpace ℂ (Fin d))
+local notation "e" => EuclideanSpace.equiv (Fin d) ℂ
+
+lemma quadratic_levi_pair {f : (Fin d → ℂ) → ℝ} (hf : ContDiff ℝ ∞ f)
+    (c x v : E) (δ : ℝ) :
+    let u : E → ℝ := fun y => f (e y) + δ*‖y-c‖^2
+    fderiv ℝ (fderiv ℝ u) x v v +
+      fderiv ℝ (fderiv ℝ u) x (Complex.I • v) (Complex.I • v) =
+        4 * inner ℝ ((PotentialKaehler.potentialMatrix f (e x)).toEuclideanLin v) v
+          + 4*δ*‖v‖^2 := by
+  let er := (EuclideanSpace.equiv (Fin d) ℂ).toContinuousLinearMap.restrictScalars ℝ
+  have he : ContDiff ℝ ∞ (f ∘ e) := hf.comp er.contDiff
+  change ContDiff ℝ ∞ (fun y : E => f (e y)) at he
+  dsimp only
+  rw [quadratic_add_hessian he, quadratic_add_hessian he]
+  have hi : inner ℝ (Complex.I • v) (Complex.I • v) = ‖v‖^2 := by
+    simp [norm_smul]
+  rw [real_inner_self_eq_norm_sq, hi]
+  have hp := levi_coord_pair hf x v
+  dsimp only [Function.comp_def] at hp
+  linarith
+
+lemma quadratic_contact_det {f : (Fin d → ℂ) → ℝ} (hf : ContDiff ℝ ∞ f)
+    (c x : E) (δ : ℝ) (G : Matrix (Fin d) (Fin d) ℂ) (hG : G.IsHermitian)
+    (hGδ : ∀ v : E, δ * ‖v‖^2 ≤ inner ℝ (G.toEuclideanLin v) v)
+    (hB : ∀ v : E, 0 ≤ fderiv ℝ (fderiv ℝ
+      (fun y : E => f (e y) + δ*‖y-c‖^2)) x v v) :
+    |(fderiv ℝ (gradient (fun y : E => f (e y) + δ*‖y-c‖^2)) x).det| ≤
+      (4:ℝ)^(2*d) * (G + PotentialKaehler.potentialMatrix f (e x)).det.re^2 := by
+  let er := (EuclideanSpace.equiv (Fin d) ℂ).toContinuousLinearMap.restrictScalars ℝ
+  have hu : ContDiff ℝ ∞ (fun y : E => f (e y) + δ*‖y-c‖^2) :=
+    (hf.comp er.contDiff).add (contDiff_const.mul
+      ((contDiff_id.sub contDiff_const).norm_sq ℝ))
+  apply gradient_det_le_hermitian hu hB _
+    (hG.add (PotentialKaehler.potentialMatrix_hermitian hf.contDiffAt))
+  intro v
+  have hp := quadratic_levi_pair hf c x v δ
+  have hb := hB (Complex.I • v)
+  have hg := hGδ v
+  rw [map_add, LinearMap.add_apply, inner_add_left]
+  linarith
+
+ 
+
+theorem local_potential_sublevel_volume
+    (μ : Measure E) [μ.IsAddHaarMeasure]
+    {f : (Fin d → ℂ) → ℝ} (hf : ContDiff ℝ ∞ f)
+    (G : E → Matrix (Fin d) (Fin d) ℂ)
+    {c : E} {r δ K : ℝ} (hr : 0 < r) (hδ : 0 < δ) (hK : 0 ≤ K)
+    (hmin : ∀ x ∈ closedBall c r, f (e c) ≤ f (e x))
+    (hG : ∀ x ∈ closedBall c r, (G x).IsHermitian)
+    (hGδ : ∀ x ∈ closedBall c r, ∀ v : E,
+      δ*‖v‖^2 ≤ inner ℝ ((G x).toEuclideanLin v) v)
+    (hdet : ∀ x ∈ closedBall c r,
+      |(G x + PotentialKaehler.potentialMatrix f (e x)).det.re| ≤ K) :
+    μ (closedBall (0:E) (δ*r/2)) ≤ ENNReal.ofReal ((4:ℝ)^(2*d)*K^2) *
+      μ {x | x ∈ closedBall c r ∧ f (e x) ≤ f (e c) + δ*r^2/2} := by
+  let u : E → ℝ := fun y => f (e y) + δ*‖y-c‖^2
+  let er := (EuclideanSpace.equiv (Fin d) ℂ).toContinuousLinearMap.restrictScalars ℝ
+  have hu : ContDiff ℝ ∞ u := (hf.comp er.contDiff).add
+    (contDiff_const.mul ((contDiff_id.sub contDiff_const).norm_sq ℝ))
+  have ha : 0 ≤ δ*r/2 := by positivity
+  have hgap : δ*r/2*r < δ*r^2 := by nlinarith [mul_pos hδ (sq_pos_of_pos hr)]
+  have hboundary : ∀ x, dist x c = r → u c + δ*r^2 ≤ u x := by
+    intro x hx
+    have hm := hmin x (by change dist x c ≤ r; exact hx.le)
+    simp only [u, ← dist_eq_norm, hx, dist_self, zero_pow (by decide : 2 ≠ 0),
+      mul_zero, add_zero]
+    linarith
+  have hi := contacts_interior hr ha hgap hboundary
+  have hlocal : μ (closedBall (0:E) (δ*r/2)) ≤
+      ENNReal.ofReal ((4:ℝ)^(2*d)*K^2) * μ (contacts u c r (δ*r/2)) := by
+    apply contact_measure_estimate μ hu hr ha hgap hboundary
+    intro x hx
+    have hd := quadratic_contact_det hf c x δ (G x) (hG x hx.1)
+      (hGδ x hx.1) (contacts_hessian_nonneg hu hx (hi hx))
+    apply hd.trans
+    apply mul_le_mul_of_nonneg_left _ (by positivity)
+    simpa only [sq_abs] using (sq_le_sq₀ (abs_nonneg _) hK).mpr (hdet x hx.1)
+  apply hlocal.trans
+  apply mul_le_mul_of_nonneg_left _ zero_le
+  apply measure_mono
+  intro x hx
+  refine ⟨hx.1, ?_⟩
+  have hs := contacts_sublevel hr.le ha hx
+  have hn : 0 ≤ δ*‖x-c‖^2 := by positivity
+  simp only [u, sub_self, norm_zero, zero_pow (by decide : 2 ≠ 0), mul_zero,
+    add_zero] at hs
+  nlinarith
+
+end PotentialABP
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/LocalRegularity.lean b/lean/OAI/Geometry/PrescribedPotential/LocalRegularity.lean
new file mode 100644
index 0000000000000000000000000000000000000000..6d7b769c477e00508396812e0702d752b9cd416f
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/LocalRegularity.lean
@@ -0,0 +1,539 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.SobolevLocalization
+import OAI.Geometry.PrescribedPotential.VariablePoisson
+
+namespace OAI
+
+/-! Local Regularity. -/
+
+section
+
+ 
+
+noncomputable section
+open Filter Topology MeasureTheory FourierTransform TemperedDistribution LineDeriv
+open scoped SchwartzMap BoundedContinuousFunction ContDiff Real ComplexOrder MatrixOrder
+
+namespace SobolevChart
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+
+lemma schwartz_distribution_injective :
+    Function.Injective (SchwartzMap.toTemperedDistributionCLM E ℂ volume) := by
+  intro f g h
+  apply SchwartzMap.injective_toLp 2 volume
+  apply l2_injective
+  simpa only [Lp.toTemperedDistributionCLM_apply,
+    Lp.toTemperedDistribution_toLp_eq] using h
+
+ 
+def schwartzCoord (s : ℝ) (f : 𝓢(E, ℂ)) : L2 E :=
+  ((memSobolev_iff_realize s (f : 𝓢'(E, ℂ))).mp f.memSobolev).choose
+
+lemma realize_schwartzCoord (s : ℝ) (f : 𝓢(E, ℂ)) :
+    realize s (schwartzCoord s f) = (f : 𝓢'(E, ℂ)) :=
+  ((memSobolev_iff_realize s (f : 𝓢'(E, ℂ))).mp f.memSobolev).choose_spec
+
+lemma schwartzCoord_add (s : ℝ) (f g : 𝓢(E, ℂ)) :
+    schwartzCoord s (f + g) = schwartzCoord s f + schwartzCoord s g := by
+  apply realize_injective s
+  simp only [map_add, realize_schwartzCoord]
+
+lemma schwartzCoord_sum {ι : Type*} (s : ℝ) (S : Finset ι) (f : ι → 𝓢(E, ℂ)) :
+    schwartzCoord s (∑ i ∈ S, f i) = ∑ i ∈ S, schwartzCoord s (f i) := by
+  apply realize_injective s
+  simp only [map_sum, realize_schwartzCoord]
+
+ 
+lemma weak_deriv_comm (v w : E) (u : 𝓢'(E, ℂ)) :
+    ∂_{v} (∂_{w} u) = ∂_{w} (∂_{v} u) := by
+  rw [FrozenPoisson.second_deriv_multiplier, FrozenPoisson.second_deriv_multiplier]
+  congr 2
+  ext x
+  ring
+
+lemma schwartz_deriv_comm (v w : E) (u : 𝓢(E, ℂ)) :
+    ∂_{v} (∂_{w} u) = ∂_{w} (∂_{v} u) := by
+  apply schwartz_distribution_injective
+  change ((∂_{v} (∂_{w} u) : 𝓢(E, ℂ)) : 𝓢'(E, ℂ)) = _
+  rw [← lineDerivOp_toTemperedDistributionCLM_eq,
+    ← lineDerivOp_toTemperedDistributionCLM_eq,
+    ← lineDerivOp_toTemperedDistributionCLM_eq,
+    ← lineDerivOp_toTemperedDistributionCLM_eq]
+  exact weak_deriv_comm v w (u : 𝓢'(E, ℂ))
+
+ 
+lemma summable_of_contracting_recurrence {a b : ℕ → ℝ} {q : ℝ}
+    (ha : ∀ n, 0 ≤ a n) (hb : ∀ n, 0 ≤ b n) (hq : q < 1)
+    (hs : Summable b) (hrecur : ∀ n, a (n+1) ≤ q * a n + b n) :
+    Summable a := by
+  apply summable_of_sum_range_le ha (c := (a 0 + ∑' n, b n) / (1 - q))
+  intro N
+  have hr := Finset.sum_le_sum (s := Finset.range N) (fun n _ => hrecur n)
+  rw [Finset.sum_add_distrib, ← Finset.mul_sum] at hr
+  have he : (∑ n ∈ Finset.range N, a (n+1)) =
+      (∑ n ∈ Finset.range N, a n) + a N - a 0 := by
+    have h := Finset.sum_range_succ' a N
+    rw [Finset.sum_range_succ] at h
+    linarith
+  rw [he] at hr
+  have hbnd : (∑ n ∈ Finset.range N, b n) ≤ ∑' n, b n :=
+    hs.sum_le_tsum (Finset.range N) (fun _ _ => hb _)
+  rw [le_div_iff₀ (sub_pos.mpr hq)]
+  nlinarith [ha N]
+
+end SobolevChart
+
+namespace FrozenPoisson
+open SobolevChart EllipticKernel
+variable {n : ℕ}
+
+def schwartzResolvent (H : Matrix (Fin n) (Fin n) ℂ) :
+    𝓢(EC n, ℂ) →L[ℂ] 𝓢(EC n, ℂ) :=
+  SchwartzMap.fourierMultiplierCLM ℂ (inverseSymbol H)
+
+lemma schwartzResolvent_distribution (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (f : 𝓢(EC n, ℂ)) :
+    SchwartzMap.toTemperedDistributionCLM (EC n) ℂ volume (schwartzResolvent H f) =
+      resolvent H (SchwartzMap.toTemperedDistributionCLM (EC n) ℂ volume f) :=
+  (fourierMultiplierCLM_toTemperedDistributionCLM_eq (inverseSymbol_temperate H hH) f).symm
+
+lemma schwartzCoord_resolvent (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (s : ℝ) (f : 𝓢(EC n, ℂ)) :
+    schwartzCoord (s + 2) (schwartzResolvent H f) =
+      hilbertResolvent H hH (schwartzCoord s f) := by
+  apply realize_injective (s + 2)
+  rw [realize_schwartzCoord, hilbertResolvent_realize, realize_schwartzCoord]
+  exact schwartzResolvent_distribution H hH f
+
+lemma weak_deriv_resolvent (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (v : EC n) (u : 𝓢'(EC n, ℂ)) :
+    ∂_{v} (resolvent H u) = resolvent H (∂_{v} u) := by
+  simp only [lineDeriv_eq_fourierMultiplierCLM, resolvent, map_smul]
+  rw [fourierMultiplierCLM_fourierMultiplierCLM_apply
+    (inverseSymbol_temperate H hH) (by fun_prop),
+    fourierMultiplierCLM_fourierMultiplierCLM_apply (by fun_prop)
+      (inverseSymbol_temperate H hH)]
+  simp only [mul_comm]
+
+lemma schwartz_deriv_resolvent (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (v : EC n) (f : 𝓢(EC n, ℂ)) :
+    ∂_{v} (schwartzResolvent H f) = schwartzResolvent H (∂_{v} f) := by
+  apply schwartz_distribution_injective
+  rw [← lineDerivOp_toTemperedDistributionCLM_eq, schwartzResolvent_distribution H hH,
+    schwartzResolvent_distribution H hH, ← lineDerivOp_toTemperedDistributionCLM_eq]
+  exact weak_deriv_resolvent H hH v (SchwartzMap.toTemperedDistributionCLM (EC n) ℂ volume f)
+
+end FrozenPoisson
+
+namespace SobolevChart
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+
+lemma schwartz_product_distribution (a f : 𝓢(E, ℂ)) :
+    ((SchwartzMap.smulLeftCLM ℂ a f : 𝓢(E, ℂ)) : 𝓢'(E, ℂ)) =
+      smulLeftCLM ℂ a (f : 𝓢'(E, ℂ)) := by
+  ext φ
+  simp only [SchwartzMap.toTemperedDistributionCLM_apply_apply,
+    smulLeftCLM_apply_apply]
+  apply integral_congr_ae
+  filter_upwards [] with x
+  simp only [SchwartzMap.smulLeftCLM_apply_apply a.hasTemperateGrowth, smul_eq_mul]
+  ring
+
+ 
+def schwartzWord : List E → (𝓢(E, ℂ) →L[ℂ] 𝓢(E, ℂ))
+  | [] => ContinuousLinearMap.id ℂ _
+  | v :: vs => lineDerivOpCLM ℂ 𝓢(E, ℂ) v ∘L schwartzWord vs
+
+def weakWord : List E → (𝓢'(E, ℂ) →L[ℂ] 𝓢'(E, ℂ))
+  | [] => ContinuousLinearMap.id ℂ _
+  | v :: vs => lineDerivOpCLM ℂ 𝓢'(E, ℂ) v ∘L weakWord vs
+
+lemma word_distribution (vs : List E) (f : 𝓢(E, ℂ)) :
+    ((schwartzWord vs f : 𝓢(E, ℂ)) : 𝓢'(E, ℂ)) = weakWord vs (f : 𝓢'(E, ℂ)) := by
+  induction vs with
+  | nil => rfl
+  | cons v vs ih =>
+    change ((∂_{v} (schwartzWord vs f) : 𝓢(E, ℂ)) : 𝓢'(E, ℂ)) = _
+    rw [← lineDerivOp_toTemperedDistributionCLM_eq, ih]
+    rfl
+
+variable {ι : Type*} [Fintype ι]
+abbrev SmoothCoefficients (ι : Type*) (E : Type*) [NormedAddCommGroup E] [NormedSpace ℝ E] :=
+  ι → ι → 𝓢(E, ℂ)
+
+def coefficientBCF (a : SmoothCoefficients ι E) (i j : ι) : E →ᵇ ℂ :=
+  (a i j).toBoundedContinuousFunction
+
+def coefficientDeriv (w : E) (a : SmoothCoefficients ι E) : SmoothCoefficients ι E :=
+  fun i j => ∂_{w} (a i j)
+
+def smoothPerturbation (v : ι → E) (a : SmoothCoefficients ι E) :
+    𝓢(E, ℂ) →L[ℂ] 𝓢(E, ℂ) :=
+  ∑ i, ∑ j, SchwartzMap.smulLeftCLM ℂ (a i j) ∘L
+    lineDerivOpCLM ℂ 𝓢(E, ℂ) (v i) ∘L lineDerivOpCLM ℂ 𝓢(E, ℂ) (v j)
+
+omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] in
+lemma smoothPerturbation_apply (v : ι → E) (a : SmoothCoefficients ι E) (f : 𝓢(E, ℂ)) :
+    smoothPerturbation v a f =
+      ∑ i, ∑ j, SchwartzMap.smulLeftCLM ℂ (a i j) (∂_{v i} (∂_{v j} f)) := by
+  simp only [smoothPerturbation, _root_.sum_apply,
+    ContinuousLinearMap.comp_apply, lineDerivOpCLM_apply]
+
+lemma smoothPerturbation_distribution (v : ι → E) (a : SmoothCoefficients ι E)
+    (f : 𝓢(E, ℂ)) :
+    ((smoothPerturbation v a f : 𝓢(E, ℂ)) : 𝓢'(E, ℂ)) =
+      (perturbation v (coefficientBCF a) (schwartzCoord 2 f) : 𝓢'(E, ℂ)) := by
+  rw [perturbation_distribution v _ (fun i j => (a i j).hasTemperateGrowth),
+    realize_schwartzCoord, smoothPerturbation_apply]
+  change SchwartzMap.toTemperedDistributionCLM E ℂ volume _ = _
+  simp only [map_sum]
+  apply Finset.sum_congr rfl
+  intro i _
+  apply Finset.sum_congr rfl
+  intro j _
+  rw [schwartz_product_distribution, ← lineDerivOp_toTemperedDistributionCLM_eq,
+    ← lineDerivOp_toTemperedDistributionCLM_eq]
+  rfl
+
+lemma schwartzCoord_perturbation (v : ι → E) (a : SmoothCoefficients ι E)
+    (f : 𝓢(E, ℂ)) :
+    schwartzCoord 0 (smoothPerturbation v a f) =
+      perturbation v (coefficientBCF a) (schwartzCoord 2 f) := by
+  apply realize_injective 0
+  rw [realize_schwartzCoord]
+  simpa [realize] using smoothPerturbation_distribution v a f
+
+lemma smoothPerturbation_bound (v : ι → E) (a : SmoothCoefficients ι E)
+    (f : 𝓢(E, ℂ)) :
+    ‖schwartzCoord 0 (smoothPerturbation v a f)‖ ≤
+      perturbationBound v (coefficientBCF a) * ‖schwartzCoord 2 f‖ := by
+  rw [schwartzCoord_perturbation]
+  exact perturbation_bound _ _ _
+
+lemma schwartz_deriv_perturbation (w : E) (v : ι → E) (a : SmoothCoefficients ι E)
+    (f : 𝓢(E, ℂ)) :
+    ∂_{w} (smoothPerturbation v a f) =
+      smoothPerturbation v a (∂_{w} f) + smoothPerturbation v (coefficientDeriv w a) f := by
+  simp only [smoothPerturbation_apply, lineDerivOp_sum, schwartz_deriv_product,
+    Finset.sum_add_distrib]
+  rw [add_comm]
+  congr 1
+  apply Finset.sum_congr rfl
+  intro i _
+  apply Finset.sum_congr rfl
+  intro j _
+  congr 1
+  rw [schwartz_deriv_comm w (v i), schwartz_deriv_comm w (v j)]
+
+end SobolevChart
+
+namespace SobolevChart
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+variable {ι : Type*} [Fintype ι]
+
+ 
+
+def commutatorTerms : List E → SmoothCoefficients ι E →
+    List (SmoothCoefficients ι E × List E)
+  | [], _ => []
+  | w :: ws, a => (coefficientDeriv w a, ws) ::
+      ((commutatorTerms ws a).map (fun t => (coefficientDeriv w t.1, t.2)) ++
+       (commutatorTerms ws a).map (fun t => (t.1, w :: t.2)))
+
+omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] [Fintype ι] in
+lemma commutatorTerms_order (ws : List E) (a : SmoothCoefficients ι E)
+    (t : SmoothCoefficients ι E × List E) (ht : t ∈ commutatorTerms ws a) :
+    t.2.length < ws.length := by
+  induction ws generalizing t with
+  | nil => simp [commutatorTerms] at ht
+  | cons w ws ih =>
+    simp only [commutatorTerms, List.mem_cons, List.mem_append, List.mem_map] at ht
+    rcases ht with rfl | ⟨b, hb, rfl⟩ | ⟨b, hb, rfl⟩
+    · simp
+    · exact (ih b hb).trans (Nat.lt_succ_self _)
+    · exact Nat.succ_lt_succ (ih b hb)
+
+lemma list_sum_add {α M : Type*} [AddCommMonoid M] (ts : List α) (f g : α → M) :
+    (ts.map (fun t => f t + g t)).sum = (ts.map f).sum + (ts.map g).sum := by
+  induction ts with
+  | nil => simp
+  | cons t ts ih => simp only [List.map_cons, List.sum_cons, ih]; abel
+
+lemma schwartz_word_perturbation (ws : List E) (v : ι → E)
+    (a : SmoothCoefficients ι E) (f : 𝓢(E, ℂ)) :
+    schwartzWord ws (smoothPerturbation v a f) =
+      smoothPerturbation v a (schwartzWord ws f) +
+        ((commutatorTerms ws a).map
+          (fun t => smoothPerturbation v t.1 (schwartzWord t.2 f))).sum := by
+  induction ws with
+  | nil => simp [schwartzWord, commutatorTerms]
+  | cons w ws ih =>
+    change ∂_{w} (schwartzWord ws (smoothPerturbation v a f)) = _
+    rw [ih, lineDerivOp_add, schwartz_deriv_perturbation]
+    have hs : ∂_{w} (((commutatorTerms ws a).map
+        (fun t => smoothPerturbation v t.1 (schwartzWord t.2 f))).sum) =
+        ((commutatorTerms ws a).map
+          (fun t => ∂_{w} (smoothPerturbation v t.1 (schwartzWord t.2 f)))).sum := by
+      exact (List.sum_map_hom _ _ (lineDerivOpCLM ℂ 𝓢(E, ℂ) w)).symm
+    rw [hs]
+    simp only [schwartz_deriv_perturbation, list_sum_add, commutatorTerms,
+      List.map_cons, List.sum_cons, List.map_append, List.sum_append, List.map_map,
+      Function.comp_def, schwartzWord, ContinuousLinearMap.comp_apply, lineDerivOpCLM_apply]
+    abel
+
+end SobolevChart
+
+namespace FrozenPoisson
+open SobolevChart EllipticKernel
+variable {n : ℕ} {ι : Type*} [Fintype ι]
+
+lemma schwartz_word_resolvent (ws : List (EC n)) (H : Matrix (Fin n) (Fin n) ℂ)
+    (hH : H.PosDef) (f : 𝓢(EC n, ℂ)) :
+    schwartzWord ws (schwartzResolvent H f) = schwartzResolvent H (schwartzWord ws f) := by
+  induction ws with
+  | nil => rfl
+  | cons w ws ih =>
+    change ∂_{w} (schwartzWord ws (schwartzResolvent H f)) = _
+    rw [ih, schwartz_deriv_resolvent H hH]
+    rfl
+
+ 
+def schwartzNeumann (H : Matrix (Fin n) (Fin n) ℂ) (v : ι → EC n)
+    (a : SmoothCoefficients ι (EC n)) : 𝓢(EC n, ℂ) →L[ℂ] 𝓢(EC n, ℂ) :=
+  schwartzResolvent H ∘L smoothPerturbation v a
+
+lemma schwartzResolvent_bound (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (f : 𝓢(EC n, ℂ)) :
+    ‖schwartzCoord 2 (schwartzResolvent H f)‖ ≤
+      ellipticBound H hH * ‖schwartzCoord 0 f‖ := by
+  have hc := schwartzCoord_resolvent H hH 0 f
+  norm_num at hc
+  rw [hc]
+  exact hilbertResolvent_bound H hH _
+
+lemma schwartzNeumann_bound (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (v : ι → EC n) (a : SmoothCoefficients ι (EC n)) (f : 𝓢(EC n, ℂ)) :
+    ‖schwartzCoord 2 (schwartzNeumann H v a f)‖ ≤
+      (ellipticBound H hH * perturbationBound v (coefficientBCF a)) *
+        ‖schwartzCoord 2 f‖ := by
+  calc
+    _ ≤ ellipticBound H hH * ‖schwartzCoord 0 (smoothPerturbation v a f)‖ :=
+      schwartzResolvent_bound H hH _
+    _ ≤ ellipticBound H hH * (perturbationBound v (coefficientBCF a) *
+        ‖schwartzCoord 2 f‖) :=
+      mul_le_mul_of_nonneg_left (smoothPerturbation_bound v a f) (ellipticBound_pos H hH).le
+    _ = _ := by ring
+
+lemma schwartz_word_neumann (ws : List (EC n)) (H : Matrix (Fin n) (Fin n) ℂ)
+    (hH : H.PosDef) (v : ι → EC n) (a : SmoothCoefficients ι (EC n)) (f : 𝓢(EC n, ℂ)) :
+    schwartzWord ws (schwartzNeumann H v a f) =
+      schwartzNeumann H v a (schwartzWord ws f) +
+        ((commutatorTerms ws a).map
+          (fun t => schwartzNeumann H v t.1 (schwartzWord t.2 f))).sum := by
+  change schwartzWord ws (schwartzResolvent H (smoothPerturbation v a f)) = _
+  rw [schwartz_word_resolvent ws H hH, schwartz_word_perturbation, map_add, map_list_sum]
+  simp only [List.map_map, Function.comp_def]
+  rfl
+
+end FrozenPoisson
+
+namespace SobolevChart
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+
+lemma schwartzCoord_zero (s : ℝ) : schwartzCoord s (0 : 𝓢(E, ℂ)) = 0 := by
+  apply realize_injective s
+  simp only [map_zero, realize_schwartzCoord]
+
+lemma schwartzCoord_list_sum (s : ℝ) (ts : List 𝓢(E, ℂ)) :
+    schwartzCoord s ts.sum = (ts.map (schwartzCoord s)).sum := by
+  induction ts with
+  | nil => simp [schwartzCoord_zero]
+  | cons t ts ih => simp only [List.sum_cons, schwartzCoord_add, ih, List.map_cons]
+
+lemma summable_norm_schwartzCoord_list {α : Type*} (s : ℝ) (ts : List α)
+    (f : α → ℕ → 𝓢(E, ℂ))
+    (hf : ∀ t ∈ ts, Summable (fun n => ‖schwartzCoord s (f t n)‖)) :
+    Summable (fun n => ‖schwartzCoord s ((ts.map (fun t => f t n)).sum)‖) := by
+  induction ts with
+  | nil => simp [schwartzCoord_zero]
+  | cons t ts ih =>
+    have ht := hf t (List.mem_cons_self ..)
+    have hts := ih (fun t ht => hf t (List.mem_cons_of_mem _ ht))
+    apply Summable.of_nonneg_of_le (fun _ => norm_nonneg _) _ (ht.add hts)
+    intro n
+    simpa only [List.map_cons, List.sum_cons, schwartzCoord_add] using
+      norm_add_le (schwartzCoord s (f t n))
+        (schwartzCoord s ((ts.map (fun t => f t n)).sum))
+
+end SobolevChart
+
+namespace FrozenPoisson
+open SobolevChart EllipticKernel
+variable {n : ℕ} {ι : Type*} [Fintype ι]
+
+def schwartzIterate (H : Matrix (Fin n) (Fin n) ℂ) (v : ι → EC n)
+    (a : SmoothCoefficients ι (EC n)) (f : 𝓢(EC n, ℂ)) (N : ℕ) : 𝓢(EC n, ℂ) :=
+  (schwartzNeumann H v a)^[N] f
+
+lemma schwartzIterate_succ (H : Matrix (Fin n) (Fin n) ℂ) (v : ι → EC n)
+    (a : SmoothCoefficients ι (EC n)) (f : 𝓢(EC n, ℂ)) (N : ℕ) :
+    schwartzIterate H v a f (N+1) = schwartzNeumann H v a (schwartzIterate H v a f N) :=
+  Function.iterate_succ_apply' _ _ _
+
+lemma summable_norm_neumann_of (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (v : ι → EC n) (a : SmoothCoefficients ι (EC n)) (f : ℕ → 𝓢(EC n, ℂ))
+    (hf : Summable (fun m => ‖schwartzCoord 2 (f m)‖)) :
+    Summable (fun m => ‖schwartzCoord 2 (schwartzNeumann H v a (f m))‖) := by
+  apply Summable.of_nonneg_of_le (fun _ => norm_nonneg _)
+    (fun m => schwartzNeumann_bound H hH v a (f m))
+  exact hf.mul_left _
+
+ 
+
+theorem neumann_summable_word (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (v : ι → EC n) (a : SmoothCoefficients ι (EC n))
+    (hsmall : perturbationBound v (coefficientBCF a) * ellipticBound H hH < 1)
+    (f : 𝓢(EC n, ℂ)) (ws : List (EC n)) :
+    Summable (fun m => ‖schwartzCoord 2 (schwartzWord ws (schwartzIterate H v a f m))‖) := by
+  induction hlen : ws.length using Nat.strong_induction_on generalizing ws with
+  | h k ih =>
+    let err : ℕ → 𝓢(EC n, ℂ) := fun m => ((commutatorTerms ws a).map
+      (fun t => schwartzNeumann H v t.1 (schwartzWord t.2 (schwartzIterate H v a f m)))).sum
+    have hs : Summable (fun m => ‖schwartzCoord 2 (err m)‖) := by
+      apply summable_norm_schwartzCoord_list
+      intro t ht
+      apply summable_norm_neumann_of H hH
+      exact ih t.2.length (hlen ▸ commutatorTerms_order ws a t ht) t.2 rfl
+    apply summable_of_contracting_recurrence (fun _ => norm_nonneg _)
+      (fun _ => norm_nonneg _) (q := ellipticBound H hH * perturbationBound v (coefficientBCF a))
+      (by simpa only [mul_comm] using hsmall) hs
+    intro m
+    rw [schwartzIterate_succ, schwartz_word_neumann ws H hH, schwartzCoord_add]
+    exact (norm_add_le _ _).trans (add_le_add
+      (schwartzNeumann_bound H hH v a _) le_rfl)
+
+end FrozenPoisson
+
+namespace SobolevChart
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+
+lemma memSobolev_word_of_summable (f : ℕ → 𝓢(E, ℂ)) (ws : List E)
+    (hf : Summable (fun m => ‖schwartzCoord 2 (f m)‖))
+    (hw : Summable (fun m => ‖schwartzCoord 2 (schwartzWord ws (f m))‖)) :
+    MemSobolev 2 2 (weakWord ws (realize 2 (∑' m, schwartzCoord 2 (f m)))) := by
+  have h₁ := (hf.of_norm.hasSum.mapL (realize (E := E) 2)).mapL (weakWord ws)
+  have h₂ := hw.of_norm.hasSum.mapL (realize (E := E) 2)
+  simp only [realize_schwartzCoord, word_distribution] at h₁ h₂
+  rw [h₁.unique h₂]
+  exact realize_memSobolev _ _
+
+lemma memSobolev_even_of_words (u : 𝓢'(E, ℂ))
+    (hu : ∀ ws : List E, MemSobolev 2 2 (weakWord ws u)) (k : ℕ) :
+    ∀ ws : List E, MemSobolev (2 * (k : ℝ)) 2 (weakWord ws u) := by
+  induction k with
+  | zero =>
+    intro ws
+    exact (hu ws).mono (by norm_num)
+  | succ k ih =>
+    intro ws
+    have hs : 2 * ((k + 1 : ℕ) : ℝ) = 2 * (k : ℝ) + 2 := by push_cast; ring
+    rw [hs]
+    apply memSobolev_of_second_basis (ih ws)
+    intro j
+    exact ih (stdOrthonormalBasis ℝ E j :: stdOrthonormalBasis ℝ E j :: ws)
+
+end SobolevChart
+
+namespace FrozenPoisson
+open SobolevChart EllipticKernel
+variable {n : ℕ} {ι : Type*} [Fintype ι]
+
+ 
+def neumannSum (H : Matrix (Fin n) (Fin n) ℂ) (v : ι → EC n)
+    (a : SmoothCoefficients ι (EC n)) (f : 𝓢(EC n, ℂ)) : L2 (EC n) :=
+  ∑' m, schwartzCoord 2 (schwartzIterate H v a f m)
+
+lemma schwartzCoord_neumann (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (v : ι → EC n) (a : SmoothCoefficients ι (EC n)) (f : 𝓢(EC n, ℂ)) :
+    schwartzCoord 2 (schwartzNeumann H v a f) =
+      hilbertResolvent H hH (perturbation v (coefficientBCF a) (schwartzCoord 2 f)) := by
+  have he := schwartzCoord_resolvent H hH 0 (smoothPerturbation v a f)
+  norm_num at he
+  rw [schwartzCoord_perturbation] at he
+  exact he
+
+lemma neumannSum_fixed (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (v : ι → EC n) (a : SmoothCoefficients ι (EC n))
+    (hsmall : perturbationBound v (coefficientBCF a) * ellipticBound H hH < 1)
+    (f : 𝓢(EC n, ℂ)) :
+    neumannSum H v a f = schwartzCoord 2 f +
+      hilbertResolvent H hH (perturbation v (coefficientBCF a) (neumannSum H v a f)) := by
+  have hs := (neumann_summable_word H hH v a hsmall f []).of_norm
+  change Summable (fun m => schwartzCoord 2 (schwartzIterate H v a f m)) at hs
+  let T := hilbertResolvent H hH ∘L perturbation v (coefficientBCF a)
+  have hmap := T.map_tsum hs
+  simp only [T, ContinuousLinearMap.comp_apply, ← schwartzCoord_neumann,
+    ← schwartzIterate_succ] at hmap
+  change (∑' m, schwartzCoord 2 (schwartzIterate H v a f m)) = _
+  rw [hs.tsum_eq_zero_add]
+  change schwartzCoord 2 f + _ = schwartzCoord 2 f + _
+  exact congrArg (schwartzCoord 2 f + ·) hmap.symm
+
+lemma equation_of_fixed (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (v : ι → EC n) (a : ι → ι → EC n →ᵇ ℂ) (u f : L2 (EC n))
+    (hu : u = hilbertResolvent H hH (f + perturbation v a u)) :
+    realize 2 u - frozenDifferential H (realize 2 u) -
+      (perturbation v a u : 𝓢'(EC n, ℂ)) = (f : 𝓢'(EC n, ℂ)) := by
+  have hr : realize 2 u = resolvent H (realize 0 (f + perturbation v a u)) := by
+    conv_lhs => rw [hu]
+    simpa only [zero_add] using hilbertResolvent_realize H hH 0 (f + perturbation v a u)
+  rw [← shifted_eq_identity_sub H hH, hr, shifted_resolvent H hH]
+  simp only [realize, neg_zero, besselPotential_zero, ContinuousLinearMap.coe_comp,
+    Function.comp_apply, ContinuousLinearMap.id_apply, Lp.toTemperedDistributionCLM_apply]
+  change Lp.toTemperedDistributionCLM ℂ volume 2 (f + perturbation v a u) -
+    Lp.toTemperedDistributionCLM ℂ volume 2 (perturbation v a u) =
+    Lp.toTemperedDistributionCLM ℂ volume 2 f
+  simp only [map_add, add_sub_cancel_right]
+
+lemma neumannSum_eq_localResolvent (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (v : ι → EC n) (a : SmoothCoefficients ι (EC n))
+    (hsmall : perturbationBound v (coefficientBCF a) * ellipticBound H hH < 1)
+    (f : 𝓢(EC n, ℂ)) :
+    neumannSum H v a (schwartzResolvent H f) =
+      localResolvent H hH v (coefficientBCF a) hsmall (schwartzCoord 0 f) := by
+  apply perturbed_unique H hH v (coefficientBCF a) hsmall
+    _ _ (schwartzCoord 0 f)
+  · apply equation_of_fixed H hH
+    rw [map_add]
+    have hr := schwartzCoord_resolvent H hH 0 f
+    norm_num at hr
+    rw [← hr]
+    exact neumannSum_fixed H hH v a hsmall (schwartzResolvent H f)
+  · exact localResolvent_equation H hH v (coefficientBCF a) hsmall _
+
+ 
+
+theorem localResolvent_schwartz_memSobolev (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (v : ι → EC n) (a : SmoothCoefficients ι (EC n))
+    (hsmall : perturbationBound v (coefficientBCF a) * ellipticBound H hH < 1)
+    (f : 𝓢(EC n, ℂ)) (k : ℕ) :
+    MemSobolev (2 * (k : ℝ)) 2
+      (realize 2 (localResolvent H hH v (coefficientBCF a) hsmall (schwartzCoord 0 f))) := by
+  rw [← neumannSum_eq_localResolvent H hH v a hsmall f]
+  apply memSobolev_even_of_words _ _ k []
+  intro ws
+  exact memSobolev_word_of_summable (schwartzIterate H v a (schwartzResolvent H f)) ws
+    (neumann_summable_word H hH v a hsmall (schwartzResolvent H f) [])
+    (neumann_summable_word H hH v a hsmall (schwartzResolvent H f) ws)
+
+end FrozenPoisson
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/LocalizedCompact.lean b/lean/OAI/Geometry/PrescribedPotential/LocalizedCompact.lean
new file mode 100644
index 0000000000000000000000000000000000000000..049f54b91379e1ae5fe1734ea5878c4b19135d58
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/LocalizedCompact.lean
@@ -0,0 +1,108 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.SobolevParametrix
+
+namespace OAI
+
+/-! Localized Compact. -/
+
+section
+
+ 
+
+noncomputable section
+open MeasureTheory Set Filter Topology FourierTransform
+open scoped BoundedContinuousFunction SchwartzMap Classical
+namespace EllipticCompact
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+open SobolevChart
+
+def localizedFun (κ : 𝓢(E, ℂ)) (s : Set E) (F : s →ᵇ ℂ) (x : E) : ℂ :=
+  if hx : x ∈ s then κ x * F ⟨x, hx⟩ else 0
+
+omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] in
+lemma localizedFun_bound (κ : 𝓢(E, ℂ)) (s : Set E) (F : s →ᵇ ℂ) (x : E) :
+    ‖localizedFun κ s F x‖ ≤ ‖F‖ * ‖κ x‖ := by
+  unfold localizedFun
+  split_ifs with hx
+  · rw [norm_mul, mul_comm]
+    exact mul_le_mul_of_nonneg_right (F.norm_coe_le_norm ⟨x,hx⟩) (norm_nonneg _)
+  · simp only [norm_zero]
+    positivity
+
+lemma localizedFun_memLp (κ : 𝓢(E, ℂ)) (s : Set E) (hs : MeasurableSet s) (F : s →ᵇ ℂ) :
+    MemLp (localizedFun κ s F) 2 (volume : Measure E) := by
+  have hm : StronglyMeasurable (localizedFun κ s F) :=
+    (show Continuous (fun x : s => κ x * F x) by fun_prop).stronglyMeasurable.dite
+      stronglyMeasurable_const hs
+  apply ((κ.memLp (p := 2) (μ := volume)).const_mul (‖F‖ : ℂ)).mono hm.aestronglyMeasurable
+  filter_upwards [] with x
+  simpa only [norm_mul, Complex.norm_real, Real.norm_eq_abs, abs_norm] using localizedFun_bound κ s F x
+
+def localizedLp (κ : 𝓢(E, ℂ)) (s : Set E) (hs : MeasurableSet s) (F : s →ᵇ ℂ) :
+    L2 E := (localizedFun_memLp κ s hs F).toLp (localizedFun κ s F)
+
+lemma localizedLp_ae (κ : 𝓢(E, ℂ)) (s : Set E) (hs : MeasurableSet s) (F : s →ᵇ ℂ) :
+    localizedLp κ s hs F =ᵐ[volume] localizedFun κ s F :=
+  (localizedFun_memLp κ s hs F).coeFn_toLp
+
+lemma localizedLp_norm (κ : 𝓢(E, ℂ)) (s : Set E) (hs : MeasurableSet s) (F : s →ᵇ ℂ) :
+    ‖localizedLp κ s hs F‖ ≤ ‖SchwartzMap.toLpCLM ℂ ℂ 2 volume κ‖ * ‖F‖ := by
+  rw [mul_comm]
+  apply Lp.norm_le_mul_norm_of_ae_le_mul
+  filter_upwards [localizedLp_ae κ s hs F,
+    κ.coeFn_toLp 2 volume] with x hF hκ
+  change ‖(localizedLp κ s hs F : E → ℂ) x‖ ≤ ‖F‖ * ‖(κ.toLp 2 : L2 E) x‖
+  rw [show (localizedLp κ s hs F : E → ℂ) x = localizedFun κ s F x from hF, hκ]
+  exact localizedFun_bound κ s F x
+
+def localizeLinear (κ : 𝓢(E, ℂ)) (s : Set E) (hs : MeasurableSet s) :
+    (s →ᵇ ℂ) →ₗ[ℂ] L2 E where
+  toFun := localizedLp κ s hs
+  map_add' := by
+    intro F G
+    apply Lp.ext
+    filter_upwards [localizedLp_ae κ s hs (F+G),
+      localizedLp_ae κ s hs F,
+      localizedLp_ae κ s hs G,
+      Lp.coeFn_add (localizedLp κ s hs F) (localizedLp κ s hs G)] with x h hF hG ha
+    rw [h, ha, Pi.add_apply, hF, hG]
+    simp only [localizedFun]
+    split_ifs <;> simp [mul_add]
+  map_smul' := by
+    intro c F
+    apply Lp.ext
+    filter_upwards [localizedLp_ae κ s hs (c • F),
+      localizedLp_ae κ s hs F,
+      Lp.coeFn_smul c (localizedLp κ s hs F)] with x h hF ha
+    simp only [RingHom.id_apply]
+    rw [h, ha, Pi.smul_apply, hF]
+    simp only [localizedFun]
+    split_ifs <;> simp [smul_eq_mul, mul_left_comm]
+
+def localizeL2 (κ : 𝓢(E, ℂ)) (s : Set E) (hs : MeasurableSet s) :
+    (s →ᵇ ℂ) →L[ℂ] L2 E :=
+  (localizeLinear κ s hs).mkContinuous ‖SchwartzMap.toLpCLM ℂ ℂ 2 volume κ‖
+    (localizedLp_norm κ s hs)
+
+lemma localizeL2_ae (κ : 𝓢(E, ℂ)) (s : Set E) (hs : MeasurableSet s) (F : s →ᵇ ℂ) :
+    localizeL2 κ s hs F =ᵐ[volume] localizedFun κ s F :=
+  localizedLp_ae κ s hs F
+
+lemma localizeL2_restrict (κ : 𝓢(E, ℂ)) (s : Set E) (hs : MeasurableSet s)
+    (hκ : Function.support κ ⊆ s) (F : E →ᵇ ℂ) :
+    localizeL2 κ s hs (F.compContinuous ⟨Subtype.val, continuous_subtype_val⟩) =ᵐ[volume]
+      fun x => κ x * F x := by
+  filter_upwards [localizeL2_ae κ s hs (F.compContinuous ⟨Subtype.val, continuous_subtype_val⟩)] with x hx
+  rw [hx, localizedFun]
+  split_ifs with h
+  · rfl
+  · have hk : κ x = 0 := not_not.mp (mt (fun hne : κ x ≠ 0 => hκ hne) h)
+    rw [hk, zero_mul]
+
+end EllipticCompact
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/LocalizedDerivations.lean b/lean/OAI/Geometry/PrescribedPotential/LocalizedDerivations.lean
new file mode 100644
index 0000000000000000000000000000000000000000..ce602b83ecade1350d74bd8e0982a0916a363c1d
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/LocalizedDerivations.lean
@@ -0,0 +1,150 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.GlobalHessian
+
+namespace OAI
+
+/-! Localized Derivations. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory LineDeriv
+open scoped ContDiff SchwartzMap Classical
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+
+namespace ChartCutoff
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  {U : Set E}
+
+def realPart (κ : ChartCutoff U) : ChartCutoff U where
+  val := SchwartzMap.postcompCLM (Complex.ofRealCLM.comp Complex.reCLM) κ.val
+  compact := by
+    apply HasCompactSupport.of_support_subset_isCompact κ.compact
+    intro x hx
+    apply subset_tsupport (κ : E → ℂ)
+    intro h
+    exact hx (by change ((κ x).re : ℂ) = 0; simp [h])
+  support_sub := by
+    apply Subset.trans _ κ.support_sub
+    apply closure_mono
+    intro x hx h
+    exact hx (by change ((κ x).re : ℂ) = 0; simp [h])
+
+@[simp] lemma realPart_apply (κ : ChartCutoff U) (x : E) : κ.realPart x = ((κ x).re : ℂ) := rfl
+
+lemma realPart_support (κ : ChartCutoff U) :
+    tsupport (κ.realPart : E → ℂ) ⊆ tsupport (κ : E → ℂ) := by
+  apply closure_mono
+  intro x hx h
+  exact hx (by simp [h])
+end ChartCutoff
+
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+def localizedDerivative (p : ι) (v : EC d) : Smooth A →ₗ[ℝ] Smooth A :=
+  (globalize (D.patch p).index (D.cutoff p).realPart).comp
+    (((lineDerivOpCLM ℂ 𝓢(EC d, ℂ) v).restrictScalars ℝ).toLinearMap.comp
+      (localizeLinear A (D.patch p).index (cutoffGlobal (D.patch p).index (D.outerCutoff p))
+        (cutoffGlobal_support _ _)))
+
+lemma localizedDerivative_bound (k : ℕ) (p : ι) (v : EC d) :
+    D.localizers.BoundedCore ((k : ℝ)+1) (k : ℝ) (D.localizedDerivative p v) := by
+  let q := (D.patch p).index
+  let ρ := cutoffGlobal q (D.outerCutoff p)
+  let hp := cutoffGlobal_support q (D.outerCutoff p)
+  obtain ⟨B,hB,hb⟩ := globalize_integer_bound D.localizers q (D.cutoff p).realPart k
+  obtain ⟨C,hC,hc⟩ := coreBound_deriv (s := (k : ℝ)+1) (t := (k : ℝ)) v (by linarith)
+  obtain ⟨E,hE,he⟩ := D.localization_integer_bound q ρ hp (k+1)
+  rw [Nat.cast_add, Nat.cast_one] at he
+  refine ⟨B*(C*E),mul_nonneg hB (mul_nonneg hC hE),fun f => ?_⟩
+  let w := localize A q ρ hp f
+  change ‖D.localizers.embed (k : ℝ) (globalize q (D.cutoff p).realPart (∂_{v} w))‖ ≤ _
+  calc
+    _ ≤ B*‖schwartzCoord (k : ℝ) (∂_{v} w)‖ := hb _
+    _ ≤ B*(C*‖schwartzCoord ((k : ℝ)+1) w‖) := mul_le_mul_of_nonneg_left (hc w) hB
+    _ ≤ B*(C*(E*‖D.localizers.embed ((k : ℝ)+1) f‖)) :=
+      mul_le_mul_of_nonneg_left (mul_le_mul_of_nonneg_left (he f) hC) hB
+    _ = _ := by ring
+
+lemma localizedDerivative_apply (p : ι) (v : EC d) (f : Smooth A) (x : X) :
+    D.localizedDerivative p v f x =
+      if x ∈ (A.euclideanChart (D.patch p).index).source then
+        (D.cutoff p).realPart (A.euclideanChart (D.patch p).index x) *
+          fderiv ℝ (f ∘ (A.euclideanChart (D.patch p).index).symm)
+            (A.euclideanChart (D.patch p).index x) v else 0 := by
+  let q := (D.patch p).index
+  let e := A.euclideanChart q
+  change globalize q (D.cutoff p).realPart
+    (∂_{v} (localize A q (cutoffGlobal q (D.outerCutoff p)) (cutoffGlobal_support _ _) f)) x = _
+  rw [globalize_apply]
+  dsimp only [q]
+  by_cases hx : x ∈ e.source
+  all_goals dsimp only [e, q] at hx
+  · simp only [ite_eq_left hx]
+    by_cases hκ : (D.cutoff p).realPart (e x) = 0
+    · rw [hκ,zero_mul,zero_mul]
+    · rw [SchwartzMap.lineDerivOp_apply_eq_fderiv]
+      have hh := localize_cutoff_eventually q (D.cutoff p).realPart (D.outerCutoff p)
+        (fun y hy => D.outerCutoff_one p y ((D.cutoff p).realPart_support hy))
+        f (e.mapsTo hx) hκ
+      rw [hh.fderiv_eq]
+  · simp only [ite_eq_right hx]
+
+lemma localizedDerivative_mul (p : ι) (v : EC d) (f h : Smooth A) :
+    D.localizedDerivative p v (f.mul h) =
+      (D.localizedDerivative p v f).mul h + f.mul (D.localizedDerivative p v h) := by
+  apply Smooth.ext
+  intro x
+  rw [Smooth.add_apply]
+  change D.localizedDerivative p v (f.mul h) x =
+    D.localizedDerivative p v f x * h x + f x * D.localizedDerivative p v h x
+  rw [D.localizedDerivative_apply, D.localizedDerivative_apply, D.localizedDerivative_apply]
+  let e := A.euclideanChart (D.patch p).index
+  by_cases hx : x ∈ e.source
+  all_goals dsimp only [e] at hx
+  · simp only [ite_eq_left hx]
+    have hf := (f.smooth (D.patch p).index).contDiffAt (e.open_target.mem_nhds (e.mapsTo hx))
+    have hh := (h.smooth (D.patch p).index).contDiffAt (e.open_target.mem_nhds (e.mapsTo hx))
+    have he : ((f.mul h) ∘ e.symm) = (fun y => f (e.symm y) * h (e.symm y)) := rfl
+    have hd := fderiv_fun_mul (hf.differentiableAt (by simp)) (hh.differentiableAt (by simp))
+    dsimp only [e, Function.comp_apply] at hd
+    rw [he]
+    dsimp only [e]
+    rw [hd]
+    simp only [add_apply, smul_apply,
+      smul_eq_mul, (A.euclideanChart (D.patch p).index).left_inv hx]
+    ring
+  · simp only [ite_eq_right hx, zero_mul, mul_zero, add_zero]
+
+lemma localizedDerivative_const (p : ι) (v : EC d) (c : ℂ) :
+    D.localizedDerivative p v (Smooth.const c) = 0 := by
+  apply Smooth.ext
+  intro x
+  rw [D.localizedDerivative_apply]
+  have he : ((Smooth.const c : Smooth A) ∘ (A.euclideanChart (D.patch p).index).symm) =
+      (fun _ : EC d => c) := rfl
+  rw [he, fderiv_const_apply]
+  split_ifs <;> simp only [zero_apply, mul_zero, Smooth.zero_apply]
+
+def completedDerivative (k : ℕ) (p : ι) (v : EC d) :
+    D.localizers.Sobolev ((k : ℝ)+1) →L[ℝ] D.localizers.Sobolev (k : ℝ) :=
+  D.localizers.extendCore ((k : ℝ)+1) (k : ℝ) (D.localizedDerivative p v)
+
+lemma completedDerivative_embed (k : ℕ) (p : ι) (v : EC d) (f : Smooth A) :
+    D.completedDerivative k p v (D.localizers.embed ((k : ℝ)+1) f) =
+      D.localizers.embed (k : ℝ) (D.localizedDerivative p v f) :=
+  D.localizers.extendCore_embed (D.localizedDerivative_bound k p v) f
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/LocalizedPullback.lean b/lean/OAI/Geometry/PrescribedPotential/LocalizedPullback.lean
new file mode 100644
index 0000000000000000000000000000000000000000..814d6981ae824426328a179250b0e40ca7f19a5e
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/LocalizedPullback.lean
@@ -0,0 +1,192 @@
+import Mathlib
+
+namespace OAI
+
+/-! Localized Pullback. -/
+
+section
+
+ 
+
+noncomputable section
+open Set MeasureTheory Topology Filter
+open scoped ENNReal ContDiff
+
+namespace GlobalElliptic
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+
+lemma map_restrict_le_of_jacobian_lower (f : E → E) (hf : Measurable f)
+    (f' : E → E →L[ℝ] E) {K : Set E} (hK : MeasurableSet K)
+    (hder : ∀ x ∈ K, HasFDerivWithinAt f (f' x) K x) (hinj : InjOn f K)
+    {δ : ℝ} (hδ : 0 < δ) (hlower : ∀ x ∈ K, δ ≤ |(f' x).det|) :
+    Measure.map f (volume.restrict K) ≤ ENNReal.ofReal δ⁻¹ • (volume : Measure E) := by
+  have hineq : (volume.restrict K).withDensity (fun _ : E => (1 : ℝ≥0∞)) ≤
+      (volume.restrict K).withDensity
+        (fun x => ENNReal.ofReal δ⁻¹ * ENNReal.ofReal |(f' x).det|) := by
+    apply withDensity_mono
+    filter_upwards [self_mem_ae_restrict hK] with x hx
+    rw [← ENNReal.ofReal_mul (by positivity)]
+    rw [← ENNReal.ofReal_one]
+    apply ENNReal.ofReal_le_ofReal
+    exact (show (1:ℝ) ≤ δ⁻¹ * |(f' x).det| by
+      rw [← div_eq_inv_mul, le_div_iff₀ hδ]
+      simpa using hlower x hx)
+  rw [withDensity_const, one_smul] at hineq
+  change volume.restrict K ≤ (volume.restrict K).withDensity
+    (ENNReal.ofReal δ⁻¹ • (fun x => ENNReal.ofReal |(f' x).det|)) at hineq
+  rw [withDensity_smul' _ _ ENNReal.ofReal_ne_top] at hineq
+  have hmap := Measure.map_mono hineq hf
+  rw [Measure.map_smul _ hf.aemeasurable,
+    map_withDensity_abs_det_fderiv_eq_addHaar volume hK.nullMeasurableSet hder hinj] at hmap
+  apply hmap.trans
+  gcongr
+  exact Measure.restrict_le_self
+
+ 
+
+theorem exists_map_restrict_bound (f : E → E) (f' : E → E →L[ℝ] E)
+    {K : Set E} (hK : IsCompact K) (hf : ContinuousOn f K)
+    (hder : ∀ x ∈ K, HasFDerivWithinAt f (f' x) K x) (hinj : InjOn f K)
+    (hcont : ContinuousOn (fun x => |(f' x).det|) K)
+    (hnonzero : ∀ x ∈ K, (f' x).det ≠ 0) :
+    ∃ C : ℝ, 0 ≤ C ∧
+      Measurable (K.indicator f) ∧
+      Measure.map (K.indicator f) (volume.restrict K) ≤ ENNReal.ofReal C • (volume : Measure E) := by
+  classical
+  have hmeas : Measurable (K.indicator f) := by
+    exact hf.measurable_piecewise continuous_zero.continuousOn hK.isClosed.measurableSet
+  by_cases hne : K.Nonempty
+  · obtain ⟨x, hx, hxmin⟩ := hK.exists_isMinOn hne hcont
+    let δ := |(f' x).det|
+    have hδ : 0 < δ := abs_pos.mpr (hnonzero x hx)
+    refine ⟨δ⁻¹, (inv_pos.mpr hδ).le, hmeas, ?_⟩
+    apply map_restrict_le_of_jacobian_lower _ hmeas f' hK.isClosed.measurableSet
+      (fun y hy => (hder y hy).congr (fun z hz => indicator_of_mem hz f)
+        (indicator_of_mem hy f))
+      (hinj.congr (fun y hy => (indicator_of_mem hy f).symm)) hδ
+    intro y hy
+    exact hxmin hy
+  · have he : K = ∅ := not_nonempty_iff_eq_empty.mp hne
+    subst K
+    exact ⟨0, le_rfl, hmeas, by simp⟩
+
+omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] in
+lemma chart_fderiv_det_ne_zero (e : OpenPartialHomeomorph E E)
+    (he : ContDiffOn ℝ ∞ e e.source) (hs : ContDiffOn ℝ ∞ e.symm e.target)
+    {x : E} (hx : x ∈ e.source) : (fderiv ℝ e x).det ≠ 0 := by
+  have h₁ := ((he.contDiffAt (e.open_source.mem_nhds hx)).differentiableAt (by simp)).hasFDerivAt
+  have h₂ := ((hs.contDiffAt (e.open_target.mem_nhds (e.mapsTo hx))).differentiableAt (by simp)).hasFDerivAt
+  have hcomp := h₂.comp x h₁
+  have hi : (fun y => e.symm (e y)) =ᶠ[nhds x] id := by
+    filter_upwards [e.open_source.mem_nhds hx] with y hy using e.left_inv hy
+  have heq : (fderiv ℝ e.symm (e x)).comp (fderiv ℝ e x) = ContinuousLinearMap.id ℝ E :=
+    (hcomp.congr_of_eventuallyEq hi.symm).unique (hasFDerivAt_id x)
+  have hdet := congrArg ContinuousLinearMap.det heq
+  change LinearMap.det ((fderiv ℝ e.symm (e x)).toLinearMap.comp (fderiv ℝ e x).toLinearMap) =
+    LinearMap.det (LinearMap.id) at hdet
+  rw [LinearMap.det_comp, LinearMap.det_id] at hdet
+  intro hz
+  change (fderiv ℝ e.symm (e x)).det * (fderiv ℝ e x).det = 1 at hdet
+  rw [hz, mul_zero] at hdet
+  exact zero_ne_one hdet
+
+ 
+
+theorem chart_map_restrict_bound (e : OpenPartialHomeomorph E E)
+    (he : ContDiffOn ℝ ∞ e e.source) (hs : ContDiffOn ℝ ∞ e.symm e.target)
+    {K : Set E} (hK : IsCompact K) (hKe : K ⊆ e.source) :
+    ∃ C : ℝ, 0 ≤ C ∧ Measurable (K.indicator e) ∧
+      Measure.map (K.indicator e) (volume.restrict K) ≤ ENNReal.ofReal C • (volume : Measure E) := by
+  apply exists_map_restrict_bound e (fderiv ℝ e) hK (e.continuousOn.mono hKe)
+    (fun x hx => (((he.contDiffAt (e.open_source.mem_nhds (hKe hx))).differentiableAt
+      (by simp)).hasFDerivAt).hasFDerivWithinAt) (e.injOn.mono hKe)
+  · exact (ContinuousLinearMap.continuous_det.comp_continuousOn
+      (he.continuousOn_fderiv_of_isOpen e.open_source (by simp))).abs.mono hKe
+  · exact fun x hx => chart_fderiv_det_ne_zero e he hs (hKe hx)
+
+end GlobalElliptic
+
+namespace MeasureTheory.Lp
+open scoped _root_.MeasureTheory _root_.MeasureTheory.Lp
+variable {α β F : Type*} [MeasurableSpace α] [MeasurableSpace β]
+  [NormedAddCommGroup F] [NormedSpace ℝ F]
+  {μ : _root_.MeasureTheory.Measure α} {ν : _root_.MeasureTheory.Measure β} {p : ℝ≥0∞} [Fact (1 ≤ p)]
+
+def extendZeroFun {K : Set α} (hK : MeasurableSet K) (u : _root_.MeasureTheory.Lp F p (μ.restrict K)) :
+    _root_.MeasureTheory.Lp F p μ :=
+  ((_root_.MeasureTheory.memLp_indicator_iff_restrict hK).mpr (_root_.MeasureTheory.Lp.memLp u)).toLp (K.indicator u)
+
+omit [NormedSpace ℝ F] [Fact (1 ≤ p)] in
+lemma coeFn_extendZeroFun {K : Set α} (hK : MeasurableSet K) (u : _root_.MeasureTheory.Lp F p (μ.restrict K)) :
+    extendZeroFun hK u =ᵐ[μ] K.indicator u := _root_.MeasureTheory.MemLp.coeFn_toLp _
+
+omit [NormedSpace ℝ F] [Fact (1 ≤ p)] in
+lemma extendZeroFun_add {K : Set α} (hK : MeasurableSet K) (u v : _root_.MeasureTheory.Lp F p (μ.restrict K)) :
+    extendZeroFun hK (u + v) = extendZeroFun hK u + extendZeroFun hK v := by
+  apply _root_.MeasureTheory.Lp.ext
+  filter_upwards [coeFn_extendZeroFun hK (u + v), coeFn_extendZeroFun hK u,
+    coeFn_extendZeroFun hK v, _root_.MeasureTheory.Lp.coeFn_add (extendZeroFun hK u) (extendZeroFun hK v),
+    (_root_.MeasureTheory.ae_restrict_iff' hK).mp (_root_.MeasureTheory.Lp.coeFn_add u v)] with x h₁ h₂ h₃ h₄ h₅
+  rw [h₁, h₄, Pi.add_apply, h₂, h₃]
+  by_cases hx : x ∈ K
+  · simp only [indicator_of_mem hx, h₅ hx, Pi.add_apply]
+  · simp only [indicator_of_notMem hx, add_zero]
+
+omit [Fact (1 ≤ p)] in
+lemma extendZeroFun_smul {K : Set α} (hK : MeasurableSet K) (c : ℝ)
+    (u : _root_.MeasureTheory.Lp F p (μ.restrict K)) : extendZeroFun hK (c • u) = c • extendZeroFun hK u := by
+  apply _root_.MeasureTheory.Lp.ext
+  filter_upwards [coeFn_extendZeroFun hK (c • u), coeFn_extendZeroFun hK u,
+    _root_.MeasureTheory.Lp.coeFn_smul c (extendZeroFun hK u),
+    (_root_.MeasureTheory.ae_restrict_iff' hK).mp (_root_.MeasureTheory.Lp.coeFn_smul c u)] with x h₁ h₂ h₃ h₄
+  rw [h₁, h₃, Pi.smul_apply, h₂]
+  by_cases hx : x ∈ K
+  · simp only [indicator_of_mem hx, h₄ hx, Pi.smul_apply]
+  · simp only [indicator_of_notMem hx, smul_zero]
+
+omit [NormedSpace ℝ F] [Fact (1 ≤ p)] in
+lemma norm_extendZeroFun {K : Set α} (hK : MeasurableSet K) (u : _root_.MeasureTheory.Lp F p (μ.restrict K)) :
+    ‖extendZeroFun hK u‖ = ‖u‖ := by
+  rw [extendZeroFun, _root_.MeasureTheory.Lp.norm_toLp, _root_.MeasureTheory.eLpNorm_indicator_eq_eLpNorm_restrict hK, _root_.MeasureTheory.Lp.norm_def]
+
+ 
+def extendZero {K : Set α} (hK : MeasurableSet K) :
+    _root_.MeasureTheory.Lp F p (μ.restrict K) →ₗᵢ[ℝ] _root_.MeasureTheory.Lp F p μ where
+  toFun := extendZeroFun hK
+  map_add' := extendZeroFun_add hK
+  map_smul' := extendZeroFun_smul hK
+  norm_map' := norm_extendZeroFun hK
+
+lemma coeFn_extendZero {K : Set α} (hK : MeasurableSet K) (u : _root_.MeasureTheory.Lp F p (μ.restrict K)) :
+    extendZero hK u =ᵐ[μ] K.indicator u := coeFn_extendZeroFun hK u
+
+ 
+def pullbackBounded (f : α → β) (hf : Measurable f) {C : ℝ≥0∞}
+    (hC : C ≠ ⊤) (h : _root_.MeasureTheory.Measure.map f μ ≤ C • ν) : _root_.MeasureTheory.Lp F p ν →L[ℝ] _root_.MeasureTheory.Lp F p μ :=
+  (_root_.MeasureTheory.Lp.compMeasurePreservingₗᵢ ℝ f ⟨hf, rfl⟩).toContinuousLinearMap ∘L
+    _root_.MeasureTheory.Lp.LpToLpOfMeasureLeSMul hC h
+
+lemma coeFn_pullbackBounded (f : α → β) (hf : Measurable f) {C : ℝ≥0∞}
+    (hC : C ≠ ⊤) (h : _root_.MeasureTheory.Measure.map f μ ≤ C • ν) (u : _root_.MeasureTheory.Lp F p ν) :
+    pullbackBounded f hf hC h u =ᵐ[μ] (fun x => u (f x)) := by
+  change _root_.MeasureTheory.Lp.compMeasurePreserving f ⟨hf, rfl⟩ (_root_.MeasureTheory.Lp.LpToLpOfMeasureLeSMul hC h u) =ᵐ[μ] _
+  apply (_root_.MeasureTheory.Lp.coeFn_compMeasurePreserving _ ⟨hf, rfl⟩).trans
+  exact _root_.MeasureTheory.ae_of_ae_map hf.aemeasurable (_root_.MeasureTheory.Lp.coeFn_LpToLpOfMeasureLeSMul hC h u)
+
+lemma pullbackBounded_norm_le (f : α → β) (hf : Measurable f) {C : ℝ≥0∞}
+    (hC : C ≠ ⊤) (h : _root_.MeasureTheory.Measure.map f μ ≤ C • ν) (u : _root_.MeasureTheory.Lp F p ν) :
+    ‖pullbackBounded f hf hC h u‖ ≤ C.toReal ^ (1 / p).toReal * ‖u‖ := by
+  change ‖_root_.MeasureTheory.Lp.compMeasurePreservingₗᵢ ℝ f ⟨hf, rfl⟩ (_root_.MeasureTheory.Lp.LpToLpOfMeasureLeSMul hC h u)‖ ≤ _
+  rw [LinearIsometry.norm_map]
+  have hb : ‖(_root_.MeasureTheory.Lp.LpToLpOfMeasureLeSMul hC h : _root_.MeasureTheory.Lp F p ν →L[ℝ] _root_.MeasureTheory.Lp F p (_root_.MeasureTheory.Measure.map f μ))‖ ≤
+      C.toReal ^ (1 / p).toReal := _root_.MeasureTheory.Lp.norm_LpToLpOfMeasureLeSMul_le hC h
+  apply ((_root_.MeasureTheory.Lp.LpToLpOfMeasureLeSMul hC h).le_opNorm u).trans
+  exact mul_le_mul_of_nonneg_right hb (norm_nonneg u)
+
+end MeasureTheory.Lp
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/LogHessianCalculus.lean b/lean/OAI/Geometry/PrescribedPotential/LogHessianCalculus.lean
new file mode 100644
index 0000000000000000000000000000000000000000..9f1c6cff3a536afc0582ca9545d2aee2a1406915
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/LogHessianCalculus.lean
@@ -0,0 +1,78 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.KaehlerClosedDerivatives
+import OAI.Geometry.PrescribedPotential.MatrixLogDet
+
+namespace OAI
+
+/-! Log Hessian Calculus. -/
+
+section
+
+ 
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace KaehlerCalculus
+variable {n : ℕ}
+
+lemma wderiv_inv {f : V n → ℂ} {z : V n}
+    (hf : DifferentiableAt ℝ f z) (hn : f z ≠ 0) (a : ℂ) (v : V n) :
+    wderiv a v (fun y => (f y)⁻¹) z = -(f z)^(-2 : ℤ)*wderiv a v f z := by
+  have h := (hasDerivAt_inv hn).comp_hasFDerivAt z hf.hasFDerivAt
+  have he : fderiv ℝ (fun y => (f y)⁻¹) z = -((f z)^2)⁻¹ • fderiv ℝ f z := h.fderiv
+  simp only [wderiv,he,_root_.smul_apply,smul_eq_mul]
+  simp only [_root_.zpow_neg,zpow_ofNat]
+  ring
+
+lemma second_wderiv_log_real {U : Set (V n)} (hU : IsOpen U)
+    {f : V n → ℝ} (hf : ContDiffOn ℝ ∞ f U) (hp : ∀ y ∈ U, f y ≠ 0)
+    {z : V n} (hz : z ∈ U) (a b : ℂ) (u v : V n) :
+    wderiv b u (wderiv a v (fun y => (Real.log (f y) : ℂ))) z =
+      (f z : ℂ)⁻¹*wderiv b u (wderiv a v (fun y => (f y : ℂ))) z -
+      (f z : ℂ)^(-2 : ℤ)*wderiv b u (fun y => (f y : ℂ)) z *
+        wderiv a v (fun y => (f y : ℂ)) z := by
+  have hs := hf.contDiffAt (hU.mem_nhds hz)
+  have hc : ContDiffAt ℝ ∞ (fun y => (f y : ℂ)) z := Complex.ofRealCLM.contDiff.contDiffAt.comp z hs
+  have he : (wderiv a v (fun y => (Real.log (f y) : ℂ))) =ᶠ[𝓝 z]
+      (fun y => (f y : ℂ)⁻¹*wderiv a v (fun y => (f y : ℂ)) y) := by
+    filter_upwards [hU.mem_nhds hz] with y hy
+    exact wderiv_log_real (hf.contDiffAt (hU.mem_nhds hy)) (hp y hy) a v
+  rw [wderiv_congr he,wderiv_mul
+    (show DifferentiableAt ℝ (fun y => (f y : ℂ)⁻¹) z from
+      (hc.differentiableAt (by simp)).inv (Complex.ofReal_ne_zero.mpr (hp z hz)))
+    ((wderiv_smooth hc a v).differentiableAt (by simp)),
+    wderiv_inv (hc.differentiableAt (by simp)) (Complex.ofReal_ne_zero.mpr (hp z hz))]
+  ring
+
+lemma dzbar_real_eq_conj {f : V n → ℝ} {z : V n}
+    (hf : DifferentiableAt ℝ f z) (v : V n) :
+    dzbar v (fun y => (f y : ℂ)) z = star (dz v (fun y => (f y : ℂ)) z) := by
+  simp only [dzbar,dz,wderiv,fderiv_ofReal hf]
+  apply Complex.ext <;> simp
+
+lemma log_hessian_diagonal {U : Set (V n)} (hU : IsOpen U)
+    {f : V n → ℝ} (hf : ContDiffOn ℝ ∞ f U) (hp : ∀ y ∈ U, f y ≠ 0)
+    {z : V n} (hz : z ∈ U) (k : Fin n) :
+    (PotentialKaehler.potentialMatrix (fun y => Real.log (f y)) z k k).re =
+      (f z)⁻¹*(PotentialKaehler.potentialMatrix f z k k).re -
+      (f z)^(-2 : ℤ)*‖dz (e k) (fun y => (f y : ℂ)) z‖^2 := by
+  have hs := hf.contDiffAt (hU.mem_nhds hz)
+  rw [potentialMatrix_eq_dzbar_dz (hs.log (hp z hz)),
+    potentialMatrix_eq_dzbar_dz hs]
+  change (wderiv Complex.I (e k) (wderiv (-Complex.I) (e k)
+    (fun y => (Real.log (f y) : ℂ))) z).re = _
+  rw [second_wderiv_log_real hU hf hp hz]
+  change ((f z : ℂ)⁻¹ * dzbar (e k) (dz (e k) (fun y => (f y : ℂ))) z -
+    (f z : ℂ)^(-2 : ℤ)*dzbar (e k) (fun y => (f y : ℂ)) z*
+      dz (e k) (fun y => (f y : ℂ)) z).re = _
+  rw [dzbar_real_eq_conj (hs.differentiableAt (by simp)),mul_assoc,Complex.star_def,
+    ← Complex.normSq_eq_conj_mul_self]
+  simp only [← Complex.ofReal_inv,← Complex.ofReal_zpow,Complex.sub_re,Complex.mul_re,Complex.ofReal_re,
+    Complex.ofReal_im,zero_mul,sub_zero,mul_zero,Complex.normSq_eq_norm_sq]
+  rfl
+end KaehlerCalculus
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/MatrixCoefficientSmooth.lean b/lean/OAI/Geometry/PrescribedPotential/MatrixCoefficientSmooth.lean
new file mode 100644
index 0000000000000000000000000000000000000000..447c6212b8e8fe8d834af71780734c49ac927701
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/MatrixCoefficientSmooth.lean
@@ -0,0 +1,53 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.MatrixHilbertCoordinates
+
+namespace OAI
+
+/-! Matrix Coefficient Smooth. -/
+
+section
+
+noncomputable section
+open Set Filter Topology Matrix Finset
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace MetricSystem
+open EllipticKernel KaehlerCalculus
+variable {n : ℕ}
+lemma W_smooth (a : ℂ) (v : V n) : ContDiff ℝ ∞ (W (n := n) a v) := by
+  unfold W
+  apply ContDiff.const_smul
+  apply ContDiff.add
+  · exact (encode n).symm.contDiff.comp (contDiff_id.clm_apply contDiff_const)
+  · exact ((encode n).symm.contDiff.comp (contDiff_id.clm_apply contDiff_const)).const_smul a
+
+lemma quadraticCoefficient_smooth {p : HM n × (EC n →L[ℝ] HM n)}
+    (hp : ((encode n).symm p.1).det ≠ 0) : ContDiffAt ℝ ∞ (quadraticCoefficient (n := n)) p := by
+  apply (encode n).contDiff.contDiffAt.comp p
+  apply ContDiffAt.sum
+  intro i _
+  apply ContDiffAt.sum
+  intro j _
+  have hi := (inverseCoefficient_smooth hp).comp p contDiffAt_fst
+  have hentry := contDiffAt_pi.mp (contDiffAt_pi.mp hi i) j
+  apply hentry.smul
+  have hW (a : ℂ) (v : V n) : ContDiffAt ℝ ∞ (fun q : HM n × (EC n →L[ℝ] HM n) => W a v q.2) p :=
+    (W_smooth a v).contDiffAt.comp p contDiffAt_snd
+  apply contDiffAt_pi.mpr
+  intro k
+  apply contDiffAt_pi.mpr
+  intro l
+  simp only [Matrix.mul_apply]
+  apply ContDiffAt.sum
+  intro s _
+  apply ContDiffAt.mul
+  · apply ContDiffAt.sum
+    intro t _
+    exact (contDiffAt_pi.mp (contDiffAt_pi.mp (hW Complex.I (e j)) k) t).mul
+      (contDiffAt_pi.mp (contDiffAt_pi.mp hi t) s)
+  · exact contDiffAt_pi.mp (contDiffAt_pi.mp (hW (-Complex.I) (e i)) s) l
+end MetricSystem
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/MatrixFrameNormalization.lean b/lean/OAI/Geometry/PrescribedPotential/MatrixFrameNormalization.lean
new file mode 100644
index 0000000000000000000000000000000000000000..6581523880518bf3ce645d6533cdc67ebb06c5e9
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/MatrixFrameNormalization.lean
@@ -0,0 +1,58 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.LinearFrameCalculus
+import OAI.Geometry.PrescribedPotential.TracePositivity
+
+namespace OAI
+
+/-! Matrix Frame Normalization. -/
+
+section
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace KaehlerCalculus
+variable {n : ℕ}
+
+def matrixFrame (B : Matrix (Fin n) (Fin n) ℂ) : V n →L[ℂ] V n :=
+  (Matrix.toLin' B).toContinuousLinearMap
+
+lemma frameMatrix_matrixFrame (B : Matrix (Fin n) (Fin n) ℂ) :
+    frameMatrix (matrixFrame B) = B := by
+  change LinearMap.toMatrix' (Matrix.toLin' B) = B
+  exact LinearMap.toMatrix'_toLin' B
+
+lemma matrixFrame_apply (B : Matrix (Fin n) (Fin n) ℂ) (v : V n) :
+    matrixFrame B v = B*ᵥv := rfl
+
+lemma normalized_frame (M : V n → Matrix (Fin n) (Fin n) ℂ) (z : V n) (hp : (M z).PosDef) :
+    ∃ (B : V n →L[ℂ] V n) (w : V n), IsUnit (frameMatrix B) ∧ B w = z ∧ pullMetric B M w = 1 := by
+  obtain ⟨C,hC,hEq⟩ := whitening (M z) hp
+  obtain ⟨w,hw⟩ := (Matrix.mulVec_surjective_iff_isUnit.mpr hC) z
+  refine ⟨matrixFrame C,w,?_,?_,?_⟩
+  · rwa [frameMatrix_matrixFrame]
+  · exact hw
+  · unfold pullMetric
+    rw [frameMatrix_matrixFrame,matrixFrame_apply,hw]
+    exact hEq
+
+lemma pullMetric_positive {U : Set (V n)} (B : V n →L[ℂ] V n)
+    (hB : IsUnit (frameMatrix B)) {M : V n → Matrix (Fin n) (Fin n) ℂ}
+    (hp : ∀ y ∈ U, (M y).PosDef) : ∀ y ∈ B ⁻¹' U, (pullMetric B M y).PosDef := by
+  intro y hy
+  exact (hp (B y) hy).conjTranspose_mul_mul_same (Matrix.mulVec_injective_iff_isUnit.mpr hB)
+
+lemma matrix_inverse_trace_congruence (B M G : Matrix (Fin n) (Fin n) ℂ) (hB : IsUnit B) :
+    ((Bᴴ*M*B)⁻¹*(Bᴴ*G*B)).trace = (M⁻¹*G).trace := by
+  have hc : B*B⁻¹ = 1 := Matrix.mul_nonsing_inv B ((Matrix.isUnit_iff_isUnit_det B).mp hB)
+  have hd : Bᴴ⁻¹*Bᴴ = 1 := by
+    rw [← Matrix.conjTranspose_nonsing_inv,← Matrix.conjTranspose_mul,hc,Matrix.conjTranspose_one]
+  simp only [Matrix.mul_inv_rev,Matrix.mul_assoc]
+  rw [← Matrix.mul_assoc Bᴴ⁻¹ Bᴴ,hd,Matrix.one_mul]
+  rw [← Matrix.mul_assoc M⁻¹,← Matrix.mul_assoc B⁻¹,Matrix.trace_mul_cycle,hc,Matrix.one_mul]
+end KaehlerCalculus
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/MatrixHilbertCoordinates.lean b/lean/OAI/Geometry/PrescribedPotential/MatrixHilbertCoordinates.lean
new file mode 100644
index 0000000000000000000000000000000000000000..6b953dc04b74951705515aeec52ceec412e10ac9
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/MatrixHilbertCoordinates.lean
@@ -0,0 +1,111 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.GlobalOperator
+import OAI.Geometry.PrescribedPotential.KaehlerClosedDerivatives
+import OAI.Geometry.PrescribedPotential.SymmetricPrincipalFrame
+
+namespace OAI
+
+/-! Matrix Hilbert Coordinates. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Matrix Finset
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace MetricSystem
+open EllipticKernel FrozenPoisson HigherJet KaehlerCalculus
+abbrev Mat (n : ℕ) := Matrix (Fin n) (Fin n) ℂ
+abbrev HM (n : ℕ) := EuclideanSpace ℂ (Fin n × Fin n)
+local instance realECIP (n : ℕ) : InnerProductSpace ℝ (EC n) := InnerProductSpace.rclikeToReal ℂ (EC n)
+local instance realHMIP (n : ℕ) : InnerProductSpace ℝ (HM n) := InnerProductSpace.rclikeToReal ℂ (HM n)
+
+variable {n : ℕ}
+def encodeLinear (n : ℕ) : Mat n ≃ₗ[ℝ] HM n where
+  toFun M := WithLp.toLp 2 (fun p => M p.1 p.2)
+  invFun u := fun i j => u (i,j)
+  left_inv _ := rfl
+  right_inv _ := rfl
+  map_add' _ _ := rfl
+  map_smul' _ _ := rfl
+
+def encode (n : ℕ) : Mat n ≃L[ℝ] HM n := (encodeLinear n).toContinuousLinearEquiv
+@[simp] lemma encode_apply (M : Mat n) (i j : Fin n) : encode n M (i,j) = M i j := rfl
+@[simp] lemma decode_apply (u : HM n) (i j : Fin n) : (encode n).symm u i j = u (i,j) := rfl
+
+ 
+def rawTrace (M : Mat n) (H : Bilin (EC n)) : ℝ :=
+  (M*PotentialKaehler.hermitianPartMatrix (pullBilin H)).trace.re
+lemma rawTrace_add (M N : Mat n) (H : Bilin (EC n)) : rawTrace (M+N) H = rawTrace M H+rawTrace N H := by
+  simp [rawTrace,add_mul,Matrix.trace_add]
+lemma rawTrace_smul (c : ℝ) (M : Mat n) (H : Bilin (EC n)) : rawTrace (c • M) H = c*rawTrace M H := by
+  simp [rawTrace,Matrix.trace_smul]
+lemma rawTrace_inv (M : Mat n) (H : Bilin (EC n)) : rawTrace M⁻¹ H = traceBilin M H := rfl
+
+variable {ι : Type*} [Fintype ι]
+def principalLinear (e : OrthonormalBasis ι ℝ (EC n)) (M : Mat n) :
+    (EC n →L[ℝ] EC n →L[ℝ] HM n) →ₗ[ℝ] HM n where
+  toFun H := ∑ i, ∑ j, ((rawTrace M (rankTwo (e i) (e j))+rawTrace M (rankTwo (e j) (e i)))/2) • H (e i) (e j)
+  map_add' H J := by simp only [_root_.add_apply,smul_add,Finset.sum_add_distrib]
+  map_smul' c H := by
+    simp only [_root_.smul_apply,Finset.smul_sum]
+    exact Finset.sum_congr rfl (fun i _ => Finset.sum_congr rfl (fun j _ => smul_comm _ _ _))
+
+lemma principalLinear_add (e : OrthonormalBasis ι ℝ (EC n)) (M N : Mat n)
+    (H : EC n →L[ℝ] EC n →L[ℝ] HM n) :
+    principalLinear e (M+N) H = principalLinear e M H+principalLinear e N H := by
+  change (∑ i, ∑ j, ((rawTrace (M+N) (rankTwo (e i) (e j))+rawTrace (M+N) (rankTwo (e j) (e i)))/2) • H (e i) (e j)) = _
+  have he (a b c d : ℝ) : (a+b+(c+d))/2 = (a+c)/2+(b+d)/2 := by ring
+  simp only [rawTrace_add,he,add_smul,Finset.sum_add_distrib]
+  rfl
+lemma principalLinear_smul (e : OrthonormalBasis ι ℝ (EC n)) (c : ℝ) (M : Mat n)
+    (H : EC n →L[ℝ] EC n →L[ℝ] HM n) :
+    principalLinear e (c • M) H = c • principalLinear e M H := by
+  change (∑ i, ∑ j, ((rawTrace (c • M) (rankTwo (e i) (e j))+rawTrace (c • M) (rankTwo (e j) (e i)))/2) • H (e i) (e j)) = _
+  have he (a b c : ℝ) : (c*a+c*b)/2 = c*((a+b)/2) := by ring
+  simp only [rawTrace_smul,he,mul_smul,← Finset.smul_sum]
+  rfl
+
+local instance hessianNormedAdd (n : ℕ) : NormedAddCommGroup (EC n →L[ℝ] EC n →L[ℝ] HM n) := inferInstance
+local instance hessianNormedSpace (n : ℕ) : NormedSpace ℝ (EC n →L[ℝ] EC n →L[ℝ] HM n) := inferInstance
+
+def principalBilinear (e : OrthonormalBasis ι ℝ (EC n)) :
+    Mat n →L[ℝ] ((EC n →L[ℝ] EC n →L[ℝ] HM n) →L[ℝ] HM n) :=
+  ({ toFun M := (principalLinear e M).toContinuousLinearMap
+     map_add' M N := by
+       apply ContinuousLinearMap.ext
+       intro H
+       exact principalLinear_add e M N H
+     map_smul' c M := by
+       apply ContinuousLinearMap.ext
+       intro H
+       exact principalLinear_smul e c M H } :
+       Mat n →ₗ[ℝ] ((EC n →L[ℝ] EC n →L[ℝ] HM n) →L[ℝ] HM n)).toContinuousLinearMap
+
+lemma principalBilinear_inv (e : OrthonormalBasis ι ℝ (EC n)) (M : Mat n)
+    (H : EC n →L[ℝ] EC n →L[ℝ] HM n) :
+    principalBilinear e M⁻¹ H = principalApply e (traceBilin M) H := rfl
+
+def inverseCoefficient (u : HM n) : Mat n := ((encode n).symm u)⁻¹
+
+lemma inverseCoefficient_smooth {u : HM n} (hu : ((encode n).symm u).det ≠ 0) :
+    ContDiffAt ℝ ∞ (inverseCoefficient (n := n)) u := by
+  let U : Set (HM n) := {u | ((encode n).symm u).det ≠ 0}
+  have hU : IsOpen U := (isClosed_eq ((encode n).symm.continuous.matrix_det) continuous_const).isOpen_compl
+  exact (MatrixSmoothGeneral.inverse (encode n).symm.contDiff.contDiffOn (fun y hy => hy)).contDiffAt (hU.mem_nhds hu)
+
+ 
+def W (a : ℂ) (v : V n) (J : EC n →L[ℝ] HM n) : Mat n :=
+  (1/2:ℂ) • ((encode n).symm (J ((coordinateEquiv n).symm v))+
+    a • (encode n).symm (J ((coordinateEquiv n).symm (Complex.I • v))))
+
+def quadraticCoefficient (p : HM n × (EC n →L[ℝ] HM n)) : HM n :=
+  encode n (∑ i, ∑ j, (inverseCoefficient p.1 i j) •
+    (W Complex.I (e j) p.2*inverseCoefficient p.1*W (-Complex.I) (e i) p.2))
+end MetricSystem
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/MatrixJetDirection.lean b/lean/OAI/Geometry/PrescribedPotential/MatrixJetDirection.lean
new file mode 100644
index 0000000000000000000000000000000000000000..beda3c2996a870f8c25b93edccd3f62fa1ecf255
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/MatrixJetDirection.lean
@@ -0,0 +1,117 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.ChernLuLocalInequality
+
+namespace OAI
+
+/-! Matrix Jet Direction. -/
+
+section
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace KaehlerCalculus
+variable {n : ℕ}
+
+lemma wderiv_direction_smul (a : ℂ) (r : ℝ) (v : V n) (f : V n → ℂ) (z : V n) :
+    wderiv a (r • v) f z = (r:ℂ)*wderiv a v f z := by
+  simp only [wderiv,smul_comm Complex.I r v,map_smul,Complex.real_smul]
+  ring
+
+lemma wderiv_second_direction_smul {f : V n → ℂ} {z : V n}
+    (hf : ContDiffAt ℝ ∞ f z) (a b : ℂ) (r : ℝ) (v : V n) :
+    wderiv a (r • v) (wderiv b (r • v) f) z =
+      (r:ℂ)^2*wderiv a v (wderiv b v f) z := by
+  have he : wderiv b (r • v) f = fun y => (r:ℂ)*wderiv b v f y :=
+    funext (fun y => wderiv_direction_smul b r v f y)
+  rw [he,wderiv_direction_smul,
+    wderiv_const_mul ((wderiv_smooth hf b v).differentiableAt (by simp))]
+  ring
+
+lemma mderiv_direction_smul (a : ℂ) (r : ℝ) (v : V n)
+    (M : V n → Matrix (Fin n) (Fin n) ℂ) (z : V n) :
+    mderiv a (r • v) M z = (r:ℂ) • mderiv a v M z := by
+  ext i j
+  exact wderiv_direction_smul a r v (fun y => M y i j) z
+
+lemma mderiv_second_direction_smul {M : V n → Matrix (Fin n) (Fin n) ℂ} {z : V n}
+    (hM : ContDiffAt ℝ ∞ M z) (a b : ℂ) (r : ℝ) (v : V n) :
+    mderiv a (r • v) (mderiv b (r • v) M) z =
+      (r:ℂ)^2 • mderiv a v (mderiv b v M) z := by
+  ext i j
+  exact wderiv_second_direction_smul (entry_smooth hM i j) a b r v
+
+lemma curvatureDiagonal_smul {G : V n → Matrix (Fin n) (Fin n) ℂ} {z : V n}
+    (hG : ContDiffAt ℝ ∞ G z) (r : ℝ) (v : V n) :
+    curvatureDiagonal G z (r • v) = (r:ℂ)^2 • curvatureDiagonal G z v := by
+  unfold curvatureDiagonal
+  rw [mderiv_second_direction_smul hG,mderiv_direction_smul]
+  simp only [Matrix.conjTranspose_smul,Complex.star_def,Complex.conj_ofReal,
+    smul_mul_assoc,mul_smul_comm,smul_smul,smul_sub,pow_two]
+
+lemma wderiv_joint_smooth {f : V n → ℂ} {z v : V n}
+    (hf : ContDiffAt ℝ ∞ f z) (a : ℂ) :
+    ContDiffAt ℝ ∞ (fun q : V n × V n => wderiv a q.2 f q.1) (z,v) := by
+  have hd := (hf.fderiv_right (m := ∞) (by simp)).comp (z,v) contDiffAt_fst
+  have hi : ContDiffAt ℝ ∞ (fun q : V n × V n => Complex.I • q.2) (z,v) :=
+    contDiffAt_snd.const_smul Complex.I
+  exact ((hd.clm_apply contDiffAt_snd).add
+    (contDiffAt_const.mul (hd.clm_apply hi))).div_const 2
+
+lemma wderiv_second_joint_continuous {U : Set (V n)} (hU : IsOpen U)
+    {f : V n → ℂ} (hf : ContDiffOn ℝ ∞ f U) (a b : ℂ) :
+    ContinuousOn (fun q : V n × V n => wderiv a q.2 (wderiv b q.2 f) q.1) (U ×ˢ univ) := by
+  intro q hq
+  have hs := hf.contDiffAt (hU.mem_nhds hq.1)
+  have hd := ((hs.fderiv_right (m := ∞) (by simp)).fderiv_right (m := ∞) (by simp)).comp q contDiffAt_fst
+  have hi : ContDiffAt ℝ ∞ (fun q : V n × V n => Complex.I • q.2) q :=
+    contDiffAt_snd.const_smul Complex.I
+  have he : (fun q : V n × V n => wderiv a q.2 (wderiv b q.2 f) q.1) =ᶠ[𝓝 q]
+      (fun q => ((fderiv ℝ (fderiv ℝ f) q.1 q.2 q.2 +
+        b*fderiv ℝ (fderiv ℝ f) q.1 q.2 (Complex.I • q.2))/2 +
+        a*((fderiv ℝ (fderiv ℝ f) q.1 (Complex.I • q.2) q.2 +
+          b*fderiv ℝ (fderiv ℝ f) q.1 (Complex.I • q.2) (Complex.I • q.2))/2))/2) := by
+    filter_upwards [continuous_fst.continuousAt.preimage_mem_nhds (hU.mem_nhds hq.1)] with p hp
+    change (fderiv ℝ (wderiv b p.2 f) p.1 p.2 + a*fderiv ℝ (wderiv b p.2 f) p.1 (Complex.I • p.2))/2 = _
+    rw [fderiv_wderiv (hf.contDiffAt (hU.mem_nhds hp)),
+      fderiv_wderiv (hf.contDiffAt (hU.mem_nhds hp))]
+  apply ContinuousAt.continuousWithinAt
+  apply ContinuousAt.congr _ he.symm
+  exact (((hd.clm_apply contDiffAt_snd).clm_apply contDiffAt_snd).add
+    (contDiffAt_const.mul ((hd.clm_apply contDiffAt_snd).clm_apply hi))).div_const 2 |>.add
+    (contDiffAt_const.mul ((((hd.clm_apply hi).clm_apply contDiffAt_snd).add
+      (contDiffAt_const.mul ((hd.clm_apply hi).clm_apply hi))).div_const 2)) |>.div_const 2 |>.continuousAt
+
+lemma curvatureDiagonal_joint_continuous {U : Set (V n)} (hU : IsOpen U)
+    {G : V n → Matrix (Fin n) (Fin n) ℂ} (hG : ContDiffOn ℝ ∞ G U)
+    (hn : ∀ z ∈ U, (G z).det ≠ 0) :
+    ContinuousOn (fun q : V n × V n => curvatureDiagonal G q.1 q.2) (U ×ˢ univ) := by
+  have hd : ContinuousOn (fun q : V n × V n => mderiv (-Complex.I) q.2 G q.1) (U ×ˢ univ) := by
+    intro q hq
+    apply ContinuousAt.continuousWithinAt
+    apply continuousAt_pi.mpr
+    intro i
+    apply continuousAt_pi.mpr
+    intro j
+    exact (wderiv_joint_smooth (entry_smooth (hG.contDiffAt (hU.mem_nhds hq.1)) i j) _).continuousAt
+  have hdd : ContinuousOn (fun q : V n × V n => mderiv Complex.I q.2 (mderiv (-Complex.I) q.2 G) q.1) (U ×ˢ univ) := by
+    apply continuousOn_pi.mpr
+    intro i
+    apply continuousOn_pi.mpr
+    intro j
+    exact wderiv_second_joint_continuous hU (contDiffOn_pi.mp (contDiffOn_pi.mp hG i) j) _ _
+  have hinv : ContinuousOn (fun q : V n × V n => (G q.1)⁻¹) (U ×ˢ univ) :=
+    (MatrixSmoothGeneral.inverse hG hn).continuousOn.comp continuous_fst.continuousOn (fun _ h => h.1)
+  have hdstar : ContinuousOn (fun q : V n × V n => (mderiv (-Complex.I) q.2 G q.1)ᴴ) (U ×ˢ univ) := by
+    apply continuousOn_pi.mpr
+    intro i
+    apply continuousOn_pi.mpr
+    intro j
+    exact (continuousOn_pi.mp (continuousOn_pi.mp hd j) i).star
+  exact hdd.sub ((hdstar.mul hinv).mul hd)
+end KaehlerCalculus
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/MatrixLimit.lean b/lean/OAI/Geometry/PrescribedPotential/MatrixLimit.lean
new file mode 100644
index 0000000000000000000000000000000000000000..09d1d7cbe424092f78aca4b42fdc73307e103fd5
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/MatrixLimit.lean
@@ -0,0 +1,42 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CircleRadialCalculus
+
+namespace OAI
+
+/-! Matrix Limit. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ComplexOrder
+namespace CompactMatrixPositivity
+variable {n : Type*} [Fintype n] [DecidableEq n]
+
+omit [DecidableEq n] in
+lemma posSemidef_of_tendsto {M : ℕ → Matrix n n ℂ} {L : Matrix n n ℂ}
+    (hL : L.IsHermitian) (hM : ∀ j, (M j).PosSemidef)
+    (ht : Tendsto M atTop (𝓝 L)) : L.PosSemidef := by
+  apply Matrix.PosSemidef.of_dotProduct_mulVec_nonneg hL
+  intro v
+  apply Complex.nonneg_iff.mpr
+  refine ⟨?_,(hL.im_star_dotProduct_mulVec_self v).symm⟩
+  have hc : Continuous (fun B : Matrix n n ℂ => (star v ⬝ᵥ (B *ᵥ v)).re) := by
+    unfold Matrix.mulVec dotProduct
+    fun_prop
+  exact ge_of_tendsto (hc.tendsto L |>.comp ht)
+    (Filter.Eventually.of_forall (fun j => (hM j).re_dotProduct_nonneg v))
+
+lemma posDef_of_tendsto_det_ne_zero {M : ℕ → Matrix n n ℂ} {L : Matrix n n ℂ}
+    (hL : L.IsHermitian) (hM : ∀ j, (M j).PosDef)
+    (ht : Tendsto M atTop (𝓝 L)) (hd : L.det ≠ 0) : L.PosDef :=
+  (posSemidef_of_tendsto hL (fun j => (hM j).posSemidef) ht).posDef_iff_det_ne_zero.mpr hd
+
+end CompactMatrixPositivity
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/MatrixLogDet.lean b/lean/OAI/Geometry/PrescribedPotential/MatrixLogDet.lean
new file mode 100644
index 0000000000000000000000000000000000000000..6c41b94e32a03e65af48221c62f421103082ed56
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/MatrixLogDet.lean
@@ -0,0 +1,135 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.MatrixWirtinger
+
+namespace OAI
+
+/-! Matrix Log Det. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace KaehlerCalculus
+variable {n : ℕ}
+local instance matrixLogDetCS (n : ℕ) : ContinuousSMul ℝ (Matrix (Fin n) (Fin n) ℂ) :=
+  inferInstanceAs (ContinuousSMul ℝ (Fin n → Fin n → ℂ))
+
+lemma determinant_smooth : ContDiff ℝ ∞ (Matrix.det : Matrix (Fin n) (Fin n) ℂ → ℂ) :=
+  contDiffOn_univ.mp (MatrixSmoothGeneral.determinant contDiffOn_id)
+
+lemma fderiv_det (H K : Matrix (Fin n) (Fin n) ℂ) (hH : H.det ≠ 0) :
+    fderiv ℝ Matrix.det H K = H.det * (H⁻¹*K).trace := by
+  have hd := determinant_smooth.differentiable (by simp) H
+  have hl : HasDerivAt (fun t : ℝ => H + t • K) K 0 := by
+    convert (hasDerivAt_const (0 : ℝ) H).fun_add
+      ((hasDerivAt_id (0 : ℝ)).smul_const K) using 1 <;> first | rfl | simp only [one_smul,zero_add]
+  have hdat : HasFDerivAt Matrix.det (fderiv ℝ Matrix.det H) (H + (0:ℝ) • K) := by
+    convert hd.hasFDerivAt using 1 <;> first | rfl | simp only [zero_smul,add_zero]
+  have h1 := hdat.comp_hasDerivAt 0 hl
+  have h2 := (MongeAmpere.hasDerivAt_det_add_smul H K (isUnit_iff_ne_zero.mpr hH)).comp_ofReal
+  convert h1.unique h2 using 1
+
+lemma fderiv_det_comp {M : V n → Matrix (Fin n) (Fin n) ℂ} {z : V n}
+    (hM : ContDiffAt ℝ ∞ M z) (hn : (M z).det ≠ 0) (v : V n) :
+    fderiv ℝ (fun y => (M y).det) z v = (M z).det*((M z)⁻¹*fderiv ℝ M z v).trace := by
+  have hd := determinant_smooth.differentiable (by simp) (M z)
+  rw [fderiv_fun_comp z hd (hM.differentiableAt (by simp)),ContinuousLinearMap.comp_apply]
+  exact fderiv_det _ _ hn
+
+lemma fderiv_matrix_apply {M : V n → Matrix (Fin n) (Fin n) ℂ} {z : V n}
+    (hM : ContDiffAt ℝ ∞ M z) (v : V n) (i j : Fin n) :
+    (fderiv ℝ M z v) i j = fderiv ℝ (fun y => M y i j) z v := by
+  have hp := (hasFDerivAt_apply j (M z i)).comp z
+    ((hasFDerivAt_apply i (M z)).comp z (hM.differentiableAt (by simp)).hasFDerivAt)
+  exact (congrArg (fun T : V n →L[ℝ] ℂ => T v) hp.fderiv).symm
+
+lemma mderiv_eq {M : V n → Matrix (Fin n) (Fin n) ℂ} {z : V n}
+    (hM : ContDiffAt ℝ ∞ M z) (a : ℂ) (v : V n) :
+    mderiv a v M z = (1/2:ℂ) • (fderiv ℝ M z v + a • fderiv ℝ M z (Complex.I • v)) := by
+  ext i j
+  simp only [mderiv,wderiv,Matrix.smul_apply,Matrix.add_apply,smul_eq_mul,
+    fderiv_matrix_apply hM]
+  ring
+
+lemma wderiv_det {M : V n → Matrix (Fin n) (Fin n) ℂ} {z : V n}
+    (hM : ContDiffAt ℝ ∞ M z) (hn : (M z).det ≠ 0) (a : ℂ) (v : V n) :
+    wderiv a v (fun y => (M y).det) z = (M z).det*((M z)⁻¹*mderiv a v M z).trace := by
+  rw [mderiv_eq hM]
+  simp only [wderiv,fderiv_det_comp hM hn,Matrix.mul_smul,Matrix.mul_add,
+    Matrix.trace_smul,Matrix.trace_add,smul_eq_mul]
+  ring
+
+lemma wderiv_log_real {f : V n → ℝ} {z : V n}
+    (hf : ContDiffAt ℝ ∞ f z) (hp : f z ≠ 0) (a : ℂ) (v : V n) :
+    wderiv a v (fun y => (Real.log (f y) : ℂ)) z =
+      (f z : ℂ)⁻¹*wderiv a v (fun y => (f y : ℂ)) z := by
+  have hl := (hf.differentiableAt (by simp)).hasFDerivAt.log hp
+  have he (u : V n) : fderiv ℝ (fun y => Real.log (f y)) z u = (f z)⁻¹*fderiv ℝ f z u := by
+    rw [hl.fderiv,_root_.smul_apply,smul_eq_mul]
+  simp only [wderiv,fderiv_ofReal ((hf.log hp).differentiableAt (by simp)),he,
+    fderiv_ofReal (hf.differentiableAt (by simp)),Complex.ofReal_mul,Complex.ofReal_inv]
+  ring
+
+lemma wderiv_logdet {U : Set (V n)} (hU : IsOpen U)
+    {M : V n → Matrix (Fin n) (Fin n) ℂ} (hM : ContDiffOn ℝ ∞ M U)
+    (hp : ∀ y ∈ U, (M y).PosDef) {z : V n} (hz : z ∈ U) (a : ℂ) (v : V n) :
+    wderiv a v (fun y => (Real.log (M y).det.re : ℂ)) z =
+      ((M z)⁻¹*mderiv a v M z).trace := by
+  have hs := hM.contDiffAt (hU.mem_nhds hz)
+  have hdr : ContDiffAt ℝ ∞ (fun y => (M y).det.re) z :=
+    Complex.reCLM.contDiff.contDiffAt.comp z (determinant_smooth.contDiffAt.comp z hs)
+  have hpos := Complex.pos_iff.mp (hp z hz).det_pos
+  rw [wderiv_log_real hdr (ne_of_gt hpos.1)]
+  have he : (fun y => ((M y).det.re : ℂ)) =ᶠ[𝓝 z] (fun y => (M y).det) := by
+    filter_upwards [hU.mem_nhds hz] with y hy
+    exact Complex.ext (by simp) (by simpa using (Complex.pos_iff.mp (hp y hy).det_pos).2)
+  rw [wderiv_congr he,wderiv_det hs (ne_of_gt (hp z hz).det_pos)]
+  have hez : ((M z).det.re : ℂ) = (M z).det := he.self_of_nhds
+  rw [← hez]
+  rw [Complex.ofReal_re,← mul_assoc,inv_mul_cancel₀ (Complex.ofReal_ne_zero.mpr (ne_of_gt hpos.1)),one_mul]
+
+lemma second_wderiv_logdet {U : Set (V n)} (hU : IsOpen U)
+    {M : V n → Matrix (Fin n) (Fin n) ℂ} (hM : ContDiffOn ℝ ∞ M U)
+    (hp : ∀ y ∈ U, (M y).PosDef) {z : V n} (hz : z ∈ U)
+    (a b : ℂ) (u v : V n) :
+    wderiv b u (wderiv a v (fun y => (Real.log (M y).det.re : ℂ))) z =
+      (-(M z)⁻¹*mderiv b u M z*(M z)⁻¹*mderiv a v M z +
+        (M z)⁻¹*mderiv b u (mderiv a v M) z).trace := by
+  have hs := hM.contDiffAt (hU.mem_nhds hz)
+  have hi := (MatrixSmoothGeneral.inverse hM
+    (fun y hy => ne_of_gt (hp y hy).det_pos)).contDiffAt (hU.mem_nhds hz)
+  have he : wderiv a v (fun y => (Real.log (M y).det.re : ℂ)) =ᶠ[𝓝 z]
+      (fun y => ((M y)⁻¹*mderiv a v M y).trace) := by
+    filter_upwards [hU.mem_nhds hz] with y hy
+    exact wderiv_logdet hU hM hp hy a v
+  have hprod : ContDiffAt ℝ ∞ (fun y => (M y)⁻¹*mderiv a v M y) z := by
+    apply contDiffAt_pi.mpr
+    intro i
+    apply contDiffAt_pi.mpr
+    intro j
+    change ContDiffAt ℝ ∞ (fun y => ∑ k, (M y)⁻¹ i k * mderiv a v M y k j) z
+    exact ContDiffAt.sum (fun k _ => (entry_smooth hi i k).mul
+      (entry_smooth (mderiv_smooth hs a v) k j))
+  rw [wderiv_congr he,mderiv_trace hprod,
+    mderiv_mul hi (mderiv_smooth hs a v),
+    mderiv_inverse hU hM (fun y hy => ne_of_gt (hp y hy).det_pos) hz]
+
+lemma second_wderiv_logdet_at_one {U : Set (V n)} (hU : IsOpen U)
+    {M : V n → Matrix (Fin n) (Fin n) ℂ} (hM : ContDiffOn ℝ ∞ M U)
+    (hp : ∀ y ∈ U, (M y).PosDef) {z : V n} (hz : z ∈ U)
+    (hMz : M z = 1) (a b : ℂ) (u v : V n) :
+    wderiv b u (wderiv a v (fun y => (Real.log (M y).det.re : ℂ))) z =
+      (mderiv b u (mderiv a v M) z).trace - (mderiv b u M z*mderiv a v M z).trace := by
+  rw [second_wderiv_logdet hU hM hp hz,hMz]
+  simp only [inv_one,neg_mul,one_mul,mul_one,trace_add,trace_neg]
+  ring
+
+end KaehlerCalculus
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/MatrixPositivity.lean b/lean/OAI/Geometry/PrescribedPotential/MatrixPositivity.lean
new file mode 100644
index 0000000000000000000000000000000000000000..ac389359257d160c93b93972e31ca74bae942d08
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/MatrixPositivity.lean
@@ -0,0 +1,71 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CircleRadialCalculus
+
+namespace OAI
+
+/-! Matrix Positivity. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ComplexOrder
+namespace CompactMatrixPositivity
+variable {n : Type*} [Fintype n]
+
+def rq (M : Matrix n n ℂ) (v : n → ℂ) : ℝ := (star v ⬝ᵥ (M *ᵥ v)).re
+
+lemma rq_smul (M : Matrix n n ℂ) (r : ℝ) (v : n → ℂ) :
+    rq M (r • v) = r*r*rq M v := by
+  unfold rq
+  rw [star_smul, star_trivial, Matrix.mulVec_smul, smul_dotProduct, dotProduct_smul,
+    smul_smul]
+  simp only [Complex.real_smul, Complex.mul_re, Complex.ofReal_re, Complex.ofReal_im,
+    zero_mul, sub_zero]
+
+lemma posDef_of_sphere {M : Matrix n n ℂ} (hM : M.IsHermitian)
+    (hp : ∀ v : n → ℂ, ‖v‖ = 1 → 0 < rq M v) : M.PosDef := by
+  apply Matrix.PosDef.of_dotProduct_mulVec_pos hM
+  intro v hv
+  apply Complex.pos_iff.mpr
+  refine ⟨?_,(hM.im_star_dotProduct_mulVec_self v).symm⟩
+  have hn : 0 < ‖v‖ := norm_pos_iff.mpr hv
+  have hw : ‖(‖v‖⁻¹ : ℝ) • v‖ = 1 := by
+    rw [norm_smul, Real.norm_eq_abs, abs_of_pos (inv_pos.mpr hn), inv_mul_cancel₀ hn.ne']
+  have hh := hp ((‖v‖⁻¹ : ℝ) • v) hw
+  rw [rq_smul] at hh
+  exact (mul_pos_iff_of_pos_left (mul_pos (inv_pos.mpr hn) (inv_pos.mpr hn))).mp hh
+
+lemma continuous_rq : Continuous (fun z : Matrix n n ℂ × (n → ℂ) => rq z.1 z.2) := by
+  unfold rq Matrix.mulVec dotProduct
+  fun_prop
+
+variable {T Y : Type*} [TopologicalSpace T] [TopologicalSpace Y]
+lemma eventually_posDef {M : T → Y → Matrix n n ℂ} {t₀ : T} {K : Set Y}
+    (hK : IsCompact K) (hcont : Continuous (fun z : T × Y => M z.1 z.2))
+    (hherm : ∀ t y, y ∈ K → (M t y).IsHermitian)
+    (hpos : ∀ y ∈ K, (M t₀ y).PosDef) :
+    ∀ᶠ t in 𝓝 t₀, ∀ y ∈ K, (M t y).PosDef := by
+  have hc : IsCompact (K ×ˢ Metric.sphere (0 : n → ℂ) 1) :=
+    hK.prod (isCompact_sphere 0 1)
+  have hloc : ∀ y ∈ K ×ˢ Metric.sphere (0 : n → ℂ) 1,
+      ∀ᶠ z : T × (Y × (n → ℂ)) in 𝓝 (t₀,y), 0 < rq (M z.1 z.2.1) z.2.2 := by
+    rintro ⟨y,v⟩ ⟨hy,hv⟩
+    have hv1 : ‖v‖ = 1 := by simpa [Metric.mem_sphere, dist_zero_right] using hv
+    have hv0 : v ≠ 0 := by intro h; simp [h] at hv1
+    have ht : Continuous (fun z : T × (Y × (n → ℂ)) => rq (M z.1 z.2.1) z.2.2) :=
+      continuous_rq.comp ((hcont.comp (continuous_fst.prodMk (continuous_fst.comp continuous_snd))).prodMk
+        (continuous_snd.comp continuous_snd))
+    exact ht.continuousAt.eventually (lt_mem_nhds ((hpos y hy).re_dotProduct_pos hv0))
+  filter_upwards [hc.eventually_forall_of_forall_eventually (P := fun t y => 0 < rq (M t y.1) y.2) hloc] with t ht y hy
+  apply posDef_of_sphere (hherm t y hy)
+  intro v hv
+  exact ht (y,v) ⟨hy,by simpa [Metric.mem_sphere, dist_zero_right] using hv⟩
+end CompactMatrixPositivity
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/MatrixRealFrames.lean b/lean/OAI/Geometry/PrescribedPotential/MatrixRealFrames.lean
new file mode 100644
index 0000000000000000000000000000000000000000..aa12d9459706292b2a69f64d2e6b1c920ce65768
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/MatrixRealFrames.lean
@@ -0,0 +1,139 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.BoundedNormalFrames
+import OAI.Geometry.PrescribedPotential.GlobalSobolev
+import OAI.Geometry.PrescribedPotential.MatrixHilbertCoordinates
+
+namespace OAI
+
+/-! Matrix Real Frames. -/
+
+section
+
+noncomputable section
+open Set Filter Topology Matrix Finset
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace MetricSystem
+open EllipticKernel FrozenPoisson HigherJet KaehlerCalculus
+variable {n : ℕ}
+local instance frameRealECIP : InnerProductSpace ℝ (EC n) := InnerProductSpace.rclikeToReal ℂ (EC n)
+
+ def realMatrix (M : Mat n) : EC n →L[ℝ] EC n :=
+  (Matrix.toEuclideanCLM (𝕜 := ℂ) (n := Fin n) M).restrictScalars ℝ
+
+lemma realMatrix_mul (M N : Mat n) : realMatrix (M*N) = (realMatrix M).comp (realMatrix N) := by
+  apply ContinuousLinearMap.ext
+  intro x
+  change Matrix.toEuclideanCLM (𝕜 := ℂ) (n := Fin n) (M*N) x = Matrix.toEuclideanCLM (𝕜 := ℂ) (n := Fin n) M (Matrix.toEuclideanCLM (𝕜 := ℂ) (n := Fin n) N x)
+  rw [map_mul]
+  rfl
+lemma realMatrix_one : realMatrix (1 : Mat n) = ContinuousLinearMap.id ℝ (EC n) := by
+  apply ContinuousLinearMap.ext
+  intro x
+  change Matrix.toEuclideanCLM (𝕜 := ℂ) (n := Fin n) (1 : Mat n) x = x
+  rw [map_one]
+  rfl
+
+lemma realMatrix_adjoint (M : Mat n) : (realMatrix M).adjoint = realMatrix Mᴴ := by
+  apply ContinuousLinearMap.ext
+  intro x
+  apply ext_inner_left ℝ
+  intro y
+  rw [ContinuousLinearMap.adjoint_inner_right]
+  change (inner ℂ (Matrix.toEuclideanCLM (𝕜 := ℂ) (n := Fin n) M y) x).re = (inner ℂ y (Matrix.toEuclideanCLM (𝕜 := ℂ) (n := Fin n) Mᴴ x)).re
+  congr 1
+  have he : Matrix.toEuclideanCLM (𝕜 := ℂ) (n := Fin n) Mᴴ = (Matrix.toEuclideanCLM (𝕜 := ℂ) (n := Fin n) M).adjoint := map_star _ M
+  rw [he,ContinuousLinearMap.adjoint_inner_right]
+
+ def frame (C D : Mat n) (hCD : C*D=1) (hDC : D*C=1) : EC n ≃L[ℝ] EC n :=
+  { toLinearEquiv :=
+    { toFun := fun x => (1/2:ℝ) • realMatrix C x
+      invFun := fun x => (2:ℝ) • realMatrix D x
+      left_inv := by
+        intro x
+        have he := congrArg (fun L : EC n →L[ℝ] EC n => L x) (realMatrix_mul D C)
+        rw [hDC,realMatrix_one] at he
+        simp only [ContinuousLinearMap.comp_apply,ContinuousLinearMap.id_apply] at he
+        simp only [map_smul,smul_smul]
+        rw [← he]
+        norm_num
+      right_inv := by
+        intro x
+        have he := congrArg (fun L : EC n →L[ℝ] EC n => L x) (realMatrix_mul C D)
+        rw [hCD,realMatrix_one] at he
+        simp only [ContinuousLinearMap.comp_apply,ContinuousLinearMap.id_apply] at he
+        simp only [map_smul,smul_smul]
+        rw [← he]
+        norm_num
+      map_add' := by intro x y; simp only [map_add,smul_add]
+      map_smul' := by intro c x; simp only [map_smul]; exact smul_comm _ _ _ }
+    continuous_toFun := ((continuous_const (y := (1/2:ℝ))).smul (realMatrix C).continuous)
+    continuous_invFun := ((continuous_const (y := (2:ℝ))).smul (realMatrix D).continuous) }
+
+lemma frame_gram {ι : Type*} [Fintype ι] (e : OrthonormalBasis ι ℝ (EC n))
+    (M C D : Mat n) (hCD : C*D=1) (hDC : D*C=1) (hM : Cᴴ*M*C=1)
+    (H : EC n →L[ℝ] EC n →L[ℝ] HM n) :
+    principalBilinear e M⁻¹ H = ∑ i, H (frame C D hCD hDC (e i)) (frame C D hCD hDC (e i)) := by
+  have hD : IsUnit D := ⟨⟨D,C,hDC,hCD⟩,rfl⟩
+  have hC : IsUnit C := ⟨⟨C,D,hCD,hDC⟩,rfl⟩
+  have hInv : M⁻¹ = C*Cᴴ := by
+    rw [unnormalize_matrix hCD hM,Matrix.mul_inv_rev,← Matrix.conjTranspose_nonsing_inv]
+    have hid : D⁻¹ = C := by
+      calc
+        D⁻¹ = D⁻¹*(D*C) := by rw [hDC,mul_one]
+        _ = C := by rw [← mul_assoc,Matrix.nonsing_inv_mul _ ((Matrix.isUnit_iff_isUnit_det _).mp hD),one_mul]
+    rw [hid]
+  let T : EC n →L[ℝ] EC n := (1/2:ℝ) • realMatrix Cᴴ
+  have hT : T.adjoint = (1/2:ℝ) • realMatrix C := by
+    dsimp [T]
+    rw [map_smul,realMatrix_adjoint,Matrix.conjTranspose_conjTranspose]
+  rw [principalBilinear_inv,principalApply_gram e (traceBilin M) T]
+  · rw [hT]
+    rfl
+  · intro v
+    have hfactor : M⁻¹ = star Cᴴ * Cᴴ := by
+      change M⁻¹ = Cᴴᴴ*Cᴴ
+      simpa only [Matrix.conjTranspose_conjTranspose] using hInv
+    have htr : traceBilin M (rankOne v) = (1/4:ℝ)*‖Matrix.toEuclideanCLM (𝕜 := ℂ) (n := Fin n) Cᴴ v‖^2 := by
+      convert trace_rankOne_factor M Cᴴ hfactor v using 1
+      congr 1
+      ext x y
+      simp only [rankOne,ContinuousLinearMap.smulRight_apply,_root_.smul_apply,realInner_apply,smul_eq_mul]
+      change (inner ℂ v x).re*(inner ℂ v y).re = _
+      simp only [PiLp.inner_apply,Complex.re_sum]
+      rfl
+    rw [htr]
+    dsimp [T,realMatrix]
+    simp only [_root_.smul_apply,norm_smul,Real.norm_eq_abs]
+    norm_num
+    ring
+
+ def realMatrixLinear (n : ℕ) : Mat n →ₗ[ℝ] (EC n →L[ℝ] EC n) where
+  toFun := realMatrix
+  map_add' := by intro M N; ext x; simp [realMatrix]
+  map_smul' := by
+    intro c M
+    apply ContinuousLinearMap.ext
+    intro x
+    change Matrix.toEuclideanCLM (𝕜 := ℂ) (n := Fin n) (c • M) x = c • Matrix.toEuclideanCLM (𝕜 := ℂ) (n := Fin n) M x
+    rw [show c • M = (c:ℂ) • M from rfl,map_smul]
+    rfl
+
+lemma frame_uniform_bound (B : ℝ) : ∃ C : ℝ, 0 ≤ C ∧ ∀ M D : Mat n,
+    ‖M‖ ≤ B → ‖D‖ ≤ B → ∀ hMD hDM,
+    ‖(frame M D hMD hDM).toContinuousLinearMap‖ ≤ C ∧
+    ‖(frame M D hMD hDM).symm.toContinuousLinearMap‖ ≤ C := by
+  have hcontinuous : Continuous (realMatrix (n := n)) := (realMatrixLinear n).continuous_of_finiteDimensional
+  obtain ⟨C,hC⟩ := (isCompact_closedBall (0:Mat n) B).exists_bound_of_continuousOn hcontinuous.continuousOn
+  refine ⟨2*max C 0,by positivity,fun M D hM hD hMD hDM => ?_⟩
+  have hm : ‖realMatrix M‖ ≤ max C 0 := (hC M (by simpa using hM)).trans (le_max_left _ _)
+  have hd : ‖realMatrix D‖ ≤ max C 0 := (hC D (by simpa using hD)).trans (le_max_left _ _)
+  change ‖(1/2:ℝ) • realMatrix M‖ ≤ 2*max C 0 ∧ ‖(2:ℝ) • realMatrix D‖ ≤ 2*max C 0
+  simp only [norm_smul,Real.norm_eq_abs]
+  rw [abs_of_pos (by norm_num : 0 < (1/2:ℝ)),abs_of_pos (by norm_num : 0 < (2:ℝ))]
+  constructor <;> nlinarith [norm_nonneg (realMatrix M)]
+end MetricSystem
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/MatrixTensorCalculus.lean b/lean/OAI/Geometry/PrescribedPotential/MatrixTensorCalculus.lean
new file mode 100644
index 0000000000000000000000000000000000000000..701612d8bd3d227c2f2e5e51144b0b02205ed4b4
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/MatrixTensorCalculus.lean
@@ -0,0 +1,69 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.WirtingerConjugate
+
+namespace OAI
+
+/-! Matrix Tensor Calculus. -/
+
+section
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace KaehlerCalculus
+variable {n : ℕ}
+
+lemma mderiv_sub {M N : V n → Matrix (Fin n) (Fin n) ℂ} {z : V n}
+    (hM : ContDiffAt ℝ ∞ M z) (hN : ContDiffAt ℝ ∞ N z) (a : ℂ) (v : V n) :
+    mderiv a v (fun y => M y-N y) z = mderiv a v M z-mderiv a v N z := by
+  simpa only [sub_eq_add_neg,mderiv_neg hN] using mderiv_add hM hN.neg a v
+
+lemma mderiv_sum {α : Type*} (s : Finset α) {M : α → V n → Matrix (Fin n) (Fin n) ℂ}
+    {z : V n} (hM : ∀ i ∈ s, ContDiffAt ℝ ∞ (M i) z) (a : ℂ) (v : V n) :
+    mderiv a v (fun y => ∑ i ∈ s, M i y) z = ∑ i ∈ s, mderiv a v (M i) z := by
+  ext i j
+  simp only [mderiv,Matrix.sum_apply]
+  exact wderiv_sum s (fun k hk => (entry_smooth (hM k hk) i j).differentiableAt (by simp)) a v
+
+lemma mderiv_smul {f : V n → ℂ} {M : V n → Matrix (Fin n) (Fin n) ℂ} {z : V n}
+    (hf : ContDiffAt ℝ ∞ f z) (hM : ContDiffAt ℝ ∞ M z) (a : ℂ) (v : V n) :
+    mderiv a v (fun y => f y • M y) z = wderiv a v f z • M z + f z • mderiv a v M z := by
+  ext i j
+  exact wderiv_mul (hf.differentiableAt (by simp))
+    ((entry_smooth hM i j).differentiableAt (by simp)) a v
+
+lemma matrix_smooth_smul {f : V n → ℂ} {M : V n → Matrix (Fin n) (Fin n) ℂ} {z : V n}
+    (hf : ContDiffAt ℝ ∞ f z) (hM : ContDiffAt ℝ ∞ M z) :
+    ContDiffAt ℝ ∞ (fun y => f y • M y) z := by
+  apply contDiffAt_pi.mpr
+  intro i
+  apply contDiffAt_pi.mpr
+  intro j
+  exact hf.mul (entry_smooth hM i j)
+
+lemma mderiv_conjTranspose {M : V n → Matrix (Fin n) (Fin n) ℂ} {z : V n}
+    (hM : ContDiffAt ℝ ∞ M z) (v : V n) :
+    mderiv (-Complex.I) v (fun y => (M y)ᴴ) z = (mderiv Complex.I v M z)ᴴ := by
+  ext i j
+  exact dz_conj ((entry_smooth hM j i).differentiableAt (by simp)) v
+
+lemma mderiv_bar_conjTranspose {M : V n → Matrix (Fin n) (Fin n) ℂ} {z : V n}
+    (hM : ContDiffAt ℝ ∞ M z) (v : V n) :
+    mderiv Complex.I v (fun y => (M y)ᴴ) z = (mderiv (-Complex.I) v M z)ᴴ := by
+  ext i j
+  exact dzbar_conj ((entry_smooth hM j i).differentiableAt (by simp)) v
+
+lemma matrix_smooth_conjTranspose {M : V n → Matrix (Fin n) (Fin n) ℂ} {z : V n}
+    (hM : ContDiffAt ℝ ∞ M z) : ContDiffAt ℝ ∞ (fun y => (M y)ᴴ) z := by
+  apply contDiffAt_pi.mpr
+  intro i
+  apply contDiffAt_pi.mpr
+  intro j
+  exact Complex.conjCLE.contDiff.contDiffAt.comp z (entry_smooth hM j i)
+
+end KaehlerCalculus
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/MatrixTraceInequalities.lean b/lean/OAI/Geometry/PrescribedPotential/MatrixTraceInequalities.lean
new file mode 100644
index 0000000000000000000000000000000000000000..0c6c71e3b6cfea92e582d119bcb66f4cd9c2c6e9
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/MatrixTraceInequalities.lean
@@ -0,0 +1,79 @@
+import Mathlib
+
+namespace OAI
+
+/-! Matrix Trace Inequalities. -/
+
+section
+
+ 
+
+noncomputable section
+open Matrix
+open scoped ComplexOrder MatrixOrder
+namespace KaehlerCalculus
+variable {n : ℕ}
+
+lemma weighted_trace_cauchy (G E : Matrix (Fin n) (Fin n) ℂ) (hG : G.PosSemidef) :
+    ‖(E*G).trace‖^2 ≤ G.trace.re * (E*Eᴴ*G).trace.re := by
+  let : SeminormedAddCommGroup (Matrix (Fin n) (Fin n) ℂ) :=
+    Matrix.toMatrixSeminormedAddCommGroup G hG
+  let : InnerProductSpace ℂ (Matrix (Fin n) (Fin n) ℂ) :=
+    Matrix.toMatrixInnerProductSpace G hG
+  have h := norm_inner_le_norm (𝕜 := ℂ) Eᴴ (1 : Matrix (Fin n) (Fin n) ℂ)
+  have hn := sq_le_sq₀ (norm_nonneg _) (mul_nonneg (norm_nonneg _) (norm_nonneg _)) |>.mpr h
+  rw [mul_pow, ← inner_self_eq_norm_sq (𝕜 := ℂ) Eᴴ, ← inner_self_eq_norm_sq (𝕜 := ℂ) (1 : Matrix (Fin n) (Fin n) ℂ)] at hn
+  change ‖(1*G*(Eᴴ)ᴴ).trace‖^2 ≤ (Eᴴ*G*(Eᴴ)ᴴ).trace.re *
+    ((1 : Matrix (Fin n) (Fin n) ℂ)*G*(1 : Matrix (Fin n) (Fin n) ℂ)ᴴ).trace.re at hn
+  simpa only [conjTranspose_conjTranspose,one_mul,conjTranspose_one,mul_one,
+    Matrix.trace_mul_comm G E,← Matrix.mul_assoc,
+    Matrix.trace_mul_cycle Eᴴ G E,mul_comm] using hn
+
+lemma trace_mul_psd_nonneg {G R : Matrix (Fin n) (Fin n) ℂ}
+    (hG : G.PosSemidef) (hR : R.PosSemidef) : 0 ≤ (R*G).trace.re := by
+  obtain ⟨B,rfl⟩ := CStarAlgebra.nonneg_iff_eq_star_mul_self.mp hG.nonneg
+  have h := (hR.mul_mul_conjTranspose_same B).trace_nonneg
+  change 0 ≤ (R*(Bᴴ*B)).trace.re
+  rw [Matrix.trace_mul_comm R (Bᴴ*B)]
+  rw [Matrix.trace_mul_cycle] at h
+  exact (Complex.nonneg_iff.mp h).1
+
+lemma trace_inverse_cycle {G : Matrix (Fin n) (Fin n) ℂ} (hG : G.PosDef)
+    (Z : Matrix (Fin n) (Fin n) ℂ) : (G⁻¹*Z*G).trace = Z.trace := by
+  rw [Matrix.trace_mul_cycle,Matrix.mul_nonsing_inv _
+    (isUnit_iff_ne_zero.mpr (ne_of_gt hG.det_pos)),one_mul]
+
+lemma trace_square_completion {G : Matrix (Fin n) (Fin n) ℂ} (hG : G.PosDef)
+    (X Y : Matrix (Fin n) (Fin n) ℂ) :
+    ((X-G⁻¹*Y)*(X-G⁻¹*Y)ᴴ*G).trace =
+      (X*Xᴴ*G).trace-(X*Yᴴ).trace-(Xᴴ*Y).trace+(Yᴴ*G⁻¹*Y).trace := by
+  have hInv : G⁻¹*G = 1 := Matrix.nonsing_inv_mul _
+    (isUnit_iff_ne_zero.mpr (ne_of_gt hG.det_pos))
+  have he : (X-G⁻¹*Y)*(X-G⁻¹*Y)ᴴ*G =
+      X*Xᴴ*G-X*Yᴴ-(G⁻¹*(Y*Xᴴ))*G+G⁻¹*(Y*Yᴴ) := by
+    rw [Matrix.conjTranspose_sub,Matrix.conjTranspose_mul,
+      Matrix.conjTranspose_nonsing_inv,hG.isHermitian.eq]
+    simp only [sub_mul,mul_sub,Matrix.mul_assoc,hInv,mul_one]
+    noncomm_ring
+  rw [he,Matrix.trace_add,Matrix.trace_sub,Matrix.trace_sub,trace_inverse_cycle hG]
+  rw [Matrix.trace_mul_comm Y Xᴴ,← Matrix.mul_assoc]
+  rw [Matrix.trace_mul_cycle G⁻¹ Y Yᴴ]
+
+lemma trace_gradient_square_bound {G : Matrix (Fin n) (Fin n) ℂ} (hG : G.PosDef)
+    (X Y : Matrix (Fin n) (Fin n) ℂ) :
+    ‖(Y-X*G).trace‖^2 ≤ G.trace.re *
+      ((X*Xᴴ*G).trace.re-(X*Yᴴ).trace.re-(Xᴴ*Y).trace.re+(Yᴴ*G⁻¹*Y).trace.re) := by
+  have h := weighted_trace_cauchy G (X-G⁻¹*Y) hG.posSemidef
+  rw [trace_square_completion hG,Complex.add_re,Complex.sub_re,Complex.sub_re] at h
+  have he : ((X-G⁻¹*Y)*G).trace = -(Y-X*G).trace := by
+    rw [sub_mul,Matrix.trace_sub,trace_inverse_cycle hG,Matrix.trace_sub]
+    ring
+  rw [he,norm_neg] at h
+  exact h
+
+end KaehlerCalculus
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/MatrixWirtinger.lean b/lean/OAI/Geometry/PrescribedPotential/MatrixWirtinger.lean
new file mode 100644
index 0000000000000000000000000000000000000000..67f22146a2eb0927ae8eb4cfdf51254a49db7ac8
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/MatrixWirtinger.lean
@@ -0,0 +1,138 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.GlobalOperator
+import OAI.Geometry.PrescribedPotential.WirtingerCalculus
+
+namespace OAI
+
+/-! Matrix Wirtinger. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace KaehlerCalculus
+variable {n : ℕ}
+local instance matrixWirtingerCS (n : ℕ) : ContinuousSMul ℝ (Matrix (Fin n) (Fin n) ℂ) :=
+  inferInstanceAs (ContinuousSMul ℝ (Fin n → Fin n → ℂ))
+
+lemma wderiv_sum {α : Type*} (s : Finset α) {f : α → V n → ℂ} {z : V n}
+    (hf : ∀ i ∈ s, DifferentiableAt ℝ (f i) z) (a : ℂ) (v : V n) :
+    wderiv a v (fun y => ∑ i ∈ s, f i y) z = ∑ i ∈ s, wderiv a v (f i) z := by
+  classical
+  simp only [wderiv, fderiv_fun_sum hf, _root_.sum_apply,
+    Finset.mul_sum]
+  rw [← Finset.sum_div, Finset.sum_add_distrib]
+
+ 
+def mderiv (a : ℂ) (v : V n) (M : V n → Matrix (Fin n) (Fin n) ℂ) (z : V n) :
+    Matrix (Fin n) (Fin n) ℂ := fun i j => wderiv a v (fun y => M y i j) z
+
+lemma entry_smooth {M : V n → Matrix (Fin n) (Fin n) ℂ} {z : V n}
+    (hM : ContDiffAt ℝ ∞ M z) (i j : Fin n) : ContDiffAt ℝ ∞ (fun y => M y i j) z :=
+  contDiffAt_pi.mp (contDiffAt_pi.mp hM i) j
+
+lemma matrix_smooth_mul {M N : V n → Matrix (Fin n) (Fin n) ℂ} {z : V n}
+    (hM : ContDiffAt ℝ ∞ M z) (hN : ContDiffAt ℝ ∞ N z) :
+    ContDiffAt ℝ ∞ (fun y => M y*N y) z := by
+  apply contDiffAt_pi.mpr
+  intro i
+  apply contDiffAt_pi.mpr
+  intro j
+  change ContDiffAt ℝ ∞ (fun y => ∑ k, M y i k*N y k j) z
+  exact ContDiffAt.sum (fun k _ => (entry_smooth hM i k).mul (entry_smooth hN k j))
+
+lemma mderiv_add {M N : V n → Matrix (Fin n) (Fin n) ℂ} {z : V n}
+    (hM : ContDiffAt ℝ ∞ M z) (hN : ContDiffAt ℝ ∞ N z) (a : ℂ) (v : V n) :
+    mderiv a v (fun y => M y+N y) z = mderiv a v M z+mderiv a v N z := by
+  ext i j
+  exact wderiv_add ((entry_smooth hM i j).differentiableAt (by simp))
+    ((entry_smooth hN i j).differentiableAt (by simp)) a v
+
+lemma mderiv_neg {M : V n → Matrix (Fin n) (Fin n) ℂ} {z : V n}
+    (hM : ContDiffAt ℝ ∞ M z) (a : ℂ) (v : V n) :
+    mderiv a v (fun y => -M y) z = -mderiv a v M z := by
+  ext i j
+  change wderiv a v (fun y => -(M y i j)) z = -wderiv a v (fun y => M y i j) z
+  have h := wderiv_const_mul ((entry_smooth hM i j).differentiableAt (by simp)) a (-1) v
+  simpa only [neg_one_mul] using h
+
+lemma mderiv_smooth {M : V n → Matrix (Fin n) (Fin n) ℂ} {z : V n}
+    (hM : ContDiffAt ℝ ∞ M z) (a : ℂ) (v : V n) :
+    ContDiffAt ℝ ∞ (mderiv a v M) z := by
+  exact contDiffAt_pi.mpr (fun i => contDiffAt_pi.mpr (fun j =>
+    wderiv_smooth (entry_smooth hM i j) a v))
+
+lemma mderiv_congr {M N : V n → Matrix (Fin n) (Fin n) ℂ} {z : V n}
+    (he : M =ᶠ[𝓝 z] N) (a : ℂ) (v : V n) : mderiv a v M z = mderiv a v N z := by
+  ext i j
+  exact wderiv_congr (he.mono (fun y hy => congrFun (congrFun hy i) j)) a v
+
+lemma mderiv_const (C : Matrix (Fin n) (Fin n) ℂ) (a : ℂ) (v z : V n) :
+    mderiv a v (fun _ => C) z = 0 := by
+  ext i j
+  exact wderiv_const a _ v z
+
+lemma mderiv_mul {M N : V n → Matrix (Fin n) (Fin n) ℂ} {z : V n}
+    (hM : ContDiffAt ℝ ∞ M z) (hN : ContDiffAt ℝ ∞ N z) (a : ℂ) (v : V n) :
+    mderiv a v (fun y => M y*N y) z = mderiv a v M z*N z+M z*mderiv a v N z := by
+  ext i j
+  change wderiv a v (fun y => ∑ k, M y i k*N y k j) z = _
+  rw [wderiv_sum Finset.univ (fun k _ => ((entry_smooth hM i k).differentiableAt (by simp)).fun_mul
+    ((entry_smooth hN k j).differentiableAt (by simp)))]
+  simp only [wderiv_mul ((entry_smooth hM _ _).differentiableAt (by simp))
+    ((entry_smooth hN _ _).differentiableAt (by simp)), Finset.sum_add_distrib,
+    Matrix.add_apply,Matrix.mul_apply,mderiv]
+
+lemma mderiv_mul_differentiable {M N : V n → Matrix (Fin n) (Fin n) ℂ} {z : V n}
+    (hM : DifferentiableAt ℝ M z) (hN : DifferentiableAt ℝ N z) (a : ℂ) (v : V n) :
+    mderiv a v (fun y => M y*N y) z = mderiv a v M z*N z+M z*mderiv a v N z := by
+  have hm (i j : Fin n) := differentiableAt_pi.mp (differentiableAt_pi.mp hM i) j
+  have hn (i j : Fin n) := differentiableAt_pi.mp (differentiableAt_pi.mp hN i) j
+  ext i j
+  change wderiv a v (fun y => ∑ k, M y i k*N y k j) z = _
+  rw [wderiv_sum Finset.univ (fun k _ => (hm i k).fun_mul (hn k j))]
+  simp only [wderiv_mul (hm _ _) (hn _ _),Finset.sum_add_distrib,
+    Matrix.add_apply,Matrix.mul_apply,mderiv]
+
+lemma mderiv_trace_differentiable {M : V n → Matrix (Fin n) (Fin n) ℂ} {z : V n}
+    (hM : DifferentiableAt ℝ M z) (a : ℂ) (v : V n) :
+    wderiv a v (fun y => (M y).trace) z = (mderiv a v M z).trace := by
+  exact wderiv_sum Finset.univ (fun i _ =>
+    differentiableAt_pi.mp (differentiableAt_pi.mp hM i) i) a v
+
+lemma mderiv_inverse {U : Set (V n)} (hU : IsOpen U)
+    {M : V n → Matrix (Fin n) (Fin n) ℂ} (hM : ContDiffOn ℝ ∞ M U)
+    (hn : ∀ y ∈ U, (M y).det ≠ 0) {z : V n} (hz : z ∈ U) (a : ℂ) (v : V n) :
+    mderiv a v (fun y => (M y)⁻¹) z = -(M z)⁻¹*mderiv a v M z*(M z)⁻¹ := by
+  have hi := (MatrixSmoothGeneral.inverse hM hn).contDiffAt (hU.mem_nhds hz)
+  have hs := hM.contDiffAt (hU.mem_nhds hz)
+  have he : (fun y => M y*(M y)⁻¹) =ᶠ[𝓝 z] (fun _ => 1) := by
+    filter_upwards [hU.mem_nhds hz] with y hy
+    exact Matrix.mul_nonsing_inv _ (isUnit_iff_ne_zero.mpr (hn y hy))
+  have hh := mderiv_congr he a v
+  rw [mderiv_mul hs hi,mderiv_const] at hh
+  have hm := congrArg (fun B => (M z)⁻¹*B) hh
+  rw [mul_add, ← mul_assoc (M z)⁻¹ (M z),
+    Matrix.nonsing_inv_mul _ (isUnit_iff_ne_zero.mpr (hn z hz)),one_mul,mul_zero] at hm
+  apply eq_neg_of_add_eq_zero_right at hm
+  simpa only [neg_mul, mul_assoc] using hm
+
+lemma mderiv_trace {M : V n → Matrix (Fin n) (Fin n) ℂ} {z : V n}
+    (hM : ContDiffAt ℝ ∞ M z) (a : ℂ) (v : V n) :
+    wderiv a v (fun y => (M y).trace) z = (mderiv a v M z).trace := by
+  exact wderiv_sum Finset.univ (fun i _ => (entry_smooth hM i i).differentiableAt (by simp)) a v
+
+lemma mderiv_comm {M : V n → Matrix (Fin n) (Fin n) ℂ} {z : V n}
+    (hM : ContDiffAt ℝ ∞ M z) (a b : ℂ) (u v : V n) :
+    mderiv a u (mderiv b v M) z = mderiv b v (mderiv a u M) z := by
+  ext i j
+  exact wderiv_comm (entry_smooth hM i j) a b u v
+end KaehlerCalculus
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/MetricFamilyBootstrap.lean b/lean/OAI/Geometry/PrescribedPotential/MetricFamilyBootstrap.lean
new file mode 100644
index 0000000000000000000000000000000000000000..bf7dcb62c23a11998a15dc6fe1d7fe32dc95eb4b
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/MetricFamilyBootstrap.lean
@@ -0,0 +1,99 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CompactCoefficientBootstrap
+import OAI.Geometry.PrescribedPotential.EncodedMetricEquation
+import OAI.Geometry.PrescribedPotential.MatrixCoefficientSmooth
+import OAI.Geometry.PrescribedPotential.MatrixRealFrames
+
+namespace OAI
+
+/-! Metric Family Bootstrap. -/
+
+section
+
+noncomputable section
+open Set Filter Topology Matrix Metric Finset
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace MetricSystem
+open EllipticKernel FrozenPoisson HigherJet KaehlerCalculus
+variable {n : ℕ} {α : Type*}
+local instance bootstrapRealECIP : InnerProductSpace ℝ (EC n) := InnerProductSpace.rclikeToReal ℂ (EC n)
+local instance bootstrapRealHMIP : InnerProductSpace ℝ (HM n) := InnerProductSpace.rclikeToReal ℂ (HM n)
+
+lemma metric_family_bootstrap {U : Set (EC n)} (hU : IsOpen U)
+    (M : α → LocalKaehlerField n) (r : α → EC n → HM n)
+    (hdom : ∀ s x, x ∈ U → coordinateEquiv n x ∈ (M s).domain)
+    (hr : ∀ s, ContDiffOn ℝ ∞ (r s) U)
+    (hric : ∀ s x, x ∈ U → encode n ((M s).ricciHessian (coordinateEquiv n x)) = r s x)
+    (hfirst : ∀ K, IsCompact K → K ⊆ U → ∃ C : ℝ, ∀ s x, x ∈ K →
+      ‖fderiv ℝ (fun y => encode n ((M s).matrix (coordinateEquiv n y))) x‖ ≤ C)
+    (hframeBound : ∀ K, IsCompact K → K ⊆ U → ∃ B : ℝ, 0 ≤ B ∧ ∀ s x, x ∈ K →
+      ∀ C D : Mat n, C*D=1 → D*C=1 → Cᴴ*(M s).matrix (coordinateEquiv n x)*C=1 → ‖C‖ ≤ B ∧ ‖D‖ ≤ B)
+    (hrbounds : ∀ m K, IsCompact K → K ⊆ U → ∃ C : ℝ, ∀ j, j ≤ m → ∀ s x, x ∈ K →
+      ‖iteratedFDeriv ℝ j (r s) x‖ ≤ C) :
+    ∀ m, 1 ≤ m → ∀ K, IsCompact K → K ⊆ U →
+      ∃ C : ℝ, 0 ≤ C ∧ ∀ s x, x ∈ K →
+        ‖iteratedFDeriv ℝ m (fun y => encode n ((M s).matrix (coordinateEquiv n y))) x‖ ≤ C := by
+  classical
+  let u : α → EC n → HM n := fun s y => encode n ((M s).matrix (coordinateEquiv n y))
+  have hu : ∀ s, ContDiffOn ℝ ∞ (u s) U := by
+    intro s x hx
+    exact ((encode n).contDiff.contDiffAt.comp x
+      (((M s).smooth.contDiffAt ((M s).isOpen.mem_nhds (hdom s x hx))).comp x
+        (coordinateEquiv n).contDiff.contDiffAt)).contDiffWithinAt
+  have hf (s : α) (x : EC n) : ∃ C D : Mat n, C*D=1 ∧ D*C=1 ∧
+      (x ∈ U → Cᴴ*(M s).matrix (coordinateEquiv n x)*C=1) := by
+    by_cases hx : x ∈ U
+    · obtain ⟨C,hC,hN⟩ := whitening _ ((M s).positive _ (hdom s x hx))
+      obtain ⟨Q,rfl⟩ := hC
+      exact ⟨Q,↑Q⁻¹,Q.mul_inv,Q.inv_mul,fun _ => hN⟩
+    · exact ⟨1,1,by simp,by simp,fun hh => (hx hh).elim⟩
+  choose C D hCD hDC hN using hf
+  let τ : α → EC n → EC n ≃L[ℝ] EC n := fun s x => frame (C s x) (D s x) (hCD s x) (hDC s x)
+  let basis := stdOrthonormalBasis ℝ (EC n)
+  apply quasilinear_compact_coefficient_bootstrap (principalBilinear basis) hU basis u r inverseCoefficient quadraticCoefficient τ hu hr
+  · intro s x hx
+    exact inverseCoefficient_smooth (by
+      change ((M s).matrix (coordinateEquiv n x)).det ≠ 0
+      exact ne_of_gt ((M s).positive _ (hdom s x hx)).det_pos)
+  · intro s x hx
+    exact quadraticCoefficient_smooth (by
+      change ((M s).matrix (coordinateEquiv n x)).det ≠ 0
+      exact ne_of_gt ((M s).positive _ (hdom s x hx)).det_pos)
+  · intro s x hx
+    exact (encoded_metric_equation basis (M s) (hdom s x hx)).trans (by rw [hric s x hx])
+  · intro s x hx H
+    exact frame_gram basis _ (C s x) (D s x) (hCD s x) (hDC s x) (hN s x hx) H
+  · intro K hK hKU
+    obtain ⟨B,hB,hbound⟩ := hframeBound K hK hKU
+    obtain ⟨R,hR,hRbound⟩ := frame_uniform_bound (n := n) B
+    refine ⟨R,hR,fun s x hx => ?_⟩
+    obtain ⟨hc,hd⟩ := hbound s x hx (C s x) (D s x) (hCD s x) (hDC s x) (hN s x (hKU hx))
+    exact hRbound _ _ hc hd _ _
+  · intro K hK hKU
+    obtain ⟨B,hB,hbound⟩ := hframeBound K hK hKU
+    obtain ⟨J,hJ⟩ := hfirst K hK hKU
+    let P : Set (HM n) := (fun p : Mat n × Mat n => encode n (p.2ᴴ*p.2)) '' boundedFramePairs B
+    have hP : IsCompact P := (boundedFramePairs_compact B).image
+      ((encode n).continuous.comp ((continuous_snd.matrix_conjTranspose).matrix_mul continuous_snd))
+    have hpositive : ∀ p ∈ P, ((encode n).symm p).PosDef := by
+      rintro p ⟨q,hq,rfl⟩
+      rw [ContinuousLinearEquiv.symm_apply_apply]
+      simpa only [mul_one] using Matrix.PosDef.one.conjTranspose_mul_mul_same
+        (Matrix.mulVec_injective_iff_isUnit.mpr (boundedFramePairs_units hq).2)
+    refine ⟨P ×ˢ closedBall (0:EC n →L[ℝ] HM n) J,hP.prod (isCompact_closedBall _ _),?_,?_⟩
+    · intro s x hx
+      refine ⟨?_,by simpa only [firstJet,mem_closedBall,dist_zero_right] using hJ s x hx⟩
+      obtain ⟨hc,hd⟩ := hbound s x hx (C s x) (D s x) (hCD s x) (hDC s x) (hN s x (hKU hx))
+      refine ⟨(C s x,D s x),⟨hc,hd,hCD s x,hDC s x⟩,?_⟩
+      change encode n ((D s x)ᴴ*D s x) = encode n ((M s).matrix (coordinateEquiv n x))
+      rw [unnormalize_matrix (hCD s x) (hN s x (hKU hx))]
+    · intro p hp
+      have hd := ne_of_gt (hpositive p.1 hp.1).det_pos
+      exact ⟨inverseCoefficient_smooth hd,quadraticCoefficient_smooth hd⟩
+  · exact hrbounds
+end MetricSystem
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/MetricInverseTrace.lean b/lean/OAI/Geometry/PrescribedPotential/MetricInverseTrace.lean
new file mode 100644
index 0000000000000000000000000000000000000000..cb8afb71e1787f6bf6d5ccd62dbf3db906d62328
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/MetricInverseTrace.lean
@@ -0,0 +1,73 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.InverseTraceFunction
+import OAI.Geometry.PrescribedPotential.MatrixFrameNormalization
+
+namespace OAI
+
+/-! Metric Inverse Trace. -/
+
+section
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace Anticanonical.SourceSmooth
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+namespace KaehlerMetric
+
+lemma derivativeMatrix_isUnit (g : KaehlerMetric A) (i j : Fin A.count) {x : X}
+    (hi : x ∈ (A.chart i).source) (hj : x ∈ (A.chart j).source) :
+    IsUnit (A.derivativeMatrix i j x) := by
+  apply (Matrix.isUnit_iff_isUnit_det _).mpr
+  apply isUnit_iff_ne_zero.mpr
+  intro he
+  have hh := congrArg Matrix.det (g.compatibility i j x hi hj)
+  rw [Matrix.det_mul,he,mul_zero] at hh
+  exact (ne_of_gt (g.positive i _ ((A.chart i).mapsTo hi)).det_pos) hh
+
+lemma inverseTrace_compatibility (g G : KaehlerMetric A) (i j : Fin A.count) {x : X}
+    (hi : x ∈ (A.chart i).source) (hj : x ∈ (A.chart j).source) :
+    KaehlerCalculus.inverseTrace (g.matrix i) (G.matrix i) (A.chart i x) =
+      KaehlerCalculus.inverseTrace (g.matrix j) (G.matrix j) (A.chart j x) := by
+  unfold KaehlerCalculus.inverseTrace
+  rw [g.compatibility i j x hi hj,G.compatibility i j x hi hj,
+    KaehlerCalculus.matrix_inverse_trace_congruence _ _ _ (g.derivativeMatrix_isUnit i j hi hj)]
+
+def inverseTraceValue (g G : KaehlerMetric A) (x : X) : ℝ :=
+  let i := (A.covers x).choose
+  KaehlerCalculus.inverseTrace (g.matrix i) (G.matrix i) (A.chart i x)
+
+lemma inverseTraceValue_local (g G : KaehlerMetric A) (i : Fin A.count) {x : X}
+    (hi : x ∈ (A.chart i).source) :
+    g.inverseTraceValue G x = KaehlerCalculus.inverseTrace (g.matrix i) (G.matrix i) (A.chart i x) := by
+  exact g.inverseTrace_compatibility G _ i (A.covers x).choose_spec hi
+
+def inverseTrace (g G : KaehlerMetric A) : SmoothRealFunction A where
+  value := g.inverseTraceValue G
+  smooth i := by
+    apply (KaehlerCalculus.inverseTrace_smooth (A.chart i).open_target (g.smooth i)
+      (g.positive i) (G.smooth i)).congr
+    intro z hz
+    dsimp only [Function.comp_apply]
+    rw [g.inverseTraceValue_local G i ((A.chart i).mapsTo_symm hz),(A.chart i).right_inv hz]
+
+lemma inverseTrace_local (g G : KaehlerMetric A) (i : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart i).target) :
+    (g.inverseTrace G).localExpression i z = KaehlerCalculus.inverseTrace (g.matrix i) (G.matrix i) z := by
+  change g.inverseTraceValue G ((A.chart i).symm z) = _
+  rw [g.inverseTraceValue_local G i ((A.chart i).mapsTo_symm hz),(A.chart i).right_inv hz]
+
+lemma inverseTrace_positive [Nonempty (Fin d)] (g G : KaehlerMetric A) (x : X) :
+    0 < (g.inverseTrace G).value x := by
+  obtain ⟨i,hi⟩ := A.covers x
+  change 0 < g.inverseTraceValue G x
+  rw [g.inverseTraceValue_local G i hi]
+  exact KaehlerCalculus.inverseTrace_pos (g.positive i _ ((A.chart i).mapsTo hi))
+    (G.positive i _ ((A.chart i).mapsTo hi))
+end KaehlerMetric
+end Anticanonical.SourceSmooth
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/MetricLocalization.lean b/lean/OAI/Geometry/PrescribedPotential/MetricLocalization.lean
new file mode 100644
index 0000000000000000000000000000000000000000..4675245820ac35addecdedd59e138244c70ba3e7
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/MetricLocalization.lean
@@ -0,0 +1,223 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.VariablePoisson
+
+namespace OAI
+
+/-! Metric Localization. -/
+
+section
+
+ 
+
+noncomputable section
+open Filter Topology Set
+open scoped ComplexOrder MatrixOrder Matrix.Norms.Elementwise ContDiff
+
+namespace MetricLocalization
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+variable {n : ℕ}
+
+lemma contDiffOn_matrix_det {U : Set E} {H : E → Matrix (Fin n) (Fin n) ℂ}
+    (hH : ContDiffOn ℝ ∞ H U) : ContDiffOn ℝ ∞ (fun z => (H z).det) U := by
+  classical
+  simp_rw [Matrix.det_apply']
+  apply ContDiffOn.sum
+  intro σ _
+  apply contDiffOn_const.mul
+  apply contDiffOn_prod
+  intro i _
+  exact contDiffOn_pi.mp (contDiffOn_pi.mp hH (σ i)) i
+
+lemma contDiffOn_matrix_adjugate {U : Set E} {H : E → Matrix (Fin n) (Fin n) ℂ}
+    (hH : ContDiffOn ℝ ∞ H U) : ContDiffOn ℝ ∞ (fun z => (H z).adjugate) U := by
+  classical
+  apply contDiffOn_pi.mpr
+  intro i
+  apply contDiffOn_pi.mpr
+  intro j
+  simp_rw [Matrix.adjugate_apply]
+  apply contDiffOn_matrix_det
+  apply contDiffOn_pi.mpr
+  intro k
+  apply contDiffOn_pi.mpr
+  intro l
+  by_cases hk : k = j
+  · subst k
+    simp only [Matrix.updateRow_self]
+    exact contDiffOn_const
+  · simp only [Matrix.updateRow_ne hk]
+    exact contDiffOn_pi.mp (contDiffOn_pi.mp hH k) l
+
+lemma contDiffOn_matrix_inv {U : Set E} {H : E → Matrix (Fin n) (Fin n) ℂ}
+    (hH : ContDiffOn ℝ ∞ H U) (hh : ∀ x ∈ U, (H x).det ≠ 0) :
+    ContDiffOn ℝ ∞ (fun z => (H z)⁻¹) U := by
+  simp_rw [Matrix.inv_def, Ring.inverse_eq_inv']
+  exact ((contDiffOn_matrix_det hH).inv hh).smul (contDiffOn_matrix_adjugate hH)
+
+lemma trace_coefficient_smooth {U : Set (EllipticKernel.EC n)}
+    {H : EllipticKernel.EC n → Matrix (Fin n) (Fin n) ℂ}
+    (hH : ContDiffOn ℝ ∞ H U) (hh : ∀ x ∈ U, (H x).PosDef)
+    (B : EllipticKernel.Bilin (EllipticKernel.EC n)) :
+    ContDiffOn ℝ ∞ (fun z => (EllipticKernel.traceBilin (H z) B : ℂ)) U := by
+  have hi := contDiffOn_matrix_inv hH (fun x hx => ne_of_gt (hh x hx).det_pos)
+  have hir : ContDiffOn ℝ ∞
+      (fun z => ((H z)⁻¹ * PotentialKaehler.hermitianPartMatrix
+        (EllipticKernel.pullBilin B)).trace.re) U := by
+    apply Complex.reCLM.contDiff.comp_contDiffOn
+    simp only [Matrix.trace, Matrix.diag_apply, Matrix.mul_apply]
+    apply ContDiffOn.sum
+    intro i _
+    apply ContDiffOn.sum
+    intro j _
+    exact (contDiffOn_pi.mp (contDiffOn_pi.mp hi i) j).mul contDiffOn_const
+  exact Complex.ofRealCLM.contDiff.comp_contDiffOn hir
+
+section Extension
+variable [FiniteDimensional ℝ E]
+variable {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F]
+
+lemma smooth_bump_smul {x : E} (ψ : ContDiffBump x) (f : E → F)
+    (hf : ∀ y ∈ tsupport ψ, ContDiffAt ℝ ∞ f y) :
+    ContDiff ℝ ∞ (fun y => ψ y • f y) := by
+  rw [contDiff_iff_contDiffAt]
+  intro y
+  by_cases hy : y ∈ tsupport ψ
+  · exact ψ.contDiff.contDiffAt.smul (hf y hy)
+  · apply (contDiffAt_const (c := (0 : F))).congr_of_eventuallyEq
+    filter_upwards [notMem_tsupport_iff_eventuallyEq.mp hy] with z hz
+    simp [hz]
+
+ 
+
+lemma small_smooth_extension {U : Set E} (hU : IsOpen U) {x : E} (hx : x ∈ U)
+    (f : E → F) (hf : ContDiffOn ℝ ∞ f U) (hfx : f x = 0)
+    {ε : ℝ} (hε : 0 < ε) :
+    ∃ a : E → F, ContDiff ℝ ∞ a ∧ HasCompactSupport a ∧
+      tsupport a ⊆ U ∧ (a =ᶠ[𝓝 x] f) ∧ ∀ y, ‖a y‖ ≤ ε := by
+  have hnear : ∀ᶠ y in 𝓝 x, ‖f y‖ < ε := by
+    have hc := (hf x hx).continuousWithinAt.continuousAt (hU.mem_nhds hx)
+    exact hc.norm.eventually (gt_mem_nhds (by simpa [hfx] using hε))
+  obtain ⟨r, hr, hsub⟩ := Metric.nhds_basis_closedBall.mem_iff.mp
+    (inter_mem (hU.mem_nhds hx) hnear)
+  let ψ : ContDiffBump x :=
+    { rIn := r / 2, rOut := r, rIn_pos := half_pos hr, rIn_lt_rOut := half_lt_self hr }
+  let a : E → F := fun y => ψ y • f y
+  have hsupp : tsupport a ⊆ tsupport ψ := tsupport_smul_subset_left _ _
+  have hψ : tsupport ψ ⊆ U ∩ {y | ‖f y‖ < ε} := by
+    simpa only [ψ, ContDiffBump.tsupport_eq] using hsub
+  refine ⟨a, ?_, ?_, hsupp.trans (hψ.trans inter_subset_left), ?_, ?_⟩
+  · apply smooth_bump_smul ψ f
+    intro y hy
+    exact (hf y (hψ hy).1).contDiffAt (hU.mem_nhds (hψ hy).1)
+  · exact ψ.hasCompactSupport.smul_right
+  · filter_upwards [ψ.eventuallyEq_one] with y hy
+    simp [a, hy]
+  · intro y
+    by_cases hy : y ∈ tsupport ψ
+    · have hf' := (hψ hy).2
+      calc
+        ‖a y‖ = |ψ y| * ‖f y‖ := norm_smul _ _
+        _ = ψ y * ‖f y‖ := by rw [abs_of_nonneg ψ.nonneg]
+        _ ≤ 1 * ‖f y‖ := mul_le_mul_of_nonneg_right ψ.le_one (norm_nonneg _)
+        _ ≤ ε := by simpa using hf'.le
+    · have hz : ψ y = 0 := image_eq_zero_of_notMem_tsupport hy
+      simp [a, hz, hε.le]
+
+end Extension
+end MetricLocalization
+
+namespace MetricLocalization
+open EllipticKernel FrozenPoisson SobolevChart
+open scoped BoundedContinuousFunction
+variable {n : ℕ} {ι : Type*} [Fintype ι]
+
+ 
+
+theorem local_coefficient_extension {U : Set (EC n)} (hU : IsOpen U)
+    (H : EC n → Matrix (Fin n) (Fin n) ℂ) (hH : ContDiffOn ℝ ∞ H U)
+    (hh : ∀ y ∈ U, (H y).PosDef) {x : EC n} (hx : x ∈ U) (v : ι → EC n) :
+    ∃ a : ι → ι → EC n →ᵇ ℂ,
+      (∀ i j, (a i j : EC n → ℂ).HasTemperateGrowth) ∧
+      (∀ i j, HasCompactSupport (a i j : EC n → ℂ)) ∧
+      perturbationBound v a * ellipticBound (H x) (hh x hx) < 1 ∧
+      ∀ᶠ y in 𝓝 x, ∀ i j,
+        a i j y = (traceBilin (H y) (rankTwo (v i) (v j)) : ℂ) -
+          (traceBilin (H x) (rankTwo (v i) (v j)) : ℂ) := by
+  classical
+  let C := ellipticBound (H x) (hh x hx)
+  have hC : 0 < C := ellipticBound_pos _ _
+  let Q : ℝ := ∑ i, ∑ j, (2 * Real.pi)^2 * ‖v i‖ * ‖v j‖
+  have hQ : 0 ≤ Q := by dsimp [Q]; positivity
+  let ε : ℝ := 1 / (1 + Q * C)
+  have hden : 0 < 1 + Q * C := by positivity
+  have hε : 0 < ε := one_div_pos.mpr hden
+  let F : EC n → ι → ι → ℂ := fun y i j =>
+    (traceBilin (H y) (rankTwo (v i) (v j)) : ℂ) -
+      (traceBilin (H x) (rankTwo (v i) (v j)) : ℂ)
+  have hF : ContDiffOn ℝ ∞ F U := by
+    apply contDiffOn_pi.mpr
+    intro i
+    apply contDiffOn_pi.mpr
+    intro j
+    exact (trace_coefficient_smooth hH hh _).sub contDiffOn_const
+  have hFx : F x = 0 := by ext i j; exact sub_self _
+  obtain ⟨b, hb, hbc, _, hbe, hbn⟩ := small_smooth_extension hU hx F hF hFx hε
+  have hbij i j : ContDiff ℝ ∞ (fun y => b y i j) :=
+    contDiff_pi.mp (contDiff_pi.mp hb i) j
+  have hbijc i j : HasCompactSupport (fun y => b y i j) :=
+    hbc.comp_left (g := fun M : ι → ι → ℂ => M i j) rfl
+  have hbound i j y : ‖b y i j‖ ≤ ε :=
+    (norm_le_pi_norm (b y i) j).trans ((norm_le_pi_norm (b y) i).trans (hbn y))
+  let a : ι → ι → EC n →ᵇ ℂ := fun i j =>
+    BoundedContinuousFunction.ofNormedAddCommGroup (fun y => b y i j)
+      (hbij i j).continuous ε (hbound i j)
+  have han i j : ‖a i j‖ ≤ ε :=
+    (BoundedContinuousFunction.norm_le hε.le).mpr (hbound i j)
+  have hp : perturbationBound v a ≤ ε * Q := by
+    simp only [perturbationBound, Q, Finset.mul_sum]
+    apply Finset.sum_le_sum
+    intro i _
+    apply Finset.sum_le_sum
+    intro j _
+    exact mul_le_mul_of_nonneg_right (han i j) (by positivity)
+  refine ⟨a, fun i j => (hbijc i j).hasTemperateGrowth (hbij i j), hbijc, ?_, ?_⟩
+  · change perturbationBound v a * C < 1
+    calc
+      _ ≤ (ε * Q) * C := mul_le_mul_of_nonneg_right hp hC.le
+      _ = (Q * C) / (1 + Q * C) := by dsimp [ε]; ring
+      _ < 1 := (div_lt_one hden).mpr (by linarith)
+  · filter_upwards [hbe] with y hy
+    intro i j
+    exact congrFun (congrFun hy i) j
+
+end MetricLocalization
+
+namespace Anticanonical.SourceSmooth
+open EllipticKernel FrozenPoisson SobolevChart
+open scoped BoundedContinuousFunction
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+
+ 
+
+theorem KaehlerMetric.local_inverse_coefficients (g : KaehlerMetric A)
+    (i : Fin A.count) {x : EC d} (hx : coordinateEquiv d x ∈ (A.chart i).target) :
+    let b := stdOrthonormalBasis ℝ (EC d)
+    let H := fun y => g.matrix i (coordinateEquiv d y)
+    ∃ a : Fin (Module.finrank ℝ (EC d)) → Fin (Module.finrank ℝ (EC d)) → EC d →ᵇ ℂ,
+      (∀ k l, (a k l : EC d → ℂ).HasTemperateGrowth) ∧
+      (∀ k l, HasCompactSupport (a k l : EC d → ℂ)) ∧
+      perturbationBound b a * ellipticBound (H x) (g.positive i _ hx) < 1 ∧
+      ∀ᶠ y in 𝓝 x, ∀ k l,
+        a k l y = (traceBilin (H y) (rankTwo (b k) (b l)) : ℂ) -
+          (traceBilin (H x) (rankTwo (b k) (b l)) : ℂ) := by
+  have hs := (g.smooth i).comp (coordinateEquiv d).contDiff.contDiffOn (fun _ h => h)
+  exact MetricLocalization.local_coefficient_extension
+    ((A.chart i).open_target.preimage (coordinateEquiv d).continuous)
+    _ hs (fun y hy => g.positive i _ hy) hx _
+
+end Anticanonical.SourceSmooth
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/MetricQuasilinearEquation.lean b/lean/OAI/Geometry/PrescribedPotential/MetricQuasilinearEquation.lean
new file mode 100644
index 0000000000000000000000000000000000000000..f5462a0625960556a7c443650cfb3acfd072f1b3
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/MetricQuasilinearEquation.lean
@@ -0,0 +1,36 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CalabiConnection
+import OAI.Geometry.PrescribedPotential.CalabiRicciContraction
+
+namespace OAI
+
+/-! Metric Quasilinear Equation. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace KaehlerCalculus.LocalKaehlerField
+variable {n : ℕ} (K : LocalKaehlerField n)
+
+lemma metric_quasilinear_equation {z : V n} (hz : z ∈ K.domain) :
+    (∑ p, ∑ q, ((K.matrix z)⁻¹ p q) •
+      mderiv Complex.I (e q) (mderiv (-Complex.I) (e p) K.matrix) z) =
+      K.ricciHessian z + ∑ p, ∑ q, ((K.matrix z)⁻¹ p q) •
+        (mderiv Complex.I (e q) K.matrix z*(K.matrix z)⁻¹*
+          mderiv (-Complex.I) (e p) K.matrix z) := by
+  have he := congrArg (fun M => K.matrix z*M) (K.curvature_contraction hz)
+  simp only [Matrix.mul_sum,Matrix.mul_smul] at he
+  simp only [K.lower_curvature_formula hz,smul_sub,Finset.sum_sub_distrib] at he
+  rw [← mul_assoc,Matrix.mul_nonsing_inv _
+    (isUnit_iff_ne_zero.mpr (ne_of_gt (K.positive z hz).det_pos)),one_mul] at he
+  exact sub_eq_iff_eq_add.mp he
+end KaehlerCalculus.LocalKaehlerField
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/NonlinearCoreCommutator.lean b/lean/OAI/Geometry/PrescribedPotential/NonlinearCoreCommutator.lean
new file mode 100644
index 0000000000000000000000000000000000000000..138aa4704a9dc900fae2f98e5bdab28ec61b5586
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/NonlinearCoreCommutator.lean
@@ -0,0 +1,132 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.NonlinearLocalCoordinates
+import OAI.Geometry.PrescribedPotential.NonlinearRemainderStrong
+import OAI.Geometry.PrescribedPotential.PotentialDensityDifferential
+import OAI.Geometry.PrescribedPotential.RealSobolev
+
+namespace OAI
+
+/-! Nonlinear Core Commutator. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Matrix LineDeriv
+open scoped ContDiff SchwartzMap Classical Matrix.Norms.Elementwise
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+lemma regularityError_source (p : ι) (v : EC d) (f : Smooth A) (i j : Fin d) {x : X}
+    (hx : x ∈ (A.euclideanChart (D.patch p).index).source)
+    (hκ : A.euclideanChart (D.patch p).index x ∈ tsupport (D.cutoff p : EC d → ℂ)) :
+    D.regularityError p v i j f x = hessianLocalizedError (D.cutoff p).realPart.val v i j
+      (localize A (D.patch p).index (cutoffGlobal (D.patch p).index (D.regularityOuter p))
+        (cutoffGlobal_support _ _) f) (A.euclideanChart (D.patch p).index x) := by
+  change globalize (D.patch p).index (D.regularityCutoff p) _ x = _
+  rw [globalize_apply,ite_eq_left hx,(D.regularityCutoff_one p hκ).eq_of_nhds,one_mul]
+  rfl
+
+lemma nonlinear_commutator_on_support (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (p : ι) (v : EC d) (φ : SmoothRealFunction A) {x : X}
+    (hx : x ∈ (A.euclideanChart (D.patch p).index).source)
+    (hκ : A.euclideanChart (D.patch p).index x ∈ tsupport (D.cutoff p : EC d → ℂ)) :
+    D.localizers.strong (k : ℝ)
+      (fderiv ℝ (D.completedVolume k hk) (D.localizers.embed ((k : ℝ)+2) (Smooth.ofReal φ))
+        (D.localizers.embed ((k : ℝ)+2) (D.localizedDerivative p v (Smooth.ofReal φ)))) x =
+      D.localizedDerivative p v (Smooth.ofReal (g.potentialDensity φ)) x +
+      D.localizers.strong (k : ℝ)
+        (D.nonlinearRemainder k hk p v (D.localizers.embed ((k : ℝ)+2) (Smooth.ofReal φ))) x := by
+  let q := (D.patch p).index
+  let e := A.euclideanChart q
+  let y := e x
+  let f := localize A q (cutoffGlobal q (D.regularityOuter p)) (cutoffGlobal_support _ _) (Smooth.ofReal φ)
+  let κ : 𝓢(EC d, ℝ) := SchwartzMap.postcompCLM Complex.reCLM (D.cutoff p).val
+  let G : EC d → Matrix (Fin d) (Fin d) ℂ := fun z => g.matrix q (coordinateEquiv d z)
+  let B : EC d → ℂ := fun z => (G z).det⁻¹
+  let ψ := (D.realLocalizedDerivative p v (RealSmooth.ofReal φ)).source
+  have hψ : Smooth.ofReal ψ = D.localizedDerivative p v (Smooth.ofReal φ) :=
+    RealSmooth.ofReal_source _
+  have ht : y ∈ e.target := e.mapsTo hx
+  have hec : coordinateEquiv d y = A.chart q x := by
+    simp only [y,e,ComplexAtlas.euclideanChart_apply,ContinuousLinearEquiv.apply_symm_apply]
+  have hf : (f : EC d → ℂ) =ᶠ[𝓝 y] Smooth.ofReal φ ∘ e.symm :=
+    localize_cutoff_nhds q (D.regularityOuter p) (Smooth.ofReal φ) ht
+      (D.regularityOuter_inner p hκ)
+  have hH : NonlinearHessian.hessianMatrix f y = φ.hessian q (A.chart q x) := by
+    rw [← hec]
+    exact (hessianMatrix_source_nhds q φ f ht hf).eq_of_nhds
+  have hck : NonlinearHessian.complexCutoff κ = (D.cutoff p).realPart.val := by
+    ext z
+    rfl
+  have hHψ : NonlinearHessian.hessianMatrix
+      (SchwartzMap.smulLeftCLM ℂ (NonlinearHessian.complexCutoff κ) (∂_{v} f)) y =
+      ψ.hessian q (A.chart q x) := by
+    rw [hck,← hec]
+    ext i j
+    apply hessianEntrySchwartz_source q ψ _ ht
+    rw [hψ]
+    exact (D.localizedDerivative_local_regularity p v (Smooth.ofReal φ) ht).symm
+  have hG : DifferentiableAt ℝ G y :=
+    ((metric_euclidean_smooth g q).contDiffAt (e.open_target.mem_nhds ht)).differentiableAt (by simp)
+  have hB : DifferentiableAt ℝ B y :=
+    ((inverseDet_euclidean_smooth g q).contDiffAt (e.open_target.mem_nhds ht)).differentiableAt (by simp)
+  have hden := density_local_schwartz g q φ f ht hf
+  have hδden : D.localizedDerivative p v (Smooth.ofReal (g.potentialDensity φ)) x =
+      (κ y : ℂ) * fderiv ℝ (fun z => B z * (G z + NonlinearHessian.hessianMatrix f z).det) y v := by
+    rw [D.localizedDerivative_apply,ite_eq_left hx,hden.fderiv_eq]
+    rfl
+  have hM (i j : Fin d) : D.regularityMetric p i j x = G y i j := by
+    have hh := (D.regularityMetric_local p i j hκ).eq_of_nhds
+    change D.regularityMetric p i j (e.symm (e x)) = G y i j at hh
+    rwa [e.left_inv hx] at hh
+  have hBv : D.regularityInverseDet p x = B y := by
+    have hh := (D.regularityInverseDet_local p hκ).eq_of_nhds
+    change D.regularityInverseDet p (e.symm (e x)) = B y at hh
+    rwa [e.left_inv hx] at hh
+  have hDM (i j : Fin d) : D.localizedDerivative p v (D.regularityMetric p i j) x =
+      (κ y : ℂ) * (fderiv ℝ G y v i j) := by
+    rw [D.localizedDerivative_apply,ite_eq_left hx,(D.regularityMetric_local p i j hκ).fderiv_eq]
+    have hd : fderiv ℝ (fun z => G z i j) y v = fderiv ℝ G y v i j := by
+      rw [fderiv_apply (differentiableAt_pi.mp hG i),fderiv_apply hG]
+      rfl
+    change (κ y : ℂ) * fderiv ℝ (fun z => G z i j) y v = _
+    rw [hd]
+  have hDB : D.localizedDerivative p v (D.regularityInverseDet p) x =
+      (κ y : ℂ) * fderiv ℝ B y v := by
+    rw [D.localizedDerivative_apply,ite_eq_left hx,(D.regularityInverseDet_local p hκ).fderiv_eq]
+    rfl
+  rw [← hψ,D.completedVolumeDerivative_source k hk φ ψ q (by simpa [q] using hx),
+    D.nonlinearRemainder_strong_embed,hδden,hBv,hDB]
+  have hmat : (fun (i j : Fin d) => D.regularityMetric p i j x +
+      D.regularityHessian p i j (Smooth.ofReal φ) x) = G y + NonlinearHessian.hessianMatrix f y := by
+    ext i j
+    rw [hM,D.regularityHessian_source p φ i j hx hκ,hH]
+    rfl
+  have herr : (fun (i j : Fin d) => D.regularityError p v i j (Smooth.ofReal φ) x -
+      D.localizedDerivative p v (D.regularityMetric p i j) x) =
+      NonlinearHessian.cutoffHessianError κ v f y - κ y • fderiv ℝ G y v := by
+    ext i j
+    rw [D.regularityError_source p v (Smooth.ofReal φ) i j hx hκ,hDM]
+    change hessianLocalizedError (D.cutoff p).realPart.val v i j f y -
+      (κ y : ℂ) * fderiv ℝ G y v i j =
+      hessianLocalizedError (NonlinearHessian.complexCutoff κ) v i j f y -
+      κ y • (fderiv ℝ G y v i j)
+    rw [hck,Complex.real_smul]
+  rw [hmat,herr]
+  have hh := NonlinearHessian.determinant_cutoff_commutator κ v f B G y hB hG
+  rw [hHψ] at hh
+  simpa only [B,G,hec,hH,add_sub_assoc] using hh
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/NonlinearHessianCommutator.lean b/lean/OAI/Geometry/PrescribedPotential/NonlinearHessianCommutator.lean
new file mode 100644
index 0000000000000000000000000000000000000000..e9eff741909fb093ea526fd7252b10a9f13c4e59
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/NonlinearHessianCommutator.lean
@@ -0,0 +1,118 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.HessianCommutator
+import OAI.Geometry.PrescribedPotential.SobolevDeterminant
+
+namespace OAI
+
+/-! Nonlinear Hessian Commutator. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Matrix LineDeriv
+open scoped ContDiff SchwartzMap Classical Matrix.Norms.Elementwise
+namespace NonlinearHessian
+open GlobalElliptic Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ}
+local instance matrixContinuousSMul (n : ℕ) : ContinuousSMul ℝ (Matrix (Fin n) (Fin n) ℂ) :=
+  inferInstanceAs (ContinuousSMul ℝ (Fin n → Fin n → ℂ))
+
+lemma determinant_contDiff (n : ℕ) :
+    ContDiff ℝ ∞ (fun M : Matrix (Fin n) (Fin n) ℂ => M.det) := by
+  simp only [Matrix.det_apply']
+  apply ContDiff.sum
+  intro σ _
+  apply contDiff_const.mul
+  apply contDiff_prod
+  intro j _
+  fun_prop
+
+def detDifferential {n : ℕ} (M : Matrix (Fin n) (Fin n) ℂ) :
+    Matrix (Fin n) (Fin n) ℂ →L[ℝ] ℂ := fderiv ℝ Matrix.det M
+
+lemma detDifferential_hasFDeriv {n : ℕ} (M : Matrix (Fin n) (Fin n) ℂ) :
+    HasFDerivAt Matrix.det (detDifferential M) M :=
+  (determinant_contDiff n).differentiable (by simp) M |>.hasFDerivAt
+
+def hessianMatrix (f : 𝓢(EC d, ℂ)) (x : EC d) : Matrix (Fin d) (Fin d) ℂ :=
+  fun i j => hessianEntrySchwartz i j f x
+
+lemma hessianMatrix_contDiff (f : 𝓢(EC d, ℂ)) : ContDiff ℝ ∞ (hessianMatrix f) := by
+  apply contDiff_pi.mpr
+  intro i
+  apply contDiff_pi.mpr
+  intro j
+  exact (hessianEntrySchwartz i j f).smooth'
+
+lemma hessianMatrix_fderiv (f : 𝓢(EC d, ℂ)) (x v : EC d) (i j : Fin d) :
+    fderiv ℝ (hessianMatrix f) x v i j = (∂_{v} (hessianEntrySchwartz i j f)) x := by
+  rw [SchwartzMap.lineDerivOp_apply_eq_fderiv]
+  change fderiv ℝ (fun y : EC d => fun (i j : Fin d) => hessianEntrySchwartz i j f y) x v i j = _
+  have H : HasFDerivAt (fun y : EC d => fun (i j : Fin d) => hessianEntrySchwartz i j f y)
+      (ContinuousLinearMap.pi fun i => ContinuousLinearMap.pi fun j =>
+        fderiv ℝ (hessianEntrySchwartz i j f) x) x := by
+    apply hasFDerivAt_pi.mpr
+    intro i
+    apply hasFDerivAt_pi.mpr
+    intro j
+    exact (hessianEntrySchwartz i j f).smooth'.differentiable (by simp) x |>.hasFDerivAt
+  exact congrArg (fun L : EC d →L[ℝ] (Fin d → Fin d → ℂ) => L v i j) H.fderiv
+
+def complexCutoff (κ : 𝓢(EC d, ℝ)) : 𝓢(EC d, ℂ) :=
+  SchwartzMap.postcompCLM Complex.ofRealCLM κ
+
+@[simp] lemma complexCutoff_apply (κ : 𝓢(EC d, ℝ)) (x : EC d) :
+    complexCutoff κ x = (κ x : ℂ) := rfl
+
+def cutoffHessianError (κ : 𝓢(EC d, ℝ)) (v : EC d) (f : 𝓢(EC d, ℂ)) (x : EC d) :
+    Matrix (Fin d) (Fin d) ℂ := fun i j => hessianLocalizedError (complexCutoff κ) v i j f x
+
+lemma hessianMatrix_cutoff (κ : 𝓢(EC d, ℝ)) (v : EC d) (f : 𝓢(EC d, ℂ)) (x : EC d) :
+    hessianMatrix (SchwartzMap.smulLeftCLM ℂ (complexCutoff κ) (∂_{v} f)) x =
+      κ x • fderiv ℝ (hessianMatrix f) x v + cutoffHessianError κ v f x := by
+  ext i j
+  change hessianEntrySchwartz i j (SchwartzMap.smulLeftCLM ℂ (complexCutoff κ) (∂_{v} f)) x = _
+  rw [hessianLocalizedError_identity]
+  simp only [_root_.add_apply]
+  rw [SchwartzMap.smulLeftCLM_apply_apply (complexCutoff κ).hasTemperateGrowth]
+  change (κ x : ℂ) • (∂_{v} (hessianEntrySchwartz i j f)) x +
+    hessianLocalizedError (complexCutoff κ) v i j f x =
+      κ x • (fderiv ℝ (hessianMatrix f) x v i j) + cutoffHessianError κ v f x i j
+  rw [hessianMatrix_fderiv]
+  simp only [Complex.real_smul]
+  rfl
+
+ 
+
+lemma determinant_cutoff_commutator
+    (κ : 𝓢(EC d, ℝ)) (v : EC d) (f : 𝓢(EC d, ℂ))
+    (B : EC d → ℂ) (G : EC d → Matrix (Fin d) (Fin d) ℂ) (x : EC d)
+    (hB : DifferentiableAt ℝ B x) (hG : DifferentiableAt ℝ G x) :
+    B x * detDifferential (G x + hessianMatrix f x)
+        (hessianMatrix (SchwartzMap.smulLeftCLM ℂ (complexCutoff κ) (∂_{v} f)) x) =
+      (κ x : ℂ) * fderiv ℝ (fun y => B y * (G y + hessianMatrix f y).det) x v +
+      B x * detDifferential (G x + hessianMatrix f x)
+        (cutoffHessianError κ v f x - κ x • fderiv ℝ G x v) -
+      (κ x : ℂ) * fderiv ℝ B x v * (G x + hessianMatrix f x).det := by
+  have hH := (hessianMatrix_contDiff f).differentiable (by simp) x
+  have hd := (detDifferential_hasFDeriv (G x + hessianMatrix f x)).comp x
+    (hG.hasFDerivAt.add hH.hasFDerivAt)
+  have hq := (hB.hasFDerivAt.mul hd).fderiv
+  change fderiv ℝ (fun y => B y * (G y + hessianMatrix f y).det) x = _ at hq
+  have hqx : fderiv ℝ (fun y => B y * (G y + hessianMatrix f y).det) x v =
+      B x * detDifferential (G x + hessianMatrix f x)
+        (fderiv ℝ G x v + fderiv ℝ (hessianMatrix f) x v) +
+      (G x + hessianMatrix f x).det * fderiv ℝ B x v := by
+    rw [hq]
+    rfl
+  rw [hqx,hessianMatrix_cutoff]
+  simp only [map_add,map_sub,map_smul,Complex.real_smul]
+  ring
+end NonlinearHessian
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/NonlinearLocalCoordinates.lean b/lean/OAI/Geometry/PrescribedPotential/NonlinearLocalCoordinates.lean
new file mode 100644
index 0000000000000000000000000000000000000000..1cd7ddf27c0d80011c3322b312be8ece975fb94a
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/NonlinearLocalCoordinates.lean
@@ -0,0 +1,99 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.NonlinearHessianCommutator
+import OAI.Geometry.PrescribedPotential.RegularityCoordinates
+import OAI.Geometry.PrescribedPotential.NonlinearRemainder
+import OAI.Geometry.PrescribedPotential.PotentialDensity
+
+namespace OAI
+
+/-! Nonlinear Local Coordinates. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Matrix LineDeriv
+open scoped ContDiff SchwartzMap Classical Matrix.Norms.Elementwise
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+
+omit [T2Space X] [CompactSpace X] in
+lemma hessianEntrySchwartz_source (q : Fin A.count) (φ : SmoothRealFunction A)
+    (f : 𝓢(EC d, ℂ)) {y : EC d} (hy : y ∈ (A.euclideanChart q).target)
+    (he : (f : EC d → ℂ) =ᶠ[𝓝 y] Smooth.ofReal φ ∘ (A.euclideanChart q).symm)
+    (i j : Fin d) : hessianEntrySchwartz i j f y = φ.hessian q (coordinateEquiv d y) i j := by
+  let u : EC d → ℝ := φ.localExpression q ∘ coordinateEquiv d
+  have hu : ContDiffAt ℝ ∞ u y := (φ.euclidean_smooth q).contDiffAt
+    (by simpa only [ComplexAtlas.euclideanChart_target] using
+      (A.euclideanChart q).open_target.mem_nhds hy)
+  have hre : (Smooth.ofReal φ ∘ (A.euclideanChart q).symm) = Complex.ofRealCLM ∘ u := rfl
+  rw [hessianEntrySchwartz_real hu _ (he.trans (Filter.Eventually.of_forall (congr_fun hre))) i j,
+    pullBilin_hessian hu]
+  have hcomp : u ∘ (coordinateEquiv d).symm = φ.localExpression q := by
+    funext z
+    simp [u]
+  rw [hcomp]
+  rfl
+
+omit [T2Space X] [CompactSpace X] in
+lemma hessianMatrix_source_nhds (q : Fin A.count) (φ : SmoothRealFunction A)
+    (f : 𝓢(EC d, ℂ)) {y : EC d} (hy : y ∈ (A.euclideanChart q).target)
+    (he : (f : EC d → ℂ) =ᶠ[𝓝 y] Smooth.ofReal φ ∘ (A.euclideanChart q).symm) :
+    NonlinearHessian.hessianMatrix f =ᶠ[𝓝 y] (fun z => φ.hessian q (coordinateEquiv d z)) := by
+  filter_upwards [(A.euclideanChart q).open_target.mem_nhds hy,he.eventually_nhds] with z hz hez
+  ext i j
+  exact hessianEntrySchwartz_source q φ f hz hez i j
+
+omit [T2Space X] [CompactSpace X] in
+lemma density_local_schwartz (g : KaehlerMetric A) (q : Fin A.count) (φ : SmoothRealFunction A)
+    (f : 𝓢(EC d, ℂ)) {y : EC d} (hy : y ∈ (A.euclideanChart q).target)
+    (he : (f : EC d → ℂ) =ᶠ[𝓝 y] Smooth.ofReal φ ∘ (A.euclideanChart q).symm) :
+    (Smooth.ofReal (g.potentialDensity φ) ∘ (A.euclideanChart q).symm) =ᶠ[𝓝 y]
+      (fun z => (g.matrix q (coordinateEquiv d z)).det⁻¹ *
+        (g.matrix q (coordinateEquiv d z) + NonlinearHessian.hessianMatrix f z).det) := by
+  filter_upwards [(A.euclideanChart q).open_target.mem_nhds hy,
+    hessianMatrix_source_nhds q φ f hy he] with z hz hez
+  change ((g.potentialDensity φ).value ((A.euclideanChart q).symm z) : ℂ) = _
+  rw [g.potentialDensity_complex φ q (by simpa using ((A.euclideanChart q).symm.mapsTo hz))]
+  have hec : A.chart q ((A.euclideanChart q).symm z) = coordinateEquiv d z := by
+    have hh := congrArg (coordinateEquiv d) ((A.euclideanChart q).right_inv hz)
+    simpa only [ComplexAtlas.euclideanChart_apply, ContinuousLinearEquiv.apply_symm_apply] using hh
+  rw [KaehlerMetric.volumePolynomial,hec,hez,div_eq_mul_inv,mul_comm]
+
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+omit [CompactSpace X] in
+lemma regularityMetric_local (p : ι) (i j : Fin d) {y : EC d}
+    (hy : y ∈ tsupport (D.cutoff p : EC d → ℂ)) :
+    (D.regularityMetric p i j ∘ (A.euclideanChart (D.patch p).index).symm) =ᶠ[𝓝 y]
+      (fun z => g.matrix (D.patch p).index (coordinateEquiv d z) i j) := by
+  have ht := (D.cutoff p).support_sub hy
+  filter_upwards [chartFunction_local (D.patch p).index (D.regularityCutoff p)
+    (fun z => g.matrix (D.patch p).index (coordinateEquiv d z) i j) _ ht,
+    D.regularityCutoff_one p hy] with z hz he
+  change D.regularityMetric p i j ((A.euclideanChart (D.patch p).index).symm z) = _ at hz ⊢
+  rw [hz,he,one_mul]
+
+omit [CompactSpace X] in
+lemma regularityInverseDet_local (p : ι) {y : EC d}
+    (hy : y ∈ tsupport (D.cutoff p : EC d → ℂ)) :
+    (D.regularityInverseDet p ∘ (A.euclideanChart (D.patch p).index).symm) =ᶠ[𝓝 y]
+      (fun z => (g.matrix (D.patch p).index (coordinateEquiv d z)).det⁻¹) := by
+  have ht := (D.cutoff p).support_sub hy
+  filter_upwards [chartFunction_local (D.patch p).index (D.regularityCutoff p)
+    (fun z => (g.matrix (D.patch p).index (coordinateEquiv d z)).det⁻¹) _ ht,
+    D.regularityCutoff_one p hy] with z hz he
+  change D.regularityInverseDet p ((A.euclideanChart (D.patch p).index).symm z) = _ at hz ⊢
+  rw [hz,he,one_mul]
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/NonlinearRemainder.lean b/lean/OAI/Geometry/PrescribedPotential/NonlinearRemainder.lean
new file mode 100644
index 0000000000000000000000000000000000000000..243200130b9719640d36d1fceacd415992e2ac71
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/NonlinearRemainder.lean
@@ -0,0 +1,87 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CircleRadialCalculus
+import OAI.Geometry.PrescribedPotential.CompletedProduct
+import OAI.Geometry.PrescribedPotential.GlobalMetricEntries
+import OAI.Geometry.PrescribedPotential.RegularityCutoffs
+import OAI.Geometry.PrescribedPotential.RegularityJets
+import OAI.Geometry.PrescribedPotential.SobolevDetDifferential
+
+namespace OAI
+
+/-! Nonlinear Remainder. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff Classical Matrix.Norms.Elementwise
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+def regularityMetric (p : ι) (i j : Fin d) : Smooth A :=
+  chartFunction (D.patch p).index (D.regularityCutoff p)
+    (fun y => g.matrix (D.patch p).index (coordinateEquiv d y) i j)
+    (contDiffOn_pi.mp (contDiffOn_pi.mp (metric_euclidean_smooth g (D.patch p).index) i) j)
+
+def regularityInverseDet (p : ι) : Smooth A :=
+  chartFunction (D.patch p).index (D.regularityCutoff p)
+    (fun y => (g.matrix (D.patch p).index (coordinateEquiv d y)).det⁻¹)
+    (inverseDet_euclidean_smooth g (D.patch p).index)
+
+def regularityMatrix (k : ℕ) (p : ι) (u : D.localizers.Sobolev ((k : ℝ)+2)) :
+    Matrix (Fin d) (Fin d) (D.localizers.Sobolev (k : ℝ)) :=
+  fun i j => D.localizers.embed (k : ℝ) (D.regularityMetric p i j) +
+    D.completedRegularityHessian k p i j u
+
+def regularityErrorMatrix (k : ℕ) (p : ι) (v : EC d)
+    (u : D.localizers.Sobolev ((k : ℝ)+2)) :
+    Matrix (Fin d) (Fin d) (D.localizers.Sobolev (k : ℝ)) :=
+  fun i j => D.completedRegularityError k p v i j u -
+    D.localizers.embed (k : ℝ) (D.localizedDerivative p v (D.regularityMetric p i j))
+
+def nonlinearRemainder (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) (p : ι) (v : EC d)
+    (u : D.localizers.Sobolev ((k : ℝ)+2)) : D.localizers.Sobolev (k : ℝ) :=
+  D.product k hk (D.localizers.embed (k : ℝ) (D.regularityInverseDet p))
+    (D.detDifferential k hk d (D.regularityMatrix k p u) (D.regularityErrorMatrix k p v u)) -
+  D.product k hk (D.localizers.embed (k : ℝ)
+    (D.localizedDerivative p v (D.regularityInverseDet p)))
+    (D.determinant k hk d (D.regularityMatrix k p u))
+
+lemma regularityMatrix_contDiff (k : ℕ) (p : ι) : ContDiff ℝ ∞ (D.regularityMatrix k p) := by
+  apply contDiff_pi.mpr
+  intro i
+  apply contDiff_pi.mpr
+  intro j
+  exact contDiff_const.add (D.completedRegularityHessian k p i j).contDiff
+
+lemma regularityErrorMatrix_contDiff (k : ℕ) (p : ι) (v : EC d) :
+    ContDiff ℝ ∞ (D.regularityErrorMatrix k p v) := by
+  apply contDiff_pi.mpr
+  intro i
+  apply contDiff_pi.mpr
+  intro j
+  exact (D.completedRegularityError k p v i j).contDiff.sub contDiff_const
+
+lemma nonlinearRemainder_contDiff (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) (p : ι) (v : EC d) :
+    ContDiff ℝ ∞ (D.nonlinearRemainder k hk p v) := by
+  apply ContDiff.sub
+  · exact (D.product k hk (D.localizers.embed (k : ℝ) (D.regularityInverseDet p))).contDiff.comp
+      (((D.detDifferential_contDiff k hk d).comp (D.regularityMatrix_contDiff k p)).clm_apply
+        (D.regularityErrorMatrix_contDiff k p v))
+  · exact (D.product k hk (D.localizers.embed (k : ℝ)
+      (D.localizedDerivative p v (D.regularityInverseDet p)))).contDiff.comp
+      ((D.determinant_map_contDiff k hk d).comp (D.regularityMatrix_contDiff k p))
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/NonlinearRemainderStrong.lean b/lean/OAI/Geometry/PrescribedPotential/NonlinearRemainderStrong.lean
new file mode 100644
index 0000000000000000000000000000000000000000..99c096ae3e9c34a1ebe40a53f507726de607bd4b
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/NonlinearRemainderStrong.lean
@@ -0,0 +1,64 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.NonlinearHessianCommutator
+import OAI.Geometry.PrescribedPotential.NonlinearRemainder
+
+namespace OAI
+
+/-! Nonlinear Remainder Strong. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff Classical Matrix.Norms.Elementwise
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+lemma nonlinearRemainder_strong_embed (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (p : ι) (v : EC d) (f : Smooth A) (x : X) :
+    D.localizers.strong (k : ℝ)
+      (D.nonlinearRemainder k hk p v (D.localizers.embed ((k : ℝ)+2) f)) x =
+      D.regularityInverseDet p x *
+        NonlinearHessian.detDifferential
+          (fun i j => D.regularityMetric p i j x + D.regularityHessian p i j f x)
+          (fun i j => D.regularityError p v i j f x -
+            D.localizedDerivative p v (D.regularityMetric p i j) x) -
+      D.localizedDerivative p v (D.regularityInverseDet p) x *
+        Matrix.det (fun i j => D.regularityMetric p i j x + D.regularityHessian p i j f x) := by
+  have hs : (Module.finrank ℝ (EC d) : ℝ) < 2*(k : ℝ) := by
+    have hh : (Module.finrank ℝ (EC d) : ℝ) < (k : ℝ) := by exact_mod_cast hk
+    linarith [Nat.cast_nonneg (α := ℝ) k]
+  simp only [nonlinearRemainder,map_sub,BoundedContinuousFunction.sub_apply,
+    D.product_strong,BoundedContinuousFunction.mul_apply,D.localizers.strong_embed _ hs]
+  erw [D.detDifferential_strong,D.determinant_strong]
+  have hM : (fun (i j : Fin d) => D.localizers.strong (k : ℝ)
+      (D.regularityMatrix k p (D.localizers.embed ((k : ℝ)+2) f) i j) x) =
+      (fun i j => D.regularityMetric p i j x + D.regularityHessian p i j f x) := by
+    funext i j
+    simp only [regularityMatrix,D.completedRegularityHessian_embed,map_add,
+      BoundedContinuousFunction.add_apply,D.localizers.strong_embed _ hs]
+    rfl
+  have hE : (fun (i j : Fin d) => D.localizers.strong (k : ℝ)
+      (D.regularityErrorMatrix k p v (D.localizers.embed ((k : ℝ)+2) f) i j) x) =
+      (fun i j => D.regularityError p v i j f x -
+        D.localizedDerivative p v (D.regularityMetric p i j) x) := by
+    funext i j
+    simp only [regularityErrorMatrix,D.completedRegularityError_embed,map_sub,
+      BoundedContinuousFunction.sub_apply,D.localizers.strong_embed _ hs]
+    rfl
+  rw [hM,hE]
+  rfl
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/NonlinearScale.lean b/lean/OAI/Geometry/PrescribedPotential/NonlinearScale.lean
new file mode 100644
index 0000000000000000000000000000000000000000..552a3807b6882256a635917d66154a60bc95d34a
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/NonlinearScale.lean
@@ -0,0 +1,159 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.RealNonlinearDerivative
+import OAI.Geometry.PrescribedPotential.RealPoissonEquivalence
+
+namespace OAI
+
+/-! Nonlinear Scale. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology
+open scoped ContDiff Classical
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace Localizers
+variable (D : Localizers A ι)
+local instance scaleNG (s : ℝ) : NormedAddCommGroup (D.RealSobolev s) :=
+  (D.realCompletion s).normedAddCommGroup
+local instance scaleNS (s : ℝ) : NormedSpace ℝ (D.RealSobolev s) :=
+  (D.realCompletion s).normedSpace
+local instance scaleTG (s : ℝ) : IsTopologicalAddGroup (D.RealSobolev s) :=
+  Submodule.isTopologicalAddGroup _
+local instance scaleCS (s : ℝ) : ContinuousSMul ℝ (D.RealSobolev s) :=
+  SMulMemClass.continuousSMul _
+
+def realLower (s t : ℝ) (hst : t ≤ s) : D.RealSobolev s →L[ℝ] D.RealSobolev t :=
+  ((D.lower s t).comp (D.realCompletion s).subtypeL).codRestrict (D.realCompletion t)
+    (fun u => D.maps_realCompletion (D.lower s t)
+      (fun f => ⟨f,D.lower_embed hst f.val⟩) u.val u.property)
+
+lemma realLower_embed {s t : ℝ} (hst : t ≤ s) (f : RealSmooth A) :
+    D.realLower s t hst (D.realEmbed s f) = D.realEmbed t f := by
+  apply Subtype.ext
+  exact D.lower_embed hst f.val
+
+lemma realLower_injective {s t : ℝ} (hst : t ≤ s) :
+    Function.Injective (D.realLower s t hst) := by
+  intro u v h
+  apply Subtype.ext
+  exact D.lower_injective hst (congrArg Subtype.val h)
+
+end Localizers
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+local instance nonlinearScaleNG (s : ℝ) : NormedAddCommGroup (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedAddCommGroup
+local instance nonlinearScaleNS (s : ℝ) : NormedSpace ℝ (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedSpace
+local instance nonlinearScaleTG (s : ℝ) : IsTopologicalAddGroup (D.localizers.RealSobolev s) :=
+  Submodule.isTopologicalAddGroup _
+local instance nonlinearScaleCS (s : ℝ) : ContinuousSMul ℝ (D.localizers.RealSobolev s) :=
+  SMulMemClass.continuousSMul _
+
+lemma realVolume_embed (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) (f : RealSmooth A) :
+    D.realVolume k hk (D.localizers.realEmbed ((k : ℝ)+2) f) =
+      D.localizers.realEmbed (k : ℝ) (RealSmooth.ofReal (g.potentialDensity f.source)) := by
+  apply Subtype.ext
+  change D.completedVolume k hk (D.localizers.embed ((k : ℝ)+2) f.val) = _
+  rw [← f.ofReal_source,D.completedVolume_embed k hk]
+  rfl
+
+lemma realVolume_lower (k l : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (hl : Module.finrank ℝ (EC d) < l) (hlk : l ≤ k)
+    (u : D.localizers.RealSobolev ((k : ℝ)+2)) :
+    D.realVolume l hl (D.localizers.realLower ((k : ℝ)+2) ((l : ℝ)+2)
+      (by exact_mod_cast Nat.add_le_add_right hlk 2) u) =
+    D.localizers.realLower (k : ℝ) (l : ℝ) (by exact_mod_cast hlk) (D.realVolume k hk u) := by
+  have hst : (l : ℝ)+2 ≤ (k : ℝ)+2 := by exact_mod_cast Nat.add_le_add_right hlk 2
+  have hkl : (l : ℝ) ≤ (k : ℝ) := by exact_mod_cast hlk
+  have he : (D.realVolume l hl) ∘ D.localizers.realLower ((k : ℝ)+2) ((l : ℝ)+2) hst =
+      D.localizers.realLower (k : ℝ) (l : ℝ) hkl ∘ D.realVolume k hk := by
+    apply (D.localizers.realEmbed_dense ((k : ℝ)+2)).equalizer
+      ((D.realVolume_continuous l hl).comp (D.localizers.realLower _ _ hst).continuous)
+      ((D.localizers.realLower _ _ hkl).continuous.comp (D.realVolume_continuous k hk))
+    funext f
+    simp only [Function.comp_apply,Localizers.realLower_embed,realVolume_embed]
+  exact congr_fun he u
+
+variable [ConnectedSpace X]
+omit [ConnectedSpace X] in
+lemma realVolumeDerivative_lower (k l : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (hl : Module.finrank ℝ (EC d) < l) (hlk : l ≤ k)
+    (u : D.localizers.RealSobolev ((k : ℝ)+2)) :
+    (D.realVolumeDerivative l hl (D.localizers.realLower ((k : ℝ)+2) ((l : ℝ)+2)
+        (by exact_mod_cast Nat.add_le_add_right hlk 2) u)).comp
+      (D.localizers.realLower ((k : ℝ)+2) ((l : ℝ)+2)
+        (by exact_mod_cast Nat.add_le_add_right hlk 2)) =
+    (D.localizers.realLower (k : ℝ) (l : ℝ) (by exact_mod_cast hlk)).comp
+      (D.realVolumeDerivative k hk u) := by
+  let L := D.localizers.realLower ((k : ℝ)+2) ((l : ℝ)+2)
+    (by exact_mod_cast Nat.add_le_add_right hlk 2)
+  let M := D.localizers.realLower (k : ℝ) (l : ℝ) (by exact_mod_cast hlk)
+  have h₁ := (D.realVolume_hasStrictFDerivAt l hl (L u)).hasFDerivAt.comp u L.hasFDerivAt
+  have h₂ := M.hasFDerivAt.comp u (D.realVolume_hasStrictFDerivAt k hk u).hasFDerivAt
+  have hf : D.realVolume l hl ∘ L = M ∘ D.realVolume k hk :=
+    funext (D.realVolume_lower k l hk hl hlk)
+  rw [hf] at h₁
+  exact h₁.unique h₂
+
+def linearizedVolume (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) (x₀ : X)
+    (u : D.localizers.RealSobolev ((k : ℝ)+2)) :
+    (D.localizers.RealSobolev ((k : ℝ)+2) × ℝ) →L[ℝ]
+      (D.localizers.RealSobolev (k : ℝ) × ℝ) :=
+  ((D.realVolumeDerivative k hk u ∘L ContinuousLinearMap.fst ℝ _ ℝ) -
+    (D.realConstants (k : ℝ) ∘L ContinuousLinearMap.snd ℝ _ ℝ)).prod
+    (D.realEvaluation ((k : ℝ)+2) x₀ ∘L ContinuousLinearMap.fst ℝ _ ℝ)
+
+omit [ConnectedSpace X] in
+lemma linearizedVolume_zero (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) (x₀ : X) :
+    D.linearizedVolume k hk x₀ 0 = D.realAugmentedL k x₀ := by
+  simp only [linearizedVolume,realVolumeDerivative_zero,realAugmentedL]
+
+omit [ConnectedSpace X] in
+lemma linearizedVolume_continuous (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) (x₀ : X) :
+    Continuous (D.linearizedVolume k hk x₀) := by
+  have hc : Continuous (fun u =>
+      (D.realVolumeDerivative k hk u ∘L ContinuousLinearMap.fst ℝ _ ℝ) -
+      (D.realConstants (k : ℝ) ∘L ContinuousLinearMap.snd ℝ _ ℝ)) := by
+    exact ((D.realVolumeDerivative_continuous k hk).clm_comp continuous_const).sub continuous_const
+  let : NormedAddCommGroup (D.localizers.RealSobolev ((k : ℝ)+2) × ℝ) := inferInstance
+  let : NormedSpace ℝ (D.localizers.RealSobolev ((k : ℝ)+2) × ℝ) := inferInstance
+  let : NormedAddCommGroup ((D.localizers.RealSobolev ((k : ℝ)+2) × ℝ) →L[ℝ]
+      D.localizers.RealSobolev (k : ℝ)) := inferInstance
+  let : NormedAddCommGroup ((D.localizers.RealSobolev ((k : ℝ)+2) × ℝ) →L[ℝ] ℝ) := inferInstance
+  exact (ContinuousLinearMap.prodₗᵢ (𝕜 := ℝ)
+    (E := D.localizers.RealSobolev ((k : ℝ)+2) × ℝ)
+    (F := D.localizers.RealSobolev (k : ℝ)) (G := ℝ) ℝ).continuous.comp
+      (hc.prodMk continuous_const)
+
+lemma linearizedVolume_invertible_nhds (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
+    (P : D.localizers.ConstantProjection) (k : ℕ)
+    (hk : Module.finrank ℝ (EC d) < k) (x₀ : X) :
+    ∀ᶠ u in 𝓝 (0 : D.localizers.RealSobolev ((k : ℝ)+2)),
+      ∃ e : (D.localizers.RealSobolev ((k : ℝ)+2) × ℝ) ≃L[ℝ]
+          (D.localizers.RealSobolev (k : ℝ) × ℝ),
+        (e : _ →L[ℝ] _) = D.linearizedVolume k hk x₀ u := by
+  have hke : (Module.finrank ℝ (EC d) : ℝ) < 2*((k : ℝ)+2) := by
+    have hh : (Module.finrank ℝ (EC d) : ℝ) < (k : ℝ) := by exact_mod_cast hk
+    linarith [Nat.cast_nonneg (α := ℝ) k]
+  let e := D.realPoissonEquiv m hm he P k hke x₀
+  have hz : (e : _ →L[ℝ] _) = D.linearizedVolume k hk x₀ 0 := by
+    rw [D.linearizedVolume_zero]
+    rfl
+  have hn := e.nhds
+  rw [hz] at hn
+  exact (D.linearizedVolume_continuous k hk x₀).continuousAt hn
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/NonlinearSobolevGain.lean b/lean/OAI/Geometry/PrescribedPotential/NonlinearSobolevGain.lean
new file mode 100644
index 0000000000000000000000000000000000000000..2581f3949a06700db4e03174f09e5b896627549c
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/NonlinearSobolevGain.lean
@@ -0,0 +1,116 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.PatchCutoffs
+import OAI.Geometry.PrescribedPotential.RealSobolev
+import OAI.Geometry.PrescribedPotential.CompletedRealification
+import OAI.Geometry.PrescribedPotential.DerivativeEllipticGain
+import OAI.Geometry.PrescribedPotential.SobolevFirstGain
+import OAI.Geometry.PrescribedPotential.WeakNonlinearCommutator
+
+namespace OAI
+
+/-! Nonlinear Sobolev Gain. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology
+open scoped ContDiff Classical
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace Localizers
+variable (L : Localizers A ι)
+lemma realLower_lower {s t r : ℝ} (hst : t ≤ s) (htr : r ≤ t) (u : L.RealSobolev s) :
+    L.realLower t r htr (L.realLower s t hst u) =
+      L.realLower s r (htr.trans hst) u := by
+  apply Subtype.ext
+  exact L.lower_lower hst htr u.val
+end Localizers
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+local instance nonlinearSobGainNG (s : ℝ) : NormedAddCommGroup (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedAddCommGroup
+local instance nonlinearSobGainNS (s : ℝ) : NormedSpace ℝ (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedSpace
+local instance nonlinearSobGainTG (s : ℝ) : IsTopologicalAddGroup (D.localizers.RealSobolev s) :=
+  Submodule.isTopologicalAddGroup _
+local instance nonlinearSobGainCS (s : ℝ) : ContinuousSMul ℝ (D.localizers.RealSobolev s) :=
+  SMulMemClass.continuousSMul _
+
+lemma real_sobolev_first_gain (k : ℕ) (u : D.localizers.RealSobolev ((k : ℝ)+1))
+    (hD : ∀ (p : ι) (v : EC d), ∃ w : D.localizers.RealSobolev ((k : ℝ)+1),
+      D.localizers.realLower ((k : ℝ)+1) (k : ℝ) (by linarith) w =
+        D.realCompletedDerivative k p v u) :
+    ∃ w : D.localizers.RealSobolev ((k+2 : ℕ) : ℝ),
+      D.localizers.realLower ((k+2 : ℕ) : ℝ) ((k : ℝ)+1) (by push_cast; linarith) w = u := by
+  have hc (p : ι) (v : EC d) : ∃ w : D.localizers.Sobolev ((k : ℝ)+1),
+      D.localizers.lower ((k : ℝ)+1) (k : ℝ) w = D.completedDerivative k p v u.val := by
+    obtain ⟨w,hw⟩ := hD p v
+    exact ⟨w.val,congrArg Subtype.val hw⟩
+  obtain ⟨w,hw⟩ := D.sobolev_first_gain k u.val hc
+  have hm : w ∈ D.localizers.realCompletion ((k+2 : ℕ) : ℝ) := by
+    apply D.mem_real_of_lower_eq (k+2) (k+1) ((k : ℝ)+1) (by push_cast; rfl) (by omega)
+    have hh : D.localizers.lower ((k+2 : ℕ) : ℝ) ((k : ℝ)+1) w ∈
+        D.localizers.realCompletion ((k : ℝ)+1) := hw.symm ▸ u.property
+    exact hh
+  exact ⟨⟨w,hm⟩,Subtype.ext hw⟩
+
+lemma derivative_lifts_reindex (k : ℕ)
+    (u : D.localizers.RealSobolev (((k+1 : ℕ) : ℝ)+2))
+    (hD : ∀ (p : ι) (v : EC d), ∃ w : D.localizers.RealSobolev (((k+1 : ℕ) : ℝ)+2),
+      D.localizers.realLower (((k+1 : ℕ) : ℝ)+2) ((k : ℝ)+2) (by push_cast; linarith) w =
+        D.potentialDerivative k p v u) :
+    ∀ (p : ι) (v : EC d), ∃ w : D.localizers.RealSobolev (((k+2 : ℕ) : ℝ)+1),
+      D.localizers.realLower (((k+2 : ℕ) : ℝ)+1) ((k+2 : ℕ) : ℝ) (by linarith) w =
+        D.realCompletedDerivative (k+2) p v
+          (D.localizers.realLower (((k+1 : ℕ) : ℝ)+2) (((k+2 : ℕ) : ℝ)+1)
+            (by push_cast; linarith) u) := by
+  intro p v
+  obtain ⟨w,hw⟩ := hD p v
+  refine ⟨D.localizers.realLower (((k+1 : ℕ) : ℝ)+2) (((k+2 : ℕ) : ℝ)+1)
+    (by push_cast; linarith) w,?_⟩
+  apply D.localizers.realLower_injective (by norm_num : (k : ℝ)+2 ≤ ((k+2 : ℕ) : ℝ))
+  rw [Localizers.realLower_lower,Localizers.realLower_lower]
+  exact hw
+
+lemma derivative_lifts_gain (k : ℕ)
+    (u : D.localizers.RealSobolev (((k+1 : ℕ) : ℝ)+2))
+    (hD : ∀ (p : ι) (v : EC d), ∃ w : D.localizers.RealSobolev (((k+1 : ℕ) : ℝ)+2),
+      D.localizers.realLower (((k+1 : ℕ) : ℝ)+2) ((k : ℝ)+2) (by push_cast; linarith) w =
+        D.potentialDerivative k p v u) :
+    ∃ w : D.localizers.RealSobolev ((k+4 : ℕ) : ℝ),
+      D.localizers.realLower ((k+4 : ℕ) : ℝ) (((k+1 : ℕ) : ℝ)+2) (by push_cast; linarith) w = u := by
+  let u' := D.localizers.realLower (((k+1 : ℕ) : ℝ)+2) (((k+2 : ℕ) : ℝ)+1)
+    (by push_cast; linarith) u
+  obtain ⟨w,hw⟩ := D.real_sobolev_first_gain (k+2) u'
+    (D.derivative_lifts_reindex k u hD)
+  refine ⟨D.localizers.realLower ((k+2+2 : ℕ) : ℝ) ((k+4 : ℕ) : ℝ) (by push_cast; linarith) w,?_⟩
+  apply D.localizers.realLower_injective
+    (by push_cast; linarith : ((k+2 : ℕ) : ℝ)+1 ≤ ((k+1 : ℕ) : ℝ)+2)
+  rw [Localizers.realLower_lower,Localizers.realLower_lower]
+  exact hw
+
+variable [ConnectedSpace X]
+ 
+
+theorem nonlinear_sobolev_gain (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) (x₀ : X) :
+    ∀ᶠ u in 𝓝 (0 : D.localizers.RealSobolev (((k+1 : ℕ) : ℝ)+2)),
+      ∀ (F : RealSmooth A),
+        D.realVolume (k+1) (by omega) u = D.localizers.realEmbed ((k+1 : ℕ) : ℝ) F →
+      ∃ w : D.localizers.RealSobolev ((k+4 : ℕ) : ℝ),
+        D.localizers.realLower ((k+4 : ℕ) : ℝ) (((k+1 : ℕ) : ℝ)+2)
+          (by push_cast; linarith) w = u := by
+  filter_upwards [D.nonlinear_first_derivative_gain k hk x₀] with u hu
+  intro F hF
+  exact D.derivative_lifts_gain k u (hu F hF)
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/NormalizedPathCompactness.lean b/lean/OAI/Geometry/PrescribedPotential/NormalizedPathCompactness.lean
new file mode 100644
index 0000000000000000000000000000000000000000..775437ee118d4e765a993759263cf8307a35b266
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/NormalizedPathCompactness.lean
@@ -0,0 +1,94 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.PatchCutoffs
+import OAI.Geometry.PrescribedPotential.RealSmoothCompactness
+import OAI.Geometry.PrescribedPotential.RealSobolev
+import OAI.Geometry.PrescribedPotential.SmoothPathLimit
+import OAI.Geometry.PrescribedPotential.VolumeNormalization
+
+namespace OAI
+
+/-! Normalized Path Compactness. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology
+open scoped ContDiff Classical
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace RealSmooth
+
+omit [T2Space X] [CompactSpace X] in
+@[simp] lemma source_ofReal (φ : SmoothRealFunction A) : (ofReal φ).source = φ := by
+  cases φ
+  rfl
+
+end RealSmooth
+namespace Localizers
+variable (D : Localizers A ι)
+
+lemma real_source_value_tendsto (f : ℕ → RealSmooth A) (fLim : RealSmooth A)
+    (hf : ∀ s : ℝ, Tendsto (fun n => D.embed s (f n).val) atTop
+      (𝓝 (D.embed s fLim.val))) (x : X) :
+    Tendsto (fun n => (f n).source.value x) atTop (𝓝 (fLim.source.value x)) := by
+  let s : ℝ := (Module.finrank ℝ (EC d) : ℝ)+1
+  have hs : (Module.finrank ℝ (EC d) : ℝ) < 2*s := by
+    dsimp only [s]
+    linarith [Nat.cast_nonneg (α := ℝ) (Module.finrank ℝ (EC d))]
+  exact (Complex.continuous_re.tendsto (fLim.val x)).comp
+    (D.smooth_value_tendsto s hs (fun n => (f n).val) fLim.val (hf s) x)
+
+end Localizers
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+ 
+
+theorem normalized_path_sequence_compact (h : SemipositiveAnticanonicalMetric A)
+    (x₀ : X) (t b : ℕ → ℝ) (tLim : ℝ) (ht : Tendsto t atTop (𝓝 tLim))
+    (hinterval : ∀ n, t n ∈ Icc 0 1) (f : ℕ → RealSmooth A)
+    (hnorm : ∀ n, (f n).source.value x₀ = 0)
+    (hsol : ∀ n, SolvesVolumePath g h (t n) (f n).source (b n))
+    (hbound : ∀ k : ℕ, ∃ C : ℝ, ∀ n : ℕ,
+      ‖D.localizers.embed (((2*k : ℕ) : ℝ)+2) (f n).val‖ ≤ C) :
+    ∃ (fLim : RealSmooth A) (bLim : ℝ) (r : ℕ → ℕ), StrictMono r ∧
+      Tendsto (b ∘ r) atTop (𝓝 bLim) ∧
+      (∀ s : ℝ, Tendsto (fun n => D.localizers.embed s (f (r n)).val) atTop
+        (𝓝 (D.localizers.embed s fLim.val))) ∧
+      fLim.source.value x₀ = 0 ∧ SolvesVolumePath g h tLim fLim.source bLim := by
+  let : Nonempty X := ⟨x₀⟩
+  let C : ℝ := ‖(⟨(prescribedForcing g h).value,
+    (prescribedForcing g h).continuous⟩ : C(X,ℝ))‖
+  have hb (n : ℕ) : b n ∈ Icc (-C) C :=
+    abs_le.mp (volumePath_scalar_bound g h (hinterval n) (hsol n))
+  obtain ⟨bLim,_,r₀,hr₀,hb₀⟩ := (isCompact_Icc : IsCompact (Icc (-C) C)).tendsto_subseq hb
+  have hbound' : ∀ k : ℕ, ∃ C : ℝ, ∀ n : ℕ,
+      ‖D.localizers.embed (((2*k : ℕ) : ℝ)+2) (f (r₀ n)).val‖ ≤ C := by
+    intro k
+    obtain ⟨K,hK⟩ := hbound k
+    exact ⟨K,fun n => hK (r₀ n)⟩
+  obtain ⟨fLim,r₁,hr₁,hf⟩ := D.real_smooth_sequence_compact (fun n => f (r₀ n)) hbound'
+  let r := r₀ ∘ r₁
+  have hr : StrictMono r := hr₀.comp hr₁
+  have hbLim : Tendsto (b ∘ r) atTop (𝓝 bLim) := hb₀.comp hr₁.tendsto_atTop
+  have hnLim : fLim.source.value x₀ = 0 := by
+    have hv := D.localizers.real_source_value_tendsto (fun n => f (r n)) fLim hf x₀
+    have he : (fun n => (f (r n)).source.value x₀) = (fun _ : ℕ => (0 : ℝ)) :=
+      funext (fun n => hnorm (r n))
+    rw [he] at hv
+    exact tendsto_nhds_unique hv tendsto_const_nhds
+  refine ⟨fLim,bLim,r,hr,hbLim,hf,hnLim,?_⟩
+  exact D.smooth_path_limit_solution h (t ∘ r) (b ∘ r) tLim bLim
+    (ht.comp hr.tendsto_atTop) hbLim (fun n => f (r n)) fLim hf (fun n => hsol (r n))
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/NormalizedPoisson.lean b/lean/OAI/Geometry/PrescribedPotential/NormalizedPoisson.lean
new file mode 100644
index 0000000000000000000000000000000000000000..e7f0d29a36fc377d41c25b2a176dc1d8ea89b308
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/NormalizedPoisson.lean
@@ -0,0 +1,94 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CompactAlternative
+import OAI.Geometry.PrescribedPotential.ComplexKernel
+import OAI.Geometry.PrescribedPotential.PatchCutoffs
+import OAI.Geometry.PrescribedPotential.FredholmKernel
+import OAI.Geometry.PrescribedPotential.GlobalRellich
+
+namespace OAI
+
+/-! Normalized Poisson. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory
+open scoped SchwartzMap ContDiff Classical
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  [ConnectedSpace X] {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+omit [ConnectedSpace X] in
+lemma resolventZero_compact (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1) :
+    IsCompactOperator (D.resolventZero m hm he) :=
+  D.lower_compact.comp_clm (D.completedResolvent m hm he)
+
+omit [ConnectedSpace X] in
+lemma fredholmOperator_compact (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
+    (P : D.localizers.ConstantProjection) : IsCompactOperator (D.fredholmOperator m hm he P) :=
+  ((D.resolventZero_compact m hm he).smul (m^2 : ℝ)).sub P.compact
+
+lemma fredholm_surjective (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
+    (P : D.localizers.ConstantProjection) :
+    Function.Surjective (1 - D.fredholmOperator m hm he P :
+      D.localizers.Sobolev 0 →L[ℝ] D.localizers.Sobolev 0) :=
+  compact_one_sub_surjective _ (D.fredholmOperator_compact m hm he P) (D.fredholm_kernel m hm he P)
+
+include D in
+theorem exists_complex_poisson (F : Smooth A) :
+    ∃ (u : Smooth A) (c : ℂ), complexL g u = F + Smooth.const c := by
+  obtain ⟨m, hm, he⟩ := D.exists_completedError_small
+  obtain ⟨P⟩ := D.localizers.constantProjection_exists
+  obtain ⟨w, hw⟩ := D.fredholm_surjective m hm he P (-D.localizers.embed 0 F)
+  obtain ⟨c, hc⟩ := P.value w
+  have hw' : w = D.localizers.embed 0 (-F - Smooth.const c) +
+      (m^2 : ℝ) • D.resolventZero m hm he w := by
+    change w - ((m^2 : ℝ) • D.resolventZero m hm he w - P.proj w) = -D.localizers.embed 0 F at hw
+    calc
+      w = -D.localizers.embed 0 F +
+          ((m^2 : ℝ) • D.resolventZero m hm he w - P.proj w) := sub_eq_iff_eq_add.mp hw
+      _ = _ := by rw [hc, map_sub, map_neg]; abel
+  obtain ⟨u, hu⟩ := D.allRegular_smooth (D.resolvent_feedback_regular m hm he w _ hw')
+  refine ⟨u, c, ?_⟩
+  have hs := (D.smooth_shifted_equation m hm he w u hu).trans hw'
+  rw [← hu, ← map_smul, ← map_add] at hs
+  have heq := D.localizers.embed_injective 0 hs
+  apply sub_right_inj.mp
+  calc
+    (m^2 : ℝ) • u - complexL g u = -F - Smooth.const c + (m^2 : ℝ) • u := heq
+    _ = (m^2 : ℝ) • u - (F + Smooth.const c) := by abel
+
+end GluingData
+end GlobalElliptic
+
+namespace Anticanonical.SourceSmooth
+open GlobalElliptic
+variable (d : ℕ) (X : Type*) [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  [ConnectedSpace X] (A : ComplexAtlas d X)
+
+ 
+
+theorem normalizedPoisson (g : KaehlerMetric A) (x₀ : X) (f : SmoothRealFunction A) :
+    HasNormalizedPoissonSolution g x₀ f := by
+  obtain ⟨S, ⟨D⟩⟩ := exists_gluingData g
+  obtain ⟨u, c, hu⟩ := D.exists_complex_poisson (Smooth.ofReal f)
+  refine ⟨(u.part Complex.reCLM).normalizeAt x₀, -c.re, SmoothRealFunction.normalizeAt_value _ _, ?_⟩
+  intro i z hz
+  rw [KaehlerMetric.linearizedMongeAmpere_normalizeAt,
+    ← g.laplacian_localExpression _ i hz]
+  change (g.laplacian (u.part Complex.reCLM)).value ((A.chart i).symm z) + -c.re = _
+  rw [← complexL_re, hu, Smooth.add_apply, Complex.add_re, Smooth.ofReal_apply, Complex.ofReal_re]
+  change f.localExpression i z + c.re + -c.re = f.localExpression i z
+  ring
+
+end Anticanonical.SourceSmooth
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/NormalizedPotentialL1.lean b/lean/OAI/Geometry/PrescribedPotential/NormalizedPotentialL1.lean
new file mode 100644
index 0000000000000000000000000000000000000000..75d5e230c4c01ea714999e307d8f90c7c4392ba8
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/NormalizedPotentialL1.lean
@@ -0,0 +1,92 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CompactIntegralPropagation
+import OAI.Geometry.PrescribedPotential.CoordinateBalls
+import OAI.Geometry.PrescribedPotential.CoordinatePotential
+import OAI.Geometry.PrescribedPotential.QuasiPSHLocalMean
+
+namespace OAI
+
+/-! Normalized Potential L1. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Metric Filter Topology MeasureTheory
+open scoped ContDiff InnerProductSpace
+namespace Anticanonical.SourceSmooth
+open EllipticKernel PotentialABP
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+local instance normalizedL1RealIP (d : ℕ) : InnerProductSpace ℝ (EC d) :=
+  InnerProductSpace.rclikeToReal ℂ (EC d)
+namespace CoordinateBall
+variable [MeasurableSpace X] [BorelSpace X] (p : CoordinateBall A)
+
+lemma local_neg_integral (g : KaehlerMetric A) : ∃ B : ℝ, 0 ≤ B ∧
+    ∀ φ : SmoothRealFunction A, g.PositivePotential φ → (∀ x, φ.value x ≤ 0) →
+    ∀ y ∈ p.source, (∫ x, -φ.value x ∂p.measure) ≤ B * (-φ.value y + 1) := by
+  obtain ⟨C, hC, hCb⟩ := p.metric_bound g
+  let V := volume.real (ball (0 : EC d) (2*p.radius))
+  let T := 9*C*p.radius^2
+  have hV : 0 ≤ V := ENNReal.toReal_nonneg
+  have hT : 0 ≤ T := by positivity
+  refine ⟨V*(1+T), mul_nonneg hV (by positivity), ?_⟩
+  intro φ hφ hneg y hy
+  have he (z) (hz : z ∈ closedBall p.center (3*p.radius)) := p.extension_eq φ hz
+  have hloc := quasiPSH_local_integral (p.extension_smooth φ) p.center p.radius_pos hC
+    (fun z hz => by rw [he z hz]; exact hneg _) (fun z hz v => p.extension_levi_lower g φ hφ hCb hz v)
+    (y := A.euclideanChart p.index y) hy.2
+  have hey : p.extension φ (A.euclideanChart p.index y) = φ.value y := by
+    rw [he _ (closedBall_subset_closedBall (by linarith [p.radius_pos])
+      (ball_subset_closedBall hy.2)), (A.euclideanChart p.index).left_inv hy.1]
+  have hei : (∫ z in ball p.center p.radius, (0:ℝ)-p.extension φ z) =
+      ∫ z in ball p.center p.radius, -φ.value ((A.euclideanChart p.index).symm z) := by
+    apply setIntegral_congr_fun isOpen_ball.measurableSet
+    intro z hz
+    dsimp only
+    rw [he z (closedBall_subset_closedBall (by linarith [p.radius_pos])
+      (ball_subset_closedBall hz)), zero_sub]
+  rw [hei, hey, zero_sub] at hloc
+  rw [p.integral_measure (fun x => -φ.value x) φ.continuous.neg]
+  apply hloc.trans
+  change V * (-φ.value y + T) ≤ V*(1+T)*(-φ.value y+1)
+  have hn : 0 ≤ -φ.value y := neg_nonneg.mpr (hneg y)
+  nlinarith [mul_nonneg hT hn, mul_nonneg hV (mul_nonneg hT hn)]
+end CoordinateBall
+
+ 
+
+theorem normalized_potential_L1 [CompactSpace X] [ConnectedSpace X]
+    [MeasurableSpace X] [BorelSpace X]
+    (g : KaehlerMetric A) (S : Finset (CoordinateBall A))
+    (hS : ∀ x, ∃ p ∈ S, x ∈ p.source) :
+    ∃ B : ℝ, ∀ φ : SmoothRealFunction A,
+      g.PositivePotential φ → (∀ x, φ.value x ≤ 0) → (∃ x, φ.value x = 0) →
+      ∀ p ∈ S, (∫ x, -φ.value x ∂p.measure) ≤ B := by
+  classical
+  let U : S → Set X := fun p => p.val.source
+  let μ : S → Measure X := fun p => p.val.measure
+  let F : (X → ℝ) → Prop := fun u => ∃ φ : SmoothRealFunction A,
+    g.PositivePotential φ ∧ (∀ x, φ.value x ≤ 0) ∧ u = fun x => -φ.value x
+  have hlocal := fun p : S => p.val.local_neg_integral g
+  choose C hC hc using hlocal
+  obtain ⟨B, hB⟩ := finite_cover_integral_bound U
+    (fun p => p.val.isOpen_source) (fun p => p.val.source_nonempty)
+    (fun x => by obtain ⟨p, hp, hx⟩ := hS x; exact ⟨⟨p, hp⟩, hx⟩) μ
+    (fun p q hn => p.val.measure_open_pos q.val.isOpen_source hn) F
+    (by rintro u ⟨φ, _, hn, rfl⟩ x; exact neg_nonneg.mpr (hn x))
+    (by
+      rintro u ⟨φ, _, _, rfl⟩ p
+      exact φ.continuous.neg.integrable_of_hasCompactSupport
+        (HasCompactSupport.of_compactSpace _)) C hC
+    (by rintro u ⟨φ, hφ, hn, rfl⟩ p y hy; exact hc p φ hφ hn y hy)
+  refine ⟨B, ?_⟩
+  intro φ hφ hn hz p hp
+  exact hB _ ⟨φ, hφ, hn, rfl⟩ (by obtain ⟨x, hx⟩ := hz; exact ⟨x, by simp [hx]⟩) ⟨p, hp⟩
+end Anticanonical.SourceSmooth
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/ParameterCoreBounds.lean b/lean/OAI/Geometry/PrescribedPotential/ParameterCoreBounds.lean
new file mode 100644
index 0000000000000000000000000000000000000000..92e7bd80eb6a21c39a75158571aba265b3beb311
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/ParameterCoreBounds.lean
@@ -0,0 +1,141 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CoreBounds
+import OAI.Geometry.PrescribedPotential.ParameterSchwartz
+
+namespace OAI
+
+/-! Parameter Core Bounds. -/
+
+section
+
+ 
+
+noncomputable section
+open Filter Topology MeasureTheory FourierTransform TemperedDistribution LineDeriv
+open scoped SchwartzMap BoundedContinuousFunction ContDiff Real ComplexOrder MatrixOrder Classical
+
+namespace SobolevChart
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+
+lemma CoreBound.list_sum {α : Type*} {s t : ℝ} (I : List α)
+    (T : α → 𝓢(E, ℂ) → 𝓢(E, ℂ)) (hh : ∀ i ∈ I, CoreBound s t (T i)) :
+    CoreBound s t (fun f => (I.map (fun i => T i f)).sum) := by
+  induction I with
+  | nil => exact ⟨0, le_rfl, by simp [schwartzCoord_zero]⟩
+  | cons i I ih =>
+    exact (hh i (List.mem_cons_self ..)).add (ih (fun j hj => hh j (List.mem_cons_of_mem _ hj)))
+
+lemma coreBound_perturbation {ι : Type*} [Fintype ι] (v : ι → E)
+    (a : SmoothCoefficients ι E) : CoreBound 2 0 (smoothPerturbation v a) :=
+  ⟨perturbationBound v (coefficientBCF a), perturbationBound_nonneg v _, smoothPerturbation_bound v a⟩
+
+ 
+lemma coreBound_of_words_base (k : ℕ) {s r : ℝ} (T : 𝓢(E, ℂ) → 𝓢(E, ℂ))
+    (h : ∀ ws : List E, ws.length ≤ 2 * k → CoreBound s r (fun f => schwartzWord ws (T f))) :
+    CoreBound s (r + 2 * (k : ℝ)) T := by
+  induction k generalizing T with
+  | zero => simpa [schwartzWord] using h [] (by simp)
+  | succ k ih =>
+    have ht := ih T (fun ws hw => h ws (by omega))
+    have hd (j : Fin (Module.finrank ℝ E)) := ih
+      (fun f => ∂_{stdOrthonormalBasis ℝ E j} (∂_{stdOrthonormalBasis ℝ E j} (T f)))
+      (fun ws hw => by
+        have hh := h (ws ++ [stdOrthonormalBasis ℝ E j, stdOrthonormalBasis ℝ E j])
+          (by simp only [List.length_append, List.length_cons, List.length_nil] ; omega)
+        simpa only [schwartzWord_append, schwartzWord, ContinuousLinearMap.comp_apply,
+          ContinuousLinearMap.id_apply, lineDerivOpCLM_apply] using hh)
+    convert coreBound_raise_two T ht hd using 1; push_cast; ring
+
+end SobolevChart
+
+namespace FrozenPoisson.ParameterRegularity
+open FrozenPoisson SobolevChart EllipticKernel
+variable {n : ℕ} {ι : Type*} [Fintype ι]
+variable (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+  (t : ℝ) (ht : 1 ≤ t) (v : ι → EC n) (a : SmoothCoefficients ι (EC n))
+  (hsmall : perturbationBound v (coefficientBCF a) * ellipticBound H hH < 1)
+
+lemma schwartzLocal_fixed (f : 𝓢(EC n, ℂ)) :
+    schwartzLocal H hH t ht v a hsmall f = schwartzResolvent H t
+      (f + smoothPerturbation v a (schwartzLocal H hH t ht v a hsmall f)) := by
+  apply schwartz_distribution_injective
+  rw [schwartzResolvent_distribution H hH t ht]
+  rw [show SchwartzMap.toTemperedDistributionCLM (EC n) ℂ volume
+    (schwartzLocal H hH t ht v a hsmall f) = _ from
+    schwartzLocalValue_distribution H hH t ht v a hsmall f]
+  have hr := parameterHilbert_realize H hH t ht 0
+    (schwartzCoord 0 f + perturbation v (coefficientBCF a)
+      (parameterLocal H hH t ht v (coefficientBCF a) hsmall (schwartzCoord 0 f)))
+  norm_num only [zero_add] at hr
+  rw [← schwartzLocal_coord H hH t ht v a hsmall f] at hr
+  rw [← schwartzCoord_perturbation, ← schwartzCoord_add, realize_schwartzCoord] at hr
+  rw [← hr, schwartzCoord_add, schwartzCoord_perturbation,
+    schwartzLocal_coord H hH t ht v a hsmall f]
+  exact congrArg (realize (E := EC n) 2)
+    (parameterLocal_fixed H hH t ht v (coefficientBCF a) hsmall (schwartzCoord 0 f))
+
+ 
+
+theorem schwartzLocal_word_bound (ws : List (EC n)) {s : ℝ} (hws : (ws.length : ℝ) ≤ s) :
+    CoreBound s 2 (fun f => schwartzWord ws (schwartzLocal H hH t ht v a hsmall f)) := by
+  induction hlen : ws.length using Nat.strong_induction_on generalizing ws s with
+  | h k ih =>
+    subst k
+    let T := schwartzLocal H hH t ht v a hsmall
+    have hlow : CoreBound s 0 (fun f => ((commutatorTerms ws a).map
+        (fun b => smoothPerturbation v b.1 (schwartzWord b.2 (T f)))).sum) := by
+      apply CoreBound.list_sum
+      intro b hb
+      exact (coreBound_perturbation v b.1).comp
+        (ih b.2.length (commutatorTerms_order ws a b hb) b.2 (le_trans (by exact_mod_cast
+          (commutatorTerms_order ws a b hb).le) hws) rfl)
+    have hforce := (coreBound_word ws (t := 0) (by linarith : 0 ≤ s - ws.length)).add hlow
+    obtain ⟨C, hC, hforce⟩ := hforce
+    let q := ellipticBound H hH * perturbationBound v (coefficientBCF a)
+    have hq : q < 1 := by simpa [q, mul_comm] using hsmall
+    have hq' : 0 < 1 - q := sub_pos.mpr hq
+    refine ⟨ellipticBound H hH * C / (1 - q),
+      div_nonneg (mul_nonneg (ellipticBound_pos H hH).le hC) hq'.le, fun f => ?_⟩
+    have he := congrArg (schwartzWord ws) (schwartzLocal_fixed H hH t ht v a hsmall f)
+    rw [schwartz_word_resolvent ws H hH t ht, map_add, schwartz_word_perturbation] at he
+    have hb := schwartzResolvent_bound H hH t ht
+      (smoothPerturbation v a (schwartzWord ws (T f)) +
+        (schwartzWord ws f + ((commutatorTerms ws a).map
+          (fun b => smoothPerturbation v b.1 (schwartzWord b.2 (T f)))).sum))
+    have he' : schwartzWord ws (T f) = schwartzResolvent H t
+        (smoothPerturbation v a (schwartzWord ws (T f)) +
+          (schwartzWord ws f + ((commutatorTerms ws a).map
+            (fun b => smoothPerturbation v b.1 (schwartzWord b.2 (T f)))).sum)) := by
+      change schwartzWord ws (T f) = _ at he
+      rw [add_left_comm] at he
+      exact he
+    rw [← he', schwartzCoord_add] at hb
+    have hbound := hb.trans (mul_le_mul_of_nonneg_left (norm_add_le _ _)
+      (ellipticBound_pos H hH).le)
+    have hp := smoothPerturbation_bound v a (schwartzWord ws (T f))
+    have hf := hforce f
+    have hfinal : (1 - q) * ‖schwartzCoord 2 (schwartzWord ws (T f))‖ ≤
+        ellipticBound H hH * C * ‖schwartzCoord s f‖ := by
+      dsimp only [q]
+      nlinarith [mul_le_mul_of_nonneg_left hp (ellipticBound_pos H hH).le,
+        mul_le_mul_of_nonneg_left hf (ellipticBound_pos H hH).le]
+    rw [div_mul_eq_mul_div, le_div_iff₀ hq']
+    simpa only [mul_comm] using hfinal
+
+ 
+
+theorem schwartzLocal_even_bound (k : ℕ) :
+    CoreBound (2 * (k : ℝ)) (2 * (k : ℝ) + 2) (schwartzLocal H hH t ht v a hsmall) := by
+  rw [add_comm]
+  apply coreBound_of_words_base k
+  intro ws hw
+  apply schwartzLocal_word_bound H hH t ht v a hsmall ws
+  exact_mod_cast hw
+
+end FrozenPoisson.ParameterRegularity
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/ParameterLocal.lean b/lean/OAI/Geometry/PrescribedPotential/ParameterLocal.lean
new file mode 100644
index 0000000000000000000000000000000000000000..58ec1908a35cecafb9c70225c0592bb7ec49eed7
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/ParameterLocal.lean
@@ -0,0 +1,177 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.ParameterResolvent
+import OAI.Geometry.PrescribedPotential.VariablePoisson
+
+namespace OAI
+
+/-! Parameter Local. -/
+
+section
+
+ 
+
+noncomputable section
+open MeasureTheory FourierTransform TemperedDistribution LineDeriv
+open scoped SchwartzMap BoundedContinuousFunction ComplexOrder MatrixOrder Real
+
+namespace FrozenPoisson
+open EllipticKernel SobolevChart
+variable {n : ℕ} {ι : Type*} [Fintype ι]
+
+lemma parameter_resolvent_perturbation_bound (H : Matrix (Fin n) (Fin n) ℂ)
+    (hH : H.PosDef) (m : ℝ) (hm : 1 ≤ m)
+    (v : ι → EC n) (a : ι → ι → EC n →ᵇ ℂ) :
+    ‖perturbation v a ∘L parameterHilbert H hH m hm‖ ≤
+      perturbationBound v a * ellipticBound H hH := by
+  apply ContinuousLinearMap.opNorm_le_bound _ (mul_nonneg (perturbationBound_nonneg v a) (ellipticBound_pos H hH).le)
+  intro u
+  change ‖perturbation v a (parameterHilbert H hH m hm u)‖ ≤ _
+  calc
+    _ ≤ perturbationBound v a * ‖parameterHilbert H hH m hm u‖ := perturbation_bound _ _ _
+    _ ≤ perturbationBound v a * (ellipticBound H hH * ‖u‖) :=
+      mul_le_mul_of_nonneg_left (parameterHilbert_bound H hH m hm u)
+        (perturbationBound_nonneg v a)
+    _ = _ := by ring
+
+def parameterCorrection (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (m : ℝ) (hm : 1 ≤ m) (v : ι → EC n) (a : ι → ι → EC n →ᵇ ℂ)
+    (hsmall : perturbationBound v a * ellipticBound H hH < 1) :
+    L2 (EC n) →L[ℂ] L2 (EC n) :=
+  ↑((Units.oneSub (perturbation v a ∘L parameterHilbert H hH m hm)
+    ((parameter_resolvent_perturbation_bound H hH m hm v a).trans_lt hsmall))⁻¹)
+
+lemma parameterCorrection_equation (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (m : ℝ) (hm : 1 ≤ m) (v : ι → EC n) (a : ι → ι → EC n →ᵇ ℂ)
+    (hsmall : perturbationBound v a * ellipticBound H hH < 1) (f : L2 (EC n)) :
+    parameterCorrection H hH m hm v a hsmall f -
+      perturbation v a (parameterHilbert H hH m hm
+        (parameterCorrection H hH m hm v a hsmall f)) = f := by
+  let U := Units.oneSub (perturbation v a ∘L parameterHilbert H hH m hm)
+    ((parameter_resolvent_perturbation_bound H hH m hm v a).trans_lt hsmall)
+  change ((↑U : L2 (EC n) →L[ℂ] L2 (EC n)) * ↑U⁻¹) f = f
+  rw [Units.mul_inv, _root_.one_apply_eq_self]
+
+lemma parameterCorrection_bound (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (m : ℝ) (hm : 1 ≤ m) (v : ι → EC n) (a : ι → ι → EC n →ᵇ ℂ)
+    (hsmall : perturbationBound v a * ellipticBound H hH < 1) (f : L2 (EC n)) :
+    ‖parameterCorrection H hH m hm v a hsmall f‖ ≤
+      ‖f‖ / (1 - perturbationBound v a * ellipticBound H hH) := by
+  let u := parameterCorrection H hH m hm v a hsmall f
+  have he : u = f + perturbation v a (parameterHilbert H hH m hm u) :=
+    sub_eq_iff_eq_add.mp (parameterCorrection_equation H hH m hm v a hsmall f)
+  have hb : ‖u‖ ≤ ‖f‖ + (perturbationBound v a * ellipticBound H hH) * ‖u‖ := by
+    calc
+      _ = ‖f + perturbation v a (parameterHilbert H hH m hm u)‖ := congrArg norm he
+      _ ≤ ‖f‖ + ‖perturbation v a (parameterHilbert H hH m hm u)‖ := norm_add_le _ _
+      _ ≤ ‖f‖ + perturbationBound v a * ‖parameterHilbert H hH m hm u‖ :=
+        add_le_add le_rfl (perturbation_bound v a (parameterHilbert H hH m hm u))
+      _ ≤ ‖f‖ + perturbationBound v a * (ellipticBound H hH * ‖u‖) :=
+        add_le_add le_rfl (mul_le_mul_of_nonneg_left (parameterHilbert_bound H hH m hm u)
+          (perturbationBound_nonneg v a))
+      _ = _ := by ring
+  change ‖u‖ ≤ _
+  rw [le_div_iff₀ (sub_pos.mpr hsmall)]
+  linarith
+
+def parameterLocal (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (m : ℝ) (hm : 1 ≤ m) (v : ι → EC n) (a : ι → ι → EC n →ᵇ ℂ)
+    (hsmall : perturbationBound v a * ellipticBound H hH < 1) :
+    L2 (EC n) →L[ℂ] L2 (EC n) :=
+  parameterHilbert H hH m hm ∘L parameterCorrection H hH m hm v a hsmall
+
+lemma parameterLocal_bound (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (m : ℝ) (hm : 1 ≤ m) (v : ι → EC n) (a : ι → ι → EC n →ᵇ ℂ)
+    (hsmall : perturbationBound v a * ellipticBound H hH < 1) (f : L2 (EC n)) :
+    ‖parameterLocal H hH m hm v a hsmall f‖ ≤
+      (ellipticBound H hH / (1 - perturbationBound v a * ellipticBound H hH)) * ‖f‖ := by
+  change ‖parameterHilbert H hH m hm (parameterCorrection H hH m hm v a hsmall f)‖ ≤ _
+  apply (parameterHilbert_bound H hH m hm _).trans
+  calc
+    _ ≤ ellipticBound H hH * (‖f‖ / (1 - perturbationBound v a * ellipticBound H hH)) :=
+      mul_le_mul_of_nonneg_left (parameterCorrection_bound H hH m hm v a hsmall f)
+        (ellipticBound_pos H hH).le
+    _ = _ := by ring
+
+lemma parameterLocal_equation (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (m : ℝ) (hm : 1 ≤ m) (v : ι → EC n) (a : ι → ι → EC n →ᵇ ℂ)
+    (hsmall : perturbationBound v a * ellipticBound H hH < 1) (f : L2 (EC n)) :
+    (m^2 : ℂ) • realize 2 (parameterLocal H hH m hm v a hsmall f) -
+      frozenDifferential H (realize 2 (parameterLocal H hH m hm v a hsmall f)) -
+      (perturbation v a (parameterLocal H hH m hm v a hsmall f) : 𝓢'(EC n, ℂ)) =
+      (f : 𝓢'(EC n, ℂ)) := by
+  let u := parameterCorrection H hH m hm v a hsmall f
+  change (m^2 : ℂ) • realize 2 (parameterHilbert H hH m hm u) -
+    frozenDifferential H (realize 2 (parameterHilbert H hH m hm u)) -
+      (perturbation v a (parameterHilbert H hH m hm u) : 𝓢'(EC n, ℂ)) = _
+  rw [parameterHilbert_equation]
+  simpa only [map_sub, Lp.toTemperedDistributionCLM_apply] using
+    congrArg (Lp.toTemperedDistributionCLM ℂ volume 2)
+      (parameterCorrection_equation H hH m hm v a hsmall f)
+
+ 
+def parameterLocalZero (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (m : ℝ) (hm : 1 ≤ m) (v : ι → EC n) (a : ι → ι → EC n →ᵇ ℂ)
+    (hsmall : perturbationBound v a * ellipticBound H hH < 1) :
+    L2 (EC n) →L[ℂ] L2 (EC n) :=
+  parameterZero H hH m (lt_of_lt_of_le zero_lt_one hm) ∘L
+    parameterCorrection H hH m hm v a hsmall
+
+lemma parameterLocalZero_distribution (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (m : ℝ) (hm : 1 ≤ m) (v : ι → EC n) (a : ι → ι → EC n →ᵇ ℂ)
+    (hsmall : perturbationBound v a * ellipticBound H hH < 1) (f : L2 (EC n)) :
+    (parameterLocalZero H hH m hm v a hsmall f : 𝓢'(EC n, ℂ)) =
+      realize 2 (parameterLocal H hH m hm v a hsmall f) := by
+  dsimp only [parameterLocalZero, parameterLocal, ContinuousLinearMap.comp_apply]
+  rw [parameterZero_distribution]
+  have h := parameterHilbert_realize H hH m hm 0
+    (parameterCorrection H hH m hm v a hsmall f)
+  simpa only [zero_add, realize, neg_zero, besselPotential_zero,
+    ContinuousLinearMap.comp_apply, ContinuousLinearMap.id_apply,
+    Lp.toTemperedDistributionCLM_apply] using h.symm
+
+lemma parameterLocalZero_bound (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (m : ℝ) (hm : 1 ≤ m) (v : ι → EC n) (a : ι → ι → EC n →ᵇ ℂ)
+    (hsmall : perturbationBound v a * ellipticBound H hH < 1) (f : L2 (EC n)) :
+    ‖parameterLocalZero H hH m hm v a hsmall f‖ ≤
+      ((m^2)⁻¹ / (1 - perturbationBound v a * ellipticBound H hH)) * ‖f‖ := by
+  apply (parameterZero_bound H hH m (lt_of_lt_of_le zero_lt_one hm) _).trans
+  calc
+    _ ≤ (m^2)⁻¹ * (‖f‖ / (1 - perturbationBound v a * ellipticBound H hH)) :=
+      mul_le_mul_of_nonneg_left (parameterCorrection_bound H hH m hm v a hsmall f) (by positivity)
+    _ = _ := by ring
+
+def parameterLocalFirst (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (m : ℝ) (hm : 1 ≤ m) (v : ι → EC n) (a : ι → ι → EC n →ᵇ ℂ)
+    (hsmall : perturbationBound v a * ellipticBound H hH < 1) (w : EC n) :
+    L2 (EC n) →L[ℂ] L2 (EC n) :=
+  parameterFirst H hH m hm w ∘L parameterCorrection H hH m hm v a hsmall
+
+lemma parameterLocalFirst_distribution (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (m : ℝ) (hm : 1 ≤ m) (v : ι → EC n) (a : ι → ι → EC n →ᵇ ℂ)
+    (hsmall : perturbationBound v a * ellipticBound H hH < 1) (w : EC n) (f : L2 (EC n)) :
+    (parameterLocalFirst H hH m hm v a hsmall w f : 𝓢'(EC n, ℂ)) =
+      ∂_{w} (realize 2 (parameterLocal H hH m hm v a hsmall f)) := by
+  dsimp only [parameterLocalFirst, ContinuousLinearMap.comp_apply]
+  rw [parameterFirst_distribution, ← parameterLocalZero_distribution,
+    parameterLocalZero, ContinuousLinearMap.comp_apply, parameterZero_distribution]
+
+lemma parameterLocalFirst_bound (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (m : ℝ) (hm : 1 ≤ m) (v : ι → EC n) (a : ι → ι → EC n →ᵇ ℂ)
+    (hsmall : perturbationBound v a * ellipticBound H hH < 1) (w : EC n) (f : L2 (EC n)) :
+    ‖parameterLocalFirst H hH m hm v a hsmall w f‖ ≤
+      ((2 * Real.pi * ‖w‖ * ellipticBound H hH) /
+        (m * (1 - perturbationBound v a * ellipticBound H hH))) * ‖f‖ := by
+  apply (parameterFirst_bound H hH m hm w _).trans
+  calc
+    _ ≤ ((2 * Real.pi * ‖w‖ * ellipticBound H hH) / m) *
+        (‖f‖ / (1 - perturbationBound v a * ellipticBound H hH)) :=
+      mul_le_mul_of_nonneg_left (parameterCorrection_bound H hH m hm v a hsmall f)
+        (by positivity [ellipticBound_pos H hH])
+    _ = _ := by simp only [div_eq_mul_inv, mul_inv_rev]; ring
+
+end FrozenPoisson
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/ParameterRegularity.lean b/lean/OAI/Geometry/PrescribedPotential/ParameterRegularity.lean
new file mode 100644
index 0000000000000000000000000000000000000000..f1e597326522d9928493992b949a78af2a2bb654
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/ParameterRegularity.lean
@@ -0,0 +1,242 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.LocalRegularity
+import OAI.Geometry.PrescribedPotential.ParameterLocal
+
+namespace OAI
+
+/-! Parameter Regularity. -/
+
+section
+
+ 
+
+noncomputable section
+open Filter Topology MeasureTheory FourierTransform TemperedDistribution LineDeriv
+open scoped SchwartzMap BoundedContinuousFunction ContDiff Real ComplexOrder MatrixOrder
+
+namespace FrozenPoisson.ParameterRegularity
+open FrozenPoisson SobolevChart EllipticKernel
+variable {n : ℕ} {ι : Type*} [Fintype ι]
+
+def schwartzResolvent (H : Matrix (Fin n) (Fin n) ℂ) (t : ℝ) :
+    𝓢(EC n, ℂ) →L[ℂ] 𝓢(EC n, ℂ) :=
+  SchwartzMap.fourierMultiplierCLM ℂ (parameterSymbol H t)
+
+lemma schwartzResolvent_distribution (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (t : ℝ) (ht : 1 ≤ t) (f : 𝓢(EC n, ℂ)) :
+    SchwartzMap.toTemperedDistributionCLM (EC n) ℂ volume (schwartzResolvent H t f) =
+      parameterResolvent H t (SchwartzMap.toTemperedDistributionCLM (EC n) ℂ volume f) :=
+  (fourierMultiplierCLM_toTemperedDistributionCLM_eq
+    (parameterSymbol_temperate H hH (lt_of_lt_of_le zero_lt_one ht)) f).symm
+
+lemma schwartzCoord_resolvent (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (t : ℝ) (ht : 1 ≤ t) (s : ℝ) (f : 𝓢(EC n, ℂ)) :
+    schwartzCoord (s + 2) (schwartzResolvent H t f) =
+      parameterHilbert H hH t ht (schwartzCoord s f) := by
+  apply realize_injective (s + 2)
+  rw [realize_schwartzCoord, parameterHilbert_realize, realize_schwartzCoord]
+  exact schwartzResolvent_distribution H hH t ht f
+
+lemma weak_deriv_resolvent (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (t : ℝ) (ht : 1 ≤ t) (v : EC n) (u : 𝓢'(EC n, ℂ)) :
+    ∂_{v} (parameterResolvent H t u) = parameterResolvent H t (∂_{v} u) := by
+  simp only [lineDeriv_eq_fourierMultiplierCLM, parameterResolvent, map_smul]
+  rw [fourierMultiplierCLM_fourierMultiplierCLM_apply
+    (parameterSymbol_temperate H hH (lt_of_lt_of_le zero_lt_one ht)) (by fun_prop),
+    fourierMultiplierCLM_fourierMultiplierCLM_apply (by fun_prop)
+      (parameterSymbol_temperate H hH (lt_of_lt_of_le zero_lt_one ht))]
+  simp only [mul_comm]
+
+lemma schwartz_deriv_resolvent (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (t : ℝ) (ht : 1 ≤ t) (v : EC n) (f : 𝓢(EC n, ℂ)) :
+    ∂_{v} (schwartzResolvent H t f) = schwartzResolvent H t (∂_{v} f) := by
+  apply schwartz_distribution_injective
+  rw [← lineDerivOp_toTemperedDistributionCLM_eq, schwartzResolvent_distribution H hH t ht,
+    schwartzResolvent_distribution H hH t ht, ← lineDerivOp_toTemperedDistributionCLM_eq]
+  exact weak_deriv_resolvent H hH t ht v (SchwartzMap.toTemperedDistributionCLM (EC n) ℂ volume f)
+
+lemma schwartz_word_resolvent (ws : List (EC n)) (H : Matrix (Fin n) (Fin n) ℂ)
+    (hH : H.PosDef) (t : ℝ) (ht : 1 ≤ t) (f : 𝓢(EC n, ℂ)) :
+    schwartzWord ws (schwartzResolvent H t f) = schwartzResolvent H t (schwartzWord ws f) := by
+  induction ws with
+  | nil => rfl
+  | cons w ws ih =>
+    change ∂_{w} (schwartzWord ws (schwartzResolvent H t f)) = _
+    rw [ih, schwartz_deriv_resolvent H hH t ht]
+    rfl
+
+ 
+def schwartzNeumann (H : Matrix (Fin n) (Fin n) ℂ) (t : ℝ) (v : ι → EC n)
+    (a : SmoothCoefficients ι (EC n)) : 𝓢(EC n, ℂ) →L[ℂ] 𝓢(EC n, ℂ) :=
+  schwartzResolvent H t ∘L smoothPerturbation v a
+
+lemma schwartzResolvent_bound (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef) (t : ℝ) (ht : 1 ≤ t)
+    (f : 𝓢(EC n, ℂ)) :
+    ‖schwartzCoord 2 (schwartzResolvent H t f)‖ ≤
+      ellipticBound H hH * ‖schwartzCoord 0 f‖ := by
+  have hc := schwartzCoord_resolvent H hH t ht 0 f
+  norm_num at hc
+  rw [hc]
+  exact parameterHilbert_bound H hH t ht _
+
+lemma schwartzNeumann_bound (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef) (t : ℝ) (ht : 1 ≤ t)
+    (v : ι → EC n) (a : SmoothCoefficients ι (EC n)) (f : 𝓢(EC n, ℂ)) :
+    ‖schwartzCoord 2 (schwartzNeumann H t v a f)‖ ≤
+      (ellipticBound H hH * perturbationBound v (coefficientBCF a)) *
+        ‖schwartzCoord 2 f‖ := by
+  calc
+    _ ≤ ellipticBound H hH * ‖schwartzCoord 0 (smoothPerturbation v a f)‖ :=
+      schwartzResolvent_bound H hH t ht _
+    _ ≤ ellipticBound H hH * (perturbationBound v (coefficientBCF a) *
+        ‖schwartzCoord 2 f‖) :=
+      mul_le_mul_of_nonneg_left (smoothPerturbation_bound v a f) (ellipticBound_pos H hH).le
+    _ = _ := by ring
+
+lemma schwartz_word_neumann (ws : List (EC n)) (H : Matrix (Fin n) (Fin n) ℂ)
+    (hH : H.PosDef) (t : ℝ) (ht : 1 ≤ t) (v : ι → EC n) (a : SmoothCoefficients ι (EC n)) (f : 𝓢(EC n, ℂ)) :
+    schwartzWord ws (schwartzNeumann H t v a f) =
+      schwartzNeumann H t v a (schwartzWord ws f) +
+        ((commutatorTerms ws a).map
+          (fun term => schwartzNeumann H t v term.1 (schwartzWord term.2 f))).sum := by
+  change schwartzWord ws (schwartzResolvent H t (smoothPerturbation v a f)) = _
+  rw [schwartz_word_resolvent ws H hH t ht, schwartz_word_perturbation, map_add, map_list_sum]
+  simp only [List.map_map, Function.comp_def]
+  rfl
+
+def schwartzIterate (H : Matrix (Fin n) (Fin n) ℂ) (t : ℝ) (v : ι → EC n)
+    (a : SmoothCoefficients ι (EC n)) (f : 𝓢(EC n, ℂ)) (N : ℕ) : 𝓢(EC n, ℂ) :=
+  (schwartzNeumann H t v a)^[N] f
+
+lemma schwartzIterate_succ (H : Matrix (Fin n) (Fin n) ℂ) (t : ℝ) (v : ι → EC n)
+    (a : SmoothCoefficients ι (EC n)) (f : 𝓢(EC n, ℂ)) (N : ℕ) :
+    schwartzIterate H t v a f (N+1) = schwartzNeumann H t v a (schwartzIterate H t v a f N) :=
+  Function.iterate_succ_apply' _ _ _
+
+lemma summable_norm_neumann_of (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef) (t : ℝ) (ht : 1 ≤ t)
+    (v : ι → EC n) (a : SmoothCoefficients ι (EC n)) (f : ℕ → 𝓢(EC n, ℂ))
+    (hf : Summable (fun m => ‖schwartzCoord 2 (f m)‖)) :
+    Summable (fun m => ‖schwartzCoord 2 (schwartzNeumann H t v a (f m))‖) := by
+  apply Summable.of_nonneg_of_le (fun _ => norm_nonneg _)
+    (fun m => schwartzNeumann_bound H hH t ht v a (f m))
+  exact hf.mul_left _
+
+ 
+
+theorem neumann_summable_word (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef) (t : ℝ) (ht : 1 ≤ t)
+    (v : ι → EC n) (a : SmoothCoefficients ι (EC n))
+    (hsmall : perturbationBound v (coefficientBCF a) * ellipticBound H hH < 1)
+    (f : 𝓢(EC n, ℂ)) (ws : List (EC n)) :
+    Summable (fun m => ‖schwartzCoord 2 (schwartzWord ws (schwartzIterate H t v a f m))‖) := by
+  induction hlen : ws.length using Nat.strong_induction_on generalizing ws with
+  | h k ih =>
+    let err : ℕ → 𝓢(EC n, ℂ) := fun m => ((commutatorTerms ws a).map
+      (fun term => schwartzNeumann H t v term.1 (schwartzWord term.2 (schwartzIterate H t v a f m)))).sum
+    have hs : Summable (fun m => ‖schwartzCoord 2 (err m)‖) := by
+      apply summable_norm_schwartzCoord_list
+      intro term hterm
+      apply summable_norm_neumann_of H hH t ht
+      exact ih term.2.length (hlen ▸ commutatorTerms_order ws a term hterm) term.2 rfl
+    apply summable_of_contracting_recurrence (fun _ => norm_nonneg _)
+      (fun _ => norm_nonneg _) (q := ellipticBound H hH * perturbationBound v (coefficientBCF a))
+      (by simpa only [mul_comm] using hsmall) hs
+    intro m
+    rw [schwartzIterate_succ, schwartz_word_neumann ws H hH t ht, schwartzCoord_add]
+    exact (norm_add_le _ _).trans (add_le_add
+      (schwartzNeumann_bound H hH t ht v a _) le_rfl)
+
+def neumannSum (H : Matrix (Fin n) (Fin n) ℂ) (t : ℝ) (v : ι → EC n)
+    (a : SmoothCoefficients ι (EC n)) (f : 𝓢(EC n, ℂ)) : L2 (EC n) :=
+  ∑' m, schwartzCoord 2 (schwartzIterate H t v a f m)
+
+lemma schwartzCoord_neumann (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef) (t : ℝ) (ht : 1 ≤ t)
+    (v : ι → EC n) (a : SmoothCoefficients ι (EC n)) (f : 𝓢(EC n, ℂ)) :
+    schwartzCoord 2 (schwartzNeumann H t v a f) =
+      parameterHilbert H hH t ht (perturbation v (coefficientBCF a) (schwartzCoord 2 f)) := by
+  have he := schwartzCoord_resolvent H hH t ht 0 (smoothPerturbation v a f)
+  norm_num at he
+  rw [schwartzCoord_perturbation] at he
+  exact he
+
+lemma neumannSum_fixed (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef) (t : ℝ) (ht : 1 ≤ t)
+    (v : ι → EC n) (a : SmoothCoefficients ι (EC n))
+    (hsmall : perturbationBound v (coefficientBCF a) * ellipticBound H hH < 1)
+    (f : 𝓢(EC n, ℂ)) :
+    neumannSum H t v a f = schwartzCoord 2 f +
+      parameterHilbert H hH t ht (perturbation v (coefficientBCF a) (neumannSum H t v a f)) := by
+  have hs := (neumann_summable_word H hH t ht v a hsmall f []).of_norm
+  change Summable (fun m => schwartzCoord 2 (schwartzIterate H t v a f m)) at hs
+  let T := parameterHilbert H hH t ht ∘L perturbation v (coefficientBCF a)
+  have hmap := T.map_tsum hs
+  simp only [T, ContinuousLinearMap.comp_apply, ← schwartzCoord_neumann,
+    ← schwartzIterate_succ] at hmap
+  change (∑' m, schwartzCoord 2 (schwartzIterate H t v a f m)) = _
+  rw [hs.tsum_eq_zero_add]
+  change schwartzCoord 2 f + _ = schwartzCoord 2 f + _
+  exact congrArg (schwartzCoord 2 f + ·) hmap.symm
+
+lemma fixed_unique (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (t : ℝ) (ht : 1 ≤ t) (v : ι → EC n) (a : ι → ι → EC n →ᵇ ℂ)
+    (hsmall : perturbationBound v a * ellipticBound H hH < 1)
+    (u w f : L2 (EC n))
+    (hu : u = parameterHilbert H hH t ht (f + perturbation v a u))
+    (hw : w = parameterHilbert H hH t ht (f + perturbation v a w)) : u = w := by
+  have he : u - w = parameterHilbert H hH t ht (perturbation v a (u - w)) := by
+    rw [map_sub, map_sub]
+    rw [map_add] at hu hw
+    calc
+      u - w = (parameterHilbert H hH t ht f + parameterHilbert H hH t ht (perturbation v a u)) -
+          (parameterHilbert H hH t ht f + parameterHilbert H hH t ht (perturbation v a w)) :=
+        congrArg₂ (· - ·) hu hw
+      _ = _ := by abel
+  have hb : ‖u - w‖ ≤ (perturbationBound v a * ellipticBound H hH) * ‖u - w‖ := by
+    calc
+      _ = ‖parameterHilbert H hH t ht (perturbation v a (u - w))‖ := congrArg norm he
+      _ ≤ _ := (parameterHilbert_bound H hH t ht _).trans
+        (by nlinarith [perturbation_bound v a (u - w), ellipticBound_pos H hH])
+  have : ‖u - w‖ = 0 := by nlinarith [norm_nonneg (u - w)]
+  exact sub_eq_zero.mp (norm_eq_zero.mp this)
+
+lemma parameterLocal_fixed (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (t : ℝ) (ht : 1 ≤ t) (v : ι → EC n) (a : ι → ι → EC n →ᵇ ℂ)
+    (hsmall : perturbationBound v a * ellipticBound H hH < 1) (f : L2 (EC n)) :
+    parameterLocal H hH t ht v a hsmall f = parameterHilbert H hH t ht
+      (f + perturbation v a (parameterLocal H hH t ht v a hsmall f)) := by
+  change parameterHilbert H hH t ht (parameterCorrection H hH t ht v a hsmall f) = _
+  congr 1
+  exact sub_eq_iff_eq_add.mp (parameterCorrection_equation H hH t ht v a hsmall f)
+
+lemma neumannSum_eq_local (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (t : ℝ) (ht : 1 ≤ t) (v : ι → EC n) (a : SmoothCoefficients ι (EC n))
+    (hsmall : perturbationBound v (coefficientBCF a) * ellipticBound H hH < 1)
+    (f : 𝓢(EC n, ℂ)) :
+    neumannSum H t v a (schwartzResolvent H t f) =
+      parameterLocal H hH t ht v (coefficientBCF a) hsmall (schwartzCoord 0 f) := by
+  apply fixed_unique H hH t ht v (coefficientBCF a) hsmall _ _ (schwartzCoord 0 f)
+  · rw [map_add]
+    have hr := schwartzCoord_resolvent H hH t ht 0 f
+    norm_num at hr
+    rw [← hr]
+    exact neumannSum_fixed H hH t ht v a hsmall (schwartzResolvent H t f)
+  · exact parameterLocal_fixed H hH t ht v (coefficientBCF a) hsmall _
+
+ 
+
+theorem parameterLocal_schwartz_memSobolev (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (t : ℝ) (ht : 1 ≤ t) (v : ι → EC n) (a : SmoothCoefficients ι (EC n))
+    (hsmall : perturbationBound v (coefficientBCF a) * ellipticBound H hH < 1)
+    (f : 𝓢(EC n, ℂ)) (k : ℕ) :
+    MemSobolev (2 * (k : ℝ)) 2
+      (realize 2 (parameterLocal H hH t ht v (coefficientBCF a) hsmall (schwartzCoord 0 f))) := by
+  rw [← neumannSum_eq_local H hH t ht v a hsmall f]
+  apply memSobolev_even_of_words _ _ k []
+  intro ws
+  exact memSobolev_word_of_summable (schwartzIterate H t v a (schwartzResolvent H t f)) ws
+    (neumann_summable_word H hH t ht v a hsmall (schwartzResolvent H t f) [])
+    (neumann_summable_word H hH t ht v a hsmall (schwartzResolvent H t f) ws)
+
+end FrozenPoisson.ParameterRegularity
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/ParameterResolvent.lean b/lean/OAI/Geometry/PrescribedPotential/ParameterResolvent.lean
new file mode 100644
index 0000000000000000000000000000000000000000..a84a060bac77d3c39a6189533c944c5a3dd88d53
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/ParameterResolvent.lean
@@ -0,0 +1,301 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.GlobalSobolev
+
+namespace OAI
+
+/-! Parameter Resolvent. -/
+
+section
+
+ 
+
+noncomputable section
+open MeasureTheory FourierTransform TemperedDistribution LineDeriv
+open scoped SchwartzMap BoundedContinuousFunction ComplexOrder MatrixOrder Real
+
+namespace FrozenPoisson
+open EllipticKernel SobolevChart
+variable {n : ℕ}
+
+ 
+def parameterSymbol (H : Matrix (Fin n) (Fin n) ℂ) (m : ℝ) (ξ : EC n) : ℂ :=
+  ((m^2 + symbol H ξ)⁻¹ : ℝ)
+
+lemma parameterSymbol_temperate (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    {m : ℝ} (hm : 0 < m) : (parameterSymbol H m).HasTemperateGrowth := by
+  obtain ⟨B, _, hB⟩ := posDef_factor H⁻¹ hH.inv
+  have he : parameterSymbol H m = fun ξ =>
+      (m^2 : ℂ)⁻¹ * (Complex.ofReal ∘
+        ((fun y : EC n => (1 + ‖y‖^2)^(-1 : ℝ)) ∘
+          ((m⁻¹ : ℝ) • factorMap B))) ξ := by
+    funext ξ
+    simp only [parameterSymbol, Function.comp_apply, Real.rpow_neg_one,
+      _root_.smul_apply, norm_smul, Real.norm_eq_abs,
+      abs_of_pos (inv_pos.mpr hm), mul_pow, symbol_eq_factorMap H B hB,
+      ← Complex.ofReal_pow, ← Complex.ofReal_inv, ← Complex.ofReal_mul]
+    congr 1
+    field_simp
+  rw [he]
+  have hg := (Function.hasTemperateGrowth_one_add_norm_sq_rpow (EC n) (-1)).comp
+    ((m⁻¹ : ℝ) • factorMap B).hasTemperateGrowth
+  exact (Function.HasTemperateGrowth.const _).mul
+    (Complex.hasTemperateGrowth_ofReal.comp hg)
+
+lemma elliptic_symbol_bound (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef) (ξ : EC n) :
+    1 + ‖ξ‖^2 ≤ ellipticBound H hH * (1 + symbol H ξ) := by
+  have hp := symbol_nonneg H hH ξ
+  have h := gainSymbol_bound H hH ξ
+  simp only [gainSymbol, inverseSymbol, ← Complex.ofReal_mul,
+    Complex.norm_real, Real.norm_eq_abs] at h
+  rw [abs_of_nonneg (by positivity), mul_inv_le_iff₀ (by linarith [symbol_nonneg H hH ξ])] at h
+  exact h
+
+lemma parameter_gain_bound (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    {m : ℝ} (hm : 1 ≤ m) (ξ : EC n) :
+    ‖(1 + ‖ξ‖^2 : ℝ) * parameterSymbol H m ξ‖ ≤ ellipticBound H hH := by
+  have hm2 : 1 ≤ m^2 := by nlinarith
+  have hp := symbol_nonneg H hH ξ
+  simp only [parameterSymbol, ← Complex.ofReal_mul, Complex.norm_real, Real.norm_eq_abs]
+  rw [abs_of_nonneg (by positivity), mul_inv_le_iff₀ (by positivity)]
+  exact (elliptic_symbol_bound H hH ξ).trans
+    (mul_le_mul_of_nonneg_left (by linarith) (ellipticBound_pos H hH).le)
+
+lemma parameter_zero_bound (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    {m : ℝ} (hm : 0 < m) (ξ : EC n) :
+    ‖parameterSymbol H m ξ‖ ≤ (m^2)⁻¹ := by
+  have hp := symbol_nonneg H hH ξ
+  simp only [parameterSymbol, Complex.norm_real, Real.norm_eq_abs,
+    abs_of_pos (inv_pos.mpr (show 0 < m^2 + symbol H ξ by positivity))]
+  exact inv_anti₀ (by positivity) (by linarith)
+
+lemma parameter_first_weight (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    {m : ℝ} (hm : 1 ≤ m) (ξ : EC n) :
+    ‖ξ‖ * (m^2 + symbol H ξ)⁻¹ ≤ ellipticBound H hH / m := by
+  let C := ellipticBound H hH
+  let D := m^2 + symbol H ξ
+  have hC : 1 ≤ C := le_max_left _ _
+  have hm0 : 0 < m := lt_of_lt_of_le zero_lt_one hm
+  have hm2 : 1 ≤ m^2 := by nlinarith
+  have hp := symbol_nonneg H hH ξ
+  have hD : 0 < D := by dsimp [D]; positivity
+  have hDm : m^2 ≤ D := by dsimp [D]; linarith
+  have hξ : ‖ξ‖^2 ≤ C * D := by
+    have hb := elliptic_symbol_bound H hH ξ
+    dsimp [D, C]
+    nlinarith [mul_nonneg (ellipticBound_pos H hH).le (sub_nonneg.mpr hm2)]
+  have hsq : (‖ξ‖ * m)^2 ≤ (C * D)^2 := by
+    calc
+      _ = ‖ξ‖^2 * m^2 := mul_pow _ _ _
+      _ ≤ (C * D) * m^2 := mul_le_mul_of_nonneg_right hξ (sq_nonneg _)
+      _ ≤ (C * D) * D := mul_le_mul_of_nonneg_left hDm (by positivity)
+      _ ≤ (C * D)^2 := by nlinarith [mul_nonneg (sub_nonneg.mpr hC) (show 0 ≤ C * D^2 by positivity)]
+  have hlin : ‖ξ‖ * m ≤ C * D := (sq_le_sq₀ (by positivity) (by positivity)).mp hsq
+  change ‖ξ‖ * D⁻¹ ≤ C / m
+  rw [le_div_iff₀ hm0]
+  rw [mul_right_comm, mul_inv_le_iff₀ hD]
+  exact hlin
+
+ 
+def parameterResolvent (H : Matrix (Fin n) (Fin n) ℂ) (m : ℝ) :
+    𝓢'(EC n, ℂ) →L[ℂ] 𝓢'(EC n, ℂ) := fourierMultiplierCLM ℂ (parameterSymbol H m)
+
+def parameterForward (H : Matrix (Fin n) (Fin n) ℂ) (m : ℝ) :
+    𝓢'(EC n, ℂ) →L[ℂ] 𝓢'(EC n, ℂ) :=
+  fourierMultiplierCLM ℂ (fun ξ => ((m^2 + symbol H ξ : ℝ) : ℂ))
+
+lemma parameterForward_temperate (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef) (m : ℝ) :
+    (fun ξ => ((m^2 + symbol H ξ : ℝ) : ℂ)).HasTemperateGrowth :=
+  Complex.hasTemperateGrowth_ofReal.comp
+    ((Function.HasTemperateGrowth.const _).add (symbol_temperate H hH))
+
+lemma parameterForward_eq (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (m : ℝ) (u : 𝓢'(EC n, ℂ)) :
+    parameterForward H m u = (m^2 : ℂ) • u - frozenDifferential H u := by
+  rw [frozenDifferential_fourier]
+  have hn : (fun ξ => ((-symbol H ξ : ℝ) : ℂ)).HasTemperateGrowth :=
+    Complex.hasTemperateGrowth_ofReal.comp (symbol_temperate H hH).neg
+  have he := multiplier_sub (Function.HasTemperateGrowth.const (m^2 : ℂ)) hn u
+  simpa only [Pi.sub_def, Complex.ofReal_neg, sub_neg_eq_add, parameterForward,
+    Complex.ofReal_add, Complex.ofReal_pow, fourierMultiplierCLM_const,
+    _root_.smul_apply, ContinuousLinearMap.id_apply] using he
+
+lemma parameterForward_resolvent (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    {m : ℝ} (hm : 0 < m) (u : 𝓢'(EC n, ℂ)) :
+    parameterForward H m (parameterResolvent H m u) = u := by
+  rw [parameterForward, parameterResolvent,
+    fourierMultiplierCLM_fourierMultiplierCLM_apply (parameterSymbol_temperate H hH hm)
+      (parameterForward_temperate H hH m)]
+  have he : parameterSymbol H m * (fun ξ => ((m^2 + symbol H ξ : ℝ) : ℂ)) =
+      fun _ => (1 : ℂ) := by
+    funext ξ
+    simp only [Pi.mul_apply, parameterSymbol, ← Complex.ofReal_mul]
+    have hp := symbol_nonneg H hH ξ
+    rw [inv_mul_cancel₀ (by positivity : m^2 + symbol H ξ ≠ 0), Complex.ofReal_one]
+  rw [he, fourierMultiplierCLM_const]
+  simp
+
+end FrozenPoisson
+
+namespace FrozenPoisson
+open EllipticKernel SobolevChart
+variable {n : ℕ}
+
+def parameterGain (H : Matrix (Fin n) (Fin n) ℂ) (m : ℝ) (ξ : EC n) : ℂ :=
+  (1 + ‖ξ‖^2 : ℝ) * parameterSymbol H m ξ
+
+lemma parameterGain_temperate (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    {m : ℝ} (hm : 0 < m) : (parameterGain H m).HasTemperateGrowth := by
+  have hi := parameterSymbol_temperate H hH hm
+  unfold parameterGain
+  fun_prop
+
+def parameterGainBCF (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (m : ℝ) (hm : 1 ≤ m) : EC n →ᵇ ℂ :=
+  BoundedContinuousFunction.ofNormedAddCommGroup (parameterGain H m)
+    (parameterGain_temperate H hH (lt_of_lt_of_le zero_lt_one hm)).1.continuous
+    (ellipticBound H hH) (parameter_gain_bound H hH hm)
+
+lemma parameterGainBCF_norm (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (m : ℝ) (hm : 1 ≤ m) : ‖parameterGainBCF H hH m hm‖ ≤ ellipticBound H hH :=
+  (BoundedContinuousFunction.norm_le (ellipticBound_pos H hH).le).mpr
+    (parameter_gain_bound H hH hm)
+
+def parameterHilbert (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (m : ℝ) (hm : 1 ≤ m) : L2 (EC n) →L[ℂ] L2 (EC n) :=
+  multiplier (parameterGainBCF H hH m hm)
+
+lemma parameterHilbert_bound (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (m : ℝ) (hm : 1 ≤ m) (u : L2 (EC n)) :
+    ‖parameterHilbert H hH m hm u‖ ≤ ellipticBound H hH * ‖u‖ :=
+  (multiplier_norm_le (parameterGainBCF H hH m hm) u).trans
+    (mul_le_mul_of_nonneg_right (parameterGainBCF_norm H hH m hm) (norm_nonneg u))
+
+lemma bessel_two_parameterResolvent (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    {m : ℝ} (hm : 0 < m) (u : 𝓢'(EC n, ℂ)) :
+    besselPotential (EC n) ℂ 2 (parameterResolvent H m u) =
+      fourierMultiplierCLM ℂ (parameterGain H m) u := by
+  rw [besselPotential, parameterResolvent,
+    fourierMultiplierCLM_fourierMultiplierCLM_apply (parameterSymbol_temperate H hH hm)
+      (by fun_prop)]
+  congr 2
+  ext ξ
+  simp [parameterGain, mul_comm]
+
+lemma parameterHilbert_realize (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (m : ℝ) (hm : 1 ≤ m) (s : ℝ) (u : L2 (EC n)) :
+    realize (s + 2) (parameterHilbert H hH m hm u) =
+      parameterResolvent H m (realize s u) := by
+  have hm0 : 0 < m := lt_of_lt_of_le zero_lt_one hm
+  change besselPotential (EC n) ℂ (-(s + 2))
+    (Lp.toTemperedDistributionCLM ℂ MeasureTheory.volume 2 (parameterHilbert H hH m hm u)) = _
+  rw [besselPotential_neg_apply_eq_iff]
+  have hc : parameterResolvent H m (realize s u) = besselPotential (EC n) ℂ (-s)
+      (parameterResolvent H m (Lp.toTemperedDistributionCLM ℂ MeasureTheory.volume 2 u)) :=
+    (bessel_multiplier_comm (parameterSymbol_temperate H hH hm0) (-s) _).symm
+  rw [hc, besselPotential_besselPotential_apply]
+  have hs : -s + (s + 2) = 2 := by ring
+  rw [hs, bessel_two_parameterResolvent H hH hm0]
+  exact (multiplier_distribution (parameterGainBCF H hH m hm)
+    (parameterGain_temperate H hH hm0) u).symm
+
+lemma parameterHilbert_equation (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (m : ℝ) (hm : 1 ≤ m) (f : L2 (EC n)) :
+    (m^2 : ℂ) • realize 2 (parameterHilbert H hH m hm f) -
+      frozenDifferential H (realize 2 (parameterHilbert H hH m hm f)) =
+        (f : 𝓢'(EC n, ℂ)) := by
+  rw [← parameterForward_eq H hH]
+  have hr := parameterHilbert_realize H hH m hm 0 f
+  simp only [zero_add] at hr
+  rw [hr, parameterForward_resolvent H hH (lt_of_lt_of_le zero_lt_one hm)]
+  simp [realize]
+
+def parameterZeroBCF (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (m : ℝ) (hm : 0 < m) : EC n →ᵇ ℂ :=
+  BoundedContinuousFunction.ofNormedAddCommGroup (parameterSymbol H m)
+    (parameterSymbol_temperate H hH hm).1.continuous (m^2)⁻¹ (parameter_zero_bound H hH hm)
+
+def parameterZero (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (m : ℝ) (hm : 0 < m) : L2 (EC n) →L[ℂ] L2 (EC n) :=
+  multiplier (parameterZeroBCF H hH m hm)
+
+lemma parameterZero_bound (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (m : ℝ) (hm : 0 < m) (f : L2 (EC n)) :
+    ‖parameterZero H hH m hm f‖ ≤ (m^2)⁻¹ * ‖f‖ := by
+  apply (multiplier_norm_le _ f).trans
+  apply mul_le_mul_of_nonneg_right _ (norm_nonneg _)
+  exact (BoundedContinuousFunction.norm_le (by positivity)).mpr (parameter_zero_bound H hH hm)
+
+lemma parameterZero_distribution (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (m : ℝ) (hm : 0 < m) (f : L2 (EC n)) :
+    (parameterZero H hH m hm f : 𝓢'(EC n, ℂ)) = parameterResolvent H m (f : 𝓢'(EC n, ℂ)) :=
+  multiplier_distribution _ (parameterSymbol_temperate H hH hm) f
+
+def parameterFirstSymbol (H : Matrix (Fin n) (Fin n) ℂ) (m : ℝ) (v ξ : EC n) : ℂ :=
+  (2 * Real.pi * Complex.I) * (inner ℝ ξ v : ℂ) * parameterSymbol H m ξ
+
+lemma parameterFirstSymbol_temperate (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    {m : ℝ} (hm : 0 < m) (v : EC n) : (parameterFirstSymbol H m v).HasTemperateGrowth := by
+  have hi := parameterSymbol_temperate H hH hm
+  unfold parameterFirstSymbol
+  fun_prop
+
+lemma parameterFirstSymbol_bound (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    {m : ℝ} (hm : 1 ≤ m) (v ξ : EC n) :
+    ‖parameterFirstSymbol H m v ξ‖ ≤ (2 * Real.pi * ‖v‖ * ellipticBound H hH) / m := by
+  have hm0 : 0 < m := lt_of_lt_of_le zero_lt_one hm
+  have hp := symbol_nonneg H hH ξ
+  have hi := abs_real_inner_le_norm ξ v
+  have hw := parameter_first_weight H hH hm ξ
+  simp only [parameterFirstSymbol, norm_mul, Complex.norm_I, mul_one,
+    Complex.norm_ofNat, Complex.norm_real, Real.norm_eq_abs, abs_of_pos Real.pi_pos,
+    parameterSymbol, abs_of_pos (inv_pos.mpr (show 0 < m^2 + symbol H ξ by positivity))]
+  calc
+    _ ≤ (2 * Real.pi) * (‖ξ‖ * ‖v‖) * (m^2 + symbol H ξ)⁻¹ := by gcongr
+    _ = (2 * Real.pi * ‖v‖) * (‖ξ‖ * (m^2 + symbol H ξ)⁻¹) := by ring
+    _ ≤ (2 * Real.pi * ‖v‖) * (ellipticBound H hH / m) := by gcongr
+    _ = _ := by ring
+
+def parameterFirstBCF (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (m : ℝ) (hm : 1 ≤ m) (v : EC n) : EC n →ᵇ ℂ :=
+  BoundedContinuousFunction.ofNormedAddCommGroup (parameterFirstSymbol H m v)
+    (parameterFirstSymbol_temperate H hH (lt_of_lt_of_le zero_lt_one hm) v).1.continuous
+    ((2 * Real.pi * ‖v‖ * ellipticBound H hH) / m) (parameterFirstSymbol_bound H hH hm v)
+
+def parameterFirst (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (m : ℝ) (hm : 1 ≤ m) (v : EC n) : L2 (EC n) →L[ℂ] L2 (EC n) :=
+  multiplier (parameterFirstBCF H hH m hm v)
+
+lemma parameterFirst_bound (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (m : ℝ) (hm : 1 ≤ m) (v : EC n) (f : L2 (EC n)) :
+    ‖parameterFirst H hH m hm v f‖ ≤ ((2 * Real.pi * ‖v‖ * ellipticBound H hH) / m) * ‖f‖ := by
+  apply (multiplier_norm_le _ f).trans
+  apply mul_le_mul_of_nonneg_right _ (norm_nonneg _)
+  exact (BoundedContinuousFunction.norm_le (by positivity [ellipticBound_pos H hH])).mpr
+    (parameterFirstSymbol_bound H hH hm v)
+
+lemma parameterFirst_distribution (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (m : ℝ) (hm : 1 ≤ m) (v : EC n) (f : L2 (EC n)) :
+    (parameterFirst H hH m hm v f : 𝓢'(EC n, ℂ)) =
+      ∂_{v} (parameterResolvent H m (f : 𝓢'(EC n, ℂ))) := by
+  have hm0 : 0 < m := lt_of_lt_of_le zero_lt_one hm
+  rw [lineDeriv_eq_fourierMultiplierCLM, parameterResolvent,
+    fourierMultiplierCLM_fourierMultiplierCLM_apply (parameterSymbol_temperate H hH hm0)
+      (by fun_prop)]
+  have ht : (parameterSymbol H m * (fun ξ => (inner ℝ ξ v : ℂ))).HasTemperateGrowth := by
+    apply (parameterSymbol_temperate H hH hm0).mul
+    fun_prop
+  rw [← _root_.smul_apply, ← fourierMultiplierCLM_smul ht, parameterFirst,
+    multiplier_distribution _ (parameterFirstSymbol_temperate H hH hm0 v)]
+  congr 2
+  ext ξ
+  change parameterFirstSymbol H m v ξ =
+    (2 * Real.pi * Complex.I) * (parameterSymbol H m ξ * (inner ℝ ξ v : ℂ))
+  unfold parameterFirstSymbol
+  ring
+
+end FrozenPoisson
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/ParameterSchwartz.lean b/lean/OAI/Geometry/PrescribedPotential/ParameterSchwartz.lean
new file mode 100644
index 0000000000000000000000000000000000000000..645c49ba22e019df569204298343c8ef149a4b9e
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/ParameterSchwartz.lean
@@ -0,0 +1,142 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.ParameterRegularity
+import OAI.Geometry.PrescribedPotential.SmoothDistributions
+
+namespace OAI
+
+/-! Parameter Schwartz. -/
+
+section
+
+ 
+
+noncomputable section
+open Filter Topology MeasureTheory FourierTransform TemperedDistribution LineDeriv
+open scoped SchwartzMap BoundedContinuousFunction ContDiff Real ComplexOrder MatrixOrder
+
+namespace SobolevChart
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+
+lemma schwartz_mul_temperate_distribution (a : 𝓢(E, ℂ)) {h : E → ℂ}
+    (hh : h.HasTemperateGrowth) :
+    ((SchwartzMap.smulLeftCLM ℂ h a : 𝓢(E, ℂ)) : 𝓢'(E, ℂ)) =
+      smulLeftCLM ℂ a (hh.toTemperedDistribution volume) := by
+  ext φ
+  simp only [SchwartzMap.toTemperedDistributionCLM_apply_apply,
+    smulLeftCLM_apply_apply, Function.HasTemperateGrowth.toTemperedDistribution_apply,
+    SchwartzMap.smulLeftCLM_apply_apply hh,
+    SchwartzMap.smulLeftCLM_apply_apply a.hasTemperateGrowth, smul_eq_mul]
+  apply integral_congr_ae
+  filter_upwards [] with x
+  ring
+
+ 
+
+def representativePerturbation {ι : Type*} [Fintype ι] (v : ι → E)
+    (a : SmoothCoefficients ι E) (u : E → ℂ) : 𝓢(E, ℂ) :=
+  ∑ i, ∑ j, SchwartzMap.smulLeftCLM ℂ
+    (directionDerivative (v i) (directionDerivative (v j) u)) (a i j)
+
+lemma representativePerturbation_distribution {ι : Type*} [Fintype ι] (v : ι → E)
+    (a : SmoothCoefficients ι E) {u : E → ℂ} (hu : u.HasTemperateGrowth) (w : L2 E)
+    (hw : hu.toTemperedDistribution volume = realize 2 w) :
+    ((representativePerturbation v a u : 𝓢(E, ℂ)) : 𝓢'(E, ℂ)) =
+      (perturbation v (coefficientBCF a) w : 𝓢'(E, ℂ)) := by
+  rw [perturbation_distribution v _ (fun i j => (a i j).hasTemperateGrowth)]
+  simp only [representativePerturbation, map_sum]
+  apply Finset.sum_congr rfl
+  intro i _
+  apply Finset.sum_congr rfl
+  intro j _
+  rw [schwartz_mul_temperate_distribution (a i j)
+    (temperate_directionDerivative (v i) (temperate_directionDerivative (v j) hu)),
+    temperateDistribution_derivative (v i) (temperate_directionDerivative (v j) hu),
+    temperateDistribution_derivative (v j) hu, hw]
+  rfl
+
+end SobolevChart
+
+namespace FrozenPoisson.ParameterRegularity
+open FrozenPoisson SobolevChart EllipticKernel
+variable {n : ℕ} {ι : Type*} [Fintype ι]
+
+ 
+
+theorem exists_schwartz_local (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (t : ℝ) (ht : 1 ≤ t) (v : ι → EC n) (a : SmoothCoefficients ι (EC n))
+    (hsmall : perturbationBound v (coefficientBCF a) * ellipticBound H hH < 1)
+    (f : 𝓢(EC n, ℂ)) :
+    ∃ u : 𝓢(EC n, ℂ), SchwartzMap.toTemperedDistributionCLM (EC n) ℂ volume u =
+      realize 2 (parameterLocal H hH t ht v (coefficientBCF a) hsmall (schwartzCoord 0 f)) := by
+  let w := parameterLocal H hH t ht v (coefficientBCF a) hsmall (schwartzCoord 0 f)
+  obtain ⟨u, hu, he⟩ := all_even_smooth_representative
+    (parameterLocal_schwartz_memSobolev H hH t ht v a hsmall f)
+  have he' : hu.toTemperedDistribution volume = realize 2 w := by
+    rw [temperateDistribution_bounded]
+    exact he
+  let F := f + representativePerturbation v a u
+  refine ⟨schwartzResolvent H t F, ?_⟩
+  rw [schwartzResolvent_distribution H hH t ht]
+  have hF : SchwartzMap.toTemperedDistributionCLM (EC n) ℂ volume F =
+      SchwartzMap.toTemperedDistributionCLM (EC n) ℂ volume f +
+      (perturbation v (coefficientBCF a) w : 𝓢'(EC n, ℂ)) := by
+    simp only [F, map_add, representativePerturbation_distribution v a hu w he']
+  rw [hF]
+  have hfixed := congrArg (realize (E := EC n) 2)
+    (parameterLocal_fixed H hH t ht v (coefficientBCF a) hsmall (schwartzCoord 0 f))
+  have hr := parameterHilbert_realize H hH t ht 0
+    (schwartzCoord 0 f + perturbation v (coefficientBCF a) w)
+  norm_num only [zero_add] at hr
+  rw [hr, map_add, realize_schwartzCoord] at hfixed
+  simpa only [realize, neg_zero, besselPotential_zero, ContinuousLinearMap.comp_apply,
+    ContinuousLinearMap.id_apply, Lp.toTemperedDistributionCLM_apply] using hfixed.symm
+
+ 
+
+def schwartzLocalValue (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (t : ℝ) (ht : 1 ≤ t) (v : ι → EC n) (a : SmoothCoefficients ι (EC n))
+    (hsmall : perturbationBound v (coefficientBCF a) * ellipticBound H hH < 1)
+    (f : 𝓢(EC n, ℂ)) : 𝓢(EC n, ℂ) :=
+  (exists_schwartz_local H hH t ht v a hsmall f).choose
+
+lemma schwartzLocalValue_distribution (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (t : ℝ) (ht : 1 ≤ t) (v : ι → EC n) (a : SmoothCoefficients ι (EC n))
+    (hsmall : perturbationBound v (coefficientBCF a) * ellipticBound H hH < 1)
+    (f : 𝓢(EC n, ℂ)) :
+    SchwartzMap.toTemperedDistributionCLM (EC n) ℂ volume
+      (schwartzLocalValue H hH t ht v a hsmall f) =
+      realize 2 (parameterLocal H hH t ht v (coefficientBCF a) hsmall (schwartzCoord 0 f)) :=
+  (exists_schwartz_local H hH t ht v a hsmall f).choose_spec
+
+def schwartzLocal (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (t : ℝ) (ht : 1 ≤ t) (v : ι → EC n) (a : SmoothCoefficients ι (EC n))
+    (hsmall : perturbationBound v (coefficientBCF a) * ellipticBound H hH < 1) :
+    𝓢(EC n, ℂ) →ₗ[ℂ] 𝓢(EC n, ℂ) where
+  toFun := schwartzLocalValue H hH t ht v a hsmall
+  map_add' f g := by
+    apply schwartz_distribution_injective
+    simp only [map_add, schwartzLocalValue_distribution, schwartzCoord_add]
+  map_smul' c f := by
+    apply schwartz_distribution_injective
+    have hc : schwartzCoord 0 (c • f) = c • schwartzCoord 0 f := by
+      apply realize_injective 0
+      simp only [map_smul, realize_schwartzCoord]
+    simp only [map_smul, schwartzLocalValue_distribution, hc, RingHom.id_apply]
+
+lemma schwartzLocal_coord (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (t : ℝ) (ht : 1 ≤ t) (v : ι → EC n) (a : SmoothCoefficients ι (EC n))
+    (hsmall : perturbationBound v (coefficientBCF a) * ellipticBound H hH < 1)
+    (f : 𝓢(EC n, ℂ)) :
+    schwartzCoord 2 (schwartzLocal H hH t ht v a hsmall f) =
+      parameterLocal H hH t ht v (coefficientBCF a) hsmall (schwartzCoord 0 f) := by
+  apply realize_injective 2
+  rw [realize_schwartzCoord]
+  exact schwartzLocalValue_distribution H hH t ht v a hsmall f
+
+end FrozenPoisson.ParameterRegularity
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/ParametrixAtlas.lean b/lean/OAI/Geometry/PrescribedPotential/ParametrixAtlas.lean
new file mode 100644
index 0000000000000000000000000000000000000000..7d46ee95b2c2d13b4ed518f2bf66236c467f309b
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/ParametrixAtlas.lean
@@ -0,0 +1,141 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.AtlasLocalization
+import OAI.Geometry.PrescribedPotential.ChartParametrix
+import OAI.Geometry.PrescribedPotential.LocalRegularity
+
+namespace OAI
+
+/-! Parametrix Atlas. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Manifold IsManifold
+open scoped ContDiff SchwartzMap BoundedContinuousFunction ComplexOrder MatrixOrder Matrix.Norms.Elementwise
+
+namespace Anticanonical.ComplexAtlas
+open EllipticKernel
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] (A : ComplexAtlas d X)
+
+def euclideanChart (i : Fin A.count) : OpenPartialHomeomorph X (EC d) :=
+  (A.chart i).trans (coordinateEquiv d).symm.toHomeomorph.toOpenPartialHomeomorph
+
+@[simp] lemma euclideanChart_source (i : Fin A.count) :
+    (A.euclideanChart i).source = (A.chart i).source := by
+  simp [euclideanChart]
+
+@[simp] lemma euclideanChart_target (i : Fin A.count) :
+    (A.euclideanChart i).target = (coordinateEquiv d) ⁻¹' (A.chart i).target := by
+  ext x
+  simp [euclideanChart]
+
+@[simp] lemma euclideanChart_apply (i : Fin A.count) (x : X) :
+    A.euclideanChart i x = (coordinateEquiv d).symm (A.chart i x) := rfl
+
+@[simp] lemma euclideanChart_symm_apply (i : Fin A.count) (y : EC d) :
+    (A.euclideanChart i).symm y = (A.chart i).symm (coordinateEquiv d y) := rfl
+
+lemma euclidean_transition_smooth (i j : Fin A.count) :
+    ContDiffOn ℝ ∞ ((A.euclideanChart i).symm.trans (A.euclideanChart j))
+      ((A.euclideanChart i).symm.trans (A.euclideanChart j)).source := by
+  let e := (A.euclideanChart i).symm.trans (A.euclideanChart j)
+  have he : (e : EC d → EC d) =
+      (coordinateEquiv d).symm ∘ A.transition i j ∘ coordinateEquiv d := rfl
+  rw [he]
+  apply (coordinateEquiv d).symm.contDiff.comp_contDiffOn
+  apply (A.real_smooth_transition i j).comp (coordinateEquiv d).contDiff.contDiffOn
+  intro z hz
+  exact ⟨by simpa using hz.1, by simpa using hz.2⟩
+
+end Anticanonical.ComplexAtlas
+
+namespace Anticanonical.SourceSmooth
+open EllipticKernel SobolevChart FrozenPoisson MetricLocalization
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+
+ 
+structure ParametrixPatch (g : KaehlerMetric A) where
+  index : Fin A.count
+  center : EC d
+  mem_target : center ∈ (A.euclideanChart index).target
+  domain : Set (EC d)
+  isOpen_domain : IsOpen domain
+  center_mem : center ∈ domain
+  domain_sub : domain ⊆ (A.euclideanChart index).target
+  coefficient : SmoothCoefficients (BasisIndex d) (EC d)
+  small : perturbationBound (stdOrthonormalBasis ℝ (EC d)) (coefficientBCF coefficient) *
+    ellipticBound (g.matrix index (coordinateEquiv d center))
+      (g.positive index _ (by simpa using mem_target)) < 1
+  equals : ∀ y ∈ domain, ∀ k l,
+    extendedCoefficient (g.matrix index (coordinateEquiv d center))
+      (coefficientBCF coefficient) k l y =
+      (traceBilin (g.matrix index (coordinateEquiv d y))
+        (rankTwo (stdOrthonormalBasis ℝ (EC d) k) (stdOrthonormalBasis ℝ (EC d) l)) : ℂ)
+
+namespace ParametrixPatch
+variable (g : KaehlerMetric A)
+
+def source (p : ParametrixPatch g) : Set X :=
+  (A.euclideanChart p.index).source ∩ (A.euclideanChart p.index) ⁻¹' p.domain
+
+lemma isOpen_source (p : ParametrixPatch g) : IsOpen p.source :=
+  (A.euclideanChart p.index).isOpen_inter_preimage p.isOpen_domain
+
+lemma source_sub (p : ParametrixPatch g) : p.source ⊆ (A.chart p.index).source :=
+  fun _ hx => by simpa using hx.1
+
+lemma exists_at (x : X) : ∃ p : ParametrixPatch g, x ∈ p.source := by
+  obtain ⟨i, hi⟩ := A.covers x
+  let e := A.euclideanChart i
+  have hxe : x ∈ e.source := by simpa [e] using hi
+  have hx : e x ∈ e.target := e.mapsTo hxe
+  let H : EC d → Matrix (Fin d) (Fin d) ℂ := fun z => g.matrix i (coordinateEquiv d z)
+  have hH : ContDiffOn ℝ ∞ H e.target :=
+    (g.smooth i).comp (coordinateEquiv d).contDiff.contDiffOn (fun z hz => by simpa [e] using hz)
+  have hh : ∀ z ∈ e.target, (H z).PosDef := fun z hz => g.positive i _ (by simpa [e] using hz)
+  let b := stdOrthonormalBasis ℝ (EC d)
+  obtain ⟨a, hat, hac, has, hag⟩ := local_coefficient_extension e.open_target H hH hh hx b
+  let a' : SmoothCoefficients (BasisIndex d) (EC d) :=
+    fun k l => (hac k l).toSchwartzMap (hat k l).1
+  have hae : coefficientBCF a' = a := by funext k l; ext y; rfl
+  obtain ⟨V, hVsub, hVo, hxV⟩ := mem_nhds_iff.mp (inter_mem (e.open_target.mem_nhds hx) hag)
+  let p : ParametrixPatch g :=
+    { index := i
+      center := e x
+      mem_target := hx
+      domain := V
+      isOpen_domain := hVo
+      center_mem := hxV
+      domain_sub := fun y hy => (hVsub hy).1
+      coefficient := a'
+      small := by rw [hae]; exact has
+      equals := by
+        intro y hy k l
+        rw [hae]
+        change (traceBilin (H (e x)) (rankTwo (b k) (b l)) : ℂ) + a k l y = _
+        rw [(hVsub hy).2 k l]
+        ring }
+  exact ⟨p, hxe, hxV⟩
+
+ 
+theorem finite_cover [CompactSpace X] :
+    ∃ S : Finset (ParametrixPatch g), ∀ x : X, ∃ p ∈ S, x ∈ p.source := by
+  classical
+  obtain ⟨S, hS⟩ := isCompact_univ.elim_finite_subcover (fun p : ParametrixPatch g => p.source)
+    (fun p => p.isOpen_source) (by
+      intro x _
+      obtain ⟨p, hp⟩ := exists_at g x
+      exact mem_iUnion.mpr ⟨p, hp⟩)
+  refine ⟨S, fun x => ?_⟩
+  obtain ⟨p, hp, hx⟩ := mem_iUnion₂.mp (hS (mem_univ x))
+  exact ⟨p, hp, hx⟩
+
+end ParametrixPatch
+end Anticanonical.SourceSmooth
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/ParametrixCommutator.lean b/lean/OAI/Geometry/PrescribedPotential/ParametrixCommutator.lean
new file mode 100644
index 0000000000000000000000000000000000000000..48394b9cf363f6e1c1dbf36a01ccdde5592722e7
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/ParametrixCommutator.lean
@@ -0,0 +1,209 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.ChartParametrix
+import OAI.Geometry.PrescribedPotential.ParameterLocal
+import OAI.Geometry.PrescribedPotential.SobolevLocalization
+
+namespace OAI
+
+/-! Parametrix Commutator. -/
+
+section
+
+ 
+
+noncomputable section
+open MeasureTheory FourierTransform TemperedDistribution LineDeriv
+open scoped SchwartzMap BoundedContinuousFunction ComplexOrder MatrixOrder Real
+
+namespace SobolevChart
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+
+omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] in
+lemma weak_second_product (v w : E) (g : 𝓢(E, ℂ)) (u : 𝓢'(E, ℂ)) :
+    ∂_{v} (∂_{w} (smulLeftCLM ℂ g u)) =
+      smulLeftCLM ℂ g (∂_{v} (∂_{w} u)) +
+        (smulLeftCLM ℂ (⇑(∂_{v} g : 𝓢(E, ℂ))) (∂_{w} u) +
+          smulLeftCLM ℂ (⇑(∂_{w} g : 𝓢(E, ℂ))) (∂_{v} u) +
+            smulLeftCLM ℂ (⇑(∂_{v} (∂_{w} g) : 𝓢(E, ℂ))) u) := by
+  rw [weak_deriv_product, lineDerivOp_add, weak_deriv_product, weak_deriv_product]
+  abel
+
+end SobolevChart
+
+namespace MetricLocalization
+open EllipticKernel FrozenPoisson SobolevChart
+variable {n : ℕ}
+
+ 
+
+def cutoffError (c : BasisIndex n → BasisIndex n → EC n →ᵇ ℂ)
+    (g : 𝓢(EC n, ℂ)) (u : L2 (EC n)) (Du : BasisIndex n → L2 (EC n)) : L2 (EC n) :=
+  ∑ k, ∑ l, multiply (c k l)
+    (multiply (∂_{stdOrthonormalBasis ℝ (EC n) k} g).toBoundedContinuousFunction (Du l) +
+     multiply (∂_{stdOrthonormalBasis ℝ (EC n) l} g).toBoundedContinuousFunction (Du k) +
+     multiply (∂_{stdOrthonormalBasis ℝ (EC n) k}
+       (∂_{stdOrthonormalBasis ℝ (EC n) l} g)).toBoundedContinuousFunction u)
+
+lemma cutoffError_distribution
+    (c : BasisIndex n → BasisIndex n → EC n →ᵇ ℂ)
+    (hc : ∀ k l, (c k l : EC n → ℂ).HasTemperateGrowth)
+    (g : 𝓢(EC n, ℂ)) (u : L2 (EC n)) (Du : BasisIndex n → L2 (EC n))
+    (hDu : ∀ k, (Du k : 𝓢'(EC n, ℂ)) = ∂_{stdOrthonormalBasis ℝ (EC n) k} (u : 𝓢'(EC n, ℂ))) :
+    (cutoffError c g u Du : 𝓢'(EC n, ℂ)) =
+      chartDifferential c (smulLeftCLM ℂ g (u : 𝓢'(EC n, ℂ))) -
+        smulLeftCLM ℂ g (chartDifferential c (u : 𝓢'(EC n, ℂ))) := by
+  classical
+  change Lp.toTemperedDistributionCLM ℂ volume 2 (cutoffError c g u Du) = _
+  simp only [cutoffError, map_sum, chartDifferential,
+    _root_.sum_apply, ContinuousLinearMap.comp_apply, lineDerivOpCLM_apply,
+    Lp.toTemperedDistributionCLM_apply, ← Finset.sum_sub_distrib]
+  apply Finset.sum_congr rfl
+  intro k _
+  apply Finset.sum_congr rfl
+  intro l _
+  rw [multiply_distribution _ (hc k l)]
+  simp only [← Lp.toTemperedDistributionCLM_apply, map_add]
+  simp only [Lp.toTemperedDistributionCLM_apply]
+  rw [multiply_distribution _ (∂_{stdOrthonormalBasis ℝ (EC n) k} g).hasTemperateGrowth,
+    multiply_distribution _ (∂_{stdOrthonormalBasis ℝ (EC n) l} g).hasTemperateGrowth,
+    multiply_distribution _ (∂_{stdOrthonormalBasis ℝ (EC n) k}
+      (∂_{stdOrthonormalBasis ℝ (EC n) l} g)).hasTemperateGrowth,
+    hDu, hDu, SobolevChart.weak_second_product]
+  simp only [map_add]
+  have he : smulLeftCLM ℂ (c k l) (smulLeftCLM ℂ g
+      (∂_{stdOrthonormalBasis ℝ (EC n) k}
+        (∂_{stdOrthonormalBasis ℝ (EC n) l} (u : 𝓢'(EC n, ℂ))))) =
+      smulLeftCLM ℂ g (smulLeftCLM ℂ (c k l)
+        (∂_{stdOrthonormalBasis ℝ (EC n) k}
+          (∂_{stdOrthonormalBasis ℝ (EC n) l} (u : 𝓢'(EC n, ℂ))))) := by
+    rw [smulLeftCLM_smulLeftCLM_apply g.hasTemperateGrowth (hc k l),
+      smulLeftCLM_smulLeftCLM_apply (hc k l) g.hasTemperateGrowth]
+    congr 2
+    ext x
+    exact mul_comm _ _
+  rw [he]
+  abel
+
+lemma cutoffError_bound
+    (c : BasisIndex n → BasisIndex n → EC n →ᵇ ℂ)
+    (g : 𝓢(EC n, ℂ)) (u : L2 (EC n)) (Du : BasisIndex n → L2 (EC n)) :
+    ‖cutoffError c g u Du‖ ≤ ∑ k, ∑ l, ‖c k l‖ *
+      (‖(∂_{stdOrthonormalBasis ℝ (EC n) k} g).toBoundedContinuousFunction‖ * ‖Du l‖ +
+       ‖(∂_{stdOrthonormalBasis ℝ (EC n) l} g).toBoundedContinuousFunction‖ * ‖Du k‖ +
+       ‖(∂_{stdOrthonormalBasis ℝ (EC n) k}
+          (∂_{stdOrthonormalBasis ℝ (EC n) l} g)).toBoundedContinuousFunction‖ * ‖u‖) := by
+  classical
+  unfold cutoffError
+  apply (norm_sum_le _ _).trans
+  apply Finset.sum_le_sum
+  intro k _
+  apply (norm_sum_le _ _).trans
+  apply Finset.sum_le_sum
+  intro l _
+  apply (multiply_norm_le _ _).trans
+  apply mul_le_mul_of_nonneg_left _ (norm_nonneg _)
+  apply (norm_add_le _ _).trans
+  apply add_le_add
+  · exact (norm_add_le _ _).trans (add_le_add (multiply_norm_le _ _) (multiply_norm_le _ _))
+  · exact multiply_norm_le _ _
+
+ 
+
+def parameterCutoffError (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (m : ℝ) (hm : 1 ≤ m)
+    (a : BasisIndex n → BasisIndex n → EC n →ᵇ ℂ)
+    (hsmall : perturbationBound (stdOrthonormalBasis ℝ (EC n)) a * ellipticBound H hH < 1)
+    (g : 𝓢(EC n, ℂ)) (f : L2 (EC n)) : L2 (EC n) :=
+  cutoffError (extendedCoefficient H a) g
+    (parameterLocalZero H hH m hm (stdOrthonormalBasis ℝ (EC n)) a hsmall f)
+    (fun k => parameterLocalFirst H hH m hm (stdOrthonormalBasis ℝ (EC n)) a hsmall
+      (stdOrthonormalBasis ℝ (EC n) k) f)
+
+lemma parameterCutoffError_distribution (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (m : ℝ) (hm : 1 ≤ m)
+    (a : BasisIndex n → BasisIndex n → EC n →ᵇ ℂ)
+    (ha : ∀ k l, (a k l : EC n → ℂ).HasTemperateGrowth)
+    (hsmall : perturbationBound (stdOrthonormalBasis ℝ (EC n)) a * ellipticBound H hH < 1)
+    (g : 𝓢(EC n, ℂ)) (f : L2 (EC n)) :
+    let u := realize 2 (parameterLocal H hH m hm (stdOrthonormalBasis ℝ (EC n)) a hsmall f)
+    (parameterCutoffError H hH m hm a hsmall g f : 𝓢'(EC n, ℂ)) =
+      chartDifferential (extendedCoefficient H a) (smulLeftCLM ℂ g u) -
+        smulLeftCLM ℂ g (chartDifferential (extendedCoefficient H a) u) := by
+  have hDu : ∀ k,
+      (parameterLocalFirst H hH m hm (stdOrthonormalBasis ℝ (EC n)) a hsmall
+        (stdOrthonormalBasis ℝ (EC n) k) f : 𝓢'(EC n, ℂ)) =
+      ∂_{stdOrthonormalBasis ℝ (EC n) k}
+        (parameterLocalZero H hH m hm (stdOrthonormalBasis ℝ (EC n)) a hsmall f : 𝓢'(EC n, ℂ)) := by
+    intro k
+    rw [parameterLocalFirst_distribution, parameterLocalZero_distribution]
+  simpa only [parameterCutoffError, parameterLocalZero_distribution] using
+    cutoffError_distribution (extendedCoefficient H a) (extendedCoefficient_temperate H a ha)
+      g _ _ hDu
+
+ 
+def cutoffErrorConstant (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (a : BasisIndex n → BasisIndex n → EC n →ᵇ ℂ) (g : 𝓢(EC n, ℂ)) : ℝ :=
+  (∑ k, ∑ l, ‖extendedCoefficient H a k l‖ *
+    (‖(∂_{stdOrthonormalBasis ℝ (EC n) k} g).toBoundedContinuousFunction‖ *
+        (2 * Real.pi * ‖stdOrthonormalBasis ℝ (EC n) l‖ * ellipticBound H hH) +
+     ‖(∂_{stdOrthonormalBasis ℝ (EC n) l} g).toBoundedContinuousFunction‖ *
+        (2 * Real.pi * ‖stdOrthonormalBasis ℝ (EC n) k‖ * ellipticBound H hH) +
+     ‖(∂_{stdOrthonormalBasis ℝ (EC n) k}
+        (∂_{stdOrthonormalBasis ℝ (EC n) l} g)).toBoundedContinuousFunction‖)) /
+    (1 - perturbationBound (stdOrthonormalBasis ℝ (EC n)) a * ellipticBound H hH)
+
+lemma parameterCutoffError_bound (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (m : ℝ) (hm : 1 ≤ m)
+    (a : BasisIndex n → BasisIndex n → EC n →ᵇ ℂ)
+    (hsmall : perturbationBound (stdOrthonormalBasis ℝ (EC n)) a * ellipticBound H hH < 1)
+    (g : 𝓢(EC n, ℂ)) (f : L2 (EC n)) :
+    ‖parameterCutoffError H hH m hm a hsmall g f‖ ≤
+      (cutoffErrorConstant H hH a g / m) * ‖f‖ := by
+  classical
+  have hm0 : 0 < m := lt_of_lt_of_le zero_lt_one hm
+  have hden : 0 < 1 - perturbationBound (stdOrthonormalBasis ℝ (EC n)) a * ellipticBound H hH :=
+    sub_pos.mpr hsmall
+  have hz : ‖parameterLocalZero H hH m hm (stdOrthonormalBasis ℝ (EC n)) a hsmall f‖ ≤
+      (1 / (m * (1 - perturbationBound (stdOrthonormalBasis ℝ (EC n)) a * ellipticBound H hH))) * ‖f‖ := by
+    apply (parameterLocalZero_bound H hH m hm _ a hsmall f).trans
+    apply mul_le_mul_of_nonneg_right _ (norm_nonneg _)
+    have hi : (m^2)⁻¹ ≤ m⁻¹ := inv_anti₀ hm0 (by nlinarith)
+    calc
+      _ ≤ m⁻¹ / (1 - perturbationBound (stdOrthonormalBasis ℝ (EC n)) a * ellipticBound H hH) :=
+        div_le_div_of_nonneg_right hi hden.le
+      _ = _ := by simp only [div_eq_mul_inv, mul_inv_rev, one_mul]; ring
+  apply (cutoffError_bound (extendedCoefficient H a) g _ _).trans
+  unfold cutoffErrorConstant
+  simp only [Finset.sum_div, Finset.sum_mul]
+  apply Finset.sum_le_sum
+  intro k _
+  apply Finset.sum_le_sum
+  intro l _
+  calc
+    _ ≤ ‖extendedCoefficient H a k l‖ *
+        (‖(∂_{stdOrthonormalBasis ℝ (EC n) k} g).toBoundedContinuousFunction‖ *
+            (((2 * Real.pi * ‖stdOrthonormalBasis ℝ (EC n) l‖ * ellipticBound H hH) /
+              (m * (1 - perturbationBound (stdOrthonormalBasis ℝ (EC n)) a * ellipticBound H hH))) * ‖f‖) +
+         ‖(∂_{stdOrthonormalBasis ℝ (EC n) l} g).toBoundedContinuousFunction‖ *
+            (((2 * Real.pi * ‖stdOrthonormalBasis ℝ (EC n) k‖ * ellipticBound H hH) /
+              (m * (1 - perturbationBound (stdOrthonormalBasis ℝ (EC n)) a * ellipticBound H hH))) * ‖f‖) +
+         ‖(∂_{stdOrthonormalBasis ℝ (EC n) k}
+            (∂_{stdOrthonormalBasis ℝ (EC n) l} g)).toBoundedContinuousFunction‖ *
+            ((1 / (m * (1 - perturbationBound (stdOrthonormalBasis ℝ (EC n)) a * ellipticBound H hH))) * ‖f‖)) := by
+      apply mul_le_mul_of_nonneg_left _ (norm_nonneg _)
+      apply add_le_add
+      · apply add_le_add
+        · exact mul_le_mul_of_nonneg_left (parameterLocalFirst_bound H hH m hm _ a hsmall _ f)
+            (norm_nonneg _)
+        · exact mul_le_mul_of_nonneg_left (parameterLocalFirst_bound H hH m hm _ a hsmall _ f)
+            (norm_nonneg _)
+      · exact mul_le_mul_of_nonneg_left hz (norm_nonneg _)
+    _ = _ := by simp only [div_eq_mul_inv, mul_inv_rev]; ring
+
+end MetricLocalization
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/PatchCutoffs.lean b/lean/OAI/Geometry/PrescribedPotential/PatchCutoffs.lean
new file mode 100644
index 0000000000000000000000000000000000000000..1a9a1138678fca0c4a0e9821811ba15d1226674b
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/PatchCutoffs.lean
@@ -0,0 +1,115 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.ChartTransport
+import OAI.Geometry.PrescribedPotential.GlobalPartition
+
+namespace OAI
+
+/-! Patch Cutoffs. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory Manifold IsManifold
+open scoped ContDiff SchwartzMap Classical
+
+namespace GlobalElliptic
+open Anticanonical EllipticKernel SobolevChart SourceSmooth
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E]
+
+ 
+
+theorem exists_chartCutoff {K U : Set E} (hK : IsCompact K) (hU : IsOpen U) (hKU : K ⊆ U) :
+    ∃ κ : ChartCutoff U, ∀ x ∈ K, κ x = 1 := by
+  obtain ⟨L, hLc, hLl, hKL, hLU⟩ := exists_compact_closed_between hK hU hKU
+  obtain ⟨f, hf, hf0, _⟩ := exists_contMDiffMap_one_nhds_of_subset_interior
+    (𝓘(ℝ, E)) hK.isClosed hKL (n := ⊤)
+  have hsm : ContDiff ℝ ∞ (fun x => (f x : ℂ)) :=
+    Complex.ofRealCLM.contDiff.comp (contMDiff_iff_contDiff.mp f.contMDiff)
+  have hsub : tsupport (fun x => (f x : ℂ)) ⊆ L := by
+    apply closure_minimal _ hLl
+    intro x hx
+    by_contra hn
+    exact hx (by simp [hf0 x hn])
+  have hc : HasCompactSupport (fun x => (f x : ℂ)) :=
+    hLc.of_isClosed_subset (isClosed_tsupport _) hsub
+  refine ⟨⟨hc.toSchwartzMap hsm, hc, hsub.trans hLU⟩, ?_⟩
+  intro x hx
+  change (f x : ℂ) = 1
+  rw [hf.self_of_nhdsSet x hx]
+  rfl
+
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X}
+
+ 
+
+structure GluingData (g : KaehlerMetric A) (ι : Type*) [Fintype ι] where
+  patch : ι → ParametrixPatch g
+  localizers : Localizers A ι
+  index_eq : ∀ p, localizers.index p = (patch p).index
+  support_patch : ∀ p, tsupport (localizers.weight p : X → ℂ) ⊆ (patch p).source
+  cutoff : ∀ p, ChartCutoff (A.euclideanChart (patch p).index).target
+  cutoff_support : ∀ p, tsupport (cutoff p : EC d → ℂ) ⊆ (patch p).domain
+  cutoff_one : ∀ p x, x ∈ tsupport (localizers.weight p : X → ℂ) →
+    cutoff p (A.euclideanChart (patch p).index x) = 1
+
+ 
+
+theorem exists_gluingData (g : KaehlerMetric A) :
+    ∃ (S : Finset (ParametrixPatch g)), Nonempty (GluingData g S) := by
+  obtain ⟨S, D, hDI, hDS⟩ := exists_patch_localizers g
+  have hc (p : S) : ∃ κ : ChartCutoff p.val.domain,
+      ∀ x ∈ tsupport (D.weight p : X → ℂ), κ (A.euclideanChart p.val.index x) = 1 := by
+    let e := A.euclideanChart p.val.index
+    have hK : IsCompact (e '' tsupport (D.weight p : X → ℂ)) :=
+      (isClosed_tsupport _).isCompact.image_of_continuousOn
+        (e.continuousOn.mono (fun x hx => (hDS p hx).1))
+    have hKU : e '' tsupport (D.weight p : X → ℂ) ⊆ p.val.domain := by
+      rintro y ⟨x, hx, rfl⟩
+      exact (hDS p hx).2
+    obtain ⟨κ, hκ⟩ := exists_chartCutoff hK p.val.isOpen_domain hKU
+    exact ⟨κ, fun x hx => hκ _ ⟨x, hx, rfl⟩⟩
+  choose κ hκ using hc
+  refine ⟨S, ⟨{
+    patch := Subtype.val
+    localizers := D
+    index_eq := hDI
+    support_patch := hDS
+    cutoff := fun p => ⟨(κ p).val, (κ p).compact, (κ p).support_sub.trans p.val.domain_sub⟩
+    cutoff_support := fun p => (κ p).support_sub
+    cutoff_one := hκ }⟩⟩
+
+namespace GluingData
+variable {ι : Type*} [Fintype ι] {g : KaehlerMetric A} (D : GluingData g ι)
+
+omit [T2Space X] [CompactSpace X]
+lemma exists_outerCutoff (p : ι) :
+    ∃ η : ChartCutoff (A.euclideanChart (D.patch p).index).target,
+      tsupport (η : EC d → ℂ) ⊆ (D.patch p).domain ∧
+      ∀ y ∈ tsupport (D.cutoff p : EC d → ℂ), η y = 1 := by
+  obtain ⟨η, hη⟩ := exists_chartCutoff (D.cutoff p).compact
+    (D.patch p).isOpen_domain (D.cutoff_support p)
+  exact ⟨⟨η.val, η.compact, η.support_sub.trans (D.patch p).domain_sub⟩, η.support_sub, hη⟩
+
+ 
+
+def outerCutoff (p : ι) : ChartCutoff (A.euclideanChart (D.patch p).index).target :=
+  (D.exists_outerCutoff p).choose
+
+lemma outerCutoff_support (p : ι) :
+    tsupport (D.outerCutoff p : EC d → ℂ) ⊆ (D.patch p).domain :=
+  (D.exists_outerCutoff p).choose_spec.1
+
+lemma outerCutoff_one (p : ι) (y : EC d) (hy : y ∈ tsupport (D.cutoff p : EC d → ℂ)) :
+    D.outerCutoff p y = 1 := (D.exists_outerCutoff p).choose_spec.2 y hy
+end GluingData
+
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/PatchDifferential.lean b/lean/OAI/Geometry/PrescribedPotential/PatchDifferential.lean
new file mode 100644
index 0000000000000000000000000000000000000000..a2d654a6d93ed45e0f5be1e9c8646eb9a98640c1
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/PatchDifferential.lean
@@ -0,0 +1,200 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.ComplexOperator
+import OAI.Geometry.PrescribedPotential.CoreBounds
+import OAI.Geometry.PrescribedPotential.GlobalParametrix
+import OAI.Geometry.PrescribedPotential.ParameterSchwartz
+import OAI.Geometry.PrescribedPotential.ParametrixCommutator
+
+namespace OAI
+
+/-! Patch Differential. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory TemperedDistribution LineDeriv
+open scoped SchwartzMap ContDiff Classical ComplexOrder MatrixOrder BoundedContinuousFunction
+
+namespace SobolevChart
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+
+lemma schwartz_temperate_product_distribution {a : E → ℂ}
+    (ha : a.HasTemperateGrowth) (f : 𝓢(E, ℂ)) :
+    SchwartzMap.toTemperedDistributionCLM E ℂ volume (SchwartzMap.smulLeftCLM ℂ a f) =
+      smulLeftCLM ℂ a (f : 𝓢'(E, ℂ)) := by
+  ext φ
+  simp only [SchwartzMap.toTemperedDistributionCLM_apply_apply,
+    smulLeftCLM_apply_apply, SchwartzMap.smulLeftCLM_apply_apply ha, smul_eq_mul]
+  apply integral_congr_ae
+  filter_upwards [] with x
+  ring
+
+omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] in
+lemma schwartz_second_deriv (f : 𝓢(E, ℂ)) (v w x : E) :
+    (∂_{v} (∂_{w} f)) x = fderiv ℝ (fderiv ℝ f) x v w := by
+  simp only [SchwartzMap.lineDerivOp_apply_eq_fderiv]
+  change fderiv ℝ (fun y => fderiv ℝ f y w) x v = _
+  exact PotentialKaehler.fderiv_eval_derivative
+    f.smooth'.contDiffAt v w
+end SobolevChart
+
+namespace MetricLocalization
+open EllipticKernel FrozenPoisson SobolevChart
+variable {n : ℕ}
+
+def schwartzDifferential (c : BasisIndex n → BasisIndex n → EC n →ᵇ ℂ) :
+    𝓢(EC n, ℂ) →L[ℂ] 𝓢(EC n, ℂ) :=
+  ∑ k, ∑ l, SchwartzMap.smulLeftCLM ℂ (c k l) ∘L
+    lineDerivOpCLM ℂ 𝓢(EC n, ℂ) (stdOrthonormalBasis ℝ (EC n) k) ∘L
+    lineDerivOpCLM ℂ 𝓢(EC n, ℂ) (stdOrthonormalBasis ℝ (EC n) l)
+
+lemma schwartzDifferential_apply
+    (c : BasisIndex n → BasisIndex n → EC n →ᵇ ℂ)
+    (hc : ∀ k l, (c k l : EC n → ℂ).HasTemperateGrowth)
+    (f : 𝓢(EC n, ℂ)) (x : EC n) :
+    schwartzDifferential c f x = ∑ k, ∑ l, c k l x *
+      fderiv ℝ (fderiv ℝ f) x (stdOrthonormalBasis ℝ (EC n) k)
+        (stdOrthonormalBasis ℝ (EC n) l) := by
+  simp only [schwartzDifferential, _root_.sum_apply,
+    ContinuousLinearMap.comp_apply, lineDerivOpCLM_apply,
+    SchwartzMap.smulLeftCLM_apply_apply (hc _ _), smul_eq_mul, schwartz_second_deriv]
+
+lemma schwartzDifferential_distribution
+    (c : BasisIndex n → BasisIndex n → EC n →ᵇ ℂ)
+    (hc : ∀ k l, (c k l : EC n → ℂ).HasTemperateGrowth) (f : 𝓢(EC n, ℂ)) :
+    SchwartzMap.toTemperedDistributionCLM (EC n) ℂ volume (schwartzDifferential c f) =
+      chartDifferential c (SchwartzMap.toTemperedDistributionCLM (EC n) ℂ volume f) := by
+  simp only [schwartzDifferential, chartDifferential, map_sum,
+    _root_.sum_apply, ContinuousLinearMap.comp_apply, lineDerivOpCLM_apply]
+  apply Finset.sum_congr rfl
+  intro k _
+  apply Finset.sum_congr rfl
+  intro l _
+  rw [schwartz_temperate_product_distribution (hc k l),
+    ← lineDerivOp_toTemperedDistributionCLM_eq, ← lineDerivOp_toTemperedDistributionCLM_eq]
+
+lemma schwartzDifferential_tsupport
+    (c : BasisIndex n → BasisIndex n → EC n →ᵇ ℂ) (f : 𝓢(EC n, ℂ)) :
+    tsupport (schwartzDifferential c f : EC n → ℂ) ⊆ tsupport f := by
+  apply closure_minimal _ (isClosed_tsupport _)
+  intro x hx
+  by_contra hn
+  apply hx
+  simp only [schwartzDifferential, _root_.sum_apply,
+    ContinuousLinearMap.comp_apply, lineDerivOpCLM_apply]
+  apply Finset.sum_eq_zero
+  intro k _
+  apply Finset.sum_eq_zero
+  intro l _
+  apply image_eq_zero_of_notMem_tsupport
+  intro hl
+  exact hn (SchwartzMap.tsupport_lineDerivOp_subset _ _
+    (SchwartzMap.tsupport_lineDerivOp_subset _ _
+      (SchwartzMap.tsupport_smulLeftCLM_subset _ _ hl).1))
+
+ 
+def schwartzCutoffError (c : BasisIndex n → BasisIndex n → EC n →ᵇ ℂ)
+    (κ : 𝓢(EC n, ℂ)) : 𝓢(EC n, ℂ) →L[ℂ] 𝓢(EC n, ℂ) :=
+  (schwartzDifferential c).comp (SchwartzMap.smulLeftCLM ℂ κ) -
+    (SchwartzMap.smulLeftCLM ℂ κ).comp (schwartzDifferential c)
+
+lemma schwartzCutoffError_distribution
+    (c : BasisIndex n → BasisIndex n → EC n →ᵇ ℂ)
+    (hc : ∀ k l, (c k l : EC n → ℂ).HasTemperateGrowth)
+    (κ f : 𝓢(EC n, ℂ)) :
+    SchwartzMap.toTemperedDistributionCLM (EC n) ℂ volume (schwartzCutoffError c κ f) =
+      chartDifferential c (smulLeftCLM ℂ κ (SchwartzMap.toTemperedDistributionCLM (EC n) ℂ volume f)) -
+        smulLeftCLM ℂ κ (chartDifferential c (SchwartzMap.toTemperedDistributionCLM (EC n) ℂ volume f)) := by
+  simp only [schwartzCutoffError, _root_.sub_apply, ContinuousLinearMap.comp_apply,
+    map_sub, schwartzDifferential_distribution c hc,
+    schwartz_temperate_product_distribution κ.hasTemperateGrowth]
+
+lemma schwartzCutoffError_tsupport
+    (c : BasisIndex n → BasisIndex n → EC n →ᵇ ℂ) (κ f : 𝓢(EC n, ℂ)) :
+    tsupport (schwartzCutoffError c κ f : EC n → ℂ) ⊆ tsupport κ := by
+  change tsupport (fun x => schwartzDifferential c (SchwartzMap.smulLeftCLM ℂ κ f) x -
+    SchwartzMap.smulLeftCLM ℂ κ (schwartzDifferential c f) x) ⊆ _
+  apply (tsupport_binop_subset (fun a b : ℂ => a - b) (sub_self 0) _ _).trans
+  apply union_subset
+  · exact (schwartzDifferential_tsupport c _).trans
+      ((SchwartzMap.tsupport_smulLeftCLM_subset _ _).trans inter_subset_right)
+  · exact (SchwartzMap.tsupport_smulLeftCLM_subset _ _).trans inter_subset_right
+
+end MetricLocalization
+
+namespace Anticanonical.SourceSmooth.ParametrixPatch
+open EllipticKernel SobolevChart FrozenPoisson ParameterRegularity MetricLocalization
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+  {g : KaehlerMetric A}
+
+abbrev coefficients (p : ParametrixPatch g) := extendedCoefficient p.matrix (coefficientBCF p.coefficient)
+
+lemma coefficients_temperate (p : ParametrixPatch g) (k l : BasisIndex d) :
+    (p.coefficients k l : EC d → ℂ).HasTemperateGrowth :=
+  extendedCoefficient_temperate p.matrix _ (fun k l => (p.coefficient k l).hasTemperateGrowth) k l
+
+def differential (p : ParametrixPatch g) := schwartzDifferential p.coefficients
+
+def cutoffError (p : ParametrixPatch g) (κ : 𝓢(EC d, ℂ)) :=
+  schwartzCutoffError p.coefficients κ
+
+lemma differential_local (p : ParametrixPatch g) (f : 𝓢(EC d, ℂ))
+    {y : EC d} (hy : y ∈ p.domain) :
+    p.differential f y = GlobalElliptic.localL g p.index f y := by
+  rw [differential, schwartzDifferential_apply _ p.coefficients_temperate]
+  unfold GlobalElliptic.localL
+  apply Finset.sum_congr rfl
+  intro k _
+  apply Finset.sum_congr rfl
+  intro l _
+  rw [p.equals y hy k l]
+
+lemma localInverse_distribution (p : ParametrixPatch g) (m : ℝ) (hm : 1 ≤ m)
+    (f : 𝓢(EC d, ℂ)) :
+    SchwartzMap.toTemperedDistributionCLM (EC d) ℂ volume (p.localInverse m hm f) =
+      realize 2 (parameterLocal p.matrix p.positive m hm (stdOrthonormalBasis ℝ (EC d))
+        (coefficientBCF p.coefficient) p.small (schwartzCoord 0 f)) :=
+  schwartzLocalValue_distribution p.matrix p.positive m hm (stdOrthonormalBasis ℝ (EC d))
+    p.coefficient p.small f
+
+lemma localInverse_equation (p : ParametrixPatch g) (m : ℝ) (hm : 1 ≤ m)
+    (f : 𝓢(EC d, ℂ)) :
+    (m^2 : ℂ) • p.localInverse m hm f - p.differential (p.localInverse m hm f) = f := by
+  apply schwartz_distribution_injective
+  simp only [map_sub, map_smul, differential, schwartzDifferential_distribution _ p.coefficients_temperate,
+    localInverse_distribution]
+  rw [chartDifferential_extension p.matrix _ (fun k l => (p.coefficient k l).hasTemperateGrowth)]
+  rw [sub_add_eq_sub_sub, parameterLocal_equation]
+  have h := realize_schwartzCoord (0 : ℝ) f
+  simpa only [realize, neg_zero, besselPotential_zero, ContinuousLinearMap.comp_apply,
+    ContinuousLinearMap.id_apply, Lp.toTemperedDistributionCLM_apply] using h
+
+lemma cutoffError_coord (p : ParametrixPatch g) (m : ℝ) (hm : 1 ≤ m)
+    (κ f : 𝓢(EC d, ℂ)) :
+    schwartzCoord 0 (p.cutoffError κ (p.localInverse m hm f)) =
+      parameterCutoffError p.matrix p.positive m hm (coefficientBCF p.coefficient) p.small κ
+        (schwartzCoord 0 f) := by
+  apply l2_injective
+  simp only [Lp.toTemperedDistributionCLM_apply]
+  rw [schwartzCoord_distribution, besselPotential_zero, ContinuousLinearMap.id_apply]
+  rw [cutoffError, schwartzCutoffError_distribution _ p.coefficients_temperate,
+    localInverse_distribution, parameterCutoffError_distribution _ _ _ _ _
+      (fun k l => (p.coefficient k l).hasTemperateGrowth)]
+
+lemma cutoffError_bound (p : ParametrixPatch g) (m : ℝ) (hm : 1 ≤ m)
+    (κ f : 𝓢(EC d, ℂ)) :
+    ‖schwartzCoord 0 (p.cutoffError κ (p.localInverse m hm f))‖ ≤
+      cutoffErrorConstant p.matrix p.positive (coefficientBCF p.coefficient) κ / m *
+        ‖schwartzCoord 0 f‖ := by
+  rw [cutoffError_coord]
+  exact parameterCutoffError_bound p.matrix p.positive m hm _ p.small κ _
+
+end Anticanonical.SourceSmooth.ParametrixPatch
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/PathCalabiBound.lean b/lean/OAI/Geometry/PrescribedPotential/PathCalabiBound.lean
new file mode 100644
index 0000000000000000000000000000000000000000..1ff139ff5f75560b48de0270186f8276e9185e4d
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/PathCalabiBound.lean
@@ -0,0 +1,99 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CalabiLocalUniform
+import OAI.Geometry.PrescribedPotential.PathCalabiRicci
+import OAI.Geometry.PrescribedPotential.PathEllipticity
+
+namespace OAI
+
+/-! Path Calabi Bound. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Matrix Metric
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace Anticanonical.SourceSmooth
+open KaehlerCalculus EllipticKernel
+local instance pathCalabiRealIP (d : ℕ) : InnerProductSpace ℝ (EC d) :=
+  InnerProductSpace.rclikeToReal ℂ (EC d)
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [CompactSpace X] [ConnectedSpace X]
+  {A : ComplexAtlas d X}
+namespace CoordinateBall
+variable (p : CoordinateBall A)
+
+theorem path_calabi_bound (g : KaehlerMetric A) (h : SemipositiveAnticanonicalMetric A) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ (t b : ℝ) (φ : SmoothRealFunction A), t ∈ Icc 0 1 →
+      ∀ hs : SolvesVolumePath g h t φ b, ∀ x ∈ p.source,
+        ((g.deform φ hs.choose).localField p.index).calabiNorm (A.chart p.index x) ≤ C := by
+  obtain ⟨R,K,hR,hK,hell⟩ := volumePath_relative_ellipticity g h
+  let ψ : ContDiffBump p.center :=
+    { rIn := p.radius, rOut := 2*p.radius, rIn_pos := p.radius_pos,
+      rIn_lt_rOut := by linarith [p.radius_pos] }
+  let L : Set (V d) := coordinateEquiv d '' closedBall p.center (3*p.radius)
+  let χ : V d → ℝ := fun z => ψ ((coordinateEquiv d).symm z)
+  have hL : IsCompact L := (isCompact_closedBall _ _).image (coordinateEquiv d).continuous
+  have hLc (z : V d) : z ∈ L ↔ (coordinateEquiv d).symm z ∈ closedBall p.center (3*p.radius) := by
+    constructor
+    · rintro ⟨y,hy,rfl⟩
+      simpa only [ContinuousLinearEquiv.symm_apply_apply] using hy
+    · intro hz
+      exact ⟨_,hz,(coordinateEquiv d).apply_symm_apply z⟩
+  have hdom : L ⊆ (A.chart p.index).target := by
+    intro z hz
+    have hh := p.closure_sub (closedBall_subset_closedBall (by linarith [p.radius_pos]) ((hLc z).mp hz))
+    simpa only [ComplexAtlas.euclideanChart_target,mem_preimage,ContinuousLinearEquiv.apply_symm_apply] using hh
+  have hχ : ContDiff ℝ ∞ χ := ψ.contDiff.comp (coordinateEquiv d).symm.contDiff
+  have hχint (z : V d) (hz : z ∈ L) (hn : χ z ≠ 0) : z ∈ interior L := by
+    have hs : (coordinateEquiv d).symm z ∈ ball p.center (2*p.radius) := by
+      rw [← ψ.support_eq]
+      exact hn
+    have hs' : (coordinateEquiv d).symm z ∈ ball p.center (3*p.radius) :=
+      ball_subset_ball (by linarith [p.radius_pos]) hs
+    apply mem_interior_iff_mem_nhds.mpr
+    apply mem_of_superset (((isOpen_ball.preimage (coordinateEquiv d).symm.continuous).mem_nhds hs'))
+    intro y hy
+    exact (hLc y).mpr (ball_subset_closedBall hy)
+  have hHG := (g.smooth p.index).continuousOn.mono hdom
+  have hH : ContinuousOn (fun q : ℝ × V d => pathLogDetHessian g h p.index q.1 q.2) (Icc 0 1 ×ˢ L) :=
+    (pathLogDetHessian_continuous g h p.index).mono (prod_mono (subset_univ _) hdom)
+  have hJ : ContinuousOn (fun q : ℝ × V d => pathRicciJet g h p.index q.1 q.2) (Icc 0 1 ×ˢ L) :=
+    (pathRicciJet_continuous g h p.index).mono (prod_mono (subset_univ _) hdom)
+  obtain ⟨C,hC,hbound⟩ := calabi_interior_uniform isCompact_Icc hL (g.matrix p.index) hHG
+    (fun z hz => g.positive _ _ (hdom hz))
+    (fun q : ℝ × V d => pathLogDetHessian g h p.index q.1 q.2)
+    (fun q : ℝ × V d => pathRicciJet g h p.index q.1 q.2) hH hJ χ hχ
+    (fun _ _ => ψ.nonneg) (fun _ _ => ψ.le_one) hχint hR.le hK.le
+  refine ⟨C,hC,?_⟩
+  intro t b φ ht hs x hx
+  have hxV : A.chart p.index x ∈ L := by
+    apply (hLc _).mpr
+    exact closedBall_subset_closedBall (by linarith [p.radius_pos]) (ball_subset_closedBall hx.2)
+  apply hbound ((g.deform φ hs.choose).localField p.index) t ht hdom
+    (fun z hz => hell t b φ ht hs p.index z (hdom hz))
+    (fun z hz => volumePath_logdet_hessian g h hs hs.choose p.index (hdom hz))
+    (fun z hz => volumePath_ricciJet g h hs hs.choose p.index (hdom hz)) _ hxV
+  exact ψ.one_of_mem_closedBall (ball_subset_closedBall hx.2)
+end CoordinateBall
+
+ 
+
+theorem volumePath_calabi_finite_cover (g : KaehlerMetric A) (h : SemipositiveAnticanonicalMetric A) :
+    ∃ (S : Finset (CoordinateBall A)) (C : ℝ), 0 ≤ C ∧
+      (∀ x, ∃ p ∈ S, x ∈ p.source) ∧
+      ∀ p ∈ S, ∀ (t b : ℝ) (φ : SmoothRealFunction A), t ∈ Icc 0 1 →
+        ∀ hs : SolvesVolumePath g h t φ b, ∀ x ∈ p.source,
+          ((g.deform φ hs.choose).localField p.index).calabiNorm (A.chart p.index x) ≤ C := by
+  classical
+  obtain ⟨S,hS⟩ := CoordinateBall.finite_cover (A := A)
+  choose C hC hb using fun p : CoordinateBall A => p.path_calabi_bound g h
+  refine ⟨S,∑ p ∈ S, C p,Finset.sum_nonneg (fun p _ => hC p),hS,?_⟩
+  intro p hp t b φ ht hs x hx
+  exact (hb p t b φ ht hs x hx).trans (Finset.single_le_sum (fun q _ => hC q) hp)
+end Anticanonical.SourceSmooth
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/PathCalabiRicci.lean b/lean/OAI/Geometry/PrescribedPotential/PathCalabiRicci.lean
new file mode 100644
index 0000000000000000000000000000000000000000..22bcc376082d303c0500444a0e5bab595ab6c100
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/PathCalabiRicci.lean
@@ -0,0 +1,86 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CalabiConnection
+import OAI.Geometry.PrescribedPotential.CalabiCovariant
+import OAI.Geometry.PrescribedPotential.CalabiTensorAlgebra
+import OAI.Geometry.PrescribedPotential.MatrixTensorCalculus
+import OAI.Geometry.PrescribedPotential.PathRicci
+
+namespace OAI
+
+/-! Path Calabi Ricci. -/
+
+section
+
+noncomputable section
+open Set Filter Topology Matrix Metric
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace Anticanonical.SourceSmooth
+open KaehlerCalculus EllipticKernel
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+
+namespace KaehlerMetric
+
+def localField (g : KaehlerMetric A) (i : Fin A.count) : LocalKaehlerField d where
+  domain := (A.chart i).target
+  isOpen := (A.chart i).open_target
+  matrix := g.matrix i
+  smooth := g.smooth i
+  positive := g.positive i
+  closed := g.closed i
+
+end KaehlerMetric
+
+def pathRicciJet (g : KaehlerMetric A) (h : SemipositiveAnticanonicalMetric A)
+    (i : Fin A.count) (t : ℝ) (z : Coordinates d) : ConnectionTensor d := fun k =>
+  ((1-t:ℝ):ℂ) • mderiv (-Complex.I) (e k)
+      (PotentialKaehler.potentialMatrix (fun y => Real.log (g.matrix i y).det.re)) z -
+    (t:ℂ) • mderiv (-Complex.I) (e k) (PotentialKaehler.potentialMatrix (h.weight i)) z
+
+lemma pathLogDetHessian_jet (g : KaehlerMetric A) (h : SemipositiveAnticanonicalMetric A)
+    (i : Fin A.count) (t : ℝ) {z : Coordinates d} (hz : z ∈ (A.chart i).target) :
+    (fun k => mderiv (-Complex.I) (e k) (pathLogDetHessian g h i t) z) = pathRicciJet g h i t z := by
+  have hf := (g.localField i).ricciHessian_smooth.contDiffAt ((A.chart i).open_target.mem_nhds hz)
+  change ContDiffAt ℝ ∞ (PotentialKaehler.potentialMatrix (fun y => Real.log (g.matrix i y).det.re)) z at hf
+  have hh := PotentialKaehler.potentialMatrix_smooth ((h.smooth i).contDiffAt ((A.chart i).open_target.mem_nhds hz))
+  funext k
+  unfold pathLogDetHessian pathRicciJet
+  rw [mderiv_sub (matrix_smooth_smul (contDiffAt_const (c := ((1-t:ℝ):ℂ))) hf) (matrix_smooth_smul (contDiffAt_const (c := (t:ℂ))) hh),
+    mderiv_smul contDiffAt_const hf,mderiv_smul contDiffAt_const hh]
+  simp only [wderiv_const,zero_smul,zero_add]
+
+lemma volumePath_ricciJet (g : KaehlerMetric A) (h : SemipositiveAnticanonicalMetric A)
+    {t b : ℝ} {φ : SmoothRealFunction A} (hs : SolvesVolumePath g h t φ b)
+    (hp : g.PositivePotential φ) (i : Fin A.count) {z : Coordinates d}
+    (hz : z ∈ (A.chart i).target) :
+    (fun k => mderiv (-Complex.I) (e k) ((g.deform φ hp).localField i).ricciHessian z) =
+      pathRicciJet g h i t z := by
+  have he : ((g.deform φ hp).localField i).ricciHessian =ᶠ[𝓝 z] pathLogDetHessian g h i t := by
+    filter_upwards [(A.chart i).open_target.mem_nhds hz] with y hy
+    exact volumePath_logdet_hessian g h hs hp i hy
+  rw [← pathLogDetHessian_jet g h i t hz]
+  funext k
+  exact mderiv_congr he _ _
+
+lemma pathRicciJet_continuous (g : KaehlerMetric A) (h : SemipositiveAnticanonicalMetric A)
+    (i : Fin A.count) : ContinuousOn (fun q : ℝ × Coordinates d => pathRicciJet g h i q.1 q.2)
+      (univ ×ˢ (A.chart i).target) := by
+  apply continuousOn_pi.mpr
+  intro k
+  have hf : ContinuousOn (mderiv (-Complex.I) (e k) (g.localField i).ricciHessian) (A.chart i).target := by
+    intro z hz
+    exact (mderiv_smooth ((g.localField i).ricciHessian_smooth.contDiffAt
+      ((A.chart i).open_target.mem_nhds hz)) _ _).continuousAt.continuousWithinAt
+  have hh : ContinuousOn (mderiv (-Complex.I) (e k) (PotentialKaehler.potentialMatrix (h.weight i))) (A.chart i).target := by
+    intro z hz
+    exact (mderiv_smooth (PotentialKaehler.potentialMatrix_smooth ((h.smooth i).contDiffAt
+      ((A.chart i).open_target.mem_nhds hz))) _ _).continuousAt.continuousWithinAt
+  exact ((Complex.continuous_ofReal.comp (continuous_const.sub continuous_fst)).continuousOn.smul
+    (hf.comp continuous_snd.continuousOn (fun _ h => h.2))).sub
+    ((Complex.continuous_ofReal.comp continuous_fst).continuousOn.smul
+      (hh.comp continuous_snd.continuousOn (fun _ h => h.2)))
+end Anticanonical.SourceSmooth
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/PathCoordinateC1.lean b/lean/OAI/Geometry/PrescribedPotential/PathCoordinateC1.lean
new file mode 100644
index 0000000000000000000000000000000000000000..9e1a55311c5c6c994beafb25234870c21b60330b
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/PathCoordinateC1.lean
@@ -0,0 +1,78 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CalabiCoordinateBound
+import OAI.Geometry.PrescribedPotential.PathCalabiBound
+import OAI.Geometry.PrescribedPotential.PathCalabiRicci
+import OAI.Geometry.PrescribedPotential.PathEllipticity
+
+namespace OAI
+
+/-! Path Coordinate C1. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Matrix Metric
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace Anticanonical.SourceSmooth
+open KaehlerCalculus EllipticKernel
+local instance pathC1RealIP (d : ℕ) : InnerProductSpace ℝ (EC d) :=
+  InnerProductSpace.rclikeToReal ℂ (EC d)
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [CompactSpace X] [ConnectedSpace X]
+  {A : ComplexAtlas d X}
+namespace CoordinateBall
+variable (p : CoordinateBall A)
+
+theorem path_coordinate_C1_bound (g : KaehlerMetric A) (h : SemipositiveAnticanonicalMetric A) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ (t b : ℝ) (φ : SmoothRealFunction A), t ∈ Icc 0 1 →
+      ∀ hs : SolvesVolumePath g h t φ b, ∀ x ∈ p.source,
+        ‖(g.deform φ hs.choose).matrix p.index (A.chart p.index x)‖ ≤ C ∧
+        ‖fderiv ℝ (fun z i j => (g.deform φ hs.choose).matrix p.index z i j)
+          (A.chart p.index x)‖ ≤ C := by
+  obtain ⟨S,hS,hcal⟩ := p.path_calabi_bound g h
+  obtain ⟨R,H,hR,hH,hell⟩ := volumePath_relative_ellipticity g h
+  let L : Set (V d) := coordinateEquiv d '' closedBall p.center p.radius
+  have hL : IsCompact L := (isCompact_closedBall _ _).image (coordinateEquiv d).continuous
+  have hLc (z : V d) : z ∈ L ↔ (coordinateEquiv d).symm z ∈ closedBall p.center p.radius := by
+    constructor
+    · rintro ⟨y,hy,rfl⟩
+      simpa only [ContinuousLinearEquiv.symm_apply_apply] using hy
+    · intro hz
+      exact ⟨_,hz,(coordinateEquiv d).apply_symm_apply z⟩
+  have hdom : L ⊆ (A.chart p.index).target := by
+    intro z hz
+    have hh := p.closure_sub (closedBall_subset_closedBall (by linarith [p.radius_pos]) ((hLc z).mp hz))
+    simpa only [ComplexAtlas.euclideanChart_target,mem_preimage,ContinuousLinearEquiv.apply_symm_apply] using hh
+  obtain ⟨C,hC,hbound⟩ := calabi_coordinate_C1_uniform hL (g.matrix p.index)
+    ((g.smooth p.index).continuousOn.mono hdom) (fun z hz => g.positive _ _ (hdom hz)) hR.le hH.le hS
+  refine ⟨C,hC,?_⟩
+  intro t b φ ht hs x hx
+  have hz : A.chart p.index x ∈ L := (hLc _).mpr (ball_subset_closedBall hx.2)
+  obtain ⟨hl,hu⟩ := hell t b φ ht hs p.index _ (hdom hz)
+  exact hbound _ hz ((g.deform φ hs.choose).localField p.index) _ (hdom hz) hl hu
+    (hcal t b φ ht hs x hx)
+end CoordinateBall
+
+ 
+theorem volumePath_coordinate_C1_finite_cover (g : KaehlerMetric A) (h : SemipositiveAnticanonicalMetric A) :
+    ∃ (S : Finset (CoordinateBall A)) (C : ℝ), 0 ≤ C ∧
+      (∀ x, ∃ p ∈ S, x ∈ p.source) ∧
+      ∀ p ∈ S, ∀ (t b : ℝ) (φ : SmoothRealFunction A), t ∈ Icc 0 1 →
+        ∀ hs : SolvesVolumePath g h t φ b, ∀ x ∈ p.source,
+          ‖(g.deform φ hs.choose).matrix p.index (A.chart p.index x)‖ ≤ C ∧
+          ‖fderiv ℝ (fun z i j => (g.deform φ hs.choose).matrix p.index z i j)
+            (A.chart p.index x)‖ ≤ C := by
+  classical
+  obtain ⟨S,hS⟩ := CoordinateBall.finite_cover (A := A)
+  choose C hC hb using fun p : CoordinateBall A => p.path_coordinate_C1_bound g h
+  refine ⟨S,∑ p ∈ S, C p,Finset.sum_nonneg (fun p _ => hC p),hS,?_⟩
+  intro p hp t b φ ht hs x hx
+  have hle := Finset.single_le_sum (fun q _ => hC q) hp
+  exact ⟨(hb p t b φ ht hs x hx).1.trans hle,(hb p t b φ ht hs x hx).2.trans hle⟩
+end Anticanonical.SourceSmooth
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/PathCoordinateDensity.lean b/lean/OAI/Geometry/PrescribedPotential/PathCoordinateDensity.lean
new file mode 100644
index 0000000000000000000000000000000000000000..ca51bed2c8c634bb351b1f6dd52539f49ae7e586
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/PathCoordinateDensity.lean
@@ -0,0 +1,90 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CoordinateBalls
+import OAI.Geometry.PrescribedPotential.VolumeNormalization
+
+namespace OAI
+
+/-! Path Coordinate Density. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Metric Filter Topology MeasureTheory Matrix
+open scoped ContDiff ComplexOrder
+namespace Anticanonical.SourceSmooth
+open EllipticKernel
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+
+lemma volumePath_det_upper [CompactSpace X] [Nonempty X]
+    (g : KaehlerMetric A) (h : SemipositiveAnticanonicalMetric A)
+    {t b : ℝ} {φ : SmoothRealFunction A} (ht : t ∈ Icc 0 1)
+    (hsol : SolvesVolumePath g h t φ b) (i : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart i).target) :
+    |(g.matrix i z + φ.hessian i z).det.re| ≤
+      Real.exp (2*‖(⟨(prescribedForcing g h).value,
+        (prescribedForcing g h).continuous⟩ : C(X,ℝ))‖) * g.volumeCoefficient i z := by
+  let F : C(X,ℝ) := ⟨(prescribedForcing g h).value, (prescribedForcing g h).continuous⟩
+  have hb := volumePath_scalar_bound g h ht hsol
+  obtain ⟨hp,he⟩ := hsol
+  have hd : 0 < (g.matrix i z + φ.hessian i z).det.re := (Complex.pos_iff.mp (hp i z hz).det_pos).1
+  have hg := g.volumeCoefficient_pos i hz
+  let x := (A.chart i).symm z
+  have hez : Real.log ((g.matrix i z + φ.hessian i z).det.re) - Real.log (g.volumeCoefficient i z) =
+      t * F x + b := by
+    have hev := he x
+    change (g.logRatio (g.deform φ hp)).localExpression i z = _ at hev
+    rw [g.logRatio_local _ i hz] at hev
+    exact hev
+  have hFx : |t * F x| ≤ ‖F‖ := by
+    rw [abs_mul, abs_of_nonneg ht.1]
+    exact (mul_le_mul_of_nonneg_right ht.2 (abs_nonneg _)).trans
+      (by simpa only [Real.norm_eq_abs, one_mul] using F.norm_coe_le_norm x)
+  have hbound : t*F x+b ≤ 2*‖F‖ := by
+    have h1 := (abs_le.mp hFx).2
+    have h2 := (abs_le.mp hb).2
+    change b ≤ ‖F‖ at h2
+    linarith
+  have heq : (g.matrix i z + φ.hessian i z).det.re = Real.exp (t*F x+b)*g.volumeCoefficient i z := by
+    calc
+      _ = Real.exp (Real.log ((g.matrix i z + φ.hessian i z).det.re)) := (Real.exp_log hd).symm
+      _ = Real.exp (t*F x+b+Real.log (g.volumeCoefficient i z)) := by congr 1; linarith
+      _ = _ := by rw [Real.exp_add, Real.exp_log hg]
+  rw [abs_of_pos hd, heq]
+  exact mul_le_mul_of_nonneg_right (Real.exp_le_exp.mpr hbound) hg.le
+
+namespace CoordinateBall
+variable (p : CoordinateBall A)
+
+lemma path_det_bound [CompactSpace X] [Nonempty X]
+    (g : KaehlerMetric A) (h : SemipositiveAnticanonicalMetric A) :
+    ∃ K : ℝ, 0 < K ∧ ∀ (t b : ℝ) (φ : SmoothRealFunction A),
+      t ∈ Icc 0 1 → SolvesVolumePath g h t φ b →
+      ∀ z ∈ closedBall p.center (3*p.radius),
+        |(g.matrix p.index (coordinateEquiv d z) + φ.hessian p.index (coordinateEquiv d z)).det.re| ≤ K := by
+  have hc : ContinuousOn (fun z : EC d => g.volumeCoefficient p.index (coordinateEquiv d z))
+      (closedBall p.center (3*p.radius)) := by
+    exact ((g.volumeCoefficient_smooth p.index).continuousOn.comp
+      (coordinateEquiv d).continuous.continuousOn (fun z hz => by
+        have hh := p.closure_sub (closedBall_subset_closedBall (by linarith [p.radius_pos]) hz)
+        simpa only [ComplexAtlas.euclideanChart_target, mem_preimage] using hh))
+  obtain ⟨B,hB⟩ := (isCompact_closedBall p.center (3*p.radius)).exists_bound_of_continuousOn hc
+  let M := ‖(⟨(prescribedForcing g h).value, (prescribedForcing g h).continuous⟩ : C(X,ℝ))‖
+  refine ⟨Real.exp (2*M)*max B 1, mul_pos (Real.exp_pos _) (lt_max_iff.mpr (Or.inr (by norm_num))), ?_⟩
+  intro t b φ ht hs z hz
+  have hzt : coordinateEquiv d z ∈ (A.chart p.index).target := by
+    have hh := p.closure_sub (closedBall_subset_closedBall (by linarith [p.radius_pos]) hz)
+    simpa only [ComplexAtlas.euclideanChart_target, mem_preimage] using hh
+  apply (volumePath_det_upper g h ht hs p.index hzt).trans
+  apply mul_le_mul_of_nonneg_left _ (Real.exp_pos _).le
+  have hbc : |g.volumeCoefficient p.index (coordinateEquiv d z)| ≤ B := by
+    simpa only [Real.norm_eq_abs] using hB z hz
+  exact (le_abs_self _).trans (hbc.trans (le_max_left _ _))
+end CoordinateBall
+end Anticanonical.SourceSmooth
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/PathEllipticity.lean b/lean/OAI/Geometry/PrescribedPotential/PathEllipticity.lean
new file mode 100644
index 0000000000000000000000000000000000000000..85d0f1fbd368aee856af3a6f2b41c9380ad4fefd
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/PathEllipticity.lean
@@ -0,0 +1,54 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.PathTraceBound
+import OAI.Geometry.PrescribedPotential.TraceEllipticity
+import OAI.Geometry.PrescribedPotential.VolumeDeterminantBounds
+
+namespace OAI
+
+/-! Path Ellipticity. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Matrix
+open scoped ComplexOrder MatrixOrder
+namespace Anticanonical.SourceSmooth
+open KaehlerCalculus
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [CompactSpace X]
+  [ConnectedSpace X] {A : ComplexAtlas d X}
+
+ 
+
+theorem volumePath_relative_ellipticity (g : KaehlerMetric A)
+    (h : SemipositiveAnticanonicalMetric A) :
+    ∃ R K : ℝ, 0 < R ∧ 0 < K ∧ ∀ (t b : ℝ) (φ : SmoothRealFunction A),
+      t ∈ Icc 0 1 → ∀ _hs : SolvesVolumePath g h t φ b,
+      ∀ i : Fin A.count, ∀ z ∈ (A.chart i).target,
+        (((R:ℂ) • (g.matrix i z+φ.hessian i z))-g.matrix i z).PosSemidef ∧
+        (((K:ℂ) • g.matrix i z)-(g.matrix i z+φ.hessian i z)).PosSemidef := by
+  obtain ⟨R,hR,hRb⟩ := volumePath_inverseTrace_bound g h
+  let R' := R+1
+  let F : C(X,ℝ) := ⟨(prescribedForcing g h).value, (prescribedForcing g h).continuous⟩
+  let K := Real.exp (2*‖F‖)*R'^(d-1)
+  have hRp : 0 < R' := by dsimp [R']; linarith
+  have hKp : 0 < K := mul_pos (Real.exp_pos _) (pow_pos hRp _)
+  refine ⟨R',K,hRp,hKp,?_⟩
+  intro t b φ ht hs i z hz
+  let M := g.deform φ hs.choose
+  have hp : (g.matrix i z+φ.hessian i z).PosDef := hs.choose i z hz
+  have htr : ((g.matrix i z+φ.hessian i z)⁻¹*g.matrix i z).trace.re ≤ R' := by
+    have hh := hRb t b φ ht hs ((A.chart i).symm z)
+    change (M.inverseTrace g).localExpression i z ≤ R at hh
+    rw [M.inverseTrace_local g i hz] at hh
+    exact hh.trans (by dsimp [R']; linarith)
+  refine ⟨inverse_trace_lower_psd hp (g.positive i z hz) htr,?_⟩
+  exact inverse_trace_relative_upper hp (g.positive i z hz) hRp.le htr
+    (volumePath_det_ratio_bounds g h ht hs.choose hs.choose_spec i hz).2
+end Anticanonical.SourceSmooth
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/PathFrameBounds.lean b/lean/OAI/Geometry/PrescribedPotential/PathFrameBounds.lean
new file mode 100644
index 0000000000000000000000000000000000000000..3d6f301c6f5c934ac4c54a66872d838a2e96c73c
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/PathFrameBounds.lean
@@ -0,0 +1,68 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CoordinateBalls
+import OAI.Geometry.PrescribedPotential.FrameCompactBounds
+import OAI.Geometry.PrescribedPotential.MatrixJetDirection
+import OAI.Geometry.PrescribedPotential.PathRicci
+
+namespace OAI
+
+/-! Path Frame Bounds. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Metric Filter Topology Matrix
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace Anticanonical.SourceSmooth
+open KaehlerCalculus EllipticKernel
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+namespace CoordinateBall
+variable (p : CoordinateBall A)
+
+lemma path_frame_bound (g : KaehlerMetric A) (h : SemipositiveAnticanonicalMetric A) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ t ∈ Icc (0:ℝ) 1, ∀ x ∈ p.source,
+      ∀ B : Matrix (Fin d) (Fin d) ℂ,
+      -C*(frameGram (g.matrix p.index (A.chart p.index x)) B)^2 ≤
+        frameResidual (pathLogDetHessian g h p.index t (A.chart p.index x))
+          (g.matrix p.index (A.chart p.index x))
+          (curvatureDiagonal (g.matrix p.index) (A.chart p.index x)) B := by
+  let K : Set (ℝ × EC d) := Icc 0 1 ×ˢ closedBall p.center p.radius
+  have hk (q : ℝ × EC d) (hq : q ∈ K) : coordinateEquiv d q.2 ∈ (A.chart p.index).target := by
+    have he := p.closure_sub (closedBall_subset_closedBall (by linarith [p.radius_pos]) hq.2)
+    simpa only [ComplexAtlas.euclideanChart_target,mem_preimage] using he
+  have hmap : Continuous (fun q : ℝ × EC d => (q.1,coordinateEquiv d q.2)) :=
+    continuous_fst.prodMk ((coordinateEquiv d).continuous.comp continuous_snd)
+  have ha : ContinuousOn (fun q : ℝ × EC d => pathLogDetHessian g h p.index q.1 (coordinateEquiv d q.2)) K :=
+    (pathLogDetHessian_continuous g h p.index).comp (f := fun q : ℝ × EC d => (q.1,coordinateEquiv d q.2)) hmap.continuousOn (fun q hq => ⟨mem_univ _,hk q hq⟩)
+  have hg : ContinuousOn (fun q : ℝ × EC d => g.matrix p.index (coordinateEquiv d q.2)) K :=
+    (g.smooth p.index).continuousOn.comp (f := fun q : ℝ × EC d => coordinateEquiv d q.2)
+      ((coordinateEquiv d).continuous.comp continuous_snd).continuousOn hk
+  have hr := curvatureDiagonal_joint_continuous (A.chart p.index).open_target (g.smooth p.index)
+    (fun z hz => ne_of_gt (g.positive p.index z hz).det_pos)
+  have hR : ContinuousOn (fun q : (ℝ × EC d) × V d =>
+      curvatureDiagonal (g.matrix p.index) (coordinateEquiv d q.1.2) q.2) (K ×ˢ univ) := by
+    apply hr.comp (f := fun q : (ℝ × EC d) × V d => (coordinateEquiv d q.1.2,q.2))
+    · exact (((coordinateEquiv d).continuous.comp (continuous_snd.comp continuous_fst)).prodMk
+        continuous_snd).continuousOn
+    · intro q hq
+      exact ⟨hk q.1 hq.1,mem_univ _⟩
+  obtain ⟨C,hC,hb⟩ := compact_frame_bound (isCompact_Icc.prod (isCompact_closedBall _ _))
+    (fun q : ℝ × EC d => pathLogDetHessian g h p.index q.1 (coordinateEquiv d q.2))
+    (fun q : ℝ × EC d => g.matrix p.index (coordinateEquiv d q.2))
+    (fun q : ℝ × EC d => curvatureDiagonal (g.matrix p.index) (coordinateEquiv d q.2))
+    ha hg (fun q hq => g.positive _ _ (hk q hq)) hR
+    (fun q hq r v => curvatureDiagonal_smul ((g.smooth p.index).contDiffAt
+      ((A.chart p.index).open_target.mem_nhds (hk q hq))) r v)
+  refine ⟨C,hC,?_⟩
+  intro t ht x hx B
+  have hh := hb (t,A.euclideanChart p.index x) ⟨ht,ball_subset_closedBall hx.2⟩ B
+  simpa only [ComplexAtlas.euclideanChart_apply,ContinuousLinearEquiv.apply_symm_apply] using hh
+end CoordinateBall
+end Anticanonical.SourceSmooth
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/PathLaplaceJets.lean b/lean/OAI/Geometry/PrescribedPotential/PathLaplaceJets.lean
new file mode 100644
index 0000000000000000000000000000000000000000..5181eb8c058429c90eb0249051540bf45b3dfcba
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/PathLaplaceJets.lean
@@ -0,0 +1,108 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.PathMetricBootstrap
+import OAI.Geometry.PrescribedPotential.UniformCompositionFamily
+
+namespace OAI
+
+/-! Path Laplace Jets. -/
+
+section
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace Anticanonical.SourceSmooth
+open KaehlerCalculus EllipticKernel MetricSystem HigherJet GlobalElliptic
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [CompactSpace X] [ConnectedSpace X]
+  {A : ComplexAtlas d X}
+local instance pathLapECIP : InnerProductSpace ℝ (EC d) := InnerProductSpace.rclikeToReal ℂ (EC d)
+local instance pathLapHMIP : InnerProductSpace ℝ (HM d) := InnerProductSpace.rclikeToReal ℂ (HM d)
+
+lemma path_metric_all_jets (g : KaehlerMetric A) (h : SemipositiveAnticanonicalMetric A)
+    (i : Fin A.count) {K : Set (EC d)} (hK : IsCompact K)
+    (hKU : K ⊆ (A.euclideanChart i).target) (m : ℕ) :
+    ∃ C : ℝ, ∀ (s : PathSolution g h) x, x ∈ K →
+      ‖iteratedFDeriv ℝ m (fun y => encode d (s.metric.matrix i (coordinateEquiv d y))) x‖ ≤ C := by
+  by_cases hm : 1 ≤ m
+  · obtain ⟨C,hC,hb⟩ := path_metric_higher_bounds g h i m hm K hK hKU
+    exact ⟨C,hb⟩
+  · have hm0 : m=0 := by omega
+    subst m
+    obtain ⟨C,hC,hb⟩ := path_metric_compact_C1 g h i hK hKU
+    refine ⟨‖encodeCLM d‖*C,fun s x hx => ?_⟩
+    rw [norm_iteratedFDeriv_zero]
+    exact ((encodeCLM d).le_opNorm _).trans (mul_le_mul_of_nonneg_left (hb s x hx).1 (norm_nonneg _))
+
+def laplaceMetricCoefficient (g : KaehlerMetric A) (i : Fin A.count)
+    (p : EC d × HM d) : ℂ :=
+  (((g.matrix i (coordinateEquiv d p.1))⁻¹ * ((encode d).symm p.2-g.matrix i (coordinateEquiv d p.1))).trace.re : ℂ)
+
+omit [CompactSpace X] [ConnectedSpace X] in
+lemma laplaceMetricCoefficient_smooth (g : KaehlerMetric A) (i : Fin A.count)
+    {x : EC d} (hx : x ∈ (A.euclideanChart i).target) (u : HM d) :
+    ContDiffAt ℝ ∞ (laplaceMetricCoefficient g i) (x,u) := by
+  have hx' : coordinateEquiv d x ∈ (A.chart i).target := by
+    simpa only [ComplexAtlas.euclideanChart_target,Set.mem_preimage] using hx
+  have hG : ContDiffAt ℝ ∞ (fun p : EC d × HM d => g.matrix i (coordinateEquiv d p.1)) (x,u) :=
+    ((g.smooth i).contDiffAt ((A.chart i).open_target.mem_nhds hx')).comp (x,u)
+      ((coordinateEquiv d).contDiff.contDiffAt.comp (x,u) contDiffAt_fst)
+  have hI := (MatrixSmoothGeneral.inverse (g.smooth i)
+    (fun z hz => ne_of_gt (g.positive i z hz).det_pos)).contDiffAt ((A.chart i).open_target.mem_nhds hx')
+  have hInv := hI.comp (x,u) ((coordinateEquiv d).contDiff.contDiffAt.comp (x,u) contDiffAt_fst)
+  have hM : ContDiffAt ℝ ∞ (fun p : EC d × HM d => (encode d).symm p.2) (x,u) :=
+    (encode d).symm.contDiff.contDiffAt.comp (x,u) contDiffAt_snd
+  change ContDiffAt ℝ ∞ (fun p : EC d × HM d =>
+    (((∑ k, ∑ l, (g.matrix i (coordinateEquiv d p.1))⁻¹ k l *
+      ((encode d).symm p.2-g.matrix i (coordinateEquiv d p.1)) l k).re) : ℂ)) (x,u)
+  apply Complex.ofRealCLM.contDiff.contDiffAt.comp (x,u)
+  apply Complex.reCLM.contDiff.contDiffAt.comp (x,u)
+  apply ContDiffAt.sum
+  intro k _
+  apply ContDiffAt.sum
+  intro l _
+  exact (contDiffAt_pi.mp (contDiffAt_pi.mp hInv k) l).mul
+    (contDiffAt_pi.mp (contDiffAt_pi.mp (hM.sub hG) l) k)
+
+omit [CompactSpace X] [ConnectedSpace X] in
+lemma path_laplacian_local (g : KaehlerMetric A) (h : SemipositiveAnticanonicalMetric A)
+    (s : PathSolution g h) (i : Fin A.count) {x : EC d} (hx : x ∈ (A.euclideanChart i).target) :
+    Smooth.ofReal (g.laplacian s.potential) ((A.euclideanChart i).symm x) =
+      laplaceMetricCoefficient g i (x,encode d (s.metric.matrix i (coordinateEquiv d x))) := by
+  have hx' : coordinateEquiv d x ∈ (A.chart i).target := by
+    simpa only [ComplexAtlas.euclideanChart_target,Set.mem_preimage] using hx
+  change ((g.laplacian s.potential).localExpression i (coordinateEquiv d x) : ℂ) = _
+  rw [g.laplacian_localExpression s.potential i hx']
+  simp only [laplaceMetricCoefficient,ContinuousLinearEquiv.symm_apply_apply,PathSolution.metric,KaehlerMetric.deform,
+    add_sub_cancel_left,KaehlerMetric.linearizedMongeAmpere]
+
+lemma path_laplacian_weighted_jets (g : KaehlerMetric A) (h : SemipositiveAnticanonicalMetric A)
+    (i : Fin A.count) (ρ : Smooth A) {K : Set (EC d)} (hK : IsCompact K)
+    (hKU : K ⊆ (A.euclideanChart i).target) (m : ℕ) :
+    ∃ C : ℝ, ∀ (s : PathSolution g h) x, x ∈ K →
+      ‖iteratedFDeriv ℝ m (fun y => ρ ((A.euclideanChart i).symm y)*
+        Smooth.ofReal (g.laplacian s.potential) ((A.euclideanChart i).symm y)) x‖ ≤ C := by
+  let u : PathSolution g h → EC d → HM d := fun s x => encode d (s.metric.matrix i (coordinateEquiv d x))
+  let a : EC d × HM d → ℂ := fun p => ρ ((A.euclideanChart i).symm p.1)*laplaceMetricCoefficient g i p
+  have hu (s : PathSolution g h) (x : EC d) (hx : x ∈ K) : ContDiffAt ℝ ∞ (u s) x := by
+    have hx' : coordinateEquiv d x ∈ (A.chart i).target := by
+      simpa only [ComplexAtlas.euclideanChart_target,Set.mem_preimage] using hKU hx
+    exact (encode d).contDiff.contDiffAt.comp x
+      (((s.metric.smooth i).contDiffAt ((A.chart i).open_target.mem_nhds hx')).comp x (coordinateEquiv d).contDiff.contDiffAt)
+  have ha (x : EC d) (hx : x ∈ K) (v : HM d) : ContDiffAt ℝ ∞ a (x,v) := by
+    have hw : ContDiffAt ℝ ∞ (fun y => ρ ((A.euclideanChart i).symm y)) x :=
+      (ρ.smooth i).contDiffAt ((A.euclideanChart i).open_target.mem_nhds (hKU hx))
+    exact (hw.comp (x,v) contDiffAt_fst).mul (laplaceMetricCoefficient_smooth g i (hKU hx) v)
+  obtain ⟨C,hC⟩ := parametric_composition_family_jets u hK hu (path_metric_all_jets g h i hK hKU) a ha m
+  refine ⟨C,fun s x hx => ?_⟩
+  have he : (fun y => ρ ((A.euclideanChart i).symm y)*Smooth.ofReal (g.laplacian s.potential) ((A.euclideanChart i).symm y))
+      =ᶠ[𝓝 x] (fun y => a (y,u s y)) := by
+    filter_upwards [(A.euclideanChart i).open_target.mem_nhds (hKU hx)] with y hy
+    exact congrArg (ρ ((A.euclideanChart i).symm y)*·) (path_laplacian_local g h s i hy)
+  rw [(he.iteratedFDeriv (𝕜 := ℝ) m).eq_of_nhds]
+  exact hC s x hx
+end Anticanonical.SourceSmooth
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/PathLaplacianSobolev.lean b/lean/OAI/Geometry/PrescribedPotential/PathLaplacianSobolev.lean
new file mode 100644
index 0000000000000000000000000000000000000000..dde988db7ff56435694d4bb46f9ed165b6d10ed3
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/PathLaplacianSobolev.lean
@@ -0,0 +1,63 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.PathLaplaceJets
+import OAI.Geometry.PrescribedPotential.SchwartzFamilyBounds
+
+namespace OAI
+
+/-! Path Laplacian Sobolev. -/
+
+section
+
+noncomputable section
+open Set Filter Topology
+open scoped ContDiff SchwartzMap
+namespace Anticanonical.SourceSmooth
+open EllipticKernel GlobalElliptic SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  [ConnectedSpace X] {A : ComplexAtlas d X}
+
+lemma path_laplacian_localized_bound (g : KaehlerMetric A) (h : SemipositiveAnticanonicalMetric A)
+    (i : Fin A.count) (ρ : Smooth A)
+    (hρ : tsupport (ρ : X → ℂ) ⊆ (A.euclideanChart i).source) (r : ℝ) :
+    ∃ C : ℝ, ∀ s : PathSolution g h,
+      ‖schwartzCoord r (localize A i ρ hρ (Smooth.ofReal (g.laplacian s.potential)))‖ ≤ C := by
+  let K := (A.euclideanChart i) '' tsupport (ρ : X → ℂ)
+  have hK : IsCompact K := (isClosed_tsupport (ρ : X → ℂ)).isCompact.image_of_continuousOn
+    ((A.euclideanChart i).continuousOn.mono hρ)
+  have hKU : K ⊆ (A.euclideanChart i).target := by
+    rintro _ ⟨x,hx,rfl⟩
+    exact (A.euclideanChart i).mapsTo (hρ hx)
+  apply schwartzCoord_family_bound (fun s : PathSolution g h =>
+    localize A i ρ hρ (Smooth.ofReal (g.laplacian s.potential))) hK
+  · intro s
+    exact closure_minimal (localizeFun_support A i ρ _) hK.isClosed
+  · intro m
+    obtain ⟨C,hC⟩ := path_laplacian_weighted_jets g h i ρ hK hKU m
+    refine ⟨C,fun s x hx => ?_⟩
+    have he : (localize A i ρ hρ (Smooth.ofReal (g.laplacian s.potential)) : EC d → ℂ)
+        =ᶠ[𝓝 x] (fun y => ρ ((A.euclideanChart i).symm y)*
+          Smooth.ofReal (g.laplacian s.potential) ((A.euclideanChart i).symm y)) := by
+      filter_upwards [(A.euclideanChart i).open_target.mem_nhds (hKU hx)] with y hy
+      exact localizeFun_apply A ρ _ hy
+    rw [(he.iteratedFDeriv (𝕜 := ℝ) m).eq_of_nhds]
+    exact hC s x hx
+
+lemma path_laplacian_sobolev_bound (g : KaehlerMetric A) (h : SemipositiveAnticanonicalMetric A)
+    {ι : Type*} [Fintype ι] (D : Localizers A ι) (r : ℝ) :
+    ∃ C : ℝ, ∀ s : PathSolution g h,
+      ‖D.embed r (Smooth.ofReal (g.laplacian s.potential))‖ ≤ C := by
+  choose C hC using fun i => path_laplacian_localized_bound g h (D.index i) (D.weight i) (D.support_sub i) r
+  let B := ∑ i, max 0 (C i)
+  have hB : 0 ≤ B := Finset.sum_nonneg fun _ _ => le_max_left _ _
+  refine ⟨B,fun s => ?_⟩
+  change ‖D.coordinates r (Smooth.ofReal (g.laplacian s.potential))‖ ≤ B
+  apply (pi_norm_le_iff_of_nonneg hB).mpr
+  intro i
+  exact (hC i s).trans ((le_max_right 0 (C i)).trans
+    (Finset.single_le_sum (fun j _ => le_max_left 0 (C j)) (Finset.mem_univ i)))
+end Anticanonical.SourceSmooth
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/PathMetricBootstrap.lean b/lean/OAI/Geometry/PrescribedPotential/PathMetricBootstrap.lean
new file mode 100644
index 0000000000000000000000000000000000000000..da2fc79419e1d6dc07500d5a7dfbadaaabec759c
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/PathMetricBootstrap.lean
@@ -0,0 +1,96 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.AffineFamilyJets
+import OAI.Geometry.PrescribedPotential.MetricFamilyBootstrap
+import OAI.Geometry.PrescribedPotential.PathMetricCompactC1
+
+namespace OAI
+
+/-! Path Metric Bootstrap. -/
+
+section
+
+noncomputable section
+open Set Filter Topology Matrix Metric Finset
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace MetricSystem
+open KaehlerCalculus EllipticKernel
+variable {n : ℕ}
+local instance pathMetricECIP : InnerProductSpace ℝ (EC n) := InnerProductSpace.rclikeToReal ℂ (EC n)
+local instance pathMetricHMIP : InnerProductSpace ℝ (HM n) := InnerProductSpace.rclikeToReal ℂ (HM n)
+def encodeCLM (n : ℕ) : (Fin n → Fin n → ℂ) →L[ℝ] HM n := (encode n).toContinuousLinearMap
+lemma encoded_fderiv_bound {M : V n → Fin n → Fin n → ℂ} {x : EC n}
+    (hM : ContDiffAt ℝ ∞ M (coordinateEquiv n x)) :
+    ‖fderiv ℝ (fun y => encode n (M (coordinateEquiv n y))) x‖ ≤
+      ‖encodeCLM n‖*‖fderiv ℝ M (coordinateEquiv n x)‖*‖(coordinateEquiv n).toContinuousLinearMap‖ := by
+  have hd := (encode n).hasFDerivAt.comp x ((hM.differentiableAt (by simp)).hasFDerivAt.comp x (coordinateEquiv n).hasFDerivAt)
+  have hn := congrArg norm hd.fderiv
+  calc
+    _ = _ := hn
+    _ ≤ _ := ((encodeCLM n).opNorm_comp_le _).trans
+      (by nlinarith [ContinuousLinearMap.opNorm_comp_le (fderiv ℝ M (coordinateEquiv n x)) (coordinateEquiv n).toContinuousLinearMap,
+        norm_nonneg (encodeCLM n)])
+end MetricSystem
+namespace Anticanonical.SourceSmooth
+open KaehlerCalculus EllipticKernel MetricSystem HigherJet
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [CompactSpace X] [ConnectedSpace X]
+  {A : ComplexAtlas d X}
+local instance pathBootstrapECIP : InnerProductSpace ℝ (EC d) := InnerProductSpace.rclikeToReal ℂ (EC d)
+local instance pathBootstrapHMIP : InnerProductSpace ℝ (HM d) := InnerProductSpace.rclikeToReal ℂ (HM d)
+
+lemma path_metric_higher_bounds (g : KaehlerMetric A) (h : SemipositiveAnticanonicalMetric A)
+    (i : Fin A.count) :
+    ∀ m, 1 ≤ m → ∀ K : Set (EC d), IsCompact K → K ⊆ (A.euclideanChart i).target →
+      ∃ C : ℝ, 0 ≤ C ∧ ∀ (s : PathSolution g h) x, x ∈ K →
+        ‖iteratedFDeriv ℝ m (fun y => encode d (s.metric.matrix i (coordinateEquiv d y))) x‖ ≤ C := by
+  let U := (A.euclideanChart i).target
+  have hU : IsOpen U := (A.euclideanChart i).open_target
+  have hc (x : EC d) : x ∈ U → coordinateEquiv d x ∈ (A.chart i).target := by
+    intro hx
+    simpa only [U,ComplexAtlas.euclideanChart_target,Set.mem_preimage] using hx
+  let f : EC d → HM d := fun x => encode d ((g.localField i).ricciHessian (coordinateEquiv d x))
+  let q : EC d → HM d := fun x => encode d (PotentialKaehler.potentialMatrix (h.weight i) (coordinateEquiv d x))
+  have hf : ContDiffOn ℝ ∞ f U := by
+    intro x hx
+    exact ((encode d).contDiff.contDiffAt.comp x (((g.localField i).ricciHessian_smooth.contDiffAt
+      ((A.chart i).open_target.mem_nhds (hc x hx))).comp x (coordinateEquiv d).contDiff.contDiffAt)).contDiffWithinAt
+  have hq : ContDiffOn ℝ ∞ q U := by
+    intro x hx
+    exact ((encode d).contDiff.contDiffAt.comp x ((PotentialKaehler.potentialMatrix_smooth
+      ((h.smooth i).contDiffAt ((A.chart i).open_target.mem_nhds (hc x hx)))).comp x
+      (coordinateEquiv d).contDiff.contDiffAt)).contDiffWithinAt
+  let r : PathSolution g h → EC d → HM d := fun s x => (1-s.t) • f x-s.t • q x
+  apply metric_family_bootstrap hU (fun s : PathSolution g h => s.metric.localField i) r
+    (fun _ x hx => hc x hx) (fun s => (hf.const_smul (1-s.t)).sub (hq.const_smul s.t))
+  · intro s x hx
+    have he := volumePath_logdet_hessian g h s.solves s.solves.choose i (hc x hx)
+    change encode d (PotentialKaehler.potentialMatrix (fun y => Real.log (s.metric.matrix i y).det.re) (coordinateEquiv d x)) = _
+    have hh := congrArg (encode d) he
+    calc
+      _ = encode d (pathLogDetHessian g h i s.t (coordinateEquiv d x)) := hh
+      _ = _ := ?_
+    change encode d ((1-s.t) • (g.localField i).ricciHessian (coordinateEquiv d x)-s.t • PotentialKaehler.potentialMatrix (h.weight i) (coordinateEquiv d x)) = _
+    rw [map_sub,map_smul,map_smul]
+  · intro K hK hKU
+    obtain ⟨C,hC,hbound⟩ := path_metric_compact_C1 g h i hK hKU
+    refine ⟨‖encodeCLM d‖*C*‖(coordinateEquiv d).toContinuousLinearMap‖,fun s x hx => ?_⟩
+    apply (encoded_fderiv_bound ((s.metric.smooth i).contDiffAt ((A.chart i).open_target.mem_nhds (hc x (hKU hx))))).trans
+    exact mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left (hbound s x hx).2 (norm_nonneg _)) (norm_nonneg _)
+  · intro K hK hKU
+    obtain ⟨R,H,hR,hH,hell⟩ := volumePath_relative_ellipticity g h
+    let L := coordinateEquiv d '' K
+    have hL : IsCompact L := hK.image (coordinateEquiv d).continuous
+    have hLU : L ⊆ (A.chart i).target := by rintro _ ⟨x,hx,rfl⟩; exact hc x (hKU hx)
+    obtain ⟨B,hB,hbound⟩ := compact_normalizing_frames hL (g.matrix i)
+      ((g.smooth i).continuousOn.mono hLU) (fun z hz => g.positive i z (hLU hz)) hR.le hH.le
+    refine ⟨B,hB,fun s x hx C D hCD hDC hN => ?_⟩
+    obtain ⟨hl,hu⟩ := hell s.t s.b s.potential s.time s.solves i _ (hc x (hKU hx))
+    exact hbound _ ⟨x,hx,rfl⟩ _ C D hl hu hCD hN
+  · intro m K hK hKU
+    obtain ⟨C,hbound⟩ := affine_family_jets hU hf hq m hK hKU
+    exact ⟨C,fun j hj s x hx => hbound j hj s.t s.time x hx⟩
+end Anticanonical.SourceSmooth
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/PathMetricCompactC1.lean b/lean/OAI/Geometry/PrescribedPotential/PathMetricCompactC1.lean
new file mode 100644
index 0000000000000000000000000000000000000000..9ec8f02bc72a55a0d7dfa328c735a4351280b36f
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/PathMetricCompactC1.lean
@@ -0,0 +1,68 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.PathCoordinateC1
+
+namespace OAI
+
+/-! Path Metric Compact C1. -/
+
+section
+
+noncomputable section
+open Set Filter Topology Matrix Metric
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace Anticanonical.SourceSmooth
+open KaehlerCalculus EllipticKernel
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [CompactSpace X] [ConnectedSpace X]
+  {A : ComplexAtlas d X}
+
+structure PathSolution (g : KaehlerMetric A) (h : SemipositiveAnticanonicalMetric A) where
+  t : ℝ
+  b : ℝ
+  potential : SmoothRealFunction A
+  time : t ∈ Icc 0 1
+  solves : SolvesVolumePath g h t potential b
+namespace PathSolution
+variable {g : KaehlerMetric A} {h : SemipositiveAnticanonicalMetric A}
+def metric (s : PathSolution g h) : KaehlerMetric A := g.deform s.potential s.solves.choose
+end PathSolution
+
+local instance compactC1RealIP : InnerProductSpace ℝ (EC d) := InnerProductSpace.rclikeToReal ℂ (EC d)
+
+lemma path_metric_compact_C1 (g : KaehlerMetric A) (h : SemipositiveAnticanonicalMetric A)
+    (i : Fin A.count) {K : Set (EC d)} (hK : IsCompact K)
+    (hKU : K ⊆ (A.euclideanChart i).target) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ (s : PathSolution g h) x, x ∈ K →
+      ‖s.metric.matrix i (coordinateEquiv d x)‖ ≤ C ∧
+      ‖fderiv ℝ (fun z k l => s.metric.matrix i z k l) (coordinateEquiv d x)‖ ≤ C := by
+  classical
+  have he (x : K) : ∃ p : CoordinateBall A, p.index = i ∧ x.val ∈ ball p.center p.radius := by
+    obtain ⟨r,hr,hs⟩ := Metric.nhds_basis_closedBall.mem_iff.mp ((A.euclideanChart i).open_target.mem_nhds (hKU x.property))
+    refine ⟨⟨i,x.val,r/4,by positivity,?_⟩,rfl,mem_ball_self (by positivity)⟩
+    convert hs using 1; ring_nf
+  choose p hp hx using he
+  obtain ⟨S,hS⟩ := hK.elim_finite_subcover (fun x : K => ball (p x).center (p x).radius)
+    (fun _ => isOpen_ball) (fun x hxK => mem_iUnion.mpr ⟨⟨x,hxK⟩,hx ⟨x,hxK⟩⟩)
+  choose C hC hbound using fun x : K => (p x).path_coordinate_C1_bound g h
+  refine ⟨∑ x ∈ S, C x,Finset.sum_nonneg (fun x _ => hC x),?_⟩
+  intro s x hxK
+  obtain ⟨q,hq,hxq⟩ := mem_iUnion₂.mp (hS hxK)
+  have ht : x ∈ (A.euclideanChart (p q).index).target := (p q).ball_sub hxq
+  let y := (A.euclideanChart (p q).index).symm x
+  have hy : y ∈ (p q).source := ⟨(A.euclideanChart _).symm.mapsTo ht,by
+    change A.euclideanChart _ ((A.euclideanChart _).symm x) ∈ ball _ _
+    rw [(A.euclideanChart _).right_inv ht]
+    exact hxq⟩
+  have hz : A.chart (p q).index y = coordinateEquiv d x := by
+    have hc := (A.euclideanChart (p q).index).right_inv ht
+    change (coordinateEquiv d).symm (A.chart (p q).index y) = x at hc
+    exact (coordinateEquiv d).symm.injective (hc.trans ((coordinateEquiv d).symm_apply_apply x).symm)
+  have hb := hbound q s.t s.b s.potential s.time s.solves y hy
+  rw [hz,hp q] at hb
+  have hc := Finset.single_le_sum (fun x _ => hC x) hq
+  exact ⟨hb.1.trans hc,hb.2.trans hc⟩
+end Anticanonical.SourceSmooth
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/PathPotentialBound.lean b/lean/OAI/Geometry/PrescribedPotential/PathPotentialBound.lean
new file mode 100644
index 0000000000000000000000000000000000000000..b6b6bbaf201909130f80b96883575834bd17f7c0
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/PathPotentialBound.lean
@@ -0,0 +1,88 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CoordinateInnerCover
+import OAI.Geometry.PrescribedPotential.CoordinateMinimumBound
+import OAI.Geometry.PrescribedPotential.NormalizedPotentialL1
+import OAI.Geometry.PrescribedPotential.PathCoordinateDensity
+import OAI.Geometry.PrescribedPotential.VolumePath
+
+namespace OAI
+
+/-! Path Potential Bound. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Metric Filter Topology MeasureTheory
+open scoped ContDiff
+namespace Anticanonical.SourceSmooth
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [CompactSpace X]
+  [ConnectedSpace X] {A : ComplexAtlas d X}
+
+theorem volumePath_oscillation_bound
+    (g : KaehlerMetric A) (h : SemipositiveAnticanonicalMetric A) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ (t b : ℝ) (φ : SmoothRealFunction A),
+      t ∈ Icc 0 1 → SolvesVolumePath g h t φ b →
+      ∀ x y, |φ.value x-φ.value y| ≤ C := by
+  classical
+  let : MeasurableSpace X := borel X
+  let : BorelSpace X := ⟨rfl⟩
+  obtain ⟨S,hS⟩ := CoordinateBall.finite_inner_cover (A := A)
+  obtain ⟨B,hB⟩ := normalized_potential_L1 g S
+    (by intro x; obtain ⟨p,hp,hx⟩ := hS x; exact ⟨p,hp,p.half_source_sub hx⟩)
+  choose K hK hKb using (fun p : S => p.val.path_det_bound g h)
+  choose C hC using (fun p : S => p.val.minimum_bound g B (hK p))
+  let R : ℝ := ∑ p : S, max (C p) 0
+  have hR : 0 ≤ R := Finset.sum_nonneg (fun _ _ => le_max_right _ _)
+  refine ⟨R,hR,?_⟩
+  intro t b φ ht hs x y
+  obtain ⟨xm,_,hmin⟩ := isCompact_univ.exists_isMinOn univ_nonempty φ.continuous.continuousOn
+  obtain ⟨xM,_,hmax⟩ := isCompact_univ.exists_isMaxOn univ_nonempty φ.continuous.continuousOn
+  let ψ := φ.normalizeAt xM
+  have hψ : g.PositivePotential ψ := by
+    intro i z hz
+    simpa only [ψ, SmoothRealFunction.hessian_normalizeAt] using hs.choose i z hz
+  have hψ0 : ∀ z, ψ.value z ≤ 0 := by
+    intro z
+    change φ.value z-φ.value xM ≤ 0
+    exact sub_nonpos.mpr (hmax (mem_univ z))
+  have hψz : ∃ z, ψ.value z = 0 := ⟨xM,φ.normalizeAt_value xM⟩
+  obtain ⟨p,hp,hxm⟩ := hS xm
+  have hlocal := hC ⟨p,hp⟩ ψ hψ0 (hB ψ hψ hψ0 hψz p hp)
+    (by
+      intro z hz
+      simpa only [ψ, SmoothRealFunction.hessian_normalizeAt] using
+        hKb ⟨p,hp⟩ t b φ ht hs z hz)
+    xm hxm (by
+      intro z
+      change φ.value xm-φ.value xM ≤ φ.value z-φ.value xM
+      exact sub_le_sub_right (hmin (mem_univ z)) _)
+  have hCR : C ⟨p,hp⟩ ≤ R := (le_max_left _ _).trans
+    (Finset.single_le_sum (fun q _ => le_max_right (C q) 0) (Finset.mem_univ (⟨p,hp⟩ : S)))
+  have hosc : φ.value xM-φ.value xm ≤ R := by
+    change -(φ.value xm-φ.value xM) ≤ C ⟨p,hp⟩ at hlocal
+    linarith
+  have hxM : φ.value x ≤ φ.value xM := hmax (mem_univ x)
+  have hyM : φ.value y ≤ φ.value xM := hmax (mem_univ y)
+  have hmx : φ.value xm ≤ φ.value x := hmin (mem_univ x)
+  have hmy : φ.value xm ≤ φ.value y := hmin (mem_univ y)
+  exact abs_le.mpr ⟨by linarith only [hosc,hyM,hmx], by linarith only [hosc,hxM,hmy]⟩
+
+ 
+
+theorem volumePath_potential_bound
+    (g : KaehlerMetric A) (h : SemipositiveAnticanonicalMetric A) (x₀ : X) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ (t b : ℝ) (φ : SmoothRealFunction A),
+      t ∈ Icc 0 1 → φ.value x₀ = 0 → SolvesVolumePath g h t φ b →
+      ∀ x, |φ.value x| ≤ C := by
+  obtain ⟨C,hC,hb⟩ := volumePath_oscillation_bound g h
+  refine ⟨C,hC,?_⟩
+  intro t b φ ht hn hs x
+  simpa only [hn,sub_zero] using hb t b φ ht hs x x₀
+end Anticanonical.SourceSmooth
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/PathRicci.lean b/lean/OAI/Geometry/PrescribedPotential/PathRicci.lean
new file mode 100644
index 0000000000000000000000000000000000000000..de2dfcde64103c839a1037149c9b91e93338a3b4
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/PathRicci.lean
@@ -0,0 +1,98 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.PotentialLinearCalculus
+import OAI.Geometry.PrescribedPotential.ScalarFrameCalculus
+import OAI.Geometry.PrescribedPotential.VolumePath
+
+namespace OAI
+
+/-! Path Ricci. -/
+
+section
+
+ 
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace Anticanonical.SourceSmooth
+open KaehlerCalculus
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+
+def pathLogDetHessian (g : KaehlerMetric A) (h : SemipositiveAnticanonicalMetric A)
+    (i : Fin A.count) (t : ℝ) (z : Coordinates d) : Matrix (Fin d) (Fin d) ℂ :=
+  ((1-t:ℝ):ℂ) • PotentialKaehler.potentialMatrix (fun y => Real.log (g.matrix i y).det.re) z -
+    (t:ℂ) • PotentialKaehler.potentialMatrix (h.weight i) z
+
+lemma volumePath_logdet_local (g : KaehlerMetric A) (h : SemipositiveAnticanonicalMetric A)
+    {t b : ℝ} {φ : SmoothRealFunction A} (hs : SolvesVolumePath g h t φ b)
+    (hp : g.PositivePotential φ) (i : Fin A.count) {z : Coordinates d}
+    (hz : z ∈ (A.chart i).target) :
+    Real.log ((g.deform φ hp).matrix i z).det.re =
+      (1-t)*Real.log (g.matrix i z).det.re-t*h.weight i z+b := by
+  have he := hs.choose_spec ((A.chart i).symm z)
+  change (g.logRatio (g.deform φ hs.choose)).localExpression i z =
+    t*(prescribedForcing g h).localExpression i z+b at he
+  rw [KaehlerMetric.logRatio_local _ _ i hz,prescribedForcing_local _ _ i hz] at he
+  change Real.log ((g.deform φ hp).matrix i z).det.re - Real.log (g.matrix i z).det.re =
+    t*(-h.weight i z-Real.log (g.matrix i z).det.re)+b at he
+  linarith
+
+lemma potentialMatrix_mul_const {f : KaehlerCalculus.V d → ℝ} {z : KaehlerCalculus.V d}
+    (hf : ContDiffAt ℝ ∞ f z) (a : ℝ) :
+    PotentialKaehler.potentialMatrix (fun y => a*f y) z =
+      (a:ℂ) • PotentialKaehler.potentialMatrix f z := by
+  have hh := potentialMatrix_affine (f := fun _ => 0) contDiffAt_const hf a
+  have hz0 : PotentialKaehler.potentialMatrix (fun _ : KaehlerCalculus.V d => (0:ℝ)) z = 0 := by
+    unfold PotentialKaehler.potentialMatrix
+    ext i j
+    simp [PotentialKaehler.hermitianPartMatrix, PotentialKaehler.hermitianPartForm, PotentialKaehler.hermitianPart]
+    rfl
+  simpa only [zero_add,hz0] using hh
+
+lemma potentialMatrix_add_const (c : ℝ) (f : KaehlerCalculus.V d → ℝ) (z : KaehlerCalculus.V d) :
+    PotentialKaehler.potentialMatrix (fun y => f y+c) z = PotentialKaehler.potentialMatrix f z := by
+  simpa only [add_comm] using potentialMatrix_const_add c f z
+
+lemma volumePath_logdet_hessian (g : KaehlerMetric A) (h : SemipositiveAnticanonicalMetric A)
+    {t b : ℝ} {φ : SmoothRealFunction A} (hs : SolvesVolumePath g h t φ b)
+    (hp : g.PositivePotential φ) (i : Fin A.count) {z : Coordinates d}
+    (hz : z ∈ (A.chart i).target) :
+    PotentialKaehler.potentialMatrix (fun y => Real.log ((g.deform φ hp).matrix i y).det.re) z =
+      pathLogDetHessian g h i t z := by
+  have he : (fun y => Real.log ((g.deform φ hp).matrix i y).det.re) =ᶠ[𝓝 z]
+      (fun y => (1-t)*Real.log (g.matrix i y).det.re-t*h.weight i y+b) := by
+    filter_upwards [(A.chart i).open_target.mem_nhds hz] with y hy
+    exact volumePath_logdet_local g h hs hp i hy
+  have hf : ContDiffAt ℝ ∞ (fun y => Real.log (g.matrix i y).det.re) z :=
+    ((g.volumeCoefficient_smooth i).log (fun y hy => ne_of_gt (g.volumeCoefficient_pos i hy))).contDiffAt
+      ((A.chart i).open_target.mem_nhds hz)
+  have hh := (h.smooth i).contDiffAt ((A.chart i).open_target.mem_nhds hz)
+  rw [PotentialKaehler.potentialMatrix_congr he,potentialMatrix_add_const,
+    potentialMatrix_sub_smul (contDiffAt_const.mul hf) hh,potentialMatrix_mul_const hf]
+  rfl
+
+lemma pathLogDetHessian_continuous (g : KaehlerMetric A) (h : SemipositiveAnticanonicalMetric A)
+    (i : Fin A.count) :
+    ContinuousOn (fun q : ℝ × Coordinates d => pathLogDetHessian g h i q.1 q.2)
+      (univ ×ˢ (A.chart i).target) := by
+  have hf : ContinuousOn (PotentialKaehler.potentialMatrix (fun y => Real.log (g.matrix i y).det.re))
+      (A.chart i).target := by
+    intro z hz
+    apply ContinuousAt.continuousWithinAt
+    apply ContDiffAt.continuousAt (𝕜 := ℝ) (n := ∞)
+    apply PotentialKaehler.potentialMatrix_smooth
+    exact ((g.volumeCoefficient_smooth i).log (fun y hy => ne_of_gt (g.volumeCoefficient_pos i hy))).contDiffAt
+      ((A.chart i).open_target.mem_nhds hz)
+  have hh : ContinuousOn (PotentialKaehler.potentialMatrix (h.weight i)) (A.chart i).target := by
+    intro z hz
+    exact (PotentialKaehler.potentialMatrix_smooth ((h.smooth i).contDiffAt
+      ((A.chart i).open_target.mem_nhds hz))).continuousAt.continuousWithinAt
+  exact ((Complex.continuous_ofReal.comp (continuous_const.sub continuous_fst)).continuousOn.smul
+    (hf.comp continuous_snd.continuousOn (fun _ h => h.2))).sub
+    ((Complex.continuous_ofReal.comp continuous_fst).continuousOn.smul
+      (hh.comp continuous_snd.continuousOn (fun _ h => h.2)))
+end Anticanonical.SourceSmooth
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/PathTraceBound.lean b/lean/OAI/Geometry/PrescribedPotential/PathTraceBound.lean
new file mode 100644
index 0000000000000000000000000000000000000000..1cda921d57d3235e5d6c5d3aa58f0689d6adc579
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/PathTraceBound.lean
@@ -0,0 +1,102 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.ChernLuMaximum
+import OAI.Geometry.PrescribedPotential.MetricInverseTrace
+import OAI.Geometry.PrescribedPotential.PathFrameBounds
+import OAI.Geometry.PrescribedPotential.PathPotentialBound
+import OAI.Geometry.PrescribedPotential.PathRicci
+
+namespace OAI
+
+/-! Path Trace Bound. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Metric Filter Topology Matrix
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace Anticanonical.SourceSmooth
+open KaehlerCalculus
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [CompactSpace X]
+  [ConnectedSpace X] [Nonempty (Fin d)] {A : ComplexAtlas d X}
+
+theorem volumePath_inverseTrace_bound_pos_dim (g : KaehlerMetric A) (h : SemipositiveAnticanonicalMetric A) :
+    ∃ R : ℝ, 0 ≤ R ∧ ∀ (t b : ℝ) (φ : SmoothRealFunction A),
+      t ∈ Icc 0 1 → ∀ hs : SolvesVolumePath g h t φ b,
+      ∀ x, ((g.deform φ hs.choose).inverseTrace g).value x ≤ R := by
+  classical
+  obtain ⟨S,hS⟩ := CoordinateBall.finite_cover (A := A)
+  choose C hC hCb using (fun p : S => p.val.path_frame_bound g h)
+  let K : ℝ := ∑ p : S, C p
+  have hK : 0 ≤ K := Finset.sum_nonneg (fun p _ => hC p)
+  have hCK (p : S) : C p ≤ K := Finset.single_le_sum (fun q _ => hC q) (Finset.mem_univ p)
+  obtain ⟨L,hL,hLb⟩ := volumePath_oscillation_bound g h
+  refine ⟨(K+1)*d*Real.exp ((K+1)*L),by positivity,?_⟩
+  intro t b φ ht hs x
+  let M := g.deform φ hs.choose
+  let f := M.inverseTrace g
+  let Q : X → ℝ := fun y => Real.log (f.value y)-(K+1)*φ.value y
+  have hfp : ∀ y, 0 < f.value y := fun y => M.inverseTrace_positive g y
+  have hQc : Continuous Q := (f.continuous.log (fun y => ne_of_gt (hfp y))).sub
+    (continuous_const.mul φ.continuous)
+  obtain ⟨xm,_,hm⟩ := isCompact_univ.exists_isMaxOn univ_nonempty hQc.continuousOn
+  obtain ⟨p,hp,hxm⟩ := hS xm
+  have hxmi : xm ∈ (A.chart p.index).source := by
+    simpa only [ComplexAtlas.euclideanChart_source] using hxm.1
+  let z := A.chart p.index xm
+  have hz : z ∈ (A.chart p.index).target := (A.chart p.index).mapsTo hxmi
+  have hb : ∀ B : Matrix (Fin d) (Fin d) ℂ,
+      -K*(frameGram (g.matrix p.index z) B)^2 ≤ frameResidual
+        (PotentialKaehler.potentialMatrix (fun y => Real.log (M.matrix p.index y).det.re) z)
+        (g.matrix p.index z) (curvatureDiagonal (g.matrix p.index) z) B := by
+    intro B
+    rw [volumePath_logdet_hessian g h hs hs.choose p.index hz]
+    have hh := hCb ⟨p,hp⟩ t ht xm hxm B
+    exact (mul_le_mul_of_nonneg_right (neg_le_neg (hCK ⟨p,hp⟩)) (sq_nonneg _)).trans hh
+  have hmax : IsLocalMax (fun y => Real.log (inverseTrace (M.matrix p.index) (g.matrix p.index) y)-
+      (K+1)*φ.localExpression p.index y) z := by
+    change ∀ᶠ y in 𝓝 z, _ ≤ _
+    filter_upwards [(A.chart p.index).open_target.mem_nhds hz] with y hy
+    have hyS := M.inverseTrace_local g p.index hy
+    have hzS := M.inverseTrace_local g p.index hz
+    rw [← hyS,← hzS]
+    change Q ((A.chart p.index).symm y) ≤ Q ((A.chart p.index).symm z)
+    rw [show (A.chart p.index).symm z = xm from (A.chart p.index).left_inv hxmi]
+    exact hm (mem_univ _)
+  have hMmax : f.value xm ≤ (K+1)*d := by
+    change M.inverseTraceValue g xm ≤ _
+    rw [M.inverseTraceValue_local g p.index hxmi]
+    exact inverseTrace_at_localMax_bound (A.chart p.index).open_target (M.smooth p.index)
+      (M.positive p.index) (M.closed p.index) (g.smooth p.index) (g.positive p.index)
+      (φ.smooth p.index) hz rfl K hb hmax
+  have hqm : Real.log (f.value x)-(K+1)*φ.value x ≤
+      Real.log (f.value xm)-(K+1)*φ.value xm := hm (mem_univ x)
+  have hφ := (abs_le.mp (hLb t b φ ht hs x xm)).2
+  have ha : 0 ≤ K+1 := by linarith
+  have hl : Real.log (f.value x) ≤ Real.log (f.value xm)+(K+1)*L := by
+    have hh := mul_le_mul_of_nonneg_left hφ ha
+    nlinarith only [hqm,hh]
+  have he := Real.exp_le_exp.mpr hl
+  rw [Real.exp_add,Real.exp_log (hfp x),Real.exp_log (hfp xm)] at he
+  exact he.trans (mul_le_mul_of_nonneg_right hMmax (Real.exp_pos _).le)
+omit [Nonempty (Fin d)] in
+ 
+theorem volumePath_inverseTrace_bound (g : KaehlerMetric A) (h : SemipositiveAnticanonicalMetric A) :
+    ∃ R : ℝ, 0 ≤ R ∧ ∀ (t b : ℝ) (φ : SmoothRealFunction A),
+      t ∈ Icc 0 1 → ∀ hs : SolvesVolumePath g h t φ b,
+      ∀ x, ((g.deform φ hs.choose).inverseTrace g).value x ≤ R := by
+  by_cases hd : d = 0
+  · subst d
+    refine ⟨0,le_rfl,?_⟩
+    intro t b φ ht hs x
+    simp [KaehlerMetric.inverseTrace,KaehlerMetric.inverseTraceValue,KaehlerCalculus.inverseTrace,
+      Matrix.trace]
+  · let : Nonempty (Fin d) := Fin.pos_iff_nonempty.mp (Nat.pos_of_ne_zero hd)
+    exact volumePath_inverseTrace_bound_pos_dim g h
+end Anticanonical.SourceSmooth
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/PointPoisson.lean b/lean/OAI/Geometry/PrescribedPotential/PointPoisson.lean
new file mode 100644
index 0000000000000000000000000000000000000000..df23d2545a3fa05453579581e359100630ae5cff
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/PointPoisson.lean
@@ -0,0 +1,133 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.GlobalStrongEmbedding
+import OAI.Geometry.PrescribedPotential.HigherMetricOperator
+import OAI.Geometry.PrescribedPotential.HigherPoisson
+import OAI.Geometry.PrescribedPotential.KaehlerClosedDerivatives
+import OAI.Geometry.PrescribedPotential.PoissonEquivalence
+
+namespace OAI
+
+/-! Point Poisson. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology
+open scoped ContDiff Classical
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  [ConnectedSpace X] {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+def pointEvaluation (s : ℝ) (x₀ : X) : D.localizers.Sobolev s →L[ℝ] ℂ :=
+  (BoundedContinuousFunction.evalCLM ℝ x₀) ∘L D.localizers.strong s
+
+omit [ConnectedSpace X] in
+lemma pointEvaluation_embed {s : ℝ} (hs : (Module.finrank ℝ (EC d) : ℝ) < 2*s)
+    (x₀ : X) (f : Smooth A) : D.pointEvaluation s x₀ (D.localizers.embed s f) = f x₀ := by
+  change D.localizers.strong s (D.localizers.embed s f) x₀ = _
+  rw [D.localizers.strong_embed s hs]
+  rfl
+
+omit [ConnectedSpace X] in
+lemma completedLOrder_constants (k : ℕ) (c : ℂ) :
+    D.completedLOrder k (D.localizers.constantsOrder ((k : ℝ)+2) c) = 0 := by
+  rw [Localizers.constantsOrder_apply, D.completedLOrder_embed, complexL_const, map_zero]
+
+omit [ConnectedSpace X] in
+lemma constantsOrder_lower {s t : ℝ} (hst : t ≤ s) (a : ℂ) :
+    D.localizers.lower s t (D.localizers.constantsOrder s a) =
+      D.localizers.constantsOrder t a := by
+  change D.localizers.lower s t (D.localizers.embed s (Smooth.const a)) =
+    D.localizers.embed t (Smooth.const a)
+  exact D.localizers.lower_embed hst (Smooth.const a)
+
+omit [ConnectedSpace X] in
+lemma completedLOrder_to_constant (k : ℕ) (u : D.localizers.Sobolev ((k : ℝ)+2)) (c : ℂ)
+    (hu : D.completedLOrder k u = D.localizers.constantsOrder (k : ℝ) c) :
+    D.completedL (D.localizers.lower ((k : ℝ)+2) 2 u) = D.localizers.constants c := by
+  have hh := congrArg (D.localizers.lower (k : ℝ) 0) hu
+  rw [D.completedLOrder_lower, D.constantsOrder_lower (Nat.cast_nonneg k)] at hh
+  exact hh
+
+lemma completedLOrder_constant_kernel (m : ℝ) (hm : 1 ≤ m)
+    (he : ‖D.completedError m hm‖ < 1) (k : ℕ)
+    (u : D.localizers.Sobolev ((k : ℝ)+2)) (c : ℂ)
+    (hu : D.completedLOrder k u = D.localizers.constantsOrder (k : ℝ) c) :
+    ∃ a : ℂ, u = D.localizers.constantsOrder ((k : ℝ)+2) a ∧ c = 0 := by
+  have hl : (2 : ℝ) ≤ (k : ℝ)+2 := by linarith [Nat.cast_nonneg (α := ℝ) k]
+  obtain ⟨a, ha, hc⟩ := D.completedL_constant_kernel m hm he
+    (D.localizers.lower ((k : ℝ)+2) 2 u) c (D.completedLOrder_to_constant k u c hu)
+  refine ⟨a, ?_, hc⟩
+  apply D.localizers.lower_injective hl
+  exact ha.trans (D.constantsOrder_lower hl a).symm
+
+def rawPoissonOrder (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
+    (P : D.localizers.ConstantProjection) (k : ℕ) := (D.poissonRaw_order m hm he P k).choose
+
+lemma rawPoissonOrder_equation (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
+    (P : D.localizers.ConstantProjection) (k : ℕ) (f : D.localizers.Sobolev (k : ℝ)) :
+    D.completedLOrder k (D.rawPoissonOrder m hm he P k f) =
+      f + D.localizers.constantsOrder (k : ℝ)
+        (D.poissonResidue m hm he P (D.localizers.lower (k : ℝ) 0 f)) := by
+  apply D.localizers.lower_injective (Nat.cast_nonneg k)
+  rw [D.completedLOrder_lower, rawPoissonOrder, (D.poissonRaw_order m hm he P k).choose_spec,
+    D.poissonRaw_equation, map_add, Localizers.constantsOrder_apply,
+    D.localizers.lower_embed (Nat.cast_nonneg k)]
+  rfl
+
+def pointPoissonOrder (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
+    (P : D.localizers.ConstantProjection) (k : ℕ) (x₀ : X) :
+    D.localizers.Sobolev (k : ℝ) →L[ℝ] D.localizers.Sobolev ((k : ℝ)+2) :=
+  D.rawPoissonOrder m hm he P k - D.localizers.constantsOrder ((k : ℝ)+2) ∘L
+    D.pointEvaluation ((k : ℝ)+2) x₀ ∘L D.rawPoissonOrder m hm he P k
+
+lemma pointPoissonOrder_equation (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
+    (P : D.localizers.ConstantProjection) (k : ℕ) (x₀ : X)
+    (f : D.localizers.Sobolev (k : ℝ)) :
+    D.completedLOrder k (D.pointPoissonOrder m hm he P k x₀ f) =
+      f + D.localizers.constantsOrder (k : ℝ)
+        (D.poissonResidue m hm he P (D.localizers.lower (k : ℝ) 0 f)) := by
+  simp only [pointPoissonOrder, _root_.sub_apply, ContinuousLinearMap.comp_apply,
+    map_sub, D.completedLOrder_constants, sub_zero, D.rawPoissonOrder_equation]
+
+lemma pointPoissonOrder_normalized (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
+    (P : D.localizers.ConstantProjection) (k : ℕ)
+    (hk : (Module.finrank ℝ (EC d) : ℝ) < 2*((k : ℝ)+2)) (x₀ : X)
+    (f : D.localizers.Sobolev (k : ℝ)) :
+    D.pointEvaluation ((k : ℝ)+2) x₀ (D.pointPoissonOrder m hm he P k x₀ f) = 0 := by
+  simp only [pointPoissonOrder, _root_.sub_apply, ContinuousLinearMap.comp_apply,
+    map_sub, Localizers.constantsOrder_apply, D.pointEvaluation_embed hk]
+  exact sub_self _
+
+lemma pointPoissonOrder_unique (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
+    (k : ℕ) (hk : (Module.finrank ℝ (EC d) : ℝ) < 2*((k : ℝ)+2)) (x₀ : X)
+    (u v : D.localizers.Sobolev ((k : ℝ)+2)) (c e : ℂ)
+    (huv : D.completedLOrder k u - D.localizers.constantsOrder (k : ℝ) c =
+      D.completedLOrder k v - D.localizers.constantsOrder (k : ℝ) e)
+    (hpt : D.pointEvaluation ((k : ℝ)+2) x₀ u = D.pointEvaluation ((k : ℝ)+2) x₀ v) :
+    u = v ∧ c = e := by
+  have hL : D.completedLOrder k (u-v) = D.localizers.constantsOrder (k : ℝ) (c-e) := by
+    rw [map_sub, map_sub]
+    calc
+      _ = (D.completedLOrder k u - D.localizers.constantsOrder (k : ℝ) c) -
+          (D.completedLOrder k v - D.localizers.constantsOrder (k : ℝ) e) +
+          (D.localizers.constantsOrder (k : ℝ) c - D.localizers.constantsOrder (k : ℝ) e) := by abel
+      _ = _ := by rw [huv, sub_self, zero_add]
+  obtain ⟨a, ha, hc⟩ := D.completedLOrder_constant_kernel m hm he k (u-v) (c-e) hL
+  have heval := congrArg (D.pointEvaluation ((k : ℝ)+2) x₀) ha
+  rw [map_sub, hpt, sub_self, Localizers.constantsOrder_apply, D.pointEvaluation_embed hk] at heval
+  have ha0 : a = 0 := heval.symm
+  rw [ha0, map_zero] at ha
+  exact ⟨sub_eq_zero.mp ha, sub_eq_zero.mp hc⟩
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/PoissonEquivalence.lean b/lean/OAI/Geometry/PrescribedPotential/PoissonEquivalence.lean
new file mode 100644
index 0000000000000000000000000000000000000000..aa08f3c0172569d852deca18490c94518057aaf5
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/PoissonEquivalence.lean
@@ -0,0 +1,142 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.BoundedPoisson
+import OAI.Geometry.PrescribedPotential.CircleRadialCalculus
+import OAI.Geometry.PrescribedPotential.ComplexKernel
+import OAI.Geometry.PrescribedPotential.QuantitativeResolvent
+
+namespace OAI
+
+/-! Poisson Equivalence. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology
+open scoped ContDiff Classical
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  [ConnectedSpace X] {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+
+omit [T2Space X] [CompactSpace X] [ConnectedSpace X] in
+lemma complexL_const (g : KaehlerMetric A) (c : ℂ) : complexL g (Smooth.const c) = 0 := by
+  ext x
+  obtain ⟨i, hi⟩ := A.covers x
+  change complexLValue g (Smooth.const c) x = 0
+  rw [complexLValue_local g _ i (by simpa using hi)]
+  change localL g i (fun _ => c) (A.euclideanChart i x) = 0
+  simp [localL]
+
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+omit [ConnectedSpace X] in
+lemma completedL_constants (c : ℂ) : D.completedL (D.localizers.constantsOrder 2 c) = 0 := by
+  rw [Localizers.constantsOrder_apply]
+  change D.localizers.extendCore 2 0 (complexL g) (D.localizers.embed 2 (Smooth.const c)) = 0
+  rw [D.localizers.extendCore_embed D.complexL_bound, complexL_const, map_zero]
+
+def poissonBounded (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
+    (P : D.localizers.ConstantProjection) : D.localizers.Sobolev 0 →L[ℝ] D.localizers.Sobolev 2 :=
+  D.poissonRaw m hm he P - D.localizers.constantsOrder 2 ∘L P.coefficient ∘L
+    D.localizers.lower 2 0 ∘L D.poissonRaw m hm he P
+
+lemma poissonBounded_equation (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
+    (P : D.localizers.ConstantProjection) (f : D.localizers.Sobolev 0) :
+    D.completedL (D.poissonBounded m hm he P f) =
+      f + D.localizers.constants (D.poissonResidue m hm he P f) := by
+  simp only [poissonBounded, _root_.sub_apply, ContinuousLinearMap.comp_apply,
+    map_sub, D.completedL_constants, sub_zero, D.poissonRaw_equation]
+
+lemma poissonBounded_normalized (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
+    (P : D.localizers.ConstantProjection) (f : D.localizers.Sobolev 0) :
+    P.coefficient (D.localizers.lower 2 0 (D.poissonBounded m hm he P f)) = 0 := by
+  simp only [poissonBounded, _root_.sub_apply, ContinuousLinearMap.comp_apply,
+    map_sub, Localizers.constantsOrder_apply,
+    D.localizers.lower_embed (by norm_num : (0:ℝ) ≤ 2)]
+  change _ - P.coefficient (D.localizers.constants _) = 0
+  rw [P.coefficient_constants, sub_self]
+
+lemma completedL_constant_kernel (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
+    (u : D.localizers.Sobolev 2) (c : ℂ) (hu : D.completedL u = D.localizers.constants c) :
+    ∃ a : ℂ, u = D.localizers.constantsOrder 2 a ∧ c = 0 := by
+  let w := D.shiftedOperator m u
+  have hr : D.resolventZero m hm he w = D.localizers.lower 2 0 u := by
+    change D.localizers.lower 2 0 (D.completedResolvent m hm he (D.shiftedOperator m u)) = _
+    rw [D.completedResolvent_left]
+  have hw : w = D.localizers.embed 0 (-Smooth.const c) + (m^2 : ℝ) • D.resolventZero m hm he w := by
+    rw [hr, map_neg]
+    change (m^2 : ℝ) • D.localizers.lower 2 0 u - D.completedL u = _
+    rw [hu]
+    change _ - D.localizers.embed 0 (Smooth.const c) = _
+    abel
+  obtain ⟨v, hv⟩ := D.allRegular_smooth (D.resolvent_feedback_regular m hm he w _ hw)
+  have hemb : D.localizers.embed 2 v = u := by
+    apply D.localizers.lower_injective (by norm_num : (0:ℝ) ≤ 2)
+    rw [D.localizers.lower_embed (by norm_num : (0:ℝ) ≤ 2), hv]
+    exact hr
+  have hl : complexL g v = Smooth.const c := by
+    apply D.localizers.embed_injective 0
+    change D.localizers.embed 0 (complexL g v) = D.localizers.constants c
+    rw [← hu, ← hemb]
+    exact (D.localizers.extendCore_embed D.complexL_bound v).symm
+  have hc := complexL_constant_rhs_zero g v hl
+  have hl0 : complexL g v = 0 := by rw [hl, hc]; rfl
+  obtain ⟨a, ha⟩ := complexL_kernel g v hl0
+  exact ⟨a, by rw [← hemb, ha]; rfl, hc⟩
+
+ 
+def augmentedL (P : D.localizers.ConstantProjection) :
+    (D.localizers.Sobolev 2 × ℂ) →L[ℝ] (D.localizers.Sobolev 0 × ℂ) :=
+  ((D.completedL ∘L ContinuousLinearMap.fst ℝ _ _) -
+    (D.localizers.constants ∘L ContinuousLinearMap.snd ℝ _ _)).prod
+    (P.coefficient ∘L D.localizers.lower 2 0 ∘L ContinuousLinearMap.fst ℝ _ _)
+
+lemma augmentedL_apply (P : D.localizers.ConstantProjection) (u : D.localizers.Sobolev 2) (c : ℂ) :
+    D.augmentedL P (u,c) = (D.completedL u - D.localizers.constants c,
+      P.coefficient (D.localizers.lower 2 0 u)) := rfl
+
+lemma augmentedL_bijective (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
+    (P : D.localizers.ConstantProjection) : Function.Bijective (D.augmentedL P) := by
+  constructor
+  · apply (LinearMap.ker_eq_bot).mp
+    apply LinearMap.ker_eq_bot'.mpr
+    rintro ⟨u,c⟩ h
+    have hl : D.completedL u = D.localizers.constants c := sub_eq_zero.mp (congrArg Prod.fst h)
+    obtain ⟨a, ha, hc⟩ := D.completedL_constant_kernel m hm he u c hl
+    have hn : P.coefficient (D.localizers.lower 2 0 u) = 0 := congrArg Prod.snd h
+    rw [ha, Localizers.constantsOrder_apply, D.localizers.lower_embed (by norm_num : (0:ℝ) ≤ 2)] at hn
+    change P.coefficient (D.localizers.constants a) = 0 at hn
+    rw [P.coefficient_constants] at hn
+    rw [ha, hn, map_zero, hc]
+    rfl
+  · rintro ⟨f,a⟩
+    refine ⟨(D.poissonBounded m hm he P f + D.localizers.constantsOrder 2 a,
+      D.poissonResidue m hm he P f), ?_⟩
+    rw [D.augmentedL_apply]
+    apply Prod.ext
+    · rw [map_add, D.completedL_constants, add_zero, D.poissonBounded_equation, add_sub_cancel_right]
+    · change P.coefficient (D.localizers.lower 2 0
+        (D.poissonBounded m hm he P f + D.localizers.constantsOrder 2 a)) = a
+      rw [map_add, map_add, D.poissonBounded_normalized, zero_add,
+        Localizers.constantsOrder_apply, D.localizers.lower_embed (by norm_num : (0:ℝ) ≤ 2)]
+      exact P.coefficient_constants a
+
+ 
+
+def poissonEquiv (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
+    (P : D.localizers.ConstantProjection) :
+    (D.localizers.Sobolev 2 × ℂ) ≃L[ℝ] (D.localizers.Sobolev 0 × ℂ) :=
+  ContinuousLinearEquiv.ofBijective (D.augmentedL P)
+    (LinearMap.ker_eq_bot.mpr (D.augmentedL_bijective m hm he P).1)
+    (LinearMap.range_eq_top.mpr (D.augmentedL_bijective m hm he P).2)
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/PositiveOpenness.lean b/lean/OAI/Geometry/PrescribedPotential/PositiveOpenness.lean
new file mode 100644
index 0000000000000000000000000000000000000000..e10eb9683ca5d97f2358a045038bf82cb8014cbc
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/PositiveOpenness.lean
@@ -0,0 +1,79 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.PatchCutoffs
+import OAI.Geometry.PrescribedPotential.RealSobolev
+import OAI.Geometry.PrescribedPotential.WeakPositive
+import OAI.Geometry.PrescribedPotential.SobolevOpenness
+
+namespace OAI
+
+/-! Positive Openness. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff Classical ComplexOrder
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+local instance positiveOpennessNG (s : ℝ) : NormedAddCommGroup (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedAddCommGroup
+local instance positiveOpennessNS (s : ℝ) : NormedSpace ℝ (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedSpace
+local instance positiveOpennessTG (s : ℝ) : IsTopologicalAddGroup (D.localizers.RealSobolev s) :=
+  Submodule.isTopologicalAddGroup _
+local instance positiveOpennessCS (s : ℝ) : ContinuousSMul ℝ (D.localizers.RealSobolev s) :=
+  SMulMemClass.continuousSMul _
+
+omit [T2Space X] [CompactSpace X] in
+lemma exists_weight_support (x : X) : ∃ p, x ∈ tsupport (D.localizers.weight p : X → ℂ) := by
+  by_contra! hn
+  have hz : ∑ p, D.localizers.weight p x = 0 := Finset.sum_eq_zero (fun p _ =>
+    image_eq_zero_of_notMem_tsupport (hn p))
+  rw [D.localizers.sum_one] at hz
+  exact one_ne_zero hz
+
+lemma weakPositive_embed_iff (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) (f : RealSmooth A) :
+    D.WeakPositive k (D.localizers.realEmbed ((k : ℝ)+2) f) ↔ g.PositivePotential f.source := by
+  constructor
+  · intro h i z hz
+    let x := (A.chart i).symm z
+    have hi : x ∈ (A.chart i).source := (A.chart i).mapsTo_symm hz
+    have hiz : A.chart i x = z := (A.chart i).right_inv hz
+    obtain ⟨p,hp⟩ := D.exists_weight_support x
+    have hj : x ∈ (A.chart (D.patch p).index).source := by simpa using D.patch_source p hp
+    have hpos := h p x hp
+    rw [D.weakMatrix_embed k hk f p hp] at hpos
+    rw [← hiz, g.compatibility i (D.patch p).index x hi hj,
+      f.source.hessian_compatibility i (D.patch p).index hi hj, ← add_mul, ← mul_add]
+    exact hpos.conjTranspose_mul_mul_same (Matrix.mulVec_injective_of_det_ne_zero
+      (g.transition_det_isUnit i (D.patch p).index hi hj).ne_zero)
+  · intro h p x hx
+    rw [D.weakMatrix_embed k hk f p hx]
+    exact h _ _ ((A.chart (D.patch p).index).mapsTo (by simpa using D.patch_source p hx))
+
+ 
+
+theorem positive_volume_eventually_solvable [ConnectedSpace X]
+    (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) (x₀ : X) :
+    ∀ᶠ f in 𝓝 (D.realConstants (k : ℝ) 1),
+      ∃ (u : D.localizers.RealSobolev ((k : ℝ)+2)) (b : ℝ),
+        D.WeakPositive k u ∧ D.realVolume k hk u = Real.exp b • f ∧
+        D.realEvaluation ((k : ℝ)+2) x₀ u = 0 := by
+  have hU : {z : D.localizers.RealSobolev ((k : ℝ)+2) × ℝ | D.WeakPositive k z.1} ∈ 𝓝 0 :=
+    (show Tendsto (Prod.fst : D.localizers.RealSobolev ((k : ℝ)+2) × ℝ → _)
+      (𝓝 0) (𝓝 0) from continuous_fst.tendsto 0) (D.weakPositive_nhds k hk)
+  exact D.volume_eventually_solvable k hk x₀ hU
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/PositivePotentialLinearization.lean b/lean/OAI/Geometry/PrescribedPotential/PositivePotentialLinearization.lean
new file mode 100644
index 0000000000000000000000000000000000000000..ad13332c4f5c7ec06bc5fb0f7c347e573a767007
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/PositivePotentialLinearization.lean
@@ -0,0 +1,115 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.NormalizedPoisson
+import OAI.Geometry.PrescribedPotential.PotentialDensity
+import OAI.Geometry.PrescribedPotential.SmoothPotentialDifference
+import OAI.Geometry.PrescribedPotential.VolumePath
+
+namespace OAI
+
+/-! Positive Potential Linearization. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff Classical ComplexOrder
+namespace Anticanonical.SourceSmooth
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+namespace KaehlerMetric
+
+lemma potentialDensity_quotient (g : KaehlerMetric A) (φ : SmoothRealFunction A)
+    (i : Fin A.count) {x : X} (hi : x ∈ (A.chart i).source) :
+    (g.potentialDensity φ).value x =
+      (g.matrix i (A.chart i x) + φ.hessian i (A.chart i x)).det.re /
+        g.volumeCoefficient i (A.chart i x) := by
+  have hz := (A.chart i).mapsTo hi
+  have hN := hermitian_det_im ((g.positive i _ hz).isHermitian.add (φ.hessian_hermitian i hz))
+  have hB := hermitian_det_im (g.positive i _ hz).isHermitian
+  have heN : ((g.matrix i (A.chart i x) + φ.hessian i (A.chart i x)).det.re : ℂ) =
+      (g.matrix i (A.chart i x) + φ.hessian i (A.chart i x)).det :=
+    Complex.ext rfl hN.symm
+  have heB : ((g.matrix i (A.chart i x)).det.re : ℂ) = (g.matrix i (A.chart i x)).det :=
+    Complex.ext rfl hB.symm
+  change g.potentialDensityValue φ x = _
+  rw [g.potentialDensityValue_local φ i hi,volumePolynomial,← heN,← heB,← Complex.ofReal_div]
+  rfl
+
+lemma potentialDensity_pos (g : KaehlerMetric A) (φ : SmoothRealFunction A)
+    (hp : g.PositivePotential φ) (x : X) : 0 < (g.potentialDensity φ).value x := by
+  obtain ⟨i,hi⟩ := A.covers x
+  rw [g.potentialDensity_quotient φ i hi]
+  exact div_pos (Complex.pos_iff.mp (hp i _ ((A.chart i).mapsTo hi)).det_pos).1
+    (g.volumeCoefficient_pos i ((A.chart i).mapsTo hi))
+
+ 
+
+lemma potentialLogDensity_directional (g : KaehlerMetric A) (φ ψ : SmoothRealFunction A)
+    (hp : g.PositivePotential φ) (x : X) :
+    HasDerivAt (fun t : ℝ => Real.log
+      ((g.potentialDensity (φ.addFunction (ψ.realSMul t))).value x))
+      (((g.deform φ hp).laplacian ψ).value x) 0 := by
+  obtain ⟨i,hi⟩ := A.covers x
+  have hz := (A.chart i).mapsTo hi
+  let H := g.matrix i (A.chart i x) + φ.hessian i (A.chart i x)
+  let K := ψ.hessian i (A.chart i x)
+  let b := g.volumeCoefficient i (A.chart i x)
+  have hH : H.PosDef := hp i _ hz
+  have hb : 0 < b := g.volumeCoefficient_pos i hz
+  have hdet : 0 < H.det.re := (Complex.pos_iff.mp hH.det_pos).1
+  have him : H.det.im = 0 := hermitian_det_im hH.isHermitian
+  have hn := (MongeAmpere.hasDerivAt_det_add_smul H K
+    (isUnit_iff_ne_zero.mpr (ne_of_gt hH.det_pos))).real_of_complex
+  have hn' : HasDerivAt (fun t : ℝ => (H + (t : ℂ) • K).det.re)
+      (H.det.re * (H⁻¹ * K).trace.re) 0 := by
+    simpa only [Complex.mul_re,him,zero_mul,sub_zero] using hn
+  have he : (fun t : ℝ => (g.potentialDensity (φ.addFunction (ψ.realSMul t))).value x) =
+      (fun t : ℝ => (H + (t : ℂ) • K).det.re / b) := by
+    funext t
+    rw [g.potentialDensity_quotient _ i hi,SmoothRealFunction.hessian_addFunction _ _ i hz,
+      SmoothRealFunction.hessian_realSMul]
+    simp only [H,K,b,add_assoc]
+  rw [show ((g.deform φ hp).laplacian ψ).value x = (H⁻¹ * K).trace.re from
+    (g.deform φ hp).laplacianValue_local ψ i hi]
+  change HasDerivAt (Real.log ∘ (fun t : ℝ =>
+    (g.potentialDensity (φ.addFunction (ψ.realSMul t))).value x)) _ 0
+  rw [he]
+  have hd := (hn'.div_const b).log (show (H + ((0 : ℝ) : ℂ) • K).det.re / b ≠ 0 from by
+    simpa using ne_of_gt (div_pos hdet hb))
+  have hc : H.det.re * (H⁻¹ * K).trace.re / b / ((H + ((0 : ℝ) : ℂ) • K).det.re / b) =
+      (H⁻¹ * K).trace.re := by
+    simp only [Complex.ofReal_zero,zero_smul,add_zero]
+    field_simp
+  rw [hc] at hd
+  exact hd.congr_of_eventuallyEq (Filter.Eventually.of_forall (fun t => rfl))
+
+end KaehlerMetric
+
+ 
+
+theorem positivePotential_linearized_solvable [T2Space X] [CompactSpace X] [ConnectedSpace X]
+    (g : KaehlerMetric A) (φ : SmoothRealFunction A) (hp : g.PositivePotential φ)
+    (x₀ : X) (f : SmoothRealFunction A) :
+    ∃ (ψ : SmoothRealFunction A) (a : ℝ), ψ.value x₀ = 0 ∧
+      ∀ x, HasDerivAt (fun t : ℝ =>
+        Real.log ((g.potentialDensity (φ.addFunction (ψ.realSMul t))).value x) + t*a)
+        (f.value x) 0 := by
+  obtain ⟨ψ,a,hn,he⟩ := normalizedPoisson d X A (g.deform φ hp) x₀ f
+  refine ⟨ψ,a,hn,fun x => ?_⟩
+  obtain ⟨i,hi⟩ := A.covers x
+  have h := (g.potentialLogDensity_directional φ ψ hp x).add ((hasDerivAt_id (0 : ℝ)).mul_const a)
+  have hr : ((g.deform φ hp).laplacian ψ).value x + a = f.value x := by
+    change (g.deform φ hp).laplacianValue ψ x + a = _
+    rw [(g.deform φ hp).laplacianValue_local ψ i hi]
+    simpa only [SmoothRealFunction.localExpression,Function.comp_apply,(A.chart i).left_inv hi]
+      using he i (A.chart i x) ((A.chart i).mapsTo hi)
+  rw [one_mul,hr] at h
+  exact h.congr_of_eventuallyEq (Filter.Eventually.of_forall (fun t => rfl))
+
+end Anticanonical.SourceSmooth
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/PotentialDensity.lean b/lean/OAI/Geometry/PrescribedPotential/PotentialDensity.lean
new file mode 100644
index 0000000000000000000000000000000000000000..a6ca87a0e4120f5a36c9a1443e19e26fb8add680
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/PotentialDensity.lean
@@ -0,0 +1,112 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.GlobalOperator
+
+namespace OAI
+
+/-! Potential Density. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff ComplexOrder Classical Matrix.Norms.Elementwise
+namespace Anticanonical.SourceSmooth
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+
+lemma hermitian_det_im {n : ℕ} {H : Matrix (Fin n) (Fin n) ℂ}
+    (hH : H.IsHermitian) : H.det.im = 0 := by
+  have hh := congrArg (fun M : Matrix (Fin n) (Fin n) ℂ => M.det.im) hH.eq
+  rw [Matrix.det_conjTranspose] at hh
+  change -H.det.im = H.det.im at hh
+  linarith
+
+namespace KaehlerMetric
+
+def volumePolynomial (g : KaehlerMetric A) (φ : SmoothRealFunction A)
+    (i : Fin A.count) (z : Coordinates d) : ℂ :=
+  (g.matrix i z + φ.hessian i z).det / (g.matrix i z).det
+
+lemma volumePolynomial_compatibility (g : KaehlerMetric A) (φ : SmoothRealFunction A)
+    (i j : Fin A.count) {x : X}
+    (hi : x ∈ (A.chart i).source) (hj : x ∈ (A.chart j).source) :
+    g.volumePolynomial φ i (A.chart i x) = g.volumePolynomial φ j (A.chart j x) := by
+  unfold volumePolynomial
+  apply (div_eq_div_iff
+    (ne_of_gt (g.positive i _ ((A.chart i).mapsTo hi)).det_pos)
+    (ne_of_gt (g.positive j _ ((A.chart j).mapsTo hj)).det_pos)).mpr
+  rw [g.compatibility i j x hi hj, φ.hessian_compatibility i j hi hj,
+    ← add_mul, ← mul_add]
+  simp only [Matrix.det_mul]
+  ring
+
+lemma volumePolynomial_im (g : KaehlerMetric A) (φ : SmoothRealFunction A)
+    (i : Fin A.count) {z : Coordinates d} (hz : z ∈ (A.chart i).target) :
+    (g.volumePolynomial φ i z).im = 0 := by
+  have hg := (g.positive i z hz).isHermitian
+  have hn := hermitian_det_im (hg.add (φ.hessian_hermitian i hz))
+  have hd := hermitian_det_im hg
+  simp only [volumePolynomial, Complex.div_im, hn, hd, zero_mul, mul_zero,
+    sub_zero, zero_div]
+
+lemma volumePolynomial_smooth (g : KaehlerMetric A) (φ : SmoothRealFunction A)
+    (i : Fin A.count) :
+    ContDiffOn ℝ ∞ (g.volumePolynomial φ i) (A.chart i).target := by
+  exact (MatrixSmoothGeneral.determinant ((g.smooth i).add (φ.hessian_smooth i))).mul
+    ((MatrixSmoothGeneral.determinant (g.smooth i)).inv
+      (fun z hz => ne_of_gt (g.positive i z hz).det_pos))
+
+def potentialDensityValue (g : KaehlerMetric A) (φ : SmoothRealFunction A) (x : X) : ℝ :=
+  (g.volumePolynomial φ (A.covers x).choose (A.chart (A.covers x).choose x)).re
+
+lemma potentialDensityValue_local (g : KaehlerMetric A) (φ : SmoothRealFunction A)
+    (i : Fin A.count) {x : X} (hi : x ∈ (A.chart i).source) :
+    g.potentialDensityValue φ x = (g.volumePolynomial φ i (A.chart i x)).re := by
+  unfold potentialDensityValue
+  rw [g.volumePolynomial_compatibility φ _ i (A.covers x).choose_spec hi]
+
+def potentialDensity (g : KaehlerMetric A) (φ : SmoothRealFunction A) : SmoothRealFunction A where
+  value := g.potentialDensityValue φ
+  smooth i := by
+    apply (Complex.reCLM.contDiff.comp_contDiffOn (g.volumePolynomial_smooth φ i)).congr
+    intro z hz
+    dsimp only [Function.comp_apply]
+    rw [g.potentialDensityValue_local φ i ((A.chart i).mapsTo_symm hz), (A.chart i).right_inv hz]
+    rfl
+
+lemma potentialDensity_complex (g : KaehlerMetric A) (φ : SmoothRealFunction A)
+    (i : Fin A.count) {x : X} (hi : x ∈ (A.chart i).source) :
+    ((g.potentialDensity φ).value x : ℂ) = g.volumePolynomial φ i (A.chart i x) := by
+  apply Complex.ext
+  · exact g.potentialDensityValue_local φ i hi
+  · exact (g.volumePolynomial_im φ i ((A.chart i).mapsTo hi)).symm
+
+lemma potentialDensity_directional (g : KaehlerMetric A) (φ : SmoothRealFunction A) (x : X) :
+    HasDerivAt (fun t : ℝ => (g.potentialDensity (φ.realSMul t)).value x)
+      ((g.laplacian φ).value x) 0 := by
+  obtain ⟨i, hi⟩ := A.covers x
+  let H := g.matrix i (A.chart i x)
+  let K := φ.hessian i (A.chart i x)
+  have hn : H.det ≠ 0 := ne_of_gt (g.positive i _ ((A.chart i).mapsTo hi)).det_pos
+  have hd := ((MongeAmpere.hasDerivAt_det_add_smul H K (isUnit_iff_ne_zero.mpr hn)).div_const H.det).real_of_complex
+  have he : H.det * (H⁻¹ * K).trace / H.det = (H⁻¹ * K).trace := by
+    rw [mul_div_cancel_left₀ _ hn]
+  rw [he] at hd
+  have hl : (g.laplacian φ).value x = (H⁻¹ * K).trace.re :=
+    g.laplacianValue_local φ i hi
+  rw [hl]
+  apply hd.congr_of_eventuallyEq
+  filter_upwards [] with t
+  change g.potentialDensityValue (φ.realSMul t) x = _
+  rw [g.potentialDensityValue_local (φ.realSMul t) i hi]
+  simp only [volumePolynomial, SmoothRealFunction.hessian_realSMul]
+  rfl
+
+end KaehlerMetric
+end Anticanonical.SourceSmooth
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/PotentialDensityDifferential.lean b/lean/OAI/Geometry/PrescribedPotential/PotentialDensityDifferential.lean
new file mode 100644
index 0000000000000000000000000000000000000000..e852b4b719b2fe6b62cd0b1694eb5ab20d759bca
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/PotentialDensityDifferential.lean
@@ -0,0 +1,107 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CompletedVolume
+import OAI.Geometry.PrescribedPotential.NonlinearHessianCommutator
+import OAI.Geometry.PrescribedPotential.PotentialDensity
+import OAI.Geometry.PrescribedPotential.SmoothPotentialDifference
+
+namespace OAI
+
+/-! Potential Density Differential. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff Classical Matrix.Norms.Elementwise
+namespace Anticanonical.SourceSmooth
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+namespace KaehlerMetric
+local instance matrixContinuousSMul (n : ℕ) : ContinuousSMul ℝ (Matrix (Fin n) (Fin n) ℂ) :=
+  inferInstanceAs (ContinuousSMul ℝ (Fin n → Fin n → ℂ))
+
+lemma potentialDensity_directional_at (g : KaehlerMetric A) (φ ψ : SmoothRealFunction A)
+    (q : Fin A.count) {x : X} (hx : x ∈ (A.chart q).source) :
+    HasDerivAt (fun t : ℝ => ((g.potentialDensity (φ.addFunction (ψ.realSMul t))).value x : ℂ))
+      ((g.matrix q (A.chart q x)).det⁻¹ *
+        NonlinearHessian.detDifferential (g.matrix q (A.chart q x) + φ.hessian q (A.chart q x))
+          (ψ.hessian q (A.chart q x))) 0 := by
+  let M := g.matrix q (A.chart q x) + φ.hessian q (A.chart q x)
+  let N := ψ.hessian q (A.chart q x)
+  have hc := ((hasDerivAt_id (0 : ℝ)).smul_const N).const_add M
+  simp only [one_smul] at hc
+  have hM : HasFDerivAt Matrix.det (NonlinearHessian.detDifferential M) (M+(0 : ℝ) • N) := by
+    simpa only [zero_smul,add_zero] using NonlinearHessian.detDifferential_hasFDeriv M
+  have hd := (hM.comp_hasDerivAt (0 : ℝ) hc).const_mul (g.matrix q (A.chart q x)).det⁻¹
+  apply hd.congr_of_eventuallyEq
+  filter_upwards [] with t
+  rw [g.potentialDensity_complex _ q hx]
+  unfold volumePolynomial
+  rw [SmoothRealFunction.hessian_addFunction _ _ q ((A.chart q).mapsTo hx),
+    SmoothRealFunction.hessian_realSMul]
+  have hs : t • N = (t : ℂ) • N := by
+    ext i j
+    exact Complex.real_smul
+  simp only [Function.comp_apply, id_eq] at *
+  rw [hs,div_eq_mul_inv,mul_comm]
+  congr 1
+  simp only [M,N,add_assoc]
+
+end KaehlerMetric
+end Anticanonical.SourceSmooth
+
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+lemma completedVolumeDerivative_source (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (φ ψ : SmoothRealFunction A) (q : Fin A.count) {x : X} (hx : x ∈ (A.chart q).source) :
+    D.localizers.strong (k : ℝ)
+      (fderiv ℝ (D.completedVolume k hk) (D.localizers.embed ((k : ℝ)+2) (Smooth.ofReal φ))
+        (D.localizers.embed ((k : ℝ)+2) (Smooth.ofReal ψ))) x =
+      (g.matrix q (A.chart q x)).det⁻¹ *
+        NonlinearHessian.detDifferential (g.matrix q (A.chart q x) + φ.hessian q (A.chart q x))
+          (ψ.hessian q (A.chart q x)) := by
+  have hs : (Module.finrank ℝ (EC d) : ℝ) < 2*(k : ℝ) := by
+    have hh : (Module.finrank ℝ (EC d) : ℝ) < (k : ℝ) := by exact_mod_cast hk
+    linarith [Nat.cast_nonneg (α := ℝ) k]
+  let u := D.localizers.embed ((k : ℝ)+2) (Smooth.ofReal φ)
+  let v := D.localizers.embed ((k : ℝ)+2) (Smooth.ofReal ψ)
+  let E : D.localizers.Sobolev (k : ℝ) →L[ℝ] ℂ :=
+    (BoundedContinuousFunction.evalCLM ℝ x).comp (D.localizers.strong (k : ℝ))
+  have hc := ((hasDerivAt_id (0 : ℝ)).smul_const v).const_add u
+  simp only [one_smul] at hc
+  have hu : HasFDerivAt (D.completedVolume k hk)
+      (fderiv ℝ (D.completedVolume k hk) u) (u+(0 : ℝ) • v) := by
+    simpa only [zero_smul,add_zero] using
+      ((D.completedVolume_contDiff k hk).differentiable (by simp) u).hasFDerivAt
+  have hd := E.hasFDerivAt.comp_hasDerivAt (0 : ℝ) (hu.comp_hasDerivAt (0 : ℝ) hc)
+  have he (t : ℝ) : u+t • v = D.localizers.embed ((k : ℝ)+2)
+      (Smooth.ofReal (φ.addFunction (ψ.realSMul t))) := by
+    dsimp only [u,v]
+    rw [← map_smul,← map_add]
+    congr 1
+    apply Smooth.ext
+    intro y
+    simp [SmoothRealFunction.addFunction,SmoothRealFunction.realSMul,Complex.real_smul]
+  have hp := g.potentialDensity_directional_at φ ψ q hx
+  apply hd.unique
+  apply hp.congr_of_eventuallyEq
+  filter_upwards [] with t
+  change E (D.completedVolume k hk (u+t • v)) = _
+  rw [he,D.completedVolume_embed]
+  change D.localizers.strong (k : ℝ) (D.localizers.embed (k : ℝ) _) x = _
+  rw [D.localizers.strong_embed _ hs]
+  rfl
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/PotentialLinearCalculus.lean b/lean/OAI/Geometry/PrescribedPotential/PotentialLinearCalculus.lean
new file mode 100644
index 0000000000000000000000000000000000000000..d625e5b3e13aa5afce4ccb08e3d67e2f7780968d
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/PotentialLinearCalculus.lean
@@ -0,0 +1,43 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.WirtingerCalculus
+
+namespace OAI
+
+/-! Potential Linear Calculus. -/
+
+section
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace KaehlerCalculus
+variable {n : ℕ}
+
+lemma potentialMatrix_affine {f g : V n → ℝ} {z : V n}
+    (hf : ContDiffAt ℝ ∞ f z) (hg : ContDiffAt ℝ ∞ g z) (a : ℝ) :
+    PotentialKaehler.potentialMatrix (fun y => f y+a*g y) z =
+      PotentialKaehler.potentialMatrix f z+(a:ℂ) • PotentialKaehler.potentialMatrix g z := by
+  have he := EllipticKernel.hessian_add_smul_sub_const hf hg a 0
+  simp only [sub_zero] at he
+  unfold PotentialKaehler.potentialMatrix
+  rw [he,EllipticKernel.hermitianPartMatrix_add,EllipticKernel.hermitianPartMatrix_smul]
+  rfl
+
+lemma potentialMatrix_sub_smul {f g : V n → ℝ} {z : V n}
+    (hf : ContDiffAt ℝ ∞ f z) (hg : ContDiffAt ℝ ∞ g z) (a : ℝ) :
+    PotentialKaehler.potentialMatrix (fun y => f y-a*g y) z =
+      PotentialKaehler.potentialMatrix f z-(a:ℂ) • PotentialKaehler.potentialMatrix g z := by
+  simpa only [neg_mul,sub_eq_add_neg,Complex.ofReal_neg,neg_smul] using potentialMatrix_affine hf hg (-a)
+
+lemma potentialMatrix_trace_sub_smul {f g : V n → ℝ} {z : V n}
+    (hf : ContDiffAt ℝ ∞ f z) (hg : ContDiffAt ℝ ∞ g z) (a : ℝ) :
+    (PotentialKaehler.potentialMatrix (fun y => f y-a*g y) z).trace.re =
+      (PotentialKaehler.potentialMatrix f z).trace.re-a*(PotentialKaehler.potentialMatrix g z).trace.re := by
+  rw [potentialMatrix_sub_smul hf hg,Matrix.trace_sub,Matrix.trace_smul]
+  simp only [Complex.sub_re,smul_eq_mul,Complex.mul_re,Complex.ofReal_re,Complex.ofReal_im,zero_mul,sub_zero]
+end KaehlerCalculus
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/PrescribedPotential.lean b/lean/OAI/Geometry/PrescribedPotential/PrescribedPotential.lean
new file mode 100644
index 0000000000000000000000000000000000000000..43880946a020e42f28ce80c6bd21058156a4dee4
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/PrescribedPotential.lean
@@ -0,0 +1,31 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.BoundedPathContinuation
+import OAI.Geometry.PrescribedPotential.UniformPathSobolev
+
+namespace OAI
+
+/-! Prescribed Potential. -/
+
+section
+
+ 
+noncomputable section
+namespace Anticanonical.SourceSmooth
+open GlobalElliptic
+
+ 
+
+theorem prescribedPotential (d : ℕ) (X : Type) [TopologicalSpace X] [T2Space X]
+    [CompactSpace X] [ConnectedSpace X] (A : ComplexAtlas d X)
+    (P : A.ProjectiveEmbedding) (h : SemipositiveAnticanonicalMetric A) :
+    HasPrescribedVolumePotential P h := by
+  obtain ⟨S,⟨D⟩⟩ := exists_gluingData (projectiveBackground P)
+  let x₀ : X := Classical.choice (inferInstance : Nonempty X)
+  exact prescribedPotential_of_path_bounds P h D x₀ (D.pathSobolevBounds h x₀)
+
+end Anticanonical.SourceSmooth
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/PrincipalComponents.lean b/lean/OAI/Geometry/PrescribedPotential/PrincipalComponents.lean
new file mode 100644
index 0000000000000000000000000000000000000000..e997d9681ff77a25220e15391615ff3f34c8797b
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/PrincipalComponents.lean
@@ -0,0 +1,85 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.ComplexHessian
+import OAI.Geometry.PrescribedPotential.MatrixHilbertCoordinates
+import OAI.Geometry.PrescribedPotential.MatrixWirtinger
+import OAI.Geometry.PrescribedPotential.ScalarPrincipalWirtinger
+
+namespace OAI
+
+/-! Principal Components. -/
+
+section
+
+noncomputable section
+open Set Filter Topology Matrix Finset
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace HigherJet
+open EllipticKernel FrozenPoisson KaehlerCalculus
+variable {E F G : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [NormedAddCommGroup F] [NormedSpace ℝ F] [NormedAddCommGroup G] [NormedSpace ℝ G]
+  {ι : Type*} [Fintype ι]
+lemma principalApply_postcomp (e : OrthonormalBasis ι ℝ E) (L : SecondOrder E)
+    (H : E →L[ℝ] E →L[ℝ] F) (_J : G) (l : F →L[ℝ] G)
+    (Q : E →L[ℝ] E →L[ℝ] G) (hQ : ∀ v w, Q v w = l (H v w)) :
+    principalApply e L Q = l (principalApply e L H) := by
+  simp only [principalApply,map_sum,map_smul,hQ]
+lemma principalApply_hessian_postcomp (e : OrthonormalBasis ι ℝ E) (L : SecondOrder E)
+    {f : E → F} {x : E} (hf : ContDiffAt ℝ ∞ f x) (l : F →L[ℝ] G) :
+    principalApply e L (fderiv ℝ (fderiv ℝ (fun y => l (f y))) x) =
+      l (principalApply e L (fderiv ℝ (fderiv ℝ f) x)) :=
+  principalApply_postcomp e L _ (0:G) l _ (fun v w => GlobalElliptic.second_postcomp_general hf l v w)
+end HigherJet
+namespace MetricSystem
+open EllipticKernel FrozenPoisson HigherJet KaehlerCalculus
+variable {n : ℕ} {ι : Type*} [Fintype ι]
+local instance componentRealECIP : InnerProductSpace ℝ (EC n) := InnerProductSpace.rclikeToReal ℂ (EC n)
+lemma complex_principal (basis : OrthonormalBasis ι ℝ (EC n)) (M : Mat n) (hM : M.PosDef)
+    {f : EC n → ℂ} {x : EC n} (hf : ContDiffAt ℝ ∞ f x) :
+    principalApply basis (traceBilin M) (fderiv ℝ (fderiv ℝ f) x) =
+      ∑ p, ∑ q, M⁻¹ p q*dzbar (e q) (dz (e p) (f ∘ (coordinateEquiv n).symm)) (coordinateEquiv n x) := by
+  have hcomp : ContDiffAt ℝ ∞ (f ∘ (coordinateEquiv n).symm) (coordinateEquiv n x) :=
+    (by simpa only [ContinuousLinearEquiv.symm_apply_apply] using hf : ContDiffAt ℝ ∞ f ((coordinateEquiv n).symm (coordinateEquiv n x))).comp _ (coordinateEquiv n).symm.contDiff.contDiffAt
+  rw [inverse_wirtinger_complex M hM hcomp]
+  apply Complex.ext
+  · have hr : ContDiffAt ℝ ∞ (fun y => (f y).re) x := Complex.reCLM.contDiff.contDiffAt.comp x hf
+    have he := scalar_principal_real basis M hr
+    have hp := principalApply_hessian_postcomp basis (traceBilin M) hf Complex.reCLM
+    simpa [Function.comp_def] using hp.symm.trans he
+  · have hi : ContDiffAt ℝ ∞ (fun y => (f y).im) x := Complex.imCLM.contDiff.contDiffAt.comp x hf
+    have he := scalar_principal_real basis M hi
+    have hp := principalApply_hessian_postcomp basis (traceBilin M) hf Complex.imCLM
+    simpa [Function.comp_def] using hp.symm.trans he
+
+ def entryCLM (i j : Fin n) : HM n →L[ℝ] ℂ :=
+  ((ContinuousLinearMap.proj j).comp (ContinuousLinearMap.proj i)).comp (encode n).symm.toContinuousLinearMap
+@[simp] lemma entryCLM_apply (i j : Fin n) (u : HM n) : entryCLM i j u = u (i,j) := rfl
+
+lemma encoded_principal (basis : OrthonormalBasis ι ℝ (EC n))
+    {M : V n → Mat n} {z : V n} (hM : ContDiffAt ℝ ∞ M z) (hp : (M z).PosDef) :
+    principalBilinear basis (M z)⁻¹
+      (fderiv ℝ (fderiv ℝ (fun x => encode n (M (coordinateEquiv n x)))) ((coordinateEquiv n).symm z)) =
+      encode n (∑ p, ∑ q, (M z)⁻¹ p q • mderiv Complex.I (e q) (mderiv (-Complex.I) (e p) M) z) := by
+  let u : EC n → HM n := fun x => encode n (M (coordinateEquiv n x))
+  let x := (coordinateEquiv n).symm z
+  have hu : ContDiffAt ℝ ∞ u x := (encode n).contDiff.contDiffAt.comp x
+    ((by simpa only [x,ContinuousLinearEquiv.apply_symm_apply] using hM : ContDiffAt ℝ ∞ M (coordinateEquiv n x)).comp x (coordinateEquiv n).contDiff.contDiffAt)
+  rw [principalBilinear_inv]
+  apply PiLp.ext
+  intro ij
+  obtain ⟨i,j⟩ := ij
+  have he := complex_principal basis (M z) hp ((entryCLM i j).contDiff.contDiffAt.comp x hu)
+  have hpcomp := principalApply_hessian_postcomp basis (traceBilin (M z)) hu (entryCLM i j)
+  have he := hpcomp.symm.trans he
+  have hf : (fun y => entryCLM i j (u y)) ∘ (coordinateEquiv n).symm = fun y => M y i j := by
+    funext y
+    simp [u]
+  change _ = ∑ p, ∑ q, (M z)⁻¹ p q*dzbar (e q) (dz (e p) ((fun y => entryCLM i j (u y)) ∘ (coordinateEquiv n).symm)) (coordinateEquiv n x) at he
+  rw [hf] at he
+  simpa only [entryCLM_apply,u,x,ContinuousLinearEquiv.apply_symm_apply,encode_apply,
+    Matrix.sum_apply,Matrix.smul_apply,smul_eq_mul,mderiv,dzbar,dz] using he
+end MetricSystem
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/QuadraticContact.lean b/lean/OAI/Geometry/PrescribedPotential/QuadraticContact.lean
new file mode 100644
index 0000000000000000000000000000000000000000..aadb30c8ff517c2289e9587bd3960719b9a3f170
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/QuadraticContact.lean
@@ -0,0 +1,85 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CircleRadialCalculus
+import OAI.Geometry.PrescribedPotential.KaehlerClosedDerivatives
+
+namespace OAI
+
+/-! Quadratic Contact. -/
+
+section
+
+ 
+
+noncomputable section
+open Matrix Set Filter Topology
+open scoped InnerProductSpace ContDiff
+namespace PotentialABP
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+
+lemma norm_sub_sq_hessian (c x v w : E) :
+    fderiv ℝ (fderiv ℝ (fun y : E => ‖y-c‖^2)) x v w = 2 * inner ℝ v w := by
+  have he : fderiv ℝ (fun y : E => ‖y-c‖^2) = fun y => (2:ℝ) • innerSL ℝ (y-c) := by
+    funext y
+    simpa [two_smul] using (((hasFDerivAt_id y).sub_const c).norm_sq).fderiv
+  rw [he]
+  have hd := ((innerSL ℝ : E →L[ℝ] E →L[ℝ] ℝ).hasFDerivAt.comp x
+      ((hasFDerivAt_id x).sub_const c)).const_smul (2:ℝ)
+  change HasFDerivAt (fun y => (2:ℝ) • innerSL ℝ (y-c)) _ x at hd
+  rw [hd.fderiv]
+  simp
+  rfl
+
+lemma quadratic_add_hessian {f : E → ℝ} (hf : ContDiff ℝ ∞ f) (δ : ℝ)
+    (c x v w : E) :
+    fderiv ℝ (fderiv ℝ (fun y => f y + δ*‖y-c‖^2)) x v w =
+      fderiv ℝ (fderiv ℝ f) x v w + 2*δ*inner ℝ v w := by
+  have hn : ContDiff ℝ ∞ (fun y : E => ‖y-c‖^2) :=
+    (contDiff_id.sub contDiff_const).norm_sq ℝ
+  have he := EllipticKernel.hessian_add_smul_sub_const (z := x) hf.contDiffAt hn.contDiffAt δ 0
+  simp only [sub_zero] at he
+  rw [he]
+  simp only [_root_.add_apply, _root_.smul_apply,
+    smul_eq_mul, norm_sub_sq_hessian]
+  ring
+
+end PotentialABP
+
+namespace PotentialABP
+variable {d : ℕ}
+local notation "E" => EuclideanSpace ℂ (Fin d)
+local instance (d : ℕ) : InnerProductSpace ℝ (EuclideanSpace ℂ (Fin d)) :=
+  InnerProductSpace.rclikeToReal ℂ (EuclideanSpace ℂ (Fin d))
+local notation "e" => EuclideanSpace.equiv (Fin d) ℂ
+
+lemma hessian_coord_comp {f : (Fin d → ℂ) → ℝ} (hf : ContDiff ℝ ∞ f)
+    (x v w : E) :
+    fderiv ℝ (fderiv ℝ (f ∘ e)) x v w =
+      fderiv ℝ (fderiv ℝ f) (e x) (e v) (e w) := by
+  let er := (EuclideanSpace.equiv (Fin d) ℂ).toContinuousLinearMap.restrictScalars ℝ
+  have her : ContDiff ℝ ∞ e := er.contDiff
+  have hd : fderiv ℝ e = fun _ => (er : E →L[ℝ] (Fin d → ℂ)) := by
+    funext y
+    exact er.hasFDerivAt.fderiv
+  rw [PotentialKaehler.second_derivative_comp her.contDiffAt hf.contDiffAt v w, hd]
+  simp
+  rfl
+
+lemma levi_coord_pair {f : (Fin d → ℂ) → ℝ} (hf : ContDiff ℝ ∞ f)
+    (x v : E) :
+    fderiv ℝ (fderiv ℝ (f ∘ e)) x v v +
+      fderiv ℝ (fderiv ℝ (f ∘ e)) x (Complex.I • v) (Complex.I • v) =
+        4 * inner ℝ ((PotentialKaehler.potentialMatrix f (e x)).toEuclideanLin v) v := by
+  rw [hessian_coord_comp hf, hessian_coord_comp hf]
+  rw [real_inner_comm, real_inner_eq_re_inner (𝕜 := ℂ), EuclideanSpace.inner_eq_star_dotProduct]
+  rw [dotProduct_comm]
+  change _ = 4 * (star (e v) ⬝ᵥ ((PotentialKaehler.potentialMatrix f (e x)) *ᵥ (e v))).re
+  rw [PotentialKaehler.potentialMatrix, PotentialKaehler.hermitianPartMatrix_pair]
+  simp only [map_smul, PotentialKaehler.hermitianPart]
+  ring
+
+end PotentialABP
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/QuantitativeResolvent.lean b/lean/OAI/Geometry/PrescribedPotential/QuantitativeResolvent.lean
new file mode 100644
index 0000000000000000000000000000000000000000..4656ec39e0afca5c5185d7a891c4a9e23fb080f9
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/QuantitativeResolvent.lean
@@ -0,0 +1,141 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CompletedResolvent
+import OAI.Geometry.PrescribedPotential.ComplexKernel
+import OAI.Geometry.PrescribedPotential.GlobalSmooth
+
+namespace OAI
+
+/-! Quantitative Resolvent. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology
+open scoped ContDiff Classical
+namespace Anticanonical.SourceSmooth.KaehlerMetric
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [CompactSpace X]
+  {A : ComplexAtlas d X}
+
+lemma positive_eigenfunction_nonpos (g : KaehlerMetric A) (φ : SmoothRealFunction A)
+    {a : ℝ} (ha : 0 < a)
+    (he : ∀ i z, z ∈ (A.chart i).target →
+      g.linearizedMongeAmpere φ i z = a * φ.localExpression i z) :
+    ∀ x, φ.value x ≤ 0 := by
+  intro y
+  let : Nonempty X := ⟨y⟩
+  obtain ⟨x, _, hx⟩ := isCompact_univ.exists_isMaxOn univ_nonempty φ.continuous.continuousOn
+  obtain ⟨i, hi⟩ := A.covers x
+  have hn := g.linearized_nonpos_at_globalMax φ (fun z => hx (mem_univ z)) i hi
+  rw [he i _ ((A.chart i).mapsTo hi)] at hn
+  have hv : φ.localExpression i (A.chart i x) = φ.value x := by
+    simp only [SmoothRealFunction.localExpression, Function.comp_apply, (A.chart i).left_inv hi]
+  rw [hv] at hn
+  exact (hx (mem_univ y)).trans (by nlinarith)
+
+lemma positive_eigenfunction_zero (g : KaehlerMetric A) (φ : SmoothRealFunction A)
+    {a : ℝ} (ha : 0 < a)
+    (he : ∀ i z, z ∈ (A.chart i).target →
+      g.linearizedMongeAmpere φ i z = a * φ.localExpression i z) :
+    ∀ x, φ.value x = 0 := by
+  have hp := g.positive_eigenfunction_nonpos φ ha he
+  have hn := g.positive_eigenfunction_nonpos (φ.realSMul (-1)) ha (by
+    intro i z hz
+    rw [g.linearizedMongeAmpere_realSMul, he i z hz]
+    change -1 * (a * φ.localExpression i z) = a * (-1 * φ.localExpression i z)
+    ring)
+  intro x
+  have hnx : -1 * φ.value x ≤ 0 := hn x
+  exact le_antisymm (hp x) (by linarith)
+
+end Anticanonical.SourceSmooth.KaehlerMetric
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [CompactSpace X]
+  {A : ComplexAtlas d X}
+
+lemma shifted_complex_kernel (g : KaehlerMetric A) {a : ℝ} (ha : 0 < a)
+    (u : Smooth A) (hu : a • u - complexL g u = 0) : u = 0 := by
+  have he : complexL g u = a • u := (sub_eq_zero.mp hu).symm
+  have hp (p : ℂ →L[ℝ] ℝ) (hpart : ∀ x,
+      p (complexL g u x) = (g.laplacian (u.part p)).value x) :
+      ∀ x, p (u x) = 0 := by
+    apply g.positive_eigenfunction_zero (u.part p) ha
+    intro i z hz
+    rw [← g.laplacian_localExpression _ i hz]
+    change (g.laplacian (u.part p)).value ((A.chart i).symm z) = _
+    rw [← hpart, he]
+    exact p.map_smul a (u ((A.chart i).symm z))
+  have hr := hp Complex.reCLM (fun x => complexL_re g u x)
+  have hi := hp Complex.imCLM (fun x => complexL_im g u x)
+  ext x
+  exact Complex.ext (hr x) (hi x)
+
+variable [T2Space X] {ι : Type*} [Fintype ι]
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+lemma shiftedOperator_embed (m : ℝ) (f : Smooth A) :
+    D.shiftedOperator m (D.localizers.embed 2 f) =
+      D.localizers.embed 0 ((m^2 : ℝ) • f - complexL g f) := by
+  simp only [shiftedOperator, completedL, sub_apply,
+    smul_apply, D.localizers.lower_embed (by norm_num : (0:ℝ) ≤ 2),
+    D.localizers.extendCore_embed D.complexL_bound, map_sub, map_smul]
+
+lemma completedResolvent_left (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
+    (v : D.localizers.Sobolev 2) :
+    D.completedResolvent m hm he (D.shiftedOperator m v) = v := by
+  have hfun : (D.completedResolvent m hm he) ∘ D.shiftedOperator m = id := by
+    apply (D.localizers.embed_dense 2).equalizer (by fun_prop) continuous_id
+    funext f
+    simp only [Function.comp_apply]
+    rw [D.shiftedOperator_embed]
+    obtain ⟨u, hu⟩ := D.allRegular_smooth (D.completedResolvent_regular m hm he
+      ((m^2 : ℝ) • f - complexL g f))
+    have hu2 : D.localizers.embed 2 u = D.completedResolvent m hm he
+        (D.localizers.embed 0 ((m^2 : ℝ) • f - complexL g f)) := by
+      apply D.localizers.lower_injective (by norm_num : (0 : ℝ) ≤ 2)
+      rw [D.localizers.lower_embed (by norm_num : (0 : ℝ) ≤ 2), hu]
+    have hs := D.completedResolvent_equation m hm he
+      (D.localizers.embed 0 ((m^2 : ℝ) • f - complexL g f))
+    rw [← hu2, D.shiftedOperator_embed] at hs
+    have hh := D.localizers.embed_injective 0 hs
+    have hk : (m^2 : ℝ) • (u-f) - complexL g (u-f) = 0 := by
+      rw [map_sub, smul_sub]
+      calc
+        (m^2 : ℝ) • u - (m^2 : ℝ) • f - (complexL g u - complexL g f) =
+            ((m^2 : ℝ) • u - complexL g u) - ((m^2 : ℝ) • f - complexL g f) := by abel
+        _ = 0 := sub_eq_zero.mpr hh
+    have hef : u = f := sub_eq_zero.mp (shifted_complex_kernel g (by positivity) (u-f) hk)
+    rw [← hu2, hef]
+    rfl
+  exact congr_fun hfun v
+
+lemma shiftedOperator_bijective (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1) :
+    Function.Bijective (D.shiftedOperator m) :=
+  ⟨Function.LeftInverse.injective (D.completedResolvent_left m hm he),
+    Function.RightInverse.surjective (D.completedResolvent_equation m hm he)⟩
+
+ 
+theorem elliptic_estimate (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
+    (u : D.localizers.Sobolev 2) :
+    ‖u‖ ≤ ‖D.completedResolvent m hm he‖ *
+      ((m^2 : ℝ) * ‖D.localizers.lower 2 0 u‖ + ‖D.completedL u‖) := by
+  calc
+    ‖u‖ = ‖D.completedResolvent m hm he (D.shiftedOperator m u)‖ :=
+      congrArg norm (D.completedResolvent_left m hm he u).symm
+    _ ≤ ‖D.completedResolvent m hm he‖ * ‖D.shiftedOperator m u‖ :=
+      (D.completedResolvent m hm he).le_opNorm _
+    _ ≤ _ := by
+      apply mul_le_mul_of_nonneg_left _ (norm_nonneg (D.completedResolvent m hm he))
+      change ‖(m^2 : ℝ) • D.localizers.lower 2 0 u - D.completedL u‖ ≤ _
+      exact (norm_sub_le _ _).trans (by rw [norm_smul, Real.norm_eq_abs, abs_of_nonneg (sq_nonneg m)])
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/QuasiPSHLocalMean.lean b/lean/OAI/Geometry/PrescribedPotential/QuasiPSHLocalMean.lean
new file mode 100644
index 0000000000000000000000000000000000000000..a1e27aff03326a0c9ce3509cf3a3dc6a276ea8d0
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/QuasiPSHLocalMean.lean
@@ -0,0 +1,110 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.BallSubmean
+import OAI.Geometry.PrescribedPotential.QuadraticContact
+
+namespace OAI
+
+/-! Quasi P S H Local Mean. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Metric MeasureTheory
+open scoped ContDiff InnerProductSpace
+namespace PotentialABP
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℂ E]
+  [FiniteDimensional ℂ E] [MeasurableSpace E] [BorelSpace E]
+local instance quasiPSHLocalMeanRealInnerProductSpace : InnerProductSpace ℝ E :=
+  InnerProductSpace.rclikeToReal ℂ E
+
+omit [InnerProductSpace ℂ E] [FiniteDimensional ℂ E] [MeasurableSpace E] [BorelSpace E] in
+lemma closedBall_small_center_subset {c y : E} {r : ℝ} (hy : y ∈ ball c r) :
+    closedBall y (2*r) ⊆ closedBall c (3*r) := by
+  intro x hx
+  have ht := dist_triangle x y c
+  have hy' : dist y c < r := hy
+  have hx' : dist x y ≤ 2*r := hx
+  change dist x c ≤ 3*r
+  linarith
+
+omit [InnerProductSpace ℂ E] [FiniteDimensional ℂ E] [MeasurableSpace E] [BorelSpace E] in
+lemma ball_small_center_subset {c y : E} {r : ℝ} (hy : y ∈ ball c r) :
+    ball c r ⊆ ball y (2*r) := by
+  intro x hx
+  have ht := dist_triangle x c y
+  have hy' : dist c y < r := by simpa [dist_comm] using hy
+  have hx' : dist x c < r := hx
+  change dist x y < 2*r
+  linarith
+
+ 
+
+theorem quasiPSH_local_integral
+    {f : E → ℝ} (hf : ContDiff ℝ ∞ f) (c : E) {r C M : ℝ}
+    (hr : 0 < r) (hC : 0 ≤ C)
+    (hupper : ∀ x ∈ closedBall c (3*r), f x ≤ M)
+    (hlevi : ∀ x ∈ closedBall c (3*r), ∀ v : E,
+      -(4*C*‖v‖^2) ≤ fderiv ℝ (fderiv ℝ f) x v v +
+        fderiv ℝ (fderiv ℝ f) x (Complex.I • v) (Complex.I • v))
+    {y : E} (hy : y ∈ ball c r) :
+    (∫ x in ball c r, M-f x) ≤
+      volume.real (ball (0:E) (2*r)) * (M-f y+9*C*r^2) := by
+  let q : E → ℝ := fun x => C*‖x-c‖^2
+  have hq : ContDiff ℝ ∞ q := contDiff_const.mul
+    ((contDiff_id.sub contDiff_const).norm_sq ℝ)
+  have hsub := closedBall_small_center_subset hy
+  have hmean := smooth_ball_submean (hf.add hq) y (by positivity : 0 ≤ 2*r) ?_
+  · have hu : IntegrableOn (fun x => M-f x) (ball y (2*r)) :=
+      ((continuous_const.sub hf.continuous).continuousOn.integrableOn_compact
+        (isCompact_closedBall y (2*r))).mono_set ball_subset_closedBall
+    have hfi : IntegrableOn f (ball y (2*r)) :=
+      (hf.continuous.continuousOn.integrableOn_compact
+        (isCompact_closedBall y (2*r))).mono_set ball_subset_closedBall
+    have hqi : IntegrableOn q (ball y (2*r)) :=
+      (hq.continuous.continuousOn.integrableOn_compact
+        (isCompact_closedBall y (2*r))).mono_set ball_subset_closedBall
+    have hvol : volume.real (ball y (2*r)) = volume.real (ball (0:E) (2*r)) := by
+      simpa only [setIntegral_const, smul_eq_mul, mul_one] using
+        (ball_integral_translate (fun _ : E => (1:ℝ)) y (2*r)).symm
+    have hqB : (∫ x in ball y (2*r), q x) ≤
+        volume.real (ball (0:E) (2*r)) * (9*C*r^2) := by
+      have hm : (∫ x in ball y (2*r), q x) ≤
+          ∫ _x in ball y (2*r), (9*C*r^2) := by
+        apply setIntegral_mono_on hqi (integrableOn_const (C := (9*C*r^2)))
+          isOpen_ball.measurableSet
+        intro x hx
+        have hb := hsub (ball_subset_closedBall hx)
+        have hn : ‖x-c‖ ≤ 3*r := by simpa [mem_closedBall, dist_eq_norm] using hb
+        have hs := (sq_le_sq₀ (norm_nonneg (x-c)) (by positivity : 0 ≤ 3*r)).mpr hn
+        change C*‖x-c‖^2 ≤ 9*C*r^2
+        nlinarith [mul_le_mul_of_nonneg_left hs hC]
+      simpa only [setIntegral_const, smul_eq_mul, hvol] using hm
+    have hm : (∫ x in ball c r, M-f x) ≤ ∫ x in ball y (2*r), M-f x := by
+      apply setIntegral_mono_set hu _ (Filter.Eventually.of_forall (ball_small_center_subset hy))
+      filter_upwards [ae_restrict_mem isOpen_ball.measurableSet] with x hx
+      exact sub_nonneg.mpr (hupper x (hsub (ball_subset_closedBall hx)))
+    have hic : IntegrableOn (fun _ : E => M) (ball y (2*r)) volume :=
+      integrableOn_const (C := M)
+    apply hm.trans
+    rw [integral_sub hic hfi,
+      setIntegral_const, smul_eq_mul, hvol]
+    rw [integral_add hfi hqi] at hmean
+    have hq0 : 0 ≤ q y := mul_nonneg hC (sq_nonneg _)
+    have hV : 0 ≤ volume.real (ball (0:E) (2*r)) := ENNReal.toReal_nonneg
+    nlinarith [mul_nonneg hV hq0]
+  · intro x hx v
+    have h := hlevi x (hsub hx) v
+    change 0 ≤ fderiv ℝ (fderiv ℝ (fun x => f x + C*‖x-c‖^2)) x v v +
+      fderiv ℝ (fderiv ℝ (fun x => f x + C*‖x-c‖^2)) x (Complex.I • v) (Complex.I • v)
+    rw [quadratic_add_hessian hf, quadratic_add_hessian hf]
+    simp only [real_inner_self_eq_norm_sq, norm_smul, Complex.norm_I, one_mul]
+    linarith
+
+end PotentialABP
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/QuasilinearFamilyBootstrap.lean b/lean/OAI/Geometry/PrescribedPotential/QuasilinearFamilyBootstrap.lean
new file mode 100644
index 0000000000000000000000000000000000000000..7fe407517877fd34670fb24e27392a507b54ebd5
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/QuasilinearFamilyBootstrap.lean
@@ -0,0 +1,99 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.QuasilinearHighResiduals
+import OAI.Geometry.PrescribedPotential.ResidualFamilyBootstrap
+
+namespace OAI
+
+/-! Quasilinear Family Bootstrap. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Finset Module
+open scoped ContDiff
+namespace HigherJet
+variable {E F G : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [NormedAddCommGroup F] [InnerProductSpace ℝ F]
+  [NormedAddCommGroup G] [NormedSpace ℝ G]
+  [FiniteDimensional ℝ E] [FiniteDimensional ℝ F]
+  {ι α : Type*} [Fintype ι]
+
+local instance hessianNormedAdd : NormedAddCommGroup (E →L[ℝ] E →L[ℝ] F) := inferInstance
+local instance hessianNormedSpace : NormedSpace ℝ (E →L[ℝ] E →L[ℝ] F) := inferInstance
+
+lemma energy_bounds_finite_norms {U : Set E} (hU : IsOpen U) (e : OrthonormalBasis ι ℝ E)
+    {u : α → E → F} (hu : ∀ a, ContDiffOn ℝ ∞ (u a) U) {K : Set E} (hKU : K ⊆ U)
+    (m : ℕ) (hlo : ∀ j, 1 ≤ j → j < m → EnergyBound (fun j a x => jetEnergy e j (u a) x) j K) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ j, 1 ≤ j → j < m → ∀ a x, x ∈ K → ‖iteratedFDeriv ℝ j (u a) x‖ ≤ C := by
+  classical
+  have he (j : Fin m) : ∃ C : ℝ, 0 ≤ C ∧ (1 ≤ j.val → ∀ a x, x ∈ K → ‖iteratedFDeriv ℝ j.val (u a) x‖ ≤ C) := by
+    by_cases hj : 1 ≤ j.val
+    · obtain ⟨R,hR,hr⟩ := hlo j hj j.isLt
+      obtain ⟨C,hC,hc⟩ := norm_iteratedFDeriv_le_jetEnergy (F := F) e.toBasis j.val
+      exact ⟨C*Real.sqrt R,by positivity,fun _ a x hx => (hc U hU (u a) (hu a) x (hKU hx)).trans
+        (mul_le_mul_of_nonneg_left (Real.sqrt_le_sqrt (hr a x hx)) hC.le)⟩
+    · exact ⟨0,le_rfl,fun h => False.elim (hj h)⟩
+  choose C hC hc using he
+  refine ⟨∑ j, C j,Finset.sum_nonneg (fun j _ => hC j),fun j hj hjm a x hx => ?_⟩
+  exact (hc ⟨j,hjm⟩ hj a x hx).trans (Finset.single_le_sum (fun i _ => hC i) (Finset.mem_univ _))
+
+lemma quasilinear_family_bootstrap (B : G →L[ℝ] (E →L[ℝ] E →L[ℝ] F) →L[ℝ] F)
+    {U : Set E} (hU : IsOpen U) (e : OrthonormalBasis ι ℝ E)
+    (u r : α → E → F) (a : F → G) (b : F × (E →L[ℝ] F) → F)
+    (τ : α → E → E ≃L[ℝ] E)
+    (hu : ∀ s, ContDiffOn ℝ ∞ (u s) U) (hr : ∀ s, ContDiffOn ℝ ∞ (r s) U)
+    (ha : ∀ s y, y ∈ U → ContDiffAt ℝ ∞ a (u s y))
+    (hb : ∀ s y, y ∈ U → ContDiffAt ℝ ∞ b (firstJet (u s) y))
+    (heq : ∀ s y, y ∈ U → B (a (u s y)) (hessian (u s) y) = r s y+b (firstJet (u s) y))
+    (hframe : ∀ s x, x ∈ U → ∀ H : E →L[ℝ] E →L[ℝ] F,
+      B (a (u s x)) H = ∑ i, H (τ s x (e i)) (τ s x (e i)))
+    (hframes : ∀ K, IsCompact K → K ⊆ U → ∃ C : ℝ, 0 ≤ C ∧ ∀ s x, x ∈ K →
+      ‖(τ s x).toContinuousLinearMap‖ ≤ C ∧ ‖(τ s x).symm.toContinuousLinearMap‖ ≤ C)
+    (hcoeff : ∀ m K, IsCompact K → K ⊆ U → ∃ C : ℝ, 1 ≤ C ∧ ‖B‖ ≤ C ∧
+      (∀ j, j ≤ m → (Fintype.card (OrderedFinpartition j) : ℝ) ≤ C) ∧ ∀ s x, x ∈ K →
+      ‖iteratedFDeriv ℝ 1 (u s) x‖ ≤ C ∧
+      (∀ j, j ≤ m → ‖iteratedFDeriv ℝ j a (u s x)‖ ≤ C) ∧
+      (∀ j, j ≤ m → ‖iteratedFDeriv ℝ j b (firstJet (u s) x)‖ ≤ C) ∧
+      (∀ j, j ≤ m → ‖iteratedFDeriv ℝ j (r s) x‖ ≤ C)) :
+    ∀ m, 1 ≤ m → ∀ K, IsCompact K → K ⊆ U →
+      ∃ C : ℝ, 0 ≤ C ∧ ∀ s x, x ∈ K → ‖iteratedFDeriv ℝ m (u s) x‖ ≤ C := by
+  apply residual_family_bootstrap hU e u τ hu hframes
+  · intro K hK hKU
+    obtain ⟨C,hC,_,_,hc⟩ := hcoeff 1 K hK hKU
+    exact ⟨C,by linarith,fun s x hx => (hc s x hx).1⟩
+  · intro K hK hKU
+    obtain ⟨C,hC,hB,hP,hc⟩ := hcoeff 2 K hK hKU
+    have hC0 : 0 ≤ C := by linarith
+    let R1 := 2*C+C^2+C^3
+    let R2 := C+3*C^2+5*C^3+C^4+C^5
+    have hR1 : 0 ≤ R1 := by dsimp [R1]; positivity
+    have hR2 : 0 ≤ R2 := by dsimp [R2]; positivity
+    refine ⟨R1+R2,add_nonneg hR1 hR2,fun s x hx => ?_⟩
+    obtain ⟨hl1,hl2⟩ := quasilinear_low_residuals B hU (hu s) (hr s) (ha s) (hb s)
+      (heq s) (hKU hx) (fun i => τ s x (e i)) (hframe s x (hKU hx)) e hC0 hB (hP 2 le_rfl)
+      (hc s x hx).1 (hc s x hx).2.1 (hc s x hx).2.2.1 (hc s x hx).2.2.2
+    exact ⟨fun σ => (hl1 σ).trans (mul_le_mul_of_nonneg_right (le_add_of_nonneg_right hR2) (by positivity)),
+      fun σ => (hl2 σ).trans (mul_le_mul_of_nonneg_right (le_add_of_nonneg_left hR1) (by positivity))⟩
+  · intro m hm K hK hKU hlo
+    obtain ⟨C0,hC0,hB,hP,hc⟩ := hcoeff m K hK hKU
+    obtain ⟨C1,hC1,hlo'⟩ := energy_bounds_finite_norms hU e hu hKU m hlo
+    let C := C0+C1
+    have hC : 1 ≤ C := by dsimp [C]; linarith
+    have hc0 : C0 ≤ C := by dsimp [C]; linarith
+    have hc1 : C1 ≤ C := by dsimp [C]; linarith
+    refine ⟨C+C^(m+2)*C+C*2^m*C^(m+2)*C,by
+      have hCnonneg : 0 ≤ C := by linarith
+      positivity,fun s x hx => ?_⟩
+    exact quasilinear_high_residual B hU (hu s) (hr s) (ha s) (hb s) (heq s) (hKU hx)
+      (fun i => τ s x (e i)) (hframe s x (hKU hx)) e hm hC (hB.trans hc0)
+      (fun j hj => (hP j hj).trans hc0) (fun j hj hjm => (hlo' j hj hjm s x hx).trans hc1)
+      (fun j hj => ((hc s x hx).2.1 j hj).trans hc0)
+      (fun j hj => ((hc s x hx).2.2.1 j hj).trans hc0) (((hc s x hx).2.2.2 m le_rfl).trans hc0)
+end HigherJet
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/QuasilinearHighResiduals.lean b/lean/OAI/Geometry/PrescribedPotential/QuasilinearHighResiduals.lean
new file mode 100644
index 0000000000000000000000000000000000000000..60a3764ffae5e6d0c2850c3b52323362ac610de0
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/QuasilinearHighResiduals.lean
@@ -0,0 +1,146 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.HigherJetComposition
+import OAI.Geometry.PrescribedPotential.QuasilinearLowResiduals
+
+namespace OAI
+
+/-! Quasilinear High Residuals. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Finset
+open scoped ContDiff
+namespace HigherJet
+variable {E F G : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [NormedAddCommGroup F] [InnerProductSpace ℝ F]
+  [NormedAddCommGroup G] [NormedSpace ℝ G]
+local instance highHessianNormed : NormedAddCommGroup (E →L[ℝ] E →L[ℝ] F) := inferInstance
+local instance highHessianSpace : NormedSpace ℝ (E →L[ℝ] E →L[ℝ] F) := inferInstance
+
+lemma coefficient_prefactor {m i : ℕ} (hi : i ≤ m) {C : ℝ} (hC : 1 ≤ C)
+    (hP : (Fintype.card (OrderedFinpartition i) : ℝ) ≤ C) :
+    (Fintype.card (OrderedFinpartition i) : ℝ)*C*C^i ≤ C^(m+2) := by
+  calc
+    _ ≤ C*C*C^i := mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hP (by linarith)) (by positivity)
+    _ = C^(i+2) := by ring
+    _ ≤ C^(m+2) := pow_le_pow_right₀ hC (by omega)
+
+lemma high_coefficient_bounds {u : E → F} {a : F → G} {b : F × (E →L[ℝ] F) → F} {x : E}
+    (hu : ContDiffAt ℝ ∞ u x) (ha : ContDiffAt ℝ ∞ a (u x))
+    (hb : ContDiffAt ℝ ∞ b (firstJet u x)) {m : ℕ} (hm : 3 ≤ m) {C : ℝ} (hC : 1 ≤ C)
+    (hP : ∀ i, i ≤ m → (Fintype.card (OrderedFinpartition i) : ℝ) ≤ C)
+    (hulow : ∀ j, 0 < j → j < m → ‖iteratedFDeriv ℝ j u x‖ ≤ C)
+    (haj : ∀ j, j ≤ m → ‖iteratedFDeriv ℝ j a (u x)‖ ≤ C)
+    (hbj : ∀ j, j ≤ m → ‖iteratedFDeriv ℝ j b (firstJet u x)‖ ≤ C) :
+    (∀ i, 0 < i → i < m → ‖iteratedFDeriv ℝ i (a ∘ u) x‖ ≤ C^(m+2)) ∧
+    ‖iteratedFDeriv ℝ m (a ∘ u) x‖ ≤ C^(m+2)*(1+‖iteratedFDeriv ℝ m u x‖) ∧
+    ‖iteratedFDeriv ℝ m (b ∘ firstJet u) x‖ ≤
+      C^(m+2)*C*(1+‖iteratedFDeriv ℝ m u x‖+‖iteratedFDeriv ℝ (m+1) u x‖) := by
+  have hC0 : 0 ≤ C := by linarith
+  refine ⟨?_,?_,?_⟩
+  · intro i hi him
+    have h := composition_top_jet_bound hu ha (m := i) (r := i+1) (by omega) hC
+      (show (1:ℝ) ≤ 1 by rfl) hC0
+      (fun j hj hji => hulow j hj (by omega))
+      (fun j hj hji => by omega) (fun j hj => haj j (by omega))
+    simpa only [mul_one] using h.trans (by simpa only [mul_one] using coefficient_prefactor (Nat.le_of_lt him) hC (hP i (Nat.le_of_lt him)))
+  · exact (composition_top_jet_bound hu ha (m := m) (r := m) (by omega) hC
+      (le_add_of_nonneg_right (norm_nonneg _)) hC0 hulow
+      (fun j hj hjm => by
+        have he : j=m := by omega
+        rw [he]
+        exact le_add_of_nonneg_left zero_le_one)
+      haj).trans (mul_le_mul_of_nonneg_right (coefficient_prefactor (le_refl m) hC (hP m le_rfl)) (by positivity))
+  · let T := C+‖iteratedFDeriv ℝ m u x‖+‖iteratedFDeriv ℝ (m+1) u x‖
+    have hT : 1 ≤ T := by dsimp [T]; linarith [norm_nonneg (iteratedFDeriv ℝ m u x),norm_nonneg (iteratedFDeriv ℝ (m+1) u x)]
+    have hlow (j : ℕ) (hj : 0 < j) (hjm : j < m-1) : ‖iteratedFDeriv ℝ j (firstJet u) x‖ ≤ C := by
+      rw [norm_firstJet_deriv hu]
+      exact max_le (hulow j hj (by omega)) (hulow (j+1) (by omega) (by omega))
+    have hhigh (j : ℕ) (hj : m-1 ≤ j) (hjm : j ≤ m) : ‖iteratedFDeriv ℝ j (firstJet u) x‖ ≤ T := by
+      rw [norm_firstJet_deriv hu]
+      by_cases he : j=m
+      · subst j
+        dsimp [T]
+        exact max_le (by linarith [norm_nonneg (iteratedFDeriv ℝ (m+1) u x)]) (by linarith [norm_nonneg (iteratedFDeriv ℝ m u x)])
+      · have he : j+1=m := by omega
+        rw [he]
+        dsimp [T]
+        exact max_le ((hulow j (by omega) (by omega)).trans (by linarith [norm_nonneg (iteratedFDeriv ℝ m u x),norm_nonneg (iteratedFDeriv ℝ (m+1) u x)]))
+          (by linarith [norm_nonneg (iteratedFDeriv ℝ (m+1) u x)])
+    apply (composition_top_jet_bound (firstJet_smoothAt hu) hb (r := m-1) (by omega) hC hT hC0 hlow hhigh hbj).trans
+    have ht : T ≤ C*(1+‖iteratedFDeriv ℝ m u x‖+‖iteratedFDeriv ℝ (m+1) u x‖) := by
+      dsimp [T]
+      nlinarith only [mul_nonneg (by linarith : 0 ≤ C-1) (norm_nonneg (iteratedFDeriv ℝ m u x)),
+        mul_nonneg (by linarith : 0 ≤ C-1) (norm_nonneg (iteratedFDeriv ℝ (m+1) u x))]
+    exact (mul_le_mul (coefficient_prefactor (le_refl m) hC (hP m le_rfl)) ht (by linarith) (by positivity)).trans_eq (by ring)
+
+variable {ι : Type*} [Fintype ι]
+lemma quasilinear_high_residual (B : G →L[ℝ] (E →L[ℝ] E →L[ℝ] F) →L[ℝ] F)
+    {U : Set E} (hU : IsOpen U) {u r : E → F} {a : F → G} {b : F × (E →L[ℝ] F) → F}
+    (hu : ContDiffOn ℝ ∞ u U) (hr : ContDiffOn ℝ ∞ r U)
+    (ha : ∀ y ∈ U, ContDiffAt ℝ ∞ a (u y))
+    (hb : ∀ y ∈ U, ContDiffAt ℝ ∞ b (firstJet u y))
+    (heq : ∀ y ∈ U, B (a (u y)) (hessian u y) = r y+b (firstJet u y))
+    {x : E} (hx : x ∈ U) (v : ι → E)
+    (hframe : ∀ H : E →L[ℝ] E →L[ℝ] F, B (a (u x)) H = ∑ i, H (v i) (v i))
+    (e : OrthonormalBasis ι ℝ E) {m : ℕ} (hm : 3 ≤ m) {C : ℝ} (hC : 1 ≤ C) (hB : ‖B‖ ≤ C)
+    (hP : ∀ i, i ≤ m → (Fintype.card (OrderedFinpartition i) : ℝ) ≤ C)
+    (hulow : ∀ j, 0 < j → j < m → ‖iteratedFDeriv ℝ j u x‖ ≤ C)
+    (haj : ∀ j, j ≤ m → ‖iteratedFDeriv ℝ j a (u x)‖ ≤ C)
+    (hbj : ∀ j, j ≤ m → ‖iteratedFDeriv ℝ j b (firstJet u x)‖ ≤ C)
+    (hrj : ‖iteratedFDeriv ℝ m r x‖ ≤ C) :
+    ∀ σ : Fin m → ι, ‖frameLaplace v (jetFamily e m u σ) x‖ ≤
+      (C+C^(m+2)*C+C*2^m*C^(m+2)*C)*(1+‖iteratedFDeriv ℝ m u x‖+‖iteratedFDeriv ℝ (m+1) u x‖) := by
+  have hC0 : 0 ≤ C := by linarith
+  have hac : ContDiffOn ℝ ∞ (a ∘ u) U := fun y hy => ((ha y hy).comp y (hu.contDiffAt (hU.mem_nhds hy))).contDiffWithinAt
+  have hbc : ContDiffOn ℝ ∞ (b ∘ firstJet u) U := fun y hy => ((hb y hy).comp y (firstJet_smoothAt (hu.contDiffAt (hU.mem_nhds hy)))).contDiffWithinAt
+  obtain ⟨halow,hatop,hbtop⟩ := high_coefficient_bounds (hu.contDiffAt (hU.mem_nhds hx)) (ha x hx) (hb x hx) hm hC hP hulow haj hbj
+  let T := 1+‖iteratedFDeriv ℝ m u x‖+‖iteratedFDeriv ℝ (m+1) u x‖
+  have hT : 1 ≤ T := by dsimp [T]; linarith [norm_nonneg (iteratedFDeriv ℝ m u x),norm_nonneg (iteratedFDeriv ℝ (m+1) u x)]
+  have hprod (i : ℕ) (hi : 0 < i) (him : i ≤ m) :
+      ‖iteratedFDeriv ℝ i (a ∘ u) x‖*‖iteratedFDeriv ℝ (m-i) (hessian u) x‖ ≤ (C^(m+2)*C)*T := by
+    rw [norm_hessian_deriv]
+    by_cases he : i=m
+    · subst i
+      have hm2 : m-m+2=2 := by omega
+      rw [hm2]
+      have h2 := hulow 2 (by decide) (by omega)
+      have hh : 1+‖iteratedFDeriv ℝ m u x‖ ≤ T := by dsimp [T]; exact le_add_of_nonneg_right (norm_nonneg _)
+      calc
+        _ ≤ (C^(m+2)*(1+‖iteratedFDeriv ℝ m u x‖))*C := mul_le_mul hatop h2 (norm_nonneg _) (by positivity)
+        _ = (C^(m+2)*C)*(1+‖iteratedFDeriv ℝ m u x‖) := by ring
+        _ ≤ _ := mul_le_mul_of_nonneg_left hh (by positivity)
+    · have hi' : i < m := by omega
+      have hh : ‖iteratedFDeriv ℝ (m-i+2) u x‖ ≤ C*T := by
+        by_cases h1 : i=1
+        · subst i
+          have he : m-1+2=m+1 := by omega
+          rw [he]
+          exact (show ‖iteratedFDeriv ℝ (m+1) u x‖ ≤ T by dsimp [T]; linarith [norm_nonneg (iteratedFDeriv ℝ m u x)]).trans (le_mul_of_one_le_left (by linarith) hC)
+        · by_cases h2 : i=2
+          · subst i
+            have he : m-2+2=m := by omega
+            rw [he]
+            exact (show ‖iteratedFDeriv ℝ m u x‖ ≤ T by dsimp [T]; linarith [norm_nonneg (iteratedFDeriv ℝ (m+1) u x)]).trans (le_mul_of_one_le_left (by linarith) hC)
+          · exact (hulow _ (by omega) (by omega)).trans (le_mul_of_one_le_right hC0 hT)
+      exact (mul_le_mul (halow i hi hi') hh (norm_nonneg _) (by positivity)).trans_eq (by ring)
+  have hcomm := frozen_product_tame_bound B hU hac (hessian_smoothOn hU hu) hx m
+    (show 0 ≤ C^(m+2)*C by positivity) (show 0 ≤ T by linarith) hprod
+  intro σ
+  have hnorm := (frozen_equation_jet_norm B hU hu (hr.add hbc) hac heq hx v hframe e m σ).trans
+    (add_le_add (norm_iteratedFDeriv_add_le (hr.contDiffAt (hU.mem_nhds hx)) (hbc.contDiffAt (hU.mem_nhds hx)) m) le_rfl)
+  have hcomm' : ‖iteratedFDeriv ℝ m (fun y => B (a (u y)-a (u x)) (hessian u y)) x‖ ≤ (C*2^m*C^(m+2)*C)*T :=
+    hcomm.trans (by nlinarith only [mul_le_mul_of_nonneg_right hB (show 0 ≤ 2^m*(C^(m+2)*C)*T by positivity)])
+  have hr' : ‖iteratedFDeriv ℝ m r x‖ ≤ C*T := hrj.trans (le_mul_of_one_le_right hC0 hT)
+  change _ ≤ (C+C^(m+2)*C+C*2^m*C^(m+2)*C)*T
+  change _ ≤ C^(m+2)*C*T at hbtop
+  exact (hnorm.trans (add_le_add (add_le_add hr' hbtop) hcomm')).trans_eq (by ring)
+end HigherJet
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/QuasilinearJetBasics.lean b/lean/OAI/Geometry/PrescribedPotential/QuasilinearJetBasics.lean
new file mode 100644
index 0000000000000000000000000000000000000000..9eaf604ecfa1aeb56ac11931b10427b9d261eeae
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/QuasilinearJetBasics.lean
@@ -0,0 +1,95 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.FrozenEllipticJets
+import OAI.Geometry.PrescribedPotential.HigherJetCommutator
+
+namespace OAI
+
+/-! Quasilinear Jet Basics. -/
+
+section
+
+ 
+noncomputable section
+open Set Filter Topology Finset
+open scoped ContDiff
+namespace HigherJet
+variable {E F G : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  [NormedAddCommGroup F] [NormedSpace ℝ F]
+  [NormedAddCommGroup G] [NormedSpace ℝ G]
+
+def firstJet (u : E → F) (x : E) : F × (E →L[ℝ] F) := (u x,fderiv ℝ u x)
+lemma firstJet_smoothOn {U : Set E} (hU : IsOpen U) {u : E → F}
+    (hu : ContDiffOn ℝ ∞ u U) : ContDiffOn ℝ ∞ (firstJet u) U :=
+  hu.prodMk (hu.fderiv_of_isOpen hU (m := ∞) (by simp))
+lemma firstJet_smoothAt {u : E → F} {x : E} (hu : ContDiffAt ℝ ∞ u x) :
+    ContDiffAt ℝ ∞ (firstJet u) x := hu.prodMk (hu.fderiv_right (m := ∞) (by simp))
+lemma norm_firstJet_deriv {u : E → F} {x : E} (hu : ContDiffAt ℝ ∞ u x) (m : ℕ) :
+    ‖iteratedFDeriv ℝ m (firstJet u) x‖ =
+      max ‖iteratedFDeriv ℝ m u x‖ ‖iteratedFDeriv ℝ (m+1) u x‖ := by
+  change ‖iteratedFDeriv ℝ m (fun y => (u y,fderiv ℝ u y)) x‖ = _
+  rw [iteratedFDeriv_prodMk hu (hu.fderiv_right (m := ∞) (by simp))
+    (show (m:ℕ∞ω) ≤ ∞ from WithTop.coe_le_coe.mpr le_top),
+    ContinuousMultilinearMap.opNorm_prod,norm_iteratedFDeriv_fderiv]
+lemma composition_first_jet_bound {f : E → F} {g : F → G} {x : E}
+    (hf : ContDiffAt ℝ ∞ f x) (hg : ContDiffAt ℝ ∞ g (f x)) :
+    ‖iteratedFDeriv ℝ 1 (g ∘ f) x‖ ≤
+      ‖iteratedFDeriv ℝ 1 g (f x)‖*‖iteratedFDeriv ℝ 1 f x‖ := by
+  simp only [norm_iteratedFDeriv_one]
+  rw [fderiv_comp x (hg.differentiableAt (by simp)) (hf.differentiableAt (by simp))]
+  exact ContinuousLinearMap.opNorm_comp_le _ _
+
+ 
+
+lemma compact_coefficient_jets {K : Set F} (hK : IsCompact K) {a : F → G}
+    (ha : ∀ y ∈ K, ContDiffAt ℝ ∞ a y) (m : ℕ) :
+    ∃ C : ℝ, 1 ≤ C ∧ ∀ j, j ≤ m → ∀ y ∈ K, ‖iteratedFDeriv ℝ j a y‖ ≤ C := by
+  have hb (j : ℕ) : ∃ C : ℝ, 0 ≤ C ∧ ∀ y ∈ K, ‖iteratedFDeriv ℝ j a y‖ ≤ C := by
+    obtain ⟨C,hC⟩ := hK.exists_bound_of_continuousOn (fun y hy =>
+      ((ha y hy).iteratedFDeriv_right (m := ∞) (i := j)
+        (by exact WithTop.coe_le_coe.mpr le_top)).continuousAt.continuousWithinAt)
+    exact ⟨max C 0,le_max_right _ _,fun y hy => (hC y hy).trans (le_max_left _ _)⟩
+  choose C hC hb using hb
+  refine ⟨1+∑ j ∈ range (m+1), C j,le_add_of_nonneg_right (Finset.sum_nonneg (fun j _ => hC j)),?_⟩
+  intro j hj y hy
+  exact (hb j y hy).trans ((Finset.single_le_sum (fun i _ => hC i)
+    (mem_range.mpr (by omega))).trans (le_add_of_nonneg_left zero_le_one))
+
+end HigherJet
+namespace HigherJet
+variable {E F G : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  [NormedAddCommGroup F] [InnerProductSpace ℝ F]
+  [NormedAddCommGroup G] [NormedSpace ℝ G]
+lemma norm_hessian_deriv (u : E → F) (x : E) (m : ℕ) :
+    ‖iteratedFDeriv ℝ m (hessian u) x‖ = ‖iteratedFDeriv ℝ (m+2) u x‖ := by
+  simp only [hessian,norm_iteratedFDeriv_fderiv]
+
+local instance hessianNormed : NormedAddCommGroup (E →L[ℝ] E →L[ℝ] F) := inferInstance
+local instance hessianSpace : NormedSpace ℝ (E →L[ℝ] E →L[ℝ] F) := inferInstance
+
+lemma frozen_first_bound (B : G →L[ℝ] (E →L[ℝ] E →L[ℝ] F) →L[ℝ] F)
+    {U : Set E} (hU : IsOpen U) {u : E → F} {a : E → G}
+    (hu : ContDiffOn ℝ ∞ u U) (ha : ContDiffOn ℝ ∞ a U) {x : E} (hx : x ∈ U) :
+    ‖iteratedFDeriv ℝ 1 (fun y => B (a y-a x) (hessian u y)) x‖ ≤
+      ‖B‖*‖iteratedFDeriv ℝ 1 a x‖*‖iteratedFDeriv ℝ 2 u x‖ := by
+  have h := frozen_product_jet_bound B hU ha (hessian_smoothOn hU hu) hx 1
+  simpa only [Finset.sum_range_succ,Finset.sum_range_zero,Nat.choose_zero_right,
+    Nat.choose_self,Nat.cast_one,ite_true,one_mul,zero_mul,zero_add,ite_false,
+    Nat.reduceSub,norm_hessian_deriv,Nat.reduceAdd,show ¬ (1 : ℕ) = 0 from by decide,mul_assoc] using h
+
+lemma frozen_second_bound (B : G →L[ℝ] (E →L[ℝ] E →L[ℝ] F) →L[ℝ] F)
+    {U : Set E} (hU : IsOpen U) {u : E → F} {a : E → G}
+    (hu : ContDiffOn ℝ ∞ u U) (ha : ContDiffOn ℝ ∞ a U) {x : E} (hx : x ∈ U) :
+    ‖iteratedFDeriv ℝ 2 (fun y => B (a y-a x) (hessian u y)) x‖ ≤
+      ‖B‖*(2*‖iteratedFDeriv ℝ 1 a x‖*‖iteratedFDeriv ℝ 3 u x‖+
+        ‖iteratedFDeriv ℝ 2 a x‖*‖iteratedFDeriv ℝ 2 u x‖) := by
+  have h := frozen_product_jet_bound B hU ha (hessian_smoothOn hU hu) hx 2
+  norm_num only [Finset.sum_range_succ,Finset.sum_range_zero,Nat.choose_zero_right,
+    Nat.choose_self,Nat.choose_one_right,ite_true,one_mul,zero_mul,zero_add,ite_false,
+    Nat.reduceSub,norm_hessian_deriv,not_false_eq_true,Nat.cast_one,Nat.cast_ofNat] at h
+  exact h
+end HigherJet
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/QuasilinearLowResiduals.lean b/lean/OAI/Geometry/PrescribedPotential/QuasilinearLowResiduals.lean
new file mode 100644
index 0000000000000000000000000000000000000000..a395a1223aa66be72aaa7eb52eb0453d7a7915aa
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/QuasilinearLowResiduals.lean
@@ -0,0 +1,137 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.DirectionalJetEnergy
+import OAI.Geometry.PrescribedPotential.QuasilinearJetBasics
+import OAI.Geometry.PrescribedPotential.QuasilinearResidualAlgebra
+import OAI.Geometry.PrescribedPotential.SecondJetComposition
+
+namespace OAI
+
+/-! Quasilinear Low Residuals. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Finset
+open scoped ContDiff
+namespace HigherJet
+variable {E F G : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [NormedAddCommGroup F] [InnerProductSpace ℝ F]
+  [NormedAddCommGroup G] [NormedSpace ℝ G]
+local instance lowHessianNormed : NormedAddCommGroup (E →L[ℝ] E →L[ℝ] F) := inferInstance
+local instance lowHessianSpace : NormedSpace ℝ (E →L[ℝ] E →L[ℝ] F) := inferInstance
+
+lemma norm_iteratedFDeriv_add_le {f g : E → F} {x : E}
+    (hf : ContDiffAt ℝ ∞ f x) (hg : ContDiffAt ℝ ∞ g x) (m : ℕ) :
+    ‖iteratedFDeriv ℝ m (fun y => f y+g y) x‖ ≤
+      ‖iteratedFDeriv ℝ m f x‖+‖iteratedFDeriv ℝ m g x‖ := by
+  change ‖iteratedFDeriv ℝ m (f+g) x‖ ≤ _
+  rw [iteratedFDeriv_add_apply (hf.of_le (show (m:ℕ∞ω) ≤ ∞ from WithTop.coe_le_coe.mpr le_top))
+    (hg.of_le (show (m:ℕ∞ω) ≤ ∞ from WithTop.coe_le_coe.mpr le_top))]
+  exact norm_add_le _ _
+
+lemma low_coefficient_bounds {u : E → F} {a : F → G} {b : F × (E →L[ℝ] F) → F} {x : E}
+    (hu : ContDiffAt ℝ ∞ u x) (ha : ContDiffAt ℝ ∞ a (u x))
+    (hb : ContDiffAt ℝ ∞ b (firstJet u x)) {C : ℝ} (hC : 0 ≤ C)
+    (hP : (Fintype.card (OrderedFinpartition 2) : ℝ) ≤ C)
+    (hu1 : ‖iteratedFDeriv ℝ 1 u x‖ ≤ C)
+    (haj : ∀ j, j ≤ 2 → ‖iteratedFDeriv ℝ j a (u x)‖ ≤ C)
+    (hbj : ∀ j, j ≤ 2 → ‖iteratedFDeriv ℝ j b (firstJet u x)‖ ≤ C) :
+    ‖iteratedFDeriv ℝ 1 (a ∘ u) x‖ ≤ C^2 ∧
+    ‖iteratedFDeriv ℝ 2 (a ∘ u) x‖ ≤ C^2*(‖iteratedFDeriv ℝ 2 u x‖+C^2) ∧
+    ‖iteratedFDeriv ℝ 1 (b ∘ firstJet u) x‖ ≤ C*(C+‖iteratedFDeriv ℝ 2 u x‖) ∧
+    ‖iteratedFDeriv ℝ 2 (b ∘ firstJet u) x‖ ≤ C^2*(‖iteratedFDeriv ℝ 2 u x‖+
+      ‖iteratedFDeriv ℝ 3 u x‖+(C+‖iteratedFDeriv ℝ 2 u x‖)^2) := by
+  have hPC : (Fintype.card (OrderedFinpartition 2) : ℝ)*C ≤ C^2 := by
+    nlinarith only [mul_le_mul_of_nonneg_right hP hC]
+  have hf1 : ‖iteratedFDeriv ℝ 1 (firstJet u) x‖ ≤ C+‖iteratedFDeriv ℝ 2 u x‖ := by
+    rw [norm_firstJet_deriv hu]
+    exact max_le ((hu1.trans (le_add_of_nonneg_right (norm_nonneg _))))
+      (le_add_of_nonneg_left hC)
+  have hf2 : ‖iteratedFDeriv ℝ 2 (firstJet u) x‖ ≤ ‖iteratedFDeriv ℝ 2 u x‖+‖iteratedFDeriv ℝ 3 u x‖ := by
+    rw [norm_firstJet_deriv hu]
+    exact max_le (le_add_of_nonneg_right (norm_nonneg _)) (le_add_of_nonneg_left (norm_nonneg _))
+  refine ⟨?_,?_,?_,?_⟩
+  · exact (composition_first_jet_bound hu ha).trans (by
+      simpa only [pow_two] using mul_le_mul (haj 1 (by decide)) hu1 (norm_nonneg _) hC)
+  · exact (composition_second_jet_bound hu ha hC haj).trans
+      (mul_le_mul hPC (add_le_add le_rfl (pow_le_pow_left₀ (norm_nonneg _) hu1 2))
+        (by positivity) (sq_nonneg C))
+  · exact (composition_first_jet_bound (firstJet_smoothAt hu) hb).trans
+      (mul_le_mul (hbj 1 (by decide)) hf1 (norm_nonneg _) hC)
+  · exact (composition_second_jet_bound (firstJet_smoothAt hu) hb hC hbj).trans
+      (mul_le_mul hPC (add_le_add hf2 (pow_le_pow_left₀ (norm_nonneg _) hf1 2))
+        (by positivity) (sq_nonneg C))
+
+variable {ι : Type*} [Fintype ι]
+lemma frozen_equation_jet_norm (B : G →L[ℝ] (E →L[ℝ] E →L[ℝ] F) →L[ℝ] F)
+    {U : Set E} (hU : IsOpen U) {u r : E → F} {a : E → G}
+    (hu : ContDiffOn ℝ ∞ u U) (hr : ContDiffOn ℝ ∞ r U)
+    (ha : ContDiffOn ℝ ∞ a U) (heq : ∀ y ∈ U, B (a y) (hessian u y) = r y)
+    {x : E} (hx : x ∈ U) (v : ι → E)
+    (hframe : ∀ H : E →L[ℝ] E →L[ℝ] F, B (a x) H = ∑ i, H (v i) (v i))
+    (e : OrthonormalBasis ι ℝ E) (m : ℕ) (σ : Fin m → ι) :
+    ‖frameLaplace v (jetFamily e m u σ) x‖ ≤ ‖iteratedFDeriv ℝ m r x‖+
+      ‖iteratedFDeriv ℝ m (fun y => B (a y-a x) (hessian u y)) x‖ := by
+  unfold jetFamily
+  rw [frozen_equation_word B hU hu hr ha heq hx v hframe]
+  apply (norm_sub_le _ _).trans
+  apply add_le_add
+  · simpa only [Function.comp_def,e.norm_eq_one,Finset.prod_const_one,mul_one] using
+      norm_word_ofFn_le hU hr m (fun i => e (σ i)) hx
+  · simpa only [Function.comp_def,e.norm_eq_one,Finset.prod_const_one,mul_one] using
+      norm_word_ofFn_le hU ((B.contDiff.comp_contDiffOn (ha.sub contDiffOn_const)).clm_apply
+        (hessian_smoothOn hU hu)) m (fun i => e (σ i)) hx
+
+lemma quasilinear_low_residuals (B : G →L[ℝ] (E →L[ℝ] E →L[ℝ] F) →L[ℝ] F)
+    {U : Set E} (hU : IsOpen U) {u r : E → F} {a : F → G} {b : F × (E →L[ℝ] F) → F}
+    (hu : ContDiffOn ℝ ∞ u U) (hr : ContDiffOn ℝ ∞ r U)
+    (ha : ∀ y ∈ U, ContDiffAt ℝ ∞ a (u y))
+    (hb : ∀ y ∈ U, ContDiffAt ℝ ∞ b (firstJet u y))
+    (heq : ∀ y ∈ U, B (a (u y)) (hessian u y) = r y+b (firstJet u y))
+    {x : E} (hx : x ∈ U) (v : ι → E)
+    (hframe : ∀ H : E →L[ℝ] E →L[ℝ] F, B (a (u x)) H = ∑ i, H (v i) (v i))
+    (e : OrthonormalBasis ι ℝ E) {C : ℝ} (hC : 0 ≤ C) (hB : ‖B‖ ≤ C)
+    (hP : (Fintype.card (OrderedFinpartition 2) : ℝ) ≤ C)
+    (hu1 : ‖iteratedFDeriv ℝ 1 u x‖ ≤ C)
+    (haj : ∀ j, j ≤ 2 → ‖iteratedFDeriv ℝ j a (u x)‖ ≤ C)
+    (hbj : ∀ j, j ≤ 2 → ‖iteratedFDeriv ℝ j b (firstJet u x)‖ ≤ C)
+    (hrj : ∀ j, j ≤ 2 → ‖iteratedFDeriv ℝ j r x‖ ≤ C) :
+    (∀ σ : Fin 1 → ι, ‖frameLaplace v (jetFamily e 1 u σ) x‖ ≤
+      (2*C+C^2+C^3)*(1+‖iteratedFDeriv ℝ 2 u x‖)) ∧
+    (∀ σ : Fin 2 → ι, ‖frameLaplace v (jetFamily e 2 u σ) x‖ ≤
+      (C+3*C^2+5*C^3+C^4+C^5)*(1+‖iteratedFDeriv ℝ 2 u x‖^2+‖iteratedFDeriv ℝ 3 u x‖)) := by
+  have hac : ContDiffOn ℝ ∞ (a ∘ u) U := fun y hy =>
+    ((ha y hy).comp y (hu.contDiffAt (hU.mem_nhds hy))).contDiffWithinAt
+  have hbc : ContDiffOn ℝ ∞ (b ∘ firstJet u) U := fun y hy =>
+    ((hb y hy).comp y (firstJet_smoothAt (hu.contDiffAt (hU.mem_nhds hy)))).contDiffWithinAt
+  obtain ⟨ha1,ha2,hb1,hb2⟩ := low_coefficient_bounds (hu.contDiffAt (hU.mem_nhds hx))
+    (ha x hx) (hb x hx) hC hP hu1 haj hbj
+  have hnorm (m : ℕ) (σ : Fin m → ι) : ‖frameLaplace v (jetFamily e m u σ) x‖ ≤
+      ‖iteratedFDeriv ℝ m r x‖+‖iteratedFDeriv ℝ m (b ∘ firstJet u) x‖+
+        ‖iteratedFDeriv ℝ m (fun y => B (a (u y)-a (u x)) (hessian u y)) x‖ := by
+    exact (frozen_equation_jet_norm B hU hu (hr.add hbc) hac heq hx v hframe e m σ).trans
+      (add_le_add (norm_iteratedFDeriv_add_le (hr.contDiffAt (hU.mem_nhds hx))
+        (hbc.contDiffAt (hU.mem_nhds hx)) m) le_rfl)
+  constructor
+  · intro σ
+    apply affine_residual_bound hC (norm_nonneg _) (hrj 1 (by decide)) hb1 _ (hnorm 1 σ)
+    have hh := frozen_first_bound B hU hu hac hx
+    exact hh.trans (by
+      calc
+        _ ≤ C*C^2*‖iteratedFDeriv ℝ 2 u x‖ :=
+          mul_le_mul_of_nonneg_right (mul_le_mul hB ha1 (norm_nonneg _) hC) (norm_nonneg _)
+        _ = _ := by ring)
+  · intro σ
+    apply quadratic_residual_bound hC (norm_nonneg _) (norm_nonneg _) (hrj 2 (by decide)) hb2 _ (hnorm 2 σ)
+    exact (frozen_second_bound B hU hu hac hx).trans (mul_le_mul hB
+      (add_le_add
+        (mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left ha1 (by norm_num)) (norm_nonneg _))
+        (mul_le_mul_of_nonneg_right ha2 (norm_nonneg _))) (by positivity) hC)
+end HigherJet
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/QuasilinearResidualAlgebra.lean b/lean/OAI/Geometry/PrescribedPotential/QuasilinearResidualAlgebra.lean
new file mode 100644
index 0000000000000000000000000000000000000000..419943ccdc780d93c2a757619096d6ae70afe42f
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/QuasilinearResidualAlgebra.lean
@@ -0,0 +1,44 @@
+import Mathlib
+
+namespace OAI
+
+/-! Quasilinear Residual Algebra. -/
+
+section
+
+ 
+namespace HigherJet
+lemma affine_residual_bound {c r A B D L : ℝ} (hc : 0 ≤ c) (hr : 0 ≤ r)
+    (hA : A ≤ c) (hB : B ≤ c*(c+r)) (hD : D ≤ c^3*r) (hL : L ≤ A+B+D) :
+    L ≤ (2*c+c^2+c^3)*(1+r) := by
+  have h1 : 0 ≤ c*r := mul_nonneg hc hr
+  have h2 : 0 ≤ c^2*r := mul_nonneg (sq_nonneg c) hr
+  have h3 : 0 ≤ c+c^3 := by positivity
+  nlinarith only [hA,hB,hD,hL,h1,h2,h3]
+
+lemma four_monomial_bound {a b c d r s : ℝ}
+    (ha : 0 ≤ a) (hb : 0 ≤ b) (hc : 0 ≤ c) (hd : 0 ≤ d) (_hr : 0 ≤ r) (hs : 0 ≤ s) :
+    a+b*r+c*r^2+d*s ≤ (a+b+c+d)*(1+r^2+s) := by
+  have h0 : 1 ≤ 1+r^2+s := by nlinarith only [sq_nonneg r,hs]
+  have h1 : r ≤ 1+r^2+s := by nlinarith only [sq_nonneg (r-1),hs]
+  have h2 : r^2 ≤ 1+r^2+s := by linarith only [hs]
+  have h3 : s ≤ 1+r^2+s := by nlinarith only [sq_nonneg r]
+  have a0 := mul_le_mul_of_nonneg_left h0 ha
+  have a1 := mul_le_mul_of_nonneg_left h1 hb
+  have a2 := mul_le_mul_of_nonneg_left h2 hc
+  have a3 := mul_le_mul_of_nonneg_left h3 hd
+  nlinarith only [a0,a1,a2,a3]
+
+lemma quadratic_residual_bound {c r s A B D L : ℝ} (hc : 0 ≤ c) (hr : 0 ≤ r) (hs : 0 ≤ s)
+    (hA : A ≤ c) (hB : B ≤ c^2*(r+s+(c+r)^2))
+    (hD : D ≤ c*(2*c^2*s+c^2*(r+c^2)*r)) (hL : L ≤ A+B+D) :
+    L ≤ (c+3*c^2+5*c^3+c^4+c^5)*(1+r^2+s) := by
+  have h := four_monomial_bound (a := c+c^4) (b := c^2+2*c^3+c^5)
+    (c := c^2+c^3) (d := c^2+2*c^3) (by positivity) (by positivity) (by positivity)
+    (by positivity) hr hs
+  nlinarith only [hA,hB,hD,hL,h]
+end HigherJet
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/RealCutoffBounds.lean b/lean/OAI/Geometry/PrescribedPotential/RealCutoffBounds.lean
new file mode 100644
index 0000000000000000000000000000000000000000..e906e13e1d9002c65db39545fdccff5e48cb9c87
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/RealCutoffBounds.lean
@@ -0,0 +1,82 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.RealFrameScalar
+
+namespace OAI
+
+/-! Real Cutoff Bounds. -/
+
+section
+
+ 
+noncomputable section
+open Set Filter Topology Finset
+open scoped ContDiff
+namespace HigherJet
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  {ι : Type*} [Fintype ι]
+
+omit [Fintype ι] in
+lemma frameGradient_const_mul {f : E → ℝ} {x : E} (hf : DifferentiableAt ℝ f x)
+    (v : ι → E) (c : ℝ) : frameGradient v (fun y => c*f y) x = c • frameGradient v f x := by
+  rw [frameGradient_mul (differentiableAt_const c) hf,frameGradient_const]
+  simp
+
+lemma frameGradient_square {f : E → ℝ} {x : E} (hf : DifferentiableAt ℝ f x) (v : ι → E) :
+    ‖frameGradient v (fun y => (f y)^2) x‖^2 = 4*(f x)^2*‖frameGradient v f x‖^2 := by
+  have he : (fun y => (f y)^2) = (fun y => f y*f y) := by funext y; ring
+  rw [he,frameGradient_mul hf hf,← add_smul,norm_smul,Real.norm_eq_abs,mul_pow,sq_abs]
+  ring
+
+lemma cutoff_uniform_frames {K : Set E} (hK : IsCompact K) (χ : E → ℝ)
+    (hχ : ContDiff ℝ ∞ χ) {B : ℝ} (hB : 0 ≤ B) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ x ∈ K, ∀ v : ι → E, (∀ i, ‖v i‖ ≤ B) →
+      -C ≤ frameLaplace v (fun y => (χ y)^2) x ∧
+      ‖frameGradient v (fun y => (χ y)^2) x‖^2 ≤ C*(χ x)^2 := by
+  obtain ⟨D,hD⟩ := hK.exists_bound_of_continuousOn (hχ.fderiv_right (m := ∞) (by simp)).continuous.continuousOn
+  obtain ⟨H,hH⟩ := hK.exists_bound_of_continuousOn
+    (((hχ.pow 2).fderiv_right (m := ∞) (by simp)).fderiv_right (m := ∞) (by simp)).continuous.continuousOn
+  let D' := max D 0
+  let H' := max H 0
+  let C := (Fintype.card ι : ℝ)*H'*B^2 + 4*(Fintype.card ι : ℝ)*D'^2*B^2
+  have hD' : 0 ≤ D' := le_max_right _ _
+  have hH' : 0 ≤ H' := le_max_right _ _
+  have hC1 : (Fintype.card ι : ℝ)*H'*B^2 ≤ C := le_add_of_nonneg_right (by positivity)
+  have hC2 : 4*(Fintype.card ι : ℝ)*D'^2*B^2 ≤ C := le_add_of_nonneg_left (by positivity)
+  refine ⟨C,by dsimp [C]; positivity,?_⟩
+  intro x hx v hv
+  have hd : ‖fderiv ℝ χ x‖ ≤ D' := (hD x hx).trans (le_max_left _ _)
+  have hh : ‖fderiv ℝ (fderiv ℝ (fun y => (χ y)^2)) x‖ ≤ H' := (hH x hx).trans (le_max_left _ _)
+  have hgrad : ‖frameGradient v χ x‖^2 ≤ (Fintype.card ι : ℝ)*D'^2*B^2 := by
+    rw [norm_frameGradient_sq]
+    calc
+      _ ≤ ∑ _i : ι, (D'*B)^2 := by
+        apply sum_le_sum
+        intro i _
+        apply pow_le_pow_left₀ (norm_nonneg _)
+        exact ((fderiv ℝ χ x).le_opNorm _).trans (mul_le_mul hd (hv i) (norm_nonneg _) hD')
+      _ = _ := by simp; ring
+  constructor
+  · have hl : ‖frameLaplace v (fun y => (χ y)^2) x‖ ≤ (Fintype.card ι : ℝ)*H'*B^2 := by
+      unfold frameLaplace
+      calc
+        _ ≤ ∑ i, ‖dir (v i) (dir (v i) (fun y => (χ y)^2)) x‖ := norm_sum_le _ _
+        _ ≤ ∑ _i : ι, H'*B^2 := by
+          apply sum_le_sum
+          intro i _
+          rw [dir_dir (hχ.contDiffAt.pow 2)]
+          exact (((fderiv ℝ (fderiv ℝ (fun y => (χ y)^2)) x) (v i)).le_opNorm _).trans
+            ((mul_le_mul
+              (((fderiv ℝ (fderiv ℝ (fun y => (χ y)^2)) x).le_opNorm _).trans
+                (mul_le_mul hh (hv i) (norm_nonneg _) hH')) (hv i) (norm_nonneg _) (by positivity)).trans_eq (by ring))
+        _ = _ := by simp; ring
+    exact (neg_le_neg (hl.trans hC1)).trans (by simpa only [Real.norm_eq_abs] using neg_abs_le (frameLaplace v (fun y => (χ y)^2) x))
+  · rw [frameGradient_square (hχ.differentiable (by simp) x)]
+    have hm := mul_le_mul_of_nonneg_left hgrad (show 0 ≤ 4*(χ x)^2 by positivity)
+    have hn := mul_le_mul_of_nonneg_right hC2 (sq_nonneg (χ x))
+    nlinarith only [hm,hn]
+end HigherJet
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/RealDeterminantOrder.lean b/lean/OAI/Geometry/PrescribedPotential/RealDeterminantOrder.lean
new file mode 100644
index 0000000000000000000000000000000000000000..b50ae92a255824ddf094e73c82354a4bb5ecb224
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/RealDeterminantOrder.lean
@@ -0,0 +1,90 @@
+import Mathlib
+
+namespace OAI
+
+/-! Real Determinant Order. -/
+
+section
+
+ 
+
+noncomputable section
+open Matrix Polynomial Filter Set
+open scoped MatrixOrder Matrix.Norms.Elementwise
+namespace PotentialABP
+variable {n : Type*} [Fintype n] [DecidableEq n]
+
+lemma real_trace_mul_nonneg {A B : Matrix n n ℝ}
+    (hA : A.PosSemidef) (hB : B.PosSemidef) : 0 ≤ (A*B).trace := by
+  let S := CFC.sqrt A
+  have hS : S.PosSemidef := (CFC.sqrt_nonneg A).posSemidef
+  have hsquare : S*S = A := by simpa [S, pow_two] using CFC.sq_sqrt A hA.nonneg
+  have htr := (hB.mul_mul_conjTranspose_same S).trace_nonneg
+  have he : (S*B*Sᴴ).trace = (A*B).trace := by
+    rw [hS.isHermitian.eq, Matrix.trace_mul_cycle, hsquare]
+  rwa [he] at htr
+
+lemma real_hasDerivAt_det_one_add_smul (M : Matrix n n ℝ) :
+    HasDerivAt (fun t : ℝ => (1+t • M).det) M.trace 0 := by
+  have hp := (Matrix.det (1 + (Polynomial.X : ℝ[X]) • M.map Polynomial.C)).hasDerivAt (0 : ℝ)
+  rw [Matrix.derivative_det_one_add_X_smul] at hp
+  simpa [eval_det, ← Matrix.smul_eq_mul_diagonal] using hp
+
+lemma real_hasDerivAt_det_add_smul (H K : Matrix n n ℝ) (hH : IsUnit H.det) :
+    HasDerivAt (fun t : ℝ => (H+t • K).det) (H.det*(H⁻¹*K).trace) 0 := by
+  have hd := (real_hasDerivAt_det_one_add_smul (H⁻¹*K)).const_mul H.det
+  apply hd.congr_of_eventuallyEq
+  filter_upwards [] with t
+  rw [← Matrix.det_mul]
+  congr 1
+  simp [Matrix.mul_add, ← Matrix.mul_assoc,
+    Matrix.mul_nonsing_inv _ hH]
+
+lemma real_hasDerivAt_det_shift (A B : Matrix n n ℝ) (t : ℝ)
+    (ht : (A+t • B).PosDef) :
+    HasDerivAt (fun s : ℝ => (A+s • B).det)
+      ((A+t • B).det*((A+t • B)⁻¹*B).trace) t := by
+  have hd := real_hasDerivAt_det_add_smul (A+t • B) B
+    (isUnit_iff_ne_zero.mpr ht.det_pos.ne')
+  have hc := hd.comp_of_eq t ((hasDerivAt_id t).sub_const t) (by simp)
+  simp only [mul_one] at hc
+  apply hc.congr_of_eventuallyEq
+  filter_upwards [] with s
+  dsimp only [Function.comp_apply,id_eq]
+  simp only [sub_smul]
+  congr 1
+  abel
+
+lemma real_det_add_posSemidef (A B : Matrix n n ℝ) (hA : A.PosDef) (hB : B.PosSemidef) :
+    A.det ≤ (A+B).det := by
+  let f : ℝ → ℝ := fun t => (A+t • B).det
+  have hp (t : ℝ) (ht : 0 ≤ t) : (A+t • B).PosDef := hA.add_posSemidef (hB.smul ht)
+  have hd (t : ℝ) (ht : 0 ≤ t) := real_hasDerivAt_det_shift A B t (hp t ht)
+  have hm : MonotoneOn f (Icc 0 1) := by
+    apply monotoneOn_of_deriv_nonneg (convex_Icc 0 1)
+    · intro t ht
+      exact (hd t ht.1).continuousAt.continuousWithinAt
+    · intro t ht
+      exact (hd t (interior_subset ht).1).differentiableAt.differentiableWithinAt
+    · intro t ht
+      rw [(hd t (interior_subset ht).1).deriv]
+      exact mul_nonneg (hp t (interior_subset ht).1).det_pos.le
+        (real_trace_mul_nonneg (hp t (interior_subset ht).1).inv.posSemidef hB)
+  simpa [f] using hm (show (0:ℝ) ∈ Icc 0 1 by constructor <;> norm_num)
+    (show (1:ℝ) ∈ Icc 0 1 by constructor <;> norm_num) (by norm_num)
+
+lemma real_det_mono {A B : Matrix n n ℝ} (hA : A.PosSemidef)
+    (hBA : (B-A).PosSemidef) : A.det ≤ B.det := by
+  by_cases ha : A.det = 0
+  · rw [ha]
+    have hb : B.PosSemidef := by simpa using hA.add hBA
+    exact hb.det_nonneg
+  · simpa using real_det_add_posSemidef A (B-A)
+      (hA.posDef_iff_det_ne_zero.mpr ha) hBA
+
+end PotentialABP
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/RealFrameScalar.lean b/lean/OAI/Geometry/PrescribedPotential/RealFrameScalar.lean
new file mode 100644
index 0000000000000000000000000000000000000000..519cca0cd5e11678f21b201c2b1d718c9ea09a53
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/RealFrameScalar.lean
@@ -0,0 +1,129 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.GlobalSobolev
+import OAI.Geometry.PrescribedPotential.RealJetEnergy
+
+namespace OAI
+
+/-! Real Frame Scalar. -/
+
+section
+
+ 
+noncomputable section
+open Set Filter Topology Finset
+open scoped ContDiff
+namespace HigherJet
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  {ι : Type*} [Fintype ι]
+
+def frameGradient (v : ι → E) (f : E → ℝ) (x : E) : EuclideanSpace ℝ ι :=
+  WithLp.toLp 2 (fun i => dir (v i) f x)
+
+lemma norm_frameGradient_sq (v : ι → E) (f : E → ℝ) (x : E) :
+    ‖frameGradient v f x‖^2 = frameGradientSq v f x := by
+  rw [PiLp.norm_sq_eq_of_L2]
+  rfl
+
+lemma dir_mul {f g : E → ℝ} {x : E} (hf : DifferentiableAt ℝ f x)
+    (hg : DifferentiableAt ℝ g x) (v : E) :
+    dir v (fun y => f y*g y) x = dir v f x*g x+f x*dir v g x := by
+  have h := congrArg (fun T : E →L[ℝ] ℝ => T v) (hf.hasFDerivAt.mul hg.hasFDerivAt).fderiv
+  change dir v (fun y => f y*g y) x = _ at h
+  simp only [add_apply, smul_apply, smul_eq_mul] at h
+  rw [h]
+  simp only [dir]
+  ring
+
+omit [Fintype ι] in
+lemma frameGradient_mul {f g : E → ℝ} {x : E} (hf : DifferentiableAt ℝ f x)
+    (hg : DifferentiableAt ℝ g x) (v : ι → E) :
+    frameGradient v (fun y => f y*g y) x =
+      g x • frameGradient v f x+f x • frameGradient v g x := by
+  ext i
+  simp only [frameGradient,PiLp.add_apply,PiLp.smul_apply,PiLp.toLp_apply,smul_eq_mul,dir_mul hf hg]
+  ring
+
+omit [Fintype ι] in
+lemma frameGradient_add {f g : E → ℝ} {x : E} (hf : DifferentiableAt ℝ f x)
+    (hg : DifferentiableAt ℝ g x) (v : ι → E) :
+    frameGradient v (fun y => f y+g y) x = frameGradient v f x+frameGradient v g x := by
+  ext i
+  exact dir_add hf hg _
+
+omit [Fintype ι] in
+lemma frameGradient_const (c : ℝ) (v : ι → E) (x : E) :
+    frameGradient v (fun _ : E => c) x = 0 := by
+  ext i
+  simp [frameGradient,dir]
+
+lemma frameLaplace_mul {U : Set E} (hU : IsOpen U) {f g : E → ℝ}
+    (hf : ContDiffOn ℝ ∞ f U) (hg : ContDiffOn ℝ ∞ g U)
+    {x : E} (hx : x ∈ U) (v : ι → E) :
+    frameLaplace v (fun y => f y*g y) x =
+      f x*frameLaplace v g x+g x*frameLaplace v f x+
+      2*inner ℝ (frameGradient v f x) (frameGradient v g x) := by
+  have hfa := (hf.contDiffAt (hU.mem_nhds hx)).differentiableAt (by simp)
+  have hga := (hg.contDiffAt (hU.mem_nhds hx)).differentiableAt (by simp)
+  have he (i : ι) : dir (v i) (fun y => f y*g y) =ᶠ[𝓝 x]
+      (fun y => dir (v i) f y*g y+f y*dir (v i) g y) := by
+    filter_upwards [hU.mem_nhds hx] with y hy
+    exact dir_mul ((hf.contDiffAt (hU.mem_nhds hy)).differentiableAt (by simp))
+      ((hg.contDiffAt (hU.mem_nhds hy)).differentiableAt (by simp)) _
+  have hd (i : ι) : dir (v i) (dir (v i) (fun y => f y*g y)) x =
+      f x*dir (v i) (dir (v i) g) x+g x*dir (v i) (dir (v i) f) x+
+      2*(dir (v i) f x*dir (v i) g x) := by
+    have hdf := ((dir_smoothOn hU hf (v i)).contDiffAt (hU.mem_nhds hx)).differentiableAt (by simp)
+    have hdg := ((dir_smoothOn hU hg (v i)).contDiffAt (hU.mem_nhds hx)).differentiableAt (by simp)
+    rw [dir_congr (he i),dir_add (f := fun y => dir (v i) f y*g y) (g := fun y => f y*dir (v i) g y)
+      (hdf.mul hga) (hfa.mul hdg),dir_mul hdf hga,dir_mul hfa hdg]
+    ring
+  simp only [frameLaplace,hd,Finset.sum_add_distrib,← Finset.mul_sum,
+    frameGradient,PiLp.inner_apply,RCLike.inner_apply,starRingEnd_apply,star_trivial]
+  congr 2
+  apply Finset.sum_congr rfl
+  intro i _
+  ring
+
+lemma frameLaplace_add {U : Set E} (hU : IsOpen U) {f g : E → ℝ}
+    (hf : ContDiffOn ℝ ∞ f U) (hg : ContDiffOn ℝ ∞ g U)
+    {x : E} (hx : x ∈ U) (v : ι → E) :
+    frameLaplace v (fun y => f y+g y) x = frameLaplace v f x+frameLaplace v g x := by
+  have he (i : ι) : dir (v i) (fun y => f y+g y) =ᶠ[𝓝 x]
+      (fun y => dir (v i) f y+dir (v i) g y) := by
+    filter_upwards [hU.mem_nhds hx] with y hy
+    exact dir_add ((hf.contDiffAt (hU.mem_nhds hy)).differentiableAt (by simp))
+      ((hg.contDiffAt (hU.mem_nhds hy)).differentiableAt (by simp)) _
+  simp only [frameLaplace,dir_congr (he _),dir_add
+    (((dir_smoothOn hU hf _).contDiffAt (hU.mem_nhds hx)).differentiableAt (by simp))
+    (((dir_smoothOn hU hg _).contDiffAt (hU.mem_nhds hx)).differentiableAt (by simp)),Finset.sum_add_distrib]
+
+lemma frameLaplace_const (c : ℝ) (v : ι → E) (x : E) :
+    frameLaplace v (fun _ : E => c) x = 0 := by
+  have he (i : ι) : dir (v i) (fun _ : E => c) = fun _ => 0 := by
+    ext y
+    simp [dir]
+  simp only [frameLaplace,he]
+  apply Finset.sum_eq_zero
+  intro i _
+  exact congrArg (fun T : E →L[ℝ] ℝ => T (v i))
+    (hasFDerivAt_const (x := x) (c := (0 : ℝ))).fderiv
+
+omit [Fintype ι] in
+lemma frameGradient_eq_zero_of_localMax {f : E → ℝ} {x : E} (h : IsLocalMax f x) (v : ι → E) :
+    frameGradient v f x = 0 := by
+  ext i
+  simp [frameGradient,dir,h.fderiv_eq_zero]
+
+lemma frameLaplace_nonpos_of_localMax {f : E → ℝ} {x : E}
+    (hf : ContDiffAt ℝ ∞ f x) (h : IsLocalMax f x) (v : ι → E) :
+    frameLaplace v f x ≤ 0 := by
+  apply Finset.sum_nonpos
+  intro i _
+  rw [dir_dir hf]
+  exact EllipticKernel.second_fderiv_nonpos_at_localMax hf h (v i)
+end HigherJet
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/RealJetEnergy.lean b/lean/OAI/Geometry/PrescribedPotential/RealJetEnergy.lean
new file mode 100644
index 0000000000000000000000000000000000000000..ccf236c66da335c03b3b92e9639fb47bdb644c63
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/RealJetEnergy.lean
@@ -0,0 +1,133 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.HigherDirectionalJets
+
+namespace OAI
+
+/-! Real Jet Energy. -/
+
+section
+
+ 
+noncomputable section
+open Set Filter Topology Finset
+open scoped ContDiff
+namespace HigherJet
+variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  [NormedAddCommGroup F] [InnerProductSpace ℝ F]
+
+lemma dir_add {f g : E → F} {x : E} (hf : DifferentiableAt ℝ f x)
+    (hg : DifferentiableAt ℝ g x) (v : E) :
+    dir v (fun y => f y+g y) x = dir v f x+dir v g x := by
+  exact congrArg (fun T : E →L[ℝ] F => T v) (hf.hasFDerivAt.add hg.hasFDerivAt).fderiv
+
+lemma dir_smul {f : E → F} {x : E} (hf : DifferentiableAt ℝ f x) (c : ℝ) (v : E) :
+    dir v (fun y => c • f y) x = c • dir v f x := by
+  exact congrArg (fun T : E →L[ℝ] F => T v) (hf.hasFDerivAt.const_smul c).fderiv
+
+lemma dir_inner {f g : E → F} {x : E} (hf : DifferentiableAt ℝ f x)
+    (hg : DifferentiableAt ℝ g x) (v : E) :
+    dir v (fun y => inner ℝ (f y) (g y)) x =
+      inner ℝ (f x) (dir v g x)+inner ℝ (dir v f x) (g x) :=
+  fderiv_inner_apply ℝ hf hg v
+
+lemma dir_norm_sq {f : E → F} {x : E} (hf : DifferentiableAt ℝ f x) (v : E) :
+    dir v (fun y => ‖f y‖^2) x = 2*inner ℝ (f x) (dir v f x) := by
+  simpa only [dir,smul_apply,ContinuousLinearMap.comp_apply,
+    innerSL_apply_apply,two_smul,two_mul,add_apply] using congrArg (fun map : E →L[ℝ] ℝ => map v) hf.hasFDerivAt.norm_sq.fderiv
+
+lemma dir_dir_norm_sq {U : Set E} (hU : IsOpen U) {f : E → F}
+    (hf : ContDiffOn ℝ ∞ f U) {x : E} (hx : x ∈ U) (v : E) :
+    dir v (dir v (fun y => ‖f y‖^2)) x =
+      2*‖dir v f x‖^2+2*inner ℝ (f x) (dir v (dir v f) x) := by
+  have he : dir v (fun y => ‖f y‖^2) =ᶠ[𝓝 x]
+      (fun y => 2*inner ℝ (f y) (dir v f y)) := by
+    filter_upwards [hU.mem_nhds hx] with y hy
+    exact dir_norm_sq ((hf.contDiffAt (hU.mem_nhds hy)).differentiableAt (by simp)) v
+  rw [dir_congr he]
+  have hfa := (hf.contDiffAt (hU.mem_nhds hx)).differentiableAt (by simp)
+  have hda := ((dir_smoothOn hU hf v).contDiffAt (hU.mem_nhds hx)).differentiableAt (by simp)
+  change dir v (fun y => (2:ℝ) • inner ℝ (f y) (dir v f y)) x = _
+  rw [dir_smul (hfa.inner ℝ hda),dir_inner hfa hda,real_inner_self_eq_norm_sq]
+  simp only [smul_eq_mul]
+  ring
+
+variable {ι κ : Type*} [Fintype ι] [Fintype κ]
+
+def frameLaplace (v : ι → E) (f : E → F) (x : E) : F := ∑ i, dir (v i) (dir (v i) f) x
+
+def frameGradientSq (v : ι → E) (f : E → F) (x : E) : ℝ := ∑ i, ‖dir (v i) f x‖^2
+
+lemma frameLaplace_norm_sq {U : Set E} (hU : IsOpen U) {f : E → F}
+    (hf : ContDiffOn ℝ ∞ f U) {x : E} (hx : x ∈ U) (v : ι → E) :
+    frameLaplace v (fun y => ‖f y‖^2) x =
+      2*frameGradientSq v f x+2*inner ℝ (f x) (frameLaplace v f x) := by
+  simp only [frameLaplace,dir_dir_norm_sq hU hf hx,Finset.sum_add_distrib,
+    ← Finset.mul_sum,inner_sum,frameGradientSq]
+
+lemma dir_sum (f : κ → E → F) {x : E} (hf : ∀ j, DifferentiableAt ℝ (f j) x) (v : E) :
+    dir v (fun y => ∑ j, f j y) x = ∑ j, dir v (f j) x := by
+  rw [show (fun y => ∑ j, f j y) = ∑ j, f j from by ext y; simp]
+  simpa only [dir,Finset.sum_apply,_root_.sum_apply] using congrArg (fun map : E →L[ℝ] F => map v)
+    ((HasFDerivAt.sum (u := Finset.univ) (fun j _ => (hf j).hasFDerivAt)).fderiv)
+
+lemma frameLaplace_sum {U : Set E} (hU : IsOpen U) (f : κ → E → F)
+    (hf : ∀ j, ContDiffOn ℝ ∞ (f j) U) {x : E} (hx : x ∈ U) (v : ι → E) :
+    frameLaplace v (fun y => ∑ j, f j y) x = ∑ j, frameLaplace v (f j) x := by
+  have hd (i : ι) : dir (v i) (fun y => ∑ j, f j y) =ᶠ[𝓝 x]
+      (fun y => ∑ j, dir (v i) (f j) y) := by
+    filter_upwards [hU.mem_nhds hx] with y hy
+    exact dir_sum f (fun j => ((hf j).contDiffAt (hU.mem_nhds hy)).differentiableAt (by simp)) _
+  simp only [frameLaplace,dir_congr (hd _),dir_sum _ (fun j =>
+    ((dir_smoothOn hU (hf j) _).contDiffAt (hU.mem_nhds hx)).differentiableAt (by simp))]
+  exact Finset.sum_comm
+
+def familyEnergy (f : κ → E → F) (x : E) : ℝ := ∑ j, ‖f j x‖^2
+
+def familyDissipation (v : ι → E) (f : κ → E → F) (x : E) : ℝ :=
+  ∑ j, frameGradientSq v (f j) x
+
+lemma familyEnergy_smoothOn {U : Set E} {f : κ → E → F}
+    (hf : ∀ j, ContDiffOn ℝ ∞ (f j) U) : ContDiffOn ℝ ∞ (familyEnergy f) U :=
+  ContDiffOn.sum (fun j _ => (hf j).norm_sq (𝕜 := ℝ))
+
+lemma familyEnergy_laplace {U : Set E} (hU : IsOpen U) {f : κ → E → F}
+    (hf : ∀ j, ContDiffOn ℝ ∞ (f j) U) {x : E} (hx : x ∈ U) (v : ι → E) :
+    frameLaplace v (familyEnergy f) x = 2*familyDissipation v f x+
+      2*∑ j, inner ℝ (f j x) (frameLaplace v (f j) x) := by
+  change frameLaplace v (fun y => ∑ j, ‖f j y‖^2) x = _
+  rw [frameLaplace_sum hU _ (fun j => (hf j).norm_sq (𝕜 := ℝ)) hx]
+  simp only [frameLaplace_norm_sq hU (hf _) hx,Finset.sum_add_distrib,
+    ← Finset.mul_sum,familyDissipation]
+
+lemma familyEnergy_gradient_bound {U : Set E} (hU : IsOpen U) {f : κ → E → F}
+    (hf : ∀ j, ContDiffOn ℝ ∞ (f j) U) {x : E} (hx : x ∈ U) (v : ι → E) :
+    frameGradientSq v (familyEnergy f) x ≤
+      4*familyEnergy f x*familyDissipation v f x := by
+  have he (i : ι) : dir (v i) (familyEnergy f) x =
+      2*∑ j, inner ℝ (f j x) (dir (v i) (f j) x) := by
+    change dir (v i) (fun y => ∑ j, ‖f j y‖^2) x = _
+    rw [dir_sum _ (fun j =>
+      (((hf j).norm_sq (𝕜 := ℝ)).contDiffAt (hU.mem_nhds hx)).differentiableAt (by simp))]
+    simp only [dir_norm_sq ((hf _).contDiffAt (hU.mem_nhds hx) |>.differentiableAt (by simp)),Finset.mul_sum]
+  have hb (i : ι) : (∑ j, inner ℝ (f j x) (dir (v i) (f j) x))^2 ≤
+      familyEnergy f x*(∑ j, ‖dir (v i) (f j) x‖^2) := by
+    let a : PiLp 2 (fun _ : κ => F) := WithLp.toLp 2 (fun j => f j x)
+    let b : PiLp 2 (fun _ : κ => F) := WithLp.toLp 2 (fun j => dir (v i) (f j) x)
+    have hh := (sq_le_sq₀ (abs_nonneg (inner ℝ a b))
+      (mul_nonneg (norm_nonneg a) (norm_nonneg b))).mpr (abs_real_inner_le_norm a b)
+    simpa only [sq_abs,mul_pow,PiLp.inner_apply,PiLp.norm_sq_eq_of_L2,
+      a,b,PiLp.toLp_apply,familyEnergy] using hh
+  calc
+    _ = ∑ i, 4*(∑ j, inner ℝ (f j x) (dir (v i) (f j) x))^2 := by
+      simp only [frameGradientSq,he,Real.norm_eq_abs,sq_abs,mul_pow]; norm_num
+    _ ≤ ∑ i, 4*familyEnergy f x*(∑ j, ‖dir (v i) (f j) x‖^2) := by
+      exact Finset.sum_le_sum (fun i _ => by nlinarith only [hb i])
+    _ = _ := by
+      rw [← Finset.mul_sum,Finset.sum_comm]
+      rfl
+end HigherJet
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/RealMapSmooth.lean b/lean/OAI/Geometry/PrescribedPotential/RealMapSmooth.lean
new file mode 100644
index 0000000000000000000000000000000000000000..77891ffafed2953b55f0deb1438091635c514ca7
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/RealMapSmooth.lean
@@ -0,0 +1,57 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CircleRadialCalculus
+import OAI.Geometry.PrescribedPotential.CompletedRealification
+import OAI.Geometry.PrescribedPotential.NonlinearRemainder
+import OAI.Geometry.PrescribedPotential.PatchCutoffs
+import OAI.Geometry.PrescribedPotential.RealNonlinearCommutator
+
+namespace OAI
+
+/-! Real Map Smooth. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology
+open scoped ContDiff Classical
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+local instance rmapNG (s : ℝ) : NormedAddCommGroup (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedAddCommGroup
+local instance rmapNS (s : ℝ) : NormedSpace ℝ (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedSpace
+local instance rmapTG (s : ℝ) : IsTopologicalAddGroup (D.localizers.RealSobolev s) :=
+  Submodule.isTopologicalAddGroup _
+local instance rmapCS (s : ℝ) : ContinuousSMul ℝ (D.localizers.RealSobolev s) :=
+  SMulMemClass.continuousSMul _
+
+variable [ConnectedSpace X]
+
+omit [ConnectedSpace X] in
+lemma realNonlinearRemainder_as_projection (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (p : ι) (v : EC d) :
+    D.realNonlinearRemainder k hk p v = fun u =>
+      D.realProjection k (D.nonlinearRemainder k hk p v u.val) := by
+  funext u
+  exact (D.realProjection_of_real k (D.realNonlinearRemainder k hk p v u)).symm
+
+omit [ConnectedSpace X] in
+lemma realNonlinearRemainder_contDiff (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (p : ι) (v : EC d) : ContDiff ℝ ∞ (D.realNonlinearRemainder k hk p v) := by
+  rw [D.realNonlinearRemainder_as_projection k hk p v]
+  exact (D.realProjection k).contDiff.comp
+    ((D.nonlinearRemainder_contDiff k hk p v).comp
+      (D.localizers.realCompletion ((k : ℝ)+2)).subtypeL.contDiff)
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/RealNonlinearCommutator.lean b/lean/OAI/Geometry/PrescribedPotential/RealNonlinearCommutator.lean
new file mode 100644
index 0000000000000000000000000000000000000000..a1ec024c2b419ef8513f766e13b1cd8e2a38562d
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/RealNonlinearCommutator.lean
@@ -0,0 +1,115 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.GlobalNonlinearCommutator
+import OAI.Geometry.PrescribedPotential.PatchCutoffs
+import OAI.Geometry.PrescribedPotential.RealNonlinearDerivative
+import OAI.Geometry.PrescribedPotential.RealSobolev
+import OAI.Geometry.PrescribedPotential.RegularityCutoffs
+
+namespace OAI
+
+/-! Real Nonlinear Commutator. -/
+
+section
+
+ 
+noncomputable section
+open Set Filter Topology
+open scoped ContDiff Classical
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+local instance realNonlinearCommutatorNormedAddCommGroup (s : ℝ) : NormedAddCommGroup (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedAddCommGroup
+local instance realNonlinearCommutatorNormedSpace (s : ℝ) : NormedSpace ℝ (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedSpace
+local instance realNonlinearCommutatorIsTopologicalAddGroup (s : ℝ) : IsTopologicalAddGroup (D.localizers.RealSobolev s) :=
+  Submodule.isTopologicalAddGroup _
+local instance realNonlinearCommutatorContinuousSMul (s : ℝ) : ContinuousSMul ℝ (D.localizers.RealSobolev s) :=
+  SMulMemClass.continuousSMul _
+
+lemma realVolumeDerivative_val (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (u w : D.localizers.RealSobolev ((k : ℝ)+2)) :
+    (D.realVolumeDerivative k hk u w).val =
+      fderiv ℝ (D.completedVolume k hk) u.val w.val := by
+  let J := (D.localizers.realCompletion ((k : ℝ)+2)).subtypeL
+  have hd := ((D.completedVolume_contDiff k hk).differentiable (by simp) u.val).hasFDerivAt.comp u
+    J.hasFDerivAt
+  change (fderiv ℝ (fun y : D.localizers.RealSobolev ((k : ℝ)+2) =>
+    D.completedVolume k hk y.val) u) w = _
+  exact congrArg (fun L : D.localizers.RealSobolev ((k : ℝ)+2) →L[ℝ]
+    D.localizers.Sobolev (k : ℝ) => L w) hd.fderiv
+
+lemma nonlinearRemainder_core_real (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (p : ι) (v : EC d) (f : RealSmooth A) :
+    D.nonlinearRemainder k hk p v (D.localizers.embed ((k : ℝ)+2) f.val) ∈
+      D.localizers.realCompletion (k : ℝ) := by
+  have hh := D.nonlinear_commutator_core k hk p v f.source
+  rw [f.ofReal_source] at hh
+  have he : D.nonlinearRemainder k hk p v (D.localizers.embed ((k : ℝ)+2) f.val) =
+      (D.realVolumeDerivative k hk (D.localizers.realEmbed ((k : ℝ)+2) f)
+        (D.localizers.realEmbed ((k : ℝ)+2) (D.realLocalizedDerivative p v f))).val -
+      D.localizers.embed (k : ℝ) (D.localizedDerivative p v (Smooth.ofReal (g.potentialDensity f.source))) := by
+    rw [D.realVolumeDerivative_val]
+    change D.nonlinearRemainder k hk p v (D.localizers.embed ((k : ℝ)+2) f.val) =
+      fderiv ℝ (D.completedVolume k hk) (D.localizers.embed ((k : ℝ)+2) f.val)
+        (D.localizers.embed ((k : ℝ)+2) (D.localizedDerivative p v f.val)) -
+      D.localizers.embed (k : ℝ) (D.localizedDerivative p v (Smooth.ofReal (g.potentialDensity f.source)))
+    rw [hh,add_sub_cancel_left]
+  rw [he]
+  apply (D.localizers.realCompletion (k : ℝ)).sub_mem
+  · exact Subtype.property _
+  · exact (D.localizers.realEmbed (k : ℝ)
+      (D.realLocalizedDerivative p v (RealSmooth.ofReal (g.potentialDensity f.source)))).property
+
+lemma nonlinearRemainder_real (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (p : ι) (v : EC d) (u : D.localizers.RealSobolev ((k : ℝ)+2)) :
+    D.nonlinearRemainder k hk p v u.val ∈ D.localizers.realCompletion (k : ℝ) := by
+  have hc : Continuous (fun u : D.localizers.RealSobolev ((k : ℝ)+2) =>
+      D.nonlinearRemainder k hk p v u.val) :=
+    (D.nonlinearRemainder_contDiff k hk p v).continuous.comp continuous_subtype_val
+  exact (D.localizers.realEmbed_dense ((k : ℝ)+2)).induction_on u
+    ((Submodule.isClosed_topologicalClosure _).preimage hc)
+    (D.nonlinearRemainder_core_real k hk p v)
+
+def realNonlinearRemainder (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (p : ι) (v : EC d) (u : D.localizers.RealSobolev ((k : ℝ)+2)) :
+    D.localizers.RealSobolev (k : ℝ) :=
+  ⟨D.nonlinearRemainder k hk p v u.val,D.nonlinearRemainder_real k hk p v u⟩
+
+lemma realNonlinearRemainder_continuous (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (p : ι) (v : EC d) : Continuous (D.realNonlinearRemainder k hk p v) :=
+  ((D.nonlinearRemainder_contDiff k hk p v).continuous.comp continuous_subtype_val).subtype_mk _
+
+lemma real_commutator_core (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (p : ι) (v : EC d) (f : RealSmooth A) :
+    D.realVolumeDerivative k hk (D.localizers.realEmbed ((k : ℝ)+2) f)
+      (D.localizers.realEmbed ((k : ℝ)+2) (D.realLocalizedDerivative p v f)) =
+    D.localizers.realEmbed (k : ℝ)
+      (D.realLocalizedDerivative p v (RealSmooth.ofReal (g.potentialDensity f.source))) +
+    D.realNonlinearRemainder k hk p v (D.localizers.realEmbed ((k : ℝ)+2) f) := by
+  apply Subtype.ext
+  rw [D.realVolumeDerivative_val]
+  have hh := D.nonlinear_commutator_core k hk p v f.source
+  rwa [f.ofReal_source] at hh
+
+def realCompletedDerivative (k : ℕ) (p : ι) (v : EC d) :
+    D.localizers.RealSobolev ((k : ℝ)+1) →L[ℝ] D.localizers.RealSobolev (k : ℝ) :=
+  ((D.completedDerivative k p v).comp (D.localizers.realCompletion ((k : ℝ)+1)).subtypeL).codRestrict
+    (D.localizers.realCompletion (k : ℝ)) (fun u => D.completedDerivative_real k p v u.val u.property)
+
+lemma realCompletedDerivative_embed (k : ℕ) (p : ι) (v : EC d) (f : RealSmooth A) :
+    D.realCompletedDerivative k p v (D.localizers.realEmbed ((k : ℝ)+1) f) =
+      D.localizers.realEmbed (k : ℝ) (D.realLocalizedDerivative p v f) := by
+  apply Subtype.ext
+  exact D.completedDerivative_embed k p v f.val
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/RealNonlinearDerivative.lean b/lean/OAI/Geometry/PrescribedPotential/RealNonlinearDerivative.lean
new file mode 100644
index 0000000000000000000000000000000000000000..63640b1f5498a0ba25cb5a21b29cb627eda37a1a
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/RealNonlinearDerivative.lean
@@ -0,0 +1,105 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.RealVolume
+
+namespace OAI
+
+/-! Real Nonlinear Derivative. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology
+open scoped ContDiff Classical
+namespace ClosedDerivative
+variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  [NormedAddCommGroup F] [NormedSpace ℝ F]
+
+lemma derivative_mem (S : Submodule ℝ F) (hS : IsClosed (S : Set F))
+    (f : E → S) {J : E →L[ℝ] F} {x : E}
+    (hJ : HasFDerivAt (fun y => (f y).val) J x) (v : E) : J v ∈ S := by
+  have hc : HasDerivAt (fun t : ℝ => x + t • v) v 0 := by
+    simpa using (hasDerivAt_id (0 : ℝ)).smul_const v |>.const_add x
+  have hJ' : HasFDerivAt (fun y => (f y).val) J (x+(0 : ℝ) • v) := by simpa using hJ
+  have hd := hJ'.comp_hasDerivAt (0 : ℝ) hc
+  apply hS.mem_of_tendsto hd.tendsto_slope
+  filter_upwards [] with t
+  exact S.smul_mem _ (S.sub_mem (f (x+t • v)).property (f (x+(0 : ℝ) • v)).property)
+
+def restrict (S : Submodule ℝ F) (hS : IsClosed (S : Set F))
+    (f : E → S) (hf : Differentiable ℝ (fun y => (f y).val)) (x : E) : E →L[ℝ] S :=
+  (fderiv ℝ (fun y => (f y).val) x).codRestrict S
+    (derivative_mem S hS f (hf x).hasFDerivAt)
+
+lemma restrict_comp (S : Submodule ℝ F) (hS : IsClosed (S : Set F))
+    (f : E → S) (hf : Differentiable ℝ (fun y => (f y).val)) (x : E) :
+    S.subtypeL ∘L restrict S hS f hf x = fderiv ℝ (fun y => (f y).val) x := by
+  ext v; rfl
+
+lemma hasStrictFDerivAt (S : Submodule ℝ F) (hS : IsClosed (S : Set F))
+    (f : E → S) (hf : ContDiff ℝ ∞ (fun y => (f y).val)) (x : E) :
+    HasStrictFDerivAt f (restrict S hS f (hf.differentiable (by simp)) x) x := by
+  apply (RealClosedDerivative.subtype_iff S f _ x).mpr
+  rw [restrict_comp]
+  exact hf.hasStrictFDerivAt (by simp)
+
+lemma restrict_continuous (S : Submodule ℝ F) (hS : IsClosed (S : Set F))
+    (f : E → S) (hf : ContDiff ℝ ∞ (fun y => (f y).val)) :
+    Continuous (restrict S hS f (hf.differentiable (by simp))) := by
+  let e : (E →L[ℝ] S) →ₗᵢ[ℝ] (E →L[ℝ] F) := S.subtypeₗᵢ.postcomp
+  have hIso : Isometry (e : (E →L[ℝ] S) → (E →L[ℝ] F)) := e.isometry
+  apply hIso.isEmbedding.continuous_iff.mpr
+  change Continuous (fun x => S.subtypeL ∘L restrict S hS f _ x)
+  simp_rw [restrict_comp]
+  exact (hf.fderiv_right (m := ∞) (by simp)).continuous
+end ClosedDerivative
+
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  [ConnectedSpace X] {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+local instance nonlinDerivNormedGroup (s : ℝ) : NormedAddCommGroup (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedAddCommGroup
+local instance nonlinDerivNormedSpace (s : ℝ) : NormedSpace ℝ (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedSpace
+local instance nonlinDerivTopologicalGroup (s : ℝ) : IsTopologicalAddGroup (D.localizers.RealSobolev s) :=
+  Submodule.isTopologicalAddGroup _
+local instance nonlinDerivContinuousSMul (s : ℝ) : ContinuousSMul ℝ (D.localizers.RealSobolev s) :=
+  SMulMemClass.continuousSMul _
+
+def realVolumeDerivative (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (u : D.localizers.RealSobolev ((k : ℝ)+2)) :
+    D.localizers.RealSobolev ((k : ℝ)+2) →L[ℝ] D.localizers.RealSobolev (k : ℝ) :=
+  ClosedDerivative.restrict (D.localizers.realCompletion (k : ℝ))
+    (Submodule.isClosed_topologicalClosure _) (D.realVolume k hk)
+    ((D.realVolume_ambient_contDiff k hk).differentiable (by simp)) u
+
+omit [ConnectedSpace X] in
+lemma realVolume_hasStrictFDerivAt (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (u : D.localizers.RealSobolev ((k : ℝ)+2)) :
+    HasStrictFDerivAt (D.realVolume k hk) (D.realVolumeDerivative k hk u) u :=
+  ClosedDerivative.hasStrictFDerivAt _ (Submodule.isClosed_topologicalClosure _)
+    _ (D.realVolume_ambient_contDiff k hk) _
+
+omit [ConnectedSpace X] in
+lemma realVolumeDerivative_continuous (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) :
+    Continuous (D.realVolumeDerivative k hk) :=
+  ClosedDerivative.restrict_continuous _ (Submodule.isClosed_topologicalClosure _)
+    _ (D.realVolume_ambient_contDiff k hk)
+
+omit [ConnectedSpace X] in
+lemma realVolumeDerivative_zero (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) :
+    D.realVolumeDerivative k hk 0 = D.realLOrder k :=
+  (D.realVolume_hasStrictFDerivAt k hk 0).hasFDerivAt.unique
+    (D.realVolume_hasStrictFDerivAt_zero k hk).hasFDerivAt
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/RealPoisson.lean b/lean/OAI/Geometry/PrescribedPotential/RealPoisson.lean
new file mode 100644
index 0000000000000000000000000000000000000000..aff58c95f98a5e0f363c6d7486c736985f3e7bd1
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/RealPoisson.lean
@@ -0,0 +1,126 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.RealSmoothOperator
+import OAI.Geometry.PrescribedPotential.PointPoisson
+
+namespace OAI
+
+/-! Real Poisson. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology
+open scoped ContDiff Classical
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  [ConnectedSpace X] {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+lemma pointPoissonOrder_realCore (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
+    (P : D.localizers.ConstantProjection) (k : ℕ)
+    (hk : (Module.finrank ℝ (EC d) : ℝ) < 2*((k : ℝ)+2)) (x₀ : X) (f : RealSmooth A) :
+    ∃ (u : RealSmooth A) (c : ℝ),
+      D.pointPoissonOrder m hm he P k x₀ (D.localizers.embed (k : ℝ) f.val) =
+        D.localizers.embed ((k : ℝ)+2) u.val ∧
+      D.poissonResidue m hm he P (D.localizers.embed 0 f.val) = (c : ℂ) := by
+  obtain ⟨u,c,hu,heq⟩ := exists_real_smooth_poisson g x₀ f
+  refine ⟨u,c,?_⟩
+  apply D.pointPoissonOrder_unique m hm he k hk x₀
+  · rw [D.pointPoissonOrder_equation, D.localizers.lower_embed (Nat.cast_nonneg k),
+      D.completedLOrder_embed, heq, map_add]
+    change (_ + D.localizers.constantsOrder (k : ℝ) _) - _ = (_ + D.localizers.constantsOrder (k : ℝ) _) - _
+    abel
+  · rw [D.pointPoissonOrder_normalized m hm he P k hk,
+      D.pointEvaluation_embed hk, hu]
+
+def realLOrder (k : ℕ) :
+    D.localizers.RealSobolev ((k : ℝ)+2) →L[ℝ] D.localizers.RealSobolev (k : ℝ) :=
+  ((D.completedLOrder k).comp (D.localizers.realCompletion ((k : ℝ)+2)).subtypeL).codRestrict
+    (D.localizers.realCompletion (k : ℝ)) (fun u =>
+      D.localizers.maps_realCompletion (D.completedLOrder k) (fun f =>
+        ⟨RealSmooth.laplace g f, D.completedLOrder_embed k f.val⟩) u.val u.property)
+
+def realConstants (s : ℝ) : ℝ →L[ℝ] D.localizers.RealSobolev s :=
+  ((D.localizers.constantsOrder s).comp Complex.ofRealCLM).codRestrict
+    (D.localizers.realCompletion s) (fun c => (D.localizers.realEmbed s (RealSmooth.const c)).property)
+
+def realEvaluation (s : ℝ) (x₀ : X) : D.localizers.RealSobolev s →L[ℝ] ℝ :=
+  Complex.reCLM ∘L D.pointEvaluation s x₀ ∘L (D.localizers.realCompletion s).subtypeL
+
+def realPointPoisson (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
+    (P : D.localizers.ConstantProjection) (k : ℕ)
+    (hk : (Module.finrank ℝ (EC d) : ℝ) < 2*((k : ℝ)+2)) (x₀ : X) :
+    D.localizers.RealSobolev (k : ℝ) →L[ℝ] D.localizers.RealSobolev ((k : ℝ)+2) :=
+  ((D.pointPoissonOrder m hm he P k x₀).comp (D.localizers.realCompletion (k : ℝ)).subtypeL).codRestrict
+    (D.localizers.realCompletion ((k : ℝ)+2)) (fun u =>
+      D.localizers.maps_realCompletion (D.pointPoissonOrder m hm he P k x₀) (fun f => by
+        obtain ⟨v,c,hv,_⟩ := D.pointPoissonOrder_realCore m hm he P k hk x₀ f
+        exact ⟨v,hv⟩) u.val u.property)
+
+lemma residue_real (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
+    (P : D.localizers.ConstantProjection) (k : ℕ)
+    (hk : (Module.finrank ℝ (EC d) : ℝ) < 2*((k : ℝ)+2)) (x₀ : X)
+    (u : D.localizers.RealSobolev (k : ℝ)) :
+    (D.poissonResidue m hm he P (D.localizers.lower (k : ℝ) 0 u.val)).im = 0 := by
+  have hh : (fun u : D.localizers.RealSobolev (k : ℝ) =>
+      (D.poissonResidue m hm he P (D.localizers.lower (k : ℝ) 0 u.val)).im) = (fun _ => 0) := by
+    apply (D.localizers.realEmbed_dense (k : ℝ)).equalizer (by fun_prop) continuous_const
+    funext f
+    dsimp only [Function.comp_apply]
+    rw [D.localizers.realEmbed_val, D.localizers.lower_embed (Nat.cast_nonneg k)]
+    obtain ⟨v,c,_,hc⟩ := D.pointPoissonOrder_realCore m hm he P k hk x₀ f
+    rw [hc, Complex.ofReal_im]
+  exact congr_fun hh u
+
+def realResidue (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
+    (P : D.localizers.ConstantProjection) (k : ℕ) : D.localizers.RealSobolev (k : ℝ) →L[ℝ] ℝ :=
+  Complex.reCLM ∘L D.poissonResidue m hm he P ∘L D.localizers.lower (k : ℝ) 0 ∘L
+    (D.localizers.realCompletion (k : ℝ)).subtypeL
+
+lemma realPointPoisson_equation (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
+    (P : D.localizers.ConstantProjection) (k : ℕ)
+    (hk : (Module.finrank ℝ (EC d) : ℝ) < 2*((k : ℝ)+2)) (x₀ : X)
+    (f : D.localizers.RealSobolev (k : ℝ)) :
+    D.realLOrder k (D.realPointPoisson m hm he P k hk x₀ f) =
+      f + D.realConstants (k : ℝ) (D.realResidue m hm he P k f) := by
+  apply Subtype.ext
+  change D.completedLOrder k (D.pointPoissonOrder m hm he P k x₀ f.val) =
+    f.val + D.localizers.constantsOrder (k : ℝ)
+      ((D.poissonResidue m hm he P (D.localizers.lower (k : ℝ) 0 f.val)).re : ℂ)
+  rw [D.pointPoissonOrder_equation]
+  congr 2
+  exact Complex.ext rfl (D.residue_real m hm he P k hk x₀ f)
+
+lemma realPointPoisson_normalized (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
+    (P : D.localizers.ConstantProjection) (k : ℕ)
+    (hk : (Module.finrank ℝ (EC d) : ℝ) < 2*((k : ℝ)+2)) (x₀ : X)
+    (f : D.localizers.RealSobolev (k : ℝ)) :
+    D.realEvaluation ((k : ℝ)+2) x₀ (D.realPointPoisson m hm he P k hk x₀ f) = 0 := by
+  change (D.pointEvaluation ((k : ℝ)+2) x₀
+    (D.pointPoissonOrder m hm he P k x₀ f.val)).re = 0
+  rw [D.pointPoissonOrder_normalized m hm he P k hk]
+  rfl
+
+omit [ConnectedSpace X] in
+lemma realLOrder_constants (k : ℕ) (c : ℝ) :
+    D.realLOrder k (D.realConstants ((k : ℝ)+2) c) = 0 :=
+  Subtype.ext (D.completedLOrder_constants k (c : ℂ))
+
+omit [ConnectedSpace X] in
+lemma realEvaluation_constants {s : ℝ} (hs : (Module.finrank ℝ (EC d) : ℝ) < 2*s)
+    (x₀ : X) (c : ℝ) : D.realEvaluation s x₀ (D.realConstants s c) = c := by
+  change (D.pointEvaluation s x₀ (D.localizers.embed s (Smooth.const (c : ℂ)))).re = c
+  rw [D.pointEvaluation_embed hs]
+  rfl
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/RealPoissonEquivalence.lean b/lean/OAI/Geometry/PrescribedPotential/RealPoissonEquivalence.lean
new file mode 100644
index 0000000000000000000000000000000000000000..6270ddab3c38929afec6a1452b34690ad81d2b33
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/RealPoissonEquivalence.lean
@@ -0,0 +1,89 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.BoundedPoisson
+import OAI.Geometry.PrescribedPotential.RealPoisson
+
+namespace OAI
+
+/-! Real Poisson Equivalence. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology
+open scoped ContDiff Classical
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  [ConnectedSpace X] {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+local instance realPoissonEquivalenceNormedAddCommGroup (s : ℝ) :
+    NormedAddCommGroup (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedAddCommGroup
+local instance realPoissonEquivalenceNormedSpace (s : ℝ) :
+    NormedSpace ℝ (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedSpace
+local instance realPoissonEquivalenceIsTopologicalAddGroup (s : ℝ) :
+    IsTopologicalAddGroup (D.localizers.RealSobolev s) :=
+  Submodule.isTopologicalAddGroup _
+local instance realPoissonEquivalenceContinuousSMul (s : ℝ) :
+    ContinuousSMul ℝ (D.localizers.RealSobolev s) :=
+  SMulMemClass.continuousSMul _
+
+def realAugmentedL (k : ℕ) (x₀ : X) :
+    (D.localizers.RealSobolev ((k : ℝ)+2) × ℝ) →L[ℝ]
+      (D.localizers.RealSobolev (k : ℝ) × ℝ) :=
+  ((D.realLOrder k ∘L ContinuousLinearMap.fst ℝ (D.localizers.RealSobolev ((k : ℝ)+2)) ℝ) -
+    (D.realConstants (k : ℝ) ∘L ContinuousLinearMap.snd ℝ (D.localizers.RealSobolev ((k : ℝ)+2)) ℝ)).prod
+    (D.realEvaluation ((k : ℝ)+2) x₀ ∘L ContinuousLinearMap.fst ℝ (D.localizers.RealSobolev ((k : ℝ)+2)) ℝ)
+
+lemma realAugmentedL_bijective (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
+    (P : D.localizers.ConstantProjection) (k : ℕ)
+    (hk : (Module.finrank ℝ (EC d) : ℝ) < 2*((k : ℝ)+2)) (x₀ : X) :
+    Function.Bijective (D.realAugmentedL k x₀) := by
+  constructor
+  · apply (LinearMap.ker_eq_bot).mp
+    apply LinearMap.ker_eq_bot'.mpr
+    rintro ⟨u,c⟩ h
+    have hl : D.realLOrder k u = D.realConstants (k : ℝ) c := sub_eq_zero.mp (congrArg Prod.fst h)
+    have hcplx : D.completedLOrder k u.val = D.localizers.constantsOrder (k : ℝ) (c : ℂ) :=
+      congrArg Subtype.val hl
+    obtain ⟨a, ha, hc⟩ := D.completedLOrder_constant_kernel m hm he k u.val (c : ℂ) hcplx
+    have hn : D.realEvaluation ((k : ℝ)+2) x₀ u = 0 := congrArg Prod.snd h
+    change (D.pointEvaluation ((k : ℝ)+2) x₀ u.val).re = 0 at hn
+    rw [ha, Localizers.constantsOrder_apply, D.pointEvaluation_embed hk] at hn
+    have hi := D.localizers.real_strong hk u x₀
+    rw [ha, Localizers.constantsOrder_apply, D.localizers.strong_embed _ hk] at hi
+    have ha0 : a = 0 := Complex.ext hn hi
+    have hu0 : u = 0 := by apply Subtype.ext; rw [ha, ha0, map_zero]; rfl
+    have hc0 : c = 0 := Complex.ofReal_eq_zero.mp hc
+    exact Prod.ext hu0 hc0
+  · rintro ⟨f,a⟩
+    refine ⟨(D.realPointPoisson m hm he P k hk x₀ f + D.realConstants ((k : ℝ)+2) a,
+      D.realResidue m hm he P k f), ?_⟩
+    apply Prod.ext
+    · change D.realLOrder k (_ + _) - _ = f
+      rw [map_add, D.realPointPoisson_equation, D.realLOrder_constants, add_zero]
+      exact add_sub_cancel_right _ _
+    · change D.realEvaluation ((k : ℝ)+2) x₀ (_ + _) = a
+      rw [map_add, D.realPointPoisson_normalized, D.realEvaluation_constants hk, zero_add]
+
+def realPoissonEquiv (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
+    (P : D.localizers.ConstantProjection) (k : ℕ)
+    (hk : (Module.finrank ℝ (EC d) : ℝ) < 2*((k : ℝ)+2)) (x₀ : X) :
+    (D.localizers.RealSobolev ((k : ℝ)+2) × ℝ) ≃L[ℝ]
+      (D.localizers.RealSobolev (k : ℝ) × ℝ) :=
+  ContinuousLinearEquiv.ofBijective (D.realAugmentedL k x₀)
+    (LinearMap.ker_eq_bot.mpr (D.realAugmentedL_bijective m hm he P k hk x₀).1)
+    (LinearMap.range_eq_top.mpr (D.realAugmentedL_bijective m hm he P k hk x₀).2)
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/RealSmoothCompactness.lean b/lean/OAI/Geometry/PrescribedPotential/RealSmoothCompactness.lean
new file mode 100644
index 0000000000000000000000000000000000000000..4f4b86ed5cb5f96bfcae25138760095538c6f518
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/RealSmoothCompactness.lean
@@ -0,0 +1,81 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.GlobalStrongEmbedding
+import OAI.Geometry.PrescribedPotential.PatchCutoffs
+import OAI.Geometry.PrescribedPotential.RealSobolev
+import OAI.Geometry.PrescribedPotential.SmoothSequenceCompactness
+
+namespace OAI
+
+/-! Real Smooth Compactness. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology
+open scoped ContDiff Classical
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace Localizers
+variable (D : Localizers A ι)
+
+lemma smooth_tendsto_real_order (f : ℕ → Smooth A) (fLim : Smooth A)
+    (hf : ∀ k : ℕ, Tendsto (fun n => D.embed (k : ℝ) (f n)) atTop
+      (𝓝 (D.embed (k : ℝ) fLim))) (s : ℝ) :
+    Tendsto (fun n => D.embed s (f n)) atTop (𝓝 (D.embed s fLim)) := by
+  obtain ⟨k,hk⟩ := exists_nat_ge s
+  have hh := D.lower_tendsto_smooth hk f (D.embed (k : ℝ) fLim) (hf k)
+  simpa only [D.lower_embed hk] using hh
+
+lemma smooth_value_tendsto (s : ℝ) (hs : (Module.finrank ℝ (EC d) : ℝ) < 2*s)
+    (f : ℕ → Smooth A) (fLim : Smooth A)
+    (hf : Tendsto (fun n => D.embed s (f n)) atTop (𝓝 (D.embed s fLim))) (x : X) :
+    Tendsto (fun n => f n x) atTop (𝓝 (fLim x)) := by
+  have hc : Continuous (fun u : D.Sobolev s => D.strong s u x) :=
+    (D.strong s).continuous.eval continuous_const
+  have ht := (hc.tendsto (D.embed s fLim)).comp hf
+  simpa only [Function.comp_def,D.strong_embed s hs,Smooth.bounded,
+    BoundedContinuousFunction.mkOfCompact_apply,ContinuousMap.coe_mk] using ht
+
+lemma smooth_limit_real (f : ℕ → RealSmooth A) (fLim : Smooth A)
+    (hf : ∀ s : ℝ, Tendsto (fun n => D.embed s (f n).val) atTop
+      (𝓝 (D.embed s fLim))) : ∀ x, (fLim x).im = 0 := by
+  intro x
+  let s : ℝ := (Module.finrank ℝ (EC d) : ℝ)+1
+  have hs : (Module.finrank ℝ (EC d) : ℝ) < 2*s := by
+    dsimp only [s]
+    linarith [Nat.cast_nonneg (α := ℝ) (Module.finrank ℝ (EC d))]
+  have hv := D.smooth_value_tendsto s hs (fun n => (f n).val) fLim (hf s) x
+  have hh := (Complex.continuous_im.tendsto (fLim x)).comp hv
+  have he : (fun n => ((f n).val x).im) = (fun _ : ℕ => (0 : ℝ)) :=
+    funext (fun n => (f n).property x)
+  change Tendsto (fun n => ((f n).val x).im) atTop (𝓝 (fLim x).im) at hh
+  rw [he] at hh
+  exact tendsto_nhds_unique hh tendsto_const_nhds
+
+end Localizers
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+ 
+
+theorem real_smooth_sequence_compact (f : ℕ → RealSmooth A)
+    (hbound : ∀ k : ℕ, ∃ C : ℝ, ∀ n : ℕ,
+      ‖D.localizers.embed (((2*k : ℕ) : ℝ)+2) (f n).val‖ ≤ C) :
+    ∃ (fLim : RealSmooth A) (r : ℕ → ℕ), StrictMono r ∧ ∀ s : ℝ,
+      Tendsto (fun n => D.localizers.embed s (f (r n)).val) atTop
+        (𝓝 (D.localizers.embed s fLim.val)) := by
+  obtain ⟨fLim,r,hr,hf⟩ := D.smooth_sequence_compact (fun n => (f n).val) hbound
+  have hf' := D.localizers.smooth_tendsto_real_order (fun n => (f (r n)).val) fLim hf
+  exact ⟨⟨fLim,D.localizers.smooth_limit_real (fun n => f (r n)) fLim hf'⟩,r,hr,hf'⟩
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/RealSmoothOperator.lean b/lean/OAI/Geometry/PrescribedPotential/RealSmoothOperator.lean
new file mode 100644
index 0000000000000000000000000000000000000000..8bcdb959220fc329757f324cebf15268e6621db7
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/RealSmoothOperator.lean
@@ -0,0 +1,82 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.NormalizedPoisson
+import OAI.Geometry.PrescribedPotential.RealSobolev
+
+namespace OAI
+
+/-! Real Smooth Operator. -/
+
+section
+
+ 
+
+noncomputable section
+namespace Anticanonical.SourceSmooth.SmoothRealFunction
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+lemma ext_value {f h : SmoothRealFunction A} (he : f.value = h.value) : f = h := by
+  cases f
+  cases h
+  cases he
+  rfl
+end Anticanonical.SourceSmooth.SmoothRealFunction
+namespace GlobalElliptic
+open Anticanonical SourceSmooth
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+
+lemma Smooth.ofReal_part_re (f : SmoothRealFunction A) :
+    (Smooth.ofReal f).part Complex.reCLM = f :=
+  SmoothRealFunction.ext_value rfl
+
+lemma Smooth.ofReal_part_im (f : SmoothRealFunction A) :
+    (Smooth.ofReal f).part Complex.imCLM = SmoothRealFunction.constant 0 :=
+  SmoothRealFunction.ext_value rfl
+
+lemma laplacian_constant (g : KaehlerMetric A) (c : ℝ) :
+    g.laplacian (SmoothRealFunction.constant c) = SmoothRealFunction.constant 0 := by
+  apply SmoothRealFunction.ext_value
+  funext x
+  obtain ⟨i, hi⟩ := A.covers x
+  change g.laplacianValue (SmoothRealFunction.constant c) x = 0
+  rw [g.laplacianValue_local _ i hi, g.linearizedMongeAmpere_constant]
+  rfl
+
+lemma complexL_ofReal (g : KaehlerMetric A) (f : SmoothRealFunction A) :
+    complexL g (Smooth.ofReal f) = Smooth.ofReal (g.laplacian f) := by
+  apply Smooth.ext
+  intro x
+  apply Complex.ext
+  · rw [complexL_re, Smooth.ofReal_part_re]
+    rfl
+  · rw [complexL_im, Smooth.ofReal_part_im, laplacian_constant]
+    rfl
+
+lemma exists_real_smooth_poisson [T2Space X] [CompactSpace X] [ConnectedSpace X]
+    (g : KaehlerMetric A) (x₀ : X) (f : RealSmooth A) :
+    ∃ (u : RealSmooth A) (c : ℝ), u.val x₀ = 0 ∧
+      complexL g u.val = f.val + Smooth.const (c : ℂ) := by
+  obtain ⟨φ, b, hφ, he⟩ := normalizedPoisson d X A g x₀ f.source
+  refine ⟨RealSmooth.ofReal φ, -b, ?_, ?_⟩
+  · change (φ.value x₀ : ℂ) = 0
+    rw [hφ, Complex.ofReal_zero]
+  · change complexL g (Smooth.ofReal φ) = _
+    rw [complexL_ofReal, ← RealSmooth.ofReal_source f]
+    apply Smooth.ext
+    intro x
+    obtain ⟨i, hi⟩ := A.covers x
+    have hh := he i (A.chart i x) ((A.chart i).mapsTo hi)
+    have hl : (g.laplacian φ).value x = g.linearizedMongeAmpere φ i (A.chart i x) :=
+      g.laplacianValue_local φ i hi
+    have hf : f.source.localExpression i (A.chart i x) = f.source.value x := by
+      change f.source.value ((A.chart i).symm (A.chart i x)) = _
+      rw [(A.chart i).left_inv hi]
+    rw [hf, ← hl] at hh
+    change ((g.laplacian φ).value x : ℂ) = (f.source.value x : ℂ) + ((-b : ℝ) : ℂ)
+    rw [← Complex.ofReal_add]
+    congr 1
+    linarith
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/RealSobolev.lean b/lean/OAI/Geometry/PrescribedPotential/RealSobolev.lean
new file mode 100644
index 0000000000000000000000000000000000000000..6bc6e2ac8e93bb30ae3d84e46d8b845d10f613a0
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/RealSobolev.lean
@@ -0,0 +1,131 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CircleRadialCalculus
+import OAI.Geometry.PrescribedPotential.ComplexKernel
+import OAI.Geometry.PrescribedPotential.GlobalStrongEmbedding
+import OAI.Geometry.PrescribedPotential.MatrixHilbertCoordinates
+
+namespace OAI
+
+/-! Real Sobolev. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology
+open scoped ContDiff Classical
+namespace ClosedCore
+variable {V W H G : Type*} [TopologicalSpace H] [TopologicalSpace G]
+lemma maps_closure_range (e : V → H) (q : W → G) (L : H → G) (hL : Continuous L)
+    (he : ∀ f, ∃ h, L (e f) = q h) :
+    ∀ u ∈ closure (range e), L u ∈ closure (range q) := by
+  apply closure_minimal _ (isClosed_closure.preimage hL)
+  rintro _ ⟨f, rfl⟩
+  obtain ⟨h, hh⟩ := he f
+  exact subset_closure ⟨h, hh.symm⟩
+end ClosedCore
+
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+
+def realSmoothSpace (A : ComplexAtlas d X) : Submodule ℝ (Smooth A) where
+  carrier := {f | ∀ x, (f x).im = 0}
+  zero_mem' := by intro x; rfl
+  add_mem' := by intro f h hf hh x; simp [hf x, hh x]
+  smul_mem' := by intro c f hf x; simp [hf x]
+
+abbrev RealSmooth (A : ComplexAtlas d X) := ↥(realSmoothSpace A)
+
+namespace RealSmooth
+
+def ofReal (f : SmoothRealFunction A) : RealSmooth A :=
+  ⟨Smooth.ofReal f, fun _ => Complex.ofReal_im _⟩
+
+def source (f : RealSmooth A) : SmoothRealFunction A := f.val.part Complex.reCLM
+
+lemma ofReal_source (f : RealSmooth A) : Smooth.ofReal f.source = f.val := by
+  apply Smooth.ext
+  intro x
+  exact Complex.ext rfl (f.property x).symm
+
+def const (r : ℝ) : RealSmooth A := ⟨Smooth.const r, fun _ => Complex.ofReal_im _⟩
+
+lemma complexL_real (g : KaehlerMetric A) (f : RealSmooth A) :
+    complexL g f.val ∈ realSmoothSpace A := by
+  intro x
+  rw [complexL_im]
+  have hi : f.val.part Complex.imCLM = SmoothRealFunction.constant 0 := by
+    cases h : f.val.part Complex.imCLM with
+    | mk v hv =>
+      have he : v = fun _ => 0 := by
+        funext x
+        have hh := congrArg (fun u : SmoothRealFunction A => u.value x) h
+        exact hh.symm.trans (f.property x)
+      subst v
+      rfl
+  rw [hi]
+  obtain ⟨i, hi⟩ := A.covers x
+  change g.laplacianValue (SmoothRealFunction.constant 0) x = 0
+  rw [g.laplacianValue_local _ i hi, g.linearizedMongeAmpere_constant]
+  rfl
+
+def laplace (g : KaehlerMetric A) : RealSmooth A →ₗ[ℝ] RealSmooth A :=
+  ((complexL g).comp (realSmoothSpace A).subtype).codRestrict (realSmoothSpace A)
+    (fun f => complexL_real g f)
+
+end RealSmooth
+
+variable [T2Space X] [CompactSpace X] {ι : Type*} [Fintype ι]
+namespace Localizers
+variable (D : Localizers A ι)
+
+def realCoreMap (s : ℝ) : RealSmooth A →ₗ[ℝ] D.Sobolev s :=
+  (D.embed s).comp (realSmoothSpace A).subtype
+
+def realCompletion (s : ℝ) : Submodule ℝ (D.Sobolev s) :=
+  (D.realCoreMap s).range.topologicalClosure
+
+abbrev RealSobolev (s : ℝ) := ↥(D.realCompletion s)
+
+instance realSobolev_complete (s : ℝ) : CompleteSpace (D.RealSobolev s) :=
+  inferInstanceAs (CompleteSpace (D.realCoreMap s).range.topologicalClosure)
+
+def realEmbed (s : ℝ) : RealSmooth A →ₗ[ℝ] D.RealSobolev s :=
+  (D.realCoreMap s).codRestrict (D.realCompletion s)
+    (fun f => (D.realCoreMap s).range.le_topologicalClosure (LinearMap.mem_range_self _ f))
+
+@[simp] lemma realEmbed_val (s : ℝ) (f : RealSmooth A) :
+    (D.realEmbed s f).val = D.embed s f.val := rfl
+
+lemma realEmbed_dense (s : ℝ) : DenseRange (D.realEmbed s) := by
+  rw [DenseRange, Subtype.dense_iff]
+  change closure ((D.realCoreMap s).range : Set (D.Sobolev s)) ⊆ _
+  apply closure_mono
+  rintro _ ⟨f, rfl⟩
+  exact ⟨D.realEmbed s f, ⟨f, rfl⟩, rfl⟩
+
+lemma maps_realCompletion {s t : ℝ} (L : D.Sobolev s →L[ℝ] D.Sobolev t)
+    (hL : ∀ f : RealSmooth A, ∃ h : RealSmooth A, L (D.embed s f.val) = D.embed t h.val) :
+    ∀ u ∈ D.realCompletion s, L u ∈ D.realCompletion t := by
+  exact ClosedCore.maps_closure_range (D.realCoreMap s) (D.realCoreMap t) L L.continuous hL
+
+lemma real_strong {s : ℝ} (hs : (Module.finrank ℝ (EC d) : ℝ) < 2*s)
+    (u : D.RealSobolev s) (x : X) : (D.strong s u.val x).im = 0 := by
+  have hc : IsClosed {v : D.Sobolev s | (D.strong s v x).im = 0} :=
+    isClosed_eq (Complex.continuous_im.comp ((BoundedContinuousFunction.lipschitz_eval_const x).continuous.comp
+      (D.strong s).continuous)) continuous_const
+  apply closure_minimal (s := ((D.realCoreMap s).range : Set (D.Sobolev s))) _ hc u.property
+  rintro _ ⟨f, rfl⟩
+  change (D.strong s (D.embed s f.val) x).im = 0
+  rw [D.strong_embed s hs]
+  exact f.property x
+
+end Localizers
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/RealVolume.lean b/lean/OAI/Geometry/PrescribedPotential/RealVolume.lean
new file mode 100644
index 0000000000000000000000000000000000000000..0994933cb2ebdafa0a1104c9a94fafda4a3c26d4
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/RealVolume.lean
@@ -0,0 +1,141 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.RealSmoothOperator
+import OAI.Geometry.PrescribedPotential.CompletedVolume
+import OAI.Geometry.PrescribedPotential.RealPoisson
+
+namespace OAI
+
+/-! Real Volume. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology
+open scoped ContDiff Classical BoundedContinuousFunction ComplexOrder
+namespace RealClosedDerivative
+variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  [NormedAddCommGroup F] [NormedSpace ℝ F]
+
+lemma subtype_iff (S : Submodule ℝ F) (f : E → S) (L : E →L[ℝ] S) (x : E) :
+    HasStrictFDerivAt f L x ↔
+      HasStrictFDerivAt (fun y => (f y).val) (S.subtypeL ∘L L) x := by
+  simp only [hasStrictFDerivAt_iff_isLittleO]
+  conv_lhs => rw [← Asymptotics.isLittleO_norm_left]
+  conv_rhs => rw [← Asymptotics.isLittleO_norm_left]
+  rfl
+end RealClosedDerivative
+
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+local instance (s : ℝ) : NormedAddCommGroup (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedAddCommGroup
+local instance (s : ℝ) : NormedSpace ℝ (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedSpace
+local instance (s : ℝ) : IsTopologicalAddGroup (D.localizers.RealSobolev s) :=
+  Submodule.isTopologicalAddGroup _
+local instance (s : ℝ) : ContinuousSMul ℝ (D.localizers.RealSobolev s) :=
+  SMulMemClass.continuousSMul _
+
+def realVolume (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (u : D.localizers.RealSobolev ((k : ℝ)+2)) : D.localizers.RealSobolev (k : ℝ) :=
+  ⟨D.completedVolume k hk u.val, D.completedVolume_real k hk u.val u.property⟩
+
+lemma realVolume_continuous (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) :
+    Continuous (D.realVolume k hk) :=
+  ((D.completedVolume_contDiff k hk).continuous.comp continuous_subtype_val).subtype_mk _
+
+lemma realVolume_ambient_contDiff (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) :
+    ContDiff ℝ ∞ (fun u : D.localizers.RealSobolev ((k : ℝ)+2) => (D.realVolume k hk u).val) :=
+  (D.completedVolume_contDiff k hk).comp (D.localizers.realCompletion ((k : ℝ)+2)).subtypeL.contDiff
+
+lemma completedVolume_real_directional (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (f : RealSmooth A) (x : X) :
+    HasDerivAt (fun t : ℝ => D.localizers.strong (k : ℝ)
+      (D.completedVolume k hk (t • D.localizers.embed ((k : ℝ)+2) f.val)) x)
+      (complexL g f.val x) 0 := by
+  have hs : (Module.finrank ℝ (EC d) : ℝ) < 2*(k : ℝ) := by
+    have hh : (Module.finrank ℝ (EC d) : ℝ) < (k : ℝ) := by exact_mod_cast hk
+    linarith [Nat.cast_nonneg (α := ℝ) k]
+  have he (t : ℝ) : t • D.localizers.embed ((k : ℝ)+2) f.val =
+      D.localizers.embed ((k : ℝ)+2) (Smooth.ofReal (f.source.realSMul t)) := by
+    rw [← map_smul, ← f.ofReal_source]
+    congr 1
+    ext y
+    simp [SmoothRealFunction.realSMul, Complex.real_smul]
+  have hd := Complex.ofRealCLM.hasFDerivAt.comp_hasDerivAt 0
+    (g.potentialDensity_directional f.source x)
+  have hl : complexL g f.val x = ((g.laplacian f.source).value x : ℂ) := by
+    rw [← f.ofReal_source, complexL_ofReal]
+    rfl
+  rw [hl]
+  apply hd.congr_of_eventuallyEq
+  filter_upwards [] with t
+  rw [he, D.completedVolume_embed, D.localizers.strong_embed _ hs]
+  rfl
+
+lemma realVolume_hasStrictFDerivAt_zero (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) :
+    HasStrictFDerivAt (D.realVolume k hk) (D.realLOrder k) 0 := by
+  let F := fun u : D.localizers.RealSobolev ((k : ℝ)+2) => (D.realVolume k hk u).val
+  let J := fderiv ℝ F 0
+  have hJ : HasStrictFDerivAt F J 0 := (D.realVolume_ambient_contDiff k hk).hasStrictFDerivAt (by simp)
+  have hs : (Module.finrank ℝ (EC d) : ℝ) < 2*(k : ℝ) := by
+    have hh : (Module.finrank ℝ (EC d) : ℝ) < (k : ℝ) := by exact_mod_cast hk
+    linarith [Nat.cast_nonneg (α := ℝ) k]
+  have hj : J = (D.localizers.realCompletion (k : ℝ)).subtypeL ∘L D.realLOrder k := by
+    apply DFunLike.coe_injective
+    apply (D.localizers.realEmbed_dense ((k : ℝ)+2)).equalizer J.continuous (by fun_prop)
+    funext f
+    apply D.localizers.strong_injective (k : ℝ) hs
+    ext x
+    let v := D.localizers.realEmbed ((k : ℝ)+2) f
+    have hc : HasDerivAt (fun t : ℝ => t • v) v 0 := by
+      simpa using (hasDerivAt_id (0 : ℝ)).smul_const v
+    have hF : HasFDerivAt F J ((0 : ℝ) • v) := by simpa using hJ.hasFDerivAt
+    have hv := hF.comp_hasDerivAt 0 hc
+    have he := (D.pointEvaluation (k : ℝ) x).hasFDerivAt.comp_hasDerivAt 0 hv
+    have hdr := D.completedVolume_real_directional k hk f x
+    have heq := he.unique hdr
+    change D.localizers.strong (k : ℝ) (J v) x = complexL g f.val x at heq
+    change D.localizers.strong (k : ℝ) (J v) x = _
+    rw [heq]
+    change complexL g f.val x = D.localizers.strong (k : ℝ)
+      (D.completedLOrder k (D.localizers.embed ((k : ℝ)+2) f.val)) x
+    rw [D.completedLOrder_embed, D.localizers.strong_embed _ hs]
+    rfl
+  apply (RealClosedDerivative.subtype_iff (D.localizers.realCompletion (k : ℝ)) _ _ _).mpr
+  rw [← hj]
+  exact hJ
+
+lemma realVolume_zero (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) :
+    D.realVolume k hk 0 = D.realConstants (k : ℝ) 1 := by
+  have h₀ : Smooth.ofReal (SmoothRealFunction.constant (A := A) 0) = 0 := by
+    ext x
+    change (↑(0 : ℝ) : ℂ) = 0
+    exact Complex.ofReal_zero
+  have hv := D.completedVolume_embed k hk (SmoothRealFunction.constant (A := A) 0)
+  rw [h₀, map_zero] at hv
+  apply Subtype.ext
+  change D.completedVolume k hk 0 = D.localizers.embed (k : ℝ) (Smooth.const (↑(1 : ℝ)))
+  rw [hv]
+  congr 1
+  ext x
+  obtain ⟨i, hi⟩ := A.covers x
+  change ((g.potentialDensity (SmoothRealFunction.constant 0)).value x : ℂ) = 1
+  rw [g.potentialDensity_complex _ i hi]
+  simp only [KaehlerMetric.volumePolynomial, SmoothRealFunction.hessian_constant, add_zero]
+  exact div_self (ne_of_gt (g.positive i _ ((A.chart i).mapsTo hi)).det_pos)
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/RegularityCoordinates.lean b/lean/OAI/Geometry/PrescribedPotential/RegularityCoordinates.lean
new file mode 100644
index 0000000000000000000000000000000000000000..339d84fdb7576717d86e734be6ab1840b6884138
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/RegularityCoordinates.lean
@@ -0,0 +1,119 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.GlobalMetricEntries
+import OAI.Geometry.PrescribedPotential.GlobalParametrixEquation
+import OAI.Geometry.PrescribedPotential.LocalizedDerivations
+import OAI.Geometry.PrescribedPotential.RegularityJets
+
+namespace OAI
+
+/-! Regularity Coordinates. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology LineDeriv
+open scoped ContDiff SchwartzMap Classical
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+
+lemma localize_cutoff_nhds (q : Fin A.count) (η : ChartCutoff (A.euclideanChart q).target)
+    (f : Smooth A) {y : EC d} (hy : y ∈ (A.euclideanChart q).target)
+    (hη : (η : EC d → ℂ) =ᶠ[𝓝 y] (fun _ => 1)) :
+    (localize A q (cutoffGlobal q η) (cutoffGlobal_support q η) f : EC d → ℂ)
+      =ᶠ[𝓝 y] f ∘ (A.euclideanChart q).symm := by
+  filter_upwards [(A.euclideanChart q).open_target.mem_nhds hy,hη] with z hz he
+  rw [localize_apply,localizeFun_apply A _ _ hz]
+  have hs : (A.euclideanChart q).symm z ∈ (A.euclideanChart q).source :=
+    (A.euclideanChart q).symm.mapsTo hz
+  change (if (A.euclideanChart q).symm z ∈ (A.euclideanChart q).source then
+    η (A.euclideanChart q ((A.euclideanChart q).symm z)) else 0) * _ = _
+  rw [ite_eq_left hs,(A.euclideanChart q).right_inv hz,he,one_mul]
+  rfl
+
+lemma hessianEntrySchwartz_congr {f h : 𝓢(EC d, ℂ)} {y : EC d}
+    (he : (f : EC d → ℂ) =ᶠ[𝓝 y] (h : EC d → ℂ)) (i j : Fin d) :
+    hessianEntrySchwartz i j f y = hessianEntrySchwartz i j h y := by
+  simp only [hessianEntrySchwartz_apply,(he.fderiv (𝕜 := ℝ)).fderiv_eq]
+
+omit [CompactSpace X] in
+lemma chartFunction_local (q : Fin A.count) (κ : ChartCutoff (A.euclideanChart q).target)
+    (f : EC d → ℂ) (hf : ContDiffOn ℝ ∞ f (A.euclideanChart q).target)
+    {y : EC d} (hy : y ∈ (A.euclideanChart q).target) :
+    (chartFunction q κ f hf ∘ (A.euclideanChart q).symm) =ᶠ[𝓝 y] (fun z => κ z * f z) := by
+  filter_upwards [(A.euclideanChart q).open_target.mem_nhds hy] with z hz
+  rw [Function.comp_apply,chartFunction_apply _ _ _ _ ((A.euclideanChart q).symm.mapsTo hz),
+    (A.euclideanChart q).right_inv hz]
+
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+omit [T2Space X] [CompactSpace X] in
+lemma cutoff_support_regularity (p : ι) :
+    tsupport (D.cutoff p : EC d → ℂ) ⊆ tsupport (D.regularityCutoff p : EC d → ℂ) := by
+  intro y hy
+  apply subset_tsupport
+  change D.regularityCutoff p y ≠ 0
+  rw [(D.regularityCutoff_one p hy).eq_of_nhds]
+  exact one_ne_zero
+
+omit [T2Space X] [CompactSpace X] in
+lemma regularityOuter_inner (p : ι) {y : EC d}
+    (hy : y ∈ tsupport (D.cutoff p : EC d → ℂ)) :
+    (D.regularityOuter p : EC d → ℂ) =ᶠ[𝓝 y] (fun _ => 1) :=
+  D.regularityOuter_one p (D.cutoff_support_regularity p hy)
+
+lemma localizedDerivative_regularity (p : ι) (v : EC d) (f : Smooth A) :
+    D.localizedDerivative p v f = globalize (D.patch p).index (D.cutoff p).realPart
+      (∂_{v} (localize A (D.patch p).index (cutoffGlobal (D.patch p).index (D.regularityOuter p))
+        (cutoffGlobal_support _ _) f)) := by
+  apply Smooth.ext
+  intro x
+  rw [D.localizedDerivative_apply,globalize_apply]
+  split_ifs with hx
+  · by_cases hκ : (D.cutoff p).realPart (A.euclideanChart (D.patch p).index x) = 0
+    · rw [hκ,zero_mul,zero_mul]
+    · have ht := (A.euclideanChart (D.patch p).index).mapsTo hx
+      have hs := (D.cutoff p).realPart_support (subset_tsupport _ hκ)
+      have he := localize_cutoff_nhds (D.patch p).index (D.regularityOuter p) f ht
+        (D.regularityOuter_inner p hs)
+      rw [SchwartzMap.lineDerivOp_apply_eq_fderiv,he.fderiv_eq]
+  · rfl
+
+lemma localizedDerivative_local_regularity (p : ι) (v : EC d) (f : Smooth A)
+    {y : EC d} (hy : y ∈ (A.euclideanChart (D.patch p).index).target) :
+    (D.localizedDerivative p v f ∘ (A.euclideanChart (D.patch p).index).symm) =ᶠ[𝓝 y]
+      (SchwartzMap.smulLeftCLM ℂ (D.cutoff p).realPart.val
+        (∂_{v} (localize A (D.patch p).index (cutoffGlobal (D.patch p).index (D.regularityOuter p))
+          (cutoffGlobal_support _ _) f)) : EC d → ℂ) := by
+  rw [D.localizedDerivative_regularity]
+  exact globalize_local _ _ _ hy
+
+lemma regularityHessian_localizedDerivative (p : ι) (v : EC d) (f : Smooth A)
+    (i j : Fin d) {x : X}
+    (hx : x ∈ (A.euclideanChart (D.patch p).index).source)
+    (hκ : A.euclideanChart (D.patch p).index x ∈ tsupport (D.cutoff p : EC d → ℂ)) :
+    D.regularityHessian p i j (D.localizedDerivative p v f) x =
+      hessianEntrySchwartz i j (SchwartzMap.smulLeftCLM ℂ (D.cutoff p).realPart.val
+        (∂_{v} (localize A (D.patch p).index (cutoffGlobal (D.patch p).index (D.regularityOuter p))
+          (cutoffGlobal_support _ _) f))) (A.euclideanChart (D.patch p).index x) := by
+  change globalize (D.patch p).index (D.regularityCutoff p)
+    (hessianEntrySchwartz i j (localize A (D.patch p).index
+      (cutoffGlobal (D.patch p).index (D.regularityOuter p)) (cutoffGlobal_support _ _)
+      (D.localizedDerivative p v f))) x = _
+  rw [globalize_apply,ite_eq_left hx,(D.regularityCutoff_one p hκ).eq_of_nhds,one_mul]
+  apply hessianEntrySchwartz_congr
+  exact (localize_cutoff_nhds _ _ _ ((A.euclideanChart (D.patch p).index).mapsTo hx)
+    (D.regularityOuter_inner p hκ)).trans
+    (D.localizedDerivative_local_regularity p v f ((A.euclideanChart (D.patch p).index).mapsTo hx))
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/RegularityCutoffs.lean b/lean/OAI/Geometry/PrescribedPotential/RegularityCutoffs.lean
new file mode 100644
index 0000000000000000000000000000000000000000..8569099bc000ddbb5849aded88e023b92d36bc90
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/RegularityCutoffs.lean
@@ -0,0 +1,124 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.LocalizedDerivations
+import OAI.Geometry.PrescribedPotential.RealSobolev
+
+namespace OAI
+
+/-! Regularity Cutoffs. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory Manifold IsManifold
+open scoped ContDiff SchwartzMap Classical
+namespace GlobalElliptic
+open Anticanonical EllipticKernel SobolevChart SourceSmooth
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E]
+
+lemma exists_chartCutoff_nhds {K U : Set E} (hK : IsCompact K) (hU : IsOpen U) (hKU : K ⊆ U) :
+    ∃ κ : ChartCutoff U, (∀ x, (κ x).im = 0) ∧
+      ∀ x ∈ K, (κ : E → ℂ) =ᶠ[𝓝 x] (fun _ => 1) := by
+  obtain ⟨L, hLc, hLl, hKL, hLU⟩ := exists_compact_closed_between hK hU hKU
+  obtain ⟨f, hf, hf0, _⟩ := exists_contMDiffMap_one_nhds_of_subset_interior
+    (𝓘(ℝ, E)) hK.isClosed hKL (n := ⊤)
+  have hsm : ContDiff ℝ ∞ (fun x => (f x : ℂ)) :=
+    Complex.ofRealCLM.contDiff.comp (contMDiff_iff_contDiff.mp f.contMDiff)
+  have hsub : tsupport (fun x => (f x : ℂ)) ⊆ L := by
+    apply closure_minimal _ hLl
+    intro x hx
+    by_contra hn
+    exact hx (by simp [hf0 x hn])
+  have hc : HasCompactSupport (fun x => (f x : ℂ)) :=
+    hLc.of_isClosed_subset (isClosed_tsupport _) hsub
+  refine ⟨⟨hc.toSchwartzMap hsm, hc, hsub.trans hLU⟩, fun _ => Complex.ofReal_im _, ?_⟩
+  intro x hx
+  filter_upwards [hf.filter_mono (nhds_le_nhdsSet hx)] with y hy
+  change (f y : ℂ) = 1
+  rw [hy]
+  rfl
+
+variable {d : ℕ} {X : Type*} [TopologicalSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+lemma exists_regularityCutoff (p : ι) :
+    ∃ η : ChartCutoff (A.euclideanChart (D.patch p).index).target,
+      (∀ x, (η x).im = 0) ∧
+      ∀ x ∈ tsupport (D.cutoff p : EC d → ℂ),
+        (η : EC d → ℂ) =ᶠ[𝓝 x] (fun _ => 1) :=
+  exists_chartCutoff_nhds (D.cutoff p).compact
+    (A.euclideanChart (D.patch p).index).open_target (D.cutoff p).support_sub
+
+def regularityCutoff (p : ι) : ChartCutoff (A.euclideanChart (D.patch p).index).target :=
+  (D.exists_regularityCutoff p).choose
+
+lemma regularityCutoff_real (p : ι) (x : EC d) : (D.regularityCutoff p x).im = 0 :=
+  (D.exists_regularityCutoff p).choose_spec.1 x
+
+lemma regularityCutoff_one (p : ι) {x : EC d}
+    (hx : x ∈ tsupport (D.cutoff p : EC d → ℂ)) :
+    (D.regularityCutoff p : EC d → ℂ) =ᶠ[𝓝 x] (fun _ => 1) :=
+  (D.exists_regularityCutoff p).choose_spec.2 x hx
+
+lemma exists_regularityOuter (p : ι) :
+    ∃ η : ChartCutoff (A.euclideanChart (D.patch p).index).target,
+      (∀ x, (η x).im = 0) ∧
+      ∀ x ∈ tsupport (D.regularityCutoff p : EC d → ℂ),
+        (η : EC d → ℂ) =ᶠ[𝓝 x] (fun _ => 1) :=
+  exists_chartCutoff_nhds (D.regularityCutoff p).compact
+    (A.euclideanChart (D.patch p).index).open_target (D.regularityCutoff p).support_sub
+
+def regularityOuter (p : ι) : ChartCutoff (A.euclideanChart (D.patch p).index).target :=
+  (D.exists_regularityOuter p).choose
+
+lemma regularityOuter_one (p : ι) {x : EC d}
+    (hx : x ∈ tsupport (D.regularityCutoff p : EC d → ℂ)) :
+    (D.regularityOuter p : EC d → ℂ) =ᶠ[𝓝 x] (fun _ => 1) :=
+  (D.exists_regularityOuter p).choose_spec.2 x hx
+
+variable [T2Space X] [CompactSpace X]
+
+lemma localizedDerivative_real (p : ι) (v : EC d) (f : RealSmooth A) :
+    D.localizedDerivative p v f.val ∈ realSmoothSpace A := by
+  intro x
+  rw [D.localizedDerivative_apply]
+  split_ifs with hx
+  · have ht := (A.euclideanChart (D.patch p).index).mapsTo hx
+    have hd : DifferentiableAt ℝ (f.val ∘ (A.euclideanChart (D.patch p).index).symm)
+        (A.euclideanChart (D.patch p).index x) :=
+      ((f.val.smooth (D.patch p).index).contDiffAt
+        ((A.euclideanChart (D.patch p).index).open_target.mem_nhds ht)).differentiableAt (by simp)
+    have he : (fun y : EC d => (f.val ((A.euclideanChart (D.patch p).index).symm y)).im) =
+        (fun _ => 0) := funext (fun y => f.property _)
+    have hh := congrArg (fun L : EC d →L[ℝ] ℝ => L v)
+      ((Complex.imCLM.hasFDerivAt.comp _ hd.hasFDerivAt).fderiv)
+    change fderiv ℝ (fun y => (f.val ((A.euclideanChart (D.patch p).index).symm y)).im)
+      (A.euclideanChart (D.patch p).index x) v = _ at hh
+    rw [he, fderiv_const_apply, zero_apply] at hh
+    simp only [ChartCutoff.realPart_apply, Complex.mul_im, Complex.ofReal_re, Complex.ofReal_im,
+      zero_mul, add_zero]
+    exact mul_eq_zero_of_right _ hh.symm
+  · rfl
+
+def realLocalizedDerivative (p : ι) (v : EC d) : RealSmooth A →ₗ[ℝ] RealSmooth A :=
+  ((D.localizedDerivative p v).comp (realSmoothSpace A).subtype).codRestrict
+    (realSmoothSpace A) (D.localizedDerivative_real p v)
+
+lemma completedDerivative_real (k : ℕ) (p : ι) (v : EC d) :
+    ∀ u ∈ D.localizers.realCompletion ((k : ℝ)+1),
+      D.completedDerivative k p v u ∈ D.localizers.realCompletion (k : ℝ) := by
+  apply D.localizers.maps_realCompletion
+  intro f
+  exact ⟨D.realLocalizedDerivative p v f,D.completedDerivative_embed k p v f.val⟩
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/RegularityJets.lean b/lean/OAI/Geometry/PrescribedPotential/RegularityJets.lean
new file mode 100644
index 0000000000000000000000000000000000000000..7ad69461680ef54bcb8a0d12ea9929a5acaf84e8
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/RegularityJets.lean
@@ -0,0 +1,136 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.GlobalHessian
+import OAI.Geometry.PrescribedPotential.HessianCommutator
+import OAI.Geometry.PrescribedPotential.RegularityCutoffs
+
+namespace OAI
+
+/-! Regularity Jets. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory
+open scoped ContDiff SchwartzMap Classical
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+
+def chartOperator (q : Fin A.count) (κ η : ChartCutoff (A.euclideanChart q).target)
+    (T : 𝓢(EC d, ℂ) →L[ℂ] 𝓢(EC d, ℂ)) : Smooth A →ₗ[ℝ] Smooth A :=
+  (globalize q κ).comp ((T.restrictScalars ℝ).toLinearMap.comp
+    (localizeLinear A q (cutoffGlobal q η) (cutoffGlobal_support q η)))
+
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+lemma chartOperator_bound (q : Fin A.count)
+    (κ η : ChartCutoff (A.euclideanChart q).target) (k l : ℕ)
+    (T : 𝓢(EC d, ℂ) →L[ℂ] 𝓢(EC d, ℂ)) (hT : CoreBound (l : ℝ) (k : ℝ) T) :
+    D.localizers.BoundedCore (l : ℝ) (k : ℝ) (chartOperator q κ η T) := by
+  obtain ⟨B,hB,hb⟩ := globalize_integer_bound D.localizers q κ k
+  obtain ⟨C,hC,hc⟩ := hT
+  obtain ⟨E,hE,he⟩ := D.localization_integer_bound q (cutoffGlobal q η) (cutoffGlobal_support q η) l
+  refine ⟨B*(C*E),mul_nonneg hB (mul_nonneg hC hE),fun f => ?_⟩
+  let u := localize A q (cutoffGlobal q η) (cutoffGlobal_support q η) f
+  change ‖D.localizers.embed (k : ℝ) (globalize q κ (T u))‖ ≤ _
+  calc
+    _ ≤ B*‖schwartzCoord (k : ℝ) (T u)‖ := hb _
+    _ ≤ B*(C*‖schwartzCoord (l : ℝ) u‖) := mul_le_mul_of_nonneg_left (hc u) hB
+    _ ≤ B*(C*(E*‖D.localizers.embed (l : ℝ) f‖)) :=
+      mul_le_mul_of_nonneg_left (mul_le_mul_of_nonneg_left (he f) hC) hB
+    _ = _ := by ring
+
+def regularityHessian (p : ι) (i j : Fin d) : Smooth A →ₗ[ℝ] Smooth A :=
+  chartOperator (D.patch p).index (D.regularityCutoff p) (D.regularityOuter p) (hessianEntrySchwartz i j)
+
+def regularityError (p : ι) (v : EC d) (i j : Fin d) : Smooth A →ₗ[ℝ] Smooth A :=
+  chartOperator (D.patch p).index (D.regularityCutoff p) (D.regularityOuter p)
+    (hessianLocalizedError (D.cutoff p).realPart.val v i j)
+
+lemma regularityHessian_bound (k : ℕ) (p : ι) (i j : Fin d) :
+    D.localizers.BoundedCore ((k : ℝ)+2) (k : ℝ) (D.regularityHessian p i j) := by
+  have hh := D.chartOperator_bound (D.patch p).index (D.regularityCutoff p) (D.regularityOuter p)
+    k (k+2) (hessianEntrySchwartz i j) (by
+      simpa only [Nat.cast_add,Nat.cast_ofNat] using hessianEntrySchwartz_bound (k : ℝ) i j)
+  simpa only [Nat.cast_add,Nat.cast_ofNat,regularityHessian] using hh
+
+lemma regularityError_bound (k : ℕ) (p : ι) (v : EC d) (i j : Fin d) :
+    D.localizers.BoundedCore ((k : ℝ)+2) (k : ℝ) (D.regularityError p v i j) := by
+  have hh := D.chartOperator_bound (D.patch p).index (D.regularityCutoff p) (D.regularityOuter p)
+    k (k+2) (hessianLocalizedError (D.cutoff p).realPart.val v i j) (by
+      simpa only [Nat.cast_add,Nat.cast_ofNat]
+        using hessianLocalizedError_bound (D.cutoff p).realPart.val v i j k)
+  simpa only [Nat.cast_add,Nat.cast_ofNat,regularityError] using hh
+
+end GluingData
+
+lemma chartHessian_source (q : Fin A.count) (κ η : ChartCutoff (A.euclideanChart q).target)
+    (hη : ∀ y ∈ tsupport (κ : EC d → ℂ), η y = 1)
+    (φ : SmoothRealFunction A) (i j : Fin d) {x : X}
+    (hx : x ∈ (A.euclideanChart q).source) :
+    chartOperator q κ η (hessianEntrySchwartz i j) (Smooth.ofReal φ) x =
+      κ (A.euclideanChart q x) * φ.hessian q (A.chart q x) i j := by
+  change globalize q κ (hessianEntrySchwartz i j
+    (localize A q (cutoffGlobal q η) (cutoffGlobal_support q η) (Smooth.ofReal φ))) x = _
+  rw [globalize_apply,ite_eq_left hx]
+  by_cases hc : κ (A.euclideanChart q x) = 0
+  · rw [hc,zero_mul,zero_mul]
+  have ht := (A.euclideanChart q).mapsTo hx
+  have he := localize_cutoff_eventually q κ η hη (Smooth.ofReal φ) ht hc
+  let u : EC d → ℝ := φ.localExpression q ∘ coordinateEquiv d
+  have hu : ContDiffAt ℝ ∞ u (A.euclideanChart q x) := (φ.euclidean_smooth q).contDiffAt
+    (by simpa only [ComplexAtlas.euclideanChart_target] using
+      (A.euclideanChart q).open_target.mem_nhds ht)
+  have hre : (fun y => Smooth.ofReal φ ((A.euclideanChart q).symm y)) = Complex.ofRealCLM ∘ u := rfl
+  rw [hessianEntrySchwartz_real hu _ (he.trans (Filter.Eventually.of_forall (congr_fun hre))) i j,
+    pullBilin_hessian hu]
+  have hcomp : u ∘ (coordinateEquiv d).symm = φ.localExpression q := by
+    funext z; simp [u]
+  rw [hcomp]
+  change κ (A.euclideanChart q x) * PotentialKaehler.hermitianPartMatrix
+    (fderiv ℝ (fderiv ℝ (φ.localExpression q))
+      (coordinateEquiv d (A.euclideanChart q x))) i j = _
+  rw [ComplexAtlas.euclideanChart_apply,ContinuousLinearEquiv.apply_symm_apply]
+  rfl
+
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+lemma regularityHessian_source (p : ι) (φ : SmoothRealFunction A) (i j : Fin d) {x : X}
+    (hx : x ∈ (A.euclideanChart (D.patch p).index).source)
+    (hκ : A.euclideanChart (D.patch p).index x ∈ tsupport (D.cutoff p : EC d → ℂ)) :
+    D.regularityHessian p i j (Smooth.ofReal φ) x =
+      φ.hessian (D.patch p).index (A.chart (D.patch p).index x) i j := by
+  rw [regularityHessian,chartHessian_source _ _ _
+    (fun y hy => (D.regularityOuter_one p hy).eq_of_nhds) φ i j hx,
+    (D.regularityCutoff_one p hκ).eq_of_nhds,one_mul]
+
+def completedRegularityHessian (k : ℕ) (p : ι) (i j : Fin d) :
+    D.localizers.Sobolev ((k : ℝ)+2) →L[ℝ] D.localizers.Sobolev (k : ℝ) :=
+  D.localizers.extendCore ((k : ℝ)+2) (k : ℝ) (D.regularityHessian p i j)
+
+def completedRegularityError (k : ℕ) (p : ι) (v : EC d) (i j : Fin d) :
+    D.localizers.Sobolev ((k : ℝ)+2) →L[ℝ] D.localizers.Sobolev (k : ℝ) :=
+  D.localizers.extendCore ((k : ℝ)+2) (k : ℝ) (D.regularityError p v i j)
+
+lemma completedRegularityHessian_embed (k : ℕ) (p : ι) (i j : Fin d) (f : Smooth A) :
+    D.completedRegularityHessian k p i j (D.localizers.embed ((k : ℝ)+2) f) =
+      D.localizers.embed (k : ℝ) (D.regularityHessian p i j f) :=
+  D.localizers.extendCore_embed (D.regularityHessian_bound k p i j) f
+
+lemma completedRegularityError_embed (k : ℕ) (p : ι) (v : EC d) (i j : Fin d) (f : Smooth A) :
+    D.completedRegularityError k p v i j (D.localizers.embed ((k : ℝ)+2) f) =
+      D.localizers.embed (k : ℝ) (D.regularityError p v i j f) :=
+  D.localizers.extendCore_embed (D.regularityError_bound k p v i j) f
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/ResidualFamilyBootstrap.lean b/lean/OAI/Geometry/PrescribedPotential/ResidualFamilyBootstrap.lean
new file mode 100644
index 0000000000000000000000000000000000000000..ab725cbb54297023954fa6b23a1023638111f88d
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/ResidualFamilyBootstrap.lean
@@ -0,0 +1,169 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.FamilyBernsteinBootstrap
+import OAI.Geometry.PrescribedPotential.JetEnergyEstimates
+import OAI.Geometry.PrescribedPotential.UniformJetControls
+
+namespace OAI
+
+/-! Residual Family Bootstrap. -/
+
+section
+
+noncomputable section
+open Set Filter Topology Finset Module
+open scoped ContDiff
+namespace HigherJet
+variable {E F : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [NormedAddCommGroup F] [InnerProductSpace ℝ F]
+  [FiniteDimensional ℝ E] [FiniteDimensional ℝ F]
+  {ι α : Type*} [Fintype ι]
+
+lemma residual_family_low_energy {U : Set E} (hU : IsOpen U)
+    (e : OrthonormalBasis ι ℝ E) (u : α → E → F) (τ : α → E → E ≃L[ℝ] E)
+    (hu : ∀ a, ContDiffOn ℝ ∞ (u a) U)
+    (hframes : ∀ K, IsCompact K → K ⊆ U → ∃ B : ℝ, 0 ≤ B ∧ ∀ a x, x ∈ K →
+      ‖(τ a x).toContinuousLinearMap‖ ≤ B ∧ ‖(τ a x).symm.toContinuousLinearMap‖ ≤ B)
+    (hfirst : ∀ K, IsCompact K → K ⊆ U → ∃ R : ℝ, 0 ≤ R ∧ ∀ a x, x ∈ K → ‖iteratedFDeriv ℝ 1 (u a) x‖ ≤ R)
+    (hlow : ∀ K, IsCompact K → K ⊆ U → ∃ R : ℝ, 0 ≤ R ∧ ∀ a x, x ∈ K →
+      (∀ σ, ‖frameLaplace (fun i => τ a x (e i)) (jetFamily e 1 (u a) σ) x‖ ≤ R*(1+‖iteratedFDeriv ℝ 2 (u a) x‖)) ∧
+      (∀ σ, ‖frameLaplace (fun i => τ a x (e i)) (jetFamily e 2 (u a) σ) x‖ ≤ R*(1+‖iteratedFDeriv ℝ 2 (u a) x‖^2+‖iteratedFDeriv ℝ 3 (u a) x‖)))
+ :
+    ∀ K, IsCompact K → K ⊆ U → ∃ R : ℝ, 0 ≤ R ∧ ∀ a x, x ∈ K →
+      familyDissipation (fun i => τ a x (e i)) (jetFamily e 1 (u a)) x-R ≤ frameLaplace (fun i => τ a x (e i)) (jetEnergy e 1 (u a)) x ∧
+      familyDissipation (fun i => τ a x (e i)) (jetFamily e 2 (u a)) x-R*(1+jetEnergy e 2 (u a) x*Real.sqrt (jetEnergy e 2 (u a) x)) ≤ frameLaplace (fun i => τ a x (e i)) (jetEnergy e 2 (u a)) x := by
+  let S := fun m a x => jetEnergy e m (u a) x
+  let D := fun m a x => familyDissipation (fun i => τ a x (e i)) (jetFamily e m (u a)) x
+  have hf : ∀ K, IsCompact K → K ⊆ U → EnergyBound S 1 K := by
+    intro K hK hKU
+    obtain ⟨R,hR,hr⟩ := hfirst K hK hKU
+    refine ⟨(Fintype.card (Fin 1 → ι) : ℝ)*R^2,by positivity,fun a x hx => ?_⟩
+    exact (jetEnergy_le_norm hU (hu a) e 1 (hKU hx)).trans
+      (mul_le_mul_of_nonneg_left (pow_le_pow_left₀ (norm_nonneg _) (hr a x hx) 2) (by positivity))
+  intro K hK hKU
+  obtain ⟨B,hB,hτ⟩ := hframes K hK hKU
+  obtain ⟨A1,hA1,hc1⟩ := uniform_jet_controls (F := F) e 1 hB
+  obtain ⟨A2,hA2,hc2⟩ := uniform_jet_controls (F := F) e 2 hB
+  obtain ⟨R0,hR0,hr0⟩ := hf K hK hKU
+  obtain ⟨R,hR,hr⟩ := hlow K hK hKU
+  let A := A1+A2+Real.sqrt R0
+  have hAA : 1 ≤ A := by dsimp [A]; linarith [Real.sqrt_nonneg R0]
+  have h1A : A1 ≤ A := by dsimp [A]; linarith [Real.sqrt_nonneg R0]
+  have h2A : A2 ≤ A := by dsimp [A]; linarith [Real.sqrt_nonneg R0]
+  let C1 := 2*(A+(Fintype.card (Fin 1 → ι)+1)*R*A)^2+(A+(Fintype.card (Fin 1 → ι)+1)*R*A)^4
+  let C2 := 4*((Fintype.card (Fin 2 → ι)+1)*R*A^2)+2*((Fintype.card (Fin 2 → ι)+1)*R*A^2)^2
+  have hC1 : 0 ≤ C1 := by dsimp [C1]; positivity
+  have hC2 : 0 ≤ C2 := by dsimp [C2]; positivity
+  refine ⟨C1+C2,add_nonneg hC1 hC2,fun a x hx => ?_⟩
+  have hcx1 := hc1 U hU (u a) (hu a) x (hKU hx) (τ a x) (hτ a x hx).1 (hτ a x hx).2
+  have hcx2 := hc2 U hU (u a) (hu a) x (hKU hx) (τ a x) (hτ a x hx).1 (hτ a x hx).2
+  have hs : Real.sqrt (S 1 a x) ≤ A := (Real.sqrt_le_sqrt (hr0 a x hx)).trans (by dsimp [A]; linarith)
+  have hh1 := energy_first_of_residual hU (hu a) e (hKU hx) (fun i => τ a x (e i)) hAA hR hs
+    (hcx1.2.1.trans (mul_le_mul_of_nonneg_right h1A (Real.sqrt_nonneg _))) (hr a x hx).1
+  have hh2 := energy_second_of_residual hU (hu a) e (hKU hx) (fun i => τ a x (e i)) hAA hR
+    (hcx2.1.trans (mul_le_mul_of_nonneg_right h2A (Real.sqrt_nonneg _)))
+    (hcx2.2.1.trans (mul_le_mul_of_nonneg_right h2A (Real.sqrt_nonneg _))) (hr a x hx).2
+  change D 1 a x-C1 ≤ _ at hh1
+  change D 2 a x-C2*(1+S 2 a x*Real.sqrt (S 2 a x)) ≤ _ at hh2
+  constructor
+  · exact (sub_le_sub_left (le_add_of_nonneg_right hC2) _).trans hh1
+  · have hq : 0 ≤ 1+S 2 a x*Real.sqrt (S 2 a x) := by have := jetEnergy_nonneg e 2 (u a) x; positivity
+    exact (sub_le_sub_left (mul_le_mul_of_nonneg_right (le_add_of_nonneg_left hC1) hq) _).trans hh2
+
+lemma residual_family_high_energy {U : Set E} (hU : IsOpen U)
+    (e : OrthonormalBasis ι ℝ E) (u : α → E → F) (τ : α → E → E ≃L[ℝ] E)
+    (hu : ∀ a, ContDiffOn ℝ ∞ (u a) U)
+    (hframes : ∀ K, IsCompact K → K ⊆ U → ∃ B : ℝ, 0 ≤ B ∧ ∀ a x, x ∈ K →
+      ‖(τ a x).toContinuousLinearMap‖ ≤ B ∧ ‖(τ a x).symm.toContinuousLinearMap‖ ≤ B)
+    (_hfirst : ∀ K, IsCompact K → K ⊆ U → ∃ R : ℝ, 0 ≤ R ∧ ∀ a x, x ∈ K → ‖iteratedFDeriv ℝ 1 (u a) x‖ ≤ R)
+    (_hlow : ∀ K, IsCompact K → K ⊆ U → ∃ R : ℝ, 0 ≤ R ∧ ∀ a x, x ∈ K →
+      (∀ σ, ‖frameLaplace (fun i => τ a x (e i)) (jetFamily e 1 (u a) σ) x‖ ≤ R*(1+‖iteratedFDeriv ℝ 2 (u a) x‖)) ∧
+      (∀ σ, ‖frameLaplace (fun i => τ a x (e i)) (jetFamily e 2 (u a) σ) x‖ ≤ R*(1+‖iteratedFDeriv ℝ 2 (u a) x‖^2+‖iteratedFDeriv ℝ 3 (u a) x‖)))
+    (hhigh : ∀ m, 3 ≤ m → ∀ K, IsCompact K → K ⊆ U →
+      (∀ j, 1 ≤ j → j < m → EnergyBound (fun j a x => jetEnergy e j (u a) x) j K) →
+      ∃ R : ℝ, 0 ≤ R ∧ ∀ a x, x ∈ K → ∀ σ,
+        ‖frameLaplace (fun i => τ a x (e i)) (jetFamily e m (u a) σ) x‖ ≤
+          R*(1+‖iteratedFDeriv ℝ m (u a) x‖+‖iteratedFDeriv ℝ (m+1) (u a) x‖)) :
+    ∀ m, 3 ≤ m → ∀ K, IsCompact K → K ⊆ U →
+      (∀ j, 1 ≤ j → j < m → EnergyBound (fun j a x => jetEnergy e j (u a) x) j K) → ∃ R : ℝ, 0 ≤ R ∧ ∀ a x, x ∈ K →
+        familyDissipation (fun i => τ a x (e i)) (jetFamily e m (u a)) x-R*(1+jetEnergy e m (u a) x) ≤ frameLaplace (fun i => τ a x (e i)) (jetEnergy e m (u a)) x := by
+  intro m hm K hK hKU hlo
+  obtain ⟨B,hB,hτ⟩ := hframes K hK hKU
+  obtain ⟨A,hA,hc⟩ := uniform_jet_controls (F := F) e m hB
+  obtain ⟨R,hR,hr⟩ := hhigh m hm K hK hKU hlo
+  refine ⟨4*((Fintype.card (Fin m → ι)+1)*R*A)+((Fintype.card (Fin m → ι)+1)*R*A)^2,by positivity,fun a x hx => ?_⟩
+  have hc' := hc U hU (u a) (hu a) x (hKU hx) (τ a x) (hτ a x hx).1 (hτ a x hx).2
+  exact energy_high_of_residual hU (hu a) e m (hKU hx) (fun i => τ a x (e i)) hA hR hc'.1 hc'.2.1 (hr a x hx)
+
+lemma residual_family_bootstrap {U : Set E} (hU : IsOpen U)
+    (e : OrthonormalBasis ι ℝ E) (u : α → E → F) (τ : α → E → E ≃L[ℝ] E)
+    (hu : ∀ a, ContDiffOn ℝ ∞ (u a) U)
+    (hframes : ∀ K, IsCompact K → K ⊆ U → ∃ B : ℝ, 0 ≤ B ∧ ∀ a x, x ∈ K →
+      ‖(τ a x).toContinuousLinearMap‖ ≤ B ∧ ‖(τ a x).symm.toContinuousLinearMap‖ ≤ B)
+    (hfirst : ∀ K, IsCompact K → K ⊆ U → ∃ R : ℝ, 0 ≤ R ∧ ∀ a x, x ∈ K → ‖iteratedFDeriv ℝ 1 (u a) x‖ ≤ R)
+    (hlow : ∀ K, IsCompact K → K ⊆ U → ∃ R : ℝ, 0 ≤ R ∧ ∀ a x, x ∈ K →
+      (∀ σ, ‖frameLaplace (fun i => τ a x (e i)) (jetFamily e 1 (u a) σ) x‖ ≤ R*(1+‖iteratedFDeriv ℝ 2 (u a) x‖)) ∧
+      (∀ σ, ‖frameLaplace (fun i => τ a x (e i)) (jetFamily e 2 (u a) σ) x‖ ≤ R*(1+‖iteratedFDeriv ℝ 2 (u a) x‖^2+‖iteratedFDeriv ℝ 3 (u a) x‖)))
+    (hhigh : ∀ m, 3 ≤ m → ∀ K, IsCompact K → K ⊆ U →
+      (∀ j, 1 ≤ j → j < m → EnergyBound (fun j a x => jetEnergy e j (u a) x) j K) →
+      ∃ R : ℝ, 0 ≤ R ∧ ∀ a x, x ∈ K → ∀ σ,
+        ‖frameLaplace (fun i => τ a x (e i)) (jetFamily e m (u a) σ) x‖ ≤
+          R*(1+‖iteratedFDeriv ℝ m (u a) x‖+‖iteratedFDeriv ℝ (m+1) (u a) x‖)) :
+    ∀ m, 1 ≤ m → ∀ K, IsCompact K → K ⊆ U →
+      ∃ R : ℝ, 0 ≤ R ∧ ∀ a x, x ∈ K → ‖iteratedFDeriv ℝ m (u a) x‖ ≤ R := by
+  let S := fun m a x => jetEnergy e m (u a) x
+  let D := fun m a x => familyDissipation (fun i => τ a x (e i)) (jetFamily e m (u a)) x
+  have hf : ∀ K, IsCompact K → K ⊆ U → EnergyBound S 1 K := by
+    intro K hK hKU
+    obtain ⟨R,hR,hr⟩ := hfirst K hK hKU
+    refine ⟨(Fintype.card (Fin 1 → ι) : ℝ)*R^2,by positivity,fun a x hx => ?_⟩
+    exact (jetEnergy_le_norm hU (hu a) e 1 (hKU hx)).trans
+      (mul_le_mul_of_nonneg_left (pow_le_pow_left₀ (norm_nonneg _) (hr a x hx) 2) (by positivity))
+  have hlowE := residual_family_low_energy hU e u τ hu hframes hfirst hlow
+  have hhighE := residual_family_high_energy hU e u τ hu hframes hfirst hlow hhigh
+  have hb : ∀ m, 1 ≤ m → ∀ K, IsCompact K → K ⊆ U → EnergyBound S m K := by
+    apply family_bernstein_bootstrap hU S D (fun a x i => τ a x (e i))
+    · exact fun m a => jetEnergy_smoothOn hU (hu a) e m
+    · exact fun m a x _ => ⟨jetEnergy_nonneg _ _ _ _,familyDissipation_nonneg _ _ _⟩
+    · exact fun m a x hx => jetEnergy_gradient_bound hU (hu a) e m hx _
+    · intro K hK hKU
+      obtain ⟨B,hB,hτ⟩ := hframes K hK hKU
+      refine ⟨B,hB,fun a x hx i => ?_⟩
+      exact ((τ a x).toContinuousLinearMap.le_opNorm (e i)).trans (by simpa only [e.norm_eq_one,mul_one] using (hτ a x hx).1)
+    · intro m _ K hK hKU
+      obtain ⟨B,hB,hτ⟩ := hframes K hK hKU
+      obtain ⟨A,hA,hc⟩ := uniform_jet_controls (F := F) e m hB
+      exact ⟨A,hA,fun a x hx => (hc U hU (u a) (hu a) x (hKU hx) (τ a x) (hτ a x hx).1 (hτ a x hx).2).2.2⟩
+    · exact hf
+    · intro m hm K hK hKU hlo
+      by_cases h1 : m=1
+      · subst m
+        obtain ⟨R,hR,hr⟩ := hlowE K hK hKU
+        exact ⟨R,hR,fun a x hx => (hr a x hx).1⟩
+      · by_cases h2 : m=2
+        · subst m
+          obtain ⟨R,hR,hr⟩ := hlowE K hK hKU
+          obtain ⟨C,hC,hc⟩ := hlo 2 (by decide) le_rfl
+          refine ⟨R*(1+C*Real.sqrt C),by positivity,fun a x hx => ?_⟩
+          have ht := mul_le_mul_of_nonneg_left (add_le_add_left
+            (mul_le_mul (hc a x hx) (Real.sqrt_le_sqrt (hc a x hx)) (Real.sqrt_nonneg _) hC) 1) hR
+          linarith only [ht,(hr a x hx).2]
+        · obtain ⟨R,hR,hr⟩ := hhighE m (by omega) K hK hKU (fun j hj hjm => hlo j hj (by omega))
+          obtain ⟨C,hC,hc⟩ := hlo m hm le_rfl
+          refine ⟨R*(1+C),by positivity,fun a x hx => ?_⟩
+          have ht := mul_le_mul_of_nonneg_left (add_le_add_left (hc a x hx) 1) hR
+          linarith only [ht,hr a x hx]
+    · intro K hK hKU
+      obtain ⟨R,hR,hr⟩ := hlowE K hK hKU
+      exact ⟨R,hR,fun a x hx => (by have hd := familyDissipation_nonneg (fun i => τ a x (e i)) (jetFamily e 2 (u a)) x; linarith only [hd,(hr a x hx).2])⟩
+    · exact hhighE
+  intro m hm K hK hKU
+  obtain ⟨R,hR,hr⟩ := hb m hm K hK hKU
+  obtain ⟨C,hC,hc⟩ := norm_iteratedFDeriv_le_jetEnergy (F := F) e.toBasis m
+  exact ⟨C*Real.sqrt R,by positivity,fun a x hx => (hc U hU (u a) (hu a) x (hKU hx)).trans
+    (mul_le_mul_of_nonneg_left (Real.sqrt_le_sqrt (hr a x hx)) hC.le)⟩
+end HigherJet
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/ScalarFrameCalculus.lean b/lean/OAI/Geometry/PrescribedPotential/ScalarFrameCalculus.lean
new file mode 100644
index 0000000000000000000000000000000000000000..7a92c59cd47f6c7c1649f1371a0ac24211144299
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/ScalarFrameCalculus.lean
@@ -0,0 +1,92 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.InverseTraceFunction
+import OAI.Geometry.PrescribedPotential.LinearFrameCalculus
+
+namespace OAI
+
+/-! Scalar Frame Calculus. -/
+
+section
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace KaehlerCalculus
+variable {n : ℕ}
+
+lemma potentialMatrix_const_add (c : ℝ) (f : V n → ℝ) (z : V n) :
+    PotentialKaehler.potentialMatrix (fun y => c+f y) z = PotentialKaehler.potentialMatrix f z := by
+  unfold PotentialKaehler.potentialMatrix
+  have he : fderiv ℝ (fun y => c+f y) = fderiv ℝ f := funext (fun _ => fderiv_const_add c)
+  rw [he]
+
+lemma potentialMatrix_linear (B : V n →L[ℂ] V n) {f : V n → ℝ} {z : V n}
+    (hf : ContDiffAt ℝ ∞ f (B z)) :
+    PotentialKaehler.potentialMatrix (fun y => f (B y)) z =
+      (frameMatrix B)ᴴ*PotentialKaehler.potentialMatrix f (B z)*frameMatrix B := by
+  have he := PotentialKaehler.potentialMatrix_comp B.contDiff.contDiffAt hf
+  have hb : fderiv ℂ (fun y => B y) z = B := B.fderiv
+  simpa only [Function.comp_def,hb,frameMatrix] using he
+
+lemma det_congruence_re (G B : Matrix (Fin n) (Fin n) ℂ) :
+    (Bᴴ*G*B).det.re = ‖B.det‖^2*G.det.re := by
+  rw [Matrix.det_mul,Matrix.det_mul,Matrix.det_conjTranspose]
+  have he : (starRingEnd ℂ) B.det*G.det*B.det =
+      ((‖B.det‖^2 : ℝ):ℂ)*G.det := by
+    rw [show (starRingEnd ℂ) B.det*G.det*B.det =
+      (B.det*(starRingEnd ℂ) B.det)*G.det by ring,Complex.mul_conj,Complex.normSq_eq_norm_sq]
+  change ((starRingEnd ℂ) B.det*G.det*B.det).re = _
+  rw [he,Complex.mul_re,Complex.ofReal_re,Complex.ofReal_im,zero_mul,sub_zero]
+
+lemma logdet_congruence (G B : Matrix (Fin n) (Fin n) ℂ)
+    (hG : G.PosDef) (hB : IsUnit B) :
+    Real.log (Bᴴ*G*B).det.re = Real.log (‖B.det‖^2) + Real.log G.det.re := by
+  have hb : B.det ≠ 0 := isUnit_iff_ne_zero.mp ((Matrix.isUnit_iff_isUnit_det B).mp hB)
+  rw [det_congruence_re]
+  exact Real.log_mul (pow_ne_zero 2 (norm_ne_zero_iff.mpr hb))
+    (ne_of_gt (Complex.pos_iff.mp hG.det_pos).1)
+
+lemma matrix_logdet_smooth {U : Set (V n)}
+    {M : V n → Matrix (Fin n) (Fin n) ℂ} (hM : ContDiffOn ℝ ∞ M U)
+    (hp : ∀ y ∈ U, (M y).PosDef) :
+    ContDiffOn ℝ ∞ (fun y => Real.log (M y).det.re) U := by
+  exact (Complex.reCLM.contDiff.comp_contDiffOn
+    (determinant_smooth.comp_contDiffOn hM)).log
+    (fun y hy => ne_of_gt (Complex.pos_iff.mp (hp y hy).det_pos).1)
+
+lemma potentialMatrix_logdet_pullMetric {U : Set (V n)} (hU : IsOpen U)
+    (B : V n →L[ℂ] V n) (hB : IsUnit (frameMatrix B))
+    {M : V n → Matrix (Fin n) (Fin n) ℂ} (hM : ContDiffOn ℝ ∞ M U)
+    (hp : ∀ y ∈ U, (M y).PosDef) {z : V n} (hz : B z ∈ U) :
+    PotentialKaehler.potentialMatrix (fun y => Real.log (pullMetric B M y).det.re) z =
+      (frameMatrix B)ᴴ * PotentialKaehler.potentialMatrix (fun y => Real.log (M y).det.re) (B z) * frameMatrix B := by
+  have he : (fun y => Real.log (pullMetric B M y).det.re) =ᶠ[𝓝 z]
+      (fun y => Real.log (‖(frameMatrix B).det‖^2) + Real.log (M (B y)).det.re) := by
+    filter_upwards [B.continuous.continuousAt.preimage_mem_nhds (hU.mem_nhds hz)] with y hy
+    exact logdet_congruence (M (B y)) (frameMatrix B) (hp (B y) hy) hB
+  rw [PotentialKaehler.potentialMatrix_congr he]
+  rw [potentialMatrix_const_add]
+  exact potentialMatrix_linear B (f := fun y => Real.log (M y).det.re)
+    (z := z) ((matrix_logdet_smooth hM hp).contDiffAt (hU.mem_nhds hz))
+
+lemma inverseTrace_pullMetric (B : V n →L[ℂ] V n) (hB : IsUnit (frameMatrix B))
+    (M G : V n → Matrix (Fin n) (Fin n) ℂ) (z : V n) :
+    inverseTrace (pullMetric B M) (pullMetric B G) z = inverseTrace M G (B z) := by
+  let C := frameMatrix B
+  have hc : C*C⁻¹ = 1 := Matrix.mul_nonsing_inv C ((Matrix.isUnit_iff_isUnit_det C).mp hB)
+  have hd : Cᴴ⁻¹*Cᴴ = 1 := by
+    rw [← Matrix.conjTranspose_nonsing_inv,← Matrix.conjTranspose_mul,hc,Matrix.conjTranspose_one]
+  unfold inverseTrace pullMetric
+  change ((Cᴴ*M (B z)*C)⁻¹*(Cᴴ*G (B z)*C)).trace.re = _
+  simp only [Matrix.mul_inv_rev,Matrix.mul_assoc]
+  rw [← Matrix.mul_assoc Cᴴ⁻¹ Cᴴ,hd,Matrix.one_mul]
+  have he : (C⁻¹*((M (B z))⁻¹*(G (B z)*C))).trace = ((M (B z))⁻¹*G (B z)).trace := by
+    rw [← Matrix.mul_assoc (M (B z))⁻¹,← Matrix.mul_assoc C⁻¹,
+      Matrix.trace_mul_cycle, hc,Matrix.one_mul]
+  rw [he]
+end KaehlerCalculus
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/ScalarPrincipalWirtinger.lean b/lean/OAI/Geometry/PrescribedPotential/ScalarPrincipalWirtinger.lean
new file mode 100644
index 0000000000000000000000000000000000000000..f72f2908a2d84a748915044250425c046b558657
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/ScalarPrincipalWirtinger.lean
@@ -0,0 +1,86 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.ComplexOperator
+import OAI.Geometry.PrescribedPotential.KaehlerClosedDerivatives
+import OAI.Geometry.PrescribedPotential.SymmetricPrincipalFrame
+import OAI.Geometry.PrescribedPotential.TracePositivity
+
+namespace OAI
+
+/-! Scalar Principal Wirtinger. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Matrix Finset
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace KaehlerCalculus
+open EllipticKernel FrozenPoisson HigherJet
+variable {n : ℕ}
+local instance realECIP : InnerProductSpace ℝ (EC n) := InnerProductSpace.rclikeToReal ℂ (EC n)
+
+lemma scalar_principal_real {ι : Type*} [Fintype ι] (e : OrthonormalBasis ι ℝ (EC n))
+    (M : Matrix (Fin n) (Fin n) ℂ) {f : EC n → ℝ} {x : EC n} (hf : ContDiffAt ℝ ∞ f x) :
+    principalApply e (traceBilin M) (fderiv ℝ (fderiv ℝ f) x) =
+      (M⁻¹*PotentialKaehler.potentialMatrix (f ∘ (coordinateEquiv n).symm) (coordinateEquiv n x)).trace.re := by
+  rw [principalApply_symmetric e _ _ (hf.isSymmSndFDerivAt (by
+    simp only [minSmoothness_of_isRCLikeNormedField]
+    exact WithTop.coe_le_coe.mpr le_top))]
+  have he : traceBilin M (fderiv ℝ (fderiv ℝ f) x) =
+      ∑ i, ∑ j, traceBilin M (rankTwo (e i) (e j))*fderiv ℝ (fderiv ℝ f) x (e i) (e j) := by
+    conv_lhs => rw [bilin_expansion e (fderiv ℝ (fderiv ℝ f) x)]
+    simp only [map_sum,map_smul,smul_eq_mul]
+    exact sum_congr rfl (fun i _ => sum_congr rfl (fun j _ => mul_comm _ _))
+  simp only [smul_eq_mul]
+  rw [← he,traceBilin_apply,pullBilin_hessian hf]
+  rfl
+
+lemma wirtinger_re_im {f : V n → ℂ} {z : V n} (hf : ContDiffAt ℝ ∞ f z) (u v : V n) :
+    dzbar u (dz v f) z =
+      dzbar u (dz v (fun y => ((f y).re:ℂ))) z +
+      Complex.I*dzbar u (dz v (fun y => ((f y).im:ℂ))) z := by
+  have hr : ContDiffAt ℝ ∞ (fun y => (f y).re) z := Complex.reCLM.contDiff.contDiffAt.comp z hf
+  have hi : ContDiffAt ℝ ∞ (fun y => (f y).im) z := Complex.imCLM.contDiff.contDiffAt.comp z hf
+  rw [dzbar_dz_real hr,dzbar_dz_real hi]
+  simp only [dzbar,dz,wderiv,fderiv_wderiv hf]
+  have hre (a b : V n) : fderiv ℝ (fderiv ℝ (fun y => (f y).re)) z a b =
+      (fderiv ℝ (fderiv ℝ f) z a b).re := GlobalElliptic.second_postcomp hf Complex.reCLM a b
+  have him (a b : V n) : fderiv ℝ (fderiv ℝ (fun y => (f y).im)) z a b =
+      (fderiv ℝ (fderiv ℝ f) z a b).im := GlobalElliptic.second_postcomp hf Complex.imCLM a b
+  simp only [PotentialKaehler.hermitianPart,hre,him]
+  apply Complex.ext <;> simp <;> ring
+
+lemma inverse_wirtinger_real (M : Matrix (Fin n) (Fin n) ℂ) (hM : M.PosDef)
+    {f : V n → ℝ} {z : V n} (hf : ContDiffAt ℝ ∞ f z) :
+    (∑ p, ∑ q, M⁻¹ p q*dzbar (e q) (dz (e p) (fun y => (f y:ℂ))) z) =
+      (((M⁻¹*PotentialKaehler.potentialMatrix f z).trace.re : ℝ):ℂ) := by
+  rw [trace_mul_real hM.inv.isHermitian (PotentialKaehler.potentialMatrix_hermitian hf)]
+  simp only [Matrix.trace]
+  apply sum_congr rfl
+  intro p _
+  apply sum_congr rfl
+  intro q _
+  change M⁻¹ p q*dzbar (e q) (dz (e p) (fun y => (f y:ℂ))) z = M⁻¹ p q*PotentialKaehler.potentialMatrix f z q p
+  rw [potentialMatrix_eq_dzbar_dz hf]
+  rfl
+
+lemma inverse_wirtinger_complex (M : Matrix (Fin n) (Fin n) ℂ) (hM : M.PosDef)
+    {f : V n → ℂ} {z : V n} (hf : ContDiffAt ℝ ∞ f z) :
+    (∑ p, ∑ q, M⁻¹ p q*dzbar (e q) (dz (e p) f) z) =
+      ((M⁻¹*PotentialKaehler.potentialMatrix (fun y => (f y).re) z).trace.re:ℂ) +
+      Complex.I*((M⁻¹*PotentialKaehler.potentialMatrix (fun y => (f y).im) z).trace.re:ℂ) := by
+  simp only [wirtinger_re_im hf,mul_add,sum_add_distrib]
+  have he : (∑ p, ∑ q, M⁻¹ p q*(Complex.I*dzbar (e q) (dz (e p) (fun y => ((f y).im:ℂ))) z)) =
+      Complex.I*(∑ p, ∑ q, M⁻¹ p q*dzbar (e q) (dz (e p) (fun y => ((f y).im:ℂ))) z) := by
+    simp only [Finset.mul_sum]
+    exact sum_congr rfl (fun p _ => sum_congr rfl (fun q _ => by ring))
+  have hr : ContDiffAt ℝ ∞ (fun y => (f y).re) z := Complex.reCLM.contDiff.contDiffAt.comp z hf
+  have hi : ContDiffAt ℝ ∞ (fun y => (f y).im) z := Complex.imCLM.contDiff.contDiffAt.comp z hf
+  rw [he,inverse_wirtinger_real M hM hr,inverse_wirtinger_real M hM hi]
+end KaehlerCalculus
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/ScalarProductLaplacian.lean b/lean/OAI/Geometry/PrescribedPotential/ScalarProductLaplacian.lean
new file mode 100644
index 0000000000000000000000000000000000000000..bee40d15628c70ed10c4a839444764ca82a771b9
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/ScalarProductLaplacian.lean
@@ -0,0 +1,93 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.WirtingerConjugate
+
+namespace OAI
+
+/-! Scalar Product Laplacian. -/
+
+section
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace KaehlerCalculus
+variable {n : ℕ}
+
+def holGradient (f : V n → ℝ) (z : V n) : EuclideanSpace ℂ (Fin n) :=
+  WithLp.toLp 2 (fun k => dz (e k) (fun y => (f y:ℂ)) z)
+
+def holGradientSquare (f : V n → ℝ) (z : V n) : ℝ := ‖holGradient f z‖^2
+
+def gradientCross (f g : V n → ℝ) (z : V n) : ℝ :=
+  2*(inner ℂ (holGradient f z) (holGradient g z)).re
+
+lemma holGradientSquare_eq (f : V n → ℝ) (z : V n) :
+    holGradientSquare f z = ∑ k, ‖dz (e k) (fun y => (f y:ℂ)) z‖^2 := by
+  rw [holGradientSquare,PiLp.norm_sq_eq_of_L2]
+  rfl
+
+lemma gradientCross_sq (f g : V n → ℝ) (z : V n) :
+    (gradientCross f g z)^2 ≤ 4*holGradientSquare f z*holGradientSquare g z := by
+  have hi := norm_inner_le_norm (𝕜 := ℂ) (holGradient f z) (holGradient g z)
+  have hr := Complex.abs_re_le_norm (inner ℂ (holGradient f z) (holGradient g z))
+  have hs := (sq_le_sq₀ (abs_nonneg _) (mul_nonneg (norm_nonneg _) (norm_nonneg _))).mpr (hr.trans hi)
+  simp only [sq_abs,mul_pow] at hs
+  unfold gradientCross holGradientSquare
+  nlinarith
+
+lemma dzbar_real {f : V n → ℝ} {z : V n} (hf : ContDiffAt ℝ ∞ f z) (v : V n) :
+    dzbar v (fun y => (f y:ℂ)) z = star (dz v (fun y => (f y:ℂ)) z) := by
+  have hd : DifferentiableAt ℝ (fun y => (f y:ℂ)) z :=
+    (Complex.ofRealCLM.contDiff.contDiffAt.comp z hf).differentiableAt (by simp)
+  have hh := dzbar_conj hd v
+  simpa only [Complex.star_def,Complex.conj_ofReal] using hh
+
+lemma potentialMatrix_trace_mul {f g : V n → ℝ} {z : V n}
+    (hf : ContDiffAt ℝ ∞ f z) (hg : ContDiffAt ℝ ∞ g z) :
+    (PotentialKaehler.potentialMatrix (fun y => f y*g y) z).trace.re =
+      f z*(PotentialKaehler.potentialMatrix g z).trace.re +
+      g z*(PotentialKaehler.potentialMatrix f z).trace.re + gradientCross f g z := by
+  have hfc : ContDiffAt ℝ ∞ (fun y => (f y:ℂ)) z := Complex.ofRealCLM.contDiff.contDiffAt.comp z hf
+  have hgc : ContDiffAt ℝ ∞ (fun y => (g y:ℂ)) z := Complex.ofRealCLM.contDiff.contDiffAt.comp z hg
+  have hd (k : Fin n) : dz (e k) (fun y => ((f y*g y:ℝ):ℂ)) =ᶠ[𝓝 z]
+      (fun y => dz (e k) (fun x => (f x:ℂ)) y*(g y:ℂ)+(f y:ℂ)*dz (e k) (fun x => (g x:ℂ)) y) := by
+    filter_upwards [(hfc.of_le (show (1 : WithTop ℕ∞) ≤ ∞ by simp)).eventually (by simp),
+      (hgc.of_le (show (1 : WithTop ℕ∞) ≤ ∞ by simp)).eventually (by simp)] with y hyF hyG
+    simp only [Complex.ofReal_mul]
+    exact wderiv_mul (hyF.differentiableAt (by simp)) (hyG.differentiableAt (by simp)) _ _
+  have hdd (k : Fin n) : (PotentialKaehler.potentialMatrix (fun y => f y*g y) z) k k =
+      (g z:ℂ)*(PotentialKaehler.potentialMatrix f z) k k +
+      (f z:ℂ)*(PotentialKaehler.potentialMatrix g z) k k +
+      dz (e k) (fun y => (f y:ℂ)) z*star (dz (e k) (fun y => (g y:ℂ)) z)+
+      star (dz (e k) (fun y => (f y:ℂ)) z)*dz (e k) (fun y => (g y:ℂ)) z := by
+    rw [potentialMatrix_eq_dzbar_dz (hf.mul hg)]
+    change wderiv Complex.I (e k) (dz (e k) _) z = _
+    rw [wderiv_congr (hd k)]
+    unfold dz
+    rw [wderiv_add
+      (((wderiv_smooth hfc (-Complex.I) (e k)).mul hgc).differentiableAt (by simp))
+      ((hfc.mul (wderiv_smooth hgc (-Complex.I) (e k))).differentiableAt (by simp)),
+      wderiv_mul ((wderiv_smooth hfc (-Complex.I) (e k)).differentiableAt (by simp)) (hgc.differentiableAt (by simp)),
+      wderiv_mul (hfc.differentiableAt (by simp)) ((wderiv_smooth hgc (-Complex.I) (e k)).differentiableAt (by simp))]
+    change dzbar (e k) (dz (e k) _) z * _ + dz (e k) _ z * dzbar (e k) _ z +
+      (dzbar (e k) _ z * dz (e k) _ z + _ * dzbar (e k) (dz (e k) _) z) = _
+    have hef : dzbar (e k) (dz (e k) (fun y => (f y:ℂ))) z = PotentialKaehler.potentialMatrix f z k k :=
+      (potentialMatrix_eq_dzbar_dz hf k k).symm
+    have heg : dzbar (e k) (dz (e k) (fun y => (g y:ℂ))) z = PotentialKaehler.potentialMatrix g z k k :=
+      (potentialMatrix_eq_dzbar_dz hg k k).symm
+    rw [dzbar_real hf,dzbar_real hg,hef,heg]
+    change _ = (g z:ℂ)*_ + (f z:ℂ)*_ + dz (e k) _ z*star (dz (e k) _ z) + star (dz (e k) _ z)*dz (e k) _ z
+    ring
+  simp only [Matrix.trace,Matrix.diag_apply,hdd,Complex.re_sum,Complex.add_re,
+    Complex.mul_re,Complex.ofReal_re,Complex.ofReal_im,zero_mul,sub_zero,
+    Finset.sum_add_distrib,← Finset.mul_sum,gradientCross,holGradient,
+    PiLp.inner_apply,RCLike.inner_apply,Complex.star_def]
+  simp only [Complex.conj_re,Complex.conj_im]
+  simp only [mul_comm]
+  ring_nf
+end KaehlerCalculus
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/SchwartzConjugation.lean b/lean/OAI/Geometry/PrescribedPotential/SchwartzConjugation.lean
new file mode 100644
index 0000000000000000000000000000000000000000..71237509b9be445d39d9b264b22dbcc181a79fe4
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/SchwartzConjugation.lean
@@ -0,0 +1,67 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CircleRadialCalculus
+import OAI.Geometry.PrescribedPotential.IntegerBounds
+
+namespace OAI
+
+/-! Schwartz Conjugation. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory LineDeriv
+open scoped ContDiff SchwartzMap Classical
+namespace SobolevChart
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+
+def schwartzConj : 𝓢(E,ℂ) →L[ℝ] 𝓢(E,ℂ) := SchwartzMap.postcompCLM Complex.conjCLE.toContinuousLinearMap
+
+@[simp] lemma schwartzConj_apply (f : 𝓢(E,ℂ)) (x : E) : schwartzConj f x = star (f x) := rfl
+
+lemma schwartzConj_deriv (v : E) (f : 𝓢(E,ℂ)) :
+    (∂_{v} (schwartzConj f) : 𝓢(E,ℂ)) = schwartzConj (∂_{v} f) := by
+  ext x
+  simp only [SchwartzMap.lineDerivOp_apply_eq_fderiv,schwartzConj_apply]
+  exact congrArg (fun L : E →L[ℝ] ℂ => L v)
+    (Complex.conjCLE.toContinuousLinearMap.hasFDerivAt.comp x
+      ((f.smooth ⊤).differentiable (by simp) x).hasFDerivAt).fderiv
+
+lemma schwartzConj_word (ws : List E) (f : 𝓢(E,ℂ)) :
+    schwartzWord ws (schwartzConj f) = schwartzConj (schwartzWord ws f) := by
+  induction ws generalizing f with
+  | nil => rfl
+  | cons v ws ih =>
+    change ∂_{v} (schwartzWord ws (schwartzConj f)) = schwartzConj (∂_{v} (schwartzWord ws f))
+    rw [ih,schwartzConj_deriv]
+
+variable [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+
+lemma schwartzConj_norm_zero (f : 𝓢(E,ℂ)) :
+    ‖schwartzCoord 0 (schwartzConj f)‖ = ‖schwartzCoord 0 f‖ := by
+  have he (u : 𝓢(E,ℂ)) : schwartzCoord 0 u = u.toLp 2 volume := by
+    apply l2_injective
+    simpa [realize] using realize_schwartzCoord 0 u
+  rw [he,he,SchwartzMap.norm_toLp,SchwartzMap.norm_toLp]
+  congr 1
+  apply eLpNorm_congr_norm_ae (schwartzConj f).continuous.aestronglyMeasurable
+    f.continuous.aestronglyMeasurable
+  exact Filter.Eventually.of_forall (fun x => by simp)
+
+lemma schwartzConj_integer_bound (k : ℕ) : CoreBound (k : ℝ) (k : ℝ) (schwartzConj (E := E)) := by
+  have hh := coreBound_of_words_integer k (s := (k : ℝ)) (t := 0) (schwartzConj (E := E)) (fun ws hws => by
+    obtain ⟨C,hC,hb⟩ := coreBound_word ws (s := (k : ℝ)) (t := 0) (by
+      have hh : (ws.length : ℝ) ≤ k := by exact_mod_cast hws
+      linarith)
+    refine ⟨C,hC,fun f => ?_⟩
+    dsimp only
+    rw [schwartzConj_word,schwartzConj_norm_zero]
+    exact hb f)
+  simpa using hh
+end SobolevChart
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/SchwartzFamilyBounds.lean b/lean/OAI/Geometry/PrescribedPotential/SchwartzFamilyBounds.lean
new file mode 100644
index 0000000000000000000000000000000000000000..cd8c70feb89da366243cea0ea55b267d6772e9b6
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/SchwartzFamilyBounds.lean
@@ -0,0 +1,62 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.LocalRegularity
+
+namespace OAI
+
+/-! Schwartz Family Bounds. -/
+
+section
+
+noncomputable section
+open Set Filter Topology MeasureTheory TemperedDistribution
+open scoped SchwartzMap ContDiff
+namespace SobolevChart
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+
+def schwartzCoordCLM (s : ℝ) : 𝓢(E,ℂ) →L[ℂ] L2 E :=
+  (SchwartzMap.toLpCLM ℂ ℂ 2 volume) ∘L
+    SchwartzMap.fourierMultiplierCLM ℂ (fun x : E => (((1+‖x‖^2)^(s/2) : ℝ) : ℂ))
+
+lemma schwartzCoordCLM_apply (s : ℝ) (f : 𝓢(E,ℂ)) :
+    schwartzCoordCLM s f = schwartzCoord s f := by
+  apply l2_injective
+  change ((SchwartzMap.fourierMultiplierCLM ℂ (fun x : E => (((1+‖x‖^2)^(s/2) : ℝ) : ℂ)) f).toLp 2 : 𝓢'(E,ℂ)) = _
+  rw [Lp.toTemperedDistribution_toLp_eq,Lp.toTemperedDistributionCLM_apply,← bessel_realize s,realize_schwartzCoord]
+  exact (fourierMultiplierCLM_toTemperedDistributionCLM_eq (by fun_prop) f).symm
+
+omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] in
+lemma schwartz_family_bounded {I : Type*} (f : I → 𝓢(E,ℂ)) {K : Set E}
+    (hK : IsCompact K) (hsupp : ∀ i, tsupport (f i : E → ℂ) ⊆ K)
+    (hjet : ∀ m : ℕ, ∃ C : ℝ, ∀ i x, x ∈ K → ‖iteratedFDeriv ℝ m (f i) x‖ ≤ C) :
+    Bornology.IsVonNBounded ℂ (Set.range f) := by
+  apply (schwartz_withSeminorms ℂ E ℂ).isVonNBounded_iff_seminorm_bounded.mpr
+  rintro ⟨a,m⟩
+  obtain ⟨R,hR,hRK⟩ := hK.isBounded.exists_pos_norm_le
+  obtain ⟨C,hC⟩ := hjet m
+  let B := R^a * max 0 C
+  have hB : 0 ≤ B := mul_nonneg (pow_nonneg hR.le a) (le_max_left _ _)
+  refine ⟨B+1,by linarith,?_⟩
+  rintro _ ⟨i,rfl⟩
+  apply lt_of_le_of_lt (SchwartzMap.seminorm_le_bound ℂ a m (f i) hB ?_) (by linarith)
+  intro x
+  by_cases hx : x ∈ K
+  · exact mul_le_mul (pow_le_pow_left₀ (norm_nonneg x) (hRK x hx) a)
+      ((hC i x hx).trans (le_max_right _ _)) (norm_nonneg _) (pow_nonneg hR.le a)
+  · have hz : iteratedFDeriv ℝ m (f i) x = 0 := Function.notMem_support.mp
+      (fun ha => hx (hsupp i (support_iteratedFDeriv_subset m ha)))
+    simpa [hz] using hB
+
+lemma schwartzCoord_family_bound {I : Type*} (f : I → 𝓢(E,ℂ)) {K : Set E}
+    (hK : IsCompact K) (hsupp : ∀ i, tsupport (f i : E → ℂ) ⊆ K)
+    (hjet : ∀ m : ℕ, ∃ C : ℝ, ∀ i x, x ∈ K → ‖iteratedFDeriv ℝ m (f i) x‖ ≤ C)
+    (s : ℝ) : ∃ C : ℝ, ∀ i, ‖schwartzCoord s (f i)‖ ≤ C := by
+  have hb := (schwartz_family_bounded f hK hsupp hjet).image (schwartzCoordCLM s)
+  obtain ⟨C,hC⟩ := (NormedSpace.isVonNBounded_iff' ℂ).mp hb
+  exact ⟨C,fun i => by simpa only [schwartzCoordCLM_apply] using hC _ ⟨f i,⟨i,rfl⟩,rfl⟩⟩
+end SobolevChart
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/SecondJetComposition.lean b/lean/OAI/Geometry/PrescribedPotential/SecondJetComposition.lean
new file mode 100644
index 0000000000000000000000000000000000000000..965ef29a0354bde1f5d311734aadb9c035df6d7a
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/SecondJetComposition.lean
@@ -0,0 +1,82 @@
+import Mathlib
+
+namespace OAI
+
+/-! Second Jet Composition. -/
+
+section
+
+ 
+
+noncomputable section
+open Finset
+open scoped ContDiff
+namespace HigherJet
+lemma positive_list_sum_two {l : List ℕ} (hp : ∀ j ∈ l, 0 < j) (hs : l.sum = 2) :
+    l = [2] ∨ l = [1,1] := by
+  cases l with
+  | nil => simp at hs
+  | cons a l =>
+    have ha := hp a (by simp)
+    cases l with
+    | nil => simp at hs; simp [hs]
+    | cons b l =>
+      have hb := hp b (by simp)
+      cases l with
+      | nil =>
+        simp only [List.sum_cons,List.sum_nil,Nat.add_zero] at hs
+        have : a=1 ∧ b=1 := by omega
+        simp [this.1,this.2]
+      | cons c l =>
+        have hc := hp c (by simp)
+        simp only [List.sum_cons] at hs
+        omega
+
+lemma OrderedFinpartition.prod_second_bound (c : OrderedFinpartition 2)
+    (a : ℕ → ℝ) (ha : ∀ i, 0 ≤ a i) :
+    ∏ i, a (c.partSize i) ≤ a 2+(a 1)^2 := by
+  have hsum : (List.ofFn c.partSize).sum = 2 := by
+    rw [List.sum_ofFn]
+    simpa using c.sum_sigma_eq_sum (fun _ => (1 : ℕ))
+  have hpos : ∀ j ∈ List.ofFn c.partSize, 0 < j := by
+    intro j hj
+    obtain ⟨i,rfl⟩ := List.mem_ofFn.mp hj
+    exact c.partSize_pos i
+  have he : ∏ i, a (c.partSize i) = ((List.ofFn c.partSize).map a).prod := by
+    rw [List.map_ofFn,List.prod_ofFn]
+    rfl
+  rw [he]
+  obtain h | h := positive_list_sum_two hpos hsum
+  · simp only [h,List.map_cons,List.map_nil,List.prod_cons,List.prod_nil,mul_one]
+    exact le_add_of_nonneg_right (sq_nonneg _)
+  · simp only [h,List.map_cons,List.map_nil,List.prod_cons,List.prod_nil,mul_one]
+    nlinarith only [ha 2]
+
+variable {E F G : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  [NormedAddCommGroup F] [NormedSpace ℝ F]
+  [NormedAddCommGroup G] [NormedSpace ℝ G]
+lemma composition_second_jet_bound {f : E → F} {g : F → G} {x : E}
+    (hf : ContDiffAt ℝ ∞ f x) (hg : ContDiffAt ℝ ∞ g (f x))
+    {C : ℝ} (hC : 0 ≤ C)
+    (hgjet : ∀ j, j ≤ 2 → ‖iteratedFDeriv ℝ j g (f x)‖ ≤ C) :
+    ‖iteratedFDeriv ℝ 2 (g ∘ f) x‖ ≤
+      (Fintype.card (OrderedFinpartition 2) : ℝ)*C*
+        (‖iteratedFDeriv ℝ 2 f x‖+‖iteratedFDeriv ℝ 1 f x‖^2) := by
+  rw [iteratedFDeriv_comp (i := 2) hg hf (by exact WithTop.coe_le_coe.mpr le_top),FormalMultilinearSeries.taylorComp]
+  calc
+    _ ≤ ∑ c : OrderedFinpartition 2, ‖c.compAlongOrderedFinpartition
+      (iteratedFDeriv ℝ c.length g (f x)) (fun i => iteratedFDeriv ℝ (c.partSize i) f x)‖ := norm_sum_le _ _
+    _ ≤ ∑ _c : OrderedFinpartition 2, C*(‖iteratedFDeriv ℝ 2 f x‖+‖iteratedFDeriv ℝ 1 f x‖^2) := by
+      apply Finset.sum_le_sum
+      intro c _
+      exact c.norm_compAlongOrderedFinpartition_le _ _ |>.trans
+        (mul_le_mul (hgjet _ c.length_le)
+          (OrderedFinpartition.prod_second_bound c (fun j => ‖iteratedFDeriv ℝ j f x‖) (fun j => norm_nonneg _))
+          (Finset.prod_nonneg (fun _ _ => norm_nonneg _)) hC)
+    _ = _ := by simp [mul_assoc]
+end HigherJet
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/SmoothChartInverse.lean b/lean/OAI/Geometry/PrescribedPotential/SmoothChartInverse.lean
new file mode 100644
index 0000000000000000000000000000000000000000..c0d1f5160be99ccdabfde533f160eb3bbff0f5d0
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/SmoothChartInverse.lean
@@ -0,0 +1,160 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.ChartParametrix
+import OAI.Geometry.PrescribedPotential.LocalRegularity
+import OAI.Geometry.PrescribedPotential.SmoothDistributions
+
+namespace OAI
+
+/-! Smooth Chart Inverse. -/
+
+section
+
+ 
+
+noncomputable section
+open Filter Set Topology MeasureTheory TemperedDistribution LineDeriv
+open scoped SchwartzMap BoundedContinuousFunction ComplexOrder MatrixOrder Matrix.Norms.Elementwise ContDiff
+
+namespace SobolevChart
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+
+lemma temperateDistribution_sum {ι : Type*} [Fintype ι] (f : ι → E → ℂ)
+    (hf : ∀ i, (f i).HasTemperateGrowth) :
+    (Function.HasTemperateGrowth.sum (s := Finset.univ) (fun i _ => hf i)).toTemperedDistribution volume =
+      ∑ i, (hf i).toTemperedDistribution volume := by
+  simp only [temperateDistribution_eq_smul_volume]
+  rw [smulLeftCLM_sum (fun i _ => hf i)]
+  simp
+
+lemma temperateDistribution_mul {f g : E → ℂ} (hf : f.HasTemperateGrowth)
+    (hg : g.HasTemperateGrowth) :
+    (hf.mul hg).toTemperedDistribution volume =
+      smulLeftCLM ℂ f (hg.toTemperedDistribution volume) := by
+  simp only [temperateDistribution_eq_smul_volume]
+  rw [smulLeftCLM_smulLeftCLM_apply hg hf, mul_comm]
+
+end SobolevChart
+
+namespace MetricLocalization
+open EllipticKernel FrozenPoisson SobolevChart
+variable {n : ℕ}
+
+def classicalDifferential (c : BasisIndex n → BasisIndex n → EC n →ᵇ ℂ)
+    (f : EC n → ℂ) : EC n → ℂ := fun x =>
+  ∑ k, ∑ l, c k l x * directionDerivative (stdOrthonormalBasis ℝ (EC n) k)
+    (directionDerivative (stdOrthonormalBasis ℝ (EC n) l) f) x
+
+lemma classicalDifferential_temperate
+    (c : BasisIndex n → BasisIndex n → EC n →ᵇ ℂ)
+    (hc : ∀ k l, (c k l : EC n → ℂ).HasTemperateGrowth)
+    {f : EC n → ℂ} (hf : f.HasTemperateGrowth) :
+    (classicalDifferential c f).HasTemperateGrowth := by
+  apply Function.HasTemperateGrowth.sum
+  intro k _
+  apply Function.HasTemperateGrowth.sum
+  intro l _
+  exact (hc k l).mul (temperate_directionDerivative _ (temperate_directionDerivative _ hf))
+
+lemma classicalDifferential_distribution
+    (c : BasisIndex n → BasisIndex n → EC n →ᵇ ℂ)
+    (hc : ∀ k l, (c k l : EC n → ℂ).HasTemperateGrowth)
+    {f : EC n → ℂ} (hf : f.HasTemperateGrowth) :
+    (classicalDifferential_temperate c hc hf).toTemperedDistribution volume =
+      chartDifferential c (hf.toTemperedDistribution volume) := by
+  let b := stdOrthonormalBasis ℝ (EC n)
+  have hg k l := (hc k l).mul
+    (temperate_directionDerivative (b k) (temperate_directionDerivative (b l) hf))
+  change (Function.HasTemperateGrowth.sum
+    (s := Finset.univ) (fun k _ => Function.HasTemperateGrowth.sum
+      (s := Finset.univ) (fun l _ => hg k l))).toTemperedDistribution volume = _
+  rw [temperateDistribution_sum
+    (fun k x => ∑ l, c k l x * directionDerivative (b k) (directionDerivative (b l) f) x)
+    (fun k => Function.HasTemperateGrowth.sum (fun l _ => hg k l))]
+  simp only [chartDifferential, sum_apply, ContinuousLinearMap.comp_apply]
+  apply Finset.sum_congr rfl
+  intro k _
+  rw [temperateDistribution_sum
+    (fun l x => c k l x * directionDerivative (b k) (directionDerivative (b l) f) x)
+    (hg k)]
+  apply Finset.sum_congr rfl
+  intro l _
+  rw [temperateDistribution_mul (hc k l)
+    (temperate_directionDerivative (b k) (temperate_directionDerivative (b l) hf)),
+    temperateDistribution_derivative (b k) (temperate_directionDerivative (b l) hf),
+    temperateDistribution_derivative (b l) hf]
+  rfl
+
+ 
+
+theorem extended_smooth_solution
+    (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (a : SmoothCoefficients (BasisIndex n) (EC n))
+    (hsmall : perturbationBound (stdOrthonormalBasis ℝ (EC n)) (coefficientBCF a) *
+      ellipticBound H hH < 1) (f : 𝓢(EC n, ℂ)) :
+    ∃ u : EC n →ᵇ ℂ, (u : EC n → ℂ).HasTemperateGrowth ∧
+      (fun x => u x - classicalDifferential (extendedCoefficient H (coefficientBCF a)) u x) = f := by
+  let v := stdOrthonormalBasis ℝ (EC n)
+  let R := localResolvent H hH v (coefficientBCF a) hsmall
+  obtain ⟨u, hu, hue⟩ := all_even_smooth_representative
+    (localResolvent_schwartz_memSobolev H hH v a hsmall f)
+  have ha k l : (coefficientBCF a k l : EC n → ℂ).HasTemperateGrowth :=
+    (a k l).hasTemperateGrowth
+  have hc := extendedCoefficient_temperate H (coefficientBCF a) ha
+  refine ⟨u, hu, ?_⟩
+  apply temperateDistribution_injective (hu.sub (classicalDifferential_temperate _ hc hu))
+    f.hasTemperateGrowth
+  rw [temperateDistribution_sub hu (classicalDifferential_temperate _ hc hu),
+    classicalDifferential_distribution _ hc hu,
+    temperateDistribution_bounded u hu, hue, temperateDistribution_schwartz]
+  rw [chartDifferential_extension H (coefficientBCF a) ha]
+  rw [← sub_sub]
+  have he := localResolvent_equation H hH v (coefficientBCF a) hsmall (schwartzCoord 0 f)
+  have hz : realize 0 (schwartzCoord 0 f) = (schwartzCoord 0 f : 𝓢'(EC n, ℂ)) := by
+    simp [realize]
+  rw [← hz, realize_schwartzCoord] at he
+  exact he
+
+ 
+
+theorem germ_smooth_solution {U : Set (EC n)} (hU : IsOpen U)
+    (H : EC n → Matrix (Fin n) (Fin n) ℂ) (hH : ContDiffOn ℝ ∞ H U)
+    (hh : ∀ y ∈ U, (H y).PosDef) {x : EC n} (hx : x ∈ U) :
+    ∃ V : Set (EC n), IsOpen V ∧ x ∈ V ∧ V ⊆ U ∧
+      ∀ f : 𝓢(EC n, ℂ), ∃ u : EC n →ᵇ ℂ,
+        (u : EC n → ℂ).HasTemperateGrowth ∧ ∀ y ∈ V,
+          u y - (∑ k, ∑ l, (traceBilin (H y)
+            (rankTwo (stdOrthonormalBasis ℝ (EC n) k)
+              (stdOrthonormalBasis ℝ (EC n) l)) : ℂ) *
+            directionDerivative (stdOrthonormalBasis ℝ (EC n) k)
+              (directionDerivative (stdOrthonormalBasis ℝ (EC n) l) u) y) = f y := by
+  let b := stdOrthonormalBasis ℝ (EC n)
+  obtain ⟨a, hat, hac, has, hag⟩ := local_coefficient_extension hU H hH hh hx b
+  let a' : SmoothCoefficients (BasisIndex n) (EC n) :=
+    fun k l => (hac k l).toSchwartzMap (hat k l).1
+  have hae : coefficientBCF a' = a := by
+    funext k l
+    ext y
+    rfl
+  have has' : perturbationBound b (coefficientBCF a') * ellipticBound (H x) (hh x hx) < 1 := by
+    rw [hae]
+    exact has
+  obtain ⟨V, hVsub, hVo, hxV⟩ := mem_nhds_iff.mp (inter_mem (hU.mem_nhds hx) hag)
+  refine ⟨V, hVo, hxV, fun y hy => (hVsub hy).1, fun f => ?_⟩
+  obtain ⟨u, hu, he⟩ := extended_smooth_solution (H x) (hh x hx) a' has' f
+  refine ⟨u, hu, fun y hy => ?_⟩
+  have hc k l : extendedCoefficient (H x) (coefficientBCF a') k l y =
+      (traceBilin (H y) (rankTwo (b k) (b l)) : ℂ) := by
+    rw [hae]
+    change (traceBilin (H x) (rankTwo (b k) (b l)) : ℂ) + a k l y = _
+    rw [(hVsub hy).2 k l]
+    ring
+  have hey := congrFun he y
+  simpa only [classicalDifferential, hc] using hey
+
+end MetricLocalization
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/SmoothDistributions.lean b/lean/OAI/Geometry/PrescribedPotential/SmoothDistributions.lean
new file mode 100644
index 0000000000000000000000000000000000000000..f93db3f1929104fda6b816b79e7d67f01af9a6ca
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/SmoothDistributions.lean
@@ -0,0 +1,114 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.SobolevEmbedding
+
+namespace OAI
+
+/-! Smooth Distributions. -/
+
+section
+
+ 
+noncomputable section
+open MeasureTheory FourierTransform TemperedDistribution LineDeriv
+open scoped SchwartzMap BoundedContinuousFunction ContDiff
+
+namespace SobolevChart
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+
+omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] in
+lemma temperate_fderiv {f : E → ℂ} (hf : f.HasTemperateGrowth) :
+    (fderiv ℝ f).HasTemperateGrowth := by
+  refine ⟨hf.1.fderiv_right (by simp), fun n => ?_⟩
+  simpa only [norm_iteratedFDeriv_fderiv] using hf.2 (n + 1)
+
+def directionDerivative (v : E) (f : E → ℂ) : E → ℂ := fun x => fderiv ℝ f x v
+
+omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] in
+lemma temperate_directionDerivative (v : E) {f : E → ℂ} (hf : f.HasTemperateGrowth) :
+    (directionDerivative v f).HasTemperateGrowth := by
+  have hd := temperate_fderiv hf
+  unfold directionDerivative
+  exact (ContinuousLinearMap.apply ℝ ℂ v).hasTemperateGrowth.comp hd
+
+lemma temperateDistribution_eq_smul_volume {f : E → ℂ} (hf : f.HasTemperateGrowth) :
+    hf.toTemperedDistribution volume = smulLeftCLM ℂ f (volume.toTemperedDistribution) := by
+  ext φ
+  simp only [Function.HasTemperateGrowth.toTemperedDistribution_apply,
+    smulLeftCLM_apply_apply, Measure.toTemperedDistribution_apply,
+    SchwartzMap.smulLeftCLM_apply hf]
+  simp only [smul_eq_mul, mul_comm]
+
+lemma temperateDistribution_sub {f g : E → ℂ} (hf : f.HasTemperateGrowth)
+    (hg : g.HasTemperateGrowth) :
+    (hf.sub hg).toTemperedDistribution volume =
+      hf.toTemperedDistribution volume - hg.toTemperedDistribution volume := by
+  simp only [temperateDistribution_eq_smul_volume]
+  rw [smulLeftCLM_sub hf hg]
+  rfl
+
+lemma temperateDistribution_schwartz (f : 𝓢(E, ℂ)) :
+    f.hasTemperateGrowth.toTemperedDistribution volume = (f : 𝓢'(E, ℂ)) := by
+  ext φ
+  simp
+
+lemma temperateDistribution_bounded (f : E →ᵇ ℂ) (hf : (f : E → ℂ).HasTemperateGrowth) :
+    hf.toTemperedDistribution volume = boundedDistribution f := by
+  ext φ
+  rw [Function.HasTemperateGrowth.toTemperedDistribution_apply volume hf φ, boundedDistribution_apply]
+
+lemma temperate_schwartz_integrable {f : E → ℂ} (hf : f.HasTemperateGrowth)
+    (φ : 𝓢(E, ℂ)) : Integrable (fun x => φ x * f x) := by
+  simpa only [SchwartzMap.smulLeftCLM_apply hf, smul_eq_mul, mul_comm] using
+    (SchwartzMap.smulLeftCLM ℂ f φ).integrable (μ := volume)
+
+ 
+
+lemma temperateDistribution_derivative (v : E) {f : E → ℂ} (hf : f.HasTemperateGrowth) :
+    (temperate_directionDerivative v hf).toTemperedDistribution volume =
+      ∂_{v} (hf.toTemperedDistribution volume) := by
+  ext φ
+  simp only [Function.HasTemperateGrowth.toTemperedDistribution_apply,
+    TemperedDistribution.lineDerivOp_apply_apply, map_neg]
+  change (∫ x, φ x * directionDerivative v f x) = -(∫ x, (∂_{v} φ) x * f x)
+  apply integral_bilinear_hasLineDerivAt_right_eq_neg_left_of_integrable
+    (B := ContinuousLinearMap.mul ℝ ℂ) (v := v)
+    (temperate_schwartz_integrable hf (∂_{v} φ))
+    (temperate_schwartz_integrable (temperate_directionDerivative v hf) φ)
+    (temperate_schwartz_integrable hf φ)
+  · intro x _
+    simpa only [SchwartzMap.lineDerivOp_apply_eq_fderiv] using
+      (SchwartzMap.hasFDerivAt φ x).hasLineDerivAt v
+  · intro x _
+    exact ((hf.1.differentiable (by simp)) x).hasFDerivAt.hasLineDerivAt v
+
+lemma temperateDistribution_eq_zero_iff {f : E → ℂ} (hf : f.HasTemperateGrowth) :
+    hf.toTemperedDistribution volume = 0 ↔ f = 0 := by
+  constructor
+  · intro h
+    apply (hf.1.continuous.ae_eq_iff_eq volume continuous_zero).mp
+    apply ae_eq_zero_of_integral_contDiff_smul_eq_zero (μ := (volume : Measure E))
+      hf.1.continuous.locallyIntegrable
+    intro g hg hc
+    let φ : 𝓢(E, ℂ) := (hc.comp_left (show Complex.ofReal 0 = 0 by rfl)).toSchwartzMap
+      (Complex.ofRealCLM.contDiff.comp hg)
+    have he := congrArg (fun t : 𝓢'(E, ℂ) => t φ) h
+    change (∫ x, ((g x : ℝ) : ℂ) • f x) = 0 at he
+    simpa only [Complex.real_smul, smul_eq_mul] using he
+  · rintro rfl
+    ext φ
+    simp
+
+lemma temperateDistribution_injective {f g : E → ℂ} (hf : f.HasTemperateGrowth)
+    (hg : g.HasTemperateGrowth)
+    (h : hf.toTemperedDistribution volume = hg.toTemperedDistribution volume) : f = g := by
+  apply sub_eq_zero.mp
+  apply (temperateDistribution_eq_zero_iff (hf.sub hg)).mp
+  rw [temperateDistribution_sub hf hg, h, sub_self]
+
+end SobolevChart
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/SmoothLocalPath.lean b/lean/OAI/Geometry/PrescribedPotential/SmoothLocalPath.lean
new file mode 100644
index 0000000000000000000000000000000000000000..9fb8baa4a59789b3e7597b73fa4711636b34047a
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/SmoothLocalPath.lean
@@ -0,0 +1,111 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.PatchCutoffs
+import OAI.Geometry.PrescribedPotential.RealSobolev
+import OAI.Geometry.PrescribedPotential.SobolevExponential
+import OAI.Geometry.PrescribedPotential.FixedScaleBootstrap
+import OAI.Geometry.PrescribedPotential.PositiveOpenness
+import OAI.Geometry.PrescribedPotential.TwoScaleRegularity
+
+namespace OAI
+
+/-! Smooth Local Path. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology
+open scoped ContDiff Classical
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  [ConnectedSpace X] {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace RealSmooth
+
+def exponentialForcing (F : RealSmooth A) (t : ℝ) : SmoothRealFunction A where
+  value := fun x => Real.exp (t * F.source.value x)
+  smooth i := by
+    exact (contDiffOn_const.mul (F.source.smooth i)).exp
+
+end RealSmooth
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+local instance smPathNG (s : ℝ) : NormedAddCommGroup (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedAddCommGroup
+local instance smPathNS (s : ℝ) : NormedSpace ℝ (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedSpace
+local instance smPathTG (s : ℝ) : IsTopologicalAddGroup (D.localizers.RealSobolev s) :=
+  Submodule.isTopologicalAddGroup _
+local instance smPathCS (s : ℝ) : ContinuousSMul ℝ (D.localizers.RealSobolev s) :=
+  SMulMemClass.continuousSMul _
+
+lemma exponentialDensity_embed (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (F : RealSmooth A) (t : ℝ) :
+    D.exponentialDensity k hk F t =
+      D.localizers.realEmbed (k : ℝ) (RealSmooth.ofReal (F.exponentialForcing t)) := by
+  have hs : (Module.finrank ℝ (EC d) : ℝ) < 2*(k : ℝ) := by
+    have hh : (Module.finrank ℝ (EC d) : ℝ) < (k : ℝ) := by exact_mod_cast hk
+    linarith [Nat.cast_nonneg (α := ℝ) k]
+  apply Subtype.ext
+  apply D.localizers.strong_injective (k : ℝ) hs
+  ext x
+  rw [D.exponentialDensity_strong]
+  change _ = D.localizers.strong (k : ℝ) (D.localizers.embed (k : ℝ) _) x
+  rw [D.localizers.strong_embed _ hs]
+  rfl
+
+include D in
+ 
+
+theorem smooth_path_eventually_solvable (F : RealSmooth A) (x₀ : X) :
+    ∀ᶠ t : ℝ in 𝓝 0, ∃ (φ : SmoothRealFunction A) (b : ℝ),
+      g.PositivePotential φ ∧ φ.value x₀ = 0 ∧
+      ∀ x, (g.potentialDensity φ).value x = Real.exp (t * F.source.value x + b) := by
+  let s := Module.finrank ℝ (EC d) + 1
+  have hs : Module.finrank ℝ (EC d) < s := by dsimp [s]; omega
+  have hk : Module.finrank ℝ (EC d) < s+1 := by omega
+  let U : Set (D.localizers.RealSobolev (((s+1 : ℕ) : ℝ)+2) × ℝ) := {z |
+    (∀ R : RealSmooth A, D.realVolume (s+1) hk z.1 = D.localizers.realEmbed ((s+1 : ℕ) : ℝ) R →
+      ∃ φ : RealSmooth A, D.localizers.realEmbed (((s+1 : ℕ) : ℝ)+2) φ = z.1) ∧
+    D.WeakPositive (s+1) z.1}
+  have hU : U ∈ 𝓝 0 := by
+    have ht : Tendsto (Prod.fst : D.localizers.RealSobolev (((s+1 : ℕ) : ℝ)+2) × ℝ → _)
+        (𝓝 0) (𝓝 0) := continuous_fst.tendsto 0
+    exact ht ((D.fixed_spatial_smoothness s hs x₀).and (D.weakPositive_nhds (s+1) hk))
+  have ht := (D.exponentialDensity_continuous (s+1) hk F).tendsto 0
+  rw [D.exponentialDensity_zero] at ht
+  filter_upwards [ht.eventually (D.volume_eventually_solvable (s+1) hk x₀ hU)] with t ht
+  obtain ⟨u,b,hu,hV,hN⟩ := ht
+  obtain ⟨hSm,hP⟩ := hu
+  have he : D.realVolume (s+1) hk u = D.localizers.realEmbed ((s+1 : ℕ) : ℝ)
+      (Real.exp b • RealSmooth.ofReal (F.exponentialForcing t)) := by
+    rw [hV,D.exponentialDensity_embed,map_smul]
+  obtain ⟨φ,hφ⟩ := hSm _ he
+  change D.localizers.realEmbed (((s+1 : ℕ) : ℝ)+2) φ = u at hφ
+  refine ⟨φ.source,b,?_,?_,?_⟩
+  · apply (D.weakPositive_embed_iff (s+1) hk φ).mp
+    rwa [hφ]
+  · rw [← hφ,D.realEvaluation_embed] at hN
+    · exact hN
+    · have hh : (Module.finrank ℝ (EC d) : ℝ) < (s : ℝ) := by exact_mod_cast hs
+      push_cast
+      linarith only [hh, Nat.cast_nonneg (α := ℝ) s]
+  · intro x
+    rw [← hφ,D.realVolume_embed,D.exponentialDensity_embed] at hV
+    have hh := congrArg (D.realEvaluation ((s+1 : ℕ) : ℝ) x) hV
+    have hpos : (Module.finrank ℝ (EC d) : ℝ) < 2*((s+1 : ℕ) : ℝ) := by
+      have hh : (Module.finrank ℝ (EC d) : ℝ) < (s : ℝ) := by exact_mod_cast hs
+      push_cast
+      linarith only [hh, Nat.cast_nonneg (α := ℝ) s]
+    rw [map_smul,D.realEvaluation_embed hpos,D.realEvaluation_embed hpos] at hh
+    change (g.potentialDensity φ.source).value x = Real.exp b * Real.exp (t * F.source.value x) at hh
+    rw [hh,Real.exp_add]
+    ring
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/SmoothPathLimit.lean b/lean/OAI/Geometry/PrescribedPotential/SmoothPathLimit.lean
new file mode 100644
index 0000000000000000000000000000000000000000..884a80a9275e859aab26897045ff2ce48f3e4c79
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/SmoothPathLimit.lean
@@ -0,0 +1,139 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.MatrixLimit
+import OAI.Geometry.PrescribedPotential.PatchCutoffs
+import OAI.Geometry.PrescribedPotential.RealSobolev
+import OAI.Geometry.PrescribedPotential.VolumePath
+import OAI.Geometry.PrescribedPotential.SmoothPathOpenness
+import OAI.Geometry.PrescribedPotential.PositiveOpenness
+import OAI.Geometry.PrescribedPotential.CompletedVolume
+import OAI.Geometry.PrescribedPotential.RealSmoothCompactness
+
+namespace OAI
+
+/-! Smooth Path Limit. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff Classical ComplexOrder
+namespace Anticanonical.SourceSmooth
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+
+lemma solvesVolumePath_density (g : KaehlerMetric A) (h : SemipositiveAnticanonicalMetric A)
+    {t b : ℝ} {φ : SmoothRealFunction A} (he : SolvesVolumePath g h t φ b) (x : X) :
+    (g.potentialDensity φ).value x = Real.exp (t * (prescribedForcing g h).value x + b) := by
+  have hh := congrArg Real.exp (he.2 x)
+  rw [g.logRatio_deform_eq_logDensity φ he.1 x,
+    Real.exp_log (g.potentialDensity_pos φ he.1 x)] at hh
+  exact hh
+
+lemma solvesVolumePath_of_density (g : KaehlerMetric A) (h : SemipositiveAnticanonicalMetric A)
+    {t b : ℝ} {φ : SmoothRealFunction A} (hp : g.PositivePotential φ)
+    (he : ∀ x, (g.potentialDensity φ).value x =
+      Real.exp (t * (prescribedForcing g h).value x + b)) : SolvesVolumePath g h t φ b := by
+  refine ⟨hp,fun x => ?_⟩
+  rw [g.logRatio_deform_eq_logDensity φ hp x,he x,Real.log_exp]
+
+end Anticanonical.SourceSmooth
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+local instance smoothLimitNG (s : ℝ) : NormedAddCommGroup (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedAddCommGroup
+local instance smoothLimitNS (s : ℝ) : NormedSpace ℝ (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedSpace
+
+lemma realMatrix_tendsto (k : ℕ) (f : ℕ → RealSmooth A) (fLim : RealSmooth A)
+    (hf : Tendsto (fun n => D.localizers.embed ((k : ℝ)+2) (f n).val) atTop
+      (𝓝 (D.localizers.embed ((k : ℝ)+2) fLim.val))) (p : ι) (x : X) :
+    Tendsto (fun n => D.weakMatrix k (D.localizers.realEmbed ((k : ℝ)+2) (f n)) p x)
+      atTop (𝓝 (D.weakMatrix k (D.localizers.realEmbed ((k : ℝ)+2) fLim) p x)) := by
+  have hr : Tendsto (fun n => D.localizers.realEmbed ((k : ℝ)+2) (f n)) atTop
+      (𝓝 (D.localizers.realEmbed ((k : ℝ)+2) fLim)) := tendsto_subtype_rng.mpr hf
+  have hc : Continuous (fun u : D.localizers.RealSobolev ((k : ℝ)+2) => D.weakMatrix k u p x) :=
+    (D.weakMatrix_continuous k p).comp (continuous_id.prodMk continuous_const)
+  exact hc.tendsto _ |>.comp hr
+
+lemma completedVolume_realSmooth_embed (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (f : RealSmooth A) :
+    D.completedVolume k hk (D.localizers.embed ((k : ℝ)+2) f.val) =
+      D.localizers.embed (k : ℝ) (Smooth.ofReal (g.potentialDensity f.source)) := by
+  rw [← f.ofReal_source,D.completedVolume_embed]
+
+lemma potentialDensity_tendsto (f : ℕ → RealSmooth A) (fLim : RealSmooth A)
+    (hf : ∀ s : ℝ, Tendsto (fun n => D.localizers.embed s (f n).val) atTop
+      (𝓝 (D.localizers.embed s fLim.val))) (x : X) :
+    Tendsto (fun n => (g.potentialDensity (f n).source).value x) atTop
+      (𝓝 ((g.potentialDensity fLim.source).value x)) := by
+  let k := Module.finrank ℝ (EC d)+1
+  have hk : Module.finrank ℝ (EC d) < k := Nat.lt_succ_self _
+  have hs : (Module.finrank ℝ (EC d) : ℝ) < 2*(k : ℝ) := by
+    have hkr : (Module.finrank ℝ (EC d) : ℝ) < (k : ℝ) := by exact_mod_cast hk
+    linarith only [hkr,Nat.cast_nonneg (α := ℝ) k]
+  have hc := (D.completedVolume_contDiff k hk).continuous.tendsto
+    (D.localizers.embed ((k : ℝ)+2) fLim.val) |>.comp (hf ((k : ℝ)+2))
+  simp only [Function.comp_def,D.completedVolume_realSmooth_embed k hk] at hc
+  have hv := D.localizers.smooth_value_tendsto (k : ℝ) hs
+    (fun n => Smooth.ofReal (g.potentialDensity (f n).source))
+    (Smooth.ofReal (g.potentialDensity fLim.source)) hc x
+  have hr := (Complex.continuous_re.tendsto
+    ((Smooth.ofReal (g.potentialDensity fLim.source)) x)).comp hv
+  simpa only [Function.comp_def,Smooth.ofReal_apply,Complex.ofReal_re] using hr
+
+lemma positivePotential_of_smooth_limit (f : ℕ → RealSmooth A) (fLim : RealSmooth A)
+    (hf : ∀ s : ℝ, Tendsto (fun n => D.localizers.embed s (f n).val) atTop
+      (𝓝 (D.localizers.embed s fLim.val)))
+    (hp : ∀ n, g.PositivePotential (f n).source)
+    (hd : ∀ x, 0 < (g.potentialDensity fLim.source).value x) : g.PositivePotential fLim.source := by
+  let k := Module.finrank ℝ (EC d)+1
+  have hk : Module.finrank ℝ (EC d) < k := Nat.lt_succ_self _
+  apply (D.weakPositive_embed_iff k hk fLim).mp
+  intro p x hx
+  have hi : x ∈ (A.chart (D.patch p).index).source := by simpa using D.patch_source p hx
+  have hdet : (D.weakMatrix k (D.localizers.realEmbed ((k : ℝ)+2) fLim) p x).det ≠ 0 := by
+    rw [D.weakMatrix_embed k hk fLim p hx]
+    intro he
+    have hh := hd x
+    rw [g.potentialDensity_quotient fLim.source (D.patch p).index hi,he,
+      Complex.zero_re,zero_div] at hh
+    exact lt_irrefl 0 hh
+  exact CompactMatrixPositivity.posDef_of_tendsto_det_ne_zero
+    (D.weakMatrix_hermitian k hk _ p hx)
+    (fun n => (D.weakPositive_embed_iff k hk (f n)).mpr (hp n) p x hx)
+    (D.realMatrix_tendsto k f fLim (hf ((k : ℝ)+2)) p x) hdet
+
+ 
+
+theorem smooth_path_limit_solution (h : SemipositiveAnticanonicalMetric A)
+    (t b : ℕ → ℝ) (tLim bLim : ℝ) (ht : Tendsto t atTop (𝓝 tLim))
+    (hb : Tendsto b atTop (𝓝 bLim)) (f : ℕ → RealSmooth A) (fLim : RealSmooth A)
+    (hf : ∀ s : ℝ, Tendsto (fun n => D.localizers.embed s (f n).val) atTop
+      (𝓝 (D.localizers.embed s fLim.val)))
+    (hsol : ∀ n, SolvesVolumePath g h (t n) (f n).source (b n)) :
+    SolvesVolumePath g h tLim fLim.source bLim := by
+  have he (x : X) : (g.potentialDensity fLim.source).value x =
+      Real.exp (tLim * (prescribedForcing g h).value x + bLim) := by
+    have hl := D.potentialDensity_tendsto f fLim hf x
+    have hn : (fun n => (g.potentialDensity (f n).source).value x) =
+        (fun n => Real.exp (t n * (prescribedForcing g h).value x + b n)) :=
+      funext (fun n => solvesVolumePath_density g h (hsol n) x)
+    rw [hn] at hl
+    exact tendsto_nhds_unique hl (Real.continuous_exp.tendsto _ |>.comp
+      ((ht.mul_const ((prescribedForcing g h).value x)).add hb))
+  apply solvesVolumePath_of_density g h _ he
+  exact D.positivePotential_of_smooth_limit f fLim hf (fun n => (hsol n).1)
+    (fun x => (he x).symm ▸ Real.exp_pos _)
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/SmoothPathOpenness.lean b/lean/OAI/Geometry/PrescribedPotential/SmoothPathOpenness.lean
new file mode 100644
index 0000000000000000000000000000000000000000..a279952080c1a69d6fdb80e59468031064db8d38
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/SmoothPathOpenness.lean
@@ -0,0 +1,111 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.PatchCutoffs
+import OAI.Geometry.PrescribedPotential.PositivePotentialLinearization
+import OAI.Geometry.PrescribedPotential.RealSobolev
+import OAI.Geometry.PrescribedPotential.SmoothLocalPath
+import OAI.Geometry.PrescribedPotential.SmoothPotentialDifference
+import OAI.Geometry.PrescribedPotential.VolumePath
+
+namespace OAI
+
+/-! Smooth Path Openness. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff Classical ComplexOrder
+namespace Anticanonical.SourceSmooth
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+namespace KaehlerMetric
+
+lemma logRatio_deform_eq_logDensity (g : KaehlerMetric A) (φ : SmoothRealFunction A)
+    (hp : g.PositivePotential φ) (x : X) :
+    (g.logRatio (g.deform φ hp)).value x = Real.log ((g.potentialDensity φ).value x) := by
+  obtain ⟨i,hi⟩ := A.covers x
+  change g.logRatioValue (g.deform φ hp) x = _
+  rw [g.logRatioValue_local _ i hi,g.potentialDensity_quotient φ i hi]
+  exact (Real.log_div
+    (ne_of_gt ((g.deform φ hp).volumeCoefficient_pos i ((A.chart i).mapsTo hi)))
+    (ne_of_gt (g.volumeCoefficient_pos i ((A.chart i).mapsTo hi)))).symm
+
+lemma positivePotential_add (g : KaehlerMetric A) (φ ψ : SmoothRealFunction A)
+    (hφ : g.PositivePotential φ) (hψ : (g.deform φ hφ).PositivePotential ψ) :
+    g.PositivePotential (φ.addFunction ψ) := by
+  intro i z hz
+  rw [SmoothRealFunction.hessian_addFunction φ ψ i hz,← add_assoc]
+  exact hψ i z hz
+
+lemma logRatio_deform_add (g : KaehlerMetric A) (φ ψ : SmoothRealFunction A)
+    (hφ : g.PositivePotential φ) (hψ : (g.deform φ hφ).PositivePotential ψ) (x : X) :
+    (g.logRatio (g.deform (φ.addFunction ψ) (g.positivePotential_add φ ψ hφ hψ))).value x =
+      (g.logRatio (g.deform φ hφ)).value x +
+      ((g.deform φ hφ).logRatio ((g.deform φ hφ).deform ψ hψ)).value x := by
+  obtain ⟨i,hi⟩ := A.covers x
+  change g.logRatioValue _ x = g.logRatioValue _ x + (g.deform φ hφ).logRatioValue _ x
+  rw [g.logRatioValue_local _ i hi,g.logRatioValue_local _ i hi,
+    (g.deform φ hφ).logRatioValue_local _ i hi]
+  simp only [volumeCoefficient,deform,
+    SmoothRealFunction.hessian_addFunction φ ψ i ((A.chart i).mapsTo hi),add_assoc]
+  ring
+
+end KaehlerMetric
+
+ 
+
+theorem smooth_log_path_eventually_solvable [T2Space X] [CompactSpace X] [ConnectedSpace X]
+    (g : KaehlerMetric A) (F : SmoothRealFunction A) (x₀ : X) :
+    ∀ᶠ t : ℝ in 𝓝 0, ∃ (φ : SmoothRealFunction A) (b : ℝ)
+      (hp : g.PositivePotential φ), φ.value x₀ = 0 ∧
+      ∀ x, (g.logRatio (g.deform φ hp)).value x = t * F.value x + b := by
+  obtain ⟨S,⟨D⟩⟩ := GlobalElliptic.exists_gluingData g
+  filter_upwards [D.smooth_path_eventually_solvable (GlobalElliptic.RealSmooth.ofReal F) x₀]
+    with t ht
+  obtain ⟨φ,b,hp,hn,he⟩ := ht
+  refine ⟨φ,b,hp,hn,fun x => ?_⟩
+  rw [g.logRatio_deform_eq_logDensity φ hp x,he x,Real.log_exp]
+  rfl
+
+ 
+
+def SmoothVolumePathTimes (g : KaehlerMetric A) (h : SemipositiveAnticanonicalMetric A)
+    (x₀ : X) : Set ℝ :=
+  {t | ∃ (φ : SmoothRealFunction A) (b : ℝ), φ.value x₀ = 0 ∧ SolvesVolumePath g h t φ b}
+
+lemma smoothVolumePathTimes_zero (g : KaehlerMetric A) (h : SemipositiveAnticanonicalMetric A)
+    (x₀ : X) : 0 ∈ SmoothVolumePathTimes g h x₀ :=
+  ⟨SmoothRealFunction.constant 0,0,rfl,volumePath_zero g h⟩
+
+lemma smoothVolumePathTimes_eventually [T2Space X] [CompactSpace X] [ConnectedSpace X]
+    (g : KaehlerMetric A) (h : SemipositiveAnticanonicalMetric A) (x₀ : X)
+    {t₀ : ℝ} (ht₀ : t₀ ∈ SmoothVolumePathTimes g h x₀) :
+    ∀ᶠ t in 𝓝 t₀, t ∈ SmoothVolumePathTimes g h x₀ := by
+  obtain ⟨φ₀,b₀,hn₀,hp₀,he₀⟩ := ht₀
+  have hs : Tendsto (fun t : ℝ => t-t₀) (𝓝 t₀) (𝓝 0) := by
+    have hc : Continuous (fun t : ℝ => t-t₀) := continuous_id.sub continuous_const
+    simpa only [sub_self] using hc.tendsto t₀
+  filter_upwards [hs.eventually (smooth_log_path_eventually_solvable
+    (g.deform φ₀ hp₀) (prescribedForcing g h) x₀)] with t ht
+  obtain ⟨ψ,a,hψ,hnψ,heψ⟩ := ht
+  refine ⟨φ₀.addFunction ψ,b₀+a,?_,g.positivePotential_add φ₀ ψ hp₀ hψ,?_⟩
+  · change φ₀.value x₀ + ψ.value x₀ = 0
+    rw [hn₀,hnψ,add_zero]
+  · intro x
+    rw [g.logRatio_deform_add φ₀ ψ hp₀ hψ,he₀ x,heψ x]
+    ring
+
+ 
+
+theorem smoothVolumePathTimes_isOpen [T2Space X] [CompactSpace X] [ConnectedSpace X]
+    (g : KaehlerMetric A) (h : SemipositiveAnticanonicalMetric A) (x₀ : X) :
+    IsOpen (SmoothVolumePathTimes g h x₀) :=
+  isOpen_iff_mem_nhds.mpr (fun _ ht => smoothVolumePathTimes_eventually g h x₀ ht)
+
+end Anticanonical.SourceSmooth
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/SmoothPotentialDifference.lean b/lean/OAI/Geometry/PrescribedPotential/SmoothPotentialDifference.lean
new file mode 100644
index 0000000000000000000000000000000000000000..2bf492594b07de562b8e04572de82c4e7e734b23
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/SmoothPotentialDifference.lean
@@ -0,0 +1,53 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.AnalyticSupport
+
+namespace OAI
+
+/-! Smooth Potential Difference. -/
+
+section
+
+ 
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff ComplexOrder Classical
+namespace Anticanonical.SourceSmooth
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+namespace SmoothRealFunction
+
+def addFunction (φ ψ : SmoothRealFunction A) : SmoothRealFunction A where
+  value x := φ.value x + ψ.value x
+  smooth i := (φ.smooth i).add (ψ.smooth i)
+
+lemma hessian_addFunction (φ ψ : SmoothRealFunction A) (i : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart i).target) :
+    (φ.addFunction ψ).hessian i z = φ.hessian i z + ψ.hessian i z := by
+  have hφ := (φ.smooth i).contDiffAt ((A.chart i).open_target.mem_nhds hz)
+  have hψ := (ψ.smooth i).contDiffAt ((A.chart i).open_target.mem_nhds hz)
+  change ContDiffAt ℝ ∞ (φ.localExpression i) z at hφ
+  change ContDiffAt ℝ ∞ (ψ.localExpression i) z at hψ
+  change PotentialKaehler.hermitianPartMatrix
+    (fderiv ℝ (fderiv ℝ (φ.localExpression i + ψ.localExpression i)) z) = _
+  rw [EllipticKernel.hessian_add hφ hψ, EllipticKernel.hermitianPartMatrix_add]
+  rfl
+
+def difference (φ ψ : SmoothRealFunction A) : SmoothRealFunction A :=
+  φ.addFunction (ψ.realSMul (-1))
+
+lemma difference_value (φ ψ : SmoothRealFunction A) (x : X) :
+    (φ.difference ψ).value x = φ.value x - ψ.value x := by
+  change φ.value x + -1 * ψ.value x = _
+  ring
+
+lemma hessian_difference (φ ψ : SmoothRealFunction A) (i : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart i).target) :
+    (φ.difference ψ).hessian i z = φ.hessian i z - ψ.hessian i z := by
+  rw [difference, hessian_addFunction φ _ i hz, hessian_realSMul]
+  simp [sub_eq_add_neg]
+end SmoothRealFunction
+end Anticanonical.SourceSmooth
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/SmoothSequenceCompactness.lean b/lean/OAI/Geometry/PrescribedPotential/SmoothSequenceCompactness.lean
new file mode 100644
index 0000000000000000000000000000000000000000..f5fbea15bac95310ef9cf5e23ba7862746853e7f
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/SmoothSequenceCompactness.lean
@@ -0,0 +1,90 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CircleRadialCalculus
+import OAI.Geometry.PrescribedPotential.EvenRellich
+import OAI.Geometry.PrescribedPotential.GlobalSmooth
+import OAI.Geometry.PrescribedPotential.PatchCutoffs
+
+namespace OAI
+
+/-! Smooth Sequence Compactness. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology
+open scoped ContDiff Classical
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace Localizers
+variable (D : Localizers A ι)
+
+lemma lower_tendsto_smooth {s t : ℝ} (hst : t ≤ s) (f : ℕ → Smooth A)
+    (u : D.Sobolev s) (hf : Tendsto (fun n => D.embed s (f n)) atTop (𝓝 u)) :
+    Tendsto (fun n => D.embed t (f n)) atTop (𝓝 (D.lower s t u)) := by
+  have hh := (D.lower s t).continuous.tendsto u |>.comp hf
+  simpa only [Function.comp_def,D.lower_embed hst] using hh
+
+end Localizers
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+lemma even_tower_subsequence (f : ℕ → Smooth A)
+    (hbound : ∀ k : ℕ, ∃ C : ℝ, ∀ n : ℕ,
+      ‖D.localizers.embed (((2*k : ℕ) : ℝ)+2) (f n)‖ ≤ C) :
+    ∃ (u : ∀ k : ℕ, D.localizers.Sobolev ((2*k : ℕ) : ℝ)) (r : ℕ → ℕ),
+      StrictMono r ∧ ∀ k : ℕ,
+        Tendsto (fun n => D.localizers.embed ((2*k : ℕ) : ℝ) (f (r n))) atTop (𝓝 (u k)) := by
+  choose C hC using hbound
+  let K (k : ℕ) : Set (D.localizers.Sobolev ((2*k : ℕ) : ℝ)) :=
+    closure (D.localizers.lower (((2*k : ℕ) : ℝ)+2) ((2*k : ℕ) : ℝ) ''
+      Metric.closedBall 0 (C k))
+  have hK (k : ℕ) : IsCompact (K k) :=
+    (D.lower_even_compact k).isCompact_closure_image_closedBall (C k)
+  have hf (n : ℕ) : (fun k : ℕ => D.localizers.embed ((2*k : ℕ) : ℝ) (f n)) ∈
+      {x | ∀ k, x k ∈ K k} := by
+    intro k
+    apply subset_closure
+    refine ⟨D.localizers.embed (((2*k : ℕ) : ℝ)+2) (f n),?_,?_⟩
+    · simpa only [Metric.mem_closedBall,dist_zero_right] using hC k n
+    · exact D.localizers.lower_embed (by linarith : ((2*k : ℕ) : ℝ) ≤ ((2*k : ℕ) : ℝ)+2) (f n)
+  obtain ⟨u,_,r,hr,hu⟩ := (isCompact_pi_infinite hK).tendsto_subseq hf
+  refine ⟨u,r,hr,fun k => ?_⟩
+  exact (continuous_apply k).tendsto u |>.comp hu
+
+ 
+
+theorem smooth_sequence_compact (f : ℕ → Smooth A)
+    (hbound : ∀ k : ℕ, ∃ C : ℝ, ∀ n : ℕ,
+      ‖D.localizers.embed (((2*k : ℕ) : ℝ)+2) (f n)‖ ≤ C) :
+    ∃ (fLim : Smooth A) (r : ℕ → ℕ), StrictMono r ∧ ∀ k : ℕ,
+      Tendsto (fun n => D.localizers.embed (k : ℝ) (f (r n))) atTop
+        (𝓝 (D.localizers.embed (k : ℝ) fLim)) := by
+  obtain ⟨u,r,hr,hu⟩ := D.even_tower_subsequence f hbound
+  let u₀ := D.localizers.lower ((2*0 : ℕ) : ℝ) 0 (u 0)
+  have h₀ : Tendsto (fun n => D.localizers.embed 0 (f (r n))) atTop (𝓝 u₀) :=
+    D.localizers.lower_tendsto_smooth (Nat.cast_nonneg _) (fun n => f (r n)) (u 0) (hu 0)
+  let v (k : ℕ) := D.localizers.lower ((2*k : ℕ) : ℝ) (k : ℝ) (u k)
+  have hv (k : ℕ) : Tendsto (fun n => D.localizers.embed (k : ℝ) (f (r n))) atTop (𝓝 (v k)) :=
+    D.localizers.lower_tendsto_smooth (by exact_mod_cast (show k ≤ 2*k by omega))
+      (fun n => f (r n)) (u k) (hu k)
+  have hv₀ (k : ℕ) : D.localizers.lower (k : ℝ) 0 (v k) = u₀ := by
+    exact tendsto_nhds_unique
+      (D.localizers.lower_tendsto_smooth (Nat.cast_nonneg k) (fun n => f (r n)) (v k) (hv k)) h₀
+  obtain ⟨fLim,hfLim⟩ := D.allRegular_smooth (fun k => ⟨v k,hv₀ k⟩)
+  refine ⟨fLim,r,hr,fun k => ?_⟩
+  have he : v k = D.localizers.embed (k : ℝ) fLim := by
+    apply D.localizers.lower_injective (Nat.cast_nonneg k)
+    rw [hv₀,D.localizers.lower_embed (Nat.cast_nonneg k),hfLim]
+  exact he ▸ hv k
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/SobolevDetDifferential.lean b/lean/OAI/Geometry/PrescribedPotential/SobolevDetDifferential.lean
new file mode 100644
index 0000000000000000000000000000000000000000..8254cf568effd03df1650b2e77adce9231a97342
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/SobolevDetDifferential.lean
@@ -0,0 +1,64 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.NonlinearHessianCommutator
+
+namespace OAI
+
+/-! Sobolev Det Differential. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff Classical Matrix.Norms.Elementwise
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+lemma determinant_map_contDiff (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) (n : ℕ) :
+    ContDiff ℝ ∞ (D.determinant k hk n) := by
+  change ContDiff ℝ ∞ (fun M : Fin n → Fin n → D.localizers.Sobolev (k : ℝ) =>
+    D.determinant k hk n M)
+  exact D.determinant_contDiff k hk n (fun i j (M : Fin n → Fin n → D.localizers.Sobolev (k : ℝ)) => M i j) (fun i j => by fun_prop)
+
+def detDifferential (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) (n : ℕ)
+    (M : Matrix (Fin n) (Fin n) (D.localizers.Sobolev (k : ℝ))) :
+    (Fin n → Fin n → D.localizers.Sobolev (k : ℝ)) →L[ℝ] D.localizers.Sobolev (k : ℝ) :=
+  fderiv ℝ (D.determinant k hk n) M
+
+lemma detDifferential_contDiff (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) (n : ℕ) :
+    ContDiff ℝ ∞ (D.detDifferential k hk n) := by
+  exact (D.determinant_map_contDiff k hk n).fderiv_right (by simp)
+
+lemma detDifferential_strong (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) (n : ℕ)
+    (M N : Matrix (Fin n) (Fin n) (D.localizers.Sobolev (k : ℝ))) (x : X) :
+    D.localizers.strong (k : ℝ) (D.detDifferential k hk n M N) x =
+      NonlinearHessian.detDifferential
+        (fun i j => D.localizers.strong (k : ℝ) (M i j) x)
+        (fun i j => D.localizers.strong (k : ℝ) (N i j) x) := by
+  let E : D.localizers.Sobolev (k : ℝ) →L[ℝ] ℂ :=
+    (BoundedContinuousFunction.evalCLM ℝ x).comp (D.localizers.strong (k : ℝ))
+  let EM : (Fin n → Fin n → D.localizers.Sobolev (k : ℝ)) →L[ℝ]
+      (Fin n → Fin n → ℂ) := ContinuousLinearMap.pi fun i => ContinuousLinearMap.pi fun j =>
+        (E.comp (ContinuousLinearMap.proj j : (Fin n → D.localizers.Sobolev (k : ℝ)) →L[ℝ]
+          D.localizers.Sobolev (k : ℝ))).comp (ContinuousLinearMap.proj i)
+  have hleft := E.hasFDerivAt.comp M
+    ((D.determinant_map_contDiff k hk n).differentiable (by simp) M).hasFDerivAt
+  have hright := (NonlinearHessian.detDifferential_hasFDeriv (EM M)).comp M EM.hasFDerivAt
+  have he : (fun u => E (D.determinant k hk n u)) = (fun u => Matrix.det (EM u)) := by
+    funext u
+    exact D.determinant_strong k hk n u x
+  have hh := hleft.unique (hright.congr_of_eventuallyEq (Filter.Eventually.of_forall (congr_fun he)))
+  exact congrArg (fun T : (Fin n → Fin n → D.localizers.Sobolev (k : ℝ)) →L[ℝ] ℂ => T N) hh
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/SobolevDeterminant.lean b/lean/OAI/Geometry/PrescribedPotential/SobolevDeterminant.lean
new file mode 100644
index 0000000000000000000000000000000000000000..fdbd06f5c6847e6c1ae608fcd2c151db5d847e63
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/SobolevDeterminant.lean
@@ -0,0 +1,88 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CompletedProduct
+import OAI.Geometry.PrescribedPotential.GlobalOperator
+import OAI.Geometry.PrescribedPotential.PatchCutoffs
+
+namespace OAI
+
+/-! Sobolev Determinant. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff Classical BoundedContinuousFunction
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+def finiteProduct (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) :
+    ∀ n : ℕ, (Fin n → D.localizers.Sobolev (k : ℝ)) → D.localizers.Sobolev (k : ℝ)
+  | 0, _ => D.localizers.embed (k : ℝ) (Smooth.const 1)
+  | n+1, f => D.product k hk (f 0) (finiteProduct k hk n (fun i => f i.succ))
+
+lemma finiteProduct_contDiff {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+    (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) (n : ℕ)
+    (f : Fin n → E → D.localizers.Sobolev (k : ℝ)) (hf : ∀ i, ContDiff ℝ ∞ (f i)) :
+    ContDiff ℝ ∞ (fun u => D.finiteProduct k hk n (fun i => f i u)) := by
+  induction n with
+  | zero => exact contDiff_const
+  | succ n ih =>
+    exact ((D.product k hk).contDiff.comp (hf 0)).clm_apply (ih _ (fun i => hf i.succ))
+
+lemma finiteProduct_strong (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) (n : ℕ)
+    (f : Fin n → D.localizers.Sobolev (k : ℝ)) (x : X) :
+    D.localizers.strong (k : ℝ) (D.finiteProduct k hk n f) x =
+      ∏ i, D.localizers.strong (k : ℝ) (f i) x := by
+  have hs : (Module.finrank ℝ (EC d) : ℝ) < 2*(k : ℝ) := by
+    have hh : (Module.finrank ℝ (EC d) : ℝ) < (k : ℝ) := by exact_mod_cast hk
+    linarith [Nat.cast_nonneg (α := ℝ) k]
+  induction n with
+  | zero =>
+    rw [finiteProduct, D.localizers.strong_embed _ hs]
+    simp only [Finset.univ_eq_empty, Finset.prod_empty]
+    rfl
+  | succ n ih =>
+    rw [finiteProduct, D.product_strong, BoundedContinuousFunction.mul_apply,
+      ih, Fin.prod_univ_succ]
+
+def determinant (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) (n : ℕ)
+    (M : Matrix (Fin n) (Fin n) (D.localizers.Sobolev (k : ℝ))) :
+    D.localizers.Sobolev (k : ℝ) :=
+  ∑ σ : Equiv.Perm (Fin n), ((↑(Equiv.Perm.sign σ) : ℤ) : ℝ) •
+    D.finiteProduct k hk n (fun i => M (σ i) i)
+
+lemma determinant_contDiff {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+    (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) (n : ℕ)
+    (M : Fin n → Fin n → E → D.localizers.Sobolev (k : ℝ))
+    (hM : ∀ i j, ContDiff ℝ ∞ (M i j)) :
+    ContDiff ℝ ∞ (fun u => D.determinant k hk n (fun i j => M i j u)) := by
+  apply ContDiff.sum
+  intro σ _
+  exact (D.finiteProduct_contDiff k hk n _ (fun i => hM (σ i) i)).const_smul _
+
+lemma determinant_strong (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) (n : ℕ)
+    (M : Matrix (Fin n) (Fin n) (D.localizers.Sobolev (k : ℝ))) (x : X) :
+    D.localizers.strong (k : ℝ) (D.determinant k hk n M) x =
+      Matrix.det (fun i j => D.localizers.strong (k : ℝ) (M i j) x) := by
+  change (Fin n → Fin n → D.localizers.Sobolev (k : ℝ)) at M
+  simp only [determinant, map_sum, map_smul, BoundedContinuousFunction.sum_apply,
+    BoundedContinuousFunction.smul_apply]
+  erw [Matrix.det_apply']
+  apply Finset.sum_congr rfl
+  intro σ _
+  rw [D.finiteProduct_strong]
+  simp only [Complex.real_smul, Complex.ofReal_intCast]
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/SobolevEmbedding.lean b/lean/OAI/Geometry/PrescribedPotential/SobolevEmbedding.lean
new file mode 100644
index 0000000000000000000000000000000000000000..0d4671bf97f623d79ee676cf5405f2cae1d9d6e0
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/SobolevEmbedding.lean
@@ -0,0 +1,160 @@
+import Mathlib
+
+namespace OAI
+
+/-! Sobolev Embedding. -/
+
+section
+
+ 
+
+noncomputable section
+open MeasureTheory FourierTransform TemperedDistribution
+open scoped SchwartzMap BoundedContinuousFunction ContDiff
+
+namespace SobolevChart
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+
+def inverseWeight (k : ℕ) : E →ᵇ ℂ :=
+  BoundedContinuousFunction.ofNormedAddCommGroup
+    (fun x : E => (((1 + ‖x‖ ^ 2) ^ (-(k : ℝ)) : ℝ) : ℂ))
+    (by
+      have ht : (fun x : E => (((1 + ‖x‖ ^ 2) ^ (-(k : ℝ)) : ℝ) : ℂ)).HasTemperateGrowth := by fun_prop
+      exact ht.1.continuous) 1 (by
+      intro x
+      change ‖(((1 + ‖x‖ ^ 2) ^ (-(k : ℝ)) : ℝ) : ℂ)‖ ≤ 1
+      rw [Complex.norm_real, Real.norm_eq_abs,
+        abs_of_pos (Real.rpow_pos_of_pos (by positivity : 0 < 1 + ‖x‖ ^ 2) _)]
+      exact Real.rpow_le_one_of_one_le_of_nonpos
+        (by nlinarith [sq_nonneg ‖x‖] : 1 ≤ 1 + ‖x‖ ^ 2) (neg_nonpos.mpr (Nat.cast_nonneg k)))
+
+lemma inverseWeight_temperate (k : ℕ) :
+    (inverseWeight (E := E) k : E → ℂ).HasTemperateGrowth := by
+  change (fun x : E => (((1 + ‖x‖ ^ 2) ^ (-(k : ℝ)) : ℝ) : ℂ)).HasTemperateGrowth
+  fun_prop
+
+lemma weight_cancel (k : ℕ) (x : E) :
+    (((1 + ‖x‖ ^ 2) ^ k : ℝ) : ℂ) * inverseWeight k x = 1 := by
+  change (((1 + ‖x‖ ^ 2) ^ k : ℝ) : ℂ) *
+    (((1 + ‖x‖ ^ 2) ^ (-(k : ℝ)) : ℝ) : ℂ) = 1
+  rw [← Complex.ofReal_mul, ← Complex.ofReal_one, Complex.ofReal_inj]
+  rw [← Real.rpow_natCast, ← Real.rpow_add (by positivity), add_neg_cancel, Real.rpow_zero]
+
+variable [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+
+lemma all_even_bessel_fourier_L1 {u : 𝓢'(E, ℂ)}
+    (hu : ∀ k : ℕ, MemSobolev (2 * (k : ℝ)) 2 u) (k : ℕ) :
+    ∃ v : Lp ℂ 1 (volume : Measure E),
+      𝓕 (besselPotential E ℂ (2 * k) u) = (v : 𝓢'(E, ℂ)) := by
+  have hm : MemSobolev (2 * ((Module.finrank ℝ E + 1 : ℕ) : ℝ)) 2
+      (besselPotential E ℂ (2 * k) u) := by
+    rw [memSobolev_besselPotential_iff]
+    convert hu (k + (Module.finrank ℝ E + 1)) using 1; push_cast; ring
+  exact hm.fourier_memL1 (by push_cast; nlinarith [Nat.cast_nonneg (α := ℝ) (Module.finrank ℝ E)])
+
+ 
+
+lemma all_even_fourier_moments {u : 𝓢'(E, ℂ)}
+    (hu : ∀ k : ℕ, MemSobolev (2 * (k : ℝ)) 2 u) :
+    ∃ v : Lp ℂ 1 (volume : Measure E),
+      𝓕 u = (v : 𝓢'(E, ℂ)) ∧
+      ∀ n : ℕ, Integrable (fun x => ‖x‖ ^ n * ‖v x‖) := by
+  obtain ⟨v, hv⟩ := all_even_bessel_fourier_L1 hu 0
+  simp only [Nat.cast_zero, mul_zero, besselPotential_zero,
+    ContinuousLinearMap.id_apply] at hv
+  refine ⟨v, hv, fun n => ?_⟩
+  obtain ⟨w, hw⟩ := all_even_bessel_fourier_L1 hu n
+  have hw' : smulLeftCLM ℂ (fun x : E => (((1 + ‖x‖ ^ 2) ^ n : ℝ) : ℂ))
+      (𝓕 u) = (w : 𝓢'(E, ℂ)) := by
+    rw [fourier_besselPotential_eq_smulLeftCLM_fourier_apply] at hw
+    simpa only [mul_div_cancel_left₀ (n : ℝ) (by norm_num : (2 : ℝ) ≠ 0),
+      Real.rpow_natCast] using hw
+  let b := inverseWeight (E := E) n
+  have hb := b.memLp_top (μ := volume)
+  have he : (hb.toLp b : Lp ℂ ⊤ volume) • w = v := by
+    apply (LinearMap.ker_eq_bot.mp (Lp.ker_toTemperedDistributionCLM_eq_bot
+      (F := ℂ) (E := E) (μ := volume) (p := 1)))
+    change (((hb.toLp b : Lp ℂ ⊤ volume) • w : Lp ℂ 1 volume) : 𝓢'(E, ℂ)) = (v : 𝓢'(E, ℂ))
+    rw [Lp.toTemperedDistribution_smul_eq (inverseWeight_temperate n) hb,
+      ← hw', smulLeftCLM_smulLeftCLM_apply (by fun_prop) (inverseWeight_temperate n)]
+    have hc : (fun x : E => (((1 + ‖x‖ ^ 2) ^ n : ℝ) : ℂ)) * (b : E → ℂ) = fun _ => 1 := by
+      ext x
+      exact weight_cancel n x
+    rw [hc, smulLeftCLM_const, one_smul, hv]
+  have he' : ∀ᵐ x, ((1 + ‖x‖ ^ 2) ^ n : ℝ) * ‖v x‖ = ‖w x‖ := by
+    have hh := Lp.coeFn_lpSMul (r := 1) (hb.toLp b : Lp ℂ ⊤ volume) w
+    rw [he] at hh
+    filter_upwards [hh, hb.coeFn_toLp] with x hx hb'
+    have hx' : (((1 + ‖x‖ ^ 2) ^ n : ℝ) : ℂ) * v x = w x := by
+      rw [hx, Pi.smul_apply', hb', smul_eq_mul, ← mul_assoc, weight_cancel, one_mul]
+    simpa only [norm_mul, Complex.norm_real, Real.norm_eq_abs,
+      abs_of_nonneg (show 0 ≤ (1 + ‖x‖ ^ 2) ^ n by positivity)] using congrArg norm hx'
+  apply (L1.integrable_coeFn w).norm.mono'
+    (((continuous_norm.pow n).aestronglyMeasurable).mul (Lp.aestronglyMeasurable v).norm)
+  filter_upwards [he'] with x hx
+  change ‖‖x‖ ^ n * ‖v x‖‖ ≤ ‖w x‖
+  rw [Real.norm_of_nonneg (mul_nonneg (pow_nonneg (norm_nonneg x) n) (norm_nonneg (v x))), ← hx]
+  apply mul_le_mul_of_nonneg_right _ (norm_nonneg _)
+  apply pow_le_pow_left₀ (norm_nonneg _)
+  nlinarith [sq_nonneg (‖x‖ - 1 / 2)]
+
+ 
+def boundedDistribution (f : E →ᵇ ℂ) : 𝓢'(E, ℂ) :=
+  (f.memLp_top (μ := volume)).toLp f
+
+lemma boundedDistribution_apply (f : E →ᵇ ℂ) (φ : 𝓢(E, ℂ)) :
+    boundedDistribution f φ = ∫ x, φ x • f x := by
+  rw [boundedDistribution, Lp.toTemperedDistribution_apply]
+  apply integral_congr_ae
+  filter_upwards [(f.memLp_top (μ := volume)).coeFn_toLp] with x hx
+  rw [hx]
+
+lemma fourier_L1_distribution (v : Lp ℂ 1 (volume : Measure E)) :
+    boundedDistribution (Real.Lp.fourierTransform v) = 𝓕 (v : 𝓢'(E, ℂ)) := by
+  ext φ
+  rw [boundedDistribution_apply, TemperedDistribution.fourier_apply,
+    Lp.toTemperedDistribution_apply]
+  symm
+  simpa using! VectorFourier.integral_fourierIntegral_smul_eq_flip
+    (L := innerₗ E) Real.continuous_fourierChar continuous_inner
+    φ.integrable (L1.integrable_coeFn v)
+
+lemma fourierInv_L1_distribution (v : Lp ℂ 1 (volume : Measure E)) :
+    boundedDistribution (Real.Lp.fourierTransformInv v) = 𝓕⁻ (v : 𝓢'(E, ℂ)) := by
+  ext φ
+  rw [boundedDistribution_apply, TemperedDistribution.fourierInv_apply,
+    Lp.toTemperedDistribution_apply]
+  symm
+  simpa [SchwartzMap.fourierInv_coe, Real.fourierInv_eq, VectorFourier.fourierIntegral, real_inner_comm] using!
+    VectorFourier.integral_fourierIntegral_smul_eq_flip
+      (μ := volume) (ν := volume) (L := -innerₗ E) Real.continuous_fourierChar continuous_inner.neg
+      φ.integrable (L1.integrable_coeFn v)
+
+lemma fourier_temperate_of_moments {f : E → ℂ} (hf : AEStronglyMeasurable f)
+    (hm : ∀ n : ℕ, Integrable (fun x => ‖x‖ ^ n * ‖f x‖)) :
+    (𝓕 f).HasTemperateGrowth := by
+  refine ⟨Real.contDiff_fourier (N := ⊤) (fun n _ => hm n), fun n => ?_⟩
+  refine ⟨0, ∫ x, ‖VectorFourier.fourierPowSMulRight (innerSL ℝ) f x n‖, fun x => ?_⟩
+  rw [Real.iteratedFDeriv_fourier (N := ⊤) (fun k _ => hm k) hf (by simp)]
+  simp only [pow_zero, mul_one]
+  exact VectorFourier.norm_fourierIntegral_le_integral_norm
+    Real.fourierChar volume (innerₗ E) _ x
+
+ 
+
+theorem all_even_smooth_representative {u : 𝓢'(E, ℂ)}
+    (hu : ∀ k : ℕ, MemSobolev (2 * (k : ℝ)) 2 u) :
+    ∃ f : E →ᵇ ℂ, (f : E → ℂ).HasTemperateGrowth ∧ boundedDistribution f = u := by
+  obtain ⟨v, hv, hm⟩ := all_even_fourier_moments hu
+  refine ⟨Real.Lp.fourierTransformInv v, ?_, ?_⟩
+  · have ht := fourier_temperate_of_moments (Lp.aestronglyMeasurable v) hm
+    change (fun x : E => 𝓕 (v : E → ℂ) (-x)).HasTemperateGrowth
+    fun_prop
+  · rw [fourierInv_L1_distribution, ← hv, fourierInv_fourier_eq]
+
+end SobolevChart
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/SobolevExponential.lean b/lean/OAI/Geometry/PrescribedPotential/SobolevExponential.lean
new file mode 100644
index 0000000000000000000000000000000000000000..a4077033f55feb0cd5aed7623f9c27c673571d9a
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/SobolevExponential.lean
@@ -0,0 +1,141 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CompletedProduct
+import OAI.Geometry.PrescribedPotential.PatchCutoffs
+import OAI.Geometry.PrescribedPotential.RealPoisson
+import OAI.Geometry.PrescribedPotential.RealSobolev
+
+namespace OAI
+
+/-! Sobolev Exponential. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology
+open scoped ContDiff Classical BoundedContinuousFunction
+namespace ExponentialEvaluation
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
+lemma apply_exp (L : E →L[ℝ] E) (q : E →L[ℝ] ℝ) (a : ℝ)
+    (hq : ∀ v, q (L v) = a * q v) (v : E) :
+    q (NormedSpace.exp L v) = Real.exp a * q v := by
+  have hp (n : ℕ) : q ((L^n) v) = a^n * q v := by
+    induction n with
+    | zero => simp
+    | succ n hn =>
+      rw [pow_succ']
+      change q (L ((L^n) v)) = _
+      rw [hq, hn, pow_succ']
+      ring
+  let e := q.comp (ContinuousLinearMap.apply ℝ E v)
+  have hl := e.hasSum (NormedSpace.exp_series_hasSum_exp' (𝕂 := ℝ) L)
+  have hr := (NormedSpace.exp_series_hasSum_exp' (𝕂 := ℝ) a).mul_right (q v)
+  rw [← Real.exp_eq_exp_ℝ] at hr
+  have ht : (fun n : ℕ => e ((n.factorial : ℝ)⁻¹ • L^n)) =
+      (fun n : ℕ => (n.factorial : ℝ)⁻¹ • a^n * q v) := by
+    funext n
+    change q (((n.factorial : ℝ)⁻¹ • L^n) v) = (n.factorial : ℝ)⁻¹ * a^n * q v
+    rw [_root_.smul_apply, map_smul, hp]
+    simp only [smul_eq_mul]
+    ring
+  rw [ht] at hl
+  exact hl.unique hr
+
+def orbit (L : E →L[ℝ] E) (v : E) (t : ℝ) : E :=
+  NormedSpace.exp (t • L) v
+
+omit [CompleteSpace E] in
+lemma orbit_zero (L : E →L[ℝ] E) (v : E) : orbit L v 0 = v := by
+  simp [orbit, NormedSpace.exp_zero]
+
+lemma orbit_continuous (L : E →L[ℝ] E) (v : E) : Continuous (orbit L v) := by
+  have ht : Continuous (fun t : ℝ => t • L) := continuous_id.smul continuous_const
+  have he : Continuous (NormedSpace.exp : (E →L[ℝ] E) → (E →L[ℝ] E)) :=
+    continuous_iff_continuousAt.mpr (fun w => (NormedSpace.exp_analytic (𝕂 := ℝ) w).continuousAt)
+  exact (ContinuousLinearMap.apply ℝ E v).continuous.comp (he.comp ht)
+end ExponentialEvaluation
+
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace RealSmooth
+
+def mul (F h : RealSmooth A) : RealSmooth A :=
+  ⟨F.val.mul h.val, fun x => by
+    change (F.val x * h.val x).im = 0
+    rw [Complex.mul_im, F.property, h.property]
+    ring⟩
+end RealSmooth
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+local instance expRealNormedGroup (s : ℝ) : NormedAddCommGroup (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedAddCommGroup
+local instance expRealNormedSpace (s : ℝ) : NormedSpace ℝ (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedSpace
+local instance expRealTopologicalGroup (s : ℝ) : IsTopologicalAddGroup (D.localizers.RealSobolev s) :=
+  Submodule.isTopologicalAddGroup _
+local instance expRealContinuousSMul (s : ℝ) : ContinuousSMul ℝ (D.localizers.RealSobolev s) :=
+  SMulMemClass.continuousSMul _
+
+def realMultiplier (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) (F : RealSmooth A) :
+    D.localizers.RealSobolev (k : ℝ) →L[ℝ] D.localizers.RealSobolev (k : ℝ) :=
+  ((D.product k hk (D.localizers.embed (k : ℝ) F.val)).comp
+    (D.localizers.realCompletion (k : ℝ)).subtypeL).codRestrict
+      (D.localizers.realCompletion (k : ℝ)) (fun u =>
+    D.localizers.maps_realCompletion (D.product k hk (D.localizers.embed (k : ℝ) F.val))
+      (fun h => ⟨F.mul h,D.product_embed k hk F.val h.val⟩) u.val u.property)
+
+lemma realMultiplier_evaluation (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (F : RealSmooth A) (u : D.localizers.RealSobolev (k : ℝ)) (x : X) :
+    D.realEvaluation (k : ℝ) x (D.realMultiplier k hk F u) =
+      F.source.value x * D.realEvaluation (k : ℝ) x u := by
+  have hs : (Module.finrank ℝ (EC d) : ℝ) < 2*(k : ℝ) := by
+    have hh : (Module.finrank ℝ (EC d) : ℝ) < (k : ℝ) := by exact_mod_cast hk
+    linarith [Nat.cast_nonneg (α := ℝ) k]
+  change (D.localizers.strong (k : ℝ)
+    (D.product k hk (D.localizers.embed (k : ℝ) F.val) u.val) x).re = _
+  rw [D.product_strong, BoundedContinuousFunction.mul_apply, D.localizers.strong_embed _ hs]
+  change (F.val x * D.localizers.strong (k : ℝ) u.val x).re = _
+  rw [Complex.mul_re, F.property, zero_mul, sub_zero]
+  rfl
+
+ 
+
+def exponentialDensity (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (F : RealSmooth A) (t : ℝ) : D.localizers.RealSobolev (k : ℝ) :=
+  ExponentialEvaluation.orbit (D.realMultiplier k hk F) (D.realConstants (k : ℝ) 1) t
+
+lemma exponentialDensity_zero (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (F : RealSmooth A) : D.exponentialDensity k hk F 0 = D.realConstants (k : ℝ) 1 := by
+  exact ExponentialEvaluation.orbit_zero _ _
+
+lemma exponentialDensity_continuous (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (F : RealSmooth A) : Continuous (D.exponentialDensity k hk F) := by
+  exact ExponentialEvaluation.orbit_continuous _ _
+
+lemma exponentialDensity_strong [ConnectedSpace X] (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (F : RealSmooth A) (t : ℝ) (x : X) :
+    D.localizers.strong (k : ℝ) (D.exponentialDensity k hk F t).val x =
+      (Real.exp (t * F.source.value x) : ℂ) := by
+  have hs : (Module.finrank ℝ (EC d) : ℝ) < 2*(k : ℝ) := by
+    have hh : (Module.finrank ℝ (EC d) : ℝ) < (k : ℝ) := by exact_mod_cast hk
+    linarith [Nat.cast_nonneg (α := ℝ) k]
+  have hh := ExponentialEvaluation.apply_exp (t • D.realMultiplier k hk F)
+    (D.realEvaluation (k : ℝ) x) (t * F.source.value x) (fun u => by
+      rw [_root_.smul_apply, map_smul, D.realMultiplier_evaluation]
+      simp only [smul_eq_mul]
+      ring) (D.realConstants (k : ℝ) 1)
+  rw [D.realEvaluation_constants hs, mul_one] at hh
+  apply Complex.ext
+  · exact hh
+  · exact D.localizers.real_strong hs (D.exponentialDensity k hk F t) x
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/SobolevFirstGain.lean b/lean/OAI/Geometry/PrescribedPotential/SobolevFirstGain.lean
new file mode 100644
index 0000000000000000000000000000000000000000..545bd10ac19043e3006a742f127957f8c438bf5b
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/SobolevFirstGain.lean
@@ -0,0 +1,89 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.SobolevReconstruction
+import OAI.Geometry.PrescribedPotential.WeakDerivativeLocalization
+
+namespace OAI
+
+/-! Sobolev First Gain. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory TemperedDistribution LineDeriv
+open scoped ContDiff SchwartzMap Classical
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+lemma completedLocalize_plus_one_embed (k : ℕ) (i : Fin A.count) (ρ : Smooth A)
+    (hρ : tsupport (ρ : X → ℂ) ⊆ (A.euclideanChart i).source) (f : Smooth A) :
+    D.completedLocalize ((k : ℝ)+1) i ρ hρ (D.localizers.embed ((k : ℝ)+1) f) =
+      schwartzCoord ((k : ℝ)+1) (localize A i ρ hρ f) := by
+  obtain ⟨C,_,hC⟩ := D.localization_integer_bound i ρ hρ (k+1)
+  rw [Nat.cast_add,Nat.cast_one] at hC
+  exact LinearMap.extendOfNorm_eq (D.localizers.embed_dense ((k : ℝ)+1)) ⟨C,hC⟩ f
+
+lemma completedLocalize_lower_one (k : ℕ) (i : Fin A.count) (ρ : Smooth A)
+    (hρ : tsupport (ρ : X → ℂ) ⊆ (A.euclideanChart i).source)
+    (u : D.localizers.Sobolev ((k : ℝ)+1)) :
+    realize (k : ℝ) (D.completedLocalize (k : ℝ) i ρ hρ
+      (D.localizers.lower ((k : ℝ)+1) (k : ℝ) u)) =
+      realize ((k : ℝ)+1) (D.completedLocalize ((k : ℝ)+1) i ρ hρ u) := by
+  have he : (fun u : D.localizers.Sobolev ((k : ℝ)+1) => realize (k : ℝ)
+      (D.completedLocalize (k : ℝ) i ρ hρ (D.localizers.lower ((k : ℝ)+1) (k : ℝ) u))) =
+      (fun u => realize ((k : ℝ)+1) (D.completedLocalize ((k : ℝ)+1) i ρ hρ u)) := by
+    apply (D.localizers.embed_dense ((k : ℝ)+1)).equalizer (by fun_prop) (by fun_prop)
+    funext f
+    dsimp only [Function.comp_apply]
+    simp only [D.localizers.lower_embed (by linarith : (k : ℝ) ≤ (k : ℝ)+1),
+      D.completedLocalize_embed,D.completedLocalize_plus_one_embed,realize_schwartzCoord]
+  exact congr_fun he u
+
+lemma local_mem_of_derivative_lifts (k : ℕ) (u : D.localizers.Sobolev ((k : ℝ)+1))
+    (hD : ∀ (p : ι) (v : EC d), ∃ w : D.localizers.Sobolev ((k : ℝ)+1),
+      D.localizers.lower ((k : ℝ)+1) (k : ℝ) w = D.completedDerivative k p v u)
+    (p : ι) :
+    MemSobolev ((k : ℝ)+2) 2 (D.localizers.distribution ((k : ℝ)+1) p u) := by
+  have hu : MemSobolev ((k : ℝ)+1) 2 (D.localizers.distribution ((k : ℝ)+1) p u) :=
+    realize_memSobolev _ _
+  have hder (v : EC d) : MemSobolev ((k : ℝ)+1) 2
+      (∂_{v} (D.localizers.distribution ((k : ℝ)+1) p u)) := by
+    obtain ⟨w,hw⟩ := hD p v
+    rw [D.distribution_derivative,D.completedLocalize_lower_one,← hw,
+      D.localizers.distribution_lower (by linarith : (k : ℝ) ≤ (k : ℝ)+1)]
+    exact (realize_memSobolev _ _).add (realize_memSobolev _ _)
+  convert memSobolev_of_first_basis hu (fun j => hder (stdOrthonormalBasis ℝ (EC d) j)) using 1
+  ring
+
+ 
+
+theorem sobolev_first_gain (k : ℕ) (u : D.localizers.Sobolev ((k : ℝ)+1))
+    (hD : ∀ (p : ι) (v : EC d), ∃ w : D.localizers.Sobolev ((k : ℝ)+1),
+      D.localizers.lower ((k : ℝ)+1) (k : ℝ) w = D.completedDerivative k p v u) :
+    ∃ w : D.localizers.Sobolev ((k+2 : ℕ) : ℝ),
+      D.localizers.lower ((k+2 : ℕ) : ℝ) ((k : ℝ)+1) w = u := by
+  have h0 : 0 ≤ (k : ℝ)+1 := by positivity
+  have hh : ∀ p, MemSobolev ((k+2 : ℕ) : ℝ) 2
+      (D.localizers.distribution 0 p (D.localizers.lower ((k : ℝ)+1) 0 u)) := by
+    intro p
+    rw [D.localizers.distribution_lower h0]
+    simpa only [Nat.cast_add,Nat.cast_ofNat] using D.local_mem_of_derivative_lifts k u hD p
+  obtain ⟨w,hw⟩ := D.sobolev_of_local_order (k+2) (D.localizers.lower ((k : ℝ)+1) 0 u) hh
+  refine ⟨w, (D.localizers.lower_injective h0) ?_⟩
+  have hc := congrArg (fun L : D.localizers.Sobolev ((k+2 : ℕ) : ℝ) →L[ℝ]
+      D.localizers.Sobolev 0 => L w)
+    (D.localizers.lower_comp (by push_cast; linarith : (k : ℝ)+1 ≤ ((k+2 : ℕ) : ℝ)) h0)
+  exact hc.trans hw
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/SobolevLocalization.lean b/lean/OAI/Geometry/PrescribedPotential/SobolevLocalization.lean
new file mode 100644
index 0000000000000000000000000000000000000000..058f6dd9fe581bf26867c69c42a8320cf5f9d4f2
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/SobolevLocalization.lean
@@ -0,0 +1,188 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.SobolevParametrix
+
+namespace OAI
+
+/-! Sobolev Localization. -/
+
+section
+
+ 
+
+noncomputable section
+open Filter Topology MeasureTheory FourierTransform TemperedDistribution LineDeriv
+open scoped SchwartzMap BoundedContinuousFunction ContDiff Laplacian Real
+
+namespace SobolevChart
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+
+lemma schwartz_deriv_product (v : E) (g φ : 𝓢(E, ℂ)) :
+    ∂_{v} (SchwartzMap.smulLeftCLM ℂ g φ) =
+      SchwartzMap.smulLeftCLM ℂ (⇑(∂_{v} g : 𝓢(E, ℂ))) φ +
+      SchwartzMap.smulLeftCLM ℂ g (∂_{v} φ) := by
+  ext x
+  have he : (SchwartzMap.smulLeftCLM ℂ g φ : E → ℂ) = fun y => g y * φ y := by
+    ext y
+    simp [SchwartzMap.smulLeftCLM_apply_apply g.hasTemperateGrowth, smul_eq_mul]
+  simp only [SchwartzMap.lineDerivOp_apply_eq_fderiv, he, add_apply,
+    SchwartzMap.smulLeftCLM_apply_apply g.hasTemperateGrowth,
+    SchwartzMap.smulLeftCLM_apply_apply (∂_{v} g).hasTemperateGrowth, smul_eq_mul]
+  rw [fderiv_fun_mul g.differentiableAt φ.differentiableAt]
+  simp only [_root_.add_apply, _root_.smul_apply, smul_eq_mul]
+  ring
+
+ 
+lemma weak_deriv_product (v : E) (g : 𝓢(E, ℂ)) (u : 𝓢'(E, ℂ)) :
+    ∂_{v} (smulLeftCLM ℂ g u) = smulLeftCLM ℂ (⇑(∂_{v} g : 𝓢(E, ℂ))) u +
+      smulLeftCLM ℂ g (∂_{v} u) := by
+  ext φ
+  simp only [TemperedDistribution.lineDerivOp_apply_apply,
+    TemperedDistribution.smulLeftCLM_apply_apply, add_apply, map_neg]
+  rw [schwartz_deriv_product, map_add]
+  abel
+
+variable [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+
+lemma multiplier_add {g h : E → ℂ} (hg : g.HasTemperateGrowth) (hh : h.HasTemperateGrowth)
+    (u : 𝓢'(E, ℂ)) :
+    fourierMultiplierCLM ℂ (g + h) u = fourierMultiplierCLM ℂ g u +
+      fourierMultiplierCLM ℂ h u := by
+  simp [fourierMultiplierCLM_apply, smulLeftCLM_add hg hh]
+
+lemma bessel_two_laplacian (u : 𝓢'(E, ℂ)) :
+    besselPotential E ℂ 2 u = u - (((2 * Real.pi)^2 : ℝ) : ℂ)⁻¹ • Δ u := by
+  have hpi : (((2 * Real.pi)^2 : ℝ) : ℂ) ≠ 0 := by exact_mod_cast (by positivity : (2 * Real.pi)^2 ≠ 0)
+  have hlo := TemperedDistribution.laplacian_eq_fourierMultiplierCLM u
+  rw [RCLike.real_smul_eq_coe_smul (K := ℂ), RCLike.ofReal_neg] at hlo
+  change Δ u = -(((2 * Real.pi)^2 : ℝ) : ℂ) • _ at hlo
+  rw [hlo, smul_smul, mul_neg, inv_mul_cancel₀ hpi, neg_one_smul,
+    sub_neg_eq_add]
+  unfold besselPotential
+  have hs : (fun x : E => (((1 + ‖x‖^2)^((2:ℝ)/2) : ℝ) : ℂ)) =
+      (fun _ : E => (1:ℂ)) + (fun x : E => ((‖x‖^2 : ℝ) : ℂ)) := by
+    ext x
+    norm_num
+  rw [hs, multiplier_add (by fun_prop) (by fun_prop)]
+  simp
+
+lemma memSobolev_sum {ι : Type*} (s : ℝ) (S : Finset ι) (u : ι → 𝓢'(E, ℂ))
+    (hu : ∀ i ∈ S, MemSobolev s 2 (u i)) : MemSobolev s 2 (∑ i ∈ S, u i) := by
+  classical
+  induction S using Finset.induction_on with
+  | empty => simp
+  | @insert i S hi ih =>
+    rw [Finset.sum_insert hi]
+    exact (hu i (Finset.mem_insert_self ..)).add
+      (ih (fun j hj => hu j (Finset.mem_insert_of_mem hj)))
+
+ 
+lemma memSobolev_of_laplacian {s : ℝ} {u : 𝓢'(E, ℂ)}
+    (hu : MemSobolev s 2 u) (hΔ : MemSobolev s 2 (Δ u)) : MemSobolev (s + 2) 2 u := by
+  rw [add_comm, ← memSobolev_besselPotential_iff]
+  rw [bessel_two_laplacian]
+  exact hu.sub (hΔ.smul _)
+
+lemma memSobolev_of_second_basis {s : ℝ} {u : 𝓢'(E, ℂ)}
+    (hu : MemSobolev s 2 u)
+    (hD : ∀ j, MemSobolev s 2 (∂_{stdOrthonormalBasis ℝ E j}
+      (∂_{stdOrthonormalBasis ℝ E j} u))) : MemSobolev (s + 2) 2 u := by
+  apply memSobolev_of_laplacian hu
+  rw [TemperedDistribution.laplacian_eq_sum (stdOrthonormalBasis ℝ E)]
+  exact memSobolev_sum _ _ _ (fun j _ => hD j)
+
+lemma memSobolev_zero_mul_schwartz (g : 𝓢(E, ℂ)) {u : 𝓢'(E, ℂ)}
+    (hu : MemSobolev 0 2 u) : MemSobolev 0 2 (smulLeftCLM ℂ g u) := by
+  obtain ⟨v, hv⟩ := memSobolev_zero_iff.mp hu
+  apply memSobolev_zero_iff.mpr
+  refine ⟨multiply g.toBoundedContinuousFunction v, ?_⟩
+  rw [hv, multiply_distribution _ g.hasTemperateGrowth]
+  rfl
+
+ 
+
+theorem memSobolev_even_mul_schwartz (k : ℕ) (g : 𝓢(E, ℂ)) {u : 𝓢'(E, ℂ)}
+    (hu : MemSobolev (2 * (k : ℝ)) 2 u) :
+    MemSobolev (2 * (k : ℝ)) 2 (smulLeftCLM ℂ g u) := by
+  induction k generalizing g u with
+  | zero => simpa using memSobolev_zero_mul_schwartz g (by simpa using hu)
+  | succ k ih =>
+    have he : 2 * ((k + 1 : ℕ) : ℝ) = 2 * (k : ℝ) + 2 := by push_cast; ring
+    rw [he] at hu ⊢
+    apply memSobolev_of_second_basis (ih g (hu.mono (by linarith)))
+    intro j
+    let v := stdOrthonormalBasis ℝ E j
+    have hu' : MemSobolev (2 * (k : ℝ)) 2 (∂_{v} u) :=
+      hu.lineDerivOp.mono (by linarith)
+    have hu'' : MemSobolev (2 * (k : ℝ)) 2 (∂_{v} (∂_{v} u)) :=
+      hu.lineDerivOp.lineDerivOp.mono (by linarith)
+    change MemSobolev _ _ (∂_{v} (∂_{v} (smulLeftCLM ℂ g u)))
+    rw [weak_deriv_product, lineDerivOp_add, weak_deriv_product, weak_deriv_product]
+    exact ((ih _ (hu.mono (by linarith))).add (ih _ hu')).add
+      ((ih _ hu').add (ih _ hu''))
+
+ 
+
+def liftOperatorValue (s t : ℝ) (T : 𝓢'(E, ℂ) →L[ℂ] 𝓢'(E, ℂ))
+    (hT : ∀ u, MemSobolev s 2 u → MemSobolev t 2 (T u)) (u : L2 E) : L2 E :=
+  ((memSobolev_iff_realize t (T (realize s u))).mp
+    (hT _ (realize_memSobolev s u))).choose
+
+lemma realize_liftOperatorValue (s t : ℝ) (T : 𝓢'(E, ℂ) →L[ℂ] 𝓢'(E, ℂ))
+    (hT : ∀ u, MemSobolev s 2 u → MemSobolev t 2 (T u)) (u : L2 E) :
+    realize t (liftOperatorValue s t T hT u) = T (realize s u) :=
+  ((memSobolev_iff_realize t _).mp (hT _ (realize_memSobolev s u))).choose_spec
+
+def liftOperatorLinear (s t : ℝ)
+    (T : 𝓢'(E, ℂ) →L[ℂ] 𝓢'(E, ℂ))
+    (hT : ∀ u, MemSobolev s 2 u → MemSobolev t 2 (T u)) : L2 E →ₗ[ℂ] L2 E where
+  toFun := liftOperatorValue s t T hT
+  map_add' u v := by
+    apply realize_injective t
+    simp only [map_add, realize_liftOperatorValue]
+  map_smul' c u := by
+    apply realize_injective t
+    simp only [map_smul, realize_liftOperatorValue, RingHom.id_apply]
+
+lemma realize_liftOperatorLinear (s t : ℝ)
+    (T : 𝓢'(E, ℂ) →L[ℂ] 𝓢'(E, ℂ))
+    (hT : ∀ u, MemSobolev s 2 u → MemSobolev t 2 (T u)) (u : L2 E) :
+    realize t (liftOperatorLinear s t T hT u) = T (realize s u) :=
+  ((memSobolev_iff_realize t _).mp (hT _ (realize_memSobolev s u))).choose_spec
+
+def liftOperator (s t : ℝ) (T : 𝓢'(E, ℂ) →L[ℂ] 𝓢'(E, ℂ))
+    (hT : ∀ u, MemSobolev s 2 u → MemSobolev t 2 (T u)) : L2 E →L[ℂ] L2 E where
+  toLinearMap := liftOperatorLinear s t T hT
+  cont := by
+    apply LinearMap.continuous_of_seq_closed_graph
+    intro u x y hu hv
+    apply realize_injective t
+    have h₁ := ((realize t).continuous.tendsto y).comp hv
+    have h₂ := (T.continuous.tendsto (realize s x)).comp
+      (((realize s).continuous.tendsto x).comp hu)
+    change Tendsto (fun n => realize t (liftOperatorLinear s t T hT (u n)))
+      atTop (nhds (realize t y)) at h₁
+    simp_rw [realize_liftOperatorLinear] at h₁
+    exact (tendsto_nhds_unique h₁ h₂).trans (realize_liftOperatorLinear s t T hT x).symm
+
+lemma realize_liftOperator (s t : ℝ)
+    (T : 𝓢'(E, ℂ) →L[ℂ] 𝓢'(E, ℂ))
+    (hT : ∀ u, MemSobolev s 2 u → MemSobolev t 2 (T u)) (u : L2 E) :
+    realize t (liftOperator s t T hT u) = T (realize s u) :=
+  realize_liftOperatorLinear s t T hT u
+
+ 
+def cutoff (k : ℕ) (g : 𝓢(E, ℂ)) : L2 E →L[ℂ] L2 E :=
+  liftOperator (2 * (k : ℝ)) (2 * (k : ℝ)) (smulLeftCLM ℂ g)
+    (fun _ hu => memSobolev_even_mul_schwartz k g hu)
+
+lemma realize_cutoff (k : ℕ) (g : 𝓢(E, ℂ)) (u : L2 E) :
+    realize (2 * (k : ℝ)) (cutoff k g u) =
+      smulLeftCLM ℂ g (realize (2 * (k : ℝ)) u) :=
+  realize_liftOperator _ _ _ _ _
+
+end SobolevChart
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/SobolevOpenness.lean b/lean/OAI/Geometry/PrescribedPotential/SobolevOpenness.lean
new file mode 100644
index 0000000000000000000000000000000000000000..5ec7ddc511c88fa06378252fb7dbd651048d285f
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/SobolevOpenness.lean
@@ -0,0 +1,84 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.PatchCutoffs
+import OAI.Geometry.PrescribedPotential.AugmentedVolume
+import OAI.Geometry.PrescribedPotential.CompletedResolvent
+
+namespace OAI
+
+/-! Sobolev Openness. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology
+open scoped ContDiff Classical
+namespace LocalNonlinearInverse
+variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
+  [NormedAddCommGroup F] [NormedSpace ℝ F] [CompleteSpace F]
+lemma image_mem_nhds {f : E → F} {L : E →L[ℝ] F} {a : E}
+    (hf : HasStrictFDerivAt f L a) (hL : Function.Surjective L)
+    {U : Set E} (hU : U ∈ 𝓝 a) : f '' U ∈ 𝓝 (f a) := by
+  rw [← hf.map_nhds_eq_of_surj (LinearMap.range_eq_top.mpr hL)]
+  change f ⁻¹' (f '' U) ∈ 𝓝 a
+  exact mem_of_superset hU (fun x hx => ⟨x,hx,rfl⟩)
+end LocalNonlinearInverse
+
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  [ConnectedSpace X] {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+local instance sobolevOpennessNormedAddCommGroup (s : ℝ) :
+    NormedAddCommGroup (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedAddCommGroup
+local instance sobolevOpennessNormedSpace (s : ℝ) : NormedSpace ℝ (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedSpace
+local instance sobolevOpennessIsTopologicalAddGroup (s : ℝ) :
+    IsTopologicalAddGroup (D.localizers.RealSobolev s) :=
+  Submodule.isTopologicalAddGroup _
+local instance sobolevOpennessContinuousSMul (s : ℝ) :
+    ContinuousSMul ℝ (D.localizers.RealSobolev s) :=
+  SMulMemClass.continuousSMul _
+
+lemma augmentedVolume_image_nhds (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) (x₀ : X)
+    {U : Set (D.localizers.RealSobolev ((k : ℝ)+2) × ℝ)} (hU : U ∈ 𝓝 0) :
+    D.augmentedVolume k hk x₀ '' U ∈ 𝓝 (D.realConstants (k : ℝ) 1, 0) := by
+  obtain ⟨m,hm,he⟩ := D.exists_completedError_small
+  obtain ⟨P⟩ := D.localizers.constantProjection_exists
+  have hs : (Module.finrank ℝ (EC d) : ℝ) < 2*((k : ℝ)+2) := by
+    have hh : (Module.finrank ℝ (EC d) : ℝ) < (k : ℝ) := by exact_mod_cast hk
+    linarith [Nat.cast_nonneg (α := ℝ) k]
+  have hi := LocalNonlinearInverse.image_mem_nhds
+    (D.augmentedVolume_hasStrictFDerivAt_zero k hk x₀)
+    (D.realAugmentedL_bijective m hm he P k hs x₀).2 hU
+  rwa [D.augmentedVolume_zero] at hi
+
+ 
+
+theorem volume_eventually_solvable (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) (x₀ : X)
+    {U : Set (D.localizers.RealSobolev ((k : ℝ)+2) × ℝ)} (hU : U ∈ 𝓝 0) :
+    ∀ᶠ f in 𝓝 (D.realConstants (k : ℝ) 1),
+      ∃ (u : D.localizers.RealSobolev ((k : ℝ)+2)) (b : ℝ),
+        (u,b) ∈ U ∧ D.realVolume k hk u = Real.exp b • f ∧
+        D.realEvaluation ((k : ℝ)+2) x₀ u = 0 := by
+  have hi := D.augmentedVolume_image_nhds k hk x₀ hU
+  have ht : Tendsto (fun f : D.localizers.RealSobolev (k : ℝ) => (f, (0 : ℝ)))
+      (𝓝 (D.realConstants (k : ℝ) 1)) (𝓝 (D.realConstants (k : ℝ) 1, 0)) :=
+    continuousAt_id.prodMk continuousAt_const
+  filter_upwards [ht hi] with f hf
+  obtain ⟨⟨u,b⟩,hu,hv⟩ := hf
+  refine ⟨u,b,hu,?_,congrArg Prod.snd hv⟩
+  have hh := congrArg (fun z => Real.exp b • z.1) hv
+  change Real.exp b • (Real.exp (-b) • D.realVolume k hk u) = Real.exp b • f at hh
+  simpa only [smul_smul, ← Real.exp_add, add_neg_cancel, Real.exp_zero, one_smul] using hh
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/SobolevParametrix.lean b/lean/OAI/Geometry/PrescribedPotential/SobolevParametrix.lean
new file mode 100644
index 0000000000000000000000000000000000000000..f4b5315a05c1fbbf61801c34fc1a8d4c39c065a9
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/SobolevParametrix.lean
@@ -0,0 +1,198 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.FrozenPoisson
+
+namespace OAI
+
+/-! Sobolev Parametrix. -/
+
+section
+
+ 
+
+noncomputable section
+open MeasureTheory FourierTransform TemperedDistribution
+open scoped SchwartzMap BoundedContinuousFunction ComplexOrder MatrixOrder Real
+
+namespace SobolevChart
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+
+abbrev L2 (E : Type*) [MeasureSpace E] := Lp ℂ 2 (volume : Measure E)
+
+ 
+
+def realize (s : ℝ) : L2 E →L[ℂ] 𝓢'(E, ℂ) :=
+  besselPotential E ℂ (-s) ∘L Lp.toTemperedDistributionCLM ℂ volume 2
+
+lemma bessel_realize (s : ℝ) (u : L2 E) :
+    besselPotential E ℂ s (realize s u) = (u : 𝓢'(E, ℂ)) := by
+  simp [realize]
+
+lemma realize_memSobolev (s : ℝ) (u : L2 E) : MemSobolev s 2 (realize s u) :=
+  ⟨u, bessel_realize s u⟩
+
+lemma l2_injective : Function.Injective (Lp.toTemperedDistributionCLM ℂ (volume : Measure E) 2) :=
+  LinearMap.ker_eq_bot.mp Lp.ker_toTemperedDistributionCLM_eq_bot
+
+lemma realize_injective (s : ℝ) : Function.Injective (realize (E := E) s) := by
+  intro u v h
+  apply l2_injective
+  rw [Lp.toTemperedDistributionCLM_apply, Lp.toTemperedDistributionCLM_apply]
+  simpa only [bessel_realize] using congrArg (besselPotential E ℂ s) h
+
+lemma memSobolev_iff_realize (s : ℝ) (u : 𝓢'(E, ℂ)) :
+    MemSobolev s 2 u ↔ ∃ v : L2 E, realize s v = u := by
+  constructor
+  · rintro ⟨v, hv⟩
+    refine ⟨v, ?_⟩
+    change besselPotential E ℂ (-s) (v : 𝓢'(E, ℂ)) = u
+    exact (besselPotential_neg_apply_eq_iff s _ u).mpr hv
+  · rintro ⟨v, rfl⟩
+    exact realize_memSobolev s v
+
+ 
+def multiply (g : E →ᵇ ℂ) : L2 E →L[ℂ] L2 E :=
+  (ContinuousLinearMap.mul ℂ ℂ).holderL volume ⊤ 2 2 (g.memLp_top.toLp g)
+
+lemma multiply_eq (g : E →ᵇ ℂ) (u : L2 E) :
+    multiply g u = (g.memLp_top.toLp g : Lp ℂ ⊤ volume) • u := by
+  apply Lp.ext
+  filter_upwards [(ContinuousLinearMap.mul ℂ ℂ).coeFn_holder
+    (r := 2) (g.memLp_top.toLp g : Lp ℂ ⊤ volume) u,
+    Lp.coeFn_lpSMul (r := 2) (g.memLp_top.toLp g : Lp ℂ ⊤ volume) u] with x h₁ h₂
+  exact h₁.trans (by simpa [Pi.smul_apply, smul_eq_mul] using h₂.symm)
+
+lemma multiply_norm_le (g : E →ᵇ ℂ) (u : L2 E) :
+    ‖multiply g u‖ ≤ ‖g‖ * ‖u‖ := by
+  rw [multiply_eq]
+  apply Lp.norm_le_mul_norm_of_ae_le_mul
+  filter_upwards [Lp.coeFn_lpSMul (r := 2)
+    (g.memLp_top.toLp g : Lp ℂ ⊤ volume) u,
+    (g.memLp_top (μ := volume)).coeFn_toLp] with x h₁ h₂
+  rw [h₁, Pi.smul_apply', h₂, norm_smul]
+  exact mul_le_mul_of_nonneg_right (g.norm_coe_le_norm x) (norm_nonneg _)
+
+lemma multiply_distribution (g : E →ᵇ ℂ) (hg : (g : E → ℂ).HasTemperateGrowth)
+    (u : L2 E) :
+    (multiply g u : 𝓢'(E, ℂ)) = smulLeftCLM ℂ g (u : 𝓢'(E, ℂ)) := by
+  rw [multiply_eq]
+  exact Lp.toTemperedDistribution_smul_eq hg g.memLp_top u
+
+ 
+def multiplier (g : E →ᵇ ℂ) : L2 E →L[ℂ] L2 E :=
+  fourierInvCLM ℂ (L2 E) ∘L multiply g ∘L fourierCLM ℂ (L2 E)
+
+lemma multiplier_norm_le (g : E →ᵇ ℂ) (u : L2 E) :
+    ‖multiplier g u‖ ≤ ‖g‖ * ‖u‖ := by
+  change ‖(Lp.fourierTransformₗᵢ E ℂ).symm (multiply g (𝓕 u))‖ ≤ _
+  rw [(Lp.fourierTransformₗᵢ E ℂ).symm.norm_map]
+  simpa only [Lp.norm_fourier_eq] using multiply_norm_le g (𝓕 u)
+
+lemma multiplier_distribution (g : E →ᵇ ℂ) (hg : (g : E → ℂ).HasTemperateGrowth)
+    (u : L2 E) :
+    (multiplier g u : 𝓢'(E, ℂ)) = fourierMultiplierCLM ℂ g (u : 𝓢'(E, ℂ)) := by
+  change ((𝓕⁻ (multiply g (𝓕 u)) : L2 E) : 𝓢'(E, ℂ)) = _
+  rw [← Lp.fourierInv_toTemperedDistribution_eq, multiply_distribution g hg,
+    ← Lp.fourier_toTemperedDistribution_eq]
+  rfl
+
+lemma bessel_multiplier_comm {g : E → ℂ} (hg : g.HasTemperateGrowth)
+    (s : ℝ) (u : 𝓢'(E, ℂ)) :
+    besselPotential E ℂ s (fourierMultiplierCLM ℂ g u) =
+      fourierMultiplierCLM ℂ g (besselPotential E ℂ s u) := by
+  rw [besselPotential, fourierMultiplierCLM_fourierMultiplierCLM_apply hg (by fun_prop),
+    fourierMultiplierCLM_fourierMultiplierCLM_apply (by fun_prop) hg, mul_comm]
+
+lemma realize_multiplier (s : ℝ) (g : E →ᵇ ℂ)
+    (hg : (g : E → ℂ).HasTemperateGrowth) (u : L2 E) :
+    realize s (multiplier g u) = fourierMultiplierCLM ℂ g (realize s u) := by
+  change besselPotential E ℂ (-s) (multiplier g u : 𝓢'(E, ℂ)) = _
+  rw [multiplier_distribution g hg, bessel_multiplier_comm hg]
+  rfl
+
+end SobolevChart
+
+namespace FrozenPoisson
+open EllipticKernel SobolevChart
+variable {n : ℕ}
+
+def gainSymbol (H : Matrix (Fin n) (Fin n) ℂ) (ξ : EC n) : ℂ :=
+  (1 + ‖ξ‖^2 : ℝ) * inverseSymbol H ξ
+
+lemma gainSymbol_temperate (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef) :
+    (gainSymbol H).HasTemperateGrowth := by
+  have hi := inverseSymbol_temperate H hH
+  unfold gainSymbol
+  fun_prop
+
+ 
+def ellipticBound (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef) : ℝ :=
+  max 1 (inverseSymbol_weighted_bounded H hH).choose
+
+lemma ellipticBound_pos (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef) :
+    0 < ellipticBound H hH := lt_of_lt_of_le zero_lt_one (le_max_left _ _)
+
+lemma gainSymbol_bound (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef) (ξ : EC n) :
+    ‖gainSymbol H ξ‖ ≤ ellipticBound H hH :=
+  ((inverseSymbol_weighted_bounded H hH).choose_spec ξ).trans (le_max_right _ _)
+
+def gainSymbolBCF (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef) : EC n →ᵇ ℂ :=
+  BoundedContinuousFunction.ofNormedAddCommGroup (gainSymbol H)
+    (gainSymbol_temperate H hH).1.continuous (ellipticBound H hH) (gainSymbol_bound H hH)
+
+lemma gainSymbolBCF_norm (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef) :
+    ‖gainSymbolBCF H hH‖ ≤ ellipticBound H hH :=
+  (BoundedContinuousFunction.norm_le (ellipticBound_pos H hH).le).mpr (gainSymbol_bound H hH)
+
+ 
+def hilbertResolvent (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef) :
+    L2 (EC n) →L[ℂ] L2 (EC n) := multiplier (gainSymbolBCF H hH)
+
+lemma hilbertResolvent_bound (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (u : L2 (EC n)) :
+    ‖hilbertResolvent H hH u‖ ≤ ellipticBound H hH * ‖u‖ :=
+  (multiplier_norm_le (gainSymbolBCF H hH) u).trans
+    (mul_le_mul_of_nonneg_right (gainSymbolBCF_norm H hH) (norm_nonneg u))
+
+lemma bessel_two_resolvent (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (u : 𝓢'(EC n, ℂ)) :
+    besselPotential (EC n) ℂ 2 (resolvent H u) =
+      fourierMultiplierCLM ℂ (gainSymbol H) u := by
+  rw [besselPotential, resolvent,
+    fourierMultiplierCLM_fourierMultiplierCLM_apply (inverseSymbol_temperate H hH)
+      (by fun_prop)]
+  congr 2
+  ext ξ
+  simp [gainSymbol, mul_comm]
+
+lemma hilbertResolvent_realize (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (s : ℝ) (u : L2 (EC n)) :
+    realize (s + 2) (hilbertResolvent H hH u) = resolvent H (realize s u) := by
+  change besselPotential (EC n) ℂ (-(s + 2))
+    (Lp.toTemperedDistributionCLM ℂ MeasureTheory.volume 2 (hilbertResolvent H hH u)) = _
+  rw [besselPotential_neg_apply_eq_iff]
+  have hc : resolvent H (realize s u) = besselPotential (EC n) ℂ (-s)
+      (resolvent H (Lp.toTemperedDistributionCLM ℂ MeasureTheory.volume 2 u)) :=
+    (bessel_multiplier_comm (inverseSymbol_temperate H hH) (-s) _).symm
+  rw [hc, besselPotential_besselPotential_apply]
+  have hs : -s + (s + 2) = 2 := by ring
+  rw [hs, bessel_two_resolvent H hH]
+  exact (multiplier_distribution (gainSymbolBCF H hH) (gainSymbol_temperate H hH) u).symm
+
+ 
+
+theorem hilbert_frozen_inverse (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (s : ℝ) (f : L2 (EC n)) :
+    let u := hilbertResolvent H hH f
+    realize (s + 2) u - frozenDifferential H (realize (s + 2) u) = realize s f ∧
+      ‖u‖ ≤ ellipticBound H hH * ‖f‖ := by
+  refine ⟨?_, hilbertResolvent_bound H hH f⟩
+  rw [← shifted_eq_identity_sub H hH, hilbertResolvent_realize,
+    shifted_resolvent H hH]
+
+end FrozenPoisson
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/SobolevProduct.lean b/lean/OAI/Geometry/PrescribedPotential/SobolevProduct.lean
new file mode 100644
index 0000000000000000000000000000000000000000..f07964104bd7108700f515bf840b8afc0b380314
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/SobolevProduct.lean
@@ -0,0 +1,174 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CutoffHigher
+import OAI.Geometry.PrescribedPotential.StrongSobolevEmbedding
+
+namespace OAI
+
+/-! Sobolev Product. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory TemperedDistribution LineDeriv
+open scoped SchwartzMap BoundedContinuousFunction ContDiff Classical
+namespace SobolevChart
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+
+lemma boundedDistribution_schwartz (f : 𝓢(E, ℂ)) :
+    boundedDistributionCLM f.toBoundedContinuousFunction = (f : 𝓢'(E, ℂ)) := by
+  ext φ
+  rw [boundedDistributionCLM_apply, boundedDistribution_apply]
+  rfl
+
+lemma strongEmbedding_schwartz (s : ℝ) (hs : (Module.finrank ℝ E : ℝ) < 2*s)
+    (f : 𝓢(E, ℂ)) : strongEmbedding s hs (schwartzCoord s f) = f.toBoundedContinuousFunction := by
+  apply boundedDistribution_injective
+  rw [strongEmbedding_distribution, realize_schwartzCoord, boundedDistribution_schwartz]
+
+lemma schwartz_word_sup_bound (ws : List E) (s : ℝ)
+    (hs : (Module.finrank ℝ E : ℝ) < 2*(s-ws.length)) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ f : 𝓢(E, ℂ),
+      ‖(schwartzWord ws f).toBoundedContinuousFunction‖ ≤ C * ‖schwartzCoord s f‖ := by
+  obtain ⟨C,hC,hb⟩ := coreBound_word ws (s := s) (t := s-ws.length) le_rfl
+  refine ⟨‖strongEmbedding (E := E) (s-ws.length) hs‖*C,
+    mul_nonneg (norm_nonneg _) hC, fun f => ?_⟩
+  rw [← strongEmbedding_schwartz (s-ws.length) hs]
+  exact ((strongEmbedding (s-ws.length) hs).le_opNorm _).trans (by
+    simpa only [mul_assoc] using mul_le_mul_of_nonneg_left (hb f)
+      (norm_nonneg (strongEmbedding (E := E) (s-ws.length) hs)))
+
+ 
+def BinaryCoreBound (s t : ℝ) (T : 𝓢(E, ℂ) → 𝓢(E, ℂ) → 𝓢(E, ℂ)) : Prop :=
+  ∃ C : ℝ, 0 ≤ C ∧ ∀ f g,
+    ‖schwartzCoord t (T f g)‖ ≤ C * (‖schwartzCoord s f‖ * ‖schwartzCoord s g‖)
+
+lemma BinaryCoreBound.add {s t : ℝ} {T U : 𝓢(E, ℂ) → 𝓢(E, ℂ) → 𝓢(E, ℂ)}
+    (hT : BinaryCoreBound s t T) (hU : BinaryCoreBound s t U) :
+    BinaryCoreBound s t (fun f g => T f g + U f g) := by
+  obtain ⟨C,hC,hc⟩ := hT
+  obtain ⟨D,hD,hd⟩ := hU
+  refine ⟨C+D, add_nonneg hC hD, fun f g => ?_⟩
+  rw [schwartzCoord_add, add_mul]
+  exact (norm_add_le _ _).trans (add_le_add (hc f g) (hd f g))
+
+lemma binaryBound_raise_one {s t : ℝ} (T : 𝓢(E, ℂ) → 𝓢(E, ℂ) → 𝓢(E, ℂ))
+    (hT : BinaryCoreBound s t T)
+    (hD : ∀ j, BinaryCoreBound s t (fun f g => ∂_{stdOrthonormalBasis ℝ E j} (T f g))) :
+    BinaryCoreBound s (t+1) T := by
+  obtain ⟨C,hC,hc⟩ := hT
+  choose D hD0 hd using hD
+  let c : ℝ := ‖(((2*Real.pi)^2 : ℝ) : ℂ)⁻¹‖
+  refine ⟨‖raiseZero (E := E) t‖*C + c*∑ j, ‖raiseDeriv t (stdOrthonormalBasis ℝ E j)‖*D j,
+    add_nonneg (mul_nonneg (norm_nonneg _) hC)
+      (mul_nonneg (norm_nonneg _) (Finset.sum_nonneg (fun j _ => mul_nonneg (norm_nonneg _) (hD0 j)))),
+    fun f g => ?_⟩
+  rw [schwartzCoord_raise_one]
+  calc
+    _ ≤ ‖raiseZero t (schwartzCoord t (T f g))‖ +
+        c*‖∑ j, raiseDeriv t (stdOrthonormalBasis ℝ E j)
+          (schwartzCoord t (∂_{stdOrthonormalBasis ℝ E j} (T f g)))‖ := by
+      simpa only [norm_smul] using norm_sub_le
+        (raiseZero t (schwartzCoord t (T f g)))
+        ((((2*Real.pi)^2 : ℝ) : ℂ)⁻¹ • ∑ j, raiseDeriv t (stdOrthonormalBasis ℝ E j)
+          (schwartzCoord t (∂_{stdOrthonormalBasis ℝ E j} (T f g))))
+    _ ≤ (‖raiseZero (E := E) t‖*C)*(‖schwartzCoord s f‖*‖schwartzCoord s g‖) +
+        c*∑ j, (‖raiseDeriv t (stdOrthonormalBasis ℝ E j)‖*D j)*
+          (‖schwartzCoord s f‖*‖schwartzCoord s g‖) := by
+      apply add_le_add
+      · exact ((raiseZero t).le_opNorm _).trans (by
+          simpa only [mul_assoc] using mul_le_mul_of_nonneg_left (hc f g)
+            (norm_nonneg (raiseZero (E := E) t)))
+      · apply mul_le_mul_of_nonneg_left _ (norm_nonneg _)
+        apply (norm_sum_le _ _).trans
+        apply Finset.sum_le_sum
+        intro j _
+        exact ((raiseDeriv t (stdOrthonormalBasis ℝ E j)).le_opNorm _).trans (by
+          simpa only [mul_assoc] using mul_le_mul_of_nonneg_left (hd j f g)
+            (norm_nonneg (raiseDeriv t (stdOrthonormalBasis ℝ E j))))
+    _ = _ := by rw [← Finset.sum_mul]; ring
+
+lemma binaryBound_of_words (k : ℕ) {s : ℝ}
+    (T : 𝓢(E, ℂ) → 𝓢(E, ℂ) → 𝓢(E, ℂ))
+    (h : ∀ ws : List E, ws.length ≤ k →
+      BinaryCoreBound s 0 (fun f g => schwartzWord ws (T f g))) :
+    BinaryCoreBound s (k : ℝ) T := by
+  induction k generalizing T with
+  | zero => simpa [schwartzWord] using h [] (by simp)
+  | succ k ih =>
+    have ht := ih T (fun ws hw => h ws (by omega))
+    have hd (j : Fin (Module.finrank ℝ E)) := ih
+      (fun f g => ∂_{stdOrthonormalBasis ℝ E j} (T f g))
+      (fun ws hw => by
+        have hh := h (ws ++ [stdOrthonormalBasis ℝ E j])
+          (by simp only [List.length_append, List.length_cons, List.length_nil]; omega)
+        simpa only [schwartzWord_append, schwartzWord, ContinuousLinearMap.comp_apply,
+          ContinuousLinearMap.id_apply, lineDerivOpCLM_apply] using hh)
+    simpa only [Nat.cast_succ] using binaryBound_raise_one T ht hd
+
+omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] in
+lemma schwartz_product_swap (f g : 𝓢(E, ℂ)) :
+    SchwartzMap.smulLeftCLM ℂ f g = SchwartzMap.smulLeftCLM ℂ g f := by
+  ext x
+  simp only [SchwartzMap.smulLeftCLM_apply_apply f.hasTemperateGrowth,
+    SchwartzMap.smulLeftCLM_apply_apply g.hasTemperateGrowth, smul_eq_mul, mul_comm]
+
+lemma binaryBound_product_words_zero (m : ℕ) (hm : Module.finrank ℝ E < m)
+    (p q : List E) (hpq : p.length+q.length ≤ m) :
+    BinaryCoreBound (m : ℝ) 0 (fun f g =>
+      SchwartzMap.smulLeftCLM ℂ (schwartzWord p f) (schwartzWord q g)) := by
+  have base (p q : List E) (hpq : p.length+q.length ≤ m) (hp : 2*p.length ≤ m) :
+      BinaryCoreBound (m : ℝ) 0 (fun f g =>
+        SchwartzMap.smulLeftCLM ℂ (schwartzWord p f) (schwartzWord q g)) := by
+    have hs : (Module.finrank ℝ E : ℝ) < 2*((m : ℝ)-p.length) := by
+      have h₁ : (Module.finrank ℝ E : ℝ) < m := by exact_mod_cast hm
+      have h₂ : 2*(p.length : ℝ) ≤ m := by exact_mod_cast hp
+      linarith
+    obtain ⟨C,hC,hc⟩ := schwartz_word_sup_bound p (m : ℝ) hs
+    obtain ⟨D,hD,hd⟩ := coreBound_word q (s := (m : ℝ)) (t := 0) (by
+      have hh : (q.length : ℝ) ≤ m := by exact_mod_cast (show q.length ≤ m by omega)
+      linarith)
+    refine ⟨C*D, mul_nonneg hC hD, fun f g => ?_⟩
+    rw [schwartzCoord_product_zero]
+    exact (multiply_norm_le _ _).trans (by
+      have hh := mul_le_mul (hc f) (hd g) (norm_nonneg _)
+        (mul_nonneg hC (norm_nonneg _))
+      convert hh using 1; ring)
+  by_cases hp : 2*p.length ≤ m
+  · exact base p q hpq hp
+  · obtain ⟨C,hC,hc⟩ := base q p (by omega) (by omega)
+    refine ⟨C,hC, fun f g => ?_⟩
+    dsimp only
+    rw [schwartz_product_swap (schwartzWord p f) (schwartzWord q g)]
+    simpa only [mul_comm] using hc g f
+
+lemma binaryBound_word_product (m : ℕ) (hm : Module.finrank ℝ E < m)
+    (ws p q : List E) (hw : ws.length+p.length+q.length ≤ m) :
+    BinaryCoreBound (m : ℝ) 0 (fun f g => schwartzWord ws
+      (SchwartzMap.smulLeftCLM ℂ (schwartzWord p f) (schwartzWord q g))) := by
+  induction ws generalizing p q with
+  | nil => exact binaryBound_product_words_zero m hm p q (by simpa using hw)
+  | cons v ws ih =>
+    have h₁ := ih (v::p) q (by simp only [List.length_cons] at *; omega)
+    have h₂ := ih p (v::q) (by simp only [List.length_cons] at *; omega)
+    convert h₁.add h₂ using 1
+    funext f g
+    change ∂_{v} (schwartzWord ws (SchwartzMap.smulLeftCLM ℂ (schwartzWord p f) (schwartzWord q g))) = _
+    rw [← GlobalElliptic.word_deriv_comm, schwartz_deriv_product, map_add]
+    rfl
+
+ 
+theorem schwartz_sobolev_product_bound (m : ℕ) (hm : Module.finrank ℝ E < m) :
+    BinaryCoreBound (m : ℝ) (m : ℝ) (fun f g : 𝓢(E, ℂ) => SchwartzMap.smulLeftCLM ℂ f g) := by
+  apply binaryBound_of_words m
+  intro ws hw
+  exact binaryBound_word_product m hm ws [] [] (by simpa using hw)
+
+end SobolevChart
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/SobolevReconstruction.lean b/lean/OAI/Geometry/PrescribedPotential/SobolevReconstruction.lean
new file mode 100644
index 0000000000000000000000000000000000000000..27f2b80d1bbf720578d9d56e692ad9bcca7eaf1f
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/SobolevReconstruction.lean
@@ -0,0 +1,97 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.LiteralWeakHessian
+import OAI.Geometry.PrescribedPotential.GlobalSmooth
+
+namespace OAI
+
+/-! Sobolev Reconstruction. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory TemperedDistribution LineDeriv
+open scoped ContDiff SchwartzMap Classical
+namespace SobolevChart
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+
+lemma memSobolev_of_first_basis {s : ℝ} {u : 𝓢'(E, ℂ)}
+    (hu : MemSobolev s 2 u)
+    (hD : ∀ j, MemSobolev s 2 (∂_{stdOrthonormalBasis ℝ E j} u)) :
+    MemSobolev (s+1) 2 u := by
+  have hh := memSobolev_of_second_basis (hu.mono (show s-1 ≤ s by linarith))
+    (fun j => (hD j).lineDerivOp (m := stdOrthonormalBasis ℝ E j))
+  convert hh using 1
+  ring
+
+def lowerCoord (s t : ℝ) (hst : t ≤ s) : L2 E →L[ℂ] L2 E :=
+  liftOperator s t (ContinuousLinearMap.id ℂ _) (fun _ hu => hu.mono hst)
+
+lemma realize_lowerCoord (s t : ℝ) (hst : t ≤ s) (u : L2 E) :
+    realize t (lowerCoord s t hst u) = realize s u :=
+  realize_liftOperator s t _ _ u
+
+lemma lowerCoord_schwartz (s t : ℝ) (hst : t ≤ s) (f : 𝓢(E,ℂ)) :
+    lowerCoord s t hst (schwartzCoord s f) = schwartzCoord t f := by
+  apply realize_injective t
+  rw [realize_lowerCoord,realize_schwartzCoord,realize_schwartzCoord]
+
+end SobolevChart
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace Localizers
+variable (D : Localizers A ι)
+
+def globalizeOrder (k : ℕ) (q : Fin A.count) (κ : ChartCutoff (A.euclideanChart q).target) :
+    L2 (EC d) →L[ℝ] D.Sobolev (k : ℝ) :=
+  ((D.embed (k : ℝ)).comp (globalize q κ)).extendOfNorm (schwartzCoordLinear (k : ℝ))
+
+lemma globalizeOrder_schwartz (k : ℕ) (q : Fin A.count)
+    (κ : ChartCutoff (A.euclideanChart q).target) (f : 𝓢(EC d,ℂ)) :
+    D.globalizeOrder k q κ (schwartzCoord (k : ℝ) f) = D.embed (k : ℝ) (globalize q κ f) := by
+  obtain ⟨C,hC,h⟩ := globalize_integer_bound D q κ k
+  exact LinearMap.extendOfNorm_eq (schwartzCoord_dense (k : ℝ)) ⟨C,h⟩ f
+
+lemma globalizeOrder_lower_zero (k : ℕ) (q : Fin A.count)
+    (κ : ChartCutoff (A.euclideanChart q).target) (u : L2 (EC d)) :
+    D.lower (k : ℝ) 0 (D.globalizeOrder k q κ u) =
+      D.globalizeL2 q κ (lowerCoord (k : ℝ) 0 (Nat.cast_nonneg k) u) := by
+  have hh : D.lower (k : ℝ) 0 ∘ D.globalizeOrder k q κ =
+      D.globalizeL2 q κ ∘ lowerCoord (k : ℝ) 0 (Nat.cast_nonneg k) := by
+    apply (schwartzCoord_dense (k : ℝ)).equalizer (by fun_prop) (by fun_prop)
+    funext f
+    simp only [Function.comp_apply,D.globalizeOrder_schwartz,D.lower_embed (Nat.cast_nonneg k),
+      lowerCoord_schwartz,D.globalizeL2_schwartz]
+  exact congr_fun hh u
+
+end Localizers
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+ 
+
+theorem sobolev_of_local_order (k : ℕ) (u : D.localizers.Sobolev 0)
+    (hu : ∀ p, MemSobolev (k : ℝ) 2 (D.localizers.distribution 0 p u)) :
+    ∃ v : D.localizers.Sobolev (k : ℝ), D.localizers.lower (k : ℝ) 0 v = u := by
+  choose f hf using fun p => (memSobolev_iff_realize (k : ℝ) _).mp (hu p)
+  refine ⟨∑ p, D.localizers.globalizeOrder k (D.patch p).index (D.cutoff p) (f p),?_⟩
+  rw [map_sum,← D.reconstruct_eq u,D.reconstruct_apply]
+  apply Finset.sum_congr rfl
+  intro p _
+  rw [D.localizers.globalizeOrder_lower_zero]
+  congr 1
+  apply realize_injective 0
+  rw [realize_lowerCoord,hf]
+  rfl
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/SourceTarget.lean b/lean/OAI/Geometry/PrescribedPotential/SourceTarget.lean
new file mode 100644
index 0000000000000000000000000000000000000000..122a62c171e4364d6068d8be765e51a3e53be05c
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/SourceTarget.lean
@@ -0,0 +1,75 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.AnalyticSupport
+
+namespace OAI
+
+/-! Source Target. -/
+
+section
+
+ 
+
+namespace Anticanonical.SourceSmooth
+open Filter
+open scoped ContDiff
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+
+lemma levi_congr {φ ψ : Coordinates d → ℝ} {z : Coordinates d}
+    (h : φ =ᶠ[nhds z] ψ) (v : Coordinates d) : levi φ z v = levi ψ z v := by
+  have h' := h.fderiv (𝕜 := ℝ)
+  simp only [levi, h'.fderiv_eq]
+
+lemma levi_sub_const (φ : Coordinates d → ℝ) (c : ℝ) (z v : Coordinates d) :
+    levi (fun w => φ w - c) z v = levi φ z v := by
+  have h : fderiv ℝ (fun w => φ w - c) = fderiv ℝ φ := by
+    funext w
+    exact fderiv_sub_const c
+  simp only [levi, h]
+
+theorem KaehlerMetric.ricci_eq_of_volume
+    (g : KaehlerMetric A) (h : SemipositiveAnticanonicalMetric A)
+    {c : ℝ} (hc : 0 < c)
+    (heq : ∀ i z, z ∈ (A.chart i).target →
+      g.volumeCoefficient i z = c * Real.exp (-h.weight i z))
+    (i : Fin A.count) {z : Coordinates d} (hz : z ∈ (A.chart i).target)
+    (v : Coordinates d) : g.ricciQuadratic i z v = levi (h.weight i) z v := by
+  have hlocal : g.ricciPotential i =ᶠ[nhds z]
+      (fun w => h.weight i w - Real.log c) := by
+    filter_upwards [(A.chart i).open_target.mem_nhds hz] with w hw
+    dsimp [KaehlerMetric.ricciPotential]
+    rw [heq i w hw, Real.log_mul (ne_of_gt hc) (ne_of_gt (Real.exp_pos _)),
+      Real.log_exp]
+    ring
+  exact (levi_congr hlocal v).trans (levi_sub_const _ _ _ _)
+
+ 
+
+def HasPrescribedVolumePotential (P : A.ProjectiveEmbedding)
+    (h : SemipositiveAnticanonicalMetric A) : Prop :=
+  ∃ (φ : SmoothRealFunction A) (c : ℝ), 0 < c ∧
+    (projectiveBackground P).PositivePotential φ ∧
+    ∀ i z, z ∈ (A.chart i).target →
+      ((projectiveBackground P).matrix i z + φ.hessian i z).det.re =
+        c * Real.exp (-h.weight i z)
+
+ 
+
+theorem prescribedRicci_of_potential (P : A.ProjectiveEmbedding)
+    (h : SemipositiveAnticanonicalMetric A) (φ : SmoothRealFunction A)
+    (hp : (projectiveBackground P).PositivePotential φ) {c : ℝ} (hc : 0 < c)
+    (heq : ∀ i z, z ∈ (A.chart i).target →
+      ((projectiveBackground P).matrix i z + φ.hessian i z).det.re =
+        c * Real.exp (-h.weight i z)) :
+    ∃ g : KaehlerMetric A, (∀ i z, g.matrix i z =
+        (projectiveBackground P).matrix i z + φ.hessian i z) ∧
+      ∀ i z, z ∈ (A.chart i).target →
+        ∀ v : Coordinates d, g.ricciQuadratic i z v = levi (h.weight i) z v := by
+  refine ⟨(projectiveBackground P).deform φ hp, fun _ _ => rfl, ?_⟩
+  intro i z hz v
+  exact KaehlerMetric.ricci_eq_of_volume ((projectiveBackground P).deform φ hp) h hc heq i hz v
+
+end Anticanonical.SourceSmooth
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/StrongDerivatives.lean b/lean/OAI/Geometry/PrescribedPotential/StrongDerivatives.lean
new file mode 100644
index 0000000000000000000000000000000000000000..19c518d183d97bbf8df27f28bb1a8e4b1e476671
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/StrongDerivatives.lean
@@ -0,0 +1,127 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.FiniteSobolevEmbedding
+import OAI.Geometry.PrescribedPotential.SobolevProduct
+
+namespace OAI
+
+/-! Strong Derivatives. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory FourierTransform TemperedDistribution LineDeriv
+open scoped SchwartzMap BoundedContinuousFunction ContDiff
+namespace SobolevChart
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+
+lemma schwartzCoord_dense (s : ℝ) : DenseRange (schwartzCoord (E := E) s) := by
+  apply (SchwartzMap.denseRange_toLpCLM (E := E) (F := ℂ) (p := 2)
+    (μ := volume) (by norm_num)).mono
+  rintro u ⟨f, rfl⟩
+  let ψ := SchwartzMap.fourierMultiplierCLM ℂ
+    (fun x : E => (((1 + ‖x‖^2)^((-s)/2) : ℝ) : ℂ)) f
+  refine ⟨ψ, ?_⟩
+  apply l2_injective
+  change (schwartzCoord s ψ : 𝓢'(E,ℂ)) = ((f.toLp 2 volume : L2 E) : 𝓢'(E,ℂ))
+  rw [schwartzCoord_distribution]
+  have hψ : (ψ : 𝓢'(E,ℂ)) = besselPotential E ℂ (-s) (f : 𝓢'(E,ℂ)) :=
+    (fourierMultiplierCLM_toTemperedDistributionCLM_eq (by fun_prop) f).symm
+  rw [hψ, besselPotential_besselPotential_apply, neg_add_cancel,
+    besselPotential_zero, ContinuousLinearMap.id_apply]
+  exact (Lp.toTemperedDistribution_toLp_eq f).symm
+
+lemma exists_schwartz_approx (s : ℝ) (u : L2 E) :
+    ∃ f : ℕ → 𝓢(E,ℂ), Tendsto (fun n => schwartzCoord s (f n)) atTop (nhds u) := by
+  obtain ⟨v,hv,ht⟩ := mem_closure_iff_seq_limit.mp (schwartzCoord_dense s u)
+  choose f hf using hv
+  refine ⟨f, ?_⟩
+  simpa only [hf] using ht
+
+ 
+def derivativeCoord (s : ℝ) (v : E) : L2 E →L[ℂ] L2 E :=
+  liftOperator s (s-1) (lineDerivOpCLM ℂ 𝓢'(E,ℂ) v) (fun _ hu => hu.lineDerivOp)
+
+lemma derivativeCoord_schwartz (s : ℝ) (v : E) (f : 𝓢(E,ℂ)) :
+    derivativeCoord s v (schwartzCoord s f) = schwartzCoord (s-1) (∂_{v} f) := by
+  apply realize_injective (s-1)
+  exact (realize_liftOperator s (s-1) _ _ _).trans (by
+    rw [realize_schwartzCoord, realize_schwartzCoord]
+    exact lineDerivOp_toTemperedDistributionCLM_eq f v)
+
+lemma strong_hasLineDerivAt (s : ℝ)
+    (hs : (Module.finrank ℝ E : ℝ) < 2*s)
+    (hs' : (Module.finrank ℝ E : ℝ) < 2*(s-1)) (u : L2 E) (x v : E) :
+    HasLineDerivAt ℝ (strongEmbedding s hs u : E → ℂ)
+      (strongEmbedding (s-1) hs' (derivativeCoord s v u) x) x v := by
+  obtain ⟨f,hf⟩ := exists_schwartz_approx s u
+  have h0 := ((strongEmbedding s hs).continuous.tendsto u).comp hf
+  have h1 := ((strongEmbedding (s-1) hs').continuous.tendsto
+    (derivativeCoord s v u)).comp (((derivativeCoord s v).continuous.tendsto u).comp hf)
+  change Tendsto (fun n => strongEmbedding (s-1) hs'
+    (derivativeCoord s v (schwartzCoord s (f n)))) atTop
+    (nhds (strongEmbedding (s-1) hs' (derivativeCoord s v u))) at h1
+  change Tendsto (fun n => strongEmbedding s hs (schwartzCoord s (f n))) atTop
+    (nhds (strongEmbedding s hs u)) at h0
+  simp_rw [derivativeCoord_schwartz, strongEmbedding_schwartz] at h1
+  simp_rw [strongEmbedding_schwartz] at h0
+  have h1' := (BoundedContinuousFunction.tendsto_iff_tendstoUniformly.mp h1).comp
+    (fun t : ℝ => x + t • v)
+  have h0' := BoundedContinuousFunction.tendsto_iff_tendstoUniformly.mp h0
+  have hd : ∀ n (t : ℝ), HasDerivAt (fun t : ℝ => f n (x + t • v))
+      ((∂_{v} (f n) : 𝓢(E,ℂ)) (x + t • v)) t := by
+    intro n t
+    simpa only [SchwartzMap.lineDerivOp_apply_eq_fderiv, one_smul, Function.comp_def, id_eq] using
+      (f n).differentiableAt.hasFDerivAt.comp_hasDerivAt t
+        ((hasDerivAt_id t).smul_const v |>.const_add x)
+  have hh := hasDerivAt_of_tendstoUniformly h1' (Eventually.of_forall hd)
+    (fun t => h0'.tendsto_at (x+t • v)) 0
+  simpa only [HasLineDerivAt, Function.comp_apply, zero_smul, add_zero] using hh
+
+lemma strong_lineDeriv (s : ℝ)
+    (hs : (Module.finrank ℝ E : ℝ) < 2*s)
+    (hs' : (Module.finrank ℝ E : ℝ) < 2*(s-1)) (u : L2 E) (x v : E) :
+    lineDeriv ℝ (strongEmbedding s hs u : E → ℂ) x v =
+      strongEmbedding (s-1) hs' (derivativeCoord s v u) x :=
+  (strong_hasLineDerivAt s hs hs' u x v).lineDeriv
+
+lemma strong_fderiv (s : ℝ)
+    (hs : (Module.finrank ℝ E : ℝ) < 2*s)
+    (hs' : (Module.finrank ℝ E : ℝ) < 2*(s-1))
+    (hN : (Module.finrank ℝ E : ℝ) < 2*(s-2)) (u : L2 E) (x v : E) :
+    fderiv ℝ (strongEmbedding s hs u : E → ℂ) x v =
+      strongEmbedding (s-1) hs' (derivativeCoord s v u) x := by
+  have hd := (strongEmbedding_contDiff s hs 1 (by simpa using hN) u).differentiable
+    (by norm_num)
+  rw [← (hd x).lineDeriv_eq_fderiv]
+  exact strong_lineDeriv s hs hs' u x v
+
+lemma strong_second_fderiv (s : ℝ)
+    (hs : (Module.finrank ℝ E : ℝ) < 2*s)
+    (h1 : (Module.finrank ℝ E : ℝ) < 2*(s-1))
+    (h2 : (Module.finrank ℝ E : ℝ) < 2*((s-1)-1))
+    (hN : (Module.finrank ℝ E : ℝ) < 2*(s-4)) (u : L2 E) (x v w : E) :
+    fderiv ℝ (fderiv ℝ (strongEmbedding s hs u : E → ℂ)) x v w =
+      strongEmbedding ((s-1)-1) h2
+        (derivativeCoord (s-1) v (derivativeCoord s w u)) x := by
+  have hf := strongEmbedding_contDiff s hs 2 (by norm_num; exact hN) u
+  have hd := (hf.fderiv_right (m := 1) (by norm_num)).differentiable (by norm_num)
+  have he : (fun y => fderiv ℝ (strongEmbedding s hs u : E → ℂ) y w) =
+      (strongEmbedding (s-1) h1 (derivativeCoord s w u) : E → ℂ) := by
+    funext y
+    exact strong_fderiv s hs h1 (by linarith) u y w
+  have hh : fderiv ℝ (fun y => fderiv ℝ (strongEmbedding s hs u : E → ℂ) y w) x v =
+      fderiv ℝ (fderiv ℝ (strongEmbedding s hs u : E → ℂ)) x v w := by
+    rw [fderiv_clm_apply (hd x) (differentiableAt_const w)]
+    simp
+  rw [← hh, he]
+  exact strong_fderiv (s-1) h1 h2 (by linarith) (derivativeCoord s w u) x v
+
+end SobolevChart
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/StrongSobolevEmbedding.lean b/lean/OAI/Geometry/PrescribedPotential/StrongSobolevEmbedding.lean
new file mode 100644
index 0000000000000000000000000000000000000000..62a152ef162fffa7de016cfc89a7ef5678812462
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/StrongSobolevEmbedding.lean
@@ -0,0 +1,103 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.SobolevEmbedding
+import OAI.Geometry.PrescribedPotential.SobolevParametrix
+
+namespace OAI
+
+/-! Strong Sobolev Embedding. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology MeasureTheory FourierTransform TemperedDistribution
+open scoped SchwartzMap BoundedContinuousFunction ContDiff
+namespace SobolevChart
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+
+def boundedToLinfLinear : (E →ᵇ ℂ) →ₗ[ℂ] Lp ℂ ⊤ (volume : Measure E) where
+  toFun f := f.memLp_top.toLp f
+  map_add' _ _ := rfl
+  map_smul' _ _ := rfl
+
+lemma boundedToLinf_norm (f : E →ᵇ ℂ) : ‖boundedToLinfLinear f‖ ≤ ‖f‖ := by
+  rw [show boundedToLinfLinear f = f.memLp_top.toLp f from rfl, Lp.norm_toLp,
+    eLpNorm_exponent_top f.memLp_top.aestronglyMeasurable]
+  exact ENNReal.toReal_le_of_le_ofReal (norm_nonneg _)
+    (eLpNormEssSup_le_of_ae_bound (Eventually.of_forall f.norm_coe_le_norm))
+
+def boundedToLinf : (E →ᵇ ℂ) →L[ℂ] Lp ℂ ⊤ (volume : Measure E) :=
+  boundedToLinfLinear.mkContinuous 1 (fun f => by simpa using boundedToLinf_norm f)
+
+lemma boundedToLinf_injective : Function.Injective (boundedToLinf (E := E)) := by
+  intro f g h
+  apply DFunLike.coe_injective
+  apply (f.continuous.ae_eq_iff_eq volume g.continuous).mp
+  exact (f.memLp_top.coeFn_toLp).symm.trans ((show
+    (f.memLp_top.toLp f : E → ℂ) =ᵐ[volume] (g.memLp_top.toLp g : E → ℂ) by rw [show
+      f.memLp_top.toLp f = g.memLp_top.toLp g from h]).trans g.memLp_top.coeFn_toLp)
+
+def boundedDistributionCLM : (E →ᵇ ℂ) →L[ℂ] 𝓢'(E, ℂ) :=
+  Lp.toTemperedDistributionCLM ℂ volume ⊤ ∘L boundedToLinf
+
+lemma boundedDistributionCLM_apply (f : E →ᵇ ℂ) :
+    boundedDistributionCLM f = boundedDistribution f := rfl
+
+lemma boundedDistribution_injective : Function.Injective (boundedDistributionCLM (E := E)) :=
+  (LinearMap.ker_eq_bot.mp (Lp.ker_toTemperedDistributionCLM_eq_bot
+    (F := ℂ) (E := E) (μ := volume) (p := ⊤))).comp boundedToLinf_injective
+
+lemma strong_representation (s : ℝ) (hs : (Module.finrank ℝ E : ℝ) < 2*s) (u : L2 E) :
+    ∃ f : E →ᵇ ℂ, boundedDistributionCLM f = realize s u := by
+  obtain ⟨v,hv⟩ := (realize_memSobolev s u).fourier_memL1 hs
+  refine ⟨Real.Lp.fourierTransformInv v, ?_⟩
+  rw [boundedDistributionCLM_apply, fourierInv_L1_distribution, ← hv, fourierInv_fourier_eq]
+
+def strongRepresentative (s : ℝ) (hs : (Module.finrank ℝ E : ℝ) < 2*s) (u : L2 E) : E →ᵇ ℂ :=
+  (strong_representation s hs u).choose
+
+lemma strongRepresentative_distribution (s : ℝ) (hs : (Module.finrank ℝ E : ℝ) < 2*s) (u : L2 E) :
+    boundedDistributionCLM (strongRepresentative s hs u) = realize s u :=
+  (strong_representation s hs u).choose_spec
+
+def strongRepresentativeLinear (s : ℝ) (hs : (Module.finrank ℝ E : ℝ) < 2*s) :
+    L2 E →ₗ[ℂ] (E →ᵇ ℂ) where
+  toFun := strongRepresentative s hs
+  map_add' u v := by
+    apply boundedDistribution_injective
+    simp only [strongRepresentative_distribution, map_add]
+  map_smul' c u := by
+    apply boundedDistribution_injective
+    simp only [strongRepresentative_distribution, map_smul, RingHom.id_apply]
+
+ 
+def strongEmbedding (s : ℝ) (hs : (Module.finrank ℝ E : ℝ) < 2*s) :
+    L2 E →L[ℂ] (E →ᵇ ℂ) where
+  toLinearMap := strongRepresentativeLinear s hs
+  cont := by
+    apply LinearMap.continuous_of_seq_closed_graph
+    intro u x y hu hv
+    apply boundedDistribution_injective
+    have h₁ := ((boundedDistributionCLM (E := E)).continuous.tendsto y).comp hv
+    have h₂ := ((realize (E := E) s).continuous.tendsto x).comp hu
+    change Tendsto (fun n => boundedDistributionCLM (strongRepresentative s hs (u n)))
+      atTop (nhds (boundedDistributionCLM y)) at h₁
+    simp_rw [strongRepresentative_distribution] at h₁
+    exact (tendsto_nhds_unique h₁ h₂).trans (strongRepresentative_distribution s hs x).symm
+
+lemma strongEmbedding_distribution (s : ℝ) (hs : (Module.finrank ℝ E : ℝ) < 2*s) (u : L2 E) :
+    boundedDistributionCLM (strongEmbedding s hs u) = realize s u :=
+  strongRepresentative_distribution s hs u
+
+lemma strongEmbedding_norm (s : ℝ) (hs : (Module.finrank ℝ E : ℝ) < 2*s) (u : L2 E) (x : E) :
+    ‖strongEmbedding s hs u x‖ ≤ ‖strongEmbedding (E := E) s hs‖ * ‖u‖ :=
+  ((strongEmbedding s hs u).norm_coe_le_norm x).trans ((strongEmbedding s hs).le_opNorm u)
+
+end SobolevChart
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/SymmetricPrincipalFrame.lean b/lean/OAI/Geometry/PrescribedPotential/SymmetricPrincipalFrame.lean
new file mode 100644
index 0000000000000000000000000000000000000000..83da261ca9a6d9d2625177cc5e950a1d0caebe7b
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/SymmetricPrincipalFrame.lean
@@ -0,0 +1,89 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.FrozenPoisson
+
+namespace OAI
+
+/-! Symmetric Principal Frame. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Finset
+open scoped ContDiff
+namespace HigherJet
+open EllipticKernel FrozenPoisson
+variable {E F : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [NormedAddCommGroup F] [NormedSpace ℝ F]
+  {ι : Type*} [Fintype ι]
+
+def principalApply (e : OrthonormalBasis ι ℝ E) (L : SecondOrder E)
+    (H : E →L[ℝ] E →L[ℝ] F) : F :=
+  ∑ i, ∑ j, ((L (rankTwo (e i) (e j))+L (rankTwo (e j) (e i)))/2) • H (e i) (e j)
+
+lemma rankOne_add (v w : E) : rankOne (v+w) = rankOne v+rankOne w+rankTwo v w+rankTwo w v := by
+  ext x y
+  simp only [rankOne,rankTwo,_root_.add_apply,ContinuousLinearMap.smulRight_apply,
+    realInner_apply,inner_add_left,_root_.smul_apply,smul_eq_mul]
+  ring
+
+lemma principal_coefficient_gram (L : SecondOrder E) (T : E →L[ℝ] E)
+    (h : ∀ v, L (rankOne v) = ‖T v‖^2) (v w : E) :
+    (L (rankTwo v w)+L (rankTwo w v))/2 = inner ℝ (T v) (T w) := by
+  have hh := h (v+w)
+  rw [rankOne_add] at hh
+  simp only [map_add,h] at hh
+  rw [norm_add_sq_real] at hh
+  linarith only [hh]
+
+lemma bilinear_basis_expansion (e : OrthonormalBasis ι ℝ E) (H : E →L[ℝ] E →L[ℝ] F) (v w : E) :
+    H v w = ∑ i, ∑ j, (inner ℝ (e i) v * inner ℝ (e j) w) • H (e i) (e j) := by
+  conv_lhs => rw [← e.sum_repr' v,← e.sum_repr' w]
+  simp only [map_sum,map_smul,_root_.sum_apply,_root_.smul_apply,
+    Finset.smul_sum,smul_smul]
+  rw [Finset.sum_comm]
+  apply sum_congr rfl
+  intro i _
+  apply sum_congr rfl
+  intro j _
+  rw [mul_comm]
+
+lemma principalApply_gram [CompleteSpace E] (e : OrthonormalBasis ι ℝ E)
+    (L : SecondOrder E) (T : E →L[ℝ] E) (h : ∀ v, L (rankOne v) = ‖T v‖^2)
+    (H : E →L[ℝ] E →L[ℝ] F) :
+    principalApply e L H = ∑ k, H (T.adjoint (e k)) (T.adjoint (e k)) := by
+  simp only [principalApply,principal_coefficient_gram L T h]
+  conv_rhs => arg 2; ext k; rw [bilinear_basis_expansion e H]
+  calc
+    _ = ∑ i, ∑ j, (∑ k, inner ℝ (e i) (T.adjoint (e k))*inner ℝ (e j) (T.adjoint (e k))) • H (e i) (e j) := by
+      apply sum_congr rfl
+      intro i _
+      apply sum_congr rfl
+      intro j _
+      congr 1
+      simp only [ContinuousLinearMap.adjoint_inner_right]
+      rw [← e.sum_inner_mul_inner (T (e i)) (T (e j))]
+      exact sum_congr rfl (fun k _ => by rw [real_inner_comm (e k) (T (e j))])
+    _ = _ := by
+      simp only [Finset.sum_smul]
+      conv_lhs => arg 2; ext i; rw [Finset.sum_comm]
+      rw [Finset.sum_comm]
+
+lemma principalApply_symmetric (e : OrthonormalBasis ι ℝ E) (L : SecondOrder E)
+    (H : E →L[ℝ] E →L[ℝ] F) (hH : ∀ v w, H v w = H w v) :
+    principalApply e L H = ∑ i, ∑ j, L (rankTwo (e i) (e j)) • H (e i) (e j) := by
+  have he : (∑ i, ∑ j, L (rankTwo (e j) (e i)) • H (e i) (e j)) =
+      ∑ i, ∑ j, L (rankTwo (e i) (e j)) • H (e i) (e j) := by
+    rw [Finset.sum_comm]
+    exact sum_congr rfl (fun i _ => sum_congr rfl (fun j _ => congrArg (fun v : F => L (rankTwo (e i) (e j)) • v) (hH _ _)))
+  simp only [principalApply,div_eq_mul_inv,mul_comm _ (2:ℝ)⁻¹,mul_smul,add_smul,Finset.sum_add_distrib,← Finset.smul_sum]
+  rw [he]
+  module
+
+end HigherJet
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/TraceEllipticity.lean b/lean/OAI/Geometry/PrescribedPotential/TraceEllipticity.lean
new file mode 100644
index 0000000000000000000000000000000000000000..9365a2b7ec0085b95b3c2c575d82520152696368
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/TraceEllipticity.lean
@@ -0,0 +1,165 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.MatrixFrameNormalization
+
+namespace OAI
+
+/-! Trace Ellipticity. -/
+
+section
+
+ 
+noncomputable section
+open Matrix
+open scoped ComplexOrder MatrixOrder Matrix.Norms.Elementwise
+namespace KaehlerCalculus
+variable {n : ℕ}
+
+lemma eig_upper_psd {M : Matrix (Fin n) (Fin n) ℂ} (hh : M.IsHermitian)
+    (C : ℝ) (he : ∀ i, hh.eigenvalues i ≤ C) : (((C:ℂ) • 1)-M).PosSemidef := by
+  have hd : (Matrix.diagonal (fun i => ((C-hh.eigenvalues i:ℝ):ℂ))).PosSemidef :=
+    Matrix.PosSemidef.diagonal (fun i => by
+      change (0:ℂ) ≤ ((C-hh.eigenvalues i:ℝ):ℂ)
+      exact_mod_cast sub_nonneg.mpr (he i))
+  have hu := hd.mul_mul_conjTranspose_same (hh.eigenvectorUnitary : Matrix (Fin n) (Fin n) ℂ)
+  have hD : Matrix.diagonal (fun i => ((C-hh.eigenvalues i:ℝ):ℂ)) =
+      (C:ℂ) • (1 : Matrix (Fin n) (Fin n) ℂ)-Matrix.diagonal (fun i => (hh.eigenvalues i:ℂ)) := by
+    simp only [Complex.ofReal_sub,Matrix.diagonal_sub,Matrix.smul_one_eq_diagonal]
+  rw [hD,Matrix.mul_sub,Matrix.sub_mul,Matrix.mul_smul,Matrix.mul_one,Matrix.smul_mul] at hu
+  have hunit : (hh.eigenvectorUnitary : Matrix (Fin n) (Fin n) ℂ)*
+      (hh.eigenvectorUnitary : Matrix (Fin n) (Fin n) ℂ)ᴴ = 1 := Unitary.coe_mul_star_self _
+  rw [hunit] at hu
+  have hs := hh.spectral_theorem
+  simp only [Unitary.conjStarAlgAut_apply,Function.comp_def] at hs
+  exact hs ▸ hu
+
+lemma trace_upper_psd {M : Matrix (Fin n) (Fin n) ℂ} (hM : M.PosSemidef) :
+    (((M.trace.re:ℂ) • 1)-M).PosSemidef := by
+  apply eig_upper_psd hM.isHermitian
+  intro i
+  rw [hM.isHermitian.trace_eq_sum_eigenvalues,Complex.re_sum]
+  exact Finset.single_le_sum (fun j _ => hM.eigenvalues_nonneg j) (Finset.mem_univ i)
+
+lemma inverse_trace_lower_psd {M G : Matrix (Fin n) (Fin n) ℂ}
+    (hM : M.PosDef) (hG : G.PosDef) {R : ℝ} (hR : (M⁻¹*G).trace.re ≤ R) :
+    (((R:ℂ) • M)-G).PosSemidef := by
+  obtain ⟨B,hB,hBM⟩ := whitening M hM
+  let N := Bᴴ*G*B
+  have hN : N.PosDef := hG.conjTranspose_mul_mul_same (Matrix.mulVec_injective_iff_isUnit.mpr hB)
+  have htr : N.trace.re = (M⁻¹*G).trace.re := by
+    have he := matrix_inverse_trace_congruence B M G hB
+    rw [hBM,inv_one,one_mul] at he
+    exact congrArg Complex.re he
+  have hn := trace_upper_psd hN.posSemidef
+  have hd : (((R-N.trace.re:ℝ):ℂ) • (1 : Matrix (Fin n) (Fin n) ℂ)).PosSemidef :=
+    Matrix.PosSemidef.smul Matrix.PosSemidef.one (by exact_mod_cast sub_nonneg.mpr (htr ▸ hR))
+  have hh := hn.add hd
+  have he : (((N.trace.re:ℂ) • (1 : Matrix (Fin n) (Fin n) ℂ))-N)+
+      ((R-N.trace.re:ℝ):ℂ) • 1 = (R:ℂ) • 1-N := by
+    rw [Complex.ofReal_sub,sub_smul]
+    abel
+  rw [he] at hh
+  have hp : (Bᴴ*((R:ℂ) • M-G)*B).PosSemidef := by
+    simpa only [mul_sub,sub_mul,mul_smul_comm,smul_mul_assoc,hBM] using hh
+  exact (hB.posSemidef_star_left_conjugate_iff).mp hp
+
+lemma inverse_trace_eigenvalues {M : Matrix (Fin n) (Fin n) ℂ} (hM : M.PosDef) :
+    M⁻¹.trace.re = ∑ i, (hM.isHermitian.eigenvalues i)⁻¹ := by
+  let U := hM.isHermitian.eigenvectorUnitary
+  have hs := hM.isHermitian.spectral_theorem
+  simp only [Unitary.conjStarAlgAut_apply,Function.comp_def] at hs
+  change M = (U : Matrix (Fin n) (Fin n) ℂ) *
+    Matrix.diagonal (fun i => (hM.isHermitian.eigenvalues i:ℂ)) *
+    (U : Matrix (Fin n) (Fin n) ℂ)ᴴ at hs
+  have hu : (U : Matrix (Fin n) (Fin n) ℂ)⁻¹ = (U : Matrix (Fin n) (Fin n) ℂ)ᴴ :=
+    Matrix.inv_eq_left_inv (Unitary.coe_star_mul_self U)
+  have hv : ((U : Matrix (Fin n) (Fin n) ℂ)ᴴ)⁻¹ = (U : Matrix (Fin n) (Fin n) ℂ) :=
+    Matrix.inv_eq_left_inv (Unitary.coe_mul_star_self U)
+  have hid : (Matrix.diagonal (fun i => (hM.isHermitian.eigenvalues i:ℂ)))⁻¹ =
+      Matrix.diagonal (fun i => ((hM.isHermitian.eigenvalues i:ℂ))⁻¹) := by
+    apply Matrix.inv_eq_left_inv
+    rw [Matrix.diagonal_mul_diagonal]
+    have he : (fun i => (hM.isHermitian.eigenvalues i:ℂ)⁻¹ * (hM.isHermitian.eigenvalues i:ℂ)) = fun _ => 1 := by
+      funext i
+      exact inv_mul_cancel₀ (Complex.ofReal_ne_zero.mpr (hM.eigenvalues_pos i).ne')
+    rw [he, Matrix.diagonal_one]
+  have hi : M⁻¹ = (U : Matrix (Fin n) (Fin n) ℂ)*
+      Matrix.diagonal (fun i => (((hM.isHermitian.eigenvalues i)⁻¹:ℝ):ℂ)) *
+      (U : Matrix (Fin n) (Fin n) ℂ)ᴴ := by
+    conv_lhs => rw [hs]
+    simp only [Matrix.mul_inv_rev, hu, hv, hid, Matrix.mul_assoc]
+    simp only [Complex.ofReal_inv]
+  have huu : (U : Matrix (Fin n) (Fin n) ℂ)ᴴ * (U : Matrix (Fin n) (Fin n) ℂ) = 1 :=
+    Unitary.coe_star_mul_self U
+  rw [hi, Matrix.trace_mul_cycle, huu, Matrix.one_mul,
+    Matrix.trace_diagonal, Complex.re_sum]
+  rfl
+
+lemma inverse_trace_det_upper {M : Matrix (Fin n) (Fin n) ℂ} (hM : M.PosDef)
+    {R D : ℝ} (_hR0 : 0 ≤ R) (hR : M⁻¹.trace.re ≤ R) (hD : M.det.re ≤ D) :
+    ((((D * R^(n-1):ℝ):ℂ) • 1)-M).PosSemidef := by
+  apply eig_upper_psd hM.isHermitian
+  intro i
+  let ell := hM.isHermitian.eigenvalues
+  have hp : ∀ j, 0 < ell j := hM.eigenvalues_pos
+  have hb : ∀ j, (ell j)⁻¹ ≤ R := by
+    intro j
+    apply le_trans _ hR
+    rw [inverse_trace_eigenvalues hM]
+    exact Finset.single_le_sum (fun k _ => inv_nonneg.mpr (hp k).le) (Finset.mem_univ j)
+  have hprod : (∏ j ∈ Finset.univ.erase i, (ell j)⁻¹) ≤ R^(n-1) := by
+    have hh := Finset.prod_le_prod₀ (s := Finset.univ.erase i)
+      (fun j _ => inv_nonneg.mpr (hp j).le) (fun j _ => hb j)
+    simpa only [Finset.prod_const, Finset.card_erase_of_mem (Finset.mem_univ i),
+      Finset.card_univ, Fintype.card_fin] using hh
+  have hdet : M.det.re = ∏ j, ell j := by
+    rw [hM.isHermitian.det_eq_prod_eigenvalues, ← RCLike.ofReal_prod]
+    rfl
+  have he : ell i = M.det.re * ∏ j ∈ Finset.univ.erase i, (ell j)⁻¹ := by
+    rw [hdet,← Finset.mul_prod_erase Finset.univ ell (Finset.mem_univ i),
+      Finset.prod_inv_distrib, mul_assoc, mul_inv_cancel₀, mul_one]
+    exact Finset.prod_ne_zero_iff.mpr (fun j _ => (hp j).ne')
+  change ell i ≤ _
+  rw [he]
+  exact mul_le_mul hD hprod
+    (Finset.prod_nonneg (fun j _ => inv_nonneg.mpr (hp j).le))
+    (le_trans (Complex.nonneg_iff.mp hM.det_pos.le).1 hD)
+
+lemma det_ratio_whitening {M G B : Matrix (Fin n) (Fin n) ℂ}
+    (hM : M.PosDef) (hG : G.PosDef) (hB : IsUnit B) (hBG : Bᴴ*G*B = 1) :
+    (Bᴴ*M*B).det.re = M.det.re/G.det.re := by
+  have hN := hM.conjTranspose_mul_mul_same (Matrix.mulVec_injective_iff_isUnit.mpr hB)
+  have he : (Bᴴ*M*B).det * G.det = M.det := by
+    have hh := congrArg Matrix.det hBG
+    simp only [Matrix.det_mul, Matrix.det_one] at hh ⊢
+    calc
+      Bᴴ.det * M.det * B.det * G.det = M.det * (Bᴴ.det * G.det * B.det) := by ring
+      _ = M.det := by rw [hh,mul_one]
+  have hr : (Bᴴ*M*B).det.re * G.det.re = M.det.re := by
+    have hh := congrArg Complex.re he
+    rw [Complex.mul_re, ← (Complex.pos_iff.mp hN.det_pos).2,
+      zero_mul,sub_zero] at hh
+    exact hh
+  exact (eq_div_iff (ne_of_gt (Complex.pos_iff.mp hG.det_pos).1)).mpr hr
+
+lemma inverse_trace_relative_upper {M G : Matrix (Fin n) (Fin n) ℂ}
+    (hM : M.PosDef) (hG : G.PosDef) {R D : ℝ}
+    (hR0 : 0 ≤ R) (hR : (M⁻¹*G).trace.re ≤ R) (hD : M.det.re/G.det.re ≤ D) :
+    ((((D*R^(n-1):ℝ):ℂ) • G)-M).PosSemidef := by
+  obtain ⟨B,hB,hBG⟩ := whitening G hG
+  let N := Bᴴ*M*B
+  have hN : N.PosDef := hM.conjTranspose_mul_mul_same (Matrix.mulVec_injective_iff_isUnit.mpr hB)
+  have htr : N⁻¹.trace.re = (M⁻¹*G).trace.re := by
+    have he := matrix_inverse_trace_congruence B M G hB
+    rw [hBG,mul_one] at he
+    exact congrArg Complex.re he
+  have hd : N.det.re ≤ D := (det_ratio_whitening hM hG hB hBG).trans_le hD
+  have hh := inverse_trace_det_upper hN hR0 (htr ▸ hR) hd
+  have he : (Bᴴ*((((D*R^(n-1):ℝ):ℂ) • G)-M)*B).PosSemidef := by
+    simpa only [mul_sub,sub_mul,mul_smul_comm,smul_mul_assoc,hBG] using hh
+  exact hB.posSemidef_star_left_conjugate_iff.mp he
+end KaehlerCalculus
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/TracePositivity.lean b/lean/OAI/Geometry/PrescribedPotential/TracePositivity.lean
new file mode 100644
index 0000000000000000000000000000000000000000..c4857b0a0c34b74162575711756143d7d456a0c0
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/TracePositivity.lean
@@ -0,0 +1,51 @@
+import Mathlib
+
+namespace OAI
+
+/-! Trace Positivity. -/
+
+section
+
+noncomputable section
+open Matrix
+open scoped ComplexOrder MatrixOrder
+namespace KaehlerCalculus
+variable {n : ℕ}
+
+lemma trace_mul_real {A B : Matrix (Fin n) (Fin n) ℂ}
+    (hA : A.IsHermitian) (hB : B.IsHermitian) :
+    ((A*B).trace.re : ℂ) = (A*B).trace := by
+  have he : star (A*B).trace = (A*B).trace := by
+    rw [← Matrix.trace_conjTranspose,Matrix.conjTranspose_mul,hA.eq,hB.eq,Matrix.trace_mul_comm]
+  apply Complex.ext
+  · simp
+  · have hi := congrArg Complex.im he
+    simp only [Complex.star_def,Complex.conj_im] at hi
+    simp only [Complex.ofReal_im]
+    linarith
+
+lemma trace_mul_pos [Nonempty (Fin n)] {A B : Matrix (Fin n) (Fin n) ℂ}
+    (hA : A.PosDef) (hB : B.PosDef) : 0 < (A*B).trace.re := by
+  obtain ⟨Q,hQ,rfl⟩ := CStarAlgebra.isStrictlyPositive_iff_eq_star_mul_self.mp hA.isStrictlyPositive
+  have h := (hB.mul_mul_conjTranspose_same (Matrix.vecMul_injective_iff_isUnit.mpr hQ)).trace_pos
+  change 0 < (Qᴴ*Q*B).trace.re
+  rw [Matrix.trace_mul_cycle] at h
+  exact (Complex.pos_iff.mp h).1
+
+lemma whitening (M : Matrix (Fin n) (Fin n) ℂ) (hM : M.PosDef) :
+    ∃ B : Matrix (Fin n) (Fin n) ℂ, IsUnit B ∧ Bᴴ*M*B = 1 := by
+  obtain ⟨Q,hQ,hM⟩ := CStarAlgebra.isStrictlyPositive_iff_eq_star_mul_self.mp hM.isStrictlyPositive
+  obtain ⟨Q,rfl⟩ := hQ
+  let R : Matrix (Fin n) (Fin n) ℂ := ↑(Q⁻¹)
+  have hQR : (Q : Matrix (Fin n) (Fin n) ℂ)*R = 1 := Q.mul_inv
+  refine ⟨R,(Q⁻¹).isUnit,?_⟩
+  rw [hM]
+  change Rᴴ*((Q : Matrix (Fin n) (Fin n) ℂ)ᴴ*(Q : Matrix (Fin n) (Fin n) ℂ))*R = 1
+  rw [Matrix.mul_assoc Rᴴ,Matrix.mul_assoc, hQR,Matrix.mul_one,
+    ← Matrix.conjTranspose_mul,hQR,Matrix.conjTranspose_one]
+end KaehlerCalculus
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/TruncatedRellich.lean b/lean/OAI/Geometry/PrescribedPotential/TruncatedRellich.lean
new file mode 100644
index 0000000000000000000000000000000000000000..12c998820c7e90b13a8e7b1e683676399f93402b
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/TruncatedRellich.lean
@@ -0,0 +1,116 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.FourierKernel
+import OAI.Geometry.PrescribedPotential.KernelCompact
+import OAI.Geometry.PrescribedPotential.LocalizedCompact
+
+namespace OAI
+
+/-! Truncated Rellich. -/
+
+section
+
+ 
+
+noncomputable section
+open MeasureTheory Set Filter Topology FourierTransform
+open scoped BoundedContinuousFunction SchwartzMap Classical ComplexInnerProductSpace
+namespace EllipticCompact
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+open SobolevChart
+
+lemma multiply_ae (g : E →ᵇ ℂ) (u : L2 E) :
+    multiply g u =ᵐ[volume] fun x => g x * u x := by
+  rw [multiply_eq]
+  filter_upwards [Lp.coeFn_lpSMul (r := 2) (g.memLp_top.toLp g : Lp ℂ ⊤ volume) u,
+    (g.memLp_top (μ := volume)).coeFn_toLp] with x h h'
+  rw [h, Pi.smul_apply', h', smul_eq_mul]
+
+lemma multiply_toLp (g f : 𝓢(E, ℂ)) :
+    multiply g.toBoundedContinuousFunction (f.toLp 2) =
+      (SchwartzMap.smulLeftCLM ℂ g f).toLp 2 := by
+  apply Lp.ext
+  filter_upwards [multiply_ae g.toBoundedContinuousFunction (f.toLp 2),
+    f.coeFn_toLp 2 volume, (SchwartzMap.smulLeftCLM ℂ g f).coeFn_toLp 2 volume] with x hm hf hp
+  rw [hm, hf, hp, SchwartzMap.smulLeftCLM_apply_apply g.hasTemperateGrowth, smul_eq_mul]
+  rfl
+
+lemma kernelSection_inner (χ f : 𝓢(E, ℂ)) (x : E) :
+    inner ℂ (kernelSection χ x) (f.toLp 2) =
+      (𝓕⁻ (SchwartzMap.smulLeftCLM ℂ χ f) : 𝓢(E, ℂ)) x := by
+  rw [L2.inner_def]
+  conv_rhs => rw [SchwartzMap.fourierInv_coe, Real.fourierInv_eq]
+  apply integral_congr_ae
+  filter_upwards [kernelSection_ae χ x, f.coeFn_toLp 2 volume] with y hχ hf
+  rw [hχ, hf, RCLike.inner_apply, kernelSectionFun, starRingEnd_apply, star_star,
+    SchwartzMap.smulLeftCLM_apply_apply χ.hasTemperateGrowth]
+  simp only [Circle.smul_def, smul_eq_mul]
+  ring
+
+def sectionsOn (χ : 𝓢(E, ℂ)) (hχ : HasCompactSupport χ)
+    (s : Set E) [CompactSpace s] : s →ᵇ (L2 E →L[ℂ] ℂ) :=
+  BoundedContinuousFunction.mkOfCompact ⟨fun x => innerSL ℂ (kernelSection χ x),
+    (innerSL ℂ).continuous.comp ((kernelSection_continuous χ hχ).comp continuous_subtype_val)⟩
+
+lemma sectionsOn_core (χ : 𝓢(E, ℂ)) (hχ : HasCompactSupport χ)
+    (s : Set E) [CompactSpace s] (f : 𝓢(E, ℂ)) :
+    kernelOperator (sectionsOn χ hχ s) (f.toLp 2) =
+      (𝓕⁻ (SchwartzMap.smulLeftCLM ℂ χ f) : 𝓢(E, ℂ)).toBoundedContinuousFunction.compContinuous
+        ⟨Subtype.val, continuous_subtype_val⟩ := by
+  ext x
+  change innerSL ℂ (kernelSection χ x) (f.toLp 2) = _
+  rw [innerSL_apply_apply]
+  exact kernelSection_inner χ f x
+
+lemma localize_sections_core (κ χ : 𝓢(E, ℂ)) (hχ : HasCompactSupport χ)
+    (s : Set E) [CompactSpace s] (hs : MeasurableSet s) (hκs : Function.support κ ⊆ s)
+    (f : 𝓢(E, ℂ)) :
+    localizeL2 κ s hs (kernelOperator (sectionsOn χ hχ s) (f.toLp 2)) =
+      (SchwartzMap.smulLeftCLM ℂ κ (𝓕⁻ (SchwartzMap.smulLeftCLM ℂ χ f))).toLp 2 := by
+  rw [sectionsOn_core]
+  apply Lp.ext
+  filter_upwards [localizeL2_restrict κ s hs hκs
+    (𝓕⁻ (SchwartzMap.smulLeftCLM ℂ χ f) : 𝓢(E, ℂ)).toBoundedContinuousFunction,
+    (SchwartzMap.smulLeftCLM ℂ κ (𝓕⁻ (SchwartzMap.smulLeftCLM ℂ χ f))).coeFn_toLp 2 volume]
+    with x h₁ h₂
+  rw [h₁, h₂, SchwartzMap.smulLeftCLM_apply_apply κ.hasTemperateGrowth, smul_eq_mul]
+  rfl
+
+lemma truncated_core (κ χ f : 𝓢(E, ℂ)) :
+    (multiply κ.toBoundedContinuousFunction ∘L
+      fourierInvCLM ℂ (L2 E) ∘L multiply χ.toBoundedContinuousFunction) (f.toLp 2) =
+    (SchwartzMap.smulLeftCLM ℂ κ (𝓕⁻ (SchwartzMap.smulLeftCLM ℂ χ f))).toLp 2 := by
+  simp only [ContinuousLinearMap.comp_apply, fourierInvCLM_apply]
+  exact (congrArg (fun u : L2 E => multiply κ.toBoundedContinuousFunction (𝓕⁻ u))
+    (multiply_toLp χ f)).trans
+      ((congrArg (multiply κ.toBoundedContinuousFunction)
+        (SchwartzMap.toLp_fourierInv_eq (SchwartzMap.smulLeftCLM ℂ χ f))).trans
+        (multiply_toLp κ (𝓕⁻ (SchwartzMap.smulLeftCLM ℂ χ f))))
+
+lemma compact_truncated (κ χ : 𝓢(E, ℂ))
+    (hκ : HasCompactSupport κ) (hχ : HasCompactSupport χ) :
+    IsCompactOperator (multiply κ.toBoundedContinuousFunction ∘L
+      fourierInvCLM ℂ (L2 E) ∘L multiply χ.toBoundedContinuousFunction) := by
+  let s := tsupport κ
+  let : CompactSpace s := isCompact_iff_compactSpace.mp hκ
+  have hs : MeasurableSet s := (isClosed_tsupport κ).measurableSet
+  let T := (localizeL2 κ s hs).comp (kernelOperator (sectionsOn χ hχ s))
+  have hT : IsCompactOperator T :=
+    (kernelOperator_compact (sectionsOn χ hχ s)).clm_comp (localizeL2 κ s hs)
+  have he : T = multiply κ.toBoundedContinuousFunction ∘L
+      fourierInvCLM ℂ (L2 E) ∘L multiply χ.toBoundedContinuousFunction := by
+    apply DFunLike.coe_injective
+    apply (SchwartzMap.denseRange_toLpCLM (E := E) (F := ℂ) (p := 2) (μ := volume) (by norm_num)).equalizer
+      T.continuous (by fun_prop)
+    funext f
+    exact (localize_sections_core κ χ hχ s hs (subset_tsupport κ) f).trans
+      (truncated_core κ χ f).symm
+  rw [← he]
+  exact hT
+
+end EllipticCompact
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/TwoScaleRegularity.lean b/lean/OAI/Geometry/PrescribedPotential/TwoScaleRegularity.lean
new file mode 100644
index 0000000000000000000000000000000000000000..10a6980a6cf256e8224178dacb86108848aeb0c4
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/TwoScaleRegularity.lean
@@ -0,0 +1,166 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.NonlinearScale
+
+namespace OAI
+
+/-! Two Scale Regularity. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology
+open scoped Classical
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  [ConnectedSpace X] {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+local instance gainNG (s : ℝ) : NormedAddCommGroup (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedAddCommGroup
+local instance gainNS (s : ℝ) : NormedSpace ℝ (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedSpace
+local instance gainTG (s : ℝ) : IsTopologicalAddGroup (D.localizers.RealSobolev s) :=
+  Submodule.isTopologicalAddGroup _
+local instance gainCS (s : ℝ) : ContinuousSMul ℝ (D.localizers.RealSobolev s) :=
+  SMulMemClass.continuousSMul _
+
+omit [ConnectedSpace X] in
+lemma realConstants_lower {s t : ℝ} (hst : t ≤ s) (c : ℝ) :
+    D.localizers.realLower s t hst (D.realConstants s c) = D.realConstants t c := by
+  apply Subtype.ext
+  exact D.constantsOrder_lower hst (c : ℂ)
+
+omit [ConnectedSpace X] in
+lemma realEvaluation_embed {s : ℝ} (hs : (Module.finrank ℝ (EC d) : ℝ) < 2*s)
+    (x₀ : X) (f : RealSmooth A) :
+    D.realEvaluation s x₀ (D.localizers.realEmbed s f) = (f.val x₀).re := by
+  change (D.pointEvaluation s x₀ (D.localizers.embed s f.val)).re = _
+  rw [D.pointEvaluation_embed hs]
+
+omit [ConnectedSpace X] in
+lemma realEvaluation_lower {s t : ℝ} (hst : t ≤ s)
+    (ht : (Module.finrank ℝ (EC d) : ℝ) < 2*t) (x₀ : X)
+    (u : D.localizers.RealSobolev s) :
+    D.realEvaluation t x₀ (D.localizers.realLower s t hst u) = D.realEvaluation s x₀ u := by
+  let : NormedAddCommGroup (D.localizers.RealSobolev s) := inferInstance
+  let : NormedSpace ℝ (D.localizers.RealSobolev s) := inferInstance
+  let : NormedAddCommGroup (D.localizers.RealSobolev t) := inferInstance
+  let : NormedSpace ℝ (D.localizers.RealSobolev t) := inferInstance
+  have hs : (Module.finrank ℝ (EC d) : ℝ) < 2*s := by linarith
+  have hf : (fun v : D.localizers.RealSobolev s =>
+      D.realEvaluation t x₀ (D.localizers.realLower s t hst v)) =
+      (fun v : D.localizers.RealSobolev s => D.realEvaluation s x₀ v) := by
+    apply (D.localizers.realEmbed_dense s).equalizer
+      ((D.realEvaluation t x₀).continuous.comp (D.localizers.realLower s t hst).continuous)
+      (D.realEvaluation s x₀).continuous
+    funext f
+    simp only [Function.comp_apply,Localizers.realLower_embed,D.realEvaluation_embed ht,
+      D.realEvaluation_embed hs]
+  exact congr_fun hf u
+
+def pairLower (s t : ℝ) (hst : t ≤ s) :
+    (D.localizers.RealSobolev s × ℝ) →L[ℝ] (D.localizers.RealSobolev t × ℝ) :=
+  (D.localizers.realLower s t hst ∘L ContinuousLinearMap.fst ℝ _ ℝ).prod
+    (ContinuousLinearMap.snd ℝ _ ℝ)
+
+omit [ConnectedSpace X] in
+lemma pairLower_apply (s t : ℝ) (hst : t ≤ s)
+    (w : D.localizers.RealSobolev s × ℝ) :
+    D.pairLower s t hst w = (D.localizers.realLower s t hst w.1,w.2) := rfl
+
+omit [ConnectedSpace X] in
+lemma linearizedVolume_apply (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) (x₀ : X)
+    (u : D.localizers.RealSobolev ((k : ℝ)+2))
+    (w : D.localizers.RealSobolev ((k : ℝ)+2) × ℝ) :
+    D.linearizedVolume k hk x₀ u w =
+      (D.realVolumeDerivative k hk u w.1 - D.realConstants (k : ℝ) w.2,
+        D.realEvaluation ((k : ℝ)+2) x₀ w.1) := by
+  simp only [linearizedVolume,ContinuousLinearMap.prod_apply,_root_.sub_apply,
+    ContinuousLinearMap.comp_apply]
+  rfl
+
+omit [ConnectedSpace X] in
+lemma linearizedVolume_lower (k l : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (hl : Module.finrank ℝ (EC d) < l) (hlk : l ≤ k) (x₀ : X)
+    (u : D.localizers.RealSobolev ((k : ℝ)+2)) (w : D.localizers.RealSobolev ((k : ℝ)+2) × ℝ) :
+    D.linearizedVolume l hl x₀ (D.localizers.realLower ((k : ℝ)+2) ((l : ℝ)+2)
+      (by exact_mod_cast Nat.add_le_add_right hlk 2) u)
+      (D.pairLower ((k : ℝ)+2) ((l : ℝ)+2) (by exact_mod_cast Nat.add_le_add_right hlk 2) w) =
+    D.pairLower (k : ℝ) (l : ℝ) (by exact_mod_cast hlk) (D.linearizedVolume k hk x₀ u w) := by
+  let : NormedAddCommGroup (D.localizers.RealSobolev ((k : ℝ)+2)) := inferInstance
+  let : NormedSpace ℝ (D.localizers.RealSobolev ((k : ℝ)+2)) := inferInstance
+  let : NormedAddCommGroup (D.localizers.RealSobolev ((l : ℝ)+2)) := inferInstance
+  let : NormedSpace ℝ (D.localizers.RealSobolev ((l : ℝ)+2)) := inferInstance
+  let : NormedAddCommGroup (D.localizers.RealSobolev (k : ℝ)) := inferInstance
+  let : NormedSpace ℝ (D.localizers.RealSobolev (k : ℝ)) := inferInstance
+  let : NormedAddCommGroup (D.localizers.RealSobolev (l : ℝ)) := inferInstance
+  let : NormedSpace ℝ (D.localizers.RealSobolev (l : ℝ)) := inferInstance
+  have hst : (l : ℝ)+2 ≤ (k : ℝ)+2 := by exact_mod_cast Nat.add_le_add_right hlk 2
+  have hkl : (l : ℝ) ≤ (k : ℝ) := by exact_mod_cast hlk
+  simp only [linearizedVolume_apply,pairLower_apply]
+  apply Prod.ext
+  · have hh := congrArg (fun L : D.localizers.RealSobolev ((k : ℝ)+2) →L[ℝ]
+        D.localizers.RealSobolev (l : ℝ) => L w.1)
+      (D.realVolumeDerivative_lower k l hk hl hlk u)
+    simp only [ContinuousLinearMap.comp_apply] at hh
+    rw [map_sub,D.realConstants_lower,hh]
+  · apply D.realEvaluation_lower hst
+    have hh : (Module.finrank ℝ (EC d) : ℝ) < (l : ℝ) := by exact_mod_cast hl
+    linarith [Nat.cast_nonneg (α := ℝ) l]
+
+ 
+
+theorem linearizedVolume_local_gain (k l : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (hl : Module.finrank ℝ (EC d) < l) (hlk : l ≤ k) (x₀ : X) :
+    ∀ᶠ u in 𝓝 (0 : D.localizers.RealSobolev ((k : ℝ)+2)),
+      ∀ (w : D.localizers.RealSobolev ((l : ℝ)+2) × ℝ)
+        (f : D.localizers.RealSobolev (k : ℝ) × ℝ),
+      D.linearizedVolume l hl x₀ (D.localizers.realLower ((k : ℝ)+2) ((l : ℝ)+2)
+          (by exact_mod_cast Nat.add_le_add_right hlk 2) u) w =
+        D.pairLower (k : ℝ) (l : ℝ) (by exact_mod_cast hlk) f →
+      ∃ v : D.localizers.RealSobolev ((k : ℝ)+2) × ℝ,
+        D.pairLower ((k : ℝ)+2) ((l : ℝ)+2) (by exact_mod_cast Nat.add_le_add_right hlk 2) v = w ∧
+        D.linearizedVolume k hk x₀ u v = f := by
+  obtain ⟨m,hm,he⟩ := D.exists_completedError_small
+  obtain ⟨P⟩ := D.localizers.constantProjection_exists
+  let : NormedAddCommGroup (D.localizers.RealSobolev ((k : ℝ)+2)) := inferInstance
+  let : NormedSpace ℝ (D.localizers.RealSobolev ((k : ℝ)+2)) := inferInstance
+  let : NormedAddCommGroup (D.localizers.RealSobolev ((l : ℝ)+2)) := inferInstance
+  let : NormedSpace ℝ (D.localizers.RealSobolev ((l : ℝ)+2)) := inferInstance
+  let : NormedAddCommGroup (D.localizers.RealSobolev (k : ℝ)) := inferInstance
+  let : NormedSpace ℝ (D.localizers.RealSobolev (k : ℝ)) := inferInstance
+  let : NormedAddCommGroup (D.localizers.RealSobolev (l : ℝ)) := inferInstance
+  let : NormedSpace ℝ (D.localizers.RealSobolev (l : ℝ)) := inferInstance
+  have hst : (l : ℝ)+2 ≤ (k : ℝ)+2 := by exact_mod_cast Nat.add_le_add_right hlk 2
+  have hkl : (l : ℝ) ≤ (k : ℝ) := by exact_mod_cast hlk
+  let L := D.localizers.realLower ((k : ℝ)+2) ((l : ℝ)+2) hst
+  have hN : Tendsto L (𝓝 0) (𝓝 0) := by simpa only [map_zero] using L.continuous.tendsto 0
+  filter_upwards [D.linearizedVolume_invertible_nhds m hm he P k hk x₀,
+    hN.eventually (D.linearizedVolume_invertible_nhds m hm he P l hl x₀)] with u hu hv
+  obtain ⟨e,heq⟩ := hu
+  obtain ⟨e',heq'⟩ := hv
+  intro w f hw
+  refine ⟨e.symm f,?_,?_⟩
+  · apply e'.injective
+    have hm₁ := congrArg (fun T : _ →L[ℝ] _ => T
+      (D.pairLower ((k : ℝ)+2) ((l : ℝ)+2) hst (e.symm f))) heq'
+    have hm₂ := congrArg (fun T : _ →L[ℝ] _ => T w) heq'
+    have hm₃ := congrArg (fun T : _ →L[ℝ] _ => T (e.symm f)) heq
+    have hei : e.toContinuousLinearMap (e.symm f) = f := e.apply_symm_apply f
+    rw [hei] at hm₃
+    exact hm₁.trans ((D.linearizedVolume_lower k l hk hl hlk x₀ u (e.symm f)).trans
+      ((congrArg (D.pairLower (k : ℝ) (l : ℝ) hkl) hm₃.symm).trans
+        (hw.symm.trans hm₂.symm)))
+  · have hh := congrArg (fun T : _ →L[ℝ] _ => T (e.symm f)) heq
+    exact hh.symm.trans (e.apply_symm_apply f)
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/UniformCompositionFamily.lean b/lean/OAI/Geometry/PrescribedPotential/UniformCompositionFamily.lean
new file mode 100644
index 0000000000000000000000000000000000000000..1b1922226ec9e103d09a97fe38c8fe9a2d75e7f2
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/UniformCompositionFamily.lean
@@ -0,0 +1,73 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CircleRadialCalculus
+import OAI.Geometry.PrescribedPotential.HigherJetComposition
+import OAI.Geometry.PrescribedPotential.QuasilinearJetBasics
+
+namespace OAI
+
+/-! Uniform Composition Family. -/
+
+section
+
+noncomputable section
+open Set Filter Topology Finset
+open scoped ContDiff
+namespace HigherJet
+variable {E F G : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  [NormedAddCommGroup F] [NormedSpace ℝ F]
+  [NormedAddCommGroup G] [NormedSpace ℝ G]
+lemma composition_family_jets {I : Type*} (f : I → E → F) {K : Set E}
+    (hf : ∀ i x, x ∈ K → ContDiffAt ℝ ∞ (f i) x)
+    (hjet : ∀ m : ℕ, ∃ C : ℝ, ∀ i x, x ∈ K → ‖iteratedFDeriv ℝ m (f i) x‖ ≤ C)
+    {Q : Set F} (hQ : IsCompact Q) (hrange : ∀ i x, x ∈ K → f i x ∈ Q)
+    (g : F → G) (hg : ∀ y ∈ Q, ContDiffAt ℝ ∞ g y) (m : ℕ) :
+    ∃ C : ℝ, ∀ i x, x ∈ K → ‖iteratedFDeriv ℝ m (g ∘ f i) x‖ ≤ C := by
+  classical
+  choose C hC using hjet
+  let B := 1+∑ j ∈ range (m+1), max 0 (C j)
+  have hB : 1 ≤ B := le_add_of_nonneg_right (Finset.sum_nonneg (fun _ _ => le_max_left _ _))
+  have hbound (i : I) (x : E) (hx : x ∈ K) (j : ℕ) (hj : j ≤ m) : ‖iteratedFDeriv ℝ j (f i) x‖ ≤ B := by
+    exact (hC j i x hx).trans ((le_max_right _ _).trans ((Finset.single_le_sum
+      (fun _ _ => le_max_left _ _) (mem_range.mpr (by omega))).trans (le_add_of_nonneg_left zero_le_one)))
+  obtain ⟨A,hA,hAj⟩ := compact_coefficient_jets hQ hg m
+  refine ⟨(Fintype.card (OrderedFinpartition m):ℝ)*A*B^m*1,fun i x hx => ?_⟩
+  exact composition_top_jet_bound (hf i x hx) (hg _ (hrange i x hx)) (r := m+1)
+    (by omega) hB le_rfl (by linarith) (fun j hj hjm => hbound i x hx j (by omega))
+    (fun j hj hjm => (by omega)) (fun j hj => hAj j hj _ (hrange i x hx))
+
+lemma pair_fixed_family_jets {I : Type*} (f : I → E → F) {v : E → G} {K : Set E}
+    (hK : IsCompact K) (hf : ∀ i x, x ∈ K → ContDiffAt ℝ ∞ (f i) x)
+    (hv : ∀ x ∈ K, ContDiffAt ℝ ∞ v x)
+    (hjet : ∀ m : ℕ, ∃ C : ℝ, ∀ i x, x ∈ K → ‖iteratedFDeriv ℝ m (f i) x‖ ≤ C)
+    (m : ℕ) : ∃ C : ℝ, ∀ i x, x ∈ K →
+      ‖iteratedFDeriv ℝ m (fun y => (v y,f i y)) x‖ ≤ C := by
+  obtain ⟨C,hC⟩ := hjet m
+  obtain ⟨B,hB,hbound⟩ := compact_coefficient_jets hK hv m
+  refine ⟨max B C,fun i x hx => ?_⟩
+  rw [iteratedFDeriv_prodMk (hv x hx) (hf i x hx)
+    (show (m:ℕ∞ω) ≤ ∞ from WithTop.coe_le_coe.mpr le_top),ContinuousMultilinearMap.opNorm_prod]
+  exact max_le_max (hbound m le_rfl x hx) (hC i x hx)
+
+lemma parametric_composition_family_jets [FiniteDimensional ℝ F] {I : Type*}
+    (f : I → E → F) {K : Set E} (hK : IsCompact K)
+    (hf : ∀ i x, x ∈ K → ContDiffAt ℝ ∞ (f i) x)
+    (hjet : ∀ m : ℕ, ∃ C : ℝ, ∀ i x, x ∈ K → ‖iteratedFDeriv ℝ m (f i) x‖ ≤ C)
+    (g : E × F → G) (hg : ∀ x ∈ K, ∀ y, ContDiffAt ℝ ∞ g (x,y)) (m : ℕ) :
+    ∃ C : ℝ, ∀ i x, x ∈ K → ‖iteratedFDeriv ℝ m (fun y => g (y,f i y)) x‖ ≤ C := by
+  obtain ⟨R,hR⟩ := hjet 0
+  let Q := K ×ˢ Metric.closedBall (0:F) (max 0 R)
+  have hQ : IsCompact Q := hK.prod (isCompact_closedBall _ _)
+  let ff : I → E → E × F := fun i x => (x,f i x)
+  have hff : ∀ i x, x ∈ K → ContDiffAt ℝ ∞ (ff i) x := fun i x hx => contDiffAt_id.prodMk (hf i x hx)
+  apply composition_family_jets ff hff (pair_fixed_family_jets f hK hf (fun _ _ => contDiffAt_id) hjet) hQ
+  · intro i x hx
+    exact ⟨hx,by simpa only [Metric.mem_closedBall,dist_zero_right,norm_iteratedFDeriv_zero] using
+      (hR i x hx).trans (le_max_right 0 R)⟩
+  · rintro ⟨x,y⟩ hxy
+    exact hg x hxy.1 y
+end HigherJet
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/UniformJetControls.lean b/lean/OAI/Geometry/PrescribedPotential/UniformJetControls.lean
new file mode 100644
index 0000000000000000000000000000000000000000..7cb3f16a436a3738f7f2cd7cdfd6665dd2ddd864
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/UniformJetControls.lean
@@ -0,0 +1,62 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.JetEnergyUniform
+
+namespace OAI
+
+/-! Uniform Jet Controls. -/
+
+section
+
+noncomputable section
+open Set Filter Topology Finset Module
+open scoped ContDiff
+namespace HigherJet
+variable {E F : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [NormedAddCommGroup F] [InnerProductSpace ℝ F]
+  [FiniteDimensional ℝ E] [FiniteDimensional ℝ F]
+  {ι : Type*} [Fintype ι]
+
+lemma sqrt_coercivity {S D q : ℝ} (_hS : 0 ≤ S) (hD : 0 ≤ D) (hq : 0 ≤ q)
+    (h : S ≤ q*D) : Real.sqrt S ≤ (1+q)*Real.sqrt D := by
+  apply (Real.sqrt_le_iff).mpr
+  refine ⟨by positivity,?_⟩
+  rw [mul_pow,Real.sq_sqrt hD]
+  have hh : q ≤ (1+q)^2 := by nlinarith only [sq_nonneg q]
+  exact h.trans (mul_le_mul_of_nonneg_right hh hD)
+
+lemma uniform_jet_controls (e : OrthonormalBasis ι ℝ E) (m : ℕ) {B : ℝ} (hB : 0 ≤ B) :
+    ∃ A : ℝ, 1 ≤ A ∧ ∀ (U : Set E), IsOpen U → ∀ u : E → F, ContDiffOn ℝ ∞ u U →
+      ∀ x ∈ U, ∀ τ : E ≃L[ℝ] E, ‖τ.toContinuousLinearMap‖ ≤ B → ‖τ.symm.toContinuousLinearMap‖ ≤ B →
+        ‖iteratedFDeriv ℝ m u x‖ ≤ A*Real.sqrt (jetEnergy e m u x) ∧
+        ‖iteratedFDeriv ℝ (m+1) u x‖ ≤ A*Real.sqrt (familyDissipation (fun i => τ (e i)) (jetFamily e m u) x) ∧
+        jetEnergy e (m+1) u x ≤ A*familyDissipation (fun i => τ (e i)) (jetFamily e m u) x ∧
+        familyDissipation (fun i => τ (e i)) (jetFamily e m u) x ≤ A*jetEnergy e (m+1) u x := by
+  obtain ⟨N,hN,hn⟩ := norm_iteratedFDeriv_le_jetEnergy (F := F) e.toBasis m
+  obtain ⟨P,hP,hp⟩ := norm_iteratedFDeriv_le_jetEnergy (F := F) e.toBasis (m+1)
+  obtain ⟨C,hC,hc⟩ := jetEnergy_frame_coercivity (F := F) e
+  obtain ⟨D,hD,hd⟩ := jetEnergy_frame_upper (F := F) e
+  let A := 1+N+P*(1+C*B^2)+C*B^2+D*B^2
+  have ht : 0 ≤ P*(1+C*B^2) := by positivity
+  have hc0 : 0 ≤ C*B^2 := by positivity
+  have hd0 : 0 ≤ D*B^2 := by positivity
+  have hnA : N ≤ A := by dsimp [A]; linarith only [hN,ht,hc0,hd0]
+  have hpA : P*(1+C*B^2) ≤ A := by dsimp [A]; nlinarith only [hN,hC,hD,sq_nonneg B,mul_nonneg hC.le (sq_nonneg B),mul_nonneg hD.le (sq_nonneg B)]
+  have hcA : C*B^2 ≤ A := by dsimp [A]; linarith only [hN,ht,hc0,hd0]
+  have hdA : D*B^2 ≤ A := by dsimp [A]; linarith only [hN,ht,hc0,hd0]
+  refine ⟨A,by dsimp [A]; linarith only [hN,ht,hc0,hd0],?_⟩
+  intro U hU u hu x hx τ hτ hτi
+  have hco := hc m u x τ B hB hτi
+  have hup := hd m u x τ B hB hτ
+  have hs := sqrt_coercivity (jetEnergy_nonneg _ _ _ _) (familyDissipation_nonneg _ _ _)
+    (show 0 ≤ C*B^2 by positivity) hco
+  refine ⟨(hn U hU u hu x hx).trans (mul_le_mul_of_nonneg_right hnA (Real.sqrt_nonneg _)),?_,
+    hco.trans (mul_le_mul_of_nonneg_right hcA (familyDissipation_nonneg _ _ _)),
+    hup.trans (mul_le_mul_of_nonneg_right hdA (jetEnergy_nonneg _ _ _ _))⟩
+  exact (hp U hU u hu x hx).trans ((mul_le_mul_of_nonneg_left hs hP.le).trans
+    (by simpa only [mul_assoc] using mul_le_mul_of_nonneg_right hpA (Real.sqrt_nonneg _)))
+end HigherJet
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/UniformPathSobolev.lean b/lean/OAI/Geometry/PrescribedPotential/UniformPathSobolev.lean
new file mode 100644
index 0000000000000000000000000000000000000000..637ada471111c306a92108a40d0905b9944422f7
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/UniformPathSobolev.lean
@@ -0,0 +1,90 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.BoundedPathContinuation
+import OAI.Geometry.PrescribedPotential.CompletedResolvent
+import OAI.Geometry.PrescribedPotential.PathLaplacianSobolev
+import OAI.Geometry.PrescribedPotential.PointPoisson
+import OAI.Geometry.PrescribedPotential.RealSmoothOperator
+
+namespace OAI
+
+/-! Uniform Path Sobolev. -/
+
+section
+
+noncomputable section
+open Set Filter Topology
+open scoped ContDiff
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  [ConnectedSpace X] {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+lemma pointPoissonOrder_laplacian (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
+    (P : D.localizers.ConstantProjection) (k : ℕ)
+    (hk : (Module.finrank ℝ (EC d) : ℝ) < 2*((k : ℝ)+2)) (x₀ : X)
+    (φ : SmoothRealFunction A) (hn : φ.value x₀ = 0) :
+    D.localizers.embed ((k : ℝ)+2) (Smooth.ofReal φ) =
+      D.pointPoissonOrder m hm he P k x₀ (D.localizers.embed (k : ℝ) (Smooth.ofReal (g.laplacian φ))) := by
+  let F := D.localizers.embed (k : ℝ) (Smooth.ofReal (g.laplacian φ))
+  apply (D.pointPoissonOrder_unique m hm he k hk x₀ _ _ 0
+    (D.poissonResidue m hm he P (D.localizers.lower (k : ℝ) 0 F)) ?_ ?_).1
+  · rw [map_zero,sub_zero,D.completedLOrder_embed,complexL_ofReal,D.pointPoissonOrder_equation]
+    exact (add_sub_cancel_right F _).symm
+  · rw [D.pointEvaluation_embed hk,D.pointPoissonOrder_normalized m hm he P k hk]
+    simp only [Smooth.ofReal_apply,hn,Complex.ofReal_zero]
+
+lemma normalized_sobolev_laplacian_estimate (m : ℝ) (hm : 1 ≤ m) (he : ‖D.completedError m hm‖ < 1)
+    (P : D.localizers.ConstantProjection) (k : ℕ)
+    (hk : (Module.finrank ℝ (EC d) : ℝ) < 2*((k : ℝ)+2)) (x₀ : X)
+    (r : ℝ) (hr : r ≤ (k : ℝ)+2) (φ : SmoothRealFunction A) (hn : φ.value x₀ = 0) :
+    ‖D.localizers.embed r (Smooth.ofReal φ)‖ ≤
+      ‖D.localizers.lower ((k : ℝ)+2) r‖ *
+        (‖D.pointPoissonOrder m hm he P k x₀‖ *
+          ‖D.localizers.embed (k : ℝ) (Smooth.ofReal (g.laplacian φ))‖) := by
+  have hv := D.pointPoissonOrder_laplacian m hm he P k hk x₀ φ hn
+  have hhigh : ‖D.localizers.embed ((k : ℝ)+2) (Smooth.ofReal φ)‖ ≤
+      ‖D.pointPoissonOrder m hm he P k x₀‖ *
+          ‖D.localizers.embed (k : ℝ) (Smooth.ofReal (g.laplacian φ))‖ := by
+    rw [hv]
+    exact (D.pointPoissonOrder m hm he P k x₀).le_opNorm _
+  calc
+    _ = ‖D.localizers.lower ((k : ℝ)+2) r (D.localizers.embed ((k : ℝ)+2) (Smooth.ofReal φ))‖ :=
+      congrArg norm (D.localizers.lower_embed hr (Smooth.ofReal φ)).symm
+    _ ≤ ‖D.localizers.lower ((k : ℝ)+2) r‖ * ‖D.localizers.embed ((k : ℝ)+2) (Smooth.ofReal φ)‖ :=
+      (D.localizers.lower ((k : ℝ)+2) r).le_opNorm _
+    _ ≤ _ := mul_le_mul_of_nonneg_left hhigh (norm_nonneg (D.localizers.lower ((k : ℝ)+2) r))
+
+ 
+
+theorem pathSobolevBounds (h : SemipositiveAnticanonicalMetric A) (x₀ : X) :
+    D.PathSobolevBounds h x₀ := by
+  obtain ⟨m,hm,he⟩ := D.exists_completedError_small
+  obtain ⟨P⟩ := D.localizers.constantProjection_exists
+  intro k
+  let j : ℕ := 2*k + Module.finrank ℝ (EC d)
+  have hj : (Module.finrank ℝ (EC d) : ℝ) < 2*((j : ℝ)+2) := by
+    dsimp [j]
+    push_cast
+    nlinarith [Nat.cast_nonneg (α := ℝ) k,Nat.cast_nonneg (α := ℝ) (Module.finrank ℝ (EC d))]
+  have hkj : (((2*k : ℕ) : ℝ)+2) ≤ ((j : ℝ)+2) := by
+    dsimp [j]
+    push_cast
+    linarith [Nat.cast_nonneg (α := ℝ) (Module.finrank ℝ (EC d))]
+  obtain ⟨C,hC⟩ := path_laplacian_sobolev_bound g h D.localizers (j : ℝ)
+  refine ⟨‖D.localizers.lower ((j : ℝ)+2) (((2*k : ℕ) : ℝ)+2)‖ *
+    (‖D.pointPoissonOrder m hm he P j x₀‖*C),fun t b φ ht hn hs => ?_⟩
+  let s : PathSolution g h := ⟨t,b,φ,ht,hs⟩
+  exact (D.normalized_sobolev_laplacian_estimate m hm he P j hj x₀
+    (((2*k : ℕ) : ℝ)+2) hkj φ hn).trans
+    (mul_le_mul_of_nonneg_left
+      (mul_le_mul_of_nonneg_left (hC s) (norm_nonneg (D.pointPoissonOrder m hm he P j x₀)))
+      (norm_nonneg (D.localizers.lower ((j : ℝ)+2) (((2*k : ℕ) : ℝ)+2))))
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/VariablePoisson.lean b/lean/OAI/Geometry/PrescribedPotential/VariablePoisson.lean
new file mode 100644
index 0000000000000000000000000000000000000000..cf79f6108b81cb2000ca9be3c76e5acb12b59187
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/VariablePoisson.lean
@@ -0,0 +1,265 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.SobolevParametrix
+
+namespace OAI
+
+/-! Variable Poisson. -/
+
+section
+
+ 
+
+noncomputable section
+open MeasureTheory FourierTransform TemperedDistribution LineDeriv
+open scoped SchwartzMap BoundedContinuousFunction ComplexOrder MatrixOrder Real
+
+namespace SobolevChart
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+
+ 
+def secondH2Symbol (v w ξ : E) : ℂ :=
+  (-(2 * Real.pi)^2 : ℂ) * (inner ℝ ξ v : ℂ) * (inner ℝ ξ w : ℂ) *
+    ((1 + ‖ξ‖^2)^(-1 : ℝ) : ℝ)
+
+lemma secondH2Symbol_temperate (v w : E) :
+    (secondH2Symbol v w).HasTemperateGrowth := by
+  unfold secondH2Symbol
+  fun_prop
+
+lemma secondH2Symbol_bound (v w ξ : E) :
+    ‖secondH2Symbol v w ξ‖ ≤ (2 * Real.pi)^2 * ‖v‖ * ‖w‖ := by
+  have hp : 0 < 1 + ‖ξ‖^2 := by positivity
+  have hv := abs_real_inner_le_norm ξ v
+  have hw := abs_real_inner_le_norm ξ w
+  have hmul := mul_le_mul hv hw (abs_nonneg _) (mul_nonneg (norm_nonneg _) (norm_nonneg _))
+  have hle : |inner ℝ ξ v| * |inner ℝ ξ w| ≤
+      (1 + ‖ξ‖^2) * (‖v‖ * ‖w‖) := by
+    have hnonneg : 0 ≤ ‖v‖ * ‖w‖ := mul_nonneg (norm_nonneg _) (norm_nonneg _)
+    nlinarith
+  simp only [secondH2Symbol, norm_mul, norm_neg, norm_pow, Complex.norm_ofNat,
+    Complex.norm_real, Real.norm_eq_abs, abs_of_pos Real.pi_pos, Real.rpow_neg_one,
+    abs_inv, abs_of_pos hp]
+  have hpi : 0 ≤ (2 * Real.pi)^2 := sq_nonneg _
+  calc
+    _ = (2 * Real.pi)^2 * ((|inner ℝ ξ v| * |inner ℝ ξ w|) / (1 + ‖ξ‖^2)) := by ring
+    _ ≤ (2 * Real.pi)^2 * (‖v‖ * ‖w‖) :=
+      mul_le_mul_of_nonneg_left ((div_le_iff₀ hp).mpr (by nlinarith [hle])) hpi
+    _ = _ := by ring
+
+def secondH2SymbolBCF (v w : E) : E →ᵇ ℂ :=
+  BoundedContinuousFunction.ofNormedAddCommGroup (secondH2Symbol v w)
+    (secondH2Symbol_temperate v w).1.continuous
+    ((2 * Real.pi)^2 * ‖v‖ * ‖w‖) (secondH2Symbol_bound v w)
+
+lemma secondH2SymbolBCF_norm (v w : E) :
+    ‖secondH2SymbolBCF v w‖ ≤ (2 * Real.pi)^2 * ‖v‖ * ‖w‖ :=
+  (BoundedContinuousFunction.norm_le (by positivity)).mpr (secondH2Symbol_bound v w)
+
+variable [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+
+def secondH2 (v w : E) : L2 E →L[ℂ] L2 E := multiplier (secondH2SymbolBCF v w)
+
+lemma secondH2_bound (v w : E) (u : L2 E) :
+    ‖secondH2 v w u‖ ≤ ((2 * Real.pi)^2 * ‖v‖ * ‖w‖) * ‖u‖ :=
+  (multiplier_norm_le (secondH2SymbolBCF v w) u).trans
+    (mul_le_mul_of_nonneg_right (secondH2SymbolBCF_norm v w) (norm_nonneg _))
+
+lemma secondH2_distribution (v w : E) (u : L2 E) :
+    (secondH2 v w u : 𝓢'(E, ℂ)) = ∂_{v} (∂_{w} (realize 2 u)) := by
+  rw [FrozenPoisson.second_deriv_multiplier]
+  change (multiplier (secondH2SymbolBCF v w) u : 𝓢'(E, ℂ)) =
+    fourierMultiplierCLM ℂ _ (besselPotential E ℂ (-2) (u : 𝓢'(E, ℂ)))
+  rw [multiplier_distribution _ (secondH2Symbol_temperate v w), besselPotential,
+    fourierMultiplierCLM_fourierMultiplierCLM_apply (by fun_prop) (by fun_prop)]
+  congr 2
+  ext ξ
+  change secondH2Symbol v w ξ = _
+  simp only [secondH2Symbol, Pi.mul_apply]
+  norm_num
+  ring
+
+variable {ι : Type*} [Fintype ι]
+
+ 
+def perturbation (v : ι → E) (a : ι → ι → E →ᵇ ℂ) : L2 E →L[ℂ] L2 E :=
+  ∑ i, ∑ j, multiply (a i j) ∘L secondH2 (v i) (v j)
+
+def perturbationBound (v : ι → E) (a : ι → ι → E →ᵇ ℂ) : ℝ :=
+  ∑ i, ∑ j, ‖a i j‖ * ((2 * Real.pi)^2 * ‖v i‖ * ‖v j‖)
+
+omit [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] in
+lemma perturbationBound_nonneg (v : ι → E) (a : ι → ι → E →ᵇ ℂ) :
+    0 ≤ perturbationBound v a := by
+  unfold perturbationBound
+  positivity
+
+lemma perturbation_bound (v : ι → E) (a : ι → ι → E →ᵇ ℂ) (u : L2 E) :
+    ‖perturbation v a u‖ ≤ perturbationBound v a * ‖u‖ := by
+  simp only [perturbation, _root_.sum_apply, ContinuousLinearMap.comp_apply,
+    perturbationBound, Finset.sum_mul]
+  apply (norm_sum_le _ _).trans
+  apply Finset.sum_le_sum
+  intro i _
+  apply (norm_sum_le _ _).trans
+  apply Finset.sum_le_sum
+  intro j _
+  calc
+    _ ≤ ‖a i j‖ * ‖secondH2 (v i) (v j) u‖ := multiply_norm_le _ _
+    _ ≤ ‖a i j‖ * (((2 * Real.pi)^2 * ‖v i‖ * ‖v j‖) * ‖u‖) :=
+      mul_le_mul_of_nonneg_left (secondH2_bound _ _ u) (norm_nonneg _)
+    _ = _ := by ring
+
+lemma perturbation_distribution (v : ι → E) (a : ι → ι → E →ᵇ ℂ)
+    (ha : ∀ i j, (a i j : E → ℂ).HasTemperateGrowth) (u : L2 E) :
+    (perturbation v a u : 𝓢'(E, ℂ)) =
+      ∑ i, ∑ j, smulLeftCLM ℂ (a i j) (∂_{v i} (∂_{v j} (realize 2 u))) := by
+  change Lp.toTemperedDistributionCLM ℂ volume 2 _ = _
+  simp only [perturbation, _root_.sum_apply, ContinuousLinearMap.comp_apply,
+    map_sum]
+  apply Finset.sum_congr rfl
+  intro i _
+  apply Finset.sum_congr rfl
+  intro j _
+  rw [Lp.toTemperedDistributionCLM_apply, multiply_distribution _ (ha i j),
+    secondH2_distribution]
+
+end SobolevChart
+
+namespace FrozenPoisson
+open EllipticKernel SobolevChart
+variable {n : ℕ} {ι : Type*} [Fintype ι]
+
+lemma perturbation_resolvent_bound (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (v : ι → EC n) (a : ι → ι → EC n →ᵇ ℂ) :
+    ‖perturbation v a ∘L hilbertResolvent H hH‖ ≤
+      perturbationBound v a * ellipticBound H hH := by
+  apply ContinuousLinearMap.opNorm_le_bound (perturbation v a ∘L hilbertResolvent H hH)
+    (mul_nonneg (perturbationBound_nonneg v a) (ellipticBound_pos H hH).le)
+  intro u
+  calc
+    _ ≤ perturbationBound v a * ‖hilbertResolvent H hH u‖ := perturbation_bound _ _ _
+    _ ≤ perturbationBound v a * (ellipticBound H hH * ‖u‖) :=
+      mul_le_mul_of_nonneg_left (hilbertResolvent_bound H hH u)
+        (perturbationBound_nonneg v a)
+    _ = _ := by ring
+
+ 
+def localResolvent (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (v : ι → EC n) (a : ι → ι → EC n →ᵇ ℂ)
+    (hsmall : perturbationBound v a * ellipticBound H hH < 1) :
+    L2 (EC n) →L[ℂ] L2 (EC n) :=
+  hilbertResolvent H hH ∘L
+    (↑((Units.oneSub (perturbation v a ∘L hilbertResolvent H hH)
+      ((perturbation_resolvent_bound H hH v a).trans_lt hsmall))⁻¹) :
+      L2 (EC n) →L[ℂ] L2 (EC n))
+
+lemma localResolvent_equation (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (v : ι → EC n) (a : ι → ι → EC n →ᵇ ℂ)
+    (hsmall : perturbationBound v a * ellipticBound H hH < 1) (f : L2 (EC n)) :
+    realize 2 (localResolvent H hH v a hsmall f) -
+      frozenDifferential H (realize 2 (localResolvent H hH v a hsmall f)) -
+      (perturbation v a (localResolvent H hH v a hsmall f) : 𝓢'(EC n, ℂ)) =
+      (f : 𝓢'(EC n, ℂ)) := by
+  let U := Units.oneSub (perturbation v a ∘L hilbertResolvent H hH)
+      ((perturbation_resolvent_bound H hH v a).trans_lt hsmall)
+  let f' := (↑U⁻¹ : L2 (EC n) →L[ℂ] L2 (EC n)) f
+  have he : f' - perturbation v a (hilbertResolvent H hH f') = f := by
+    change ((↑U : L2 (EC n) →L[ℂ] L2 (EC n)) * ↑U⁻¹) f = f
+    rw [Units.mul_inv, one_apply_eq_self]
+  have hr := (hilbert_frozen_inverse H hH 0 f').1
+  norm_num at hr
+  change realize 2 (hilbertResolvent H hH f') -
+    frozenDifferential H (realize 2 (hilbertResolvent H hH f')) -
+      (perturbation v a (hilbertResolvent H hH f') : 𝓢'(EC n, ℂ)) = _
+  rw [hr]
+  have hz (g : L2 (EC n)) : realize 0 g = (g : 𝓢'(EC n, ℂ)) := by simp [realize]
+  rw [hz]
+  simpa only [map_sub, Lp.toTemperedDistributionCLM_apply] using
+    congrArg (Lp.toTemperedDistributionCLM ℂ volume 2) he
+
+lemma fixed_for_perturbed (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (v : ι → EC n) (a : ι → ι → EC n →ᵇ ℂ) (u f : L2 (EC n))
+    (hu : realize 2 u - frozenDifferential H (realize 2 u) -
+      (perturbation v a u : 𝓢'(EC n, ℂ)) = (f : 𝓢'(EC n, ℂ))) :
+    u = hilbertResolvent H hH (f + perturbation v a u) := by
+  apply realize_injective 2
+  have hs : 2 = (0 : ℝ) + 2 := by norm_num
+  rw [hs, hilbertResolvent_realize H hH 0]
+  have he : shifted H (realize 2 u) =
+      realize 0 (f + perturbation v a u) := by
+    rw [shifted_eq_identity_sub H hH]
+    simpa [realize, map_add, sub_eq_iff_eq_add] using sub_eq_iff_eq_add.mp hu
+  rw [← he, resolvent_shifted H hH]
+  norm_num
+
+lemma perturbed_unique (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (v : ι → EC n) (a : ι → ι → EC n →ᵇ ℂ)
+    (hsmall : perturbationBound v a * ellipticBound H hH < 1)
+    (u w f : L2 (EC n))
+    (hu : realize 2 u - frozenDifferential H (realize 2 u) -
+      (perturbation v a u : 𝓢'(EC n, ℂ)) = (f : 𝓢'(EC n, ℂ)))
+    (hw : realize 2 w - frozenDifferential H (realize 2 w) -
+      (perturbation v a w : 𝓢'(EC n, ℂ)) = (f : 𝓢'(EC n, ℂ))) : u = w := by
+  have huf := fixed_for_perturbed H hH v a u f hu
+  have hwf := fixed_for_perturbed H hH v a w f hw
+  have he : u - w = hilbertResolvent H hH (perturbation v a (u - w)) := by
+    rw [map_sub, map_sub]
+    rw [map_add] at huf hwf
+    calc
+      u - w = (hilbertResolvent H hH f + hilbertResolvent H hH (perturbation v a u)) -
+          (hilbertResolvent H hH f + hilbertResolvent H hH (perturbation v a w)) :=
+        congrArg₂ (· - ·) huf hwf
+      _ = _ := by abel
+  have hn : ‖u - w‖ ≤ ellipticBound H hH * perturbationBound v a * ‖u - w‖ := by
+    calc
+      _ = ‖hilbertResolvent H hH (perturbation v a (u - w))‖ := congrArg norm he
+      _ ≤ _ := (hilbertResolvent_bound H hH _).trans
+        (by nlinarith [perturbation_bound v a (u - w), ellipticBound_pos H hH])
+  have : ‖u - w‖ = 0 := by nlinarith [norm_nonneg (u - w)]
+  exact sub_eq_zero.mp (norm_eq_zero.mp this)
+
+lemma localResolvent_bound (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (v : ι → EC n) (a : ι → ι → EC n →ᵇ ℂ)
+    (hsmall : perturbationBound v a * ellipticBound H hH < 1) (f : L2 (EC n)) :
+    ‖localResolvent H hH v a hsmall f‖ ≤
+      (ellipticBound H hH / (1 - perturbationBound v a * ellipticBound H hH)) * ‖f‖ := by
+  let u := localResolvent H hH v a hsmall f
+  have he := fixed_for_perturbed H hH v a u f (localResolvent_equation H hH v a hsmall f)
+  have hn : ‖u‖ ≤ ellipticBound H hH *
+      (‖f‖ + perturbationBound v a * ‖u‖) := by
+    calc
+      ‖u‖ = ‖hilbertResolvent H hH (f + perturbation v a u)‖ := congrArg norm he
+      _ ≤ ellipticBound H hH * ‖f + perturbation v a u‖ := hilbertResolvent_bound H hH _
+      _ ≤ _ := mul_le_mul_of_nonneg_left
+        ((norm_add_le _ _).trans (add_le_add le_rfl (perturbation_bound v a u)))
+          (ellipticBound_pos H hH).le
+  have hp : 0 < 1 - perturbationBound v a * ellipticBound H hH := sub_pos.mpr hsmall
+  change ‖u‖ ≤ _
+  rw [div_mul_eq_mul_div, le_div_iff₀ hp]
+  nlinarith
+
+ 
+
+theorem variable_sobolev_resolvent (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (v : ι → EC n) (a : ι → ι → EC n →ᵇ ℂ)
+    (ha : ∀ i j, (a i j : EC n → ℂ).HasTemperateGrowth)
+    (hsmall : perturbationBound v a * ellipticBound H hH < 1) (f : L2 (EC n)) :
+    ∃! u : L2 (EC n),
+      realize 2 u - frozenDifferential H (realize 2 u) -
+        ∑ i, ∑ j, smulLeftCLM ℂ (a i j) (∂_{v i} (∂_{v j} (realize 2 u))) =
+          (f : 𝓢'(EC n, ℂ)) := by
+  refine ⟨localResolvent H hH v a hsmall f, ?_, ?_⟩
+  · dsimp only
+    rw [← perturbation_distribution v a ha]
+    exact localResolvent_equation H hH v a hsmall f
+  · intro u hu
+    rw [← perturbation_distribution v a ha] at hu
+    exact perturbed_unique H hH v a hsmall u _ f hu
+      (localResolvent_equation H hH v a hsmall f)
+
+end FrozenPoisson
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/VolumeDeterminantBounds.lean b/lean/OAI/Geometry/PrescribedPotential/VolumeDeterminantBounds.lean
new file mode 100644
index 0000000000000000000000000000000000000000..efb370b81d82077f2466d3253e91b3c9899c487c
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/VolumeDeterminantBounds.lean
@@ -0,0 +1,64 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.VolumeNormalization
+
+namespace OAI
+
+/-! Volume Determinant Bounds. -/
+
+section
+
+ 
+
+noncomputable section
+open Matrix Filter Set Topology
+open scoped ComplexOrder Classical ContDiff
+namespace Anticanonical.SourceSmooth
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+
+lemma volumePath_logRatio_bound [CompactSpace X] [Nonempty X]
+    (g : KaehlerMetric A) (h : SemipositiveAnticanonicalMetric A)
+    {t b : ℝ} {φ : SmoothRealFunction A} (ht : t ∈ Icc 0 1)
+    (hp : g.PositivePotential φ)
+    (he : ∀ x, (g.logRatio (g.deform φ hp)).value x = t*(prescribedForcing g h).value x+b)
+    (x : X) : |(g.logRatio (g.deform φ hp)).value x| ≤
+      2*‖(⟨(prescribedForcing g h).value, (prescribedForcing g h).continuous⟩ : C(X,ℝ))‖ := by
+  let F : C(X,ℝ) := ⟨(prescribedForcing g h).value, (prescribedForcing g h).continuous⟩
+  have hb := volumePath_scalar_bound g h ht ⟨hp,he⟩
+  have hf : |t*(prescribedForcing g h).value x| ≤ ‖F‖ := by
+    rw [abs_mul, abs_of_nonneg ht.1]
+    apply (mul_le_mul_of_nonneg_right ht.2 (abs_nonneg _)).trans
+    simpa [F, Real.norm_eq_abs] using F.norm_coe_le_norm x
+  rw [he x]
+  exact (abs_add_le _ _).trans (by change _ ≤ 2*‖F‖; linarith)
+
+ 
+
+theorem volumePath_det_ratio_bounds [CompactSpace X] [Nonempty X]
+    (g : KaehlerMetric A) (h : SemipositiveAnticanonicalMetric A)
+    {t b : ℝ} {φ : SmoothRealFunction A} (ht : t ∈ Icc 0 1)
+    (hp : g.PositivePotential φ)
+    (he : ∀ x, (g.logRatio (g.deform φ hp)).value x = t*(prescribedForcing g h).value x+b)
+    (i : Fin A.count) {z : Coordinates d} (hz : z ∈ (A.chart i).target) :
+    let M := ‖(⟨(prescribedForcing g h).value, (prescribedForcing g h).continuous⟩ : C(X,ℝ))‖;
+    Real.exp (-2*M) ≤ (g.matrix i z + φ.hessian i z).det.re / (g.matrix i z).det.re ∧
+    (g.matrix i z + φ.hessian i z).det.re / (g.matrix i z).det.re ≤ Real.exp (2*M) := by
+  let M := ‖(⟨(prescribedForcing g h).value, (prescribedForcing g h).continuous⟩ : C(X,ℝ))‖
+  let x := (A.chart i).symm z
+  have hi : x ∈ (A.chart i).source := (A.chart i).mapsTo_symm hz
+  have hl := volumePath_logRatio_bound g h ht hp he x
+  change |g.logRatioValue (g.deform φ hp) x| ≤ 2*M at hl
+  rw [g.logRatioValue_local _ i hi, (A.chart i).right_inv hz] at hl
+  have heq : (g.matrix i z + φ.hessian i z).det.re / (g.matrix i z).det.re =
+      Real.exp (Real.log ((g.deform φ hp).volumeCoefficient i z) - Real.log (g.volumeCoefficient i z)) := by
+    rw [Real.exp_sub, Real.exp_log ((g.deform φ hp).volumeCoefficient_pos i hz),
+      Real.exp_log (g.volumeCoefficient_pos i hz)]
+    rfl
+  rw [heq]
+  obtain ⟨hlo,hup⟩ := abs_le.mp hl
+  exact ⟨Real.exp_le_exp.mpr (by simpa only [neg_mul] using hlo), Real.exp_le_exp.mpr hup⟩
+end Anticanonical.SourceSmooth
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/VolumeNormalization.lean b/lean/OAI/Geometry/PrescribedPotential/VolumeNormalization.lean
new file mode 100644
index 0000000000000000000000000000000000000000..73674643d308039b0af95587353f357c3730644a
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/VolumeNormalization.lean
@@ -0,0 +1,100 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.DeterminantOrder
+import OAI.Geometry.PrescribedPotential.GlobalSobolev
+import OAI.Geometry.PrescribedPotential.VolumePath
+
+namespace OAI
+
+/-! Volume Normalization. -/
+
+section
+
+ 
+
+noncomputable section
+open Matrix Filter Set Topology
+open scoped ComplexOrder Classical ContDiff
+namespace Anticanonical.SourceSmooth
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+namespace SmoothRealFunction
+
+lemma negative_hessian_posSemidef_at_globalMax (φ : SmoothRealFunction A) {x : X}
+    (hx : ∀ y, φ.value y ≤ φ.value x) (i : Fin A.count) (hi : x ∈ (A.chart i).source) :
+    (-φ.hessian i (A.chart i x)).PosSemidef := by
+  have hs := (φ.smooth i).contDiffAt
+    ((A.chart i).open_target.mem_nhds ((A.chart i).mapsTo hi))
+  have hm : IsLocalMax (φ.localExpression i) (A.chart i x) := by
+    apply Filter.Eventually.of_forall
+    intro z
+    simpa [localExpression, (A.chart i).left_inv hi] using hx ((A.chart i).symm z)
+  exact EllipticKernel.negative_potentialMatrix_posSemidef_at_localMax hs hm
+
+lemma hessian_posSemidef_at_globalMin (φ : SmoothRealFunction A) {x : X}
+    (hx : ∀ y, φ.value x ≤ φ.value y) (i : Fin A.count) (hi : x ∈ (A.chart i).source) :
+    (φ.hessian i (A.chart i x)).PosSemidef := by
+  have hm : ∀ y, (φ.realSMul (-1)).value y ≤ (φ.realSMul (-1)).value x := by
+    intro y
+    change -1 * φ.value y ≤ -1 * φ.value x
+    linarith [hx y]
+  have hp := (φ.realSMul (-1)).negative_hessian_posSemidef_at_globalMax hm i hi
+  simpa [hessian_realSMul] using hp
+end SmoothRealFunction
+namespace KaehlerMetric
+
+lemma logRatio_deform_nonpos_at_globalMax (g : KaehlerMetric A) (φ : SmoothRealFunction A)
+    (hp : g.PositivePotential φ) {x : X} (hx : ∀ y, φ.value y ≤ φ.value x) :
+    (g.logRatio (g.deform φ hp)).value x ≤ 0 := by
+  obtain ⟨i,hi⟩ := A.covers x
+  have hz := (A.chart i).mapsTo hi
+  have hd : (g.matrix i (A.chart i x) + φ.hessian i (A.chart i x)).det.re ≤
+      (g.matrix i (A.chart i x)).det.re := by
+    apply MongeAmpere.det_re_mono (hp i _ hz)
+    simpa only [sub_add_cancel_left] using φ.negative_hessian_posSemidef_at_globalMax hx i hi
+  change g.logRatioValue (g.deform φ hp) x ≤ 0
+  rw [g.logRatioValue_local _ i hi]
+  apply sub_nonpos.mpr
+  apply Real.log_le_log ((g.deform φ hp).volumeCoefficient_pos i hz)
+  exact hd
+
+lemma logRatio_deform_nonneg_at_globalMin (g : KaehlerMetric A) (φ : SmoothRealFunction A)
+    (hp : g.PositivePotential φ) {x : X} (hx : ∀ y, φ.value x ≤ φ.value y) :
+    0 ≤ (g.logRatio (g.deform φ hp)).value x := by
+  obtain ⟨i,hi⟩ := A.covers x
+  have hz := (A.chart i).mapsTo hi
+  have hd := MongeAmpere.det_re_add_posSemidef _ _ (g.positive i _ hz)
+    (φ.hessian_posSemidef_at_globalMin hx i hi)
+  change 0 ≤ g.logRatioValue (g.deform φ hp) x
+  rw [g.logRatioValue_local _ i hi]
+  apply sub_nonneg.mpr
+  exact Real.log_le_log (g.volumeCoefficient_pos i hz) hd
+end KaehlerMetric
+
+ 
+
+theorem volumePath_scalar_bound [CompactSpace X] [Nonempty X]
+    (g : KaehlerMetric A) (h : SemipositiveAnticanonicalMetric A)
+    {t b : ℝ} {φ : SmoothRealFunction A} (ht : t ∈ Icc 0 1)
+    (hsol : SolvesVolumePath g h t φ b) :
+    |b| ≤ ‖(⟨(prescribedForcing g h).value, (prescribedForcing g h).continuous⟩ : C(X,ℝ))‖ := by
+  let F : C(X,ℝ) := ⟨(prescribedForcing g h).value, (prescribedForcing g h).continuous⟩
+  obtain ⟨hp,he⟩ := hsol
+  have hbound (x : X) : |t * (prescribedForcing g h).value x| ≤ ‖F‖ := by
+    rw [abs_mul, abs_of_nonneg ht.1]
+    exact (mul_le_mul_of_nonneg_right ht.2 (abs_nonneg _)).trans (by
+      simpa [F, Real.norm_eq_abs] using F.norm_coe_le_norm x)
+  obtain ⟨x,_,hx⟩ := isCompact_univ.exists_isMaxOn univ_nonempty φ.continuous.continuousOn
+  obtain ⟨y,_,hy⟩ := isCompact_univ.exists_isMinOn univ_nonempty φ.continuous.continuousOn
+  have hmax := g.logRatio_deform_nonpos_at_globalMax φ hp (fun z => hx (mem_univ z))
+  have hmin := g.logRatio_deform_nonneg_at_globalMin φ hp (fun z => hy (mem_univ z))
+  rw [he x] at hmax
+  rw [he y] at hmin
+  obtain ⟨hlx,hux⟩ := abs_le.mp (hbound x)
+  obtain ⟨hly,huy⟩ := abs_le.mp (hbound y)
+  apply abs_le.mpr
+  constructor <;> linarith
+end Anticanonical.SourceSmooth
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/VolumePath.lean b/lean/OAI/Geometry/PrescribedPotential/VolumePath.lean
new file mode 100644
index 0000000000000000000000000000000000000000..02c10617b9abf92839e6ef0a60f2f3ee41ae0cea
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/VolumePath.lean
@@ -0,0 +1,178 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.SourceTarget
+
+namespace OAI
+
+/-! Volume Path. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff ComplexOrder Classical
+namespace Anticanonical.SourceSmooth
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+namespace KaehlerMetric
+
+lemma volumeCoefficient_pos (g : KaehlerMetric A) (i : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart i).target) : 0 < g.volumeCoefficient i z :=
+  (Complex.pos_iff.mp (g.positive i z hz).det_pos).1
+
+lemma volumeCoefficient_smooth (g : KaehlerMetric A) (i : Fin A.count) :
+    ContDiffOn ℝ ∞ (g.volumeCoefficient i) (A.chart i).target := by
+  apply Complex.reCLM.contDiff.comp_contDiffOn
+  simp only [Matrix.det_apply']
+  apply ContDiffOn.sum
+  intro σ _
+  apply contDiffOn_const.mul
+  apply contDiffOn_prod
+  intro j _
+  exact contDiffOn_pi.mp (contDiffOn_pi.mp (g.smooth i) (σ j)) j
+
+lemma volumeCoefficient_compatibility (g : KaehlerMetric A) (i j : Fin A.count) {x : X}
+    (hi : x ∈ (A.chart i).source) (hj : x ∈ (A.chart j).source) :
+    g.volumeCoefficient i (A.chart i x) = ‖A.jacobian i j x‖^2 *
+      g.volumeCoefficient j (A.chart j x) := by
+  unfold volumeCoefficient
+  rw [g.compatibility i j x hi hj, Matrix.det_mul, Matrix.det_mul, Matrix.det_conjTranspose]
+  change ((starRingEnd ℂ) (A.jacobian i j x) *
+    (g.matrix j (A.chart j x)).det * A.jacobian i j x).re = _
+  have he : (starRingEnd ℂ) (A.jacobian i j x) *
+      (g.matrix j (A.chart j x)).det * A.jacobian i j x =
+      ((‖A.jacobian i j x‖^2 : ℝ) : ℂ) * (g.matrix j (A.chart j x)).det := by
+    rw [show (starRingEnd ℂ) (A.jacobian i j x) *
+      (g.matrix j (A.chart j x)).det * A.jacobian i j x =
+      (A.jacobian i j x * (starRingEnd ℂ) (A.jacobian i j x)) *
+      (g.matrix j (A.chart j x)).det by ring, Complex.mul_conj,
+      Complex.normSq_eq_norm_sq]
+  rw [he, Complex.mul_re, Complex.ofReal_re, Complex.ofReal_im, zero_mul, sub_zero]
+
+lemma logVolume_compatibility (g : KaehlerMetric A) (i j : Fin A.count) {x : X}
+    (hi : x ∈ (A.chart i).source) (hj : x ∈ (A.chart j).source) :
+    Real.log (g.volumeCoefficient i (A.chart i x)) =
+      Real.log (‖A.jacobian i j x‖^2) + Real.log (g.volumeCoefficient j (A.chart j x)) := by
+  rw [g.volumeCoefficient_compatibility i j hi hj]
+  apply Real.log_mul
+  · have hpos := g.volumeCoefficient_pos i ((A.chart i).mapsTo hi)
+    rw [g.volumeCoefficient_compatibility i j hi hj] at hpos
+    exact left_ne_zero_of_mul (ne_of_gt hpos)
+  · exact ne_of_gt (g.volumeCoefficient_pos j ((A.chart j).mapsTo hj))
+
+ 
+def logRatioValue (g₀ g : KaehlerMetric A) (x : X) : ℝ :=
+  let i := (A.covers x).choose;
+  Real.log (g.volumeCoefficient i (A.chart i x)) -
+    Real.log (g₀.volumeCoefficient i (A.chart i x))
+
+lemma logRatioValue_local (g₀ g : KaehlerMetric A) (i : Fin A.count) {x : X}
+    (hi : x ∈ (A.chart i).source) :
+    g₀.logRatioValue g x = Real.log (g.volumeCoefficient i (A.chart i x)) -
+      Real.log (g₀.volumeCoefficient i (A.chart i x)) := by
+  dsimp only [logRatioValue]
+  rw [g.logVolume_compatibility _ i (A.covers x).choose_spec hi,
+    g₀.logVolume_compatibility _ i (A.covers x).choose_spec hi]
+  ring
+
+def logRatio (g₀ g : KaehlerMetric A) : SmoothRealFunction A where
+  value := g₀.logRatioValue g
+  smooth i := by
+    have hg := (g.volumeCoefficient_smooth i).log
+      (fun z hz => ne_of_gt (g.volumeCoefficient_pos i hz))
+    have h₀ := (g₀.volumeCoefficient_smooth i).log
+      (fun z hz => ne_of_gt (g₀.volumeCoefficient_pos i hz))
+    apply (hg.sub h₀).congr
+    intro z hz
+    dsimp only [Function.comp_apply]
+    rw [g₀.logRatioValue_local g i ((A.chart i).mapsTo_symm hz), (A.chart i).right_inv hz]
+
+lemma logRatio_local (g₀ g : KaehlerMetric A) (i : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart i).target) :
+    (g₀.logRatio g).localExpression i z =
+      Real.log (g.volumeCoefficient i z) - Real.log (g₀.volumeCoefficient i z) := by
+  change g₀.logRatioValue g ((A.chart i).symm z) = _
+  rw [g₀.logRatioValue_local g i ((A.chart i).mapsTo_symm hz), (A.chart i).right_inv hz]
+
+end KaehlerMetric
+
+ 
+
+def prescribedForcingValue (g₀ : KaehlerMetric A)
+    (h : SemipositiveAnticanonicalMetric A) (x : X) : ℝ :=
+  let i := (A.covers x).choose;
+  -h.weight i (A.chart i x) - Real.log (g₀.volumeCoefficient i (A.chart i x))
+
+lemma prescribedForcingValue_local (g₀ : KaehlerMetric A)
+    (h : SemipositiveAnticanonicalMetric A) (i : Fin A.count) {x : X}
+    (hi : x ∈ (A.chart i).source) :
+    prescribedForcingValue g₀ h x =
+      -h.weight i (A.chart i x) - Real.log (g₀.volumeCoefficient i (A.chart i x)) := by
+  dsimp only [prescribedForcingValue]
+  have hw := h.compatibility (A.covers x).choose i x (A.covers x).choose_spec hi
+  rw [g₀.logVolume_compatibility _ i (A.covers x).choose_spec hi]
+  linarith
+
+def prescribedForcing (g₀ : KaehlerMetric A)
+    (h : SemipositiveAnticanonicalMetric A) : SmoothRealFunction A where
+  value := prescribedForcingValue g₀ h
+  smooth i := by
+    apply ((h.smooth i).neg.sub ((g₀.volumeCoefficient_smooth i).log
+      (fun z hz => ne_of_gt (g₀.volumeCoefficient_pos i hz)))).congr
+    intro z hz
+    dsimp only [Function.comp_apply]
+    rw [prescribedForcingValue_local g₀ h i ((A.chart i).mapsTo_symm hz), (A.chart i).right_inv hz]
+
+lemma prescribedForcing_local (g₀ : KaehlerMetric A)
+    (h : SemipositiveAnticanonicalMetric A) (i : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart i).target) :
+    (prescribedForcing g₀ h).localExpression i z =
+      -h.weight i z - Real.log (g₀.volumeCoefficient i z) := by
+  change prescribedForcingValue g₀ h ((A.chart i).symm z) = _
+  rw [prescribedForcingValue_local g₀ h i ((A.chart i).mapsTo_symm hz), (A.chart i).right_inv hz]
+
+ 
+
+def SolvesVolumePath (g₀ : KaehlerMetric A) (h : SemipositiveAnticanonicalMetric A)
+    (t : ℝ) (φ : SmoothRealFunction A) (b : ℝ) : Prop :=
+  ∃ hp : g₀.PositivePotential φ, ∀ x,
+    (g₀.logRatio (g₀.deform φ hp)).value x = t * (prescribedForcing g₀ h).value x + b
+
+lemma volumePath_zero (g₀ : KaehlerMetric A) (h : SemipositiveAnticanonicalMetric A) :
+    SolvesVolumePath g₀ h 0 (SmoothRealFunction.constant 0) 0 := by
+  have hp : g₀.PositivePotential (SmoothRealFunction.constant 0) := by
+    intro i z hz
+    rw [SmoothRealFunction.hessian_constant, add_zero]
+    exact g₀.positive i z hz
+  refine ⟨hp, fun x => ?_⟩
+  obtain ⟨i,hi⟩ := A.covers x
+  change g₀.logRatioValue (g₀.deform _ hp) x = _
+  rw [g₀.logRatioValue_local _ i hi]
+  simp [KaehlerMetric.volumeCoefficient, KaehlerMetric.deform,
+    SmoothRealFunction.hessian_constant]
+
+ 
+
+lemma volumePath_one (P : A.ProjectiveEmbedding) (h : SemipositiveAnticanonicalMetric A)
+    (φ : SmoothRealFunction A) (b : ℝ)
+    (hsol : SolvesVolumePath (projectiveBackground P) h 1 φ b) :
+    HasPrescribedVolumePotential P h := by
+  obtain ⟨hp, heq⟩ := hsol
+  refine ⟨φ, Real.exp b, Real.exp_pos b, hp, fun i z hz => ?_⟩
+  have hev := heq ((A.chart i).symm z)
+  change ((projectiveBackground P).logRatio ((projectiveBackground P).deform φ hp)).localExpression i z =
+    1 * (prescribedForcing (projectiveBackground P) h).localExpression i z + b at hev
+  rw [KaehlerMetric.logRatio_local _ _ i hz, prescribedForcing_local _ _ i hz, one_mul] at hev
+  have hv : Real.log (((projectiveBackground P).deform φ hp).volumeCoefficient i z) = b - h.weight i z := by
+    linarith
+  have he := congrArg Real.exp hv
+  rw [Real.exp_log (((projectiveBackground P).deform φ hp).volumeCoefficient_pos i hz),
+    sub_eq_add_neg, Real.exp_add] at he
+  exact he
+
+end Anticanonical.SourceSmooth
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/WeakDerivativeLocalization.lean b/lean/OAI/Geometry/PrescribedPotential/WeakDerivativeLocalization.lean
new file mode 100644
index 0000000000000000000000000000000000000000..6decd9072c626ac48eeb0bb4b352c882f873625f
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/WeakDerivativeLocalization.lean
@@ -0,0 +1,75 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CircleRadialCalculus
+import OAI.Geometry.PrescribedPotential.PatchCutoffs
+import OAI.Geometry.PrescribedPotential.DerivativeLocalization
+import OAI.Geometry.PrescribedPotential.LiteralWeakHessian
+
+namespace OAI
+
+/-! Weak Derivative Localization. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory TemperedDistribution LineDeriv
+open scoped ContDiff SchwartzMap Classical
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+lemma distribution_embed_forcing (s : ℝ) (p : ι) (f : Smooth A) :
+    D.localizers.distribution s p (D.localizers.embed s f) =
+      SchwartzMap.toTemperedDistributionCLM (EC d) ℂ volume (D.forcing p f) := D.localizers.distribution_embed s p f
+
+lemma completedLocalize_lower (k l : ℕ) (hlk : l ≤ k) (i : Fin A.count) (ρ : Smooth A)
+    (hρ : tsupport (ρ : X → ℂ) ⊆ (A.euclideanChart i).source)
+    (u : D.localizers.Sobolev (k : ℝ)) :
+    realize (l : ℝ) (D.completedLocalize (l : ℝ) i ρ hρ
+      (D.localizers.lower (k : ℝ) (l : ℝ) u)) =
+      realize (k : ℝ) (D.completedLocalize (k : ℝ) i ρ hρ u) := by
+  have hkl : (l : ℝ) ≤ (k : ℝ) := by exact_mod_cast hlk
+  have he : (fun u : D.localizers.Sobolev (k : ℝ) => realize (l : ℝ)
+      (D.completedLocalize (l : ℝ) i ρ hρ (D.localizers.lower (k : ℝ) (l : ℝ) u))) =
+      (fun u => realize (k : ℝ) (D.completedLocalize (k : ℝ) i ρ hρ u)) := by
+    apply (D.localizers.embed_dense (k : ℝ)).equalizer (by fun_prop) (by fun_prop)
+    funext f
+    dsimp only [Function.comp_apply]
+    simp only [D.localizers.lower_embed hkl,D.completedLocalize_embed,realize_schwartzCoord]
+  exact congr_fun he u
+
+lemma distribution_derivative (k : ℕ) (p : ι) (v : EC d)
+    (u : D.localizers.Sobolev ((k : ℝ)+1)) :
+    (∂_{v} (D.localizers.distribution ((k : ℝ)+1) p u) : 𝓢'(EC d,ℂ)) =
+      realize (k : ℝ) (D.completedLocalize (k : ℝ) (D.patch p).index
+        (D.localizedDerivative p v (D.localizers.weight p)) (D.derivativeWeight_support p v)
+        (D.localizers.lower ((k : ℝ)+1) (k : ℝ) u)) +
+      D.localizers.distribution (k : ℝ) p (D.completedDerivative k p v u) := by
+  have he : (fun u : D.localizers.Sobolev ((k : ℝ)+1) =>
+      (∂_{v} (D.localizers.distribution ((k : ℝ)+1) p u) : 𝓢'(EC d,ℂ))) =
+      (fun u => realize (k : ℝ) (D.completedLocalize (k : ℝ) (D.patch p).index
+        (D.localizedDerivative p v (D.localizers.weight p)) (D.derivativeWeight_support p v)
+        (D.localizers.lower ((k : ℝ)+1) (k : ℝ) u)) +
+      D.localizers.distribution (k : ℝ) p (D.completedDerivative k p v u)) := by
+    apply (D.localizers.embed_dense ((k : ℝ)+1)).equalizer
+      ((lineDerivOpCLM ℂ 𝓢'(EC d,ℂ) v).continuous.comp
+        (D.localizers.distribution ((k : ℝ)+1) p).continuous) (by fun_prop)
+    funext f
+    dsimp only [Function.comp_apply]
+    rw [D.distribution_embed_forcing,D.localizers.lower_embed (by linarith : (k : ℝ) ≤ (k : ℝ)+1),
+      D.completedLocalize_embed,realize_schwartzCoord,D.completedDerivative_embed,
+      D.distribution_embed_forcing,lineDerivOpCLM_apply,
+      lineDerivOp_toTemperedDistributionCLM_eq,D.forcing_derivative,map_add]
+  exact congr_fun he u
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/WeakNonlinearCommutator.lean b/lean/OAI/Geometry/PrescribedPotential/WeakNonlinearCommutator.lean
new file mode 100644
index 0000000000000000000000000000000000000000..5ee7e0498516a69b74ac3a0d336efeb17f62594e
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/WeakNonlinearCommutator.lean
@@ -0,0 +1,129 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.NonlinearScale
+import OAI.Geometry.PrescribedPotential.RealNonlinearCommutator
+
+namespace OAI
+
+/-! Weak Nonlinear Commutator. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology
+open scoped ContDiff Classical
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+local instance weakNonlinearCommutatorNormedAddCommGroup (s : ℝ) : NormedAddCommGroup (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedAddCommGroup
+local instance weakNonlinearCommutatorNormedSpace (s : ℝ) : NormedSpace ℝ (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedSpace
+local instance weakNonlinearCommutatorIsTopologicalAddGroup (s : ℝ) : IsTopologicalAddGroup (D.localizers.RealSobolev s) :=
+  Submodule.isTopologicalAddGroup _
+local instance weakNonlinearCommutatorContinuousSMul (s : ℝ) : ContinuousSMul ℝ (D.localizers.RealSobolev s) :=
+  SMulMemClass.continuousSMul _
+
+lemma realNonlinearRemainder_lower (k l : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (hl : Module.finrank ℝ (EC d) < l) (hlk : l ≤ k) (p : ι) (v : EC d)
+    (u : D.localizers.RealSobolev ((k : ℝ)+2)) :
+    D.localizers.realLower (k : ℝ) (l : ℝ) (by exact_mod_cast hlk)
+      (D.realNonlinearRemainder k hk p v u) =
+    D.realNonlinearRemainder l hl p v
+      (D.localizers.realLower ((k : ℝ)+2) ((l : ℝ)+2)
+        (by exact_mod_cast Nat.add_le_add_right hlk 2) u) := by
+  have hst : (l : ℝ)+2 ≤ (k : ℝ)+2 := by exact_mod_cast Nat.add_le_add_right hlk 2
+  have hkl : (l : ℝ) ≤ (k : ℝ) := by exact_mod_cast hlk
+  let L := D.localizers.realLower ((k : ℝ)+2) ((l : ℝ)+2) hst
+  let M := D.localizers.realLower (k : ℝ) (l : ℝ) hkl
+  have he : M ∘ D.realNonlinearRemainder k hk p v =
+      D.realNonlinearRemainder l hl p v ∘ L := by
+    apply (D.localizers.realEmbed_dense ((k : ℝ)+2)).equalizer
+      (M.continuous.comp (D.realNonlinearRemainder_continuous k hk p v))
+      ((D.realNonlinearRemainder_continuous l hl p v).comp L.continuous)
+    funext f
+    dsimp only [Function.comp_apply]
+    have hh := congrArg M (D.real_commutator_core k hk p v f)
+    have hd := congrArg (fun T : D.localizers.RealSobolev ((k : ℝ)+2) →L[ℝ]
+        D.localizers.RealSobolev (l : ℝ) => T
+          (D.localizers.realEmbed ((k : ℝ)+2) (D.realLocalizedDerivative p v f)))
+      (D.realVolumeDerivative_lower k l hk hl hlk (D.localizers.realEmbed ((k : ℝ)+2) f))
+    simp only [ContinuousLinearMap.comp_apply,Localizers.realLower_embed] at hd
+    change M (D.realVolumeDerivative k hk (D.localizers.realEmbed ((k : ℝ)+2) f)
+      (D.localizers.realEmbed ((k : ℝ)+2) (D.realLocalizedDerivative p v f))) = _ at hh
+    rw [← hd,D.real_commutator_core l hl p v f,map_add] at hh
+    dsimp only [M] at hh
+    rw [Localizers.realLower_embed] at hh
+    change _ = D.localizers.realEmbed (l : ℝ)
+      (D.realLocalizedDerivative p v (RealSmooth.ofReal (g.potentialDensity f.source))) +
+      M (D.realNonlinearRemainder k hk p v (D.localizers.realEmbed ((k : ℝ)+2) f)) at hh
+    have hh' := add_left_cancel hh
+    change M (D.realNonlinearRemainder k hk p v (D.localizers.realEmbed ((k : ℝ)+2) f)) = _
+    rw [show L (D.localizers.realEmbed ((k : ℝ)+2) f) = D.localizers.realEmbed ((l : ℝ)+2) f
+      from D.localizers.realLower_embed hst f]
+    exact hh'.symm
+  exact congr_fun he u
+
+def potentialDerivative (k : ℕ) (p : ι) (v : EC d) :
+    D.localizers.RealSobolev (((k+1 : ℕ) : ℝ)+2) →L[ℝ]
+      D.localizers.RealSobolev ((k : ℝ)+2) :=
+  D.localizers.realLower ((k+2 : ℕ) : ℝ) ((k : ℝ)+2) (by norm_num) ∘L
+    D.realCompletedDerivative (k+2) p v ∘L
+    D.localizers.realLower (((k+1 : ℕ) : ℝ)+2) (((k+2 : ℕ) : ℝ)+1) (by push_cast; linarith)
+
+lemma potentialDerivative_embed (k : ℕ) (p : ι) (v : EC d) (f : RealSmooth A) :
+    D.potentialDerivative k p v (D.localizers.realEmbed (((k+1 : ℕ) : ℝ)+2) f) =
+      D.localizers.realEmbed ((k : ℝ)+2) (D.realLocalizedDerivative p v f) := by
+  simp only [potentialDerivative,ContinuousLinearMap.comp_apply,
+    Localizers.realLower_embed,realCompletedDerivative_embed]
+
+def densityDerivative (k : ℕ) (p : ι) (v : EC d) :
+    D.localizers.RealSobolev ((k+1 : ℕ) : ℝ) →L[ℝ] D.localizers.RealSobolev (k : ℝ) :=
+  D.realCompletedDerivative k p v ∘L
+    D.localizers.realLower ((k+1 : ℕ) : ℝ) ((k : ℝ)+1) (by norm_num)
+
+lemma densityDerivative_embed (k : ℕ) (p : ι) (v : EC d) (f : RealSmooth A) :
+    D.densityDerivative k p v (D.localizers.realEmbed ((k+1 : ℕ) : ℝ) f) =
+      D.localizers.realEmbed (k : ℝ) (D.realLocalizedDerivative p v f) := by
+  simp only [densityDerivative,ContinuousLinearMap.comp_apply,
+    Localizers.realLower_embed,realCompletedDerivative_embed]
+
+ 
+
+theorem nonlinear_commutator_weak (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (p : ι) (v : EC d) (u : D.localizers.RealSobolev (((k+1 : ℕ) : ℝ)+2)) :
+    D.realVolumeDerivative k hk
+      (D.localizers.realLower (((k+1 : ℕ) : ℝ)+2) ((k : ℝ)+2) (by push_cast; linarith) u)
+      (D.potentialDerivative k p v u) =
+    D.densityDerivative k p v (D.realVolume (k+1) (by omega) u) +
+    D.localizers.realLower ((k+1 : ℕ) : ℝ) (k : ℝ) (by push_cast; linarith)
+      (D.realNonlinearRemainder (k+1) (by omega) p v u) := by
+  let L := D.localizers.realLower (((k+1 : ℕ) : ℝ)+2) ((k : ℝ)+2) (by push_cast; linarith)
+  let M := D.localizers.realLower ((k+1 : ℕ) : ℝ) (k : ℝ) (by push_cast; linarith)
+  have he : (fun u => D.realVolumeDerivative k hk (L u) (D.potentialDerivative k p v u)) =
+      (fun u => D.densityDerivative k p v (D.realVolume (k+1) (by omega) u) +
+        M (D.realNonlinearRemainder (k+1) (by omega) p v u)) := by
+    apply (D.localizers.realEmbed_dense (((k+1 : ℕ) : ℝ)+2)).equalizer
+      (((D.realVolumeDerivative_continuous k hk).comp L.continuous).clm_apply
+        (D.potentialDerivative k p v).continuous)
+      (((D.densityDerivative k p v).continuous.comp (D.realVolume_continuous (k+1) (by omega))).add
+        (M.continuous.comp (D.realNonlinearRemainder_continuous (k+1) (by omega) p v)))
+    funext f
+    dsimp only [L,M,Function.comp_apply,Pi.add_apply]
+    rw [Localizers.realLower_embed,D.potentialDerivative_embed,D.realVolume_embed,
+      D.densityDerivative_embed,D.realNonlinearRemainder_lower (k+1) k (by omega) hk (by omega),
+      Localizers.realLower_embed]
+    exact D.real_commutator_core k hk p v f
+  exact congr_fun he u
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/WeakPositive.lean b/lean/OAI/Geometry/PrescribedPotential/WeakPositive.lean
new file mode 100644
index 0000000000000000000000000000000000000000..dccb633e4fb6d085a75a96eadeaf687c94e1574c
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/WeakPositive.lean
@@ -0,0 +1,112 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.GlobalHessian
+import OAI.Geometry.PrescribedPotential.GlobalMetricEntries
+import OAI.Geometry.PrescribedPotential.MatrixPositivity
+import OAI.Geometry.PrescribedPotential.PatchCutoffs
+import OAI.Geometry.PrescribedPotential.RealSobolev
+
+namespace OAI
+
+/-! Weak Positive. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff Classical ComplexOrder
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+local instance weakPositiveNG (s : ℝ) : NormedAddCommGroup (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedAddCommGroup
+local instance weakPositiveNS (s : ℝ) : NormedSpace ℝ (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedSpace
+local instance weakPositiveTG (s : ℝ) : IsTopologicalAddGroup (D.localizers.RealSobolev s) :=
+  Submodule.isTopologicalAddGroup _
+local instance weakPositiveCS (s : ℝ) : ContinuousSMul ℝ (D.localizers.RealSobolev s) :=
+  SMulMemClass.continuousSMul _
+
+def weakMatrix (k : ℕ) (u : D.localizers.RealSobolev ((k : ℝ)+2)) (p : ι) (x : X) :
+    Matrix (Fin d) (Fin d) ℂ := fun i j => D.metricEntry p i j x +
+      D.localizers.strong (k : ℝ) (D.completedHessian k p i j u.val) x
+
+lemma weakMatrix_continuous (k : ℕ) (p : ι) :
+    Continuous (fun z : D.localizers.RealSobolev ((k : ℝ)+2) × X => D.weakMatrix k z.1 p z.2) := by
+  apply continuous_pi
+  intro i
+  apply continuous_pi
+  intro j
+  apply ((D.metricEntry p i j).continuous.comp continuous_snd).add
+  have hc : Continuous (fun z : D.localizers.RealSobolev ((k : ℝ)+2) × X =>
+      D.localizers.strong (k : ℝ) (D.completedHessian k p i j z.1.val)) :=
+    (D.localizers.strong (k : ℝ)).continuous.comp ((D.completedHessian k p i j).continuous.comp
+      (continuous_subtype_val.comp continuous_fst))
+  exact hc.eval continuous_snd
+
+lemma weakMatrix_embed (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) (f : RealSmooth A)
+    (p : ι) {x : X} (hx : x ∈ tsupport (D.localizers.weight p : X → ℂ)) :
+    D.weakMatrix k (D.localizers.realEmbed ((k : ℝ)+2) f) p x =
+      g.matrix (D.patch p).index (A.chart (D.patch p).index x) +
+        f.source.hessian (D.patch p).index (A.chart (D.patch p).index x) := by
+  have hs : (Module.finrank ℝ (EC d) : ℝ) < 2*(k : ℝ) := by
+    have hh : (Module.finrank ℝ (EC d) : ℝ) < (k : ℝ) := by exact_mod_cast hk
+    linarith [Nat.cast_nonneg (α := ℝ) k]
+  ext i j
+  change D.metricEntry p i j x + D.localizers.strong (k : ℝ)
+    (D.completedHessian k p i j (D.localizers.embed ((k : ℝ)+2) f.val)) x = _
+  rw [D.completedHessian_embed, D.localizers.strong_embed _ hs]
+  change D.metricEntry p i j x + D.localizedHessian p i j f.val x = _
+  rw [D.metricEntry_apply p i j hx, ← f.ofReal_source, D.localizedHessian_source p i j f.source hx]
+  rfl
+
+lemma weakMatrix_hermitian (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (u : D.localizers.RealSobolev ((k : ℝ)+2)) (p : ι) {x : X}
+    (hx : x ∈ tsupport (D.localizers.weight p : X → ℂ)) : (D.weakMatrix k u p x).IsHermitian := by
+  have hc : Continuous (fun v : D.localizers.RealSobolev ((k : ℝ)+2) => D.weakMatrix k v p x) :=
+    (D.weakMatrix_continuous k p).comp (continuous_id.prodMk continuous_const)
+  have he : (fun v : D.localizers.RealSobolev ((k : ℝ)+2) => (D.weakMatrix k v p x)ᴴ) =
+      (fun v => D.weakMatrix k v p x) := by
+    apply (D.localizers.realEmbed_dense ((k : ℝ)+2)).equalizer hc.matrix_conjTranspose hc
+    funext f
+    dsimp only [Function.comp_apply]
+    rw [D.weakMatrix_embed k hk f p hx]
+    have hs : A.chart (D.patch p).index x ∈ (A.chart (D.patch p).index).target :=
+      (A.chart (D.patch p).index).mapsTo (by simpa using D.patch_source p hx)
+    exact (g.positive _ _ hs).isHermitian.add (f.source.hessian_hermitian _ hs)
+  exact congr_fun he u
+
+lemma weakMatrix_zero (k : ℕ) (p : ι) {x : X}
+    (hx : x ∈ tsupport (D.localizers.weight p : X → ℂ)) :
+    D.weakMatrix k 0 p x = g.matrix (D.patch p).index (A.chart (D.patch p).index x) := by
+  ext i j
+  change D.metricEntry p i j x + D.localizers.strong (k : ℝ) (D.completedHessian k p i j 0) x = _
+  rw [map_zero, map_zero]
+  exact (add_zero _).trans (D.metricEntry_apply p i j hx)
+
+def WeakPositive (k : ℕ) (u : D.localizers.RealSobolev ((k : ℝ)+2)) : Prop :=
+  ∀ p x, x ∈ tsupport (D.localizers.weight p : X → ℂ) → (D.weakMatrix k u p x).PosDef
+
+lemma weakPositive_nhds (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) :
+    ∀ᶠ u in 𝓝 (0 : D.localizers.RealSobolev ((k : ℝ)+2)), D.WeakPositive k u := by
+  apply Filter.eventually_all.mpr
+  intro p
+  apply CompactMatrixPositivity.eventually_posDef (isClosed_tsupport _).isCompact
+    (D.weakMatrix_continuous k p)
+  · intro u x hx
+    exact D.weakMatrix_hermitian k hk u p hx
+  · intro x hx
+    rw [D.weakMatrix_zero k p hx]
+    exact g.positive _ _ ((A.chart (D.patch p).index).mapsTo (by simpa using D.patch_source p hx))
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/WirtingerCalculus.lean b/lean/OAI/Geometry/PrescribedPotential/WirtingerCalculus.lean
new file mode 100644
index 0000000000000000000000000000000000000000..641569478caf6a64ff05e8cd51f6998a6d7c3472
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/WirtingerCalculus.lean
@@ -0,0 +1,122 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.AnalyticSupport
+
+namespace OAI
+
+/-! Wirtinger Calculus. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology
+open scoped ContDiff
+namespace KaehlerCalculus
+variable {n : ℕ}
+abbrev V (n : ℕ) := Fin n → ℂ
+
+def wderiv (a : ℂ) (v : V n) (f : V n → ℂ) (z : V n) : ℂ :=
+  (fderiv ℝ f z v + a * fderiv ℝ f z (Complex.I • v))/2
+
+def dz (v : V n) (f : V n → ℂ) : V n → ℂ := wderiv (-Complex.I) v f
+def dzbar (v : V n) (f : V n → ℂ) : V n → ℂ := wderiv Complex.I v f
+
+lemma wderiv_smooth {f : V n → ℂ} {z : V n}
+    (hf : ContDiffAt ℝ ∞ f z) (a : ℂ) (v : V n) :
+    ContDiffAt ℝ ∞ (wderiv a v f) z := by
+  have hd := hf.fderiv_right (m := ∞) (by simp)
+  exact ((hd.clm_apply contDiffAt_const).add
+    (contDiffAt_const.mul (hd.clm_apply contDiffAt_const))).div_const 2
+
+lemma wderiv_congr {f g : V n → ℂ} {z : V n} (h : f =ᶠ[𝓝 z] g) (a : ℂ) (v : V n) :
+    wderiv a v f z = wderiv a v g z := by
+  simp only [wderiv,h.fderiv_eq]
+
+lemma wderiv_add {f g : V n → ℂ} {z : V n}
+    (hf : DifferentiableAt ℝ f z) (hg : DifferentiableAt ℝ g z) (a : ℂ) (v : V n) :
+    wderiv a v (fun y => f y+g y) z = wderiv a v f z+wderiv a v g z := by
+  simp only [wderiv,fderiv_fun_add hf hg,add_apply]
+  ring
+
+lemma wderiv_const_mul {f : V n → ℂ} {z : V n}
+    (hf : DifferentiableAt ℝ f z) (a c : ℂ) (v : V n) :
+    wderiv a v (fun y => c*f y) z = c*wderiv a v f z := by
+  simp only [wderiv,fderiv_const_mul hf,smul_apply,smul_eq_mul]
+  ring
+
+lemma wderiv_mul {f g : V n → ℂ} {z : V n}
+    (hf : DifferentiableAt ℝ f z) (hg : DifferentiableAt ℝ g z) (a : ℂ) (v : V n) :
+    wderiv a v (fun y => f y*g y) z =
+      wderiv a v f z*g z+f z*wderiv a v g z := by
+  simp only [wderiv,fderiv_fun_mul hf hg,add_apply,
+    smul_apply,smul_eq_mul]
+  ring
+
+lemma wderiv_const (a c : ℂ) (v z : V n) : wderiv a v (fun _ => c) z = 0 := by
+  simp [wderiv]
+
+lemma fderiv_wderiv {f : V n → ℂ} {z : V n}
+    (hf : ContDiffAt ℝ ∞ f z) (a : ℂ) (u v : V n) :
+    fderiv ℝ (wderiv a v f) z u =
+      (fderiv ℝ (fderiv ℝ f) z u v +
+        a*fderiv ℝ (fderiv ℝ f) z u (Complex.I • v))/2 := by
+  have hd (w : V n) : DifferentiableAt ℝ (fun y => fderiv ℝ f y w) z :=
+    ((hf.fderiv_right (m := ∞) (by simp)).clm_apply contDiffAt_const).differentiableAt (by simp)
+  unfold wderiv
+  simp only [div_eq_mul_inv]
+  rw [fderiv_mul_const ((hd v).fun_add ((hd (Complex.I • v)).const_mul a)),
+    fderiv_fun_add (hd v) ((hd (Complex.I • v)).const_mul a),fderiv_const_mul (hd (Complex.I • v))]
+  simp only [smul_apply,add_apply,smul_eq_mul,
+    PotentialKaehler.fderiv_eval_derivative hf]
+  ring
+
+lemma wderiv_comm {f : V n → ℂ} {z : V n}
+    (hf : ContDiffAt ℝ ∞ f z) (a b : ℂ) (u v : V n) :
+    wderiv a u (wderiv b v f) z = wderiv b v (wderiv a u f) z := by
+  simp only [wderiv, fderiv_wderiv hf]
+  have hs := hf.isSymmSndFDerivAt (by simp only [minSmoothness_of_isRCLikeNormedField]; change ((2 : ℕ∞) : WithTop ℕ∞) ≤ ((⊤ : ℕ∞) : WithTop ℕ∞); exact WithTop.coe_le_coe.mpr le_top)
+  rw [hs u v, hs u (Complex.I • v), hs (Complex.I • u) v,
+    hs (Complex.I • u) (Complex.I • v)]
+  ring
+
+lemma fderiv_ofReal {f : V n → ℝ} {z : V n}
+    (hf : DifferentiableAt ℝ f z) (v : V n) :
+    fderiv ℝ (fun y => (f y : ℂ)) z v = (fderiv ℝ f z v : ℂ) := by
+  exact congrArg (fun T : V n →L[ℝ] ℂ => T v)
+    ((Complex.ofRealCLM.hasFDerivAt.comp z hf.hasFDerivAt).fderiv)
+
+lemma snd_fderiv_ofReal {f : V n → ℝ} {z : V n}
+    (hf : ContDiffAt ℝ ∞ f z) (u v : V n) :
+    fderiv ℝ (fderiv ℝ (fun y => (f y : ℂ))) z u v =
+      (fderiv ℝ (fderiv ℝ f) z u v : ℂ) := by
+  have hc : ContDiffAt ℝ ∞ (fun y => (f y : ℂ)) z := Complex.ofRealCLM.contDiff.contDiffAt.comp z hf
+  rw [← PotentialKaehler.fderiv_eval_derivative hc]
+  have he : (fun y => fderiv ℝ (fun y => (f y : ℂ)) y v) =ᶠ[𝓝 z]
+      (fun y => (fderiv ℝ f y v : ℂ)) := by
+    filter_upwards [(hf.of_le (show (1 : WithTop ℕ∞) ≤ ∞ by simp)).eventually (by decide)] with y hy
+    exact fderiv_ofReal (hy.differentiableAt (by simp)) v
+  rw [he.fderiv_eq, fderiv_ofReal
+    (((hf.fderiv_right (m := ∞) (by simp)).clm_apply contDiffAt_const).differentiableAt (by simp)),
+    PotentialKaehler.fderiv_eval_derivative hf]
+
+lemma dzbar_dz_real {f : V n → ℝ} {z : V n}
+    (hf : ContDiffAt ℝ ∞ f z) (u v : V n) :
+    dzbar u (dz v (fun y => (f y : ℂ))) z =
+      PotentialKaehler.hermitianPart (fderiv ℝ (fderiv ℝ f) z) u v := by
+  have hc : ContDiffAt ℝ ∞ (fun y => (f y : ℂ)) z := Complex.ofRealCLM.contDiff.contDiffAt.comp z hf
+  simp only [dzbar,dz,wderiv,fderiv_wderiv hc,snd_fderiv_ofReal hf]
+  apply Complex.ext <;> simp [PotentialKaehler.hermitianPart] <;> ring
+
+lemma potentialMatrix_eq_dzbar_dz {f : V n → ℝ} {z : V n}
+    (hf : ContDiffAt ℝ ∞ f z) (i j : Fin n) :
+    PotentialKaehler.potentialMatrix f z i j =
+      dzbar (Pi.single i 1) (dz (Pi.single j 1) (fun y => (f y : ℂ))) z := by
+  rw [dzbar_dz_real hf]
+  exact PotentialKaehler.hermitianPartMatrix_apply _ i j
+end KaehlerCalculus
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedPotential/WirtingerConjugate.lean b/lean/OAI/Geometry/PrescribedPotential/WirtingerConjugate.lean
new file mode 100644
index 0000000000000000000000000000000000000000..a99677ecd6776906edaa620cb31c97c54a6b6471
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedPotential/WirtingerConjugate.lean
@@ -0,0 +1,64 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.KaehlerClosedDerivatives
+import OAI.Geometry.PrescribedPotential.MatrixWirtinger
+
+namespace OAI
+
+/-! Wirtinger Conjugate. -/
+
+section
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace KaehlerCalculus
+variable {n : ℕ}
+
+lemma fderiv_conj {f : V n → ℂ} {z : V n} (hf : DifferentiableAt ℝ f z) (v : V n) :
+    fderiv ℝ (fun y => star (f y)) z v = star (fderiv ℝ f z v) := by
+  exact congrArg (fun T : V n →L[ℝ] ℂ => T v)
+    ((Complex.conjCLE.toContinuousLinearMap.hasFDerivAt.comp z hf.hasFDerivAt).fderiv)
+
+lemma dzbar_conj {f : V n → ℂ} {z : V n} (hf : DifferentiableAt ℝ f z) (v : V n) :
+    dzbar v (fun y => star (f y)) z = star (dz v f z) := by
+  simp only [dzbar,dz,wderiv,fderiv_conj hf]
+  simp
+
+lemma dz_conj {f : V n → ℂ} {z : V n} (hf : DifferentiableAt ℝ f z) (v : V n) :
+    dz v (fun y => star (f y)) z = star (dzbar v f z) := by
+  simp only [dzbar,dz,wderiv,fderiv_conj hf]
+  simp
+
+lemma mderiv_bar_eq_conjTranspose {U : Set (V n)} (hU : IsOpen U)
+    {M : V n → Matrix (Fin n) (Fin n) ℂ} (hM : ContDiffOn ℝ ∞ M U)
+    (hh : ∀ y ∈ U, (M y).IsHermitian) {z : V n} (hz : z ∈ U) (v : V n) :
+    mderiv Complex.I v M z = (mderiv (-Complex.I) v M z)ᴴ := by
+  ext i j
+  have he : (fun y => M y i j) =ᶠ[𝓝 z] (fun y => star (M y j i)) := by
+    filter_upwards [hU.mem_nhds hz] with y hy
+    exact ((hh y hy).apply i j).symm
+  change dzbar v (fun y => M y i j) z = star (dz v (fun y => M y j i) z)
+  rw [show dzbar v (fun y => M y i j) z = dzbar v (fun y => star (M y j i)) z
+    from wderiv_congr he _ _]
+  exact dzbar_conj ((entry_smooth (hM.contDiffAt (hU.mem_nhds hz)) j i).differentiableAt (by simp)) v
+
+lemma closed_dzbar_symmetry {U : Set (V n)} (hU : IsOpen U)
+    {M : V n → Matrix (Fin n) (Fin n) ℂ}
+    (hM : ContDiffOn ℝ ∞ M U) (hh : ∀ y ∈ U, (M y).IsHermitian)
+    (hclosed : ∀ y ∈ U, ∀ u v w : V n,
+      fderiv ℝ (fun q => Anticanonical.ComplexAtlas.fundamentalForm (M q) v w) y u +
+      fderiv ℝ (fun q => Anticanonical.ComplexAtlas.fundamentalForm (M q) w u) y v +
+      fderiv ℝ (fun q => Anticanonical.ComplexAtlas.fundamentalForm (M q) u v) y w = 0)
+    {z : V n} (hz : z ∈ U) (i j k : Fin n) :
+    dzbar (e k) (fun y => M y i j) z = dzbar (e i) (fun y => M y k j) z := by
+  have h1 := congrFun (congrFun (mderiv_bar_eq_conjTranspose hU hM hh hz (e k)) i) j
+  have h2 := congrFun (congrFun (mderiv_bar_eq_conjTranspose hU hM hh hz (e i)) k) j
+  change dzbar (e k) (fun y => M y i j) z = star (dz (e k) (fun y => M y j i) z) at h1
+  change dzbar (e i) (fun y => M y k j) z = star (dz (e i) (fun y => M y j k) z) at h2
+  rw [h1,h2,closed_dz_symmetry hU hM hh hclosed hz j i k]
+end KaehlerCalculus
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/ArrayWirtinger.lean b/lean/OAI/Geometry/PrescribedRicci/ArrayWirtinger.lean
new file mode 100644
index 0000000000000000000000000000000000000000..fc1bad100e23fbed04d4d1dd2b9059ee9e451eb8
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/ArrayWirtinger.lean
@@ -0,0 +1,88 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.CalabiTensorAlgebra
+import OAI.Geometry.PrescribedRicci.ChernPairDifferential
+import OAI.Geometry.PrescribedRicci.KahlerGradientEnergy
+import OAI.Geometry.PrescribedRicci.MatrixWirtinger
+
+namespace OAI
+
+/-! Array Wirtinger. -/
+
+noncomputable section
+open Matrix Filter Set Topology
+open scoped ContDiff ComplexOrder Kronecker
+namespace Anticanonical.SourceSmooth.KaehlerMetric
+open MongeAmpere
+variable {d : ℕ} {n : Type*}
+
+lemma holArray_eq_holDerivative {f : Coordinates d → Matrix (Fin d) (Fin d) ℂ}
+    {z : Coordinates d} (hf : DifferentiableAt ℝ f z) (a : Fin d) :
+    holArray f z a = holDerivative f z a := by
+  ext i j
+  exact holDeriv_entry hf a i j
+
+lemma barArray_eq_barDerivative {f : Coordinates d → Matrix (Fin d) (Fin d) ℂ}
+    {z : Coordinates d} (hf : DifferentiableAt ℝ f z) (a : Fin d) :
+    barArray f z a = barDerivative f z a := by
+  ext i j
+  exact barDeriv_entry hf a i j
+
+lemma holArray_const (K : Matrix n n ℂ) (z : Coordinates d) (a : Fin d) :
+    holArray (fun _ => K) z a = 0 := by
+  ext i j
+  simp [holArray,holDeriv]
+
+lemma barArray_const (K : Matrix n n ℂ) (z : Coordinates d) (a : Fin d) :
+    barArray (fun _ => K) z a = 0 := by
+  ext i j
+  simp [barArray,barDeriv]
+
+lemma holArray_transpose (K : Coordinates d → Matrix n n ℂ) (z : Coordinates d) (a : Fin d) :
+    holArray (fun y => (K y).transpose) z a = (holArray K z a).transpose := rfl
+
+lemma barArray_transpose (K : Coordinates d → Matrix n n ℂ) (z : Coordinates d) (a : Fin d) :
+    barArray (fun y => (K y).transpose) z a = (barArray K z a).transpose := rfl
+
+lemma barArray_eq_adjoint_hol {K : Coordinates d → Matrix n n ℂ} {z : Coordinates d}
+    (hK : ∀ᶠ y in nhds z, (K y).IsHermitian) (a : Fin d) :
+    barArray K z a = (holArray K z a)ᴴ := by
+  ext i j
+  change barDeriv (fun y => K y i j) z a = star (holDeriv (fun y => K y j i) z a)
+  have he : (fun y => K y i j) =ᶠ[nhds z] (fun y => star (K y j i)) := by
+    filter_upwards [hK] with y hy
+    exact (Matrix.IsHermitian.ext_iff.mp hy i j).symm
+  rw [barDeriv_congr he,barDeriv_star]
+
+lemma holArray_tensor3 {L U V : Coordinates d → Matrix n n ℂ} {z : Coordinates d}
+    (hL : ∀ i j, DifferentiableAt ℝ (fun y => L y i j) z)
+    (hU : ∀ i j, DifferentiableAt ℝ (fun y => U y i j) z)
+    (hV : ∀ i j, DifferentiableAt ℝ (fun y => V y i j) z) (a : Fin d) :
+    holArray (fun y => tensor3 (L y) (U y) (V y)) z a =
+      tensor3 (holArray L z a) (U z) (V z)+
+      tensor3 (L z) (holArray U z a) (V z)+tensor3 (L z) (U z) (holArray V z a) := by
+  ext i j
+  change holDeriv (fun y => L y i.1 j.1*(U y i.2.1 j.2.1*V y i.2.2 j.2.2)) z a = _
+  rw [holDeriv_mul (hL _ _) ((hU _ _).fun_mul (hV _ _)),
+    holDeriv_mul (hU _ _) (hV _ _)]
+  simp only [Matrix.add_apply,tensor3_apply,holArray]
+  ring
+
+lemma barArray_tensor3 {L U V : Coordinates d → Matrix n n ℂ} {z : Coordinates d}
+    (hL : ∀ i j, DifferentiableAt ℝ (fun y => L y i j) z)
+    (hU : ∀ i j, DifferentiableAt ℝ (fun y => U y i j) z)
+    (hV : ∀ i j, DifferentiableAt ℝ (fun y => V y i j) z) (a : Fin d) :
+    barArray (fun y => tensor3 (L y) (U y) (V y)) z a =
+      tensor3 (barArray L z a) (U z) (V z)+
+      tensor3 (L z) (barArray U z a) (V z)+tensor3 (L z) (U z) (barArray V z a) := by
+  ext i j
+  change barDeriv (fun y => L y i.1 j.1*(U y i.2.1 j.2.1*V y i.2.2 j.2.2)) z a = _
+  rw [barDeriv_mul (hL _ _) ((hU _ _).fun_mul (hV _ _)),
+    barDeriv_mul (hU _ _) (hV _ _)]
+  simp only [Matrix.add_apply,tensor3_apply,barArray]
+  ring
+
+end Anticanonical.SourceSmooth.KaehlerMetric
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/BoundedSmoothPoisson.lean b/lean/OAI/Geometry/PrescribedRicci/BoundedSmoothPoisson.lean
new file mode 100644
index 0000000000000000000000000000000000000000..1b89363232809b25a9b335b85c6219c9f57e71cf
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/BoundedSmoothPoisson.lean
@@ -0,0 +1,60 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.BoundedPoisson
+import OAI.Geometry.PrescribedPotential.ComplexOperator
+import OAI.Geometry.PrescribedPotential.PatchCutoffs
+
+namespace OAI
+
+/-! Bounded Smooth Poisson. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory
+open scoped SchwartzMap ContDiff Classical
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  [ConnectedSpace X] {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+theorem exists_complex_poisson_bounded : ∃ C : ℝ, 0 ≤ C ∧ ∀ F : Smooth A,
+    ∃ (u : Smooth A) (c : ℂ), complexL g u = F + Smooth.const c ∧
+      ‖D.localizers.embed 0 u‖ ≤ C * ‖D.localizers.embed 0 F‖ := by
+  obtain ⟨m, hm, he⟩ := D.exists_completedError_small
+  obtain ⟨P⟩ := D.localizers.constantProjection_exists
+  let T := D.resolventZero m hm he ∘L D.poissonInput m hm he P
+  refine ⟨‖T‖, norm_nonneg T, fun F => ?_⟩
+  let w := D.poissonInput m hm he P (D.localizers.embed 0 F)
+  have hT := T.le_opNorm (D.localizers.embed 0 F)
+  let c := D.poissonResidue m hm he P (D.localizers.embed 0 F)
+  have hc : P.proj w = D.localizers.embed 0 (Smooth.const c) :=
+    (P.coefficient_spec w).symm
+  have hw : w = D.localizers.embed 0 (-F - Smooth.const c) +
+      (m^2 : ℝ) • D.resolventZero m hm he w := by
+    have hh := D.poissonInput_equation m hm he P (D.localizers.embed 0 F)
+    change w = -D.localizers.embed 0 F - P.proj w + _ at hh
+    rw [hc] at hh
+    simpa only [map_sub, map_neg] using hh
+  obtain ⟨u, hu⟩ := D.allRegular_smooth (D.resolvent_feedback_regular m hm he w _ hw)
+  refine ⟨u, c, ?_, ?_⟩
+  · have hs := (D.smooth_shifted_equation m hm he w u hu).trans hw
+    rw [← hu, ← map_smul, ← map_add] at hs
+    have hh := D.localizers.embed_injective 0 hs
+    apply sub_right_inj.mp
+    calc
+      (m^2 : ℝ) • u - complexL g u = -F - Smooth.const c + (m^2 : ℝ) • u := hh
+      _ = (m^2 : ℝ) • u - (F + Smooth.const c) := by abel
+  · rw [hu]
+    simpa only [T, ContinuousLinearMap.comp_apply, w] using hT
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/CalabiBochner.lean b/lean/OAI/Geometry/PrescribedRicci/CalabiBochner.lean
new file mode 100644
index 0000000000000000000000000000000000000000..67b81b9cb5f52d52903d48aa52ecc72872dbc19b
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/CalabiBochner.lean
@@ -0,0 +1,62 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.CalabiRough
+import OAI.Geometry.PrescribedRicci.CalabiTensorAlgebra
+import OAI.Geometry.PrescribedRicci.ContractedBochner
+import OAI.Geometry.PrescribedRicci.MatrixWirtinger
+
+namespace OAI
+
+/-! Calabi Bochner. -/
+
+section
+
+ 
+
+noncomputable section
+open Matrix Set Filter Topology
+open scoped ComplexOrder ContDiff MatrixOrder
+namespace Anticanonical.SourceSmooth.KaehlerMetric
+open MongeAmpere
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+local notation "TI" => TensorIndex (Fin d)
+
+def calabiEnergy (g : KaehlerMetric A) (q : Fin A.count) (z : Coordinates d) : ℝ :=
+  (tensorEnergy (g.matrix q z)⁻¹ (g.calabiMetric q z) (barVector (g.calabiTensor q) z)).re+
+  (tensorEnergy (g.matrix q z)⁻¹.transpose (g.calabiMetric q z)
+    (covHol (g.calabiConnection q) (g.calabiTensor q) z)).re
+
+def calabiRoughTerm (g : KaehlerMetric A) (q : Fin A.count) (z : Coordinates d) : TI → ℂ :=
+  fun i => (-(g.matrix q z)⁻¹*holDerivative (g.curvatureRicci q) z i.1+
+    ((g.matrix q z)⁻¹*g.curvatureRicci q z)*g.chernConnection q z i.1) i.2.1 i.2.2
+
+lemma calabiEnergy_nonneg (g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) : 0 ≤ g.calabiEnergy q z := by
+  unfold calabiEnergy
+  exact add_nonneg (tensorEnergy_nonneg (g.positive q z hz).inv.posSemidef
+    (g.calabiMetric_posDef q hz).posSemidef _)
+    (tensorEnergy_nonneg (g.positive q z hz).inv.transpose.posSemidef
+      (g.calabiMetric_posDef q hz).posSemidef _)
+
+lemma calabiNorm_laplacian (g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) :
+    ((g.matrix q z)⁻¹*PotentialKaehler.potentialMatrix (g.calabiNorm q) z).trace.re =
+      g.calabiEnergy q z+
+      2*(hermPair (g.calabiMetric q z) (g.calabiTensor q z) (g.calabiRoughTerm q z)).re+
+      (hermPair (g.calabiMetric q z) (g.calabiTensor q z)
+        ((tensorConnection (-((g.matrix q z)⁻¹*g.curvatureRicci q z)))*ᵥ
+          g.calabiTensor q z)).re := by
+  have h := contracted_bochners_identity (A.chart q).open_target hz
+    (fun y hy => g.calabiMetric_smooth q hy) (fun y hy => g.calabiConnection_smooth q hy)
+    (fun y hy => g.calabiTensor_smooth q hy)
+    (fun y hy => (g.calabiMetric_posDef q hy).isHermitian)
+    (fun y hy => g.calabiMetric_hol q hy) (fun y hy => g.calabiMetric_bar q hy)
+    (g.matrix q z)⁻¹ (g.positive q z hz).inv.isHermitian
+  rw [g.calabiTensor_rough q hz,g.calabiConnection_contracted_curvature q hz] at h
+  exact h
+
+end Anticanonical.SourceSmooth.KaehlerMetric
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/CalabiCutoffCalculus.lean b/lean/OAI/Geometry/PrescribedRicci/CalabiCutoffCalculus.lean
new file mode 100644
index 0000000000000000000000000000000000000000..5cc73bf877c497557af260ac4981edb55f69dc42
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/CalabiCutoffCalculus.lean
@@ -0,0 +1,114 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.CalabiTensorBounds
+import OAI.Geometry.PrescribedRicci.KahlerGradientEnergy
+import OAI.Geometry.PrescribedRicci.VolumePathRicci
+
+namespace OAI
+
+/-! Calabi Cutoff Calculus. -/
+
+section
+
+ 
+noncomputable section
+open Matrix Set Filter Topology
+open scoped ContDiff ComplexOrder MatrixOrder Matrix.Norms.Elementwise
+namespace Anticanonical.SourceSmooth.KaehlerMetric
+open MongeAmpere
+variable {d : ℕ}
+
+def localLap (H : Matrix (Fin d) (Fin d) ℂ) (f : Coordinates d → ℝ)
+    (z : Coordinates d) : ℝ := (H⁻¹*PotentialKaehler.potentialMatrix f z).trace.re
+
+lemma localLap_affine {f h : Coordinates d → ℝ} {z : Coordinates d}
+    (hf : ContDiffAt ℝ ∞ f z) (hh : ContDiffAt ℝ ∞ h z)
+    (H : Matrix (Fin d) (Fin d) ℂ) (a : ℝ) :
+    localLap H (fun y => f y+a*h y) z = localLap H f z+a*localLap H h z := by
+  have he := PotentialKaehler.potentialMatrix_affine hf hh a 0
+  simp only [add_zero] at he
+  unfold localLap
+  rw [he,mul_add,Matrix.mul_smul,trace_add,trace_smul,Complex.add_re]
+  simp only [smul_eq_mul,Complex.mul_re,Complex.ofReal_re,Complex.ofReal_im,zero_mul,sub_zero]
+
+lemma localLap_abs_bound (H : Matrix (Fin d) (Fin d) ℂ) (f : Coordinates d → ℝ)
+    (z : Coordinates d) : |localLap H f z| ≤
+      (d:ℝ)^2*‖H⁻¹‖*‖PotentialKaehler.potentialMatrix f z‖ := by
+  apply (Complex.abs_re_le_norm _).trans
+  exact (trace_entry_bound (fun i j => norm_entry_le_entrywise_sup_norm
+    (H⁻¹*PotentialKaehler.potentialMatrix f z) (i:=i) (j:=j))).trans
+    ((mul_le_mul_of_nonneg_left (matrix_norm_mul_le _ _) (by positivity)).trans_eq (by
+      simp only [Fintype.card_fin]; ring))
+
+lemma gradientPair_norm_bound (H : Matrix (Fin d) (Fin d) ℂ) (p q : Fin d → ℂ) :
+    ‖gradientPair H p q‖ ≤ (d:ℝ)^2*‖H⁻¹‖*‖p‖*‖q‖ := by
+  unfold gradientPair
+  apply (norm_sum_le _ _).trans
+  calc
+    _ ≤ ∑ _j : Fin d, ∑ _i : Fin d, ‖H⁻¹‖*‖p‖*‖q‖ := by
+      apply Finset.sum_le_sum
+      intro j _
+      apply (norm_sum_le _ _).trans
+      apply Finset.sum_le_sum
+      intro i _
+      simp only [norm_mul,norm_star]
+      exact mul_le_mul (mul_le_mul (norm_entry_le_entrywise_sup_norm _)
+        (norm_le_pi_norm p i) (norm_nonneg _) (norm_nonneg _))
+        (norm_le_pi_norm q j) (norm_nonneg _) (by positivity)
+    _ = _ := by simp only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul]; ring
+
+lemma holRealDeriv_square {χ : Coordinates d → ℝ} {z : Coordinates d}
+    (hχ : DifferentiableAt ℝ χ z) :
+    holRealDeriv (fun y => χ y*χ y) z = (2*(χ z : ℂ)) • holRealDeriv χ z := by
+  funext a
+  rw [holRealDeriv_mul hχ hχ]
+  simp only [Pi.smul_apply,smul_eq_mul]
+  ring
+
+lemma norm_pi_sq_bound {ι : Type*} [Fintype ι] {v : ι → ℂ} {B : ℝ}
+    (hB : 0 ≤ B) (hv : ∀ i, ‖v i‖^2 ≤ B) : ‖v‖^2 ≤ B := by
+  have hn : ‖v‖ ≤ Real.sqrt B := (pi_norm_le_iff_of_nonneg (Real.sqrt_nonneg B)).mpr
+    (fun i => (Real.le_sqrt (norm_nonneg _) hB).mpr (hv i))
+  exact ((sq_le_sq₀ (norm_nonneg _) (Real.sqrt_nonneg _)).mpr hn).trans_eq (Real.sq_sqrt hB)
+
+lemma square_absorb {a x y : ℝ} (_ha : 0 ≤ a) (hx : 0 ≤ x) (hy : 0 ≤ y)
+    (hb : a^2 ≤ x*y) : a ≤ (x+y)/2 := by
+  nlinarith [sq_nonneg (x-y)]
+
+lemma cutoff_cross_bound (H : Matrix (Fin d) (Fin d) ℂ)
+    {χ S : Coordinates d → ℝ} {z : Coordinates d}
+    (hχ : DifferentiableAt ℝ χ z) {M B K s E : ℝ}
+    (hM : 0 ≤ M) (hB : 0 ≤ B) (hK : 0 ≤ K) (hs : 0 ≤ s) (hE : 0 ≤ E)
+    (hI : ‖H⁻¹‖ ≤ M) (hDχ : ‖holRealDeriv χ z‖ ≤ B)
+    (hDS : ‖holRealDeriv S z‖^2 ≤ K*s*E) :
+    2*(gradientPair H (holRealDeriv (fun y => χ y*χ y) z) (holRealDeriv S z)).re ≥
+      -(χ z)^2*E/2-((4*(d:ℝ)^2*M*B)^2*K)*s/2 := by
+  let L := 4*(d:ℝ)^2*M*B
+  have hL : 0 ≤ L := by dsimp [L]; positivity
+  have hn : 2*‖gradientPair H (holRealDeriv (fun y => χ y*χ y) z)
+      (holRealDeriv S z)‖ ≤ L*|χ z| *‖holRealDeriv S z‖ := by
+    apply (mul_le_mul_of_nonneg_left (gradientPair_norm_bound _ _ _) (by norm_num)).trans
+    rw [holRealDeriv_square hχ,norm_smul,norm_mul]
+    norm_num only [Complex.norm_real,Real.norm_eq_abs,Complex.norm_ofNat]
+    calc
+      _ ≤ 2*((d:ℝ)^2*M*(2*|χ z| *B)*‖holRealDeriv S z‖) := by
+        gcongr
+      _ = _ := by dsimp [L]; ring
+  have hsq : (L*|χ z| *‖holRealDeriv S z‖)^2 ≤ ((χ z)^2*E)*(L^2*K*s) := by
+    have hh := mul_le_mul_of_nonneg_left hDS (show 0 ≤ L^2*(χ z)^2 by positivity)
+    calc
+      _ = L^2*(χ z)^2*‖holRealDeriv S z‖^2 := by rw [mul_pow,mul_pow,sq_abs]
+      _ ≤ L^2*(χ z)^2*(K*s*E) := hh
+      _ = _ := by ring
+  have ha := square_absorb (show 0 ≤ L*|χ z| *‖holRealDeriv S z‖ by positivity)
+    (show 0 ≤ (χ z)^2*E by positivity) (show 0 ≤ L^2*K*s by positivity) hsq
+  have hr := (abs_le.mp (Complex.abs_re_le_norm
+    (gradientPair H (holRealDeriv (fun y => χ y*χ y) z) (holRealDeriv S z)))).1
+  change _ ≥ -(χ z)^2*E/2-(L^2*K)*s/2
+  linarith
+
+end Anticanonical.SourceSmooth.KaehlerMetric
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/CalabiCutoffDifferential.lean b/lean/OAI/Geometry/PrescribedRicci/CalabiCutoffDifferential.lean
new file mode 100644
index 0000000000000000000000000000000000000000..4bc7d3ca1ace67f0fe714489799409ad6dd24fa9
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/CalabiCutoffDifferential.lean
@@ -0,0 +1,67 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.CalabiCutoffCalculus
+
+namespace OAI
+
+/-! Calabi Cutoff Differential. -/
+
+section
+
+ 
+
+noncomputable section
+open Matrix Set Filter Topology
+open scoped ContDiff ComplexOrder MatrixOrder Matrix.Norms.Elementwise
+namespace Anticanonical.SourceSmooth.KaehlerMetric
+variable {d : ℕ}
+
+def cutoffCoefficient (d : ℕ) (M B C K δ : ℝ) : ℝ :=
+  (B^2*C+(d:ℝ)^2*M*B+(4*(d:ℝ)^2*M*B)^2*K/2+1)/δ
+
+lemma cutoffCoefficient_pos {M B C K δ : ℝ} (hM : 0 ≤ M) (hB : 0 ≤ B)
+    (hC : 0 ≤ C) (hK : 0 ≤ K) (hδ : 0 < δ) :
+    0 < cutoffCoefficient d M B C K δ := by unfold cutoffCoefficient; positivity
+
+lemma cutoff_test_lower_bound (H : Matrix (Fin d) (Fin d) ℂ) (hH : H.PosDef)
+    {χ S T : Coordinates d → ℝ} {z : Coordinates d}
+    (hχ : ContDiffAt ℝ ∞ χ z) (hS : ContDiffAt ℝ ∞ S z) (hT : ContDiffAt ℝ ∞ T z)
+    {M B C K δ R E : ℝ} (hM : 0 ≤ M) (hB : 0 ≤ B) (hC : 0 ≤ C)
+    (hK : 0 ≤ K) (hδ : 0 < δ) (hE : 0 ≤ E) (hSz : 0 ≤ S z)
+    (hI : ‖H⁻¹‖ ≤ M) (hχB : |χ z| ≤ B) (hDχ : ‖holRealDeriv χ z‖ ≤ B)
+    (hDDχ : ‖PotentialKaehler.potentialMatrix (fun y => χ y*χ y) z‖ ≤ B)
+    (hDS : ‖holRealDeriv S z‖^2 ≤ K*S z*E)
+    (hLS : E-C*(S z+1) ≤ localLap H S z)
+    (hLT : δ*S z-R ≤ localLap H T z) :
+    S z-B^2*C-cutoffCoefficient d M B C K δ*R ≤
+      localLap H (fun y => χ y*χ y*S y+cutoffCoefficient d M B C K δ*T y) z := by
+  let a := cutoffCoefficient d M B C K δ
+  have ha : 0 ≤ a := (cutoffCoefficient_pos hM hB hC hK hδ).le
+  have had : a*δ = B^2*C+(d:ℝ)^2*M*B+(4*(d:ℝ)^2*M*B)^2*K/2+1 := by
+    dsimp [a,cutoffCoefficient]
+    exact div_mul_cancel₀ _ hδ.ne'
+  have hc2 : (χ z)^2 ≤ B^2 := by
+    simpa only [sq_abs] using (sq_le_sq₀ (abs_nonneg _) hB).mpr hχB
+  have hχlap : -(d:ℝ)^2*M*B ≤ localLap H (fun y => χ y*χ y) z := by
+    have hh := (localLap_abs_bound H (fun y => χ y*χ y) z).trans
+      (mul_le_mul (mul_le_mul_of_nonneg_left hI (sq_nonneg _)) hDDχ (norm_nonneg _) (by positivity))
+    convert! (abs_le.mp hh).1 using 1
+    ring
+  have hcross := cutoff_cross_bound H (hχ.differentiableAt (by simp)) hM hB hK hSz hE hI hDχ hDS
+  have h1 := mul_le_mul_of_nonneg_left hLS (sq_nonneg (χ z))
+  have h2 := mul_le_mul_of_nonneg_left hχlap hSz
+  have h3 := mul_le_mul_of_nonneg_left hLT ha
+  have h4 := mul_le_mul_of_nonneg_right hc2 (show 0 ≤ C*(S z+1) by positivity)
+  have h5 : 0 ≤ (χ z)^2*E := mul_nonneg (sq_nonneg _) hE
+  rw [localLap_affine ((hχ.mul hχ).mul hS) hT,localLap,
+    trace_hessian_product (hχ.mul hχ) hS H hH.isHermitian]
+  change S z-B^2*C-a*R ≤ χ z*χ z*localLap H S z+
+    S z*localLap H (fun y => χ y*χ y) z+
+    2*(gradientPair H (holRealDeriv (fun y => χ y*χ y) z) (holRealDeriv S z)).re+a*localLap H T z
+  nlinarith [congrArg (fun t : ℝ => t*S z) had]
+
+end Anticanonical.SourceSmooth.KaehlerMetric
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/CalabiGradient.lean b/lean/OAI/Geometry/PrescribedRicci/CalabiGradient.lean
new file mode 100644
index 0000000000000000000000000000000000000000..76c2e0ff17303bd9fd9fe882c28608cdcf86e1fd
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/CalabiGradient.lean
@@ -0,0 +1,106 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.CalabiBochner
+import OAI.Geometry.PrescribedRicci.CalabiLower
+import OAI.Geometry.PrescribedRicci.CalabiTensorBounds
+import OAI.Geometry.PrescribedRicci.TensorEnergyBounds
+
+namespace OAI
+
+/-! Calabi Gradient. -/
+
+section
+
+ 
+
+noncomputable section
+open Matrix Set Filter Topology
+open scoped ComplexOrder ContDiff MatrixOrder Matrix.Norms.Elementwise
+namespace Anticanonical.SourceSmooth.KaehlerMetric
+open MongeAmpere
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+local notation "TI" => TensorIndex (Fin d)
+
+lemma calabiNorm_hol (g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) (a : Fin d) :
+    holRealDeriv (g.calabiNorm q) z a =
+      hermPair (g.calabiMetric q z) (barVector (g.calabiTensor q) z a) (g.calabiTensor q z)+
+      hermPair (g.calabiMetric q z) (g.calabiTensor q z)
+        (covHol (g.calabiConnection q) (g.calabiTensor q) z a) := by
+  rw [← holDeriv_ofReal ((g.calabiNorm_smooth q hz).differentiableAt (by simp))]
+  have he : (fun y => (g.calabiNorm q y : ℂ)) =ᶠ[nhds z]
+      fun y => hermPair (g.calabiMetric q y) (g.calabiTensor q y) (g.calabiTensor q y) := by
+    filter_upwards [(A.chart q).open_target.mem_nhds hz] with y hy
+    exact hermPair_self_ofReal (g.calabiMetric_posDef q hy).isHermitian _
+  rw [holDeriv_congr he]
+  exact hermPair_hol_compatible (fun i j => (g.calabiMetric_smooth q hz i j).differentiableAt (by simp))
+    (fun i => (g.calabiTensor_smooth q hz i).differentiableAt (by simp))
+    (fun i => (g.calabiTensor_smooth q hz i).differentiableAt (by simp)) a (g.calabiMetric_hol q hz a)
+
+lemma calabiDerivative_sq_bound (g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) {M : ℝ} (hM : 0 ≤ M)
+    (hH : ‖g.matrix q z‖ ≤ M) (hI : ‖(g.matrix q z)⁻¹‖ ≤ M) (a : Fin d) :
+    ‖barVector (g.calabiTensor q) z a‖^2+
+      ‖covHol (g.calabiConnection q) (g.calabiTensor q) z a‖^2 ≤ M^4*g.calabiEnergy q z := by
+  have hi : ((g.matrix q z)⁻¹)⁻¹ = g.matrix q z := Matrix.inv_eq_right_inv
+    (Matrix.nonsing_inv_mul _ (isUnit_iff_ne_zero.mpr (g.positive q z hz).det_pos.ne'))
+  have hiT : ((g.matrix q z)⁻¹.transpose)⁻¹ = (g.matrix q z).transpose := by
+    rw [← Matrix.transpose_nonsing_inv,hi]
+  have hki := tensorMetric_inv_norm_bound (g.positive q z hz) hM hH hI
+  have h1 := vectorNorm_sq_le_tensorEnergy (g.positive q z hz).inv (g.calabiMetric_posDef q hz)
+    (barVector (g.calabiTensor q) z) a
+  have h2 := vectorNorm_sq_le_tensorEnergy (g.positive q z hz).inv.transpose (g.calabiMetric_posDef q hz)
+    (covHol (g.calabiConnection q) (g.calabiTensor q) z) a
+  rw [hi] at h1
+  rw [hiT,Matrix.norm_transpose] at h2
+  have hcoeff : ‖g.matrix q z‖*‖(g.calabiMetric q z)⁻¹‖ ≤ M^4 := by
+    exact (mul_le_mul hH hki (norm_nonneg _) hM).trans_eq (by ring)
+  have hp1 := tensorEnergy_nonneg (g.positive q z hz).inv.posSemidef
+    (g.calabiMetric_posDef q hz).posSemidef (barVector (g.calabiTensor q) z)
+  have hp2 := tensorEnergy_nonneg (g.positive q z hz).inv.transpose.posSemidef
+    (g.calabiMetric_posDef q hz).posSemidef
+      (covHol (g.calabiConnection q) (g.calabiTensor q) z)
+  apply (add_le_add (h1.trans (mul_le_mul_of_nonneg_right hcoeff hp1))
+    (h2.trans (mul_le_mul_of_nonneg_right hcoeff hp2))).trans_eq
+  unfold calabiEnergy
+  ring
+
+lemma calabiNorm_gradient_sq_bound (g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) {M : ℝ} (hM : 0 ≤ M)
+    (hH : ‖g.matrix q z‖ ≤ M) (hI : ‖(g.matrix q z)⁻¹‖ ≤ M) (a : Fin d) :
+    ‖holRealDeriv (g.calabiNorm q) z a‖^2 ≤
+      2*(Fintype.card TI : ℝ)^4*M^13*g.calabiNorm q z*g.calabiEnergy q z := by
+  let f := g.calabiTensor q z
+  let u := barVector (g.calabiTensor q) z a
+  let v := covHol (g.calabiConnection q) (g.calabiTensor q) z a
+  let N : ℝ := Fintype.card TI
+  have hK : ‖g.calabiMetric q z‖ ≤ M^3 := tensorMetric_norm_bound hM hH hI
+  have hS := g.calabiNorm_nonneg q hz
+  have hE := g.calabiEnergy_nonneg q hz
+  have hf := g.calabiTensor_sq_bound q hz hM hH hI
+  have hd := g.calabiDerivative_sq_bound q hz hM hH hI a
+  have hb : ‖holRealDeriv (g.calabiNorm q) z a‖ ≤ N^2*M^3*‖f‖*(‖u‖+‖v‖) := by
+    rw [g.calabiNorm_hol q hz]
+    apply (norm_add_le _ _).trans
+    have hp1 := (pair_norm_le (g.calabiMetric q z) u f).trans
+      (mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right
+        (mul_le_mul_of_nonneg_left hK (sq_nonneg N)) (norm_nonneg u)) (norm_nonneg f))
+    have hp2 := (pair_norm_le (g.calabiMetric q z) f v).trans
+      (mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right
+        (mul_le_mul_of_nonneg_left hK (sq_nonneg N)) (norm_nonneg f)) (norm_nonneg v))
+    exact (add_le_add hp1 hp2).trans_eq (by ring)
+  have hsq := (sq_le_sq₀ (norm_nonneg _) (show 0 ≤ N^2*M^3*‖f‖*(‖u‖+‖v‖) by positivity)).mpr hb
+  have huv : (‖u‖+‖v‖)^2 ≤ 2*(‖u‖^2+‖v‖^2) := by nlinarith [sq_nonneg (‖u‖-‖v‖)]
+  calc
+    _ ≤ (N^2*M^3*‖f‖*(‖u‖+‖v‖))^2 := hsq
+    _ = N^4*M^6*‖f‖^2*(‖u‖+‖v‖)^2 := by ring
+    _ ≤ N^4*M^6*(M^3*g.calabiNorm q z)*(2*(M^4*g.calabiEnergy q z)) :=
+      mul_le_mul (mul_le_mul_of_nonneg_left hf (by positivity))
+        (huv.trans (mul_le_mul_of_nonneg_left hd (by norm_num))) (sq_nonneg _) (by positivity)
+    _ = _ := by dsimp [N]; ring
+
+end Anticanonical.SourceSmooth.KaehlerMetric
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/CalabiLocalMaximum.lean b/lean/OAI/Geometry/PrescribedRicci/CalabiLocalMaximum.lean
new file mode 100644
index 0000000000000000000000000000000000000000..d9cb6d7a2287f9f188ab15fce192ad1abeeaa639
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/CalabiLocalMaximum.lean
@@ -0,0 +1,119 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.CalabiCutoffDifferential
+import OAI.Geometry.PrescribedRicci.CalabiGradient
+import OAI.Geometry.PrescribedRicci.CalabiLower
+import OAI.Geometry.PrescribedRicci.CalabiTraceBound
+
+namespace OAI
+
+/-! Calabi Local Maximum. -/
+
+section
+
+ 
+
+noncomputable section
+open Matrix Set Filter Topology
+open scoped ContDiff ComplexOrder MatrixOrder Matrix.Norms.Elementwise
+namespace Anticanonical.SourceSmooth.KaehlerMetric
+open MongeAmpere
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+
+lemma flatTrace_nonneg (g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) : 0 ≤ g.flatTrace q z :=
+  (Complex.nonneg_iff.mp (g.positive q z hz).posSemidef.trace_nonneg).1
+
+lemma compact_cutoff_derivative_bound {L : Set (Coordinates d)} (hL : IsCompact L)
+    (χ : Coordinates d → ℝ) (hχ : ContDiff ℝ ∞ χ) :
+    ∃ B : ℝ, 0 < B ∧ ∀ z ∈ L, |χ z| ≤ B ∧ ‖holRealDeriv χ z‖ ≤ B ∧
+      ‖PotentialKaehler.potentialMatrix (fun y => χ y*χ y) z‖ ≤ B := by
+  let size : Coordinates d → ℝ := fun z => |χ z|+‖holRealDeriv χ z‖+
+    ‖PotentialKaehler.potentialMatrix (fun y => χ y*χ y) z‖
+  have hs : Continuous size := by
+    have hd : ContDiff ℝ ∞ (fun z => holRealDeriv χ z) := contDiff_iff_contDiffAt.mpr
+      (fun z => contDiffAt_pi.mpr (fun a => contDiffAt_holRealDeriv hχ.contDiffAt a))
+    have hdd : ContDiff ℝ ∞ (fun z => PotentialKaehler.potentialMatrix (fun y => χ y*χ y) z) :=
+      contDiff_iff_contDiffAt.mpr (fun z => PotentialKaehler.potentialMatrix_smooth (hχ.mul hχ).contDiffAt)
+    exact (hχ.continuous.abs.add hd.continuous.norm).add hdd.continuous.norm
+  obtain ⟨b,hb⟩ := hL.bddAbove_image hs.continuousOn
+  refine ⟨1+max 0 b,by positivity,?_⟩
+  intro z hz
+  have hh := (hb (mem_image_of_mem _ hz)).trans (le_max_right (0:ℝ) b)
+  dsimp [size] at hh
+  refine ⟨?_,?_,?_⟩ <;> linarith [abs_nonneg (χ z),norm_nonneg (holRealDeriv χ z),
+    norm_nonneg (PotentialKaehler.potentialMatrix (fun y => χ y*χ y) z)]
+
+ 
+
+theorem calabi_bound_on_compact (q : Fin A.count) {L : Set (Coordinates d)}
+    (hL : IsCompact L) (hLt : L ⊆ (A.chart q).target)
+    (χ : Coordinates d → ℝ) (hχ : ContDiff ℝ ∞ χ)
+    (hχL : tsupport χ ⊆ interior L) {M R : ℝ} (hM : 0 ≤ M) (hR : 0 ≤ R) :
+    ∃ P : ℝ, 0 < P ∧ ∀ g : KaehlerMetric A,
+      (∀ z ∈ L, ‖g.matrix q z‖ ≤ M ∧ ‖(g.matrix q z)⁻¹‖ ≤ M ∧
+        ‖g.curvatureRicci q z‖ ≤ R ∧ ∀ a, ‖holDerivative (g.curvatureRicci q) z a‖ ≤ R) →
+      ∀ z ∈ L, χ z = 1 → g.calabiNorm q z ≤ P := by
+  obtain ⟨C,hC,hCal⟩ := calabiNorm_lower_bound d M R hM hR
+  obtain ⟨B,hB,hcut⟩ := compact_cutoff_derivative_bound hL χ hχ
+  let K := 2*(Fintype.card (TensorIndex (Fin d)):ℝ)^4*M^13
+  let δ := (1+(Fintype.card (TensorIndex (Fin d)):ℝ)^2*M^5)⁻¹
+  have hK : 0 ≤ K := by dsimp [K]; positivity
+  have hδ : 0 < δ := by dsimp [δ]; positivity
+  let a := cutoffCoefficient d M B C K δ
+  have ha : 0 < a := cutoffCoefficient_pos hM hB.le hC.le hK hδ
+  let Q := B^2*C+a*((d:ℝ)*R)
+  have hQ : 0 ≤ Q := by dsimp [Q]; positivity
+  refine ⟨1+B^2*Q+a*((d:ℝ)*M),by positivity,?_⟩
+  intro g hbounds z hz hzχ
+  let f : Coordinates d → ℝ := fun y => χ y*χ y*g.calabiNorm q y+a*g.flatTrace q y
+  have hfs (y : Coordinates d) (hy : y ∈ L) : ContDiffAt ℝ ∞ f y :=
+    ((hχ.contDiffAt.mul hχ.contDiffAt).mul (g.calabiNorm_smooth q (hLt hy))).add
+      (contDiffAt_const.mul (g.flatTrace_smooth q (hLt hy)))
+  obtain ⟨x,hx,hmax⟩ := hL.exists_isMaxOn ⟨z,hz⟩ (fun y hy => (hfs y hy).continuousAt.continuousWithinAt)
+  have hzle : g.calabiNorm q z ≤ f z := by
+    change _ ≤ χ z*χ z*g.calabiNorm q z+a*g.flatTrace q z
+    rw [hzχ]
+    have ht := mul_nonneg ha.le (g.flatTrace_nonneg q (hLt hz))
+    nlinarith
+  have hmaxbound : f x ≤ B^2*Q+a*((d:ℝ)*M) := by
+    have ht := g.flatTrace_bound q x (hbounds x hx).1
+    by_cases h0 : χ x = 0
+    · change χ x*χ x*g.calabiNorm q x+a*g.flatTrace q x ≤ _
+      rw [h0]
+      have hh := mul_le_mul_of_nonneg_left ht ha.le
+      nlinarith [mul_nonneg (sq_nonneg B) hQ]
+    · have hn : L ∈ nhds x := mem_interior_iff_mem_nhds.mp (hχL (subset_tsupport _ h0))
+      have hlocal : IsLocalMax f x := Filter.eventually_of_mem hn (fun y hy => hmax hy)
+      have hnonpos := EllipticKernel.trace_potentialMatrix_nonpos_at_localMax
+        (hfs x hx) hlocal (g.matrix q x) (g.positive q x (hLt hx))
+      have hb := hbounds x hx
+      have hgrad : ‖holRealDeriv (g.calabiNorm q) x‖^2 ≤ K*g.calabiNorm q x*g.calabiEnergy q x :=
+        norm_pi_sq_bound (by positivity [g.calabiNorm_nonneg q (hLt hx),g.calabiEnergy_nonneg q (hLt hx)])
+          (fun j => g.calabiNorm_gradient_sq_bound q (hLt hx) hM hb.1 hb.2.1 j)
+      have hlow := cutoff_test_lower_bound (g.matrix q x) (g.positive q x (hLt hx))
+        hχ.contDiffAt (g.calabiNorm_smooth q (hLt hx)) (g.flatTrace_smooth q (hLt hx))
+        hM hB.le hC.le hK hδ (g.calabiEnergy_nonneg q (hLt hx)) (g.calabiNorm_nonneg q (hLt hx))
+        hb.2.1 (hcut x hx).1 (hcut x hx).2.1 (hcut x hx).2.2 hgrad
+        (hCal g q x (hLt hx) hb.1 hb.2.1 hb.2.2.1 hb.2.2.2)
+        (g.flatTrace_lower_bound q (hLt hx) hM hb.1 hb.2.1 hb.2.2.1)
+      have hS : g.calabiNorm q x ≤ Q := by
+        change g.calabiNorm q x-B^2*C-a*((d:ℝ)*R) ≤ localLap (g.matrix q x) f x at hlow
+        change localLap (g.matrix q x) f x ≤ 0 at hnonpos
+        dsimp [Q]
+        linarith
+      have hcx : χ x*χ x ≤ B^2 := by
+        have hh := (sq_le_sq₀ (abs_nonneg (χ x)) hB.le).mpr (hcut x hx).1
+        rw [sq_abs] at hh
+        simpa only [pow_two] using hh
+      have hh := mul_le_mul hcx hS (g.calabiNorm_nonneg q (hLt hx)) (sq_nonneg B)
+      have hhT := mul_le_mul_of_nonneg_left ht ha.le
+      change χ x*χ x*g.calabiNorm q x+a*g.flatTrace q x ≤ _
+      linarith
+  exact (hzle.trans (hmax hz)).trans (hmaxbound.trans (by linarith))
+
+end Anticanonical.SourceSmooth.KaehlerMetric
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/CalabiLower.lean b/lean/OAI/Geometry/PrescribedRicci/CalabiLower.lean
new file mode 100644
index 0000000000000000000000000000000000000000..b84d38d5393ee489b762b094741ac29c2a0439ec
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/CalabiLower.lean
@@ -0,0 +1,146 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.CalabiBochner
+import OAI.Geometry.PrescribedRicci.CalabiTensorBounds
+import OAI.Geometry.PrescribedRicci.HermitianCoercivity
+import OAI.Geometry.PrescribedRicci.MatrixWirtinger
+
+namespace OAI
+
+/-! Calabi Lower. -/
+
+section
+
+ 
+noncomputable section
+open Matrix Set Filter Topology
+open scoped ComplexOrder ContDiff MatrixOrder Matrix.Norms.Elementwise
+namespace Anticanonical.SourceSmooth.KaehlerMetric
+open MongeAmpere
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+local notation "TI" => TensorIndex (Fin d)
+
+lemma chernConnection_norm_le_calabiTensor (g : KaehlerMetric A) (q : Fin A.count)
+    (z : Coordinates d) (a : Fin d) : ‖g.chernConnection q z a‖ ≤ ‖g.calabiTensor q z‖ := by
+  apply matrix_norm_le_of_entries (norm_nonneg _)
+  intro i j
+  exact norm_le_pi_norm (g.calabiTensor q z) (a,i,j)
+
+lemma calabiTensor_sq_bound (g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) {M : ℝ} (hM : 0 ≤ M)
+    (hH : ‖g.matrix q z‖ ≤ M) (hI : ‖(g.matrix q z)⁻¹‖ ≤ M) :
+    ‖g.calabiTensor q z‖^2 ≤ M^3*g.calabiNorm q z := by
+  exact (norm_sq_le_hermPair (g.calabiMetric_posDef q hz) _).trans
+    (mul_le_mul_of_nonneg_right (tensorMetric_inv_norm_bound (g.positive q z hz) hM hH hI)
+      (g.calabiNorm_nonneg q hz))
+
+lemma calabiRoughTerm_bound (g : KaehlerMetric A) (q : Fin A.count) (z : Coordinates d)
+    {M R : ℝ} (hM : 0 ≤ M) (hR : 0 ≤ R) (hI : ‖(g.matrix q z)⁻¹‖ ≤ M)
+    (hRc : ‖g.curvatureRicci q z‖ ≤ R)
+    (hDR : ∀ a, ‖holDerivative (g.curvatureRicci q) z a‖ ≤ R) :
+    ‖g.calabiRoughTerm q z‖ ≤ (d:ℝ)*M*R+(d:ℝ)^2*M*R*‖g.calabiTensor q z‖ := by
+  have hn : (0:ℝ) ≤ d := Nat.cast_nonneg _
+  have hb : ‖(g.matrix q z)⁻¹*g.curvatureRicci q z‖ ≤ (d:ℝ)*M*R := by
+    apply (matrix_norm_mul_le _ _).trans
+    simp only [Fintype.card_fin]
+    exact mul_le_mul (mul_le_mul_of_nonneg_left hI hn) hRc (norm_nonneg _) (by positivity)
+  apply (pi_norm_le_iff_of_nonneg (by positivity)).mpr
+  intro i
+  change ‖(-(g.matrix q z)⁻¹*holDerivative (g.curvatureRicci q) z i.1+
+    ((g.matrix q z)⁻¹*g.curvatureRicci q z)*g.chernConnection q z i.1) i.2.1 i.2.2‖ ≤ _
+  apply (norm_entry_le_entrywise_sup_norm _).trans
+  apply (norm_add_le _ _).trans
+  have h1 : ‖-(g.matrix q z)⁻¹*holDerivative (g.curvatureRicci q) z i.1‖ ≤ (d:ℝ)*M*R := by
+    rw [neg_mul,norm_neg]
+    apply (matrix_norm_mul_le _ _).trans
+    simp only [Fintype.card_fin]
+    exact mul_le_mul (mul_le_mul_of_nonneg_left hI hn) (hDR i.1) (norm_nonneg _) (by positivity)
+  have h2 : ‖((g.matrix q z)⁻¹*g.curvatureRicci q z)*g.chernConnection q z i.1‖ ≤
+      (d:ℝ)^2*M*R*‖g.calabiTensor q z‖ := by
+    apply (matrix_norm_mul_le _ _).trans
+    simp only [Fintype.card_fin]
+    calc
+      _ ≤ (d:ℝ)*((d:ℝ)*M*R)*‖g.calabiTensor q z‖ :=
+        mul_le_mul (mul_le_mul_of_nonneg_left hb hn)
+          (g.chernConnection_norm_le_calabiTensor q z i.1) (norm_nonneg _) (by positivity)
+      _ = _ := by ring
+  exact add_le_add h1 h2
+
+lemma tensorCurvature_bound (g : KaehlerMetric A) (q : Fin A.count) (z : Coordinates d)
+    {M R : ℝ} (hM : 0 ≤ M) (_hR : 0 ≤ R) (hI : ‖(g.matrix q z)⁻¹‖ ≤ M)
+    (hRc : ‖g.curvatureRicci q z‖ ≤ R) :
+    ‖tensorConnection (-((g.matrix q z)⁻¹*g.curvatureRicci q z))‖ ≤ 3*(d:ℝ)*M*R := by
+  apply (tensorConnection_norm_le _).trans
+  rw [norm_neg]
+  have h := matrix_norm_mul_le (g.matrix q z)⁻¹ (g.curvatureRicci q z)
+  simp only [Fintype.card_fin] at h
+  calc
+    _ ≤ 3*((d:ℝ)*M*R) := mul_le_mul_of_nonneg_left
+      (h.trans (mul_le_mul (mul_le_mul_of_nonneg_left hI (Nat.cast_nonneg _)) hRc
+        (norm_nonneg _) (by positivity))) (by norm_num)
+    _ = _ := by ring
+
+ 
+
+theorem calabiNorm_lower_bound (d : ℕ) (M R : ℝ) (hM : 0 ≤ M) (hR : 0 ≤ R) :
+    ∃ C : ℝ, 0 < C ∧ ∀ {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+      (g : KaehlerMetric A) (q : Fin A.count) (z : Coordinates d), z ∈ (A.chart q).target →
+      ‖g.matrix q z‖ ≤ M → ‖(g.matrix q z)⁻¹‖ ≤ M → ‖g.curvatureRicci q z‖ ≤ R →
+      (∀ a, ‖holDerivative (g.curvatureRicci q) z a‖ ≤ R) →
+      g.calabiEnergy q z-C*(g.calabiNorm q z+1) ≤
+        ((g.matrix q z)⁻¹*PotentialKaehler.potentialMatrix (g.calabiNorm q) z).trace.re := by
+  let N : ℝ := Fintype.card (TensorIndex (Fin d))
+  let A₀ := N^2*M^3*((d:ℝ)*M*R+(d:ℝ)^2*M*R)
+  let B₀ := N^2*M^3*(N*(3*(d:ℝ)*M*R))
+  have hN : 0 ≤ N := Nat.cast_nonneg _
+  have hA : 0 ≤ A₀ := by dsimp [A₀]; positivity
+  have hB : 0 ≤ B₀ := by dsimp [B₀]; positivity
+  refine ⟨1+2*A₀*(1+2*M^3)+B₀*M^3,by positivity,?_⟩
+  intro X _ A g q z hz hH hI hRc hDR
+  let f := g.calabiTensor q z
+  let S := g.calabiNorm q z
+  have hS : 0 ≤ S := g.calabiNorm_nonneg q hz
+  have hf : ‖f‖^2 ≤ M^3*S := g.calabiTensor_sq_bound q hz hM hH hI
+  have hK : ‖g.calabiMetric q z‖ ≤ M^3 := tensorMetric_norm_bound hM hH hI
+  have hr : ‖g.calabiRoughTerm q z‖ ≤ ((d:ℝ)*M*R+(d:ℝ)^2*M*R)*(1+‖f‖) := by
+    apply (g.calabiRoughTerm_bound q z hM hR hI hRc hDR).trans
+    dsimp [f]
+    nlinarith [mul_nonneg (show 0 ≤ (d:ℝ)*M*R by positivity) (norm_nonneg (g.calabiTensor q z)),
+      show (0:ℝ) ≤ (d:ℝ)^2*M*R by positivity]
+  have hpair1 : ‖hermPair (g.calabiMetric q z) f (g.calabiRoughTerm q z)‖ ≤ A₀*(1+2*M^3*S) := by
+    apply (pair_norm_le _ _ _).trans
+    calc
+      _ ≤ N^2*M^3*‖f‖*(((d:ℝ)*M*R+(d:ℝ)^2*M*R)*(1+‖f‖)) :=
+        mul_le_mul (mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left hK (sq_nonneg N))
+          (norm_nonneg f)) hr (norm_nonneg _) (by positivity)
+      _ = A₀*(‖f‖+‖f‖^2) := by dsimp [A₀]; ring
+      _ ≤ A₀*(1+2*M^3*S) := mul_le_mul_of_nonneg_left (by nlinarith [sq_nonneg (‖f‖-1)]) hA
+  have ha : ‖tensorConnection (-((g.matrix q z)⁻¹*g.curvatureRicci q z))*ᵥ f‖ ≤
+      N*(3*(d:ℝ)*M*R)*‖f‖ := by
+    exact (mulVec_norm_le _ _).trans (mul_le_mul_of_nonneg_right
+      (mul_le_mul_of_nonneg_left (g.tensorCurvature_bound q z hM hR hI hRc) hN) (norm_nonneg f))
+  have hpair2 : ‖hermPair (g.calabiMetric q z) f
+      (tensorConnection (-((g.matrix q z)⁻¹*g.curvatureRicci q z))*ᵥ f)‖ ≤ B₀*M^3*S := by
+    apply (pair_norm_le _ _ _).trans
+    calc
+      _ ≤ N^2*M^3*‖f‖*(N*(3*(d:ℝ)*M*R)*‖f‖) :=
+        mul_le_mul (mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left hK (sq_nonneg N))
+          (norm_nonneg f)) ha (norm_nonneg _) (by positivity)
+      _ = B₀*‖f‖^2 := by dsimp [B₀]; ring
+      _ ≤ B₀*(M^3*S) := mul_le_mul_of_nonneg_left hf hB
+      _ = _ := by ring
+  rw [g.calabiNorm_laplacian q hz]
+  have hp1 := (abs_le.mp (Complex.abs_re_le_norm (hermPair (g.calabiMetric q z) f (g.calabiRoughTerm q z)))).1
+  have hp2 := (abs_le.mp (Complex.abs_re_le_norm (hermPair (g.calabiMetric q z) f
+    (tensorConnection (-((g.matrix q z)⁻¹*g.curvatureRicci q z))*ᵥ f)))).1
+  change g.calabiEnergy q z-(1+2*A₀*(1+2*M^3)+B₀*M^3)*(S+1) ≤ _
+  change _ ≤ g.calabiEnergy q z+2*(hermPair (g.calabiMetric q z) f (g.calabiRoughTerm q z)).re+
+    (hermPair (g.calabiMetric q z) f (tensorConnection (-((g.matrix q z)⁻¹*g.curvatureRicci q z))*ᵥ f)).re
+  nlinarith [mul_nonneg hA hS, mul_nonneg hB (pow_nonneg hM 3),
+    mul_nonneg hA (pow_nonneg hM 3)]
+
+end Anticanonical.SourceSmooth.KaehlerMetric
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/CalabiMetricC1.lean b/lean/OAI/Geometry/PrescribedRicci/CalabiMetricC1.lean
new file mode 100644
index 0000000000000000000000000000000000000000..ccdb05449bf80d789624a91789e9ba17a4db2918
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/CalabiMetricC1.lean
@@ -0,0 +1,129 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.ArrayWirtinger
+import OAI.Geometry.PrescribedRicci.CalabiLower
+import OAI.Geometry.PrescribedRicci.CalabiTensorBounds
+import OAI.Geometry.PrescribedRicci.CompactPathBounds
+import OAI.Geometry.PrescribedRicci.UniformCalabi
+import OAI.Geometry.PrescribedPotential.VolumePath
+
+namespace OAI
+
+/-! Calabi Metric C1. -/
+
+section
+
+ 
+
+noncomputable section
+open Matrix Set Filter Topology
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace Anticanonical.SourceSmooth.KaehlerMetric
+open MongeAmpere
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+local notation "Mat" => Matrix (Fin d) (Fin d) ℂ
+
+lemma real_direction_of_hol_bar (f : Coordinates d → Mat) (z : Coordinates d) (a : Fin d) :
+    fderiv ℝ f z (coordinateVector a) = holDerivative f z a+barDerivative f z a := by
+  ext i j
+  simp only [holDerivative,barDerivative,Matrix.add_apply,Matrix.smul_apply,Matrix.sub_apply,smul_eq_mul]
+  ring
+
+lemma imaginary_direction_of_hol_bar (f : Coordinates d → Mat) (z : Coordinates d) (a : Fin d) :
+    fderiv ℝ f z (Complex.I • coordinateVector a) = Complex.I • (holDerivative f z a-barDerivative f z a) := by
+  ext i j
+  simp only [holDerivative,barDerivative,Matrix.add_apply,Matrix.smul_apply,Matrix.sub_apply,smul_eq_mul]
+  ring_nf
+  simp
+
+lemma norm_real_linear_map_bound {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F]
+    (L : Coordinates d →L[ℝ] F) {B : ℝ} (hB : 0 ≤ B)
+    (hre : ∀ a, ‖L (coordinateVector a)‖ ≤ B)
+    (him : ∀ a, ‖L (Complex.I • coordinateVector a)‖ ≤ B) :
+    ‖L‖ ≤ (2*(d:ℝ))*B := by
+  apply L.opNorm_le_bound (by positivity)
+  intro v
+  have he : v = ∑ j, ((v j).re • coordinateVector j + (v j).im • (Complex.I • coordinateVector j)) := by
+    ext k
+    simp only [coordinateVector,Finset.sum_apply,Pi.add_apply,Pi.smul_apply,smul_eq_mul,
+      Complex.real_smul,Pi.single_apply]
+    simp [mul_ite,Finset.sum_add_distrib]
+  calc
+    ‖L v‖ = ‖∑ j, ((v j).re • L (coordinateVector j)+(v j).im • L (Complex.I • coordinateVector j))‖ := by
+      conv_lhs => rw [he,map_sum]
+      simp only [map_add,map_smul]
+    _ ≤ ∑ j : Fin d, (‖(v j).re‖*‖L (coordinateVector j)‖+‖(v j).im‖*‖L (Complex.I • coordinateVector j)‖) := by
+      apply (norm_sum_le _ _).trans
+      apply Finset.sum_le_sum
+      intro j _
+      simpa only [norm_smul] using norm_add_le ((v j).re • L (coordinateVector j)) ((v j).im • L (Complex.I • coordinateVector j))
+    _ ≤ ∑ _j : Fin d, (‖v‖*B+‖v‖*B) := by
+      apply Finset.sum_le_sum
+      intro j _
+      exact add_le_add
+        (mul_le_mul ((Complex.abs_re_le_norm _).trans (norm_le_pi_norm v j)) (hre j) (norm_nonneg _) (norm_nonneg _))
+        (mul_le_mul ((Complex.abs_im_le_norm _).trans (norm_le_pi_norm v j)) (him j) (norm_nonneg _) (norm_nonneg _))
+    _ = _ := by simp only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul]; ring
+
+lemma metric_fderiv_bound (g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) {M P : ℝ} (hM : 0 ≤ M) (hP : 0 ≤ P)
+    (hH : ‖g.matrix q z‖ ≤ M) (hI : ‖(g.matrix q z)⁻¹‖ ≤ M)
+    (hCal : g.calabiNorm q z ≤ P) (i j : Fin d) :
+    ‖fderiv ℝ (fun y => g.matrix q y i j) z‖ ≤ 4*(d:ℝ)^2*M*(1+M^3*P) := by
+  let B := 1+M^3*P
+  have hT2 := (g.calabiTensor_sq_bound q hz hM hH hI).trans
+    (mul_le_mul_of_nonneg_left hCal (pow_nonneg hM 3))
+  have hT : ‖g.calabiTensor q z‖ ≤ B := by
+    dsimp [B]
+    nlinarith only [hT2,sq_nonneg (‖g.calabiTensor q z‖-1)]
+  have hg : DifferentiableAt ℝ (g.matrix q) z :=
+    ((g.smooth q).contDiffAt ((A.chart q).open_target.mem_nhds hz)).differentiableAt (by simp)
+  have hhol (a : Fin d) : ‖holDerivative (g.matrix q) z a‖ ≤ (d:ℝ)*M*B := by
+    rw [← holArray_eq_holDerivative hg,g.metric_hol_connection q hz]
+    apply (matrix_norm_mul_le _ _).trans
+    simp only [Fintype.card_fin]
+    exact mul_le_mul
+      (mul_le_mul_of_nonneg_left hH (Nat.cast_nonneg d))
+      ((g.chernConnection_norm_le_calabiTensor q z a).trans hT) (norm_nonneg _) (by positivity)
+  have hbar (a : Fin d) : ‖barDerivative (g.matrix q) z a‖ ≤ (d:ℝ)*M*B := by
+    rw [← barArray_eq_barDerivative hg,barArray_eq_adjoint_hol,
+      Matrix.norm_conjTranspose,holArray_eq_holDerivative hg]
+    · exact hhol a
+    · filter_upwards [(A.chart q).open_target.mem_nhds hz] with y hy
+      exact (g.positive q y hy).isHermitian
+  have hB : 0 ≤ 2*(d:ℝ)*M*B := by dsimp [B]; positivity
+  have hre (a : Fin d) : ‖fderiv ℝ (fun y => g.matrix q y i j) z (coordinateVector a)‖ ≤ 2*(d:ℝ)*M*B := by
+    rw [fderiv_entry hg,real_direction_of_hol_bar]
+    exact (norm_entry_le_entrywise_sup_norm _).trans ((norm_add_le _ _).trans (by linarith only [hhol a,hbar a]))
+  have him (a : Fin d) : ‖fderiv ℝ (fun y => g.matrix q y i j) z (Complex.I • coordinateVector a)‖ ≤ 2*(d:ℝ)*M*B := by
+    rw [fderiv_entry hg,imaginary_direction_of_hol_bar]
+    apply (norm_entry_le_entrywise_sup_norm _).trans
+    rw [norm_smul,Complex.norm_I,one_mul]
+    exact (norm_sub_le _ _).trans (by linarith only [hhol a,hbar a])
+  have hh := norm_real_linear_map_bound (fderiv ℝ (fun y => g.matrix q y i j) z) hB hre him
+  convert hh using 1
+  dsimp only [B]
+  ring
+
+variable [T2Space X] [CompactSpace X] [ConnectedSpace X]
+
+theorem volumePath_metric_C1_on_compact (g : KaehlerMetric A)
+    (line : SemipositiveAnticanonicalMetric A) (hd : 2 ≤ d) (q : Fin A.count)
+    {K : Set (Coordinates d)} (hK : IsCompact K) (hKt : K ⊆ (A.chart q).target) :
+    ∃ P : ℝ, 0 < P ∧ ∀ (φ : SmoothRealFunction A) (hp : g.PositivePotential φ),
+      g.integral φ.value = 0 → ∀ (t b : ℝ), t ∈ Icc (0:ℝ) 1 →
+      (∀ x, (g.logRatio (g.deform φ hp)).value x = t*(prescribedForcing g line).value x+b) →
+      ∀ z ∈ K, ∀ i j : Fin d, ‖fderiv ℝ (fun y => (g.deform φ hp).matrix q y i j) z‖ ≤ P := by
+  obtain ⟨M,hM,hmetric⟩ := g.volumePath_metric_bounds_on_compact line hd q hK hKt
+  obtain ⟨P,hP,hcal⟩ := g.volumePath_calabi_bound_on_compact line hd q hK hKt
+  refine ⟨1+4*(d:ℝ)^2*M*(1+M^3*P),by positivity,?_⟩
+  intro φ hp hm t b ht heq z hz i j
+  have hb := hmetric φ hp hm t b ht heq z hz
+  have hh := metric_fderiv_bound (g.deform φ hp) q (hKt hz) hM.le hP.le hb.1 hb.2
+    (hcal φ hp hm t b ht heq z hz) i j
+  linarith only [hh]
+end Anticanonical.SourceSmooth.KaehlerMetric
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/CalabiRough.lean b/lean/OAI/Geometry/PrescribedRicci/CalabiRough.lean
new file mode 100644
index 0000000000000000000000000000000000000000..195413d1d2981bb19d4ba6444edeb0983ce71ce8
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/CalabiRough.lean
@@ -0,0 +1,162 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.CalabiTensorAlgebra
+import OAI.Geometry.PrescribedRicci.CalabiTensorMetric
+import OAI.Geometry.PrescribedRicci.ChernBianchi
+import OAI.Geometry.PrescribedRicci.ChernPairDifferential
+import OAI.Geometry.PrescribedRicci.ChernRough
+import OAI.Geometry.PrescribedRicci.KahlerGradientEnergy
+import OAI.Geometry.PrescribedRicci.MatrixWirtinger
+import OAI.Geometry.PrescribedRicci.MatrixWirtingerExtra
+
+namespace OAI
+
+/-! Calabi Rough. -/
+
+section
+
+ 
+
+noncomputable section
+open Matrix Filter Set Topology
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace Anticanonical.SourceSmooth.KaehlerMetric
+open MongeAmpere
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+local notation "Mat" => Matrix (Fin d) (Fin d) ℂ
+local notation "TI" => TensorIndex (Fin d)
+
+lemma holDeriv_neg (f : Coordinates d → ℂ) (z : Coordinates d) (a : Fin d) :
+    holDeriv (fun y => -f y) z a = -holDeriv f z a := by
+  unfold holDeriv
+  rw [fderiv_fun_neg]
+  simp only [_root_.neg_apply]
+  ring
+
+lemma barDeriv_neg (f : Coordinates d → ℂ) (z : Coordinates d) (a : Fin d) :
+    barDeriv (fun y => -f y) z a = -barDeriv f z a := by
+  unfold barDeriv
+  rw [fderiv_fun_neg]
+  simp only [_root_.neg_apply]
+  ring
+
+lemma calabiTensor_bar (g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) (b : Fin d) (i : TI) :
+    barVector (g.calabiTensor q) z b i = -g.chernCurvature q z b i.1 i.2.1 i.2.2 := by
+  change barDeriv (fun y => g.chernConnection q y i.1 i.2.1 i.2.2) z b = _
+  rw [barDeriv_entry ((g.chernConnection_smooth q hz i.1).differentiableAt (by simp)),
+    g.chernConnection_bar q hz]
+  rfl
+
+lemma calabiTensor_covHol_bar (g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) (u b a k j : Fin d) :
+    covHol (g.calabiConnection q) (fun y => barVector (g.calabiTensor q) y b) z u (a,k,j) =
+      -(holDerivative (fun y => g.chernCurvature q y b a) z u+
+        g.chernConnection q z u*g.chernCurvature q z b a-
+        g.chernCurvature q z b a*g.chernConnection q z u-
+        ∑ v, g.chernConnection q z u v a • g.chernCurvature q z b v) k j := by
+  have he : (fun y => barVector (g.calabiTensor q) y b (a,k,j)) =ᶠ[nhds z]
+      (fun y => -g.chernCurvature q y b a k j) := by
+    filter_upwards [(A.chart q).open_target.mem_nhds hz] with y hy
+    exact g.calabiTensor_bar q hy b (a,k,j)
+  change holDeriv (fun y => barVector (g.calabiTensor q) y b (a,k,j)) z u+
+    (tensorConnection (g.chernConnection q z u)*ᵥ barVector (g.calabiTensor q) z b) (a,k,j) = _
+  rw [holDeriv_congr he,holDeriv_neg,
+    holDeriv_entry ((g.chernCurvature_smooth q hz b a).differentiableAt (by simp)),
+    tensorConnection_mulVec]
+  simp_rw [g.calabiTensor_bar q hz]
+  simp only [Matrix.add_apply,Matrix.sub_apply,Matrix.mul_apply,Matrix.sum_apply,
+    Matrix.smul_apply,smul_eq_mul,mul_neg,Finset.sum_neg_distrib]
+  simp only [mul_comm]
+  ring
+
+lemma calabiTensor_rough (g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) :
+    ∑ u, ∑ b, ((g.matrix q z)⁻¹ u b) •
+      covHol (g.calabiConnection q) (fun y => barVector (g.calabiTensor q) y b) z u =
+      fun i : TI => (-(g.matrix q z)⁻¹*holDerivative (g.curvatureRicci q) z i.1+
+        ((g.matrix q z)⁻¹*g.curvatureRicci q z)*g.chernConnection q z i.1) i.2.1 i.2.2 := by
+  funext ⟨a,k,j⟩
+  simp only [Finset.sum_apply,Pi.smul_apply,smul_eq_mul]
+  simp_rw [g.calabiTensor_covHol_bar q hz]
+  have he := congrArg (fun M : Mat => -(M k j)) (g.chernCurvature_rough q hz a)
+  simp only [Matrix.sum_apply,Matrix.smul_apply,Matrix.sub_apply,Matrix.add_apply,
+    Matrix.neg_apply,neg_mul,smul_eq_mul,mul_neg,Finset.sum_neg_distrib] at he ⊢
+  convert he using 1
+  ring
+
+lemma barDeriv_mul_const {f : Coordinates d → ℂ} {z : Coordinates d}
+    (hf : DifferentiableAt ℝ f z) (c : ℂ) (b : Fin d) :
+    barDeriv (fun y => f y*c) z b = barDeriv f z b*c := by
+  rw [barDeriv_mul hf (differentiableAt_const c)]
+  simp [barDeriv]
+
+lemma barDeriv_const_mul {f : Coordinates d → ℂ} {z : Coordinates d}
+    (hf : DifferentiableAt ℝ f z) (c : ℂ) (b : Fin d) :
+    barDeriv (fun y => c*f y) z b = c*barDeriv f z b := by
+  rw [barDeriv_mul (differentiableAt_const c) hf]
+  simp [barDeriv]
+
+lemma barArray_tensorConnection {G : Coordinates d → Mat} {z : Coordinates d}
+    (hG : DifferentiableAt ℝ G z) (b : Fin d) :
+    barArray (fun y => tensorConnection (G y)) z b = tensorConnection (barDerivative G z b) := by
+  have hg (i j : Fin d) := differentiableAt_pi.mp (differentiableAt_pi.mp hG i) j
+  have hconst (c : ℂ) : DifferentiableAt ℝ (fun _ : Coordinates d => c) z :=
+    differentiableAt_const c
+  ext i j
+  let f1 : Coordinates d → ℂ := fun y => -G y j.1 i.1*((1:Mat) i.2.1 j.2.1*(1:Mat) i.2.2 j.2.2)
+  let f2 : Coordinates d → ℂ := fun y => (1:Mat) i.1 j.1*(G y i.2.1 j.2.1*(1:Mat) i.2.2 j.2.2)
+  let f3 : Coordinates d → ℂ := fun y => (1:Mat) i.1 j.1*((1:Mat) i.2.1 j.2.1*(-G y j.2.2 i.2.2))
+  have h1 : DifferentiableAt ℝ f1 z := (hg _ _).neg.fun_mul (hconst _)
+  have h2 : DifferentiableAt ℝ f2 z := (hconst _).fun_mul ((hg _ _).fun_mul (hconst _))
+  have h3 : DifferentiableAt ℝ f3 z := (hconst _).fun_mul ((hconst _).fun_mul (hg _ _).neg)
+  have h12 : DifferentiableAt ℝ (fun y => f1 y+f2 y) z := h1.fun_add h2
+  change barDeriv (fun y => f1 y+f2 y+f3 y) z b = _
+  rw [barDeriv_add h12 h3,barDeriv_add h1 h2]
+  dsimp only [f1,f2,f3]
+  rw [barDeriv_mul_const (f := fun y => -G y j.1 i.1) (hg _ _).neg,
+    barDeriv_const_mul (f := fun y => G y i.2.1 j.2.1*(1:Mat) i.2.2 j.2.2)
+      ((hg _ _).fun_mul (hconst _)),
+    barDeriv_mul_const (f := fun y => G y i.2.1 j.2.1) (hg _ _),
+    barDeriv_const_mul (f := fun y => (1:Mat) i.2.1 j.2.1*(-G y j.2.2 i.2.2))
+      ((hconst _).fun_mul (hg _ _).neg),
+    barDeriv_const_mul (f := fun y => -G y j.2.2 i.2.2) (hg _ _).neg]
+  simp only [barDeriv_neg,barDeriv_entry hG,Matrix.add_apply,tensorConnection,tensor3_apply,
+    Matrix.transpose_apply,Matrix.neg_apply]
+
+lemma tensorConnection_sum_smul {ι : Type*} [Fintype ι] (c : ι → ℂ) (M : ι → Mat) :
+    tensorConnection (∑ i, c i • M i) = ∑ i, c i • tensorConnection (M i) := by
+  ext a b
+  simp only [tensorConnection,tensor3_apply,Matrix.transpose_apply,Matrix.neg_apply,
+    Matrix.add_apply,Matrix.sum_apply,Matrix.smul_apply,smul_eq_mul,
+    Finset.mul_sum,Finset.sum_mul,mul_add,neg_mul,mul_neg,
+    ← Finset.sum_neg_distrib,← Finset.sum_add_distrib]
+  apply Finset.sum_congr rfl
+  intro i _
+  ring
+
+lemma calabiConnection_contracted_curvature (g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) :
+    ∑ a, ∑ b, ((g.matrix q z)⁻¹ a b) •
+      barArray (fun y => g.calabiConnection q y a) z b =
+        tensorConnection (-((g.matrix q z)⁻¹*g.curvatureRicci q z)) := by
+  have hb (a b : Fin d) : barArray (fun y => g.calabiConnection q y a) z b =
+      tensorConnection (-g.chernCurvature q z b a) := by
+    exact (barArray_tensorConnection
+      ((g.chernConnection_smooth q hz a).differentiableAt (by simp)) b).trans
+        (congrArg tensorConnection (g.chernConnection_bar q hz a b))
+  simp_rw [hb]
+  have hs := tensorConnection_sum_smul
+    (fun ab : Fin d × Fin d => (g.matrix q z)⁻¹ ab.1 ab.2)
+    (fun ab : Fin d × Fin d => -g.chernCurvature q z ab.2 ab.1)
+  simp only [Fintype.sum_prod_type] at hs
+  rw [← hs]
+  congr 1
+  simp only [smul_neg,Finset.sum_neg_distrib,
+    g.chernCurvature_contraction q hz]
+
+end Anticanonical.SourceSmooth.KaehlerMetric
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/CalabiTensorAlgebra.lean b/lean/OAI/Geometry/PrescribedRicci/CalabiTensorAlgebra.lean
new file mode 100644
index 0000000000000000000000000000000000000000..eb202ee197afc340c54a65bdd0bd825a9b51d241
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/CalabiTensorAlgebra.lean
@@ -0,0 +1,77 @@
+import Mathlib
+
+namespace OAI
+
+/-! Calabi Tensor Algebra. -/
+
+noncomputable section
+open Matrix
+open scoped Kronecker ComplexOrder MatrixOrder
+namespace MongeAmpere
+variable {n : Type*} [Fintype n] [DecidableEq n]
+
+abbrev TensorIndex (n : Type*) := n × (n × n)
+
+def tensor3 (L U V : Matrix n n ℂ) : Matrix (TensorIndex n) (TensorIndex n) ℂ :=
+  L ⊗ₖ (U ⊗ₖ V)
+
+omit [Fintype n] [DecidableEq n] in
+lemma tensor3_apply (L U V : Matrix n n ℂ) (i j : TensorIndex n) :
+    tensor3 L U V i j = L i.1 j.1*(U i.2.1 j.2.1*V i.2.2 j.2.2) := rfl
+
+omit [DecidableEq n] in
+lemma tensor3_mul (L U V L' U' V' : Matrix n n ℂ) :
+    tensor3 L U V*tensor3 L' U' V' = tensor3 (L*L') (U*U') (V*V') := by
+  simp only [tensor3,← Matrix.mul_kronecker_mul]
+
+omit [Fintype n] in
+lemma tensor3_one : tensor3 (1 : Matrix n n ℂ) 1 1 = 1 := by
+  simp only [tensor3,Matrix.one_kronecker_one]
+
+lemma tensor3_mulVec_left (L : Matrix n n ℂ) (T : TensorIndex n → ℂ) (a k j : n) :
+    (tensor3 L 1 1*ᵥ T) (a,k,j) = ∑ v, L a v*T (v,k,j) := by
+  classical
+  simp [Matrix.mulVec,dotProduct,Fintype.sum_prod_type,tensor3_apply,
+    Matrix.one_apply,mul_ite,ite_mul]
+
+lemma tensor3_mulVec_middle (U : Matrix n n ℂ) (T : TensorIndex n → ℂ) (a k j : n) :
+    (tensor3 1 U 1*ᵥ T) (a,k,j) = ∑ v, U k v*T (a,v,j) := by
+  classical
+  simp [Matrix.mulVec,dotProduct,Fintype.sum_prod_type,tensor3_apply,
+    Matrix.one_apply,mul_ite,ite_mul]
+
+lemma tensor3_mulVec_right (V : Matrix n n ℂ) (T : TensorIndex n → ℂ) (a k j : n) :
+    (tensor3 1 1 V*ᵥ T) (a,k,j) = ∑ v, V j v*T (a,k,v) := by
+  classical
+  simp [Matrix.mulVec,dotProduct,Fintype.sum_prod_type,tensor3_apply,
+    Matrix.one_apply,mul_ite,ite_mul]
+
+def tensorMetric (H : Matrix n n ℂ) : Matrix (TensorIndex n) (TensorIndex n) ℂ :=
+  tensor3 H⁻¹.transpose H H⁻¹.transpose
+
+def tensorConnection (G : Matrix n n ℂ) : Matrix (TensorIndex n) (TensorIndex n) ℂ :=
+  tensor3 (-G.transpose) 1 1+tensor3 1 G 1+tensor3 1 1 (-G.transpose)
+
+lemma tensorMetric_posDef {H : Matrix n n ℂ} (hH : H.PosDef) : (tensorMetric H).PosDef :=
+  hH.inv.transpose.kronecker (hH.kronecker hH.inv.transpose)
+
+lemma tensorConnection_mulVec (G : Matrix n n ℂ) (T : TensorIndex n → ℂ) (a k j : n) :
+    (tensorConnection G*ᵥ T) (a,k,j) = ∑ v, G k v*T (a,v,j)-
+      (∑ v, G v a*T (v,k,j))-(∑ v, G v j*T (a,k,v)) := by
+  simp only [tensorConnection,Matrix.add_mulVec,Pi.add_apply,tensor3_mulVec_left,
+    tensor3_mulVec_middle,tensor3_mulVec_right,Matrix.neg_apply,Matrix.transpose_apply,
+    neg_mul,Finset.sum_neg_distrib]
+  ring
+
+lemma tensorMetric_mul_connection (H G : Matrix n n ℂ) :
+    tensorMetric H*tensorConnection G =
+      tensor3 (H⁻¹.transpose*(-G.transpose)) H H⁻¹.transpose+
+      tensor3 H⁻¹.transpose (H*G) H⁻¹.transpose+
+      tensor3 H⁻¹.transpose H (H⁻¹.transpose*(-G.transpose)) := by
+  simp only [tensorMetric,tensorConnection,Matrix.mul_add,tensor3_mul,mul_one]
+
+end MongeAmpere
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/CalabiTensorBounds.lean b/lean/OAI/Geometry/PrescribedRicci/CalabiTensorBounds.lean
new file mode 100644
index 0000000000000000000000000000000000000000..202a69c733870abd144934df1281e6bc395c7b62
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/CalabiTensorBounds.lean
@@ -0,0 +1,104 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.CalabiTensorAlgebra
+import OAI.Geometry.PrescribedRicci.MatrixCompactBounds
+import OAI.Geometry.PrescribedRicci.MatrixComponentBounds
+
+namespace OAI
+
+/-! Calabi Tensor Bounds. -/
+
+noncomputable section
+open Matrix
+open scoped ComplexOrder MatrixOrder Matrix.Norms.Elementwise
+namespace MongeAmpere
+variable {n : Type*} [Fintype n] [DecidableEq n]
+
+omit [DecidableEq n] in
+lemma tensor3_norm_le (L U V : Matrix n n ℂ) :
+    ‖tensor3 L U V‖ ≤ ‖L‖*‖U‖*‖V‖ := by
+  apply matrix_norm_le_of_entries (by positivity)
+  intro i j
+  rw [tensor3_apply,norm_mul,norm_mul]
+  calc
+    _ ≤ ‖L‖*(‖U‖*‖V‖) := mul_le_mul (norm_entry_le_entrywise_sup_norm L)
+      (mul_le_mul (norm_entry_le_entrywise_sup_norm U) (norm_entry_le_entrywise_sup_norm V)
+        (norm_nonneg _) (norm_nonneg _)) (mul_nonneg (norm_nonneg _) (norm_nonneg _)) (norm_nonneg _)
+    _ = _ := by ring
+
+lemma tensorMetric_inv {H : Matrix n n ℂ} (hH : H.PosDef) :
+    (tensorMetric H)⁻¹ = tensor3 H.transpose H⁻¹ H.transpose := by
+  apply Matrix.inv_eq_right_inv
+  rw [tensorMetric,tensor3_mul,← Matrix.transpose_mul,
+    Matrix.mul_nonsing_inv _ (isUnit_iff_ne_zero.mpr hH.det_pos.ne'),
+    Matrix.transpose_one,tensor3_one]
+
+lemma tensorMetric_norm_bound {H : Matrix n n ℂ} {M : ℝ}
+    (hM : 0 ≤ M) (hH : ‖H‖ ≤ M) (hI : ‖H⁻¹‖ ≤ M) : ‖tensorMetric H‖ ≤ M^3 := by
+  apply (tensor3_norm_le _ _ _).trans
+  rw [Matrix.norm_transpose]
+  calc
+    _ ≤ M*M*M := mul_le_mul
+      (mul_le_mul hI hH (norm_nonneg _) hM) hI (norm_nonneg _) (mul_nonneg hM hM)
+    _ = _ := by ring
+
+lemma tensorMetric_inv_norm_bound {H : Matrix n n ℂ} (hHp : H.PosDef) {M : ℝ}
+    (hM : 0 ≤ M) (hH : ‖H‖ ≤ M) (hI : ‖H⁻¹‖ ≤ M) : ‖(tensorMetric H)⁻¹‖ ≤ M^3 := by
+  rw [tensorMetric_inv hHp]
+  apply (tensor3_norm_le _ _ _).trans
+  rw [Matrix.norm_transpose]
+  calc
+    _ ≤ M*M*M := mul_le_mul
+      (mul_le_mul hH hI (norm_nonneg _) hM) hH (norm_nonneg _) (mul_nonneg hM hM)
+    _ = _ := by ring
+
+lemma tensorConnection_norm_le (G : Matrix n n ℂ) : ‖tensorConnection G‖ ≤ 3*‖G‖ := by
+  have hone : ‖(1 : Matrix n n ℂ)‖ ≤ 1 := by
+    apply matrix_norm_le_of_entries zero_le_one
+    intro i j
+    simp only [Matrix.one_apply]
+    split_ifs <;> simp
+  have h1 : ‖tensor3 (-G.transpose) 1 1‖ ≤ ‖G‖ := by
+    apply (tensor3_norm_le _ _ _).trans
+    simp only [norm_neg,Matrix.norm_transpose]
+    nlinarith [norm_nonneg (1 : Matrix n n ℂ),norm_nonneg G,
+      mul_le_mul_of_nonneg_left hone (norm_nonneg G),
+      mul_le_mul_of_nonneg_left hone (mul_nonneg (norm_nonneg G) (norm_nonneg (1:Matrix n n ℂ)))]
+  have h2 : ‖tensor3 1 G 1‖ ≤ ‖G‖ := by
+    apply (tensor3_norm_le _ _ _).trans
+    nlinarith [norm_nonneg (1 : Matrix n n ℂ),norm_nonneg G,
+      mul_le_mul_of_nonneg_right hone (norm_nonneg G),
+      mul_le_mul_of_nonneg_left hone (mul_nonneg (norm_nonneg (1:Matrix n n ℂ)) (norm_nonneg G))]
+  have h3 : ‖tensor3 1 1 (-G.transpose)‖ ≤ ‖G‖ := by
+    apply (tensor3_norm_le _ _ _).trans
+    simp only [norm_neg,Matrix.norm_transpose]
+    have ho : ‖(1 : Matrix n n ℂ)‖*‖(1 : Matrix n n ℂ)‖ ≤ 1 :=
+      (mul_le_mul hone hone (norm_nonneg _) zero_le_one).trans_eq (one_mul 1)
+    simpa using mul_le_mul_of_nonneg_right ho (norm_nonneg G)
+  exact (norm_add_le _ _).trans ((add_le_add ((norm_add_le _ _).trans
+    (add_le_add h1 h2)) h3).trans_eq (by ring))
+
+omit [DecidableEq n] in
+lemma matrix_norm_mul_le (A B : Matrix n n ℂ) :
+    ‖A*B‖ ≤ (Fintype.card n:ℝ)*‖A‖*‖B‖ := by
+  exact matrix_norm_le_of_entries (by positivity)
+    (entry_mul_bound (norm_nonneg A) (fun _ _ => norm_entry_le_entrywise_sup_norm A)
+      (fun _ _ => norm_entry_le_entrywise_sup_norm B))
+
+omit [DecidableEq n] in
+lemma mulVec_norm_le (A : Matrix n n ℂ) (v : n → ℂ) :
+    ‖A*ᵥ v‖ ≤ (Fintype.card n:ℝ)*‖A‖*‖v‖ := by
+  apply (pi_norm_le_iff_of_nonneg (by positivity)).mpr
+  intro i
+  calc
+    _ ≤ ∑ j, ‖A i j*v j‖ := norm_sum_le _ _
+    _ ≤ ∑ _j : n, ‖A‖*‖v‖ := Finset.sum_le_sum (fun j _ => by
+      rw [norm_mul]
+      exact mul_le_mul (norm_entry_le_entrywise_sup_norm A) (norm_le_pi_norm v j)
+        (norm_nonneg _) (norm_nonneg _))
+    _ = _ := by simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul]; ring
+
+end MongeAmpere
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/CalabiTensorMetric.lean b/lean/OAI/Geometry/PrescribedRicci/CalabiTensorMetric.lean
new file mode 100644
index 0000000000000000000000000000000000000000..2d29eb7508fbcb57424cf5a088c04b87f027b440
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/CalabiTensorMetric.lean
@@ -0,0 +1,117 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.ArrayWirtinger
+import OAI.Geometry.PrescribedRicci.ChernConnection
+
+namespace OAI
+
+/-! Calabi Tensor Metric. -/
+
+noncomputable section
+open Matrix Filter Set Topology
+open scoped ContDiff ComplexOrder Kronecker Matrix.Norms.Elementwise
+namespace Anticanonical.SourceSmooth.KaehlerMetric
+open MongeAmpere
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+local notation "Mat" => Matrix (Fin d) (Fin d) ℂ
+local notation "TI" => TensorIndex (Fin d)
+
+def calabiTensor (g : KaehlerMetric A) (q : Fin A.count) (z : Coordinates d) : TI → ℂ :=
+  fun v => g.chernConnection q z v.1 v.2.1 v.2.2
+
+def calabiMetric (g : KaehlerMetric A) (q : Fin A.count) (z : Coordinates d) : Matrix TI TI ℂ :=
+  tensorMetric (g.matrix q z)
+
+def calabiConnection (g : KaehlerMetric A) (q : Fin A.count) (z : Coordinates d)
+    (a : Fin d) : Matrix TI TI ℂ := tensorConnection (g.chernConnection q z a)
+
+def calabiNorm (g : KaehlerMetric A) (q : Fin A.count) (z : Coordinates d) : ℝ :=
+  (hermPair (g.calabiMetric q z) (g.calabiTensor q z) (g.calabiTensor q z)).re
+
+lemma calabiTensor_smooth (g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) (i : TI) :
+    ContDiffAt ℝ ∞ (fun y => g.calabiTensor q y i) z :=
+  contDiffAt_pi.mp (contDiffAt_pi.mp (g.chernConnection_smooth q hz i.1) i.2.1) i.2.2
+
+lemma calabiMetric_posDef (g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) : (g.calabiMetric q z).PosDef :=
+  tensorMetric_posDef (g.positive q z hz)
+
+lemma calabiNorm_nonneg (g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) : 0 ≤ g.calabiNorm q z :=
+  hermPair_nonneg (g.calabiMetric_posDef q hz).posSemidef _
+
+lemma calabiMetric_smooth (g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) (i j : TI) :
+    ContDiffAt ℝ ∞ (fun y => g.calabiMetric q y i j) z := by
+  have hg := (g.smooth q).contDiffAt ((A.chart q).open_target.mem_nhds hz)
+  have hi : ContDiffAt ℝ ∞ (fun y => (g.matrix q y)⁻¹) z :=
+    ((MongeAmpere.contDiffAt_inv _ (g.positive q z hz).det_pos.ne').restrict_scalars ℝ).comp z hg
+  exact (contDiffAt_pi.mp (contDiffAt_pi.mp hi j.1) i.1).mul
+    ((contDiffAt_pi.mp (contDiffAt_pi.mp hg i.2.1) j.2.1).mul
+      (contDiffAt_pi.mp (contDiffAt_pi.mp hi j.2.2) i.2.2))
+
+lemma calabiConnection_smooth (g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) (a : Fin d) (i j : TI) :
+    ContDiffAt ℝ ∞ (fun y => g.calabiConnection q y a i j) z := by
+  have hc (u v : Fin d) : ContDiffAt ℝ ∞ (fun y => g.chernConnection q y a u v) z :=
+    contDiffAt_pi.mp (contDiffAt_pi.mp (g.chernConnection_smooth q hz a) u) v
+  exact (((hc j.1 i.1).neg.mul (contDiffAt_const.mul contDiffAt_const)).add
+    (contDiffAt_const.mul ((hc i.2.1 j.2.1).mul contDiffAt_const))).add
+    (contDiffAt_const.mul (contDiffAt_const.mul (hc j.2.2 i.2.2).neg))
+
+lemma calabiNorm_smooth (g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) :
+    ContDiffAt ℝ ∞ (g.calabiNorm q) z :=
+  Complex.reCLM.contDiff.contDiffAt.comp z
+    (hermPair_smooth (g.calabiMetric_smooth q hz) (g.calabiTensor_smooth q hz) (g.calabiTensor_smooth q hz))
+
+lemma metric_hol_connection (g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) (a : Fin d) :
+    holArray (g.matrix q) z a = g.matrix q z*g.chernConnection q z a := by
+  rw [holArray_eq_holDerivative
+    (((g.smooth q).contDiffAt ((A.chart q).open_target.mem_nhds hz)).differentiableAt (by simp))]
+  rw [chernConnection,← mul_assoc,Matrix.mul_nonsing_inv _
+      (isUnit_iff_ne_zero.mpr (g.positive q z hz).det_pos.ne'),one_mul]
+
+lemma inverse_metric_hol_connection (g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) (a : Fin d) :
+    holArray (fun y => (g.matrix q y)⁻¹.transpose) z a =
+      (g.matrix q z)⁻¹.transpose*(-(g.chernConnection q z a).transpose) := by
+  have hg := (g.smooth q).contDiffAt ((A.chart q).open_target.mem_nhds hz)
+  have hi : ContDiffAt ℝ ∞ (fun y => (g.matrix q y)⁻¹) z :=
+    ((MongeAmpere.contDiffAt_inv _ (g.positive q z hz).det_pos.ne').restrict_scalars ℝ).comp z hg
+  rw [holArray_transpose,holArray_eq_holDerivative (hi.differentiableAt (by simp)),
+    holDerivative_inv (hg.differentiableAt (by simp))
+      (isUnit_iff_ne_zero.mpr (g.positive q z hz).det_pos.ne')]
+  simp only [chernConnection,Matrix.transpose_neg,Matrix.transpose_mul,mul_neg,mul_assoc]
+
+lemma calabiMetric_hol (g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) (a : Fin d) :
+    holArray (g.calabiMetric q) z a = g.calabiMetric q z*g.calabiConnection q z a := by
+  have hg := (g.smooth q).contDiffAt ((A.chart q).open_target.mem_nhds hz)
+  have hi : ContDiffAt ℝ ∞ (fun y => (g.matrix q y)⁻¹) z :=
+    ((MongeAmpere.contDiffAt_inv _ (g.positive q z hz).det_pos.ne').restrict_scalars ℝ).comp z hg
+  have hg' (i j : Fin d) : DifferentiableAt ℝ (fun y => g.matrix q y i j) z :=
+    (contDiffAt_pi.mp (contDiffAt_pi.mp hg i) j).differentiableAt (by simp)
+  have hi' (i j : Fin d) : DifferentiableAt ℝ (fun y => (g.matrix q y)⁻¹.transpose i j) z :=
+    (contDiffAt_pi.mp (contDiffAt_pi.mp hi j) i).differentiableAt (by simp)
+  change holArray (fun y => tensor3 (g.matrix q y)⁻¹.transpose (g.matrix q y)
+    (g.matrix q y)⁻¹.transpose) z a = _
+  rw [holArray_tensor3 hi' hg' hi',g.metric_hol_connection q hz,
+    g.inverse_metric_hol_connection q hz]
+  exact (tensorMetric_mul_connection _ _).symm
+
+lemma calabiMetric_bar (g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) (a : Fin d) :
+    barArray (g.calabiMetric q) z a = (g.calabiConnection q z a)ᴴ*g.calabiMetric q z := by
+  have hh : ∀ᶠ y in nhds z, (g.calabiMetric q y).IsHermitian := by
+    filter_upwards [(A.chart q).open_target.mem_nhds hz] with y hy
+    exact (g.calabiMetric_posDef q hy).isHermitian
+  rw [barArray_eq_adjoint_hol hh,g.calabiMetric_hol q hz,Matrix.conjTranspose_mul,
+    (g.calabiMetric_posDef q hz).isHermitian.eq]
+
+end Anticanonical.SourceSmooth.KaehlerMetric
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/CalabiTrace.lean b/lean/OAI/Geometry/PrescribedRicci/CalabiTrace.lean
new file mode 100644
index 0000000000000000000000000000000000000000..eae68cf853a505cc88d044dfcac5cc8a6cf3ec00
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/CalabiTrace.lean
@@ -0,0 +1,113 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.CalabiTensorAlgebra
+import OAI.Geometry.PrescribedRicci.CalabiTensorMetric
+import OAI.Geometry.PrescribedRicci.ChernConnection
+import OAI.Geometry.PrescribedRicci.HermitianTensorEnergy
+import OAI.Geometry.PrescribedRicci.KahlerGradientEnergy
+
+namespace OAI
+
+/-! Calabi Trace. -/
+
+noncomputable section
+open Matrix Set Filter Topology
+open scoped ComplexOrder ContDiff MatrixOrder Kronecker Matrix.Norms.Elementwise
+namespace MongeAmpere
+variable {n : Type*} [Fintype n] [DecidableEq n]
+lemma hermPair_kronecker_one_trace (H U V : Matrix n n ℂ) :
+    hermPair (H ⊗ₖ (1 : Matrix n n ℂ)) (fun p => U p.1 p.2) (fun p => V p.1 p.2) =
+      (Uᴴ*H*V).trace := by
+  classical
+  simp only [hermPair_expand,Fintype.sum_prod_type,Matrix.kronecker_apply,Matrix.one_apply,
+    mul_ite,ite_mul,mul_one,mul_zero,zero_mul,Finset.sum_ite_eq,Finset.mem_univ,ite_true,
+    Matrix.trace,Matrix.diag_apply,Matrix.mul_apply,Matrix.conjTranspose_apply,Finset.sum_mul]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro j _
+  rw [Finset.sum_comm]
+
+omit [DecidableEq n] in
+lemma trace_real {H : Matrix n n ℂ} (hH : H.IsHermitian) : (H.trace.re : ℂ) = H.trace := by
+  have he : star H.trace = H.trace := by rw [← trace_conjTranspose,hH.eq]
+  have hi := congrArg Complex.im he
+  apply Complex.ext
+  · rfl
+  · simp only [Complex.ofReal_im]
+    simp only [Complex.star_def,Complex.conj_im] at hi
+    linarith
+end MongeAmpere
+namespace Anticanonical.SourceSmooth.KaehlerMetric
+open MongeAmpere
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+local notation "TI" => TensorIndex (Fin d)
+
+def flatTrace (g : KaehlerMetric A) (q : Fin A.count) (z : Coordinates d) : ℝ :=
+  (g.matrix q z).trace.re
+
+lemma flatTrace_smooth (g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) : ContDiffAt ℝ ∞ (g.flatTrace q) z := by
+  have hs := (g.smooth q).contDiffAt ((A.chart q).open_target.mem_nhds hz)
+  exact Complex.reCLM.contDiff.contDiffAt.comp z (ContDiffAt.sum (fun i _ =>
+    contDiffAt_pi.mp (contDiffAt_pi.mp hs i) i))
+
+lemma flatTrace_hol (g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) (a : Fin d) :
+    holRealDeriv (g.flatTrace q) z a = (holDerivative (g.matrix q) z a).trace := by
+  rw [← holDeriv_ofReal ((g.flatTrace_smooth q hz).differentiableAt (by simp))]
+  have he : (fun y => (g.flatTrace q y : ℂ)) =ᶠ[nhds z] fun y => (g.matrix q y).trace := by
+    filter_upwards [(A.chart q).open_target.mem_nhds hz] with y hy
+    exact trace_real (g.positive q y hy).isHermitian
+  rw [holDeriv_congr he,holDeriv_trace
+    (((g.smooth q).contDiffAt ((A.chart q).open_target.mem_nhds hz)).differentiableAt (by simp))]
+
+lemma flatTrace_hessian (g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) (b a : Fin d) :
+    PotentialKaehler.potentialMatrix (g.flatTrace q) z b a =
+      ((g.chernConnection q z b)ᴴ*g.matrix q z*g.chernConnection q z a).trace-
+        (g.curvatureMatrix q z b a).trace := by
+  rw [← barDeriv_holRealDeriv (g.flatTrace_smooth q hz)]
+  have he : (fun y => holRealDeriv (g.flatTrace q) y a) =ᶠ[nhds z]
+      fun y => (holDerivative (g.matrix q) y a).trace := by
+    filter_upwards [(A.chart q).open_target.mem_nhds hz] with y hy
+    exact g.flatTrace_hol q hy a
+  rw [barDeriv_congr he,barDeriv_trace ((contDiffAt_holDerivative
+    ((g.smooth q).contDiffAt ((A.chart q).open_target.mem_nhds hz)) a).differentiableAt (by simp))]
+  have hp : (g.chernConnection q z b)ᴴ*g.matrix q z*g.chernConnection q z a =
+      barDerivative (g.matrix q) z b*(g.matrix q z)⁻¹*holDerivative (g.matrix q) z a := by
+    simp only [chernConnection,Matrix.conjTranspose_mul,g.holDerivative_adjoint q hz,
+      (g.positive q z hz).inv.isHermitian.eq]
+    rw [mul_assoc,Matrix.mul_nonsing_inv_cancel_left _ _
+      (isUnit_iff_ne_zero.mpr (g.positive q z hz).det_pos.ne')]
+  rw [hp]
+  simp only [curvatureMatrix,trace_add,trace_neg]
+  abel
+
+lemma flatTrace_laplacian (g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) :
+    ((g.matrix q z)⁻¹*PotentialKaehler.potentialMatrix (g.flatTrace q) z).trace.re =
+      (hermPair (tensor3 (g.matrix q z)⁻¹.transpose (g.matrix q z) 1)
+        (g.calabiTensor q z) (g.calabiTensor q z)).re-(g.curvatureRicci q z).trace.re := by
+  have he : ((g.matrix q z)⁻¹*PotentialKaehler.potentialMatrix (g.flatTrace q) z).trace =
+      hermPair (tensor3 (g.matrix q z)⁻¹.transpose (g.matrix q z) 1)
+        (g.calabiTensor q z) (g.calabiTensor q z)-(g.curvatureRicci q z).trace := by
+    change (∑ u, ∑ b, (g.matrix q z)⁻¹ u b*PotentialKaehler.potentialMatrix (g.flatTrace q) z b u) = _
+    simp_rw [g.flatTrace_hessian q hz]
+    simp only [mul_sub,Finset.sum_sub_distrib]
+    congr 1
+    · have ht := tensorEnergy_eq (g.matrix q z)⁻¹.transpose
+        ((g.matrix q z) ⊗ₖ (1 : Matrix (Fin d) (Fin d) ℂ))
+        (fun a (p : Fin d × Fin d) => g.chernConnection q z a p.1 p.2)
+      change _ = hermPair (tensor3 (g.matrix q z)⁻¹.transpose (g.matrix q z) 1)
+        (g.calabiTensor q z) (g.calabiTensor q z) at ht
+      rw [← ht,← tensorEnergy_flip]
+      simp only [hermPair_kronecker_one_trace]
+    · have hh := congrArg Matrix.trace (g.curvature_contraction q hz)
+      simp only [trace_sum,trace_smul,smul_eq_mul] at hh
+      exact hh
+  exact congrArg Complex.re he
+
+end Anticanonical.SourceSmooth.KaehlerMetric
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/CalabiTraceBound.lean b/lean/OAI/Geometry/PrescribedRicci/CalabiTraceBound.lean
new file mode 100644
index 0000000000000000000000000000000000000000..8fae80e4bfc78064efb69e5e7af611178d0e264f
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/CalabiTraceBound.lean
@@ -0,0 +1,106 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.AnalyticSupport
+import OAI.Geometry.PrescribedRicci.CalabiTensorBounds
+import OAI.Geometry.PrescribedRicci.CalabiTrace
+import OAI.Geometry.PrescribedRicci.HermitianCoercivity
+
+namespace OAI
+
+/-! Calabi Trace Bound. -/
+
+noncomputable section
+open Matrix Set Filter Topology
+open scoped ComplexOrder ContDiff MatrixOrder Kronecker Matrix.Norms.Elementwise
+namespace MongeAmpere
+variable {n : Type*} [Fintype n] [DecidableEq n]
+lemma flatTensor_inv {H : Matrix n n ℂ} (hH : H.PosDef) :
+    (tensor3 H⁻¹.transpose H 1)⁻¹ = tensor3 H.transpose H⁻¹ 1 := by
+  apply Matrix.inv_eq_right_inv
+  rw [tensor3_mul,← Matrix.transpose_mul,
+    Matrix.mul_nonsing_inv _ (isUnit_iff_ne_zero.mpr hH.det_pos.ne'),
+    Matrix.transpose_one,one_mul,tensor3_one]
+
+lemma entrywise_norm_one_le : ‖(1 : Matrix n n ℂ)‖ ≤ 1 := by
+  apply matrix_norm_le_of_entries zero_le_one
+  intro i j
+  simp only [Matrix.one_apply]
+  split_ifs <;> simp
+
+lemma flatTensor_inv_norm {H : Matrix n n ℂ} (hHp : H.PosDef) {M : ℝ}
+    (hM : 0 ≤ M) (hH : ‖H‖ ≤ M) (hI : ‖H⁻¹‖ ≤ M) :
+    ‖(tensor3 H⁻¹.transpose H 1)⁻¹‖ ≤ M^2 := by
+  rw [flatTensor_inv hHp]
+  apply (tensor3_norm_le _ _ _).trans
+  rw [Matrix.norm_transpose]
+  exact (mul_le_mul (mul_le_mul hH hI (norm_nonneg _) hM) entrywise_norm_one_le
+    (norm_nonneg _) (mul_nonneg hM hM)).trans_eq (by ring)
+end MongeAmpere
+namespace Anticanonical.SourceSmooth.KaehlerMetric
+open MongeAmpere
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+local notation "TI" => TensorIndex (Fin d)
+
+lemma calabiNorm_le_flatEnergy (g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) {M : ℝ} (hM : 0 ≤ M)
+    (hH : ‖g.matrix q z‖ ≤ M) (hI : ‖(g.matrix q z)⁻¹‖ ≤ M) :
+    g.calabiNorm q z ≤ (Fintype.card TI:ℝ)^2*M^5*
+      (hermPair (tensor3 (g.matrix q z)⁻¹.transpose (g.matrix q z) 1)
+        (g.calabiTensor q z) (g.calabiTensor q z)).re := by
+  let f := g.calabiTensor q z
+  let K := tensor3 (g.matrix q z)⁻¹.transpose (g.matrix q z) 1
+  have hK : K.PosDef := (g.positive q z hz).inv.transpose.kronecker
+    ((g.positive q z hz).kronecker Matrix.PosDef.one)
+  have hE : 0 ≤ (hermPair K f f).re := hermPair_nonneg hK.posSemidef _
+  have hf : ‖f‖^2 ≤ M^2*(hermPair K f f).re := (norm_sq_le_hermPair hK f).trans
+    (mul_le_mul_of_nonneg_right (flatTensor_inv_norm (g.positive q z hz) hM hH hI) hE)
+  have hb : g.calabiNorm q z ≤ (Fintype.card TI:ℝ)^2*M^3*‖f‖^2 := by
+    apply (Complex.re_le_norm (hermPair (g.calabiMetric q z) f f)).trans
+    apply (pair_norm_le _ _ _).trans
+    have hh := tensorMetric_norm_bound hM hH hI
+    calc
+      _ ≤ (Fintype.card TI:ℝ)^2*M^3*‖f‖*‖f‖ :=
+        mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right
+          (mul_le_mul_of_nonneg_left hh (sq_nonneg _)) (norm_nonneg _)) (norm_nonneg _)
+      _ = _ := by ring
+  apply hb.trans
+  calc
+    _ ≤ (Fintype.card TI:ℝ)^2*M^3*(M^2*(hermPair K f f).re) :=
+      mul_le_mul_of_nonneg_left hf (by positivity)
+    _ = _ := by dsimp [K,f]; ring
+
+lemma flatTrace_lower_bound (g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) {M R : ℝ} (hM : 0 ≤ M)
+    (hH : ‖g.matrix q z‖ ≤ M) (hI : ‖(g.matrix q z)⁻¹‖ ≤ M)
+    (hR : ‖g.curvatureRicci q z‖ ≤ R) :
+    (1+(Fintype.card TI:ℝ)^2*M^5)⁻¹*g.calabiNorm q z-(d:ℝ)*R ≤
+      ((g.matrix q z)⁻¹*PotentialKaehler.potentialMatrix (g.flatTrace q) z).trace.re := by
+  let E := (hermPair (tensor3 (g.matrix q z)⁻¹.transpose (g.matrix q z) 1)
+    (g.calabiTensor q z) (g.calabiTensor q z)).re
+  have hE : 0 ≤ E := hermPair_nonneg
+    ((g.positive q z hz).inv.transpose.kronecker
+      ((g.positive q z hz).kronecker Matrix.PosDef.one)).posSemidef _
+  have hb := g.calabiNorm_le_flatEnergy q hz hM hH hI
+  have hδ : 0 < 1+(Fintype.card TI:ℝ)^2*M^5 := by positivity
+  have hS : (1+(Fintype.card TI:ℝ)^2*M^5)⁻¹*g.calabiNorm q z ≤ E := by
+    rw [← div_eq_inv_mul,div_le_iff₀ hδ]
+    change g.calabiNorm q z ≤ E*(1+(Fintype.card TI:ℝ)^2*M^5)
+    change g.calabiNorm q z ≤ (Fintype.card TI:ℝ)^2*M^5*E at hb
+    nlinarith
+  have hRc : (g.curvatureRicci q z).trace.re ≤ (d:ℝ)*R := by
+    apply (Complex.re_le_norm _).trans
+    simpa only [Fintype.card_fin] using trace_entry_bound (fun i j =>
+      (norm_entry_le_entrywise_sup_norm (g.curvatureRicci q z) (i:=i) (j:=j)).trans hR)
+  rw [g.flatTrace_laplacian q hz]
+  exact sub_le_sub hS hRc
+
+lemma flatTrace_bound (g : KaehlerMetric A) (q : Fin A.count) (z : Coordinates d)
+    {M : ℝ} (hH : ‖g.matrix q z‖ ≤ M) : g.flatTrace q z ≤ (d:ℝ)*M := by
+  apply (Complex.re_le_norm _).trans
+  simpa only [Fintype.card_fin] using trace_entry_bound (fun i j =>
+    (norm_entry_le_entrywise_sup_norm (g.matrix q z) (i:=i) (j:=j)).trans hH)
+
+end Anticanonical.SourceSmooth.KaehlerMetric
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/ChartSobolevAlgebra.lean b/lean/OAI/Geometry/PrescribedRicci/ChartSobolevAlgebra.lean
new file mode 100644
index 0000000000000000000000000000000000000000..955dfe34580413caa0b16aa4db18b9e8ef231884
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/ChartSobolevAlgebra.lean
@@ -0,0 +1,141 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.BoundedBilinearCompletion
+import OAI.Geometry.PrescribedPotential.SobolevReconstruction
+
+namespace OAI
+
+/-! Chart Sobolev Algebra. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory TemperedDistribution LineDeriv
+open scoped SchwartzMap BoundedContinuousFunction ContDiff Classical
+namespace SobolevChart
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+
+def schwartzProductLinear : 𝓢(E,ℂ) →ₗ[ℝ] 𝓢(E,ℂ) →ₗ[ℝ] 𝓢(E,ℂ) where
+  toFun f := (SchwartzMap.smulLeftCLM ℂ f).restrictScalars ℝ |>.toLinearMap
+  map_add' f g := by
+    ext h x
+    change (SchwartzMap.smulLeftCLM ℂ (f+g) h) x =
+      (SchwartzMap.smulLeftCLM ℂ f h) x + (SchwartzMap.smulLeftCLM ℂ g h) x
+    rw [SchwartzMap.smulLeftCLM_apply_apply (f+g).hasTemperateGrowth,
+      SchwartzMap.smulLeftCLM_apply_apply f.hasTemperateGrowth,
+      SchwartzMap.smulLeftCLM_apply_apply g.hasTemperateGrowth]
+    simp only [add_apply,smul_eq_mul]
+    ring
+  map_smul' c f := by
+    ext h x
+    change (SchwartzMap.smulLeftCLM ℂ (c • f) h) x =
+      c • (SchwartzMap.smulLeftCLM ℂ f h) x
+    rw [SchwartzMap.smulLeftCLM_apply_apply (c • f).hasTemperateGrowth,
+      SchwartzMap.smulLeftCLM_apply_apply f.hasTemperateGrowth]
+    simp only [smul_apply,smul_eq_mul,Complex.real_smul]
+    ring
+
+def productCoord (k : ℕ) (hk : Module.finrank ℝ E < k) : L2 E →L[ℝ] L2 E →L[ℝ] L2 E :=
+  BoundedBilinearCompletion.extend (schwartzCoordLinear (k:ℝ)) (schwartzCoord_dense (k:ℝ))
+    schwartzProductLinear (schwartz_sobolev_product_bound k hk).choose_spec.2
+
+lemma productCoord_schwartz (k : ℕ) (hk : Module.finrank ℝ E < k) (f g : 𝓢(E,ℂ)) :
+    productCoord k hk (schwartzCoord (k:ℝ) f) (schwartzCoord (k:ℝ) g) =
+      schwartzCoord (k:ℝ) (SchwartzMap.smulLeftCLM ℂ f g) :=
+  BoundedBilinearCompletion.extend_eq _ (schwartzCoord_dense (k:ℝ)) _
+    (schwartz_sobolev_product_bound k hk).choose_spec.1
+    (schwartz_sobolev_product_bound k hk).choose_spec.2 f g
+
+lemma productCoord_strong (k : ℕ) (hk : Module.finrank ℝ E < k)
+    (hs : (Module.finrank ℝ E:ℝ) < 2*(k:ℝ)) (u v : L2 E) :
+    strongEmbedding (k:ℝ) hs (productCoord k hk u v) =
+      strongEmbedding (k:ℝ) hs u * strongEmbedding (k:ℝ) hs v := by
+  have core (f : 𝓢(E,ℂ)) : (fun v => strongEmbedding (k:ℝ) hs
+      (productCoord k hk (schwartzCoord (k:ℝ) f) v)) =
+      (fun v => strongEmbedding (k:ℝ) hs (schwartzCoord (k:ℝ) f) * strongEmbedding (k:ℝ) hs v) := by
+    apply (schwartzCoord_dense (k:ℝ)).equalizer (by fun_prop) (by fun_prop)
+    funext g
+    simp only [Function.comp_apply,productCoord_schwartz,strongEmbedding_schwartz]
+    ext x
+    exact SchwartzMap.smulLeftCLM_apply_apply f.hasTemperateGrowth g x
+  have he : (fun u => strongEmbedding (k:ℝ) hs (productCoord k hk u v)) =
+      (fun u => strongEmbedding (k:ℝ) hs u * strongEmbedding (k:ℝ) hs v) := by
+    apply (schwartzCoord_dense (k:ℝ)).equalizer (by fun_prop) (by fun_prop)
+    funext f
+    exact congr_fun (core f) v
+  exact congr_fun he u
+
+lemma productCoord_lower_zero_core (k : ℕ) (hk : Module.finrank ℝ E < k)
+    (hs : (Module.finrank ℝ E:ℝ) < 2*(k:ℝ)) (f : 𝓢(E,ℂ)) (v : L2 E) :
+    lowerCoord (k:ℝ) 0 (Nat.cast_nonneg k) (productCoord k hk (schwartzCoord (k:ℝ) f) v) =
+      multiply (strongEmbedding (k:ℝ) hs (schwartzCoord (k:ℝ) f))
+        (lowerCoord (k:ℝ) 0 (Nat.cast_nonneg k) v) := by
+  have he : (fun v : L2 E => lowerCoord (k:ℝ) 0 (Nat.cast_nonneg k)
+      (productCoord k hk (schwartzCoord (k:ℝ) f) v)) =
+      (fun v => multiply (strongEmbedding (k:ℝ) hs (schwartzCoord (k:ℝ) f))
+        (lowerCoord (k:ℝ) 0 (Nat.cast_nonneg k) v)) := by
+    apply (schwartzCoord_dense (k:ℝ)).equalizer
+      ((lowerCoord (k:ℝ) 0 (Nat.cast_nonneg k)).continuous.comp
+        (productCoord k hk (schwartzCoord (k:ℝ) f)).continuous)
+      ((multiply (strongEmbedding (k:ℝ) hs (schwartzCoord (k:ℝ) f))).continuous.comp
+        (lowerCoord (k:ℝ) 0 (Nat.cast_nonneg k)).continuous)
+    funext g
+    simp only [Function.comp_apply,productCoord_schwartz,
+      lowerCoord_schwartz,strongEmbedding_schwartz]
+    exact schwartzCoord_product_zero f g
+  exact congr_fun he v
+
+lemma productCoord_lower_zero (k : ℕ) (hk : Module.finrank ℝ E < k)
+    (hs : (Module.finrank ℝ E:ℝ) < 2*(k:ℝ)) (u v : L2 E) :
+    lowerCoord (k:ℝ) 0 (Nat.cast_nonneg k) (productCoord k hk u v) =
+      multiply (strongEmbedding (k:ℝ) hs u) (lowerCoord (k:ℝ) 0 (Nat.cast_nonneg k) v) := by
+  have hc : Continuous (fun u : L2 E => multiply (strongEmbedding (k:ℝ) hs u)
+      (lowerCoord (k:ℝ) 0 (Nat.cast_nonneg k) v)) :=
+    by
+      change Continuous (fun u : L2 E => (ContinuousLinearMap.mul ℂ ℂ).holderL volume ⊤ 2 2
+        (boundedToLinf (strongEmbedding (k:ℝ) hs u)) (lowerCoord (k:ℝ) 0 (Nat.cast_nonneg k) v))
+      exact (((ContinuousLinearMap.mul ℂ ℂ).holderL volume ⊤ 2 2).continuous.comp
+        (boundedToLinf.continuous.comp (strongEmbedding (k:ℝ) hs).continuous)).clm_apply continuous_const
+  have he : (fun u => lowerCoord (k:ℝ) 0 (Nat.cast_nonneg k) (productCoord k hk u v)) =
+      (fun u => multiply (strongEmbedding (k:ℝ) hs u) (lowerCoord (k:ℝ) 0 (Nat.cast_nonneg k) v)) := by
+    apply (schwartzCoord_dense (k:ℝ)).equalizer
+      ((lowerCoord (k:ℝ) 0 (Nat.cast_nonneg k)).continuous.comp
+        ((productCoord k hk).continuous.clm_apply continuous_const)) hc
+    funext f
+    exact productCoord_lower_zero_core k hk hs f v
+  exact congr_fun he u
+
+lemma strongEmbedding_lower {s t : ℝ} (hst : t ≤ s)
+    (hs : (Module.finrank ℝ E:ℝ) < 2*s) (ht : (Module.finrank ℝ E:ℝ) < 2*t)
+    (u : L2 E) : strongEmbedding t ht (lowerCoord s t hst u) = strongEmbedding s hs u := by
+  apply boundedDistribution_injective
+  rw [strongEmbedding_distribution,realize_lowerCoord,strongEmbedding_distribution]
+
+lemma productCoord_lower (k l : ℕ) (hk : Module.finrank ℝ E < k)
+    (hl : Module.finrank ℝ E < l) (hlk : l ≤ k) (u v : L2 E) :
+    lowerCoord (k:ℝ) (l:ℝ) (by exact_mod_cast hlk) (productCoord k hk u v) =
+      productCoord l hl (lowerCoord (k:ℝ) (l:ℝ) (by exact_mod_cast hlk) u)
+        (lowerCoord (k:ℝ) (l:ℝ) (by exact_mod_cast hlk) v) := by
+  have core (f : 𝓢(E,ℂ)) : (fun v => lowerCoord (k:ℝ) (l:ℝ) (by exact_mod_cast hlk)
+      (productCoord k hk (schwartzCoord (k:ℝ) f) v)) =
+      (fun v => productCoord l hl (lowerCoord (k:ℝ) (l:ℝ) (by exact_mod_cast hlk) (schwartzCoord (k:ℝ) f))
+        (lowerCoord (k:ℝ) (l:ℝ) (by exact_mod_cast hlk) v)) := by
+    apply (schwartzCoord_dense (k:ℝ)).equalizer (by fun_prop) (by fun_prop)
+    funext g
+    simp only [Function.comp_apply,productCoord_schwartz,lowerCoord_schwartz]
+  have he : (fun u => lowerCoord (k:ℝ) (l:ℝ) (by exact_mod_cast hlk) (productCoord k hk u v)) =
+      (fun u => productCoord l hl (lowerCoord (k:ℝ) (l:ℝ) (by exact_mod_cast hlk) u)
+        (lowerCoord (k:ℝ) (l:ℝ) (by exact_mod_cast hlk) v)) := by
+    apply (schwartzCoord_dense (k:ℝ)).equalizer (by fun_prop) (by fun_prop)
+    funext f
+    exact congr_fun (core f) v
+  exact congr_fun he u
+
+end SobolevChart
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/ChernBianchi.lean b/lean/OAI/Geometry/PrescribedRicci/ChernBianchi.lean
new file mode 100644
index 0000000000000000000000000000000000000000..96fdd5265810174cc5dfed2e9dde0dbb0e99c2bb
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/ChernBianchi.lean
@@ -0,0 +1,94 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.ChernConnection
+
+namespace OAI
+
+/-! Chern Bianchi. -/
+
+noncomputable section
+open Matrix Filter Set Topology
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace Anticanonical.SourceSmooth.KaehlerMetric
+variable {d : ℕ}
+local notation "Mat" => Matrix (Fin d) (Fin d) ℂ
+
+lemma holDerivative_neg (f : Coordinates d → Mat) (z : Coordinates d) (a : Fin d) :
+    holDerivative (fun y => -f y) z a = -holDerivative f z a := by
+  unfold holDerivative
+  have he : fderiv ℝ (fun y => -f y) z = -fderiv ℝ f z := by
+    exact fderiv_fun_neg
+  rw [he]
+  ext i j
+  simp only [_root_.neg_apply,Matrix.smul_apply,Matrix.sub_apply,Matrix.neg_apply,smul_eq_mul]
+  ring
+
+variable {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+
+lemma chernCurvature_smooth (g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) (b a : Fin d) :
+    ContDiffAt ℝ ∞ (fun y => g.chernCurvature q y b a) z := by
+  have hg := (g.smooth q).contDiffAt ((A.chart q).open_target.mem_nhds hz)
+  have hi : ContDiffAt ℝ ∞ (fun y => (g.matrix q y)⁻¹) z :=
+    ((MongeAmpere.contDiffAt_inv _ (g.positive q z hz).det_pos.ne').restrict_scalars ℝ).comp z hg
+  unfold chernCurvature curvatureMatrix
+  exact matrix_mul_contDiffAt hi
+    ((contDiffAt_barDerivative (contDiffAt_holDerivative hg a) b).neg.add
+      (matrix_mul_contDiffAt (matrix_mul_contDiffAt (contDiffAt_barDerivative hg b) hi)
+        (contDiffAt_holDerivative hg a)))
+
+lemma chernConnection_hol_bar (g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) (b a u : Fin d) :
+    holDerivative (fun y => barDerivative (fun w => g.chernConnection q w a) y b) z u =
+      -holDerivative (fun y => g.chernCurvature q y b a) z u := by
+  have he : (fun y => barDerivative (fun w => g.chernConnection q w a) y b) =ᶠ[nhds z]
+      (fun y => -g.chernCurvature q y b a) := by
+    filter_upwards [(A.chart q).open_target.mem_nhds hz] with y hy
+    exact g.chernConnection_bar q hy a b
+  rw [holDerivative_congr he,holDerivative_neg]
+
+lemma chern_bianchi (g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) (b a u : Fin d) :
+    holDerivative (fun y => g.chernCurvature q y b a) z u +
+      g.chernConnection q z u*g.chernCurvature q z b a -
+      g.chernCurvature q z b a*g.chernConnection q z u =
+    holDerivative (fun y => g.chernCurvature q y b u) z a +
+      g.chernConnection q z a*g.chernCurvature q z b u -
+      g.chernCurvature q z b u*g.chernConnection q z a := by
+  have hc (v : Fin d) := g.chernConnection_smooth q hz v
+  have he : (fun y => holDerivative (fun w => g.chernConnection q w a) y u-
+      holDerivative (fun w => g.chernConnection q w u) y a) =ᶠ[nhds z]
+      (fun y => g.chernConnection q y a*g.chernConnection q y u-
+        g.chernConnection q y u*g.chernConnection q y a) := by
+    filter_upwards [(A.chart q).open_target.mem_nhds hz] with y hy
+    exact g.chernConnection_maurerCartan q hy a u
+  have hb := barDerivative_congr he b
+  rw [barDerivative_sub ((contDiffAt_holDerivative (hc a) u).differentiableAt (by simp))
+      ((contDiffAt_holDerivative (hc u) a).differentiableAt (by simp)),
+    bar_hol_comm (hc a),bar_hol_comm (hc u),g.chernConnection_hol_bar q hz,
+    g.chernConnection_hol_bar q hz] at hb
+  rw [barDerivative_sub (matrix_mul_differentiableAt ((hc a).differentiableAt (by simp))
+      ((hc u).differentiableAt (by simp)))
+      (matrix_mul_differentiableAt ((hc u).differentiableAt (by simp))
+      ((hc a).differentiableAt (by simp))),
+    barDerivative_mul ((hc a).differentiableAt (by simp)) ((hc u).differentiableAt (by simp)),
+    barDerivative_mul ((hc u).differentiableAt (by simp)) ((hc a).differentiableAt (by simp)),
+    g.chernConnection_bar q hz,g.chernConnection_bar q hz] at hb
+  simp only [neg_mul,mul_neg] at hb
+  have hh := congrArg (fun M : Mat => M+
+    (holDerivative (fun y => g.chernCurvature q y b a) z u+
+      g.chernConnection q z a*g.chernCurvature q z b u-
+      g.chernCurvature q z b u*g.chernConnection q z a)) hb
+  convert hh.symm using 1 <;> abel
+
+lemma chernCurvature_contraction (g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) :
+    ∑ u, ∑ b, ((g.matrix q z)⁻¹ u b) • g.chernCurvature q z b u =
+      (g.matrix q z)⁻¹*g.curvatureRicci q z := by
+  rw [← g.curvature_contraction q hz]
+  simp only [chernCurvature,Matrix.mul_sum,Matrix.mul_smul]
+
+end Anticanonical.SourceSmooth.KaehlerMetric
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/ChernBochner.lean b/lean/OAI/Geometry/PrescribedRicci/ChernBochner.lean
new file mode 100644
index 0000000000000000000000000000000000000000..f686d643770b39b4df750355558d635d3fa6edb5
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/ChernBochner.lean
@@ -0,0 +1,79 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.ChernPairDifferential
+import OAI.Geometry.PrescribedRicci.KahlerGradientEnergy
+
+namespace OAI
+
+/-! Chern Bochner. -/
+
+noncomputable section
+open Matrix Filter Set Topology
+open scoped ContDiff ComplexOrder
+namespace Anticanonical.SourceSmooth.KaehlerMetric
+open MongeAmpere
+variable {d : ℕ} {n : Type*} [Fintype n]
+
+lemma bar_covHol {C : Coordinates d → Fin d → Matrix n n ℂ} {f : Coordinates d → n → ℂ}
+    {z : Coordinates d} (hC : ∀ a i j, ContDiffAt ℝ ∞ (fun y => C y a i j) z)
+    (hf : ∀ i, ContDiffAt ℝ ∞ (fun y => f y i) z) (a b : Fin d) :
+    barVector (fun y => covHol C f y a) z b =
+      covHol C (fun y => barVector f y b) z a+
+        barArray (fun y => C y a) z b*ᵥ f z := by
+  ext i
+  have hc (j : n) : DifferentiableAt ℝ (fun y => C y a i j*f y j) z :=
+    ((hC a i j).differentiableAt (by simp)).fun_mul ((hf j).differentiableAt (by simp))
+  change barDeriv (fun y => holDeriv (fun w => f w i) y a + ∑ j, C y a i j*f y j) z b =
+    holDeriv (fun y => barDeriv (fun w => f w i) y b) z a+
+      ∑ j, C z a i j*barDeriv (fun y => f y j) z b+
+        ∑ j, barDeriv (fun y => C y a i j) z b*f z j
+  rw [barDeriv_add ((contDiffAt_holDeriv (hf i) a).differentiableAt (by simp))
+      (DifferentiableAt.fun_sum (u:=Finset.univ) (fun j _ => hc j)),
+    barDeriv_sum hc,scalar_bar_hol_comm (hf i)]
+  simp_rw [barDeriv_mul ((hC a i _).differentiableAt (by simp)) ((hf _).differentiableAt (by simp))]
+  rw [Finset.sum_add_distrib]
+  abel
+
+lemma hermPair_bochners_identity {s : Set (Coordinates d)} (hs : IsOpen s)
+    {K : Coordinates d → Matrix n n ℂ} {C : Coordinates d → Fin d → Matrix n n ℂ}
+    {f : Coordinates d → n → ℂ} {z : Coordinates d} (hz : z ∈ s)
+    (hK : ∀ y ∈ s, ∀ i j, ContDiffAt ℝ ∞ (fun w => K w i j) y)
+    (hC : ∀ y ∈ s, ∀ a i j, ContDiffAt ℝ ∞ (fun w => C w a i j) y)
+    (hf : ∀ y ∈ s, ∀ i, ContDiffAt ℝ ∞ (fun w => f w i) y)
+    (hhol : ∀ y ∈ s, ∀ a, holArray K y a = K y*C y a)
+    (hbar : ∀ y ∈ s, ∀ a, barArray K y a = (C y a)ᴴ*K y) (a b : Fin d) :
+    barDeriv (fun y => holDeriv (fun w => hermPair (K w) (f w) (f w)) y a) z b =
+      hermPair (K z) (covHol C (fun y => barVector f y a) z b) (f z)+
+      hermPair (K z) (barVector f z a) (barVector f z b)+
+      hermPair (K z) (covHol C f z b) (covHol C f z a)+
+      hermPair (K z) (f z) (covHol C (fun y => barVector f y b) z a)+
+      hermPair (K z) (f z) (barArray (fun y => C y a) z b*ᵥ f z) := by
+  have hKz := hK z hz
+  have hfz := hf z hz
+  have hbarf (u : Fin d) (i : n) : ContDiffAt ℝ ∞ (fun y => barVector f y u i) z :=
+    contDiffAt_barDeriv (hfz i) u
+  have hcovf (u : Fin d) := covHol_smooth (hC z hz) hfz u
+  have he : (fun y => holDeriv (fun w => hermPair (K w) (f w) (f w)) y a) =ᶠ[nhds z]
+      (fun y => hermPair (K y) (barVector f y a) (f y)+
+        hermPair (K y) (f y) (covHol C f y a)) := by
+    filter_upwards [hs.mem_nhds hz] with y hy
+    exact hermPair_hol_compatible
+      (fun i j => (hK y hy i j).differentiableAt (by simp))
+      (fun i => (hf y hy i).differentiableAt (by simp))
+      (fun i => (hf y hy i).differentiableAt (by simp)) a (hhol y hy a)
+  rw [barDeriv_congr he]
+  rw [barDeriv_add ((hermPair_smooth hKz (hbarf a) hfz).differentiableAt (by simp))
+    ((hermPair_smooth hKz hfz (hcovf a)).differentiableAt (by simp))]
+  rw [hermPair_bar_compatible (fun i j => (hKz i j).differentiableAt (by simp))
+      (fun i => (hbarf a i).differentiableAt (by simp))
+      (fun i => (hfz i).differentiableAt (by simp)) b (hbar z hz b),
+    hermPair_bar_compatible (fun i j => (hKz i j).differentiableAt (by simp))
+      (fun i => (hfz i).differentiableAt (by simp))
+      (fun i => (hcovf a i).differentiableAt (by simp)) b (hbar z hz b),
+    bar_covHol (hC z hz) hfz,hermPair_add_right]
+  abel
+
+end Anticanonical.SourceSmooth.KaehlerMetric
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/ChernConnection.lean b/lean/OAI/Geometry/PrescribedRicci/ChernConnection.lean
new file mode 100644
index 0000000000000000000000000000000000000000..0a6f83a14228552850b4a0262a39b1a1e3ae6986
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/ChernConnection.lean
@@ -0,0 +1,120 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.ChernLuAlgebra
+import OAI.Geometry.PrescribedRicci.MatrixWirtingerExtra
+import OAI.Geometry.PrescribedRicci.MatrixWirtingerSecond
+
+namespace OAI
+
+/-! Chern Connection. -/
+
+noncomputable section
+open Matrix Filter Set Topology
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace Anticanonical.SourceSmooth.KaehlerMetric
+variable {d : ℕ}
+local notation "Mat" => Matrix (Fin d) (Fin d) ℂ
+
+lemma holDerivative_congr {f g : Coordinates d → Mat} {z : Coordinates d}
+    (he : f =ᶠ[nhds z] g) (a : Fin d) : holDerivative f z a = holDerivative g z a := by
+  unfold holDerivative
+  rw [he.fderiv_eq]
+
+lemma holDerivative_sub {f g : Coordinates d → Mat} {z : Coordinates d}
+    (hf : DifferentiableAt ℝ f z) (hg : DifferentiableAt ℝ g z) (a : Fin d) :
+    holDerivative (fun y => f y-g y) z a = holDerivative f z a-holDerivative g z a := by
+  unfold holDerivative
+  have he : fderiv ℝ (fun y => f y-g y) z = fderiv ℝ f z-fderiv ℝ g z := by
+    convert! (hf.hasFDerivAt.sub hg.hasFDerivAt).fderiv using 1
+  rw [he]
+  simp only [_root_.sub_apply,smul_sub]
+  abel
+
+lemma hol_hol_comm {f : Coordinates d → Mat} {z : Coordinates d}
+    (hf : ContDiffAt ℝ ∞ f z) (a b : Fin d) :
+    holDerivative (fun y => holDerivative f y a) z b =
+      holDerivative (fun y => holDerivative f y b) z a := by
+  change (2:ℂ)⁻¹ • (fderiv ℝ (fun y => holDerivative f y a) z (coordinateVector b) -
+    Complex.I • fderiv ℝ (fun y => holDerivative f y a) z (Complex.I • coordinateVector b)) =
+    (2:ℂ)⁻¹ • (fderiv ℝ (fun y => holDerivative f y b) z (coordinateVector a) -
+    Complex.I • fderiv ℝ (fun y => holDerivative f y b) z (Complex.I • coordinateVector a))
+  rw [fderiv_holDerivative hf,fderiv_holDerivative hf,
+    fderiv_holDerivative hf,fderiv_holDerivative hf]
+  have hs (u v : Coordinates d) : fderiv ℝ (fderiv ℝ f) z u v = fderiv ℝ (fderiv ℝ f) z v u := by
+    exact hf.isSymmSndFDerivAt (by
+      simp only [minSmoothness_of_isRCLikeNormedField]
+      change ((2:ℕ∞):WithTop ℕ∞) ≤ ((⊤:ℕ∞):WithTop ℕ∞)
+      exact WithTop.coe_le_coe.mpr le_top) u v
+  rw [hs (coordinateVector a) (coordinateVector b),
+    hs (coordinateVector a) (Complex.I • coordinateVector b),
+    hs (Complex.I • coordinateVector a) (coordinateVector b),
+    hs (Complex.I • coordinateVector a) (Complex.I • coordinateVector b)]
+  ext i j
+  simp only [Matrix.smul_apply,Matrix.sub_apply,smul_eq_mul]
+  ring
+
+variable {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+
+def chernConnection (g : KaehlerMetric A) (q : Fin A.count) (z : Coordinates d) (a : Fin d) : Mat :=
+  (g.matrix q z)⁻¹*holDerivative (g.matrix q) z a
+
+def chernCurvature (g : KaehlerMetric A) (q : Fin A.count) (z : Coordinates d)
+    (b a : Fin d) : Mat := (g.matrix q z)⁻¹*g.curvatureMatrix q z b a
+
+lemma chernConnection_smooth (g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) (a : Fin d) :
+    ContDiffAt ℝ ∞ (fun y => g.chernConnection q y a) z := by
+  have hg := (g.smooth q).contDiffAt ((A.chart q).open_target.mem_nhds hz)
+  exact matrix_mul_contDiffAt
+    (((MongeAmpere.contDiffAt_inv _ (g.positive q z hz).det_pos.ne').restrict_scalars ℝ).comp z hg)
+    (contDiffAt_holDerivative hg a)
+
+lemma chernConnection_torsion (g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) (a k j : Fin d) :
+    g.chernConnection q z a k j = g.chernConnection q z j k a := by
+  simp only [chernConnection,Matrix.mul_apply]
+  apply Finset.sum_congr rfl
+  intro u _
+  rw [g.holDerivative_symm q hz a u j]
+
+lemma chernConnection_bar (g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) (a b : Fin d) :
+    barDerivative (fun y => g.chernConnection q y a) z b = -g.chernCurvature q z b a := by
+  have hg := (g.smooth q).contDiffAt ((A.chart q).open_target.mem_nhds hz)
+  have hi : ContDiffAt ℝ ∞ (fun y => (g.matrix q y)⁻¹) z :=
+    ((MongeAmpere.contDiffAt_inv _ (g.positive q z hz).det_pos.ne').restrict_scalars ℝ).comp z hg
+  unfold chernConnection
+  rw [barDerivative_mul (hi.differentiableAt (by simp))
+    ((contDiffAt_holDerivative hg a).differentiableAt (by simp)),
+    barDerivative_inv (hg.differentiableAt (by simp)) (isUnit_iff_ne_zero.mpr (g.positive q z hz).det_pos.ne')]
+  simp only [chernCurvature,curvatureMatrix,mul_add,neg_mul,mul_neg,mul_assoc]
+  abel
+
+lemma chernConnection_hol (g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) (a u : Fin d) :
+    holDerivative (fun y => g.chernConnection q y a) z u =
+      -(g.chernConnection q z u*g.chernConnection q z a)+
+        (g.matrix q z)⁻¹*holDerivative (fun y => holDerivative (g.matrix q) y a) z u := by
+  have hg := (g.smooth q).contDiffAt ((A.chart q).open_target.mem_nhds hz)
+  have hi : ContDiffAt ℝ ∞ (fun y => (g.matrix q y)⁻¹) z :=
+    ((MongeAmpere.contDiffAt_inv _ (g.positive q z hz).det_pos.ne').restrict_scalars ℝ).comp z hg
+  unfold chernConnection
+  rw [holDerivative_mul (hi.differentiableAt (by simp))
+    ((contDiffAt_holDerivative hg a).differentiableAt (by simp)),
+    holDerivative_inv (hg.differentiableAt (by simp)) (isUnit_iff_ne_zero.mpr (g.positive q z hz).det_pos.ne')]
+  simp only [neg_mul,mul_assoc]
+
+lemma chernConnection_maurerCartan (g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) (a u : Fin d) :
+    holDerivative (fun y => g.chernConnection q y a) z u -
+      holDerivative (fun y => g.chernConnection q y u) z a =
+      g.chernConnection q z a*g.chernConnection q z u-
+        g.chernConnection q z u*g.chernConnection q z a := by
+  rw [g.chernConnection_hol q hz,g.chernConnection_hol q hz,
+    hol_hol_comm ((g.smooth q).contDiffAt ((A.chart q).open_target.mem_nhds hz)) a u]
+  abel
+
+end Anticanonical.SourceSmooth.KaehlerMetric
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/ChernLuAlgebra.lean b/lean/OAI/Geometry/PrescribedRicci/ChernLuAlgebra.lean
new file mode 100644
index 0000000000000000000000000000000000000000..9699bf8f0e8f6776c9fbf177bbe3332455fc9f57
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/ChernLuAlgebra.lean
@@ -0,0 +1,93 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.MatrixWirtinger
+import OAI.Geometry.PrescribedRicci.MatrixWirtingerSecond
+
+namespace OAI
+
+/-! Chern Lu Algebra. -/
+
+noncomputable section
+open Matrix Filter Set Topology
+open scoped ContDiff ComplexOrder MatrixOrder Matrix.Norms.Elementwise
+namespace MongeAmpere
+variable {n : Type*} [Fintype n] [DecidableEq n]
+
+lemma trace_lu_identity (B G Pg Qg Ph Qh Rg Rh : Matrix n n ℂ)
+    (hG : IsUnit G.det) :
+    (B*Rg-B*Qh*B*Pg-B*Ph*B*Qg-B*Rh*B*G+
+      B*Qh*B*Ph*B*G+B*Ph*B*Qh*B*G).trace =
+    (-(B*(-Rg+Qg*G⁻¹*Pg))).trace +
+      (B*(Qg-Qh*B*G)*G⁻¹*(Pg-G*B*Ph)).trace +
+      (B*(-Rh+Qh*B*Ph)*B*G).trace := by
+  have he : B*(Qg-Qh*B*G)*G⁻¹*(Pg-G*B*Ph) =
+      B*Qg*G⁻¹*Pg-B*Qg*B*Ph-B*Qh*B*Pg+B*Qh*B*G*B*Ph := by
+    simp only [mul_sub,sub_mul]
+    simp only [mul_assoc,Matrix.mul_nonsing_inv_cancel_left G _ hG,
+      Matrix.nonsing_inv_mul_cancel_left G _ hG]
+    noncomm_ring
+  rw [he]
+  simp only [mul_add,add_mul,mul_neg,neg_mul,trace_add,trace_sub,trace_neg]
+  have h1 : (B*Ph*B*Qg).trace = (B*Qg*B*Ph).trace := by
+    simpa only [mul_assoc] using Matrix.trace_mul_comm (B*Ph) (B*Qg)
+  have h2 : (B*Ph*B*Qh*B*G).trace = (B*Qh*B*G*B*Ph).trace := by
+    simpa only [mul_assoc] using Matrix.trace_mul_comm (B*Ph) (B*Qh*B*G)
+  simp only [mul_assoc] at h1 h2 ⊢
+  linear_combination h2 - h1
+
+end MongeAmpere
+namespace Anticanonical.SourceSmooth.KaehlerMetric
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+local notation "Mat" => Matrix (Fin d) (Fin d) ℂ
+
+def curvatureMatrix (g : KaehlerMetric A) (q : Fin A.count) (z : Coordinates d)
+    (b a : Fin d) : Mat :=
+  -barDerivative (fun y => holDerivative (g.matrix q) y a) z b +
+    barDerivative (g.matrix q) z b*(g.matrix q z)⁻¹*holDerivative (g.matrix q) z a
+
+lemma curvatureMatrix_exchange (g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) (a b i j : Fin d) :
+    g.curvatureMatrix q z b a i j = g.curvatureMatrix q z i j b a := by
+  unfold curvatureMatrix
+  simp only [Matrix.add_apply,Matrix.neg_apply]
+  rw [g.secondDerivative_row_symm q hz a b i j,
+    g.secondDerivative_col_symm q hz a i b j]
+  congr 1
+  simp only [Matrix.mul_apply]
+  apply Finset.sum_congr rfl
+  intro k _
+  rw [g.holDerivative_symm q hz a k j]
+  congr 1
+  apply Finset.sum_congr rfl
+  intro l _
+  rw [g.barDerivative_symm q hz b i l]
+
+def curvatureRicci (g : KaehlerMetric A) (q : Fin A.count) (z : Coordinates d) : Mat :=
+  fun b a => ((g.matrix q z)⁻¹*g.curvatureMatrix q z b a).trace
+
+lemma curvature_contraction (g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) :
+    ∑ a, ∑ b, ((g.matrix q z)⁻¹ a b) • g.curvatureMatrix q z b a =
+      g.curvatureRicci q z := by
+  ext i j
+  simp only [Matrix.sum_apply,Matrix.smul_apply,smul_eq_mul,curvatureRicci,
+    Matrix.trace,Matrix.diag_apply,Matrix.mul_apply]
+  apply Finset.sum_congr rfl
+  intro a _
+  apply Finset.sum_congr rfl
+  intro b _
+  rw [g.curvatureMatrix_exchange q hz a b i j]
+
+lemma curvature_trace_contraction (g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) (G : Mat) :
+    ∑ a, ∑ b, (g.matrix q z)⁻¹ a b *
+      ((g.matrix q z)⁻¹*g.curvatureMatrix q z b a*(g.matrix q z)⁻¹*G).trace =
+      ((g.matrix q z)⁻¹*g.curvatureRicci q z*(g.matrix q z)⁻¹*G).trace := by
+  rw [← g.curvature_contraction q hz]
+  simp only [Matrix.mul_sum,Matrix.sum_mul,Matrix.mul_smul,Matrix.smul_mul,
+    trace_sum,trace_smul,smul_eq_mul]
+
+end Anticanonical.SourceSmooth.KaehlerMetric
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/ChernLuConnection.lean b/lean/OAI/Geometry/PrescribedRicci/ChernLuConnection.lean
new file mode 100644
index 0000000000000000000000000000000000000000..3ac232a7594189e96fda2e694c0b58e63647b311
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/ChernLuConnection.lean
@@ -0,0 +1,151 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.CurvatureRicci
+import OAI.Geometry.PrescribedRicci.KahlerGradientEnergy
+import OAI.Geometry.PrescribedRicci.KaehlerTrace
+import OAI.Geometry.PrescribedRicci.MatrixWirtingerExtra
+
+namespace OAI
+
+/-! Chern Lu Connection. -/
+
+section
+
+ 
+
+noncomputable section
+open Matrix Filter Set Topology
+open scoped ContDiff ComplexOrder MatrixOrder Matrix.Norms.Elementwise
+namespace MongeAmpere
+variable {n : Type*} [Fintype n] [DecidableEq n]
+
+lemma trace_lu_W_identity (B G Pg Qg Ph Qh Rg Rh : Matrix n n ℂ)
+    (hG : IsUnit G.det) :
+    (-(B*Qh*B)*(Pg-G*B*Ph)+
+      B*(Rg-(Qg*B*Ph-G*B*Qh*B*Ph+G*B*Rh))).trace =
+    (-(B*(-Rg+Qg*G⁻¹*Pg))).trace +
+      (B*(Qg-Qh*B*G)*G⁻¹*(Pg-G*B*Ph)).trace +
+      (B*(-Rh+Qh*B*Ph)*B*G).trace := by
+  have he : B*(Qg-Qh*B*G)*G⁻¹*(Pg-G*B*Ph) =
+      B*Qg*G⁻¹*Pg-B*Qg*B*Ph-B*Qh*B*Pg+B*Qh*B*G*B*Ph := by
+    simp only [mul_sub,sub_mul]
+    simp only [mul_assoc,Matrix.mul_nonsing_inv_cancel_left G _ hG,
+      Matrix.nonsing_inv_mul_cancel_left G _ hG]
+    noncomm_ring
+  rw [he]
+  simp only [mul_add,add_mul,mul_sub,mul_neg,neg_mul,trace_add,trace_sub,trace_neg]
+  have h1 : (B*G*B*Qh*B*Ph).trace = (B*Qh*B*Ph*B*G).trace := by
+    simpa only [mul_assoc] using Matrix.trace_mul_comm (B*G) (B*Qh*B*Ph)
+  have h2 : (B*G*B*Rh).trace = (B*Rh*B*G).trace := by
+    simpa only [mul_assoc] using Matrix.trace_mul_comm (B*G) (B*Rh)
+  simp only [mul_assoc] at h1 h2 ⊢
+  linear_combination h1-h2
+
+end MongeAmpere
+namespace Anticanonical.SourceSmooth.KaehlerMetric
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+local notation "Mat" => Matrix (Fin d) (Fin d) ℂ
+
+def connectionDifference (h g : KaehlerMetric A) (q : Fin A.count)
+    (z : Coordinates d) (a : Fin d) : Mat :=
+  holDerivative (g.matrix q) z a - g.matrix q z*(h.matrix q z)⁻¹*holDerivative (h.matrix q) z a
+
+lemma connectionDifference_smooth (h g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) (a : Fin d) :
+    ContDiffAt ℝ ∞ (fun y => h.connectionDifference g q y a) z := by
+  have hg := (g.smooth q).contDiffAt ((A.chart q).open_target.mem_nhds hz)
+  have hh := (h.smooth q).contDiffAt ((A.chart q).open_target.mem_nhds hz)
+  have hi : ContDiffAt ℝ ∞ (fun y => (h.matrix q y)⁻¹) z :=
+    ((MongeAmpere.contDiffAt_inv _ (h.positive q z hz).det_pos.ne').restrict_scalars ℝ).comp z hh
+  exact (contDiffAt_holDerivative hg a).sub
+    (matrix_mul_contDiffAt (matrix_mul_contDiffAt hg hi) (contDiffAt_holDerivative hh a))
+
+lemma connectionDifference_adjoint (h g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) (a : Fin d) :
+    (h.connectionDifference g q z a)ᴴ =
+      barDerivative (g.matrix q) z a - barDerivative (h.matrix q) z a*(h.matrix q z)⁻¹*g.matrix q z := by
+  unfold connectionDifference
+  rw [conjTranspose_sub,conjTranspose_mul,conjTranspose_mul,
+    g.holDerivative_adjoint q hz,h.holDerivative_adjoint q hz,
+    (h.positive q z hz).inv.isHermitian.eq,(g.positive q z hz).isHermitian.eq]
+  simp only [mul_assoc]
+
+lemma holRealDeriv_traceMetric (h g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) (a : Fin d) :
+    holRealDeriv ((h.traceMetric g).localExpression q) z a =
+      ((h.matrix q z)⁻¹*h.connectionDifference g q z a).trace := by
+  have hs : ContDiffAt ℝ ∞ ((h.traceMetric g).localExpression q) z :=
+    ((h.traceMetric g).smooth q).contDiffAt ((A.chart q).open_target.mem_nhds hz)
+  have hg := (g.smooth q).contDiffAt ((A.chart q).open_target.mem_nhds hz)
+  have hh := (h.smooth q).contDiffAt ((A.chart q).open_target.mem_nhds hz)
+  have hi : ContDiffAt ℝ ∞ (fun y => (h.matrix q y)⁻¹) z :=
+    ((MongeAmpere.contDiffAt_inv _ (h.positive q z hz).det_pos.ne').restrict_scalars ℝ).comp z hh
+  have he : (fun y => (((h.traceMetric g).localExpression q y):ℂ)) =ᶠ[nhds z]
+      (fun y => ((h.matrix q y)⁻¹*g.matrix q y).trace) := by
+    filter_upwards [(A.chart q).open_target.mem_nhds hz] with y hy
+    rw [h.traceMetric_local g q hy]
+    exact MongeAmpere.trace_mul_real (h.positive q y hy).inv.isHermitian (g.positive q y hy).isHermitian
+  rw [← holDeriv_ofReal (hs.differentiableAt (by simp)),holDeriv_congr he,
+    holDeriv_trace (matrix_mul_differentiableAt (hi.differentiableAt (by simp)) (hg.differentiableAt (by simp))),
+    holDerivative_mul (hi.differentiableAt (by simp)) (hg.differentiableAt (by simp)),
+    holDerivative_inv (hh.differentiableAt (by simp)) (isUnit_iff_ne_zero.mpr (h.positive q z hz).det_pos.ne')]
+  simp only [connectionDifference,neg_mul,mul_sub,trace_add,trace_sub,trace_neg]
+  have hc : ((h.matrix q z)⁻¹*holDerivative (h.matrix q) z a*(h.matrix q z)⁻¹*g.matrix q z).trace =
+      ((h.matrix q z)⁻¹*g.matrix q z*(h.matrix q z)⁻¹*holDerivative (h.matrix q) z a).trace := by
+    simpa only [mul_assoc] using Matrix.trace_mul_comm
+      ((h.matrix q z)⁻¹*holDerivative (h.matrix q) z a) ((h.matrix q z)⁻¹*g.matrix q z)
+  simp only [mul_assoc] at hc ⊢
+  linear_combination -hc
+
+lemma barDerivative_connectionDifference (h g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) (b a : Fin d) :
+    barDerivative (fun y => h.connectionDifference g q y a) z b =
+      barDerivative (fun y => holDerivative (g.matrix q) y a) z b -
+        (barDerivative (g.matrix q) z b*(h.matrix q z)⁻¹*holDerivative (h.matrix q) z a -
+          g.matrix q z*(h.matrix q z)⁻¹*barDerivative (h.matrix q) z b*(h.matrix q z)⁻¹*holDerivative (h.matrix q) z a +
+          g.matrix q z*(h.matrix q z)⁻¹*barDerivative (fun y => holDerivative (h.matrix q) y a) z b) := by
+  have hg := (g.smooth q).contDiffAt ((A.chart q).open_target.mem_nhds hz)
+  have hh := (h.smooth q).contDiffAt ((A.chart q).open_target.mem_nhds hz)
+  have hi : ContDiffAt ℝ ∞ (fun y => (h.matrix q y)⁻¹) z :=
+    ((MongeAmpere.contDiffAt_inv _ (h.positive q z hz).det_pos.ne').restrict_scalars ℝ).comp z hh
+  have pd := (contDiffAt_holDerivative hh a).differentiableAt (by simp)
+  have gd := hg.differentiableAt (by simp)
+  have id := hi.differentiableAt (by simp)
+  unfold connectionDifference
+  rw [barDerivative_sub ((contDiffAt_holDerivative hg a).differentiableAt (by simp))
+      (matrix_mul_differentiableAt (matrix_mul_differentiableAt gd id) pd),
+    barDerivative_mul (matrix_mul_differentiableAt gd id) pd,
+    barDerivative_mul gd id,
+    barDerivative_inv (hh.differentiableAt (by simp)) (isUnit_iff_ne_zero.mpr (h.positive q z hz).det_pos.ne')]
+  simp only [mul_neg,neg_mul,add_mul,mul_assoc,sub_eq_add_neg]
+
+lemma hessian_traceMetric (h g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) (b a : Fin d) :
+    (h.traceMetric g).hessian q z b a =
+      (-((h.matrix q z)⁻¹*g.curvatureMatrix q z b a)).trace +
+        ((h.matrix q z)⁻¹*(h.connectionDifference g q z b)ᴴ*(g.matrix q z)⁻¹*h.connectionDifference g q z a).trace +
+        ((h.matrix q z)⁻¹*h.curvatureMatrix q z b a*(h.matrix q z)⁻¹*g.matrix q z).trace := by
+  have hh := (h.smooth q).contDiffAt ((A.chart q).open_target.mem_nhds hz)
+  have hi : ContDiffAt ℝ ∞ (fun y => (h.matrix q y)⁻¹) z :=
+    ((MongeAmpere.contDiffAt_inv _ (h.positive q z hz).det_pos.ne').restrict_scalars ℝ).comp z hh
+  have hw := (h.connectionDifference_smooth g q hz a).differentiableAt (by simp)
+  have he : (fun y => holRealDeriv ((h.traceMetric g).localExpression q) y a) =ᶠ[nhds z]
+      (fun y => ((h.matrix q y)⁻¹*h.connectionDifference g q y a).trace) := by
+    filter_upwards [(A.chart q).open_target.mem_nhds hz] with y hy
+    exact h.holRealDeriv_traceMetric g q hy a
+  have hs : ContDiffAt ℝ ∞ ((h.traceMetric g).localExpression q) z :=
+    ((h.traceMetric g).smooth q).contDiffAt ((A.chart q).open_target.mem_nhds hz)
+  change PotentialKaehler.potentialMatrix ((h.traceMetric g).localExpression q) z b a = _
+  rw [← barDeriv_holRealDeriv hs,
+    barDeriv_congr he,barDeriv_trace (matrix_mul_differentiableAt (hi.differentiableAt (by simp)) hw),
+    barDerivative_mul (hi.differentiableAt (by simp)) hw,
+    barDerivative_inv (hh.differentiableAt (by simp)) (isUnit_iff_ne_zero.mpr (h.positive q z hz).det_pos.ne'),
+    h.barDerivative_connectionDifference g q hz,h.connectionDifference_adjoint g q hz]
+  exact MongeAmpere.trace_lu_W_identity _ _ _ _ _ _ _ _
+    (isUnit_iff_ne_zero.mpr (g.positive q z hz).det_pos.ne')
+
+end Anticanonical.SourceSmooth.KaehlerMetric
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/ChernLuDifferential.lean b/lean/OAI/Geometry/PrescribedRicci/ChernLuDifferential.lean
new file mode 100644
index 0000000000000000000000000000000000000000..9b875b37d055cb5df44dd2ad0b9c1957fe241f73
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/ChernLuDifferential.lean
@@ -0,0 +1,97 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.ChernLuGradient
+import OAI.Geometry.PrescribedRicci.KahlerGradientEnergy
+import OAI.Geometry.PrescribedRicci.ChernLuConnection
+import OAI.Geometry.PrescribedRicci.GlobalKahlerEnergy
+import OAI.Geometry.PrescribedRicci.KaehlerLogTrace
+
+namespace OAI
+
+/-! Chern Lu Differential. -/
+
+section
+
+ 
+
+noncomputable section
+open Matrix Filter Set Topology
+open scoped ContDiff ComplexOrder MatrixOrder Matrix.Norms.Elementwise
+namespace Anticanonical.SourceSmooth.KaehlerMetric
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+
+def connectionEnergy (h g : KaehlerMetric A) (q : Fin A.count) (z : Coordinates d) : ℝ :=
+  (∑ a, ∑ b, (h.matrix q z)⁻¹ a b *
+    ((h.matrix q z)⁻¹*(h.connectionDifference g q z b)ᴴ*(g.matrix q z)⁻¹*h.connectionDifference g q z a).trace).re
+
+def targetCurvatureTerm (h g : KaehlerMetric A) (q : Fin A.count) (z : Coordinates d) : ℝ :=
+  (∑ a, ∑ b, (h.matrix q z)⁻¹ a b *
+    ((h.matrix q z)⁻¹*g.curvatureMatrix q z b a).trace).re
+
+def sourceRicciTerm (h g : KaehlerMetric A) (q : Fin A.count) (z : Coordinates d) : ℝ :=
+  ((h.matrix q z)⁻¹*h.curvatureRicci q z*(h.matrix q z)⁻¹*g.matrix q z).trace.re
+
+lemma laplacian_traceMetric (h g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) :
+    (h.laplacian (h.traceMetric g)).localExpression q z =
+      -h.targetCurvatureTerm g q z + h.connectionEnergy g q z + h.sourceRicciTerm g q z := by
+  have hl := h.laplacianValue_local (h.traceMetric g) q ((A.chart q).mapsTo_symm hz)
+  change (h.laplacian (h.traceMetric g)).localExpression q z = _ at hl
+  rw [(A.chart q).right_inv hz] at hl
+  rw [hl]
+  simp only [linearizedMongeAmpere,Matrix.trace,Matrix.diag_apply,Matrix.mul_apply]
+  simp only [h.hessian_traceMetric g q hz,trace_neg,mul_add,mul_neg,
+    Finset.sum_add_distrib,Finset.sum_neg_distrib,Complex.add_re,Complex.neg_re]
+  change -h.targetCurvatureTerm g q z+h.connectionEnergy g q z+_ = _
+  congr 1
+  exact congrArg Complex.re (h.curvature_trace_contraction q hz (g.matrix q z))
+
+lemma trace_gradient_bound (h g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) :
+    (h.energy (h.traceMetric g) (h.traceMetric g)).localExpression q z ≤
+      (h.traceMetric g).localExpression q z * h.connectionEnergy g q z := by
+  rw [h.energy_local (h.traceMetric g) (h.traceMetric g) q hz,h.traceMetric_local g q hz]
+  unfold gradientPair connectionEnergy
+  simp only [h.holRealDeriv_traceMetric g q hz]
+  exact MongeAmpere.chernLu_gradient_bound _ _ (h.positive q z hz).inv.posSemidef
+    (g.positive q z hz) (h.connectionDifference g q z)
+
+lemma chernLu_pointwise (h g : KaehlerMetric A) (hd : 0 < d) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) :
+    ((h.traceMetric g).localExpression q z)⁻¹ *
+      (-h.targetCurvatureTerm g q z+h.sourceRicciTerm g q z) ≤
+        (h.laplacian (h.logTrace g hd)).localExpression q z := by
+  have hs : 0 < (h.traceMetric g).localExpression q z :=
+    h.traceMetric_pos g hd ((A.chart q).symm z)
+  have hg := h.trace_gradient_bound g q hz
+  rw [h.laplacian_logTrace g hd q hz,h.laplacian_traceMetric g q hz]
+  have hc := mul_le_mul_of_nonneg_left hg (inv_nonneg.mpr (sq_nonneg ((h.traceMetric g).localExpression q z)))
+  have he : (((h.traceMetric g).localExpression q z)^2)⁻¹ *
+      ((h.traceMetric g).localExpression q z*h.connectionEnergy g q z) =
+      ((h.traceMetric g).localExpression q z)⁻¹ * h.connectionEnergy g q z := by
+    field_simp
+  rw [he] at hc
+  linarith
+
+lemma chernLu_of_component_bounds (h g : KaehlerMetric A) (hd : 0 < d) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) {C R : ℝ}
+    (hC : h.targetCurvatureTerm g q z ≤ C*((h.traceMetric g).localExpression q z)^2)
+    (hR : -R*((h.traceMetric g).localExpression q z)^2 ≤ h.sourceRicciTerm g q z) :
+    -(C+R)*(h.traceMetric g).localExpression q z ≤
+      (h.laplacian (h.logTrace g hd)).localExpression q z := by
+  have hs : 0 < (h.traceMetric g).localExpression q z :=
+    h.traceMetric_pos g hd ((A.chart q).symm z)
+  apply le_trans _ (h.chernLu_pointwise g hd q hz)
+  have hh : -(C+R)*((h.traceMetric g).localExpression q z)^2 ≤
+      -h.targetCurvatureTerm g q z+h.sourceRicciTerm g q z := by linarith
+  have hm := mul_le_mul_of_nonneg_left hh (inv_pos.mpr hs).le
+  have he : ((h.traceMetric g).localExpression q z)⁻¹ *
+      (-(C+R)*((h.traceMetric g).localExpression q z)^2) =
+      -(C+R)*(h.traceMetric g).localExpression q z := by field_simp
+  rwa [he] at hm
+
+end Anticanonical.SourceSmooth.KaehlerMetric
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/ChernLuGradient.lean b/lean/OAI/Geometry/PrescribedRicci/ChernLuGradient.lean
new file mode 100644
index 0000000000000000000000000000000000000000..e5cf98375671927e637943ce154c3b78a10fd57d
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/ChernLuGradient.lean
@@ -0,0 +1,116 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.TraceCauchySchwarz
+
+namespace OAI
+
+/-! Chern Lu Gradient. -/
+
+noncomputable section
+open Matrix
+open scoped ComplexOrder MatrixOrder
+namespace MongeAmpere
+variable {n m : Type*} [Fintype n] [DecidableEq n] [Fintype m] [DecidableEq m]
+
+omit [DecidableEq n] [DecidableEq m] in
+lemma tracePair_sum (B K : Matrix n n ℂ) (W : m → Matrix n n ℂ) (v : m → ℂ) :
+    (B*(∑ i, v i • W i)ᴴ*K*(∑ j, v j • W j)).trace =
+      ∑ i, ∑ j, star (v i)*v j*(B*(W i)ᴴ*K*W j).trace := by
+  simp only [conjTranspose_sum,conjTranspose_smul,Matrix.sum_mul,
+    Matrix.mul_smul,Matrix.smul_mul,trace_sum,trace_smul,smul_eq_mul,Finset.mul_sum]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro i _
+  apply Finset.sum_congr rfl
+  intro j _
+  ring
+
+omit [DecidableEq n] [DecidableEq m] in
+lemma trace_sum_smul (B : Matrix n n ℂ) (W : m → Matrix n n ℂ) (v : m → ℂ) :
+    (B*(∑ i, v i • W i)).trace = ∑ i, v i*(B*W i).trace := by
+  simp only [Matrix.mul_sum,Matrix.mul_smul,trace_sum,trace_smul,smul_eq_mul]
+
+omit [DecidableEq m] in
+lemma trace_schur_posSemidef (B G : Matrix n n ℂ) (hB : B.PosSemidef)
+    (hG : G.PosDef) (W : m → Matrix n n ℂ) :
+    Matrix.PosSemidef (fun i j => ((B*G).trace.re : ℂ)*(B*(W i)ᴴ*G⁻¹*W j).trace -
+      star (B*W i).trace*(B*W j).trace : Matrix m m ℂ) := by
+  let S := (B*G).trace.re
+  let N : Matrix m m ℂ := fun i j => (S:ℂ)*(B*(W i)ᴴ*G⁻¹*W j).trace -
+    star (B*W i).trace*(B*W j).trace
+  have hs (i j : m) : star (B*(W j)ᴴ*G⁻¹*W i).trace = (B*(W i)ᴴ*G⁻¹*W j).trace :=
+    (tracePairCore B G⁻¹ hB hG.inv.posSemidef).conj_inner_symm _ _
+  apply Matrix.posSemidef_iff_dotProduct_mulVec.mpr
+  constructor
+  · apply Matrix.IsHermitian.ext
+    intro i j
+    change star (N j i) = N i j
+    dsimp only [N]
+    rw [star_sub,star_mul,star_mul,star_star,hs]
+    simp only [Complex.star_def,Complex.conj_ofReal]
+    ring
+  · intro v
+    let C := ∑ i, v i • W i
+    have he : star v ⬝ᵥ (N *ᵥ v) = (S:ℂ)*(B*Cᴴ*G⁻¹*C).trace -
+        star (B*C).trace*(B*C).trace := by
+      dsimp only [C]
+      rw [tracePair_sum,trace_sum_smul]
+      simp only [dotProduct,mulVec,Pi.star_apply,N,Finset.mul_sum,Finset.sum_mul,
+        Finset.sum_sub_distrib,mul_sub,sub_mul,star_sum,star_mul]
+      congr 1
+      · apply Finset.sum_congr rfl
+        intro i _
+        apply Finset.sum_congr rfl
+        intro j _
+        ring
+      · rw [Finset.sum_comm]
+        apply Finset.sum_congr rfl
+        intro i _
+        apply Finset.sum_congr rfl
+        intro j _
+        ring
+    change 0 ≤ star v ⬝ᵥ (N *ᵥ v)
+    rw [he]
+    have hq := (tracePairCore B G⁻¹ hB hG.inv.posSemidef).conj_inner_symm C C
+    have hqi : (B*Cᴴ*G⁻¹*C).trace.im = 0 := by
+      have hi := congrArg Complex.im hq
+      simp only [Complex.conj_im] at hi
+      change -(B*Cᴴ*G⁻¹*C).trace.im = (B*Cᴴ*G⁻¹*C).trace.im at hi
+      linarith
+    have hCS := trace_cauchySchwarz B G C hB hG
+    apply Complex.nonneg_iff.mpr
+    constructor
+    · simp only [Complex.sub_re,Complex.mul_re,Complex.ofReal_re,Complex.ofReal_im,zero_mul,sub_zero,
+        Complex.star_def,Complex.conj_re,Complex.conj_im]
+      rw [Complex.sq_norm,Complex.normSq_apply] at hCS
+      dsimp only [S]
+      nlinarith
+    · simp only [Complex.sub_im,Complex.mul_im,Complex.ofReal_re,Complex.ofReal_im,zero_mul,add_zero,
+        hqi,mul_zero,Complex.star_def,Complex.conj_re,Complex.conj_im]
+      ring
+
+lemma chernLu_gradient_bound (B G : Matrix n n ℂ) (hB : B.PosSemidef)
+    (hG : G.PosDef) (W : n → Matrix n n ℂ) :
+    (∑ a, ∑ b, B a b*star (B*W b).trace*(B*W a).trace).re ≤
+      (B*G).trace.re * (∑ a, ∑ b, B a b*(B*(W b)ᴴ*G⁻¹*W a).trace).re := by
+  have hn := EllipticKernel.trace_mul_nonneg hB (trace_schur_posSemidef B G hB hG W)
+  simp only [Matrix.trace,Matrix.diag_apply,Matrix.mul_apply] at hn
+  have he : (∑ a, ∑ b, B a b *
+      (((B*G).trace.re:ℂ)*(B*(W b)ᴴ*G⁻¹*W a).trace-star (B*W b).trace*(B*W a).trace)) =
+      ((B*G).trace.re:ℂ)*(∑ a, ∑ b, B a b*(B*(W b)ᴴ*G⁻¹*W a).trace) -
+        (∑ a, ∑ b, B a b*star (B*W b).trace*(B*W a).trace) := by
+    simp only [Finset.mul_sum,← Finset.sum_sub_distrib]
+    apply Finset.sum_congr rfl
+    intro a _
+    apply Finset.sum_congr rfl
+    intro b _
+    ring
+  change 0 ≤ (∑ a, ∑ b, B a b *
+      (((B*G).trace.re:ℂ)*(B*(W b)ᴴ*G⁻¹*W a).trace-star (B*W b).trace*(B*W a).trace)).re at hn
+  rw [he,Complex.sub_re,Complex.re_ofReal_mul] at hn
+  linarith
+
+end MongeAmpere
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/ChernPairDifferential.lean b/lean/OAI/Geometry/PrescribedRicci/ChernPairDifferential.lean
new file mode 100644
index 0000000000000000000000000000000000000000..5d440467128578e88391f2c9c89ac0bbbf65cbe5
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/ChernPairDifferential.lean
@@ -0,0 +1,121 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.HermitianPair
+import OAI.Geometry.PrescribedRicci.LocalKahlerEnergy
+import OAI.Geometry.PrescribedRicci.WirtingerScalar
+
+namespace OAI
+
+/-! Chern Pair Differential. -/
+
+noncomputable section
+open Matrix Filter Set Topology
+open scoped ContDiff ComplexOrder
+namespace Anticanonical.SourceSmooth.KaehlerMetric
+open MongeAmpere
+variable {d : ℕ} {n : Type*} [Fintype n]
+
+def holVector (f : Coordinates d → n → ℂ) (z : Coordinates d) (a : Fin d) : n → ℂ :=
+  fun i => holDeriv (fun y => f y i) z a
+
+def barVector (f : Coordinates d → n → ℂ) (z : Coordinates d) (a : Fin d) : n → ℂ :=
+  fun i => barDeriv (fun y => f y i) z a
+
+def holArray (K : Coordinates d → Matrix n n ℂ) (z : Coordinates d) (a : Fin d) : Matrix n n ℂ :=
+  fun i j => holDeriv (fun y => K y i j) z a
+
+def barArray (K : Coordinates d → Matrix n n ℂ) (z : Coordinates d) (a : Fin d) : Matrix n n ℂ :=
+  fun i j => barDeriv (fun y => K y i j) z a
+
+lemma hermPair_smooth {K : Coordinates d → Matrix n n ℂ} {f g : Coordinates d → n → ℂ}
+    {z : Coordinates d} (hK : ∀ i j, ContDiffAt ℝ ∞ (fun y => K y i j) z)
+    (hf : ∀ i, ContDiffAt ℝ ∞ (fun y => f y i) z)
+    (hg : ∀ i, ContDiffAt ℝ ∞ (fun y => g y i) z) :
+    ContDiffAt ℝ ∞ (fun y => hermPair (K y) (f y) (g y)) z := by
+  simp only [hermPair_expand]
+  apply ContDiffAt.sum
+  intro i _
+  apply ContDiffAt.sum
+  intro j _
+  exact ((((starL' ℝ : ℂ ≃L[ℝ] ℂ).contDiff.contDiffAt).comp z (hf i)).mul (hK i j)).mul (hg j)
+
+lemma hermPair_hol {K : Coordinates d → Matrix n n ℂ} {f g : Coordinates d → n → ℂ}
+    {z : Coordinates d} (hK : ∀ i j, DifferentiableAt ℝ (fun y => K y i j) z)
+    (hf : ∀ i, DifferentiableAt ℝ (fun y => f y i) z)
+    (hg : ∀ i, DifferentiableAt ℝ (fun y => g y i) z) (a : Fin d) :
+    holDeriv (fun y => hermPair (K y) (f y) (g y)) z a =
+      hermPair (K z) (barVector f z a) (g z)+hermPair (holArray K z a) (f z) (g z)+
+        hermPair (K z) (f z) (holVector g z a) := by
+  have hterm (i j : n) : DifferentiableAt ℝ (fun y => star (f y i)*K y i j*g y j) z :=
+    (((hf i).star).fun_mul (hK i j)).fun_mul (hg j)
+  have hrow (i : n) : DifferentiableAt ℝ (fun y => ∑ j, star (f y i)*K y i j*g y j) z :=
+    DifferentiableAt.fun_sum (u:=Finset.univ) (fun j _ => hterm i j)
+  simp only [hermPair_expand]
+  rw [holDeriv_sum hrow]
+  simp_rw [holDeriv_sum (fun j => hterm _ j)]
+  have he (i j : n) : holDeriv (fun y => star (f y i)*K y i j*g y j) z a =
+      (holDeriv (fun y => star (f y i)) z a*K z i j+star (f z i)*holDeriv (fun y => K y i j) z a)*g z j+
+        (star (f z i)*K z i j)*holDeriv (fun y => g y j) z a := by
+    rw [holDeriv_mul (((hf i).star).fun_mul (hK i j)) (hg j),
+      holDeriv_mul (hf i).star (hK i j)]
+  simp_rw [he,holDeriv_star]
+  simp only [holArray,holVector,barVector,add_mul,Finset.sum_add_distrib]
+
+lemma hermPair_bar {K : Coordinates d → Matrix n n ℂ} {f g : Coordinates d → n → ℂ}
+    {z : Coordinates d} (hK : ∀ i j, DifferentiableAt ℝ (fun y => K y i j) z)
+    (hf : ∀ i, DifferentiableAt ℝ (fun y => f y i) z)
+    (hg : ∀ i, DifferentiableAt ℝ (fun y => g y i) z) (a : Fin d) :
+    barDeriv (fun y => hermPair (K y) (f y) (g y)) z a =
+      hermPair (K z) (holVector f z a) (g z)+hermPair (barArray K z a) (f z) (g z)+
+        hermPair (K z) (f z) (barVector g z a) := by
+  have hterm (i j : n) : DifferentiableAt ℝ (fun y => star (f y i)*K y i j*g y j) z :=
+    (((hf i).star).fun_mul (hK i j)).fun_mul (hg j)
+  have hrow (i : n) : DifferentiableAt ℝ (fun y => ∑ j, star (f y i)*K y i j*g y j) z :=
+    DifferentiableAt.fun_sum (u:=Finset.univ) (fun j _ => hterm i j)
+  simp only [hermPair_expand]
+  rw [barDeriv_sum hrow]
+  simp_rw [barDeriv_sum (fun j => hterm _ j)]
+  have he (i j : n) : barDeriv (fun y => star (f y i)*K y i j*g y j) z a =
+      (barDeriv (fun y => star (f y i)) z a*K z i j+star (f z i)*barDeriv (fun y => K y i j) z a)*g z j+
+        (star (f z i)*K z i j)*barDeriv (fun y => g y j) z a := by
+    rw [barDeriv_mul (((hf i).star).fun_mul (hK i j)) (hg j),
+      barDeriv_mul (hf i).star (hK i j)]
+  simp_rw [he,barDeriv_star]
+  simp only [barArray,holVector,barVector,add_mul,Finset.sum_add_distrib]
+
+def covHol (C : Coordinates d → Fin d → Matrix n n ℂ) (f : Coordinates d → n → ℂ)
+    (z : Coordinates d) (a : Fin d) : n → ℂ := holVector f z a+C z a*ᵥ f z
+
+lemma covHol_smooth {C : Coordinates d → Fin d → Matrix n n ℂ} {f : Coordinates d → n → ℂ}
+    {z : Coordinates d} (hC : ∀ a i j, ContDiffAt ℝ ∞ (fun y => C y a i j) z)
+    (hf : ∀ i, ContDiffAt ℝ ∞ (fun y => f y i) z) (a : Fin d) (i : n) :
+    ContDiffAt ℝ ∞ (fun y => covHol C f y a i) z := by
+  change ContDiffAt ℝ ∞ (fun y => holDeriv (fun w => f w i) y a+∑ j, C y a i j*f y j) z
+  exact (contDiffAt_holDeriv (hf i) a).add
+    (ContDiffAt.sum (fun j _ => (hC a i j).mul (hf j)))
+
+lemma hermPair_hol_compatible {K : Coordinates d → Matrix n n ℂ}
+    {C : Coordinates d → Fin d → Matrix n n ℂ} {f g : Coordinates d → n → ℂ}
+    {z : Coordinates d} (hK : ∀ i j, DifferentiableAt ℝ (fun y => K y i j) z)
+    (hf : ∀ i, DifferentiableAt ℝ (fun y => f y i) z)
+    (hg : ∀ i, DifferentiableAt ℝ (fun y => g y i) z) (a : Fin d)
+    (hcomp : holArray K z a = K z*C z a) :
+    holDeriv (fun y => hermPair (K y) (f y) (g y)) z a =
+      hermPair (K z) (barVector f z a) (g z)+hermPair (K z) (f z) (covHol C g z a) := by
+  rw [hermPair_hol hK hf hg,hcomp,covHol,hermPair_add_right,← hermPair_mulVec_right]
+  abel
+
+lemma hermPair_bar_compatible {K : Coordinates d → Matrix n n ℂ}
+    {C : Coordinates d → Fin d → Matrix n n ℂ} {f g : Coordinates d → n → ℂ}
+    {z : Coordinates d} (hK : ∀ i j, DifferentiableAt ℝ (fun y => K y i j) z)
+    (hf : ∀ i, DifferentiableAt ℝ (fun y => f y i) z)
+    (hg : ∀ i, DifferentiableAt ℝ (fun y => g y i) z) (a : Fin d)
+    (hcomp : barArray K z a = (C z a)ᴴ*K z) :
+    barDeriv (fun y => hermPair (K y) (f y) (g y)) z a =
+      hermPair (K z) (covHol C f z a) (g z)+hermPair (K z) (f z) (barVector g z a) := by
+  rw [hermPair_bar hK hf hg,hcomp,covHol,hermPair_add_left,← hermPair_mulVec_left]
+
+end Anticanonical.SourceSmooth.KaehlerMetric
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/ChernRough.lean b/lean/OAI/Geometry/PrescribedRicci/ChernRough.lean
new file mode 100644
index 0000000000000000000000000000000000000000..442a93a3831396450a79c7dc4306dea76d0192e0
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/ChernRough.lean
@@ -0,0 +1,144 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.ChernConnection
+import OAI.Geometry.PrescribedRicci.ChernBianchi
+import OAI.Geometry.PrescribedRicci.CurvatureCompactBounds
+import OAI.Geometry.PrescribedRicci.WirtingerScalar
+
+namespace OAI
+
+/-! Chern Rough. -/
+
+section
+
+ 
+noncomputable section
+open Matrix Filter Set Topology
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace Anticanonical.SourceSmooth.KaehlerMetric
+open MongeAmpere
+variable {d : ℕ}
+local notation "Mat" => Matrix (Fin d) (Fin d) ℂ
+
+lemma holDerivative_sum_smul {ι : Type*} [Fintype ι]
+    {f : ι → Coordinates d → ℂ} {M : ι → Coordinates d → Mat} {z : Coordinates d}
+    (hf : ∀ i, DifferentiableAt ℝ (f i) z) (hM : ∀ i, DifferentiableAt ℝ (M i) z)
+    (a : Fin d) :
+    holDerivative (fun y => ∑ i, f i y • M i y) z a =
+      ∑ i, (holDeriv (f i) z a • M i z + f i z • holDerivative (M i) z a) := by
+  have hm (i : ι) (k j : Fin d) := (differentiableAt_pi.mp (differentiableAt_pi.mp (hM i) k) j)
+  have ht (i : ι) : DifferentiableAt ℝ (fun y => f i y • M i y) z := (hf i).smul (hM i)
+  have hs : DifferentiableAt ℝ (fun y => ∑ i, f i y • M i y) z :=
+    DifferentiableAt.fun_sum (u:=Finset.univ) (fun i _ => ht i)
+  ext k j
+  rw [← holDeriv_entry hs]
+  simp only [Matrix.sum_apply,Matrix.smul_apply,smul_eq_mul]
+  rw [holDeriv_sum (fun i => (hf i).fun_mul (hm i k j))]
+  apply Finset.sum_congr rfl
+  intro i _
+  rw [holDeriv_mul (hf i) (hm i k j),holDeriv_entry (hM i)]
+  rfl
+
+variable {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+
+lemma inverse_derivative_curvature_correction (g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) (a : Fin d) :
+    ∑ u, ∑ b, holDerivative (fun y => (g.matrix q y)⁻¹) z a u b • g.chernCurvature q z b u =
+      -(∑ u, ∑ b, ((g.matrix q z)⁻¹ u b) •
+          (∑ v, (g.chernConnection q z u v a) • g.chernCurvature q z b v)) := by
+  have hg := (g.smooth q).contDiffAt ((A.chart q).open_target.mem_nhds hz)
+  rw [holDerivative_inv (hg.differentiableAt (by simp))
+    (isUnit_iff_ne_zero.mpr (g.positive q z hz).det_pos.ne')]
+  change ∑ u, ∑ b, (-(g.chernConnection q z a*(g.matrix q z)⁻¹)) u b •
+    g.chernCurvature q z b u = _
+  simp only [Matrix.neg_apply,Matrix.mul_apply,neg_smul,Finset.sum_neg_distrib,
+    Finset.sum_smul,Finset.smul_sum,smul_smul]
+  congr 1
+  conv_lhs =>
+    rw [Finset.sum_comm]
+    arg 2
+    ext b
+    rw [Finset.sum_comm]
+  conv_lhs => rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro u _
+  apply Finset.sum_congr rfl
+  intro b _
+  apply Finset.sum_congr rfl
+  intro v _
+  rw [g.chernConnection_torsion q hz a v u]
+  congr 1
+  ring
+
+lemma chernCurvature_rough (g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) (a : Fin d) :
+    ∑ u, ∑ b, ((g.matrix q z)⁻¹ u b) •
+      (holDerivative (fun y => g.chernCurvature q y b a) z u +
+        g.chernConnection q z u*g.chernCurvature q z b a-
+        g.chernCurvature q z b a*g.chernConnection q z u-
+        ∑ v, g.chernConnection q z u v a • g.chernCurvature q z b v) =
+      (g.matrix q z)⁻¹*holDerivative (g.curvatureRicci q) z a-
+        ((g.matrix q z)⁻¹*g.curvatureRicci q z)*g.chernConnection q z a := by
+  have hg := (g.smooth q).contDiffAt ((A.chart q).open_target.mem_nhds hz)
+  have hi : ContDiffAt ℝ ∞ (fun y => (g.matrix q y)⁻¹) z :=
+    ((MongeAmpere.contDiffAt_inv _ (g.positive q z hz).det_pos.ne').restrict_scalars ℝ).comp z hg
+  have hI (u b : Fin d) : DifferentiableAt ℝ (fun y => (g.matrix q y)⁻¹ u b) z :=
+    (contDiffAt_pi.mp (contDiffAt_pi.mp hi u) b).differentiableAt (by simp)
+  have hR (b u : Fin d) := (g.chernCurvature_smooth q hz b u).differentiableAt (by simp)
+  have hr := (g.curvatureRicci_smooth q hz).differentiableAt (by simp)
+  have he : (fun y => ∑ ub : Fin d × Fin d,
+      (g.matrix q y)⁻¹ ub.1 ub.2 • g.chernCurvature q y ub.2 ub.1) =ᶠ[nhds z]
+        (fun y => (g.matrix q y)⁻¹*g.curvatureRicci q y) := by
+    filter_upwards [(A.chart q).open_target.mem_nhds hz] with y hy
+    simpa only [Fintype.sum_prod_type] using g.chernCurvature_contraction q hy
+  have hd := holDerivative_congr he a
+  have hdL := holDerivative_sum_smul
+    (f:=fun ub : Fin d × Fin d => fun y => (g.matrix q y)⁻¹ ub.1 ub.2)
+    (M:=fun ub : Fin d × Fin d => fun y => g.chernCurvature q y ub.2 ub.1)
+    (fun ub => hI ub.1 ub.2) (fun ub => hR ub.2 ub.1) a
+  rw [hdL] at hd
+  have hdR := holDerivative_mul (f:=fun y => (g.matrix q y)⁻¹)
+    (g:=g.curvatureRicci q) (hi.differentiableAt (by simp)) hr a
+  rw [hdR] at hd
+  have hdi := holDerivative_inv (hg.differentiableAt (by simp))
+    (isUnit_iff_ne_zero.mpr (g.positive q z hz).det_pos.ne') a
+  rw [hdi] at hd
+  simp only [Fintype.sum_prod_type,Finset.sum_add_distrib] at hd
+  simp_rw [holDeriv_entry (hi.differentiableAt (by simp))] at hd
+  rw [g.inverse_derivative_curvature_correction q hz] at hd
+  simp only [neg_mul,mul_assoc] at hd
+  rw [← mul_assoc (g.matrix q z)⁻¹ (holDerivative (g.matrix q) z a)
+    ((g.matrix q z)⁻¹*g.curvatureRicci q z)] at hd
+  change -(∑ u, ∑ b, ((g.matrix q z)⁻¹ u b) •
+      (∑ v, g.chernConnection q z u v a • g.chernCurvature q z b v)) +
+      (∑ u, ∑ b, ((g.matrix q z)⁻¹ u b) •
+        holDerivative (fun y => g.chernCurvature q y b u) z a) =
+      -(g.chernConnection q z a*((g.matrix q z)⁻¹*g.curvatureRicci q z))+
+        (g.matrix q z)⁻¹*holDerivative (g.curvatureRicci q) z a at hd
+  conv_lhs =>
+    arg 2
+    ext u
+    arg 2
+    ext b
+    rw [g.chern_bianchi q hz]
+  have hleft : (∑ u, ∑ b, ((g.matrix q z)⁻¹ u b) •
+      (g.chernConnection q z a*g.chernCurvature q z b u)) =
+        g.chernConnection q z a*((g.matrix q z)⁻¹*g.curvatureRicci q z) := by
+    rw [← g.chernCurvature_contraction q hz]
+    simp only [Matrix.mul_sum,Matrix.mul_smul]
+  have hright : (∑ u, ∑ b, ((g.matrix q z)⁻¹ u b) •
+      (g.chernCurvature q z b u*g.chernConnection q z a)) =
+        ((g.matrix q z)⁻¹*g.curvatureRicci q z)*g.chernConnection q z a := by
+    rw [← g.chernCurvature_contraction q hz]
+    simp only [Matrix.sum_mul,Matrix.smul_mul]
+  simp only [smul_sub,smul_add,Finset.sum_sub_distrib,Finset.sum_add_distrib,hleft,hright]
+  have hh := congrArg (fun M : Mat => M+
+    g.chernConnection q z a*((g.matrix q z)⁻¹*g.curvatureRicci q z)-
+      ((g.matrix q z)⁻¹*g.curvatureRicci q z)*g.chernConnection q z a) hd
+  convert hh using 1 <;> abel
+
+end Anticanonical.SourceSmooth.KaehlerMetric
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/CofactorBilin.lean b/lean/OAI/Geometry/PrescribedRicci/CofactorBilin.lean
new file mode 100644
index 0000000000000000000000000000000000000000..94d56b756109be247cc4baf9780b32710db47544
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/CofactorBilin.lean
@@ -0,0 +1,76 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.FiniteCofactor
+
+namespace OAI
+
+/-! Cofactor Bilin. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff SchwartzMap Classical BoundedContinuousFunction Matrix.Norms.Elementwise
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart FrozenPoisson
+variable {d : ℕ}
+local instance : NormedAddCommGroup (Bilin (EC d)) := inferInstance
+local instance : NormedSpace ℝ (Bilin (EC d)) := inferInstance
+
+def cofactorBilinLinear (c : ℂ) (M : Matrix (Fin d) (Fin d) ℂ) : Bilin (EC d) →ₗ[ℝ] ℝ where
+  toFun B := (c * NonlinearHessian.detDifferential M (GluingData.cofactorMatrix B)).re
+  map_add' B C := by
+    simp only [GluingData.cofactorMatrix,pullBilin_add,hermitianPartMatrix_add,map_add,mul_add,Complex.add_re]
+  map_smul' r B := by
+    simp only [GluingData.cofactorMatrix,pullBilin_smul,hermitianPartMatrix_smul,map_smul,
+      RingHom.id_apply,Complex.real_smul,smul_eq_mul]
+    simp only [Complex.mul_re,Complex.mul_im,
+      Complex.ofReal_re,Complex.ofReal_im,zero_mul,sub_zero,add_zero]
+    ring
+
+def cofactorBilin (c : ℂ) (M : Matrix (Fin d) (Fin d) ℂ) : SecondOrder (EC d) :=
+  by
+    change Bilin (EC d) →L[ℝ] ℝ
+    exact (cofactorBilinLinear c M).toContinuousLinearMap
+
+lemma cofactorBilin_apply (c : ℂ) (M : Matrix (Fin d) (Fin d) ℂ) (B : Bilin (EC d)) :
+    cofactorBilin c M B = (c * NonlinearHessian.detDifferential M (GluingData.cofactorMatrix B)).re := rfl
+
+lemma cofactorBilin_expansion (c : ℂ) (M : Matrix (Fin d) (Fin d) ℂ) (B : Bilin (EC d)) :
+    (cofactorBilin c M B : ℂ) = ∑ i, ∑ j,
+      (cofactorBilin c M (rankTwo (stdOrthonormalBasis ℝ (EC d) i)
+        (stdOrthonormalBasis ℝ (EC d) j)) : ℂ) *
+        (B (stdOrthonormalBasis ℝ (EC d) i) (stdOrthonormalBasis ℝ (EC d) j) : ℂ) := by
+  conv_lhs => rw [bilin_expansion (stdOrthonormalBasis ℝ (EC d)) B]
+  simp only [map_sum,map_smul,smul_eq_mul,Complex.ofReal_sum,Complex.ofReal_mul]
+  apply Finset.sum_congr rfl
+  intro i _
+  apply Finset.sum_congr rfl
+  intro j _
+  ring
+
+lemma cofactor_smooth_expression {U : EC d → ℝ} {y : EC d}
+    (hU : ContDiffAt ℝ ∞ U y) (f : 𝓢(EC d,ℂ))
+    (he : (f : EC d → ℂ) =ᶠ[𝓝 y] Complex.ofRealCLM ∘ U)
+    (c : ℂ) (M : Matrix (Fin d) (Fin d) ℂ) :
+    ((c * NonlinearHessian.detDifferential M (fun i j => hessianEntrySchwartz i j f y)).re : ℂ) =
+      ∑ i, ∑ j, (cofactorBilin c M (rankTwo (stdOrthonormalBasis ℝ (EC d) i)
+        (stdOrthonormalBasis ℝ (EC d) j)) : ℂ) *
+        fderiv ℝ (fderiv ℝ f) y (stdOrthonormalBasis ℝ (EC d) i) (stdOrthonormalBasis ℝ (EC d) j) := by
+  simp_rw [hessianEntrySchwartz_real hU f he]
+  change (cofactorBilin c M (fderiv ℝ (fderiv ℝ U) y) : ℂ) = _
+  rw [cofactorBilin_expansion]
+  apply Finset.sum_congr rfl
+  intro i _
+  apply Finset.sum_congr rfl
+  intro j _
+  rw [(he.fderiv (𝕜:=ℝ)).fderiv_eq,second_postcomp_general hU]
+  rfl
+
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/CofactorCore.lean b/lean/OAI/Geometry/PrescribedRicci/CofactorCore.lean
new file mode 100644
index 0000000000000000000000000000000000000000..e7868d7aa993acebd6103887191f41eec90e419f
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/CofactorCore.lean
@@ -0,0 +1,136 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.FiniteCofactor
+import OAI.Geometry.PrescribedRicci.FiniteCofactorPatch
+
+namespace OAI
+
+/-! Cofactor Core. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff SchwartzMap Classical BoundedContinuousFunction Matrix.Norms.Elementwise
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart FrozenPoisson MetricLocalization
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+def cofactorComplexCore (p : ι) (B : Bilin (EC d)) (f : Smooth A) : Smooth A :=
+  ⟨fun x => D.inverseDet p x * NonlinearHessian.detDifferential
+      (fun i j => D.metricEntry p i j x + D.localizedHessian p i j f x) (cofactorMatrix B),
+    fun q => by
+      apply (D.inverseDet p).smooth q |>.mul
+      apply ContDiffOn.clm_apply
+      · apply ((NonlinearHessian.determinant_contDiff d).fderiv_right (m:=∞) (by simp)).comp_contDiffOn
+        apply contDiffOn_pi.mpr
+        intro i
+        apply contDiffOn_pi.mpr
+        intro j
+        exact ((D.metricEntry p i j).smooth q).add ((D.localizedHessian p i j f).smooth q)
+      · exact contDiffOn_const⟩
+
+def cofactorScalarCore (p : ι) (B : Bilin (EC d)) (f : Smooth A) : Smooth A :=
+  Smooth.realify (D.cofactorComplexCore p B f)
+
+lemma cofactorScalarCore_apply (p : ι) (B : Bilin (EC d)) (f : Smooth A) (x : X) :
+    D.cofactorScalarCore p B f x =
+      ((D.inverseDet p x * NonlinearHessian.detDifferential
+        (fun i j => D.metricEntry p i j x + D.localizedHessian p i j f x) (cofactorMatrix B)).re : ℂ) := by
+  exact Smooth.realify_apply _ x
+
+lemma cofactorScalar_embed (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (p : ι) (B : Bilin (EC d)) (f : Smooth A) :
+    D.cofactorScalar k hk p B (D.localizers.embed ((k:ℝ)+2) f) =
+      D.localizers.embed (k:ℝ) (D.cofactorScalarCore p B f) := by
+  have hs : (Module.finrank ℝ (EC d):ℝ) < 2*(k:ℝ) := by
+    have hh : (Module.finrank ℝ (EC d):ℝ) < (k:ℝ) := by exact_mod_cast hk
+    linarith [Nat.cast_nonneg (α:=ℝ) k]
+  apply D.localizers.strong_injective (k:ℝ) hs
+  ext x
+  rw [D.cofactorScalar_strong,D.localizers.strong_embed _ hs]
+  simp_rw [D.completedHessian_embed,D.localizers.strong_embed _ hs]
+  exact (D.cofactorScalarCore_apply p B f x).symm
+
+def cofactorLocalCore (p : ι) (i j : BasisIndex d) (f : Smooth A) : 𝓢(EC d,ℂ) :=
+  (D.patch p).coefficient i j +
+    localize A (D.patch p).index (cutoffGlobal (D.patch p).index (D.cutoff p))
+      (cutoffGlobal_support _ _)
+      (D.cofactorScalarCore p (rankTwo (stdOrthonormalBasis ℝ (EC d) i)
+        (stdOrthonormalBasis ℝ (EC d) j)) f -
+       D.cofactorScalarCore p (rankTwo (stdOrthonormalBasis ℝ (EC d) i)
+        (stdOrthonormalBasis ℝ (EC d) j)) 0)
+
+lemma cofactorLocal_embed (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (p : ι) (i j : BasisIndex d) (f : Smooth A) :
+    D.cofactorLocal k hk p i j (D.localizers.embed ((k:ℝ)+2) f) =
+      schwartzCoord (k:ℝ) (D.cofactorLocalCore p i j f) := by
+  rw [cofactorLocal,D.cofactorScalar_embed]
+  have h0 := D.cofactorScalar_embed k hk p
+    (rankTwo (stdOrthonormalBasis ℝ (EC d) i) (stdOrthonormalBasis ℝ (EC d) j)) 0
+  rw [map_zero] at h0
+  rw [h0,← map_sub,D.completedLocalize_embed,← schwartzCoord_add]
+  rfl
+
+lemma cofactorBCF_embed (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (hs : (Module.finrank ℝ (EC d):ℝ) < 2*(k:ℝ)) (p : ι) (f : Smooth A) :
+    D.cofactorBCF k hk hs p (D.localizers.embed ((k:ℝ)+2) f) =
+      coefficientBCF (D.cofactorLocalCore p · · f) := by
+  funext i j
+  rw [cofactorBCF,D.cofactorLocal_embed,strongEmbedding_schwartz]
+  rfl
+
+lemma cofactorScalarCore_zero_source (p : ι) (B : Bilin (EC d)) {x : X}
+    (hx : x ∈ tsupport (D.localizers.weight p : X → ℂ)) :
+    D.cofactorScalarCore p B 0 x =
+      (traceBilin (g.matrix (D.patch p).index (A.chart (D.patch p).index x)) B : ℂ) := by
+  let k := Module.finrank ℝ (EC d)+1
+  have hk : Module.finrank ℝ (EC d) < k := by omega
+  have hs : (Module.finrank ℝ (EC d):ℝ) < 2*(k:ℝ) := by
+    dsimp [k]
+    push_cast
+    linarith [Nat.cast_nonneg (α:=ℝ) (Module.finrank ℝ (EC d))]
+  have hh := D.cofactorScalar_zero_source k hk p B hx
+  have he := D.cofactorScalar_embed k hk p B 0
+  rw [map_zero] at he
+  rw [he,D.localizers.strong_embed _ hs] at hh
+  exact hh
+
+lemma cofactorLocalCore_source (p : ι) (i j : BasisIndex d) (f : Smooth A) {x : X}
+    (hx : x ∈ tsupport (D.localizers.weight p : X → ℂ)) :
+    extendedCoefficient (D.patch p).matrix (coefficientBCF (D.cofactorLocalCore p · · f)) i j
+      (A.euclideanChart (D.patch p).index x) =
+      D.cofactorScalarCore p (rankTwo (stdOrthonormalBasis ℝ (EC d) i)
+        (stdOrthonormalBasis ℝ (EC d) j)) f x := by
+  have hs := D.patch_source p hx
+  have hy := (A.euclideanChart (D.patch p).index).mapsTo hs
+  have he := (A.euclideanChart (D.patch p).index).left_inv hs
+  change (traceBilin (D.patch p).matrix (rankTwo _ _) : ℂ) +
+    D.cofactorLocalCore p i j f (A.euclideanChart (D.patch p).index x) = _
+  rw [cofactorLocalCore,_root_.add_apply,localize_apply,localizeFun_apply A _ _ hy,he]
+  have hc : cutoffGlobal (D.patch p).index (D.cutoff p) x = 1 := by
+    change (if x ∈ (A.euclideanChart (D.patch p).index).source then _ else _) = _
+    rw [ite_eq_left hs,D.cutoff_one p x hx]
+  rw [hc,one_mul]
+  change _ + ((D.patch p).coefficient i j (A.euclideanChart (D.patch p).index x) +
+    (D.cofactorScalarCore p _ f x-D.cofactorScalarCore p _ 0 x)) = _
+  rw [← add_assoc]
+  have hp := (D.patch p).equals (A.euclideanChart (D.patch p).index x)
+    (D.support_patch p hx).2 i j
+  change (traceBilin (D.patch p).matrix (rankTwo _ _) : ℂ) +
+    (D.patch p).coefficient i j (A.euclideanChart (D.patch p).index x) = _ at hp
+  rw [hp,D.cofactorScalarCore_zero_source p _ hx]
+  simp only [ComplexAtlas.euclideanChart_apply,ContinuousLinearEquiv.apply_symm_apply]
+  ring
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/CofactorCoreEquation.lean b/lean/OAI/Geometry/PrescribedRicci/CofactorCoreEquation.lean
new file mode 100644
index 0000000000000000000000000000000000000000..9761780af2b34dbe5a5809b8003da3a978c71cc3
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/CofactorCoreEquation.lean
@@ -0,0 +1,81 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.PatchCutoffs
+import OAI.Geometry.PrescribedPotential.PotentialDensityDifferential
+import OAI.Geometry.PrescribedPotential.RealNonlinearCommutator
+import OAI.Geometry.PrescribedPotential.RealSobolev
+import OAI.Geometry.PrescribedPotential.SobolevProduct
+import OAI.Geometry.PrescribedRicci.CofactorLocalization
+
+namespace OAI
+
+/-! Cofactor Core Equation. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Matrix _root_.MeasureTheory _root_.OAI.MeasureTheory
+open scoped ContDiff SchwartzMap Classical BoundedContinuousFunction Matrix.Norms.Elementwise
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart FrozenPoisson MetricLocalization
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  [ConnectedSpace X] {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+local instance cofactorCoreEquationNG (s : ℝ) : NormedAddCommGroup (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedAddCommGroup
+local instance cofactorCoreEquationNS (s : ℝ) : NormedSpace ℝ (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedSpace
+local instance cofactorCoreEquationTG (s : ℝ) : IsTopologicalAddGroup (D.localizers.RealSobolev s) :=
+  Submodule.isTopologicalAddGroup _
+local instance cofactorCoreEquationCS (s : ℝ) : ContinuousSMul ℝ (D.localizers.RealSobolev s) :=
+  SMulMemClass.continuousSMul _
+
+omit [ConnectedSpace X] in
+lemma cofactor_core_forcing (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (p : ι) (φ ψ : RealSmooth A) :
+    schwartzCoord (k:ℝ) (SchwartzMap.smulLeftCLM ℂ (D.weightCore p)
+      (schwartzDifferential (extendedCoefficient (D.patch p).matrix
+        (coefficientBCF (D.cofactorLocalCore p · · φ.val))) (D.outerCore p ψ.val))) =
+      (D.realVolumeDerivative k hk (D.localizers.realEmbed ((k:ℝ)+2) φ)
+        (D.localizers.realEmbed ((k:ℝ)+2) ψ)).val.val p := by
+  have hs : (Module.finrank ℝ (EC d):ℝ) < 2*(k:ℝ) := by
+    have hh : (Module.finrank ℝ (EC d):ℝ) < (k:ℝ) := by exact_mod_cast hk
+    linarith [Nat.cast_nonneg (α:=ℝ) k]
+  apply realize_injective (k:ℝ)
+  rw [← strongEmbedding_distribution (k:ℝ) hs,← strongEmbedding_distribution (k:ℝ) hs]
+  congr 1
+  ext y
+  rw [strongEmbedding_schwartz,SchwartzMap.toBoundedContinuousFunction_apply,D.localizers.strong_localized (k:ℝ) hs,
+    SchwartzMap.smulLeftCLM_apply_apply (D.weightCore p).hasTemperateGrowth,smul_eq_mul,
+    D.weightCore_apply,D.index_eq]
+  split_ifs with hy
+  · by_cases hw : D.localizers.weight p ((A.euclideanChart (D.patch p).index).symm y) = 0
+    · rw [hw,zero_mul,zero_mul]
+    · have hx := subset_tsupport (D.localizers.weight p : X → ℂ) hw
+      have he := (A.euclideanChart (D.patch p).index).right_inv hy
+      have hh := D.cofactorDifferential_source p φ.source ψ.source hx
+      rw [φ.ofReal_source,ψ.ofReal_source,he] at hh
+      rw [hh]
+      congr 1
+      have hv := D.completedVolumeDerivative_source k hk φ.source ψ.source (D.patch p).index
+        (x:=((A.euclideanChart (D.patch p).index).symm y)) (by simpa using D.patch_source p hx)
+      rw [φ.ofReal_source,ψ.ofReal_source] at hv
+      rw [← hv]
+      have hd := D.realVolumeDerivative_val k hk
+        (D.localizers.realEmbed ((k:ℝ)+2) φ) (D.localizers.realEmbed ((k:ℝ)+2) ψ)
+      simp only [Localizers.realEmbed_val] at hd
+      simp only [← hd]
+      apply Complex.ext
+      · rfl
+      · exact (D.localizers.real_strong hs _ _).symm
+  · exact zero_mul _
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/CofactorLocalization.lean b/lean/OAI/Geometry/PrescribedRicci/CofactorLocalization.lean
new file mode 100644
index 0000000000000000000000000000000000000000..0fcdb200c016db266e4c48f137a8aa7d0ceb2490
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/CofactorLocalization.lean
@@ -0,0 +1,112 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.CofactorBilin
+import OAI.Geometry.PrescribedRicci.CofactorCore
+import OAI.Geometry.PrescribedPotential.GlobalHessian
+
+namespace OAI
+
+/-! Cofactor Localization. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Matrix _root_.MeasureTheory _root_.OAI.MeasureTheory
+open scoped ContDiff SchwartzMap Classical BoundedContinuousFunction Matrix.Norms.Elementwise
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart FrozenPoisson MetricLocalization
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+def outerCore (p : ι) (f : Smooth A) : 𝓢(EC d,ℂ) :=
+  localize A (D.patch p).index (cutoffGlobal (D.patch p).index (D.outerCutoff p))
+    (cutoffGlobal_support _ _) f
+
+def weightCore (p : ι) : 𝓢(EC d,ℂ) := D.forcing p (Smooth.const 1)
+
+lemma outerCore_eventually (p : ι) (f : Smooth A) {x : X}
+    (hx : x ∈ tsupport (D.localizers.weight p : X → ℂ)) :
+    (D.outerCore p f : EC d → ℂ) =ᶠ[𝓝 (A.euclideanChart (D.patch p).index x)]
+      f ∘ (A.euclideanChart (D.patch p).index).symm :=
+  localize_cutoff_eventually _ (D.cutoff p) (D.outerCutoff p) (D.outerCutoff_one p) f
+    ((A.euclideanChart _).mapsTo (D.patch_source p hx))
+    (by rw [D.cutoff_one p x hx]; exact one_ne_zero)
+
+lemma outerHessian_source (p : ι) (i j : Fin d) (φ : SmoothRealFunction A) {x : X}
+    (hx : x ∈ tsupport (D.localizers.weight p : X → ℂ)) :
+    hessianEntrySchwartz i j (D.outerCore p (Smooth.ofReal φ))
+      (A.euclideanChart (D.patch p).index x) =
+      φ.hessian (D.patch p).index (A.chart (D.patch p).index x) i j := by
+  have hh := D.localizedHessian_source p i j φ hx
+  change globalize (D.patch p).index (D.cutoff p)
+    (hessianEntrySchwartz i j (D.outerCore p (Smooth.ofReal φ))) x = _ at hh
+  rw [globalize_apply,ite_eq_left (D.patch_source p hx),D.cutoff_one p x hx,one_mul] at hh
+  exact hh
+
+lemma cofactorDifferential_source (p : ι) (φ ψ : SmoothRealFunction A) {x : X}
+    (hx : x ∈ tsupport (D.localizers.weight p : X → ℂ)) :
+    schwartzDifferential
+      (extendedCoefficient (D.patch p).matrix (coefficientBCF (D.cofactorLocalCore p · · (Smooth.ofReal φ))))
+      (D.outerCore p (Smooth.ofReal ψ)) (A.euclideanChart (D.patch p).index x) =
+      (((g.matrix (D.patch p).index (A.chart (D.patch p).index x)).det⁻¹ *
+        NonlinearHessian.detDifferential
+          (g.matrix (D.patch p).index (A.chart (D.patch p).index x) +
+            φ.hessian (D.patch p).index (A.chart (D.patch p).index x))
+          (ψ.hessian (D.patch p).index (A.chart (D.patch p).index x))).re : ℂ) := by
+  let q := (D.patch p).index
+  let e := A.euclideanChart q
+  let U := ψ.localExpression q ∘ coordinateEquiv d
+  have hs := D.patch_source p hx
+  have ht := e.mapsTo hs
+  have hU : ContDiffAt ℝ ∞ U (e x) := (ψ.euclidean_smooth q).contDiffAt
+    (by simpa only [e,ComplexAtlas.euclideanChart_target] using e.open_target.mem_nhds ht)
+  have he : (D.outerCore p (Smooth.ofReal ψ) : EC d → ℂ) =ᶠ[𝓝 (e x)] Complex.ofRealCLM ∘ U :=
+    D.outerCore_eventually p (Smooth.ofReal ψ) hx
+  rw [schwartzDifferential_apply _ (extendedCoefficient_temperate _ _
+    (fun i j => (D.cofactorLocalCore p i j (Smooth.ofReal φ)).hasTemperateGrowth))]
+  simp_rw [D.cofactorLocalCore_source p _ _ _ hx,D.cofactorScalarCore_apply,
+    D.metricEntry_apply p _ _ hx,D.inverseDet_apply p hx,D.localizedHessian_source p _ _ φ hx]
+  have hh := cofactor_smooth_expression hU (D.outerCore p (Smooth.ofReal ψ)) he
+    (g.matrix q (A.chart q x)).det⁻¹
+    (g.matrix q (A.chart q x)+φ.hessian q (A.chart q x))
+  dsimp only [q,e] at hh
+  simp_rw [D.outerHessian_source p _ _ ψ hx] at hh
+  exact hh.symm
+
+lemma weightCore_apply (p : ι) (y : EC d) : D.weightCore p y =
+    if y ∈ (A.euclideanChart (D.patch p).index).target then
+      D.localizers.weight p ((A.euclideanChart (D.patch p).index).symm y) else 0 := by
+  change localize A (D.localizers.index p) (D.localizers.weight p) (D.localizers.support_sub p)
+    (Smooth.const 1) y = _
+  simp only [localize_apply,localizeFun,D.index_eq]
+  have hc (z : X) : (Smooth.const (A:=A) 1) z = 1 := rfl
+  simp only [hc,mul_one]
+
+lemma weightCore_outerCore (p : ι) (f : Smooth A) :
+    SchwartzMap.smulLeftCLM ℂ (D.weightCore p) (D.outerCore p f) = D.forcing p f := by
+  ext y
+  rw [SchwartzMap.smulLeftCLM_apply_apply (D.weightCore p).hasTemperateGrowth,smul_eq_mul,
+    D.weightCore_apply]
+  change _ = localizeFun A (D.localizers.index p) (D.localizers.weight p) f y
+  simp only [localizeFun,D.index_eq]
+  split_ifs with hy
+  · by_cases hw : D.localizers.weight p ((A.euclideanChart (D.patch p).index).symm y) = 0
+    · rw [hw,zero_mul,zero_mul]
+    · have hx := subset_tsupport (D.localizers.weight p : X → ℂ) hw
+      have he := (A.euclideanChart (D.patch p).index).right_inv hy
+      have hh := (D.outerCore_eventually p f hx).self_of_nhds
+      rw [he] at hh
+      rw [hh]
+      rfl
+  · exact zero_mul _
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/CofactorWeakEquation.lean b/lean/OAI/Geometry/PrescribedRicci/CofactorWeakEquation.lean
new file mode 100644
index 0000000000000000000000000000000000000000..562e417ef728829a270329f89d48ef79d5f5f54b
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/CofactorWeakEquation.lean
@@ -0,0 +1,197 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.NonlinearScale
+import OAI.Geometry.PrescribedRicci.CofactorCore
+import OAI.Geometry.PrescribedRicci.CofactorCoreEquation
+import OAI.Geometry.PrescribedRicci.RoughCutoff
+import OAI.Geometry.PrescribedRicci.TameApproximation
+
+namespace OAI
+
+/-! Cofactor Weak Equation. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Matrix _root_.MeasureTheory _root_.OAI.MeasureTheory TemperedDistribution
+open scoped ContDiff SchwartzMap Classical BoundedContinuousFunction Matrix.Norms.Elementwise
+namespace MetricLocalization
+open EllipticKernel SobolevChart FrozenPoisson
+lemma cofactor_cutoff_distribution {n : ℕ} (H : Matrix (Fin n) (Fin n) ℂ)
+    (a : SmoothCoefficients (BasisIndex n) (EC n)) (κ f : 𝓢(EC n,ℂ)) :
+    frozenDifferential H (SchwartzMap.toTemperedDistributionCLM (EC n) ℂ volume
+      (SchwartzMap.smulLeftCLM ℂ κ f)) +
+      (Lp.toTemperedDistributionCLM ℂ volume 2
+        (perturbation (stdOrthonormalBasis ℝ (EC n)) (coefficientBCF a)
+          (schwartzCoord 2 (SchwartzMap.smulLeftCLM ℂ κ f)))) =
+    SchwartzMap.toTemperedDistributionCLM (EC n) ℂ volume
+      (SchwartzMap.smulLeftCLM ℂ κ (schwartzDifferential (extendedCoefficient H (coefficientBCF a)) f)) +
+    SchwartzMap.toTemperedDistributionCLM (EC n) ℂ volume
+      (schwartzCutoffError (extendedCoefficient H (coefficientBCF a)) κ f) := by
+  change frozenDifferential H (SchwartzMap.toTemperedDistributionCLM (EC n) ℂ volume
+    (SchwartzMap.smulLeftCLM ℂ κ f)) +
+      ((perturbation (stdOrthonormalBasis ℝ (EC n)) (coefficientBCF a)
+        (schwartzCoord 2 (SchwartzMap.smulLeftCLM ℂ κ f)) : L2 (EC n)) : 𝓢'(EC n,ℂ)) = _
+  rw [← realize_schwartzCoord 2 (SchwartzMap.smulLeftCLM ℂ κ f),
+    ← chartDifferential_extension H (coefficientBCF a) (fun i j => (a i j).hasTemperateGrowth),
+    realize_schwartzCoord,← schwartzDifferential_distribution _
+      (extendedCoefficient_temperate H (coefficientBCF a) (fun i j => (a i j).hasTemperateGrowth)),← map_add]
+  congr 1
+  change schwartzDifferential (extendedCoefficient H (coefficientBCF a))
+    (SchwartzMap.smulLeftCLM ℂ κ f) =
+    SchwartzMap.smulLeftCLM ℂ κ (schwartzDifferential (extendedCoefficient H (coefficientBCF a)) f) +
+      (schwartzDifferential (extendedCoefficient H (coefficientBCF a)) (SchwartzMap.smulLeftCLM ℂ κ f) -
+       SchwartzMap.smulLeftCLM ℂ κ (schwartzDifferential (extendedCoefficient H (coefficientBCF a)) f))
+  abel
+end MetricLocalization
+
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart FrozenPoisson MetricLocalization
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+local instance cofactorWeakEquationNG (s : ℝ) : NormedAddCommGroup (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedAddCommGroup
+local instance cofactorWeakEquationNS (s : ℝ) : NormedSpace ℝ (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedSpace
+local instance cofactorWeakEquationTG (s : ℝ) : IsTopologicalAddGroup (D.localizers.RealSobolev s) :=
+  Submodule.isTopologicalAddGroup _
+local instance cofactorWeakEquationCS (s : ℝ) : ContinuousSMul ℝ (D.localizers.RealSobolev s) :=
+  SMulMemClass.continuousSMul _
+
+def outerCompleted (s : ℝ) (p : ι) : D.localizers.Sobolev s →L[ℝ] L2 (EC d) :=
+  D.completedLocalize s (D.patch p).index (cutoffGlobal (D.patch p).index (D.outerCutoff p))
+    (cutoffGlobal_support _ _)
+
+lemma outerCompleted_embed (k : ℕ) (p : ι) (f : Smooth A) :
+    D.outerCompleted (k:ℝ) p (D.localizers.embed (k:ℝ) f) =
+      schwartzCoord (k:ℝ) (D.outerCore p f) :=
+  D.completedLocalize_embed k _ _ _ f
+
+lemma outerCompleted_embed_add_two (k : ℕ) (p : ι) (f : Smooth A) :
+    D.outerCompleted ((k:ℝ)+2) p (D.localizers.embed ((k:ℝ)+2) f) =
+      schwartzCoord ((k:ℝ)+2) (D.outerCore p f) := by
+  have hh := D.outerCompleted_embed (k+2) p f
+  have he : ((k+2:ℕ):ℝ) = (k:ℝ)+2 := by push_cast; rfl
+  rw [he] at hh
+  exact hh
+
+abbrev CofactorPair (k : ℕ) :=
+  D.localizers.RealSobolev (((k+1:ℕ):ℝ)+2) × D.localizers.RealSobolev ((k:ℝ)+2)
+
+def cofactorWeakLeft (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (hs : (Module.finrank ℝ (EC d):ℝ) < 2*((k+1:ℕ):ℝ)) (p : ι)
+    (z : D.CofactorPair k) : 𝓢'(EC d,ℂ) :=
+  frozenDifferential (D.patch p).matrix (D.localizers.distribution ((k:ℝ)+2) p z.2.val) +
+    (Lp.toTemperedDistributionCLM ℂ volume 2)
+      (perturbation (stdOrthonormalBasis ℝ (EC d)) (D.cofactorBCF (k+1) (by omega) hs p z.1.val)
+        (lowerCoord ((k:ℝ)+2) 2 (by linarith [Nat.cast_nonneg (α:=ℝ) k]) (z.2.val.val p)))
+
+def cofactorWeakRight (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) (p : ι)
+    (z : D.CofactorPair k) : 𝓢'(EC d,ℂ) :=
+  D.localizers.distribution (k:ℝ) p ((D.realVolumeDerivative k hk
+    (D.localizers.realLower (((k+1:ℕ):ℝ)+2) ((k:ℝ)+2) (by push_cast; linarith) z.1) z.2).val) +
+  realize ((k+1:ℕ):ℝ) (roughCutoff (D.patch p).matrix (k+1) (by omega) (D.weightCore p)
+    (fun i j => D.cofactorLocal (k+1) (by omega) p i j z.1.val)
+    (lowerCoord ((k:ℝ)+2) (((k+1:ℕ):ℝ)+1) (by push_cast; linarith)
+      (D.outerCompleted ((k:ℝ)+2) p z.2.val)))
+
+lemma cofactorWeakLeft_continuous (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (hs : (Module.finrank ℝ (EC d):ℝ) < 2*((k+1:ℕ):ℝ)) (p : ι) :
+    Continuous (D.cofactorWeakLeft k hk hs p) := by
+  have hw : Continuous (fun z : D.CofactorPair k => z.2.val) :=
+    continuous_subtype_val.comp continuous_snd
+  have hwp : Continuous (fun z : D.CofactorPair k => z.2.val.val p) :=
+    (continuous_apply p).comp (continuous_subtype_val.comp hw)
+  have hu : Continuous (fun z : D.CofactorPair k => z.1.val) :=
+    continuous_subtype_val.comp continuous_fst
+  have hc : Continuous (fun z : D.CofactorPair k => D.cofactorBCF (k+1) (by omega) hs p z.1.val) :=
+    (D.cofactorBCF_continuous (k+1) (by omega) hs p).comp hu
+  have hl : Continuous (fun z : D.CofactorPair k =>
+      lowerCoord ((k:ℝ)+2) 2 (by linarith [Nat.cast_nonneg (α:=ℝ) k]) (z.2.val.val p)) :=
+    (lowerCoord ((k:ℝ)+2) 2 (by linarith [Nat.cast_nonneg (α:=ℝ) k])).continuous.comp hwp
+  have hp := (continuous_perturbation (stdOrthonormalBasis ℝ (EC d))).comp (hc.prodMk hl)
+  exact ((frozenDifferential (D.patch p).matrix).continuous.comp
+    ((D.localizers.distribution ((k:ℝ)+2) p).continuous.comp hw)).add
+      ((Lp.toTemperedDistributionCLM ℂ volume 2).continuous.comp hp)
+
+lemma cofactorCutoff_continuous (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) (p : ι) :
+    Continuous (fun z : D.CofactorPair k =>
+      roughCutoff (D.patch p).matrix (k+1) (by omega) (D.weightCore p)
+        (fun i j => D.cofactorLocal (k+1) (by omega) p i j z.1.val)
+        (lowerCoord ((k:ℝ)+2) (((k+1:ℕ):ℝ)+1) (by push_cast; linarith)
+          (D.outerCompleted ((k:ℝ)+2) p z.2.val))) := by
+  have ha : Continuous (fun z : D.CofactorPair k =>
+      fun i j => D.cofactorLocal (k+1) (by omega) p i j z.1.val) := by
+    apply continuous_pi
+    intro i
+    apply continuous_pi
+    intro j
+    exact (D.cofactorLocal_continuous (k+1) (by omega) p i j).comp
+      (continuous_subtype_val.comp continuous_fst)
+  have hw : Continuous (fun z : D.CofactorPair k =>
+      lowerCoord ((k:ℝ)+2) (((k+1:ℕ):ℝ)+1) (by push_cast; linarith)
+        (D.outerCompleted ((k:ℝ)+2) p z.2.val)) :=
+    (lowerCoord ((k:ℝ)+2) (((k+1:ℕ):ℝ)+1) (by push_cast; linarith)).continuous.comp
+      ((D.outerCompleted ((k:ℝ)+2) p).continuous.comp (continuous_subtype_val.comp continuous_snd))
+  have hp := ha.prodMk hw
+  have hc := roughCutoff_continuous (D.patch p).matrix (k+1) (by omega) (D.weightCore p)
+  have hr := hc.comp hp
+  exact hr
+
+lemma cofactorWeakRight_continuous (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) (p : ι) :
+    Continuous (D.cofactorWeakRight k hk p) := by
+  have hu : Continuous (fun z : D.CofactorPair k =>
+      D.localizers.realLower (((k+1:ℕ):ℝ)+2) ((k:ℝ)+2) (by push_cast; linarith) z.1) :=
+    (D.localizers.realLower (((k+1:ℕ):ℝ)+2) ((k:ℝ)+2) (by push_cast; linarith)).continuous.comp continuous_fst
+  have hv : Continuous (fun z : D.CofactorPair k => D.realVolumeDerivative k hk
+      (D.localizers.realLower (((k+1:ℕ):ℝ)+2) ((k:ℝ)+2) (by push_cast; linarith) z.1) z.2) :=
+    ((D.realVolumeDerivative_continuous k hk).comp hu).clm_apply continuous_snd
+  have h₁ := (D.localizers.distribution (k:ℝ) p).continuous.comp (continuous_subtype_val.comp hv)
+  exact h₁.add ((realize ((k+1:ℕ):ℝ)).continuous.comp (D.cofactorCutoff_continuous k hk p))
+
+lemma cofactorWeak_core (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (hs : (Module.finrank ℝ (EC d):ℝ) < 2*((k+1:ℕ):ℝ)) (p : ι) (φ ψ : RealSmooth A) :
+    D.cofactorWeakLeft k hk hs p (D.localizers.realEmbed (((k+1:ℕ):ℝ)+2) φ,
+      D.localizers.realEmbed ((k:ℝ)+2) ψ) =
+    D.cofactorWeakRight k hk p (D.localizers.realEmbed (((k+1:ℕ):ℝ)+2) φ,
+      D.localizers.realEmbed ((k:ℝ)+2) ψ) := by
+  dsimp only [cofactorWeakLeft,cofactorWeakRight,Prod.fst,Prod.snd,Localizers.realEmbed_val]
+  rw [D.cofactorBCF_embed,D.outerCompleted_embed_add_two,lowerCoord_schwartz,
+    Localizers.realLower_embed]
+  simp_rw [D.cofactorLocal_embed]
+  rw [roughCutoff_schwartz,realize_schwartzCoord]
+  change _ = realize (k:ℝ) ((D.realVolumeDerivative k hk
+    (D.localizers.realEmbed ((k:ℝ)+2) φ) (D.localizers.realEmbed ((k:ℝ)+2) ψ)).val.val p) + _
+  rw [← D.cofactor_core_forcing k hk p φ ψ,realize_schwartzCoord,
+    D.localizers.distribution_embed]
+  change frozenDifferential _ (SchwartzMap.toTemperedDistributionCLM (EC d) ℂ volume (D.forcing p ψ.val)) +
+    ((Lp.toTemperedDistributionCLM ℂ volume 2) (perturbation _ _
+      (lowerCoord ((k:ℝ)+2) 2 (by linarith [Nat.cast_nonneg (α:=ℝ) k])
+        (schwartzCoord ((k:ℝ)+2) (D.forcing p ψ.val))))) = _
+  rw [lowerCoord_schwartz,← D.weightCore_outerCore p ψ.val]
+  exact cofactor_cutoff_distribution (D.patch p).matrix (D.cofactorLocalCore p · · φ.val)
+    (D.weightCore p) (D.outerCore p ψ.val)
+
+lemma cofactor_weak_equation (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (hs : (Module.finrank ℝ (EC d):ℝ) < 2*((k+1:ℕ):ℝ)) (p : ι)
+    (u : D.localizers.RealSobolev (((k+1:ℕ):ℝ)+2))
+    (w : D.localizers.RealSobolev ((k:ℝ)+2)) :
+    D.cofactorWeakLeft k hk hs p (u,w) = D.cofactorWeakRight k hk p (u,w) := by
+  have he : D.cofactorWeakLeft k hk hs p = D.cofactorWeakRight k hk p := by
+    apply ((D.localizers.realEmbed_dense (((k+1:ℕ):ℝ)+2)).prodMap
+      (D.localizers.realEmbed_dense ((k:ℝ)+2))).equalizer
+      (D.cofactorWeakLeft_continuous k hk hs p) (D.cofactorWeakRight_continuous k hk p)
+    funext z
+    exact D.cofactorWeak_core k hk hs p z.1 z.2
+  exact congr_fun he (u,w)
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/CompactParameterIntegral.lean b/lean/OAI/Geometry/PrescribedRicci/CompactParameterIntegral.lean
new file mode 100644
index 0000000000000000000000000000000000000000..003f421dbdf89df1b6ffebbceaf4c10c4eedc331
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/CompactParameterIntegral.lean
@@ -0,0 +1,46 @@
+import Mathlib
+
+namespace OAI
+
+/-! Compact Parameter Integral. -/
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace KahlerCalculus
+variable {E : Type*} [TopologicalSpace E] [T2Space E]
+  [MeasurableSpace E] [BorelSpace E] {μ : Measure E} [IsFiniteMeasureOnCompacts μ]
+
+lemma hasDerivAt_integral_compact {F F' : ℝ → E → ℝ} {K : Set E}
+    (hK : IsCompact K) (hF : Continuous (Function.uncurry F))
+    (hF' : Continuous (Function.uncurry F'))
+    (hs : ∀ t, Function.support (F t) ⊆ K)
+    (hs' : ∀ t, Function.support (F' t) ⊆ K)
+    (hd : ∀ t z, HasDerivAt (fun s => F s z) (F' t z) t) (t : ℝ) :
+    HasDerivAt (fun s => ∫ z, F s z ∂μ) (∫ z, F' t z ∂μ) t := by
+  have hfc (s : ℝ) : Continuous (F s) := hF.comp (continuous_const.prodMk continuous_id)
+  have hfc' (s : ℝ) : Continuous (F' s) := hF'.comp (continuous_const.prodMk continuous_id)
+  have hfi (s : ℝ) : Integrable (F s) μ := (hfc s).integrable_of_hasCompactSupport
+    (HasCompactSupport.of_support_subset_isCompact hK (hs s))
+  obtain ⟨C, hC⟩ := (isCompact_Icc (a := t - 1) (b := t + 1)).prod hK
+    |>.exists_bound_of_continuousOn hF'.continuousOn
+  let B : E → ℝ := K.indicator (fun _ => max C 0)
+  have hB : Integrable B μ := by
+    apply (integrable_indicator_iff hK.measurableSet).mpr
+    exact integrableOn_const hK.measure_lt_top.ne
+  refine (hasDerivAt_integral_of_dominated_loc_of_deriv_le
+    (Ioo_mem_nhds (by linarith : t - 1 < t) (by linarith : t < t + 1))
+    (Filter.Eventually.of_forall (fun s => (hfc s).aestronglyMeasurable)) (hfi t)
+    (hfc' t).aestronglyMeasurable ?_ hB ?_).2
+  · filter_upwards [] with z s hst
+    by_cases hz : z ∈ K
+    · exact (hC (s, z) ⟨⟨hst.1.le, hst.2.le⟩, hz⟩).trans
+        (by simpa only [B, Set.indicator_of_mem hz] using le_max_left C 0)
+    · have he : F' s z = 0 := Function.notMem_support.mp (fun hn => hz (hs' s hn))
+      simp only [he, norm_zero, B, Set.indicator_of_notMem hz, le_refl]
+  · exact Filter.Eventually.of_forall (fun z s _ => hd s z)
+
+end KahlerCalculus
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/CompactPathBounds.lean b/lean/OAI/Geometry/PrescribedRicci/CompactPathBounds.lean
new file mode 100644
index 0000000000000000000000000000000000000000..6867a283e8c883a8ebc5b3d26f8d128c2a8e2d69
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/CompactPathBounds.lean
@@ -0,0 +1,91 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.GlobalSobolev
+import OAI.Geometry.PrescribedRicci.MatrixCompactBounds
+import OAI.Geometry.PrescribedRicci.TraceEntryBounds
+import OAI.Geometry.PrescribedPotential.VolumePath
+import OAI.Geometry.PrescribedRicci.CurvatureCompactBounds
+import OAI.Geometry.PrescribedRicci.KahlerVolumeVariation
+import OAI.Geometry.PrescribedRicci.MongeAmpereL2
+import OAI.Geometry.PrescribedRicci.TraceMaximum
+
+namespace OAI
+
+/-! Compact Path Bounds. -/
+
+section
+
+ 
+
+noncomputable section
+open Matrix Set Filter Topology
+open scoped ContDiff ComplexOrder MatrixOrder Matrix.Norms.Elementwise
+namespace Anticanonical.SourceSmooth.KaehlerMetric
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  [ConnectedSpace X] {A : ComplexAtlas d X}
+
+theorem volumePath_metric_bounds_on_compact (g : KaehlerMetric A) (line : SemipositiveAnticanonicalMetric A)
+    (hd : 2 ≤ d) (q : Fin A.count) {L : Set (Coordinates d)}
+    (hL : IsCompact L) (hLt : L ⊆ (A.chart q).target) : ∃ K : ℝ, 0 < K ∧
+    ∀ (φ : SmoothRealFunction A) (hp : g.PositivePotential φ), g.integral φ.value = 0 →
+    ∀ (t b : ℝ), t ∈ Icc (0:ℝ) 1 →
+    (∀ x, (g.logRatio (g.deform φ hp)).value x = t*(prescribedForcing g line).value x+b) →
+    ∀ z ∈ L, ‖(g.deform φ hp).matrix q z‖ ≤ K ∧
+      ‖((g.deform φ hp).matrix q z)⁻¹‖ ≤ K := by
+  have hd' : 0 < d := by omega
+  let : Nonempty (Fin d) := Fin.pos_iff_nonempty.mp hd'
+  obtain ⟨T,hT,htrace⟩ := g.volumePath_trace_bound line hd
+  obtain ⟨m,hm⟩ := hL.bddAbove_image
+    ((g.backgroundSize_continuousOn (prescribedForcing g line) q).mono hLt)
+  let M := max 1 m
+  have hM : 1 ≤ M := le_max_left _ _
+  have hbg (z : Coordinates d) (hz : z ∈ L) :
+      g.BackgroundComponentBound (prescribedForcing g line) q z M :=
+    g.backgroundComponents_of_size _ q z ((hm (mem_image_of_mem _ hz)).trans (le_max_right _ _))
+  have hM0 : 0 ≤ M := by linarith
+  obtain ⟨D,hD,hdet⟩ := MongeAmpere.compact_matrix_bound
+    (show Continuous (fun H : Matrix (Fin d) (Fin d) ℂ => H.det.re) from
+      Complex.continuous_re.comp continuous_id.matrix_det) M
+  obtain ⟨Q,hQ,hmetric⟩ := MongeAmpere.metric_norm_bound_from_inverse (n:=Fin d) (M*T)
+  let E := Real.exp (2*‖(⟨(prescribedForcing g line).value,
+    (prescribedForcing g line).continuous⟩ : C(X,ℝ))‖)
+  have hE : 0 < E := Real.exp_pos _
+  refine ⟨1+M*T+E*D*Q,by positivity,?_⟩
+  intro φ hp hmean t b ht heq z hzL
+  have hz := hLt hzL
+  let x := (A.chart q).symm z
+  have hxs := (A.chart q).map_target hz
+  have hxz : A.chart q x = z := (A.chart q).right_inv hz
+  let H := (g.deform φ hp).matrix q z
+  have hH := (g.deform φ hp).positive q z hz
+  have hbi : ‖H⁻¹‖ ≤ M*T := by
+    apply MongeAmpere.matrix_norm_le_of_entries (mul_nonneg hM0 hT.le)
+    intro i j
+    have hi := MongeAmpere.entry_le_relative_trace H⁻¹ (g.matrix q z) hH.inv
+      (g.positive q _ hz) hM0
+      (fun i => (Complex.re_le_norm _).trans ((hbg z hzL).2.1 i i)) i j
+    have hs := htrace φ hp hmean t b ht heq x
+    change (g.deform φ hp).traceMetricValue g x ≤ T at hs
+    rw [(g.deform φ hp).traceMetricValue_local g q hxs,hxz] at hs
+    exact hi.trans (mul_le_mul_of_nonneg_left hs hM0)
+  have hdh : H.det.re ≤ E*D := by
+    have hv := g.density_mul_volume φ q hz
+    change (g.potentialDensity φ).value x*(g.matrix q z).det.re = H.det.re at hv
+    rw [← hv]
+    have hρ := (g.volumePath_density_bounds line ht hp heq x).2
+    change (g.potentialDensity φ).value x ≤ E at hρ
+    exact mul_le_mul hρ
+      (hdet _ (MongeAmpere.matrix_norm_le_of_entries hM0 (hbg z hzL).1))
+      (Complex.pos_iff.mp (g.positive q _ hz).det_pos).1.le hE.le
+  constructor
+  · have hb := (hmetric H hH hbi).trans (mul_le_mul_of_nonneg_right hdh hQ.le)
+    change ‖H‖ ≤ _
+    nlinarith [mul_nonneg hM0 hT.le]
+  · change ‖H⁻¹‖ ≤ _
+    nlinarith [mul_nonneg (mul_nonneg hE.le hD.le) hQ.le]
+
+end Anticanonical.SourceSmooth.KaehlerMetric
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/CompactPathRicci.lean b/lean/OAI/Geometry/PrescribedRicci/CompactPathRicci.lean
new file mode 100644
index 0000000000000000000000000000000000000000..1d34a50e6208287d4c74cc3ca8070f49406da966
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/CompactPathRicci.lean
@@ -0,0 +1,102 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.ChernLuAlgebra
+import OAI.Geometry.PrescribedRicci.CurvatureCompactBounds
+import OAI.Geometry.PrescribedRicci.VolumePathRicci
+import OAI.Geometry.PrescribedPotential.VolumePath
+
+namespace OAI
+
+/-! Compact Path Ricci. -/
+
+section
+
+ 
+
+noncomputable section
+open Matrix Set Filter Topology
+open scoped ContDiff ComplexOrder MatrixOrder Matrix.Norms.Elementwise
+namespace Anticanonical.SourceSmooth.KaehlerMetric
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+
+lemma volumePath_holDerivative_congr {f h : Coordinates d → Matrix (Fin d) (Fin d) ℂ}
+    {z : Coordinates d} (he : f =ᶠ[nhds z] h) (a : Fin d) :
+    holDerivative f z a = holDerivative h z a := by unfold holDerivative; rw [he.fderiv_eq]
+
+lemma holDerivative_sub_smul {f h : Coordinates d → Matrix (Fin d) (Fin d) ℂ}
+    {z : Coordinates d} (hf : DifferentiableAt ℝ f z) (hh : DifferentiableAt ℝ h z)
+    (t : ℂ) (a : Fin d) :
+    holDerivative (fun y => f y-t • h y) z a = holDerivative f z a-t • holDerivative h z a := by
+  unfold holDerivative
+  have he : fderiv ℝ (fun y => f y-t • h y) z = fderiv ℝ f z-t • fderiv ℝ h z := by
+    convert! (hf.hasFDerivAt.sub (hh.hasFDerivAt.const_smul t)).fderiv using 1
+  rw [he]
+  simp only [_root_.sub_apply,_root_.smul_apply,smul_sub,
+    smul_smul]
+  module
+
+lemma volumePath_ricci_hol (g : KaehlerMetric A) (line : SemipositiveAnticanonicalMetric A)
+    {t b : ℝ} {φ : SmoothRealFunction A} (hp : g.PositivePotential φ)
+    (heq : ∀ x, (g.logRatio (g.deform φ hp)).value x =
+      t*(prescribedForcing g line).value x+b)
+    (q : Fin A.count) {z : Coordinates d} (hz : z ∈ (A.chart q).target) (a : Fin d) :
+    holDerivative ((g.deform φ hp).curvatureRicci q) z a =
+      holDerivative (g.curvatureRicci q) z a-
+        (t:ℂ) • holDerivative ((prescribedForcing g line).hessian q) z a := by
+  have he := Filter.eventually_of_mem ((A.chart q).open_target.mem_nhds hz)
+    (fun y hy => g.volumePath_ricci line hp heq q hy)
+  rw [volumePath_holDerivative_congr he,holDerivative_sub_smul
+    ((g.curvatureRicci_smooth q hz).differentiableAt (by simp))
+    ((((prescribedForcing g line).hessian_smooth q).contDiffAt
+      ((A.chart q).open_target.mem_nhds hz)).differentiableAt (by simp))]
+
+lemma volumePath_ricci_bounds_on_compact (g : KaehlerMetric A)
+    (line : SemipositiveAnticanonicalMetric A) (q : Fin A.count) {L : Set (Coordinates d)}
+    (hL : IsCompact L) (hLt : L ⊆ (A.chart q).target) :
+    ∃ R : ℝ, 0 < R ∧ ∀ (φ : SmoothRealFunction A) (hp : g.PositivePotential φ)
+      (t b : ℝ), t ∈ Icc (0:ℝ) 1 →
+      (∀ x, (g.logRatio (g.deform φ hp)).value x = t*(prescribedForcing g line).value x+b) →
+      ∀ z ∈ L, ‖(g.deform φ hp).curvatureRicci q z‖ ≤ R ∧
+        ∀ a, ‖holDerivative ((g.deform φ hp).curvatureRicci q) z a‖ ≤ R := by
+  let F := prescribedForcing g line
+  let size : Coordinates d → ℝ := fun z => ‖g.curvatureRicci q z‖+‖F.hessian q z‖+
+    ‖holDerivative (g.curvatureRicci q) z‖+‖holDerivative (F.hessian q) z‖
+  have hs : ContinuousOn size L := by
+    intro z hz
+    have hR := g.curvatureRicci_smooth q (hLt hz)
+    have hF := (F.hessian_smooth q).contDiffAt ((A.chart q).open_target.mem_nhds (hLt hz))
+    have hdR : ContDiffAt ℝ ∞ (fun y => holDerivative (g.curvatureRicci q) y) z :=
+      contDiffAt_pi.mpr (fun a => contDiffAt_holDerivative hR a)
+    have hdF : ContDiffAt ℝ ∞ (fun y => holDerivative (F.hessian q) y) z :=
+      contDiffAt_pi.mpr (fun a => contDiffAt_holDerivative hF a)
+    exact (((hR.continuousAt.norm.add hF.continuousAt.norm).add hdR.continuousAt.norm).add
+      hdF.continuousAt.norm).continuousWithinAt
+  obtain ⟨r,hr⟩ := hL.bddAbove_image hs
+  refine ⟨1+max 0 r,by positivity,?_⟩
+  intro φ hp t b ht heq z hz
+  have htr : ‖(t:ℂ)‖ ≤ 1 := by simpa only [Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg ht.1] using ht.2
+  have hb : size z ≤ 1+max 0 r := (hr (mem_image_of_mem _ hz)).trans (by linarith [le_max_right (0:ℝ) r])
+  constructor
+  · rw [g.volumePath_ricci line hp heq q (hLt hz)]
+    apply (norm_sub_le _ _).trans
+    rw [norm_smul]
+    have hn := mul_le_mul_of_nonneg_right htr (norm_nonneg (F.hessian q z))
+    change ‖g.curvatureRicci q z‖+‖(t:ℂ)‖*‖F.hessian q z‖ ≤ _
+    dsimp [size] at hb
+    linarith [norm_nonneg (holDerivative (g.curvatureRicci q) z),norm_nonneg (holDerivative (F.hessian q) z)]
+  · intro a
+    rw [g.volumePath_ricci_hol line hp heq q (hLt hz) a]
+    apply (norm_sub_le _ _).trans
+    rw [norm_smul]
+    have hn := mul_le_mul_of_nonneg_right htr (norm_nonneg (holDerivative (F.hessian q) z a))
+    have h1 := norm_le_pi_norm (holDerivative (g.curvatureRicci q) z) a
+    have h2 := norm_le_pi_norm (holDerivative (F.hessian q) z) a
+    change ‖holDerivative (g.curvatureRicci q) z a‖+‖(t:ℂ)‖*‖holDerivative (F.hessian q) z a‖ ≤ _
+    dsimp [size] at hb
+    linarith [norm_nonneg (g.curvatureRicci q z),norm_nonneg (F.hessian q z)]
+
+end Anticanonical.SourceSmooth.KaehlerMetric
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/ComplexIntegrationByParts.lean b/lean/OAI/Geometry/PrescribedRicci/ComplexIntegrationByParts.lean
new file mode 100644
index 0000000000000000000000000000000000000000..8903f72ec77def90f027c5e47794ba84724a8106
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/ComplexIntegrationByParts.lean
@@ -0,0 +1,132 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.GlobalSobolev
+import OAI.Geometry.PrescribedRicci.KahlerCofactorDivergence
+
+namespace OAI
+
+/-! Complex Integration By Parts. -/
+
+noncomputable section
+open Matrix Filter Set Topology MeasureTheory
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace Anticanonical.SourceSmooth
+namespace KaehlerMetric
+variable {d : ℕ}
+
+def barDeriv (f : Coordinates d → ℂ) (z : Coordinates d) (j : Fin d) : ℂ :=
+  (2 : ℂ)⁻¹ * (fderiv ℝ f z (coordinateVector j) +
+    Complex.I * fderiv ℝ f z (Complex.I • coordinateVector j))
+
+def holDeriv (f : Coordinates d → ℂ) (z : Coordinates d) (j : Fin d) : ℂ :=
+  (2 : ℂ)⁻¹ * (fderiv ℝ f z (coordinateVector j) -
+    Complex.I * fderiv ℝ f z (Complex.I • coordinateVector j))
+
+lemma contDiff_deriv_direction {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+    [NormedAddCommGroup F] [NormedSpace ℝ F] {f : E → F}
+    (hf : ContDiff ℝ ∞ f) (v : E) : ContDiff ℝ ∞ (fun z => fderiv ℝ f z v) :=
+  (hf.fderiv_right (by simp)).clm_apply contDiff_const
+
+lemma contDiff_barDeriv {f : Coordinates d → ℂ} (hf : ContDiff ℝ ∞ f) (j : Fin d) :
+    ContDiff ℝ ∞ (fun z => barDeriv f z j) :=
+  contDiff_const.mul ((contDiff_deriv_direction hf _).add
+    (contDiff_const.mul (contDiff_deriv_direction hf _)))
+
+lemma contDiff_holDeriv {f : Coordinates d → ℂ} (hf : ContDiff ℝ ∞ f) (j : Fin d) :
+    ContDiff ℝ ∞ (fun z => holDeriv f z j) :=
+  contDiff_const.mul ((contDiff_deriv_direction hf _).sub
+    (contDiff_const.mul (contDiff_deriv_direction hf _)))
+
+lemma compactSupport_barDeriv {f : Coordinates d → ℂ} (hf : HasCompactSupport f)
+    (j : Fin d) : HasCompactSupport (fun z => barDeriv f z j) := by
+  exact ((hf.fderiv_apply ℝ (coordinateVector j)).add
+    (hf.fderiv_apply ℝ (Complex.I • coordinateVector j)).mul_left).mul_left
+
+lemma integrable_mul_deriv {f g : Coordinates d → ℂ}
+    (hf : ContDiff ℝ ∞ f) (hg : ContDiff ℝ ∞ g) (hc : HasCompactSupport f)
+    (v : Coordinates d) : Integrable (fun z => f z * fderiv ℝ g z v) :=
+  (hf.continuous.mul (contDiff_deriv_direction hg v).continuous).integrable_of_hasCompactSupport
+    hc.mul_right
+
+lemma integrable_deriv_mul {f g : Coordinates d → ℂ}
+    (hf : ContDiff ℝ ∞ f) (hg : ContDiff ℝ ∞ g) (hc : HasCompactSupport f)
+    (v : Coordinates d) : Integrable (fun z => fderiv ℝ f z v * g z) :=
+  ((contDiff_deriv_direction hf v).continuous.mul hg.continuous).integrable_of_hasCompactSupport
+    (hc.fderiv_apply ℝ v).mul_right
+
+lemma contDiffAt_deriv_direction {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+    [NormedAddCommGroup F] [NormedSpace ℝ F] {f : E → F} {z : E}
+    (hf : ContDiffAt ℝ ∞ f z) (v : E) : ContDiffAt ℝ ∞ (fun y => fderiv ℝ f y v) z :=
+  (hf.fderiv_right (by simp)).clm_apply contDiffAt_const
+
+lemma continuous_mul_on_tsupport {E : Type*} [TopologicalSpace E] {f g : E → ℂ}
+    (hf : Continuous f) (hg : ∀ z ∈ tsupport f, ContinuousAt g z) :
+    Continuous (fun z => f z * g z) := by
+  rw [continuous_iff_continuousAt]
+  intro z
+  by_cases hz : z ∈ tsupport f
+  · exact hf.continuousAt.mul (hg z hz)
+  · apply (continuousAt_const (y := (0 : ℂ))).congr
+    filter_upwards [notMem_tsupport_iff_eventuallyEq.mp hz] with y hy
+    simp only [hy, zero_mul, Pi.zero_apply]
+
+lemma contDiff_mul_on_tsupport {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+    {f g : E → ℂ} (hf : ContDiff ℝ ∞ f)
+    (hg : ∀ z ∈ tsupport f, ContDiffAt ℝ ∞ g z) :
+    ContDiff ℝ ∞ (fun z => f z * g z) := by
+  rw [contDiff_iff_contDiffAt]
+  intro z
+  by_cases hz : z ∈ tsupport f
+  · exact hf.contDiffAt.mul (hg z hz)
+  · apply (contDiffAt_const (c := (0 : ℂ))).congr_of_eventuallyEq
+    filter_upwards [notMem_tsupport_iff_eventuallyEq.mp hz] with y hy
+    simp only [hy, zero_mul, Pi.zero_apply]
+
+lemma integrable_mul_deriv_local {f g : Coordinates d → ℂ}
+    (hf : ContDiff ℝ ∞ f) (hg : ∀ z ∈ tsupport f, ContDiffAt ℝ ∞ g z)
+    (hc : HasCompactSupport f) (v : Coordinates d) :
+    Integrable (fun z => f z * fderiv ℝ g z v) :=
+  (continuous_mul_on_tsupport hf.continuous (fun z hz =>
+    (contDiffAt_deriv_direction (hg z hz) v).continuousAt)).integrable_of_hasCompactSupport
+      hc.mul_right
+
+lemma integrable_deriv_mul_local {f g : Coordinates d → ℂ}
+    (hf : ContDiff ℝ ∞ f) (hg : ∀ z ∈ tsupport f, ContDiffAt ℝ ∞ g z)
+    (hc : HasCompactSupport f) (v : Coordinates d) :
+    Integrable (fun z => fderiv ℝ f z v * g z) :=
+  (continuous_mul_on_tsupport (contDiff_deriv_direction hf v).continuous (fun z hz =>
+    (hg z (tsupport_fderiv_apply_subset ℝ v hz)).continuousAt)).integrable_of_hasCompactSupport
+      (hc.fderiv_apply ℝ v).mul_right
+
+lemma integral_mul_barDeriv_local {f g : Coordinates d → ℂ}
+    (hf : ContDiff ℝ ∞ f) (hg : ∀ z ∈ tsupport f, ContDiffAt ℝ ∞ g z)
+    (hc : HasCompactSupport f) (j : Fin d) :
+    (∫ z, f z * barDeriv g z j) = -(∫ z, barDeriv f z j * g z) := by
+  have hv (v : Coordinates d) : (∫ z, f z * fderiv ℝ g z v) =
+      -(∫ z, fderiv ℝ f z v * g z) :=
+    integral_mul_fderiv_eq_neg_fderiv_mul_of_integrable
+      (integrable_deriv_mul_local hf hg hc v) (integrable_mul_deriv_local hf hg hc v)
+      ((continuous_mul_on_tsupport hf.continuous (fun z hz => (hg z hz).continuousAt)).integrable_of_hasCompactSupport hc.mul_right)
+      (fun z _ => hf.differentiable (by simp) z)
+      (fun z hz => (hg z hz).differentiableAt (by simp))
+  have he (z : Coordinates d) : f z * barDeriv g z j =
+      (2 : ℂ)⁻¹ * (f z * fderiv ℝ g z (coordinateVector j)) +
+      ((2 : ℂ)⁻¹ * Complex.I) * (f z * fderiv ℝ g z (Complex.I • coordinateVector j)) := by
+    unfold barDeriv
+    ring
+  have he' (z : Coordinates d) : barDeriv f z j * g z =
+      (2 : ℂ)⁻¹ * (fderiv ℝ f z (coordinateVector j) * g z) +
+      ((2 : ℂ)⁻¹ * Complex.I) * (fderiv ℝ f z (Complex.I • coordinateVector j) * g z) := by
+    unfold barDeriv
+    ring
+  simp_rw [he, he']
+  rw [integral_add ((integrable_mul_deriv_local hf hg hc _).const_mul _) ((integrable_mul_deriv_local hf hg hc _).const_mul _),
+    integral_add ((integrable_deriv_mul_local hf hg hc _).const_mul _) ((integrable_deriv_mul_local hf hg hc _).const_mul _)]
+  simp only [integral_const_mul, hv]
+  ring
+
+end KaehlerMetric
+end Anticanonical.SourceSmooth
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/ComplexJet.lean b/lean/OAI/Geometry/PrescribedRicci/ComplexJet.lean
new file mode 100644
index 0000000000000000000000000000000000000000..f617707e4048d9dfd13cef3703874ab469e0cd7c
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/ComplexJet.lean
@@ -0,0 +1,116 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.FamilyJetNorm
+
+namespace OAI
+
+/-! Complex Jet. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+open scoped ContDiff ENNReal Classical BigOperators
+namespace TameInterpolation
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+variable {ι κ : Type*} [Fintype ι] [Nonempty ι] [Fintype κ] [Nonempty κ]
+
+def cdir (v : E) (f : E → ℂ) (x : E) : ℂ := fderiv ℝ f x v
+
+omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] in
+lemma cdir_smooth (v : E) {f : E → ℂ} (hf : ContDiff ℝ ∞ f) : ContDiff ℝ ∞ (cdir v f) :=
+  (hf.fderiv_right (by simp)).clm_apply contDiff_const
+
+omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] in
+lemma cdir_compact (v : E) {f : E → ℂ} (hf : HasCompactSupport f) :
+    HasCompactSupport (cdir v f) :=
+  (hf.fderiv ℝ).comp_left (g:=fun L : E →L[ℝ] ℂ => L v) (by simp)
+
+def cjet (e : ι → E) (f : E → ℂ) : {j : ℕ} → (Fin j → ι) → E → ℂ
+  | 0,_ => f
+  | _j+1,w => cdir (e (w 0)) (cjet e f (w ∘ Fin.succ))
+
+omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+  [Fintype ι] [Nonempty ι] in
+lemma cjet_smooth (e : ι → E) {f : E → ℂ} (hf : ContDiff ℝ ∞ f)
+    {j : ℕ} (w : Fin j → ι) : ContDiff ℝ ∞ (cjet e f w) := by
+  induction j with
+  | zero => exact hf
+  | succ j ih => exact cdir_smooth _ (ih _)
+
+omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+  [Fintype ι] [Nonempty ι] in
+lemma cjet_compact (e : ι → E) {f : E → ℂ} (hf : HasCompactSupport f)
+    {j : ℕ} (w : Fin j → ι) : HasCompactSupport (cjet e f w) := by
+  induction j with
+  | zero => exact hf
+  | succ j ih => exact cdir_compact _ (ih _)
+
+omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+  [Fintype ι] [Nonempty ι] in
+lemma cjet_real_map (e : ι → E) {f : E → ℂ} (hf : ContDiff ℝ ∞ f)
+    (L : ℂ →L[ℝ] ℝ) {j : ℕ} (w : Fin j → ι) :
+    jet e (fun x => L (f x)) w = fun x => L (cjet e f w x) := by
+  induction j with
+  | zero => rfl
+  | succ j ih =>
+    simp only [jet,cjet,ih]
+    funext x
+    dsimp only [dir,cdir]
+    exact congrArg (fun T : E →L[ℝ] ℝ => T (e (w 0)))
+      (L.hasFDerivAt.comp x ((cjet_smooth e hf _).differentiable (by simp) x).hasFDerivAt).fderiv
+
+def realComponents (f : κ → E → ℂ) : κ × Bool → E → ℝ :=
+  fun a x => if a.2 then (f a.1 x).im else (f a.1 x).re
+
+omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+  [Fintype κ] [Nonempty κ] in
+lemma realComponents_smooth (f : κ → E → ℂ) (hf : ∀ a, ContDiff ℝ ∞ (f a))
+    (a : κ × Bool) : ContDiff ℝ ∞ (realComponents f a) := by
+  rcases a with ⟨a,b⟩; cases b
+  · exact Complex.reCLM.contDiff.comp (hf a)
+  · exact Complex.imCLM.contDiff.comp (hf a)
+
+omit [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E]
+  [BorelSpace E] [Fintype κ] [Nonempty κ] in
+lemma realComponents_compact (f : κ → E → ℂ) (hf : ∀ a, HasCompactSupport (f a))
+    (a : κ × Bool) : HasCompactSupport (realComponents f a) := by
+  rcases a with ⟨a,b⟩; cases b
+  · exact (hf a).comp_left (g:=Complex.re) rfl
+  · exact (hf a).comp_left (g:=Complex.im) rfl
+
+lemma lpNorm_complex_le (f : E → ℂ) {p : ℝ≥0∞} (hp : 1 ≤ p)
+    (hr : MemLp (fun x => (f x).re) p volume) (hi : MemLp (fun x => (f x).im) p volume) :
+    lpNorm f p volume ≤ lpNorm (fun x => (f x).re) p volume + lpNorm (fun x => (f x).im) p volume := by
+  have hh := lpNorm_mono_real (hr.norm.add hi.norm) (f:=f)
+    (fun x => by simpa only [Real.norm_eq_abs,Pi.add_apply] using Complex.norm_le_abs_re_add_abs_im (f x))
+  apply hh.trans
+  simpa only [lpNorm_norm hr.aestronglyMeasurable,lpNorm_norm hi.aestronglyMeasurable] using lpNorm_add_le hr.norm (g:=fun x => ‖(f x).im‖) hp
+
+lemma cjet_lpNorm_le (e : ι → E) (f : κ → E → ℂ)
+    (hf : ∀ a, ContDiff ℝ ∞ (f a)) (hc : ∀ a, HasCompactSupport (f a))
+    (a : κ) {j : ℕ} (w : Fin j → ι) {p : ℝ≥0∞} (hp : 1 ≤ p) :
+    lpNorm (cjet e (f a) w) p volume ≤ 2*familyJetNorm e (realComponents f) j p := by
+  have hr : ContDiff ℝ ∞ (fun x => (cjet e (f a) w x).re) := Complex.reCLM.contDiff.comp (cjet_smooth e (hf a) w)
+  have hi : ContDiff ℝ ∞ (fun x => (cjet e (f a) w x).im) := Complex.imCLM.contDiff.comp (cjet_smooth e (hf a) w)
+  have hcr := (cjet_compact e (hc a) w).comp_left (g:=Complex.re) rfl
+  have hci := (cjet_compact e (hc a) w).comp_left (g:=Complex.im) rfl
+  have hl := lpNorm_complex_le _ hp (hr.continuous.memLp_of_hasCompactSupport hcr) (hi.continuous.memLp_of_hasCompactSupport hci)
+  have h1 := (lpNorm_jet_le e (realComponents f (a,false)) w p).trans (jetNorm_le_family e (realComponents f) (a,false) j p)
+  have h2 := (lpNorm_jet_le e (realComponents f (a,true)) w p).trans (jetNorm_le_family e (realComponents f) (a,true) j p)
+  change lpNorm (jet e (fun x => Complex.reCLM (f a x)) w) p volume ≤ _ at h1
+  change lpNorm (jet e (fun x => Complex.imCLM (f a x)) w) p volume ≤ _ at h2
+  rw [cjet_real_map e (hf a) Complex.reCLM] at h1
+  rw [cjet_real_map e (hf a) Complex.imCLM] at h2
+  change lpNorm (fun x => (cjet e (f a) w x).re) p volume ≤ _ at h1
+  change lpNorm (fun x => (cjet e (f a) w x).im) p volume ≤ _ at h2
+  linarith only [hl,h1,h2]
+
+end TameInterpolation
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/ComplexJetProduct.lean b/lean/OAI/Geometry/PrescribedRicci/ComplexJetProduct.lean
new file mode 100644
index 0000000000000000000000000000000000000000..79163a412abdfe84246318e0153191c4a33948aa
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/ComplexJetProduct.lean
@@ -0,0 +1,93 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.ComplexJet
+import OAI.Geometry.PrescribedRicci.HolderFiniteProduct
+
+namespace OAI
+
+/-! Complex Jet Product. -/
+
+section
+
+ 
+noncomputable section
+open Set MeasureTheory
+open scoped ContDiff ENNReal Classical BigOperators
+namespace TameInterpolation
+variable {Y : Type*} [MeasurableSpace Y] {μ : Measure Y}
+
+lemma lpNorm_complex_mul_le {p q r : ℝ≥0∞} [ENNReal.HolderTriple p q r]
+    {f g : Y → ℂ} (hf : MemLp f p μ) (hg : MemLp g q μ) :
+    lpNorm (fun x => f x*g x) r μ ≤ lpNorm f p μ*lpNorm g q μ := by
+  have he := eLpNorm_smul_le_mul_eLpNorm (p:=p) (q:=q) (r:=r)
+    hf.aestronglyMeasurable hg.aestronglyMeasurable
+  change eLpNorm (fun x => f x*g x) r μ ≤ eLpNorm f p μ * eLpNorm g q μ at he
+  have hh := ENNReal.toReal_mono (ENNReal.mul_ne_top hf.eLpNorm_ne_top hg.eLpNorm_ne_top) he
+  simpa only [ENNReal.toReal_mul,toReal_eLpNorm] using hh
+
+lemma lpNorm_complex_finite_prod {ι : Type*} (s : Finset ι) (f : ι → Y → ℂ) (p : ι → ℝ≥0∞)
+    (hf : ∀ i ∈ s, MemLp (f i) (p i) μ) :
+    lpNorm (fun x => ∏ i ∈ s, f i x) (∑ i ∈ s, (p i)⁻¹)⁻¹ μ ≤
+      ∏ i ∈ s, lpNorm (f i) (p i) μ := by
+  induction s using Finset.cons_induction with
+  | empty =>
+    by_cases hμ : μ = 0
+    · simp [hμ]
+    · simp [lpNorm_const, hμ]
+  | cons i s hi ih =>
+    have hfi : MemLp (f i) (p i) μ := hf i (Finset.mem_cons_self ..)
+    have hfs : ∀ a ∈ s, MemLp (f a) (p a) μ := fun a ha => hf a (Finset.mem_cons_of_mem ha)
+    have hprod := MemLp.fun_prod hfs
+    let : ENNReal.HolderTriple (p i) (∑ a ∈ s, (p a)⁻¹)⁻¹ (∑ a ∈ Finset.cons i s hi, (p a)⁻¹)⁻¹ :=
+      ⟨by simp only [inv_inv,Finset.sum_cons]⟩
+    simp only [Finset.prod_cons]
+    exact (lpNorm_complex_mul_le hfi hprod).trans (mul_le_mul_of_nonneg_left (ih hfs) lpNorm_nonneg)
+
+lemma exponent_one_le {m j : ℕ} (hm : 0 < m) (hj : j ≤ m) : 1 ≤ exponent m j := by
+  by_cases hj0 : j = 0
+  · subst j; simp [exponent_zero]
+  · rw [exponent_of_pos hm (Nat.pos_of_ne_zero hj0)]
+    apply ENNReal.one_le_ofReal.mpr
+    apply (le_div_iff₀ (by exact_mod_cast (Nat.pos_of_ne_zero hj0) : 0 < (j:ℝ))).mpr
+    have hh : (j:ℝ) ≤ (m:ℝ) := by exact_mod_cast hj
+    have h0 : 0 ≤ (m:ℝ) := by positivity
+    linarith only [hh,h0]
+
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+variable {ι κ : Type*} [Fintype ι] [Nonempty ι] [Fintype κ] [Nonempty κ]
+
+theorem cjet_product_L2 (e : ι → E) (f : κ → E → ℂ)
+    (hf : ∀ a, ContDiff ℝ ∞ (f a)) (hc : ∀ a, HasCompactSupport (f a))
+    (m : ℕ) (hm : 0 < m) {ν : Type*} (s : Finset ν)
+    (a : ν → κ) (j : ν → ℕ) (w : (i : ν) → Fin (j i) → ι)
+    (hs : ∑ i ∈ s, j i = m) :
+    lpNorm (fun x => ∏ i ∈ s, cjet e (f (a i)) (w i) x) 2 volume ≤
+      2^s.card * ((2*(m:ℝ))^(∑ i ∈ s, j i*(m-j i)) *
+        (familyJetNorm e (realComponents f) 0 ∞)^(s.card-1) *
+          familyJetNorm e (realComponents f) m 2) := by
+  have hi (i : ν) : MemLp (cjet e (f (a i)) (w i)) (exponent m (j i)) volume :=
+    (cjet_smooth e (hf (a i)) (w i)).continuous.memLp_of_hasCompactSupport
+      (cjet_compact e (hc (a i)) (w i))
+  have hh := lpNorm_complex_finite_prod (μ:=volume) s
+    (fun i => cjet e (f (a i)) (w i)) (fun i => exponent m (j i)) (fun i _ => hi i)
+  rw [exponent_sum_inverse s j m hm hs] at hh
+  apply hh.trans
+  have hprod : (∏ i ∈ s, lpNorm (cjet e (f (a i)) (w i)) (exponent m (j i)) volume) ≤
+      ∏ i ∈ s, 2*familyJetNorm e (realComponents f) (j i) (exponent m (j i)) := by
+    apply Finset.prod_le_prod₀
+    · intro i _; exact lpNorm_nonneg
+    · intro i hmem
+      have hj : j i ≤ m := by
+        rw [← hs]; exact Finset.single_le_sum (fun _ _ => Nat.zero_le _) hmem
+      exact cjet_lpNorm_le e f hf hc _ _ (exponent_one_le hm hj)
+  apply hprod.trans
+  rw [Finset.prod_mul_distrib,Finset.prod_const]
+  exact mul_le_mul_of_nonneg_left
+    (family_jet_tame_many e (realComponents f) (realComponents_smooth f hf)
+      (realComponents_compact f hc) m hm s j hs) (by positivity)
+end TameInterpolation
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/ContractedBochner.lean b/lean/OAI/Geometry/PrescribedRicci/ContractedBochner.lean
new file mode 100644
index 0000000000000000000000000000000000000000..838a31b824460eb23ad8eeed088cc901f8719348
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/ContractedBochner.lean
@@ -0,0 +1,87 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.ChernBochner
+import OAI.Geometry.PrescribedRicci.HermitianTensorEnergy
+import OAI.Geometry.PrescribedRicci.MatrixWirtingerExtra
+
+namespace OAI
+
+/-! Contracted Bochner. -/
+
+noncomputable section
+open Matrix Set Filter Topology
+open scoped ComplexOrder ContDiff MatrixOrder
+namespace MongeAmpere
+variable {n : Type*} [Fintype n]
+lemma hermPair_self_ofReal {K : Matrix n n ℂ} (hK : K.IsHermitian) (v : n → ℂ) :
+    ((hermPair K v v).re : ℂ) = hermPair K v v := by
+  have he := congrArg Complex.im (hermPair_conj K hK v v)
+  apply Complex.ext
+  · rfl
+  · simp only [Complex.ofReal_im]
+    simp only [Complex.star_def,Complex.conj_im] at he
+    linarith
+end MongeAmpere
+namespace Anticanonical.SourceSmooth.KaehlerMetric
+open MongeAmpere
+variable {d : ℕ} {n : Type*} [Fintype n]
+
+lemma hessian_hermPair {s : Set (Coordinates d)} (hs : IsOpen s)
+    {K : Coordinates d → Matrix n n ℂ} {f : Coordinates d → n → ℂ}
+    {z : Coordinates d} (hz : z ∈ s)
+    (hK : ∀ y ∈ s, (K y).IsHermitian)
+    (hKs : ∀ y ∈ s, ∀ i j, ContDiffAt ℝ ∞ (fun w => K w i j) y)
+    (hf : ∀ y ∈ s, ∀ i, ContDiffAt ℝ ∞ (fun w => f w i) y) (a b : Fin d) :
+    PotentialKaehler.potentialMatrix (fun y => (hermPair (K y) (f y) (f y)).re) z b a =
+      barDeriv (fun y => holDeriv (fun w => hermPair (K w) (f w) (f w)) y a) z b := by
+  have hdr (y : Coordinates d) (hy : y ∈ s) :
+      ContDiffAt ℝ ∞ (fun w => (hermPair (K w) (f w) (f w)).re) y :=
+    Complex.reCLM.contDiff.contDiffAt.comp y (hermPair_smooth (hKs y hy) (hf y hy) (hf y hy))
+  rw [← barDeriv_holRealDeriv (hdr z hz)]
+  apply barDeriv_congr
+  filter_upwards [hs.mem_nhds hz] with y hy
+  rw [← holDeriv_ofReal ((hdr y hy).differentiableAt (by simp))]
+  apply holDeriv_congr
+  filter_upwards [hs.mem_nhds hy] with w hw
+  exact hermPair_self_ofReal (hK w hw) _
+
+lemma contracted_bochners_identity {s : Set (Coordinates d)} (hs : IsOpen s)
+    {K : Coordinates d → Matrix n n ℂ} {C : Coordinates d → Fin d → Matrix n n ℂ}
+    {f : Coordinates d → n → ℂ} {z : Coordinates d} (hz : z ∈ s)
+    (hK : ∀ y ∈ s, ∀ i j, ContDiffAt ℝ ∞ (fun w => K w i j) y)
+    (hC : ∀ y ∈ s, ∀ a i j, ContDiffAt ℝ ∞ (fun w => C w a i j) y)
+    (hf : ∀ y ∈ s, ∀ i, ContDiffAt ℝ ∞ (fun w => f w i) y)
+    (hKh : ∀ y ∈ s, (K y).IsHermitian)
+    (hhol : ∀ y ∈ s, ∀ a, holArray K y a = K y*C y a)
+    (hbar : ∀ y ∈ s, ∀ a, barArray K y a = (C y a)ᴴ*K y)
+    (B : Matrix (Fin d) (Fin d) ℂ) (hB : B.IsHermitian) :
+    (B*PotentialKaehler.potentialMatrix (fun y => (hermPair (K y) (f y) (f y)).re) z).trace.re =
+      (tensorEnergy B (K z) (barVector f z)).re+
+      (tensorEnergy B.transpose (K z) (covHol C f z)).re+
+      2*(hermPair (K z) (f z)
+        (∑ a, ∑ b, B a b • covHol C (fun y => barVector f y b) z a)).re+
+      (hermPair (K z) (f z)
+        ((∑ a, ∑ b, B a b • barArray (fun y => C y a) z b)*ᵥ f z)).re := by
+  have he (a b : Fin d) := hermPair_bochners_identity hs hz hK hC hf hhol hbar a b
+  simp only [Matrix.trace,Matrix.diag_apply,Matrix.mul_apply]
+  simp_rw [hessian_hermPair hs hz hKh hK hf,he]
+  simp only [mul_add,Finset.sum_add_distrib]
+  rw [hermPair_weighted_rough_left B hB,hermPair_weighted_rough_right,
+    hermPair_weighted_curvature,tensorEnergy_flip]
+  change (hermPair (K z)
+      (∑ a, ∑ b, B a b • covHol C (fun y => barVector f y b) z a) (f z)+
+    tensorEnergy B (K z) (barVector f z)+
+    tensorEnergy B.transpose (K z) (covHol C f z)+
+    hermPair (K z) (f z)
+      (∑ a, ∑ b, B a b • covHol C (fun y => barVector f y b) z a)+
+    hermPair (K z) (f z)
+      ((∑ a, ∑ b, B a b • barArray (fun y => C y a) z b)*ᵥ f z)).re = _
+  have hc := congrArg Complex.re (hermPair_conj (K z) (hKh z hz) (f z)
+    (∑ a, ∑ b, B a b • covHol C (fun y => barVector f y b) z a))
+  simp only [Complex.add_re,Complex.star_def,Complex.conj_re] at hc ⊢
+  linarith
+
+end Anticanonical.SourceSmooth.KaehlerMetric
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/CoordinateVolumeComparison.lean b/lean/OAI/Geometry/PrescribedRicci/CoordinateVolumeComparison.lean
new file mode 100644
index 0000000000000000000000000000000000000000..a5ea33f7a4ef5800449334c5a5552d06084630ce
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/CoordinateVolumeComparison.lean
@@ -0,0 +1,95 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.GlobalRellich
+import OAI.Geometry.PrescribedRicci.GlobalKahlerIntegral
+
+namespace OAI
+
+/-! Coordinate Volume Comparison. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory
+open scoped SchwartzMap ContDiff Classical
+namespace EllipticKernel
+
+instance coordinateMap_haar (d : ℕ) :
+    (Measure.map (coordinateEquiv d) (volume : Measure (EC d))).IsAddHaarMeasure :=
+  (coordinateEquiv d).isAddHaarMeasure_map _
+
+def coordinateVolumeFactor (d : ℕ) : ℝ :=
+  Measure.addHaarScalarFactor (Measure.map (coordinateEquiv d) (volume : Measure (EC d))) volume
+
+lemma coordinateVolumeFactor_pos (d : ℕ) : 0 < coordinateVolumeFactor d :=
+  Measure.addHaarScalarFactor_pos_of_isAddHaarMeasure _ _
+
+lemma integral_coordinates (d : ℕ) {F : Anticanonical.Coordinates d → ℝ} (hF : Continuous F) :
+    (∫ z : EC d, F (coordinateEquiv d z)) = coordinateVolumeFactor d * ∫ z, F z := by
+  rw [← integral_map (coordinateEquiv d).continuous.measurable.aemeasurable hF.aestronglyMeasurable,
+    Measure.isAddLeftInvariant_eq_smul (Measure.map (coordinateEquiv d) volume) volume,
+    integral_smul_nnreal_measure]
+  rfl
+
+end EllipticKernel
+namespace SobolevChart
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+
+lemma schwartzCoord_zero_eq (f : 𝓢(E, ℂ)) : schwartzCoord 0 f = f.toLp 2 := by
+  apply realize_injective 0
+  rw [realize_schwartzCoord, realize_zero]
+  exact (Lp.toTemperedDistribution_toLp_eq f).symm
+
+lemma norm_sq_schwartzCoord_zero (f : 𝓢(E, ℂ)) :
+    ‖schwartzCoord 0 f‖ ^ 2 = ∫ z, ‖f z‖ ^ 2 := by
+  rw [schwartzCoord_zero_eq]
+  have hi := inner_self_eq_norm_sq_to_K (𝕜 := ℂ) (f.toLp 2)
+  rw [SchwartzMap.inner_toL2_toL2_eq f f volume] at hi
+  simp only [inner_self_eq_norm_sq_to_K, ← RCLike.ofReal_pow] at hi
+  rw [integral_ofReal] at hi
+  exact (RCLike.ofReal_injective hi).symm
+
+end SobolevChart
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X}
+
+lemma norm_localize_coordinates (i : Fin A.count) (ρ f : Smooth A)
+    (hs : tsupport (ρ : X → ℂ) ⊆ (A.euclideanChart i).source) (z : EC d) :
+    ‖localize A i ρ hs f z‖ ^ 2 =
+      KaehlerMetric.localizeFunction i (fun x => ‖ρ x * f x‖ ^ 2) (coordinateEquiv d z) := by
+  by_cases hz : z ∈ (A.euclideanChart i).target
+  · have hz' : coordinateEquiv d z ∈ (A.chart i).target := by simpa using hz
+    simp only [localize_apply, localizeFun, ite_eq_left hz,
+      KaehlerMetric.localizeFunction, ite_eq_left hz']
+    rfl
+  · have hz' : coordinateEquiv d z ∉ (A.chart i).target := by simpa using hz
+    simp only [localize_apply, localizeFun, ite_eq_right hz,
+      KaehlerMetric.localizeFunction, ite_eq_right hz', norm_zero, zero_pow (by decide : 2 ≠ 0)]
+
+lemma norm_sq_localize_integral (i : Fin A.count) (ρ f : Smooth A)
+    (hs : tsupport (ρ : X → ℂ) ⊆ (A.euclideanChart i).source) :
+    ‖schwartzCoord 0 (localize A i ρ hs f)‖ ^ 2 =
+      coordinateVolumeFactor d * ∫ z, KaehlerMetric.localizeFunction i (fun x => ‖ρ x * f x‖ ^ 2) z := by
+  rw [norm_sq_schwartzCoord_zero]
+  simp_rw [norm_localize_coordinates]
+  apply integral_coordinates
+  apply (KaehlerMetric.localizeFunction_continuous_compact i
+    ((ρ.continuous.mul f.continuous).norm.pow 2) ?_).1
+  refine Subset.trans (closure_minimal ?_ (isClosed_tsupport (ρ : X → ℂ))) (by simpa using hs)
+  intro x hx
+  apply subset_tsupport
+  change ρ x ≠ 0
+  intro h
+  have : ‖ρ x * f x‖ ^ 2 = 0 := by rw [h, zero_mul, norm_zero, zero_pow (by decide : 2 ≠ 0)]
+  exact hx this
+
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/CurvatureCompactBounds.lean b/lean/OAI/Geometry/PrescribedRicci/CurvatureCompactBounds.lean
new file mode 100644
index 0000000000000000000000000000000000000000..3fd7f2402bc1a6c6a9c698d2ebff71d4a6f3fe7f
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/CurvatureCompactBounds.lean
@@ -0,0 +1,123 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.GlobalKahlerIntegral
+import OAI.Geometry.PrescribedRicci.MatrixWirtinger
+import OAI.Geometry.PrescribedRicci.MatrixWirtingerExtra
+import OAI.Geometry.PrescribedRicci.VolumePathRicci
+
+namespace OAI
+
+/-! Curvature Compact Bounds. -/
+
+section
+
+ 
+
+noncomputable section
+open Matrix Filter Set Topology
+open scoped ContDiff ComplexOrder MatrixOrder Matrix.Norms.Elementwise
+namespace Anticanonical.SourceSmooth.KaehlerMetric
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+
+lemma curvatureMatrix_smooth (g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) (b a : Fin d) :
+    ContDiffAt ℝ ∞ (fun y => g.curvatureMatrix q y b a) z := by
+  have hs := (g.smooth q).contDiffAt ((A.chart q).open_target.mem_nhds hz)
+  have hi : ContDiffAt ℝ ∞ (fun y => (g.matrix q y)⁻¹) z :=
+    ((MongeAmpere.contDiffAt_inv _ (g.positive q z hz).det_pos.ne').restrict_scalars ℝ).comp z hs
+  exact (contDiffAt_barDerivative (contDiffAt_holDerivative hs a) b).neg.add
+    (matrix_mul_contDiffAt (matrix_mul_contDiffAt (contDiffAt_barDerivative hs b) hi)
+      (contDiffAt_holDerivative hs a))
+
+lemma curvatureRicci_smooth (g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) :
+    ContDiffAt ℝ ∞ (g.curvatureRicci q) z := by
+  apply (PotentialKaehler.potentialMatrix_smooth (g.ricciPotential_contDiffAt q hz)).congr_of_eventuallyEq
+  filter_upwards [(A.chart q).open_target.mem_nhds hz] with y hy
+  exact g.curvatureRicci_eq q hy
+
+def backgroundSize (g : KaehlerMetric A) (F : SmoothRealFunction A)
+    (q : Fin A.count) (z : Coordinates d) : ℝ :=
+  1+‖g.matrix q z‖+‖(g.matrix q z)⁻¹‖+‖g.curvatureRicci q z‖+
+    ‖F.hessian q z‖+‖g.curvatureMatrix q z‖
+
+def BackgroundComponentBound (g : KaehlerMetric A) (F : SmoothRealFunction A)
+    (q : Fin A.count) (z : Coordinates d) (M : ℝ) : Prop :=
+  (∀ i j, ‖g.matrix q z i j‖ ≤ M) ∧
+  (∀ i j, ‖(g.matrix q z)⁻¹ i j‖ ≤ M) ∧
+  (∀ i j, ‖g.curvatureRicci q z i j‖ ≤ M) ∧
+  (∀ i j, ‖F.hessian q z i j‖ ≤ M) ∧
+  (∀ a b i j, ‖g.curvatureMatrix q z b a i j‖ ≤ M)
+
+lemma backgroundSize_continuousOn (g : KaehlerMetric A) (F : SmoothRealFunction A)
+    (q : Fin A.count) : ContinuousOn (g.backgroundSize F q) (A.chart q).target := by
+  intro z hz
+  have hs := (g.smooth q).contDiffAt ((A.chart q).open_target.mem_nhds hz)
+  have hi : ContDiffAt ℝ ∞ (fun y => (g.matrix q y)⁻¹) z :=
+    ((MongeAmpere.contDiffAt_inv _ (g.positive q z hz).det_pos.ne').restrict_scalars ℝ).comp z hs
+  have hR : ContDiffAt ℝ ∞ (g.curvatureMatrix q) z := by
+    apply contDiffAt_pi.mpr
+    intro b
+    apply contDiffAt_pi.mpr
+    intro a
+    exact g.curvatureMatrix_smooth q hz b a
+  exact (((((continuousAt_const.add hs.continuousAt.norm).add hi.continuousAt.norm).add
+    (g.curvatureRicci_smooth q hz).continuousAt.norm).add
+      ((F.hessian_smooth q).contDiffAt ((A.chart q).open_target.mem_nhds hz)).continuousAt.norm).add
+        hR.continuousAt.norm).continuousWithinAt
+
+lemma backgroundComponents_of_size (g : KaehlerMetric A) (F : SmoothRealFunction A)
+    (q : Fin A.count) (z : Coordinates d) {M : ℝ} (hb : g.backgroundSize F q z ≤ M) :
+    g.BackgroundComponentBound F q z M := by
+  have h1 : ‖g.matrix q z‖ ≤ M := by unfold backgroundSize at hb; linarith [norm_nonneg ((g.matrix q z)⁻¹),norm_nonneg (g.curvatureRicci q z),norm_nonneg (F.hessian q z),norm_nonneg (g.curvatureMatrix q z)]
+  have h2 : ‖(g.matrix q z)⁻¹‖ ≤ M := by unfold backgroundSize at hb; linarith [norm_nonneg (g.matrix q z),norm_nonneg (g.curvatureRicci q z),norm_nonneg (F.hessian q z),norm_nonneg (g.curvatureMatrix q z)]
+  have h3 : ‖g.curvatureRicci q z‖ ≤ M := by unfold backgroundSize at hb; linarith [norm_nonneg (g.matrix q z),norm_nonneg ((g.matrix q z)⁻¹),norm_nonneg (F.hessian q z),norm_nonneg (g.curvatureMatrix q z)]
+  have h4 : ‖F.hessian q z‖ ≤ M := by unfold backgroundSize at hb; linarith [norm_nonneg (g.matrix q z),norm_nonneg ((g.matrix q z)⁻¹),norm_nonneg (g.curvatureRicci q z),norm_nonneg (g.curvatureMatrix q z)]
+  have h5 : ‖g.curvatureMatrix q z‖ ≤ M := by unfold backgroundSize at hb; linarith [norm_nonneg (g.matrix q z),norm_nonneg ((g.matrix q z)⁻¹),norm_nonneg (g.curvatureRicci q z),norm_nonneg (F.hessian q z)]
+  refine ⟨fun i j => ?_,fun i j => ?_,fun i j => ?_,fun i j => ?_,fun a b i j => ?_⟩
+  · exact (norm_le_pi_norm _ j).trans ((norm_le_pi_norm _ i).trans h1)
+  · exact (norm_le_pi_norm _ j).trans ((norm_le_pi_norm _ i).trans h2)
+  · exact (norm_le_pi_norm _ j).trans ((norm_le_pi_norm _ i).trans h3)
+  · exact (norm_le_pi_norm _ j).trans ((norm_le_pi_norm _ i).trans h4)
+  · exact (norm_le_pi_norm _ j).trans ((norm_le_pi_norm _ i).trans
+      ((norm_le_pi_norm _ a).trans ((norm_le_pi_norm _ b).trans h5)))
+
+variable [T2Space X] [CompactSpace X]
+
+lemma compact_background_bounds (g : KaehlerMetric A) (F : SmoothRealFunction A) :
+    ∃ M : ℝ, 1 ≤ M ∧ ∀ q : Fin A.count, ∀ x ∈ tsupport (chartPartition (A:=A) q).value,
+      g.BackgroundComponentBound F q (A.chart q x) M := by
+  have hb (q : Fin A.count) : ∃ M : ℝ, 1 ≤ M ∧ ∀ x ∈ tsupport (chartPartition (A:=A) q).value,
+      g.backgroundSize F q (A.chart q x) ≤ M := by
+    let K := (A.chart q) '' tsupport (chartPartition (A:=A) q).value
+    have hK : IsCompact K := (isClosed_tsupport _).isCompact.image_of_continuousOn
+      ((A.chart q).continuousOn.mono (chartPartition_support q))
+    have hKt : K ⊆ (A.chart q).target := by
+      rintro z ⟨x,hx,rfl⟩
+      exact (A.chart q).mapsTo (chartPartition_support q hx)
+    obtain ⟨b,hb⟩ := hK.bddAbove_image ((g.backgroundSize_continuousOn F q).mono hKt)
+    refine ⟨max 1 b,le_max_left _ _,fun x hx => ?_⟩
+    exact (hb (mem_image_of_mem _ (mem_image_of_mem _ hx))).trans (le_max_right _ _)
+  choose M hM hMb using hb
+  refine ⟨1+∑ q, M q,?_,fun q x hx => g.backgroundComponents_of_size F q (A.chart q x) ?_⟩
+  · have hn : 0 ≤ ∑ q, M q := Finset.sum_nonneg (fun q _ => (by norm_num : (0:ℝ)≤1).trans (hM q))
+    linarith
+  · have hle : M q ≤ ∑ j, M j := Finset.single_le_sum
+      (fun j _ => (by norm_num : (0:ℝ)≤1).trans (hM j)) (Finset.mem_univ q)
+    exact (hMb q x hx).trans (by linarith)
+
+lemma exists_partition_chart (x : X) : ∃ q : Fin A.count,
+    x ∈ tsupport (chartPartition (A:=A) q).value := by
+  by_contra! hh
+  have hz (q : Fin A.count) : (chartPartition (A:=A) q).value x = 0 := by
+    by_contra hn
+    exact hh q (subset_tsupport _ hn)
+  have he := chartPartition_sum (A:=A) x
+  simp only [hz,Finset.sum_const_zero] at he
+  norm_num at he
+
+end Anticanonical.SourceSmooth.KaehlerMetric
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/CurvatureRicci.lean b/lean/OAI/Geometry/PrescribedRicci/CurvatureRicci.lean
new file mode 100644
index 0000000000000000000000000000000000000000..7e32db4ee85a027b09b22ea6587c248b322123d6
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/CurvatureRicci.lean
@@ -0,0 +1,114 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.ChernLuAlgebra
+import OAI.Geometry.PrescribedPotential.VolumePath
+
+namespace OAI
+
+/-! Curvature Ricci. -/
+
+section
+
+ 
+
+noncomputable section
+open Matrix Filter Set Topology
+open scoped ContDiff ComplexOrder MatrixOrder Matrix.Norms.Elementwise
+namespace MongeAmpere
+variable {d : ℕ}
+local notation "Mat" => Matrix (Fin d) (Fin d) ℂ
+
+lemma contDiffAt_logdetReal (H : Mat) (hH : H.PosDef) :
+    ContDiffAt ℝ ∞ (fun M : Mat => Real.log M.det.re) H := by
+  exact ((Complex.reCLM.contDiff.contDiffAt).comp H
+    ((contDiff_det.restrict_scalars ℝ).contDiffAt)).log
+      (ne_of_gt (Complex.pos_iff.mp hH.det_pos).1)
+
+lemma fderiv_logdetReal (H K : Mat) (hH : H.PosDef) :
+    fderiv ℝ (fun M : Mat => Real.log M.det.re) H K = (H⁻¹*K).trace.re := by
+  have hp : HasDerivAt (fun t : ℝ => H + (t:ℂ) • K) K 0 := by
+    convert! ((hasDerivAt_id (0 : ℝ)).smul_const K).const_add H using 1
+    simp only [one_smul]
+  have hd := ((contDiffAt_logdetReal H hH).differentiableAt (by simp)).hasFDerivAt
+  have hd' : HasFDerivAt (fun M : Mat => Real.log M.det.re)
+      (fderiv ℝ (fun M : Mat => Real.log M.det.re) H) (H + ((0:ℝ):ℂ) • K) := by
+    convert! hd using 1
+    simp only [Complex.ofReal_zero,zero_smul,add_zero]
+  have hf := hd'.comp_hasDerivAt 0 hp
+  exact hf.unique (hasDerivAt_logdet_add_smul H K hH)
+
+lemma trace_mul_real {H K : Mat} (hH : H.IsHermitian) (hK : K.IsHermitian) :
+    ((H*K).trace.re : ℂ) = (H*K).trace := by
+  have he : star (H*K).trace = (H*K).trace := by
+    rw [← trace_conjTranspose,conjTranspose_mul,hH.eq,hK.eq,trace_mul_comm]
+  have hi := congrArg Complex.im he
+  change -(H*K).trace.im = (H*K).trace.im at hi
+  apply Complex.ext
+  · rfl
+  · simp only [Complex.ofReal_im]; linarith
+
+end MongeAmpere
+namespace Anticanonical.SourceSmooth.KaehlerMetric
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+local notation "Mat" => Matrix (Fin d) (Fin d) ℂ
+
+lemma fderiv_logVolume (g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) (v : Coordinates d) :
+    fderiv ℝ (fun y => Real.log (g.volumeCoefficient q y)) z v =
+      ((g.matrix q z)⁻¹*fderiv ℝ (g.matrix q) z v).trace.re := by
+  have hg := ((g.smooth q).contDiffAt ((A.chart q).open_target.mem_nhds hz)).differentiableAt (by simp)
+  have hh := (((MongeAmpere.contDiffAt_logdetReal _ (g.positive q z hz)).differentiableAt (by simp)).hasFDerivAt.comp z hg.hasFDerivAt).fderiv
+  have he := congrArg (fun L : Coordinates d →L[ℝ] ℝ => L v) hh
+  exact he.trans (MongeAmpere.fderiv_logdetReal _ _ (g.positive q z hz))
+
+lemma holRealDeriv_logVolume (g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) (a : Fin d) :
+    holRealDeriv (fun y => Real.log (g.volumeCoefficient q y)) z a =
+      ((g.matrix q z)⁻¹*holDerivative (g.matrix q) z a).trace := by
+  change (2:ℂ)⁻¹ * ((fderiv ℝ (fun y => Real.log (g.volumeCoefficient q y)) z (coordinateVector a):ℂ) -
+    Complex.I * (fderiv ℝ (fun y => Real.log (g.volumeCoefficient q y)) z (Complex.I • coordinateVector a):ℂ)) = _
+  rw [g.fderiv_logVolume q hz,g.fderiv_logVolume q hz,
+    MongeAmpere.trace_mul_real (g.positive q z hz).inv.isHermitian (g.derivative_hermitian q hz _),
+    MongeAmpere.trace_mul_real (g.positive q z hz).inv.isHermitian (g.derivative_hermitian q hz _)]
+  simp only [holDerivative,Matrix.mul_smul,mul_sub,trace_smul,trace_sub,smul_eq_mul]
+
+lemma curvatureRicci_eq (g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) :
+    g.curvatureRicci q z = PotentialKaehler.potentialMatrix (g.ricciPotential q) z := by
+  have hs := (g.smooth q).contDiffAt ((A.chart q).open_target.mem_nhds hz)
+  have hi := ((MongeAmpere.contDiffAt_inv _ (g.positive q z hz).det_pos.ne').restrict_scalars ℝ).comp z hs
+  have hl := (g.volumeCoefficient_smooth q).contDiffAt ((A.chart q).open_target.mem_nhds hz)
+  have hl' := hl.log (g.volumeCoefficient_pos q hz).ne'
+  ext b a
+  have he : (fun y => holRealDeriv (g.ricciPotential q) y a) =ᶠ[nhds z]
+      (fun y => -((g.matrix q y)⁻¹*holDerivative (g.matrix q) y a).trace) := by
+    filter_upwards [(A.chart q).open_target.mem_nhds hz] with y hy
+    have hh := (g.volumeCoefficient_smooth q).contDiffAt ((A.chart q).open_target.mem_nhds hy)
+    have hh' := (hh.log (g.volumeCoefficient_pos q hy).ne').differentiableAt (by simp)
+    change holRealForm a (fderiv ℝ (fun w => -Real.log (g.volumeCoefficient q w)) y) = _
+    rw [fderiv_fun_neg,map_neg]
+    exact congrArg Neg.neg (g.holRealDeriv_logVolume q hy a)
+  change ((g.matrix q z)⁻¹*g.curvatureMatrix q z b a).trace =
+    PotentialKaehler.potentialMatrix (g.ricciPotential q) z b a
+  have hric : ContDiffAt ℝ ∞ (g.ricciPotential q) z := hl'.neg
+  rw [← barDeriv_holRealDeriv hric]
+  have hb : barDeriv (fun y => holRealDeriv (g.ricciPotential q) y a) z b =
+      -barDeriv (fun y => ((g.matrix q y)⁻¹*holDerivative (g.matrix q) y a).trace) z b := by
+    unfold barDeriv
+    rw [he.fderiv_eq,fderiv_fun_neg]
+    simp only [_root_.neg_apply]; ring
+  have hid : DifferentiableAt ℝ (fun y => (g.matrix q y)⁻¹) z := hi.differentiableAt (by simp)
+  have hpd := (contDiffAt_holDerivative hs a).differentiableAt (by simp)
+  have hprod : DifferentiableAt ℝ (fun y => (g.matrix q y)⁻¹*holDerivative (g.matrix q) y a) z :=
+    (MongeAmpere.matrixMulCLM.hasFDerivAt_of_bilinear hid.hasFDerivAt hpd.hasFDerivAt).differentiableAt
+  rw [hb,barDeriv_trace hprod,
+    barDerivative_mul hid hpd,
+    barDerivative_inv (hs.differentiableAt (by simp)) (isUnit_iff_ne_zero.mpr (g.positive q z hz).det_pos.ne')]
+  simp only [curvatureMatrix,mul_add,mul_neg,neg_mul,trace_add,trace_neg,mul_assoc]
+  abel
+
+end Anticanonical.SourceSmooth.KaehlerMetric
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/CutoffJetBound.lean b/lean/OAI/Geometry/PrescribedRicci/CutoffJetBound.lean
new file mode 100644
index 0000000000000000000000000000000000000000..467a7dbe03cb53dcccc6a77973e87413896796fc
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/CutoffJetBound.lean
@@ -0,0 +1,102 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.ChartTransport
+import OAI.Geometry.PrescribedRicci.JetTrim
+
+namespace OAI
+
+/-! Cutoff Jet Bound. -/
+
+section
+
+ 
+noncomputable section
+open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory LineDeriv
+open scoped ContDiff SchwartzMap Classical ENNReal
+namespace GlobalElliptic
+open SobolevChart TameInterpolation
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+  {ι κ Z : Type*} [Fintype ι] [Nonempty ι] [Fintype κ] [Nonempty κ]
+
+lemma initialJet_zero_bound (e : ι → E) (F : κ → 𝓢(E,ℂ)) {B : ℝ} (hB : 0 ≤ B)
+    (h0 : ∀ a x, ‖F a x‖ ≤ B)
+    (h1 : ∀ a i x, ‖fderiv ℝ (F a) x (e i)‖ ≤ B) :
+    familyJetNorm e (realComponents (initialJet e (fun a x => F a x))) 0 ∞ ≤ B := by
+  obtain ⟨⟨⟨a,i⟩,b⟩,ha⟩ := familyJetNorm_attained e (realComponents (initialJet e (fun a x => F a x))) 0 ∞
+  rw [ha]
+  obtain ⟨w,hw⟩ := jetNorm_attained e (realComponents (initialJet e (fun a x => F a x)) ((a,i),b)) 0 ∞
+  rw [hw]
+  have hh (x : E) : ‖initialJet e (fun a x => F a x) (a,i) x‖ ≤ B := by
+    cases i with
+    | none => exact h0 a x
+    | some i => exact h1 a i x
+  have hmeas := (realComponents_smooth (initialJet e (fun a x => F a x))
+    (initialJet_smooth e _ (fun a => (F a).smooth')) ((a,i),b)).continuous.aestronglyMeasurable (μ:=volume)
+  change lpNorm (realComponents (initialJet e (fun a x => F a x)) ((a,i),b)) ∞ volume ≤ B
+  rw [← toReal_eLpNorm,eLpNorm_exponent_top hmeas]
+  apply (ENNReal.toReal_mono ENNReal.ofReal_ne_top (eLpNormEssSup_le_of_ae_bound ?_)).trans_eq
+    (ENNReal.toReal_ofReal hB)
+  filter_upwards [] with x
+  change ‖if b then (initialJet e (fun a x => F a x) (a,i) x).im else
+    (initialJet e (fun a x => F a x) (a,i) x).re‖ ≤ B
+  cases b
+  · exact (Complex.abs_re_le_norm _).trans (hh x)
+  · exact (Complex.abs_im_le_norm _).trans (hh x)
+
+omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] [Nonempty ι] in
+lemma ChartCutoff.uniform_C1 {U : Set E} (σ : ChartCutoff U) (hU : IsOpen U)
+    (e : ι → E) (f : Z → E → ℂ) (hs : ∀ z, ContDiffOn ℝ ∞ (f z) U)
+    {M P : ℝ} (hM : 0 ≤ M) (hP : 0 ≤ P)
+    (hm : ∀ z x, x ∈ tsupport (σ : E → ℂ) → ‖f z x‖ ≤ M)
+    (hp : ∀ z x, x ∈ tsupport (σ : E → ℂ) → ∀ i, ‖fderiv ℝ (f z) x (e i)‖ ≤ P) :
+    ∃ B : ℝ, 0 ≤ B ∧ ∀ z, (∀ x, ‖σ.product hU (hs z) x‖ ≤ B) ∧
+      (∀ i x, ‖fderiv ℝ (σ.product hU (hs z)) x (e i)‖ ≤ B) := by
+  let K := ‖σ.val.toBoundedContinuousFunction‖
+  let Q := ∑ i, ‖(∂_{e i} σ.val).toBoundedContinuousFunction‖
+  have hK : 0 ≤ K := norm_nonneg _
+  have hQ : 0 ≤ Q := Finset.sum_nonneg fun _ _ => norm_nonneg _
+  have hB : 0 ≤ K*M+(Q*M+K*P) := add_nonneg (mul_nonneg hK hM)
+    (add_nonneg (mul_nonneg hQ hM) (mul_nonneg hK hP))
+  refine ⟨K*M+(Q*M+K*P),hB,fun z => ⟨?_,?_⟩⟩
+  · intro x
+    by_cases hx : x ∈ tsupport (σ : E → ℂ)
+    · rw [ChartCutoff.product_apply,norm_mul]
+      have h := mul_le_mul (σ.val.toBoundedContinuousFunction.norm_coe_le_norm x) (hm z x hx)
+        (norm_nonneg _) hK
+      change ‖σ x‖*‖f z x‖ ≤ K*M at h
+      nlinarith only [h,mul_nonneg hQ hM,mul_nonneg hK hP]
+    · rw [ChartCutoff.product_apply,image_eq_zero_of_notMem_tsupport hx,zero_mul,norm_zero]
+      exact hB
+  · intro i x
+    by_cases hx : x ∈ tsupport (σ : E → ℂ)
+    · have hf := ((hs z).contDiffAt (hU.mem_nhds (σ.support_sub hx))).differentiableAt (by simp)
+      have he : fderiv ℝ (σ.product hU (hs z)) x (e i) =
+          σ x * fderiv ℝ (f z) x (e i) + fderiv ℝ σ.val x (e i) * f z x := by
+        change fderiv ℝ (fun y => σ.val y * f z y) x (e i) = _
+        rw [fderiv_fun_mul (σ.val.differentiableAt) hf]
+        simp only [_root_.add_apply,_root_.smul_apply,smul_eq_mul]
+        ring
+      rw [he]
+      have hq : ‖fderiv ℝ σ.val x (e i)‖ ≤ Q := by
+        have hh := (∂_{e i} σ.val).toBoundedContinuousFunction.norm_coe_le_norm x
+        change ‖(∂_{e i} σ.val) x‖ ≤ _ at hh
+        rw [SchwartzMap.lineDerivOp_apply_eq_fderiv] at hh
+        exact hh.trans (Finset.single_le_sum (fun j _ => norm_nonneg (∂_{e j} σ.val).toBoundedContinuousFunction) (Finset.mem_univ i))
+      have hk : ‖σ x‖ ≤ K := σ.val.toBoundedContinuousFunction.norm_coe_le_norm x
+      have hsum : ‖σ x * fderiv ℝ (f z) x (e i) + fderiv ℝ σ.val x (e i) * f z x‖ ≤ K*P+Q*M := by
+        apply (norm_add_le _ _).trans
+        simp only [norm_mul]
+        exact add_le_add (mul_le_mul hk (hp z x hx i) (norm_nonneg _) hK)
+          (mul_le_mul hq (hm z x hx) (norm_nonneg _) hQ)
+      exact hsum.trans (by nlinarith only [mul_nonneg hK hM])
+    · have he : (σ.product hU (hs z) : E → ℂ) =ᶠ[𝓝 x] 0 := by
+        filter_upwards [notMem_tsupport_iff_eventuallyEq.mp hx] with y hy
+        simp only [ChartCutoff.product_apply]; rw [hy]; simp
+      rw [he.fderiv_eq]
+      simpa using hB
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/DeterminantJet.lean b/lean/OAI/Geometry/PrescribedRicci/DeterminantJet.lean
new file mode 100644
index 0000000000000000000000000000000000000000..f0f85cff653462cf360f764cdf2ac2be2233d4c2
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/DeterminantJet.lean
@@ -0,0 +1,131 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.JetRemainder
+import OAI.Geometry.PrescribedRicci.MatrixDifferentialNew
+import OAI.Geometry.PrescribedPotential.NonlinearHessianCommutator
+
+namespace OAI
+
+/-! Determinant Jet. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Finset Matrix
+open scoped ContDiff Classical BigOperators Matrix.Norms.Elementwise
+namespace TameInterpolation
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+variable {ι : Type*} {n : ℕ}
+
+omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] in
+lemma cword_finite_sum (e : ι → E) (ws : List ι) {ν : Type*} (s : Finset ν)
+    (f : ν → E → ℂ) (hf : ∀ i ∈ s, ContDiff ℝ ∞ (f i)) :
+    cword e (fun x => ∑ i ∈ s, f i x) ws = fun x => ∑ i ∈ s, cword e (f i) ws x := by
+  induction ws with
+  | nil => rfl
+  | cons a ws ih =>
+    change cdir (e a) (cword e (fun x => ∑ i ∈ s, f i x) ws) = _
+    rw [ih,cdir_finite_sum e a s _ (fun i hi => cword_smooth e (hf i hi) ws)]
+    rfl
+
+omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] in
+lemma cword_const_mul (e : ι → E) (ws : List ι) (c : ℂ) (f : E → ℂ)
+    (hf : ContDiff ℝ ∞ f) :
+    cword e (fun x => c*f x) ws = fun x => c*cword e f ws x := by
+  induction ws with
+  | nil => rfl
+  | cons a ws ih =>
+    change cdir (e a) (cword e (fun x => c*f x) ws) = _
+    rw [ih]
+    funext x
+    unfold cdir
+    rw [fderiv_const_mul ((cword_smooth e hf ws).differentiable (by simp) x)]
+    rfl
+
+omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] in
+lemma monoEval_single (e : ι → E) (f : Fin n → E → ℂ) (ws : List ι) (i : Fin n) (x : E) :
+    monoEval e f (singleState ws i) x = cword e (f i) ws x * ∏ j ∈ Finset.univ.erase i, f j x := by
+  unfold monoEval
+  rw [← Finset.mul_prod_erase Finset.univ _ (Finset.mem_univ i)]
+  simp only [singleState,Function.update_self]
+  congr 1
+  apply Finset.prod_congr rfl
+  intro j hj
+  rw [Function.update_of_ne (Finset.mem_erase.mp hj).1]
+  rfl
+
+def detPrincipal (M N : Matrix (Fin n) (Fin n) ℂ) : ℂ :=
+  ∑ σ : Equiv.Perm (Fin n), (Equiv.Perm.sign σ : ℂ) *
+    ∑ i, N (σ i) i * ∏ j ∈ Finset.univ.erase i, M (σ j) j
+
+lemma detPrincipal_eq (M N : Matrix (Fin n) (Fin n) ℂ) :
+    detPrincipal M N = NonlinearHessian.detDifferential M N := by
+  have hp : HasDerivAt (fun t : ℂ => (M+t•N).det) (detPrincipal M N) 0 := by
+    have hent (i j : Fin n) : HasDerivAt (fun t : ℂ => M i j+t*N i j) (N i j) 0 := by
+      simpa using ((hasDerivAt_id (0:ℂ)).mul_const (N i j)).const_add (M i j)
+    have hprod (σ : Equiv.Perm (Fin n)) :=
+      HasDerivAt.finsetProd (u:=Finset.univ) (fun i _ => hent (σ i) i)
+    have hs := HasDerivAt.fun_sum (u:=Finset.univ)
+      (fun σ _ => hprod σ |>.const_mul (Equiv.Perm.sign σ : ℂ))
+    convert! hs using 1
+    · funext t
+      simp only [Matrix.det_apply',Matrix.add_apply,Matrix.smul_apply,smul_eq_mul,
+        Finset.prod_apply]
+    · unfold detPrincipal
+      simp only [zero_mul,add_zero,smul_eq_mul,mul_comm]
+  have hc : HasDerivAt (fun t : ℂ => M+t•N) N 0 := by
+    convert! ((hasDerivAt_id (0:ℂ)).smul_const N).const_add M using 1
+    simp only [one_smul]
+  have hd := (MongeAmpere.differentiable_det (M+(0:ℂ)•N)).hasFDerivAt.comp_hasDerivAt 0 hc
+  have hd' : HasDerivAt (fun t : ℂ => (M+t•N).det)
+      (fderiv ℂ (fun K : Matrix (Fin n) (Fin n) ℂ => K.det) M N) 0 := by
+    convert! hd using 1
+    simp only [zero_smul,add_zero]
+    rfl
+  have hr := ((MongeAmpere.differentiable_det M).hasFDerivAt.restrictScalars ℝ).fderiv
+  have hre := congrArg (fun L : Matrix (Fin n) (Fin n) ℂ →L[ℝ] ℂ => L N) hr
+  exact (hp.unique hd').trans hre.symm
+
+def detJetRemainder (e : ι → E) (H : E → Matrix (Fin n) (Fin n) ℂ) (ws : List ι) (x : E) : ℂ :=
+  ∑ σ : Equiv.Perm (Fin n), (Equiv.Perm.sign σ : ℂ) *
+    ((monoRemainder ws).map (fun s => monoEval e (fun i x => H x (σ i) i) s x)).sum
+
+omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] in
+theorem cword_det_remainder (e : ι → E) (H : E → Matrix (Fin n) (Fin n) ℂ)
+    (hH : ∀ i j, ContDiff ℝ ∞ (fun x => H x i j)) (ws : List ι) (hw : ws ≠ []) (x : E) :
+    cword e (fun x => (H x).det) ws x =
+      NonlinearHessian.detDifferential (H x) (fun i j => cword e (fun y => H y i j) ws x) +
+        detJetRemainder e H ws x := by
+  have he : (fun x => (H x).det) = (fun x => ∑ σ : Equiv.Perm (Fin n),
+      (Equiv.Perm.sign σ : ℂ)*∏ i, H x (σ i) i) := by
+    funext x; exact Matrix.det_apply' (H x)
+  rw [he,cword_finite_sum e ws _ _ (by
+    intro σ _; exact contDiff_const.mul (contDiff_prod fun i _ => hH (σ i) i))]
+  simp_rw [cword_const_mul e ws _ _ (contDiff_prod fun i _ => hH _ i)]
+  simp_rw [cword_product_remainder e _ (fun i => hH _ i) ws hw]
+  erw [← detPrincipal_eq]
+  unfold detPrincipal detJetRemainder
+  rw [← Finset.sum_add_distrib]
+  apply Finset.sum_congr rfl
+  intro σ _
+  simp_rw [monoEval_single]
+  ring
+
+omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] in
+theorem cword_det_trace (e : ι → E) (H : E → Matrix (Fin n) (Fin n) ℂ)
+    (hH : ∀ i j, ContDiff ℝ ∞ (fun x => H x i j)) (ws : List ι) (hw : ws ≠ []) (x : E)
+    (hinv : IsUnit (H x).det) :
+    cword e (fun x => (H x).det) ws x =
+      (H x).det * ((H x)⁻¹ * (show Matrix (Fin n) (Fin n) ℂ from fun i j => cword e (fun y => H y i j) ws x)).trace +
+        detJetRemainder e H ws x := by
+  rw [cword_det_remainder e H hH ws hw x]
+  congr 1
+  exact MongeAmpere.real_fderiv_det_apply _ _ hinv
+end TameInterpolation
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/DimensionOnePotential.lean b/lean/OAI/Geometry/PrescribedRicci/DimensionOnePotential.lean
new file mode 100644
index 0000000000000000000000000000000000000000..78d7cb7cfbaf1d47aeca3a4c5522271f6afa66bc
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/DimensionOnePotential.lean
@@ -0,0 +1,111 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.NormalizedPoisson
+import OAI.Geometry.PrescribedPotential.VolumePath
+
+namespace OAI
+
+/-! Dimension One Potential. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff Classical ComplexOrder
+namespace MongeAmpere
+
+lemma singleton_det_add (H K : Matrix (Fin 1) (Fin 1) ℂ) (hH : H.PosDef) :
+    (H + K).det.re = H.det.re * (1 + (H⁻¹ * K).trace.re) := by
+  have hr : ((H 0 0).re : ℂ) = H 0 0 :=
+    Complex.conj_eq_iff_re.mp (hH.isHermitian.apply 0 0)
+  have hn : (H 0 0).re ≠ 0 := ne_of_gt (Complex.pos_iff.mp hH.diag_pos).1
+  simp only [Matrix.det_fin_one, Matrix.add_apply, Complex.add_re,
+    Matrix.trace_fin_one, Matrix.mul_apply, Fin.sum_univ_one,
+    Matrix.inv_def, Matrix.adjugate_fin_one, Matrix.det_fin_one]
+  rw [← hr]
+  simp [Complex.mul_re, Complex.inv_re, Complex.inv_im, Complex.normSq, Ring.inverse_eq_inv]
+  field_simp
+
+lemma singleton_posDef_of_det (H : Matrix (Fin 1) (Fin 1) ℂ)
+    (hH : H.IsHermitian) (hp : 0 < H.det.re) : H.PosDef := by
+  have hd : H = Matrix.diagonal (fun _ : Fin 1 => H 0 0) := by
+    ext i j
+    fin_cases i
+    fin_cases j
+    simp
+  rw [hd, Matrix.posDef_diagonal_iff]
+  intro i
+  apply Complex.pos_iff.mpr
+  refine ⟨?_, ?_⟩
+  · simpa only [Matrix.det_fin_one] using hp
+  · exact (Complex.conj_eq_iff_im.mp (hH.apply 0 0)).symm
+end MongeAmpere
+
+namespace Anticanonical.SourceSmooth
+variable {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  [ConnectedSpace X] {A : ComplexAtlas 1 X}
+
+ 
+
+theorem dimensionOne_positiveDensity (g : KaehlerMetric A) (q : SmoothRealFunction A)
+    (hq : ∀ x, 0 < q.value x) :
+    ∃ (φ : SmoothRealFunction A) (c : ℝ), 0 < c ∧ g.PositivePotential φ ∧
+      ∀ i z, z ∈ (A.chart i).target →
+        (g.matrix i z + φ.hessian i z).det.re =
+          c * q.localExpression i z * g.volumeCoefficient i z := by
+  let x₀ : X := Classical.arbitrary X
+  obtain ⟨u,a,hn,he⟩ := normalizedPoisson 1 X A g x₀ q
+  have ha : 0 < a := by
+    obtain ⟨x,_,hx⟩ := isCompact_univ.exists_isMaxOn Set.univ_nonempty u.continuous.continuousOn
+    obtain ⟨i,hi⟩ := A.covers x
+    have hm := g.linearized_nonpos_at_globalMax u (fun y => hx (mem_univ y)) i hi
+    have heq := he i (A.chart i x) ((A.chart i).mapsTo hi)
+    have hqe : q.localExpression i (A.chart i x) = q.value x := by
+      simp only [SmoothRealFunction.localExpression, Function.comp_apply, (A.chart i).left_inv hi]
+    rw [hqe] at heq
+    linarith [hq x]
+  let φ := u.realSMul a⁻¹
+  have heq (i : Fin A.count) (z : Coordinates 1) (hz : z ∈ (A.chart i).target) :
+      (g.matrix i z + φ.hessian i z).det.re =
+        a⁻¹ * q.localExpression i z * g.volumeCoefficient i z := by
+    rw [MongeAmpere.singleton_det_add _ _ (g.positive i z hz)]
+    change g.volumeCoefficient i z * (1 + g.linearizedMongeAmpere φ i z) = _
+    rw [show φ = u.realSMul a⁻¹ from rfl, g.linearizedMongeAmpere_realSMul]
+    have hh := he i z hz
+    have hae : a ≠ 0 := ne_of_gt ha
+    field_simp
+    rw [add_comm a _,hh]
+  refine ⟨φ,a⁻¹,inv_pos.mpr ha,?_,heq⟩
+  intro i z hz
+  apply MongeAmpere.singleton_posDef_of_det
+  · exact (g.positive i z hz).isHermitian.add (φ.hessian_hermitian i hz)
+  · rw [heq i z hz]
+    exact mul_pos (mul_pos (inv_pos.mpr ha) (hq ((A.chart i).symm z)))
+      (g.volumeCoefficient_pos i hz)
+
+ 
+
+theorem prescribedPotential_dimensionOne (X : Type) [TopologicalSpace X] [T2Space X]
+    [CompactSpace X] [ConnectedSpace X] (A : ComplexAtlas 1 X)
+    (P : A.ProjectiveEmbedding) (h : SemipositiveAnticanonicalMetric A) :
+    HasPrescribedVolumePotential P h := by
+  let g := projectiveBackground P
+  let F := prescribedForcing g h
+  let q : SmoothRealFunction A := {
+    value := fun x => Real.exp (F.value x)
+    smooth := fun i => Real.contDiff_exp.contDiffOn.comp (F.smooth i) (fun _ _ => mem_univ _) }
+  obtain ⟨φ,c,hc,hp,he⟩ := dimensionOne_positiveDensity g q (fun x => Real.exp_pos _)
+  refine ⟨φ,c,hc,hp,fun i z hz => ?_⟩
+  rw [he i z hz]
+  change c * Real.exp (F.localExpression i z) * g.volumeCoefficient i z = _
+  rw [prescribedForcing_local g h i hz, Real.exp_sub,
+    Real.exp_log (g.volumeCoefficient_pos i hz)]
+  field_simp [ne_of_gt (g.volumeCoefficient_pos i hz)]
+
+end Anticanonical.SourceSmooth
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/DimensionZeroPotential.lean b/lean/OAI/Geometry/PrescribedRicci/DimensionZeroPotential.lean
new file mode 100644
index 0000000000000000000000000000000000000000..af4828c8c15ada59c7a78ec8e3d3e9372892cb9a
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/DimensionZeroPotential.lean
@@ -0,0 +1,48 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.NormalizedPoisson
+import OAI.Geometry.PrescribedPotential.VolumePath
+
+namespace OAI
+
+/-! Dimension Zero Potential. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff Classical ComplexOrder
+namespace Anticanonical.SourceSmooth
+variable {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  [ConnectedSpace X] {A : ComplexAtlas 0 X}
+
+lemma dimensionZero_forcing_constant (g : KaehlerMetric A)
+    (line : SemipositiveAnticanonicalMetric A) :
+    ∃ c : ℝ, ∀ x, (prescribedForcing g line).value x = c := by
+  obtain ⟨u,c,_,he⟩ := normalizedPoisson 0 X A g (Classical.arbitrary X) (prescribedForcing g line)
+  refine ⟨c,fun x => ?_⟩
+  obtain ⟨i,hi⟩ := A.covers x
+  have h := he i (A.chart i x) ((A.chart i).mapsTo hi)
+  have hz : g.linearizedMongeAmpere u i (A.chart i x) = 0 := by
+    simp [KaehlerMetric.linearizedMongeAmpere,Matrix.trace]
+  rw [hz,zero_add] at h
+  have hx : (prescribedForcing g line).localExpression i (A.chart i x) =
+      (prescribedForcing g line).value x := by
+    simp only [SmoothRealFunction.localExpression,Function.comp_apply,(A.chart i).left_inv hi]
+  exact (h.trans hx).symm
+
+lemma volumePath_zeroDim_one (g : KaehlerMetric A) (line : SemipositiveAnticanonicalMetric A) :
+    ∃ (φ : SmoothRealFunction A) (b : ℝ), SolvesVolumePath g line 1 φ b := by
+  obtain ⟨c,hc⟩ := dimensionZero_forcing_constant g line
+  obtain ⟨hp,hzero⟩ := volumePath_zero g line
+  refine ⟨SmoothRealFunction.constant 0,-c,hp,fun x => ?_⟩
+  rw [hzero x,hc x]
+  ring
+
+end Anticanonical.SourceSmooth
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/DirectionalInterpolation.lean b/lean/OAI/Geometry/PrescribedRicci/DirectionalInterpolation.lean
new file mode 100644
index 0000000000000000000000000000000000000000..baaf59fd4a35b12e0f2cc7a728a971f42a53b43e
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/DirectionalInterpolation.lean
@@ -0,0 +1,83 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.LpPowerBounds
+import OAI.Geometry.PrescribedRicci.PowerIntegration
+
+namespace OAI
+
+/-! Directional Interpolation. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+open scoped ContDiff ENNReal
+namespace TameInterpolation
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+
+lemma directional_lp_interpolation {p : ℝ} (hp : 2 < p)
+    (α β r s : ℝ≥0∞) [ENNReal.HolderTriple α β s] [ENNReal.HolderTriple s r 1]
+    (hr : r * ENNReal.ofReal (p-2) = ENNReal.ofReal p)
+    (v : E) {f : E → ℝ} (hf : ContDiff ℝ ∞ f) (hc : HasCompactSupport f) :
+    (lpNorm (dir v f) (ENNReal.ofReal p) volume)^2 ≤
+      (p-1)*lpNorm f α volume*lpNorm (dir v (dir v f)) β volume := by
+  let a := dir v f
+  let b := dir v a
+  have ha : ContDiff ℝ ∞ a := dir_smooth v hf
+  have hb : ContDiff ℝ ∞ b := dir_smooth v ha
+  have hac : HasCompactSupport a := dir_compact v hc
+  have hbc : HasCompactSupport b := dir_compact v hac
+  have hfa : MemLp f α := hf.continuous.memLp_of_hasCompactSupport hc
+  have hbb : MemLp b β := hb.continuous.memLp_of_hasCompactSupport hbc
+  have hfb : MemLp (fun x => f x*b x) s := hfa.fun_mul hbb
+  have hpc : Continuous (fun x => |a x|^(p-2)) :=
+    ha.continuous.abs.rpow_const (fun _ => Or.inr (by linarith))
+  have hps : HasCompactSupport (fun x => |a x|^(p-2)) :=
+    hac.comp_left (g:=fun x : ℝ => |x|^(p-2)) (by simp [Real.zero_rpow (by linarith : p-2 ≠ 0)])
+  have hpr : MemLp (fun x => |a x|^(p-2)) r := hpc.memLp_of_hasCompactSupport hps
+  have hpw : lpNorm (fun x => |a x|^(p-2)) r volume =
+      (lpNorm a (ENNReal.ofReal p) volume)^(p-2) := by
+    simpa only [Real.norm_eq_abs,hr] using lpNorm_norm_rpow a ha.continuous.aestronglyMeasurable r (by linarith : 0 < p-2)
+  have hI : |∫ x, f x*|a x|^(p-2)*b x| ≤
+      (lpNorm f α volume*lpNorm b β volume)*(lpNorm a (ENNReal.ofReal p) volume)^(p-2) := by
+    have he : (fun x => f x*|a x|^(p-2)*b x) = (fun x => (f x*b x)*|a x|^(p-2)) := by
+      funext x; ring
+    rw [he]
+    calc
+      _ ≤ lpNorm (fun x => (f x*b x)*|a x|^(p-2)) 1 volume :=
+        integral_norm_le_lpNorm (hfb.aestronglyMeasurable.mul hpr.aestronglyMeasurable)
+      _ ≤ lpNorm (fun x => f x*b x) s volume * lpNorm (fun x => |a x|^(p-2)) r volume := lpNorm_mul_le hfb hpr
+      _ ≤ (lpNorm f α volume*lpNorm b β volume)*lpNorm (fun x => |a x|^(p-2)) r volume :=
+        mul_le_mul_of_nonneg_right (lpNorm_mul_le hfa hbb) lpNorm_nonneg
+      _ = _ := by rw [hpw]
+  let Y := lpNorm a (ENNReal.ofReal p) volume
+  have hY : 0 ≤ Y := lpNorm_nonneg
+  have he := power_dir_integration hp v hf hc
+  have hpow : Y^p ≤ ((p-1)*lpNorm f α volume*lpNorm b β volume)*Y^(p-2) := by
+    calc
+      Y^p = ∫ x, |a x|^p := by simpa [Y,Real.norm_eq_abs] using lpNorm_rpow_integral a ha.continuous.aestronglyMeasurable (by linarith : 0 < p)
+      _ = -(p-1) * ∫ x, f x*|a x|^(p-2)*b x := he
+      _ ≤ (p-1)*|∫ x, f x*|a x|^(p-2)*b x| := by
+        nlinarith [neg_le_abs (∫ x, f x*|a x|^(p-2)*b x)]
+      _ ≤ (p-1)*((lpNorm f α volume*lpNorm b β volume)*Y^(p-2)) :=
+        mul_le_mul_of_nonneg_left hI (by linarith)
+      _ = _ := by ring
+  by_cases hz : Y = 0
+  · change Y^2 ≤ _
+    rw [hz,zero_pow (by decide : 2 ≠ 0)]
+    exact mul_nonneg (mul_nonneg (by linarith) lpNorm_nonneg) lpNorm_nonneg
+  · have hYp : 0 < Y := lt_of_le_of_ne hY (Ne.symm hz)
+    have hyr : Y^p = Y^2*Y^(p-2) := by
+      rw [← Real.rpow_natCast Y 2,← Real.rpow_add hYp]
+      congr 1
+      ring
+    rw [hyr] at hpow
+    exact (mul_le_mul_iff_left₀ (Real.rpow_pos_of_pos hYp (p-2))).mp hpow
+end TameInterpolation
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/DiscreteInterpolation.lean b/lean/OAI/Geometry/PrescribedRicci/DiscreteInterpolation.lean
new file mode 100644
index 0000000000000000000000000000000000000000..af2df4d170eec1b10107da9a35a6fe14c8b3916f
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/DiscreteInterpolation.lean
@@ -0,0 +1,123 @@
+import Mathlib
+
+namespace OAI
+
+/-! Discrete Interpolation. -/
+
+noncomputable section
+open Set Filter Topology
+namespace TameInterpolation
+
+lemma discrete_maximum (m : ℕ) (z : ℕ → ℝ) (h0 : z 0 ≤ 0) (hm : z m ≤ 0)
+    (hr : ∀ j, j+2 ≤ m → 2*z (j+1) ≤ z j+z (j+2)) :
+    ∀ j, j ≤ m → z j ≤ 0 := by
+  let S := Finset.range (m+1)
+  have hS : S.Nonempty := ⟨0,by simp [S]⟩
+  let M := S.sup' hS z
+  have hupper (j : ℕ) (hj : j ≤ m) : z j ≤ M :=
+    Finset.le_sup' z (by simpa [S] using Nat.lt_succ_of_le hj)
+  have hm0 : M ≤ 0 := by
+    by_contra! hM
+    obtain ⟨i,hi,he⟩ := Finset.exists_mem_eq_sup' hS z
+    have hex : ∃ i, i ≤ m ∧ z i = M := ⟨i,by simpa [S] using hi,he.symm⟩
+    let q := Nat.find hex
+    have hq := Nat.find_spec hex
+    change q ≤ m ∧ z q = M at hq
+    have hq0 : 0 < q := by
+      by_contra! hn
+      have hzero : q = 0 := by omega
+      rw [hzero] at hq
+      linarith only [h0,hq.2,hM]
+    have hqm : q < m := by
+      by_contra! hn
+      have heq : q = m := by omega
+      rw [heq] at hq
+      linarith only [hm,hq.2,hM]
+    have hrec := hr (q-1) (by omega)
+    have hleft := hupper (q-1) (by omega)
+    have hright := hupper (q+1) (by omega)
+    have he1 : q-1+1 = q := by omega
+    have he2 : q-1+2 = q+1 := by omega
+    rw [he1,he2,hq.2] at hrec
+    have hprev : z (q-1) = M := by linarith only [hrec,hleft,hright]
+    have hmin := Nat.find_min' hex (show q-1 ≤ m ∧ z (q-1) = M from ⟨by omega,hprev⟩)
+    change q ≤ q-1 at hmin
+    omega
+  exact fun j hj => (hupper j hj).trans hm0
+
+lemma positive_sequence_product (m j : ℕ) (hm : 0 < m) (hj : j ≤ m)
+    (a : ℕ → ℝ) (ha : ∀ i, 0 < a i) (C : ℝ) (hC : 0 < C)
+    (hr : ∀ i, i+2 ≤ m → (a (i+1))^2 ≤ C*a i*a (i+2)) :
+    a j*a (m-j) ≤ C^(j*(m-j))*a 0*a m := by
+  have hm' : (m:ℝ) ≠ 0 := by exact_mod_cast (Nat.ne_of_gt hm)
+  let z : ℕ → ℝ := fun i => Real.log (a i) - ((i:ℝ)/(m:ℝ))*Real.log (a m) -
+    (1-(i:ℝ)/(m:ℝ))*Real.log (a 0) - ((i:ℝ)*((m:ℝ)-(i:ℝ))/2)*Real.log C
+  have hz0 : z 0 ≤ 0 := by simp [z]
+  have hzm : z m ≤ 0 := by simp [z,hm']
+  have hzr (i : ℕ) (hi : i+2 ≤ m) : 2*z (i+1) ≤ z i+z (i+2) := by
+    have hlog := Real.log_le_log (sq_pos_of_pos (ha (i+1))) (hr i hi)
+    rw [Real.log_pow,Real.log_mul (mul_ne_zero hC.ne' (ha i).ne') (ha (i+2)).ne',
+      Real.log_mul hC.ne' (ha i).ne'] at hlog
+    have he : z i+z (i+2)-2*z (i+1) =
+        Real.log (a i)+Real.log (a (i+2))-2*Real.log (a (i+1))+Real.log C := by
+      dsimp [z]
+      push_cast
+      ring
+    norm_num only [Nat.cast_ofNat] at hlog
+    linarith only [hlog,he]
+  have h1 := discrete_maximum m z hz0 hzm hzr j hj
+  have h2 := discrete_maximum m z hz0 hzm hzr (m-j) (by omega)
+  have he : z j+z (m-j) = Real.log (a j)+Real.log (a (m-j))-
+      Real.log (a 0)-Real.log (a m)-((j*(m-j):ℕ):ℝ)*Real.log C := by
+    dsimp [z]
+    rw [Nat.cast_sub hj,Nat.cast_mul,Nat.cast_sub hj]
+    field_simp
+    ring
+  have hlog : Real.log (a j*a (m-j)) ≤ Real.log (C^(j*(m-j))*a 0*a m) := by
+    rw [Real.log_mul (ha j).ne' (ha (m-j)).ne',
+      Real.log_mul (mul_ne_zero (pow_ne_zero _ hC.ne') (ha 0).ne') (ha m).ne',
+      Real.log_mul (pow_ne_zero _ hC.ne') (ha 0).ne',Real.log_pow]
+    linarith only [h1,h2,he]
+  have hh := Real.exp_le_exp.mpr hlog
+  rw [Real.exp_log (mul_pos (ha j) (ha (m-j))),
+    Real.exp_log (mul_pos (mul_pos (pow_pos hC _) (ha 0)) (ha m))] at hh
+  exact hh
+
+lemma clamp_recurrence (m : ℕ) (a : ℕ → ℝ) (ha : ∀ i, 0 ≤ a i)
+    (C : ℝ) (hC : 1 ≤ C)
+    (hr : ∀ i, i+2 ≤ m → (a (i+1))^2 ≤ C*a i*a (i+2))
+    (ε : ℝ) (hε : 0 < ε) :
+    ∀ i, i+2 ≤ m → (max (a (i+1)) ε)^2 ≤ C*max (a i) ε*max (a (i+2)) ε := by
+  intro i hi
+  by_cases hmid : a (i+1) ≤ ε
+  · rw [max_eq_right hmid]
+    have hprod : ε*ε ≤ max (a i) ε*max (a (i+2)) ε :=
+      mul_le_mul (le_max_right _ _) (le_max_right _ _) hε.le (hε.le.trans (le_max_right _ _))
+    nlinarith [mul_nonneg hε.le hε.le]
+  · rw [max_eq_left (le_of_not_ge hmid)]
+    exact (hr i hi).trans (by gcongr <;> first | exact le_max_left _ _ | exact ha _)
+
+theorem nonnegative_sequence_product (m j : ℕ) (hm : 0 < m) (hj : j ≤ m)
+    (a : ℕ → ℝ) (ha : ∀ i, 0 ≤ a i) (C : ℝ) (hC : 1 ≤ C)
+    (hr : ∀ i, i+2 ≤ m → (a (i+1))^2 ≤ C*a i*a (i+2)) :
+    a j*a (m-j) ≤ C^(j*(m-j))*a 0*a m := by
+  have hbound (ε : ℝ) (hε : 0 < ε) :
+      a j*a (m-j) ≤ C^(j*(m-j))*max (a 0) ε*max (a m) ε := by
+    apply (mul_le_mul (le_max_left _ _) (le_max_left _ _) (ha _) (le_trans (ha _) (le_max_left _ _))).trans
+    exact positive_sequence_product m j hm hj (fun i => max (a i) ε)
+      (fun i => hε.trans_le (le_max_right _ _)) C (by linarith only [hC])
+      (clamp_recurrence m a ha C hC hr ε hε)
+  have hcont : Continuous (fun ε : ℝ => C^(j*(m-j))*max (a 0) ε*max (a m) ε) :=
+    (continuous_const.mul (continuous_const.max continuous_id)).mul (continuous_const.max continuous_id)
+  have hlim : Tendsto (fun ε : ℝ => C^(j*(m-j))*max (a 0) ε*max (a m) ε)
+      (𝓝[>] 0) (𝓝 (C^(j*(m-j))*a 0*a m)) := by
+    simpa only [max_eq_left (ha 0),max_eq_left (ha m)] using
+      (hcont.tendsto 0).mono_left nhdsWithin_le_nhds
+  apply ge_of_tendsto hlim
+  filter_upwards [self_mem_nhdsWithin] with ε hε
+  exact hbound ε hε
+end TameInterpolation
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/DiscreteManyProduct.lean b/lean/OAI/Geometry/PrescribedRicci/DiscreteManyProduct.lean
new file mode 100644
index 0000000000000000000000000000000000000000..5bf8fc72691d45965be6a208dce75fb3b8387463
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/DiscreteManyProduct.lean
@@ -0,0 +1,111 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.DiscreteInterpolation
+
+namespace OAI
+
+/-! Discrete Many Product. -/
+
+noncomputable section
+open Set Filter Topology
+open scoped BigOperators
+namespace TameInterpolation
+
+lemma positive_sequence_log_bound (m : ℕ) (hm : 0 < m)
+    (a : ℕ → ℝ) (ha : ∀ i, 0 < a i) (C : ℝ) (hC : 0 < C)
+    (hr : ∀ i, i+2 ≤ m → (a (i+1))^2 ≤ C*a i*a (i+2))
+    (j : ℕ) (hj : j ≤ m) :
+    Real.log (a j) ≤ ((j:ℝ)/(m:ℝ))*Real.log (a m) +
+      (1-(j:ℝ)/(m:ℝ))*Real.log (a 0) + ((j:ℝ)*((m:ℝ)-(j:ℝ))/2)*Real.log C := by
+  have hm' : (m:ℝ) ≠ 0 := by exact_mod_cast (Nat.ne_of_gt hm)
+  let z : ℕ → ℝ := fun i => Real.log (a i) - ((i:ℝ)/(m:ℝ))*Real.log (a m) -
+    (1-(i:ℝ)/(m:ℝ))*Real.log (a 0) - ((i:ℝ)*((m:ℝ)-(i:ℝ))/2)*Real.log C
+  have hz0 : z 0 ≤ 0 := by simp [z]
+  have hzm : z m ≤ 0 := by simp [z,hm']
+  have hzr (i : ℕ) (hi : i+2 ≤ m) : 2*z (i+1) ≤ z i+z (i+2) := by
+    have hlog := Real.log_le_log (sq_pos_of_pos (ha (i+1))) (hr i hi)
+    rw [Real.log_pow,Real.log_mul (mul_ne_zero hC.ne' (ha i).ne') (ha (i+2)).ne',
+      Real.log_mul hC.ne' (ha i).ne'] at hlog
+    have he : z i+z (i+2)-2*z (i+1) =
+        Real.log (a i)+Real.log (a (i+2))-2*Real.log (a (i+1))+Real.log C := by
+      dsimp [z]
+      push_cast
+      ring
+    norm_num only [Nat.cast_ofNat] at hlog
+    linarith only [hlog,he]
+  have h := discrete_maximum m z hz0 hzm hzr j hj
+  dsimp [z] at h
+  linarith only [h]
+
+lemma positive_sequence_many (m : ℕ) (hm : 0 < m)
+    (a : ℕ → ℝ) (ha : ∀ i, 0 < a i) (C : ℝ) (hC : 1 ≤ C)
+    (hr : ∀ i, i+2 ≤ m → (a (i+1))^2 ≤ C*a i*a (i+2))
+    {ι : Type*} (s : Finset ι) (j : ι → ℕ) (hs : ∑ i ∈ s, j i = m) :
+    ∏ i ∈ s, a (j i) ≤ C^(∑ i ∈ s, j i*(m-j i))*a 0^(s.card-1)*a m := by
+  have hC0 : 0 < C := by linarith only [hC]
+  have hm' : (m:ℝ) ≠ 0 := by exact_mod_cast (Nat.ne_of_gt hm)
+  have hsn : s.Nonempty := by
+    by_contra h
+    rw [Finset.not_nonempty_iff_eq_empty.mp h,Finset.sum_empty] at hs
+    omega
+  have hj (i : ι) (hi : i ∈ s) : j i ≤ m := by
+    rw [← hs]
+    exact Finset.single_le_sum (fun _ _ => Nat.zero_le _) hi
+  have hc : (s.card:ℝ)-1 = ((s.card-1:ℕ):ℝ) := by
+    rw [Nat.cast_sub (by simpa using hsn.card_pos),Nat.cast_one]
+  have hs' : ∑ i ∈ s, (j i:ℝ)/(m:ℝ) = 1 := by
+    rw [← Finset.sum_div,← Nat.cast_sum,hs,div_self hm']
+  have he1 : ∑ i ∈ s, (j i:ℝ)/(m:ℝ)*Real.log (a m) = Real.log (a m) := by
+    rw [← Finset.sum_mul,hs',one_mul]
+  have he2 : ∑ i ∈ s, (1-(j i:ℝ)/(m:ℝ))*Real.log (a 0) = ((s.card-1:ℕ):ℝ)*Real.log (a 0) := by
+    rw [← Finset.sum_mul,Finset.sum_sub_distrib,hs']
+    simp only [Finset.sum_const,nsmul_eq_mul,mul_one,hc]
+  have he3 : ∑ i ∈ s, ((j i:ℝ)*((m:ℝ)-(j i:ℝ))/2)*Real.log C ≤
+      ((∑ i ∈ s, j i*(m-j i):ℕ):ℝ)*Real.log C := by
+    rw [Nat.cast_sum,Finset.sum_mul]
+    apply Finset.sum_le_sum
+    intro i hi
+    rw [Nat.cast_mul,Nat.cast_sub (hj i hi)]
+    have hnon : 0 ≤ (j i:ℝ)*((m:ℝ)-(j i:ℝ)) := by
+      exact mul_nonneg (by positivity) (sub_nonneg.mpr (by exact_mod_cast hj i hi))
+    have hlogC : 0 ≤ Real.log C := Real.log_nonneg hC
+    nlinarith only [hnon,hlogC,mul_nonneg hnon hlogC]
+  have hlog : ∑ i ∈ s, Real.log (a (j i)) ≤
+      ((∑ i ∈ s, j i*(m-j i):ℕ):ℝ)*Real.log C + ((s.card-1:ℕ):ℝ)*Real.log (a 0)+Real.log (a m) := by
+    have hh := Finset.sum_le_sum (fun i hi => positive_sequence_log_bound m hm a ha C hC0 hr (j i) (hj i hi))
+    rw [Finset.sum_add_distrib,Finset.sum_add_distrib,he1,he2] at hh
+    linarith only [hh,he3]
+  have hl : Real.log (∏ i ∈ s, a (j i)) = ∑ i ∈ s, Real.log (a (j i)) :=
+    Real.log_prod (fun _ _ => (ha _).ne')
+  have hp : 0 < ∏ i ∈ s, a (j i) := Finset.prod_pos (fun _ _ => ha _)
+  have hrp : 0 < C^(∑ i ∈ s, j i*(m-j i))*a 0^(s.card-1)*a m := mul_pos (mul_pos (pow_pos hC0 _) (pow_pos (ha 0) _)) (ha m)
+  have hrlog : Real.log (C^(∑ i ∈ s, j i*(m-j i))*a 0^(s.card-1)*a m) =
+      ((∑ i ∈ s, j i*(m-j i):ℕ):ℝ)*Real.log C + ((s.card-1:ℕ):ℝ)*Real.log (a 0)+Real.log (a m) := by
+    rw [Real.log_mul (mul_ne_zero (pow_ne_zero _ hC0.ne') (pow_ne_zero _ (ha 0).ne')) (ha m).ne',
+      Real.log_mul (pow_ne_zero _ hC0.ne') (pow_ne_zero _ (ha 0).ne'),Real.log_pow,Real.log_pow]
+  rw [← hl,← hrlog] at hlog
+  exact (Real.log_le_log_iff hp hrp).mp hlog
+
+lemma nonnegative_sequence_many (m : ℕ) (hm : 0 < m)
+    (a : ℕ → ℝ) (ha : ∀ i, 0 ≤ a i) (C : ℝ) (hC : 1 ≤ C)
+    (hr : ∀ i, i+2 ≤ m → (a (i+1))^2 ≤ C*a i*a (i+2))
+    {ι : Type*} (s : Finset ι) (j : ι → ℕ) (hs : ∑ i ∈ s, j i = m) :
+    ∏ i ∈ s, a (j i) ≤ C^(∑ i ∈ s, j i*(m-j i))*a 0^(s.card-1)*a m := by
+  have hbound (ε : ℝ) (hε : 0 < ε) :
+      ∏ i ∈ s, a (j i) ≤ C^(∑ i ∈ s, j i*(m-j i))*(max (a 0) ε)^(s.card-1)*max (a m) ε := by
+    apply (Finset.prod_le_prod₀ (fun i _ => ha (j i)) (fun i _ => le_max_left (a (j i)) ε)).trans
+    exact positive_sequence_many m hm (fun i => max (a i) ε)
+      (fun i => hε.trans_le (le_max_right _ _)) C hC (clamp_recurrence m a ha C hC hr ε hε) s j hs
+  have hcont : Continuous (fun ε : ℝ => C^(∑ i ∈ s, j i*(m-j i))*(max (a 0) ε)^(s.card-1)*max (a m) ε) := by
+    fun_prop
+  have hlim : Tendsto (fun ε : ℝ => C^(∑ i ∈ s, j i*(m-j i))*(max (a 0) ε)^(s.card-1)*max (a m) ε)
+      (𝓝[>] 0) (𝓝 (C^(∑ i ∈ s, j i*(m-j i))*a 0^(s.card-1)*a m)) := by
+    simpa only [max_eq_left (ha 0),max_eq_left (ha m)] using
+      (hcont.tendsto 0).mono_left nhdsWithin_le_nhds
+  apply ge_of_tendsto hlim
+  filter_upwards [self_mem_nhdsWithin] with ε hε
+  exact hbound ε hε
+end TameInterpolation
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/FamilyJetNorm.lean b/lean/OAI/Geometry/PrescribedRicci/FamilyJetNorm.lean
new file mode 100644
index 0000000000000000000000000000000000000000..41e80e3e04ef9c63dace67b339a104f1023ba50f
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/FamilyJetNorm.lean
@@ -0,0 +1,73 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.DiscreteManyProduct
+import OAI.Geometry.PrescribedRicci.JetTameProduct
+
+namespace OAI
+
+/-! Family Jet Norm. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+open scoped ContDiff ENNReal Classical BigOperators
+namespace TameInterpolation
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+variable {ι κ : Type*} [Fintype ι] [Nonempty ι] [Fintype κ] [Nonempty κ]
+
+def familyJetNorm (e : ι → E) (f : κ → E → ℝ) (j : ℕ) (p : ℝ≥0∞) : ℝ :=
+  Finset.univ.sup' Finset.univ_nonempty (fun a : κ => jetNorm e (f a) j p)
+
+lemma jetNorm_le_family (e : ι → E) (f : κ → E → ℝ) (a : κ) (j : ℕ) (p : ℝ≥0∞) :
+    jetNorm e (f a) j p ≤ familyJetNorm e f j p := by
+  exact Finset.le_sup' (fun a : κ => jetNorm e (f a) j p) (Finset.mem_univ a)
+
+lemma familyJetNorm_nonneg (e : ι → E) (f : κ → E → ℝ) (j : ℕ) (p : ℝ≥0∞) :
+    0 ≤ familyJetNorm e f j p :=
+  (jetNorm_nonneg e (f (Classical.choice ‹Nonempty κ›)) j p).trans (jetNorm_le_family _ _ _ _ _)
+
+lemma familyJetNorm_attained (e : ι → E) (f : κ → E → ℝ) (j : ℕ) (p : ℝ≥0∞) :
+    ∃ a : κ, familyJetNorm e f j p = jetNorm e (f a) j p := by
+  obtain ⟨a,_,ha⟩ := Finset.exists_mem_eq_sup' (s:=Finset.univ) Finset.univ_nonempty
+    (fun a : κ => jetNorm e (f a) j p)
+  exact ⟨a,ha⟩
+
+lemma familyJetNorm_recurrence (e : ι → E) (f : κ → E → ℝ)
+    (hf : ∀ a, ContDiff ℝ ∞ (f a)) (hc : ∀ a, HasCompactSupport (f a))
+    (m : ℕ) (hm : 0 < m) (j : ℕ) (hj : j+2 ≤ m) :
+    (familyJetNorm e f (j+1) (exponent m (j+1)))^2 ≤
+    (2*(m:ℝ))*familyJetNorm e f j (exponent m j)*familyJetNorm e f (j+2) (exponent m (j+2)) := by
+  obtain ⟨a,ha⟩ := familyJetNorm_attained e f (j+1) (exponent m (j+1))
+  rw [ha]
+  apply (jet_exponent_recurrence e (hf a) (hc a) m hm j hj).trans
+  exact mul_le_mul
+    (mul_le_mul_of_nonneg_left (jetNorm_le_family _ _ _ _ _) (by positivity))
+    (jetNorm_le_family _ _ _ _ _) (jetNorm_nonneg _ _ _ _) (mul_nonneg (by positivity) (familyJetNorm_nonneg _ _ _ _))
+
+lemma exponent_top {m : ℕ} (hm : 0 < m) : exponent m m = 2 := by
+  rw [exponent_of_pos hm hm]
+  have hmn : (m:ℝ) ≠ 0 := by exact_mod_cast (Nat.ne_of_gt hm)
+  rw [mul_div_cancel_right₀ _ hmn]
+  norm_num
+
+theorem family_jet_tame_many (e : ι → E) (f : κ → E → ℝ)
+    (hf : ∀ a, ContDiff ℝ ∞ (f a)) (hc : ∀ a, HasCompactSupport (f a))
+    (m : ℕ) (hm : 0 < m) {ν : Type*} (s : Finset ν) (j : ν → ℕ)
+    (hs : ∑ i ∈ s, j i = m) :
+    ∏ i ∈ s, familyJetNorm e f (j i) (exponent m (j i)) ≤
+    (2*(m:ℝ))^(∑ i ∈ s, j i*(m-j i))*(familyJetNorm e f 0 ∞)^(s.card-1)*familyJetNorm e f m 2 := by
+  have hC : 1 ≤ 2*(m:ℝ) := by
+    have : (1:ℝ) ≤ (m:ℝ) := by exact_mod_cast hm
+    linarith only [this]
+  simpa only [exponent_zero,exponent_top hm] using nonnegative_sequence_many m hm
+    (fun i => familyJetNorm e f i (exponent m i)) (fun _ => familyJetNorm_nonneg _ _ _ _)
+    (2*(m:ℝ)) hC (familyJetNorm_recurrence e f hf hc m hm) s j hs
+end TameInterpolation
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/FiniteCofactor.lean b/lean/OAI/Geometry/PrescribedRicci/FiniteCofactor.lean
new file mode 100644
index 0000000000000000000000000000000000000000..8db392b4b04ba015db768985c29091c1ac576e22
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/FiniteCofactor.lean
@@ -0,0 +1,136 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CompletedRealification
+import OAI.Geometry.PrescribedPotential.GlobalConjugation
+import OAI.Geometry.PrescribedPotential.GlobalHessian
+import OAI.Geometry.PrescribedPotential.GlobalMetricEntries
+import OAI.Geometry.PrescribedRicci.MatrixDifferentialNew
+import OAI.Geometry.PrescribedPotential.NonlinearHessianCommutator
+import OAI.Geometry.PrescribedPotential.SobolevDetDifferential
+
+namespace OAI
+
+/-! Finite Cofactor. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff SchwartzMap Classical BoundedContinuousFunction ComplexOrder MatrixOrder Matrix.Norms.Elementwise
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace Localizers
+variable (L : Localizers A ι)
+lemma strong_lower {s t : ℝ} (hst : t ≤ s)
+    (hs : (Module.finrank ℝ (EC d):ℝ) < 2*s)
+    (ht : (Module.finrank ℝ (EC d):ℝ) < 2*t) (u : L.Sobolev s) :
+    L.strong t (L.lower s t u) = L.strong s u := by
+  have he : (fun u => L.strong t (L.lower s t u)) = (fun u => L.strong s u) := by
+    apply (L.embed_dense s).equalizer (by fun_prop) (by fun_prop)
+    funext f
+    simp only [Function.comp_apply,L.lower_embed hst,L.strong_embed s hs,L.strong_embed t ht]
+  exact congr_fun he u
+end Localizers
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+lemma completedHessian_lower (k l : ℕ) (hlk : l ≤ k) (p : ι) (i j : Fin d)
+    (u : D.localizers.Sobolev ((k:ℝ)+2)) :
+    D.completedHessian l p i j (D.localizers.lower ((k:ℝ)+2) ((l:ℝ)+2) u) =
+      D.localizers.lower (k:ℝ) (l:ℝ) (D.completedHessian k p i j u) := by
+  have hkl : (l:ℝ) ≤ (k:ℝ) := by exact_mod_cast hlk
+  have he : (fun u => D.completedHessian l p i j (D.localizers.lower ((k:ℝ)+2) ((l:ℝ)+2) u)) =
+      (fun u => D.localizers.lower (k:ℝ) (l:ℝ) (D.completedHessian k p i j u)) := by
+    apply (D.localizers.embed_dense ((k:ℝ)+2)).equalizer (by fun_prop) (by fun_prop)
+    funext f
+    simp only [Function.comp_apply,D.completedHessian_embed,
+      D.localizers.lower_embed (add_le_add hkl le_rfl),D.localizers.lower_embed hkl]
+  exact congr_fun he u
+
+lemma completedRealify_strong (k : ℕ)
+    (hk : (Module.finrank ℝ (EC d):ℝ) < 2*(k:ℝ)) (u : D.localizers.Sobolev (k:ℝ)) (x : X) :
+    D.localizers.strong (k:ℝ) (D.completedRealify k u) x =
+      ((D.localizers.strong (k:ℝ) u x).re : ℂ) := by
+  have he : (fun u => D.localizers.strong (k:ℝ) (D.completedRealify k u) x) =
+      (fun u => ((D.localizers.strong (k:ℝ) u x).re : ℂ)) := by
+    apply (D.localizers.embed_dense (k:ℝ)).equalizer (by fun_prop) (by fun_prop)
+    funext f
+    simp only [Function.comp_apply,D.completedRealify_embed,D.localizers.strong_embed _ hk]
+    exact Smooth.realify_apply f x
+  exact congr_fun he u
+
+def cofactorMatrix (B : Bilin (EC d)) : Matrix (Fin d) (Fin d) ℂ :=
+  PotentialKaehler.hermitianPartMatrix (pullBilin B)
+
+def cofactorScalar (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) (p : ι)
+    (B : Bilin (EC d)) (u : D.localizers.Sobolev ((k:ℝ)+2)) : D.localizers.Sobolev (k:ℝ) :=
+  D.completedRealify k (D.product k hk (D.localizers.embed (k:ℝ) (D.inverseDet p))
+    (D.detDifferential k hk d
+      (fun i j => D.localizers.embed (k:ℝ) (D.metricEntry p i j) + D.completedHessian k p i j u)
+      (fun i j => D.localizers.embed (k:ℝ) (Smooth.const (cofactorMatrix B i j)))))
+
+lemma cofactorScalar_continuous (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (p : ι) (B : Bilin (EC d)) : Continuous (D.cofactorScalar k hk p B) := by
+  apply (D.completedRealify k).continuous.comp
+  apply (D.product k hk (D.localizers.embed (k:ℝ) (D.inverseDet p))).continuous.comp
+  apply Continuous.clm_apply
+  · exact (D.detDifferential_contDiff k hk d).continuous.comp (by fun_prop)
+  · fun_prop
+
+lemma cofactorScalar_strong (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (p : ι) (B : Bilin (EC d)) (u : D.localizers.Sobolev ((k:ℝ)+2)) (x : X) :
+    D.localizers.strong (k:ℝ) (D.cofactorScalar k hk p B u) x =
+      ((D.inverseDet p x * NonlinearHessian.detDifferential
+        (fun i j => D.metricEntry p i j x + D.localizers.strong (k:ℝ) (D.completedHessian k p i j u) x)
+        (cofactorMatrix B)).re : ℂ) := by
+  have hs : (Module.finrank ℝ (EC d):ℝ) < 2*(k:ℝ) := by
+    have hh : (Module.finrank ℝ (EC d):ℝ) < k := by exact_mod_cast hk
+    linarith [Nat.cast_nonneg (α:=ℝ) k]
+  rw [cofactorScalar,D.completedRealify_strong k hs,D.product_strong,
+    BoundedContinuousFunction.mul_apply,D.localizers.strong_embed _ hs]
+  erw [D.detDifferential_strong]
+  simp only [map_add,BoundedContinuousFunction.add_apply,D.localizers.strong_embed _ hs]
+  rfl
+
+lemma cofactorScalar_zero_source (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (p : ι) (B : Bilin (EC d)) {x : X}
+    (hx : x ∈ tsupport (D.localizers.weight p : X → ℂ)) :
+    D.localizers.strong (k:ℝ) (D.cofactorScalar k hk p B 0) x =
+      (traceBilin (g.matrix (D.patch p).index (A.chart (D.patch p).index x)) B : ℂ) := by
+  rw [D.cofactorScalar_strong]
+  simp only [map_zero,BoundedContinuousFunction.coe_zero,Pi.zero_apply,add_zero]
+  simp_rw [D.metricEntry_apply p _ _ hx]
+  rw [D.inverseDet_apply p hx]
+  have hp := g.positive (D.patch p).index _ ((A.chart (D.patch p).index).mapsTo (by simpa using D.patch_source p hx))
+  change (((((g.matrix (D.patch p).index (A.chart (D.patch p).index x)).det)⁻¹ *
+    fderiv ℝ Matrix.det (g.matrix (D.patch p).index (A.chart (D.patch p).index x)) (cofactorMatrix B)).re : ℂ)) = _
+  rw [MongeAmpere.real_fderiv_det_apply _ _
+    (isUnit_iff_ne_zero.mpr hp.det_pos.ne')]
+  rw [← mul_assoc,inv_mul_cancel₀ hp.det_pos.ne',one_mul]
+  rfl
+
+lemma cofactorScalar_strong_lower (k l : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (hl : Module.finrank ℝ (EC d) < l) (hlk : l ≤ k) (p : ι) (B : Bilin (EC d))
+    (u : D.localizers.Sobolev ((k:ℝ)+2)) (x : X) :
+    D.localizers.strong (l:ℝ) (D.cofactorScalar l hl p B
+      (D.localizers.lower ((k:ℝ)+2) ((l:ℝ)+2) u)) x =
+    D.localizers.strong (k:ℝ) (D.cofactorScalar k hk p B u) x := by
+  have hs : (Module.finrank ℝ (EC d):ℝ) < 2*(l:ℝ) := by
+    have hh : (Module.finrank ℝ (EC d):ℝ) < l := by exact_mod_cast hl
+    linarith [Nat.cast_nonneg (α:=ℝ) l]
+  have hkl : (l:ℝ) ≤ (k:ℝ) := by exact_mod_cast hlk
+  have ht : (Module.finrank ℝ (EC d):ℝ) < 2*(k:ℝ) := by linarith
+  rw [D.cofactorScalar_strong,D.cofactorScalar_strong]
+  simp_rw [D.completedHessian_lower k l hlk,
+    D.localizers.strong_lower hkl ht hs]
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/FiniteCofactorPatch.lean b/lean/OAI/Geometry/PrescribedRicci/FiniteCofactorPatch.lean
new file mode 100644
index 0000000000000000000000000000000000000000..6d51e981f7ae3d08435dc52b2008e8886b8e4017
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/FiniteCofactorPatch.lean
@@ -0,0 +1,130 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.BoundedPoisson
+import OAI.Geometry.PrescribedPotential.GlobalHessian
+import OAI.Geometry.PrescribedPotential.LiteralWeakHessian
+import OAI.Geometry.PrescribedRicci.TameApproximation
+import OAI.Geometry.PrescribedRicci.FiniteCofactor
+
+namespace OAI
+
+/-! Finite Cofactor Patch. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff SchwartzMap Classical BoundedContinuousFunction ComplexOrder MatrixOrder
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart FrozenPoisson MetricLocalization
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+def cofactorLocal (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (p : ι) (i j : BasisIndex d) (u : D.localizers.Sobolev ((k:ℝ)+2)) : L2 (EC d) :=
+  schwartzCoord (k:ℝ) ((D.patch p).coefficient i j) +
+    D.completedLocalize (k:ℝ) (D.patch p).index (cutoffGlobal (D.patch p).index (D.cutoff p))
+      (cutoffGlobal_support _ _)
+      (D.cofactorScalar k hk p (rankTwo (stdOrthonormalBasis ℝ (EC d) i)
+        (stdOrthonormalBasis ℝ (EC d) j)) u -
+       D.cofactorScalar k hk p (rankTwo (stdOrthonormalBasis ℝ (EC d) i)
+        (stdOrthonormalBasis ℝ (EC d) j)) 0)
+
+lemma cofactorLocal_continuous (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (p : ι) (i j : BasisIndex d) : Continuous (D.cofactorLocal k hk p i j) := by
+  exact continuous_const.add ((D.completedLocalize _ _ _ _).continuous.comp
+    ((D.cofactorScalar_continuous k hk p _).sub continuous_const))
+
+lemma cofactorLocal_zero (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (p : ι) (i j : BasisIndex d) :
+    D.cofactorLocal k hk p i j 0 = schwartzCoord (k:ℝ) ((D.patch p).coefficient i j) := by
+  simp [cofactorLocal]
+
+def cofactorBCF (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (hs : (Module.finrank ℝ (EC d):ℝ) < 2*(k:ℝ)) (p : ι)
+    (u : D.localizers.Sobolev ((k:ℝ)+2)) : BasisIndex d → BasisIndex d → EC d →ᵇ ℂ :=
+  fun i j => strongEmbedding (k:ℝ) hs (D.cofactorLocal k hk p i j u)
+
+lemma cofactorBCF_continuous (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (hs : (Module.finrank ℝ (EC d):ℝ) < 2*(k:ℝ)) (p : ι) :
+    Continuous (D.cofactorBCF k hk hs p) := by
+  apply continuous_pi
+  intro i
+  apply continuous_pi
+  intro j
+  exact (strongEmbedding _ hs).continuous.comp (D.cofactorLocal_continuous k hk p i j)
+
+lemma cofactorBCF_zero (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (hs : (Module.finrank ℝ (EC d):ℝ) < 2*(k:ℝ)) (p : ι) :
+    D.cofactorBCF k hk hs p 0 = coefficientBCF (D.patch p).coefficient := by
+  funext i j
+  rw [cofactorBCF,D.cofactorLocal_zero,strongEmbedding_schwartz]
+  rfl
+
+lemma cofactorBCF_lower (k l : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (hl : Module.finrank ℝ (EC d) < l) (hlk : l ≤ k)
+    (hk₂ : (Module.finrank ℝ (EC d):ℝ) < 2*(k:ℝ))
+    (hl₂ : (Module.finrank ℝ (EC d):ℝ) < 2*(l:ℝ)) (p : ι)
+    (u : D.localizers.Sobolev ((k:ℝ)+2)) :
+    D.cofactorBCF l hl hl₂ p (D.localizers.lower ((k:ℝ)+2) ((l:ℝ)+2) u) =
+      D.cofactorBCF k hk hk₂ p u := by
+  funext i j
+  ext y
+  simp only [cofactorBCF,cofactorLocal,map_add,BoundedContinuousFunction.add_apply,
+    strongEmbedding_schwartz]
+  rw [D.completedLocalize_strong l hl₂,D.completedLocalize_strong k hk₂]
+  split_ifs
+  · simp only [map_sub,BoundedContinuousFunction.sub_apply]
+    rw [D.cofactorScalar_strong_lower k l hk hl hlk]
+    have hh := D.cofactorScalar_strong_lower k l hk hl hlk p
+      (rankTwo (stdOrthonormalBasis ℝ (EC d) i) (stdOrthonormalBasis ℝ (EC d) j)) 0
+      ((A.euclideanChart (D.patch p).index).symm y)
+    rw [map_zero] at hh
+    rw [hh]
+  · rfl
+
+def CofactorSmall (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (hs : (Module.finrank ℝ (EC d):ℝ) < 2*(k:ℝ))
+    (u : D.localizers.Sobolev ((k:ℝ)+2)) : Prop :=
+  ∀ p, perturbationBound (stdOrthonormalBasis ℝ (EC d)) (D.cofactorBCF k hk hs p u) *
+    ellipticBound (g.matrix (D.patch p).index (coordinateEquiv d (D.patch p).center))
+      (g.positive _ _ (by simpa using (D.patch p).mem_target)) < 1
+
+lemma cofactor_small_nhds (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (hs : (Module.finrank ℝ (EC d):ℝ) < 2*(k:ℝ)) :
+    ∀ᶠ u in 𝓝 (0 : D.localizers.Sobolev ((k:ℝ)+2)), D.CofactorSmall k hk hs u := by
+  apply eventually_all.mpr
+  intro p
+  have hc := ((continuous_perturbationBound (stdOrthonormalBasis ℝ (EC d))).comp
+    (D.cofactorBCF_continuous k hk hs p)).mul_const
+      (ellipticBound (g.matrix (D.patch p).index (coordinateEquiv d (D.patch p).center))
+        (g.positive _ _ (by simpa using (D.patch p).mem_target)))
+  apply hc.continuousAt (gt_mem_nhds ?_)
+  simp only [Function.comp_apply]
+  rw [D.cofactorBCF_zero]
+  exact (D.patch p).small
+
+ 
+
+lemma cofactor_small_of_lower (k l : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (hl : Module.finrank ℝ (EC d) < l) (hlk : l ≤ k)
+    (hk₂ : (Module.finrank ℝ (EC d):ℝ) < 2*(k:ℝ))
+    (hl₂ : (Module.finrank ℝ (EC d):ℝ) < 2*(l:ℝ))
+    (u : D.localizers.Sobolev ((k:ℝ)+2))
+    (hu : D.CofactorSmall l hl hl₂ (D.localizers.lower ((k:ℝ)+2) ((l:ℝ)+2) u)) :
+    D.CofactorSmall k hk hk₂ u := by
+  intro p
+  have hh := hu p
+  rw [D.cofactorBCF_lower k l hk hl hlk hk₂ hl₂] at hh
+  exact hh
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/FixedCofactorBootstrap.lean b/lean/OAI/Geometry/PrescribedRicci/FixedCofactorBootstrap.lean
new file mode 100644
index 0000000000000000000000000000000000000000..664e13c683703a07db3de5da619a05ff04b4de32
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/FixedCofactorBootstrap.lean
@@ -0,0 +1,153 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.FiniteCofactorPatch
+import OAI.Geometry.PrescribedRicci.FixedCofactorStep
+import OAI.Geometry.PrescribedPotential.GlobalSmooth
+import OAI.Geometry.PrescribedPotential.NonlinearSobolevGain
+import OAI.Geometry.PrescribedPotential.PatchCutoffs
+import OAI.Geometry.PrescribedPotential.RealSobolev
+
+namespace OAI
+
+/-! Fixed Cofactor Bootstrap. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology
+open scoped ContDiff Classical
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+local instance fixedCofactorBootNG (s : ℝ) : NormedAddCommGroup (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedAddCommGroup
+local instance fixedCofactorBootNS (s : ℝ) : NormedSpace ℝ (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedSpace
+local instance fixedCofactorBootTG (s : ℝ) : IsTopologicalAddGroup (D.localizers.RealSobolev s) :=
+  Submodule.isTopologicalAddGroup _
+local instance fixedCofactorBootCS (s : ℝ) : ContinuousSMul ℝ (D.localizers.RealSobolev s) :=
+  SMulMemClass.continuousSMul _
+
+lemma lift_over_lift {s t r : ℝ} (hst : t ≤ s) (htr : r ≤ t)
+    (u : D.localizers.RealSobolev r) (v : D.localizers.RealSobolev t)
+    (hv : D.localizers.realLower t r htr v = u)
+    (h : ∃ z : D.localizers.RealSobolev s, D.localizers.realLower s t hst z = v) :
+    ∃ z : D.localizers.RealSobolev s, D.localizers.realLower s r (htr.trans hst) z = u := by
+  obtain ⟨z,hz⟩ := h
+  refine ⟨z,?_⟩
+  rw [← D.localizers.realLower_lower hst htr z,hz,hv]
+
+lemma cofactor_small_lifted (k l : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (hl : Module.finrank ℝ (EC d) < l) (hkl : k ≤ l)
+    (hs : (Module.finrank ℝ (EC d):ℝ) < 2*(k:ℝ))
+    (ht : (Module.finrank ℝ (EC d):ℝ) < 2*(l:ℝ))
+    (u : D.localizers.RealSobolev ((k:ℝ)+2))
+    (v : D.localizers.RealSobolev ((l:ℝ)+2))
+    (hv : D.localizers.realLower ((l:ℝ)+2) ((k:ℝ)+2) (by exact_mod_cast Nat.add_le_add_right hkl 2) v = u)
+    (hu : D.CofactorSmall k hk hs u.val) : D.CofactorSmall l hl ht v.val := by
+  apply D.cofactor_small_of_lower l k hl hk hkl ht hs v.val
+  have he := congrArg Subtype.val hv
+  change D.localizers.lower _ _ v.val = u.val at he
+  rw [he]
+  exact hu
+
+lemma cofactor_lift_step (k l : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (hl : Module.finrank ℝ (EC d) < l) (hkl : k ≤ l)
+    (hs : (Module.finrank ℝ (EC d):ℝ) < 2*((k+1:ℕ):ℝ))
+    (ht : (Module.finrank ℝ (EC d):ℝ) < 2*((l+1:ℕ):ℝ))
+    (u : D.localizers.RealSobolev (((k+1:ℕ):ℝ)+2))
+    (v : D.localizers.RealSobolev (((l+1:ℕ):ℝ)+2))
+    (hv : D.localizers.realLower (((l+1:ℕ):ℝ)+2) (((k+1:ℕ):ℝ)+2)
+      (by exact_mod_cast Nat.add_le_add_right (Nat.add_le_add_right hkl 1) 2) v = u)
+    (hu : D.CofactorSmall (k+1) (by omega) hs u.val) (F : RealSmooth A)
+    (hF : D.realVolume (k+1) (by omega) u = D.localizers.realEmbed ((k+1:ℕ):ℝ) F) :
+    ∃ z : D.localizers.RealSobolev (((l+2:ℕ):ℝ)+2),
+      D.localizers.realLower (((l+2:ℕ):ℝ)+2) (((k+1:ℕ):ℝ)+2) (by push_cast; exact_mod_cast Nat.add_le_add_right (by omega : k+1 ≤ l+2) 2) z = u := by
+  have hsmall : D.CofactorSmall (l+1) (by omega) ht v.val := D.cofactor_small_lifted (k+1) (l+1) (by omega) (by omega) (by omega) hs ht u v hv hu
+  have hEq : D.realVolume (l+1) (by omega) v = D.localizers.realEmbed ((l+1:ℕ):ℝ) F := D.cofactor_lift_equation (k+1) (l+1) (by omega) (by omega) (by omega) u v hv F hF
+  have hgain := D.cofactor_nonlinear_step l hl ht v hsmall F hEq
+  obtain ⟨z,hz⟩ := hgain
+  refine ⟨z,?_⟩
+  have hst : (((l+1:ℕ):ℝ)+2) ≤ (((l+2:ℕ):ℝ)+2) := by push_cast; linarith only
+  have htr : (((k+1:ℕ):ℝ)+2) ≤ (((l+1:ℕ):ℝ)+2) := by exact_mod_cast Nat.add_le_add_right (Nat.add_le_add_right hkl 1) 2
+  rw [← D.localizers.realLower_lower (s:=((l+2:ℕ):ℝ)+2) (t:=((l+1:ℕ):ℝ)+2)
+    (r:=((k+1:ℕ):ℝ)+2) hst htr z, hz, hv]
+
+lemma cofactor_solution_lifts (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (hs : (Module.finrank ℝ (EC d):ℝ) < 2*((k+1:ℕ):ℝ))
+    (u : D.localizers.RealSobolev (((k+1:ℕ):ℝ)+2))
+    (hu : D.CofactorSmall (k+1) (by omega) hs u.val) (F : RealSmooth A)
+    (hF : D.realVolume (k+1) (by omega) u = D.localizers.realEmbed ((k+1:ℕ):ℝ) F) :
+    ∀ n : ℕ, ∃ z : D.localizers.RealSobolev (((k+n+1:ℕ):ℝ)+2),
+      D.localizers.realLower (((k+n+1:ℕ):ℝ)+2) (((k+1:ℕ):ℝ)+2)
+        (by push_cast; linarith only [Nat.cast_nonneg (α:=ℝ) n]) z = u := by
+  intro n
+  induction n with
+  | zero =>
+    refine ⟨u,?_⟩
+    apply Subtype.ext
+    change D.localizers.lower _ _ u.val = u.val
+    rw [D.localizers.lower_self,ContinuousLinearMap.id_apply]
+  | succ n ih =>
+    obtain ⟨v,hv⟩ := ih
+    have ht : (Module.finrank ℝ (EC d):ℝ) < 2*((k+n+1:ℕ):ℝ) :=
+      hs.trans_le (by exact_mod_cast Nat.mul_le_mul_left 2 (by omega : k+1 ≤ k+n+1))
+    have hh := D.cofactor_lift_step k (k+n) hk (by omega) (by omega) hs ht u v hv hu F hF
+    exact D.realLift_transport _ _ _ _ _ (by congr 2) u hh
+
+lemma realLift_order_witness (s t : ℝ) (k : ℕ) (hs : 0 ≤ s) (hst : s ≤ t)
+    (hkt : (k:ℝ) ≤ t) (u : D.localizers.RealSobolev s) (v : D.localizers.RealSobolev t)
+    (hv : D.localizers.realLower t s hst v = u) :
+    ∃ w : D.localizers.Sobolev (k:ℝ), D.localizers.lower (k:ℝ) 0 w =
+      D.localizers.lower s 0 u.val := by
+  refine ⟨D.localizers.lower t (k:ℝ) v.val, ?_⟩
+  rw [D.localizers.lower_lower hkt (Nat.cast_nonneg k)]
+  rw [← D.localizers.lower_lower hst hs v.val]
+  exact congrArg (D.localizers.lower s 0) (congrArg Subtype.val hv)
+
+lemma cofactor_solution_allRegular (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (hs : (Module.finrank ℝ (EC d):ℝ) < 2*((k+1:ℕ):ℝ))
+    (u : D.localizers.RealSobolev (((k+1:ℕ):ℝ)+2))
+    (hu : D.CofactorSmall (k+1) (by omega) hs u.val) (F : RealSmooth A)
+    (hF : D.realVolume (k+1) (by omega) u = D.localizers.realEmbed ((k+1:ℕ):ℝ) F) :
+    D.localizers.AllRegular (D.localizers.lower (((k+1:ℕ):ℝ)+2) 0 u.val) := by
+  intro n
+  have hb : 0 ≤ ((k+1:ℕ):ℝ)+2 := by positivity
+  have hnb : ((k+1:ℕ):ℝ)+2 ≤ ((k+n+1:ℕ):ℝ)+2 := by
+    push_cast; linarith only [Nat.cast_nonneg (α:=ℝ) n]
+  have hnv : (n:ℝ) ≤ ((k+n+1:ℕ):ℝ)+2 := by
+    push_cast; linarith only [Nat.cast_nonneg (α:=ℝ) k]
+  obtain ⟨v,hv⟩ := D.cofactor_solution_lifts k hk hs u hu F hF n
+  exact D.realLift_order_witness _ _ n hb hnb hnv u v hv
+
+ 
+
+theorem cofactor_solution_smooth (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (hs : (Module.finrank ℝ (EC d):ℝ) < 2*((k+1:ℕ):ℝ))
+    (u : D.localizers.RealSobolev (((k+1:ℕ):ℝ)+2))
+    (hu : D.CofactorSmall (k+1) (by omega) hs u.val) (F : RealSmooth A)
+    (hF : D.realVolume (k+1) (by omega) u = D.localizers.realEmbed ((k+1:ℕ):ℝ) F) :
+    ∃ φ : RealSmooth A, D.localizers.realEmbed (((k+1:ℕ):ℝ)+2) φ = u := by
+  obtain ⟨f,hf⟩ := D.allRegular_smooth (D.cofactor_solution_allRegular k hk hs u hu F hF)
+  have he : D.localizers.embed (((k+1:ℕ):ℝ)+2) f = u.val := by
+    apply D.localizers.lower_injective (by positivity : 0 ≤ ((k+1:ℕ):ℝ)+2)
+    rw [D.localizers.lower_embed (by positivity)]
+    exact hf
+  have hb : (Module.finrank ℝ (EC d):ℝ) < 2*(((k+1:ℕ):ℝ)+2) := by linarith only [hs]
+  have hr (x : X) : (f x).im = 0 := by
+    have hh := D.localizers.real_strong hb u x
+    rw [← he,D.localizers.strong_embed _ hb] at hh
+    exact hh
+  exact ⟨⟨f,hr⟩,Subtype.ext he⟩
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/FixedCofactorStep.lean b/lean/OAI/Geometry/PrescribedRicci/FixedCofactorStep.lean
new file mode 100644
index 0000000000000000000000000000000000000000..ecbf78f1af393aa05984a39a9a2f88cb06193d90
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/FixedCofactorStep.lean
@@ -0,0 +1,78 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.DerivativeEllipticGain
+import OAI.Geometry.PrescribedPotential.NonlinearScale
+import OAI.Geometry.PrescribedPotential.NonlinearSobolevGain
+import OAI.Geometry.PrescribedPotential.WeakNonlinearCommutator
+import OAI.Geometry.PrescribedRicci.FiniteCofactorPatch
+import OAI.Geometry.PrescribedRicci.TameCofactorGain
+
+namespace OAI
+
+/-! Fixed Cofactor Step. -/
+
+section
+
+ 
+noncomputable section
+open Set Filter Topology
+open scoped ContDiff Classical
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  [ConnectedSpace X] {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+local instance fixedCofactorStepNG (s : ℝ) : NormedAddCommGroup (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedAddCommGroup
+local instance fixedCofactorStepNS (s : ℝ) : NormedSpace ℝ (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedSpace
+local instance fixedCofactorStepTG (s : ℝ) : IsTopologicalAddGroup (D.localizers.RealSobolev s) :=
+  Submodule.isTopologicalAddGroup _
+local instance fixedCofactorStepCS (s : ℝ) : ContinuousSMul ℝ (D.localizers.RealSobolev s) :=
+  SMulMemClass.continuousSMul _
+
+omit [ConnectedSpace X] in
+lemma realLift_transport (s t r : ℝ) (hs : r ≤ s) (ht : r ≤ t) (hst : s = t)
+    (u : D.localizers.RealSobolev r)
+    (h : ∃ z : D.localizers.RealSobolev s, D.localizers.realLower s r hs z = u) :
+    ∃ z : D.localizers.RealSobolev t, D.localizers.realLower t r ht z = u := by
+  subst t
+  exact h
+
+omit [ConnectedSpace X] in
+theorem cofactor_nonlinear_step (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (hs : (Module.finrank ℝ (EC d):ℝ) < 2*((k+1:ℕ):ℝ))
+    (u : D.localizers.RealSobolev (((k+1:ℕ):ℝ)+2))
+    (hu : D.CofactorSmall (k+1) (by omega) hs u.val)
+    (F : RealSmooth A)
+    (hF : D.realVolume (k+1) (by omega) u = D.localizers.realEmbed ((k+1:ℕ):ℝ) F) :
+    ∃ z : D.localizers.RealSobolev (((k+2:ℕ):ℝ)+2),
+      D.localizers.realLower (((k+2:ℕ):ℝ)+2) (((k+1:ℕ):ℝ)+2) (by push_cast; linarith) z = u := by
+  have hD (p : ι) (v : EC d) := D.cofactor_global_gain k hk hs u hu
+    (D.potentialDerivative k p v u)
+    (D.localizers.realEmbed ((k+1:ℕ):ℝ) (D.realLocalizedDerivative p v F) +
+      D.realNonlinearRemainder (k+1) (by omega) p v u)
+    (D.nonlinear_commutator_solution k hk p v u F hF)
+  have hh := D.derivative_lifts_gain k u hD
+  have he : ((k+4:ℕ):ℝ) = ((k+2:ℕ):ℝ)+2 := by push_cast; ring
+  exact D.realLift_transport _ _ _ _ _ he u hh
+
+ 
+omit [ConnectedSpace X] in
+lemma cofactor_lift_equation (k l : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (hl : Module.finrank ℝ (EC d) < l) (hkl : k ≤ l)
+    (u : D.localizers.RealSobolev ((k:ℝ)+2))
+    (v : D.localizers.RealSobolev ((l:ℝ)+2))
+    (hv : D.localizers.realLower ((l:ℝ)+2) ((k:ℝ)+2) (by exact_mod_cast Nat.add_le_add_right hkl 2) v = u)
+    (F : RealSmooth A) (hF : D.realVolume k hk u = D.localizers.realEmbed (k:ℝ) F) :
+    D.realVolume l hl v = D.localizers.realEmbed (l:ℝ) F := by
+  apply D.localizers.realLower_injective (by exact_mod_cast hkl : (k:ℝ) ≤ (l:ℝ))
+  rw [← D.realVolume_lower l k hl hk hkl, hv, hF,Localizers.realLower_embed]
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/GlobalKahlerEnergy.lean b/lean/OAI/Geometry/PrescribedRicci/GlobalKahlerEnergy.lean
new file mode 100644
index 0000000000000000000000000000000000000000..ecd11dcb4c9c42c4978d7b1f92ec694ce9fa3a89
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/GlobalKahlerEnergy.lean
@@ -0,0 +1,137 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.KahlerGradientEnergy
+import OAI.Geometry.PrescribedRicci.KahlerIntegration
+import OAI.Geometry.PrescribedPotential.GlobalOperator
+import OAI.Geometry.PrescribedPotential.SmoothPotentialDifference
+
+namespace OAI
+
+/-! Global Kahler Energy. -/
+
+section
+
+ 
+
+noncomputable section
+open Matrix Filter Set Topology MeasureTheory
+open scoped ContDiff ComplexOrder Classical
+namespace Anticanonical.SourceSmooth
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+namespace KaehlerMetric
+
+ 
+def energy (g : KaehlerMetric A) (ψ φ : SmoothRealFunction A) : SmoothRealFunction A where
+  value x := ((g.laplacian (ψ.product φ)).value x -
+    ψ.value x * (g.laplacian φ).value x - φ.value x * (g.laplacian ψ).value x) / 2
+  smooth i := (((g.laplacian (ψ.product φ)).smooth i).sub
+    ((ψ.smooth i).mul ((g.laplacian φ).smooth i))).sub
+      ((φ.smooth i).mul ((g.laplacian ψ).smooth i)) |>.div_const 2
+
+lemma energy_local (g : KaehlerMetric A) (ψ φ : SmoothRealFunction A)
+    (i : Fin A.count) {z : Coordinates d} (hz : z ∈ (A.chart i).target) :
+    (g.energy ψ φ).localExpression i z =
+      (gradientPair (g.matrix i z) (holRealDeriv (ψ.localExpression i) z)
+        (holRealDeriv (φ.localExpression i) z)).re := by
+  have hp := trace_hessian_product (φ := ψ.localExpression i) (ψ := φ.localExpression i)
+    ((ψ.smooth i).contDiffAt ((A.chart i).open_target.mem_nhds hz))
+    ((φ.smooth i).contDiffAt ((A.chart i).open_target.mem_nhds hz))
+    (g.matrix i z) (g.positive i z hz).isHermitian
+  change ((g.laplacian (ψ.product φ)).localExpression i z -
+    ψ.localExpression i z * (g.laplacian φ).localExpression i z -
+    φ.localExpression i z * (g.laplacian ψ).localExpression i z) / 2 = _
+  rw [g.laplacian_localExpression _ i hz, g.laplacian_localExpression _ i hz,
+    g.laplacian_localExpression _ i hz]
+  change (((g.matrix i z)⁻¹ * PotentialKaehler.potentialMatrix
+      (fun y => ψ.localExpression i y * φ.localExpression i y) z).trace.re -
+    ψ.localExpression i z * ((g.matrix i z)⁻¹ * φ.hessian i z).trace.re -
+    φ.localExpression i z * ((g.matrix i z)⁻¹ * ψ.hessian i z).trace.re) / 2 = _
+  rw [hp]
+  change (_ - _ * ((g.matrix i z)⁻¹ * PotentialKaehler.potentialMatrix (φ.localExpression i) z).trace.re -
+    _ * ((g.matrix i z)⁻¹ * PotentialKaehler.potentialMatrix (ψ.localExpression i) z).trace.re) / 2 = _
+  ring
+
+lemma energy_nonneg (g : KaehlerMetric A) (φ : SmoothRealFunction A) (x : X) :
+    0 ≤ (g.energy φ φ).value x := by
+  obtain ⟨i, hi⟩ := A.covers x
+  have hz := (A.chart i).mapsTo hi
+  have he := g.energy_local φ φ i hz
+  simp only [SmoothRealFunction.localExpression, Function.comp_apply, (A.chart i).left_inv hi] at he
+  rw [he]
+  exact gradientPair_self_nonneg _ (g.positive i _ hz) _
+
+lemma energy_symm (g : KaehlerMetric A) (ψ φ : SmoothRealFunction A) :
+    (g.energy ψ φ).value = (g.energy φ ψ).value := by
+  funext x
+  obtain ⟨i, hi⟩ := A.covers x
+  have hz := (A.chart i).mapsTo hi
+  have he := g.energy_local ψ φ i hz
+  have he' := g.energy_local φ ψ i hz
+  simp only [SmoothRealFunction.localExpression, Function.comp_apply, (A.chart i).left_inv hi] at he he'
+  rw [he, he', ← gradientPair_star _ (g.positive i _ hz).isHermitian]
+  rfl
+
+lemma holRealDeriv_zero_of_notMem_tsupport (ψ : SmoothRealFunction A) (i : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart i).target)
+    (hs : (A.chart i).symm z ∉ tsupport ψ.value) : holRealDeriv (ψ.localExpression i) z = 0 := by
+  have he : ψ.localExpression i =ᶠ[𝓝 z] (fun _ => 0) :=
+    (notMem_tsupport_iff_eventuallyEq.mp hs).comp_tendsto
+      (((A.chart i).continuousOn_symm z hz).continuousAt ((A.chart i).open_target.mem_nhds hz))
+  funext j
+  unfold holRealDeriv
+  rw [he.fderiv_eq]
+  simp only [fderiv_const_apply, map_zero, Pi.zero_apply]
+
+lemma energy_support (g : KaehlerMetric A) (ψ φ : SmoothRealFunction A) :
+    tsupport (g.energy ψ φ).value ⊆ tsupport ψ.value := by
+  apply closure_minimal _ (isClosed_tsupport _)
+  intro x hx
+  by_contra hs
+  obtain ⟨i, hi⟩ := A.covers x
+  have hz := (A.chart i).mapsTo hi
+  have he := g.energy_local ψ φ i hz
+  have hd := holRealDeriv_zero_of_notMem_tsupport ψ i hz
+    (by simpa only [(A.chart i).left_inv hi] using hs)
+  simp only [SmoothRealFunction.localExpression, Function.comp_apply, (A.chart i).left_inv hi] at he
+  dsimp only [SmoothRealFunction.localExpression] at hd
+  rw [hd] at he
+  simp only [gradientPair, Pi.zero_apply, star_zero, mul_zero, zero_mul, Finset.sum_const_zero,
+    Complex.zero_re] at he
+  exact hx he
+
+lemma holRealDeriv_add {f h : Coordinates d → ℝ} {z : Coordinates d}
+    (hf : DifferentiableAt ℝ f z) (hh : DifferentiableAt ℝ h z) :
+    holRealDeriv (fun y => f y + h y) z = holRealDeriv f z + holRealDeriv h z := by
+  funext j
+  unfold holRealDeriv
+  rw [fderiv_fun_add hf hh, map_add]
+  rfl
+
+lemma gradientPair_add_left (H : Matrix (Fin d) (Fin d) ℂ) (p q r : Fin d → ℂ) :
+    gradientPair H (p + q) r = gradientPair H p r + gradientPair H q r := by
+  simp [gradientPair, Pi.add_apply, mul_add, add_mul, Finset.sum_add_distrib]
+
+lemma energy_add_left (g : KaehlerMetric A) (ψ χ φ : SmoothRealFunction A) :
+    (g.energy (ψ.addFunction χ) φ).value =
+      fun x => (g.energy ψ φ).value x + (g.energy χ φ).value x := by
+  funext x
+  obtain ⟨i, hi⟩ := A.covers x
+  have hz := (A.chart i).mapsTo hi
+  have he := g.energy_local (ψ.addFunction χ) φ i hz
+  have he₁ := g.energy_local ψ φ i hz
+  have he₂ := g.energy_local χ φ i hz
+  simp only [SmoothRealFunction.localExpression, Function.comp_apply, (A.chart i).left_inv hi] at he he₁ he₂
+  rw [he, he₁, he₂]
+  change (gradientPair _ (holRealDeriv (fun y => ψ.localExpression i y + χ.localExpression i y) _)
+    _).re = _
+  rw [holRealDeriv_add (f := ψ.localExpression i) (h := χ.localExpression i) (((ψ.smooth i).contDiffAt ((A.chart i).open_target.mem_nhds hz)).differentiableAt (by simp))
+    (((χ.smooth i).contDiffAt ((A.chart i).open_target.mem_nhds hz)).differentiableAt (by simp)),
+    gradientPair_add_left, Complex.add_re]
+  rfl
+
+end KaehlerMetric
+end Anticanonical.SourceSmooth
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/GlobalKahlerIntegral.lean b/lean/OAI/Geometry/PrescribedRicci/GlobalKahlerIntegral.lean
new file mode 100644
index 0000000000000000000000000000000000000000..c7e3d2d067bb9d0464027a563389d9d1577a36d7
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/GlobalKahlerIntegral.lean
@@ -0,0 +1,186 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.KahlerIntegration
+
+namespace OAI
+
+/-! Global Kahler Integral. -/
+
+section
+
+ 
+
+noncomputable section
+open Matrix Filter Set Topology MeasureTheory
+open scoped ContDiff ComplexOrder Classical
+namespace Anticanonical.SourceSmooth
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+
+namespace KaehlerMetric
+
+def localizeFunction (i : Fin A.count) (f : X → ℝ) (z : Coordinates d) : ℝ :=
+  if z ∈ (A.chart i).target then f ((A.chart i).symm z) else 0
+
+lemma localizeFunction_support (i : Fin A.count) (f : X → ℝ) :
+    Function.support (localizeFunction (A := A) i f) ⊆ (A.chart i) '' tsupport f := by
+  intro z hz
+  change localizeFunction i f z ≠ 0 at hz
+  by_cases ht : z ∈ (A.chart i).target
+  · have hh : f ((A.chart i).symm z) ≠ 0 := by simpa [localizeFunction, ht] using hz
+    exact ⟨(A.chart i).symm z, subset_tsupport _ hh, (A.chart i).right_inv ht⟩
+  · exact (hz (ite_eq_right ht)).elim
+
+lemma localizeFunction_continuous_compact [T2Space X] [CompactSpace X]
+    (i : Fin A.count) {f : X → ℝ} (hf : Continuous f)
+    (hs : tsupport f ⊆ (A.chart i).source) :
+    Continuous (localizeFunction (A := A) i f) ∧
+      HasCompactSupport (localizeFunction (A := A) i f) ∧
+      tsupport (localizeFunction (A := A) i f) ⊆ (A.chart i).target := by
+  let e := A.chart i
+  let K := e '' tsupport f
+  have hK : IsCompact K := (isClosed_tsupport _).isCompact.image_of_continuousOn
+    (e.continuousOn.mono hs)
+  have hsub : Function.support (localizeFunction i f) ⊆ K := localizeFunction_support i f
+  have hts : tsupport (localizeFunction i f) ⊆ K := closure_minimal hsub hK.isClosed
+  refine ⟨?_, HasCompactSupport.of_support_subset_isCompact hK hsub, ?_⟩
+  · rw [continuous_iff_continuousAt]
+    intro z
+    by_cases hz : z ∈ e.target
+    · apply (hf.continuousAt.comp ((e.continuousOn_symm z hz).continuousAt
+        (e.open_target.mem_nhds hz))).congr
+      filter_upwards [e.open_target.mem_nhds hz] with y hy
+      exact (ite_eq_left hy).symm
+    · have hzK : z ∉ K := by
+        rintro ⟨x, hx, rfl⟩
+        exact hz (e.mapsTo (hs hx))
+      apply (continuousAt_const (y := (0 : ℝ))).congr
+      filter_upwards [hK.isClosed.isOpen_compl.mem_nhds hzK] with y hy
+      exact (Function.notMem_support.mp (fun h => hy (hsub h))).symm
+  · rintro z hz
+    obtain ⟨x, hx, rfl⟩ := hts hz
+    exact e.mapsTo (hs hx)
+
+lemma continuous_real_mul_on_tsupport {E : Type*} [TopologicalSpace E] {f h : E → ℝ}
+    (hf : Continuous f) (hh : ∀ z ∈ tsupport f, ContinuousAt h z) :
+    Continuous (fun z => f z * h z) := by
+  rw [continuous_iff_continuousAt]
+  intro z
+  by_cases hz : z ∈ tsupport f
+  · exact hf.continuousAt.mul (hh z hz)
+  · apply (continuousAt_const (y := (0 : ℝ))).congr
+    filter_upwards [notMem_tsupport_iff_eventuallyEq.mp hz] with y hy
+    simp only [hy, zero_mul, Pi.zero_apply]
+
+lemma chart_integrable [T2Space X] [CompactSpace X] (g : KaehlerMetric A)
+    (i : Fin A.count) {f : X → ℝ} (hf : Continuous f)
+    (hs : tsupport f ⊆ (A.chart i).source) :
+    IntegrableOn (fun z => f ((A.chart i).symm z) * g.volumeCoefficient i z)
+      (A.chart i).target := by
+  obtain ⟨hl, hc, ht⟩ := localizeFunction_continuous_compact i hf hs
+  have hi : Integrable (fun z => localizeFunction i f z * g.volumeCoefficient i z) :=
+    (continuous_real_mul_on_tsupport hl (fun z hz =>
+    ((g.volumeCoefficient_smooth i).continuousOn z (ht hz)).continuousAt
+      ((A.chart i).open_target.mem_nhds (ht hz)))).integrable_of_hasCompactSupport hc.mul_right
+  apply hi.integrableOn.congr_fun _ (A.chart i).open_target.measurableSet
+  intro z hz
+  simp only [localizeFunction, ite_eq_left hz]
+
+lemma chartIntegral_linear_add [T2Space X] [CompactSpace X] (g : KaehlerMetric A)
+    (i : Fin A.count) {f h : X → ℝ} (hf : Continuous f) (hh : Continuous h)
+    (hsf : tsupport f ⊆ (A.chart i).source) (hsh : tsupport h ⊆ (A.chart i).source) :
+    g.chartIntegral i (fun x => f x + h x) = g.chartIntegral i f + g.chartIntegral i h := by
+  simp only [chartIntegral, add_mul]
+  exact integral_add (g.chart_integrable i hf hsf) (g.chart_integrable i hh hsh)
+
+lemma chartIntegral_sum [T2Space X] [CompactSpace X] (g : KaehlerMetric A)
+    (i : Fin A.count) {ι : Type*} (s : Finset ι) (f : ι → X → ℝ)
+    (hf : ∀ k ∈ s, Continuous (f k)) (hs : ∀ k ∈ s, tsupport (f k) ⊆ (A.chart i).source) :
+    g.chartIntegral i (fun x => ∑ k ∈ s, f k x) = ∑ k ∈ s, g.chartIntegral i (f k) := by
+  simp only [chartIntegral, Finset.sum_mul]
+  exact integral_finsetSum s (fun k hk => g.chart_integrable i (hf k hk) (hs k hk))
+
+def chartPartition [T2Space X] [CompactSpace X] : Fin A.count → SmoothRealFunction A :=
+  A.exists_smooth_partition.choose
+
+lemma chartPartition_nonneg [T2Space X] [CompactSpace X] (i : Fin A.count) (x : X) :
+    0 ≤ (chartPartition (A := A) i).value x := A.exists_smooth_partition.choose_spec.1 i x
+
+lemma chartPartition_sum [T2Space X] [CompactSpace X] (x : X) :
+    ∑ i, (chartPartition (A := A) i).value x = 1 := A.exists_smooth_partition.choose_spec.2.1 x
+
+lemma chartPartition_support [T2Space X] [CompactSpace X] (i : Fin A.count) :
+    tsupport (chartPartition (A := A) i).value ⊆ (A.chart i).source :=
+  A.exists_smooth_partition.choose_spec.2.2 i
+
+ 
+
+def integral [T2Space X] [CompactSpace X] (g : KaehlerMetric A) (f : X → ℝ) : ℝ :=
+  ∑ i, g.chartIntegral i (fun x => (chartPartition i).value x * f x)
+
+lemma integral_chart [T2Space X] [CompactSpace X] (g : KaehlerMetric A)
+    (j : Fin A.count) {f : X → ℝ} (hf : Continuous f)
+    (hs : tsupport f ⊆ (A.chart j).source) : g.integral f = g.chartIntegral j f := by
+  unfold integral
+  have he : ∀ i, g.chartIntegral i (fun x => (chartPartition i).value x * f x) =
+      g.chartIntegral j (fun x => (chartPartition i).value x * f x) := by
+    intro i
+    apply g.chartIntegral_eq
+    · exact fun x hx => chartPartition_support i
+        (subset_tsupport _ (Function.support_mul_subset_left _ _ hx))
+    · exact fun x hx => hs (subset_tsupport _ (Function.support_mul_subset_right _ _ hx))
+  simp_rw [he]
+  rw [← g.chartIntegral_sum j Finset.univ (fun i x => (chartPartition i).value x * f x)
+    (fun i _ => (chartPartition i).continuous.fun_mul hf)
+    (fun i _ => tsupport_mul_subset_right.trans hs)]
+  congr 1
+  funext x
+  rw [← Finset.sum_mul, chartPartition_sum, one_mul]
+
+lemma integral_add [T2Space X] [CompactSpace X] (g : KaehlerMetric A)
+    {f h : X → ℝ} (hf : Continuous f) (hh : Continuous h) :
+    g.integral (fun x => f x + h x) = g.integral f + g.integral h := by
+  unfold integral
+  rw [← Finset.sum_add_distrib]
+  apply Finset.sum_congr rfl
+  intro i _
+  simp_rw [mul_add]
+  exact g.chartIntegral_linear_add i ((chartPartition i).continuous.mul hf)
+    ((chartPartition i).continuous.mul hh)
+    (tsupport_mul_subset_left.trans (chartPartition_support i))
+    (tsupport_mul_subset_left.trans (chartPartition_support i))
+
+lemma integral_const_mul [T2Space X] [CompactSpace X] (g : KaehlerMetric A)
+    (c : ℝ) (f : X → ℝ) : g.integral (fun x => c * f x) = c * g.integral f := by
+  simp only [integral, chartIntegral, Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro i _
+  simp_rw [show ∀ z, (chartPartition i).value ((A.chart i).symm z) * (c * f ((A.chart i).symm z)) *
+      g.volumeCoefficient i z = c * ((chartPartition i).value ((A.chart i).symm z) *
+      f ((A.chart i).symm z) * g.volumeCoefficient i z) by intro z; ring]
+  exact MeasureTheory.integral_const_mul c _
+
+lemma integral_sum [T2Space X] [CompactSpace X] (g : KaehlerMetric A)
+    {ι : Type*} (s : Finset ι) (f : ι → X → ℝ) (hf : ∀ k ∈ s, Continuous (f k)) :
+    g.integral (fun x => ∑ k ∈ s, f k x) = ∑ k ∈ s, g.integral (f k) := by
+  simp only [integral, Finset.mul_sum]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro i _
+  exact g.chartIntegral_sum i s _ (fun k hk => (chartPartition i).continuous.mul (hf k hk))
+    (fun k _ => tsupport_mul_subset_left.trans (chartPartition_support i))
+
+lemma integral_nonneg [T2Space X] [CompactSpace X] (g : KaehlerMetric A)
+    {f : X → ℝ} (hf : ∀ x, 0 ≤ f x) : 0 ≤ g.integral f := by
+  apply Finset.sum_nonneg
+  intro i _
+  apply setIntegral_nonneg (A.chart i).open_target.measurableSet
+  intro z hz
+  exact mul_nonneg (mul_nonneg (chartPartition_nonneg i _) (hf _))
+    (g.volumeCoefficient_pos i hz).le
+
+end KaehlerMetric
+end Anticanonical.SourceSmooth
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/GlobalSmoothCompact.lean b/lean/OAI/Geometry/PrescribedRicci/GlobalSmoothCompact.lean
new file mode 100644
index 0000000000000000000000000000000000000000..8b386e1903c14361f482e99265c61e14e90eac7e
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/GlobalSmoothCompact.lean
@@ -0,0 +1,114 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.GlobalRellich
+import OAI.Geometry.PrescribedPotential.PatchCutoffs
+import OAI.Geometry.PrescribedRicci.SobolevInterpolation
+
+namespace OAI
+
+/-! Global Smooth Compact. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology
+open scoped ContDiff SchwartzMap Classical
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace Localizers
+variable (D : Localizers A ι)
+lemma sobolev_interpolation (r s t : ℝ) (hst : s < t) {ε : ℝ} (hε : 0 < ε) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ f : Smooth A,
+      ‖D.embed s f‖ ≤ ε*‖D.embed t f‖+C*‖D.embed r f‖ := by
+  obtain ⟨C,hC,hc⟩ := schwartz_sobolev_interpolation (E:=EC d) r s t hst hε
+  refine ⟨C,hC,fun f => ?_⟩
+  change ‖D.coordinates s f‖ ≤ ε*‖D.coordinates t f‖+C*‖D.coordinates r f‖
+  apply (pi_norm_le_iff_of_nonneg (by positivity)).mpr
+  intro p
+  exact (hc (localize A (D.index p) (D.weight p) (D.support_sub p) f)).trans
+    (add_le_add (mul_le_mul_of_nonneg_left (norm_le_pi_norm (D.coordinates t f) p) hε.le)
+      (mul_le_mul_of_nonneg_left (norm_le_pi_norm (D.coordinates r f) p) hC))
+
+lemma cauchy_of_low (f : ℕ → Smooth A) (s t : ℝ) (hst : s < t)
+    (hbound : ∃ B : ℝ, 0 ≤ B ∧ ∀ n, ‖D.embed t (f n)‖ ≤ B)
+    (hc : CauchySeq (fun n => D.embed 0 (f n))) :
+    CauchySeq (fun n => D.embed s (f n)) := by
+  obtain ⟨B,hB,hb⟩ := hbound
+  apply Metric.cauchySeq_iff.mpr
+  intro ε hε
+  let η := ε/(4*(B+1))
+  have hη : 0 < η := div_pos hε (by positivity)
+  obtain ⟨C,hC,hI⟩ := D.sobolev_interpolation 0 s t hst hη
+  obtain ⟨N,hN⟩ := Metric.cauchySeq_iff.mp hc (ε/(4*(C+1))) (div_pos hε (by positivity))
+  refine ⟨N,fun m hm n hn => ?_⟩
+  have hhigh : ‖D.embed t (f m-f n)‖ ≤ 2*B := by
+    rw [map_sub]
+    exact (norm_sub_le _ _).trans (by linarith [hb m,hb n])
+  have hlow : ‖D.embed 0 (f m-f n)‖ < ε/(4*(C+1)) := by
+    simpa only [map_sub,dist_eq_norm] using hN m hm n hn
+  rw [dist_eq_norm,← map_sub]
+  have h1 : η*(2*B) < ε/2 := by
+    dsimp [η]
+    rw [div_mul_eq_mul_div]
+    apply (div_lt_iff₀ (by positivity : 0 < 4*(B+1))).mpr
+    nlinarith
+  have h2 : C*(ε/(4*(C+1))) < ε/2 := by
+    rw [← mul_div_assoc]
+    apply (div_lt_iff₀ (by positivity : 0 < 4*(C+1))).mpr
+    nlinarith
+  calc
+    _ ≤ η*‖D.embed t (f m-f n)‖+C*‖D.embed 0 (f m-f n)‖ := hI _
+    _ ≤ η*(2*B)+C*(ε/(4*(C+1))) := add_le_add
+      (mul_le_mul_of_nonneg_left hhigh hη.le) (mul_le_mul_of_nonneg_left hlow.le hC)
+    _ < ε := by linarith
+end Localizers
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+theorem smooth_limit_of_all_bounds (f : ℕ → Smooth A)
+    (hb : ∀ k : ℕ, ∃ B : ℝ, 0 ≤ B ∧ ∀ n, ‖D.localizers.embed (k:ℝ) (f n)‖ ≤ B)
+    (u : D.localizers.Sobolev 0)
+    (hu : Tendsto (fun n => D.localizers.embed 0 (f n)) atTop (𝓝 u)) :
+    ∃ F : Smooth A, ∀ k : ℕ, Tendsto (fun n => D.localizers.embed (k:ℝ) (f n))
+      atTop (𝓝 (D.localizers.embed (k:ℝ) F)) := by
+  have hc (k : ℕ) : CauchySeq (fun n => D.localizers.embed (k:ℝ) (f n)) :=
+    D.localizers.cauchy_of_low f (k:ℝ) ((k+1:ℕ):ℝ) (by exact_mod_cast Nat.lt_succ_self k)
+      (hb (k+1)) hu.cauchySeq
+  choose v hv using fun k => cauchySeq_tendsto_of_complete (hc k)
+  have hlow (k : ℕ) : D.localizers.lower (k:ℝ) 0 (v k) = u := by
+    have hh := ((D.localizers.lower (k:ℝ) 0).continuous.tendsto (v k)).comp (hv k)
+    simp only [Function.comp_def,D.localizers.lower_embed (Nat.cast_nonneg k)] at hh
+    exact tendsto_nhds_unique hh hu
+  obtain ⟨F,hF⟩ := D.allRegular_smooth (show D.localizers.AllRegular u from fun k => ⟨v k,hlow k⟩)
+  refine ⟨F,fun k => ?_⟩
+  have he : v k = D.localizers.embed (k:ℝ) F := by
+    apply D.localizers.lower_injective (Nat.cast_nonneg k)
+    rw [hlow,D.localizers.lower_embed (Nat.cast_nonneg k),hF]
+  exact he ▸ hv k
+
+theorem smooth_subsequence (f : ℕ → Smooth A)
+    (hb : ∀ k : ℕ, ∃ B : ℝ, 0 ≤ B ∧ ∀ n, ‖D.localizers.embed (k:ℝ) (f n)‖ ≤ B) :
+    ∃ F : Smooth A, ∃ ν : ℕ → ℕ, StrictMono ν ∧ ∀ k : ℕ,
+      Tendsto (fun n => D.localizers.embed (k:ℝ) (f (ν n))) atTop
+        (𝓝 (D.localizers.embed (k:ℝ) F)) := by
+  obtain ⟨B,hB,hb2⟩ := hb 2
+  let K := closure (D.localizers.lower 2 0 '' Metric.closedBall 0 B)
+  have hK : IsCompact K := D.lower_compact.isCompact_closure_image_closedBall B
+  have hfK (n : ℕ) : D.localizers.embed 0 (f n) ∈ K := by
+    apply subset_closure
+    refine ⟨D.localizers.embed 2 (f n),?_,D.localizers.lower_embed (by norm_num) (f n)⟩
+    simpa only [Metric.mem_closedBall,dist_zero_right,Nat.cast_ofNat] using hb2 n
+  obtain ⟨u,_,ν,hν,hu⟩ := hK.tendsto_subseq hfK
+  obtain ⟨F,hF⟩ := D.smooth_limit_of_all_bounds (f ∘ ν)
+    (fun k => by obtain ⟨B,hB,hb⟩ := hb k; exact ⟨B,hB,fun n => hb (ν n)⟩) u hu
+  exact ⟨F,ν,hν,hF⟩
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/HermitianCoercivity.lean b/lean/OAI/Geometry/PrescribedRicci/HermitianCoercivity.lean
new file mode 100644
index 0000000000000000000000000000000000000000..f49a08ba6d2760cc76dda7eb014fce541cf1490e
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/HermitianCoercivity.lean
@@ -0,0 +1,67 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.HermitianPair
+
+namespace OAI
+
+/-! Hermitian Coercivity. -/
+
+noncomputable section
+open Matrix
+open scoped ComplexOrder MatrixOrder Matrix.Norms.Elementwise
+namespace MongeAmpere
+variable {n : Type*} [Fintype n] [DecidableEq n]
+
+lemma hermPair_single (v : n → ℂ) (i : n) :
+    hermPair (1 : Matrix n n ℂ) (Pi.single i 1) v = v i := by
+  simp [hermPair,dotProduct,Pi.single_apply]
+
+lemma hermPair_dual_coordinate {K : Matrix n n ℂ} (hK : K.PosDef) (v : n → ℂ) (i : n) :
+    hermPair K (K⁻¹*ᵥ Pi.single i 1) v = v i := by
+  rw [hermPair_mulVec_left,hK.inv.isHermitian.eq,Matrix.nonsing_inv_mul _
+    (isUnit_iff_ne_zero.mpr hK.det_pos.ne'),hermPair_single]
+
+lemma component_sq_le_hermPair {K : Matrix n n ℂ} (hK : K.PosDef) (v : n → ℂ) (i : n) :
+    ‖v i‖^2 ≤ (K⁻¹ i i).re*(hermPair K v v).re := by
+  have he := hermPair_cauchySchwarz K hK.posSemidef (K⁻¹*ᵥ Pi.single i 1) v
+  rw [hermPair_dual_coordinate hK,hermPair_dual_coordinate hK] at he
+  simpa [Matrix.mulVec,dotProduct,Pi.single_apply] using he
+
+lemma norm_sq_le_hermPair {K : Matrix n n ℂ} (hK : K.PosDef) (v : n → ℂ) :
+    ‖v‖^2 ≤ ‖K⁻¹‖*(hermPair K v v).re := by
+  have hs := hermPair_nonneg hK.posSemidef v
+  have hprod : 0 ≤ ‖K⁻¹‖*(hermPair K v v).re := mul_nonneg (norm_nonneg _) hs
+  have hc (i : n) : ‖v i‖^2 ≤ ‖K⁻¹‖*(hermPair K v v).re := by
+    apply (component_sq_le_hermPair hK v i).trans
+    exact mul_le_mul_of_nonneg_right ((Complex.re_le_norm _).trans
+      (norm_entry_le_entrywise_sup_norm (K⁻¹))) hs
+  have hv : ‖v‖ ≤ Real.sqrt (‖K⁻¹‖*(hermPair K v v).re) := by
+    apply (pi_norm_le_iff_of_nonneg (Real.sqrt_nonneg _)).mpr
+    intro i
+    exact (Real.le_sqrt (norm_nonneg _) hprod).mpr (hc i)
+  have hh := sq_le_sq₀ (norm_nonneg v) (Real.sqrt_nonneg (‖K⁻¹‖*(hermPair K v v).re))
+  exact (hh.mpr hv).trans_eq (Real.sq_sqrt hprod)
+
+omit [DecidableEq n] in
+lemma pair_norm_le (K : Matrix n n ℂ) (v w : n → ℂ) :
+    ‖hermPair K v w‖ ≤ (Fintype.card n : ℝ)^2*‖K‖*‖v‖*‖w‖ := by
+  rw [hermPair_expand]
+  apply (norm_sum_le _ _).trans
+  calc
+    ∑ i, ‖∑ j, star (v i)*K i j*w j‖ ≤
+        ∑ i : n, ∑ j : n, ‖v‖*‖K‖*‖w‖ := by
+      apply Finset.sum_le_sum
+      intro i _
+      apply (norm_sum_le _ _).trans
+      apply Finset.sum_le_sum
+      intro j _
+      simp only [norm_mul,norm_star]
+      exact mul_le_mul (mul_le_mul (norm_le_pi_norm v i)
+        (norm_entry_le_entrywise_sup_norm K) (norm_nonneg _) (norm_nonneg _))
+        (norm_le_pi_norm w j) (norm_nonneg _) (mul_nonneg (norm_nonneg _) (norm_nonneg _))
+    _ = _ := by simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul]; ring
+
+end MongeAmpere
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/HermitianPair.lean b/lean/OAI/Geometry/PrescribedRicci/HermitianPair.lean
new file mode 100644
index 0000000000000000000000000000000000000000..a84cbad79255273c456fff6fef16c52e0f55759f
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/HermitianPair.lean
@@ -0,0 +1,72 @@
+import Mathlib
+
+namespace OAI
+
+/-! Hermitian Pair. -/
+
+noncomputable section
+open Matrix
+open scoped ComplexOrder MatrixOrder
+namespace MongeAmpere
+variable {n : Type*} [Fintype n]
+
+def hermPair (K : Matrix n n ℂ) (v w : n → ℂ) : ℂ := star v ⬝ᵥ (K *ᵥ w)
+
+lemma hermPair_expand (K : Matrix n n ℂ) (v w : n → ℂ) :
+    hermPair K v w = ∑ i, ∑ j, star (v i)*K i j*w j := by
+  simp only [hermPair,dotProduct,Matrix.mulVec,Finset.mul_sum,mul_assoc,Pi.star_apply]
+
+lemma hermPair_add_left (K : Matrix n n ℂ) (u v w : n → ℂ) :
+    hermPair K (u+v) w = hermPair K u w+hermPair K v w := by
+  simp only [hermPair,star_add,add_dotProduct]
+
+lemma hermPair_add_right (K : Matrix n n ℂ) (u v w : n → ℂ) :
+    hermPair K u (v+w) = hermPair K u v+hermPair K u w := by
+  simp only [hermPair,Matrix.mulVec_add,dotProduct_add]
+
+lemma hermPair_smul_left (K : Matrix n n ℂ) (u v : n → ℂ) (c : ℂ) :
+    hermPair K (c • u) v = star c*hermPair K u v := by
+  simp only [hermPair,star_smul,smul_dotProduct,smul_eq_mul]
+
+lemma hermPair_smul_right (K : Matrix n n ℂ) (u v : n → ℂ) (c : ℂ) :
+    hermPair K u (c • v) = c*hermPair K u v := by
+  simp only [hermPair,Matrix.mulVec_smul,dotProduct_smul,smul_eq_mul]
+
+lemma hermPair_conj (K : Matrix n n ℂ) (hK : K.IsHermitian) (u v : n → ℂ) :
+    star (hermPair K u v) = hermPair K v u := by
+  simp only [hermPair]
+  rw [star_dotProduct,star_star,star_mulVec,hK.eq]
+  exact (dotProduct_mulVec (star v) K u).symm
+
+lemma hermPair_mulVec_left (K C : Matrix n n ℂ) (u v : n → ℂ) :
+    hermPair K (C *ᵥ u) v = hermPair (Cᴴ*K) u v := by
+  simp only [hermPair,star_mulVec,← dotProduct_mulVec,Matrix.mulVec_mulVec]
+
+lemma hermPair_mulVec_right (K C : Matrix n n ℂ) (u v : n → ℂ) :
+    hermPair K u (C *ᵥ v) = hermPair (K*C) u v := by
+  simp only [hermPair,Matrix.mulVec_mulVec]
+
+lemma hermPair_nonneg {K : Matrix n n ℂ} (hK : K.PosSemidef) (u : n → ℂ) :
+    0 ≤ (hermPair K u u).re := hK.re_dotProduct_nonneg u
+
+@[instance_reducible] def hermPairCore (K : Matrix n n ℂ) (hK : K.PosSemidef) :
+    PreInnerProductSpace.Core ℂ (n → ℂ) where
+  inner := hermPair K
+  conj_inner_symm x y := hermPair_conj K hK.isHermitian y x
+  re_inner_nonneg := hermPair_nonneg hK
+  add_left := hermPair_add_left K
+  smul_left x y c := hermPair_smul_left K x y c
+
+lemma hermPair_cauchySchwarz (K : Matrix n n ℂ) (hK : K.PosSemidef) (u v : n → ℂ) :
+    ‖hermPair K u v‖^2 ≤ (hermPair K u u).re*(hermPair K v v).re := by
+  let := hermPairCore K hK
+  have h := InnerProductSpace.Core.inner_mul_inner_self_le (𝕜:=ℂ) u v
+  change ‖hermPair K u v‖*‖hermPair K v u‖ ≤ (hermPair K u u).re*(hermPair K v v).re at h
+  rw [← hermPair_conj K hK.isHermitian u v,norm_star] at h
+  simpa only [pow_two] using h
+
+end MongeAmpere
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/HermitianTensorEnergy.lean b/lean/OAI/Geometry/PrescribedRicci/HermitianTensorEnergy.lean
new file mode 100644
index 0000000000000000000000000000000000000000..a5fd610221b6db545bfed29390820e0ac66fc4d0
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/HermitianTensorEnergy.lean
@@ -0,0 +1,95 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.HermitianPair
+
+namespace OAI
+
+/-! Hermitian Tensor Energy. -/
+
+noncomputable section
+open Matrix
+open scoped ComplexOrder MatrixOrder Kronecker
+namespace MongeAmpere
+variable {n m : Type*} [Fintype n] [Fintype m]
+
+lemma hermPair_sum_left {ι : Type*} (s : Finset ι) (K : Matrix n n ℂ) (V : ι → n → ℂ) (w : n → ℂ) :
+    hermPair K (∑ i ∈ s, V i) w = ∑ i ∈ s, hermPair K (V i) w := by
+  classical
+  induction s using Finset.induction_on with
+  | empty => simp [hermPair]
+  | @insert i s hi ih => simp only [Finset.sum_insert hi,hermPair_add_left,ih]
+
+lemma hermPair_sum_right {ι : Type*} (s : Finset ι) (K : Matrix n n ℂ) (v : n → ℂ) (W : ι → n → ℂ) :
+    hermPair K v (∑ i ∈ s, W i) = ∑ i ∈ s, hermPair K v (W i) := by
+  classical
+  induction s using Finset.induction_on with
+  | empty => simp [hermPair]
+  | @insert i s hi ih => simp only [Finset.sum_insert hi,hermPair_add_right,ih]
+
+def tensorEnergy (B : Matrix m m ℂ) (K : Matrix n n ℂ) (V : m → n → ℂ) : ℂ :=
+  ∑ a, ∑ b, B a b*hermPair K (V a) (V b)
+
+lemma tensorEnergy_eq (B : Matrix m m ℂ) (K : Matrix n n ℂ) (V : m → n → ℂ) :
+    tensorEnergy B K V = hermPair (B ⊗ₖ K) (fun i => V i.1 i.2) (fun i => V i.1 i.2) := by
+  simp only [tensorEnergy,hermPair_expand,Fintype.sum_prod_type,Matrix.kronecker_apply,
+    Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro a _
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro i _
+  apply Finset.sum_congr rfl
+  intro b _
+  apply Finset.sum_congr rfl
+  intro j _
+  ring
+
+lemma tensorEnergy_nonneg {B : Matrix m m ℂ} {K : Matrix n n ℂ}
+    (hB : B.PosSemidef) (hK : K.PosSemidef) (V : m → n → ℂ) :
+    0 ≤ (tensorEnergy B K V).re := by
+  classical
+  rw [tensorEnergy_eq]
+  exact hermPair_nonneg (hB.kronecker hK) _
+
+lemma hermPair_weighted_rough_left (B : Matrix m m ℂ) (hB : B.IsHermitian)
+    (K : Matrix n n ℂ) (P : m → m → n → ℂ) (f : n → ℂ) :
+    ∑ a, ∑ b, B a b*hermPair K (P b a) f =
+      hermPair K (∑ a, ∑ b, B a b • P a b) f := by
+  classical
+  simp only [hermPair_sum_left,hermPair_smul_left]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro a _
+  apply Finset.sum_congr rfl
+  intro b _
+  rw [show star (B a b) = B b a from Matrix.IsHermitian.ext_iff.mp hB b a]
+
+lemma hermPair_weighted_rough_right (B : Matrix m m ℂ) (K : Matrix n n ℂ)
+    (P : m → m → n → ℂ) (f : n → ℂ) :
+    ∑ a, ∑ b, B a b*hermPair K f (P a b) =
+      hermPair K f (∑ a, ∑ b, B a b • P a b) := by
+  classical
+  simp only [hermPair_sum_right,hermPair_smul_right]
+
+lemma hermPair_weighted_curvature (B : Matrix m m ℂ) (K : Matrix n n ℂ)
+    (R : m → m → Matrix n n ℂ) (f : n → ℂ) :
+    ∑ a, ∑ b, B a b*hermPair K f (R a b*ᵥ f) =
+      hermPair K f ((∑ a, ∑ b, B a b • R a b)*ᵥ f) := by
+  classical
+  simp only [Matrix.sum_mulVec,Matrix.smul_mulVec,hermPair_sum_right,hermPair_smul_right]
+
+lemma tensorEnergy_flip (B : Matrix m m ℂ) (K : Matrix n n ℂ) (V : m → n → ℂ) :
+    ∑ a, ∑ b, B a b*hermPair K (V b) (V a) = tensorEnergy B.transpose K V := by
+  rw [Finset.sum_comm]
+  rfl
+
+lemma hermPair_two_re (K : Matrix n n ℂ) (hK : K.IsHermitian) (v w : n → ℂ) :
+    (hermPair K v w+hermPair K w v).re = 2*(hermPair K w v).re := by
+  rw [← hermPair_conj K hK w v]
+  simp only [Complex.add_re,Complex.star_def,Complex.conj_re]
+  ring
+
+end MongeAmpere
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/HolderFiniteProduct.lean b/lean/OAI/Geometry/PrescribedRicci/HolderFiniteProduct.lean
new file mode 100644
index 0000000000000000000000000000000000000000..502fc5ccf902abdc25e9ceffa5a0a6564536aec1
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/HolderFiniteProduct.lean
@@ -0,0 +1,50 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.InterpolationExponents
+import OAI.Geometry.PrescribedRicci.LpPowerBounds
+
+namespace OAI
+
+/-! Holder Finite Product. -/
+
+section
+
+ 
+noncomputable section
+open MeasureTheory
+open scoped ENNReal BigOperators Classical
+namespace TameInterpolation
+variable {Y : Type*} [MeasurableSpace Y] {μ : Measure Y}
+
+lemma lpNorm_finite_prod {ι : Type*} (s : Finset ι) (f : ι → Y → ℝ) (p : ι → ℝ≥0∞)
+    (hf : ∀ i ∈ s, MemLp (f i) (p i) μ) :
+    lpNorm (fun x => ∏ i ∈ s, f i x) (∑ i ∈ s, (p i)⁻¹)⁻¹ μ ≤
+      ∏ i ∈ s, lpNorm (f i) (p i) μ := by
+  induction s using Finset.cons_induction with
+  | empty =>
+    by_cases hμ : μ = 0
+    · simp [hμ]
+    · simp [lpNorm_const, hμ]
+  | cons i s hi ih =>
+    have hfi : MemLp (f i) (p i) μ := hf i (Finset.mem_cons_self ..)
+    have hfs : ∀ a ∈ s, MemLp (f a) (p a) μ := fun a ha => hf a (Finset.mem_cons_of_mem ha)
+    have hprod := MemLp.fun_prod hfs
+    let : ENNReal.HolderTriple (p i) (∑ a ∈ s, (p a)⁻¹)⁻¹ (∑ a ∈ Finset.cons i s hi, (p a)⁻¹)⁻¹ :=
+      ⟨by simp only [inv_inv,Finset.sum_cons]⟩
+    simp only [Finset.prod_cons]
+    exact (lpNorm_mul_le hfi hprod).trans (mul_le_mul_of_nonneg_left (ih hfs) lpNorm_nonneg)
+
+lemma exponent_sum_inverse {ι : Type*} (s : Finset ι) (j : ι → ℕ) (m : ℕ)
+    (hm : 0 < m) (hs : ∑ i ∈ s, j i = m) :
+    (∑ i ∈ s, (exponent m (j i))⁻¹)⁻¹ = 2 := by
+  simp only [exponent,inv_inv]
+  rw [← ENNReal.ofReal_sum_of_nonneg (fun i _ => by positivity),← Finset.sum_div,← Nat.cast_sum,hs]
+  have hm' : (m:ℝ) ≠ 0 := by exact_mod_cast (Nat.ne_of_gt hm)
+  have he : (m:ℝ)/(2*(m:ℝ)) = 1/2 := by field_simp
+  rw [he,← ENNReal.ofReal_inv_of_pos (by norm_num : (0:ℝ) < 1/2)]
+  norm_num
+end TameInterpolation
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/IntegratedKahlerEnergy.lean b/lean/OAI/Geometry/PrescribedRicci/IntegratedKahlerEnergy.lean
new file mode 100644
index 0000000000000000000000000000000000000000..7da506c85ef533197fe866eaa4614e4d1db72829
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/IntegratedKahlerEnergy.lean
@@ -0,0 +1,154 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.GlobalKahlerEnergy
+import OAI.Geometry.PrescribedRicci.KahlerGradientEnergy
+import OAI.Geometry.PrescribedRicci.KahlerIntegration
+
+namespace OAI
+
+/-! Integrated Kahler Energy. -/
+
+section
+
+ 
+
+noncomputable section
+open Matrix Filter Set Topology MeasureTheory
+open scoped ContDiff ComplexOrder Classical
+namespace Anticanonical.SourceSmooth
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+namespace KaehlerMetric
+
+lemma determinant_eq_volume (g : KaehlerMetric A) (i : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart i).target) :
+    (g.matrix i z).det = (g.volumeCoefficient i z : ℂ) := by
+  apply Complex.ext
+  · rfl
+  · exact (Complex.pos_iff.mp (g.positive i z hz).det_pos).2.symm
+
+lemma holRealDeriv_zero_off_support {q : Coordinates d → ℝ} {z : Coordinates d}
+    (hz : z ∉ tsupport q) : holRealDeriv q z = 0 := by
+  have hd : fderiv ℝ q z = 0 := image_eq_zero_of_notMem_tsupport
+    (fun h => hz (tsupport_fderiv_subset ℝ h))
+  funext j
+  simp only [holRealDeriv, hd, map_zero, Pi.zero_apply]
+
+lemma integral_real_cofactor (g : KaehlerMetric A) (φ : SmoothRealFunction A)
+    (i : Fin A.count) {q : Coordinates d → ℝ} (hq : ContDiff ℝ ∞ q)
+    (hc : HasCompactSupport q) (hs : tsupport q ⊆ (A.chart i).target) :
+    (∫ z, q z * g.volumeCoefficient i z * ((g.matrix i z)⁻¹ * φ.hessian i z).trace.re) =
+      -(∫ z, g.volumeCoefficient i z *
+        (gradientPair (g.matrix i z) (holRealDeriv q z) (holRealDeriv (φ.localExpression i) z)).re) := by
+  let qc : Coordinates d → ℂ := fun z => (q z : ℂ)
+  have hqc : ContDiff ℝ ∞ qc := Complex.ofRealCLM.contDiff.comp hq
+  have hcc : HasCompactSupport qc := hc.comp_left Complex.ofReal_zero
+  have hsc : tsupport qc ⊆ (A.chart i).target := (tsupport_comp_subset Complex.ofReal_zero q).trans hs
+  have hd (j : Fin d) : ∀ z ∈ tsupport qc,
+      ContDiffAt ℝ ∞ (fun y => g.cofactorFlux φ i y j) z :=
+    fun z hz => g.contDiffAt_cofactorFlux φ i (hsc hz) j
+  let F := fun z => qc z * ((g.matrix i z).det * ((g.matrix i z)⁻¹ * φ.hessian i z).trace)
+  let G := fun z => ∑ j, barDeriv qc z j * g.cofactorFlux φ i z j
+  have hF : Integrable F := by
+    have hh := integrable_finsetSum Finset.univ (fun j _ => integrable_mul_barDeriv_local hqc (hd j) hcc j)
+    apply hh.congr (Filter.Eventually.of_forall _)
+    intro z
+    simp only [← Finset.mul_sum]
+    by_cases hz : z ∈ tsupport qc
+    · rw [g.cofactorFlux_divergence φ i (hsc hz)]
+    · simp only [F, image_eq_zero_of_notMem_tsupport hz, zero_mul]
+  have hG : Integrable G := integrable_finsetSum Finset.univ
+    (fun j _ => integrable_barDeriv_mul_local hqc (hd j) hcc j)
+  have he := congrArg Complex.re (g.integral_cofactorFlux φ i hqc hcc hsc)
+  have hFe : ∀ z, (F z).re = q z * g.volumeCoefficient i z *
+      ((g.matrix i z)⁻¹ * φ.hessian i z).trace.re := by
+    intro z
+    by_cases hz : z ∈ tsupport q
+    · simp only [F, qc, g.determinant_eq_volume i (hs hz), Complex.mul_re,
+        Complex.ofReal_re, Complex.ofReal_im, zero_mul, sub_zero, Complex.mul_im,
+        add_zero]
+      ring
+    · simp only [F, qc, image_eq_zero_of_notMem_tsupport hz, Complex.ofReal_zero,
+        zero_mul, Complex.zero_re]
+  have hGe : ∀ z, (G z).re = g.volumeCoefficient i z *
+      (gradientPair (g.matrix i z) (holRealDeriv q z) (holRealDeriv (φ.localExpression i) z)).re := by
+    intro z
+    have hb : ∀ j, barDeriv qc z j = star (holRealDeriv q z j) :=
+      fun j => barDeriv_ofReal (hq.differentiable (by simp) z) j
+    have hc' : G z = (g.matrix i z).det *
+        gradientPair (g.matrix i z) (holRealDeriv q z) (holRealDeriv (φ.localExpression i) z) := by
+      simp only [G, cofactorFlux, hb, MongeAmpere.weightedInverse, Matrix.smul_apply,
+        smul_eq_mul, gradientPair, Finset.mul_sum]
+      rw [Finset.sum_comm]
+      apply Finset.sum_congr rfl
+      intro k _
+      apply Finset.sum_congr rfl
+      intro j _
+      ring
+    rw [hc']
+    by_cases hz : z ∈ tsupport q
+    · rw [g.determinant_eq_volume i (hs hz)]
+      simp only [Complex.mul_re, Complex.ofReal_re, Complex.ofReal_im, zero_mul, sub_zero]
+    · rw [holRealDeriv_zero_off_support hz]
+      simp only [gradientPair, Pi.zero_apply, star_zero, mul_zero, zero_mul,
+        Finset.sum_const_zero, Complex.zero_re]
+  change (∫ z, F z).re = (-(∫ z, G z)).re at he
+  have hrF := Complex.reCLM.integral_comp_comm hF
+  have hrG := Complex.reCLM.integral_comp_comm hG
+  change (∫ z, (F z).re) = (∫ z, F z).re at hrF
+  change (∫ z, (G z).re) = (∫ z, G z).re at hrG
+  rw [← hrF, Complex.neg_re, ← hrG] at he
+  simpa only [hFe, hGe] using he
+
+lemma chart_energy_identity [T2Space X] [CompactSpace X] (g : KaehlerMetric A)
+    (ψ φ : SmoothRealFunction A) (i : Fin A.count)
+    (hs : tsupport ψ.value ⊆ (A.chart i).source) :
+    g.chartIntegral i (fun x => ψ.value x * (g.laplacian φ).value x) =
+      -g.chartIntegral i (g.energy ψ φ).value := by
+  obtain ⟨hq, hc, ht⟩ := ψ.localized_smooth_compact i hs
+  have he := g.integral_real_cofactor φ i hq hc ht
+  have hl : (fun z => ψ.localized i z * g.volumeCoefficient i z *
+      ((g.matrix i z)⁻¹ * φ.hessian i z).trace.re) =
+      (A.chart i).target.indicator (fun z =>
+        ψ.value ((A.chart i).symm z) * (g.laplacian φ).value ((A.chart i).symm z) *
+          g.volumeCoefficient i z) := by
+    funext z
+    by_cases hz : z ∈ (A.chart i).target
+    · rw [Set.indicator_of_mem hz, ψ.localized_eq hz]
+      have hp := g.laplacian_localExpression φ i hz
+      change (g.laplacian φ).value ((A.chart i).symm z) = _ at hp
+      rw [hp]
+      change _ * _ * _ = _ * ((g.matrix i z)⁻¹ * φ.hessian i z).trace.re * _
+      dsimp only [SmoothRealFunction.localExpression, Function.comp_apply]
+      ring
+    · simp only [SmoothRealFunction.localized, ite_eq_right hz, Set.indicator_of_notMem hz, zero_mul]
+  have hr : (fun z => g.volumeCoefficient i z *
+      (gradientPair (g.matrix i z) (holRealDeriv (ψ.localized i) z)
+        (holRealDeriv (φ.localExpression i) z)).re) =
+      (A.chart i).target.indicator (fun z => (g.energy ψ φ).value ((A.chart i).symm z) *
+        g.volumeCoefficient i z) := by
+    funext z
+    by_cases hz : z ∈ (A.chart i).target
+    · rw [Set.indicator_of_mem hz]
+      have hloc : ψ.localized i =ᶠ[𝓝 z] ψ.localExpression i := by
+        filter_upwards [(A.chart i).open_target.mem_nhds hz] with y hy using ψ.localized_eq hy
+      have hd : holRealDeriv (ψ.localized i) z = holRealDeriv (ψ.localExpression i) z := by
+        funext j
+        unfold holRealDeriv
+        rw [hloc.fderiv_eq]
+      rw [hd]
+      have hh := g.energy_local ψ φ i hz
+      change (g.energy ψ φ).value ((A.chart i).symm z) = _ at hh
+      rw [hh, mul_comm]
+    · rw [Set.indicator_of_notMem hz, holRealDeriv_zero_off_support (fun h => hz (ht h))]
+      simp only [gradientPair, Pi.zero_apply, star_zero, mul_zero, zero_mul,
+        Finset.sum_const_zero, Complex.zero_re]
+  rw [hl, hr, integral_indicator (A.chart i).open_target.measurableSet,
+    integral_indicator (A.chart i).open_target.measurableSet] at he
+  exact he
+
+end KaehlerMetric
+end Anticanonical.SourceSmooth
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/InterpolationExponents.lean b/lean/OAI/Geometry/PrescribedRicci/InterpolationExponents.lean
new file mode 100644
index 0000000000000000000000000000000000000000..1a5d977198f3bea9113a252bb08108550d6d1344
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/InterpolationExponents.lean
@@ -0,0 +1,64 @@
+import Mathlib
+
+namespace OAI
+
+/-! Interpolation Exponents. -/
+
+noncomputable section
+open scoped ENNReal
+namespace TameInterpolation
+
+def exponent (m j : ℕ) : ℝ≥0∞ := (ENNReal.ofReal ((j:ℝ)/(2*(m:ℝ))))⁻¹
+
+lemma exponent_zero (m : ℕ) : exponent m 0 = ∞ := by simp [exponent]
+
+lemma exponent_of_pos {m j : ℕ} (hm : 0 < m) (hj : 0 < j) :
+    exponent m j = ENNReal.ofReal (2*(m:ℝ)/(j:ℝ)) := by
+  have hm' : 0 < (m:ℝ) := by exact_mod_cast hm
+  have hj' : 0 < (j:ℝ) := by exact_mod_cast hj
+  rw [exponent,← ENNReal.ofReal_inv_of_pos (div_pos hj' (by positivity))]
+  congr 1
+  field_simp
+
+lemma holder_inverse (a b : ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b) :
+    ENNReal.HolderTriple (ENNReal.ofReal a)⁻¹ (ENNReal.ofReal b)⁻¹ (ENNReal.ofReal (a+b))⁻¹ := by
+  constructor
+  simp only [inv_inv,ENNReal.ofReal_add ha hb]
+
+lemma exponent_middle_gt {m j : ℕ} (hj : j+1 < m) : 2 < 2*(m:ℝ)/((j+1:ℕ):ℝ) := by
+  have hj' : 0 < ((j+1:ℕ):ℝ) := by positivity
+  apply (lt_div_iff₀ hj').mpr
+  have hh : ((j+1:ℕ):ℝ) < (m:ℝ) := by exact_mod_cast hj
+  linarith only [hh]
+
+lemma exponent_holder_step {m j : ℕ} (hj : j+1 < m) :
+    ENNReal.HolderTriple (exponent m j) (exponent m (j+2))
+      (ENNReal.ofReal (2/(2*(m:ℝ)/((j+1:ℕ):ℝ)) ))⁻¹ := by
+  have hm : 0 < (m:ℝ) := by exact_mod_cast (by omega : 0 < m)
+  have hj0 : (j+1:ℝ) ≠ 0 := by positivity
+  have he : 2/(2*(m:ℝ)/((j+1:ℕ):ℝ)) = (j:ℝ)/(2*(m:ℝ))+((j+2:ℕ):ℝ)/(2*(m:ℝ)) := by
+    push_cast
+    field_simp
+    ring
+  rw [he]
+  exact holder_inverse _ _ (by positivity) (by positivity)
+
+lemma interpolation_holder_conjugate {p : ℝ} (hp : 2 < p) :
+    ENNReal.HolderTriple (ENNReal.ofReal (2/p))⁻¹ (ENNReal.ofReal ((p-2)/p))⁻¹ 1 := by
+  constructor
+  simp only [inv_inv,inv_one,← ENNReal.ofReal_add (by positivity : 0 ≤ 2/p) (by positivity : 0 ≤ (p-2)/p)]
+  rw [show 2/p+(p-2)/p = 1 by field_simp; ring]
+  exact ENNReal.ofReal_one
+
+lemma interpolation_power_exponent {p : ℝ} (hp : 2 < p) :
+    (ENNReal.ofReal ((p-2)/p))⁻¹ * ENNReal.ofReal (p-2) = ENNReal.ofReal p := by
+  have hp0 : 0 < p := by linarith only [hp]
+  have hp2 : 0 < p-2 := by linarith only [hp]
+  rw [← ENNReal.ofReal_inv_of_pos (div_pos hp2 hp0),← ENNReal.ofReal_mul (inv_nonneg.mpr (div_nonneg hp2.le hp0.le))]
+  congr 1
+  field_simp
+end TameInterpolation
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/JetInterpolation.lean b/lean/OAI/Geometry/PrescribedRicci/JetInterpolation.lean
new file mode 100644
index 0000000000000000000000000000000000000000..30c3045e60937d6bb3cca2dec7f79fd0a0c42f91
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/JetInterpolation.lean
@@ -0,0 +1,80 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.DirectionalInterpolation
+
+namespace OAI
+
+/-! Jet Interpolation. -/
+
+section
+
+ 
+noncomputable section
+open Set Filter Topology MeasureTheory
+open scoped ContDiff ENNReal
+namespace TameInterpolation
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+variable {ι : Type*} [Fintype ι] [Nonempty ι]
+
+def jet (e : ι → E) (f : E → ℝ) : {j : ℕ} → (Fin j → ι) → E → ℝ
+  | 0,_ => f
+  | _+1,w => dir (e (w 0)) (jet e f (w ∘ Fin.succ))
+
+omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] [Fintype ι] [Nonempty ι] in
+lemma jet_smooth (e : ι → E) {f : E → ℝ} (hf : ContDiff ℝ ∞ f)
+    {j : ℕ} (w : Fin j → ι) : ContDiff ℝ ∞ (jet e f w) := by
+  induction j with
+  | zero => exact hf
+  | succ j ih => exact dir_smooth _ (ih _)
+
+omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] [Fintype ι] [Nonempty ι] in
+lemma jet_compact (e : ι → E) {f : E → ℝ} (hf : HasCompactSupport f)
+    {j : ℕ} (w : Fin j → ι) : HasCompactSupport (jet e f w) := by
+  induction j with
+  | zero => exact hf
+  | succ j ih => exact dir_compact _ (ih _)
+
+def jetNorm (e : ι → E) (f : E → ℝ) (j : ℕ) (p : ℝ≥0∞) : ℝ :=
+  Finset.univ.sup' Finset.univ_nonempty (fun w : Fin j → ι => lpNorm (jet e f w) p volume)
+
+lemma lpNorm_jet_le (e : ι → E) (f : E → ℝ) {j : ℕ} (w : Fin j → ι) (p : ℝ≥0∞) :
+    lpNorm (jet e f w) p volume ≤ jetNorm e f j p := by
+  exact Finset.le_sup' (fun w : Fin j → ι => lpNorm (jet e f w) p volume) (Finset.mem_univ w)
+
+lemma jetNorm_nonneg (e : ι → E) (f : E → ℝ) (j : ℕ) (p : ℝ≥0∞) :
+    0 ≤ jetNorm e f j p :=
+  le_trans lpNorm_nonneg (lpNorm_jet_le e f (fun _ => Classical.choice ‹Nonempty ι›) p)
+
+lemma jetNorm_attained (e : ι → E) (f : E → ℝ) (j : ℕ) (p : ℝ≥0∞) :
+    ∃ w : Fin j → ι, jetNorm e f j p = lpNorm (jet e f w) p volume := by
+  obtain ⟨w,_,hw⟩ := Finset.exists_mem_eq_sup' (s:=Finset.univ) Finset.univ_nonempty
+    (fun w : Fin j → ι => lpNorm (jet e f w) p volume)
+  exact ⟨w,hw⟩
+
+lemma jetNorm_step (e : ι → E) {f : E → ℝ} (hf : ContDiff ℝ ∞ f) (hc : HasCompactSupport f)
+    (j : ℕ) {p : ℝ} (hp : 2 < p) (α β r s : ℝ≥0∞)
+    [ENNReal.HolderTriple α β s] [ENNReal.HolderTriple s r 1]
+    (hr : r * ENNReal.ofReal (p-2) = ENNReal.ofReal p) :
+    (jetNorm e f (j+1) (ENNReal.ofReal p))^2 ≤
+      (p-1)*jetNorm e f j α*jetNorm e f (j+2) β := by
+  obtain ⟨w,hw⟩ := jetNorm_attained e f (j+1) (ENNReal.ofReal p)
+  rw [hw]
+  let w₀ : Fin j → ι := w ∘ Fin.succ
+  have he : dir (e (w 0)) (dir (e (w 0)) (jet e f w₀)) =
+      jet e f (Fin.cons (w 0) (Fin.cons (w 0) w₀)) := by
+    simp only [jet,Fin.cons_zero,Fin.cons_succ,Function.comp_def]
+  have h := directional_lp_interpolation hp α β r s hr (e (w 0))
+    (jet_smooth e hf w₀) (jet_compact e hc w₀)
+  rw [he] at h
+  apply h.trans
+  apply mul_le_mul
+  · exact mul_le_mul_of_nonneg_left (lpNorm_jet_le e f w₀ α) (by linarith only [hp])
+  · exact lpNorm_jet_le e f _ β
+  · exact lpNorm_nonneg
+  · exact mul_nonneg (by linarith only [hp]) (jetNorm_nonneg e f j α)
+end TameInterpolation
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/JetPolynomial.lean b/lean/OAI/Geometry/PrescribedRicci/JetPolynomial.lean
new file mode 100644
index 0000000000000000000000000000000000000000..1857ce8922dd17bda25c8b2f54afacef91ac5b00
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/JetPolynomial.lean
@@ -0,0 +1,131 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.ComplexJet
+
+namespace OAI
+
+/-! Jet Polynomial. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Finset
+open scoped ContDiff Classical BigOperators
+namespace TameInterpolation
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+variable {ι κ : Type*} [Fintype κ] [DecidableEq κ]
+
+def cword (e : ι → E) (f : E → ℂ) : List ι → E → ℂ
+  | [] => f
+  | a::as => cdir (e a) (cword e f as)
+
+lemma cword_smooth (e : ι → E) {f : E → ℂ} (hf : ContDiff ℝ ∞ f) (as : List ι) :
+    ContDiff ℝ ∞ (cword e f as) := by
+  induction as with
+  | nil => exact hf
+  | cons a as ih => exact cdir_smooth _ ih
+
+lemma cword_compact (e : ι → E) {f : E → ℂ} (hf : HasCompactSupport f) (as : List ι) :
+    HasCompactSupport (cword e f as) := by
+  induction as with
+  | nil => exact hf
+  | cons a as ih => exact cdir_compact _ ih
+
+def monoEval (e : ι → E) (f : κ → E → ℂ) (s : κ → List ι) (x : E) : ℂ :=
+  ∏ i, cword e (f i) (s i) x
+
+def stepState (a : ι) (s : κ → List ι) (i : κ) : κ → List ι :=
+  Function.update s i (a::s i)
+
+def stateOrder (s : κ → List ι) : ℕ := ∑ i, (s i).length
+
+def activeSet (s : κ → List ι) : Finset κ := Finset.univ.filter fun i => s i ≠ []
+
+lemma stateOrder_step (a : ι) (s : κ → List ι) (i : κ) :
+    stateOrder (stepState a s i) = stateOrder s+1 := by
+  unfold stateOrder
+  rw [← Finset.add_sum_erase Finset.univ (fun j => (stepState a s i j).length) (Finset.mem_univ i),
+    ← Finset.add_sum_erase Finset.univ (fun j => (s j).length) (Finset.mem_univ i)]
+  simp only [stepState,Function.update_self,List.length_cons]
+  have he : ∑ j ∈ Finset.univ.erase i, (Function.update s i (a::s i) j).length =
+      ∑ j ∈ Finset.univ.erase i, (s j).length := by
+    apply Finset.sum_congr rfl
+    intro j hj
+    rw [Function.update_of_ne (by simpa using (Finset.mem_erase.mp hj).1)]
+  rw [he]; omega
+
+lemma activeSet_step (a : ι) (s : κ → List ι) (i : κ) :
+    activeSet (stepState a s i) = insert i (activeSet s) := by
+  apply Finset.ext
+  intro j
+  change j ∈ Finset.univ.filter (fun j => stepState a s i j ≠ []) ↔
+    j ∈ insert i (Finset.univ.filter (fun j => s j ≠ []))
+  rw [Finset.mem_filter,Finset.mem_insert,Finset.mem_filter]
+  simp only [Finset.mem_univ,true_and]
+  by_cases hj : j = i
+  · subst j; simp [stepState]
+  · simp [stepState,hj]
+
+omit [DecidableEq κ] in
+lemma monoEval_smooth (e : ι → E) (f : κ → E → ℂ) (hf : ∀ i, ContDiff ℝ ∞ (f i))
+    (s : κ → List ι) : ContDiff ℝ ∞ (monoEval e f s) := by
+  apply contDiff_prod
+  intro i _
+  exact cword_smooth e (hf i) _
+
+lemma cdir_monoEval (e : ι → E) (f : κ → E → ℂ) (hf : ∀ i, ContDiff ℝ ∞ (f i))
+    (a : ι) (s : κ → List ι) :
+    cdir (e a) (monoEval e f s) = fun x => ∑ i, monoEval e f (stepState a s i) x := by
+  funext x
+  unfold cdir monoEval
+  rw [fderiv_finsetProd (fun i _ => (cword_smooth e (hf i) _).differentiable (by simp) x)]
+  simp only [_root_.sum_apply,smul_apply,smul_eq_mul]
+  apply Finset.sum_congr rfl
+  intro i _
+  rw [← Finset.mul_prod_erase Finset.univ
+    (fun j => cword e (f j) (stepState a s i j) x) (Finset.mem_univ i)]
+  simp only [stepState,Function.update_self,cword,cdir]
+  have he : ∏ j ∈ Finset.univ.erase i, cword e (f j) (Function.update s i (a::s i) j) x =
+      ∏ j ∈ Finset.univ.erase i, cword e (f j) (s j) x := by
+    apply Finset.prod_congr rfl
+    intro j hj
+    rw [Function.update_of_ne (by simpa using (Finset.mem_erase.mp hj).1)]
+  rw [he]
+  ring
+
+lemma cdir_finite_sum (e : ι → E) (a : ι) {ν : Type*} (s : Finset ν)
+    (f : ν → E → ℂ) (hf : ∀ i ∈ s, ContDiff ℝ ∞ (f i)) :
+    cdir (e a) (fun x => ∑ i ∈ s, f i x) = fun x => ∑ i ∈ s, cdir (e a) (f i) x := by
+  funext x
+  unfold cdir
+  rw [fderiv_fun_sum (fun i hi => (hf i hi).differentiable (by simp) x)]
+  simp only [_root_.sum_apply]
+
+lemma cdir_list_sum (e : ι → E) (a : ι) {ν : Type*} (s : List ν)
+    (f : ν → E → ℂ) (hf : ∀ i ∈ s, ContDiff ℝ ∞ (f i)) :
+    cdir (e a) (fun x => (s.map (fun i => f i x)).sum) =
+      fun x => (s.map (fun i => cdir (e a) (f i) x)).sum := by
+  induction s with
+  | nil => funext x; simp [cdir]
+  | cons i s ih =>
+    have hsum : ContDiff ℝ ∞ (fun x => (s.map (fun j => f j x)).sum) := by
+      have hh : ∀ j ∈ s, ContDiff ℝ ∞ (f j) := fun j hj => hf j (List.mem_cons_of_mem i hj)
+      clear ih hf
+      induction s with
+      | nil => exact contDiff_const
+      | cons j s ih =>
+        exact (hh j (List.mem_cons_self ..)).add (ih (fun k hk => hh k (List.mem_cons_of_mem j hk)))
+    simp only [List.map_cons,List.sum_cons]
+    funext x
+    unfold cdir
+    rw [fderiv_fun_add ((hf i (List.mem_cons_self ..)).differentiable (by simp) x)
+      (hsum.differentiable (by simp) x),add_apply]
+    exact congrArg (fun z : ℂ => fderiv ℝ (f i) x (e a)+z)
+      (congrFun (ih (fun j hj => hf j (List.mem_cons_of_mem i hj))) x)
+end TameInterpolation
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/JetProductL2.lean b/lean/OAI/Geometry/PrescribedRicci/JetProductL2.lean
new file mode 100644
index 0000000000000000000000000000000000000000..ef63e8b8e4f5cf4dde99f42ba6330b816103c069
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/JetProductL2.lean
@@ -0,0 +1,47 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.FamilyJetNorm
+import OAI.Geometry.PrescribedRicci.HolderFiniteProduct
+
+namespace OAI
+
+/-! Jet Product L2. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+open scoped ContDiff ENNReal Classical BigOperators
+namespace TameInterpolation
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+variable {ι κ : Type*} [Fintype ι] [Nonempty ι] [Fintype κ] [Nonempty κ]
+
+theorem jet_product_L2 (e : ι → E) (f : κ → E → ℝ)
+    (hf : ∀ a, ContDiff ℝ ∞ (f a)) (hc : ∀ a, HasCompactSupport (f a))
+    (m : ℕ) (hm : 0 < m) {ν : Type*} (s : Finset ν)
+    (a : ν → κ) (j : ν → ℕ) (w : (i : ν) → Fin (j i) → ι)
+    (hs : ∑ i ∈ s, j i = m) :
+    lpNorm (fun x => ∏ i ∈ s, jet e (f (a i)) (w i) x) 2 volume ≤
+      (2*(m:ℝ))^(∑ i ∈ s, j i*(m-j i)) *
+        (familyJetNorm e f 0 ∞)^(s.card-1) * familyJetNorm e f m 2 := by
+  have hi (i : ν) : MemLp (jet e (f (a i)) (w i)) (exponent m (j i)) volume :=
+    (jet_smooth e (hf (a i)) (w i)).continuous.memLp_of_hasCompactSupport
+      (jet_compact e (hc (a i)) (w i))
+  have hh := lpNorm_finite_prod (μ:=volume) s
+    (fun i => jet e (f (a i)) (w i)) (fun i => exponent m (j i)) (fun i _ => hi i)
+  rw [exponent_sum_inverse s j m hm hs] at hh
+  apply hh.trans
+  apply le_trans _ (family_jet_tame_many e f hf hc m hm s j hs)
+  apply Finset.prod_le_prod₀
+  · intro i _; exact lpNorm_nonneg
+  · intro i _
+    exact (lpNorm_jet_le e (f (a i)) (w i) (exponent m (j i))).trans
+      (jetNorm_le_family e f (a i) (j i) (exponent m (j i)))
+end TameInterpolation
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/JetRemainder.lean b/lean/OAI/Geometry/PrescribedRicci/JetRemainder.lean
new file mode 100644
index 0000000000000000000000000000000000000000..3d3447e22d5735d6ff10a82c67286ea48e988fe2
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/JetRemainder.lean
@@ -0,0 +1,173 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.JetPolynomial
+
+namespace OAI
+
+/-! Jet Remainder. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Finset
+open scoped ContDiff Classical BigOperators
+namespace TameInterpolation
+variable {ι κ : Type*} [Fintype κ] [DecidableEq κ]
+
+def singleState (ws : List ι) (i : κ) : κ → List ι := Function.update (fun _ => []) i ws
+
+def mixedStep (a : ι) (ws : List ι) : List (κ → List ι) :=
+  Finset.univ.toList.flatMap fun i => (Finset.univ.erase i).toList.map fun j =>
+    stepState a (singleState ws i) j
+
+def monoRemainder : List ι → List (κ → List ι)
+  | [] => []
+  | [_] => []
+  | a::b::ws => (monoRemainder (b::ws)).flatMap
+      (fun s => Finset.univ.toList.map (stepState a s)) ++ mixedStep a (b::ws)
+
+lemma singleState_order (ws : List ι) (i : κ) : stateOrder (singleState ws i) = ws.length := by
+  unfold stateOrder
+  rw [← Finset.add_sum_erase Finset.univ (fun j => (singleState ws i j).length) (Finset.mem_univ i)]
+  simp only [singleState,Function.update_self]
+  have he : (∑ j ∈ Finset.univ.erase i, (Function.update (fun _ : κ => ([]:List ι)) i ws j).length) = 0 := by
+    apply Finset.sum_eq_zero
+    intro j hj
+    rw [Function.update_of_ne (Finset.mem_erase.mp hj).1]
+    rfl
+  rw [he,Nat.add_zero]
+
+lemma singleState_active {ws : List ι} (hw : ws ≠ []) (i : κ) :
+    activeSet (singleState ws i) = {i} := by
+  apply Finset.ext
+  intro j
+  change j ∈ Finset.univ.filter (fun j => singleState ws i j ≠ []) ↔ j ∈ ({i}:Finset κ)
+  rw [Finset.mem_filter,Finset.mem_singleton]
+  simp only [Finset.mem_univ,true_and]
+  by_cases hj : j = i
+  · subst j; simp [singleState,hw]
+  · simp [singleState,hj]
+
+omit [Fintype κ] in
+lemma singleState_step_same (a : ι) (ws : List ι) (i : κ) :
+    stepState a (singleState ws i) i = singleState (a::ws) i := by
+  simp only [stepState,singleState,Function.update_self,Function.update_idem]
+
+lemma monoRemainder_order (ws : List ι) (s : κ → List ι) (hs : s ∈ monoRemainder ws) :
+    stateOrder s = ws.length ∧ 2 ≤ (activeSet s).card := by
+  induction ws generalizing s with
+  | nil => simp [monoRemainder] at hs
+  | cons a ws ih =>
+    cases ws with
+    | nil => simp [monoRemainder] at hs
+    | cons b ws =>
+      simp only [monoRemainder,List.mem_append] at hs
+      rcases hs with hs | hs
+      · obtain ⟨s',hs',hs⟩ := List.mem_flatMap.mp hs
+        obtain ⟨i,_,rfl⟩ := List.mem_map.mp hs
+        have hh := ih s' hs'
+        constructor
+        · simp only [stateOrder_step,hh.1,List.length_cons]
+        · rw [activeSet_step]
+          exact hh.2.trans (Finset.card_mono (Finset.subset_insert _ _))
+      · obtain ⟨i,_,hs⟩ := List.mem_flatMap.mp hs
+        obtain ⟨j,hj,rfl⟩ := List.mem_map.mp hs
+        have hji : j ≠ i := (Finset.mem_erase.mp (Finset.mem_toList.mp hj)).1
+        constructor
+        · simp only [stateOrder_step,singleState_order,List.length_cons]
+        · rw [activeSet_step,singleState_active (by simp),
+            Finset.card_insert_of_notMem (by simpa using hji),Finset.card_singleton]
+
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+
+lemma sum_flatMap_map {α β γ : Type*} [AddCommMonoid γ]
+    (s : List α) (g : α → List β) (f : β → γ) :
+    ((s.flatMap g).map f).sum = (s.map (fun a => ((g a).map f).sum)).sum := by
+  induction s with
+  | nil => rfl
+  | cons a s ih => simp only [List.flatMap_cons,List.map_append,List.sum_append,List.map_cons,List.sum_cons,ih]
+
+lemma mixedStep_sum (e : ι → E) (f : κ → E → ℂ) (a : ι) (ws : List ι) (x : E) :
+    ((mixedStep a ws).map (fun s => monoEval e f s x)).sum =
+      ∑ i, ∑ j ∈ Finset.univ.erase i, monoEval e f (stepState a (singleState ws i) j) x := by
+  unfold mixedStep
+  rw [sum_flatMap_map]
+  simp
+
+lemma cdir_principal (e : ι → E) (f : κ → E → ℂ) (hf : ∀ i, ContDiff ℝ ∞ (f i))
+    (a : ι) (ws : List ι) :
+    cdir (e a) (fun x => ∑ i, monoEval e f (singleState ws i) x) = fun x =>
+      (∑ i, monoEval e f (singleState (a::ws) i) x)+
+        ((mixedStep a ws).map (fun s => monoEval e f s x)).sum := by
+  rw [cdir_finite_sum e a _ _ (fun i _ => monoEval_smooth e f hf _)]
+  simp_rw [cdir_monoEval e f hf]
+  funext x
+  rw [mixedStep_sum,← Finset.sum_add_distrib]
+  apply Finset.sum_congr rfl
+  intro i _
+  rw [← Finset.add_sum_erase Finset.univ
+    (fun j => monoEval e f (stepState a (singleState ws i) j) x) (Finset.mem_univ i),
+    singleState_step_same]
+
+lemma monoRemainder_step_sum (e : ι → E) (f : κ → E → ℂ)
+    (hf : ∀ i, ContDiff ℝ ∞ (f i)) (a b : ι) (ws : List ι) :
+    cdir (e a) (fun x => ((monoRemainder (b::ws)).map (fun s => monoEval e f s x)).sum) =
+      fun x => (((monoRemainder (b::ws)).flatMap
+        (fun s => Finset.univ.toList.map (stepState a s))).map (fun s => monoEval e f s x)).sum := by
+  rw [cdir_list_sum e a _ _ (fun s _ => monoEval_smooth e f hf s)]
+  simp_rw [cdir_monoEval e f hf]
+  funext x
+  rw [sum_flatMap_map]
+  simp
+
+omit [DecidableEq κ] in
+lemma smooth_mono_list_sum (e : ι → E) (f : κ → E → ℂ)
+    (hf : ∀ i, ContDiff ℝ ∞ (f i)) (ss : List (κ → List ι)) :
+    ContDiff ℝ ∞ (fun x => (ss.map (fun s => monoEval e f s x)).sum) := by
+  induction ss with
+  | nil => exact contDiff_const
+  | cons s ss ih => exact (monoEval_smooth e f hf s).add ih
+
+theorem cword_product_remainder (e : ι → E) (f : κ → E → ℂ)
+    (hf : ∀ i, ContDiff ℝ ∞ (f i)) (ws : List ι) (hw : ws ≠ []) :
+    cword e (fun x => ∏ i, f i x) ws = fun x =>
+      (∑ i, monoEval e f (singleState ws i) x)+
+        ((monoRemainder ws).map (fun s => monoEval e f s x)).sum := by
+  induction ws with
+  | nil => exact (hw rfl).elim
+  | cons a ws ih =>
+    cases ws with
+    | nil =>
+      change cdir (e a) (monoEval e f (fun _ => [])) = _
+      rw [cdir_monoEval e f hf]
+      funext x
+      simp only [monoRemainder,List.map_nil,List.sum_nil,add_zero]
+      apply Finset.sum_congr rfl
+      intro i _
+      rfl
+    | cons b ws =>
+      have hi := ih (by simp)
+      change cdir (e a) (cword e (fun x => ∏ i, f i x) (b::ws)) = _
+      rw [hi]
+      have h1 : ContDiff ℝ ∞ (fun x => ∑ i, monoEval e f (singleState (b::ws) i) x) := by
+        apply ContDiff.sum; intro i _; exact monoEval_smooth e f hf _
+      have h2 := smooth_mono_list_sum e f hf (monoRemainder (b::ws))
+      have hadd : cdir (e a) (fun x => (∑ i, monoEval e f (singleState (b::ws) i) x)+
+          ((monoRemainder (b::ws)).map (fun s => monoEval e f s x)).sum) =
+          fun x => cdir (e a) (fun x => ∑ i, monoEval e f (singleState (b::ws) i) x) x+
+            cdir (e a) (fun x => ((monoRemainder (b::ws)).map (fun s => monoEval e f s x)).sum) x := by
+        funext x
+        unfold cdir
+        rw [fderiv_fun_add (h1.differentiable (by simp) x) (h2.differentiable (by simp) x)]
+        rfl
+      rw [hadd,cdir_principal e f hf,monoRemainder_step_sum e f hf]
+      funext x
+      simp only [monoRemainder,List.map_append,List.sum_append]
+      abel
+end TameInterpolation
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/JetRemainderBound.lean b/lean/OAI/Geometry/PrescribedRicci/JetRemainderBound.lean
new file mode 100644
index 0000000000000000000000000000000000000000..759ff457aed1b2c64d6155be5875ed615d9f5c90
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/JetRemainderBound.lean
@@ -0,0 +1,119 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.JetTrim
+
+namespace OAI
+
+/-! Jet Remainder Bound. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Finset MeasureTheory
+open scoped ContDiff Classical BigOperators ENNReal
+namespace TameInterpolation
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+variable {ι κ : Type*} [Fintype ι] [Nonempty ι] [Fintype κ] [Nonempty κ] [DecidableEq κ]
+
+omit [Nonempty κ] in
+lemma lpNorm_complex_prod_L2_Linf (f : κ → E → ℂ) (i : κ)
+    (hf2 : MemLp (f i) 2 volume) (hfi : ∀ j, MemLp (f j) ∞ volume)
+    {A B : ℝ} (_hA : 0 ≤ A) (hB : 0 ≤ B)
+    (hfa : ∀ j, lpNorm (f j) ∞ volume ≤ A) (hfb : lpNorm (f i) 2 volume ≤ B) :
+    lpNorm (fun x => ∏ j, f j x) 2 volume ≤ B*A^(Fintype.card κ-1) := by
+  let p : κ → ℝ≥0∞ := fun j => if j = i then 2 else ∞
+  have hp : ∀ j, MemLp (f j) (p j) volume := by
+    intro j; by_cases hj : j = i
+    · subst j; simpa [p] using hf2
+    · simpa [p,hj] using hfi j
+  have hs : (∑ j, (p j)⁻¹)⁻¹ = 2 := by
+    rw [← Finset.add_sum_erase Finset.univ (fun j => (p j)⁻¹) (Finset.mem_univ i)]
+    have he : ∑ j ∈ Finset.univ.erase i, (p j)⁻¹ = 0 := by
+      apply Finset.sum_eq_zero
+      intro j hj
+      simp [p,(Finset.mem_erase.mp hj).1]
+    simp [he,p]
+  have hh := lpNorm_complex_finite_prod (μ:=volume) Finset.univ f p (fun j _ => hp j)
+  rw [hs] at hh
+  apply hh.trans
+  rw [← Finset.mul_prod_erase Finset.univ (fun j => lpNorm (f j) (p j) volume) (Finset.mem_univ i)]
+  have hi : p i = 2 := by simp [p]
+  rw [hi]
+  apply mul_le_mul hfb _ (Finset.prod_nonneg fun _ _ => lpNorm_nonneg) hB
+  have he : ∏ j ∈ Finset.univ.erase i, lpNorm (f j) (p j) volume ≤
+      ∏ _j ∈ Finset.univ.erase i, A := by
+    apply Finset.prod_le_prod₀
+    · intro j _; exact lpNorm_nonneg
+    · intro j hj
+      simpa [p,(Finset.mem_erase.mp hj).1] using hfa j
+  simpa using he
+
+lemma monoEval_tame_zero (e : ι → E) (f : κ → E → ℂ)
+    (hf : ∀ i, ContDiff ℝ ∞ (f i)) (hc : ∀ i, HasCompactSupport (f i))
+    (s : κ → List ι) (hm : ∑ i, (trimWord (s i)).length = 0) :
+    lpNorm (monoEval e f s) 2 volume ≤
+      (2*familyJetNorm e (realComponents (initialJet e f)) 0 2) *
+        (2*familyJetNorm e (realComponents (initialJet e f)) 0 ∞)^(Fintype.card κ-1) := by
+  have hzero (i : κ) : trimWord (s i) = [] := by
+    apply List.length_eq_zero_iff.mp
+    exact Nat.eq_zero_of_le_zero (hm ▸ Finset.single_le_sum (f:=fun i => (trimWord (s i)).length) (fun _ _ => Nat.zero_le _) (Finset.mem_univ i))
+  let g := initialJet e f
+  let a : κ → κ × Option ι := fun i => (i,baseDirection (s i))
+  have he : monoEval e f s = fun x => ∏ i, g (a i) x := by
+    funext x; unfold monoEval
+    apply Finset.prod_congr rfl
+    intro i _
+    rw [cword_trim e f i (s i),hzero i]
+    rfl
+  rw [he]
+  have hg := initialJet_smooth e f hf
+  have hgc := initialJet_compact e f hc
+  let i : κ := Classical.choice inferInstance
+  apply lpNorm_complex_prod_L2_Linf (fun i => g (a i)) i
+    ((hg _).continuous.memLp_of_hasCompactSupport (hgc _))
+    (fun j => (hg _).continuous.memLp_of_hasCompactSupport (hgc _))
+    (mul_nonneg (by norm_num) (familyJetNorm_nonneg ..))
+    (mul_nonneg (by norm_num) (familyJetNorm_nonneg ..))
+  · intro j
+    exact cjet_lpNorm_le e g hg hgc (a j) (Fin.elim0 : Fin 0 → ι) (by simp)
+  · exact cjet_lpNorm_le e g hg hgc (a i) (Fin.elim0 : Fin 0 → ι) (by norm_num)
+
+theorem monoRemainder_uniform_bound {Z : Type*} (e : ι → E) (F : Z → κ → E → ℂ)
+    (hf : ∀ z i, ContDiff ℝ ∞ (F z i)) (hc : ∀ z i, HasCompactSupport (F z i))
+    (ws : List ι) (s : κ → List ι) (hs : s ∈ monoRemainder ws)
+    {A : ℝ} (hA : 0 ≤ A)
+    (h0 : ∀ z, familyJetNorm e (realComponents (initialJet e (F z))) 0 ∞ ≤ A)
+    (h2 : ∀ j ≤ ws.length-2, ∃ B : ℝ, 0 ≤ B ∧
+      ∀ z, familyJetNorm e (realComponents (initialJet e (F z))) j 2 ≤ B) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ z, lpNorm (monoEval e (F z) s) 2 volume ≤ C := by
+  let m := ∑ i, (trimWord (s i)).length
+  have hmle : m ≤ ws.length-2 := remainder_trimmed_order hs
+  obtain ⟨B,hB,hb⟩ := h2 m hmle
+  by_cases hm : m = 0
+  · refine ⟨(2*B)*(2*A)^(Fintype.card κ-1),by positivity,fun z => ?_⟩
+    apply (monoEval_tame_zero e (F z) (hf z) (hc z) s hm).trans
+    apply mul_le_mul
+    · exact mul_le_mul_of_nonneg_left (by simpa [hm] using hb z) (by norm_num)
+    · exact pow_le_pow_left₀ (mul_nonneg (by norm_num) (familyJetNorm_nonneg ..))
+        (mul_le_mul_of_nonneg_left (h0 z) (by norm_num)) _
+    · exact pow_nonneg (mul_nonneg (by norm_num) (familyJetNorm_nonneg ..)) _
+    · positivity
+  · let K : ℝ := 2^(Fintype.card κ) * (2*(m:ℝ))^(∑ i, (trimWord (s i)).length*(m-(trimWord (s i)).length))
+    have hK : 0 ≤ K := by dsimp [K]; positivity
+    refine ⟨K*A^(Fintype.card κ-1)*B,by positivity,fun z => ?_⟩
+    apply (monoEval_tame_positive e (F z) (hf z) (hc z) s (Nat.pos_of_ne_zero hm)).trans
+    simp only [← mul_assoc]
+    change K * _ * _ ≤ K * _ * _
+    apply mul_le_mul
+    · exact mul_le_mul_of_nonneg_left (pow_le_pow_left₀ (familyJetNorm_nonneg ..) (h0 z) _) hK
+    · exact hb z
+    · exact familyJetNorm_nonneg ..
+    · positivity
+end TameInterpolation
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/JetSobolev.lean b/lean/OAI/Geometry/PrescribedRicci/JetSobolev.lean
new file mode 100644
index 0000000000000000000000000000000000000000..36975c443a44a6bfa81b28cc102151ab5af08e38
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/JetSobolev.lean
@@ -0,0 +1,132 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CoreBounds
+import OAI.Geometry.PrescribedRicci.JetTrim
+
+namespace OAI
+
+/-! Jet Sobolev. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Finset MeasureTheory LineDeriv TemperedDistribution
+open scoped ContDiff Classical BigOperators ENNReal SchwartzMap
+namespace TameInterpolation
+open SobolevChart EllipticKernel
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+variable {ι κ : Type*}
+
+lemma cdir_schwartz (v : E) (f : 𝓢(E,ℂ)) : cdir v f = (∂_{v} f : 𝓢(E,ℂ)) := by
+  funext x
+  rw [SchwartzMap.lineDerivOp_apply_eq_fderiv]
+  rfl
+
+lemma cword_schwartz (e : ι → E) (f : 𝓢(E,ℂ)) (ws : List ι) :
+    cword e f ws = (schwartzWord (ws.map e) f : E → ℂ) := by
+  induction ws with
+  | nil => rfl
+  | cons a ws ih =>
+    change cdir (e a) (cword e f ws) = _
+    rw [ih,cdir_schwartz]
+    rfl
+
+lemma cjet_schwartz (e : ι → E) (f : 𝓢(E,ℂ)) {j : ℕ} (w : Fin j → ι) :
+    cjet e f w = (schwartzWord ((List.ofFn w).map e) f : E → ℂ) := by
+  induction j with
+  | zero => rfl
+  | succ j ih =>
+    simp only [cjet,List.ofFn_succ,List.map_cons,schwartzWord,ContinuousLinearMap.comp_apply]
+    rw [ih,cdir_schwartz]
+    rfl
+
+variable [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+
+lemma lpNorm_schwartz_zero (f : 𝓢(E,ℂ)) : lpNorm (f : E → ℂ) 2 volume = ‖schwartzCoord 0 f‖ := by
+  have he : schwartzCoord 0 f = f.toLp 2 volume := by
+    apply l2_injective
+    simpa [realize] using realize_schwartzCoord 0 f
+  rw [he,SchwartzMap.norm_toLp,toReal_eLpNorm]
+
+lemma lpNorm_realComponent_le (f : E → ℂ) {p : ℝ≥0∞} (hf : MemLp f p volume) (b : Bool) :
+    lpNorm (fun x => if b then (f x).im else (f x).re) p volume ≤ lpNorm f p volume := by
+  apply (lpNorm_mono_real hf.norm (fun x => ?_)).trans_eq (lpNorm_norm hf.aestronglyMeasurable p)
+  cases b
+  · exact Complex.abs_re_le_norm (f x)
+  · exact Complex.abs_im_le_norm (f x)
+
+variable [Fintype ι] [Nonempty ι] [Fintype κ] [Nonempty κ]
+
+theorem familyJetNorm_sobolev_bound (e : ι → E) (j : ℕ) (s : ℝ) (hs : (j:ℝ) ≤ s) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ F : κ → 𝓢(E,ℂ),
+      familyJetNorm e (realComponents (fun i => (F i : E → ℂ))) j 2 ≤
+        C*∑ i, ‖schwartzCoord s (F i)‖ := by
+  have hb (w : Fin j → ι) : CoreBound s 0 (schwartzWord ((List.ofFn w).map e)) :=
+    coreBound_word _ (by simpa using sub_nonneg.mpr hs)
+  choose C hC hbound using hb
+  refine ⟨∑ w, C w,Finset.sum_nonneg fun w _ => hC w,fun F => ?_⟩
+  obtain ⟨⟨i,b⟩,ha⟩ := familyJetNorm_attained e (realComponents (fun i => (F i : E → ℂ))) j 2
+  obtain ⟨w,hw⟩ := jetNorm_attained e (realComponents (fun i => (F i : E → ℂ)) (i,b)) j 2
+  rw [ha,hw]
+  have he : jet e (realComponents (fun i => (F i : E → ℂ)) (i,b)) w =
+      fun x => if b then (cjet e (F i) w x).im else (cjet e (F i) w x).re := by
+    cases b
+    · exact cjet_real_map e (F i).smooth' Complex.reCLM w
+    · exact cjet_real_map e (F i).smooth' Complex.imCLM w
+  rw [he]
+  have hmem : MemLp (cjet e (F i) w) 2 volume := by rw [cjet_schwartz]; exact (schwartzWord _ _).memLp _ _
+  apply (lpNorm_realComponent_le _ hmem b).trans
+  rw [cjet_schwartz,lpNorm_schwartz_zero]
+  apply (hbound w (F i)).trans
+  exact mul_le_mul (Finset.single_le_sum (fun w _ => hC w) (Finset.mem_univ w))
+    (Finset.single_le_sum (f:=fun i => ‖schwartzCoord s (F i)‖) (fun _ _ => norm_nonneg _) (Finset.mem_univ i))
+    (norm_nonneg _) (Finset.sum_nonneg fun w _ => hC w)
+
+def initialSchwartz (e : ι → E) (F : κ → 𝓢(E,ℂ)) (q : κ × Option ι) : 𝓢(E,ℂ) :=
+  match q.2 with
+  | none => F q.1
+  | some a => ∂_{e a} (F q.1)
+
+omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+  [Fintype ι] [Nonempty ι] [Fintype κ] [Nonempty κ] in
+lemma initialSchwartz_coe (e : ι → E) (F : κ → 𝓢(E,ℂ)) (q : κ × Option ι) :
+    (initialSchwartz e F q : E → ℂ) = initialJet e (fun i => (F i : E → ℂ)) q := by
+  rcases q with ⟨i,q⟩; cases q
+  · rfl
+  · exact (cdir_schwartz _ _).symm
+
+theorem initialJet_sobolev_bound (e : ι → E) (j : ℕ) (s : ℝ) (hs : (j:ℝ)+1 ≤ s) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ F : κ → 𝓢(E,ℂ),
+      familyJetNorm e (realComponents (initialJet e (fun i => (F i : E → ℂ)))) j 2 ≤
+        C*∑ i, ‖schwartzCoord s (F i)‖ := by
+  obtain ⟨B,hB,hb⟩ := familyJetNorm_sobolev_bound (κ:=κ×Option ι) e j (j:ℝ) le_rfl
+  have hq (q : Option ι) : ∃ C : ℝ, 0 ≤ C ∧ ∀ f : 𝓢(E,ℂ),
+      ‖schwartzCoord (j:ℝ) (initialSchwartz e (fun _ : κ => f) (Classical.choice inferInstance,q))‖ ≤
+        C*‖schwartzCoord s f‖ := by
+    cases q with
+    | none => exact CoreBound.id (by linarith)
+    | some a => exact coreBound_deriv (e a) (by linarith)
+  choose C hC hcb using hq
+  refine ⟨B*∑ q : κ×Option ι, C q.2,mul_nonneg hB (Finset.sum_nonneg fun q _ => hC q.2),fun F => ?_⟩
+  have he : (fun q => (initialSchwartz e F q : E → ℂ)) = initialJet e (fun i => (F i : E → ℂ)) :=
+    funext (initialSchwartz_coe e F)
+  rw [← he]
+  apply (hb (initialSchwartz e F)).trans
+  rw [mul_assoc,Finset.sum_mul]
+  apply mul_le_mul_of_nonneg_left _ hB
+  apply Finset.sum_le_sum
+  intro q _
+  have hqq := hcb q.2 (F q.1)
+  have heq : initialSchwartz e (fun _ : κ => F q.1) (Classical.choice inferInstance,q.2) = initialSchwartz e F q := by
+    cases q with
+    | mk i a => cases a <;> rfl
+  rw [heq] at hqq
+  exact hqq.trans (mul_le_mul_of_nonneg_left
+    (Finset.single_le_sum (f:=fun i => ‖schwartzCoord s (F i)‖) (fun _ _ => norm_nonneg _) (Finset.mem_univ q.1)) (hC q.2))
+end TameInterpolation
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/JetTameProduct.lean b/lean/OAI/Geometry/PrescribedRicci/JetTameProduct.lean
new file mode 100644
index 0000000000000000000000000000000000000000..762e648def937b508ff79d26daeefab85c9b59a4
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/JetTameProduct.lean
@@ -0,0 +1,64 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.DiscreteInterpolation
+import OAI.Geometry.PrescribedRicci.InterpolationExponents
+import OAI.Geometry.PrescribedRicci.JetInterpolation
+
+namespace OAI
+
+/-! Jet Tame Product. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+open scoped ContDiff ENNReal
+namespace TameInterpolation
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+variable {ι : Type*} [Fintype ι] [Nonempty ι]
+
+lemma jet_exponent_recurrence (e : ι → E) {f : E → ℝ}
+    (hf : ContDiff ℝ ∞ f) (hc : HasCompactSupport f) (m : ℕ) (hm : 0 < m)
+    (j : ℕ) (hj : j+2 ≤ m) :
+    (jetNorm e f (j+1) (exponent m (j+1)))^2 ≤
+    (2*(m:ℝ))*jetNorm e f j (exponent m j)*jetNorm e f (j+2) (exponent m (j+2)) := by
+  let p : ℝ := 2*(m:ℝ)/((j+1:ℕ):ℝ)
+  have hp : 2 < p := exponent_middle_gt (by omega : j+1 < m)
+  let := exponent_holder_step (by omega : j+1 < m)
+  let := interpolation_holder_conjugate hp
+  have h := jetNorm_step e hf hc j hp (exponent m j) (exponent m (j+2))
+    (ENNReal.ofReal ((p-2)/p))⁻¹ (ENNReal.ofReal (2/p))⁻¹ (interpolation_power_exponent hp)
+  rw [exponent_of_pos hm (by omega : 0 < j+1)]
+  apply h.trans
+  have hpm : p ≤ 2*(m:ℝ) := by
+    apply (div_le_iff₀ (by positivity : 0 < ((j+1:ℕ):ℝ))).mpr
+    have hj1 : (1:ℝ) ≤ ((j+1:ℕ):ℝ) := by exact_mod_cast (by omega : 1 ≤ j+1)
+    nlinarith only [hj1,Nat.cast_nonneg (α:=ℝ) m]
+  exact mul_le_mul_of_nonneg_right
+    (mul_le_mul_of_nonneg_right (by linarith only [hpm]) (jetNorm_nonneg _ _ _ _)) (jetNorm_nonneg _ _ _ _)
+
+theorem jet_tame_product (e : ι → E) {f : E → ℝ}
+    (hf : ContDiff ℝ ∞ f) (hc : HasCompactSupport f) (m j : ℕ)
+    (hm : 0 < m) (hj : j ≤ m) :
+    jetNorm e f j (exponent m j)*jetNorm e f (m-j) (exponent m (m-j)) ≤
+    (2*(m:ℝ))^(j*(m-j))*jetNorm e f 0 ∞*jetNorm e f m 2 := by
+  have h := nonnegative_sequence_product m j hm hj
+    (fun i => jetNorm e f i (exponent m i)) (fun _ => jetNorm_nonneg _ _ _ _)
+    (2*(m:ℝ)) (by
+      have : (1:ℝ) ≤ (m:ℝ) := by exact_mod_cast hm
+      linarith only [this])
+    (fun i hi => jet_exponent_recurrence e hf hc m hm i hi)
+  have he : exponent m m = 2 := by
+    rw [exponent_of_pos hm hm]
+    have hmn : (m:ℝ) ≠ 0 := by exact_mod_cast (Nat.ne_of_gt hm)
+    rw [mul_div_cancel_right₀ _ hmn]
+    norm_num
+  simpa only [exponent_zero,he] using h
+end TameInterpolation
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/JetTrim.lean b/lean/OAI/Geometry/PrescribedRicci/JetTrim.lean
new file mode 100644
index 0000000000000000000000000000000000000000..8f267f0d342080bb889d21e59e49b971ece7c478
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/JetTrim.lean
@@ -0,0 +1,123 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.ComplexJetProduct
+import OAI.Geometry.PrescribedRicci.JetRemainder
+
+namespace OAI
+
+/-! Jet Trim. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Finset MeasureTheory
+open scoped ContDiff Classical BigOperators ENNReal
+namespace TameInterpolation
+variable {ι κ : Type*}
+
+def trimWord : List ι → List ι
+  | [] => []
+  | [_] => []
+  | a::b::ws => a::trimWord (b::ws)
+def baseDirection : List ι → Option ι
+  | [] => none
+  | [a] => some a
+  | _::b::ws => baseDirection (b::ws)
+
+lemma trimWord_length (ws : List ι) : (trimWord ws).length = ws.length-1 := by
+  induction ws with
+  | nil => rfl
+  | cons a ws ih => cases ws <;> simp_all [trimWord]
+
+lemma trimWord_length_add (ws : List ι) :
+    (trimWord ws).length + (if ws = [] then 0 else 1) = ws.length := by
+  rw [trimWord_length]
+  cases ws <;> simp
+
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+
+def initialJet (e : ι → E) (f : κ → E → ℂ) (q : κ × Option ι) : E → ℂ :=
+  match q.2 with
+  | none => f q.1
+  | some a => cdir (e a) (f q.1)
+
+lemma initialJet_smooth (e : ι → E) (f : κ → E → ℂ)
+    (hf : ∀ i, ContDiff ℝ ∞ (f i)) (q : κ × Option ι) :
+    ContDiff ℝ ∞ (initialJet e f q) := by
+  rcases q with ⟨i,q⟩; cases q
+  · exact hf i
+  · exact cdir_smooth _ (hf i)
+
+lemma initialJet_compact (e : ι → E) (f : κ → E → ℂ)
+    (hf : ∀ i, HasCompactSupport (f i)) (q : κ × Option ι) :
+    HasCompactSupport (initialJet e f q) := by
+  rcases q with ⟨i,q⟩; cases q
+  · exact hf i
+  · exact cdir_compact _ (hf i)
+
+lemma cword_trim (e : ι → E) (f : κ → E → ℂ) (i : κ) (ws : List ι) :
+    cword e (f i) ws = cword e (initialJet e f (i,baseDirection ws)) (trimWord ws) := by
+  induction ws with
+  | nil => rfl
+  | cons a ws ih =>
+    cases ws with
+    | nil => rfl
+    | cons b ws => exact congrArg (cdir (e a)) ih
+
+lemma cword_eq_cjet (e : ι → E) (f : E → ℂ) (ws : List ι) :
+    cword e f ws = cjet e f ws.get := by
+  induction ws with
+  | nil => rfl
+  | cons a ws ih =>
+    change cdir (e a) (cword e f ws) = cdir (e a) (cjet e f (fun i => (a::ws).get i.succ))
+    rw [ih]
+    rfl
+
+variable [Fintype κ] [DecidableEq κ]
+omit [DecidableEq κ] in
+lemma trimmed_order (s : κ → List ι) :
+    (∑ i, (trimWord (s i)).length) + (activeSet s).card = stateOrder s := by
+  have hcard : (activeSet s).card = ∑ i, if s i = [] then 0 else 1 := by
+    classical
+    simp [activeSet,Finset.card_filter,ite_not]
+  rw [hcard,← Finset.sum_add_distrib]
+  exact Finset.sum_congr rfl (fun i _ => trimWord_length_add (s i))
+
+lemma remainder_trimmed_order {ws : List ι} {s : κ → List ι} (hs : s ∈ monoRemainder ws) :
+    ∑ i, (trimWord (s i)).length ≤ ws.length-2 := by
+  have h := monoRemainder_order ws s hs
+  have he := trimmed_order s
+  omega
+
+variable [Fintype ι] [Nonempty ι] [Nonempty κ]
+variable [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+
+omit [DecidableEq κ] in
+theorem monoEval_tame_positive (e : ι → E) (f : κ → E → ℂ)
+    (hf : ∀ i, ContDiff ℝ ∞ (f i)) (hc : ∀ i, HasCompactSupport (f i))
+    (s : κ → List ι) (hm : 0 < ∑ i, (trimWord (s i)).length) :
+    let m := ∑ i, (trimWord (s i)).length
+    lpNorm (monoEval e f s) 2 volume ≤
+      2^(Fintype.card κ) * ((2*(m:ℝ))^(∑ i, (trimWord (s i)).length*(m-(trimWord (s i)).length)) *
+        familyJetNorm e (realComponents (initialJet e f)) 0 ∞ ^ (Fintype.card κ-1) *
+          familyJetNorm e (realComponents (initialJet e f)) m 2) := by
+  dsimp only
+  have he : monoEval e f s = fun x => ∏ i, cjet e
+      (initialJet e f (i,baseDirection (s i))) (trimWord (s i)).get x := by
+    funext x
+    unfold monoEval
+    apply Finset.prod_congr rfl
+    intro i _
+    rw [cword_trim e f i (s i),cword_eq_cjet]
+  rw [he]
+  exact cjet_product_L2 e (initialJet e f) (initialJet_smooth e f hf)
+    (initialJet_compact e f hc) _ hm Finset.univ
+    (fun i => (i,baseDirection (s i))) (fun i => (trimWord (s i)).length)
+    (fun i => (trimWord (s i)).get) rfl
+end TameInterpolation
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/KaehlerLogTrace.lean b/lean/OAI/Geometry/PrescribedRicci/KaehlerLogTrace.lean
new file mode 100644
index 0000000000000000000000000000000000000000..0a258ca2b5dc4e0e1b13e04069ac1fd4aad919f4
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/KaehlerLogTrace.lean
@@ -0,0 +1,112 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.KahlerEnergyChain
+import OAI.Geometry.PrescribedRicci.KaehlerTrace
+
+namespace OAI
+
+/-! Kaehler Log Trace. -/
+
+section
+
+ 
+noncomputable section
+open Matrix Filter Set Topology
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace Anticanonical.SourceSmooth.KaehlerMetric
+variable {d : ℕ}
+
+lemma holRealDeriv_log {φ : Coordinates d → ℝ} {z : Coordinates d}
+    (hφ : DifferentiableAt ℝ φ z) (hp : φ z ≠ 0) (a : Fin d) :
+    holRealDeriv (fun y => Real.log (φ y)) z a =
+      ((φ z)⁻¹:ℝ) * holRealDeriv φ z a := by
+  have he := congrFun (holRealDeriv_comp hφ (Real.differentiableAt_log hp)) a
+  simpa only [Function.comp_def,Real.deriv_log] using he
+
+lemma holRealDeriv_inv {φ : Coordinates d → ℝ} {z : Coordinates d}
+    (hφ : DifferentiableAt ℝ φ z) (hp : φ z ≠ 0) (a : Fin d) :
+    holRealDeriv (fun y => (φ y)⁻¹) z a =
+      (-((φ z)^2)⁻¹:ℝ) * holRealDeriv φ z a := by
+  have he := congrFun (holRealDeriv_comp hφ (differentiableAt_inv hp)) a
+  simpa only [Function.comp_def,deriv_inv] using he
+
+lemma potentialMatrix_log {φ : Coordinates d → ℝ} {z : Coordinates d}
+    (hφ : ContDiffAt ℝ ∞ φ z) (hp : φ z ≠ 0) (b a : Fin d) :
+    PotentialKaehler.potentialMatrix (fun y => Real.log (φ y)) z b a =
+      ((φ z)⁻¹:ℝ)*PotentialKaehler.potentialMatrix φ z b a -
+        (((φ z)^2)⁻¹:ℝ)*star (holRealDeriv φ z b)*holRealDeriv φ z a := by
+  rw [← barDeriv_holRealDeriv (hφ.log hp)]
+  have he : (fun y => holRealDeriv (fun w => Real.log (φ w)) y a) =ᶠ[nhds z]
+      (fun y => (((φ y)⁻¹:ℝ):ℂ)*holRealDeriv φ y a) := by
+    filter_upwards [(hφ.of_le (show (1:ℕ∞ω) ≤ ∞ by simp)).eventually (by simp),
+      hφ.continuousAt.eventually (eventually_ne_nhds hp)] with y hy hn
+    exact holRealDeriv_log (hy.differentiableAt (by simp)) hn a
+  have hd := hφ.differentiableAt (by simp)
+  have hi : DifferentiableAt ℝ (fun y => (φ y)⁻¹) z := hd.inv hp
+  have hic : DifferentiableAt ℝ (fun y => (((φ y)⁻¹:ℝ):ℂ)) z :=
+    Complex.ofRealCLM.differentiableAt.comp z hi
+  rw [barDeriv_congr he,barDeriv_mul hic
+      ((contDiffAt_holRealDeriv hφ a).differentiableAt (by simp)),
+    barDeriv_ofReal hi,barDeriv_holRealDeriv hφ,holRealDeriv_inv hd hp]
+  simp only [star_mul,Complex.star_def,Complex.ofReal_neg,map_neg,Complex.conj_ofReal]
+  ring
+
+variable {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+
+def logTrace (h g : KaehlerMetric A) (hd : 0 < d) : SmoothRealFunction A where
+  value x := Real.log ((h.traceMetric g).value x)
+  smooth q := ((h.traceMetric g).smooth q).log (fun z _hz =>
+    (h.traceMetric_pos g hd ((A.chart q).symm z)).ne')
+
+lemma laplacian_logTrace (h g : KaehlerMetric A) (hd : 0 < d) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) :
+    (h.laplacian (h.logTrace g hd)).localExpression q z =
+      ((h.traceMetric g).localExpression q z)⁻¹ * (h.laplacian (h.traceMetric g)).localExpression q z -
+      (((h.traceMetric g).localExpression q z)^2)⁻¹ *
+        (h.energy (h.traceMetric g) (h.traceMetric g)).localExpression q z := by
+  have hs := ((h.traceMetric g).smooth q).contDiffAt ((A.chart q).open_target.mem_nhds hz)
+  have hp : (h.traceMetric g).localExpression q z ≠ 0 :=
+    (h.traceMetric_pos g hd ((A.chart q).symm z)).ne'
+  have hx := (A.chart q).mapsTo_symm hz
+  have hlocφ := h.laplacianValue_local (h.traceMetric g) q hx
+  have hlocl := h.laplacianValue_local (h.logTrace g hd) q hx
+  have hloce := h.energy_local (h.traceMetric g) (h.traceMetric g) q hz
+  change (h.laplacianValue (h.logTrace g hd)) ((A.chart q).symm z) = _
+  rw [hlocl,(A.chart q).right_inv hz]
+  change _ = ((h.traceMetric g).localExpression q z)⁻¹ *
+    h.laplacianValue (h.traceMetric g) ((A.chart q).symm z) - _
+  rw [hlocφ,(A.chart q).right_inv hz]
+  change _ = _ - _ * (h.energy (h.traceMetric g) (h.traceMetric g)).value ((A.chart q).symm z)
+  change (h.energy (h.traceMetric g) (h.traceMetric g)).value ((A.chart q).symm z) = _ at hloce
+  rw [hloce]
+  simp only [linearizedMongeAmpere,Matrix.trace,Matrix.diag_apply,Matrix.mul_apply,gradientPair]
+  have hm (b a : Fin d) : (h.logTrace g hd).hessian q z b a =
+      (((h.traceMetric g).localExpression q z)⁻¹:ℝ)*(h.traceMetric g).hessian q z b a -
+        ((((h.traceMetric g).localExpression q z)^2)⁻¹:ℝ)*
+          star (holRealDeriv ((h.traceMetric g).localExpression q) z b)*
+          holRealDeriv ((h.traceMetric g).localExpression q) z a :=
+    potentialMatrix_log hs hp b a
+  simp only [hm,mul_sub,Finset.sum_sub_distrib]
+  have he (c : ℝ) (f : Fin d → Fin d → ℂ) :
+      (∑ a, ∑ b, (h.matrix q z)⁻¹ a b*((c:ℂ)*f b a)).re =
+        c*(∑ a, ∑ b, (h.matrix q z)⁻¹ a b*f b a).re := by
+    have hh : (∑ a, ∑ b, (h.matrix q z)⁻¹ a b*((c:ℂ)*f b a)) =
+        (c:ℂ)*(∑ a, ∑ b, (h.matrix q z)⁻¹ a b*f b a) := by
+      simp only [Finset.mul_sum]
+      apply Finset.sum_congr rfl
+      intro a _
+      apply Finset.sum_congr rfl
+      intro b _
+      ring
+    rw [hh,Complex.mul_re,Complex.ofReal_re,Complex.ofReal_im,zero_mul,sub_zero]
+  rw [Complex.sub_re]
+  apply congrArg₂ (fun a b : ℝ => a-b)
+  · exact he ((h.traceMetric g).localExpression q z)⁻¹ ((h.traceMetric g).hessian q z)
+  · simpa only [mul_assoc] using he (((h.traceMetric g).localExpression q z)^2)⁻¹
+      (fun b a => star (holRealDeriv ((h.traceMetric g).localExpression q) z b)*holRealDeriv ((h.traceMetric g).localExpression q) z a)
+
+end Anticanonical.SourceSmooth.KaehlerMetric
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/KaehlerTrace.lean b/lean/OAI/Geometry/PrescribedRicci/KaehlerTrace.lean
new file mode 100644
index 0000000000000000000000000000000000000000..4ad27efd457fbd0bc7305511488441c72f7333be
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/KaehlerTrace.lean
@@ -0,0 +1,110 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.GlobalOperator
+
+namespace OAI
+
+/-! Kaehler Trace. -/
+
+noncomputable section
+open Matrix Filter Set Topology
+open scoped ContDiff ComplexOrder MatrixOrder Matrix.Norms.Elementwise
+namespace MongeAmpere
+variable {n : Type*} [Fintype n] [DecidableEq n]
+
+lemma trace_mul_pos [Nonempty n] {H G : Matrix n n ℂ}
+    (hH : H.PosDef) (hG : G.PosDef) : 0 < (H*G).trace.re := by
+  let S := CFC.sqrt H
+  have hS : S.PosSemidef := (CFC.sqrt_nonneg H).posSemidef
+  have hs : S*S=H := by simpa [S,pow_two] using CFC.sq_sqrt H hH.posSemidef.nonneg
+  have hSn : S.det ≠ 0 := by
+    intro hz
+    have he := congrArg Matrix.det hs
+    rw [det_mul,hz,zero_mul] at he
+    exact hH.det_pos.ne' he.symm
+  have hp := hG.conjTranspose_mul_mul_same (Matrix.mulVec_injective_of_det_ne_zero hSn)
+  have ht := (Complex.pos_iff.mp hp.trace_pos).1
+  have he : (Sᴴ*G*S).trace = (H*G).trace := by
+    rw [hS.isHermitian.eq,trace_mul_cycle,hs]
+  rwa [he] at ht
+
+end MongeAmpere
+namespace Anticanonical.SourceSmooth.KaehlerMetric
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+
+lemma traceMetric_compatibility (g h : KaehlerMetric A) (i j : Fin A.count)
+    {x : X} (hi : x ∈ (A.chart i).source) (hj : x ∈ (A.chart j).source) :
+    ((g.matrix i (A.chart i x))⁻¹*h.matrix i (A.chart i x)).trace =
+      ((g.matrix j (A.chart j x))⁻¹*h.matrix j (A.chart j x)).trace := by
+  let D := A.derivativeMatrix i j x
+  have hD : IsUnit D.det := g.transition_det_isUnit i j hi hj
+  have hs : IsUnit Dᴴ.det := by rw [det_conjTranspose]; exact hD.star
+  rw [g.compatibility i j x hi hj,h.compatibility i j x hi hj]
+  change ((Dᴴ*g.matrix j (A.chart j x)*D)⁻¹*(Dᴴ*h.matrix j (A.chart j x)*D)).trace = _
+  rw [Matrix.mul_inv_rev,Matrix.mul_inv_rev]
+  simp only [mul_assoc,nonsing_inv_mul_cancel_left Dᴴ _ hs]
+  have he : D⁻¹ * ((g.matrix j (A.chart j x))⁻¹ * (h.matrix j (A.chart j x) * D)) =
+      D⁻¹ * ((g.matrix j (A.chart j x))⁻¹ * h.matrix j (A.chart j x)) * D := by simp only [mul_assoc]
+  rw [he, trace_mul_cycle, mul_nonsing_inv D hD, one_mul]
+
+def traceMetricValue (g h : KaehlerMetric A) (x : X) : ℝ :=
+  (((g.matrix (A.covers x).choose (A.chart (A.covers x).choose x))⁻¹)*
+    h.matrix (A.covers x).choose (A.chart (A.covers x).choose x)).trace.re
+
+lemma traceMetricValue_local (g h : KaehlerMetric A) (i : Fin A.count)
+    {x : X} (hx : x ∈ (A.chart i).source) :
+    g.traceMetricValue h x = ((g.matrix i (A.chart i x))⁻¹*h.matrix i (A.chart i x)).trace.re :=
+  congrArg Complex.re (g.traceMetric_compatibility h _ i (A.covers x).choose_spec hx)
+
+lemma traceMetric_local_smooth (g h : KaehlerMetric A) (i : Fin A.count) :
+    ContDiffOn ℝ ∞ (fun z => ((g.matrix i z)⁻¹*h.matrix i z).trace.re) (A.chart i).target := by
+  apply Complex.reCLM.contDiff.comp_contDiffOn
+  simp only [Matrix.trace,Matrix.diag_apply,Matrix.mul_apply]
+  apply ContDiffOn.sum
+  intro j _
+  apply ContDiffOn.sum
+  intro k _
+  exact (contDiffOn_pi.mp (contDiffOn_pi.mp (MatrixSmoothGeneral.inverse (g.smooth i)
+    (fun z hz => (g.positive i z hz).det_pos.ne')) j) k).mul
+    (contDiffOn_pi.mp (contDiffOn_pi.mp (h.smooth i) k) j)
+
+def traceMetric (g h : KaehlerMetric A) : SmoothRealFunction A where
+  value := g.traceMetricValue h
+  smooth i := by
+    apply (g.traceMetric_local_smooth h i).congr
+    intro z hz
+    dsimp only [Function.comp_apply]
+    rw [g.traceMetricValue_local h i ((A.chart i).mapsTo_symm hz),(A.chart i).right_inv hz]
+
+lemma traceMetric_local (g h : KaehlerMetric A) (i : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart i).target) :
+    (g.traceMetric h).localExpression i z = ((g.matrix i z)⁻¹*h.matrix i z).trace.re := by
+  change g.traceMetricValue h ((A.chart i).symm z) = _
+  rw [g.traceMetricValue_local h i ((A.chart i).mapsTo_symm hz),(A.chart i).right_inv hz]
+
+lemma traceMetric_pos (g h : KaehlerMetric A) (hd : 0 < d) (x : X) :
+    0 < (g.traceMetric h).value x := by
+  let : Nonempty (Fin d) := ⟨⟨0,hd⟩⟩
+  obtain ⟨i,hi⟩ := A.covers x
+  change 0 < g.traceMetricValue h x
+  rw [g.traceMetricValue_local h i hi]
+  exact MongeAmpere.trace_mul_pos (g.positive i _ ((A.chart i).mapsTo hi)).inv
+    (h.positive i _ ((A.chart i).mapsTo hi))
+
+lemma laplacian_deform (g : KaehlerMetric A) (φ : SmoothRealFunction A)
+    (hp : g.PositivePotential φ) (x : X) :
+    ((g.deform φ hp).laplacian φ).value x = d-((g.deform φ hp).traceMetric g).value x := by
+  obtain ⟨i,hi⟩ := A.covers x
+  let H := (g.deform φ hp).matrix i (A.chart i x)
+  have hH : IsUnit H.det := isUnit_iff_ne_zero.mpr ((g.deform φ hp).positive i _ ((A.chart i).mapsTo hi)).det_pos.ne'
+  change (g.deform φ hp).laplacianValue φ x = (d:ℝ)-(g.deform φ hp).traceMetricValue g x
+  rw [(g.deform φ hp).laplacianValue_local φ i hi,(g.deform φ hp).traceMetricValue_local g i hi]
+  unfold linearizedMongeAmpere
+  have he : φ.hessian i (A.chart i x) = H-g.matrix i (A.chart i x) := by dsimp [H,deform]; abel
+  rw [he,mul_sub,Matrix.nonsing_inv_mul _ hH,trace_sub,Complex.sub_re,trace_one]
+  simp
+
+end Anticanonical.SourceSmooth.KaehlerMetric
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/KahlerCauchySchwarz.lean b/lean/OAI/Geometry/PrescribedRicci/KahlerCauchySchwarz.lean
new file mode 100644
index 0000000000000000000000000000000000000000..f67988a48fa4bed5f71c0ad4e40f0d3bd157f6f4
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/KahlerCauchySchwarz.lean
@@ -0,0 +1,113 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.GlobalSobolev
+import OAI.Geometry.PrescribedPotential.SmoothPotentialDifference
+import OAI.Geometry.PrescribedRicci.KahlerEnergyChain
+import OAI.Geometry.PrescribedRicci.KahlerGreenIdentity
+import OAI.Geometry.PrescribedRicci.MongeAmpereEnergy
+
+namespace OAI
+
+/-! Kahler Cauchy Schwarz. -/
+
+section
+
+ 
+
+noncomputable section
+open Matrix Filter Set Topology MeasureTheory
+open scoped ContDiff ComplexOrder Classical
+namespace Anticanonical.SourceSmooth
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+namespace KaehlerMetric
+
+lemma energy_realSMul_left (g : KaehlerMetric A) (ψ φ : SmoothRealFunction A)
+    (a : ℝ) (x : X) : (g.energy (ψ.realSMul a) φ).value x = a * (g.energy ψ φ).value x := by
+  change (g.energy (ψ.compose (fun t => a * t) (contDiff_const.mul contDiff_id)) φ).value x = _
+  have hd : deriv (fun t => a * t) (ψ.value x) = a := by
+    simpa only [id_eq, mul_one] using ((hasDerivAt_id (ψ.value x)).const_mul a).deriv
+  rw [g.energy_comp_left, hd]
+
+lemma energy_constant_left (g : KaehlerMetric A) (φ : SmoothRealFunction A) (a : ℝ) (x : X) :
+    (g.energy (SmoothRealFunction.constant a) φ).value x = 0 := by
+  change (g.energy ((SmoothRealFunction.constant (A := A) 0).compose (fun _ => a) contDiff_const) φ).value x = _
+  rw [g.energy_comp_left, deriv_const, zero_mul]
+
+variable [T2Space X] [CompactSpace X]
+
+lemma integral_laplacian (g : KaehlerMetric A) (φ : SmoothRealFunction A) :
+    g.integral (g.laplacian φ).value = 0 := by
+  have h := g.green_identity (SmoothRealFunction.constant 1) φ
+  have he : (g.energy (SmoothRealFunction.constant 1) φ).value = fun _ => 0 :=
+    funext (g.energy_constant_left φ 1)
+  rw [he, g.integral_zero, neg_zero] at h
+  simpa only [SmoothRealFunction.constant, one_mul] using h
+
+lemma integral_cauchySchwarz (g : KaehlerMetric A) {f h : X → ℝ}
+    (hf : Continuous f) (hh : Continuous h) :
+    (g.integral (fun x => f x * h x)) ^ 2 ≤
+      g.integral (fun x => f x ^ 2) * g.integral (fun x => h x ^ 2) := by
+  have hq (t : ℝ) : 0 ≤ g.integral (fun x => f x ^ 2) * (t*t) +
+      (2 * g.integral (fun x => f x * h x)) * t + g.integral (fun x => h x ^ 2) := by
+    calc
+      0 ≤ g.integral (fun x => (t * f x + h x) ^ 2) := g.integral_nonneg (fun _ => sq_nonneg _)
+      _ = _ := by
+        have he : (fun x => (t*f x+h x)^2) =
+            fun x => (t*t) * (f x)^2 + (2*t) * (f x*h x) + (h x)^2 := by funext x; ring
+        rw [he, g.integral_add
+          (f := fun x => (t*t) * f x ^ 2 + (2*t) * (f x*h x))
+          (h := fun x => h x ^ 2)
+          ((continuous_const.fun_mul (hf.pow 2)).fun_add (continuous_const.fun_mul (hf.fun_mul hh)))
+          (hh.pow 2), g.integral_add
+          (f := fun x => (t*t) * f x ^ 2) (h := fun x => (2*t) * (f x*h x))
+          (continuous_const.fun_mul (hf.pow 2)) (continuous_const.fun_mul (hf.fun_mul hh)),
+          g.integral_const_mul, g.integral_const_mul]
+        ring
+  have hd := discrim_le_zero hq
+  unfold discrim at hd
+  nlinarith
+
+omit [T2Space X] [CompactSpace X] in
+lemma energy_add_right (g : KaehlerMetric A) (φ ψ χ : SmoothRealFunction A) :
+    (g.energy φ (ψ.addFunction χ)).value =
+      fun x => (g.energy φ ψ).value x + (g.energy φ χ).value x := by
+  rw [g.energy_symm, g.energy_add_left, g.energy_symm ψ φ, g.energy_symm χ φ]
+
+omit [T2Space X] [CompactSpace X] in
+lemma energy_realSMul_right (g : KaehlerMetric A) (ψ φ : SmoothRealFunction A)
+    (a : ℝ) (x : X) : (g.energy ψ (φ.realSMul a)).value x = a * (g.energy ψ φ).value x := by
+  rw [g.energy_symm, g.energy_realSMul_left, g.energy_symm φ ψ]
+
+lemma energy_cauchySchwarz (g : KaehlerMetric A) (φ ψ : SmoothRealFunction A) :
+    (g.integral (g.energy φ ψ).value) ^ 2 ≤
+      g.integral (g.energy φ φ).value * g.integral (g.energy ψ ψ).value := by
+  have he (t : ℝ) (x : X) :
+      (g.energy ((φ.realSMul t).addFunction ψ) ((φ.realSMul t).addFunction ψ)).value x =
+        (t*t) * (g.energy φ φ).value x + (2*t) * (g.energy φ ψ).value x + (g.energy ψ ψ).value x := by
+    simp only [g.energy_add_left, g.energy_add_right, g.energy_realSMul_left,
+      g.energy_realSMul_right, g.energy_symm ψ φ]
+    ring
+  have hq (t : ℝ) : 0 ≤ g.integral (g.energy φ φ).value * (t*t) +
+      (2 * g.integral (g.energy φ ψ).value) * t + g.integral (g.energy ψ ψ).value := by
+    calc
+      0 ≤ g.integral (g.energy ((φ.realSMul t).addFunction ψ) ((φ.realSMul t).addFunction ψ)).value :=
+        g.integral_nonneg (g.energy_nonneg _)
+      _ = _ := by
+        rw [show (g.energy ((φ.realSMul t).addFunction ψ) ((φ.realSMul t).addFunction ψ)).value =
+          (fun x => (t*t) * (g.energy φ φ).value x + (2*t) * (g.energy φ ψ).value x +
+            (g.energy ψ ψ).value x) from funext (he t)]
+        rw [g.integral_add ((continuous_const.fun_mul (g.energy φ φ).continuous).fun_add
+          (continuous_const.fun_mul (g.energy φ ψ).continuous)) (g.energy ψ ψ).continuous,
+          g.integral_add (continuous_const.fun_mul (g.energy φ φ).continuous)
+          (continuous_const.fun_mul (g.energy φ ψ).continuous), g.integral_const_mul, g.integral_const_mul]
+        ring
+  have hd := discrim_le_zero hq
+  unfold discrim at hd
+  nlinarith
+
+end KaehlerMetric
+end Anticanonical.SourceSmooth
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/KahlerChartGradient.lean b/lean/OAI/Geometry/PrescribedRicci/KahlerChartGradient.lean
new file mode 100644
index 0000000000000000000000000000000000000000..772a31866cf50d90c8c1cc35c88b53ce084df2e3
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/KahlerChartGradient.lean
@@ -0,0 +1,115 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.GlobalKahlerEnergy
+import OAI.Geometry.PrescribedRicci.GlobalKahlerIntegral
+import OAI.Geometry.PrescribedRicci.KahlerGradientCoercivity
+import OAI.Geometry.PrescribedRicci.KahlerLocalL2Comparison
+
+namespace OAI
+
+/-! Kahler Chart Gradient. -/
+
+section
+
+ 
+noncomputable section
+open Matrix Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory
+open scoped ContDiff Classical
+namespace Anticanonical.SourceSmooth.KaehlerMetric
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X}
+
+omit [T2Space X] [CompactSpace X] in
+lemma formEnergy_localized (g : KaehlerMetric A) (ψ : SmoothRealFunction A)
+    (i : Fin A.count) {z : Coordinates d} (hz : z ∈ (A.chart i).target) :
+    formEnergy (g.matrix i z) (fderiv ℝ (ψ.localized i) z) =
+      (g.energy ψ ψ).value ((A.chart i).symm z) := by
+  have he : ψ.localized i =ᶠ[𝓝 z] ψ.localExpression i := by
+    filter_upwards [(A.chart i).open_target.mem_nhds hz] with y hy using ψ.localized_eq hy
+  rw [he.fderiv_eq]
+  exact (g.energy_local ψ ψ i hz).symm
+
+omit [T2Space X] [CompactSpace X] in
+lemma localized_gradient_pointwise (g : KaehlerMetric A) (i : Fin A.count)
+    (ψ : SmoothRealFunction A) {K : Set (Coordinates d)}
+    (ht : K ⊆ (A.chart i).target) (hqK : tsupport (ψ.localized i) ⊆ K)
+    {a v : ℝ} (hv : 0 ≤ v)
+    (hgrad : ∀ z ∈ K, ∀ L : Coordinates d →L[ℝ] ℝ,
+      a * ‖L‖ ^ 2 ≤ formEnergy (g.matrix i z) L)
+    (hvol : ∀ z ∈ K, v ≤ g.volumeCoefficient i z) (z : Coordinates d) :
+    (a*v) * ‖fderiv ℝ (ψ.localized i) z‖ ^ 2 ≤
+      localizeFunction i (g.energy ψ ψ).value z * g.volumeCoefficient i z := by
+  by_cases hz : z ∈ tsupport (ψ.localized i)
+  · have hzK := hqK hz
+    have hzT := ht hzK
+    have hG := hgrad z hzK (fderiv ℝ (ψ.localized i) z)
+    rw [g.formEnergy_localized ψ i hzT] at hG
+    simp only [localizeFunction, ite_eq_left hzT]
+    calc
+      _ = v * (a * ‖fderiv ℝ (ψ.localized i) z‖^2) := by ring
+      _ ≤ v * (g.energy ψ ψ).value ((A.chart i).symm z) := mul_le_mul_of_nonneg_left hG hv
+      _ ≤ g.volumeCoefficient i z * (g.energy ψ ψ).value ((A.chart i).symm z) :=
+        mul_le_mul_of_nonneg_right (hvol z hzK) (g.energy_nonneg ψ _)
+      _ = _ := mul_comm _ _
+  · have hd : fderiv ℝ (ψ.localized i) z = 0 :=
+      image_eq_zero_of_notMem_tsupport (fun hh => hz (tsupport_fderiv_subset ℝ hh))
+    rw [hd, norm_zero, zero_pow (by omega : (2:ℕ) ≠ 0), mul_zero]
+    by_cases ht : z ∈ (A.chart i).target
+    · simp only [localizeFunction, ite_eq_left ht]
+      exact mul_nonneg (g.energy_nonneg ψ _) (g.volumeCoefficient_pos i ht).le
+    · simp only [localizeFunction, ite_eq_right ht, zero_mul, le_refl]
+
+lemma localized_gradient_integrable (ψ : SmoothRealFunction A) (i : Fin A.count)
+    (hψs : tsupport ψ.value ⊆ (A.chart i).source) :
+    Integrable (fun z : Coordinates d => ‖fderiv ℝ (ψ.localized i) z‖^2) := by
+  obtain ⟨hq,hqc,hqt⟩ := ψ.localized_smooth_compact i hψs
+  have hD : Continuous (fun z => ‖fderiv ℝ (ψ.localized i) z‖^2) :=
+    (hq.continuous_fderiv (by simp)).norm.pow 2
+  have hDc : HasCompactSupport (fun z => ‖fderiv ℝ (ψ.localized i) z‖^2) :=
+    HasCompactSupport.comp_left (f := fderiv ℝ (ψ.localized i))
+      (g := fun L : Coordinates d →L[ℝ] ℝ => ‖L‖^2) (hqc.fderiv ℝ) (by simp)
+  exact hD.integrable_of_hasCompactSupport hDc
+
+lemma localized_gradient_integral_le (g : KaehlerMetric A) (i : Fin A.count)
+    (ψ : SmoothRealFunction A) (hψs : tsupport ψ.value ⊆ (A.chart i).source)
+    {c : ℝ} (hc : 0 < c)
+    (hpoint : ∀ z : Coordinates d, c * ‖fderiv ℝ (ψ.localized i) z‖ ^ 2 ≤
+      localizeFunction i (g.energy ψ ψ).value z * g.volumeCoefficient i z) :
+    (∫ z : Coordinates d, ‖fderiv ℝ (ψ.localized i) z‖ ^ 2) ≤
+      c⁻¹ * g.integral (g.energy ψ ψ).value := by
+  have hEs : tsupport (g.energy ψ ψ).value ⊆ (A.chart i).source :=
+    (g.energy_support ψ ψ).trans hψs
+  have hint := g.chart_weighted_integrable i (g.energy ψ ψ).continuous hEs
+  have hDi := localized_gradient_integrable ψ i hψs
+  have hh := MeasureTheory.integral_mono (hDi.const_mul c) hint hpoint
+  rw [MeasureTheory.integral_const_mul] at hh
+  have he : (∫ z, localizeFunction i (g.energy ψ ψ).value z * g.volumeCoefficient i z) =
+      g.integral (g.energy ψ ψ).value := by
+    rw [g.integral_chart i (g.energy ψ ψ).continuous hEs, g.chartIntegral_eq_localize]
+  rw [he] at hh
+  calc
+    _ = c⁻¹ * (c * ∫ z : Coordinates d, ‖fderiv ℝ (ψ.localized i) z‖ ^ 2) := by
+      rw [← mul_assoc, inv_mul_cancel₀ hc.ne', one_mul]
+    _ ≤ _ := mul_le_mul_of_nonneg_left hh (inv_pos.mpr hc).le
+
+lemma chart_gradient_L2_bound (g : KaehlerMetric A) (i : Fin A.count)
+    {K : Set X} (hK : IsCompact K) (hs : K ⊆ (A.chart i).source) :
+    ∃ C : ℝ, 0 < C ∧ ∀ ψ : SmoothRealFunction A, tsupport ψ.value ⊆ K →
+      (∫ z : Coordinates d, ‖fderiv ℝ (ψ.localized i) z‖ ^ 2) ≤
+        C * g.integral (g.energy ψ ψ).value := by
+  let K' := (A.chart i) '' K
+  have hK' : IsCompact K' := hK.image_of_continuousOn ((A.chart i).continuousOn.mono hs)
+  have ht : K' ⊆ (A.chart i).target := by rintro z ⟨x,hx,rfl⟩; exact (A.chart i).mapsTo (hs hx)
+  obtain ⟨a,ha,hgrad⟩ := g.compact_gradient_coercivity i hK' ht
+  obtain ⟨v,b,hv,hb,hvol⟩ := g.compact_volume_bounds i hK' ht
+  refine ⟨(a*v)⁻¹, inv_pos.mpr (mul_pos ha hv), fun ψ hψ => ?_⟩
+  have hqK : tsupport (ψ.localized i) ⊆ K' :=
+    closure_minimal ((ψ.localized_support i).trans (image_mono hψ)) hK'.isClosed
+  apply g.localized_gradient_integral_le i ψ (hψ.trans hs) (mul_pos ha hv)
+  exact g.localized_gradient_pointwise i ψ ht hqK hv.le hgrad (fun z hz => (hvol z hz).1)
+
+end Anticanonical.SourceSmooth.KaehlerMetric
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/KahlerChartLp.lean b/lean/OAI/Geometry/PrescribedRicci/KahlerChartLp.lean
new file mode 100644
index 0000000000000000000000000000000000000000..a56751ada6a1694c13b4c6f9a846b794bda955e9
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/KahlerChartLp.lean
@@ -0,0 +1,91 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.GlobalKahlerIntegral
+import OAI.Geometry.PrescribedRicci.KahlerLocalL2Comparison
+import OAI.Geometry.PrescribedRicci.KahlerVolumeMeasure
+
+namespace OAI
+
+/-! Kahler Chart Lp. -/
+
+section
+
+ 
+noncomputable section
+open Matrix Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory
+open scoped ContDiff Classical NNReal ENNReal
+namespace Anticanonical.SourceSmooth.KaehlerMetric
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X}
+
+lemma chart_integral_upper (g : KaehlerMetric A) (i : Fin A.count)
+    {K : Set X} (hK : IsCompact K) (hs : K ⊆ (A.chart i).source) :
+    ∃ b : ℝ, 0 < b ∧ ∀ f : X → ℝ, Continuous f → (∀ x, 0 ≤ f x) →
+      tsupport f ⊆ K → g.integral f ≤ b * ∫ z : Coordinates d, localizeFunction i f z := by
+  let K' := (A.chart i) '' K
+  have hK' : IsCompact K' := hK.image_of_continuousOn ((A.chart i).continuousOn.mono hs)
+  have ht : K' ⊆ (A.chart i).target := by rintro z ⟨x,hx,rfl⟩; exact (A.chart i).mapsTo (hs hx)
+  obtain ⟨a,b,ha,hb,hab⟩ := g.compact_volume_bounds i hK' ht
+  refine ⟨b,hb,fun f hf hn hfs => ?_⟩
+  have hss := hfs.trans hs
+  obtain ⟨hl,hc,hlt⟩ := localizeFunction_continuous_compact i hf hss
+  have hi : Integrable (localizeFunction i f) volume := hl.integrable_of_hasCompactSupport hc
+  have hwi := g.chart_weighted_integrable i hf hss
+  have hsup : Function.support (localizeFunction i f) ⊆ K' :=
+    (localizeFunction_support i f).trans (image_mono hfs)
+  rw [g.integral_chart i hf hss, g.chartIntegral_eq_localize, ← MeasureTheory.integral_const_mul]
+  apply MeasureTheory.integral_mono hwi (hi.const_mul b)
+  intro z
+  by_cases hz : z ∈ Function.support (localizeFunction i f)
+  · have hp : 0 ≤ localizeFunction i f z := by
+      unfold localizeFunction
+      split_ifs <;> simp_all only [le_refl]
+    simpa only [mul_comm] using mul_le_mul_of_nonneg_left (hab z (hsup hz)).2 hp
+  · simp only [Function.notMem_support.mp hz, zero_mul, mul_zero, le_refl]
+
+omit [T2Space X] [CompactSpace X] in
+lemma tsupport_norm_rpow {f : X → ℝ} {q : ℝ} (hq : 0 < q) :
+    tsupport (fun x => ‖f x‖ ^ q) ⊆ tsupport f := by
+  apply closure_minimal _ (isClosed_tsupport _)
+  intro x hx
+  apply subset_tsupport
+  intro hz
+  exact hx (by simp [hz, Real.zero_rpow hq.ne'])
+
+omit [T2Space X] [CompactSpace X] in
+lemma localize_norm_rpow (i : Fin A.count) (f : X → ℝ) {q : ℝ} (hq : 0 < q) :
+    localizeFunction i (fun x => ‖f x‖ ^ q) = fun z => ‖localizeFunction i f z‖ ^ q := by
+  funext z
+  unfold localizeFunction
+  split_ifs <;> simp [Real.zero_rpow hq.ne']
+
+variable [MeasurableSpace X] [BorelSpace X]
+lemma chart_lpNorm_bound (g : KaehlerMetric A) (i : Fin A.count)
+    {K : Set X} (hK : IsCompact K) (hs : K ⊆ (A.chart i).source) :
+    ∃ b : ℝ, 0 < b ∧ ∀ (ψ : SmoothRealFunction A), tsupport ψ.value ⊆ K →
+      ∀ q : ℝ≥0, 0 < q → lpNorm ψ.value q g.volumeMeasure ≤
+        b ^ (q : ℝ)⁻¹ * lpNorm (ψ.localized i) q volume := by
+  obtain ⟨b,hb,hbnd⟩ := g.chart_integral_upper i hK hs
+  refine ⟨b,hb,fun ψ hψ q hq => ?_⟩
+  have hqR : (0:ℝ) < q := hq
+  have hc : Continuous (fun x => ‖ψ.value x‖ ^ (q:ℝ)) :=
+    ψ.continuous.norm.rpow_const (fun _ => Or.inr hqR.le)
+  have hpow := hbnd (fun x => ‖ψ.value x‖ ^ (q:ℝ)) hc
+    (fun x => Real.rpow_nonneg (norm_nonneg _) _) ((tsupport_norm_rpow hqR).trans hψ)
+  rw [g.integral_volumeMeasure hc |>.symm, localize_norm_rpow i ψ.value hqR] at hpow
+  have hloc : localizeFunction i ψ.value = ψ.localized i := rfl
+  rw [hloc] at hpow
+  obtain ⟨hl,hcl,_⟩ := ψ.localized_smooth_compact i (hψ.trans hs)
+  rw [lpNorm_nnreal_eq_integral_norm_rpow hq.ne' ψ.continuous.aestronglyMeasurable,
+    lpNorm_nnreal_eq_integral_norm_rpow hq.ne' hl.continuous.aestronglyMeasurable]
+  calc
+    _ ≤ (b * ∫ z, ‖ψ.localized i z‖ ^ (q:ℝ)) ^ (q:ℝ)⁻¹ :=
+      Real.rpow_le_rpow (MeasureTheory.integral_nonneg (fun x => Real.rpow_nonneg (norm_nonneg _) _)) hpow
+        (inv_nonneg.mpr hqR.le)
+    _ = _ := Real.mul_rpow hb.le (MeasureTheory.integral_nonneg (fun z => Real.rpow_nonneg (norm_nonneg _) _))
+
+end Anticanonical.SourceSmooth.KaehlerMetric
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/KahlerCofactorDivergence.lean b/lean/OAI/Geometry/PrescribedRicci/KahlerCofactorDivergence.lean
new file mode 100644
index 0000000000000000000000000000000000000000..10c0ec4803b6605bae5f6c661869173749428408
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/KahlerCofactorDivergence.lean
@@ -0,0 +1,193 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.MatrixDifferentialNew
+
+namespace OAI
+
+/-! Kahler Cofactor Divergence. -/
+
+noncomputable section
+open Matrix Filter Set Topology
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace Anticanonical.SourceSmooth
+namespace KaehlerMetric
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+
+abbrev coordinateVector (j : Fin d) : Coordinates d := Pi.single j 1
+
+lemma pair_coordinate (M : Matrix (Fin d) (Fin d) ℂ) (j k : Fin d) :
+    star (coordinateVector j) ⬝ᵥ (M *ᵥ coordinateVector k) = M j k := by
+  change star (Pi.single j 1) ⬝ᵥ (M *ᵥ Pi.single k 1) = _
+  rw [Pi.star_single, star_one, single_dotProduct, one_mul]
+  change M j ⬝ᵥ Pi.single k 1 = M j k
+  rw [dotProduct_single, mul_one]
+
+lemma fundamentalForm_coordinate (M : Matrix (Fin d) (Fin d) ℂ) (j k : Fin d) :
+    ComplexAtlas.fundamentalForm M (coordinateVector j) (coordinateVector k) = (M j k).im := by
+  simp only [ComplexAtlas.fundamentalForm, pair_coordinate]
+
+lemma fundamentalForm_I_coordinate (M : Matrix (Fin d) (Fin d) ℂ) (j k : Fin d) :
+    ComplexAtlas.fundamentalForm M (Complex.I • coordinateVector j) (coordinateVector k) =
+      -(M j k).re := by
+  simp only [ComplexAtlas.fundamentalForm, star_smul, smul_dotProduct, pair_coordinate]
+  simp [smul_eq_mul]
+
+lemma fundamentalForm_coordinate_I (M : Matrix (Fin d) (Fin d) ℂ) (j k : Fin d) :
+    ComplexAtlas.fundamentalForm M (coordinateVector j) (Complex.I • coordinateVector k) =
+      (M j k).re := by
+  simp only [ComplexAtlas.fundamentalForm, mulVec_smul, dotProduct_smul, pair_coordinate]
+  simp [smul_eq_mul]
+
+lemma fundamentalForm_I_I (M : Matrix (Fin d) (Fin d) ℂ) (j k : Fin d) :
+    ComplexAtlas.fundamentalForm M (Complex.I • coordinateVector j)
+      (Complex.I • coordinateVector k) = (M j k).im := by
+  simp only [ComplexAtlas.fundamentalForm, star_smul, mulVec_smul, smul_dotProduct,
+    dotProduct_smul, pair_coordinate]
+  simp [smul_eq_mul, Complex.mul_im]
+
+def fundamentalFormCLM (u v : Coordinates d) : Matrix (Fin d) (Fin d) ℂ →L[ℝ] ℝ :=
+  LinearMap.toContinuousLinearMap
+    { toFun := fun M => ComplexAtlas.fundamentalForm M u v
+      map_add' := by
+        intro M N
+        exact fundamentalForm_add M N u v
+      map_smul' := by
+        intro c M
+        simp [ComplexAtlas.fundamentalForm, smul_mulVec, dotProduct_smul] }
+
+lemma fundamentalForm_fderiv (g : KaehlerMetric A) (i : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart i).target) (u v w : Coordinates d) :
+    fderiv ℝ (fun y => ComplexAtlas.fundamentalForm (g.matrix i y) v w) z u =
+      ComplexAtlas.fundamentalForm (fderiv ℝ (g.matrix i) z u) v w := by
+  have hg := ((g.smooth i).contDiffAt ((A.chart i).open_target.mem_nhds hz)).differentiableAt
+    (by simp)
+  exact congrArg (fun L : Coordinates d →L[ℝ] ℝ => L u)
+    (((fundamentalFormCLM v w).hasFDerivAt.comp z hg.hasFDerivAt).fderiv)
+
+lemma derivative_closed (g : KaehlerMetric A) (i : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart i).target) (u v w : Coordinates d) :
+    ComplexAtlas.fundamentalForm (fderiv ℝ (g.matrix i) z u) v w +
+      ComplexAtlas.fundamentalForm (fderiv ℝ (g.matrix i) z v) w u +
+      ComplexAtlas.fundamentalForm (fderiv ℝ (g.matrix i) z w) u v = 0 := by
+  simpa only [g.fundamentalForm_fderiv i hz] using g.closed i z hz u v w
+
+def conjugateTransposeCLM : Matrix (Fin d) (Fin d) ℂ →L[ℝ] Matrix (Fin d) (Fin d) ℂ :=
+  LinearMap.toContinuousLinearMap
+    { toFun := Matrix.conjTranspose
+      map_add' := Matrix.conjTranspose_add
+      map_smul' := by intro c M; simp }
+
+lemma derivative_hermitian (g : KaehlerMetric A) (i : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart i).target) (u : Coordinates d) :
+    (fderiv ℝ (g.matrix i) z u).IsHermitian := by
+  have he : (fun y => (g.matrix i y)ᴴ) =ᶠ[nhds z] g.matrix i := by
+    filter_upwards [(A.chart i).open_target.mem_nhds hz] with y hy
+    exact (g.positive i y hy).isHermitian
+  have hg := ((g.smooth i).contDiffAt ((A.chart i).open_target.mem_nhds hz)).differentiableAt
+    (by simp)
+  have hd := congrArg (fun L : Coordinates d →L[ℝ] Matrix (Fin d) (Fin d) ℂ => L u)
+    he.fderiv_eq
+  have hs := ((conjugateTransposeCLM (d := d)).hasFDerivAt.comp z hg.hasFDerivAt).fderiv
+  have hs' : fderiv ℝ (fun y => (g.matrix i y)ᴴ) z u =
+      (fderiv ℝ (g.matrix i) z u)ᴴ :=
+    congrArg (fun L : Coordinates d →L[ℝ] Matrix (Fin d) (Fin d) ℂ => L u) hs
+  exact hs'.symm.trans hd
+
+def barDerivative (f : Coordinates d → Matrix (Fin d) (Fin d) ℂ)
+    (z : Coordinates d) (j : Fin d) : Matrix (Fin d) (Fin d) ℂ :=
+  (2 : ℂ)⁻¹ • (fderiv ℝ f z (coordinateVector j) +
+    Complex.I • fderiv ℝ f z (Complex.I • coordinateVector j))
+
+lemma barDerivative_symm (g : KaehlerMetric A) (a : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart a).target) (i j k : Fin d) :
+    barDerivative (g.matrix a) z i j k = barDerivative (g.matrix a) z j i k := by
+  have h1 := g.derivative_closed a hz (coordinateVector i) (coordinateVector j)
+    (Complex.I • coordinateVector k)
+  have h2 := g.derivative_closed a hz (Complex.I • coordinateVector i)
+    (Complex.I • coordinateVector j) (Complex.I • coordinateVector k)
+  have h3 := g.derivative_closed a hz (coordinateVector i) (coordinateVector j)
+    (coordinateVector k)
+  have h4 := g.derivative_closed a hz (Complex.I • coordinateVector i)
+    (Complex.I • coordinateVector j) (coordinateVector k)
+  simp only [fundamentalForm_coordinate, fundamentalForm_I_coordinate,
+    fundamentalForm_coordinate_I, fundamentalForm_I_I] at h1 h2 h3 h4
+  have hr := congrArg Complex.re ((g.derivative_hermitian a hz (coordinateVector j)).apply i k)
+  have hi := congrArg Complex.im ((g.derivative_hermitian a hz (coordinateVector j)).apply i k)
+  have hIr := congrArg Complex.re
+    ((g.derivative_hermitian a hz (Complex.I • coordinateVector j)).apply i k)
+  have hIi := congrArg Complex.im
+    ((g.derivative_hermitian a hz (Complex.I • coordinateVector j)).apply i k)
+  simp only [Complex.star_def, Complex.conj_re, Complex.conj_im] at hr hi hIr hIi
+  apply Complex.ext
+  · simp only [barDerivative, Matrix.smul_apply, Matrix.add_apply, smul_eq_mul,
+      Complex.mul_re, Complex.mul_im, Complex.add_re, Complex.add_im]
+    norm_num
+    linarith
+  · simp only [barDerivative, Matrix.smul_apply, Matrix.add_apply, smul_eq_mul,
+      Complex.mul_re, Complex.mul_im, Complex.add_re, Complex.add_im]
+    norm_num
+    linarith
+
+lemma barDerivative_comp {f : Coordinates d → Matrix (Fin d) (Fin d) ℂ}
+    {F : Matrix (Fin d) (Fin d) ℂ → Matrix (Fin d) (Fin d) ℂ}
+    {z : Coordinates d} (hf : DifferentiableAt ℝ f z)
+    (hF : DifferentiableAt ℂ F (f z)) (j : Fin d) :
+    barDerivative (F ∘ f) z j = fderiv ℂ F (f z) (barDerivative f z j) := by
+  have hd := ((hF.hasFDerivAt.restrictScalars ℝ).comp z hf.hasFDerivAt).fderiv
+  unfold barDerivative
+  have h1 := congrArg (fun L : Coordinates d →L[ℝ] Matrix (Fin d) (Fin d) ℂ =>
+    L (coordinateVector j)) hd
+  have h2 := congrArg (fun L : Coordinates d →L[ℝ] Matrix (Fin d) (Fin d) ℂ =>
+    L (Complex.I • coordinateVector j)) hd
+  calc
+    _ = (2 : ℂ)⁻¹ • (fderiv ℂ F (f z) (fderiv ℝ f z (coordinateVector j)) +
+        Complex.I • fderiv ℂ F (f z) (fderiv ℝ f z (Complex.I • coordinateVector j))) :=
+      congrArg₂ (fun U V : Matrix (Fin d) (Fin d) ℂ => (2 : ℂ)⁻¹ • (U + Complex.I • V)) h1 h2
+    _ = _ := by
+      simp only [map_smul (fderiv ℂ F (f z)), map_add (fderiv ℂ F (f z))]
+
+private lemma sum_swap_outer_inner (f : Fin d → Fin d → Fin d → ℂ) :
+    ∑ i, ∑ j, ∑ k, f i j k = ∑ k, ∑ j, ∑ i, f i j k := by
+  calc
+    _ = ∑ j, ∑ i, ∑ k, f i j k := Finset.sum_comm
+    _ = ∑ j, ∑ k, ∑ i, f i j k := by
+      apply Finset.sum_congr rfl
+      intro j _
+      exact Finset.sum_comm
+    _ = _ := Finset.sum_comm
+
+lemma cofactor_cancellation (B : Matrix (Fin d) (Fin d) ℂ)
+    (C : Fin d → Matrix (Fin d) (Fin d) ℂ)
+    (hC : ∀ i j k, C i j k = C j i k) (k : Fin d) :
+    ∑ i, ((B * C i).trace * B k i - (B * C i * B) k i) = 0 := by
+  rw [Finset.sum_sub_distrib, sub_eq_zero]
+  simp only [Matrix.trace, Matrix.diag, Matrix.mul_apply, Finset.sum_mul]
+  rw [sum_swap_outer_inner]
+  apply Finset.sum_congr rfl
+  intro i _
+  apply Finset.sum_congr rfl
+  intro j _
+  apply Finset.sum_congr rfl
+  intro l _
+  rw [hC l i j]
+  ring
+
+theorem weightedInverse_divergence (g : KaehlerMetric A) (a : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart a).target) (k : Fin d) :
+    ∑ i, barDerivative (MongeAmpere.weightedInverse ∘ g.matrix a) z i k i = 0 := by
+  have hg := ((g.smooth a).contDiffAt ((A.chart a).open_target.mem_nhds hz)).differentiableAt
+    (by simp)
+  have hdet : IsUnit (g.matrix a z).det :=
+    isUnit_iff_ne_zero.mpr (ne_of_gt (g.positive a z hz).det_pos)
+  simp only [barDerivative_comp hg (MongeAmpere.differentiableAt_weightedInverse _ hdet),
+    MongeAmpere.fderiv_weightedInverse_apply _ _ hdet, Matrix.smul_apply,
+    Matrix.sub_apply, smul_eq_mul]
+  rw [← Finset.mul_sum]
+  rw [cofactor_cancellation _ _ (g.barDerivative_symm a hz)]
+  exact mul_zero _
+
+end KaehlerMetric
+end Anticanonical.SourceSmooth
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/KahlerEnergyChain.lean b/lean/OAI/Geometry/PrescribedRicci/KahlerEnergyChain.lean
new file mode 100644
index 0000000000000000000000000000000000000000..73ab14731ea71b89f8a3207576fd105adc60db6f
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/KahlerEnergyChain.lean
@@ -0,0 +1,76 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.KahlerGradientEnergy
+import OAI.Geometry.PrescribedRicci.GlobalKahlerEnergy
+
+namespace OAI
+
+/-! Kahler Energy Chain. -/
+
+section
+
+ 
+noncomputable section
+open Matrix Filter Set Topology MeasureTheory
+open scoped ContDiff ComplexOrder Classical
+namespace Anticanonical.SourceSmooth
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+
+namespace SmoothRealFunction
+
+def compose (φ : SmoothRealFunction A) (F : ℝ → ℝ) (hF : ContDiff ℝ ∞ F) :
+    SmoothRealFunction A where
+  value x := F (φ.value x)
+  smooth i := hF.comp_contDiffOn (φ.smooth i)
+
+end SmoothRealFunction
+namespace KaehlerMetric
+
+lemma holRealDeriv_comp {φ : Coordinates d → ℝ} {F : ℝ → ℝ} {z : Coordinates d}
+    (hφ : DifferentiableAt ℝ φ z) (hF : DifferentiableAt ℝ F (φ z)) :
+    holRealDeriv (F ∘ φ) z = fun j => ((deriv F (φ z) : ℝ) : ℂ) * holRealDeriv φ z j := by
+  funext j
+  unfold holRealDeriv
+  rw [(hF.hasDerivAt.comp_hasFDerivAt z hφ.hasFDerivAt).fderiv]
+  rw [map_smul]
+  rfl
+
+lemma gradientPair_real_smul_left (H : Matrix (Fin d) (Fin d) ℂ)
+    (p q : Fin d → ℂ) (a : ℝ) :
+    gradientPair H (fun i => (a : ℂ) * p i) q = (a : ℂ) * gradientPair H p q := by
+  simp only [gradientPair, star_mul, Complex.star_def, Complex.conj_ofReal, Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro j _
+  apply Finset.sum_congr rfl
+  intro i _
+  ring
+
+lemma energy_comp_left (g : KaehlerMetric A) (ψ φ : SmoothRealFunction A)
+    (F : ℝ → ℝ) (hF : ContDiff ℝ ∞ F) (x : X) :
+    (g.energy (ψ.compose F hF) φ).value x =
+      deriv F (ψ.value x) * (g.energy ψ φ).value x := by
+  obtain ⟨i, hi⟩ := A.covers x
+  have hz := (A.chart i).mapsTo hi
+  have he := g.energy_local (ψ.compose F hF) φ i hz
+  have he' := g.energy_local ψ φ i hz
+  simp only [SmoothRealFunction.localExpression, Function.comp_apply, (A.chart i).left_inv hi] at he he'
+  rw [he, he']
+  change (gradientPair _ (holRealDeriv (F ∘ ψ.localExpression i) _) _).re = _
+  rw [holRealDeriv_comp (φ := ψ.localExpression i) (((ψ.smooth i).contDiffAt ((A.chart i).open_target.mem_nhds hz)).differentiableAt (by simp))
+      (hF.differentiable (by simp) _), gradientPair_real_smul_left]
+  simp only [Complex.mul_re, Complex.ofReal_re, Complex.ofReal_im, zero_mul, sub_zero,
+    SmoothRealFunction.localExpression, Function.comp_apply, (A.chart i).left_inv hi]
+
+lemma energy_comp_self (g : KaehlerMetric A) (φ : SmoothRealFunction A)
+    (F : ℝ → ℝ) (hF : ContDiff ℝ ∞ F) (x : X) :
+    (g.energy (φ.compose F hF) (φ.compose F hF)).value x =
+      (deriv F (φ.value x)) ^ 2 * (g.energy φ φ).value x := by
+  rw [g.energy_comp_left, g.energy_symm φ (φ.compose F hF), g.energy_comp_left]
+  ring
+
+end KaehlerMetric
+end Anticanonical.SourceSmooth
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/KahlerEnergyCutoff.lean b/lean/OAI/Geometry/PrescribedRicci/KahlerEnergyCutoff.lean
new file mode 100644
index 0000000000000000000000000000000000000000..83b7c4132deae2fd3856d8c1916a4feed6851d82
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/KahlerEnergyCutoff.lean
@@ -0,0 +1,69 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.GlobalSobolev
+import OAI.Geometry.PrescribedRicci.KahlerEnergyProduct
+import OAI.Geometry.PrescribedRicci.MongeAmpereEnergy
+
+namespace OAI
+
+/-! Kahler Energy Cutoff. -/
+
+section
+
+ 
+
+noncomputable section
+open Matrix Set Filter Topology MeasureTheory
+open scoped ContDiff Classical
+namespace Anticanonical.SourceSmooth.KaehlerMetric
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X}
+
+omit [T2Space X] in
+lemma cutoff_energy_bound (g : KaehlerMetric A) (ρ : SmoothRealFunction A) :
+    ∃ C : ℝ, 0 < C ∧ ∀ (φ : SmoothRealFunction A) (x : X),
+      (g.energy (ρ.product φ) (ρ.product φ)).value x ≤
+        C * ((g.energy φ φ).value x + φ.value x ^ 2) := by
+  let F : C(X,ℝ) := ⟨ρ.value,ρ.continuous⟩
+  let E : C(X,ℝ) := ⟨(g.energy ρ ρ).value,(g.energy ρ ρ).continuous⟩
+  let C : ℝ := 2*(‖F‖^2+‖E‖+1)
+  have hC : 0 < C := by dsimp [C]; positivity
+  refine ⟨C,hC,fun φ x => ?_⟩
+  have hF : ρ.value x ^ 2 ≤ ‖F‖^2 := by
+    have hh := F.norm_coe_le_norm x
+    have hs := (sq_le_sq₀ (norm_nonneg _) (norm_nonneg _)).2 hh
+    simpa only [Real.norm_eq_abs, sq_abs, F, ContinuousMap.coe_mk] using hs
+  have hE : (g.energy ρ ρ).value x ≤ ‖E‖ := by
+    exact (le_abs_self _).trans (by simpa only [Real.norm_eq_abs, E, ContinuousMap.coe_mk] using E.norm_coe_le_norm x)
+  have hcoefF : 2*ρ.value x^2 ≤ C := by dsimp [C]; nlinarith [norm_nonneg E]
+  have hcoefE : 2*(g.energy ρ ρ).value x ≤ C := by dsimp [C]; nlinarith [sq_nonneg ‖F‖]
+  calc
+    _ ≤ 2*ρ.value x^2*(g.energy φ φ).value x + 2*φ.value x^2*(g.energy ρ ρ).value x :=
+      g.energy_product_le ρ φ x
+    _ = (2*ρ.value x^2)*(g.energy φ φ).value x + (2*(g.energy ρ ρ).value x)*φ.value x^2 := by ring
+    _ ≤ C*(g.energy φ φ).value x + C*φ.value x^2 :=
+      add_le_add (mul_le_mul_of_nonneg_right hcoefF (g.energy_nonneg φ x))
+        (mul_le_mul_of_nonneg_right hcoefE (sq_nonneg _))
+    _ = _ := by ring
+
+lemma cutoff_integral_energy_bound (g : KaehlerMetric A) (ρ : SmoothRealFunction A) :
+    ∃ C : ℝ, 0 < C ∧ ∀ φ : SmoothRealFunction A,
+      g.integral (g.energy (ρ.product φ) (ρ.product φ)).value ≤
+        C * (g.integral (g.energy φ φ).value + g.integral (fun x => φ.value x^2)) := by
+  obtain ⟨C,hC,hbound⟩ := g.cutoff_energy_bound ρ
+  refine ⟨C,hC,fun φ => ?_⟩
+  have hh := g.integral_mono (g.energy (ρ.product φ) (ρ.product φ)).continuous
+    (continuous_const.fun_mul ((g.energy φ φ).continuous.fun_add (φ.continuous.pow 2))) (hbound φ)
+  rwa [g.integral_const_mul, g.integral_add (g.energy φ φ).continuous (φ.continuous.pow 2)] at hh
+
+omit [T2Space X] [CompactSpace X] in
+lemma product_support_left (ρ φ : SmoothRealFunction A) :
+    tsupport (ρ.product φ).value ⊆ tsupport ρ.value := by
+  apply closure_minimal _ (isClosed_tsupport _)
+  exact (Function.support_mul_subset_left _ _).trans subset_closure
+
+end Anticanonical.SourceSmooth.KaehlerMetric
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/KahlerEnergyProduct.lean b/lean/OAI/Geometry/PrescribedRicci/KahlerEnergyProduct.lean
new file mode 100644
index 0000000000000000000000000000000000000000..55637d50e1502e3335bd0e2e5c798f6aaf7aac01
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/KahlerEnergyProduct.lean
@@ -0,0 +1,86 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.GlobalKahlerEnergy
+import OAI.Geometry.PrescribedRicci.KahlerEnergyChain
+
+namespace OAI
+
+/-! Kahler Energy Product. -/
+
+section
+
+ 
+noncomputable section
+open Matrix Set Filter Topology MeasureTheory
+open scoped ContDiff ComplexOrder Classical
+namespace Anticanonical.SourceSmooth.KaehlerMetric
+variable {d : ℕ}
+
+lemma gradientPair_add_right (H : Matrix (Fin d) (Fin d) ℂ) (p q r : Fin d → ℂ) :
+    gradientPair H p (q+r) = gradientPair H p q + gradientPair H p r := by
+  simp [gradientPair, Pi.add_apply, mul_add, Finset.sum_add_distrib]
+
+lemma gradientPair_real_smul_right (H : Matrix (Fin d) (Fin d) ℂ)
+    (p q : Fin d → ℂ) (a : ℝ) :
+    gradientPair H p (fun i => (a : ℂ)*q i) = (a : ℂ)*gradientPair H p q := by
+  simp only [gradientPair, Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro j _
+  apply Finset.sum_congr rfl
+  intro i _
+  ring
+
+lemma gradientPair_smul_self (H : Matrix (Fin d) (Fin d) ℂ) (p : Fin d → ℂ) (a : ℝ) :
+    (gradientPair H (fun i => (a:ℂ)*p i) (fun i => (a:ℂ)*p i)).re =
+      a^2 * (gradientPair H p p).re := by
+  rw [gradientPair_real_smul_left, gradientPair_real_smul_right]
+  simp only [Complex.mul_re, Complex.ofReal_re, Complex.ofReal_im, zero_mul, sub_zero]
+  ring
+
+lemma gradientPair_add_self_le (H : Matrix (Fin d) (Fin d) ℂ) (hH : H.PosDef)
+    (p q : Fin d → ℂ) :
+    (gradientPair H (p+q) (p+q)).re ≤
+      2 * (gradientPair H p p).re + 2 * (gradientPair H q q).re := by
+  have hh := gradientPair_self_nonneg H hH (p + fun i => ((-1:ℝ):ℂ)*q i)
+  simp only [gradientPair_add_left, gradientPair_add_right,
+    gradientPair_real_smul_left, gradientPair_real_smul_right,
+    Complex.add_re, Complex.mul_re, Complex.ofReal_re, Complex.ofReal_im,
+    zero_mul, sub_zero] at hh ⊢
+  nlinarith
+
+variable {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+lemma energy_product_le (g : KaehlerMetric A) (ψ φ : SmoothRealFunction A) (x : X) :
+    (g.energy (ψ.product φ) (ψ.product φ)).value x ≤
+      2 * ψ.value x ^ 2 * (g.energy φ φ).value x +
+      2 * φ.value x ^ 2 * (g.energy ψ ψ).value x := by
+  obtain ⟨i,hi⟩ := A.covers x
+  have hz := (A.chart i).mapsTo hi
+  have he := g.energy_local (ψ.product φ) (ψ.product φ) i hz
+  have heψ := g.energy_local ψ ψ i hz
+  have heφ := g.energy_local φ φ i hz
+  simp only [SmoothRealFunction.localExpression, Function.comp_apply,
+    (A.chart i).left_inv hi] at he heψ heφ
+  rw [he,heψ,heφ]
+  have hd : holRealDeriv ((ψ.product φ).localExpression i) ((A.chart i) x) =
+      (fun j => (ψ.value x : ℂ) * holRealDeriv (φ.localExpression i) ((A.chart i) x) j) +
+      (fun j => (φ.value x : ℂ) * holRealDeriv (ψ.localExpression i) ((A.chart i) x) j) := by
+    funext j
+    change holRealDeriv (fun y => ψ.localExpression i y * φ.localExpression i y) _ j = _
+    rw [holRealDeriv_mul (φ := ψ.localExpression i) (ψ := φ.localExpression i) (((ψ.smooth i).contDiffAt ((A.chart i).open_target.mem_nhds hz)).differentiableAt (by simp))
+      (((φ.smooth i).contDiffAt ((A.chart i).open_target.mem_nhds hz)).differentiableAt (by simp))]
+    simp only [SmoothRealFunction.localExpression, Function.comp_apply,
+      (A.chart i).left_inv hi, Pi.add_apply]
+  change (gradientPair _ (holRealDeriv ((ψ.product φ).localExpression i) _)
+    (holRealDeriv ((ψ.product φ).localExpression i) _)).re ≤ _
+  rw [hd]
+  have hh := gradientPair_add_self_le _ (g.positive i _ hz)
+    (fun j => (ψ.value x : ℂ)*holRealDeriv (φ.localExpression i) ((A.chart i) x) j)
+    (fun j => (φ.value x : ℂ)*holRealDeriv (ψ.localExpression i) ((A.chart i) x) j)
+  rw [gradientPair_smul_self,gradientPair_smul_self] at hh
+  simpa only [mul_assoc, SmoothRealFunction.localExpression] using hh
+
+end Anticanonical.SourceSmooth.KaehlerMetric
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/KahlerGradientCoercivity.lean b/lean/OAI/Geometry/PrescribedRicci/KahlerGradientCoercivity.lean
new file mode 100644
index 0000000000000000000000000000000000000000..680ce9b89dbc704ab690028bc0fa7aa4a0295c32
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/KahlerGradientCoercivity.lean
@@ -0,0 +1,145 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.KahlerGradientEnergy
+
+namespace OAI
+
+/-! Kahler Gradient Coercivity. -/
+
+noncomputable section
+open Matrix Set Filter Topology MeasureTheory
+open scoped ContDiff ComplexOrder Classical Matrix.Norms.Elementwise
+namespace Anticanonical.SourceSmooth.KaehlerMetric
+variable {d : ℕ}
+
+lemma holRealForm_injective : Function.Injective
+    (fun L : Coordinates d →L[ℝ] ℝ => fun j => holRealForm j L) := by
+  suffices hz : ∀ L : Coordinates d →L[ℝ] ℝ, (∀ j, holRealForm j L = 0) → L = 0 by
+    intro L M he
+    have hh := hz (L-M) (by intro j; simp only [map_sub]; exact sub_eq_zero.mpr (congrFun he j))
+    exact sub_eq_zero.mp hh
+  intro L hL
+  have hre (j : Fin d) : L (coordinateVector j) = 0 := by
+    have hh := congrArg Complex.re (hL j)
+    simp [holRealForm] at hh
+    linarith
+  have him (j : Fin d) : L (Complex.I • coordinateVector j) = 0 := by
+    have hh := congrArg Complex.im (hL j)
+    simp [holRealForm] at hh
+    linarith
+  apply ContinuousLinearMap.ext
+  intro v
+  have he : v = ∑ j, ((v j).re • coordinateVector j + (v j).im • (Complex.I • coordinateVector j)) := by
+    ext k
+    simp only [coordinateVector, Finset.sum_apply, Pi.add_apply, Pi.smul_apply, smul_eq_mul,
+      Complex.real_smul, Pi.single_apply]
+    simp [mul_ite, Finset.sum_add_distrib]
+  rw [he, map_sum]
+  simp only [map_add, map_smul, hre, him, smul_zero, add_zero, Finset.sum_const_zero,
+    _root_.zero_apply]
+
+def formEnergy (H : Matrix (Fin d) (Fin d) ℂ) (L : Coordinates d →L[ℝ] ℝ) : ℝ :=
+  (gradientPair H (fun j => holRealForm j L) (fun j => holRealForm j L)).re
+
+lemma formEnergy_pos (H : Matrix (Fin d) (Fin d) ℂ) (hH : H.PosDef)
+    (L : Coordinates d →L[ℝ] ℝ) (hL : L ≠ 0) : 0 < formEnergy H L := by
+  let p := fun j => holRealForm j L
+  have hp : p ≠ 0 := by
+    intro hh
+    apply hL
+    apply holRealForm_injective
+    funext j
+    change holRealForm j L = holRealForm j 0
+    rw [map_zero]
+    exact congrFun hh j
+  have hps : star p ≠ 0 := by simpa using hp
+  have hh := hH.inv.re_dotProduct_pos hps
+  have he : gradientPair H p p = star (star p) ⬝ᵥ (H⁻¹ *ᵥ star p) := by
+    simp only [gradientPair, dotProduct, Matrix.mulVec, Pi.star_apply, star_star, Finset.mul_sum]
+    apply Finset.sum_congr rfl
+    intro j _
+    apply Finset.sum_congr rfl
+    intro i _
+    ring
+  change 0 < (gradientPair H p p).re
+  rw [he]
+  exact hh
+
+lemma formEnergy_smul (H : Matrix (Fin d) (Fin d) ℂ) (c : ℝ)
+    (L : Coordinates d →L[ℝ] ℝ) : formEnergy H (c • L) = c^2 * formEnergy H L := by
+  unfold formEnergy gradientPair
+  simp only [map_smul, Complex.real_smul, star_mul, Complex.star_def, Complex.conj_ofReal]
+  have he : (∑ j, ∑ i, H⁻¹ j i * (star ((c : ℂ) * holRealForm i L)) * ((c : ℂ) * holRealForm j L)) =
+      (c : ℂ)^2 * ∑ j, ∑ i, H⁻¹ j i * star (holRealForm i L) * holRealForm j L := by
+    simp only [star_mul, Complex.star_def, Complex.conj_ofReal, Finset.mul_sum]
+    apply Finset.sum_congr rfl
+    intro j _
+    apply Finset.sum_congr rfl
+    intro i _
+    ring
+  simp only [star_mul, Complex.star_def, Complex.conj_ofReal] at he
+  rw [he, ← Complex.ofReal_pow]
+  simp only [Complex.mul_re, Complex.ofReal_re, Complex.ofReal_im, zero_mul, sub_zero]
+
+variable {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+
+lemma continuousOn_formEnergy (g : KaehlerMetric A) (i : Fin A.count) :
+    ContinuousOn (fun z : Coordinates d × (Coordinates d →L[ℝ] ℝ) => formEnergy (g.matrix i z.1) z.2)
+      ((A.chart i).target ×ˢ Set.univ) := by
+  have hH : ContinuousOn (fun z => (g.matrix i z)⁻¹) (A.chart i).target := by
+    intro z hz
+    exact (MongeAmpere.contDiffAt_inv _ (ne_of_gt (g.positive i z hz).det_pos)).continuousAt.comp_continuousWithinAt
+      ((g.smooth i).continuousOn z hz)
+  unfold formEnergy gradientPair
+  apply Complex.continuous_re.comp_continuousOn
+  apply continuousOn_finsetSum
+  intro j _
+  apply continuousOn_finsetSum
+  intro k _
+  exact (((continuousOn_pi.mp (continuousOn_pi.mp hH j) k).comp continuous_fst.continuousOn
+    (fun z hz => hz.1)).mul ((holRealForm k).continuous.comp continuous_snd).star.continuousOn).mul
+      ((holRealForm j).continuous.comp continuous_snd).continuousOn
+
+lemma compact_gradient_coercivity (g : KaehlerMetric A) (i : Fin A.count)
+    {K : Set (Coordinates d)} (hK : IsCompact K) (hs : K ⊆ (A.chart i).target) :
+    ∃ a : ℝ, 0 < a ∧ ∀ z ∈ K, ∀ L : Coordinates d →L[ℝ] ℝ,
+      a * ‖L‖^2 ≤ formEnergy (g.matrix i z) L := by
+  let S : Set (Coordinates d × (Coordinates d →L[ℝ] ℝ)) := K ×ˢ Metric.sphere 0 1
+  have hc : IsCompact S := hK.prod (isCompact_sphere 0 1)
+  have hcF := (g.continuousOn_formEnergy i).mono (show S ⊆ (A.chart i).target ×ˢ univ from
+    fun z hz => ⟨hs hz.1, mem_univ _⟩)
+  by_cases hn : S.Nonempty
+  · obtain ⟨⟨z₀,L₀⟩,hm,hlow⟩ := hc.exists_isMinOn hn hcF
+    have hL₀ : L₀ ≠ 0 := by
+      intro h
+      have := hm.2
+      simp [h] at this
+    refine ⟨formEnergy (g.matrix i z₀) L₀, formEnergy_pos _ (g.positive i z₀ (hs hm.1)) L₀ hL₀, ?_⟩
+    intro z hz L
+    by_cases hL : L = 0
+    · simp [hL, formEnergy, gradientPair]
+    · have hnorm : 0 < ‖L‖ := norm_pos_iff.mpr hL
+      have hnL : ‖(‖L‖⁻¹ : ℝ) • L‖ = 1 := by
+        rw [norm_smul, Real.norm_eq_abs, abs_of_pos (inv_pos.mpr hnorm), inv_mul_cancel₀ hnorm.ne']
+      have hh := hlow (show (z,(‖L‖⁻¹ : ℝ) • L) ∈ S from
+        ⟨hz, by simpa only [Metric.mem_sphere, dist_zero_right] using hnL⟩)
+      change formEnergy (g.matrix i z₀) L₀ ≤ formEnergy (g.matrix i z) ((‖L‖⁻¹ : ℝ) • L) at hh
+      rw [formEnergy_smul] at hh
+      have ht := mul_le_mul_of_nonneg_right hh (sq_nonneg ‖L‖)
+      have he : ‖L‖⁻¹ ^2 * formEnergy (g.matrix i z) L * ‖L‖^2 = formEnergy (g.matrix i z) L := by
+        calc
+          _ = formEnergy (g.matrix i z) L * (‖L‖⁻¹ ^2 * ‖L‖^2) := by ring
+          _ = _ := by rw [← mul_pow, inv_mul_cancel₀ hnorm.ne']; simp
+      rwa [he] at ht
+  · refine ⟨1,by norm_num,fun z hz L => ?_⟩
+    by_cases hL : L = 0
+    · simp [hL, formEnergy, gradientPair]
+    · have hnorm : 0 < ‖L‖ := norm_pos_iff.mpr hL
+      have hnL : ‖(‖L‖⁻¹ : ℝ) • L‖ = 1 := by
+        rw [norm_smul, Real.norm_eq_abs, abs_of_pos (inv_pos.mpr hnorm), inv_mul_cancel₀ hnorm.ne']
+      exact (hn ⟨(z,(‖L‖⁻¹ : ℝ) • L),hz,by simpa only [Metric.mem_sphere, dist_zero_right] using hnL⟩).elim
+
+end Anticanonical.SourceSmooth.KaehlerMetric
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/KahlerGradientEnergy.lean b/lean/OAI/Geometry/PrescribedRicci/KahlerGradientEnergy.lean
new file mode 100644
index 0000000000000000000000000000000000000000..2b99c98cb0f930c616d39b747a3315f7100fca1f
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/KahlerGradientEnergy.lean
@@ -0,0 +1,126 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.LocalKahlerEnergy
+
+namespace OAI
+
+/-! Kahler Gradient Energy. -/
+
+noncomputable section
+open Matrix Filter Set Topology MeasureTheory
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise Classical
+namespace Anticanonical.SourceSmooth.KaehlerMetric
+variable {d : ℕ}
+
+lemma barDeriv_congr {f h : Coordinates d → ℂ} {z : Coordinates d}
+    (he : f =ᶠ[𝓝 z] h) (i : Fin d) : barDeriv f z i = barDeriv h z i := by
+  unfold barDeriv
+  rw [he.fderiv_eq]
+
+lemma barDeriv_add {f h : Coordinates d → ℂ} {z : Coordinates d}
+    (hf : DifferentiableAt ℝ f z) (hh : DifferentiableAt ℝ h z) (i : Fin d) :
+    barDeriv (fun y => f y + h y) z i = barDeriv f z i + barDeriv h z i := by
+  unfold barDeriv
+  rw [fderiv_fun_add hf hh]
+  simp only [_root_.add_apply]
+  ring
+
+lemma barDeriv_ofReal {φ : Coordinates d → ℝ} {z : Coordinates d}
+    (hφ : DifferentiableAt ℝ φ z) (i : Fin d) :
+    barDeriv (fun y => (φ y : ℂ)) z i = star (holRealDeriv φ z i) := by
+  have hd := (Complex.ofRealCLM.hasFDerivAt.comp z hφ.hasFDerivAt).fderiv
+  change fderiv ℝ (fun y => (φ y : ℂ)) z = _ at hd
+  unfold barDeriv holRealDeriv holRealForm
+  rw [hd]
+  apply Complex.ext <;> simp
+
+lemma holRealDeriv_mul {φ ψ : Coordinates d → ℝ} {z : Coordinates d}
+    (hφ : DifferentiableAt ℝ φ z) (hψ : DifferentiableAt ℝ ψ z) (j : Fin d) :
+    holRealDeriv (fun y => φ y * ψ y) z j =
+      (φ z : ℂ) * holRealDeriv ψ z j + (ψ z : ℂ) * holRealDeriv φ z j := by
+  unfold holRealDeriv
+  rw [fderiv_fun_mul hφ hψ, map_add, map_smul, map_smul]
+  simp only [Complex.real_smul]
+
+lemma potentialMatrix_mul {φ ψ : Coordinates d → ℝ} {z : Coordinates d}
+    (hφ : ContDiffAt ℝ ∞ φ z) (hψ : ContDiffAt ℝ ∞ ψ z) (i j : Fin d) :
+    PotentialKaehler.potentialMatrix (fun y => φ y * ψ y) z i j =
+      (φ z : ℂ) * PotentialKaehler.potentialMatrix ψ z i j +
+      (ψ z : ℂ) * PotentialKaehler.potentialMatrix φ z i j +
+      star (holRealDeriv φ z i) * holRealDeriv ψ z j +
+      star (holRealDeriv ψ z i) * holRealDeriv φ z j := by
+  rw [← barDeriv_holRealDeriv (hφ.mul hψ)]
+  have hev : (fun y => holRealDeriv (fun w => φ w * ψ w) y j) =ᶠ[𝓝 z]
+      fun y => (φ y : ℂ) * holRealDeriv ψ y j + (ψ y : ℂ) * holRealDeriv φ y j := by
+    filter_upwards [(hφ.of_le (show (1 : ℕ∞ω) ≤ ∞ by simp)).eventually (by simp),
+      (hψ.of_le (show (1 : ℕ∞ω) ≤ ∞ by simp)).eventually (by simp)] with y hyφ hyψ
+    exact holRealDeriv_mul (hyφ.differentiableAt (by simp)) (hyψ.differentiableAt (by simp)) j
+  rw [barDeriv_congr hev]
+  have hdφ := hφ.differentiableAt (by simp)
+  have hdψ := hψ.differentiableAt (by simp)
+  have hcφ : DifferentiableAt ℝ (fun y => (φ y : ℂ)) z :=
+    Complex.ofRealCLM.differentiableAt.comp z hdφ
+  have hcψ : DifferentiableAt ℝ (fun y => (ψ y : ℂ)) z :=
+    Complex.ofRealCLM.differentiableAt.comp z hdψ
+  have hhφ := (contDiffAt_holRealDeriv hφ j).differentiableAt (by simp)
+  have hhψ := (contDiffAt_holRealDeriv hψ j).differentiableAt (by simp)
+  rw [barDeriv_add (hcφ.fun_mul hhψ) (hcψ.fun_mul hhφ),
+    barDeriv_mul hcφ hhψ, barDeriv_mul hcψ hhφ,
+    barDeriv_ofReal hdφ, barDeriv_ofReal hdψ,
+    barDeriv_holRealDeriv hψ, barDeriv_holRealDeriv hφ]
+  ring
+
+def gradientPair (H : Matrix (Fin d) (Fin d) ℂ) (p q : Fin d → ℂ) : ℂ :=
+  ∑ j, ∑ i, H⁻¹ j i * star (p i) * q j
+
+lemma gradientPair_star (H : Matrix (Fin d) (Fin d) ℂ) (hH : H.IsHermitian)
+    (p q : Fin d → ℂ) : star (gradientPair H p q) = gradientPair H q p := by
+  have hi := hH.inv
+  unfold gradientPair
+  simp only [star_sum, star_mul, star_star]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro j _
+  apply Finset.sum_congr rfl
+  intro i _
+  have he : star (H⁻¹ i j) = H⁻¹ j i := congrFun (congrFun hi.eq j) i
+  rw [he]
+  ring
+
+lemma gradientPair_self_nonneg (H : Matrix (Fin d) (Fin d) ℂ) (hH : H.PosDef)
+    (p : Fin d → ℂ) : 0 ≤ (gradientPair H p p).re := by
+  have hp := hH.inv.posSemidef.dotProduct_mulVec_nonneg (star p)
+  have he : gradientPair H p p = star (star p) ⬝ᵥ (H⁻¹ *ᵥ star p) := by
+    simp only [gradientPair, dotProduct, Matrix.mulVec, Pi.star_apply, star_star, Finset.mul_sum]
+    apply Finset.sum_congr rfl
+    intro j _
+    apply Finset.sum_congr rfl
+    intro i _
+    ring
+  rw [he]
+  exact (Complex.nonneg_iff.mp hp).1
+
+lemma trace_hessian_product {φ ψ : Coordinates d → ℝ} {z : Coordinates d}
+    (hφ : ContDiffAt ℝ ∞ φ z) (hψ : ContDiffAt ℝ ∞ ψ z)
+    (H : Matrix (Fin d) (Fin d) ℂ) (hH : H.IsHermitian) :
+    ((H⁻¹ * PotentialKaehler.potentialMatrix (fun y => φ y * ψ y) z).trace).re =
+      φ z * ((H⁻¹ * PotentialKaehler.potentialMatrix ψ z).trace).re +
+      ψ z * ((H⁻¹ * PotentialKaehler.potentialMatrix φ z).trace).re +
+      2 * (gradientPair H (holRealDeriv φ z) (holRealDeriv ψ z)).re := by
+  have he : (H⁻¹ * PotentialKaehler.potentialMatrix (fun y => φ y * ψ y) z).trace =
+      (φ z : ℂ) * (H⁻¹ * PotentialKaehler.potentialMatrix ψ z).trace +
+      (ψ z : ℂ) * (H⁻¹ * PotentialKaehler.potentialMatrix φ z).trace +
+      gradientPair H (holRealDeriv φ z) (holRealDeriv ψ z) +
+      gradientPair H (holRealDeriv ψ z) (holRealDeriv φ z) := by
+    simp only [Matrix.trace, Matrix.diag, Matrix.mul_apply, potentialMatrix_mul hφ hψ,
+      gradientPair, Finset.mul_sum, mul_add, Finset.sum_add_distrib]
+    simp only [mul_assoc, mul_left_comm, mul_comm]
+  rw [he, ← gradientPair_star H hH]
+  simp only [Complex.add_re, Complex.mul_re, Complex.ofReal_re, Complex.ofReal_im,
+    zero_mul, sub_zero, Complex.star_def, Complex.conj_re]
+  ring
+
+end Anticanonical.SourceSmooth.KaehlerMetric
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/KahlerGreenIdentity.lean b/lean/OAI/Geometry/PrescribedRicci/KahlerGreenIdentity.lean
new file mode 100644
index 0000000000000000000000000000000000000000..1412bc414951de98bcfaa1f027c9c15b787a6850
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/KahlerGreenIdentity.lean
@@ -0,0 +1,99 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.GlobalKahlerIntegral
+import OAI.Geometry.PrescribedRicci.IntegratedKahlerEnergy
+import OAI.Geometry.PrescribedRicci.KahlerGradientEnergy
+
+namespace OAI
+
+/-! Kahler Green Identity. -/
+
+section
+
+ 
+
+noncomputable section
+open Matrix Filter Set Topology MeasureTheory
+open scoped ContDiff ComplexOrder Classical
+namespace Anticanonical.SourceSmooth
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+namespace KaehlerMetric
+
+lemma holRealDeriv_sum {ι : Type*} (s : Finset ι) {f : ι → Coordinates d → ℝ}
+    {z : Coordinates d} (hf : ∀ k ∈ s, DifferentiableAt ℝ (f k) z) :
+    holRealDeriv (fun y => ∑ k ∈ s, f k y) z = fun j => ∑ k ∈ s, holRealDeriv (f k) z j := by
+  funext j
+  unfold holRealDeriv
+  rw [fderiv_fun_sum hf, map_sum]
+
+lemma gradientPair_sum_left {ι : Type*} (s : Finset ι) (H : Matrix (Fin d) (Fin d) ℂ)
+    (p : ι → Fin d → ℂ) (q : Fin d → ℂ) :
+    gradientPair H (fun j => ∑ k ∈ s, p k j) q = ∑ k ∈ s, gradientPair H (p k) q := by
+  simp only [gradientPair, star_sum, Finset.mul_sum, Finset.sum_mul]
+  simp_rw [Finset.sum_comm (s := Finset.univ) (t := s)]
+
+lemma energy_partition [T2Space X] [CompactSpace X] (g : KaehlerMetric A)
+    (ψ φ : SmoothRealFunction A) (x : X) :
+    ∑ i, (g.energy ((chartPartition i).product ψ) φ).value x = (g.energy ψ φ).value x := by
+  obtain ⟨a, ha⟩ := A.covers x
+  have hz := (A.chart a).mapsTo ha
+  have he (i : Fin A.count) := g.energy_local ((chartPartition i).product ψ) φ a hz
+  have he' := g.energy_local ψ φ a hz
+  simp only [SmoothRealFunction.localExpression, Function.comp_apply, (A.chart a).left_inv ha] at he he'
+  simp_rw [he, he', ← Complex.re_sum]
+  rw [← gradientPair_sum_left]
+  have hs : (fun y => ∑ i, ((chartPartition i).product ψ).localExpression a y) = ψ.localExpression a := by
+    funext y
+    change (∑ i, (chartPartition i).value ((A.chart a).symm y) * ψ.value ((A.chart a).symm y)) = _
+    rw [← Finset.sum_mul, chartPartition_sum, one_mul]
+    rfl
+  have hd := holRealDeriv_sum Finset.univ
+    (f := fun i => ((chartPartition i).product ψ).localExpression a)
+    (fun i _ => ((((chartPartition i).product ψ).smooth a).contDiffAt
+      ((A.chart a).open_target.mem_nhds hz)).differentiableAt (by simp))
+  rw [hs] at hd
+  exact congrArg (fun p => (gradientPair (g.matrix a (A.chart a x)) p
+    (holRealDeriv (φ.value ∘ (A.chart a).symm) (A.chart a x))).re) hd.symm
+
+ 
+theorem green_identity [T2Space X] [CompactSpace X] (g : KaehlerMetric A)
+    (ψ φ : SmoothRealFunction A) :
+    g.integral (fun x => ψ.value x * (g.laplacian φ).value x) =
+      -g.integral (g.energy ψ φ).value := by
+  have hs (i : Fin A.count) : tsupport ((chartPartition i).product ψ).value ⊆ (A.chart i).source :=
+    tsupport_mul_subset_left.trans (chartPartition_support i)
+  have he (i : Fin A.count) :
+      g.chartIntegral i (fun x => (chartPartition i).value x * (ψ.value x * (g.laplacian φ).value x)) =
+        -g.integral (g.energy ((chartPartition i).product ψ) φ).value := by
+    rw [g.integral_chart i (g.energy ((chartPartition i).product ψ) φ).continuous
+      ((g.energy_support _ _).trans (hs i))]
+    have hh := g.chart_energy_identity ((chartPartition i).product ψ) φ i (hs i)
+    convert hh using 1
+    congr 1
+    funext x
+    change _ = ((chartPartition i).value x * ψ.value x) * (g.laplacian φ).value x
+    ring
+  change (∑ i, g.chartIntegral i (fun x => (chartPartition i).value x *
+    (ψ.value x * (g.laplacian φ).value x))) = _
+  simp_rw [he]
+  rw [Finset.sum_neg_distrib, ← g.integral_sum Finset.univ _
+    (fun i _ => (g.energy ((chartPartition i).product ψ) φ).continuous)]
+  simp_rw [g.energy_partition ψ φ]
+
+lemma integral_laplacian_mul_symm [T2Space X] [CompactSpace X] (g : KaehlerMetric A)
+    (ψ φ : SmoothRealFunction A) :
+    g.integral (fun x => ψ.value x * (g.laplacian φ).value x) =
+      g.integral (fun x => φ.value x * (g.laplacian ψ).value x) := by
+  rw [g.green_identity, g.green_identity, g.energy_symm]
+
+lemma integral_laplacian_self_nonpos [T2Space X] [CompactSpace X] (g : KaehlerMetric A)
+    (φ : SmoothRealFunction A) : g.integral (fun x => φ.value x * (g.laplacian φ).value x) ≤ 0 := by
+  rw [g.green_identity]
+  exact neg_nonpos.mpr (g.integral_nonneg (g.energy_nonneg φ))
+
+end KaehlerMetric
+end Anticanonical.SourceSmooth
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/KahlerIntegralPos.lean b/lean/OAI/Geometry/PrescribedRicci/KahlerIntegralPos.lean
new file mode 100644
index 0000000000000000000000000000000000000000..7d56a352a6457e50566dd33d295abbe9a2646d65
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/KahlerIntegralPos.lean
@@ -0,0 +1,81 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.GlobalKahlerIntegral
+import OAI.Geometry.PrescribedRicci.KahlerLocalL2Comparison
+import OAI.Geometry.PrescribedRicci.KahlerVolumeMeasure
+
+namespace OAI
+
+/-! Kahler Integral Pos. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory
+open scoped ContDiff Classical
+namespace Anticanonical.SourceSmooth.KaehlerMetric
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X}
+
+lemma integral_pos (g : KaehlerMetric A) {f : X → ℝ} (hf : Continuous f)
+    (hn : ∀ x, 0 ≤ f x) {x : X} (hx : 0 < f x) : 0 < g.integral f := by
+  obtain ⟨i,hi⟩ : ∃ i : Fin A.count, 0 < (chartPartition i).value x := by
+    by_contra! hh
+    have hsum := Finset.sum_nonpos (s:=Finset.univ) (f:=fun i : Fin A.count => (chartPartition i).value x) (fun i _ => hh i)
+    rw [chartPartition_sum] at hsum
+    linarith
+  let F : X → ℝ := fun y => (chartPartition i).value y * f y
+  have hF : Continuous F := (chartPartition i).continuous.fun_mul hf
+  have hFs : tsupport F ⊆ (A.chart i).source := tsupport_mul_subset_left.trans (chartPartition_support i)
+  obtain ⟨hl,hc,ht⟩ := localizeFunction_continuous_compact i hF hFs
+  let H : Coordinates d → ℝ := fun z => localizeFunction i F z * g.volumeCoefficient i z
+  have hH : Continuous H := continuous_real_mul_on_tsupport hl (fun z hz =>
+    ((g.volumeCoefficient_smooth i).continuousOn z (ht hz)).continuousAt ((A.chart i).open_target.mem_nhds (ht hz)))
+  have hHn (z : Coordinates d) : 0 ≤ H z := by
+    by_cases hz : z ∈ (A.chart i).target
+    · dsimp [H,localizeFunction]; rw [ite_eq_left hz]
+      exact mul_nonneg (mul_nonneg (chartPartition_nonneg i _) (hn _)) (g.volumeCoefficient_pos i hz).le
+    · simp [H,localizeFunction,hz]
+  have hxs : x ∈ (A.chart i).source := chartPartition_support i (subset_tsupport _ (ne_of_gt hi))
+  have hz := (A.chart i).mapsTo hxs
+  have hHx : 0 < H ((A.chart i) x) := by
+    dsimp only [H,localizeFunction]
+    rw [ite_eq_left hz,(A.chart i).left_inv hxs]
+    exact mul_pos (mul_pos hi hx) (g.volumeCoefficient_pos i hz)
+  have hI : 0 < g.chartIntegral i F := by
+    rw [g.chartIntegral_eq_localize]
+    exact hH.integral_pos_of_hasCompactSupport_nonneg_nonzero hc.mul_right hHn (ne_of_gt hHx)
+  apply hI.trans_le
+  change g.chartIntegral i F ≤ ∑ j, g.chartIntegral j (fun y => (chartPartition j).value y*f y)
+  apply Finset.single_le_sum (f:=fun j : Fin A.count => g.chartIntegral j (fun y => (chartPartition j).value y*f y)) _ (Finset.mem_univ i)
+  intro j _
+  apply setIntegral_nonneg (A.chart j).open_target.measurableSet
+  intro z hz
+  exact mul_nonneg (mul_nonneg (chartPartition_nonneg j _) (hn _)) (g.volumeCoefficient_pos j hz).le
+
+variable [MeasurableSpace X] [BorelSpace X]
+
+instance volumeMeasure_openPos (g : KaehlerMetric A) : g.volumeMeasure.IsOpenPosMeasure where
+  open_pos U hU hUn := by
+    obtain ⟨x,hx⟩ := hUn
+    obtain ⟨f,hf1,_,hfs,hfn⟩ := exists_continuousMap_one_of_isCompact_subset_isOpen
+      (isCompact_singleton (x:=x)) hU (singleton_subset_iff.mpr hx)
+    have hpos := g.integral_pos f.continuous (fun y => (hfn y).1)
+      (show 0 < f x by rw [hf1 (mem_singleton x)]; norm_num)
+    intro hzero
+    have hae : f =ᵐ[g.volumeMeasure] (fun _ => (0:ℝ)) := by
+      apply ae_iff.mpr
+      apply measure_mono_null _ hzero
+      intro y hy
+      exact hfs (subset_tsupport f hy)
+    have hz : g.integral f = 0 := by
+      rw [← g.integral_volumeMeasure f.continuous, integral_congr_ae hae, MeasureTheory.integral_zero]
+    linarith
+
+end Anticanonical.SourceSmooth.KaehlerMetric
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/KahlerIntegration.lean b/lean/OAI/Geometry/PrescribedRicci/KahlerIntegration.lean
new file mode 100644
index 0000000000000000000000000000000000000000..f6744619bde20064e6c4df825b4929173961e1b9
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/KahlerIntegration.lean
@@ -0,0 +1,157 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.AnalyticSupport
+import OAI.Geometry.PrescribedPotential.AtlasLocalization
+import OAI.Geometry.PrescribedPotential.VolumePath
+
+namespace OAI
+
+/-! Kahler Integration. -/
+
+section
+
+ 
+
+noncomputable section
+open Matrix Filter Set Topology MeasureTheory
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise Classical
+namespace Anticanonical
+namespace ComplexAtlas
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] (A : ComplexAtlas d X)
+
+def transitionHomeomorph (i j : Fin A.count) :
+    OpenPartialHomeomorph (Coordinates d) (Coordinates d) :=
+  (A.chart i).symm.trans (A.chart j)
+
+lemma transitionHomeomorph_source (i j : Fin A.count) :
+    (A.transitionHomeomorph i j).source = coordinateOverlap (A.chart i) (A.chart j) := rfl
+
+lemma transition_fderiv_det (i j : Fin A.count) {z : Coordinates d}
+    (hz : z ∈ coordinateOverlap (A.chart i) (A.chart j)) :
+    |(fderiv ℝ (A.transition i j) z).det| =
+      ‖A.jacobian i j ((A.chart i).symm z)‖ ^ 2 := by
+  have hd := (A.holomorphic i j).differentiableAt
+    ((A.isOpen_coordinateOverlap i j).mem_nhds hz)
+  change DifferentiableAt ℂ (A.transition i j) z at hd
+  rw [hd.fderiv_restrictScalars ℝ]
+  simp only [ContinuousLinearMap.det, ContinuousLinearMap.coe_restrictScalars,
+    LinearMap.det_restrictScalars, Algebra.norm_complex_eq, jacobian, (A.chart i).right_inv hz.1,
+    LinearMap.det_toMatrix']
+  change |Complex.normSq _| = _
+  rw [Complex.normSq_eq_norm_sq, abs_of_nonneg (sq_nonneg _)]
+
+lemma transition_hasFDerivAt (i j : Fin A.count) {z : Coordinates d}
+    (hz : z ∈ (A.transitionHomeomorph i j).source) :
+    HasFDerivAt (A.transitionHomeomorph i j) (fderiv ℝ (A.transition i j) z) z :=
+  ((A.real_smooth_transition i j).contDiffAt
+    ((A.isOpen_coordinateOverlap i j).mem_nhds hz)).differentiableAt (by simp) |>.hasFDerivAt
+
+end ComplexAtlas
+namespace SourceSmooth
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+namespace SmoothRealFunction
+
+def product (φ ψ : SmoothRealFunction A) : SmoothRealFunction A where
+  value x := φ.value x * ψ.value x
+  smooth i := (φ.smooth i).mul (ψ.smooth i)
+
+ 
+
+def localized (φ : SmoothRealFunction A) (i : Fin A.count) (z : Coordinates d) : ℝ :=
+  if z ∈ (A.chart i).target then φ.localExpression i z else 0
+
+lemma localized_eq {φ : SmoothRealFunction A} {i : Fin A.count} {z : Coordinates d}
+    (hz : z ∈ (A.chart i).target) : φ.localized i z = φ.localExpression i z := ite_eq_left hz
+
+lemma localized_support (φ : SmoothRealFunction A) (i : Fin A.count) :
+    Function.support (φ.localized i) ⊆ (A.chart i) '' tsupport φ.value := by
+  intro z hz
+  change φ.localized i z ≠ 0 at hz
+  by_cases ht : z ∈ (A.chart i).target
+  · have hh : φ.value ((A.chart i).symm z) ≠ 0 := by
+      simpa only [localized_eq ht, localExpression, Function.comp_apply] using hz
+    exact ⟨(A.chart i).symm z, subset_tsupport _ hh, (A.chart i).right_inv ht⟩
+  · exact (hz (ite_eq_right ht)).elim
+
+lemma localized_smooth_compact [T2Space X] [CompactSpace X]
+    (φ : SmoothRealFunction A) (i : Fin A.count)
+    (hφ : tsupport φ.value ⊆ (A.chart i).source) :
+    ContDiff ℝ ∞ (φ.localized i) ∧ HasCompactSupport (φ.localized i) ∧
+      tsupport (φ.localized i) ⊆ (A.chart i).target := by
+  let e := A.chart i
+  let K := e '' tsupport φ.value
+  have hK : IsCompact K := (isClosed_tsupport _).isCompact.image_of_continuousOn
+    (e.continuousOn.mono hφ)
+  have hs : Function.support (φ.localized i) ⊆ K := φ.localized_support i
+  have hts : tsupport (φ.localized i) ⊆ K := closure_minimal hs hK.isClosed
+  refine ⟨?_, HasCompactSupport.of_support_subset_isCompact hK hs, ?_⟩
+  · rw [contDiff_iff_contDiffAt]
+    intro z
+    by_cases hz : z ∈ e.target
+    · apply ((φ.smooth i).contDiffAt (e.open_target.mem_nhds hz)).congr_of_eventuallyEq
+      filter_upwards [e.open_target.mem_nhds hz] with y hy using φ.localized_eq hy
+    · have hzK : z ∉ K := by
+        rintro ⟨x, hx, rfl⟩
+        exact hz (e.mapsTo (hφ hx))
+      apply (contDiffAt_const (c := (0 : ℝ))).congr_of_eventuallyEq
+      filter_upwards [hK.isClosed.isOpen_compl.mem_nhds hzK] with y hy
+      exact Function.notMem_support.mp (fun h => hy (hs h))
+  · rintro z hz
+    obtain ⟨x, hx, rfl⟩ := hts hz
+    exact e.mapsTo (hφ hx)
+
+end SmoothRealFunction
+namespace KaehlerMetric
+
+def chartIntegral (g : KaehlerMetric A) (i : Fin A.count) (f : X → ℝ) : ℝ :=
+  ∫ z in (A.chart i).target, f ((A.chart i).symm z) * g.volumeCoefficient i z
+
+ 
+
+theorem chartIntegral_eq (g : KaehlerMetric A) (i j : Fin A.count) (f : X → ℝ)
+    (hi : Function.support f ⊆ (A.chart i).source)
+    (hj : Function.support f ⊆ (A.chart j).source) :
+    g.chartIntegral i f = g.chartIntegral j f := by
+  let e := A.transitionHomeomorph i j
+  have hs (a b : Fin A.count) (hb : Function.support f ⊆ (A.chart b).source) :
+      (∫ z in (A.transitionHomeomorph a b).source,
+        f ((A.chart a).symm z) * g.volumeCoefficient a z) = g.chartIntegral a f := by
+    symm
+    apply setIntegral_eq_of_subset_of_forall_sdiff_eq_zero (A.chart a).open_target.measurableSet
+      (fun _ h => h.1)
+    intro z hz
+    have hn : (A.chart a).symm z ∉ (A.chart b).source := by
+      intro h
+      exact hz.2 ⟨hz.1, h⟩
+    have hf : f ((A.chart a).symm z) = 0 := Function.notMem_support.mp (fun h => hn (hb h))
+    rw [hf, zero_mul]
+  rw [← hs i j hj, ← hs j i hi]
+  have heq := integral_target_eq_integral_abs_det_fderiv_smul (μ := (volume : Measure (Coordinates d)))
+    (f := e) (f' := fun z => fderiv ℝ (A.transition i j) z)
+    (fun z hz => A.transition_hasFDerivAt i j hz)
+    (fun z => f ((A.chart j).symm z) * g.volumeCoefficient j z)
+  have htarget : e.target = (A.transitionHomeomorph j i).source := by
+    ext z
+    rfl
+  rw [htarget] at heq
+  rw [heq]
+  apply setIntegral_congr_fun e.open_source.measurableSet
+  intro z hz
+  have hzi : (A.chart i).symm z ∈ (A.chart i).source := (A.chart i).mapsTo_symm hz.1
+  have hzj : (A.chart i).symm z ∈ (A.chart j).source := hz.2
+  have hc := g.volumeCoefficient_compatibility i j hzi hzj
+  rw [(A.chart i).right_inv hz.1] at hc
+  change f ((A.chart i).symm z) * g.volumeCoefficient i z =
+    |(fderiv ℝ (A.transition i j) z).det| *
+      (f ((A.chart j).symm (A.chart j ((A.chart i).symm z))) *
+        g.volumeCoefficient j (A.chart j ((A.chart i).symm z)))
+  rw [A.transition_fderiv_det i j hz, (A.chart j).left_inv hzj, hc]
+  ring
+
+end KaehlerMetric
+end SourceSmooth
+end Anticanonical
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/KahlerL2Comparison.lean b/lean/OAI/Geometry/PrescribedRicci/KahlerL2Comparison.lean
new file mode 100644
index 0000000000000000000000000000000000000000..8455c11cb625eeabed5e6a1be1250287661712da
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/KahlerL2Comparison.lean
@@ -0,0 +1,126 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.KahlerLocalL2Comparison
+import OAI.Geometry.PrescribedRicci.MongeAmpereEnergy
+
+namespace OAI
+
+/-! Kahler L2 Comparison. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory
+open scoped SchwartzMap ContDiff Classical
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X}
+
+lemma finite_norm_sq_le_sum {ι : Type*} [Fintype ι] {E : ι → Type*}
+    [∀ i, SeminormedAddCommGroup (E i)] (f : ∀ i, E i) : ‖f‖ ^ 2 ≤ ∑ i, ‖f i‖ ^ 2 := by
+  have hn : 0 ≤ ∑ i, ‖f i‖ ^ 2 := Finset.sum_nonneg (fun _ _ => sq_nonneg _)
+  have hb : ‖f‖ ≤ Real.sqrt (∑ i, ‖f i‖ ^ 2) := by
+    apply (pi_norm_le_iff_of_nonneg (Real.sqrt_nonneg _)).2
+    intro i
+    exact (Real.le_sqrt (norm_nonneg _) hn).2
+      (Finset.single_le_sum (fun j _ => sq_nonneg ‖f j‖) (Finset.mem_univ i))
+  exact (Real.le_sqrt (norm_nonneg _) hn).1 hb
+
+omit [T2Space X] in
+lemma smooth_square_bounded (ρ : Smooth A) : ∃ M : ℝ, 0 < M ∧ ∀ x, ‖ρ x‖ ^ 2 ≤ M := by
+  obtain ⟨b,hb⟩ := isCompact_univ.bddAbove_image (ρ.continuous.norm.pow 2).continuousOn
+  refine ⟨max b 1, lt_of_lt_of_le (by norm_num) (le_max_right _ _), fun x => ?_⟩
+  exact (hb (mem_image_of_mem _ (mem_univ x))).trans (le_max_left _ _)
+
+namespace Localizers
+variable {ι : Type*} [Fintype ι] (D : Localizers A ι)
+
+lemma norm_coordinate_le (f : Smooth A) (i : ι) :
+    ‖schwartzCoord 0 (localize A (D.index i) (D.weight i) (D.support_sub i) f)‖ ≤
+      ‖D.embed 0 f‖ := norm_le_pi_norm (D.coordinates 0 f) i
+
+ 
+theorem kahler_L2_comparison (g : KaehlerMetric A) : ∃ C B : ℝ, 0 < C ∧ 0 < B ∧
+    ∀ f : Smooth A,
+      ‖D.embed 0 f‖ ^ 2 ≤ C * g.integral (fun x => ‖f x‖ ^ 2) ∧
+      g.integral (fun x => ‖f x‖ ^ 2) ≤ B * ‖D.embed 0 f‖ ^ 2 := by
+  have hl (i : ι) := local_weighted_L2_comparison g (D.index i) (D.weight i) (D.support_sub i)
+  choose a b ha hb hab using hl
+  choose M hM hMb using (fun i : ι => smooth_square_bounded (D.weight i))
+  let c := coordinateVolumeFactor d
+  have hc : 0 < c := coordinateVolumeFactor_pos d
+  let C := 1 + ∑ i, c * M i / a i
+  let B := 1 + (Fintype.card ι : ℝ) * ∑ i, b i / c
+  have hC : 0 < C := by
+    have hh : 0 ≤ ∑ i, c * M i / a i := Finset.sum_nonneg
+      (fun i _ => div_nonneg (mul_nonneg hc.le (hM i).le) (ha i).le)
+    dsimp [C]; linarith
+  have hB : 0 < B := by
+    have hh : 0 ≤ (Fintype.card ι : ℝ) * ∑ i, b i / c := mul_nonneg (Nat.cast_nonneg _)
+      (Finset.sum_nonneg (fun i _ => div_nonneg (hb i).le hc.le))
+    dsimp [B]; linarith
+  refine ⟨C,B,hC,hB,fun f => ?_⟩
+  let I := g.integral (fun x => ‖f x‖ ^ 2)
+  let Iw (i : ι) := g.integral (fun x => ‖D.weight i x * f x‖ ^ 2)
+  have hI : 0 ≤ I := g.integral_nonneg (fun _ => sq_nonneg _)
+  have hIw (i : ι) : Iw i ≤ M i * I := by
+    rw [← g.integral_const_mul]
+    apply g.integral_mono ((D.weight i).continuous.fun_mul f.continuous |>.norm.pow 2)
+      (continuous_const.mul (f.continuous.norm.pow 2))
+    intro x
+    change ‖D.weight i x * f x‖ ^ 2 ≤ M i * ‖f x‖ ^ 2
+    rw [norm_mul, mul_pow]
+    exact mul_le_mul_of_nonneg_right (hMb i x) (sq_nonneg _)
+  have hp (i : ι) :
+      ‖schwartzCoord 0 (localize A (D.index i) (D.weight i) (D.support_sub i) f)‖ ^ 2 ≤
+        (c * M i / a i) * I := by
+    calc
+      _ ≤ (c * Iw i) / a i := (le_div_iff₀ (ha i)).2 (by rw [mul_comm]; exact (hab i f).1)
+      _ ≤ (c * (M i * I)) / a i := div_le_div_of_nonneg_right
+        (mul_le_mul_of_nonneg_left (hIw i) hc.le) (ha i).le
+      _ = _ := by ring
+  have hq (i : ι) : Iw i ≤ (b i / c) * ‖D.embed 0 f‖ ^ 2 := by
+    calc
+      _ ≤ (b i * ‖schwartzCoord 0 (localize A (D.index i) (D.weight i) (D.support_sub i) f)‖ ^ 2) / c :=
+        (le_div_iff₀ hc).2 (by rw [mul_comm]; exact (hab i f).2)
+      _ ≤ (b i * ‖D.embed 0 f‖ ^ 2) / c := div_le_div_of_nonneg_right
+        (mul_le_mul_of_nonneg_left ((sq_le_sq₀ (norm_nonneg _) (norm_nonneg _)).2
+          (D.norm_coordinate_le f i)) (hb i).le) hc.le
+      _ = _ := by ring
+  constructor
+  · calc
+      _ = ‖D.coordinates 0 f‖ ^ 2 := rfl
+      _ ≤ ∑ i, ‖D.coordinates 0 f i‖ ^ 2 := finite_norm_sq_le_sum _
+      _ ≤ ∑ i, (c * M i / a i) * I := Finset.sum_le_sum (fun i _ => hp i)
+      _ = (∑ i, c * M i / a i) * I := (Finset.sum_mul _ _ _).symm
+      _ ≤ C * I := mul_le_mul_of_nonneg_right (by dsimp [C]; linarith) hI
+  · have hi : I ≤ (Fintype.card ι : ℝ) * ∑ i, Iw i := by
+      dsimp only [I, Iw]
+      rw [← g.integral_sum Finset.univ (fun i x => ‖D.weight i x * f x‖ ^ 2) (fun i _ =>
+        ((D.weight i).continuous.fun_mul f.continuous).norm.pow 2), ← g.integral_const_mul]
+      apply g.integral_mono (f.continuous.norm.pow 2)
+        (continuous_const.mul (continuous_finsetSum _ (fun i _ =>
+          ((D.weight i).continuous.fun_mul f.continuous).norm.pow 2)))
+      intro x
+      have hs : ∑ i, D.weight i x * f x = f x := by rw [← Finset.sum_mul, D.sum_one, one_mul]
+      calc
+        ‖f x‖ ^ 2 = ‖∑ i, D.weight i x * f x‖ ^ 2 := by rw [hs]
+        _ ≤ (∑ i, ‖D.weight i x * f x‖) ^ 2 :=
+          (sq_le_sq₀ (norm_nonneg _) (Finset.sum_nonneg (fun i _ => norm_nonneg _))).2 (norm_sum_le _ _)
+        _ ≤ _ := sq_sum_le_card_mul_sum_sq
+    calc
+      I ≤ (Fintype.card ι : ℝ) * ∑ i, Iw i := hi
+      _ ≤ (Fintype.card ι : ℝ) * ∑ i, (b i / c) * ‖D.embed 0 f‖ ^ 2 :=
+        mul_le_mul_of_nonneg_left (Finset.sum_le_sum (fun i _ => hq i)) (Nat.cast_nonneg _)
+      _ = ((Fintype.card ι : ℝ) * ∑ i, b i / c) * ‖D.embed 0 f‖ ^ 2 := by rw [← Finset.sum_mul]; ring
+      _ ≤ B * ‖D.embed 0 f‖ ^ 2 := mul_le_mul_of_nonneg_right (by dsimp [B]; linarith) (sq_nonneg _)
+
+end Localizers
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/KahlerLocalL2Comparison.lean b/lean/OAI/Geometry/PrescribedRicci/KahlerLocalL2Comparison.lean
new file mode 100644
index 0000000000000000000000000000000000000000..c61bca8ea09b3f43d2a520520410a112114b0183
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/KahlerLocalL2Comparison.lean
@@ -0,0 +1,122 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.CoordinateVolumeComparison
+import OAI.Geometry.PrescribedRicci.GlobalKahlerIntegral
+
+namespace OAI
+
+/-! Kahler Local L2 Comparison. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory
+open scoped SchwartzMap ContDiff Classical
+namespace Anticanonical.SourceSmooth
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+namespace KaehlerMetric
+
+lemma chartIntegral_eq_localize (g : KaehlerMetric A) (i : Fin A.count) (f : X → ℝ) :
+    g.chartIntegral i f = ∫ z, localizeFunction i f z * g.volumeCoefficient i z := by
+  rw [chartIntegral, ← integral_indicator (A.chart i).open_target.measurableSet]
+  apply integral_congr_ae
+  filter_upwards [] with z
+  by_cases hz : z ∈ (A.chart i).target
+  · simp only [Set.indicator_of_mem hz, localizeFunction, ite_eq_left hz]
+  · simp only [Set.indicator_of_notMem hz, localizeFunction, ite_eq_right hz, zero_mul]
+
+lemma chart_weighted_integrable [T2Space X] [CompactSpace X] (g : KaehlerMetric A)
+    (i : Fin A.count) {f : X → ℝ} (hf : Continuous f) (hs : tsupport f ⊆ (A.chart i).source) :
+    Integrable (fun z => localizeFunction i f z * g.volumeCoefficient i z) := by
+  obtain ⟨hl,hc,ht⟩ := localizeFunction_continuous_compact i hf hs
+  exact (continuous_real_mul_on_tsupport hl (fun z hz =>
+    ((g.volumeCoefficient_smooth i).continuousOn z (ht hz)).continuousAt
+      ((A.chart i).open_target.mem_nhds (ht hz)))).integrable_of_hasCompactSupport hc.mul_right
+
+lemma compact_volume_bounds (g : KaehlerMetric A) (i : Fin A.count)
+    {K : Set (Coordinates d)} (hK : IsCompact K) (hs : K ⊆ (A.chart i).target) :
+    ∃ a b : ℝ, 0 < a ∧ 0 < b ∧ ∀ z ∈ K, a ≤ g.volumeCoefficient i z ∧ g.volumeCoefficient i z ≤ b := by
+  have hc := (g.volumeCoefficient_smooth i).continuousOn.mono hs
+  obtain ⟨b,hb⟩ := hK.bddAbove_image hc
+  by_cases hn : K.Nonempty
+  · obtain ⟨z,hz,hm⟩ := hK.exists_isMinOn hn hc
+    refine ⟨g.volumeCoefficient i z, max b 1, g.volumeCoefficient_pos i (hs hz),
+      lt_of_lt_of_le (by norm_num : (0:ℝ)<1) (le_max_right _ _), fun y hy => ⟨hm hy, ?_⟩⟩
+    exact (hb (mem_image_of_mem _ hy)).trans (le_max_left _ _)
+  · refine ⟨1,1,by norm_num,by norm_num,fun z hz => ?_⟩
+    exact (hn ⟨z,hz⟩).elim
+
+end KaehlerMetric
+end Anticanonical.SourceSmooth
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X}
+
+omit [T2Space X] [CompactSpace X] in
+lemma square_weight_support (ρ f : Smooth A) :
+    tsupport (fun x => ‖ρ x * f x‖ ^ 2) ⊆ tsupport (ρ : X → ℂ) := by
+  apply closure_minimal _ (isClosed_tsupport _)
+  intro x hx
+  apply subset_tsupport
+  change ρ x ≠ 0
+  intro h
+  exact hx (by simp [h])
+
+lemma local_weighted_L2_comparison (g : KaehlerMetric A) (i : Fin A.count) (ρ : Smooth A)
+    (hs : tsupport (ρ : X → ℂ) ⊆ (A.euclideanChart i).source) :
+    ∃ a b : ℝ, 0 < a ∧ 0 < b ∧ ∀ f : Smooth A,
+      a * ‖schwartzCoord 0 (localize A i ρ hs f)‖ ^ 2 ≤
+        coordinateVolumeFactor d * g.integral (fun x => ‖ρ x * f x‖ ^ 2) ∧
+      coordinateVolumeFactor d * g.integral (fun x => ‖ρ x * f x‖ ^ 2) ≤
+        b * ‖schwartzCoord 0 (localize A i ρ hs f)‖ ^ 2 := by
+  have hs' : tsupport (ρ : X → ℂ) ⊆ (A.chart i).source := by simpa using hs
+  let K := (A.chart i) '' tsupport (ρ : X → ℂ)
+  have hK : IsCompact K := (isClosed_tsupport _).isCompact.image_of_continuousOn
+    ((A.chart i).continuousOn.mono hs')
+  have hKt : K ⊆ (A.chart i).target := by rintro z ⟨x,hx,rfl⟩; exact (A.chart i).mapsTo (hs' hx)
+  obtain ⟨a,b,ha,hb,hab⟩ := g.compact_volume_bounds i hK hKt
+  refine ⟨a,b,ha,hb,fun f => ?_⟩
+  let F (x : X) := ‖ρ x * f x‖ ^ 2
+  have hF : Continuous F := (ρ.continuous.fun_mul f.continuous).norm.pow 2
+  have hFs : tsupport F ⊆ (A.chart i).source := (square_weight_support ρ f).trans hs'
+  have hraw := KaehlerMetric.localizeFunction_continuous_compact i hF hFs
+  have hi : Integrable (KaehlerMetric.localizeFunction i F) (volume : Measure (Coordinates d)) :=
+    hraw.1.integrable_of_hasCompactSupport hraw.2.1
+  have hwi := g.chart_weighted_integrable i hF hFs
+  have hsK : Function.support (KaehlerMetric.localizeFunction i F) ⊆ K :=
+    (KaehlerMetric.localizeFunction_support i F).trans (image_mono (square_weight_support ρ f))
+  have hpos (z : Coordinates d) : 0 ≤ KaehlerMetric.localizeFunction i F z := by
+    unfold KaehlerMetric.localizeFunction
+    split_ifs <;> positivity
+  have hl : a * ∫ z, KaehlerMetric.localizeFunction i F z ≤ g.integral F := by
+    rw [g.integral_chart i hF hFs, g.chartIntegral_eq_localize, ← integral_const_mul]
+    apply MeasureTheory.integral_mono (hi.const_mul a) hwi
+    intro z
+    by_cases hz : z ∈ Function.support (KaehlerMetric.localizeFunction i F)
+    · simpa only [mul_comm] using mul_le_mul_of_nonneg_left (hab z (hsK hz)).1 (hpos z)
+    · simp only [Function.notMem_support.mp hz, mul_zero, zero_mul, le_refl]
+  have hu : g.integral F ≤ b * ∫ z, KaehlerMetric.localizeFunction i F z := by
+    rw [g.integral_chart i hF hFs, g.chartIntegral_eq_localize, ← integral_const_mul]
+    apply MeasureTheory.integral_mono hwi (hi.const_mul b)
+    intro z
+    by_cases hz : z ∈ Function.support (KaehlerMetric.localizeFunction i F)
+    · simpa only [mul_comm] using mul_le_mul_of_nonneg_left (hab z (hsK hz)).2 (hpos z)
+    · simp only [Function.notMem_support.mp hz, mul_zero, zero_mul, le_refl]
+  rw [norm_sq_localize_integral]
+  constructor
+  · calc
+      _ = coordinateVolumeFactor d * (a * ∫ z, KaehlerMetric.localizeFunction i F z) := by ring
+      _ ≤ _ := mul_le_mul_of_nonneg_left hl (coordinateVolumeFactor_pos d).le
+  · calc
+      _ ≤ coordinateVolumeFactor d * (b * ∫ z, KaehlerMetric.localizeFunction i F z) :=
+        mul_le_mul_of_nonneg_left hu (coordinateVolumeFactor_pos d).le
+      _ = _ := by ring
+
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/KahlerPoincare.lean b/lean/OAI/Geometry/PrescribedRicci/KahlerPoincare.lean
new file mode 100644
index 0000000000000000000000000000000000000000..87f7ef5e3195e45d32aa6cd0384dab555cad993f
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/KahlerPoincare.lean
@@ -0,0 +1,74 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.AnalyticSupport
+import OAI.Geometry.PrescribedRicci.GlobalKahlerEnergy
+import OAI.Geometry.PrescribedRicci.KahlerCauchySchwarz
+import OAI.Geometry.PrescribedRicci.KahlerGreenIdentity
+import OAI.Geometry.PrescribedRicci.KahlerPoissonL2
+
+namespace OAI
+
+/-! Kahler Poincare. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory
+open scoped ContDiff Classical
+namespace Anticanonical.SourceSmooth
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  [ConnectedSpace X] {A : ComplexAtlas d X}
+namespace KaehlerMetric
+
+theorem poincare (g : KaehlerMetric A) : ∃ P : ℝ, 0 < P ∧
+    ∀ φ : SmoothRealFunction A, g.integral φ.value = 0 →
+      g.integral (fun x => φ.value x ^ 2) ≤ P * g.integral (g.energy φ φ).value := by
+  obtain ⟨K,hK,hpoisson⟩ := g.exists_meanZero_poisson_L2
+  refine ⟨K+1, by linarith, fun φ hφ => ?_⟩
+  obtain ⟨u,hLu,hU⟩ := hpoisson φ hφ
+  let Iφ := g.integral (fun x => φ.value x ^ 2)
+  let E := g.integral (g.energy φ φ).value
+  let Eu := g.integral (g.energy u u).value
+  have hI : 0 ≤ Iφ := g.integral_nonneg (fun _ => sq_nonneg _)
+  have hE : 0 ≤ E := g.integral_nonneg (g.energy_nonneg _)
+  have hEu : 0 ≤ Eu := g.integral_nonneg (g.energy_nonneg _)
+  have hgreenU : g.integral (fun x => u.value x * φ.value x) = -Eu := by
+    have hh := g.green_identity u u
+    rw [hLu] at hh
+    exact hh
+  have hgreen : Iφ = - g.integral (g.energy φ u).value := by
+    have hh := g.green_identity φ u
+    rw [hLu] at hh
+    simpa only [Iφ, pow_two] using hh
+  have hEuSq : Eu ^ 2 ≤ K * Iφ ^ 2 := by
+    have hh := g.integral_cauchySchwarz u.continuous φ.continuous
+    rw [hgreenU, neg_sq] at hh
+    calc
+      Eu ^ 2 ≤ g.integral (fun x => u.value x ^ 2) * Iφ := hh
+      _ ≤ (K * Iφ) * Iφ := mul_le_mul_of_nonneg_right hU hI
+      _ = _ := by ring
+  have hEuBound : Eu ≤ (K+1) * Iφ := by
+    apply (sq_le_sq₀ hEu (mul_nonneg (by linarith) hI)).1
+    calc
+      Eu ^ 2 ≤ K * Iφ ^ 2 := hEuSq
+      _ ≤ (K+1)^2 * Iφ^2 := mul_le_mul_of_nonneg_right (by nlinarith [sq_nonneg K]) (sq_nonneg Iφ)
+      _ = _ := by ring
+  have hfinal : Iφ ^ 2 ≤ (K+1) * E * Iφ := by
+    calc
+      Iφ ^ 2 = (g.integral (g.energy φ u).value) ^ 2 := by rw [hgreen, neg_sq]
+      _ ≤ E * Eu := g.energy_cauchySchwarz φ u
+      _ ≤ E * ((K+1)*Iφ) := mul_le_mul_of_nonneg_left hEuBound hE
+      _ = _ := by ring
+  change Iφ ≤ (K+1)*E
+  by_cases hz : Iφ = 0
+  · rw [hz]; exact mul_nonneg (by linarith) hE
+  · exact (mul_le_mul_iff_right₀ (lt_of_le_of_ne hI (Ne.symm hz))).1 (by simpa only [pow_two, mul_assoc, mul_comm, mul_left_comm] using hfinal)
+
+end KaehlerMetric
+end Anticanonical.SourceSmooth
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/KahlerPoissonL2.lean b/lean/OAI/Geometry/PrescribedRicci/KahlerPoissonL2.lean
new file mode 100644
index 0000000000000000000000000000000000000000..161678b372a521cb8b5a4bbac40cf6f71e517af9
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/KahlerPoissonL2.lean
@@ -0,0 +1,93 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.ComplexKernel
+import OAI.Geometry.PrescribedPotential.PatchCutoffs
+import OAI.Geometry.PrescribedRicci.BoundedSmoothPoisson
+import OAI.Geometry.PrescribedRicci.KahlerCauchySchwarz
+import OAI.Geometry.PrescribedRicci.KahlerL2Comparison
+import OAI.Geometry.PrescribedRicci.MongeAmpereEnergy
+
+namespace OAI
+
+/-! Kahler Poisson L2. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory
+open scoped SchwartzMap ContDiff Classical
+namespace Anticanonical.SourceSmooth
+open GlobalElliptic
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X}
+namespace KaehlerMetric
+
+lemma integral_const (g : KaehlerMetric A) (c : ℝ) :
+    g.integral (fun _ => c) = c * g.integral (fun _ => 1) := by
+  simpa only [mul_one] using g.integral_const_mul c (fun _ => 1)
+
+lemma volume_pos [Nonempty X] (g : KaehlerMetric A) : 0 < g.integral (fun _ => 1) := by
+  obtain ⟨S,⟨D⟩⟩ := exists_gluingData g
+  obtain ⟨C,B,hC,hB,hbound⟩ := D.localizers.kahler_L2_comparison g
+  have hn : D.localizers.embed 0 (Smooth.const 1) ≠ 0 := by
+    intro hh
+    have he : Smooth.const (A := A) 1 = 0 := D.localizers.embed_injective 0 (by simpa using hh)
+    have hx := congrArg (fun f : Smooth A => f (Classical.arbitrary X)) he
+    exact one_ne_zero hx
+  have hnorm : 0 < ‖D.localizers.embed 0 (Smooth.const 1)‖ ^ 2 := sq_pos_of_pos (norm_pos_iff.mpr hn)
+  have hb := (hbound (Smooth.const 1)).1
+  change ‖D.localizers.embed 0 (Smooth.const 1)‖ ^ 2 ≤ C * g.integral (fun _ => ‖(1 : ℂ)‖ ^ 2) at hb
+  simp only [norm_one, one_pow] at hb
+  exact (mul_pos_iff_of_pos_left hC).mp (hnorm.trans_le hb)
+
+variable [ConnectedSpace X]
+
+ 
+
+theorem exists_meanZero_poisson_L2 (g : KaehlerMetric A) : ∃ K : ℝ, 0 < K ∧
+    ∀ f : SmoothRealFunction A, g.integral f.value = 0 → ∃ u : SmoothRealFunction A,
+      (g.laplacian u).value = f.value ∧
+      g.integral (fun x => u.value x ^ 2) ≤ K * g.integral (fun x => f.value x ^ 2) := by
+  obtain ⟨S,⟨D⟩⟩ := exists_gluingData g
+  obtain ⟨C,B,hC,hB,hbound⟩ := D.localizers.kahler_L2_comparison g
+  obtain ⟨Q,hQ,hpoisson⟩ := D.exists_complex_poisson_bounded
+  let K := 1 + B * Q ^ 2 * C
+  have hK : 0 < K := by dsimp [K]; positivity
+  refine ⟨K,hK,fun f hf => ?_⟩
+  obtain ⟨v,c,hv,hvn⟩ := hpoisson (Smooth.ofReal f)
+  let u := v.part Complex.reCLM
+  have he (x : X) : (g.laplacian u).value x = f.value x + c.re := by
+    rw [← complexL_re, hv, Smooth.add_apply, Complex.add_re, Smooth.ofReal_apply, Complex.ofReal_re]
+    rfl
+  have hz : c.re = 0 := by
+    have hi := g.integral_laplacian u
+    rw [show (g.laplacian u).value = (fun x => f.value x + c.re) from funext he,
+      g.integral_add f.continuous continuous_const, hf, zero_add, g.integral_const] at hi
+    exact (mul_eq_zero.mp hi).resolve_right (ne_of_gt g.volume_pos)
+  have hI : 0 ≤ g.integral (fun x => f.value x ^ 2) := g.integral_nonneg (fun _ => sq_nonneg _)
+  refine ⟨u, funext (fun x => by rw [he x,hz,add_zero]), ?_⟩
+  have hr : g.integral (fun x => u.value x ^ 2) ≤ g.integral (fun x => ‖v x‖ ^ 2) := by
+    apply g.integral_mono (u.continuous.pow 2) (v.continuous.norm.pow 2)
+    intro x
+    change (v x).re ^ 2 ≤ ‖v x‖ ^ 2
+    simpa only [sq_abs] using (sq_le_sq₀ (abs_nonneg _) (norm_nonneg _)).2 (Complex.abs_re_le_norm (v x))
+  have hnv : ‖D.localizers.embed 0 v‖ ^ 2 ≤ Q ^ 2 * ‖D.localizers.embed 0 (Smooth.ofReal f)‖ ^ 2 := by
+    simpa only [mul_pow] using (sq_le_sq₀ (norm_nonneg _) (mul_nonneg hQ (norm_nonneg _))).2 hvn
+  have hnf : ‖D.localizers.embed 0 (Smooth.ofReal f)‖ ^ 2 ≤ C * g.integral (fun x => f.value x ^ 2) := by
+    simpa only [Smooth.ofReal_apply, Complex.norm_real, Real.norm_eq_abs, sq_abs] using (hbound (Smooth.ofReal f)).1
+  calc
+    _ ≤ g.integral (fun x => ‖v x‖ ^ 2) := hr
+    _ ≤ B * ‖D.localizers.embed 0 v‖ ^ 2 := (hbound v).2
+    _ ≤ B * (Q ^ 2 * ‖D.localizers.embed 0 (Smooth.ofReal f)‖ ^ 2) := mul_le_mul_of_nonneg_left hnv hB.le
+    _ ≤ B * (Q ^ 2 * (C * g.integral (fun x => f.value x ^ 2))) :=
+      mul_le_mul_of_nonneg_left (mul_le_mul_of_nonneg_left hnf (sq_nonneg Q)) hB.le
+    _ ≤ K * g.integral (fun x => f.value x ^ 2) := by dsimp [K]; nlinarith
+
+end KaehlerMetric
+end Anticanonical.SourceSmooth
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/KahlerSobolev.lean b/lean/OAI/Geometry/PrescribedRicci/KahlerSobolev.lean
new file mode 100644
index 0000000000000000000000000000000000000000..f9e47d049d9308dd3546d830f153d83212dd3d8f
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/KahlerSobolev.lean
@@ -0,0 +1,87 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.GlobalKahlerIntegral
+import OAI.Geometry.PrescribedRicci.KahlerEnergyCutoff
+import OAI.Geometry.PrescribedRicci.KahlerSobolevLocal
+
+namespace OAI
+
+/-! Kahler Sobolev. -/
+
+section
+
+ 
+
+noncomputable section
+open Matrix Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory
+open scoped ContDiff Classical NNReal ENNReal
+namespace Anticanonical.SourceSmooth.KaehlerMetric
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  [MeasurableSpace X] [BorelSpace X] {A : ComplexAtlas d X}
+
+lemma partition_lpNorm_le (g : KaehlerMetric A) (φ : SmoothRealFunction A)
+    {q : ℝ≥0} (hq : 1 ≤ q) :
+    lpNorm φ.value q g.volumeMeasure ≤
+      ∑ i, lpNorm ((chartPartition i).product φ).value q g.volumeMeasure := by
+  have he : φ.value = ∑ i : Fin A.count, ((chartPartition i).product φ).value := by
+    funext x
+    simp only [Finset.sum_apply, SmoothRealFunction.product]
+    rw [← Finset.sum_mul, chartPartition_sum, one_mul]
+  rw [he]
+  apply lpNorm_sum_le
+  · intro i _
+    exact ((chartPartition i).product φ).continuous.memLp_of_hasCompactSupport
+      (HasCompactSupport.of_compactSpace _)
+  · exact_mod_cast hq
+
+lemma partition_lpNorm_sq_le (g : KaehlerMetric A) (φ : SmoothRealFunction A)
+    {q : ℝ≥0} (hq : 1 ≤ q) :
+    (lpNorm φ.value q g.volumeMeasure)^2 ≤
+      (A.count:ℝ) * ∑ i, (lpNorm ((chartPartition i).product φ).value q g.volumeMeasure)^2 := by
+  have hh := g.partition_lpNorm_le φ hq
+  have hs := (sq_le_sq₀ lpNorm_nonneg (Finset.sum_nonneg (fun _ _ => lpNorm_nonneg))).mpr hh
+  refine hs.trans ?_
+  simpa only [one_mul, one_pow, Finset.sum_const, Finset.card_univ, Fintype.card_fin,
+    nsmul_eq_mul, mul_one] using Finset.sum_mul_sq_le_sq_mul_sq Finset.univ
+      (fun _ : Fin A.count => (1:ℝ))
+      (fun i => lpNorm ((chartPartition i).product φ).value q g.volumeMeasure)
+
+ 
+
+theorem sobolev (g : KaehlerMetric A) (hd : 2 ≤ d) :
+    ∃ q : ℝ≥0, 2 < q ∧ ∃ S : ℝ, 0 < S ∧ ∀ φ : SmoothRealFunction A,
+      (lpNorm φ.value q g.volumeMeasure)^2 ≤
+        S * (g.integral (g.energy φ φ).value + g.integral (fun x => φ.value x^2)) := by
+  obtain ⟨q,hq,hqe⟩ := critical_exponent hd
+  have hc (i : Fin A.count) := g.chart_sobolev i
+    (isClosed_tsupport (chartPartition (A:=A) i).value).isCompact
+    (chartPartition_support i) hq (by omega : 0 < d) hqe
+  choose C hC hClocal using hc
+  have hb (i : Fin A.count) := g.cutoff_integral_energy_bound (chartPartition i)
+  choose B hB hBcut using hb
+  let T : ℝ := (A.count:ℝ) * ∑ i, C i * B i
+  have hT : 0 ≤ T := mul_nonneg (Nat.cast_nonneg _) (Finset.sum_nonneg (fun i _ => mul_nonneg (hC i).le (hB i).le))
+  refine ⟨q,hq,1+T,by linarith,fun φ => ?_⟩
+  let J := g.integral (g.energy φ φ).value + g.integral (fun x => φ.value x^2)
+  have hJ : 0 ≤ J := add_nonneg (g.integral_nonneg (g.energy_nonneg φ))
+    (g.integral_nonneg (fun _ => sq_nonneg _))
+  have hlocal (i : Fin A.count) :
+      (lpNorm ((chartPartition i).product φ).value q g.volumeMeasure)^2 ≤ C i * B i * J := by
+    have hh := hClocal i ((chartPartition i).product φ) (product_support_left _ _)
+    calc
+      _ ≤ C i * g.integral (g.energy ((chartPartition i).product φ) ((chartPartition i).product φ)).value := hh
+      _ ≤ C i * (B i * J) := mul_le_mul_of_nonneg_left (hBcut i φ) (hC i).le
+      _ = _ := by ring
+  calc
+    _ ≤ (A.count:ℝ) * ∑ i, (lpNorm ((chartPartition i).product φ).value q g.volumeMeasure)^2 :=
+      g.partition_lpNorm_sq_le φ (le_trans (by norm_num) hq.le)
+    _ ≤ (A.count:ℝ) * ∑ i, C i * B i * J :=
+      mul_le_mul_of_nonneg_left (Finset.sum_le_sum (fun i _ => hlocal i)) (Nat.cast_nonneg _)
+    _ = T * J := by rw [← Finset.sum_mul]; simp only [T, mul_assoc]
+    _ ≤ _ := by change T*J ≤ (1+T)*J; nlinarith
+
+end Anticanonical.SourceSmooth.KaehlerMetric
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/KahlerSobolevLocal.lean b/lean/OAI/Geometry/PrescribedRicci/KahlerSobolevLocal.lean
new file mode 100644
index 0000000000000000000000000000000000000000..76f9c2271d903f939dadbeec71b01519928c3ff4
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/KahlerSobolevLocal.lean
@@ -0,0 +1,102 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.AnalyticSupport
+import OAI.Geometry.PrescribedRicci.KahlerChartGradient
+import OAI.Geometry.PrescribedRicci.KahlerChartLp
+
+namespace OAI
+
+/-! Kahler Sobolev Local. -/
+
+section
+
+ 
+noncomputable section
+open Matrix Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory
+open scoped ContDiff Classical NNReal ENNReal
+namespace Anticanonical.SourceSmooth.KaehlerMetric
+variable {d : ℕ}
+
+lemma coordinates_finrank : Module.finrank ℝ (Coordinates d) = 2*d := by
+  simp [Coordinates, Module.finrank_pi_fintype, Complex.finrank_real_complex, mul_comm]
+
+lemma critical_exponent (hd : 2 ≤ d) :
+    ∃ q : ℝ≥0, 2 < q ∧ (q:ℝ)⁻¹ = (2:ℝ)⁻¹ - ((2*d:ℕ):ℝ)⁻¹ := by
+  have hdR : (2:ℝ) ≤ d := by exact_mod_cast hd
+  have hp : 0 < (d:ℝ)-1 := by linarith
+  let q : ℝ≥0 := ⟨2*(d:ℝ)/((d:ℝ)-1), (div_pos (by positivity) hp).le⟩
+  refine ⟨q, ?_, ?_⟩
+  · change (2:ℝ) < 2*(d:ℝ)/((d:ℝ)-1)
+    apply (lt_div_iff₀ hp).2
+    linarith
+  · change (2*(d:ℝ)/((d:ℝ)-1))⁻¹ = _
+    push_cast
+    field_simp
+
+lemma lpNorm_two_sq {E : Type*} [NormedAddCommGroup E]
+    {Y : Type*} [MeasurableSpace Y] {μ : Measure Y} {f : Y → E}
+    (hf : AEStronglyMeasurable f μ) :
+    (lpNorm f 2 μ) ^ 2 = ∫ x, ‖f x‖^2 ∂μ := by
+  rw [show (2:ℝ≥0∞) = ((2:ℝ≥0):ℝ≥0∞) by norm_num,
+    lpNorm_nnreal_eq_integral_norm_rpow (by norm_num : (2:ℝ≥0) ≠ 0) hf]
+  simp only [NNReal.coe_ofNat, Real.rpow_two]
+  simpa only [one_div, Nat.cast_ofNat] using Real.rpow_inv_natCast_pow (x := ∫ x, ‖f x‖^2 ∂μ) (n := 2)
+    (MeasureTheory.integral_nonneg (fun x => sq_nonneg ‖f x‖)) (by norm_num)
+
+lemma euclidean_sobolev_sq {q : ℝ≥0} (hd : 0 < d)
+    (he : (q:ℝ)⁻¹ = (2:ℝ)⁻¹ - ((2*d:ℕ):ℝ)⁻¹)
+    {u : Coordinates d → ℝ} (hu : ContDiff ℝ ∞ u) (huc : HasCompactSupport u) :
+    (lpNorm u q volume)^2 ≤
+      ((eLpNormLESNormFDerivOfEqInnerConst (volume : Measure (Coordinates d)) 2 : ℝ≥0):ℝ)^2 *
+      ∫ z, ‖fderiv ℝ u z‖^2 := by
+  have hn : 0 < Module.finrank ℝ (Coordinates d) := by rw [coordinates_finrank]; omega
+  have he' : (q:ℝ)⁻¹ = (((2:ℝ≥0):ℝ))⁻¹ - (Module.finrank ℝ (Coordinates d) : ℝ)⁻¹ := by
+    simpa only [coordinates_finrank, NNReal.coe_ofNat] using he
+  have hD := hu.continuous_fderiv (by simp)
+  have hm : MemLp (fderiv ℝ u) 2 volume := hD.memLp_of_hasCompactSupport (huc.fderiv ℝ)
+  have hh := eLpNorm_le_eLpNorm_fderiv_of_eq_inner (volume : Measure (Coordinates d))
+    (hu.of_le (by simp)) huc (by norm_num : (1:ℝ≥0) ≤ 2) hn he'
+  have hhR := ENNReal.toReal_mono (ENNReal.mul_ne_top ENNReal.coe_ne_top hm.eLpNorm_ne_top) hh
+  rw [ENNReal.toReal_mul, ENNReal.coe_toReal,
+    toReal_eLpNorm, toReal_eLpNorm] at hhR
+  have hsq := (sq_le_sq₀ lpNorm_nonneg (mul_nonneg (NNReal.coe_nonneg _) lpNorm_nonneg)).mpr hhR
+  rw [mul_pow, lpNorm_two_sq hm.aestronglyMeasurable] at hsq
+  exact hsq
+
+variable {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  [MeasurableSpace X] [BorelSpace X] {A : ComplexAtlas d X}
+
+lemma chart_sobolev (g : KaehlerMetric A) (i : Fin A.count)
+    {K : Set X} (hK : IsCompact K) (hs : K ⊆ (A.chart i).source)
+    {q : ℝ≥0} (hq : 2 < q) (hd : 0 < d)
+    (he : (q:ℝ)⁻¹ = (2:ℝ)⁻¹ - ((2*d:ℕ):ℝ)⁻¹) :
+    ∃ C : ℝ, 0 < C ∧ ∀ ψ : SmoothRealFunction A, tsupport ψ.value ⊆ K →
+      (lpNorm ψ.value q g.volumeMeasure)^2 ≤ C * g.integral (g.energy ψ ψ).value := by
+  obtain ⟨b,hb,hbnd⟩ := g.chart_lpNorm_bound i hK hs
+  obtain ⟨a,ha,habnd⟩ := g.chart_gradient_L2_bound i hK hs
+  let S : ℝ := eLpNormLESNormFDerivOfEqInnerConst (volume : Measure (Coordinates d)) 2
+  let B : ℝ := b ^ (q:ℝ)⁻¹
+  have hB : 0 ≤ B := Real.rpow_nonneg hb.le _
+  have hcoef : 0 ≤ B^2*S^2*a := mul_nonneg (mul_nonneg (sq_nonneg _) (sq_nonneg _)) ha.le
+  refine ⟨1+B^2*S^2*a,by linarith,fun ψ hψ => ?_⟩
+  have hq0 : 0 < q := lt_trans (by norm_num) hq
+  have hl := hbnd ψ hψ q hq0
+  have hls := (sq_le_sq₀ lpNorm_nonneg (mul_nonneg hB lpNorm_nonneg)).mpr hl
+  obtain ⟨hu,huc,_⟩ := ψ.localized_smooth_compact i (hψ.trans hs)
+  have hsobo := euclidean_sobolev_sq hd he hu huc
+  have hgr := habnd ψ hψ
+  have hEn := g.integral_nonneg (g.energy_nonneg ψ)
+  calc
+    _ ≤ B^2 * (lpNorm (ψ.localized i) q volume)^2 := by simpa only [mul_pow] using hls
+    _ ≤ B^2 * (S^2 * ∫ z, ‖fderiv ℝ (ψ.localized i) z‖^2) :=
+      mul_le_mul_of_nonneg_left hsobo (sq_nonneg _)
+    _ = (B^2*S^2) * ∫ z, ‖fderiv ℝ (ψ.localized i) z‖^2 := by ring
+    _ ≤ (B^2*S^2) * (a*g.integral (g.energy ψ ψ).value) :=
+      mul_le_mul_of_nonneg_left hgr (mul_nonneg (sq_nonneg _) (sq_nonneg _))
+    _ ≤ _ := by nlinarith
+
+end Anticanonical.SourceSmooth.KaehlerMetric
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/KahlerVolumeMeasure.lean b/lean/OAI/Geometry/PrescribedRicci/KahlerVolumeMeasure.lean
new file mode 100644
index 0000000000000000000000000000000000000000..5bae1731d3390020f0ef9f721c31bba701d86461
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/KahlerVolumeMeasure.lean
@@ -0,0 +1,50 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.AnalyticSupport
+import OAI.Geometry.PrescribedRicci.MongeAmpereEnergy
+
+namespace OAI
+
+/-! Kahler Volume Measure. -/
+
+section
+
+ 
+noncomputable section
+open Matrix Filter Set Topology MeasureTheory
+open scoped ContDiff ComplexOrder Classical CompactlySupported
+namespace Anticanonical.SourceSmooth
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X}
+namespace KaehlerMetric
+
+def integralPositive (g : KaehlerMetric A) : C_c(X, ℝ) →ₚ[ℝ] ℝ where
+  toFun f := g.integral f
+  map_add' f h := g.integral_add f.continuous h.continuous
+  map_smul' c f := g.integral_const_mul c f
+  monotone' f h hfh := g.integral_mono f.continuous h.continuous hfh
+
+variable [MeasurableSpace X] [BorelSpace X]
+
+def volumeMeasure (g : KaehlerMetric A) : Measure X :=
+  RealRMK.rieszMeasure g.integralPositive
+
+instance volumeMeasure_regular (g : KaehlerMetric A) : g.volumeMeasure.Regular := by
+  dsimp [volumeMeasure]
+  infer_instance
+
+instance volumeMeasure_finite (g : KaehlerMetric A) : IsFiniteMeasure g.volumeMeasure := by
+  dsimp [volumeMeasure]
+  infer_instance
+
+lemma integral_volumeMeasure (g : KaehlerMetric A) {f : X → ℝ} (hf : Continuous f) :
+    ∫ x, f x ∂g.volumeMeasure = g.integral f := by
+  exact RealRMK.integral_rieszMeasure g.integralPositive
+    ⟨⟨f, hf⟩, HasCompactSupport.of_compactSpace f⟩
+
+end KaehlerMetric
+end Anticanonical.SourceSmooth
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/KahlerVolumeVariation.lean b/lean/OAI/Geometry/PrescribedRicci/KahlerVolumeVariation.lean
new file mode 100644
index 0000000000000000000000000000000000000000..c53f1c2d9b2f6e1264f706a6063a50a76ddda31d
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/KahlerVolumeVariation.lean
@@ -0,0 +1,201 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.CompactParameterIntegral
+import OAI.Geometry.PrescribedRicci.GlobalKahlerEnergy
+import OAI.Geometry.PrescribedRicci.KahlerGreenIdentity
+import OAI.Geometry.PrescribedPotential.PotentialDensity
+
+namespace OAI
+
+/-! Kahler Volume Variation. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Matrix MeasureTheory
+open scoped ContDiff ComplexOrder Classical Matrix.Norms.Elementwise
+namespace Anticanonical.SourceSmooth
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+namespace KaehlerMetric
+
+lemma PositivePotential.segment {g : KaehlerMetric A} {φ : SmoothRealFunction A}
+    (hp : g.PositivePotential φ) {s : ℝ} (hs : s ∈ Icc 0 1) :
+    g.PositivePotential (φ.realSMul s) := by
+  intro i z hz
+  rw [SmoothRealFunction.hessian_realSMul]
+  by_cases h1 : s = 1
+  · simpa only [h1, Complex.ofReal_one, one_smul] using hp i z hz
+  have hc : (0 : ℂ) < ((1 - s : ℝ) : ℂ) :=
+    Complex.pos_iff.mpr ⟨sub_pos.mpr (lt_of_le_of_ne hs.2 h1), rfl⟩
+  have hq := (hp i z hz).posSemidef.smul
+    (show (0 : ℂ) ≤ (s : ℂ) from Complex.nonneg_iff.mpr ⟨hs.1, rfl⟩)
+  have he := ((g.positive i z hz).smul hc).add_posSemidef hq
+  convert he using 1
+  ext j k
+  simp only [Matrix.add_apply, Matrix.smul_apply, smul_eq_mul, Complex.ofReal_sub, Complex.ofReal_one]
+  ring
+
+lemma continuous_weighted_parameter {E : Type*} [TopologicalSpace E]
+    {w : E → ℝ} (hw : Continuous w) {F : ℝ × E → ℝ}
+    (hF : ∀ p, p.2 ∈ tsupport w → ContinuousAt F p) :
+    Continuous (fun p : ℝ × E => w p.2 * F p) := by
+  apply continuous_real_mul_on_tsupport (hw.comp continuous_snd)
+  intro p hp
+  exact hF p (tsupport_comp_subset_preimage w continuous_snd hp)
+
+lemma chart_det_variation [T2Space X] [CompactSpace X]
+    (g : KaehlerMetric A) (φ : SmoothRealFunction A) (i : Fin A.count)
+    {w : X → ℝ} (hw : Continuous w) (hs : tsupport w ⊆ (A.chart i).source) (s : ℝ) :
+    HasDerivAt (fun t => ∫ z, localizeFunction i w z *
+      (g.matrix i z + t • φ.hessian i z).det.re)
+      (∫ z, localizeFunction i w z *
+        (fderiv ℝ (fun M : Matrix (Fin d) (Fin d) ℂ => M.det)
+          (g.matrix i z + s • φ.hessian i z) (φ.hessian i z)).re) s := by
+  obtain ⟨hl, hc, ht⟩ := localizeFunction_continuous_compact i hw hs
+  let F (t : ℝ) (z : Coordinates d) := localizeFunction i w z *
+    (g.matrix i z + t • φ.hessian i z).det.re
+  let F' (t : ℝ) (z : Coordinates d) := localizeFunction i w z *
+    (fderiv ℝ (fun M : Matrix (Fin d) (Fin d) ℂ => M.det)
+      (g.matrix i z + t • φ.hessian i z) (φ.hessian i z)).re
+  have hM (p : ℝ × Coordinates d) (hp : p.2 ∈ tsupport (localizeFunction i w)) :
+      ContinuousAt (fun q : ℝ × Coordinates d => g.matrix i q.2 + q.1 • φ.hessian i q.2) p := by
+    have hg := ((g.smooth i).contDiffAt ((A.chart i).open_target.mem_nhds (ht hp))).continuousAt
+    have hf := ((φ.hessian_smooth i).contDiffAt ((A.chart i).open_target.mem_nhds (ht hp))).continuousAt
+    exact (hg.comp continuousAt_snd).add
+      (continuousAt_fst.smul (hf.comp continuousAt_snd))
+  have hF : Continuous (Function.uncurry F) := by
+    apply continuous_weighted_parameter hl
+    intro p hp
+    exact Complex.continuous_re.continuousAt.comp
+      (MongeAmpere.contDiff_det.continuous.continuousAt.comp (hM p hp))
+  have hF' : Continuous (Function.uncurry F') := by
+    apply continuous_weighted_parameter hl
+    intro p hp
+    have hf := ((φ.hessian_smooth i).contDiffAt ((A.chart i).open_target.mem_nhds (ht hp))).continuousAt
+    exact Complex.continuous_re.continuousAt.comp
+      ((((MongeAmpere.contDiff_det.restrict_scalars ℝ).continuous_fderiv (by simp)).continuousAt.comp
+        (hM p hp)).clm_apply (hf.comp continuousAt_snd))
+  exact KahlerCalculus.hasDerivAt_integral_compact hc hF hF'
+    (fun t => (Function.support_mul_subset_left _ _).trans (subset_tsupport _))
+    (fun t => (Function.support_mul_subset_left _ _).trans (subset_tsupport _))
+    (fun t z => by
+      have hh := (hasDerivAt_id t).smul_const (φ.hessian i z) |>.const_add (g.matrix i z)
+      have hd := ((MongeAmpere.differentiable_det _).restrictScalars ℝ).hasFDerivAt.comp_hasDerivAt t hh
+      have hr := Complex.reCLM.hasFDerivAt.comp_hasDerivAt t hd
+      convert! hr.const_mul (localizeFunction i w z) using 1
+      simp only [F', one_smul, id_eq, Complex.reCLM_apply]
+      rfl) s
+
+lemma density_mul_volume (g : KaehlerMetric A) (φ : SmoothRealFunction A)
+    (i : Fin A.count) {z : Coordinates d} (hz : z ∈ (A.chart i).target) :
+    (g.potentialDensity φ).value ((A.chart i).symm z) * g.volumeCoefficient i z =
+      (g.matrix i z + φ.hessian i z).det.re := by
+  have he := g.potentialDensity_complex φ i ((A.chart i).mapsTo_symm hz)
+  rw [(A.chart i).right_inv hz] at he
+  have hn := (g.positive i z hz).det_pos.ne'
+  have hh := congrArg (fun c : ℂ => (c * (g.matrix i z).det).re) he
+  simp only [volumePolynomial, div_mul_cancel₀ _ hn, Complex.mul_re, Complex.ofReal_re,
+    Complex.ofReal_im, zero_mul, sub_zero] at hh
+  exact hh
+
+lemma integral_density [T2Space X] [CompactSpace X] (g : KaehlerMetric A)
+    (φ : SmoothRealFunction A) (hp : g.PositivePotential φ) (w : X → ℝ) :
+    g.integral (fun x => w x * (g.potentialDensity φ).value x) = (g.deform φ hp).integral w := by
+  apply Finset.sum_congr rfl
+  intro i _
+  apply setIntegral_congr_fun (A.chart i).open_target.measurableSet
+  intro z hz
+  change (chartPartition i).value ((A.chart i).symm z) *
+      (w ((A.chart i).symm z) * (g.potentialDensity φ).value ((A.chart i).symm z)) *
+        g.volumeCoefficient i z = _
+  rw [mul_assoc, mul_assoc, density_mul_volume g φ i hz]
+  exact (mul_assoc _ _ _).symm
+
+lemma real_smul_hessian (φ : SmoothRealFunction A) (s : ℝ) (i : Fin A.count) (z : Coordinates d) :
+    s • φ.hessian i z = (φ.realSMul s).hessian i z := by
+  rw [SmoothRealFunction.hessian_realSMul]
+  ext j k
+  exact Complex.real_smul
+
+lemma integral_density_full [T2Space X] [CompactSpace X] (g : KaehlerMetric A)
+    (φ : SmoothRealFunction A) (w : X → ℝ) :
+    g.integral (fun x => w x * (g.potentialDensity φ).value x) =
+      ∑ i, ∫ z, localizeFunction i (fun x => (chartPartition i).value x * w x) z *
+        (g.matrix i z + φ.hessian i z).det.re := by
+  apply Finset.sum_congr rfl
+  intro i _
+  unfold chartIntegral
+  rw [← integral_indicator (A.chart i).open_target.measurableSet]
+  apply integral_congr_ae
+  filter_upwards [] with z
+  by_cases hz : z ∈ (A.chart i).target
+  · rw [Set.indicator_of_mem hz]
+    simp only [localizeFunction, ite_eq_left hz]
+    rw [mul_assoc, mul_assoc, density_mul_volume g φ i hz, ← mul_assoc]
+  · simp only [Set.indicator_of_notMem hz, localizeFunction, ite_eq_right hz, zero_mul]
+
+lemma full_firstVariation_eq [T2Space X] [CompactSpace X]
+    (g : KaehlerMetric A) (φ : SmoothRealFunction A) (s : ℝ)
+    (hp : g.PositivePotential (φ.realSMul s)) (i : Fin A.count) (w : X → ℝ) :
+    (∫ z, localizeFunction i w z *
+      (fderiv ℝ (fun M : Matrix (Fin d) (Fin d) ℂ => M.det)
+        (g.matrix i z + s • φ.hessian i z) (φ.hessian i z)).re) =
+      (g.deform (φ.realSMul s) hp).chartIntegral i
+        (fun x => w x * ((g.deform (φ.realSMul s) hp).laplacian φ).value x) := by
+  let gs := g.deform (φ.realSMul s) hp
+  change _ = ∫ z in (A.chart i).target,
+    (w ((A.chart i).symm z) * (gs.laplacian φ).value ((A.chart i).symm z)) * gs.volumeCoefficient i z
+  rw [← integral_indicator (A.chart i).open_target.measurableSet]
+  apply integral_congr_ae
+  filter_upwards [] with z
+  by_cases hz : z ∈ (A.chart i).target
+  · rw [Set.indicator_of_mem hz]
+    simp only [localizeFunction, ite_eq_left hz, real_smul_hessian]
+    change w ((A.chart i).symm z) * (fderiv ℝ (fun M : Matrix (Fin d) (Fin d) ℂ => M.det)
+      (gs.matrix i z) (φ.hessian i z)).re = _
+    rw [MongeAmpere.real_fderiv_det_apply _ _ (isUnit_iff_ne_zero.mpr (gs.positive i z hz).det_pos.ne')]
+    have hl := gs.laplacian_localExpression φ i hz
+    change (gs.laplacian φ).value ((A.chart i).symm z) = _ at hl
+    rw [hl, Complex.mul_re, hermitian_det_im (gs.positive i z hz).isHermitian, zero_mul, sub_zero]
+    change _ = (_ * ((gs.matrix i z)⁻¹ * φ.hessian i z).trace.re) * (gs.matrix i z).det.re
+    ring
+  · simp only [Set.indicator_of_notMem hz, localizeFunction, ite_eq_right hz, zero_mul]
+
+ 
+
+theorem volume_firstVariation [T2Space X] [CompactSpace X]
+    (g : KaehlerMetric A) (φ ψ : SmoothRealFunction A) (s : ℝ)
+    (hp : g.PositivePotential (φ.realSMul s)) :
+    HasDerivAt (fun t => g.integral (fun x => ψ.value x *
+      (g.potentialDensity (φ.realSMul t)).value x))
+      (-(g.deform (φ.realSMul s) hp).integral
+        ((g.deform (φ.realSMul s) hp).energy ψ φ).value) s := by
+  have hh := HasDerivAt.fun_sum (u := Finset.univ) (fun i _ => g.chart_det_variation φ i
+    ((chartPartition i).continuous.fun_mul ψ.continuous)
+    (tsupport_mul_subset_left.trans (chartPartition_support i)) s)
+  simp_rw [g.full_firstVariation_eq φ s hp] at hh
+  have he : (∑ i, (g.deform (φ.realSMul s) hp).chartIntegral i
+      (fun x => ((chartPartition i).value x * ψ.value x) *
+        ((g.deform (φ.realSMul s) hp).laplacian φ).value x)) =
+      -(g.deform (φ.realSMul s) hp).integral ((g.deform (φ.realSMul s) hp).energy ψ φ).value := by
+    rw [← (g.deform (φ.realSMul s) hp).green_identity ψ φ]
+    unfold integral
+    congr 1
+    funext i
+    congr 1
+    funext x
+    ring
+  rw [he] at hh
+  convert! hh using 1
+  funext t
+  rw [g.integral_density_full]
+  simp only [real_smul_hessian]
+
+end KaehlerMetric
+end Anticanonical.SourceSmooth
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/LocalDeterminantBound.lean b/lean/OAI/Geometry/PrescribedRicci/LocalDeterminantBound.lean
new file mode 100644
index 0000000000000000000000000000000000000000..bd7003b44b4da1537b98b03a4d7c32fe3c0560d3
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/LocalDeterminantBound.lean
@@ -0,0 +1,101 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.LocalDeterminantEquation
+import OAI.Geometry.PrescribedRicci.UniformDeterminantRemainder
+import OAI.Geometry.PrescribedRicci.UniformL2
+
+namespace OAI
+
+/-! Local Determinant Bound. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Matrix _root_.MeasureTheory _root_.OAI.MeasureTheory LineDeriv
+open scoped ContDiff SchwartzMap Classical Matrix.Norms.Elementwise BoundedContinuousFunction ENNReal
+namespace TameInterpolation
+open SobolevChart
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+variable {ι : Type*} [Fintype ι] [Nonempty ι] {n : ℕ} [Nonempty (Fin n)] {Z : Type*}
+omit [Fintype ι] [Nonempty ι] in
+lemma detJetRemainder_memLp (e : ι → E) (F : Fin n → Fin n → 𝓢(E,ℂ))
+    (hc : ∀ i j, HasCompactSupport (F i j : E → ℂ)) (ws : List ι) :
+    MemLp (detJetRemainder e (fun x i j => F i j x) ws) 2 volume := by
+  have hs (σ : Equiv.Perm (Fin n)) : ∀ L : List (Fin n → List ι),
+      MemLp (fun x => (L.map (fun s => monoEval e (fun i x => F (σ i) i x) s x)).sum) 2 volume := by
+    intro L
+    induction L with
+    | nil => simp
+    | cons a L ih =>
+      exact ((monoEval_smooth e _ (fun i => (F (σ i) i).smooth') a).continuous.memLp_of_hasCompactSupport
+        (monoEval_compact e _ (fun i => hc (σ i) i) a)).add ih
+  have hh := memLp_finsetSum Finset.univ (fun σ _ => (hs σ (monoRemainder ws)).const_mul (Equiv.Perm.sign σ : ℂ))
+  convert! hh using 1
+
+lemma detJetRemainder_uniformL2 (e : ι → E) (F : Z → Fin n → Fin n → 𝓢(E,ℂ))
+    (hc : ∀ z i j, HasCompactSupport (F z i j : E → ℂ)) (ws : List ι)
+    (h0 : ∀ σ : Equiv.Perm (Fin n), ∃ A : ℝ, 0 ≤ A ∧ ∀ z,
+      familyJetNorm e (realComponents (initialJet e (fun i x => F z (σ i) i x))) 0 ∞ ≤ A)
+    (hS : ∀ i j, UniformSobolev ((ws.length:ℝ)-1) (fun z => F z i j)) :
+    UniformL2 (fun z => detJetRemainder e (fun x i j => F z i j x) ws) :=
+  ⟨fun z => detJetRemainder_memLp e (F z) (hc z) ws,detJetRemainder_uniform_bound e F hc ws h0 hS⟩
+end TameInterpolation
+
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart FrozenPoisson MetricLocalization TameInterpolation
+variable {d : ℕ} {Z : Type*}
+lemma norm_trace_mul_le_sum (M N : Matrix (Fin d) (Fin d) ℂ) {B : ℝ}
+    (hB : ∀ i j, ‖M i j‖ ≤ B) :
+    ‖(M*N).trace‖ ≤ B*∑ i, ∑ j, ‖N j i‖ := by
+  change ‖∑ i, ∑ j, M i j*N j i‖ ≤ _
+  rw [Finset.mul_sum]
+  apply (norm_sum_le _ _).trans
+  apply Finset.sum_le_sum
+  intro i _
+  rw [Finset.mul_sum]
+  exact (norm_sum_le _ _).trans (Finset.sum_le_sum (fun j _ => by
+    rw [norm_mul]
+    exact mul_le_mul_of_nonneg_right (hB i j) (norm_nonneg _)))
+
+lemma determinant_word_cutoff_bound
+    {ι : Type*} (e : ι → EC d) (ws : List ι) (hw : ws ≠ [])
+    (κ f : 𝓢(EC d,ℂ)) (G H : Fin d → Fin d → 𝓢(EC d,ℂ))
+    (ρ : 𝓢(EC d,ℂ)) (y : EC d)
+    (he : ∀ i j, (H i j : EC d → ℂ) =ᶠ[𝓝 y]
+      (fun x => G i j x+hessianEntrySchwartz i j f x))
+    (hρ : (fun x => (show Matrix (Fin d) (Fin d) ℂ from fun i j => H i j x).det) =ᶠ[𝓝 y] ρ)
+    (hinv : IsUnit (show Matrix (Fin d) (Fin d) ℂ from fun i j => H i j y).det)
+    {B J K : ℝ} (hB : 0 ≤ B) (hJ : 0 ≤ J)
+    (hb : ∀ i j, ‖(show Matrix (Fin d) (Fin d) ℂ from fun i j => H i j y)⁻¹ i j‖ ≤ B)
+    (hj : ‖(show Matrix (Fin d) (Fin d) ℂ from fun i j => H i j y).det⁻¹‖ ≤ J)
+    (hk : ‖κ y‖ ≤ K) :
+    ‖(((show Matrix (Fin d) (Fin d) ℂ from fun i j => H i j y)⁻¹) *
+      (show Matrix (Fin d) (Fin d) ℂ from fun i j => hessianEntrySchwartz i j
+        (SchwartzMap.smulLeftCLM ℂ κ (schwartzWord (ws.map e) f)) y)).trace‖ ≤
+      K*J*(‖schwartzWord (ws.map e) ρ y‖+‖detJetRemainder e (fun x i j => H i j x) ws y‖)+
+      B*∑ i, ∑ j, (‖wordHessianError κ (ws.map e) j i f y‖+K*‖schwartzWord (ws.map e) (G j i) y‖) := by
+  rw [determinant_word_cutoff_trace e ws hw κ f G H ρ y he hρ hinv]
+  apply (norm_add_le _ _).trans
+  apply add_le_add
+  · rw [norm_mul,norm_mul]
+    exact mul_le_mul (mul_le_mul hk hj (norm_nonneg _) (le_trans (norm_nonneg _) hk))
+      (norm_sub_le _ _) (norm_nonneg _) (mul_nonneg (le_trans (norm_nonneg _) hk) hJ)
+  · apply (norm_trace_mul_le_sum _ _ hb).trans
+    apply mul_le_mul_of_nonneg_left _ hB
+    apply Finset.sum_le_sum
+    intro i _
+    apply Finset.sum_le_sum
+    intro j _
+    change ‖wordHessianError κ (ws.map e) j i f y-κ y*schwartzWord (ws.map e) (G j i) y‖ ≤ _
+    apply (norm_sub_le _ _).trans
+    apply add_le_add_right
+    rw [norm_mul]
+    exact mul_le_mul_of_nonneg_right hk (norm_nonneg _)
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/LocalDeterminantEquation.lean b/lean/OAI/Geometry/PrescribedRicci/LocalDeterminantEquation.lean
new file mode 100644
index 0000000000000000000000000000000000000000..71c21b2872c3dd5652f9f0d4a720922266cb894f
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/LocalDeterminantEquation.lean
@@ -0,0 +1,132 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.CofactorBilin
+import OAI.Geometry.PrescribedRicci.DeterminantJet
+import OAI.Geometry.PrescribedRicci.SchwartzHessianWords
+
+namespace OAI
+
+/-! Local Determinant Equation. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Matrix LineDeriv
+open scoped ContDiff SchwartzMap Classical Matrix.Norms.Elementwise BoundedContinuousFunction
+namespace TameInterpolation
+open SobolevChart
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] {ι : Type*}
+omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] in
+lemma cword_eventuallyEq (e : ι → E) (ws : List ι) {f g : E → ℂ} {x : E}
+    (h : f =ᶠ[𝓝 x] g) : cword e f ws =ᶠ[𝓝 x] cword e g ws := by
+  induction ws with
+  | nil => exact h
+  | cons a ws ih =>
+    filter_upwards [ih.fderiv (𝕜:=ℝ)] with y hy
+    exact congrArg (fun L : E →L[ℝ] ℂ => L (e a)) hy
+omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] in
+lemma cword_add (e : ι → E) (ws : List ι) (f g : E → ℂ)
+    (hf : ContDiff ℝ ∞ f) (hg : ContDiff ℝ ∞ g) :
+    cword e (fun x => f x+g x) ws = fun x => cword e f ws x+cword e g ws x := by
+  induction ws with
+  | nil => rfl
+  | cons a ws ih =>
+    change cdir (e a) (cword e (fun x => f x+g x) ws) = _
+    rw [ih]
+    funext x
+    unfold cdir
+    rw [fderiv_fun_add ((cword_smooth e hf ws).differentiable (by simp) x)
+      ((cword_smooth e hg ws).differentiable (by simp) x)]
+    rfl
+omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] in
+lemma cwordSchwartz_general (e : ι → E) (f : 𝓢(E,ℂ)) (ws : List ι) :
+    cword e f ws = (schwartzWord (ws.map e) f : E → ℂ) := by
+  induction ws with
+  | nil => rfl
+  | cons a ws ih =>
+    change cdir (e a) (cword e f ws) = _
+    rw [ih,cdir_schwartz]
+    rfl
+end TameInterpolation
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart FrozenPoisson MetricLocalization TameInterpolation
+variable {d : ℕ}
+lemma cofactorBilin_inverse_det (H : Matrix (Fin d) (Fin d) ℂ) (h : IsUnit H.det) :
+    cofactorBilin H.det⁻¹ H = traceBilin H := by
+  ext B
+  rw [cofactorBilin_apply,NonlinearHessian.detDifferential,MongeAmpere.real_fderiv_det_apply H _ h]
+  rw [← mul_assoc,inv_mul_cancel₀ h.ne_zero,one_mul]
+  rfl
+
+lemma schwartz_real_trace_expression (H : Matrix (Fin d) (Fin d) ℂ) (h : IsUnit H.det)
+    (f : 𝓢(EC d,ℂ)) (hr : ∀ x, (f x).im = 0) (y : EC d) :
+    ∑ i, ∑ j, (traceBilin H (rankTwo (stdOrthonormalBasis ℝ (EC d) i)
+      (stdOrthonormalBasis ℝ (EC d) j)) : ℂ) *
+      fderiv ℝ (fderiv ℝ f) y (stdOrthonormalBasis ℝ (EC d) i) (stdOrthonormalBasis ℝ (EC d) j) =
+      (((H⁻¹ * (show Matrix (Fin d) (Fin d) ℂ from fun i j => hessianEntrySchwartz i j f y)).trace).re : ℂ) := by
+  have he : (f : EC d → ℂ) =ᶠ[𝓝 y] Complex.ofRealCLM ∘ (fun x => (f x).re) := by
+    filter_upwards [] with x
+    exact Complex.ext rfl (by simpa using hr x)
+  have hh := cofactor_smooth_expression (Complex.reCLM.contDiff.comp f.smooth').contDiffAt f he H.det⁻¹ H
+  rw [cofactorBilin_inverse_det H h,NonlinearHessian.detDifferential] at hh
+  erw [MongeAmpere.real_fderiv_det_apply H _ h,
+    ← mul_assoc,inv_mul_cancel₀ h.ne_zero,one_mul] at hh
+  exact hh.symm
+
+lemma wordHessian_trace_cutoff (κ f : 𝓢(EC d,ℂ)) (ws : List (EC d))
+    (G H : Fin d → Fin d → 𝓢(EC d,ℂ)) (y : EC d)
+    (he : ∀ i j, (H i j : EC d → ℂ) =ᶠ[𝓝 y]
+      (fun x => G i j x+hessianEntrySchwartz i j f x)) :
+    (show Matrix (Fin d) (Fin d) ℂ from fun i j => hessianEntrySchwartz i j (SchwartzMap.smulLeftCLM ℂ κ (schwartzWord ws f)) y) =
+      κ y • (show Matrix (Fin d) (Fin d) ℂ from fun i j => schwartzWord ws (H i j) y) +
+        (show Matrix (Fin d) (Fin d) ℂ from fun i j => wordHessianError κ ws i j f y-κ y*schwartzWord ws (G i j) y) := by
+  ext i j
+  change _ = κ y * schwartzWord ws (H i j) y + (wordHessianError κ ws i j f y-κ y*schwartzWord ws (G i j) y)
+  rw [wordHessianError_identity]
+  simp only [_root_.add_apply,SchwartzMap.smulLeftCLM_apply_apply κ.hasTemperateGrowth,
+    smul_eq_mul]
+  have hh := (cword_eventuallyEq id ws (he i j)).eq_of_nhds
+  erw [cword_add id ws (G i j : EC d → ℂ) (hessianEntrySchwartz i j f : EC d → ℂ) (G i j).smooth' (hessianEntrySchwartz i j f).smooth'] at hh
+  simp only [cwordSchwartz_general,List.map_id] at hh
+  rw [hh]
+  ring
+
+ 
+
+theorem determinant_word_cutoff_trace
+    {ι : Type*} (e : ι → EC d) (ws : List ι) (hw : ws ≠ [])
+    (κ f : 𝓢(EC d,ℂ)) (G H : Fin d → Fin d → 𝓢(EC d,ℂ))
+    (ρ : 𝓢(EC d,ℂ)) (y : EC d)
+    (he : ∀ i j, (H i j : EC d → ℂ) =ᶠ[𝓝 y]
+      (fun x => G i j x+hessianEntrySchwartz i j f x))
+    (hρ : (fun x => (show Matrix (Fin d) (Fin d) ℂ from fun i j => H i j x).det) =ᶠ[𝓝 y] ρ)
+    (hinv : IsUnit (show Matrix (Fin d) (Fin d) ℂ from fun i j => H i j y).det) :
+    (((show Matrix (Fin d) (Fin d) ℂ from fun i j => H i j y)⁻¹) *
+      (show Matrix (Fin d) (Fin d) ℂ from fun i j => hessianEntrySchwartz i j (SchwartzMap.smulLeftCLM ℂ κ (schwartzWord (ws.map e) f)) y)).trace =
+      κ y * (show Matrix (Fin d) (Fin d) ℂ from fun i j => H i j y).det⁻¹ *
+        (schwartzWord (ws.map e) ρ y-detJetRemainder e (fun x i j => H i j x) ws y) +
+      (((show Matrix (Fin d) (Fin d) ℂ from fun i j => H i j y)⁻¹) *
+        (show Matrix (Fin d) (Fin d) ℂ from fun i j => wordHessianError κ (ws.map e) i j f y-κ y*schwartzWord (ws.map e) (G i j) y)).trace := by
+  let M : Matrix (Fin d) (Fin d) ℂ := fun i j => H i j y
+  have hd := cword_det_trace e (fun x i j => H i j x) (fun i j => (H i j).smooth') ws hw y hinv
+  have hdρ := (cword_eventuallyEq e ws hρ).eq_of_nhds
+  rw [cwordSchwartz_general] at hdρ
+  rw [hdρ] at hd
+  simp only [cwordSchwartz_general] at hd
+  have ht : ((M⁻¹ * (show Matrix (Fin d) (Fin d) ℂ from fun i j => schwartzWord (ws.map e) (H i j) y)).trace) =
+      M.det⁻¹ * (schwartzWord (ws.map e) ρ y-detJetRemainder e (fun x i j => H i j x) ws y) := by
+    have hh := congrArg (fun z : ℂ => M.det⁻¹*z) (sub_eq_iff_eq_add.mpr hd)
+    simpa only [M,← mul_assoc,inv_mul_cancel₀ hinv.ne_zero,one_mul] using hh.symm
+  rw [wordHessian_trace_cutoff κ f (ws.map e) G H y he]
+  rw [Matrix.mul_add,Matrix.mul_smul,trace_add,trace_smul]
+  change κ y * (M⁻¹ * _).trace + _ = _
+  rw [ht]
+  ring
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/LocalDeterminantGain.lean b/lean/OAI/Geometry/PrescribedRicci/LocalDeterminantGain.lean
new file mode 100644
index 0000000000000000000000000000000000000000..edad29ad06dd84986a114b6bfe77089ad9b71221
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/LocalDeterminantGain.lean
@@ -0,0 +1,117 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.LocalDeterminantBound
+
+namespace OAI
+
+/-! Local Determinant Gain. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Matrix _root_.MeasureTheory _root_.OAI.MeasureTheory LineDeriv
+open scoped ContDiff SchwartzMap Classical Matrix.Norms.Elementwise BoundedContinuousFunction ENNReal
+open scoped ComplexOrder MatrixOrder
+namespace SobolevChart
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] {Z : Type*}
+lemma UniformSobolev.neg {s : ℝ} {F : Z → 𝓢(E,ℂ)} (hF : UniformSobolev s F) :
+    UniformSobolev s (fun z => -F z) := by
+  obtain ⟨C,hC,hc⟩ := hF
+  refine ⟨C,hC,fun z => ?_⟩
+  have he : schwartzCoord s (-F z) = -schwartzCoord s (F z) := by
+    simpa only [neg_one_smul] using schwartzCoord_smul s (-1) (F z)
+  change ‖schwartzCoord s (-F z)‖ ≤ C
+  rw [he,norm_neg]
+  exact hc z
+
+lemma UniformSobolev.sub {s : ℝ} {F G : Z → 𝓢(E,ℂ)}
+    (hF : UniformSobolev s F) (hG : UniformSobolev s G) :
+    UniformSobolev s (fun z => F z-G z) := by
+  simpa only [sub_eq_add_neg] using hF.add hG.neg
+end SobolevChart
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart FrozenPoisson MetricLocalization TameInterpolation
+variable {d : ℕ} [Nonempty (Fin d)] {Z : Type*}
+
+lemma uniform_determinant_word_differential
+    {ι : Type*} [Fintype ι] [Nonempty ι] (e : ι → EC d) (ws : List ι) (hw : ws ≠ [])
+    (κ : 𝓢(EC d,ℂ)) (f ρ : Z → 𝓢(EC d,ℂ)) (G H : Z → Fin d → Fin d → 𝓢(EC d,ℂ))
+    (c : Z → BasisIndex d → BasisIndex d → EC d →ᵇ ℂ)
+    (hc : ∀ z i j, (c z i j : EC d → ℂ).HasTemperateGrowth)
+    (hec : ∀ z y, y ∈ tsupport (κ : EC d → ℂ) → ∀ i j,
+      c z i j y = (traceBilin (fun k l => H z k l y)
+        (rankTwo (stdOrthonormalBasis ℝ (EC d) i) (stdOrthonormalBasis ℝ (EC d) j)) : ℂ))
+    (hr : ∀ z y, (SchwartzMap.smulLeftCLM ℂ κ (schwartzWord (ws.map e) (f z)) y).im = 0)
+    (he : ∀ z y, y ∈ tsupport (κ : EC d → ℂ) → ∀ i j,
+      (H z i j : EC d → ℂ) =ᶠ[𝓝 y] (fun x => G z i j x+hessianEntrySchwartz i j (f z) x))
+    (heρ : ∀ z y, y ∈ tsupport (κ : EC d → ℂ) →
+      (fun x => (show Matrix (Fin d) (Fin d) ℂ from fun i j => H z i j x).det) =ᶠ[𝓝 y] ρ z)
+    (hinv : ∀ z y, y ∈ tsupport (κ : EC d → ℂ) →
+      IsUnit (show Matrix (Fin d) (Fin d) ℂ from fun i j => H z i j y).det)
+    {B J : ℝ} (hB : 0 ≤ B) (hJ : 0 ≤ J)
+    (hb : ∀ z y, y ∈ tsupport (κ : EC d → ℂ) → ∀ i j,
+      ‖(show Matrix (Fin d) (Fin d) ℂ from fun i j => H z i j y)⁻¹ i j‖ ≤ B)
+    (hj : ∀ z y, y ∈ tsupport (κ : EC d → ℂ) →
+      ‖(show Matrix (Fin d) (Fin d) ℂ from fun i j => H z i j y).det⁻¹‖ ≤ J)
+    (hF : UniformSobolev ((ws.length:ℝ)+1) f)
+    (hρ : UniformSobolev (ws.length:ℝ) ρ)
+    (hG : ∀ i j, UniformSobolev (ws.length:ℝ) (fun z => G z i j))
+    (hHc : ∀ z i j, HasCompactSupport (H z i j : EC d → ℂ))
+    (hH0 : ∀ σ : Equiv.Perm (Fin d), ∃ A : ℝ, 0 ≤ A ∧ ∀ z,
+      familyJetNorm e (realComponents (initialJet e (fun i x => H z (σ i) i x))) 0 ∞ ≤ A)
+    (hHS : ∀ i j, UniformSobolev ((ws.length:ℝ)-1) (fun z => H z i j)) :
+    UniformSobolev 0 (fun z => schwartzDifferential (c z)
+      (SchwartzMap.smulLeftCLM ℂ κ (schwartzWord (ws.map e) (f z)))) := by
+  let K := ‖κ.toBoundedContinuousFunction‖
+  have hK : 0 ≤ K := norm_nonneg _
+  have hk (y) : ‖κ y‖ ≤ K := κ.toBoundedContinuousFunction.norm_coe_le_norm y
+  let R (z : Z) := detJetRemainder e (fun x i j => H z i j x) ws
+  have hR : UniformL2 R := detJetRemainder_uniformL2 e H hHc ws hH0 hHS
+  have hp : UniformL2 (fun z x => ‖schwartzWord (ws.map e) (ρ z) x‖) :=
+    (hρ.word (ws.map e) (by simp)).uniformL2.norm
+  have herr (i j) : UniformL2 (fun z x => ‖wordHessianError κ (ws.map e) i j (f z) x‖) := by
+    apply UniformL2.norm
+    apply UniformSobolev.uniformL2
+    apply hF.of_coreBound
+    simpa only [List.length_map] using wordHessianError_bound κ (ws.map e) i j
+  have hbg (i j) : UniformL2 (fun z x => K*‖schwartzWord (ws.map e) (G z i j) x‖) := by
+    simpa only [smul_eq_mul] using ((hG i j).word (ws.map e) (by simp)).uniformL2.norm.smul K
+  have hmajor : UniformL2 (fun z y =>
+      K*J*(‖schwartzWord (ws.map e) (ρ z) y‖+‖R z y‖)+
+        B*∑ i, ∑ j, (‖wordHessianError κ (ws.map e) j i (f z) y‖+
+          K*‖schwartzWord (ws.map e) (G z j i) y‖)) := by
+    simpa only [smul_eq_mul] using ((hp.add hR.norm).smul (K*J)).add
+      ((UniformL2.sum Finset.univ (fun i => fun z y => ∑ j, (‖wordHessianError κ (ws.map e) j i (f z) y‖+
+        K*‖schwartzWord (ws.map e) (G z j i) y‖)) (fun i _ =>
+          UniformL2.sum Finset.univ _ (fun j _ => (herr j i).add (hbg j i)))).smul B)
+  apply UniformSobolev.zero_of_majorant hmajor
+  intro z y
+  by_cases hy : y ∈ tsupport (κ : EC d → ℂ)
+  · rw [schwartzDifferential_apply _ (hc z)]
+    have hsum : (∑ i, ∑ j, c z i j y *
+        fderiv ℝ (fderiv ℝ (SchwartzMap.smulLeftCLM ℂ κ (schwartzWord (ws.map e) (f z)))) y
+          (stdOrthonormalBasis ℝ (EC d) i) (stdOrthonormalBasis ℝ (EC d) j)) =
+        (((show Matrix (Fin d) (Fin d) ℂ from fun i j => H z i j y)⁻¹ *
+          (show Matrix (Fin d) (Fin d) ℂ from fun i j => hessianEntrySchwartz i j
+            (SchwartzMap.smulLeftCLM ℂ κ (schwartzWord (ws.map e) (f z))) y)).trace.re : ℂ) := by
+      simp_rw [hec z y hy]
+      exact schwartz_real_trace_expression _ (hinv z y hy) _ (hr z) y
+    rw [hsum,Complex.norm_real,Real.norm_eq_abs]
+    exact (Complex.abs_re_le_norm _).trans
+      (determinant_word_cutoff_bound e ws hw κ (f z) (G z) (H z) (ρ z) y
+        (he z y hy) (heρ z y hy) (hinv z y hy) hB hJ (hb z y hy) (hj z y hy) (hk y))
+  · have hzero : schwartzDifferential (c z) (SchwartzMap.smulLeftCLM ℂ κ (schwartzWord (ws.map e) (f z))) y = 0 :=
+      image_eq_zero_of_notMem_tsupport (fun ht => hy (((schwartzDifferential_tsupport _ _).trans
+        ((SchwartzMap.tsupport_smulLeftCLM_subset _ _).trans inter_subset_right)) ht))
+    rw [hzero,norm_zero]
+    exact add_nonneg (mul_nonneg (mul_nonneg hK hJ) (add_nonneg (norm_nonneg _) (norm_nonneg _)))
+      (mul_nonneg hB (Finset.sum_nonneg fun i _ => Finset.sum_nonneg fun j _ =>
+        add_nonneg (norm_nonneg _) (mul_nonneg hK (norm_nonneg _))))
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/LocalKahlerEnergy.lean b/lean/OAI/Geometry/PrescribedRicci/LocalKahlerEnergy.lean
new file mode 100644
index 0000000000000000000000000000000000000000..1446a7d95bf91a2b897d2b3e9275d74542164774
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/LocalKahlerEnergy.lean
@@ -0,0 +1,220 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.ComplexIntegrationByParts
+
+namespace OAI
+
+/-! Local Kahler Energy. -/
+
+noncomputable section
+open Matrix Filter Set Topology MeasureTheory
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace Anticanonical.SourceSmooth
+namespace KaehlerMetric
+variable {d : ℕ}
+
+def holRealForm (j : Fin d) : (Coordinates d →L[ℝ] ℝ) →L[ℝ] ℂ :=
+  LinearMap.toContinuousLinearMap
+    { toFun := fun L => (2 : ℂ)⁻¹ * ((L (coordinateVector j) : ℂ) -
+        Complex.I * (L (Complex.I • coordinateVector j) : ℂ))
+      map_add' := by intro L M; simp; ring
+      map_smul' := by intro c L; simp [smul_eq_mul, Complex.real_smul]; ring }
+
+def holRealDeriv (φ : Coordinates d → ℝ) (z : Coordinates d) (j : Fin d) : ℂ :=
+  holRealForm j (fderiv ℝ φ z)
+
+lemma contDiffAt_holRealDeriv {φ : Coordinates d → ℝ} {z : Coordinates d}
+    (hφ : ContDiffAt ℝ ∞ φ z) (j : Fin d) :
+    ContDiffAt ℝ ∞ (fun y => holRealDeriv φ y j) z :=
+  (holRealForm j).contDiff.contDiffAt.comp z (hφ.fderiv_right (by simp))
+
+lemma fderiv_holRealDeriv {φ : Coordinates d → ℝ} {z : Coordinates d}
+    (hφ : ContDiffAt ℝ ∞ φ z) (j : Fin d) (v : Coordinates d) :
+    fderiv ℝ (fun y => holRealDeriv φ y j) z v =
+      holRealForm j (fderiv ℝ (fderiv ℝ φ) z v) := by
+  have hd := ((holRealForm j).hasFDerivAt.comp z
+    ((hφ.fderiv_right (m := ∞) (by simp)).differentiableAt (by simp)).hasFDerivAt).fderiv
+  exact congrArg (fun L : Coordinates d →L[ℝ] ℂ => L v) hd
+
+lemma barDeriv_holRealDeriv {φ : Coordinates d → ℝ} {z : Coordinates d}
+    (hφ : ContDiffAt ℝ ∞ φ z) (i j : Fin d) :
+    barDeriv (fun y => holRealDeriv φ y j) z i = PotentialKaehler.potentialMatrix φ z i j := by
+  unfold barDeriv
+  rw [fderiv_holRealDeriv hφ, fderiv_holRealDeriv hφ]
+  simp only [PotentialKaehler.potentialMatrix, PotentialKaehler.hermitianPartMatrix_apply,
+    PotentialKaehler.hermitianPart, holRealForm]
+  apply Complex.ext <;> simp <;> ring
+
+lemma barDeriv_mul {f h : Coordinates d → ℂ} {z : Coordinates d}
+    (hf : DifferentiableAt ℝ f z) (hh : DifferentiableAt ℝ h z) (i : Fin d) :
+    barDeriv (fun y => f y * h y) z i =
+      barDeriv f z i * h z + f z * barDeriv h z i := by
+  unfold barDeriv
+  rw [fderiv_fun_mul hf hh]
+  simp only [_root_.add_apply, _root_.smul_apply, smul_eq_mul]
+  ring
+
+def entryCLM (i j : Fin d) : Matrix (Fin d) (Fin d) ℂ →L[ℝ] ℂ :=
+  (ContinuousLinearMap.proj j : (Fin d → ℂ) →L[ℝ] ℂ).comp
+    (ContinuousLinearMap.proj i : Matrix (Fin d) (Fin d) ℂ →L[ℝ] (Fin d → ℂ))
+
+lemma fderiv_entry {f : Coordinates d → Matrix (Fin d) (Fin d) ℂ} {z : Coordinates d}
+    (hf : DifferentiableAt ℝ f z) (i j : Fin d) (v : Coordinates d) :
+    fderiv ℝ (fun y => f y i j) z v = fderiv ℝ f z v i j := by
+  exact congrArg (fun L : Coordinates d →L[ℝ] ℂ => L v)
+    (((entryCLM i j).hasFDerivAt.comp z hf.hasFDerivAt).fderiv)
+
+lemma barDeriv_entry {f : Coordinates d → Matrix (Fin d) (Fin d) ℂ} {z : Coordinates d}
+    (hf : DifferentiableAt ℝ f z) (i j k : Fin d) :
+    barDeriv (fun y => f y j k) z i = barDerivative f z i j k := by
+  unfold barDeriv barDerivative
+  rw [fderiv_entry hf, fderiv_entry hf]
+  rfl
+
+lemma barDeriv_sum {ι : Type*} [Fintype ι] {f : ι → Coordinates d → ℂ}
+    {z : Coordinates d} (hf : ∀ k, DifferentiableAt ℝ (f k) z) (i : Fin d) :
+    barDeriv (fun y => ∑ k, f k y) z i = ∑ k, barDeriv (f k) z i := by
+  unfold barDeriv
+  rw [fderiv_fun_sum (fun k _ => hf k)]
+  simp only [_root_.sum_apply, Finset.mul_sum, mul_add, Finset.sum_add_distrib]
+
+lemma contDiffAt_barDeriv {f : Coordinates d → ℂ} {z : Coordinates d}
+    (hf : ContDiffAt ℝ ∞ f z) (i : Fin d) :
+    ContDiffAt ℝ ∞ (fun y => barDeriv f y i) z :=
+  contDiffAt_const.mul ((contDiffAt_deriv_direction hf _).add
+    (contDiffAt_const.mul (contDiffAt_deriv_direction hf _)))
+
+variable {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+
+lemma contDiffAt_weightedInverse_matrix (g : KaehlerMetric A) (a : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart a).target) :
+    ContDiffAt ℝ ∞ (MongeAmpere.weightedInverse ∘ g.matrix a) z := by
+  have hd : IsUnit (g.matrix a z).det := isUnit_iff_ne_zero.mpr
+    (ne_of_gt (g.positive a z hz).det_pos)
+  exact ((MongeAmpere.contDiffAt_weightedInverse _ hd).restrict_scalars ℝ).comp z
+    ((g.smooth a).contDiffAt ((A.chart a).open_target.mem_nhds hz))
+
+lemma weightedInverse_entry_divergence (g : KaehlerMetric A) (a : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart a).target) (k : Fin d) :
+    ∑ i, barDeriv (fun y => MongeAmpere.weightedInverse (g.matrix a y) k i) z i = 0 := by
+  have hd := (g.contDiffAt_weightedInverse_matrix a hz).differentiableAt (by simp)
+  calc
+    _ = ∑ i, barDerivative (MongeAmpere.weightedInverse ∘ g.matrix a) z i k i :=
+      Finset.sum_congr rfl (fun i _ => barDeriv_entry hd i k i)
+    _ = 0 := g.weightedInverse_divergence a hz k
+
+def cofactorFlux (g : KaehlerMetric A) (φ : SmoothRealFunction A)
+    (a : Fin A.count) (z : Coordinates d) (i : Fin d) : ℂ :=
+  ∑ j, MongeAmpere.weightedInverse (g.matrix a z) j i * holRealDeriv (φ.localExpression a) z j
+
+lemma contDiffAt_cofactorFlux (g : KaehlerMetric A) (φ : SmoothRealFunction A)
+    (a : Fin A.count) {z : Coordinates d} (hz : z ∈ (A.chart a).target) (i : Fin d) :
+    ContDiffAt ℝ ∞ (fun y => g.cofactorFlux φ a y i) z := by
+  apply ContDiffAt.sum
+  intro j _
+  exact ((entryCLM j i).contDiff.contDiffAt.comp z
+    (g.contDiffAt_weightedInverse_matrix a hz)).mul
+      (contDiffAt_holRealDeriv
+        ((φ.smooth a).contDiffAt ((A.chart a).open_target.mem_nhds hz)) j)
+
+theorem cofactorFlux_divergence (g : KaehlerMetric A) (φ : SmoothRealFunction A)
+    (a : Fin A.count) {z : Coordinates d} (hz : z ∈ (A.chart a).target) :
+    ∑ i, barDeriv (fun y => g.cofactorFlux φ a y i) z i =
+      (g.matrix a z).det * ((g.matrix a z)⁻¹ * φ.hessian a z).trace := by
+  have hφ := (φ.smooth a).contDiffAt ((A.chart a).open_target.mem_nhds hz)
+  have hQ := g.contDiffAt_weightedInverse_matrix a hz
+  have hent (j i : Fin d) : DifferentiableAt ℝ
+      (fun y => MongeAmpere.weightedInverse (g.matrix a y) j i) z :=
+    ((entryCLM j i).contDiff.contDiffAt.comp z hQ).differentiableAt (by simp)
+  have hhol (j : Fin d) := (contDiffAt_holRealDeriv hφ j).differentiableAt (by simp)
+  have he (i : Fin d) : barDeriv (fun y => g.cofactorFlux φ a y i) z i =
+      ∑ j, (barDeriv (fun y => MongeAmpere.weightedInverse (g.matrix a y) j i) z i *
+        holRealDeriv (φ.localExpression a) z j +
+        MongeAmpere.weightedInverse (g.matrix a z) j i * φ.hessian a z i j) := by
+    calc
+      _ = ∑ j, barDeriv (fun y => MongeAmpere.weightedInverse (g.matrix a y) j i *
+          holRealDeriv (φ.localExpression a) y j) z i :=
+        barDeriv_sum (f := fun j y => MongeAmpere.weightedInverse (g.matrix a y) j i *
+          holRealDeriv (φ.localExpression a) y j) (fun j => (hent j i).mul (hhol j)) i
+      _ = _ := by
+        apply Finset.sum_congr rfl
+        intro j _
+        exact (barDeriv_mul (hent j i) (hhol j) i).trans
+          (congrArg (fun c : ℂ => barDeriv
+            (fun y => MongeAmpere.weightedInverse (g.matrix a y) j i) z i *
+              holRealDeriv (φ.localExpression a) z j +
+              MongeAmpere.weightedInverse (g.matrix a z) j i * c)
+            (barDeriv_holRealDeriv hφ i j))
+  simp_rw [he, Finset.sum_add_distrib]
+  have hc : (∑ i, ∑ j,
+      barDeriv (fun y => MongeAmpere.weightedInverse (g.matrix a y) j i) z i *
+        holRealDeriv (φ.localExpression a) z j) = 0 := by
+    rw [Finset.sum_comm]
+    simp only [← Finset.sum_mul, g.weightedInverse_entry_divergence a hz, zero_mul,
+      Finset.sum_const_zero]
+  rw [hc, zero_add]
+  simp only [MongeAmpere.weightedInverse, Matrix.smul_apply, smul_eq_mul,
+    Matrix.trace, Matrix.diag, Matrix.mul_apply, Finset.mul_sum]
+  rw [Finset.sum_comm]
+  apply Finset.sum_congr rfl
+  intro i _
+  apply Finset.sum_congr rfl
+  intro j _
+  ring
+
+lemma integrable_mul_barDeriv_local {f h : Coordinates d → ℂ}
+    (hf : ContDiff ℝ ∞ f) (hh : ∀ z ∈ tsupport f, ContDiffAt ℝ ∞ h z)
+    (hc : HasCompactSupport f) (i : Fin d) :
+    Integrable (fun z => f z * barDeriv h z i) := by
+  have he : (fun z => f z * barDeriv h z i) =
+      fun z => (2 : ℂ)⁻¹ * (f z * fderiv ℝ h z (coordinateVector i)) +
+        ((2 : ℂ)⁻¹ * Complex.I) * (f z * fderiv ℝ h z (Complex.I • coordinateVector i)) := by
+    funext z
+    unfold barDeriv
+    ring
+  rw [he]
+  exact ((integrable_mul_deriv_local hf hh hc _).const_mul _).add
+    ((integrable_mul_deriv_local hf hh hc _).const_mul _)
+
+lemma integrable_barDeriv_mul_local {f h : Coordinates d → ℂ}
+    (hf : ContDiff ℝ ∞ f) (hh : ∀ z ∈ tsupport f, ContDiffAt ℝ ∞ h z)
+    (hc : HasCompactSupport f) (i : Fin d) :
+    Integrable (fun z => barDeriv f z i * h z) := by
+  have he : (fun z => barDeriv f z i * h z) =
+      fun z => (2 : ℂ)⁻¹ * (fderiv ℝ f z (coordinateVector i) * h z) +
+        ((2 : ℂ)⁻¹ * Complex.I) * (fderiv ℝ f z (Complex.I • coordinateVector i) * h z) := by
+    funext z
+    unfold barDeriv
+    ring
+  rw [he]
+  exact ((integrable_deriv_mul_local hf hh hc _).const_mul _).add
+    ((integrable_deriv_mul_local hf hh hc _).const_mul _)
+
+theorem integral_cofactorFlux (g : KaehlerMetric A) (φ : SmoothRealFunction A)
+    (a : Fin A.count) {ψ : Coordinates d → ℂ}
+    (hψ : ContDiff ℝ ∞ ψ) (hc : HasCompactSupport ψ)
+    (hs : tsupport ψ ⊆ (A.chart a).target) :
+    (∫ z, ψ z * ((g.matrix a z).det * ((g.matrix a z)⁻¹ * φ.hessian a z).trace)) =
+      -(∫ z, ∑ i, barDeriv ψ z i * g.cofactorFlux φ a z i) := by
+  have hd (i : Fin d) : ∀ z ∈ tsupport ψ,
+      ContDiffAt ℝ ∞ (fun y => g.cofactorFlux φ a y i) z :=
+    fun z hz => g.contDiffAt_cofactorFlux φ a (hs hz) i
+  have hleft : (fun z => ψ z * ((g.matrix a z).det *
+      ((g.matrix a z)⁻¹ * φ.hessian a z).trace)) =
+      fun z => ∑ i, ψ z * barDeriv (fun y => g.cofactorFlux φ a y i) z i := by
+    funext z
+    rw [← Finset.mul_sum]
+    by_cases hz : z ∈ tsupport ψ
+    · rw [g.cofactorFlux_divergence φ a (hs hz)]
+    · simp [image_eq_zero_of_notMem_tsupport hz]
+  rw [hleft, integral_finsetSum _ (fun i _ =>
+    integrable_mul_barDeriv_local hψ (hd i) hc i)]
+  simp_rw [integral_mul_barDeriv_local hψ (hd _) hc]
+  rw [Finset.sum_neg_distrib, integral_finsetSum _ (fun i _ =>
+    integrable_barDeriv_mul_local hψ (hd i) hc i)]
+
+end KaehlerMetric
+end Anticanonical.SourceSmooth
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/LocalizedPathData.lean b/lean/OAI/Geometry/PrescribedRicci/LocalizedPathData.lean
new file mode 100644
index 0000000000000000000000000000000000000000..57fdd9264a7755acd1f2f8ed039b7caa9e9b8038
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/LocalizedPathData.lean
@@ -0,0 +1,120 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.GlobalMetricEntries
+import OAI.Geometry.PrescribedRicci.PathUniformForcing
+
+namespace OAI
+
+/-! Localized Path Data. -/
+
+section
+
+ 
+noncomputable section
+open Set Filter Topology Matrix LineDeriv
+open scoped ContDiff SchwartzMap Classical ComplexOrder MatrixOrder Matrix.Norms.Elementwise
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X}
+
+def cutMetric (g : KaehlerMetric A) (q : Fin A.count)
+    (σ : ChartCutoff (A.euclideanChart q).target) (i j : Fin d) : 𝓢(EC d,ℂ) :=
+  σ.product (A.euclideanChart q).open_target
+    (contDiffOn_pi.mp (contDiffOn_pi.mp (GluingData.metric_euclidean_smooth g q) i) j)
+
+omit [T2Space X] [CompactSpace X] in
+lemma cutMetric_apply (g : KaehlerMetric A) (q : Fin A.count)
+    (σ : ChartCutoff (A.euclideanChart q).target) (i j : Fin d) (y : EC d) :
+    cutMetric g q σ i j y = σ y*g.matrix q (coordinateEquiv d y) i j := rfl
+
+lemma localize_cutoff_germ (q : Fin A.count) (τ : ChartCutoff (A.euclideanChart q).target)
+    (F : Smooth A) {y : EC d} (_hy : y ∈ (A.euclideanChart q).target)
+    (hτ : (τ : EC d → ℂ) =ᶠ[𝓝 y] fun _ => 1) :
+    (localize A q (cutoffGlobal q τ) (cutoffGlobal_support q τ) F : EC d → ℂ) =ᶠ[𝓝 y]
+      F ∘ (A.euclideanChart q).symm := by
+  rw [localize_cutoff_product]
+  filter_upwards [hτ] with x hx
+  simp only [ChartCutoff.product_apply,hx,one_mul,Function.comp_apply]
+
+lemma localized_hessian_germ (q : Fin A.count) (τ : ChartCutoff (A.euclideanChart q).target)
+    (φ : SmoothRealFunction A) {y : EC d} (hy : y ∈ (A.euclideanChart q).target)
+    (hτ : (τ : EC d → ℂ) =ᶠ[𝓝 y] fun _ => 1) (i j : Fin d) :
+    (hessianEntrySchwartz i j (localize A q (cutoffGlobal q τ) (cutoffGlobal_support q τ)
+      (Smooth.ofReal φ)) : EC d → ℂ) =ᶠ[𝓝 y]
+        (fun x => φ.hessian q (coordinateEquiv d x) i j) := by
+  have hev := (localize_cutoff_germ q τ (Smooth.ofReal φ) hy hτ).eventuallyEq_nhds
+  filter_upwards [hev,(A.euclideanChart q).open_target.mem_nhds hy] with x hx hxt
+  let u : EC d → ℝ := φ.localExpression q ∘ coordinateEquiv d
+  have hu : ContDiffAt ℝ ∞ u x := (φ.euclidean_smooth q).contDiffAt
+    (by simpa only [ComplexAtlas.euclideanChart_target] using (A.euclideanChart q).open_target.mem_nhds hxt)
+  have he : (Smooth.ofReal φ : X → ℂ) ∘ (A.euclideanChart q).symm = Complex.ofRealCLM ∘ u := rfl
+  rw [he] at hx
+  rw [hessianEntrySchwartz_real hu _ hx i j,pullBilin_hessian hu]
+  have hcomp : u ∘ (coordinateEquiv d).symm = φ.localExpression q := by
+    funext z; simp [u]
+  rw [hcomp]
+  rfl
+
+lemma cutMetric_deform (g : KaehlerMetric A) (q : Fin A.count)
+    (σ τ : ChartCutoff (A.euclideanChart q).target)
+    (hτ : ∀ y ∈ tsupport (σ : EC d → ℂ), (τ : EC d → ℂ) =ᶠ[𝓝 y] fun _ => 1)
+    (φ : SmoothRealFunction A) (hp : g.PositivePotential φ) (i j : Fin d) :
+    cutMetric (g.deform φ hp) q σ i j = cutMetric g q σ i j+
+      SchwartzMap.smulLeftCLM ℂ σ.val (hessianEntrySchwartz i j
+        (localize A q (cutoffGlobal q τ) (cutoffGlobal_support q τ) (Smooth.ofReal φ))) := by
+  ext y
+  simp only [_root_.add_apply, SchwartzMap.smulLeftCLM_apply_apply σ.val.hasTemperateGrowth, smul_eq_mul]
+  by_cases hy : y ∈ tsupport (σ : EC d → ℂ)
+  · have hh := (localized_hessian_germ q τ φ (σ.support_sub hy) (hτ y hy) i j).eq_of_nhds
+    change σ y*(g.matrix q (coordinateEquiv d y) i j+φ.hessian q (coordinateEquiv d y) i j) =
+      σ y*g.matrix q (coordinateEquiv d y) i j+σ y*_
+    rw [hh,mul_add]
+  · have hh : σ y = 0 := image_eq_zero_of_notMem_tsupport hy
+    change σ y*_ = σ y*_+σ y*_
+    rw [hh]; ring
+
+lemma cutMetric_deform_germ (g : KaehlerMetric A) (q : Fin A.count)
+    (σ τ : ChartCutoff (A.euclideanChart q).target) (φ : SmoothRealFunction A) (hp : g.PositivePotential φ)
+    {y : EC d} (hy : y ∈ (A.euclideanChart q).target)
+    (hσ : (σ : EC d → ℂ) =ᶠ[𝓝 y] fun _ => 1)
+    (hτ : (τ : EC d → ℂ) =ᶠ[𝓝 y] fun _ => 1) (i j : Fin d) :
+    (cutMetric (g.deform φ hp) q σ i j : EC d → ℂ) =ᶠ[𝓝 y]
+      (fun x => cutMetric g q σ i j x+hessianEntrySchwartz i j
+        (localize A q (cutoffGlobal q τ) (cutoffGlobal_support q τ) (Smooth.ofReal φ)) x) := by
+  filter_upwards [hσ,localized_hessian_germ q τ φ hy hτ i j] with x hx hh
+  simp only [cutMetric_apply,hx,one_mul]
+  rw [hh]
+  rfl
+
+lemma schwartzWord_real {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+    [FiniteDimensional ℝ E] (ws : List E) (F : 𝓢(E,ℂ)) (hF : ∀ x, (F x).im = 0) :
+    ∀ x, (schwartzWord ws F x).im = 0 := by
+  induction ws with
+  | nil => exact hF
+  | cons v ws ih =>
+    intro x
+    let G := schwartzWord ws F
+    have he : (fun y => (G y).im) = fun _ => (0:ℝ) := funext ih
+    have hh := congrArg (fun L : E →L[ℝ] ℝ => L v)
+      ((Complex.imCLM.hasFDerivAt.comp x G.differentiableAt.hasFDerivAt).fderiv)
+    change fderiv ℝ (fun y => (G y).im) x v = (fderiv ℝ G x v).im at hh
+    rw [he] at hh
+    simp only [schwartzWord,ContinuousLinearMap.comp_apply,lineDerivOpCLM_apply]
+    rw [SchwartzMap.lineDerivOp_apply_eq_fderiv]
+    change (fderiv ℝ G x v).im = 0
+    rw [← hh]
+    simp
+
+lemma localize_cutoff_real (q : Fin A.count) (τ : ChartCutoff (A.euclideanChart q).target)
+    (hτ : ∀ y, (τ y).im = 0) (φ : SmoothRealFunction A) :
+    ∀ y, (localize A q (cutoffGlobal q τ) (cutoffGlobal_support q τ) (Smooth.ofReal φ) y).im = 0 := by
+  intro y
+  rw [localize_cutoff_product,ChartCutoff.product_apply]
+  change (τ y*(φ.value ((A.euclideanChart q).symm y):ℂ)).im = 0
+  simp only [Complex.mul_im,Complex.ofReal_re,Complex.ofReal_im,hτ,zero_mul,mul_zero,add_zero]
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/LpPowerBounds.lean b/lean/OAI/Geometry/PrescribedRicci/LpPowerBounds.lean
new file mode 100644
index 0000000000000000000000000000000000000000..15a4e9dd67db7c07112ce50371ff7f3f66776c4d
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/LpPowerBounds.lean
@@ -0,0 +1,51 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.GlobalKahlerIntegral
+
+namespace OAI
+
+/-! Lp Power Bounds. -/
+
+section
+
+ 
+
+noncomputable section
+open MeasureTheory
+open scoped ENNReal
+namespace TameInterpolation
+variable {Y : Type*} [MeasurableSpace Y] {μ : Measure Y}
+
+lemma lpNorm_mul_le {p q r : ℝ≥0∞} [ENNReal.HolderTriple p q r]
+    {f g : Y → ℝ} (hf : MemLp f p μ) (hg : MemLp g q μ) :
+    lpNorm (fun x => f x*g x) r μ ≤ lpNorm f p μ*lpNorm g q μ := by
+  have he := eLpNorm_smul_le_mul_eLpNorm (p:=p) (q:=q) (r:=r)
+    hf.aestronglyMeasurable hg.aestronglyMeasurable
+  change eLpNorm (fun x => f x*g x) r μ ≤ eLpNorm f p μ * eLpNorm g q μ at he
+  have hh := ENNReal.toReal_mono (ENNReal.mul_ne_top hf.eLpNorm_ne_top hg.eLpNorm_ne_top) he
+  simpa only [ENNReal.toReal_mul,toReal_eLpNorm] using hh
+
+lemma lpNorm_norm_rpow {E : Type*} [NormedAddCommGroup E]
+    (f : Y → E) (hf : AEStronglyMeasurable f μ) (p : ℝ≥0∞) {q : ℝ} (hq : 0 < q) :
+    lpNorm (fun x => ‖f x‖^q) p μ = (lpNorm f (p*ENNReal.ofReal q) μ)^q := by
+  rw [← toReal_eLpNorm,eLpNorm_norm_rpow f hf hq,
+    ← ENNReal.toReal_rpow,toReal_eLpNorm]
+
+lemma lpNorm_rpow_integral {E : Type*} [NormedAddCommGroup E]
+    (f : Y → E) (hf : AEStronglyMeasurable f μ) {p : ℝ} (hp : 0 < p) :
+    (lpNorm f (ENNReal.ofReal p) μ)^p = ∫ x, ‖f x‖^p ∂μ := by
+  rw [lpNorm_eq_integral_norm_rpow_toReal (by positivity : ENNReal.ofReal p ≠ 0) ENNReal.ofReal_ne_top hf,
+    ENNReal.toReal_ofReal hp.le]
+  rw [← Real.rpow_mul (integral_nonneg fun x => Real.rpow_nonneg (norm_nonneg _) _)]
+  simp [hp.ne']
+
+lemma integral_norm_le_lpNorm {f : Y → ℝ} (hf : AEStronglyMeasurable f μ) :
+    |∫ x, f x ∂μ| ≤ lpNorm f 1 μ := by
+  rw [lpNorm_one_eq_integral_norm hf]
+  exact norm_integral_le_integral_norm f
+
+end TameInterpolation
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/MatrixCofactorOrder.lean b/lean/OAI/Geometry/PrescribedRicci/MatrixCofactorOrder.lean
new file mode 100644
index 0000000000000000000000000000000000000000..fcd3a48c007b9e9e8ccd2dc9879cb426c4d02aa6
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/MatrixCofactorOrder.lean
@@ -0,0 +1,114 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.DeterminantOrder
+
+namespace OAI
+
+/-! Matrix Cofactor Order. -/
+
+section
+
+ 
+
+noncomputable section
+open Matrix Filter Set Topology
+open scoped ComplexOrder Matrix.Norms.Elementwise Classical
+namespace MongeAmpere
+variable {d : ℕ}
+local notation "Mat" => Matrix (Fin d) (Fin d) ℂ
+
+def cofactorQuadratic (H : Mat) (v : Fin d → ℂ) : ℝ :=
+  (H.det * (star v ⬝ᵥ (H⁻¹ *ᵥ v))).re
+
+lemma det_add_rankOne (H : Mat) (hH : IsUnit H.det) (v : Fin d → ℂ) (t : ℂ) :
+    (H + t • Matrix.vecMulVec v (star v)).det = H.det + t * (H.det * (star v ⬝ᵥ (H⁻¹ *ᵥ v))) := by
+  have hu : t • Matrix.vecMulVec v (star v) =
+      Matrix.replicateCol Unit (t • v) * Matrix.replicateRow Unit (star v) := by
+    ext i j
+    simp [Matrix.mul_apply, Matrix.vecMulVec, mul_assoc]
+  rw [hu, Matrix.det_add_replicateCol_mul_replicateRow hH]
+  rw [Matrix.det_unique (A := (1 + Matrix.replicateRow Unit (star v) * H⁻¹ *
+    Matrix.replicateCol Unit (t • v)))]
+  have hv : (Matrix.replicateRow Unit (star v) * H⁻¹ * Matrix.replicateCol Unit (t • v)) () () =
+      t * (star v ⬝ᵥ (H⁻¹ *ᵥ v)) := by
+    simp only [Matrix.mul_apply, Matrix.replicateRow_apply, Matrix.replicateCol_apply,
+      Pi.smul_apply, smul_eq_mul, Matrix.mulVec, dotProduct, Finset.mul_sum, Finset.sum_mul]
+    rw [Finset.sum_comm]
+    apply Finset.sum_congr rfl
+    intro i _
+    apply Finset.sum_congr rfl
+    intro j _
+    ring
+  change H.det * (1 + (Matrix.replicateRow Unit (star v) * H⁻¹ * Matrix.replicateCol Unit (t • v)) () ()) = _
+  rw [hv]
+  ring
+
+lemma det_re_add_rankOne (H : Mat) (hH : H.PosDef) (v : Fin d → ℂ) (t : ℝ) :
+    (H + (t : ℂ) • Matrix.vecMulVec v (star v)).det.re = H.det.re + t * cofactorQuadratic H v := by
+  rw [det_add_rankOne H (isUnit_iff_ne_zero.mpr hH.det_pos.ne')]
+  simp only [Complex.add_re, Complex.mul_re, Complex.ofReal_re, Complex.ofReal_im,
+    zero_mul, sub_zero, cofactorQuadratic]
+
+ 
+
+lemma cofactorQuadratic_mono {H K : Mat} (hH : H.PosDef) (hK : K.PosDef)
+    (hKH : (K - H).PosSemidef) (v : Fin d → ℂ) :
+    cofactorQuadratic H v ≤ cofactorQuadratic K v := by
+  have hb (t : ℝ) (ht : 0 ≤ t) : H.det.re + t * cofactorQuadratic H v ≤
+      K.det.re + t * cofactorQuadratic K v := by
+    have ht' := (Matrix.posSemidef_vecMulVec_self_star v).smul
+      (show (0 : ℂ) ≤ (t : ℂ) from Complex.nonneg_iff.mpr ⟨ht, rfl⟩)
+    have hh : ((K + (t : ℂ) • Matrix.vecMulVec v (star v)) -
+        (H + (t : ℂ) • Matrix.vecMulVec v (star v))).PosSemidef := by
+      convert hKH using 1
+      abel
+    have hm := det_re_mono (hH.add_posSemidef ht') hh
+    simpa only [det_re_add_rankOne H hH, det_re_add_rankOne K hK] using hm
+  by_contra hn
+  have hp : 0 < cofactorQuadratic H v - cofactorQuadratic K v := sub_pos.mpr (lt_of_not_ge hn)
+  let t := (|K.det.re - H.det.re| + 1) / (cofactorQuadratic H v - cofactorQuadratic K v)
+  have ht : 0 ≤ t := by dsimp [t]; positivity
+  have he : t * (cofactorQuadratic H v - cofactorQuadratic K v) = |K.det.re - H.det.re| + 1 :=
+    div_mul_cancel₀ _ hp.ne'
+  have hh := hb t ht
+  nlinarith [le_abs_self (K.det.re - H.det.re)]
+
+lemma cofactorQuadratic_nonneg (H : Mat) (hH : H.PosDef) (v : Fin d → ℂ) :
+    0 ≤ cofactorQuadratic H v := by
+  exact (Complex.nonneg_iff.mp (mul_nonneg hH.det_pos.le
+    (hH.inv.posSemidef.dotProduct_mulVec_nonneg v))).1
+
+lemma cofactorQuadratic_smul (H : Mat) (hH : H.PosDef) (v : Fin d → ℂ)
+    (hd : 0 < d) (c : ℝ) (hc : 0 < c) :
+    cofactorQuadratic ((c : ℂ) • H) v = c ^ (d - 1) * cofactorQuadratic H v := by
+  have hn : (c : ℂ) ≠ 0 := Complex.ofReal_ne_zero.mpr hc.ne'
+  let : Invertible (c : ℂ) := invertibleOfNonzero hn
+  have hi := Matrix.inv_smul H (c : ℂ) (isUnit_iff_ne_zero.mpr hH.det_pos.ne')
+  have hp : (c : ℂ) ^ d * (c : ℂ)⁻¹ = (c : ℂ) ^ (d - 1) := by
+    conv_lhs => rw [show d = (d - 1) + 1 by omega, pow_succ]
+    rw [mul_assoc, mul_inv_cancel₀ hn, mul_one]
+  simp only [cofactorQuadratic, Matrix.det_smul, Fintype.card_fin, hi, invOf_eq_inv,
+    Matrix.smul_mulVec, dotProduct_smul, smul_eq_mul]
+  rw [show (c : ℂ) ^ d * H.det * ((c : ℂ)⁻¹ * (star v ⬝ᵥ (H⁻¹ *ᵥ v))) =
+    ((c : ℂ) ^ d * (c : ℂ)⁻¹) * (H.det * (star v ⬝ᵥ (H⁻¹ *ᵥ v))) by ring, hp,
+    ← Complex.ofReal_pow]
+  simp only [Complex.mul_re, Complex.ofReal_re, Complex.ofReal_im, zero_mul, sub_zero]
+
+lemma cofactorQuadratic_segment {H K : Mat} (hH : H.PosDef) (hK : K.PosDef)
+    (v : Fin d → ℂ) (hd : 0 < d) {s : ℝ} (hs : s ∈ Set.Ico 0 1) :
+    (1 - s) ^ (d - 1) * cofactorQuadratic H v ≤
+      cofactorQuadratic (((1 - s : ℝ) : ℂ) • H + (s : ℂ) • K) v := by
+  have hc : (0 : ℂ) < ((1 - s : ℝ) : ℂ) := Complex.pos_iff.mpr ⟨sub_pos.mpr hs.2, rfl⟩
+  have hp := hH.smul hc
+  have hq := hK.posSemidef.smul
+    (show (0 : ℂ) ≤ (s : ℂ) from Complex.nonneg_iff.mpr ⟨hs.1, rfl⟩)
+  have hm := cofactorQuadratic_mono hp (hp.add_posSemidef hq)
+    (show ((((1 - s : ℝ) : ℂ) • H + (s : ℂ) • K) - ((1 - s : ℝ) : ℂ) • H).PosSemidef by
+      simpa only [add_sub_cancel_left] using hq) v
+  rwa [cofactorQuadratic_smul H hH v hd (1 - s) (sub_pos.mpr hs.2)] at hm
+
+end MongeAmpere
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/MatrixCompactBounds.lean b/lean/OAI/Geometry/PrescribedRicci/MatrixCompactBounds.lean
new file mode 100644
index 0000000000000000000000000000000000000000..633662bb36f9ad1f589cd15a72c86f716875c3b3
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/MatrixCompactBounds.lean
@@ -0,0 +1,56 @@
+import Mathlib
+
+namespace OAI
+
+/-! Matrix Compact Bounds. -/
+
+noncomputable section
+open Matrix Set
+open scoped Matrix.Norms.Elementwise ComplexOrder MatrixOrder
+namespace MongeAmpere
+variable {n : Type*} [Fintype n]
+local notation "Mat" => Matrix n n ℂ
+
+lemma compact_matrix_bound {f : Mat → ℝ} (hf : Continuous f) (B : ℝ) :
+    ∃ Q : ℝ, 0 < Q ∧ ∀ H : Mat, ‖H‖ ≤ B → f H ≤ Q := by
+  obtain ⟨b,hb⟩ := (isCompact_closedBall (0:Mat) B).bddAbove_image hf.continuousOn
+  refine ⟨max b 1,lt_of_lt_of_le zero_lt_one (le_max_right _ _),fun H hH => ?_⟩
+  exact (hb (mem_image_of_mem _ (by simpa only [Metric.mem_closedBall,dist_zero_right] using hH))).trans
+    (le_max_left _ _)
+
+lemma matrix_norm_le_of_entries {H : Mat} {B : ℝ} (hB : 0 ≤ B)
+    (hH : ∀ i j, ‖H i j‖ ≤ B) : ‖H‖ ≤ B := by
+  apply (pi_norm_le_iff_of_nonneg hB).mpr
+  intro i
+  exact (pi_norm_le_iff_of_nonneg hB).mpr (hH i)
+
+variable [DecidableEq n]
+
+lemma posDef_eq_det_smul_adjugate_inv (H : Mat) (hH : H.PosDef) :
+    H = H.det • H⁻¹.adjugate := by
+  let : Invertible H := Matrix.invertibleOfIsUnitDet H (isUnit_iff_ne_zero.mpr hH.det_pos.ne')
+  calc
+    H = H⁻¹⁻¹ := (Matrix.inv_inv_of_invertible H).symm
+    _ = (Ring.inverse H⁻¹.det) • H⁻¹.adjugate := Matrix.inv_def _
+    _ = H.det • H⁻¹.adjugate := by rw [Matrix.det_nonsing_inv]; simp only [Ring.inverse_eq_inv',inv_inv]
+
+lemma posDef_norm_det (H : Mat) (hH : H.PosDef) : ‖H.det‖ = H.det.re := by
+  have he : H.det = (H.det.re:ℂ) := Complex.ext rfl (by simpa using (Complex.pos_iff.mp hH.det_pos).2.symm)
+  rw [he,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg (Complex.pos_iff.mp hH.det_pos).1.le]
+  rfl
+
+lemma metric_norm_bound_from_inverse (B : ℝ) : ∃ Q : ℝ, 0 < Q ∧
+    ∀ H : Mat, H.PosDef → ‖H⁻¹‖ ≤ B → ‖H‖ ≤ H.det.re*Q := by
+  obtain ⟨Q,hQ,hb⟩ := compact_matrix_bound (show Continuous (fun H : Mat => ‖H.adjugate‖) from
+    (continuous_id.matrix_adjugate : Continuous (fun H : Mat => H.adjugate)).norm) B
+  refine ⟨Q,hQ,fun H hH hi => ?_⟩
+  calc
+    ‖H‖ = ‖H.det • H⁻¹.adjugate‖ := congrArg norm (posDef_eq_det_smul_adjugate_inv H hH)
+    _ = H.det.re*‖H⁻¹.adjugate‖ := by rw [norm_smul,posDef_norm_det H hH]
+    _ ≤ _ := mul_le_mul_of_nonneg_left (hb _ hi) (Complex.pos_iff.mp hH.det_pos).1.le
+
+end MongeAmpere
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/MatrixComponentBounds.lean b/lean/OAI/Geometry/PrescribedRicci/MatrixComponentBounds.lean
new file mode 100644
index 0000000000000000000000000000000000000000..fbc1ef7ac192d2133d85e6ba870df81eb1e8a1ea
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/MatrixComponentBounds.lean
@@ -0,0 +1,76 @@
+import Mathlib
+
+namespace OAI
+
+/-! Matrix Component Bounds. -/
+
+noncomputable section
+open Matrix
+open scoped ComplexOrder MatrixOrder
+namespace MongeAmpere
+variable {n : Type*} [Fintype n]
+local notation "Mat" => Matrix n n ℂ
+
+lemma entry_mul_bound {A B : Mat} {a b : ℝ} (ha : 0 ≤ a)
+    (hA : ∀ i j, ‖A i j‖ ≤ a) (hB : ∀ i j, ‖B i j‖ ≤ b) (i j : n) :
+    ‖(A*B) i j‖ ≤ (Fintype.card n:ℝ)*a*b := by
+  rw [Matrix.mul_apply]
+  calc
+    _ ≤ ∑ k, ‖A i k*B k j‖ := norm_sum_le _ _
+    _ ≤ ∑ _k : n, a*b := Finset.sum_le_sum (fun k _ => by
+      rw [norm_mul]
+      exact mul_le_mul (hA i k) (hB k j) (norm_nonneg _) ha)
+    _ = _ := by simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul]; ring
+
+lemma trace_entry_bound {A : Mat} {a : ℝ} (hA : ∀ i j, ‖A i j‖ ≤ a) :
+    ‖A.trace‖ ≤ (Fintype.card n:ℝ)*a := by
+  calc
+    _ ≤ ∑ i, ‖A i i‖ := norm_sum_le _ _
+    _ ≤ ∑ _i : n, a := Finset.sum_le_sum (fun i _ => hA i i)
+    _ = _ := by simp
+
+lemma trace_mul_entry_bound {A B : Mat} {a b : ℝ} (ha : 0 ≤ a)
+    (hA : ∀ i j, ‖A i j‖ ≤ a) (hB : ∀ i j, ‖B i j‖ ≤ b) :
+    ‖(A*B).trace‖ ≤ (Fintype.card n:ℝ)^2*a*b := by
+  calc
+    _ ≤ (Fintype.card n:ℝ)*((Fintype.card n:ℝ)*a*b) :=
+      trace_entry_bound (entry_mul_bound ha hA hB)
+    _ = _ := by ring
+
+lemma trace_four_entry_bound {A B C D : Mat} {a b c e : ℝ}
+    (ha : 0 ≤ a) (hb : 0 ≤ b) (hc : 0 ≤ c)
+    (hA : ∀ i j, ‖A i j‖ ≤ a) (hB : ∀ i j, ‖B i j‖ ≤ b)
+    (hC : ∀ i j, ‖C i j‖ ≤ c) (hD : ∀ i j, ‖D i j‖ ≤ e) :
+    ‖(A*B*C*D).trace‖ ≤ (Fintype.card n:ℝ)^4*a*b*c*e := by
+  have hn : (0:ℝ) ≤ Fintype.card n := Nat.cast_nonneg _
+  have hab := entry_mul_bound ha hA hB
+  have habc := entry_mul_bound (mul_nonneg (mul_nonneg hn ha) hb) hab hC
+  calc
+    _ ≤ (Fintype.card n:ℝ)^2*((Fintype.card n:ℝ)*((Fintype.card n:ℝ)*a*b)*c)*e :=
+      trace_mul_entry_bound (by positivity) habc hD
+    _ = _ := by ring
+
+lemma contracted_curvature_bound {B : Mat} {R : n → n → Mat} {a r : ℝ}
+    (ha : 0 ≤ a) (hB : ∀ i j, ‖B i j‖ ≤ a)
+    (hR : ∀ a b i j, ‖R b a i j‖ ≤ r) :
+    ‖∑ a, ∑ b, B a b*(B*R b a).trace‖ ≤ (Fintype.card n:ℝ)^4*a^2*r := by
+  have ht (i j : n) : ‖(B*R j i).trace‖ ≤ (Fintype.card n:ℝ)^2*a*r :=
+    trace_mul_entry_bound ha hB (hR i j)
+  calc
+    _ ≤ ∑ i, ‖∑ j, B i j*(B*R j i).trace‖ := norm_sum_le _ _
+    _ ≤ ∑ i, ∑ j, ‖B i j*(B*R j i).trace‖ :=
+      Finset.sum_le_sum (fun i _ => norm_sum_le _ _)
+    _ ≤ ∑ _i : n, ∑ _j : n, a*((Fintype.card n:ℝ)^2*a*r) := by
+      apply Finset.sum_le_sum
+      intro i _
+      apply Finset.sum_le_sum
+      intro j _
+      rw [norm_mul]
+      exact mul_le_mul (hB i j) (ht i j) (norm_nonneg _) ha
+    _ = _ := by simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul]; ring
+
+end MongeAmpere
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/MatrixDifferentialNew.lean b/lean/OAI/Geometry/PrescribedRicci/MatrixDifferentialNew.lean
new file mode 100644
index 0000000000000000000000000000000000000000..ac793474296477fbfcb3a13bfd42c3a2715ff60d
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/MatrixDifferentialNew.lean
@@ -0,0 +1,157 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.AnalyticSupport
+
+namespace OAI
+
+/-! Matrix Differential New. -/
+
+noncomputable section
+open Matrix Filter Topology
+open scoped ContDiff Matrix.Norms.Elementwise
+namespace MongeAmpere
+variable {d : ℕ}
+local notation "Mat" => Matrix (Fin d) (Fin d) ℂ
+
+lemma contDiff_det : ContDiff ℂ ∞ (fun H : Mat => H.det) := by
+  simp only [Matrix.det_apply']
+  apply ContDiff.sum
+  intro σ _
+  apply contDiff_const.mul
+  apply contDiff_prod
+  intro j _
+  exact contDiff_apply_apply ℂ ℂ (σ j) j
+
+lemma differentiable_det : Differentiable ℂ (fun H : Mat => H.det) :=
+  contDiff_det.differentiable (by simp)
+
+lemma fderiv_det_apply (H K : Mat) (hH : IsUnit H.det) :
+    fderiv ℂ (fun M : Mat => M.det) H K = H.det * (H⁻¹ * K).trace := by
+  have hp : HasDerivAt (fun t : ℂ => H + t • K) K 0 := by
+    convert! ((hasDerivAt_id (0 : ℂ)).smul_const K).const_add H using 1
+    simp only [one_smul]
+  have hf := (differentiable_det (H + (0 : ℂ) • K)).hasFDerivAt.comp_hasDerivAt 0 hp
+  have hf' : HasDerivAt (fun t : ℂ => (H + t • K).det)
+      (fderiv ℂ (fun M : Mat => M.det) H K) 0 := by
+    convert! hf using 1
+    simp only [zero_smul, add_zero]
+    rfl
+  exact hf'.unique (hasDerivAt_det_add_smul H K hH)
+
+lemma real_fderiv_det_apply (H K : Mat) (hH : IsUnit H.det) :
+    fderiv ℝ (fun M : Mat => M.det) H K = H.det * (H⁻¹ * K).trace := by
+  have hd := ((differentiable_det H).hasFDerivAt.restrictScalars ℝ).fderiv
+  have he := congrArg (fun L : Mat →L[ℝ] ℂ => L K) hd
+  exact he.trans (fderiv_det_apply H K hH)
+
+lemma contDiff_adjugate : ContDiff ℂ ∞ (fun H : Mat => H.adjugate) := by
+  apply contDiff_pi.mpr
+  intro i
+  apply contDiff_pi.mpr
+  intro j
+  simp only [Matrix.adjugate_apply]
+  apply contDiff_det.comp
+  apply contDiff_pi.mpr
+  intro k
+  apply contDiff_pi.mpr
+  intro l
+  by_cases hk : k = j
+  · subst k
+    simp only [Matrix.updateRow_self]
+    exact contDiff_const
+  · simp only [Matrix.updateRow_ne hk]
+    exact contDiff_apply_apply ℂ ℂ k l
+
+lemma differentiableAt_inv (H : Mat) (hH : H.det ≠ 0) :
+    DifferentiableAt ℂ (fun M : Mat => M⁻¹) H := by
+  simp only [Matrix.inv_def, Ring.inverse_eq_inv']
+  exact ((differentiable_det H).inv hH).smul
+    (contDiff_adjugate.differentiable (by simp) H)
+
+def matrixMulCLM : Mat →L[ℝ] Mat →L[ℝ] Mat :=
+  LinearMap.toContinuousLinearMap
+    { toFun := fun M => LinearMap.toContinuousLinearMap
+        { toFun := fun N => M * N
+          map_add' := fun N P => Matrix.mul_add M N P
+          map_smul' := by intro c N; exact Matrix.mul_smul M c N }
+      map_add' := by
+        intro M N
+        apply ContinuousLinearMap.ext
+        intro P
+        exact Matrix.add_mul M N P
+      map_smul' := by
+        intro c M
+        apply ContinuousLinearMap.ext
+        intro N
+        exact Matrix.smul_mul c M N }
+
+lemma real_fderiv_matrix_mul {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+    {f h : E → Mat} {x : E} (hf : DifferentiableAt ℝ f x)
+    (hh : DifferentiableAt ℝ h x) (u : E) :
+    fderiv ℝ (fun y => f y * h y) x u =
+      f x * fderiv ℝ h x u + fderiv ℝ f x u * h x := by
+  have he := (matrixMulCLM.hasFDerivAt_of_bilinear hf.hasFDerivAt hh.hasFDerivAt).fderiv
+  exact congrArg (fun L : E →L[ℝ] Mat => L u) he
+
+lemma real_fderiv_inv_apply (H K : Mat) (hH : IsUnit H.det) :
+    fderiv ℝ (fun M : Mat => M⁻¹) H K = -(H⁻¹ * K * H⁻¹) := by
+  have hi := (differentiableAt_inv H (isUnit_iff_ne_zero.mp hH)).restrictScalars ℝ
+  have he : (fun M : Mat => M * M⁻¹) =ᶠ[nhds H] (fun _ => 1) := by
+    filter_upwards [(isOpen_ne.preimage differentiable_det.continuous).mem_nhds
+      (isUnit_iff_ne_zero.mp hH)] with M hM
+    exact Matrix.mul_nonsing_inv M (isUnit_iff_ne_zero.mpr hM)
+  have hd := congrArg (fun L : Mat →L[ℝ] Mat => L K) he.fderiv_eq
+  have hp := real_fderiv_matrix_mul (differentiableAt_id (𝕜 := ℝ)) hi K
+  have hid : (fderiv ℝ (id : Mat → Mat) H) K = K :=
+    congrArg (fun L : Mat →L[ℝ] Mat => L K) (fderiv_id (𝕜 := ℝ) (x := H))
+  have hz : H * fderiv ℝ (fun M : Mat => M⁻¹) H K + K * H⁻¹ = 0 := by
+    have hp' : (fderiv ℝ (fun M : Mat => M * M⁻¹) H) K =
+        H * fderiv ℝ (fun M : Mat => M⁻¹) H K + K * H⁻¹ :=
+      hp.trans (congrArg (fun N : Mat =>
+        H * fderiv ℝ (fun M : Mat => M⁻¹) H K + N * H⁻¹) hid)
+    have hc : fderiv ℝ (fun _ : Mat => (1 : Mat)) H = 0 := fderiv_const_apply _
+    exact hp'.symm.trans (hd.trans (congrArg (fun L : Mat →L[ℝ] Mat => L K) hc))
+
+  have hl := congrArg (fun M : Mat => H⁻¹ * M) hz
+  rw [Matrix.mul_add, ← Matrix.mul_assoc, Matrix.nonsing_inv_mul _ hH,
+    Matrix.one_mul, Matrix.mul_zero, ← Matrix.mul_assoc] at hl
+  exact eq_neg_of_add_eq_zero_left hl
+
+lemma fderiv_inv_apply (H K : Mat) (hH : IsUnit H.det) :
+    fderiv ℂ (fun M : Mat => M⁻¹) H K = -(H⁻¹ * K * H⁻¹) := by
+  have hd := ((differentiableAt_inv H (isUnit_iff_ne_zero.mp hH)).hasFDerivAt.restrictScalars ℝ).fderiv
+  have he := congrArg (fun L : Mat →L[ℝ] Mat => L K) hd
+  exact he.symm.trans (real_fderiv_inv_apply H K hH)
+
+lemma contDiffAt_inv (H : Mat) (hH : H.det ≠ 0) :
+    ContDiffAt ℂ ∞ (fun M : Mat => M⁻¹) H := by
+  simp only [Matrix.inv_def, Ring.inverse_eq_inv']
+  exact (contDiff_det.contDiffAt.inv hH).smul contDiff_adjugate.contDiffAt
+
+def weightedInverse (H : Mat) : Mat := H.det • H⁻¹
+
+lemma contDiffAt_weightedInverse (H : Mat) (hH : IsUnit H.det) :
+    ContDiffAt ℂ ∞ weightedInverse H :=
+  contDiff_det.contDiffAt.smul (contDiffAt_inv H (isUnit_iff_ne_zero.mp hH))
+
+lemma differentiableAt_weightedInverse (H : Mat) (hH : IsUnit H.det) :
+    DifferentiableAt ℂ weightedInverse H :=
+  (differentiable_det H).smul (differentiableAt_inv H (isUnit_iff_ne_zero.mp hH))
+
+lemma fderiv_weightedInverse_apply (H K : Mat) (hH : IsUnit H.det) :
+    fderiv ℂ weightedInverse H K =
+      H.det • ((H⁻¹ * K).trace • H⁻¹ - H⁻¹ * K * H⁻¹) := by
+  have he := ((differentiable_det H).hasFDerivAt.smul
+    (differentiableAt_inv H (isUnit_iff_ne_zero.mp hH)).hasFDerivAt).fderiv
+  have he' := congrArg (fun L : Mat →L[ℂ] Mat => L K) he
+  change fderiv ℂ weightedInverse H K =
+    H.det • (fderiv ℂ (fun M : Mat => M⁻¹) H K) +
+      (fderiv ℂ (fun M : Mat => M.det) H K) • H⁻¹ at he'
+  rw [fderiv_det_apply H K hH, fderiv_inv_apply H K hH] at he'
+  rw [he', smul_sub, smul_neg, mul_smul]
+  abel
+
+end MongeAmpere
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/MatrixWirtinger.lean b/lean/OAI/Geometry/PrescribedRicci/MatrixWirtinger.lean
new file mode 100644
index 0000000000000000000000000000000000000000..8bfa506c548f5b3d3506f82bf1f5a78b2c6be1ab
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/MatrixWirtinger.lean
@@ -0,0 +1,129 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.LocalKahlerEnergy
+
+namespace OAI
+
+/-! Matrix Wirtinger. -/
+
+noncomputable section
+open Matrix Filter Set Topology
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace Anticanonical.SourceSmooth.KaehlerMetric
+variable {d : ℕ}
+local notation "Mat" => Matrix (Fin d) (Fin d) ℂ
+
+def holDerivative (f : Coordinates d → Mat) (z : Coordinates d) (a : Fin d) : Mat :=
+  (2:ℂ)⁻¹ • (fderiv ℝ f z (coordinateVector a) -
+    Complex.I • fderiv ℝ f z (Complex.I • coordinateVector a))
+
+lemma holDeriv_entry {f : Coordinates d → Mat} {z : Coordinates d}
+    (hf : DifferentiableAt ℝ f z) (a i j : Fin d) :
+    holDeriv (fun y => f y i j) z a = holDerivative f z a i j := by
+  unfold holDeriv holDerivative
+  rw [fderiv_entry hf,fderiv_entry hf]
+  rfl
+
+lemma contDiffAt_barDerivative {f : Coordinates d → Mat} {z : Coordinates d}
+    (hf : ContDiffAt ℝ ∞ f z) (a : Fin d) :
+    ContDiffAt ℝ ∞ (fun y => barDerivative f y a) z := by
+  unfold barDerivative
+  exact ContDiffAt.smul (contDiffAt_const (c := (2:ℂ)⁻¹))
+    ((contDiffAt_deriv_direction hf (coordinateVector a)).add
+      (ContDiffAt.smul (contDiffAt_const (c := Complex.I))
+        (contDiffAt_deriv_direction hf (Complex.I • coordinateVector a))))
+
+lemma contDiffAt_holDerivative {f : Coordinates d → Mat} {z : Coordinates d}
+    (hf : ContDiffAt ℝ ∞ f z) (a : Fin d) :
+    ContDiffAt ℝ ∞ (fun y => holDerivative f y a) z := by
+  unfold holDerivative
+  exact ContDiffAt.smul (contDiffAt_const (c := (2:ℂ)⁻¹))
+    ((contDiffAt_deriv_direction hf (coordinateVector a)).sub
+      (ContDiffAt.smul (contDiffAt_const (c := Complex.I))
+        (contDiffAt_deriv_direction hf (Complex.I • coordinateVector a))))
+
+lemma barDerivative_mul {f g : Coordinates d → Mat} {z : Coordinates d}
+    (hf : DifferentiableAt ℝ f z) (hg : DifferentiableAt ℝ g z) (a : Fin d) :
+    barDerivative (fun y => f y*g y) z a = barDerivative f z a*g z + f z*barDerivative g z a := by
+  unfold barDerivative
+  rw [MongeAmpere.real_fderiv_matrix_mul hf hg,MongeAmpere.real_fderiv_matrix_mul hf hg]
+  simp only [smul_add,Matrix.smul_mul,Matrix.mul_smul,mul_add,add_mul]
+  abel
+
+lemma holDerivative_mul {f g : Coordinates d → Mat} {z : Coordinates d}
+    (hf : DifferentiableAt ℝ f z) (hg : DifferentiableAt ℝ g z) (a : Fin d) :
+    holDerivative (fun y => f y*g y) z a = holDerivative f z a*g z + f z*holDerivative g z a := by
+  unfold holDerivative
+  rw [MongeAmpere.real_fderiv_matrix_mul hf hg,MongeAmpere.real_fderiv_matrix_mul hf hg]
+  simp only [smul_add,smul_sub,Matrix.smul_mul,Matrix.mul_smul,mul_sub,sub_mul]
+  abel
+
+lemma barDerivative_inv {f : Coordinates d → Mat} {z : Coordinates d}
+    (hf : DifferentiableAt ℝ f z) (hi : IsUnit (f z).det) (a : Fin d) :
+    barDerivative (fun y => (f y)⁻¹) z a = - ((f z)⁻¹*barDerivative f z a*(f z)⁻¹) := by
+  have hh := barDerivative_comp hf (MongeAmpere.differentiableAt_inv (f z) (isUnit_iff_ne_zero.mp hi)) a
+  exact hh.trans (MongeAmpere.fderiv_inv_apply _ _ hi)
+
+lemma holDerivative_comp {f : Coordinates d → Mat} {F : Mat → Mat} {z : Coordinates d}
+    (hf : DifferentiableAt ℝ f z) (hF : DifferentiableAt ℂ F (f z)) (a : Fin d) :
+    holDerivative (F ∘ f) z a = fderiv ℂ F (f z) (holDerivative f z a) := by
+  have hd := ((hF.hasFDerivAt.restrictScalars ℝ).comp z hf.hasFDerivAt).fderiv
+  unfold holDerivative
+  have h1 := congrArg (fun L : Coordinates d →L[ℝ] Mat => L (coordinateVector a)) hd
+  have h2 := congrArg (fun L : Coordinates d →L[ℝ] Mat => L (Complex.I • coordinateVector a)) hd
+  calc
+    _ = (2:ℂ)⁻¹ • (fderiv ℂ F (f z) (fderiv ℝ f z (coordinateVector a)) -
+        Complex.I • fderiv ℂ F (f z) (fderiv ℝ f z (Complex.I • coordinateVector a))) :=
+      congrArg₂ (fun U V : Mat => (2:ℂ)⁻¹ • (U-Complex.I • V)) h1 h2
+    _ = _ := by simp only [map_smul,map_sub]
+
+lemma holDerivative_inv {f : Coordinates d → Mat} {z : Coordinates d}
+    (hf : DifferentiableAt ℝ f z) (hi : IsUnit (f z).det) (a : Fin d) :
+    holDerivative (fun y => (f y)⁻¹) z a = - ((f z)⁻¹*holDerivative f z a*(f z)⁻¹) := by
+  have hh := holDerivative_comp hf (MongeAmpere.differentiableAt_inv (f z) (isUnit_iff_ne_zero.mp hi)) a
+  exact hh.trans (MongeAmpere.fderiv_inv_apply _ _ hi)
+
+def matrixTraceCLM : Mat →L[ℝ] ℂ := (Matrix.traceLinearMap (Fin d) ℝ ℂ).toContinuousLinearMap
+
+lemma fderiv_trace {f : Coordinates d → Mat} {z : Coordinates d}
+    (hf : DifferentiableAt ℝ f z) (v : Coordinates d) :
+    fderiv ℝ (fun y => (f y).trace) z v = (fderiv ℝ f z v).trace := by
+  exact congrArg (fun L : Coordinates d →L[ℝ] ℂ => L v)
+    (((matrixTraceCLM (d:=d)).hasFDerivAt.comp z hf.hasFDerivAt).fderiv)
+
+lemma barDeriv_trace {f : Coordinates d → Mat} {z : Coordinates d}
+    (hf : DifferentiableAt ℝ f z) (a : Fin d) :
+    barDeriv (fun y => (f y).trace) z a = (barDerivative f z a).trace := by
+  unfold barDeriv barDerivative
+  rw [fderiv_trace hf,fderiv_trace hf]
+  simp only [trace_smul,trace_add,smul_eq_mul]
+
+lemma holDeriv_trace {f : Coordinates d → Mat} {z : Coordinates d}
+    (hf : DifferentiableAt ℝ f z) (a : Fin d) :
+    holDeriv (fun y => (f y).trace) z a = (holDerivative f z a).trace := by
+  unfold holDeriv holDerivative
+  rw [fderiv_trace hf,fderiv_trace hf]
+  simp only [trace_smul,trace_sub,smul_eq_mul]
+
+variable {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+
+lemma holDerivative_adjoint (g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) (a : Fin d) :
+    (holDerivative (g.matrix q) z a)ᴴ = barDerivative (g.matrix q) z a := by
+  unfold holDerivative barDerivative
+  rw [conjTranspose_smul,conjTranspose_sub,conjTranspose_smul,
+    (g.derivative_hermitian q hz _).eq,(g.derivative_hermitian q hz _).eq]
+  simp only [map_inv₀,map_ofNat,Complex.star_def,Complex.conj_I,neg_smul,sub_neg_eq_add]
+
+lemma holDerivative_symm (g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) (a i j : Fin d) :
+    holDerivative (g.matrix q) z a i j = holDerivative (g.matrix q) z j i a := by
+  have hh := congrArg (fun M : Mat => star (M j i)) (g.holDerivative_adjoint q hz a)
+  have hh' := congrArg (fun M : Mat => star (M a i)) (g.holDerivative_adjoint q hz j)
+  simp only [conjTranspose_apply,star_star] at hh hh'
+  rw [hh,hh',g.barDerivative_symm q hz a j i]
+
+end Anticanonical.SourceSmooth.KaehlerMetric
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/MatrixWirtingerExtra.lean b/lean/OAI/Geometry/PrescribedRicci/MatrixWirtingerExtra.lean
new file mode 100644
index 0000000000000000000000000000000000000000..1e44cff9e2b5f804cd59f62a538e5d0825c8e100
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/MatrixWirtingerExtra.lean
@@ -0,0 +1,66 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.LocalKahlerEnergy
+
+namespace OAI
+
+/-! Matrix Wirtinger Extra. -/
+
+noncomputable section
+open Matrix Filter Set Topology
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace Anticanonical.SourceSmooth.KaehlerMetric
+variable {d : ℕ}
+local notation "Mat" => Matrix (Fin d) (Fin d) ℂ
+
+lemma matrix_mul_differentiableAt {f g : Coordinates d → Mat} {z : Coordinates d}
+    (hf : DifferentiableAt ℝ f z) (hg : DifferentiableAt ℝ g z) :
+    DifferentiableAt ℝ (fun y => f y*g y) z :=
+  (MongeAmpere.matrixMulCLM.hasFDerivAt_of_bilinear hf.hasFDerivAt hg.hasFDerivAt).differentiableAt
+
+lemma matrix_mul_contDiffAt {f g : Coordinates d → Mat} {z : Coordinates d}
+    (hf : ContDiffAt ℝ ∞ f z) (hg : ContDiffAt ℝ ∞ g z) :
+    ContDiffAt ℝ ∞ (fun y => f y*g y) z := by
+  apply contDiffAt_pi.mpr
+  intro i
+  apply contDiffAt_pi.mpr
+  intro j
+  simp only [Matrix.mul_apply]
+  apply ContDiffAt.sum
+  intro k _
+  exact (contDiffAt_pi.mp (contDiffAt_pi.mp hf i) k).mul
+    (contDiffAt_pi.mp (contDiffAt_pi.mp hg k) j)
+
+lemma barDerivative_congr {f g : Coordinates d → Mat} {z : Coordinates d}
+    (he : f =ᶠ[nhds z] g) (a : Fin d) : barDerivative f z a = barDerivative g z a := by
+  unfold barDerivative
+  rw [he.fderiv_eq]
+
+lemma barDerivative_sub {f g : Coordinates d → Mat} {z : Coordinates d}
+    (hf : DifferentiableAt ℝ f z) (hg : DifferentiableAt ℝ g z) (a : Fin d) :
+    barDerivative (fun y => f y-g y) z a = barDerivative f z a - barDerivative g z a := by
+  unfold barDerivative
+  have he : fderiv ℝ (fun y => f y-g y) z = fderiv ℝ f z - fderiv ℝ g z := by
+    convert! (hf.hasFDerivAt.sub hg.hasFDerivAt).fderiv using 1
+  rw [he]
+  simp only [_root_.sub_apply,smul_sub,smul_add]
+  abel
+
+lemma holDeriv_congr {f g : Coordinates d → ℂ} {z : Coordinates d}
+    (he : f =ᶠ[nhds z] g) (a : Fin d) : holDeriv f z a = holDeriv g z a := by
+  unfold holDeriv
+  rw [he.fderiv_eq]
+
+lemma holDeriv_ofReal {φ : Coordinates d → ℝ} {z : Coordinates d}
+    (hφ : DifferentiableAt ℝ φ z) (a : Fin d) :
+    holDeriv (fun y => (φ y:ℂ)) z a = holRealDeriv φ z a := by
+  have hd := (Complex.ofRealCLM.hasFDerivAt.comp z hφ.hasFDerivAt).fderiv
+  change fderiv ℝ (fun y => (φ y:ℂ)) z = _ at hd
+  unfold holDeriv holRealDeriv holRealForm
+  rw [hd]
+  rfl
+
+end Anticanonical.SourceSmooth.KaehlerMetric
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/MatrixWirtingerSecond.lean b/lean/OAI/Geometry/PrescribedRicci/MatrixWirtingerSecond.lean
new file mode 100644
index 0000000000000000000000000000000000000000..6987592a54b7b0a0206eb8b8dcf0d336c33afde2
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/MatrixWirtingerSecond.lean
@@ -0,0 +1,107 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.MatrixWirtinger
+
+namespace OAI
+
+/-! Matrix Wirtinger Second. -/
+
+noncomputable section
+open Matrix Filter Set Topology
+open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise
+namespace Anticanonical.SourceSmooth.KaehlerMetric
+variable {d : ℕ}
+local notation "Mat" => Matrix (Fin d) (Fin d) ℂ
+
+def barMatrixForm (a : Fin d) : (Coordinates d →L[ℝ] Mat) →L[ℝ] Mat :=
+  (2:ℂ)⁻¹ • ((ContinuousLinearMap.apply ℝ Mat (coordinateVector a)) +
+    Complex.I • (ContinuousLinearMap.apply ℝ Mat (Complex.I • coordinateVector a)))
+
+def holMatrixForm (a : Fin d) : (Coordinates d →L[ℝ] Mat) →L[ℝ] Mat :=
+  (2:ℂ)⁻¹ • ((ContinuousLinearMap.apply ℝ Mat (coordinateVector a)) -
+    Complex.I • (ContinuousLinearMap.apply ℝ Mat (Complex.I • coordinateVector a)))
+
+lemma fderiv_barDerivative {f : Coordinates d → Mat} {z : Coordinates d}
+    (hf : ContDiffAt ℝ ∞ f z) (a : Fin d) (v : Coordinates d) :
+    fderiv ℝ (fun y => barDerivative f y a) z v =
+      (2:ℂ)⁻¹ • (fderiv ℝ (fderiv ℝ f) z v (coordinateVector a) +
+        Complex.I • fderiv ℝ (fderiv ℝ f) z v (Complex.I • coordinateVector a)) := by
+  exact congrArg (fun L : Coordinates d →L[ℝ] Mat => L v)
+    (((barMatrixForm a).hasFDerivAt.comp z
+      ((hf.fderiv_right (m:=∞) (by simp)).differentiableAt (by simp)).hasFDerivAt).fderiv)
+
+lemma fderiv_holDerivative {f : Coordinates d → Mat} {z : Coordinates d}
+    (hf : ContDiffAt ℝ ∞ f z) (a : Fin d) (v : Coordinates d) :
+    fderiv ℝ (fun y => holDerivative f y a) z v =
+      (2:ℂ)⁻¹ • (fderiv ℝ (fderiv ℝ f) z v (coordinateVector a) -
+        Complex.I • fderiv ℝ (fderiv ℝ f) z v (Complex.I • coordinateVector a)) := by
+  exact congrArg (fun L : Coordinates d →L[ℝ] Mat => L v)
+    (((holMatrixForm a).hasFDerivAt.comp z
+      ((hf.fderiv_right (m:=∞) (by simp)).differentiableAt (by simp)).hasFDerivAt).fderiv)
+
+lemma bar_hol_comm {f : Coordinates d → Mat} {z : Coordinates d}
+    (hf : ContDiffAt ℝ ∞ f z) (a b : Fin d) :
+    barDerivative (fun y => holDerivative f y a) z b =
+      holDerivative (fun y => barDerivative f y b) z a := by
+  unfold barDerivative holDerivative
+  have ha := fderiv_holDerivative hf a (coordinateVector b)
+  have hb := fderiv_holDerivative hf a (Complex.I • coordinateVector b)
+  have hc := fderiv_barDerivative hf b (coordinateVector a)
+  have hd := fderiv_barDerivative hf b (Complex.I • coordinateVector a)
+  change fderiv ℝ (fun y => (2:ℂ)⁻¹ • (fderiv ℝ f y (coordinateVector a)-
+    Complex.I • fderiv ℝ f y (Complex.I • coordinateVector a))) z (coordinateVector b) = _ at ha
+  change fderiv ℝ (fun y => (2:ℂ)⁻¹ • (fderiv ℝ f y (coordinateVector a)-
+    Complex.I • fderiv ℝ f y (Complex.I • coordinateVector a))) z (Complex.I • coordinateVector b) = _ at hb
+  change fderiv ℝ (fun y => (2:ℂ)⁻¹ • (fderiv ℝ f y (coordinateVector b)+
+    Complex.I • fderiv ℝ f y (Complex.I • coordinateVector b))) z (coordinateVector a) = _ at hc
+  change fderiv ℝ (fun y => (2:ℂ)⁻¹ • (fderiv ℝ f y (coordinateVector b)+
+    Complex.I • fderiv ℝ f y (Complex.I • coordinateVector b))) z (Complex.I • coordinateVector a) = _ at hd
+  rw [ha,hb,hc,hd]
+  have hs (u v : Coordinates d) : fderiv ℝ (fderiv ℝ f) z u v = fderiv ℝ (fderiv ℝ f) z v u := by
+    exact hf.isSymmSndFDerivAt (by
+      simp only [minSmoothness_of_isRCLikeNormedField]
+      change ((2:ℕ∞):WithTop ℕ∞) ≤ ((⊤:ℕ∞):WithTop ℕ∞)
+      exact WithTop.coe_le_coe.mpr le_top) u v
+  rw [hs (coordinateVector a) (coordinateVector b),
+    hs (coordinateVector a) (Complex.I • coordinateVector b),
+    hs (Complex.I • coordinateVector a) (coordinateVector b),
+    hs (Complex.I • coordinateVector a) (Complex.I • coordinateVector b)]
+  ext i j
+  simp only [Matrix.smul_apply,Matrix.add_apply,Matrix.sub_apply,smul_eq_mul]
+  ring
+
+variable {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+
+lemma secondDerivative_col_symm (g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) (a b i j : Fin d) :
+    barDerivative (fun y => holDerivative (g.matrix q) y a) z b i j =
+      barDerivative (fun y => holDerivative (g.matrix q) y j) z b i a := by
+  have hs := (g.smooth q).contDiffAt ((A.chart q).open_target.mem_nhds hz)
+  have he : (fun y => holDerivative (g.matrix q) y a i j) =ᶠ[nhds z]
+      (fun y => holDerivative (g.matrix q) y j i a) := by
+    filter_upwards [(A.chart q).open_target.mem_nhds hz] with y hy
+    exact g.holDerivative_symm q hy a i j
+  rw [← barDeriv_entry ((contDiffAt_holDerivative hs a).differentiableAt (by simp)),
+    ← barDeriv_entry ((contDiffAt_holDerivative hs j).differentiableAt (by simp))]
+  unfold barDeriv
+  rw [he.fderiv_eq]
+
+lemma secondDerivative_row_symm (g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) (a b i j : Fin d) :
+    barDerivative (fun y => holDerivative (g.matrix q) y a) z b i j =
+      barDerivative (fun y => holDerivative (g.matrix q) y a) z i b j := by
+  have hs := (g.smooth q).contDiffAt ((A.chart q).open_target.mem_nhds hz)
+  rw [bar_hol_comm hs,bar_hol_comm hs]
+  have he : (fun y => barDerivative (g.matrix q) y b i j) =ᶠ[nhds z]
+      (fun y => barDerivative (g.matrix q) y i b j) := by
+    filter_upwards [(A.chart q).open_target.mem_nhds hz] with y hy
+    exact g.barDerivative_symm q hy b i j
+  rw [← holDeriv_entry ((contDiffAt_barDerivative hs b).differentiableAt (by simp)),
+    ← holDeriv_entry ((contDiffAt_barDerivative hs i).differentiableAt (by simp))]
+  unfold holDeriv
+  rw [he.fderiv_eq]
+
+end Anticanonical.SourceSmooth.KaehlerMetric
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/MomentBoundPointwise.lean b/lean/OAI/Geometry/PrescribedRicci/MomentBoundPointwise.lean
new file mode 100644
index 0000000000000000000000000000000000000000..bea289ae6018980e94659bacfa64cd0d7e807b54
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/MomentBoundPointwise.lean
@@ -0,0 +1,46 @@
+import Mathlib
+
+namespace OAI
+
+/-! Moment Bound Pointwise. -/
+
+noncomputable section
+open Set Filter Topology MeasureTheory
+namespace Anticanonical.SourceSmooth
+
+lemma continuous_le_of_moments {X : Type*} [TopologicalSpace X] [MeasurableSpace X]
+    [BorelSpace X] {μ : Measure X} [IsFiniteMeasure μ] [μ.IsOpenPosMeasure]
+    {w : X → ℝ} (hw : Continuous w) (hwn : ∀ x, 0 ≤ w x)
+    (p : ℕ → ℝ) (hp : ∀ n, 0 < p n) (hpt : Tendsto p atTop atTop)
+    {B : ℝ} (hB : 0 ≤ B)
+    (hi : ∀ n, Integrable (fun x => w x ^ p n) μ)
+    (hb : ∀ n, ∫ x, w x ^ p n ∂μ ≤ B ^ p n) : ∀ x, w x ≤ B := by
+  intro x
+  by_contra! hx
+  obtain ⟨r,hrB,hrx⟩ := exists_between hx
+  have hr : 0 < r := lt_of_le_of_lt hB hrB
+  let U := {y | r < w y}
+  have hU : IsOpen U := isOpen_lt continuous_const hw
+  have hUm : 0 < μ.real U := ENNReal.toReal_pos (ne_of_gt (hU.measure_pos μ ⟨x,hrx⟩)) (by finiteness)
+  have hbound (n : ℕ) : μ.real U ≤ (B/r)^(p n) := by
+    have hs : U ⊆ {y | r^(p n) ≤ w y^(p n)} := fun y hy =>
+      Real.rpow_le_rpow hr.le hy.le (hp n).le
+    have hm := mul_meas_ge_le_integral_of_nonneg
+      (ae_of_all μ (fun y => Real.rpow_nonneg (hwn y) (p n))) (hi n) (r^(p n))
+    have hh : r^(p n)*μ.real U ≤ B^(p n) :=
+      (mul_le_mul_of_nonneg_left (measureReal_mono hs) (Real.rpow_nonneg hr.le _)).trans (hm.trans (hb n))
+    rw [Real.div_rpow hB hr.le]
+    apply (le_div_iff₀ (Real.rpow_pos_of_pos hr _)).mpr
+    simpa only [mul_comm] using hh
+  have hlim : Tendsto (fun n => (B/r)^(p n)) atTop (𝓝 0) :=
+    (tendsto_rpow_atTop_of_base_lt_one (B/r)
+      (by have hh := div_nonneg hB hr.le; linarith)
+      ((div_lt_one hr).mpr hrB)).comp hpt
+  have hz : μ.real U ≤ 0 := ge_of_tendsto hlim (Eventually.of_forall hbound)
+  exact (not_le_of_gt hUm) hz
+
+end Anticanonical.SourceSmooth
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/MongeAmpereC0.lean b/lean/OAI/Geometry/PrescribedRicci/MongeAmpereC0.lean
new file mode 100644
index 0000000000000000000000000000000000000000..6e64e55c179014ff356572584b6dfa6cd13a7b83
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/MongeAmpereC0.lean
@@ -0,0 +1,99 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.MomentBoundPointwise
+import OAI.Geometry.PrescribedRicci.MongeAmpereMoments
+import OAI.Geometry.PrescribedRicci.MongeAmpereL2
+import OAI.Geometry.PrescribedRicci.KahlerIntegralPos
+import OAI.Geometry.PrescribedPotential.VolumePath
+
+namespace OAI
+
+/-! Monge Ampere C0. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory
+open scoped ContDiff Classical
+namespace Anticanonical.SourceSmooth.KaehlerMetric
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  [ConnectedSpace X] {A : ComplexAtlas d X}
+
+ 
+
+theorem mongeAmpere_C0_bound (g : KaehlerMetric A) (hd : 2 ≤ d)
+    (D : ℝ) (hD : 0 ≤ D) : ∃ C : ℝ, 0 ≤ C ∧
+    ∀ φ : SmoothRealFunction A, g.PositivePotential φ → g.integral φ.value = 0 →
+      (∀ x, |1-(g.potentialDensity φ).value x| ≤ D) → ∀ x, |φ.value x| ≤ C := by
+  let : MeasurableSpace X := borel X
+  let : BorelSpace X := ⟨rfl⟩
+  obtain ⟨L,hL,hL2⟩ := g.mongeAmpere_L2_bound (by omega : 0 < d) D hD
+  obtain ⟨ν,B,hν,hB,hM⟩ := g.mongeAmpere_moment_bounds hd D hD
+  let V := g.integral (fun _ => (1:ℝ))
+  have hV : 0 ≤ V := g.integral_nonneg (fun _ => zero_le_one)
+  let M : ℝ := (V+L)^(1/2:ℝ)
+  have hMn : 0 ≤ M := Real.rpow_nonneg (add_nonneg hV hL) _
+  refine ⟨B*M,mul_nonneg hB hMn,fun φ hp hm hρ => ?_⟩
+  have h2 : g.powerMoment φ 2 ≤ M := by
+    have hI : g.powerIntegral φ 2 = V + g.integral (fun x => φ.value x^2) := by
+      simp only [powerIntegral, div_self (by norm_num : (2:ℝ) ≠ 0), Real.rpow_one]
+      exact g.integral_add continuous_const (φ.continuous.pow 2)
+    change (g.powerIntegral φ 2)^(1/2:ℝ) ≤ (V+L)^(1/2:ℝ)
+    apply Real.rpow_le_rpow (g.powerIntegral_nonneg φ 2) _ (by norm_num)
+    rw [hI]
+    have hh := hL2 φ hp hm hρ
+    linarith
+  let w : X → ℝ := fun x => Real.sqrt (1+φ.value x^2)
+  have hw : Continuous w := (continuous_const.add (φ.continuous.pow 2)).sqrt
+  have hwn (x : X) : 0 ≤ w x := Real.sqrt_nonneg _
+  let p : ℕ → ℝ := fun n => 2*ν^n
+  have hp0 (n : ℕ) : 0 < p n := mul_pos (by norm_num) (pow_pos (lt_trans zero_lt_one hν) n)
+  have hpt : Tendsto p atTop atTop := (tendsto_pow_atTop_atTop_of_one_lt hν).const_mul_atTop (by norm_num)
+  have hwp (n : ℕ) : (fun x => w x^(p n)) = (fun x => (1+φ.value x^2)^((p n)/2)) := by
+    funext x
+    dsimp only [w]
+    rw [Real.sqrt_eq_rpow, ← Real.rpow_mul (by positivity : 0 ≤ 1+φ.value x^2)]
+    congr 1; ring
+  have hi (n : ℕ) : Integrable (fun x => w x ^ p n) g.volumeMeasure := by
+    rw [hwp]
+    exact (powerIntegrand_continuous φ _).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)
+  have hmom (n : ℕ) : ∫ x, w x^p n ∂g.volumeMeasure ≤ (B*M)^p n := by
+    rw [hwp, g.integral_volumeMeasure (powerIntegrand_continuous φ _)]
+    have hh : g.powerMoment φ (p n) ≤ B*M :=
+      (hM φ hp hρ n).trans (mul_le_mul_of_nonneg_left h2 hB)
+    have hpow := Real.rpow_le_rpow (g.powerMoment_nonneg φ (p n)) hh (hp0 n).le
+    dsimp only [powerMoment] at hpow
+    rw [← Real.rpow_mul (g.powerIntegral_nonneg φ (p n)), one_div_mul_cancel (ne_of_gt (hp0 n)), Real.rpow_one] at hpow
+    exact hpow
+  have hpoint := continuous_le_of_moments hw hwn p hp0 hpt (mul_nonneg hB hMn) hi hmom
+  intro x
+  calc
+    |φ.value x| = Real.sqrt (φ.value x^2) := (Real.sqrt_sq_eq_abs _).symm
+    _ ≤ w x := Real.sqrt_le_sqrt (by linarith)
+    _ ≤ _ := hpoint x
+
+ 
+theorem volumePath_C0_bound (g : KaehlerMetric A) (h : SemipositiveAnticanonicalMetric A)
+    (hd : 2 ≤ d) : ∃ C : ℝ, 0 ≤ C ∧
+    ∀ (φ : SmoothRealFunction A) (hp : g.PositivePotential φ), g.integral φ.value = 0 →
+      ∀ (t b : ℝ), t ∈ Icc 0 1 →
+        (∀ x, (g.logRatio (g.deform φ hp)).value x = t*(prescribedForcing g h).value x+b) →
+          ∀ x, |φ.value x| ≤ C := by
+  let M := ‖(⟨(prescribedForcing g h).value, (prescribedForcing g h).continuous⟩ : C(X,ℝ))‖
+  let D := 1+Real.exp (2*M)
+  obtain ⟨C,hC,hbound⟩ := g.mongeAmpere_C0_bound hd D (by dsimp [D]; positivity)
+  refine ⟨C,hC,fun φ hp hm t b ht he => hbound φ hp hm (fun x => ?_)⟩
+  have hh := g.volumePath_density_bounds h ht hp he x
+  have hpρ : 0 ≤ (g.potentialDensity φ).value x := (Real.exp_pos _).le.trans hh.1
+  calc
+    |1-(g.potentialDensity φ).value x| ≤ |(1:ℝ)|+|(g.potentialDensity φ).value x| := abs_sub _ _
+    _ = 1+(g.potentialDensity φ).value x := by rw [abs_one, abs_of_nonneg hpρ]
+    _ ≤ _ := by dsimp [D,M]; linarith [hh.2]
+
+end Anticanonical.SourceSmooth.KaehlerMetric
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/MongeAmpereEnergy.lean b/lean/OAI/Geometry/PrescribedRicci/MongeAmpereEnergy.lean
new file mode 100644
index 0000000000000000000000000000000000000000..00789428344e80c0fa439ace15c22a320cbd725f
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/MongeAmpereEnergy.lean
@@ -0,0 +1,196 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.GlobalKahlerEnergy
+import OAI.Geometry.PrescribedRicci.GlobalKahlerIntegral
+import OAI.Geometry.PrescribedRicci.MatrixCofactorOrder
+import OAI.Geometry.PrescribedPotential.PotentialDensity
+import OAI.Geometry.PrescribedRicci.KahlerVolumeVariation
+
+namespace OAI
+
+/-! Monge Ampere Energy. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Matrix MeasureTheory
+open scoped ContDiff ComplexOrder Classical Matrix.Norms.Elementwise
+namespace Anticanonical.SourceSmooth
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+namespace KaehlerMetric
+
+lemma integral_sub [T2Space X] [CompactSpace X] (g : KaehlerMetric A)
+    {f h : X → ℝ} (hf : Continuous f) (hh : Continuous h) :
+    g.integral (fun x => f x - h x) = g.integral f - g.integral h := by
+  have he : (fun x => f x - h x) = fun x => f x + (-1) * h x := by funext x; ring
+  rw [he, g.integral_add hf (show Continuous (fun x => (-1) * h x) from continuous_const.fun_mul hh), g.integral_const_mul]
+  ring
+
+lemma integral_mono [T2Space X] [CompactSpace X] (g : KaehlerMetric A)
+    {f h : X → ℝ} (hf : Continuous f) (hh : Continuous h) (hle : ∀ x, f x ≤ h x) :
+    g.integral f ≤ g.integral h := by
+  have he := g.integral_nonneg (fun x => sub_nonneg.mpr (hle x))
+  rw [g.integral_sub hh hf] at he
+  exact sub_nonneg.mp he
+
+lemma integral_zero [T2Space X] [CompactSpace X] (g : KaehlerMetric A) :
+    g.integral (fun _ => 0) = 0 := by simp [integral, chartIntegral]
+
+lemma cofactorQuadratic_eq_gradientPair (H : Matrix (Fin d) (Fin d) ℂ)
+    (hH : H.IsHermitian) (p : Fin d → ℂ) :
+    MongeAmpere.cofactorQuadratic H (star p) = H.det.re * (gradientPair H p p).re := by
+  have he : star (star p) ⬝ᵥ (H⁻¹ *ᵥ star p) = gradientPair H p p := by
+    simp only [gradientPair, dotProduct, Matrix.mulVec, Pi.star_apply, star_star, Finset.mul_sum]
+    apply Finset.sum_congr rfl
+    intro j _
+    apply Finset.sum_congr rfl
+    intro i _
+    ring
+  rw [MongeAmpere.cofactorQuadratic, he, Complex.mul_re, hermitian_det_im hH, zero_mul, sub_zero]
+
+lemma energy_segment_coercive (g : KaehlerMetric A) (φ : SmoothRealFunction A)
+    (hp : g.PositivePotential φ) (hd : 0 < d) (s : ℝ) (hs : s ∈ Ico 0 1) (x : X) :
+    (1 - s) ^ (d - 1) * (g.energy φ φ).value x ≤
+      (g.potentialDensity (φ.realSMul s)).value x *
+        ((g.deform (φ.realSMul s) (hp.segment ⟨hs.1, hs.2.le⟩)).energy φ φ).value x := by
+  let gs := g.deform (φ.realSMul s) (hp.segment ⟨hs.1, hs.2.le⟩)
+  obtain ⟨i, hi⟩ := A.covers x
+  have hz := (A.chart i).mapsTo hi
+  let z := A.chart i x
+  let p := holRealDeriv (φ.localExpression i) z
+  have hem : (((1 - s : ℝ) : ℂ) • g.matrix i z + (s : ℂ) •
+      (g.matrix i z + φ.hessian i z)) = gs.matrix i z := by
+    change _ = g.matrix i z + (φ.realSMul s).hessian i z
+    rw [SmoothRealFunction.hessian_realSMul]
+    ext j k
+    simp only [Matrix.add_apply, Matrix.smul_apply, smul_eq_mul, Complex.ofReal_sub, Complex.ofReal_one]
+    ring
+  have hm := MongeAmpere.cofactorQuadratic_segment (g.positive i z hz) (hp i z hz) (star p) hd hs
+  rw [hem, cofactorQuadratic_eq_gradientPair _ (g.positive i z hz).isHermitian,
+    cofactorQuadratic_eq_gradientPair _ (gs.positive i z hz).isHermitian] at hm
+  have he := g.energy_local φ φ i hz
+  have he' := gs.energy_local φ φ i hz
+  simp only [SmoothRealFunction.localExpression, Function.comp_apply, (A.chart i).left_inv hi] at he he'
+  change (g.energy φ φ).value x = (gradientPair (g.matrix i z) p p).re at he
+  change (gs.energy φ φ).value x = (gradientPair (gs.matrix i z) p p).re at he'
+  rw [← he, ← he'] at hm
+  have hv := g.density_mul_volume (φ.realSMul s) i hz
+  rw [(A.chart i).left_inv hi] at hv
+  change (g.potentialDensity (φ.realSMul s)).value x * (g.matrix i z).det.re = (gs.matrix i z).det.re at hv
+  apply (mul_le_mul_iff_of_pos_left (g.volumeCoefficient_pos i hz)).mp
+  change (g.matrix i z).det.re * _ ≤ (g.matrix i z).det.re * _
+  calc
+    _ = (1 - s) ^ (d - 1) * ((g.matrix i z).det.re * (g.energy φ φ).value x) := by ring
+    _ ≤ (gs.matrix i z).det.re * (gs.energy φ φ).value x := hm
+    _ = _ := by rw [← hv]; ring
+
+lemma integral_energy_segment_coercive [T2Space X] [CompactSpace X]
+    (g : KaehlerMetric A) (φ : SmoothRealFunction A) (hp : g.PositivePotential φ)
+    (hd : 0 < d) (s : ℝ) (hs : s ∈ Ico 0 1) :
+    (1 - s) ^ (d - 1) * g.integral (g.energy φ φ).value ≤
+      (g.deform (φ.realSMul s) (hp.segment ⟨hs.1, hs.2.le⟩)).integral
+        ((g.deform (φ.realSMul s) (hp.segment ⟨hs.1, hs.2.le⟩)).energy φ φ).value := by
+  let gs := g.deform (φ.realSMul s) (hp.segment ⟨hs.1, hs.2.le⟩)
+  rw [← g.integral_const_mul, ← g.integral_density (φ.realSMul s) (hp.segment ⟨hs.1, hs.2.le⟩)]
+  apply g.integral_mono (continuous_const.mul (g.energy φ φ).continuous)
+    ((gs.energy φ φ).continuous.mul (g.potentialDensity _).continuous)
+  intro x
+  change (1 - s) ^ (d - 1) * (g.energy φ φ).value x ≤
+    (gs.energy φ φ).value x * (g.potentialDensity (φ.realSMul s)).value x
+  simpa only [mul_comm] using g.energy_segment_coercive φ hp hd s hs x
+
+ 
+
+lemma integral_energy_half_segment [T2Space X] [CompactSpace X]
+    (g : KaehlerMetric A) (φ : SmoothRealFunction A) (hp : g.PositivePotential φ)
+    (hd : 0 < d) (s : ℝ) (hs : s ∈ Icc 0 (1/2)) :
+    (1/2 : ℝ) ^ (d - 1) * g.integral (g.energy φ φ).value ≤
+      (g.deform (φ.realSMul s) (hp.segment ⟨hs.1, by linarith [hs.2]⟩)).integral
+        ((g.deform (φ.realSMul s) (hp.segment ⟨hs.1, by linarith [hs.2]⟩)).energy φ φ).value := by
+  have hs' : s ∈ Ico 0 1 := ⟨hs.1, by linarith [hs.2]⟩
+  refine le_trans ?_ (g.integral_energy_segment_coercive φ hp hd s hs')
+  exact mul_le_mul_of_nonneg_right
+    (pow_le_pow_left₀ (by norm_num) (by linarith [hs.2]) (d - 1))
+    (g.integral_nonneg (g.energy_nonneg φ))
+
+lemma potentialDensity_smul_zero (g : KaehlerMetric A) (φ : SmoothRealFunction A) (x : X) :
+    (g.potentialDensity (φ.realSMul 0)).value x = 1 := by
+  obtain ⟨i, hi⟩ := A.covers x
+  change g.potentialDensityValue _ x = _
+  rw [g.potentialDensityValue_local _ i hi]
+  simp only [volumePolynomial, SmoothRealFunction.hessian_realSMul, Complex.ofReal_zero,
+    zero_smul, add_zero, div_self (g.positive i _ ((A.chart i).mapsTo hi)).det_pos.ne', Complex.one_re]
+
+lemma potentialDensity_smul_one (g : KaehlerMetric A) (φ : SmoothRealFunction A) (x : X) :
+    (g.potentialDensity (φ.realSMul 1)).value x = (g.potentialDensity φ).value x := by
+  obtain ⟨i, hi⟩ := A.covers x
+  change g.potentialDensityValue _ x = g.potentialDensityValue _ x
+  rw [g.potentialDensityValue_local _ i hi, g.potentialDensityValue_local _ i hi]
+  simp only [volumePolynomial, SmoothRealFunction.hessian_realSMul, Complex.ofReal_one, one_smul]
+
+ 
+
+theorem mongeAmpere_energy [T2Space X] [CompactSpace X]
+    (g : KaehlerMetric A) (φ : SmoothRealFunction A) (hp : g.PositivePotential φ) (hd : 0 < d) :
+    (1/2 : ℝ) ^ d * g.integral (g.energy φ φ).value ≤
+      g.integral φ.value - g.integral (fun x => φ.value x * (g.potentialDensity φ).value x) := by
+  let f (t : ℝ) := g.integral (fun x => φ.value x * (g.potentialDensity (φ.realSMul t)).value x)
+  have hf (s : ℝ) (hs : s ∈ Icc 0 1) := g.volume_firstVariation φ φ s (hp.segment hs)
+  have hfc : ContinuousOn f (Icc 0 1) := fun s hs => (hf s hs).continuousAt.continuousWithinAt
+  have hfd : DifferentiableOn ℝ f (interior (Icc 0 1)) :=
+    fun s hs => (hf s (interior_subset hs)).differentiableAt.differentiableWithinAt
+  have ha : AntitoneOn f (Icc 0 1) := antitoneOn_of_deriv_nonpos (convex_Icc 0 1) hfc hfd (by
+    intro s hs
+    rw [(hf s (interior_subset hs)).deriv]
+    exact neg_nonpos.mpr ((g.deform (φ.realSMul s) (hp.segment (interior_subset hs))).integral_nonneg
+      ((g.deform (φ.realSMul s) (hp.segment (interior_subset hs))).energy_nonneg φ)))
+  have hsub : Icc (0 : ℝ) (1/2) ⊆ Icc 0 1 := fun s hs => ⟨hs.1, by linarith [hs.2]⟩
+  have hdh : DifferentiableOn ℝ f (interior (Icc 0 (1/2))) :=
+    fun s hs => (hf s (hsub (interior_subset hs))).differentiableAt.differentiableWithinAt
+  have hh := (convex_Icc (0 : ℝ) (1/2)).image_sub_le_mul_sub_of_deriv_le
+    (hfc.mono hsub) hdh (C := -((1/2 : ℝ) ^ (d - 1) * g.integral (g.energy φ φ).value))
+    (by
+      intro s hs
+      rw [(hf s (hsub (interior_subset hs))).deriv]
+      exact neg_le_neg (g.integral_energy_half_segment φ hp hd s (interior_subset hs)))
+    0 (by norm_num) (1/2) (by norm_num) (by norm_num)
+  have he := ha (show (1/2 : ℝ) ∈ Icc 0 1 by norm_num) (show (1 : ℝ) ∈ Icc 0 1 by norm_num) (by norm_num)
+  have h0 : f 0 = g.integral φ.value := by
+    dsimp only [f]
+    simp_rw [g.potentialDensity_smul_zero, mul_one]
+  have h1 : f 1 = g.integral (fun x => φ.value x * (g.potentialDensity φ).value x) := by
+    dsimp only [f]
+    simp_rw [g.potentialDensity_smul_one]
+  have hpow : (1/2 : ℝ) ^ d = (1/2 : ℝ) ^ (d - 1) * (1/2) := by
+    conv_lhs => rw [show d = (d - 1) + 1 by omega, pow_succ]
+  rw [h0] at hh
+  rw [h1] at he
+  rw [hpow]
+  nlinarith
+
+lemma mongeAmpere_energy_density_bound [T2Space X] [CompactSpace X]
+    (g : KaehlerMetric A) (φ : SmoothRealFunction A) (hp : g.PositivePotential φ) (hd : 0 < d)
+    (D : ℝ) (hD : ∀ x, |1 - (g.potentialDensity φ).value x| ≤ D) :
+    (1/2 : ℝ) ^ d * g.integral (g.energy φ φ).value ≤ D * g.integral (fun x => |φ.value x|) := by
+  refine (g.mongeAmpere_energy φ hp hd).trans ?_
+  rw [← g.integral_sub φ.continuous (φ.continuous.fun_mul (g.potentialDensity φ).continuous),
+    ← g.integral_const_mul]
+  apply g.integral_mono (φ.continuous.sub (φ.continuous.fun_mul (g.potentialDensity φ).continuous))
+    (continuous_const.mul φ.continuous.abs)
+  intro x
+  change φ.value x - φ.value x * (g.potentialDensity φ).value x ≤ D * |φ.value x|
+  calc
+    _ = φ.value x * (1 - (g.potentialDensity φ).value x) := by ring
+    _ ≤ |φ.value x * (1 - (g.potentialDensity φ).value x)| := le_abs_self _
+    _ = |φ.value x| * |1 - (g.potentialDensity φ).value x| := abs_mul _ _
+    _ ≤ |φ.value x| * D := mul_le_mul_of_nonneg_left (hD x) (abs_nonneg _)
+    _ = _ := mul_comm _ _
+
+end KaehlerMetric
+end Anticanonical.SourceSmooth
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/MongeAmpereL2.lean b/lean/OAI/Geometry/PrescribedRicci/MongeAmpereL2.lean
new file mode 100644
index 0000000000000000000000000000000000000000..29dd0b6c7a56e5a3e047bb6f901807c19a9e4a12
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/MongeAmpereL2.lean
@@ -0,0 +1,114 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.KahlerPoincare
+import OAI.Geometry.PrescribedRicci.KahlerVolumeVariation
+import OAI.Geometry.PrescribedRicci.MongeAmpereEnergy
+import OAI.Geometry.PrescribedPotential.VolumeDeterminantBounds
+
+namespace OAI
+
+/-! Monge Ampere L2. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory
+open scoped ContDiff Classical
+namespace Anticanonical.SourceSmooth
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  [ConnectedSpace X] {A : ComplexAtlas d X}
+namespace KaehlerMetric
+
+theorem mongeAmpere_L2_bound (g : KaehlerMetric A) (hd : 0 < d)
+    (D : ℝ) (hD : 0 ≤ D) : ∃ C : ℝ, 0 ≤ C ∧
+    ∀ φ : SmoothRealFunction A, g.PositivePotential φ → g.integral φ.value = 0 →
+      (∀ x, |1 - (g.potentialDensity φ).value x| ≤ D) →
+      g.integral (fun x => φ.value x ^ 2) ≤ C := by
+  obtain ⟨P,hP,hpoincare⟩ := g.poincare
+  let c : ℝ := (1/2) ^ d
+  have hc : 0 < c := pow_pos (by norm_num) _
+  let V := g.integral (fun _ => 1)
+  have hV : 0 ≤ V := g.integral_nonneg (fun _ => zero_le_one)
+  refine ⟨(P*D)^2 * V / c^2, div_nonneg (mul_nonneg (sq_nonneg _) hV) (sq_nonneg _), ?_⟩
+  intro φ hp hmean hρ
+  let Iφ := g.integral (fun x => φ.value x ^ 2)
+  let Eφ := g.integral (g.energy φ φ).value
+  let Lφ := g.integral (fun x => |φ.value x|)
+  have hI : 0 ≤ Iφ := g.integral_nonneg (fun _ => sq_nonneg _)
+  have hL : 0 ≤ Lφ := g.integral_nonneg (fun _ => abs_nonneg _)
+  have hpi : Iφ ≤ P * Eφ := hpoincare φ hmean
+  have hma : c * Eφ ≤ D * Lφ := g.mongeAmpere_energy_density_bound φ hp hd D hρ
+  have hcs : Lφ ^ 2 ≤ Iφ * V := by
+    have hh := g.integral_cauchySchwarz φ.continuous.abs (continuous_const (y := (1 : ℝ)))
+    simpa only [one_pow, mul_one, sq_abs] using hh
+  have hfirst : c * Iφ ≤ (P*D) * Lφ := by
+    calc
+      c * Iφ ≤ c * (P * Eφ) := mul_le_mul_of_nonneg_left hpi hc.le
+      _ = P * (c * Eφ) := by ring
+      _ ≤ P * (D * Lφ) := mul_le_mul_of_nonneg_left hma hP.le
+      _ = _ := by ring
+  have hs : (c*Iφ)^2 ≤ ((P*D)*Lφ)^2 :=
+    (sq_le_sq₀ (mul_nonneg hc.le hI) (mul_nonneg (mul_nonneg hP.le hD) hL)).2 hfirst
+  have hprod : (c^2 * Iφ) * Iφ ≤ ((P*D)^2 * V) * Iφ := by
+    calc
+      (c^2 * Iφ) * Iφ = (c*Iφ)^2 := by ring
+      _ ≤ ((P*D)*Lφ)^2 := hs
+      _ = (P*D)^2 * Lφ^2 := by ring
+      _ ≤ (P*D)^2 * (Iφ*V) := mul_le_mul_of_nonneg_left hcs (sq_nonneg _)
+      _ = _ := by ring
+  change Iφ ≤ (P*D)^2 * V / c^2
+  by_cases hz : Iφ = 0
+  · rw [hz]; exact div_nonneg (mul_nonneg (sq_nonneg _) hV) (sq_nonneg _)
+  · apply (le_div_iff₀ (sq_pos_of_pos hc)).2
+    have hh := (mul_le_mul_iff_left₀ (lt_of_le_of_ne hI (Ne.symm hz))).1 hprod
+    simpa only [mul_comm] using hh
+
+omit [T2Space X] in
+lemma volumePath_density_bounds (g : KaehlerMetric A) (h : SemipositiveAnticanonicalMetric A)
+    {t b : ℝ} {φ : SmoothRealFunction A} (ht : t ∈ Icc 0 1)
+    (hp : g.PositivePotential φ)
+    (he : ∀ x, (g.logRatio (g.deform φ hp)).value x = t*(prescribedForcing g h).value x+b)
+    (x : X) :
+    let M := ‖(⟨(prescribedForcing g h).value, (prescribedForcing g h).continuous⟩ : C(X,ℝ))‖;
+    Real.exp (-2*M) ≤ (g.potentialDensity φ).value x ∧
+      (g.potentialDensity φ).value x ≤ Real.exp (2*M) := by
+  obtain ⟨i,hi⟩ := A.covers x
+  have hz := (A.chart i).mapsTo hi
+  have hh := volumePath_det_ratio_bounds g h ht hp he i hz
+  have hdens := g.density_mul_volume φ i hz
+  rw [(A.chart i).left_inv hi] at hdens
+  have hv := g.volumeCoefficient_pos i hz
+  change 0 < (g.matrix i (A.chart i x)).det.re at hv
+  have heq : (g.potentialDensity φ).value x =
+      (g.matrix i (A.chart i x) + φ.hessian i (A.chart i x)).det.re /
+        (g.matrix i (A.chart i x)).det.re := (eq_div_iff (ne_of_gt hv)).2 hdens
+  rw [heq]
+  exact hh
+
+ 
+
+theorem volumePath_L2_bound (g : KaehlerMetric A) (h : SemipositiveAnticanonicalMetric A)
+    (hd : 0 < d) : ∃ C : ℝ, 0 ≤ C ∧
+    ∀ (t b : ℝ) (φ : SmoothRealFunction A) (hp : g.PositivePotential φ), t ∈ Icc 0 1 →
+      g.integral φ.value = 0 →
+      (∀ x, (g.logRatio (g.deform φ hp)).value x = t*(prescribedForcing g h).value x+b) →
+      g.integral (fun x => φ.value x ^ 2) ≤ C := by
+  let M := ‖(⟨(prescribedForcing g h).value, (prescribedForcing g h).continuous⟩ : C(X,ℝ))‖
+  obtain ⟨C,hC,hbound⟩ := g.mongeAmpere_L2_bound hd (1 + Real.exp (2*M)) (by positivity)
+  refine ⟨C,hC,fun t b φ hp ht hmean he => hbound φ hp hmean ?_⟩
+  intro x
+  obtain ⟨hlo,hup⟩ := g.volumePath_density_bounds h ht hp he x
+  change Real.exp (-2*M) ≤ _ at hlo
+  change _ ≤ Real.exp (2*M) at hup
+  have hρ : 0 ≤ (g.potentialDensity φ).value x := (Real.exp_pos _).le.trans hlo
+  apply abs_le.mpr
+  constructor <;> linarith [Real.exp_pos (2*M)]
+
+end KaehlerMetric
+end Anticanonical.SourceSmooth
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/MongeAmpereMoments.lean b/lean/OAI/Geometry/PrescribedRicci/MongeAmpereMoments.lean
new file mode 100644
index 0000000000000000000000000000000000000000..7850f9b47d8fa216ca83d2784505c08d9a79be0d
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/MongeAmpereMoments.lean
@@ -0,0 +1,134 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.MongeAmperePowerEnergy
+import OAI.Geometry.PrescribedRicci.MoserIteration
+import OAI.Geometry.PrescribedRicci.KahlerSobolev
+import OAI.Geometry.PrescribedRicci.KahlerVolumeMeasure
+
+namespace OAI
+
+/-! Monge Ampere Moments. -/
+
+section
+
+ 
+noncomputable section
+open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory
+open scoped ContDiff Classical NNReal ENNReal
+namespace Anticanonical.SourceSmooth.KaehlerMetric
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  [MeasurableSpace X] [BorelSpace X] {A : ComplexAtlas d X}
+
+def powerIntegral (g : KaehlerMetric A) (φ : SmoothRealFunction A) (p : ℝ) : ℝ :=
+  g.integral (fun x => (1+φ.value x^2)^(p/2))
+def powerMoment (g : KaehlerMetric A) (φ : SmoothRealFunction A) (p : ℝ) : ℝ :=
+  (g.powerIntegral φ p)^(1/p)
+
+omit [T2Space X] [CompactSpace X] [MeasurableSpace X] [BorelSpace X] in
+lemma powerIntegrand_continuous (φ : SmoothRealFunction A) (p : ℝ) :
+    Continuous (fun x => (1+φ.value x^2)^(p/2)) :=
+  (continuous_const.add (φ.continuous.pow 2)).rpow_const (fun x => Or.inl (show 1+φ.value x^2 ≠ 0 from ne_of_gt (by positivity)))
+omit [MeasurableSpace X] [BorelSpace X] in
+lemma powerIntegral_nonneg (g : KaehlerMetric A) (φ : SmoothRealFunction A) (p : ℝ) :
+    0 ≤ g.powerIntegral φ p := g.integral_nonneg (fun _ => Real.rpow_nonneg (by positivity) _)
+omit [MeasurableSpace X] [BorelSpace X] in
+lemma powerMoment_nonneg (g : KaehlerMetric A) (φ : SmoothRealFunction A) (p : ℝ) :
+    0 ≤ g.powerMoment φ p := Real.rpow_nonneg (g.powerIntegral_nonneg φ p) _
+
+lemma moserPower_lpNorm_sq (g : KaehlerMetric A) (φ : SmoothRealFunction A)
+    {q : ℝ≥0} (hq : 0 < q) (p : ℝ) :
+    (lpNorm (φ.compose (moserPower p) (moserPower_smooth p)).value q g.volumeMeasure)^2 =
+      (g.powerIntegral φ (p*(q:ℝ)/2))^(2/(q:ℝ)) := by
+  rw [lpNorm_nnreal_eq_integral_norm_rpow (ne_of_gt hq)
+    (φ.compose (moserPower p) (moserPower_smooth p)).continuous.aestronglyMeasurable]
+  have he : (fun x => ‖(φ.compose (moserPower p) (moserPower_smooth p)).value x‖^(q:ℝ)) =
+      (fun x => (1+φ.value x^2)^((p*(q:ℝ)/2)/2)) := by
+    funext x
+    rw [show (φ.compose (moserPower p) (moserPower_smooth p)).value x = moserPower p (φ.value x) from rfl,
+      moserPower_rpow]
+    congr 1; ring
+  rw [he, g.integral_volumeMeasure (powerIntegrand_continuous φ _)]
+  change ((g.powerIntegral φ (p*(q:ℝ)/2))^((q:ℝ)⁻¹))^2 = _
+  rw [← Real.rpow_two, ← Real.rpow_mul (g.powerIntegral_nonneg φ _)]
+  congr 1; ring
+
+ 
+theorem mongeAmpere_moment_step (g : KaehlerMetric A) (hd : 2 ≤ d)
+    (D : ℝ) (hD : 0 ≤ D) : ∃ ν C : ℝ, 1 < ν ∧ 0 < C ∧
+    ∀ φ : SmoothRealFunction A, g.PositivePotential φ →
+      (∀ x, |1-(g.potentialDensity φ).value x| ≤ D) →
+      ∀ p : ℝ, 2 ≤ p → g.powerMoment φ (p*ν) ≤
+        (C*p^2)^(1/p) * g.powerMoment φ p := by
+  obtain ⟨q,hq,S,hS,hsob⟩ := g.sobolev hd
+  have hqR : (2:ℝ) < q := by exact_mod_cast hq
+  let ν : ℝ := (q:ℝ)/2
+  let K : ℝ := D/(1/2:ℝ)^d
+  let C : ℝ := S*(K+1)
+  have hK : 0 ≤ K := div_nonneg hD (by positivity)
+  have hC : 0 < C := mul_pos hS (by linarith)
+  refine ⟨ν,C,by dsimp [ν]; linarith,hC,?_⟩
+  intro φ hp hρ p hpow
+  let ψ := φ.compose (moserPower p) (moserPower_smooth p)
+  have hs := hsob ψ
+  have he := g.mongeAmpere_power_energy (by omega : 0 < d) D hD φ hp hρ p hpow
+  have heI : g.integral (fun x => (moserPower p (φ.value x))^2) = g.powerIntegral φ p := by
+    simp only [moserPower_sq, powerIntegral]
+  have hsI : g.integral (fun x => ψ.value x^2) = g.powerIntegral φ p := heI
+  rw [heI] at he
+  rw [hsI, g.moserPower_lpNorm_sq φ (lt_trans (by norm_num) hq) p] at hs
+  have hp0 : 0 < p := by linarith
+  have hc : S*((p^2/4)*K+1) ≤ C*p^2 := by
+    dsimp only [C]
+    have hp2 : 1 ≤ p^2 := by nlinarith
+    have hh : (p^2/4)*K+1 ≤ (K+1)*p^2 := by nlinarith [mul_nonneg hK (sq_nonneg p)]
+    nlinarith [mul_le_mul_of_nonneg_left hh hS.le]
+  have hh : (g.powerIntegral φ (p*ν))^(1/ν) ≤ C*p^2*g.powerIntegral φ p := by
+    have hνeq : 1/ν = 2/(q:ℝ) := by dsimp [ν]; ring
+    have hpe : p*ν = p*(q:ℝ)/2 := by dsimp [ν]; ring
+    rw [hνeq,hpe]
+    calc
+      _ ≤ S*(g.integral (g.energy ψ ψ).value + g.powerIntegral φ p) := hs
+      _ ≤ S*((p^2/4)*K*g.powerIntegral φ p + g.powerIntegral φ p) := mul_le_mul_of_nonneg_left (by dsimp only [ψ,K]; linarith only [he]) hS.le
+      _ = S*((p^2/4)*K+1)*g.powerIntegral φ p := by ring
+      _ ≤ _ := mul_le_mul_of_nonneg_right hc (g.powerIntegral_nonneg φ p)
+  have hν0 : 0 < ν := by dsimp [ν]; positivity
+  have hh' := Real.rpow_le_rpow (Real.rpow_nonneg (g.powerIntegral_nonneg φ (p*ν)) _) hh (le_of_lt (one_div_pos.mpr hp0))
+  rw [← Real.rpow_mul (g.powerIntegral_nonneg φ (p*ν)),
+    Real.mul_rpow (mul_nonneg hC.le (sq_nonneg p)) (g.powerIntegral_nonneg φ p)] at hh'
+  convert hh' using 1 <;> dsimp only [powerMoment]
+  congr 1
+  field_simp
+
+ 
+theorem mongeAmpere_moment_bounds (g : KaehlerMetric A) (hd : 2 ≤ d)
+    (D : ℝ) (hD : 0 ≤ D) : ∃ ν B : ℝ, 1 < ν ∧ 0 ≤ B ∧
+    ∀ φ : SmoothRealFunction A, g.PositivePotential φ →
+      (∀ x, |1-(g.potentialDensity φ).value x| ≤ D) →
+      ∀ n : ℕ, g.powerMoment φ (2*ν^n) ≤ B * g.powerMoment φ 2 := by
+  obtain ⟨ν,C,hν,hC,hstep⟩ := g.mongeAmpere_moment_step hd D hD
+  let a (n : ℕ) := (C*(2*ν^n)^2)^(1/(2*ν^n))
+  let u (n : ℕ) := ∏ j ∈ Finset.range n, a j
+  have hu : ∀ n, 0 ≤ u n := fun n => Finset.prod_nonneg (fun j _ => Real.rpow_nonneg (by positivity) _)
+  have hs (n : ℕ) : u (n+1) ≤ (C*(2*ν^n)^2)^(1/(2*ν^n))*u n := by simp only [u, Finset.prod_range_succ, a]; rw [mul_comm]
+  obtain ⟨B,hB,hbound⟩ := moser_sequence_bound ν C 2 hν hC u hu hs (by norm_num)
+  refine ⟨ν,B,hν,hB,fun φ hp hρ n => ?_⟩
+  have hi (n : ℕ) : g.powerMoment φ (2*ν^n) ≤ u n * g.powerMoment φ 2 := by
+    induction n with
+    | zero => simp [u]
+    | succ n ih =>
+      have hn : 2 ≤ 2*ν^n := by nlinarith [one_le_pow₀ (n:=n) hν.le]
+      have he : 2*ν^(n+1) = (2*ν^n)*ν := by rw [pow_succ]; ring
+      rw [he]
+      calc
+        _ ≤ a n * g.powerMoment φ (2*ν^n) := hstep φ hp hρ (2*ν^n) hn
+        _ ≤ a n * (u n * g.powerMoment φ 2) := mul_le_mul_of_nonneg_left ih (Real.rpow_nonneg (by positivity) _)
+        _ = _ := by dsimp only [u]; rw [Finset.prod_range_succ]; ring
+  apply (hi n).trans
+  apply mul_le_mul_of_nonneg_right _ (g.powerMoment_nonneg φ 2)
+  simpa only [u, Finset.range_zero, Finset.prod_empty, mul_one] using hbound n
+
+end Anticanonical.SourceSmooth.KaehlerMetric
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/MongeAmperePowerEnergy.lean b/lean/OAI/Geometry/PrescribedRicci/MongeAmperePowerEnergy.lean
new file mode 100644
index 0000000000000000000000000000000000000000..2165170cf962e5f0cdb358bb0ff3dca59dc0297d
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/MongeAmperePowerEnergy.lean
@@ -0,0 +1,155 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.GlobalSobolev
+import OAI.Geometry.PrescribedRicci.WeightedMongeAmpereEnergy
+
+namespace OAI
+
+/-! Monge Ampere Power Energy. -/
+
+section
+
+ 
+noncomputable section
+open Set Filter Topology MeasureTheory
+open scoped ContDiff Classical
+namespace Anticanonical.SourceSmooth
+
+private def powerBase (t : ℝ) : ℝ := 1+t^2
+private lemma powerBase_pos (t : ℝ) : 0 < powerBase t := by dsimp [powerBase]; positivity
+private lemma powerBase_smooth : ContDiff ℝ ∞ powerBase := by unfold powerBase; fun_prop
+private lemma powerBase_deriv (t : ℝ) : HasDerivAt powerBase (2*t) t := by
+  convert (hasDerivAt_const t (1:ℝ)).add ((hasDerivAt_id t).pow 2) using 1 <;> first | rfl | norm_num [powerBase]
+
+def moserTest (p t : ℝ) : ℝ := t * powerBase t ^ (p/2-1)
+def moserPower (p t : ℝ) : ℝ := powerBase t ^ (p/4)
+
+lemma moserTest_smooth (p : ℝ) : ContDiff ℝ ∞ (moserTest p) :=
+  contDiff_id.mul (powerBase_smooth.rpow_const_of_ne (fun t => ne_of_gt (powerBase_pos t)))
+lemma moserPower_smooth (p : ℝ) : ContDiff ℝ ∞ (moserPower p) :=
+  powerBase_smooth.rpow_const_of_ne (fun t => ne_of_gt (powerBase_pos t))
+
+lemma moserTest_deriv (p t : ℝ) : deriv (moserTest p) t =
+    powerBase t ^ (p/2-1) + t*((p/2-1)*powerBase t ^ (p/2-1-1)*(2*t)) := by
+  have h := ((hasDerivAt_id t).mul ((powerBase_deriv t).rpow_const (p:=p/2-1) (Or.inl (ne_of_gt (powerBase_pos t))))).deriv
+  change deriv (moserTest p) t = _ at h
+  rw [h]
+  dsimp only [id_eq]
+  ring
+
+lemma moserPower_deriv (p t : ℝ) : deriv (moserPower p) t =
+    (p/4)*powerBase t ^ (p/4-1)*(2*t) := by
+  have h := ((powerBase_deriv t).rpow_const (p:=p/4) (Or.inl (ne_of_gt (powerBase_pos t)))).deriv
+  change deriv (moserPower p) t = _ at h
+  rw [h]
+  ring
+
+lemma moserTest_deriv_lower {p : ℝ} (hp : 2 ≤ p) (t : ℝ) :
+    powerBase t ^ (p/2-1) ≤ deriv (moserTest p) t := by
+  rw [moserTest_deriv]
+  have ha : 0 ≤ p/2-1 := by linarith
+  have hb := Real.rpow_nonneg (powerBase_pos t).le (p/2-1-1)
+  nlinarith [mul_nonneg (mul_nonneg ha hb) (sq_nonneg t)]
+
+lemma moserTest_deriv_nonneg {p : ℝ} (hp : 2 ≤ p) (t : ℝ) :
+    0 ≤ deriv (moserTest p) t :=
+  (Real.rpow_nonneg (powerBase_pos t).le _).trans (moserTest_deriv_lower hp t)
+
+lemma moserPower_deriv_sq_le {p : ℝ} (hp : 2 ≤ p) (t : ℝ) :
+    (deriv (moserPower p) t)^2 ≤ (p^2/4) * deriv (moserTest p) t := by
+  have hb := powerBase_pos t
+  have he : (powerBase t ^ (p/4-1))^2 = powerBase t ^ (p/2-2) := by
+    rw [← Real.rpow_two, ← Real.rpow_mul hb.le]
+    congr 1; ring
+  have he' : powerBase t ^ (p/2-2) * powerBase t = powerBase t ^ (p/2-1) := by
+    conv_lhs => rhs; rw [← Real.rpow_one (powerBase t)]
+    rw [← Real.rpow_add hb]
+    congr 1; ring
+  calc
+    _ = (p^2/4) * (powerBase t ^ (p/2-2) * t^2) := by rw [moserPower_deriv, mul_pow, mul_pow, he]; ring
+    _ ≤ (p^2/4) * (powerBase t ^ (p/2-2) * powerBase t) := by
+      gcongr
+      dsimp [powerBase]; linarith
+    _ = (p^2/4) * powerBase t ^ (p/2-1) := by rw [he']
+    _ ≤ _ := mul_le_mul_of_nonneg_left (moserTest_deriv_lower hp t) (by positivity)
+
+lemma abs_moserTest_le (p t : ℝ) : |moserTest p t| ≤ (moserPower p t)^2 := by
+  have hb := powerBase_pos t
+  have hab : |t| ≤ powerBase t := by
+    dsimp [powerBase]
+    nlinarith [sq_nonneg (|t|-1), sq_abs t]
+  have he : powerBase t * powerBase t ^ (p/2-1) = (moserPower p t)^2 := by
+    unfold moserPower
+    rw [← Real.rpow_two, ← Real.rpow_mul hb.le]
+    conv_lhs => lhs; rw [← Real.rpow_one (powerBase t)]
+    rw [← Real.rpow_add hb]
+    congr 1; ring
+  calc
+    _ = |t| * powerBase t ^ (p/2-1) := by rw [moserTest, abs_mul, abs_of_nonneg (Real.rpow_nonneg hb.le _)]
+    _ ≤ powerBase t * powerBase t ^ (p/2-1) := mul_le_mul_of_nonneg_right hab (Real.rpow_nonneg hb.le _)
+    _ = _ := he
+
+lemma moserPower_sq (p t : ℝ) : (moserPower p t)^2 = (1+t^2)^(p/2) := by
+  unfold moserPower powerBase
+  rw [← Real.rpow_two, ← Real.rpow_mul (by positivity : (0:ℝ) ≤ 1+t^2)]
+  congr 1; ring
+
+lemma moserPower_rpow (p q t : ℝ) : ‖moserPower p t‖^q = (1+t^2)^(p*q/4) := by
+  unfold moserPower powerBase
+  rw [Real.norm_eq_abs, abs_of_pos (Real.rpow_pos_of_pos (by positivity) _),
+    ← Real.rpow_mul (by positivity : (0:ℝ) ≤ 1+t^2)]
+  congr 1; ring
+
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X}
+namespace KaehlerMetric
+
+theorem mongeAmpere_power_energy (g : KaehlerMetric A) (hd : 0 < d)
+    (D : ℝ) (_hD : 0 ≤ D) (φ : SmoothRealFunction A) (hp : g.PositivePotential φ)
+    (hρ : ∀ x, |1 - (g.potentialDensity φ).value x| ≤ D)
+    (p : ℝ) (hpow : 2 ≤ p) :
+    g.integral (g.energy (φ.compose (moserPower p) (moserPower_smooth p))
+      (φ.compose (moserPower p) (moserPower_smooth p))).value ≤
+        (p^2/4) * (D / (1/2:ℝ)^d) * g.integral (fun x => (moserPower p (φ.value x))^2) := by
+  let F := moserTest p
+  let U := moserPower p
+  have hF := moserTest_smooth p
+  have hU := moserPower_smooth p
+  have hma := g.mongeAmpere_weighted_energy φ hp hd F hF (moserTest_deriv_nonneg hpow)
+  have hFc : Continuous (fun x => F (φ.value x)) := hF.continuous.comp φ.continuous
+  have hUc : Continuous (fun x => (U (φ.value x))^2) := (hU.continuous.comp φ.continuous).pow 2
+  have hbound : g.integral (fun x => F (φ.value x)) -
+      g.integral (fun x => F (φ.value x) * (g.potentialDensity φ).value x) ≤
+        D * g.integral (fun x => (U (φ.value x))^2) := by
+    rw [← g.integral_sub hFc (hFc.fun_mul (g.potentialDensity φ).continuous), ← g.integral_const_mul]
+    apply g.integral_mono (hFc.sub (hFc.fun_mul (g.potentialDensity φ).continuous)) (continuous_const.fun_mul hUc)
+    intro x
+    calc
+      _ = F (φ.value x) * (1-(g.potentialDensity φ).value x) := by simp only [Pi.sub_apply]; ring
+      _ ≤ |F (φ.value x) * (1-(g.potentialDensity φ).value x)| := le_abs_self _
+      _ = |F (φ.value x)| * |1-(g.potentialDensity φ).value x| := abs_mul _ _
+      _ ≤ (U (φ.value x))^2 * D := mul_le_mul (abs_moserTest_le p _) (hρ x) (abs_nonneg _) (sq_nonneg _)
+      _ = _ := mul_comm _ _
+  have hE : g.integral (g.energy (φ.compose U hU) (φ.compose U hU)).value ≤
+      (p^2/4) * g.integral (fun x => deriv F (φ.value x) * (g.energy φ φ).value x) := by
+    rw [← g.integral_const_mul]
+    apply g.integral_mono (g.energy _ _).continuous
+      (continuous_const.fun_mul (((hF.continuous_deriv (by simp)).comp φ.continuous).fun_mul (g.energy φ φ).continuous))
+    intro x
+    rw [g.energy_comp_self]
+    have hh := mul_le_mul_of_nonneg_right (moserPower_deriv_sq_le hpow (φ.value x)) (g.energy_nonneg φ x)
+    simpa only [U, Function.comp_apply, mul_assoc] using hh
+  have hc : 0 < (1/2:ℝ)^d := by positivity
+  have hma' : g.integral (fun x => deriv F (φ.value x) * (g.energy φ φ).value x) ≤
+      (D / (1/2:ℝ)^d) * g.integral (fun x => (U (φ.value x))^2) := by
+    rw [div_mul_eq_mul_div]
+    apply (le_div_iff₀ hc).mpr
+    simpa only [mul_comm] using hma.trans hbound
+  exact hE.trans (by simpa only [mul_assoc] using mul_le_mul_of_nonneg_left hma' (show 0 ≤ p^2/4 by positivity))
+
+end KaehlerMetric
+end Anticanonical.SourceSmooth
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/MoserIteration.lean b/lean/OAI/Geometry/PrescribedRicci/MoserIteration.lean
new file mode 100644
index 0000000000000000000000000000000000000000..5b3243ef5e2184498d0ae675aad94cf2848c8964
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/MoserIteration.lean
@@ -0,0 +1,54 @@
+import Mathlib
+
+namespace OAI
+
+/-! Moser Iteration. -/
+
+noncomputable section
+open Filter Finset Topology
+namespace Anticanonical.SourceSmooth
+
+theorem moser_sequence_bound (ν C a : ℝ) (hν : 1 < ν) (hC : 0 < C)
+    (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n)
+    (hstep : ∀ n, u (n+1) ≤ (C*(a*ν^n)^2) ^ (1/(a*ν^n)) * u n)
+    (ha : 0 < a) : ∃ B : ℝ, 0 ≤ B ∧ ∀ n, u n ≤ B * u 0 := by
+  let b : ℕ → ℝ := fun n => (Real.log C + 2*Real.log a + 2*(n:ℝ)*Real.log ν) / (a*ν^n)
+  have hν0 : 0 < ν := lt_trans zero_lt_one hν
+  have hb (n : ℕ) : (C*(a*ν^n)^2) ^ (1/(a*ν^n)) = Real.exp (b n) := by
+    rw [Real.rpow_def_of_pos (mul_pos hC (sq_pos_of_pos (mul_pos ha (pow_pos hν0 _)) ))]
+    rw [Real.log_mul (ne_of_gt hC) (ne_of_gt (sq_pos_of_pos (mul_pos ha (pow_pos hν0 _)))),
+      Real.log_pow, Real.log_mul (ne_of_gt ha) (ne_of_gt (pow_pos hν0 _)), Real.log_pow]
+    congr 1; dsimp [b]; ring
+  have hs : Summable b := by
+    have hi : |ν⁻¹| < 1 := by rw [abs_of_pos (inv_pos.mpr hν0)]; exact inv_lt_one_of_one_lt₀ hν
+    have hg := summable_geometric_of_abs_lt_one hi
+    have hn : Summable (fun n : ℕ => (n:ℝ)*ν⁻¹^n) := by
+      simpa only [pow_one] using summable_pow_mul_geometric_of_norm_lt_one 1 (show ‖ν⁻¹‖ < 1 by simpa only [Real.norm_eq_abs] using hi)
+    apply (((hg.mul_left (Real.log C + 2*Real.log a)).add (hn.mul_left (2*Real.log ν))).div_const a).congr
+    intro n
+    dsimp only [b]
+    simp only [inv_pow, div_eq_mul_inv, mul_inv_rev]
+    ring
+  have hsabs := hs.abs
+  refine ⟨Real.exp (∑' n, |b n|), (Real.exp_pos _).le, ?_⟩
+  have hind (n : ℕ) : u n ≤ Real.exp (∑ k ∈ range n, b k) * u 0 := by
+    induction n with
+    | zero => simp
+    | succ n ih =>
+      calc
+        u (n+1) ≤ Real.exp (b n) * u n := by simpa only [hb] using hstep n
+        _ ≤ Real.exp (b n) * (Real.exp (∑ k ∈ range n, b k) * u 0) := mul_le_mul_of_nonneg_left ih (Real.exp_pos _).le
+        _ = _ := by rw [sum_range_succ, Real.exp_add]; ring
+  intro n
+  apply (hind n).trans
+  apply mul_le_mul_of_nonneg_right _ (hu 0)
+  apply Real.exp_le_exp.mpr
+  calc
+    ∑ k ∈ range n, b k ≤ ∑ k ∈ range n, |b k| := sum_le_sum (fun _ _ => le_abs_self _)
+    _ ≤ ∑' k, |b k| := Summable.sum_le_tsum (range n) (fun k _ => abs_nonneg (b k)) hsabs
+
+end Anticanonical.SourceSmooth
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/NormalizeVolumePath.lean b/lean/OAI/Geometry/PrescribedRicci/NormalizeVolumePath.lean
new file mode 100644
index 0000000000000000000000000000000000000000..770d2bb9ff2bbcfadf77843825f22432be7655e1
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/NormalizeVolumePath.lean
@@ -0,0 +1,69 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.SmoothPotentialDifference
+import OAI.Geometry.PrescribedPotential.VolumePath
+import OAI.Geometry.PrescribedRicci.KahlerIntegralPos
+import OAI.Geometry.PrescribedRicci.PathUniformSobolevBase
+
+namespace OAI
+
+/-! Normalize Volume Path. -/
+
+section
+
+ 
+noncomputable section
+open Set Filter Topology
+open scoped ContDiff Classical
+namespace Anticanonical.SourceSmooth
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  [ConnectedSpace X] {A : ComplexAtlas d X}
+namespace KaehlerMetric
+omit [T2Space X] [CompactSpace X] [ConnectedSpace X] in
+lemma positive_add_constant (g : KaehlerMetric A) (φ : SmoothRealFunction A)
+    (hp : g.PositivePotential φ) (c : ℝ) :
+    g.PositivePotential (φ.addFunction (SmoothRealFunction.constant c)) := by
+  intro i z hz
+  rw [SmoothRealFunction.hessian_addFunction _ _ i hz,SmoothRealFunction.hessian_constant,add_zero]
+  exact hp i z hz
+
+omit [T2Space X] [CompactSpace X] [ConnectedSpace X] in
+lemma logRatio_add_constant (g : KaehlerMetric A) (φ : SmoothRealFunction A)
+    (hp : g.PositivePotential φ) (c : ℝ) (x : X) :
+    (g.logRatio (g.deform (φ.addFunction (SmoothRealFunction.constant c))
+      (g.positive_add_constant φ hp c))).value x = (g.logRatio (g.deform φ hp)).value x := by
+  obtain ⟨i,hi⟩ := A.covers x
+  change g.logRatioValue _ x = g.logRatioValue _ x
+  rw [g.logRatioValue_local _ i hi,g.logRatioValue_local _ i hi]
+  simp only [volumeCoefficient,deform,
+    SmoothRealFunction.hessian_addFunction _ _ i ((A.chart i).mapsTo hi),
+    SmoothRealFunction.hessian_constant,add_zero]
+
+lemma normalizeVolumePath (g : KaehlerMetric A) (line : SemipositiveAnticanonicalMetric A)
+    {t : ℝ} (ht : t ∈ Icc (0:ℝ) 1) {φ : SmoothRealFunction A} {b : ℝ}
+    (hsol : SolvesVolumePath g line t φ b) :
+    ∃ z : g.NormalizedPathSolution line, z.time = t ∧ z.constant = b := by
+  obtain ⟨hp,he⟩ := hsol
+  let V := g.integral (fun _ => (1:ℝ))
+  obtain ⟨x₀⟩ := (inferInstance : Nonempty X)
+  have hV : 0 < V := g.integral_pos continuous_const (fun _ => zero_le_one) (x:=x₀) zero_lt_one
+  let c := -g.integral φ.value/V
+  let ψ := φ.addFunction (SmoothRealFunction.constant c)
+  have hm : g.integral ψ.value = 0 := by
+    change g.integral (fun x => φ.value x+c) = 0
+    rw [g.integral_add φ.continuous continuous_const]
+    have hc : g.integral (fun _ : X => c) = c*V := by
+      simpa only [mul_one] using g.integral_const_mul c (fun _ => (1:ℝ))
+    rw [hc]
+    dsimp [c]
+    rw [div_mul_cancel₀ _ hV.ne']
+    ring
+  refine ⟨⟨ψ,g.positive_add_constant φ hp c,hm,t,ht,b,?_⟩,rfl,rfl⟩
+  intro x
+  rw [g.logRatio_add_constant φ hp c x,he x]
+end KaehlerMetric
+end Anticanonical.SourceSmooth
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/PathLocalGain.lean b/lean/OAI/Geometry/PrescribedRicci/PathLocalGain.lean
new file mode 100644
index 0000000000000000000000000000000000000000..de68e806182f6b9b777682d7f98c9c220527a7f7
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/PathLocalGain.lean
@@ -0,0 +1,235 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.LocalDeterminantGain
+import OAI.Geometry.PrescribedRicci.MatrixCompactBounds
+import OAI.Geometry.PrescribedRicci.PathLocalizedDensity
+import OAI.Geometry.PrescribedRicci.PathLocalizedMetricBounds
+import OAI.Geometry.PrescribedRicci.PathUniformAtlas
+import OAI.Geometry.PrescribedRicci.UniformLocalGain
+
+namespace OAI
+
+/-! Path Local Gain. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff SchwartzMap Classical ComplexOrder MatrixOrder Matrix.Norms.Elementwise
+namespace SobolevChart
+lemma UniformSobolev.cutoff_integer {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+    [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] {Z : Type*}
+    (κ : 𝓢(E,ℂ)) (k : ℕ) {f : Z → 𝓢(E,ℂ)} (hf : UniformSobolev (k:ℝ) f) :
+    UniformSobolev (k:ℝ) (fun z => SchwartzMap.smulLeftCLM ℂ κ (f z)) :=
+  hf.of_coreBound (coreBound_product_integer κ k)
+lemma UniformSobolev.cutoff_zero {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+    [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] {Z : Type*}
+    (κ : 𝓢(E,ℂ)) {f : Z → 𝓢(E,ℂ)} (hf : UniformSobolev 0 f) :
+    UniformSobolev 0 (fun z => SchwartzMap.smulLeftCLM ℂ κ (f z)) := by
+  exact hf.of_coreBound (by simpa only [Nat.cast_zero] using coreBound_product_integer κ 0)
+lemma UniformSobolev.cutoff_two {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+    [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] {Z : Type*}
+    (κ : 𝓢(E,ℂ)) {f : Z → 𝓢(E,ℂ)} (hf : UniformSobolev 2 f) :
+    UniformSobolev 2 (fun z => SchwartzMap.smulLeftCLM ℂ κ (f z)) := by
+  exact hf.of_coreBound (by simpa only [Nat.cast_ofNat] using coreBound_product_integer κ 2)
+end SobolevChart
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart MetricLocalization FrozenPoisson
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  [ConnectedSpace X] {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+  {g : KaehlerMetric A}
+namespace GluingData
+variable (D : GluingData g ι)
+
+lemma cutoff_nested_one {U : Set (EC d)} (κ σ τ : ChartCutoff U)
+    (hσ : ∀ y ∈ tsupport (κ : EC d → ℂ), (σ : EC d → ℂ) =ᶠ[𝓝 y] fun _ => 1)
+    (hτ : ∀ y ∈ tsupport (σ : EC d → ℂ), (τ : EC d → ℂ) =ᶠ[𝓝 y] fun _ => 1) :
+    ∀ y ∈ tsupport (κ : EC d → ℂ), (τ : EC d → ℂ) =ᶠ[𝓝 y] fun _ => 1 := by
+  intro y hy
+  apply hτ
+  apply subset_tsupport
+  change σ y ≠ 0
+  rw [(hσ y hy).eq_of_nhds]
+  exact one_ne_zero
+
+omit [ConnectedSpace X] in
+lemma path_cutoff_identity (q : Fin A.count) (κ τ : ChartCutoff (A.euclideanChart q).target)
+    (hτ : ∀ y ∈ tsupport (κ : EC d → ℂ), (τ : EC d → ℂ) =ᶠ[𝓝 y] fun _ => 1)
+    (φ : SmoothRealFunction A) :
+    localize A q (cutoffGlobal q κ) (cutoffGlobal_support q κ) (Smooth.ofReal φ) =
+      SchwartzMap.smulLeftCLM ℂ κ.val
+        (localize A q (cutoffGlobal q τ) (cutoffGlobal_support q τ) (Smooth.ofReal φ)) := by
+  ext y
+  rw [localize_cutoff_product,ChartCutoff.product_apply,
+    SchwartzMap.smulLeftCLM_apply_apply κ.val.hasTemperateGrowth,smul_eq_mul]
+  by_cases hy : y ∈ tsupport (κ : EC d → ℂ)
+  · rw [(localize_cutoff_germ q τ (Smooth.ofReal φ) (κ.support_sub hy) (hτ y hy)).eq_of_nhds]
+  · simp only [image_eq_zero_of_notMem_tsupport hy,zero_mul]
+
+omit [ConnectedSpace X] in
+lemma path_cutoff_bound (line : SemipositiveAnticanonicalMetric A) (q : Fin A.count)
+    (τ : ChartCutoff (A.euclideanChart q).target) (k : ℕ)
+    (hprev : ∃ C : ℝ, 0 ≤ C ∧ ∀ z : g.NormalizedPathSolution line,
+      ‖D.localizers.embed (k:ℝ) (Smooth.ofReal z.potential)‖ ≤ C) :
+    UniformSobolev (k:ℝ) (fun z : g.NormalizedPathSolution line =>
+      localize A q (cutoffGlobal q τ) (cutoffGlobal_support q τ) (Smooth.ofReal z.potential)) := by
+  obtain ⟨C,hC,hb⟩ := hprev
+  obtain ⟨B,hB,hlocal⟩ := D.localization_integer_bound q
+    (cutoffGlobal q τ) (cutoffGlobal_support q τ) k
+  exact ⟨B*C,mul_nonneg hB hC,fun z => (hlocal (Smooth.ofReal z.potential)).trans
+    (mul_le_mul_of_nonneg_left (hb z) hB)⟩
+
+include D in
+lemma path_local_word_differential (line : SemipositiveAnticanonicalMetric A) (hd : 2 ≤ d)
+    (p : UniformPathPatch g line)
+    (κ σ τ : ChartCutoff (A.euclideanChart p.index).target)
+    (hκp : tsupport (κ : EC d → ℂ) ⊆ p.domain)
+    (hκr : ∀ y, (κ y).im = 0) (hτr : ∀ y, (τ y).im = 0)
+    (hσ : ∀ y ∈ tsupport (κ : EC d → ℂ), (σ : EC d → ℂ) =ᶠ[𝓝 y] fun _ => 1)
+    (hτ : ∀ y ∈ tsupport (σ : EC d → ℂ), (τ : EC d → ℂ) =ᶠ[𝓝 y] fun _ => 1)
+    (ws : List (BasisIndex d)) (hw0 : ws ≠ [])
+    (hF : UniformSobolev ((ws.length:ℝ)+1) (fun z : g.NormalizedPathSolution line =>
+      localize A p.index (cutoffGlobal p.index τ) (cutoffGlobal_support p.index τ) (Smooth.ofReal z.potential))) :
+    UniformSobolev 0 (fun z : g.NormalizedPathSolution line =>
+      schwartzDifferential (extendedCoefficient
+        ((g.deform z.potential z.positive).matrix p.index (coordinateEquiv d p.center))
+        (coefficientBCF (p.coefficient z)))
+        (SchwartzMap.smulLeftCLM ℂ κ.val (schwartzWord (ws.map (stdOrthonormalBasis ℝ (EC d)))
+          (localize A p.index (cutoffGlobal p.index τ) (cutoffGlobal_support p.index τ) (Smooth.ofReal z.potential))))) := by
+  let : Nonempty (Fin d) := ⟨⟨0,by omega⟩⟩
+  let : Nonempty (BasisIndex d) := ⟨⟨0,Module.finrank_pos⟩⟩
+  let f (z : g.NormalizedPathSolution line) :=
+    localize A p.index (cutoffGlobal p.index τ) (cutoffGlobal_support p.index τ) (Smooth.ofReal z.potential)
+  let F (z : g.NormalizedPathSolution line) := SchwartzMap.smulLeftCLM ℂ κ.val
+    (schwartzWord (ws.map (stdOrthonormalBasis ℝ (EC d))) (f z))
+  have hτκ := cutoff_nested_one κ σ τ hσ hτ
+  let H (z : g.NormalizedPathSolution line) := cutMetric (g.deform z.potential z.positive) p.index σ
+  let G (_ : g.NormalizedPathSolution line) := cutMetric g p.index σ
+  let c (z : g.NormalizedPathSolution line) :=
+    extendedCoefficient ((g.deform z.potential z.positive).matrix p.index (coordinateEquiv d p.center))
+      (coefficientBCF (p.coefficient z))
+  have hc (z : g.NormalizedPathSolution line) (i j) : (c z i j : EC d → ℂ).HasTemperateGrowth :=
+    extendedCoefficient_temperate _ _ (fun i j => ((p.coefficient z) i j).hasTemperateGrowth) i j
+  have heH (z : g.NormalizedPathSolution line) (y : EC d) (hy : y ∈ tsupport (κ : EC d → ℂ)) :
+      (show Matrix (Fin d) (Fin d) ℂ from fun i j => H z i j y) =
+      (g.deform z.potential z.positive).matrix p.index (coordinateEquiv d y) := by
+    ext i j
+    change σ y*_ = _
+    rw [(hσ y hy).eq_of_nhds,one_mul]
+  let K := coordinateEquiv d '' tsupport (κ : EC d → ℂ)
+  have hKt : K ⊆ (A.chart p.index).target := by
+    rintro _ ⟨y,hy,rfl⟩
+    simpa using κ.support_sub hy
+  obtain ⟨B,hB,hb⟩ := g.volumePath_metric_bounds_on_compact line hd p.index
+    (κ.compact.image (coordinateEquiv d).continuous) hKt
+  obtain ⟨J,hJ,hj⟩ := MongeAmpere.compact_matrix_bound
+    (show Continuous (fun M : Matrix (Fin d) (Fin d) ℂ => ‖M.det‖) from continuous_id.matrix_det.norm) B
+  change UniformSobolev 0 (fun z => schwartzDifferential (c z) (F z))
+  apply uniform_determinant_word_differential (stdOrthonormalBasis ℝ (EC d)) ws hw0
+    κ.val f (cutPathDensity g line p.index σ) G H c hc
+  · intro z y hy i j
+    rw [heH z y hy]
+    exact p.equals z y (hκp hy) i j
+  · intro z y
+    rw [SchwartzMap.smulLeftCLM_apply_apply κ.val.hasTemperateGrowth]
+    change (κ y*schwartzWord _ (f z) y).im = 0
+    rw [Complex.mul_im,hκr,schwartzWord_real _ _ (localize_cutoff_real p.index τ hτr z.potential)]
+    ring
+  · intro z y hy i j
+    exact cutMetric_deform_germ g p.index σ τ z.potential z.positive
+      (κ.support_sub hy) (hσ y hy) (hτκ y hy) i j
+  · intro z y hy
+    exact cutPathDensity_germ line p.index σ z (κ.support_sub hy) (hσ y hy)
+  · intro z y hy
+    rw [heH z y hy]
+    exact isUnit_iff_ne_zero.mpr ((g.deform z.potential z.positive).positive p.index _
+      (by simpa using κ.support_sub hy)).det_pos.ne'
+  · exact hB.le
+  · exact hJ.le
+  · intro z y hy i j
+    rw [heH z y hy]
+    exact (norm_entry_le_entrywise_sup_norm _).trans
+      (hb z.potential z.positive z.mean_zero z.time z.constant z.time_mem z.equation _
+        (mem_image_of_mem _ hy)).2
+  · intro z y hy
+    rw [heH z y hy,← Ring.inverse_eq_inv,← Matrix.det_nonsing_inv]
+    exact hj _ (hb z.potential z.positive z.mean_zero z.time z.constant z.time_mem z.equation _
+      (mem_image_of_mem _ hy)).2
+  · exact hF
+  · exact cutPathDensity_uniform D line p.index σ ws.length
+  · intro i j
+    exact UniformSobolev.const _ _
+  · intro z i j
+    exact σ.product_compact _
+  · exact cutMetric_uniform_initialJet line hd p.index σ
+  · intro i j
+    have hn : 1 ≤ ws.length := List.length_pos_iff.mpr hw0
+    have hncast : ((ws.length-1:ℕ):ℝ) = (ws.length:ℝ)-1 := by rw [Nat.cast_sub hn]; norm_num
+    have hlow : UniformSobolev (((ws.length-1:ℕ):ℝ)+2) f := hF.mono (by rw [hncast]; linarith)
+    simpa only [hncast] using cutMetric_uniform_sobolev line p.index σ τ hτ (ws.length-1) hlow i j
+
+include D in
+lemma path_local_word_H2 (line : SemipositiveAnticanonicalMetric A) (hd : 2 ≤ d)
+    (p : UniformPathPatch g line)
+    (κ σ τ : ChartCutoff (A.euclideanChart p.index).target)
+    (hκp : tsupport (κ : EC d → ℂ) ⊆ p.domain)
+    (hκr : ∀ y, (κ y).im = 0) (hτr : ∀ y, (τ y).im = 0)
+    (hσ : ∀ y ∈ tsupport (κ : EC d → ℂ), (σ : EC d → ℂ) =ᶠ[𝓝 y] fun _ => 1)
+    (hτ : ∀ y ∈ tsupport (σ : EC d → ℂ), (τ : EC d → ℂ) =ᶠ[𝓝 y] fun _ => 1)
+    (ws : List (BasisIndex d))
+    (hF : UniformSobolev ((ws.length:ℝ)+1) (fun z : g.NormalizedPathSolution line =>
+      localize A p.index (cutoffGlobal p.index τ) (cutoffGlobal_support p.index τ) (Smooth.ofReal z.potential)))
+    (hF2 : UniformSobolev 2 (fun z : g.NormalizedPathSolution line =>
+      localize A p.index (cutoffGlobal p.index τ) (cutoffGlobal_support p.index τ) (Smooth.ofReal z.potential))) :
+    UniformSobolev 2 (fun z : g.NormalizedPathSolution line =>
+      SchwartzMap.smulLeftCLM ℂ κ.val (schwartzWord (ws.map (stdOrthonormalBasis ℝ (EC d)))
+        (localize A p.index (cutoffGlobal p.index τ) (cutoffGlobal_support p.index τ) (Smooth.ofReal z.potential)))) := by
+  have hword := hF.word (t:=0) (ws.map (stdOrthonormalBasis ℝ (EC d)))
+    (by simp only [List.length_map]; linarith)
+  have hfword := hword.cutoff_zero κ.val
+  by_cases hw0 : ws = []
+  · subst ws
+    simpa only [List.map_nil,schwartzWord,ContinuousLinearMap.id_apply] using
+      hF2.cutoff_two κ.val
+  have hD := D.path_local_word_differential line hd p κ σ τ hκp hκr hτr hσ hτ ws hw0 hF
+  exact uniform_schwartz_H2_of_differential (Z:=g.NormalizedPathSolution line) (n:=d) p.bound_pos.le
+    (fun z => (g.deform z.potential z.positive).matrix p.index (coordinateEquiv d p.center))
+    (fun z => (g.deform z.potential z.positive).positive p.index _ (by simpa using p.center_target))
+    p.matrix_bound (fun z => coefficientBCF (p.coefficient z))
+    (fun z i j => (p.coefficient z i j).hasTemperateGrowth) p.small _ hfword hD
+
+lemma path_local_gain (line : SemipositiveAnticanonicalMetric A) (hd : 2 ≤ d)
+    (p : UniformPathPatch g line)
+    (κ σ τ : ChartCutoff (A.euclideanChart p.index).target)
+    (hκp : tsupport (κ : EC d → ℂ) ⊆ p.domain)
+    (hκr : ∀ y, (κ y).im = 0) (hτr : ∀ y, (τ y).im = 0)
+    (hσ : ∀ y ∈ tsupport (κ : EC d → ℂ), (σ : EC d → ℂ) =ᶠ[𝓝 y] fun _ => 1)
+    (hτ : ∀ y ∈ tsupport (σ : EC d → ℂ), (τ : EC d → ℂ) =ᶠ[𝓝 y] fun _ => 1)
+    (k : ℕ) (hk : 1 ≤ k)
+    (hprev : ∃ C : ℝ, 0 ≤ C ∧ ∀ z : g.NormalizedPathSolution line,
+      ‖D.localizers.embed ((k:ℝ)+1) (Smooth.ofReal z.potential)‖ ≤ C) :
+    UniformSobolev ((k:ℝ)+2) (fun z : g.NormalizedPathSolution line =>
+      localize A p.index (cutoffGlobal p.index κ) (cutoffGlobal_support p.index κ)
+        (Smooth.ofReal z.potential)) := by
+  let f (z : g.NormalizedPathSolution line) :=
+    localize A p.index (cutoffGlobal p.index τ) (cutoffGlobal_support p.index τ) (Smooth.ofReal z.potential)
+  have hF : UniformSobolev ((k:ℝ)+1) f := by
+    have he : (k:ℝ)+1 = ((k+1:ℕ):ℝ) := by simp
+    rw [he] at hprev ⊢
+    exact D.path_cutoff_bound line p.index τ (k+1) hprev
+  have heF (z : g.NormalizedPathSolution line) := path_cutoff_identity p.index κ τ
+    (cutoff_nested_one κ σ τ hσ hτ) z.potential
+  simp_rw [heF]
+  apply uniform_cutoff_word_gain κ.val k hF
+  intro ws hw
+  have hlen : (ws.length:ℝ) ≤ k := by exact_mod_cast hw
+  exact D.path_local_word_H2 line hd p κ σ τ hκp hκr hτr hσ hτ ws
+    (hF.mono (by linarith)) (hF.mono (by exact_mod_cast (show 2 ≤ k+1 by omega)))
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/PathLocalizedDensity.lean b/lean/OAI/Geometry/PrescribedRicci/PathLocalizedDensity.lean
new file mode 100644
index 0000000000000000000000000000000000000000..4a196752a6fbfcbfb3fe1bd25dc5af8862da6b2a
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/PathLocalizedDensity.lean
@@ -0,0 +1,102 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.PositivePotentialLinearization
+import OAI.Geometry.PrescribedRicci.LocalizedPathData
+import OAI.Geometry.PrescribedRicci.PositivePathOpenness
+
+namespace OAI
+
+/-! Path Localized Density. -/
+
+section
+
+ 
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff SchwartzMap Classical ComplexOrder MatrixOrder Matrix.Norms.Elementwise
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  [ConnectedSpace X] {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+  {g : KaehlerMetric A}
+
+omit [T2Space X] [CompactSpace X] [ConnectedSpace X] in
+lemma metric_det_euclidean_smooth (g : KaehlerMetric A) (q : Fin A.count) :
+    ContDiffOn ℝ ∞ (fun y : EC d => (g.matrix q (coordinateEquiv d y)).det)
+      (A.euclideanChart q).target := by
+  simp only [Matrix.det_apply']
+  apply ContDiffOn.sum
+  intro π _
+  apply contDiffOn_const.mul
+  apply contDiffOn_prod
+  intro i _
+  exact contDiffOn_pi.mp (contDiffOn_pi.mp (GluingData.metric_euclidean_smooth g q) (π i)) i
+
+def cutPathDensity (g : KaehlerMetric A) (line : SemipositiveAnticanonicalMetric A)
+    (q : Fin A.count) (σ : ChartCutoff (A.euclideanChart q).target)
+    (z : g.NormalizedPathSolution line) : 𝓢(EC d,ℂ) :=
+  localize A q
+    (cutoffGlobal q (σ.mulOn (A.euclideanChart q).open_target (metric_det_euclidean_smooth g q)))
+    (cutoffGlobal_support q _)
+    ((RealSmooth.ofReal (prescribedForcing g line)).expAffine z.time z.constant).val
+
+omit [ConnectedSpace X] in
+lemma cutPathDensity_apply (line : SemipositiveAnticanonicalMetric A)
+    (q : Fin A.count) (σ : ChartCutoff (A.euclideanChart q).target)
+    (z : g.NormalizedPathSolution line) (y : EC d) :
+    cutPathDensity g line q σ z y = σ y*(g.matrix q (coordinateEquiv d y)).det*
+      (Real.exp (z.time*(prescribedForcing g line).value ((A.euclideanChart q).symm y)+z.constant):ℂ) := by
+  rw [cutPathDensity,localize_cutoff_product]
+  rfl
+
+lemma cutPathDensity_uniform (D : GluingData g ι) (line : SemipositiveAnticanonicalMetric A)
+    (q : Fin A.count) (σ : ChartCutoff (A.euclideanChart q).target) (k : ℕ) :
+    UniformSobolev (k:ℝ) (cutPathDensity g line q σ) :=
+  D.uniform_localize k _ (D.volumePath_forcing_uniform line k) q _ _
+
+omit [ConnectedSpace X] in
+lemma path_determinant_equation (line : SemipositiveAnticanonicalMetric A)
+    (z : g.NormalizedPathSolution line) (q : Fin A.count) {y : EC d}
+    (hy : y ∈ (A.euclideanChart q).target) :
+    ((g.deform z.potential z.positive).matrix q (coordinateEquiv d y)).det =
+      (g.matrix q (coordinateEquiv d y)).det *
+        (Real.exp (z.time*(prescribedForcing g line).value ((A.euclideanChart q).symm y)+z.constant):ℂ) := by
+  let x := (A.euclideanChart q).symm y
+  have hyt : coordinateEquiv d y ∈ (A.chart q).target := by simpa using hy
+  have hx : x ∈ (A.chart q).source := by simpa [x] using (A.euclideanChart q).symm.mapsTo hy
+  have hxy : A.chart q x = coordinateEquiv d y :=
+    (A.chart q).right_inv hyt
+  have hp := g.potentialDensity_pos z.potential z.positive x
+  have heq := z.equation x
+  rw [g.logRatio_eq_log_density z.potential z.positive x] at heq
+  have he : (g.potentialDensity z.potential).value x =
+      Real.exp (z.time*(prescribedForcing g line).value x+z.constant) := by
+    rw [← heq,Real.exp_log hp]
+  have hd := g.potentialDensity_complex z.potential q hx
+  rw [hxy,he] at hd
+  change (Real.exp (_):ℂ) = _/_ at hd
+  have hn : (g.matrix q (coordinateEquiv d y)).det ≠ 0 := (g.positive q _ hyt).det_pos.ne'
+  rw [eq_div_iff hn] at hd
+  exact hd.symm.trans (mul_comm _ _)
+
+omit [ConnectedSpace X] in
+lemma cutPathDensity_germ (line : SemipositiveAnticanonicalMetric A)
+    (q : Fin A.count) (σ : ChartCutoff (A.euclideanChart q).target)
+    (z : g.NormalizedPathSolution line) {y : EC d} (hy : y ∈ (A.euclideanChart q).target)
+    (hσ : (σ : EC d → ℂ) =ᶠ[𝓝 y] fun _ => 1) :
+    (fun x => (show Matrix (Fin d) (Fin d) ℂ from fun i j =>
+      cutMetric (g.deform z.potential z.positive) q σ i j x).det) =ᶠ[𝓝 y]
+      cutPathDensity g line q σ z := by
+  filter_upwards [hσ,(A.euclideanChart q).open_target.mem_nhds hy] with x hx hxt
+  have hH : (show Matrix (Fin d) (Fin d) ℂ from fun i j =>
+      cutMetric (g.deform z.potential z.positive) q σ i j x) =
+      (g.deform z.potential z.positive).matrix q (coordinateEquiv d x) := by
+    ext i j
+    rw [cutMetric_apply,hx,one_mul]
+  rw [hH,cutPathDensity_apply,hx,one_mul]
+  exact path_determinant_equation line z q hxt
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/PathLocalizedMetricBounds.lean b/lean/OAI/Geometry/PrescribedRicci/PathLocalizedMetricBounds.lean
new file mode 100644
index 0000000000000000000000000000000000000000..ced5d0a89708a19101cf151abf745870f7163138
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/PathLocalizedMetricBounds.lean
@@ -0,0 +1,108 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CutoffHigher
+import OAI.Geometry.PrescribedRicci.CalabiMetricC1
+import OAI.Geometry.PrescribedRicci.CompactPathBounds
+import OAI.Geometry.PrescribedRicci.CutoffJetBound
+import OAI.Geometry.PrescribedRicci.LocalizedPathData
+
+namespace OAI
+
+/-! Path Localized Metric Bounds. -/
+
+section
+
+ 
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff SchwartzMap Classical ComplexOrder MatrixOrder Matrix.Norms.Elementwise ENNReal
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart TameInterpolation MetricLocalization
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  [ConnectedSpace X] {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+  {g : KaehlerMetric A}
+
+omit [ConnectedSpace X] in
+lemma cutMetric_uniform_sobolev (line : SemipositiveAnticanonicalMetric A) (q : Fin A.count)
+    (σ τ : ChartCutoff (A.euclideanChart q).target)
+    (hτ : ∀ y ∈ tsupport (σ : EC d → ℂ), (τ : EC d → ℂ) =ᶠ[𝓝 y] fun _ => 1)
+    (k : ℕ) (hF : UniformSobolev ((k:ℝ)+2) (fun z : g.NormalizedPathSolution line =>
+      localize A q (cutoffGlobal q τ) (cutoffGlobal_support q τ) (Smooth.ofReal z.potential))) (i j : Fin d) :
+    UniformSobolev (k:ℝ) (fun z : g.NormalizedPathSolution line =>
+      cutMetric (g.deform z.potential z.positive) q σ i j) := by
+  simp_rw [cutMetric_deform g q σ τ hτ]
+  exact (UniformSobolev.const _ _).add
+    ((hF.of_coreBound (hessianEntrySchwartz_bound (k:ℝ) i j)).of_coreBound
+      (coreBound_product_integer σ.val k))
+
+lemma cutMetric_uniform_C1 (line : SemipositiveAnticanonicalMetric A) (hd : 2 ≤ d)
+    (q : Fin A.count) (σ : ChartCutoff (A.euclideanChart q).target) :
+    ∃ B : ℝ, 0 ≤ B ∧ ∀ z : g.NormalizedPathSolution line, ∀ i j : Fin d,
+      (∀ y, ‖cutMetric (g.deform z.potential z.positive) q σ i j y‖ ≤ B) ∧
+      (∀ a y, ‖fderiv ℝ (cutMetric (g.deform z.potential z.positive) q σ i j) y
+        (stdOrthonormalBasis ℝ (EC d) a)‖ ≤ B) := by
+  let : Nonempty (Fin d) := ⟨⟨0,by omega⟩⟩
+  let : Nonempty (BasisIndex d) := ⟨⟨0,Module.finrank_pos⟩⟩
+  let K := coordinateEquiv d '' tsupport (σ : EC d → ℂ)
+  have hK : IsCompact K := σ.compact.image (coordinateEquiv d).continuous
+  have hKt : K ⊆ (A.chart q).target := by
+    rintro _ ⟨y,hy,rfl⟩
+    simpa only [ComplexAtlas.euclideanChart_target, mem_preimage] using σ.support_sub hy
+  obtain ⟨M,hM,hm⟩ := g.volumePath_metric_bounds_on_compact line hd q hK hKt
+  obtain ⟨P,hP,hp⟩ := g.volumePath_metric_C1_on_compact line hd q hK hKt
+  let Q := P*‖(coordinateEquiv d).toContinuousLinearMap‖
+  have hQ : 0 ≤ Q := mul_nonneg hP.le (norm_nonneg _)
+  let f (z : g.NormalizedPathSolution line) (i j : Fin d) (y : EC d) :=
+    (g.deform z.potential z.positive).matrix q (coordinateEquiv d y) i j
+  have hs (z : g.NormalizedPathSolution line) (i j : Fin d) :
+      ContDiffOn ℝ ∞ (f z i j) (A.euclideanChart q).target :=
+    contDiffOn_pi.mp (contDiffOn_pi.mp (GluingData.metric_euclidean_smooth _ q) i) j
+  have h0 (i j : Fin d) (z : g.NormalizedPathSolution line) (y : EC d)
+      (hy : y ∈ tsupport (σ : EC d → ℂ)) : ‖f z i j y‖ ≤ M :=
+    (norm_entry_le_entrywise_sup_norm ((g.deform z.potential z.positive).matrix q (coordinateEquiv d y))).trans
+      (hm z.potential z.positive z.mean_zero z.time z.constant z.time_mem z.equation _
+        (mem_image_of_mem _ hy)).1
+  have h1 (i j : Fin d) (z : g.NormalizedPathSolution line) (y : EC d)
+      (hy : y ∈ tsupport (σ : EC d → ℂ)) (a : BasisIndex d) :
+      ‖fderiv ℝ (f z i j) y (stdOrthonormalBasis ℝ (EC d) a)‖ ≤ Q := by
+    let H := fun x => (g.deform z.potential z.positive).matrix q x i j
+    have hdy : DifferentiableAt ℝ H (coordinateEquiv d y) :=
+      ((contDiffOn_pi.mp (contDiffOn_pi.mp ((g.deform z.potential z.positive).smooth q) i) j).contDiffAt
+        ((A.chart q).open_target.mem_nhds (hKt (mem_image_of_mem _ hy)))).differentiableAt (by simp)
+    have he : fderiv ℝ (f z i j) y = (fderiv ℝ H (coordinateEquiv d y)).comp
+        (coordinateEquiv d).toContinuousLinearMap :=
+      (hdy.hasFDerivAt.comp y (coordinateEquiv d).hasFDerivAt).fderiv
+    rw [he,ContinuousLinearMap.comp_apply]
+    apply (ContinuousLinearMap.le_opNorm _ _).trans
+    apply (mul_le_mul (hp z.potential z.positive z.mean_zero z.time z.constant z.time_mem z.equation _
+      (mem_image_of_mem _ hy) i j)
+      ((coordinateEquiv d).toContinuousLinearMap.le_opNorm _)
+      (norm_nonneg _) hP.le).trans_eq
+    simp only [OrthonormalBasis.norm_eq_one,mul_one]
+    rfl
+  have hb (i j : Fin d) := σ.uniform_C1 (A.euclideanChart q).open_target
+    (stdOrthonormalBasis ℝ (EC d)) (fun z => f z i j) (fun z => hs z i j)
+    hM.le hQ (h0 i j) (h1 i j)
+  choose B hB hb using hb
+  refine ⟨∑ i, ∑ j, B i j, Finset.sum_nonneg fun i _ => Finset.sum_nonneg fun j _ => hB i j,?_⟩
+  intro z i j
+  have hle : B i j ≤ ∑ i, ∑ j, B i j :=
+    (Finset.single_le_sum (fun j _ => hB i j) (Finset.mem_univ j)).trans
+      (Finset.single_le_sum (fun i _ => Finset.sum_nonneg fun j _ => hB i j) (Finset.mem_univ i))
+  exact ⟨fun y => ((hb i j z).1 y).trans hle,fun a y => ((hb i j z).2 a y).trans hle⟩
+
+lemma cutMetric_uniform_initialJet [Nonempty (Fin d)] [Nonempty (BasisIndex d)]
+    (line : SemipositiveAnticanonicalMetric A) (hd : 2 ≤ d)
+    (q : Fin A.count) (σ : ChartCutoff (A.euclideanChart q).target) (π : Equiv.Perm (Fin d)) :
+    ∃ B : ℝ, 0 ≤ B ∧ ∀ z : g.NormalizedPathSolution line,
+      familyJetNorm (stdOrthonormalBasis ℝ (EC d)) (realComponents
+        (initialJet (stdOrthonormalBasis ℝ (EC d))
+          (fun i y => cutMetric (g.deform z.potential z.positive) q σ (π i) i y))) 0 ∞ ≤ B := by
+  obtain ⟨B,hB,hb⟩ := cutMetric_uniform_C1 (g:=g) line hd q σ
+  exact ⟨B,hB,fun z => initialJet_zero_bound _ _ hB (fun i => (hb z (π i) i).1)
+    (fun i => (hb z (π i) i).2)⟩
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/PathUniformAtlas.lean b/lean/OAI/Geometry/PrescribedRicci/PathUniformAtlas.lean
new file mode 100644
index 0000000000000000000000000000000000000000..88f25c6eabf7b5582b0d5598383deee9569b45a9
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/PathUniformAtlas.lean
@@ -0,0 +1,146 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.BoundedPoisson
+import OAI.Geometry.PrescribedRicci.PathUniformSobolevBase
+import OAI.Geometry.PrescribedRicci.UniformFrozenEstimate
+import OAI.Geometry.PrescribedRicci.UniformPathPatches
+
+namespace OAI
+
+/-! Path Uniform Atlas. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Manifold IsManifold Matrix
+open scoped ContDiff SchwartzMap BoundedContinuousFunction ComplexOrder MatrixOrder Matrix.Norms.Elementwise Classical
+namespace Anticanonical.SourceSmooth
+open EllipticKernel SobolevChart FrozenPoisson MetricLocalization GlobalElliptic
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  [ConnectedSpace X] {A : ComplexAtlas d X}
+
+structure UniformPathPatch (g : KaehlerMetric A) (line : SemipositiveAnticanonicalMetric A) where
+  index : Fin A.count
+  center : EC d
+  domain : Set (EC d)
+  isOpen_domain : IsOpen domain
+  center_mem : center ∈ domain
+  domain_sub : domain ⊆ (A.euclideanChart index).target
+  bound : ℝ
+  bound_pos : 0 < bound
+  matrix_bound : ∀ z : g.NormalizedPathSolution line,
+    ‖(g.deform z.potential z.positive).matrix index (coordinateEquiv d center)‖ ≤ bound
+  coefficient : g.NormalizedPathSolution line → SmoothCoefficients (BasisIndex d) (EC d)
+  small : ∀ z, perturbationBound (stdOrthonormalBasis ℝ (EC d)) (coefficientBCF (coefficient z)) *
+    uniformEllipticBound d bound ≤ 1/2
+  equals : ∀ z, ∀ y ∈ domain, ∀ k l,
+    extendedCoefficient ((g.deform z.potential z.positive).matrix index (coordinateEquiv d center))
+      (coefficientBCF (coefficient z)) k l y =
+      (traceBilin ((g.deform z.potential z.positive).matrix index (coordinateEquiv d y))
+        (rankTwo (stdOrthonormalBasis ℝ (EC d) k) (stdOrthonormalBasis ℝ (EC d) l)) : ℂ)
+
+namespace UniformPathPatch
+variable {g : KaehlerMetric A} {line : SemipositiveAnticanonicalMetric A}
+def source (p : UniformPathPatch g line) : Set X :=
+  (A.euclideanChart p.index).source ∩ (A.euclideanChart p.index) ⁻¹' p.domain
+omit [ConnectedSpace X] in
+lemma isOpen_source (p : UniformPathPatch g line) : IsOpen p.source :=
+  (A.euclideanChart p.index).isOpen_inter_preimage p.isOpen_domain
+omit [ConnectedSpace X] in
+lemma source_sub (p : UniformPathPatch g line) : p.source ⊆ (A.euclideanChart p.index).source :=
+  fun _ hx => hx.1
+omit [ConnectedSpace X] in
+lemma center_target (p : UniformPathPatch g line) : p.center ∈ (A.euclideanChart p.index).target :=
+  p.domain_sub p.center_mem
+
+theorem exists_at (g : KaehlerMetric A) (line : SemipositiveAnticanonicalMetric A)
+    (hd : 2 ≤ d) (x : X) : ∃ p : UniformPathPatch g line, x ∈ p.source := by
+  obtain ⟨i,hi⟩ := A.covers x
+  let e := A.euclideanChart i
+  have hxe : x ∈ e.source := by simpa [e] using hi
+  have hx : e x ∈ e.target := e.mapsTo hxe
+  obtain ⟨ε,hε,hball⟩ := Metric.mem_nhds_iff.mp (e.open_target.mem_nhds hx)
+  have hc : Metric.closedBall (e x) (ε/2) ⊆ e.target :=
+    (Metric.closedBall_subset_ball (by linarith : ε/2 < ε)).trans hball
+  obtain ⟨M,r,hM,hr,hrR,hpatch⟩ := g.volumePath_uniform_patch line hd i (e x)
+    (by linarith : 0 < ε/2) hc (stdOrthonormalBasis ℝ (EC d))
+  have hp (z : g.NormalizedPathSolution line) :=
+    hpatch z.potential z.positive z.mean_zero z.time z.constant z.time_mem z.equation
+  choose hb a has hac hsmall heq using hp
+  let coeff (z : g.NormalizedPathSolution line) : SmoothCoefficients (BasisIndex d) (EC d) :=
+    fun k l => (hac z k l).toSchwartzMap (has z k l)
+  have he (z : g.NormalizedPathSolution line) : coefficientBCF (coeff z) = a z := by
+    funext k l; ext y; rfl
+  let p : UniformPathPatch g line := {
+    index := i
+    center := e x
+    domain := Metric.ball (e x) (r/2)
+    isOpen_domain := Metric.isOpen_ball
+    center_mem := Metric.mem_ball_self (by linarith)
+    domain_sub := fun y hy => hc ((Metric.closedBall_subset_closedBall hrR)
+      ((Metric.ball_subset_closedBall).trans (Metric.closedBall_subset_closedBall (by linarith : r/2 ≤ r)) hy))
+    bound := M
+    bound_pos := hM
+    matrix_bound := hb
+    coefficient := coeff
+    small := by intro z; rw [he]; exact hsmall z
+    equals := by
+      intro z y hy k l
+      rw [he]
+      change (traceBilin _ _ : ℂ)+a z k l y = _
+      rw [heq z y (Metric.mem_closedBall.mpr (Metric.mem_ball.mp hy).le) k l]
+      ring }
+  exact ⟨p,hxe,Metric.mem_ball_self (by linarith)⟩
+
+theorem finite_cover (g : KaehlerMetric A) (line : SemipositiveAnticanonicalMetric A) (hd : 2 ≤ d) :
+    ∃ S : Finset (UniformPathPatch g line), ∀ x : X, ∃ p ∈ S, x ∈ p.source := by
+  obtain ⟨S,hS⟩ := isCompact_univ.elim_finite_subcover (fun p : UniformPathPatch g line => p.source)
+    (fun p => p.isOpen_source) (by
+      intro x _
+      obtain ⟨p,hp⟩ := exists_at g line hd x
+      exact mem_iUnion.mpr ⟨p,hp⟩)
+  refine ⟨S,fun x => ?_⟩
+  obtain ⟨p,hp,hx⟩ := mem_iUnion₂.mp (hS (mem_univ x))
+  exact ⟨p,hp,hx⟩
+end UniformPathPatch
+
+theorem exists_uniform_path_localizers (g : KaehlerMetric A)
+    (line : SemipositiveAnticanonicalMetric A) (hd : 2 ≤ d) :
+    ∃ (S : Finset (UniformPathPatch g line)) (D : Localizers A S),
+      (∀ p : S, D.index p = p.val.index) ∧
+      (∀ p : S, tsupport (D.weight p : X → ℂ) ⊆ p.val.source) := by
+  obtain ⟨S,hS⟩ := UniformPathPatch.finite_cover g line hd
+  let := A.chartedSpace
+  let := A.isManifold
+  obtain ⟨ρ,hρ⟩ := SmoothPartitionOfUnity.exists_isSubordinate
+    𝓘(ℝ,Coordinates d) (s:=(Set.univ : Set X)) isClosed_univ
+    (fun p : S => p.val.source) (fun p => p.val.isOpen_source) (by
+      intro x _
+      obtain ⟨p,hp,hx⟩ := hS x
+      exact mem_iUnion.mpr ⟨⟨p,hp⟩,hx⟩)
+  let f (p : S) : SmoothRealFunction A := ⟨ρ p,(A.contMDiff_iff (ρ p)).mp (ρ p).contMDiff⟩
+  let D : Localizers A S := {
+    index := fun p => p.val.index
+    weight := fun p => Smooth.ofReal (f p)
+    support_sub := by
+      intro p
+      rw [Smooth.ofReal_tsupport]
+      exact (hρ p).trans p.val.source_sub
+    sum_one := by
+      intro x
+      change (∑ p : S,((ρ p x : ℝ):ℂ)) = 1
+      rw [← Complex.ofReal_sum]
+      exact_mod_cast (show ∑ p : S,ρ p x = 1 by
+        simpa only [finsum_eq_sum_of_fintype] using ρ.sum_eq_one (mem_univ x)) }
+  refine ⟨S,D,fun _ => rfl,?_⟩
+  intro p
+  change tsupport (Smooth.ofReal (f p) : X → ℂ) ⊆ _
+  rw [Smooth.ofReal_tsupport]
+  exact hρ p
+end Anticanonical.SourceSmooth
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/PathUniformForcing.lean b/lean/OAI/Geometry/PrescribedRicci/PathUniformForcing.lean
new file mode 100644
index 0000000000000000000000000000000000000000..61881471e8b8db3efefd8048d9ec0aad3f528b15
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/PathUniformForcing.lean
@@ -0,0 +1,97 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.GlobalHessian
+import OAI.Geometry.PrescribedPotential.RealSobolev
+import OAI.Geometry.PrescribedPotential.SobolevExponential
+import OAI.Geometry.PrescribedRicci.PathUniformSobolevBase
+import OAI.Geometry.PrescribedRicci.SmoothPositivePath
+import OAI.Geometry.PrescribedRicci.UniformSchwartz
+import OAI.Geometry.PrescribedPotential.VolumeNormalization
+
+namespace OAI
+
+/-! Path Uniform Forcing. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology
+open scoped ContDiff Classical SchwartzMap
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  [ConnectedSpace X] {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+local instance pathForcingNG (s : ℝ) : NormedAddCommGroup (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedAddCommGroup
+local instance pathForcingNS (s : ℝ) : NormedSpace ℝ (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedSpace
+local instance pathForcingTG (s : ℝ) : IsTopologicalAddGroup (D.localizers.RealSobolev s) :=
+  Submodule.isTopologicalAddGroup _
+local instance pathForcingCS (s : ℝ) : ContinuousSMul ℝ (D.localizers.RealSobolev s) :=
+  SMulMemClass.continuousSMul _
+
+lemma expAffine_uniform_bound (F : RealSmooth A) (B : ℝ) (k : ℕ) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ t ∈ Icc (0:ℝ) 1, ∀ b : ℝ, b ≤ B →
+      ‖D.localizers.embed (k:ℝ) (F.expAffine t b).val‖ ≤ C := by
+  let m := max k (Module.finrank ℝ (EC d)+1)
+  have hm : Module.finrank ℝ (EC d) < m := lt_of_lt_of_le (Nat.lt_succ_self _) (le_max_right _ _)
+  obtain ⟨Q,hQ⟩ := isCompact_Icc.bddAbove_image
+    ((D.exponentialDensity_continuous m hm F).norm.continuousOn)
+  have hq (t : ℝ) (ht : t ∈ Icc (0:ℝ) 1) : ‖D.exponentialDensity m hm F t‖ ≤ max Q 0 :=
+    (hQ (mem_image_of_mem _ ht)).trans (le_max_left _ _)
+  let L := D.localizers.lower (m:ℝ) (k:ℝ)
+  refine ⟨‖L‖*(Real.exp B*max Q 0),by positivity,fun t ht b hb => ?_⟩
+  have hh : ‖D.localizers.embed (m:ℝ) (F.expAffine t b).val‖ ≤ Real.exp B*max Q 0 := by
+    change ‖D.localizers.realEmbed (m:ℝ) (F.expAffine t b)‖ ≤ _
+    rw [← D.exponentialDensity_embedViaEnergy m hm F t b,norm_smul,Real.norm_eq_abs,abs_of_pos (Real.exp_pos b)]
+    exact mul_le_mul (Real.exp_le_exp.mpr hb) (hq t ht) (norm_nonneg _) (Real.exp_pos B).le
+  have he : L (D.localizers.embed (m:ℝ) (F.expAffine t b).val) =
+      D.localizers.embed (k:ℝ) (F.expAffine t b).val :=
+    D.localizers.lower_embed (by exact_mod_cast (le_max_left k (Module.finrank ℝ (EC d)+1))) _
+  rw [← he]
+  exact (L.le_opNorm _).trans (mul_le_mul_of_nonneg_left hh (norm_nonneg _))
+
+lemma volumePath_forcing_uniform (line : SemipositiveAnticanonicalMetric A) (k : ℕ) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ z : g.NormalizedPathSolution line,
+      ‖D.localizers.embed (k:ℝ)
+        ((RealSmooth.ofReal (prescribedForcing g line)).expAffine z.time z.constant).val‖ ≤ C := by
+  let F := RealSmooth.ofReal (prescribedForcing g line)
+  let B := ‖(⟨(prescribedForcing g line).value,(prescribedForcing g line).continuous⟩ : C(X,ℝ))‖
+  obtain ⟨C,hC,hb⟩ := D.expAffine_uniform_bound F B k
+  refine ⟨C,hC,fun z => hb z.time z.time_mem z.constant ?_⟩
+  exact (le_abs_self _).trans (volumePath_scalar_bound g line z.time_mem ⟨z.positive,z.equation⟩)
+
+omit [ConnectedSpace X] in
+lemma uniform_localize {Z : Type*} (k : ℕ) (F : Z → Smooth A)
+    (hb : ∃ C : ℝ, 0 ≤ C ∧ ∀ z, ‖D.localizers.embed (k:ℝ) (F z)‖ ≤ C)
+    (i : Fin A.count) (ρ : Smooth A) (hρ : tsupport (ρ : X → ℂ) ⊆ (A.euclideanChart i).source) :
+    UniformSobolev (k:ℝ) (fun z => localize A i ρ hρ (F z)) := by
+  obtain ⟨C,hC,hc⟩ := hb
+  obtain ⟨B,hB,hb⟩ := D.localization_integer_bound i ρ hρ k
+  exact ⟨B*C,mul_nonneg hB hC,fun z => (hb _).trans (mul_le_mul_of_nonneg_left (hc z) hB)⟩
+end GluingData
+
+omit [ConnectedSpace X] in
+lemma localize_cutoff_product (i : Fin A.count) (κ : ChartCutoff (A.euclideanChart i).target) (F : Smooth A) :
+    localize A i (cutoffGlobal i κ) (cutoffGlobal_support i κ) F =
+      κ.product (A.euclideanChart i).open_target (F.smooth i) := by
+  ext y
+  by_cases hy : y ∈ (A.euclideanChart i).target
+  · rw [localize_apply,localizeFun_apply A _ _ hy]
+    have hx : (A.euclideanChart i).symm y ∈ (A.euclideanChart i).source := (A.euclideanChart i).symm.mapsTo hy
+    change (if (A.euclideanChart i).symm y ∈ (A.euclideanChart i).source then _ else 0)*_ = _
+    rw [ite_eq_left hx,(A.euclideanChart i).right_inv hy]
+    rfl
+  · have hκ : κ y = 0 := image_eq_zero_of_notMem_tsupport (fun hs => hy (κ.support_sub hs))
+    rw [localize_apply]
+    change (if y ∈ (A.euclideanChart i).target then _ else 0) = _
+    rw [ite_eq_right hy,ChartCutoff.product_apply,hκ,zero_mul]
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/PathUniformHigher.lean b/lean/OAI/Geometry/PrescribedRicci/PathUniformHigher.lean
new file mode 100644
index 0000000000000000000000000000000000000000..17923cd6f7e902f1347077d9b403c69093df9340
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/PathUniformHigher.lean
@@ -0,0 +1,133 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.PathLocalGain
+import OAI.Geometry.PrescribedRicci.PathUniformAtlas
+import OAI.Geometry.PrescribedRicci.PathUniformForcing
+import OAI.Geometry.PrescribedPotential.RegularityCutoffs
+
+namespace OAI
+
+/-! Path Uniform Higher. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff SchwartzMap Classical ComplexOrder MatrixOrder Matrix.Norms.Elementwise
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  [ConnectedSpace X] {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+  {g : KaehlerMetric A}
+
+omit [ConnectedSpace X] in
+lemma weighted_cutoff_reconstruction (i : Fin A.count) (w F : Smooth A)
+    (hw : tsupport (w : X → ℂ) ⊆ (A.euclideanChart i).source)
+    (κ : ChartCutoff (A.euclideanChart i).target)
+    (hκ : ∀ x ∈ tsupport (w : X → ℂ), κ (A.euclideanChart i x) = 1) :
+    globalize i (κ.mulOn (A.euclideanChart i).open_target (w.smooth i))
+      (localize A i (cutoffGlobal i κ) (cutoffGlobal_support i κ) F) = w.mul F := by
+  ext x
+  rw [globalize_apply,localize_cutoff_product]
+  by_cases hx : x ∈ (A.euclideanChart i).source
+  · rw [ite_eq_left hx]
+    change (κ (A.euclideanChart i x)*w ((A.euclideanChart i).symm (A.euclideanChart i x)))*
+      (κ (A.euclideanChart i x)*F ((A.euclideanChart i).symm (A.euclideanChart i x))) = w x*F x
+    rw [(A.euclideanChart i).left_inv hx]
+    by_cases hwx : w x = 0
+    · rw [hwx]; ring
+    · rw [hκ x (subset_tsupport _ hwx),one_mul,one_mul]
+  · rw [ite_eq_right hx]
+    change 0 = w x*F x
+    rw [image_eq_zero_of_notMem_tsupport (fun hs => hx (hw hs)),zero_mul]
+
+namespace GluingData
+variable (D : GluingData g ι)
+
+lemma path_patch_gain (line : SemipositiveAnticanonicalMetric A) (hd : 2 ≤ d)
+    (p : UniformPathPatch g line) (κ : ChartCutoff (A.euclideanChart p.index).target)
+    (hκp : tsupport (κ : EC d → ℂ) ⊆ p.domain) (hκr : ∀ y, (κ y).im = 0)
+    (k : ℕ) (hk : 1 ≤ k)
+    (hprev : ∃ C : ℝ, 0 ≤ C ∧ ∀ z : g.NormalizedPathSolution line,
+      ‖D.localizers.embed ((k:ℝ)+1) (Smooth.ofReal z.potential)‖ ≤ C) :
+    UniformSobolev ((k:ℝ)+2) (fun z : g.NormalizedPathSolution line =>
+      localize A p.index (cutoffGlobal p.index κ) (cutoffGlobal_support p.index κ) (Smooth.ofReal z.potential)) := by
+  obtain ⟨σ,_,hσ⟩ := exists_chartCutoff_nhds κ.compact (A.euclideanChart p.index).open_target κ.support_sub
+  obtain ⟨τ,hτr,hτ⟩ := exists_chartCutoff_nhds σ.compact (A.euclideanChart p.index).open_target σ.support_sub
+  exact D.path_local_gain line hd p κ σ τ hκp hκr hτr hσ hτ k hk hprev
+
+lemma volumePath_uniform_gain (line : SemipositiveAnticanonicalMetric A) (hd : 2 ≤ d)
+    (k : ℕ) (hk : 1 ≤ k)
+    (hprev : ∃ C : ℝ, 0 ≤ C ∧ ∀ z : g.NormalizedPathSolution line,
+      ‖D.localizers.embed ((k:ℝ)+1) (Smooth.ofReal z.potential)‖ ≤ C) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ z : g.NormalizedPathSolution line,
+      ‖D.localizers.embed ((k:ℝ)+2) (Smooth.ofReal z.potential)‖ ≤ C := by
+  obtain ⟨S,L,hLI,hLS⟩ := exists_uniform_path_localizers g line hd
+  have hc (p : S) : ∃ κ : ChartCutoff (A.euclideanChart p.val.index).target,
+      tsupport (κ : EC d → ℂ) ⊆ p.val.domain ∧ (∀ y, (κ y).im = 0) ∧
+      ∀ x ∈ tsupport (L.weight p : X → ℂ), κ (A.euclideanChart p.val.index x) = 1 := by
+    let e := A.euclideanChart p.val.index
+    have hK : IsCompact (e '' tsupport (L.weight p : X → ℂ)) :=
+      (isClosed_tsupport _).isCompact.image_of_continuousOn
+        (e.continuousOn.mono (fun x hx => (hLS p hx).1))
+    have hKU : e '' tsupport (L.weight p : X → ℂ) ⊆ p.val.domain := by
+      rintro y ⟨x,hx,rfl⟩
+      exact (hLS p hx).2
+    obtain ⟨κ,hκr,hκ⟩ := exists_chartCutoff_nhds hK p.val.isOpen_domain hKU
+    exact ⟨⟨κ.val,κ.compact,κ.support_sub.trans p.val.domain_sub⟩,κ.support_sub,hκr,
+      fun x hx => (hκ _ ⟨x,hx,rfl⟩).eq_of_nhds⟩
+  choose κ hκp hκr hκ using hc
+  have hlocal (p : S) := D.path_patch_gain line hd p.val (κ p) (hκp p) (hκr p) k hk hprev
+  choose B hB hb using hlocal
+  let η (p : S) := (κ p).mulOn (A.euclideanChart p.val.index).open_target ((L.weight p).smooth p.val.index)
+  have hglob (p : S) := globalize_integer_bound D.localizers p.val.index (η p) (k+2)
+  choose Q hQ hq using hglob
+  have hsum (F : Smooth A) : F = ∑ p : S, globalize p.val.index (η p)
+      (localize A p.val.index (cutoffGlobal p.val.index (κ p)) (cutoffGlobal_support p.val.index (κ p)) F) := by
+    have he (p : S) := weighted_cutoff_reconstruction p.val.index (L.weight p) F
+      (by simpa only [hLI p] using L.support_sub p) (κ p) (hκ p)
+    simp_rw [show ∀ p : S, globalize p.val.index (η p)
+      (localize A p.val.index (cutoffGlobal p.val.index (κ p)) (cutoffGlobal_support p.val.index (κ p)) F) =
+      (L.weight p).mul F from he]
+    ext x
+    change F.val x = (∑ p : S, (L.weight p).mul F).val x
+    simp only [Submodule.coe_sum,Finset.sum_apply]
+    change F x = ∑ p : S, L.weight p x*F x
+    rw [← Finset.sum_mul,L.sum_one,one_mul]
+  refine ⟨∑ p, Q p*B p,Finset.sum_nonneg (fun p _ => mul_nonneg (hQ p) (hB p)),fun z => ?_⟩
+  rw [hsum (Smooth.ofReal z.potential),map_sum]
+  apply (norm_sum_le _ _).trans
+  apply Finset.sum_le_sum
+  intro p _
+  have hh := hq p (localize A p.val.index (cutoffGlobal p.val.index (κ p))
+    (cutoffGlobal_support p.val.index (κ p)) (Smooth.ofReal z.potential))
+  rw [Nat.cast_add,Nat.cast_ofNat] at hh
+  exact hh.trans (mul_le_mul_of_nonneg_left (hb p z) (hQ p))
+
+theorem volumePath_uniform_all (line : SemipositiveAnticanonicalMetric A) (hd : 2 ≤ d) :
+    ∀ k : ℕ, ∃ C : ℝ, 0 ≤ C ∧ ∀ z : g.NormalizedPathSolution line,
+      ‖D.localizers.embed (k:ℝ) (Smooth.ofReal z.potential)‖ ≤ C := by
+  have hhigh : ∀ k : ℕ, ∃ C : ℝ, 0 ≤ C ∧ ∀ z : g.NormalizedPathSolution line,
+      ‖D.localizers.embed ((k:ℝ)+2) (Smooth.ofReal z.potential)‖ ≤ C := by
+    intro n
+    induction n with
+    | zero =>
+      rw [Nat.cast_zero,zero_add]
+      exact D.volumePath_uniform_H2 line hd
+    | succ n ih =>
+      have he : ((n+1:ℕ):ℝ)+1 = (n:ℝ)+2 := by push_cast; ring
+      have h := D.volumePath_uniform_gain line hd (n+1) (by omega) (by rw [he]; exact ih)
+      exact h
+  intro k
+  obtain ⟨C,hC,hc⟩ := hhigh k
+  obtain ⟨B,hB,hb⟩ := D.localizers.boundedCore_lower (by linarith : (k:ℝ) ≤ (k:ℝ)+2)
+  refine ⟨B*C,mul_nonneg hB hC,fun z => ?_⟩
+  exact (hb (Smooth.ofReal z.potential)).trans (mul_le_mul_of_nonneg_left (hc z) hB)
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/PathUniformSobolevBase.lean b/lean/OAI/Geometry/PrescribedRicci/PathUniformSobolevBase.lean
new file mode 100644
index 0000000000000000000000000000000000000000..311f2c5072569eb2be74d8ac9686d249e0eb0707
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/PathUniformSobolevBase.lean
@@ -0,0 +1,114 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CompletedResolvent
+import OAI.Geometry.PrescribedPotential.QuantitativeResolvent
+import OAI.Geometry.PrescribedRicci.CurvatureCompactBounds
+import OAI.Geometry.PrescribedRicci.KahlerL2Comparison
+import OAI.Geometry.PrescribedRicci.MatrixComponentBounds
+import OAI.Geometry.PrescribedRicci.MongeAmpereC0
+import OAI.Geometry.PrescribedRicci.UniformMetricBounds
+import OAI.Geometry.PrescribedPotential.RealSmoothOperator
+import OAI.Geometry.PrescribedPotential.VolumePath
+
+namespace OAI
+
+/-! Path Uniform Sobolev Base. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff Classical Matrix.Norms.Elementwise
+namespace Anticanonical.SourceSmooth.KaehlerMetric
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  [ConnectedSpace X] {A : ComplexAtlas d X}
+
+structure NormalizedPathSolution (g : KaehlerMetric A) (line : SemipositiveAnticanonicalMetric A) where
+  potential : SmoothRealFunction A
+  positive : g.PositivePotential potential
+  mean_zero : g.integral potential.value = 0
+  time : ℝ
+  time_mem : time ∈ Icc (0:ℝ) 1
+  constant : ℝ
+  equation : ∀ x, (g.logRatio (g.deform potential positive)).value x =
+    time*(prescribedForcing g line).value x+constant
+
+lemma volumePath_background_laplacian_bound (g : KaehlerMetric A)
+    (line : SemipositiveAnticanonicalMetric A) (hd : 2 ≤ d) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ z : g.NormalizedPathSolution line, ∀ x,
+      |(g.laplacian z.potential).value x| ≤ C := by
+  obtain ⟨K,hK,hk⟩ := g.volumePath_metric_bounds line hd
+  obtain ⟨M,hM,hm⟩ := g.compact_background_bounds (prescribedForcing g line)
+  have hM0 : 0 ≤ M := by linarith
+  refine ⟨(d:ℝ)^2*M*(K+M),by positivity,fun z x => ?_⟩
+  obtain ⟨q,hq⟩ := exists_partition_chart (A:=A) x
+  have hx := chartPartition_support q hq
+  have hb := hm q x hq
+  have hH := (hk z.potential z.positive z.mean_zero z.time z.constant z.time_mem z.equation q x hq).1
+  have hp (i j : Fin d) : ‖z.potential.hessian q (A.chart q x) i j‖ ≤ K+M := by
+    have he : z.potential.hessian q (A.chart q x) i j =
+        (g.deform z.potential z.positive).matrix q (A.chart q x) i j-g.matrix q (A.chart q x) i j := by
+      change _ = (g.matrix q (A.chart q x) i j+_)-_
+      ring
+    rw [he]
+    apply (norm_sub_le _ _).trans
+    exact add_le_add ((norm_le_pi_norm _ j).trans ((norm_le_pi_norm _ i).trans hH)) (hb.1 i j)
+  change |g.laplacianValue z.potential x| ≤ _
+  rw [g.laplacianValue_local z.potential q hx]
+  exact (Complex.abs_re_le_norm _).trans (by
+    simpa only [Fintype.card_fin] using MongeAmpere.trace_mul_entry_bound hM0 hb.2.1 hp)
+end Anticanonical.SourceSmooth.KaehlerMetric
+
+namespace GlobalElliptic
+open Anticanonical SourceSmooth SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  [ConnectedSpace X] {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+
+omit [ConnectedSpace X] in
+lemma Localizers.uniform_zero_of_bound (D : Localizers A ι) (g : KaehlerMetric A)
+    {Z : Type*} (F : Z → Smooth A) {M : ℝ} (hM : 0 ≤ M)
+    (hb : ∀ z x, ‖F z x‖ ≤ M) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ z, ‖D.embed 0 (F z)‖ ≤ C := by
+  obtain ⟨B,Q,hB,hQ,hbq⟩ := D.kahler_L2_comparison g
+  let C : ℝ := B*(M^2*g.integral (fun _ => (1:ℝ)))
+  have hC : 0 ≤ C := mul_nonneg hB.le (mul_nonneg (sq_nonneg _) (g.integral_nonneg (fun _ => zero_le_one)))
+  refine ⟨Real.sqrt C,Real.sqrt_nonneg _,fun z => (Real.le_sqrt (norm_nonneg _) hC).mpr ?_⟩
+  apply (hbq (F z)).1.trans
+  apply mul_le_mul_of_nonneg_left _ hB.le
+  rw [← g.integral_const_mul (M^2) (fun _ => (1:ℝ))]
+  apply g.integral_mono ((F z).continuous.norm.pow 2) continuous_const
+  intro x
+  change ‖F z x‖^2 ≤ M^2*1
+  rw [mul_one]
+  exact (sq_le_sq₀ (norm_nonneg _) hM).mpr (hb z x)
+
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+theorem volumePath_uniform_H2 (line : SemipositiveAnticanonicalMetric A) (hd : 2 ≤ d) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ z : g.NormalizedPathSolution line,
+      ‖D.localizers.embed 2 (Smooth.ofReal z.potential)‖ ≤ C := by
+  obtain ⟨M,hM,hm⟩ := g.volumePath_C0_bound line hd
+  obtain ⟨B,hB,hb⟩ := D.localizers.uniform_zero_of_bound g
+    (fun z : g.NormalizedPathSolution line => Smooth.ofReal z.potential) hM (fun z x => by
+      simpa only [Smooth.ofReal_apply,Complex.norm_real,Real.norm_eq_abs] using
+        hm z.potential z.positive z.mean_zero z.time z.constant z.time_mem z.equation x)
+  obtain ⟨L,hL,hl⟩ := g.volumePath_background_laplacian_bound line hd
+  obtain ⟨J,hJ,hj⟩ := D.localizers.uniform_zero_of_bound g
+    (fun z : g.NormalizedPathSolution line => Smooth.ofReal (g.laplacian z.potential)) hL (fun z x => by
+      simpa only [Smooth.ofReal_apply,Complex.norm_real,Real.norm_eq_abs] using hl z x)
+  obtain ⟨m,hm,he⟩ := D.exists_completedError_small
+  refine ⟨‖D.completedResolvent m hm he‖*(m^2*B+J),by positivity,fun z => ?_⟩
+  have h := D.elliptic_estimate m hm he (D.localizers.embed 2 (Smooth.ofReal z.potential))
+  rw [D.localizers.lower_embed (by norm_num : (0:ℝ) ≤ 2),
+    completedL,D.localizers.extendCore_embed D.complexL_bound,complexL_ofReal] at h
+  exact h.trans (mul_le_mul_of_nonneg_left (add_le_add
+    (mul_le_mul_of_nonneg_left (hb z) (sq_nonneg m)) (hj z)) (norm_nonneg (D.completedResolvent m hm he)))
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/PositiveOpenness.lean b/lean/OAI/Geometry/PrescribedRicci/PositiveOpenness.lean
new file mode 100644
index 0000000000000000000000000000000000000000..077d0109351a02ec6d8f2197e0d332570a92fd38
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/PositiveOpenness.lean
@@ -0,0 +1,79 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.PatchCutoffs
+import OAI.Geometry.PrescribedPotential.RealSobolev
+import OAI.Geometry.PrescribedPotential.WeakPositive
+import OAI.Geometry.PrescribedPotential.SobolevOpenness
+
+namespace OAI
+
+/-! Positive Openness. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff Classical ComplexOrder
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+local instance positiveOpenNG (s : ℝ) : NormedAddCommGroup (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedAddCommGroup
+local instance positiveOpenNS (s : ℝ) : NormedSpace ℝ (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedSpace
+local instance positiveOpenTG (s : ℝ) : IsTopologicalAddGroup (D.localizers.RealSobolev s) :=
+  Submodule.isTopologicalAddGroup _
+local instance positiveOpenCS (s : ℝ) : ContinuousSMul ℝ (D.localizers.RealSobolev s) :=
+  SMulMemClass.continuousSMul _
+
+omit [T2Space X] [CompactSpace X] in
+lemma exists_weight_supportViaEnergy (x : X) : ∃ p, x ∈ tsupport (D.localizers.weight p : X → ℂ) := by
+  by_contra! hn
+  have hz : ∑ p, D.localizers.weight p x = 0 := Finset.sum_eq_zero (fun p _ =>
+    image_eq_zero_of_notMem_tsupport (hn p))
+  rw [D.localizers.sum_one] at hz
+  exact one_ne_zero hz
+
+lemma weakPositive_embed_iffViaEnergy (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) (f : RealSmooth A) :
+    D.WeakPositive k (D.localizers.realEmbed ((k : ℝ)+2) f) ↔ g.PositivePotential f.source := by
+  constructor
+  · intro h i z hz
+    let x := (A.chart i).symm z
+    have hi : x ∈ (A.chart i).source := (A.chart i).mapsTo_symm hz
+    have hiz : A.chart i x = z := (A.chart i).right_inv hz
+    obtain ⟨p,hp⟩ := D.exists_weight_supportViaEnergy x
+    have hj : x ∈ (A.chart (D.patch p).index).source := by simpa using D.patch_source p hp
+    have hpos := h p x hp
+    rw [D.weakMatrix_embed k hk f p hp] at hpos
+    rw [← hiz, g.compatibility i (D.patch p).index x hi hj,
+      f.source.hessian_compatibility i (D.patch p).index hi hj, ← add_mul, ← mul_add]
+    exact hpos.conjTranspose_mul_mul_same (Matrix.mulVec_injective_of_det_ne_zero
+      (g.transition_det_isUnit i (D.patch p).index hi hj).ne_zero)
+  · intro h p x hx
+    rw [D.weakMatrix_embed k hk f p hx]
+    exact h _ _ ((A.chart (D.patch p).index).mapsTo (by simpa using D.patch_source p hx))
+
+ 
+
+theorem positive_volume_eventually_solvableViaEnergy [ConnectedSpace X]
+    (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) (x₀ : X) :
+    ∀ᶠ f in 𝓝 (D.realConstants (k : ℝ) 1),
+      ∃ (u : D.localizers.RealSobolev ((k : ℝ)+2)) (b : ℝ),
+        D.WeakPositive k u ∧ D.realVolume k hk u = Real.exp b • f ∧
+        D.realEvaluation ((k : ℝ)+2) x₀ u = 0 := by
+  have hU : {z : D.localizers.RealSobolev ((k : ℝ)+2) × ℝ | D.WeakPositive k z.1} ∈ 𝓝 0 :=
+    (show Tendsto (Prod.fst : D.localizers.RealSobolev ((k : ℝ)+2) × ℝ → _)
+      (𝓝 0) (𝓝 0) from continuous_fst.tendsto 0) (D.weakPositive_nhds k hk)
+  exact D.volume_eventually_solvable k hk x₀ hU
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/PositivePathClosedness.lean b/lean/OAI/Geometry/PrescribedRicci/PositivePathClosedness.lean
new file mode 100644
index 0000000000000000000000000000000000000000..d1d0e2e77c45f1099c8beafad4d9700900e731bd
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/PositivePathClosedness.lean
@@ -0,0 +1,73 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.PositivePathOpenness
+import OAI.Geometry.PrescribedPotential.VolumeNormalization
+import OAI.Geometry.PrescribedRicci.NormalizeVolumePath
+import OAI.Geometry.PrescribedRicci.PositivePathLimit
+import OAI.Geometry.PrescribedRicci.SmoothPathLimit
+
+namespace OAI
+
+/-! Positive Path Closedness. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology
+open scoped ContDiff Classical
+namespace Anticanonical.SourceSmooth
+open GlobalElliptic
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  [ConnectedSpace X] {A : ComplexAtlas d X}
+
+theorem volumePath_soluble_limit (g : KaehlerMetric A) (line : SemipositiveAnticanonicalMetric A)
+    (hd : 2 ≤ d) (t : ℕ → ℝ) (u : ℝ)
+    (ht : ∀ n, t n ∈ Icc (0:ℝ) 1)
+    (hsol : ∀ n, ∃ (φ : SmoothRealFunction A) (b : ℝ), SolvesVolumePath g line (t n) φ b)
+    (hu : Tendsto t atTop (𝓝 u)) :
+    ∃ (φ : SmoothRealFunction A) (b : ℝ), SolvesVolumePath g line u φ b := by
+  choose φ b hφ using hsol
+  choose z hz _ using fun n => g.normalizeVolumePath line (ht n) (hφ n)
+  obtain ⟨S,⟨D⟩⟩ := exists_gluingData g
+  obtain ⟨ψ,ν,hν,hψ⟩ := D.path_smooth_subsequence line hd z
+  let M := ‖(⟨(prescribedForcing g line).value,(prescribedForcing g line).continuous⟩ : C(X,ℝ))‖
+  have hb (n : ℕ) : (z (ν n)).constant ∈ Icc (-M) M := by
+    exact abs_le.mp (volumePath_scalar_bound g line (z (ν n)).time_mem
+      ⟨(z (ν n)).positive,(z (ν n)).equation⟩)
+  obtain ⟨c,_,μ,hμ,hc⟩ := isCompact_Icc.tendsto_subseq hb
+  refine ⟨ψ,c,D.path_limit_solves line (fun n => z (ν (μ n))) ψ u c
+    (fun k => (hψ k).comp hμ.tendsto_atTop) ?_ hc⟩
+  have he : (fun n => (z (ν (μ n))).time) = t ∘ ν ∘ μ := by
+    funext n
+    exact hz _
+  rw [he]
+  exact (hu.comp hν.tendsto_atTop).comp hμ.tendsto_atTop
+
+theorem volumePath_soluble_isClosed (g : KaehlerMetric A) (line : SemipositiveAnticanonicalMetric A)
+    (hd : 2 ≤ d) :
+    IsClosed {t : Icc (0:ℝ) 1 | ∃ (φ : SmoothRealFunction A) (b : ℝ),
+      SolvesVolumePath g line t.val φ b} := by
+  apply IsSeqClosed.isClosed
+  intro t u ht hu
+  exact volumePath_soluble_limit g line hd (fun n => (t n).val) u.val
+    (fun n => (t n).property) ht (continuous_subtype_val.continuousAt.tendsto.comp hu)
+
+theorem volumePath_one_exists (g : KaehlerMetric A) (line : SemipositiveAnticanonicalMetric A)
+    (hd : 2 ≤ d) :
+    ∃ (φ : SmoothRealFunction A) (b : ℝ), SolvesVolumePath g line 1 φ b := by
+  let S : Set (Icc (0:ℝ) 1) := {t | ∃ (φ : SmoothRealFunction A) (b : ℝ),
+    SolvesVolumePath g line t.val φ b}
+  have hS : IsClopen S := ⟨volumePath_soluble_isClosed g line hd,
+    (volumePath_soluble_isOpen g line).preimage continuous_subtype_val⟩
+  let : PreconnectedSpace (Icc (0:ℝ) 1) := Subtype.preconnectedSpace isPreconnected_Icc
+  have he : S = univ := hS.eq_univ ⟨⟨0,by norm_num⟩,SmoothRealFunction.constant 0,0,
+    volumePath_zero g line⟩
+  have h1 : (⟨1,by norm_num⟩ : Icc (0:ℝ) 1) ∈ S := by rw [he]; trivial
+  exact h1
+end Anticanonical.SourceSmooth
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/PositivePathLimit.lean b/lean/OAI/Geometry/PrescribedRicci/PositivePathLimit.lean
new file mode 100644
index 0000000000000000000000000000000000000000..c890c58bf88ed72b4eab92553f9e701237138ce6
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/PositivePathLimit.lean
@@ -0,0 +1,140 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.PathLocalizedDensity
+import OAI.Geometry.PrescribedRicci.SmoothPathLimit
+
+namespace OAI
+
+/-! Positive Path Limit. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff Classical ComplexOrder
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  [ConnectedSpace X] {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+  {g : KaehlerMetric A}
+
+omit [ConnectedSpace X] in
+lemma path_determinant_source (line : SemipositiveAnticanonicalMetric A)
+    (z : g.NormalizedPathSolution line) (q : Fin A.count) {x : X}
+    (hx : x ∈ (A.chart q).source) :
+    ((g.deform z.potential z.positive).matrix q (A.chart q x)).det =
+      (g.matrix q (A.chart q x)).det *
+        (Real.exp (z.time*(prescribedForcing g line).value x+z.constant):ℂ) := by
+  have hxe : x ∈ (A.euclideanChart q).source := by simpa using hx
+  have h := path_determinant_equation line z q ((A.euclideanChart q).mapsTo hxe)
+  have he : coordinateEquiv d (A.euclideanChart q x) = A.chart q x :=
+    (coordinateEquiv d).apply_symm_apply _
+  rw [he,(A.euclideanChart q).left_inv hxe] at h
+  exact h
+
+namespace GluingData
+variable (D : GluingData g ι)
+
+omit [ConnectedSpace X] in
+lemma path_limit_matrix (line : SemipositiveAnticanonicalMetric A)
+    (z : ℕ → g.NormalizedPathSolution line) (φ : SmoothRealFunction A)
+    (hlim : ∀ k : ℕ, Tendsto (fun n => D.localizers.embed (k:ℝ) (Smooth.ofReal (z n).potential))
+      atTop (𝓝 (D.localizers.embed (k:ℝ) (Smooth.ofReal φ))))
+    (p : ι) {x : X} (hx : x ∈ tsupport (D.localizers.weight p : X → ℂ)) :
+    Tendsto (fun n => (g.deform (z n).potential (z n).positive).matrix
+      (D.patch p).index (A.chart (D.patch p).index x)) atTop
+      (𝓝 (g.matrix (D.patch p).index (A.chart (D.patch p).index x)+
+        φ.hessian (D.patch p).index (A.chart (D.patch p).index x))) := by
+  apply tendsto_pi_nhds.mpr
+  intro i
+  apply tendsto_pi_nhds.mpr
+  intro j
+  exact tendsto_const_nhds.add (D.hessian_of_sobolev _ φ hlim p hx i j)
+
+omit [ConnectedSpace X] in
+lemma path_limit_determinant (line : SemipositiveAnticanonicalMetric A)
+    (z : ℕ → g.NormalizedPathSolution line) (φ : SmoothRealFunction A) (t b : ℝ)
+    (hlim : ∀ k : ℕ, Tendsto (fun n => D.localizers.embed (k:ℝ) (Smooth.ofReal (z n).potential))
+      atTop (𝓝 (D.localizers.embed (k:ℝ) (Smooth.ofReal φ))))
+    (ht : Tendsto (fun n => (z n).time) atTop (𝓝 t))
+    (hb : Tendsto (fun n => (z n).constant) atTop (𝓝 b))
+    (p : ι) {x : X} (hx : x ∈ tsupport (D.localizers.weight p : X → ℂ)) :
+    (g.matrix (D.patch p).index (A.chart (D.patch p).index x)+
+      φ.hessian (D.patch p).index (A.chart (D.patch p).index x)).det =
+      (g.matrix (D.patch p).index (A.chart (D.patch p).index x)).det *
+        (Real.exp (t*(prescribedForcing g line).value x+b):ℂ) := by
+  have hs : x ∈ (A.chart (D.patch p).index).source := by simpa using D.patch_source p hx
+  have hdet := (show Continuous (fun M : Matrix (Fin d) (Fin d) ℂ => M.det) from
+    continuous_id.matrix_det).continuousAt.tendsto.comp (D.path_limit_matrix line z φ hlim p hx)
+  have hex := Real.continuous_exp.continuousAt.tendsto.comp
+    ((ht.mul_const ((prescribedForcing g line).value x)).add hb)
+  have hc := Complex.continuous_ofReal.continuousAt.tendsto.comp hex
+  have hr := hc.const_mul ((g.matrix (D.patch p).index (A.chart (D.patch p).index x)).det)
+  have he (n : ℕ) := path_determinant_source line (z n) (D.patch p).index hs
+  simp only [Function.comp_def] at hdet hr
+  simp_rw [he] at hdet
+  exact tendsto_nhds_unique hdet hr
+
+omit [ConnectedSpace X] in
+lemma path_limit_positive (line : SemipositiveAnticanonicalMetric A)
+    (z : ℕ → g.NormalizedPathSolution line) (φ : SmoothRealFunction A) (t b : ℝ)
+    (hlim : ∀ k : ℕ, Tendsto (fun n => D.localizers.embed (k:ℝ) (Smooth.ofReal (z n).potential))
+      atTop (𝓝 (D.localizers.embed (k:ℝ) (Smooth.ofReal φ))))
+    (ht : Tendsto (fun n => (z n).time) atTop (𝓝 t))
+    (hb : Tendsto (fun n => (z n).constant) atTop (𝓝 b)) : g.PositivePotential φ := by
+  have hpos (p : ι) (x : X) (hx : x ∈ tsupport (D.localizers.weight p : X → ℂ)) :
+      (g.matrix (D.patch p).index (A.chart (D.patch p).index x)+
+        φ.hessian (D.patch p).index (A.chart (D.patch p).index x)).PosDef := by
+    have hs : x ∈ (A.chart (D.patch p).index).source := by simpa using D.patch_source p hx
+    have hz := (A.chart (D.patch p).index).mapsTo hs
+    apply (Matrix.PosSemidef.posDef_iff_det_ne_zero ?_).mpr
+    · rw [D.path_limit_determinant line z φ t b hlim ht hb p hx]
+      exact mul_ne_zero (g.positive _ _ hz).det_pos.ne' (Complex.ofReal_ne_zero.mpr (Real.exp_ne_zero _))
+    · apply Matrix.PosSemidef.of_dotProduct_mulVec_nonneg
+        ((g.positive _ _ hz).isHermitian.add (φ.hessian_hermitian _ hz))
+      intro v
+      have hc : Continuous (fun M : Matrix (Fin d) (Fin d) ℂ => star v ⬝ᵥ (M *ᵥ v)) := by fun_prop
+      have h := hc.continuousAt.tendsto.comp (D.path_limit_matrix line z φ hlim p hx)
+      exact isClosed_Ici.mem_of_tendsto h (Eventually.of_forall (fun n =>
+        ((z n).positive _ _ hz).posSemidef.dotProduct_mulVec_nonneg v))
+  intro i y hy
+  let x := (A.chart i).symm y
+  have hi : x ∈ (A.chart i).source := (A.chart i).mapsTo_symm hy
+  have hiy : A.chart i x = y := (A.chart i).right_inv hy
+  obtain ⟨p,hp⟩ := D.exists_weight_supportViaEnergy x
+  have hj : x ∈ (A.chart (D.patch p).index).source := by simpa using D.patch_source p hp
+  rw [← hiy,g.compatibility i (D.patch p).index x hi hj,
+    φ.hessian_compatibility i (D.patch p).index hi hj,← add_mul,← mul_add]
+  exact (hpos p x hp).conjTranspose_mul_mul_same (Matrix.mulVec_injective_of_det_ne_zero
+    (g.transition_det_isUnit i (D.patch p).index hi hj).ne_zero)
+
+omit [ConnectedSpace X] in
+lemma path_limit_solves (line : SemipositiveAnticanonicalMetric A)
+    (z : ℕ → g.NormalizedPathSolution line) (φ : SmoothRealFunction A) (t b : ℝ)
+    (hlim : ∀ k : ℕ, Tendsto (fun n => D.localizers.embed (k:ℝ) (Smooth.ofReal (z n).potential))
+      atTop (𝓝 (D.localizers.embed (k:ℝ) (Smooth.ofReal φ))))
+    (ht : Tendsto (fun n => (z n).time) atTop (𝓝 t))
+    (hb : Tendsto (fun n => (z n).constant) atTop (𝓝 b)) : SolvesVolumePath g line t φ b := by
+  let hp := D.path_limit_positive line z φ t b hlim ht hb
+  refine ⟨hp,fun x => ?_⟩
+  obtain ⟨p,hpX⟩ := D.exists_weight_supportViaEnergy x
+  have hs : x ∈ (A.chart (D.patch p).index).source := by simpa using D.patch_source p hpX
+  let q := (D.patch p).index
+  have hz := (A.chart q).mapsTo hs
+  have he := congrArg Complex.re (D.path_limit_determinant line z φ t b hlim ht hb p hpX)
+  have hv : (g.deform φ hp).volumeCoefficient q (A.chart q x) =
+      g.volumeCoefficient q (A.chart q x)*Real.exp (t*(prescribedForcing g line).value x+b) := by
+    simpa only [KaehlerMetric.volumeCoefficient,KaehlerMetric.deform,
+      Complex.mul_re,Complex.ofReal_re,Complex.ofReal_im,mul_zero,sub_zero] using he
+  change g.logRatioValue (g.deform φ hp) x = _
+  rw [g.logRatioValue_local _ q hs,hv,
+    Real.log_mul (g.volumeCoefficient_pos q hz).ne' (Real.exp_ne_zero _),Real.log_exp]
+  ring
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/PositivePathOpenness.lean b/lean/OAI/Geometry/PrescribedRicci/PositivePathOpenness.lean
new file mode 100644
index 0000000000000000000000000000000000000000..b8c125f6e016a5707551620513ad816ba6f137f0
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/PositivePathOpenness.lean
@@ -0,0 +1,81 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.PatchCutoffs
+import OAI.Geometry.PrescribedPotential.PositivePotentialLinearization
+import OAI.Geometry.PrescribedPotential.RealSobolev
+import OAI.Geometry.PrescribedPotential.SmoothPotentialDifference
+import OAI.Geometry.PrescribedPotential.VolumePath
+import OAI.Geometry.PrescribedRicci.SmoothPositivePath
+
+namespace OAI
+
+/-! Positive Path Openness. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology
+open scoped ContDiff Classical ComplexOrder
+namespace Anticanonical.SourceSmooth
+open GlobalElliptic
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  [ConnectedSpace X] {A : ComplexAtlas d X}
+namespace KaehlerMetric
+
+omit [T2Space X] [CompactSpace X] [ConnectedSpace X] in
+lemma logRatio_eq_log_density (g : KaehlerMetric A) (φ : SmoothRealFunction A)
+    (hp : g.PositivePotential φ) (x : X) :
+    (g.logRatio (g.deform φ hp)).value x = Real.log ((g.potentialDensity φ).value x) := by
+  obtain ⟨i,hi⟩ := A.covers x
+  rw [g.potentialDensity_quotient φ i hi]
+  change g.logRatioValue (g.deform φ hp) x = _
+  rw [g.logRatioValue_local _ i hi,Real.log_div
+    (Complex.pos_iff.mp (hp i _ ((A.chart i).mapsTo hi)).det_pos).1.ne'
+    (g.volumeCoefficient_pos i ((A.chart i).mapsTo hi)).ne']
+  rfl
+
+omit [T2Space X] [CompactSpace X] [ConnectedSpace X] in
+lemma positive_add_of_deformed (g : KaehlerMetric A) (φ ψ : SmoothRealFunction A)
+    (hp : g.PositivePotential φ) (hq : (g.deform φ hp).PositivePotential ψ) :
+    g.PositivePotential (φ.addFunction ψ) := by
+  intro i z hz
+  rw [SmoothRealFunction.hessian_addFunction _ _ i hz,← add_assoc]
+  exact hq i z hz
+
+omit [T2Space X] [CompactSpace X] [ConnectedSpace X] in
+lemma logRatio_add_deformed (g : KaehlerMetric A) (φ ψ : SmoothRealFunction A)
+    (hp : g.PositivePotential φ) (hq : (g.deform φ hp).PositivePotential ψ) (x : X) :
+    (g.logRatio (g.deform (φ.addFunction ψ) (g.positive_add_of_deformed φ ψ hp hq))).value x =
+    (g.logRatio (g.deform φ hp)).value x +
+    ((g.deform φ hp).logRatio ((g.deform φ hp).deform ψ hq)).value x := by
+  obtain ⟨i,hi⟩ := A.covers x
+  change g.logRatioValue _ x = g.logRatioValue _ x + (g.deform φ hp).logRatioValue _ x
+  rw [g.logRatioValue_local _ i hi,g.logRatioValue_local _ i hi,
+    (g.deform φ hp).logRatioValue_local _ i hi]
+  simp only [volumeCoefficient,deform,SmoothRealFunction.hessian_addFunction _ _ i ((A.chart i).mapsTo hi),add_assoc]
+  ring
+end KaehlerMetric
+
+theorem volumePath_soluble_isOpen (g : KaehlerMetric A) (h : SemipositiveAnticanonicalMetric A) :
+    IsOpen {t : ℝ | ∃ (φ : SmoothRealFunction A) (b : ℝ), SolvesVolumePath g h t φ b} := by
+  apply isOpen_iff_mem_nhds.mpr
+  rintro t ⟨φ,b,hp,he⟩
+  obtain ⟨x₀⟩ := (inferInstance : Nonempty X)
+  obtain ⟨S,⟨D⟩⟩ := exists_gluingData (g.deform φ hp)
+  have hn := D.smooth_positive_path_eventually (RealSmooth.ofReal (prescribedForcing g h)) x₀
+  have hsub : Tendsto (fun u : ℝ => u-t) (𝓝 t) (𝓝 0) := by
+    simpa only [sub_self] using (show Continuous (fun u : ℝ => u-t) from continuous_id.sub continuous_const).tendsto t
+  filter_upwards [hsub.eventually hn] with u hu
+  obtain ⟨ψ,c,hq,_,hden⟩ := hu
+  refine ⟨φ.addFunction ψ,b+c,g.positive_add_of_deformed φ ψ hp hq,fun x => ?_⟩
+  rw [g.logRatio_add_deformed φ ψ hp hq x,he x,
+    (g.deform φ hp).logRatio_eq_log_density ψ hq x,hden x,Real.log_exp]
+  change t*(prescribedForcing g h).value x+b+((u-t)*(prescribedForcing g h).value x+c) = _
+  ring
+end Anticanonical.SourceSmooth
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/PowerDual.lean b/lean/OAI/Geometry/PrescribedRicci/PowerDual.lean
new file mode 100644
index 0000000000000000000000000000000000000000..304288b2b63b58fb581d8d0d4390d970227d5f72
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/PowerDual.lean
@@ -0,0 +1,61 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.GlobalSobolev
+
+namespace OAI
+
+/-! Power Dual. -/
+
+noncomputable section
+open Filter Topology Asymptotics
+namespace TameInterpolation
+
+def powerDual (p x : ℝ) : ℝ := x * |x| ^ (p-2)
+
+lemma powerDual_zero (p : ℝ) : powerDual p 0 = 0 := by simp [powerDual]
+
+lemma powerDual_hasDerivAt {p : ℝ} (hp : 2 < p) (x : ℝ) :
+    HasDerivAt (powerDual p) ((p-1)*|x|^(p-2)) x := by
+  by_cases hx : x = 0
+  · subst x
+    have hq : 0 < p-2 := by linarith
+    have hF : HasFDerivAt (powerDual p) (0 : ℝ →L[ℝ] ℝ) 0 := by
+      refine .of_isLittleO ?_
+      simp only [powerDual,abs_zero,zero_mul,zero_apply,sub_zero]
+      have hsmall : (fun y : ℝ => |y|^(p-2)) =o[𝓝 0] (fun _ => (1:ℝ)) := by
+        apply (isLittleO_const_iff one_ne_zero).mpr
+        have hc : ContinuousAt (fun y : ℝ => |y|^(p-2)) 0 :=
+          continuousAt_id.abs.rpow_const (Or.inr hq.le)
+        simpa only [abs_zero,Real.zero_rpow hq.ne'] using hc.tendsto
+      simpa only [mul_one,sub_zero] using (isBigO_refl (fun y : ℝ => y) (𝓝 0)).mul_isLittleO hsmall
+    simpa [hq.ne'] using hF.hasDerivAt
+  · have hd := (hasDerivAt_id x).mul ((hasDerivAt_abs hx).rpow_const (p:=p-2) (Or.inl (abs_ne_zero.mpr hx)))
+    have ha : 0 < |x| := abs_pos.mpr hx
+    have he : |x|^(p-2) = |x|^(p-2-1)*|x| := by
+      rw [← Real.rpow_add_one ha.ne']
+      congr 1
+      ring
+    have hs : x*(SignType.sign x : ℝ) = |x| := by
+      simpa only [mul_comm] using sign_mul_self x
+    have hval : (p-1)*|x|^(p-2) = 1*|x|^(p-2)+x*((SignType.sign x:ℝ)*(p-2)*|x|^(p-2-1)) := by
+      rw [he]
+      calc
+        _ = |x|^(p-2-1)*|x|+ (x*(SignType.sign x:ℝ))*(p-2)*|x|^(p-2-1) := by rw [hs]; ring
+        _ = _ := by ring
+    rw [hval]
+    exact hd
+
+lemma powerDual_continuous {p : ℝ} (hp : 2 < p) : Continuous (powerDual p) :=
+  (show Differentiable ℝ (powerDual p) from fun x => (powerDual_hasDerivAt hp x).differentiableAt).continuous
+
+lemma powerDual_mul_self {p : ℝ} (hp : 2 < p) (x : ℝ) :
+    powerDual p x*x = |x|^p := by
+  by_cases hx : x = 0
+  · simp [hx,powerDual,Real.zero_rpow (by linarith : p ≠ 0)]
+  · change x * |x|^(p-2) * x = |x|^p
+    conv_rhs => rw [show p = (p-2)+2 by ring,Real.rpow_add (abs_pos.mpr hx),Real.rpow_two,sq_abs]
+    ring
+end TameInterpolation
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/PowerIntegration.lean b/lean/OAI/Geometry/PrescribedRicci/PowerIntegration.lean
new file mode 100644
index 0000000000000000000000000000000000000000..772f95f5209d4b0b408ca90f25e62881ff64c1ff
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/PowerIntegration.lean
@@ -0,0 +1,76 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.GlobalKahlerIntegral
+import OAI.Geometry.PrescribedRicci.PowerDual
+
+namespace OAI
+
+/-! Power Integration. -/
+
+section
+
+ 
+noncomputable section
+open Set Filter Topology MeasureTheory
+open scoped ContDiff
+namespace TameInterpolation
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+
+def dir (v : E) (f : E → ℝ) (x : E) : ℝ := fderiv ℝ f x v
+
+omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] in
+lemma dir_smooth (v : E) {f : E → ℝ} (hf : ContDiff ℝ ∞ f) : ContDiff ℝ ∞ (dir v f) :=
+  (hf.fderiv_right (by simp)).clm_apply contDiff_const
+
+omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] in
+lemma dir_compact (v : E) {f : E → ℝ} (hf : HasCompactSupport f) :
+    HasCompactSupport (dir v f) :=
+  (hf.fderiv ℝ).comp_left (g:=fun L : E →L[ℝ] ℝ => L v) (by simp : (fun L : E →L[ℝ] ℝ => L v) 0 = 0)
+
+lemma power_dir_integration {p : ℝ} (hp : 2 < p) (v : E)
+    {f : E → ℝ} (hf : ContDiff ℝ ∞ f) (hc : HasCompactSupport f) :
+    (∫ x, |dir v f x|^p) = -(p-1) *
+      ∫ x, f x * |dir v f x|^(p-2) * dir v (dir v f) x := by
+  let a := dir v f
+  let b := dir v a
+  have ha : ContDiff ℝ ∞ a := dir_smooth v hf
+  have hb : ContDiff ℝ ∞ b := dir_smooth v ha
+  have hac : HasCompactSupport a := dir_compact v hc
+  have hbc : HasCompactSupport b := dir_compact v hac
+  have hpow : Continuous (fun x => |a x|^(p-2)) := ha.continuous.abs.rpow_const (fun _ => Or.inr (by linarith))
+  have hdual : Continuous (fun x => powerDual p (a x)) := (powerDual_continuous hp).comp ha.continuous
+  have hdval (x : E) : fderiv ℝ (fun x => powerDual p (a x)) x v = (p-1)*|a x|^(p-2)*b x := by
+    have h := (powerDual_hasDerivAt hp (a x)).hasFDerivAt.comp x (ha.differentiable (by simp) x).hasFDerivAt
+    have he := congrArg (fun L : E →L[ℝ] ℝ => L v) h.fderiv
+    dsimp only [Function.comp_def] at he
+    rw [he]
+    simp only [ContinuousLinearMap.comp_apply]
+    change (fderiv ℝ a x v) * ((p-1)*|a x|^(p-2)) = _
+    exact mul_comm _ _
+  have h1 : Integrable (fun x => fderiv ℝ f x v * powerDual p (a x)) :=
+    (ha.continuous.mul hdual).integrable_of_hasCompactSupport hac.mul_right
+  have h2 : Integrable (fun x => f x * fderiv ℝ (fun x => powerDual p (a x)) x v) := by
+    simp_rw [hdval]
+    exact (hf.continuous.mul ((continuous_const.mul hpow).mul hb.continuous)).integrable_of_hasCompactSupport hc.mul_right
+  have h3 : Integrable (fun x => f x * powerDual p (a x)) :=
+    (hf.continuous.mul hdual).integrable_of_hasCompactSupport hc.mul_right
+  have he := integral_mul_fderiv_eq_neg_fderiv_mul_of_integrable h1 h2 h3
+    (fun x _ => hf.differentiable (by simp) x)
+    (fun x _ => ((powerDual_hasDerivAt hp (a x)).differentiableAt.comp x (ha.differentiable (by simp) x)))
+  have hright : (fun x => fderiv ℝ f x v * powerDual p (a x)) = (fun x => |a x|^p) := by
+    funext x
+    exact (mul_comm _ _).trans (powerDual_mul_self hp (a x))
+  have hleft : (fun x => f x * fderiv ℝ (fun x => powerDual p (a x)) x v) =
+      (fun x => (p-1)*(f x*|a x|^(p-2)*b x)) := by
+    funext x
+    rw [hdval]
+    ring
+  rw [hright,hleft,integral_const_mul] at he
+  dsimp only [a,b] at he
+  linarith
+end TameInterpolation
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/PrescribedPotential.lean b/lean/OAI/Geometry/PrescribedRicci/PrescribedPotential.lean
new file mode 100644
index 0000000000000000000000000000000000000000..723185aa4dab98cd9eefa1f47e74a7b60ada05d5
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/PrescribedPotential.lean
@@ -0,0 +1,34 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.DimensionOnePotential
+import OAI.Geometry.PrescribedRicci.DimensionZeroPotential
+import OAI.Geometry.PrescribedRicci.PositivePathClosedness
+
+namespace OAI
+
+/-! Prescribed Potential. -/
+
+section
+
+ 
+
+noncomputable section
+namespace Anticanonical.SourceSmooth
+
+theorem prescribedPotentialViaEnergy (d : ℕ) (X : Type) [TopologicalSpace X] [T2Space X]
+    [CompactSpace X] [ConnectedSpace X] (A : ComplexAtlas d X)
+    (P : A.ProjectiveEmbedding) (h : SemipositiveAnticanonicalMetric A) :
+    HasPrescribedVolumePotential P h := by
+  rcases d with _ | d
+  · obtain ⟨φ,b,hφ⟩ := volumePath_zeroDim_one (projectiveBackground P) h
+    exact volumePath_one P h φ b hφ
+  rcases d with _ | d
+  · exact prescribedPotential_dimensionOne X A P h
+  · obtain ⟨φ,b,hφ⟩ := volumePath_one_exists (projectiveBackground P) h (by omega)
+    exact volumePath_one P h φ b hφ
+
+end Anticanonical.SourceSmooth
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/PrescribedRicci.lean b/lean/OAI/Geometry/PrescribedRicci/PrescribedRicci.lean
new file mode 100644
index 0000000000000000000000000000000000000000..b8fe4956670c4c506c5374b649a3b73330416dd6
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/PrescribedRicci.lean
@@ -0,0 +1,182 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.PrescribedPotential
+
+namespace OAI
+
+/-! Anticanonical sections and semipositive prescribed Ricci curvature. -/
+
+noncomputable section
+open Set
+open scoped ContDiff
+
+namespace Anticanonical.ComplexAtlas
+
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] (A : ComplexAtlas d X)
+
+ 
+
+def anticanonicalSections (m : ℤ) :
+    Submodule ℂ (Fin A.count → Coordinates d → ℂ) where
+  carrier := {f |
+    (∀ i, DifferentiableOn ℂ (f i) (A.chart i).target) ∧
+    (∀ i z, z ∉ (A.chart i).target → f i z = 0) ∧
+    (∀ i j x, x ∈ (A.chart i).source → x ∈ (A.chart j).source →
+      f j (A.chart j x) = A.jacobian i j x ^ m * f i (A.chart i x))}
+  zero_mem' := by
+    refine ⟨fun _ => differentiableOn_const _, fun _ _ _ => rfl, ?_⟩
+    intro i j x hi hj
+    simp
+  add_mem' := by
+    intro f g hf hg
+    refine ⟨fun i => (hf.1 i).add (hg.1 i), ?_, ?_⟩
+    · intro i z hz
+      change f i z + g i z = 0
+      rw [hf.2.1 i z hz, hg.2.1 i z hz, add_zero]
+    · intro i j x hi hj
+      change f j (A.chart j x) + g j (A.chart j x) =
+        A.jacobian i j x ^ m * (f i (A.chart i x) + g i (A.chart i x))
+      rw [hf.2.2 i j x hi hj, hg.2.2 i j x hi hj, mul_add]
+  smul_mem' := by
+    intro c f hf
+    refine ⟨fun i => (hf.1 i).const_smul c, ?_, ?_⟩
+    · intro i z hz
+      change c • f i z = 0
+      rw [hf.2.1 i z hz, smul_zero]
+    · intro i j x hi hj
+      change c * f j (A.chart j x) = A.jacobian i j x ^ m * (c * f i (A.chart i x))
+      rw [hf.2.2 i j x hi hj]
+      ring
+
+ 
+
+abbrev H0Anticanonical (m : ℤ) := ↥(A.anticanonicalSections m)
+
+end Anticanonical.ComplexAtlas
+
+namespace Anticanonical.ComplexAtlas
+
+variable {d : ℕ} {X : Type*} [TopologicalSpace X]
+    {A : ComplexAtlas d X} {m : ℤ}
+
+ 
+
+theorem section_eq_zero_iff (s : A.H0Anticanonical m) :
+    s = 0 ↔ ∀ i z, z ∈ (A.chart i).target → s.val i z = 0 := by
+  constructor
+  · intro hs
+    subst hs
+    intro i z hz
+    rfl
+  · intro hs
+    apply Subtype.ext
+    funext i z
+    by_cases hz : z ∈ (A.chart i).target
+    · exact hs i z hz
+    · exact s.property.2.1 i z hz
+
+ 
+
+theorem section_ne_zero_iff (s : A.H0Anticanonical m) :
+    s ≠ 0 ↔ ∃ i z, z ∈ (A.chart i).target ∧ s.val i z ≠ 0 := by
+  rw [ne_eq, section_eq_zero_iff]
+  simp only [not_forall, exists_prop]
+
+ 
+
+theorem section_ne_zero_iff_on_source (s : A.H0Anticanonical m) :
+    s ≠ 0 ↔ ∃ (x : X) (i : Fin A.count),
+      x ∈ (A.chart i).source ∧ s.val i (A.chart i x) ≠ 0 := by
+  rw [section_ne_zero_iff]
+  constructor
+  · rintro ⟨i,z,hz,hs⟩
+    refine ⟨(A.chart i).symm z,i,(A.chart i).mapsTo_symm hz,?_⟩
+    simpa only [(A.chart i).right_inv hz] using hs
+  · rintro ⟨x,i,hx,hs⟩
+    exact ⟨i,A.chart i x,(A.chart i).mapsTo hx,hs⟩
+
+ 
+
+def unitSectionOfTrivialTransitions (A : ComplexAtlas d X) (m : ℤ)
+    (hJ : ∀ i j x, x ∈ (A.chart i).source → x ∈ (A.chart j).source →
+      A.jacobian i j x ^ m = 1) : A.H0Anticanonical m := by
+  classical
+  refine ⟨fun i z => if z ∈ (A.chart i).target then 1 else 0, ?_, ?_, ?_⟩
+  · intro i
+    apply (differentiableOn_const (1 : ℂ)).congr
+    intro z hz
+    simp only [ite_eq_left hz]
+  · intro i z hz
+    simp only [ite_eq_right hz]
+  · intro i j x hi hj
+    simp only [ite_eq_left ((A.chart i).mapsTo hi), ite_eq_left ((A.chart j).mapsTo hj),
+      hJ i j x hi hj, mul_one]
+
+theorem unitSectionOfTrivialTransitions_ne_zero [Nonempty X]
+    (A : ComplexAtlas d X) (m : ℤ)
+    (hJ : ∀ i j x, x ∈ (A.chart i).source → x ∈ (A.chart j).source →
+      A.jacobian i j x ^ m = 1) : A.unitSectionOfTrivialTransitions m hJ ≠ 0 := by
+  classical
+  obtain ⟨x⟩ := ‹Nonempty X›
+  obtain ⟨i,hi⟩ := A.covers x
+  apply (section_ne_zero_iff_on_source _).mpr
+  refine ⟨x,i,hi,?_⟩
+  simp only [unitSectionOfTrivialTransitions, ite_eq_left ((A.chart i).mapsTo hi), ne_eq,
+    one_ne_zero, not_false_eq_true]
+
+ 
+
+def degreeZeroSection (A : ComplexAtlas d X) : A.H0Anticanonical 0 :=
+  A.unitSectionOfTrivialTransitions 0 (by intros; exact zpow_zero _)
+
+ 
+@[simp] theorem jacobian_dim_zero (A : ComplexAtlas 0 X) (i j : Fin A.count) (x : X) :
+    A.jacobian i j x = 1 := by
+  simp [jacobian, Matrix.det_isEmpty]
+
+ 
+
+def dimensionZeroSection (A : ComplexAtlas 0 X) (m : ℤ) : A.H0Anticanonical m :=
+  A.unitSectionOfTrivialTransitions m (by intros; simp)
+
+theorem dimensionZeroSection_ne_zero [Nonempty X] (A : ComplexAtlas 0 X) (m : ℤ) :
+    A.dimensionZeroSection m ≠ 0 :=
+  A.unitSectionOfTrivialTransitions_ne_zero m _
+
+end Anticanonical.ComplexAtlas
+
+namespace Anticanonical
+
+ 
+
+theorem headline_dimensionZero (X : Type) [TopologicalSpace X] [Nonempty X]
+    (A : ComplexAtlas 0 X) :
+    ∃ m : ℤ, 0 < m ∧ ∃ s : A.H0Anticanonical m, s ≠ 0 := by
+  exact ⟨1, by norm_num, A.dimensionZeroSection 1, A.dimensionZeroSection_ne_zero 1⟩
+
+end Anticanonical
+
+namespace Anticanonical.SourceSmooth
+
+ 
+
+theorem prescribedRicciFromSemipositive (d : ℕ) (X : Type)
+    [TopologicalSpace X] [T2Space X] [CompactSpace X] [ConnectedSpace X]
+    (A : ComplexAtlas d X) (P : A.ProjectiveEmbedding)
+    (h : SemipositiveAnticanonicalMetric A) :
+    ∃ g : KaehlerMetric A,
+      (∀ i z, z ∈ (A.chart i).target →
+        ∀ v : Coordinates d, g.ricciQuadratic i z v = levi (h.weight i) z v) ∧
+      (∀ i z, z ∈ (A.chart i).target →
+        ∀ v : Coordinates d, 0 ≤ g.ricciQuadratic i z v) := by
+  obtain ⟨φ,c,hc,hp,he⟩ := prescribedPotentialViaEnergy d X A P h
+  obtain ⟨g,_,hRic⟩ := prescribedRicci_of_potential P h φ hp hc he
+  refine ⟨g,hRic,?_⟩
+  intro i z hz v
+  rw [hRic i z hz v]
+  exact h.semipositive i z hz v
+
+end Anticanonical.SourceSmooth
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/RoughCutoff.lean b/lean/OAI/Geometry/PrescribedRicci/RoughCutoff.lean
new file mode 100644
index 0000000000000000000000000000000000000000..02dc1a566c4ffa3264fb096ad2eafcd5b16b27bf
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/RoughCutoff.lean
@@ -0,0 +1,112 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.ChartSobolevAlgebra
+
+namespace OAI
+
+/-! Rough Cutoff. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory TemperedDistribution LineDeriv
+open scoped SchwartzMap BoundedContinuousFunction ContDiff Classical ComplexOrder MatrixOrder
+namespace SobolevChart
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+
+def cutoffTerm (k : ℕ) (hk : Module.finrank ℝ E < k) (κ : 𝓢(E,ℂ))
+    (v w : E) (u : L2 E) : L2 E :=
+  productCoord k hk (schwartzCoord (k:ℝ) (∂_{v} κ)) (derivativeCoord ((k:ℝ)+1) w u) +
+  productCoord k hk (schwartzCoord (k:ℝ) (∂_{w} κ)) (derivativeCoord ((k:ℝ)+1) v u) +
+  productCoord k hk (schwartzCoord (k:ℝ) (∂_{v} (∂_{w} κ)))
+    (lowerCoord ((k:ℝ)+1) (k:ℝ) (by linarith) u)
+
+lemma cutoffTerm_continuous (k : ℕ) (hk : Module.finrank ℝ E < k) (κ : 𝓢(E,ℂ))
+    (v w : E) : Continuous (cutoffTerm k hk κ v w) := by
+  unfold cutoffTerm
+  fun_prop
+
+lemma cutoffTerm_schwartz (k : ℕ) (hk : Module.finrank ℝ E < k) (κ f : 𝓢(E,ℂ))
+    (v w : E) : cutoffTerm k hk κ v w (schwartzCoord ((k:ℝ)+1) f) =
+    schwartzCoord (k:ℝ)
+      (SchwartzMap.smulLeftCLM ℂ (∂_{v} κ : 𝓢(E,ℂ)) (∂_{w} f) +
+        SchwartzMap.smulLeftCLM ℂ (∂_{w} κ : 𝓢(E,ℂ)) (∂_{v} f) +
+        SchwartzMap.smulLeftCLM ℂ (∂_{v} (∂_{w} κ) : 𝓢(E,ℂ)) f) := by
+  simp only [cutoffTerm,derivativeCoord_schwartz,add_sub_cancel_right,
+    lowerCoord_schwartz,productCoord_schwartz,schwartzCoord_add]
+
+end SobolevChart
+namespace MetricLocalization
+open EllipticKernel SobolevChart FrozenPoisson
+variable {n : ℕ}
+
+def roughCutoff (H : Matrix (Fin n) (Fin n) ℂ) (k : ℕ)
+    (hk : Module.finrank ℝ (EC n) < k) (κ : 𝓢(EC n,ℂ))
+    (a : BasisIndex n → BasisIndex n → L2 (EC n)) (u : L2 (EC n)) : L2 (EC n) :=
+  ∑ i, ∑ j, (
+    (traceBilin H (rankTwo (stdOrthonormalBasis ℝ (EC n) i)
+      (stdOrthonormalBasis ℝ (EC n) j)) : ℂ) •
+        cutoffTerm k hk κ (stdOrthonormalBasis ℝ (EC n) i) (stdOrthonormalBasis ℝ (EC n) j) u +
+    productCoord k hk (a i j)
+      (cutoffTerm k hk κ (stdOrthonormalBasis ℝ (EC n) i) (stdOrthonormalBasis ℝ (EC n) j) u))
+
+lemma roughCutoff_continuous (H : Matrix (Fin n) (Fin n) ℂ) (k : ℕ)
+    (hk : Module.finrank ℝ (EC n) < k) (κ : 𝓢(EC n,ℂ)) :
+    Continuous (fun p : (BasisIndex n → BasisIndex n → L2 (EC n)) × L2 (EC n) =>
+      roughCutoff H k hk κ p.1 p.2) := by
+  unfold roughCutoff
+  apply continuous_finsetSum
+  intro i _
+  apply continuous_finsetSum
+  intro j _
+  let T := (BasisIndex n → BasisIndex n → L2 (EC n)) × L2 (EC n)
+  have hc : Continuous (fun p : T => cutoffTerm k hk κ
+      (stdOrthonormalBasis ℝ (EC n) i) (stdOrthonormalBasis ℝ (EC n) j) p.2) :=
+    (cutoffTerm_continuous k hk κ _ _).comp continuous_snd
+  have ha : Continuous (fun p : T => p.1 i j) :=
+    (continuous_apply j).comp ((continuous_apply i).comp continuous_fst)
+  exact (hc.const_smul (traceBilin H (rankTwo (stdOrthonormalBasis ℝ (EC n) i)
+    (stdOrthonormalBasis ℝ (EC n) j)) : ℂ)).add (((productCoord k hk).continuous.comp ha).clm_apply hc)
+
+lemma roughCutoff_schwartz (H : Matrix (Fin n) (Fin n) ℂ) (k : ℕ)
+    (hk : Module.finrank ℝ (EC n) < k) (κ f : 𝓢(EC n,ℂ))
+    (a : SmoothCoefficients (BasisIndex n) (EC n)) :
+    roughCutoff H k hk κ (fun i j => schwartzCoord (k:ℝ) (a i j))
+      (schwartzCoord ((k:ℝ)+1) f) =
+    schwartzCoord (k:ℝ) (schwartzCutoffError (extendedCoefficient H (coefficientBCF a)) κ f) := by
+  rw [schwartzCutoffError_firstOrder _
+    (extendedCoefficient_temperate H _ (fun i j => (a i j).hasTemperateGrowth))]
+  simp only [roughCutoff,cutoffTerm_schwartz,productCoord_schwartz,
+    schwartzCoord_sum]
+  apply Finset.sum_congr rfl
+  intro i _
+  apply Finset.sum_congr rfl
+  intro j _
+  rw [← schwartzCoord_smul,← schwartzCoord_add]
+  congr 1
+  ext x
+  simp only [add_apply,smul_apply,
+    SchwartzMap.smulLeftCLM_apply_apply
+      (SchwartzMap.smulLeftCLM ℂ _ _).hasTemperateGrowth]
+  simp only [add_apply,smul_eq_mul,
+    SchwartzMap.smulLeftCLM_apply_apply (extendedCoefficient_temperate H (coefficientBCF a)
+      (fun i j => (a i j).hasTemperateGrowth) i j),
+    SchwartzMap.smulLeftCLM_apply_apply (a i j).hasTemperateGrowth,
+    SchwartzMap.smulLeftCLM_apply_apply (∂_{stdOrthonormalBasis ℝ (EC n) i} κ).hasTemperateGrowth,
+    SchwartzMap.smulLeftCLM_apply_apply (∂_{stdOrthonormalBasis ℝ (EC n) j} κ).hasTemperateGrowth,
+    SchwartzMap.smulLeftCLM_apply_apply (∂_{stdOrthonormalBasis ℝ (EC n) i}
+      (∂_{stdOrthonormalBasis ℝ (EC n) j} κ)).hasTemperateGrowth]
+  simp only [extendedCoefficient,BoundedContinuousFunction.add_apply,BoundedContinuousFunction.const_apply]
+  change _ = ((traceBilin H (rankTwo _ _) : ℂ)+a i j x) * _ * _ +
+    ((traceBilin H (rankTwo _ _) : ℂ)+a i j x) * _ * _ +
+    ((traceBilin H (rankTwo _ _) : ℂ)+a i j x) * _ * _
+  ring
+
+end MetricLocalization
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/RoughEllipticGain.lean b/lean/OAI/Geometry/PrescribedRicci/RoughEllipticGain.lean
new file mode 100644
index 0000000000000000000000000000000000000000..6d8b9f0873285ee620ce13d94a722a9687bbb935
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/RoughEllipticGain.lean
@@ -0,0 +1,77 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.TameRoughInverse
+
+namespace OAI
+
+/-! Rough Elliptic Gain. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory TemperedDistribution LineDeriv
+open scoped SchwartzMap BoundedContinuousFunction ContDiff Classical ComplexOrder MatrixOrder
+namespace FrozenPoisson
+open SobolevChart EllipticKernel
+variable {n : ℕ} {ι : Type*} [Fintype ι]
+
+lemma parameterResolvent_forward (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    {t : ℝ} (ht : 0 < t) (u : 𝓢'(EC n,ℂ)) :
+    parameterResolvent H t (parameterForward H t u) = u := by
+  rw [parameterResolvent,parameterForward,
+    fourierMultiplierCLM_fourierMultiplierCLM_apply (parameterForward_temperate H hH t)
+      (parameterSymbol_temperate H hH ht)]
+  have he : (fun ξ => ((t^2+symbol H ξ : ℝ):ℂ)) * parameterSymbol H t = fun _ => (1:ℂ) := by
+    funext ξ
+    simp only [Pi.mul_apply,parameterSymbol,← Complex.ofReal_mul]
+    have hp := symbol_nonneg H hH ξ
+    rw [mul_inv_cancel₀ (by positivity : t^2+symbol H ξ ≠ 0),Complex.ofReal_one]
+  rw [he,fourierMultiplierCLM_const]
+  simp
+
+lemma fixed_of_differential_equation (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (t : ℝ) (ht : 1 ≤ t) (v : ι → EC n) (a : ι → ι → EC n →ᵇ ℂ)
+    (u f : L2 (EC n))
+    (he : (t^2:ℂ) • realize 2 u - frozenDifferential H (realize 2 u) -
+      ((perturbation v a u : L2 (EC n)) : 𝓢'(EC n,ℂ)) = (f : 𝓢'(EC n,ℂ))) :
+    u = parameterHilbert H hH t ht (f+perturbation v a u) := by
+  apply realize_injective 2
+  have hr := parameterHilbert_realize H hH t ht 0 (f+perturbation v a u)
+  simp only [zero_add] at hr
+  rw [hr]
+  have hf : realize 0 (f+perturbation v a u) =
+      (f : 𝓢'(EC n,ℂ)) + ((perturbation v a u : L2 (EC n)) : 𝓢'(EC n,ℂ)) := by
+    simp [realize, ← Lp.toTemperedDistributionCLM_apply, map_add]
+  rw [hf,← sub_eq_iff_eq_add.mp he,← parameterForward_eq H hH]
+  exact (parameterResolvent_forward H hH (lt_of_lt_of_le zero_lt_one ht) _).symm
+
+ 
+
+theorem rough_equation_memSobolev (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (t : ℝ) (ht : 1 ≤ t) (v : ι → EC n) (k : ℕ)
+    (hk : Module.finrank ℝ (EC n)+1 < k)
+    (hs : (Module.finrank ℝ (EC n):ℝ) < 2*(k:ℝ))
+    (a : ι → ι → L2 (EC n)) (f u : L2 (EC n))
+    (hsmall : perturbationBound v (fun i j => strongEmbedding (k:ℝ) hs (a i j))*ellipticBound H hH < 1)
+    (he : (t^2:ℂ) • realize 2 u - frozenDifferential H (realize 2 u) -
+      ((perturbation v (fun i j => strongEmbedding (k:ℝ) hs (a i j)) u : L2 (EC n)) : 𝓢'(EC n,ℂ)) =
+        realize (k:ℝ) f) :
+    MemSobolev ((k:ℝ)+2) 2 (realize 2 u) := by
+  let f0 := lowerCoord (k:ℝ) 0 (Nat.cast_nonneg k) f
+  have hf0 : (f0 : 𝓢'(EC n,ℂ)) = realize (k:ℝ) f := by
+    have hh := realize_lowerCoord (k:ℝ) 0 (Nat.cast_nonneg k) f
+    simpa [realize,f0] using hh
+  have hu := fixed_of_differential_equation H hH t ht v _ u f0 (he.trans hf0.symm)
+  have heq := ParameterRegularity.fixed_unique H hH t ht v _ hsmall u
+    (parameterLocal H hH t ht v _ hsmall f0) f0 hu
+    (ParameterRegularity.parameterLocal_fixed H hH t ht v _ hsmall f0)
+  rw [heq]
+  exact ParameterRegularity.rough_local_memSobolev H hH t ht v k hk hs a f hsmall
+
+end FrozenPoisson
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/SchwartzHessianWords.lean b/lean/OAI/Geometry/PrescribedRicci/SchwartzHessianWords.lean
new file mode 100644
index 0000000000000000000000000000000000000000..4d72f9c3a94c51021dbd7604214c72fb0e478f25
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/SchwartzHessianWords.lean
@@ -0,0 +1,130 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.ComplexHessian
+import OAI.Geometry.PrescribedPotential.CutoffHigher
+import OAI.Geometry.PrescribedRicci.UniformSchwartz
+
+namespace OAI
+
+/-! Schwartz Hessian Words. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory LineDeriv
+open scoped ContDiff SchwartzMap Classical
+namespace SobolevChart
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+
+lemma schwartzWord_deriv (ws : List E) (v : E) (f : 𝓢(E,ℂ)) :
+    schwartzWord ws (∂_{v} f) = ∂_{v} (schwartzWord ws f) := by
+  induction ws with
+  | nil => rfl
+  | cons a ws ih =>
+    change ∂_{a} (schwartzWord ws (∂_{v} f)) = ∂_{v} (∂_{a} (schwartzWord ws f))
+    rw [ih,schwartz_deriv_comm]
+
+def wordSecondError (κ : 𝓢(E,ℂ)) (ws : List E) (a b : E) :
+    𝓢(E,ℂ) → 𝓢(E,ℂ) := fun f =>
+  SchwartzMap.smulLeftCLM ℂ (∂_{a} κ : 𝓢(E,ℂ)) (∂_{b} (schwartzWord ws f)) +
+  SchwartzMap.smulLeftCLM ℂ (∂_{b} κ : 𝓢(E,ℂ)) (∂_{a} (schwartzWord ws f)) +
+  SchwartzMap.smulLeftCLM ℂ (∂_{a} (∂_{b} κ) : 𝓢(E,ℂ)) (schwartzWord ws f)
+
+lemma wordSecondError_identity (κ f : 𝓢(E,ℂ)) (ws : List E) (a b : E) :
+    ∂_{a} (∂_{b} (SchwartzMap.smulLeftCLM ℂ κ (schwartzWord ws f))) =
+      SchwartzMap.smulLeftCLM ℂ κ (schwartzWord ws (∂_{a} (∂_{b} f))) +
+        wordSecondError κ ws a b f := by
+  rw [schwartz_second_product,schwartzWord_deriv,schwartzWord_deriv]
+  rfl
+
+lemma wordSecondError_bound (κ : 𝓢(E,ℂ)) (ws : List E) (a b : E) :
+    CoreBound ((ws.length:ℝ)+1) 0 (wordSecondError κ ws a b) := by
+  have hw : CoreBound ((ws.length:ℝ)+1) 1 (schwartzWord ws) := coreBound_word ws (by linarith)
+  have hd (v : E) : CoreBound ((ws.length:ℝ)+1) 0 (fun f => ∂_{v} (schwartzWord ws f)) :=
+    (coreBound_deriv v (s:=1) (t:=0) (by norm_num)).comp hw
+  have h1 : CoreBound ((ws.length:ℝ)+1) 0 (fun f =>
+      SchwartzMap.smulLeftCLM ℂ (∂_{a} κ : 𝓢(E,ℂ)) (∂_{b} (schwartzWord ws f))) :=
+    (coreBound_product_zero (∂_{a} κ)).comp (hd b)
+  have h2 : CoreBound ((ws.length:ℝ)+1) 0 (fun f =>
+      SchwartzMap.smulLeftCLM ℂ (∂_{b} κ : 𝓢(E,ℂ)) (∂_{a} (schwartzWord ws f))) :=
+    (coreBound_product_zero (∂_{b} κ)).comp (hd a)
+  have h3 : CoreBound ((ws.length:ℝ)+1) 0 (fun f =>
+      SchwartzMap.smulLeftCLM ℂ (∂_{a} (∂_{b} κ) : 𝓢(E,ℂ)) (schwartzWord ws f)) :=
+    (coreBound_product_zero (∂_{a} (∂_{b} κ))).comp
+      (coreBound_word ws (by linarith : (0:ℝ) ≤ (ws.length:ℝ)+1-ws.length))
+  exact (h1.add h2).add h3
+
+lemma UniformSobolev.raise_one {Z : Type*} {s : ℝ} {F : Z → 𝓢(E,ℂ)}
+    (hF : UniformSobolev s F)
+    (hD : ∀ j, UniformSobolev s (fun z => ∂_{stdOrthonormalBasis ℝ E j} (F z))) :
+    UniformSobolev (s+1) F := by
+  obtain ⟨C,hC,hc⟩ := hF
+  choose B hB hb using hD
+  let c : ℝ := ‖(((2*Real.pi)^2:ℝ):ℂ)⁻¹‖
+  refine ⟨‖raiseZero (E:=E) s‖*C+c*∑ j, ‖raiseDeriv s (stdOrthonormalBasis ℝ E j)‖*B j,
+    add_nonneg (mul_nonneg (norm_nonneg _) hC)
+      (mul_nonneg (norm_nonneg _) (Finset.sum_nonneg fun j _ => mul_nonneg (norm_nonneg _) (hB j))),fun z => ?_⟩
+  rw [schwartzCoord_raise_one]
+  apply (norm_sub_le _ _).trans
+  rw [norm_smul]
+  apply add_le_add
+  · exact ((raiseZero s).le_opNorm _).trans (mul_le_mul_of_nonneg_left (hc z) (norm_nonneg _))
+  · apply mul_le_mul_of_nonneg_left _ (norm_nonneg _)
+    exact (norm_sum_le _ _).trans (Finset.sum_le_sum (fun j _ =>
+      ((raiseDeriv s (stdOrthonormalBasis ℝ E j)).le_opNorm _).trans
+        (mul_le_mul_of_nonneg_left (hb j z) (norm_nonneg _))))
+
+lemma UniformSobolev.of_words {Z : Type*} (k : ℕ) {s : ℝ} {F : Z → 𝓢(E,ℂ)}
+    (h : ∀ ws : List E, ws.length ≤ k → UniformSobolev s (fun z => schwartzWord ws (F z))) :
+    UniformSobolev (s+k) F := by
+  induction k generalizing F with
+  | zero => simpa [schwartzWord] using h [] (by simp)
+  | succ k ih =>
+    have hv := ih (fun ws hw => h ws (by omega))
+    have hd (j : Fin (Module.finrank ℝ E)) := ih
+      (F:=fun z => ∂_{stdOrthonormalBasis ℝ E j} (F z)) (fun ws hw => by
+        have hh := h (ws++[stdOrthonormalBasis ℝ E j]) (by simp only [List.length_append,List.length_cons,List.length_nil]; omega)
+        simpa only [schwartzWord_append,schwartzWord,ContinuousLinearMap.comp_apply,
+          ContinuousLinearMap.id_apply,lineDerivOpCLM_apply] using hh)
+    convert hv.raise_one hd using 1
+    push_cast
+    ring
+end SobolevChart
+
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ}
+
+lemma hessianEntrySchwartz_word (ws : List (EC d)) (i j : Fin d) (f : 𝓢(EC d,ℂ)) :
+    hessianEntrySchwartz i j (schwartzWord ws f) = schwartzWord ws (hessianEntrySchwartz i j f) := by
+  simp only [hessianEntrySchwartz,smul_apply,add_apply,
+    ContinuousLinearMap.comp_apply,lineDerivOpCLM_apply,map_add,map_smul,schwartzWord_deriv]
+
+def wordHessianError (κ : 𝓢(EC d,ℂ)) (ws : List (EC d)) (i j : Fin d)
+    (f : 𝓢(EC d,ℂ)) : 𝓢(EC d,ℂ) :=
+  (1/4:ℂ) • (wordSecondError κ ws (hessianDirection i) (hessianDirection j) f +
+    wordSecondError κ ws (hessianIDirection i) (hessianIDirection j) f +
+    Complex.I • (wordSecondError κ ws (hessianIDirection i) (hessianDirection j) f +
+      (-1:ℂ) • wordSecondError κ ws (hessianDirection i) (hessianIDirection j) f))
+
+lemma wordHessianError_identity (κ f : 𝓢(EC d,ℂ)) (ws : List (EC d)) (i j : Fin d) :
+    hessianEntrySchwartz i j (SchwartzMap.smulLeftCLM ℂ κ (schwartzWord ws f)) =
+      SchwartzMap.smulLeftCLM ℂ κ (schwartzWord ws (hessianEntrySchwartz i j f)) +
+        wordHessianError κ ws i j f := by
+  simp only [hessianEntrySchwartz,wordHessianError,smul_apply,
+    add_apply,ContinuousLinearMap.comp_apply,lineDerivOpCLM_apply,
+    wordSecondError_identity,map_add,map_smul]
+  module
+
+lemma wordHessianError_bound (κ : 𝓢(EC d,ℂ)) (ws : List (EC d)) (i j : Fin d) :
+    CoreBound ((ws.length:ℝ)+1) 0 (wordHessianError κ ws i j) :=
+  ((wordSecondError_bound κ ws _ _).add (wordSecondError_bound κ ws _ _) |>.add
+    (((wordSecondError_bound κ ws _ _).add ((wordSecondError_bound κ ws _ _).smul (-1))).smul Complex.I)).smul (1/4)
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/SmoothLocalInverse.lean b/lean/OAI/Geometry/PrescribedRicci/SmoothLocalInverse.lean
new file mode 100644
index 0000000000000000000000000000000000000000..b265dcfa3b2500b3429bd8059f380606f27f909c
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/SmoothLocalInverse.lean
@@ -0,0 +1,106 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.SobolevExponential
+import OAI.Geometry.PrescribedPotential.AugmentedVolume
+import OAI.Geometry.PrescribedPotential.ClosedRangeSmooth
+
+namespace OAI
+
+/-! Smooth Local Inverse. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology
+open scoped ContDiff Classical
+namespace ExponentialEvaluation
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
+lemma orbit_contDiff (L : E →L[ℝ] E) (v : E) : ContDiff ℝ ∞ (orbit L v) := by
+  have he : ContDiff ℝ ∞ (NormedSpace.exp : (E →L[ℝ] E) → (E →L[ℝ] E)) :=
+    contDiff_iff_contDiffAt.mpr (fun w => (NormedSpace.exp_analytic (𝕂 := ℝ) w).contDiffAt)
+  exact (ContinuousLinearMap.apply ℝ E v).contDiff.comp
+    (he.comp (contDiff_id.smul contDiff_const))
+end ExponentialEvaluation
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  [ConnectedSpace X] {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+local instance inverseRealNormedGroup (s : ℝ) : NormedAddCommGroup (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedAddCommGroup
+local instance inverseRealNormedSpace (s : ℝ) : NormedSpace ℝ (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedSpace
+local instance inverseRealTopologicalGroup (s : ℝ) : IsTopologicalAddGroup (D.localizers.RealSobolev s) :=
+  Submodule.isTopologicalAddGroup _
+local instance inverseRealContinuousSMul (s : ℝ) : ContinuousSMul ℝ (D.localizers.RealSobolev s) :=
+  SMulMemClass.continuousSMul _
+
+omit [ConnectedSpace X] in
+lemma exponentialDensity_contDiff (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (F : RealSmooth A) : ContDiff ℝ ∞ (D.exponentialDensity k hk F) :=
+  ExponentialEvaluation.orbit_contDiff _ _
+
+ 
+theorem exists_volume_localInverse (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) (x₀ : X) :
+    ∃ R : (D.localizers.RealSobolev (k : ℝ) × ℝ) →
+        (D.localizers.RealSobolev ((k : ℝ)+2) × ℝ),
+      R (D.realConstants (k : ℝ) 1,0) = 0 ∧
+      ContDiffAt ℝ ∞ R (D.realConstants (k : ℝ) 1,0) ∧
+      (∀ᶠ y in 𝓝 (D.realConstants (k : ℝ) 1,0), D.augmentedVolume k hk x₀ (R y) = y) ∧
+      (∀ᶠ u in 𝓝 0, R (D.augmentedVolume k hk x₀ u) = u) := by
+  obtain ⟨m,hm,he⟩ := D.exists_completedError_small
+  obtain ⟨P⟩ := D.localizers.constantProjection_exists
+  have hs : (Module.finrank ℝ (EC d) : ℝ) < 2*((k : ℝ)+2) := by
+    have hh : (Module.finrank ℝ (EC d) : ℝ) < (k : ℝ) := by exact_mod_cast hk
+    linarith [Nat.cast_nonneg (α := ℝ) k]
+  let L := D.realPoissonEquiv m hm he P k hs x₀
+  have hL : HasFDerivAt (D.augmentedVolume k hk x₀) (L : _ →L[ℝ] _) 0 :=
+    (D.augmentedVolume_hasStrictFDerivAt_zero k hk x₀).hasFDerivAt
+  have hf := (D.augmentedVolume_contDiff k hk x₀).contDiffAt (x := 0)
+  have hn : (∞ : ℕ∞ω) ≠ 0 := by simp
+  let R := hf.localInverse hL hn
+  refine ⟨R,?_,?_,?_,?_⟩
+  · simpa only [D.augmentedVolume_zero] using hf.localInverse_apply_image hL hn
+  · simpa only [D.augmentedVolume_zero] using hf.to_localInverse hL hn
+  · have hh : ∀ᶠ y in 𝓝 (D.augmentedVolume k hk x₀ 0),
+        D.augmentedVolume k hk x₀ (R y) = y :=
+      (hf.hasStrictFDerivAt' hL hn).eventually_right_inverse
+    rwa [D.augmentedVolume_zero] at hh
+  · exact (hf.hasStrictFDerivAt' hL hn).eventually_left_inverse
+
+ 
+
+theorem exists_smooth_sobolev_path (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (x₀ : X) (F : RealSmooth A) :
+    ∃ γ : ℝ → D.localizers.RealSobolev ((k : ℝ)+2) × ℝ,
+      γ 0 = 0 ∧ ContDiffAt ℝ ∞ γ 0 ∧
+      ∀ᶠ t in 𝓝 0,
+        D.realVolume k hk (γ t).1 = Real.exp (γ t).2 • D.exponentialDensity k hk F t ∧
+        D.realEvaluation ((k : ℝ)+2) x₀ (γ t).1 = 0 := by
+  obtain ⟨R,hR,hr,hh,_⟩ := D.exists_volume_localInverse k hk x₀
+  let f : ℝ → D.localizers.RealSobolev (k : ℝ) × ℝ := fun t =>
+    (D.exponentialDensity k hk F t,0)
+  have hf : ContDiff ℝ ∞ f := (D.exponentialDensity_contDiff k hk F).prodMk contDiff_const
+  have hz : f 0 = (D.realConstants (k : ℝ) 1,0) := by
+    simp only [f,D.exponentialDensity_zero]
+  refine ⟨R ∘ f,by rw [Function.comp_apply,hz,hR],?_,?_⟩
+  · apply ContDiffAt.comp 0 _ hf.contDiffAt
+    rwa [hz]
+  · have ht : Tendsto f (𝓝 0) (𝓝 (D.realConstants (k : ℝ) 1,0)) := by
+      simpa only [hz] using hf.continuous.tendsto 0
+    filter_upwards [ht.eventually hh] with t ht
+    refine ⟨?_,congrArg Prod.snd ht⟩
+    have he := congrArg (fun z => Real.exp (R (f t)).2 • z.1) ht
+    change Real.exp (R (f t)).2 •
+      (Real.exp (-(R (f t)).2) • D.realVolume k hk (R (f t)).1) = _ at he
+    simpa only [Function.comp_apply, f, smul_smul, ← Real.exp_add, add_neg_cancel,
+      Real.exp_zero, one_smul] using he
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/SmoothPathLimit.lean b/lean/OAI/Geometry/PrescribedRicci/SmoothPathLimit.lean
new file mode 100644
index 0000000000000000000000000000000000000000..8bde6477ca4acd67ace5323aaf56ac6967e507c0
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/SmoothPathLimit.lean
@@ -0,0 +1,84 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.PatchCutoffs
+import OAI.Geometry.PrescribedPotential.RealSobolev
+import OAI.Geometry.PrescribedPotential.GlobalHessian
+import OAI.Geometry.PrescribedRicci.GlobalSmoothCompact
+import OAI.Geometry.PrescribedRicci.PathUniformHigher
+
+namespace OAI
+
+/-! Smooth Path Limit. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff Classical ComplexOrder
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+
+lemma pointwise_of_sobolev (f : ℕ → Smooth A) (F : Smooth A)
+    (hlim : ∀ k : ℕ, Tendsto (fun n => D.localizers.embed (k:ℝ) (f n)) atTop
+      (𝓝 (D.localizers.embed (k:ℝ) F))) (x : X) :
+    Tendsto (fun n => f n x) atTop (𝓝 (F x)) := by
+  let k := Module.finrank ℝ (EC d)+1
+  have hs : (Module.finrank ℝ (EC d):ℝ) < 2*(k:ℝ) := by
+    exact_mod_cast (by dsimp [k]; omega : Module.finrank ℝ (EC d) < 2*k)
+  have hl := ((D.localizers.strong (k:ℝ)).continuous.tendsto _).comp (hlim k)
+  simp only [Function.comp_def,D.localizers.strong_embed _ hs] at hl
+  exact ((BoundedContinuousFunction.lipschitz_eval_const x).continuous.tendsto _).comp hl
+
+lemma real_of_sobolev (f : ℕ → SmoothRealFunction A) (F : Smooth A)
+    (hlim : ∀ k : ℕ, Tendsto (fun n => D.localizers.embed (k:ℝ) (Smooth.ofReal (f n))) atTop
+      (𝓝 (D.localizers.embed (k:ℝ) F))) : ∀ x, (F x).im = 0 := by
+  intro x
+  have h := Complex.continuous_im.continuousAt.tendsto.comp (D.pointwise_of_sobolev _ F hlim x)
+  have hz : Tendsto (fun n => (Smooth.ofReal (f n) x).im) atTop (𝓝 (0:ℝ)) := tendsto_const_nhds
+  exact tendsto_nhds_unique h hz
+
+lemma hessian_of_sobolev (f : ℕ → SmoothRealFunction A) (φ : SmoothRealFunction A)
+    (hlim : ∀ k : ℕ, Tendsto (fun n => D.localizers.embed (k:ℝ) (Smooth.ofReal (f n))) atTop
+      (𝓝 (D.localizers.embed (k:ℝ) (Smooth.ofReal φ))))
+    (p : ι) {x : X} (hx : x ∈ tsupport (D.localizers.weight p : X → ℂ)) (i j : Fin d) :
+    Tendsto (fun n => (f n).hessian (D.patch p).index (A.chart (D.patch p).index x) i j)
+      atTop (𝓝 (φ.hessian (D.patch p).index (A.chart (D.patch p).index x) i j)) := by
+  have hl (k : ℕ) := ((D.completedHessian k p i j).continuous.tendsto _).comp (show
+      Tendsto (fun n => D.localizers.embed ((k:ℝ)+2) (Smooth.ofReal (f n))) atTop
+        (𝓝 (D.localizers.embed ((k:ℝ)+2) (Smooth.ofReal φ))) from by
+          have h := hlim (k+2)
+          rw [Nat.cast_add,Nat.cast_ofNat] at h
+          exact h)
+  have hh (k : ℕ) : Tendsto (fun n => D.localizers.embed (k:ℝ)
+      (D.localizedHessian p i j (Smooth.ofReal (f n)))) atTop
+      (𝓝 (D.localizers.embed (k:ℝ) (D.localizedHessian p i j (Smooth.ofReal φ)))) := by
+    simpa only [Function.comp_def,D.completedHessian_embed] using hl k
+  have h := D.pointwise_of_sobolev _ _ hh x
+  simpa only [D.localizedHessian_source p i j _ hx] using h
+
+variable [ConnectedSpace X]
+theorem path_smooth_subsequence (line : SemipositiveAnticanonicalMetric A) (hd : 2 ≤ d)
+    (z : ℕ → g.NormalizedPathSolution line) :
+    ∃ φ : SmoothRealFunction A, ∃ ν : ℕ → ℕ, StrictMono ν ∧ ∀ k : ℕ,
+      Tendsto (fun n => D.localizers.embed (k:ℝ) (Smooth.ofReal (z (ν n)).potential)) atTop
+        (𝓝 (D.localizers.embed (k:ℝ) (Smooth.ofReal φ))) := by
+  obtain ⟨F,ν,hν,hF⟩ := D.smooth_subsequence (fun n => Smooth.ofReal (z n).potential)
+    (fun k => by
+      obtain ⟨C,hC,hc⟩ := D.volumePath_uniform_all line hd k
+      exact ⟨C,hC,fun n => hc (z n)⟩)
+  let R : RealSmooth A := ⟨F,D.real_of_sobolev (fun n => (z (ν n)).potential) F hF⟩
+  refine ⟨R.source,ν,hν,?_⟩
+  rw [R.ofReal_source]
+  exact hF
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/SmoothPositivePath.lean b/lean/OAI/Geometry/PrescribedRicci/SmoothPositivePath.lean
new file mode 100644
index 0000000000000000000000000000000000000000..669a73de3f3ec2c0794da6ee1d1a805b6e1cdfd4
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/SmoothPositivePath.lean
@@ -0,0 +1,104 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.NonlinearScale
+import OAI.Geometry.PrescribedPotential.PatchCutoffs
+import OAI.Geometry.PrescribedPotential.RealSobolev
+import OAI.Geometry.PrescribedPotential.TwoScaleRegularity
+import OAI.Geometry.PrescribedPotential.WeakPositive
+import OAI.Geometry.PrescribedRicci.FiniteCofactorPatch
+import OAI.Geometry.PrescribedRicci.FixedCofactorBootstrap
+import OAI.Geometry.PrescribedRicci.PositiveOpenness
+import OAI.Geometry.PrescribedRicci.SmoothLocalInverse
+
+namespace OAI
+
+/-! Smooth Positive Path. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology
+open scoped ContDiff Classical
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  [ConnectedSpace X] {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace RealSmooth
+
+def expAffine (F : RealSmooth A) (t b : ℝ) : RealSmooth A := ofReal {
+  value := fun x => Real.exp (t*F.source.value x+b)
+  smooth i := ((contDiffOn_const.mul (F.source.smooth i)).add contDiffOn_const).exp }
+
+omit [T2Space X] [CompactSpace X] [ConnectedSpace X] in
+lemma expAffine_apply (F : RealSmooth A) (t b : ℝ) (x : X) :
+    (F.expAffine t b).val x = (Real.exp (t*F.source.value x+b) : ℂ) := rfl
+end RealSmooth
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+local instance smoothPathNG (s : ℝ) : NormedAddCommGroup (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedAddCommGroup
+local instance smoothPathNS (s : ℝ) : NormedSpace ℝ (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedSpace
+local instance smoothPathTG (s : ℝ) : IsTopologicalAddGroup (D.localizers.RealSobolev s) :=
+  Submodule.isTopologicalAddGroup _
+local instance smoothPathCS (s : ℝ) : ContinuousSMul ℝ (D.localizers.RealSobolev s) :=
+  SMulMemClass.continuousSMul _
+
+lemma exponentialDensity_embedViaEnergy (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (F : RealSmooth A) (t b : ℝ) :
+    Real.exp b • D.exponentialDensity k hk F t = D.localizers.realEmbed (k:ℝ) (F.expAffine t b) := by
+  have hs : (Module.finrank ℝ (EC d):ℝ) < 2*(k:ℝ) := by
+    have hh : (Module.finrank ℝ (EC d):ℝ) < (k:ℝ) := by exact_mod_cast hk
+    linarith only [hh,Nat.cast_nonneg (α:=ℝ) k]
+  apply Subtype.ext
+  apply D.localizers.strong_injective (k:ℝ) hs
+  ext x
+  change D.localizers.strong (k:ℝ) (Real.exp b • (D.exponentialDensity k hk F t).val) x =
+    D.localizers.strong (k:ℝ) (D.localizers.embed (k:ℝ) (F.expAffine t b).val) x
+  rw [ContinuousLinearMap.map_smul_of_tower,D.localizers.strong_embed _ hs]
+  simp only [BoundedContinuousFunction.smul_apply,D.exponentialDensity_strong,
+    Complex.real_smul]
+  change (Real.exp b:ℂ) * (Real.exp (t*F.source.value x):ℂ) = (F.expAffine t b).val x
+  rw [RealSmooth.expAffine_apply,Real.exp_add,Complex.ofReal_mul]
+  ring
+
+include D in
+ 
+
+theorem smooth_positive_path_eventually (F : RealSmooth A) (x₀ : X) :
+    ∀ᶠ t : ℝ in 𝓝 0, ∃ (φ : SmoothRealFunction A) (b : ℝ),
+      g.PositivePotential φ ∧ φ.value x₀ = 0 ∧
+      ∀ x, (g.potentialDensity φ).value x = Real.exp (t*F.source.value x+b) := by
+  let k := Module.finrank ℝ (EC d)+1
+  have hk : Module.finrank ℝ (EC d) < k := by dsimp [k]; omega
+  have hk1 : Module.finrank ℝ (EC d) < k+1 := by omega
+  have hs : (Module.finrank ℝ (EC d):ℝ) < 2*((k+1:ℕ):ℝ) := by
+    dsimp [k]; push_cast; linarith only [Nat.cast_nonneg (α:=ℝ) (Module.finrank ℝ (EC d))]
+  obtain ⟨γ,hγ,hγc,hγeq⟩ := D.exists_smooth_sobolev_path (k+1) hk1 x₀ F
+  have hfst : Tendsto (fun t => (γ t).1) (𝓝 0) (𝓝 (0 : D.localizers.RealSobolev (((k+1:ℕ):ℝ)+2))) := by
+    simpa only [hγ,Prod.fst_zero] using hγc.continuousAt.fst.tendsto
+  have hval : Tendsto (fun t => (γ t).1.val) (𝓝 0) (𝓝 (0 : D.localizers.Sobolev (((k+1:ℕ):ℝ)+2))) :=
+    continuous_subtype_val.tendsto 0 |>.comp hfst
+  filter_upwards [hγeq,hfst.eventually (D.weakPositive_nhds (k+1) hk1),
+    hval.eventually (D.cofactor_small_nhds (k+1) hk1 hs)] with t he hp hc
+  obtain ⟨hEq,hn⟩ := he
+  rw [D.exponentialDensity_embedViaEnergy] at hEq
+  obtain ⟨φ,hφ⟩ := D.cofactor_solution_smooth k hk hs (γ t).1 hc (F.expAffine t (γ t).2) hEq
+  refine ⟨φ.source,(γ t).2,?_,?_,?_⟩
+  · rw [← hφ] at hp
+    exact (D.weakPositive_embed_iffViaEnergy (k+1) hk1 φ).mp hp
+  · rw [← hφ,D.realEvaluation_embed (by linarith only [hs] : (Module.finrank ℝ (EC d):ℝ) < 2*(((k+1:ℕ):ℝ)+2))] at hn
+    exact hn
+  · rw [← hφ,D.realVolume_embed] at hEq
+    have hcore := D.localizers.embed_injective ((k+1:ℕ):ℝ) (congrArg Subtype.val hEq)
+    intro x
+    have hv := congrArg (fun f : Smooth A => (f x).re) hcore
+    exact hv
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/SobolevCommutatorBound.lean b/lean/OAI/Geometry/PrescribedRicci/SobolevCommutatorBound.lean
new file mode 100644
index 0000000000000000000000000000000000000000..f7a9ce209045e971580bbad621f63e313af2f352
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/SobolevCommutatorBound.lean
@@ -0,0 +1,165 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.SobolevProduct
+
+namespace OAI
+
+/-! Sobolev Commutator Bound. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory TemperedDistribution LineDeriv
+open scoped SchwartzMap BoundedContinuousFunction ContDiff Classical
+namespace SobolevChart
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+
+lemma product_words_asymmetric_bound (s t : ℝ) (p q : List E)
+    (hp : (p.length : ℝ) ≤ s) (hq : (q.length : ℝ) ≤ t)
+    (hd : (Module.finrank ℝ E : ℝ) < (s-p.length)+(t-q.length)) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ f g : 𝓢(E,ℂ),
+      ‖schwartzCoord 0 (SchwartzMap.smulLeftCLM ℂ (schwartzWord p f) (schwartzWord q g))‖ ≤
+        C * (‖schwartzCoord s f‖ * ‖schwartzCoord t g‖) := by
+  have base (s t : ℝ) (p q : List E) (hq : (q.length : ℝ) ≤ t)
+      (hd : (Module.finrank ℝ E : ℝ) < 2*(s-p.length)) :
+      ∃ C : ℝ, 0 ≤ C ∧ ∀ f g : 𝓢(E,ℂ),
+        ‖schwartzCoord 0 (SchwartzMap.smulLeftCLM ℂ (schwartzWord p f) (schwartzWord q g))‖ ≤
+          C * (‖schwartzCoord s f‖ * ‖schwartzCoord t g‖) := by
+    obtain ⟨C,hC,hc⟩ := schwartz_word_sup_bound p s hd
+    obtain ⟨D,hD,hd⟩ := coreBound_word q (s:=t) (t:=0) (by linarith)
+    refine ⟨C*D,mul_nonneg hC hD,fun f g => ?_⟩
+    rw [schwartzCoord_product_zero]
+    exact (multiply_norm_le _ _).trans (by
+      have h := mul_le_mul (hc f) (hd g) (norm_nonneg _) (mul_nonneg hC (norm_nonneg _))
+      convert h using 1
+      ring)
+  by_cases h : (Module.finrank ℝ E : ℝ) < 2*(s-p.length)
+  · exact base s t p q hq h
+  · obtain ⟨C,hC,hb⟩ := base t s q p hp (by linarith)
+    refine ⟨C,hC,fun f g => ?_⟩
+    rw [schwartz_product_swap]
+    simpa only [mul_comm] using hb g f
+
+ 
+
+lemma second_order_commutator_product_bound (k : ℕ)
+    (hk : Module.finrank ℝ E + 1 < k) (p q : List E)
+    (hp : 0 < p.length) (hpq : p.length+q.length ≤ k) (v w : E) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ a f : 𝓢(E,ℂ),
+      ‖schwartzCoord 0 (SchwartzMap.smulLeftCLM ℂ (schwartzWord p a)
+        (∂_{v} (∂_{w} (schwartzWord q f))))‖ ≤
+        C * (‖schwartzCoord (k:ℝ) a‖ * ‖schwartzCoord ((k:ℝ)+1) f‖) := by
+  have h := product_words_asymmetric_bound (E:=E) (k:ℝ) ((k:ℝ)+1) p (v::w::q)
+    (by exact_mod_cast (show p.length ≤ k by omega))
+    (by simp only [List.length_cons]; exact_mod_cast (show q.length+1+1 ≤ k+1 by omega))
+    (by
+      simp only [List.length_cons,Nat.cast_add,Nat.cast_one]
+      have h1 : (Module.finrank ℝ E:ℝ)+1 < k := by exact_mod_cast hk
+      have h2 : (p.length:ℝ)+q.length ≤ k := by exact_mod_cast hpq
+      linarith)
+  exact h
+
+ 
+
+def commutatorWords : List E → List (List E × List E)
+  | [] => []
+  | v::ws => ( [v],ws ) ::
+      ((commutatorWords ws).map (fun t => (v::t.1,t.2)) ++
+       (commutatorWords ws).map (fun t => (t.1,v::t.2)))
+
+omit [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
+  [MeasurableSpace E] [BorelSpace E] in
+lemma commutatorWords_orders (ws : List E) (t : List E × List E)
+    (ht : t ∈ commutatorWords ws) :
+    0 < t.1.length ∧ t.1.length+t.2.length = ws.length := by
+  induction ws generalizing t with
+  | nil => simp [commutatorWords] at ht
+  | cons v ws ih =>
+    simp only [commutatorWords,List.mem_cons,List.mem_append,List.mem_map] at ht
+    rcases ht with rfl | ⟨t,ht,rfl⟩ | ⟨t,ht,rfl⟩
+    · simp [Nat.add_comm]
+    · obtain ⟨h1,h2⟩ := ih t ht
+      simp only [List.length_cons]
+      omega
+    · obtain ⟨h1,h2⟩ := ih t ht
+      simp only [List.length_cons]
+      omega
+
+omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] in
+lemma schwartz_word_product_commutator (ws : List E) (a f : 𝓢(E,ℂ)) :
+    schwartzWord ws (SchwartzMap.smulLeftCLM ℂ a f) =
+      SchwartzMap.smulLeftCLM ℂ a (schwartzWord ws f) +
+      ((commutatorWords ws).map (fun t =>
+        SchwartzMap.smulLeftCLM ℂ (schwartzWord t.1 a) (schwartzWord t.2 f))).sum := by
+  induction ws with
+  | nil => simp [schwartzWord,commutatorWords]
+  | cons v ws ih =>
+    change ∂_{v} (schwartzWord ws (SchwartzMap.smulLeftCLM ℂ a f)) = _
+    rw [ih,lineDerivOp_add,schwartz_deriv_product]
+    have he : ∂_{v} (((commutatorWords ws).map (fun t =>
+        SchwartzMap.smulLeftCLM ℂ (schwartzWord t.1 a) (schwartzWord t.2 f))).sum) =
+        ((commutatorWords ws).map (fun t => ∂_{v}
+          (SchwartzMap.smulLeftCLM ℂ (schwartzWord t.1 a) (schwartzWord t.2 f)))).sum :=
+      (List.sum_map_hom _ _ (lineDerivOpCLM ℂ 𝓢(E,ℂ) v)).symm
+    rw [he]
+    simp only [schwartz_deriv_product,list_sum_add,commutatorWords,List.map_cons,
+      List.sum_cons,List.map_append,List.sum_append,List.map_map,Function.comp_def,
+      schwartzWord,ContinuousLinearMap.comp_apply,ContinuousLinearMap.id_apply,
+      lineDerivOpCLM_apply]
+    abel
+
+lemma second_order_commutator_bound (k : ℕ) (hk : Module.finrank ℝ E + 1 < k)
+    (ws : List E) (hws : ws.length ≤ k) (v w : E) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ a f : 𝓢(E,ℂ),
+      ‖schwartzCoord 0 (schwartzWord ws
+        (SchwartzMap.smulLeftCLM ℂ a (∂_{v} (∂_{w} f))) -
+        SchwartzMap.smulLeftCLM ℂ a (∂_{v} (∂_{w} (schwartzWord ws f))))‖ ≤
+        C * (‖schwartzCoord (k:ℝ) a‖ * ‖schwartzCoord ((k:ℝ)+1) f‖) := by
+  have hcomm (q : List E) (f : 𝓢(E,ℂ)) :
+      schwartzWord q (∂_{v} (∂_{w} f)) = ∂_{v} (∂_{w} (schwartzWord q f)) := by
+    rw [GlobalElliptic.word_deriv_comm,GlobalElliptic.word_deriv_comm]
+  have hb (t : List E × List E) (ht : t ∈ commutatorWords ws) :=
+    second_order_commutator_product_bound k hk t.1 t.2
+      (commutatorWords_orders ws t ht).1
+      (by rw [(commutatorWords_orders ws t ht).2]; exact hws) v w
+  classical
+  let C : List E × List E → ℝ := fun t =>
+    if ht : t ∈ commutatorWords ws then (hb t ht).choose else 0
+  have hC (t) (ht : t ∈ commutatorWords ws) : 0 ≤ C t := by
+    simp only [C,dite_eq_left ht]
+    exact (hb t ht).choose_spec.1
+  have hB (t) (ht : t ∈ commutatorWords ws) (a f : 𝓢(E,ℂ)) :
+      ‖schwartzCoord 0 (SchwartzMap.smulLeftCLM ℂ (schwartzWord t.1 a)
+        (∂_{v} (∂_{w} (schwartzWord t.2 f))))‖ ≤
+        C t * (‖schwartzCoord (k:ℝ) a‖ * ‖schwartzCoord ((k:ℝ)+1) f‖) := by
+    simp only [C,dite_eq_left ht]
+    exact (hb t ht).choose_spec.2 a f
+  refine ⟨((commutatorWords ws).map C).sum,List.sum_nonneg (by simpa only
+    [List.mem_map,forall_exists_index,and_imp,forall_apply_eq_imp_iff₂] using hC),fun a f => ?_⟩
+  rw [schwartz_word_product_commutator,hcomm,add_sub_cancel_left,schwartzCoord_list_sum]
+  simp_rw [hcomm]
+  calc
+    _ ≤ ((commutatorWords ws).map (fun t => ‖schwartzCoord 0
+      (SchwartzMap.smulLeftCLM ℂ (schwartzWord t.1 a) (∂_{v} (∂_{w} (schwartzWord t.2 f))))‖)).sum :=
+      by
+        induction commutatorWords ws with
+        | nil => simp
+        | cons t ts ih =>
+          simp only [List.map_cons,List.sum_cons]
+          exact (norm_add_le _ _).trans (add_le_add le_rfl ih)
+    _ ≤ ((commutatorWords ws).map (fun t => C t *
+        (‖schwartzCoord (k:ℝ) a‖ * ‖schwartzCoord ((k:ℝ)+1) f‖))).sum :=
+      List.sum_le_sum (fun t ht => hB t ht a f)
+    _ = _ := by
+      induction commutatorWords ws with
+      | nil => simp
+      | cons t ts ih => simp only [List.map_cons,List.sum_cons,ih,add_mul]
+
+end SobolevChart
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/SobolevFamilyBounds.lean b/lean/OAI/Geometry/PrescribedRicci/SobolevFamilyBounds.lean
new file mode 100644
index 0000000000000000000000000000000000000000..8a91b0e4b7653a07bb8a9bdaab0f1f5b7f7c2cc0
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/SobolevFamilyBounds.lean
@@ -0,0 +1,74 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.IntegerBounds
+
+namespace OAI
+
+/-! Sobolev Family Bounds. -/
+
+noncomputable section
+open Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory TemperedDistribution LineDeriv
+open scoped SchwartzMap ContDiff Classical Laplacian
+namespace SobolevChart
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+
+def FamilyBound {α : Type*} (t : ℝ) (R : α → ℝ) (T : α → 𝓢(E,ℂ)) : Prop :=
+  ∃ C : ℝ, 0 ≤ C ∧ ∀ x, ‖schwartzCoord t (T x)‖ ≤ C * R x
+
+lemma familyBound_raise_one {t : ℝ} {α : Type*} {R : α → ℝ} (T : α → 𝓢(E, ℂ))
+    (hT : FamilyBound t R T)
+    (hD : ∀ j, FamilyBound t R (fun f => ∂_{stdOrthonormalBasis ℝ E j} (T f))) :
+    FamilyBound (t + 1) R T := by
+  obtain ⟨C, hC, hc⟩ := hT
+  choose D hD0 hdb using hD
+  let c : ℝ := ‖(((2 * Real.pi)^2 : ℝ) : ℂ)⁻¹‖
+  refine ⟨‖raiseZero (E := E) t‖ * C + c * ∑ j, ‖raiseDeriv t (stdOrthonormalBasis ℝ E j)‖ * D j,
+    add_nonneg (mul_nonneg (norm_nonneg _) hC)
+      (mul_nonneg (norm_nonneg _) (Finset.sum_nonneg (fun j _ => mul_nonneg (norm_nonneg _) (hD0 j)))), fun f => ?_⟩
+  rw [schwartzCoord_raise_one]
+  calc
+    _ ≤ ‖raiseZero t (schwartzCoord t (T f))‖ +
+        c * ‖∑ j, raiseDeriv t (stdOrthonormalBasis ℝ E j)
+          (schwartzCoord t (∂_{stdOrthonormalBasis ℝ E j} (T f)))‖ := by
+      simpa only [norm_smul] using norm_sub_le
+        (raiseZero t (schwartzCoord t (T f)))
+        ((((2 * Real.pi)^2 : ℝ) : ℂ)⁻¹ • ∑ j, raiseDeriv t (stdOrthonormalBasis ℝ E j)
+          (schwartzCoord t (∂_{stdOrthonormalBasis ℝ E j} (T f))))
+    _ ≤ (‖raiseZero (E := E) t‖ * C) * R f +
+        c * ∑ j, (‖raiseDeriv t (stdOrthonormalBasis ℝ E j)‖ * D j) * R f := by
+      apply add_le_add
+      · exact ((raiseZero t).le_opNorm _).trans (by
+          simpa only [mul_assoc] using
+            (mul_le_mul_of_nonneg_left (hc f) (norm_nonneg (raiseZero (E := E) t))))
+      · apply mul_le_mul_of_nonneg_left _ (norm_nonneg _)
+        apply (norm_sum_le _ _).trans
+        apply Finset.sum_le_sum
+        intro j _
+        exact ((raiseDeriv t (stdOrthonormalBasis ℝ E j)).le_opNorm _).trans (by
+          simpa only [mul_assoc] using mul_le_mul_of_nonneg_left (hdb j f)
+            (norm_nonneg (raiseDeriv t (stdOrthonormalBasis ℝ E j))))
+    _ = _ := by rw [← Finset.sum_mul]; ring
+
+lemma familyBound_of_words_integer (k : ℕ) {t : ℝ} {α : Type*} {R : α → ℝ} (T : α → 𝓢(E, ℂ))
+    (h : ∀ ws : List E, ws.length ≤ k → FamilyBound t R (fun f => schwartzWord ws (T f))) :
+    FamilyBound (t + k) R T := by
+  induction k generalizing T with
+  | zero => simpa [schwartzWord] using h [] (by simp)
+  | succ k ih =>
+    have ht := ih T (fun ws hw => h ws (by omega))
+    have hd (j : Fin (Module.finrank ℝ E)) := ih
+      (fun f => ∂_{stdOrthonormalBasis ℝ E j} (T f))
+      (fun ws hw => by
+        have hh := h (ws ++ [stdOrthonormalBasis ℝ E j])
+          (by simp only [List.length_append, List.length_cons, List.length_nil]; omega)
+        simpa only [schwartzWord_append, schwartzWord, ContinuousLinearMap.comp_apply,
+          ContinuousLinearMap.id_apply, lineDerivOpCLM_apply] using hh)
+    convert familyBound_raise_one T ht hd using 1
+    push_cast
+    ring
+
+end SobolevChart
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/SobolevInterpolation.lean b/lean/OAI/Geometry/PrescribedRicci/SobolevInterpolation.lean
new file mode 100644
index 0000000000000000000000000000000000000000..52a6ea0b0a26ee9e29d16362d94b08f05e7b9af5
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/SobolevInterpolation.lean
@@ -0,0 +1,119 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.CoreBounds
+
+namespace OAI
+
+/-! Sobolev Interpolation. -/
+
+noncomputable section
+open Set Filter Topology MeasureTheory FourierTransform TemperedDistribution
+open scoped SchwartzMap BoundedContinuousFunction ContDiff Classical
+namespace SobolevChart
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+
+lemma rpow_epsilon_bound (r s t : ℝ) (hst : s < t) {ε : ℝ} (hε : 0 < ε) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ x : ℝ, 1 ≤ x →
+      x^(s/2) ≤ ε*x^(t/2)+C*x^(r/2) := by
+  have htend : Tendsto (fun x : ℝ => x^((s-t)/2)) atTop (nhds 0) := by
+    convert tendsto_rpow_neg_atTop (show 0 < (t-s)/2 by linarith) using 1
+    funext x
+    congr 1
+    ring
+  obtain ⟨R,hR⟩ := eventually_atTop.mp (htend.eventually (gt_mem_nhds hε))
+  let B : ℝ := max 1 R
+  have hc : ContinuousOn (fun x : ℝ => x^((s-r)/2)) (Icc 1 B) := by
+    apply ContinuousOn.rpow_const continuousOn_id
+    intro x hx
+    left
+    change x ≠ 0
+    linarith [hx.1]
+  obtain ⟨C,hC⟩ := isCompact_Icc.exists_bound_of_continuousOn hc
+  refine ⟨max 0 C,le_max_left _ _,fun x hx => ?_⟩
+  have hx0 : 0 < x := by linarith
+  have he (a b : ℝ) : x^((a-b)/2)*x^(b/2)=x^(a/2) := by
+    rw [← Real.rpow_add hx0]
+    congr 1
+    ring
+  by_cases hxb : x ≤ B
+  · have hb := hC x ⟨hx,hxb⟩
+    rw [Real.norm_eq_abs,abs_of_nonneg (Real.rpow_nonneg hx0.le _)] at hb
+    have hmul := mul_le_mul_of_nonneg_right (hb.trans (le_max_right 0 C))
+      (Real.rpow_nonneg hx0.le (r/2))
+    rw [he s r] at hmul
+    exact hmul.trans (le_add_of_nonneg_left (mul_nonneg hε.le (Real.rpow_nonneg hx0.le _)))
+  · have ht := (hR x ((le_max_right 1 R).trans (le_of_not_ge hxb)))
+    have hmul := mul_le_mul_of_nonneg_right ht.le (Real.rpow_nonneg hx0.le (t/2))
+    rw [he s t] at hmul
+    exact hmul.trans (le_add_of_nonneg_right (mul_nonneg (le_max_left _ _) (Real.rpow_nonneg hx0.le _)))
+
+def weightedFourier (s : ℝ) (f : 𝓢(E,ℂ)) : 𝓢(E,ℂ) :=
+  SchwartzMap.smulLeftCLM ℂ (fun x : E => ((1+‖x‖^2)^(s/2):ℝ)) (𝓕 f)
+
+lemma weightedFourier_apply (s : ℝ) (f : 𝓢(E,ℂ)) (x : E) :
+    weightedFourier s f x = (((1+‖x‖^2)^(s/2):ℝ):ℂ) * 𝓕 f x := by
+  rw [weightedFourier,SchwartzMap.smulLeftCLM_apply_apply (by fun_prop)]
+  rfl
+
+lemma fourier_schwartzCoord (s : ℝ) (f : 𝓢(E,ℂ)) :
+    𝓕 (schwartzCoord s f) = SchwartzMap.toLpCLM ℂ ℂ 2 volume (weightedFourier s f) := by
+  apply l2_injective
+  change ((𝓕 (schwartzCoord s f) : L2 E) : 𝓢'(E,ℂ)) =
+    (((weightedFourier s f).toLp 2 volume : L2 E) : 𝓢'(E,ℂ))
+  rw [← Lp.fourier_toTemperedDistribution_eq,schwartzCoord_distribution,
+    fourier_besselPotential_eq_smulLeftCLM_fourier_apply]
+  rw [Lp.toTemperedDistribution_toLp_eq]
+  rw [fourier_toTemperedDistributionCLM_eq]
+  ext q
+  simp only [smulLeftCLM_apply_apply,SchwartzMap.toTemperedDistributionCLM_apply_apply]
+  apply integral_congr_ae
+  filter_upwards [] with x
+  simp only [SchwartzMap.smulLeftCLM_apply_apply (by fun_prop :
+    (fun x : E => (((1+‖x‖^2)^(s/2):ℝ):ℂ)).HasTemperateGrowth),
+    weightedFourier_apply,smul_eq_mul]
+  ring
+
+lemma schwartzCoord_norm_weightedFourier (s : ℝ) (f : 𝓢(E,ℂ)) :
+    ‖schwartzCoord s f‖ = ‖SchwartzMap.toLpCLM ℂ ℂ 2 volume (weightedFourier s f)‖ := by
+  rw [← fourier_schwartzCoord,Lp.norm_fourier_eq]
+
+theorem schwartz_sobolev_interpolation (r s t : ℝ) (hst : s < t)
+    {ε : ℝ} (hε : 0 < ε) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ f : 𝓢(E,ℂ),
+      ‖schwartzCoord s f‖ ≤ ε*‖schwartzCoord t f‖+C*‖schwartzCoord r f‖ := by
+  obtain ⟨C,hC,hb⟩ := rpow_epsilon_bound r s t hst hε
+  refine ⟨C,hC,fun f => ?_⟩
+  let W := fun a : ℝ => SchwartzMap.toLpCLM ℂ ℂ 2 volume (weightedFourier a f)
+  have hn : ‖W s‖ ≤ ‖(ε:ℂ) • W t+(C:ℂ) • W r‖ := by
+    apply Lp.norm_le_norm_of_ae_le
+    filter_upwards [(weightedFourier s f).coeFn_toLp (p:=2) (μ:=volume),
+      (weightedFourier t f).coeFn_toLp (p:=2) (μ:=volume),
+      (weightedFourier r f).coeFn_toLp (p:=2) (μ:=volume),
+      Lp.coeFn_add ((ε:ℂ) • W t) ((C:ℂ) • W r),
+      Lp.coeFn_smul (ε:ℂ) (W t), Lp.coeFn_smul (C:ℂ) (W r)] with x hs ht hr hadd hsm hsm'
+    change ‖W s x‖ ≤ _
+    rw [hadd]
+    simp only [Pi.add_apply]
+    rw [hsm,hsm']
+    change ‖SchwartzMap.toLpCLM ℂ ℂ 2 volume (weightedFourier s f) x‖ ≤
+      ‖(ε:ℂ) * SchwartzMap.toLpCLM ℂ ℂ 2 volume (weightedFourier t f) x+
+        (C:ℂ)*SchwartzMap.toLpCLM ℂ ℂ 2 volume (weightedFourier r f) x‖
+    simp only [SchwartzMap.toLpCLM_apply]
+    rw [hs,ht,hr]
+    simp only [weightedFourier_apply]
+    simp only [← mul_assoc]
+    rw [← add_mul,← Complex.ofReal_mul,← Complex.ofReal_mul,← Complex.ofReal_add]
+    simp only [norm_mul,Complex.norm_real,Real.norm_eq_abs]
+    have hx : 0 ≤ 1+‖x‖^2 := by positivity
+    rw [abs_of_nonneg (Real.rpow_nonneg hx _),abs_of_nonneg (add_nonneg
+      (mul_nonneg hε.le (Real.rpow_nonneg hx _)) (mul_nonneg hC (Real.rpow_nonneg hx _)))]
+    exact mul_le_mul_of_nonneg_right (hb _ (by nlinarith [sq_nonneg ‖x‖])) (norm_nonneg _)
+  have hn' := hn.trans (norm_add_le _ _)
+  simp only [norm_smul,Complex.norm_real,Real.norm_eq_abs,abs_of_pos hε,abs_of_nonneg hC] at hn'
+  simpa only [W,← schwartzCoord_norm_weightedFourier] using hn'
+
+end SobolevChart
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/TameApproximation.lean b/lean/OAI/Geometry/PrescribedRicci/TameApproximation.lean
new file mode 100644
index 0000000000000000000000000000000000000000..3026a519f4f6429f684b17f26019c1a65e73c433
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/TameApproximation.lean
@@ -0,0 +1,119 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.TameStability
+
+namespace OAI
+
+/-! Tame Approximation. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory TemperedDistribution LineDeriv
+open scoped SchwartzMap BoundedContinuousFunction ContDiff Classical ComplexOrder MatrixOrder
+namespace SobolevChart
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+variable {ι : Type*} [Fintype ι]
+
+lemma schwartzCoord_sub (s : ℝ) (f g : 𝓢(E,ℂ)) :
+    schwartzCoord s (f-g) = schwartzCoord s f-schwartzCoord s g := by
+  apply realize_injective s
+  simp only [map_sub,realize_schwartzCoord]
+
+lemma continuous_multiply : Continuous (fun p : (E →ᵇ ℂ) × L2 E => multiply p.1 p.2) := by
+  change Continuous (fun p : (E →ᵇ ℂ) × L2 E =>
+    (ContinuousLinearMap.mul ℂ ℂ).holderL volume ⊤ 2 2 (boundedToLinf p.1) p.2)
+  fun_prop
+
+omit [Fintype ι] in
+lemma continuous_perturbation_entry (v : ι → E) (i j : ι) :
+    Continuous (fun p : (ι → ι → E →ᵇ ℂ) × L2 E =>
+      multiply (p.1 i j) (secondH2 (v i) (v j) p.2)) := by
+  have ha : Continuous (fun p : (ι → ι → E →ᵇ ℂ) × L2 E => p.1 i j) :=
+    (continuous_apply j).comp ((continuous_apply i).comp continuous_fst)
+  have hu : Continuous (fun p : (ι → ι → E →ᵇ ℂ) × L2 E =>
+      secondH2 (v i) (v j) p.2) :=
+    (secondH2 (v i) (v j)).continuous.comp continuous_snd
+  change Continuous (fun p : (ι → ι → E →ᵇ ℂ) × L2 E =>
+    (ContinuousLinearMap.mul ℂ ℂ).holderL volume ⊤ 2 2
+      (boundedToLinf (p.1 i j)) (secondH2 (v i) (v j) p.2))
+  exact (((ContinuousLinearMap.mul ℂ ℂ).holderL volume ⊤ 2 2).continuous.comp
+    (boundedToLinf.continuous.comp ha)).clm_apply hu
+
+lemma perturbation_apply_identity (v : ι → E) (a : ι → ι → E →ᵇ ℂ) (u : L2 E) :
+    perturbation v a u = ∑ i, ∑ j, multiply (a i j) (secondH2 (v i) (v j) u) := by
+  simp only [perturbation,_root_.sum_apply,ContinuousLinearMap.comp_apply]
+
+lemma continuous_perturbation (v : ι → E) :
+    Continuous (fun p : (ι → ι → E →ᵇ ℂ) × L2 E => perturbation v p.1 p.2) := by
+  simp only [perturbation_apply_identity]
+  exact continuous_finsetSum Finset.univ (fun i _ =>
+    continuous_finsetSum Finset.univ (fun j _ => continuous_perturbation_entry v i j))
+
+omit [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E]
+  [BorelSpace E] in
+lemma continuous_perturbationBound (v : ι → E) :
+    Continuous (perturbationBound v) := by
+  unfold perturbationBound
+  fun_prop
+
+end SobolevChart
+namespace FrozenPoisson.ParameterRegularity
+open SobolevChart EllipticKernel
+variable {n : ℕ} {ι : Type*} [Fintype ι]
+
+lemma tame_high_cauchy (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (t : ℝ) (ht : 1 ≤ t) (v : ι → EC n) (k : ℕ)
+    (hk : Module.finrank ℝ (EC n)+1 < k) (M : ℝ) (hM : 0 ≤ M)
+    (θ : ℝ) (hθ : θ < 1) (x : ℕ → TameData H t v k M θ (ellipticBound H hH))
+    (hf : CauchySeq (fun l => schwartzCoord (k:ℝ) (x l).f))
+    (ha : ∀ i j, CauchySeq (fun l => schwartzCoord (k:ℝ) ((x l).a i j))) :
+    CauchySeq (fun l => schwartzCoord ((k:ℝ)+2) (x l).u) := by
+  obtain ⟨J,hJ,hj⟩ := tame_solution_bound H hH t ht v k hk M hM θ hθ
+  obtain ⟨R,hR,hfR⟩ := hf.isBounded_range.exists_pos_norm_le
+  have hu (l : ℕ) : ‖schwartzCoord ((k:ℝ)+2) (x l).u‖ ≤ J*R :=
+    (hj (x l)).trans (mul_le_mul_of_nonneg_left (hfR _ (mem_range_self l)) hJ)
+  obtain ⟨C,B,hC,hB,hcb⟩ := tame_stability H hH t ht v k hk M hM θ hθ
+  obtain ⟨bf,hbf,hbf',htf⟩ := cauchySeq_iff_le_tendsto_0.mp hf
+  choose ba hba hba' hta using fun i j => cauchySeq_iff_le_tendsto_0.mp (ha i j)
+  apply cauchySeq_of_le_tendsto_0 (fun N => C*(bf N+B*(∑ i, ∑ j, ba i j N)*(J*R)))
+  · intro l m N hl hm
+    have h0 := hcb (x l) (x m)
+    simp only [schwartzCoord_sub] at h0
+    rw [dist_eq_norm]
+    apply h0.trans
+    apply mul_le_mul_of_nonneg_left _ hC
+    apply add_le_add
+    · simpa only [dist_eq_norm] using hbf' l m N hl hm
+    · have ha' : (∑ i, ∑ j, ‖schwartzCoord (k:ℝ) ((x l).a i j)-
+          schwartzCoord (k:ℝ) ((x m).a i j)‖) ≤ ∑ i, ∑ j, ba i j N := by
+        apply Finset.sum_le_sum
+        intro i _
+        apply Finset.sum_le_sum
+        intro j _
+        simpa only [dist_eq_norm] using hba' i j l m N hl hm
+      exact mul_le_mul (mul_le_mul_of_nonneg_left ha' hB) (hu m) (norm_nonneg _)
+        (mul_nonneg hB (Finset.sum_nonneg (fun i _ => Finset.sum_nonneg (fun j _ => hba i j N))))
+  · have ha0 : Tendsto (fun N => ∑ i, ∑ j, ba i j N) atTop (nhds (0:ℝ)) := by
+      simpa using tendsto_finsetSum _ (fun i _ => tendsto_finsetSum _ (fun j _ => hta i j))
+    simpa using (htf.add ((ha0.const_mul B).mul_const (J*R))).const_mul C
+
+lemma tame_data_low_equation (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (t : ℝ) (ht : 1 ≤ t) (v : ι → EC n) (k : ℕ) (M θ : ℝ)
+    (x : TameData H t v k M θ (ellipticBound H hH)) :
+    schwartzCoord 2 x.u = parameterHilbert H hH t ht
+      (schwartzCoord 0 x.f + perturbation v (coefficientBCF x.a) (schwartzCoord 2 x.u)) := by
+  have hh := congrArg (schwartzCoord 2) x.equation
+  have hr := schwartzCoord_resolvent H hH t ht 0 (x.f+smoothPerturbation v x.a x.u)
+  norm_num only [zero_add] at hr
+  rw [hr,schwartzCoord_add,schwartzCoord_perturbation] at hh
+  exact hh
+
+end FrozenPoisson.ParameterRegularity
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/TameCofactorGain.lean b/lean/OAI/Geometry/PrescribedRicci/TameCofactorGain.lean
new file mode 100644
index 0000000000000000000000000000000000000000..857bd28ec9940138d23b8c00561e5f9cef1fe23f
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/TameCofactorGain.lean
@@ -0,0 +1,125 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.CofactorWeakEquation
+import OAI.Geometry.PrescribedPotential.GlobalCompletion
+import OAI.Geometry.PrescribedRicci.RoughEllipticGain
+
+namespace OAI
+
+/-! Tame Cofactor Gain. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Matrix _root_.MeasureTheory _root_.OAI.MeasureTheory TemperedDistribution
+open scoped ContDiff SchwartzMap Classical BoundedContinuousFunction Matrix.Norms.Elementwise
+namespace GlobalElliptic
+open Anticanonical SourceSmooth EllipticKernel SobolevChart FrozenPoisson MetricLocalization
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  {A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
+namespace GluingData
+variable {g : KaehlerMetric A} (D : GluingData g ι)
+local instance tameCofactorGainNG (s : ℝ) : NormedAddCommGroup (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedAddCommGroup
+local instance tameCofactorGainNS (s : ℝ) : NormedSpace ℝ (D.localizers.RealSobolev s) :=
+  (D.localizers.realCompletion s).normedSpace
+local instance tameCofactorGainTG (s : ℝ) : IsTopologicalAddGroup (D.localizers.RealSobolev s) :=
+  Submodule.isTopologicalAddGroup _
+local instance tameCofactorGainCS (s : ℝ) : ContinuousSMul ℝ (D.localizers.RealSobolev s) :=
+  SMulMemClass.continuousSMul _
+
+lemma cofactor_local_gain (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (hs : (Module.finrank ℝ (EC d):ℝ) < 2*((k+1:ℕ):ℝ))
+    (u : D.localizers.RealSobolev (((k+1:ℕ):ℝ)+2))
+    (hu : D.CofactorSmall (k+1) (by omega) hs u.val)
+    (w : D.localizers.RealSobolev ((k:ℝ)+2))
+    (F : D.localizers.RealSobolev ((k+1:ℕ):ℝ))
+    (hF : D.realVolumeDerivative k hk
+      (D.localizers.realLower (((k+1:ℕ):ℝ)+2) ((k:ℝ)+2) (by push_cast; linarith) u) w =
+      D.localizers.realLower ((k+1:ℕ):ℝ) (k:ℝ) (by push_cast; linarith) F) (p : ι) :
+    MemSobolev (((k+1:ℕ):ℝ)+2) 2 (D.localizers.distribution ((k:ℝ)+2) p w.val) := by
+  let a := fun i j => D.cofactorLocal (k+1) (by omega) p i j u.val
+  let E := roughCutoff (D.patch p).matrix (k+1) (by omega) (D.weightCore p) a
+    (lowerCoord ((k:ℝ)+2) (((k+1:ℕ):ℝ)+1) (by push_cast; linarith)
+      (D.outerCompleted ((k:ℝ)+2) p w.val))
+  let f : L2 (EC d) := lowerCoord ((k:ℝ)+2) ((k+1:ℕ):ℝ) (by push_cast; linarith)
+    (w.val.val p)-F.val.val p-E
+  let w₂ := lowerCoord ((k:ℝ)+2) 2 (by linarith [Nat.cast_nonneg (α:=ℝ) k]) (w.val.val p)
+  have he := D.cofactor_weak_equation k hk hs p u w
+  dsimp only [cofactorWeakLeft,cofactorWeakRight,Prod.fst,Prod.snd] at he
+  rw [hF] at he
+  have hFl : D.localizers.distribution (k:ℝ) p
+      ((D.localizers.realLower ((k+1:ℕ):ℝ) (k:ℝ) (by push_cast; linarith) F).val) =
+      D.localizers.distribution ((k+1:ℕ):ℝ) p F.val :=
+    D.localizers.distribution_lower (by push_cast; linarith) p F.val
+  rw [hFl] at he
+  change frozenDifferential (D.patch p).matrix (realize ((k:ℝ)+2) (w.val.val p)) +
+    ((perturbation (stdOrthonormalBasis ℝ (EC d))
+      (D.cofactorBCF (k+1) (by omega) hs p u.val) w₂ : L2 (EC d)) : 𝓢'(EC d,ℂ)) =
+      realize ((k+1:ℕ):ℝ) (F.val.val p) + realize ((k+1:ℕ):ℝ) E at he
+  have hx : (1^2:ℂ) • realize 2 w₂ - frozenDifferential (D.patch p).matrix (realize 2 w₂) -
+      ((perturbation (stdOrthonormalBasis ℝ (EC d))
+        (fun i j => strongEmbedding ((k+1:ℕ):ℝ) hs (a i j)) w₂ : L2 (EC d)) : 𝓢'(EC d,ℂ)) =
+      realize ((k+1:ℕ):ℝ) f := by
+    simp only [one_pow,one_smul,f,map_sub,realize_lowerCoord,w₂]
+    change realize ((k:ℝ)+2) (w.val.val p) -
+      frozenDifferential (D.patch p).matrix (realize ((k:ℝ)+2) (w.val.val p)) -
+      ((perturbation (stdOrthonormalBasis ℝ (EC d)) (D.cofactorBCF (k+1) (by omega) hs p u.val)
+        (lowerCoord ((k:ℝ)+2) 2 (by linarith [Nat.cast_nonneg (α:=ℝ) k]) (w.val.val p)) : L2 (EC d)) : 𝓢'(EC d,ℂ)) = _
+    rw [sub_sub,he]
+    abel
+  have hg := rough_equation_memSobolev (D.patch p).matrix (D.patch p).positive 1 le_rfl
+    (stdOrthonormalBasis ℝ (EC d)) (k+1) (by omega) hs a f w₂ (hu p) hx
+  simpa only [w₂,realize_lowerCoord,Localizers.distribution_apply] using hg
+
+lemma real_sobolev_of_local_order (k l : ℕ) (hkl : l ≤ k)
+    (w : D.localizers.RealSobolev (l:ℝ))
+    (hw : ∀ p, MemSobolev (k:ℝ) 2 (D.localizers.distribution (l:ℝ) p w.val)) :
+    ∃ z : D.localizers.RealSobolev (k:ℝ),
+      D.localizers.realLower (k:ℝ) (l:ℝ) (by exact_mod_cast hkl) z = w := by
+  let w₀ := D.localizers.lower (l:ℝ) 0 w.val
+  have hh (p : ι) : MemSobolev (k:ℝ) 2 (D.localizers.distribution 0 p w₀) := by
+    rw [D.localizers.distribution_lower (Nat.cast_nonneg l)]
+    exact hw p
+  obtain ⟨z,hz⟩ := D.sobolev_of_local_order k w₀ hh
+  have he : D.localizers.lower (k:ℝ) (l:ℝ) z = w.val := by
+    apply D.localizers.lower_injective (Nat.cast_nonneg l)
+    rw [D.localizers.lower_lower (by exact_mod_cast hkl) (Nat.cast_nonneg l)]
+    exact hz
+  have hm := D.mem_real_of_lower k l hkl z (he.symm ▸ w.property)
+  exact ⟨⟨z,hm⟩,Subtype.ext he⟩
+
+lemma real_sobolev_of_local_order_eq (k l : ℕ) (hkl : l ≤ k) (s t : ℝ)
+    (hs : (k:ℝ) = s) (ht : (l:ℝ) = t)
+    (w : D.localizers.RealSobolev t)
+    (hw : ∀ p, MemSobolev s 2 (D.localizers.distribution t p w.val)) :
+    ∃ z : D.localizers.RealSobolev s,
+      D.localizers.realLower s t (by rw [← hs,← ht]; exact_mod_cast hkl) z = w := by
+  subst s; subst t
+  exact D.real_sobolev_of_local_order k l hkl w hw
+
+ 
+
+theorem cofactor_global_gain (k : ℕ) (hk : Module.finrank ℝ (EC d) < k)
+    (hs : (Module.finrank ℝ (EC d):ℝ) < 2*((k+1:ℕ):ℝ))
+    (u : D.localizers.RealSobolev (((k+1:ℕ):ℝ)+2))
+    (hu : D.CofactorSmall (k+1) (by omega) hs u.val)
+    (w : D.localizers.RealSobolev ((k:ℝ)+2))
+    (F : D.localizers.RealSobolev ((k+1:ℕ):ℝ))
+    (hF : D.realVolumeDerivative k hk
+      (D.localizers.realLower (((k+1:ℕ):ℝ)+2) ((k:ℝ)+2) (by push_cast; linarith) u) w =
+      D.localizers.realLower ((k+1:ℕ):ℝ) (k:ℝ) (by push_cast; linarith) F) :
+    ∃ z : D.localizers.RealSobolev (((k+1:ℕ):ℝ)+2),
+      D.localizers.realLower (((k+1:ℕ):ℝ)+2) ((k:ℝ)+2) (by push_cast; linarith) z = w := by
+  exact D.real_sobolev_of_local_order_eq (k+3) (k+2) (by omega)
+    (((k+1:ℕ):ℝ)+2) ((k:ℝ)+2) (by push_cast; ring) (by push_cast; rfl)
+    w (D.cofactor_local_gain k hk hs u hu w F hF)
+
+end GluingData
+end GlobalElliptic
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/TameLimit.lean b/lean/OAI/Geometry/PrescribedRicci/TameLimit.lean
new file mode 100644
index 0000000000000000000000000000000000000000..f56fbc9602e6700b8ef4f57b79ff4f173adfbd7f
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/TameLimit.lean
@@ -0,0 +1,91 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.SobolevReconstruction
+import OAI.Geometry.PrescribedRicci.TameApproximation
+
+namespace OAI
+
+/-! Tame Limit. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory TemperedDistribution LineDeriv
+open scoped SchwartzMap BoundedContinuousFunction ContDiff Classical ComplexOrder MatrixOrder
+namespace SobolevChart
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+variable {ι : Type*} [Fintype ι]
+
+omit [Fintype ι] in
+lemma coefficientBCF_tendsto (k : ℝ) (hk : (Module.finrank ℝ E : ℝ) < 2*k)
+    (a : ℕ → SmoothCoefficients ι E) (b : ι → ι → L2 E)
+    (ha : ∀ i j, Tendsto (fun l => schwartzCoord k (a l i j)) atTop (nhds (b i j))) :
+    Tendsto (fun l => coefficientBCF (a l)) atTop
+      (nhds (fun i j => strongEmbedding k hk (b i j))) := by
+  apply tendsto_pi_nhds.mpr
+  intro i
+  apply tendsto_pi_nhds.mpr
+  intro j
+  have hh := ((strongEmbedding k hk).continuous.tendsto (b i j)).comp (ha i j)
+  simpa only [Function.comp_def,strongEmbedding_schwartz,coefficientBCF] using hh
+
+end SobolevChart
+namespace FrozenPoisson.ParameterRegularity
+open SobolevChart EllipticKernel
+variable {n : ℕ} {ι : Type*} [Fintype ι]
+
+lemma tame_limit_equation (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (t : ℝ) (ht : 1 ≤ t) (v : ι → EC n) (k : ℕ)
+    (hk : (Module.finrank ℝ (EC n):ℝ) < 2*(k:ℝ)) (M θ : ℝ)
+    (x : ℕ → TameData H t v k M θ (ellipticBound H hH))
+    (a : ι → ι → L2 (EC n)) (f u : L2 (EC n))
+    (ha : ∀ i j, Tendsto (fun l => schwartzCoord (k:ℝ) ((x l).a i j)) atTop (nhds (a i j)))
+    (hf : Tendsto (fun l => schwartzCoord (k:ℝ) (x l).f) atTop (nhds f))
+    (hu : Tendsto (fun l => schwartzCoord ((k:ℝ)+2) (x l).u) atTop (nhds u)) :
+    lowerCoord ((k:ℝ)+2) 2 (by simp) u = parameterHilbert H hH t ht
+      (lowerCoord (k:ℝ) 0 (Nat.cast_nonneg k) f +
+        perturbation v (fun i j => strongEmbedding (k:ℝ) hk (a i j))
+          (lowerCoord ((k:ℝ)+2) 2 (by simp) u)) := by
+  have hf0 := ((lowerCoord (k:ℝ) 0 (Nat.cast_nonneg k)).continuous.tendsto f).comp hf
+  have hu2 := ((lowerCoord ((k:ℝ)+2) 2 (by simp)).continuous.tendsto u).comp hu
+  simp only [Function.comp_def,lowerCoord_schwartz] at hf0 hu2
+  have ha0 := coefficientBCF_tendsto (k:ℝ) hk (fun l => (x l).a) a ha
+  have hp := ((continuous_perturbation v).tendsto
+    ((fun i j => strongEmbedding (k:ℝ) hk (a i j)),
+      lowerCoord ((k:ℝ)+2) 2 (by simp) u)).comp (ha0.prodMk_nhds hu2)
+  have hr := ((parameterHilbert H hH t ht).continuous.tendsto _).comp (hf0.add hp)
+  have he l := tame_data_low_equation H hH t ht v k M θ (x l)
+  simp only [Function.comp_def,← he] at hr
+  exact tendsto_nhds_unique hu2 hr
+
+ 
+
+theorem tame_limit_representation (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (t : ℝ) (ht : 1 ≤ t) (v : ι → EC n) (k : ℕ)
+    (hk : Module.finrank ℝ (EC n)+1 < k)
+    (hs : (Module.finrank ℝ (EC n):ℝ) < 2*(k:ℝ))
+    (M : ℝ) (hM : 0 ≤ M) (θ : ℝ) (hθ : θ < 1)
+    (x : ℕ → TameData H t v k M θ (ellipticBound H hH))
+    (a : ι → ι → L2 (EC n)) (f : L2 (EC n))
+    (ha : ∀ i j, Tendsto (fun l => schwartzCoord (k:ℝ) ((x l).a i j)) atTop (nhds (a i j)))
+    (hf : Tendsto (fun l => schwartzCoord (k:ℝ) (x l).f) atTop (nhds f))
+    (hsmall : perturbationBound v (fun i j => strongEmbedding (k:ℝ) hs (a i j))*ellipticBound H hH < 1) :
+    MemSobolev ((k:ℝ)+2) 2 (realize 2 (parameterLocal H hH t ht v
+      (fun i j => strongEmbedding (k:ℝ) hs (a i j)) hsmall (lowerCoord (k:ℝ) 0 (Nat.cast_nonneg k) f))) := by
+  have hc := tame_high_cauchy H hH t ht v k hk M hM θ hθ x hf.cauchySeq
+    (fun i j => (ha i j).cauchySeq)
+  obtain ⟨u,hu⟩ := cauchySeq_tendsto_of_complete hc
+  have he := tame_limit_equation H hH t ht v k hs M θ x a f u ha hf hu
+  have he' := fixed_unique H hH t ht v _ hsmall _ _ _ he
+    (parameterLocal_fixed H hH t ht v _ hsmall _)
+  rw [← he',realize_lowerCoord]
+  exact realize_memSobolev _ _
+
+end FrozenPoisson.ParameterRegularity
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/TameLocalInverse.lean b/lean/OAI/Geometry/PrescribedRicci/TameLocalInverse.lean
new file mode 100644
index 0000000000000000000000000000000000000000..97c4206e209375b0af5d011c1322ec3b97a56593
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/TameLocalInverse.lean
@@ -0,0 +1,209 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.SobolevCommutatorBound
+import OAI.Geometry.PrescribedRicci.SobolevFamilyBounds
+import OAI.Geometry.PrescribedRicci.SobolevInterpolation
+
+namespace OAI
+
+/-! Tame Local Inverse. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory TemperedDistribution LineDeriv
+open scoped SchwartzMap BoundedContinuousFunction ContDiff Classical ComplexOrder MatrixOrder
+namespace SobolevChart
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+variable {ι : Type*} [Fintype ι]
+
+lemma perturbation_commutator_bound (k : ℕ) (hk : Module.finrank ℝ E + 1 < k)
+    (ws : List E) (hws : ws.length ≤ k) (v : ι → E) {M : ℝ} (hM : 0 ≤ M) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ (a : SmoothCoefficients ι E),
+      (∀ i j, ‖schwartzCoord (k:ℝ) (a i j)‖ ≤ M) → ∀ u : 𝓢(E,ℂ),
+      ‖schwartzCoord 0 (schwartzWord ws (smoothPerturbation v a u) -
+        smoothPerturbation v a (schwartzWord ws u))‖ ≤ C * ‖schwartzCoord ((k:ℝ)+1) u‖ := by
+  choose C hC hB using fun i j => second_order_commutator_bound k hk ws hws (v i) (v j)
+  refine ⟨(∑ i, ∑ j, C i j)*M, mul_nonneg (Finset.sum_nonneg (fun i _ =>
+    Finset.sum_nonneg (fun j _ => hC i j))) hM, fun a ha u => ?_⟩
+  have he : schwartzWord ws (smoothPerturbation v a u) -
+      smoothPerturbation v a (schwartzWord ws u) =
+      ∑ i, ∑ j, (schwartzWord ws (SchwartzMap.smulLeftCLM ℂ (a i j) (∂_{v i} (∂_{v j} u))) -
+        SchwartzMap.smulLeftCLM ℂ (a i j) (∂_{v i} (∂_{v j} (schwartzWord ws u)))) := by
+    simp only [smoothPerturbation_apply,map_sum,Finset.sum_sub_distrib]
+  rw [he,schwartzCoord_sum]
+  simp_rw [schwartzCoord_sum]
+  calc
+    _ ≤ ∑ i, ∑ j, C i j * (M*‖schwartzCoord ((k:ℝ)+1) u‖) := by
+      apply (norm_sum_le _ _).trans
+      apply Finset.sum_le_sum
+      intro i _
+      apply (norm_sum_le _ _).trans
+      apply Finset.sum_le_sum
+      intro j _
+      exact (hB i j (a i j) u).trans (mul_le_mul_of_nonneg_left
+        (mul_le_mul_of_nonneg_right (ha i j) (norm_nonneg _)) (hC i j))
+    _ = _ := by simp_rw [← Finset.sum_mul]; ring
+
+end SobolevChart
+
+namespace FrozenPoisson.ParameterRegularity
+open SobolevChart EllipticKernel
+variable {n : ℕ} {ι : Type*} [Fintype ι]
+
+ 
+
+structure TameData (H : Matrix (Fin n) (Fin n) ℂ) (t : ℝ)
+    (v : ι → EC n) (k : ℕ) (M θ E : ℝ) where
+  a : SmoothCoefficients ι (EC n)
+  f : 𝓢(EC n,ℂ)
+  u : 𝓢(EC n,ℂ)
+  coefficient_bound : ∀ i j, ‖schwartzCoord (k:ℝ) (a i j)‖ ≤ M
+  small : perturbationBound v (coefficientBCF a)*E ≤ θ
+  equation : u = schwartzResolvent H t (f + smoothPerturbation v a u)
+
+lemma tame_word_bound (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (t : ℝ) (ht : 1 ≤ t) (v : ι → EC n) (k : ℕ)
+    (hk : Module.finrank ℝ (EC n)+1 < k) (M : ℝ) (hM : 0 ≤ M)
+    (θ : ℝ) (hθ : θ < 1) (ws : List (EC n)) (hws : ws.length ≤ k) :
+    FamilyBound 2 (fun x : TameData H t v k M θ (ellipticBound H hH) =>
+      ‖schwartzCoord (k:ℝ) x.f‖ + ‖schwartzCoord ((k:ℝ)+1) x.u‖)
+      (fun x => schwartzWord ws x.u) := by
+  have hw : (ws.length : ℝ) ≤ k := by exact_mod_cast hws
+  obtain ⟨C,hC,hc⟩ := coreBound_word ws (s:=(k:ℝ)) (t:=0) (by linarith)
+  obtain ⟨D,hD,hd⟩ := perturbation_commutator_bound k hk ws hws v hM
+  let B := ellipticBound H hH
+  have hB : 0 ≤ B := (ellipticBound_pos H hH).le
+  refine ⟨B*(C+D)/(1-θ),div_nonneg (mul_nonneg hB (add_nonneg hC hD)) (by linarith),fun x => ?_⟩
+  let z := schwartzWord ws x.u
+  let e := schwartzWord ws (smoothPerturbation v x.a x.u)-smoothPerturbation v x.a z
+  have he : z = schwartzResolvent H t (smoothPerturbation v x.a z +
+      (schwartzWord ws x.f + e)) := by
+    have he := congrArg (schwartzWord ws) x.equation
+    rw [schwartz_word_resolvent ws H hH t ht,map_add] at he
+    change z = _ at he
+    apply he.trans
+    congr 1
+    dsimp only [e]
+    abel
+  have hr := schwartzResolvent_bound H hH t ht
+    (smoothPerturbation v x.a z+(schwartzWord ws x.f+e))
+  rw [← he,schwartzCoord_add,schwartzCoord_add] at hr
+  have hg := (norm_add_le (schwartzCoord 0 (smoothPerturbation v x.a z))
+    (schwartzCoord 0 (schwartzWord ws x.f)+schwartzCoord 0 e)).trans
+      (add_le_add le_rfl (norm_add_le _ _))
+  have hh := hr.trans (mul_le_mul_of_nonneg_left hg hB)
+  have hp := smoothPerturbation_bound v x.a z
+  have hf := hc x.f
+  have hc' := hd x.a x.coefficient_bound x.u
+  have hs := mul_le_mul_of_nonneg_right x.small (norm_nonneg (schwartzCoord 2 z))
+  have hy : (1-θ)*‖schwartzCoord 2 z‖ ≤
+      B*(C+D)*(‖schwartzCoord (k:ℝ) x.f‖+‖schwartzCoord ((k:ℝ)+1) x.u‖) := by
+    have h0 := mul_le_mul_of_nonneg_left hp hB
+    have h1 := mul_le_mul_of_nonneg_left hf hB
+    have h2 := mul_le_mul_of_nonneg_left hc' hB
+    change ‖schwartzCoord 0 e‖ ≤ _ at hc'
+    dsimp only [B] at *
+    nlinarith [mul_nonneg (mul_nonneg hB hC) (norm_nonneg (schwartzCoord ((k:ℝ)+1) x.u)),
+      mul_nonneg (mul_nonneg hB hD) (norm_nonneg (schwartzCoord (k:ℝ) x.f))]
+  rw [div_mul_eq_mul_div,le_div_iff₀ (show 0 < 1-θ by linarith)]
+  simpa only [z,mul_comm] using hy
+
+lemma tame_preestimate (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (t : ℝ) (ht : 1 ≤ t) (v : ι → EC n) (k : ℕ)
+    (hk : Module.finrank ℝ (EC n)+1 < k) (M : ℝ) (hM : 0 ≤ M)
+    (θ : ℝ) (hθ : θ < 1) :
+    FamilyBound ((k:ℝ)+2) (fun x : TameData H t v k M θ (ellipticBound H hH) =>
+      ‖schwartzCoord (k:ℝ) x.f‖ + ‖schwartzCoord ((k:ℝ)+1) x.u‖) (fun x => x.u) := by
+  rw [add_comm]
+  exact familyBound_of_words_integer k _ (fun ws hws => tame_word_bound H hH t ht v k hk M hM θ hθ ws hws)
+
+ 
+
+theorem tame_estimate (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (t : ℝ) (ht : 1 ≤ t) (v : ι → EC n) (k : ℕ)
+    (hk : Module.finrank ℝ (EC n)+1 < k) (M : ℝ) (hM : 0 ≤ M)
+    (θ : ℝ) (hθ : θ < 1) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ x : TameData H t v k M θ (ellipticBound H hH),
+      ‖schwartzCoord ((k:ℝ)+2) x.u‖ ≤
+        C*(‖schwartzCoord (k:ℝ) x.f‖+‖schwartzCoord 2 x.u‖) := by
+  obtain ⟨C,hC,hc⟩ := tame_preestimate H hH t ht v k hk M hM θ hθ
+  let D := C+1
+  have hD : 0 < D := by dsimp [D]; linarith
+  let ε := (2*D)⁻¹
+  have hε : 0 < ε := inv_pos.mpr (mul_pos (by norm_num) hD)
+  have he : D*ε = 1/2 := by dsimp [ε]; field_simp
+  obtain ⟨J,hJ,hj⟩ := schwartz_sobolev_interpolation (E:=EC n) 2 ((k:ℝ)+1) ((k:ℝ)+2)
+    (by linarith) hε
+  refine ⟨2*D*(1+J),by positivity,fun x => ?_⟩
+  have h1 := hj x.u
+  have h0 := (hc x).trans (mul_le_mul_of_nonneg_right
+    (show C ≤ D by dsimp [D]; linarith)
+    (add_nonneg (norm_nonneg _) (norm_nonneg _)))
+  have h2 := mul_le_mul_of_nonneg_left h1 hD.le
+  have h3 : D*(ε*‖schwartzCoord ((k:ℝ)+2) x.u‖) =
+      (1/2)*‖schwartzCoord ((k:ℝ)+2) x.u‖ := by rw [← mul_assoc,he]
+  nlinarith [mul_nonneg (mul_nonneg hD.le hJ) (norm_nonneg (schwartzCoord (k:ℝ) x.f)),
+    mul_nonneg hD.le (norm_nonneg (schwartzCoord 2 x.u))]
+
+private lemma combine_estimate {a f u C B : ℝ} (hC : 0 ≤ C)
+    (h : a ≤ C*(f+u)) (hu : u ≤ B*f) : a ≤ C*(1+B)*f := by
+  have hh := mul_le_mul_of_nonneg_left hu hC
+  nlinarith
+
+lemma tame_low_estimate (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (t : ℝ) (ht : 1 ≤ t) (v : ι → EC n) (k : ℕ) (M θ : ℝ) (hθ : θ < 1) :
+    ∃ B : ℝ, 0 ≤ B ∧ ∀ x : TameData H t v k M θ (ellipticBound H hH),
+      ‖schwartzCoord 2 x.u‖ ≤ B*‖schwartzCoord (k:ℝ) x.f‖ := by
+  obtain ⟨J,hJ,hj⟩ := coreBound_word ([] : List (EC n)) (s:=(k:ℝ)) (t:=0) (by simp)
+  let B := ellipticBound H hH*J/(1-θ)
+  have hB : 0 ≤ B := div_nonneg (mul_nonneg (ellipticBound_pos H hH).le hJ) (by linarith)
+  refine ⟨B,hB,fun x => ?_⟩
+  have hh := schwartzResolvent_bound H hH t ht (x.f+smoothPerturbation v x.a x.u)
+  rw [← x.equation,schwartzCoord_add] at hh
+  have hh' := hh.trans (mul_le_mul_of_nonneg_left (norm_add_le _ _) (ellipticBound_pos H hH).le)
+  have hp := mul_le_mul_of_nonneg_left (smoothPerturbation_bound v x.a x.u) (ellipticBound_pos H hH).le
+  have hsmall := mul_le_mul_of_nonneg_right x.small (norm_nonneg (schwartzCoord 2 x.u))
+  have hforce := mul_le_mul_of_nonneg_left (hj x.f) (ellipticBound_pos H hH).le
+  dsimp only [B]
+  rw [div_mul_eq_mul_div,le_div_iff₀ (show 0 < 1-θ by linarith)]
+  simp only [schwartzWord,ContinuousLinearMap.id_apply] at hforce
+  nlinarith
+
+ 
+
+theorem tame_solution_bound (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (t : ℝ) (ht : 1 ≤ t) (v : ι → EC n) (k : ℕ)
+    (hk : Module.finrank ℝ (EC n)+1 < k) (M : ℝ) (hM : 0 ≤ M)
+    (θ : ℝ) (hθ : θ < 1) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ x : TameData H t v k M θ (ellipticBound H hH),
+      ‖schwartzCoord ((k:ℝ)+2) x.u‖ ≤ C*‖schwartzCoord (k:ℝ) x.f‖ := by
+  obtain ⟨C,hC,hc⟩ := tame_estimate H hH t ht v k hk M hM θ hθ
+  obtain ⟨B,hB,hb⟩ := tame_low_estimate H hH t ht v k M θ hθ
+  refine ⟨C*(1+B),mul_nonneg hC (by linarith),fun x => ?_⟩
+  exact combine_estimate hC (hc x) (hb x)
+
+ 
+
+theorem tame_inverse_bound (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (t : ℝ) (ht : 1 ≤ t) (v : ι → EC n) (k : ℕ)
+    (hk : Module.finrank ℝ (EC n)+1 < k) (M : ℝ) (hM : 0 ≤ M)
+    (θ : ℝ) (hθ : θ < 1) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ (a : SmoothCoefficients ι (EC n))
+      (_ha : ∀ i j, ‖schwartzCoord (k:ℝ) (a i j)‖ ≤ M)
+      (hs : perturbationBound v (coefficientBCF a)*ellipticBound H hH ≤ θ),
+      ∀ f : 𝓢(EC n,ℂ), ‖schwartzCoord ((k:ℝ)+2)
+        (schwartzLocal H hH t ht v a (hs.trans_lt hθ) f)‖ ≤ C*‖schwartzCoord (k:ℝ) f‖ := by
+  obtain ⟨C,hC,hc⟩ := tame_solution_bound H hH t ht v k hk M hM θ hθ
+  refine ⟨C,hC,fun a ha hs f => ?_⟩
+  exact hc ⟨a,f,schwartzLocal H hH t ht v a (hs.trans_lt hθ) f,ha,hs,
+    schwartzLocal_fixed H hH t ht v a (hs.trans_lt hθ) f⟩
+
+end FrozenPoisson.ParameterRegularity
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/TameRoughInverse.lean b/lean/OAI/Geometry/PrescribedRicci/TameRoughInverse.lean
new file mode 100644
index 0000000000000000000000000000000000000000..0a18ef95ba6c5d5c1c4ba66d262113b4d39bd902
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/TameRoughInverse.lean
@@ -0,0 +1,82 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.TameLimit
+
+namespace OAI
+
+/-! Tame Rough Inverse. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory TemperedDistribution LineDeriv
+open scoped SchwartzMap BoundedContinuousFunction ContDiff Classical ComplexOrder MatrixOrder
+namespace FrozenPoisson.ParameterRegularity
+open SobolevChart EllipticKernel
+variable {n : ℕ} {ι : Type*} [Fintype ι]
+
+lemma exists_tame_approximants (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (t : ℝ) (ht : 1 ≤ t) (v : ι → EC n) (k : ℕ)
+    (hs : (Module.finrank ℝ (EC n):ℝ) < 2*(k:ℝ))
+    (a : ι → ι → L2 (EC n)) (f : L2 (EC n))
+    (hsmall : perturbationBound v (fun i j => strongEmbedding (k:ℝ) hs (a i j))*ellipticBound H hH < 1) :
+    ∃ (M θ : ℝ) (_ : 0 ≤ M) (_ : θ < 1)
+      (x : ℕ → TameData H t v k M θ (ellipticBound H hH)),
+      (∀ i j, Tendsto (fun l => schwartzCoord (k:ℝ) ((x l).a i j)) atTop (nhds (a i j))) ∧
+      Tendsto (fun l => schwartzCoord (k:ℝ) (x l).f) atTop (nhds f) := by
+  choose aa haa using fun i j => exists_schwartz_approx (k:ℝ) (a i j)
+  obtain ⟨ff,hff⟩ := exists_schwartz_approx (k:ℝ) f
+  let asq : ℕ → SmoothCoefficients ι (EC n) := fun l i j => aa i j l
+  let p := perturbationBound v (fun i j => strongEmbedding (k:ℝ) hs (a i j))*ellipticBound H hH
+  let θ := (p+1)/2
+  have hθp : p < θ := by dsimp [θ,p]; linarith
+  have hθ : θ < 1 := by dsimp [θ,p]; linarith
+  have ha0 := coefficientBCF_tendsto (k:ℝ) hs asq a haa
+  have hp : Tendsto (fun l => perturbationBound v (coefficientBCF (asq l))*ellipticBound H hH)
+      atTop (nhds p) := ((continuous_perturbationBound v).tendsto _ |>.comp ha0).mul_const _
+  have hsm : ∀ᶠ l in atTop, perturbationBound v (coefficientBCF (asq l))*ellipticBound H hH ≤ θ :=
+    (hp.eventually (gt_mem_nhds hθp)).mono (fun _ h => h.le)
+  let M := (∑ i, ∑ j, ‖a i j‖)+1
+  have hM : 0 ≤ M := by dsimp [M]; positivity
+  have hMa (i j : ι) : ‖a i j‖+1 ≤ M := by
+    have hij : ‖a i j‖ ≤ ∑ i, ∑ j, ‖a i j‖ :=
+      (Finset.single_le_sum (fun l _ => norm_nonneg (a i l)) (Finset.mem_univ j)).trans
+        (Finset.single_le_sum (fun l _ => Finset.sum_nonneg (fun m _ => norm_nonneg (a l m))) (Finset.mem_univ i))
+    exact add_le_add hij le_rfl
+  have hbd : ∀ᶠ l in atTop, ∀ i j, ‖schwartzCoord (k:ℝ) (asq l i j)‖ ≤ M := by
+    apply eventually_all.mpr
+    intro i
+    apply eventually_all.mpr
+    intro j
+    have hh := (haa i j).norm.eventually (gt_mem_nhds (show ‖a i j‖ < ‖a i j‖+1 by linarith))
+    exact hh.mono (fun _ hl => hl.le.trans (hMa i j))
+  obtain ⟨N,hN⟩ := eventually_atTop.mp (hbd.and hsm)
+  have hn (l : ℕ) := hN (l+N) (Nat.le_add_left N l)
+  let x (l : ℕ) : TameData H t v k M θ (ellipticBound H hH) :=
+    ⟨asq (l+N),ff (l+N),schwartzLocal H hH t ht v (asq (l+N)) ((hn l).2.trans_lt hθ) (ff (l+N)),
+      (hn l).1,(hn l).2,
+      schwartzLocal_fixed H hH t ht v (asq (l+N)) ((hn l).2.trans_lt hθ) (ff (l+N))⟩
+  refine ⟨M,θ,hM,hθ,x,fun i j => ?_,?_⟩
+  · exact (haa i j).comp (tendsto_add_atTop_nat N)
+  · exact hff.comp (tendsto_add_atTop_nat N)
+
+ 
+
+theorem rough_local_memSobolev (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (t : ℝ) (ht : 1 ≤ t) (v : ι → EC n) (k : ℕ)
+    (hk : Module.finrank ℝ (EC n)+1 < k)
+    (hs : (Module.finrank ℝ (EC n):ℝ) < 2*(k:ℝ))
+    (a : ι → ι → L2 (EC n)) (f : L2 (EC n))
+    (hsmall : perturbationBound v (fun i j => strongEmbedding (k:ℝ) hs (a i j))*ellipticBound H hH < 1) :
+    MemSobolev ((k:ℝ)+2) 2 (realize 2 (parameterLocal H hH t ht v
+      (fun i j => strongEmbedding (k:ℝ) hs (a i j)) hsmall (lowerCoord (k:ℝ) 0 (Nat.cast_nonneg k) f))) := by
+  obtain ⟨M,θ,hM,hθ,x,ha,hf⟩ := exists_tame_approximants H hH t ht v k hs a f hsmall
+  exact tame_limit_representation H hH t ht v k hk hs M hM θ hθ x a f ha hf hsmall
+
+end FrozenPoisson.ParameterRegularity
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/TameStability.lean b/lean/OAI/Geometry/PrescribedRicci/TameStability.lean
new file mode 100644
index 0000000000000000000000000000000000000000..33d3e77a30a4cc0f91650c9a3ee0fb514619190f
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/TameStability.lean
@@ -0,0 +1,123 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.TameLocalInverse
+
+namespace OAI
+
+/-! Tame Stability. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology _root_.MeasureTheory _root_.OAI.MeasureTheory TemperedDistribution LineDeriv
+open scoped SchwartzMap BoundedContinuousFunction ContDiff Classical ComplexOrder MatrixOrder
+namespace SobolevChart
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+variable {ι : Type*} [Fintype ι]
+
+omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] in
+lemma smoothPerturbation_coeff_sub (v : ι → E) (a b : SmoothCoefficients ι E)
+    (u : 𝓢(E,ℂ)) : smoothPerturbation v (a-b) u =
+      smoothPerturbation v a u - smoothPerturbation v b u := by
+  ext x
+  simp only [smoothPerturbation_apply,_root_.sum_apply,_root_.sub_apply,
+    Pi.sub_apply]
+  simp only [SchwartzMap.smulLeftCLM_apply_apply (a _ _).hasTemperateGrowth,
+    SchwartzMap.smulLeftCLM_apply_apply (b _ _).hasTemperateGrowth,
+    SchwartzMap.smulLeftCLM_apply_apply (a _ _ - b _ _).hasTemperateGrowth,
+    _root_.sub_apply,smul_eq_mul,sub_mul,Finset.sum_sub_distrib]
+
+lemma perturbation_high_bound (k : ℕ) (hk : Module.finrank ℝ E < k) (v : ι → E) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ (a : SmoothCoefficients ι E) (u : 𝓢(E,ℂ)),
+      ‖schwartzCoord (k:ℝ) (smoothPerturbation v a u)‖ ≤
+        C * (∑ i, ∑ j, ‖schwartzCoord (k:ℝ) (a i j)‖) *
+          ‖schwartzCoord ((k:ℝ)+2) u‖ := by
+  obtain ⟨C,hC,hc⟩ := schwartz_sobolev_product_bound (E:=E) k hk
+  choose D hD hd using fun i j => coreBound_word [v i,v j] (s:=(k:ℝ)+2) (t:=(k:ℝ)) (by simp)
+  let B := ∑ i, ∑ j, D i j
+  have hB : 0 ≤ B := Finset.sum_nonneg (fun i _ => Finset.sum_nonneg (fun j _ => hD i j))
+  have hDB (i j : ι) : D i j ≤ B :=
+    (Finset.single_le_sum (fun l _ => hD i l) (Finset.mem_univ j)).trans
+      (Finset.single_le_sum (fun l _ => Finset.sum_nonneg (fun m _ => hD l m)) (Finset.mem_univ i))
+  refine ⟨C*B,mul_nonneg hC hB,fun a u => ?_⟩
+  rw [smoothPerturbation_apply,schwartzCoord_sum]
+  simp_rw [schwartzCoord_sum]
+  calc
+    _ ≤ ∑ i, ∑ j, C*B*‖schwartzCoord (k:ℝ) (a i j)‖*‖schwartzCoord ((k:ℝ)+2) u‖ := by
+      apply (norm_sum_le _ _).trans
+      apply Finset.sum_le_sum
+      intro i _
+      apply (norm_sum_le _ _).trans
+      apply Finset.sum_le_sum
+      intro j _
+      apply (hc (a i j) (∂_{v i} (∂_{v j} u))).trans
+      have hb := (hd i j u).trans (mul_le_mul_of_nonneg_right (hDB i j) (norm_nonneg _))
+      change ‖schwartzCoord (k:ℝ) (∂_{v i} (∂_{v j} u))‖ ≤ _ at hb
+      have hh := mul_le_mul_of_nonneg_left hb (mul_nonneg hC (norm_nonneg (schwartzCoord (k:ℝ) (a i j))))
+      convert hh using 1 <;> ring
+    _ = _ := by simp_rw [← Finset.sum_mul,← Finset.mul_sum]
+
+end SobolevChart
+namespace FrozenPoisson.ParameterRegularity
+open SobolevChart EllipticKernel
+variable {n : ℕ} {ι : Type*} [Fintype ι]
+
+lemma tame_difference_equation (H : Matrix (Fin n) (Fin n) ℂ) (t : ℝ)
+    (v : ι → EC n) (k : ℕ) (M θ E : ℝ) (x y : TameData H t v k M θ E) :
+    x.u-y.u = schwartzResolvent H t
+      ((x.f-y.f+smoothPerturbation v (x.a-y.a) y.u) + smoothPerturbation v x.a (x.u-y.u)) := by
+  rw [smoothPerturbation_coeff_sub,map_sub]
+  have he : x.f-y.f+(smoothPerturbation v x.a y.u-smoothPerturbation v y.a y.u)+
+      (smoothPerturbation v x.a x.u-smoothPerturbation v x.a y.u) =
+      (x.f+smoothPerturbation v x.a x.u)-(y.f+smoothPerturbation v y.a y.u) := by abel
+  rw [he,map_sub,← x.equation,← y.equation]
+
+lemma tame_difference_bound (H : Matrix (Fin n) (Fin n) ℂ) (t : ℝ)
+    (v : ι → EC n) (k : ℕ) (M θ E C : ℝ)
+    (hc : ∀ x : TameData H t v k M θ E,
+      ‖schwartzCoord ((k:ℝ)+2) x.u‖ ≤ C*‖schwartzCoord (k:ℝ) x.f‖)
+    (x y : TameData H t v k M θ E) :
+    ‖schwartzCoord ((k:ℝ)+2) (x.u-y.u)‖ ≤ C*‖schwartzCoord (k:ℝ)
+      (x.f-y.f+smoothPerturbation v (x.a-y.a) y.u)‖ :=
+  hc ⟨x.a,x.f-y.f+smoothPerturbation v (x.a-y.a) y.u,x.u-y.u,
+    x.coefficient_bound,x.small,tame_difference_equation H t v k M θ E x y⟩
+
+lemma high_forcing_difference_bound (k : ℕ) (v : ι → EC n) (B : ℝ)
+    (hb : ∀ (a : SmoothCoefficients ι (EC n)) (u : 𝓢(EC n,ℂ)),
+      ‖schwartzCoord (k:ℝ) (smoothPerturbation v a u)‖ ≤
+        B*(∑ i, ∑ j, ‖schwartzCoord (k:ℝ) (a i j)‖)*‖schwartzCoord ((k:ℝ)+2) u‖)
+    (a b : SmoothCoefficients ι (EC n)) (f g u : 𝓢(EC n,ℂ)) :
+    ‖schwartzCoord (k:ℝ) (f-g+smoothPerturbation v (a-b) u)‖ ≤
+      ‖schwartzCoord (k:ℝ) (f-g)‖+
+        B*(∑ i, ∑ j, ‖schwartzCoord (k:ℝ) (a i j-b i j)‖)*‖schwartzCoord ((k:ℝ)+2) u‖ := by
+  rw [schwartzCoord_add]
+  have h1 := norm_add_le (schwartzCoord (k:ℝ) (f-g))
+    (schwartzCoord (k:ℝ) (smoothPerturbation v (a-b) u))
+  have h2 := add_le_add (le_refl ‖schwartzCoord (k:ℝ) (f-g)‖) (hb (a-b) u)
+  exact h1.trans h2
+
+ 
+
+theorem tame_stability (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    (t : ℝ) (ht : 1 ≤ t) (v : ι → EC n) (k : ℕ)
+    (hk : Module.finrank ℝ (EC n)+1 < k) (M : ℝ) (hM : 0 ≤ M)
+    (θ : ℝ) (hθ : θ < 1) :
+    ∃ C B : ℝ, 0 ≤ C ∧ 0 ≤ B ∧ ∀ x y : TameData H t v k M θ (ellipticBound H hH),
+      ‖schwartzCoord ((k:ℝ)+2) (x.u-y.u)‖ ≤ C *
+        (‖schwartzCoord (k:ℝ) (x.f-y.f)‖ +
+          B * (∑ i, ∑ j, ‖schwartzCoord (k:ℝ) (x.a i j-y.a i j)‖) *
+            ‖schwartzCoord ((k:ℝ)+2) y.u‖) := by
+  obtain ⟨C,hC,hc⟩ := tame_solution_bound H hH t ht v k hk M hM θ hθ
+  obtain ⟨B,hB,hb⟩ := perturbation_high_bound k (by omega) v
+  refine ⟨C,B,hC,hB,fun x y => ?_⟩
+  exact (tame_difference_bound H t v k M θ _ C hc x y).trans
+    (mul_le_mul_of_nonneg_left (high_forcing_difference_bound k v B hb x.a y.a x.f y.f y.u) hC)
+
+end FrozenPoisson.ParameterRegularity
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/TensorEnergyBounds.lean b/lean/OAI/Geometry/PrescribedRicci/TensorEnergyBounds.lean
new file mode 100644
index 0000000000000000000000000000000000000000..0f9f40e33a1dce220441e361b0d26977a28639ae
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/TensorEnergyBounds.lean
@@ -0,0 +1,54 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.HermitianCoercivity
+import OAI.Geometry.PrescribedRicci.HermitianTensorEnergy
+import OAI.Geometry.PrescribedRicci.MatrixCompactBounds
+
+namespace OAI
+
+/-! Tensor Energy Bounds. -/
+
+noncomputable section
+open Matrix
+open scoped ComplexOrder MatrixOrder Kronecker Matrix.Norms.Elementwise
+namespace MongeAmpere
+variable {n m : Type*} [Fintype n] [Fintype m] [DecidableEq n] [DecidableEq m]
+
+omit [DecidableEq n] [DecidableEq m] in
+lemma kronecker_norm_le (B : Matrix m m ℂ) (K : Matrix n n ℂ) : ‖B ⊗ₖ K‖ ≤ ‖B‖*‖K‖ := by
+  apply matrix_norm_le_of_entries (by positivity)
+  intro i j
+  rw [Matrix.kronecker_apply,norm_mul]
+  exact mul_le_mul (norm_entry_le_entrywise_sup_norm B) (norm_entry_le_entrywise_sup_norm K)
+    (norm_nonneg _) (norm_nonneg _)
+
+lemma kronecker_inv {B : Matrix m m ℂ} {K : Matrix n n ℂ} (hB : B.PosDef) (hK : K.PosDef) :
+    (B ⊗ₖ K)⁻¹ = B⁻¹ ⊗ₖ K⁻¹ := by
+  apply Matrix.inv_eq_right_inv
+  rw [← Matrix.mul_kronecker_mul,
+    Matrix.mul_nonsing_inv _ (isUnit_iff_ne_zero.mpr hB.det_pos.ne'),
+    Matrix.mul_nonsing_inv _ (isUnit_iff_ne_zero.mpr hK.det_pos.ne'),Matrix.one_kronecker_one]
+
+lemma vectorNorm_sq_le_tensorEnergy {B : Matrix m m ℂ} {K : Matrix n n ℂ}
+    (hB : B.PosDef) (hK : K.PosDef) (V : m → n → ℂ) (a : m) :
+    ‖V a‖^2 ≤ ‖B⁻¹‖*‖K⁻¹‖*(tensorEnergy B K V).re := by
+  have hE := tensorEnergy_nonneg hB.posSemidef hK.posSemidef V
+  have hbound : ∀ i : n, ‖V a i‖^2 ≤ ‖B⁻¹‖*‖K⁻¹‖*(tensorEnergy B K V).re := by
+    intro i
+    have h := component_sq_le_hermPair (hB.kronecker hK) (fun p : m × n => V p.1 p.2) (a,i)
+    rw [← tensorEnergy_eq,kronecker_inv hB hK] at h
+    apply h.trans
+    apply mul_le_mul_of_nonneg_right _ hE
+    exact (Complex.re_le_norm _).trans ((norm_entry_le_entrywise_sup_norm _).trans
+      (kronecker_norm_le _ _))
+  have hp : 0 ≤ ‖B⁻¹‖*‖K⁻¹‖*(tensorEnergy B K V).re := by positivity
+  have hv : ‖V a‖ ≤ Real.sqrt (‖B⁻¹‖*‖K⁻¹‖*(tensorEnergy B K V).re) := by
+    apply (pi_norm_le_iff_of_nonneg (Real.sqrt_nonneg _)).mpr
+    intro i
+    exact (Real.le_sqrt (norm_nonneg _) hp).mpr (hbound i)
+  exact ((sq_le_sq₀ (norm_nonneg _) (Real.sqrt_nonneg _)).mpr hv).trans_eq (Real.sq_sqrt hp)
+
+end MongeAmpere
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/TraceCauchySchwarz.lean b/lean/OAI/Geometry/PrescribedRicci/TraceCauchySchwarz.lean
new file mode 100644
index 0000000000000000000000000000000000000000..31d08bc142c62b2541ce7f94436afaa8ae135d0f
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/TraceCauchySchwarz.lean
@@ -0,0 +1,51 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.AnalyticSupport
+
+namespace OAI
+
+/-! Trace Cauchy Schwarz. -/
+
+noncomputable section
+open Matrix
+open scoped ComplexOrder MatrixOrder
+namespace MongeAmpere
+variable {n : Type*} [Fintype n] [DecidableEq n]
+
+@[instance_reducible] def tracePairCore (B K : Matrix n n ℂ) (hB : B.PosSemidef) (hK : K.PosSemidef) :
+    PreInnerProductSpace.Core ℂ (Matrix n n ℂ) where
+  inner U V := (B*Uᴴ*K*V).trace
+  conj_inner_symm U V := by
+    change star (B*Vᴴ*K*U).trace = _
+    rw [← trace_conjTranspose]
+    simp only [conjTranspose_mul,conjTranspose_conjTranspose,hB.isHermitian.eq,hK.isHermitian.eq]
+    simp only [← mul_assoc]
+    simpa only [mul_assoc] using (Matrix.trace_mul_comm (Uᴴ*K*V) B)
+  re_inner_nonneg U := by
+    change 0 ≤ (B*Uᴴ*K*U).trace.re
+    simpa only [mul_assoc] using
+      EllipticKernel.trace_mul_nonneg hB (hK.conjTranspose_mul_mul_same U)
+  add_left U V W := by simp only [conjTranspose_add,mul_add,add_mul,trace_add]
+  smul_left U V c := by simp only [conjTranspose_smul,Matrix.mul_smul,Matrix.smul_mul,trace_smul,smul_eq_mul]; rfl
+
+lemma trace_cauchySchwarz (B G W : Matrix n n ℂ) (hB : B.PosSemidef)
+    (hG : G.PosDef) :
+    ‖(B*W).trace‖^2 ≤ (B*G).trace.re * (B*Wᴴ*G⁻¹*W).trace.re := by
+  let := tracePairCore B G⁻¹ hB hG.inv.posSemidef
+  have hh := InnerProductSpace.Core.inner_mul_inner_self_le (𝕜 := ℂ) G W
+  have hunit : IsUnit G.det := isUnit_iff_ne_zero.mpr hG.det_pos.ne'
+  have h1 : (B*Gᴴ*G⁻¹*W).trace = (B*W).trace := by
+    rw [hG.isHermitian.eq,mul_assoc B G,Matrix.mul_nonsing_inv G hunit,mul_one]
+  have h2 : (B*Wᴴ*G⁻¹*G).trace = star (B*W).trace := by
+    rw [mul_assoc (B*Wᴴ),Matrix.nonsing_inv_mul G hunit,mul_one,← trace_conjTranspose]
+    rw [conjTranspose_mul,hB.isHermitian.eq,trace_mul_comm]
+  have h3 : (B*Gᴴ*G⁻¹*G).trace = (B*G).trace := by
+    rw [hG.isHermitian.eq,mul_assoc B G,Matrix.mul_nonsing_inv G hunit,mul_one]
+  change ‖(B*Gᴴ*G⁻¹*W).trace‖ * ‖(B*Wᴴ*G⁻¹*G).trace‖ ≤
+    (B*Gᴴ*G⁻¹*G).trace.re * (B*Wᴴ*G⁻¹*W).trace.re at hh
+  simpa only [h1,h2,h3,norm_star,pow_two] using hh
+
+end MongeAmpere
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/TraceEntryBounds.lean b/lean/OAI/Geometry/PrescribedRicci/TraceEntryBounds.lean
new file mode 100644
index 0000000000000000000000000000000000000000..9caea9f37644b94bbed1d6221a9a575068703052
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/TraceEntryBounds.lean
@@ -0,0 +1,66 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.KaehlerTrace
+import OAI.Geometry.PrescribedRicci.TraceCauchySchwarz
+
+namespace OAI
+
+/-! Trace Entry Bounds. -/
+
+noncomputable section
+open Matrix
+open scoped ComplexOrder MatrixOrder
+namespace MongeAmpere
+variable {n : Type*} [Fintype n] [DecidableEq n]
+
+lemma trace_single_quadratic (B K : Matrix n n ℂ) (i j : n) :
+    (B*(Matrix.single i j 1)ᴴ*K*Matrix.single i j 1).trace = K i i*B j j := by
+  rw [Matrix.conjTranspose_single]
+  simp only [star_one]
+  rw [mul_assoc B (Matrix.single j i 1),mul_assoc B (Matrix.single j i 1*K),
+    Matrix.single_mul_mul_single,one_mul,mul_one,Matrix.trace_mul_single]
+  simp only [op_smul_eq_mul]
+  exact mul_comm _ _
+
+lemma entry_sq_le_relative_trace (B G : Matrix n n ℂ) (hB : B.PosSemidef)
+    (hG : G.PosDef) (i j : n) :
+    ‖B j i‖^2 ≤ (B*G).trace.re * (G⁻¹ i i).re * (B j j).re := by
+  have hc := trace_cauchySchwarz B G (Matrix.single i j 1) hB hG
+  rw [Matrix.trace_mul_single,op_smul_eq_mul,mul_one,trace_single_quadratic] at hc
+  have him := (Complex.nonneg_iff.mp (hG.inv.posSemidef.diag_nonneg (i:=i))).2.symm
+  simpa only [Complex.mul_re,him,zero_mul,sub_zero,mul_assoc] using hc
+
+lemma diag_le_relative_trace (B G : Matrix n n ℂ) (hB : B.PosDef)
+    (hG : G.PosDef) (i : n) :
+    (B i i).re ≤ (B*G).trace.re * (G⁻¹ i i).re := by
+  have hb := (Complex.pos_iff.mp (hB.diag_pos (i:=i))).1
+  have hc := entry_sq_le_relative_trace B G hB.posSemidef hG i i
+  have hi := (Complex.pos_iff.mp (hB.diag_pos (i:=i))).2.symm
+  rw [Complex.sq_norm,Complex.normSq_apply,hi,mul_zero,add_zero] at hc
+  nlinarith
+
+lemma entry_le_relative_trace [Nonempty n] (B G : Matrix n n ℂ)
+    (hB : B.PosDef) (hG : G.PosDef) {M : ℝ} (hM : 0 ≤ M)
+    (hdiag : ∀ i, (G⁻¹ i i).re ≤ M) (i j : n) :
+    ‖B i j‖ ≤ M*(B*G).trace.re := by
+  have hs := (trace_mul_pos hB hG).le
+  have hb := (Complex.pos_iff.mp (hB.diag_pos (i:=i))).1.le
+  have hd := diag_le_relative_trace B G hB hG i
+  have hd' : (B i i).re ≤ M*(B*G).trace.re := by
+    calc
+      _ ≤ (B*G).trace.re*(G⁻¹ i i).re := hd
+      _ ≤ (B*G).trace.re*M := mul_le_mul_of_nonneg_left (hdiag i) hs
+      _ = _ := mul_comm _ _
+  have hc := entry_sq_le_relative_trace B G hB.posSemidef hG j i
+  have hx : (B*G).trace.re*(G⁻¹ j j).re*(B i i).re ≤ (M*(B*G).trace.re)^2 := by
+    calc
+      _ ≤ ((B*G).trace.re*M)*(B i i).re := mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left (hdiag j) hs) hb
+      _ ≤ ((B*G).trace.re*M)*(M*(B*G).trace.re) := mul_le_mul_of_nonneg_left hd' (mul_nonneg hs hM)
+      _ = _ := by ring
+  have hpos := mul_nonneg hM hs
+  nlinarith [norm_nonneg (B i j)]
+
+end MongeAmpere
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/TraceMaximum.lean b/lean/OAI/Geometry/PrescribedRicci/TraceMaximum.lean
new file mode 100644
index 0000000000000000000000000000000000000000..8c61b842b4a89aa4b1d6ca3ea4d50972b863681a
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/TraceMaximum.lean
@@ -0,0 +1,96 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.KaehlerLogTrace
+import OAI.Geometry.PrescribedRicci.KaehlerTrace
+import OAI.Geometry.PrescribedRicci.MongeAmpereC0
+import OAI.Geometry.PrescribedRicci.UniformChernLu
+import OAI.Geometry.PrescribedPotential.SmoothPotentialDifference
+import OAI.Geometry.PrescribedPotential.VolumePath
+
+namespace OAI
+
+/-! Trace Maximum. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Matrix
+open scoped ContDiff ComplexOrder MatrixOrder
+namespace Anticanonical.SourceSmooth.KaehlerMetric
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+
+lemma laplacian_addFunction (g : KaehlerMetric A) (φ ψ : SmoothRealFunction A) (x : X) :
+    (g.laplacian (φ.addFunction ψ)).value x = (g.laplacian φ).value x+(g.laplacian ψ).value x := by
+  obtain ⟨q,hq⟩ := A.covers x
+  change g.laplacianValue _ x = g.laplacianValue _ x+g.laplacianValue _ x
+  rw [g.laplacianValue_local _ q hq,g.laplacianValue_local φ q hq,g.laplacianValue_local ψ q hq]
+  simp only [linearizedMongeAmpere,φ.hessian_addFunction ψ q ((A.chart q).mapsTo hq),
+    mul_add,trace_add,Complex.add_re]
+
+lemma laplacian_realSMul (g : KaehlerMetric A) (φ : SmoothRealFunction A) (c : ℝ) (x : X) :
+    (g.laplacian (φ.realSMul c)).value x = c*(g.laplacian φ).value x :=
+  g.linearizedMongeAmpere_realSMul φ c _ _
+
+lemma laplacian_nonpos_at_globalMax (g : KaehlerMetric A) (φ : SmoothRealFunction A)
+    {x : X} (hx : ∀ y, φ.value y ≤ φ.value x) : (g.laplacian φ).value x ≤ 0 :=
+  g.linearized_nonpos_at_globalMax φ hx _ (A.covers x).choose_spec
+
+variable [CompactSpace X] [Nonempty X]
+
+lemma trace_bound_of_C0_chernLu (g : KaehlerMetric A) (hd : 0 < d)
+    (φ : SmoothRealFunction A) (hp : g.PositivePotential φ) {C B : ℝ}
+    (hC : 0 ≤ C) (hB : ∀ x, |φ.value x| ≤ B)
+    (hLu : ∀ x, -C*((g.deform φ hp).traceMetric g).value x ≤
+      ((g.deform φ hp).laplacian ((g.deform φ hp).logTrace g hd)).value x) :
+    ∀ x, ((g.deform φ hp).traceMetric g).value x ≤
+      ((C+1)*(d:ℝ))*Real.exp (2*(C+1)*B) := by
+  let h := g.deform φ hp
+  let f := (h.logTrace g hd).addFunction (φ.realSMul (-(C+1)))
+  obtain ⟨x₀,_,hm⟩ := isCompact_univ.exists_isMaxOn univ_nonempty f.continuous.continuousOn
+  have hmax : ∀ y, f.value y ≤ f.value x₀ := fun y => hm (mem_univ y)
+  have hlap := h.laplacian_nonpos_at_globalMax f hmax
+  rw [laplacian_addFunction,laplacian_realSMul,g.laplacian_deform φ hp] at hlap
+  have hS₀ : (h.traceMetric g).value x₀ ≤ (C+1)*(d:ℝ) := by
+    have hh := hLu x₀
+    change -C*(h.traceMetric g).value x₀ ≤ _ at hh
+    nlinarith
+  intro x
+  have hlog : Real.log ((h.traceMetric g).value x) ≤
+      Real.log ((h.traceMetric g).value x₀)+2*(C+1)*B := by
+    have hh := hmax x
+    change Real.log ((h.traceMetric g).value x)+(-(C+1))*φ.value x ≤
+      Real.log ((h.traceMetric g).value x₀)+(-(C+1))*φ.value x₀ at hh
+    have hx := abs_le.mp (hB x)
+    have hy := abs_le.mp (hB x₀)
+    nlinarith [mul_nonneg (by linarith : 0 ≤ C+1) (sub_nonneg.mpr hx.2),
+      mul_nonneg (by linarith : 0 ≤ C+1) (by linarith : 0 ≤ φ.value x₀+B)]
+  have he := Real.exp_le_exp.mpr hlog
+  rw [Real.exp_add,Real.exp_log (h.traceMetric_pos g hd x),
+    Real.exp_log (h.traceMetric_pos g hd x₀)] at he
+  exact he.trans (mul_le_mul_of_nonneg_right hS₀ (Real.exp_pos _).le)
+
+variable [T2Space X] [ConnectedSpace X]
+
+ 
+
+theorem volumePath_trace_bound (g : KaehlerMetric A) (line : SemipositiveAnticanonicalMetric A)
+    (hd : 2 ≤ d) : ∃ K : ℝ, 0 < K ∧
+    ∀ (φ : SmoothRealFunction A) (hp : g.PositivePotential φ), g.integral φ.value = 0 →
+    ∀ (t b : ℝ), t ∈ Icc (0:ℝ) 1 →
+    (∀ x, (g.logRatio (g.deform φ hp)).value x = t*(prescribedForcing g line).value x+b) →
+    ∀ x, ((g.deform φ hp).traceMetric g).value x ≤ K := by
+  have hd' : 0 < d := by omega
+  obtain ⟨C,hC,hLu⟩ := g.uniform_chernLu line hd'
+  obtain ⟨B,hB,hC0⟩ := g.volumePath_C0_bound line hd
+  refine ⟨((C+1)*(d:ℝ))*Real.exp (2*(C+1)*B),by positivity,?_⟩
+  intro φ hp hm t b ht heq
+  exact g.trace_bound_of_C0_chernLu hd' φ hp hC (hC0 φ hp hm t b ht heq)
+    (hLu t b φ hp heq ht)
+
+end Anticanonical.SourceSmooth.KaehlerMetric
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/UniformCalabi.lean b/lean/OAI/Geometry/PrescribedRicci/UniformCalabi.lean
new file mode 100644
index 0000000000000000000000000000000000000000..b7516199317054c3ef287f7cd014d94c7634ab20
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/UniformCalabi.lean
@@ -0,0 +1,67 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.CalabiLocalMaximum
+import OAI.Geometry.PrescribedRicci.CalabiTensorMetric
+import OAI.Geometry.PrescribedRicci.CompactPathBounds
+import OAI.Geometry.PrescribedRicci.CompactPathRicci
+
+namespace OAI
+
+/-! Uniform Calabi. -/
+
+section
+
+ 
+
+noncomputable section
+open Matrix Set Filter Topology Manifold IsManifold
+open scoped ContDiff ComplexOrder MatrixOrder Matrix.Norms.Elementwise
+namespace Anticanonical.SourceSmooth.KaehlerMetric
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+
+lemma exists_nested_coordinateCutoff {K U : Set (Coordinates d)}
+    (hK : IsCompact K) (hU : IsOpen U) (hKU : K ⊆ U) :
+    ∃ (χ : Coordinates d → ℝ) (L : Set (Coordinates d)), ContDiff ℝ ∞ χ ∧
+      IsCompact L ∧ L ⊆ U ∧ tsupport χ ⊆ interior L ∧ K ⊆ L ∧ ∀ z ∈ K, χ z = 1 := by
+  obtain ⟨K₁,hK₁,hK₁c,hKK₁,hK₁U⟩ := exists_compact_closed_between hK hU hKU
+  obtain ⟨f,hf,hf0,_⟩ := exists_contMDiffMap_one_nhds_of_subset_interior
+    (𝓘(ℝ,Coordinates d)) hK.isClosed hKK₁ (n:=⊤)
+  have hsm : ContDiff ℝ ∞ (fun z => f z) := contMDiff_iff_contDiff.mp f.contMDiff
+  have hsup : tsupport (fun z => f z) ⊆ K₁ := by
+    apply closure_minimal _ hK₁c
+    intro z hz
+    by_contra hn
+    exact hz (hf0 z hn)
+  obtain ⟨L,hL,_,hK₁L,hLU⟩ := exists_compact_closed_between hK₁ hU hK₁U
+  refine ⟨f,L,hsm,hL,hLU,hsup.trans hK₁L,?_,?_⟩
+  · exact (hKK₁.trans interior_subset).trans (hK₁L.trans interior_subset)
+  · intro z hz
+    exact hf.self_of_nhdsSet z hz
+
+variable [T2Space X] [CompactSpace X] [ConnectedSpace X]
+
+ 
+
+theorem volumePath_calabi_bound_on_compact (g : KaehlerMetric A)
+    (line : SemipositiveAnticanonicalMetric A) (hd : 2 ≤ d) (q : Fin A.count)
+    {K : Set (Coordinates d)} (hK : IsCompact K) (hKt : K ⊆ (A.chart q).target) :
+    ∃ P : ℝ, 0 < P ∧ ∀ (φ : SmoothRealFunction A) (hp : g.PositivePotential φ),
+      g.integral φ.value = 0 → ∀ (t b : ℝ), t ∈ Icc (0:ℝ) 1 →
+      (∀ x, (g.logRatio (g.deform φ hp)).value x = t*(prescribedForcing g line).value x+b) →
+      ∀ z ∈ K, (g.deform φ hp).calabiNorm q z ≤ P := by
+  obtain ⟨χ,L,hχ,hL,hLt,hχL,hKL,hχK⟩ := exists_nested_coordinateCutoff hK (A.chart q).open_target hKt
+  obtain ⟨M,hM,hmetric⟩ := g.volumePath_metric_bounds_on_compact line hd q hL hLt
+  obtain ⟨R,hR,hRicci⟩ := g.volumePath_ricci_bounds_on_compact line q hL hLt
+  obtain ⟨P,hP,hCalabi⟩ := calabi_bound_on_compact q hL hLt χ hχ hχL hM.le hR.le
+  refine ⟨P,hP,?_⟩
+  intro φ hp hm t b ht heq z hz
+  apply hCalabi (g.deform φ hp) (fun y hy => ?_) z (hKL hz) (hχK z hz)
+  have h1 := hmetric φ hp hm t b ht heq y hy
+  have h2 := hRicci φ hp t b ht heq y hy
+  exact ⟨h1.1,h1.2,h2.1,h2.2⟩
+
+end Anticanonical.SourceSmooth.KaehlerMetric
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/UniformChernLu.lean b/lean/OAI/Geometry/PrescribedRicci/UniformChernLu.lean
new file mode 100644
index 0000000000000000000000000000000000000000..13f54a708ff9d6b44de58774c3def00845a1593f
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/UniformChernLu.lean
@@ -0,0 +1,104 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.ChernLuAlgebra
+import OAI.Geometry.PrescribedRicci.ChernLuDifferential
+import OAI.Geometry.PrescribedRicci.CurvatureCompactBounds
+import OAI.Geometry.PrescribedRicci.KaehlerLogTrace
+import OAI.Geometry.PrescribedRicci.MatrixComponentBounds
+import OAI.Geometry.PrescribedRicci.TraceEntryBounds
+import OAI.Geometry.PrescribedPotential.VolumePath
+
+namespace OAI
+
+/-! Uniform Chern Lu. -/
+
+section
+
+ 
+
+noncomputable section
+open Matrix Filter Set Topology
+open scoped ContDiff ComplexOrder MatrixOrder
+namespace Anticanonical.SourceSmooth.KaehlerMetric
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+
+lemma volumePath_ricci_bound (g : KaehlerMetric A) (line : SemipositiveAnticanonicalMetric A)
+    {t b : ℝ} {φ : SmoothRealFunction A} (hp : g.PositivePotential φ)
+    (heq : ∀ x, (g.logRatio (g.deform φ hp)).value x =
+      t*(prescribedForcing g line).value x+b)
+    (ht : t ∈ Icc (0:ℝ) 1) (q : Fin A.count) {z : Coordinates d}
+    (hz : z ∈ (A.chart q).target) {M : ℝ} (hM : 0 ≤ M)
+    (hB : g.BackgroundComponentBound (prescribedForcing g line) q z M) (i j : Fin d) :
+    ‖(g.deform φ hp).curvatureRicci q z i j‖ ≤ 2*M := by
+  rw [g.volumePath_ricci line hp heq q hz]
+  change ‖g.curvatureRicci q z i j-(t:ℂ)*(prescribedForcing g line).hessian q z i j‖ ≤ _
+  calc
+    _ ≤ ‖g.curvatureRicci q z i j‖+‖(t:ℂ)*(prescribedForcing g line).hessian q z i j‖ := norm_sub_le _ _
+    _ ≤ M+M := add_le_add (hB.2.2.1 i j) (by
+      rw [norm_mul,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg ht.1]
+      exact (mul_le_mul_of_nonneg_left (hB.2.2.2.1 i j) ht.1).trans
+        (by nlinarith [ht.2]))
+    _ = _ := by ring
+
+lemma chernLu_of_background_bound (g : KaehlerMetric A) (line : SemipositiveAnticanonicalMetric A)
+    (hd : 0 < d) {t b : ℝ} {φ : SmoothRealFunction A} (hp : g.PositivePotential φ)
+    (heq : ∀ x, (g.logRatio (g.deform φ hp)).value x =
+      t*(prescribedForcing g line).value x+b)
+    (ht : t ∈ Icc (0:ℝ) 1) (q : Fin A.count) {z : Coordinates d}
+    (hz : z ∈ (A.chart q).target) {M : ℝ} (hM : 0 ≤ M)
+    (hB : g.BackgroundComponentBound (prescribedForcing g line) q z M) :
+    -((d:ℝ)^4*M^3+2*(d:ℝ)^4*M^4)*((g.deform φ hp).traceMetric g).localExpression q z ≤
+      ((g.deform φ hp).laplacian ((g.deform φ hp).logTrace g hd)).localExpression q z := by
+  let : Nonempty (Fin d) := Fin.pos_iff_nonempty.mp hd
+  let h := g.deform φ hp
+  let S := (h.traceMetric g).localExpression q z
+  have hS : 0 < S := h.traceMetric_pos g hd ((A.chart q).symm z)
+  have hi (i j : Fin d) : ‖(h.matrix q z)⁻¹ i j‖ ≤ M*S := by
+    dsimp [S]
+    rw [h.traceMetric_local g q hz]
+    exact MongeAmpere.entry_le_relative_trace _ _ (h.positive q z hz).inv (g.positive q z hz)
+      hM (fun i => (Complex.re_le_norm _).trans (hB.2.1 i i)) i j
+  have hcurv : h.targetCurvatureTerm g q z ≤ ((d:ℝ)^4*M^3)*S^2 := by
+    change (∑ a, ∑ b, (h.matrix q z)⁻¹ a b*((h.matrix q z)⁻¹*g.curvatureMatrix q z b a).trace).re ≤ _
+    apply (Complex.re_le_norm _).trans
+    have hh := MongeAmpere.contracted_curvature_bound (mul_nonneg hM hS.le) hi hB.2.2.2.2
+    simp only [Fintype.card_fin] at hh
+    convert hh using 1
+    first | rfl | ring
+  have hricci : -(2*(d:ℝ)^4*M^4)*S^2 ≤ h.sourceRicciTerm g q z := by
+    have hh := MongeAmpere.trace_four_entry_bound (mul_nonneg hM hS.le)
+      (mul_nonneg (by norm_num : (0:ℝ)≤2) hM) (mul_nonneg hM hS.le)
+      hi (g.volumePath_ricci_bound line hp heq ht q hz hM hB) hi hB.1
+    simp only [Fintype.card_fin] at hh
+    have hh' : ‖((h.matrix q z)⁻¹*h.curvatureRicci q z*(h.matrix q z)⁻¹*g.matrix q z).trace‖ ≤
+        (2*(d:ℝ)^4*M^4)*S^2 := by
+      convert hh using 1
+      first | rfl | ring
+    have hl := (abs_le.mp (Complex.abs_re_le_norm ((h.matrix q z)⁻¹*h.curvatureRicci q z*(h.matrix q z)⁻¹*g.matrix q z).trace)).1
+    dsimp [sourceRicciTerm]
+    linarith
+  exact h.chernLu_of_component_bounds g hd q hz hcurv hricci
+
+variable [T2Space X] [CompactSpace X]
+
+lemma uniform_chernLu (g : KaehlerMetric A) (line : SemipositiveAnticanonicalMetric A)
+    (hd : 0 < d) : ∃ C : ℝ, 0 ≤ C ∧
+    ∀ (t b : ℝ) (φ : SmoothRealFunction A) (hp : g.PositivePotential φ),
+    (∀ x, (g.logRatio (g.deform φ hp)).value x = t*(prescribedForcing g line).value x+b) →
+    t ∈ Icc (0:ℝ) 1 → ∀ x,
+    -C*((g.deform φ hp).traceMetric g).value x ≤
+      ((g.deform φ hp).laplacian ((g.deform φ hp).logTrace g hd)).value x := by
+  obtain ⟨M,hM,hB⟩ := g.compact_background_bounds (prescribedForcing g line)
+  refine ⟨(d:ℝ)^4*M^3+2*(d:ℝ)^4*M^4,by positivity,?_⟩
+  intro t b φ hp heq ht x
+  obtain ⟨q,hq⟩ := exists_partition_chart (A:=A) x
+  have hx := chartPartition_support q hq
+  have hh := g.chernLu_of_background_bound line hd hp heq ht q ((A.chart q).mapsTo hx)
+    (by linarith) (hB q x hq)
+  simpa only [SmoothRealFunction.localExpression,Function.comp_apply,(A.chart q).left_inv hx] using hh
+
+end Anticanonical.SourceSmooth.KaehlerMetric
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/UniformCoefficientExtension.lean b/lean/OAI/Geometry/PrescribedRicci/UniformCoefficientExtension.lean
new file mode 100644
index 0000000000000000000000000000000000000000..da8ab4465769ea12bcafb119fd90b18e65c6b228
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/UniformCoefficientExtension.lean
@@ -0,0 +1,120 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.GlobalSobolev
+import OAI.Geometry.PrescribedPotential.MetricLocalization
+import OAI.Geometry.PrescribedRicci.UniformFrozenEstimate
+
+namespace OAI
+
+/-! Uniform Coefficient Extension. -/
+
+noncomputable section
+open Matrix Set Filter Topology
+open scoped ContDiff ComplexOrder MatrixOrder Matrix.Norms.Elementwise BoundedContinuousFunction
+namespace MetricLocalization
+open EllipticKernel FrozenPoisson SobolevChart
+variable {n : ℕ} {ι : Type*} [Fintype ι]
+
+omit [Fintype ι] in
+theorem coefficient_extension_on_bump {U : Set (EC n)} (hU : IsOpen U)
+    (H : EC n → Matrix (Fin n) (Fin n) ℂ) (hH : ContDiffOn ℝ ∞ H U)
+    (hh : ∀ y ∈ U, (H y).PosDef) {x : EC n} (_hx : x ∈ U)
+    (v : ι → EC n) (ψ : ContDiffBump x) (hψ : tsupport ψ ⊆ U)
+    {ε : ℝ} (hε : 0 ≤ ε)
+    (hb : ∀ y ∈ tsupport ψ, ∀ i j,
+      |traceBilin (H y) (rankTwo (v i) (v j))-traceBilin (H x) (rankTwo (v i) (v j))| ≤ ε) :
+    ∃ a : ι → ι → EC n →ᵇ ℂ,
+      (∀ i j, ContDiff ℝ ∞ (a i j : EC n → ℂ)) ∧
+      (∀ i j, HasCompactSupport (a i j : EC n → ℂ)) ∧
+      (∀ i j, ‖a i j‖ ≤ ε) ∧
+      ∀ y ∈ Metric.closedBall x ψ.rIn, ∀ i j,
+        a i j y = (traceBilin (H y) (rankTwo (v i) (v j)) : ℂ)-
+          (traceBilin (H x) (rankTwo (v i) (v j)) : ℂ) := by
+  let F : ι → ι → EC n → ℂ := fun i j y =>
+    (traceBilin (H y) (rankTwo (v i) (v j)) : ℂ)-
+      (traceBilin (H x) (rankTwo (v i) (v j)) : ℂ)
+  let b : ι → ι → EC n → ℂ := fun i j y => ψ y • F i j y
+  have hF (i j : ι) : ContDiffOn ℝ ∞ (F i j) U :=
+    (trace_coefficient_smooth hH hh _).sub contDiffOn_const
+  have hs (i j : ι) : ContDiff ℝ ∞ (b i j) := by
+    apply smooth_bump_smul
+    intro y hy
+    exact (hF i j y (hψ hy)).contDiffAt (hU.mem_nhds (hψ hy))
+  have hc (i j : ι) : HasCompactSupport (b i j) := ψ.hasCompactSupport.smul_right
+  have hn (i j : ι) (y : EC n) : ‖b i j y‖ ≤ ε := by
+    by_cases hy : y ∈ tsupport ψ
+    · calc
+        _ = ψ y*‖F i j y‖ := by simp [b,abs_of_nonneg ψ.nonneg]
+        _ ≤ ‖F i j y‖ := by simpa using mul_le_mul_of_nonneg_right ψ.le_one (norm_nonneg (F i j y))
+        _ ≤ ε := by simpa only [F,← Complex.ofReal_sub,Complex.norm_real,Real.norm_eq_abs] using hb y hy i j
+    · simp [b,image_eq_zero_of_notMem_tsupport hy,hε]
+  let a : ι → ι → EC n →ᵇ ℂ := fun i j =>
+    BoundedContinuousFunction.ofNormedAddCommGroup (b i j) (hs i j).continuous ε (hn i j)
+  refine ⟨a,hs,hc,fun i j => (BoundedContinuousFunction.norm_le hε).mpr (hn i j),?_⟩
+  intro y hy i j
+  change ψ y • F i j y = F i j y
+  rw [ψ.one_of_mem_closedBall hy,one_smul]
+
+lemma perturbationBound_le_of_norm (v : ι → EC n) (a : ι → ι → EC n →ᵇ ℂ)
+    {ε : ℝ} (ha : ∀ i j, ‖a i j‖ ≤ ε) :
+    perturbationBound v a ≤ ε*(∑ i, ∑ j, (2*Real.pi)^2*‖v i‖*‖v j‖) := by
+  simp only [perturbationBound,Finset.mul_sum]
+  apply Finset.sum_le_sum
+  intro i _
+  apply Finset.sum_le_sum
+  intro j _
+  exact mul_le_mul_of_nonneg_right (ha i j) (by positivity)
+
+theorem uniform_coefficient_radius (v : ι → EC n) {M P R : ℝ}
+    (hM : 0 ≤ M) (hP : 0 ≤ P) (hR : 0 < R) :
+    ∃ r : ℝ, 0 < r ∧ r ≤ R ∧
+      ∀ (x : EC n) (U : Set (EC n)), IsOpen U → Metric.closedBall x R ⊆ U →
+      ∀ (H : EC n → Matrix (Fin n) (Fin n) ℂ), ContDiffOn ℝ ∞ H U →
+      (∀ y ∈ U, (H y).PosDef) →
+      (∀ y ∈ Metric.closedBall x R, ∀ i j,
+        |traceBilin (H y) (rankTwo (v i) (v j))-traceBilin (H x) (rankTwo (v i) (v j))| ≤ P*‖y-x‖) →
+      ∃ a : ι → ι → EC n →ᵇ ℂ,
+        (∀ i j, ContDiff ℝ ∞ (a i j : EC n → ℂ)) ∧
+        (∀ i j, HasCompactSupport (a i j : EC n → ℂ)) ∧
+        perturbationBound v a*uniformEllipticBound n M ≤ 1/2 ∧
+        ∀ y ∈ Metric.closedBall x (r/2), ∀ i j,
+          a i j y = (traceBilin (H y) (rankTwo (v i) (v j)) : ℂ)-
+            (traceBilin (H x) (rankTwo (v i) (v j)) : ℂ) := by
+  let C := uniformEllipticBound n M
+  let Q : ℝ := ∑ i, ∑ j, (2*Real.pi)^2*‖v i‖*‖v j‖
+  have hC : 0 < C := uniformEllipticBound_pos hM
+  have hQ : 0 ≤ Q := by dsimp only [Q]; positivity
+  let ε : ℝ := 1/(2*(1+Q*C))
+  have hε : 0 < ε := by dsimp only [ε]; positivity
+  let r : ℝ := min R (ε/(P+1))
+  have hr : 0 < r := lt_min hR (div_pos hε (by linarith only [hP]))
+  have hrR : r ≤ R := min_le_left ..
+  have hpr : P*r ≤ ε := by
+    have h1 := (le_div_iff₀ (by linarith only [hP] : 0 < P+1)).mp (min_le_right R (ε/(P+1)))
+    change r*(P+1) ≤ ε at h1
+    nlinarith only [h1,hr]
+  refine ⟨r,hr,hrR,?_⟩
+  intro x U hU hsub H hH hh hosc
+  let ψ : ContDiffBump x :=
+    { rIn := r/2, rOut := r, rIn_pos := half_pos hr, rIn_lt_rOut := half_lt_self hr }
+  have hs : tsupport ψ ⊆ Metric.closedBall x R := by
+    rw [ContDiffBump.tsupport_eq]
+    exact Metric.closedBall_subset_closedBall hrR
+  obtain ⟨a,ha,hac,han,heq⟩ := coefficient_extension_on_bump hU H hH hh
+    (hsub (Metric.mem_closedBall_self hR.le)) v ψ (hs.trans hsub) hε.le (by
+      intro y hy i j
+      apply (hosc y (hs hy) i j).trans
+      apply (mul_le_mul_of_nonneg_left _ hP).trans hpr
+      simpa only [ContDiffBump.tsupport_eq,Metric.mem_closedBall,dist_eq_norm,ψ] using hy)
+  refine ⟨a,ha,hac,?_,heq⟩
+  apply (mul_le_mul_of_nonneg_right (perturbationBound_le_of_norm v a han) hC.le).trans
+  change ε*Q*C ≤ 1/2
+  have hd : 0 < 2*(1+Q*C) := by positivity
+  dsimp only [ε]
+  rw [div_mul_eq_mul_div,div_mul_eq_mul_div,one_mul]
+  apply (div_le_iff₀ hd).mpr
+  nlinarith only [mul_nonneg hQ hC.le]
+end MetricLocalization
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/UniformCoefficientOscillation.lean b/lean/OAI/Geometry/PrescribedRicci/UniformCoefficientOscillation.lean
new file mode 100644
index 0000000000000000000000000000000000000000..2576b706841ab269242b605e147418a662b0a58a
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/UniformCoefficientOscillation.lean
@@ -0,0 +1,100 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.CalabiMetricC1
+import OAI.Geometry.PrescribedRicci.CalabiTensorBounds
+import OAI.Geometry.PrescribedPotential.VolumePath
+
+namespace OAI
+
+/-! Uniform Coefficient Oscillation. -/
+
+section
+
+ 
+
+noncomputable section
+open Matrix Set Filter Topology
+open scoped ContDiff ComplexOrder MatrixOrder Matrix.Norms.Elementwise
+namespace MongeAmpere
+variable {n : Type*} [Fintype n] [DecidableEq n]
+
+omit [DecidableEq n] in
+lemma matrix_trace_norm_le (B : Matrix n n ℂ) : ‖B.trace‖ ≤ (Fintype.card n:ℝ)*‖B‖ := by
+  apply (norm_sum_le _ _).trans
+  apply (Finset.sum_le_sum (fun i _ => norm_entry_le_entrywise_sup_norm B (i:=i) (j:=i))).trans_eq
+  simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
+
+lemma inverse_difference_bound (H K : Matrix n n ℂ) (hH : H.PosDef) (hK : K.PosDef)
+    {M : ℝ} (hM : 0 ≤ M) (hI : ‖H⁻¹‖ ≤ M) (hJ : ‖K⁻¹‖ ≤ M) :
+    ‖H⁻¹-K⁻¹‖ ≤ (Fintype.card n:ℝ)^2*M^2*‖H-K‖ := by
+  have hHU : IsUnit H := (Matrix.isUnit_iff_isUnit_det H).mpr (isUnit_iff_ne_zero.mpr hH.det_pos.ne')
+  have hKU : IsUnit K := (Matrix.isUnit_iff_isUnit_det K).mpr (isUnit_iff_ne_zero.mpr hK.det_pos.ne')
+  rw [Matrix.inv_sub_inv ⟨fun _ => hKU,fun _ => hHU⟩]
+  have h1 := matrix_norm_mul_le (H⁻¹*(K-H)) K⁻¹
+  have h2 := matrix_norm_mul_le H⁻¹ (K-H)
+  calc
+    _ ≤ (Fintype.card n:ℝ)*((Fintype.card n:ℝ)*M*‖K-H‖)*M :=
+      h1.trans (mul_le_mul
+        (mul_le_mul_of_nonneg_left (h2.trans (mul_le_mul_of_nonneg_right
+          (mul_le_mul_of_nonneg_left hI (Nat.cast_nonneg _)) (norm_nonneg _))) (Nat.cast_nonneg _))
+        hJ (norm_nonneg _) (by positivity))
+    _ = _ := by rw [norm_sub_rev]; ring
+end MongeAmpere
+
+namespace EllipticKernel
+open MongeAmpere
+variable {n : ℕ}
+
+lemma traceBilin_difference_bound (H K : Matrix (Fin n) (Fin n) ℂ) (B : Bilin (EC n)) :
+    |traceBilin H B-traceBilin K B| ≤ (n:ℝ)^2*‖H⁻¹-K⁻¹‖*
+      ‖PotentialKaehler.hermitianPartMatrix (pullBilin B)‖ := by
+  rw [traceBilin_apply,traceBilin_apply,← Complex.sub_re,← Matrix.trace_sub,← Matrix.sub_mul]
+  apply (Complex.abs_re_le_norm _).trans
+  apply (matrix_trace_norm_le _).trans
+  apply (mul_le_mul_of_nonneg_left (matrix_norm_mul_le _ _) (Nat.cast_nonneg _)).trans_eq
+  simp only [Fintype.card_fin]; ring
+end EllipticKernel
+
+namespace Anticanonical.SourceSmooth.KaehlerMetric
+open MongeAmpere EllipticKernel
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  [ConnectedSpace X] {A : ComplexAtlas d X}
+
+theorem volumePath_metric_lipschitz_on_compact (g : KaehlerMetric A)
+    (line : SemipositiveAnticanonicalMetric A) (hd : 2 ≤ d) (q : Fin A.count)
+    {K : Set (Coordinates d)} (hK : IsCompact K) (hKc : Convex ℝ K) (hKt : K ⊆ (A.chart q).target) :
+    ∃ P : ℝ, 0 < P ∧ ∀ (φ : SmoothRealFunction A) (hp : g.PositivePotential φ),
+      g.integral φ.value = 0 → ∀ (t b : ℝ), t ∈ Icc (0:ℝ) 1 →
+      (∀ x, (g.logRatio (g.deform φ hp)).value x = t*(prescribedForcing g line).value x+b) →
+      ∀ x ∈ K, ∀ y ∈ K,
+      ‖(g.deform φ hp).matrix q y-(g.deform φ hp).matrix q x‖ ≤ P*‖y-x‖ ∧
+      ‖((g.deform φ hp).matrix q y)⁻¹-((g.deform φ hp).matrix q x)⁻¹‖ ≤ P*‖y-x‖ := by
+  obtain ⟨M,hM,hmetric⟩ := g.volumePath_metric_bounds_on_compact line hd q hK hKt
+  obtain ⟨P,hP,hder⟩ := g.volumePath_metric_C1_on_compact line hd q hK hKt
+  refine ⟨P+(d:ℝ)^2*M^2*P,by positivity,?_⟩
+  intro φ hp hm t b ht heq x hx y hy
+  let H := (g.deform φ hp).matrix q
+  have hdiff : ‖H y-H x‖ ≤ P*‖y-x‖ := by
+    apply matrix_norm_le_of_entries (by positivity)
+    intro i j
+    apply Convex.norm_image_sub_le_of_norm_fderiv_le (𝕜:=ℝ) (s:=K) _ _ hKc hx hy
+    · intro z hz
+      exact differentiableAt_pi.mp (differentiableAt_pi.mp
+        ((((g.deform φ hp).smooth q).contDiffAt ((A.chart q).open_target.mem_nhds (hKt hz))).differentiableAt
+          (by simp)) i) j
+    · intro z hz
+      exact hder φ hp hm t b ht heq z hz i j
+  have hi := inverse_difference_bound (H y) (H x)
+    ((g.deform φ hp).positive q y (hKt hy)) ((g.deform φ hp).positive q x (hKt hx)) hM.le
+    (hmetric φ hp hm t b ht heq y hy).2 (hmetric φ hp hm t b ht heq x hx).2
+  simp only [Fintype.card_fin] at hi
+  have hi' := hi.trans (mul_le_mul_of_nonneg_left hdiff (by positivity))
+  constructor
+  · exact hdiff.trans (mul_le_mul_of_nonneg_right (le_add_of_nonneg_right (by positivity)) (norm_nonneg _))
+  · change ‖(H y)⁻¹-(H x)⁻¹‖ ≤ _
+    nlinarith only [hi',mul_nonneg hP.le (norm_nonneg (y-x))]
+end Anticanonical.SourceSmooth.KaehlerMetric
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/UniformDeterminantRemainder.lean b/lean/OAI/Geometry/PrescribedRicci/UniformDeterminantRemainder.lean
new file mode 100644
index 0000000000000000000000000000000000000000..d9298d31eb385dfd55ed00678fd680c41ce7e108
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/UniformDeterminantRemainder.lean
@@ -0,0 +1,119 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.DeterminantJet
+import OAI.Geometry.PrescribedRicci.JetRemainderBound
+import OAI.Geometry.PrescribedRicci.UniformSchwartz
+
+namespace OAI
+
+/-! Uniform Determinant Remainder. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Finset _root_.MeasureTheory _root_.OAI.MeasureTheory
+open scoped ContDiff Classical BigOperators ENNReal SchwartzMap
+namespace TameInterpolation
+open SobolevChart
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+variable {ι : Type*} [Fintype ι] [Nonempty ι] {n : ℕ} [Nonempty (Fin n)] {Z : Type*}
+
+omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] [Fintype ι] [Nonempty ι] in
+lemma monoEval_compact (e : ι → E) (f : Fin n → E → ℂ)
+    (hc : ∀ i, HasCompactSupport (f i)) (s : Fin n → List ι) :
+    HasCompactSupport (monoEval e f s) := by
+  let i : Fin n := Classical.choice inferInstance
+  apply (cword_compact e (hc i) (s i)).mono
+  intro x hx
+  apply Finset.prod_ne_zero_iff.mp hx i (Finset.mem_univ i)
+
+lemma list_uniformLp {ν : Type*} (L : List ν) (F : ν → Z → E → ℂ)
+    (hf : ∀ a ∈ L, ∀ z, MemLp (F a z) 2 volume)
+    (hb : ∀ a ∈ L, ∃ C : ℝ, 0 ≤ C ∧ ∀ z, lpNorm (F a z) 2 volume ≤ C) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ z, lpNorm (fun x => (L.map (fun a => F a z x)).sum) 2 volume ≤ C := by
+  induction L with
+  | nil => exact ⟨0,le_rfl,fun z => by simp⟩
+  | cons a L ih =>
+    obtain ⟨C,hC,hc⟩ := hb a (by simp)
+    obtain ⟨B,hB,hb'⟩ := ih (fun b h => hf b (List.mem_cons_of_mem a h))
+      (fun b h => hb b (List.mem_cons_of_mem a h))
+    refine ⟨C+B,add_nonneg hC hB,fun z => ?_⟩
+    simp only [List.map_cons,List.sum_cons]
+    exact (lpNorm_add_le (hf a (by simp) z) (by norm_num)).trans (add_le_add (hc z) (hb' z))
+
+lemma detJetRemainder_uniform_bound (e : ι → E)
+    (F : Z → Fin n → Fin n → 𝓢(E,ℂ))
+    (hc : ∀ z i j, HasCompactSupport (F z i j : E → ℂ)) (ws : List ι)
+    (h0 : ∀ σ : Equiv.Perm (Fin n), ∃ A : ℝ, 0 ≤ A ∧ ∀ z,
+      familyJetNorm e (realComponents (initialJet e (fun i x => F z (σ i) i x))) 0 ∞ ≤ A)
+    (hS : ∀ i j, UniformSobolev ((ws.length:ℝ)-1) (fun z => F z i j)) :
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ z,
+      lpNorm (detJetRemainder e (fun x i j => F z i j x) ws) 2 volume ≤ C := by
+  have hσ (σ : Equiv.Perm (Fin n)) : ∃ C : ℝ, 0 ≤ C ∧ ∀ z,
+      lpNorm (fun x => ((monoRemainder ws).map
+        (fun s => monoEval e (fun i x => F z (σ i) i x) s x)).sum) 2 volume ≤ C := by
+    obtain ⟨A,hA,ha⟩ := h0 σ
+    apply list_uniformLp
+    · intro s hs z
+      exact (monoEval_smooth e _ (fun i => (F z (σ i) i).smooth') s).continuous.memLp_of_hasCompactSupport
+        (monoEval_compact e _ (fun i => hc z (σ i) i) s)
+    · intro s hs
+      apply monoRemainder_uniform_bound e _ (fun z i => (F z (σ i) i).smooth')
+        (fun z i => hc z (σ i) i) ws s hs hA ha
+      intro j hj
+      have hlen : 2 ≤ ws.length := by
+        cases ws with
+        | nil => simp [monoRemainder] at hs
+        | cons a ws =>
+          cases ws with
+          | nil => simp [monoRemainder] at hs
+          | cons b ws => simp
+      have hjs : (j:ℝ)+1 ≤ (ws.length:ℝ)-1 := by
+        have hh : j+2 ≤ ws.length := by omega
+        have hh' : (j:ℝ)+2 ≤ (ws.length:ℝ) := by exact_mod_cast hh
+        linarith only [hh']
+      obtain ⟨B,hB,hb⟩ := initialJet_sobolev_bound (κ:=Fin n) e j ((ws.length:ℝ)-1) hjs
+      choose C hC hcb using fun i => hS (σ i) i
+      refine ⟨B*∑ i, C i,mul_nonneg hB (Finset.sum_nonneg fun i _ => hC i),fun z => ?_⟩
+      exact (hb (fun i => F z (σ i) i)).trans (mul_le_mul_of_nonneg_left
+        (Finset.sum_le_sum fun i _ => hcb i z) hB)
+  choose C hC hcb using hσ
+  refine ⟨∑ σ : Equiv.Perm (Fin n), ‖(Equiv.Perm.sign σ : ℂ)‖*C σ,
+    Finset.sum_nonneg (fun σ _ => mul_nonneg (norm_nonneg _) (hC σ)),fun z => ?_⟩
+  unfold detJetRemainder
+  have hm (σ : Equiv.Perm (Fin n)) : MemLp (fun x => (Equiv.Perm.sign σ : ℂ)*
+      ((monoRemainder ws).map (fun s => monoEval e (fun i x => F z (σ i) i x) s x)).sum) 2 volume := by
+    have hs : ∀ L : List (Fin n → List ι), MemLp (fun x => (L.map
+        (fun s => monoEval e (fun i x => F z (σ i) i x) s x)).sum) 2 volume := by
+      intro L
+      induction L with
+      | nil => simp
+      | cons a L ih =>
+        exact ((monoEval_smooth e _ (fun i => (F z (σ i) i).smooth') a).continuous.memLp_of_hasCompactSupport
+          (monoEval_compact e _ (fun i => hc z (σ i) i) a)).add ih
+    exact (hs _).const_mul _
+  have hh := lpNorm_sum_le (s:=Finset.univ) (fun σ _ => hm σ) (by norm_num : (1:ℝ≥0∞) ≤ 2)
+  have heval : (∑ σ : Equiv.Perm (Fin n), fun x => (Equiv.Perm.sign σ : ℂ)*((monoRemainder ws).map
+      (fun s => monoEval e (fun i x => F z (σ i) i x) s x)).sum) =
+      (fun x => ∑ σ : Equiv.Perm (Fin n), (Equiv.Perm.sign σ : ℂ)*((monoRemainder ws).map
+      (fun s => monoEval e (fun i x => F z (σ i) i x) s x)).sum) := by
+    funext x
+    exact Finset.sum_apply _ _ _
+  rw [heval] at hh
+  apply hh.trans
+  apply Finset.sum_le_sum
+  intro σ _
+  have he : (fun x => (Equiv.Perm.sign σ : ℂ)*((monoRemainder ws).map
+      (fun s => monoEval e (fun i x => F z (σ i) i x) s x)).sum) =
+      (Equiv.Perm.sign σ : ℂ) • (fun x => ((monoRemainder ws).map
+      (fun s => monoEval e (fun i x => F z (σ i) i x) s x)).sum) := rfl
+  rw [he,lpNorm_const_smul]
+  exact mul_le_mul_of_nonneg_left (hcb σ z) (norm_nonneg _)
+end TameInterpolation
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/UniformFrozenEstimate.lean b/lean/OAI/Geometry/PrescribedRicci/UniformFrozenEstimate.lean
new file mode 100644
index 0000000000000000000000000000000000000000..c8b09123d75f1dac4decca2edaa517244ac09ebf
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/UniformFrozenEstimate.lean
@@ -0,0 +1,71 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.HermitianCoercivity
+import OAI.Geometry.PrescribedPotential.SobolevParametrix
+
+namespace OAI
+
+/-! Uniform Frozen Estimate. -/
+
+noncomputable section
+open Matrix MeasureTheory TemperedDistribution
+open scoped ComplexOrder MatrixOrder Matrix.Norms.Elementwise SchwartzMap Real
+namespace FrozenPoisson
+open EllipticKernel SobolevChart MongeAmpere
+variable {n : ℕ}
+
+lemma trace_rankOne_hermPair (H : Matrix (Fin n) (Fin n) ℂ) (ξ : EC n) :
+    traceBilin H (rankOne ξ) = (1/4:ℝ)*(hermPair H⁻¹ (coordinateEquiv n ξ) (coordinateEquiv n ξ)).re := by
+  rw [traceBilin_apply,pull_rankOne,Matrix.mul_smul,Matrix.trace_smul]
+  simp only [Complex.smul_re,smul_eq_mul]
+  congr 1
+  rw [Matrix.mul_vecMulVec,Matrix.trace_vecMulVec,dotProduct_comm]
+  rfl
+
+lemma coordinate_coercivity (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    {M : ℝ} (hM : ‖H‖ ≤ M) (ξ : EC n) :
+    ‖ξ‖^2 ≤ (n:ℝ)*M*(hermPair H⁻¹ (coordinateEquiv n ξ) (coordinateEquiv n ξ)).re := by
+  let : Invertible H := Matrix.invertibleOfIsUnitDet H (isUnit_iff_ne_zero.mpr hH.det_pos.ne')
+  have hp := hermPair_nonneg hH.inv.posSemidef (coordinateEquiv n ξ)
+  have hc (i : Fin n) : ‖ξ i‖^2 ≤ M*(hermPair H⁻¹ (coordinateEquiv n ξ) (coordinateEquiv n ξ)).re := by
+    apply (component_sq_le_hermPair hH.inv (coordinateEquiv n ξ) i).trans
+    rw [Matrix.inv_inv_of_invertible]
+    exact mul_le_mul_of_nonneg_right ((Complex.re_le_norm _).trans
+      ((norm_entry_le_entrywise_sup_norm H).trans hM)) hp
+  rw [PiLp.norm_sq_eq_of_L2]
+  apply (Finset.sum_le_sum (fun i _ => hc i)).trans_eq
+  simp only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul,mul_assoc]
+
+def uniformEllipticBound (n : ℕ) (M : ℝ) : ℝ := 1+(n:ℝ)*M/Real.pi^2
+
+lemma uniformEllipticBound_pos {M : ℝ} (hM : 0 ≤ M) :
+    0 < uniformEllipticBound n M := by unfold uniformEllipticBound; positivity
+
+lemma gainSymbol_uniform_bound (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    {M : ℝ} (hM : 0 ≤ M) (hHM : ‖H‖ ≤ M) (ξ : EC n) :
+    ‖gainSymbol H ξ‖ ≤ uniformEllipticBound n M := by
+  have hc := coordinate_coercivity H hH hHM ξ
+  have hs : symbol H ξ = Real.pi^2*(hermPair H⁻¹ (coordinateEquiv n ξ) (coordinateEquiv n ξ)).re := by
+    rw [symbol,trace_rankOne_hermPair]; ring
+  have he : ‖ξ‖^2 ≤ ((n:ℝ)*M/Real.pi^2)*symbol H ξ := by
+    rw [hs]
+    convert hc using 1
+    field_simp
+  have hp := symbol_nonneg H hH ξ
+  simp only [gainSymbol,inverseSymbol,← Complex.ofReal_mul,Complex.norm_real,Real.norm_eq_abs]
+  rw [abs_of_nonneg (by positivity),mul_inv_le_iff₀ (by positivity)]
+  have hq : 0 ≤ (n:ℝ)*M/Real.pi^2 := by positivity
+  dsimp [uniformEllipticBound]
+  nlinarith only [he,hp,hq]
+
+lemma hilbertResolvent_uniform_bound (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    {M : ℝ} (hM : 0 ≤ M) (hHM : ‖H‖ ≤ M) (u : L2 (EC n)) :
+    ‖hilbertResolvent H hH u‖ ≤ uniformEllipticBound n M*‖u‖ := by
+  apply (multiplier_norm_le (gainSymbolBCF H hH) u).trans
+  apply mul_le_mul_of_nonneg_right _ (norm_nonneg u)
+  exact (BoundedContinuousFunction.norm_le (uniformEllipticBound_pos hM).le).mpr
+    (gainSymbol_uniform_bound H hH hM hHM)
+end FrozenPoisson
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/UniformL2.lean b/lean/OAI/Geometry/PrescribedRicci/UniformL2.lean
new file mode 100644
index 0000000000000000000000000000000000000000..ffb089b653edfab0eb3f459fec71ce6440d17fe6
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/UniformL2.lean
@@ -0,0 +1,73 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.UniformSchwartz
+
+namespace OAI
+
+/-! Uniform L2. -/
+
+section
+
+ 
+noncomputable section
+open Set MeasureTheory
+open scoped ContDiff SchwartzMap ENNReal Classical
+namespace SobolevChart
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] {Z : Type*}
+variable {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V]
+
+def UniformL2 (F : Z → E → V) : Prop :=
+  (∀ z, MemLp (F z) 2 volume) ∧ ∃ C : ℝ, 0 ≤ C ∧ ∀ z, lpNorm (F z) 2 volume ≤ C
+
+lemma UniformSobolev.uniformL2 {F : Z → 𝓢(E,ℂ)} (hF : UniformSobolev 0 F) :
+    UniformL2 (fun z => (F z : E → ℂ)) :=
+  ⟨fun z => (F z).memLp _ _,(uniformSobolev_zero_iff F).mp hF⟩
+
+omit [NormedSpace ℝ V] in
+lemma UniformL2.norm {F : Z → E → V} (h : UniformL2 F) : UniformL2 (fun z x => ‖F z x‖) := by
+  refine ⟨fun z => (h.1 z).norm,?_⟩
+  obtain ⟨C,hC,hc⟩ := h.2
+  exact ⟨C,hC,fun z => by rw [lpNorm_norm (h.1 z).aestronglyMeasurable]; exact hc z⟩
+
+omit [NormedSpace ℝ V] in
+lemma UniformL2.add {F G : Z → E → V} (hF : UniformL2 F) (hG : UniformL2 G) :
+    UniformL2 (fun z x => F z x+G z x) := by
+  obtain ⟨C,hC,hc⟩ := hF.2
+  obtain ⟨B,hB,hb⟩ := hG.2
+  refine ⟨fun z => (hF.1 z).add (hG.1 z),C+B,add_nonneg hC hB,fun z => ?_⟩
+  exact (lpNorm_add_le (hF.1 z) (by norm_num)).trans (add_le_add (hc z) (hb z))
+
+lemma UniformL2.smul {F : Z → E → V} (hF : UniformL2 F) (c : ℝ) :
+    UniformL2 (fun z x => c • F z x) := by
+  obtain ⟨C,hC,hc⟩ := hF.2
+  refine ⟨fun z => (hF.1 z).const_smul c,‖c‖*C,mul_nonneg (norm_nonneg _) hC,fun z => ?_⟩
+  change lpNorm (c • F z) 2 volume ≤ _
+  rw [lpNorm_const_smul]
+  exact mul_le_mul_of_nonneg_left (hc z) (norm_nonneg _)
+
+omit [NormedSpace ℝ V] in
+lemma UniformL2.zero : UniformL2 (fun _ : Z => (0 : E → V)) :=
+  ⟨fun _ => MemLp.zero,0,le_rfl,fun _ => by simp⟩
+
+omit [NormedSpace ℝ V] in
+lemma UniformL2.sum {ι : Type*} (S : Finset ι) (F : ι → Z → E → V)
+    (h : ∀ i ∈ S, UniformL2 (F i)) : UniformL2 (fun z x => ∑ i ∈ S, F i z x) := by
+  induction S using Finset.induction_on with
+  | empty =>
+    simp only [Finset.sum_empty]
+    exact UniformL2.zero
+  | @insert a S ha ih =>
+    simpa only [Finset.sum_insert ha] using
+      (h a (Finset.mem_insert_self a S)).add (ih (fun i hi => h i (Finset.mem_insert_of_mem hi)))
+
+lemma UniformSobolev.zero_of_majorant {F : Z → 𝓢(E,ℂ)} {G : Z → E → ℝ}
+    (hG : UniformL2 G) (h : ∀ z x, ‖F z x‖ ≤ G z x) : UniformSobolev 0 F := by
+  obtain ⟨C,hC,hc⟩ := hG.2
+  apply (uniformSobolev_zero_iff F).mpr
+  exact ⟨C,hC,fun z => (lpNorm_mono_real (hG.1 z) (h z)).trans (hc z)⟩
+end SobolevChart
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/UniformLocalGain.lean b/lean/OAI/Geometry/PrescribedRicci/UniformLocalGain.lean
new file mode 100644
index 0000000000000000000000000000000000000000..599fade983b064fe06a909095460b41b610a50d8
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/UniformLocalGain.lean
@@ -0,0 +1,96 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.LocalDeterminantGain
+import OAI.Geometry.PrescribedRicci.SchwartzHessianWords
+import OAI.Geometry.PrescribedRicci.SobolevCommutatorBound
+import OAI.Geometry.PrescribedRicci.UniformSchwartz
+
+namespace OAI
+
+/-! Uniform Local Gain. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Filter Topology Matrix _root_.MeasureTheory _root_.OAI.MeasureTheory LineDeriv
+open scoped ContDiff SchwartzMap Classical Matrix.Norms.Elementwise BoundedContinuousFunction
+open scoped ComplexOrder MatrixOrder
+namespace FrozenPoisson
+open SobolevChart EllipticKernel MetricLocalization
+variable {n : ℕ} {Z : Type*}
+lemma uniform_schwartz_H2_of_differential {M : ℝ} (hM : 0 ≤ M)
+    (H : Z → Matrix (Fin n) (Fin n) ℂ) (hH : ∀ z, (H z).PosDef)
+    (hHM : ∀ z, ‖H z‖ ≤ M)
+    (a : Z → BasisIndex n → BasisIndex n → EC n →ᵇ ℂ)
+    (ha : ∀ z i j, (a z i j : EC n → ℂ).HasTemperateGrowth)
+    (hsmall : ∀ z, perturbationBound (stdOrthonormalBasis ℝ (EC n)) (a z)*uniformEllipticBound n M ≤ 1/2)
+    (F : Z → 𝓢(EC n,ℂ)) (hF : UniformSobolev 0 F)
+    (hD : UniformSobolev 0 (fun z => schwartzDifferential (extendedCoefficient (H z) (a z)) (F z))) :
+    UniformSobolev 2 F := by
+  apply uniform_schwartz_H2 hM H hH hHM a ha hsmall F
+    (fun z => F z-schwartzDifferential (extendedCoefficient (H z) (a z)) (F z)) (hF.sub hD)
+  intro z
+  rw [map_sub,schwartzDifferential_distribution _ (extendedCoefficient_temperate _ _ (ha z))]
+end FrozenPoisson
+namespace SobolevChart
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] {Z : Type*}
+lemma UniformSobolev.of_basis_words (k : ℕ) {s : ℝ} {F : Z → 𝓢(E,ℂ)}
+    (h : ∀ ws : List (Fin (Module.finrank ℝ E)), ws.length ≤ k →
+      UniformSobolev s (fun z => schwartzWord (ws.map (stdOrthonormalBasis ℝ E)) (F z))) :
+    UniformSobolev (s+k) F := by
+  induction k generalizing F with
+  | zero => simpa [schwartzWord] using h [] (by simp)
+  | succ k ih =>
+    have hv := ih (fun ws hw => h ws (by omega))
+    have hd (j : Fin (Module.finrank ℝ E)) := ih
+      (F:=fun z => ∂_{stdOrthonormalBasis ℝ E j} (F z)) (fun ws hw => by
+        have hh := h (ws++[j]) (by simp only [List.length_append,List.length_cons,List.length_nil]; omega)
+        simpa only [List.map_append,List.map_cons,List.map_nil,schwartzWord_append,schwartzWord,ContinuousLinearMap.comp_apply,
+          ContinuousLinearMap.id_apply,lineDerivOpCLM_apply] using hh)
+    convert hv.raise_one hd using 1
+    push_cast
+    ring
+
+lemma UniformSobolev.list_sum {ν : Type*} (L : List ν) {s : ℝ} (F : ν → Z → 𝓢(E,ℂ))
+    (h : ∀ a ∈ L, UniformSobolev s (F a)) :
+    UniformSobolev s (fun z => (L.map (fun a => F a z)).sum) := by
+  induction L with
+  | nil => simpa using UniformSobolev.const (Z:=Z) s (0 : 𝓢(E,ℂ))
+  | cons a L ih =>
+    simpa only [List.map_cons,List.sum_cons] using
+      (h a (List.mem_cons_self)).add (ih (fun b hb => h b (List.mem_cons_of_mem a hb)))
+
+lemma uniform_cutoff_commutator (κ : 𝓢(E,ℂ)) (ws : List E)
+    {F : Z → 𝓢(E,ℂ)} (hF : UniformSobolev ((ws.length:ℝ)+1) F) :
+    UniformSobolev 2 (fun z => schwartzWord ws (SchwartzMap.smulLeftCLM ℂ κ (F z))-
+      SchwartzMap.smulLeftCLM ℂ κ (schwartzWord ws (F z))) := by
+  simp_rw [schwartz_word_product_commutator,add_sub_cancel_left]
+  apply UniformSobolev.list_sum
+  intro p hp
+  have hh := commutatorWords_orders ws p hp
+  apply UniformSobolev.of_coreBound (hF.word (t:=2) p.2 (by
+    have hlen : (p.2.length:ℝ)+1 ≤ ws.length := by exact_mod_cast (show p.2.length+1 ≤ ws.length by omega)
+    linarith))
+  exact coreBound_product_integer (schwartzWord p.1 κ) 2
+
+lemma uniform_cutoff_word_gain (κ : 𝓢(E,ℂ)) (k : ℕ) {F : Z → 𝓢(E,ℂ)}
+    (hF : UniformSobolev ((k:ℝ)+1) F)
+    (h : ∀ ws : List (Fin (Module.finrank ℝ E)), ws.length ≤ k →
+      UniformSobolev 2 (fun z => SchwartzMap.smulLeftCLM ℂ κ
+        (schwartzWord (ws.map (stdOrthonormalBasis ℝ E)) (F z)))) :
+    UniformSobolev ((k:ℝ)+2) (fun z => SchwartzMap.smulLeftCLM ℂ κ (F z)) := by
+  have hh := UniformSobolev.of_basis_words (F:=fun z => SchwartzMap.smulLeftCLM ℂ κ (F z)) k (s:=2) (fun ws hw => by
+    have hcom := uniform_cutoff_commutator κ (ws.map (stdOrthonormalBasis ℝ E))
+      (hF.mono (by simp only [List.length_map]; exact_mod_cast (show ws.length+1 ≤ k+1 by omega)))
+    convert (h ws hw).add hcom using 1
+    funext z
+    abel)
+  simpa only [add_comm] using hh
+end SobolevChart
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/UniformLocalH2.lean b/lean/OAI/Geometry/PrescribedRicci/UniformLocalH2.lean
new file mode 100644
index 0000000000000000000000000000000000000000..552d97d8e144b803c09ab1eb1cda46e982bb7a56
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/UniformLocalH2.lean
@@ -0,0 +1,49 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.UniformFrozenEstimate
+import OAI.Geometry.PrescribedPotential.VariablePoisson
+
+namespace OAI
+
+/-! Uniform Local H2. -/
+
+noncomputable section
+open Matrix MeasureTheory TemperedDistribution
+open scoped ComplexOrder MatrixOrder Matrix.Norms.Elementwise SchwartzMap Real BoundedContinuousFunction
+namespace FrozenPoisson
+open EllipticKernel SobolevChart
+variable {n : ℕ} {ι : Type*} [Fintype ι]
+
+theorem uniform_H2_absorb (H : Matrix (Fin n) (Fin n) ℂ) (hH : H.PosDef)
+    {M : ℝ} (hM : 0 ≤ M) (hHM : ‖H‖ ≤ M)
+    (v : ι → EC n) (a : ι → ι → EC n →ᵇ ℂ)
+    (hsmall : perturbationBound v a * uniformEllipticBound n M ≤ 1/2)
+    (u f : L2 (EC n))
+    (he : realize 2 u - frozenDifferential H (realize 2 u) -
+      (perturbation v a u : 𝓢'(EC n,ℂ)) = (f : 𝓢'(EC n,ℂ))) :
+    ‖u‖ ≤ 2*uniformEllipticBound n M*‖f‖ := by
+  have he' : shifted H (realize 2 u) =
+      ((f+perturbation v a u : L2 (EC n)) : 𝓢'(EC n,ℂ)) := by
+    rw [shifted_eq_identity_sub H hH]
+    simpa only [← Lp.toTemperedDistributionCLM_apply,map_add] using (sub_eq_iff_eq_add.mp he)
+  have hu : u = hilbertResolvent H hH (f+perturbation v a u) := by
+    apply realize_injective 2
+    have hr := hilbertResolvent_realize H hH 0 (f+perturbation v a u)
+    simp only [zero_add] at hr
+    have hzero : realize 0 (f+perturbation v a u) = ((f+perturbation v a u : L2 (EC n)) : 𝓢'(EC n,ℂ)) := by
+      simp only [realize,neg_zero,besselPotential_zero,ContinuousLinearMap.comp_apply,
+        ContinuousLinearMap.id_apply,Lp.toTemperedDistributionCLM_apply]
+    rw [hr,hzero,← he',resolvent_shifted H hH]
+  have hC := (uniformEllipticBound_pos (n:=n) hM).le
+  have h1 := hilbertResolvent_uniform_bound H hH hM hHM (f+perturbation v a u)
+  rw [← hu] at h1
+  have h2 := norm_add_le f (perturbation v a u)
+  have h3 := perturbation_bound v a u
+  have h4 := mul_le_mul_of_nonneg_left (h2.trans (add_le_add_right h3 _)) hC
+  have h5 := mul_le_mul_of_nonneg_right hsmall (norm_nonneg u)
+  nlinarith only [h1,h4,h5]
+
+end FrozenPoisson
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/UniformMetricBounds.lean b/lean/OAI/Geometry/PrescribedRicci/UniformMetricBounds.lean
new file mode 100644
index 0000000000000000000000000000000000000000..29dfc4f533111f63ec68539476fd6c4b997572aa
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/UniformMetricBounds.lean
@@ -0,0 +1,84 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.GlobalKahlerIntegral
+import OAI.Geometry.PrescribedPotential.GlobalSobolev
+import OAI.Geometry.PrescribedRicci.MatrixCompactBounds
+import OAI.Geometry.PrescribedRicci.TraceEntryBounds
+import OAI.Geometry.PrescribedRicci.TraceMaximum
+import OAI.Geometry.PrescribedPotential.VolumePath
+
+namespace OAI
+
+/-! Uniform Metric Bounds. -/
+
+section
+
+ 
+
+noncomputable section
+open Matrix Set Filter Topology
+open scoped ComplexOrder MatrixOrder Matrix.Norms.Elementwise
+namespace Anticanonical.SourceSmooth.KaehlerMetric
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  [ConnectedSpace X] {A : ComplexAtlas d X}
+
+ 
+
+theorem volumePath_metric_bounds (g : KaehlerMetric A) (line : SemipositiveAnticanonicalMetric A)
+    (hd : 2 ≤ d) : ∃ K : ℝ, 0 < K ∧
+    ∀ (φ : SmoothRealFunction A) (hp : g.PositivePotential φ), g.integral φ.value = 0 →
+    ∀ (t b : ℝ), t ∈ Icc (0:ℝ) 1 →
+    (∀ x, (g.logRatio (g.deform φ hp)).value x = t*(prescribedForcing g line).value x+b) →
+    ∀ q : Fin A.count, ∀ x ∈ tsupport (chartPartition (A:=A) q).value,
+      ‖(g.deform φ hp).matrix q (A.chart q x)‖ ≤ K ∧
+      ‖((g.deform φ hp).matrix q (A.chart q x))⁻¹‖ ≤ K := by
+  have hd' : 0 < d := by omega
+  let : Nonempty (Fin d) := Fin.pos_iff_nonempty.mp hd'
+  obtain ⟨T,hT,htrace⟩ := g.volumePath_trace_bound line hd
+  obtain ⟨M,hM,hbg⟩ := g.compact_background_bounds (prescribedForcing g line)
+  have hM0 : 0 ≤ M := by linarith
+  obtain ⟨D,hD,hdet⟩ := MongeAmpere.compact_matrix_bound
+    (show Continuous (fun H : Matrix (Fin d) (Fin d) ℂ => H.det.re) from
+      Complex.continuous_re.comp continuous_id.matrix_det) M
+  obtain ⟨Q,hQ,hmetric⟩ := MongeAmpere.metric_norm_bound_from_inverse (n:=Fin d) (M*T)
+  let E := Real.exp (2*‖(⟨(prescribedForcing g line).value,
+    (prescribedForcing g line).continuous⟩ : C(X,ℝ))‖)
+  have hE : 0 < E := Real.exp_pos _
+  refine ⟨1+M*T+E*D*Q,by positivity,?_⟩
+  intro φ hp hm t b ht heq q x hx
+  have hxs := chartPartition_support q hx
+  have hz := (A.chart q).mapsTo hxs
+  let H := (g.deform φ hp).matrix q (A.chart q x)
+  have hH := (g.deform φ hp).positive q (A.chart q x) hz
+  have hbi : ‖H⁻¹‖ ≤ M*T := by
+    apply MongeAmpere.matrix_norm_le_of_entries (mul_nonneg hM0 hT.le)
+    intro i j
+    have hi := MongeAmpere.entry_le_relative_trace H⁻¹ (g.matrix q (A.chart q x)) hH.inv
+      (g.positive q _ hz) hM0
+      (fun i => (Complex.re_le_norm _).trans ((hbg q x hx).2.1 i i)) i j
+    have hs := htrace φ hp hm t b ht heq x
+    change (g.deform φ hp).traceMetricValue g x ≤ T at hs
+    rw [(g.deform φ hp).traceMetricValue_local g q hxs] at hs
+    exact hi.trans (mul_le_mul_of_nonneg_left hs hM0)
+  have hdh : H.det.re ≤ E*D := by
+    have hv := g.density_mul_volume φ q hz
+    rw [(A.chart q).left_inv hxs] at hv
+    change (g.potentialDensity φ).value x*(g.matrix q (A.chart q x)).det.re = H.det.re at hv
+    rw [← hv]
+    have hρ := (g.volumePath_density_bounds line ht hp heq x).2
+    change (g.potentialDensity φ).value x ≤ E at hρ
+    exact mul_le_mul hρ
+      (hdet _ (MongeAmpere.matrix_norm_le_of_entries hM0 (hbg q x hx).1))
+      (Complex.pos_iff.mp (g.positive q _ hz).det_pos).1.le hE.le
+  constructor
+  · have hb := (hmetric H hH hbi).trans (mul_le_mul_of_nonneg_right hdh hQ.le)
+    change ‖H‖ ≤ _
+    nlinarith [mul_nonneg hM0 hT.le]
+  · change ‖H⁻¹‖ ≤ _
+    nlinarith [mul_nonneg (mul_nonneg hE.le hD.le) hQ.le]
+
+end Anticanonical.SourceSmooth.KaehlerMetric
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/UniformPathPatches.lean b/lean/OAI/Geometry/PrescribedRicci/UniformPathPatches.lean
new file mode 100644
index 0000000000000000000000000000000000000000..229c721ad2ea2b07be28387466325ddc13e429c7
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/UniformPathPatches.lean
@@ -0,0 +1,86 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.UniformCoefficientExtension
+import OAI.Geometry.PrescribedRicci.UniformCoefficientOscillation
+
+namespace OAI
+
+/-! Uniform Path Patches. -/
+
+section
+
+ 
+
+noncomputable section
+open Matrix Set Filter Topology
+open scoped ContDiff ComplexOrder MatrixOrder Matrix.Norms.Elementwise BoundedContinuousFunction
+namespace Anticanonical.SourceSmooth.KaehlerMetric
+open MongeAmpere EllipticKernel FrozenPoisson MetricLocalization SobolevChart
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
+  [ConnectedSpace X] {A : ComplexAtlas d X}
+
+theorem volumePath_uniform_patch (g : KaehlerMetric A) (line : SemipositiveAnticanonicalMetric A)
+    (hd : 2 ≤ d) (q : Fin A.count) (x : EC d) {R : ℝ} (hR : 0 < R)
+    (hsub : Metric.closedBall x R ⊆ (A.euclideanChart q).target)
+    {ι : Type*} [Fintype ι] (v : ι → EC d) :
+    ∃ M r : ℝ, 0 < M ∧ 0 < r ∧ r ≤ R ∧
+      ∀ (φ : SmoothRealFunction A) (hp : g.PositivePotential φ), g.integral φ.value = 0 →
+      ∀ t b : ℝ, t ∈ Icc (0:ℝ) 1 →
+      (∀ y, (g.logRatio (g.deform φ hp)).value y = t*(prescribedForcing g line).value y+b) →
+      ‖(g.deform φ hp).matrix q (coordinateEquiv d x)‖ ≤ M ∧
+      ∃ a : ι → ι → EC d →ᵇ ℂ,
+        (∀ i j, ContDiff ℝ ∞ (a i j : EC d → ℂ)) ∧
+        (∀ i j, HasCompactSupport (a i j : EC d → ℂ)) ∧
+        perturbationBound v a*uniformEllipticBound d M ≤ 1/2 ∧
+        ∀ y ∈ Metric.closedBall x (r/2), ∀ i j,
+          a i j y = (traceBilin ((g.deform φ hp).matrix q (coordinateEquiv d y)) (rankTwo (v i) (v j)) : ℂ)-
+            (traceBilin ((g.deform φ hp).matrix q (coordinateEquiv d x)) (rankTwo (v i) (v j)) : ℂ) := by
+  let K : Set (Coordinates d) := coordinateEquiv d '' Metric.closedBall x R
+  have hK : IsCompact K := (isCompact_closedBall x R).image (coordinateEquiv d).continuous
+  have hKc : Convex ℝ K := (convex_closedBall x R).linear_image (coordinateEquiv d).toLinearMap
+  have hKt : K ⊆ (A.chart q).target := by
+    rintro z ⟨y,hy,rfl⟩
+    simpa using hsub hy
+  have hxK : coordinateEquiv d x ∈ K := ⟨x,Metric.mem_closedBall_self hR.le,rfl⟩
+  obtain ⟨M,hM,hmetric⟩ := g.volumePath_metric_bounds_on_compact line hd q hK hKt
+  obtain ⟨P,hP,hlip⟩ := g.volumePath_metric_lipschitz_on_compact line hd q hK hKc hKt
+  let Q : ℝ := ∑ i, ∑ j, ‖PotentialKaehler.hermitianPartMatrix (pullBilin (rankTwo (v i) (v j)))‖
+  have hQ : 0 ≤ Q := by dsimp only [Q]; positivity
+  have hBQ (i j : ι) : ‖PotentialKaehler.hermitianPartMatrix (pullBilin (rankTwo (v i) (v j)))‖ ≤ Q := by
+    apply (Finset.single_le_sum (fun _ _ => norm_nonneg _) (Finset.mem_univ j)).trans
+    exact Finset.single_le_sum (f:=fun i => ∑ j, ‖PotentialKaehler.hermitianPartMatrix (pullBilin (rankTwo (v i) (v j)))‖)
+      (fun _ _ => Finset.sum_nonneg fun _ _ => norm_nonneg _) (Finset.mem_univ i)
+  let B : ℝ := (d:ℝ)^2*P*‖(coordinateEquiv d).toContinuousLinearMap‖*Q
+  have hB : 0 ≤ B := by dsimp only [B]; positivity
+  obtain ⟨r,hr,hrR,hpatch⟩ := uniform_coefficient_radius v hM.le hB hR
+  refine ⟨M,r,hM,hr,hrR,?_⟩
+  intro φ hp hm t b ht heq
+  refine ⟨(hmetric φ hp hm t b ht heq _ hxK).1,?_⟩
+  let H : EC d → Matrix (Fin d) (Fin d) ℂ := fun y => (g.deform φ hp).matrix q (coordinateEquiv d y)
+  have hH : ContDiffOn ℝ ∞ H (A.euclideanChart q).target := by
+    apply ((g.deform φ hp).smooth q).comp (coordinateEquiv d).contDiff.contDiffOn
+    intro y hy
+    simpa using hy
+  have hh : ∀ y ∈ (A.euclideanChart q).target, (H y).PosDef := by
+    intro y hy
+    exact (g.deform φ hp).positive q _ (by simpa using hy)
+  apply hpatch x (A.euclideanChart q).target (A.euclideanChart q).open_target hsub H hH hh
+  intro y hy i j
+  have hyK : coordinateEquiv d y ∈ K := ⟨y,hy,rfl⟩
+  have hI := (hlip φ hp hm t b ht heq _ hxK _ hyK).2
+  have hc : ‖coordinateEquiv d y-coordinateEquiv d x‖ ≤
+      ‖(coordinateEquiv d).toContinuousLinearMap‖*‖y-x‖ := by
+    rw [← map_sub]
+    exact (coordinateEquiv d).toContinuousLinearMap.le_opNorm (y-x)
+  have hI' : ‖(H y)⁻¹-(H x)⁻¹‖ ≤ P*‖(coordinateEquiv d).toContinuousLinearMap‖*‖y-x‖ := by
+    exact hI.trans (by simpa only [mul_assoc] using mul_le_mul_of_nonneg_left hc hP.le)
+  apply (traceBilin_difference_bound (H y) (H x) (rankTwo (v i) (v j))).trans
+  apply (mul_le_mul (mul_le_mul_of_nonneg_left hI' (by positivity)) (hBQ i j)
+    (norm_nonneg _) (by positivity)).trans_eq
+  dsimp only [B]
+  ring
+end Anticanonical.SourceSmooth.KaehlerMetric
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/UniformSchwartz.lean b/lean/OAI/Geometry/PrescribedRicci/UniformSchwartz.lean
new file mode 100644
index 0000000000000000000000000000000000000000..3c3ec4ee76ea052c593c98e23d03200b050cd95e
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/UniformSchwartz.lean
@@ -0,0 +1,114 @@
+import Mathlib
+import OAI.Geometry.PrescribedPotential.ChartParametrix
+import OAI.Geometry.PrescribedRicci.JetSobolev
+import OAI.Geometry.PrescribedRicci.UniformLocalH2
+
+namespace OAI
+
+/-! Uniform Schwartz. -/
+
+section
+
+ 
+
+noncomputable section
+open Set Finset MeasureTheory LineDeriv TemperedDistribution
+open scoped ContDiff Classical BigOperators ENNReal SchwartzMap BoundedContinuousFunction Real
+open scoped ComplexOrder MatrixOrder Matrix.Norms.Elementwise
+namespace SobolevChart
+variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] {Z : Type*}
+
+def UniformSobolev (s : ℝ) (F : Z → 𝓢(E,ℂ)) : Prop :=
+  ∃ C : ℝ, 0 ≤ C ∧ ∀ z, ‖schwartzCoord s (F z)‖ ≤ C
+
+lemma UniformSobolev.of_coreBound {s t : ℝ} {F : Z → 𝓢(E,ℂ)}
+    (hF : UniformSobolev s F) {T : 𝓢(E,ℂ) → 𝓢(E,ℂ)} (hT : CoreBound s t T) :
+    UniformSobolev t (fun z => T (F z)) := by
+  obtain ⟨C,hC,hc⟩ := hF
+  obtain ⟨B,hB,hb⟩ := hT
+  exact ⟨B*C,mul_nonneg hB hC,fun z => (hb _).trans (mul_le_mul_of_nonneg_left (hc z) hB)⟩
+
+lemma UniformSobolev.mono {s t : ℝ} {F : Z → 𝓢(E,ℂ)}
+    (hF : UniformSobolev s F) (hst : t ≤ s) : UniformSobolev t F :=
+  hF.of_coreBound (T:=id) (CoreBound.id hst)
+
+lemma UniformSobolev.add {s : ℝ} {F G : Z → 𝓢(E,ℂ)}
+    (hF : UniformSobolev s F) (hG : UniformSobolev s G) : UniformSobolev s (fun z => F z+G z) := by
+  obtain ⟨C,hC,hc⟩ := hF
+  obtain ⟨B,hB,hb⟩ := hG
+  refine ⟨C+B,add_nonneg hC hB,fun z => ?_⟩
+  rw [schwartzCoord_add]
+  exact (norm_add_le _ _).trans (add_le_add (hc z) (hb z))
+
+lemma UniformSobolev.const (s : ℝ) (f : 𝓢(E,ℂ)) : UniformSobolev s (fun _ : Z => f) :=
+  ⟨‖schwartzCoord s f‖,norm_nonneg _,fun _ => le_rfl⟩
+
+lemma UniformSobolev.sum {s : ℝ} {κ : Type*} (I : Finset κ)
+    (F : κ → Z → 𝓢(E,ℂ)) (hF : ∀ i ∈ I, UniformSobolev s (F i)) :
+    UniformSobolev s (fun z => ∑ i ∈ I, F i z) := by
+  induction I using Finset.induction_on with
+  | empty => simpa using UniformSobolev.const (Z:=Z) s (0 : 𝓢(E,ℂ))
+  | @insert i I hi ih =>
+    simp only [Finset.sum_insert hi]
+    exact (hF i (Finset.mem_insert_self i I)).add (ih (fun j hj => hF j (Finset.mem_insert_of_mem hj)))
+
+lemma UniformSobolev.deriv {s t : ℝ} {F : Z → 𝓢(E,ℂ)}
+    (hF : UniformSobolev s F) (v : E) (hst : t ≤ s-1) :
+    UniformSobolev t (fun z => ∂_{v} (F z)) := hF.of_coreBound (coreBound_deriv v hst)
+
+lemma UniformSobolev.word {s t : ℝ} {F : Z → 𝓢(E,ℂ)}
+    (hF : UniformSobolev s F) (ws : List E) (hst : t ≤ s-ws.length) :
+    UniformSobolev t (fun z => schwartzWord ws (F z)) := hF.of_coreBound (coreBound_word ws hst)
+
+lemma uniformSobolev_zero_iff (F : Z → 𝓢(E,ℂ)) : UniformSobolev 0 F ↔
+    ∃ C : ℝ, 0 ≤ C ∧ ∀ z, lpNorm (F z : E → ℂ) 2 volume ≤ C := by
+  simp only [UniformSobolev,TameInterpolation.lpNorm_schwartz_zero]
+
+lemma uniformSobolev_raise_two {s : ℝ} {F : Z → 𝓢(E,ℂ)}
+    (hF : UniformSobolev s F)
+    (hD : ∀ j, UniformSobolev s (fun z =>
+      ∂_{stdOrthonormalBasis ℝ E j} (∂_{stdOrthonormalBasis ℝ E j} (F z)))) :
+    UniformSobolev (s+2) F := by
+  obtain ⟨C,hC,hc⟩ := hF
+  choose B hB hb using hD
+  refine ⟨C+‖(((2*Real.pi)^2:ℝ):ℂ)⁻¹‖*∑ j, B j,add_nonneg hC (mul_nonneg (norm_nonneg _) (Finset.sum_nonneg fun j _ => hB j)),fun z => ?_⟩
+  apply (schwartzCoord_raise_two_norm s (F z)).trans
+  exact add_le_add (hc z) (mul_le_mul_of_nonneg_left
+    (Finset.sum_le_sum (fun j _ => hb j z)) (norm_nonneg _))
+end SobolevChart
+
+namespace FrozenPoisson
+open SobolevChart EllipticKernel MetricLocalization
+variable {n : ℕ} {Z : Type*}
+
+theorem uniform_schwartz_H2 {M : ℝ} (hM : 0 ≤ M)
+    (H : Z → Matrix (Fin n) (Fin n) ℂ) (hH : ∀ z, (H z).PosDef)
+    (hHM : ∀ z, ‖H z‖ ≤ M)
+    (a : Z → BasisIndex n → BasisIndex n → EC n →ᵇ ℂ)
+    (ha : ∀ z i j, (a z i j : EC n → ℂ).HasTemperateGrowth)
+    (hsmall : ∀ z, perturbationBound (stdOrthonormalBasis ℝ (EC n)) (a z) *
+      uniformEllipticBound n M ≤ 1/2)
+    (F G : Z → 𝓢(EC n,ℂ)) (hG : UniformSobolev 0 G)
+    (he : ∀ z, (SchwartzMap.toTemperedDistributionCLM (EC n) ℂ volume (F z)) -
+      chartDifferential (extendedCoefficient (H z) (a z)) (SchwartzMap.toTemperedDistributionCLM (EC n) ℂ volume (F z)) =
+      (SchwartzMap.toTemperedDistributionCLM (EC n) ℂ volume (G z))) : UniformSobolev 2 F := by
+  obtain ⟨C,hC,hc⟩ := hG
+  have hU := (uniformEllipticBound_pos (n:=n) hM).le
+  refine ⟨2*uniformEllipticBound n M*C,by positivity,fun z => ?_⟩
+  have hEq : realize 2 (schwartzCoord 2 (F z)) - frozenDifferential (H z) (realize 2 (schwartzCoord 2 (F z))) -
+      (perturbation (stdOrthonormalBasis ℝ (EC n)) (a z) (schwartzCoord 2 (F z)) : 𝓢'(EC n,ℂ)) =
+      (schwartzCoord 0 (G z) : 𝓢'(EC n,ℂ)) := by
+    rw [sub_sub,← chartDifferential_extension (H z) (a z) (ha z),realize_schwartzCoord]
+    have h0 : (schwartzCoord 0 (G z) : 𝓢'(EC n,ℂ)) = (SchwartzMap.toTemperedDistributionCLM (EC n) ℂ volume (G z)) := by
+      rw [schwartzCoord_distribution,besselPotential_zero,ContinuousLinearMap.id_apply]
+    rw [h0]
+    exact he z
+  exact (uniform_H2_absorb (H z) (hH z) hM (hHM z) _ _ (hsmall z) _ _ hEq).trans
+    (mul_le_mul_of_nonneg_left (hc z) (by positivity))
+end FrozenPoisson
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/VolumePathRicci.lean b/lean/OAI/Geometry/PrescribedRicci/VolumePathRicci.lean
new file mode 100644
index 0000000000000000000000000000000000000000..623871e7768bc610f32054eeac1aae17559a7412
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/VolumePathRicci.lean
@@ -0,0 +1,85 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.CurvatureRicci
+import OAI.Geometry.PrescribedPotential.VolumePath
+
+namespace OAI
+
+/-! Volume Path Ricci. -/
+
+section
+
+ 
+noncomputable section
+open Matrix Filter Set Topology
+open scoped ContDiff ComplexOrder MatrixOrder Matrix.Norms.Elementwise
+namespace PotentialKaehler
+variable {d : ℕ}
+
+lemma potentialMatrix_add_const (f : V d → ℝ) (z : V d) (c : ℝ) :
+    potentialMatrix (fun y => f y+c) z = potentialMatrix f z := by
+  unfold potentialMatrix
+  have he : fderiv ℝ (fun y => f y+c) = fderiv ℝ f := by
+    funext y
+    exact fderiv_add_const c
+  rw [he]
+
+lemma potentialMatrix_affine {f g : V d → ℝ} {z : V d}
+    (hf : ContDiffAt ℝ ∞ f z) (hg : ContDiffAt ℝ ∞ g z) (t c : ℝ) :
+    potentialMatrix (fun y => f y+t*g y+c) z =
+      potentialMatrix f z + (t:ℂ) • potentialMatrix g z := by
+  rw [potentialMatrix_add_const]
+  change hermitianPartMatrix (fderiv ℝ (fderiv ℝ (f+t • g)) z) = _
+  have ht : ContDiffAt ℝ ∞ (t • g) z := hg.const_smul t
+  rw [EllipticKernel.hessian_add hf ht,
+    fderiv_const_smul_field (𝕜:=ℝ) (f:=g) t,
+    fderiv_const_smul_field (𝕜:=ℝ) (f:=fderiv ℝ g) t,
+    EllipticKernel.hermitianPartMatrix_add]
+  congr 1
+  ext a b
+  simp only [hermitianPartMatrix_apply,Matrix.smul_apply]
+  apply Complex.ext <;>
+    simp [potentialMatrix,hermitianPartMatrix_apply,hermitianPart,Pi.smul_apply,smul_eq_mul,Complex.mul_re,Complex.mul_im] <;> ring
+
+end PotentialKaehler
+namespace Anticanonical.SourceSmooth
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+namespace KaehlerMetric
+
+lemma ricciPotential_contDiffAt (g : KaehlerMetric A) (q : Fin A.count)
+    {z : Coordinates d} (hz : z ∈ (A.chart q).target) :
+    ContDiffAt ℝ ∞ (g.ricciPotential q) z :=
+  (((g.volumeCoefficient_smooth q).contDiffAt ((A.chart q).open_target.mem_nhds hz)).log
+    (g.volumeCoefficient_pos q hz).ne').neg
+
+lemma volumePath_ricci (g : KaehlerMetric A) (line : SemipositiveAnticanonicalMetric A)
+    {t b : ℝ} {φ : SmoothRealFunction A} (hp : g.PositivePotential φ)
+    (heq : ∀ x, (g.logRatio (g.deform φ hp)).value x =
+      t*(prescribedForcing g line).value x+b)
+    (q : Fin A.count) {z : Coordinates d} (hz : z ∈ (A.chart q).target) :
+    (g.deform φ hp).curvatureRicci q z =
+      g.curvatureRicci q z - (t:ℂ) • (prescribedForcing g line).hessian q z := by
+  have he : (g.deform φ hp).ricciPotential q =ᶠ[nhds z]
+      (fun y => g.ricciPotential q y+(-t)*(prescribedForcing g line).localExpression q y+(-b)) := by
+    filter_upwards [(A.chart q).open_target.mem_nhds hz] with y hy
+    have hh := heq ((A.chart q).symm y)
+    change (g.logRatio (g.deform φ hp)).localExpression q y =
+      t*(prescribedForcing g line).localExpression q y+b at hh
+    rw [g.logRatio_local _ q hy] at hh
+    unfold ricciPotential
+    linarith
+  rw [(g.deform φ hp).curvatureRicci_eq q hz,
+    PotentialKaehler.potentialMatrix_congr he,
+    PotentialKaehler.potentialMatrix_affine (g.ricciPotential_contDiffAt q hz)
+      (show ContDiffAt ℝ ∞ ((prescribedForcing g line).localExpression q) z from
+        ((prescribedForcing g line).smooth q).contDiffAt ((A.chart q).open_target.mem_nhds hz)),
+    g.curvatureRicci_eq q hz]
+  simp only [Complex.ofReal_neg,neg_smul,sub_eq_add_neg]
+  rfl
+
+end KaehlerMetric
+end Anticanonical.SourceSmooth
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/WeightedMongeAmpereEnergy.lean b/lean/OAI/Geometry/PrescribedRicci/WeightedMongeAmpereEnergy.lean
new file mode 100644
index 0000000000000000000000000000000000000000..164fdd6ef551fa5bc1306e004c4d22b5581cfee4
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/WeightedMongeAmpereEnergy.lean
@@ -0,0 +1,110 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.GlobalKahlerEnergy
+import OAI.Geometry.PrescribedRicci.GlobalKahlerIntegral
+import OAI.Geometry.PrescribedRicci.KahlerEnergyChain
+import OAI.Geometry.PrescribedRicci.MongeAmpereEnergy
+
+namespace OAI
+
+/-! Weighted Monge Ampere Energy. -/
+
+section
+
+ 
+noncomputable section
+open Set Filter Topology Matrix MeasureTheory
+open scoped ContDiff ComplexOrder Classical
+namespace Anticanonical.SourceSmooth
+variable {d : ℕ} {X : Type*} [TopologicalSpace X] {A : ComplexAtlas d X}
+namespace KaehlerMetric
+
+lemma integral_weighted_energy_segment [T2Space X] [CompactSpace X]
+    (g : KaehlerMetric A) (φ : SmoothRealFunction A) (hp : g.PositivePotential φ)
+    (hd : 0 < d) (W : X → ℝ) (hW : Continuous W) (hWpos : ∀ x, 0 ≤ W x)
+    (s : ℝ) (hs : s ∈ Ico 0 1) :
+    (1 - s) ^ (d - 1) * g.integral (fun x => W x * (g.energy φ φ).value x) ≤
+      (g.deform (φ.realSMul s) (hp.segment ⟨hs.1, hs.2.le⟩)).integral
+        (fun x => W x * ((g.deform (φ.realSMul s) (hp.segment ⟨hs.1, hs.2.le⟩)).energy φ φ).value x) := by
+  let gs := g.deform (φ.realSMul s) (hp.segment ⟨hs.1, hs.2.le⟩)
+  rw [← g.integral_const_mul, ← g.integral_density (φ.realSMul s) (hp.segment ⟨hs.1, hs.2.le⟩)]
+  apply g.integral_mono (continuous_const.fun_mul (hW.fun_mul (g.energy φ φ).continuous))
+    ((hW.fun_mul (gs.energy φ φ).continuous).fun_mul (g.potentialDensity _).continuous)
+  intro x
+  have he := mul_le_mul_of_nonneg_left (g.energy_segment_coercive φ hp hd s hs x) (hWpos x)
+  change (1-s)^(d-1) * (W x * (g.energy φ φ).value x) ≤
+    (W x * (gs.energy φ φ).value x) * (g.potentialDensity (φ.realSMul s)).value x
+  nlinarith only [he]
+
+lemma integral_weighted_energy_half_segment [T2Space X] [CompactSpace X]
+    (g : KaehlerMetric A) (φ : SmoothRealFunction A) (hp : g.PositivePotential φ)
+    (hd : 0 < d) (W : X → ℝ) (hW : Continuous W) (hWpos : ∀ x, 0 ≤ W x)
+    (s : ℝ) (hs : s ∈ Icc 0 (1/2)) :
+    (1/2 : ℝ) ^ (d - 1) * g.integral (fun x => W x * (g.energy φ φ).value x) ≤
+      (g.deform (φ.realSMul s) (hp.segment ⟨hs.1, by linarith [hs.2]⟩)).integral
+        (fun x => W x * ((g.deform (φ.realSMul s) (hp.segment ⟨hs.1, by linarith [hs.2]⟩)).energy φ φ).value x) := by
+  have hs' : s ∈ Ico 0 1 := ⟨hs.1, by linarith [hs.2]⟩
+  refine le_trans ?_ (g.integral_weighted_energy_segment φ hp hd W hW hWpos s hs')
+  exact mul_le_mul_of_nonneg_right
+    (pow_le_pow_left₀ (by norm_num) (by linarith [hs.2]) (d - 1))
+    (g.integral_nonneg (fun x => mul_nonneg (hWpos x) (g.energy_nonneg φ x)))
+
+ 
+
+theorem mongeAmpere_weighted_energy [T2Space X] [CompactSpace X]
+    (g : KaehlerMetric A) (φ : SmoothRealFunction A) (hp : g.PositivePotential φ) (hd : 0 < d)
+    (F : ℝ → ℝ) (hF : ContDiff ℝ ∞ F) (hFpos : ∀ x, 0 ≤ deriv F x) :
+    (1/2 : ℝ) ^ d * g.integral (fun x => deriv F (φ.value x) * (g.energy φ φ).value x) ≤
+      g.integral (fun x => F (φ.value x)) -
+        g.integral (fun x => F (φ.value x) * (g.potentialDensity φ).value x) := by
+  let ψ := φ.compose F hF
+  let W (x : X) := deriv F (φ.value x)
+  have hW : Continuous W := (hF.continuous_deriv (by simp)).comp φ.continuous
+  let f (t : ℝ) := g.integral (fun x => ψ.value x * (g.potentialDensity (φ.realSMul t)).value x)
+  have hf (s : ℝ) (hs : s ∈ Icc 0 1) := g.volume_firstVariation φ ψ s (hp.segment hs)
+  have hfc : ContinuousOn f (Icc 0 1) := fun s hs => (hf s hs).continuousAt.continuousWithinAt
+  have hfd : DifferentiableOn ℝ f (interior (Icc 0 1)) :=
+    fun s hs => (hf s (interior_subset hs)).differentiableAt.differentiableWithinAt
+  have hae (s : ℝ) (hs : s ∈ Icc 0 1) :
+      ((g.deform (φ.realSMul s) (hp.segment hs)).energy ψ φ).value =
+        fun x => W x * ((g.deform (φ.realSMul s) (hp.segment hs)).energy φ φ).value x := by
+    funext x
+    exact (g.deform (φ.realSMul s) (hp.segment hs)).energy_comp_left φ φ F hF x
+  have ha : AntitoneOn f (Icc 0 1) := antitoneOn_of_deriv_nonpos (convex_Icc 0 1) hfc hfd (by
+    intro s hs
+    rw [(hf s (interior_subset hs)).deriv, hae s (interior_subset hs)]
+    exact neg_nonpos.mpr ((g.deform (φ.realSMul s) (hp.segment (interior_subset hs))).integral_nonneg
+      (fun x => mul_nonneg (hFpos _) ((g.deform (φ.realSMul s) (hp.segment (interior_subset hs))).energy_nonneg φ x))))
+  have hsub : Icc (0 : ℝ) (1/2) ⊆ Icc 0 1 := fun s hs => ⟨hs.1, by linarith [hs.2]⟩
+  have hdh : DifferentiableOn ℝ f (interior (Icc 0 (1/2))) :=
+    fun s hs => (hf s (hsub (interior_subset hs))).differentiableAt.differentiableWithinAt
+  have hh := (convex_Icc (0 : ℝ) (1/2)).image_sub_le_mul_sub_of_deriv_le
+    (hfc.mono hsub) hdh
+    (C := -((1/2 : ℝ) ^ (d - 1) * g.integral (fun x => W x * (g.energy φ φ).value x)))
+    (by
+      intro s hs
+      rw [(hf s (hsub (interior_subset hs))).deriv, hae s (hsub (interior_subset hs))]
+      exact neg_le_neg (g.integral_weighted_energy_half_segment φ hp hd W hW (fun x => hFpos _) s (interior_subset hs)))
+    0 (by norm_num) (1/2) (by norm_num) (by norm_num)
+  have he := ha (show (1/2 : ℝ) ∈ Icc 0 1 by norm_num) (show (1 : ℝ) ∈ Icc 0 1 by norm_num) (by norm_num)
+  have h0 : f 0 = g.integral ψ.value := by
+    dsimp only [f]
+    simp_rw [g.potentialDensity_smul_zero, mul_one]
+  have h1 : f 1 = g.integral (fun x => ψ.value x * (g.potentialDensity φ).value x) := by
+    dsimp only [f]
+    simp_rw [g.potentialDensity_smul_one]
+  have hpow : (1/2 : ℝ) ^ d = (1/2 : ℝ) ^ (d - 1) * (1/2) := by
+    conv_lhs => rw [show d = (d - 1) + 1 by omega, pow_succ]
+  rw [h0] at hh
+  rw [h1] at he
+  rw [hpow]
+  change ((1/2 : ℝ) ^ (d - 1) * (1/2)) * g.integral (fun x => W x * (g.energy φ φ).value x) ≤
+    g.integral ψ.value - g.integral (fun x => ψ.value x * (g.potentialDensity φ).value x)
+  nlinarith
+
+end KaehlerMetric
+end Anticanonical.SourceSmooth
+
+end
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/PrescribedRicci/WirtingerScalar.lean b/lean/OAI/Geometry/PrescribedRicci/WirtingerScalar.lean
new file mode 100644
index 0000000000000000000000000000000000000000..ce24a6e8fa8eb386c317dca1f744a827838ab871
--- /dev/null
+++ b/lean/OAI/Geometry/PrescribedRicci/WirtingerScalar.lean
@@ -0,0 +1,106 @@
+import Mathlib
+import OAI.Geometry.PrescribedRicci.ComplexIntegrationByParts
+
+namespace OAI
+
+/-! Wirtinger Scalar. -/
+
+noncomputable section
+open Matrix Filter Set Topology
+open scoped ContDiff ComplexOrder
+namespace Anticanonical.SourceSmooth.KaehlerMetric
+variable {d : ℕ}
+
+lemma holDeriv_mul {f g : Coordinates d → ℂ} {z : Coordinates d}
+    (hf : DifferentiableAt ℝ f z) (hg : DifferentiableAt ℝ g z) (a : Fin d) :
+    holDeriv (fun y => f y*g y) z a = holDeriv f z a*g z+f z*holDeriv g z a := by
+  unfold holDeriv
+  rw [fderiv_fun_mul hf hg]
+  simp only [_root_.add_apply,_root_.smul_apply,smul_eq_mul]
+  ring
+
+lemma holDeriv_add {f g : Coordinates d → ℂ} {z : Coordinates d}
+    (hf : DifferentiableAt ℝ f z) (hg : DifferentiableAt ℝ g z) (a : Fin d) :
+    holDeriv (fun y => f y+g y) z a = holDeriv f z a+holDeriv g z a := by
+  unfold holDeriv
+  rw [fderiv_fun_add hf hg]
+  simp only [_root_.add_apply]
+  ring
+
+lemma holDeriv_sum {ι : Type*} [Fintype ι] {f : ι → Coordinates d → ℂ}
+    {z : Coordinates d} (hf : ∀ i, DifferentiableAt ℝ (f i) z) (a : Fin d) :
+    holDeriv (fun y => ∑ i, f i y) z a = ∑ i, holDeriv (f i) z a := by
+  unfold holDeriv
+  rw [fderiv_fun_sum (fun i _ => hf i)]
+  simp only [_root_.sum_apply,← Finset.sum_sub_distrib,Finset.mul_sum]
+
+lemma holDeriv_star (f : Coordinates d → ℂ) (z : Coordinates d) (a : Fin d) :
+    holDeriv (fun y => star (f y)) z a = star (barDeriv f z a) := by
+  unfold holDeriv barDeriv
+  rw [fderiv_star]
+  simp only [ContinuousLinearMap.comp_apply,ContinuousLinearEquiv.coe_coe,starL'_apply,
+    map_mul,map_add,map_inv₀,map_ofNat,Complex.star_def,Complex.conj_I]
+  ring
+
+lemma barDeriv_star (f : Coordinates d → ℂ) (z : Coordinates d) (a : Fin d) :
+    barDeriv (fun y => star (f y)) z a = star (holDeriv f z a) := by
+  have h := holDeriv_star (fun y => star (f y)) z a
+  simpa only [star_star] using congrArg star h.symm
+
+lemma contDiffAt_holDeriv {f : Coordinates d → ℂ} {z : Coordinates d}
+    (hf : ContDiffAt ℝ ∞ f z) (a : Fin d) :
+    ContDiffAt ℝ ∞ (fun y => holDeriv f y a) z := by
+  unfold holDeriv
+  exact contDiffAt_const.mul (((hf.fderiv_right (m:=∞) (by simp)).clm_apply contDiffAt_const).sub
+    (contDiffAt_const.mul ((hf.fderiv_right (m:=∞) (by simp)).clm_apply contDiffAt_const)))
+
+def scalarHolForm (a : Fin d) : (Coordinates d →L[ℝ] ℂ) →L[ℝ] ℂ :=
+  (2:ℂ)⁻¹ • (ContinuousLinearMap.apply ℝ ℂ (coordinateVector a) -
+    Complex.I • ContinuousLinearMap.apply ℝ ℂ (Complex.I • coordinateVector a))
+
+def scalarBarForm (a : Fin d) : (Coordinates d →L[ℝ] ℂ) →L[ℝ] ℂ :=
+  (2:ℂ)⁻¹ • (ContinuousLinearMap.apply ℝ ℂ (coordinateVector a) +
+    Complex.I • ContinuousLinearMap.apply ℝ ℂ (Complex.I • coordinateVector a))
+
+lemma fderiv_holDeriv {f : Coordinates d → ℂ} {z : Coordinates d}
+    (hf : ContDiffAt ℝ ∞ f z) (a : Fin d) (v : Coordinates d) :
+    fderiv ℝ (fun y => holDeriv f y a) z v =
+      (2:ℂ)⁻¹ * (fderiv ℝ (fderiv ℝ f) z v (coordinateVector a) -
+        Complex.I*fderiv ℝ (fderiv ℝ f) z v (Complex.I • coordinateVector a)) := by
+  exact congrArg (fun L : Coordinates d →L[ℝ] ℂ => L v)
+    (((scalarHolForm a).hasFDerivAt.comp z
+      ((hf.fderiv_right (m:=∞) (by simp)).differentiableAt (by simp)).hasFDerivAt).fderiv)
+
+lemma fderiv_barDeriv {f : Coordinates d → ℂ} {z : Coordinates d}
+    (hf : ContDiffAt ℝ ∞ f z) (a : Fin d) (v : Coordinates d) :
+    fderiv ℝ (fun y => barDeriv f y a) z v =
+      (2:ℂ)⁻¹ * (fderiv ℝ (fderiv ℝ f) z v (coordinateVector a) +
+        Complex.I*fderiv ℝ (fderiv ℝ f) z v (Complex.I • coordinateVector a)) := by
+  exact congrArg (fun L : Coordinates d →L[ℝ] ℂ => L v)
+    (((scalarBarForm a).hasFDerivAt.comp z
+      ((hf.fderiv_right (m:=∞) (by simp)).differentiableAt (by simp)).hasFDerivAt).fderiv)
+
+lemma scalar_bar_hol_comm {f : Coordinates d → ℂ} {z : Coordinates d}
+    (hf : ContDiffAt ℝ ∞ f z) (a b : Fin d) :
+    barDeriv (fun y => holDeriv f y a) z b = holDeriv (fun y => barDeriv f y b) z a := by
+  change (2:ℂ)⁻¹ * (fderiv ℝ (fun y => holDeriv f y a) z (coordinateVector b) +
+    Complex.I*fderiv ℝ (fun y => holDeriv f y a) z (Complex.I • coordinateVector b)) =
+    (2:ℂ)⁻¹ * (fderiv ℝ (fun y => barDeriv f y b) z (coordinateVector a) -
+    Complex.I*fderiv ℝ (fun y => barDeriv f y b) z (Complex.I • coordinateVector a))
+  rw [fderiv_holDeriv hf,fderiv_holDeriv hf,fderiv_barDeriv hf,fderiv_barDeriv hf]
+  have hs (u v : Coordinates d) : fderiv ℝ (fderiv ℝ f) z u v = fderiv ℝ (fderiv ℝ f) z v u := by
+    exact hf.isSymmSndFDerivAt (by
+      simp only [minSmoothness_of_isRCLikeNormedField]
+      change ((2:ℕ∞):WithTop ℕ∞) ≤ ((⊤:ℕ∞):WithTop ℕ∞)
+      exact WithTop.coe_le_coe.mpr le_top) u v
+  rw [hs (coordinateVector a) (coordinateVector b),
+    hs (coordinateVector a) (Complex.I • coordinateVector b),
+    hs (Complex.I • coordinateVector a) (coordinateVector b),
+    hs (Complex.I • coordinateVector a) (Complex.I • coordinateVector b)]
+  ring
+
+end Anticanonical.SourceSmooth.KaehlerMetric
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/ProjectionBodies/AffineCovariance.lean b/lean/OAI/Geometry/ProjectionBodies/AffineCovariance.lean
new file mode 100644
index 0000000000000000000000000000000000000000..f827b39a4216f68083ee5390678dc90d5190821d
--- /dev/null
+++ b/lean/OAI/Geometry/ProjectionBodies/AffineCovariance.lean
@@ -0,0 +1,210 @@
+import OAI.Geometry.ProjectionBodies.Normalization
+
+namespace OAI
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology
+open scoped NNReal RealInnerProductSpace Pointwise
+namespace PettyProjection
+variable {n : ℕ}
+
+lemma unitSegment_smul (y : Space n) (a : ℝ) : unitSegment (a • y)=a • unitSegment y := by
+  rw [unitSegment,unitSegment,← image_smul,image_image]
+  congr 1
+  funext t
+  exact smul_comm t a y
+
+lemma unitSegment_image (T : Space n →ₗ[ℝ] Space n) (y : Space n) :
+    T '' unitSegment y=unitSegment (T y) := by
+  rw [unitSegment,unitSegment,image_image]
+  congr 1
+  funext a
+  exact map_smul T a y
+
+lemma unitSegment_zero : unitSegment (0:Space n)={0} := by
+  ext x
+  simp only [unitSegment,mem_image,mem_Icc,smul_zero,mem_singleton_iff]
+  constructor
+  · rintro ⟨a,ha,hax⟩;exact hax.symm
+  · rintro rfl;exact ⟨0,by norm_num, rfl⟩
+
+lemma segment_volume_support [NeZero n] {K : Set (Space n)}
+    (hK : IsCompact K) (hc : Convex ℝ K) (h0 : K∈𝓝 (0:Space n)) (y : Space n) :
+    volume.real (K+unitSegment y)=volume.real K+2*support (projectionBody K) y := by
+  have hP := projectionBody_body hK hc h0
+  have hne := hP.2.2.mono interior_subset
+  by_cases hy : y=0
+  · rw [hy,unitSegment_zero,add_singleton]
+    simp only [add_zero,image_id']
+    have hs0 : support (projectionBody K) 0=0 := by
+      simpa using support_smul hP.1 hne (r :=0) le_rfl (0:Space n)
+    simp only [hs0,mul_zero,add_zero]
+  · let u : Spherical.Sphere n := ⟨‖y‖⁻¹ • y,by simp [norm_smul,hy]⟩
+    have he : y=‖y‖ • (u:Space n) := by simp [u,smul_smul,hy]
+    conv_lhs => rw [he,unitSegment_smul]
+    rw [segment_volume u hK hc (norm_nonneg y)]
+    conv_rhs => rhs;rhs;rw [he,support_smul hP.1 hne (norm_nonneg y),projectionBody_support_unit hK hc h0 u]
+    ring
+
+lemma linearEquiv_nhds (T : Space n ≃L[ℝ] Space n) {K : Set (Space n)}
+    (h0 : K∈𝓝 (0:Space n)) : T '' K∈𝓝 (0:Space n) := by
+  have hh := T.toHomeomorph.isOpenMap.image_mem_nhds h0
+  change T '' K∈𝓝 (T 0) at hh
+  simpa only [map_zero] using hh
+
+lemma projection_support_linearEquiv [NeZero n] (T : Space n ≃L[ℝ] Space n)
+    {K : Set (Space n)} (hK : IsCompact K) (hc : Convex ℝ K) (h0 : K∈𝓝 (0:Space n))
+    (y : Space n) : support (projectionBody (T '' K)) y=
+      |LinearMap.det T.toLinearMap| * support (projectionBody K) (T.symm y) := by
+  have him : T '' (K+unitSegment (T.symm y))=T '' K+unitSegment y := by
+    change T.toLinearMap '' (K+unitSegment (T.symm y))=T.toLinearMap '' K+unitSegment y
+    rw [image_add T.toLinearMap,unitSegment_image T.toLinearMap]
+    change T.toLinearMap '' K+unitSegment (T (T.symm y))=_
+    rw [T.apply_symm_apply]
+  have hh := volumeReal_linearEquiv T (K+unitSegment (T.symm y))
+  rw [him,segment_volume_support (hK.image T.continuous) (hc.linear_image T.toLinearMap)
+    (linearEquiv_nhds T h0),segment_volume_support hK hc h0,volumeReal_linearEquiv] at hh
+  linarith
+
+end PettyProjection
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology
+open scoped RealInnerProductSpace Pointwise
+namespace PettyProjection
+variable {n : ℕ}
+
+lemma adjoint_inverse_comp (T : Space n ≃L[ℝ] Space n) :
+    T.toContinuousLinearMap.adjoint.toLinearMap.comp T.symm.toContinuousLinearMap.adjoint.toLinearMap=LinearMap.id := by
+  apply LinearMap.ext
+  intro x
+  apply ext_inner_right ℝ
+  intro y
+  change ⟪T.toContinuousLinearMap.adjoint (T.symm.toContinuousLinearMap.adjoint x),y⟫=⟪x,y⟫
+  rw [ContinuousLinearMap.adjoint_inner_left,ContinuousLinearMap.adjoint_inner_left]
+  simp
+
+/-- Invertibility of the transpose uses the transpose of the inverse. -/
+def adjointEquiv (T : Space n ≃L[ℝ] Space n) : Space n ≃L[ℝ] Space n :=
+  (LinearEquiv.ofLinearMap T.toContinuousLinearMap.adjoint.toLinearMap
+    T.symm.toContinuousLinearMap.adjoint.toLinearMap
+    (adjoint_inverse_comp T) (adjoint_inverse_comp T.symm)).toContinuousLinearEquiv
+
+lemma adjointEquiv_clm (T : Space n ≃L[ℝ] Space n) :
+    (adjointEquiv T).toContinuousLinearMap=T.toContinuousLinearMap.adjoint := rfl
+
+lemma det_adjoint (A : Space n →L[ℝ] Space n) :
+    LinearMap.det A.adjoint.toLinearMap=LinearMap.det A.toLinearMap := by
+  let b := stdOrthonormalBasis ℝ (Space n)
+  rw [← LinearMap.det_toMatrix b.toBasis,← ContinuousLinearMap.adjoint_toLinearMap,
+    LinearMap.toMatrix_adjoint,Matrix.det_conjTranspose,LinearMap.det_toMatrix]
+  exact star_trivial _
+
+lemma adjointEquiv_det (T : Space n ≃L[ℝ] Space n) :
+    LinearMap.det (adjointEquiv T).toLinearMap=LinearMap.det T.toLinearMap := det_adjoint _
+
+end PettyProjection
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology
+open scoped NNReal RealInnerProductSpace Pointwise
+namespace PettyProjection
+variable {n : ℕ} [NeZero n]
+
+/-- The actual unpolarized projection body transforms by the determinant and
+inverse transpose, proved from orthogonal shadow volumes via segment addition. -/
+theorem projectionBody_linearEquiv (T : Space n ≃L[ℝ] Space n)
+    {K : Set (Space n)} (hK : IsCompact K) (hc : Convex ℝ K) (h0 : K∈𝓝 (0:Space n)) :
+    projectionBody (T '' K)=|LinearMap.det T.toLinearMap| • (adjointEquiv T.symm '' projectionBody K) := by
+  let P := projectionBody K
+  let S := adjointEquiv T.symm
+  let d := |LinearMap.det T.toLinearMap|
+  have hd : 0<d := abs_pos.mpr (linearEquiv_det_ne_zero T)
+  have hP : IsConvexBody P := projectionBody_body hK hc h0
+  have hS := linearEquiv_body S hP
+  have hR := smul_body hS hd.ne'
+  have hL := projectionBody_body (hK.image T.continuous) (hc.linear_image T.toLinearMap) (linearEquiv_nhds T h0)
+  apply (support_eq_iff hL.1 hL.2.1 (hL.2.2.mono interior_subset) hR.1 hR.2.1 (hR.2.2.mono interior_subset)).mp
+  funext y
+  rw [projection_support_linearEquiv T hK hc h0]
+  change d*support P (T.symm y)=support (d • (S '' P)) y
+  rw [support_smul_set hS.1 (hS.2.2.mono interior_subset) hd.le]
+  change d*support P (T.symm y)=d*support (S.toContinuousLinearMap '' P) y
+  rw [support_linear_image hP.1 (hP.2.2.mono interior_subset),adjointEquiv_clm,
+    ContinuousLinearMap.adjoint_adjoint]
+  rfl
+
+lemma projection_volume_linearEquiv (T : Space n ≃L[ℝ] Space n)
+    {K : Set (Space n)} (hK : IsCompact K) (hc : Convex ℝ K) (h0 : K∈𝓝 (0:Space n)) :
+    volume.real (projectionBody (T '' K))=|LinearMap.det T.toLinearMap|^(n-1)*volume.real (projectionBody K) := by
+  rw [projectionBody_linearEquiv T hK hc h0,volumeReal_smul_nonneg (abs_nonneg _),
+    finrank_euclideanSpace_fin,volumeReal_linearEquiv,adjointEquiv_det]
+  rw [show T.symm.toLinearMap=(T.toLinearEquiv.symm : Space n →ₗ[ℝ] Space n) from rfl,
+    LinearEquiv.det_coe_symm,abs_inv]
+  have hd := abs_pos.mpr (linearEquiv_det_ne_zero T)
+  have he : n=(n-1)+1 := by have := NeZero.pos n;omega
+  have hp := congrArg (fun k : ℕ => |LinearMap.det T.toLinearMap|^k) he
+  simp only [pow_succ] at hp
+  rw [hp]
+  field_simp
+
+lemma projectionRatio_linearEquiv (T : Space n ≃L[ℝ] Space n)
+    {K : Set (Space n)} (hK : IsCompact K) (hc : Convex ℝ K) (h0 : K∈𝓝 (0:Space n)) :
+    projectionRatio (T '' K)=projectionRatio K := by
+  rw [projectionRatio,projection_volume_linearEquiv T hK hc h0,volumeReal_linearEquiv,
+    projectionRatio,mul_pow]
+  have hd := abs_pos.mpr (linearEquiv_det_ne_zero T)
+  field_simp
+
+end PettyProjection
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology
+open scoped NNReal RealInnerProductSpace Pointwise
+namespace PettyProjection
+variable {n : ℕ}
+
+lemma projectionBody_unitBall : projectionBody (unitBall n)=kappa (n-1) • unitBall n := by
+  have he : projectionBody (unitBall n)=wulff (fun _ : Spherical.Sphere n => kappa (n-1)) := by
+    ext x
+    constructor
+    · intro hx u
+      have hh := hx (u:Space n) (Spherical.norm_coe u)
+      rwa [shadowVolume_unitBall u] at hh
+    · intro hx u hu
+      let v : Spherical.Sphere n := ⟨u,by simpa only [mem_sphere,dist_zero_right] using hu⟩
+      have hh : ⟪u,x⟫≤kappa (n-1) := hx v
+      have hv : shadowVolume (unitBall n) u=kappa (n-1) := shadowVolume_unitBall v
+      rwa [hv]
+  rw [he]
+  have hb : wulff (fun _ : Spherical.Sphere n => kappa (n-1))=closedBall (0:Space n) (kappa (n-1)) :=
+    Subset.antisymm (wulff_subset_closedBall (kappa_pos (n-1)).le (fun _ => le_rfl))
+      (closedBall_subset_wulff (fun _ => le_rfl))
+  rw [hb,unitBall,smul_closedBall' (kappa_pos (n-1)).ne']
+  simp only [smul_zero,Real.norm_of_nonneg (kappa_pos (n-1)).le,mul_one]
+
+lemma projectionRatio_unitBall {n : ℕ} (hn : 2≤n) : projectionRatio (unitBall n)=pettyConstant n := by
+  let : NeZero n := ⟨by omega⟩
+  rw [projectionRatio_normalized hn rfl,projectionBody_unitBall,
+    normalizedVolume_smul (kappa_pos (n-1)).le,normalizedVolume_of_volume (K :=unitBall n) rfl,mul_one]
+  rfl
+
+/-- All affine ellipsoids, including arbitrary translates, attain equality. -/
+theorem ellipsoid_projectionRatio {n : ℕ} (hn : 2≤n) {K : Set (Space n)}
+    (hK : IsEllipsoid K) : projectionRatio K=pettyConstant n := by
+  let : NeZero n := ⟨by omega⟩
+  obtain ⟨a,T,rfl⟩ := hK
+  have he : (fun x : Space n => a+T x) '' unitBall n=(fun y => a+y) '' (T '' unitBall n) := by
+    rw [image_image]
+  rw [he,projectionRatio_translate]
+  change projectionRatio (T.toContinuousLinearEquiv '' unitBall n)=_
+  rw [projectionRatio_linearEquiv T.toContinuousLinearEquiv unitBall_body.1 unitBall_body.2.1 unitBall_nhds,
+    projectionRatio_unitBall hn]
+
+end PettyProjection
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/ProjectionBodies/AffinePosition.lean b/lean/OAI/Geometry/ProjectionBodies/AffinePosition.lean
new file mode 100644
index 0000000000000000000000000000000000000000..f8ca56ff9ee2cdcd042ac6d020d2825c9b422704
--- /dev/null
+++ b/lean/OAI/Geometry/ProjectionBodies/AffinePosition.lean
@@ -0,0 +1,492 @@
+import OAI.Geometry.ProjectionBodies.PositionMatrices
+
+namespace OAI
+
+universe uX
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology
+open scoped RealInnerProductSpace
+namespace PettyProjection
+open Spherical (Sphere mean sigma gaugeField gaugeMoment absolutePair)
+
+/-- The genuine unpolarized cosine pairing of two (not necessarily even) measures. -/
+def measurePair {n : ℕ} (μ ν : Measure (Space n)) : ℝ :=
+  ∫ x,∫ y,|⟪x,y⟫| ∂ν ∂μ
+
+def Represents {n : ℕ} (μ : Measure (Space n)) (g : Seminorm ℝ (Space n)) (k : ℕ) (s : ℝ) : Prop :=
+  ∀ (φ : Space n → ℝ),Continuous φ → (∀ x,φ (-x)=φ x) →
+    (∀ r : ℝ,0 < r → ∀ x,φ (r • x)=r*φ x) →
+      (∫ x,φ x ∂μ)=s*mean (fun u : Sphere n => φ (gaugeField g k u))
+
+lemma represents_pair {n : ℕ} [NeZero n] {μ ν : Measure (Space n)}
+    (hν : Integrable (fun x : Space n => ‖x‖) ν)
+    (f g : Seminorm ℝ (Space n)) {k : ℕ} (hk : 0 < k) {s t : ℝ}
+    (hf : Represents μ f k s) (hg : Represents ν g k t) :
+    measurePair μ ν=s*t*absolutePair (gaugeField f k) (gaugeField g k) := by
+  have hcos : measurePair μ ν=∫ x,cosineSeminorm ν hν x ∂μ := by
+    unfold measurePair
+    apply integral_congr_ae
+    filter_upwards [] with x
+    change (∫ y,|⟪x,y⟫| ∂ν)=∫ y,|⟪y,x⟫| ∂ν
+    simp only [real_inner_comm x]
+  rw [hcos,hf _ (Spherical.seminorm_continuous n _) (fun x => map_neg_eq_map _ x)
+    (fun r hr x => by rw [map_smul_eq_mul,Real.norm_of_nonneg hr.le])]
+  have he (u : Sphere n) : cosineSeminorm ν hν (gaugeField f k u)=
+      t*mean (fun v : Sphere n => |⟪gaugeField f k u,gaugeField g k v⟫|) := by
+    have hh := hg (fun x => |⟪gaugeField f k u,x⟫|) (by fun_prop)
+      (fun x => by simp) (fun r hr x => by simp only [inner_smul_right,abs_mul,abs_of_pos hr])
+    change (∫ x,|⟪x,gaugeField f k u⟫| ∂ν)=_
+    simpa only [real_inner_comm (gaugeField f k u)] using hh
+  simp_rw [he]
+  rw [mean,integral_const_mul,mul_assoc]
+  congr 1
+  congr 1
+  exact (integral_prod _ (Spherical.gaugeField_absolute_integrable f g hk)).symm
+
+namespace RelativeGauge
+variable {n : ℕ} [NeZero n] {μ : Measure (Space n)}
+
+lemma scaled_represents
+    (hμ : Integrable (fun x : Space n => ‖x‖) μ)
+    (hq : ∀ x : Space n,x≠0 → 0 < cosineSeminorm μ hμ x)
+    (g : RelativeGauge (⊤ : Submodule ℝ (Space n))) (hgn : supportFunctional μ g.body=1)
+    (hmin : ∀ q : RelativeGauge (⊤ : Submodule ℝ (Space n)),supportFunctional μ q.body=1 →
+      Spherical.objective (regime n) g.toSeminorm≤Spherical.objective (regime n) q.toSeminorm)
+    {a : ℝ} (ha : 0 < a) :
+    Represents μ (g.scale a ha).toSeminorm (Scalar.exponent n)
+      (supportFunctional μ (g.scale a ha).body/gaugeMoment (g.scale a ha).toSeminorm (Scalar.exponent n-1)) := by
+  intro φ hφ he hh
+  have ht : 0 < gaugeMoment (g.scale a ha).toSeminorm (Scalar.exponent n-1) :=
+    Position.mean_power_pos _ ((g.scale a ha).ne_zero top_ne_bot) _
+  have h := scaled_field_representation hμ hq g hgn hmin ha φ hφ he hh
+  rw [div_mul_eq_mul_div]
+  exact (eq_div_iff ht.ne').mpr h
+
+end RelativeGauge
+end PettyProjection
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology
+open scoped RealInnerProductSpace
+namespace PettyProjection.Position
+open Spherical (Sphere norm_coe)
+
+lemma compact_inward {X : Type uX} [TopologicalSpace X] [CompactSpace X]
+    {a b : X → ℝ} (ha : Continuous a) (hb : Continuous b)
+    (ha0 : ∀ x,0≤a x) (hab : ∀ x,a x=0 → b x<0) :
+    ∃ ε : ℝ,0<ε ∧ ∀ x,0<a x-ε*b x := by
+  let K : Set X := {x | 0≤b x}
+  have hK : IsCompact K := (isClosed_le continuous_const hb).isCompact
+  by_cases hne : K.Nonempty
+  · obtain ⟨x,hx,hmin⟩ := hK.exists_isMinOn hne ha.continuousOn
+    have hax : 0<a x := lt_of_le_of_ne (ha0 x) (fun h => (hab x h.symm).not_ge hx)
+    obtain ⟨M,hM,hbound⟩ := (isCompact_univ.image hb).isBounded.exists_pos_norm_le
+    let ε := a x/(2*M)
+    have hε : 0<ε := div_pos hax (by positivity)
+    refine ⟨ε,hε,fun y => ?_⟩
+    by_cases hy : y∈K
+    · have hmy : a x≤a y := hmin hy
+      have hbM : b y≤M := (le_abs_self _).trans (by simpa only [Real.norm_eq_abs] using hbound (b y) ⟨y,mem_univ _,rfl⟩)
+      have hh : ε*M=a x/2 := by dsimp [ε]; field_simp
+      have hle := mul_le_mul_of_nonneg_left hbM hε.le
+      rw [hh] at hle
+      linarith
+    · have hby : b y<0 := lt_of_not_ge hy
+      have hneg := mul_neg_of_pos_of_neg hε hby
+      linarith [ha0 y]
+  · refine ⟨1,zero_lt_one,fun y => ?_⟩
+    have hby : b y<0 := by
+      by_contra h
+      exact hne ⟨y,le_of_not_gt h⟩
+    linarith [ha0 y]
+
+lemma inward_perturbation {n : ℕ} [NeZero n] {A F : Mat n} (hA : A∈domain n)
+    (hF : Symmetric F) (htr : tr F=0)
+    (hker : ∀ e : Sphere n,op A e=0 → quad F e<0) :
+    ∃ ε : ℝ,0<ε ∧ A-ε • F∈domain n ∧ ∀ x : Space n,x≠0 → 0<quad (A-ε • F) x := by
+  have hca : Continuous (fun u : Sphere n => quad A u) :=
+    by simp_rw [quad_eq]; fun_prop
+  have hcf : Continuous (fun u : Sphere n => quad F u) :=
+    by simp_rw [quad_eq]; fun_prop
+  obtain ⟨ε,hε,hp⟩ := compact_inward hca hcf (fun u => hA.1.2 u)
+    (fun u hu => hker u (op_kernel_of_quad_zero hA.1 hu))
+  have hq : ∀ x : Space n,x≠0 → 0<quad (A-ε • F) x := by
+    intro x hx
+    let u : Sphere n := ⟨‖x‖⁻¹ • x,by rw [mem_sphere_zero_iff_norm,norm_smul,Real.norm_of_nonneg (inv_nonneg.mpr (norm_nonneg _)),inv_mul_cancel₀ (norm_ne_zero_iff.mpr hx)]⟩
+    have hh := hp u
+    rw [← quad_smul,← quad_sub] at hh
+    have he : quad (A-ε • F) u=‖x‖⁻¹^2*quad (A-ε • F) x := by
+      simp only [quad,u,map_smul,inner_smul_left,inner_smul_right,conj_trivial]
+      ring
+    rw [he] at hh
+    exact (mul_pos_iff.mp hh).resolve_right (fun h => (sq_nonneg ‖x‖⁻¹).not_gt h.1) |>.2
+  refine ⟨ε,hε,⟨⟨?_,?_⟩,?_⟩,hq⟩
+  · intro i j
+    change A (i,j)-ε*F (i,j)=A (j,i)-ε*F (j,i)
+    rw [hA.1.1 i j,hF i j]
+  · intro x
+    by_cases hx : x=0
+    · simp [hx,quad]
+    · exact (hq x hx).le
+  · change (∑ i,(A (i,i)-ε*F (i,i)))=1
+    rw [Finset.sum_sub_distrib,← Finset.mul_sum]
+    change tr A-ε*tr F=1
+    rw [hA.2,htr,mul_zero,sub_zero]
+
+/-- The topological mechanism. The field is genuine and continuous;
+its negativity on every null direction forces an interior zero. -/
+theorem traceOne_field_zero {n : ℕ} [NeZero n] (F : {A : Mat n // A∈domain n} → Mat n)
+    (hFc : Continuous F) (hFs : ∀ A,Symmetric (F A)) (hFt : ∀ A,tr (F A)=0)
+    (hFk : ∀ A (e : Sphere n),op A.val e=0 → quad (F A) e<0) :
+    ∃ A : {A : Mat n // A∈domain n},F A=0 ∧ Function.Injective (op A.val) := by
+  let p := FixedPoint.nearest (domain n) domain_nonempty domain_compact.isComplete domain_convex
+  have hp (x : Mat n) : p x∈domain n := FixedPoint.nearest_mem _ _ _ _ x
+  have hpc : Continuous p := (FixedPoint.nearest_lipschitz _ _ _ _).continuous
+  let f : Mat n → Mat n := fun x => p (x-F ⟨p x,hp x⟩)
+  have hfc : Continuous f := hpc.comp (continuous_id.sub (hFc.comp (hpc.subtype_mk hp)))
+  obtain ⟨A,hA,hfix⟩ := FixedPoint.fixedPoint_compact_convex domain_nonempty domain_compact domain_convex
+    hfc.continuousOn (fun x _ => hp _)
+  let a : {A : Mat n // A∈domain n} := ⟨A,hA⟩
+  have hpa : p A=A := FixedPoint.nearest_eq_of_mem _ _ _ _ hA
+  have hfix' : p (A-F a)=A := by simpa only [f,hpa] using hfix
+  have hi := FixedPoint.nearest_inner (domain n) domain_nonempty domain_compact.isComplete domain_convex (A-F a)
+  change ∀ z∈domain n,⟪A-F a-p (A-F a),z-p (A-F a)⟫≤0 at hi
+  rw [hfix'] at hi
+  obtain ⟨ε,hε,hB,_⟩ := inward_perturbation hA (hFs a) (hFt a) (hFk a)
+  have hh := hi (A-ε • F a) hB
+  have hf0 : F a=0 := by
+    simp only [sub_sub_cancel_left,inner_neg_left,inner_neg_right,inner_smul_right,real_inner_self_eq_norm_sq] at hh
+    have hs : ‖F a‖^2≤0 := nonpos_of_mul_nonpos_left (by nlinarith only [hh]) hε
+    have hn : ‖F a‖=0 := by nlinarith [norm_nonneg (F a)]
+    exact norm_eq_zero.mp hn
+  refine ⟨a,hf0,?_⟩
+  apply (op A).ker_eq_bot.mp
+  apply LinearMap.ker_eq_bot'.mpr
+  intro x hx
+  change op A x=0 at hx
+  by_contra hx0
+  let e : Sphere n := ⟨‖x‖⁻¹ • x,by rw [mem_sphere_zero_iff_norm,norm_smul,Real.norm_of_nonneg (inv_nonneg.mpr (norm_nonneg _)),inv_mul_cancel₀ (norm_ne_zero_iff.mpr hx0)]⟩
+  have he : op A e=0 := by change op A (‖x‖⁻¹ • x)=0; rw [map_smul,hx,smul_zero]
+  have hh := hFk a e he
+  rw [hf0] at hh
+  simp only [quad_eq] at hh
+  simp at hh
+end PettyProjection.Position
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology
+open scoped RealInnerProductSpace
+namespace PettyProjection.Position
+open Spherical (Sphere mean seminorm_continuous)
+open RelativeGauge
+variable {n : ℕ} [NeZero n] {μ : Measure (Space n)}
+
+abbrev Domain (n : ℕ) := {A : Mat n // A∈domain n}
+instance domain_compactSpace : CompactSpace (Domain n) := isCompact_iff_compactSpace.mp domain_compact
+
+def domainMap (A : Domain n) : NonzeroMap n := ⟨op A.val,domain_range_ne_bot A.property⟩
+omit [NeZero n] in
+lemma domainMap_continuous : Continuous (domainMap : Domain n → NonzeroMap n) :=
+  (op_continuous.comp continuous_subtype_val).subtype_mk _
+
+lemma optimizer_kernel (hμ : Integrable (fun x : Space n => ‖x‖) μ)
+    (hq : ∀ x : Space n,x≠0 → 0<cosineSeminorm μ hμ x) (b : Bool) (A : Domain n)
+    (e : Space n) (he : op A.val e=0) : optimizer hμ hq b (domainMap A) e=0 := by
+  let V := LinearMap.range (op A.val).toLinearMap
+  have hp : V.starProjection e=0 := by
+    apply Submodule.eq_starProjection_of_mem_of_inner_eq_zero (V.zero_mem)
+    rintro y ⟨z,rfl⟩
+    simp only [sub_zero]
+    have hs := symmetric_op A.property.1.1 e z
+    change ⟪op A.val e,z⟫=⟪e,op A.val z⟫ at hs
+    rw [he,inner_zero_left] at hs
+    exact hs.symm
+  have hh := (optimizer hμ hq b (domainMap A)).projection e
+  change (optimizer hμ hq b (domainMap A)) (V.starProjection e)=_ at hh
+  rw [hp] at hh
+  exact hh.symm.trans (map_zero _)
+
+lemma optimizer_joint_continuous (hμ : Integrable (fun x : Space n => ‖x‖) μ)
+    (hq : ∀ x : Space n,x≠0 → 0<cosineSeminorm μ hμ x) (b : Bool) :
+    Continuous (fun p : Domain n × Sphere n => optimizer hμ hq b (domainMap p.1) p.2) := by
+  exact ((optimizerSphere_continuous hμ hq b).comp domainMap_continuous).fst'.eval continuous_snd
+
+def positionField (hμ : Integrable (fun x : Space n => ‖x‖) μ)
+    (hq : ∀ x : Space n,x≠0 → 0<cosineSeminorm μ hμ x) (b : Bool) (k : ℕ) (A : Domain n) : Mat n :=
+  momentField (optimizer hμ hq b (domainMap A)).toSeminorm k
+
+lemma positionField_continuous (hμ : Integrable (fun x : Space n => ‖x‖) μ)
+    (hq : ∀ x : Space n,x≠0 → 0<cosineSeminorm μ hμ x) (b : Bool) (k : ℕ) :
+    Continuous (positionField hμ hq b k) := by
+  have hg := optimizer_joint_continuous hμ hq b
+  have hm : Continuous (fun A : Domain n => mean (fun u : Sphere n => optimizer hμ hq b (domainMap A) u^k)) := by
+    simpa only [Measure.restrict_univ,mean] using
+      (continuous_parametric_integral_of_continuous (μ := Spherical.sigma n) (hg.pow k) isCompact_univ)
+  have hij (i j : Fin n) : Continuous (fun A : Domain n => mean (fun u : Sphere n =>
+      optimizer hμ hq b (domainMap A) u^k*(u:Space n) i*(u:Space n) j)) := by
+    have hc : Continuous (fun p : Domain n × Sphere n =>
+        optimizer hμ hq b (domainMap p.1) p.2^k*(p.2:Space n) i*(p.2:Space n) j) :=
+      (hg.pow k).mul (by fun_prop) |>.mul (by fun_prop)
+    simpa only [Measure.restrict_univ,mean] using
+      (continuous_parametric_integral_of_continuous (μ := Spherical.sigma n) hc isCompact_univ)
+  apply (PiLp.continuous_toLp 2 _).comp
+  apply continuous_pi
+  intro ij
+  exact ((hij ij.1 ij.2).div hm (fun A =>
+    (mean_power_pos _ ((optimizer hμ hq b (domainMap A)).ne_zero (domainMap A).property) k).ne')).sub continuous_const
+
+/-- A genuine nonsingular affine position for the variational optimizer. The
+trace-one representative is used here; determinant normalization is a scalar
+change and is performed in the global assembly. -/
+theorem exists_optimizer_position (hμ : Integrable (fun x : Space n => ‖x‖) μ)
+    (hq : ∀ x : Space n,x≠0 → 0<cosineSeminorm μ hμ x) (b : Bool) {k : ℕ} (hk : 0<k) :
+    ∃ A : Domain n,Function.Injective (op A.val) ∧
+      momentField (optimizer hμ hq b (domainMap A)).toSeminorm k=0 := by
+  obtain ⟨A,hA,hinj⟩ := traceOne_field_zero (positionField hμ hq b k)
+    (positionField_continuous hμ hq b k)
+    (fun A => field_symmetric _ k)
+    (fun A => field_trace _ ((optimizer hμ hq b (domainMap A)).ne_zero (domainMap A).property) k)
+    (fun A e he => field_kernel ((optimizer hμ hq b (domainMap A)).ne_zero (domainMap A).property)
+      hk e (optimizer_kernel hμ hq b A e he))
+  exact ⟨A,hinj,hA⟩
+end PettyProjection.Position
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology
+open scoped RealInnerProductSpace Pointwise
+namespace PettyProjection
+variable {n : ℕ}
+
+lemma volumeReal_linearEquiv (T : Space n ≃L[ℝ] Space n) (K : Set (Space n)) :
+    volume.real (T '' K)=|LinearMap.det T.toLinearMap| * volume.real K := by
+  unfold Measure.real
+  rw [Measure.addHaar_image_continuousLinearEquiv,ENNReal.toReal_mul,ENNReal.toReal_ofReal (abs_nonneg _)]
+
+lemma linearEquiv_body (T : Space n ≃L[ℝ] Space n) {K : Set (Space n)} (hK : IsConvexBody K) :
+    IsConvexBody (T '' K) := by
+  refine ⟨hK.1.image T.continuous,hK.2.1.linear_image T.toLinearMap,?_⟩
+  change (interior (T.toHomeomorph '' K)).Nonempty
+  rw [← T.toHomeomorph.image_interior]
+  exact hK.2.2.image T
+
+lemma linearEquiv_det_ne_zero (T : Space n ≃L[ℝ] Space n) : LinearMap.det T.toLinearMap≠0 :=
+  T.toLinearEquiv.isUnit_det'.ne_zero
+
+def volumeFactor (T : Space n ≃L[ℝ] Space n) : ℝ := |LinearMap.det T.toLinearMap|^((n:ℝ)⁻¹)
+lemma volumeFactor_pos (T : Space n ≃L[ℝ] Space n) : 0 < volumeFactor T :=
+  Real.rpow_pos_of_pos (abs_pos.mpr (linearEquiv_det_ne_zero T)) _
+
+lemma normalizedVolume_linearEquiv (T : Space n ≃L[ℝ] Space n) (K : Set (Space n)) :
+    normalizedVolume (T '' K)=volumeFactor T*normalizedVolume K := by
+  rw [normalizedVolume,volumeRoot,volumeReal_linearEquiv,
+    Real.mul_rpow (abs_nonneg _) measureReal_nonneg]
+  unfold volumeFactor normalizedVolume volumeRoot
+  ring
+
+lemma volumeFactor_symm (T : Space n ≃L[ℝ] Space n) : volumeFactor T.symm=(volumeFactor T)⁻¹ := by
+  unfold volumeFactor
+  rw [show T.symm.toLinearMap=(T.toLinearEquiv.symm : Space n →ₗ[ℝ] Space n) from rfl,
+    LinearEquiv.det_coe_symm,abs_inv,Real.inv_rpow (abs_nonneg _)]
+
+lemma symmetric_equiv_symm (T : Space n ≃L[ℝ] Space n) (hT : T.toLinearMap.IsSymmetric) :
+    T.symm.toLinearMap.IsSymmetric := by
+  intro x y
+  change ⟪T.symm x,y⟫=⟪x,T.symm y⟫
+  have hh := hT (T.symm x) (T.symm y)
+  change ⟪T (T.symm x),T.symm y⟫=⟪T.symm x,T (T.symm y)⟫ at hh
+  simpa only [T.apply_symm_apply] using hh.symm
+
+lemma functional_map_symmetric {μ : Measure (Space n)} (T : Space n ≃L[ℝ] Space n)
+    (hT : T.toLinearMap.IsSymmetric) {K : Set (Space n)} (hK : IsCompact K) (hne : K.Nonempty) :
+    supportFunctional (μ.map T) K=supportFunctional μ (T '' K) := by
+  change supportFunctional (μ.map T.toContinuousLinearMap) K=_
+  rw [functional_map T.toContinuousLinearMap hK]
+  unfold supportFunctional
+  apply integral_congr_ae
+  filter_upwards [] with x
+  change support K (T.toContinuousLinearMap x)=support (T.toContinuousLinearMap '' K) x
+  rw [support_linear_image hK hne T.toContinuousLinearMap,hT.clm_adjoint_eq]
+
+lemma measurePair_cosine {μ ν : Measure (Space n)}
+    (hν : Integrable (fun x : Space n => ‖x‖) ν) :
+    measurePair μ ν=∫ x,cosineSeminorm ν hν x ∂μ := by
+  unfold measurePair
+  apply integral_congr_ae
+  filter_upwards [] with x
+  change (∫ y,|⟪x,y⟫| ∂ν)=∫ y,|⟪y,x⟫| ∂ν
+  simp only [real_inner_comm x]
+
+lemma measurePair_map_symmetric {μ ν : Measure (Space n)}
+    (hν : Integrable (fun x : Space n => ‖x‖) ν)
+    (T : Space n ≃L[ℝ] Space n) (hT : T.toLinearMap.IsSymmetric) :
+    measurePair (μ.map T) (ν.map T.symm)=measurePair μ ν := by
+  change measurePair (μ.map T.toContinuousLinearMap) (ν.map T.symm.toContinuousLinearMap)=_
+  rw [measurePair_cosine (norm_integrable_map hν T.symm.toContinuousLinearMap),
+    integral_map T.toContinuousLinearMap.continuous.measurable.aemeasurable (Spherical.seminorm_continuous n _).aestronglyMeasurable,
+    measurePair_cosine hν]
+  apply integral_congr_ae
+  filter_upwards [] with x
+  rw [cosine_map ν hν T.symm.toContinuousLinearMap,(symmetric_equiv_symm T hT).clm_adjoint_eq]
+  simp
+
+end PettyProjection
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology
+open scoped RealInnerProductSpace
+namespace PettyProjection.Position
+open Spherical
+variable {n : ℕ} [NeZero n]
+
+lemma mean_inner_sq (x : Space n) : (n:ℝ)*mean (fun u : Sphere n => ⟪x,(u:Space n)⟫^2)=‖x‖^2 := by
+  by_cases hx : x=0
+  · simp [hx,mean]
+  let e : Sphere n := ⟨‖x‖⁻¹ • x,by rw [mem_sphere_zero_iff_norm,norm_smul,Real.norm_of_nonneg (inv_nonneg.mpr (norm_nonneg _)),inv_mul_cancel₀ (norm_ne_zero_iff.mpr hx)]⟩
+  have he := sphere_kernel_moment (0 : Seminorm ℝ (Space n)) 0 e (by simp)
+  simp only [Nat.cast_zero,add_zero,pow_zero,one_mul,mean_const] at he
+  have hh : mean (fun u : Sphere n => ⟪(e:Space n),(u:Space n)⟫^2)=
+      ‖x‖⁻¹^2*mean (fun u : Sphere n => ⟪x,(u:Space n)⟫^2) := by
+    simp only [e,inner_smul_left,conj_trivial,mul_pow,mean,integral_const_mul]
+  rw [hh] at he
+  have hn := norm_ne_zero_iff.mpr hx
+  field_simp at he
+  nlinarith [sq_pos_of_ne_zero hn]
+
+omit [NeZero n] in
+lemma symmetric_eq_zero_of_quad {A : Mat n} (hA : Symmetric A) (hq : ∀ x,quad A x=0) : A=0 := by
+  have hd (i : Fin n) : A (i,i)=0 := by rw [← quad_basis,hq]
+  apply WithLp.ofLp_injective
+  funext ij
+  have hh := hq (EuclideanSpace.single ij.1 1+EuclideanSpace.single ij.2 1)
+  have he := quad_basis_add A ij.1 ij.2 1 1
+  simp only [one_smul,one_pow,one_mul,hd,hA ij.2 ij.1] at he
+  change A ij=0
+  linarith
+
+lemma constantMomentField_zero : momentField (0 : Seminorm ℝ (Space n)) 0=0 := by
+  apply symmetric_eq_zero_of_quad (field_symmetric _ _) (fun x => ?_)
+  rw [field_quad]
+  simp only [pow_zero,one_mul,mean_const,div_one]
+  have hm := mean_inner_sq x
+  have hn : (n:ℝ)≠0 := Nat.cast_ne_zero.mpr (NeZero.ne n)
+  rw [sub_eq_zero,eq_div_iff hn]
+  linarith
+
+lemma coordinateProduct_mean (i j : Fin n) : mean (coordinateProduct n i j)=(if i=j then 1 else 0)/(n:ℝ) := by
+  have hh := congrArg (fun A : Mat n => A (i,j)) (constantMomentField_zero (n := n))
+  change mean (fun u : Sphere n => (0 : Seminorm ℝ (Space n)) u^0*(u:Space n) i*(u:Space n) j)/
+    mean (fun u : Sphere n => (0 : Seminorm ℝ (Space n)) u^0)-(if i=j then 1 else 0)/(n:ℝ)=0 at hh
+  simp only [pow_zero,one_mul,mean_const,div_one] at hh
+  exact sub_eq_zero.mp hh
+
+/-- The matrix isotropy condition gives precisely the actual H₂ projection,
+not a replacement cancellation condition. -/
+lemma field_zero_harmonic {n : ℕ} (hn : 2≤n) [NeZero n] (g : Seminorm ℝ (Space n))
+    (hg : g≠0) (k : ℕ) (hF : momentField g k=0) :
+    (harmonicSpace n 2).starProjection (toH n
+      (restrictContinuous (fun x => g x^k) ((seminorm_continuous n g).pow k)))=0 := by
+  let f : C(Sphere n,ℝ) := restrictContinuous (fun x => g x^k) ((seminorm_continuous n g).pow k)
+  let c := mean (fun u : Sphere n => g u^k)
+  let v := toH n f-toH n (ContinuousMap.const (Sphere n) c)
+  have ho : v∈(lowSpace n)ᗮ := by
+    apply (Submodule.mem_orthogonal _ _).mpr
+    intro w hw
+    induction hw using Submodule.span_induction with
+    | mem w hw =>
+      obtain ⟨⟨i,j⟩,rfl⟩ := hw
+      rw [real_inner_comm,show v=toH n (f-ContinuousMap.const (Sphere n) c) by simp [v],inner_toH]
+      change mean (fun u : Sphere n => (g u^k-c)*((u:Space n) i*(u:Space n) j))=0
+      have hij := congrArg (fun A : Mat n => A (i,j)) hF
+      change mean (fun u : Sphere n => g u^k*(u:Space n) i*(u:Space n) j)/c-(if i=j then 1 else 0)/(n:ℝ)=0 at hij
+      have hc : c≠0 := (mean_power_pos g hg k).ne'
+      have hv := (div_eq_iff hc).mp (sub_eq_zero.mp hij)
+      change _=0
+      simp_rw [sub_mul]
+      have hi : Integrable (fun u : Sphere n => g u^k*((u:Space n) i*(u:Space n) j)) (sigma n) :=
+        continuous_integrable (((seminorm_continuous n g).comp continuous_subtype_val).pow k |>.mul
+          (by fun_prop : Continuous (fun u : Sphere n => (u:Space n) i*(u:Space n) j)))
+      have hi' : Integrable (fun u : Sphere n => c*((u:Space n) i*(u:Space n) j)) (sigma n) :=
+        continuous_integrable (by fun_prop)
+      rw [mean,integral_sub hi hi',integral_const_mul]
+      change mean (fun u : Sphere n => g u^k*((u:Space n) i*(u:Space n) j))-c*mean (coordinateProduct n i j)=0
+      rw [coordinateProduct_mean]
+      have he : mean (fun u : Sphere n => g u^k*((u:Space n) i*(u:Space n) j))=
+          mean (fun u : Sphere n => g u^k*(u:Space n) i*(u:Space n) j) := by congr 1;funext u;ring
+      rw [he,hv]; ring
+    | zero => simp
+    | add x y _ _ hx hy => simp only [inner_add_left,hx,hy,add_zero]
+    | smul a x _ hx => simp only [inner_smul_left,conj_trivial,hx,mul_zero]
+  apply Submodule.eq_starProjection_of_mem_of_inner_eq_zero (Submodule.zero_mem _)
+  intro w hw
+  simp only [sub_zero]
+  have hvol := (Submodule.mem_orthogonal _ _).mp ho w (harmonicSpace_two_le_lowSpace n hw)
+  have hconst : ⟪toH n (ContinuousMap.const (Sphere n) c),w⟫=0 :=
+    harmonic_inner_ne hn (d := 0) (by decide) (constant_mem_harmonicSpace_zero n c) hw
+  change ⟪toH n f,w⟫=0
+  dsimp only [v] at hvol
+  have hconst' : ⟪w,toH n (ContinuousMap.const (Sphere n) c)⟫=0 :=
+    (real_inner_comm _ _).trans hconst
+  rw [inner_sub_right,hconst',sub_zero] at hvol
+  exact (real_inner_comm _ _).trans hvol
+end PettyProjection.Position
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology
+open scoped RealInnerProductSpace
+namespace PettyProjection
+namespace RelativeGauge
+variable {n : ℕ} [NeZero n]
+
+def IsMinimizer (μ : Measure (Space n)) (b : Bool) (g : RelativeGauge (⊤ : Submodule ℝ (Space n))) : Prop :=
+  supportFunctional μ g.body=1 ∧
+    ∀ q : RelativeGauge (⊤ : Submodule ℝ (Space n)),supportFunctional μ q.body=1 →
+      Spherical.objective b g.toSeminorm≤Spherical.objective b q.toSeminorm
+
+omit [NeZero n] in
+lemma copy_minimizer {V : Submodule ℝ (Space n)} (hV : V=⊤)
+    {μ : Measure (Space n)} {b : Bool} (g : RelativeGauge V)
+    (hgn : supportFunctional μ g.body=1)
+    (hmin : ∀ q : RelativeGauge V,supportFunctional μ q.body=1 →
+      Spherical.objective b g.toSeminorm≤Spherical.objective b q.toSeminorm) :
+    ∃ f : RelativeGauge (⊤ : Submodule ℝ (Space n)),f.toSeminorm=g.toSeminorm ∧ IsMinimizer μ b f := by
+  subst V
+  exact ⟨g,rfl,hgn,hmin⟩
+
+/-- Actual full-dimensional position of the logarithmic/linear optimizer,
+with no determinant normalization needed in the later weighted assembly. -/
+theorem exists_symmetric_position {μ : Measure (Space n)}
+    (hμ : Integrable (fun x : Space n => ‖x‖) μ)
+    (hq : ∀ x : Space n,x≠0 → 0 < cosineSeminorm μ hμ x) :
+    ∃ T : Space n ≃L[ℝ] Space n,T.toLinearMap.IsSymmetric ∧
+      ∃ g : RelativeGauge (⊤ : Submodule ℝ (Space n)),IsMinimizer (μ.map T) (regime n) g ∧
+        Position.momentField g.toSeminorm (Scalar.exponent n)=0 := by
+  obtain ⟨A,hinj,hF⟩ := Position.exists_optimizer_position hμ hq (regime n) (exponent_pos n)
+  have hsurj : Function.Surjective (Position.op A.val) :=
+    LinearMap.surjective_of_injective (f := (Position.op A.val).toLinearMap) hinj
+  have hfull : LinearMap.range (Position.op A.val).toLinearMap=⊤ := LinearMap.range_eq_top.mpr hsurj
+  let T : Space n ≃L[ℝ] Space n :=
+    (LinearEquiv.ofBijective (Position.op A.val).toLinearMap ⟨hinj,hsurj⟩).toContinuousLinearEquiv
+  obtain ⟨g,hg,hmin⟩ := copy_minimizer hfull (optimizer hμ hq (regime n) (Position.domainMap A))
+    (optimizer_normalized hμ hq (regime n) (Position.domainMap A))
+    (optimizer_minimal hμ hq (regime n) (Position.domainMap A))
+  refine ⟨T,Position.symmetric_op A.property.1.1,g,hmin,?_⟩
+  rw [hg]
+  exact hF
+
+lemma exists_full_minimizer {μ : Measure (Space n)}
+    (hμ : Integrable (fun x : Space n => ‖x‖) μ)
+    (hq : ∀ x : Space n,x≠0 → 0 < cosineSeminorm μ hμ x) (b : Bool) :
+    ∃ g : RelativeGauge (⊤ : Submodule ℝ (Space n)),IsMinimizer μ b g :=
+  exists_minimizer hμ top_ne_bot (fun x _ hx => hq x hx) b
+
+end RelativeGauge
+end PettyProjection
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/ProjectionBodies/BilinearEquality.lean b/lean/OAI/Geometry/ProjectionBodies/BilinearEquality.lean
new file mode 100644
index 0000000000000000000000000000000000000000..c1e879ee372eb60d96b724237c2c3e4294c01a9d
--- /dev/null
+++ b/lean/OAI/Geometry/ProjectionBodies/BilinearEquality.lean
@@ -0,0 +1,312 @@
+import OAI.Geometry.ProjectionBodies.AffinePosition
+
+namespace OAI
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology
+open scoped RealInnerProductSpace
+namespace PettyProjection
+open Spherical (Sphere mean sigma gaugeMoment)
+namespace RelativeGauge
+variable {n : ℕ} [NeZero n] {μ : Measure (Space n)}
+
+omit [NeZero n] in
+lemma scale_momentField (g : RelativeGauge (⊤ : Submodule ℝ (Space n)))
+    {a : ℝ} (ha : 0<a) (k : ℕ) :
+    Position.momentField (g.scale a ha).toSeminorm k=Position.momentField g.toSeminorm k := by
+  apply WithLp.ofLp_injective
+  funext ij
+  change mean (fun u : Sphere n => (a*g u)^k*(u:Space n) ij.1*(u:Space n) ij.2)/
+    mean (fun u : Sphere n => (a*g u)^k)-_=mean (fun u : Sphere n => g u^k*(u:Space n) ij.1*(u:Space n) ij.2)/
+    mean (fun u : Sphere n => g u^k)-_
+  have he : mean (fun u : Sphere n => (a*g u)^k*(u:Space n) ij.1*(u:Space n) ij.2)=
+      a^k*mean (fun u : Sphere n => g u^k*(u:Space n) ij.1*(u:Space n) ij.2) := by
+    simp_rw [mul_pow,mul_assoc]
+    exact integral_const_mul _ _
+  have he' : mean (fun u : Sphere n => (a*g u)^k)=a^k*mean (fun u : Sphere n => g u^k) := by
+    simp only [mul_pow,mean,integral_const_mul]
+  rw [he,he',mul_div_mul_left _ _ (pow_ne_zero _ ha.ne')]
+
+lemma volumeNormalize_harmonic (hn : 2≤n) (g : RelativeGauge (⊤ : Submodule ℝ (Space n)))
+    (hF : Position.momentField g.toSeminorm (Scalar.exponent n)=0) :
+    (Spherical.harmonicSpace n 2).starProjection (Spherical.toH n
+      (Spherical.restrictContinuous (fun x => g.volumeNormalize x^Scalar.exponent n)
+        ((Spherical.seminorm_continuous n g.volumeNormalize.toSeminorm).pow _)))=0 := by
+  apply Position.field_zero_harmonic hn _ (g.volumeNormalize.ne_zero top_ne_bot)
+  rw [volumeNormalize,g.scale_momentField,hF]
+
+lemma volumeNormalize_representation
+    (hμ : Integrable (fun x : Space n => ‖x‖) μ)
+    (hq : ∀ x : Space n,x≠0 → 0<cosineSeminorm μ hμ x)
+    (g : RelativeGauge (⊤ : Submodule ℝ (Space n))) (hg : IsMinimizer μ (regime n) g) :
+    ∃ s : ℝ,0<s ∧ Represents μ g.volumeNormalize.toSeminorm (Scalar.exponent n) s ∧
+      supportFunctional μ g.volumeNormalize.body=s*gaugeMoment g.volumeNormalize.toSeminorm (Scalar.exponent n-1) := by
+  have hm := Position.mean_power_pos g.volumeNormalize.toSeminorm (g.volumeNormalize.ne_zero top_ne_bot) (Scalar.exponent n-1)
+  have hv := normalizedVolume_pos g.compact g.top_interior
+  have hf : 0<supportFunctional μ g.volumeNormalize.body := by
+    rw [volumeNormalize,g.functional_scale,hg.1,mul_one]
+    exact inv_pos.mpr hv
+  refine ⟨_,div_pos hf hm,scaled_represents hμ hq g hg.1 hg.2 hv,?_⟩
+  exact (div_mul_cancel₀ _ hm.ne').symm
+
+end RelativeGauge
+end PettyProjection
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology
+open scoped RealInnerProductSpace Pointwise
+namespace PettyProjection
+open Spherical (Sphere mean sigma gaugeMoment cosineConstant)
+open RelativeGauge
+
+lemma normalizedVolume_of_volume {n : ℕ} {K : Set (Space n)} (hK : volume.real K=kappa n) :
+    normalizedVolume K=1 := by
+  rw [normalizedVolume_formula,hK,div_self (kappa_pos n).ne',Real.one_rpow]
+
+lemma cosine_equiv_pos {n : ℕ} {μ : Measure (Space n)}
+    (hμ : Integrable (fun x : Space n => ‖x‖) μ)
+    (hq : ∀ x : Space n,x≠0 → 0< cosineSeminorm μ hμ x)
+    (T : Space n ≃L[ℝ] Space n) :
+    ∀ x : Space n,x≠0 → 0< cosineSeminorm (μ.map T) (norm_integrable_map hμ T.toContinuousLinearMap) x := by
+  intro x hx
+  exact cosine_map_pos hμ hq T.toContinuousLinearMap x ⟨T.symm x,T.apply_symm_apply x⟩ hx
+
+/-- The genuine bilinear inequality, with its global rigidity
+consequence. The hypotheses are the support-functional bounds supplied by
+Brunn--Minkowski, not the desired projection-volume conclusion. -/
+theorem measure_bilinear {n : ℕ} (hn : 4≤ n) [NeZero n] {μ ν : Measure (Space n)}
+    (hμ : Integrable (fun x : Space n => ‖x‖) μ)
+    (hν : Integrable (fun x : Space n => ‖x‖) ν)
+    (hqμ : ∀ x : Space n,x≠0 → 0< cosineSeminorm μ hμ x)
+    (hqν : ∀ x : Space n,x≠0 → 0< cosineSeminorm ν hν x)
+    (hbμ : ∀ M : Set (Space n),IsConvexBody M → normalizedVolume M≤ supportFunctional μ M)
+    (hbν : ∀ M : Set (Space n),IsConvexBody M → normalizedVolume M≤ supportFunctional ν M) :
+    cosineConstant n≤ measurePair μ ν ∧
+      (measurePair μ ν=cosineConstant n →
+        ∃ T : Space n ≃L[ℝ] Space n,supportFunctional μ (T '' unitBall n)=normalizedVolume (T '' unitBall n)) := by
+  obtain ⟨T,hT,g,hg,hF⟩ := exists_symmetric_position hμ hqμ
+  let μ' := μ.map T
+  let ν' := ν.map T.symm
+  have hiμ : Integrable (fun x : Space n => ‖x‖) μ' := norm_integrable_map hμ T.toContinuousLinearMap
+  have hiν : Integrable (fun x : Space n => ‖x‖) ν' := norm_integrable_map hν T.symm.toContinuousLinearMap
+  have hpμ : ∀ x : Space n,x≠0 → 0< cosineSeminorm μ' hiμ x := cosine_equiv_pos hμ hqμ T
+  have hpν : ∀ x : Space n,x≠0 → 0< cosineSeminorm ν' hiν x := cosine_equiv_pos hν hqν T.symm
+  obtain ⟨f,hf⟩ := exists_full_minimizer hiν hpν (regime n)
+  obtain ⟨s,hs,hsrep,hse⟩ := volumeNormalize_representation hiμ hpμ g hg
+  obtain ⟨t,ht,htrep,hte⟩ := volumeNormalize_representation hiν hpν f hf
+  let α := volumeFactor T
+  have hα : 0<α := volumeFactor_pos T
+  have hgbody : IsConvexBody g.volumeNormalize.body :=
+    ⟨g.volumeNormalize.compact,g.volumeNormalize.convex,g.volumeNormalize.top_interior⟩
+  have hfbody : IsConvexBody f.volumeNormalize.body :=
+    ⟨f.volumeNormalize.compact,f.volumeNormalize.convex,f.volumeNormalize.top_interior⟩
+  have hgs : α≤ supportFunctional μ' g.volumeNormalize.body := by
+    have h := hbμ _ (linearEquiv_body T hgbody)
+    rw [normalizedVolume_linearEquiv,normalizedVolume_of_volume g.volumeNormalize_body_volume,mul_one,
+      ← functional_map_symmetric T hT hgbody.1 (hgbody.2.2.mono interior_subset)] at h
+    exact h
+  have hft : α⁻¹≤ supportFunctional ν' f.volumeNormalize.body := by
+    have h := hbν _ (linearEquiv_body T.symm hfbody)
+    rw [normalizedVolume_linearEquiv,normalizedVolume_of_volume f.volumeNormalize_body_volume,mul_one,
+      volumeFactor_symm,← functional_map_symmetric T.symm (symmetric_equiv_symm T hT)
+        hfbody.1 (hfbody.2.2.mono interior_subset)] at h
+    exact h
+  rw [hse] at hgs
+  rw [hte] at hft
+  have hsbound : 1≤(s/α)*gaugeMoment g.volumeNormalize.toSeminorm (Scalar.exponent n-1) := by
+    rw [div_mul_eq_mul_div,le_div_iff₀ hα,one_mul]
+    exact hgs
+  have htbound : 1≤(t*α)*gaugeMoment f.volumeNormalize.toSeminorm (Scalar.exponent n-1) := by
+    have hh := mul_le_mul_of_nonneg_left hft hα.le
+    rw [mul_inv_cancel₀ hα.ne'] at hh
+    nlinarith [hh]
+  obtain ⟨hlow,heq⟩ := Spherical.normalized_norm_pair hn g.volumeNormalize.toSeminorm
+    f.volumeNormalize.toSeminorm g.volumeNormalize.top_pos f.volumeNormalize.top_pos
+    g.volumeNormalize_moment f.volumeNormalize_moment (g.volumeNormalize_harmonic (by omega) hF)
+    (div_pos hs hα) (mul_pos ht hα) hsbound htbound
+  have hsprod : (s/α)*(t*α)=s*t := by field_simp
+  rw [hsprod] at hlow heq
+  have hpair := represents_pair hiν g.volumeNormalize.toSeminorm f.volumeNormalize.toSeminorm
+    (exponent_pos n) hsrep htrep
+  have hPinv : measurePair μ' ν'=measurePair μ ν := measurePair_map_symmetric hν T hT
+  rw [hPinv] at hpair
+  refine ⟨hpair.symm ▸ hlow,?_⟩
+  intro h
+  obtain ⟨hg1,_,hs1,_⟩ := heq (hpair.symm.trans h)
+  have hbody : g.volumeNormalize.body=unitBall n := full_body_of_sphere_one _ hg1
+  have hmoment : gaugeMoment g.volumeNormalize.toSeminorm (Scalar.exponent n-1)=1 := by
+    simp only [gaugeMoment]
+    simp_rw [hg1,one_pow]
+    exact Spherical.mean_const n 1
+  have hsα : s=α := (div_eq_one_iff_eq hα.ne').mp hs1
+  rw [hbody,hmoment,mul_one,hsα] at hse
+  refine ⟨T,?_⟩
+  have hball : IsConvexBody (unitBall n) := hbody ▸ hgbody
+  rw [← functional_map_symmetric T hT hball.1 (hball.2.2.mono interior_subset),
+    hse,normalizedVolume_linearEquiv,normalizedVolume_of_volume (K := unitBall n) rfl,mul_one]
+
+end PettyProjection
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology
+open scoped NNReal RealInnerProductSpace Pointwise
+namespace PettyProjection
+open Spherical (Sphere norm_coe seminorm_continuous)
+variable {n : ℕ} [NeZero n] {K : Set (Space n)}
+
+lemma gaugeMeasure_norm_integrable (hK : IsCompact K) (hc : Convex ℝ K)
+    (h0 : K∈𝓝 (0:Space n)) : Integrable (fun x : Space n => ‖x‖) (gaugeProjectionMeasure K) := by
+  let := gaugeMeasure_finite hK hc h0
+  exact boundedMeasure_integrable_continuous (gaugeMeasure_bounded hc h0) continuous_norm
+
+lemma gaugeMeasure_cosine_pos (hK : IsCompact K) (hc : Convex ℝ K)
+    (h0 : K∈𝓝 (0:Space n)) :
+    ∀ x : Space n,x≠0 → 0< cosineSeminorm (gaugeProjectionMeasure K) (gaugeMeasure_norm_integrable hK hc h0) x := by
+  intro x hx
+  let u : Sphere n := ⟨‖x‖⁻¹ • x,by simp [norm_smul,hx]⟩
+  have hu := gaugeMeasure_shadow hK hc h0 u
+  have hh := shadowVolume_pos hK h0 (u:Space n)
+  have hn : 0 < (n:ℝ)*kappa n := mul_pos (Nat.cast_pos.mpr (NeZero.pos n)) (kappa_pos n)
+  have hp : 0< cosineSeminorm (gaugeProjectionMeasure K) (gaugeMeasure_norm_integrable hK hc h0) u := by
+    change 0 < ∫ y,|⟪y,(u:Space n)⟫| ∂gaugeProjectionMeasure K
+    nlinarith
+  change 0< cosineSeminorm _ _ (‖x‖⁻¹ • x) at hp
+  rw [map_smul_eq_mul,Real.norm_of_nonneg (inv_nonneg.mpr (norm_nonneg _))] at hp
+  exact (mul_pos_iff.mp hp).elim (fun h => h.2) (fun h => False.elim (not_lt_of_ge (inv_nonneg.mpr (norm_nonneg x)) h.1))
+
+def projectionSeminorm (hK : IsCompact K) (hc : Convex ℝ K) (h0 : K∈𝓝 (0:Space n)) : Seminorm ℝ (Space n) := by
+  let q := cosineSeminorm (gaugeProjectionMeasure K) (gaugeMeasure_norm_integrable hK hc h0)
+  let c : ℝ := (n : ℝ)*kappa n/2
+  have hc : 0 ≤ c := div_nonneg (mul_nonneg (Nat.cast_nonneg _) (kappa_pos n).le) (by norm_num)
+  exact Seminorm.of (fun x => c*q x)
+    (fun x y => by
+      simpa only [mul_add] using mul_le_mul_of_nonneg_left (map_add_le_add q x y) hc)
+    (fun r x => by rw [map_smul_eq_mul];ring)
+
+lemma projectionSeminorm_unit (hK : IsCompact K) (hc : Convex ℝ K) (h0 : K∈𝓝 (0:Space n))
+    (u : Sphere n) : projectionSeminorm hK hc h0 u=shadowVolume K u := by
+  have hh := gaugeMeasure_shadow hK hc h0 u
+  change (n*kappa n/2)*(∫ x,|⟪x,(u:Space n)⟫| ∂gaugeProjectionMeasure K)=_
+  linarith
+
+lemma projectionSeminorm_pos (hK : IsCompact K) (hc : Convex ℝ K) (h0 : K∈𝓝 (0:Space n))
+    (u : Sphere n) : 0< projectionSeminorm hK hc h0 u := by
+  rw [projectionSeminorm_unit]
+  exact shadowVolume_pos hK h0 u
+
+lemma projectionBody_eq_wulff (hK : IsCompact K) (hc : Convex ℝ K) (h0 : K∈𝓝 (0:Space n)) :
+    projectionBody K=wulff (fun u : Sphere n => projectionSeminorm hK hc h0 u) := by
+  ext x
+  constructor
+  · intro hx u
+    change ⟪(u:Space n),x⟫ ≤ projectionSeminorm hK hc h0 u
+    rw [projectionSeminorm_unit]
+    exact hx u (norm_coe u)
+  · intro hx u hu
+    let v : Sphere n := ⟨u,by simpa only [mem_sphere,dist_zero_right] using hu⟩
+    have h : ⟪u,x⟫ ≤ projectionSeminorm hK hc h0 v := hx v
+    rw [projectionSeminorm_unit] at h
+    exact h
+
+lemma projectionBody_body (hK : IsCompact K) (hc : Convex ℝ K) (h0 : K∈𝓝 (0:Space n)) :
+    IsConvexBody (projectionBody K) := by
+  rw [projectionBody_eq_wulff hK hc h0]
+  exact wulff_body ⟨fun u : Sphere n => projectionSeminorm hK hc h0 (u:Space n),(seminorm_continuous n (projectionSeminorm hK hc h0)).comp continuous_subtype_val⟩
+    (projectionSeminorm_pos hK hc h0)
+
+lemma projectionBody_nhds (hK : IsCompact K) (hc : Convex ℝ K) (h0 : K∈𝓝 (0:Space n)) :
+    projectionBody K∈𝓝 (0:Space n) := by
+  rw [projectionBody_eq_wulff hK hc h0]
+  exact wulff_nhds ⟨fun u : Sphere n => projectionSeminorm hK hc h0 (u:Space n),(seminorm_continuous n (projectionSeminorm hK hc h0)).comp continuous_subtype_val⟩
+    (projectionSeminorm_pos hK hc h0)
+
+lemma projectionBody_support (hK : IsCompact K) (hc : Convex ℝ K) (h0 : K∈𝓝 (0:Space n))
+    (x : Space n) : support (projectionBody K) x=
+      (n*kappa n/2)*∫ y,|⟪y,x⟫| ∂gaugeProjectionMeasure K := by
+  rw [projectionBody_eq_wulff hK hc h0,support_wulff_seminorm _ (projectionSeminorm_pos hK hc h0)]
+  rfl
+
+lemma projectionBody_support_unit (hK : IsCompact K) (hc : Convex ℝ K) (h0 : K∈𝓝 (0:Space n))
+    (u : Sphere n) : support (projectionBody K) u=shadowVolume K u := by
+  rw [projectionBody_eq_wulff hK hc h0,support_wulff_seminorm _ (projectionSeminorm_pos hK hc h0),
+    projectionSeminorm_unit]
+
+end PettyProjection
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology
+open scoped NNReal RealInnerProductSpace Pointwise Gradient
+namespace PettyProjection
+open Spherical (Sphere norm_coe mean)
+variable {n : ℕ} [NeZero n]
+
+omit [NeZero n] in
+lemma unitBall_body : IsConvexBody (unitBall n) := by
+  exact ⟨isCompact_closedBall _ _,convex_closedBall _ _,⟨0,mem_interior_iff_mem_nhds.mpr (closedBall_mem_nhds _ (by norm_num))⟩⟩
+
+omit [NeZero n] in
+lemma unitBall_nhds : unitBall n∈𝓝 (0:Space n) := closedBall_mem_nhds _ (by norm_num)
+
+omit [NeZero n] in
+lemma shadowVolume_unitBall (u : Sphere n) : shadowVolume (unitBall n) u=kappa (n-1) := by
+  have him : (perpendicular (u:Space n)).orthogonalProjectionOnto '' unitBall n=
+      closedBall (0:perpendicular (u:Space n)) 1 := by
+    ext z
+    constructor
+    · rintro ⟨x,hx,rfl⟩
+      simp only [unitBall,mem_closedBall,dist_zero_right] at hx ⊢
+      exact ((perpendicular (u:Space n)).norm_orthogonalProjectionOnto_apply_le x).trans hx
+    · intro hz
+      refine ⟨z,?_,Submodule.orthogonalProjectionOnto_mem_subspace_eq_self z⟩
+      simpa [unitBall] using hz
+  let e : perpendicular (u:Space n) ≃ₗᵢ[ℝ] Space (n-1) :=
+    ((stdOrthonormalBasis ℝ (perpendicular (u:Space n))).reindex (finCongr (perpendicular_finrank u))).repr
+  unfold shadowVolume
+  rw [him]
+  have he : e ⁻¹' unitBall (n-1)=closedBall (0:perpendicular (u:Space n)) 1 := by
+    ext z
+    simp [unitBall,dist_zero_right]
+  have hh := e.measurePreserving.measure_preimage (isClosed_closedBall.measurableSet.nullMeasurableSet : NullMeasurableSet (unitBall (n-1)) volume)
+  change volume (e ⁻¹' unitBall (n-1)) = volume (unitBall (n-1)) at hh
+  rw [he] at hh
+  exact congrArg ENNReal.toReal hh
+
+omit [NeZero n] in
+lemma gradient_norm_unit (u : Sphere n) : ∇ (fun x : Space n => ‖x‖) (u:Space n)=(u:Space n) := by
+  have hd := (hasStrictFDerivAt_norm_sq (u:Space n)).hasFDerivAt.sqrt (by rw [norm_coe];norm_num)
+  simp only [norm_coe,one_pow,Real.sqrt_one,mul_one] at hd
+  have hs : (1/(2:ℝ)) • (2 • innerSL ℝ (u:Space n))=innerSL ℝ (u:Space n) := by
+    ext v
+    simp only [smul_apply,two_smul,smul_eq_mul,add_apply]
+    ring
+  rw [hs] at hd
+  have hd' : HasFDerivAt (fun x : Space n => ‖x‖) (innerSL ℝ (u:Space n)) (u:Space n) := by
+    simpa only [Real.sqrt_sq (norm_nonneg _)] using hd
+  apply (InnerProductSpace.toDual ℝ (Space n)).injective
+  rw [toDual_gradient,hd'.fderiv]
+  rfl
+
+lemma ball_cosine_constant : 2*kappa (n-1)=(n:ℝ)*kappa n*Spherical.cosineConstant n := by
+  let u : Sphere n := Spherical.pole n
+  have hh := shadow_gauge_formula (K :=unitBall n) unitBall_body.1 unitBall_body.2.1 unitBall_nhds u
+  have hg : gauge (unitBall n)=(fun x : Space n => ‖x‖) := by
+    funext x
+    simp [unitBall,gauge_closedBall (E :=Space n) (by norm_num : (0:ℝ)≤1) x]
+  have he : (fun v : Sphere n => (gauge (unitBall n) (v:Space n))⁻¹^n*
+      |⟪∇ (gauge (unitBall n)) (v:Space n),(u:Space n)⟫|)=
+      (fun v : Sphere n => |⟪(u:Space n),(v:Space n)⟫|) := by
+    funext v
+    rw [hg,gradient_norm_unit]
+    change (‖(v:Space n)‖)⁻¹^n*|⟪(v:Space n),(u:Space n)⟫|=_
+    rw [norm_coe]
+    simp only [inv_one,one_pow,one_mul,real_inner_comm (v:Space n)]
+  rw [shadowVolume_unitBall,he,Spherical.mean_abs_inner] at hh
+  exact hh
+
+end PettyProjection
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/ProjectionBodies/Brouwer.lean b/lean/OAI/Geometry/ProjectionBodies/Brouwer.lean
new file mode 100644
index 0000000000000000000000000000000000000000..80b0a5d5d054f5a7c3641d8a342344292ddc404b
--- /dev/null
+++ b/lean/OAI/Geometry/ProjectionBodies/Brouwer.lean
@@ -0,0 +1,184 @@
+import OAI.Geometry.ProjectionBodies.FixedPointCalculus
+
+namespace OAI
+
+universe uE
+
+noncomputable section
+open Set Metric Filter Topology MeasureTheory
+open scoped RealInnerProductSpace
+namespace PettyProjection.FixedPoint
+variable {E : Type uE} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+
+lemma contDiffAt_rayRetraction {f : E → E} {x : E} (hf : ContDiffAt ℝ 1 f x)
+    (hc : ‖f x‖<1) (hne : x≠f x) :
+    ContDiffAt ℝ 1 (fun y => rayRetraction (f y) y) x := by
+  have hv : ContDiffAt ℝ 1 (fun y => y-f y) x := contDiffAt_id.sub hf
+  have ha : ContDiffAt ℝ 1 (fun y => ‖y-f y‖^2) x := hv.norm_sq (𝕜 := ℝ)
+  have hb : ContDiffAt ℝ 1 (fun y => ⟪f y,y-f y⟫) x := hf.inner ℝ hv
+  have hc2 : ContDiffAt ℝ 1 (fun y => ‖f y‖^2) x := hf.norm_sq (𝕜 := ℝ)
+  have ha0 : 0<‖x-f x‖^2 := sq_pos_of_pos (norm_pos_iff.mpr (sub_ne_zero.mpr hne))
+  have hcc : ‖f x‖^2<1 := by nlinarith [norm_nonneg (f x)]
+  have hd0 : 0<⟪f x,x-f x⟫^2+‖x-f x‖^2*(1-‖f x‖^2) := by
+    nlinarith [sq_nonneg ⟪f x,x-f x⟫]
+  have hroot := (((hb.pow 2).add (ha.mul (contDiffAt_const.sub hc2))).sqrt
+    (ne_of_gt hd0)).sub hb
+  have ht := hroot.div ha (ne_of_gt ha0)
+  exact hf.add (ht.smul hv)
+
+lemma smooth_fixedPoint_ball [FiniteDimensional ℝ E] {f : E → E}
+    (hf : ContDiff ℝ 1 f) (hb : ∀ x∈closedBall (0:E) 1, ‖f x‖<1) :
+    ∃ x∈closedBall (0:E) 1, f x=x := by
+  by_contra h
+  have hne (x : E) (hx : x∈closedBall (0:E) 1) : x≠f x := by
+    intro he
+    exact h ⟨x,hx,he.symm⟩
+  let r : E → E := fun x => rayRetraction (f x) x
+  have hr (x : E) (hx : x∈closedBall (0:E) 1) : ContDiffAt ℝ 1 r x :=
+    contDiffAt_rayRetraction hf.contDiffAt (hb x hx) (hne x hx)
+  borelize E
+  apply no_smooth_retraction (.addHaar : Measure E)
+    (r' := fderiv ℝ r)
+    (fun x hx => ((hr x hx).differentiableAt (by norm_num)).hasFDerivAt.hasFDerivWithinAt)
+    (fun x hx => ((hr x hx).continuousAt_fderiv (by norm_num)).continuousWithinAt)
+  · intro x hx
+    exact rayRetraction_norm (hb x hx) (hne x hx)
+  · intro x hx
+    exact rayRetraction_boundary (hb x (by simpa only [mem_closedBall,dist_zero_right,hx] using le_rfl (a := (1:ℝ)))) hx
+
+end PettyProjection.FixedPoint
+end
+
+noncomputable section
+open Set Metric Topology
+namespace PettyProjection.FixedPoint
+variable {E : Type uE} [NormedAddCommGroup E]
+section Approximation
+variable [NormedSpace ℝ E]
+
+lemma scaled_approximation {u v : E} {ε : ℝ} (hε : 0<ε) (hv : ‖v‖≤1)
+    (huv : dist u v<ε) :
+    ‖(1+ε)⁻¹ • u‖<1 ∧ dist ((1+ε)⁻¹ • u) v<2*ε := by
+  let a : ℝ := (1+ε)⁻¹
+  have hden : 0<1+ε := by linarith
+  have ha : 0<a := inv_pos.mpr hden
+  have hae : a*(1+ε)=1 := inv_mul_cancel₀ (ne_of_gt hden)
+  have ha1 : a≤1 := by nlinarith
+  have hae2 : 1-a≤ε := by nlinarith
+  have hua : ‖u‖<1+ε := by
+    have hh := norm_le_norm_add_norm_sub v u
+    rw [← dist_eq_norm,dist_comm v u] at hh
+    linarith
+  constructor
+  · rw [norm_smul,Real.norm_eq_abs,abs_of_pos ha]
+    nlinarith
+  · have ht : dist (a • u) (a • v) < ε := by
+      rw [dist_smul₀,Real.norm_eq_abs,abs_of_pos ha]
+      nlinarith [dist_nonneg (x := u) (y := v)]
+    have hs : dist (a • v) v≤ε := by
+      rw [dist_eq_norm]
+      have he : a • v-v=(a-1) • v := by rw [sub_smul,one_smul]
+      rw [he,norm_smul,Real.norm_eq_abs,abs_of_nonpos (sub_nonpos.mpr ha1)]
+      nlinarith [norm_nonneg v]
+    have htri := dist_triangle (a • u) (a • v) v
+    change dist (a • u) v<2*ε
+    linarith
+
+end Approximation
+
+variable [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
+
+lemma uniformlyContinuous_ball_extension {f : E → E}
+    (hf : ContinuousOn f (closedBall (0:E) 1)) :
+    UniformContinuous (fun x => f (ballNearest x)) := by
+  let : CompactSpace (closedBall (0:E) 1) := isCompact_iff_compactSpace.mp (isCompact_closedBall _ _)
+  have hfc : Continuous (fun x : closedBall (0:E) 1 => f x) :=
+    continuousOn_iff_continuous_domRestrict.mp hf
+  have hp : UniformContinuous (fun x : E => (⟨ballNearest x,ballNearest_mem x⟩ : closedBall (0:E) 1)) :=
+    ballNearest_lipschitz.uniformContinuous.subtype_mk _
+  exact (CompactSpace.uniformContinuous_of_continuous hfc).comp hp
+
+end PettyProjection.FixedPoint
+end
+
+noncomputable section
+open Set Metric Topology
+namespace PettyProjection.FixedPoint
+variable {E : Type uE} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
+
+/-- The full continuous finite-dimensional unit-ball fixed-point theorem,
+obtained from polynomial Jacobian integration, not from a topological axiom. -/
+lemma fixedPoint_unitBall {f : E → E} (hf : ContinuousOn f (closedBall (0:E) 1))
+    (hball : MapsTo f (closedBall (0:E) 1) (closedBall (0:E) 1)) :
+    ∃ x∈closedBall (0:E) 1, f x=x := by
+  by_contra hfix
+  have hgap : ContinuousOn (fun x => ‖f x-x‖) (closedBall (0:E) 1) :=
+    (hf.sub continuousOn_id).norm
+  obtain ⟨x₀,hx₀,hmin⟩ := (isCompact_closedBall (0:E) 1).exists_isMinOn
+    (nonempty_closedBall.mpr (by norm_num)) hgap
+  let δ : ℝ := ‖f x₀-x₀‖
+  have hδ : 0<δ := by
+    apply norm_pos_iff.mpr
+    intro he
+    exact hfix ⟨x₀,hx₀,sub_eq_zero.mp he⟩
+  let ε : ℝ := δ/4
+  have hε : 0<ε := div_pos hδ (by norm_num)
+  obtain ⟨g,hg,hga⟩ := (uniformlyContinuous_ball_extension hf).exists_contDiff_dist_le hε
+  have happ (x : E) (hx : x∈closedBall (0:E) 1) : dist (g x) (f x)<ε := by
+    have he : ballNearest x=x := ballNearest_eq
+      (by simpa only [mem_closedBall,dist_zero_right] using hx)
+    simpa only [he] using hga x
+  have hn (x : E) (hx : x∈closedBall (0:E) 1) : ‖f x‖≤1 := by
+    simpa only [mem_closedBall,dist_zero_right] using hball hx
+  let q : E → E := fun x => (1+ε)⁻¹ • g x
+  have hq : ContDiff ℝ 1 q := (hg.of_le (by norm_num)).const_smul _
+  have hqb (x : E) (hx : x∈closedBall (0:E) 1) : ‖q x‖<1 :=
+    (scaled_approximation hε (hn x hx) (happ x hx)).1
+  obtain ⟨x,hx,hqx⟩ := smooth_fixedPoint_ball hq hqb
+  have hdist := (scaled_approximation hε (hn x hx) (happ x hx)).2
+  change dist (q x) (f x)<2*ε at hdist
+  rw [hqx,dist_eq_norm,norm_sub_rev] at hdist
+  have hh : δ≤‖f x-x‖ := hmin hx
+  dsimp [ε] at hdist
+  linarith
+
+end PettyProjection.FixedPoint
+end
+
+noncomputable section
+open Set Metric Topology
+namespace PettyProjection.FixedPoint
+variable {E : Type uE} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
+
+/-- Brouwer on any nonempty compact convex set, derived from the genuine
+unit-ball theorem by the nearest-point extension. -/
+theorem fixedPoint_compact_convex {K : Set E} (hne : K.Nonempty)
+    (hk : IsCompact K) (hc : Convex ℝ K) {f : E → E}
+    (hf : ContinuousOn f K) (hmap : MapsTo f K K) : ∃ x∈K, f x=x := by
+  obtain ⟨R,hR,hbound⟩ := hk.isBounded.exists_pos_norm_le
+  let p : E → E := nearest K hne hk.isComplete hc
+  have hp (x : E) : p x∈K := nearest_mem K hne hk.isComplete hc x
+  have hpc : Continuous p := (nearest_lipschitz K hne hk.isComplete hc).continuous
+  let F : E → E := fun x => R⁻¹ • f (p (R • x))
+  have hFc : Continuous F := by
+    apply Continuous.const_smul
+    exact hf.comp_continuous (hpc.comp (continuous_const.smul continuous_id)) (fun x => hp _)
+  have hFB : MapsTo F (closedBall (0:E) 1) (closedBall (0:E) 1) := by
+    intro x hx
+    rw [mem_closedBall,dist_zero_right]
+    change ‖R⁻¹ • f (p (R • x))‖≤1
+    rw [norm_smul,Real.norm_eq_abs,abs_of_pos (inv_pos.mpr hR)]
+    exact (inv_mul_le_iff₀ hR).mpr (by simpa using hbound _ (hmap (hp _)))
+  obtain ⟨x,_,hfix⟩ := fixedPoint_unitBall hFc.continuousOn hFB
+  have he : f (p (R • x))=R • x := by
+    have hh := congrArg (fun y : E => R • y) hfix
+    simpa only [F,smul_smul,mul_inv_cancel₀ hR.ne',one_smul] using hh
+  have hxK : R • x∈K := he ▸ hmap (hp _)
+  refine ⟨R • x,hxK,?_⟩
+  have hpfix : p (R • x)=R • x := nearest_eq_of_mem K hne hk.isComplete hc hxK
+  rwa [hpfix] at he
+
+end PettyProjection.FixedPoint
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/ProjectionBodies/BrunnMinkowski.lean b/lean/OAI/Geometry/ProjectionBodies/BrunnMinkowski.lean
new file mode 100644
index 0000000000000000000000000000000000000000..8358d76a37fb7dad5618a8d5672e447054b10188
--- /dev/null
+++ b/lean/OAI/Geometry/ProjectionBodies/BrunnMinkowski.lean
@@ -0,0 +1,315 @@
+import OAI.Geometry.ProjectionBodies.SliceTransport
+
+namespace OAI
+
+universe uE uF
+
+noncomputable section
+open Set MeasureTheory Filter Topology
+open scoped Pointwise
+namespace PettyProjection
+section
+variable {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E]
+    [FiniteDimensional ℝ E] [MeasureSpace E] [BorelSpace E]
+    [Measure.IsAddHaarMeasure (volume : Measure E)]
+
+/-- The dimension-induction predicate, used only with a proved lower-dimensional
+case. It is not an extra hypothesis on the projection-volume main theorem. -/
+def BrunnMinkowskiRoot (E : Type uE) [NormedAddCommGroup E] [NormedSpace ℝ E]
+    [MeasureSpace E] (m : ℕ) : Prop :=
+  ∀ K M : Set E, IsCompact K → Convex ℝ K → (interior K).Nonempty →
+    IsCompact M → Convex ℝ M → (interior M).Nonempty →
+    ∀ a b : ℝ, 0 ≤ a → 0 ≤ b → a+b=1 →
+      a*(volume.real K)^((m:ℝ)⁻¹)+b*(volume.real M)^((m:ℝ)⁻¹) ≤
+        (volume.real (a • K+b • M))^((m:ℝ)⁻¹)
+
+omit [FiniteDimensional ℝ E] [BorelSpace E] in
+lemma sliceDensity_root_concave {m : ℕ} (hm : BrunnMinkowskiRoot E m)
+    {K : Set (E × ℝ)} (hK : IsCompact K) (hc : Convex ℝ K)
+    (hi : (interior K).Nonempty) {l r : ℝ}
+    (hl : l ∈ Prod.snd '' K) (hr : r ∈ Prod.snd '' K) :
+    ConcaveOn ℝ (Ioo l r) (fun s => (sliceDensity K s)^((m:ℝ)⁻¹)) := by
+  refine ⟨convex_Ioo l r,?_⟩
+  intro s hs t ht a b ha hb hab
+  have hlower := hm (horizontalSlice K s) (horizontalSlice K t)
+    (horizontalSlice_compact hK s) (horizontalSlice_convex hc s)
+    (horizontalSlice_interior_nonempty hc hi hl hr hs)
+    (horizontalSlice_compact hK t) (horizontalSlice_convex hc t)
+    (horizontalSlice_interior_nonempty hc hi hl hr ht) a b ha hb hab
+  exact hlower.trans (Real.rpow_le_rpow measureReal_nonneg
+    (measureReal_mono (horizontalSlice_combo_subset hc ha hb hab s t)
+      (horizontalSlice_compact hK (a*s+b*t)).measure_ne_top) (by positivity))
+
+omit [FiniteDimensional ℝ E] [BorelSpace E] in
+lemma sliceDensity_continuous {m : ℕ} (hm0 : 0 < m) (hm : BrunnMinkowskiRoot E m)
+    {K : Set (E × ℝ)} (hK : IsCompact K) (hc : Convex ℝ K)
+    (hi : (interior K).Nonempty) {l r : ℝ}
+    (hl : l ∈ Prod.snd '' K) (hr : r ∈ Prod.snd '' K) :
+    ContinuousOn (sliceDensity K) (Ioo l r) := by
+  have h := (ConcaveOn.continuousOn isOpen_Ioo
+    (sliceDensity_root_concave hm hK hc hi hl hr)).pow m
+  convert h using 1
+  funext s
+  exact (Real.rpow_inv_natCast_pow (sliceDensity_nonneg K s) hm0.ne').symm
+
+/-- The actual convex-body product step, not a replacement by abstract densities. -/
+theorem brunnMinkowski_product_unit {m : ℕ} (hm0 : 0 < m) (hm : BrunnMinkowskiRoot E m)
+    {K M : Set (E × ℝ)} (hK : IsCompact K) (hcK : Convex ℝ K)
+    (hiK : (interior K).Nonempty) (hM : IsCompact M) (hcM : Convex ℝ M)
+    (hiM : (interior M).Nonempty) (hvK : volume.real K=1) (hvM : volume.real M=1)
+    {a b : ℝ} (ha : 0 < a) (hb : 0 < b) (hab : a+b=1) :
+    1 ≤ volume.real (a • K+b • M) := by
+  obtain ⟨l,r,hlr,hpr⟩ := vertical_projection_interval hK hcK hiK
+  obtain ⟨L,R,hLR,hPR⟩ := vertical_projection_interval hM hcM hiM
+  have hl : l ∈ Prod.snd '' K := by rw [hpr]; exact left_mem_Icc.mpr hlr.le
+  have hr : r ∈ Prod.snd '' K := by rw [hpr]; exact right_mem_Icc.mpr hlr.le
+  have hL : L ∈ Prod.snd '' M := by rw [hPR]; exact left_mem_Icc.mpr hLR.le
+  have hR : R ∈ Prod.snd '' M := by rw [hPR]; exact right_mem_Icc.mpr hLR.le
+  have hC : IsCompact (a • K+b • M) := (hK.smul a).add (hM.smul b)
+  have hbounds : Prod.snd '' (a • K+b • M) ⊆ Icc (a*l+b*L) (a*r+b*R) := by
+    rintro s ⟨_,⟨_,⟨x,hx,rfl⟩,_,⟨y,hy,rfl⟩,rfl⟩,rfl⟩
+    have hx' : x.2 ∈ Icc l r := by rw [← hpr]; exact mem_image_of_mem _ hx
+    have hy' : y.2 ∈ Icc L R := by rw [← hPR]; exact mem_image_of_mem _ hy
+    change a*l+b*L ≤ a*x.2+b*y.2 ∧ a*x.2+b*y.2 ≤ a*r+b*R
+    exact ⟨add_le_add (mul_le_mul_of_nonneg_left hx'.1 ha.le) (mul_le_mul_of_nonneg_left hy'.1 hb.le),
+      add_le_add (mul_le_mul_of_nonneg_left hx'.2 ha.le) (mul_le_mul_of_nonneg_left hy'.2 hb.le)⟩
+  rw [← sliceDensity_interval_integral hC (by nlinarith) hbounds]
+  apply slice_transport hm0 ha hb hab hlr hLR
+    (sliceDensity_continuous hm0 hm hK hcK hiK hl hr)
+    (sliceDensity_continuous hm0 hm hM hcM hiM hL hR)
+    (sliceDensity_integrable hK).intervalIntegrable
+    (sliceDensity_integrable hM).intervalIntegrable
+    (fun s hs => sliceDensity_pos hK hcK hiK hl hr hs)
+    (fun s hs => sliceDensity_pos hM hcM hiM hL hR hs)
+  · rw [sliceDensity_interval_integral hK hlr.le (by rw [hpr]),hvK]
+  · rw [sliceDensity_interval_integral hM hLR.le (by rw [hPR]),hvM]
+  · exact (sliceDensity_integrable hC).intervalIntegrable
+  · intro s hs t ht
+    have hroot := hm (horizontalSlice K s) (horizontalSlice M t)
+      (horizontalSlice_compact hK s) (horizontalSlice_convex hcK s)
+      (horizontalSlice_interior_nonempty hcK hiK hl hr hs)
+      (horizontalSlice_compact hM t) (horizontalSlice_convex hcM t)
+      (horizontalSlice_interior_nonempty hcM hiM hL hR ht) a b ha.le hb.le hab
+    have hf0 := sliceDensity_nonneg K s
+    have hg0 := sliceDensity_nonneg M t
+    have hroot' := pow_le_pow_left₀ (by positivity : 0 ≤ a*(sliceDensity K s)^((m:ℝ)⁻¹)+b*(sliceDensity M t)^((m:ℝ)⁻¹)) hroot m
+    rw [Real.rpow_inv_natCast_pow measureReal_nonneg hm0.ne'] at hroot'
+    exact hroot'.trans (measureReal_mono (horizontalSlice_minkowski_subset K M a b s t)
+      (horizontalSlice_compact hC _).measure_ne_top)
+
+end
+end PettyProjection
+end
+
+noncomputable section
+open Set MeasureTheory Filter Topology
+open scoped Pointwise
+namespace PettyProjection
+section
+variable {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E]
+    [FiniteDimensional ℝ E] [MeasureSpace E] [BorelSpace E]
+    [Measure.IsAddHaarMeasure (volume : Measure E)]
+
+lemma volumeReal_smul_nonneg {r : ℝ} (hr : 0 ≤ r) (K : Set E) :
+    volume.real (r • K)=r^(Module.finrank ℝ E)*volume.real K := by
+  rw [Measure.real,Measure.addHaar_smul_of_nonneg volume hr,ENNReal.toReal_mul,
+    ENNReal.toReal_ofReal (pow_nonneg hr _)]
+  rfl
+
+omit [NormedSpace ℝ E] [FiniteDimensional ℝ E] [BorelSpace E] in
+lemma volumeReal_pos_of_body {K : Set E} (hK : IsCompact K) (hi : (interior K).Nonempty) :
+    0 < volume.real K :=
+  ENNReal.toReal_pos (ne_of_gt (lt_of_lt_of_le (isOpen_interior.measure_pos volume hi)
+    (measure_mono interior_subset))) hK.measure_ne_top
+
+/-- Usual normalization from the unit-volume form. The exponent is the actual
+finite dimension of the Haar space, and the bodies remain arbitrary. -/
+theorem brunnMinkowski_of_unit {d : ℕ} (hd : 0 < d) (hDim : Module.finrank ℝ E=d)
+    (hunit : ∀ K M : Set E, IsCompact K → Convex ℝ K → (interior K).Nonempty →
+      IsCompact M → Convex ℝ M → (interior M).Nonempty →
+      volume.real K=1 → volume.real M=1 → ∀ a b : ℝ, 0 < a → 0 < b → a+b=1 →
+      1 ≤ volume.real (a • K+b • M)) : BrunnMinkowskiRoot E d := by
+  intro K M hK hcK hiK hM hcM hiM a b ha hb hab
+  obtain rfl | ha := ha.eq_or_lt
+  · have hb1 : b=1 := by linarith
+    simp [hb1,zero_smul_set (hiK.mono interior_subset)]
+  obtain rfl | hb := hb.eq_or_lt
+  · have ha1 : a=1 := by linarith
+    simp [ha1,zero_smul_set (hiM.mono interior_subset)]
+  let p := (volume.real K)^((d:ℝ)⁻¹)
+  let q := (volume.real M)^((d:ℝ)⁻¹)
+  let D := a*p+b*q
+  have hp : 0 < p := Real.rpow_pos_of_pos (volumeReal_pos_of_body hK hiK) _
+  have hq : 0 < q := Real.rpow_pos_of_pos (volumeReal_pos_of_body hM hiM) _
+  have hD : 0 < D := by dsimp [D]; positivity
+  have hpm : p^d=volume.real K := Real.rpow_inv_natCast_pow measureReal_nonneg hd.ne'
+  have hqm : q^d=volume.real M := Real.rpow_inv_natCast_pow measureReal_nonneg hd.ne'
+  have hUK : volume.real (p⁻¹ • K)=1 := by
+    rw [volumeReal_smul_nonneg (inv_nonneg.mpr hp.le),hDim,inv_pow,hpm]
+    exact inv_mul_cancel₀ (volumeReal_pos_of_body hK hiK).ne'
+  have hUM : volume.real (q⁻¹ • M)=1 := by
+    rw [volumeReal_smul_nonneg (inv_nonneg.mpr hq.le),hDim,inv_pow,hqm]
+    exact inv_mul_cancel₀ (volumeReal_pos_of_body hM hiM).ne'
+  have hiUK : (interior (p⁻¹ • K)).Nonempty := by
+    rw [interior_smul₀ (inv_ne_zero hp.ne')]
+    exact hiK.image _
+  have hiUM : (interior (q⁻¹ • M)).Nonempty := by
+    rw [interior_smul₀ (inv_ne_zero hq.ne')]
+    exact hiM.image _
+  have he := hunit (p⁻¹ • K) (q⁻¹ • M) (hK.smul _) (hcK.smul _) hiUK
+    (hM.smul _) (hcM.smul _) hiUM hUK hUM (a*p/D) (b*q/D)
+    (by positivity) (by positivity) (by rw [← add_div]; exact div_self hD.ne')
+  have hsets : (a*p/D) • (p⁻¹ • K)+(b*q/D) • (q⁻¹ • M)=D⁻¹ • (a • K+b • M) := by
+    rw [smul_add,smul_smul,smul_smul,smul_smul,smul_smul]
+    congr 2 <;> field_simp
+  rw [hsets,volumeReal_smul_nonneg (inv_nonneg.mpr hD.le),hDim,inv_pow] at he
+  have hpow : D^d ≤ volume.real (a • K+b • M) := by
+    have hx := mul_le_mul_of_nonneg_left he (pow_nonneg hD.le d)
+    simpa only [mul_one,← mul_assoc,mul_inv_cancel₀ (pow_ne_zero _ hD.ne'),one_mul] using hx
+  have hr := Real.rpow_le_rpow (pow_nonneg hD.le d) hpow (by positivity : 0 ≤ (d:ℝ)⁻¹)
+  rw [← Real.rpow_natCast,← Real.rpow_mul hD.le,mul_inv_cancel₀ (by exact_mod_cast hd.ne'),Real.rpow_one] at hr
+  exact hr
+
+end
+end PettyProjection
+end
+
+noncomputable section
+open Set MeasureTheory Filter Topology
+open scoped Pointwise
+namespace PettyProjection
+section
+variable {E : Type uE} {F : Type uF} [NormedAddCommGroup E] [NormedSpace ℝ E]
+    [NormedAddCommGroup F] [NormedSpace ℝ F]
+    [MeasureSpace E] [BorelSpace E] [MeasureSpace F] [BorelSpace F]
+
+omit [MeasureSpace E] [BorelSpace E] [MeasureSpace F] [BorelSpace F] in
+lemma linearEquiv_preimage_combo (e : E ≃L[ℝ] F) (K M : Set F) (a b : ℝ) :
+    e ⁻¹' (a • K+b • M)=a • (e ⁻¹' K)+b • (e ⁻¹' M) := by
+  ext z
+  constructor
+  · rintro ⟨_,⟨x,hx,rfl⟩,_,⟨y,hy,rfl⟩,he⟩
+    refine ⟨a • e.symm x,⟨e.symm x,by simpa using hx,rfl⟩,
+      b • e.symm y,⟨e.symm y,by simpa using hy,rfl⟩,?_⟩
+    apply e.injective
+    simpa using he
+  · rintro ⟨_,⟨x,hx,rfl⟩,_,⟨y,hy,rfl⟩,rfl⟩
+    exact ⟨a • e x,⟨e x,hx,rfl⟩,b • e y,⟨e y,hy,rfl⟩,by simp⟩
+
+omit [BorelSpace E] in
+lemma BrunnMinkowskiRoot.equiv {m : ℕ} (hm : BrunnMinkowskiRoot E m)
+    (e : E ≃L[ℝ] F) (he : MeasurePreserving e) : BrunnMinkowskiRoot F m := by
+  intro K M hK hcK hiK hM hcM hiM a b ha hb hab
+  have hPK : IsCompact (e ⁻¹' K) := e.toHomeomorph.isCompact_preimage.mpr hK
+  have hPM : IsCompact (e ⁻¹' M) := e.toHomeomorph.isCompact_preimage.mpr hM
+  have hiPK : (interior (e ⁻¹' K)).Nonempty := by
+    change (interior (e.toHomeomorph ⁻¹' K)).Nonempty
+    rw [← e.toHomeomorph.preimage_interior]
+    obtain ⟨x,hx⟩ := hiK
+    exact ⟨e.symm x,by simpa using hx⟩
+  have hiPM : (interior (e ⁻¹' M)).Nonempty := by
+    change (interior (e.toHomeomorph ⁻¹' M)).Nonempty
+    rw [← e.toHomeomorph.preimage_interior]
+    obtain ⟨x,hx⟩ := hiM
+    exact ⟨e.symm x,by simpa using hx⟩
+  have h := hm (e ⁻¹' K) (e ⁻¹' M) hPK (hcK.linear_preimage e.toLinearMap) hiPK
+    hPM (hcM.linear_preimage e.toLinearMap) hiPM a b ha hb hab
+  have hm' (S : Set F) (hS : IsCompact S) : volume.real (e ⁻¹' S)=volume.real S :=
+    congrArg ENNReal.toReal (he.measure_preimage hS.measurableSet.nullMeasurableSet)
+  rw [hm' K hK,hm' M hM,← linearEquiv_preimage_combo,hm' _ ((hK.smul _).add (hM.smul _))] at h
+  exact h
+end
+
+/-- The one-dimensional base case, with no symmetry or origin condition. -/
+lemma brunnMinkowski_real : BrunnMinkowskiRoot ℝ 1 := by
+  intro K M hK hcK hiK hM hcM hiM a b ha hb hab
+  obtain rfl | ha := ha.eq_or_lt
+  · have hb1 : b=1 := by linarith
+    simp [hb1,zero_smul_set (hiK.mono interior_subset)]
+  obtain rfl | hb := hb.eq_or_lt
+  · have ha1 : a=1 := by linarith
+    simp [ha1,zero_smul_set (hiM.mono interior_subset)]
+  rw [compact_convex_real_interval hK hcK (hiK.mono interior_subset),
+    compact_convex_real_interval hM hcM (hiM.mono interior_subset)]
+  have hKlr : sInf K ≤ sSup K := (hK.isGLB_sInf (hiK.mono interior_subset)).1
+    (hK.sSup_mem (hiK.mono interior_subset))
+  have hMlr : sInf M ≤ sSup M := (hM.isGLB_sInf (hiM.mono interior_subset)).1
+    (hM.sSup_mem (hiM.mono interior_subset))
+  have hKa : a*sInf K ≤ a*sSup K := mul_le_mul_of_nonneg_left hKlr ha.le
+  have hMb : b*sInf M ≤ b*sSup M := mul_le_mul_of_nonneg_left hMlr hb.le
+  rw [LinearOrderedField.smul_Icc ha,LinearOrderedField.smul_Icc hb,Icc_add_Icc hKa hMb]
+  simp only [Nat.cast_one,inv_one,Real.rpow_one,Real.volume_real_Icc]
+  rw [max_eq_left (sub_nonneg.mpr hKlr),max_eq_left (sub_nonneg.mpr hMlr),
+    max_eq_left (sub_nonneg.mpr (add_le_add hKa hMb))]
+  nlinarith
+
+/-- Standard coordinate splitting, with explicit measure evidence. -/
+def finSplitLinear (n : ℕ) : (Fin (n+1) → ℝ) ≃ₗ[ℝ] ((Fin n → ℝ) × ℝ) where
+  toEquiv := (MeasurableEquiv.piFinSuccAbove (fun _ : Fin (n+1) => ℝ) 0).toEquiv.trans
+    (Equiv.prodComm _ _)
+  map_add' _ _ := rfl
+  map_smul' _ _ := rfl
+
+def finSplit (n : ℕ) : (Fin (n+1) → ℝ) ≃L[ℝ] ((Fin n → ℝ) × ℝ) :=
+  (finSplitLinear n).toContinuousLinearEquiv
+
+lemma finSplit_measurePreserving (n : ℕ) : MeasurePreserving (finSplit n) := by
+  have hs : MeasurePreserving (Prod.swap : ℝ × (Fin n → ℝ) → (Fin n → ℝ) × ℝ) := by
+    simpa only [Measure.volume_eq_prod] using
+      (Measure.measurePreserving_swap (μ := (volume : Measure ℝ)) (ν := (volume : Measure (Fin n → ℝ))))
+  exact hs.comp (volume_preserving_piFinSuccAbove (fun _ : Fin (n+1) => ℝ) 0)
+
+def finOneLinear : (Fin 1 → ℝ) ≃ₗ[ℝ] ℝ := LinearEquiv.funUnique (Fin 1) ℝ ℝ
+
+def finOne : (Fin 1 → ℝ) ≃L[ℝ] ℝ := finOneLinear.toContinuousLinearEquiv
+
+lemma finOne_measurePreserving : MeasurePreserving finOne :=
+  volume_preserving_piUnique (fun _ : Fin 1 => ℝ)
+
+/-- Brunn--Minkowski in all positive Cartesian dimensions, obtained by the
+actual convex-body slice induction. -/
+theorem brunnMinkowski_fin (n : ℕ) : BrunnMinkowskiRoot (Fin (n+1) → ℝ) (n+1) := by
+  induction n with
+  | zero =>
+    exact brunnMinkowski_real.equiv finOne.symm
+      (MeasurePreserving.symm finOne.toHomeomorph.toMeasurableEquiv finOne_measurePreserving)
+  | succ n ih =>
+    let : Measure.IsAddHaarMeasure (volume : Measure ((Fin (n+1) → ℝ) × ℝ)) := by
+      rw [Measure.volume_eq_prod]
+      infer_instance
+    have hprod : BrunnMinkowskiRoot ((Fin (n+1) → ℝ) × ℝ) (n+1+1) := by
+      apply brunnMinkowski_of_unit (by omega)
+      · simp
+      · intro K M hK hcK hiK hM hcM hiM hvK hvM a b ha hb hab
+        exact brunnMinkowski_product_unit (by omega) ih hK hcK hiK hM hcM hiM hvK hvM ha hb hab
+    exact hprod.equiv (finSplit (n+1)).symm
+      (MeasurePreserving.symm (finSplit (n+1)).toHomeomorph.toMeasurableEquiv
+        (finSplit_measurePreserving (n+1)))
+
+/-- Standard Euclidean volume version used for the projection-body theorem. -/
+theorem brunnMinkowski_space {n : ℕ} (hn : 0 < n) : BrunnMinkowskiRoot (Space n) n := by
+  obtain ⟨m,rfl⟩ := Nat.exists_eq_succ_of_ne_zero hn.ne'
+  exact (brunnMinkowski_fin m).equiv
+    (PiLp.continuousLinearEquiv 2 ℝ (fun _ : Fin (m+1) => ℝ)).symm
+    (PiLp.volume_preserving_toLp (Fin (m+1)))
+
+end PettyProjection
+end
+
+noncomputable section
+open Set MeasureTheory Filter Topology
+namespace PettyProjection
+
+/-- Brunn--Minkowski for the canonical Euclidean volume on any finite-dimensional
+real inner-product space. -/
+theorem brunnMinkowski_innerProduct {E : Type uE} [NormedAddCommGroup E]
+    [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+    (hE : 0 < Module.finrank ℝ E) : BrunnMinkowskiRoot E (Module.finrank ℝ E) := by
+  exact (brunnMinkowski_space hE).equiv
+    (stdOrthonormalBasis ℝ E).repr.symm.toContinuousLinearEquiv
+    (stdOrthonormalBasis ℝ E).measurePreserving_repr_symm
+
+end PettyProjection
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/ProjectionBodies/CosineCalculus.lean b/lean/OAI/Geometry/ProjectionBodies/CosineCalculus.lean
new file mode 100644
index 0000000000000000000000000000000000000000..d58e0d35d8fd09570ff3fdb1ef9a3c1bde440c88
--- /dev/null
+++ b/lean/OAI/Geometry/ProjectionBodies/CosineCalculus.lean
@@ -0,0 +1,285 @@
+import OAI.Geometry.ProjectionBodies.NormEstimates
+
+namespace OAI
+
+noncomputable section
+open Set Filter Topology Function
+namespace PettyProjection.Spherical
+
+/-- Positive part to a natural power; the endpoint is fixed to zero. -/
+def posPower (r : ℕ) (x : ℝ) : ℝ := if 0 < x then x^r else 0
+
+lemma posPower_hasDerivAt {r : ℕ} (hr : 2 ≤ r) (x : ℝ) :
+    HasDerivAt (posPower r) ((r : ℝ)*posPower (r-1) x) x := by
+  have hd (z : ℝ) : HasDerivAt (fun a : ℝ => (a-1)^r) ((r : ℝ)*(z-1)^(r-1)) z := by
+    convert! ((hasDerivAt_id z).sub_const 1).pow r using 1
+    simp
+  have hp := Scalar.paste_hasDerivAt (fun z _ => hd z)
+    (by simp [show r ≠ 0 by omega]) (by simp [show r-1 ≠ 0 by omega]) (x+1)
+  have h := hp.comp x ((hasDerivAt_id x).add_const 1)
+  convert! h using 1
+  · funext z; simp [posPower]
+  · simp [posPower, mul_ite]
+
+lemma posPower_contDiff_one {r : ℕ} (hr : 2 ≤ r) : ContDiff ℝ 1 (posPower r) := by
+  have hd (z : ℝ) : HasDerivAt (fun a : ℝ => (a-1)^r) ((r : ℝ)*(z-1)^(r-1)) z := by
+    convert! ((hasDerivAt_id z).sub_const 1).pow r using 1
+    simp
+  have h := Scalar.paste_contDiff_one (fun z _ => hd z)
+    (by simp [show r ≠ 0 by omega]) (by simp [show r-1 ≠ 0 by omega])
+    (by fun_prop)
+  have hc := h.comp (contDiff_id.add contDiff_const : ContDiff ℝ 1 (fun x : ℝ => x+1))
+  convert! hc using 1
+  ext x
+  simp [posPower]
+
+lemma posPower_contDiff_two {r : ℕ} (hr : 3 ≤ r) : ContDiff ℝ 2 (posPower r) := by
+  have he : deriv (posPower r) = fun x => (r : ℝ)*posPower (r-1) x :=
+    funext fun x => (posPower_hasDerivAt (by omega) x).deriv
+  change ContDiff ℝ (1+1) (posPower r)
+  rw [contDiff_succ_iff_deriv]
+  refine ⟨fun x => (posPower_hasDerivAt (by omega) x).differentiableAt, ?_, ?_⟩
+  · simp
+  · rw [he]
+    exact contDiff_const.mul (posPower_contDiff_one (by omega))
+
+lemma abs_pow_posPower {r : ℕ} (hr : 0 < r) (x : ℝ) :
+    |x|^r = posPower r x + posPower r (-x) := by
+  rcases lt_trichotomy x 0 with hx | rfl | hx
+  · simp [posPower, abs_of_neg hx, not_lt.mpr hx.le, neg_pos.mpr hx]
+  · simp [posPower, hr.ne']
+  · simp [posPower, abs_of_pos hx, hx, not_lt.mpr (neg_nonpos.mpr hx.le)]
+
+lemma abs_pow_contDiff_two {r : ℕ} (hr : 3 ≤ r) : ContDiff ℝ 2 (fun x : ℝ => |x|^r) := by
+  have he : (fun x : ℝ => |x|^r) = fun x => posPower r x+posPower r (-x) :=
+    funext (abs_pow_posPower (by omega))
+  rw [he]
+  exact (posPower_contDiff_two hr).add ((posPower_contDiff_two hr).comp contDiff_neg)
+
+lemma abs_pow_hasDerivAt {r : ℕ} (hr : 2 ≤ r) (x : ℝ) :
+    HasDerivAt (fun x : ℝ => |x|^r) ((r : ℝ)*x*|x|^(r-2)) x := by
+  have h := (posPower_hasDerivAt hr x).add ((posPower_hasDerivAt hr (-x)).comp x (hasDerivAt_neg x))
+  have he : (fun x : ℝ => |x|^r) = fun x => posPower r x+posPower r (-x) :=
+    funext (abs_pow_posPower (by omega))
+  rw [he]
+  convert! h using 1
+  have hp : r-1 = (r-2)+1 := by omega
+  rcases lt_trichotomy x 0 with hx | rfl | hx
+  · simp only [posPower, ite_eq_right (not_lt.mpr hx.le), ite_eq_left (neg_pos.mpr hx),
+      abs_of_neg hx, hp, pow_succ]
+    ring
+  · simp [posPower]
+  · simp only [posPower, ite_eq_left hx, ite_eq_right (not_lt.mpr (neg_nonpos.mpr hx.le)),
+      abs_of_pos hx, hp, pow_succ]
+    ring
+
+lemma abs_pow_second_deriv {r : ℕ} (hr : 3 ≤ r) (x : ℝ) :
+    HasDerivAt (fun x : ℝ => (r : ℝ)*x*|x|^(r-2))
+      ((r : ℝ)*((r : ℝ)-1)*|x|^(r-2)) x := by
+  have he : (fun x : ℝ => (r : ℝ)*x*|x|^(r-2)) =
+      fun x => (r : ℝ)*(posPower (r-1) x - posPower (r-1) (-x)) := by
+    funext z
+    have hp : r-1 = (r-2)+1 := by omega
+    rcases lt_trichotomy z 0 with hz | rfl | hz
+    · simp only [posPower, ite_eq_right (not_lt.mpr hz.le), ite_eq_left (neg_pos.mpr hz),
+        abs_of_neg hz, hp, pow_succ]; ring
+    · simp [posPower]
+    · simp only [posPower, ite_eq_left hz, ite_eq_right (not_lt.mpr (neg_nonpos.mpr hz.le)),
+        abs_of_pos hz, hp, pow_succ]; ring
+  rw [he]
+  have h := ((posPower_hasDerivAt (show 2 ≤ r-1 by omega) x).sub
+    ((posPower_hasDerivAt (show 2 ≤ r-1 by omega) (-x)).comp x (hasDerivAt_neg x))).const_mul (r : ℝ)
+  convert! h using 1
+  rw [show r-1-1 = r-2 by omega, Nat.cast_sub (by omega), Nat.cast_one,
+    abs_pow_posPower (show 0 < r-2 by omega)]
+  ring
+
+end PettyProjection.Spherical
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology Function
+open scoped ENNReal RealInnerProductSpace
+namespace PettyProjection.Spherical
+
+lemma inner_generator {n : ℕ} (u x : Space n) (i j : Fin n) :
+    ⟪u, generator n i j x⟫ = x i*u j-x j*u i := by
+  simp [generator_apply, inner_sub_right, inner_smul_right,
+    EuclideanSpace.inner_single_right]
+
+lemma sum_inner_generator_sq {n : ℕ} (u x : Space n) :
+    ∑ i : Fin n, ∑ j : Fin n, ⟪u, generator n i j x⟫^2 =
+      2*(‖u‖^2*‖x‖^2-⟪u,x⟫^2) := by
+  simp_rw [inner_generator]
+  have he (i j : Fin n) : (x i*u j-x j*u i)^2 =
+      x i^2*u j^2+x j^2*u i^2-2*(x i*u i)*(x j*u j) := by ring
+  simp_rw [he, Finset.sum_sub_distrib, Finset.sum_add_distrib,
+    ← Finset.mul_sum, ← Finset.sum_mul]
+  rw [← EuclideanSpace.real_norm_sq_eq, ← EuclideanSpace.real_norm_sq_eq]
+  simp only [← Finset.mul_sum]
+  rw [← EuclideanSpace.real_norm_sq_eq]
+  have hi : ⟪u,x⟫ = ∑ i, x i*u i := by
+    simp [EuclideanSpace.inner_eq_star_dotProduct, dotProduct]
+  rw [hi]
+  ring
+
+lemma rotDeriv_scalar_linear {n : ℕ} (T : Space n →L[ℝ] Space n)
+    (L : Space n →L[ℝ] ℝ) {f f' : ℝ → ℝ}
+    (hf : ∀ t, HasDerivAt f (f' t) t) (x : Space n) :
+    rotDeriv T (f ∘ L) x = f' (L x)*L (T x) := by
+  have h := (hf (L x)).comp_hasFDerivAt x L.hasFDerivAt
+  simp only [rotDeriv, h.fderiv, smul_apply, smul_eq_mul]
+
+lemma rotDeriv_scalar_linear_twice {n : ℕ} (T : Space n →L[ℝ] Space n)
+    (L : Space n →L[ℝ] ℝ) {f f' f'' : ℝ → ℝ}
+    (hf : ∀ t, HasDerivAt f (f' t) t)
+    (hf' : ∀ t, HasDerivAt f' (f'' t) t) (x : Space n) :
+    rotDeriv T (rotDeriv T (f ∘ L)) x =
+      f'' (L x)*(L (T x))^2+f' (L x)*L (T (T x)) := by
+  have he : rotDeriv T (f ∘ L) = fun y => f' (L y)*L (T y) :=
+    funext (rotDeriv_scalar_linear T L hf)
+  rw [he]
+  have hd := ((hf' (L x)).comp_hasFDerivAt x L.hasFDerivAt).mul
+    (L.comp T).hasFDerivAt
+  change fderiv ℝ ((f' ∘ L) * (fun y => (L.comp T) y)) x (T x) = _
+  rw [hd.fderiv]
+  simp only [add_apply, smul_apply, smul_eq_mul, ContinuousLinearMap.comp_apply, comp_apply]
+  ring
+
+lemma casimir_scalar_inner {n : ℕ} (u x : Space n) {f f' f'' : ℝ → ℝ}
+    (hf : ∀ t, HasDerivAt f (f' t) t)
+    (hf' : ∀ t, HasDerivAt f' (f'' t) t) :
+    casimir (fun y => f ⟪u,y⟫) x =
+      f'' ⟪u,x⟫*(‖u‖^2*‖x‖^2-⟪u,x⟫^2)-
+        ((n : ℝ)-1)*f' ⟪u,x⟫*⟪u,x⟫ := by
+  change casimir (f ∘ (innerSL ℝ u)) x = _
+  unfold casimir
+  simp_rw [rotDeriv_scalar_linear_twice _ (innerSL ℝ u) hf hf',
+    innerSL_apply_apply, Finset.sum_add_distrib, ← Finset.mul_sum]
+  rw [sum_inner_generator_sq]
+  simp_rw [← inner_sum, sum_generator_square, inner_smul_right]
+  ring
+
+lemma posPower_mul {r : ℕ} (x : ℝ) :
+    posPower r x*x = posPower (r+1) x := by
+  by_cases hx : 0 < x <;> simp [posPower, hx, pow_succ]
+
+lemma posPower_mul_sq {r : ℕ} (x : ℝ) :
+    posPower r x*x^2 = posPower (r+2) x := by
+  rw [pow_two, ← mul_assoc, posPower_mul, posPower_mul]
+
+lemma casimir_posPower_inner {n r : ℕ} (hr : 3 ≤ r) (u x : Sphere n) :
+    casimir (fun y => posPower r ⟪(u : Space n),y⟫) x =
+      (r : ℝ)*((r : ℝ)-1)*posPower (r-2) ⟪(u : Space n),(x : Space n)⟫ -
+      eigenvalue n r*posPower r ⟪(u : Space n),(x : Space n)⟫ := by
+  have hd (t : ℝ) : HasDerivAt (fun z => (r : ℝ)*posPower (r-1) z)
+      ((r : ℝ)*((r : ℝ)-1)*posPower (r-2) t) t := by
+    convert! (posPower_hasDerivAt (show 2 ≤ r-1 by omega) t).const_mul (r : ℝ) using 1
+    rw [show r-1-1 = r-2 by omega, Nat.cast_sub (by omega), Nat.cast_one]
+    ring
+  rw [casimir_scalar_inner _ _ (posPower_hasDerivAt (by omega)) hd,
+    norm_coe, norm_coe]
+  simp only [one_pow, mul_one]
+  have h₁ := posPower_mul (r := r-1) ⟪(u : Space n),(x : Space n)⟫
+  have h₂ := posPower_mul_sq (r := r-2) ⟪(u : Space n),(x : Space n)⟫
+  rw [show r-1+1 = r by omega] at h₁
+  rw [show r-2+2 = r by omega] at h₂
+  unfold eigenvalue
+  calc
+    _ = (r : ℝ)*((r : ℝ)-1)*posPower (r-2) ⟪(u : Space n),(x : Space n)⟫ -
+      (r : ℝ)*((r : ℝ)-1)*(posPower (r-2) ⟪(u : Space n),(x : Space n)⟫ * ⟪(u : Space n),(x : Space n)⟫^2) -
+      ((n : ℝ)-1)*(r : ℝ)*(posPower (r-1) ⟪(u : Space n),(x : Space n)⟫ * ⟪(u : Space n),(x : Space n)⟫) := by ring
+    _ = _ := by rw [h₁, h₂]; ring
+
+lemma mean_casimir_comm {n : ℕ} [NeZero n] {f g : Space n → ℝ}
+    (hf : ContDiff ℝ 2 f) (hg : ContDiff ℝ 2 g) :
+    mean (fun u : Sphere n => f u*casimir g u) =
+      mean (fun u : Sphere n => g u*casimir f u) := by
+  rw [mean_casimir (hf.of_le (by norm_num)) hg,
+    mean_casimir (hg.of_le (by norm_num)) hf, energy_symm]
+
+end PettyProjection.Spherical
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology Function
+open scoped ENNReal RealInnerProductSpace Gradient
+namespace PettyProjection.Spherical
+
+lemma sum_inner_generator_mul {n : ℕ} (u w x : Space n) :
+    ∑ i : Fin n, ∑ j : Fin n, ⟪u,generator n i j x⟫*⟪w,generator n i j x⟫ =
+      2*(‖x‖^2*⟪u,w⟫-⟪u,x⟫*⟪w,x⟫) := by
+  simp_rw [inner_generator]
+  have he (i j : Fin n) : (x i*u j-x j*u i)*(x i*w j-x j*w i) =
+      x i^2*(u j*w j)+x j^2*(u i*w i)-(x i*w i)*(x j*u j)-(x i*u i)*(x j*w j) := by ring
+  simp_rw [he, Finset.sum_sub_distrib, Finset.sum_add_distrib,
+    ← Finset.mul_sum, ← Finset.sum_mul]
+  have hi (a b : Space n) : ⟪a,b⟫ = ∑ i, a i*b i := by
+    simp [EuclideanSpace.inner_eq_star_dotProduct, dotProduct, mul_comm]
+  rw [hi u w, hi u x, hi w x, EuclideanSpace.real_norm_sq_eq]
+  simp_rw [mul_comm (u _) (x _), mul_comm (w _) (x _)]
+  simp only [← Finset.mul_sum]
+  ring
+
+/-- Genuine tangential gradient on the unit sphere, expressed ambiently. -/
+def sphereGradient {n : ℕ} (f : Space n → ℝ) (x : Space n) : Space n :=
+  ∇ f x - (fderiv ℝ f x x) • x
+
+lemma continuous_gradient_of_contDiff {n : ℕ} {f : Space n → ℝ} (hf : ContDiff ℝ 1 f) :
+    Continuous (gradient f) :=
+  (InnerProductSpace.toDual ℝ (Space n)).symm.continuous.comp (hf.continuous_fderiv (by norm_num))
+
+lemma sphereGradient_continuous {n : ℕ} {f : Space n → ℝ} (hf : ContDiff ℝ 1 f) :
+    Continuous (sphereGradient f) := by
+  exact (continuous_gradient_of_contDiff hf).sub
+    (((hf.continuous_fderiv (by norm_num)).clm_apply continuous_id).smul continuous_id)
+
+lemma sphereGradient_inner {n : ℕ} (f : Space n → ℝ) (x w : Space n) :
+    ⟪sphereGradient f x,w⟫ = fderiv ℝ f x w - fderiv ℝ f x x*⟪x,w⟫ := by
+  simp only [sphereGradient, inner_sub_left, inner_smul_left, inner_gradient_left, starRingEnd_apply, star_trivial]
+
+lemma sphereGradient_tangent {n : ℕ} (f : Space n → ℝ) (x : Sphere n) :
+    ⟪sphereGradient f x,(x : Space n)⟫ = 0 := by
+  rw [sphereGradient_inner, real_inner_self_eq_norm_sq, norm_coe]
+  simp
+
+lemma sum_rotDeriv_inner {n : ℕ} (f : Space n → ℝ) (x : Sphere n) (w : Space n) :
+    (1/2 : ℝ)*(∑ i : Fin n, ∑ j : Fin n,
+      rotDeriv (generator n i j) f x*⟪w,generator n i j x⟫) = ⟪sphereGradient f x,w⟫ := by
+  simp only [rotDeriv, ← inner_gradient_left]
+  rw [sum_inner_generator_mul, norm_coe, sphereGradient_inner,
+    inner_gradient_left, inner_gradient_left, real_inner_comm w (x : Space n)]
+  ring
+
+lemma mean_gradient_casimir {n : ℕ} [NeZero n] {f g : Space n → ℝ}
+    (hf : ContDiff ℝ 1 f) (hg : ContDiff ℝ 2 g) :
+    mean (fun x : Sphere n => ⟪sphereGradient f x,sphereGradient g x⟫) =
+      -mean (fun x : Sphere n => f x*casimir g x) := by
+  rw [mean_casimir hf hg, neg_neg]
+  have hs (x : Sphere n) : ⟪sphereGradient f x,sphereGradient g x⟫ =
+      (1/2 : ℝ)*∑ i : Fin n, ∑ j : Fin n,
+        rotDeriv (generator n i j) f x*rotDeriv (generator n i j) g x := by
+    have h := sum_rotDeriv_inner f x (∇ g x)
+    simp only [inner_gradient_left] at h
+    change ⟪sphereGradient f x, ∇ g x - (fderiv ℝ g x x) • (x : Space n)⟫ = _
+    rw [inner_sub_right, inner_smul_right, sphereGradient_tangent, mul_zero, sub_zero]
+    exact h.symm
+  simp_rw [hs]
+  rw [mean, integral_const_mul]
+  congr 1
+  rw [integral_finsetSum]
+  · apply Finset.sum_congr rfl
+    intro i _
+    exact integral_finsetSum _ (fun j _ => continuous_integrable
+      (((continuous_rotDeriv _ hf).comp continuous_subtype_val).mul
+        ((continuous_rotDeriv _ (hg.of_le (by norm_num))).comp continuous_subtype_val)))
+  · intro i _
+    apply continuous_integrable
+    exact continuous_finsetSum _ fun j _ =>
+      ((continuous_rotDeriv _ hf).comp continuous_subtype_val).mul
+        ((continuous_rotDeriv _ (hg.of_le (by norm_num))).comp continuous_subtype_val)
+
+end PettyProjection.Spherical
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/ProjectionBodies/CosineSpectrum.lean b/lean/OAI/Geometry/ProjectionBodies/CosineSpectrum.lean
new file mode 100644
index 0000000000000000000000000000000000000000..f0926a3d6570093710987aa051bb0bfcb805287e
--- /dev/null
+++ b/lean/OAI/Geometry/ProjectionBodies/CosineSpectrum.lean
@@ -0,0 +1,441 @@
+import OAI.Geometry.ProjectionBodies.SignCalculus
+
+namespace OAI
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology Function
+open scoped ENNReal RealInnerProductSpace
+namespace PettyProjection.Spherical
+
+lemma posPower_eq_max_pow {r : ℕ} (hr : 0 < r) (x : ℝ) :
+    posPower r x = max x 0 ^ r := by
+  by_cases hx : 0 < x
+  · simp [posPower, hx, max_eq_left hx.le]
+  · simp [posPower, hx, max_eq_right (not_lt.mp hx), hr.ne']
+
+/-- Positive-hemisphere power kernel, including the constant zeroth power. -/
+def higherKernel {n : ℕ} (r : ℕ) (u : Sphere n) (f : Sphere n → ℝ) : ℝ :=
+  mean (fun x => (max ⟪(u : Space n),(x : Space n)⟫ 0)^r*f x)
+
+lemma higherKernel_const_pos {n : ℕ} [NeZero n] (r : ℕ) (u : Sphere n) :
+    0 < higherKernel r u (fun _ => 1) := by
+  have hc : Continuous (fun x : Sphere n => max ⟪(u : Space n),(x : Space n)⟫ 0 ^ r) := by
+    fun_prop
+  simp only [higherKernel, mul_one, mean]
+  exact integral_pos_of_integrable_nonneg_nonzero hc (continuous_integrable hc)
+    (fun x => pow_nonneg (le_max_right _ _) _) (x := u)
+    (by rw [real_inner_self_eq_norm_sq, norm_coe]; norm_num)
+
+lemma higherKernel_recurrence {n r d : ℕ} [NeZero n] (hr : 3 ≤ r)
+    (u : Sphere n) {f : Space n → ℝ} (hf : ContDiff ℝ 2 f)
+    (heig : ∀ x : Sphere n, casimir f x = -eigenvalue n d*f x) :
+    (eigenvalue n r-eigenvalue n d)*higherKernel r u (fun x => f x) =
+      (r : ℝ)*((r : ℝ)-1)*higherKernel (r-2) u (fun x => f x) := by
+  let g : Space n → ℝ := fun y => posPower r ⟪(u : Space n),y⟫
+  have hg : ContDiff ℝ 2 g := (posPower_contDiff_two hr).comp (innerSL ℝ (u : Space n)).contDiff
+  have h := mean_casimir_comm hf hg
+  have he (x : Sphere n) : f x*casimir g x =
+      (r : ℝ)*((r : ℝ)-1)*((max ⟪(u : Space n),(x : Space n)⟫ 0)^(r-2)*f x) -
+      eigenvalue n r*((max ⟪(u : Space n),(x : Space n)⟫ 0)^r*f x) := by
+    rw [show casimir g x = _ from casimir_posPower_inner hr u x,
+      posPower_eq_max_pow (show 0 < r-2 by omega), posPower_eq_max_pow (show 0 < r by omega)]
+    ring
+  simp_rw [he, heig, g, posPower_eq_max_pow (show 0 < r by omega)] at h
+  have hct (r : ℕ) : Continuous (fun x : Sphere n => max ⟪(u : Space n),(x : Space n)⟫ 0 ^ r * f x) := by
+    exact ((continuous_const.inner continuous_subtype_val).max continuous_const |>.pow r).mul
+      (hf.continuous.comp continuous_subtype_val)
+  have hh : (fun x : Sphere n => max ⟪(u : Space n),(x : Space n)⟫ 0 ^ r *
+      (-eigenvalue n d*f x)) = fun x : Sphere n => -eigenvalue n d*
+      (max ⟪(u : Space n),(x : Space n)⟫ 0 ^ r*f x) := by funext x; ring
+  rw [hh, mean, integral_sub
+    ((continuous_integrable (hct (r-2))).const_mul _) ((continuous_integrable (hct r)).const_mul _),
+    integral_const_mul, integral_const_mul, mean, integral_const_mul] at h
+  change (r : ℝ)*((r : ℝ)-1)*higherKernel (r-2) u (fun x => f x) -
+    eigenvalue n r*higherKernel r u (fun x => f x) =
+    -eigenvalue n d*higherKernel r u (fun x => f x) at h
+  linarith
+
+lemma higherKernel_const_recurrence {n r : ℕ} [NeZero n] (hr : 3 ≤ r) (u : Sphere n) :
+    eigenvalue n r*higherKernel r u (fun _ => 1) =
+      (r : ℝ)*((r : ℝ)-1)*higherKernel (r-2) u (fun _ => 1) := by
+  have he (x : Sphere n) : casimir (fun _ : Space n => (1 : ℝ)) x = -eigenvalue n 0*1 := by
+    have hc (T : Space n →L[ℝ] Space n) : rotDeriv T (fun _ : Space n => (1 : ℝ)) = fun _ => 0 :=
+      funext (rotDeriv_const T 1)
+    simp [casimir, hc, rotDeriv_const, eigenvalue]
+  simpa [eigenvalue] using higherKernel_recurrence hr u contDiff_const he
+
+lemma higherKernel_peak {n : ℕ} [NeZero n] (u : Sphere n) {f : Sphere n → ℝ}
+    (hf : Continuous f) :
+    Tendsto (fun r : ℕ => higherKernel r u f / higherKernel r u (fun _ => 1))
+      atTop (𝓝 (f u)) := by
+  have hc : Continuous (fun x : Sphere n => max ⟪(u : Space n),(x : Space n)⟫ 0) := by fun_prop
+  have hmax (x : Sphere n) (_ : x ∈ (univ : Set (Sphere n))) (hne : x ≠ u) :
+      max ⟪(u : Space n),(x : Space n)⟫ 0 < max ⟪(u : Space n),(u : Space n)⟫ 0 := by
+    rw [real_inner_self_eq_norm_sq, norm_coe]
+    norm_num only [one_pow, max_eq_left zero_le_one]
+    apply max_lt
+    · exact (inner_lt_one_iff_real_of_norm_eq_one (norm_coe u) (norm_coe x)).mpr
+        (by intro h; exact hne (Subtype.ext h.symm))
+    · norm_num
+  have h := tendsto_setIntegral_pow_smul_of_unique_maximum_of_isCompact_of_continuousOn
+    (μ := sigma n) isCompact_univ hc.continuousOn hmax
+    (fun x _ => le_max_right _ _) (by rw [real_inner_self_eq_norm_sq, norm_coe]; norm_num)
+    (by simp) hf.continuousOn
+  simpa [smul_eq_mul, div_eq_mul_inv, higherKernel, mean, mul_comm] using h
+
+end PettyProjection.Spherical
+end
+
+noncomputable section
+open Filter Topology
+namespace PettyProjection.Spherical
+
+/-- Normalized multiplier of the positive-power kernel on degree `2*j`. -/
+def higherFactor (n j : ℕ) (s : ℝ) : ℝ :=
+  ∏ a ∈ Finset.range j, (s-2*a)/(s+n+2*a)
+
+lemma higherFactor_zero (n : ℕ) (s : ℝ) : higherFactor n 0 s = 1 := by
+  simp [higherFactor]
+
+lemma higherFactor_succ (n j : ℕ) (s : ℝ) :
+    higherFactor n (j+1) s = higherFactor n j s*((s-2*j)/(s+n+2*j)) := by
+  simp only [higherFactor, Finset.prod_range_succ]
+
+lemma higherFactor_odd_ne_zero {n : ℕ} (hn : 2 ≤ n) (j m : ℕ) :
+    higherFactor n j (2*m+1) ≠ 0 := by
+  apply Finset.prod_ne_zero_iff.mpr
+  intro a _
+  apply div_ne_zero
+  · apply sub_ne_zero.mpr
+    have h : (2*m+1 : ℕ) ≠ 2*a := by omega
+    exact_mod_cast h
+  · have hn' : (2 : ℝ) ≤ n := by exact_mod_cast hn
+    have hm : (0 : ℝ) ≤ m := Nat.cast_nonneg _
+    have ha : (0 : ℝ) ≤ a := Nat.cast_nonneg _
+    linarith
+
+lemma higherFactor_step {n : ℕ} (hn : 2 ≤ n) (j : ℕ) {s : ℝ} (hs : 3 ≤ s) :
+    (s-2*j)*(s+2*j+n-2)*higherFactor n j s =
+      s*(s+n-2)*higherFactor n j (s-2) := by
+  have hn' : (2 : ℝ) ≤ n := by exact_mod_cast hn
+  induction j with
+  | zero => simp [higherFactor]
+  | succ j ih =>
+    rw [higherFactor_succ, higherFactor_succ]
+    have hj : (0 : ℝ) ≤ j := Nat.cast_nonneg _
+    have h₁ : s+n+2*j ≠ 0 := by linarith
+    have h₂ : s-2+n+2*j ≠ 0 := by linarith
+    push_cast
+    field_simp
+    linear_combination (s-2*j-2)*(s+n+2*j)*ih
+
+lemma higherFactor_tendsto (n j : ℕ) :
+    Tendsto (fun m : ℕ => higherFactor n j (2*m+1)) atTop (𝓝 1) := by
+  induction j with
+  | zero => simpa only [higherFactor_zero] using tendsto_const_nhds
+  | succ j ih =>
+    have h : Tendsto (fun m : ℕ => ((2*(m : ℝ)+1)-2*j)/((2*(m : ℝ)+1)+n+2*j)) atTop (𝓝 1) := by
+      have h := tendsto_add_mul_div_add_mul_atTop_nhds (1-2*(j : ℝ))
+        (1+n+2*(j : ℝ)) (2 : ℝ) (show (2 : ℝ) ≠ 0 by norm_num)
+      convert! h using 1
+      · funext m; congr 1 <;> ring
+      · norm_num
+    simpa only [higherFactor_succ, mul_one] using ih.mul h
+
+end PettyProjection.Spherical
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology Function
+open scoped ENNReal RealInnerProductSpace
+namespace PettyProjection.Spherical
+
+lemma eigenvalue_sub_even {n j : ℕ} (s : ℕ) :
+    eigenvalue n s-eigenvalue n (2*j) =
+      ((s : ℝ)-2*j)*((s : ℝ)+2*j+n-2) := by
+  simp only [eigenvalue, Nat.cast_mul, Nat.cast_ofNat]
+  ring
+
+lemma eigenvalue_odd_sub_even_ne_zero {n : ℕ} (hn : 2 ≤ n) (m j : ℕ) :
+    eigenvalue n (2*m+1)-eigenvalue n (2*j) ≠ 0 := by
+  rw [eigenvalue_sub_even]
+  apply mul_ne_zero
+  · apply sub_ne_zero.mpr
+    have h : (2*m+1 : ℕ) ≠ 2*j := by omega
+    exact_mod_cast h
+  · have hn' : (2 : ℝ) ≤ n := by exact_mod_cast hn
+    have hm : (0 : ℝ) ≤ m := Nat.cast_nonneg _
+    have hj : (0 : ℝ) ≤ j := Nat.cast_nonneg _
+    push_cast
+    linarith
+
+lemma higherKernel_normalized_recurrence {n r d : ℕ} [NeZero n] (hr : 3 ≤ r)
+    (u : Sphere n) {f : Space n → ℝ} (hf : ContDiff ℝ 2 f)
+    (heig : ∀ x : Sphere n, casimir f x = -eigenvalue n d*f x) :
+    (eigenvalue n r-eigenvalue n d)*(higherKernel r u (fun x => f x)/higherKernel r u (fun _ => 1)) =
+      eigenvalue n r*(higherKernel (r-2) u (fun x => f x)/higherKernel (r-2) u (fun _ => 1)) := by
+  have h := higherKernel_recurrence hr u hf heig
+  have hz := higherKernel_const_recurrence hr u
+  have h₁ := (higherKernel_const_pos r u).ne'
+  have h₂ := (higherKernel_const_pos (r-2) u).ne'
+  field_simp
+  linear_combination higherKernel (r-2) u (fun _ => 1)*h -
+    higherKernel (r-2) u (fun x => f x)*hz
+
+lemma higherKernel_eigenfunction {n j : ℕ} (hn : 2 ≤ n) [NeZero n]
+    (u : Sphere n) {f : Space n → ℝ} (hf : ContDiff ℝ 2 f)
+    (heig : ∀ x : Sphere n, casimir f x = -eigenvalue n (2*j)*f x) :
+    higherKernel 1 u (fun x => f x)/higherKernel 1 u (fun _ => 1) =
+      higherFactor n j 1*f u := by
+  let A (m : ℕ) := higherKernel (2*m+1) u (fun x => f x)/higherKernel (2*m+1) u (fun _ => 1)
+  let F (m : ℕ) := higherFactor n j (2*m+1)
+  have hinv (m : ℕ) : A m*F 0 = A 0*F m := by
+    induction m with
+    | zero => rfl
+    | succ m ih =>
+      have hc := eigenvalue_odd_sub_even_ne_zero hn (m+1) j
+      have hA := higherKernel_normalized_recurrence (show 3 ≤ 2*(m+1)+1 by omega) u hf heig
+      rw [show 2*(m+1)+1-2 = 2*m+1 by omega] at hA
+      change (eigenvalue n (2*(m+1)+1)-eigenvalue n (2*j))*A (m+1) =
+        eigenvalue n (2*(m+1)+1)*A m at hA
+      have hF : (eigenvalue n (2*(m+1)+1)-eigenvalue n (2*j))*F (m+1) =
+          eigenvalue n (2*(m+1)+1)*F m := by
+        have hm : (0 : ℝ) ≤ m := Nat.cast_nonneg _
+        have h := higherFactor_step hn j (s := 2*(m+1)+1) (by linarith)
+        have he : (2*((m : ℝ)+1)+1)-2 = 2*m+1 := by ring
+        rw [he] at h
+        rw [eigenvalue_sub_even]
+        simpa only [F, eigenvalue, Nat.cast_add,
+          Nat.cast_mul, Nat.cast_one, Nat.cast_ofNat] using h
+      apply mul_left_cancel₀ hc
+      calc
+        _ = ((eigenvalue n (2*(m+1)+1)-eigenvalue n (2*j))*A (m+1))*F 0 := by ring
+        _ = eigenvalue n (2*(m+1)+1)*(A m*F 0) := by rw [hA]; ring
+        _ = eigenvalue n (2*(m+1)+1)*(A 0*F m) := by rw [ih]
+        _ = A 0*((eigenvalue n (2*(m+1)+1)-eigenvalue n (2*j))*F (m+1)) := by rw [hF]; ring
+        _ = _ := by ring
+  have ht : Tendsto (fun m : ℕ => 2*m+1) atTop atTop := by
+    apply Filter.tendsto_atTop.mpr
+    intro b
+    filter_upwards [eventually_ge_atTop b] with m hm
+    omega
+  have hA : Tendsto A atTop (𝓝 (f u)) :=
+    (higherKernel_peak u (hf.continuous.comp continuous_subtype_val)).comp ht
+  have hF : Tendsto F atTop (𝓝 1) := higherFactor_tendsto n j
+  have hlim₁ := hA.mul (tendsto_const_nhds (x := F 0))
+  have hlim₂ := (tendsto_const_nhds (x := A 0)).mul hF
+  have heq : (fun m => A m*F 0) = (fun m => A 0*F m) := funext hinv
+  rw [heq] at hlim₁
+  have h := tendsto_nhds_unique hlim₁ hlim₂
+  simpa [A, F, mul_comm] using h.symm
+
+lemma higherKernel_harmonic {n j : ℕ} (hn : 2 ≤ n) [NeZero n]
+    (u : Sphere n) {p : MvPolynomial (Fin n) ℝ} (hp : p.IsHomogeneous (2*j))
+    (hΔ : laplacePolynomial p = 0) :
+    higherKernel 1 u (fun x => polynomialFunction p x)/higherKernel 1 u (fun _ => 1) =
+      higherFactor n j 1*polynomialFunction p u := by
+  exact higherKernel_eigenfunction hn u ((polynomialFunction_contDiff p).of_le (by norm_num))
+    (fun x => casimir_harmonic hp hΔ x)
+
+end PettyProjection.Spherical
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology Function
+open scoped ENNReal RealInnerProductSpace
+namespace PettyProjection.Spherical
+
+def pole (n : ℕ) [NeZero n] : Sphere n :=
+  ⟨EuclideanSpace.single (0 : Fin n) 1, by simp⟩
+
+lemma higherKernel_const_rotation {n : ℕ} (r : ℕ) (u : Sphere n)
+    (U : Space n ≃ₗᵢ[ℝ] Space n) :
+    higherKernel r (sphereMap U u) (fun _ => 1) = higherKernel r u (fun _ => 1) := by
+  have h := mean_rotation U (fun x : Sphere n => max ⟪U (u : Space n),(x : Space n)⟫ 0 ^ r)
+  change mean (fun x : Sphere n => max ⟪U (u : Space n),U (x : Space n)⟫ 0 ^ r) = _ at h
+  simp only [U.inner_map_map] at h
+  simp only [higherKernel, mul_one]
+  change mean (fun x : Sphere n => max ⟪U (u : Space n),(x : Space n)⟫ 0 ^ r) = _
+  exact h.symm
+
+lemma higherKernel_const_independent {n : ℕ} (r : ℕ) (u v : Sphere n) :
+    higherKernel r u (fun _ => 1) = higherKernel r v (fun _ => 1) := by
+  let U := (ℝ ∙ ((u : Space n)-(v : Space n)))ᗮ.reflection
+  have h : sphereMap U u = v := by
+    apply Subtype.ext
+    exact Submodule.reflection_sub ((norm_coe u).trans (norm_coe v).symm)
+  rw [← h, higherKernel_const_rotation]
+
+/-- The true normalized cosine kernel's constant, not an assumed multiplier. -/
+def cosineConstant (n : ℕ) [NeZero n] : ℝ :=
+  2*higherKernel 1 (pole n) (fun _ => 1)
+
+lemma cosineConstant_pos (n : ℕ) [NeZero n] : 0 < cosineConstant n :=
+  mul_pos (by norm_num) (higherKernel_const_pos 1 (pole n))
+
+lemma mean_pos_inner_even {n : ℕ} [NeZero n] (u : Sphere n) {f : Space n → ℝ}
+    (heven : Function.Even f) (hf : Continuous f) :
+    mean (fun x : Sphere n => |⟪(u : Space n),(x : Space n)⟫| *f x) =
+      2*higherKernel 1 u (fun x => f x) := by
+  have h := mean_rotation (LinearIsometryEquiv.neg ℝ : Space n ≃ₗᵢ[ℝ] Space n)
+    (fun x : Sphere n => max ⟪(u : Space n),(x : Space n)⟫ 0*f x)
+  have he : (fun x : Sphere n => max ⟪(u : Space n),((sphereMap (LinearIsometryEquiv.neg ℝ)) x : Space n)⟫ 0 *
+      f (sphereMap (LinearIsometryEquiv.neg ℝ) x)) =
+      (fun x : Sphere n => max (-⟪(u : Space n),(x : Space n)⟫) 0*f x) := by
+    funext x
+    change max ⟪(u : Space n),-(x : Space n)⟫ 0*f (-(x : Space n)) = _
+    rw [inner_neg_right, heven]
+  rw [he] at h
+  have hk (t : ℝ) : |t| = max t 0+max (-t) 0 := by
+    rcases le_total 0 t with ht | ht <;> simp [abs_of_nonneg, abs_of_nonpos, ht]
+  simp only [higherKernel, pow_one]
+  simp_rw [hk, add_mul]
+  rw [mean, integral_add]
+  · change mean (fun x : Sphere n => max ⟪(u : Space n),(x : Space n)⟫ 0*f x) +
+      mean (fun x : Sphere n => max (-⟪(u : Space n),(x : Space n)⟫) 0*f x) = _
+    rw [h]
+    ring
+  · exact continuous_integrable (by fun_prop)
+  · exact continuous_integrable (by fun_prop)
+
+lemma mean_abs_inner {n : ℕ} [NeZero n] (u : Sphere n) :
+    mean (fun x : Sphere n => |⟪(u : Space n),(x : Space n)⟫|) = cosineConstant n := by
+  have h := mean_pos_inner_even u (f := fun _ => 1) (fun _ => rfl) continuous_const
+  simpa only [mul_one, higherKernel_const_independent 1 u (pole n), cosineConstant] using h
+
+/-- The actual cosine integral, with its constant fixed geometrically. -/
+def cosine {n : ℕ} [NeZero n] (f : Sphere n → ℝ) (u : Sphere n) : ℝ :=
+  mean (fun x : Sphere n => |⟪(u : Space n),(x : Space n)⟫| *f x)/cosineConstant n
+
+lemma cosine_continuous {n : ℕ} [NeZero n] {f : Sphere n → ℝ} (hf : Continuous f) :
+    Continuous (cosine f) := by
+  apply Continuous.div_const
+  have h := continuous_parametric_integral_of_continuous (μ := sigma n)
+    (f := fun u x : Sphere n => |⟪(u : Space n),(x : Space n)⟫| *f x)
+    (by fun_prop) isCompact_univ
+  simpa only [Measure.restrict_univ, mean] using h
+
+lemma cosine_const {n : ℕ} [NeZero n] (c : ℝ) : cosine (fun _ : Sphere n => c) = fun _ => c := by
+  funext u
+  rw [cosine, mean, integral_mul_const]
+  change mean (fun x : Sphere n => |⟪(u : Space n),(x : Space n)⟫|)*c/cosineConstant n = c
+  rw [mean_abs_inner]
+  field_simp [(cosineConstant_pos n).ne']
+
+lemma polynomialFunction_even {n d : ℕ} {p : MvPolynomial (Fin n) ℝ}
+    (hp : p.IsHomogeneous d) (hd : Even d) : Function.Even (polynomialFunction p) := by
+  intro x
+  simpa only [neg_one_smul, hd.neg_one_pow, one_mul] using
+    polynomialFunction_smul_argument hp (-1) x
+
+lemma cosine_harmonic {n j : ℕ} (hn : 2 ≤ n) [NeZero n]
+    {p : MvPolynomial (Fin n) ℝ} (hp : p.IsHomogeneous (2*j))
+    (hΔ : laplacePolynomial p = 0) (u : Sphere n) :
+    cosine (fun x => polynomialFunction p x) u = higherFactor n j 1*polynomialFunction p u := by
+  rw [cosine, mean_pos_inner_even u (polynomialFunction_even hp (even_two_mul j))
+    (polynomialFunction_contDiff p).continuous, cosineConstant,
+    ← higherKernel_const_independent 1 u (pole n), mul_div_mul_left _ _ (by norm_num : (2 : ℝ) ≠ 0)]
+  exact higherKernel_harmonic hn u hp hΔ
+
+end PettyProjection.Spherical
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology Function
+open scoped ENNReal RealInnerProductSpace
+namespace PettyProjection.Spherical
+
+def cosineMap {n : ℕ} [NeZero n] (f : C(Sphere n, ℝ)) : C(Sphere n, ℝ) :=
+  ⟨cosine f, cosine_continuous f.continuous⟩
+
+lemma cosine_selfadj {n : ℕ} [NeZero n] {f g : Sphere n → ℝ}
+    (hf : Continuous f) (hg : Continuous g) :
+    mean (fun u => f u*cosine g u) = mean (fun u => cosine f u*g u) := by
+  have hi : Integrable (fun z : Sphere n × Sphere n =>
+      f z.1 * (|⟪(z.1 : Space n),(z.2 : Space n)⟫| * g z.2)) ((sigma n).prod (sigma n)) :=
+    (by fun_prop : Continuous _).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)
+  have h := integral_integral_swap (f := fun u v : Sphere n => f u*(|⟪(u : Space n),(v : Space n)⟫| * g v)) hi
+  simp only [cosine, mean, ← mul_div_assoc, div_mul_eq_mul_div, integral_div, ← integral_const_mul]
+  rw [h]
+  congr 1
+  apply integral_congr_ae
+  filter_upwards with u
+  rw [← integral_mul_const]
+  apply integral_congr_ae
+  filter_upwards with v
+  rw [real_inner_comm (u : Space n) (v : Space n)]
+  ring
+
+lemma mean_cosine {n : ℕ} [NeZero n] {f : Sphere n → ℝ} (hf : Continuous f) :
+    mean (cosine f) = mean f := by
+  have h := cosine_selfadj (f := fun _ => 1) continuous_const hf
+  simpa only [one_mul, cosine_const] using h
+
+lemma cosine_neg {n : ℕ} [NeZero n] (f : Sphere n → ℝ) (u : Sphere n) :
+    cosine f (sphereMap (LinearIsometryEquiv.neg ℝ) u) = cosine f u := by
+  unfold cosine
+  congr 1
+  congr 1
+  funext x
+  change |⟪-(u : Space n),(x : Space n)⟫| * f x = _
+  rw [inner_neg_left, abs_neg]
+
+lemma mean_sphere_even_mul_odd {n d : ℕ} {f : Sphere n → ℝ}
+    (heven : ∀ u, f (sphereMap (LinearIsometryEquiv.neg ℝ) u) = f u)
+    {p : MvPolynomial (Fin n) ℝ} (hp : p.IsHomogeneous d) (hd : Odd d) :
+    mean (fun u : Sphere n => f u*polynomialFunction p u) = 0 := by
+  have h := mean_rotation (LinearIsometryEquiv.neg ℝ : Space n ≃ₗᵢ[ℝ] Space n)
+    (fun u : Sphere n => f u*polynomialFunction p u)
+  have he : (fun u : Sphere n => f (sphereMap (LinearIsometryEquiv.neg ℝ) u)*
+      polynomialFunction p (sphereMap (LinearIsometryEquiv.neg ℝ) u)) =
+      (fun u : Sphere n => -(f u*polynomialFunction p u)) := by
+    funext u
+    rw [heven]
+    change f u*polynomialFunction p (-(u : Space n)) = _
+    rw [polynomialFunction_odd hp hd, mul_neg]
+  rw [he, mean, integral_neg] at h
+  change -mean (fun u : Sphere n => f u*polynomialFunction p u) = _ at h
+  linarith
+
+lemma cosine_harmonic_odd {n d : ℕ} [NeZero n] {p : MvPolynomial (Fin n) ℝ}
+    (hp : p.IsHomogeneous d) (hd : Odd d) (u : Sphere n) :
+    cosine (fun x => polynomialFunction p x) u = 0 := by
+  rw [cosine, mean_even_mul_odd (f := fun x => |⟪(u : Space n),x⟫|) _ hp hd, zero_div]
+  intro x
+  simp only [inner_neg_right, abs_neg]
+
+lemma cosine_projection_even {n j : ℕ} (hn : 2 ≤ n) [NeZero n] (f : C(Sphere n, ℝ)) :
+    (harmonicSpace n (2*j)).starProjection (toH n (cosineMap f)) =
+      higherFactor n j 1 • (harmonicSpace n (2*j)).starProjection (toH n f) := by
+  apply Submodule.eq_starProjection_of_mem_of_inner_eq_zero
+    ((harmonicSpace n (2*j)).smul_mem _ ((harmonicSpace n (2*j)).starProjection_apply_mem _))
+  rintro v ⟨p, hp, rfl⟩
+  rw [inner_sub_left, real_inner_smul_left, Submodule.inner_starProjection_left_eq_right]
+  rw [Submodule.starProjection_eq_self_iff.mpr (show polynomialL2 n p ∈ harmonicSpace n (2*j) from ⟨p,hp,rfl⟩)]
+  change ⟪toH n (cosineMap f),toH n (polynomialRestriction n p)⟫ -
+    higherFactor n j 1 * ⟪toH n f,toH n (polynomialRestriction n p)⟫ = 0
+  rw [inner_toH, inner_toH]
+  change mean (fun u : Sphere n => cosine f u*(polynomialRestriction n p) u) - _ = 0
+  rw [← cosine_selfadj f.continuous (polynomialRestriction n p).continuous]
+  have he : (fun u : Sphere n => f u*cosine (polynomialRestriction n p) u) =
+      (fun u : Sphere n => higherFactor n j 1*(f u*polynomialFunction p u)) := by
+    funext u
+    rw [show cosine (polynomialRestriction n p) u = _ from cosine_harmonic hn hp.1 hp.2 u]
+    ring
+  rw [he, mean, integral_const_mul]
+  change _ - higherFactor n j 1 * (∫ u : Sphere n, f u * polynomialFunction p u ∂sigma n) = 0
+  exact sub_self _
+
+lemma cosine_projection_odd {n d : ℕ} [NeZero n] (f : C(Sphere n, ℝ)) (hd : Odd d) :
+    (harmonicSpace n d).starProjection (toH n (cosineMap f)) = 0 := by
+  apply Submodule.eq_starProjection_of_mem_of_inner_eq_zero (Submodule.zero_mem _)
+  rintro v ⟨p, hp, rfl⟩
+  rw [sub_zero]
+  change ⟪toH n (cosineMap f),toH n (polynomialRestriction n p)⟫ = 0
+  rw [inner_toH]
+  exact mean_sphere_even_mul_odd (cosine_neg f) hp.1 hd
+
+end PettyProjection.Spherical
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/ProjectionBodies/FixedPointCalculus.lean b/lean/OAI/Geometry/ProjectionBodies/FixedPointCalculus.lean
new file mode 100644
index 0000000000000000000000000000000000000000..25890938f81bf468d3fa8971fc7712da4697002e
--- /dev/null
+++ b/lean/OAI/Geometry/ProjectionBodies/FixedPointCalculus.lean
@@ -0,0 +1,591 @@
+import OAI.Geometry.ProjectionBodies.VariationalRepresentation
+
+namespace OAI
+
+universe uE uX uι uα
+
+noncomputable section
+namespace PettyProjection.FixedPoint
+open scoped RealInnerProductSpace
+
+def rayRoot (a b c : ℝ) : ℝ := (Real.sqrt (b^2+a*(1-c))-b)/a
+lemma rayRoot_positive {a b c : ℝ} (ha : 0<a) (hc : c<1) : 0<rayRoot a b c := by
+  have hd : 0<b^2+a*(1-c) := by nlinarith [sq_nonneg b]
+  have hs := Real.sq_sqrt hd.le
+  have hs0 := Real.sqrt_nonneg (b^2+a*(1-c))
+  have hb : b<Real.sqrt (b^2+a*(1-c)) := by nlinarith
+  exact div_pos (sub_pos.mpr hb) ha
+
+lemma rayRoot_quadratic {a b c : ℝ} (ha : 0<a) (hc : c<1) :
+    a*rayRoot a b c^2+2*b*rayRoot a b c+c=1 := by
+  have hd : 0<b^2+a*(1-c) := by nlinarith [sq_nonneg b]
+  have hs := Real.sq_sqrt hd.le
+  dsimp [rayRoot]
+  field_simp
+  nlinarith
+
+lemma rayRoot_eq_one {a b c : ℝ} (ha : 0<a) (hc : c<1) (he : c+2*b+a=1) :
+    rayRoot a b c=1 := by
+  have hpos := rayRoot_positive (b := b) ha hc
+  have hq := rayRoot_quadratic (b := b) ha hc
+  have hfactor : (rayRoot a b c-1)*(a*rayRoot a b c+(1-c))=0 := by nlinarith
+  have hfacpos : 0<a*rayRoot a b c+(1-c) := by positivity
+  exact sub_eq_zero.mp ((mul_eq_zero.mp hfactor).resolve_right (ne_of_gt hfacpos))
+
+variable {E : Type uE} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+
+def rayRetraction (c x : E) : E :=
+  c+rayRoot (‖x-c‖^2) ⟪c,x-c⟫ (‖c‖^2) • (x-c)
+
+lemma norm_add_smul_sq (c v : E) (t : ℝ) :
+    ‖c+t • v‖^2=‖c‖^2+2*t*⟪c,v⟫+t^2*‖v‖^2 := by
+  simp only [← real_inner_self_eq_norm_sq,inner_add_left,inner_add_right,
+    inner_smul_left,inner_smul_right,conj_trivial]
+  rw [real_inner_comm v c]
+  ring
+
+lemma rayRetraction_norm {c x : E} (hc : ‖c‖<1) (hne : x≠c) : ‖rayRetraction c x‖=1 := by
+  have ha : 0<‖x-c‖^2 := sq_pos_of_pos (norm_pos_iff.mpr (sub_ne_zero.mpr hne))
+  have hcc : ‖c‖^2<1 := by nlinarith [norm_nonneg c]
+  have hq := rayRoot_quadratic (b := ⟪c,x-c⟫) ha hcc
+  have he : ‖rayRetraction c x‖^2=1 := by
+    rw [rayRetraction,norm_add_smul_sq]
+    nlinarith
+  nlinarith [norm_nonneg (rayRetraction c x)]
+
+lemma rayRetraction_boundary {c x : E} (hc : ‖c‖<1) (hx : ‖x‖=1) : rayRetraction c x=x := by
+  have hne : x≠c := by intro h; rw [← h,hx] at hc; linarith
+  have ha : 0<‖x-c‖^2 := sq_pos_of_pos (norm_pos_iff.mpr (sub_ne_zero.mpr hne))
+  have hcc : ‖c‖^2<1 := by nlinarith [norm_nonneg c]
+  have hh := norm_add_smul_sq c (x-c) 1
+  simp only [one_smul,add_sub_cancel,one_pow,one_mul,mul_one,hx] at hh
+  have he := rayRoot_eq_one ha hcc hh.symm
+  simp only [rayRetraction,he,one_smul,add_sub_cancel]
+
+end PettyProjection.FixedPoint
+end
+
+noncomputable section
+open Set MeasureTheory Polynomial
+open scoped BigOperators
+namespace PettyProjection.FixedPoint
+
+variable {X : Type uX} [TopologicalSpace X]
+
+def ContinuousCoeffs (p : X → Polynomial ℝ) : Prop :=
+  ∀ k, Continuous (fun x => (p x).coeff k)
+
+lemma continuousCoeffs_const (p : Polynomial ℝ) : ContinuousCoeffs (fun _ : X => p) :=
+  fun _ => continuous_const
+
+lemma continuousCoeffs_C {f : X → ℝ} (hf : Continuous f) :
+    ContinuousCoeffs (fun x => Polynomial.C (f x)) := by
+  intro k
+  by_cases hk : k = 0
+  · subst k; simpa only [coeff_C_zero] using hf
+  · simpa only [coeff_C, ite_eq_right hk] using (continuous_const : Continuous (fun _ : X => (0 : ℝ)))
+
+lemma ContinuousCoeffs.add {p q : X → Polynomial ℝ}
+    (hp : ContinuousCoeffs p) (hq : ContinuousCoeffs q) :
+    ContinuousCoeffs (fun x => p x + q x) := by
+  intro k
+  simp only [coeff_add]
+  exact continuous_add.comp ((hp k).prodMk (hq k))
+
+lemma ContinuousCoeffs.mul {p q : X → Polynomial ℝ}
+    (hp : ContinuousCoeffs p) (hq : ContinuousCoeffs q) :
+    ContinuousCoeffs (fun x => p x * q x) := by
+  intro k
+  simp only [coeff_mul]
+  exact continuous_finsetSum _ (fun ij _ => (hp ij.1).mul (hq ij.2))
+
+lemma ContinuousCoeffs.sum {ι : Type uι} (s : Finset ι) {p : ι → X → Polynomial ℝ}
+    (hp : ∀ i ∈ s, ContinuousCoeffs (p i)) :
+    ContinuousCoeffs (fun x => ∑ i ∈ s, p i x) := by
+  intro k
+  simp only [finsetSum_coeff]
+  exact continuous_finsetSum _ (fun i hi => hp i hi k)
+
+lemma ContinuousCoeffs.prod {ι : Type uι} (s : Finset ι) {p : ι → X → Polynomial ℝ}
+    (hp : ∀ i ∈ s, ContinuousCoeffs (p i)) :
+    ContinuousCoeffs (fun x => ∏ i ∈ s, p i x) := by
+  classical
+  induction s using Finset.induction_on with
+  | empty => simpa using continuousCoeffs_const (X := X) (1 : Polynomial ℝ)
+  | @insert i s hi ih =>
+      simp only [Finset.prod_insert hi]
+      exact (hp i (Finset.mem_insert_self _ _)).mul
+        (ih (fun j hj => hp j (Finset.mem_insert_of_mem hj)))
+
+lemma ContinuousCoeffs.det {ι : Type uι} [Fintype ι] [DecidableEq ι]
+    {p : X → Matrix ι ι (Polynomial ℝ)}
+    (hp : ∀ i j, ContinuousCoeffs (fun x => p x i j)) :
+    ContinuousCoeffs (fun x => (p x).det) := by
+  classical
+  simp only [Matrix.det_apply]
+  apply ContinuousCoeffs.sum
+  intro e _
+  have hprod := ContinuousCoeffs.prod Finset.univ (fun i _ => hp (e i) i)
+  intro k
+  simp only [coeff_smul]
+  exact continuous_smul.comp (continuous_const.prodMk (hprod k))
+
+/-- Integrating a uniformly bounded-degree polynomial coefficient by coefficient. -/
+def integralPolynomial {α : Type uα} [MeasurableSpace α] (μ : Measure α)
+    (p : α → Polynomial ℝ) (d : ℕ) : Polynomial ℝ :=
+  ∑ k ∈ Finset.range (d+1), Polynomial.C (∫ x, (p x).coeff k ∂μ) * Polynomial.X^k
+
+lemma eval_integralPolynomial {α : Type uα} [MeasurableSpace α] (μ : Measure α)
+    (p : α → Polynomial ℝ) (d : ℕ) (hp : ∀ x, (p x).natDegree ≤ d)
+    (hi : ∀ k ≤ d, Integrable (fun x => (p x).coeff k) μ) (t : ℝ) :
+    (integralPolynomial μ p d).eval t = ∫ x, (p x).eval t ∂μ := by
+  have he (x : α) : (p x).eval t = ∑ k ∈ Finset.range (d+1), (p x).coeff k*t^k := by
+    exact Polynomial.eval_eq_sum_range' (Nat.lt_succ_of_le (hp x)) t
+  simp only [integralPolynomial, eval_finsetSum, eval_mul, eval_C, eval_pow, eval_X]
+  simp_rw [he]
+  rw [integral_finsetSum]
+  · apply Finset.sum_congr rfl
+    intro k hk
+    exact (integral_mul_const _ _).symm
+  · intro k hk
+    exact (hi k (Nat.le_of_lt_succ (Finset.mem_range.mp hk))).mul_const _
+
+lemma exists_integral_det_polynomial {ι : Type uι} [Fintype ι] [DecidableEq ι]
+    {α : Type uα} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α]
+    (μ : Measure α) [IsFiniteMeasure μ] [CompactSpace α]
+    (A B : α → Matrix ι ι ℝ) (hA : Continuous A) (hB : Continuous B) :
+    ∃ p : Polynomial ℝ, ∀ t : ℝ,
+      p.eval t = ∫ x, (t • A x + B x).det ∂μ := by
+  let q : α → Polynomial ℝ := fun x =>
+    Matrix.det ((Polynomial.X : Polynomial ℝ) • (A x).map Polynomial.C + (B x).map Polynomial.C)
+  have hq : ContinuousCoeffs q := by
+    apply ContinuousCoeffs.det
+    intro i j
+    apply ContinuousCoeffs.add
+    · exact (continuousCoeffs_const Polynomial.X).mul
+        (continuousCoeffs_C ((continuous_apply j).comp ((continuous_apply i).comp hA)))
+    · exact continuousCoeffs_C ((continuous_apply j).comp ((continuous_apply i).comp hB))
+  have hdeg (x : α) : (q x).natDegree ≤ Fintype.card ι :=
+    Polynomial.natDegree_det_X_add_C_le (A x) (B x)
+  refine ⟨integralPolynomial μ q (Fintype.card ι), fun t => ?_⟩
+  rw [eval_integralPolynomial μ q _ hdeg (fun k _ => (by simpa only [integrableOn_univ] using (hq k).continuousOn.integrableOn_compact' (μ := μ) isCompact_univ MeasurableSet.univ))]
+  apply integral_congr_ae
+  filter_upwards with x
+  change Polynomial.eval t (Matrix.det _) = _
+  change (Polynomial.evalRingHom t) _ = _
+  rw [RingHom.map_det]
+  congr 1
+  ext i j
+  simp only [RingHom.mapMatrix_apply, Matrix.map_apply, Matrix.add_apply, Matrix.smul_apply, smul_eq_mul, map_add, Polynomial.coe_evalRingHom, eval_mul, eval_X, eval_C]
+
+lemma exists_integralOn_det_polynomial {ι : Type uι} [Fintype ι] [DecidableEq ι]
+    {α : Type uα} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [T2Space α]
+    (μ : Measure α) [IsFiniteMeasureOnCompacts μ] {s : Set α} (hs : IsCompact s)
+    (A B : α → Matrix ι ι ℝ) (hA : ContinuousOn A s) (hB : ContinuousOn B s) :
+    ∃ p : Polynomial ℝ, ∀ t : ℝ,
+      p.eval t = ∫ x in s, (t • A x + B x).det ∂μ := by
+  let q : α → Polynomial ℝ := fun x =>
+    Matrix.det ((Polynomial.X : Polynomial ℝ) • (A x).map Polynomial.C + (B x).map Polynomial.C)
+  have hq : ContinuousCoeffs (fun x : s => q x) := by
+    apply ContinuousCoeffs.det
+    intro i j
+    apply ContinuousCoeffs.add
+    · exact (continuousCoeffs_const Polynomial.X).mul
+        (continuousCoeffs_C ((continuous_apply j).comp ((continuous_apply i).comp
+          (continuousOn_iff_continuous_domRestrict.mp hA))))
+    · exact continuousCoeffs_C ((continuous_apply j).comp ((continuous_apply i).comp
+        (continuousOn_iff_continuous_domRestrict.mp hB)))
+  have hdeg (x : α) : (q x).natDegree ≤ Fintype.card ι :=
+    Polynomial.natDegree_det_X_add_C_le (A x) (B x)
+  refine ⟨integralPolynomial (μ.restrict s) q (Fintype.card ι), fun t => ?_⟩
+  rw [eval_integralPolynomial (μ.restrict s) q _ hdeg
+    (fun k _ => ContinuousOn.integrableOn_compact hs
+      (continuousOn_iff_continuous_domRestrict.mpr (hq k)))]
+  apply integral_congr_ae
+  filter_upwards with x
+  change (Polynomial.evalRingHom t) (Matrix.det _) = _
+  rw [RingHom.map_det]
+  congr 1
+  ext i j
+  simp only [RingHom.mapMatrix_apply, Matrix.map_apply, Matrix.add_apply, Matrix.smul_apply,
+    smul_eq_mul, map_add, Polynomial.coe_evalRingHom, eval_mul, eval_X, eval_C]
+
+lemma exists_integralOn_clm_det_polynomial
+    {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E]
+    {α : Type uα} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] [T2Space α]
+    (μ : Measure α) [IsFiniteMeasureOnCompacts μ] {s : Set α} (hs : IsCompact s)
+    (A B : α → (E →L[ℝ] E)) (hA : ContinuousOn A s) (hB : ContinuousOn B s) :
+    ∃ p : Polynomial ℝ, ∀ t : ℝ,
+      p.eval t = ∫ x in s, (t • A x + B x).det ∂μ := by
+  classical
+  let b := Module.finBasis ℝ E
+  let M := (LinearMap.toMatrix b b).toLinearMap.comp (ContinuousLinearMap.coeLM ℝ)
+  have hM := M.continuous_of_finiteDimensional
+  obtain ⟨p,hp⟩ := exists_integralOn_det_polynomial μ hs
+    (fun x => M (A x)) (fun x => M (B x)) (hM.comp_continuousOn hA) (hM.comp_continuousOn hB)
+  refine ⟨p,fun t => (hp t).trans ?_⟩
+  apply integral_congr_ae
+  filter_upwards with x
+  rw [← map_smul M,← map_add M]
+  exact LinearMap.det_toMatrix b _
+
+end PettyProjection.FixedPoint
+end
+
+noncomputable section
+open Set Metric Function
+open scoped NNReal
+namespace PettyProjection.FixedPoint
+variable {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
+
+lemma bijective_add_small {z : E → E} {C : ℝ≥0} (hz : LipschitzWith C z)
+    {t : ℝ} (ht : ‖t‖₊*C<1) : Bijective (fun x => x+t • z x) := by
+  have hc (y : E) : ContractingWith (‖t‖₊*C) (fun x => y-t • z x) := by
+    refine ⟨ht, ?_⟩
+    apply LipschitzWith.of_dist_le_mul
+    intro x w
+    simp only [dist_sub_left, dist_smul₀, NNReal.coe_mul, coe_nnnorm]
+    calc
+      ‖t‖ * dist (z x) (z w) ≤ ‖t‖ * (C * dist x w) :=
+        mul_le_mul_of_nonneg_left (hz.dist_le_mul x w) (norm_nonneg t)
+      _ = _ := by ring
+  constructor
+  · intro x y hxy
+    apply (hc (x+t • z x)).fixedPoint_unique'
+    · change x+t • z x-t • z x=x
+      abel
+    · change x+t • z x-t • z y=y
+      change x+t • z x=y+t • z y at hxy
+      rw [hxy]
+      abel
+  · intro y
+    obtain ⟨x,hx,-⟩ := (hc y).exists_fixedPoint (0:E) (edist_ne_top _ _)
+    refine ⟨x,?_⟩
+    change y-t • z x=x at hx
+    exact eq_sub_iff_add_eq.mp hx.symm
+
+lemma image_ball_add_small {z : E → E} {C : ℝ≥0} (hz : LipschitzWith C z)
+    (hzero : ∀ x, 1≤‖x‖ → z x=0) {t : ℝ} (ht : ‖t‖₊*C<1) :
+    (fun x => x+t • z x) '' ball (0:E) 1 = ball (0:E) 1 := by
+  have hbi := bijective_add_small hz ht
+  have hout (x : E) (hx : x∉ball (0:E) 1) : x+t • z x=x := by
+    have hxn : 1≤‖x‖ := by simpa only [mem_ball,dist_zero_right,not_lt] using hx
+    rw [hzero x hxn,smul_zero,add_zero]
+  apply Subset.antisymm
+  · rintro y ⟨x,hx,rfl⟩
+    by_contra hy
+    have he : x+t • z x=x := hbi.1 (hout (x+t • z x) hy)
+    exact hy (by change x+t • z x∈_; rw [he]; exact hx)
+  · intro y hy
+    obtain ⟨x,hxy⟩ := hbi.2 y
+    refine ⟨x,?_,hxy⟩
+    by_contra hx
+    change x+t • z x=y at hxy
+    rw [hout x hx] at hxy
+    exact hx (hxy.symm ▸ hy)
+
+lemma image_closedBall_add_small {z : E → E} {C : ℝ≥0} (hz : LipschitzWith C z)
+    (hzero : ∀ x, 1≤‖x‖ → z x=0) {t : ℝ} (ht : ‖t‖₊*C<1) :
+    (fun x => x+t • z x) '' closedBall (0:E) 1 = closedBall (0:E) 1 := by
+  have hbi := bijective_add_small hz ht
+  have hout (x : E) (hx : x∉closedBall (0:E) 1) : x+t • z x=x := by
+    have hxn : 1<‖x‖ := by simpa only [mem_closedBall,dist_zero_right,not_le] using hx
+    rw [hzero x hxn.le,smul_zero,add_zero]
+  apply Subset.antisymm
+  · rintro y ⟨x,hx,rfl⟩
+    by_contra hy
+    have he : x+t • z x=x := hbi.1 (hout (x+t • z x) hy)
+    exact hy (by change x+t • z x∈_; rw [he]; exact hx)
+  · intro y hy
+    obtain ⟨x,hxy⟩ := hbi.2 y
+    refine ⟨x,?_,hxy⟩
+    by_contra hx
+    change x+t • z x=y at hxy
+    rw [hout x hx] at hxy
+    exact hx (hxy.symm ▸ hy)
+
+end PettyProjection.FixedPoint
+end
+
+noncomputable section
+open Set Metric Filter Topology
+open scoped RealInnerProductSpace
+namespace PettyProjection.FixedPoint
+
+variable {E : Type uE} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+
+lemma nearest_unique {K : Set E} {x p q : E}
+    (hp : ∀ z ∈ K, ⟪x-p,z-p⟫ ≤ 0) (hq : ∀ z ∈ K, ⟪x-q,z-q⟫ ≤ 0)
+    (hpK : p∈K) (hqK : q∈K) : p=q := by
+  have h1 := hp q hqK
+  have h2 := hq p hpK
+  have he : ⟪x-p,q-p⟫+⟪x-q,p-q⟫ = ‖p-q‖^2 := by
+    rw [← real_inner_self_eq_norm_sq]
+    simp only [inner_sub_left, inner_sub_right]
+    ring
+  have hn : ‖p-q‖^2 ≤ 0 := by linarith
+  have hz : ‖p-q‖=0 := by nlinarith [norm_nonneg (p-q)]
+  exact sub_eq_zero.mp (norm_eq_zero.mp hz)
+
+/-- The genuine nearest point; no fixed-point principle is used here. -/
+def nearest (K : Set E) (hne : K.Nonempty) (hcomplete : IsComplete K)
+    (hc : Convex ℝ K) (x : E) : E :=
+  Classical.choose (exists_norm_eq_iInf_of_complete_convex hne hcomplete hc x)
+
+lemma nearest_mem (K : Set E) (hne : K.Nonempty) (hcomplete : IsComplete K)
+    (hc : Convex ℝ K) (x : E) : nearest K hne hcomplete hc x∈K :=
+  (Classical.choose_spec (exists_norm_eq_iInf_of_complete_convex hne hcomplete hc x)).1
+
+lemma nearest_inner (K : Set E) (hne : K.Nonempty) (hcomplete : IsComplete K)
+    (hc : Convex ℝ K) (x : E) :
+    ∀ z∈K, ⟪x-nearest K hne hcomplete hc x,z-nearest K hne hcomplete hc x⟫ ≤ 0 :=
+  (norm_eq_iInf_iff_real_inner_le_zero hc (nearest_mem K hne hcomplete hc x)).mp
+    (Classical.choose_spec (exists_norm_eq_iInf_of_complete_convex hne hcomplete hc x)).2
+
+lemma nearest_eq_of_mem (K : Set E) (hne : K.Nonempty) (hcomplete : IsComplete K)
+    (hc : Convex ℝ K) {x : E} (hx : x∈K) : nearest K hne hcomplete hc x=x := by
+  apply nearest_unique (nearest_inner K hne hcomplete hc x) _
+    (nearest_mem K hne hcomplete hc x) hx
+  intro z hz
+  simp only [sub_self,inner_zero_left,le_refl]
+
+lemma nearest_eq_of_inner (K : Set E) (hne : K.Nonempty) (hcomplete : IsComplete K)
+    (hc : Convex ℝ K) {x p : E} (hp : p∈K)
+    (hi : ∀ z∈K,⟪x-p,z-p⟫≤0) : nearest K hne hcomplete hc x=p :=
+  nearest_unique (nearest_inner K hne hcomplete hc x) hi
+    (nearest_mem K hne hcomplete hc x) hp
+
+lemma nearest_lipschitz (K : Set E) (hne : K.Nonempty) (hcomplete : IsComplete K)
+    (hc : Convex ℝ K) : LipschitzWith 1 (nearest K hne hcomplete hc) := by
+  apply LipschitzWith.of_dist_le_mul
+  intro x y
+  let p := nearest K hne hcomplete hc x
+  let q := nearest K hne hcomplete hc y
+  have h1 := nearest_inner K hne hcomplete hc x q (nearest_mem K hne hcomplete hc y)
+  have h2 := nearest_inner K hne hcomplete hc y p (nearest_mem K hne hcomplete hc x)
+  change ⟪x-p,q-p⟫≤0 at h1
+  change ⟪y-q,p-q⟫≤0 at h2
+  have he : ⟪x-p,q-p⟫+⟪y-q,p-q⟫ = ‖p-q‖^2-⟪x-y,p-q⟫ := by
+    rw [← real_inner_self_eq_norm_sq]
+    simp only [inner_sub_left,inner_sub_right]
+    ring
+  have hle : ‖p-q‖^2 ≤ ‖x-y‖*‖p-q‖ := by
+    have hcs := real_inner_le_norm (x-y) (p-q)
+    linarith
+  change dist p q ≤ (1:ℝ)*dist x y
+  simp only [one_mul,dist_eq_norm]
+  by_cases hz : ‖p-q‖=0
+  · rw [hz]; exact norm_nonneg _
+  · nlinarith [lt_of_le_of_ne (norm_nonneg (p-q)) (Ne.symm hz)]
+
+variable [CompleteSpace E]
+
+def ballNearest (x : E) : E := nearest (closedBall (0:E) 1)
+  (nonempty_closedBall.mpr (by norm_num)) isClosed_closedBall.isComplete (convex_closedBall _ _) x
+
+lemma ballNearest_mem (x : E) : ballNearest x∈closedBall (0:E) 1 :=
+  nearest_mem _ _ _ _ x
+
+lemma ballNearest_eq {x : E} (hx : ‖x‖≤1) : ballNearest x=x :=
+  nearest_eq_of_mem _ _ _ _ (by simpa only [mem_closedBall,dist_zero_right] using hx)
+
+lemma ballNearest_lipschitz : LipschitzWith 1 (ballNearest : E → E) :=
+  nearest_lipschitz (closedBall (0:E) 1)
+    (nonempty_closedBall.mpr (by norm_num)) isClosed_closedBall.isComplete (convex_closedBall _ _)
+
+lemma ballNearest_norm {x : E} (hx : 1≤‖x‖) : ‖ballNearest x‖=1 := by
+  have hx0 : ‖x‖≠0 := by linarith
+  let p : E := ‖x‖⁻¹ • x
+  have hp : ‖p‖=1 := by simp [p,norm_smul,hx0]
+  have hxp : x=‖x‖ • p := by simp [p,hx0]
+  have he : ballNearest x=p := by
+    apply nearest_eq_of_inner (closedBall (0:E) 1) _ _ _
+    · simpa only [mem_closedBall,dist_zero_right,hp] using le_rfl (a := (1:ℝ))
+    · intro z hz
+      have hsub : x-p=(‖x‖-1) • p := by rw [sub_smul,one_smul,← hxp]
+      rw [hsub,inner_smul_left,inner_sub_right,real_inner_self_eq_norm_sq,hp]
+      have hcs := real_inner_le_norm p z
+      have hzn : ‖z‖≤1 := by simpa only [mem_closedBall,dist_zero_right] using hz
+      have hi : ⟪p,z⟫-1≤0 := by rw [hp,one_mul] at hcs; linarith
+      simpa using mul_nonpos_of_nonneg_of_nonpos (sub_nonneg.mpr hx) hi
+  rw [he,hp]
+
+end PettyProjection.FixedPoint
+end
+
+noncomputable section
+open Set Metric Filter Topology MeasureTheory
+open scoped RealInnerProductSpace NNReal
+namespace PettyProjection.FixedPoint
+variable {E : Type uE} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E]
+
+omit [FiniteDimensional ℝ E] in
+lemma ball_uniqueDiff : UniqueDiffOn ℝ (closedBall (0:E) 1) := by
+  apply uniqueDiffOn_convex (convex_closedBall _ _)
+  rw [interior_closedBall (0:E) (by norm_num : (1:ℝ)≠0)]
+  exact nonempty_ball.mpr (by norm_num)
+
+lemma sphere_derivative_det_zero {r : E → E} {r' : E → E →L[ℝ] E}
+    (hd : ∀ x∈closedBall (0:E) 1, HasFDerivWithinAt r (r' x) (closedBall (0:E) 1) x)
+    (hnorm : ∀ x∈closedBall (0:E) 1, ‖r x‖=1) {x : E} (hx : x∈closedBall (0:E) 1) :
+    (r' x).det=0 := by
+  have hi := (hd x hx).inner ℝ (hd x hx)
+  have hc : HasFDerivWithinAt (fun y => ⟪r y,r y⟫) (0 : E →L[ℝ] ℝ)
+      (closedBall (0:E) 1) x := by
+    apply (hasFDerivWithinAt_const (1:ℝ) x (closedBall (0:E) 1)).congr
+    · intro y hy
+      change ⟪r y,r y⟫=1
+      rw [real_inner_self_eq_norm_sq,hnorm y hy,one_pow]
+    · rw [real_inner_self_eq_norm_sq,hnorm x hx,one_pow]
+  have he := (ball_uniqueDiff x hx).eq hi hc
+  have hort (v : E) : ⟪r x,r' x v⟫=0 := by
+    have h := congrArg (fun L : E →L[ℝ] ℝ => L v) he
+    simp only [ContinuousLinearMap.comp_apply,ContinuousLinearMap.prod_apply,
+      fderivInnerCLM_apply,zero_apply] at h
+    have hsymm := real_inner_comm (r x) (r' x v)
+    linarith
+  change ((r' x) : E →ₗ[ℝ] E).det=0
+  apply LinearMap.det_eq_zero_iff_ker_ne_bot.mpr
+  intro hker
+  have hsurj := (LinearMap.injective_iff_surjective).mp (LinearMap.ker_eq_bot.mp hker)
+  obtain ⟨v,hv⟩ := hsurj (r x)
+  change r' x v=r x at hv
+  have hz := hort v
+  rw [hv,real_inner_self_eq_norm_sq,hnorm x hx] at hz
+  norm_num at hz
+
+lemma retraction_extension {r : E → E} {r' : E → E →L[ℝ] E}
+    (hd : ∀ x∈closedBall (0:E) 1, HasFDerivWithinAt r (r' x) (closedBall (0:E) 1) x)
+    (hc : ContinuousOn r' (closedBall (0:E) 1))
+    (hb : ∀ x : E, ‖x‖=1 → r x=x) :
+    ∃ (z : E → E) (C : ℝ≥0), LipschitzWith C z ∧
+      (∀ x∈closedBall (0:E) 1, z x=r x-x) ∧ (∀ x,1≤‖x‖ → z x=0) := by
+  have hcN := hc.nnnorm
+  obtain ⟨C,hC⟩ := (isCompact_closedBall (0:E) 1).bddAbove_image hcN
+  have hr : LipschitzOnWith C r (closedBall (0:E) 1) :=
+    (convex_closedBall _ _).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le hd
+      (fun x hx => hC (mem_image_of_mem _ hx))
+  let z : E → E := fun x => r (ballNearest x)-ballNearest x
+  refine ⟨z,C+1,?_,?_,?_⟩
+  · apply LipschitzWith.of_dist_le_mul
+    intro x y
+    calc
+      dist (z x) (z y) ≤ dist (r (ballNearest x)) (r (ballNearest y)) +
+          dist (ballNearest x) (ballNearest y) := dist_sub_sub_le _ _ _ _
+      _ ≤ C * dist (ballNearest x) (ballNearest y) + dist (ballNearest x) (ballNearest y) :=
+        add_le_add (hr.dist_le_mul _ (ballNearest_mem x) _ (ballNearest_mem y)) le_rfl
+      _ = (C+1:ℝ) * dist (ballNearest x) (ballNearest y) := by ring
+      _ ≤ (C+1:ℝ) * dist x y := by
+        gcongr
+        simpa only [NNReal.coe_one,one_mul] using ballNearest_lipschitz.dist_le_mul x y
+      _ = _ := by simp only [NNReal.coe_add,NNReal.coe_one]
+  · intro x hx
+    have he : ballNearest x=x := ballNearest_eq (by simpa only [mem_closedBall,dist_zero_right] using hx)
+    simp only [z,he]
+  · intro x hx
+    simp only [z,hb _ (ballNearest_norm hx),sub_self]
+
+end PettyProjection.FixedPoint
+end
+
+noncomputable section
+open Set Metric Filter Topology MeasureTheory
+open scoped RealInnerProductSpace NNReal
+namespace PettyProjection.FixedPoint
+variable {E : Type uE} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
+  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
+
+omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] in
+lemma eventually_pos_det_homotopy {A : E → E →L[ℝ] E} {K : Set E}
+    (hK : IsCompact K) (hA : ContinuousOn A K) :
+    ∀ᶠ t : ℝ in 𝓝 0, ∀ x∈K, 0<(t • A x+ContinuousLinearMap.id ℝ E).det := by
+  let : CompactSpace K := isCompact_iff_compactSpace.mp hK
+  have hAc : Continuous (fun x : K => A x) := continuousOn_iff_continuous_domRestrict.mp hA
+  have hL : Continuous (fun z : ℝ × K => z.1 • A z.2+ContinuousLinearMap.id ℝ E) := by
+    exact continuous_add.comp ((continuous_smul.comp
+      (continuous_fst.prodMk (hAc.comp continuous_snd))).prodMk continuous_const)
+  have hD := ContinuousLinearMap.continuous_det.comp hL
+  have he : ∀ᶠ t : ℝ in 𝓝 0, ∀ x:K, 0<(t • A x+ContinuousLinearMap.id ℝ E).det := by
+    have hh := (isCompact_univ : IsCompact (univ : Set K)).eventually_forall_of_forall_eventually
+      (x₀ := (0:ℝ)) (P := fun t x => 0<(t • A x+ContinuousLinearMap.id ℝ E).det)
+    apply Filter.Eventually.mono (hh ?_)
+    · intro t ht x
+      exact ht x (mem_univ x)
+    · intro x hx
+      apply (hD.continuousAt (x := (0,x))).eventually
+      apply Ioi_mem_nhds
+      simp [ContinuousLinearMap.det]
+  filter_upwards [he] with t ht x hx
+  exact ht ⟨x,hx⟩
+
+/-- Milnor's elementary determinant-polynomial obstruction to a differentiable
+retraction of a ball onto its sphere. Only Jacobian change of variables and the
+Banach contraction principle are needed. -/
+lemma no_smooth_retraction (μ : Measure E) [μ.IsAddHaarMeasure]
+    {r : E → E} {r' : E → E →L[ℝ] E}
+    (hd : ∀ x∈closedBall (0:E) 1, HasFDerivWithinAt r (r' x) (closedBall (0:E) 1) x)
+    (hc : ContinuousOn r' (closedBall (0:E) 1))
+    (hnorm : ∀ x∈closedBall (0:E) 1, ‖r x‖=1)
+    (hb : ∀ x : E, ‖x‖=1 → r x=x) : False := by
+  let K := closedBall (0:E) 1
+  let I : E →L[ℝ] E := ContinuousLinearMap.id ℝ E
+  obtain ⟨z,C,hz,hzK,hzout⟩ := retraction_extension hd hc hb
+  have hzD (x : E) (hx : x∈K) : HasFDerivWithinAt z (r' x-I) K x := by
+    apply ((hd x hx).sub (hasFDerivWithinAt_id x K)).congr
+    · intro y hy
+      exact hzK y hy
+    · exact hzK x hx
+  have hA : ContinuousOn (fun x => r' x-I) K := hc.sub continuousOn_const
+  obtain ⟨p,hp⟩ := exists_integralOn_clm_det_polynomial μ (isCompact_closedBall (0:E) 1)
+    (fun x => r' x-I) (fun _ => I) hA continuousOn_const
+  have hsmall : ∀ᶠ t : ℝ in 𝓝 0, ‖t‖₊*C<1 := by
+    have hh : Continuous (fun t : ℝ => ‖t‖₊*C) := by fun_prop
+    exact (hh.continuousAt (x := 0)).eventually (Iio_mem_nhds (by simp))
+  have hpos := eventually_pos_det_homotopy (isCompact_closedBall (0:E) 1) hA
+  have he : ∀ᶠ t : ℝ in 𝓝 0, p.eval t=μ.real K := by
+    filter_upwards [hsmall,hpos] with t ht hpt
+    rw [hp]
+    have hFD (x : E) (hx : x∈K) :
+        HasFDerivWithinAt (fun x => x+t • z x) (t • (r' x-I)+I) K x := by
+      convert ((hzD x hx).const_smul t).add (hasFDerivWithinAt_id x K) using 1
+      ext y
+      simp only [Pi.add_apply,Pi.smul_apply,id_eq,add_comm]
+    have hJac := integral_image_eq_integral_abs_det_fderiv_smul μ isClosed_closedBall.measurableSet
+      hFD (bijective_add_small hz ht).1.injOn (fun _ : E => (1:ℝ))
+    rw [image_closedBall_add_small hz hzout ht] at hJac
+    have heq : (∫ x in K, |(t • (r' x-I)+I).det| • (1:ℝ) ∂μ) =
+        ∫ x in K, (t • (r' x-I)+I).det ∂μ := by
+      apply setIntegral_congr_fun isClosed_closedBall.measurableSet
+      intro x hx
+      change |(t • (r' x-I)+I).det| • (1:ℝ)=(t • (r' x-I)+I).det
+      rw [abs_of_pos (hpt x hx)]
+      simp only [smul_eq_mul,mul_one,I]
+    rw [heq] at hJac
+    simpa only [integral_const,smul_eq_mul,mul_one,Measure.real,Measure.restrict_apply_univ,K] using hJac.symm
+  have hpoly : p=Polynomial.C (μ.real K) := by
+    apply Polynomial.eq_of_infinite_eval_eq
+    apply infinite_of_mem_nhds (0:ℝ)
+    change {t | p.eval t=μ.real K}∈𝓝 (0:ℝ) at he
+    simpa only [Polynomial.eval_C] using he
+  have hzero : p.eval 1=0 := by
+    rw [hp]
+    calc
+      (∫ x in closedBall (0:E) 1, ((1:ℝ) • (r' x-I)+I).det ∂μ) =
+          ∫ _ in closedBall (0:E) 1, (0:ℝ) ∂μ := by
+        apply setIntegral_congr_fun isClosed_closedBall.measurableSet
+        intro x hx
+        change ((1:ℝ) • (r' x-I)+I).det=0
+        simp only [one_smul,sub_add_cancel]
+        exact sphere_derivative_det_zero hd hnorm hx
+      _ = 0 := integral_zero _ _
+  rw [hpoly,Polynomial.eval_C] at hzero
+  have hpositive : 0<μ.real K := by
+    exact ENNReal.toReal_pos (ne_of_gt (measure_closedBall_pos μ (0:E) zero_lt_one))
+      (measure_closedBall_lt_top.ne)
+  linarith
+
+end PettyProjection.FixedPoint
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/ProjectionBodies/GaussianMeasures.lean b/lean/OAI/Geometry/ProjectionBodies/GaussianMeasures.lean
new file mode 100644
index 0000000000000000000000000000000000000000..30d5c001b6745f15db79707f3b2fcf9cd9e42340
--- /dev/null
+++ b/lean/OAI/Geometry/ProjectionBodies/GaussianMeasures.lean
@@ -0,0 +1,260 @@
+import OAI.Geometry.ProjectionBodies.Brouwer
+
+namespace OAI
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology Function
+open scoped ENNReal NNReal RealInnerProductSpace
+namespace PettyProjection
+open Spherical (Sphere norm_coe surface sigma mean)
+
+/-- The actual one-dimensional radial Gaussian moment. -/
+def gaussianRadial (d : ℕ) : ℝ := ∫ r in Ioi (0:ℝ),r^d*Real.exp (-(r^2))
+
+lemma gaussianRadial_eq (d : ℕ) : gaussianRadial d=(1/2:ℝ)*Real.Gamma (((d:ℝ)+1)/2) := by
+  simpa only [gaussianRadial,Real.rpow_natCast,Real.rpow_two] using
+    integral_rpow_mul_exp_neg_rpow (p := (2:ℝ)) (q := (d:ℝ)) (by norm_num) (by linarith [Nat.cast_nonneg (α := ℝ) d])
+
+lemma gaussianRadial_pos (d : ℕ) : 0<gaussianRadial d := by
+  rw [gaussianRadial_eq]
+  exact mul_pos (by norm_num) (Real.Gamma_pos_of_pos (by positivity))
+
+lemma gaussianRadial_add_two (d : ℕ) :
+    gaussianRadial (d+2)=((d:ℝ)+1)/2*gaussianRadial d := by
+  rw [gaussianRadial_eq,gaussianRadial_eq]
+  have he : (((d+2:ℕ):ℝ)+1)/2=((d:ℝ)+1)/2+1 := by push_cast;ring
+  rw [he,Real.Gamma_add_one (by positivity)]
+  ring
+
+lemma integral_radial_gaussian (m k : ℕ) :
+    (∫ r : Ioi (0:ℝ),r.val^k*Real.exp (-(r.val^2)) ∂Measure.volumeIoiPow m)=gaussianRadial (m+k) := by
+  simp only [Measure.volumeIoiPow,ENNReal.ofReal]
+  rw [integral_withDensity_eq_integral_smul]
+  · rw [integral_subtype_comap measurableSet_Ioi (fun r : ℝ => (r^m).toNNReal • (r^k*Real.exp (-(r^2))))]
+    unfold gaussianRadial
+    apply setIntegral_congr_fun measurableSet_Ioi
+    intro x hx
+    dsimp only
+    rw [NNReal.smul_def,Real.coe_toNNReal _ (pow_nonneg hx.le _),pow_add]
+    ring
+  · exact (measurable_subtype_coe.pow_const _).real_toNNReal
+
+lemma surface_integral_eq_mean {n : ℕ} [NeZero n] (f : Sphere n → ℝ) :
+    (∫ u,f u ∂surface n)=(n:ℝ)*kappa n*mean f := by
+  rw [mean,sigma,integral_smul_measure,ENNReal.toReal_inv,smul_eq_mul]
+  change _=(n:ℝ)*kappa n*(((surface n).real univ)⁻¹* _)
+  rw [surface_real_univ]
+  field_simp [(kappa_pos n).ne',(Nat.cast_pos.mpr (NeZero.pos n) : (0:ℝ)<n).ne']
+
+/-- Polar separation for any positively homogeneous integrand. No smoothness
+or differentiability is used, and the Bochner integral has its ordinary meaning. -/
+lemma homogeneous_gaussian_integral {n : ℕ} [NeZero n] (k : ℕ) (f : Space n → ℝ)
+    (hh : ∀ r : ℝ,0<r → ∀ x,f (r • x)=r^k*f x) :
+    (∫ x,f x*Real.exp (-(‖x‖^2)))=
+      (n:ℝ)*kappa n*mean (fun u : Sphere n => f u)*gaussianRadial (n-1+k) := by
+  let F : Space n → ℝ := fun x => f x*Real.exp (-(‖x‖^2))
+  let ν := Measure.volumeIoiPow (Module.finrank ℝ (Space n)-1)
+  have he : (∫ x,F x)=∫ x : ({0}ᶜ : Set (Space n)),F x ∂volume.comap Subtype.val := by
+    rw [integral_subtype_comap (measurableSet_singleton _).compl,restrict_compl_singleton]
+  rw [show (∫ x,f x*Real.exp (-(‖x‖^2)))=∫ x,F x from rfl,he]
+  have hm := (volume : Measure (Space n)).measurePreserving_homeomorphUnitSphereProd
+  rw [← (hm.symm (homeomorphUnitSphereProd (Space n)).toMeasurableEquiv).integral_comp
+    (homeomorphUnitSphereProd (Space n)).symm.measurableEmbedding (fun x => F x)]
+  have hfun : (fun z : Sphere n × Ioi (0:ℝ) => F ↑((homeomorphUnitSphereProd (Space n)).symm z))=
+      (fun z => f z.1*(z.2.val^k*Real.exp (-(z.2.val^2)))) := by
+    funext z
+    rw [homeomorphUnitSphereProd_symm_apply_coe]
+    dsimp only [F]
+    rw [hh _ z.2.property,norm_smul,Real.norm_of_nonneg z.2.property.le,norm_coe,mul_one]
+    ring
+  change (∫ z : Sphere n × Ioi (0:ℝ),F ↑((homeomorphUnitSphereProd (Space n)).symm z) ∂(surface n).prod ν)=_
+  rw [hfun]
+  rw [integral_prod_mul (fun u : Sphere n => f u) (fun r : Ioi (0:ℝ) => r.val^k*Real.exp (-(r.val^2))),
+    surface_integral_eq_mean]
+  dsimp only [ν]
+  rw [integral_radial_gaussian]
+  simp only [finrank_euclideanSpace_fin]
+
+end PettyProjection
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology
+open scoped NNReal RealInnerProductSpace
+namespace PettyProjection
+variable {n : ℕ}
+
+lemma norm_integrable_map {μ : Measure (Space n)}
+    (hμ : Integrable (fun x : Space n => ‖x‖) μ) (A : Space n →L[ℝ] Space n) :
+    Integrable (fun x : Space n => ‖x‖) (μ.map A) := by
+  apply (integrable_map_measure continuous_norm.aestronglyMeasurable A.continuous.measurable.aemeasurable).mpr
+  exact (hμ.const_mul ‖A‖).mono' (continuous_norm.comp A.continuous).aestronglyMeasurable
+    (Eventually.of_forall (fun x => by simpa only [Function.comp_apply,Real.norm_eq_abs,abs_norm] using A.le_opNorm x))
+
+lemma cosine_map (μ : Measure (Space n)) (hμ : Integrable (fun x : Space n => ‖x‖) μ)
+    (A : Space n →L[ℝ] Space n) (x : Space n) :
+    cosineSeminorm (μ.map A) (norm_integrable_map hμ A) x=cosineSeminorm μ hμ (A.adjoint x) := by
+  change (∫ y,|⟪y,x⟫| ∂μ.map A)=∫ y,|⟪y,A.adjoint x⟫| ∂μ
+  rw [integral_map A.continuous.measurable.aemeasurable (by fun_prop)]
+  congr 1
+  funext y
+  rw [ContinuousLinearMap.adjoint_inner_right]
+
+lemma adjoint_ne_zero_on_range (A : Space n →L[ℝ] Space n) {x : Space n}
+    (hx : x∈LinearMap.range A.toLinearMap) (hx0 : x≠0) : A.adjoint x≠0 := by
+  intro h
+  obtain ⟨y,rfl⟩ := hx
+  change A.adjoint (A y)=0 at h
+  change A y≠0 at hx0
+  have hh := ContinuousLinearMap.adjoint_inner_right A y (A y)
+  rw [h,inner_zero_right,real_inner_self_eq_norm_sq] at hh
+  exact hx0 (norm_eq_zero.mp (by nlinarith [norm_nonneg (A y)]))
+
+lemma cosine_map_pos {μ : Measure (Space n)}
+    (hμ : Integrable (fun x : Space n => ‖x‖) μ)
+    (hq : ∀ x : Space n,x≠0 → 0<cosineSeminorm μ hμ x)
+    (A : Space n →L[ℝ] Space n) (x : Space n)
+    (hx : x∈LinearMap.range A.toLinearMap) (hx0 : x≠0) :
+    0<cosineSeminorm (μ.map A) (norm_integrable_map hμ A) x := by
+  rw [cosine_map]
+  exact hq _ (adjoint_ne_zero_on_range A hx hx0)
+
+lemma functional_map {μ : Measure (Space n)} (A : Space n →L[ℝ] Space n)
+    {K : Set (Space n)} (hK : IsCompact K) :
+    supportFunctional (μ.map A) K=∫ x,support K (A x) ∂μ := by
+  exact integral_map A.continuous.measurable.aemeasurable (support_continuous hK).aestronglyMeasurable
+
+lemma cosine_map_projection {μ : Measure (Space n)}
+    (hμ : Integrable (fun x : Space n => ‖x‖) μ) (A : Space n →L[ℝ] Space n) (x : Space n) :
+    cosineSeminorm (μ.map A) (norm_integrable_map hμ A)
+      ((LinearMap.range A.toLinearMap).starProjection x)=
+        cosineSeminorm (μ.map A) (norm_integrable_map hμ A) x := by
+  change (∫ y,|⟪y,(LinearMap.range A.toLinearMap).starProjection x⟫| ∂μ.map A)=∫ y,|⟪y,x⟫| ∂μ.map A
+  rw [integral_map A.continuous.measurable.aemeasurable (by fun_prop),
+    integral_map A.continuous.measurable.aemeasurable (by fun_prop)]
+  congr 1
+  funext y
+  rw [← Submodule.inner_starProjection_left_eq_right,
+    (LinearMap.range A.toLinearMap).starProjection_eq_self_iff.mpr
+      (show A y∈LinearMap.range A.toLinearMap from ⟨y,rfl⟩)]
+
+lemma functional_map_continuous {μ : Measure (Space n)}
+    (hμ : Integrable (fun x : Space n => ‖x‖) μ) {K : Set (Space n)}
+    (hK : IsCompact K) (hne : K.Nonempty) :
+    Continuous (fun A : Space n →L[ℝ] Space n => supportFunctional (μ.map A) K) := by
+  simp_rw [functional_map _ hK]
+  obtain ⟨R,hR,hbound⟩ := hK.isBounded.exists_pos_norm_le
+  have hlip := support_lipschitz hK hne (R := ⟨R,hR.le⟩) hbound
+  apply continuous_iff_continuousAt.mpr
+  intro A
+  apply continuousAt_of_dominated (bound := fun x => R*(‖A‖+1)*‖x‖) <;> try infer_instance
+  · filter_upwards [] with B
+    exact (support_continuous hK).aestronglyMeasurable.comp_measurable B.continuous.measurable
+  · filter_upwards [continuous_norm.continuousAt.eventually (gt_mem_nhds (show ‖A‖<‖A‖+1 by linarith))] with B hB
+    filter_upwards [] with x
+    have hz : support K 0=0 := by
+      obtain ⟨z,hz,he⟩ := support_attained hK hne 0
+      simpa only [inner_zero_left] using he
+    have hh := hlip.norm_sub_le (B x) 0
+    rw [hz,sub_zero,sub_zero] at hh
+    exact hh.trans ((mul_le_mul_of_nonneg_left (B.le_opNorm x) hR.le).trans
+      (by
+        calc
+          R*(‖B‖*‖x‖) ≤ R*((‖A‖+1)*‖x‖) := mul_le_mul_of_nonneg_left
+            (mul_le_mul_of_nonneg_right hB.le (norm_nonneg _)) hR.le
+          _ = R*(‖A‖+1)*‖x‖ := by ring))
+  · exact hμ.const_mul (R*(‖A‖+1))
+  · filter_upwards [] with x
+    exact ((support_continuous hK).comp (by fun_prop : Continuous (fun B : Space n →L[ℝ] Space n => B x))).continuousAt
+
+end PettyProjection
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology
+open scoped NNReal RealInnerProductSpace Pointwise
+namespace PettyProjection
+open Spherical (seminorm_continuous)
+variable {n : ℕ}
+
+def projectedUnit (V : Submodule ℝ (Space n)) (g : Seminorm ℝ (Space n)) : Set (Space n) :=
+  closure (V.starProjection '' seminormUnit g)
+
+lemma projectedUnit_sub (V : Submodule ℝ (Space n)) (g : Seminorm ℝ (Space n)) :
+    projectedUnit V g⊆V := by
+  apply closure_minimal _ V.closed_of_finiteDimensional
+  rintro _ ⟨x,_,rfl⟩
+  exact V.starProjection_apply_mem x
+
+lemma projectedUnit_convex (V : Submodule ℝ (Space n)) (g : Seminorm ℝ (Space n)) :
+    Convex ℝ (projectedUnit V g) := (seminormUnit_convex g).linear_image V.starProjection.toLinearMap |>.closure
+
+lemma projectedUnit_balanced (V : Submodule ℝ (Space n)) (g : Seminorm ℝ (Space n)) :
+    Balanced ℝ (projectedUnit V g) := by
+  apply Balanced.closure
+  intro a ha y hy
+  obtain ⟨z,⟨w,hw,rfl⟩,rfl⟩ := hy
+  refine ⟨a • w,(seminormUnit_balanced g) a ha ⟨w,hw,rfl⟩,?_⟩
+  exact map_smul V.starProjection a w
+
+lemma projectedUnit_inner (V : Submodule ℝ (Space n)) (g : Seminorm ℝ (Space n)) :
+    ∃ r : ℝ,0<r ∧ ∀ x∈V,‖x‖≤r → x∈projectedUnit V g := by
+  obtain ⟨r,hr,hin⟩ := Metric.mem_nhds_iff.mp (seminormUnit_nhds g)
+  refine ⟨r/2,by positivity,fun x hx hn => subset_closure ⟨x,hin ?_,?_⟩⟩
+  · rw [mem_ball,dist_zero_right]; linarith
+  · exact V.starProjection_eq_self_iff.mpr hx
+
+lemma projectedUnit_compact (V : Submodule ℝ (Space n)) (g : Seminorm ℝ (Space n))
+    {c : ℝ} (hc : 0<c) (hl : ∀ x,c*‖V.starProjection x‖≤g x) :
+    IsCompact (projectedUnit V g) := by
+  apply (isCompact_closedBall (0:Space n) c⁻¹).of_isClosed_subset isClosed_closure
+  apply closure_minimal _ isClosed_closedBall
+  rintro _ ⟨x,hx,rfl⟩
+  rw [mem_closedBall,dist_zero_right,inv_eq_one_div,le_div_iff₀ hc]
+  change g x≤1 at hx
+  nlinarith [hl x]
+
+namespace RelativeGauge
+variable {V : Submodule ℝ (Space n)}
+
+def project (V : Submodule ℝ (Space n)) (g : Seminorm ℝ (Space n))
+    {c : ℝ} (hc : 0<c) (hl : ∀ x,c*‖V.starProjection x‖≤g x) : RelativeGauge V :=
+  ofBody (projectedUnit_compact V g hc hl) (projectedUnit_convex V g)
+    (projectedUnit_balanced V g) (projectedUnit_inner V g)
+
+lemma project_body (V : Submodule ℝ (Space n)) (g : Seminorm ℝ (Space n))
+    {c : ℝ} (hc : 0<c) (hl : ∀ x,c*‖V.starProjection x‖≤g x) :
+    (project V g hc hl).body=projectedUnit V g :=
+  body_ofBody _ _ _ (projectedUnit_sub V g) _
+
+lemma project_le (V : Submodule ℝ (Space n)) (g : Seminorm ℝ (Space n))
+    {c : ℝ} (hc : 0<c) (hl : ∀ x,c*‖V.starProjection x‖≤g x) (x : Space n) :
+    project V g hc hl x≤g x := by
+  change gauge (V.starProjection ⁻¹' projectedUnit V g) x≤g x
+  rw [← gauge_seminormUnit g x]
+  apply gauge_mono (absorbent_nhds_zero (seminormUnit_nhds g))
+  intro y hy
+  exact subset_closure ⟨y,hy,rfl⟩
+
+lemma support_cylinder_bound (g : RelativeGauge V) (u : Space n) (hu : u∈V) (z : Space n) :
+    ⟪u,z⟫ / max 1 (g z) ≤ support g.body u := by
+  have ha : 0 < max 1 (g z) := lt_of_lt_of_le zero_lt_one (le_max_left _ _)
+  let y := (max 1 (g z))⁻¹ • V.starProjection z
+  have hy : y∈g.body := by
+    refine ⟨V.smul_mem _ (V.starProjection_apply_mem z),?_⟩
+    change g.toSeminorm ((max 1 (g z))⁻¹ • V.starProjection z)≤1
+    rw [map_smul_eq_mul,Real.norm_of_nonneg (inv_nonneg.mpr ha.le),inv_mul_le_iff₀ ha,mul_one]
+    change g (V.starProjection z)≤ max 1 (g z)
+    rw [g.projection]
+    exact le_max_right _ _
+  have hh := inner_le_support g.compact hy u
+  dsimp only [y] at hh
+  rw [inner_smul_right,← Submodule.inner_starProjection_left_eq_right,
+    V.starProjection_eq_self_iff.mpr hu] at hh
+  simpa only [div_eq_mul_inv,mul_comm] using hh
+
+end RelativeGauge
+end PettyProjection
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/ProjectionBodies/Homothety.lean b/lean/OAI/Geometry/ProjectionBodies/Homothety.lean
new file mode 100644
index 0000000000000000000000000000000000000000..ebd37aa0be8f5737edd50bc1237d6284a39e3c3d
--- /dev/null
+++ b/lean/OAI/Geometry/ProjectionBodies/Homothety.lean
@@ -0,0 +1,566 @@
+import OAI.Geometry.ProjectionBodies.BrunnMinkowski
+
+namespace OAI
+
+universe uE
+
+noncomputable section
+open Set MeasureTheory Filter Topology
+namespace PettyProjection
+
+lemma quantile_first_moment {l r : ℝ} {f Q : ℝ → ℝ}
+    (hQc : Continuous Q) (hQ0 : Q 0=l) (hQ1 : Q 1=r)
+    (hQd : ∀ s ∈ Ioo (0:ℝ) 1, HasDerivAt Q (f (Q s))⁻¹ s)
+    (hfp : ∀ s ∈ Ioo (0:ℝ) 1, 0 < f (Q s)) :
+    (∫ s in (0:ℝ)..1, Q s)=(∫ x in l..r, x*f x) := by
+  have hdu : ∀ s ∈ Ioo (min (0:ℝ) 1) (max (0:ℝ) 1), HasDerivAt Q (f (Q s))⁻¹ s := by
+    simpa only [min_eq_left zero_le_one,max_eq_right zero_le_one] using hQd
+  have hpu : ∀ s ∈ Ioo (min (0:ℝ) 1) (max (0:ℝ) 1), 0 ≤ (f (Q s))⁻¹ := by
+    simpa only [min_eq_left zero_le_one,max_eq_right zero_le_one] using
+      fun s hs => (inv_pos.mpr (hfp s hs)).le
+  have he := intervalIntegral.integral_comp_mul_deriv_of_deriv_nonneg
+    (g := fun x => x*f x) hQc.continuousOn hdu hpu
+  rw [hQ0,hQ1] at he
+  refine (intervalIntegral.integral_congr_ae_restrict ?_).trans he
+  rw [uIoc_of_le (by norm_num : (0:ℝ) ≤ 1),← Measure.restrict_congr_set Ioo_ae_eq_Ioc]
+  filter_upwards [ae_restrict_mem measurableSet_Ioo] with s hs
+  simp [(hfp s hs).ne']
+
+/-- Equality in slice transport forces the two full marginal distributions to
+be translates. In particular both projection endpoints differ by the difference
+of their first moments. -/
+theorem slice_transport_equality {m : ℕ} (hm : 0 < m) {a b l r L R : ℝ}
+    (ha : 0 < a) (hb : 0 < b) (hab : a+b=1) (hlr : l<r) (hLR : L<R)
+    {f g h : ℝ → ℝ}
+    (hf : ContinuousOn f (Ioo l r)) (hg : ContinuousOn g (Ioo L R))
+    (hfi : IntervalIntegrable f volume l r) (hgi : IntervalIntegrable g volume L R)
+    (hfp : ∀ x ∈ Ioo l r, 0 < f x) (hgp : ∀ x ∈ Ioo L R, 0 < g x)
+    (hfI : (∫ x in l..r, f x)=1) (hgI : (∫ x in L..R, g x)=1)
+    (hhi : IntervalIntegrable h volume (a*l+b*L) (a*r+b*R))
+    (hbound : ∀ x ∈ Ioo l r, ∀ y ∈ Ioo L R,
+      (a*(f x)^((m:ℝ)⁻¹)+b*(g y)^((m:ℝ)⁻¹))^m ≤ h (a*x+b*y))
+    (hI : (∫ z in (a*l+b*L)..(a*r+b*R), h z)=1) :
+    l-L=(∫ x in l..r, x*f x)-(∫ y in L..R, y*g y) ∧
+    r-R=(∫ x in l..r, x*f x)-(∫ y in L..R, y*g y) := by
+  obtain ⟨Q,hQc,hQ0,hQ1,hQmem,hQd⟩ := exists_interval_quantile hlr hf hfi hfp hfI
+  obtain ⟨P,hPc,hP0,hP1,hPmem,hPd⟩ := exists_interval_quantile hLR hg hgi hgp hgI
+  let z : ℝ → ℝ := fun s => a*Q s+b*P s
+  let z' : ℝ → ℝ := fun s => a/(f (Q s))+b/(g (P s))
+  have hzc : Continuous z := by dsimp [z]; fun_prop
+  have hzd : ∀ s ∈ Ioo (0:ℝ) 1, HasDerivAt z (z' s) s := by
+    intro s hs
+    convert ((hQd s hs).2.const_mul a).add ((hPd s hs).2.const_mul b) using 1
+    all_goals rfl
+  have hzp : ∀ s ∈ Ioo (0:ℝ) 1, 0<z' s := by
+    intro s hs
+    exact add_pos (div_pos ha (hfp _ (hQd s hs).1)) (div_pos hb (hgp _ (hPd s hs).1))
+  have hz0 : z 0=a*l+b*L := by dsimp [z]; rw [hQ0,hP0]
+  have hz1 : z 1=a*r+b*R := by dsimp [z]; rw [hQ1,hP1]
+  have hzz : z 0 ≤ z 1 := by rw [hz0,hz1]; nlinarith
+  have hhi : IntervalIntegrable h volume (z 0) (z 1) := by
+    rwa [hz0,hz1]
+  have hzdu : ∀ s ∈ Ioo (min (0:ℝ) 1) (max (0:ℝ) 1), HasDerivAt z (z' s) s := by
+    simpa only [min_eq_left zero_le_one,max_eq_right zero_le_one] using hzd
+  have hzpu : ∀ s ∈ Ioo (min (0:ℝ) 1) (max (0:ℝ) 1), 0≤z' s := by
+    simpa only [min_eq_left zero_le_one,max_eq_right zero_le_one] using fun s hs => (hzp s hs).le
+  have hint : IntervalIntegrable (fun s => h (z s)*z' s) volume 0 1 :=
+    (intervalIntegral.integrable_comp_mul_deriv_iff_of_deriv_nonneg hzc.continuousOn hzdu hzpu).mpr hhi
+  have hsub := intervalIntegral.integral_comp_mul_deriv_of_deriv_nonneg
+    (g := h) hzc.continuousOn hzdu hzpu
+  change (∫ s in (0:ℝ)..1, h (z s)*z' s) = _ at hsub
+  have hpt : ∀ s ∈ Ioo (0:ℝ) 1, 1 ≤ h (z s)*z' s := by
+    intro s hs
+    let p := (f (Q s))^((m:ℝ)⁻¹)
+    let q := (g (P s))^((m:ℝ)⁻¹)
+    have hp : 0<p := Real.rpow_pos_of_pos (hfp _ (hQd s hs).1) _
+    have hq : 0<q := Real.rpow_pos_of_pos (hgp _ (hPd s hs).1) _
+    have hpm : p^m=f (Q s) := Real.rpow_inv_natCast_pow (hfp _ (hQd s hs).1).le hm.ne'
+    have hqm : q^m=g (P s) := Real.rpow_inv_natCast_pow (hgp _ (hPd s hs).1).le hm.ne'
+    have ht := (transport_scalar hm ha hb hab hp hq).1
+    rw [hpm,hqm] at ht
+    exact ht.trans (mul_le_mul_of_nonneg_right
+      (hbound _ (hQd s hs).1 _ (hPd s hs).1) (hzp s hs).le)
+  have hae : (fun _ : ℝ => (1:ℝ)) ≤ᵐ[volume.restrict (Icc (0:ℝ) 1)] (fun s => h (z s)*z' s) := by
+    rw [← Measure.restrict_congr_set Ioo_ae_eq_Icc]
+    exact (ae_restrict_iff' measurableSet_Ioo).mpr (ae_of_all _ hpt)
+  have hInt : (∫ s in Icc (0:ℝ) 1, h (z s)*z' s)=1 := by
+    rw [integral_Icc_eq_integral_Ioc,← intervalIntegral.integral_of_le (by norm_num : (0:ℝ) ≤ 1),
+      hsub,hz0,hz1,hI]
+  have hOne : (∫ _ in Icc (0:ℝ) 1, (1:ℝ))=1 := by simp
+  have hAe : (fun _ : ℝ => (1:ℝ)) =ᵐ[volume.restrict (Icc (0:ℝ) 1)]
+      (fun s => h (z s)*z' s) :=
+    (integral_eq_iff_of_ae_le (integrableOn_const isCompact_Icc.measure_ne_top)
+      ((intervalIntegrable_iff_integrableOn_Icc_of_le (by norm_num : (0:ℝ) ≤ 1)).mp hint) hae).mp
+        (hOne.trans hInt.symm)
+  have hAe' : (fun s => f (Q s)) =ᵐ[volume.restrict (Ioo (0:ℝ) 1)] (fun s => g (P s)) := by
+    rw [← Measure.restrict_congr_set Ioo_ae_eq_Icc] at hAe
+    filter_upwards [hAe,ae_restrict_mem measurableSet_Ioo] with s hs hsmem
+    let p := (f (Q s))^((m:ℝ)⁻¹)
+    let q := (g (P s))^((m:ℝ)⁻¹)
+    have hp : 0 < p := Real.rpow_pos_of_pos (hfp _ (hQd s hsmem).1) _
+    have hq : 0 < q := Real.rpow_pos_of_pos (hgp _ (hPd s hsmem).1) _
+    have hpm : p^m=f (Q s) := Real.rpow_inv_natCast_pow (hfp _ (hQd s hsmem).1).le hm.ne'
+    have hqm : q^m=g (P s) := Real.rpow_inv_natCast_pow (hgp _ (hPd s hsmem).1).le hm.ne'
+    have ht := transport_scalar hm ha hb hab hp hq
+    have hle : (a*p+b*q)^m*(a/(p^m)+b/(q^m)) ≤ 1 := by
+      rw [hpm,hqm,hs]
+      exact mul_le_mul_of_nonneg_right (hbound _ (hQd s hsmem).1 _ (hPd s hsmem).1) (hzp s hsmem).le
+    have hpq : p=q := ht.2.mp (le_antisymm hle ht.1)
+    exact hpm.symm.trans ((congrArg (fun x : ℝ => x^m) hpq).trans hqm)
+  have hfg : EqOn (fun s => f (Q s)) (fun s => g (P s)) (Ioo (0:ℝ) 1) :=
+    Measure.eqOn_open_of_ae_eq hAe' isOpen_Ioo
+      (hf.comp hQc.continuousOn (fun s hs => (hQd s hs).1))
+      (hg.comp hPc.continuousOn (fun s hs => (hPd s hs).1))
+  obtain ⟨c,hc⟩ := IsOpen.exists_eq_add_of_deriv_eq isOpen_Ioo (convex_Ioo (0:ℝ) 1).isPreconnected
+    (fun s hs => (hQd s hs).2.differentiableAt.differentiableWithinAt)
+    (fun s hs => (hPd s hs).2.differentiableAt.differentiableWithinAt)
+    (fun s hs => by
+      rw [(hQd s hs).2.deriv,(hPd s hs).2.deriv]
+      exact congrArg Inv.inv (hfg hs))
+  have hcc : EqOn Q (fun s => P s+c) (Icc (0:ℝ) 1) := by
+    simpa only [closure_Ioo (by norm_num : (0:ℝ) ≠ 1)] using hc.closure hQc (hPc.add continuous_const)
+  have hQInt := quantile_first_moment hQc hQ0 hQ1 (fun s hs => (hQd s hs).2)
+    (fun s hs => hfp _ (hQd s hs).1)
+  have hPInt := quantile_first_moment hPc hP0 hP1 (fun s hs => (hPd s hs).2)
+    (fun s hs => hgp _ (hPd s hs).1)
+  have hcInt : (∫ s in (0:ℝ)..1, Q s)=(∫ s in (0:ℝ)..1, P s)+c := by
+    rw [intervalIntegral.integral_congr (fun s hs => hcc (by simpa only [uIcc_of_le zero_le_one] using hs)),
+      intervalIntegral.integral_add (hPc.intervalIntegrable _ _) intervalIntegrable_const,
+      intervalIntegral.integral_const]
+    simp
+  have hc0 := hcc (show (0:ℝ) ∈ Icc 0 1 by norm_num)
+  have hc1 := hcc (show (1:ℝ) ∈ Icc 0 1 by norm_num)
+  dsimp only at hc0 hc1
+  rw [hQ0,hP0] at hc0
+  rw [hQ1,hP1] at hc1
+  rw [hQInt,hPInt] at hcInt
+  constructor <;> linarith
+
+end PettyProjection
+end
+
+noncomputable section
+open Set MeasureTheory Filter Topology
+open scoped Pointwise
+namespace PettyProjection
+section
+variable {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E]
+    [FiniteDimensional ℝ E] [MeasureSpace E] [BorelSpace E]
+    [Measure.IsAddHaarMeasure (volume : Measure E)]
+
+omit [NormedSpace ℝ E] [FiniteDimensional ℝ E] [Measure.IsAddHaarMeasure (volume : Measure E)] in
+lemma sliceDensity_moment_indicator {K : Set (E × ℝ)} (hK : IsCompact K) (s : ℝ) :
+    (∫ x : E, K.indicator Prod.snd (x,s))=s*sliceDensity K s := by
+  have he : (fun x : E => K.indicator Prod.snd (x,s)) =
+      (horizontalSlice K s).indicator (fun _ => s) := by
+    funext x
+    by_cases hx : (x,s) ∈ K
+    · rw [Set.indicator_of_mem hx,Set.indicator_of_mem (show x ∈ horizontalSlice K s from hx)]
+    · rw [Set.indicator_of_notMem hx,Set.indicator_of_notMem (show x ∉ horizontalSlice K s from hx)]
+  rw [he,integral_indicator (horizontalSlice_compact hK s).measurableSet,setIntegral_const]
+  simp only [smul_eq_mul,sliceDensity,mul_comm]
+
+lemma sliceDensity_moment_integrable {K : Set (E × ℝ)} (hK : IsCompact K) :
+    Integrable (fun s => s*sliceDensity K s) := by
+  have hi : Integrable (K.indicator Prod.snd) (volume.prod volume) := by
+    rw [← Measure.volume_eq_prod]
+    exact (continuous_snd.continuousOn.integrableOn_compact hK).integrable_indicator hK.measurableSet
+  exact hi.integral_prod_right.congr (Eventually.of_forall fun s => sliceDensity_moment_indicator hK s)
+
+lemma sliceDensity_moment {K : Set (E × ℝ)} (hK : IsCompact K) :
+    (∫ s, s*sliceDensity K s)=(∫ x in K, x.2) := by
+  have hi : Integrable (K.indicator Prod.snd) (volume.prod volume) := by
+    rw [← Measure.volume_eq_prod]
+    exact (continuous_snd.continuousOn.integrableOn_compact hK).integrable_indicator hK.measurableSet
+  have he := integral_prod_symm (K.indicator Prod.snd) hi
+  simp_rw [sliceDensity_moment_indicator hK] at he
+  rw [← he,← Measure.volume_eq_prod,integral_indicator hK.measurableSet]
+
+lemma sliceDensity_interval_moment {K : Set (E × ℝ)} (hK : IsCompact K) {l r : ℝ}
+    (hlr : l≤r) (hp : Prod.snd '' K ⊆ Icc l r) :
+    (∫ s in l..r, s*sliceDensity K s)=(∫ x in K, x.2) := by
+  rw [intervalIntegral.integral_of_le hlr,← integral_Icc_eq_integral_Ioc]
+  rw [setIntegral_eq_integral_of_forall_compl_eq_zero]
+  · exact sliceDensity_moment hK
+  · intro s hs
+    rw [sliceDensity_zero_of_notMem (fun h => hs (hp h)),mul_zero]
+
+end
+end PettyProjection
+end
+
+noncomputable section
+open Set MeasureTheory Filter Topology
+open scoped Pointwise
+namespace PettyProjection
+section
+variable {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E]
+    [FiniteDimensional ℝ E] [MeasureSpace E] [BorelSpace E]
+    [Measure.IsAddHaarMeasure (volume : Measure E)]
+
+/-- Equality in the actual product-body step identifies both extreme heights
+with the difference of the (unit-volume) vertical first moments. -/
+theorem brunnMinkowski_product_unit_equality {m : ℕ} (hm0 : 0 < m) (hm : BrunnMinkowskiRoot E m)
+    {K M : Set (E × ℝ)} (hK : IsCompact K) (hcK : Convex ℝ K)
+    (hiK : (interior K).Nonempty) (hM : IsCompact M) (hcM : Convex ℝ M)
+    (hiM : (interior M).Nonempty) (hvK : volume.real K=1) (hvM : volume.real M=1)
+    {a b : ℝ} (ha : 0 < a) (hb : 0 < b) (hab : a+b=1)
+    (hvC : volume.real (a • K+b • M)=1)
+    {l r L R : ℝ} (hlr : l<r) (hLR : L<R)
+    (hpr : Prod.snd '' K=Icc l r) (hPR : Prod.snd '' M=Icc L R) :
+    l-L=(∫ x in K, x.2)-(∫ x in M, x.2) ∧
+    r-R=(∫ x in K, x.2)-(∫ x in M, x.2) := by
+  have hl : l ∈ Prod.snd '' K := by rw [hpr]; exact left_mem_Icc.mpr hlr.le
+  have hr : r ∈ Prod.snd '' K := by rw [hpr]; exact right_mem_Icc.mpr hlr.le
+  have hL : L ∈ Prod.snd '' M := by rw [hPR]; exact left_mem_Icc.mpr hLR.le
+  have hR : R ∈ Prod.snd '' M := by rw [hPR]; exact right_mem_Icc.mpr hLR.le
+  have hC : IsCompact (a • K+b • M) := (hK.smul a).add (hM.smul b)
+  have hbounds : Prod.snd '' (a • K+b • M) ⊆ Icc (a*l+b*L) (a*r+b*R) := by
+    rintro s ⟨_,⟨_,⟨x,hx,rfl⟩,_,⟨y,hy,rfl⟩,rfl⟩,rfl⟩
+    have hx' : x.2 ∈ Icc l r := by rw [← hpr]; exact mem_image_of_mem _ hx
+    have hy' : y.2 ∈ Icc L R := by rw [← hPR]; exact mem_image_of_mem _ hy
+    change a*l+b*L ≤ a*x.2+b*y.2 ∧ a*x.2+b*y.2 ≤ a*r+b*R
+    exact ⟨add_le_add (mul_le_mul_of_nonneg_left hx'.1 ha.le) (mul_le_mul_of_nonneg_left hy'.1 hb.le),
+      add_le_add (mul_le_mul_of_nonneg_left hx'.2 ha.le) (mul_le_mul_of_nonneg_left hy'.2 hb.le)⟩
+  have hmain : l-L=(∫ s in l..r, s*sliceDensity K s)-(∫ t in L..R, t*sliceDensity M t) ∧
+      r-R=(∫ s in l..r, s*sliceDensity K s)-(∫ t in L..R, t*sliceDensity M t) := by
+    apply slice_transport_equality hm0 ha hb hab hlr hLR
+      (sliceDensity_continuous hm0 hm hK hcK hiK hl hr)
+      (sliceDensity_continuous hm0 hm hM hcM hiM hL hR)
+      (sliceDensity_integrable hK).intervalIntegrable
+      (sliceDensity_integrable hM).intervalIntegrable
+      (fun s hs => sliceDensity_pos hK hcK hiK hl hr hs)
+      (fun s hs => sliceDensity_pos hM hcM hiM hL hR hs)
+    · rw [sliceDensity_interval_integral hK hlr.le (by rw [hpr]),hvK]
+    · rw [sliceDensity_interval_integral hM hLR.le (by rw [hPR]),hvM]
+    · exact (sliceDensity_integrable hC).intervalIntegrable
+    · intro s hs t ht
+      have hroot := hm (horizontalSlice K s) (horizontalSlice M t)
+        (horizontalSlice_compact hK s) (horizontalSlice_convex hcK s)
+        (horizontalSlice_interior_nonempty hcK hiK hl hr hs)
+        (horizontalSlice_compact hM t) (horizontalSlice_convex hcM t)
+        (horizontalSlice_interior_nonempty hcM hiM hL hR ht) a b ha.le hb.le hab
+      have hf0 := sliceDensity_nonneg K s
+      have hg0 := sliceDensity_nonneg M t
+      have hroot' := pow_le_pow_left₀ (by positivity : 0 ≤ a*(sliceDensity K s)^((m:ℝ)⁻¹)+b*(sliceDensity M t)^((m:ℝ)⁻¹)) hroot m
+      rw [Real.rpow_inv_natCast_pow measureReal_nonneg hm0.ne'] at hroot'
+      exact hroot'.trans (measureReal_mono (horizontalSlice_minkowski_subset K M a b s t)
+        (horizontalSlice_compact hC _).measure_ne_top)
+    · rw [sliceDensity_interval_integral hC (by nlinarith) hbounds,hvC]
+  rwa [sliceDensity_interval_moment hK hlr.le (by rw [hpr]),
+    sliceDensity_interval_moment hM hLR.le (by rw [hPR])] at hmain
+
+end
+end PettyProjection
+end
+
+noncomputable section
+open Set MeasureTheory Filter Topology Function
+open scoped Pointwise RealInnerProductSpace
+namespace PettyProjection
+open Spherical (Sphere norm_coe)
+
+/-- The unnormalized Euclidean first moment. It is the barycenter for unit
+volume bodies, the only place where it is used below. -/
+def firstMoment {n : ℕ} (K : Set (Space n)) : Space n := ∫ x in K, x
+
+lemma shadowContinuousEquiv_inner {n : ℕ} (u : Sphere n)
+    (z : perpendicular (u : Space n) × ℝ) :
+    ⟪(u : Space n),shadowContinuousEquiv u z⟫=z.2 := by
+  rw [shadowContinuousEquiv_apply,inner_add_right,perpendicular_inner,
+    real_inner_smul_right,real_inner_self_eq_norm_sq,norm_coe]
+  ring
+
+lemma shadow_vertical_image {n : ℕ} (u : Sphere n) (K : Set (Space n)) :
+    Prod.snd '' (shadowContinuousEquiv u ⁻¹' K)=(fun x => ⟪(u : Space n),x⟫) '' K := by
+  ext s
+  constructor
+  · rintro ⟨z,hz,rfl⟩
+    exact ⟨shadowContinuousEquiv u z,hz,shadowContinuousEquiv_inner u z⟩
+  · rintro ⟨x,hx,rfl⟩
+    refine ⟨(shadowContinuousEquiv u).symm x,by simpa using hx,?_⟩
+    rw [← shadowContinuousEquiv_inner u,ContinuousLinearEquiv.apply_symm_apply]
+
+lemma shadow_firstMoment {n : ℕ} (u : Sphere n) {K : Set (Space n)} (hK : IsCompact K) :
+    (∫ z in shadowContinuousEquiv u ⁻¹' K, z.2)=⟪(u : Space n),firstMoment K⟫ := by
+  have he : MeasurePreserving (shadowContinuousEquiv u) := shadowMeasurableEquiv_measurePreserving u
+  have hi := (he.restrict_preimage hK.measurableSet).integral_comp
+    (shadowContinuousEquiv u).toHomeomorph.measurableEmbedding (fun x => ⟪(u : Space n),x⟫)
+  simp_rw [shadowContinuousEquiv_inner] at hi
+  rw [hi]
+  exact integral_inner (continuous_id.continuousOn.integrableOn_compact hK) _
+
+lemma perpendicular_finrank {n : ℕ} (u : Sphere n) :
+    Module.finrank ℝ (perpendicular (u : Space n))=n-1 := by
+  have hu : (u : Space n) ≠ 0 := by intro h; simpa [h] using norm_coe u
+  have hr := (Submodule.span ℝ {(u : Space n)}).finrank_add_finrank_orthogonal
+  rw [finrank_span_singleton hu] at hr
+  have he : Module.finrank ℝ (Space n)=n := by simp [Space]
+  rw [he] at hr
+  change Module.finrank ℝ ((Submodule.span ℝ {(u : Space n)})ᗮ)=n-1
+  omega
+
+/-- Equality for an intermediate Minkowski combination of unit-volume bodies
+identifies every directional support difference with the first-moment shift. -/
+theorem brunnMinkowski_unit_support_equality {n : ℕ} (hn : 2 ≤ n)
+    {K M : Set (Space n)} (hK : IsCompact K) (hcK : Convex ℝ K)
+    (hiK : (interior K).Nonempty) (hM : IsCompact M) (hcM : Convex ℝ M)
+    (hiM : (interior M).Nonempty) (hvK : volume.real K=1) (hvM : volume.real M=1)
+    {a b : ℝ} (ha : 0 < a) (hb : 0 < b) (hab : a+b=1)
+    (hvC : volume.real (a • K+b • M)=1) (u : Sphere n) :
+    support K u-support M u=⟪(u : Space n),firstMoment K-firstMoment M⟫ := by
+  let e := shadowContinuousEquiv u
+  have he : MeasurePreserving e := shadowMeasurableEquiv_measurePreserving u
+  have hPK : IsCompact (e ⁻¹' K) := e.toHomeomorph.isCompact_preimage.mpr hK
+  have hPM : IsCompact (e ⁻¹' M) := e.toHomeomorph.isCompact_preimage.mpr hM
+  have hiPK : (interior (e ⁻¹' K)).Nonempty := by
+    change (interior (e.toHomeomorph ⁻¹' K)).Nonempty
+    rw [← e.toHomeomorph.preimage_interior]
+    obtain ⟨x,hx⟩ := hiK
+    exact ⟨e.symm x,by simpa using hx⟩
+  have hiPM : (interior (e ⁻¹' M)).Nonempty := by
+    change (interior (e.toHomeomorph ⁻¹' M)).Nonempty
+    rw [← e.toHomeomorph.preimage_interior]
+    obtain ⟨x,hx⟩ := hiM
+    exact ⟨e.symm x,by simpa using hx⟩
+  have hcPK := hcK.linear_preimage e.toLinearMap
+  have hcPM := hcM.linear_preimage e.toLinearMap
+  obtain ⟨l,r,hlr,hpr⟩ := vertical_projection_interval hPK hcPK hiPK
+  obtain ⟨L,R,hLR,hPR⟩ := vertical_projection_interval hPM hcPM hiPM
+  have hv (S : Set (Space n)) (hS : IsCompact S) : volume.real (e ⁻¹' S)=volume.real S :=
+    congrArg ENNReal.toReal (he.measure_preimage hS.measurableSet.nullMeasurableSet)
+  have hm0 : 0 < Module.finrank ℝ (perpendicular (u : Space n)) := by
+    rw [perpendicular_finrank]; omega
+  have heq := brunnMinkowski_product_unit_equality hm0 (brunnMinkowski_innerProduct hm0)
+    hPK hcPK hiPK hPM hcPM hiPM ((hv K hK).trans hvK) ((hv M hM).trans hvM) ha hb hab
+    (by rw [← linearEquiv_preimage_combo,hv _ ((hK.smul _).add (hM.smul _)),hvC]) hlr hLR hpr hPR
+  have hr : support K u=r := by
+    unfold support
+    rw [← shadow_vertical_image u K,hpr,csSup_Icc hlr.le]
+  have hR : support M u=R := by
+    unfold support
+    rw [← shadow_vertical_image u M,hPR,csSup_Icc hLR.le]
+  rw [hr,hR,heq.2,inner_sub_right]
+  change (∫ z in shadowContinuousEquiv u ⁻¹' K, z.2)-
+      (∫ z in shadowContinuousEquiv u ⁻¹' M, z.2)=_
+  rw [shadow_firstMoment u hK,shadow_firstMoment u hM]
+
+lemma support_translate {n : ℕ} {K : Set (Space n)} (hK : IsCompact K)
+    (hKn : K.Nonempty) (c u : Space n) :
+    support ((fun x => c+x) '' K) u=⟪u,c⟫+support K u := by
+  apply le_antisymm
+  · apply (support_le_iff (hK.image (by fun_prop)) (hKn.image _)).mpr
+    rintro _ ⟨x,hx,rfl⟩
+    rw [inner_add_right]
+    exact add_le_add_right (inner_le_support hK hx u) _
+  · obtain ⟨x,hx,he⟩ := support_attained hK hKn u
+    rw [he,← inner_add_right]
+    exact inner_le_support (hK.image (by fun_prop)) (mem_image_of_mem _ hx) u
+
+/-- The unrestricted translated-body equality case of the unit-volume
+Brunn--Minkowski inequality (no origin symmetry or boundary regularity). -/
+theorem brunnMinkowski_unit_equality {n : ℕ} (hn : 2 ≤ n)
+    {K M : Set (Space n)} (hK : IsCompact K) (hcK : Convex ℝ K)
+    (hiK : (interior K).Nonempty) (hM : IsCompact M) (hcM : Convex ℝ M)
+    (hiM : (interior M).Nonempty) (hvK : volume.real K=1) (hvM : volume.real M=1)
+    {a b : ℝ} (ha : 0 < a) (hb : 0 < b) (hab : a+b=1)
+    (hvC : volume.real (a • K+b • M)=1) :
+    K=(fun x => firstMoment K-firstMoment M+x) '' M := by
+  have hKn := hiK.mono interior_subset
+  have hMn := hiM.mono interior_subset
+  apply (support_eq_iff hK hcK hKn (hM.image (by fun_prop))
+    (hcM.translate _) (hMn.image _)).mp
+  funext x
+  rw [support_translate hM hMn]
+  by_cases hx : x=0
+  · subst x
+    simp [support,hKn,hMn]
+  · let u : Sphere n := ⟨‖x‖⁻¹ • x,by simp [norm_smul,hx]⟩
+    have hxu : x=‖x‖ • (u : Space n) := by
+      change x=‖x‖ • (‖x‖⁻¹ • x)
+      rw [smul_smul,mul_inv_cancel₀ (norm_ne_zero_iff.mpr hx),one_smul]
+    have he := brunnMinkowski_unit_support_equality hn hK hcK hiK hM hcM hiM hvK hvM ha hb hab hvC u
+    rw [hxu,support_smul hK hKn (norm_nonneg x),support_smul hM hMn (norm_nonneg x),real_inner_smul_left]
+    rw [← he]
+    ring
+
+end PettyProjection
+end
+
+noncomputable section
+open Set MeasureTheory Filter Topology
+open scoped Pointwise
+namespace PettyProjection
+
+def volumeRoot {n : ℕ} (K : Set (Space n)) : ℝ := (volume.real K)^((n:ℝ)⁻¹)
+
+lemma volumeRoot_pos {n : ℕ} {K : Set (Space n)} (hK : IsCompact K)
+    (hi : (interior K).Nonempty) : 0 < volumeRoot K :=
+  Real.rpow_pos_of_pos (volumeReal_pos_of_body hK hi) _
+
+lemma volumeRoot_pow {n : ℕ} (hn : 0 < n) (K : Set (Space n)) :
+    volumeRoot K^n=volume.real K :=
+  Real.rpow_inv_natCast_pow measureReal_nonneg hn.ne'
+
+lemma volumeRoot_smul {n : ℕ} (hn : 0 < n) {r : ℝ} (hr : 0 ≤ r) (K : Set (Space n)) :
+    volumeRoot (r • K)=r*volumeRoot K := by
+  unfold volumeRoot
+  rw [volumeReal_smul_nonneg hr]
+  have hd : Module.finrank ℝ (Space n)=n := by simp [Space]
+  rw [hd,Real.mul_rpow (pow_nonneg hr _) measureReal_nonneg,← Real.rpow_natCast,
+    ← Real.rpow_mul hr, mul_inv_cancel₀ (by exact_mod_cast hn.ne'),Real.rpow_one]
+
+lemma interior_add_nonempty_left {n : ℕ} {K M : Set (Space n)}
+    (hiK : (interior K).Nonempty) (hM : M.Nonempty) : (interior (K+M)).Nonempty := by
+  obtain ⟨y,hy⟩ := hM
+  obtain ⟨x,hx⟩ := hiK
+  refine ⟨x+y,?_⟩
+  exact (isOpen_interior.add_right.subset_interior_iff.mpr
+    (add_subset_add interior_subset Subset.rfl)) (add_mem_add hx hy)
+
+/-- The homogeneous, additive form of Brunn--Minkowski. -/
+theorem brunnMinkowski_add {n : ℕ} (hn : 0 < n)
+    {K M : Set (Space n)} (hK : IsCompact K) (hcK : Convex ℝ K)
+    (hiK : (interior K).Nonempty) (hM : IsCompact M) (hcM : Convex ℝ M)
+    (hiM : (interior M).Nonempty) :
+    volumeRoot K+volumeRoot M ≤ volumeRoot (K+M) := by
+  have h := brunnMinkowski_space hn K M hK hcK hiK hM hcM hiM
+    (1/2) (1/2) (by norm_num) (by norm_num) (by norm_num)
+  change (1/2)*volumeRoot K+(1/2)*volumeRoot M ≤ volumeRoot ((1/2:ℝ) • K+(1/2:ℝ) • M) at h
+  rw [← smul_add,volumeRoot_smul hn (by norm_num)] at h
+  linarith
+
+/-- Equality in the homogeneous additive form, retaining arbitrary translation. -/
+theorem brunnMinkowski_add_equality {n : ℕ} (hn : 2 ≤ n)
+    {K M : Set (Space n)} (hK : IsCompact K) (hcK : Convex ℝ K)
+    (hiK : (interior K).Nonempty) (hM : IsCompact M) (hcM : Convex ℝ M)
+    (hiM : (interior M).Nonempty)
+    (hv : volumeRoot (K+M)=volumeRoot K+volumeRoot M) :
+    ∃ z : Space n, ∃ c : ℝ, 0 < c ∧ K=(fun x => z+c • x) '' M := by
+  let p := volumeRoot K
+  let q := volumeRoot M
+  have hp : 0 < p := volumeRoot_pos hK hiK
+  have hq : 0 < q := volumeRoot_pos hM hiM
+  have hpq : 0 < p+q := add_pos hp hq
+  have hd : Module.finrank ℝ (Space n)=n := by simp [Space]
+  have hUK : volume.real (p⁻¹ • K)=1 := by
+    rw [volumeReal_smul_nonneg (inv_nonneg.mpr hp.le),hd,inv_pow,volumeRoot_pow (by omega)]
+    exact inv_mul_cancel₀ (volumeReal_pos_of_body hK hiK).ne'
+  have hUM : volume.real (q⁻¹ • M)=1 := by
+    rw [volumeReal_smul_nonneg (inv_nonneg.mpr hq.le),hd,inv_pow,volumeRoot_pow (by omega)]
+    exact inv_mul_cancel₀ (volumeReal_pos_of_body hM hiM).ne'
+  have hiUK : (interior (p⁻¹ • K)).Nonempty := by
+    rw [interior_smul₀ (inv_ne_zero hp.ne')]
+    exact hiK.image _
+  have hiUM : (interior (q⁻¹ • M)).Nonempty := by
+    rw [interior_smul₀ (inv_ne_zero hq.ne')]
+    exact hiM.image _
+  have hsets : (p/(p+q)) • (p⁻¹ • K)+(q/(p+q)) • (q⁻¹ • M)=(p+q)⁻¹ • (K+M) := by
+    rw [smul_add,smul_smul,smul_smul]
+    congr 2 <;> field_simp
+  have hunit : volume.real ((p/(p+q)) • (p⁻¹ • K)+(q/(p+q)) • (q⁻¹ • M))=1 := by
+    rw [hsets,volumeReal_smul_nonneg (inv_nonneg.mpr hpq.le),hd,inv_pow,
+      ← volumeRoot_pow (by omega) (K+M),hv]
+    exact inv_mul_cancel₀ (pow_ne_zero _ hpq.ne')
+  have he := brunnMinkowski_unit_equality hn (hK.smul _) (hcK.smul _) hiUK
+    (hM.smul _) (hcM.smul _) hiUM hUK hUM (div_pos hp hpq) (div_pos hq hpq)
+    (by rw [← add_div,div_self hpq.ne']) hunit
+  let d := firstMoment (p⁻¹ • K)-firstMoment (q⁻¹ • M)
+  refine ⟨p • d,p/q,div_pos hp hq,?_⟩
+  calc
+    K = p • (p⁻¹ • K) := by rw [smul_smul,mul_inv_cancel₀ hp.ne',one_smul]
+    _ = p • ((fun x => d+x) '' (q⁻¹ • M)) := by rw [he]
+    _ = (fun x => p • d+(p/q) • x) '' M := by
+      rw [← image_smul,image_image,← image_smul,image_image]
+      congr 1
+      funext x
+      simp only [smul_add,smul_smul,div_eq_mul_inv]
+
+end PettyProjection
+end
+
+noncomputable section
+open Set MeasureTheory Filter Topology
+open scoped Pointwise
+namespace PettyProjection
+
+lemma minkowski_parameter_combo {n : ℕ} {K M : Set (Space n)}
+    (hcK : Convex ℝ K) (hcM : Convex ℝ M) {s t a b : ℝ}
+    (hs : 0 ≤ s) (ht : 0 ≤ t) (ha : 0 ≤ a) (hb : 0 ≤ b) (hab : a+b=1) :
+    a • (K+s • M)+b • (K+t • M)=K+(a*s+b*t) • M := by
+  rw [smul_add,smul_add,smul_smul,smul_smul,add_add_add_comm,
+    ← hcK.add_smul ha hb,hab,one_smul,← hcM.add_smul (mul_nonneg ha hs) (mul_nonneg hb ht)]
+
+lemma minkowski_volumeRoot_concave {n : ℕ} (hn : 0 < n)
+    {K M : Set (Space n)} (hK : IsCompact K) (hcK : Convex ℝ K)
+    (hiK : (interior K).Nonempty) (hM : IsCompact M) (hcM : Convex ℝ M)
+    (hMn : M.Nonempty) : ConcaveOn ℝ (Ici 0) (fun t : ℝ => volumeRoot (K+t • M)) := by
+  refine ⟨convex_Ici _,?_⟩
+  intro s hs t ht a b ha hb hab
+  have hSM : (s • M).Nonempty := hMn.image _
+  have hTM : (t • M).Nonempty := hMn.image _
+  have h := brunnMinkowski_space hn (K+s • M) (K+t • M)
+    (hK.add (hM.smul _)) (hcK.add (hcM.smul _)) (interior_add_nonempty_left hiK hSM)
+    (hK.add (hM.smul _)) (hcK.add (hcM.smul _)) (interior_add_nonempty_left hiK hTM) a b ha hb hab
+  rw [minkowski_parameter_combo hcK hcM hs ht ha hb hab] at h
+  exact h
+
+lemma volumeRoot_diff_factor {n : ℕ} (hn : 0 < n) {K : Set (Space n)}
+    (hv : 0 < volume.real K) :
+    ((n:ℝ)*volumeRoot K^(n-1))*((n:ℝ)⁻¹*(volume.real K)^((n:ℝ)⁻¹-1))=1 := by
+  have hn0 : (n:ℝ) ≠ 0 := by exact_mod_cast hn.ne'
+  rw [Real.rpow_sub hv,Real.rpow_one]
+  change (n*volumeRoot K^(n-1))*((n:ℝ)⁻¹*(volumeRoot K/volume.real K))=1
+  have hp : volumeRoot K^(n-1)*volumeRoot K=volume.real K := by
+    rw [← pow_succ,Nat.sub_add_cancel hn,volumeRoot_pow hn]
+  field_simp
+  nlinarith [hp]
+
+/-- The sharp first-variation inequality together with its global translated
+homothety equality implication, derived from actual Brunn--Minkowski concavity. -/
+theorem minkowski_derivative_bound_equality {n : ℕ} (hn : 2 ≤ n)
+    {K M : Set (Space n)} (hK : IsCompact K) (hcK : Convex ℝ K)
+    (hiK : (interior K).Nonempty) (hM : IsCompact M) (hcM : Convex ℝ M)
+    (hiM : (interior M).Nonempty) {d : ℝ}
+    (hd : HasDerivWithinAt (fun t : ℝ => volume.real (K+t • M)) d (Ici 0) 0) :
+    (n:ℝ)*volumeRoot K^(n-1)*volumeRoot M ≤ d ∧
+      (d=(n:ℝ)*volumeRoot K^(n-1)*volumeRoot M →
+        ∃ z : Space n, ∃ c : ℝ, 0 < c ∧ K=(fun x => z+c • x) '' M) := by
+  have hn0 : 0 < n := by omega
+  have hnR : 0 < (n:ℝ) := by exact_mod_cast hn0
+  have hMn := hiM.mono interior_subset
+  have hzero : K+(0:ℝ) • M=K := by simp [zero_smul_set hMn]
+  have hvK := volumeReal_pos_of_body hK hiK
+  have hf := hd.rpow_const (p := (n:ℝ)⁻¹) (Or.inl (by rw [hzero]; exact hvK.ne'))
+  rw [hzero] at hf
+  have hc := minkowski_volumeRoot_concave hn0 hK hcK hiK hM hcM hMn
+  have hs := hc.slope_le_of_hasDerivWithinAt (by norm_num : (0:ℝ) ∈ Ici 0) (by norm_num : (1:ℝ) ∈ Ici 0)
+    (by norm_num : (0:ℝ) < 1) hf
+  rw [slope_def_field,hzero,one_smul,sub_zero,div_one] at hs
+  have hlow := brunnMinkowski_add hn0 hK hcK hiK hM hcM hiM
+  let D : ℝ := n*volumeRoot K^(n-1)
+  let C : ℝ := (n:ℝ)⁻¹*(volume.real K)^((n:ℝ)⁻¹-1)
+  have hD : 0 < D := mul_pos hnR (pow_pos (volumeRoot_pos hK hiK) _)
+  have hDC : D*C=1 := volumeRoot_diff_factor hn0 hvK
+  have hs' : D*(volumeRoot (K+M)-volumeRoot K) ≤ d := by
+    have he : D*(d*C)=d := by calc
+      _ = d*(D*C) := by ring
+      _ = d := by rw [hDC,mul_one]
+    rw [← he]
+    apply mul_le_mul_of_nonneg_left _ hD.le
+    simpa only [C,mul_assoc] using hs
+  constructor
+  · change D*volumeRoot M ≤ d
+    exact (mul_le_mul_of_nonneg_left (by linarith : volumeRoot M ≤ volumeRoot (K+M)-volumeRoot K) hD.le).trans hs'
+  · intro heq
+    have hv : volumeRoot (K+M)=volumeRoot K+volumeRoot M := by
+      change d=D*volumeRoot M at heq
+      rw [heq] at hs'
+      have ht := (mul_le_mul_iff_right₀ hD).mp hs'
+      linarith
+    exact brunnMinkowski_add_equality hn hK hcK hiK hM hcM hiM hv
+
+end PettyProjection
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/ProjectionBodies/Main.lean b/lean/OAI/Geometry/ProjectionBodies/Main.lean
new file mode 100644
index 0000000000000000000000000000000000000000..ea4c5cc4c021e62cc1f8a7c37e9f97ac18549dd0
--- /dev/null
+++ b/lean/OAI/Geometry/ProjectionBodies/Main.lean
@@ -0,0 +1,25 @@
+import OAI.Geometry.ProjectionBodies.AffineCovariance
+
+namespace OAI
+
+noncomputable section
+open Set MeasureTheory
+namespace PettyProjection
+
+/-- The exact unrestricted inequality and its global equality characterization.
+-/
+theorem petty_projection_volume (n : ℕ) (hn : 4 ≤ n)
+    (K : Set (Space n)) (hK : IsConvexBody K) :
+    pettyConstant n ≤ projectionRatio K ∧
+      (projectionRatio K = pettyConstant n ↔ IsEllipsoid K) := by
+  obtain ⟨hbound,hrigid⟩ := projection_lower_bound hn hK
+  exact ⟨hbound,hrigid,ellipsoid_projectionRatio (by omega)⟩
+
+theorem mainClaim_proved : MainClaim := petty_projection_volume
+
+theorem MainClaim_proved : MainClaim := mainClaim_proved
+
+end PettyProjection
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/ProjectionBodies/Model.lean b/lean/OAI/Geometry/ProjectionBodies/Model.lean
new file mode 100644
index 0000000000000000000000000000000000000000..6040bc270af377b018da0eacc3639ea6a70a365d
--- /dev/null
+++ b/lean/OAI/Geometry/ProjectionBodies/Model.lean
@@ -0,0 +1,51 @@
+import Mathlib
+
+namespace OAI
+
+noncomputable section
+open Set MeasureTheory
+open scoped RealInnerProductSpace
+
+namespace PettyProjection
+
+abbrev Space (n : ℕ) := EuclideanSpace ℝ (Fin n)
+
+def IsConvexBody {n : ℕ} (K : Set (Space n)) : Prop :=
+  IsCompact K ∧ Convex ℝ K ∧ (interior K).Nonempty
+
+def unitBall (n : ℕ) : Set (Space n) := Metric.closedBall 0 1
+
+def kappa (n : ℕ) : ℝ := volume.real (unitBall n)
+
+def perpendicular {n : ℕ} (u : Space n) : Submodule ℝ (Space n) :=
+  (Submodule.span ℝ {u})ᗮ
+
+/-- Volume in the induced Euclidean structure on the orthogonal hyperplane. -/
+def shadowVolume {n : ℕ} (K : Set (Space n)) (u : Space n) : ℝ :=
+  (volume : Measure (perpendicular u)).real
+    ((perpendicular u).orthogonalProjectionOnto '' K)
+
+/-- The body with support function equal to the volumes of orthogonal shadows,
+expressed as its defining intersection of supporting half-spaces. -/
+def projectionBody {n : ℕ} (K : Set (Space n)) : Set (Space n) :=
+  {x | ∀ u : Space n, ‖u‖ = 1 → ⟪u, x⟫ ≤ shadowVolume K u}
+
+def projectionRatio {n : ℕ} (K : Set (Space n)) : ℝ :=
+  volume.real (projectionBody K) / volume.real K ^ (n - 1)
+
+def pettyConstant (n : ℕ) : ℝ :=
+  kappa (n - 1) ^ n * kappa n ^ (2 - (n : ℤ))
+
+def IsEllipsoid {n : ℕ} (K : Set (Space n)) : Prop :=
+  ∃ (a : Space n) (T : Space n ≃ₗ[ℝ] Space n),
+    K = (fun x => a + T x) '' unitBall n
+
+def MainClaim : Prop :=
+  ∀ (n : ℕ), 4 ≤ n → ∀ (K : Set (Space n)), IsConvexBody K →
+    pettyConstant n ≤ projectionRatio K ∧
+      (projectionRatio K = pettyConstant n ↔ IsEllipsoid K)
+
+end PettyProjection
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/ProjectionBodies/NormCurvature.lean b/lean/OAI/Geometry/ProjectionBodies/NormCurvature.lean
new file mode 100644
index 0000000000000000000000000000000000000000..ff13a8053b1340b71d70af7d55bb64120dd7098f
--- /dev/null
+++ b/lean/OAI/Geometry/ProjectionBodies/NormCurvature.lean
@@ -0,0 +1,361 @@
+import OAI.Geometry.ProjectionBodies.SphericalGap
+
+namespace OAI
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology Function
+open scoped ENNReal NNReal RealInnerProductSpace Pointwise
+namespace PettyProjection
+open Spherical (Sphere norm_coe)
+
+/-- Intersection of the actual supporting constraints, with no evenness or
+convexity condition on the constraint function. -/
+def wulff {n : ℕ} (s : Sphere n → ℝ) : Set (Space n) :=
+  {x | ∀ u : Sphere n, ⟪(u : Space n),x⟫ ≤ s u}
+
+lemma wulff_closed {n : ℕ} (s : Sphere n → ℝ) : IsClosed (wulff s) := by
+  unfold wulff
+  rw [ofPred_forall]
+  exact isClosed_iInter (fun _ => isClosed_le (by fun_prop) continuous_const)
+
+lemma wulff_convex {n : ℕ} (s : Sphere n → ℝ) : Convex ℝ (wulff s) := by
+  intro x hx y hy a b ha hb hab u
+  rw [inner_add_right,real_inner_smul_right,real_inner_smul_right]
+  calc
+    _ ≤ a*s u+b*s u := add_le_add (mul_le_mul_of_nonneg_left (hx u) ha)
+      (mul_le_mul_of_nonneg_left (hy u) hb)
+    _ = s u := by rw [← add_mul,hab,one_mul]
+
+lemma closedBall_subset_wulff {n : ℕ} {s : Sphere n → ℝ} {r : ℝ}
+    (hr : ∀ u, r ≤ s u) : closedBall (0 : Space n) r ⊆ wulff s := by
+  intro x hx u
+  calc
+    _ ≤ ‖(u : Space n)‖*‖x‖ := real_inner_le_norm _ _
+    _ = ‖x‖ := by rw [norm_coe,one_mul]
+    _ ≤ r := by simpa only [mem_closedBall,dist_zero_right] using hx
+    _ ≤ s u := hr u
+
+lemma wulff_subset_closedBall {n : ℕ} {s : Sphere n → ℝ} {R : ℝ}
+    (hR : 0 ≤ R) (hs : ∀ u, s u ≤ R) : wulff s ⊆ closedBall (0 : Space n) R := by
+  intro x hx
+  rw [mem_closedBall,dist_zero_right]
+  by_cases h : x = 0
+  · simpa [h] using hR
+  · let u : Sphere n := ⟨‖x‖⁻¹ • x, by
+      rw [mem_sphere,dist_zero_right,norm_smul,Real.norm_eq_abs,abs_inv,abs_norm,
+        inv_mul_cancel₀ (norm_ne_zero_iff.mpr h)]⟩
+    have hu : ⟪(u : Space n),x⟫ = ‖x‖ := by
+      dsimp only [u]
+      rw [real_inner_smul_left,real_inner_self_eq_norm_sq]
+      field_simp
+    exact hu ▸ (hx u).trans (hs u)
+
+lemma positive_constraint_bounds {n : ℕ} [NeZero n] (s : C(Sphere n,ℝ))
+    (hs : ∀ u, 0 < s u) : ∃ m M : ℝ, 0 < m ∧ 0 < M ∧ ∀ u, m ≤ s u ∧ s u ≤ M := by
+  have hne : (univ : Set (Sphere n)).Nonempty := by
+    obtain ⟨u,hu⟩ := (NormedSpace.sphere_nonempty (x := (0 : Space n))).mpr (by norm_num : (0:ℝ) ≤ 1)
+    exact ⟨⟨u,hu⟩,mem_univ _⟩
+  obtain ⟨u,_,hu⟩ := isCompact_univ.exists_isMinOn hne s.continuous.continuousOn
+  obtain ⟨v,_,hv⟩ := isCompact_univ.exists_isMaxOn hne s.continuous.continuousOn
+  exact ⟨s u,s v,hs u,hs v,fun w => ⟨hu (mem_univ w),hv (mem_univ w)⟩⟩
+
+lemma wulff_nhds {n : ℕ} [NeZero n] (s : C(Sphere n,ℝ)) (hs : ∀ u, 0 < s u) :
+    wulff s ∈ 𝓝 (0 : Space n) := by
+  obtain ⟨m,M,hm,_,hb⟩ := positive_constraint_bounds s hs
+  exact Metric.mem_nhds_iff.mpr ⟨m,hm,ball_subset_closedBall.trans
+    (closedBall_subset_wulff (fun u => (hb u).1))⟩
+
+lemma wulff_compact {n : ℕ} [NeZero n] (s : C(Sphere n,ℝ)) (hs : ∀ u, 0 < s u) :
+    IsCompact (wulff s) := by
+  obtain ⟨m,M,_,hM,hb⟩ := positive_constraint_bounds s hs
+  exact (isCompact_closedBall (0 : Space n) M).of_isClosed_subset (wulff_closed _)
+    (wulff_subset_closedBall hM.le (fun u => (hb u).2))
+
+lemma wulff_body {n : ℕ} [NeZero n] (s : C(Sphere n,ℝ)) (hs : ∀ u, 0 < s u) :
+    IsConvexBody (wulff s) :=
+  ⟨wulff_compact s hs,wulff_convex s,⟨0,mem_interior_iff_mem_nhds.mpr (wulff_nhds s hs)⟩⟩
+
+lemma wulff_support {n : ℕ} [NeZero n] {K : Set (Space n)}
+    (hK : IsCompact K) (hc : Convex ℝ K) (hKn : K.Nonempty) :
+    wulff (fun u : Sphere n => support K u) = K := by
+  ext x
+  rw [mem_iff_inner_le_support hK hc hKn]
+  constructor
+  · intro h u
+    by_cases hu : u = 0
+    · subst u
+      obtain ⟨y,hy,he⟩ := support_attained hK hKn 0
+      simpa only [inner_zero_left] using he.ge
+    · let v : Sphere n := ⟨‖u‖⁻¹ • u, by
+        rw [mem_sphere,dist_zero_right,norm_smul,Real.norm_eq_abs,abs_inv,abs_norm,
+          inv_mul_cancel₀ (norm_ne_zero_iff.mpr hu)]⟩
+      have he : u = ‖u‖ • (v : Space n) := by dsimp [v]; rw [smul_inv_smul₀ (norm_ne_zero_iff.mpr hu)]
+      rw [he,real_inner_smul_left,support_smul hK hKn (norm_nonneg _)]
+      exact mul_le_mul_of_nonneg_left (h v) (norm_nonneg _)
+  · intro h u; exact h u
+
+/-- Compact coefficient set of the maximum formula. -/
+def wulffCoefficients {n : ℕ} (s : Sphere n → ℝ) : Set (Space n) :=
+  range (fun u : Sphere n => (s u)⁻¹ • (u : Space n))
+
+def wulffMax {n : ℕ} (s : Sphere n → ℝ) (x : Space n) : ℝ :=
+  support (wulffCoefficients s) x
+
+lemma wulffCoefficients_compact {n : ℕ} (s : C(Sphere n,ℝ)) (hs : ∀ u, 0 < s u) :
+    IsCompact (wulffCoefficients s) :=
+  isCompact_range ((s.continuous.inv₀ (fun u => (hs u).ne')).smul continuous_subtype_val)
+
+lemma wulffCoefficients_nonempty {n : ℕ} [NeZero n] (s : Sphere n → ℝ) :
+    (wulffCoefficients s).Nonempty := by
+  obtain ⟨u,hu⟩ := (NormedSpace.sphere_nonempty (x := (0 : Space n))).mpr (by norm_num : (0:ℝ) ≤ 1)
+  exact ⟨_,⟨⟨u,hu⟩,rfl⟩⟩
+
+lemma wulffMax_attained {n : ℕ} [NeZero n] (s : C(Sphere n,ℝ)) (hs : ∀ u, 0 < s u)
+    (x : Space n) : ∃ u : Sphere n, wulffMax s x = ⟪x,(u : Space n)⟫ / s u := by
+  obtain ⟨y,⟨u,rfl⟩,he⟩ := support_attained (wulffCoefficients_compact s hs)
+    (wulffCoefficients_nonempty s) x
+  exact ⟨u,by simpa only [wulffMax,real_inner_smul_right,div_eq_inv_mul] using he⟩
+
+lemma le_wulffMax {n : ℕ} (s : C(Sphere n,ℝ)) (hs : ∀ u, 0 < s u)
+    (x : Space n) (u : Sphere n) : ⟪x,(u : Space n)⟫ / s u ≤ wulffMax s x := by
+  simpa only [wulffMax,real_inner_smul_right,div_eq_inv_mul] using
+    inner_le_support (wulffCoefficients_compact s hs) (mem_range_self u) x
+
+lemma wulffMax_pos {n : ℕ} (s : C(Sphere n,ℝ)) (hs : ∀ u, 0 < s u)
+    {x : Space n} (hx : x ≠ 0) : 0 < wulffMax s x := by
+  let u : Sphere n := ⟨‖x‖⁻¹ • x, by
+    rw [mem_sphere,dist_zero_right,norm_smul,Real.norm_eq_abs,abs_inv,abs_norm,
+      inv_mul_cancel₀ (norm_ne_zero_iff.mpr hx)]⟩
+  have he : ⟪x,(u : Space n)⟫ = ‖x‖ := by
+    dsimp only [u]
+    rw [real_inner_smul_right,real_inner_self_eq_norm_sq]
+    field_simp
+  apply lt_of_lt_of_le _ (le_wulffMax s hs x u)
+  rw [he]
+  exact div_pos (norm_pos_iff.mpr hx) (hs u)
+
+lemma wulffMax_zero {n : ℕ} [NeZero n] (s : C(Sphere n,ℝ)) (hs : ∀ u, 0 < s u) :
+    wulffMax s (0 : Space n) = 0 := by
+  obtain ⟨u,he⟩ := wulffMax_attained s hs 0
+  simpa only [inner_zero_left,zero_div] using he
+
+lemma mem_smul_wulff {n : ℕ} {s : Sphere n → ℝ} {a : ℝ} (ha : 0 < a) {x : Space n} :
+    x ∈ a • wulff s ↔ ∀ u : Sphere n, ⟪x,(u : Space n)⟫ ≤ a*s u := by
+  rw [mem_smul_set_iff_inv_smul_mem₀ ha.ne']
+  change (∀ u : Sphere n, ⟪(u : Space n),a⁻¹ • x⟫ ≤ s u) ↔ _
+  simp_rw [real_inner_smul_right,real_inner_comm (y := x),← div_eq_inv_mul,div_le_iff₀ ha]
+  simp only [mul_comm]
+
+lemma gauge_wulff_eq_max {n : ℕ} [NeZero n] (s : C(Sphere n,ℝ)) (hs : ∀ u, 0 < s u)
+    (x : Space n) : gauge (wulff s) x = wulffMax s x := by
+  by_cases hx : x = 0
+  · subst x; rw [gauge_zero,wulffMax_zero s hs]
+  have hp := wulffMax_pos s hs hx
+  have hmem : x ∈ wulffMax s x • wulff s := by
+    apply (mem_smul_wulff hp).mpr
+    intro u
+    have h := (div_le_iff₀ (hs u)).mp (le_wulffMax s hs x u)
+    simpa only [mul_comm] using h
+  apply le_antisymm (gauge_le_of_mem hp.le hmem)
+  rw [gauge_def]
+  refine le_csInf (s := {r : ℝ | r ∈ Ioi 0 ∧ x ∈ r • wulff s})
+    ⟨wulffMax s x,hp,hmem⟩ ?_
+  rintro a ⟨ha,hax⟩
+  obtain ⟨u,he⟩ := wulffMax_attained s hs x
+  rw [he]
+  exact (div_le_iff₀ (hs u)).mpr ((mem_smul_wulff ha).mp hax u)
+
+lemma gauge_ae_differentiable_sphere {n : ℕ} {K : Set (Space n)}
+    (hc : Convex ℝ K) (h0 : K ∈ 𝓝 (0 : Space n)) :
+    ∀ᵐ (u : Sphere n) ∂Spherical.sigma n, DifferentiableAt ℝ (gauge K) (u : Space n) := by
+  obtain ⟨C,hC⟩ := hc.lipschitz_gauge h0
+  apply Spherical.sphere_ae_of_radial hC.ae_differentiableAt
+  intro u r hr hd
+  have hh : DifferentiableAt ℝ (fun x : Space n => r⁻¹ * gauge K (r • x)) (u : Space n) :=
+    (hd.comp (u : Space n) (differentiableAt_id.const_smul r)).const_mul r⁻¹
+  have he : (fun x : Space n => r⁻¹ * gauge K (r • x)) = gauge K := by
+    funext x
+    rw [gauge_smul_of_nonneg (α := ℝ) hr.le,smul_eq_mul,← mul_assoc,
+      inv_mul_cancel₀ hr.ne',one_mul]
+  rwa [he] at hh
+
+end PettyProjection
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology
+open scoped ENNReal RealInnerProductSpace Pointwise
+namespace PettyProjection.Spherical
+
+lemma rotDeriv_comp_scalar {n : ℕ} (T : Space n →L[ℝ] Space n)
+    {f : Space n → ℝ} {G : ℝ → ℝ} (hf : Differentiable ℝ f)
+    (hG : Differentiable ℝ G) (x : Space n) :
+    rotDeriv T (G ∘ f) x = deriv G (f x) * rotDeriv T f x := by
+  have h := (hG (f x)).hasDerivAt.comp_hasFDerivAt x (hf x).hasFDerivAt
+  simp only [rotDeriv, h.fderiv, smul_apply, smul_eq_mul]
+
+lemma energy_comp_relation {n : ℕ} (f : Space n → ℝ) (G I : ℝ → ℝ) (b : ℝ)
+    (hf : Differentiable ℝ f) (hG : Differentiable ℝ G) (hI : Differentiable ℝ I)
+    (hrel : ∀ z, deriv G z ^ 2 = b ^ 2 * deriv I z) :
+    energy (G ∘ f) (G ∘ f) = b ^ 2 * energy (I ∘ f) f := by
+  unfold energy
+  simp_rw [rotDeriv_comp_scalar _ hf hG, rotDeriv_comp_scalar _ hf hI]
+  have he (i j : Fin n) :
+      (fun u : Sphere n => deriv G (f u) * rotDeriv (generator n i j) f u *
+        (deriv G (f u) * rotDeriv (generator n i j) f u)) =
+      (fun u : Sphere n => b ^ 2 *
+        (deriv I (f u) * rotDeriv (generator n i j) f u * rotDeriv (generator n i j) f u)) := by
+    funext u
+    calc
+      _ = deriv G (f u) ^ 2 * (rotDeriv (generator n i j) f u) ^ 2 := by ring
+      _ = _ := by rw [hrel]; ring
+  simp_rw [he, mean, integral_const_mul]
+  simp only [← Finset.mul_sum]
+  ring
+
+lemma smooth_convex_energy_bound {n : ℕ} [NeZero n]
+    {f : Space n → ℝ} {G I : ℝ → ℝ} {b : ℝ}
+    (hf : ContDiff ℝ 2 f) (hc : ConvexOn ℝ univ f)
+    (hG : ContDiff ℝ 1 G) (hI : ContDiff ℝ 1 I)
+    (hpos : ∀ z, 0 ≤ I z) (hrel : ∀ z, deriv G z ^ 2 = b ^ 2 * deriv I z) :
+    energy (G ∘ f) (G ∘ f) ≤ ((n : ℝ) - 1) * b ^ 2 *
+      mean (fun u : Sphere n => I (f u) * fderiv ℝ f u u) := by
+  have hfc : ContDiff ℝ 1 f := hf.of_le (by norm_num)
+  have hr := energy_comp_relation f G I b (hf.differentiable (by norm_num))
+    (hG.differentiable (by norm_num)) (hI.differentiable (by norm_num)) hrel
+  rw [hr]
+  have hibp := mean_casimir (hI.comp hfc) hf
+  have hcont : Continuous (fun u : Sphere n => I (f u) * casimir f u) := by
+    apply (hI.continuous.comp (hf.continuous.comp continuous_subtype_val)).mul
+    unfold casimir
+    exact continuous_const.mul (continuous_finsetSum _ fun i _ =>
+      continuous_finsetSum _ fun j _ =>
+        (continuous_rotDeriv _ (contDiff_rotDeriv _ hf)).comp continuous_subtype_val)
+  have hcont' : Continuous (fun u : Sphere n =>
+      -((n : ℝ) - 1) * (I (f u) * fderiv ℝ f u u)) :=
+    continuous_const.mul ((hI.continuous.comp (hf.continuous.comp continuous_subtype_val)).mul
+      ((hf.continuous_fderiv (by norm_num)).comp continuous_subtype_val |>.clm_apply continuous_subtype_val))
+  have hle : mean (fun u : Sphere n => -((n : ℝ) - 1) * (I (f u) * fderiv ℝ f u u)) ≤
+      mean (fun u : Sphere n => I (f u) * casimir f u) := by
+    apply integral_mono (continuous_integrable hcont') (continuous_integrable hcont)
+    intro u
+    have h := mul_le_mul_of_nonneg_left (convex_casimir_lower hf hc u) (hpos (f u))
+    nlinarith
+  rw [mean, integral_const_mul] at hle
+  change -((n : ℝ) - 1) * mean (fun u : Sphere n => I (f u) * fderiv ℝ f u u) ≤ _ at hle
+  simp only [Function.comp_apply] at hibp
+  rw [hibp] at hle
+  have h := mul_le_mul_of_nonneg_left hle (sq_nonneg b)
+  nlinarith
+
+lemma smooth_convex_curvature {n : ℕ} (hn : 2 ≤ n) [NeZero n]
+    {f : Space n → ℝ} {G I : ℝ → ℝ} {b : ℝ}
+    (hf : ContDiff ℝ 2 f) (hc : ConvexOn ℝ univ f) (heven : Function.Even f)
+    (hG : ContDiff ℝ 1 G) (hI : ContDiff ℝ 1 I)
+    (hpos : ∀ z, 0 ≤ I z) (hrel : ∀ z, deriv G z ^ 2 = b ^ 2 * deriv I z) :
+    (2 * n : ℝ) * variance (fun u : Sphere n => G (f u)) + (2 * n + 8 : ℝ) *
+      ‖Q n (toH n (restrictContinuous (G ∘ f) (hG.continuous.comp hf.continuous)))‖ ^ 2 ≤
+      ((n : ℝ) - 1) * b ^ 2 * mean (fun u : Sphere n => I (f u) * fderiv ℝ f u u) := by
+  have he : Function.Even (G ∘ f) := fun x => by simp only [Function.comp_apply]; rw [heven x]
+  exact (even_spectral_gap hn (hG.comp (hf.of_le (by norm_num))) he).trans
+    (smooth_convex_energy_bound hf hc hG hI hpos hrel)
+
+end PettyProjection.Spherical
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology Function
+open scoped NNReal ENNReal RealInnerProductSpace Pointwise Convolution
+namespace PettyProjection.Spherical
+
+lemma continuous_mean {n : ℕ} [NeZero n] :
+    Continuous (fun f : C(Sphere n, ℝ) => mean f) := by
+  have he : (fun f : C(Sphere n, ℝ) => mean f) =
+      (fun f => ⟪toH n f, toH n (ContinuousMap.const (Sphere n) 1)⟫) := by
+    funext f
+    rw [inner_toH]
+    simp only [ContinuousMap.const_apply, mul_one]
+  rw [he]
+  exact (toH n).continuous.inner continuous_const
+
+lemma continuous_variance {n : ℕ} [NeZero n] :
+    Continuous (fun f : C(Sphere n, ℝ) => variance f) := by
+  have he : (fun f : C(Sphere n, ℝ) => variance f) =
+      (fun f => ‖toH n f‖ ^ 2 - (mean f) ^ 2) := by
+    funext f
+    rw [norm_toH_sq]
+    rfl
+  rw [he]
+  exact ((toH n).continuous.norm.pow 2).sub (continuous_mean.pow 2)
+
+def shrinkingBump (n j : ℕ) : ContDiffBump (0 : Space n) where
+  rIn := (1 / ((j : ℝ) + 1)) / 2
+  rOut := 1 / ((j : ℝ) + 1)
+  rIn_pos := by positivity
+  rIn_lt_rOut := by
+    have h : 0 < 1 / ((j : ℝ) + 1) := by positivity
+    linarith
+
+lemma seminorm_approximation {n : ℕ} (g : Seminorm ℝ (Space n)) :
+    ∃ F R : ℕ → C(Sphere n, ℝ),
+      (∀ j, F j = restrictContinuous (normSmooth g (shrinkingBump n j))
+        (normSmooth_contDiff g _).continuous) ∧
+      (∀ j u, R j u = fderiv ℝ (normSmooth g (shrinkingBump n j)) u u) ∧
+      Tendsto F atTop (𝓝 (restrictContinuous g (seminorm_continuous n g))) ∧
+      Tendsto R atTop (𝓝 (restrictContinuous g (seminorm_continuous n g))) := by
+  let F : ℕ → C(Sphere n, ℝ) := fun j => restrictContinuous
+    (normSmooth g (shrinkingBump n j)) (normSmooth_contDiff g _).continuous
+  let R : ℕ → C(Sphere n, ℝ) := fun j => ⟨fun u =>
+    fderiv ℝ (normSmooth g (shrinkingBump n j)) u u,
+    ((normSmooth_contDiff g _).continuous_fderiv (by norm_num)).comp continuous_subtype_val
+      |>.clm_apply continuous_subtype_val⟩
+  refine ⟨F, R, fun _ => rfl, fun _ _ => rfl, ?_, ?_⟩
+  · obtain ⟨C, hC⟩ := seminorm_lipschitz n g
+    apply tendsto_iff_dist_tendsto_zero.mpr
+    exact squeeze_zero (fun _ => dist_nonneg) (fun j =>
+      (ContinuousMap.dist_le (mul_nonneg C.coe_nonneg (shrinkingBump n j).rOut_pos.le)).mpr
+        (fun u => normSmooth_dist_le g _ hC u))
+      (by simpa only [mul_zero, shrinkingBump] using
+        (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).const_mul (C : ℝ))
+  · obtain ⟨C, hC⟩ := seminorm_lipschitz n g
+    apply tendsto_iff_dist_tendsto_zero.mpr
+    exact squeeze_zero (fun _ => dist_nonneg) (fun j =>
+      (ContinuousMap.dist_le (mul_nonneg (mul_nonneg (by norm_num) C.coe_nonneg)
+        (shrinkingBump n j).rOut_pos.le)).mpr
+        (fun u => normSmooth_radial_dist_le g _ hC u))
+      (by simpa only [mul_zero, shrinkingBump] using
+        (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).const_mul (2 * C : ℝ))
+
+/-- The nonsmooth curvature inequality, proved by actual even convolution of
+an arbitrary seminorm. No smoothness, homogeneity of approximants, or assumed
+spectral interface is required. -/
+theorem seminorm_curvature {n : ℕ} (hn : 2 ≤ n) [NeZero n]
+    (g : Seminorm ℝ (Space n)) {G I : ℝ → ℝ} {b : ℝ}
+    (hG : ContDiff ℝ 1 G) (hI : ContDiff ℝ 1 I)
+    (hpos : ∀ z, 0 ≤ I z) (hrel : ∀ z, deriv G z ^ 2 = b ^ 2 * deriv I z) :
+    (2 * n : ℝ) * variance (fun u : Sphere n => G (g u)) + (2 * n + 8 : ℝ) *
+      ‖Q n (toH n (restrictContinuous (G ∘ g) (hG.continuous.comp (seminorm_continuous n g))))‖ ^ 2 ≤
+      ((n : ℝ) - 1) * b ^ 2 * mean (fun u : Sphere n => I (g u) * g u) := by
+  obtain ⟨F, R, hF, hR, hFt, hRt⟩ := seminorm_approximation g
+  let Gc : C(ℝ, ℝ) := ⟨G, hG.continuous⟩
+  let Ic : C(ℝ, ℝ) := ⟨I, hI.continuous⟩
+  have hGt := Gc.continuous_postcomp.tendsto _ |>.comp hFt
+  have hIt := Ic.continuous_postcomp.tendsto _ |>.comp hFt
+  have hl := (continuous_variance.tendsto _ |>.comp hGt).const_mul (2 * n : ℝ)
+  have hq := (((Q n).continuous.comp (toH n).continuous).tendsto _ |>.comp hGt).norm.pow 2
+  have hr := (continuous_mean.tendsto _ |>.comp (hIt.mul hRt)).const_mul
+    (((n : ℝ) - 1) * b ^ 2)
+  apply le_of_tendsto_of_tendsto' (hl.add (hq.const_mul (2 * n + 8 : ℝ))) hr
+  intro j
+  have h := smooth_convex_curvature hn (normSmooth_contDiff g (shrinkingBump n j))
+    (normSmooth_convex g _) (normSmooth_even g _) hG hI hpos hrel
+  simp only [Function.comp_apply, hF, hR, Gc, Ic, ContinuousMap.coe_mk,
+    ContinuousMap.coe_mul, Pi.mul_def,
+    ContinuousMap.comp, restrictContinuous]
+  convert h using 1
+  rfl
+
+end PettyProjection.Spherical
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/ProjectionBodies/NormEstimates.lean b/lean/OAI/Geometry/ProjectionBodies/NormEstimates.lean
new file mode 100644
index 0000000000000000000000000000000000000000..c57446ef451acb3f9187e2731237f2cd2b30a4d0
--- /dev/null
+++ b/lean/OAI/Geometry/ProjectionBodies/NormEstimates.lean
@@ -0,0 +1,392 @@
+import OAI.Geometry.ProjectionBodies.ScalarCalculus
+
+namespace OAI
+
+universe uE
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology Function
+open scoped ENNReal RealInnerProductSpace
+namespace PettyProjection.Spherical
+
+lemma mean_add {n : ℕ} [NeZero n] {f g : Sphere n → ℝ} (hf : Continuous f) (hg : Continuous g) :
+    mean (fun u => f u + g u) = mean f + mean g :=
+  integral_add (continuous_integrable hf) (continuous_integrable hg)
+
+lemma mean_sub {n : ℕ} [NeZero n] {f g : Sphere n → ℝ} (hf : Continuous f) (hg : Continuous g) :
+    mean (fun u => f u - g u) = mean f - mean g :=
+  integral_sub (continuous_integrable hf) (continuous_integrable hg)
+
+lemma mean_const_mul {n : ℕ} (c : ℝ) (f : Sphere n → ℝ) :
+    mean (fun u => c * f u) = c * mean f := integral_const_mul c f
+
+lemma mean_mul_const {n : ℕ} (c : ℝ) (f : Sphere n → ℝ) :
+    mean (fun u => f u * c) = mean f * c := integral_mul_const c f
+
+lemma mean_div_const {n : ℕ} (c : ℝ) (f : Sphere n → ℝ) :
+    mean (fun u => f u / c) = mean f / c := integral_div c f
+
+lemma mean_mono {n : ℕ} [NeZero n] {f g : Sphere n → ℝ} (hf : Continuous f) (hg : Continuous g)
+    (h : ∀ u, f u ≤ g u) : mean f ≤ mean g :=
+  integral_mono (continuous_integrable hf) (continuous_integrable hg) h
+
+lemma mean_nonneg {n : ℕ} {f : Sphere n → ℝ} (h : ∀ u, 0 ≤ f u) : 0 ≤ mean f :=
+  integral_nonneg h
+
+lemma mean_centered_square {n : ℕ} [NeZero n] (f : C(Sphere n, ℝ)) :
+    mean (fun u => (f u - mean f) ^ 2) = variance f := by
+  have hi : (fun u => (f u - mean f) ^ 2) =
+      (fun u => (f u) ^ 2 - 2 * mean f * f u + (mean f) ^ 2) := by
+    funext u; ring
+  rw [hi, mean_add (f := fun u => (f u)^2 - 2 * mean f * f u)
+      (g := fun _ => (mean f)^2)
+      ((f.continuous.pow 2).sub (continuous_const.mul f.continuous)) continuous_const,
+    mean_sub (f := fun u => (f u)^2) (g := fun u => 2 * mean f * f u)
+      (f.continuous.pow 2) (continuous_const.mul f.continuous),
+    mean_const_mul, mean_const, variance]
+  ring
+
+lemma norm_Q_le_variance {n : ℕ} [NeZero n] (f : C(Sphere n, ℝ)) :
+    ‖Q n (toH n f)‖ ^ 2 ≤ variance f := by
+  simpa only [mean_centered_square] using norm_Q_le_centered n f (mean f)
+
+lemma weighted_square_bound {a b w : ℝ} (hw : 0 < w) (hw1 : w < 1) :
+    (a+b)^2 ≤ a^2/w + b^2/(1-w) := by
+  have h1 : 0 < 1-w := sub_pos.mpr hw1
+  apply (mul_le_mul_iff_of_pos_right (mul_pos hw h1)).mp
+  have he : (a^2/w + b^2/(1-w)) * (w*(1-w)) = a^2*(1-w)+b^2*w := by
+    field_simp
+  rw [he]
+  nlinarith only [sq_nonneg ((1-w)*a-w*b)]
+
+lemma weighted_norm_bound {E : Type uE} [SeminormedAddCommGroup E] (a b : E)
+    {w : ℝ} (hw : 0 < w) (hw1 : w < 1) :
+    ‖a+b‖^2 ≤ ‖a‖^2/w + ‖b‖^2/(1-w) :=
+  (sq_le_sq₀ (norm_nonneg _) (add_nonneg (norm_nonneg _) (norm_nonneg _))).mpr
+    (norm_add_le a b) |>.trans (weighted_square_bound hw hw1)
+
+lemma scalar_correction {n : ℕ} (hn : 4 ≤ n) {m t : ℝ}
+    (hm : 0 ≤ m) (ht : 0 ≤ t) (htb : t ≤ Scalar.beta n * m) :
+    Scalar.coefficient n * ((2*n : ℝ) * t^2 / (Scalar.weight n * (2*n+8)) -
+      (m-t)^2/(1-Scalar.weight n)) ≤
+      (2*(Scalar.exponent n : ℝ)-1)/(Scalar.exponent n : ℝ)^2 * m^2 := by
+  have hb := (Scalar.beta_pos n).le
+  have hb1 := (Scalar.beta_lt_one n).le
+  have hw := Scalar.weight_pos n
+  have hw1 := Scalar.weight_lt_one n
+  have hden : 0 < Scalar.weight n * (2*n+8 : ℝ) := mul_pos hw (by positivity)
+  have hs1 : t^2 ≤ (Scalar.beta n)^2 * m^2 := by
+    simpa only [mul_pow] using (sq_le_sq₀ ht (mul_nonneg hb hm)).mpr htb
+  have ht1 : t ≤ m := htb.trans (by nlinarith)
+  have hs2 : (1-Scalar.beta n)^2*m^2 ≤ (m-t)^2 := by
+    have hle : (1-Scalar.beta n)*m ≤ m-t := by nlinarith
+    simpa only [mul_pow] using (sq_le_sq₀ (mul_nonneg (sub_nonneg.mpr hb1) hm)
+      (sub_nonneg.mpr ht1)).mpr hle
+  have h1 := div_le_div_of_nonneg_right
+    (mul_le_mul_of_nonneg_left hs1 (by positivity : 0 ≤ (2*n : ℝ))) hden.le
+  have h2 := div_le_div_of_nonneg_right hs2 (sub_nonneg.mpr hw1.le)
+  have h3 := mul_le_mul_of_nonneg_left (sub_le_sub h1 h2) (Scalar.coefficient_pos n).le
+  have h4 := mul_le_mul_of_nonneg_right (Scalar.mean_bound n hn) (sq_nonneg m)
+  apply h3.trans
+  convert h4 using 1; ring
+
+lemma scalar_curvature {n : ℕ} (hn : 4 ≤ n) [NeZero n] (g : Seminorm ℝ (Space n)) :
+    (2 * n : ℝ) * variance (fun u : Sphere n => Scalar.G n (g u)) + (2 * n + 8 : ℝ) *
+      ‖Q n (toH n (restrictContinuous (Scalar.G n ∘ g)
+        ((Scalar.contDiff_G hn).continuous.comp (seminorm_continuous n g))))‖ ^ 2 ≤
+      ((n : ℝ) - 1) * Scalar.beta n ^ 2 *
+        mean (fun u : Sphere n => Scalar.I n (g u) * g u) :=
+  seminorm_curvature (by omega) g (Scalar.contDiff_G hn) (Scalar.contDiff_I hn)
+    (Scalar.I_nonneg hn) (Scalar.derivative_relation hn)
+
+end PettyProjection.Spherical
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology Function
+open scoped ENNReal RealInnerProductSpace
+namespace PettyProjection.Spherical
+open Scalar
+
+lemma continuous_e_comp {n : ℕ} (f : C(Sphere n, ℝ)) (hf : ∀ u, 0 < f u) :
+    Continuous (fun u => e n (f u)) :=
+  (((f.continuous.inv₀ (fun u => (hf u).ne')).pow n).sub continuous_const).div_const _
+
+lemma continuous_M_comp {n : ℕ} (f : C(Sphere n, ℝ)) (hf : ∀ u, 0 < f u) :
+    Continuous (fun u => M n (f u)) :=
+  ((f.continuous.pow _).sub continuous_const).add
+    (continuous_const.mul (continuous_e_comp f hf))
+
+lemma mean_e_zero {n : ℕ} [NeZero n] (f : C(Sphere n, ℝ)) (hf : ∀ u, 0 < f u)
+    (hnorm : mean (fun u => (f u)⁻¹ ^ n) = 1) : mean (fun u => e n (f u)) = 0 := by
+  unfold e
+  rw [mean_div_const, mean_sub (f := fun u => (f u)⁻¹^n) (g := fun _ => 1) ((f.continuous.inv₀ (fun u => (hf u).ne')).pow n)
+    continuous_const, mean_const, hnorm, sub_self, zero_div]
+
+lemma mean_M_of_normalized {n : ℕ} [NeZero n] (f : C(Sphere n, ℝ)) (hf : ∀ u, 0 < f u)
+    (hnorm : mean (fun u => (f u)⁻¹ ^ n) = 1) :
+    mean (fun u => M n (f u)) = mean (fun u => (f u) ^ exponent n) - 1 := by
+  unfold M
+  rw [mean_add (f := fun u => (f u)^exponent n - 1)
+      (g := fun u => (exponent n : ℝ) * e n (f u))
+      ((f.continuous.pow _).sub continuous_const) (continuous_const.mul (continuous_e_comp f hf)),
+    mean_sub (f := fun u => (f u)^exponent n) (g := fun _ => 1) (f.continuous.pow _) continuous_const, mean_const, mean_const_mul,
+    mean_e_zero f hf hnorm, mul_zero, add_zero]
+
+lemma mean_W_of_normalized {n : ℕ} [NeZero n] (f : C(Sphere n, ℝ)) (hf : ∀ u, 0 < f u)
+    (hnorm : mean (fun u => (f u)⁻¹ ^ n) = 1) :
+    mean (fun u => W n (f u)) = mean (fun u => (f u)^exponent n) -
+      mean (fun u => (f u)^(exponent n-1)) := by
+  unfold W
+  rw [mean_add (f := fun u => (f u)^exponent n - (f u)^(exponent n-1))
+      (g := fun u => e n (f u)) ((f.continuous.pow _).sub (f.continuous.pow _))
+      (continuous_e_comp f hf),
+    mean_sub (f := fun u => (f u)^exponent n) (g := fun u => (f u)^(exponent n-1)) (f.continuous.pow _) (f.continuous.pow _), mean_e_zero f hf hnorm, add_zero]
+
+lemma mean_first_power_ge_one {n : ℕ} (hn : 0 < n) [NeZero n]
+    (f : C(Sphere n, ℝ)) (hf : ∀ u, 0 < f u)
+    (hnorm : mean (fun u => (f u)⁻¹ ^ n) = 1) : 1 ≤ mean f := by
+  have hi := mean_mono (f := fun u => 1-f u) (g := fun u => e n (f u)) (continuous_const.sub f.continuous) (continuous_e_comp f hf)
+    (fun u => e_lower hn (hf u))
+  rw [mean_sub (f := fun _ => 1) (g := f) continuous_const f.continuous, mean_const, mean_e_zero f hf hnorm] at hi
+  linarith
+
+lemma norm_mean_ranges {n : ℕ} (hn : 4 ≤ n) [NeZero n]
+    (f : C(Sphere n, ℝ)) (hf : ∀ u, 0 < f u)
+    (hnorm : mean (fun u => (f u)⁻¹ ^ n) = 1) :
+    let m := mean (fun u => (f u) ^ exponent n) - 1
+    let ell := mean (fun u => (f u) ^ (exponent n - 1)) - 1
+    let t := mean (fun u => G n (f u))
+    0 ≤ m ∧ 0 ≤ ell ∧ ell ≤ ((exponent n : ℝ) - 1) / exponent n * m ∧
+      0 ≤ t ∧ t ≤ beta n * m := by
+  dsimp only
+  have hm : 0 ≤ mean (fun u => (f u)^exponent n) - 1 := by
+    rw [← mean_M_of_normalized f hf hnorm]
+    exact mean_nonneg (fun u => M_nonneg (by omega) (hf u))
+  have ht : 0 ≤ mean (fun u => G n (f u)) :=
+    mean_nonneg (fun u => (G_bounds (by omega) (hf u)).1)
+  have htb := mean_mono (f := fun u => G n (f u)) (g := fun u => beta n*M n (f u)) ((contDiff_G hn).continuous.comp f.continuous)
+    (continuous_const.mul (continuous_M_comp f hf)) (fun u => (G_bounds (by omega) (hf u)).2)
+  rw [mean_const_mul, mean_M_of_normalized f hf hnorm] at htb
+  refine ⟨hm, ?_, ?_, ht, htb⟩
+  · by_cases hn4 : n = 4
+    · simp [exponent, hn4, mean_const]
+    · simpa only [exponent, ite_eq_right hn4, Nat.reduceSub, pow_one, sub_nonneg] using
+        mean_first_power_ge_one (by omega) f hf hnorm
+  · by_cases hn4 : n = 4
+    · simp [exponent, hn4, mean_const]
+    · simp only [exponent, ite_eq_right hn4, Nat.reduceSub, Nat.cast_ofNat, pow_one]
+      have h := mean_mono (f := fun u => 2*f u) (g := fun u => (f u)^2+1)
+        (continuous_const.mul f.continuous) ((f.continuous.pow 2).add continuous_const)
+        (fun u => by nlinarith only [sq_nonneg (f u - 1)])
+      rw [mean_const_mul, mean_add (f := fun u =>(f u)^2) (g := fun _ => 1) (f.continuous.pow 2) continuous_const, mean_const] at h
+      linarith
+
+end PettyProjection.Spherical
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology Function
+open scoped ENNReal RealInnerProductSpace
+namespace PettyProjection.Spherical
+open Scalar
+
+lemma continuous_D {n : ℕ} (hn : 4 ≤ n) : Continuous (D n) :=
+  ((continuous_id.pow _).sub continuous_const).sub (contDiff_G hn).continuous
+
+lemma continuous_L {n : ℕ} (hn : 4 ≤ n) : Continuous (L n) :=
+  continuous_const.mul (((((continuous_const.mul continuous_id).mul
+    (contDiff_I hn).continuous).sub (continuous_const.mul ((contDiff_G hn).continuous.pow 2))).div_const _).add
+      (((continuous_D hn).pow 2).div_const _))
+
+lemma continuous_W_comp {n : ℕ} (f : C(Sphere n, ℝ)) (hf : ∀ u, 0 < f u) :
+    Continuous (fun u => W n (f u)) :=
+  ((f.continuous.pow _).sub (f.continuous.pow _)).add (continuous_e_comp f hf)
+
+lemma mean_D {n : ℕ} (hn : 4 ≤ n) [NeZero n] (f : C(Sphere n, ℝ)) :
+    mean (fun u => D n (f u)) = mean (fun u => (f u)^exponent n) - 1 -
+      mean (fun u => G n (f u)) := by
+  unfold D
+  rw [mean_sub (f := fun u => (f u)^exponent n - 1) (g := fun u => G n (f u))
+    ((f.continuous.pow _).sub continuous_const) ((contDiff_G hn).continuous.comp f.continuous),
+    mean_sub (f := fun u => (f u)^exponent n) (g := fun _ => 1)
+      (f.continuous.pow _) continuous_const, mean_const]
+
+lemma mean_L {n : ℕ} (hn : 4 ≤ n) [NeZero n] (f : C(Sphere n, ℝ)) :
+    mean (fun u => L n (f u)) = coefficient n *
+      ((((n : ℝ)-1)*beta n^2 * mean (fun u => I n (f u)*f u) -
+        2*n*mean (fun u => (G n (f u))^2))/(weight n*(2*n+8)) +
+          mean (fun u => (D n (f u))^2)/(1-weight n)) := by
+  have hG := (contDiff_G hn).continuous.comp f.continuous
+  have hI := (contDiff_I hn).continuous.comp f.continuous
+  have hD := (continuous_D hn).comp f.continuous
+  unfold L
+  rw [mean_const_mul,
+    mean_add (f := fun u => (((n : ℝ)-1)*beta n^2*f u*I n (f u)-2*n*(G n (f u))^2)/(weight n*(2*n+8)))
+      (g := fun u => (D n (f u))^2/(1-weight n))
+      (((continuous_const.mul f.continuous).mul hI |>.sub (continuous_const.mul (hG.pow 2))).div_const _)
+      ((hD.pow 2).div_const _),
+    mean_div_const, mean_div_const,
+    mean_sub (f := fun u => ((n : ℝ)-1)*beta n^2*f u*I n (f u))
+      (g := fun u => 2*n*(G n (f u))^2)
+      ((continuous_const.mul f.continuous).mul hI) (continuous_const.mul (hG.pow 2)),
+    mean_const_mul]
+  congr 2
+  congr 1
+  have he : (fun u : Sphere n => ((n : ℝ)-1)*beta n^2*f u*I n (f u)) =
+      (fun u => ((n : ℝ)-1)*beta n^2*(I n (f u)*f u)) := by funext u; ring
+  rw [he, mean_const_mul]
+
+lemma Q_power_split {n : ℕ} (hn : 4 ≤ n) [NeZero n] (f : C(Sphere n, ℝ)) :
+    Q n (toH n ⟨fun u => (f u)^exponent n, f.continuous.pow _⟩) =
+      Q n (toH n ⟨fun u => G n (f u), (contDiff_G hn).continuous.comp f.continuous⟩) +
+      Q n (toH n ⟨fun u => D n (f u), (continuous_D hn).comp f.continuous⟩) := by
+  have he : (⟨fun u => (f u)^exponent n, f.continuous.pow _⟩ : C(Sphere n, ℝ)) =
+      ContinuousMap.const (Sphere n) 1 +
+      (⟨fun u => G n (f u), (contDiff_G hn).continuous.comp f.continuous⟩ : C(Sphere n, ℝ)) +
+      ⟨fun u => D n (f u), (continuous_D hn).comp f.continuous⟩ := by
+    ext u; simp only [ContinuousMap.add_apply, ContinuousMap.const_apply, ContinuousMap.coe_mk, D]; ring
+  rw [he, map_add, map_add, map_add, map_add, Q_constant, zero_add]
+
+/-- Exact expectation reduction in the proof of the norm theorem. -/
+lemma norm_reduction {n : ℕ} (hn : 4 ≤ n) [NeZero n] (g : Seminorm ℝ (Space n)) :
+    coefficient n * ‖Q n (toH n (restrictContinuous (fun x => g x ^ exponent n)
+      ((seminorm_continuous n g).pow _)))‖^2 ≤
+      mean (fun u : Sphere n => L n (g u)) + coefficient n *
+        (2*n*(mean (fun u : Sphere n => G n (g u)))^2/(weight n*(2*n+8)) -
+          (mean (fun u : Sphere n => D n (g u)))^2/(1-weight n)) := by
+  let f := restrictContinuous g (seminorm_continuous n g)
+  let F : C(Sphere n, ℝ) := ⟨fun u => (f u)^exponent n, f.continuous.pow _⟩
+  let GG : C(Sphere n, ℝ) := ⟨fun u => G n (f u), (contDiff_G hn).continuous.comp f.continuous⟩
+  let DD : C(Sphere n, ℝ) := ⟨fun u => D n (f u), (continuous_D hn).comp f.continuous⟩
+  have hs : Q n (toH n F) = Q n (toH n GG) + Q n (toH n DD) := Q_power_split hn f
+  have hq := weighted_norm_bound (Q n (toH n GG)) (Q n (toH n DD))
+    (weight_pos n) (weight_lt_one n)
+  rw [← hs] at hq
+  have hd := norm_Q_le_variance DD
+  have hc := scalar_curvature hn g
+  change (2*n : ℝ)*variance GG + (2*n+8 : ℝ)*‖Q n (toH n GG)‖^2 ≤
+    ((n : ℝ)-1)*beta n^2 * mean (fun u => I n (f u)*f u) at hc
+  have hp : 0 < (2*n+8 : ℝ) := by positivity
+  have hcg : ‖Q n (toH n GG)‖^2 ≤
+      (((n : ℝ)-1)*beta n^2 * mean (fun u => I n (f u)*f u) -
+        (2*n : ℝ)*variance GG)/(2*n+8) := (le_div_iff₀ hp).mpr (by linarith)
+  have hbound := hq.trans (add_le_add (div_le_div_of_nonneg_right hcg (weight_pos n).le)
+    (div_le_div_of_nonneg_right hd (sub_nonneg.mpr (weight_lt_one n).le)))
+  have hbound' := mul_le_mul_of_nonneg_left hbound (coefficient_pos n).le
+  change coefficient n * ‖Q n (toH n F)‖^2 ≤ _
+  refine hbound'.trans_eq ?_
+  rw [show mean (fun u : Sphere n => L n (g u)) = mean (fun u => L n (f u)) from rfl,
+    mean_L hn f]
+  change coefficient n *
+      (((((n : ℝ)-1)*beta n^2 * mean (fun u => I n (f u)*f u) -
+        (2*n : ℝ)*(mean (fun u => (G n (f u))^2) - (mean (fun u => G n (f u)))^2)) /
+        (2*n+8))/weight n +
+      (mean (fun u => (D n (f u))^2) - (mean (fun u => D n (f u)))^2)/(1-weight n)) = _
+  dsimp only [f, restrictContinuous, ContinuousMap.coe_mk]
+  simp only [div_eq_mul_inv, mul_inv_rev]
+  ring
+
+lemma mean_square_remainder {k : ℝ} (hk : 0 < k) {m ell : ℝ}
+    (hm : 0 ≤ m) (hl : 0 ≤ ell) (hkm : 0 ≤ k-1) (h : ell ≤ (k-1)/k*m) :
+    (2*k-1)/k^2*m^2 ≤ m^2-ell^2 := by
+  have hs : ell^2 ≤ ((k-1)/k*m)^2 :=
+    (sq_le_sq₀ hl (mul_nonneg (div_nonneg hkm hk.le) hm)).mpr h
+  have he : (2*k-1)/k^2*m^2 = m^2-((k-1)/k*m)^2 := by field_simp; ring
+  rw [he]; linarith
+
+end PettyProjection.Spherical
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology Function
+open scoped ENNReal RealInnerProductSpace
+namespace PettyProjection.Spherical
+open Scalar
+
+lemma scalar_expectation {n : ℕ} (hn : 4 ≤ n) [NeZero n]
+    (f : C(Sphere n, ℝ)) (hf : ∀ u, 0 < f u) :
+    mean (fun u => L n (f u)) ≤ 2*mean (fun u => W n (f u)) ∧
+      (mean (fun u => L n (f u)) = 2*mean (fun u => W n (f u)) ↔ ∀ u, f u = 1) := by
+  have hL := (continuous_L hn).comp f.continuous
+  have hW : Continuous (fun u => 2*W n (f u)) := continuous_const.mul (continuous_W_comp f hf)
+  have hp : ∀ u, L n (f u) ≤ 2*W n (f u) := fun u => (pointwise n hn (hf u)).1
+  have hm := mean_mono (f := fun u => L n (f u)) (g := fun u => 2*W n (f u)) hL hW hp
+  rw [mean_const_mul] at hm
+  refine ⟨hm, ?_, ?_⟩
+  · intro he
+    have hae : (fun u => L n (f u)) =ᵐ[sigma n] (fun u => 2*W n (f u)) := by
+      apply (integral_eq_iff_of_ae_le (continuous_integrable hL) (continuous_integrable hW)
+        (Eventually.of_forall hp)).mp
+      change mean (fun u => L n (f u)) = mean (fun u => 2*W n (f u))
+      rw [mean_const_mul]; exact he
+    have heq := (hL.ae_eq_iff_eq (sigma n) hW).mp hae
+    intro u
+    exact (pointwise n hn (hf u)).2.mp (congrFun heq u)
+  · intro he
+    have heq : (fun u => L n (f u)) = (fun u => 2*W n (f u)) := by
+      funext u; exact (pointwise n hn (hf u)).2.mpr (he u)
+    rw [heq, mean_const_mul]
+
+/-- The full arbitrary-norm estimate, including strictness.
+It uses the actual spherical measure and the actual quadratic-removing Q. -/
+theorem norm_estimate {n : ℕ} (hn : 4 ≤ n) [NeZero n]
+    (g : Seminorm ℝ (Space n)) (hg : ∀ u : Sphere n, 0 < g u)
+    (hnorm : mean (fun u : Sphere n => (g u)⁻¹^n) = 1) :
+    coefficient n * ‖Q n (toH n (restrictContinuous (fun x => g x ^ exponent n)
+      ((seminorm_continuous n g).pow _)))‖^2 ≤
+      (mean (fun u : Sphere n => (g u)^exponent n))^2 -
+        (mean (fun u : Sphere n => (g u)^(exponent n-1)))^2 ∧
+    (coefficient n * ‖Q n (toH n (restrictContinuous (fun x => g x ^ exponent n)
+      ((seminorm_continuous n g).pow _)))‖^2 =
+      (mean (fun u : Sphere n => (g u)^exponent n))^2 -
+        (mean (fun u : Sphere n => (g u)^(exponent n-1)))^2 ↔ ∀ u : Sphere n, g u = 1) := by
+  let f := restrictContinuous g (seminorm_continuous n g)
+  let m := mean (fun u => (f u)^exponent n)-1
+  let ell := mean (fun u => (f u)^(exponent n-1))-1
+  let t := mean (fun u => G n (f u))
+  obtain ⟨hm, hl, hlm, ht, htb⟩ := norm_mean_ranges hn f hg hnorm
+  have hcorr := scalar_correction hn hm ht htb
+  have hk : 0 < (exponent n : ℝ) := by unfold exponent; split_ifs <;> norm_num
+  have hkm : 0 ≤ (exponent n : ℝ)-1 := by unfold exponent; split_ifs <;> norm_num
+  have hrem := mean_square_remainder hk hm hl hkm hlm
+  have hred := norm_reduction hn g
+  have hred' : coefficient n * ‖Q n (toH n (restrictContinuous (fun x => g x^exponent n)
+      ((seminorm_continuous n g).pow _)))‖^2 ≤
+      mean (fun u => L n (f u)) + m^2-ell^2 := by
+    have hd : mean (fun u : Sphere n => D n (g u)) = m-t := mean_D hn f
+    rw [hd] at hred
+    exact hred.trans (by dsimp only [m, ell, t, f, restrictContinuous, ContinuousMap.coe_mk] at *; linarith)
+  obtain ⟨hsc, hsce⟩ := scalar_expectation hn f hg
+  have hw : mean (fun u => W n (f u)) = m-ell := by
+    rw [mean_W_of_normalized f hg hnorm]
+    dsimp only [m, ell]; ring
+  have hend : 2*(m-ell)+m^2-ell^2 =
+      (mean (fun u : Sphere n => (g u)^exponent n))^2 -
+        (mean (fun u : Sphere n => (g u)^(exponent n-1)))^2 := by
+    dsimp only [m, ell, f, restrictContinuous, ContinuousMap.coe_mk]; ring
+  have hfinal : coefficient n * ‖Q n (toH n (restrictContinuous (fun x => g x^exponent n)
+      ((seminorm_continuous n g).pow _)))‖^2 ≤
+      (mean (fun u : Sphere n => (g u)^exponent n))^2 -
+        (mean (fun u : Sphere n => (g u)^(exponent n-1)))^2 :=
+    hred'.trans (by rw [← hend]; rw [hw] at hsc; linarith)
+  refine ⟨hfinal, ?_, ?_⟩
+  · intro he
+    apply hsce.mp
+    rw [hw]
+    rw [← hend] at he
+    linarith
+  · intro he
+    have hpow : restrictContinuous (fun x => g x^exponent n) ((seminorm_continuous n g).pow _) =
+        ContinuousMap.const (Sphere n) 1 := by ext u; simp [restrictContinuous, he u]
+    rw [hpow, Q_constant, norm_zero, zero_pow (by norm_num : 2 ≠ 0), mul_zero]
+    have hem : (fun u : Sphere n => (g u)^exponent n) = (fun _ => (1 : ℝ)) := by
+      funext u; simp [he u]
+    have hel : (fun u : Sphere n => (g u)^(exponent n-1)) = (fun _ => (1 : ℝ)) := by
+      funext u; simp [he u]
+    rw [hem, hel, sub_self]
+
+end PettyProjection.Spherical
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/ProjectionBodies/NormRigidity.lean b/lean/OAI/Geometry/ProjectionBodies/NormRigidity.lean
new file mode 100644
index 0000000000000000000000000000000000000000..7503df81068307041b037e386e376b983f89af1e
--- /dev/null
+++ b/lean/OAI/Geometry/ProjectionBodies/NormRigidity.lean
@@ -0,0 +1,601 @@
+import OAI.Geometry.ProjectionBodies.SignSpectrum
+
+namespace OAI
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology Function
+open scoped NNReal ENNReal RealInnerProductSpace Gradient
+namespace PettyProjection.Spherical
+
+lemma norm_gradient_eq {n : ℕ} (f : Space n → ℝ) (x : Space n) :
+    ‖∇ f x‖ = ‖fderiv ℝ f x‖ :=
+  (InnerProductSpace.toDual ℝ (Space n)).symm.norm_map _
+
+lemma gradient_pow {n : ℕ} {f : Space n → ℝ} {x : Space n}
+    (hf : DifferentiableAt ℝ f x) (k : ℕ) :
+    ∇ (fun y => f y^k) x = ((k : ℝ)*f x^(k-1)) • ∇ f x := by
+  apply (InnerProductSpace.toDual ℝ (Space n)).injective
+  rw [toDual_gradient, map_smul, toDual_gradient]
+  simpa only [nsmul_eq_mul] using (hf.hasFDerivAt.pow k).fderiv
+
+lemma signField_eq_gradient {n : ℕ} (k : ℝ) (f : Space n → ℝ) (u : Sphere n) :
+    signField k f u = f u • (u : Space n) +
+      k⁻¹ • (∇ f u - ⟪∇ f u,(u : Space n)⟫ • (u : Space n)) := by
+  rw [inner_gradient_left]
+  rfl
+
+lemma norm_signField_le {n : ℕ} (k : ℝ) (f : Space n → ℝ) (u : Sphere n) :
+    ‖signField k f u‖ ≤ |f u| + 2*|k⁻¹| *‖fderiv ℝ f u‖ := by
+  rw [signField_eq_gradient]
+  calc
+    _ ≤ ‖f u • (u : Space n)‖ + ‖k⁻¹ • (∇ f u - ⟪∇ f u,(u : Space n)⟫ • (u : Space n))‖ := norm_add_le _ _
+    _ ≤ |f u| + |k⁻¹| *(‖∇ f u‖ + |⟪∇ f u,(u : Space n)⟫|) := by
+      simp only [norm_smul, norm_coe, mul_one, Real.norm_eq_abs]
+      gcongr
+      have h := norm_sub_le (∇ f u) (⟪∇ f u,(u : Space n)⟫ • (u : Space n))
+      simpa only [norm_smul, norm_coe, mul_one, Real.norm_eq_abs] using h
+    _ ≤ |f u| + |k⁻¹| *(‖∇ f u‖ + ‖∇ f u‖) := by
+      gcongr
+      simpa only [norm_coe, mul_one] using abs_real_inner_le_norm (∇ f u) (u : Space n)
+    _ = _ := by rw [norm_gradient_eq]; ring
+
+lemma normSmooth_powerField_bound {n : ℕ} (g : Seminorm ℝ (Space n)) (k : ℕ) :
+    ∃ B : ℝ, 0 ≤ B ∧ ∀ j (u : Sphere n),
+      ‖signField (k : ℝ) (fun x => normSmooth g (shrinkingBump n j) x^k) u‖ ≤ B := by
+  obtain ⟨C,hC⟩ := seminorm_lipschitz n g
+  let M : ℝ := 2*C
+  have hM : 0 ≤ M := by positivity
+  have hv (j : ℕ) (u : Sphere n) : |normSmooth g (shrinkingBump n j) u| ≤ M := by
+    have hg : |g u| ≤ C := by
+      have h := hC.dist_le_mul (u : Space n) 0
+      simpa only [map_zero, dist_zero_right, norm_coe, mul_one, Real.norm_eq_abs] using h
+    have he := normSmooth_dist_le g (shrinkingBump n j) hC u
+    have hr : (shrinkingBump n j).rOut ≤ 1 := by
+      change 1 / ((j : ℝ)+1) ≤ 1
+      exact (div_le_one (by positivity)).mpr (by linarith [Nat.cast_nonneg (α := ℝ) j])
+    have hc := mul_le_mul_of_nonneg_left hr C.coe_nonneg
+    have ht := abs_add_le (normSmooth g (shrinkingBump n j) u-g u) (g u)
+    rw [sub_add_cancel] at ht
+    rw [Real.dist_eq] at he
+    dsimp [M]
+    linarith
+  have hd (j : ℕ) (u : Sphere n) :
+      ‖fderiv ℝ (fun x => normSmooth g (shrinkingBump n j) x^k) u‖ ≤
+        (k : ℝ) * M^(k-1) * C := by
+    have hdiff := (normSmooth_contDiff g (shrinkingBump n j)).differentiable (by norm_num) (u : Space n)
+    rw [(hdiff.hasFDerivAt.pow k).fderiv, norm_smul, Real.norm_eq_abs,
+      nsmul_eq_mul, abs_mul, abs_of_nonneg (Nat.cast_nonneg k : (0 : ℝ) ≤ k), abs_pow]
+    exact mul_le_mul (mul_le_mul_of_nonneg_left (pow_le_pow_left₀ (abs_nonneg _) (hv j u) _) (Nat.cast_nonneg _))
+      (norm_fderiv_le_of_lipschitz ℝ (normSmooth_lipschitz g _ hC)) (norm_nonneg _) (by positivity)
+  refine ⟨M^k + 2*|(k : ℝ)⁻¹| *((k : ℝ)*M^(k-1)*C), by positivity, ?_⟩
+  intro j u
+  exact (norm_signField_le _ _ u).trans (add_le_add
+    (by simpa only [abs_pow] using pow_le_pow_left₀ (abs_nonneg _) (hv j u) k)
+    (mul_le_mul_of_nonneg_left (hd j u) (by positivity)))
+
+lemma normSmooth_powerField_tendsto {n : ℕ} (g : Seminorm ℝ (Space n)) (k : ℕ) :
+    ∀ᵐ (u : Sphere n) ∂sigma n,
+      Tendsto (fun j => signField (k : ℝ) (fun x => normSmooth g (shrinkingBump n j) x^k) u)
+        atTop (𝓝 (signField (k : ℝ) (fun x => g x^k) u)) := by
+  filter_upwards [seminorm_ae_differentiable_sphere g] with u hu
+  have hg : Tendsto (fun j => ∇ (fun x => normSmooth g (shrinkingBump n j) x^k) u)
+      atTop (𝓝 (∇ (fun x => g x^k) u)) := by
+    simp_rw [gradient_pow ((normSmooth_contDiff g _).differentiable (by norm_num) (u : Space n)) k, gradient_pow hu k]
+    exact (((normSmooth_tendsto g u).pow (k-1)).const_mul (k : ℝ)).smul (normSmooth_gradient_tendsto g hu)
+  simp_rw [signField_eq_gradient]
+  exact ((normSmooth_tendsto g u).pow k).smul_const (u : Space n) |>.add
+    ((hg.sub ((hg.inner (tendsto_const_nhds (x := (u : Space n)))).smul_const (u : Space n))).const_smul (k : ℝ)⁻¹)
+
+end PettyProjection.Spherical
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology Function
+open scoped ENNReal RealInnerProductSpace
+namespace PettyProjection.Spherical
+
+lemma signedPair_tendsto {n : ℕ} [NeZero n] {F G : ℕ → Sphere n → Space n}
+    {F₀ G₀ : Sphere n → Space n} {B C : ℝ} (hB : 0 ≤ B)
+    (hF : ∀ j, Continuous (F j)) (hG : ∀ j, Continuous (G j))
+    (hFB : ∀ j u, ‖F j u‖ ≤ B) (hGC : ∀ j u, ‖G j u‖ ≤ C)
+    (hFlim : ∀ᵐ u ∂sigma n, Tendsto (fun j => F j u) atTop (𝓝 (F₀ u)))
+    (hGlim : ∀ᵐ u ∂sigma n, Tendsto (fun j => G j u) atTop (𝓝 (G₀ u))) :
+    Tendsto (fun j => signedPair (F j) (G j)) atTop (𝓝 (signedPair F₀ G₀)) := by
+  apply tendsto_integral_of_dominated_convergence (fun _ => B*C)
+  · intro j
+    exact (signedPair_integrable (hF j) (hG j)).aestronglyMeasurable
+  · exact integrable_const _
+  · intro j
+    filter_upwards with z
+    rw [Real.norm_eq_abs, abs_mul]
+    exact (mul_le_mul (abs_sign_le _) (abs_real_inner_le_norm _ _) (abs_nonneg _) (by norm_num)).trans
+      (by simpa only [one_mul] using mul_le_mul (hFB j z.1) (hGC j z.2) (norm_nonneg _) hB)
+  · filter_upwards [Measure.quasiMeasurePreserving_fst.ae hFlim,
+      Measure.quasiMeasurePreserving_snd.ae hGlim] with z hzF hzG
+    exact (hzF.inner hzG).const_mul _
+
+lemma ae_bound_of_limit {n : ℕ} {F : ℕ → Sphere n → Space n}
+    {F₀ : Sphere n → Space n} {B : ℝ} (hFB : ∀ j u, ‖F j u‖ ≤ B)
+    (hFlim : ∀ᵐ u ∂sigma n, Tendsto (fun j => F j u) atTop (𝓝 (F₀ u))) :
+    ∀ᵐ u ∂sigma n, ‖F₀ u‖ ≤ B := by
+  filter_upwards [hFlim] with u hu
+  exact le_of_tendsto hu.norm (Eventually.of_forall fun j => hFB j u)
+
+lemma signedPair_integrable_of_ae_bound {n : ℕ} [NeZero n] {F G : Sphere n → Space n}
+    {B C : ℝ} (hB : 0 ≤ B)
+    (hF : AEStronglyMeasurable F (sigma n)) (hG : AEStronglyMeasurable G (sigma n))
+    (hFB : ∀ᵐ u ∂sigma n, ‖F u‖ ≤ B) (hGC : ∀ᵐ u ∂sigma n, ‖G u‖ ≤ C) :
+    Integrable (fun z : Sphere n × Sphere n => Real.sign ⟪(z.1 : Space n),(z.2 : Space n)⟫ *
+      ⟪F z.1,G z.2⟫) ((sigma n).prod (sigma n)) := by
+  apply (integrable_const (B*C)).mono'
+  · exact ((sign_measurable.comp (by fun_prop : Continuous
+      (fun z : Sphere n × Sphere n => ⟪(z.1 : Space n),(z.2 : Space n)⟫)).measurable).aestronglyMeasurable).mul
+        ((hF.comp_quasiMeasurePreserving Measure.quasiMeasurePreserving_fst).inner
+          (hG.comp_quasiMeasurePreserving Measure.quasiMeasurePreserving_snd))
+  · filter_upwards [Measure.quasiMeasurePreserving_fst.ae hFB,
+      Measure.quasiMeasurePreserving_snd.ae hGC] with z hzF hzG
+    rw [Real.norm_eq_abs, abs_mul]
+    exact (mul_le_mul (abs_sign_le _) (abs_real_inner_le_norm _ _) (abs_nonneg _) (by norm_num)).trans
+      (by simpa only [one_mul] using mul_le_mul hzF hzG (norm_nonneg _) hB)
+
+end PettyProjection.Spherical
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology Function
+open scoped ENNReal RealInnerProductSpace
+namespace PettyProjection.Spherical
+
+lemma inner_Q_Q_expand {n : ℕ} (hn : 2 ≤ n) [NeZero n] (f g : C(Sphere n, ℝ)) :
+    ⟪Q n (toH n f),Q n (toH n g)⟫ = mean (fun u => f u*g u) - mean f*mean g -
+      ⟪(harmonicSpace n 2).starProjection (toH n f),(harmonicSpace n 2).starProjection (toH n g)⟫ := by
+  rw [inner_Q_Q, Q_eq_sub_projections hn, inner_sub_left, inner_sub_left,
+    harmonicSpace_zero_projection, inner_toH, inner_toH]
+  have h : mean (fun u => (mean f)*g u) = mean f*mean g := integral_const_mul (mean f) g
+  change _ - mean (fun u => mean f*g u) - _ = _
+  rw [h]
+  congr 1
+  have hs := Submodule.inner_starProjection_left_eq_right (harmonicSpace n 2)
+    (toH n f) ((harmonicSpace n 2).starProjection (toH n g))
+  rw [Submodule.starProjection_eq_self_iff.mpr ((harmonicSpace n 2).starProjection_apply_mem _)] at hs
+  exact (Submodule.inner_starProjection_left_eq_right _ _ _).trans hs.symm
+
+lemma sign_spectral_estimate_with_two {n : ℕ} (hn : 2 ≤ n) [NeZero n]
+    {a : ℝ} (ha : 0 ≤ a) (f : C(Sphere n, ℝ)) {g : Space n → ℝ}
+    (hg : ContDiff ℝ 2 g) :
+    mean f*mean (fun x : Sphere n => g x) -
+      ((1+a*(4*(n+2)))/((n+1)*(n+3)))*‖Q n (toH n f)‖*
+        ‖Q n (toH n (restrictContinuous g hg.continuous))‖ -
+      |signFactor n 1 a| * ‖(harmonicSpace n 2).starProjection (toH n f)‖ *
+        ‖(harmonicSpace n 2).starProjection (toH n (restrictContinuous g hg.continuous))‖ ≤
+      mean (fun x => f x*signOperator a g hg x) := by
+  have hi := inner_Q_Q_expand hn f (signOperator a g hg)
+  rw [mean_signOperator] at hi
+  have hfat := signOperator_projection_even (j := 1) hn a hg
+  norm_num only [Nat.mul_one] at hfat
+  rw [hfat, real_inner_smul_right] at hi
+  have hcs := neg_le_of_abs_le (abs_real_inner_le_norm
+    (Q n (toH n f)) (Q n (toH n (signOperator a g hg))))
+  have hb := mul_le_mul_of_nonneg_left (norm_Q_signOperator_le hn ha hg)
+    (norm_nonneg (Q n (toH n f)))
+  have hab := mul_le_mul_of_nonneg_left
+    (abs_real_inner_le_norm ((harmonicSpace n 2).starProjection (toH n f))
+      ((harmonicSpace n 2).starProjection (toH n (restrictContinuous g hg.continuous))))
+      (abs_nonneg (signFactor n 1 a))
+  rw [← abs_mul] at hab
+  have htwo := neg_le_of_abs_le hab
+  rw [hi] at hcs
+  nlinarith
+
+lemma sign_form_estimate_with_two {n : ℕ} (hn : 2 ≤ n) [NeZero n]
+    {k : ℝ} (hk : 0 < k) {f g : Space n → ℝ} (hf : ContDiff ℝ 2 f) (hg : ContDiff ℝ 2 g) :
+    mean (fun u : Sphere n => f u)*mean (fun u : Sphere n => g u) -
+      ((1+(2/k+((n : ℝ)-1)/k^2)*(4*(n+2)))/((n+1)*(n+3)))*
+        ‖Q n (toH n (restrictContinuous f hf.continuous))‖*
+        ‖Q n (toH n (restrictContinuous g hg.continuous))‖ -
+      |signFactor n 1 (2/k+((n : ℝ)-1)/k^2)| *
+        ‖(harmonicSpace n 2).starProjection (toH n (restrictContinuous f hf.continuous))‖ *
+        ‖(harmonicSpace n 2).starProjection (toH n (restrictContinuous g hg.continuous))‖ ≤
+      signedPair (signField k f) (signField k g)/cosineConstant n := by
+  rw [signedPair_fields hn k hf hg, mul_div_cancel_left₀ _ (cosineConstant_pos n).ne']
+  have hn' : (1 : ℝ) ≤ n := by exact_mod_cast (by omega : 1 ≤ n)
+  exact sign_spectral_estimate_with_two hn (a := 2/k+((n : ℝ)-1)/k^2) (by positivity) (restrictContinuous f hf.continuous) hg
+
+end PettyProjection.Spherical
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology Function
+open scoped NNReal ENNReal RealInnerProductSpace Gradient
+namespace PettyProjection.Spherical
+
+lemma normSmooth_power_restrict_tendsto {n : ℕ} (g : Seminorm ℝ (Space n)) (k : ℕ) :
+    Tendsto (fun j => restrictContinuous (fun x => normSmooth g (shrinkingBump n j) x^k)
+      ((normSmooth_contDiff g _).continuous.pow k)) atTop
+      (𝓝 (restrictContinuous (fun x => g x^k) ((seminorm_continuous n g).pow k))) := by
+  obtain ⟨F,R,hF,_,hFt,_⟩ := seminorm_approximation g
+  have h := hFt.pow k
+  simp only [hF] at h
+  convert h using 1 <;> congr 1
+
+lemma norm_signedPair_tendsto {n : ℕ} [NeZero n]
+    (f g : Seminorm ℝ (Space n)) (k : ℕ) :
+    Tendsto (fun j => signedPair
+      (signField (k : ℝ) (fun x => normSmooth f (shrinkingBump n j) x^k))
+      (signField (k : ℝ) (fun x => normSmooth g (shrinkingBump n j) x^k))) atTop
+      (𝓝 (signedPair (signField (k : ℝ) (fun x => f x^k))
+        (signField (k : ℝ) (fun x => g x^k)))) := by
+  obtain ⟨B,hB,hFB⟩ := normSmooth_powerField_bound f k
+  obtain ⟨C,_,hGC⟩ := normSmooth_powerField_bound g k
+  exact signedPair_tendsto hB
+    (fun j => signField_continuous _ ((normSmooth_contDiff f _).pow k |>.of_le (by norm_num)))
+    (fun j => signField_continuous _ ((normSmooth_contDiff g _).pow k |>.of_le (by norm_num)))
+    hFB hGC (normSmooth_powerField_tendsto f k) (normSmooth_powerField_tendsto g k)
+
+/-- The sign-transform lower bound for powers of arbitrary nonsmooth norms. -/
+theorem norm_sign_form {n : ℕ} (hn : 2 ≤ n) [NeZero n]
+    (f g : Seminorm ℝ (Space n)) {k : ℕ} (hk : 0 < k)
+    (h2 : (harmonicSpace n 2).starProjection (toH n
+      (restrictContinuous (fun x => f x^k) ((seminorm_continuous n f).pow k))) = 0) :
+    mean (fun u : Sphere n => f u^k)*mean (fun u : Sphere n => g u^k) -
+      ((1+(2/(k : ℝ)+((n : ℝ)-1)/(k : ℝ)^2)*(4*(n+2)))/((n+1)*(n+3)))*
+        ‖Q n (toH n (restrictContinuous (fun x => f x^k) ((seminorm_continuous n f).pow k)))‖*
+        ‖Q n (toH n (restrictContinuous (fun x => g x^k) ((seminorm_continuous n g).pow k)))‖ ≤
+      signedPair (signField (k : ℝ) (fun x => f x^k))
+        (signField (k : ℝ) (fun x => g x^k))/cosineConstant n := by
+  have hF := normSmooth_power_restrict_tendsto f k
+  have hG := normSmooth_power_restrict_tendsto g k
+  have hmF := (continuous_mean.tendsto _).comp hF
+  have hmG := (continuous_mean.tendsto _).comp hG
+  have hqF := (((Q n).continuous.comp (toH n).continuous).norm.tendsto _).comp hF
+  have hqG := (((Q n).continuous.comp (toH n).continuous).norm.tendsto _).comp hG
+  have hpF := ((((harmonicSpace n 2).starProjection.continuous.comp (toH n).continuous).norm).tendsto _).comp hF
+  have hpG := ((((harmonicSpace n 2).starProjection.continuous.comp (toH n).continuous).norm).tendsto _).comp hG
+  have hleft := ((hmF.mul hmG).sub ((hqF.const_mul
+    ((1+(2/(k : ℝ)+((n : ℝ)-1)/(k : ℝ)^2)*(4*(n+2)))/((n+1)*(n+3)))).mul hqG)).sub
+      ((hpF.const_mul |signFactor n 1 (2/(k : ℝ)+((n : ℝ)-1)/(k : ℝ)^2)|).mul hpG)
+  simp only [Function.comp_apply] at hleft
+  rw [h2, norm_zero, mul_zero, zero_mul, sub_zero] at hleft
+  exact le_of_tendsto_of_tendsto hleft ((norm_signedPair_tendsto f g k).div_const (cosineConstant n))
+    (Eventually.of_forall fun j => sign_form_estimate_with_two hn (by exact_mod_cast hk)
+      ((normSmooth_contDiff f _).pow k |>.of_le (by norm_num))
+      ((normSmooth_contDiff g _).pow k |>.of_le (by norm_num)))
+
+end PettyProjection.Spherical
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology Function
+open scoped ENNReal RealInnerProductSpace
+namespace PettyProjection.Spherical
+
+lemma reverse_lorentz {a b x y : ℝ} (ha : 1 ≤ a) (hb : 1 ≤ b)
+    (hx : 0 ≤ x) (hy : 0 ≤ y) (hx2 : x^2 ≤ a^2-1) (hy2 : y^2 ≤ b^2-1) :
+    1 ≤ a*b-x*y := by
+  have hab : 0 ≤ a*b-1 := by nlinarith [mul_nonneg (sub_nonneg.mpr ha) (sub_nonneg.mpr hb)]
+  have he : (x*y)^2 ≤ (a*b-1)^2 := by
+    have hp := mul_le_mul hx2 hy2 (sq_nonneg y) (by nlinarith : 0 ≤ a^2-1)
+    nlinarith [sq_nonneg (a-b)]
+  have hxy : 0 ≤ x*y := mul_nonneg hx hy
+  nlinarith [sq_nonneg (a*b-1-x*y)]
+
+lemma reverse_lorentz_strict {a b x y : ℝ} (ha : 1 ≤ a) (hb : 1 ≤ b)
+    (hx : 0 ≤ x) (hy : 0 ≤ y) (hx2 : x^2 ≤ a^2-1) (hy2 : y^2 ≤ b^2-1)
+    (hxs : 1 < a → x^2 < a^2-1) (hys : 1 < b → y^2 < b^2-1)
+    (he : a*b-x*y = 1) : a = 1 ∧ b = 1 := by
+  have hprod : x*y = a*b-1 := by linarith
+  by_cases hA : a = 1
+  · have hx0 : x = 0 := by rw [hA] at hx2; nlinarith [sq_nonneg x]
+    rw [hA,hx0] at he
+    constructor; exact hA; nlinarith
+  have ha' : 1 < a := lt_of_le_of_ne ha (Ne.symm hA)
+  by_cases hB : b = 1
+  · have hy0 : y = 0 := by rw [hB] at hy2; nlinarith [sq_nonneg y]
+    rw [hB,hy0] at he
+    exact False.elim (hA (by nlinarith))
+  have hb' : 1 < b := lt_of_le_of_ne hb (Ne.symm hB)
+  have hpa : 0 < a^2-1 := by nlinarith
+  have hpb : 0 < b^2-1 := by nlinarith
+  have hp : x^2*y^2 < (a^2-1)*(b^2-1) :=
+    (mul_le_mul_of_nonneg_left hy2 (sq_nonneg x)).trans_lt
+      (mul_lt_mul_of_pos_right (hxs ha') hpb)
+  have hbstrict := hys hb'
+  nlinarith [sq_nonneg (a-b), sq_nonneg (x*y-(a*b-1))]
+
+lemma coefficient_sign_factor {n : ℕ} (hn : 4 ≤ n) :
+    ((1+(2/(Scalar.exponent n : ℝ)+((n : ℝ)-1)/(Scalar.exponent n : ℝ)^2)*
+      (4*(n+2)))/((n+1)*(n+3))) = Scalar.coefficient n := by
+  unfold Scalar.exponent Scalar.coefficient
+  split_ifs with h
+  · subst n; norm_num
+  · have h1 : (n : ℝ)+1 ≠ 0 := by positivity
+    have h3 : (n : ℝ)+3 ≠ 0 := by positivity
+    push_cast
+    field_simp
+    ring
+
+end PettyProjection.Spherical
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology Function
+open scoped ENNReal RealInnerProductSpace
+namespace PettyProjection.Spherical
+open Scalar
+
+def gaugeMoment {n : ℕ} (g : Seminorm ℝ (Space n)) (k : ℕ) : ℝ :=
+  mean (fun u : Sphere n => g u^k)
+
+def gaugeTail {n : ℕ} [NeZero n] (g : Seminorm ℝ (Space n)) (k : ℕ) : ℝ :=
+  ‖Q n (toH n (restrictContinuous (fun x => g x^k) ((seminorm_continuous n g).pow k)))‖
+
+lemma gaugeMoment_ranges {n : ℕ} (hn : 4 ≤ n) [NeZero n]
+    (g : Seminorm ℝ (Space n)) (hg : ∀ u : Sphere n, 0 < g u)
+    (hnorm : mean (fun u : Sphere n => (g u)⁻¹^n) = 1) :
+    1 ≤ gaugeMoment g (exponent n-1) ∧ gaugeMoment g (exponent n-1) ≤ gaugeMoment g (exponent n) := by
+  obtain ⟨hm,ht,htm,_,_⟩ := norm_mean_ranges hn (restrictContinuous g (seminorm_continuous n g)) hg hnorm
+  have hk : 0 < (exponent n : ℝ) := by unfold exponent; split_ifs <;> norm_num
+  have hc : ((exponent n : ℝ)-1)/(exponent n : ℝ) ≤ 1 := (div_le_one hk).mpr (by linarith)
+  change 0 ≤ gaugeMoment g (exponent n)-1 at hm
+  change 0 ≤ gaugeMoment g (exponent n-1)-1 at ht
+  change gaugeMoment g (exponent n-1)-1 ≤ _*(gaugeMoment g (exponent n)-1) at htm
+  constructor; linarith
+  have hp := mul_le_mul_of_nonneg_right hc hm
+  linarith
+
+lemma gaugeTail_of_one {n : ℕ} [NeZero n] (g : Seminorm ℝ (Space n)) (k : ℕ)
+    (hg : ∀ u : Sphere n, g u = 1) : gaugeTail g k = 0 := by
+  have he : restrictContinuous (fun x => g x^k) ((seminorm_continuous n g).pow k) =
+      ContinuousMap.const (Sphere n) 1 := by ext u; simp [restrictContinuous,hg u]
+  unfold gaugeTail
+  rw [he,Q_constant,norm_zero]
+
+lemma scaled_norm_energy {n : ℕ} (hn : 4 ≤ n) [NeZero n]
+    (g : Seminorm ℝ (Space n)) (hg : ∀ u : Sphere n, 0 < g u)
+    (hnorm : mean (fun u : Sphere n => (g u)⁻¹^n) = 1)
+    {s : ℝ} (hs : 0 < s) (hst : 1 ≤ s*gaugeMoment g (exponent n-1)) :
+    let a := s*gaugeMoment g (exponent n)
+    let e := s*Real.sqrt (coefficient n)*gaugeTail g (exponent n)
+    1 ≤ a ∧ 0 ≤ e ∧ e^2 ≤ a^2-1 ∧ (1 < a → e^2 < a^2-1) := by
+  dsimp only
+  obtain ⟨ht,htm⟩ := gaugeMoment_ranges hn g hg hnorm
+  have ha : 1 ≤ s*gaugeMoment g (exponent n) := hst.trans (mul_le_mul_of_nonneg_left htm hs.le)
+  have he : 0 ≤ s*Real.sqrt (coefficient n)*gaugeTail g (exponent n) := by
+    apply mul_nonneg (mul_nonneg hs.le (Real.sqrt_nonneg _)); exact norm_nonneg _
+  have hc := coefficient_pos n
+  have hid : (s*Real.sqrt (coefficient n)*gaugeTail g (exponent n))^2 =
+      s^2*(coefficient n*gaugeTail g (exponent n)^2) := by
+    rw [mul_pow,mul_pow,Real.sq_sqrt hc.le]; ring
+  obtain ⟨hnrm,hnrme⟩ := norm_estimate hn g hg hnorm
+  change coefficient n*gaugeTail g (exponent n)^2 ≤
+      gaugeMoment g (exponent n)^2-gaugeMoment g (exponent n-1)^2 at hnrm
+  have hsq : 1 ≤ (s*gaugeMoment g (exponent n-1))^2 := by nlinarith
+  have herr : (s*Real.sqrt (coefficient n)*gaugeTail g (exponent n))^2 ≤
+      (s*gaugeMoment g (exponent n))^2-1 := by
+    rw [hid]
+    calc
+      _ ≤ s^2*(gaugeMoment g (exponent n)^2-gaugeMoment g (exponent n-1)^2) :=
+        mul_le_mul_of_nonneg_left hnrm (sq_nonneg s)
+      _ = (s*gaugeMoment g (exponent n))^2-(s*gaugeMoment g (exponent n-1))^2 := by ring
+      _ ≤ _ := sub_le_sub_left hsq _
+  refine ⟨ha,he,herr,?_⟩
+  intro ha'
+  by_cases hconst : ∀ u : Sphere n, g u = 1
+  · rw [gaugeTail_of_one g _ hconst]
+    nlinarith
+  have hstrict : coefficient n*gaugeTail g (exponent n)^2 <
+      gaugeMoment g (exponent n)^2-gaugeMoment g (exponent n-1)^2 := by
+    apply lt_of_le_of_ne hnrm
+    intro heq
+    exact hconst (hnrme.mp heq)
+  rw [hid]
+  calc
+    _ < s^2*(gaugeMoment g (exponent n)^2-gaugeMoment g (exponent n-1)^2) :=
+      mul_lt_mul_of_pos_left hstrict (sq_pos_of_pos hs)
+    _ = (s*gaugeMoment g (exponent n))^2-(s*gaugeMoment g (exponent n-1))^2 := by ring
+    _ ≤ _ := sub_le_sub_left hsq _
+
+lemma normalized_one_eq {n : ℕ} (hn : 4 ≤ n) [NeZero n]
+    (g : Seminorm ℝ (Space n)) (hg : ∀ u : Sphere n, 0 < g u)
+    (hnorm : mean (fun u : Sphere n => (g u)⁻¹^n) = 1)
+    {s : ℝ} (hs : 0 < s) (hst : 1 ≤ s*gaugeMoment g (exponent n-1))
+    (ha : s*gaugeMoment g (exponent n) = 1) :
+    (∀ u : Sphere n, g u = 1) ∧ s = 1 := by
+  have hne := norm_estimate hn g hg hnorm
+  have ht := (gaugeMoment_ranges hn g hg hnorm).1
+  have hmt : gaugeMoment g (exponent n) ≤ gaugeMoment g (exponent n-1) := by
+    apply (mul_le_mul_iff_right₀ hs).mp
+    simpa only [mul_comm s] using (ha.le.trans hst)
+  have hm : 0 ≤ gaugeMoment g (exponent n) := by nlinarith [ha]
+  have heq : coefficient n*gaugeTail g (exponent n)^2 =
+      gaugeMoment g (exponent n)^2-gaugeMoment g (exponent n-1)^2 := by
+    have hineq : coefficient n*gaugeTail g (exponent n)^2 ≤
+        gaugeMoment g (exponent n)^2-gaugeMoment g (exponent n-1)^2 := hne.1
+    apply le_antisymm hineq
+    have hc := mul_nonneg (coefficient_pos n).le (sq_nonneg (gaugeTail g (exponent n)))
+    nlinarith
+  have hconst := hne.2.mp heq
+  refine ⟨hconst,?_⟩
+  have hmone : gaugeMoment g (exponent n) = 1 := by
+    unfold gaugeMoment
+    have he : (fun u : Sphere n => g u^exponent n) = (fun _ => (1 : ℝ)) := by funext u; simp [hconst u]
+    rw [he,mean_const]
+  simpa only [hmone,mul_one] using ha
+
+end PettyProjection.Spherical
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology Function
+open scoped NNReal ENNReal RealInnerProductSpace Gradient
+namespace PettyProjection.Spherical
+
+lemma seminorm_euler {n : ℕ} (g : Seminorm ℝ (Space n)) {x : Space n}
+    (hd : DifferentiableAt ℝ (g : Space n → ℝ) x) :
+    fderiv ℝ g x x = g x := by
+  have he : (fun t : ℝ => g (t • x)) =ᶠ[𝓝 (1 : ℝ)] (fun t => t * g x) := by
+    filter_upwards [eventually_gt_nhds (zero_lt_one : (0 : ℝ) < 1)] with t ht
+    rw [map_smul_eq_mul, Real.norm_eq_abs, abs_of_pos ht]
+  have hx : HasFDerivAt (g : Space n → ℝ) (fderiv ℝ g x) ((1 : ℝ) • x) := by simpa using hd.hasFDerivAt
+  have hc := hx.comp_hasDerivAt (1 : ℝ) ((hasDerivAt_id (1 : ℝ)).smul_const x)
+  simp only [one_smul] at hc
+  have hl := (hasDerivAt_id (1 : ℝ)).mul_const (g x)
+  simp only [one_mul] at hl
+  exact (hc.congr_of_eventuallyEq he.symm).unique hl
+
+def gaugeField {n : ℕ} (g : Seminorm ℝ (Space n)) (k : ℕ) (u : Sphere n) : Space n :=
+  g u^(k-1) • ∇ (g : Space n → ℝ) u
+
+lemma signField_power_eq {n : ℕ} (g : Seminorm ℝ (Space n)) {k : ℕ} (hk : 0 < k)
+    {u : Sphere n} (hd : DifferentiableAt ℝ (g : Space n → ℝ) (u : Space n)) :
+    signField (k : ℝ) (fun x => g x^k) u = gaugeField g k u := by
+  rw [signField_eq_gradient, gradient_pow hd k]
+  rw [real_inner_smul_left, inner_gradient_left, seminorm_euler g hd]
+  have hk0 : (k : ℝ) ≠ 0 := by exact_mod_cast hk.ne'
+  have hp : g u^(k-1)*g u = g u^k := by
+    rw [← pow_succ]; congr 1; omega
+  rw [smul_sub, smul_smul, smul_smul]
+  have h1 : (k : ℝ)⁻¹*((k : ℝ)*g u^(k-1)) = g u^(k-1) := by field_simp
+  have h2 : (k : ℝ)⁻¹*((k : ℝ)*g u^(k-1)*g u) = g u^k := by
+    calc
+      _ = ((k : ℝ)⁻¹*((k : ℝ)*g u^(k-1)))*g u := by ring
+      _ = _ := by rw [h1,hp]
+  rw [h1,h2]
+  dsimp only [gaugeField]
+  abel
+
+lemma signField_power_ae_eq {n : ℕ} (g : Seminorm ℝ (Space n)) {k : ℕ} (hk : 0 < k) :
+    signField (k : ℝ) (fun x => g x^k) =ᵐ[sigma n] gaugeField g k := by
+  filter_upwards [seminorm_ae_differentiable_sphere g] with u hu
+  exact signField_power_eq g hk hu
+
+lemma gaugeField_measurable_bound {n : ℕ} (g : Seminorm ℝ (Space n)) {k : ℕ} (hk : 0 < k) :
+    AEStronglyMeasurable (gaugeField g k) (sigma n) ∧
+      ∃ B : ℝ, 0 ≤ B ∧ ∀ᵐ u ∂sigma n, ‖gaugeField g k u‖ ≤ B := by
+  have hlim := normSmooth_powerField_tendsto g k
+  have he := signField_power_ae_eq g hk
+  have hm := aestronglyMeasurable_of_tendsto_ae (u := atTop)
+    (fun j => (signField_continuous (k : ℝ)
+      ((normSmooth_contDiff g _).pow k |>.of_le (by norm_num))).aestronglyMeasurable) hlim
+  obtain ⟨B,hB,hbound⟩ := normSmooth_powerField_bound g k
+  refine ⟨hm.congr he, B,hB, ?_⟩
+  filter_upwards [he, ae_bound_of_limit hbound hlim] with u hu hb
+  rw [← hu]; exact hb
+
+lemma signedPair_congr_ae {n : ℕ} {F G F' G' : Sphere n → Space n}
+    (hF : F =ᵐ[sigma n] F') (hG : G =ᵐ[sigma n] G') :
+    signedPair F G = signedPair F' G' := by
+  have : IsFiniteMeasure (sigma n) := by unfold sigma; infer_instance
+  apply integral_congr_ae
+  filter_upwards [Measure.quasiMeasurePreserving_fst.ae hF,
+    Measure.quasiMeasurePreserving_snd.ae hG] with z hzF hzG
+  rw [hzF,hzG]
+
+lemma signedPair_gaugeField {n : ℕ} (f g : Seminorm ℝ (Space n)) {k : ℕ} (hk : 0 < k) :
+    signedPair (signField (k : ℝ) (fun x => f x^k)) (signField (k : ℝ) (fun x => g x^k)) =
+      signedPair (gaugeField f k) (gaugeField g k) :=
+  signedPair_congr_ae (signField_power_ae_eq f hk) (signField_power_ae_eq g hk)
+
+def absolutePair {n : ℕ} (F G : Sphere n → Space n) : ℝ :=
+  ∫ z : Sphere n × Sphere n, |⟪F z.1,G z.2⟫| ∂(sigma n).prod (sigma n)
+
+lemma gaugeField_absolute_integrable {n : ℕ} [NeZero n]
+    (f g : Seminorm ℝ (Space n)) {k : ℕ} (hk : 0 < k) :
+    Integrable (fun z : Sphere n × Sphere n =>
+      |⟪gaugeField f k z.1,gaugeField g k z.2⟫|) ((sigma n).prod (sigma n)) := by
+  obtain ⟨hf,B,hB,hFB⟩ := gaugeField_measurable_bound f hk
+  obtain ⟨hg,C,_,hGC⟩ := gaugeField_measurable_bound g hk
+  apply (integrable_const (B*C)).mono'
+  · have hm : AEStronglyMeasurable (fun z : Sphere n × Sphere n =>
+        ⟪gaugeField f k z.1,gaugeField g k z.2⟫) ((sigma n).prod (sigma n)) :=
+      (hf.comp_quasiMeasurePreserving Measure.quasiMeasurePreserving_fst).inner
+        (hg.comp_quasiMeasurePreserving Measure.quasiMeasurePreserving_snd)
+    simpa only [Real.norm_eq_abs] using hm.norm
+  · filter_upwards [Measure.quasiMeasurePreserving_fst.ae hFB,
+      Measure.quasiMeasurePreserving_snd.ae hGC] with z hzF hzG
+    simpa only [Real.norm_eq_abs, abs_abs] using
+      (abs_real_inner_le_norm _ _).trans (mul_le_mul hzF hzG (norm_nonneg _) hB)
+
+lemma signedPair_gaugeField_le_absolute {n : ℕ} [NeZero n]
+    (f g : Seminorm ℝ (Space n)) {k : ℕ} (hk : 0 < k) :
+    signedPair (gaugeField f k) (gaugeField g k) ≤
+      absolutePair (gaugeField f k) (gaugeField g k) := by
+  obtain ⟨hf,B,hB,hFB⟩ := gaugeField_measurable_bound f hk
+  obtain ⟨hg,C,_,hGC⟩ := gaugeField_measurable_bound g hk
+  apply integral_mono (signedPair_integrable_of_ae_bound hB hf hg hFB hGC)
+    (gaugeField_absolute_integrable f g hk)
+  intro z
+  exact (le_abs_self _).trans (by rw [abs_mul]; exact mul_le_of_le_one_left (abs_nonneg _) (abs_sign_le _))
+
+end PettyProjection.Spherical
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology Function
+open scoped ENNReal RealInnerProductSpace
+namespace PettyProjection.Spherical
+open Scalar
+
+lemma norm_absolutePair_lower {n : ℕ} (hn : 4 ≤ n) [NeZero n]
+    (f g : Seminorm ℝ (Space n))
+    (h2 : (harmonicSpace n 2).starProjection (toH n
+      (restrictContinuous (fun x => f x^exponent n) ((seminorm_continuous n f).pow _))) = 0) :
+    gaugeMoment f (exponent n)*gaugeMoment g (exponent n) -
+      coefficient n*gaugeTail f (exponent n)*gaugeTail g (exponent n) ≤
+      absolutePair (gaugeField f (exponent n)) (gaugeField g (exponent n))/cosineConstant n := by
+  have hk : 0 < exponent n := by unfold exponent; split_ifs <;> norm_num
+  have h := norm_sign_form (by omega) f g hk h2
+  rw [coefficient_sign_factor hn, signedPair_gaugeField f g hk] at h
+  exact h.trans (div_le_div_of_nonneg_right (signedPair_gaugeField_le_absolute f g hk)
+    (cosineConstant_pos n).le)
+
+/-- Spherical form of the full bilinear estimate, before construction of the
+variational representing bodies. Both norms can be nonsmooth. -/
+theorem normalized_norm_pair {n : ℕ} (hn : 4 ≤ n) [NeZero n]
+    (f g : Seminorm ℝ (Space n)) (hf : ∀ u : Sphere n, 0 < f u) (hg : ∀ u : Sphere n, 0 < g u)
+    (hfn : mean (fun u : Sphere n => (f u)⁻¹^n) = 1)
+    (hgn : mean (fun u : Sphere n => (g u)⁻¹^n) = 1)
+    (h2 : (harmonicSpace n 2).starProjection (toH n
+      (restrictContinuous (fun x => f x^exponent n) ((seminorm_continuous n f).pow _))) = 0)
+    {s t : ℝ} (hs : 0 < s) (ht : 0 < t)
+    (hsf : 1 ≤ s*gaugeMoment f (exponent n-1)) (htg : 1 ≤ t*gaugeMoment g (exponent n-1)) :
+    cosineConstant n ≤ s*t*absolutePair (gaugeField f (exponent n)) (gaugeField g (exponent n)) ∧
+    (s*t*absolutePair (gaugeField f (exponent n)) (gaugeField g (exponent n)) = cosineConstant n →
+      (∀ u : Sphere n, f u = 1) ∧ (∀ u : Sphere n, g u = 1) ∧ s = 1 ∧ t = 1) := by
+  let a := s*gaugeMoment f (exponent n)
+  let b := t*gaugeMoment g (exponent n)
+  let x := s*Real.sqrt (coefficient n)*gaugeTail f (exponent n)
+  let y := t*Real.sqrt (coefficient n)*gaugeTail g (exponent n)
+  obtain ⟨ha,hx,hx2,hxs⟩ := scaled_norm_energy hn f hf hfn hs hsf
+  obtain ⟨hb,hy,hy2,hys⟩ := scaled_norm_energy hn g hg hgn ht htg
+  have hlor := reverse_lorentz ha hb hx hy hx2 hy2
+  have hid : a*b-x*y = s*t*(gaugeMoment f (exponent n)*gaugeMoment g (exponent n)-
+      coefficient n*gaugeTail f (exponent n)*gaugeTail g (exponent n)) := by
+    calc
+      _ = s*t*(gaugeMoment f (exponent n)*gaugeMoment g (exponent n)-
+          (Real.sqrt (coefficient n))^2*gaugeTail f (exponent n)*gaugeTail g (exponent n)) := by
+        dsimp only [a,b,x,y]; ring
+      _ = _ := by rw [Real.sq_sqrt (coefficient_pos n).le]
+  have hlow : a*b-x*y ≤ s*t*
+      (absolutePair (gaugeField f (exponent n)) (gaugeField g (exponent n))/cosineConstant n) := by
+    rw [hid]
+    exact mul_le_mul_of_nonneg_left (norm_absolutePair_lower hn f g h2) (mul_nonneg hs.le ht.le)
+  have hbound : cosineConstant n ≤ s*t*absolutePair
+      (gaugeField f (exponent n)) (gaugeField g (exponent n)) := by
+    have h := hlor.trans hlow
+    rw [← mul_div_assoc] at h
+    simpa only [one_mul] using (le_div_iff₀ (cosineConstant_pos n)).mp h
+  refine ⟨hbound,?_⟩
+  intro heq
+  have hupper : a*b-x*y ≤ 1 := by
+    rw [← mul_div_assoc,heq,div_self (cosineConstant_pos n).ne'] at hlow
+    exact hlow
+  obtain ⟨ha1,hb1⟩ := reverse_lorentz_strict ha hb hx hy hx2 hy2 hxs hys (le_antisymm hupper hlor)
+  obtain ⟨hf1,hs1⟩ := normalized_one_eq hn f hf hfn hs hsf ha1
+  obtain ⟨hg1,ht1⟩ := normalized_one_eq hn g hg hgn ht htg hb1
+  exact ⟨hf1,hg1,hs1,ht1⟩
+
+end PettyProjection.Spherical
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/ProjectionBodies/Normalization.lean b/lean/OAI/Geometry/ProjectionBodies/Normalization.lean
new file mode 100644
index 0000000000000000000000000000000000000000..9a6a5e13b34afd52724d269017c06aa19199bf63
--- /dev/null
+++ b/lean/OAI/Geometry/ProjectionBodies/Normalization.lean
@@ -0,0 +1,294 @@
+import OAI.Geometry.ProjectionBodies.BilinearEquality
+
+namespace OAI
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology
+open scoped NNReal RealInnerProductSpace Pointwise
+namespace PettyProjection
+variable {n : ℕ}
+
+lemma smul_body {K : Set (Space n)} (hK : IsConvexBody K) {a : ℝ} (ha : a≠0) :
+    IsConvexBody (a • K) := by
+  refine ⟨hK.1.smul a,hK.2.1.smul a,?_⟩
+  rw [interior_smul₀ ha]
+  exact hK.2.2.image _
+
+lemma translate_body {K : Set (Space n)} (hK : IsConvexBody K) (a : Space n) :
+    IsConvexBody ((fun x => a+x) '' K) := by
+  refine ⟨hK.1.image (by fun_prop),hK.2.1.translate a,?_⟩
+  change (interior ((Homeomorph.addLeft a) '' K)).Nonempty
+  rw [← (Homeomorph.addLeft a).image_interior]
+  exact hK.2.2.image _
+
+lemma isEllipsoid_linear_image {K : Set (Space n)} (hK : IsEllipsoid K)
+    (T : Space n ≃ₗ[ℝ] Space n) : IsEllipsoid (T '' K) := by
+  obtain ⟨a,S,rfl⟩ := hK
+  refine ⟨T a,S.trans T,?_⟩
+  simp only [image_image]
+  congr 1
+  funext x
+  exact map_add T a (S x)
+
+lemma isEllipsoid_translate {K : Set (Space n)} (hK : IsEllipsoid K) (b : Space n) :
+    IsEllipsoid ((fun x => b+x) '' K) := by
+  obtain ⟨a,T,rfl⟩ := hK
+  refine ⟨b+a,T,?_⟩
+  simp only [image_image]
+  congr 1
+  funext x
+  exact (add_assoc b a (T x)).symm
+
+lemma isEllipsoid_smul {K : Set (Space n)} (hK : IsEllipsoid K) {c : ℝ} (hc : c≠0) :
+    IsEllipsoid (c • K) := by
+  exact isEllipsoid_linear_image hK (LinearEquiv.smulOfNeZero ℝ (Space n) c hc)
+
+lemma ellipsoid_homothety (T : Space n ≃L[ℝ] Space n) (a : Space n) {c : ℝ} (hc : c≠0) :
+    IsEllipsoid ((fun x => a+c • x) '' (T '' unitBall n)) := by
+  have he : IsEllipsoid (T '' unitBall n) := ⟨0,T.toLinearEquiv,by simp⟩
+  have hh := isEllipsoid_translate (isEllipsoid_smul he hc) a
+  change IsEllipsoid ((fun x => a+x) '' ((fun x => c • x) '' (T '' unitBall n))) at hh
+  simpa only [image_image,Function.comp_def] using hh
+
+def normalizeBody (K : Set (Space n)) : Set (Space n) := (normalizedVolume K)⁻¹ • K
+
+lemma normalizeBody_body {K : Set (Space n)} (hK : IsConvexBody K) : IsConvexBody (normalizeBody K) :=
+  smul_body hK (inv_ne_zero (normalizedVolume_pos hK.1 hK.2.2).ne')
+
+lemma normalizeBody_nhds {K : Set (Space n)} (hK : IsCompact K) (h0 : K∈𝓝 (0:Space n)) :
+    normalizeBody K∈𝓝 (0:Space n) := by
+  apply (set_smul_mem_nhds_zero_iff (inv_ne_zero (normalizedVolume_pos hK ⟨0,mem_interior_iff_mem_nhds.mpr h0⟩).ne')).mpr h0
+
+lemma normalizeBody_volume [NeZero n] {K : Set (Space n)} (hK : IsConvexBody K) : volume.real (normalizeBody K)=kappa n := by
+  have hp := normalizedVolume_pos hK.1 hK.2.2
+  rw [normalizeBody,volumeReal_smul_nonneg (inv_nonneg.mpr hp.le),finrank_euclideanSpace_fin,inv_pow,normalizedVolume_pow]
+  field_simp [(volumeReal_pos_of_body hK.1 hK.2.2).ne',(kappa_pos n).ne']
+
+lemma normalizeBody_ellipsoid {K : Set (Space n)} (hK : IsConvexBody K)
+    (he : IsEllipsoid (normalizeBody K)) : IsEllipsoid K := by
+  have hh := isEllipsoid_smul he (normalizedVolume_pos hK.1 hK.2.2).ne'
+  simpa only [normalizeBody,smul_smul,mul_inv_cancel₀ (normalizedVolume_pos hK.1 hK.2.2).ne',one_smul] using hh
+
+end PettyProjection
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology
+open scoped NNReal RealInnerProductSpace Pointwise
+namespace PettyProjection
+variable {n : ℕ}
+
+lemma measurePair_comm {μ ν : Measure (Space n)} [SFinite μ] [SFinite ν]
+    (hμ : Integrable (fun x : Space n => ‖x‖) μ)
+    (hν : Integrable (fun x : Space n => ‖x‖) ν) : measurePair μ ν=measurePair ν μ := by
+  have hi : Integrable (fun z : Space n × Space n => |⟪z.1,z.2⟫|) (μ.prod ν) := by
+    apply (hμ.mul_prod hν).mono' (by fun_prop)
+    exact Filter.Eventually.of_forall (fun z => by simpa only [Real.norm_eq_abs,abs_abs] using abs_real_inner_le_norm z.1 z.2)
+  rw [measurePair,integral_integral_swap hi]
+  unfold measurePair
+  simp_rw [real_inner_comm]
+
+/-- Main geometric estimate and equality rigidity at the natural volume
+normalization; both the input and the projection body remain nonsmooth. -/
+theorem normalized_projection {n : ℕ} (hn : 4≤ n) [NeZero n] {K : Set (Space n)}
+    (hK : IsCompact K) (hc : Convex ℝ K) (h0 : K∈𝓝 (0:Space n))
+    (hv : volume.real K=kappa n) :
+    kappa (n-1)≤ normalizedVolume (projectionBody K) ∧
+      (normalizedVolume (projectionBody K)=kappa (n-1) → IsEllipsoid K) := by
+  let P := projectionBody K
+  let r := normalizedVolume P
+  have hP : IsConvexBody P := projectionBody_body hK hc h0
+  have hP0 : P∈𝓝 (0:Space n) := projectionBody_nhds hK hc h0
+  have hr : 0< r := normalizedVolume_pos hP.1 hP.2.2
+  let D := normalizeBody P
+  have hD : IsConvexBody D := normalizeBody_body hP
+  have hD0 : D∈𝓝 (0:Space n) := normalizeBody_nhds hP.1 hP0
+  have hDv : volume.real D=kappa n := normalizeBody_volume hP
+  let μ := gaugeProjectionMeasure K
+  let ν := gaugeProjectionMeasure D
+  let := gaugeMeasure_finite hK hc h0
+  let := gaugeMeasure_finite hD.1 hD.2.1 hD0
+  have hμ : Integrable (fun x : Space n => ‖x‖) μ := gaugeMeasure_norm_integrable hK hc h0
+  have hν : Integrable (fun x : Space n => ‖x‖) ν := gaugeMeasure_norm_integrable hD.1 hD.2.1 hD0
+  obtain ⟨hlow,heq⟩ := measure_bilinear hn hμ hν (gaugeMeasure_cosine_pos hK hc h0)
+    (gaugeMeasure_cosine_pos hD.1 hD.2.1 hD0)
+    (fun M hM => (gaugeMeasure_functional (by omega) hK hc h0 hv hM.1 hM.2.1 hM.2.2).1)
+    (fun M hM => (gaugeMeasure_functional (by omega) hD.1 hD.2.1 hD0 hDv hM.1 hM.2.1 hM.2.2).1)
+  have hself : supportFunctional ν D=1 := by
+    rw [gaugeMeasure_self hD.1 hD.2.1 hD0,hDv,div_self (kappa_pos n).ne']
+  have hs : supportFunctional ν D=r⁻¹*((n:ℝ)*kappa n/2)*measurePair μ ν := by
+    change (∫ x,support (r⁻¹ • P) x ∂ν)=_
+    simp_rw [support_smul_set hP.1 (hP.2.2.mono interior_subset) (inv_nonneg.mpr hr.le)]
+    dsimp only [P]
+    simp_rw [projectionBody_support hK hc h0]
+    rw [integral_const_mul,integral_const_mul]
+    change r⁻¹*((n:ℝ)*kappa n/2*(∫ x,cosineSeminorm μ hμ x ∂ν))=_
+    rw [← measurePair_cosine hμ,measurePair_comm hν hμ]
+    ring
+  have hpair : (n:ℝ)*kappa n*measurePair μ ν=2*r := by
+    rw [hself] at hs
+    have h := congrArg (fun x : ℝ => (2*r)*x) hs
+    field_simp at h
+    nlinarith [h]
+  have hb := ball_cosine_constant (n :=n)
+  have hnk : 0<(n:ℝ)*kappa n := mul_pos (Nat.cast_pos.mpr (NeZero.pos n)) (kappa_pos n)
+  refine ⟨?_,?_⟩
+  · change kappa (n-1)≤ r
+    have h := mul_le_mul_of_nonneg_left hlow hnk.le
+    nlinarith
+  · intro he
+    have hp : measurePair μ ν=Spherical.cosineConstant n := by
+      change r=kappa (n-1) at he
+      apply (mul_left_cancel₀ hnk.ne')
+      rw [hpair,he,← hb]
+    obtain ⟨T,hT⟩ := heq hp
+    have hE : IsConvexBody (T '' unitBall n) := linearEquiv_body T unitBall_body
+    obtain ⟨z,c,hcpos,hset⟩ := (gaugeMeasure_functional (by omega) hK hc h0 hv hE.1 hE.2.1 hE.2.2).2 hT
+    rw [hset]
+    exact ellipsoid_homothety T z hcpos.ne'
+
+end PettyProjection
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology
+open scoped NNReal RealInnerProductSpace Pointwise
+namespace PettyProjection
+variable {n : ℕ}
+
+lemma volumeReal_translate (K : Set (Space n)) (a : Space n) :
+    volume.real ((fun x => a+x) '' K)=volume.real K := by
+  unfold Measure.real
+  rw [image_add_left,measure_preimage_add]
+
+lemma normalizedVolume_translate (K : Set (Space n)) (a : Space n) :
+    normalizedVolume ((fun x => a+x) '' K)=normalizedVolume K := by
+  rw [normalizedVolume_formula,volumeReal_translate,← normalizedVolume_formula]
+
+lemma shadowVolume_translate (K : Set (Space n)) (a u : Space n) :
+    shadowVolume ((fun x => a+x) '' K) u=shadowVolume K u := by
+  have he : (perpendicular u).orthogonalProjectionOnto '' ((fun x => a+x) '' K)=
+      (fun z => (perpendicular u).orthogonalProjectionOnto a+z) ''
+        ((perpendicular u).orthogonalProjectionOnto '' K) := by
+    simp only [image_image]
+    congr 1
+    funext x
+    exact map_add _ a x
+  unfold shadowVolume Measure.real
+  rw [he,image_add_left,measure_preimage_add]
+
+lemma projectionBody_translate (K : Set (Space n)) (a : Space n) :
+    projectionBody ((fun x => a+x) '' K)=projectionBody K := by
+  ext x
+  simp only [projectionBody,mem_ofPred_eq,shadowVolume_translate]
+
+lemma projectionRatio_translate (K : Set (Space n)) (a : Space n) :
+    projectionRatio ((fun x => a+x) '' K)=projectionRatio K := by
+  rw [projectionRatio,projectionBody_translate,volumeReal_translate,projectionRatio]
+
+lemma shadowVolume_smul (K : Set (Space n)) {a : ℝ} (ha : 0≤ a) (u : Spherical.Sphere n) :
+    shadowVolume (a • K) u=a^(n-1)*shadowVolume K u := by
+  have he : (perpendicular (u:Space n)).orthogonalProjectionOnto '' (a • K)=
+      a • ((perpendicular (u:Space n)).orthogonalProjectionOnto '' K) := by
+    change _ '' ((fun x : Space n => a • x) '' K)=
+      (fun z : perpendicular (u:Space n) => a • z) '' (_ '' K)
+    simp only [image_image]
+    congr 1
+    funext x
+    exact map_smul _ a x
+  unfold shadowVolume
+  rw [he,volumeReal_smul_nonneg ha,perpendicular_finrank]
+
+lemma projectionBody_smul (K : Set (Space n)) {a : ℝ} (ha : 0< a) :
+    projectionBody (a • K)=a^(n-1) • projectionBody K := by
+  have hp : 0< a^(n-1) := pow_pos ha _
+  ext x
+  rw [mem_smul_set_iff_inv_smul_mem₀ hp.ne']
+  change (∀ u : Space n,‖u‖=1 → ⟪u,x⟫≤ shadowVolume (a • K) u) ↔
+    (∀ u : Space n,‖u‖=1 → ⟪u,(a^(n-1))⁻¹ • x⟫≤ shadowVolume K u)
+  apply forall_congr'
+  intro u
+  apply forall_congr'
+  intro hu
+  have hv := shadowVolume_smul K ha.le (⟨u,by simpa only [mem_sphere,dist_zero_right] using hu⟩ : Spherical.Sphere n)
+  change shadowVolume (a • K) u=a^(n-1)*shadowVolume K u at hv
+  rw [hv,inner_smul_right]
+  rw [show (a^(n-1))⁻¹*⟪u,x⟫=⟪u,x⟫/(a^(n-1)) by ring,div_le_iff₀ hp]
+  rw [mul_comm (a^(n-1))]
+
+lemma projectionRatio_smul (K : Set (Space n))
+    {a : ℝ} (ha : 0< a) : projectionRatio (a • K)=projectionRatio K := by
+  rw [projectionRatio,projectionBody_smul K ha,volumeReal_smul_nonneg (pow_nonneg ha.le _),
+    volumeReal_smul_nonneg ha.le,finrank_euclideanSpace_fin,projectionRatio,mul_pow]
+  have he : (a^(n-1))^n=(a^n)^(n-1) := by rw [← pow_mul,← pow_mul,Nat.mul_comm]
+  rw [he]
+  field_simp [ha.ne']
+
+end PettyProjection
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology
+open scoped NNReal RealInnerProductSpace Pointwise
+namespace PettyProjection
+variable {n : ℕ}
+
+lemma projectionRatio_normalized {n : ℕ} (hn : 2≤n) [NeZero n] {K : Set (Space n)}
+    (hv : volume.real K=kappa n) :
+    projectionRatio K=normalizedVolume (projectionBody K)^n*kappa n^(2-(n:ℤ)) := by
+  have hn1 : n-1=(n-2)+1 := by omega
+  have hn2 : (2-(n:ℤ))=-((n-2:ℕ):ℤ) := by omega
+  have hvP : volume.real (projectionBody K)=normalizedVolume (projectionBody K)^n*kappa n := by
+    exact (div_eq_iff (kappa_pos n).ne').mp (normalizedVolume_pow (projectionBody K)).symm
+  rw [projectionRatio,hv,hvP,hn1,pow_succ,hn2,zpow_neg,zpow_natCast]
+  field_simp [(kappa_pos n).ne']
+
+lemma normalized_projection_ratio {n : ℕ} (hn : 4≤n) [NeZero n] {K : Set (Space n)}
+    (hK : IsCompact K) (hc : Convex ℝ K) (h0 : K∈𝓝 (0:Space n))
+    (hv : volume.real K=kappa n) :
+    pettyConstant n≤projectionRatio K ∧ (projectionRatio K=pettyConstant n → IsEllipsoid K) := by
+  obtain ⟨hb,he⟩ := normalized_projection hn hK hc h0 hv
+  have hkp := kappa_pos (n-1)
+  have hpow := zpow_pos (kappa_pos n) (2-(n:ℤ))
+  rw [projectionRatio_normalized (by omega) hv]
+  unfold pettyConstant
+  refine ⟨mul_le_mul_of_nonneg_right (pow_le_pow_left₀ hkp.le hb n) hpow.le,?_⟩
+  intro h
+  apply he
+  apply (pow_left_inj₀ (hkp.le.trans hb) hkp.le (NeZero.ne n)).mp
+  exact mul_right_cancel₀ hpow.ne' h
+
+lemma centered_projection_lower_bound {n : ℕ} (hn : 4≤n) [NeZero n] {K : Set (Space n)}
+    (hK : IsCompact K) (hc : Convex ℝ K) (h0 : K∈𝓝 (0:Space n)) :
+    pettyConstant n≤projectionRatio K ∧ (projectionRatio K=pettyConstant n → IsEllipsoid K) := by
+  have hbody : IsConvexBody K := ⟨hK,hc,⟨0,mem_interior_iff_mem_nhds.mpr h0⟩⟩
+  have hM := normalizeBody_body hbody
+  have hM0 := normalizeBody_nhds hK h0
+  obtain ⟨hb,he⟩ := normalized_projection_ratio hn hM.1 hM.2.1 hM0 (normalizeBody_volume hbody)
+  have hr : projectionRatio (normalizeBody K)=projectionRatio K :=
+    projectionRatio_smul K (inv_pos.mpr (normalizedVolume_pos hK hbody.2.2))
+  rw [hr] at hb he
+  exact ⟨hb,fun hh => normalizeBody_ellipsoid hbody (he hh)⟩
+
+/-- No origin, symmetry, localness, or boundary smoothness hypothesis is used. -/
+theorem projection_lower_bound {n : ℕ} (hn : 4≤n) {K : Set (Space n)}
+    (hK : IsConvexBody K) :
+    pettyConstant n≤projectionRatio K ∧ (projectionRatio K=pettyConstant n → IsEllipsoid K) := by
+  let : NeZero n := ⟨by omega⟩
+  obtain ⟨a,ha⟩ := hK.2.2
+  let L := (fun x => -a+x) '' K
+  have hL : IsConvexBody L := translate_body hK (-a)
+  have hL0 : L∈𝓝 (0:Space n) := by
+    have hh := (Homeomorph.addLeft (-a)).isOpenMap.image_mem_nhds (mem_interior_iff_mem_nhds.mp ha)
+    simpa [L] using hh
+  obtain ⟨hb,he⟩ := centered_projection_lower_bound hn hL.1 hL.2.1 hL0
+  have hr : projectionRatio L=projectionRatio K := projectionRatio_translate K (-a)
+  rw [hr] at hb he
+  refine ⟨hb,fun hh => ?_⟩
+  have h := isEllipsoid_translate (he hh) a
+  simpa only [L,image_image,Function.comp_def,add_neg_cancel_left,image_id'] using h
+
+end PettyProjection
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/ProjectionBodies/PositionMatrices.lean b/lean/OAI/Geometry/ProjectionBodies/PositionMatrices.lean
new file mode 100644
index 0000000000000000000000000000000000000000..a7491655a9be8c27fd232bfa517bc38df0f0ccf4
--- /dev/null
+++ b/lean/OAI/Geometry/ProjectionBodies/PositionMatrices.lean
@@ -0,0 +1,421 @@
+import OAI.Geometry.ProjectionBodies.RankContinuity
+
+namespace OAI
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology
+open scoped NNReal RealInnerProductSpace
+namespace PettyProjection
+open Spherical (Sphere objective)
+namespace RelativeGauge
+variable {n : ℕ} [NeZero n] {μ : Measure (Space n)}
+
+abbrev NonzeroMap (n : ℕ) := {A : Space n →L[ℝ] Space n // LinearMap.range A.toLinearMap≠⊥}
+
+def optimizer (hμ : Integrable (fun x : Space n => ‖x‖) μ)
+    (hq : ∀ x : Space n,x≠0 → 0< cosineSeminorm μ hμ x) (b : Bool) (A : NonzeroMap n) :
+    RelativeGauge (LinearMap.range A.val.toLinearMap) :=
+  (exists_minimizer (norm_integrable_map hμ A) A.property (cosine_map_pos hμ hq A) b).choose
+
+lemma optimizer_normalized (hμ : Integrable (fun x : Space n => ‖x‖) μ)
+    (hq : ∀ x : Space n,x≠0 → 0< cosineSeminorm μ hμ x) (b : Bool) (A : NonzeroMap n) :
+    supportFunctional (μ.map A.val) (optimizer hμ hq b A).body=1 :=
+  (exists_minimizer (norm_integrable_map hμ A) A.property (cosine_map_pos hμ hq A) b).choose_spec.1
+
+lemma optimizer_minimal (hμ : Integrable (fun x : Space n => ‖x‖) μ)
+    (hq : ∀ x : Space n,x≠0 → 0< cosineSeminorm μ hμ x) (b : Bool) (A : NonzeroMap n)
+    (g : RelativeGauge (LinearMap.range A.val.toLinearMap)) (hg : supportFunctional (μ.map A.val) g.body=1) :
+    objective b (optimizer hμ hq b A).toSeminorm≤ objective b g.toSeminorm :=
+  (exists_minimizer (norm_integrable_map hμ A) A.property (cosine_map_pos hμ hq A) b).choose_spec.2 g hg
+
+def optimizerSphere (hμ : Integrable (fun x : Space n => ‖x‖) μ)
+    (hq : ∀ x : Space n,x≠0 → 0< cosineSeminorm μ hμ x) (b : Bool) (A : NonzeroMap n) :
+    BoundedContinuousFunction (Sphere n) ℝ :=
+  BoundedContinuousFunction.mkOfCompact ⟨fun u => optimizer hμ hq b A u,
+    (Spherical.seminorm_continuous n _).comp continuous_subtype_val⟩
+
+/-- Rank-continuity of the actual canonical variational optimizer. The
+codomain carries the uniform norm on the entire ambient sphere. -/
+theorem optimizerSphere_continuous (hμ : Integrable (fun x : Space n => ‖x‖) μ)
+    (hq : ∀ x : Space n,x≠0 → 0< cosineSeminorm μ hμ x) (b : Bool) :
+    Continuous (optimizerSphere hμ hq b) := by
+  apply continuous_iff_seqContinuous.mpr
+  intro Aj A hA
+  apply Filter.tendsto_of_subseq_tendsto
+  intro ns hns
+  have hAv : Tendsto (fun i => (Aj (ns i)).val) atTop (𝓝 A.val) :=
+    (continuous_subtype_val.tendsto A).comp (hA.comp hns)
+  obtain ⟨φ,_,ht⟩ := minimizers_subsequence hμ hq hAv A.property (fun i => (Aj (ns i)).property) b
+    (fun i => optimizer hμ hq b (Aj (ns i))) (fun i => optimizer_normalized hμ hq b (Aj (ns i)))
+    (fun i => optimizer_minimal hμ hq b (Aj (ns i))) (optimizer hμ hq b A)
+    (optimizer_normalized hμ hq b A) (optimizer_minimal hμ hq b A)
+  refine ⟨φ,BoundedContinuousFunction.tendsto_iff_tendstoUniformly.mpr ?_⟩
+  exact ht
+
+end RelativeGauge
+end PettyProjection
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology Function
+open scoped ENNReal NNReal RealInnerProductSpace
+namespace PettyProjection
+open Spherical (Sphere norm_coe surface sigma mean)
+
+lemma gaussian_real_second_moment :
+    (∫ t : ℝ,t^2*Real.exp (-(t^2)))=(1/2:ℝ)*(∫ t : ℝ,Real.exp (-(t^2))) := by
+  have h0 := integral_fun_norm_addHaar (volume : Measure ℝ) (fun r : ℝ => Real.exp (-(r^2)))
+  have h2 := integral_fun_norm_addHaar (volume : Measure ℝ) (fun r : ℝ => r^2*Real.exp (-(r^2)))
+  simp only [Module.finrank_self,Nat.sub_self,pow_zero,one_smul,Real.norm_eq_abs,sq_abs] at h0 h2
+  have he : (∫ r in Ioi (0:ℝ),Real.exp (-(r^2)))=gaussianRadial 0 := by simp [gaussianRadial]
+  change _=volume.real (ball (0:ℝ) 1)*gaussianRadial 2 at h2
+  rw [he] at h0
+  change _=volume.real (ball (0:ℝ) 1)*gaussianRadial 0 at h0
+  rw [h2,h0,show (2:ℕ)=0+2 from rfl,gaussianRadial_add_two]
+  norm_num
+  ring
+
+lemma seminorm_add_kernel {n : ℕ} (g : Seminorm ℝ (Space n)) {e : Space n}
+    (he : g e=0) (x : Space n) (t : ℝ) : g (x+t • e)=g x := by
+  have h1 := map_add_le_add g x (t • e)
+  have h2 := map_add_le_add g (x+t • e) (-(t • e))
+  have hz : g (t • e)=0 := by rw [map_smul_eq_mul,he,mul_zero]
+  rw [hz,add_zero] at h1
+  rw [add_neg_cancel_right,map_neg_eq_map,hz,add_zero] at h2
+  exact le_antisymm h1 h2
+
+lemma gaussian_shadow_norm {n : ℕ} (e : Sphere n) (z : perpendicular (e:Space n)) (t : ℝ) :
+    ‖(z:Space n)+t • (e:Space n)‖^2=‖z‖^2+t^2 := by
+  rw [norm_add_sq_real,inner_smul_right,real_inner_comm,perpendicular_inner,norm_smul,
+    Real.norm_eq_abs,mul_pow,norm_coe,one_pow,mul_one,sq_abs]
+  simp
+
+/-- Genuine Gaussian independence in a seminorm kernel, proved by the
+measure-preserving orthogonal coordinate map and Fubini product integrals. -/
+lemma gaussian_kernel_moment {n : ℕ} (g : Seminorm ℝ (Space n)) (k : ℕ)
+    (e : Sphere n) (he : g e=0) :
+    (∫ x : Space n,g x^k*⟪(e:Space n),x⟫^2*Real.exp (-(‖x‖^2)))=
+      (1/2:ℝ)*(∫ x : Space n,g x^k*Real.exp (-(‖x‖^2))) := by
+  let F : perpendicular (e:Space n) → ℝ := fun z => g z^k*Real.exp (-(‖z‖^2))
+  have he0 : (fun z : perpendicular (e:Space n) × ℝ =>
+      g (shadowMeasurableEquiv e z)^k*Real.exp (-(‖shadowMeasurableEquiv e z‖^2)))=
+      (fun z => F z.1*Real.exp (-(z.2^2))) := by
+    funext z
+    change g ((z.1:Space n)+z.2 • (e:Space n))^k*Real.exp (-‖(z.1:Space n)+z.2 • (e:Space n)‖^2)=_
+    rw [seminorm_add_kernel g he,gaussian_shadow_norm,neg_add,Real.exp_add]
+    dsimp only [F]
+    ring
+  have he2 : (fun z : perpendicular (e:Space n) × ℝ =>
+      g (shadowMeasurableEquiv e z)^k*⟪(e:Space n),shadowMeasurableEquiv e z⟫^2*
+        Real.exp (-(‖shadowMeasurableEquiv e z‖^2)))=
+      (fun z => F z.1*(z.2^2*Real.exp (-(z.2^2)))) := by
+    funext z
+    change g ((z.1:Space n)+z.2 • (e:Space n))^k*⟪(e:Space n),(z.1:Space n)+z.2 • (e:Space n)⟫^2*
+      Real.exp (-‖(z.1:Space n)+z.2 • (e:Space n)‖^2)=_
+    rw [seminorm_add_kernel g he,gaussian_shadow_norm,neg_add,Real.exp_add,
+      inner_add_right,perpendicular_inner,inner_smul_right,real_inner_self_eq_norm_sq,norm_coe]
+    dsimp only [F]
+    ring
+  rw [← (shadowMeasurableEquiv_measurePreserving e).integral_comp (shadowMeasurableEquiv e).measurableEmbedding
+    (fun x : Space n => g x^k*⟪(e:Space n),x⟫^2*Real.exp (-(‖x‖^2))),
+    ← (shadowMeasurableEquiv_measurePreserving e).integral_comp (shadowMeasurableEquiv e).measurableEmbedding
+    (fun x : Space n => g x^k*Real.exp (-(‖x‖^2)))]
+  rw [he0,he2,Measure.volume_eq_prod,
+    integral_prod_mul F (fun t : ℝ => t^2*Real.exp (-(t^2))),
+    integral_prod_mul F (fun t : ℝ => Real.exp (-(t^2))),gaussian_real_second_moment]
+  ring
+
+/-- The boundary moment ratio, for every seminorm and every natural
+homogeneity degree, including all singular and nonsmooth cases. -/
+theorem sphere_kernel_moment {n : ℕ} [NeZero n] (g : Seminorm ℝ (Space n)) (k : ℕ)
+    (e : Sphere n) (he : g e=0) :
+    ((n:ℝ)+k)*mean (fun u : Sphere n => g u^k*⟪(e:Space n),(u:Space n)⟫^2)=
+      mean (fun u : Sphere n => g u^k) := by
+  have hh0 : ∀ r : ℝ,0<r → ∀ x : Space n,g (r • x)^k=r^k*g x^k := by
+    intro r hr x
+    rw [map_smul_eq_mul,Real.norm_of_nonneg hr.le,mul_pow]
+  have hh2 : ∀ r : ℝ,0<r → ∀ x : Space n,g (r • x)^k*⟪(e:Space n),r • x⟫^2=
+      r^(k+2)*(g x^k*⟪(e:Space n),x⟫^2) := by
+    intro r hr x
+    rw [hh0 r hr,inner_smul_right,mul_pow,pow_add]
+    ring
+  have hG := gaussian_kernel_moment g k e he
+  rw [homogeneous_gaussian_integral (k+2) _ hh2,homogeneous_gaussian_integral k _ hh0] at hG
+  rw [show n-1+(k+2)=(n-1+k)+2 by omega,gaussianRadial_add_two] at hG
+  have hn : ((n-1+k:ℕ):ℝ)+1=(n:ℝ)+k := by
+    have hh : n-1+k+1=n+k := by have := NeZero.pos n;omega
+    exact_mod_cast hh
+  rw [hn] at hG
+  have hc : (n:ℝ)*kappa n*gaussianRadial (n-1+k)≠0 :=
+    mul_ne_zero (mul_ne_zero (Nat.cast_ne_zero.mpr (NeZero.ne n)) (kappa_pos n).ne') (gaussianRadial_pos _).ne'
+  apply (mul_left_cancel₀ hc)
+  nlinarith only [hG]
+
+end PettyProjection
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology
+open scoped RealInnerProductSpace
+namespace PettyProjection.Position
+abbrev Mat (n : ℕ) := EuclideanSpace ℝ (Fin n × Fin n)
+variable {n : ℕ}
+def matrix (A : Mat n) : Matrix (Fin n) (Fin n) ℝ := fun i j => A (i,j)
+def op (A : Mat n) : Space n →L[ℝ] Space n := (matrix A).toEuclideanLin.toContinuousLinearMap
+def tr (A : Mat n) : ℝ := ∑ i,A (i,i)
+def quad (A : Mat n) (x : Space n) : ℝ := ⟪x,op A x⟫
+def Symmetric (A : Mat n) : Prop := ∀ i j,A (i,j)=A (j,i)
+def PSD (A : Mat n) : Prop := Symmetric A ∧ ∀ x,0≤quad A x
+def domain (n : ℕ) : Set (Mat n) := {A | PSD A ∧ tr A=1}
+
+lemma op_apply (A : Mat n) (x : Space n) (i : Fin n) : op A x i=∑ j,A (i,j)*x j := rfl
+lemma quad_eq (A : Mat n) (x : Space n) : quad A x=∑ i,∑ j,A (i,j)*x i*x j := by
+  simp only [quad,EuclideanSpace.inner_eq_star_dotProduct,dotProduct,star_trivial,op_apply,Finset.sum_mul]
+  congr 1; funext i
+  apply Finset.sum_congr rfl
+  intro j _; ring
+lemma op_continuous : Continuous (op : Mat n → Space n →L[ℝ] Space n) := by
+  let L : Mat n →ₗ[ℝ] (Space n →L[ℝ] Space n) :=
+    { toFun := op
+      map_add' := by intro A B; ext x i; simp [op_apply,add_mul,Finset.sum_add_distrib]
+      map_smul' := by intro a A; ext x i; simp [op_apply,Finset.mul_sum,mul_assoc] }
+  exact L.continuous_of_finiteDimensional
+lemma quad_continuous : Continuous (fun p : Mat n × Space n => quad p.1 p.2) := by
+  exact continuous_snd.inner (op_continuous.fst'.clm_apply continuous_snd)
+lemma tr_continuous : Continuous (tr : Mat n → ℝ) := by
+  unfold tr
+  fun_prop
+lemma quad_add (A B : Mat n) (x : Space n) : quad (A+B) x=quad A x+quad B x := by
+  simp [quad_eq,add_mul,Finset.sum_add_distrib]
+lemma quad_smul (a : ℝ) (A : Mat n) (x : Space n) : quad (a • A) x=a*quad A x := by
+  simp [quad_eq,mul_assoc,Finset.mul_sum]
+lemma quad_sub (A B : Mat n) (x : Space n) : quad (A-B) x=quad A x-quad B x := by
+  simp [quad_eq,sub_mul,Finset.sum_sub_distrib]
+lemma symmetric_op {A : Mat n} (hA : Symmetric A) : (op A).toLinearMap.IsSymmetric := by
+  apply Matrix.isSymmetric_toEuclideanLin_iff.mpr
+  ext i j
+  exact hA j i
+lemma op_kernel_of_quad_zero {A : Mat n} (hA : PSD A) {x : Space n} (hx : quad A x=0) : op A x=0 := by
+  have hs := symmetric_op hA.1
+  have hxy (y : Space n) : ⟪y,op A x⟫=0 := by
+    have hh (t : ℝ) := hA.2 (x+t • y)
+    have he (t : ℝ) : quad A (x+t • y)=2*t*⟪y,op A x⟫+t^2*quad A y := by
+      simp only [quad,map_add,map_smul,inner_add_left,inner_add_right,inner_smul_left,
+        inner_smul_right,conj_trivial] at *
+      have hi : ⟪x,op A y⟫=⟪y,op A x⟫ := by
+        have h := hs x y
+        change ⟪op A x,y⟫=⟪x,op A y⟫ at h
+        rw [← h,real_inner_comm]
+      rw [hx,hi]; ring
+    simp_rw [he] at hh
+    let a := quad A y
+    let b := ⟪y,op A x⟫
+    have ha : 0≤a := hA.2 y
+    have han : a+1≠0 := by linarith
+    have hv := mul_nonneg (hh (-b/(a+1))) (sq_nonneg (a+1))
+    have heval : (2*(-b/(a+1))*b+(-b/(a+1))^2*a)*(a+1)^2= -b^2*(a+2) := by
+      field_simp
+      ring
+    change 0≤(2*(-b/(a+1))*b+(-b/(a+1))^2*a)*(a+1)^2 at hv
+    rw [heval] at hv
+    change b=0
+    nlinarith [sq_nonneg b,mul_nonneg ha (sq_nonneg b)]
+  exact inner_self_eq_zero.mp (hxy (op A x))
+
+lemma symmetric_closed : IsClosed {A : Mat n | Symmetric A} := by
+  have he : {A : Mat n | Symmetric A}=⋂ i : Fin n,⋂ j : Fin n,{A : Mat n | A (i,j)=A (j,i)} := by
+    ext A; simp [Symmetric]
+  rw [he]
+  apply isClosed_iInter
+  intro i
+  apply isClosed_iInter
+  intro j
+  exact isClosed_eq (by fun_prop) (by fun_prop)
+lemma quad_nonneg_closed (x : Space n) : IsClosed {A : Mat n | 0≤quad A x} := by
+  have hc : Continuous (fun A : Mat n => quad A x) :=
+    by simp_rw [quad_eq]; fun_prop
+  exact isClosed_le (continuous_const : Continuous (fun _ : Mat n => (0:ℝ))) hc
+lemma psd_closed : IsClosed {A : Mat n | PSD A} := by
+  have hq : IsClosed (⋂ x : Space n,{A : Mat n | 0≤quad A x}) :=
+    isClosed_iInter (fun x : Space n => quad_nonneg_closed x)
+  have h := (symmetric_closed (n := n)).inter hq
+  convert h using 1
+  ext A
+  simp only [mem_inter_iff,mem_ofPred_eq,mem_iInter,PSD]
+lemma domain_closed : IsClosed (domain n) :=
+  (psd_closed (n := n)).inter (isClosed_eq (tr_continuous (n := n)) continuous_const)
+lemma domain_convex : Convex ℝ (domain n) := by
+  intro A hA B hB a b ha hb hab
+  refine ⟨⟨?_,?_⟩,?_⟩
+  · intro i j
+    change a*A (i,j)+b*B (i,j)=a*A (j,i)+b*B (j,i)
+    rw [hA.1.1 i j,hB.1.1 i j]
+  · intro x
+    rw [quad_add,quad_smul,quad_smul]
+    exact add_nonneg (mul_nonneg ha (hA.1.2 x)) (mul_nonneg hb (hB.1.2 x))
+  · change (∑ i,(a*A (i,i)+b*B (i,i)))=1
+    rw [Finset.sum_add_distrib,← Finset.mul_sum,← Finset.mul_sum]
+    change a*tr A+b*tr B=1
+    rw [hA.2,hB.2,mul_one,mul_one,hab]
+
+lemma quad_basis (A : Mat n) (i : Fin n) : quad A (EuclideanSpace.single i 1)=A (i,i) := by
+  simp [quad_eq]
+lemma quad_basis_add (A : Mat n) (i j : Fin n) (a b : ℝ) :
+    quad A (a • EuclideanSpace.single i 1+b • EuclideanSpace.single j 1)=
+      a^2*A (i,i)+a*b*(A (i,j)+A (j,i))+b^2*A (j,j) := by
+  simp [quad_eq,add_mul,mul_add,Finset.sum_add_distrib]
+  ring
+lemma domain_entry_bound {A : Mat n} (hA : A∈domain n) (i j : Fin n) : |A (i,j)|≤1 := by
+  have hd (i : Fin n) : 0≤A (i,i) := by simpa only [quad_basis] using hA.1.2 (EuclideanSpace.single i 1)
+  have hd1 (i : Fin n) : A (i,i)≤1 := by
+    rw [← hA.2]
+    exact Finset.single_le_sum (fun j _ => hd j) (Finset.mem_univ i)
+  have hp := hA.1.2 ((1:ℝ) • EuclideanSpace.single i 1+(1:ℝ) • EuclideanSpace.single j 1)
+  have hm := hA.1.2 ((1:ℝ) • EuclideanSpace.single i 1+(-1:ℝ) • EuclideanSpace.single j 1)
+  rw [quad_basis_add,hA.1.1 j i] at hp hm
+  exact abs_le.mpr ⟨by nlinarith [hd1 i,hd1 j],by nlinarith [hd1 i,hd1 j]⟩
+lemma domain_compact : IsCompact (domain n) := by
+  apply (isCompact_closedBall (0:Mat n) (n:ℝ)).of_isClosed_subset domain_closed
+  intro A hA
+  rw [mem_closedBall,dist_zero_right]
+  have hn : (0:ℝ)≤n := Nat.cast_nonneg _
+  rw [← sq_le_sq₀ (norm_nonneg A) hn,EuclideanSpace.norm_sq_eq]
+  calc
+    ∑ i : Fin n × Fin n,‖A i‖^2 ≤ ∑ _ : Fin n × Fin n,(1:ℝ) := by
+      apply Finset.sum_le_sum
+      intro i _
+      have hh := domain_entry_bound hA i.1 i.2
+      dsimp only [Real.norm_eq_abs]
+      nlinarith [abs_nonneg (A i)]
+    _ = (n:ℝ)^2 := by simp [sq]
+lemma domain_nonempty [NeZero n] : (domain n).Nonempty := by
+  let A : Mat n := WithLp.toLp 2 (fun ij => if ij.1=ij.2 then (n:ℝ)⁻¹ else 0)
+  refine ⟨A,⟨?_,?_⟩,?_⟩
+  · intro i j
+    dsimp [A]
+    simp only [eq_comm]
+  · intro x
+    rw [quad_eq]
+    dsimp [A]
+    simp only [ite_mul,zero_mul,Finset.sum_ite_eq,Finset.mem_univ,ite_true]
+    apply Finset.sum_nonneg
+    intro i _
+    simpa only [sq,mul_assoc] using mul_nonneg (inv_nonneg.mpr (Nat.cast_nonneg n : (0:ℝ)≤n)) (sq_nonneg (x i))
+  · simp [tr,A,NeZero.ne n]
+lemma domain_range_ne_bot {A : Mat n} (hA : A∈domain n) : LinearMap.range (op A).toLinearMap≠⊥ := by
+  intro h
+  have hz : op A=0 := by
+    apply ContinuousLinearMap.ext
+    intro x
+    exact (Submodule.mem_bot ℝ).mp (h ▸ LinearMap.mem_range_self _ x)
+  have hdiag (i : Fin n) : A (i,i)=0 := by rw [← quad_basis,quad,hz]; simp
+  have : tr A=0 := by simp [tr,hdiag]
+  linarith [hA.2]
+end PettyProjection.Position
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology
+open scoped RealInnerProductSpace
+namespace PettyProjection.Position
+open Spherical (Sphere mean sigma norm_coe continuous_integrable seminorm_continuous)
+variable {n : ℕ} [NeZero n]
+
+def identity (n : ℕ) : Mat n := WithLp.toLp 2 (fun ij => if ij.1=ij.2 then 1 else 0)
+def momentField (g : Seminorm ℝ (Space n)) (k : ℕ) : Mat n :=
+  WithLp.toLp 2 (fun ij => mean (fun u : Sphere n => g u^k*(u:Space n) ij.1*(u:Space n) ij.2)/
+    mean (fun u : Sphere n => g u^k)-(if ij.1=ij.2 then 1 else 0)/(n:ℝ))
+
+lemma mean_power_pos (g : Seminorm ℝ (Space n)) (hg : g≠0) (k : ℕ) :
+    0 < mean (fun u : Sphere n => g u^k) := by
+  obtain ⟨R,e,hR,_,hb⟩ := Spherical.seminorm_coordinate_minorant g hg
+  have he := hb e
+  simp only [real_inner_self_eq_norm_sq,norm_coe,one_pow,abs_one,mul_one] at he
+  have hc : Continuous (fun u : Sphere n => g u^k) := ((seminorm_continuous n g).comp continuous_subtype_val).pow k
+  apply integral_pos_of_integrable_nonneg_nonzero (x := e) hc (continuous_integrable hc)
+    (fun u => pow_nonneg (apply_nonneg g (u:Space n)) k)
+  exact (pow_pos (hR.trans_le he) k).ne'
+
+omit [NeZero n] in
+lemma field_symmetric (g : Seminorm ℝ (Space n)) (k : ℕ) : Symmetric (momentField g k) := by
+  intro i j
+  change mean _ / _ - _ / _=mean _ / _ - _ / _
+  congr 2
+  · congr 1; funext u; ring
+  · simp only [eq_comm]
+
+lemma field_trace (g : Seminorm ℝ (Space n)) (hg : g≠0) (k : ℕ) : tr (momentField g k)=0 := by
+  have hc (i : Fin n) : Continuous (fun u : Sphere n => g u^k*(u:Space n) i*(u:Space n) i) := by
+    exact (((seminorm_continuous n g).comp continuous_subtype_val).pow k).mul (by fun_prop) |>.mul (by fun_prop)
+  have hsum : (∑ i : Fin n,mean (fun u : Sphere n => g u^k*(u:Space n) i*(u:Space n) i))=
+      mean (fun u : Sphere n => g u^k) := by
+    rw [← Spherical.mean_sum _ (fun i _ => hc i)]
+    congr 1
+    funext u
+    have hh := EuclideanSpace.real_norm_sq_eq (u:Space n)
+    rw [norm_coe,one_pow] at hh
+    calc
+      (∑ i : Fin n,g u^k*(u:Space n) i*(u:Space n) i)=g u^k*(∑ i : Fin n,(u:Space n) i^2) := by
+        rw [Finset.mul_sum]; apply Finset.sum_congr rfl; intro i _; ring
+      _ = g u^k := by rw [← hh,mul_one]
+  simp only [tr,momentField,ite_true]
+  change (∑ i : Fin n,(mean (fun u : Sphere n => g u^k*(u:Space n) i*(u:Space n) i)/mean (fun u : Sphere n => g u^k)-1/(n:ℝ)))=0
+  rw [Finset.sum_sub_distrib,← Finset.sum_div,hsum,div_self (mean_power_pos g hg k).ne']
+  simp [NeZero.ne n]
+
+lemma field_quad (g : Seminorm ℝ (Space n)) (k : ℕ) (x : Space n) :
+    quad (momentField g k) x=mean (fun u : Sphere n => g u^k*⟪x,(u:Space n)⟫^2)/
+      mean (fun u : Sphere n => g u^k)-‖x‖^2/(n:ℝ) := by
+  have hc (i j : Fin n) : Continuous (fun u : Sphere n =>
+      g u^k*(u:Space n) i*(u:Space n) j*x i*x j) := by
+    exact (((((seminorm_continuous n g).comp continuous_subtype_val).pow k).mul (by fun_prop)).mul (by fun_prop)).mul continuous_const |>.mul continuous_const
+  have hnum : (∑ i : Fin n,∑ j : Fin n,
+      mean (fun u : Sphere n => g u^k*(u:Space n) i*(u:Space n) j)*x i*x j)=
+      mean (fun u : Sphere n => g u^k*⟪x,(u:Space n)⟫^2) := by
+    have hi (i : Fin n) : (∑ j : Fin n,mean (fun u : Sphere n => g u^k*(u:Space n) i*(u:Space n) j)*x i*x j)=
+        mean (fun u : Sphere n => ∑ j : Fin n,g u^k*(u:Space n) i*(u:Space n) j*x i*x j) := by
+      simp_rw [mean,← integral_mul_const]
+      exact (integral_finsetSum _ (fun j _ => continuous_integrable (hc i j))).symm
+    simp_rw [hi]
+    rw [← Spherical.mean_sum _ (fun i _ => continuous_finsetSum _ (fun j _ => hc i j))]
+    congr 1
+    funext u
+    simp only [EuclideanSpace.inner_eq_star_dotProduct,dotProduct,star_trivial,sq,Finset.sum_mul,Finset.mul_sum]
+    apply Finset.sum_congr rfl
+    intro i _
+    apply Finset.sum_congr rfl
+    intro j _
+    ring
+  rw [quad_eq]
+  simp only [momentField,sub_mul,Finset.sum_sub_distrib]
+  have hdiv : (∑ i : Fin n,∑ j : Fin n,
+      mean (fun u : Sphere n => g u^k*(u:Space n) i*(u:Space n) j)/mean (fun u : Sphere n => g u^k)*x i*x j)=
+      mean (fun u : Sphere n => g u^k*⟪x,(u:Space n)⟫^2)/mean (fun u : Sphere n => g u^k) := by
+    simp_rw [div_mul_eq_mul_div,← Finset.sum_div]
+    rw [hnum]
+  rw [hdiv]
+  congr 1
+  simp only [ite_div,zero_div,ite_mul,zero_mul,Finset.sum_ite_eq,Finset.mem_univ,ite_true]
+  rw [EuclideanSpace.real_norm_sq_eq]
+  rw [Finset.sum_div]
+  simp only [one_div,sq]
+  apply Finset.sum_congr rfl
+  intro i _
+  ring
+
+lemma field_kernel {g : Seminorm ℝ (Space n)} (hg : g≠0) {k : ℕ} (hk : 0<k)
+    (e : Sphere n) (he : g e=0) : quad (momentField g k) e<0 := by
+  rw [field_quad,norm_coe,one_pow]
+  have hm := sphere_kernel_moment g k e he
+  have hp := mean_power_pos g hg k
+  have hn : (0:ℝ)<n := Nat.cast_pos.mpr (NeZero.pos n)
+  have hk' : (0:ℝ)<k := Nat.cast_pos.mpr hk
+  apply sub_neg.mpr
+  apply (div_lt_iff₀ hp).mpr
+  rw [one_div,mul_comm,← div_eq_mul_inv,lt_div_iff₀ hn]
+  have hnon : 0 < mean (fun u : Sphere n => g u^k*⟪(e:Space n),(u:Space n)⟫^2) := by nlinarith
+  nlinarith
+end PettyProjection.Position
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/ProjectionBodies/RankContinuity.lean b/lean/OAI/Geometry/ProjectionBodies/RankContinuity.lean
new file mode 100644
index 0000000000000000000000000000000000000000..429abb033dad96814620cd9687c167cd49ce78e7
--- /dev/null
+++ b/lean/OAI/Geometry/ProjectionBodies/RankContinuity.lean
@@ -0,0 +1,507 @@
+import OAI.Geometry.ProjectionBodies.GaussianMeasures
+
+namespace OAI
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology
+open scoped NNReal RealInnerProductSpace
+namespace PettyProjection
+open Spherical (Sphere)
+variable {n : ℕ}
+
+/-- Actual lower-semicontinuity at a singular change of range, formulated
+with a bounded projected cylinder instead of extended real support functions. -/
+lemma support_min_tendsto {V : Submodule ℝ (Space n)}
+    {Vj : ℕ → Submodule ℝ (Space n)} (gj : ∀ i,RelativeGauge (Vj i))
+    (g : Seminorm ℝ (Space n)) {c : ℝ} (hc : 0<c)
+    (hl : ∀ x,c*‖V.starProjection x‖≤g x)
+    (hg : ∀ x,Tendsto (fun i => gj i x) atTop (𝓝 (g x)))
+    {uj : ℕ → Space n} {u : Space n} (hu : u∈V) (huj : ∀ i,uj i∈Vj i)
+    (ht : Tendsto uj atTop (𝓝 u)) :
+    Tendsto (fun i => min (support (gj i).body (uj i))
+      (support (RelativeGauge.project V g hc hl).body u)) atTop
+      (𝓝 (support (RelativeGauge.project V g hc hl).body u)) := by
+  let f := RelativeGauge.project V g hc hl
+  apply tendsto_order.mpr
+  constructor
+  · intro r hr
+    have hex : ∃ z∈seminormUnit g,r<⟪u,z⟫ := by
+      by_contra! hn
+      have hb : support f.body u≤r := by
+        apply (support_le_iff f.compact f.nonempty).mpr
+        intro y hy
+        rw [RelativeGauge.project_body] at hy
+        have hs : projectedUnit V g⊆{y | ⟪u,y⟫≤r} := by
+          apply closure_minimal _ (isClosed_le (by fun_prop) continuous_const)
+          rintro _ ⟨z,hz,rfl⟩
+          change ⟪u,V.starProjection z⟫≤r
+          rw [← Submodule.inner_starProjection_left_eq_right,V.starProjection_eq_self_iff.mpr hu]
+          exact hn z hz
+        exact hs hy
+      exact (not_le_of_gt hr) hb
+    obtain ⟨z,hz,hzr⟩ := hex
+    have hm : max 1 (g z)=1 := max_eq_left hz
+    have hd : Tendsto (fun i => ⟪uj i,z⟫ / max 1 (gj i z)) atTop (𝓝 ⟪u,z⟫) := by
+      have hh := (ht.inner (tendsto_const_nhds (x := z))).div (tendsto_const_nhds.max (hg z)) (by rw [hm]; norm_num)
+      change Tendsto (fun i => ⟪uj i,z⟫ / max 1 (gj i z)) atTop (𝓝 (⟪u,z⟫ / max 1 (g z))) at hh
+      simpa only [hm,div_one] using hh
+    filter_upwards [hd.eventually (lt_mem_nhds hzr)] with i hi
+    exact lt_min (hi.trans_le ((gj i).support_cylinder_bound (uj i) (huj i) z)) hr
+  · intro r hr
+    exact Eventually.of_forall (fun i => (min_le_right _ _).trans_lt hr)
+
+lemma projected_limit_normalization {μ : Measure (Space n)}
+    (hμ : Integrable (fun x : Space n => ‖x‖) μ)
+    {A : Space n →L[ℝ] Space n} {Aj : ℕ → Space n →L[ℝ] Space n}
+    (hA : ∀ x,Tendsto (fun i => Aj i x) atTop (𝓝 (A x)))
+    (gj : ∀ i,RelativeGauge (LinearMap.range (Aj i).toLinearMap))
+    (hjn : ∀ i,supportFunctional (μ.map (Aj i)) (gj i).body=1)
+    (g : Seminorm ℝ (Space n))
+    (hg : ∀ x,Tendsto (fun i => gj i x) atTop (𝓝 (g x)))
+    {c : ℝ} (hc : 0<c)
+    (hl : ∀ x,c*‖(LinearMap.range A.toLinearMap).starProjection x‖≤g x) :
+    supportFunctional (μ.map A) (RelativeGauge.project (LinearMap.range A.toLinearMap) g hc hl).body≤1 := by
+  let V := LinearMap.range A.toLinearMap
+  let f := RelativeGauge.project V g hc hl
+  let H : Space n → ℝ := fun x => support f.body (A x)
+  let Hj : ℕ → Space n → ℝ := fun i x => support (gj i).body (Aj i x)
+  have hi : Integrable H μ := by
+    exact (integrable_map_measure (support_continuous f.compact).aestronglyMeasurable
+      A.continuous.measurable.aemeasurable).mp
+        (support_integrable (norm_integrable_map hμ A) f.compact f.nonempty)
+  have hij (i : ℕ) : Integrable (Hj i) μ := by
+    exact (integrable_map_measure (support_continuous (gj i).compact).aestronglyMeasurable
+      (Aj i).continuous.measurable.aemeasurable).mp
+        (support_integrable (norm_integrable_map hμ (Aj i)) (gj i).compact (gj i).nonempty)
+  have hpos (x : Space n) : 0≤H x := support_nonneg f.compact f.zero_mem _
+  have hjpos (i : ℕ) (x : Space n) : 0≤Hj i x := support_nonneg (gj i).compact (gj i).zero_mem _
+  have ht : Tendsto (fun i => ∫ x,min (Hj i x) (H x) ∂μ) atTop (𝓝 (∫ x,H x ∂μ)) := by
+    apply tendsto_integral_of_dominated_convergence H
+    · intro i
+      exact ((hij i).inf hi).aestronglyMeasurable
+    · exact hi
+    · intro i
+      filter_upwards [] with x
+      rw [Real.norm_eq_abs,abs_of_nonneg (le_min (hjpos i x) (hpos x))]
+      exact min_le_right _ _
+    · filter_upwards [] with x
+      exact support_min_tendsto gj g hc hl hg (show A x∈V from ⟨x,rfl⟩)
+        (fun i => show Aj i x∈LinearMap.range (Aj i).toLinearMap from ⟨x,rfl⟩) (hA x)
+  have hb (i : ℕ) : (∫ x,min (Hj i x) (H x) ∂μ)≤1 := by
+    have hh := integral_mono ((hij i).inf hi) (hij i) (fun _ => min_le_left _ _)
+    have he : (∫ x,Hj i x ∂μ)=1 := by rw [← functional_map _ (gj i).compact,hjn]
+    exact hh.trans he.le
+  rw [functional_map _ f.compact]
+  exact le_of_tendsto ht (Eventually.of_forall hb)
+
+end PettyProjection
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology
+open scoped NNReal RealInnerProductSpace Pointwise
+namespace PettyProjection
+open Spherical (Sphere objective)
+namespace RelativeGauge
+variable {n : ℕ}
+
+lemma full_lower (f : RelativeGauge (⊤ : Submodule ℝ (Space n))) :
+    ∃ c : ℝ,0< c ∧ ∀ x,c*‖x‖≤ f x := by
+  obtain ⟨c,hc,hb⟩ := f.lower
+  exact ⟨c,hc,fun x => by simpa only [Submodule.starProjection_top,ContinuousLinearMap.id_apply] using hb x⟩
+
+def projectFull (V : Submodule ℝ (Space n))
+    (f : RelativeGauge (⊤ : Submodule ℝ (Space n))) : RelativeGauge V :=
+  project V f.toSeminorm f.full_lower.choose_spec.1 (fun x =>
+    (mul_le_mul_of_nonneg_left (V.norm_starProjection_apply_le x) f.full_lower.choose_spec.1.le).trans
+      (f.full_lower.choose_spec.2 x))
+
+lemma projectFull_le (V : Submodule ℝ (Space n))
+    (f : RelativeGauge (⊤ : Submodule ℝ (Space n))) (x : Space n) : projectFull V f x≤ f x :=
+  project_le _ _ _ _ _
+
+lemma projectFull_body (V : Submodule ℝ (Space n))
+    (f : RelativeGauge (⊤ : Submodule ℝ (Space n))) :
+    (projectFull V f).body=V.starProjection '' f.body := by
+  rw [projectFull,project_body,projectedUnit,← f.top_body]
+  exact (f.compact.image V.starProjection.continuous).isClosed.closure_eq
+
+lemma support_projectFull (V : Submodule ℝ (Space n))
+    (f : RelativeGauge (⊤ : Submodule ℝ (Space n))) {u : Space n} (hu : u∈V) :
+    support (projectFull V f).body u=support f.body u := by
+  have he (x : Space n) : ⟪u,V.starProjection x⟫=⟪u,x⟫ := by
+    rw [← Submodule.inner_starProjection_left_eq_right,V.starProjection_eq_self_iff.mpr hu]
+  apply le_antisymm
+  · apply (support_le_iff (projectFull V f).compact (projectFull V f).nonempty).mpr
+    intro x hx
+    rw [projectFull_body] at hx
+    obtain ⟨y,hy,rfl⟩ := hx
+    rw [he]
+    exact inner_le_support f.compact hy u
+  · obtain ⟨x,hx,hxe⟩ := support_attained f.compact f.nonempty u
+    rw [hxe,← he]
+    apply inner_le_support (projectFull V f).compact
+    rw [projectFull_body]
+    exact mem_image_of_mem _ hx
+
+lemma functional_projectFull {μ : Measure (Space n)} (A : Space n →L[ℝ] Space n)
+    (f : RelativeGauge (⊤ : Submodule ℝ (Space n))) :
+    supportFunctional (μ.map A) (projectFull (LinearMap.range A.toLinearMap) f).body=
+      supportFunctional (μ.map A) f.body := by
+  rw [functional_map _ (projectFull _ f).compact,functional_map _ f.compact]
+  congr 1
+  funext x
+  exact support_projectFull _ f (show A x∈LinearMap.range A.toLinearMap from ⟨x,rfl⟩)
+
+lemma minimizer_full_comparison [NeZero n] {μ : Measure (Space n)}
+    (hμ : Integrable (fun x : Space n => ‖x‖) μ)
+    (hq : ∀ x : Space n,x≠0 → 0< cosineSeminorm μ hμ x)
+    (A : Space n →L[ℝ] Space n) (hA : LinearMap.range A.toLinearMap≠⊥) (b : Bool)
+    (g : RelativeGauge (LinearMap.range A.toLinearMap))
+    (hmin : ∀ q : RelativeGauge (LinearMap.range A.toLinearMap),supportFunctional (μ.map A) q.body=1 →
+      objective b g.toSeminorm≤ objective b q.toSeminorm)
+    (f : RelativeGauge (⊤ : Submodule ℝ (Space n))) :
+    ∃ ha : 0< supportFunctional (μ.map A) f.body,
+      objective b g.toSeminorm≤ objective b (f.scale _ ha).toSeminorm := by
+  let p := projectFull (LinearMap.range A.toLinearMap) f
+  have hp := p.functional_pos hA (norm_integrable_map hμ A) (cosine_map_pos hμ hq A)
+  have he : supportFunctional (μ.map A) p.body=supportFunctional (μ.map A) f.body := functional_projectFull A f
+  have ha : 0< supportFunctional (μ.map A) f.body := he ▸ hp
+  refine ⟨ha,?_⟩
+  have hpn : supportFunctional (μ.map A) (p.scale _ ha).body=1 := by
+    rw [p.functional_scale _ ha (μ.map A),he,inv_mul_cancel₀ ha.ne']
+  apply (hmin _ hpn).trans
+  apply Spherical.objective_mono b ((p.scale _ ha).ne_zero hA) ((f.scale _ ha).ne_zero top_ne_bot)
+  intro x
+  exact mul_le_mul_of_nonneg_left (projectFull_le _ f x) ha.le
+
+end RelativeGauge
+end PettyProjection
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology
+open scoped NNReal RealInnerProductSpace
+namespace PettyProjection
+open Spherical (Sphere objective)
+namespace RelativeGauge
+variable {n : ℕ} {V : Submodule ℝ (Space n)}
+
+/-- A genuine full-dimensional approximation of an arbitrary relative gauge.
+The sum regularization avoids any constant-rank assumption. -/
+def thicken (g : RelativeGauge V) (ε : ℝ) (hε : 0<ε) :
+    RelativeGauge (⊤ : Submodule ℝ (Space n)) where
+  toSeminorm := Seminorm.of (fun x => g x+ε*‖x‖)
+    (fun x y => by
+      have hg := map_add_le_add g.toSeminorm x y
+      have hn := mul_le_mul_of_nonneg_left (norm_add_le x y) hε.le
+      linarith)
+    (fun r x => by rw [map_smul_eq_mul,norm_smul]; ring)
+  projection x := by simp only [Submodule.starProjection_top,ContinuousLinearMap.id_apply]
+  lower := ⟨ε,hε,fun x => by
+    change ε*‖(⊤ : Submodule ℝ (Space n)).starProjection x‖≤ g x+ε*‖x‖
+    simp only [Submodule.starProjection_top,ContinuousLinearMap.id_apply]
+    linarith [apply_nonneg g.toSeminorm x]⟩
+
+lemma thicken_apply (g : RelativeGauge V) {ε : ℝ} (hε : 0<ε) (x : Space n) :
+    thicken g ε hε x=g x+ε*‖x‖ := rfl
+
+lemma thicken_ge (g : RelativeGauge V) {ε : ℝ} (hε : 0<ε) (x : Space n) :
+    g x≤ thicken g ε hε x := le_add_of_nonneg_right (mul_nonneg hε.le (norm_nonneg x))
+
+lemma support_thicken_le (g : RelativeGauge V) {ε : ℝ} (hε : 0<ε)
+    {u : Space n} (hu : u∈V) : support (thicken g ε hε).body u≤ support g.body u := by
+  apply (support_le_iff (thicken g ε hε).compact (thicken g ε hε).nonempty).mpr
+  intro x hx
+  have hpx : V.starProjection x∈g.body := by
+    refine ⟨V.starProjection_apply_mem x,?_⟩
+    change g (V.starProjection x)≤1
+    rw [g.projection]
+    exact (g.thicken_ge hε x).trans hx.2
+  have hh := inner_le_support g.compact hpx u
+  rwa [← Submodule.inner_starProjection_left_eq_right,V.starProjection_eq_self_iff.mpr hu] at hh
+
+lemma functional_thicken_le {μ : Measure (Space n)}
+    (hμ : Integrable (fun x : Space n => ‖x‖) μ)
+    (A : Space n →L[ℝ] Space n) (g : RelativeGauge (LinearMap.range A.toLinearMap))
+    {ε : ℝ} (hε : 0<ε) :
+    supportFunctional (μ.map A) (g.thicken ε hε).body≤ supportFunctional (μ.map A) g.body := by
+  rw [functional_map _ (g.thicken ε hε).compact,functional_map _ g.compact]
+  have hi (f : Set (Space n)) (hf : IsCompact f) (hfn : f.Nonempty) :=
+    (integrable_map_measure (support_continuous hf).aestronglyMeasurable A.continuous.measurable.aemeasurable).mp
+      (support_integrable (norm_integrable_map hμ A) hf hfn)
+  exact integral_mono (hi _ (g.thicken ε hε).compact (g.thicken ε hε).nonempty) (hi _ g.compact g.nonempty)
+    (fun x => support_thicken_le g hε (show A x∈LinearMap.range A.toLinearMap from ⟨x,rfl⟩))
+
+lemma objective_thicken_tendsto [NeZero n] (g : RelativeGauge V) (hV : V≠⊥) (b : Bool) :
+    Tendsto (fun i : ℕ => objective b (g.thicken (1/((i:ℝ)+1)) (by positivity)).toSeminorm)
+      atTop (𝓝 (objective b g.toSeminorm)) := by
+  obtain ⟨x,hx,hx0⟩ := (Submodule.ne_bot_iff V).mp hV
+  let u : Sphere n := ⟨‖x‖⁻¹ • x,by simp [norm_smul,hx0]⟩
+  have huV : (u:Space n)∈V := V.smul_mem _ hx
+  have hun : (u:Space n)≠0 := by intro he; have hh := Spherical.norm_coe u; simp [he] at hh
+  have hp := g.pos huV hun
+  let F : C(Sphere n,ℝ) := ⟨fun u => g u,(Spherical.seminorm_continuous n g.toSeminorm).comp continuous_subtype_val⟩
+  refine Spherical.objective_tendsto b hp (fun i => ⟨u,g.thicken_ge (by positivity) u⟩)
+    (R := ‖F‖+1) ?_ (g.ne_zero hV) ?_
+  · intro i v
+    rw [thicken_apply,Spherical.norm_coe,mul_one]
+    have hf : g v≤‖F‖ := (le_abs_self _).trans (F.norm_coe_le_norm v)
+    have hi : (1:ℝ)/((i:ℝ)+1)≤1 := (div_le_one (by positivity)).mpr (by linarith [Nat.cast_nonneg (α := ℝ) i])
+    linarith
+  · intro v
+    have ht := (tendsto_const_nhds (x := g v)).add ((tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).mul_const ‖(v:Space n)‖)
+    simpa only [thicken_apply,zero_mul,add_zero] using ht
+
+end RelativeGauge
+end PettyProjection
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology
+open scoped NNReal RealInnerProductSpace
+namespace PettyProjection
+open Spherical (Sphere objective)
+namespace RelativeGauge
+variable {n : ℕ} [NeZero n] {μ : Measure (Space n)}
+
+omit [NeZero n] in
+lemma cosine_map_tendsto (hμ : Integrable (fun x : Space n => ‖x‖) μ)
+    {Aj : ℕ → Space n →L[ℝ] Space n} {A : Space n →L[ℝ] Space n}
+    (hA : Tendsto Aj atTop (𝓝 A)) (x : Space n) :
+    Tendsto (fun i => cosineSeminorm (μ.map (Aj i)) (norm_integrable_map hμ (Aj i)) x)
+      atTop (𝓝 (cosineSeminorm (μ.map A) (norm_integrable_map hμ A) x)) := by
+  simp_rw [cosine_map μ hμ]
+  have hc : Continuous (fun B : Space n →L[ℝ] Space n => cosineSeminorm μ hμ (B.adjoint x)) :=
+    (Spherical.seminorm_continuous n _).comp (by fun_prop)
+  exact (hc.tendsto A).comp hA
+
+omit [NeZero n] in
+lemma normalized_limit_lower (hμ : Integrable (fun x : Space n => ‖x‖) μ)
+    {Aj : ℕ → Space n →L[ℝ] Space n} {A : Space n →L[ℝ] Space n}
+    (hA : Tendsto Aj atTop (𝓝 A))
+    (gj : ∀ i,RelativeGauge (LinearMap.range (Aj i).toLinearMap))
+    (hn : ∀ i,supportFunctional (μ.map (Aj i)) (gj i).body=1)
+    {g : Seminorm ℝ (Space n)} (hg : ∀ x,Tendsto (fun i => gj i x) atTop (𝓝 (g x))) (x : Space n) :
+    cosineSeminorm (μ.map A) (norm_integrable_map hμ A) x≤ g x := by
+  apply le_of_tendsto_of_tendsto (cosine_map_tendsto hμ hA x) (hg x)
+  apply Eventually.of_forall
+  intro i
+  have hh := (gj i).normalized_lower (norm_integrable_map hμ (Aj i)) (hn i) x
+  rwa [cosine_map_projection hμ] at hh
+
+lemma full_comparison_limit (hμ : Integrable (fun x : Space n => ‖x‖) μ)
+    (hq : ∀ x : Space n,x≠0 → 0< cosineSeminorm μ hμ x)
+    {Aj : ℕ → Space n →L[ℝ] Space n} {A : Space n →L[ℝ] Space n}
+    (hA : Tendsto Aj atTop (𝓝 A)) (hAn : LinearMap.range A.toLinearMap≠⊥)
+    (hAj : ∀ i,LinearMap.range (Aj i).toLinearMap≠⊥) (b : Bool)
+    (gj : ∀ i,RelativeGauge (LinearMap.range (Aj i).toLinearMap))
+    (hm : ∀ i (q : RelativeGauge (LinearMap.range (Aj i).toLinearMap)),supportFunctional (μ.map (Aj i)) q.body=1 → objective b (gj i).toSeminorm≤ objective b q.toSeminorm)
+    {g : Seminorm ℝ (Space n)}
+    (hobj : Tendsto (fun i => objective b (gj i).toSeminorm) atTop (𝓝 (objective b g)))
+    (f : RelativeGauge (⊤ : Submodule ℝ (Space n))) :
+    ∃ ha : 0< supportFunctional (μ.map A) f.body,
+      objective b g≤ objective b (f.scale _ ha).toSeminorm := by
+  obtain ⟨gA,hgn,hgm⟩ := exists_minimizer (norm_integrable_map hμ A) hAn (cosine_map_pos hμ hq A) b
+  obtain ⟨ha,_⟩ := minimizer_full_comparison hμ hq A hAn b gA hgm f
+  refine ⟨ha,?_⟩
+  have hscale : ∀ {a : ℝ} (ha : 0< a),objective b (f.scale a ha).toSeminorm=
+      if b then Real.log a+objective b f.toSeminorm else a*objective b f.toSeminorm := by
+    intro a ha
+    exact Spherical.objective_scale b (f.ne_zero top_ne_bot) ha (fun _ => rfl)
+  rw [hscale]
+  have ht := ((functional_map_continuous hμ f.compact f.nonempty).tendsto A).comp hA
+  have htr : Tendsto (fun i => if b then Real.log (supportFunctional (μ.map (Aj i)) f.body)+objective b f.toSeminorm
+      else supportFunctional (μ.map (Aj i)) f.body*objective b f.toSeminorm)
+      atTop (𝓝 (if b then Real.log (supportFunctional (μ.map A) f.body)+objective b f.toSeminorm
+        else supportFunctional (μ.map A) f.body*objective b f.toSeminorm)) := by
+    cases b
+    · exact ht.mul_const _
+    · exact (ht.log ha.ne').add_const _
+  apply le_of_tendsto_of_tendsto hobj htr
+  apply Eventually.of_forall
+  intro i
+  obtain ⟨hai,hi⟩ := minimizer_full_comparison hμ hq (Aj i) (hAj i) b (gj i) (hm i) f
+  rwa [hscale] at hi
+
+/-- Every seminorm limit of actual minimizers is the actual minimizer in the
+limiting range, including arbitrary drops of rank. -/
+theorem minimizer_limit_identify (hμ : Integrable (fun x : Space n => ‖x‖) μ)
+    (hq : ∀ x : Space n,x≠0 → 0< cosineSeminorm μ hμ x)
+    {Aj : ℕ → Space n →L[ℝ] Space n} {A : Space n →L[ℝ] Space n}
+    (hA : Tendsto Aj atTop (𝓝 A)) (hAn : LinearMap.range A.toLinearMap≠⊥)
+    (hAj : ∀ i,LinearMap.range (Aj i).toLinearMap≠⊥) (b : Bool)
+    (gj : ∀ i,RelativeGauge (LinearMap.range (Aj i).toLinearMap))
+    (hn : ∀ i,supportFunctional (μ.map (Aj i)) (gj i).body=1)
+    (hm : ∀ i (q : RelativeGauge (LinearMap.range (Aj i).toLinearMap)),supportFunctional (μ.map (Aj i)) q.body=1 → objective b (gj i).toSeminorm≤ objective b q.toSeminorm)
+    (gA : RelativeGauge (LinearMap.range A.toLinearMap)) (hgn : supportFunctional (μ.map A) gA.body=1)
+    (hgm : ∀ q : RelativeGauge (LinearMap.range A.toLinearMap),supportFunctional (μ.map A) q.body=1 → objective b gA.toSeminorm≤ objective b q.toSeminorm)
+    {g : Seminorm ℝ (Space n)} (hg : ∀ x,Tendsto (fun i => gj i x) atTop (𝓝 (g x)))
+    (hobj : Tendsto (fun i => objective b (gj i).toSeminorm) atTop (𝓝 (objective b g))) :
+    g=gA.toSeminorm := by
+  let V := LinearMap.range A.toLinearMap
+  have hμA := norm_integrable_map hμ A
+  have hqA := cosine_map_pos hμ hq A
+  have hqle := normalized_limit_lower hμ hA gj hn hg
+  obtain ⟨c,hc,hcb⟩ := seminorm_positive_subspace_bound V hAn (cosineSeminorm (μ.map A) hμA) hqA
+  have hl (x : Space n) : c*‖V.starProjection x‖≤ g x := by
+    have hh := hcb x
+    rw [cosine_map_projection hμ] at hh
+    exact hh.trans (hqle x)
+  let p := project V g hc hl
+  have hpnle : supportFunctional (μ.map A) p.body≤1 := by
+    apply projected_limit_normalization hμ _ gj hn g hg hc hl
+    intro x
+    exact ((show Continuous (fun B : Space n →L[ℝ] Space n => B x) by fun_prop).tendsto A).comp hA
+  have hp := p.functional_pos hAn hμA hqA
+  let q := p.scale _ hp
+  have hqn : supportFunctional (μ.map A) q.body=1 := by
+    rw [functional_scale p _ hp (μ.map A),inv_mul_cancel₀ hp.ne']
+  have hqg (x : Space n) : q x≤ g x := by
+    exact (mul_le_of_le_one_left (apply_nonneg p.toSeminorm x) hpnle).trans (project_le V g hc hl x)
+  have hg0 : g≠0 := by
+    intro he
+    have hzero : q.toSeminorm=0 := by ext x; exact le_antisymm (by simpa only [he,zero_apply] using hqg x) (apply_nonneg q.toSeminorm x)
+    exact q.ne_zero hAn hzero
+  have hupper : objective b g≤ objective b gA.toSeminorm := by
+    apply ge_of_tendsto (gA.objective_thicken_tendsto hAn b)
+    apply Eventually.of_forall
+    intro i
+    let ε : ℝ := 1/((i:ℝ)+1)
+    have hε : 0<ε := by dsimp [ε]; positivity
+    let f := gA.thicken ε hε
+    obtain ⟨ha,hh⟩ := full_comparison_limit hμ hq hA hAn hAj b gj hm hobj f
+    have hle : supportFunctional (μ.map A) f.body≤1 := by
+      exact (functional_thicken_le hμ A gA hε).trans hgn.le
+    apply hh.trans
+    apply Spherical.objective_mono b ((f.scale _ ha).ne_zero top_ne_bot) (f.ne_zero top_ne_bot)
+    intro x
+    exact mul_le_of_le_one_left (apply_nonneg f.toSeminorm x) hle
+  have hlower := hgm q hqn
+  have hmid := Spherical.objective_mono b (q.ne_zero hAn) hg0 hqg
+  have heq : objective b q.toSeminorm=objective b g := le_antisymm hmid (hupper.trans hlower)
+  have hqeq := Spherical.objective_mono_eq b (q.ne_zero hAn) hg0 hqg heq
+  have huniq := minimizer_unique hμA hAn b hgn hqn hgm (le_antisymm hlower (hmid.trans hupper))
+  rw [← hqeq,← huniq]
+
+end RelativeGauge
+end PettyProjection
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology
+open scoped NNReal RealInnerProductSpace
+namespace PettyProjection
+open Spherical (Sphere objective)
+namespace RelativeGauge
+variable {n : ℕ} [NeZero n] {μ : Measure (Space n)}
+
+omit [NeZero n] in
+lemma full_functional_pos (hμ : Integrable (fun x : Space n => ‖x‖) μ)
+    (hq : ∀ x : Space n,x≠0 → 0< cosineSeminorm μ hμ x)
+    (A : Space n →L[ℝ] Space n) (hA : LinearMap.range A.toLinearMap≠⊥)
+    (f : RelativeGauge (⊤ : Submodule ℝ (Space n))) : 0< supportFunctional (μ.map A) f.body := by
+  have hp := (projectFull (LinearMap.range A.toLinearMap) f).functional_pos hA
+    (norm_integrable_map hμ A) (cosine_map_pos hμ hq A)
+  rwa [functional_projectFull] at hp
+
+lemma minimizers_uniform_bound (hμ : Integrable (fun x : Space n => ‖x‖) μ)
+    (hq : ∀ x : Space n,x≠0 → 0< cosineSeminorm μ hμ x)
+    {Aj : ℕ → Space n →L[ℝ] Space n} {A : Space n →L[ℝ] Space n}
+    (hA : Tendsto Aj atTop (𝓝 A)) (hAn : LinearMap.range A.toLinearMap≠⊥)
+    (hAj : ∀ i,LinearMap.range (Aj i).toLinearMap≠⊥) (b : Bool)
+    (gj : ∀ i,RelativeGauge (LinearMap.range (Aj i).toLinearMap))
+    (hm : ∀ i (q : RelativeGauge (LinearMap.range (Aj i).toLinearMap)),supportFunctional (μ.map (Aj i)) q.body=1 →
+      objective b (gj i).toSeminorm≤ objective b q.toSeminorm) :
+    ∃ R : ℝ,0< R ∧ ∀ i (u : Sphere n),gj i u≤ R := by
+  let f := ball (⊤ : Submodule ℝ (Space n))
+  let a : (Space n →L[ℝ] Space n) → ℝ := fun B => supportFunctional (μ.map B) f.body
+  have ha := full_functional_pos hμ hq A hAn f
+  have ht : Tendsto (fun i => a (Aj i)) atTop (𝓝 (a A)) :=
+    ((functional_map_continuous hμ f.compact f.nonempty).tendsto A).comp hA
+  let v : ℕ → ℝ := fun i => if b then Real.log (a (Aj i))+objective b f.toSeminorm
+    else a (Aj i)*objective b f.toSeminorm
+  have hvt : Tendsto v atTop (𝓝 (if b then Real.log (a A)+objective b f.toSeminorm else a A*objective b f.toSeminorm)) := by
+    cases b
+    · exact ht.mul_const _
+    · exact (ht.log ha.ne').add_const _
+  obtain ⟨M,hM⟩ := (Metric.isBounded_range_of_tendsto v hvt).bddAbove
+  obtain ⟨R,hR,hRb⟩ := Spherical.objective_coercive (n := n) b M
+  refine ⟨R,hR,fun i u => hRb _ ((gj i).ne_zero (hAj i)) ?_ u⟩
+  obtain ⟨hai,hh⟩ := minimizer_full_comparison hμ hq (Aj i) (hAj i) b (gj i) (hm i) f
+  have he := Spherical.objective_scale b (g := f.toSeminorm) (f := (f.scale _ hai).toSeminorm)
+    (f.ne_zero top_ne_bot) hai (fun _ => rfl)
+  rw [he] at hh
+  exact hh.trans (hM (mem_range_self i))
+
+lemma objective_tendsto_of_pointwise {gj : ℕ → Seminorm ℝ (Space n)} {g : Seminorm ℝ (Space n)}
+    (hg : g≠0) {R : ℝ} (hR : ∀ i (u : Sphere n),gj i u≤ R)
+    (ht : ∀ x,Tendsto (fun i => gj i x) atTop (𝓝 (g x))) (b : Bool) :
+    Tendsto (fun i => objective b (gj i)) atTop (𝓝 (objective b g)) := by
+  have hex : ∃ u : Sphere n,0< g u := by
+    by_contra! hn
+    have he : g=0 := by
+      ext x
+      rw [← Spherical.radialExtension_seminorm g x]
+      have hz : (fun u : Sphere n => g u)=(fun _ => 0) := by
+        funext u; exact le_antisymm (hn u) (apply_nonneg g _)
+      rw [hz]
+      unfold Spherical.radialExtension
+      split_ifs <;> simp
+    exact hg he
+  obtain ⟨u,hu⟩ := hex
+  have hh := (ht u).eventually (lt_mem_nhds (show g u/2< g u by linarith))
+  obtain ⟨N,hN⟩ := eventually_atTop.mp hh
+  apply (tendsto_add_atTop_iff_nat N).mp
+  apply Spherical.objective_tendsto b (show 0< g u/2 by linarith)
+    (fun i => ⟨u,(hN (i+N) (Nat.le_add_left N i)).le⟩) (fun i => hR (i+N)) hg
+  intro v
+  exact (ht v).comp (tendsto_add_atTop_nat N)
+
+omit [NeZero n] in
+lemma limit_ne_zero (hμ : Integrable (fun x : Space n => ‖x‖) μ)
+    (hq : ∀ x : Space n,x≠0 → 0< cosineSeminorm μ hμ x)
+    {Aj : ℕ → Space n →L[ℝ] Space n} {A : Space n →L[ℝ] Space n}
+    (hA : Tendsto Aj atTop (𝓝 A)) (hAn : LinearMap.range A.toLinearMap≠⊥)
+    (gj : ∀ i,RelativeGauge (LinearMap.range (Aj i).toLinearMap))
+    (hn : ∀ i,supportFunctional (μ.map (Aj i)) (gj i).body=1)
+    {g : Seminorm ℝ (Space n)} (ht : ∀ x,Tendsto (fun i => gj i x) atTop (𝓝 (g x))) : g≠0 := by
+  obtain ⟨x,hx,hx0⟩ := (Submodule.ne_bot_iff _).mp hAn
+  have hp := cosine_map_pos hμ hq A x hx hx0
+  have hh := normalized_limit_lower hμ hA gj hn ht x
+  intro he
+  rw [he,zero_apply] at hh
+  linarith
+
+/-- Sequential uniform compactness, and identification, for the actual
+optimizer family across all nonzero linear maps, without a rank restriction. -/
+theorem minimizers_subsequence (hμ : Integrable (fun x : Space n => ‖x‖) μ)
+    (hq : ∀ x : Space n,x≠0 → 0< cosineSeminorm μ hμ x)
+    {Aj : ℕ → Space n →L[ℝ] Space n} {A : Space n →L[ℝ] Space n}
+    (hA : Tendsto Aj atTop (𝓝 A)) (hAn : LinearMap.range A.toLinearMap≠⊥)
+    (hAj : ∀ i,LinearMap.range (Aj i).toLinearMap≠⊥) (b : Bool)
+    (gj : ∀ i,RelativeGauge (LinearMap.range (Aj i).toLinearMap))
+    (hn : ∀ i,supportFunctional (μ.map (Aj i)) (gj i).body=1)
+    (hm : ∀ i (q : RelativeGauge (LinearMap.range (Aj i).toLinearMap)),supportFunctional (μ.map (Aj i)) q.body=1 →
+      objective b (gj i).toSeminorm≤ objective b q.toSeminorm)
+    (gA : RelativeGauge (LinearMap.range A.toLinearMap)) (hgn : supportFunctional (μ.map A) gA.body=1)
+    (hgm : ∀ q : RelativeGauge (LinearMap.range A.toLinearMap),supportFunctional (μ.map A) q.body=1 →
+      objective b gA.toSeminorm≤ objective b q.toSeminorm) :
+    ∃ φ : ℕ → ℕ,StrictMono φ ∧
+      TendstoUniformly (fun i (u : Sphere n) => gj (φ i) u) (fun u => gA u) atTop := by
+  obtain ⟨R,hR,hRb⟩ := minimizers_uniform_bound hμ hq hA hAn hAj b gj hm
+  obtain ⟨g,φ,hφ,hu,ht⟩ := Spherical.bounded_seminorm_subsequence (R := ⟨R,hR.le⟩)
+    (fun i => (gj i).toSeminorm) hRb
+  have hA' := hA.comp hφ.tendsto_atTop
+  have hg := limit_ne_zero hμ hq hA' hAn (fun i => gj (φ i)) (fun i => hn (φ i)) ht
+  have ho := objective_tendsto_of_pointwise hg (fun i => hRb (φ i)) ht b
+  have he := minimizer_limit_identify hμ hq hA' hAn (fun i => hAj (φ i)) b
+    (fun i => gj (φ i)) (fun i => hn (φ i)) (fun i => hm (φ i)) gA hgn hgm ht ho
+  refine ⟨φ,hφ,?_⟩
+  rwa [he] at hu
+
+end RelativeGauge
+end PettyProjection
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/ProjectionBodies/RelativeGauges.lean b/lean/OAI/Geometry/ProjectionBodies/RelativeGauges.lean
new file mode 100644
index 0000000000000000000000000000000000000000..03575d84ffbd4f52a5c60449cd2f048bd7c4df1c
--- /dev/null
+++ b/lean/OAI/Geometry/ProjectionBodies/RelativeGauges.lean
@@ -0,0 +1,590 @@
+import OAI.Geometry.ProjectionBodies.SeminormBounds
+
+namespace OAI
+
+noncomputable section
+open Set Metric Filter Topology
+open scoped NNReal RealInnerProductSpace Pointwise
+namespace PettyProjection
+open Spherical (Sphere seminorm_continuous)
+
+/-- A genuine extended gauge of a relative symmetric convex body. The lower
+bound is only on its supporting subspace, so cylinders are retained. -/
+structure RelativeGauge {n : ℕ} (V : Submodule ℝ (Space n)) where
+  toSeminorm : Seminorm ℝ (Space n)
+  projection : ∀ x,toSeminorm (V.starProjection x)=toSeminorm x
+  lower : ∃ c : ℝ,0<c ∧ ∀ x,c*‖V.starProjection x‖≤toSeminorm x
+
+instance {n : ℕ} {V : Submodule ℝ (Space n)} : CoeFun (RelativeGauge V) (fun _ => Space n → ℝ) :=
+  ⟨fun g => g.toSeminorm⟩
+
+namespace RelativeGauge
+variable {n : ℕ} {V : Submodule ℝ (Space n)}
+
+def body (g : RelativeGauge V) : Set (Space n) := (V : Set (Space n)) ∩ seminormUnit g.toSeminorm
+
+lemma zero_mem (g : RelativeGauge V) : 0∈g.body := ⟨V.zero_mem,seminormUnit_zero _⟩
+lemma nonempty (g : RelativeGauge V) : g.body.Nonempty := ⟨0,g.zero_mem⟩
+lemma convex (g : RelativeGauge V) : Convex ℝ g.body := V.convex.inter (seminormUnit_convex _)
+lemma closed (g : RelativeGauge V) : IsClosed g.body := V.closed_of_finiteDimensional.inter (seminormUnit_closed _)
+lemma neg_mem (g : RelativeGauge V) {x : Space n} (hx : x∈g.body) : -x∈g.body := by
+  exact ⟨V.neg_mem hx.1,by
+    change g.toSeminorm (-x)≤1
+    rw [map_neg_eq_map]
+    exact hx.2⟩
+
+lemma compact (g : RelativeGauge V) : IsCompact g.body := by
+  obtain ⟨c,hc,hbound⟩ := g.lower
+  apply (isCompact_closedBall (0:Space n) c⁻¹).of_isClosed_subset g.closed
+  intro x hx
+  rw [mem_closedBall,dist_zero_right,inv_eq_one_div,le_div_iff₀ hc]
+  have hb := hbound x
+  rw [V.starProjection_eq_self_iff.mpr hx.1] at hb
+  have hh : g x≤1 := hx.2
+  nlinarith
+
+lemma ne_zero (g : RelativeGauge V) (hV : V≠⊥) : g.toSeminorm≠0 := by
+  obtain ⟨x,hx,hx0⟩ := (Submodule.ne_bot_iff V).mp hV
+  obtain ⟨c,hc,hb⟩ := g.lower
+  intro h
+  have hh := hb x
+  rw [V.starProjection_eq_self_iff.mpr hx,h] at hh
+  have : 0<c*‖x‖ := mul_pos hc (norm_pos_iff.mpr hx0)
+  exact this.not_ge hh
+
+lemma pos (g : RelativeGauge V) {x : Space n} (hx : x∈V) (hx0 : x≠0) : 0<g x := by
+  obtain ⟨c,hc,hb⟩ := g.lower
+  have hh := hb x
+  rw [V.starProjection_eq_self_iff.mpr hx] at hh
+  exact (mul_pos hc (norm_pos_iff.mpr hx0)).trans_le hh
+
+lemma radial_mem (g : RelativeGauge V) {x : Space n} (hx : x∈V) (hx0 : x≠0) :
+    (g x)⁻¹ • x∈g.body := by
+  refine ⟨V.smul_mem _ hx,?_⟩
+  change g.toSeminorm ((g x)⁻¹ • x)≤1
+  rw [map_smul_eq_mul,Real.norm_of_nonneg (inv_nonneg.mpr (apply_nonneg _ _)),
+    inv_mul_cancel₀ (g.pos hx hx0).ne']
+
+lemma inner_le_gauge_support (g : RelativeGauge V) (u x : Space n) :
+    ⟪u,V.starProjection x⟫≤g x*support g.body u := by
+  by_cases hx : V.starProjection x=0
+  · rw [hx,inner_zero_right]
+    exact mul_nonneg (apply_nonneg _ _) (support_nonneg g.compact g.zero_mem u)
+  have hp := g.pos (V.starProjection_apply_mem x) hx
+  have hh := inner_le_support g.compact (g.radial_mem (V.starProjection_apply_mem x) hx) u
+  rw [inner_smul_right,inv_mul_le_iff₀ hp,g.projection] at hh
+  simpa only [mul_comm] using hh
+
+lemma cosine_le_gauge_functional (g : RelativeGauge V) {μ : MeasureTheory.Measure (Space n)}
+    (hμ : MeasureTheory.Integrable (fun x : Space n => ‖x‖) μ) (x : Space n) :
+    cosineSeminorm μ hμ (V.starProjection x)≤g x*supportFunctional μ g.body := by
+  by_cases hx : V.starProjection x=0
+  · rw [hx,map_zero]
+    exact mul_nonneg (apply_nonneg _ _) (functional_nonneg g.compact g.zero_mem)
+  have hp := g.pos (V.starProjection_apply_mem x) hx
+  have hh := cosine_le_functional hμ g.compact (fun y hy => g.neg_mem hy)
+    (g.radial_mem (V.starProjection_apply_mem x) hx)
+  rw [map_smul_eq_mul,Real.norm_of_nonneg (inv_nonneg.mpr hp.le),inv_mul_le_iff₀ hp,
+    g.projection] at hh
+  simpa only [mul_comm] using hh
+
+/-- Every positive dilation is again the exact relative gauge. -/
+def scale (g : RelativeGauge V) (a : ℝ) (ha : 0<a) : RelativeGauge V where
+  toSeminorm := Seminorm.of (fun x => a*g x)
+    (fun x y => by simpa only [mul_add] using mul_le_mul_of_nonneg_left (map_add_le_add g.toSeminorm x y) ha.le)
+    (fun r x => by rw [map_smul_eq_mul]; ring)
+  projection x := by change a*g (V.starProjection x)=a*g x; rw [g.projection]
+  lower := by
+    obtain ⟨c,hc,hb⟩ := g.lower
+    refine ⟨a*c,mul_pos ha hc,fun x => ?_⟩
+    change a*c*‖V.starProjection x‖≤a*g x
+    simpa only [mul_assoc] using mul_le_mul_of_nonneg_left (hb x) ha.le
+
+lemma scale_apply (g : RelativeGauge V) (a : ℝ) (ha : 0<a) (x : Space n) :
+    g.scale a ha x=a*g x := rfl
+
+lemma body_scale (g : RelativeGauge V) (a : ℝ) (ha : 0<a) :
+    (g.scale a ha).body=a⁻¹ • g.body := by
+  ext x
+  constructor
+  · rintro ⟨hx,hg⟩
+    refine ⟨a • x,⟨V.smul_mem a hx,?_⟩,inv_smul_smul₀ ha.ne' x⟩
+    change g.toSeminorm (a • x)≤1
+    rw [map_smul_eq_mul,Real.norm_of_nonneg ha.le]
+    exact hg
+  · rintro ⟨y,hy,rfl⟩
+    refine ⟨V.smul_mem _ hy.1,?_⟩
+    change a*g.toSeminorm (a⁻¹ • y)≤1
+    rw [map_smul_eq_mul,Real.norm_of_nonneg (inv_nonneg.mpr ha.le),← mul_assoc,
+      mul_inv_cancel₀ ha.ne',one_mul]
+    exact hy.2
+
+lemma functional_scale (g : RelativeGauge V) (a : ℝ) (ha : 0<a)
+    (μ : MeasureTheory.Measure (Space n)) :
+    supportFunctional μ (g.scale a ha).body=a⁻¹*supportFunctional μ g.body := by
+  rw [g.body_scale a ha,functional_smul g.compact g.nonempty (inv_nonneg.mpr ha.le)]
+
+end RelativeGauge
+end PettyProjection
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology
+open scoped NNReal RealInnerProductSpace
+namespace PettyProjection.Spherical
+
+/-- Variational objective: the logarithmic endpoint (k=1), or the linear (k=2). -/
+def objectiveIntegrand (logarithmic : Bool) (t : ℝ) : ℝ :=
+  if logarithmic then Real.log t else t
+
+def objective {n : ℕ} (logarithmic : Bool) (g : Seminorm ℝ (Space n)) : ℝ :=
+  mean (fun u : Sphere n => objectiveIntegrand logarithmic (g u))
+
+lemma seminorm_integrable {n : ℕ} [NeZero n] (g : Seminorm ℝ (Space n)) :
+    Integrable (fun u : Sphere n => g u) (sigma n) :=
+  continuous_integrable ((seminorm_continuous n g).comp continuous_subtype_val)
+
+lemma objective_integrable {n : ℕ} [NeZero n] (b : Bool) {g : Seminorm ℝ (Space n)} (hg : g≠0) :
+    Integrable (fun u : Sphere n => objectiveIntegrand b (g u)) (sigma n) := by
+  cases b
+  · exact seminorm_integrable g
+  · exact seminorm_log_integrable hg
+
+lemma objectiveIntegrand_strictMono (b : Bool) : StrictMonoOn (objectiveIntegrand b) (Ioi 0) := by
+  cases b
+  · exact fun _ _ _ _ h => h
+  · exact Real.strictMonoOn_log
+
+lemma objective_mono {n : ℕ} [NeZero n] (b : Bool) {g f : Seminorm ℝ (Space n)}
+    (hg : g≠0) (hf : f≠0) (hle : ∀ x,g x≤f x) : objective b g≤objective b f := by
+  apply integral_mono_ae (objective_integrable b hg) (objective_integrable b hf)
+  filter_upwards [seminorm_pos_ae_sphere hg,seminorm_pos_ae_sphere hf] with u hu hv
+  exact (objectiveIntegrand_strictMono b).monotoneOn hu hv (hle u)
+
+lemma objective_mono_eq {n : ℕ} [NeZero n] (b : Bool) {g f : Seminorm ℝ (Space n)}
+    (hg : g≠0) (hf : f≠0) (hle : ∀ x,g x≤f x) (heq : objective b g=objective b f) : g=f := by
+  have hnn : 0≤ᵐ[sigma n] (fun u : Sphere n => objectiveIntegrand b (f u)-objectiveIntegrand b (g u)) := by
+    filter_upwards [seminorm_pos_ae_sphere hg,seminorm_pos_ae_sphere hf] with u hu hv
+    exact sub_nonneg.mpr ((objectiveIntegrand_strictMono b).monotoneOn hu hv (hle u))
+  have hi := (objective_integrable b hf).sub (objective_integrable b hg)
+  have hz : (∫ u : Sphere n,objectiveIntegrand b (f u)-objectiveIntegrand b (g u) ∂sigma n)=0 := by
+    rw [integral_sub (objective_integrable b hf) (objective_integrable b hg)]
+    exact sub_eq_zero.mpr heq.symm
+  have hae := (integral_eq_zero_iff_of_nonneg_ae hnn hi).mp hz
+  have he : (fun u : Sphere n => f u)=ᵐ[sigma n] (fun u : Sphere n => g u) := by
+    filter_upwards [hae,seminorm_pos_ae_sphere hg,seminorm_pos_ae_sphere hf] with u hu hgu hfu
+    exact (objectiveIntegrand_strictMono b).injOn hfu hgu (sub_eq_zero.mp hu)
+  have hpoint := Measure.eq_of_ae_eq he ((seminorm_continuous n f).comp continuous_subtype_val)
+    ((seminorm_continuous n g).comp continuous_subtype_val)
+  ext x
+  rw [← radialExtension_seminorm g x,← radialExtension_seminorm f x,hpoint]
+
+lemma objective_scale {n : ℕ} [NeZero n] (b : Bool) {g f : Seminorm ℝ (Space n)}
+    (hg : g≠0) {a : ℝ} (ha : 0<a) (he : ∀ x,f x=a*g x) :
+    objective b f=if b then Real.log a+objective b g else a*objective b g := by
+  cases b
+  · change (∫ u : Sphere n,f u ∂sigma n)=a*∫ u : Sphere n,g u ∂sigma n
+    simp_rw [he]; exact integral_const_mul _ _
+  · have hae : (fun u : Sphere n => Real.log (f u))=ᵐ[sigma n]
+        (fun u : Sphere n => Real.log a+Real.log (g u)) := by
+      filter_upwards [seminorm_pos_ae_sphere hg] with u hu
+      rw [he,Real.log_mul ha.ne' hu.ne']
+    change (∫ u : Sphere n,Real.log (f u) ∂sigma n)=Real.log a+∫ u : Sphere n,Real.log (g u) ∂sigma n
+    rw [integral_congr_ae hae,integral_add (integrable_const _) (seminorm_log_integrable hg),integral_const]
+    simp only [probReal_univ,one_smul]
+
+lemma coordinate_mean_pos (n : ℕ) [NeZero n] :
+    0 < mean (fun u : Sphere n => |⟪(pole n:Space n),(u:Space n)⟫|) := by
+  let f := fun u : Sphere n => |⟪(pole n:Space n),(u:Space n)⟫|
+  have hc : Continuous f := by fun_prop
+  apply integral_pos_of_integrable_nonneg_nonzero (x := pole n) hc (continuous_integrable hc) (fun _ => abs_nonneg _)
+  dsimp [f]
+  simp only [real_inner_self_eq_norm_sq,norm_coe,one_pow,abs_one,one_ne_zero,not_false_eq_true]
+
+/-- Objective sublevels impose a common Lipschitz bound, including cylinders. -/
+lemma objective_coercive {n : ℕ} [NeZero n] (b : Bool) (M : ℝ) :
+    ∃ R : ℝ,0<R ∧ ∀ g : Seminorm ℝ (Space n),g≠0 → objective b g≤M → ∀ u : Sphere n,g u≤R := by
+  let c := mean (fun u : Sphere n => Real.log |⟪(pole n:Space n),(u:Space n)⟫|)
+  let d := mean (fun u : Sphere n => |⟪(pole n:Space n),(u:Space n)⟫|)
+  have hd : 0<d := coordinate_mean_pos n
+  cases b
+  · refine ⟨max 1 (M/d),lt_of_lt_of_le zero_lt_one (le_max_left _ _),fun g hg hM u => ?_⟩
+    obtain ⟨R,e,hR,hm,hb⟩ := seminorm_coordinate_minorant g hg
+    have hlow : R*d≤objective false g := by
+      have hi : Integrable (fun u : Sphere n => |⟪(e:Space n),(u:Space n)⟫|) (sigma n) :=
+        continuous_integrable (by fun_prop : Continuous (fun u : Sphere n => |⟪(e:Space n),(u:Space n)⟫|))
+      have h := integral_mono (hi.const_mul R) (seminorm_integrable g) (fun u => by
+        simpa only [real_inner_comm] using hb u)
+      rw [integral_const_mul] at h
+      change R*mean _≤_ at h
+      rw [mean_inner_independent (fun t => |t|) e (pole n)] at h
+      exact h
+    exact (hm u).trans (((le_div_iff₀ hd).mpr (hlow.trans hM)).trans (le_max_right _ _))
+  · refine ⟨Real.exp (M-c),Real.exp_pos _,fun g hg hM u => ?_⟩
+    obtain ⟨R,e,hR,hm,hb⟩ := seminorm_coordinate_minorant g hg
+    have hlow := seminorm_log_coercivity hR hg hb
+    change mean (fun u : Sphere n => Real.log (g u))≤M at hM
+    have hh : Real.log R≤M-c := by change Real.log R+c≤_ at hlow; linarith
+    exact (hm u).trans ((Real.log_le_iff_le_exp hR).mp hh)
+
+lemma objective_tendsto {n : ℕ} [NeZero n] (b : Bool)
+    {g : ℕ → Seminorm ℝ (Space n)} {f : Seminorm ℝ (Space n)} {r R : ℝ}
+    (hr : 0<r) (hlo : ∀ i,∃ u : Sphere n,r≤g i u) (hhi : ∀ i (u : Sphere n),g i u≤R)
+    (hf : f≠0) (hc : ∀ u : Sphere n,Tendsto (fun i => g i u) atTop (𝓝 (f u))) :
+    Tendsto (fun i => objective b (g i)) atTop (𝓝 (objective b f)) := by
+  cases b
+  · apply tendsto_integral_of_dominated_convergence (fun _ => |R|)
+    · exact fun i => (seminorm_integrable (g i)).aestronglyMeasurable
+    · exact integrable_const _
+    · intro i
+      exact Eventually.of_forall (fun u => by
+        change ‖g i u‖≤|R|
+        rw [Real.norm_of_nonneg (apply_nonneg (g i) _)]
+        exact (hhi i u).trans (le_abs_self _))
+    · exact Eventually.of_forall hc
+  · exact seminorm_log_integral_tendsto hr hlo hhi hf hc
+
+end PettyProjection.Spherical
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology
+open scoped NNReal RealInnerProductSpace Pointwise
+namespace PettyProjection
+open Spherical (Sphere seminorm_continuous)
+
+lemma seminorm_positive_subspace_bound {n : ℕ} (V : Submodule ℝ (Space n)) (hV : V≠⊥)
+    (g : Seminorm ℝ (Space n)) (hg : ∀ x∈V,x≠0 → 0<g x) :
+    ∃ c : ℝ,0<c ∧ ∀ x,c*‖V.starProjection x‖≤g (V.starProjection x) := by
+  let S : Set (Space n) := (V : Set (Space n)) ∩ sphere 0 1
+  obtain ⟨v,hv,hv0⟩ := (Submodule.ne_bot_iff V).mp hV
+  have hne : S.Nonempty := ⟨‖v‖⁻¹ • v,V.smul_mem _ hv,by simp [norm_smul,hv0]⟩
+  have hc : IsCompact S := (isCompact_sphere (0:Space n) 1).inter_left V.closed_of_finiteDimensional
+  obtain ⟨u,hu,hmin⟩ := hc.exists_isMinOn hne (seminorm_continuous n g).continuousOn
+  have hu1 : ‖u‖=1 := by simpa only [mem_sphere,dist_zero_right] using hu.2
+  have hu0 : u≠0 := by intro h; simp [h] at hu1
+  refine ⟨g u,hg u hu.1 hu0,fun x => ?_⟩
+  by_cases hx : V.starProjection x=0
+  · simp [hx]
+  have hn : 0<‖V.starProjection x‖ := norm_pos_iff.mpr hx
+  have hunit : ‖V.starProjection x‖⁻¹ • V.starProjection x∈S :=
+    ⟨V.smul_mem _ (V.starProjection_apply_mem x),by simp [norm_smul,hn.ne']⟩
+  have hh : g u≤g (‖V.starProjection x‖⁻¹ • V.starProjection x) := hmin hunit
+  rw [map_smul_eq_mul,Real.norm_of_nonneg (inv_nonneg.mpr hn.le)] at hh
+  have h := mul_le_mul_of_nonneg_left hh hn.le
+  rw [← mul_assoc,mul_inv_cancel₀ hn.ne',one_mul] at h
+  simpa only [mul_comm] using h
+
+namespace RelativeGauge
+variable {n : ℕ} {V : Submodule ℝ (Space n)}
+
+/-- Extended gauge of the unit ball in V. -/
+def ball (V : Submodule ℝ (Space n)) : RelativeGauge V where
+  toSeminorm := (normSeminorm ℝ (Space n)).comp V.starProjection.toLinearMap
+  projection x := by
+    change ‖V.starProjection (V.starProjection x)‖=‖V.starProjection x‖
+    rw [V.starProjection_eq_self_iff.mpr (V.starProjection_apply_mem x)]
+  lower := ⟨1,zero_lt_one,fun _ => by simp⟩
+
+lemma ball_apply (V : Submodule ℝ (Space n)) (x : Space n) : ball V x=‖V.starProjection x‖ := rfl
+
+lemma support_ball (V : Submodule ℝ (Space n)) (u : Space n) :
+    support (ball V).body u=‖V.starProjection u‖ := by
+  apply le_antisymm
+  · apply (support_le_iff (ball V).compact (ball V).nonempty).mpr
+    intro x hx
+    have hx1 : ‖x‖≤1 := by
+      have h : ‖V.starProjection x‖≤1 := hx.2
+      rwa [V.starProjection_eq_self_iff.mpr hx.1] at h
+    calc
+      ⟪u,x⟫ = ⟪V.starProjection u,x⟫ := by
+        rw [Submodule.inner_starProjection_left_eq_right,V.starProjection_eq_self_iff.mpr hx.1]
+      _ ≤ ‖V.starProjection u‖*‖x‖ := real_inner_le_norm _ _
+      _ ≤ ‖V.starProjection u‖ := mul_le_of_le_one_right (norm_nonneg _) hx1
+  · by_cases hu : V.starProjection u=0
+    · rw [hu,norm_zero]; exact support_nonneg (ball V).compact (ball V).zero_mem u
+    have hp : 0<‖V.starProjection u‖ := norm_pos_iff.mpr hu
+    have hz := (ball V).radial_mem (V.starProjection_apply_mem u) hu
+    have he : ball V (V.starProjection u)=‖V.starProjection u‖ := by
+      rw [ball_apply,V.starProjection_eq_self_iff.mpr (V.starProjection_apply_mem u)]
+    have hh := inner_le_support (ball V).compact hz u
+    have hi : ⟪u,V.starProjection u⟫=‖V.starProjection u‖^2 := by
+      have h := Submodule.inner_starProjection_left_eq_right V u (V.starProjection u)
+      rw [V.starProjection_eq_self_iff.mpr (V.starProjection_apply_mem u),real_inner_self_eq_norm_sq] at h
+      exact h.symm
+    rw [he,inner_smul_right,hi] at hh
+    have hmul : ‖V.starProjection u‖⁻¹*‖V.starProjection u‖^2=‖V.starProjection u‖ := by
+      field_simp
+    rwa [hmul] at hh
+
+lemma cylinder_nhds {K : Set (Space n)}
+    (hinner : ∃ r : ℝ,0<r ∧ ∀ x∈V,‖x‖≤r → x∈K) :
+    V.starProjection ⁻¹' K∈𝓝 (0:Space n) := by
+  obtain ⟨r,hr,hin⟩ := hinner
+  apply Filter.mem_of_superset (Metric.ball_mem_nhds (0:Space n) hr)
+  intro x hx
+  apply hin _ (V.starProjection_apply_mem x)
+  exact (V.norm_starProjection_apply_le x).trans (by
+    have hh : ‖x‖<r := by simpa only [mem_ball,dist_zero_right] using hx
+    exact hh.le)
+
+/-- Recover the relative extended gauge from an actual relative convex body. -/
+def ofBody {K : Set (Space n)} (hK : IsCompact K) (hc : Convex ℝ K) (hb : Balanced ℝ K)
+    (hinner : ∃ r : ℝ,0<r ∧ ∀ x∈V,‖x‖≤r → x∈K) : RelativeGauge V := by
+  let Z := V.starProjection ⁻¹' K
+  have hZc : Convex ℝ Z := hc.linear_preimage V.starProjection.toLinearMap
+  have hZb : Balanced ℝ Z := by
+    intro a ha y hy
+    obtain ⟨z,hz,rfl⟩ := hy
+    change V.starProjection (a • z)∈K
+    rw [map_smul]
+    exact hb a ha ⟨V.starProjection z,hz,rfl⟩
+  have hZn : Z∈𝓝 (0:Space n) := cylinder_nhds hinner
+  let g := gaugeSeminorm hZb hZc (absorbent_nhds_zero hZn)
+  refine ⟨g,?_,?_⟩
+  · intro x
+    change gauge Z (V.starProjection x)=gauge Z x
+    simp only [gauge_def',Z,mem_preimage,map_smul,
+      V.starProjection_eq_self_iff.mpr (V.starProjection_apply_mem x)]
+  · obtain ⟨R,hR,hbound⟩ := hK.isBounded.exists_pos_norm_le
+    refine ⟨R⁻¹,inv_pos.mpr hR,fun x => ?_⟩
+    change R⁻¹*‖V.starProjection x‖≤gauge Z x
+    unfold gauge
+    apply le_csInf (absorbent_nhds_zero hZn).gauge_set_nonempty
+    rintro r ⟨hr,y,hy,rfl⟩
+    rw [map_smul,norm_smul,Real.norm_of_nonneg hr.le]
+    have hh : ‖V.starProjection y‖≤R := hbound _ hy
+    calc
+      R⁻¹*(r*‖V.starProjection y‖)≤R⁻¹*(r*R) := mul_le_mul_of_nonneg_left (mul_le_mul_of_nonneg_left hh hr.le) (inv_nonneg.mpr hR.le)
+      _ = r := by field_simp
+
+lemma ofBody_apply {K : Set (Space n)} (hK : IsCompact K) (hc : Convex ℝ K) (hb : Balanced ℝ K)
+    (hinner : ∃ r : ℝ,0<r ∧ ∀ x∈V,‖x‖≤r → x∈K) (x : Space n) :
+    ofBody hK hc hb hinner x=gauge (V.starProjection ⁻¹' K) x := rfl
+
+lemma body_ofBody {K : Set (Space n)} (hK : IsCompact K) (hc : Convex ℝ K) (hb : Balanced ℝ K)
+    (hsub : K⊆V) (hinner : ∃ r : ℝ,0<r ∧ ∀ x∈V,‖x‖≤r → x∈K) :
+    (ofBody hK hc hb hinner).body=K := by
+  ext x
+  change (x∈V ∧ gauge (V.starProjection ⁻¹' K) x≤1) ↔ x∈K
+  rw [gauge_le_one_iff_mem_closure (s := V.starProjection ⁻¹' K) (hc.linear_preimage V.starProjection.toLinearMap)
+    (cylinder_nhds hinner),(hK.isClosed.preimage V.starProjection.continuous).closure_eq]
+  constructor
+  · rintro ⟨hx,hy⟩
+    change V.starProjection x∈K at hy
+    rwa [V.starProjection_eq_self_iff.mpr hx] at hy
+  · intro hx
+    refine ⟨hsub hx,?_⟩
+    change V.starProjection x∈K
+    rwa [V.starProjection_eq_self_iff.mpr (hsub hx)]
+
+end RelativeGauge
+end PettyProjection
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology
+open scoped NNReal RealInnerProductSpace Pointwise
+namespace PettyProjection
+open Spherical (Sphere seminorm_continuous)
+
+lemma support_mono_compact {n : ℕ} {K L : Set (Space n)} (hK : IsCompact K)
+    (hKn : K.Nonempty) (hL : IsCompact L) (hsub : K⊆L) (u : Space n) :
+    support K u≤ support L u := by
+  obtain ⟨x,hx,he⟩ := support_attained hK hKn u
+  rw [he]
+  exact inner_le_support hL (hsub hx) u
+
+lemma functional_mono_compact {n : ℕ} {μ : Measure (Space n)}
+    (hμ : Integrable (fun x : Space n => ‖x‖) μ)
+    {K L : Set (Space n)} (hK : IsCompact K) (hKn : K.Nonempty) (hL : IsCompact L) (hsub : K⊆L) :
+    supportFunctional μ K≤ supportFunctional μ L :=
+  integral_mono (support_integrable hμ hK hKn) (support_integrable hμ hL (hKn.mono hsub))
+    (support_mono_compact hK hKn hL hsub)
+
+namespace RelativeGauge
+variable {n : ℕ} {V : Submodule ℝ (Space n)}
+
+lemma balanced (g : RelativeGauge V) : Balanced ℝ g.body := by
+  intro a ha y hy
+  obtain ⟨z,hz,rfl⟩ := hy
+  refine ⟨V.smul_mem _ hz.1,?_⟩
+  exact (seminormUnit_balanced g.toSeminorm) a ha ⟨z,hz.2,rfl⟩
+
+lemma inner_ball (g : RelativeGauge V) : ∃ r : ℝ,0<r ∧ ∀ x∈V,‖x‖≤r → x∈g.body := by
+  obtain ⟨ε,hε,hεg⟩ := Metric.mem_nhds_iff.mp (seminormUnit_nhds g.toSeminorm)
+  refine ⟨ε/2,by positivity,fun x hx hnorm => ⟨hx,hεg ?_⟩⟩
+  rw [mem_ball,dist_zero_right]
+  linarith
+
+lemma body_subset_smul (g f : RelativeGauge V) {a : ℝ} (ha : 0<a)
+    (hle : ∀ x,f x≤a*g x) : g.body⊆a • f.body := by
+  intro x hx
+  refine ⟨a⁻¹ • x,⟨V.smul_mem _ hx.1,?_⟩,smul_inv_smul₀ ha.ne' x⟩
+  change f.toSeminorm (a⁻¹ • x)≤1
+  rw [map_smul_eq_mul,Real.norm_of_nonneg (inv_nonneg.mpr ha.le),inv_mul_le_iff₀ ha,mul_one]
+  exact (hle x).trans (mul_le_of_le_one_right ha.le hx.2)
+
+lemma functional_le_scale (g f : RelativeGauge V) {μ : Measure (Space n)}
+    (hμ : Integrable (fun x : Space n => ‖x‖) μ) {a : ℝ} (ha : 0<a)
+    (hle : ∀ x,f x≤a*g x) : supportFunctional μ g.body≤a*supportFunctional μ f.body := by
+  have hh := functional_mono_compact hμ g.compact g.nonempty (f.compact.smul a)
+    (g.body_subset_smul f ha hle)
+  rwa [functional_smul f.compact f.nonempty ha.le] at hh
+
+lemma functional_pos (g : RelativeGauge V) (hV : V≠⊥) {μ : Measure (Space n)}
+    (hμ : Integrable (fun x : Space n => ‖x‖) μ)
+    (hq : ∀ x∈V,x≠0 → 0<cosineSeminorm μ hμ x) : 0 < supportFunctional μ g.body := by
+  obtain ⟨x,hx,hx0⟩ := (Submodule.ne_bot_iff V).mp hV
+  have hy := g.radial_mem hx hx0
+  have hqy := hq _ hy.1 (smul_ne_zero (inv_ne_zero (g.pos hx hx0).ne') hx0)
+  exact hqy.trans_le (cosine_le_functional hμ g.compact (fun _ h => g.neg_mem h) hy)
+
+lemma normalized_lower (g : RelativeGauge V) {μ : Measure (Space n)}
+    (hμ : Integrable (fun x : Space n => ‖x‖) μ) (hn : supportFunctional μ g.body=1) (x : Space n) :
+    cosineSeminorm μ hμ (V.starProjection x)≤g x := by
+  have hh := g.cosine_le_gauge_functional hμ x
+  rwa [hn,mul_one] at hh
+
+lemma support_bound (g : RelativeGauge V) {c : ℝ} (hc : 0<c)
+    (hl : ∀ x,c*‖V.starProjection x‖≤g x) (u : Space n) : support g.body u≤c⁻¹*‖u‖ := by
+  apply (support_le_iff g.compact g.nonempty).mpr
+  intro x hx
+  have hh := hl x
+  rw [V.starProjection_eq_self_iff.mpr hx.1] at hh
+  have hb : ‖x‖≤c⁻¹ := by
+    rw [inv_eq_one_div,le_div_iff₀ hc]
+    have hx1 : g x≤1 := hx.2
+    nlinarith
+  exact (real_inner_le_norm _ _).trans (by nlinarith [norm_nonneg u])
+
+lemma functional_bound (g : RelativeGauge V) {μ : Measure (Space n)}
+    (hμ : Integrable (fun x : Space n => ‖x‖) μ) {c : ℝ} (hc : 0<c)
+    (hl : ∀ x,c*‖V.starProjection x‖≤g x) : supportFunctional μ g.body≤c⁻¹*(∫ x,‖x‖ ∂μ) := by
+  have hh := integral_mono (support_integrable hμ g.compact g.nonempty) (hμ.const_mul c⁻¹)
+    (g.support_bound hc hl)
+  rwa [integral_const_mul] at hh
+
+lemma seminorm_close_ambient [NeZero n] (g f : Seminorm ℝ (Space n)) {ε : ℝ}
+    (hclose : ∀ u : Sphere n,|g u-f u|≤ε) (x : Space n) : |g x-f x|≤ε*‖x‖ := by
+  by_cases hx : x=0
+  · simp [hx]
+  have hp : 0<‖x‖ := norm_pos_iff.mpr hx
+  let u : Sphere n := ⟨‖x‖⁻¹ • x,by simp [norm_smul,hp.ne']⟩
+  have hh := hclose u
+  change |g (‖x‖⁻¹ • x)-f (‖x‖⁻¹ • x)|≤ε at hh
+  rw [map_smul_eq_mul,map_smul_eq_mul,Real.norm_of_nonneg (inv_nonneg.mpr hp.le),← mul_sub,
+    abs_mul,abs_of_pos (inv_pos.mpr hp),inv_mul_le_iff₀ hp] at hh
+  simpa only [mul_comm] using hh
+
+lemma close_le_scale [NeZero n] (g f : RelativeGauge V) {c ε : ℝ} (hc : 0<c) (hε : 0≤ε)
+    (hl : ∀ x,c*‖V.starProjection x‖≤g x)
+    (hclose : ∀ u : Sphere n,|g u-f u|≤ε) (x : Space n) : f x≤(1+ε/c)*g x := by
+  have hh := seminorm_close_ambient g.toSeminorm f.toSeminorm hclose (V.starProjection x)
+  rw [g.projection,f.projection] at hh
+  have hdiff := (abs_le.mp hh).1
+  have hprod := mul_le_mul_of_nonneg_left (hl x) (div_nonneg hε hc.le)
+  have he : (ε/c)*(c*‖V.starProjection x‖)=ε*‖V.starProjection x‖ := by field_simp
+  rw [he] at hprod
+  nlinarith
+
+lemma functional_close [NeZero n] (g f : RelativeGauge V) {μ : Measure (Space n)}
+    (hμ : Integrable (fun x : Space n => ‖x‖) μ) {c ε : ℝ} (hc : 0<c) (hε : 0≤ε)
+    (hg : ∀ x,c*‖V.starProjection x‖≤g x) (hf : ∀ x,c*‖V.starProjection x‖≤f x)
+    (hclose : ∀ u : Sphere n,|g u-f u|≤ε) :
+    |supportFunctional μ g.body-supportFunctional μ f.body|≤(ε/c)*(c⁻¹*(∫ x,‖x‖ ∂μ)) := by
+  have ha : 0<1+ε/c := by positivity
+  have h1 := g.functional_le_scale f hμ ha (g.close_le_scale f hc hε hg hclose)
+  have hclose' : ∀ u : Sphere n,|f u-g u|≤ε := fun u => by rw [abs_sub_comm]; exact hclose u
+  have h2 := f.functional_le_scale g hμ ha (f.close_le_scale g hc hε hf hclose')
+  have hb1 := mul_le_mul_of_nonneg_left (g.functional_bound hμ hc hg) (div_nonneg hε hc.le)
+  have hb2 := mul_le_mul_of_nonneg_left (f.functional_bound hμ hc hf) (div_nonneg hε hc.le)
+  rw [abs_le]
+  constructor <;> nlinarith
+
+end RelativeGauge
+end PettyProjection
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology
+open scoped NNReal RealInnerProductSpace
+namespace PettyProjection
+open Spherical (Sphere)
+namespace RelativeGauge
+variable {n : ℕ} [NeZero n] {V : Submodule ℝ (Space n)}
+
+lemma functional_tendsto {g : ℕ → RelativeGauge V} {f : RelativeGauge V}
+    {μ : Measure (Space n)} (hμ : Integrable (fun x : Space n => ‖x‖) μ)
+    {c : ℝ} (hc : 0<c) (hg : ∀ i x,c*‖V.starProjection x‖≤g i x)
+    (hf : ∀ x,c*‖V.starProjection x‖≤f x)
+    (hunif : TendstoUniformly (fun i (u : Sphere n) => g i u) (fun u => f u) atTop) :
+    Tendsto (fun i => supportFunctional μ (g i).body) atTop (𝓝 (supportFunctional μ f.body)) := by
+  let C := c⁻¹*(c⁻¹*(∫ x,‖x‖ ∂μ))
+  have hC : 0≤C := mul_nonneg (inv_nonneg.mpr hc.le)
+    (mul_nonneg (inv_nonneg.mpr hc.le) (integral_nonneg (fun _ => norm_nonneg _)))
+  apply Metric.tendsto_atTop.mpr
+  intro ε hε
+  have hδ : 0<ε/(C+1) := div_pos hε (by linarith)
+  have hu := (Metric.tendstoUniformly_iff.mp hunif) (ε/(C+1)) hδ
+  obtain ⟨N,hN⟩ := eventually_atTop.mp hu
+  refine ⟨N,fun i hi => ?_⟩
+  have hh := (g i).functional_close f hμ hc hδ.le (hg i) hf (fun u => by
+    have h := hN i hi u
+    rw [Real.dist_eq] at h
+    simpa only [abs_sub_comm] using h.le)
+  rw [Real.dist_eq]
+  have he : (ε/(C+1)/c)*(c⁻¹*(∫ x,‖x‖ ∂μ))=(ε/(C+1))*C := by dsimp [C]; ring
+  rw [he] at hh
+  apply hh.trans_lt
+  have ht : C/(C+1)<1 := (div_lt_one (by linarith : 0<C+1)).mpr (by linarith)
+  have hm := mul_lt_mul_of_pos_left ht hε
+  simpa only [div_eq_mul_inv,mul_assoc,mul_comm,mul_left_comm,mul_one] using hm
+
+/-- Compactness of fixed-subspace normalized gauges. All constants arise
+from the actual cosine norm and objective, not regularity assumptions. -/
+theorem normalized_subsequence {μ : Measure (Space n)}
+    (hμ : Integrable (fun x : Space n => ‖x‖) μ) (hV : V≠⊥)
+    (hq : ∀ x∈V,x≠0 → 0<cosineSeminorm μ hμ x)
+    (b : Bool) (g : ℕ → RelativeGauge V) (hn : ∀ i,supportFunctional μ (g i).body=1)
+    {M : ℝ} (hM : ∀ i,Spherical.objective b (g i).toSeminorm≤M) :
+    ∃ f : RelativeGauge V, ∃ φ : ℕ → ℕ, StrictMono φ ∧ supportFunctional μ f.body=1 ∧
+      TendstoUniformly (fun i (u : Sphere n) => g (φ i) u) (fun u => f u) atTop ∧
+      Tendsto (fun i => Spherical.objective b (g (φ i)).toSeminorm) atTop
+        (𝓝 (Spherical.objective b f.toSeminorm)) := by
+  obtain ⟨c,hc,hcb⟩ := seminorm_positive_subspace_bound V hV (cosineSeminorm μ hμ) hq
+  have hlo (i : ℕ) (x : Space n) : c*‖V.starProjection x‖≤g i x :=
+    (hcb x).trans ((g i).normalized_lower hμ (hn i) x)
+  obtain ⟨R,hR,hRb⟩ := Spherical.objective_coercive (n := n) b M
+  have hhi (i : ℕ) (u : Sphere n) : g i u≤R := hRb _ ((g i).ne_zero hV) (hM i) u
+  obtain ⟨q,φ,hφ,hu,hpoint⟩ := Spherical.bounded_seminorm_subsequence
+    (R := ⟨R,hR.le⟩) (fun i => (g i).toSeminorm) hhi
+  have hqp : ∀ x,q (V.starProjection x)=q x := by
+    intro x
+    apply tendsto_nhds_unique (hpoint (V.starProjection x))
+    simpa only [(g _).projection] using hpoint x
+  have hql (x : Space n) : c*‖V.starProjection x‖≤q x :=
+    ge_of_tendsto (hpoint x) (Eventually.of_forall (fun i => hlo (φ i) x))
+  let f : RelativeGauge V := ⟨q,hqp,⟨c,hc,hql⟩⟩
+  have hf : supportFunctional μ f.body=1 := by
+    have ht := functional_tendsto (f := f) hμ hc (fun i => hlo (φ i)) hql hu
+    have hc1 : Tendsto (fun i => supportFunctional μ (g (φ i)).body) atTop (𝓝 (1:ℝ)) := by
+      simp_rw [hn]
+      exact tendsto_const_nhds
+    exact tendsto_nhds_unique ht hc1
+  obtain ⟨x,hx,hx0⟩ := (Submodule.ne_bot_iff V).mp hV
+  let u : Sphere n := ⟨‖x‖⁻¹ • x,by simp [norm_smul,hx0]⟩
+  have huV : (u:Space n)∈V := V.smul_mem _ hx
+  have hlow (i : ℕ) : ∃ v : Sphere n,c≤g i v := by
+    refine ⟨u,?_⟩
+    have h := hlo i u
+    rwa [V.starProjection_eq_self_iff.mpr huV,Spherical.norm_coe,mul_one] at h
+  refine ⟨f,φ,hφ,hf,hu,?_⟩
+  exact Spherical.objective_tendsto b hc (fun i => hlow (φ i)) (fun i => hhi (φ i))
+    (f.ne_zero hV) (fun u => hpoint u)
+
+end RelativeGauge
+end PettyProjection
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/ProjectionBodies/ScalarBounds.lean b/lean/OAI/Geometry/ProjectionBodies/ScalarBounds.lean
new file mode 100644
index 0000000000000000000000000000000000000000..364add29bef118cb2b4c50b09f453b672261928b
--- /dev/null
+++ b/lean/OAI/Geometry/ProjectionBodies/ScalarBounds.lean
@@ -0,0 +1,770 @@
+import OAI.Geometry.ProjectionBodies.NormCurvature
+
+namespace OAI
+
+noncomputable section
+namespace PettyProjection.Scalar
+
+def exponent (n : ℕ) : ℕ := if n = 4 then 1 else 2
+
+def coefficient (n : ℕ) : ℝ :=
+  if n = 4 then 121 / 35 else
+    ((n : ℝ)^2 + 5 * n + 7) / (((n : ℝ) + 1) * ((n : ℝ) + 3))
+
+def beta (n : ℕ) : ℝ := if n = 4 then 1 / 2 else 2 / 3
+
+def weight (n : ℕ) : ℝ :=
+  if n = 4 then 3 / 14 else if n = 5 ∨ n = 6 then 1 / 3 else 1 / 2
+
+def e (n : ℕ) (z : ℝ) : ℝ := ((z⁻¹)^n - 1) / n
+
+def M (n : ℕ) (z : ℝ) : ℝ := z ^ exponent n - 1 + exponent n * e n z
+
+def W (n : ℕ) (z : ℝ) : ℝ := z ^ exponent n - z ^ (exponent n - 1) + e n z
+
+def G (n : ℕ) (z : ℝ) : ℝ := if 1 < z then beta n * M n z else 0
+
+def D (n : ℕ) (z : ℝ) : ℝ := z ^ exponent n - 1 - G n z
+
+/-- Closed form of the integral of M'(a)^2 over [1,z], or zero below 1. -/
+def I (n : ℕ) (z : ℝ) : ℝ :=
+  if 1 < z then
+    (exponent n : ℝ)^2 *
+      ((z ^ (2 * exponent n - 1) - 1) / (2 * exponent n - 1 : ℕ) -
+        2 * (1 - (z⁻¹) ^ (n + 1 - exponent n)) / (n + 1 - exponent n : ℕ) +
+        (1 - (z⁻¹) ^ (2 * n + 1)) / (2 * n + 1 : ℕ))
+  else 0
+
+def L (n : ℕ) (z : ℝ) : ℝ :=
+  coefficient n *
+    ((((n : ℝ) - 1) * beta n ^ 2 * z * I n z - 2 * n * G n z ^ 2) /
+        (weight n * (2 * n + 8)) + D n z ^ 2 / (1 - weight n))
+
+/-- A second-order lower bound for inverse powers, valid in every dimension.
+This is the low-z convexity estimate. -/
+lemma inverse_pow_lower (n : ℕ) {z : ℝ} (hz : 0 < z) (hz1 : z ≤ 1) :
+    1 + n * (1 - z) + (n : ℝ) * (n + 1) / 2 * (1 - z)^2 ≤ (z⁻¹)^n := by
+  have hx : 0 ≤ 1 - z := sub_nonneg.mpr hz1
+  have hinv : 1 + (1 - z) + (1 - z)^2 ≤ z⁻¹ := by
+    rw [← one_div z]
+    apply (le_div_iff₀ hz).2
+    nlinarith [mul_nonneg (sq_nonneg (1-z)) hx]
+  induction n with
+  | zero => simp
+  | succ n ih =>
+    rw [pow_succ]
+    have hprod := mul_le_mul ih hinv (by positivity : 0 ≤ 1 + (1-z) + (1-z)^2)
+      (by positivity : 0 ≤ (z⁻¹)^n)
+    push_cast
+    calc
+      1 + ((n : ℝ) + 1) * (1 - z) + ((n : ℝ) + 1) * ((n : ℝ) + 1 + 1) / 2 * (1 - z)^2
+          ≤ (1 + (n : ℝ) * (1-z) + (n : ℝ) * (n+1) / 2 * (1-z)^2) *
+            (1 + (1-z) + (1-z)^2) := by
+              nlinarith [mul_nonneg (by positivity : 0 ≤ (n : ℝ))
+                  (mul_nonneg (sq_nonneg (1-z)) hx),
+                mul_nonneg (by positivity : 0 ≤ (n : ℝ) * (n+1) / 2)
+                  (mul_nonneg (sq_nonneg (1-z)) hx),
+                mul_nonneg (by positivity : 0 ≤ (n : ℝ) * (n+1) / 2)
+                  (sq_nonneg ((1-z)^2))]
+      _ ≤ _ := hprod
+
+lemma coefficient_pos (n : ℕ) : 0 < coefficient n := by
+  unfold coefficient
+  split_ifs <;> positivity
+
+lemma beta_pos (n : ℕ) : 0 < beta n := by
+  unfold beta
+  split_ifs <;> norm_num
+
+lemma beta_lt_one (n : ℕ) : beta n < 1 := by
+  unfold beta
+  split_ifs <;> norm_num
+
+lemma weight_pos (n : ℕ) : 0 < weight n := by
+  unfold weight
+  split_ifs <;> norm_num
+
+lemma weight_lt_one (n : ℕ) : weight n < 1 := by
+  unfold weight
+  split_ifs <;> norm_num
+
+/-- Exact positivity margin on the low-z interval in dimensions at least seven. -/
+lemma large_local_margin {r : ℝ} (hr : 7 ≤ r) :
+    0 < r + 3 - 8 * (r^2 + 5*r + 7) / ((r+1)*(r+3)) := by
+  have hden : 0 < (r+1)*(r+3) := by positivity
+  have ht : 0 ≤ r - 7 := by linarith
+  have hpoly : 0 < r^3 - r^2 - 25*r - 47 := by
+    have hcert : r^3-r^2-25*r-47 =
+        (r-7)^3 + 20*(r-7)^2 + 108*(r-7) + 72 := by ring
+    rw [hcert]
+    positivity
+  have heq : r+3-8*(r^2+5*r+7)/((r+1)*(r+3)) =
+      (r^3-r^2-25*r-47)/((r+1)*(r+3)) := by
+    field_simp
+    ring
+  rw [heq]
+  exact div_pos hpoly hden
+
+/-- Scalar mean bound, including every dimension. -/
+lemma mean_bound (n : ℕ) (hn : 4 ≤ n) :
+    coefficient n * (2*n*beta n^2 / (weight n * (2*n+8)) -
+      (1-beta n)^2 / (1-weight n)) ≤
+      (2*(exponent n : ℝ)-1) / (exponent n : ℝ)^2 := by
+  by_cases h4 : n = 4
+  · subst n
+    norm_num [coefficient, beta, weight, exponent]
+  by_cases h5 : n = 5
+  · subst n
+    norm_num [coefficient, beta, weight, exponent]
+  by_cases h6 : n = 6
+  · subst n
+    norm_num [coefficient, beta, weight, exponent]
+  have hn7 : 7 ≤ n := by omega
+  have hr : 7 ≤ (n : ℝ) := by exact_mod_cast hn7
+  simp only [coefficient, beta, weight, exponent, h4, h5, h6,
+    ite_false, or_self, Nat.cast_ofNat]
+  have h1 : (n : ℝ) + 1 ≠ 0 := by positivity
+  have h3 : (n : ℝ) + 3 ≠ 0 := by positivity
+  have h4' : (n : ℝ) + 4 ≠ 0 := by positivity
+  have hd : 0 < 36*((n : ℝ)+1)*(n+3)*(n+4) := by positivity
+  have hcert : 3/4 - ( ((n : ℝ)^2+5*n+7) / ((n+1)*(n+3)) *
+      (2*n*(2/3)^2 / (1/2*(2*n+8)) - (1-2/3)^2/(1-1/2))) =
+      (3*(n : ℝ)^3+128*n^2+505*n+548) / (36*(n+1)*(n+3)*(n+4)) := by
+    field_simp
+    ring
+  have hpos : 0 < 3*(n : ℝ)^3+128*n^2+505*n+548 := by positivity
+  have := div_pos hpos hd
+  rw [← hcert] at this
+  norm_num only [OfNat.ofNat_ne_zero, Nat.reduceAdd, Nat.reduceMul] at *
+  linarith
+
+lemma local_margin (n : ℕ) (hn : 4 ≤ n) :
+    coefficient n * (exponent n : ℝ)^2 / (1 - weight n) <
+      n + 2 * (exponent n : ℝ) - 1 := by
+  by_cases h4 : n = 4
+  · subst n
+    norm_num [coefficient, weight, exponent]
+  by_cases h5 : n = 5
+  · subst n
+    norm_num [coefficient, weight, exponent]
+  by_cases h6 : n = 6
+  · subst n
+    norm_num [coefficient, weight, exponent]
+  have hn7 : 7 ≤ n := by omega
+  have hr : 7 ≤ (n : ℝ) := by exact_mod_cast hn7
+  have h := large_local_margin hr
+  simp only [coefficient, weight, exponent, h4, h5, h6, ite_false, or_self,
+    Nat.cast_ofNat]
+  convert (sub_pos.mp h) using 1 <;> ring
+
+lemma low_W_bound (n : ℕ) (hn : 4 ≤ n) {z : ℝ} (hz : 0 < z) (hz1 : z ≤ 1) :
+    (n + 2 * (exponent n : ℝ) - 1) * (z-1)^2 ≤ 2 * W n z := by
+  have hn0 : 0 < (n : ℝ) := by exact_mod_cast (show 0 < n by omega)
+  have hp := inverse_pow_lower n hz hz1
+  have he : -(z-1) + ((n : ℝ)+1)/2 * (z-1)^2 ≤ e n z := by
+    unfold e
+    apply (le_div_iff₀ hn0).2
+    nlinarith [hp]
+  by_cases h4 : n = 4
+  · subst n
+    norm_num only [W, exponent, ite_true, pow_one, Nat.reduceSub, pow_zero,
+      Nat.cast_one, Nat.cast_ofNat] at ⊢ he
+    nlinarith
+  · simp only [W, exponent, h4, ite_false, Nat.reduceSub, pow_one, Nat.cast_ofNat]
+    nlinarith
+
+lemma low_D_square (n : ℕ) {z : ℝ} (hz : 0 < z) (hz1 : z ≤ 1) :
+    (z ^ exponent n - 1)^2 ≤ (exponent n : ℝ)^2 * (z-1)^2 := by
+  by_cases h4 : n = 4
+  · simp [exponent, h4]
+  · simp only [exponent, h4, ite_false, Nat.cast_ofNat]
+    have hs : (z+1)^2 ≤ 4 := by nlinarith
+    have := mul_le_mul_of_nonneg_left hs (sq_nonneg (z-1))
+    nlinarith
+
+lemma low_pointwise (n : ℕ) (hn : 4 ≤ n) {z : ℝ} (hz : 0 < z) (hz1 : z ≤ 1) :
+    L n z ≤ 2 * W n z ∧ (L n z = 2 * W n z ↔ z = 1) := by
+  have hnot : ¬ 1 < z := not_lt.mpr hz1
+  have hwt : 0 < 1 - weight n := sub_pos.mpr (weight_lt_one n)
+  have hc : 0 < coefficient n / (1-weight n) := div_pos (coefficient_pos n) hwt
+  have hL : L n z = coefficient n / (1-weight n) * (z ^ exponent n - 1)^2 := by
+    simp [L, D, G, I, hnot, div_mul_eq_mul_div]
+    ring
+  by_cases heq : z = 1
+  · subst z
+    simp [L, G, I, D, W, e]
+  have hsq : 0 < (z-1)^2 := sq_pos_of_ne_zero (sub_ne_zero.mpr heq)
+  have hlt : L n z < 2 * W n z := calc
+    L n z = coefficient n / (1-weight n) * (z ^ exponent n - 1)^2 := hL
+    _ ≤ coefficient n / (1-weight n) * ((exponent n : ℝ)^2 * (z-1)^2) :=
+      mul_le_mul_of_nonneg_left (low_D_square n hz hz1) hc.le
+    _ = (coefficient n * (exponent n : ℝ)^2 / (1-weight n)) * (z-1)^2 := by ring
+    _ < (n + 2 * (exponent n : ℝ) - 1) * (z-1)^2 :=
+      mul_lt_mul_of_pos_right (local_margin n hn) hsq
+    _ ≤ 2 * W n z := low_W_bound n hn hz hz1
+  exact ⟨hlt.le, iff_of_false (ne_of_lt hlt) heq⟩
+
+def P4 (z : ℝ) : ℝ := 462 * z^10 - 983 * z^9 + 93 * z^8 + 3399 * z^5 - 3378 * z^4 + 407
+
+lemma certificate_4 {z : ℝ} (hz : 1 < z) :
+    2880 * z^8 * (2 * W 4 z - L 4 z) = P4 z := by
+  have hz0 : z ≠ 0 := by linarith
+  simp only [W, L, D, G, I, M, e, exponent, coefficient, weight, beta, P4]
+  norm_num [hz]
+  field_simp
+  ring
+
+def U4 (x : ℝ) : ℝ := 17052 * x^4 - 6711 * x^2 - 1446 * x + 1728
+
+lemma U4_pos (x : ℝ) : 0 < U4 x := by
+  have h1 : 1446*x ≤ 400 + 1400*x^2 := by
+    nlinarith [sq_nonneg (2800*x-1446)]
+  have h2 : 8111*x^2 ≤ 1000 + 17052*x^4 := by
+    nlinarith [sq_nonneg (34104*x^2-8111)]
+  dsimp [U4]
+  linarith
+
+lemma P4_pos {z : ℝ} (hz : 1 < z) : 0 < P4 z := by
+  let x := z-1
+  have hx : 0 < x := sub_pos.mpr hz
+  have heq : P4 z = x^2 * (U4 x + (462 * x^8 + 3637 * x^7 + 12036 * x^6 + 20796 * x^5 + 1173 * x^3)) := by
+    dsimp [P4, U4, x]
+    ring
+  rw [heq]
+  have hu := U4_pos x
+  positivity
+
+lemma high_pointwise_4 {z : ℝ} (hz : 1 < z) : L 4 z < 2 * W 4 z := by
+  have h := P4_pos hz
+  rw [← certificate_4 hz] at h
+  have hp : 0 < (2880 : ℝ) * z^8 := by positivity
+  exact sub_pos.mp ((mul_pos_iff_of_pos_left hp).mp h)
+
+def P5 (z : ℝ) : ℝ := 151525 * z^14 - 273570 * z^12 - 680800 * z^11 + 938157 * z^10 + 225720 * z^7 - 462264 * z^5 + 101232
+
+lemma certificate_5 {z : ℝ} (hz : 1 < z) :
+    712800 * z^10 * (2 * W 5 z - L 5 z) = P5 z := by
+  have hz0 : z ≠ 0 := by linarith
+  simp only [W, L, D, G, I, M, e, exponent, coefficient, weight, beta, P5]
+  norm_num [hz]
+  field_simp
+  ring
+
+def U5 (x : ℝ) : ℝ := 86314305 * x^4 - 5802775 * x^2 - 1505900 * x + 623700
+
+lemma U5_pos (x : ℝ) : 0 < U5 x := by
+  have h1 : 1505900*x ≤ 200000 + 2900000*x^2 := by
+    nlinarith [sq_nonneg (5800000*x-1505900)]
+  have h2 : 8702775*x^2 ≤ 230000 + 86314305*x^4 := by
+    nlinarith [sq_nonneg (172628610*x^2-8702775)]
+  dsimp [U5]
+  linarith
+
+lemma P5_pos {z : ℝ} (hz : 1 < z) : 0 < P5 z := by
+  let x := z-1
+  have hx : 0 < x := sub_pos.mpr hz
+  have heq : P5 z = x^2 * (U5 x + (151525 * x^12 + 2121350 * x^11 + 13515205 * x^10 + 51191460 * x^9 + 127070262 * x^8 + 215105220 * x^7 + 249497490 * x^6 + 191506920 * x^5 + 12849430 * x^3)) := by
+    dsimp [P5, U5, x]
+    ring
+  rw [heq]
+  have hu := U5_pos x
+  positivity
+
+lemma high_pointwise_5 {z : ℝ} (hz : 1 < z) : L 5 z < 2 * W 5 z := by
+  have h := P5_pos hz
+  rw [← certificate_5 hz] at h
+  have hp : 0 < (712800 : ℝ) * z^10 := by positivity
+  exact sub_pos.mp ((mul_pos_iff_of_pos_left hp).mp h)
+
+def P6 (z : ℝ) : ℝ := 48399 * z^16 - 75894 * z^14 - 218004 * z^13 + 285961 * z^12 + 56940 * z^8 - 127478 * z^6 + 30076
+
+lemma certificate_6 {z : ℝ} (hz : 1 < z) :
+    221130 * z^12 * (2 * W 6 z - L 6 z) = P6 z := by
+  have hz0 : z ≠ 0 := by linarith
+  simp only [W, L, D, G, I, M, e, exponent, coefficient, weight, beta, P6]
+  norm_num [hz]
+  field_simp
+  ring
+
+def U6 (x : ℝ) : ℝ := 51269452 * x^4 - 132249 * x^2 + 452790
+
+lemma U6_pos (x : ℝ) : 0 < U6 x := by
+  dsimp [U6]
+  nlinarith [sq_nonneg (102538904*x^2-132249)]
+
+lemma P6_pos {z : ℝ} (hz : 1 < z) : 0 < P6 z := by
+  let x := z-1
+  have hx : 0 < x := sub_pos.mpr hz
+  have heq : P6 z = x^2 * (U6 x + (48399 * x^14 + 774384 * x^13 + 5731986 * x^12 + 25822920 * x^11 + 78631735 * x^10 + 170208636 * x^9 + 268133580 * x^8 + 308783332 * x^7 + 256021935 * x^6 + 146058120 * x^5 + 7800780 * x^3 + 679380 * x)) := by
+    dsimp [P6, U6, x]
+    ring
+  rw [heq]
+  have hu := U6_pos x
+  positivity
+
+lemma high_pointwise_6 {z : ℝ} (hz : 1 < z) : L 6 z < 2 * W 6 z := by
+  have h := P6_pos hz
+  rw [← certificate_6 hz] at h
+  have hp : 0 < (221130 : ℝ) * z^12 := by positivity
+  exact sub_pos.mp ((mul_pos_iff_of_pos_left hp).mp h)
+
+def P7 (z : ℝ) : ℝ := 3185 * z^18 + 77910 * z^16 - 185472 * z^15 + 101385 * z^14 + 40040 * z^9 - 43080 * z^7 + 6032
+
+lemma certificate_7 {z : ℝ} (hz : 1 < z) :
+    138600 * z^14 * (2 * W 7 z - L 7 z) = P7 z := by
+  have hz0 : z ≠ 0 := by linarith
+  simp only [W, L, D, G, I, M, e, exponent, coefficient, weight, beta, P7]
+  norm_num [hz]
+  field_simp
+  ring
+
+lemma P7_pos {z : ℝ} (hz : 1 < z) : 0 < P7 z := by
+  let x := z-1
+  have hx : 0 < x := sub_pos.mpr hz
+  have heq : P7 z = x^2 * (3185 * x^16 + 57330 * x^15 + 565215 * x^14 + 3660048 * x^13 + 16414605 * x^12 + 52863510 * x^11 + 125758815 * x^10 + 225405180 * x^9 + 307786479 * x^8 + 320870550 * x^7 + 253378125 * x^6 + 148489200 * x^5 + 62263215 * x^4 + 17740674 * x^3 + 3396645 * x^2 + 598500 * x + 124740) := by
+    dsimp [P7, x]
+    ring
+  rw [heq]
+  positivity
+
+lemma high_pointwise_7 {z : ℝ} (hz : 1 < z) : L 7 z < 2 * W 7 z := by
+  have h := P7_pos hz
+  rw [← certificate_7 hz] at h
+  have hp : 0 < (138600 : ℝ) * z^14 := by positivity
+  exact sub_pos.mp ((mul_pos_iff_of_pos_left hp).mp h)
+
+def P8 (z : ℝ) : ℝ := 5032 * z^20 + 106896 * z^18 - 245128 * z^17 + 131937 * z^16 + 45288 * z^10 - 52683 * z^8 + 8658
+
+lemma certificate_8 {z : ℝ} (hz : 1 < z) :
+    181764 * z^16 * (2 * W 8 z - L 8 z) = P8 z := by
+  have hz0 : z ≠ 0 := by linarith
+  simp only [W, L, D, G, I, M, e, exponent, coefficient, weight, beta, P8]
+  norm_num [hz]
+  field_simp
+  ring
+
+lemma P8_pos {z : ℝ} (hz : 1 < z) : 0 < P8 z := by
+  let x := z-1
+  have hx : 0 < x := sub_pos.mpr hz
+  have heq : P8 z = x^2 * (5032 * x^18 + 100640 * x^17 + 1062976 * x^16 + 7415480 * x^15 + 36699889 * x^14 + 134016848 * x^13 + 371287480 * x^12 + 796445648 * x^11 + 1341571660 * x^10 + 1789629712 * x^9 + 1896594800 * x^8 + 1593208720 * x^7 + 1052388995 * x^6 + 539061976 * x^5 + 210340388 * x^4 + 61812136 * x^3 + 14025170 * x^2 + 2645608 * x + 369036) := by
+    dsimp [P8, x]
+    ring
+  rw [heq]
+  positivity
+
+lemma high_pointwise_8 {z : ℝ} (hz : 1 < z) : L 8 z < 2 * W 8 z := by
+  have h := P8_pos hz
+  rw [← certificate_8 hz] at h
+  have hp : 0 < (181764 : ℝ) * z^16 := by positivity
+  exact sub_pos.mp ((mul_pos_iff_of_pos_left hp).mp h)
+
+def P9 (z : ℝ) : ℝ := 17955 * z^22 + 344826 * z^20 - 771336 * z^19 + 409763 * z^18 + 124488 * z^11 - 153472 * z^9 + 27776
+
+lemma certificate_9 {z : ℝ} (hz : 1 < z) :
+    568620 * z^18 * (2 * W 9 z - L 9 z) = P9 z := by
+  have hz0 : z ≠ 0 := by linarith
+  simp only [W, L, D, G, I, M, e, exponent, coefficient, weight, beta, P9]
+  norm_num [hz]
+  field_simp
+  ring
+
+lemma P9_pos {z : ℝ} (hz : 1 < z) : 0 < P9 z := by
+  let x := z-1
+  have hx : 0 < x := sub_pos.mpr hz
+  have heq : P9 z = x^2 * (17955 * x^20 + 395010 * x^19 + 4492431 * x^18 + 33775884 * x^17 + 182612144 * x^16 + 741405888 * x^15 + 2325627540 * x^14 + 5752968096 * x^13 + 11391705882 * x^12 + 18245048652 * x^11 + 23789022114 * x^10 + 25324311168 * x^9 + 21996503100 * x^8 + 15523044480 * x^7 + 8829652644 * x^6 + 4002804432 * x^5 + 1428705963 * x^4 + 398939634 * x^3 + 87942807 * x^2 + 15343236 * x + 1781676) := by
+    dsimp [P9, x]
+    ring
+  rw [heq]
+  positivity
+
+lemma high_pointwise_9 {z : ℝ} (hz : 1 < z) : L 9 z < 2 * W 9 z := by
+  have h := P9_pos hz
+  rw [← certificate_9 hz] at h
+  have hp : 0 < (568620 : ℝ) * z^18 := by positivity
+  exact sub_pos.mp ((mul_pos_iff_of_pos_left hp).mp h)
+
+open Finset
+
+/-- The exact coefficient comparison used in the all-dimensional scalar estimate. -/
+lemma choose_double (n j : ℕ) : 2^j * n.choose j ≤ (2*n).choose j := by
+  induction n generalizing j with
+  | zero => cases j <;> simp
+  | succ n ih =>
+    cases j with
+    | zero => simp
+    | succ j =>
+      rw [Nat.choose_succ_succ]
+      calc
+        2^(j+1)*(n.choose j + n.choose (j+1)) =
+            2*(2^j*n.choose j) + 2^(j+1)*n.choose (j+1) := by ring
+        _ ≤ 2*(2*n).choose j + (2*n).choose (j+1) :=
+          Nat.add_le_add (Nat.mul_le_mul_left 2 (ih j)) (ih (j+1))
+        _ ≤ (2*(n+1)).choose (j+1) := by
+          have hh := Nat.choose_le_succ (2*n) j
+          have heq : 2*(n+1) = (2*n+1)+1 := by omega
+          rw [heq, Nat.choose_succ_succ, Nat.choose_succ_succ]
+          simp only [Nat.succ_eq_add_one] at hh ⊢
+          omega
+
+lemma choose_double_cubic {n j : ℕ} (hj : 3 ≤ j) :
+    8 * (n.choose j : ℝ) ≤ ((2*n).choose j : ℝ) := by
+  have hp : 8 ≤ 2^j := by
+    have := Nat.pow_le_pow_right (by omega : 0 < 2) hj
+    norm_num at this ⊢
+    exact this
+  exact_mod_cast le_trans (Nat.mul_le_mul_right (n.choose j) hp) (choose_double n j)
+
+/-- Taylor remainder of the n-th power after its quadratic term. -/
+def tail (n : ℕ) (x : ℝ) : ℝ :=
+  ∑ j ∈ range (n+1), if 3 ≤ j then (n.choose j : ℝ)*x^j else 0
+
+lemma tail_nonneg (n : ℕ) {x : ℝ} (hx : 0 ≤ x) : 0 ≤ tail n x := by
+  apply sum_nonneg
+  intro j _
+  split_ifs <;> positivity
+
+lemma tail_expansion (n : ℕ) (hn : 2 ≤ n) (x : ℝ) :
+    (1+x)^n = 1 + n*x + (n.choose 2 : ℝ)*x^2 + tail n x := by
+  have heq : n+1 = 3 + (n-2) := by omega
+  have hp := add_pow x (1:ℝ) n
+  simp only [one_pow,mul_one] at hp
+  rw [add_comm] at hp
+  rw [hp]
+  simp only [tail]
+  rw [heq, sum_range_add, sum_range_add]
+  have hsum : (∑ j ∈ range (n-2), x^(3+j)*(n.choose (3+j) : ℝ)) =
+      ∑ j ∈ range (n-2), if 3 ≤ 3+j then (n.choose (3+j) : ℝ)*x^(3+j) else 0 := by
+    apply sum_congr rfl
+    intro j _
+    simp [mul_comm]
+  rw [hsum]
+  norm_num [sum_range_succ, Nat.choose_one_right]
+  ring
+
+lemma tail_double (n : ℕ) {x : ℝ} (hx : 0 ≤ x) :
+    8 * tail n x ≤ tail (2*n) x := by
+  let f : ℕ → ℝ := fun j => if 3 ≤ j then (n.choose j : ℝ)*x^j else 0
+  have hzero : ∀ j, n+1 ≤ j → f j = 0 := by
+    intro j hj
+    unfold f
+    rw [Nat.choose_eq_zero_of_lt (by omega : n < j)]
+    simp
+  have hext : tail n x = ∑ j ∈ range (2*n+1), f j := by
+    change (∑ j ∈ range (n+1), f j) = _
+    have heq : 2*n+1 = (n+1)+n := by omega
+    rw [heq, sum_range_add (f := f) (n+1) n]
+    have : ∑ j ∈ range n, f (n+1+j) = 0 := by
+      apply sum_eq_zero
+      intro j _
+      exact hzero _ (by omega)
+    change (∑ j ∈ range (n+1), f j) = _
+    rw [this,add_zero]
+  rw [hext, mul_sum]
+  unfold tail
+  apply sum_le_sum
+  intro j _
+  dsimp [f]
+  split_ifs with hj
+  · have h := mul_le_mul_of_nonneg_right (choose_double_cubic (n:=n) hj)
+      (pow_nonneg hx j)
+    nlinarith only [h]
+  · simp
+
+/-- Positivity mechanism for the all-dimensional polynomial.
+Cancellation of its first two coefficients leaves a positive quadratic term;
+all later coefficients are controlled by the doubled binomial comparison. -/
+lemma shifted_polynomial_pos (n : ℕ) (hn : 2 ≤ n) {x : ℝ} (hx : 0 < x)
+    (a b c d e u v w t : ℝ)
+    (hzero : a+u+t=0) (hone : 2*n*a+b+n*u+v=0)
+    (htwo : 0 < ((2*n).choose 2 : ℝ)*a+2*n*b+c+(n.choose 2 : ℝ)*u+n*v+w)
+    (ha : 0 ≤ a) (hau : 0 ≤ 8*a+u)
+    (hb : 0 ≤ b) (hc : 0 ≤ c) (hd : 0 ≤ d) (he : 0 ≤ e)
+    (hv : 0 ≤ v) (hw : 0 ≤ w) :
+    0 < (1+x)^(2*n)*(a+b*x+c*x^2+d*x^3+e*x^4)+
+      (1+x)^n*(u+v*x+w*x^2)+t := by
+  have htn := tail_nonneg n hx.le
+  have ht2n := tail_nonneg (2*n) hx.le
+  have hdouble : 0 ≤ tail (2*n) x - 8*tail n x := sub_nonneg.mpr (tail_double n hx.le)
+  have hmain : 0 < (((2*n).choose 2 : ℝ)*a+2*n*b+c+(n.choose 2 : ℝ)*u+n*v+w)*x^2 :=
+    mul_pos htwo (sq_pos_of_pos hx)
+  have hrest : 0 ≤
+      a*(tail (2*n) x-8*tail n x) + (8*a+u)*tail n x +
+      b*x*(((2*n).choose 2 : ℝ)*x^2+tail (2*n) x) +
+      c*x^2*(2*n*x+((2*n).choose 2 : ℝ)*x^2+tail (2*n) x) +
+      d*x^3*(1+x)^(2*n) + e*x^4*(1+x)^(2*n) +
+      v*x*((n.choose 2 : ℝ)*x^2+tail n x) +
+      w*x^2*(n*x+(n.choose 2 : ℝ)*x^2+tail n x) := by positivity
+  have hidentity : (1+x)^(2*n)*(a+b*x+c*x^2+d*x^3+e*x^4)+
+      (1+x)^n*(u+v*x+w*x^2)+t =
+      (((2*n).choose 2 : ℝ)*a+2*n*b+c+(n.choose 2 : ℝ)*u+n*v+w)*x^2 +
+      (a*(tail (2*n) x-8*tail n x) + (8*a+u)*tail n x +
+      b*x*(((2*n).choose 2 : ℝ)*x^2+tail (2*n) x) +
+      c*x^2*(2*n*x+((2*n).choose 2 : ℝ)*x^2+tail (2*n) x) +
+      d*x^3*(1+x)^(2*n) + e*x^4*(1+x)^(2*n) +
+      v*x*((n.choose 2 : ℝ)*x^2+tail n x) +
+      w*x^2*(n*x+(n.choose 2 : ℝ)*x^2+tail n x)) := by
+    rw [tail_expansion n hn x, tail_expansion (2*n) (by omega) x]
+    have ht : t = -a-u := by linarith
+    have hb' : b = -2*n*a-n*u-v := by linarith
+    rw [ht, hb']
+    push_cast
+    ring
+  rw [hidentity]
+  exact add_pos_of_pos_of_nonneg hmain hrest
+
+def largeC (r : ℝ) := (r^2+5*r+7)/((r+1)*(r+3))
+def H (r : ℝ) := 18*(r+4)/largeC r
+
+def A4 (r : ℝ) := (2*r-8)/3
+def A2 (r : ℝ) := H r-12*r-32-64/r
+def A1 (r : ℝ) := -H r+16*(r-1)/3+32-16*(r-1)/(2*r+1)
+def A0 (r : ℝ) := -H r/r+6*r+40+64/r-128/r^2
+def B2 (r : ℝ) := 16+64/r
+def B0 (r : ℝ) := H r/r-48-64/r+256/r^2
+def T0 (r : ℝ) := 16*(r-1)/(2*r+1)-128/r^2
+
+def q0 (r : ℝ) := A4 r+A2 r+A1 r+A0 r
+def q1 (r : ℝ) := 4*A4 r+2*A2 r+A1 r
+def q2 (r : ℝ) := 6*A4 r+A2 r
+def q3 (r : ℝ) := 4*A4 r
+def q4 (r : ℝ) := A4 r
+def r0 (r : ℝ) := B2 r+B0 r
+def r1 (r : ℝ) := 2*B2 r
+def r2 (r : ℝ) := B2 r
+
+def V0 (r : ℝ) := 34*r^5-95*r^4-1666*r^3-5701*r^2-8052*r-2688
+def V1 (r : ℝ) := 2*r^5+9*r^4-162*r^3-845*r^2-1220*r-448
+def V2 (r : ℝ) := 5*r^3+8*r^2-41*r-56
+def R0 (r : ℝ) := 7*r^4+8*r^3-187*r^2-748*r-896
+
+lemma V0_pos {r : ℝ} (hr : 10 ≤ r) : 0 < V0 r := by
+  have ht : 0 ≤ r-10 := by linarith
+  have heq : V0 r = 34*(r-10)^5+1605*(r-10)^4+28534*(r-10)^3+
+      227319*(r-10)^2+698128*(r-10)+130692 := by unfold V0; ring
+  rw [heq]
+  positivity
+
+lemma V1_pos {r : ℝ} (hr : 10 ≤ r) : 0 < V1 r := by
+  have ht : 0 ≤ r-10 := by linarith
+  have heq : V1 r = 2*(r-10)^5+109*(r-10)^4+2198*(r-10)^3+
+      19695*(r-10)^2+69280*(r-10)+30852 := by unfold V1; ring
+  rw [heq]
+  positivity
+
+lemma V2_pos {r : ℝ} (hr : 7 ≤ r) : 0 < V2 r := by
+  have ht : 0 ≤ r-7 := by linarith
+  have heq : V2 r = 5*(r-7)^3+113*(r-7)^2+806*(r-7)+1764 := by unfold V2; ring
+  rw [heq]
+  positivity
+
+lemma R0_pos {r : ℝ} (hr : 7 ≤ r) : 0 < R0 r := by
+  have ht : 0 ≤ r-7 := by linarith
+  have heq : R0 r = 7*(r-7)^4+204*(r-7)^3+2039*(r-7)^2+7414*(r-7)+4256 := by
+    unfold R0
+    ring
+  rw [heq]
+  positivity
+
+lemma q0_identity {r : ℝ} (hr : 0 < r) :
+    q0 r+r0 r/8 = V0 r/(4*r^2*(2*r+1)*(r^2+5*r+7)) := by
+  unfold q0 r0 A4 A2 A1 A0 B2 B0 H largeC V0
+  field_simp
+  ring
+
+lemma q1_identity {r : ℝ} (hr : 0 < r) :
+    q1 r = 2*V1 r/(r*(2*r+1)*(r^2+5*r+7)) := by
+  unfold q1 A4 A2 A1 H largeC V1
+  field_simp
+  ring
+
+lemma q2_identity {r : ℝ} (hr : 0 < r) :
+    q2 r = 2*(r+4)*V2 r/(r*(r^2+5*r+7)) := by
+  unfold q2 A4 A2 H largeC V2
+  field_simp
+  ring
+
+lemma r0_identity {r : ℝ} (hr : 0 < r) :
+    r0 r = -2*R0 r/(r^2*(r^2+5*r+7)) := by
+  unfold r0 B2 B0 H largeC R0
+  field_simp
+  ring
+
+lemma large_signs {r : ℝ} (hr : 10 ≤ r) :
+    0 < q0 r ∧ 0 < 8*q0 r+r0 r ∧ 0 < q1 r ∧ 0 < q2 r ∧
+    0 < q3 r ∧ 0 < q4 r ∧ 0 < r1 r ∧ 0 < r2 r := by
+  have hr0 : 0 < r := by linarith
+  have hr7 : 7 ≤ r := by linarith
+  have hs : 0 < q0 r+r0 r/8 := by
+    rw [q0_identity hr0]
+    exact div_pos (V0_pos hr) (by positivity)
+  have hneg : r0 r < 0 := by
+    rw [r0_identity hr0]
+    exact div_neg_of_neg_of_pos (by nlinarith [R0_pos hr7]) (by positivity)
+  have h8 : 0 < 2*r-8 := by linarith
+  refine ⟨by linarith, by linarith, ?_, ?_, ?_, ?_, ?_, ?_⟩
+  · rw [q1_identity hr0]
+    exact div_pos (mul_pos (by norm_num) (V1_pos hr)) (by positivity)
+  · rw [q2_identity hr0]
+    exact div_pos (mul_pos (by positivity) (V2_pos hr7)) (by positivity)
+  · unfold q3 A4
+    positivity
+  · unfold q4 A4
+    positivity
+  · unfold r1 B2
+    positivity
+  · unfold r2 B2
+    positivity
+
+lemma constant_vanish {r : ℝ} (hr : 0 < r) : q0 r + r0 r + T0 r = 0 := by
+  unfold q0 r0 A4 A2 A1 A0 B2 B0 T0 H largeC
+  field_simp
+  ring
+
+lemma linear_vanish {r : ℝ} (hr : 0 < r) :
+    2*r*q0 r + q1 r + r*r0 r + r1 r = 0 := by
+  unfold q0 q1 r0 r1 A4 A2 A1 A0 B2 B0 H largeC
+  field_simp
+  ring
+
+lemma quadratic_identity {r : ℝ} (hr : 0 < r) :
+    r*(2*r-1)*q0 r + 2*r*q1 r + q2 r + r*(r-1)/2*r0 r+r*r1 r+r2 r =
+      9*(r+4)/largeC r*(r+3-8*largeC r) := by
+  unfold q0 q1 q2 r0 r1 r2 A4 A2 A1 A0 B2 B0 H largeC
+  field_simp
+  ring
+
+lemma large_polynomial_pos (n : ℕ) (hn : 10 ≤ n) {z : ℝ} (hz : 1 < z) :
+    0 < z^(2*n)*(A4 n*z^4+A2 n*z^2+A1 n*z+A0 n)+
+      z^n*(B2 n*z^2+B0 n)+T0 n := by
+  have hr : 10 ≤ (n : ℝ) := by exact_mod_cast hn
+  have hr0 : 0 < (n : ℝ) := by linarith
+  obtain ⟨hq0,hq0r0,hq1,hq2,hq3,hq4,hr1,hr2⟩ := large_signs hr
+  have hc : 0 < largeC n := by unfold largeC; positivity
+  have hmargin : 0 < (n : ℝ)+3-8*largeC n := by
+    have hm := large_local_margin (r := (n : ℝ)) (by linarith)
+    unfold largeC
+    convert hm using 1; ring
+  have htwo : 0 < (((2*n).choose 2 : ℕ) : ℝ)*q0 n+2*n*q1 n+q2 n+
+      (n.choose 2 : ℝ)*r0 n+n*r1 n+r2 n := by
+    rw [Nat.cast_choose_two ℝ, Nat.cast_choose_two ℝ]
+    push_cast
+    have heq : (2*(n:ℝ)*(2*n-1)/2*q0 n+2*n*q1 n+q2 n+
+        n*(n-1)/2*r0 n+n*r1 n+r2 n) =
+        9*(n+4)/largeC n*(n+3-8*largeC n) := by
+      convert quadratic_identity hr0 using 1; ring
+    rw [heq]
+    positivity
+  have hh := shifted_polynomial_pos n (by omega) (sub_pos.mpr hz)
+    (q0 n) (q1 n) (q2 n) (q3 n) (q4 n) (r0 n) (r1 n) (r2 n) (T0 n)
+    (constant_vanish hr0) (linear_vanish hr0) htwo hq0.le hq0r0.le
+    hq1.le hq2.le hq3.le hq4.le hr1.le hr2.le
+  have heq : z^(2*n)*(A4 n*z^4+A2 n*z^2+A1 n*z+A0 n)+
+      z^n*(B2 n*z^2+B0 n)+T0 n =
+      (1+(z-1))^(2*n)*(q0 n+q1 n*(z-1)+q2 n*(z-1)^2+q3 n*(z-1)^3+q4 n*(z-1)^4)+
+      (1+(z-1))^n*(r0 n+r1 n*(z-1)+r2 n*(z-1)^2)+T0 n := by
+    rw [show 1+(z-1)=z by ring]
+    unfold q0 q1 q2 q3 q4 r0 r1 r2
+    ring
+  rw [heq]
+  exact hh
+
+lemma z_mul_inverse_pow_succ {z : ℝ} (hz : z ≠ 0) (m : ℕ) :
+    z*(z⁻¹)^(m+1) = (z⁻¹)^m := by
+  rw [pow_succ', ← mul_assoc, mul_inv_cancel₀ hz, one_mul]
+
+lemma large_I_formula (n : ℕ) (hn : 7 ≤ n) {z : ℝ} (hz : 1 < z) :
+    z*I n z = 4*((z^4-z)/3 - 2*(z-z^2*(z⁻¹)^n)/((n:ℝ)-1) +
+      (z-((z⁻¹)^n)^2)/(2*n+1)) := by
+  have h4 : n ≠ 4 := by omega
+  have hz0 : z ≠ 0 := by linarith
+  have hp : (z⁻¹)^(n+1-2) = z*(z⁻¹)^n := by
+    have hv := z_mul_inverse_pow_succ hz0 (n+1-2)
+    have he : n+1-2+1=n := by omega
+    rw [he] at hv
+    exact hv.symm
+  have hd : z*(z⁻¹)^(2*n+1) = ((z⁻¹)^n)^2 := by
+    rw [z_mul_inverse_pow_succ hz0, show 2*n=n*2 by omega, pow_mul]
+  have hcast : ((n+1-2 : ℕ) : ℝ) = (n:ℝ)-1 := by
+    rw [Nat.cast_sub (by omega), Nat.cast_add]
+    norm_num
+    ring
+  simp only [I, exponent, h4, ite_false, hz, ite_true, Nat.cast_ofNat,
+    Nat.reduceMul, Nat.reduceSub, Nat.cast_add, Nat.cast_mul, Nat.cast_one, hcast]
+  rw [hp]
+  have hr1 : (n:ℝ)-1 ≠ 0 := by
+    have hr : 7 ≤ (n:ℝ) := by exact_mod_cast hn
+    linarith
+  have hr2 : 2*(n:ℝ)+1 ≠ 0 := by positivity
+  field_simp
+  simp only [inv_eq_one_div] at hd
+  linear_combination -12 * ((n : ℝ) - 1) * hd
+
+lemma large_gap_formula (n : ℕ) (hn : 7 ≤ n) {z : ℝ} (hz : 1 < z) :
+    9*((n:ℝ)+4)/coefficient n*(2*W n z-L n z) =
+      A4 n*z^4+A2 n*z^2+A1 n*z+A0 n+
+      (z⁻¹)^n*(B2 n*z^2+B0 n)+((z⁻¹)^n)^2*T0 n := by
+  have h4 : n ≠ 4 := by omega
+  have h56 : ¬(n=5 ∨ n=6) := by omega
+  have hr : 7 ≤ (n:ℝ) := by exact_mod_cast hn
+  have hr0 : 0 < (n:ℝ) := by linarith
+  have hc : coefficient n = largeC n := by
+    simp [coefficient, largeC, h4]
+  have hc0 : largeC n ≠ 0 := by
+    unfold largeC
+    positivity
+  have hI := large_I_formula n hn hz
+  unfold L
+  simp only [W, D, G, M, e, exponent, beta, weight, h4, h56, ite_false, hz,
+    ite_true, Nat.cast_ofNat, Nat.reduceSub, pow_one]
+  rw [hc]
+  have heq : ((n:ℝ)-1)*(2/3)^2*z*I n z = ((n:ℝ)-1)*(2/3)^2*(z*I n z) := by ring
+  rw [heq, hI]
+  have hr1 : (n:ℝ)-1 ≠ 0 := by linarith
+  unfold A4 A2 A1 A0 B2 B0 T0 H largeC
+  field_simp [hr1]
+  ring
+
+lemma large_cleared_gap (n : ℕ) (hn : 7 ≤ n) {z : ℝ} (hz : 1 < z) :
+    9*((n:ℝ)+4)/coefficient n*z^(2*n)*(2*W n z-L n z) =
+      z^(2*n)*(A4 n*z^4+A2 n*z^2+A1 n*z+A0 n)+
+      z^n*(B2 n*z^2+B0 n)+T0 n := by
+  have hz0 : z ≠ 0 := by linarith
+  have hzpow : z^n ≠ 0 := pow_ne_zero n hz0
+  calc
+    _ = z^(2*n)*(9*((n:ℝ)+4)/coefficient n*(2*W n z-L n z)) := by ring
+    _ = _ := by
+      rw [large_gap_formula n hn hz, show 2*n=n*2 by omega, pow_mul, inv_pow]
+      field_simp
+
+lemma high_pointwise_large (n : ℕ) (hn : 10 ≤ n) {z : ℝ} (hz : 1 < z) :
+    L n z < 2*W n z := by
+  have hp := large_polynomial_pos n hn hz
+  rw [← large_cleared_gap n (by omega) hz] at hp
+  have hc : 0 < 9*((n:ℝ)+4)/coefficient n*z^(2*n) := by
+    have hc := coefficient_pos n
+    positivity
+  exact sub_pos.mp ((mul_pos_iff_of_pos_left hc).mp hp)
+
+/-- The pointwise scalar estimate, including its exact equality
+condition, in every dimension covered by the main theorem. -/
+theorem pointwise (n : ℕ) (hn : 4 ≤ n) {z : ℝ} (hz : 0 < z) :
+    L n z ≤ 2*W n z ∧ (L n z = 2*W n z ↔ z=1) := by
+  by_cases hz1 : z ≤ 1
+  · exact low_pointwise n hn hz hz1
+  have hz' : 1 < z := lt_of_not_ge hz1
+  have hlt : L n z < 2*W n z := by
+    by_cases hn10 : 10 ≤ n
+    · exact high_pointwise_large n hn10 hz'
+    interval_cases n
+    · exact high_pointwise_4 hz'
+    · exact high_pointwise_5 hz'
+    · exact high_pointwise_6 hz'
+    · exact high_pointwise_7 hz'
+    · exact high_pointwise_8 hz'
+    · exact high_pointwise_9 hz'
+  exact ⟨hlt.le, iff_of_false (ne_of_lt hlt) (ne_of_gt hz')⟩
+
+end PettyProjection.Scalar
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/ProjectionBodies/ScalarCalculus.lean b/lean/OAI/Geometry/ProjectionBodies/ScalarCalculus.lean
new file mode 100644
index 0000000000000000000000000000000000000000..7b199c7ed15a1152735e8ebf01e65e54269e0beb
--- /dev/null
+++ b/lean/OAI/Geometry/ProjectionBodies/ScalarCalculus.lean
@@ -0,0 +1,245 @@
+import OAI.Geometry.ProjectionBodies.ScalarBounds
+
+namespace OAI
+
+noncomputable section
+open Set Filter Topology
+namespace PettyProjection.Scalar
+
+lemma hasDerivAt_inverse_pow (n : ℕ) {z : ℝ} (hz : z ≠ 0) :
+    HasDerivAt (fun x : ℝ => (x⁻¹) ^ n) (-(n : ℝ) * (z⁻¹) ^ (n + 1)) z := by
+  have h := hasDerivAt_zpow (-(n : ℤ)) z (Or.inl hz)
+  have he : -(n : ℤ) - 1 = -((n + 1 : ℕ) : ℤ) := by omega
+  simpa only [he, zpow_neg, zpow_natCast, inv_pow, Int.cast_neg, Int.cast_natCast] using h
+
+lemma hasDerivAt_e {n : ℕ} (hn : n ≠ 0) {z : ℝ} (hz : z ≠ 0) :
+    HasDerivAt (e n) (-(z⁻¹) ^ (n + 1)) z := by
+  have h := ((hasDerivAt_inverse_pow n hz).sub_const 1).div_const (n : ℝ)
+  have hn' : (n : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr hn
+  convert! h using 1
+  field_simp
+
+lemma hasDerivAt_M {n : ℕ} (hn : n ≠ 0) {z : ℝ} (hz : z ≠ 0) :
+    HasDerivAt (M n) ((exponent n : ℝ) * (z ^ (exponent n - 1) - (z⁻¹) ^ (n + 1))) z := by
+  have h := ((hasDerivAt_pow (exponent n) z).sub_const 1).add
+    ((hasDerivAt_e hn hz).const_mul (exponent n : ℝ))
+  convert! h using 1
+  ring
+
+lemma M_one (n : ℕ) : M n 1 = 0 := by simp [M, e]
+
+lemma paste_hasDerivAt {f f' : ℝ → ℝ}
+    (hf : ∀ x, 1 ≤ x → HasDerivAt f (f' x) x) (hf1 : f 1 = 0) (hf'1 : f' 1 = 0)
+    (x : ℝ) :
+    HasDerivAt (fun z => if 1 < z then f z else 0) (if 1 < x then f' x else 0) x := by
+  rcases lt_trichotomy x 1 with hx | rfl | hx
+  · have he : (fun z => if 1 < z then f z else 0) =ᶠ[𝓝 x] (fun _ => 0) := by
+      filter_upwards [eventually_lt_nhds hx] with z hz
+      simp only [ite_eq_right (not_lt.mpr hz.le)]
+    simpa only [ite_eq_right (not_lt.mpr hx.le)] using (hasDerivAt_const x (0 : ℝ)).congr_of_eventuallyEq he
+  · have hl : HasDerivWithinAt (fun z => if 1 < z then f z else 0) 0 (Iic 1) 1 := by
+      apply (hasDerivAt_const 1 (0 : ℝ)).hasDerivWithinAt.congr
+      · intro z hz
+        simp only [ite_eq_right (not_lt.mpr (show z ≤ 1 from hz))]
+      · simp
+    have hr : HasDerivWithinAt (fun z => if 1 < z then f z else 0) 0 (Ici 1) 1 := by
+      have h := (hf 1 le_rfl).hasDerivWithinAt (s := Ici 1)
+      rw [hf'1] at h
+      apply h.congr
+      · intro z hz
+        rcases lt_or_eq_of_le (show 1 ≤ z from hz) with hz | rfl
+        · simp only [ite_eq_left hz]
+        · simp [hf1]
+      · simp [hf1]
+    have h := hl.union hr
+    rw [Iic_union_Ici, hasDerivWithinAt_univ] at h
+    simpa using h
+  · have he : (fun z => if 1 < z then f z else 0) =ᶠ[𝓝 x] f := by
+      filter_upwards [eventually_gt_nhds hx] with z hz
+      simp only [ite_eq_left hz]
+    simpa only [ite_eq_left hx] using (hf x hx.le).congr_of_eventuallyEq he
+
+lemma paste_contDiff_one {f f' : ℝ → ℝ}
+    (hf : ∀ x, 1 ≤ x → HasDerivAt f (f' x) x) (hf1 : f 1 = 0) (hf'1 : f' 1 = 0)
+    (hcont : ContinuousOn f' (Ici 1)) :
+    ContDiff ℝ 1 (fun z => if 1 < z then f z else 0) := by
+  have hd := paste_hasDerivAt hf hf1 hf'1
+  apply contDiff_one_iff_deriv.mpr
+  constructor
+  · exact fun x => (hd x).differentiableAt
+  · have he : deriv (fun z => if 1 < z then f z else 0) =
+        (fun z => if z ≤ 1 then 0 else f' z) := by
+      funext z
+      rw [(hd z).deriv]
+      simp only [← not_le, ite_not]
+    rw [he]
+    exact continuous_if_le continuous_id continuous_const continuousOn_const hcont
+      (fun x hx => by change x = 1 at hx; subst x; exact hf'1.symm)
+
+end PettyProjection.Scalar
+end
+
+noncomputable section
+open Set Filter Topology
+namespace PettyProjection.Scalar
+
+def Mprime (n : ℕ) (z : ℝ) : ℝ :=
+  (exponent n : ℝ) * (z ^ (exponent n - 1) - (z⁻¹) ^ (n + 1))
+
+def Iformula (n : ℕ) (z : ℝ) : ℝ :=
+  (exponent n : ℝ)^2 *
+    ((z ^ (2 * exponent n - 1) - 1) / (2 * exponent n - 1 : ℕ) -
+      2 * (1 - (z⁻¹) ^ (n + 1 - exponent n)) / (n + 1 - exponent n : ℕ) +
+      (1 - (z⁻¹) ^ (2 * n + 1)) / (2 * n + 1 : ℕ))
+
+lemma hasDerivAt_Iformula_four {z : ℝ} (hz : z ≠ 0) :
+    HasDerivAt (Iformula 4) (Mprime 4 z ^ 2) z := by
+  have h := ((((hasDerivAt_pow 1 z).sub_const 1).div_const 1).sub
+    (((hasDerivAt_inverse_pow 4 hz).const_sub 1).const_mul 2 |>.div_const 4)).add
+    (((hasDerivAt_inverse_pow 9 hz).const_sub 1).div_const 9)
+  convert! h using 1
+  · funext x
+    simp [Iformula, exponent]
+  · simp only [Mprime, exponent]
+    norm_num
+    rw [show 10 = 5 * 2 by omega, pow_mul]
+    ring
+
+lemma hasDerivAt_Iformula_other {n : ℕ} (hn : 4 ≤ n) (hne : n ≠ 4)
+    {z : ℝ} (hz : z ≠ 0) :
+    HasDerivAt (Iformula n) (Mprime n z ^ 2) z := by
+  have hn1 : (n - 1 : ℝ) ≠ 0 := by
+    have : (4 : ℝ) ≤ n := by exact_mod_cast hn
+    linarith
+  have hn2 : (2 * n + 1 : ℝ) ≠ 0 := by positivity
+  have h := (((((hasDerivAt_pow 3 z).sub_const 1).div_const 3).sub
+    (((hasDerivAt_inverse_pow (n-1) hz).const_sub 1).const_mul 2 |>.div_const (n-1 : ℝ))).add
+    (((hasDerivAt_inverse_pow (2*n+1) hz).const_sub 1).div_const (2*n+1 : ℝ))).const_mul 4
+  have he : n + 1 - 2 = n - 1 := by omega
+  have he' : n - 1 + 1 = n := by omega
+  have hncast : ((n - 1 : ℕ) : ℝ) = n - 1 := by rw [Nat.cast_sub (by omega)]; norm_num
+  have hi := z_mul_inverse_pow_succ hz n
+  have hp : (z⁻¹) ^ (2*n+1+1) = ((z⁻¹) ^ (n+1)) ^ 2 := by
+    rw [show 2*n+1+1 = (n+1)*2 by omega, pow_mul]
+  convert! h using 1
+  · funext x
+    simp only [Iformula, exponent, ite_eq_right hne, Nat.reduceMul, Nat.reduceSub,
+      Nat.cast_ofNat, he, hncast, Nat.cast_add, Nat.cast_mul, Nat.cast_one, Pi.add_apply, Pi.sub_apply]
+    norm_num
+  · simp only [Mprime, exponent, ite_eq_right hne]
+    norm_num only [Nat.cast_ofNat, Nat.reduceSub, Nat.reducePow, pow_one,
+      Nat.cast_add, Nat.cast_mul, hncast, he', hp]
+    field_simp
+    simp only [one_div]
+    nlinarith only [hi]
+
+lemma hasDerivAt_Iformula {n : ℕ} (hn : 4 ≤ n) {z : ℝ} (hz : z ≠ 0) :
+    HasDerivAt (Iformula n) (Mprime n z ^ 2) z := by
+  by_cases hne : n = 4
+  · subst n; exact hasDerivAt_Iformula_four hz
+  · exact hasDerivAt_Iformula_other hn hne hz
+
+lemma Mprime_one (n : ℕ) : Mprime n 1 = 0 := by simp [Mprime]
+lemma Iformula_one (n : ℕ) : Iformula n 1 = 0 := by simp [Iformula]
+
+lemma continuousOn_Mprime (n : ℕ) : ContinuousOn (Mprime n) (Ici 1) := by
+  apply continuousOn_const.mul
+  apply (continuousOn_id.pow _).sub
+  exact (continuousOn_id.inv₀ (fun x hx => ne_of_gt (lt_of_lt_of_le zero_lt_one hx))).pow _
+
+lemma contDiff_G {n : ℕ} (hn : 4 ≤ n) : ContDiff ℝ 1 (G n) := by
+  apply paste_contDiff_one (f' := fun z => beta n * Mprime n z)
+  · intro x hx
+    exact (hasDerivAt_M (by omega) (ne_of_gt (lt_of_lt_of_le zero_lt_one hx))).const_mul _
+  · simp [M_one]
+  · simp [Mprime_one]
+  · exact continuousOn_const.mul (continuousOn_Mprime n)
+
+lemma contDiff_I {n : ℕ} (hn : 4 ≤ n) : ContDiff ℝ 1 (I n) := by
+  exact paste_contDiff_one (f' := fun z => Mprime n z ^ 2)
+    (fun x hx => hasDerivAt_Iformula hn (ne_of_gt (lt_of_lt_of_le zero_lt_one hx)))
+    (Iformula_one n) (by simp [Mprime_one]) ((continuousOn_Mprime n).pow 2)
+
+lemma deriv_G {n : ℕ} (hn : 4 ≤ n) (z : ℝ) :
+    deriv (G n) z = if 1 < z then beta n * Mprime n z else 0 :=
+  (paste_hasDerivAt (f' := fun z => beta n * Mprime n z)
+    (fun x hx => (hasDerivAt_M (by omega) (ne_of_gt (lt_of_lt_of_le zero_lt_one hx))).const_mul _)
+    (by simp [M_one]) (by simp [Mprime_one]) z).deriv
+
+lemma deriv_I {n : ℕ} (hn : 4 ≤ n) (z : ℝ) :
+    deriv (I n) z = if 1 < z then Mprime n z ^ 2 else 0 :=
+  (paste_hasDerivAt (f' := fun z => Mprime n z ^ 2)
+    (fun x hx => hasDerivAt_Iformula hn (ne_of_gt (lt_of_lt_of_le zero_lt_one hx)))
+    (Iformula_one n) (by simp [Mprime_one]) z).deriv
+
+lemma derivative_relation {n : ℕ} (hn : 4 ≤ n) (z : ℝ) :
+    deriv (G n) z ^ 2 = beta n ^ 2 * deriv (I n) z := by
+  rw [deriv_G hn, deriv_I hn]
+  split_ifs <;> ring
+
+lemma I_nonneg {n : ℕ} (hn : 4 ≤ n) (z : ℝ) : 0 ≤ I n z := by
+  by_cases hz : 1 < z
+  · have hmono : Monotone (I n) := by
+      apply monotone_of_deriv_nonneg ((contDiff_I hn).differentiable (by norm_num))
+      intro x
+      rw [deriv_I hn]
+      split_ifs <;> positivity
+    have h := hmono hz.le
+    simpa [I] using h
+  · simp [I, hz]
+
+end PettyProjection.Scalar
+end
+
+noncomputable section
+open Set
+namespace PettyProjection.Scalar
+
+lemma convex_tangent_bound {f : ℝ → ℝ} {S : Set ℝ} {x y f' : ℝ}
+    (hc : ConvexOn ℝ S f) (hx : x ∈ S) (hy : y ∈ S) (hd : HasDerivAt f f' x) :
+    f x + f' * (y - x) ≤ f y := by
+  rcases lt_trichotomy x y with h | rfl | h
+  · have hh := hc.le_slope_of_hasDerivAt hx hy h hd
+    rw [slope_def_field, le_div_iff₀ (sub_pos.mpr h)] at hh
+    linarith
+  · simp
+  · have hh := hc.slope_le_of_hasDerivAt hy hx h hd
+    rw [slope_def_field, div_le_iff₀ (sub_pos.mpr h)] at hh
+    nlinarith
+
+lemma inverse_pow_tangent (n : ℕ) {z : ℝ} (hz : 0 < z) :
+    1 + (n : ℝ) * (1 - z) ≤ (z⁻¹) ^ n := by
+  have hc : ConvexOn ℝ (Ioi 0) (fun x : ℝ => (x⁻¹) ^ n) := by
+    simpa only [zpow_neg, zpow_natCast, inv_pow] using (convexOn_zpow (-(n : ℤ)) (𝕜 := ℝ))
+  have hd : HasDerivAt (fun x : ℝ => (x⁻¹) ^ n) (-(n : ℝ)) 1 := by
+    simpa only [inv_one, one_pow, mul_one] using hasDerivAt_inverse_pow n (one_ne_zero : (1 : ℝ) ≠ 0)
+  have h := convex_tangent_bound hc (show (1 : ℝ) ∈ Ioi 0 by norm_num) hz hd
+  simp only [inv_one, one_pow] at h
+  nlinarith only [h]
+
+lemma e_lower {n : ℕ} (hn : 0 < n) {z : ℝ} (hz : 0 < z) : 1 - z ≤ e n z := by
+  have hn' : (0 : ℝ) < n := by exact_mod_cast hn
+  rw [e, le_div_iff₀ hn']
+  nlinarith only [inverse_pow_tangent n hz]
+
+lemma M_nonneg {n : ℕ} (hn : 0 < n) {z : ℝ} (hz : 0 < z) : 0 ≤ M n z := by
+  have h := e_lower hn hz
+  unfold M exponent
+  split_ifs <;> norm_num <;> nlinarith [sq_nonneg (z-1)]
+
+lemma G_bounds {n : ℕ} (hn : 0 < n) {z : ℝ} (hz : 0 < z) :
+    0 ≤ G n z ∧ G n z ≤ beta n * M n z := by
+  have hm := M_nonneg hn hz
+  have hb := (beta_pos n).le
+  unfold G
+  split_ifs <;> constructor <;> nlinarith
+
+lemma W_mean_expression (n : ℕ) (z : ℝ) :
+    W n z = (z ^ exponent n - 1) - (z ^ (exponent n - 1) - 1) + e n z := by
+  unfold W
+  ring
+
+end PettyProjection.Scalar
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/ProjectionBodies/SeminormBounds.lean b/lean/OAI/Geometry/ProjectionBodies/SeminormBounds.lean
new file mode 100644
index 0000000000000000000000000000000000000000..9ce897849663cee470a12e81863a6aadb16b5659
--- /dev/null
+++ b/lean/OAI/Geometry/ProjectionBodies/SeminormBounds.lean
@@ -0,0 +1,670 @@
+import OAI.Geometry.ProjectionBodies.NormRigidity
+
+namespace OAI
+
+universe uα uι
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology Function
+open scoped ENNReal NNReal RealInnerProductSpace
+namespace PettyProjection.Spherical
+
+lemma seminorm_le_max_norm {n : ℕ} (g : Seminorm ℝ (Space n)) {R : ℝ}
+    (hmax : ∀ u : Sphere n, g u ≤ R) (x : Space n) : g x ≤ R*‖x‖ := by
+  by_cases hx : x=0
+  · simp [hx]
+  have hx0 : ‖x‖ ≠ 0 := norm_ne_zero_iff.mpr hx
+  let u : Sphere n := ⟨‖x‖⁻¹ • x, by simp [norm_smul,hx0]⟩
+  have hu := hmax u
+  have hgu : g x=‖x‖*g u := by
+    change g x=‖x‖*g (‖x‖⁻¹ • x)
+    rw [map_smul_eq_mul,Real.norm_of_nonneg (inv_nonneg.mpr (norm_nonneg _))]
+    field_simp
+  rw [hgu]
+  nlinarith [norm_nonneg x]
+
+/-- Exact coordinate minorant used for logarithmic coercivity and continuous
+extension to singular matrix ranks. -/
+lemma seminorm_coordinate_minorant {n : ℕ} [NeZero n] (g : Seminorm ℝ (Space n))
+    (hg : g ≠ 0) : ∃ (R : ℝ) (e : Sphere n), 0 < R ∧
+      (∀ u : Sphere n, g u ≤ R) ∧
+      (∀ x : Space n, R*|⟪x,(e:Space n)⟫| ≤ g x) := by
+  have hne : (sphere (0 : Space n) 1).Nonempty := by
+    obtain ⟨x,hx⟩ := exists_norm_eq (Space n) (by norm_num : (0:ℝ)≤1)
+    exact ⟨x,by simpa [mem_sphere,dist_zero_right]⟩
+  obtain ⟨v,hv,hm⟩ := (isCompact_sphere (0 : Space n) 1).exists_isMaxOn hne
+    (seminorm_continuous n g).continuousOn
+  have hv1 : ‖v‖=1 := by simpa [mem_sphere,dist_zero_right] using hv
+  let R := g v
+  have hR : 0≤R := apply_nonneg g v
+  have hmax (u : Sphere n) : g u ≤ R := hm u.property
+  have hbound := seminorm_le_max_norm g hmax
+  have hRp : 0<R := by
+    by_contra h
+    have hz : R=0 := le_antisymm (le_of_not_gt h) hR
+    apply hg
+    ext x
+    exact le_antisymm (by simpa [hz] using hbound x) (apply_nonneg g x)
+  let S := Submodule.span ℝ {v}
+  let f : Module.Dual ℝ S := R • ((innerSL ℝ v).toLinearMap.comp S.subtype)
+  have hf (y : S) : |f y| ≤ g (y:Space n) := by
+    obtain ⟨a,ha⟩ := Submodule.mem_span_singleton.mp y.property
+    change |R*⟪v,(y:Space n)⟫| ≤ g (y:Space n)
+    rw [← ha,inner_smul_right,real_inner_self_eq_norm_sq,hv1,map_smul_eq_mul]
+    simp only [one_pow,mul_one,Real.norm_eq_abs,abs_mul,abs_of_nonneg hR]
+    dsimp only [R]
+    rw [mul_comm]
+  obtain ⟨φ,hφ,hφbound⟩ := Module.Dual.exists_continuous_extension_of_le_seminorm_real
+    S f (seminorm_continuous n g) (fun y => (le_abs_self _).trans (hf y))
+  have hφv : φ v=R := by
+    have h := hφ ⟨v,Submodule.mem_span_singleton_self v⟩
+    simpa [f,real_inner_self_eq_norm_sq,hv1] using h
+  have hφnorm : ‖φ‖=R := by
+    apply le_antisymm
+    · exact φ.opNorm_le_bound hR (fun x => by simpa only [Real.norm_eq_abs] using (hφbound x).trans (hbound x))
+    · have h := φ.le_opNorm v
+      rw [hφv,hv1,Real.norm_of_nonneg hR,mul_one] at h
+      exact h
+  let w := (InnerProductSpace.toDual ℝ (Space n)).symm φ
+  have hw : ‖w‖=R := by simpa only [w,LinearIsometryEquiv.norm_map] using hφnorm
+  let e : Sphere n := ⟨R⁻¹ • w,by
+    simp only [mem_sphere,dist_zero_right,norm_smul,Real.norm_of_nonneg (inv_nonneg.mpr hR),hw]
+    exact inv_mul_cancel₀ hRp.ne'⟩
+  refine ⟨R,e,hRp,hmax,fun x => ?_⟩
+  change R*|⟪x,R⁻¹ • w⟫| ≤ g x
+  rw [inner_smul_right,abs_mul,abs_of_pos (inv_pos.mpr hRp),← mul_assoc,
+    mul_inv_cancel₀ hRp.ne',one_mul,real_inner_comm]
+  simpa only [w,InnerProductSpace.toDual_symm_apply] using hφbound x
+
+end PettyProjection.Spherical
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology Function
+open scoped ENNReal NNReal RealInnerProductSpace
+namespace PettyProjection.Spherical
+
+lemma intervalIntegrable_abs_rpow {p a b : ℝ} (hp : -1<p) :
+    IntervalIntegrable (fun t : ℝ => |t|^p) volume a b := by
+  apply intervalIntegrable_of_even (fun t => by rw [abs_neg])
+  intro x hx
+  apply (intervalIntegral.intervalIntegrable_rpow' hp).congr
+  intro t ht
+  rw [uIoc_of_le hx.le] at ht
+  dsimp only
+  rw [abs_of_nonneg ht.1.le]
+
+lemma inner_rpow_integrable_ball {n : ℕ} (e : Sphere n) {p : ℝ} (hp : -1<p) :
+    IntegrableOn (fun x : Space n => |⟪(e:Space n),x⟫|^p) (ball 0 1) := by
+  let B := ball (0 : perpendicular (e:Space n)) 1
+  let : IsFiniteMeasure (volume.restrict B) := ⟨by
+    rw [Measure.restrict_apply_univ]
+    exact (isBounded_ball : Bornology.IsBounded B).measure_lt_top⟩
+  have hI : Integrable (fun z : perpendicular (e:Space n) × ℝ => |z.2|^p)
+      ((volume.restrict B).prod (volume.restrict (Icc (-1:ℝ) 1))) := by
+    have h := (integrableOn_Icc_iff_integrableOn_Ioc (f := fun t : ℝ => |t|^p)).mpr
+      (intervalIntegrable_abs_rpow (a := -1) (b := 1) hp).1
+    exact h.comp_snd _
+  rw [Measure.prod_restrict,← Measure.volume_eq_prod] at hI
+  have hsub : (shadowMeasurableEquiv e) ⁻¹' ball (0:Space n) 1 ⊆ B ×ˢ Icc (-1:ℝ) 1 := by
+    rintro ⟨z,t⟩ h
+    have hnorm : ‖shadowCoords e (WithLp.toLp 2 (z,t))‖<1 := by
+      simpa [shadowMeasurableEquiv] using h
+    rw [(shadowCoords e).norm_map] at hnorm
+    have hz := WithLp.norm_fst_le (p := (2:ℝ≥0∞)) _ (WithLp.toLp 2 (z,t))
+    have ht := WithLp.norm_snd_le (p := (2:ℝ≥0∞)) _ (WithLp.toLp 2 (z,t))
+    refine ⟨?_,?_⟩
+    · simpa only [B,mem_ball,dist_zero_right,WithLp.toLp_fst] using hz.trans_lt hnorm
+    · have habs : |t|<1 := by exact ht.trans_lt hnorm
+      exact ⟨(neg_lt_of_abs_lt habs).le,(lt_of_abs_lt habs).le⟩
+  change IntegrableOn (fun z : perpendicular (e:Space n) × ℝ => |z.2|^p) (B ×ˢ Icc (-1:ℝ) 1) at hI
+  have hI' := hI.mono_set hsub
+  have heq : (fun z : perpendicular (e:Space n) × ℝ => |⟪(e:Space n),shadowMeasurableEquiv e z⟫|^p)=
+      (fun z => |z.2|^p) := by
+    funext z
+    change |⟪(e:Space n),(z.1:Space n)+z.2 • (e:Space n)⟫|^p=_
+    rw [inner_add_right,perpendicular_inner,inner_smul_right,real_inner_self_eq_norm_sq,norm_coe]
+    simp
+  exact ((shadowMeasurableEquiv_measurePreserving e).integrableOn_comp_preimage
+    (shadowMeasurableEquiv e).measurableEmbedding).mp (by simpa only [comp_def,heq] using hI')
+
+lemma sphere_integrable_of_homogeneous {n : ℕ} [NeZero n] {f : Space n → ℝ} {p : ℝ}
+    (hf : IntegrableOn f (ball 0 1))
+    (hh : ∀ (r : ℝ),0<r → ∀ x,f (r • x)=r^p*f x) :
+    Integrable (fun u : Sphere n => f u) (sigma n) := by
+  let ν := Measure.volumeIoiPow (Module.finrank ℝ (Space n)-1)
+  let I : Set (Ioi (0:ℝ)) := Iio ⟨1,by norm_num⟩
+  have hνI : ν I ≠ 0 := by
+    have h := Measure.volumeIoiPow_apply_Iio (Module.finrank ℝ (Space n)-1)
+      (⟨1,by norm_num⟩ : Ioi (0:ℝ))
+    change ν I=_ at h
+    rw [h]
+    apply (ENNReal.ofReal_pos.mpr _).ne'
+    positivity
+  let F : Space n → ℝ := (ball (0:Space n) 1).indicator f
+  have hF : Integrable F := hf.integrable_indicator measurableSet_ball
+  have hs := (integrableOn_iff_comap_subtypeVal (measurableSet_singleton (0:Space n)).compl).mp
+    (hF.integrableOn (s := ({0}ᶜ : Set (Space n))))
+  have hm := (volume : Measure (Space n)).measurePreserving_homeomorphUnitSphereProd
+  have hprod := (hm.symm (homeomorphUnitSphereProd (Space n)).toMeasurableEquiv).integrable_comp_emb
+    (homeomorphUnitSphereProd (Space n)).symm.measurableEmbedding |>.mpr hs
+  have hAE : ∀ᵐ r ∂ν, Integrable (fun u : Sphere n => F (r.1 • (u:Space n))) (surface n) := by
+    have h := hprod.prod_left_ae
+    change ∀ᵐ r ∂ν, Integrable (fun u => F ↑((homeomorphUnitSphereProd (Space n)).symm (u,r))) (surface n) at h
+    simpa only [homeomorphUnitSphereProd_symm_apply_coe] using h
+  obtain ⟨r,hr,hir⟩ := Measure.exists_mem_of_measure_ne_zero_of_ae hνI (ae_restrict_of_ae hAE)
+  have hr1 : (r:ℝ)<1 := hr
+  have hp : r.1^p ≠ 0 := (Real.rpow_pos_of_pos r.property p).ne'
+  have hfun : (fun u : Sphere n => F (r.1 • (u:Space n)))=(fun u : Sphere n => r.1^p*f u) := by
+    funext u
+    have hu : r.1 • (u:Space n) ∈ ball (0:Space n) 1 := by
+      simpa only [mem_ball,dist_zero_right,norm_smul,Real.norm_of_nonneg r.property.le,norm_coe,mul_one] using hr1
+    dsimp only [F]
+    rw [indicator_of_mem hu,hh r r.property]
+  rw [hfun] at hir
+  have hi := (integrable_const_mul_iff (isUnit_iff_ne_zero.mpr hp) (fun u : Sphere n => f u)).mp hir
+  exact hi.smul_measure (ENNReal.inv_ne_top.mpr (Measure.measure_univ_ne_zero.mpr (surface_ne_zero n)))
+
+lemma inner_rpow_integrable_sphere {n : ℕ} [NeZero n] (e : Sphere n) {p : ℝ} (hp : -1<p) :
+    Integrable (fun u : Sphere n => |⟪(e:Space n),(u:Space n)⟫|^p) (sigma n) := by
+  apply sphere_integrable_of_homogeneous (p := p) (inner_rpow_integrable_ball e hp)
+  intro r hr x
+  rw [inner_smul_right,abs_mul,abs_of_pos hr,Real.mul_rpow hr.le (abs_nonneg _)]
+
+lemma log_sq_le_negative_rpow {t : ℝ} (ht : 0<t) (h1 : t≤1) :
+    (Real.log t)^2 ≤ 16*t^(-(1/2:ℝ)) := by
+  have hl : 0 ≤ -Real.log t := neg_nonneg.mpr (Real.log_nonpos ht.le h1)
+  have h := Real.log_le_rpow_div (inv_nonneg.mpr ht.le) (by norm_num : (0:ℝ)<1/4)
+  rw [Real.log_inv] at h
+  have hsq := sq_le_sq₀ hl (by positivity : 0 ≤ (t⁻¹)^(1/4:ℝ)/(1/4:ℝ)) |>.mpr h
+  have hp : ((t⁻¹)^(1/4:ℝ))^2=t^(-(1/2:ℝ)) := by
+    rw [← Real.rpow_natCast,← Real.rpow_mul (inv_nonneg.mpr ht.le),Real.inv_rpow ht.le]
+    norm_num
+    rw [Real.rpow_neg ht.le]
+  rw [div_pow,neg_sq, hp] at hsq
+  norm_num at hsq
+  nlinarith
+
+lemma log_inner_sq_integrable {n : ℕ} [NeZero n] (e : Sphere n) :
+    Integrable (fun u : Sphere n => (Real.log |⟪(e:Space n),(u:Space n)⟫|)^2) (sigma n) := by
+  apply ((inner_rpow_integrable_sphere e (by norm_num : (-1:ℝ)< -(1/2:ℝ))).const_mul 16).mono'
+    ((measurable_const.inner measurable_subtype_coe).abs.log.pow_const 2).aestronglyMeasurable
+  filter_upwards with u
+  rw [Real.norm_of_nonneg (sq_nonneg _)]
+  by_cases h : |⟪(e:Space n),(u:Space n)⟫|=0
+  · simp only [h,Real.log_zero,zero_pow (by norm_num : (2:ℕ)≠0)]
+    positivity
+  · apply log_sq_le_negative_rpow (lt_of_le_of_ne (abs_nonneg _) (Ne.symm h))
+    simpa only [norm_coe,mul_one] using (abs_real_inner_le_norm (e:Space n) (u:Space n))
+
+lemma log_inner_integrable {n : ℕ} [NeZero n] (e : Sphere n) :
+    Integrable (fun u : Sphere n => Real.log |⟪(e:Space n),(u:Space n)⟫|) (sigma n) := by
+  apply ((log_inner_sq_integrable e).add (integrable_const (1:ℝ))).mono'
+    ((measurable_const.inner measurable_subtype_coe).abs.log.aestronglyMeasurable)
+  filter_upwards with u
+  rw [Real.norm_eq_abs]
+  let x := Real.log |⟪(e:Space n),(u:Space n)⟫|
+  change |x| ≤ x^2+1
+  nlinarith [sq_nonneg (|x|-1),sq_abs x]
+
+end PettyProjection.Spherical
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology Function
+open scoped ENNReal NNReal RealInnerProductSpace
+namespace PettyProjection.Spherical
+
+lemma inner_ne_zero_ae_sphere {n : ℕ} (e : Sphere n) :
+    ∀ᵐ (u : Sphere n) ∂sigma n, ⟪(e:Space n),(u:Space n)⟫ ≠ 0 := by
+  apply sphere_ae_of_radial (P := fun x : Space n => ⟪(e:Space n),x⟫ ≠ 0)
+  · have hR : ∀ᵐ t : ℝ ∂volume, t ≠ 0 := by
+      simpa only [ae_iff,not_not,Set.ofPred_eq_eq_singleton] using (measure_singleton (0:ℝ))
+    have hp : ∀ᵐ z : perpendicular (e:Space n) × ℝ ∂volume, z.2 ≠ 0 := by
+      rw [Measure.volume_eq_prod]
+      exact (Measure.ae_prod_iff_ae_ae (by measurability)).mpr (Eventually.of_forall (fun _ => hR))
+    have he : ∀ᵐ x : Space n ∂volume, ((shadowMeasurableEquiv e).symm x).2 ≠ 0 :=
+      ((shadowMeasurableEquiv_measurePreserving e).symm (shadowMeasurableEquiv e)).quasiMeasurePreserving.ae hp
+    filter_upwards [he] with x hx
+    have hy := (shadowMeasurableEquiv e).apply_symm_apply x
+    have hid (z : perpendicular (e:Space n) × ℝ) :
+        ⟪(e:Space n),shadowMeasurableEquiv e z⟫=z.2 := by
+      change ⟪(e:Space n),(z.1:Space n)+z.2 • (e:Space n)⟫=z.2
+      rw [inner_add_right,perpendicular_inner,inner_smul_right,real_inner_self_eq_norm_sq,norm_coe]
+      simp
+    rwa [← hid,hy] at hx
+  · intro u r _ h
+    rw [inner_smul_right,mul_ne_zero_iff] at h
+    exact h.2
+
+lemma mean_inner_independent {n : ℕ} (f : ℝ → ℝ) (e v : Sphere n) :
+    mean (fun u : Sphere n => f ⟪(e:Space n),(u:Space n)⟫) =
+      mean (fun u : Sphere n => f ⟪(v:Space n),(u:Space n)⟫) := by
+  let U := (ℝ ∙ ((e:Space n)-(v:Space n)))ᗮ.reflection
+  have hev : U e=v := Submodule.reflection_sub ((norm_coe e).trans (norm_coe v).symm)
+  have h := mean_rotation U (fun u : Sphere n => f ⟪U (e:Space n),(u:Space n)⟫)
+  change mean (fun u : Sphere n => f ⟪U (e:Space n),U (u:Space n)⟫)=_ at h
+  simp only [U.inner_map_map] at h
+  simpa only [hev] using h
+
+lemma seminorm_pos_ae_sphere {n : ℕ} [NeZero n] {g : Seminorm ℝ (Space n)} (hg : g ≠ 0) :
+    ∀ᵐ (u : Sphere n) ∂sigma n, 0 < g u := by
+  obtain ⟨R,e,hR,_,hb⟩ := seminorm_coordinate_minorant g hg
+  filter_upwards [inner_ne_zero_ae_sphere e] with u hu
+  exact (mul_pos hR (abs_pos.mpr (by simpa only [real_inner_comm] using hu))).trans_le (hb u)
+
+lemma log_sq_coercive_bound {a R t : ℝ} (ha : 0<a) (ha1 : a≤1) (hR : 0<R)
+    (htl : R*a≤t) (htu : t≤R) :
+    (Real.log t)^2 ≤ 2*(Real.log R)^2+2*(Real.log a)^2 := by
+  have ht := (mul_pos hR ha).trans_le htl
+  have hl0 := Real.log_le_log (mul_pos hR ha) htl
+  rw [Real.log_mul hR.ne' ha.ne'] at hl0
+  have hl1 := Real.log_le_log ht htu
+  have ha0 : Real.log a≤0 := Real.log_nonpos ha.le ha1
+  have habs : |Real.log t|≤|Real.log R|+|Real.log a| := by
+    rw [abs_le]
+    constructor
+    · rw [abs_of_nonpos ha0]
+      linarith [neg_abs_le (Real.log R)]
+    · linarith [le_abs_self (Real.log R),abs_nonneg (Real.log a)]
+  have hs := sq_le_sq₀ (abs_nonneg _) (by positivity : 0≤|Real.log R|+|Real.log a|) |>.mpr habs
+  nlinarith [sq_nonneg (|Real.log R|-|Real.log a|),sq_abs (Real.log R),sq_abs (Real.log a),sq_abs (Real.log t)]
+
+lemma seminorm_log_sq_integrable {n : ℕ} [NeZero n] {g : Seminorm ℝ (Space n)} (hg : g ≠ 0) :
+    Integrable (fun u : Sphere n => (Real.log (g u))^2) (sigma n) := by
+  obtain ⟨R,e,hR,hm,hb⟩ := seminorm_coordinate_minorant g hg
+  apply ((integrable_const (2*(Real.log R)^2)).add ((log_inner_sq_integrable e).const_mul 2)).mono'
+    (((seminorm_continuous n g).comp continuous_subtype_val).measurable.log.pow_const 2).aestronglyMeasurable
+  filter_upwards [inner_ne_zero_ae_sphere e] with u hu
+  rw [Real.norm_of_nonneg (sq_nonneg _)]
+  apply log_sq_coercive_bound (abs_pos.mpr hu) _ hR _ (hm u)
+  · simpa only [norm_coe,mul_one] using abs_real_inner_le_norm (e:Space n) (u:Space n)
+  · simpa only [real_inner_comm] using hb u
+
+lemma seminorm_log_integrable {n : ℕ} [NeZero n] {g : Seminorm ℝ (Space n)} (hg : g ≠ 0) :
+    Integrable (fun u : Sphere n => Real.log (g u)) (sigma n) := by
+  apply ((seminorm_log_sq_integrable hg).add (integrable_const (1:ℝ))).mono'
+    (((seminorm_continuous n g).comp continuous_subtype_val).measurable.log.aestronglyMeasurable)
+  filter_upwards with u
+  rw [Real.norm_eq_abs]
+  change |Real.log (g u)| ≤ (Real.log (g u))^2+1
+  nlinarith [sq_nonneg (|Real.log (g u)|-1),sq_abs (Real.log (g u))]
+
+lemma seminorm_log_coercivity {n : ℕ} [NeZero n] {g : Seminorm ℝ (Space n)}
+    {R : ℝ} {e : Sphere n} (hR : 0<R) (hg : g ≠ 0)
+    (hb : ∀ x : Space n,R*|⟪x,(e:Space n)⟫|≤g x) :
+    Real.log R+mean (fun u : Sphere n => Real.log |⟪(pole n:Space n),(u:Space n)⟫|) ≤
+      mean (fun u : Sphere n => Real.log (g u)) := by
+  have hi := integral_mono_ae ((integrable_const (Real.log R)).add (log_inner_integrable e))
+    (seminorm_log_integrable hg) (by
+      filter_upwards [inner_ne_zero_ae_sphere e] with u hu
+      have h := hb u
+      rw [real_inner_comm] at h
+      simpa only [Pi.add_apply,Real.log_mul hR.ne' (abs_pos.mpr hu).ne'] using
+        Real.log_le_log (mul_pos hR (abs_pos.mpr hu)) h)
+  simp only [Pi.add_apply] at hi
+  rw [integral_add (integrable_const _) (log_inner_integrable e),integral_const] at hi
+  simp only [probReal_univ,one_smul] at hi
+  change (Real.log R)+mean (fun u : Sphere n => Real.log |⟪(e:Space n),(u:Space n)⟫|)≤_ at hi
+  rwa [mean_inner_independent (fun t => Real.log |t|) e (pole n)] at hi
+
+end PettyProjection.Spherical
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology Function
+open scoped ENNReal NNReal RealInnerProductSpace
+namespace PettyProjection.Spherical
+
+lemma unifIntegrable_of_sq_bound {α : Type uα} {ι : Type uι} [MeasurableSpace α] {μ : Measure α}
+    {f : ι → α → ℝ} {C : ℝ} (hC : 0≤C)
+    (hi : ∀ i,Integrable (f i) μ) (hsq : ∀ i,Integrable (fun x => (f i x)^2) μ)
+    (hb : ∀ i,(∫ x,(f i x)^2 ∂μ)≤C) : UnifIntegrable f 1 μ := by
+  apply unifIntegrable_of le_rfl (by norm_num) (fun i => (hi i).aestronglyMeasurable)
+  intro ε hε
+  by_cases hεtop : ε = ∞
+  · exact ⟨0, fun _ => hεtop.symm ▸ le_top⟩
+  have hεreal : 0 < ε.toReal := ENNReal.toReal_pos hε.ne' hεtop
+  rw [← ENNReal.ofReal_toReal hεtop]
+  let M : ℝ := C/ε.toReal+1
+  have hM : 0<M := by dsimp [M]; positivity
+  refine ⟨⟨M,hM.le⟩,fun i => ?_⟩
+  let T := {x | (⟨M,hM.le⟩ : ℝ≥0)≤‖f i x‖₊}
+  have hind : Integrable (T.indicator (f i)) μ := (hi i).indicator₀
+    (nullMeasurableSet_le aemeasurable_const (hi i).aestronglyMeasurable.aemeasurable.nnnorm)
+  change eLpNorm (T.indicator (f i)) 1 μ ≤ ENNReal.ofReal ε.toReal
+  rw [eLpNorm_one_eq_lintegral_enorm hind.aestronglyMeasurable,
+    ← ofReal_integral_norm_eq_lintegral_enorm hind]
+  apply ENNReal.ofReal_le_ofReal
+  calc
+    _ ≤ ∫ x,(f i x)^2/M ∂μ := by
+      apply integral_mono hind.norm ((hsq i).div_const M)
+      intro x
+      change ‖T.indicator (f i) x‖ ≤ (f i x)^2/M
+      by_cases hx : x∈T
+      · rw [indicator_of_mem hx,Real.norm_eq_abs]
+        have hx' : M≤|f i x| := hx
+        apply (le_div_iff₀ hM).mpr
+        nlinarith [sq_abs (f i x)]
+      · rw [indicator_of_notMem hx,norm_zero]
+        positivity
+    _ = (∫ x,(f i x)^2 ∂μ)/M := by simp only [div_eq_mul_inv,integral_mul_const]
+    _ ≤ C/M := div_le_div_of_nonneg_right (hb i) hM.le
+    _ ≤ ε.toReal := by
+      apply (div_le_iff₀ hM).mpr
+      dsimp [M]
+      rw [mul_add,mul_div_cancel₀ _ hεreal.ne']
+      linarith
+
+lemma tendsto_integral_of_sq_bound {α : Type uα} [MeasurableSpace α] {μ : Measure α}
+    [IsFiniteMeasure μ] {f : ℕ → α → ℝ} {g : α → ℝ} {C : ℝ} (hC : 0≤C)
+    (hi : ∀ i,Integrable (f i) μ) (hsq : ∀ i,Integrable (fun x => (f i x)^2) μ)
+    (hb : ∀ i,(∫ x,(f i x)^2 ∂μ)≤C) (hg : Integrable g μ)
+    (hc : ∀ᵐ x ∂μ,Tendsto (fun i => f i x) atTop (𝓝 (g x))) :
+    Tendsto (fun i => ∫ x,f i x ∂μ) atTop (𝓝 (∫ x,g x ∂μ)) := by
+  apply tendsto_integral_of_L1' g (Eventually.of_forall hi)
+  exact tendsto_Lp_finite_of_tendsto_ae le_rfl (by norm_num)
+    (fun i => (hi i).aestronglyMeasurable) (memLp_one_iff_integrable.mpr hg)
+    (unifIntegrable_of_sq_bound hC hi hsq hb) hc
+
+lemma seminorm_log_sq_bound {n : ℕ} [NeZero n] {g : Seminorm ℝ (Space n)}
+    {r R : ℝ} (hr : 0<r) (hlo : ∃ u : Sphere n,r≤g u) (hhi : ∀ u : Sphere n,g u≤R) :
+    mean (fun u : Sphere n => (Real.log (g u))^2) ≤
+      2*(|Real.log r|+|Real.log R|)^2+
+        2*mean (fun u : Sphere n => (Real.log |⟪(pole n:Space n),(u:Space n)⟫|)^2) := by
+  have hg : g ≠ 0 := by
+    rintro rfl
+    obtain ⟨u,hu⟩ := hlo
+    simpa using hr.not_ge hu
+  obtain ⟨a,e,ha,hm,hb⟩ := seminorm_coordinate_minorant g hg
+  have hla : r≤a := by obtain ⟨u,hu⟩ := hlo; exact hu.trans (hm u)
+  have hha : a≤R := by
+    have h := hb e
+    rw [real_inner_self_eq_norm_sq,norm_coe] at h
+    norm_num at h
+    exact h.trans (hhi e)
+  have hlogs : |Real.log a|≤|Real.log r|+|Real.log R| := by
+    have h1 := Real.log_le_log hr hla
+    have h2 := Real.log_le_log ha hha
+    rw [abs_le]
+    constructor
+    · linarith [neg_abs_le (Real.log r),abs_nonneg (Real.log R)]
+    · linarith [le_abs_self (Real.log R),abs_nonneg (Real.log r)]
+  have hlogs2 : (Real.log a)^2≤(|Real.log r|+|Real.log R|)^2 := by
+    simpa only [sq_abs] using sq_le_sq₀ (abs_nonneg _) (by positivity) |>.mpr hlogs
+  have h := integral_mono_ae (seminorm_log_sq_integrable hg)
+    ((integrable_const (2*(Real.log a)^2)).add ((log_inner_sq_integrable e).const_mul 2)) (by
+      filter_upwards [inner_ne_zero_ae_sphere e] with u hu
+      change (Real.log (g u))^2≤2*(Real.log a)^2+2*(Real.log |⟪(e:Space n),(u:Space n)⟫|)^2
+      apply log_sq_coercive_bound (abs_pos.mpr hu) _ ha _ (hm u)
+      · simpa only [norm_coe,mul_one] using abs_real_inner_le_norm (e:Space n) (u:Space n)
+      · simpa only [real_inner_comm] using hb u)
+  simp only [Pi.add_apply] at h
+  rw [integral_add (integrable_const _) ((log_inner_sq_integrable e).const_mul 2),
+    integral_const,integral_const_mul,probReal_univ,one_smul] at h
+  change mean _ ≤ _+2*mean _ at h
+  rw [mean_inner_independent (fun t => (Real.log |t|)^2) e (pole n)] at h
+  linarith
+
+lemma seminorm_log_integral_tendsto {n : ℕ} [NeZero n]
+    {g : ℕ → Seminorm ℝ (Space n)} {f : Seminorm ℝ (Space n)} {r R : ℝ}
+    (hr : 0<r) (hlo : ∀ i,∃ u : Sphere n,r≤g i u)
+    (hhi : ∀ i (u : Sphere n),g i u≤R) (hf : f ≠ 0)
+    (hc : ∀ u : Sphere n,Tendsto (fun i => g i u) atTop (𝓝 (f u))) :
+    Tendsto (fun i => mean (fun u : Sphere n => Real.log (g i u))) atTop
+      (𝓝 (mean (fun u : Sphere n => Real.log (f u)))) := by
+  have hg (i : ℕ) : g i ≠ 0 := by
+    intro h
+    obtain ⟨u,hu⟩ := hlo i
+    rw [h] at hu
+    exact hr.not_ge hu
+  apply tendsto_integral_of_sq_bound (C := 2*(|Real.log r|+|Real.log R|)^2+
+    2*mean (fun u : Sphere n => (Real.log |⟪(pole n:Space n),(u:Space n)⟫|)^2))
+  · have hp : 0 ≤ mean (fun u : Sphere n => (Real.log |⟪(pole n:Space n),(u:Space n)⟫|)^2) :=
+      integral_nonneg (fun _ => sq_nonneg _)
+    positivity
+  · exact fun i => seminorm_log_integrable (hg i)
+  · exact fun i => seminorm_log_sq_integrable (hg i)
+  · exact fun i => seminorm_log_sq_bound hr (hlo i) (hhi i)
+  · exact seminorm_log_integrable hf
+  · filter_upwards [seminorm_pos_ae_sphere hf] with u hu
+    exact (Real.continuousAt_log hu.ne').tendsto.comp (hc u)
+
+end PettyProjection.Spherical
+end
+
+noncomputable section
+open Set Metric Filter Topology
+open scoped NNReal RealInnerProductSpace
+namespace PettyProjection.Spherical
+
+lemma seminorm_lipschitz_of_sphere_bound {n : ℕ} (g : Seminorm ℝ (Space n))
+    {R : ℝ≥0} (hR : ∀ u : Sphere n,g u≤R) : LipschitzWith R g := by
+  apply LipschitzWith.of_dist_le_mul
+  intro x y
+  change |g x-g y|≤(R:ℝ)*dist x y
+  exact (abs_sub_map_le_sub g x y).trans (by
+    simpa only [dist_eq_norm] using seminorm_le_max_norm g hR (x-y))
+
+def radialExtension {n : ℕ} (f : Sphere n → ℝ) (x : Space n) : ℝ :=
+  if hx : x=0 then 0 else ‖x‖*f ⟨‖x‖⁻¹ • x,by
+    simp only [mem_sphere,dist_zero_right,norm_smul,Real.norm_of_nonneg
+      (inv_nonneg.mpr (norm_nonneg _))]
+    exact inv_mul_cancel₀ (norm_ne_zero_iff.mpr hx)⟩
+
+lemma radialExtension_sphere {n : ℕ} (f : Sphere n → ℝ) (u : Sphere n) :
+    radialExtension f u=f u := by
+  have hu0 : (u:Space n)≠0 := by intro h; have := norm_coe u; simp [h] at this
+  simp only [radialExtension,dite_eq_right hu0,norm_coe,inv_one,one_smul,one_mul]
+
+lemma radialExtension_seminorm {n : ℕ} (g : Seminorm ℝ (Space n)) (x : Space n) :
+    radialExtension (fun u : Sphere n => g u) x=g x := by
+  by_cases hx : x=0
+  · simp [radialExtension,hx]
+  · simp only [radialExtension,dite_eq_right hx,map_smul_eq_mul,Real.norm_of_nonneg
+      (inv_nonneg.mpr (norm_nonneg _))]
+    rw [← mul_assoc,mul_inv_cancel₀ (norm_ne_zero_iff.mpr hx),one_mul]
+
+lemma radialExtension_tendsto {n : ℕ} {g : ℕ → Seminorm ℝ (Space n)}
+    {f : Sphere n → ℝ} (h : ∀ u : Sphere n,Tendsto (fun i => g i u) atTop (𝓝 (f u)))
+    (x : Space n) : Tendsto (fun i => g i x) atTop (𝓝 (radialExtension f x)) := by
+  have he (i : ℕ) : g i x=radialExtension (fun u : Sphere n => g i u) x :=
+    (radialExtension_seminorm (g i) x).symm
+  simp_rw [he]
+  unfold radialExtension
+  split_ifs with hx
+  · exact tendsto_const_nhds
+  · exact tendsto_const_nhds.mul (h _)
+
+/-- Uniform sphere compactness of bounded seminorms, including all singular
+limits. The limiting algebraic object is proved to be a seminorm. -/
+theorem bounded_seminorm_subsequence {n : ℕ} (g : ℕ → Seminorm ℝ (Space n))
+    {R : ℝ≥0} (hR : ∀ i (u : Sphere n),g i u≤R) :
+    ∃ f : Seminorm ℝ (Space n), ∃ φ : ℕ → ℕ, StrictMono φ ∧
+      TendstoUniformly (fun i (u : Sphere n) => g (φ i) u) (fun u => f u) atTop ∧
+      (∀ x,Tendsto (fun i => g (φ i) x) atTop (𝓝 (f x))) := by
+  let F : ℕ → (BoundedContinuousFunction (Sphere n) ℝ) := fun i =>
+    BoundedContinuousFunction.mkOfCompact ⟨fun u => g i u,
+      (seminorm_continuous n (g i)).comp continuous_subtype_val⟩
+  have hL (i : ℕ) : LipschitzWith R (F i) := by
+    apply LipschitzWith.of_dist_le_mul
+    intro x y
+    exact (seminorm_lipschitz_of_sphere_bound (g i) (hR i)).dist_le_mul (x:Space n) (y:Space n)
+  have hc : IsCompact (closure (range F)) := by
+    apply BoundedContinuousFunction.arzela_ascoli (Icc (0:ℝ) R) isCompact_Icc
+    · intro f u hf
+      obtain ⟨i,rfl⟩ := hf
+      exact ⟨apply_nonneg (g i) (u:Space n),hR i u⟩
+    · apply UniformEquicontinuous.equicontinuous
+      apply LipschitzWith.uniformEquicontinuous _ R
+      intro f
+      obtain ⟨i,hi⟩ := f.property
+      simpa only [hi] using hL i
+  obtain ⟨f,_,φ,hφ,hconv⟩ := hc.tendsto_subseq (fun i => subset_closure (mem_range_self i))
+  have hunif : TendstoUniformly (fun i => (F (φ i) : Sphere n → ℝ)) f atTop :=
+    BoundedContinuousFunction.tendsto_iff_tendstoUniformly.mp hconv
+  have hpoint := radialExtension_tendsto (g := fun i => g (φ i)) hunif.tendsto_at
+  let q : Seminorm ℝ (Space n) := Seminorm.of (radialExtension f)
+    (fun x y => le_of_tendsto_of_tendsto (hpoint (x+y)) ((hpoint x).add (hpoint y))
+      (Eventually.of_forall (fun i => map_add_le_add (g (φ i)) x y)))
+    (fun a x => tendsto_nhds_unique (hpoint (a • x)) (by
+      simp_rw [map_smul_eq_mul]
+      exact tendsto_const_nhds.mul (hpoint x)))
+  refine ⟨q,φ,hφ,?_,hpoint⟩
+  have he : (fun u : Sphere n => q u)=(f : Sphere n → ℝ) := by
+    funext u; exact radialExtension_sphere f u
+  rw [he]
+  exact hunif
+
+end PettyProjection.Spherical
+end
+
+noncomputable section
+open Set Metric Filter Topology
+open scoped NNReal RealInnerProductSpace Pointwise
+namespace PettyProjection
+open Spherical (Sphere seminorm_continuous)
+
+/-- Closed unit set of the actual seminorm. It is allowed to be a cylinder. -/
+def seminormUnit {n : ℕ} (g : Seminorm ℝ (Space n)) : Set (Space n) := {x | g x≤1}
+
+lemma seminormUnit_zero {n : ℕ} (g : Seminorm ℝ (Space n)) : 0∈seminormUnit g := by
+  simp [seminormUnit]
+
+lemma seminormUnit_closed {n : ℕ} (g : Seminorm ℝ (Space n)) : IsClosed (seminormUnit g) :=
+  isClosed_le (seminorm_continuous n g) continuous_const
+
+lemma seminormUnit_convex {n : ℕ} (g : Seminorm ℝ (Space n)) : Convex ℝ (seminormUnit g) := by
+  simpa only [seminormUnit,← g.closedBall_zero_eq] using g.convex_closedBall 0 1
+
+lemma seminormUnit_balanced {n : ℕ} (g : Seminorm ℝ (Space n)) : Balanced ℝ (seminormUnit g) := by
+  simpa only [seminormUnit,← g.closedBall_zero_eq] using g.balanced_closedBall_zero 1
+
+lemma seminormUnit_nhds {n : ℕ} (g : Seminorm ℝ (Space n)) : seminormUnit g∈𝓝 (0:Space n) := by
+  have ho : IsOpen {x : Space n | g x<1} := isOpen_lt (seminorm_continuous n g) continuous_const
+  exact Filter.mem_of_superset (ho.mem_nhds (by simp)) (fun x hx => show g x≤1 from hx.le)
+
+lemma gauge_seminormUnit {n : ℕ} (g : Seminorm ℝ (Space n)) (x : Space n) :
+    gauge (seminormUnit g) x=g x := by
+  apply le_antisymm
+  · apply le_of_forall_pos_le_add
+    intro ε hε
+    have hp : 0<g x+ε := add_pos_of_nonneg_of_pos (apply_nonneg g x) hε
+    apply gauge_le_of_mem hp.le
+    refine ⟨(g x+ε)⁻¹ • x,?_,smul_inv_smul₀ hp.ne' x⟩
+    change g ((g x+ε)⁻¹ • x)≤1
+    rw [map_smul_eq_mul,Real.norm_of_nonneg (inv_nonneg.mpr hp.le),inv_mul_le_iff₀ hp]
+    linarith
+  · unfold gauge
+    apply le_csInf (absorbent_nhds_zero (seminormUnit_nhds g)).gauge_set_nonempty
+    rintro r ⟨hr,y,hy,rfl⟩
+    rw [map_smul_eq_mul,Real.norm_of_nonneg hr.le]
+    exact mul_le_of_le_one_right hr.le hy
+
+lemma seminormUnit_compact {n : ℕ} (g : Seminorm ℝ (Space n)) {c : ℝ}
+    (hc : 0<c) (hbound : ∀ x,c*‖x‖≤g x) : IsCompact (seminormUnit g) := by
+  apply (isCompact_closedBall (0:Space n) c⁻¹).of_isClosed_subset (seminormUnit_closed g)
+  intro x hx
+  rw [mem_closedBall,dist_zero_right,inv_eq_one_div,le_div_iff₀ hc]
+  change g x≤1 at hx
+  nlinarith [hbound x]
+
+lemma seminorm_pos_uniform {n : ℕ} [NeZero n] (g : Seminorm ℝ (Space n))
+    (hg : ∀ u : Sphere n,0<g u) : ∃ c : ℝ, 0<c ∧ ∀ x,c*‖x‖≤g x := by
+  let : Nonempty (Sphere n) := ⟨Spherical.pole n⟩
+  obtain ⟨u,_,hu⟩ := isCompact_univ.exists_isMinOn (Set.univ_nonempty : (univ : Set (Sphere n)).Nonempty)
+    ((seminorm_continuous n g).comp continuous_subtype_val).continuousOn
+  refine ⟨g u,hg u,fun x => ?_⟩
+  by_cases hx : x=0
+  · simp [hx]
+  · let v : Sphere n := ⟨‖x‖⁻¹ • x,by simp [norm_smul,norm_ne_zero_iff.mpr hx]⟩
+    have hh : g u≤g v := hu (mem_univ v)
+    change g u≤g (‖x‖⁻¹ • x) at hh
+    rw [map_smul_eq_mul,Real.norm_of_nonneg (inv_nonneg.mpr (norm_nonneg _))] at hh
+    have hp : 0<‖x‖ := norm_pos_iff.mpr hx
+    have hh' := mul_le_mul_of_nonneg_left hh hp.le
+    rw [← mul_assoc,mul_inv_cancel₀ hp.ne',one_mul] at hh'
+    simpa only [mul_comm] using hh'
+
+end PettyProjection
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology
+open scoped NNReal ENNReal RealInnerProductSpace Pointwise
+namespace PettyProjection
+open Spherical (Sphere)
+
+lemma integrable_abs_inner {n : ℕ} {μ : Measure (Space n)}
+    (hμ : Integrable (fun x : Space n => ‖x‖) μ) (y : Space n) :
+    Integrable (fun x => |⟪x,y⟫|) μ := by
+  apply (hμ.mul_const ‖y‖).mono' (by fun_prop)
+  exact Eventually.of_forall (fun x => by simpa only [Real.norm_eq_abs,abs_abs] using abs_real_inner_le_norm x y)
+
+/-- The actual cosine-transform norm furnished by a spanning measure. -/
+def cosineSeminorm {n : ℕ} (μ : Measure (Space n))
+    (hμ : Integrable (fun x : Space n => ‖x‖) μ) : Seminorm ℝ (Space n) :=
+  Seminorm.of (fun y => ∫ x,|⟪x,y⟫| ∂μ)
+    (fun y z => by
+      rw [← integral_add (integrable_abs_inner hμ y) (integrable_abs_inner hμ z)]
+      apply integral_mono (integrable_abs_inner hμ (y+z))
+        ((integrable_abs_inner hμ y).add (integrable_abs_inner hμ z))
+      intro x
+      change |⟪x,y+z⟫|≤ |⟪x,y⟫|+|⟪x,z⟫|
+      rw [inner_add_right]
+      exact abs_add_le _ _)
+    (fun a y => by
+      simp only [inner_smul_right,abs_mul,Real.norm_eq_abs,integral_const_mul])
+
+lemma support_integrable {n : ℕ} {μ : Measure (Space n)}
+    (hμ : Integrable (fun x : Space n => ‖x‖) μ)
+    {K : Set (Space n)} (hK : IsCompact K) (hne : K.Nonempty) :
+    Integrable (support K) μ := by
+  obtain ⟨R,hR,hbound⟩ := hK.isBounded.exists_pos_norm_le
+  apply (hμ.const_mul R).mono' (support_continuous hK).aestronglyMeasurable
+  refine Eventually.of_forall (fun x => ?_)
+  obtain ⟨y,hy,he⟩ := support_attained hK hne x
+  rw [Real.norm_eq_abs,he]
+  exact (abs_real_inner_le_norm x y).trans (by nlinarith [norm_nonneg x,hbound y hy])
+
+lemma abs_inner_le_support {n : ℕ} {K : Set (Space n)} (hK : IsCompact K)
+    (hsym : ∀ y∈K,-y∈K) {y : Space n} (hy : y∈K) (x : Space n) :
+    |⟪x,y⟫|≤ support K x := by
+  rw [abs_le]
+  constructor
+  · have hh := inner_le_support hK (hsym y hy) x
+    rw [inner_neg_right] at hh
+    linarith
+  · exact inner_le_support hK hy x
+
+lemma cosine_le_functional {n : ℕ} {μ : Measure (Space n)}
+    (hμ : Integrable (fun x : Space n => ‖x‖) μ)
+    {K : Set (Space n)} (hK : IsCompact K) (hsym : ∀ y∈K,-y∈K)
+    {y : Space n} (hy : y∈K) : cosineSeminorm μ hμ y≤ supportFunctional μ K := by
+  exact integral_mono (integrable_abs_inner hμ y) (support_integrable hμ hK ⟨y,hy⟩)
+    (abs_inner_le_support hK hsym hy)
+
+lemma functional_smul {n : ℕ} {μ : Measure (Space n)}
+    {K : Set (Space n)} (hK : IsCompact K) (hne : K.Nonempty) {a : ℝ} (ha : 0≤a) :
+    supportFunctional μ (a • K)=a*supportFunctional μ K := by
+  unfold supportFunctional
+  simp_rw [support_smul_set hK hne ha]
+  exact integral_const_mul _ _
+
+lemma functional_add {n : ℕ} {μ : Measure (Space n)}
+    (hμ : Integrable (fun x : Space n => ‖x‖) μ)
+    {K L : Set (Space n)} (hK : IsCompact K) (hKn : K.Nonempty)
+    (hL : IsCompact L) (hLn : L.Nonempty) :
+    supportFunctional μ (K+L)=supportFunctional μ K+supportFunctional μ L := by
+  unfold supportFunctional
+  simp_rw [support_minkowski hK hKn hL hLn]
+  exact integral_add (support_integrable hμ hK hKn) (support_integrable hμ hL hLn)
+
+lemma functional_nonneg {n : ℕ} {μ : Measure (Space n)}
+    {K : Set (Space n)} (hK : IsCompact K) (hK0 : 0∈K) : 0≤ supportFunctional μ K :=
+  integral_nonneg (fun x => support_nonneg hK hK0 x)
+
+end PettyProjection
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/ProjectionBodies/ShadowSlices.lean b/lean/OAI/Geometry/ProjectionBodies/ShadowSlices.lean
new file mode 100644
index 0000000000000000000000000000000000000000..cdb038e7a916d44ffb899725cd525239617e6cda
--- /dev/null
+++ b/lean/OAI/Geometry/ProjectionBodies/ShadowSlices.lean
@@ -0,0 +1,292 @@
+import OAI.Geometry.ProjectionBodies.WulffVariation
+
+namespace OAI
+
+universe uE
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology Function
+open scoped ENNReal NNReal RealInnerProductSpace Pointwise
+namespace PettyProjection
+
+lemma compact_convex_real_interval {C : Set ℝ} (hC : IsCompact C) (hc : Convex ℝ C)
+    (hn : C.Nonempty) : C = Icc (sInf C) (sSup C) := by
+  apply Subset.antisymm
+  · exact fun x hx => ⟨csInf_le hC.bddBelow hx,le_csSup hC.bddAbove hx⟩
+  · exact (convex_iff_ordConnected.mp hc).out (hC.sInf_mem hn) (hC.sSup_mem hn)
+
+lemma compact_convex_real_add_interval_volume {C : Set ℝ} (hC : IsCompact C)
+    (hc : Convex ℝ C) (hn : C.Nonempty) {t : ℝ} (ht : 0 ≤ t) :
+    volume (C+Icc (-t) t) = volume C+ENNReal.ofReal (2*t) := by
+  rw [compact_convex_real_interval hC hc hn]
+  have hab : sInf C ≤ sSup C := (hC.isGLB_sInf hn).1 (hC.sSup_mem hn)
+  rw [Icc_add_Icc hab (by linarith),Real.volume_Icc,Real.volume_Icc,
+    ← ENNReal.ofReal_add (sub_nonneg.mpr hab) (by positivity)]
+  congr 1
+  ring
+
+section Product
+variable {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E]
+    [MeasureSpace E] [BorelSpace E]
+
+omit [NormedSpace ℝ E] [MeasureSpace E] [BorelSpace E] in
+lemma compact_fiber {K : Set (E × ℝ)} (hK : IsCompact K) (x : E) :
+    IsCompact (Prod.mk x ⁻¹' K) :=
+  (IsClosedEmbedding.of_isEmbedding_isClosedMap (isEmbedding_prodMkRight x) (isClosedMap_prodMk_left x)).isCompact_preimage hK
+
+omit [MeasureSpace E] [BorelSpace E] in
+lemma convex_fiber {K : Set (E × ℝ)} (hc : Convex ℝ K) (x : E) :
+    Convex ℝ (Prod.mk x ⁻¹' K) := by
+  intro a ha b hb s t hs ht hst
+  have h := hc ha hb hs ht hst
+  change (x,s • a+t • b) ∈ K
+  simpa only [Prod.smul_mk,Prod.mk_add_mk,← add_smul,hst,one_smul] using h
+
+omit [NormedSpace ℝ E] [MeasureSpace E] [BorelSpace E] in
+lemma fiber_vertical_add (K : Set (E × ℝ)) (I : Set ℝ) (x : E) :
+    Prod.mk x ⁻¹' (K+(Prod.mk (0:E) '' I)) = (Prod.mk x ⁻¹' K)+I := by
+  ext r
+  constructor
+  · rintro ⟨z,hz,_,⟨a,ha,rfl⟩,he⟩
+    have hx : z.1 = x := by simpa only [Prod.fst_add,add_zero] using congrArg Prod.fst he
+    refine ⟨z.2,?_,a,ha,?_⟩
+    · change (x,z.2) ∈ K
+      simpa only [← hx,Prod.mk.eta] using hz
+    · exact congrArg Prod.snd he
+  · rintro ⟨a,ha,b,hb,rfl⟩
+    exact ⟨(x,a),ha,(0,b),⟨b,hb,rfl⟩,by simp only [Prod.mk_add_mk,add_zero]⟩
+
+omit [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasureSpace E] [BorelSpace E] in
+lemma fiber_nonempty_iff (K : Set (E × ℝ)) (x : E) :
+    (Prod.mk x ⁻¹' K).Nonempty ↔ x ∈ Prod.fst '' K := by
+  constructor
+  · rintro ⟨r,hr⟩; exact ⟨(x,r),hr,rfl⟩
+  · rintro ⟨⟨y,r⟩,hr,rfl⟩; exact ⟨r,hr⟩
+
+/-- The actual Fubini segment-addition formula in orthogonal product coordinates. -/
+theorem vertical_segment_volume {K : Set (E × ℝ)} (hK : IsCompact K)
+    (hc : Convex ℝ K) {t : ℝ} (ht : 0 ≤ t) :
+    volume (K+Prod.mk (0:E) '' Icc (-t) t) =
+      volume K+ENNReal.ofReal (2*t)*volume (Prod.fst '' K) := by
+  have hC : IsCompact (K+Prod.mk (0:E) '' Icc (-t) t) :=
+    hK.add (isCompact_Icc.image (by fun_prop))
+  have hP : MeasurableSet (Prod.fst '' K) := (hK.image continuous_fst).measurableSet
+  rw [Measure.volume_eq_prod,Measure.prod_apply hC.measurableSet,
+    Measure.prod_apply hK.measurableSet]
+  have he : ∀ x : E, volume (Prod.mk x ⁻¹' (K+Prod.mk (0:E) '' Icc (-t) t)) =
+      volume (Prod.mk x ⁻¹' K)+(Prod.fst '' K).indicator (fun _ => ENNReal.ofReal (2*t)) x := by
+    intro x
+    rw [fiber_vertical_add]
+    by_cases hn : (Prod.mk x ⁻¹' K).Nonempty
+    · rw [indicator_of_mem ((fiber_nonempty_iff K x).mp hn)]
+      exact compact_convex_real_add_interval_volume (compact_fiber hK x) (convex_fiber hc x) hn ht
+    · rw [not_nonempty_iff_eq_empty.mp hn,empty_add,measure_empty,
+        indicator_of_notMem (fun h => hn ((fiber_nonempty_iff K x).mpr h)),add_zero]
+  simp_rw [he]
+  rw [lintegral_add_left (measurable_measure_prodMk_left hK.measurableSet),
+    lintegral_indicator hP,lintegral_const,Measure.restrict_apply_univ]
+
+end Product
+end PettyProjection
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology Function
+open scoped ENNReal NNReal RealInnerProductSpace Pointwise
+namespace PettyProjection
+open Spherical (Sphere norm_coe)
+
+lemma perpendicular_inner {n : ℕ} (u : Space n) (z : perpendicular u) : ⟪u,(z : Space n)⟫ = 0 :=
+  Submodule.mem_orthogonal_singleton_iff_inner_right.mp z.property
+
+/-- Orthogonal coordinates with the last coordinate in a specified unit direction. -/
+def shadowCoordLinear {n : ℕ} (u : Sphere n) :
+    WithLp 2 (perpendicular (u : Space n) × ℝ) →ₗ[ℝ] Space n where
+  toFun z := (WithLp.ofLp z).1+(WithLp.ofLp z).2 • (u : Space n)
+  map_add' z w := by
+    simp only [WithLp.ofLp_add,Prod.fst_add,Prod.snd_add,Submodule.coe_add,add_smul]
+    abel
+  map_smul' a z := by
+    simp only [WithLp.ofLp_smul,Prod.smul_fst,Prod.smul_snd,Submodule.coe_smul_of_tower,
+      smul_eq_mul,smul_add,smul_smul,RingHom.id_apply]
+
+lemma shadowCoordLinear_inner {n : ℕ} (u : Sphere n)
+    (z w : WithLp 2 (perpendicular (u : Space n) × ℝ)) :
+    ⟪shadowCoordLinear u z,shadowCoordLinear u w⟫ = ⟪z,w⟫ := by
+  change ⟪((WithLp.ofLp z).1 : Space n)+(WithLp.ofLp z).2 • (u : Space n),
+    ((WithLp.ofLp w).1 : Space n)+(WithLp.ofLp w).2 • (u : Space n)⟫ = _
+  simp only [inner_add_left,inner_add_right,real_inner_smul_left,real_inner_smul_right]
+  rw [real_inner_self_eq_norm_sq,norm_coe,one_pow,mul_one,
+    perpendicular_inner,Submodule.mem_orthogonal_singleton_iff_inner_left.mp (WithLp.ofLp z).1.property]
+  simp only [mul_zero,add_zero,zero_add,WithLp.prod_inner_apply,Submodule.coe_inner,Real.inner_apply]
+  ring
+
+lemma shadowCoordLinear_surjective {n : ℕ} (u : Sphere n) : Surjective (shadowCoordLinear u) := by
+  intro x
+  have hz : x-⟪(u : Space n),x⟫ • (u : Space n) ∈ perpendicular (u : Space n) := by
+    apply Submodule.mem_orthogonal_singleton_iff_inner_right.mpr
+    rw [inner_sub_right,real_inner_smul_right,real_inner_self_eq_norm_sq,norm_coe,one_pow,mul_one,sub_self]
+  refine ⟨WithLp.toLp 2 (⟨_,hz⟩,⟪(u : Space n),x⟫),?_⟩
+  change x-_+_ = x
+  exact sub_add_cancel _ _
+
+def shadowCoords {n : ℕ} (u : Sphere n) :
+    WithLp 2 (perpendicular (u : Space n) × ℝ) ≃ₗᵢ[ℝ] Space n :=
+  LinearIsometryEquiv.ofSurjective
+    (LinearMap.isometryOfInner (shadowCoordLinear u) (shadowCoordLinear_inner u))
+    (shadowCoordLinear_surjective u)
+
+lemma shadowCoords_apply {n : ℕ} (u : Sphere n) (z : perpendicular (u : Space n)) (t : ℝ) :
+    shadowCoords u (WithLp.toLp 2 (z,t)) = z+t • (u : Space n) := rfl
+
+lemma shadowCoords_projection {n : ℕ} (u : Sphere n)
+    (z : perpendicular (u : Space n)) (t : ℝ) :
+    (perpendicular (u : Space n)).orthogonalProjectionOnto
+      (shadowCoords u (WithLp.toLp 2 (z,t))) = z := by
+  rw [shadowCoords_apply,map_add,map_smul,Submodule.orthogonalProjectionOnto_mem_subspace_eq_self]
+  have hz : (perpendicular (u : Space n)).orthogonalProjectionOnto (u : Space n) = 0 := by
+    exact Submodule.orthogonalProjectionOnto_orthogonalComplement_singleton_eq_zero (u : Space n)
+  rw [hz,smul_zero,add_zero]
+
+/-- The coordinate change preserves the actual standard Euclidean measures. -/
+def shadowMeasurableEquiv {n : ℕ} (u : Sphere n) :
+    (perpendicular (u : Space n) × ℝ) ≃ᵐ Space n :=
+  (MeasurableEquiv.toLp 2 _).trans (shadowCoords u).toMeasurableEquiv
+
+lemma shadowMeasurableEquiv_measurePreserving {n : ℕ} (u : Sphere n) :
+    MeasurePreserving (shadowMeasurableEquiv u) :=
+  (WithLp.volume_preserving_toLp _ _).trans (shadowCoords u).measurePreserving
+
+end PettyProjection
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology Function
+open scoped ENNReal NNReal RealInnerProductSpace Pointwise Gradient
+namespace PettyProjection
+open Spherical (Sphere norm_coe sigma mean)
+
+def unitSegment {n : ℕ} (y : Space n) : Set (Space n) :=
+  (fun a : ℝ => a • y) '' Icc (-1) 1
+
+lemma unitSegment_compact {n : ℕ} (y : Space n) : IsCompact (unitSegment y) :=
+  isCompact_Icc.image (by fun_prop)
+
+lemma unitSegment_nonempty {n : ℕ} (y : Space n) : (unitSegment y).Nonempty :=
+  ⟨0,⟨0,by norm_num,zero_smul ℝ y⟩⟩
+
+lemma unitSegment_convex {n : ℕ} (y : Space n) : Convex ℝ (unitSegment y) :=
+  (convex_Icc (-1:ℝ) 1).linear_image (LinearMap.toSpanSingleton ℝ (Space n) y)
+
+lemma unitSegment_support {n : ℕ} (y x : Space n) : support (unitSegment y) x = |⟪x,y⟫| := by
+  apply le_antisymm
+  · apply (support_le_iff (unitSegment_compact y) (unitSegment_nonempty y)).mpr
+    rintro _ ⟨a,ha,rfl⟩
+    rw [real_inner_smul_right]
+    calc
+      _ ≤ |a*⟪x,y⟫| := le_abs_self _
+      _ = |a| * |⟪x,y⟫| := abs_mul _ _
+      _ ≤ 1 * |⟪x,y⟫| := mul_le_mul_of_nonneg_right (abs_le.mpr ha) (abs_nonneg _)
+      _ = _ := one_mul _
+  · rw [abs_le]
+    constructor
+    · have h := inner_le_support (unitSegment_compact y)
+        (mem_image_of_mem (fun a : ℝ => a • y) (left_mem_Icc.mpr (by norm_num : (-1:ℝ) ≤ 1))) x
+      simp only [real_inner_smul_right,neg_one_mul] at h
+      linarith
+    · have h := inner_le_support (unitSegment_compact y)
+        (mem_image_of_mem (fun a : ℝ => a • y) (right_mem_Icc.mpr (by norm_num : (-1:ℝ) ≤ 1))) x
+      simpa only [real_inner_smul_right,one_mul] using h
+
+lemma smul_unitSegment {n : ℕ} (y : Space n) {t : ℝ} (ht : 0 ≤ t) :
+    t • unitSegment y = (fun a : ℝ => a • y) '' Icc (-t) t := by
+  have he : Icc (-t) t = (fun a : ℝ => t*a) '' Icc (-1) 1 := by
+    simpa only [mul_neg,mul_one] using (image_mul_left_Icc ht (by norm_num : (-1:ℝ) ≤ 1)).symm
+  rw [he,image_image,unitSegment,← image_smul,image_image]
+  congr 1
+  funext a
+  exact smul_smul t a y
+
+def shadowContinuousEquiv {n : ℕ} (u : Sphere n) :
+    (perpendicular (u : Space n) × ℝ) ≃L[ℝ] Space n :=
+  (WithLp.prodContinuousLinearEquiv 2 ℝ (perpendicular (u : Space n)) ℝ).symm.trans
+    (shadowCoords u).toContinuousLinearEquiv
+
+lemma shadowContinuousEquiv_apply {n : ℕ} (u : Sphere n) (z : perpendicular (u : Space n) × ℝ) :
+    shadowContinuousEquiv u z = z.1+z.2 • (u : Space n) := rfl
+
+lemma shadow_preimage_projection {n : ℕ} (u : Sphere n) (K : Set (Space n)) :
+    Prod.fst '' (shadowContinuousEquiv u ⁻¹' K) =
+      (perpendicular (u : Space n)).orthogonalProjectionOnto '' K := by
+  ext z
+  constructor
+  · rintro ⟨⟨w,t⟩,hw,rfl⟩
+    exact ⟨shadowContinuousEquiv u (w,t),hw,shadowCoords_projection u w t⟩
+  · rintro ⟨x,hx,rfl⟩
+    let q := (shadowContinuousEquiv u).symm x
+    have hq : shadowContinuousEquiv u q = x := (shadowContinuousEquiv u).apply_symm_apply x
+    refine ⟨q,?_,?_⟩
+    · change shadowContinuousEquiv u q ∈ K
+      rwa [hq]
+    · rw [← hq]
+      exact (shadowCoords_projection u q.1 q.2).symm
+
+lemma shadow_preimage_add_segment {n : ℕ} (u : Sphere n) (K : Set (Space n))
+    {t : ℝ} (ht : 0 ≤ t) :
+    shadowContinuousEquiv u ⁻¹' (K+t • unitSegment (u : Space n)) =
+      (shadowContinuousEquiv u ⁻¹' K) + Prod.mk 0 '' Icc (-t) t := by
+  rw [smul_unitSegment _ ht]
+  ext z
+  constructor
+  · rintro ⟨x,hx,_,⟨a,ha,rfl⟩,he⟩
+    refine ⟨(shadowContinuousEquiv u).symm x,by simpa using hx,(0,a),⟨a,ha,rfl⟩,?_⟩
+    apply (shadowContinuousEquiv u).injective
+    rw [map_add,ContinuousLinearEquiv.apply_symm_apply,shadowContinuousEquiv_apply,Submodule.coe_zero,zero_add]
+    exact he
+  · rintro ⟨x,hx,_,⟨a,ha,rfl⟩,rfl⟩
+    change shadowContinuousEquiv u (x+(0,a)) ∈ K+(fun a : ℝ => a • (u : Space n)) '' Icc (-t) t
+    rw [map_add,shadowContinuousEquiv_apply u (0,a),Submodule.coe_zero,zero_add]
+    exact add_mem_add hx (mem_image_of_mem _ ha)
+
+/-- Adding a centered segment increases volume by its length times the actual
+orthogonal shadow volume. No boundary regularity is used. -/
+theorem segment_volume {n : ℕ} (u : Sphere n) {K : Set (Space n)}
+    (hK : IsCompact K) (hc : Convex ℝ K) {t : ℝ} (ht : 0 ≤ t) :
+    volume.real (K+t • unitSegment (u : Space n)) =
+      volume.real K+2*t*shadowVolume K u := by
+  let e := shadowContinuousEquiv u
+  have hC : IsCompact (e ⁻¹' K) := e.toHomeomorph.isCompact_preimage.mpr hK
+  have hcv : Convex ℝ (e ⁻¹' K) := hc.linear_preimage e.toLinearMap
+  have h := vertical_segment_volume hC hcv ht
+  rw [← shadow_preimage_add_segment u K ht,shadow_preimage_projection] at h
+  have hm := shadowMeasurableEquiv_measurePreserving u
+  have hm' (B : Set (Space n)) (hB : MeasurableSet B) : volume (e ⁻¹' B) = volume B :=
+    hm.measure_preimage hB.nullMeasurableSet
+  rw [hm' _ (hK.add ((unitSegment_compact _).smul t)).measurableSet,hm' _ hK.measurableSet] at h
+  have hr := congrArg ENNReal.toReal h
+  rw [ENNReal.toReal_add hK.measure_ne_top (ENNReal.mul_ne_top ENNReal.ofReal_ne_top
+    (hK.image (perpendicular (u : Space n)).orthogonalProjectionOnto.continuous).measure_ne_top),
+    ENNReal.toReal_mul,ENNReal.toReal_ofReal (by positivity)] at hr
+  exact hr
+
+/-- Gauge representation of actual orthogonal shadow volume. -/
+theorem shadow_gauge_formula {n : ℕ} [NeZero n] {K : Set (Space n)}
+    (hK : IsCompact K) (hc : Convex ℝ K) (h0 : K ∈ 𝓝 (0 : Space n)) (u : Sphere n) :
+    2*shadowVolume K u = n*kappa n*mean (fun v : Sphere n =>
+      (gauge K (v : Space n))⁻¹^n*|⟪∇ (gauge K) (v : Space n),(u : Space n)⟫|) := by
+  have hd := minkowski_volume_derivative hK hc h0 (unitSegment_compact (u : Space n))
+    (unitSegment_convex (u : Space n)) (unitSegment_nonempty (u : Space n))
+  simp_rw [unitSegment_support] at hd
+  have hlin : HasDerivAt (fun t : ℝ => volume.real K+2*t*shadowVolume K u)
+      (2*shadowVolume K u) 0 := by
+    apply (((hasDerivAt_id (0:ℝ)).const_mul (2:ℝ)).mul_const (shadowVolume K u)).const_add (volume.real K) |>.congr_deriv
+    ring
+  have hd' := hlin.hasDerivWithinAt (s := Ici (0:ℝ)) |>.congr
+    (fun t ht => segment_volume u hK hc ht) (segment_volume u hK hc le_rfl)
+  exact (hd'.derivWithin (uniqueDiffWithinAt_Ici (0:ℝ))).symm.trans
+    (hd.derivWithin (uniqueDiffWithinAt_Ici (0:ℝ)))
+
+end PettyProjection
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/ProjectionBodies/SignCalculus.lean b/lean/OAI/Geometry/ProjectionBodies/SignCalculus.lean
new file mode 100644
index 0000000000000000000000000000000000000000..e6f5f0770df85f2b249238da2c9527277dc6e3e2
--- /dev/null
+++ b/lean/OAI/Geometry/ProjectionBodies/SignCalculus.lean
@@ -0,0 +1,429 @@
+import OAI.Geometry.ProjectionBodies.SupportMeasures
+
+namespace OAI
+
+noncomputable section
+open Set Filter Topology
+namespace PettyProjection.Spherical
+
+def softAbs (ε t : ℝ) : ℝ := Real.sqrt (t^2+ε)
+def softSign (ε t : ℝ) : ℝ := t/softAbs ε t
+def softSlope (ε t : ℝ) : ℝ := ε/(softAbs ε t)^3
+
+lemma softAbs_pos {ε : ℝ} (hε : 0 < ε) (t : ℝ) : 0 < softAbs ε t :=
+  Real.sqrt_pos.mpr (by positivity)
+
+lemma softAbs_sq {ε : ℝ} (hε : 0 < ε) (t : ℝ) : (softAbs ε t)^2 = t^2+ε :=
+  Real.sq_sqrt (by positivity)
+
+lemma softAbs_contDiff {ε : ℝ} (hε : 0 < ε) : ContDiff ℝ ⊤ (softAbs ε) := by
+  exact ((contDiff_id.pow 2).add contDiff_const).sqrt (fun t => ne_of_gt (by positivity))
+
+lemma softSign_contDiff {ε : ℝ} (hε : 0 < ε) : ContDiff ℝ ⊤ (softSign ε) :=
+  contDiff_id.div (softAbs_contDiff hε) (fun t => (softAbs_pos hε t).ne')
+
+lemma softAbs_hasDerivAt {ε : ℝ} (hε : 0 < ε) (t : ℝ) :
+    HasDerivAt (softAbs ε) (softSign ε t) t := by
+  have h := ((hasDerivAt_pow 2 t).add_const ε).sqrt (by positivity : t^2+ε ≠ 0)
+  convert! h using 1
+  simp only [softSign, softAbs, Nat.cast_ofNat]
+  ring
+
+lemma softSign_hasDerivAt {ε : ℝ} (hε : 0 < ε) (t : ℝ) :
+    HasDerivAt (softSign ε) (softSlope ε t) t := by
+  have h := (hasDerivAt_id t).div (softAbs_hasDerivAt hε t) (softAbs_pos hε t).ne'
+  convert! h using 1
+  dsimp [softSlope, softSign]
+  field_simp [(softAbs_pos hε t).ne']
+  nlinarith [softAbs_sq hε t]
+
+lemma abs_le_softAbs {ε : ℝ} (hε : 0 < ε) (t : ℝ) : |t| ≤ softAbs ε t := by
+  have hs := softAbs_sq hε t
+  have hp := (softAbs_pos hε t).le
+  nlinarith [sq_abs t, abs_nonneg t]
+
+lemma softAbs_le {ε : ℝ} (hε : 0 < ε) (t : ℝ) : softAbs ε t ≤ |t|+Real.sqrt ε := by
+  have hs := softAbs_sq hε t
+  have hp := (softAbs_pos hε t).le
+  have he := Real.sq_sqrt hε.le
+  have hn := mul_nonneg (abs_nonneg t) (Real.sqrt_nonneg ε)
+  nlinarith [sq_abs t, abs_nonneg t, Real.sqrt_nonneg ε]
+
+lemma abs_softSign_le {ε : ℝ} (hε : 0 < ε) (t : ℝ) : |softSign ε t| ≤ 1 := by
+  rw [softSign, abs_div, abs_of_pos (softAbs_pos hε t), div_le_one (softAbs_pos hε t)]
+  exact abs_le_softAbs hε t
+
+lemma abs_softCorrection_le {ε : ℝ} (hε : 0 < ε) (t : ℝ) : |t*softSlope ε t| ≤ 1 := by
+  have hp := softAbs_pos hε t
+  have hs : ε ≤ (softAbs ε t)^2 := by nlinarith [softAbs_sq hε t, sq_nonneg t]
+  have hm := mul_le_mul (abs_le_softAbs hε t) hs hε.le hp.le
+  rw [softSlope, ← mul_div_assoc, abs_div, abs_of_pos (pow_pos hp _),
+    abs_mul, abs_of_pos hε, div_le_one (pow_pos hp _)]
+  nlinarith
+
+lemma real_sign_eq (t : ℝ) : Real.sign t = t/|t| := by
+  rcases lt_trichotomy t 0 with ht | rfl | ht
+  · rw [Real.sign_of_neg ht, abs_of_neg ht]
+    field_simp [ht.ne]
+  · simp
+  · rw [Real.sign_of_pos ht, abs_of_pos ht, div_self ht.ne']
+
+lemma softAbs_tendsto (t : ℝ) :
+    Tendsto (fun m : ℕ => softAbs (1/((m : ℝ)+1)) t) atTop (𝓝 |t|) := by
+  have h : Tendsto (fun m : ℕ => t^2 + 1/((m : ℝ)+1)) atTop (𝓝 (t^2+0)) :=
+    tendsto_const_nhds.add tendsto_one_div_add_atTop_nhds_zero_nat
+  simp only [add_zero] at h
+  simpa only [softAbs, Function.comp_def, Real.sqrt_sq_eq_abs] using
+    (Real.continuous_sqrt.tendsto (t^2)).comp h
+
+lemma softSign_tendsto (t : ℝ) :
+    Tendsto (fun m : ℕ => softSign (1/((m : ℝ)+1)) t) atTop (𝓝 (Real.sign t)) := by
+  by_cases ht : t = 0
+  · subst t
+    simpa only [softSign, zero_div, Real.sign_zero] using tendsto_const_nhds
+  · rw [real_sign_eq]
+    exact tendsto_const_nhds.div (softAbs_tendsto t) (abs_ne_zero.mpr ht)
+
+lemma softCorrection_tendsto (t : ℝ) :
+    Tendsto (fun m : ℕ => t*softSlope (1/((m : ℝ)+1)) t) atTop (𝓝 0) := by
+  by_cases ht : t = 0
+  · subst t
+    simp only [zero_mul]
+    exact tendsto_const_nhds
+  · have h := (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).div
+      ((softAbs_tendsto t).pow 3) (pow_ne_zero 3 (abs_ne_zero.mpr ht))
+    simpa only [softSlope, Pi.div_apply, zero_div, mul_zero] using h.const_mul t
+
+end PettyProjection.Spherical
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology Function
+open scoped ENNReal RealInnerProductSpace Gradient
+namespace PettyProjection.Spherical
+
+lemma casimir_linear {n : ℕ} (w x : Space n) :
+    casimir (fun y => ⟪w,y⟫) x = -((n : ℝ)-1)*⟪w,x⟫ := by
+  simpa only [id_eq, zero_mul, mul_one, zero_sub, neg_mul] using
+    casimir_scalar_inner w x (fun t => hasDerivAt_id t)
+      (fun t => hasDerivAt_const t (1 : ℝ))
+
+lemma sphereGradient_linear_inner {n : ℕ} (w : Space n) (x : Sphere n) (v : Space n) :
+    ⟪sphereGradient (fun y => ⟪w,y⟫) x,v⟫ = ⟪w,v⟫-⟪w,(x : Space n)⟫*⟪(x : Space n),v⟫ := by
+  rw [sphereGradient_inner]
+  change (fderiv ℝ (innerSL ℝ w) x) v - (fderiv ℝ (innerSL ℝ w) x) x * _ = _
+  rw [(innerSL ℝ w).fderiv]
+  rfl
+
+lemma mean_sphereGradient_inner {n : ℕ} [NeZero n] {f : Space n → ℝ}
+    (hf : ContDiff ℝ 1 f) (w : Space n) :
+    mean (fun x : Sphere n => ⟪sphereGradient f x,w⟫) =
+      ((n : ℝ)-1)*mean (fun x : Sphere n => f x*⟪(x : Space n),w⟫) := by
+  have h := mean_gradient_casimir hf ((innerSL ℝ w).contDiff (n := 2))
+  change mean (fun x : Sphere n => ⟪sphereGradient f x,sphereGradient (fun y => ⟪w,y⟫) x⟫) =
+    -mean (fun x : Sphere n => f x*casimir (fun y => ⟪w,y⟫) x) at h
+  have hg (x : Sphere n) : ⟪sphereGradient f x,sphereGradient (fun y => ⟪w,y⟫) x⟫ =
+      ⟪sphereGradient f x,w⟫ := by
+    have ht : ⟪(x : Space n),sphereGradient f x⟫ = 0 := by
+      rw [real_inner_comm, sphereGradient_tangent]
+    rw [real_inner_comm, sphereGradient_linear_inner, ht, mul_zero, sub_zero, real_inner_comm]
+  simp_rw [hg, casimir_linear] at h
+  rw [h]
+  simp_rw [real_inner_comm w]
+  rw [mean, mean, ← integral_const_mul, ← integral_neg]
+  congr 1
+  funext x
+  ring
+
+lemma sphereGradient_product_scalar_inner {n : ℕ} {f : Space n → ℝ}
+    (hf : ContDiff ℝ 1 f) {ρ ρ' : ℝ → ℝ}
+    (hρ : ∀ t, HasDerivAt ρ (ρ' t) t) (v x w : Space n) :
+    ⟪sphereGradient (fun y => f y*ρ ⟪v,y⟫) x,w⟫ =
+      ρ ⟪v,x⟫*⟪sphereGradient f x,w⟫+
+      f x*ρ' ⟪v,x⟫*(⟪v,w⟫-⟪v,x⟫*⟪x,w⟫) := by
+  rw [sphereGradient_inner, sphereGradient_inner]
+  have hd := (hf.differentiable (by norm_num) x).hasFDerivAt.mul
+    ((hρ ⟪v,x⟫).comp_hasFDerivAt x (innerSL ℝ v).hasFDerivAt)
+  change HasFDerivAt (fun y => f y*ρ ⟪v,y⟫) _ x at hd
+  rw [hd.fderiv]
+  simp only [add_apply, smul_apply, smul_eq_mul, innerSL_apply_apply, comp_apply]
+  ring
+
+lemma mean_weighted_gradient {n : ℕ} [NeZero n] {f : Space n → ℝ}
+    (hf : ContDiff ℝ 1 f) {ρ ρ' : ℝ → ℝ}
+    (hρ : ContDiff ℝ 1 ρ) (hρ' : Continuous ρ') (hd : ∀ t, HasDerivAt ρ (ρ' t) t)
+    (v w : Space n) :
+    mean (fun x : Sphere n => ρ ⟪v,(x : Space n)⟫*⟪sphereGradient f x,w⟫) =
+      ((n : ℝ)-1)*mean (fun x : Sphere n => ρ ⟪v,(x : Space n)⟫*f x*⟪(x : Space n),w⟫) -
+      mean (fun x : Sphere n => f x*ρ' ⟪v,(x : Space n)⟫*
+        (⟪v,w⟫-⟪v,(x : Space n)⟫*⟪(x : Space n),w⟫)) := by
+  have h := mean_sphereGradient_inner (hf.mul (hρ.comp (innerSL ℝ v).contDiff)) w
+  change mean (fun x : Sphere n => ⟪sphereGradient (fun y => f y*ρ ⟪v,y⟫) x,w⟫) =
+    ((n : ℝ)-1)*mean (fun x : Sphere n => (f x*ρ ⟪v,(x : Space n)⟫)*⟪(x : Space n),w⟫) at h
+  simp_rw [sphereGradient_product_scalar_inner hf hd] at h
+  have h₁ : Continuous (fun x : Sphere n => ρ ⟪v,(x : Space n)⟫*⟪sphereGradient f x,w⟫) :=
+    (hρ.continuous.comp (continuous_const.inner continuous_subtype_val)).mul
+      (((sphereGradient_continuous hf).comp continuous_subtype_val).inner continuous_const)
+  have h₂ : Continuous (fun x : Sphere n => f x*ρ' ⟪v,(x : Space n)⟫*
+      (⟪v,w⟫-⟪v,(x : Space n)⟫*⟪(x : Space n),w⟫)) := by fun_prop
+  rw [mean, integral_add (continuous_integrable h₁) (continuous_integrable h₂)] at h
+  change mean _ + mean _ = _ at h
+  have he : (fun x : Sphere n => (f x*ρ ⟪v,(x : Space n)⟫)*⟪(x : Space n),w⟫) =
+      (fun x : Sphere n => ρ ⟪v,(x : Space n)⟫*f x*⟪(x : Space n),w⟫) := by
+    funext x; ring
+  rw [he] at h
+  linarith
+
+end PettyProjection.Spherical
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology Function
+open scoped ENNReal RealInnerProductSpace Gradient
+namespace PettyProjection.Spherical
+
+lemma sign_measurable : Measurable Real.sign := by
+  have he : Real.sign = fun t : ℝ => t/|t| := funext real_sign_eq
+  rw [he]
+  exact measurable_id.div measurable_id.abs
+
+lemma abs_sign_le (t : ℝ) : |Real.sign t| ≤ 1 := by
+  rw [real_sign_eq, abs_div, abs_abs]
+  by_cases ht : t = 0
+  · simp [ht]
+  · rw [div_self (abs_ne_zero.mpr ht)]
+
+lemma sign_mul_self (t : ℝ) : Real.sign t*t = |t| := by
+  rw [real_sign_eq]
+  by_cases ht : t = 0
+  · simp [ht]
+  · field_simp [abs_ne_zero.mpr ht]
+    exact (sq_abs t).symm
+
+lemma sign_weight_integrable {n : ℕ} [NeZero n] (t f : Sphere n → ℝ)
+    (ht : Continuous t) (hf : Continuous f) :
+    Integrable (fun x => Real.sign (t x)*f x) (sigma n) := by
+  apply (continuous_integrable hf).bdd_mul (sign_measurable.comp ht.measurable).aestronglyMeasurable
+  filter_upwards with x
+  exact abs_sign_le (t x)
+
+lemma mean_softSign_tendsto {n : ℕ} [NeZero n] (t f : Sphere n → ℝ)
+    (ht : Continuous t) (hf : Continuous f) :
+    Tendsto (fun m : ℕ => mean (fun x => softSign (1/((m : ℝ)+1)) (t x)*f x)) atTop
+      (𝓝 (mean (fun x => Real.sign (t x)*f x))) := by
+  apply tendsto_integral_of_dominated_convergence (fun x => |f x|)
+  · intro m
+    exact (((softSign_contDiff (by positivity : 0 < 1/((m : ℝ)+1))).continuous.comp ht).mul hf).aestronglyMeasurable
+  · exact continuous_integrable hf.abs
+  · intro m
+    filter_upwards with x
+    rw [Real.norm_eq_abs, abs_mul]
+    simpa only [one_mul] using mul_le_mul_of_nonneg_right
+      (abs_softSign_le (by positivity : 0 < 1/((m : ℝ)+1)) (t x)) (abs_nonneg (f x))
+  · filter_upwards with x
+    exact (softSign_tendsto (t x)).mul_const (f x)
+
+lemma mean_softCorrection_tendsto {n : ℕ} [NeZero n] (t f : Sphere n → ℝ)
+    (ht : Continuous t) (hf : Continuous f) :
+    Tendsto (fun m : ℕ => mean (fun x => (t x*softSlope (1/((m : ℝ)+1)) (t x))*f x)) atTop
+      (𝓝 0) := by
+  have h : Tendsto (fun m : ℕ => mean (fun x => (t x*softSlope (1/((m : ℝ)+1)) (t x))*f x)) atTop
+      (𝓝 (mean (fun _ : Sphere n => 0))) := by
+    apply tendsto_integral_of_dominated_convergence (fun x => |f x|)
+    · intro m
+      have hc : Continuous (softSlope (1/((m : ℝ)+1))) :=
+        continuous_const.div ((softAbs_contDiff (by positivity)).continuous.pow 3)
+          (fun y => pow_ne_zero _ (softAbs_pos (by positivity) y).ne')
+      exact ((ht.mul (hc.comp ht)).mul hf).aestronglyMeasurable
+    · exact continuous_integrable hf.abs
+    · intro m
+      filter_upwards with x
+      rw [Real.norm_eq_abs, abs_mul]
+      simpa only [one_mul] using mul_le_mul_of_nonneg_right
+        (abs_softCorrection_le (by positivity : 0 < 1/((m : ℝ)+1)) (t x)) (abs_nonneg (f x))
+    · filter_upwards with x
+      simpa only [zero_mul] using (softCorrection_tendsto (t x)).mul_const (f x)
+  simpa only [mean, integral_zero] using h
+
+lemma mean_softAbs_tendsto {n : ℕ} [NeZero n] (t f : Sphere n → ℝ)
+    (ht : Continuous t) (hf : Continuous f) :
+    Tendsto (fun m : ℕ => mean (fun x => softAbs (1/((m : ℝ)+1)) (t x)*f x)) atTop
+      (𝓝 (mean (fun x => |t x| *f x))) := by
+  apply tendsto_integral_of_dominated_convergence (fun x => (|t x|+1)*|f x|)
+  · intro m
+    exact (((softAbs_contDiff (by positivity : 0 < 1/((m : ℝ)+1))).continuous.comp ht).mul hf).aestronglyMeasurable
+  · exact continuous_integrable ((ht.abs.add continuous_const).mul hf.abs)
+  · intro m
+    filter_upwards with x
+    have hε : Real.sqrt (1/((m : ℝ)+1)) ≤ 1 := by
+      apply (Real.sqrt_le_one).mpr
+      apply (div_le_one (by positivity : 0 < (m : ℝ)+1)).mpr
+      linarith [Nat.cast_nonneg (α := ℝ) m]
+    rw [Real.norm_eq_abs, abs_mul, abs_of_pos (softAbs_pos (by positivity) (t x))]
+    exact mul_le_mul_of_nonneg_right ((softAbs_le (by positivity) (t x)).trans (by linarith)) (abs_nonneg (f x))
+  · filter_upwards with x
+    exact (softAbs_tendsto (t x)).mul_const (f x)
+
+end PettyProjection.Spherical
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology Function
+open scoped ENNReal RealInnerProductSpace Gradient
+namespace PettyProjection.Spherical
+
+lemma mean_sign_gradient_tangent {n : ℕ} [NeZero n] {f : Space n → ℝ}
+    (hf : ContDiff ℝ 1 f) (v w : Space n) (hvw : ⟪v,w⟫ = 0) :
+    mean (fun x : Sphere n => Real.sign ⟪v,(x : Space n)⟫*⟪sphereGradient f x,w⟫) =
+      ((n : ℝ)-1)*mean (fun x : Sphere n => Real.sign ⟪v,(x : Space n)⟫*f x*⟪(x : Space n),w⟫) := by
+  let t : Sphere n → ℝ := fun x => ⟪v,(x : Space n)⟫
+  have ht : Continuous t := continuous_const.inner continuous_subtype_val
+  have hF : Continuous (fun x : Sphere n => ⟪sphereGradient f x,w⟫) :=
+    ((sphereGradient_continuous hf).comp continuous_subtype_val).inner continuous_const
+  have hG : Continuous (fun x : Sphere n => f x*⟪(x : Space n),w⟫) := by fun_prop
+  have he (m : ℕ) :
+      mean (fun x : Sphere n => softSign (1/((m : ℝ)+1)) (t x)*⟪sphereGradient f x,w⟫) =
+      ((n : ℝ)-1)*mean (fun x : Sphere n => softSign (1/((m : ℝ)+1)) (t x)*(f x*⟪(x : Space n),w⟫)) +
+        mean (fun x : Sphere n => (t x*softSlope (1/((m : ℝ)+1)) (t x))*(f x*⟪(x : Space n),w⟫)) := by
+    have hp : 0 < 1/((m : ℝ)+1) := by positivity
+    have hc : Continuous (softSlope (1/((m : ℝ)+1))) :=
+      continuous_const.div ((softAbs_contDiff hp).continuous.pow 3)
+        (fun y => pow_ne_zero _ (softAbs_pos hp y).ne')
+    have h := mean_weighted_gradient hf ((softSign_contDiff hp).of_le (by norm_num)) hc
+      (softSign_hasDerivAt hp) v w
+    rw [h]
+    simp only [hvw, zero_sub]
+    rw [mean, mean, mean, mean, sub_eq_add_neg, ← integral_neg]
+    congr 1
+    · congr 1
+      apply integral_congr_ae
+      filter_upwards with x
+      dsimp [t]; ring
+    · apply integral_congr_ae
+      filter_upwards with x
+      dsimp [t]; ring
+  have h₁ := mean_softSign_tendsto t _ ht hF
+  have h₂ := ((mean_softSign_tendsto t _ ht hG).const_mul ((n : ℝ)-1)).add
+    (mean_softCorrection_tendsto t _ ht hG)
+  have heq := tendsto_nhds_unique h₁ (h₂.congr' (Filter.Eventually.of_forall (fun m => (he m).symm)))
+  simpa only [add_zero, t, mul_assoc] using heq
+
+lemma sphereGradient_scalar_inner {n : ℕ} {ρ ρ' : ℝ → ℝ}
+    (hρ : ∀ t, HasDerivAt ρ (ρ' t) t) (v x w : Space n) :
+    ⟪sphereGradient (fun y => ρ ⟪v,y⟫) x,w⟫ =
+      ρ' ⟪v,x⟫*(⟪v,w⟫-⟪v,x⟫*⟪x,w⟫) := by
+  have hd := (hρ ⟪v,x⟫).comp_hasFDerivAt x (innerSL ℝ v).hasFDerivAt
+  change HasFDerivAt (fun y => ρ ⟪v,y⟫) _ x at hd
+  rw [sphereGradient_inner, hd.fderiv]
+  simp only [smul_apply, smul_eq_mul, innerSL_apply_apply]
+  ring
+
+lemma mean_sign_gradient_radial {n : ℕ} [NeZero n] {f : Space n → ℝ}
+    (hf : ContDiff ℝ 2 f) (v : Space n) :
+    mean (fun x : Sphere n => Real.sign ⟪v,(x : Space n)⟫*⟪sphereGradient f x,v⟫) =
+      -mean (fun x : Sphere n => |⟪v,(x : Space n)⟫| * casimir f x) := by
+  let t : Sphere n → ℝ := fun x => ⟪v,(x : Space n)⟫
+  have ht : Continuous t := continuous_const.inner continuous_subtype_val
+  have hF : Continuous (fun x : Sphere n => ⟪sphereGradient f x,v⟫) :=
+    ((sphereGradient_continuous (hf.of_le (by norm_num))).comp continuous_subtype_val).inner continuous_const
+  have hG : Continuous (fun x : Sphere n => casimir f x) :=
+    by
+      unfold casimir
+      exact continuous_const.mul (continuous_finsetSum _ fun i _ =>
+        continuous_finsetSum _ fun j _ =>
+          (continuous_rotDeriv _ (contDiff_rotDeriv _ hf)).comp continuous_subtype_val)
+  have he (m : ℕ) :
+      mean (fun x : Sphere n => softSign (1/((m : ℝ)+1)) (t x)*⟪sphereGradient f x,v⟫) =
+      -mean (fun x : Sphere n => softAbs (1/((m : ℝ)+1)) (t x)*casimir f x) := by
+    have hp : 0 < 1/((m : ℝ)+1) := by positivity
+    have h := mean_gradient_casimir
+      ((show ContDiff ℝ 1 (softAbs (1/((m : ℝ)+1))) from (softAbs_contDiff hp).of_le le_top).comp (innerSL ℝ v).contDiff) hf
+    change mean (fun x : Sphere n => ⟪sphereGradient (fun y => softAbs (1/((m : ℝ)+1)) ⟪v,y⟫) x,sphereGradient f x⟫) = _ at h
+    have hg (x : Sphere n) :
+        ⟪sphereGradient (fun y => softAbs (1/((m : ℝ)+1)) ⟪v,y⟫) x,sphereGradient f x⟫ =
+          softSign (1/((m : ℝ)+1)) (t x)*⟪sphereGradient f x,v⟫ := by
+      rw [sphereGradient_scalar_inner (softAbs_hasDerivAt hp)]
+      have hx : ⟪(x : Space n),sphereGradient f x⟫ = 0 := by rw [real_inner_comm, sphereGradient_tangent]
+      rw [hx, mul_zero, sub_zero, real_inner_comm v]
+    simp_rw [hg] at h
+    exact h
+  have h₁ := mean_softSign_tendsto t _ ht hF
+  have h₂ := (mean_softAbs_tendsto t _ ht hG).neg
+  exact tendsto_nhds_unique h₁ (h₂.congr' (Filter.Eventually.of_forall (fun m => (he m).symm)))
+
+end PettyProjection.Spherical
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology Function
+open scoped ENNReal RealInnerProductSpace Gradient
+namespace PettyProjection.Spherical
+
+def signedPair {n : ℕ} (F G : Sphere n → Space n) : ℝ :=
+  ∫ z : Sphere n × Sphere n, Real.sign ⟪(z.1 : Space n),(z.2 : Space n)⟫ *
+    ⟪F z.1,G z.2⟫ ∂(sigma n).prod (sigma n)
+
+lemma signedPair_integrable {n : ℕ} [NeZero n] {F G : Sphere n → Space n}
+    (hF : Continuous F) (hG : Continuous G) :
+    Integrable (fun z : Sphere n × Sphere n => Real.sign ⟪(z.1 : Space n),(z.2 : Space n)⟫ *
+      ⟪F z.1,G z.2⟫) ((sigma n).prod (sigma n)) := by
+  have hi : Integrable (fun z : Sphere n × Sphere n => ⟪F z.1,G z.2⟫)
+      ((sigma n).prod (sigma n)) :=
+    (by fun_prop : Continuous _).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)
+  apply hi.bdd_mul (sign_measurable.comp (by fun_prop : Continuous
+    (fun z : Sphere n × Sphere n => ⟪(z.1 : Space n),(z.2 : Space n)⟫)).measurable).aestronglyMeasurable
+  filter_upwards with z
+  exact abs_sign_le _
+
+lemma signedPair_eq {n : ℕ} [NeZero n] {F G : Sphere n → Space n}
+    (hF : Continuous F) (hG : Continuous G) :
+    signedPair F G = mean (fun u => mean (fun v =>
+      Real.sign ⟪(u : Space n),(v : Space n)⟫ * ⟪F u,G v⟫)) :=
+  integral_prod _ (signedPair_integrable hF hG)
+
+lemma signedPair_eq_swap {n : ℕ} [NeZero n] {F G : Sphere n → Space n}
+    (hF : Continuous F) (hG : Continuous G) :
+    signedPair F G = mean (fun v => mean (fun u =>
+      Real.sign ⟪(u : Space n),(v : Space n)⟫ * ⟪F u,G v⟫)) := by
+  rw [signedPair_eq hF hG]
+  exact integral_integral_swap (signedPair_integrable hF hG)
+
+lemma signedPair_symm {n : ℕ} [NeZero n] {F G : Sphere n → Space n}
+    (hF : Continuous F) (hG : Continuous G) : signedPair F G = signedPair G F := by
+  rw [signedPair_eq hF hG, signedPair_eq_swap hG hF]
+  congr 1
+  funext u
+  congr 1
+  funext v
+  rw [real_inner_comm (u : Space n) (v : Space n), real_inner_comm (F u) (G v)]
+
+lemma signedPair_add_left {n : ℕ} [NeZero n] {F G J : Sphere n → Space n}
+    (hF : Continuous F) (hG : Continuous G) (hJ : Continuous J) :
+    signedPair (fun u => F u+G u) J = signedPair F J+signedPair G J := by
+  unfold signedPair
+  simp_rw [inner_add_left, mul_add]
+  exact integral_add (signedPair_integrable hF hJ) (signedPair_integrable hG hJ)
+
+lemma signedPair_add_right {n : ℕ} [NeZero n] {F G J : Sphere n → Space n}
+    (hF : Continuous F) (hG : Continuous G) (hJ : Continuous J) :
+    signedPair F (fun u => G u+J u) = signedPair F G+signedPair F J := by
+  unfold signedPair
+  simp_rw [inner_add_right, mul_add]
+  exact integral_add (signedPair_integrable hF hG) (signedPair_integrable hF hJ)
+
+lemma signedPair_smul_left {n : ℕ} (F G : Sphere n → Space n) (c : ℝ) :
+    signedPair (fun u => c • F u) G = c*signedPair F G := by
+  unfold signedPair
+  simp_rw [real_inner_smul_left, mul_left_comm _ c]
+  exact integral_const_mul c _
+
+lemma signedPair_smul_right {n : ℕ} (F G : Sphere n → Space n) (c : ℝ) :
+    signedPair F (fun u => c • G u) = c*signedPair F G := by
+  unfold signedPair
+  simp_rw [real_inner_smul_right, mul_left_comm _ c]
+  exact integral_const_mul c _
+
+end PettyProjection.Spherical
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/ProjectionBodies/SignSpectrum.lean b/lean/OAI/Geometry/ProjectionBodies/SignSpectrum.lean
new file mode 100644
index 0000000000000000000000000000000000000000..e55ee67a91ff5678e2a66969cf059edd861482e3
--- /dev/null
+++ b/lean/OAI/Geometry/ProjectionBodies/SignSpectrum.lean
@@ -0,0 +1,557 @@
+import OAI.Geometry.ProjectionBodies.CosineSpectrum
+
+namespace OAI
+
+universe uE
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology Function
+open scoped ENNReal RealInnerProductSpace
+namespace PettyProjection.Spherical
+
+def signFactor (n j : ℕ) (a : ℝ) : ℝ :=
+  higherFactor n j 1*(1+a*eigenvalue n (2*j))
+
+lemma signFactor_abs_step {n j : ℕ} (hn : 2 ≤ n) (hj : 1 ≤ j) {a : ℝ} (ha : 0 ≤ a) :
+    |signFactor n (j+1) a| ≤ |signFactor n j a| := by
+  have hn' : (2 : ℝ) ≤ n := by exact_mod_cast hn
+  have hj' : (1 : ℝ) ≤ j := by exact_mod_cast hj
+  have hden : 0 < 1+(n : ℝ)+2*j := by linarith
+  have he : 0 ≤ eigenvalue n (2*j) := by
+    unfold eigenvalue
+    push_cast
+    apply mul_nonneg (by positivity)
+    linarith
+  have he' : 0 ≤ eigenvalue n (2*(j+1)) := by
+    unfold eigenvalue
+    push_cast
+    apply mul_nonneg (by positivity)
+    linarith
+  have hb : (2*(j : ℝ)-1)*(1+a*eigenvalue n (2*(j+1))) ≤
+      (1+(n : ℝ)+2*j)*(1+a*eigenvalue n (2*j)) := by
+    have hpos : 0 ≤ a*(((n : ℝ)-2)*(2*j)*(2*j+n)+2*n) := by positivity
+    simp only [eigenvalue, Nat.cast_mul, Nat.cast_ofNat, Nat.cast_add, Nat.cast_one]
+    nlinarith
+  simp only [signFactor, higherFactor_succ, abs_mul]
+  rw [abs_div, abs_of_pos hden, abs_of_nonpos (by linarith : 1-2*(j : ℝ) ≤ 0), neg_sub,
+    abs_of_nonneg (by positivity : 0 ≤ 1+a*eigenvalue n (2*(j+1))),
+    abs_of_nonneg (by positivity : 0 ≤ 1+a*eigenvalue n (2*j))]
+  rw [mul_assoc]
+  apply mul_le_mul_of_nonneg_left _ (abs_nonneg _)
+  rw [div_mul_eq_mul_div, div_le_iff₀ hden]
+  simpa only [mul_comm] using hb
+
+lemma signFactor_abs_le_four {n j : ℕ} (hn : 2 ≤ n) (hj : 2 ≤ j) {a : ℝ} (ha : 0 ≤ a) :
+    |signFactor n j a| ≤ (1+a*(4*(n+2)))/((n+1)*(n+3)) := by
+  have hstep : |signFactor n j a| ≤ |signFactor n 2 a| := by
+    induction j, hj using Nat.le_induction with
+    | base => exact le_rfl
+    | succ j hj ih => exact (signFactor_abs_step hn (by omega) ha).trans ih
+  have hn' : (2 : ℝ) ≤ n := by exact_mod_cast hn
+  have he : signFactor n 2 a = -(1+a*(4*(n+2)))/((n+1)*(n+3)) := by
+    norm_num [signFactor, higherFactor, Finset.prod_range_succ, eigenvalue]
+    field_simp
+    ring
+  rw [he, abs_of_nonpos (by exact div_nonpos_of_nonpos_of_nonneg (neg_nonpos.mpr (by positivity)) (by positivity) : -(1+a*(4*(n+2)))/((n+1)*(n+3)) ≤ 0)] at hstep
+  simpa only [neg_div, neg_neg] using hstep
+
+end PettyProjection.Spherical
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology MvPolynomial
+open scoped ENNReal RealInnerProductSpace
+namespace PettyProjection.Spherical
+
+lemma homogeneous_two_mem_harmonic_sup {n : ℕ} (hn : 2 ≤ n) [NeZero n]
+    {p : MvPolynomial (Fin n) ℝ} (hp : p.IsHomogeneous 2) :
+    polynomialL2 n p ∈ harmonicSpace n 0 ⊔ harmonicSpace n 2 := by
+  have hs := homogeneous_mem_fischerSpan hn 2 hp
+  clear hp
+  induction hs using Submodule.span_induction with
+  | mem x hx =>
+    obtain ⟨j,l,p,hd,hp,hΔ,rfl⟩ := hx
+    have hr : polynomialL2 n (radiusPolynomial n ^ j * p) = polynomialL2 n p := by
+      change toH n (polynomialRestriction n _) = toH n (polynomialRestriction n p)
+      rw [map_mul, map_pow, polynomialRestriction_radius, one_pow, one_mul]
+    rw [hr]
+    have hl : l = 0 ∨ l = 2 := by omega
+    rcases hl with rfl | rfl
+    · exact Submodule.mem_sup_left ⟨p,⟨hp,hΔ⟩,rfl⟩
+    · exact Submodule.mem_sup_right ⟨p,⟨hp,hΔ⟩,rfl⟩
+  | zero => simp
+  | add p q _ _ hp hq => simpa only [map_add] using Submodule.add_mem _ hp hq
+  | smul c p _ hp => simpa only [map_smul] using Submodule.smul_mem _ c hp
+
+lemma lowSpace_eq {n : ℕ} (hn : 2 ≤ n) [NeZero n] :
+    lowSpace n = harmonicSpace n 0 ⊔ harmonicSpace n 2 := by
+  apply le_antisymm
+  · apply Submodule.span_le.mpr
+    rintro _ ⟨⟨i,j⟩,rfl⟩
+    change toH n (coordinateProduct n i j) ∈ _
+    rw [← polynomialL2_X_mul_X]
+    exact homogeneous_two_mem_harmonic_sup hn ((isHomogeneous_X ℝ i).mul (isHomogeneous_X ℝ j))
+  · apply sup_le
+    · intro v hv
+      obtain ⟨c,rfl⟩ := harmonicSpace_zero_constant hv
+      exact constant_mem_lowSpace n c
+    · exact harmonicSpace_two_le_lowSpace n
+
+lemma harmonic_inner_ne {n : ℕ} (hn : 2 ≤ n) [NeZero n] {d e : ℕ} (hde : d ≠ e)
+    {v w : H n} (hv : v ∈ harmonicSpace n d) (hw : w ∈ harmonicSpace n e) :
+    ⟪v,w⟫ = 0 := by
+  exact harmonicSpace_orthogonal hn hde ⟨v,hv⟩ ⟨w,hw⟩
+
+lemma lowSpace_projection {n : ℕ} (hn : 2 ≤ n) [NeZero n] (v : H n) :
+    (lowSpace n).starProjection v = (harmonicSpace n 0).starProjection v +
+      (harmonicSpace n 2).starProjection v := by
+  apply Submodule.eq_starProjection_of_mem_of_inner_eq_zero
+  · rw [lowSpace_eq hn]
+    exact Submodule.add_mem _ (Submodule.mem_sup_left ((harmonicSpace n 0).starProjection_apply_mem v))
+      (Submodule.mem_sup_right ((harmonicSpace n 2).starProjection_apply_mem v))
+  · intro w hw
+    rw [lowSpace_eq hn, Submodule.mem_sup] at hw
+    obtain ⟨w0,hw0,w2,hw2,rfl⟩ := hw
+    rw [inner_add_right]
+    have h0 : ⟪(harmonicSpace n 2).starProjection v,w0⟫ = 0 :=
+      harmonic_inner_ne hn (by decide) ((harmonicSpace n 2).starProjection_apply_mem v) hw0
+    have h2 : ⟪(harmonicSpace n 0).starProjection v,w2⟫ = 0 :=
+      harmonic_inner_ne hn (by decide) ((harmonicSpace n 0).starProjection_apply_mem v) hw2
+    simp only [inner_sub_left, inner_add_left, h0, h2, add_zero, zero_add]
+    rw [Submodule.inner_starProjection_left_eq_right,
+      Submodule.starProjection_eq_self_iff.mpr hw0,
+      Submodule.inner_starProjection_left_eq_right,
+      Submodule.starProjection_eq_self_iff.mpr hw2]
+    ring
+
+lemma Q_eq_sub_projections {n : ℕ} (hn : 2 ≤ n) [NeZero n] (v : H n) :
+    Q n v = v - (harmonicSpace n 0).starProjection v - (harmonicSpace n 2).starProjection v := by
+  rw [Q_eq_sub, lowSpace_projection hn, sub_add_eq_sub_sub]
+
+lemma harmonic_projection_of_mem_ne {n : ℕ} (hn : 2 ≤ n) [NeZero n] {d e : ℕ} (hde : d ≠ e)
+    {v : H n} (hv : v ∈ harmonicSpace n e) : (harmonicSpace n d).starProjection v = 0 := by
+  apply Submodule.eq_starProjection_of_mem_of_inner_eq_zero (Submodule.zero_mem _)
+  intro w hw
+  rw [sub_zero]
+  exact harmonic_inner_ne hn hde.symm hv hw
+
+lemma harmonic_projection_Q {n d : ℕ} (hn : 2 ≤ n) [NeZero n] (v : H n) :
+    (harmonicSpace n d).starProjection (Q n v) =
+      if d = 0 ∨ d = 2 then 0 else (harmonicSpace n d).starProjection v := by
+  rw [Q_eq_sub_projections hn, map_sub, map_sub]
+  by_cases h0 : d = 0
+  · subst d
+    rw [ite_eq_left (Or.inl rfl), Submodule.starProjection_eq_self_iff.mpr ((harmonicSpace n _).starProjection_apply_mem v),
+      harmonic_projection_of_mem_ne hn (by decide) ((harmonicSpace n 2).starProjection_apply_mem v)]
+    simp
+  · by_cases h2 : d = 2
+    · subst d
+      rw [ite_eq_left (Or.inr rfl), Submodule.starProjection_eq_self_iff.mpr ((harmonicSpace n _).starProjection_apply_mem v),
+        harmonic_projection_of_mem_ne hn (by decide) ((harmonicSpace n 0).starProjection_apply_mem v)]
+      simp
+    · rw [ite_eq_right (not_or.mpr ⟨h0,h2⟩),
+        harmonic_projection_of_mem_ne hn h0 ((harmonicSpace n 0).starProjection_apply_mem v),
+        harmonic_projection_of_mem_ne hn h2 ((harmonicSpace n 2).starProjection_apply_mem v)]
+      simp
+
+end PettyProjection.Spherical
+end
+
+noncomputable section
+open MeasureTheory Set Filter Topology
+open scoped ENNReal RealInnerProductSpace
+namespace PettyProjection
+variable {E : Type uE} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E]
+variable {F : ℕ → Submodule ℝ E} [∀ i, CompleteSpace (F i)]
+
+lemma hilbertSum_norm_le_of_projection_bound
+    (h : IsHilbertSum ℝ (fun i => F i) (fun i => (F i).subtypeₗᵢ))
+    (x y : E) {c : ℝ} (hc : 0 ≤ c)
+    (hp : ∀ i, ‖(F i).starProjection y‖ ≤ c*‖(F i).starProjection x‖) :
+    ‖y‖ ≤ c*‖x‖ := by
+  have hsq : ‖y‖^2 ≤ c^2*‖x‖^2 := by
+    apply hasSum_le _ (hilbertSum_hasSum_projection_norm_sq h y)
+      (HasSum.mul_left (c^2) (hilbertSum_hasSum_projection_norm_sq h x))
+    intro i
+    have h := sq_le_sq₀ (norm_nonneg ((F i).starProjection y))
+      (mul_nonneg hc (norm_nonneg ((F i).starProjection x))) |>.mpr (hp i)
+    simpa only [mul_pow] using h
+  nlinarith [norm_nonneg y, mul_nonneg hc (norm_nonneg x)]
+
+omit [CompleteSpace E] in
+lemma inner_projection_both (K : Submodule ℝ E) [CompleteSpace K] (x y : E) :
+    ⟪K.starProjection x,K.starProjection y⟫ = ⟪x,K.starProjection y⟫ := by
+  rw [Submodule.inner_starProjection_left_eq_right,
+    Submodule.starProjection_eq_self_iff.mpr (K.starProjection_apply_mem y)]
+
+lemma hilbertSum_hasSum_inner
+    (h : IsHilbertSum ℝ (fun i => F i) (fun i => (F i).subtypeₗᵢ)) (x y : E) :
+    HasSum (fun i => ⟪(F i).starProjection x,(F i).starProjection y⟫) ⟪x,y⟫ := by
+  simp only [inner_projection_both]
+  exact (innerSL ℝ x).hasSum (hilbertSum_hasSum_projection h y)
+
+lemma hilbertSum_multiplier_selfadj
+    (h : IsHilbertSum ℝ (fun i => F i) (fun i => (F i).subtypeₗᵢ))
+    (x y tx ty : E) (c : ℕ → ℝ)
+    (hx : ∀ i, (F i).starProjection tx = c i • (F i).starProjection x)
+    (hy : ∀ i, (F i).starProjection ty = c i • (F i).starProjection y) :
+    ⟪x,ty⟫ = ⟪tx,y⟫ := by
+  have h₁ := hilbertSum_hasSum_inner h x ty
+  have h₂ := hilbertSum_hasSum_inner h tx y
+  simp only [hx, hy, real_inner_smul_left, real_inner_smul_right] at h₁ h₂
+  exact h₁.unique h₂
+
+end PettyProjection
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology Function
+open scoped ENNReal RealInnerProductSpace
+namespace PettyProjection.Spherical
+
+lemma casimir_continuous {n : ℕ} {f : Space n → ℝ} (hf : ContDiff ℝ 2 f) :
+    Continuous (casimir f) := by
+  unfold casimir
+  exact continuous_const.mul (continuous_finsetSum _ fun i _ =>
+    continuous_finsetSum _ fun j _ => continuous_rotDeriv _ (contDiff_rotDeriv _ hf))
+
+def casimirMap {n : ℕ} (f : Space n → ℝ) (hf : ContDiff ℝ 2 f) : C(Sphere n, ℝ) :=
+  restrictContinuous (casimir f) (casimir_continuous hf)
+
+lemma casimir_projection {n d : ℕ} [NeZero n] {f : Space n → ℝ} (hf : ContDiff ℝ 2 f) :
+    (harmonicSpace n d).starProjection (toH n (casimirMap f hf)) =
+      -eigenvalue n d • (harmonicSpace n d).starProjection (toH n (restrictContinuous f hf.continuous)) := by
+  apply Submodule.eq_starProjection_of_mem_of_inner_eq_zero
+    ((harmonicSpace n d).smul_mem _ ((harmonicSpace n d).starProjection_apply_mem _))
+  rintro v ⟨p,hp,rfl⟩
+  rw [inner_sub_left, real_inner_smul_left, Submodule.inner_starProjection_left_eq_right,
+    Submodule.starProjection_eq_self_iff.mpr (show polynomialL2 n p ∈ harmonicSpace n d from ⟨p,hp,rfl⟩)]
+  change ⟪toH n (casimirMap f hf),toH n (polynomialRestriction n p)⟫ -
+    (-eigenvalue n d)*⟪toH n (restrictContinuous f hf.continuous),toH n (polynomialRestriction n p)⟫ = 0
+  rw [inner_toH, inner_toH]
+  change mean (fun x : Sphere n => casimir f x*polynomialFunction p x) -
+    (-eigenvalue n d)*mean (fun x : Sphere n => f x*polynomialFunction p x) = 0
+  simp_rw [mul_comm (casimir f _)]
+  rw [mean_casimir_comm ((polynomialFunction_contDiff p).of_le le_top) hf]
+  simp_rw [casimir_harmonic hp.1 hp.2, mul_left_comm (f _) (-eigenvalue n d)]
+  rw [mean, integral_const_mul]
+  exact sub_self _
+
+def signOperator {n : ℕ} [NeZero n] (a : ℝ) (f : Space n → ℝ) (hf : ContDiff ℝ 2 f) :
+    C(Sphere n, ℝ) := cosineMap (restrictContinuous f hf.continuous - a • casimirMap f hf)
+
+lemma signOperator_projection_even {n j : ℕ} (hn : 2 ≤ n) [NeZero n]
+    (a : ℝ) {f : Space n → ℝ} (hf : ContDiff ℝ 2 f) :
+    (harmonicSpace n (2*j)).starProjection (toH n (signOperator a f hf)) =
+      signFactor n j a • (harmonicSpace n (2*j)).starProjection (toH n (restrictContinuous f hf.continuous)) := by
+  rw [signOperator, cosine_projection_even hn, map_sub, map_smul, map_sub, map_smul,
+    casimir_projection hf, smul_smul, smul_sub, smul_smul, ← sub_smul]
+  congr 1
+  unfold signFactor
+  ring
+
+lemma signOperator_projection_odd {n d : ℕ} [NeZero n]
+    (a : ℝ) {f : Space n → ℝ} (hf : ContDiff ℝ 2 f) (hd : Odd d) :
+    (harmonicSpace n d).starProjection (toH n (signOperator a f hf)) = 0 :=
+  cosine_projection_odd _ hd
+
+lemma norm_Q_signOperator_le {n : ℕ} (hn : 2 ≤ n) [NeZero n]
+    {a : ℝ} (ha : 0 ≤ a) {f : Space n → ℝ} (hf : ContDiff ℝ 2 f) :
+    ‖Q n (toH n (signOperator a f hf))‖ ≤
+      ((1+a*(4*(n+2)))/((n+1)*(n+3)))*‖Q n (toH n (restrictContinuous f hf.continuous))‖ := by
+  apply hilbertSum_norm_le_of_projection_bound (harmonicSpace_isHilbertSum hn) _ _ (by positivity)
+  intro d
+  rw [harmonic_projection_Q hn, harmonic_projection_Q hn]
+  split_ifs with hd
+  · simp
+  · by_cases he : Even d
+    · obtain ⟨j,hj⟩ := he
+      have hde : d = 2*j := by omega
+      rw [hde]
+      have hj2 : 2 ≤ j := by omega
+      rw [signOperator_projection_even hn, norm_smul, Real.norm_eq_abs]
+      exact mul_le_mul_of_nonneg_right (signFactor_abs_le_four hn hj2 ha) (norm_nonneg _)
+    · rw [signOperator_projection_odd a hf (Nat.not_even_iff_odd.mp he), norm_zero]
+      positivity
+
+lemma signOperator_selfadj {n : ℕ} (hn : 2 ≤ n) [NeZero n] (a : ℝ)
+    {f g : Space n → ℝ} (hf : ContDiff ℝ 2 f) (hg : ContDiff ℝ 2 g) :
+    mean (fun x : Sphere n => f x*signOperator a g hg x) =
+      mean (fun x : Sphere n => signOperator a f hf x*g x) := by
+  change mean (fun x => restrictContinuous f hf.continuous x*signOperator a g hg x) =
+    mean (fun x => signOperator a f hf x*restrictContinuous g hg.continuous x)
+  rw [← inner_toH n (restrictContinuous f hf.continuous) (signOperator a g hg),
+    ← inner_toH n (signOperator a f hf) (restrictContinuous g hg.continuous)]
+  let c : ℕ → ℝ := fun d => if Even d then signFactor n (d/2) a else 0
+  apply hilbertSum_multiplier_selfadj (harmonicSpace_isHilbertSum hn) _ _ _ _ c
+  all_goals
+    intro d
+    by_cases he : Even d
+    · obtain ⟨j,hj⟩ := he
+      have hd : d = 2*j := by omega
+      rw [hd]
+      simp only [c, even_two_mul, ↓reduceIte, Nat.mul_div_cancel_left _ (by decide : 0 < 2)]
+      exact signOperator_projection_even hn a _
+    · simp only [c, he, ↓reduceIte, zero_smul]
+      exact signOperator_projection_odd a _ (Nat.not_even_iff_odd.mp he)
+
+end PettyProjection.Spherical
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology Function
+open scoped ENNReal RealInnerProductSpace
+namespace PettyProjection.Spherical
+
+lemma cosine_sub_smul {n : ℕ} [NeZero n] {f g : Sphere n → ℝ}
+    (hf : Continuous f) (hg : Continuous g) (a : ℝ) (u : Sphere n) :
+    cosine (fun x => f x-a*g x) u = cosine f u-a*cosine g u := by
+  unfold cosine mean
+  simp_rw [mul_sub, mul_left_comm _ a]
+  rw [integral_sub (continuous_integrable (by fun_prop)) (continuous_integrable (by fun_prop)),
+    integral_const_mul]
+  ring
+
+lemma signOperator_apply {n : ℕ} [NeZero n] (a : ℝ) {f : Space n → ℝ}
+    (hf : ContDiff ℝ 2 f) (u : Sphere n) :
+    signOperator a f hf u = cosine (fun x : Sphere n => f x) u -
+      a*cosine (fun x : Sphere n => casimir f x) u := by
+  exact cosine_sub_smul (hf.continuous.comp continuous_subtype_val)
+    ((casimir_continuous hf).comp continuous_subtype_val) a u
+
+lemma casimir_const {n : ℕ} (c : ℝ) : casimir (fun _ : Space n => c) = 0 := by
+  have hr (X : Space n →L[ℝ] Space n) : rotDeriv X (fun _ => c) = 0 := by
+    ext x
+    simp [rotDeriv]
+  ext x
+  simp [casimir, hr, rotDeriv]
+
+lemma mean_casimir_zero {n : ℕ} [NeZero n] {f : Space n → ℝ} (hf : ContDiff ℝ 2 f) :
+    mean (fun x : Sphere n => casimir f x) = 0 := by
+  have h := mean_casimir_comm (f := fun _ => 1) contDiff_const hf
+  simpa only [one_mul, casimir_const, Pi.zero_apply, mul_zero, mean, integral_zero] using h
+
+lemma mean_signOperator {n : ℕ} [NeZero n] (a : ℝ) {f : Space n → ℝ}
+    (hf : ContDiff ℝ 2 f) : mean (signOperator a f hf) = mean (fun x : Sphere n => f x) := by
+  rw [signOperator, show (cosineMap (restrictContinuous f hf.continuous - a • casimirMap f hf) : Sphere n → ℝ) =
+      cosine (restrictContinuous f hf.continuous - a • casimirMap f hf) from rfl,
+    mean_cosine (ContinuousMap.continuous _)]
+  change mean (fun x : Sphere n => f x-a*casimir f x) = _
+  unfold mean
+  rw [integral_sub (f := fun x : Sphere n => f x) (g := fun x : Sphere n => a*casimir f x)
+    (continuous_integrable (hf.continuous.comp continuous_subtype_val))
+    (continuous_integrable (continuous_const.mul ((casimir_continuous hf).comp continuous_subtype_val))),
+    integral_const_mul]
+  change mean (fun x : Sphere n => f x) - a*mean (fun x : Sphere n => casimir f x) = _
+  rw [mean_casimir_zero hf, mul_zero, sub_zero]
+  rfl
+
+lemma mean_signOperator_expand {n : ℕ} [NeZero n] (a : ℝ) {f g : Space n → ℝ}
+    (hf : Continuous f) (hg : ContDiff ℝ 2 g) :
+    mean (fun x : Sphere n => f x*signOperator a g hg x) =
+      mean (fun x : Sphere n => f x*cosine (fun y : Sphere n => g y) x) -
+        a*mean (fun x : Sphere n => f x*cosine (fun y : Sphere n => casimir g y) x) := by
+  simp_rw [signOperator_apply, mul_sub, mul_left_comm (f _) a]
+  exact (integral_sub
+    (continuous_integrable ((hf.comp continuous_subtype_val).mul (cosine_continuous (hg.continuous.comp continuous_subtype_val))))
+    (continuous_integrable (continuous_const.mul ((hf.comp continuous_subtype_val).mul
+      (cosine_continuous ((casimir_continuous hg).comp continuous_subtype_val)))))).trans
+    (by
+      change _ - (∫ x : Sphere n, a * (f x * cosine (fun y : Sphere n => casimir g y) x) ∂sigma n) = _
+      rw [integral_const_mul]
+      rfl)
+
+lemma cosine_casimir_selfadj {n : ℕ} (hn : 2 ≤ n) [NeZero n]
+    {f g : Space n → ℝ} (hf : ContDiff ℝ 2 f) (hg : ContDiff ℝ 2 g) :
+    mean (fun x : Sphere n => f x*cosine (fun y : Sphere n => casimir g y) x) =
+      mean (fun x : Sphere n => g x*cosine (fun y : Sphere n => casimir f y) x) := by
+  have h := signOperator_selfadj hn 1 hf hg
+  simp_rw [mul_comm (signOperator 1 f hf _)] at h
+  rw [mean_signOperator_expand 1 hf.continuous hg, mean_signOperator_expand 1 hg.continuous hf] at h
+  have hc := cosine_selfadj (hf.continuous.comp continuous_subtype_val)
+    (hg.continuous.comp continuous_subtype_val)
+  simp_rw [mul_comm (cosine _ _)] at hc
+  change (mean (fun x : Sphere n => f x * cosine (fun y : Sphere n => g y) x)) =
+    mean (fun x : Sphere n => g x * cosine (fun y : Sphere n => f y) x) at hc
+  linarith
+
+end PettyProjection.Spherical
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology Function
+open scoped ENNReal RealInnerProductSpace
+namespace PettyProjection.Spherical
+
+lemma inner_Q_Q {n : ℕ} [NeZero n] (x y : H n) :
+    ⟪Q n x,Q n y⟫ = ⟪Q n x,y⟫ := by
+  change ⟪(lowSpace n)ᗮ.starProjection x,(lowSpace n)ᗮ.starProjection y⟫ = _
+  rw [Submodule.inner_starProjection_left_eq_right,
+    Submodule.starProjection_eq_self_iff.mpr ((lowSpace n)ᗮ.starProjection_apply_mem y)]
+  exact (Submodule.inner_starProjection_left_eq_right _ _ _).symm
+
+lemma inner_Q_Q_of_degree_two_zero {n : ℕ} (hn : 2 ≤ n) [NeZero n]
+    (f g : C(Sphere n, ℝ)) (h2 : (harmonicSpace n 2).starProjection (toH n f) = 0) :
+    ⟪Q n (toH n f),Q n (toH n g)⟫ = mean (fun u => f u*g u) - mean f*mean g := by
+  rw [inner_Q_Q, Q_eq_sub_projections hn, h2, sub_zero, inner_sub_left,
+    harmonicSpace_zero_projection, inner_toH, inner_toH]
+  change _ - mean (fun u => mean f*g u) = _
+  congr 1
+  exact integral_const_mul (mean f) g
+
+lemma sign_spectral_estimate {n : ℕ} (hn : 2 ≤ n) [NeZero n]
+    {a : ℝ} (ha : 0 ≤ a) (f : C(Sphere n, ℝ)) {g : Space n → ℝ}
+    (hg : ContDiff ℝ 2 g) (h2 : (harmonicSpace n 2).starProjection (toH n f) = 0) :
+    mean f*mean (fun x : Sphere n => g x) -
+      ((1+a*(4*(n+2)))/((n+1)*(n+3)))*‖Q n (toH n f)‖*
+        ‖Q n (toH n (restrictContinuous g hg.continuous))‖ ≤
+      mean (fun x => f x*signOperator a g hg x) := by
+  have hi := inner_Q_Q_of_degree_two_zero hn f (signOperator a g hg) h2
+  rw [mean_signOperator] at hi
+  have hcs := neg_le_of_abs_le (abs_real_inner_le_norm
+    (Q n (toH n f)) (Q n (toH n (signOperator a g hg))))
+  have hb := mul_le_mul_of_nonneg_left (norm_Q_signOperator_le hn ha hg)
+    (norm_nonneg (Q n (toH n f)))
+  rw [hi] at hcs
+  nlinarith
+
+end PettyProjection.Spherical
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology Function
+open scoped ENNReal RealInnerProductSpace Gradient
+namespace PettyProjection.Spherical
+
+def radialField {n : ℕ} (f : Space n → ℝ) (u : Sphere n) : Space n := f u • (u : Space n)
+def tangentField {n : ℕ} (f : Space n → ℝ) (u : Sphere n) : Space n := sphereGradient f u
+
+def signField {n : ℕ} (k : ℝ) (f : Space n → ℝ) (u : Sphere n) : Space n :=
+  radialField f u + k⁻¹ • tangentField f u
+
+lemma radialField_continuous {n : ℕ} {f : Space n → ℝ} (hf : Continuous f) :
+    Continuous (radialField f) := (hf.comp continuous_subtype_val).smul continuous_subtype_val
+
+lemma tangentField_continuous {n : ℕ} {f : Space n → ℝ} (hf : ContDiff ℝ 1 f) :
+    Continuous (tangentField f) := (sphereGradient_continuous hf).comp continuous_subtype_val
+
+lemma signedPair_radial_radial {n : ℕ} [NeZero n] {f g : Space n → ℝ}
+    (hf : Continuous f) (hg : Continuous g) :
+    signedPair (radialField f) (radialField g) = cosineConstant n *
+      mean (fun u : Sphere n => f u*cosine (fun v : Sphere n => g v) u) := by
+  rw [signedPair_eq (radialField_continuous hf) (radialField_continuous hg)]
+  have hi (u v : Sphere n) :
+      Real.sign ⟪(u : Space n),(v : Space n)⟫ * ⟪radialField f u,radialField g v⟫ =
+        f u*(|⟪(u : Space n),(v : Space n)⟫| * g v) := by
+    simp only [radialField, real_inner_smul_left, real_inner_smul_right]
+    calc
+      _ = (Real.sign ⟪(u : Space n),(v : Space n)⟫ * ⟪(u : Space n),(v : Space n)⟫)*f u*g v := by ring
+      _ = _ := by rw [sign_mul_self]; ring
+  simp_rw [hi, mean, integral_const_mul]
+  simp only [cosine, mean, ← mul_div_assoc, integral_div]
+  field_simp [(cosineConstant_pos n).ne']
+
+lemma signedPair_tangent_radial {n : ℕ} [NeZero n] {f g : Space n → ℝ}
+    (hf : ContDiff ℝ 2 f) (hg : Continuous g) :
+    signedPair (tangentField f) (radialField g) = -cosineConstant n *
+      mean (fun u : Sphere n => g u*cosine (fun v : Sphere n => casimir f v) u) := by
+  rw [signedPair_eq_swap (tangentField_continuous (hf.of_le (by norm_num))) (radialField_continuous hg)]
+  have hi (u v : Sphere n) :
+      Real.sign ⟪(u : Space n),(v : Space n)⟫ * ⟪tangentField f u,radialField g v⟫ =
+        g v*(Real.sign ⟪(v : Space n),(u : Space n)⟫*⟪sphereGradient f u,(v : Space n)⟫) := by
+    simp only [radialField, tangentField, real_inner_smul_right]
+    rw [real_inner_comm (u : Space n) (v : Space n)]
+    ring
+  simp_rw [hi, mean, integral_const_mul]
+  change mean (fun v : Sphere n => g v * mean (fun u : Sphere n =>
+    Real.sign ⟪(v : Space n),(u : Space n)⟫ * ⟪sphereGradient f u,(v : Space n)⟫)) = _
+  simp_rw [mean_sign_gradient_radial hf]
+  simp only [cosine, mean, ← mul_div_assoc, integral_div, mul_neg]
+  rw [integral_neg]
+  have hb := (cosineConstant_pos n).ne'
+  field_simp
+
+lemma signedPair_tangent_tangent {n : ℕ} [NeZero n] {f g : Space n → ℝ}
+    (hf : ContDiff ℝ 1 f) (hg : ContDiff ℝ 1 g) :
+    signedPair (tangentField f) (tangentField g) = ((n : ℝ)-1)*
+      signedPair (radialField f) (tangentField g) := by
+  rw [signedPair_eq_swap (tangentField_continuous hf) (tangentField_continuous hg),
+    signedPair_eq_swap (radialField_continuous hf.continuous) (tangentField_continuous hg)]
+  have hi (v : Sphere n) :
+      mean (fun u : Sphere n => Real.sign ⟪(u : Space n),(v : Space n)⟫ * ⟪tangentField f u,tangentField g v⟫) =
+        ((n : ℝ)-1)*mean (fun u : Sphere n => Real.sign ⟪(u : Space n),(v : Space n)⟫ * ⟪radialField f u,tangentField g v⟫) := by
+    have h := mean_sign_gradient_tangent hf (v : Space n) (sphereGradient g v)
+      (by rw [real_inner_comm]; exact sphereGradient_tangent g v)
+    convert h using 1
+    · congr 1
+      funext u
+      rw [real_inner_comm (u : Space n) (v : Space n)]
+      rfl
+    · congr 1
+      congr 1
+      funext u
+      simp only [radialField, tangentField, real_inner_smul_left]
+      rw [real_inner_comm (u : Space n) (v : Space n)]
+      ring
+  simp_rw [hi]
+  exact integral_const_mul _ _
+
+end PettyProjection.Spherical
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology Function
+open scoped ENNReal RealInnerProductSpace Gradient
+namespace PettyProjection.Spherical
+
+lemma signField_continuous {n : ℕ} (k : ℝ) {f : Space n → ℝ} (hf : ContDiff ℝ 1 f) :
+    Continuous (signField k f) :=
+  (radialField_continuous hf.continuous).add ((tangentField_continuous hf).const_smul k⁻¹)
+
+lemma signedPair_fields {n : ℕ} (hn : 2 ≤ n) [NeZero n]
+    (k : ℝ) {f g : Space n → ℝ} (hf : ContDiff ℝ 2 f) (hg : ContDiff ℝ 2 g) :
+    signedPair (signField k f) (signField k g) = cosineConstant n *
+      mean (fun u : Sphere n => f u*signOperator (2/k+((n : ℝ)-1)/k^2) g hg u) := by
+  have hf1 := hf.of_le (by norm_num : (1 : WithTop ℕ∞) ≤ 2)
+  have hg1 := hg.of_le (by norm_num : (1 : WithTop ℕ∞) ≤ 2)
+  have hrf := radialField_continuous hf.continuous
+  have hrg := radialField_continuous hg.continuous
+  have htf := tangentField_continuous hf1
+  have htg := tangentField_continuous hg1
+  have hktf : Continuous (fun u => k⁻¹ • tangentField f u) := htf.const_smul k⁻¹
+  have hktg : Continuous (fun u => k⁻¹ • tangentField g u) := htg.const_smul k⁻¹
+  have hrt : signedPair (radialField f) (tangentField g) = -cosineConstant n *
+      mean (fun u : Sphere n => f u*cosine (fun v : Sphere n => casimir g v) u) := by
+    rw [signedPair_symm hrf htg]
+    exact signedPair_tangent_radial hg hf.continuous
+  have htr : signedPair (tangentField f) (radialField g) = -cosineConstant n *
+      mean (fun u : Sphere n => f u*cosine (fun v : Sphere n => casimir g v) u) := by
+    rw [signedPair_tangent_radial hf hg.continuous, cosine_casimir_selfadj hn hg hf]
+  change signedPair (fun u => radialField f u+k⁻¹ • tangentField f u)
+    (fun u => radialField g u+k⁻¹ • tangentField g u) = _
+  rw [signedPair_add_left hrf hktf (show Continuous (fun u => radialField g u+k⁻¹ • tangentField g u) from signField_continuous k hg1),
+    signedPair_add_right hrf hrg hktg,
+    signedPair_add_right hktf hrg hktg,
+    signedPair_smul_right, signedPair_smul_left, signedPair_smul_left, signedPair_smul_right,
+    signedPair_radial_radial hf.continuous hg.continuous,
+    signedPair_tangent_tangent hf1 hg1, hrt, htr,
+    mean_signOperator_expand _ hf.continuous hg]
+  simp only [div_eq_mul_inv]
+  ring
+
+lemma sign_form_estimate {n : ℕ} (hn : 2 ≤ n) [NeZero n]
+    {k : ℝ} (hk : 0 < k) {f g : Space n → ℝ} (hf : ContDiff ℝ 2 f) (hg : ContDiff ℝ 2 g)
+    (h2 : (harmonicSpace n 2).starProjection (toH n (restrictContinuous f hf.continuous)) = 0) :
+    mean (fun u : Sphere n => f u)*mean (fun u : Sphere n => g u) -
+      ((1+(2/k+((n : ℝ)-1)/k^2)*(4*(n+2)))/((n+1)*(n+3)))*
+        ‖Q n (toH n (restrictContinuous f hf.continuous))‖*
+        ‖Q n (toH n (restrictContinuous g hg.continuous))‖ ≤
+      signedPair (signField k f) (signField k g)/cosineConstant n := by
+  rw [signedPair_fields hn k hf hg, mul_div_cancel_left₀ _ (cosineConstant_pos n).ne']
+  have hn' : (1 : ℝ) ≤ n := by exact_mod_cast (by omega : 1 ≤ n)
+  exact sign_spectral_estimate hn (by positivity) _ hg h2
+
+end PettyProjection.Spherical
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/ProjectionBodies/SliceTransport.lean b/lean/OAI/Geometry/ProjectionBodies/SliceTransport.lean
new file mode 100644
index 0000000000000000000000000000000000000000..a80e7e5896c4a49ce4d887b9b2194fc2dec312b3
--- /dev/null
+++ b/lean/OAI/Geometry/ProjectionBodies/SliceTransport.lean
@@ -0,0 +1,381 @@
+import OAI.Geometry.ProjectionBodies.ShadowSlices
+
+namespace OAI
+
+universe uE
+
+noncomputable section
+open Set MeasureTheory Filter Topology
+namespace PettyProjection
+
+/-- The strict scalar inequality in the slice transport proof of
+Brunn--Minkowski. The variables `p,q` are the positive slice-volume roots. -/
+lemma transport_scalar {m : ℕ} (hm : 0 < m) {a b p q : ℝ}
+    (ha : 0 < a) (hb : 0 < b) (hab : a+b=1) (hp : 0 < p) (hq : 0 < q) :
+    1 ≤ (a*p+b*q)^m*(a/(p^m)+b/(q^m)) ∧
+      ((a*p+b*q)^m*(a/(p^m)+b/(q^m))=1 ↔ p=q) := by
+  let D := p^a*q^b
+  have hD : 0 < D := mul_pos (Real.rpow_pos_of_pos hp _) (Real.rpow_pos_of_pos hq _)
+  have hX : D ≤ a*p+b*q := Real.geom_mean_le_arith_mean2_weighted ha.le hb.le hp.le hq.le hab
+  have hXi : D < a*p+b*q ↔ p ≠ q :=
+    Real.geom_mean_lt_arith_mean2_weighted_iff_of_pos ha hb hp.le hq.le hab
+  have hY : (D^m)⁻¹ ≤ a/(p^m)+b/(q^m) := by
+    have hh := Real.geom_mean_le_arith_mean2_weighted ha.le hb.le
+      (inv_nonneg.mpr (pow_nonneg hp.le m)) (inv_nonneg.mpr (pow_nonneg hq.le m)) hab
+    convert hh using 1
+    · dsimp [D]
+      rw [mul_pow,mul_inv,Real.inv_rpow (pow_nonneg hp.le m),
+        Real.inv_rpow (pow_nonneg hq.le m)]
+      rw [← Real.rpow_natCast,← Real.rpow_natCast,← Real.rpow_mul hp.le,
+        ← Real.rpow_mul hq.le]
+      rw [← Real.rpow_natCast,← Real.rpow_natCast,← Real.rpow_mul hp.le,
+        ← Real.rpow_mul hq.le]
+      simp only [mul_comm]
+    · simp only [div_eq_mul_inv]
+  have hYP : 0 < a/(p^m)+b/(q^m) := by positivity
+  have hX0 : 0 < a*p+b*q := by positivity
+  have hle : 1 ≤ (a*p+b*q)^m*(a/(p^m)+b/(q^m)) := by
+    calc
+      1 = D^m*(D^m)⁻¹ := (mul_inv_cancel₀ (pow_ne_zero _ hD.ne')).symm
+      _ ≤ (a*p+b*q)^m*(a/(p^m)+b/(q^m)) :=
+        mul_le_mul (pow_le_pow_left₀ hD.le hX m) hY (inv_nonneg.mpr (pow_nonneg hD.le _))
+          (pow_nonneg hX0.le _)
+  refine ⟨hle,⟨fun he => ?_,fun he => ?_⟩⟩
+  · by_contra hn
+    have hlt : D^m < (a*p+b*q)^m := pow_lt_pow_left₀ (hXi.mpr hn) hD.le hm.ne'
+    have hh : 1 < (a*p+b*q)^m*(a/(p^m)+b/(q^m)) := calc
+      1 = D^m*(D^m)⁻¹ := (mul_inv_cancel₀ (pow_ne_zero _ hD.ne')).symm
+      _ ≤ D^m*(a/(p^m)+b/(q^m)) := mul_le_mul_of_nonneg_left hY (pow_nonneg hD.le _)
+      _ < _ := mul_lt_mul_of_pos_right hlt hYP
+    linarith
+  · subst q
+    rw [← add_mul,hab,one_mul,← add_div,hab]
+    exact mul_one_div_cancel (pow_ne_zero _ hp.ne')
+
+end PettyProjection
+end
+
+noncomputable section
+open Set MeasureTheory Filter Topology
+namespace PettyProjection
+
+/-- The inverse cumulative mass coordinate on a compact interval. Zero density
+at the two endpoints is allowed; the derivative is needed only in the interior. -/
+theorem exists_interval_quantile {l r : ℝ} (hlr : l < r) {f : ℝ → ℝ}
+    (hf : ContinuousOn f (Ioo l r)) (hI : IntervalIntegrable f volume l r)
+    (hpos : ∀ x ∈ Ioo l r, 0 < f x)
+    (hmass : (∫ x in l..r, f x)=1) :
+    ∃ Q : ℝ → ℝ, Continuous Q ∧ Q 0=l ∧ Q 1=r ∧
+      (∀ s ∈ Icc (0:ℝ) 1, Q s ∈ Icc l r) ∧
+      (∀ s ∈ Ioo (0:ℝ) 1, Q s ∈ Ioo l r ∧
+        HasDerivAt Q (f (Q s))⁻¹ s) := by
+  let F : ℝ → ℝ := fun x => ∫ y in l..x, f y
+  have hFc : ContinuousOn F (Icc l r) := by
+    simpa only [uIcc_of_le hlr.le] using
+      (intervalIntegral.continuousOn_primitive_interval' hI left_mem_uIcc)
+  have hd : ∀ x ∈ Ioo l r, HasDerivAt F (f x) x := by
+    intro x hx
+    have hc : ContinuousAt f x := hf.continuousAt (Ioo_mem_nhds hx.1 hx.2)
+    exact intervalIntegral.integral_hasDerivAt_right (hI.mono_set (by
+      simp only [uIcc_of_le hlr.le,uIcc_of_le hx.1.le]
+      exact Icc_subset_Icc_right hx.2.le))
+      (hf.stronglyMeasurableAtFilter isOpen_Ioo x hx) hc
+  have hmono : StrictMonoOn F (Icc l r) := by
+    apply strictMonoOn_of_deriv_pos (convex_Icc l r) hFc
+    intro x hx
+    rw [interior_Icc] at hx
+    rw [(hd x hx).deriv]
+    exact hpos x hx
+  have hFl : F l=0 := intervalIntegral.integral_same
+  have hFr : F r=1 := hmass
+  have hIm : F '' Icc l r = Icc (0:ℝ) 1 := by
+    simpa only [hFl,hFr] using hFc.image_Icc_of_monotoneOn hlr.le hmono.monotoneOn
+  let e : Icc l r ≃o Icc (0:ℝ) 1 :=
+    (hmono.orderIso F (Icc l r)).trans (Set.orderIsoOfEq _ _ hIm)
+  let Q : ℝ → ℝ := fun s => (e.symm (projIcc 0 1 (by norm_num) s)).val
+  have hQc : Continuous Q := continuous_subtype_val.comp
+    (e.symm.continuous.comp continuous_projIcc)
+  have hQmem : ∀ s, Q s ∈ Icc l r := fun s => (e.symm (projIcc 0 1 (by norm_num) s)).property
+  have hFQ : ∀ s ∈ Icc (0:ℝ) 1, F (Q s)=s := by
+    intro s hs
+    have he := congrArg Subtype.val (e.apply_symm_apply (projIcc 0 1 (by norm_num) s))
+    change F (Q s) = (projIcc 0 1 (by norm_num) s).val at he
+    simpa only [projIcc_of_mem (by norm_num : (0:ℝ) ≤ 1) hs] using he
+  have hQ0 : Q 0=l := hmono.injOn (hQmem 0) (left_mem_Icc.mpr hlr.le)
+    ((hFQ 0 (by norm_num)).trans hFl.symm)
+  have hQ1 : Q 1=r := hmono.injOn (hQmem 1) (right_mem_Icc.mpr hlr.le)
+    ((hFQ 1 (by norm_num)).trans hFr.symm)
+  refine ⟨Q,hQc,hQ0,hQ1,fun s _ => hQmem s,?_⟩
+  intro s hs
+  have hQi : Q s ∈ Ioo l r := by
+    refine ⟨lt_of_le_of_ne (hQmem s).1 ?_,lt_of_le_of_ne (hQmem s).2 ?_⟩
+    · intro he
+      have hy := hFQ s ⟨hs.1.le,hs.2.le⟩
+      rw [← he,hFl] at hy
+      linarith [hs.1]
+    · intro he
+      have hy := hFQ s ⟨hs.1.le,hs.2.le⟩
+      rw [he,hFr] at hy
+      linarith [hs.2]
+  refine ⟨hQi,HasDerivAt.of_local_left_inverse hQc.continuousAt (hd _ hQi)
+    (hpos _ hQi).ne' ?_⟩
+  filter_upwards [Ioo_mem_nhds hs.1 hs.2] with t ht
+  exact hFQ t ⟨ht.1.le,ht.2.le⟩
+
+end PettyProjection
+end
+
+noncomputable section
+open Set MeasureTheory Filter Topology
+namespace PettyProjection
+
+/-- The analytic step of the slice proof. All three functions here are actual
+slice densities when this result is applied to convex bodies. -/
+theorem slice_transport {m : ℕ} (hm : 0 < m) {a b l r L R : ℝ}
+    (ha : 0 < a) (hb : 0 < b) (hab : a+b=1) (hlr : l<r) (hLR : L<R)
+    {f g h : ℝ → ℝ}
+    (hf : ContinuousOn f (Ioo l r)) (hg : ContinuousOn g (Ioo L R))
+    (hfi : IntervalIntegrable f volume l r) (hgi : IntervalIntegrable g volume L R)
+    (hfp : ∀ x ∈ Ioo l r, 0<f x) (hgp : ∀ x ∈ Ioo L R, 0<g x)
+    (hfI : (∫ x in l..r, f x)=1) (hgI : (∫ x in L..R, g x)=1)
+    (hhi : IntervalIntegrable h volume (a*l+b*L) (a*r+b*R))
+    (hbound : ∀ x ∈ Ioo l r, ∀ y ∈ Ioo L R,
+      (a*(f x)^((m:ℝ)⁻¹)+b*(g y)^((m:ℝ)⁻¹))^m ≤ h (a*x+b*y)) :
+    1 ≤ ∫ z in (a*l+b*L)..(a*r+b*R), h z := by
+  obtain ⟨Q,hQc,hQ0,hQ1,hQmem,hQd⟩ := exists_interval_quantile hlr hf hfi hfp hfI
+  obtain ⟨P,hPc,hP0,hP1,hPmem,hPd⟩ := exists_interval_quantile hLR hg hgi hgp hgI
+  let z : ℝ → ℝ := fun s => a*Q s+b*P s
+  let z' : ℝ → ℝ := fun s => a/(f (Q s))+b/(g (P s))
+  have hzc : Continuous z := by dsimp [z]; fun_prop
+  have hzd : ∀ s ∈ Ioo (0:ℝ) 1, HasDerivAt z (z' s) s := by
+    intro s hs
+    convert ((hQd s hs).2.const_mul a).add ((hPd s hs).2.const_mul b) using 1
+    all_goals rfl
+  have hzp : ∀ s ∈ Ioo (0:ℝ) 1, 0<z' s := by
+    intro s hs
+    exact add_pos (div_pos ha (hfp _ (hQd s hs).1)) (div_pos hb (hgp _ (hPd s hs).1))
+  have hz0 : z 0=a*l+b*L := by dsimp [z]; rw [hQ0,hP0]
+  have hz1 : z 1=a*r+b*R := by dsimp [z]; rw [hQ1,hP1]
+  have hzz : z 0 ≤ z 1 := by rw [hz0,hz1]; nlinarith
+  have hhi : IntervalIntegrable h volume (z 0) (z 1) := by
+    rwa [hz0,hz1]
+  have hzdu : ∀ s ∈ Ioo (min (0:ℝ) 1) (max (0:ℝ) 1), HasDerivAt z (z' s) s := by
+    simpa only [min_eq_left zero_le_one,max_eq_right zero_le_one] using hzd
+  have hzpu : ∀ s ∈ Ioo (min (0:ℝ) 1) (max (0:ℝ) 1), 0≤z' s := by
+    simpa only [min_eq_left zero_le_one,max_eq_right zero_le_one] using fun s hs => (hzp s hs).le
+  have hint : IntervalIntegrable (fun s => h (z s)*z' s) volume 0 1 :=
+    (intervalIntegral.integrable_comp_mul_deriv_iff_of_deriv_nonneg hzc.continuousOn hzdu hzpu).mpr hhi
+  have hsub := intervalIntegral.integral_comp_mul_deriv_of_deriv_nonneg
+    (g := h) hzc.continuousOn hzdu hzpu
+  change (∫ s in (0:ℝ)..1, h (z s)*z' s) = _ at hsub
+  have hpt : ∀ s ∈ Ioo (0:ℝ) 1, 1 ≤ h (z s)*z' s := by
+    intro s hs
+    let p := (f (Q s))^((m:ℝ)⁻¹)
+    let q := (g (P s))^((m:ℝ)⁻¹)
+    have hp : 0<p := Real.rpow_pos_of_pos (hfp _ (hQd s hs).1) _
+    have hq : 0<q := Real.rpow_pos_of_pos (hgp _ (hPd s hs).1) _
+    have hpm : p^m=f (Q s) := Real.rpow_inv_natCast_pow (hfp _ (hQd s hs).1).le hm.ne'
+    have hqm : q^m=g (P s) := Real.rpow_inv_natCast_pow (hgp _ (hPd s hs).1).le hm.ne'
+    have ht := (transport_scalar hm ha hb hab hp hq).1
+    rw [hpm,hqm] at ht
+    exact ht.trans (mul_le_mul_of_nonneg_right
+      (hbound _ (hQd s hs).1 _ (hPd s hs).1) (hzp s hs).le)
+  have hae : (fun _ : ℝ => (1:ℝ)) ≤ᵐ[volume.restrict (Icc (0:ℝ) 1)] (fun s => h (z s)*z' s) := by
+    rw [← Measure.restrict_congr_set Ioo_ae_eq_Icc]
+    exact (ae_restrict_iff' measurableSet_Ioo).mpr (ae_of_all _ hpt)
+  have hi := intervalIntegral.integral_mono_ae_restrict (by norm_num : (0:ℝ)≤1)
+    intervalIntegrable_const hint hae
+  rw [intervalIntegral.integral_const] at hi
+  norm_num at hi
+  rw [hsub,hz0,hz1] at hi
+  exact hi
+
+end PettyProjection
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology Function
+open scoped Pointwise
+namespace PettyProjection
+section
+variable {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E]
+
+def horizontalSlice (K : Set (E × ℝ)) (s : ℝ) : Set E := (fun y => (y,s)) ⁻¹' K
+
+omit [NormedSpace ℝ E] in
+lemma horizontalSlice_compact {K : Set (E × ℝ)} (hK : IsCompact K) (s : ℝ) :
+    IsCompact (horizontalSlice K s) :=
+  (IsClosedEmbedding.of_isEmbedding_isClosedMap (isEmbedding_prodMkLeft s)
+    (isClosedMap_prodMk_right s)).isCompact_preimage hK
+
+lemma horizontalSlice_convex {K : Set (E × ℝ)} (hK : Convex ℝ K) (s : ℝ) :
+    Convex ℝ (horizontalSlice K s) := by
+  intro x hx y hy a b ha hb hab
+  have h := hK hx hy ha hb hab
+  change (a • x+b • y,s) ∈ K
+  simpa only [Prod.smul_mk,Prod.mk_add_mk,← add_smul,hab,one_smul] using h
+
+omit [NormedAddCommGroup E] [NormedSpace ℝ E] in
+lemma horizontalSlice_nonempty_iff {K : Set (E × ℝ)} {s : ℝ} :
+    (horizontalSlice K s).Nonempty ↔ s ∈ Prod.snd '' K := by
+  constructor
+  · rintro ⟨x,hx⟩; exact ⟨(x,s),hx,rfl⟩
+  · rintro ⟨⟨x,t⟩,hx,rfl⟩; exact ⟨x,hx⟩
+
+lemma horizontalSlice_minkowski_subset (K M : Set (E × ℝ)) (a b s t : ℝ) :
+    a • horizontalSlice K s+b • horizontalSlice M t ⊆
+      horizontalSlice (a • K+b • M) (a*s+b*t) := by
+  rintro _ ⟨_,⟨x,hx,rfl⟩,_,⟨y,hy,rfl⟩,rfl⟩
+  exact ⟨a • (x,s),smul_mem_smul_set hx,b • (y,t),smul_mem_smul_set hy,rfl⟩
+
+lemma horizontalSlice_combo_subset {K : Set (E × ℝ)} (hK : Convex ℝ K)
+    {a b : ℝ} (ha : 0≤a) (hb : 0≤b) (hab : a+b=1) (s t : ℝ) :
+    a • horizontalSlice K s+b • horizontalSlice K t ⊆
+      horizontalSlice K (a*s+b*t) := by
+  rintro _ ⟨_,⟨x,hx,rfl⟩,_,⟨y,hy,rfl⟩,rfl⟩
+  exact hK hx hy ha hb hab
+
+omit [NormedSpace ℝ E] in
+lemma interior_slice_of_interior {K : Set (E × ℝ)} {x : E} {s : ℝ}
+    (hx : (x,s) ∈ interior K) : x ∈ interior (horizontalSlice K s) :=
+  preimage_interior_subset_interior_preimage (by fun_prop) hx
+
+/-- A projection point strictly between two attained heights is a height of an
+interior point. No smoothness of the convex body is imposed. -/
+lemma interior_point_at_height {K : Set (E × ℝ)} (hK : Convex ℝ K)
+    (hi : (interior K).Nonempty) {l r s : ℝ}
+    (hl : l ∈ Prod.snd '' K) (hr : r ∈ Prod.snd '' K) (hs : s ∈ Ioo l r) :
+    ∃ x : E, (x,s) ∈ interior K := by
+  obtain ⟨⟨x0,s0⟩,h0⟩ := hi
+  obtain ⟨⟨xl,sl⟩,hxl,hl⟩ := hl
+  obtain ⟨⟨xr,sr⟩,hxr,hr⟩ := hr
+  change sl=l at hl
+  change sr=r at hr
+  subst sl; subst sr
+  obtain h | h | h := lt_trichotomy s s0
+  · let t := (s0-s)/(s0-l)
+    have hd : 0<s0-l := by linarith [hs.1]
+    have ht0 : 0≤t := (div_pos (by linarith) hd).le
+    have ht1 : t<1 := (div_lt_one hd).mpr (by linarith [hs.1])
+    have hc := hK.combo_interior_self_mem_interior h0 hxl (by linarith : 0<1-t) ht0 (by ring : (1-t)+t=1)
+    refine ⟨(1-t) • x0+t • xl,?_⟩
+    convert hc using 1
+    ext
+    · rfl
+    · dsimp [t]
+      field_simp
+      ring
+  · subst s
+    exact ⟨x0,h0⟩
+  · let t := (s-s0)/(r-s0)
+    have hd : 0<r-s0 := by linarith [hs.2]
+    have ht0 : 0≤t := (div_pos (by linarith) hd).le
+    have ht1 : t<1 := (div_lt_one hd).mpr (by linarith [hs.2])
+    have hc := hK.combo_interior_self_mem_interior h0 hxr (by linarith : 0<1-t) ht0 (by ring : (1-t)+t=1)
+    refine ⟨(1-t) • x0+t • xr,?_⟩
+    convert hc using 1
+    ext
+    · rfl
+    · dsimp [t]
+      field_simp
+      ring
+
+lemma horizontalSlice_interior_nonempty {K : Set (E × ℝ)} (hK : Convex ℝ K)
+    (hi : (interior K).Nonempty) {l r s : ℝ}
+    (hl : l ∈ Prod.snd '' K) (hr : r ∈ Prod.snd '' K) (hs : s ∈ Ioo l r) :
+    (interior (horizontalSlice K s)).Nonempty := by
+  obtain ⟨x,hx⟩ := interior_point_at_height hK hi hl hr hs
+  exact ⟨x,interior_slice_of_interior hx⟩
+
+lemma vertical_projection_interval {K : Set (E × ℝ)} (hK : IsCompact K)
+    (hc : Convex ℝ K) (hi : (interior K).Nonempty) :
+    ∃ l r : ℝ, l<r ∧ Prod.snd '' K=Icc l r := by
+  have hn : K.Nonempty := hi.mono interior_subset
+  refine ⟨sInf (Prod.snd '' K),sSup (Prod.snd '' K),?_,
+    compact_convex_real_interval (hK.image continuous_snd)
+      (hc.linear_image (LinearMap.snd ℝ E ℝ)) (hn.image _)⟩
+  have hInt : (interior (Prod.snd '' K)).Nonempty :=
+    (hi.image Prod.snd).mono (isOpenMap_snd.image_interior_subset K)
+  rw [compact_convex_real_interval (hK.image continuous_snd)
+    (hc.linear_image (LinearMap.snd ℝ E ℝ)) (hn.image _),interior_Icc] at hInt
+  exact nonempty_Ioo.mp hInt
+
+end
+end PettyProjection
+end
+
+noncomputable section
+open Set MeasureTheory Filter Topology Function
+open scoped Pointwise
+namespace PettyProjection
+section
+variable {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E]
+    [FiniteDimensional ℝ E] [MeasureSpace E] [BorelSpace E]
+    [Measure.IsAddHaarMeasure (volume : Measure E)]
+
+def sliceDensity (K : Set (E × ℝ)) (s : ℝ) : ℝ := volume.real (horizontalSlice K s)
+
+omit [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E]
+  [BorelSpace E] [Measure.IsAddHaarMeasure (volume : Measure E)] in
+lemma sliceDensity_nonneg (K : Set (E × ℝ)) (s : ℝ) : 0 ≤ sliceDensity K s := measureReal_nonneg
+
+omit [FiniteDimensional ℝ E] [BorelSpace E] in
+lemma sliceDensity_pos {K : Set (E × ℝ)} (hK : IsCompact K)
+    (hc : Convex ℝ K) (hi : (interior K).Nonempty) {l r s : ℝ}
+    (hl : l ∈ Prod.snd '' K) (hr : r ∈ Prod.snd '' K) (hs : s ∈ Ioo l r) :
+    0 < sliceDensity K s := by
+  apply ENNReal.toReal_pos (ne_of_gt ?_) (horizontalSlice_compact hK s).measure_ne_top
+  exact lt_of_lt_of_le (isOpen_interior.measure_pos volume
+    (horizontalSlice_interior_nonempty hc hi hl hr hs)) (measure_mono interior_subset)
+
+omit [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E]
+  [BorelSpace E] [Measure.IsAddHaarMeasure (volume : Measure E)] in
+lemma sliceDensity_zero_of_notMem {K : Set (E × ℝ)} {s : ℝ}
+    (hs : s ∉ Prod.snd '' K) : sliceDensity K s=0 := by
+  have he : horizontalSlice K s=∅ := not_nonempty_iff_eq_empty.mp
+    (fun h => hs (horizontalSlice_nonempty_iff.mp h))
+  simp [sliceDensity,he]
+
+omit [NormedSpace ℝ E] [FiniteDimensional ℝ E] [Measure.IsAddHaarMeasure (volume : Measure E)] in
+lemma sliceDensity_indicator_integral {K : Set (E × ℝ)} (hK : IsCompact K) (s : ℝ) :
+    (∫ x : E, K.indicator (fun _ => (1:ℝ)) (x,s))=sliceDensity K s := by
+  have he : (fun x : E => K.indicator (fun _ => (1:ℝ)) (x,s)) =
+      (horizontalSlice K s).indicator (fun _ => (1:ℝ)) := by
+    funext x
+    by_cases hx : (x,s) ∈ K
+    · rw [Set.indicator_of_mem hx,Set.indicator_of_mem (show x ∈ horizontalSlice K s from hx)]
+    · rw [Set.indicator_of_notMem hx,Set.indicator_of_notMem (show x ∉ horizontalSlice K s from hx)]
+  rw [he,integral_indicator (horizontalSlice_compact hK s).measurableSet,setIntegral_const]
+  simp only [smul_eq_mul,mul_one,sliceDensity]
+
+lemma sliceDensity_integrable {K : Set (E × ℝ)} (hK : IsCompact K) :
+    Integrable (sliceDensity K) := by
+  have hi : Integrable (K.indicator (fun _ => (1:ℝ))) (volume.prod volume) := by
+    rw [← Measure.volume_eq_prod]
+    exact (integrableOn_const hK.measure_ne_top).integrable_indicator hK.measurableSet
+  exact hi.integral_prod_right.congr (Eventually.of_forall fun s => sliceDensity_indicator_integral hK s)
+
+lemma sliceDensity_integral {K : Set (E × ℝ)} (hK : IsCompact K) :
+    (∫ s, sliceDensity K s)=volume.real K := by
+  have hi : Integrable (K.indicator (fun _ => (1:ℝ))) (volume.prod volume) := by
+    rw [← Measure.volume_eq_prod]
+    exact (integrableOn_const hK.measure_ne_top).integrable_indicator hK.measurableSet
+  have he := integral_prod_symm (K.indicator (fun _ => (1:ℝ))) hi
+  simp_rw [sliceDensity_indicator_integral hK] at he
+  rw [← he,← Measure.volume_eq_prod,integral_indicator hK.measurableSet,setIntegral_const]
+  simp only [smul_eq_mul,mul_one]
+
+lemma sliceDensity_interval_integral {K : Set (E × ℝ)} (hK : IsCompact K) {l r : ℝ}
+    (hlr : l≤r) (hp : Prod.snd '' K ⊆ Icc l r) :
+    (∫ s in l..r, sliceDensity K s)=volume.real K := by
+  rw [intervalIntegral.integral_of_le hlr,← integral_Icc_eq_integral_Ioc]
+  rw [setIntegral_eq_integral_of_forall_compl_eq_zero]
+  · exact sliceDensity_integral hK
+  · intro s hs
+    exact sliceDensity_zero_of_notMem (fun h => hs (hp h))
+
+end
+end PettyProjection
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/ProjectionBodies/SphericalCore.lean b/lean/OAI/Geometry/ProjectionBodies/SphericalCore.lean
new file mode 100644
index 0000000000000000000000000000000000000000..db55df52dfa4de3a266ed5c899b51b4c46a3058d
--- /dev/null
+++ b/lean/OAI/Geometry/ProjectionBodies/SphericalCore.lean
@@ -0,0 +1,1013 @@
+import OAI.Geometry.ProjectionBodies.Model
+
+namespace OAI
+
+universe uι
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology
+open scoped ENNReal RealInnerProductSpace Pointwise
+
+namespace PettyProjection.Spherical
+
+abbrev Space (n : ℕ) := EuclideanSpace ℝ (Fin n)
+abbrev Sphere (n : ℕ) := sphere (0 : Space n) 1
+
+def surface (n : ℕ) : Measure (Sphere n) :=
+  (volume : Measure (Space n)).toSphere
+
+def sigma (n : ℕ) : Measure (Sphere n) :=
+  (surface n univ)⁻¹ • surface n
+
+instance surface_finite (n : ℕ) : IsFiniteMeasure (surface n) := by
+  unfold surface
+  infer_instance
+
+lemma surface_ne_zero (n : ℕ) [NeZero n] : surface n ≠ 0 := by
+  exact Measure.toSphere_ne_zero _
+
+instance sigma_probability (n : ℕ) [NeZero n] : IsProbabilityMeasure (sigma n) := by
+  constructor
+  rw [sigma, Measure.smul_apply, smul_eq_mul]
+  exact ENNReal.inv_mul_cancel
+    (Measure.measure_univ_ne_zero.mpr (surface_ne_zero n)) (measure_ne_top _ _)
+
+instance surface_open_pos (n : ℕ) : (surface n).IsOpenPosMeasure := by
+  unfold surface
+  infer_instance
+
+instance sigma_open_pos (n : ℕ) : (sigma n).IsOpenPosMeasure := by
+  exact Measure.isOpenPosMeasure_smul _ (ENNReal.inv_ne_zero.mpr (measure_ne_top _ _))
+
+/-- Spherical means in the probability normalization. -/
+def mean {n : ℕ} (f : Sphere n → ℝ) : ℝ := ∫ u, f u ∂sigma n
+
+lemma mean_const (n : ℕ) [NeZero n] (c : ℝ) : mean (fun _ : Sphere n => c) = c := by
+  simp [mean]
+
+instance sphere_compact (n : ℕ) : CompactSpace (Sphere n) :=
+  isCompact_iff_compactSpace.mp (isCompact_sphere (0 : Space n) 1)
+
+lemma norm_coe {n : ℕ} (u : Sphere n) : ‖(u : Space n)‖ = 1 := by
+  simpa only [mem_sphere, dist_zero_right] using u.property
+
+/-- Continuous functions are put into the actual spherical L² space. -/
+abbrev H (n : ℕ) := Lp ℝ 2 (sigma n)
+
+def toH (n : ℕ) [NeZero n] : C(Sphere n, ℝ) →L[ℝ] H n :=
+  ContinuousMap.toLp 2 (sigma n) ℝ
+
+/-- Restrictions of ambient coordinate quadratics. Their span contains the
+constants on the unit sphere and all degree-two spherical harmonics. -/
+def coordinateProduct (n : ℕ) (i j : Fin n) : C(Sphere n, ℝ) where
+  toFun u := (u : Space n) i * (u : Space n) j
+  continuous_toFun := by fun_prop
+
+def lowSpace (n : ℕ) [NeZero n] : Submodule ℝ (H n) :=
+  Submodule.span ℝ (Set.range fun ij : Fin n × Fin n =>
+    toH n (coordinateProduct n ij.1 ij.2))
+
+instance lowSpace_finite (n : ℕ) [NeZero n] : FiniteDimensional ℝ (lowSpace n) :=
+  FiniteDimensional.span_of_finite ℝ (Set.finite_range _)
+
+/-- The complement of the constants and the quadratics. For even input this
+is exactly the even tail of degree at least four. -/
+def Q (n : ℕ) [NeZero n] : H n →L[ℝ] H n :=
+  (lowSpace n)ᗮ.starProjection
+
+lemma sum_coordinateProduct (n : ℕ) :
+    ∑ i : Fin n, coordinateProduct n i i = ContinuousMap.const (Sphere n) 1 := by
+  ext u
+  have h := EuclideanSpace.real_norm_sq_eq (u : Space n)
+  rw [norm_coe, one_pow] at h
+  simpa [coordinateProduct, sq] using h.symm
+
+lemma constant_mem_lowSpace (n : ℕ) [NeZero n] (c : ℝ) :
+    toH n (ContinuousMap.const (Sphere n) c) ∈ lowSpace n := by
+  have h1 : toH n (ContinuousMap.const (Sphere n) 1) ∈ lowSpace n := by
+    rw [← sum_coordinateProduct, map_sum]
+    apply Submodule.sum_mem
+    intro i _
+    exact Submodule.subset_span ⟨(i, i), rfl⟩
+  have hf : ContinuousMap.const (Sphere n) c =
+      c • ContinuousMap.const (Sphere n) (1 : ℝ) := by ext; simp
+  rw [hf, map_smul]
+  exact (lowSpace n).smul_mem c h1
+
+lemma Q_eq_sub (n : ℕ) [NeZero n] (f : H n) :
+    Q n f = f - (lowSpace n).starProjection f :=
+  Submodule.starProjection_orthogonal_val f
+
+lemma Q_constant (n : ℕ) [NeZero n] (c : ℝ) :
+    Q n (toH n (ContinuousMap.const (Sphere n) c)) = 0 := by
+  rw [Q_eq_sub, Submodule.starProjection_eq_self_iff.mpr (constant_mem_lowSpace n c),
+    sub_self]
+
+lemma norm_Q_le (n : ℕ) [NeZero n] (f : H n) : ‖Q n f‖ ≤ ‖f‖ :=
+  (lowSpace n)ᗮ.norm_starProjection_apply_le f
+
+lemma inner_toH (n : ℕ) [NeZero n] (f g : C(Sphere n, ℝ)) :
+    ⟪toH n f, toH n g⟫ = mean (fun u => f u * g u) := by
+  simpa [toH, mean, mul_comm] using ContinuousMap.inner_toLp (sigma n) f g
+
+lemma norm_toH_sq (n : ℕ) [NeZero n] (f : C(Sphere n, ℝ)) :
+    ‖toH n f‖ ^ 2 = mean (fun u => (f u) ^ 2) := by
+  rw [← real_inner_self_eq_norm_sq, inner_toH]
+  simp [sq]
+
+lemma norm_Q_le_centered (n : ℕ) [NeZero n] (f : C(Sphere n, ℝ)) (c : ℝ) :
+    ‖Q n (toH n f)‖ ^ 2 ≤ mean (fun u => (f u - c) ^ 2) := by
+  have h := norm_Q_le n (toH n (f - ContinuousMap.const (Sphere n) c))
+  have hQ : Q n (toH n (f - ContinuousMap.const (Sphere n) c)) = Q n (toH n f) := by
+    simp only [map_sub, Q_constant, sub_zero]
+  rw [hQ] at h
+  have hh := (sq_le_sq₀ (norm_nonneg (Q n (toH n f)))
+    (norm_nonneg (toH n (f - ContinuousMap.const (Sphere n) c)))).mpr h
+  simpa only [norm_toH_sq, ContinuousMap.sub_apply, ContinuousMap.const_apply] using hh
+
+/-- Orthogonal changes of variables on the actual unit sphere. -/
+def sphereMap {n : ℕ} (U : Space n ≃ₗᵢ[ℝ] Space n) : Sphere n ≃ₜ Sphere n where
+  toFun u := ⟨U u, by simpa only [mem_sphere, dist_zero_right, U.norm_map] using norm_coe u⟩
+  invFun u := ⟨U.symm u, by simpa only [mem_sphere, dist_zero_right, U.symm.norm_map] using norm_coe u⟩
+  left_inv u := by ext; simp
+  right_inv u := by ext; simp
+  continuous_toFun := (U.continuous.comp continuous_subtype_val).subtype_mk _
+  continuous_invFun := (U.symm.continuous.comp continuous_subtype_val).subtype_mk _
+
+lemma cone_preimage {n : ℕ} (U : Space n ≃ₗᵢ[ℝ] Space n) (s : Set (Sphere n)) :
+    Ioo (0 : ℝ) 1 • (Subtype.val '' (sphereMap U ⁻¹' s)) =
+      U ⁻¹' (Ioo (0 : ℝ) 1 • (Subtype.val '' s)) := by
+  ext x
+  constructor
+  · rintro ⟨r, hr, y, ⟨u, hu, rfl⟩, rfl⟩
+    exact ⟨r, hr, U u, ⟨sphereMap U u, hu, rfl⟩, by simp⟩
+  · rintro ⟨r, hr, y, ⟨v, hv, rfl⟩, heq⟩
+    refine ⟨r, hr, U.symm v, ⟨(sphereMap U).symm v, ?_, rfl⟩, ?_⟩
+    · simpa only [mem_preimage, Homeomorph.apply_symm_apply] using hv
+    · have hh := congrArg U.symm heq
+      simpa only [map_smul, U.symm_apply_apply] using hh
+
+lemma surface_preserving {n : ℕ} (U : Space n ≃ₗᵢ[ℝ] Space n) :
+    MeasurePreserving (sphereMap U) (surface n) (surface n) := by
+  refine ⟨(sphereMap U).measurable, ?_⟩
+  ext s hs
+  rw [Measure.map_apply (sphereMap U).measurable hs, surface,
+    Measure.toSphere_apply' _ ((sphereMap U).measurable hs),
+    Measure.toSphere_apply' _ hs, cone_preimage]
+  rw [U.measurePreserving.measure_preimage_emb U.toHomeomorph.measurableEmbedding]
+
+lemma sigma_preserving {n : ℕ} (U : Space n ≃ₗᵢ[ℝ] Space n) :
+    MeasurePreserving (sphereMap U) (sigma n) (sigma n) :=
+  (surface_preserving U).smul_measure _
+
+lemma mean_rotation {n : ℕ} (U : Space n ≃ₗᵢ[ℝ] Space n) (f : Sphere n → ℝ) :
+    mean (fun u => f (sphereMap U u)) = mean f :=
+  (sigma_preserving U).integral_comp (sphereMap U).measurableEmbedding f
+
+local instance (n : ℕ) : CompleteSpace (Space n →L[ℝ] Space n) :=
+  FiniteDimensional.complete ℝ _
+
+local instance (n : ℕ) : NormedAlgebra ℚ (Space n →L[ℝ] Space n) :=
+  NormedAlgebra.restrictScalars ℚ ℝ _
+
+/-- Exponentiating an actual skew-adjoint operator produces rotations. -/
+def rotation {n : ℕ} (T : Space n →L[ℝ] Space n) (hT : star T = -T) (t : ℝ) :
+    Space n ≃ₗᵢ[ℝ] Space n :=
+  Unitary.linearIsometryEquiv ⟨NormedSpace.exp (t • T),
+    NormedSpace.exp_mem_unitary_of_mem_skewAdjoint (by
+      rw [skewAdjoint.mem_iff, star_smul, star_trivial, hT, smul_neg])⟩
+
+lemma rotation_apply {n : ℕ} (T : Space n →L[ℝ] Space n) (hT : star T = -T)
+    (t : ℝ) (x : Space n) : rotation T hT t x = NormedSpace.exp (t • T) x := rfl
+
+lemma rotation_zero {n : ℕ} (T : Space n →L[ℝ] Space n) (hT : star T = -T)
+    (x : Space n) : rotation T hT 0 x = x := by
+  rw [rotation_apply]
+  have hz : (0 : ℝ) • T = 0 := zero_smul ℝ T
+  rw [hz, NormedSpace.exp_zero]
+  rfl
+
+lemma hasDerivAt_rotation {n : ℕ} (T : Space n →L[ℝ] Space n) (hT : star T = -T)
+    (t : ℝ) (x : Space n) :
+    HasDerivAt (fun s => rotation T hT s x) (T (rotation T hT t x)) t := by
+  simpa only [mul_apply_eq_comp, map_zero, add_zero, rotation_apply] using
+    (hasDerivAt_exp_smul_const' T t).clm_apply (hasDerivAt_const t x)
+
+lemma continuous_integrable {n : ℕ} [NeZero n] {f : Sphere n → ℝ}
+    (hf : Continuous f) : Integrable f (sigma n) := by
+  simpa only [integrableOn_univ] using
+    hf.continuousOn.integrableOn_compact (μ := sigma n) isCompact_univ
+
+/-- Infinitesimal rotation invariance, with differentiation justified using
+uniform derivative bounds on the compact sphere. This supplies spherical
+integration by parts without postulating it as an interface. -/
+lemma mean_fderiv_skew {n : ℕ} [NeZero n] (T : Space n →L[ℝ] Space n)
+    (hT : star T = -T) (f : Space n → ℝ) (hf : ContDiff ℝ 1 f) :
+    mean (fun u : Sphere n => fderiv ℝ f u (T u)) = 0 := by
+  let df : C(Sphere n, Space n →L[ℝ] ℝ) :=
+    ⟨fun u => fderiv ℝ f u, (hf.continuous_fderiv (by norm_num)).comp continuous_subtype_val⟩
+  let F : ℝ → Sphere n → ℝ := fun t u => f (rotation T hT t u)
+  let F' : ℝ → Sphere n → ℝ :=
+    fun t u => fderiv ℝ f (rotation T hT t u) (T (rotation T hT t u))
+  have hFc (t : ℝ) : Continuous (F t) :=
+    hf.continuous.comp ((rotation T hT t).continuous.comp continuous_subtype_val)
+  have hF'c (t : ℝ) : Continuous (F' t) := by
+    exact ((hf.continuous_fderiv (by norm_num)).comp
+      ((rotation T hT t).continuous.comp continuous_subtype_val)).clm_apply
+      (T.continuous.comp ((rotation T hT t).continuous.comp continuous_subtype_val))
+  have hbound (t : ℝ) (u : Sphere n) : ‖F' t u‖ ≤ ‖df‖ * ‖T‖ := by
+    let v : Sphere n := sphereMap (rotation T hT t) u
+    calc
+      ‖F' t u‖ ≤ ‖fderiv ℝ f v‖ * ‖T v‖ := ContinuousLinearMap.le_opNorm _ _
+      _ ≤ ‖df‖ * (‖T‖ * ‖(v : Space n)‖) :=
+        mul_le_mul (ContinuousMap.norm_coe_le_norm df v)
+          (T.le_opNorm v) (norm_nonneg _) (norm_nonneg df)
+      _ = ‖df‖ * ‖T‖ := by rw [norm_coe, mul_one]
+  have hderiv (t : ℝ) (u : Sphere n) : HasDerivAt (fun s => F s u) (F' t u) t := by
+    exact (hf.differentiable (by norm_num) (rotation T hT t u)).hasFDerivAt.comp_hasDerivAt t
+      (hasDerivAt_rotation T hT t u)
+  have hI := hasDerivAt_integral_of_dominated_loc_of_deriv_le
+    (μ := sigma n) (x₀ := (0 : ℝ)) (s := univ) (F := F) (F' := F')
+    (bound := fun _ => ‖df‖ * ‖T‖) (Filter.univ_mem)
+    (Filter.Eventually.of_forall fun t => (hFc t).aestronglyMeasurable)
+    (continuous_integrable (hFc 0)) (hF'c 0).aestronglyMeasurable
+    (ae_of_all _ fun u t _ => hbound t u) (integrable_const _)
+    (ae_of_all _ fun u t _ => hderiv t u)
+  have hconst : (fun t => ∫ u, F t u ∂sigma n) = fun _ : ℝ => mean (fun u : Sphere n => f u) := by
+    funext t
+    exact mean_rotation (rotation T hT t) (fun u : Sphere n => f u)
+  rw [hconst] at hI
+  have hz := hI.2.unique (hasDerivAt_const (0 : ℝ) (mean (fun u : Sphere n => f u)))
+  simpa only [F', rotation_zero, mean] using hz
+
+/-- Derivative along an infinitesimal linear rotation. -/
+def rotDeriv {n : ℕ} (T : Space n →L[ℝ] Space n) (f : Space n → ℝ) : Space n → ℝ :=
+  fun x => fderiv ℝ f x (T x)
+
+lemma continuous_rotDeriv {n : ℕ} (T : Space n →L[ℝ] Space n)
+    {f : Space n → ℝ} (hf : ContDiff ℝ 1 f) : Continuous (rotDeriv T f) :=
+  (hf.continuous_fderiv (by norm_num)).clm_apply T.continuous
+
+lemma contDiff_rotDeriv {n : ℕ} (T : Space n →L[ℝ] Space n)
+    {f : Space n → ℝ} (hf : ContDiff ℝ 2 f) : ContDiff ℝ 1 (rotDeriv T f) :=
+  (hf.fderiv_right (by norm_num)).clm_apply T.contDiff
+
+lemma rotDeriv_mul {n : ℕ} (T : Space n →L[ℝ] Space n)
+    {f g : Space n → ℝ} (hf : Differentiable ℝ f) (hg : Differentiable ℝ g) (x : Space n) :
+    rotDeriv T (fun y => f y * g y) x =
+      f x * rotDeriv T g x + g x * rotDeriv T f x := by
+  simp only [rotDeriv, fderiv_fun_mul (hf x) (hg x), add_apply,
+    smul_apply, smul_eq_mul]
+
+/-- Genuine spherical integration by parts for each rotation generator. -/
+lemma mean_rotDeriv_mul {n : ℕ} [NeZero n] (T : Space n →L[ℝ] Space n)
+    (hT : star T = -T) {f g : Space n → ℝ}
+    (hf : ContDiff ℝ 1 f) (hg : ContDiff ℝ 1 g) :
+    mean (fun u : Sphere n => f u * rotDeriv T g u) =
+      - mean (fun u : Sphere n => g u * rotDeriv T f u) := by
+  have h := mean_fderiv_skew T hT (fun x => f x * g x) (hf.mul hg)
+  change mean (fun u : Sphere n => rotDeriv T (fun x => f x * g x) u) = 0 at h
+  simp_rw [rotDeriv_mul T (hf.differentiable (by norm_num))
+    (hg.differentiable (by norm_num))] at h
+  have hi₁ := continuous_integrable
+    ((hf.continuous.comp continuous_subtype_val).mul
+      ((continuous_rotDeriv T hg).comp continuous_subtype_val))
+  have hi₂ := continuous_integrable
+    ((hg.continuous.comp continuous_subtype_val).mul
+      ((continuous_rotDeriv T hf).comp continuous_subtype_val))
+  change Integrable (fun u : Sphere n => f u * rotDeriv T g u) (sigma n) at hi₁
+  change Integrable (fun u : Sphere n => g u * rotDeriv T f u) (sigma n) at hi₂
+  rw [mean, integral_add hi₁ hi₂] at h
+  exact eq_neg_of_add_eq_zero_left h
+
+/-- The standard skew generator in the i,j plane. -/
+def generator (n : ℕ) (i j : Fin n) : Space n →L[ℝ] Space n :=
+  InnerProductSpace.rankOne ℝ (EuclideanSpace.single j 1) (EuclideanSpace.single i 1) -
+    InnerProductSpace.rankOne ℝ (EuclideanSpace.single i 1) (EuclideanSpace.single j 1)
+
+lemma generator_skew (n : ℕ) (i j : Fin n) :
+    star (generator n i j) = -generator n i j := by
+  simp only [generator, ContinuousLinearMap.star_eq_adjoint, map_sub,
+    InnerProductSpace.adjoint_rankOne, neg_sub]
+
+lemma generator_apply {n : ℕ} (i j : Fin n) (x : Space n) :
+    generator n i j x = x i • EuclideanSpace.single j 1 - x j • EuclideanSpace.single i 1 := by
+  simp [generator, EuclideanSpace.inner_single_left]
+
+lemma generator_apply_coord {n : ℕ} (i j a : Fin n) (x : Space n) :
+    (generator n i j x) a = x i * (if j = a then 1 else 0) -
+      x j * (if i = a then 1 else 0) := by
+  simp [generator_apply, PiLp.single_apply, eq_comm]
+
+lemma generator_self {n : ℕ} (i : Fin n) : generator n i i = 0 := by
+  simp [generator]
+
+lemma generator_swap {n : ℕ} (i j : Fin n) : generator n j i = -generator n i j := by
+  simp [generator, neg_sub]
+
+/-- The Dirichlet form, expressed without local coordinates by rotations. -/
+def energy {n : ℕ} (f g : Space n → ℝ) : ℝ :=
+  (1 / 2 : ℝ) * ∑ i : Fin n, ∑ j : Fin n,
+    mean (fun u : Sphere n => rotDeriv (generator n i j) f u *
+      rotDeriv (generator n i j) g u)
+
+/-- The spherical Laplacian expressed by its rotation generators. -/
+def casimir {n : ℕ} (f : Space n → ℝ) : Space n → ℝ := fun x =>
+  (1 / 2 : ℝ) * ∑ i : Fin n, ∑ j : Fin n,
+    rotDeriv (generator n i j) (rotDeriv (generator n i j) f) x
+
+lemma energy_symm {n : ℕ} (f g : Space n → ℝ) : energy f g = energy g f := by
+  simp only [energy, mul_comm]
+
+lemma energy_nonneg {n : ℕ} (f : Space n → ℝ) : 0 ≤ energy f f := by
+  apply mul_nonneg (by norm_num)
+  apply Finset.sum_nonneg
+  intro i _
+  apply Finset.sum_nonneg
+  intro j _
+  exact integral_nonneg fun _ => mul_self_nonneg _
+
+lemma mean_sum {n : ℕ} [NeZero n] {ι : Type uι} (s : Finset ι)
+    {f : ι → Sphere n → ℝ} (hf : ∀ i ∈ s, Continuous (f i)) :
+    mean (fun u => ∑ i ∈ s, f i u) = ∑ i ∈ s, mean (f i) := by
+  exact integral_finsetSum s fun i hi => continuous_integrable (hf i hi)
+
+lemma mean_casimir {n : ℕ} [NeZero n] {f g : Space n → ℝ}
+    (hf : ContDiff ℝ 1 f) (hg : ContDiff ℝ 2 g) :
+    mean (fun u : Sphere n => f u * casimir g u) = -energy f g := by
+  have hc (i j : Fin n) : Continuous (fun u : Sphere n =>
+      f u * rotDeriv (generator n i j) (rotDeriv (generator n i j) g) u) :=
+    (hf.continuous.comp continuous_subtype_val).mul
+      ((continuous_rotDeriv _ (contDiff_rotDeriv _ hg)).comp continuous_subtype_val)
+  have heq : (fun u : Sphere n => f u * casimir g u) =
+      fun u : Sphere n => (1 / 2 : ℝ) * ∑ i : Fin n, ∑ j : Fin n,
+        f u * rotDeriv (generator n i j) (rotDeriv (generator n i j) g) u := by
+    funext u
+    simp only [casimir, mul_left_comm (f u) (1 / 2 : ℝ), Finset.mul_sum]
+  rw [heq, mean, integral_const_mul]
+  change (1 / 2 : ℝ) * mean (fun u : Sphere n => ∑ i : Fin n, ∑ j : Fin n,
+      f u * rotDeriv (generator n i j) (rotDeriv (generator n i j) g) u) = _
+  rw [mean_sum _ (fun i _ => continuous_finsetSum _ fun j _ => hc i j)]
+  rw [show -energy f g = (1 / 2 : ℝ) * ∑ i : Fin n, ∑ j : Fin n,
+      -mean (fun u : Sphere n => rotDeriv (generator n i j) f u *
+        rotDeriv (generator n i j) g u) by
+    simp [energy, Finset.sum_neg_distrib]]
+  congr 1
+  apply Finset.sum_congr rfl
+  intro i _
+  rw [mean_sum _ (fun j _ => hc i j)]
+  apply Finset.sum_congr rfl
+  intro j _
+  rw [mean_rotDeriv_mul _ (generator_skew n i j) hf (contDiff_rotDeriv _ hg)]
+  simp only [mul_comm]
+
+/-- Ambient evaluation of a polynomial, using the Euclidean coordinates. -/
+def polynomialFunction {n : ℕ} (p : MvPolynomial (Fin n) ℝ) : Space n → ℝ :=
+  fun x => MvPolynomial.eval (fun i => x i) p
+
+lemma polynomialFunction_contDiff {n : ℕ} (p : MvPolynomial (Fin n) ℝ) :
+    ContDiff ℝ ⊤ (polynomialFunction p) := by
+  unfold polynomialFunction
+  induction p using MvPolynomial.induction_on with
+  | C c => simpa only [MvPolynomial.eval_C] using (contDiff_const : ContDiff ℝ ⊤ (fun _ : Space n => c))
+  | add p q hp hq => simpa only [MvPolynomial.eval_add] using hp.add hq
+  | mul_X p i hp =>
+    have hi : ContDiff ℝ ⊤ (fun x : Space n => x i) := (EuclideanSpace.proj (𝕜 := ℝ) i).contDiff
+    simpa only [MvPolynomial.eval_mul, MvPolynomial.eval_X] using hp.mul hi
+
+/-- Polynomial rotation differentiation, a derivation preserving degree. -/
+def polynomialRotation {n : ℕ} (i j : Fin n) (p : MvPolynomial (Fin n) ℝ) :
+    MvPolynomial (Fin n) ℝ :=
+  MvPolynomial.X i * MvPolynomial.pderiv j p - MvPolynomial.X j * MvPolynomial.pderiv i p
+
+lemma rotDeriv_add {n : ℕ} (T : Space n →L[ℝ] Space n)
+    {f g : Space n → ℝ} (hf : Differentiable ℝ f) (hg : Differentiable ℝ g) (x : Space n) :
+    rotDeriv T (fun y => f y + g y) x = rotDeriv T f x + rotDeriv T g x := by
+  simp only [rotDeriv, fderiv_fun_add (hf x) (hg x), add_apply]
+
+lemma rotDeriv_const {n : ℕ} (T : Space n →L[ℝ] Space n) (c : ℝ) (x : Space n) :
+    rotDeriv T (fun _ => c) x = 0 := by
+  simp [rotDeriv]
+
+lemma rotDeriv_coordinate {n : ℕ} (i j a : Fin n) (x : Space n) :
+    rotDeriv (generator n i j) (fun y => y a) x =
+      x i * (if j = a then 1 else 0) - x j * (if i = a then 1 else 0) := by
+  change fderiv ℝ (EuclideanSpace.proj a) x (generator n i j x) = _
+  rw [ContinuousLinearMap.fderiv]
+  exact generator_apply_coord i j a x
+
+lemma polynomialRotation_add {n : ℕ} (i j : Fin n) (p q : MvPolynomial (Fin n) ℝ) :
+    polynomialRotation i j (p + q) = polynomialRotation i j p + polynomialRotation i j q := by
+  simp only [polynomialRotation, map_add]
+  ring
+
+lemma polynomialRotation_mul {n : ℕ} (i j : Fin n) (p q : MvPolynomial (Fin n) ℝ) :
+    polynomialRotation i j (p * q) =
+      p * polynomialRotation i j q + q * polynomialRotation i j p := by
+  simp only [polynomialRotation, MvPolynomial.pderiv_mul]
+  ring
+
+lemma polynomialRotation_C {n : ℕ} (i j : Fin n) (c : ℝ) :
+    polynomialRotation i j (MvPolynomial.C c) = 0 := by
+  simp [polynomialRotation]
+
+lemma polynomialFunction_add {n : ℕ} (p q : MvPolynomial (Fin n) ℝ) :
+    polynomialFunction (p + q) = fun x => polynomialFunction p x + polynomialFunction q x := by
+  funext x
+  exact MvPolynomial.eval_add
+
+lemma polynomialFunction_mul {n : ℕ} (p q : MvPolynomial (Fin n) ℝ) :
+    polynomialFunction (p * q) = fun x => polynomialFunction p x * polynomialFunction q x := by
+  funext x
+  exact MvPolynomial.eval_mul
+
+lemma polynomialFunction_X {n : ℕ} (i : Fin n) :
+    polynomialFunction (MvPolynomial.X i) = fun x : Space n => x i := by
+  funext x
+  exact MvPolynomial.eval_X _
+
+lemma polynomialFunction_C {n : ℕ} (c : ℝ) :
+    polynomialFunction (MvPolynomial.C c : MvPolynomial (Fin n) ℝ) = fun _ => c := by
+  funext x
+  exact MvPolynomial.eval_C _
+
+/-- The formal polynomial generator agrees with actual differentiation. -/
+lemma rotDeriv_polynomialFunction {n : ℕ} (i j : Fin n) (p : MvPolynomial (Fin n) ℝ) :
+    rotDeriv (generator n i j) (polynomialFunction p) =
+      polynomialFunction (polynomialRotation i j p) := by
+  induction p using MvPolynomial.induction_on with
+  | C c =>
+    rw [polynomialFunction_C, polynomialRotation_C]
+    funext x
+    simp [rotDeriv_const, polynomialFunction]
+  | add p q hp hq =>
+    rw [polynomialFunction_add, polynomialRotation_add, polynomialFunction_add]
+    funext x
+    rw [rotDeriv_add _ ((polynomialFunction_contDiff p).differentiable (by norm_num))
+      ((polynomialFunction_contDiff q).differentiable (by norm_num)), hp, hq]
+  | mul_X p a hp =>
+    rw [polynomialFunction_mul, polynomialRotation_mul, polynomialFunction_add,
+      polynomialFunction_mul, polynomialFunction_mul]
+    funext x
+    rw [rotDeriv_mul _ ((polynomialFunction_contDiff p).differentiable (by norm_num))
+      ((polynomialFunction_contDiff (MvPolynomial.X a)).differentiable (by norm_num)), hp]
+    congr 1
+    rw [polynomialFunction_X, rotDeriv_coordinate]
+    by_cases hia : i = a <;> by_cases hja : j = a <;>
+      simp [polynomialFunction, polynomialRotation, hia, hja, eq_comm]
+
+open MvPolynomial
+
+/-- Radius squared as an ambient polynomial. -/
+def radiusPolynomial (n : ℕ) : MvPolynomial (Fin n) ℝ := ∑ i : Fin n, X i ^ 2
+
+def eulerPolynomial {n : ℕ} (p : MvPolynomial (Fin n) ℝ) : MvPolynomial (Fin n) ℝ :=
+  ∑ i : Fin n, X i * pderiv i p
+
+def laplacePolynomial {n : ℕ} (p : MvPolynomial (Fin n) ℝ) : MvPolynomial (Fin n) ℝ :=
+  ∑ i : Fin n, pderiv i (pderiv i p)
+
+lemma polynomialRotation_twice {n : ℕ} (i j : Fin n) (p : MvPolynomial (Fin n) ℝ) :
+    polynomialRotation i j (polynomialRotation i j p) =
+      X i ^ 2 * pderiv j (pderiv j p) + X j ^ 2 * pderiv i (pderiv i p) -
+      X i * X j * pderiv j (pderiv i p) - X i * X j * pderiv i (pderiv j p) -
+      X i * pderiv i p - X j * pderiv j p +
+      (if i = j then 2 * (X i * pderiv i p) else 0) := by
+  by_cases h : i = j
+  · subst j
+    simp only [polynomialRotation, sub_self, map_zero, mul_zero, ite_true]
+    ring
+  · simp only [polynomialRotation, map_sub, MvPolynomial.pderiv_mul,
+      pderiv_X, Pi.single_apply, h, Ne.symm h, ite_false, ite_true, one_mul, zero_mul]
+    ring
+
+lemma eulerPolynomial_twice {n : ℕ} (p : MvPolynomial (Fin n) ℝ) :
+    eulerPolynomial (eulerPolynomial p) = eulerPolynomial p +
+      ∑ i : Fin n, ∑ j : Fin n, X i * X j * pderiv i (pderiv j p) := by
+  have h (i : Fin n) : (∑ j : Fin n, X i *
+      (pderiv i (X j) * pderiv j p + X j * pderiv i (pderiv j p))) =
+      X i * pderiv i p + ∑ j : Fin n, X i * X j * pderiv i (pderiv j p) := by
+    simp only [mul_add, ← mul_assoc, Finset.sum_add_distrib]
+    congr 1
+    simp [pderiv_X, Pi.single_apply, eq_comm]
+  simp only [eulerPolynomial, map_sum, MvPolynomial.pderiv_mul, Finset.mul_sum]
+  simp_rw [h, Finset.sum_add_distrib]
+
+lemma sum_polynomialRotation_twice {n : ℕ} (p : MvPolynomial (Fin n) ℝ) :
+    ∑ i : Fin n, ∑ j : Fin n, polynomialRotation i j (polynomialRotation i j p) =
+      2 * (radiusPolynomial n * laplacePolynomial p) -
+      2 * eulerPolynomial (eulerPolynomial p) -
+      2 * ((n : MvPolynomial (Fin n) ℝ) - 2) * eulerPolynomial p := by
+  have hA : (∑ i : Fin n, ∑ j : Fin n, X i ^ 2 * pderiv j (pderiv j p)) =
+      radiusPolynomial n * laplacePolynomial p := by
+    rw [radiusPolynomial, laplacePolynomial, Finset.sum_mul]
+    simp only [Finset.mul_sum]
+  have hB : (∑ i : Fin n, ∑ j : Fin n, X j ^ 2 * pderiv i (pderiv i p)) =
+      radiusPolynomial n * laplacePolynomial p := by
+    rw [Finset.sum_comm]
+    exact hA
+  have hC : (∑ i : Fin n, ∑ j : Fin n, X i * X j * pderiv j (pderiv i p)) =
+      ∑ i : Fin n, ∑ j : Fin n, X i * X j * pderiv i (pderiv j p) := by
+    rw [Finset.sum_comm]
+    apply Finset.sum_congr rfl
+    intro i _
+    apply Finset.sum_congr rfl
+    intro j _
+    rw [mul_comm (X j) (X i)]
+  have hD : (∑ i : Fin n, ∑ j : Fin n, X i * pderiv i p) =
+      (n : MvPolynomial (Fin n) ℝ) * eulerPolynomial p := by
+    simp [eulerPolynomial, nsmul_eq_mul, Finset.mul_sum]
+  have hE : (∑ i : Fin n, ∑ j : Fin n, X j * pderiv j p) =
+      (n : MvPolynomial (Fin n) ℝ) * eulerPolynomial p := by
+    rw [Finset.sum_comm]
+    exact hD
+  have hF : (∑ i : Fin n, ∑ j : Fin n,
+      if i = j then 2 * (X i * pderiv i p) else 0) = 2 * eulerPolynomial p := by
+    simp [eulerPolynomial, Finset.mul_sum]
+  simp only [polynomialRotation_twice, Finset.sum_add_distrib, Finset.sum_sub_distrib]
+  rw [hA, hB, hC, hD, hE, hF, eulerPolynomial_twice]
+  ring
+
+/-- The spherical Laplace eigenvalues. -/
+def eigenvalue (n d : ℕ) : ℝ := (d : ℝ) * ((d : ℝ) + n - 2)
+
+lemma eulerPolynomial_nsmul {n : ℕ} (d : ℕ) (p : MvPolynomial (Fin n) ℝ) :
+    eulerPolynomial (d • p) = d • eulerPolynomial p := by
+  simp only [eulerPolynomial, map_nsmul]
+  simp only [nsmul_eq_mul]
+  simp_rw [mul_left_comm (X _) (d : MvPolynomial (Fin n) ℝ)]
+  rw [← Finset.mul_sum]
+
+lemma eulerPolynomial_homogeneous {n d : ℕ} {p : MvPolynomial (Fin n) ℝ}
+    (hp : p.IsHomogeneous d) : eulerPolynomial p = d • p :=
+  hp.sum_X_mul_pderiv
+
+lemma sum_polynomialRotation_homogeneous {n d : ℕ} {p : MvPolynomial (Fin n) ℝ}
+    (hp : p.IsHomogeneous d) :
+    ∑ i : Fin n, ∑ j : Fin n, polynomialRotation i j (polynomialRotation i j p) =
+      2 * (radiusPolynomial n * laplacePolynomial p - C (eigenvalue n d) * p) := by
+  have he := eulerPolynomial_homogeneous hp
+  have hee : eulerPolynomial (eulerPolynomial p) = d • (d • p) := by
+    rw [he, eulerPolynomial_nsmul, he]
+  rw [sum_polynomialRotation_twice, hee, he]
+  simp only [nsmul_eq_mul, eigenvalue, map_mul, map_sub, map_add, map_natCast, map_ofNat]
+  ring
+
+lemma polynomialFunction_radius {n : ℕ} (x : Space n) :
+    polynomialFunction (radiusPolynomial n) x = ‖x‖ ^ 2 := by
+  unfold polynomialFunction radiusPolynomial
+  simp only [map_sum, map_pow, eval_X]
+  exact (EuclideanSpace.real_norm_sq_eq x).symm
+
+/-- Polar Laplacian identity, derived from finite rotation sums and Euler's
+identity rather than assumed as a spherical calculus interface. -/
+lemma casimir_polynomialFunction {n d : ℕ} {p : MvPolynomial (Fin n) ℝ}
+    (hp : p.IsHomogeneous d) (x : Space n) :
+    casimir (polynomialFunction p) x =
+      ‖x‖ ^ 2 * polynomialFunction (laplacePolynomial p) x -
+        eigenvalue n d * polynomialFunction p x := by
+  have hs := congrArg (MvPolynomial.eval (fun i => x i))
+    (sum_polynomialRotation_homogeneous hp)
+  simp only [map_sum, map_mul, map_sub, eval_C, map_ofNat] at hs
+  simp only [casimir, rotDeriv_polynomialFunction, polynomialFunction]
+  rw [hs]
+  have hr := polynomialFunction_radius x
+  unfold polynomialFunction at hr
+  rw [hr]
+  ring
+
+lemma casimir_harmonic {n d : ℕ} {p : MvPolynomial (Fin n) ℝ}
+    (hp : p.IsHomogeneous d) (hΔp : laplacePolynomial p = 0) (x : Space n) :
+    casimir (polynomialFunction p) x = -eigenvalue n d * polynomialFunction p x := by
+  rw [casimir_polynomialFunction hp, hΔp]
+  simp [polynomialFunction]
+
+lemma eigenvalue_strictMono {n : ℕ} (hn : 2 ≤ n) : StrictMono (eigenvalue n) := by
+  intro d e hde
+  have hn' : (2 : ℝ) ≤ n := by exact_mod_cast hn
+  have hde' : (d : ℝ) < e := by exact_mod_cast hde
+  have hd : (0 : ℝ) ≤ d := Nat.cast_nonneg _
+  have he : (0 : ℝ) ≤ e := Nat.cast_nonneg _
+  unfold eigenvalue
+  nlinarith
+
+/-- Distinct homogeneous harmonic degrees are orthogonal under actual
+normalized surface measure. -/
+lemma harmonic_orthogonal {n d e : ℕ} (hn : 2 ≤ n)
+    {p q : MvPolynomial (Fin n) ℝ} (hp : p.IsHomogeneous d) (hq : q.IsHomogeneous e)
+    (hΔp : laplacePolynomial p = 0) (hΔq : laplacePolynomial q = 0) (hde : d ≠ e) :
+    mean (fun u : Sphere n => polynomialFunction p u * polynomialFunction q u) = 0 := by
+  have : NeZero n := ⟨by omega⟩
+  have h₁ := mean_casimir (f := polynomialFunction p) (g := polynomialFunction q)
+    ((polynomialFunction_contDiff p).of_le le_top) ((polynomialFunction_contDiff q).of_le le_top)
+  have h₂ := mean_casimir (f := polynomialFunction q) (g := polynomialFunction p)
+    ((polynomialFunction_contDiff q).of_le le_top) ((polynomialFunction_contDiff p).of_le le_top)
+  simp_rw [casimir_harmonic hq hΔq, mul_left_comm (polynomialFunction p _) (-eigenvalue n e)] at h₁
+  simp_rw [casimir_harmonic hp hΔp, mul_left_comm (polynomialFunction q _) (-eigenvalue n d),
+    mul_comm (polynomialFunction q _) (polynomialFunction p _)] at h₂
+  rw [mean, integral_const_mul] at h₁ h₂
+  rw [energy_symm (polynomialFunction q) (polynomialFunction p)] at h₂
+  have hne : eigenvalue n d ≠ eigenvalue n e := (eigenvalue_strictMono hn).injective.ne hde
+  unfold mean
+  have hz : (eigenvalue n d - eigenvalue n e) *
+      (∫ u : Sphere n, polynomialFunction p u * polynomialFunction q u ∂sigma n) = 0 := by
+    nlinarith [h₁, h₂]
+  exact (mul_eq_zero.mp hz).resolve_left (sub_ne_zero.mpr hne)
+
+lemma laplacePolynomial_add {n : ℕ} (p q : MvPolynomial (Fin n) ℝ) :
+    laplacePolynomial (p + q) = laplacePolynomial p + laplacePolynomial q := by
+  simp [laplacePolynomial, Finset.sum_add_distrib]
+
+lemma laplacePolynomial_sub {n : ℕ} (p q : MvPolynomial (Fin n) ℝ) :
+    laplacePolynomial (p - q) = laplacePolynomial p - laplacePolynomial q := by
+  simp [laplacePolynomial, Finset.sum_sub_distrib]
+
+lemma laplacePolynomial_C_mul {n : ℕ} (c : ℝ) (p : MvPolynomial (Fin n) ℝ) :
+    laplacePolynomial (C c * p) = C c * laplacePolynomial p := by
+  simp [laplacePolynomial, Finset.mul_sum]
+
+lemma laplacePolynomial_homogeneous {n d : ℕ} {p : MvPolynomial (Fin n) ℝ}
+    (hp : p.IsHomogeneous d) : (laplacePolynomial p).IsHomogeneous (d - 2) := by
+  apply IsHomogeneous.sum
+  intro i _
+  convert hp.pderiv.pderiv using 1
+
+lemma pderiv_radiusPolynomial {n : ℕ} (i : Fin n) :
+    pderiv i (radiusPolynomial n) = 2 * X i := by
+  simp [radiusPolynomial, pderiv_X, Pi.single_apply, eq_comm]
+
+lemma radiusPolynomial_homogeneous (n : ℕ) : (radiusPolynomial n).IsHomogeneous 2 := by
+  apply IsHomogeneous.sum
+  intro i _
+  exact isHomogeneous_X_pow i 2
+
+lemma laplacePolynomial_radius_mul {n : ℕ} (p : MvPolynomial (Fin n) ℝ) :
+    laplacePolynomial (radiusPolynomial n * p) =
+      radiusPolynomial n * laplacePolynomial p + 4 * eulerPolynomial p + 2 * n * p := by
+  have hd (i : Fin n) : pderiv i (pderiv i (radiusPolynomial n * p)) =
+      radiusPolynomial n * pderiv i (pderiv i p) + 4 * (X i * pderiv i p) + 2 * p := by
+    simp only [pderiv_mul, pderiv_radiusPolynomial, map_add, pderiv_X_self]
+    have htwo : pderiv i (2 : MvPolynomial (Fin n) ℝ) = 0 := (pderiv i).map_natCast 2
+    rw [htwo]
+    ring
+  simp only [laplacePolynomial, hd, Finset.sum_add_distrib, ← Finset.mul_sum,
+    Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul, eulerPolynomial]
+  ring
+
+lemma laplacePolynomial_radius_mul_homogeneous {n d : ℕ} {p : MvPolynomial (Fin n) ℝ}
+    (hp : p.IsHomogeneous d) :
+    laplacePolynomial (radiusPolynomial n * p) =
+      radiusPolynomial n * laplacePolynomial p + C (4 * (d : ℝ) + 2 * n) * p := by
+  rw [laplacePolynomial_radius_mul, eulerPolynomial_homogeneous hp]
+  simp only [nsmul_eq_mul, map_add, map_mul, map_ofNat, map_natCast]
+  ring
+
+/-- The nonzero radial coefficient in the finite Fischer decomposition. -/
+def radialCoefficient (n l j : ℕ) : ℝ := 2 * (j + 1 : ℕ) * (2 * l + 2 * (j + 1 : ℕ) + n - 2)
+
+lemma radialCoefficient_pos {n : ℕ} (hn : 2 ≤ n) (l j : ℕ) :
+    0 < radialCoefficient n l j := by
+  have hn' : (2 : ℝ) ≤ n := by exact_mod_cast hn
+  have hl : (0 : ℝ) ≤ l := Nat.cast_nonneg _
+  have hj : (0 : ℝ) ≤ j := Nat.cast_nonneg _
+  unfold radialCoefficient
+  push_cast
+  apply mul_pos (by positivity)
+  linarith
+
+lemma radius_pow_mul_homogeneous {n l : ℕ} {p : MvPolynomial (Fin n) ℝ}
+    (hp : p.IsHomogeneous l) (j : ℕ) :
+    ((radiusPolynomial n) ^ j * p).IsHomogeneous (2 * j + l) :=
+  ((radiusPolynomial_homogeneous n).pow j).mul hp
+
+lemma laplacePolynomial_radial_harmonic {n l : ℕ} {p : MvPolynomial (Fin n) ℝ}
+    (hp : p.IsHomogeneous l) (hΔp : laplacePolynomial p = 0) (j : ℕ) :
+    laplacePolynomial (radiusPolynomial n ^ (j + 1) * p) =
+      C (radialCoefficient n l j) * (radiusPolynomial n ^ j * p) := by
+  have hr (a : ℕ) : radiusPolynomial n ^ (a + 1) * p =
+      radiusPolynomial n * (radiusPolynomial n ^ a * p) := by rw [pow_succ']; ring
+  induction j with
+  | zero =>
+      simp only [Nat.zero_add, pow_one, pow_zero, one_mul]
+      rw [laplacePolynomial_radius_mul_homogeneous hp, hΔp, mul_zero, zero_add]
+      congr 2
+      simp only [radialCoefficient]
+      ring
+  | succ j ih =>
+      rw [hr (j + 1), laplacePolynomial_radius_mul_homogeneous (radius_pow_mul_homogeneous hp (j + 1)), ih]
+      rw [hr j]
+      simp only [radialCoefficient, Nat.cast_add, Nat.cast_mul, Nat.cast_ofNat,
+        map_mul, map_add, map_sub, map_ofNat, map_natCast]
+      ring
+
+lemma laplacePolynomial_smul {n : ℕ} (c : ℝ) (p : MvPolynomial (Fin n) ℝ) :
+    laplacePolynomial (c • p) = c • laplacePolynomial p := by
+  simpa only [← C_mul'] using laplacePolynomial_C_mul c p
+
+lemma laplacePolynomial_eq_zero_small {n d : ℕ} {p : MvPolynomial (Fin n) ℝ}
+    (hp : p.IsHomogeneous d) (hd : d < 2) : laplacePolynomial p = 0 := by
+  apply Finset.sum_eq_zero
+  intro i _
+  have h₀ : (pderiv i p).IsHomogeneous 0 := by
+    convert hp.pderiv using 1
+    omega
+  rw [← totalDegree_zero_iff_isHomogeneous, totalDegree_eq_zero_iff_eq_C] at h₀
+  rw [h₀, pderiv_C]
+
+/-- Algebraic span of radius powers times homogeneous harmonics of fixed total degree. -/
+def fischerSpan (n d : ℕ) : Submodule ℝ (MvPolynomial (Fin n) ℝ) :=
+  Submodule.span ℝ {q | ∃ j l p, 2 * j + l = d ∧ p.IsHomogeneous l ∧
+    laplacePolynomial p = 0 ∧ q = radiusPolynomial n ^ j * p}
+
+lemma fischerSpan_le_homogeneous (n d : ℕ) :
+    fischerSpan n d ≤ homogeneousSubmodule (Fin n) ℝ d := by
+  apply Submodule.span_le.mpr
+  rintro q ⟨j, l, p, hd, hp, _, rfl⟩
+  exact hd ▸ radius_pow_mul_homogeneous hp j
+
+lemma harmonic_mem_fischerSpan {n d : ℕ} {p : MvPolynomial (Fin n) ℝ}
+    (hp : p.IsHomogeneous d) (hΔp : laplacePolynomial p = 0) : p ∈ fischerSpan n d := by
+  exact Submodule.subset_span ⟨0, d, p, by omega, hp, hΔp, by simp⟩
+
+lemma fischerSpan_laplace_preimage {n d : ℕ} (hn : 2 ≤ n) {p : MvPolynomial (Fin n) ℝ}
+    (hp : p ∈ fischerSpan n d) :
+    ∃ q ∈ fischerSpan n (d + 2), laplacePolynomial q = p := by
+  induction hp using Submodule.span_induction with
+  | mem x hx =>
+      obtain ⟨j, l, p, hd, hhom, hΔ, rfl⟩ := hx
+      refine ⟨(radialCoefficient n l j)⁻¹ • (radiusPolynomial n ^ (j + 1) * p), ?_, ?_⟩
+      · apply Submodule.smul_mem
+        exact Submodule.subset_span ⟨j + 1, l, p, by omega, hhom, hΔ, rfl⟩
+      · rw [laplacePolynomial_smul, laplacePolynomial_radial_harmonic hhom hΔ]
+        rw [C_mul', smul_smul, inv_mul_cancel₀ (ne_of_gt (radialCoefficient_pos hn l j)), one_smul]
+  | zero => exact ⟨0, Submodule.zero_mem _, by simp [laplacePolynomial]⟩
+  | add x y _ _ hx hy =>
+      obtain ⟨q, hq, hΔq⟩ := hx
+      obtain ⟨r, hr, hΔr⟩ := hy
+      exact ⟨q + r, Submodule.add_mem _ hq hr, by rw [laplacePolynomial_add, hΔq, hΔr]⟩
+  | smul c x _ hx =>
+      obtain ⟨q, hq, hΔq⟩ := hx
+      exact ⟨c • q, Submodule.smul_mem _ c hq, by rw [laplacePolynomial_smul, hΔq]⟩
+
+/-- Finite Fischer decomposition: the induction on the degree. -/
+lemma homogeneous_mem_fischerSpan {n : ℕ} (hn : 2 ≤ n) (d : ℕ)
+    {p : MvPolynomial (Fin n) ℝ} (hp : p.IsHomogeneous d) : p ∈ fischerSpan n d := by
+  induction d using Nat.strong_induction_on generalizing p with
+  | h d ih =>
+      by_cases hd : d < 2
+      · exact harmonic_mem_fischerSpan hp (laplacePolynomial_eq_zero_small hp hd)
+      · have hdp := ih (d - 2) (by omega) (laplacePolynomial_homogeneous hp)
+        obtain ⟨q, hq, hΔq⟩ := fischerSpan_laplace_preimage hn hdp
+        have hd' : d - 2 + 2 = d := by omega
+        rw [hd'] at hq
+        have hhomq := fischerSpan_le_homogeneous n d hq
+        have hrem : p - q ∈ fischerSpan n d := harmonic_mem_fischerSpan (hp.sub hhomq)
+          (by rw [laplacePolynomial_sub, hΔq, sub_self])
+        simpa only [sub_add_cancel] using (fischerSpan n d).add_mem hrem hq
+
+/-- Actual polynomial restrictions, as continuous functions on the sphere. -/
+def polynomialRestriction (n : ℕ) : MvPolynomial (Fin n) ℝ →ₐ[ℝ] C(Sphere n, ℝ) where
+  toFun p := ⟨fun u => polynomialFunction p u,
+    (polynomialFunction_contDiff p).continuous.comp continuous_subtype_val⟩
+  map_zero' := by
+    ext u
+    change MvPolynomial.eval (fun i => (u : Space n) i) 0 = 0
+    exact map_zero _
+  map_one' := by
+    ext u
+    change MvPolynomial.eval (fun i => (u : Space n) i) 1 = 1
+    exact map_one _
+  map_add' p q := by ext u; exact MvPolynomial.eval_add
+  map_mul' p q := by ext u; exact MvPolynomial.eval_mul
+  commutes' c := by ext u; exact MvPolynomial.eval_C _
+
+lemma polynomialRestriction_apply {n : ℕ} (p : MvPolynomial (Fin n) ℝ) (u : Sphere n) :
+    polynomialRestriction n p u = polynomialFunction p u := rfl
+
+lemma polynomialRestriction_radius (n : ℕ) : polynomialRestriction n (radiusPolynomial n) = 1 := by
+  ext u
+  change polynomialFunction (radiusPolynomial n) u = 1
+  rw [polynomialFunction_radius, norm_coe, one_pow]
+
+lemma polynomialRestriction_separates (n : ℕ) : (polynomialRestriction n).range.SeparatesPoints := by
+  intro u v huv
+  have h : ∃ i : Fin n, (u : Space n) i ≠ (v : Space n) i := by
+    by_contra! h
+    apply huv
+    apply Subtype.ext
+    exact PiLp.ext h
+  obtain ⟨i, hi⟩ := h
+  refine ⟨polynomialRestriction n (X i), ⟨polynomialRestriction n (X i), ⟨X i, rfl⟩, rfl⟩, ?_⟩
+  simpa only [polynomialRestriction_apply, polynomialFunction_X] using hi
+
+lemma polynomialRestriction_dense (n : ℕ) : DenseRange (polynomialRestriction n) := by
+  apply dense_iff_closure_eq.mpr
+  change closure (↑(polynomialRestriction n).range : Set C(Sphere n, ℝ)) = univ
+  rw [← Subalgebra.topologicalClosure_coe,
+    ContinuousMap.subalgebra_topologicalClosure_eq_top_of_separatesPoints _
+      (polynomialRestriction_separates n)]
+  rfl
+
+def polynomialL2 (n : ℕ) [NeZero n] : MvPolynomial (Fin n) ℝ →ₗ[ℝ] H n :=
+  (toH n).toLinearMap.comp (polynomialRestriction n).toLinearMap
+
+lemma polynomialL2_dense (n : ℕ) [NeZero n] : DenseRange (polynomialL2 n) :=
+  (ContinuousMap.toLp_denseRange ℝ (sigma n) ℝ (by norm_num : (2 : ℝ≥0∞) ≠ ⊤)).comp
+    (polynomialRestriction_dense n) (toH n).continuous
+
+def laplaceLinear (n : ℕ) : MvPolynomial (Fin n) ℝ →ₗ[ℝ] MvPolynomial (Fin n) ℝ where
+  toFun := laplacePolynomial
+  map_add' := laplacePolynomial_add
+  map_smul' := laplacePolynomial_smul
+
+def harmonicPolynomials (n d : ℕ) : Submodule ℝ (MvPolynomial (Fin n) ℝ) :=
+  homogeneousSubmodule (Fin n) ℝ d ⊓ (laplaceLinear n).ker
+
+instance harmonicPolynomials_finite (n d : ℕ) : FiniteDimensional ℝ (harmonicPolynomials n d) := by
+  have : FiniteDimensional ℝ (homogeneousSubmodule (Fin n) ℝ d) :=
+    Module.Finite.of_fg (homogeneousSubmodule_fg (Fin n) ℝ d)
+  exact Submodule.finiteDimensional_of_le (show harmonicPolynomials n d ≤
+    homogeneousSubmodule (Fin n) ℝ d from inf_le_left)
+
+/-- The genuine homogeneous harmonic subspaces of spherical L². -/
+def harmonicSpace (n d : ℕ) [NeZero n] : Submodule ℝ (H n) :=
+  (harmonicPolynomials n d).map (polynomialL2 n)
+
+instance harmonicSpace_finite (n d : ℕ) [NeZero n] : FiniteDimensional ℝ (harmonicSpace n d) := by
+  apply Module.Finite.of_fg
+  exact ((Submodule.fg_iff_finiteDimensional (harmonicPolynomials n d)).mpr inferInstance).map _
+
+lemma harmonicSpace_orthogonal {n : ℕ} (hn : 2 ≤ n) [NeZero n] :
+    OrthogonalFamily ℝ (fun d => harmonicSpace n d) (fun d => (harmonicSpace n d).subtypeₗᵢ) := by
+  intro d e hde p q
+  obtain ⟨P, hP, hPeq⟩ := p.property
+  obtain ⟨Q, hQ, hQeq⟩ := q.property
+  change ⟪(p : H n), (q : H n)⟫ = 0
+  rw [← hPeq, ← hQeq]
+  change ⟪toH n (polynomialRestriction n P), toH n (polynomialRestriction n Q)⟫ = 0
+  rw [inner_toH]
+  exact harmonic_orthogonal hn hP.1 hQ.1 hP.2 hQ.2 hde
+
+lemma polynomialL2_mem_iSup_of_fischer {n d : ℕ} [NeZero n] {p : MvPolynomial (Fin n) ℝ}
+    (hp : p ∈ fischerSpan n d) : polynomialL2 n p ∈ ⨆ l, harmonicSpace n l := by
+  induction hp using Submodule.span_induction with
+  | mem x hx =>
+      obtain ⟨j, l, p, _, hhom, hΔ, rfl⟩ := hx
+      have hr : polynomialL2 n (radiusPolynomial n ^ j * p) = polynomialL2 n p := by
+        change toH n (polynomialRestriction n _) = toH n (polynomialRestriction n p)
+        rw [map_mul, map_pow, polynomialRestriction_radius, one_pow, one_mul]
+      rw [hr]
+      exact Submodule.mem_iSup_of_mem l (Submodule.mem_map.mpr ⟨p, ⟨hhom, hΔ⟩, rfl⟩)
+  | zero => simp
+  | add p q _ _ hp hq => simpa only [map_add] using Submodule.add_mem _ hp hq
+  | smul c p _ hp => simpa only [map_smul] using Submodule.smul_mem _ c hp
+
+lemma polynomialL2_mem_iSup {n : ℕ} (hn : 2 ≤ n) [NeZero n] (p : MvPolynomial (Fin n) ℝ) :
+    polynomialL2 n p ∈ ⨆ l, harmonicSpace n l := by
+  rw [← sum_homogeneousComponent p, map_sum]
+  apply Submodule.sum_mem
+  intro d _
+  exact polynomialL2_mem_iSup_of_fischer
+    (homogeneous_mem_fischerSpan hn d (homogeneousComponent_isHomogeneous d p))
+
+/-- Completeness under the actual measure, using Stone--Weierstrass and finite
+Fischer decomposition, with no assumed spectral interface. -/
+lemma harmonicSpace_total {n : ℕ} (hn : 2 ≤ n) [NeZero n] :
+    (⨆ l, harmonicSpace n l).topologicalClosure = ⊤ := by
+  apply SetLike.coe_injective
+  rw [Submodule.topologicalClosure_coe, Submodule.top_coe]
+  apply dense_iff_closure_eq.mp
+  exact (polynomialL2_dense n).mono (by rintro _ ⟨p, rfl⟩; exact polynomialL2_mem_iSup hn p)
+
+lemma energy_add_right {n : ℕ} [NeZero n] {f g h : Space n → ℝ}
+    (hf : ContDiff ℝ 1 f) (hg : ContDiff ℝ 1 g) (hh : ContDiff ℝ 1 h) :
+    energy f (fun x => g x + h x) = energy f g + energy f h := by
+  unfold energy
+  rw [← mul_add, ← Finset.sum_add_distrib]
+  congr 1
+  apply Finset.sum_congr rfl
+  intro i _
+  rw [← Finset.sum_add_distrib]
+  apply Finset.sum_congr rfl
+  intro j _
+  simp_rw [rotDeriv_add _ (hg.differentiable (by norm_num))
+    (hh.differentiable (by norm_num)), mul_add]
+  exact integral_add
+    (continuous_integrable (((continuous_rotDeriv _ hf).mul (continuous_rotDeriv _ hg)).comp
+      continuous_subtype_val))
+    (continuous_integrable (((continuous_rotDeriv _ hf).mul (continuous_rotDeriv _ hh)).comp
+      continuous_subtype_val))
+
+lemma energy_add_left {n : ℕ} [NeZero n] {f g h : Space n → ℝ}
+    (hf : ContDiff ℝ 1 f) (hg : ContDiff ℝ 1 g) (hh : ContDiff ℝ 1 h) :
+    energy (fun x => f x + g x) h = energy f h + energy g h := by
+  simpa only [energy_symm] using energy_add_right hh hf hg
+
+lemma rotDeriv_const_mul {n : ℕ} (T : Space n →L[ℝ] Space n)
+    {f : Space n → ℝ} (hf : Differentiable ℝ f) (c : ℝ) (x : Space n) :
+    rotDeriv T (fun y => c * f y) x = c * rotDeriv T f x := by
+  unfold rotDeriv
+  rw [fderiv_const_mul (hf x) c]
+  rfl
+
+lemma energy_const_mul_right {n : ℕ} {f g : Space n → ℝ}
+    (hg : Differentiable ℝ g) (c : ℝ) :
+    energy f (fun x => c * g x) = c * energy f g := by
+  simp only [energy, rotDeriv_const_mul _ hg c, mul_left_comm _ c, mean, integral_const_mul,
+    ← Finset.mul_sum]
+
+lemma energy_const_mul_left {n : ℕ} {f g : Space n → ℝ}
+    (hf : Differentiable ℝ f) (c : ℝ) :
+    energy (fun x => c * f x) g = c * energy f g := by
+  simpa only [energy_symm] using energy_const_mul_right (f := g) hf c
+
+lemma energy_sub_self {n : ℕ} [NeZero n] {f g : Space n → ℝ}
+    (hf : ContDiff ℝ 1 f) (hg : ContDiff ℝ 1 g) :
+    energy (fun x => f x - g x) (fun x => f x - g x) =
+      energy f f - 2 * energy f g + energy g g := by
+  have hsub : (fun x => f x - g x) = fun x => f x + (-1) * g x := by funext x; ring
+  rw [hsub, energy_add_left hf (contDiff_const.mul hg) (hf.add (contDiff_const.mul hg)),
+    energy_add_right hf hf (contDiff_const.mul hg),
+    energy_const_mul_left (hg.differentiable (by norm_num)),
+    energy_add_right hg hf (contDiff_const.mul hg),
+    energy_const_mul_right (hg.differentiable (by norm_num)),
+    energy_const_mul_right (hg.differentiable (by norm_num)), energy_symm g f]
+  ring
+
+lemma energy_zero_right {n : ℕ} (f : Space n → ℝ) : energy f (fun _ => 0) = 0 := by
+  simp [energy, rotDeriv_const, mean]
+
+lemma energy_finset_sum_right {n : ℕ} [NeZero n] {ι : Type uι} (s : Finset ι)
+    {f : Space n → ℝ} {g : ι → Space n → ℝ}
+    (hf : ContDiff ℝ 1 f) (hg : ∀ i, ContDiff ℝ 1 (g i)) :
+    energy f (fun x => ∑ i ∈ s, g i x) = ∑ i ∈ s, energy f (g i) := by
+  classical
+  induction s using Finset.induction_on with
+  | empty => simp [energy_zero_right]
+  | @insert a s ha ih =>
+      simp only [Finset.sum_insert ha]
+      rw [energy_add_right hf (hg a) (ContDiff.sum fun i _ => hg i), ih]
+
+/-- Integration by parts against each genuine harmonic eigenfunction. -/
+lemma energy_harmonic_right {n d : ℕ} [NeZero n] {f : Space n → ℝ}
+    {p : MvPolynomial (Fin n) ℝ} (hf : ContDiff ℝ 1 f) (hp : p.IsHomogeneous d)
+    (hΔp : laplacePolynomial p = 0) :
+    energy f (polynomialFunction p) = eigenvalue n d *
+      mean (fun u : Sphere n => f u * polynomialFunction p u) := by
+  have h := mean_casimir hf ((polynomialFunction_contDiff p).of_le le_top)
+  simp_rw [casimir_harmonic hp hΔp, mul_left_comm (f _) (-eigenvalue n d)] at h
+  rw [mean, integral_const_mul] at h
+  unfold mean
+  linarith
+
+lemma polynomialFunction_smul_argument {n d : ℕ} {p : MvPolynomial (Fin n) ℝ}
+    (hp : p.IsHomogeneous d) (c : ℝ) (x : Space n) :
+    polynomialFunction p (c • x) = c ^ d * polynomialFunction p x := by
+  unfold polynomialFunction
+  simp only [MvPolynomial.eval_eq, PiLp.smul_apply, smul_eq_mul, mul_pow,
+    Finset.prod_mul_distrib, Finset.prod_pow_eq_pow_sum, Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro a ha
+  rw [← hp.degree_eq_sum_deg_support ha]
+  ring
+
+lemma harmonicSpace_zero_constant {n : ℕ} [NeZero n] {v : H n} (hv : v ∈ harmonicSpace n 0) :
+    ∃ c : ℝ, v = toH n (ContinuousMap.const (Sphere n) c) := by
+  obtain ⟨p, hp, rfl⟩ := hv
+  have hhom : p.IsHomogeneous 0 := hp.1
+  rw [← totalDegree_zero_iff_isHomogeneous,
+    totalDegree_eq_zero_iff_eq_C] at hhom
+  refine ⟨p.coeff 0, ?_⟩
+  rw [hhom]
+  simp only [coeff_C]
+  change toH n (polynomialRestriction n (C _)) = _
+  congr 1
+  ext u
+  exact congrFun (polynomialFunction_C _) _
+
+lemma constant_mem_harmonicSpace_zero (n : ℕ) [NeZero n] (c : ℝ) :
+    toH n (ContinuousMap.const (Sphere n) c) ∈ harmonicSpace n 0 := by
+  refine Submodule.mem_map.mpr ⟨C c, ⟨isHomogeneous_C _ _, ?_⟩, ?_⟩
+  · simp [LinearMap.mem_ker, laplaceLinear, laplacePolynomial]
+  · change toH n (polynomialRestriction n (C c)) = _
+    congr 1
+    ext u
+    exact congrFun (polynomialFunction_C _) _
+
+lemma harmonicSpace_zero_projection (n : ℕ) [NeZero n] (f : C(Sphere n, ℝ)) :
+    (harmonicSpace n 0).starProjection (toH n f) =
+      toH n (ContinuousMap.const (Sphere n) (mean f)) := by
+  apply Submodule.eq_starProjection_of_mem_of_inner_eq_zero (constant_mem_harmonicSpace_zero n _)
+  intro v hv
+  obtain ⟨c, rfl⟩ := harmonicSpace_zero_constant hv
+  rw [← map_sub, inner_toH]
+  change mean (fun u : Sphere n => (f u - mean f) * c) = 0
+  unfold mean
+  rw [integral_mul_const, integral_sub (continuous_integrable f.continuous)
+    (integrable_const _)]
+  simp
+
+end PettyProjection.Spherical
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/ProjectionBodies/SphericalGap.lean b/lean/OAI/Geometry/ProjectionBodies/SphericalGap.lean
new file mode 100644
index 0000000000000000000000000000000000000000..3962b950bc7f50d60e39455fc5eb7b731d7a6ee6
--- /dev/null
+++ b/lean/OAI/Geometry/ProjectionBodies/SphericalGap.lean
@@ -0,0 +1,527 @@
+import OAI.Geometry.ProjectionBodies.Support
+
+namespace OAI
+
+universe uE
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology MvPolynomial
+open scoped ENNReal RealInnerProductSpace Pointwise
+namespace PettyProjection.Spherical
+
+def restrictContinuous {n : ℕ} (f : Space n → ℝ) (hf : Continuous f) : C(Sphere n, ℝ) :=
+  ⟨fun u => f u, hf.comp continuous_subtype_val⟩
+
+lemma polynomialFunction_odd {n d : ℕ} {p : MvPolynomial (Fin n) ℝ}
+    (hp : p.IsHomogeneous d) (hd : Odd d) : Function.Odd (polynomialFunction p) := by
+  intro x
+  simpa only [neg_one_smul, hd.neg_one_pow, neg_one_mul] using
+    polynomialFunction_smul_argument hp (-1) x
+
+lemma mean_even_mul_odd {n d : ℕ} {p : MvPolynomial (Fin n) ℝ}
+    {f : Space n → ℝ} (heven : Function.Even f) (hp : p.IsHomogeneous d) (hd : Odd d) :
+    mean (fun u : Sphere n => f u * polynomialFunction p u) = 0 := by
+  have h := mean_rotation (LinearIsometryEquiv.neg ℝ : Space n ≃ₗᵢ[ℝ] Space n)
+    (fun u : Sphere n => f u * polynomialFunction p u)
+  have hh : (fun u : Sphere n =>
+      f (sphereMap (LinearIsometryEquiv.neg ℝ) u) *
+        polynomialFunction p (sphereMap (LinearIsometryEquiv.neg ℝ) u)) =
+      (fun u : Sphere n => -(f u * polynomialFunction p u)) := by
+    funext u
+    change f (-↑u) * polynomialFunction p (-↑u) = _
+    rw [heven, polynomialFunction_odd hp hd, mul_neg]
+  rw [hh, mean, integral_neg] at h
+  unfold mean at h ⊢
+  linarith
+
+lemma projection_odd_zero {n d : ℕ} [NeZero n] {f : Space n → ℝ}
+    (hf : Continuous f) (heven : Function.Even f) (hd : Odd d) :
+    (harmonicSpace n d).starProjection (toH n (restrictContinuous f hf)) = 0 := by
+  apply Submodule.eq_starProjection_of_mem_of_inner_eq_zero (Submodule.zero_mem _)
+  rintro v ⟨p, hp, rfl⟩
+  rw [sub_zero]
+  change ⟪toH n (restrictContinuous f hf), toH n (polynomialRestriction n p)⟫ = 0
+  rw [inner_toH]
+  exact mean_even_mul_odd heven hp.1 hd
+
+/-- A chosen true polynomial representative of a harmonic L² vector. -/
+def harmonicRepresentative {n d : ℕ} [NeZero n] (v : harmonicSpace n d) : MvPolynomial (Fin n) ℝ :=
+  Classical.choose v.property
+
+lemma harmonicRepresentative_homogeneous {n d : ℕ} [NeZero n] (v : harmonicSpace n d) :
+    (harmonicRepresentative v).IsHomogeneous d :=
+  (Classical.choose_spec v.property).1.1
+
+lemma harmonicRepresentative_laplace {n d : ℕ} [NeZero n] (v : harmonicSpace n d) :
+    laplacePolynomial (harmonicRepresentative v) = 0 :=
+  (Classical.choose_spec v.property).1.2
+
+lemma harmonicRepresentative_toH {n d : ℕ} [NeZero n] (v : harmonicSpace n d) :
+    polynomialL2 n (harmonicRepresentative v) = (v : H n) :=
+  (Classical.choose_spec v.property).2
+
+lemma inner_projection_self {n d : ℕ} [NeZero n] (v : H n) :
+    ⟪v, (harmonicSpace n d).starProjection v⟫ = ‖(harmonicSpace n d).starProjection v‖ ^ 2 := by
+  rw [← real_inner_self_eq_norm_sq, Submodule.inner_starProjection_left_eq_right]
+  rw [Submodule.starProjection_eq_self_iff.mpr
+    ((harmonicSpace n d).starProjection_apply_mem v)]
+
+lemma energy_harmonicRepresentative_right {n d : ℕ} [NeZero n]
+    {f : Space n → ℝ} (hf : ContDiff ℝ 1 f) (v : harmonicSpace n d) :
+    energy f (polynomialFunction (harmonicRepresentative v)) =
+      eigenvalue n d * ⟪toH n (restrictContinuous f hf.continuous), (v : H n)⟫ := by
+  rw [energy_harmonic_right hf (harmonicRepresentative_homogeneous v)
+    (harmonicRepresentative_laplace v)]
+  rw [← harmonicRepresentative_toH v]
+  change _ = eigenvalue n d * ⟪toH n (restrictContinuous f hf.continuous),
+    toH n (polynomialRestriction n (harmonicRepresentative v))⟫
+  rw [inner_toH]
+  rfl
+
+lemma sum_inner_projection {n : ℕ} (hn : 2 ≤ n) [NeZero n] (s : Finset ℕ)
+    (v : H n) {e : ℕ} (he : e ∈ s) :
+    ⟪∑ d ∈ s, (harmonicSpace n d).starProjection v,
+        (harmonicSpace n e).starProjection v⟫ =
+      ‖(harmonicSpace n e).starProjection v‖ ^ 2 := by
+  rw [sum_inner]
+  rw [Finset.sum_eq_single e]
+  · exact real_inner_self_eq_norm_sq _
+  · intro d _ hde
+    have h := harmonicSpace_orthogonal hn hde
+      ((harmonicSpace n d).orthogonalProjectionOnto v)
+      ((harmonicSpace n e).orthogonalProjectionOnto v)
+    simpa only [Submodule.coe_subtypeₗᵢ, Submodule.coe_subtype,
+      Submodule.coe_orthogonalProjectionOnto_apply] using h
+  · exact fun h => False.elim (h he)
+
+/-- The finite Bessel bound, established by subtracting a finite harmonic
+polynomial sum. No infinite series is differentiated. -/
+lemma energy_bessel_finset {n : ℕ} (hn : 2 ≤ n) [NeZero n]
+    {f : Space n → ℝ} (hf : ContDiff ℝ 1 f) (s : Finset ℕ) :
+    ∑ d ∈ s, eigenvalue n d * ‖(harmonicSpace n d).starProjection
+      (toH n (restrictContinuous f hf.continuous))‖ ^ 2 ≤ energy f f := by
+  let v := toH n (restrictContinuous f hf.continuous)
+  let vs (d : ℕ) := (harmonicSpace n d).orthogonalProjectionOnto v
+  let ps (d : ℕ) := harmonicRepresentative (vs d)
+  let g : Space n → ℝ := fun x => ∑ d ∈ s, polynomialFunction (ps d) x
+  have hpd (d : ℕ) : ContDiff ℝ 1 (polynomialFunction (ps d)) :=
+    (polynomialFunction_contDiff _).of_le le_top
+  have hg : ContDiff ℝ 1 g := ContDiff.sum fun d _ => hpd d
+  have hG : toH n (restrictContinuous g hg.continuous) =
+      ∑ d ∈ s, (harmonicSpace n d).starProjection v := by
+    have hr : restrictContinuous g hg.continuous = ∑ d ∈ s, polynomialRestriction n (ps d) := by
+      ext u
+      simp [restrictContinuous, g, polynomialRestriction_apply]
+    rw [hr, map_sum]
+    apply Finset.sum_congr rfl
+    intro d _
+    exact (harmonicRepresentative_toH (vs d)).trans
+      ((harmonicSpace n d).coe_orthogonalProjectionOnto_apply v)
+  have hfg : energy f g =
+      ∑ d ∈ s, eigenvalue n d * ‖(harmonicSpace n d).starProjection v‖ ^ 2 := by
+    rw [energy_finset_sum_right s hf hpd]
+    apply Finset.sum_congr rfl
+    intro d _
+    rw [energy_harmonicRepresentative_right hf (vs d)]
+    exact congrArg (eigenvalue n d * ·) (inner_projection_self v)
+  have hgg : energy g g =
+      ∑ d ∈ s, eigenvalue n d * ‖(harmonicSpace n d).starProjection v‖ ^ 2 := by
+    rw [energy_finset_sum_right s hg hpd]
+    apply Finset.sum_congr rfl
+    intro d hd
+    rw [energy_harmonicRepresentative_right hg (vs d), hG]
+    exact congrArg (eigenvalue n d * ·) (sum_inner_projection hn s v hd)
+  have hpos := energy_nonneg (fun x => f x - g x)
+  rw [energy_sub_self hf hg, hfg, hgg] at hpos
+  linarith
+
+lemma eigenvalue_nonneg {n : ℕ} (hn : 2 ≤ n) (d : ℕ) : 0 ≤ eigenvalue n d := by
+  have hn' : (2 : ℝ) ≤ n := by exact_mod_cast hn
+  have hd : (0 : ℝ) ≤ d := Nat.cast_nonneg d
+  unfold eigenvalue
+  exact mul_nonneg hd (by linarith)
+
+lemma eigenvalue_four_le {n d : ℕ} (hn : 2 ≤ n) (hd : 4 ≤ d) :
+    eigenvalue n 4 ≤ eigenvalue n d := by
+  have hn' : (2 : ℝ) ≤ n := by exact_mod_cast hn
+  have hd' : (4 : ℝ) ≤ d := by exact_mod_cast hd
+  have hprod := mul_nonneg (sub_nonneg.mpr hd') (sub_nonneg.mpr hn')
+  norm_num [eigenvalue]
+  nlinarith
+
+lemma Q_zero_of_mem_lowSpace {n : ℕ} [NeZero n] {v : H n} (hv : v ∈ lowSpace n) :
+    Q n v = 0 := by
+  rw [Q_eq_sub, Submodule.starProjection_eq_self_iff.mpr hv, sub_self]
+
+lemma projection_zero_mem_lowSpace {n : ℕ} [NeZero n] (v : H n) :
+    (harmonicSpace n 0).starProjection v ∈ lowSpace n := by
+  obtain ⟨c, hc⟩ := harmonicSpace_zero_constant ((harmonicSpace n 0).starProjection_apply_mem v)
+  rw [hc]
+  exact constant_mem_lowSpace n c
+
+lemma projection_inner_ne {n : ℕ} (hn : 2 ≤ n) [NeZero n] (v : H n)
+    {d e : ℕ} (hde : d ≠ e) :
+    ⟪(harmonicSpace n d).starProjection v, (harmonicSpace n e).starProjection v⟫ = 0 := by
+  have h := harmonicSpace_orthogonal hn hde
+    ((harmonicSpace n d).orthogonalProjectionOnto v)
+    ((harmonicSpace n e).orthogonalProjectionOnto v)
+  simpa only [Submodule.coe_subtypeₗᵢ, Submodule.coe_subtype,
+    Submodule.coe_orthogonalProjectionOnto_apply] using h
+
+lemma norm_Q_sq_le_remainder {n : ℕ} (hn : 2 ≤ n) [NeZero n] (v : H n) :
+    ‖Q n v‖ ^ 2 ≤ ‖v‖ ^ 2 - ‖(harmonicSpace n 0).starProjection v‖ ^ 2 -
+      ‖(harmonicSpace n 2).starProjection v‖ ^ 2 := by
+  let p0 := (harmonicSpace n 0).starProjection v
+  let p2 := (harmonicSpace n 2).starProjection v
+  have h0 : Q n p0 = 0 := Q_zero_of_mem_lowSpace (projection_zero_mem_lowSpace v)
+  have h2 : Q n p2 = 0 := Q_zero_of_mem_lowSpace
+    (harmonicSpace_two_le_lowSpace n ((harmonicSpace n 2).starProjection_apply_mem v))
+  have hQ : Q n (v - p0 - p2) = Q n v := by simp only [map_sub, h0, h2, sub_zero]
+  have h := norm_Q_le n (v - p0 - p2)
+  rw [hQ] at h
+  have hsq := sq_le_sq₀ (norm_nonneg (Q n v)) (norm_nonneg (v - p0 - p2)) |>.mpr h
+  have hnorm : ‖v - p0 - p2‖ ^ 2 = ‖v‖ ^ 2 - ‖p0‖ ^ 2 - ‖p2‖ ^ 2 := by
+    rw [norm_sub_sq_real, norm_sub_sq_real, inner_sub_left,
+      inner_projection_self v, inner_projection_self v,
+      projection_inner_ne hn v (by decide : (0 : ℕ) ≠ 2)]
+    ring
+  exact hsq.trans_eq hnorm
+
+lemma energy_spectral_lower {n : ℕ} (hn : 2 ≤ n) [NeZero n]
+    {f : Space n → ℝ} (hf : ContDiff ℝ 1 f) (heven : Function.Even f) :
+    let v := toH n (restrictContinuous f hf.continuous)
+    eigenvalue n 4 * ‖v‖ ^ 2 - eigenvalue n 4 * ‖(harmonicSpace n 0).starProjection v‖ ^ 2 -
+      (eigenvalue n 4 - eigenvalue n 2) * ‖(harmonicSpace n 2).starProjection v‖ ^ 2 ≤
+        energy f f := by
+  let v := toH n (restrictContinuous f hf.continuous)
+  let a (d : ℕ) := ‖(harmonicSpace n d).starProjection v‖ ^ 2
+  let b (d : ℕ) := eigenvalue n 4 * a d -
+    (if d = 0 then eigenvalue n 4 * a 0 else 0) -
+    (if d = 2 then (eigenvalue n 4 - eigenvalue n 2) * a 2 else 0)
+  have ha : HasSum a (‖v‖ ^ 2) :=
+    hilbertSum_hasSum_projection_norm_sq (harmonicSpace_isHilbertSum hn) v
+  have hb : HasSum b (eigenvalue n 4 * ‖v‖ ^ 2 - eigenvalue n 4 * a 0 -
+      (eigenvalue n 4 - eigenvalue n 2) * a 2) :=
+    ((ha.mul_left (eigenvalue n 4)).sub (hasSum_ite_eq 0 _)).sub (hasSum_ite_eq 2 _)
+  have hb_le (d : ℕ) : b d ≤ eigenvalue n d * a d := by
+    by_cases h0 : d = 0
+    · subst d; simp [b, eigenvalue]
+    by_cases h2 : d = 2
+    · subst d; simp only [b, ite_eq_right (by decide : (2 : ℕ) ≠ 0), ite_true, sub_zero]; ring_nf; exact le_rfl
+    by_cases h1 : d = 1
+    · subst d
+      have hz : a 1 = 0 := by
+        simp [a, v, projection_odd_zero hf.continuous heven (by decide : Odd 1)]
+      simp [b, hz]
+    by_cases h3 : d = 3
+    · subst d
+      have hz : a 3 = 0 := by
+        simp [a, v, projection_odd_zero hf.continuous heven (by decide : Odd 3)]
+      simp [b, hz]
+    have hd : 4 ≤ d := by omega
+    simpa only [b, ite_eq_right h0, ite_eq_right h2, sub_zero] using
+      mul_le_mul_of_nonneg_right (eigenvalue_four_le hn hd) (sq_nonneg _)
+  apply le_of_tendsto hb
+  apply Eventually.of_forall
+  intro s
+  exact (Finset.sum_le_sum fun d _ => hb_le d).trans (energy_bessel_finset hn hf s)
+
+/-- The mean-subtracted variance in the probability normalization. -/
+def variance {n : ℕ} (f : Sphere n → ℝ) : ℝ := mean (fun u => f u ^ 2) - (mean f) ^ 2
+
+/-- The genuine degree-two-removed even spherical spectral gap, under the
+canonical probability measure. `energy` is the integrated squared tangential
+derivative represented by the coordinate rotation generators. -/
+lemma even_spectral_gap {n : ℕ} (hn : 2 ≤ n) [NeZero n]
+    {f : Space n → ℝ} (hf : ContDiff ℝ 1 f) (heven : Function.Even f) :
+    (2 * n : ℝ) * variance (fun u : Sphere n => f u) + (2 * n + 8 : ℝ) *
+      ‖Q n (toH n (restrictContinuous f hf.continuous))‖ ^ 2 ≤ energy f f := by
+  let F := restrictContinuous f hf.continuous
+  let v := toH n F
+  have h0 : ‖(harmonicSpace n 0).starProjection v‖ ^ 2 = (mean F) ^ 2 := by
+    rw [harmonicSpace_zero_projection n F, norm_toH_sq]
+    simpa only [ContinuousMap.const_apply] using mean_const n ((mean F) ^ 2)
+  have hvar : variance (fun u : Sphere n => f u) = ‖v‖ ^ 2 - (mean F) ^ 2 := by
+    rw [norm_toH_sq]
+    rfl
+  have h := energy_spectral_lower hn hf heven
+  have hQ := norm_Q_sq_le_remainder hn v
+  change eigenvalue n 4 * ‖v‖ ^ 2 - eigenvalue n 4 * ‖(harmonicSpace n 0).starProjection v‖ ^ 2 -
+    (eigenvalue n 4 - eigenvalue n 2) * ‖(harmonicSpace n 2).starProjection v‖ ^ 2 ≤ energy f f at h
+  rw [h0] at h hQ
+  rw [hvar]
+  have hmul := mul_le_mul_of_nonneg_left hQ (by positivity : (0 : ℝ) ≤ 2 * n + 8)
+  norm_num [eigenvalue] at h
+  nlinarith
+
+end PettyProjection.Spherical
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology
+open scoped ENNReal RealInnerProductSpace Pointwise
+namespace PettyProjection.Spherical
+
+lemma line_convex {n : ℕ} {f : Space n → ℝ} (hc : ConvexOn ℝ univ f)
+    (x y : Space n) : ConvexOn ℝ univ (fun t : ℝ => f (x + t • y)) := by
+  constructor
+  · exact convex_univ
+  · intro a _ b _ s t hs ht hst
+    have h := hc.2 (mem_univ (x + a • y)) (mem_univ (x + b • y)) hs ht hst
+    have he : x + (s * a + t * b) • y = s • (x + a • y) + t • (x + b • y) := by
+      calc
+        _ = (s + t) • x + (s * a + t * b) • y := by rw [hst, one_smul]
+        _ = _ := by module
+    simpa only [smul_eq_mul, ← he] using h
+
+lemma line_hasDerivAt {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] (x y : E) (t : ℝ) :
+    HasDerivAt (fun s : ℝ => x + s • y) y t := by
+  have h : HasDerivAt (fun s : ℝ => s • y) y t := by
+    simpa only [id_eq, one_smul] using (hasDerivAt_id t).smul_const y
+  exact h.const_add x
+
+lemma line_second_hasDerivAt {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E]
+    {f : E → ℝ} (hf : ContDiff ℝ 2 f) (x y : E) :
+    HasDerivAt (fun t : ℝ => fderiv ℝ f (x + t • y) y)
+      (fderiv ℝ (fderiv ℝ f) x y y) 0 := by
+  have hfd : ContDiff ℝ 1 (fderiv ℝ f) := hf.fderiv_right (by norm_num)
+  have hh : HasFDerivAt (fderiv ℝ f) (fderiv ℝ (fderiv ℝ f) x) x :=
+    (hfd.differentiable (by norm_num) x).hasFDerivAt
+  have hl : HasDerivAt (fun t : ℝ => x + t • y) y 0 := line_hasDerivAt x y 0
+  have hdx : HasDerivAt (fun t : ℝ => fderiv ℝ f (x + t • y))
+      (fderiv ℝ (fderiv ℝ f) x y) 0 := by
+    have hh' : HasFDerivAt (fderiv ℝ f) (fderiv ℝ (fderiv ℝ f) x) (x + (0 : ℝ) • y) := by
+      simpa only [zero_smul, add_zero] using hh
+    exact hh'.comp_hasDerivAt 0 hl
+  have hd : HasDerivAt (fun t : ℝ => fderiv ℝ f (x + t • y) y)
+      (fderiv ℝ (fderiv ℝ f) x y y + fderiv ℝ f (x + (0 : ℝ) • y) 0) 0 :=
+    hdx.clm_apply (hasDerivAt_const 0 y)
+  simpa only [map_zero, add_zero] using hd
+
+lemma hessian_nonneg_of_convex {n : ℕ} {f : Space n → ℝ}
+    (hf : ContDiff ℝ 2 f) (hc : ConvexOn ℝ univ f) (x y : Space n) :
+    0 ≤ fderiv ℝ (fderiv ℝ f) x y y := by
+  have hdf (t : ℝ) : HasDerivAt (fun s : ℝ => f (x + s • y))
+      (fderiv ℝ f (x + t • y) y) t :=
+    (hf.differentiable (by norm_num) _).hasFDerivAt.comp_hasDerivAt t (line_hasDerivAt x y t)
+  have hmono : Monotone (fun t : ℝ => fderiv ℝ f (x + t • y) y) := by
+    have h := (line_convex hc x y).monotoneOn_deriv (fun t _ => (hdf t).differentiableAt)
+    have heq : deriv (fun t : ℝ => f (x + t • y)) =
+        (fun t => fderiv ℝ f (x + t • y) y) := funext fun t => (hdf t).deriv
+    rw [heq] at h
+    exact monotoneOn_univ.mp h
+  rw [← (line_second_hasDerivAt hf x y).deriv]
+  exact hmono.deriv_nonneg
+
+lemma rotDeriv_rotDeriv {n : ℕ} (T : Space n →L[ℝ] Space n) {f : Space n → ℝ}
+    (hf : ContDiff ℝ 2 f) (x : Space n) :
+    rotDeriv T (rotDeriv T f) x =
+      fderiv ℝ (fderiv ℝ f) x (T x) (T x) + fderiv ℝ f x (T (T x)) := by
+  have hfd : ContDiff ℝ 1 (fderiv ℝ f) := hf.fderiv_right (by norm_num)
+  have hd := (hfd.differentiable (by norm_num) x).hasFDerivAt.clm_apply T.hasFDerivAt
+  change fderiv ℝ (fun y => fderiv ℝ f y (T y)) x (T x) = _
+  rw [hd.fderiv]
+  simp only [add_apply, ContinuousLinearMap.comp_apply,
+    ContinuousLinearMap.flip_apply, add_comm]
+
+lemma generator_square_coord {n : ℕ} (i j a : Fin n) (x : Space n) :
+    (generator n i j (generator n i j x)) a =
+      2 * (if i = j then if i = a then x a else 0 else 0) -
+        (if j = a then x a else 0) - (if i = a then x a else 0) := by
+  by_cases hij : i = j <;> by_cases hia : i = a <;> by_cases hja : j = a <;>
+    simp_all [generator_apply_coord, eq_comm]; ring
+
+lemma sum_generator_square {n : ℕ} (x : Space n) :
+    ∑ i : Fin n, ∑ j : Fin n, generator n i j (generator n i j x) =
+      (-2 * ((n : ℝ) - 1)) • x := by
+  ext a
+  simp only [WithLp.ofLp_sum, Finset.sum_apply, generator_square_coord, PiLp.smul_apply, smul_eq_mul,
+    Finset.sum_sub_distrib]
+  simp only [← Finset.mul_sum, Finset.sum_ite_eq, Finset.mem_univ, ite_true,
+    Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul, Finset.sum_ite_eq', Finset.mem_univ, ite_true]
+  ring
+
+lemma convex_casimir_lower {n : ℕ} {f : Space n → ℝ}
+    (hf : ContDiff ℝ 2 f) (hc : ConvexOn ℝ univ f) (x : Space n) :
+    -((n : ℝ) - 1) * fderiv ℝ f x x ≤ casimir f x := by
+  have h (i j : Fin n) : fderiv ℝ f x (generator n i j (generator n i j x)) ≤
+      rotDeriv (generator n i j) (rotDeriv (generator n i j) f) x := by
+    rw [rotDeriv_rotDeriv _ hf]
+    linarith [hessian_nonneg_of_convex hf hc x (generator n i j x)]
+  have hs := Finset.sum_le_sum (fun i (_ : i ∈ Finset.univ) =>
+    Finset.sum_le_sum (fun j (_ : j ∈ Finset.univ) => h i j))
+  have hh := mul_le_mul_of_nonneg_left hs (by norm_num : (0 : ℝ) ≤ 1 / 2)
+  simp only [← map_sum, sum_generator_square, map_smul, smul_eq_mul] at hh
+  unfold casimir
+  nlinarith
+
+end PettyProjection.Spherical
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology Function
+open scoped NNReal ENNReal RealInnerProductSpace Pointwise Convolution
+namespace PettyProjection.Spherical
+
+lemma seminorm_continuous (n : ℕ) (g : Seminorm ℝ (Space n)) : Continuous g :=
+  continuousOn_univ.mp (g.convexOn.continuousOn isOpen_univ)
+
+lemma seminorm_lipschitz (n : ℕ) (g : Seminorm ℝ (Space n)) :
+    ∃ C : ℝ≥0, LipschitzWith C g := by
+  obtain ⟨C, hC, hbound⟩ := g.bound_of_continuous_normedSpace (seminorm_continuous n g)
+  refine ⟨⟨C, hC.le⟩, LipschitzWith.of_dist_le_mul fun x y => ?_⟩
+  change |g x - g y| ≤ C * dist x y
+  exact (abs_sub_map_le_sub g x y).trans (by simpa only [dist_eq_norm] using hbound (x - y))
+
+def normSmooth {n : ℕ} (g : Seminorm ℝ (Space n)) (φ : ContDiffBump (0 : Space n)) :
+    Space n → ℝ := φ.normed volume ⋆[ContinuousLinearMap.lsmul ℝ ℝ, volume] (g : Space n → ℝ)
+
+lemma normSmooth_contDiff {n : ℕ} (g : Seminorm ℝ (Space n)) (φ : ContDiffBump (0 : Space n)) :
+    ContDiff ℝ 2 (normSmooth g φ) :=
+  φ.hasCompactSupport_normed.contDiff_convolution_left (ContinuousLinearMap.lsmul ℝ ℝ)
+    φ.contDiff_normed (seminorm_continuous n g).locallyIntegrable
+
+lemma normSmooth_even {n : ℕ} (g : Seminorm ℝ (Space n)) (φ : ContDiffBump (0 : Space n)) :
+    Function.Even (normSmooth g φ) := by
+  intro x
+  exact convolution_neg_of_neg_eq (ContinuousLinearMap.lsmul ℝ ℝ) (Eventually.of_forall φ.normed_neg)
+    (Eventually.of_forall (fun x => map_neg_eq_map g x))
+
+lemma normSmooth_integrable {n : ℕ} (g : Seminorm ℝ (Space n)) (φ : ContDiffBump (0 : Space n))
+    (x : Space n) : Integrable (fun y => φ.normed volume y * g (x - y)) volume :=
+  φ.hasCompactSupport_normed.convolutionExists_left_of_continuous_right
+    (ContinuousLinearMap.lsmul ℝ ℝ) φ.continuous_normed.locallyIntegrable (seminorm_continuous n g) x
+
+lemma normSmooth_convex {n : ℕ} (g : Seminorm ℝ (Space n)) (φ : ContDiffBump (0 : Space n)) :
+    ConvexOn ℝ univ (normSmooth g φ) := by
+  constructor
+  · exact convex_univ
+  · intro x _ y _ a b ha hb hab
+    have hi₁ := (normSmooth_integrable g φ x).const_mul a
+    have hi₂ := (normSmooth_integrable g φ y).const_mul b
+    change (∫ z, φ.normed volume z * g (a • x + b • y - z)) ≤
+        a * (∫ z, φ.normed volume z * g (x - z)) + b * (∫ z, φ.normed volume z * g (y - z))
+    rw [← integral_const_mul, ← integral_const_mul, ← integral_add hi₁ hi₂]
+    apply integral_mono (normSmooth_integrable g φ (a • x + b • y)) (hi₁.add hi₂)
+    intro z
+    have he : a • x + b • y - z = a • (x - z) + b • (y - z) := by
+      calc
+        _ = a • x + b • y - (a + b) • z := by rw [hab, one_smul]
+        _ = _ := by module
+    have h := mul_le_mul_of_nonneg_left
+      (g.convexOn.2 (mem_univ (x-z)) (mem_univ (y-z)) ha hb hab) (φ.nonneg_normed (μ := volume) z)
+    simp only [Pi.add_apply]
+    rw [he]
+    simpa only [smul_eq_mul, mul_add, mul_left_comm] using h
+
+lemma normSmooth_dist_le {n : ℕ} (g : Seminorm ℝ (Space n)) (φ : ContDiffBump (0 : Space n))
+    {C : ℝ≥0} (hC : LipschitzWith C g) (x : Space n) :
+    dist (normSmooth g φ x) (g x) ≤ C * φ.rOut := by
+  apply φ.dist_normed_convolution_le (seminorm_continuous n g).aestronglyMeasurable
+  intro y hy
+  exact (hC.dist_le_mul y x).trans (mul_le_mul_of_nonneg_left (mem_ball.mp hy).le C.coe_nonneg)
+
+lemma convex_radial_bounds {n : ℕ} {f : Space n → ℝ}
+    (hf : Differentiable ℝ f) (hc : ConvexOn ℝ univ f) (x : Space n) :
+    f x - f 0 ≤ fderiv ℝ f x x ∧ fderiv ℝ f x x ≤ f ((2 : ℝ) • x) - f x := by
+  have hd : HasDerivAt (fun t : ℝ => f (x + t • x)) (fderiv ℝ f x x) 0 := by
+    have h : HasFDerivAt f (fderiv ℝ f x) (x + (0 : ℝ) • x) := by
+      simpa only [zero_smul, add_zero] using (hf x).hasFDerivAt
+    exact h.comp_hasDerivAt 0 (line_hasDerivAt x x 0)
+  have hl := (line_convex hc x x).slope_le_of_hasDerivAt (mem_univ (-1 : ℝ)) (mem_univ 0)
+    (by norm_num : (-1 : ℝ) < 0) hd
+  have hr := (line_convex hc x x).le_slope_of_hasDerivAt (mem_univ 0) (mem_univ (1 : ℝ))
+    (by norm_num : (0 : ℝ) < 1) hd
+  simpa only [slope, vsub_eq_sub, neg_smul, one_smul, add_neg_cancel,
+    zero_smul, add_zero, two_smul, zero_sub, neg_neg, sub_zero, inv_one, smul_eq_mul, one_mul] using And.intro hl hr
+
+lemma normSmooth_radial_dist_le {n : ℕ} (g : Seminorm ℝ (Space n)) (φ : ContDiffBump (0 : Space n))
+    {C : ℝ≥0} (hC : LipschitzWith C g) (x : Space n) :
+    dist (fderiv ℝ (normSmooth g φ) x x) (g x) ≤ 2 * C * φ.rOut := by
+  have h := convex_radial_bounds ((normSmooth_contDiff g φ).differentiable (by norm_num))
+    (normSmooth_convex g φ) x
+  have hx := normSmooth_dist_le g φ hC x
+  have h0 := normSmooth_dist_le g φ hC 0
+  have h2 := normSmooth_dist_le g φ hC ((2 : ℝ) • x)
+  rw [Real.dist_eq, abs_le] at hx h0 h2 ⊢
+  simp only [map_zero, map_smul_eq_mul, Real.norm_eq_abs, abs_of_pos (by norm_num : (0 : ℝ) < 2)] at h0 h2
+  constructor <;> linarith
+
+end PettyProjection.Spherical
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology
+open scoped ENNReal NNReal RealInnerProductSpace Pointwise
+namespace PettyProjection.Spherical
+
+lemma sphere_ae_of_radial {n : ℕ} {P : Space n → Prop}
+    (hP : ∀ᵐ x ∂(volume : Measure (Space n)), P x)
+    (hr : ∀ (u : Sphere n) (r : ℝ), 0 < r → P (r • (u : Space n)) → P u) :
+    ∀ᵐ (u : Sphere n) ∂sigma n, P (u : Space n) := by
+  let ν := Measure.volumeIoiPow (Module.finrank ℝ (Space n) - 1)
+  have hν : ν ≠ 0 := by
+    intro h
+    have he := Measure.volumeIoiPow_apply_Iio (Module.finrank ℝ (Space n) - 1)
+      (⟨1, mem_Ioi.mpr one_pos⟩ : Ioi (0 : ℝ))
+    change ν _ = _ at he
+    rw [h] at he
+    have hp : 0 < (1 : ℝ) ^ (Module.finrank ℝ (Space n) - 1 + 1) /
+        ((Module.finrank ℝ (Space n) - 1 : ℕ) + 1) := by positivity
+    exact (ne_of_gt (ENNReal.ofReal_pos.mpr hp)) he.symm
+  let : NeZero ν := ⟨hν⟩
+  have hs : ∀ᵐ (x : ({0}ᶜ : Set (Space n))) ∂(volume : Measure (Space n)).comap
+      (Subtype.val : ({0}ᶜ : Set (Space n)) → Space n), P (x : Space n) :=
+    (ae_restrict_iff_subtype (measurableSet_singleton (0 : Space n)).compl).mp
+      (ae_restrict_of_ae hP)
+  have hm := (volume : Measure (Space n)).measurePreserving_homeomorphUnitSphereProd
+  have hi := hm.symm (homeomorphUnitSphereProd (Space n)).toMeasurableEquiv
+  have hh := hi.quasiMeasurePreserving.ae hs
+  have hh' : ∀ᵐ z ∂(surface n).prod ν, P (z.2.1 • (z.1 : Space n)) := by
+    change ∀ᵐ z ∂(surface n).prod ν, P (((homeomorphUnitSphereProd (Space n)).symm z : ({0}ᶜ : Set (Space n))) : Space n) at hh
+    simpa only [homeomorphUnitSphereProd_symm_apply_coe] using hh
+  apply Measure.ae_smul_measure (μ := surface n) _ ((surface n univ)⁻¹)
+  filter_upwards [Measure.ae_ae_of_ae_prod hh'] with u hu
+  obtain ⟨r, hr'⟩ := hu.exists
+  exact hr u r r.property hr'
+
+lemma seminorm_ae_differentiable_sphere {n : ℕ} (g : Seminorm ℝ (Space n)) :
+    ∀ᵐ (u : Sphere n) ∂sigma n, DifferentiableAt ℝ (g : Space n → ℝ) (u : Space n) := by
+  obtain ⟨C, hC⟩ := seminorm_lipschitz n g
+  apply sphere_ae_of_radial hC.ae_differentiableAt
+  intro u r hr hd
+  have hc := (hd.comp (u : Space n) (differentiableAt_id.const_smul r)).const_smul r⁻¹
+  have he : (r⁻¹ • ((g : Space n → ℝ) ∘ (r • id))) = (g : Space n → ℝ) := by
+    ext x
+    simp only [Pi.smul_apply, Function.comp_apply, id_eq, map_smul_eq_mul,
+      Real.norm_eq_abs, abs_of_pos hr, smul_eq_mul]
+    rw [← mul_assoc, inv_mul_cancel₀ hr.ne', one_mul]
+  change DifferentiableAt ℝ (r⁻¹ • ((g : Space n → ℝ) ∘ (r • id))) (u : Space n) at hc
+  rw [he] at hc
+  exact hc
+
+lemma normSmooth_lipschitz {n : ℕ} (g : Seminorm ℝ (Space n))
+    (φ : ContDiffBump (0 : Space n)) {C : ℝ≥0} (hC : LipschitzWith C g) :
+    LipschitzWith C (normSmooth g φ) := by
+  apply LipschitzWith.of_dist_le_mul
+  intro x y
+  change ‖(∫ z, φ.normed volume z * g (x - z)) -
+    (∫ z, φ.normed volume z * g (y - z))‖ ≤ _
+  rw [← integral_sub (normSmooth_integrable g φ x) (normSmooth_integrable g φ y)]
+  calc
+    _ ≤ ∫ z, ‖φ.normed volume z * g (x - z) - φ.normed volume z * g (y - z)‖ := norm_integral_le_integral_norm _
+    _ ≤ ∫ z, φ.normed volume z * (C * dist x y) := by
+      apply integral_mono
+        ((normSmooth_integrable g φ x).sub (normSmooth_integrable g φ y)).norm
+        (φ.integrable_normed.mul_const _)
+      intro z
+      change ‖φ.normed volume z * g (x-z) - φ.normed volume z * g (y-z)‖ ≤ _
+      rw [← mul_sub, norm_mul, Real.norm_of_nonneg (φ.nonneg_normed z)]
+      apply mul_le_mul_of_nonneg_left _ (φ.nonneg_normed z)
+      simpa [dist_eq_norm] using hC.dist_le_mul (x-z) (y-z)
+    _ = C * dist x y := by rw [integral_mul_const, φ.integral_normed, one_mul]
+
+end PettyProjection.Spherical
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/ProjectionBodies/Support.lean b/lean/OAI/Geometry/ProjectionBodies/Support.lean
new file mode 100644
index 0000000000000000000000000000000000000000..7cbc6169d70b913bc829fdfbf4475ca21247615b
--- /dev/null
+++ b/lean/OAI/Geometry/ProjectionBodies/Support.lean
@@ -0,0 +1,223 @@
+import OAI.Geometry.ProjectionBodies.SphericalCore
+
+namespace OAI
+
+universe uE
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology Function
+open scoped ENNReal NNReal RealInnerProductSpace Pointwise
+namespace PettyProjection
+
+/-- The actual support function of a compact body, not a surrogate gauge. -/
+def support {n : ℕ} (K : Set (Space n)) (u : Space n) : ℝ :=
+  sSup ((fun x => ⟪u,x⟫) '' K)
+
+lemma support_attained {n : ℕ} {K : Set (Space n)} (hK : IsCompact K) (hKn : K.Nonempty)
+    (u : Space n) : ∃ x ∈ K, support K u = ⟪u,x⟫ :=
+  hK.exists_sSup_image_eq hKn (by fun_prop)
+
+lemma inner_le_support {n : ℕ} {K : Set (Space n)} (hK : IsCompact K)
+    {x : Space n} (hx : x ∈ K) (u : Space n) : ⟪u,x⟫ ≤ support K u := by
+  exact le_csSup (hK.bddAbove_image (by fun_prop)) (mem_image_of_mem _ hx)
+
+lemma support_le_iff {n : ℕ} {K : Set (Space n)} (hK : IsCompact K) (hKn : K.Nonempty)
+    {u : Space n} {a : ℝ} : support K u ≤ a ↔ ∀ x ∈ K, ⟪u,x⟫ ≤ a := by
+  constructor
+  · intro h x hx; exact (inner_le_support hK hx u).trans h
+  · intro h
+    obtain ⟨x,hx,he⟩ := support_attained hK hKn u
+    rw [he]; exact h x hx
+
+lemma support_continuous {n : ℕ} {K : Set (Space n)} (hK : IsCompact K) :
+    Continuous (support K) :=
+  hK.continuous_sSup (by fun_prop)
+
+lemma support_lipschitz {n : ℕ} {K : Set (Space n)} (hK : IsCompact K) (hKn : K.Nonempty)
+    {R : ℝ≥0} (hR : ∀ x ∈ K, ‖x‖ ≤ R) : LipschitzWith R (support K) := by
+  apply LipschitzWith.of_le_add_mul
+  intro u v
+  obtain ⟨x,hx,he⟩ := support_attained hK hKn u
+  rw [he]
+  have hi := inner_le_support hK hx v
+  have hbound : ⟪u-v,x⟫ ≤ R*dist u v := by
+    calc
+      _ ≤ ‖u-v‖*‖x‖ := real_inner_le_norm _ _
+      _ ≤ ‖u-v‖*R := mul_le_mul_of_nonneg_left (hR x hx) (norm_nonneg _)
+      _ = _ := by rw [dist_eq_norm]; ring
+  rw [inner_sub_left] at hbound
+  linarith
+
+lemma support_smul {n : ℕ} {K : Set (Space n)} (hK : IsCompact K) (hKn : K.Nonempty)
+    {r : ℝ} (hr : 0 ≤ r) (u : Space n) : support K (r • u) = r*support K u := by
+  apply le_antisymm
+  · apply (support_le_iff hK hKn).mpr
+    intro x hx
+    rw [real_inner_smul_left]
+    exact mul_le_mul_of_nonneg_left (inner_le_support hK hx u) hr
+  · obtain ⟨x,hx,he⟩ := support_attained hK hKn u
+    rw [he]
+    simpa only [real_inner_smul_left] using inner_le_support hK hx (r • u)
+
+lemma support_add_le {n : ℕ} {K : Set (Space n)} (hK : IsCompact K) (hKn : K.Nonempty)
+    (u v : Space n) : support K (u+v) ≤ support K u+support K v := by
+  apply (support_le_iff hK hKn).mpr
+  intro x hx
+  rw [inner_add_left]
+  exact add_le_add (inner_le_support hK hx u) (inner_le_support hK hx v)
+
+lemma support_nonneg {n : ℕ} {K : Set (Space n)} (hK : IsCompact K) (h0 : 0 ∈ K)
+    (u : Space n) : 0 ≤ support K u := by
+  simpa only [inner_zero_right] using inner_le_support hK h0 u
+
+lemma support_minkowski {n : ℕ} {K L : Set (Space n)}
+    (hK : IsCompact K) (hKn : K.Nonempty) (hL : IsCompact L) (hLn : L.Nonempty)
+    (u : Space n) : support (K+L) u = support K u+support L u := by
+  apply le_antisymm
+  · apply (support_le_iff (hK.add hL) (hKn.add hLn)).mpr
+    rintro _ ⟨x,hx,y,hy,rfl⟩
+    rw [inner_add_right]
+    exact add_le_add (inner_le_support hK hx u) (inner_le_support hL hy u)
+  · obtain ⟨x,hx,hxe⟩ := support_attained hK hKn u
+    obtain ⟨y,hy,hye⟩ := support_attained hL hLn u
+    rw [hxe,hye,← inner_add_right]
+    exact inner_le_support (hK.add hL) (add_mem_add hx hy) u
+
+lemma support_linear_image {n : ℕ} {K : Set (Space n)} (hK : IsCompact K) (hKn : K.Nonempty)
+    (T : Space n →L[ℝ] Space n) (u : Space n) :
+    support (T '' K) u = support K (T.adjoint u) := by
+  apply le_antisymm
+  · apply (support_le_iff (hK.image T.continuous) (hKn.image T)).mpr
+    rintro _ ⟨x,hx,rfl⟩
+    rw [← ContinuousLinearMap.adjoint_inner_left]
+    exact inner_le_support hK hx _
+  · obtain ⟨x,hx,he⟩ := support_attained hK hKn (T.adjoint u)
+    rw [he,ContinuousLinearMap.adjoint_inner_left]
+    exact inner_le_support (hK.image T.continuous) (mem_image_of_mem T hx) u
+
+lemma mem_iff_inner_le_support {n : ℕ} {K : Set (Space n)}
+    (hK : IsCompact K) (hc : Convex ℝ K) (hKn : K.Nonempty) {x : Space n} :
+    x ∈ K ↔ ∀ u : Space n, ⟪u,x⟫ ≤ support K u := by
+  constructor
+  · intro hx u; exact inner_le_support hK hx u
+  · intro hx
+    by_contra hn
+    obtain ⟨f,a,hfa,hax⟩ := geometric_hahn_banach_closed_point hc hK.isClosed hn
+    let u := (InnerProductSpace.toDual ℝ (Space n)).symm f
+    have hu (y : Space n) : ⟪u,y⟫ = f y := by
+      change (InnerProductSpace.toDual ℝ (Space n)) u y = f y
+      rw [LinearIsometryEquiv.apply_symm_apply]
+    obtain ⟨y,hy,he⟩ := support_attained hK hKn u
+    have h := hx u
+    rw [he,hu,hu] at h
+    exact (not_lt_of_ge h) ((hfa y hy).trans hax)
+
+lemma support_eq_iff {n : ℕ} {K L : Set (Space n)}
+    (hK : IsCompact K) (hcK : Convex ℝ K) (hKn : K.Nonempty)
+    (hL : IsCompact L) (hcL : Convex ℝ L) (hLn : L.Nonempty) :
+    support K = support L ↔ K = L := by
+  constructor
+  · intro h
+    ext x
+    rw [mem_iff_inner_le_support hK hcK hKn,mem_iff_inner_le_support hL hcL hLn,h]
+  · rintro rfl; rfl
+
+end PettyProjection
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology
+open scoped ENNReal RealInnerProductSpace Pointwise
+namespace PettyProjection.Spherical
+
+lemma harmonicSpace_isHilbertSum {n : ℕ} (hn : 2 ≤ n) [NeZero n] :
+    IsHilbertSum ℝ (fun d : ℕ => harmonicSpace n d)
+      (fun d : ℕ => (harmonicSpace n d).subtypeₗᵢ) := by
+  let F : ℕ → Submodule ℝ (H n) := fun d => harmonicSpace n d
+  have horth : OrthogonalFamily ℝ (fun d => F d) (fun d => (F d).subtypeₗᵢ) :=
+    harmonicSpace_orthogonal hn
+  have htotal : (⨆ d, F d).topologicalClosure = ⊤ := harmonicSpace_total hn
+  let : ∀ d, CompleteSpace (F d) := fun d => FiniteDimensional.complete ℝ (harmonicSpace n d)
+  exact IsHilbertSum.mkInternal F horth htotal.ge
+
+open MvPolynomial
+
+lemma polynomialL2_X_mul_X (n : ℕ) [NeZero n] (i j : Fin n) :
+    polynomialL2 n (X i * X j) = toH n (coordinateProduct n i j) := by
+  change toH n (polynomialRestriction n (X i * X j)) = _
+  congr 1
+  ext u
+  simp [polynomialRestriction_apply, polynomialFunction, coordinateProduct]
+
+lemma polynomialL2_X_mul_mem_lowSpace {n : ℕ} [NeZero n] (i : Fin n)
+    {p : MvPolynomial (Fin n) ℝ} (hp : p.IsHomogeneous 1) :
+    polynomialL2 n (X i * p) ∈ lowSpace n := by
+  have hmem : p ∈ Submodule.span ℝ (Set.range (X : Fin n → MvPolynomial (Fin n) ℝ)) := by
+    rw [← homogeneousSubmodule_one_eq_span_X]
+    exact hp
+  clear hp
+  induction hmem using Submodule.span_induction with
+  | mem q hq =>
+      obtain ⟨j, rfl⟩ := hq
+      rw [polynomialL2_X_mul_X]
+      exact Submodule.subset_span ⟨(i,j), rfl⟩
+  | zero => simp
+  | add p q _ _ hp hq =>
+      simpa only [mul_add, map_add] using Submodule.add_mem (lowSpace n) hp hq
+  | smul c p _ hp =>
+      simpa only [mul_smul_comm, map_smul] using Submodule.smul_mem (lowSpace n) c hp
+
+lemma homogeneous_two_mem_lowSpace {n : ℕ} [NeZero n] {p : MvPolynomial (Fin n) ℝ}
+    (hp : p.IsHomogeneous 2) : polynomialL2 n p ∈ lowSpace n := by
+  have hsum : polynomialL2 n (∑ i : Fin n, X i * pderiv i p) ∈ lowSpace n := by
+    rw [map_sum]
+    exact Submodule.sum_mem _ fun i _ => polynomialL2_X_mul_mem_lowSpace i hp.pderiv
+  have htwo : (2 : ℝ) • polynomialL2 n p ∈ lowSpace n := by
+    simpa only [hp.sum_X_mul_pderiv, two_nsmul, map_add, two_smul ℝ] using hsum
+  exact (Submodule.smul_mem_iff (lowSpace n) (by norm_num : (2 : ℝ) ≠ 0)).mp htwo
+
+lemma harmonicSpace_two_le_lowSpace (n : ℕ) [NeZero n] :
+    harmonicSpace n 2 ≤ lowSpace n := by
+  rintro v ⟨p, hp, rfl⟩
+  exact homogeneous_two_mem_lowSpace hp.1
+
+end PettyProjection.Spherical
+end
+
+noncomputable section
+open MeasureTheory Set Filter Topology
+open scoped ENNReal RealInnerProductSpace
+namespace PettyProjection
+
+variable {E : Type uE} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E]
+variable {F : ℕ → Submodule ℝ E} [∀ i, CompleteSpace (F i)]
+
+lemma hilbertSum_component (h : IsHilbertSum ℝ (fun i => F i) (fun i => (F i).subtypeₗᵢ))
+    (x : E) (i : ℕ) :
+    (F i).starProjection x = (h.linearIsometryEquiv x i : E) := by
+  apply Submodule.eq_starProjection_of_mem_of_inner_eq_zero (h.linearIsometryEquiv x i).property
+  intro w hw
+  have hh := h.linearIsometryEquiv.symm.inner_map_map (h.linearIsometryEquiv x)
+    (lp.single 2 i (⟨w,hw⟩ : F i))
+  rw [LinearIsometryEquiv.symm_apply_apply, h.linearIsometryEquiv_symm_apply_single,
+    lp.inner_single_right] at hh
+  change ⟪x, w⟫ = ⟪(h.linearIsometryEquiv x i : E), w⟫ at hh
+  rw [inner_sub_left, hh, sub_self]
+
+lemma hilbertSum_hasSum_projection (h : IsHilbertSum ℝ (fun i => F i) (fun i => (F i).subtypeₗᵢ))
+    (x : E) : HasSum (fun i => (F i).starProjection x) x := by
+  have hh := h.hasSum_linearIsometryEquiv_symm (h.linearIsometryEquiv x)
+  rw [LinearIsometryEquiv.symm_apply_apply] at hh
+  simpa only [hilbertSum_component h, Submodule.coe_subtypeₗᵢ, Submodule.coe_subtype] using hh
+
+lemma hilbertSum_hasSum_projection_norm_sq
+    (h : IsHilbertSum ℝ (fun i => F i) (fun i => (F i).subtypeₗᵢ)) (x : E) :
+    HasSum (fun i => ‖(F i).starProjection x‖ ^ 2) (‖x‖ ^ 2) := by
+  have hh := lp.hasSum_norm (by norm_num : 0 < (2 : ℝ≥0∞).toReal) (h.linearIsometryEquiv x)
+  simpa only [ENNReal.toReal_ofNat, Real.rpow_two, LinearIsometryEquiv.norm_map,
+    hilbertSum_component h, Submodule.norm_coe] using hh
+
+end PettyProjection
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/ProjectionBodies/SupportMeasures.lean b/lean/OAI/Geometry/ProjectionBodies/SupportMeasures.lean
new file mode 100644
index 0000000000000000000000000000000000000000..9a485fbd7a20fa027513e2c8499adeac45423954
--- /dev/null
+++ b/lean/OAI/Geometry/ProjectionBodies/SupportMeasures.lean
@@ -0,0 +1,287 @@
+import OAI.Geometry.ProjectionBodies.Homothety
+
+namespace OAI
+
+universe uι
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology Function
+open scoped ENNReal NNReal RealInnerProductSpace Pointwise Gradient
+namespace PettyProjection
+open Spherical (Sphere norm_coe sigma mean)
+
+lemma gauge_pos_sphere {n : ℕ} [NeZero n] {K : Set (Space n)}
+    (hK : IsCompact K) (hc : Convex ℝ K) (h0 : K ∈ 𝓝 (0 : Space n))
+    (u : Sphere n) : 0 < gauge K (u : Space n) := by
+  let s : C(Sphere n,ℝ) := ⟨fun u => support K u,(support_continuous hK).comp continuous_subtype_val⟩
+  have hs : ∀ u,0 < s u := support_pos_of_nhds hK h0
+  have hw : wulff s = K := wulff_support hK hc ⟨0,mem_of_mem_nhds h0⟩
+  rw [← hw,gauge_wulff_eq_max s hs]
+  exact wulffMax_pos s hs (by intro h; simpa [h] using norm_coe u)
+
+def gaugeDensity {n : ℕ} (K : Set (Space n)) (u : Sphere n) : ENNReal :=
+  ENNReal.ofReal ((gauge K (u : Space n))⁻¹^n)
+
+def gaugeProjectionMeasure {n : ℕ} (K : Set (Space n)) : Measure (Space n) :=
+  Measure.map (fun u : Sphere n => ∇ (gauge K) (u : Space n))
+    ((sigma n).withDensity (gaugeDensity K))
+
+lemma gaugeDensity_measurable {n : ℕ} {K : Set (Space n)}
+    (hc : Convex ℝ K) (h0 : K ∈ 𝓝 (0 : Space n)) : Measurable (gaugeDensity K) :=
+  ENNReal.measurable_ofReal.comp ((((continuous_gauge hc h0).measurable.comp
+    measurable_subtype_coe).inv).pow_const n)
+
+lemma gaugeDensity_toReal {n : ℕ} (K : Set (Space n)) (u : Sphere n) :
+    (gaugeDensity K u).toReal=(gauge K (u : Space n))⁻¹^n := by
+  exact ENNReal.toReal_ofReal (pow_nonneg (inv_nonneg.mpr (gauge_nonneg _)) _)
+
+lemma gaugeMeasure_integral {n : ℕ} {K : Set (Space n)}
+    (hc : Convex ℝ K) (h0 : K ∈ 𝓝 (0 : Space n)) {f : Space n → ℝ}
+    (hf : StronglyMeasurable f) :
+    ∫ x,f x ∂gaugeProjectionMeasure K =
+      mean (fun u : Sphere n => (gauge K (u : Space n))⁻¹^n*f (∇ (gauge K) (u : Space n))) := by
+  have hm : Measurable (fun u : Sphere n => ∇ (gauge K) (u : Space n)) :=
+    (measurable_gradient _).comp measurable_subtype_coe
+  rw [gaugeProjectionMeasure,integral_map_of_stronglyMeasurable hm hf,
+    integral_withDensity_eq_integral_toReal_smul (gaugeDensity_measurable hc h0)
+      (Eventually.of_forall (fun _ => ENNReal.ofReal_lt_top))]
+  simp only [gaugeDensity_toReal,smul_eq_mul,mean]
+
+lemma gaugeMeasure_finite {n : ℕ} [NeZero n] {K : Set (Space n)}
+    (hK : IsCompact K) (hc : Convex ℝ K) (h0 : K ∈ 𝓝 (0 : Space n)) :
+    IsFiniteMeasure (gaugeProjectionMeasure K) := by
+  have hp := gauge_pos_sphere hK hc h0
+  have hcont : Continuous (fun u : Sphere n => (gauge K (u : Space n))⁻¹^n) :=
+    (((continuous_gauge hc h0).comp continuous_subtype_val).inv₀ (fun u => (hp u).ne')).pow n
+  have hi := hcont.integrable_of_hasCompactSupport (μ := sigma n) (HasCompactSupport.of_compactSpace _)
+  have hf : ∫⁻ u, gaugeDensity K u ∂sigma n ≠ ⊤ := by
+    unfold gaugeDensity
+    rw [← ofReal_integral_eq_lintegral_ofReal hi
+      (Eventually.of_forall (fun u => (pow_nonneg (inv_nonneg.mpr (hp u).le) n)))]
+    exact ENNReal.ofReal_ne_top
+  have := isFiniteMeasure_withDensity hf
+  exact Measure.isFiniteMeasure_map _ _
+
+lemma gaugeMeasure_mass {n : ℕ} [NeZero n] {K : Set (Space n)}
+    (hK : IsCompact K) (hc : Convex ℝ K) (h0 : K ∈ 𝓝 (0 : Space n)) :
+    (gaugeProjectionMeasure K).real univ=volume.real K/kappa n := by
+  have hi := gaugeMeasure_integral hc h0 (f := fun _ => 1) stronglyMeasurable_const
+  simp only [integral_const,smul_eq_mul,mul_one] at hi
+  rw [hi,gauge_volume hc h0 (gauge_pos_sphere hK hc h0)]
+  field_simp [(kappa_pos n).ne']
+
+lemma gaugeMeasure_probability {n : ℕ} [NeZero n] {K : Set (Space n)}
+    (hK : IsCompact K) (hc : Convex ℝ K) (h0 : K ∈ 𝓝 (0 : Space n))
+    (hv : volume.real K=kappa n) : IsProbabilityMeasure (gaugeProjectionMeasure K) := by
+  constructor
+  apply (ENNReal.toReal_eq_one_iff _).mp
+  rw [show (gaugeProjectionMeasure K univ).toReal=(gaugeProjectionMeasure K).real univ from rfl,
+    gaugeMeasure_mass hK hc h0,hv,div_self (kappa_pos n).ne']
+
+lemma gaugeMeasure_bounded {n : ℕ} {K : Set (Space n)}
+    (hc : Convex ℝ K) (h0 : K ∈ 𝓝 (0 : Space n)) :
+    ∃ B : ℝ, 0 ≤ B ∧ ∀ᵐ x ∂gaugeProjectionMeasure K, ‖x‖ ≤ B := by
+  obtain ⟨C,hC⟩ := hc.lipschitz_gauge h0
+  refine ⟨C,C.coe_nonneg,?_⟩
+  have hm : Measurable (fun u : Sphere n => ∇ (gauge K) (u : Space n)) :=
+    (measurable_gradient _).comp measurable_subtype_coe
+  apply (ae_map_iff hm.aemeasurable
+    (measurableSet_le measurable_norm measurable_const)).mpr
+  apply Eventually.of_forall
+  intro u
+  have he : ‖∇ (gauge K) (u : Space n)‖=‖fderiv ℝ (gauge K) (u : Space n)‖ :=
+    (InnerProductSpace.toDual ℝ (Space n)).symm.norm_map _
+  rw [he]
+  exact norm_fderiv_le_of_lipschitz ℝ hC
+
+lemma boundedMeasure_integrable_continuous {n : ℕ} {μ : Measure (Space n)} [IsFiniteMeasure μ]
+    (hb : ∃ B : ℝ, 0 ≤ B ∧ ∀ᵐ x ∂μ, ‖x‖ ≤ B) {f : Space n → ℝ}
+    (hf : Continuous f) : Integrable f μ := by
+  obtain ⟨B,_,hB⟩ := hb
+  obtain ⟨C,hC⟩ := (isCompact_closedBall (0 : Space n) B).exists_bound_of_continuousOn hf.continuousOn
+  apply (integrable_const C).mono' hf.aestronglyMeasurable
+  filter_upwards [hB] with x hx
+  exact hC x (by simpa only [mem_closedBall,dist_zero_right] using hx)
+
+lemma gaugeMeasure_shadow {n : ℕ} [NeZero n] {K : Set (Space n)}
+    (hK : IsCompact K) (hc : Convex ℝ K) (h0 : K ∈ 𝓝 (0 : Space n)) (u : Sphere n) :
+    2*shadowVolume K u=n*kappa n*∫ x,|⟪x,(u : Space n)⟫| ∂gaugeProjectionMeasure K := by
+  rw [gaugeMeasure_integral hc h0 (by fun_prop)]
+  exact shadow_gauge_formula hK hc h0 u
+
+lemma gaugeMeasure_firstVariation {n : ℕ} [NeZero n] {K M : Set (Space n)}
+    (hK : IsCompact K) (hcK : Convex ℝ K) (h0 : K ∈ 𝓝 (0 : Space n))
+    (hM : IsCompact M) (hcM : Convex ℝ M) (hMn : M.Nonempty) :
+    HasDerivWithinAt (fun t : ℝ => volume.real (K+t • M))
+      (n*kappa n*∫ x,support M x ∂gaugeProjectionMeasure K) (Ici 0) 0 := by
+  rw [gaugeMeasure_integral hcK h0 (support_continuous hM).stronglyMeasurable]
+  exact minkowski_volume_derivative hK hcK h0 hM hcM hMn
+
+end PettyProjection
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology Function
+open scoped ENNReal NNReal RealInnerProductSpace Pointwise Gradient
+namespace PettyProjection
+open Spherical (Sphere norm_coe sigma mean)
+
+def supportFunctional {n : ℕ} (μ : Measure (Space n)) (K : Set (Space n)) : ℝ :=
+  ∫ x,support K x ∂μ
+
+def normalizedVolume {n : ℕ} (K : Set (Space n)) : ℝ :=
+  volumeRoot K/(kappa n)^((n:ℝ)⁻¹)
+
+lemma normalizedVolume_formula {n : ℕ} (K : Set (Space n)) :
+    normalizedVolume K=(volume.real K/kappa n)^((n:ℝ)⁻¹) := by
+  rw [Real.div_rpow measureReal_nonneg (kappa_pos n).le]
+  rfl
+
+lemma gaugeMeasure_functional {n : ℕ} (hn : 2 ≤ n) [NeZero n]
+    {K M : Set (Space n)} (hK : IsCompact K) (hcK : Convex ℝ K)
+    (h0 : K ∈ 𝓝 (0 : Space n)) (hv : volume.real K=kappa n)
+    (hM : IsCompact M) (hcM : Convex ℝ M) (hiM : (interior M).Nonempty) :
+    normalizedVolume M ≤ supportFunctional (gaugeProjectionMeasure K) M ∧
+      (supportFunctional (gaugeProjectionMeasure K) M=normalizedVolume M →
+        ∃ z : Space n, ∃ c : ℝ, 0 < c ∧ K=(fun x => z+c • x) '' M) := by
+  have hiK : (interior K).Nonempty := ⟨0,mem_interior_iff_mem_nhds.mpr h0⟩
+  have hV := gaugeMeasure_firstVariation hK hcK h0 hM hcM (hiM.mono interior_subset)
+  obtain ⟨hlow,heq⟩ := minkowski_derivative_bound_equality hn hK hcK hiK hM hcM hiM hV
+  have hnR : 0 < (n:ℝ) := by exact_mod_cast (show 0<n by omega)
+  have hp : 0 < (kappa n)^((n:ℝ)⁻¹) := Real.rpow_pos_of_pos (kappa_pos n) _
+  have hroot : volumeRoot K=(kappa n)^((n:ℝ)⁻¹) := by unfold volumeRoot; rw [hv]
+  have hmul : n*kappa n*normalizedVolume M=(n:ℝ)*volumeRoot K^(n-1)*volumeRoot M := by
+    have hpow := volumeRoot_pow (show 0<n by omega) K
+    have hsplit : volumeRoot K^(n-1)*volumeRoot K=kappa n := by
+      rw [← pow_succ,Nat.sub_add_cancel (by omega),hpow,hv]
+    rw [normalizedVolume,← hroot,← hsplit]
+    field_simp [(volumeRoot_pos hK hiK).ne']
+  refine ⟨?_,fun h => heq ?_⟩
+  · rw [← hmul] at hlow
+    exact (mul_le_mul_iff_right₀ (mul_pos hnR (kappa_pos n))).mp hlow
+  · change n*kappa n*supportFunctional (gaugeProjectionMeasure K) M=_
+    rw [h,hmul]
+
+lemma gaugeMeasure_self {n : ℕ} [NeZero n] {K : Set (Space n)}
+    (hK : IsCompact K) (hc : Convex ℝ K) (h0 : K ∈ 𝓝 (0 : Space n)) :
+    supportFunctional (gaugeProjectionMeasure K) K=volume.real K/kappa n := by
+  have hKn : K.Nonempty := ⟨0,mem_of_mem_nhds h0⟩
+  have hV := gaugeMeasure_firstVariation hK hc h0 hK hc hKn
+  have hd : HasDerivAt (fun t : ℝ => (1+t)^n*volume.real K) (n*volume.real K) 0 := by
+    simpa only [id_eq,Pi.pow_apply,add_zero,one_pow,mul_one] using
+      (((hasDerivAt_id (0:ℝ)).const_add 1).pow n).mul_const (volume.real K)
+  have he (t : ℝ) (ht : t ∈ Ici (0:ℝ)) :
+      volume.real (K+t • K)=(1+t)^n*volume.real K := by
+    have he : K+t • K=(1+t) • K := by simpa only [one_smul] using (hc.add_smul (by norm_num : (0:ℝ)≤1) ht).symm
+    rw [he,volumeReal_smul_nonneg (by have := ht; change 0 ≤ t at this; linarith : 0≤1+t)]
+    simp only [finrank_euclideanSpace_fin]
+  have hd' := hd.hasDerivWithinAt (s := Ici (0:ℝ)) |>.congr he (he 0 (by norm_num))
+  have hdEq := (hV.derivWithin (uniqueDiffWithinAt_Ici (0:ℝ))).symm.trans
+    (hd'.derivWithin (uniqueDiffWithinAt_Ici (0:ℝ)))
+  change n*kappa n*supportFunctional (gaugeProjectionMeasure K) K=n*volume.real K at hdEq
+  have hn : (n:ℝ) ≠ 0 := by exact_mod_cast (NeZero.ne n)
+  apply (eq_div_iff (kappa_pos n).ne').mpr
+  rw [mul_assoc] at hdEq
+  simpa only [mul_comm] using (mul_left_cancel₀ hn hdEq)
+
+lemma shadowVolume_pos {n : ℕ} {K : Set (Space n)}
+    (hK : IsCompact K) (h0 : K ∈ 𝓝 (0 : Space n)) (u : Space n) : 0 < shadowVolume K u := by
+  obtain ⟨r,hr,hrK⟩ := Metric.mem_nhds_iff.mp h0
+  have hball : ball (0 : perpendicular u) r ⊆ (perpendicular u).orthogonalProjectionOnto '' K := by
+    intro z hz
+    refine ⟨z,hrK ?_,?_⟩
+    · simpa only [mem_ball,dist_zero_right,Submodule.norm_coe] using hz
+    · exact Submodule.orthogonalProjectionOnto_mem_subspace_eq_self z
+  have hpos : 0 < (volume : Measure (perpendicular u))
+      ((perpendicular u).orthogonalProjectionOnto '' K) :=
+    (measure_ball_pos volume 0 hr).trans_le (measure_mono hball)
+  exact ENNReal.toReal_pos hpos.ne'
+    (hK.image (perpendicular u).orthogonalProjectionOnto.continuous).measure_ne_top
+
+end PettyProjection
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology Function
+open scoped NNReal ENNReal RealInnerProductSpace Gradient
+namespace PettyProjection.Spherical
+
+lemma convex_deriv_limit {ι : Type uι} {l : Filter ι} {f : ι → ℝ → ℝ}
+    {g : ℝ → ℝ} {d : ι → ℝ} {a : ℝ}
+    (hf : ∀ j, ConvexOn ℝ univ (f j))
+    (hd : ∀ j, HasDerivAt (f j) (d j) 0)
+    (hg : HasDerivAt g a 0)
+    (hlim : ∀ t, Tendsto (fun j => f j t) l (𝓝 (g t))) :
+    Tendsto d l (𝓝 a) := by
+  have hs (t : ℝ) : Tendsto (fun j => slope (f j) 0 t) l (𝓝 (slope g 0 t)) := by
+    simpa only [slope, vsub_eq_sub, smul_eq_mul] using
+      ((hlim t).sub (hlim 0)).const_mul (t - 0)⁻¹
+  rw [tendsto_order]
+  constructor
+  · intro b hb
+    have hh : ∀ᶠ t in 𝓝[<] (0 : ℝ), b < slope g 0 t :=
+      (hg.tendsto_slope.mono_left (nhdsWithin_mono _ (by grind : Iio (0 : ℝ) ⊆ {0}ᶜ))).eventually
+        (eventually_gt_nhds hb)
+    have ht : ∀ᶠ t in 𝓝[<] (0 : ℝ), t < 0 := self_mem_nhdsWithin
+    obtain ⟨t, ht, hbt⟩ := (ht.and hh).exists
+    filter_upwards [(hs t).eventually (eventually_gt_nhds hbt)] with j hj
+    have hj' := (hf j).slope_le_of_hasDerivAt (mem_univ t) (mem_univ 0) ht (hd j)
+    rw [slope_comm] at hj'
+    exact hj.trans_le hj'
+  · intro b hb
+    have hh : ∀ᶠ t in 𝓝[>] (0 : ℝ), slope g 0 t < b :=
+      (hg.tendsto_slope.mono_left (nhdsWithin_mono _ (by grind : Ioi (0 : ℝ) ⊆ {0}ᶜ))).eventually
+        (eventually_lt_nhds hb)
+    have ht : ∀ᶠ t in 𝓝[>] (0 : ℝ), 0 < t := self_mem_nhdsWithin
+    obtain ⟨t, ht, hbt⟩ := (ht.and hh).exists
+    filter_upwards [(hs t).eventually (eventually_lt_nhds hbt)] with j hj
+    exact ((hf j).le_slope_of_hasDerivAt (mem_univ 0) (mem_univ t) ht (hd j)).trans_lt hj
+
+lemma convex_fderiv_apply_limit {n : ℕ} {ι : Type uι} {l : Filter ι}
+    {f : ι → Space n → ℝ} {g : Space n → ℝ} {x : Space n}
+    (hf : ∀ j, ConvexOn ℝ univ (f j)) (hd : ∀ j, DifferentiableAt ℝ (f j) x)
+    (hg : DifferentiableAt ℝ g x)
+    (hlim : ∀ y, Tendsto (fun j => f j y) l (𝓝 (g y))) (w : Space n) :
+    Tendsto (fun j => fderiv ℝ (f j) x w) l (𝓝 (fderiv ℝ g x w)) := by
+  apply convex_deriv_limit (fun j => line_convex (hf j) x w)
+  · intro j
+    have h : HasFDerivAt (f j) (fderiv ℝ (f j) x) (x + (0 : ℝ) • w) := by
+      simpa only [zero_smul, add_zero] using (hd j).hasFDerivAt
+    exact h.comp_hasDerivAt 0 (line_hasDerivAt x w 0)
+  · have h : HasFDerivAt g (fderiv ℝ g x) (x + (0 : ℝ) • w) := by
+      simpa only [zero_smul, add_zero] using hg.hasFDerivAt
+    exact h.comp_hasDerivAt 0 (line_hasDerivAt x w 0)
+  · intro t
+    exact hlim (x + t • w)
+
+lemma convex_gradient_limit {n : ℕ} {ι : Type uι} {l : Filter ι}
+    {f : ι → Space n → ℝ} {g : Space n → ℝ} {x : Space n}
+    (hf : ∀ j, ConvexOn ℝ univ (f j)) (hd : ∀ j, DifferentiableAt ℝ (f j) x)
+    (hg : DifferentiableAt ℝ g x)
+    (hlim : ∀ y, Tendsto (fun j => f j y) l (𝓝 (g y))) :
+    Tendsto (fun j => ∇ (f j) x) l (𝓝 (∇ g x)) := by
+  have hp : Tendsto (fun j i => (∇ (f j) x) i) l (𝓝 (fun i => (∇ g x) i)) := by
+    apply tendsto_pi_nhds.mpr
+    intro i
+    have h := convex_fderiv_apply_limit hf hd hg hlim (EuclideanSpace.single i 1)
+    simpa only [← inner_gradient_left, EuclideanSpace.inner_single_right, one_mul, conj_trivial] using h
+  exact (PiLp.continuous_toLp 2 (fun _ : Fin n => ℝ)).continuousAt.tendsto.comp hp
+
+lemma normSmooth_tendsto {n : ℕ} (g : Seminorm ℝ (Space n)) (x : Space n) :
+    Tendsto (fun j => normSmooth g (shrinkingBump n j) x) atTop (𝓝 (g x)) := by
+  obtain ⟨C, hC⟩ := seminorm_lipschitz n g
+  apply tendsto_iff_dist_tendsto_zero.mpr
+  exact squeeze_zero (fun _ => dist_nonneg) (fun j => normSmooth_dist_le g _ hC x)
+    (by simpa only [mul_zero, shrinkingBump] using
+      (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).const_mul (C : ℝ))
+
+lemma normSmooth_gradient_tendsto {n : ℕ} (g : Seminorm ℝ (Space n)) {x : Space n}
+    (hx : DifferentiableAt ℝ (g : Space n → ℝ) x) :
+    Tendsto (fun j => ∇ (normSmooth g (shrinkingBump n j)) x) atTop (𝓝 (∇ (g : Space n → ℝ) x)) :=
+  convex_gradient_limit (fun j => normSmooth_convex g _)
+    (fun j => (normSmooth_contDiff g _).differentiable (by norm_num) x) hx (normSmooth_tendsto g)
+
+end PettyProjection.Spherical
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/ProjectionBodies/VariationalMinimizers.lean b/lean/OAI/Geometry/ProjectionBodies/VariationalMinimizers.lean
new file mode 100644
index 0000000000000000000000000000000000000000..6e30bc191463cef5f0aaf239ee5dd444736908af
--- /dev/null
+++ b/lean/OAI/Geometry/ProjectionBodies/VariationalMinimizers.lean
@@ -0,0 +1,537 @@
+import OAI.Geometry.ProjectionBodies.RelativeGauges
+
+namespace OAI
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology
+open scoped NNReal RealInnerProductSpace
+namespace PettyProjection
+namespace RelativeGauge
+variable {n : ℕ} [NeZero n] {V : Submodule ℝ (Space n)}
+
+omit [NeZero n] in
+lemma exists_normalized {μ : Measure (Space n)}
+    (hμ : Integrable (fun x : Space n => ‖x‖) μ) (hV : V≠⊥)
+    (hq : ∀ x∈V,x≠0 → 0<cosineSeminorm μ hμ x) :
+    ∃ g : RelativeGauge V,supportFunctional μ g.body=1 := by
+  let g := ball V
+  have hp := g.functional_pos hV hμ hq
+  refine ⟨g.scale _ hp,?_⟩
+  rw [functional_scale g _ hp μ,inv_mul_cancel₀ hp.ne']
+
+/-- The actual fixed-subspace variational minimizer, including singular V. -/
+theorem exists_minimizer {μ : Measure (Space n)}
+    (hμ : Integrable (fun x : Space n => ‖x‖) μ) (hV : V≠⊥)
+    (hq : ∀ x∈V,x≠0 → 0<cosineSeminorm μ hμ x) (b : Bool) :
+    ∃ g : RelativeGauge V,supportFunctional μ g.body=1 ∧
+      ∀ f : RelativeGauge V,supportFunctional μ f.body=1 →
+        Spherical.objective b g.toSeminorm≤Spherical.objective b f.toSeminorm := by
+  let S : Set ℝ := {r | ∃ g : RelativeGauge V,supportFunctional μ g.body=1 ∧ Spherical.objective b g.toSeminorm=r}
+  obtain ⟨g₀,hg₀⟩ := exists_normalized hμ hV hq
+  have hS : S.Nonempty := ⟨_,g₀,hg₀,rfl⟩
+  let q := (cosineSeminorm μ hμ).comp V.starProjection.toLinearMap
+  have hq0 : q≠0 := by
+    obtain ⟨x,hx,hx0⟩ := (Submodule.ne_bot_iff V).mp hV
+    intro he
+    have hp := hq x hx hx0
+    have hz : q x=0 := by rw [he]; rfl
+    change cosineSeminorm μ hμ (V.starProjection x)=0 at hz
+    rw [V.starProjection_eq_self_iff.mpr hx] at hz
+    linarith
+  have hSb : BddBelow S := by
+    refine ⟨Spherical.objective b q,?_⟩
+    rintro _ ⟨g,hg,rfl⟩
+    exact Spherical.objective_mono b hq0 (g.ne_zero hV) (g.normalized_lower hμ hg)
+  obtain ⟨r,hr,ht,hm⟩ := exists_seq_tendsto_sInf hS hSb
+  choose g hgn hgr using hm
+  obtain ⟨f,φ,hφ,hfn,hu,hfo⟩ := normalized_subsequence hμ hV hq b g hgn
+    (M := r 0) (fun i => by rw [hgr i]; exact hr (Nat.zero_le i))
+  have he : Spherical.objective b f.toSeminorm=sInf S := by
+    apply tendsto_nhds_unique hfo
+    have hh := ht.comp hφ.tendsto_atTop
+    simpa only [Function.comp_def,← hgr] using hh
+  refine ⟨f,hfn,fun g hg => ?_⟩
+  rw [he]
+  exact csInf_le hSb ⟨g,hg,rfl⟩
+
+end RelativeGauge
+end PettyProjection
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology
+open scoped NNReal RealInnerProductSpace Pointwise
+namespace PettyProjection
+namespace RelativeGauge
+variable {n : ℕ} {V : Submodule ℝ (Space n)}
+
+def midBody (g f : RelativeGauge V) : Set (Space n) := (1/2:ℝ) • (g.body+f.body)
+lemma midBody_compact (g f : RelativeGauge V) : IsCompact (midBody g f) :=
+  (g.compact.add f.compact).smul _
+lemma midBody_convex (g f : RelativeGauge V) : Convex ℝ (midBody g f) :=
+  (g.convex.add f.convex).smul _
+lemma midBody_balanced (g f : RelativeGauge V) : Balanced ℝ (midBody g f) :=
+  (g.balanced.add f.balanced).smul _
+lemma midBody_sub (g f : RelativeGauge V) : midBody g f⊆V := by
+  rintro x ⟨y,⟨a,ha,b,hb,rfl⟩,rfl⟩
+  exact V.smul_mem _ (V.add_mem ha.1 hb.1)
+lemma midBody_inner (g f : RelativeGauge V) :
+    ∃ r : ℝ,0<r ∧ ∀ x∈V,‖x‖≤r → x∈midBody g f := by
+  obtain ⟨r,hr,hin⟩ := g.inner_ball
+  refine ⟨r/2,by positivity,fun x hx hh => ?_⟩
+  refine ⟨(2:ℝ) • x,⟨(2:ℝ) • x,hin _ (V.smul_mem _ hx) ?_,0,f.zero_mem,by simp⟩,by simp [smul_smul]⟩
+  rw [norm_smul,Real.norm_of_nonneg (by norm_num : (0:ℝ)≤2)]
+  linarith
+
+def midpoint (g f : RelativeGauge V) : RelativeGauge V :=
+  ofBody (midBody_compact g f) (midBody_convex g f) (midBody_balanced g f) (midBody_inner g f)
+lemma body_midpoint (g f : RelativeGauge V) : (midpoint g f).body=midBody g f :=
+  body_ofBody _ _ _ (midBody_sub g f) _
+lemma functional_midpoint (g f : RelativeGauge V) {μ : Measure (Space n)}
+    (hμ : Integrable (fun x : Space n => ‖x‖) μ) :
+    supportFunctional μ (midpoint g f).body=(supportFunctional μ g.body+supportFunctional μ f.body)/2 := by
+  rw [body_midpoint,midBody,functional_smul (g.compact.add f.compact) (g.nonempty.add f.nonempty) (by norm_num : (0:ℝ)≤1/2),
+    functional_add hμ g.compact g.nonempty f.compact f.nonempty]
+  ring
+
+lemma midpoint_harmonic (g f : RelativeGauge V) {x : Space n} (hx : V.starProjection x≠0) :
+    midpoint g f x≤((g x)⁻¹+(f x)⁻¹)⁻¹*2 := by
+  let y := V.starProjection x
+  have hy : y∈V := V.starProjection_apply_mem x
+  have hg : 0<g x := by simpa only [y,g.projection] using g.pos hy hx
+  have hf : 0<f x := by simpa only [y,f.projection] using f.pos hy hx
+  have hmem : ((g x)⁻¹/2+(f x)⁻¹/2) • y∈(midpoint g f).body := by
+    rw [body_midpoint]
+    refine ⟨(g x)⁻¹ • y+(f x)⁻¹ • y,⟨_,?_,_,?_,rfl⟩,?_⟩
+    · simpa only [y,g.projection] using g.radial_mem hy hx
+    · simpa only [y,f.projection] using f.radial_mem hy hx
+    · simp only [smul_smul,← add_smul]; congr 1; ring
+  have hpos : 0<(g x)⁻¹/2+(f x)⁻¹/2 := by positivity
+  have hh : (midpoint g f).toSeminorm (((g x)⁻¹/2+(f x)⁻¹/2) • y)≤1 := hmem.2
+  rw [map_smul_eq_mul,Real.norm_of_nonneg hpos.le,(midpoint g f).projection] at hh
+  have hb := (le_div_iff₀ hpos).mpr (by simpa only [mul_comm] using hh)
+  have he : 1/((g x)⁻¹/2+(f x)⁻¹/2)=((g x)⁻¹+(f x)⁻¹)⁻¹*2 := by
+    rw [show (g x)⁻¹/2+(f x)⁻¹/2=((g x)⁻¹+(f x)⁻¹)/2 by ring,one_div,inv_div]
+    ring
+  exact hb.trans_eq he
+
+end RelativeGauge
+end PettyProjection
+end
+
+noncomputable section
+open Set
+namespace PettyProjection.Spherical
+
+lemma harmonic_objective {a d c : ℝ} (ha : 0<a) (hd : 0<d) (hc : 0<c)
+    (hle : c≤(a⁻¹+d⁻¹)⁻¹*2) (b : Bool) :
+    objectiveIntegrand b c≤(objectiveIntegrand b a+objectiveIntegrand b d)/2 ∧
+      (objectiveIntegrand b c=(objectiveIntegrand b a+objectiveIntegrand b d)/2 → a=d) := by
+  have hsum : 0<a+d := add_pos ha hd
+  have he : (a⁻¹+d⁻¹)⁻¹*2=2*a*d/(a+d) := by field_simp; ring
+  have hmul : c*(a+d)≤2*a*d := (le_div_iff₀ hsum).mp (by rwa [← he])
+  have havg : c≤(a+d)/2 := by nlinarith [sq_nonneg (a-d)]
+  cases b
+  · change c≤(a+d)/2 ∧ (c=(a+d)/2 → a=d)
+    refine ⟨havg,fun hh => ?_⟩
+    nlinarith [sq_nonneg (a-d)]
+  · change Real.log c≤(Real.log a+Real.log d)/2 ∧
+      (Real.log c=(Real.log a+Real.log d)/2 → a=d)
+    have hmean : 0<(a⁻¹+d⁻¹)/2 := by positivity
+    have he' : (a⁻¹+d⁻¹)⁻¹*2=((a⁻¹+d⁻¹)/2)⁻¹ := by field_simp
+    have hlog : Real.log c≤ -Real.log ((a⁻¹+d⁻¹)/2) := by
+      have hh := Real.strictMonoOn_log.monotoneOn hc (inv_pos.mpr hmean) (by rwa [← he'])
+      simpa only [Real.log_inv] using hh
+    have hj : (1/2:ℝ)*Real.log a⁻¹+(1/2:ℝ)*Real.log d⁻¹≤Real.log ((1/2:ℝ)*a⁻¹+(1/2:ℝ)*d⁻¹) :=
+      (strictConcaveOn_log_Ioi).concaveOn.2 (inv_pos.mpr ha) (inv_pos.mpr hd) (by norm_num) (by norm_num) (by norm_num)
+    simp only [Real.log_inv] at hj
+    have hemean : (1/2:ℝ)*a⁻¹+(1/2:ℝ)*d⁻¹=(a⁻¹+d⁻¹)/2 := by ring
+    rw [hemean] at hj
+    refine ⟨by linarith,fun hh => ?_⟩
+    by_contra hne
+    have hi : a⁻¹≠d⁻¹ := by intro he; exact hne (inv_injective he)
+    have hs : (1/2:ℝ)*Real.log a⁻¹+(1/2:ℝ)*Real.log d⁻¹<Real.log ((1/2:ℝ)*a⁻¹+(1/2:ℝ)*d⁻¹) :=
+      strictConcaveOn_log_Ioi.2 (inv_pos.mpr ha) (inv_pos.mpr hd) hi (by norm_num) (by norm_num) (by norm_num)
+    simp only [Real.log_inv,hemean] at hs
+    linarith
+
+end PettyProjection.Spherical
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology
+open scoped NNReal RealInnerProductSpace
+namespace PettyProjection
+open Spherical (Sphere sigma objective objectiveIntegrand objective_integrable)
+namespace RelativeGauge
+variable {n : ℕ} [NeZero n] {V : Submodule ℝ (Space n)}
+
+omit [NeZero n] in
+@[ext] lemma ext {g f : RelativeGauge V} (h : g.toSeminorm=f.toSeminorm) : g=f := by
+  cases g; cases f; cases h; rfl
+
+lemma midpoint_objective {g f : RelativeGauge V} (hV : V≠⊥) (b : Bool) :
+    objective b (midpoint g f).toSeminorm≤(objective b g.toSeminorm+objective b f.toSeminorm)/2 ∧
+      (objective b (midpoint g f).toSeminorm=(objective b g.toSeminorm+objective b f.toSeminorm)/2 → g=f) := by
+  let h := midpoint g f
+  have hpos := Spherical.seminorm_pos_ae_sphere (g.ne_zero hV)
+  have hposf := Spherical.seminorm_pos_ae_sphere (f.ne_zero hV)
+  have hposh := Spherical.seminorm_pos_ae_sphere (h.ne_zero hV)
+  have hmul : ∀ᵐ u : Sphere n ∂sigma n,
+      objectiveIntegrand b (h u)≤(objectiveIntegrand b (g u)+objectiveIntegrand b (f u))/2 ∧
+        (objectiveIntegrand b (h u)=(objectiveIntegrand b (g u)+objectiveIntegrand b (f u))/2 → g u=f u) := by
+    filter_upwards [hpos,hposf,hposh] with u hu hv hw
+    have hx : V.starProjection (u:Space n)≠0 := by
+      intro he
+      have hz := g.projection u
+      rw [he,map_zero] at hz
+      linarith
+    exact Spherical.harmonic_objective hu hv hw (g.midpoint_harmonic f hx) b
+  have hi : Integrable (fun u : Sphere n => (objectiveIntegrand b (g u)+objectiveIntegrand b (f u))/2) (sigma n) :=
+    ((objective_integrable b (g.ne_zero hV)).add (objective_integrable b (f.ne_zero hV))).div_const 2
+  have hh := objective_integrable b (h.ne_zero hV)
+  have havg : (∫ u : Sphere n,(objectiveIntegrand b (g u)+objectiveIntegrand b (f u))/2 ∂sigma n)=
+      (objective b g.toSeminorm+objective b f.toSeminorm)/2 := by
+    rw [integral_div,integral_add (objective_integrable b (g.ne_zero hV)) (objective_integrable b (f.ne_zero hV))]
+    rfl
+  have hle := integral_mono_ae hh hi (hmul.mono (fun _ h => h.1))
+  refine ⟨by simpa only [havg,objective,Spherical.mean,h] using hle,fun he => ?_⟩
+  have hae := (integral_eq_iff_of_ae_le hh hi (hmul.mono (fun _ h => h.1))).mp (by
+    rw [havg]; exact he)
+  have hae' : (fun u : Sphere n => g u)=ᵐ[sigma n] (fun u : Sphere n => f u) := by
+    filter_upwards [hae,hmul] with u hu hv
+    exact hv.2 hu
+  have hp := Measure.eq_of_ae_eq hae' ((Spherical.seminorm_continuous n g.toSeminorm).comp continuous_subtype_val)
+    ((Spherical.seminorm_continuous n f.toSeminorm).comp continuous_subtype_val)
+  apply ext
+  ext x
+  rw [← Spherical.radialExtension_seminorm g.toSeminorm x,← Spherical.radialExtension_seminorm f.toSeminorm x,hp]
+
+lemma minimizer_unique {μ : Measure (Space n)}
+    (hμ : Integrable (fun x : Space n => ‖x‖) μ) (hV : V≠⊥) (b : Bool)
+    {g f : RelativeGauge V} (hgn : supportFunctional μ g.body=1) (hfn : supportFunctional μ f.body=1)
+    (hmin : ∀ q : RelativeGauge V,supportFunctional μ q.body=1 → objective b g.toSeminorm≤objective b q.toSeminorm)
+    (heq : objective b g.toSeminorm=objective b f.toSeminorm) : g=f := by
+  have hn : supportFunctional μ (midpoint g f).body=1 := by
+    rw [functional_midpoint g f hμ,hgn,hfn]; norm_num
+  have hm := hmin (midpoint g f) hn
+  obtain ⟨hle,he⟩ := midpoint_objective (g := g) (f := f) hV b
+  apply he
+  rw [← heq] at hle ⊢
+  linarith
+
+end RelativeGauge
+end PettyProjection
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology Function
+open scoped ENNReal NNReal RealInnerProductSpace Gradient
+namespace PettyProjection
+open Spherical (Sphere norm_coe sigma mean objectiveIntegrand)
+
+lemma objective_param_lipschitz {n : ℕ} [NeZero n] (b : Bool)
+    (s φ : C(Sphere n,ℝ)) (hs : ∀ u,0 < s u) :
+    ∃ T ∈ 𝓝 (0 : ℝ), ∃ C : ℝ≥0, ∀ u : Sphere n,
+      LipschitzOnWith C (fun t : ℝ => objectiveIntegrand b (gauge (wulff ⇑(s+t • φ)) (u : Space n))) T := by
+  obtain ⟨m,M,hm,hM,he⟩ := positive_constraints_eventually s φ hs
+  let T : Set ℝ := {t | ∀ u, m ≤ (s+t • φ) u ∧ (s+t • φ) u ≤ M}
+  have hT : T ∈ 𝓝 (0 : ℝ) := he
+  have hc : ContDiffOn ℝ 1 (objectiveIntegrand b) (Icc M⁻¹ m⁻¹) := by
+    cases b
+    · exact contDiff_id.contDiffOn
+    · intro a ha
+      exact (Real.contDiffAt_log.mpr ((inv_pos.mpr hM).trans_le ha.1).ne').contDiffWithinAt
+  obtain ⟨D,hD⟩ := hc.exists_lipschitzOnWith (by norm_num) (convex_Icc _ _) isCompact_Icc
+  let C : ℝ≥0 := ⟨‖φ‖/m^2,by positivity⟩
+  refine ⟨T,hT,D*C,fun u => ?_⟩
+  apply hD.comp (wulff_gauge_param_lipschitz s φ hm (fun t ht v => (ht v).1) u)
+  exact fun t ht => wulff_sphere_gauge_bounds (s+t • φ) hm ht u
+
+lemma objectiveIntegrand_continuousOn (b : Bool) : ContinuousOn (objectiveIntegrand b) (Ioi 0) := by
+  cases b
+  · exact continuous_id.continuousOn
+  · exact Real.continuousOn_log.mono (fun _ h => ne_of_gt h)
+
+lemma objective_gauge_derivative {n : ℕ} [NeZero n] (b : Bool) (s : C(Sphere n,ℝ))
+    (hs : ∀ u,0 < s u) (φ : Space n → ℝ) (hφ : Continuous φ)
+    (hh : ∀ (a : ℝ),0 < a → ∀ x,φ (a • x)=a*φ x)
+    {u : Sphere n} (hd : DifferentiableAt ℝ (gauge (wulff s)) (u : Space n)) :
+    HasDerivAt (fun t : ℝ => objectiveIntegrand b (gauge (wulff ⇑(s+t • (⟨fun v : Sphere n => φ v,
+      hφ.comp continuous_subtype_val⟩ : C(Sphere n,ℝ)))) (u : Space n)))
+      (- (if b then 1 else gauge (wulff s) (u:Space n))*φ (∇ (gauge (wulff s)) (u : Space n))) 0 := by
+  have hp := gauge_pos_sphere (wulff_compact s hs) (wulff_convex _) (wulff_nhds s hs) u
+  have hder := wulff_derivative s hs φ hφ hh hd
+  cases b
+  · exact hder
+  · have hdlog := hder.log (by simpa only [zero_smul,add_zero] using hp.ne')
+    apply hdlog.congr_deriv
+    simp only [zero_smul,add_zero]
+    change -gauge (wulff s) (u:Space n)*φ _/gauge (wulff s) (u:Space n)= -1*φ _
+    field_simp
+
+/-- Two-sided first variation of both variational objectives, with no smoothness
+assumption on the optimizer. -/
+theorem wulff_objective_derivative {n : ℕ} [NeZero n] (b : Bool) (s : C(Sphere n,ℝ))
+    (hs : ∀ u,0 < s u) (φ : Space n → ℝ) (hφ : Continuous φ)
+    (hh : ∀ (a : ℝ),0 < a → ∀ x,φ (a • x)=a*φ x) :
+    HasDerivAt (fun t : ℝ => mean (fun u : Sphere n => objectiveIntegrand b
+      (gauge (wulff ⇑(s+t • (⟨fun v : Sphere n => φ v,
+        hφ.comp continuous_subtype_val⟩ : C(Sphere n,ℝ)))) (u:Space n))))
+      (-mean (fun u : Sphere n => (if b then 1 else gauge (wulff s) (u:Space n))*
+        φ (∇ (gauge (wulff s)) (u:Space n)))) 0 := by
+  let φs : C(Sphere n,ℝ) := ⟨fun u => φ u,hφ.comp continuous_subtype_val⟩
+  let F : ℝ → Sphere n → ℝ := fun t u => objectiveIntegrand b (gauge (wulff ⇑(s+t • φs)) (u:Space n))
+  let F' : Sphere n → ℝ := fun u => -(if b then 1 else gauge (wulff s) (u:Space n))*φ (∇ (gauge (wulff s)) (u:Space n))
+  have hg := continuous_gauge (wulff_convex s) (wulff_nhds s hs)
+  have hp := gauge_pos_sphere (wulff_compact s hs) (wulff_convex _) (wulff_nhds s hs)
+  have hc : Continuous (F 0) := by
+    simp only [F,zero_smul,add_zero]
+    exact (objectiveIntegrand_continuousOn b).comp_continuous (hg.comp continuous_subtype_val) hp
+  obtain ⟨T,hT,C,hC⟩ := objective_param_lipschitz b s φs hs
+  have hFm : ∀ᶠ t in 𝓝 (0:ℝ), AEStronglyMeasurable (F t) (sigma n) := by
+    filter_upwards [positive_constraints_nhds s φs hs] with t ht
+    have hgt := continuous_gauge (wulff_convex (s+t • φs)) (wulff_nhds (s+t • φs) ht)
+    have hpt := gauge_pos_sphere (wulff_compact (s+t • φs) ht) (wulff_convex _) (wulff_nhds (s+t • φs) ht)
+    exact ((objectiveIntegrand_continuousOn b).comp_continuous (hgt.comp continuous_subtype_val) hpt).aestronglyMeasurable
+  have hF'm : AEStronglyMeasurable F' (sigma n) := by
+    apply Measurable.aestronglyMeasurable
+    apply Measurable.mul
+    · cases b <;> simp only [Bool.false_eq_true,ite_false,ite_true]
+      · exact (hg.measurable.comp measurable_subtype_coe).neg
+      · exact measurable_const
+    · exact hφ.measurable.comp ((measurable_gradient _).comp measurable_subtype_coe)
+  have hder : ∀ᵐ u ∂sigma n, HasDerivAt (fun t => F t u) (F' u) 0 := by
+    filter_upwards [gauge_ae_differentiable_sphere (wulff_convex s) (wulff_nhds s hs)] with u hu
+    exact objective_gauge_derivative b s hs φ hφ hh hu
+  have hI := (hasDerivAt_integral_of_dominated_loc_of_lip hT hFm
+    (Spherical.continuous_integrable hc) hF'm
+    (Eventually.of_forall (fun u => by simpa only [Real.nnabs_coe] using hC u))
+    (integrable_const (C:ℝ)) hder).2
+  apply hI.congr_deriv
+  dsimp only [F',mean]
+  simp only [neg_mul,integral_neg]
+
+end PettyProjection
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology
+open scoped NNReal RealInnerProductSpace Pointwise
+namespace PettyProjection
+open Spherical (Sphere)
+
+lemma support_even {n : ℕ} {K : Set (Space n)} (hK : IsCompact K)
+    (hne : K.Nonempty) (hsym : ∀ x∈K,-x∈K) (u : Space n) : support K (-u)=support K u := by
+  apply le_antisymm
+  · apply (support_le_iff hK hne).mpr
+    intro x hx
+    have hh := inner_le_support hK (hsym x hx) u
+    simpa only [inner_neg_left,inner_neg_right] using hh
+  · apply (support_le_iff hK hne).mpr
+    intro x hx
+    have hh := inner_le_support hK (hsym x hx) (-u)
+    simpa only [inner_neg_left,inner_neg_right,neg_neg] using hh
+
+lemma wulff_balanced {n : ℕ} (s : Sphere n → ℝ)
+    (hs : ∀ u : Sphere n,s ⟨-(u:Space n),by simp⟩=s u) : Balanced ℝ (wulff s) := by
+  apply (balanced_iff_neg_mem (wulff_convex s)).mpr
+  intro x hx u
+  have hh := hx ⟨-(u:Space n),by simp⟩
+  change ⟪(u:Space n),-x⟫≤ s u
+  simpa only [inner_neg_left,inner_neg_right,hs] using hh
+
+namespace RelativeGauge
+variable {n : ℕ}
+
+lemma top_body (g : RelativeGauge (⊤ : Submodule ℝ (Space n))) : g.body=seminormUnit g.toSeminorm := by
+  ext x; simp [body]
+lemma top_nhds (g : RelativeGauge (⊤ : Submodule ℝ (Space n))) : g.body∈𝓝 (0:Space n) := by
+  rw [g.top_body]; exact seminormUnit_nhds _
+lemma top_gauge (g : RelativeGauge (⊤ : Submodule ℝ (Space n))) (x : Space n) : gauge g.body x=g x := by
+  rw [g.top_body]; exact gauge_seminormUnit _ _
+lemma full_inner {K : Set (Space n)} (h0 : K∈𝓝 (0:Space n)) :
+    ∃ r : ℝ,0<r ∧ ∀ x∈(⊤ : Submodule ℝ (Space n)),‖x‖≤r → x∈K := by
+  obtain ⟨r,hr,hin⟩ := Metric.mem_nhds_iff.mp h0
+  refine ⟨r/2,by positivity,fun x _ hx => hin ?_⟩
+  rw [mem_ball,dist_zero_right]; linarith
+
+def ofFullBody {K : Set (Space n)} (hK : IsCompact K) (hc : Convex ℝ K)
+    (hb : Balanced ℝ K) (h0 : K∈𝓝 (0:Space n)) : RelativeGauge (⊤ : Submodule ℝ (Space n)) :=
+  ofBody hK hc hb (full_inner h0)
+lemma body_ofFullBody {K : Set (Space n)} (hK : IsCompact K) (hc : Convex ℝ K)
+    (hb : Balanced ℝ K) (h0 : K∈𝓝 (0:Space n)) : (ofFullBody hK hc hb h0).body=K :=
+  body_ofBody _ _ _ (fun _ _ => Submodule.mem_top) _
+lemma ofFullBody_apply {K : Set (Space n)} (hK : IsCompact K) (hc : Convex ℝ K)
+    (hb : Balanced ℝ K) (h0 : K∈𝓝 (0:Space n)) (x : Space n) : ofFullBody hK hc hb h0 x=gauge K x := by
+  rw [← top_gauge,body_ofFullBody]
+
+end RelativeGauge
+
+lemma homogeneous_integrable {n : ℕ} [NeZero n] {μ : Measure (Space n)}
+    (hμ : Integrable (fun x : Space n => ‖x‖) μ) {φ : Space n → ℝ} (hφ : Continuous φ)
+    (hh : ∀ a : ℝ,0<a → ∀ x,φ (a • x)=a*φ x) : Integrable φ μ := by
+  let φs : C(Sphere n,ℝ) := ⟨fun u => φ u,hφ.comp continuous_subtype_val⟩
+  have hzero : φ 0=0 := by
+    have hz := hh 2 (by norm_num) 0
+    simp only [smul_zero] at hz
+    linarith
+  apply (hμ.const_mul ‖φs‖).mono' hφ.aestronglyMeasurable
+  apply Eventually.of_forall
+  intro x
+  by_cases hx : x=0
+  · simp [hx,hzero]
+  have hp := norm_pos_iff.mpr hx
+  let u : Sphere n := ⟨‖x‖⁻¹ • x,by simp [norm_smul,hx]⟩
+  have he : x=‖x‖ • (u:Space n) := by simp [u,smul_smul,hx]
+  rw [he,hh _ hp,norm_mul,Real.norm_of_nonneg hp.le]
+  have hbound := φs.norm_coe_le_norm u
+  change |φ (u:Space n)|≤‖φs‖ at hbound
+  have heq : ‖‖x‖ • (u:Space n)‖=‖x‖ := by simp [norm_smul]
+  rw [heq]
+  simpa only [Real.norm_eq_abs,mul_comm] using mul_le_mul_of_nonneg_left hbound hp.le
+
+end PettyProjection
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology
+open scoped NNReal RealInnerProductSpace Pointwise
+namespace PettyProjection
+open Spherical (Sphere norm_coe seminorm_continuous)
+
+/-- Hahn--Banach attainment for the actual Wulff body of a seminorm. -/
+lemma seminorm_supporting_vector {n : ℕ} (q : Seminorm ℝ (Space n)) (u : Sphere n) :
+    ∃ z : Space n,(∀ x : Space n,|⟪x,z⟫|≤q x) ∧ ⟪(u:Space n),z⟫=q u := by
+  let V := Submodule.span ℝ {(u:Space n)}
+  let f : Module.Dual ℝ V := q u • ((innerSL ℝ (u:Space n)).toLinearMap.comp V.subtype)
+  have hf (x : V) : |f x|≤q (x:Space n) := by
+    obtain ⟨a,ha⟩ := Submodule.mem_span_singleton.mp x.property
+    change |q u*⟪(u:Space n),(x:Space n)⟫|≤q (x:Space n)
+    rw [← ha,inner_smul_right,real_inner_self_eq_norm_sq,norm_coe,map_smul_eq_mul]
+    simp only [one_pow,mul_one,Real.norm_eq_abs,abs_mul,abs_of_nonneg (apply_nonneg q _)]
+    rw [mul_comm]
+  obtain ⟨φ,hφ,hb⟩ := Module.Dual.exists_continuous_extension_of_le_seminorm_real
+    V f (seminorm_continuous n q) (fun x => (le_abs_self _).trans (hf x))
+  let z := (InnerProductSpace.toDual ℝ (Space n)).symm φ
+  have hz (x : Space n) : ⟪x,z⟫=φ x := by
+    rw [real_inner_comm]
+    exact InnerProductSpace.toDual_symm_apply
+  refine ⟨z,fun x => by rw [hz];exact hb x,?_⟩
+  rw [hz]
+  have h := hφ ⟨(u:Space n),Submodule.mem_span_singleton_self (u:Space n)⟩
+  simpa [f,real_inner_self_eq_norm_sq,norm_coe] using h
+
+lemma support_wulff_seminorm_unit {n : ℕ} [NeZero n] (q : Seminorm ℝ (Space n))
+    (hq : ∀ u : Sphere n,0<q u) (u : Sphere n) :
+    support (wulff (fun v : Sphere n => q v)) u=q u := by
+  let s : C(Sphere n,ℝ) := ⟨fun u => q u,(seminorm_continuous n q).comp continuous_subtype_val⟩
+  have hc := wulff_compact s hq
+  have hn : (wulff s).Nonempty := ⟨0,mem_of_mem_nhds (wulff_nhds s hq)⟩
+  apply le_antisymm
+  · exact (support_le_iff hc hn).mpr (fun x hx => hx u)
+  · obtain ⟨z,hz,he⟩ := seminorm_supporting_vector q u
+    have hmem : z∈wulff s := fun v => (le_abs_self _).trans (hz v)
+    rw [← he]
+    exact inner_le_support hc hmem u
+
+lemma support_wulff_seminorm {n : ℕ} [NeZero n] (q : Seminorm ℝ (Space n))
+    (hq : ∀ u : Sphere n,0<q u) (x : Space n) :
+    support (wulff (fun v : Sphere n => q v)) x=q x := by
+  let s : C(Sphere n,ℝ) := ⟨fun u => q u,(seminorm_continuous n q).comp continuous_subtype_val⟩
+  have hc := wulff_compact s hq
+  have hn : (wulff s).Nonempty := ⟨0,mem_of_mem_nhds (wulff_nhds s hq)⟩
+  change IsCompact (wulff (fun v : Sphere n => q v)) at hc
+  change (wulff (fun v : Sphere n => q v)).Nonempty at hn
+  by_cases hx : x=0
+  · subst x
+    obtain ⟨z,_,he⟩ := support_attained hc hn 0
+    simpa only [inner_zero_left,map_zero] using he
+  · let u : Sphere n := ⟨‖x‖⁻¹ • x,by simp [norm_smul,hx]⟩
+    have hxu : x=‖x‖ • (u:Space n) := by simp [u,smul_smul,norm_ne_zero_iff.mpr hx]
+    rw [hxu,support_smul hc hn (norm_nonneg _),support_wulff_seminorm_unit q hq]
+    simpa only [Real.norm_of_nonneg (norm_nonneg x)] using (map_smul_eq_mul q ‖x‖ (u:Space n)).symm
+
+end PettyProjection
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology
+open scoped NNReal RealInnerProductSpace
+namespace PettyProjection
+open Spherical (Sphere)
+
+lemma homogeneous_zero {n : ℕ} {φ : Space n → ℝ}
+    (hh : ∀ a : ℝ,0<a → ∀ x,φ (a • x)=a*φ x) : φ 0=0 := by
+  have hh := hh 2 (by norm_num) 0
+  simp only [smul_zero] at hh
+  linarith
+
+lemma support_wulff_bound {n : ℕ} [NeZero n] {K : Set (Space n)}
+    (hK : IsCompact K) (hne : K.Nonempty) (φ : Space n → ℝ) (hφ : Continuous φ)
+    (hh : ∀ a : ℝ,0<a → ∀ x,φ (a • x)=a*φ x) (t : ℝ)
+    (hp : ∀ u : Sphere n,0 < support K u+t*φ u) (x : Space n) :
+    support (wulff (fun u : Sphere n => support K u+t*φ u)) x≤ support K x+t*φ x := by
+  let s : C(Sphere n,ℝ) := ⟨fun u => support K u+t*φ u,((support_continuous hK).comp continuous_subtype_val).add (continuous_const.mul (hφ.comp continuous_subtype_val))⟩
+  have hW := wulff_compact s hp
+  have hWn : (wulff s).Nonempty := ⟨0,mem_of_mem_nhds (wulff_nhds s hp)⟩
+  apply (support_le_iff hW hWn).mpr
+  intro y hy
+  by_cases hx : x=0
+  · subst x
+    rw [homogeneous_zero hh,inner_zero_left]
+    have hz : support K 0=0 := by
+      obtain ⟨z,hz,he⟩ := support_attained hK hne 0
+      simpa only [inner_zero_left] using he
+    simp only [hz,mul_zero,add_zero,le_refl]
+  have hxpos := norm_pos_iff.mpr hx
+  let u : Sphere n := ⟨‖x‖⁻¹ • x,by simp [norm_smul,hx]⟩
+  have hu := hy u
+  have he : x=‖x‖ • (u:Space n) := by simp [u,smul_smul,hx]
+  calc
+    ⟪x,y⟫ = ‖x‖*⟪(u:Space n),y⟫ := by
+      conv_lhs => rw [he,inner_smul_left]
+      rfl
+    _ ≤ ‖x‖*(support K u+t*φ u) := mul_le_mul_of_nonneg_left hu hxpos.le
+    _ = support K x+t*φ x := by
+      conv_rhs => rw [he,support_smul hK hne hxpos.le,hh _ hxpos]
+      ring
+
+lemma functional_wulff_bound {n : ℕ} [NeZero n] {μ : Measure (Space n)}
+    (hμ : Integrable (fun x : Space n => ‖x‖) μ) {K : Set (Space n)}
+    (hK : IsCompact K) (hne : K.Nonempty) (φ : Space n → ℝ) (hφ : Continuous φ)
+    (hh : ∀ a : ℝ,0<a → ∀ x,φ (a • x)=a*φ x) (t : ℝ)
+    (hp : ∀ u : Sphere n,0 < support K u+t*φ u) :
+    supportFunctional μ (wulff (fun u : Sphere n => support K u+t*φ u))≤
+      supportFunctional μ K+t*(∫ x,φ x ∂μ) := by
+  let s : C(Sphere n,ℝ) := ⟨fun u => support K u+t*φ u,((support_continuous hK).comp continuous_subtype_val).add (continuous_const.mul (hφ.comp continuous_subtype_val))⟩
+  have hi := support_integrable hμ (wulff_compact s hp) ⟨0,mem_of_mem_nhds (wulff_nhds s hp)⟩
+  have hφi := homogeneous_integrable hμ hφ hh
+  have hh := integral_mono hi ((support_integrable hμ hK hne).add (hφi.const_mul t))
+    (support_wulff_bound hK hne φ hφ hh t hp)
+  simp only [Pi.add_apply] at hh
+  rwa [integral_add (support_integrable hμ hK hne) (hφi.const_mul t),integral_const_mul] at hh
+
+namespace RelativeGauge
+variable {n : ℕ} [NeZero n] {V : Submodule ℝ (Space n)}
+lemma minimizer_scaled_lower {μ : Measure (Space n)}
+    (hμ : Integrable (fun x : Space n => ‖x‖) μ) (hV : V≠⊥)
+    (hq : ∀ x∈V,x≠0 → 0<cosineSeminorm μ hμ x) (b : Bool)
+    {g : RelativeGauge V}
+    (hmin : ∀ q : RelativeGauge V,supportFunctional μ q.body=1 →
+      Spherical.objective b g.toSeminorm≤Spherical.objective b q.toSeminorm)
+    (f : RelativeGauge V) {a : ℝ} (ha : 0<a) (hle : supportFunctional μ f.body≤a) :
+    Spherical.objective b g.toSeminorm≤Spherical.objective b (f.scale a ha).toSeminorm := by
+  have hp := f.functional_pos hV hμ hq
+  have hn : supportFunctional μ (f.scale _ hp).body=1 := by
+    rw [f.functional_scale _ hp μ,inv_mul_cancel₀ hp.ne']
+  apply (hmin _ hn).trans
+  apply Spherical.objective_mono b ((f.scale _ hp).ne_zero hV) ((f.scale _ ha).ne_zero hV)
+  intro x
+  exact mul_le_mul_of_nonneg_right hle (apply_nonneg f.toSeminorm x)
+
+end RelativeGauge
+end PettyProjection
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/ProjectionBodies/VariationalRepresentation.lean b/lean/OAI/Geometry/ProjectionBodies/VariationalRepresentation.lean
new file mode 100644
index 0000000000000000000000000000000000000000..567426d1ef0a7772e88c43276bf434cd935ac233
--- /dev/null
+++ b/lean/OAI/Geometry/ProjectionBodies/VariationalRepresentation.lean
@@ -0,0 +1,265 @@
+import OAI.Geometry.ProjectionBodies.VariationalMinimizers
+
+namespace OAI
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology
+open scoped NNReal RealInnerProductSpace Gradient
+namespace PettyProjection
+open Spherical (Sphere sigma mean objective objectiveIntegrand)
+namespace RelativeGauge
+variable {n : ℕ} [NeZero n]
+
+/-- Actual Euler equation for the normalized variational optimizer. The test
+is only continuous, even and degree-one homogeneous, not assumed convex. -/
+theorem normalized_representation {μ : Measure (Space n)}
+    (hμ : Integrable (fun x : Space n => ‖x‖) μ)
+    (hq : ∀ x : Space n,x≠0 → 0<cosineSeminorm μ hμ x) (b : Bool)
+    (g : RelativeGauge (⊤ : Submodule ℝ (Space n))) (hgn : supportFunctional μ g.body=1)
+    (hmin : ∀ q : RelativeGauge (⊤ : Submodule ℝ (Space n)),supportFunctional μ q.body=1 →
+      objective b g.toSeminorm≤objective b q.toSeminorm)
+    (φ : Space n → ℝ) (hφ : Continuous φ) (heven : ∀ x,φ (-x)=φ x)
+    (hh : ∀ a : ℝ,0<a → ∀ x,φ (a • x)=a*φ x) :
+    (∫ x,φ x ∂μ)*(if b then 1 else objective false g.toSeminorm)=
+      mean (fun u : Sphere n => (if b then 1 else g u)*φ (∇ (fun x => g x) (u:Space n))) := by
+  let s : C(Sphere n,ℝ) := ⟨fun u => support g.body u,(support_continuous g.compact).comp continuous_subtype_val⟩
+  let φs : C(Sphere n,ℝ) := ⟨fun u => φ u,hφ.comp continuous_subtype_val⟩
+  have hs : ∀ u,0<s u := support_pos_of_nhds g.compact g.top_nhds
+  have hw : wulff s=g.body := wulff_support g.compact g.convex g.nonempty
+  have hge : gauge (wulff s)=(fun x => g x) := by ext x; rw [hw,g.top_gauge]
+  let I := ∫ x,φ x ∂μ
+  let F : ℝ → ℝ := fun t => mean (fun u : Sphere n => objectiveIntegrand b (gauge (wulff ⇑(s+t • φs)) (u:Space n)))
+  let D := mean (fun u : Sphere n => (if b then 1 else g u)*φ (∇ (fun x => g x) (u:Space n)))
+  let H : ℝ → ℝ := fun t => if b then Real.log (1+t*I)+F t else (1+t*I)*F t
+  have hF0 : F 0=objective b g.toSeminorm := by
+    simp only [F,zero_smul,add_zero,hge,objective]
+  have hFder : HasDerivAt F (-D) 0 := by
+    have h := wulff_objective_derivative b s hs φ hφ hh
+    rw [hge] at h
+    exact h
+  have hlin : HasDerivAt (fun t : ℝ => 1+t*I) I 0 := by
+    simpa only [one_mul,id_eq] using (((hasDerivAt_id (0:ℝ)).mul_const I).const_add 1)
+  have hH0 : H 0=objective b g.toSeminorm := by
+    dsimp only [H]; rw [zero_mul,add_zero,Real.log_one,zero_add,one_mul]; split <;> exact hF0
+  have hHder : HasDerivAt H (I*(if b then 1 else objective false g.toSeminorm)-D) 0 := by
+    cases b
+    · have hd := hlin.mul hFder
+      simpa only [Pi.mul_def,H,Bool.false_eq_true,ite_false,zero_mul,add_zero,one_mul,hF0,mul_neg,one_mul,sub_eq_add_neg] using hd
+    · have hd := (hlin.log (by norm_num)).add hFder
+      simpa only [Pi.add_def,H,ite_true,zero_mul,add_zero,div_one,mul_one,sub_eq_add_neg] using hd
+  have hHmin : IsLocalMin H 0 := by
+    have hpos : ∀ᶠ t in 𝓝 (0:ℝ),0<1+t*I := by
+      have hc : Continuous (fun t : ℝ => 1+t*I) := by fun_prop
+      have hh := hc.continuousAt.eventually (lt_mem_nhds (by norm_num : (0:ℝ)<1+0*I))
+      exact hh
+    filter_upwards [positive_constraints_nhds s φs hs,hpos] with t ht htp
+    rw [hH0]
+    have hsym : ∀ u : Sphere n,(s+t • φs) ⟨-(u:Space n),by simp⟩=(s+t • φs) u := by
+      intro u
+      change support g.body (-(u:Space n))+t*φ (-(u:Space n))=support g.body u+t*φ u
+      rw [support_even g.compact g.nonempty (fun _ h => g.neg_mem h),heven]
+    let f := ofFullBody (wulff_compact (s+t • φs) ht) (wulff_convex _) (wulff_balanced _ hsym) (wulff_nhds (s+t • φs) ht)
+    have hfb : f.body=wulff ⇑(s+t • φs) := body_ofFullBody _ _ _ _
+    have hbound : supportFunctional μ f.body≤1+t*I := by
+      rw [hfb]
+      have hb := functional_wulff_bound hμ g.compact g.nonempty φ hφ hh t ht
+      rw [hgn] at hb
+      exact hb
+    have hlo := minimizer_scaled_lower hμ (show (⊤ : Submodule ℝ (Space n))≠⊥ from top_ne_bot)
+      (fun x _ hx => hq x hx) b hmin f htp hbound
+    have hscale := Spherical.objective_scale b (g := f.toSeminorm)
+      (f := (f.scale (1+t*I) htp).toSeminorm) (f.ne_zero top_ne_bot) htp (fun _ => rfl)
+    rw [hscale] at hlo
+    have hfobj : objective b f.toSeminorm=F t := by
+      unfold objective
+      congr 1
+      funext u
+      rw [← f.top_gauge,hfb]
+    rw [hfobj] at hlo
+    exact hlo
+  have hz := hHmin.hasDerivAt_eq_zero hHder
+  exact sub_eq_zero.mp hz
+
+end RelativeGauge
+end PettyProjection
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology
+open scoped NNReal RealInnerProductSpace Pointwise Gradient
+namespace PettyProjection
+open Spherical (Sphere norm_coe mean sigma)
+
+lemma normalizedVolume_pos {n : ℕ} {K : Set (Space n)} (hK : IsCompact K)
+    (hi : (interior K).Nonempty) : 0 < normalizedVolume K :=
+  div_pos (volumeRoot_pos hK hi) (Real.rpow_pos_of_pos (kappa_pos n) _)
+
+lemma normalizedVolume_pow {n : ℕ} [NeZero n] (K : Set (Space n)) :
+    normalizedVolume K^n=volume.real K/kappa n := by
+  rw [normalizedVolume_formula]
+  exact Real.rpow_inv_natCast_pow (div_nonneg measureReal_nonneg (kappa_pos n).le) (NeZero.ne n)
+
+lemma normalizedVolume_smul {n : ℕ} [NeZero n] {a : ℝ} (ha : 0 ≤ a) (K : Set (Space n)) :
+    normalizedVolume (a • K)=a*normalizedVolume K := by
+  rw [normalizedVolume,volumeRoot_smul (NeZero.pos n) ha,normalizedVolume]
+  ring
+
+namespace RelativeGauge
+variable {n : ℕ} [NeZero n]
+
+omit [NeZero n] in
+lemma top_pos (g : RelativeGauge (⊤ : Submodule ℝ (Space n))) (u : Sphere n) : 0 < g u :=
+  g.pos (Submodule.mem_top) (by intro h;simpa [h] using norm_coe u)
+
+omit [NeZero n] in
+lemma top_interior (g : RelativeGauge (⊤ : Submodule ℝ (Space n))) : (interior g.body).Nonempty :=
+  ⟨0,mem_interior_iff_mem_nhds.mpr g.top_nhds⟩
+
+def volumeNormalize (g : RelativeGauge (⊤ : Submodule ℝ (Space n))) :
+    RelativeGauge (⊤ : Submodule ℝ (Space n)) :=
+  g.scale (normalizedVolume g.body) (normalizedVolume_pos g.compact g.top_interior)
+
+lemma volumeNormalize_body_volume (g : RelativeGauge (⊤ : Submodule ℝ (Space n))) :
+    volume.real g.volumeNormalize.body=kappa n := by
+  have ha := normalizedVolume_pos g.compact g.top_interior
+  rw [volumeNormalize,g.body_scale,volumeReal_smul_nonneg (inv_nonneg.mpr ha.le),
+    finrank_euclideanSpace_fin,inv_pow,normalizedVolume_pow]
+  field_simp [(volumeReal_pos_of_body g.compact g.top_interior).ne',(kappa_pos n).ne']
+
+lemma volumeNormalize_moment (g : RelativeGauge (⊤ : Submodule ℝ (Space n))) :
+    mean (fun u : Sphere n => (g.volumeNormalize u)⁻¹^n)=1 := by
+  have hh := gauge_volume g.volumeNormalize.convex g.volumeNormalize.top_nhds
+    (fun u => by rw [g.volumeNormalize.top_gauge];exact g.volumeNormalize.top_pos u)
+  simp_rw [g.volumeNormalize.top_gauge] at hh
+  rw [g.volumeNormalize_body_volume] at hh
+  exact (mul_left_cancel₀ (kappa_pos n).ne' (by simpa only [mul_one] using hh.symm))
+
+omit [NeZero n] in
+lemma full_body_of_sphere_one (g : RelativeGauge (⊤ : Submodule ℝ (Space n)))
+    (hg : ∀ u : Sphere n,g u=1) : g.body=unitBall n := by
+  have he (x : Space n) : g x=‖x‖ := by
+    by_cases hx : x=0
+    · simp [hx]
+    let u : Sphere n := ⟨‖x‖⁻¹ • x,by simp [norm_smul,hx]⟩
+    have h := hg u
+    change g.toSeminorm (‖x‖⁻¹ • x)=1 at h
+    rw [map_smul_eq_mul,Real.norm_of_nonneg (inv_nonneg.mpr (norm_nonneg _))] at h
+    have hn := norm_ne_zero_iff.mpr hx
+    field_simp at h
+    exact h
+  ext x
+  rw [g.top_body]
+  change g x≤1 ↔ x∈closedBall 0 1
+  rw [he,mem_closedBall,dist_zero_right]
+
+def fullOfRange {A : Space n →L[ℝ] Space n} (hA : LinearMap.range A.toLinearMap=⊤)
+    (g : RelativeGauge (LinearMap.range A.toLinearMap)) : RelativeGauge (⊤ : Submodule ℝ (Space n)) where
+  toSeminorm := g.toSeminorm
+  projection x := by rw [Submodule.starProjection_top];rfl
+  lower := by simpa only [hA] using g.lower
+
+omit [NeZero n] in
+lemma fullOfRange_body {A : Space n →L[ℝ] Space n} (hA : LinearMap.range A.toLinearMap=⊤)
+    (g : RelativeGauge (LinearMap.range A.toLinearMap)) : (fullOfRange hA g).body=g.body := by
+  simp only [body,fullOfRange,hA]
+
+end RelativeGauge
+end PettyProjection
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology
+open scoped RealInnerProductSpace Pointwise Gradient
+namespace PettyProjection
+open Spherical (Sphere mean sigma gaugeField gaugeMoment)
+
+def regime (n : ℕ) : Bool := decide (n=4)
+lemma exponent_pos (n : ℕ) : 0 < Scalar.exponent n := by unfold Scalar.exponent;split_ifs <;> norm_num
+
+namespace RelativeGauge
+variable {n : ℕ} [NeZero n] {μ : Measure (Space n)}
+
+omit [NeZero n] in
+lemma scale_field_ae (g : RelativeGauge (⊤ : Submodule ℝ (Space n))) {a : ℝ} (ha : 0 < a)
+    {k : ℕ} (hk : 0 < k) :
+    gaugeField (g.scale a ha).toSeminorm k =ᵐ[sigma n] (fun u => a^k • gaugeField g.toSeminorm k u) := by
+  filter_upwards [Spherical.seminorm_ae_differentiable_sphere g.toSeminorm] with u hu
+  have hgrad : ∇ (fun x => g.scale a ha x) (u:Space n)=a • ∇ (fun x => g x) (u:Space n) := by
+    apply (InnerProductSpace.toDual ℝ (Space n)).injective
+    rw [toDual_gradient,map_smul,toDual_gradient]
+    exact fderiv_const_mul hu a
+  change (a*g u)^(k-1) • ∇ (fun x => g.scale a ha x) (u:Space n)=_
+  rw [hgrad,mul_pow,smul_smul]
+  change (a^(k-1)*g u^(k-1)*a) • _=a^k • (g u^(k-1) • _)
+  rw [smul_smul]
+  congr 1
+  have hp : a^k=a^(k-1)*a := by
+    conv_lhs => rw [show k=(k-1)+1 by omega,pow_succ]
+  rw [hp]
+  ring
+
+omit [NeZero n] in
+lemma scale_moment (g : RelativeGauge (⊤ : Submodule ℝ (Space n))) {a : ℝ} (ha : 0 < a) (k : ℕ) :
+    gaugeMoment (g.scale a ha).toSeminorm k=a^k*gaugeMoment g.toSeminorm k := by
+  change mean (fun u : Sphere n => (a*g u)^k)=_
+  simp only [mul_pow,gaugeMoment,mean,integral_const_mul]
+
+lemma minimizer_field_representation
+    (hμ : Integrable (fun x : Space n => ‖x‖) μ)
+    (hq : ∀ x : Space n,x≠0 → 0 < cosineSeminorm μ hμ x)
+    (g : RelativeGauge (⊤ : Submodule ℝ (Space n))) (hgn : supportFunctional μ g.body=1)
+    (hmin : ∀ q : RelativeGauge (⊤ : Submodule ℝ (Space n)),supportFunctional μ q.body=1 →
+      Spherical.objective (regime n) g.toSeminorm≤Spherical.objective (regime n) q.toSeminorm)
+    (φ : Space n → ℝ) (hφ : Continuous φ) (heven : ∀ x,φ (-x)=φ x)
+    (hh : ∀ a : ℝ,0 < a → ∀ x,φ (a • x)=a*φ x) :
+    (∫ x,φ x ∂μ)*gaugeMoment g.toSeminorm (Scalar.exponent n-1)=
+      mean (fun u : Sphere n => φ (gaugeField g.toSeminorm (Scalar.exponent n) u)) := by
+  have hv := normalized_representation hμ hq (regime n) g hgn hmin φ hφ heven hh
+  have he : mean (fun u : Sphere n => φ (gaugeField g.toSeminorm (Scalar.exponent n) u))=
+      mean (fun u : Sphere n => (if regime n then 1 else g u)*φ (∇ (fun x => g x) (u:Space n))) := by
+    congr 1
+    funext u
+    by_cases hn : n=4
+    · simp [gaugeField,Scalar.exponent,regime,hn]
+    · simp only [gaugeField,Scalar.exponent,ite_eq_right hn,Nat.reduceSub,pow_one,
+        regime,decide_eq_false hn,Bool.false_eq_true,ite_false]
+      exact hh _ (g.top_pos u) _
+  rw [he]
+  convert hv using 1
+  congr 1
+  by_cases hn : n=4
+  · simp [gaugeMoment,Scalar.exponent,regime,hn,Spherical.mean_const]
+  · simp [gaugeMoment,Scalar.exponent,regime,hn,Spherical.objective,Spherical.objectiveIntegrand]
+
+lemma scaled_field_representation
+    (hμ : Integrable (fun x : Space n => ‖x‖) μ)
+    (hq : ∀ x : Space n,x≠0 → 0 < cosineSeminorm μ hμ x)
+    (g : RelativeGauge (⊤ : Submodule ℝ (Space n))) (hgn : supportFunctional μ g.body=1)
+    (hmin : ∀ q : RelativeGauge (⊤ : Submodule ℝ (Space n)),supportFunctional μ q.body=1 →
+      Spherical.objective (regime n) g.toSeminorm≤Spherical.objective (regime n) q.toSeminorm)
+    {a : ℝ} (ha : 0 < a)
+    (φ : Space n → ℝ) (hφ : Continuous φ) (heven : ∀ x,φ (-x)=φ x)
+    (hh : ∀ r : ℝ,0 < r → ∀ x,φ (r • x)=r*φ x) :
+    (∫ x,φ x ∂μ)*gaugeMoment (g.scale a ha).toSeminorm (Scalar.exponent n-1)=
+      supportFunctional μ (g.scale a ha).body*
+        mean (fun u : Sphere n => φ (gaugeField (g.scale a ha).toSeminorm (Scalar.exponent n) u)) := by
+  have he : mean (fun u : Sphere n => φ (gaugeField (g.scale a ha).toSeminorm (Scalar.exponent n) u))=
+      a^(Scalar.exponent n)*mean (fun u : Sphere n => φ (gaugeField g.toSeminorm (Scalar.exponent n) u)) := by
+    simp only [mean]
+    rw [← integral_const_mul]
+    apply integral_congr_ae
+    filter_upwards [g.scale_field_ae ha (exponent_pos n)] with u hu
+    rw [hu,hh _ (pow_pos ha _)]
+  rw [g.scale_moment ha,g.functional_scale,hgn,mul_one,he]
+  have hr := minimizer_field_representation hμ hq g hgn hmin φ hφ heven hh
+  rw [← hr]
+  have hk : a^(Scalar.exponent n)=a^(Scalar.exponent n-1)*a := by
+    rw [← pow_succ];congr 1;have := exponent_pos n;omega
+  rw [hk]
+  field_simp
+
+end RelativeGauge
+end PettyProjection
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/ProjectionBodies/WulffVariation.lean b/lean/OAI/Geometry/ProjectionBodies/WulffVariation.lean
new file mode 100644
index 0000000000000000000000000000000000000000..c7cc11572d37940ab4f2c7308013760fdb956872
--- /dev/null
+++ b/lean/OAI/Geometry/ProjectionBodies/WulffVariation.lean
@@ -0,0 +1,576 @@
+import OAI.Geometry.ProjectionBodies.CosineCalculus
+
+namespace OAI
+
+universe uU
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology Function
+open scoped ENNReal NNReal RealInnerProductSpace Gradient
+namespace PettyProjection
+open Spherical (Sphere norm_coe)
+
+lemma support_gradient_of_active {n : ℕ} {C : Set (Space n)}
+    (hC : IsCompact C) {x y : Space n} (hy : y ∈ C)
+    (hxy : support C x = ⟪x,y⟫)
+    (hd : DifferentiableAt ℝ (support C) x) : ∇ (support C) x = y := by
+  let l : Space n →L[ℝ] ℝ := InnerProductSpace.toDual ℝ (Space n) y
+  have he (z : Space n) : l z = ⟪z,y⟫ := by
+    change ⟪y,z⟫ = ⟪z,y⟫
+    exact real_inner_comm z y
+  have hl : IsLocalMin (fun z => support C z-l z) x := by
+    apply Eventually.of_forall
+    intro z
+    dsimp only
+    rw [he,he,hxy,sub_self]
+    exact sub_nonneg.mpr (inner_le_support hC hy z)
+  have hz := hl.hasFDerivAt_eq_zero (hd.hasFDerivAt.sub l.hasFDerivAt)
+  have hd' : fderiv ℝ (support C) x = l := sub_eq_zero.mp hz
+  apply ext_inner_right ℝ
+  intro z
+  rw [inner_gradient_left,hd',he]
+  exact real_inner_comm _ _
+
+lemma wulff_gradient_of_active {n : ℕ} [NeZero n] (s : C(Sphere n,ℝ))
+    (hs : ∀ u, 0 < s u) {x : Space n} {u : Sphere n}
+    (hu : wulffMax s x = ⟪x,(u : Space n)⟫ / s u)
+    (hd : DifferentiableAt ℝ (gauge (wulff s)) x) :
+    ∇ (gauge (wulff s)) x = (s u)⁻¹ • (u : Space n) := by
+  have he : gauge (wulff s) = support (wulffCoefficients s) :=
+    funext (gauge_wulff_eq_max s hs)
+  rw [he] at hd ⊢
+  apply support_gradient_of_active (wulffCoefficients_compact s hs) (mem_range_self u) _ hd
+  simpa only [real_inner_smul_right,div_eq_inv_mul,wulffMax] using hu
+
+lemma wulff_active_unique {n : ℕ} [NeZero n] (s : C(Sphere n,ℝ))
+    (hs : ∀ u, 0 < s u) {x : Space n} (hd : DifferentiableAt ℝ (gauge (wulff s)) x)
+    {u v : Sphere n} (hu : wulffMax s x = ⟪x,(u : Space n)⟫/s u)
+    (hv : wulffMax s x = ⟪x,(v : Space n)⟫/s v) : u = v := by
+  have he := (wulff_gradient_of_active s hs hu hd).symm.trans (wulff_gradient_of_active s hs hv hd)
+  have hn := congrArg norm he
+  simp only [norm_smul,norm_coe,mul_one,Real.norm_eq_abs,abs_inv,abs_of_pos (hs u),
+    abs_of_pos (hs v)] at hn
+  have hsuv : s u = s v := inv_injective hn
+  apply Subtype.ext
+  rw [hsuv] at he
+  exact (smul_right_injective _ (inv_ne_zero (hs v).ne')) he
+
+end PettyProjection
+end
+
+noncomputable section
+open Set Filter Topology
+namespace PettyProjection
+
+/-- Convergence of maximizers at a unique maximum, with a compact parameter
+space and joint continuity only at the limiting parameter. -/
+lemma selected_max_tendsto {U : Type uU} [TopologicalSpace U] [CompactSpace U]
+    {f : ℝ → U → ℝ} {a : ℝ → U} {u₀ : U}
+    (hc : ∀ u : U, ContinuousAt (fun p : ℝ × U => f p.1 p.2) (0,u))
+    (hm : ∀ᶠ t in 𝓝 (0 : ℝ), ∀ u, f t u ≤ f t (a t))
+    (hu : ∀ u, (∀ v, f 0 v ≤ f 0 u) → u = u₀) :
+    Tendsto a (𝓝 0) (𝓝 u₀) := by
+  apply tendsto_nhds_of_unique_mapClusterPt
+  intro u hu'
+  obtain ⟨l,hl,hat⟩ := mapClusterPt_iff_ultrafilter.mp hu'
+  apply hu
+  intro v
+  have ht : Tendsto (fun t : ℝ => t) (l : Filter ℝ) (𝓝 0) := hl
+  have hv : Tendsto (fun t => f t v) (l : Filter ℝ) (𝓝 (f 0 v)) :=
+    (hc v).tendsto.comp (ht.prodMk_nhds tendsto_const_nhds)
+  have hav : Tendsto (fun t => f t (a t)) (l : Filter ℝ) (𝓝 (f 0 u)) :=
+    (hc u).tendsto.comp (ht.prodMk_nhds hat)
+  exact le_of_tendsto_of_tendsto hv hav (hl (hm.mono (fun t h => h v)))
+
+lemma quotient_bounds {a b z t : ℝ} (ht : t ≠ 0) (hl : t*a ≤ z) (hh : z ≤ t*b) :
+    min a b ≤ z/t ∧ z/t ≤ max a b := by
+  have he : z/t*t = z := div_mul_cancel₀ z ht
+  rcases lt_or_gt_of_ne ht with ht | ht
+  · have hb : b ≤ z/t := by nlinarith
+    have ha : z/t ≤ a := by nlinarith
+    exact ⟨(min_le_right a b).trans hb,ha.trans (le_max_left a b)⟩
+  · have ha : a ≤ z/t := by nlinarith
+    have hb : z/t ≤ b := by nlinarith
+    exact ⟨(min_le_left a b).trans ha,hb.trans (le_max_right a b)⟩
+
+/-- Envelope differentiation from an exact continuous difference quotient.
+The statement does not require a smooth boundary or a smooth maximizer. -/
+lemma hasDerivAt_selected_max {U : Type uU} [TopologicalSpace U] [CompactSpace U]
+    {f d : ℝ → U → ℝ} {q : ℝ → ℝ} {a : ℝ → U} {u₀ : U}
+    (hc : ∀ u : U, ContinuousAt (fun p : ℝ × U => f p.1 p.2) (0,u))
+    (hd : ContinuousAt (fun p : ℝ × U => d p.1 p.2) (0,u₀))
+    (hq : ∀ᶠ t in 𝓝 (0 : ℝ), q t = f t (a t) ∧ ∀ u, f t u ≤ q t)
+    (hq₀ : q 0 = f 0 u₀) (hm₀ : ∀ u, f 0 u ≤ q 0)
+    (hu : ∀ u, (∀ v, f 0 v ≤ f 0 u) → u = u₀)
+    (he : ∀ᶠ t in 𝓝 (0 : ℝ), ∀ u, f t u-f 0 u = t*d t u) :
+    HasDerivAt q (d 0 u₀) 0 := by
+  have ha : Tendsto a (𝓝 0) (𝓝 u₀) := selected_max_tendsto hc
+    (hq.mono (fun t h u => h.1 ▸ h.2 u)) hu
+  have hd₀ : Tendsto (fun t => d t u₀) (𝓝 0) (𝓝 (d 0 u₀)) :=
+    hd.tendsto.comp (tendsto_id.prodMk_nhds tendsto_const_nhds)
+  have hda : Tendsto (fun t => d t (a t)) (𝓝 0) (𝓝 (d 0 u₀)) :=
+    hd.tendsto.comp (tendsto_id.prodMk_nhds ha)
+  have hlo : Tendsto (fun t => min (d t u₀) (d t (a t))) (𝓝 0) (𝓝 (d 0 u₀)) := by
+    simpa only [min_self] using hd₀.min hda
+  have hhi : Tendsto (fun t => max (d t u₀) (d t (a t))) (𝓝 0) (𝓝 (d 0 u₀)) := by
+    simpa only [max_self] using hd₀.max hda
+  rw [hasDerivAt_iff_tendsto_slope_zero]
+  simp only [zero_add,smul_eq_mul,← div_eq_inv_mul]
+  have hb : ∀ᶠ t in 𝓝[≠] (0 : ℝ), min (d t u₀) (d t (a t)) ≤ (q t-q 0)/t ∧
+      (q t-q 0)/t ≤ max (d t u₀) (d t (a t)) := by
+    filter_upwards [hq.filter_mono nhdsWithin_le_nhds,he.filter_mono nhdsWithin_le_nhds,
+      self_mem_nhdsWithin] with t ht he ht0
+    have htn : t ≠ 0 := ht0
+    have hl : t*d t u₀ ≤ q t-q 0 := by
+      rw [← he u₀,hq₀]
+      exact sub_le_sub_right (ht.2 u₀) _
+    have hh : q t-q 0 ≤ t*d t (a t) := by
+      rw [← he (a t),ht.1]
+      exact sub_le_sub_left (hm₀ (a t)) _
+    exact quotient_bounds htn hl hh
+  exact tendsto_of_tendsto_of_tendsto_of_le_of_le'
+    (hlo.mono_left nhdsWithin_le_nhds) (hhi.mono_left nhdsWithin_le_nhds)
+    (hb.mono (fun _ h => h.1)) (hb.mono (fun _ h => h.2))
+
+end PettyProjection
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology Function
+open scoped ENNReal NNReal RealInnerProductSpace Gradient
+namespace PettyProjection
+open Spherical (Sphere norm_coe)
+
+lemma positive_constraints_eventually {n : ℕ} [NeZero n]
+    (s φ : C(Sphere n,ℝ)) (hs : ∀ u, 0 < s u) :
+    ∃ m M : ℝ, 0 < m ∧ 0 < M ∧
+      (∀ᶠ t in 𝓝 (0 : ℝ), ∀ u, m ≤ (s+t • φ) u ∧ (s+t • φ) u ≤ M) := by
+  obtain ⟨a,b,ha,hb,hab⟩ := positive_constraint_bounds s hs
+  refine ⟨a/2,b+a/2,by positivity,by positivity,?_⟩
+  have ht : ∀ᶠ t in 𝓝 (0 : ℝ), |t| *(‖φ‖+1) < a/2 := by
+    have h : Tendsto (fun t : ℝ => |t| *(‖φ‖+1)) (𝓝 0) (𝓝 0) := by
+      simpa using (continuous_abs.tendsto (0:ℝ)).mul_const (‖φ‖+1)
+    exact h.eventually (eventually_lt_nhds (by positivity))
+  filter_upwards [ht] with t ht u
+  have hφ : |φ u| ≤ ‖φ‖ := φ.norm_coe_le_norm u
+  have hbound : |t*φ u| ≤ |t| *(‖φ‖+1) := by
+    rw [abs_mul]
+    exact mul_le_mul_of_nonneg_left (hφ.trans (by linarith)) (abs_nonneg _)
+  have hlo := neg_abs_le (t*φ u)
+  have hhi := le_abs_self (t*φ u)
+  have hh := hab u
+  change a/2 ≤ s u+t*φ u ∧ s u+t*φ u ≤ b+a/2
+  constructor <;> linarith
+
+lemma positive_constraints_nhds {n : ℕ} [NeZero n]
+    (s φ : C(Sphere n,ℝ)) (hs : ∀ u, 0 < s u) :
+    ∀ᶠ t in 𝓝 (0 : ℝ), ∀ u, 0 < (s+t • φ) u := by
+  obtain ⟨m,M,hm,_,he⟩ := positive_constraints_eventually s φ hs
+  exact he.mono (fun _ h u => hm.trans_le (h u).1)
+
+lemma wulff_derivative_active {n : ℕ} [NeZero n]
+    (s φ : C(Sphere n,ℝ)) (hs : ∀ u, 0 < s u) {x : Space n}
+    (hd : DifferentiableAt ℝ (gauge (wulff s)) x) {u₀ : Sphere n}
+    (hu₀ : wulffMax s x = ⟪x,(u₀ : Space n)⟫/s u₀) :
+    HasDerivAt (fun t : ℝ => gauge (wulff ⇑(s+t • φ)) x)
+      (-(gauge (wulff s) x)*φ u₀/s u₀) 0 := by
+  classical
+  let f : ℝ → Sphere n → ℝ := fun t u => ⟪x,(u : Space n)⟫/(s u+t*φ u)
+  let d : ℝ → Sphere n → ℝ := fun t u => -(⟪x,(u : Space n)⟫*φ u)/(s u*(s u+t*φ u))
+  let a : ℝ → Sphere n := fun t => if hp : ∀ u, 0 < (s+t • φ) u then
+    (wulffMax_attained (s+t • φ) hp x).choose else u₀
+  have hpos := positive_constraints_nhds s φ hs
+  have hq : ∀ᶠ t in 𝓝 (0 : ℝ), gauge (wulff ⇑(s+t • φ)) x = f t (a t) ∧
+      ∀ u, f t u ≤ gauge (wulff ⇑(s+t • φ)) x := by
+    filter_upwards [hpos] with t ht
+    rw [gauge_wulff_eq_max (s+t • φ) ht]
+    constructor
+    · dsimp only [a]
+      rw [dite_eq_left ht]
+      exact (wulffMax_attained (s+t • φ) ht x).choose_spec
+    · intro u; exact le_wulffMax (s+t • φ) ht x u
+  have hq0 : gauge (wulff (s+(0:ℝ) • φ)) x = f 0 u₀ := by
+    simpa only [zero_smul,add_zero,f,zero_mul] using (gauge_wulff_eq_max s hs x).trans hu₀
+  have hm0 : ∀ u, f 0 u ≤ gauge (wulff (s+(0:ℝ) • φ)) x := by
+    intro u
+    simpa only [zero_smul,add_zero,f,zero_mul,gauge_wulff_eq_max s hs x] using le_wulffMax s hs x u
+  have hc (u : Sphere n) : ContinuousAt (fun p : ℝ × Sphere n => f p.1 p.2) (0,u) := by
+    apply ContinuousAt.div
+    · fun_prop
+    · fun_prop
+    · simpa using (hs u).ne'
+  have hdc : ContinuousAt (fun p : ℝ × Sphere n => d p.1 p.2) (0,u₀) := by
+    apply ContinuousAt.div
+    · fun_prop
+    · fun_prop
+    · simpa using mul_ne_zero (hs u₀).ne' (hs u₀).ne'
+  have hu : ∀ u, (∀ v, f 0 v ≤ f 0 u) → u = u₀ := by
+    intro u h
+    apply wulff_active_unique s hs hd _ hu₀
+    apply le_antisymm
+    · simpa only [f,zero_mul,add_zero,← hu₀] using h u₀
+    · exact le_wulffMax s hs x u
+  have he : ∀ᶠ t in 𝓝 (0 : ℝ), ∀ u, f t u-f 0 u = t*d t u := by
+    filter_upwards [hpos] with t ht u
+    have hst : s u+t*φ u ≠ 0 := (ht u).ne'
+    dsimp only [f,d]
+    simp only [zero_mul,add_zero]
+    field_simp [hst,(hs u).ne']
+    ring
+  have h := hasDerivAt_selected_max hc hdc hq hq0 hm0 hu he
+  have he0 : d 0 u₀ = -(gauge (wulff s) x)*φ u₀/s u₀ := by
+    rw [gauge_wulff_eq_max s hs x,hu₀]
+    dsimp [d]
+    ring
+  rwa [he0] at h
+
+/-- Wulff variation against an actual continuous positively homogeneous test.
+No symmetry or boundary smoothness is assumed. -/
+theorem wulff_derivative {n : ℕ} [NeZero n] (s : C(Sphere n,ℝ))
+    (hs : ∀ u, 0 < s u) (φ : Space n → ℝ) (hφ : Continuous φ)
+    (hh : ∀ (a : ℝ), 0 < a → ∀ x, φ (a • x) = a*φ x)
+    {x : Space n} (hd : DifferentiableAt ℝ (gauge (wulff s)) x) :
+    HasDerivAt (fun t : ℝ => gauge (wulff (s+t • (⟨fun u : Sphere n => φ u,
+      hφ.comp continuous_subtype_val⟩ : C(Sphere n,ℝ)))) x)
+      (-(gauge (wulff s) x)*φ (∇ (gauge (wulff s)) x)) 0 := by
+  obtain ⟨u,hu⟩ := wulffMax_attained s hs x
+  have he := wulff_gradient_of_active s hs hu hd
+  have h := wulff_derivative_active s ⟨fun u : Sphere n => φ u,hφ.comp continuous_subtype_val⟩ hs hd hu
+  rw [he,hh _ (inv_pos.mpr (hs u))]
+  convert h using 1
+  dsimp
+  ring
+
+end PettyProjection
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology Function
+open scoped ENNReal NNReal RealInnerProductSpace
+namespace PettyProjection
+open Spherical (Sphere norm_coe surface sigma mean)
+
+lemma kappa_pos (n : ℕ) : 0 < kappa n := by
+  unfold kappa unitBall
+  exact ENNReal.toReal_pos (measure_closedBall_pos volume 0 (by norm_num)).ne'
+    (measure_closedBall_lt_top.ne)
+
+lemma surface_real_univ (n : ℕ) [NeZero n] : (surface n).real univ = n*kappa n := by
+  rw [surface,Measure.toSphere_real_apply_univ]
+  simp only [finrank_euclideanSpace_fin]
+  congr 1
+  exact congrArg ENNReal.toReal (Measure.addHaar_closedBall_eq_addHaar_ball volume (0 : Space n) (1:ℝ)).symm
+
+/-- Polar integration of an arbitrary convex neighborhood of zero; the body
+need not be symmetric or have a smooth boundary. -/
+lemma gauge_volume_lintegral {n : ℕ} [NeZero n] {K : Set (Space n)}
+    (hc : Convex ℝ K) (h0 : K ∈ 𝓝 (0 : Space n))
+    (hp : ∀ u : Sphere n, 0 < gauge K (u : Space n)) :
+    volume K = ∫⁻ u : Sphere n, ENNReal.ofReal ((gauge K (u : Space n))⁻¹ ^ n / n)
+      ∂surface n := by
+  let ν := Measure.volumeIoiPow (Module.finrank ℝ (Space n)-1)
+  let A : Set ({0}ᶜ : Set (Space n)) := Subtype.val ⁻¹' interior K
+  let S : Set (Sphere n × Ioi (0:ℝ)) :=
+    {z | z.2.1 < (gauge K (z.1 : Space n))⁻¹}
+  have hA : MeasurableSet A := isOpen_interior.measurableSet.preimage measurable_subtype_coe
+  have hS : MeasurableSet S := by
+    apply isOpen_lt (by fun_prop) ?_ |>.measurableSet
+    exact ((continuous_gauge hc h0).comp (continuous_subtype_val.comp continuous_fst)).inv₀
+      (fun z => (hp z.1).ne')
+  have he : (homeomorphUnitSphereProd (Space n)).symm ⁻¹' A = S := by
+    ext z
+    change (↑((homeomorphUnitSphereProd (Space n)).symm z) : Space n) ∈ interior K ↔ _
+    rw [homeomorphUnitSphereProd_symm_apply_coe,← gauge_lt_one_iff_mem_interior hc h0,
+      gauge_smul_of_nonneg (α := ℝ) z.2.property.le]
+    change (z.2 : ℝ)*gauge K (z.1 : Space n) < 1 ↔ (z.2 : ℝ) < (gauge K (z.1 : Space n))⁻¹
+    rw [inv_eq_one_div,lt_div_iff₀ (hp z.1)]
+  have hm := ((volume : Measure (Space n)).measurePreserving_homeomorphUnitSphereProd).symm
+    (homeomorphUnitSphereProd (Space n)).toMeasurableEquiv
+  have hm' := hm.measure_preimage hA.nullMeasurableSet
+  change (surface n).prod ν ((homeomorphUnitSphereProd (Space n)).symm ⁻¹' A) = _ at hm'
+  rw [he, ((MeasurableEmbedding.subtype_coe (measurableSet_singleton (0 : Space n)).compl)).comap_apply] at hm'
+  have hi : Subtype.val '' A = interior K \ {0} := by
+    ext x
+    constructor
+    · rintro ⟨y,hy,rfl⟩; exact ⟨hy,y.property⟩
+    · intro hx; exact ⟨⟨x,hx.2⟩,hx.1,rfl⟩
+  rw [hi,measure_sdiff_null (measure_singleton _),
+    measure_interior_of_null_frontier (hc.addHaar_frontier volume)] at hm'
+  rw [← hm',Measure.prod_apply hS]
+  apply lintegral_congr
+  intro u
+  have hu : Prod.mk u ⁻¹' S = Iio (⟨(gauge K (u : Space n))⁻¹,inv_pos.mpr (hp u)⟩ : Ioi (0:ℝ)) := rfl
+  rw [hu,Measure.volumeIoiPow_apply_Iio]
+  have hn : n-1+1 = n := Nat.sub_add_cancel (NeZero.one_le : 1 ≤ n)
+  have hnR := congrArg (fun j : ℕ => (j : ℝ)) hn
+  simp only [Nat.cast_add,Nat.cast_one] at hnR
+  simp only [finrank_euclideanSpace_fin,hn,hnR]
+
+lemma gauge_volume {n : ℕ} [NeZero n] {K : Set (Space n)}
+    (hc : Convex ℝ K) (h0 : K ∈ 𝓝 (0 : Space n))
+    (hp : ∀ u : Sphere n, 0 < gauge K (u : Space n)) :
+    volume.real K = kappa n * mean (fun u : Sphere n => (gauge K (u : Space n))⁻¹ ^ n) := by
+  have hcont : Continuous (fun u : Sphere n => (gauge K (u : Space n))⁻¹ ^ n) :=
+    (((continuous_gauge hc h0).comp continuous_subtype_val).inv₀ (fun u => (hp u).ne')).pow n
+  have hnn : ∀ u : Sphere n, 0 ≤ (gauge K (u : Space n))⁻¹ ^ n := fun u => pow_nonneg (inv_nonneg.mpr (hp u).le) n
+  have hv := congrArg ENNReal.toReal (gauge_volume_lintegral hc h0 hp)
+  rw [← ofReal_integral_eq_lintegral_ofReal (hcont.div_const n |>.integrable_of_hasCompactSupport
+    (HasCompactSupport.of_compactSpace _)) (Eventually.of_forall (fun u => div_nonneg (hnn u) (Nat.cast_nonneg n))),
+    ENNReal.toReal_ofReal (integral_nonneg (fun u => div_nonneg (hnn u) (Nat.cast_nonneg n))),
+    integral_div] at hv
+  change volume.real K = _ at hv
+  rw [hv,mean,sigma,integral_smul_measure,ENNReal.toReal_inv,smul_eq_mul]
+  change _ = kappa n * (((surface n).real univ)⁻¹ * _)
+  rw [surface_real_univ]
+  field_simp [(kappa_pos n).ne',(Nat.cast_pos.mpr (NeZero.pos n) : (0:ℝ)<n).ne']
+
+lemma wulff_volume {n : ℕ} [NeZero n] (s : C(Sphere n,ℝ)) (hs : ∀ u, 0 < s u) :
+    volume.real (wulff s) = kappa n * mean (fun u : Sphere n => (gauge (wulff s) (u : Space n))⁻¹ ^ n) := by
+  apply gauge_volume (wulff_convex s) (wulff_nhds s hs)
+  intro u
+  rw [gauge_wulff_eq_max s hs]
+  exact wulffMax_pos s hs (by intro h; simpa [h] using norm_coe u)
+
+end PettyProjection
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology Function
+open scoped ENNReal NNReal RealInnerProductSpace
+namespace PettyProjection
+open Spherical (Sphere norm_coe)
+
+lemma wulff_sphere_gauge_bounds {n : ℕ} [NeZero n] (s : C(Sphere n,ℝ))
+    {m M : ℝ} (hm : 0 < m) (hs : ∀ u, m ≤ s u ∧ s u ≤ M)
+    (u : Sphere n) : M⁻¹ ≤ gauge (wulff s) (u : Space n) ∧
+      gauge (wulff s) (u : Space n) ≤ m⁻¹ := by
+  have hp : ∀ v, 0 < s v := fun v => hm.trans_le (hs v).1
+  rw [gauge_wulff_eq_max s hp]
+  constructor
+  · have h := le_wulffMax s hp (u : Space n) u
+    rw [real_inner_self_eq_norm_sq,norm_coe,one_pow,one_div] at h
+    exact (inv_anti₀ (hp u) (hs u).2).trans h
+  · obtain ⟨v,hv⟩ := wulffMax_attained s hp (u : Space n)
+    rw [hv]
+    have hi : ⟪(u : Space n),(v : Space n)⟫ ≤ 1 := by
+      simpa only [norm_coe,one_mul] using real_inner_le_norm (u : Space n) (v : Space n)
+    calc
+      _ ≤ 1 / s v := div_le_div_of_nonneg_right hi (hp v).le
+      _ ≤ m⁻¹ := by simpa only [one_div] using inv_anti₀ hm (hs v).1
+
+lemma quotient_perturb_bound {a b c t r m B : ℝ} (hm : 0 < m)
+    (ht : m ≤ b+t*c) (hr : m ≤ b+r*c) (ha : |a| ≤ 1) (hc : |c| ≤ B) :
+    |a/(b+t*c)-a/(b+r*c)| ≤ (B/m^2)*|t-r| := by
+  have ht0 := hm.trans_le ht
+  have hr0 := hm.trans_le hr
+  have hB : 0 ≤ B := (abs_nonneg c).trans hc
+  have he : a/(b+t*c)-a/(b+r*c) = a*c*(r-t)/((b+t*c)*(b+r*c)) := by
+    rw [div_sub_div _ _ ht0.ne' hr0.ne']
+    congr 1
+    ring
+  rw [he,abs_div,abs_mul,abs_mul,abs_mul,abs_of_pos ht0,abs_of_pos hr0,abs_sub_comm r t]
+  calc
+    _ ≤ (1*B)*|t-r|/(m*m) := by gcongr
+    _ = _ := by ring
+
+lemma wulff_gauge_param_sub_le {n : ℕ} [NeZero n] (s φ : C(Sphere n,ℝ))
+    {m t r : ℝ} (hm : 0 < m) (ht : ∀ u, m ≤ (s+t • φ) u)
+    (hr : ∀ u, m ≤ (s+r • φ) u) (u : Sphere n) :
+    gauge (wulff ⇑(s+t • φ)) (u : Space n)-gauge (wulff ⇑(s+r • φ)) (u : Space n) ≤
+      (‖φ‖/m^2)*|t-r| := by
+  have htp : ∀ v, 0 < (s+t • φ) v := fun v => hm.trans_le (ht v)
+  have hrp : ∀ v, 0 < (s+r • φ) v := fun v => hm.trans_le (hr v)
+  rw [gauge_wulff_eq_max (s+t • φ) htp,gauge_wulff_eq_max (s+r • φ) hrp]
+  obtain ⟨v,hv⟩ := wulffMax_attained (s+t • φ) htp (u : Space n)
+  rw [hv]
+  have hi : |⟪(u : Space n),(v : Space n)⟫| ≤ 1 := by
+    simpa only [norm_coe,one_mul] using abs_real_inner_le_norm (u : Space n) (v : Space n)
+  calc
+    _ ≤ ⟪(u : Space n),(v : Space n)⟫/(s+t • φ) v - ⟪(u : Space n),(v : Space n)⟫/(s+r • φ) v :=
+      sub_le_sub_left (le_wulffMax (s+r • φ) hrp _ v) _
+    _ ≤ |⟪(u : Space n),(v : Space n)⟫/(s+t • φ) v - ⟪(u : Space n),(v : Space n)⟫/(s+r • φ) v| := le_abs_self _
+    _ ≤ _ := quotient_perturb_bound hm (ht v) (hr v) hi (φ.norm_coe_le_norm v)
+
+lemma wulff_gauge_param_lipschitz {n : ℕ} [NeZero n] (s φ : C(Sphere n,ℝ))
+    {m : ℝ} (hm : 0 < m) {T : Set ℝ} (hT : ∀ t ∈ T, ∀ u, m ≤ (s+t • φ) u)
+    (u : Sphere n) : LipschitzOnWith ⟨‖φ‖/m^2,by positivity⟩
+      (fun t : ℝ => gauge (wulff ⇑(s+t • φ)) (u : Space n)) T := by
+  apply LipschitzOnWith.of_dist_le_mul
+  intro t ht r hr
+  change |gauge (wulff ⇑(s+t • φ)) (u : Space n)-gauge (wulff ⇑(s+r • φ)) (u : Space n)| ≤ (‖φ‖/m^2)*|t-r|
+  apply abs_le.mpr
+  constructor
+  · have h := wulff_gauge_param_sub_le s φ hm (hT r hr) (hT t ht) u
+    rw [abs_sub_comm r t] at h
+    linarith
+  · exact wulff_gauge_param_sub_le s φ hm (hT t ht) (hT r hr) u
+
+end PettyProjection
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology Function
+open scoped ENNReal NNReal RealInnerProductSpace Gradient
+namespace PettyProjection
+open Spherical (Sphere norm_coe sigma mean)
+
+lemma measurable_gradient {n : ℕ} (f : Space n → ℝ) : Measurable (gradient f) := by
+  exact (InnerProductSpace.toDual ℝ (Space n)).symm.continuous.measurable.comp (measurable_fderiv ℝ f)
+
+lemma wulff_invpow_param_lipschitz {n : ℕ} [NeZero n]
+    (s φ : C(Sphere n,ℝ)) (hs : ∀ u,0 < s u) :
+    ∃ T ∈ 𝓝 (0 : ℝ), ∃ C : ℝ≥0, ∀ u : Sphere n,
+      LipschitzOnWith C (fun t : ℝ => (gauge (wulff ⇑(s+t • φ)) (u : Space n))⁻¹ ^ n) T := by
+  obtain ⟨m,M,hm,hM,he⟩ := positive_constraints_eventually s φ hs
+  let T : Set ℝ := {t | ∀ u, m ≤ (s+t • φ) u ∧ (s+t • φ) u ≤ M}
+  have hT : T ∈ 𝓝 (0 : ℝ) := he
+  have hc : ContDiffOn ℝ 1 (fun a : ℝ => a⁻¹^n) (Icc M⁻¹ m⁻¹) := by
+    intro a ha
+    exact ((contDiffAt_id.inv ((inv_pos.mpr hM).trans_le ha.1).ne').pow n).contDiffWithinAt
+  obtain ⟨D,hD⟩ := hc.exists_lipschitzOnWith (by norm_num) (convex_Icc _ _) isCompact_Icc
+  let C : ℝ≥0 := ⟨‖φ‖/m^2,by positivity⟩
+  refine ⟨T,hT,D*C,?_⟩
+  intro u
+  apply hD.comp (wulff_gauge_param_lipschitz s φ hm (fun t ht v => (ht v).1) u)
+  intro t ht
+  exact wulff_sphere_gauge_bounds (s+t • φ) hm ht u
+
+lemma wulff_invpow_derivative {n : ℕ} [NeZero n] (s : C(Sphere n,ℝ))
+    (hs : ∀ u,0 < s u) (φ : Space n → ℝ) (hφ : Continuous φ)
+    (hh : ∀ (a : ℝ),0 < a → ∀ x,φ (a • x)=a*φ x)
+    {u : Sphere n} (hd : DifferentiableAt ℝ (gauge (wulff s)) (u : Space n)) :
+    HasDerivAt (fun t : ℝ => (gauge (wulff ⇑(s+t • (⟨fun v : Sphere n => φ v,
+      hφ.comp continuous_subtype_val⟩ : C(Sphere n,ℝ)))) (u : Space n))⁻¹^n)
+      (n*(gauge (wulff s) (u : Space n))⁻¹^n*φ (∇ (gauge (wulff s)) (u : Space n))) 0 := by
+  have hp : 0 < gauge (wulff s) (u : Space n) := by
+    rw [gauge_wulff_eq_max s hs]
+    exact wulffMax_pos s hs (by intro h; simpa [h] using norm_coe u)
+  have h := ((wulff_derivative s hs φ hφ hh hd).inv
+    (by simpa only [zero_smul,add_zero] using hp.ne')).pow n
+  simp only [zero_smul,add_zero] at h
+  apply h.congr_deriv
+  simp only [Pi.inv_apply,zero_smul,add_zero]
+  let q := gauge (wulff s) (u : Space n)
+  have hn : n-1+1 = n := Nat.sub_add_cancel (NeZero.one_le : 1 ≤ n)
+  have he : (q⁻¹)^(n-1)*q⁻¹ = (q⁻¹)^n := by rw [← pow_succ,hn]
+  change (n:ℝ)*q⁻¹^(n-1)*(-(-q*_)/q^2) = (n:ℝ)*q⁻¹^n*_
+  rw [← he]
+  field_simp [show q ≠ 0 from hp.ne']
+
+/-- Actual two-sided first variation in positive Wulff constraints. This is
+valid for arbitrary continuous degree-one tests, not only support functions. -/
+theorem wulff_volume_derivative {n : ℕ} [NeZero n] (s : C(Sphere n,ℝ))
+    (hs : ∀ u,0 < s u) (φ : Space n → ℝ) (hφ : Continuous φ)
+    (hh : ∀ (a : ℝ),0 < a → ∀ x,φ (a • x)=a*φ x) :
+    HasDerivAt (fun t : ℝ => volume.real (wulff ⇑(s+t • (⟨fun u : Sphere n => φ u,
+      hφ.comp continuous_subtype_val⟩ : C(Sphere n,ℝ)))))
+      (n*kappa n*mean (fun u : Sphere n => (gauge (wulff s) (u : Space n))⁻¹^n*
+        φ (∇ (gauge (wulff s)) (u : Space n)))) 0 := by
+  let φs : C(Sphere n,ℝ) := ⟨fun u => φ u,hφ.comp continuous_subtype_val⟩
+  let F : ℝ → Sphere n → ℝ := fun t u => (gauge (wulff ⇑(s+t • φs)) (u : Space n))⁻¹^n
+  let F' : Sphere n → ℝ := fun u => (n:ℝ)*(gauge (wulff s) (u : Space n))⁻¹^n*
+    φ (∇ (gauge (wulff s)) (u : Space n))
+  have hg : Continuous (gauge (wulff s)) := continuous_gauge (wulff_convex s) (wulff_nhds s hs)
+  have hgp : ∀ u : Sphere n,0 < gauge (wulff s) (u : Space n) := by
+    intro u
+    rw [gauge_wulff_eq_max s hs]
+    exact wulffMax_pos s hs (by intro h; simpa [h] using norm_coe u)
+  have hc : Continuous (F 0) := by
+    change Continuous (fun u : Sphere n => (gauge (wulff ⇑(s+0 • φs)) (u : Space n))⁻¹^n)
+    simp only [zero_smul,add_zero]
+    exact ((hg.comp continuous_subtype_val).inv₀ (fun u => (hgp u).ne')).pow n
+  obtain ⟨T,hT,C,hC⟩ := wulff_invpow_param_lipschitz s φs hs
+  have hFm : ∀ᶠ t in 𝓝 (0:ℝ), AEStronglyMeasurable (F t) (sigma n) := by
+    filter_upwards [positive_constraints_nhds s φs hs] with t ht
+    exact (((continuous_gauge (wulff_convex _) (wulff_nhds (s+t • φs) ht)).comp
+      continuous_subtype_val).measurable.inv.pow_const n).aestronglyMeasurable
+  have hF'm : AEStronglyMeasurable F' (sigma n) := by
+    apply Measurable.aestronglyMeasurable
+    exact (measurable_const.mul ((hg.measurable.comp measurable_subtype_coe).inv.pow_const n)).mul
+      (hφ.measurable.comp ((measurable_gradient _).comp measurable_subtype_coe))
+  have hder : ∀ᵐ u ∂sigma n, HasDerivAt (fun t => F t u) (F' u) 0 := by
+    filter_upwards [gauge_ae_differentiable_sphere (wulff_convex s) (wulff_nhds s hs)] with u hu
+    exact wulff_invpow_derivative s hs φ hφ hh hu
+  have hI := (hasDerivAt_integral_of_dominated_loc_of_lip hT hFm
+    (hc.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)) hF'm
+    (Eventually.of_forall (fun u => by simpa only [Real.nnabs_coe] using hC u))
+    (integrable_const (C:ℝ)) hder).2
+  have he : (fun t => volume.real (wulff ⇑(s+t • φs))) =ᶠ[𝓝 (0:ℝ)]
+      (fun t => kappa n * ∫ u,F t u ∂sigma n) := by
+    filter_upwards [positive_constraints_nhds s φs hs] with t ht
+    exact wulff_volume (s+t • φs) ht
+  have h := (hI.const_mul (kappa n)).congr_of_eventuallyEq he
+  apply h.congr_deriv
+  dsimp only [F']
+  simp_rw [mul_assoc (n:ℝ)]
+  rw [integral_const_mul]
+  dsimp only [mean]
+  ring
+
+end PettyProjection
+end
+
+noncomputable section
+open Set MeasureTheory Metric Filter Topology Function
+open scoped ENNReal NNReal RealInnerProductSpace Pointwise Gradient
+namespace PettyProjection
+open Spherical (Sphere norm_coe sigma mean)
+
+lemma support_smul_set {n : ℕ} {K : Set (Space n)} (hK : IsCompact K) (hKn : K.Nonempty)
+    {t : ℝ} (ht : 0 ≤ t) (u : Space n) : support (t • K) u = t*support K u := by
+  have hn : (t • K).Nonempty := by obtain ⟨x,hx⟩ := hKn; exact ⟨t • x,smul_mem_smul_set hx⟩
+  apply le_antisymm
+  · apply (support_le_iff (hK.smul t) hn).mpr
+    rintro _ ⟨x,hx,rfl⟩
+    rw [real_inner_smul_right]
+    exact mul_le_mul_of_nonneg_left (inner_le_support hK hx u) ht
+  · obtain ⟨x,hx,he⟩ := support_attained hK hKn u
+    rw [he]
+    simpa only [real_inner_smul_right] using
+      inner_le_support (hK.smul t) (smul_mem_smul_set hx) u
+
+lemma support_pos_of_nhds {n : ℕ} {K : Set (Space n)} (hK : IsCompact K)
+    (h0 : K ∈ 𝓝 (0 : Space n)) (u : Sphere n) : 0 < support K u := by
+  obtain ⟨r,hr,hrK⟩ := Metric.mem_nhds_iff.mp h0
+  have hx : (r/2) • (u : Space n) ∈ K := by
+    apply hrK
+    rw [mem_ball,dist_zero_right,norm_smul,Real.norm_eq_abs,abs_of_pos (by positivity),norm_coe,mul_one]
+    linarith
+  have h := inner_le_support hK hx (u : Space n)
+  rw [real_inner_smul_right,real_inner_self_eq_norm_sq,norm_coe,one_pow,mul_one] at h
+  linarith
+
+/-- First-variation identity for actual Minkowski addition; the added
+set is allowed to have any dimension. -/
+theorem minkowski_volume_derivative {n : ℕ} [NeZero n] {K M : Set (Space n)}
+    (hK : IsCompact K) (hcK : Convex ℝ K) (h0 : K ∈ 𝓝 (0 : Space n))
+    (hM : IsCompact M) (hcM : Convex ℝ M) (hMn : M.Nonempty) :
+    HasDerivWithinAt (fun t : ℝ => volume.real (K+t • M))
+      (n*kappa n*mean (fun u : Sphere n => (gauge K (u : Space n))⁻¹^n*
+        support M (∇ (gauge K) (u : Space n)))) (Ici 0) 0 := by
+  have hKn : K.Nonempty := ⟨0,mem_of_mem_nhds h0⟩
+  let s : C(Sphere n,ℝ) := ⟨fun u => support K u,(support_continuous hK).comp continuous_subtype_val⟩
+  have hs : ∀ u,0 < s u := support_pos_of_nhds hK h0
+  let φ : C(Sphere n,ℝ) := ⟨fun u => support M u,(support_continuous hM).comp continuous_subtype_val⟩
+  have he : ∀ t : ℝ, 0 ≤ t → wulff ⇑(s+t • φ) = K+t • M := by
+    intro t ht
+    have hf : ⇑(s+t • φ) = fun u : Sphere n => support (K+t • M) u := by
+      funext u
+      simp only [ContinuousMap.coe_add,ContinuousMap.coe_smul,Pi.add_apply,Pi.smul_apply,smul_eq_mul]
+      change support K u+t*support M u = support (K+t • M) u
+      rw [support_minkowski hK hKn (hM.smul t)
+        (by obtain ⟨x,hx⟩ := hMn; exact ⟨t • x,smul_mem_smul_set hx⟩),
+        support_smul_set hM hMn ht]
+    rw [hf]
+    exact wulff_support (hK.add (hM.smul t)) (hcK.add (hcM.smul t)) (hKn.add (by obtain ⟨x,hx⟩ := hMn; exact ⟨t • x,smul_mem_smul_set hx⟩))
+  have h := (wulff_volume_derivative s hs (support M) (support_continuous hM)
+    (fun a ha x => support_smul hM hMn ha.le x)).hasDerivWithinAt (s := Ici (0:ℝ))
+  have hw : wulff s = K := wulff_support hK hcK hKn
+  rw [hw] at h
+  exact h.congr (fun t ht => congrArg volume.real (he t ht).symm)
+    (congrArg volume.real (he 0 le_rfl).symm)
+
+end PettyProjection
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/ProjectionBody/AffineChartMeasure.lean b/lean/OAI/Geometry/ProjectionBody/AffineChartMeasure.lean
new file mode 100644
index 0000000000000000000000000000000000000000..275835c55127afe970e3517b235bddf90f6ecf80
--- /dev/null
+++ b/lean/OAI/Geometry/ProjectionBody/AffineChartMeasure.lean
@@ -0,0 +1,56 @@
+import OAI.Geometry.ProjectionBody.NullHyperplane
+import Mathlib.Analysis.InnerProductSpace.NormDet
+
+namespace OAI
+
+noncomputable section
+open Set MeasureTheory Measure
+
+namespace ProjectionCounterexample
+
+variable {U V : Type*}
+  [NormedAddCommGroup U] [InnerProductSpace ℝ U] [FiniteDimensional ℝ U]
+  [MeasurableSpace U] [BorelSpace U]
+  [NormedAddCommGroup V] [InnerProductSpace ℝ V]
+  [MeasurableSpace V] [BorelSpace V]
+
+/-- The volume of a projected affine chart, with its genuine linear Jacobian. -/
+theorem measure_projected_affine_chart (L : U →ₗ[ℝ] V) (c : V)
+    (q : V →ₗ[ℝ] V) (s : Set U) :
+    μHE[Module.finrank ℝ U] (q '' ((fun x => c + L x) '' s)) =
+      ENNReal.ofReal (q.comp L).normDet * volume s := by
+  have himage : q '' ((fun x => c + L x) '' s) =
+      (fun y => q c + y) '' (q.comp L '' s) := by
+    simp only [Set.image_image]
+    congr 1
+    funext x
+    exact map_add q c (L x)
+  rw [himage, (isometry_add_left _).euclideanHausdorffMeasure_image,
+    (q.comp L).euclideanHausdorffMeasure_image_eq_normDet_mul_volume]
+
+/-- Intersecting a facet chart with a genuinely transverse affine equation
+has zero facet-dimensional measure, including after any linear projection. -/
+theorem measure_projected_transverse_intersection (L : U →ₗ[ℝ] V) (c : V)
+    (q : V →ₗ[ℝ] V) (g : V →ₗ[ℝ] ℝ) (r : ℝ) (S T : Set V)
+    (hS : S ⊆ (fun x => c + L x) '' (univ : Set U))
+    (hT : T ⊆ {y | g y = r}) (hg : g.comp L ≠ 0) :
+    μHE[Module.finrank ℝ U] (q '' (S ∩ T)) = 0 := by
+  have hsub : q '' (S ∩ T) ⊆
+      q '' ((fun x => c + L x) '' {x | (g.comp L) x = r - g c}) := by
+    rintro z ⟨y, ⟨hyS, hyT⟩, rfl⟩
+    obtain ⟨x, _, rfl⟩ := hS hyS
+    refine ⟨c + L x, ⟨x, ?_, rfl⟩, rfl⟩
+    have heq := hT hyT
+    change g (c + L x) = r at heq
+    rw [map_add] at heq
+    change g (L x) = r - g c
+    exact eq_sub_of_add_eq' heq
+  apply measure_mono_null hsub
+  rw [measure_projected_affine_chart,
+    volume_linear_level_eq_zero (g.comp L) hg, mul_zero]
+
+end ProjectionCounterexample
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/ProjectionBody/Arithmetic.lean b/lean/OAI/Geometry/ProjectionBody/Arithmetic.lean
new file mode 100644
index 0000000000000000000000000000000000000000..8574469264644c5037817e571790dfd60ca86e88
--- /dev/null
+++ b/lean/OAI/Geometry/ProjectionBody/Arithmetic.lean
@@ -0,0 +1,44 @@
+import Mathlib.Basic.Real.Basic
+import Mathlib.Data.Nat.Factorial.Basic
+import Mathlib.Algebra.BigOperators.Ring.Finset
+import Mathlib.Tactic.NormNum
+import Mathlib.Tactic.Linarith
+
+namespace OAI
+
+/-! Numerical consequences, to be used only after the geometric volume identities. -/
+
+namespace ProjectionCounterexample
+
+/-- Algebraic double-counting identity used after the visible-face measure sum.
+The balance condition is later proved from the actual simplex facet vectors. -/
+theorem sum_positive_eq_half_sum_abs {ι : Type*} [Fintype ι]
+    (a : ι → ℝ) (ha : ∑ i, a i = 0) :
+    (∑ i with 0 < a i, a i) = (∑ i, |a i|) / 2 := by
+  classical
+  have hi (i : ι) : 2 * (if 0 < a i then a i else 0) = |a i| + a i := by
+    split_ifs with h
+    · rw [abs_of_pos h]
+      linarith
+    · rw [abs_of_nonpos (le_of_not_gt h)]
+      linarith
+  have hs : 2 * (∑ i, if 0 < a i then a i else 0) = ∑ i, |a i| := by
+    rw [Finset.mul_sum]
+    simp_rw [hi]
+    rw [Finset.sum_add_distrib, ha, add_zero]
+  rw [Finset.sum_filter]
+  linarith
+
+theorem numerical_product_ratio :
+    (((11 : ℝ) * 10 ^ 10 / (Nat.factorial 10 : ℝ)) ^ 2) /
+        ((21 : ℝ) * 20 ^ 20 / (Nat.factorial 20 : ℝ)) =
+      (22355476 : ℝ) / 22020096 := by
+  norm_num [Nat.factorial]
+
+theorem numerical_strict_excess :
+    (1 : ℝ) < (22355476 : ℝ) / 22020096 := by
+  norm_num
+
+end ProjectionCounterexample
+
+end OAI
diff --git a/lean/OAI/Geometry/ProjectionBody/BrightnessSum.lean b/lean/OAI/Geometry/ProjectionBody/BrightnessSum.lean
new file mode 100644
index 0000000000000000000000000000000000000000..d8d7f2618dfcc603ab6a78a52a9a157f3919efd7
--- /dev/null
+++ b/lean/OAI/Geometry/ProjectionBody/BrightnessSum.lean
@@ -0,0 +1,48 @@
+import OAI.Geometry.ProjectionBody.FacetMeasure
+import OAI.Geometry.ProjectionBody.Arithmetic
+import Mathlib.Basic.ENNReal.BigOperators
+
+namespace OAI
+
+noncomputable section
+open Set MeasureTheory
+
+namespace ProjectionCounterexample
+
+variable {ι V W : Type*} [Fintype ι]
+  [AddCommGroup V] [Module ℝ V] [AddCommGroup W] [Module ℝ W]
+  [MeasurableSpace W]
+
+/-- Convert the genuine facet-measure sum to the balanced absolute-slope formula.
+Every geometric input, including each projected facet measure, remains explicit. -/
+theorem real_measure_image_halfspaceBody
+    (a : ι → V →ₗ[ℝ] ℝ) (b : ι → ℝ) (q : V →ₗ[ℝ] W) (μ : Measure W)
+    {u : V} (hqu : q u = 0) (hu : ∃ i, 0 < a i u)
+    (hker : LinearMap.ker q = Submodule.span ℝ {u})
+    (hmeas : ∀ i, MeasurableSet (q '' boundingFace a b i))
+    (hnull : ∀ i j, i ≠ j →
+      μ (q '' (boundingFace a b i ∩ boundingFace a b j)) = 0)
+    (c : ℝ) (hc : 0 ≤ c) (hbalance : ∑ i, a i u = 0)
+    (hfacet : ∀ i, μ (q '' boundingFace a b i) = ENNReal.ofReal (c * |a i u|)) :
+    (μ (q '' halfspaceBody a b)).toReal = c * (∑ i, |a i u|) / 2 := by
+  classical
+  rw [measure_image_halfspaceBody_eq_sum_visible a b q μ hqu hu hker hmeas hnull]
+  simp_rw [hfacet]
+  rw [ENNReal.toReal_sum (fun _ _ => ENNReal.ofReal_ne_top)]
+  simp_rw [ENNReal.toReal_ofReal (mul_nonneg hc (abs_nonneg _))]
+  have hs : (∑ i : {i : ι // 0 < a i u}, |a i.1 u|) =
+      (∑ i with 0 < a i u, a i u) := by
+    calc
+      (∑ i : {i : ι // 0 < a i u}, |a i.1 u|) =
+          ∑ i : {i : ι // 0 < a i u}, a i.1 u := by
+        apply Finset.sum_congr rfl
+        intro i _
+        exact abs_of_pos i.2
+      _ = _ := (Finset.sum_subtype _ (by simp) (fun i => a i u)).symm
+  rw [← Finset.mul_sum, hs, sum_positive_eq_half_sum_abs _ hbalance, mul_div_assoc]
+
+end ProjectionCounterexample
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/ProjectionBody/ConvexGeometry.lean b/lean/OAI/Geometry/ProjectionBody/ConvexGeometry.lean
new file mode 100644
index 0000000000000000000000000000000000000000..814d46cf5d7edd9ec56c63e2d75e13697e5d5a27
--- /dev/null
+++ b/lean/OAI/Geometry/ProjectionBody/ConvexGeometry.lean
@@ -0,0 +1,119 @@
+import OAI.Geometry.ProjectionBody.GeometryDefinitions
+import Mathlib.Analysis.Convex.Topology
+import Mathlib.Analysis.Normed.Affine.AddTorsorBases
+import Mathlib.Analysis.InnerProductSpace.PiL2
+
+namespace OAI
+
+open Set
+
+namespace ProjectionCounterexample
+
+noncomputable section
+
+theorem simplex_convex (d : ℕ) : Convex ℝ (simplex d) :=
+  convex_convexHull ℝ _
+
+theorem simplex_compact (d : ℕ) : IsCompact (simplex d) :=
+  ((finite_range _).insert 0).isCompact_convexHull ℝ
+
+theorem simplex_interior_nonempty (d : ℕ) : (interior (simplex d)).Nonempty := by
+  rw [(simplex_convex d).interior_nonempty_iff_affineSpan_eq_top]
+  rw [simplex, affineSpan_convexHull]
+  apply SetLike.coe_injective
+  rw [affineSpan_insert_zero]
+  have hb : (EuclideanSpace.basisFun (Fin d) ℝ : Fin d → E d) =
+      (fun i : Fin d => EuclideanSpace.single i (1 : ℝ)) := by
+    funext i
+    simp only [EuclideanSpace.basisFun_apply]
+  have hspan : Submodule.span ℝ (range fun i : Fin d => EuclideanSpace.single i (1 : ℝ)) = ⊤ := by
+    simpa only [OrthonormalBasis.coe_toBasis, hb] using
+      (EuclideanSpace.basisFun (Fin d) ℝ).toBasis.span_eq
+  exact congrArg (fun S : Submodule ℝ (E d) => (S : Set (E d))) hspan
+
+theorem simplex_eq_coordinates (d : ℕ) :
+    simplex d = {x | (∀ i, 0 ≤ x i) ∧ ∑ i, x i ≤ 1} := by
+  classical
+  apply Subset.antisymm
+  · apply convexHull_min
+    · rintro x (rfl | ⟨i, rfl⟩)
+      · simp
+      · constructor
+        · intro j
+          simp only [PiLp.single_apply]
+          split_ifs
+          · exact zero_le_one
+          · exact le_rfl
+        · simp [PiLp.single_apply]
+    · intro x hx y hy a b ha hb hab
+      constructor
+      · intro i
+        exact add_nonneg (mul_nonneg ha (hx.1 i)) (mul_nonneg hb (hy.1 i))
+      · change ∑ i, (a * x i + b * y i) ≤ 1
+        rw [Finset.sum_add_distrib, ← Finset.mul_sum, ← Finset.mul_sum]
+        calc
+          a * ∑ i, x i + b * ∑ i, y i ≤ a * 1 + b * 1 :=
+            add_le_add (mul_le_mul_of_nonneg_left hx.2 ha)
+              (mul_le_mul_of_nonneg_left hy.2 hb)
+          _ = 1 := by simpa using hab
+  · intro x hx
+    let w : Option (Fin d) → ℝ := fun o =>
+      match o with
+      | none => 1 - ∑ i, x i
+      | some i => x i
+    let z : Option (Fin d) → E d := fun o =>
+      match o with
+      | none => 0
+      | some i => EuclideanSpace.single i 1
+    apply mem_convexHull_of_exists_fintype w z
+    · intro o
+      cases o with
+      | none => exact sub_nonneg.mpr hx.2
+      | some i => exact hx.1 i
+    · simp [w, Fintype.sum_option]
+    · intro o
+      cases o with
+      | none => exact mem_insert 0 _
+      | some i => exact mem_insert_of_mem 0 (mem_range_self i)
+    · simpa [w, z, Fintype.sum_option, EuclideanSpace.basisFun_apply] using
+        (EuclideanSpace.basisFun (Fin d) ℝ).sum_repr x
+
+def blockEquiv : E 20 ≃L[ℝ] E 10 × E 10 :=
+  EuclideanSpace.finAddEquivProd (𝕜 := ℝ) (n := 10) (m := 10)
+
+theorem blockEquiv_apply (x : E 20) :
+    blockEquiv x =
+      (WithLp.toLp 2 fun i : Fin 10 => x (Fin.castAdd 10 i),
+       WithLp.toLp 2 fun i : Fin 10 => x (Fin.natAdd 10 i)) := by
+  rfl
+
+theorem productBody_eq_preimage :
+    productBody = blockEquiv ⁻¹' (simplex 10 ×ˢ simplex 10) := by
+  rfl
+
+theorem productBody_compact : IsCompact productBody := by
+  rw [productBody_eq_preimage]
+  exact blockEquiv.toHomeomorph.isCompact_preimage.mpr
+    ((simplex_compact 10).prod (simplex_compact 10))
+
+theorem productBody_convex : Convex ℝ productBody := by
+  rw [productBody_eq_preimage]
+  exact ((simplex_convex 10).prod (simplex_convex 10)).linear_preimage
+    blockEquiv.toLinearMap
+
+theorem productBody_interior_nonempty : (interior productBody).Nonempty := by
+  rw [productBody_eq_preimage]
+  change (interior (blockEquiv.toHomeomorph ⁻¹' (simplex 10 ×ˢ simplex 10))).Nonempty
+  rw [← blockEquiv.toHomeomorph.preimage_interior, interior_prod_eq]
+  obtain ⟨x, hx⟩ := simplex_interior_nonempty 10
+  exact ⟨blockEquiv.symm (x, x), by simpa using And.intro hx hx⟩
+
+theorem productBody_convexBody :
+    IsCompact productBody ∧ Convex ℝ productBody ∧ (interior productBody).Nonempty :=
+  ⟨productBody_compact, productBody_convex, productBody_interior_nonempty⟩
+
+end
+
+end ProjectionCounterexample
+
+end OAI
diff --git a/lean/OAI/Geometry/ProjectionBody/Counterexample.lean b/lean/OAI/Geometry/ProjectionBody/Counterexample.lean
new file mode 100644
index 0000000000000000000000000000000000000000..a5091dd2a0c7e0c12d2751e6a4b92da496e46f58
--- /dev/null
+++ b/lean/OAI/Geometry/ProjectionBody/Counterexample.lean
@@ -0,0 +1,105 @@
+import OAI.Geometry.ProjectionBody.ProductBrightness
+import OAI.Geometry.ProjectionBody.ProductZonotopeBridge
+import OAI.Geometry.Zonotope.EuclideanVolume
+import OAI.Geometry.ProjectionBody.Arithmetic
+
+namespace OAI
+
+/-!
+The geometric projection formulas identify the two actual projection bodies.
+Their proved Euclidean volumes are substituted into the normalized ratio before
+the final exact arithmetic is performed.
+-/
+
+noncomputable section
+open Set MeasureTheory
+open scoped RealInnerProductSpace
+
+namespace ProjectionCounterexample
+
+theorem simplex_twenty_brightness {u : E 20} (hu : ‖u‖ = 1) :
+    projectionVolume (simplex 20) u =
+      (1 / (Nat.factorial 19 : ℝ)) * DiagonalZonotope.support (WithLp.ofLp u) := by
+  rw [simplex_projectionVolume (n := 19) (by decide) hu]
+  unfold DiagonalZonotope.support
+  ring
+
+theorem product_brightness {u : E 20} (hu : ‖u‖ = 1) :
+    projectionVolume productBody u =
+      (1 / ((Nat.factorial 9 : ℝ) * Nat.factorial 10)) * blockSupport (n := 10) (m := 10) u := by
+  rw [product_projectionVolume hu]
+  change (((∑ i, |firstBlock u i|) + |∑ i, firstBlock u i|) +
+      ((∑ i, |secondBlock u i|) + |∑ i, secondBlock u i|)) /
+      (2 * ((Nat.factorial 9 : ℝ) * Nat.factorial 10)) =
+    (1 / ((Nat.factorial 9 : ℝ) * Nat.factorial 10)) *
+      (((∑ i, |firstBlock u i|) + |∑ i, firstBlock u i|) / 2 +
+        ((∑ i, |secondBlock u i|) + |∑ i, secondBlock u i|) / 2)
+  ring
+
+theorem simplex_twenty_projectionBody :
+    projectionBody (simplex 20) =
+      DiagonalZonotope.euclidean (1 / (Nat.factorial 19 : ℝ)) :=
+  projectionBody_eq_zonotope (simplex 20) (1 / (Nat.factorial 19 : ℝ))
+    (by positivity) (fun _ => simplex_twenty_brightness)
+
+theorem product_projectionBody :
+    projectionBody productBody = DiagonalZonotope.euclideanProduct 10 10
+      (1 / ((Nat.factorial 9 : ℝ) * Nat.factorial 10)) :=
+  projectionBody_eq_productZonotope (n := 10) (m := 10) productBody
+    (1 / ((Nat.factorial 9 : ℝ) * Nat.factorial 10))
+    (by positivity) (fun _ => product_brightness)
+
+theorem simplex_twenty_projection_support :
+    IsConvexBody (projectionBody (simplex 20)) ∧
+      ∀ u : E 20, ‖u‖ = 1 →
+        ∃ x ∈ projectionBody (simplex 20), ⟪u, x⟫ = projectionVolume (simplex 20) u :=
+  projection_zonotope_is_body_and_attains (simplex 20) (1 / (Nat.factorial 19 : ℝ))
+    (by positivity)
+    (fun _ => simplex_twenty_brightness)
+
+theorem product_projection_support :
+    IsConvexBody (projectionBody productBody) ∧
+      ∀ u : E 20, ‖u‖ = 1 →
+        ∃ x ∈ projectionBody productBody, ⟪u, x⟫ = projectionVolume productBody u :=
+  projection_productZonotope_is_body_and_attains (n := 10) (m := 10) productBody
+    (1 / ((Nat.factorial 9 : ℝ) * Nat.factorial 10)) (by positivity)
+    (fun _ => product_brightness)
+
+theorem simplex_twenty_ratio :
+    ratio (simplex 20) = ((21 : ℝ) * 20 ^ 20) / (Nat.factorial 20 : ℝ) := by
+  unfold ratio bodyVolume
+  rw [simplex_twenty_projectionBody,
+    DiagonalZonotope.volume_euclidean_toReal _ (by positivity), volume_simplex]
+  norm_num [Nat.factorial]
+
+theorem product_ratio_value :
+    ratio productBody = (((11 : ℝ) * 10 ^ 10) / (Nat.factorial 10 : ℝ)) ^ 2 := by
+  unfold ratio bodyVolume
+  rw [product_projectionBody,
+    DiagonalZonotope.volume_euclideanProduct_toReal 10 10 _ (by positivity), volume_productBody]
+  norm_num [Nat.factorial]
+
+theorem product_to_simplex_ratio :
+    ratio productBody / ratio (simplex 20) = (22355476 : ℝ) / 22020096 := by
+  rw [product_ratio_value, simplex_twenty_ratio]
+  exact numerical_product_ratio
+
+theorem product_exceeds_simplex : ratio (simplex 20) < ratio productBody := by
+  have hpos : 0 < ratio (simplex 20) := by
+    rw [simplex_twenty_ratio]
+    positivity
+  have hratio : 1 < ratio productBody / ratio (simplex 20) := by
+    rw [product_to_simplex_ratio]
+    exact numerical_strict_excess
+  simpa using (lt_div_iff₀ hpos).mp hratio
+
+theorem universal_simplex_upper_bound_false :
+    ¬ (∀ P : Set (E 20), IsConvexBody P → ratio P ≤ ratio (simplex 20)) := by
+  intro h
+  exact (not_le_of_gt product_exceeds_simplex) (h productBody productBody_convexBody)
+
+end ProjectionCounterexample
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/ProjectionBody/Definitions.lean b/lean/OAI/Geometry/ProjectionBody/Definitions.lean
new file mode 100644
index 0000000000000000000000000000000000000000..8b90df38132e92c7d9e551017d59c8682c6e50c5
--- /dev/null
+++ b/lean/OAI/Geometry/ProjectionBody/Definitions.lean
@@ -0,0 +1,44 @@
+import OAI.Geometry.ProjectionBody.GeometryDefinitions
+import Mathlib.Analysis.InnerProductSpace.Projection.Basic
+import Mathlib.Geometry.Euclidean.Volume.Measure
+
+namespace OAI
+
+/-!
+Geometric definitions for the product counterexample.
+The projection measure is the canonically normalized Euclidean Hausdorff
+measure of dimension `d - 1`, not ambient `d`-dimensional measure.
+-/
+
+noncomputable section
+
+open Set MeasureTheory
+open scoped RealInnerProductSpace
+
+namespace ProjectionCounterexample
+
+/-- Orthogonal projection onto the hyperplane normal to `u`. -/
+def project {d : ℕ} (u : E d) : E d →L[ℝ] E d :=
+  (ℝ ∙ u)ᗮ.starProjection
+
+/-- Actual (d−1)-dimensional Euclidean volume of the orthogonal projection. -/
+def projectionVolume {d : ℕ} (P : Set (E d)) (u : E d) : ℝ :=
+  (Measure.euclideanHausdorffMeasure (d - 1) (project u '' P)).toReal
+
+/-- The projection body, defined entirely by its directional projection volumes.
+The bounds are the normalized Hausdorff measures of orthogonal projections. -/
+def projectionBody {d : ℕ} (P : Set (E d)) : Set (E d) :=
+  {x | ∀ u : E d, ‖u‖ = 1 → ⟪u, x⟫ ≤ projectionVolume P u}
+
+/-- Lebesgue volume in the ambient Euclidean space. -/
+def bodyVolume {d : ℕ} (P : Set (E d)) : ℝ := (volume P).toReal
+
+/-- The normalized projection-body volume. -/
+def ratio {d : ℕ} (P : Set (E d)) : ℝ :=
+  bodyVolume (projectionBody P) / bodyVolume P ^ (d - 1)
+
+end ProjectionCounterexample
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/ProjectionBody/FacetCoordinates.lean b/lean/OAI/Geometry/ProjectionBody/FacetCoordinates.lean
new file mode 100644
index 0000000000000000000000000000000000000000..79dbed7033ea895890190cb2cd013eeae0d9b39f
--- /dev/null
+++ b/lean/OAI/Geometry/ProjectionBody/FacetCoordinates.lean
@@ -0,0 +1,108 @@
+import OAI.Geometry.ProjectionBody.ConvexGeometry
+import Mathlib.Algebra.BigOperators.Fin
+
+namespace OAI
+
+noncomputable section
+open Set
+
+namespace ProjectionCounterexample
+
+/-- The coordinate isometry inserting a zero in slot `i`. -/
+def insertZero {n : ℕ} (i : Fin (n + 1)) : E n →ₗᵢ[ℝ] E (n + 1) where
+  toFun x := WithLp.toLp 2 (i.insertNth 0 x)
+  map_add' x y := by
+    ext j
+    induction j using i.succAboveCases <;> simp
+  map_smul' c x := by
+    ext j
+    induction j using i.succAboveCases <;> simp
+  norm_map' x := by
+    apply (sq_eq_sq₀ (norm_nonneg _) (norm_nonneg _)).mp
+    rw [EuclideanSpace.real_norm_sq_eq, EuclideanSpace.real_norm_sq_eq,
+      Fin.sum_univ_succAbove _ i]
+    simp
+
+@[simp] theorem insertZero_same {n : ℕ} (i : Fin (n + 1)) (x : E n) :
+    insertZero i x i = 0 := by simp [insertZero]
+
+@[simp] theorem insertZero_succAbove {n : ℕ} (i : Fin (n + 1)) (x : E n)
+    (j : Fin n) : insertZero i x (i.succAbove j) = x j := by simp [insertZero]
+
+theorem sum_insertZero {n : ℕ} (i : Fin (n + 1)) (x : E n) :
+    ∑ j, insertZero i x j = ∑ j, x j := by
+  rw [Fin.sum_univ_succAbove _ i]
+  simp
+
+/-- A coordinate facet is an actual isometric image of the lower-dimensional simplex. -/
+theorem coordinate_facet_eq_image {n : ℕ} (i : Fin (n + 1)) :
+    {x ∈ simplex (n + 1) | x i = 0} = insertZero i '' simplex n := by
+  simp_rw [simplex_eq_coordinates]
+  apply Subset.antisymm
+  · rintro x ⟨⟨hx, hs⟩, hxi⟩
+    let y : E n := WithLp.toLp 2 (fun j => x (i.succAbove j))
+    have hy : (∀ j, 0 ≤ y j) ∧ ∑ j, y j ≤ 1 := by
+      refine ⟨fun j => hx _, ?_⟩
+      rw [Fin.sum_univ_succAbove _ i, hxi, zero_add] at hs
+      exact hs
+    refine ⟨y, hy, ?_⟩
+    ext j
+    induction j using i.succAboveCases
+    · simpa using hxi.symm
+    · simp [y]
+  · rintro x ⟨y, ⟨hy, hs⟩, rfl⟩
+    refine ⟨⟨?_, ?_⟩, by simp⟩
+    · intro j
+      induction j using i.succAboveCases
+      · simp
+      · simpa using hy _
+    · simpa [sum_insertZero] using hs
+
+/-- The affine chart on the facet whose coordinate sum is one. -/
+def sumFacetMap (n : ℕ) (x : E n) : E (n + 1) :=
+  WithLp.toLp 2 (Fin.snoc (fun j => x j) (1 - ∑ j, x j))
+
+@[simp] theorem sumFacetMap_castSucc (n : ℕ) (x : E n) (i : Fin n) :
+    sumFacetMap n x i.castSucc = x i := by simp [sumFacetMap]
+
+@[simp] theorem sumFacetMap_last (n : ℕ) (x : E n) :
+    sumFacetMap n x (Fin.last n) = 1 - ∑ j, x j := by simp [sumFacetMap]
+
+@[simp] theorem sum_sumFacetMap (n : ℕ) (x : E n) :
+    ∑ j, sumFacetMap n x j = 1 := by
+  rw [Fin.sum_univ_castSucc]
+  simp
+
+/-- The non-coordinate facet is an actual affine image of the lower-dimensional simplex. -/
+theorem sum_facet_eq_image (n : ℕ) :
+    {x ∈ simplex (n + 1) | ∑ j, x j = 1} = sumFacetMap n '' simplex n := by
+  simp_rw [simplex_eq_coordinates]
+  apply Subset.antisymm
+  · rintro x ⟨⟨hx, _⟩, hs⟩
+    let y : E n := WithLp.toLp 2 (fun j : Fin n => x j.castSucc)
+    have hsum : ∑ j, y j = 1 - x (Fin.last n) := by
+      rw [Fin.sum_univ_castSucc] at hs
+      dsimp [y]
+      linarith
+    have hy : (∀ j, 0 ≤ y j) ∧ ∑ j, y j ≤ 1 := by
+      refine ⟨fun j => hx _, ?_⟩
+      rw [hsum]
+      linarith [hx (Fin.last n)]
+    refine ⟨y, hy, ?_⟩
+    ext j
+    induction j using Fin.lastCases
+    · simp [hsum]
+    · simp [y]
+  · rintro x ⟨y, ⟨hy, hs⟩, rfl⟩
+    refine ⟨⟨?_, by simp⟩, by simp⟩
+    intro j
+    induction j using Fin.lastCases
+    · simp only [sumFacetMap_last]
+      exact sub_nonneg.mpr hs
+    · simpa using hy _
+
+end ProjectionCounterexample
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/ProjectionBody/FacetCover.lean b/lean/OAI/Geometry/ProjectionBody/FacetCover.lean
new file mode 100644
index 0000000000000000000000000000000000000000..5dcf9f4f627182d544263738fe63d0c879c2a2fa
--- /dev/null
+++ b/lean/OAI/Geometry/ProjectionBody/FacetCover.lean
@@ -0,0 +1,119 @@
+import Mathlib.Data.Finset.Max
+import Mathlib.Basic.Real.Basic
+import Mathlib.LinearAlgebra.Span.Defs
+import Mathlib.Algebra.Module.Submodule.Ker
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.Positivity
+
+namespace OAI
+
+/-! Finite-halfspace geometry underlying the visible-facet projection formula. -/
+
+noncomputable section
+open Set
+
+namespace ProjectionCounterexample
+
+variable {ι V W : Type*} [Fintype ι]
+  [AddCommGroup V] [Module ℝ V] [AddCommGroup W] [Module ℝ W]
+
+/-- A finite intersection of closed halfspaces. -/
+def halfspaceBody (a : ι → V →ₗ[ℝ] ℝ) (b : ι → ℝ) : Set V :=
+  {x | ∀ i, a i x ≤ b i}
+
+/-- The part of the body on the bounding hyperplane of inequality `i`. -/
+def boundingFace (a : ι → V →ₗ[ℝ] ℝ) (b : ι → ℝ) (i : ι) : Set V :=
+  {x | x ∈ halfspaceBody a b ∧ a i x = b i}
+
+/-- Moving in a direction with a positive halfspace slope reaches a visible face. -/
+theorem ray_reaches_visible_face (a : ι → V →ₗ[ℝ] ℝ) (b : ι → ℝ)
+    {y u : V} (hy : y ∈ halfspaceBody a b) (hu : ∃ i, 0 < a i u) :
+    ∃ (t : ℝ) (i : ι), 0 ≤ t ∧ 0 < a i u ∧
+      y + t • u ∈ boundingFace a b i := by
+  classical
+  let s := Finset.univ.filter fun i => 0 < a i u
+  have hs : s.Nonempty := by
+    obtain ⟨i, hi⟩ := hu
+    exact ⟨i, Finset.mem_filter.mpr ⟨Finset.mem_univ i, hi⟩⟩
+  obtain ⟨i, hi, hmin⟩ :=
+    s.exists_min_image (fun j => (b j - a j y) / a j u) hs
+  have hiu : 0 < a i u := (Finset.mem_filter.mp hi).2
+  let t := (b i - a i y) / a i u
+  have ht : 0 ≤ t := div_nonneg (sub_nonneg.mpr (hy i)) hiu.le
+  refine ⟨t, i, ht, hiu, ?_, ?_⟩
+  · intro j
+    rw [map_add, map_smul, smul_eq_mul]
+    by_cases hju : 0 < a j u
+    · have hle : t ≤ (b j - a j y) / a j u :=
+        hmin j (Finset.mem_filter.mpr ⟨Finset.mem_univ j, hju⟩)
+      have := (le_div_iff₀ hju).mp hle
+      linarith
+    · have hm : t * a j u ≤ 0 := mul_nonpos_of_nonneg_of_nonpos ht (le_of_not_gt hju)
+      have := hy j
+      linarith
+  · rw [map_add, map_smul, smul_eq_mul]
+    have hm : t * a i u = b i - a i y := div_mul_cancel₀ _ (ne_of_gt hiu)
+    linarith
+
+/-- The projected body is covered by the projections of its visible faces. -/
+theorem image_halfspaceBody_eq_visible_faces
+    (a : ι → V →ₗ[ℝ] ℝ) (b : ι → ℝ) (q : V →ₗ[ℝ] W)
+    {u : V} (hqu : q u = 0) (hu : ∃ i, 0 < a i u) :
+    q '' halfspaceBody a b =
+      ⋃ i : {i : ι // 0 < a i u}, q '' boundingFace a b i.1 := by
+  ext x
+  constructor
+  · rintro ⟨y, hy, rfl⟩
+    obtain ⟨t, i, _, hiu, hface⟩ := ray_reaches_visible_face a b hy hu
+    apply Set.mem_iUnion.mpr
+    refine ⟨⟨i, hiu⟩, y + t • u, hface, ?_⟩
+    simp [map_add, map_smul, hqu]
+  · intro hx
+    obtain ⟨i, y, hy, rfl⟩ := Set.mem_iUnion.mp hx
+    exact ⟨y, hy.1, rfl⟩
+
+omit [Fintype ι] in
+/-- Two visible boundary points on the same line parallel to the direction coincide.
+This will put overlaps of projected visible faces in their common lower-dimensional face. -/
+theorem visible_face_line_unique
+    (a : ι → V →ₗ[ℝ] ℝ) (b : ι → ℝ)
+    {i j : ι} {y z u : V} {t : ℝ}
+    (hy : y ∈ boundingFace a b i) (hz : z ∈ boundingFace a b j)
+    (hi : 0 < a i u) (hj : 0 < a j u) (hline : z = y + t • u) : z = y := by
+  have hti : t * a i u ≤ 0 := by
+    have h := hz.1 i
+    rw [hline, map_add, map_smul, smul_eq_mul, hy.2] at h
+    linarith
+  have htj : 0 ≤ t * a j u := by
+    have h := hy.1 j
+    have he := hz.2
+    rw [hline, map_add, map_smul, smul_eq_mul] at he
+    linarith
+  have ht0 : t = 0 := by nlinarith
+  simpa [ht0] using hline
+
+omit [Fintype ι] in
+/-- Overlaps of visible-face shadows are precisely shadows of the face intersections. -/
+theorem image_visible_faces_inter
+    (a : ι → V →ₗ[ℝ] ℝ) (b : ι → ℝ) (q : V →ₗ[ℝ] W)
+    {u : V} (hker : LinearMap.ker q = Submodule.span ℝ {u})
+    {i j : ι} (hi : 0 < a i u) (hj : 0 < a j u) :
+    (q '' boundingFace a b i) ∩ (q '' boundingFace a b j) =
+      q '' (boundingFace a b i ∩ boundingFace a b j) := by
+  apply Set.Subset.antisymm
+  · rintro x ⟨⟨y, hy, hyq⟩, ⟨z, hz, hzq⟩⟩
+    have hdiff : z - y ∈ Submodule.span ℝ {u} := by
+      rw [← hker, LinearMap.mem_ker, map_sub, hyq, hzq, sub_self]
+    obtain ⟨t, ht⟩ := Submodule.mem_span_singleton.mp hdiff
+    have hline : z = y + t • u := by
+      rw [ht]
+      simp [sub_eq_add_neg]
+    have hzy := visible_face_line_unique a b hy hz hi hj hline
+    exact ⟨y, ⟨hy, hzy ▸ hz⟩, hyq⟩
+  · exact Set.image_inter_subset q _ _
+
+end ProjectionCounterexample
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/ProjectionBody/FacetMeasure.lean b/lean/OAI/Geometry/ProjectionBody/FacetMeasure.lean
new file mode 100644
index 0000000000000000000000000000000000000000..a0cbc8b42ff0691c0099ebfa7db134d87aa2ee9f
--- /dev/null
+++ b/lean/OAI/Geometry/ProjectionBody/FacetMeasure.lean
@@ -0,0 +1,41 @@
+import OAI.Geometry.ProjectionBody.FacetCover
+import Mathlib.MeasureTheory.Measure.Basic
+
+namespace OAI
+
+noncomputable section
+open Set MeasureTheory
+
+namespace ProjectionCounterexample
+
+variable {ι V W : Type*} [Fintype ι]
+  [AddCommGroup V] [Module ℝ V] [AddCommGroup W] [Module ℝ W]
+  [MeasurableSpace W]
+
+/-- Finite additivity for shadows of visible faces. The geometric hypotheses
+are explicit: the later applications must prove measurability and nullity of
+the projected intersections. No facet-volume formula is assumed here. -/
+theorem measure_image_halfspaceBody_eq_sum_visible
+    (a : ι → V →ₗ[ℝ] ℝ) (b : ι → ℝ) (q : V →ₗ[ℝ] W) (μ : Measure W)
+    {u : V} (hqu : q u = 0) (hu : ∃ i, 0 < a i u)
+    (hker : LinearMap.ker q = Submodule.span ℝ {u})
+    (hmeas : ∀ i, MeasurableSet (q '' boundingFace a b i))
+    (hnull : ∀ i j, i ≠ j →
+      μ (q '' (boundingFace a b i ∩ boundingFace a b j)) = 0) :
+    μ (q '' halfspaceBody a b) =
+      ∑ i : {i : ι // 0 < a i u}, μ (q '' boundingFace a b i.1) := by
+  classical
+  rw [image_halfspaceBody_eq_visible_faces a b q hqu hu]
+  have hd : Pairwise (fun i j : {i : ι // 0 < a i u} =>
+      AEDisjoint μ (q '' boundingFace a b i.1) (q '' boundingFace a b j.1)) := by
+    intro i j hij
+    change μ ((q '' boundingFace a b i.1) ∩ (q '' boundingFace a b j.1)) = 0
+    rw [image_visible_faces_inter a b q hker i.2 j.2]
+    exact hnull i j (fun h => hij (Subtype.ext h))
+  rw [measure_iUnion₀ hd (fun i => (hmeas i.1).nullMeasurableSet), tsum_fintype]
+
+end ProjectionCounterexample
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/ProjectionBody/GeometryDefinitions.lean b/lean/OAI/Geometry/ProjectionBody/GeometryDefinitions.lean
new file mode 100644
index 0000000000000000000000000000000000000000..50f7919f82d39701548562e285bd29459eea731d
--- /dev/null
+++ b/lean/OAI/Geometry/ProjectionBody/GeometryDefinitions.lean
@@ -0,0 +1,37 @@
+import Mathlib.Analysis.Convex.Hull
+import Mathlib.Analysis.InnerProductSpace.PiL2
+
+namespace OAI
+
+noncomputable section
+open Set
+
+namespace ProjectionCounterexample
+
+abbrev E (d : ℕ) := EuclideanSpace ℝ (Fin d)
+
+/-- The convex hull of the origin and the `d` standard unit vectors. -/
+def simplex (d : ℕ) : Set (E d) :=
+  convexHull ℝ (insert 0 (range fun i : Fin d => EuclideanSpace.single i (1 : ℝ)))
+
+/-- The first ten coordinates in the standard orthogonal decomposition of R²⁰. -/
+def firstBlock (x : E 20) : E 10 :=
+  WithLp.toLp 2 fun i : Fin 10 => x (Fin.castAdd 10 i)
+
+/-- The last ten coordinates in the standard orthogonal decomposition of R²⁰. -/
+def secondBlock (x : E 20) : E 10 :=
+  WithLp.toLp 2 fun i : Fin 10 => x (Fin.natAdd 10 i)
+
+/-- The genuine Cartesian product T₁₀ × T₁₀ in Euclidean R²⁰. -/
+def productBody : Set (E 20) :=
+  {x | firstBlock x ∈ simplex 10 ∧ secondBlock x ∈ simplex 10}
+
+/-- Compact, convex, and full-dimensional in its specified Euclidean space. -/
+def IsConvexBody {d : ℕ} (P : Set (E d)) : Prop :=
+  IsCompact P ∧ Convex ℝ P ∧ (interior P).Nonempty
+
+end ProjectionCounterexample
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/ProjectionBody/GraphChart.lean b/lean/OAI/Geometry/ProjectionBody/GraphChart.lean
new file mode 100644
index 0000000000000000000000000000000000000000..554300305e8e0742fc7f4e30fae32d61c908d652
--- /dev/null
+++ b/lean/OAI/Geometry/ProjectionBody/GraphChart.lean
@@ -0,0 +1,67 @@
+import OAI.Geometry.ProjectionBody.FacetCoordinates
+import Mathlib.Tactic.Abel
+
+namespace OAI
+
+noncomputable section
+open scoped RealInnerProductSpace
+
+namespace ProjectionCounterexample
+
+/-- A graph chart with one coordinate determined by a linear equation. -/
+def graphLinear {n : ℕ} (i : Fin (n + 1)) (a : E n) : E n →ₗ[ℝ] E (n + 1) where
+  toFun x := insertZero i x - ⟪a, x⟫ • EuclideanSpace.single i 1
+  map_add' x y := by
+    simp only [map_add, inner_add_right, add_smul]
+    abel
+  map_smul' c x := by
+    simp only [map_smul, real_inner_smul_right, smul_sub, smul_smul, RingHom.id_apply]
+
+/-- The corresponding unnormalized normal has coordinate `i` equal to one. -/
+def graphNormal {n : ℕ} (i : Fin (n + 1)) (a : E n) : E (n + 1) :=
+  insertZero i a + EuclideanSpace.single i 1
+
+@[simp] theorem graphNormal_same {n : ℕ} (i : Fin (n + 1)) (a : E n) :
+    graphNormal i a i = 1 := by simp [graphNormal]
+
+theorem graphNormal_ne_zero {n : ℕ} (i : Fin (n + 1)) (a : E n) :
+    graphNormal i a ≠ 0 := by
+  intro h
+  have := congrArg (fun x : E (n + 1) => x i) h
+  simp at this
+
+@[simp] theorem graphLinear_same {n : ℕ} (i : Fin (n + 1)) (a x : E n) :
+    graphLinear i a x i = -⟪a, x⟫ := by simp [graphLinear]
+
+@[simp] theorem graphLinear_succAbove {n : ℕ} (i : Fin (n + 1))
+    (a x : E n) (j : Fin n) :
+    graphLinear i a x (i.succAbove j) = x j := by simp [graphLinear]
+
+theorem graphLinear_inner {n : ℕ} (i : Fin (n + 1)) (a x y : E n) :
+    ⟪graphLinear i a x, graphLinear i a y⟫ = ⟪x, y⟫ + ⟪a, x⟫ * ⟪a, y⟫ := by
+  simp [graphLinear, inner_sub_left, inner_sub_right, real_inner_smul_left,
+    real_inner_smul_right, EuclideanSpace.inner_single_left,
+    EuclideanSpace.inner_single_right, (insertZero i).inner_map_map, mul_comm]
+
+theorem graphNormal_inner_graphLinear {n : ℕ} (i : Fin (n + 1)) (a x : E n) :
+    ⟪graphNormal i a, graphLinear i a x⟫ = 0 := by
+  simp [graphNormal, graphLinear, inner_add_left, inner_sub_right,
+    real_inner_smul_right, EuclideanSpace.inner_single_left,
+    EuclideanSpace.inner_single_right, (insertZero i).inner_map_map]
+
+theorem graphNormal_norm_sq {n : ℕ} (i : Fin (n + 1)) (a : E n) :
+    ‖graphNormal i a‖ ^ 2 = 1 + ‖a‖ ^ 2 := by
+  calc
+    ‖graphNormal i a‖ ^ 2 = ⟪graphNormal i a, graphNormal i a⟫ :=
+      (real_inner_self_eq_norm_sq _).symm
+    _ = 1 + ‖a‖ ^ 2 := by
+      unfold graphNormal
+      rw [inner_add_left, inner_add_right, inner_add_right]
+      simp [EuclideanSpace.inner_single_left, EuclideanSpace.inner_single_right,
+        add_comm]
+
+end ProjectionCounterexample
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/ProjectionBody/GraphJacobian.lean b/lean/OAI/Geometry/ProjectionBody/GraphJacobian.lean
new file mode 100644
index 0000000000000000000000000000000000000000..0593dc655dbace4eca300e2caa87a1da2e80f287
--- /dev/null
+++ b/lean/OAI/Geometry/ProjectionBody/GraphJacobian.lean
@@ -0,0 +1,40 @@
+import OAI.Geometry.ProjectionBody.GraphChart
+import Mathlib.Analysis.InnerProductSpace.NormDet
+import Mathlib.LinearAlgebra.Matrix.SchurComplement
+
+namespace OAI
+
+noncomputable section
+open scoped RealInnerProductSpace
+
+namespace ProjectionCounterexample
+
+/-- The Jacobian of a graph chart is the length of its unnormalized normal. -/
+theorem graphLinear_normDet {n : ℕ} (i : Fin (n + 1)) (a : E n) :
+    (graphLinear i a).normDet = ‖graphNormal i a‖ := by
+  classical
+  let b := EuclideanSpace.basisFun (Fin n) ℝ
+  have hGram : Matrix.gram ℝ (fun j => graphLinear i a (b j)) =
+      1 + Matrix.replicateCol Unit (fun j => a j) *
+        Matrix.replicateRow Unit (fun j => a j) := by
+    ext j k
+    rw [Matrix.gram_apply, graphLinear_inner]
+    simp [b, EuclideanSpace.basisFun_apply, eq_comm,
+      EuclideanSpace.inner_single_right, Matrix.add_apply, Matrix.one_apply,
+      Matrix.mul_apply, Matrix.replicateCol_apply, Matrix.replicateRow_apply]
+  have hsquare : (graphLinear i a).normDet ^ 2 = ‖graphNormal i a‖ ^ 2 := by
+    calc
+      (graphLinear i a).normDet ^ 2 =
+          (Matrix.gram ℝ (fun j => graphLinear i a (b j))).det :=
+        (graphLinear i a).normDet_sq_eq_det_gram b
+      _ = ‖graphNormal i a‖ ^ 2 := by
+        rw [hGram, Matrix.det_one_add_replicateCol_mul_replicateRow,
+          graphNormal_norm_sq, EuclideanSpace.real_norm_sq_eq]
+        simp [dotProduct, pow_two]
+  exact (sq_eq_sq₀ (graphLinear i a).normDet_nonneg (norm_nonneg _)).mp hsquare
+
+end ProjectionCounterexample
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/ProjectionBody/GraphProjection.lean b/lean/OAI/Geometry/ProjectionBody/GraphProjection.lean
new file mode 100644
index 0000000000000000000000000000000000000000..b69c9feab19f37fe3ded1070eb9dbc44a0ad6f3c
--- /dev/null
+++ b/lean/OAI/Geometry/ProjectionBody/GraphProjection.lean
@@ -0,0 +1,46 @@
+import OAI.Geometry.ProjectionBody.GraphJacobian
+import OAI.Geometry.ProjectionBody.HyperplaneJacobian
+import OAI.Geometry.ProjectionBody.Definitions
+import Mathlib.Tactic.FieldSimp
+
+namespace OAI
+
+noncomputable section
+open scoped RealInnerProductSpace
+
+namespace ProjectionCounterexample
+
+/-- Projection of a graph chart cancels the length of its unnormalized normal. -/
+theorem graphProjection_normDet {n : ℕ} (i : Fin (n + 1)) (a : E n)
+    {u : E (n + 1)} (hu : ‖u‖ = 1) :
+    ((project u).toLinearMap.comp (graphLinear i a)).normDet =
+      |⟪graphNormal i a, u⟫| := by
+  let w := graphNormal i a
+  have hw : ‖w‖ ≠ 0 := norm_ne_zero_iff.mpr (graphNormal_ne_zero i a)
+  let v := ‖w‖⁻¹ • w
+  have hv : ‖v‖ = 1 := by
+    simp [v, norm_smul, hw]
+  have hmem (x : E n) : ⟪v, graphLinear i a x⟫ = 0 := by
+    rw [show v = ‖w‖⁻¹ • graphNormal i a from rfl, real_inner_smul_left,
+      graphNormal_inner_graphLinear, mul_zero]
+  change ((perpendicularTo u).starProjection.toLinearMap.comp (graphLinear i a)).normDet = _
+  rw [normDet_project_comp (graphLinear i a) hv hu hmem (by simp),
+    graphLinear_normDet]
+  change |⟪‖w‖⁻¹ • w, u⟫| * ‖w‖ = |⟪w, u⟫|
+  rw [real_inner_smul_left, abs_mul, abs_inv, abs_of_nonneg (norm_nonneg w)]
+  field_simp [hw]
+
+theorem coordinateProjection_normDet {n : ℕ} (i : Fin (n + 1))
+    {u : E (n + 1)} (hu : ‖u‖ = 1) :
+    ((project u).toLinearMap.comp (insertZero i).toLinearMap).normDet = |u i| := by
+  have hL : graphLinear i 0 = (insertZero i).toLinearMap := by
+    ext x j
+    simp [graphLinear]
+  have h := graphProjection_normDet i 0 hu
+  simpa [hL, graphNormal, EuclideanSpace.inner_single_left] using h
+
+end ProjectionCounterexample
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/ProjectionBody/HyperplaneJacobian.lean b/lean/OAI/Geometry/ProjectionBody/HyperplaneJacobian.lean
new file mode 100644
index 0000000000000000000000000000000000000000..310f51440ff3b2314029dac2a778f32f69c8defa
--- /dev/null
+++ b/lean/OAI/Geometry/ProjectionBody/HyperplaneJacobian.lean
@@ -0,0 +1,160 @@
+import Mathlib.Analysis.InnerProductSpace.NormDet
+import Mathlib.LinearAlgebra.Matrix.SchurComplement
+import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.Ring
+
+namespace OAI
+
+open scoped RealInnerProductSpace
+
+namespace ProjectionCounterexample
+
+noncomputable section
+
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V]
+  [FiniteDimensional ℝ V]
+
+abbrev perpendicularTo (v : V) : Submodule ℝ V := (ℝ ∙ v)ᗮ
+
+/-- Restrict the actual orthogonal projection to the source tangent hyperplane. -/
+def hyperplaneProjection (n u : V) : perpendicularTo n →ₗ[ℝ] V :=
+  (perpendicularTo u).starProjection.toLinearMap.comp (perpendicularTo n).subtype
+
+theorem perpendicularTo_finrank {n : V} (hn : ‖n‖ = 1) :
+    Module.finrank ℝ (perpendicularTo n) = Module.finrank ℝ V - 1 := by
+  have hn0 : n ≠ 0 := by
+    intro h
+    rw [h, norm_zero] at hn
+    exact zero_ne_one hn
+  have h := (ℝ ∙ n).finrank_add_finrank_orthogonal
+  have h' := congrArg (fun k : ℕ => k - 1) h
+  simpa only [finrank_span_singleton hn0, Nat.add_sub_cancel_left] using h'
+
+omit [FiniteDimensional ℝ V] in
+theorem perpendicularTo_projection_apply {u : V} (hu : ‖u‖ = 1) (x : V) :
+    (perpendicularTo u).starProjection x = x - ⟪u, x⟫ • u := by
+  rw [Submodule.starProjection_orthogonal_val,
+    Submodule.starProjection_unit_singleton ℝ hu]
+
+omit [FiniteDimensional ℝ V] in
+theorem inner_perpendicularTo_projection {u : V} (hu : ‖u‖ = 1) (x y : V) :
+    ⟪(perpendicularTo u).starProjection x, (perpendicularTo u).starProjection y⟫ =
+      ⟪x, y⟫ - ⟪u, x⟫ * ⟪u, y⟫ := by
+  rw [perpendicularTo_projection_apply hu, perpendicularTo_projection_apply hu]
+  simp only [inner_sub_left, inner_sub_right, real_inner_smul_left,
+    real_inner_smul_right, real_inner_self_eq_norm_sq, hu, one_pow, mul_one]
+  rw [real_inner_comm x u]
+  ring
+
+/-- The projected facet's area factor is the absolute inner product of unit normals. -/
+theorem hyperplaneProjection_normDet {n u : V} (hn : ‖n‖ = 1) (hu : ‖u‖ = 1) :
+    (hyperplaneProjection n u).normDet = |⟪n, u⟫| := by
+  classical
+  let b := stdOrthonormalBasis ℝ (perpendicularTo n)
+  let a : Fin (Module.finrank ℝ (perpendicularTo n)) → ℝ :=
+    fun i => ⟪u, (b i : V)⟫
+  let f := hyperplaneProjection n u
+  have hGram : Matrix.gram ℝ (fun i => f (b i)) =
+      1 + Matrix.replicateCol Unit (-a) * Matrix.replicateRow Unit a := by
+    ext i j
+    rw [Matrix.gram_apply]
+    change ⟪(perpendicularTo u).starProjection (b i : V),
+      (perpendicularTo u).starProjection (b j : V)⟫ = _
+    rw [inner_perpendicularTo_projection hu]
+    have hbij : ⟪(b i : V), (b j : V)⟫ = if i = j then 1 else 0 :=
+      b.inner_eq_ite i j
+    rw [hbij]
+    simp [Matrix.add_apply, Matrix.one_apply, Matrix.mul_apply,
+      Matrix.replicateCol_apply, Matrix.replicateRow_apply, a, sub_eq_add_neg]
+  have hparseval : ∑ i, (a i) ^ 2 =
+      ‖(perpendicularTo n).orthogonalProjectionOnto u‖ ^ 2 := by
+    calc
+      ∑ i, (a i) ^ 2 =
+          ∑ i, ⟪b i, (perpendicularTo n).orthogonalProjectionOnto u⟫ ^ 2 := by
+        apply Finset.sum_congr rfl
+        intro i _
+        congr 1
+        rw [Submodule.inner_orthogonalProjectionOnto_eq_of_mem_left]
+        exact real_inner_comm _ _
+      _ = ‖(perpendicularTo n).orthogonalProjectionOnto u‖ ^ 2 :=
+        b.sum_sq_inner_right _
+  have hnormal : ‖(ℝ ∙ n).starProjection u‖ ^ 2 = ⟪n, u⟫ ^ 2 := by
+    rw [Submodule.starProjection_unit_singleton ℝ hn]
+    simp [norm_smul, hn, Real.norm_eq_abs, sq_abs]
+  have hsum : ∑ i, (a i) ^ 2 = 1 - ⟪n, u⟫ ^ 2 := by
+    have hp := Submodule.norm_sq_eq_add_norm_sq_starProjection u (ℝ ∙ n)
+    rw [hu, one_pow, hnormal] at hp
+    change 1 = ⟪n, u⟫ ^ 2 + ‖(perpendicularTo n).orthogonalProjectionOnto u‖ ^ 2 at hp
+    rw [← hparseval] at hp
+    linarith
+  have hsquare : f.normDet ^ 2 = ⟪n, u⟫ ^ 2 := by
+    calc
+      f.normDet ^ 2 = (Matrix.gram ℝ (fun i => f (b i))).det :=
+        f.normDet_sq_eq_det_gram b
+      _ = 1 + a ⬝ᵥ (-a) := by
+        rw [hGram, Matrix.det_one_add_replicateCol_mul_replicateRow]
+      _ = 1 - ∑ i, (a i) ^ 2 := by
+        simp [dotProduct, pow_two, Finset.sum_neg_distrib, sub_eq_add_neg]
+      _ = ⟪n, u⟫ ^ 2 := by rw [hsum]; ring
+  apply (sq_eq_sq₀ f.normDet_nonneg (abs_nonneg _)).mp
+  simpa using hsquare
+
+/-- The same area factor for any linear chart into the source tangent hyperplane.
+The chart may be singular; the norm determinant identity handles that case as well. -/
+theorem normDet_project_comp {U : Type*} [NormedAddCommGroup U] [InnerProductSpace ℝ U]
+    [FiniteDimensional ℝ U] (L : U →ₗ[ℝ] V) {n u : V}
+    (hn : ‖n‖ = 1) (hu : ‖u‖ = 1)
+    (hmem : ∀ x, ⟪n, L x⟫ = 0)
+    (hrank : Module.finrank ℝ U = Module.finrank ℝ V - 1) :
+    ((perpendicularTo u).starProjection.toLinearMap.comp L).normDet =
+      |⟪n, u⟫| * L.normDet := by
+  let L' : U →ₗ[ℝ] perpendicularTo n := L.codRestrict (perpendicularTo n)
+    (fun x => Submodule.mem_orthogonal_singleton_iff_inner_right.mpr (hmem x))
+  have hdim : Module.finrank ℝ U = Module.finrank ℝ (perpendicularTo n) :=
+    hrank.trans (perpendicularTo_finrank hn).symm
+  have hL : L'.normDet = L.normDet := by
+    dsimp only [L']
+    exact LinearMap.normDet_codRestrict
+      (fun x => Submodule.mem_orthogonal_singleton_iff_inner_right.mpr (hmem x))
+  calc
+    ((perpendicularTo u).starProjection.toLinearMap.comp L).normDet =
+        ((hyperplaneProjection n u).comp L').normDet := rfl
+    _ = (hyperplaneProjection n u).normDet * L'.normDet :=
+      LinearMap.normDet_comp_of_finrank_eq L' (hyperplaneProjection n u) hdim
+    _ = |⟪n, u⟫| * L.normDet := by rw [hyperplaneProjection_normDet hn hu, hL]
+
+open MeasureTheory Measure in
+theorem hyperplaneProjection_measure_image [MeasurableSpace V] [BorelSpace V]
+    {n u : V} (hn : ‖n‖ = 1) (hu : ‖u‖ = 1) (s : Set (perpendicularTo n)) :
+    μHE[Module.finrank ℝ (perpendicularTo n)] (hyperplaneProjection n u '' s) =
+      ENNReal.ofReal |⟪n, u⟫| * volume s := by
+  rw [(hyperplaneProjection n u).euclideanHausdorffMeasure_image_eq_normDet_mul_volume,
+    hyperplaneProjection_normDet hn hu]
+
+open MeasureTheory Measure in
+theorem hyperplaneProjection_measure_translated [MeasurableSpace V] [BorelSpace V]
+    {n u : V} (hn : ‖n‖ = 1) (hu : ‖u‖ = 1) (c : V)
+    (s : Set (perpendicularTo n)) :
+    μHE[Module.finrank ℝ (perpendicularTo n)]
+        ((perpendicularTo u).starProjection '' ((fun x : perpendicularTo n => c + (x : V)) '' s)) =
+      ENNReal.ofReal |⟪n, u⟫| *
+        μHE[Module.finrank ℝ (perpendicularTo n)] ((fun x : perpendicularTo n => c + (x : V)) '' s) := by
+  have hshift : Isometry (fun x : perpendicularTo n => c + (x : V)) :=
+    (isometry_add_left c).comp isometry_subtype_coe
+  have himage :
+      (perpendicularTo u).starProjection '' ((fun x : perpendicularTo n => c + (x : V)) '' s) =
+        (fun y : V => (perpendicularTo u).starProjection c + y) ''
+          (hyperplaneProjection n u '' s) := by
+    rw [Set.image_image, Set.image_image]
+    congr 1
+    funext x
+    exact map_add (perpendicularTo u).starProjection c (x : V)
+  rw [himage, (isometry_add_left _).euclideanHausdorffMeasure_image,
+    hyperplaneProjection_measure_image hn hu, hshift.euclideanHausdorffMeasure_image,
+    InnerProductSpace.euclideanHausdorffMeasure_eq_volume]
+
+end
+
+end ProjectionCounterexample
+
+end OAI
diff --git a/lean/OAI/Geometry/ProjectionBody/Main.lean b/lean/OAI/Geometry/ProjectionBody/Main.lean
new file mode 100644
index 0000000000000000000000000000000000000000..d1085a30fce91d81810972bbcc607eef94b1715c
--- /dev/null
+++ b/lean/OAI/Geometry/ProjectionBody/Main.lean
@@ -0,0 +1 @@
+import OAI.Geometry.ProjectionBody.Specification
diff --git a/lean/OAI/Geometry/ProjectionBody/NullHyperplane.lean b/lean/OAI/Geometry/ProjectionBody/NullHyperplane.lean
new file mode 100644
index 0000000000000000000000000000000000000000..2aea52dba4ef87668fc892cfc1f129c8478daf3d
--- /dev/null
+++ b/lean/OAI/Geometry/ProjectionBody/NullHyperplane.lean
@@ -0,0 +1,35 @@
+import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
+
+namespace OAI
+
+noncomputable section
+open Set MeasureTheory
+
+namespace ProjectionCounterexample
+
+/-- Every level set of a nonzero real linear functional has zero ambient volume. -/
+theorem volume_linear_level_eq_zero
+    {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V]
+    [FiniteDimensional ℝ V] [MeasureSpace V] [BorelSpace V]
+    [Measure.IsAddHaarMeasure (volume : Measure V)]
+    (f : V →ₗ[ℝ] ℝ) (hf : f ≠ 0) (c : ℝ) :
+    volume {x : V | f x = c} = 0 := by
+  let H : AffineSubspace ℝ V := (affineSpan ℝ ({c} : Set ℝ)).comap f.toAffineMap
+  have hmem (x : V) : x ∈ H ↔ f x = c := by
+    simp [H]
+  have hproper : H ≠ ⊤ := by
+    intro htop
+    have heq (x : V) : f x = c := (hmem x).mp (by rw [htop]; trivial)
+    have hc : c = 0 := by simpa using (heq 0).symm
+    apply hf
+    ext x
+    simpa [hc] using heq x
+  have hset : (H : Set V) = {x : V | f x = c} := Set.ext hmem
+  rw [← hset]
+  exact Measure.addHaar_affineSubspace volume H hproper
+
+end ProjectionCounterexample
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/ProjectionBody/ProductBrightness.lean b/lean/OAI/Geometry/ProjectionBody/ProductBrightness.lean
new file mode 100644
index 0000000000000000000000000000000000000000..3c976e6033ace1004f388b008c62557de7e35986
--- /dev/null
+++ b/lean/OAI/Geometry/ProjectionBody/ProductBrightness.lean
@@ -0,0 +1,72 @@
+import OAI.Geometry.ProjectionBody.ProductFacetProjection
+import OAI.Geometry.ProjectionBody.SimplexBrightness
+import OAI.Geometry.ProjectionBody.ProductVolumes
+
+namespace OAI
+
+noncomputable section
+open Set MeasureTheory Measure
+
+namespace ProjectionCounterexample
+
+theorem product_projected_facet_volume (i : Bool × Option (Fin 10))
+    {u : E 20} (hu : ‖u‖ = 1) :
+    μHE[19] (project u '' boundingFace productFunctional productLevel i) =
+      ENNReal.ofReal ((1 / ((Nat.factorial 9 : ℝ) * Nat.factorial 10)) *
+        |productFunctional i u|) := by
+  rw [product_facet_eq_affine_image]
+  have h := measure_projected_affine_chart (productFacetLinear i)
+    (productFacetCenter i) (project u).toLinearMap (productFacetDomain i.1)
+  simp only [finrank_euclideanSpace_fin, ContinuousLinearMap.coe_coe] at h
+  rw [h, productFacetProjection_normDet i hu, volume_productFacetDomain,
+    ← ENNReal.ofReal_mul (abs_nonneg _), mul_comm]
+
+theorem product_projected_facet_measurable
+    (i : Bool × Option (Fin 10)) (u : E 20) :
+    MeasurableSet (project u '' boundingFace productFunctional productLevel i) := by
+  rw [product_facet_eq_affine_image]
+  apply IsCompact.measurableSet
+  apply IsCompact.image _ (project u).continuous
+  apply (productFacetDomain_compact i.1).image
+  exact continuous_const.add (productFacetLinear i).continuous_of_finiteDimensional
+
+theorem product_projected_facet_intersection_null
+    (i j : Bool × Option (Fin 10)) (hij : i ≠ j) (u : E 20) :
+    μHE[19] (project u ''
+      (boundingFace productFunctional productLevel i ∩
+        boundingFace productFunctional productLevel j)) = 0 := by
+  have hS : boundingFace productFunctional productLevel i ⊆
+      (fun x => productFacetCenter i + productFacetLinear i x) '' (univ : Set (E 19)) := by
+    rw [product_facet_eq_affine_image]
+    exact Set.image_mono (subset_univ _)
+  simpa only [finrank_euclideanSpace_fin, ContinuousLinearMap.coe_coe] using
+    measure_projected_transverse_intersection (productFacetLinear i)
+      (productFacetCenter i) (project u).toLinearMap (productFunctional j) (productLevel j)
+      _ _ hS (fun _ h => h.2) (product_facet_transverse i j hij)
+
+/-- Actual orthogonal projection volume of the twenty-dimensional product body. -/
+theorem product_projectionVolume {u : E 20} (hu : ‖u‖ = 1) :
+    projectionVolume productBody u =
+      (((∑ i, |firstBlock u i|) + |∑ i, firstBlock u i|) +
+        ((∑ i, |secondBlock u i|) + |∑ i, secondBlock u i|)) /
+      (2 * ((Nat.factorial 9 : ℝ) * Nat.factorial 10)) := by
+  have hu0 : u ≠ 0 := by intro h; simp [h] at hu
+  have h := real_measure_image_halfspaceBody
+    productFunctional productLevel (project u).toLinearMap μHE[19]
+    (project_self u) (product_has_positive_slope hu0) (project_ker u)
+    (fun i => product_projected_facet_measurable i u)
+    (fun i j hij => product_projected_facet_intersection_null i j hij u)
+    (1 / ((Nat.factorial 9 : ℝ) * Nat.factorial 10)) (by positivity)
+    (product_functional_balance u) (fun i => product_projected_facet_volume i hu)
+  rw [← productBody_eq_halfspaceBody, product_absolute_slopes] at h
+  simp only [ContinuousLinearMap.coe_coe] at h
+  unfold projectionVolume
+  change (μHE[19] (project u '' productBody)).toReal = _
+  rw [h]
+  ring
+
+end ProjectionCounterexample
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/ProjectionBody/ProductFacetCharts.lean b/lean/OAI/Geometry/ProjectionBody/ProductFacetCharts.lean
new file mode 100644
index 0000000000000000000000000000000000000000..a54c923cd00d33ecc6cbe2d05433174c34d8be2d
--- /dev/null
+++ b/lean/OAI/Geometry/ProjectionBody/ProductFacetCharts.lean
@@ -0,0 +1,187 @@
+import OAI.Geometry.ProjectionBody.ProductFacets
+import OAI.Geometry.ProjectionBody.SimplexFacetCharts
+import OAI.Geometry.Zonotope.Product
+
+namespace OAI
+
+/-! Explicit nineteen-dimensional affine charts on the twenty-two product facets. -/
+noncomputable section
+open Set
+
+namespace ProjectionCounterexample
+
+/-- The product-simplex parameter domain for a facet in the selected block. -/
+def productFacetDomain : Bool → Set (E 19)
+  | false => (@EuclideanSpace.finAddEquivProd ℝ _ 9 10) ⁻¹'
+      (simplex 9 ×ˢ simplex 10)
+  | true => (@EuclideanSpace.finAddEquivProd ℝ _ 10 9) ⁻¹'
+      (simplex 10 ×ˢ simplex 9)
+
+/-- Linear part of the affine product facet chart. -/
+def productFacetLinear : Bool × Option (Fin 10) → E 19 →ₗ[ℝ] E 20
+  | (false, i) => (@EuclideanSpace.finAddEquivProd ℝ _ 10 10).symm.toLinearMap.comp
+      (((simplexFacetLinear 9 i).prodMap (LinearMap.id : E 10 →ₗ[ℝ] E 10)).comp
+        (@EuclideanSpace.finAddEquivProd ℝ _ 9 10).toLinearMap)
+  | (true, i) => (@EuclideanSpace.finAddEquivProd ℝ _ 10 10).symm.toLinearMap.comp
+      (((LinearMap.id : E 10 →ₗ[ℝ] E 10).prodMap (simplexFacetLinear 9 i)).comp
+        (@EuclideanSpace.finAddEquivProd ℝ _ 10 9).toLinearMap)
+
+def productFacetCenter : Bool × Option (Fin 10) → E 20
+  | (false, i) => (@EuclideanSpace.finAddEquivProd ℝ _ 10 10).symm (simplexFacetCenter 9 i, 0)
+  | (true, i) => (@EuclideanSpace.finAddEquivProd ℝ _ 10 10).symm (0, simplexFacetCenter 9 i)
+
+@[simp] lemma firstBlock_finAdd (x : E 20) :
+    ((@EuclideanSpace.finAddEquivProd ℝ _ 10 10) x).1 = firstBlock x := rfl
+
+@[simp] lemma secondBlock_finAdd (x : E 20) :
+    ((@EuclideanSpace.finAddEquivProd ℝ _ 10 10) x).2 = secondBlock x := rfl
+
+lemma productFacetChart_first (i : Option (Fin 10)) (x : E 19) :
+    (@EuclideanSpace.finAddEquivProd ℝ _ 10 10)
+      (productFacetCenter (false, i) + productFacetLinear (false, i) x) =
+      (simplexFacetCenter 9 i + simplexFacetLinear 9 i
+        ((@EuclideanSpace.finAddEquivProd ℝ _ 9 10) x).1,
+       ((@EuclideanSpace.finAddEquivProd ℝ _ 9 10) x).2) := by
+  simp [productFacetCenter, productFacetLinear, map_add, LinearMap.prodMap_apply]
+  rfl
+
+lemma productFacetChart_second (i : Option (Fin 10)) (x : E 19) :
+    (@EuclideanSpace.finAddEquivProd ℝ _ 10 10)
+      (productFacetCenter (true, i) + productFacetLinear (true, i) x) =
+      (((@EuclideanSpace.finAddEquivProd ℝ _ 10 9) x).1,
+       simplexFacetCenter 9 i + simplexFacetLinear 9 i
+        ((@EuclideanSpace.finAddEquivProd ℝ _ 10 9) x).2) := by
+  simp [productFacetCenter, productFacetLinear, map_add, LinearMap.prodMap_apply]
+  rfl
+
+/-- Each product facet is exactly the image of the appropriate product simplex. -/
+theorem product_facet_eq_affine_image (i : Bool × Option (Fin 10)) :
+    boundingFace productFunctional productLevel i =
+      (fun x => productFacetCenter i + productFacetLinear i x) '' productFacetDomain i.1 := by
+  rcases i with ⟨b, i⟩
+  cases b
+  · rw [product_boundingFace_first, simplex_facet_eq_affine_image]
+    ext x
+    constructor
+    · rintro ⟨⟨a, ha, heq⟩, hb⟩
+      let z : E 19 := (@EuclideanSpace.finAddEquivProd ℝ _ 9 10).symm (a, secondBlock x)
+      refine ⟨z, ?_, ?_⟩
+      · change (@EuclideanSpace.finAddEquivProd ℝ _ 9 10) z ∈ simplex 9 ×ˢ simplex 10
+        simpa only [z, ContinuousLinearEquiv.apply_symm_apply, Set.mem_prod] using And.intro ha hb
+      · apply (@EuclideanSpace.finAddEquivProd ℝ _ 10 10).injective
+        rw [productFacetChart_first]
+        simp only [z, ContinuousLinearEquiv.apply_symm_apply]
+        apply Prod.ext
+        · exact heq
+        · rfl
+    · rintro ⟨z, hz, rfl⟩
+      have heq := productFacetChart_first i z
+      have hfirst := congrArg Prod.fst heq
+      have hsecond := congrArg Prod.snd heq
+      rw [firstBlock_finAdd] at hfirst
+      rw [secondBlock_finAdd] at hsecond
+      change ((@EuclideanSpace.finAddEquivProd ℝ _ 9 10) z).1 ∈ simplex 9 ∧
+        ((@EuclideanSpace.finAddEquivProd ℝ _ 9 10) z).2 ∈ simplex 10 at hz
+      refine ⟨⟨_, hz.1, hfirst.symm⟩, ?_⟩
+      rw [hsecond]
+      exact hz.2
+  · rw [product_boundingFace_second, simplex_facet_eq_affine_image]
+    ext x
+    constructor
+    · rintro ⟨ha, ⟨b, hb, heq⟩⟩
+      let z : E 19 := (@EuclideanSpace.finAddEquivProd ℝ _ 10 9).symm (firstBlock x, b)
+      refine ⟨z, ?_, ?_⟩
+      · change (@EuclideanSpace.finAddEquivProd ℝ _ 10 9) z ∈ simplex 10 ×ˢ simplex 9
+        simpa only [z, ContinuousLinearEquiv.apply_symm_apply, Set.mem_prod] using And.intro ha hb
+      · apply (@EuclideanSpace.finAddEquivProd ℝ _ 10 10).injective
+        rw [productFacetChart_second]
+        simp only [z, ContinuousLinearEquiv.apply_symm_apply]
+        apply Prod.ext
+        · rfl
+        · exact heq
+    · rintro ⟨z, hz, rfl⟩
+      have heq := productFacetChart_second i z
+      have hfirst := congrArg Prod.fst heq
+      have hsecond := congrArg Prod.snd heq
+      rw [firstBlock_finAdd] at hfirst
+      rw [secondBlock_finAdd] at hsecond
+      change ((@EuclideanSpace.finAddEquivProd ℝ _ 10 9) z).1 ∈ simplex 10 ∧
+        ((@EuclideanSpace.finAddEquivProd ℝ _ 10 9) z).2 ∈ simplex 9 at hz
+      refine ⟨?_, ⟨_, hz.2, hsecond.symm⟩⟩
+      rw [hfirst]
+      exact hz.1
+
+@[simp] lemma productFacetLinear_first_first (i : Option (Fin 10)) (z : E 9) (w : E 10) :
+    firstBlock (productFacetLinear (false, i)
+      ((@EuclideanSpace.finAddEquivProd ℝ _ 9 10).symm (z, w))) = simplexFacetLinear 9 i z := by
+  change ((@EuclideanSpace.finAddEquivProd ℝ _ 10 10) (productFacetLinear (false, i) _)).1 = _
+  simp [productFacetLinear, LinearMap.prodMap_apply]
+
+@[simp] lemma productFacetLinear_first_second (i : Option (Fin 10)) (z : E 9) (w : E 10) :
+    secondBlock (productFacetLinear (false, i)
+      ((@EuclideanSpace.finAddEquivProd ℝ _ 9 10).symm (z, w))) = w := by
+  change ((@EuclideanSpace.finAddEquivProd ℝ _ 10 10) (productFacetLinear (false, i) _)).2 = _
+  simp [productFacetLinear, LinearMap.prodMap_apply]
+
+@[simp] lemma productFacetLinear_second_first (i : Option (Fin 10)) (z : E 10) (w : E 9) :
+    firstBlock (productFacetLinear (true, i)
+      ((@EuclideanSpace.finAddEquivProd ℝ _ 10 9).symm (z, w))) = z := by
+  change ((@EuclideanSpace.finAddEquivProd ℝ _ 10 10) (productFacetLinear (true, i) _)).1 = _
+  simp [productFacetLinear, LinearMap.prodMap_apply]
+
+@[simp] lemma productFacetLinear_second_second (i : Option (Fin 10)) (z : E 10) (w : E 9) :
+    secondBlock (productFacetLinear (true, i)
+      ((@EuclideanSpace.finAddEquivProd ℝ _ 10 9).symm (z, w))) = simplexFacetLinear 9 i w := by
+  change ((@EuclideanSpace.finAddEquivProd ℝ _ 10 10) (productFacetLinear (true, i) _)).2 = _
+  simp [productFacetLinear, LinearMap.prodMap_apply]
+
+private lemma simplexFunctional_ten_ne_zero (j : Option (Fin 10)) :
+    simplexFunctional 10 j ≠ 0 := by
+  intro heq
+  cases j with
+  | none =>
+    have h := LinearMap.congr_fun heq (EuclideanSpace.single (0 : Fin 10) 1)
+    simp at h
+  | some j =>
+    have h := LinearMap.congr_fun heq (EuclideanSpace.single j 1)
+    simp at h
+
+/-- Every other defining hyperplane cuts a product facet chart transversely. -/
+theorem product_facet_transverse (i j : Bool × Option (Fin 10)) (hij : i ≠ j) :
+    (productFunctional j).comp (productFacetLinear i) ≠ 0 := by
+  rcases i with ⟨bi, i⟩
+  rcases j with ⟨bj, j⟩
+  intro heq
+  cases bi <;> cases bj
+  · apply simplex_facet_transverse (n := 9) (by decide) i j
+      (fun h => hij (congrArg (fun k => (false, k)) h))
+    ext z
+    have h := LinearMap.congr_fun heq
+      ((@EuclideanSpace.finAddEquivProd ℝ _ 9 10).symm (z, 0))
+    simpa only [LinearMap.comp_apply, productFunctional_first,
+      productFacetLinear_first_first, LinearMap.zero_apply] using h
+  · apply simplexFunctional_ten_ne_zero j
+    ext w
+    have h := LinearMap.congr_fun heq
+      ((@EuclideanSpace.finAddEquivProd ℝ _ 9 10).symm (0, w))
+    simpa only [LinearMap.comp_apply, productFunctional_second,
+      productFacetLinear_first_second, LinearMap.zero_apply] using h
+  · apply simplexFunctional_ten_ne_zero j
+    ext z
+    have h := LinearMap.congr_fun heq
+      ((@EuclideanSpace.finAddEquivProd ℝ _ 10 9).symm (z, 0))
+    simpa only [LinearMap.comp_apply, productFunctional_first,
+      productFacetLinear_second_first, LinearMap.zero_apply] using h
+  · apply simplex_facet_transverse (n := 9) (by decide) i j
+      (fun h => hij (congrArg (fun k => (true, k)) h))
+    ext w
+    have h := LinearMap.congr_fun heq
+      ((@EuclideanSpace.finAddEquivProd ℝ _ 10 9).symm (0, w))
+    simpa only [LinearMap.comp_apply, productFunctional_second,
+      productFacetLinear_second_second, LinearMap.zero_apply] using h
+
+end ProjectionCounterexample
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/ProjectionBody/ProductFacetGraphs.lean b/lean/OAI/Geometry/ProjectionBody/ProductFacetGraphs.lean
new file mode 100644
index 0000000000000000000000000000000000000000..e1bbaa1bcbca8aa723fce8f3636981faa196da97
--- /dev/null
+++ b/lean/OAI/Geometry/ProjectionBody/ProductFacetGraphs.lean
@@ -0,0 +1,213 @@
+import OAI.Geometry.ProjectionBody.ProductFacetCharts
+import OAI.Geometry.ProjectionBody.ProductFacetIndices
+
+namespace OAI
+
+noncomputable section
+open scoped RealInnerProductSpace
+
+namespace ProjectionCounterexample
+
+lemma firstBlock_productFacetLinear_first (i : Option (Fin 10)) (x : E 19) :
+    firstBlock (productFacetLinear (false, i) x) =
+      simplexFacetLinear 9 i ((@EuclideanSpace.finAddEquivProd ℝ _ 9 10) x).1 := by
+  change ((@EuclideanSpace.finAddEquivProd ℝ _ 10 10) (productFacetLinear (false, i) x)).1 = _
+  simp [productFacetLinear, LinearMap.prodMap_apply]
+
+lemma secondBlock_productFacetLinear_first (i : Option (Fin 10)) (x : E 19) :
+    secondBlock (productFacetLinear (false, i) x) =
+      ((@EuclideanSpace.finAddEquivProd ℝ _ 9 10) x).2 := by
+  change ((@EuclideanSpace.finAddEquivProd ℝ _ 10 10) (productFacetLinear (false, i) x)).2 = _
+  simp [productFacetLinear, LinearMap.prodMap_apply]
+
+lemma firstBlock_productFacetLinear_second (i : Option (Fin 10)) (x : E 19) :
+    firstBlock (productFacetLinear (true, i) x) =
+      ((@EuclideanSpace.finAddEquivProd ℝ _ 10 9) x).1 := by
+  change ((@EuclideanSpace.finAddEquivProd ℝ _ 10 10) (productFacetLinear (true, i) x)).1 = _
+  simp [productFacetLinear, LinearMap.prodMap_apply]
+
+lemma secondBlock_productFacetLinear_second (i : Option (Fin 10)) (x : E 19) :
+    secondBlock (productFacetLinear (true, i) x) =
+      simplexFacetLinear 9 i ((@EuclideanSpace.finAddEquivProd ℝ _ 10 9) x).2 := by
+  change ((@EuclideanSpace.finAddEquivProd ℝ _ 10 10) (productFacetLinear (true, i) x)).2 = _
+  simp [productFacetLinear, LinearMap.prodMap_apply]
+
+lemma productFacetLinear_first_coordinate (i : Fin 10) :
+    productFacetLinear (false, some i) = (insertZero (i.castAdd 10)).toLinearMap := by
+  ext x j
+  induction j using (i.castAdd 10).succAboveCases
+  · change firstBlock (productFacetLinear (false, some i) x) i = _
+    rw [firstBlock_productFacetLinear_first]
+    simp [simplexFacetLinear]
+  · rename_i j
+    simp only [LinearIsometry.coe_toLinearMap]
+    rw [insertZero_succAbove]
+    induction j using Fin.addCases (m := 9) (n := 10) with
+    | left j =>
+      rw [first_succAbove_first]
+      change firstBlock (productFacetLinear (false, some i) x) (i.succAbove j) = _
+      rw [firstBlock_productFacetLinear_first]
+      simp only [simplexFacetLinear, LinearIsometry.coe_toLinearMap, insertZero_succAbove]
+      rfl
+    | right j =>
+      rw [first_succAbove_second]
+      change secondBlock (productFacetLinear (false, some i) x) j = _
+      rw [secondBlock_productFacetLinear_first]
+      rfl
+
+lemma productFacetLinear_second_coordinate (i : Fin 10) :
+    productFacetLinear (true, some i) = (insertZero (i.natAdd 10)).toLinearMap := by
+  ext x j
+  induction j using (i.natAdd 10).succAboveCases
+  · change secondBlock (productFacetLinear (true, some i) x) i = _
+    rw [secondBlock_productFacetLinear_second]
+    simp [simplexFacetLinear]
+  · rename_i j
+    simp only [LinearIsometry.coe_toLinearMap]
+    rw [insertZero_succAbove]
+    induction j using Fin.addCases (m := 10) (n := 9) with
+    | left j =>
+      rw [second_succAbove_first]
+      change firstBlock (productFacetLinear (true, some i) x) j = _
+      rw [firstBlock_productFacetLinear_second]
+      rfl
+    | right j =>
+      rw [second_succAbove_second]
+      change secondBlock (productFacetLinear (true, some i) x) (i.succAbove j) = _
+      rw [secondBlock_productFacetLinear_second]
+      simp only [simplexFacetLinear, LinearIsometry.coe_toLinearMap, insertZero_succAbove]
+      rfl
+
+/-- Graph coefficient vector for the first block's sum facet. -/
+def firstSumGraph : E 19 := (@EuclideanSpace.finAddEquivProd ℝ _ 9 10).symm (allOnes 9, 0)
+
+/-- Graph coefficient vector for the second block's sum facet. -/
+def secondSumGraph : E 19 := (@EuclideanSpace.finAddEquivProd ℝ _ 10 9).symm (0, allOnes 9)
+
+lemma inner_firstSumGraph (x : E 19) :
+    ⟪firstSumGraph, x⟫ = ⟪allOnes 9, ((@EuclideanSpace.finAddEquivProd ℝ _ 9 10) x).1⟫ := by
+  rw [DiagonalZonotope.inner_finAdd 9 10]
+  have h : (@EuclideanSpace.finAddEquivProd ℝ _ 9 10) firstSumGraph =
+      (allOnes 9, 0) := (@EuclideanSpace.finAddEquivProd ℝ _ 9 10).apply_symm_apply _
+  rw [h]
+  simp only [inner_zero_left, add_zero]
+
+lemma inner_secondSumGraph (x : E 19) :
+    ⟪secondSumGraph, x⟫ = ⟪allOnes 9, ((@EuclideanSpace.finAddEquivProd ℝ _ 10 9) x).2⟫ := by
+  rw [DiagonalZonotope.inner_finAdd 10 9]
+  have h : (@EuclideanSpace.finAddEquivProd ℝ _ 10 9) secondSumGraph =
+      (0, allOnes 9) := (@EuclideanSpace.finAddEquivProd ℝ _ 10 9).apply_symm_apply _
+  rw [h]
+  simp only [inner_zero_left, zero_add]
+
+lemma productFacetLinear_first_sum :
+    productFacetLinear (false, none) = graphLinear ((Fin.last 9).castAdd 10) firstSumGraph := by
+  ext x j
+  induction j using ((Fin.last 9).castAdd 10).succAboveCases
+  · rw [graphLinear_same, inner_firstSumGraph]
+    change firstBlock (productFacetLinear (false, none) x) (Fin.last 9) = _
+    rw [firstBlock_productFacetLinear_first]
+    simp [simplexFacetLinear]
+  · rename_i j
+    rw [graphLinear_succAbove]
+    induction j using Fin.addCases (m := 9) (n := 10) with
+    | left j =>
+      rw [first_succAbove_first]
+      change firstBlock (productFacetLinear (false, none) x) ((Fin.last 9).succAbove j) = _
+      rw [firstBlock_productFacetLinear_first]
+      simp only [simplexFacetLinear, graphLinear_succAbove]
+      rfl
+    | right j =>
+      rw [first_succAbove_second]
+      change secondBlock (productFacetLinear (false, none) x) j = _
+      rw [secondBlock_productFacetLinear_first]
+      rfl
+
+lemma productFacetLinear_second_sum :
+    productFacetLinear (true, none) = graphLinear ((Fin.last 9).natAdd 10) secondSumGraph := by
+  ext x j
+  induction j using ((Fin.last 9).natAdd 10).succAboveCases
+  · rw [graphLinear_same, inner_secondSumGraph]
+    change secondBlock (productFacetLinear (true, none) x) (Fin.last 9) = _
+    rw [secondBlock_productFacetLinear_second]
+    simp [simplexFacetLinear]
+  · rename_i j
+    rw [graphLinear_succAbove]
+    induction j using Fin.addCases (m := 10) (n := 9) with
+    | left j =>
+      rw [second_succAbove_first]
+      change firstBlock (productFacetLinear (true, none) x) j = _
+      rw [firstBlock_productFacetLinear_second]
+      rfl
+    | right j =>
+      rw [second_succAbove_second]
+      change secondBlock (productFacetLinear (true, none) x) ((Fin.last 9).succAbove j) = _
+      rw [secondBlock_productFacetLinear_second]
+      simp only [simplexFacetLinear, graphLinear_succAbove]
+      rfl
+
+@[simp] lemma blockJoin_first {n m : ℕ} (x : E n) (y : E m) (i : Fin n) :
+    (@EuclideanSpace.finAddEquivProd ℝ _ n m).symm (x, y) (i.castAdd m) = x i := by
+  exact congrArg (fun z : E n × E m => z.1 i)
+    ((@EuclideanSpace.finAddEquivProd ℝ _ n m).apply_symm_apply (x, y))
+
+@[simp] lemma blockJoin_second {n m : ℕ} (x : E n) (y : E m) (i : Fin m) :
+    (@EuclideanSpace.finAddEquivProd ℝ _ n m).symm (x, y) (i.natAdd n) = y i := by
+  exact congrArg (fun z : E n × E m => z.2 i)
+    ((@EuclideanSpace.finAddEquivProd ℝ _ n m).apply_symm_apply (x, y))
+
+lemma graphNormal_succAbove {n : ℕ} (i : Fin (n + 1)) (a : E n) (j : Fin n) :
+    graphNormal i a (i.succAbove j) = a j := by
+  simp [graphNormal]
+
+lemma firstSumGraph_normal :
+    graphNormal ((Fin.last 9).castAdd 10) firstSumGraph =
+      (@EuclideanSpace.finAddEquivProd ℝ _ 10 10).symm (allOnes 10, 0) := by
+  ext j
+  induction j using ((Fin.last 9).castAdd 10).succAboveCases
+  · rw [graphNormal_same, blockJoin_first]
+    rfl
+  · rename_i j
+    rw [graphNormal_succAbove]
+    induction j using Fin.addCases (m := 9) (n := 10) with
+    | left j =>
+      rw [first_succAbove_first (Fin.last 9) j, firstSumGraph, blockJoin_first, blockJoin_first]
+      rfl
+    | right j =>
+      rw [first_succAbove_second (Fin.last 9) j, firstSumGraph, blockJoin_second, blockJoin_second]
+
+lemma secondSumGraph_normal :
+    graphNormal ((Fin.last 9).natAdd 10) secondSumGraph =
+      (@EuclideanSpace.finAddEquivProd ℝ _ 10 10).symm (0, allOnes 10) := by
+  ext j
+  induction j using ((Fin.last 9).natAdd 10).succAboveCases
+  · rw [graphNormal_same, blockJoin_second]
+    rfl
+  · rename_i j
+    rw [graphNormal_succAbove]
+    induction j using Fin.addCases (m := 10) (n := 9) with
+    | left j =>
+      rw [second_succAbove_first (Fin.last 9) j, secondSumGraph, blockJoin_first, blockJoin_first]
+    | right j =>
+      rw [second_succAbove_second (Fin.last 9) j, secondSumGraph, blockJoin_second, blockJoin_second]
+      rfl
+
+lemma firstSumGraph_normal_inner (u : E 20) :
+    ⟪graphNormal ((Fin.last 9).castAdd 10) firstSumGraph, u⟫ =
+      simplexFunctional 10 none (firstBlock u) := by
+  rw [firstSumGraph_normal, DiagonalZonotope.inner_finAdd 10 10,
+    ContinuousLinearEquiv.apply_symm_apply]
+  simp only [inner_zero_left, add_zero, inner_allOnes, firstBlock_finAdd, simplexFunctional_none]
+
+lemma secondSumGraph_normal_inner (u : E 20) :
+    ⟪graphNormal ((Fin.last 9).natAdd 10) secondSumGraph, u⟫ =
+      simplexFunctional 10 none (secondBlock u) := by
+  rw [secondSumGraph_normal, DiagonalZonotope.inner_finAdd 10 10,
+    ContinuousLinearEquiv.apply_symm_apply]
+  simp only [inner_zero_left, zero_add, inner_allOnes, secondBlock_finAdd, simplexFunctional_none]
+
+end ProjectionCounterexample
+
+end
+
+end OAI
diff --git a/lean/OAI/Geometry/ProjectionBody/ProductFacetIndices.lean b/lean/OAI/Geometry/ProjectionBody/ProductFacetIndices.lean
new file mode 100644
index 0000000000000000000000000000000000000000..5865c965f6331624bb785bc37a52021938ce6264
--- /dev/null
+++ b/lean/OAI/Geometry/ProjectionBody/ProductFacetIndices.lean
@@ -0,0 +1,38 @@
+import Mathlib.Data.Fin.Tuple.Basic
+
+namespace OAI
+
+namespace ProjectionCounterexample
+
+lemma first_succAbove_first (i : Fin 10) (j : Fin 9) :
+    (i.castAdd 10).succAbove (j.castAdd 10) = (i.succAbove j).castAdd 10 := by
+  ext
+  simp only [Fin.succAbove, Fin.lt_def, Fin.val_castSucc, Fin.val_castAdd]
+  split_ifs <;> rfl
+
+lemma first_succAbove_second (i : Fin 10) (j : Fin 10) :
+    (i.castAdd 10).succAbove (j.natAdd 9) = j.natAdd 10 := by
+  ext
+  simp only [Fin.succAbove, Fin.lt_def, Fin.val_castSucc, Fin.val_castAdd, Fin.val_natAdd]
+  split_ifs with h
+  · omega
+  · simp only [Fin.val_succ, Fin.val_natAdd]
+    omega
+
+lemma second_succAbove_first (i : Fin 10) (j : Fin 10) :
+    (i.natAdd 10).succAbove (j.castAdd 9) = j.castAdd 10 := by
+  ext
+  simp only [Fin.succAbove, Fin.lt_def, Fin.val_castSucc, Fin.val_castAdd, Fin.val_natAdd]
+  split_ifs with h
+  · rfl
+  · omega
+
+lemma second_succAbove_second (i : Fin 10) (j : Fin 9) :
+    (i.natAdd 10).succAbove (j.natAdd 10) = (i.succAbove j).natAdd 10 := by
+  ext
+  simp only [Fin.succAbove, Fin.lt_def, Fin.val_castSucc, Fin.val_natAdd]
+  split_ifs <;> simp_all only [Fin.val_castSucc, Fin.val_succ, Fin.val_natAdd] <;> omega
+
+end ProjectionCounterexample
+
+end OAI